{"text": "import polyhedral_lattice.cosimplicial\nimport polyhedral_lattice.Hom\n\nopen_locale nnreal\n\nnamespace PolyhedralLattice\n\nopen pseudo_normed_group polyhedral_lattice.conerve (L obj lift' lift'_w)\n\nvariables (r' : \u211d\u22650) (\u039b : PolyhedralLattice)\nvariables (M : ProFiltPseuNormGrpWithTinv r') (N : \u2115) [fact (0 < N)]\nvariables (m : \u2115) (g\u2080 : \u039b \u2192+ M)\nvariables (g : fin (m + 1) \u2192 (\u039b.rescaled_power N \u2192+ M))\nvariables (hg : \u2200 i l, (g i) (\u039b.diagonal_embedding N l) = g\u2080 l)\n\nlemma cosimplicial_lift_mk (l) :\n  \u039b.cosimplicial_lift N m g\u2080 g hg (quotient_add_group.mk l) = finsupp.lift_add_hom g l :=\nbegin\n  dsimp only [cosimplicial_lift, lift'],\n  have := quotient_add_group.lift_mk'\n    (L (\u039b.diagonal_embedding N) (m + 1))\n    (lift'_w (\u039b.diagonal_embedding N) m g\u2080 g hg _) l,\n  exact this,\nend\n\nlemma cosimplicial_lift_mem_filtration (c : \u211d\u22650)\n  (H : \u2200 i, g i \u2208 filtration ((\u039b.rescaled_power N) \u2192+ M) c) :\n  cosimplicial_lift \u039b N m g\u2080 g hg \u2208 filtration (obj (\u039b.diagonal_embedding N) (m + 1) \u2192+ M) c :=\nbegin\n  intros c' l' hl',\n  rw [semi_normed_group.mem_filtration_iff] at hl',\n  obtain \u27e8l, rfl, hl\u27e9 := polyhedral_lattice.norm_lift _ l',\n  erw cosimplicial_lift_mk,\n  rw [finsupp.lift_add_hom_apply, finsupp.sum_fintype],\n  swap, { intro, rw add_monoid_hom.map_zero },\n  simp only [\u2190 coe_nnnorm, nnreal.eq_iff] at hl,\n  erw [finsupp.nnnorm_def, finsupp.sum_fintype] at hl,\n  swap, { intro, rw nnnorm_zero },\n  rw \u2190 hl at hl',\n  replace hl' := mul_le_mul' (le_refl c) hl',\n  rw [finset.mul_sum] at hl',\n  apply filtration_mono hl',\n  apply sum_mem_filtration,\n  rintro i -,\n  apply H,\n  exact semi_normed_group.mem_filtration_nnnorm (l i),\nend\n\nend PolyhedralLattice\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/polyhedral_lattice/cosimplicial_extra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.39999754232136026}}
{"text": "import Mathlib.Data.Fintype.Card\nimport Mathlib.Data.Fintype.Sum\nimport Mathlib.Data.Fintype.Sigma\nimport Mathlib.Data.Fintype.BigOperators\nimport Lean4bits.defs\nimport Mathlib.Tactic.Zify\nimport Mathlib.Tactic.Ring\n\nopen Sum\n\nvariable {\u03b1 \u03b2 \u03b1' \u03b2' : Type} {\u03b3 : \u03b2 \u2192 Type}\n\ndef propagateAux (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool) \u00d7 Bool\n  | 0 => next_bit init_carry (fun i => x i 0)\n  | n+1 => next_bit (propagateAux init_carry next_bit x n).1 (fun i => x i (n+1))\n\ndef propagate (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) : Bool :=\n  (propagateAux init_carry next_bit x i).2\n\n@[simp] def propagateCarry (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool)\n  | 0 => next_bit init_carry (fun i => x i 0)\n  | n+1 => next_bit (propagateCarry init_carry next_bit x n) (fun i => x i (n+1))\n\n@[simp] def propagateCarry2 (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool)\n  | 0 => init_carry\n  | n+1 => next_bit (propagateCarry2 init_carry next_bit x n) (fun i => x i n)\n\nlemma propagateCarry2_succ (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2200 (n : \u2115),\n    propagateCarry2 init_carry next_bit x (n+1) =\n    propagateCarry init_carry next_bit x n\n  | 0 => rfl\n  | n+1 => by rw [propagateCarry2, propagateCarry2_succ _ _ _ n, propagateCarry]\n\n@[simp] lemma propagateAux_fst_eq_carry (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2200 n : \u2115,\n    (propagateAux init_carry next_bit x n).1 =\n    propagateCarry init_carry (fun c b => (next_bit c b).1) x n\n  | 0 => rfl\n  | n+1 => by rw [propagateAux, propagateCarry, propagateAux_fst_eq_carry _ _ _ n]\n\n@[simp] lemma propagate_zero (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n    (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    propagate init_carry next_bit x 0 = (next_bit init_carry (fun i => x i 0)).2 :=\n  rfl\n\nlemma propagate_succ (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) :\n    propagate init_carry next_bit x (i+1) = (next_bit\n      (propagateCarry init_carry (fun c b => (next_bit c b).1) x i)\n      (\u03bb j => x j (i+1))).2 :=\n  by rw [\u2190 propagateAux_fst_eq_carry]; rfl\n\nlemma propagate_succ2 (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) :\n    propagate init_carry next_bit x (i+1) = (next_bit\n      (propagateCarry2 init_carry (\u03bb c b => (next_bit c b).1) x (i+1))\n      (\u03bb j => x j (i+1))).2 :=\n  by rw [propagateCarry2_succ, \u2190 propagateAux_fst_eq_carry]; rfl\n\nlemma propagateCarry_propagate {\u03b4 : \u03b2 \u2192 Type} {\u03b2' : Type}\n      (f : \u2200 a, \u03b4 a \u2192 \u03b2') : \u2200 (n : \u2115) (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (init_carry_x : \u2200 a, \u03b3 a \u2192 Bool)\n    (next_bit_x : \u2200 a (_carry : \u03b3 a \u2192 Bool) (_bits : \u03b4 a \u2192 Bool),\n      (\u03b3 a \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool),\n    propagateCarry init_carry next_bit (\u03bb a => propagate (init_carry_x a)\n      (next_bit_x a) (\u03bb d => x (f a d))) n =\n    propagateCarry\n      (\u03bb a : \u03b1 \u2295 (\u03a3 a, \u03b3 a) => Sum.elim init_carry (\u03bb b : \u03a3 a, \u03b3 a =>\n        init_carry_x b.1 b.2) a)\n      (\u03bb (carry : (\u03b1 \u2295 (\u03a3 a, \u03b3 a)) \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n    -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n        let f : \u2200 (a : \u03b2), (\u03b3 a \u2192 Bool) \u00d7 Bool := \u03bb a => next_bit_x a\n          (\u03bb d => carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (f a d))\n        let g : (\u03b1 \u2192 Bool) := (next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n        Sum.elim g (\u03bb x => (f x.1).1 x.2))\n      x n \u2218 inl\n  | 0, init_carry, next_bit, init_carry_x, next_bit_x, x => rfl\n  | n+1, init_carry, next_bit, init_carry_x, next_bit_x, x => by\n    have := propagateCarry_propagate f n\n    simp only [propagateCarry, propagate_succ, elim_inl, Nat.add] at *\n    conv_lhs => simp only [this]\n    clear this\n    dsimp\n    congr\n    ext a\n    dsimp\n    congr\n    ext b\n    dsimp [propagateCarry, propagate_succ, elim_inl, Nat.add]\n    congr\n    dsimp\n    induction' n with n ih\n    . simp\n    . simp [ih]\n\nlemma propagate_propagate {\u03b4 : \u03b2 \u2192 Type} {\u03b2' : Type}\n      (f : \u2200 a, \u03b4 a \u2192 \u03b2') : \u2200 (n : \u2115) (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (init_carry_x : \u2200 a, \u03b3 a \u2192 Bool)\n    (next_bit_x : \u2200 a (_carry : \u03b3 a \u2192 Bool) (_bits : \u03b4 a \u2192 Bool),\n      (\u03b3 a \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool),\n    propagate init_carry next_bit (\u03bb a => propagate (init_carry_x a)\n      (next_bit_x a) (\u03bb d => x (f a d))) n =\n    propagate\n      (\u03bb a : \u03b1 \u2295 (\u03a3 a, \u03b3 a) => Sum.elim init_carry (\u03bb b : \u03a3 a, \u03b3 a =>\n        init_carry_x b.1 b.2) a)\n      (\u03bb (carry : (\u03b1 \u2295 (\u03a3 a, \u03b3 a)) \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n        -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n        let f : \u2200 (a : \u03b2), (\u03b3 a \u2192 Bool) \u00d7 Bool := \u03bb a => next_bit_x a (\u03bb d =>\n          carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (f a d))\n        let g : (\u03b1 \u2192 Bool) \u00d7 Bool := (next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n        (Sum.elim g.1 (\u03bb x => (f x.1).1 x.2), g.2)\n      )\n    x n\n  | 0, init_carry, next_bit, init_carry_x, next_bit_x, x => rfl\n  | n+1, init_carry, next_bit, init_carry_x, next_bit_x, x => by\n    simp only [propagate_succ]\n    rw [propagateCarry_propagate]\n    congr\n    ext\n    congr\n    induction' n with n ih\n    . simp\n    . simp [ih]\n\nlemma propagateCarry_changeVars {\u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool) (i : \u2115)\n    (changeVars : \u03b2 \u2192 \u03b2') :\n    propagateCarry init_carry next_bit (\u03bb b => x (changeVars b)) i =\n    propagateCarry init_carry (\u03bb (carry : \u03b1 \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n      next_bit carry (\u03bb b => bits (changeVars b))) x i := by\n  induction i\n  . simp\n  . simp [*]\n\nlemma propagate_changeVars {\u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool) (i : \u2115)\n    (changeVars : \u03b2 \u2192 \u03b2') :\n    propagate init_carry next_bit (\u03bb b => x (changeVars b)) i =\n    propagate init_carry (\u03bb (carry : \u03b1 \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n      next_bit carry (\u03bb b => bits (changeVars b))) x i := by\n  induction' i with i ih\n  . rfl\n  . simp only [propagate_succ, propagateCarry_changeVars, ih]\n\nopen Term\n\n@[simp] def arity : Term \u2192 \u2115\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| Term.and t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.or t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.xor t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.not t => arity t\n| ls t => arity t\n| add t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| sub t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| neg t => arity t\n| incr t => arity t\n| decr t => arity t\n\n@[simp] def Term.evalFin : \u2200 (t : Term) (_vars : Fin (arity t) \u2192 \u2115 \u2192 Bool), \u2115 \u2192 Bool\n| var n, vars => vars (Fin.last n)\n| zero, _vars => zeroSeq\n| one, _vars => oneSeq\n| negOne, _vars => negOneSeq\n| Term.and t\u2081 t\u2082, vars =>\n  andSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| Term.or t\u2081 t\u2082, vars =>\n  orSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| Term.xor t\u2081 t\u2082, vars =>\n  xorSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| not t, vars => notSeq (Term.evalFin t vars)\n| ls t, vars => lsSeq (Term.evalFin t vars)\n| add t\u2081 t\u2082, vars =>\n  addSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| sub t\u2081 t\u2082, vars =>\n  subSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| neg t, vars => negSeq (Term.evalFin t vars)\n| incr t, vars => incrSeq (Term.evalFin t vars)\n| decr t, vars => decrSeq (Term.evalFin t vars)\n\nlemma evalFin_eq_eval (t : Term) (vars : \u2115 \u2192 \u2115 \u2192 Bool) :\n    Term.evalFin t (fun i => vars i) = Term.eval t vars := by\n  induction t <;>\n  dsimp [Term.evalFin, Term.eval, arity] at * <;> simp [*]\n\n\nlemma id_eq_propagate (x : \u2115 \u2192 Bool) :\n    x = propagate Empty.elim (\u03bb _ (y : Unit \u2192 Bool) => (Empty.elim, y ())) (\u03bb _ => x) := by\n  ext n; cases n <;> rfl\n\nlemma zero_eq_propagate :\n    zeroSeq = propagate Empty.elim (\u03bb (_ _ : Empty \u2192 Bool) => (Empty.elim, false)) Empty.elim := by\n  ext n; cases n <;> rfl\n\nlemma one_eq_propagate :\n    oneSeq = propagate (\u03bb _ : Unit => true)\n      (\u03bb f (_ : Empty \u2192 Bool) => (\u03bb _ => false, f ())) Empty.elim := by\n  ext n\n  match n with\n  | 0 => rfl\n  | 1 => rfl\n  | n+2 => simp [oneSeq, propagate_succ]\n\nlemma and_eq_propagate (x y : \u2115 \u2192 Bool) :\n    andSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, y true && y false)) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, andSeq]\n\nlemma or_eq_propagate (x y : \u2115 \u2192 Bool) :\n    orSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, y true || y false)) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, orSeq]\n\nlemma xor_eq_propagate (x y : \u2115 \u2192 Bool) :\n    xorSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, xor (y true) (y false))) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, xorSeq]\n\nlemma not_eq_propagate (x : \u2115 \u2192 Bool) :\n    notSeq x = propagate Empty.elim (\u03bb _ (y : Unit \u2192 Bool) => (Empty.elim, !(y ()))) (\u03bb _ => x) := by\n  ext n; cases n <;> simp [propagate, propagateAux, notSeq]\n\nlemma ls_eq_propagate (x : \u2115 \u2192 Bool) :\n    lsSeq x = propagate (\u03bb _ : Unit => false)\n      (\u03bb (carry x : Unit \u2192 Bool) => (x, carry ())) (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => rfl\n  | 1 => rfl\n  | n+2 => simp [lsSeq, propagate_succ]\n\nlemma addSeqAux_eq_propagateCarry (x y : \u2115 \u2192 Bool) (n : \u2115) :\n    (addSeqAux x y n).2 = propagateCarry (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        \u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()))\n    (\u03bb b => cond b x y) n () := by\n  induction n <;> simp [addSeqAux, *]\n\nlemma add_eq_propagate (x y : \u2115 \u2192 Bool) :\n    addSeq x y = propagate (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))))\n    (\u03bb b => cond b x y) := by\n  ext n\n  match n with\n  | 0 => simp [addSeq, addSeqAux]\n  | 1 => simp [addSeq, addSeqAux, propagate, propagateAux]\n  | n+2 => simp [addSeq, addSeqAux, addSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma subSeqAux_eq_propagateCarry (x y : \u2115 \u2192 Bool) (n : \u2115) :\n    (subSeqAux x y n).2 = propagateCarry (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        \u03bb _ => (!(bits true) && (bits false)) ||\n          (!(_root_.xor (bits true) (bits false))) && carry ())\n    (\u03bb b => cond b x y) n () := by\n  induction n <;> simp [subSeqAux, *]\n\nlemma sub_eq_propagate (x y : \u2115 \u2192 Bool) :\n    subSeq x y = propagate (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (!(bits true) && (bits false)) ||\n          ((!(_root_.xor (bits true) (bits false))) && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))))\n    (\u03bb b => cond b x y) := by\n  ext n\n  match n with\n  | 0 => simp [subSeq, subSeqAux]\n  | 1 => simp [subSeq, subSeqAux, propagate, propagateAux]\n  | n+2 => simp [subSeq, subSeqAux, subSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma negSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (negSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (!(bits ())) && (carry ()))\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [negSeqAux, *]\n\nlemma neg_eq_propagate (x : \u2115 \u2192 Bool) :\n    negSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (!(bits ())) && (carry ()), _root_.xor (!(bits ())) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [negSeq, negSeqAux]\n  | 1 => simp [negSeq, negSeqAux, propagate, propagateAux]\n  | n+2 => simp [negSeq, negSeqAux, negSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma incrSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (incrSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (bits ()) && carry ())\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [incrSeqAux, *]\n\nlemma incr_eq_propagate (x : \u2115 \u2192 Bool) :\n    incrSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (bits ()) && carry (), _root_.xor (bits ()) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [incrSeq, incrSeqAux]\n  | 1 => simp [incrSeq, incrSeqAux, propagate, propagateAux]\n  | n+2 => simp [incrSeq, incrSeqAux, incrSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma decrSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (decrSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (!(bits ())) && carry ())\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [decrSeqAux, *]\n\nlemma decr_eq_propagate (x : \u2115 \u2192 Bool) :\n    decrSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (!(bits ())) && carry (), _root_.xor (bits ()) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [decrSeq, decrSeqAux]\n  | 1 => simp [decrSeq, decrSeqAux, propagate, propagateAux]\n  | n+2 => simp [decrSeq, decrSeqAux, decrSeqAux_eq_propagateCarry, propagate_succ]\n\nstructure PropagateStruc (arity : Type) : Type 1 :=\n  ( \u03b1  : Type )\n  [ i : Fintype \u03b1 ]\n  ( init_carry : \u03b1 \u2192 Bool )\n  ( next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : arity \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool )\n\nattribute [instance] PropagateStruc.i\n\nnamespace PropagateStruc\n\nvariable {arity : Type} (p : PropagateStruc arity)\n\ndef eval : (arity \u2192 \u2115 \u2192 Bool) \u2192 \u2115 \u2192 Bool :=\n  propagate p.init_carry p.next_bit\n\ndef changeVars {arity2 : Type} (changeVars : arity \u2192 arity2) :\n    PropagateStruc arity2 :=\n  { \u03b1 := p.\u03b1,\n    i := p.i,\n    init_carry := p.init_carry,\n    next_bit := \u03bb carry bits => p.next_bit carry (fun i => bits (changeVars i)) }\n\ndef compose [Fintype arity]\n  (new_arity : Type)\n    (q_arity : arity \u2192 Type)\n    (vars : \u2200 (a : arity), q_arity a \u2192 new_arity)\n    (q : \u2200 (a : arity), PropagateStruc (q_arity a)) :\n    PropagateStruc (new_arity) :=\n  { \u03b1 := p.\u03b1 \u2295 (\u03a3 a, (q a).\u03b1),\n    i := by letI := p.i;  infer_instance,\n    init_carry := Sum.elim p.init_carry (\u03bb x => (q x.1).init_carry x.2),\n    next_bit := \u03bb carry bits =>\n      let f : \u2200 (a : arity), ((q a).\u03b1 \u2192 Bool) \u00d7 Bool := \u03bb a => (q a).next_bit (\u03bb d =>\n          carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (vars a d))\n      let g : (p.\u03b1 \u2192 Bool) \u00d7 Bool := (p.next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n      (Sum.elim g.1 (\u03bb x => (f x.1).1 x.2), g.2) }\n\nlemma eval_compose [Fintype arity]\n    (new_arity : Type)\n    (q_arity : arity \u2192 Type)\n    (vars : \u2200 (a : arity), q_arity a \u2192 new_arity)\n    (q : \u2200 (a : arity), PropagateStruc (q_arity a))\n    (x : new_arity \u2192 \u2115 \u2192 Bool):\n    (p.compose new_arity q_arity vars q).eval x =\n    p.eval (\u03bb a => (q a).eval (fun i => x (vars _ i))) := by\n  ext n; simp only [eval, compose, propagate_propagate]\n\n\ndef and : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits true && bits false) }\n\n@[simp] lemma eval_and (x : Bool \u2192 \u2115 \u2192 Bool) : and.eval x = andSeq (x true) (x false) := by\n  ext n; cases n <;> simp [and, andSeq, eval, propagate_succ]\n\ndef or : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits true || bits false) }\n\n@[simp] lemma eval_or (x : Bool \u2192 \u2115 \u2192 Bool) : or.eval x = orSeq (x true) (x false) := by\n  ext n; cases n <;> simp [or, orSeq, eval, propagate_succ]\n\ndef xor : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, _root_.xor (bits true) (bits false)) }\n\n@[simp] lemma eval_xor (x : Bool \u2192 \u2115 \u2192 Bool) : xor.eval x = xorSeq (x true) (x false) := by\n  ext n; cases n <;> simp [xor, xorSeq, eval, propagate_succ]\n\n  def add : PropagateStruc Bool :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))) }\n\n@[simp] lemma eval_add (x : Bool \u2192 \u2115 \u2192 Bool) : add.eval x = addSeq (x true) (x false) := by\n  dsimp [add, eval]\n  rw [add_eq_propagate]\n  congr\n  funext b\n  cases b; rfl\n  simp\n  congr\n  funext i\n  cases i <;> simp\n\n\ndef sub : PropagateStruc Bool :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (!(bits true) && (bits false)) ||\n          ((!(_root_.xor (bits true) (bits false))) && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))) }\n\n@[simp] lemma eval_sub (x : Bool \u2192 \u2115 \u2192 Bool) : sub.eval x = subSeq (x true) (x false) := by\n  dsimp [sub, eval]\n  rw [sub_eq_propagate]\n  congr\n  funext b\n  cases b;  rfl\n  simp\n  congr\n  funext i\n  cases i <;> simp\n\ndef neg : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n      (\u03bb _ => (!(bits ())) && (carry ()), _root_.xor (!(bits ())) (carry ())) }\n\n@[simp] lemma eval_neg (x : Unit \u2192 \u2115 \u2192 Bool) : neg.eval x = negSeq (x ()) := by\n  dsimp [neg, eval]\n  rw [neg_eq_propagate]\n\ndef not : PropagateStruc Unit :=\n{ \u03b1 := Empty,\n  i := by infer_instance,\n  init_carry := Empty.elim,\n  next_bit := \u03bb _carry bits => (Empty.elim, !(bits ())) }\n\n@[simp] lemma eval_not (x : Unit \u2192 \u2115 \u2192 Bool) : not.eval x = notSeq (x ()) := by\n  ext n; cases n <;> simp [not, notSeq, eval, propagate_succ]\n\ndef zero : PropagateStruc (Fin 0) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry _bits => (Empty.elim, false) }\n\n@[simp] lemma eval_zero (x : Fin 0 \u2192 \u2115 \u2192 Bool) : zero.eval x = zeroSeq := by\n  ext n; cases n <;> simp [zero, zeroSeq, eval, propagate_succ]\n\ndef one : PropagateStruc (Fin 0) :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry _bits => (\u03bb _ => false, carry ()) }\n\n@[simp] lemma eval_one (x : Fin 0 \u2192 \u2115 \u2192 Bool) : one.eval x = oneSeq := by\n  ext n; cases n <;> simp [one, oneSeq, eval, propagate_succ2, @eq_comm _ false]\n\ndef negOne : PropagateStruc (Fin 0) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry _bits => (Empty.elim, true) }\n\n@[simp] lemma eval_negOne (x : Fin 0 \u2192 \u2115 \u2192 Bool) : negOne.eval x = negOneSeq := by\n  ext n; cases n <;> simp [negOne, negOneSeq, eval, propagate_succ2]\n\ndef ls : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb carry bits => (bits, carry ()) }\n\n@[simp] lemma eval_ls (x : Unit \u2192 \u2115 \u2192 Bool) : ls.eval x = lsSeq (x ()) := by\n  ext n; cases n <;> simp [ls, lsSeq, eval, propagate_succ2]\n\ndef var (n : \u2115) : PropagateStruc (Fin (n+1)) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits (Fin.last n)) }\n\n@[simp] lemma eval_var (n : \u2115) (x : Fin (n+1) \u2192 \u2115 \u2192 Bool) : (var n).eval x = x (Fin.last n) := by\n  ext m; cases m <;> simp [var, eval, propagate_succ]\n\ndef incr : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry bits => (\u03bb _ => bits () && carry (), _root_.xor (bits ()) (carry ())) }\n\n@[simp] lemma eval_incr (x : Unit \u2192 \u2115 \u2192 Bool) : incr.eval x = incrSeq (x ()) := by\n  dsimp [incr, eval]\n  rw [incr_eq_propagate]\n\ndef decr : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry bits => (\u03bb _ => !(bits ()) && carry (), _root_.xor (bits ()) (carry ())) }\n\n@[simp] lemma eval_decr (x : Unit \u2192 \u2115 \u2192 Bool) : decr.eval x = decrSeq (x ()) := by\n  dsimp [decr, eval]\n  rw [decr_eq_propagate]\n\nend PropagateStruc\n\nstructure PropagateSolution (t : Term) extends PropagateStruc (Fin (arity t)) :=\n( good : t.evalFin = toPropagateStruc.eval )\n\ndef composeUnary\n    (p : PropagateStruc Unit)\n    {t : Term}\n    (q : PropagateSolution t) :\n    PropagateStruc (Fin (arity t)) :=\n  p.compose\n    (Fin (arity t))\n    _\n    (\u03bb _ => id)\n    (\u03bb _ => q.toPropagateStruc)\ndef X := @Bool.casesOn\ndef composeBinary\n    (p : PropagateStruc Bool)\n    {t\u2081 t\u2082 : Term}\n    (q\u2081 : PropagateSolution t\u2081)\n    (q\u2082 : PropagateSolution t\u2082) :\n    PropagateStruc (Fin (max (arity t\u2081) (arity t\u2082))) :=\n  p.compose (Fin (max (arity t\u2081) (arity t\u2082)))\n    (\u03bb b => Fin (cond b (arity t\u2081) (arity t\u2082)))\n    (\u03bb b i => Fin.castLe (by cases b <;> simp) i)\n    (\u03bb b => match b with\n      | true => q\u2081.toPropagateStruc\n      | false => q\u2082.toPropagateStruc)\n\n@[simp] lemma composeUnary_eval\n    (p : PropagateStruc Unit)\n    {t : Term}\n    (q : PropagateSolution t)\n    (x : Fin (arity t) \u2192 \u2115 \u2192 Bool) :\n    (composeUnary p q).eval x = p.eval (\u03bb _ => t.evalFin x) := by\n  rw [composeUnary, PropagateStruc.eval_compose, q.good]; rfl\n\n@[simp] lemma composeBinary_eval\n    (p : PropagateStruc Bool)\n    {t\u2081 t\u2082 : Term}\n    (q\u2081 : PropagateSolution t\u2081)\n    (q\u2082 : PropagateSolution t\u2082)\n    (x : Fin (max (arity t\u2081) (arity t\u2082)) \u2192 \u2115 \u2192 Bool) :\n    (composeBinary p q\u2081 q\u2082).eval x = p.eval\n      (\u03bb b => cond b (t\u2081.evalFin (fun i => x (Fin.castLe (by simp) i)))\n                  (t\u2082.evalFin (fun i => x (Fin.castLe (by simp) i)))) := by\n  rw [composeBinary, PropagateStruc.eval_compose, q\u2081.good, q\u2082.good]\n  congr\n  ext b\n  cases b <;> dsimp <;> congr <;> funext b <;> cases b <;> simp\n\ninstance {\u03b1 \u03b2 : Type} [Fintype \u03b1] [Fintype \u03b2] (b : Bool) :\n    Fintype (cond b \u03b1 \u03b2) :=\n  by cases b <;> simp <;> infer_instance\n\nlemma cond_propagate {\u03b1 \u03b1' \u03b2 \u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (init_carry' : \u03b1' \u2192 Bool)\n    (next_bit' : \u2200 (_carry : \u03b1' \u2192 Bool) (_bits : \u03b2' \u2192 Bool),\n      (\u03b1' \u2192 Bool) \u00d7 Bool)\n    {\u03b3 : Type} (f\u03b2 : \u03b2 \u2192 \u03b3) (f\u03b2' : \u03b2' \u2192 \u03b3)\n    (x : \u03b3 \u2192 \u2115 \u2192 Bool) (b : Bool) :\n    cond b (propagate init_carry next_bit (\u03bb b => (x (f\u03b2 b))))\n      (propagate init_carry' next_bit' (\u03bb b => (x (f\u03b2' b)))) =\n    propagate (show cond b \u03b1 \u03b1' \u2192 Bool from Bool.rec init_carry' init_carry b)\n      (show \u2200 (_carry : cond b \u03b1 \u03b1' \u2192 Bool) (_bits : cond b \u03b2 \u03b2' \u2192 Bool),\n          (cond b \u03b1 \u03b1' \u2192 Bool) \u00d7 Bool\n        from Bool.rec next_bit' next_bit b)\n      (show cond b \u03b2 \u03b2' \u2192 \u2115 \u2192 Bool from Bool.rec (\u03bb b => (x (f\u03b2' b))) (\u03bb b => (x (f\u03b2 b))) b) :=\n  by cases b <;> rfl\n\ndef termEvalEqPropagate : \u2200 (t : Term),\n  PropagateSolution t\n| var n =>\n  { toPropagateStruc := PropagateStruc.var n,\n    good := by ext; simp [Term.evalFin] }\n| zero =>\n  { toPropagateStruc := PropagateStruc.zero,\n    good := by ext; simp [Term.evalFin] }\n| one =>\n  { toPropagateStruc := PropagateStruc.one,\n    good := by ext; simp [Term.evalFin] }\n| negOne =>\n  { toPropagateStruc := PropagateStruc.negOne,\n    good := by ext; simp [Term.evalFin] }\n| Term.and t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.and q\u2081 q\u2082,\n    good := by ext; simp }\n| Term.or t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.or q\u2081 q\u2082,\n    good := by ext; simp }\n| Term.xor t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.xor q\u2081 q\u2082,\n    good := by ext; simp }\n| ls t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.ls q,\n    good := by ext; simp }\n| Term.not t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.not q,\n    good := by ext; simp }\n| add t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.add q\u2081 q\u2082,\n    good := by ext; simp }\n| sub t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.sub q\u2081 q\u2082,\n    good := by ext; simp }\n| neg t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.neg q,\n    good := by ext; simp }\n| incr t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.incr q,\n    good := by ext; simp }\n| decr t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.decr q,\n    good := by ext; simp }\n\nvariable\n  (init_carry : \u03b1 \u2192 Bool)\n  (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool))\n  (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool)\n\nvariable [Fintype \u03b1] [Fintype \u03b1']\n\nopen Fintype\n\nlemma exists_repeat_carry (seq : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    \u2203 n m : Fin (2 ^ (card \u03b1) + 1),\n      propagateCarry2 init_carry next_carry seq n =\n      propagateCarry2 init_carry next_carry seq m \u2227\n      n < m := by\n  by_contra h\n  haveI := Classical.decEq \u03b1\n  push_neg at h\n  have := \u03bb a b hab => (le_antisymm (h a b hab) (h b a hab.symm)).symm\n  have := Fintype.card_le_of_injective _ this\n  simp at this\n\nlemma propagateCarry2_eq_of_seq_eq_lt (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool))\n    (i : \u2115) (h : \u2200 (b) (j) (_hj : j < i), seq\u2081 b j = seq\u2082 b j) :\n    propagateCarry2 init_carry next_carry seq\u2081 i =\n      propagateCarry2 init_carry next_carry seq\u2082 i := by\n  induction' i with i ih\n  { simp [propagateCarry2] }\n  { simp only [propagateCarry2, h _ i (Nat.lt_succ_self i)]\n    rw [ih]\n    exact \u03bb b j hj => h b j (Nat.lt_succ_of_lt hj) }\n\nlemma propagate_eq_of_seq_eq_le (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (i : \u2115) (h : \u2200 (b) (j) (_hj : j \u2264 i), seq\u2081 b j = seq\u2082 b j) :\n    propagate init_carry next_bit seq\u2081 i =\n      propagate init_carry next_bit seq\u2082 i := by\n  cases i\n  { simp [propagate_zero, h _ 0 (le_refl _)] }\n  { simp only [propagate_succ2, propagate_succ2, h _ _ (le_refl _)]\n    congr 2\n    apply propagateCarry2_eq_of_seq_eq_lt\n    exact \u03bb b j hj => h b j (le_of_lt hj) }\n\n\nlemma propagateCarry2_eq_of_carry_eq (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (m n : \u2115)\n    (h\u2081 : propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 m =\n        propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 n) (x : \u2115)\n    (h\u2083 : \u2200 y b, y \u2264 x \u2192 seq\u2081 b (m + y) = seq\u2082 b (n + y))  :\n    propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 (m + x) =\n    propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 (n + x) := by\n  induction' x with x ih generalizing seq\u2081 seq\u2082\n  { simp [*] at * }\n  { simp only [propagateCarry2, Nat.add_eq, h\u2083 x _ (Nat.le_succ _)] at *\n    rw [ih]\n    assumption\n    exact \u03bb y b h => h\u2083 y b (Nat.le_succ_of_le h) }\n\nlemma propagate_eq_of_carry_eq (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (m n : \u2115)\n    (h\u2081 : propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 m =\n        propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 n) (x : \u2115)\n    (h\u2083 : \u2200 y b, y \u2264 x \u2192 seq\u2081 b (m + y) = seq\u2082 b (n + y))  :\n    propagate init_carry next_bit seq\u2081 (m + x) =\n    propagate init_carry next_bit seq\u2082 (n + x) := by\n  cases x\n  { cases m\n    { cases n\n      { simp [h\u2083 0 _ (le_refl _), propagateCarry2, *] at * }\n      { simp [*, h\u2083 0 _ (le_refl _), propagate_succ2] at *\n        rw [\u2190 h\u2081] } }\n    { cases n\n      { simp [*, propagate_succ2] at *\n        have := fun i => h\u2083 0 i rfl\n        dsimp at this\n        simp [this]\n        simp [h\u2081] }\n      { rw [propagate_succ2, h\u2081, propagate_succ2]\n        have := h\u2083 0\n        simp [*] at * } } }\n  { erw [Nat.add_succ, propagate_succ2, propagate_succ2, Nat.add_eq, Nat.add_eq]\n    simp [\u2190 Nat.succ_eq_add_one, \u2190 Nat.add_succ, h\u2083 _ _ (le_refl _)]\n    congr\n    . apply propagateCarry2_eq_of_carry_eq\n      . assumption\n      . exact \u03bb y b h => h\u2083 y b (Nat.le_succ_of_le h)\n    . funext i\n      rw [h\u2083]\n      exact Nat.le_succ _ }\n\nlemma propagateCarry_propagateCarry_add (x : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    \u2200 (init_carry : \u03b1 \u2192 Bool)\n      (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool)),\n    \u2200 n i : \u2115,\n    propagateCarry2 (propagateCarry2 init_carry next_carry x n)\n      next_carry (\u03bb b k => x b (k + n)) i =\n    propagateCarry2 init_carry next_carry x (i + n)\n  | init_carry, _next_carry, 0, 0 => by simp [propagateCarry2]\n  | init_carry, next_carr, n+1, 0 =>\n    by simp [propagateCarry, propagateCarry2_succ]\n  | init_carry, next_carry, n, i+1 => by\n    rw [propagateCarry2, add_assoc,\n      propagateCarry_propagateCarry_add _ _ _ _ i]\n    simp only [Nat.one_add, Nat.add_one, Nat.succ_add, Nat.add_succ,\n      add_zero, propagateCarry2, zero_add]\n\n\nlemma exists_repeat : \u2200 (seq : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (n : \u2115),\n    \u2203 (m : \u2115) (_hm : m < 2 ^ (card \u03b1)) (seq2 : \u03b2 \u2192 \u2115 \u2192 Bool),\n      propagate init_carry next_bit seq2 m = propagate init_carry next_bit seq n\n  | seq, n => by\n    by_cases hn2 : n < 2 ^ card \u03b1\n    { exact \u27e8n, hn2, seq, rfl\u27e9 }\n    { rcases exists_repeat_carry\n        (propagateCarry2 init_carry (\u03bb c b => (next_bit c b).1) seq\n          (n - 2 ^ card \u03b1))\n        (\u03bb carry bits => (next_bit  carry bits).1)\n        (\u03bb b i => seq b (i + (n - 2^ (card \u03b1)))) with \u27e8a, b, h\u2081, h\u2082\u27e9\n      simp only [propagateCarry_propagateCarry_add] at h\u2081\n      rcases have _wf : n - (b - a) < n :=\n         Nat.sub_lt (lt_of_lt_of_le (pow_pos (by norm_num) _) (le_of_not_lt hn2)) (Nat.sub_pos_of_lt h\u2082)\n        exists_repeat (\u03bb c i => if i < a + (n - 2 ^ card \u03b1) then seq c i else\n          seq c (i + (b - a))) (n - (b - a)) with \u27e8m, hmle, seq2, hm\u27e9\n      use m; use hmle; use seq2\n      rw [hm]; clear hm\n      have h1 : n - (b - a) = (a + (n - 2 ^ (card \u03b1))) + (2 ^ card \u03b1 - b) := by\n      { zify\n        rw [Nat.cast_sub, Nat.cast_sub, Nat.cast_sub, Nat.cast_sub]\n        ring_nf\n        exact Nat.le_of_lt_succ b.2\n        simp [*] at *\n        exact hn2\n        exact le_of_lt h\u2082\n        exact le_trans (Nat.sub_le _ _) (le_trans (Nat.le_of_lt_succ b.2)\n          (le_of_not_lt hn2)) }\n      rw [h1]\n      have h2 : n = (b + (n - 2 ^ card \u03b1)) + (2 ^ card \u03b1 - b) := by\n      { zify\n        rw [Nat.cast_sub, Nat.cast_sub]\n        ring\n        exact Nat.le_of_lt_succ b.2\n        simp [*] at *\n        exact hn2 }\n      conv_rhs => rw [h2]\n      refine' propagate_eq_of_carry_eq _ _ _ _ _ _ _ _ _\n      { have _h : \u2191b + (n - 2 ^ card \u03b1) = (a + (n - 2 ^ card \u03b1)) + (b - a) := by\n        { zify\n          rw [Nat.cast_sub, Nat.cast_sub]\n          ring_nf\n          exact le_of_lt h\u2082\n          exact le_of_not_lt hn2 }\n        rw [\u2190 h\u2081]\n        apply propagateCarry2_eq_of_seq_eq_lt\n        simp (config := { contextual := true }) }\n      { intro y c _hc\n        simp only [add_lt_iff_neg_left, not_lt_zero', if_false]\n        congr 1\n        zify\n        rw [Nat.cast_sub, Nat.cast_sub]\n        ring\n        exact le_of_lt h\u2082\n        exact le_of_not_lt hn2 } }\n\n\nlemma propagate_eq_zero_iff (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool) :\n    (\u2200 seq, propagate init_carry next_bit seq = zeroSeq) \u2194\n    (\u2200 seq, \u2200 i < 2 ^ (card \u03b1), propagate init_carry next_bit seq i = false) := by\n  constructor\n  { intro h i _\n    simp [h, zeroSeq] }\n  { intro h seq\n    funext i\n    rcases exists_repeat init_carry next_bit seq i with \u27e8j, hj, seq2, hseq2\u27e9\n    rw [\u2190 hseq2, h seq2 j hj, zeroSeq] }\n\nlemma eq_iff_xorSeq_eq_zero (seq\u2081 seq\u2082 : \u2115 \u2192 Bool) :\n    (\u2200 i, seq\u2081 i = seq\u2082 i) \u2194 (\u2200 i, xorSeq seq\u2081 seq\u2082 i = zeroSeq i) := by\n  simp [Function.funext_iff, xorSeq, zeroSeq]\n  constructor\n  { intro i _; simp [*] }\n  { intro h a\n    specialize h a\n    revert h\n    cases (seq\u2081 a) <;> cases (seq\u2082 a) <;> simp [*] at * }\n\nlemma eval_eq_iff_xorSeq_eq_zero (t\u2081 t\u2082 : Term) :\n    t\u2081.eval = t\u2082.eval \u2194 (t\u2081.xor t\u2082).evalFin = \u03bb _ => zeroSeq := by\n  simp only [Function.funext_iff, Term.eval, Term.evalFin,\n    \u2190 eq_iff_xorSeq_eq_zero, \u2190 evalFin_eq_eval]\n  constructor\n  { intro h seq n\n    have := h (\u03bb j => if hj : j < (arity (t\u2081.xor t\u2082)) then seq \u27e8j, hj\u27e9 else \u03bb _ => false) n\n    simp at this\n    convert this }\n  { intro h seq m\n    exact h (\u03bb j => seq j) _ }\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Bitfid/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.39999754232136026}}
{"text": "import category_theory.opposites\nimport category_theory.hom_functor\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.pullbacks\nimport topology.opens\nimport topology.category.Top.opens\n\nopen opposite\nopen category_theory\n\n\nset_option pp.universes true\n\nsection sieves\n\nuniverses v u\nvariables {C : Type u} [\ud835\udc9e : category.{v} C]\ninclude \ud835\udc9e\n\n /- Maybe define sieve as a subfunctor? (but then I have to define subfunctor...)\n \n structure subfunctor (F : C \u2964 D) :=\n(G : C \u2964 D)\n(obj : \u2200 (c : C), ) -/\n\n/- Could potentially simplify hom definition by using hom_obj in hom_functor.lean somehow...-/\nstructure sieve (X : C) := \n(map : \u03a0 (Y : C), set (Y \u27f6 X))\n(comp : \u2200 (Y Z: C) (g : Y \u27f6 Z) (f \u2208 map Z), g \u226b f \u2208 map Y)\n\ninstance sieve_partial_order {U : C} : partial_order (sieve U) :=\n    {   le := (\u03bb S, \u03bb T, S.map \u2264 T.map),\n        le_refl := by tidy,\n        le_trans := by tidy,\n        le_antisymm := by {intros a b hab hba, cases a, cases b, tidy, }\n    }\n\ninstance sieve_semilattice_inf {U : C} : lattice.semilattice_inf (sieve U) :=\n    { inf := \u03bb S, \u03bb T, \u27e8S.map \u2293 T.map, by {\n        intros Y Z g f hf,\n        have h1 : (S.map \u2293 T.map) \u2264 S.map := lattice.inf_le_left,\n        have hS := S.comp Y Z g f (h1 Z hf),\n        have h2 : (S.map \u2293 T.map) \u2264 T.map := lattice.inf_le_right,\n        have hT := T.comp Y Z g f (h2 Z hf),\n        exact \u27e8hS, hT\u27e9, }\u27e9,\n      inf_le_left := by tidy,\n      inf_le_right := by tidy,\n      le_inf := by tidy,\n      ..sieve_partial_order\n    }\n\ninstance sieve_semilattice_sup {U : C} : lattice.semilattice_sup (sieve U) :=\n    { sup := \u03bb S, \u03bb T, \u27e8S.map \u2294 T.map, by {\n        intros Y Z g f hf,\n        cases hf with hS hT, {\n        have h := S.comp Y Z g f hS,\n        have h1 : S.map \u2264 (S.map \u2294 T.map) := lattice.le_sup_left,\n        exact (h1 Y) h, }, {\n        have h := T.comp Y Z g f hT,\n        have h1 : T.map \u2264 (S.map \u2294 T.map) := lattice.le_sup_right,\n        exact (h1 Y) h, } }\u27e9,\n      le_sup_left := by {tidy, left, exact a_2,},\n      le_sup_right := by {tidy, right, exact a_2,},\n      sup_le := by tidy,\n      ..sieve_partial_order\n    }\n\n--inf and sup of sets of sieves SGA 4 I 4.3.2.\ninstance sieve_complete_lattice {U : C} : lattice.complete_lattice (sieve U) :=\n{\n    top := \u27e8 \u03bb Y, {a | true }, by tidy \u27e9,\n    le_top := by tidy,\n    bot := \u27e8 \u03bb Y, \u2205, by tidy \u27e9,\n    bot_le := by tidy,\n    Inf := \u03bb A, \u27e8 \u03bb V, {f : V \u27f6 U | \u2200 S : sieve U, S \u2208 A \u2192 f \u2208 S.map V}, by {\n        intros Y Z g f hf S hS,\n        apply S.comp,\n        exact hf S hS,\n    }\u27e9, \n    Inf_le := by {\n        intros A S hS V f hf,\n        exact hf S hS,\n    },\n    le_Inf := by {\n        intros A S hS V f hf,\n        intros T hT,\n        exact (hS T hT) V hf,\n    },\n    Sup := \u03bb A, \u27e8 \u03bb V, {f : V \u27f6 U | \u2203 S : sieve U, S \u2208 A \u2227 f \u2208 S.map V}, by {\n        intros Y Z g f hf,\n        cases hf with S hS,\n        existsi S,\n        split,\n        exact hS.1,\n        apply S.comp,\n        exact hS.2,\n    }\u27e9,\n    Sup_le := by {\n        intros A S hS V f hf,\n        cases hf with T hT,\n        exact hS T hT.1 V hT.2,\n    },\n    le_Sup := by {\n        intros S T hT V f hf,\n        split,\n        split,\n        exact hT,\n        exact hf,\n    },\n    ..sieve_semilattice_inf,\n    ..sieve_semilattice_sup,\n}\n\n--SGA 4 I 4.3.3. sieve generated by family of morphisms \ndef sieve_gen_by {X : C} (fa : \u03a0 Y : C, set (Y \u27f6 X)) : sieve X := \n    lattice.Inf {S : sieve X | fa \u2264 S.map }\n\ndef id_sieve (X : C) : sieve X := \u27e8\u03bb (Y : C), {f | true}, by tidy\u27e9\n\ndef pullback_sieve {X Y : C} (f : Y \u27f6 X) (S : sieve X) : sieve Y \n    := \u27e8\u03bb Z, {g | g \u226b f \u2208 S.map Z}, by {tidy, apply S.comp, exact H}\u27e9\n\nlemma pullback_id_sieve {X Y : C} (f : Y \u27f6 X) \n    : pullback_sieve f (id_sieve X) = id_sieve Y := by tidy\n\nlemma sieve_ext {X : C} {S T : sieve X} : S.map = T.map \u2192 S = T\n    := by {cases S, intro H, cases T, tidy, }\n\nstructure sieve_domain {X : C} (S : sieve X) :=\n(Y : C)\n(f : Y \u27f6 X)\n(in_cover : f \u2208 S.map Y)\n\nomit \ud835\udc9e\nend sieves\n", "meta": {"author": "ImperialCollegeLondon", "repo": "condensed-sets", "sha": "e308291646396003dbed3896e5fbb40cb57c7050", "save_path": "github-repos/lean/ImperialCollegeLondon-condensed-sets", "path": "github-repos/lean/ImperialCollegeLondon-condensed-sets/condensed-sets-e308291646396003dbed3896e5fbb40cb57c7050/src/sieve.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3999975349934194}}
{"text": "-- /-\n-- Copyright (c) 2019 The Flypitch Project. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n\n-- Authors: Jesse Han, Floris van Doorn\n-- -/\n-- /- A development of first-order logic in Lean.\n\n-- * The object theory uses classical logic\n-- * We use de Bruijn variables.\n-- * We use a deep embedding of the logic, i.e. the type of terms and formulas is inductively defined.\n-- * There is no well-formedness predicate; all elements of type \"term\" are well-formed.\n-- -/\n\n-- import to_mathlib\n\n-- open nat set\n-- universe variables u v\n\n-- local notation h :: t  := dvector.cons h t\n-- local notation `[` l:(foldr `, ` (h t, dvector.cons h t) dvector.nil `]`:0) := l\n\n-- namespace fol\n\n-- /- realizers of variables are just maps \u2115 \u2192 S. We need some operations on them -/\n\n-- /-- Given a valuation v, a nat n, and an x : S, return v truncated to its first n values, with the rest of the values replaced by x. --/\n-- def subst_realize {S : Type u} (v : \u2115 \u2192 S) (x : S) (n k : \u2115) : S :=\n-- if k < n then v k else if n < k then v (k - 1) else x\n\n-- notation v `[`:95 x ` // `:95 n `]`:0 := fol.subst_realize v x n\n\n-- /-- --/\n-- @[simp] lemma subst_realize_lt {S : Type u} (v : \u2115 \u2192 S) (x : S) {n k : \u2115} (H : k < n) :\n--   v[x // n] k = v k :=\n-- by simp only [H, subst_realize, if_true, eq_self_iff_true]\n\n-- @[simp] lemma subst_realize_gt {S : Type u} (v : \u2115 \u2192 S) (x : S) {n k : \u2115} (H : n < k) :\n--   v[x // n] k = v (k-1) :=\n-- have h : \u00ac(k < n), from lt_asymm H,\n-- by simp only [*, subst_realize, if_true, eq_self_iff_true, if_false]\n\n-- @[simp] lemma subst_realize_var_eq {S : Type u} (v : \u2115 \u2192 S) (x : S) (n : \u2115) : v[x // n] n = x :=\n-- by simp only [subst_realize, lt_irrefl, eq_self_iff_true, if_false]\n\n-- lemma subst_realize_congr {S : Type u} {v v' : \u2115 \u2192 S} (hv : \u2200k, v k = v' k) (x : S) (n k : \u2115) :\n--  v [x // n] k = v' [x // n] k :=\n-- by apply decidable.lt_by_cases k n; intro h;\n--    simp only [*, subst_realize_lt, subst_realize_gt, subst_realize_var_eq, eq_self_iff_true]\n\n-- lemma subst_realize2 {S : Type u} (v : \u2115 \u2192 S) (x x' : S) (n\u2081 n\u2082 k : \u2115) :\n--   v [x' // n\u2081 + n\u2082] [x // n\u2081] k = v [x // n\u2081] [x' // n\u2081 + n\u2082 + 1] k :=\n-- begin\n--     apply decidable.lt_by_cases k n\u2081; intro h,\n--     { have : k < n\u2081 + n\u2082, from lt_of_le_of_lt (k.le_add_right n\u2082) (add_lt_add_right h n\u2082),\n--       have : k < n\u2081 + n\u2082 + 1, from lt.step this,\n--       simp only [*, fol.subst_realize_lt, eq_self_iff_true] },\n--     { have : k < n\u2082 + (k + 1), from nat.lt_add_left _ _ n\u2082 (lt.base k),\n--       subst h, simp [*, -add_comm] },\n--     apply decidable.lt_by_cases k (n\u2081 + n\u2082 + 1); intro h',\n--     { have : k - 1 < n\u2081 + n\u2082, from (nat.sub_lt_right_iff_lt_add (one_le_of_lt h)).2 h',\n--       simp [*, -add_comm, -add_assoc] },\n--     { subst h', simp [h, -add_comm, -add_assoc] },\n--     { have : n\u2081 + n\u2082 < k - 1, from nat.lt_sub_right_of_add_lt h',\n--       have : n\u2081 < k - 1, from lt_of_le_of_lt (n\u2081.le_add_right n\u2082) this,\n--       simp only [*, fol.subst_realize_gt, eq_self_iff_true] }\n-- end\n\n-- lemma subst_realize2_0 {S : Type u} (v : \u2115 \u2192 S) (x x' : S) (n k : \u2115) :\n--   v [x' // n] [x // 0] k = v [x // 0] [x' // n + 1] k :=\n-- let h := subst_realize2 v x x' 0 n k in by simp only [zero_add] at h; exact h\n\n-- lemma subst_realize_irrel {S : Type u} {v\u2081 v\u2082 : \u2115 \u2192 S} {n : \u2115} (hv : \u2200k < n, v\u2081 k = v\u2082 k) (x : S)\n--   {k : \u2115} (hk : k < n + 1) : v\u2081[x // 0] k = v\u2082[x // 0] k :=\n-- begin\n--   cases k, refl, have h : 0 < succ k, from zero_lt_succ k, simp [h, hv k (lt_of_succ_lt_succ hk)]\n-- end\n\n-- lemma lift_subst_realize_cancel {S : Type u} (v : \u2115 \u2192 S) (k : \u2115) :\n--   (\u03bbn, v (n + 1))[v 0 // 0] k = v k :=\n-- begin\n--   cases k, refl, have h : 0 < succ k, from zero_lt_succ k, simp [h],\n-- end\n\n-- lemma subst_fin_realize_eq {S : Type u} {n} {v\u2081 : dvector S n} {v\u2082 : \u2115 \u2192 S}\n--   (hv : \u2200k (hk : k < n), v\u2081.nth k hk = v\u2082 k) (x : S) (k : \u2115) (hk : k < n+1) :\n--     (x::v\u2081).nth k hk = v\u2082[x // 0] k :=\n-- begin\n--   cases k, refl,\n--   have h : 0 < succ k, from zero_lt_succ k,\n--   have h' : (0 : fin (n+1)).val < (fin.mk (succ k) hk).val, from h,\n--   rw [subst_realize_gt v\u2082 x h, dvector.nth], apply hv\n-- end\n\n-- structure Language : Type (u+1) :=\n-- (functions : \u2115 \u2192 Type u) (relations : \u2115 \u2192 Type u)\n\n-- def Language.constants (L : Language) := L.functions 0\n\n-- variable (L : Language.{u})\n\n-- /- preterm L l is a partially applied term. if applied to n terms, it becomes a term.\n-- * Every element of preterm L 0 is a well-formed term.\n-- * We use this encoding to avoid mutual or nested inductive types, since those are not too convenient to work with in Lean. -/\n-- inductive preterm : \u2115 \u2192 Type u\n-- | var {} : \u2200 (k : \u2115), preterm 0\n-- | func : \u2200 {l : \u2115} (f : L.functions l), preterm l\n-- | app : \u2200 {l : \u2115} (t : preterm (l + 1)) (s : preterm 0), preterm l\n-- export preterm\n\n-- @[reducible] def term := preterm L 0\n\n-- variable {L}\n-- prefix `&`:max := fol.preterm.var\n\n-- @[simp] def apps : \u2200{l}, preterm L l \u2192 dvector (term L) l \u2192 term L\n-- | _ t []       := t\n-- | _ t (t'::ts) := apps (app t t') ts\n\n-- -- @[simp] def apps' : \u2200{l l'}, preterm L (l'+l) \u2192 dvector (term L) l \u2192 preterm L l'\n-- -- | _ _ t []       := t\n-- -- | _ _ t (t'::ts) := apps' (app t t') ts\n\n-- -- @[simp] def rev_apps : \u2200{l l'}, preterm L (l+l) \u2192 dvector (term L) l' \u2192 preterm L l\n-- -- | _ _ t []       := sorry\n-- -- | l _ t (@dvector.cons _ l' t' ts) := app (@rev_apps (l+1) l' t ts) t'\n\n-- @[simp] lemma apps_zero (t : term L) (ts : dvector (term L) 0) : apps t ts = t :=\n-- by cases ts; refl\n\n-- lemma apps_eq_app {l} (t : preterm L (l+1)) (s : term L) (ts : dvector (term L) l) :\n--   \u2203t' s', apps t (s::ts) = app t' s' :=\n-- begin\n--   induction ts generalizing s, exact \u27e8t, s, rfl\u27e9, exact ts_ih (app t s) ts_x\n-- end\n\n-- namespace preterm\n-- @[simp] def change_arity' : \u2200{l l'} (h : l = l') (t : preterm L l), preterm L l'\n-- | _ _ h &k          := by induction h; exact &k\n-- | _ _ h (func f)    := func (by induction h; exact f)\n-- | _ _ h (app t\u2081 t\u2082) := app (change_arity' (congr_arg succ h) t\u2081) t\u2082\n\n-- @[simp] lemma change_arity'_rfl : \u2200{l} (t : preterm L l), change_arity' rfl t = t\n-- | _ &k          := by refl\n-- | _ (func f)    := by refl\n-- | _ (app t\u2081 t\u2082) := by dsimp; simp*\n\n-- end preterm\n\n-- -- lemma apps'_concat {l l'} (t : preterm L (l'+(l+1))) (s : term L) (ts : dvector (term L) l) :\n-- --   apps' t (ts.concat s) = app (apps' (t.change_arity' (by simp)) ts) s :=\n-- -- begin\n-- --   induction ts generalizing s,\n-- --   { simp },\n-- --   { apply ts_ih (app t ts_x) s }\n-- -- end\n\n-- lemma apps_ne_var {l} {f : L.functions l} {ts : dvector (term L) l} {k : \u2115} :\n--   apps (func f) ts \u2260 &k :=\n-- begin\n--   intro h, cases ts, injection h,\n--   rcases apps_eq_app (func f) ts_x ts_xs with \u27e8t, s, h'\u27e9, cases h.symm.trans h'\n-- end\n\n-- lemma apps_inj' {l} {t t' : preterm L l} {ts ts' : dvector (term L) l}\n--   (h : apps t ts = apps t' ts') : t = t' \u2227 ts = ts' :=\n-- begin\n--   induction ts; cases ts',\n--   { exact \u27e8h, rfl\u27e9 },\n--   { rcases ts_ih h with \u27e8\u27e8rfl, rfl\u27e9, rfl\u27e9, exact \u27e8rfl, rfl\u27e9 }\n-- end\n\n-- -- lemma apps_inj_length {l l'} {f : L.functions l} {f' : L.functions l'}\n-- --   {ts : dvector (term L) l} {ts' : dvector (term L) l'}\n-- --   (h : apps (func f) ts = apps (func f') ts') : l = l' :=\n-- -- begin\n-- --   sorry\n-- -- end\n\n-- -- lemma apps'_inj_length {l\u2081 l\u2082 l'} {f : L.functions (l' + l\u2081)} {f' : L.functions (l' + l\u2082)}\n-- --   {ts : dvector (term L) l\u2081} {ts' : dvector (term L) l\u2082}\n-- --   (h : apps' (func f) ts = apps' (func f') ts') : l\u2081 = l\u2082 :=\n-- -- begin\n-- --   sorry\n-- --   -- induction ts generalizing l'; cases ts',\n-- --   -- { refl },\n-- --   -- { rcases apps'_eq_app (func f') ts'_x ts'_xs with \u27e8t, s, h'\u27e9, cases h.trans h' },\n-- --   -- { rcases apps'_eq_app (func f) ts_x ts_xs with \u27e8t, s, h'\u27e9, cases h.symm.trans h' },\n-- --   -- { rcases apps'_eq_app (func f) ts_x ts_xs with \u27e8t\u2081, s\u2081, h\u2081\u27e9,\n-- --   --   rcases apps'_eq_app (func f') ts'_x ts'_xs with \u27e8t\u2082, s\u2082, h\u2082\u27e9,\n-- --   --    }\n-- -- end\n\n-- lemma apps_inj {l} {f f' : L.functions l} {ts ts' : dvector (term L) l}\n--   (h : apps (func f) ts = apps (func f') ts') : f = f' \u2227 ts = ts' :=\n-- by rcases apps_inj' h with \u27e8h', rfl\u27e9; cases h'; exact \u27e8rfl, rfl\u27e9\n\n-- def term_of_function {l} (f : L.functions l) : arity' (term L) (term L) l :=\n-- arity'.of_dvector_map $ apps (func f)\n\n-- @[elab_as_eliminator] def term.rec {C : term L \u2192 Sort v}\n--   (hvar : \u2200(k : \u2115), C &k)\n--   (hfunc : \u03a0 {l} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : \u2200t, ts.pmem t \u2192 C t),\n--     C (apps (func f) ts)) : \u2200(t : term L), C t :=\n-- have h : \u2200{l} (t : preterm L l) (ts : dvector (term L) l) (ih_ts : \u2200s, ts.pmem s \u2192 C s),\n--   C (apps t ts),\n-- begin\n--   intros, induction t; try {rw ts.zero_eq},\n--   { apply hvar },\n--   { apply hfunc t_f ts ih_ts },\n--   { apply t_ih_t (t_s::ts), intros t ht,\n--     cases ht,\n--     { induction ht, apply t_ih_s ([]), intros s hs, cases hs },\n--     { exact ih_ts t ht }},\n-- end,\n-- \u03bbt, h t ([]) (by intros s hs; cases hs)\n\n-- @[elab_as_eliminator] def term.elim' {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C) :\n--   \u2200{l} (t : preterm L l) (ts : dvector (term L) l) (ih_ts : dvector C l), C\n-- | _ &k ts ih_ts        := hvar k\n-- | _ (func f) ts ih_ts  := hfunc f ts ih_ts\n-- | _ (app t s) ts ih_ts := term.elim' t (s::ts) (term.elim' s ([]) ([])::ih_ts)\n\n-- @[elab_as_eliminator] def term.elim {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C) :\n--   \u2200(t : term L), C :=\n-- \u03bbt, term.elim' hvar hfunc t ([]) ([])\n\n-- lemma term.elim'_apps {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C)\n--   {l} (t : preterm L l) (ts : dvector (term L) l) :\n--   @term.elim' L C hvar hfunc 0 (apps t ts) ([]) ([]) = @term.elim' L C hvar hfunc l t ts\n--   (ts.map $ term.elim hvar hfunc) :=\n-- begin\n--   induction ts,\n--   { refl },\n--   { dsimp only [dvector.map, apps], rw [ts_ih], refl }\n-- end\n\n-- lemma term.elim_apps {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C)\n--   {l} (f : L.functions l) (ts : dvector (term L) l) :\n--   @term.elim L C hvar hfunc (apps (func f) ts) = hfunc f ts (ts.map $ @term.elim L C hvar hfunc) :=\n-- by dsimp only [term.elim]; rw term.elim'_apps; refl\n\n-- /- lift_term_at _ t n m raises variables in t which are at least m by n -/\n-- @[simp] def lift_term_at : \u2200 {l}, preterm L l \u2192 \u2115 \u2192 \u2115 \u2192 preterm L l\n-- | _ &k          n m := &(if m \u2264 k then k+n else k)\n-- | _ (func f)    n m := func f\n-- | _ (app t\u2081 t\u2082) n m := app (lift_term_at t\u2081 n m) (lift_term_at t\u2082 n m)\n\n-- notation t ` \u2191' `:90 n ` # `:90 m:90 := fol.lift_term_at t n m -- input \u2191 with \\u or \\upa\n\n-- -- @[simp] lemma lift_term_var_le {k n m} (h : m \u2264 k) : &k \u2191' n # m = (&(k+n) : term L) := dif_pos h\n-- -- @[simp] lemma lift_term_var_gt {k n m} (h : \u00ac(m \u2264 k)) : &k \u2191' n # m = (&k : term L) := dif_neg h\n-- -- @[simp] lemma lift_term_at_func {l} (f : L.functions l) (n m) : func f \u2191' n # m = func f := by refl\n-- -- @[simp] lemma lift_term_at_app {l} (t : preterm L (l+1)) (s : preterm L 0) (n m) :\n-- --   app t s \u2191' n # m = app (t \u2191' n # m) (s \u2191' n # m) := by refl\n\n-- @[reducible] def lift_term {l} (t : preterm L l) (n : \u2115) : preterm L l := t \u2191' n # 0\n-- infix ` \u2191 `:100 := fol.lift_term -- input \u2191' with \\u or \\upa\n-- @[reducible, simp] def lift_term1 {l} (t : preterm L l) : preterm L l := t \u2191 1\n\n-- @[simp] lemma lift_term_def {l} (t : preterm L l) (n : \u2115) : t \u2191' n # 0 = t \u2191 n := by refl\n\n-- lemma injective_lift_term_at : \u2200 {l} {n m : \u2115},\n--   function.injective (\u03bb(t : preterm L l), lift_term_at t n m)\n-- | _ n m &k &k' h :=\n--   by by_cases h\u2081 : m \u2264 k; by_cases h\u2082 : m \u2264 k'; simp [h\u2081, h\u2082] at h;\n--      congr;[assumption, skip, skip, assumption]; exfalso; try {apply h\u2081};\n--      try {apply h\u2082}; subst h; apply le_trans (by assumption) (le_add_left _ _)\n-- | _ n m &k (func f')            h := by cases h\n-- | _ n m &k (app t\u2081' t\u2082')        h := by cases h\n-- | _ n m (func f) &k'            h := by cases h\n-- | _ n m (func f) (func f')      h := h\n-- | _ n m (func f) (app t\u2081' t\u2082')  h := by cases h\n-- | _ n m (app t\u2081 t\u2082) &k'         h := by cases h\n-- | _ n m (app t\u2081 t\u2082) (func f')   h := by cases h\n-- | _ n m (app t\u2081 t\u2082) (app t\u2081' t\u2082') h :=\n--   begin injection h, congr; apply injective_lift_term_at; assumption end\n\n-- @[simp] lemma lift_term_at_zero : \u2200 {l} (t : preterm L l) (m : \u2115), t \u2191' 0 # m = t\n-- | _ &k          m := by simp [lift_term_at]\n-- | _ (func f)    m := by refl\n-- | _ (app t\u2081 t\u2082) m := by dsimp; congr; apply lift_term_at_zero\n\n-- @[simp] lemma lift_term_zero {l} (t : preterm L l) : t \u2191 0 = t := lift_term_at_zero t 0\n\n-- /- the following lemmas simplify iterated lifts, depending on the size of m' -/\n-- lemma lift_term_at2_small : \u2200 {l} (t : preterm L l) (n n') {m m'}, m' \u2264 m \u2192\n--   (t \u2191' n # m) \u2191' n' # m' = (t \u2191' n' # m') \u2191' n # (m + n')\n-- | _ &k          n n' m m' H :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2081 : m' \u2264 k := le_trans H h,\n--       have h\u2082 : m' \u2264 k + n, from le_trans h\u2081 (k.le_add_right n),\n--       simp [*, -add_assoc, -add_comm], simp },\n--     { have h\u2081 : \u00acm + n' \u2264 k + n', from \u03bb h', h (le_of_add_le_add_right h'),\n--       have h\u2082 : \u00acm + n' \u2264 k, from \u03bb h', h\u2081 (le_trans h' (k.le_add_right n')),\n--       by_cases h' : m' \u2264 k; simp [*, -add_comm, -add_assoc] }\n--   end\n-- | _ (func f)    n n' m m' H := by refl\n-- | _ (app t\u2081 t\u2082) n n' m m' H :=\n--   begin dsimp; congr1; apply lift_term_at2_small; assumption end\n\n-- lemma lift_term_at2_medium : \u2200 {l} (t : preterm L l) {n} (n') {m m'}, m \u2264 m' \u2192 m' \u2264 m+n \u2192\n--   (t \u2191' n # m) \u2191' n' # m' = t \u2191' (n+n') # m\n-- | _ &k          n n' m m' H\u2081 H\u2082 :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2081 : m' \u2264 k + n, from le_trans H\u2082 (add_le_add_right h n), simp [*, -add_comm], },\n--     { have h\u2081 : \u00acm' \u2264 k, from \u03bb h', h (le_trans H\u2081 h'), simp [*, -add_comm, -add_assoc] }\n--   end\n-- | _ (func f)    n n' m m' H\u2081 H\u2082 := by refl\n-- | _ (app t\u2081 t\u2082) n n' m m' H\u2081 H\u2082 :=\n--   begin dsimp; congr1; apply lift_term_at2_medium; assumption end\n\n-- lemma lift_term2_medium {l} (t : preterm L l) {n} (n') {m'} (h : m' \u2264 n) :\n--   (t \u2191 n) \u2191' n' # m' = t \u2191 (n+n') :=\n-- lift_term_at2_medium t n' m'.zero_le (by simp*)\n\n-- lemma lift_term2 {l} (t : preterm L l) (n n') : (t \u2191 n) \u2191 n' = t \u2191 (n+n') :=\n-- lift_term2_medium t n' n.zero_le\n\n-- lemma lift_term_at2_eq {l} (t : preterm L l) (n n' m : \u2115) :\n--   (t \u2191' n # m) \u2191' n' # (m+n) = t \u2191' (n+n') # m :=\n-- lift_term_at2_medium t n' (m.le_add_right n) (le_refl _)\n\n-- lemma lift_term_at2_large {l} (t : preterm L l) {n} (n') {m m'} (H : m + n \u2264 m') :\n--   (t \u2191' n # m) \u2191' n' # m' = (t \u2191' n' # (m'-n)) \u2191' n # m :=\n-- have H\u2081 : n \u2264 m', from le_trans (n.le_add_left m) H,\n-- have H\u2082 : m \u2264 m' - n, from nat.le_sub_right_of_add_le H,\n-- begin rw fol.lift_term_at2_small t n' n H\u2082, rw [nat.sub_add_cancel], exact H\u2081 end\n\n-- @[simp] lemma lift_term_var0 (n : \u2115) : &0 \u2191 n = (&n : term L) :=\n-- by have h : 0 \u2264 0 := le_refl 0; rw [\u2190lift_term_def]; simp [h, -lift_term_def]\n\n-- @[simp] lemma lift_term_at_apps {l} (t : preterm L l) (ts : dvector (term L) l) (n m : \u2115) :\n--   (apps t ts) \u2191' n # m = apps (t \u2191' n # m) (ts.map $ \u03bbx, x \u2191' n # m) :=\n-- by induction ts generalizing t;[refl, apply ts_ih (app t ts_x)]\n\n-- @[simp] lemma lift_term_apps {l} (t : preterm L l) (ts : dvector (term L) l) (n : \u2115) :\n--   (apps t ts) \u2191 n = apps (t \u2191 n) (ts.map $ \u03bbx, x \u2191 n) :=\n-- lift_term_at_apps t ts n 0\n\n-- /- subst_term t s n substitutes s for (&n) and reduces the level of all variables above n by 1 -/\n-- def subst_term : \u2200 {l}, preterm L l \u2192 term L \u2192 \u2115 \u2192 preterm L l\n-- | _ &k          s n := subst_realize var (s \u2191 n) n k\n-- | _ (func f)    s n := func f\n-- | _ (app t\u2081 t\u2082) s n := app (subst_term t\u2081 s n) (subst_term t\u2082 s n)\n\n-- notation t `[`:max s ` // `:95 n `]`:0 := fol.subst_term t s n\n\n-- @[simp] lemma subst_term_var_lt (s : term L) {k n : \u2115} (H : k < n) : &k[s // n] = &k :=\n-- by simp only [H, fol.subst_term, fol.subst_realize_lt, eq_self_iff_true]\n\n-- @[simp] lemma subst_term_var_gt (s : term L) {k n : \u2115} (H : n < k) : &k[s // n] = &(k-1) :=\n-- by simp only [H, fol.subst_term, fol.subst_realize_gt, eq_self_iff_true]\n\n-- @[simp] lemma subst_term_var_eq (s : term L) (n : \u2115) : &n[s // n] = s \u2191' n # 0 :=\n-- by simp [subst_term]\n\n-- lemma subst_term_var0 (s : term L) : &0[s // 0] = s := by simp\n\n-- @[simp] lemma subst_term_func {l} (f : L.functions l) (s : term L) (n : \u2115) :\n--   (func f)[s // n] = func f :=\n-- by refl\n\n-- @[simp] lemma subst_term_app {l} (t\u2081 : preterm L (l+1)) (t\u2082 s : term L) (n : \u2115) :\n--   (app t\u2081 t\u2082)[s // n] = app (t\u2081[s // n]) (t\u2082[s // n]) :=\n-- by refl\n\n-- @[simp] lemma subst_term_apps {l} (t : preterm L l) (ts : dvector (term L) l) (s : term L)\n--   (n : \u2115) : (apps t ts)[s // n] = apps (t[s // n]) (ts.map $ \u03bbx, x[s // n]) :=\n-- by induction ts generalizing t;[refl, apply ts_ih (app t ts_x)]\n\n-- /- the following lemmas simplify first lifting and then substituting, depending on the size\n--   of the substituted variable -/\n-- lemma lift_at_subst_term_large : \u2200{l} (t : preterm L l) (s : term L) {n\u2081} (n\u2082) {m}, m \u2264 n\u2081 \u2192\n--  (t \u2191' n\u2082 # m)[s // n\u2081+n\u2082] = (t [s // n\u2081]) \u2191' n\u2082 # m\n-- | _ &k          s n\u2081 n\u2082 m h :=\n--   begin\n--     apply decidable.lt_by_cases k n\u2081; intro h\u2082,\n--     { have : k < n\u2081 + n\u2082, from lt_of_le_of_lt (k.le_add_right n\u2082) (by simp*),\n--       by_cases m \u2264 k; simp* },\n--     { subst h\u2082, simp [*, lift_term2_medium] },\n--     { have h\u2082 : m < k, by apply lt_of_le_of_lt; assumption,\n--       have : m \u2264 k - 1, from nat.le_sub_right_of_add_le (succ_le_of_lt h\u2082),\n--       have : m \u2264 k, from le_of_lt h\u2082,\n--       have : 1 \u2264 k, from one_le_of_lt h\u2082,\n--       simp [*, nat.add_sub_swap this n\u2082, -add_assoc, -add_comm] }\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m h := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m h := by simp*\n\n-- lemma lift_subst_term_large {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 n\u2082)[s // n\u2081+n\u2082] = (t [s // n\u2081]) \u2191 n\u2082 :=\n-- lift_at_subst_term_large t s n\u2082 n\u2081.zero_le\n\n-- lemma lift_subst_term_large' {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 n\u2082)[s // n\u2082+n\u2081] = (t [s // n\u2081]) \u2191 n\u2082 :=\n-- by rw [add_comm]; apply lift_subst_term_large\n\n-- lemma lift_at_subst_term_medium : \u2200{l} (t : preterm L l) (s : term L) {n\u2081 n\u2082 m}, m \u2264 n\u2082 \u2192\n--   n\u2082 \u2264 m + n\u2081 \u2192 (t \u2191' n\u2081+1 # m)[s // n\u2082] = t \u2191' n\u2081 # m\n-- | _ &k          s n\u2081 n\u2082 m h\u2081 h\u2082 :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2083 : n\u2082 < k + (n\u2081 + 1), from lt_succ_of_le (le_trans h\u2082 (add_le_add_right h _)),\n--       simp [*, add_sub_cancel_right] },\n--     { have h\u2083 : k < n\u2082, from lt_of_lt_of_le (lt_of_not_ge h) h\u2081, simp* }\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m h\u2081 h\u2082 := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m h\u2081 h\u2082 := by simp*\n\n-- lemma lift_subst_term_medium {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 ((n\u2081 + n\u2082) + 1))[s // n\u2081] = t \u2191 (n\u2081 + n\u2082) :=\n-- lift_at_subst_term_medium t s n\u2081.zero_le (by rw [zero_add]; exact n\u2081.le_add_right n\u2082)\n\n-- lemma lift_at_subst_term_eq {l} (t : preterm L l) (s : term L) (n : \u2115) : (t \u2191' 1 # n)[s // n] = t :=\n-- begin rw [lift_at_subst_term_medium t s, lift_term_at_zero]; refl end\n\n-- @[simp] lemma lift_term1_subst_term {l} (t : preterm L l) (s : term L) : (t \u2191 1)[s // 0] = t :=\n-- lift_at_subst_term_eq t s 0\n\n-- lemma lift_at_subst_term_small : \u2200{l} (t : preterm L l) (s : term L) (n\u2081 n\u2082 m),\n--  (t \u2191' n\u2081 # (m + n\u2082 + 1))[s \u2191' n\u2081 # m // n\u2082] = (t [s // n\u2082]) \u2191' n\u2081 # (m + n\u2082)\n-- | _ &k          s n\u2081 n\u2082 m :=\n--   begin\n--     by_cases h : m + n\u2082 + 1 \u2264 k,\n--     { change m + n\u2082 + 1 \u2264 k at h,\n--       have h\u2082 : n\u2082 < k := lt_of_le_of_lt (le_add_left n\u2082 m) (lt_of_succ_le h),\n--       have h\u2083 : n\u2082 < k + n\u2081 := by apply nat.lt_add_right; exact h\u2082,\n--       have h\u2084 : m + n\u2082 \u2264 k - 1 := nat.le_sub_right_of_add_le h,\n--       simp [*, -add_comm, -add_assoc, nat.add_sub_swap (one_le_of_lt h\u2082)] },\n--     { change \u00ac(m + n\u2082 + 1 \u2264 k) at h,\n--       apply decidable.lt_by_cases k n\u2082; intro h\u2082,\n--       { have h\u2083 : \u00ac(m + n\u2082 \u2264 k) := \u03bbh', not_le_of_gt h\u2082 (le_trans (le_add_left n\u2082 m) h'),\n--         simp [h, h\u2082, h\u2083, -add_comm, -add_assoc] },\n--       { subst h\u2082,\n--         have h\u2083 : \u00ac(k + m + 1 \u2264 k) := by rw [add_comm k m]; exact h,\n--         simp [h, h\u2083, -add_comm, -add_assoc],\n--         exact lift_term_at2_small _ _ _ m.zero_le },\n--       { have h\u2083 : \u00ac(m + n\u2082 \u2264 k - 1) :=\n--           \u03bbh', h $ (nat.le_sub_right_iff_add_le $ one_le_of_lt h\u2082).mp h',\n--         simp [h, h\u2082, h\u2083, -add_comm, -add_assoc] }}\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m := by simp [*, -add_assoc, -add_comm]\n\n-- lemma subst_term2 : \u2200{l} (t : preterm L l) (s\u2081 s\u2082 : term L) (n\u2081 n\u2082),\n--   t [s\u2081 // n\u2081] [s\u2082 // n\u2081 + n\u2082] = t [s\u2082 // n\u2081 + n\u2082 + 1] [s\u2081[s\u2082 // n\u2082] // n\u2081]\n-- | _ &k          s\u2081 s\u2082 n\u2081 n\u2082 :=\n--   begin -- can we use subst_realize2 here?\n--     apply decidable.lt_by_cases k n\u2081; intro h,\n--     { have : k < n\u2081 + n\u2082, from lt_of_le_of_lt (k.le_add_right n\u2082) (by simp*),\n--       have : k < n\u2081 + n\u2082 + 1, from lt.step this,\n--       simp only [*, eq_self_iff_true, fol.subst_term_var_lt] },\n--     { have : k < k + (n\u2082 + 1), from lt_succ_of_le (le_add_right _ n\u2082),\n--       subst h, simp [*, lift_subst_term_large', -add_comm] },\n--     apply decidable.lt_by_cases k (n\u2081 + n\u2082 + 1); intro h',\n--     { have : k - 1 < n\u2081 + n\u2082, from (nat.sub_lt_right_iff_lt_add (one_le_of_lt h)).2 h',\n--       simp [*, -add_comm, -add_assoc] },\n--     { subst h', simp [h, lift_subst_term_medium, -add_comm, -add_assoc] },\n--     { have : n\u2081 + n\u2082 < k - 1, from nat.lt_sub_right_of_add_lt h',\n--       have : n\u2081 < k - 1, from lt_of_le_of_lt (n\u2081.le_add_right n\u2082) this,\n--       simp only [*, eq_self_iff_true, fol.subst_term_var_gt] }\n--   end\n-- | _ (func f)    s\u2081 s\u2082 n\u2081 n\u2082 := rfl\n-- | _ (app t\u2081 t\u2082) s\u2081 s\u2082 n\u2081 n\u2082 := by simp*\n\n-- lemma subst_term2_0 {l} (t : preterm L l) (s\u2081 s\u2082 : term L) (n) :\n--   t [s\u2081 // 0] [s\u2082 // n] = t [s\u2082 // n + 1] [s\u2081[s\u2082 // n] // 0] :=\n-- let h := subst_term2 t s\u2081 s\u2082 0 n in by simp only [zero_add] at h; exact h\n\n-- lemma lift_subst_term_cancel : \u2200{l} (t : preterm L l) (n : \u2115), (t \u2191' 1 # (n+1))[&0 // n] = t\n-- | _ &k          n :=\n--   begin\n--     apply decidable.lt_by_cases n k; intro h,\n--     { change n+1 \u2264 k at h, have h' : n < k+1, from lt.step (lt_of_succ_le h), simp [h, h'] },\n--     { have h' : \u00ac(k+1 \u2264 k), from not_succ_le_self k, simp [h, h'] },\n--     { have h' : \u00ac(n+1 \u2264 k) := not_le_of_lt (lt.step h), simp [h, h'] }\n--   end\n-- | _ (func f)    n := rfl\n-- | _ (app t\u2081 t\u2082) n := by dsimp; simp [*]\n\n\n-- /- Probably useful facts about substitution which we should add when needed:\n-- (forall M N i j k, ( M [ j \u2190 N] ) \u2191' k # (j+i) = (M \u2191' k # (S (j+i))) [ j \u2190 (N \u2191' k # i ) ])\n-- subst_travers : (forall M N P n, (M [\u2190 N]) [n \u2190 P] = (M [n+1 \u2190 P])[\u2190 N[n\u2190 P]])\n-- erasure_lem3 : (forall n m t, m>n->#m = (#m \u2191' 1 # (S n)) [n \u2190 t]).\n-- lift_is_lift_sublemma : forall j v, j<v->exists w,#v=w\u21911#j.\n-- lift_is_lift : (forall N A n i j,N \u2191' i # n=A \u2191' 1 # j -> j<n -> exists M,N=M \u2191' 1 # j)\n-- subst_is_lift : (forall N T A n j, N [n \u2190 T]=A\u2191' 1#j->j<n->exists M,N=M\u2191' 1#j)\n-- -/\n\n-- /- preformula l is a partially applied formula. if applied to n terms, it becomes a formula.\n--   * We only have implication as binary connective. Since we use classical logic, we can define\n--     the other connectives from implication and falsum.\n--   * Similarly, universal quantification is our only quantifier.\n--   * We could make `falsum` and `equal` into elements of rel. However, if we do that, then we cannot make the interpretation of them in a model definitionally what we want.\n-- -/\n-- variable (L)\n-- inductive preformula : \u2115 \u2192 Type u\n-- | falsum {} : preformula 0\n-- | equal (t\u2081 t\u2082 : term L) : preformula 0\n-- | rel {l : \u2115} (R : L.relations l) : preformula l\n-- | apprel {l : \u2115} (f : preformula (l + 1)) (t : term L) : preformula l\n-- | imp (f\u2081 f\u2082 : preformula 0) : preformula 0\n-- | all (f : preformula 0) : preformula 0\n-- export preformula\n-- @[reducible] def formula := preformula L 0\n-- variable {L}\n\n-- notation `\u22a5` := fol.preformula.falsum -- input: \\bot\n-- infix ` \u2243 `:88 := fol.preformula.equal -- input \\~- or \\simeq\n-- infixr ` \u27f9 `:62 := fol.preformula.imp -- input \\==>\n-- prefix `\u2200'`:110 := fol.preformula.all\n-- def not   (f : formula L)     : formula L := f \u27f9 \u22a5\n-- prefix `\u223c`:max := fol.not -- input \\~, the ASCII character ~ has too low precedence\n-- notation `\u22a4` := \u223c\u22a5 -- input: \\top\n-- def and   (f\u2081 f\u2082 : formula L) : formula L := \u223c(f\u2081 \u27f9 \u223cf\u2082)\n-- infixr ` \u2293 ` := fol.and -- input: \\sqcap\n-- def or    (f\u2081 f\u2082 : formula L) : formula L := \u223cf\u2081 \u27f9 f\u2082\n-- infixr ` \u2294 ` := fol.or -- input: \\sqcup\n-- def biimp (f\u2081 f\u2082 : formula L) : formula L := (f\u2081 \u27f9 f\u2082) \u2293 (f\u2082 \u27f9 f\u2081)\n-- infix ` \u21d4 `:61 := fol.biimp -- input \\<=>\n-- def ex    (f : formula L)     : formula L := \u223c \u2200' \u223cf\n-- prefix `\u2203'`:110 := fol.ex -- input \\ex\n\n-- @[simp] def apps_rel : \u2200{l} (f : preformula L l) (ts : dvector (term L) l), formula L\n-- | 0     f []      := f\n-- | (n+1) f (t::ts) := apps_rel (apprel f t) ts\n\n-- @[simp] lemma apps_rel_zero (f : formula L) (ts : dvector (term L) 0) : apps_rel f ts = f :=\n-- by cases ts; refl\n\n-- -- lemma apps_rel_ne_falsum {l} {R : L.relations l} {ts : dvector (term L) l} :\n-- --   apps_rel (rel R) ts \u2260 \u22a5 :=\n-- -- by induction l; cases ts; [{cases ts_xs, intro h, injection h}, apply l_ih]\n\n-- -- lemma apps_rel_ne_falsum {l} {f : preformula L (l+1)} {ts : dvector (term L) (l+1)} :\n-- --   apps_rel f ts \u2260 \u22a5 :=\n-- -- by induction l; cases ts; [{cases ts_xs, intro h, injection h}, apply l_ih]\n-- -- lemma apps_rel_ne_equal {l} {f : preformula L (l+1)} {ts : dvector (term L) (l+1)}\n-- --   {t\u2081 t\u2082 : term L} : apps_rel f ts \u2260 t\u2081 \u2243 t\u2082 :=\n-- -- by induction l; cases ts; [{cases ts_xs, intro h, injection h}, apply l_ih]\n-- -- lemma apps_rel_ne_imp {l} {f : preformula L (l+1)} {ts : dvector (term L) (l+1)}\n-- --   {f\u2081 f\u2082 : formula L} : apps_rel f ts \u2260 f\u2081 \u27f9 f\u2082 :=\n-- -- by induction l; cases ts; [{cases ts_xs, intro h, injection h}, apply l_ih]\n-- -- lemma apps_rel_ne_all {l} {f : preformula L (l+1)} {ts : dvector (term L) (l+1)}\n-- --   {f' : formula L} : apps_rel f ts \u2260 \u2200' f' :=\n-- -- by induction l; cases ts; [{cases ts_xs, intro h, injection h}, apply l_ih]\n\n-- def formula_of_relation {l} (R : L.relations l) : arity' (term L) (formula L) l :=\n-- arity'.of_dvector_map $ apps_rel (rel R)\n\n-- @[elab_as_eliminator] def formula.rec' {C : formula L \u2192 Sort v}\n--   (hfalsum : C \u22a5)\n--   (hequal : \u03a0 (t\u2081 t\u2082 : term L), C (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {{l}} (R : L.relations l) (ts : dvector (term L) l), C (apps_rel (rel R) ts))\n--   (himp : \u03a0 {{f\u2081 f\u2082 : formula L}} (ih\u2081 : C f\u2081) (ih\u2082 : C f\u2082), C (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {{f : formula L}} (ih : C f), C (\u2200' f)) :\n--   \u2200{l} (f : preformula L l) (ts : dvector (term L) l), C (apps_rel f ts)\n-- | _ falsum       ts := by cases ts; exact hfalsum\n-- | _ (t\u2081 \u2243 t\u2082)    ts := by cases ts; apply hequal\n-- | _ (rel R)      ts := by apply hrel\n-- | _ (apprel f t) ts := by apply formula.rec' f (t::ts)\n-- | _ (f\u2081 \u27f9 f\u2082)   ts := by cases ts; exact himp (formula.rec' f\u2081 ([])) (formula.rec' f\u2082 ([]))\n-- | _ (\u2200' f)       ts := by cases ts; exact hall (formula.rec' f ([]))\n\n-- @[elab_as_eliminator] def formula.rec {C : formula L \u2192 Sort v}\n--   (hfalsum : C \u22a5)\n--   (hequal : \u03a0 (t\u2081 t\u2082 : term L), C (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {{l}} (R : L.relations l) (ts : dvector (term L) l), C (apps_rel (rel R) ts))\n--   (himp : \u03a0 {{f\u2081 f\u2082 : formula L}} (ih\u2081 : C f\u2081) (ih\u2082 : C f\u2082), C (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {{f : formula L}} (ih : C f), C (\u2200' f)) : \u2200f, C f :=\n-- \u03bbf, formula.rec' hfalsum hequal hrel himp hall f ([])\n\n-- @[simp] def formula.rec'_apps_rel {C : formula L \u2192 Sort v}\n--   (hfalsum : C \u22a5)\n--   (hequal : \u03a0 (t\u2081 t\u2082 : term L), C (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {{l}} (R : L.relations l) (ts : dvector (term L) l), C (apps_rel (rel R) ts))\n--   (himp : \u03a0 {{f\u2081 f\u2082 : formula L}} (ih\u2081 : C f\u2081) (ih\u2082 : C f\u2082), C (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {{f : formula L}} (ih : C f), C (\u2200' f))\n--   {l} (f : preformula L l) (ts : dvector (term L) l) :\n--   @formula.rec' L C hfalsum hequal hrel himp hall 0 (apps_rel f ts) ([]) =\n--   @formula.rec' L C hfalsum hequal hrel himp hall l f ts :=\n-- begin\n--   induction ts,\n--   { refl },\n--   { dsimp only [dvector.map, apps_rel], rw [ts_ih], refl }\n-- end\n\n-- @[simp] def formula.rec_apps_rel {C : formula L \u2192 Sort v}\n--   (hfalsum : C \u22a5)\n--   (hequal : \u03a0 (t\u2081 t\u2082 : term L), C (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {{l}} (R : L.relations l) (ts : dvector (term L) l), C (apps_rel (rel R) ts))\n--   (himp : \u03a0 {{f\u2081 f\u2082 : formula L}} (ih\u2081 : C f\u2081) (ih\u2082 : C f\u2082), C (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {{f : formula L}} (ih : C f), C (\u2200' f))\n--   {l} (R : L.relations l) (ts : dvector (term L) l) :\n--   @formula.rec L C hfalsum hequal hrel himp hall (apps_rel (rel R) ts) = hrel R ts :=\n-- by dsimp only [formula.rec]; rw formula.rec'_apps_rel; refl\n\n-- @[simp] def lift_formula_at : \u2200 {l}, preformula L l \u2192 \u2115 \u2192 \u2115 \u2192 preformula L l\n-- | _ falsum       n m := falsum\n-- | _ (t\u2081 \u2243 t\u2082)    n m := lift_term_at t\u2081 n m \u2243 lift_term_at t\u2082 n m\n-- | _ (rel R)      n m := rel R\n-- | _ (apprel f t) n m := apprel (lift_formula_at f n m) (lift_term_at t n m)\n-- | _ (f\u2081 \u27f9 f\u2082)   n m := lift_formula_at f\u2081 n m \u27f9 lift_formula_at f\u2082 n m\n-- | _ (\u2200' f)       n m := \u2200' lift_formula_at f n (m+1)\n\n-- notation f ` \u2191' `:90 n ` # `:90 m:90 := fol.lift_formula_at f n m -- input \u2191' with \\upa\n\n-- @[reducible] def lift_formula {l} (f : preformula L l) (n : \u2115) : preformula L l := f \u2191' n # 0\n-- infix ` \u2191 `:100 := fol.lift_formula -- input \u2191' with \\upa\n-- @[reducible, simp] def lift_formula1 {l} (f : preformula L l) : preformula L l := f \u2191 1\n\n-- @[simp] lemma lift_formula_def {l} (f : preformula L l) (n : \u2115) : f \u2191' n # 0 = f \u2191 n := by refl\n-- @[simp] lemma lift_formula1_not (n : \u2115) (f : formula L) : \u223cf \u2191 n  = \u223c(f \u2191 n) := by refl\n\n-- lemma injective_lift_formula_at {l} {n m : \u2115} :\n--   function.injective (\u03bb (f : preformula L l), lift_formula_at f n m) :=\n-- begin\n--   intros f f' H, induction f generalizing m; cases f'; injection H,\n--   { simp only [injective_lift_term_at h_1, injective_lift_term_at h_2, eq_self_iff_true, and_self] },\n--   { simp only [f_ih h_1, injective_lift_term_at h_2, eq_self_iff_true, and_self] },\n--   { simp only [f_ih_f\u2081 h_1, f_ih_f\u2082 h_2, eq_self_iff_true, and_self] },\n--   { simp only [f_ih h_1, eq_self_iff_true] }\n-- end\n\n-- @[simp] lemma lift_formula_at_zero : \u2200 {l} (f : preformula L l) (m : \u2115), f \u2191' 0 # m = f\n-- | _ falsum       m := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    m := by simp\n-- | _ (rel R)      m := by refl\n-- | _ (apprel f t) m := by simp; apply lift_formula_at_zero\n-- | _ (f\u2081 \u27f9 f\u2082)   m := by dsimp; congr1; apply lift_formula_at_zero\n-- | _ (\u2200' f)       m := by simp; apply lift_formula_at_zero\n\n-- /- the following lemmas simplify iterated lifts, depending on the size of m' -/\n-- lemma lift_formula_at2_small : \u2200 {l} (f : preformula L l) (n n') {m m'}, m' \u2264 m \u2192\n--   (f \u2191' n # m) \u2191' n' # m' = (f \u2191' n' # m') \u2191' n # (m + n')\n-- | _ falsum       n n' m m' H := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    n n' m m' H := by simp [lift_term_at2_small, H]\n-- | _ (rel R)      n n' m m' H := by refl\n-- | _ (apprel f t) n n' m m' H :=\n--   by simp [lift_term_at2_small, H, -add_comm]; apply lift_formula_at2_small; assumption\n-- | _ (f\u2081 \u27f9 f\u2082)   n n' m m' H := by dsimp; congr1; apply lift_formula_at2_small; assumption\n-- | _ (\u2200' f)       n n' m m' H :=\n--   by simp [lift_term_at2_small, H, lift_formula_at2_small f n n' (add_le_add_right H 1)]\n\n-- lemma lift_formula_at2_medium : \u2200 {l} (f : preformula L l) (n n') {m m'}, m \u2264 m' \u2192 m' \u2264 m+n \u2192\n--   (f \u2191' n # m) \u2191' n' # m' = f \u2191' (n+n') # m\n-- | _ falsum       n n' m m' H\u2081 H\u2082 := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    n n' m m' H\u2081 H\u2082 := by simp [*, lift_term_at2_medium]\n-- | _ (rel R)      n n' m m' H\u2081 H\u2082 := by refl\n-- | _ (apprel f t) n n' m m' H\u2081 H\u2082 := by simp [*, lift_term_at2_medium, -add_comm]\n-- | _ (f\u2081 \u27f9 f\u2082)   n n' m m' H\u2081 H\u2082 := by simp*\n-- | _ (\u2200' f)       n n' m m' H\u2081 H\u2082 :=\n--   have m' + 1 \u2264 (m + 1) + n, from le_trans (add_le_add_right H\u2082 1) (by simp), by simp*\n\n-- lemma lift_formula_at2_eq {l} (f : preformula L l) (n n' m : \u2115) :\n--   (f \u2191' n # m) \u2191' n' # (m+n) = f \u2191' (n+n') # m :=\n-- lift_formula_at2_medium f n n' (m.le_add_right n) (le_refl _)\n\n-- lemma lift_formula_at2_large {l} (f : preformula L l) (n n') {m m'} (H : m + n \u2264 m') :\n--   (f \u2191' n # m) \u2191' n' # m' = (f \u2191' n' # (m'-n)) \u2191' n # m :=\n-- have H\u2081 : n \u2264 m', from le_trans (n.le_add_left m) H,\n-- have H\u2082 : m \u2264 m' - n, from nat.le_sub_right_of_add_le H,\n-- begin rw lift_formula_at2_small f n' n H\u2082, rw [nat.sub_add_cancel], exact H\u2081 end\n\n-- @[simp] lemma lift_formula_at_apps_rel {l} (f : preformula L l) (ts : dvector (term L) l)\n--   (n m : \u2115) : (apps_rel f ts) \u2191' n # m = apps_rel (f \u2191' n # m) (ts.map $ \u03bbx, x \u2191' n # m) :=\n-- by induction ts generalizing f;[refl, apply ts_ih (apprel f ts_x)]\n\n-- @[simp] lemma lift_formula_apps_rel {l} (f : preformula L l) (ts : dvector (term L) l)\n--   (n : \u2115) : (apps_rel f ts) \u2191 n = apps_rel (f \u2191 n) (ts.map $ \u03bbx, x \u2191 n) :=\n-- lift_formula_at_apps_rel f ts n 0\n\n-- @[simp] def subst_formula : \u2200 {l}, preformula L l \u2192 term L \u2192 \u2115 \u2192 preformula L l\n-- | _ falsum       s n := falsum\n-- | _ (t\u2081 \u2243 t\u2082)    s n := subst_term t\u2081 s n \u2243 subst_term t\u2082 s n\n-- | _ (rel R)      s n := rel R\n-- | _ (apprel f t) s n := apprel (subst_formula f s n) (subst_term t s n)\n-- | _ (f\u2081 \u27f9 f\u2082)   s n := subst_formula f\u2081 s n \u27f9 subst_formula f\u2082 s n\n-- | _ (\u2200' f)       s n := \u2200' subst_formula f s (n+1)\n\n-- notation f `[`:95 s ` // `:95 n `]`:0 := fol.subst_formula f s n\n\n-- lemma subst_formula_equal (t\u2081 t\u2082 s : term L) (n : \u2115) :\n--   (t\u2081 \u2243 t\u2082)[s // n] = t\u2081[s // n] \u2243 (t\u2082[s // n]) :=\n-- by refl\n\n-- @[simp] lemma subst_formula_biimp (f\u2081 f\u2082 : formula L) (s : term L) (n : \u2115) :\n--   (f\u2081 \u21d4 f\u2082)[s // n] = f\u2081[s // n] \u21d4 (f\u2082[s // n]) :=\n-- by refl\n\n-- lemma lift_at_subst_formula_large : \u2200{l} (f : preformula L l) (s : term L) {n\u2081} (n\u2082) {m}, m \u2264 n\u2081 \u2192\n--   (f \u2191' n\u2082 # m)[s // n\u2081+n\u2082] = (f [s // n\u2081]) \u2191' n\u2082 # m\n-- | _ falsum       s n\u2081 n\u2082 m h := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    s n\u2081 n\u2082 m h := by simp [*, lift_at_subst_term_large]\n-- | _ (rel R)      s n\u2081 n\u2082 m h := by refl\n-- | _ (apprel f t) s n\u2081 n\u2082 m h := by simp [*, lift_at_subst_term_large]\n-- | _ (f\u2081 \u27f9 f\u2082)   s n\u2081 n\u2082 m h := by simp*\n-- | _ (\u2200' f)       s n\u2081 n\u2082 m h :=\n--   by have := lift_at_subst_formula_large f s n\u2082 (add_le_add_right h 1); simp at this; simp*\n\n-- lemma lift_subst_formula_large {l} (f : preformula L l) (s : term L) {n\u2081 n\u2082} :\n--   (f \u2191 n\u2082)[s // n\u2081+n\u2082] = (f [s // n\u2081]) \u2191 n\u2082 :=\n-- lift_at_subst_formula_large f s n\u2082 n\u2081.zero_le\n\n-- lemma lift_subst_formula_large' {l} (f : preformula L l) (s : term L) {n\u2081 n\u2082} :\n--   (f \u2191 n\u2082)[s // n\u2082+n\u2081] = (f [s // n\u2081]) \u2191 n\u2082 :=\n-- by rw [add_comm]; apply lift_subst_formula_large\n\n-- lemma lift_at_subst_formula_medium : \u2200{l} (f : preformula L l) (s : term L) {n\u2081 n\u2082 m}, m \u2264 n\u2082 \u2192\n--   n\u2082 \u2264 m + n\u2081 \u2192 (f \u2191' n\u2081+1 # m)[s // n\u2082] = f \u2191' n\u2081 # m\n-- | _ falsum       s n\u2081 n\u2082 m h\u2081 h\u2082 := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    s n\u2081 n\u2082 m h\u2081 h\u2082 := by simp [*, lift_at_subst_term_medium]\n-- | _ (rel R)      s n\u2081 n\u2082 m h\u2081 h\u2082 := by refl\n-- | _ (apprel f t) s n\u2081 n\u2082 m h\u2081 h\u2082 := by simp [*, lift_at_subst_term_medium]\n-- | _ (f\u2081 \u27f9 f\u2082)   s n\u2081 n\u2082 m h\u2081 h\u2082 := by simp*\n-- | _ (\u2200' f)       s n\u2081 n\u2082 m h\u2081 h\u2082 :=\n--   begin\n--     have h : n\u2082 + 1 \u2264 (m + 1) + n\u2081, from le_trans (add_le_add_right h\u2082 1) (by simp),\n--     have := lift_at_subst_formula_medium f s (add_le_add_right h\u2081 1) h,\n--     simp only [fol.subst_formula, fol.lift_formula_at] at this, simp*\n--   end\n\n-- lemma lift_subst_formula_medium {l} (f : preformula L l) (s : term L) (n\u2081 n\u2082) :\n--   (f \u2191 ((n\u2081 + n\u2082) + 1))[s // n\u2081] = f \u2191 (n\u2081 + n\u2082) :=\n-- lift_at_subst_formula_medium f s n\u2081.zero_le (by rw [zero_add]; exact n\u2081.le_add_right n\u2082)\n\n-- lemma lift_at_subst_formula_eq {l} (f : preformula L l) (s : term L) (n : \u2115) :\n--   (f \u2191' 1 # n)[s // n] = f :=\n-- begin rw [lift_at_subst_formula_medium f s, lift_formula_at_zero]; refl end\n\n-- @[simp] lemma lift_formula1_subst {l} (f : preformula L l) (s : term L) : (f \u2191 1)[s // 0] = f :=\n-- lift_at_subst_formula_eq f s 0\n\n-- lemma lift_at_subst_formula_small : \u2200{l} (f : preformula L l) (s : term L) (n\u2081 n\u2082 m),\n--  (f \u2191' n\u2081 # (m + n\u2082 + 1))[s \u2191' n\u2081 # m // n\u2082] = (f [s // n\u2082]) \u2191' n\u2081 # (m + n\u2082)\n-- | _ falsum       s n\u2081 n\u2082 m := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    s n\u2081 n\u2082 m :=\n--     by dsimp; simp only [lift_at_subst_term_small, eq_self_iff_true, and_self]\n-- | _ (rel R)      s n\u2081 n\u2082 m := by refl\n-- | _ (apprel f t) s n\u2081 n\u2082 m :=\n--     by dsimp; simp only [*, lift_at_subst_term_small, eq_self_iff_true, and_self]\n-- | _ (f\u2081 \u27f9 f\u2082)   s n\u2081 n\u2082 m :=\n--     by dsimp; simp only [*, lift_at_subst_term_small, eq_self_iff_true, and_self]\n-- | _ (\u2200' f)       s n\u2081 n\u2082 m :=\n--     by have := lift_at_subst_formula_small f s n\u2081 (n\u2082+1) m; dsimp; simp at this \u22a2; exact this\n\n-- lemma lift_at_subst_formula_small0 {l} (f : preformula L l) (s : term L) (n\u2081 m) :\n--  (f \u2191' n\u2081 # (m + 1))[s \u2191' n\u2081 # m // 0] = (f [s // 0]) \u2191' n\u2081 # m :=\n-- lift_at_subst_formula_small f s n\u2081 0 m\n\n-- lemma subst_formula2 : \u2200{l} (f : preformula L l) (s\u2081 s\u2082 : term L) (n\u2081 n\u2082),\n--   f [s\u2081 // n\u2081] [s\u2082 // n\u2081 + n\u2082] = f [s\u2082 // n\u2081 + n\u2082 + 1] [s\u2081[s\u2082 // n\u2082] // n\u2081]\n-- | _ falsum       s\u2081 s\u2082 n\u2081 n\u2082 := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    s\u2081 s\u2082 n\u2081 n\u2082 := by simp [*, subst_term2]\n-- | _ (rel R)      s\u2081 s\u2082 n\u2081 n\u2082 := by refl\n-- | _ (apprel f t) s\u2081 s\u2082 n\u2081 n\u2082 := by simp [*, subst_term2]\n-- | _ (f\u2081 \u27f9 f\u2082)   s\u2081 s\u2082 n\u2081 n\u2082 := by simp*\n-- | _ (\u2200' f)       s\u2081 s\u2082 n\u2081 n\u2082 :=\n--   by simp*; rw [add_comm n\u2082 1, \u2190add_assoc, subst_formula2 f s\u2081 s\u2082 (n\u2081 + 1) n\u2082]; simp\n\n-- lemma subst_formula2_zero {l} (f : preformula L l) (s\u2081 s\u2082 : term L) (n) :\n--   f [s\u2081 // 0] [s\u2082 // n] = f [s\u2082 // n + 1] [s\u2081[s\u2082 // n] // 0] :=\n-- let h := subst_formula2 f s\u2081 s\u2082 0 n in by simp only [fol.subst_formula, zero_add] at h; exact h\n\n-- lemma lift_subst_formula_cancel : \u2200{l} (f : preformula L l) (n : \u2115), (f \u2191' 1 # (n+1))[&0 // n] = f\n-- | _ falsum       n := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    n := by simp [*, lift_subst_term_cancel]\n-- | _ (rel R)      n := by refl\n-- | _ (apprel f t) n := by simp [*, lift_subst_term_cancel]\n-- | _ (f\u2081 \u27f9 f\u2082)   n := by simp*\n-- | _ (\u2200' f)       n := by simp*\n\n-- @[simp] lemma subst_formula_apps_rel {l} (f : preformula L l) (ts : dvector (term L) l) (s : term L)\n--   (n : \u2115): (apps_rel f ts)[s // n] = apps_rel (f[s // n]) (ts.map $ \u03bbx, x[s // n]) :=\n-- by induction ts generalizing f;[refl, apply ts_ih (apprel f ts_x)]\n\n-- @[simp] def count_quantifiers : \u2200 {l}, preformula L l \u2192 \u2115\n-- | _ falsum       := 0\n-- | _ (t\u2081 \u2243 t\u2082)    := 0\n-- | _ (rel R)      := 0\n-- | _ (apprel f t) := 0\n-- | _ (f\u2081 \u27f9 f\u2082)   := count_quantifiers f\u2081 + count_quantifiers f\u2082\n-- | _ (\u2200' f)       := count_quantifiers f + 1\n\n-- @[simp] def count_quantifiers_succ {l} (f : preformula L (l+1)) : count_quantifiers f = 0 :=\n-- by cases f; refl\n\n-- @[simp] lemma count_quantifiers_subst : \u2200 {l} (f : preformula L l) (s : term L) (n : \u2115),\n--   count_quantifiers (f[s // n]) = count_quantifiers f\n-- | _ falsum       s n := by refl\n-- | _ (t\u2081 \u2243 t\u2082)    s n := by refl\n-- | _ (rel R)      s n := by refl\n-- | _ (apprel f t) s n := by refl\n-- | _ (f\u2081 \u27f9 f\u2082)   s n := by simp*\n-- | _ (\u2200' f)       s n := by simp*\n\n-- def quantifier_free {l} : preformula L l \u2192 Prop := \u03bb f, count_quantifiers f = 0\n\n-- /- Provability\n-- * to decide: should \u0393 be a list or a set (or finset)?\n-- * We use natural deduction as our deduction system, since that is most convenient to work with.\n-- * All rules are motivated to work well with backwards reasoning.\n-- -/\n-- inductive prf : set (formula L) \u2192 formula L \u2192 Type u\n-- | axm     {\u0393 A} (h : A \u2208 \u0393) : prf \u0393 A\n-- | impI    {\u0393 : set $ formula L} {A B} (h : prf (insert A \u0393) B) : prf \u0393 (A \u27f9 B)\n-- | impE    {\u0393} (A) {B} (h\u2081 : prf \u0393 (A \u27f9 B)) (h\u2082 : prf \u0393 A) : prf \u0393 B\n-- | falsumE {\u0393 : set $ formula L} {A} (h : prf (insert \u223cA \u0393) \u22a5) : prf \u0393 A\n-- | allI    {\u0393 A} (h : prf (lift_formula1 '' \u0393) A) : prf \u0393 (\u2200' A)\n-- | allE\u2082   {\u0393} A t (h : prf \u0393 (\u2200' A)) : prf \u0393 (A[t // 0])\n-- | ref     (\u0393 t) : prf \u0393 (t \u2243 t)\n-- | subst\u2082  {\u0393} (s t f) (h\u2081 : prf \u0393 (s \u2243 t)) (h\u2082 : prf \u0393 (f[s // 0])) : prf \u0393 (f[t // 0])\n\n-- export prf\n-- infix ` \u22a2 `:51 := fol.prf -- input: \\|- or \\vdash\n\n-- def provable (T : set $ formula L) (f : formula L) := nonempty (T \u22a2 f)\n-- infix ` \u22a2' `:51 := fol.provable -- input: \\|- or \\vdash\n\n-- def allE {\u0393} (A : formula L) (t) {B} (H\u2081 : \u0393 \u22a2 \u2200' A) (H\u2082 : A[t // 0] = B) : \u0393 \u22a2 B :=\n-- by induction H\u2082; exact allE\u2082 A t H\u2081\n\n-- def subst {\u0393} {s t} (f\u2081 : formula L) {f\u2082} (H\u2081 : \u0393 \u22a2 s \u2243 t) (H\u2082 : \u0393 \u22a2 f\u2081[s // 0])\n--   (H\u2083 : f\u2081[t // 0] = f\u2082) : \u0393 \u22a2 f\u2082 :=\n-- by induction H\u2083; exact subst\u2082 s t f\u2081 H\u2081 H\u2082\n\n-- def axm1 {\u0393 : set (formula L)} {A : formula L} : insert A \u0393 \u22a2 A := by apply axm; left; refl\n-- def axm2 {\u0393 : set (formula L)} {A B : formula L} : insert A (insert B \u0393) \u22a2 B :=\n-- by apply axm; right; left; refl\n\n-- def weakening {\u0393 \u0394} {f : formula L} (H\u2081 : \u0393 \u2286 \u0394) (H\u2082 : \u0393 \u22a2 f) : \u0394 \u22a2 f :=\n-- begin\n--   induction H\u2082 generalizing \u0394,\n--   { apply axm, exact H\u2081 H\u2082_h, },\n--   { apply impI, apply H\u2082_ih, apply insert_subset_insert, apply H\u2081 },\n--   { apply impE, apply H\u2082_ih_h\u2081, assumption, apply H\u2082_ih_h\u2082, assumption },\n--   { apply falsumE, apply H\u2082_ih, apply insert_subset_insert, apply H\u2081 },\n--   { apply allI, apply H\u2082_ih, apply image_subset _ H\u2081 },\n--   { apply allE\u2082, apply H\u2082_ih, assumption },\n--   { apply ref },\n--   { apply subst\u2082, apply H\u2082_ih_h\u2081, assumption, apply H\u2082_ih_h\u2082, assumption },\n-- end\n\n-- def prf_lift {\u0393} {f : formula L} (n m : \u2115) (H : \u0393 \u22a2 f) : (\u03bbf', f' \u2191' n # m) '' \u0393 \u22a2 f \u2191' n # m :=\n-- begin\n--   induction H generalizing m,\n--   { apply axm, apply mem_image_of_mem _ H_h },\n--   { apply impI, have h := @H_ih m, rw [image_insert_eq] at h, exact h },\n--   { apply impE, apply H_ih_h\u2081, apply H_ih_h\u2082 },\n--   { apply falsumE, have h := @H_ih m, rw [image_insert_eq] at h, exact h },\n--   { apply allI, rw [image_image], have h := @H_ih (m+1), rw [image_image] at h,\n--     apply cast _ h, congr1, apply image_congr', intro f', symmetry,\n--     exact lift_formula_at2_small f' _ _ m.zero_le },\n--   { apply allE _ _ (H_ih m), apply lift_at_subst_formula_small0 },\n--   { apply ref },\n--   { apply subst _ (H_ih_h\u2081 m),\n--     { have h := @H_ih_h\u2082 m, rw [\u2190lift_at_subst_formula_small0] at h, exact h},\n--     rw [lift_at_subst_formula_small0] },\n-- end\n\n-- def substitution {\u0393} {f : formula L} (t n) (H : \u0393 \u22a2 f) : (\u03bbx, x[t // n]) '' \u0393 \u22a2 f[t // n] :=\n-- begin\n--   induction H generalizing n,\n--   { apply axm, apply mem_image_of_mem _ H_h },\n--   { apply impI, have h := H_ih n, rw [image_insert_eq] at h, exact h },\n--   { apply impE, apply H_ih_h\u2081, apply H_ih_h\u2082 },\n--   { apply falsumE, have h := H_ih n, rw [image_insert_eq] at h, exact h },\n--   { apply allI, rw [image_image], have h := @H_ih (n+1), rw [image_image] at h,\n--     apply cast _ h, congr1, apply image_congr', intro,\n--     apply lift_subst_formula_large },\n--   { apply allE _ _ (H_ih n), symmetry, apply subst_formula2_zero },\n--   { apply ref },\n--   { apply subst _ (H_ih_h\u2081 n), { have h := @H_ih_h\u2082 n, rw [subst_formula2_zero] at h, exact h},\n--     rw [subst_formula2_zero] },\n-- end\n\n-- def reflect_prf_lift1 {\u0393} {f : formula L} (h : lift_formula1 '' \u0393 \u22a2 f \u2191 1) : \u0393 \u22a2 f :=\n-- begin\n--   have := substitution &0 0 h, simp [image_image] at this, exact this\n-- end\n\n-- -- def reflect_prf_lift {\u0393} {f : formula L} (n m : \u2115) :\n-- --   (\u03bbf' : formula L, f' \u2191' n # m) '' \u0393 \u22a2 f \u2191' n # m \u2192 \u0393 \u22a2 f :=\n-- -- begin\n-- --   induction n,\n-- --   { rw [lift_zero] },\n-- --   { }\n-- -- end\n\n-- def weakening1 {\u0393} {f\u2081 f\u2082 : formula L} (H : \u0393 \u22a2 f\u2082) : insert f\u2081 \u0393 \u22a2 f\u2082 :=\n-- weakening (subset_insert f\u2081 \u0393) H\n\n\n\n-- def weakening2 {\u0393} {f\u2081 f\u2082 f\u2083 : formula L} (H : insert f\u2081 \u0393 \u22a2 f\u2082) : insert f\u2081 (insert f\u2083 \u0393) \u22a2 f\u2082 :=\n-- weakening (insert_subset_insert (subset_insert _ \u0393)) H\n\n-- def deduction {\u0393} {A B : formula L} (H : \u0393 \u22a2 A \u27f9 B) : insert A \u0393 \u22a2 B :=\n-- impE A (weakening1 H) axm1\n\n-- def exfalso {\u0393} {A : formula L} (H : \u0393 \u22a2 falsum) : \u0393 \u22a2 A :=\n-- falsumE (weakening1 H)\n\n-- def exfalso' {\u0393} {A : formula L} (H : \u0393 \u22a2' falsum) : \u0393 \u22a2' A :=\n-- by {fapply nonempty.map, exact \u0393 \u22a2 falsum, exact exfalso, exact H}\n\n-- def notI {\u0393} {A : formula L} (H : \u0393 \u22a2 A \u27f9 falsum) : \u0393 \u22a2 \u223c A :=\n--   by {rw[not], assumption}\n\n-- def andI {\u0393} {f\u2081 f\u2082 : formula L} (H\u2081 : \u0393 \u22a2 f\u2081) (H\u2082 : \u0393 \u22a2 f\u2082) : \u0393 \u22a2 f\u2081 \u2293 f\u2082 :=\n-- begin\n--   apply impI, apply impE f\u2082,\n--   { apply impE f\u2081, apply axm1, exact weakening1 H\u2081 },\n--   { exact weakening1 H\u2082 }\n-- end\n\n-- def andE1 {\u0393 f\u2081} (f\u2082 : formula L) (H : \u0393 \u22a2 f\u2081 \u2293 f\u2082) : \u0393 \u22a2 f\u2081 :=\n-- begin\n--   apply falsumE, apply impE _ (weakening1 H), apply impI, apply exfalso,\n--   apply impE f\u2081; [apply axm2, apply axm1]\n-- end\n\n-- def andE2 {\u0393} (f\u2081 : formula L) {f\u2082} (H : \u0393 \u22a2 f\u2081 \u2293 f\u2082) : \u0393 \u22a2 f\u2082 :=\n-- begin apply falsumE, apply impE _ (weakening1 H), apply impI, apply axm2 end\n\n-- def orI1 {\u0393} {A B : formula L} (H : \u0393 \u22a2 A) : \u0393 \u22a2 A \u2294 B :=\n-- begin apply impI, apply exfalso, refine impE _ _ (weakening1 H), apply axm1 end\n\n-- def orI2 {\u0393} {A B : formula L} (H : \u0393 \u22a2 B) : \u0393 \u22a2 A \u2294 B :=\n-- impI $ weakening1 H\n\n-- def orE {\u0393} {A B C : formula L} (H\u2081 : \u0393 \u22a2 A \u2294 B) (H\u2082 : insert A \u0393 \u22a2 C) (H\u2083 : insert B \u0393 \u22a2 C) :\n--   \u0393 \u22a2 C :=\n-- begin\n--   apply falsumE, apply impE C, { apply axm1 },\n--   apply impE B, { apply impI, exact weakening2 H\u2083 },\n--   apply impE _ (weakening1 H\u2081),\n--   apply impI (impE _ axm2 (weakening2 H\u2082))\n-- end\n\n-- def biimpI {\u0393} {f\u2081 f\u2082 : formula L} (H\u2081 : insert f\u2081 \u0393 \u22a2 f\u2082) (H\u2082 : insert f\u2082 \u0393 \u22a2 f\u2081) : \u0393 \u22a2 f\u2081 \u21d4 f\u2082 :=\n-- by apply andI; apply impI; assumption\n\n-- def biimpE1 {\u0393} {f\u2081 f\u2082 : formula L} (H : \u0393 \u22a2 f\u2081 \u21d4 f\u2082) : insert f\u2081 \u0393 \u22a2 f\u2082 := deduction (andE1 _ H)\n-- def biimpE2 {\u0393} {f\u2081 f\u2082 : formula L} (H : \u0393 \u22a2 f\u2081 \u21d4 f\u2082) : insert f\u2082 \u0393 \u22a2 f\u2081 := deduction (andE2 _ H)\n\n-- def exI {\u0393 f} (t : term L) (H : \u0393 \u22a2 f [t // 0]) : \u0393 \u22a2 \u2203' f :=\n-- begin\n--   apply impI,\n--   apply impE (f[t // 0]) _ (weakening1 H),\n--   apply allE\u2082 \u223cf t axm1,\n-- end\n\n-- def exE {\u0393} {f\u2081 f\u2082 : formula L} (H\u2081 : \u0393 \u22a2 \u2203' f\u2081)\n--   (H\u2082 : insert f\u2081 (lift_formula1 '' \u0393) \u22a2 lift_formula1 f\u2082) : \u0393 \u22a2 f\u2082 :=\n-- begin\n--   apply falsumE, apply impE _ (weakening1 H\u2081), apply allI, apply impI,\n--   rw [image_insert_eq], apply impE _ axm2, apply weakening2 H\u2082\n-- end\n\n-- def ex_not_of_not_all {\u0393} {f : formula L} (H : \u0393 \u22a2 \u223c \u2200' f) : \u0393 \u22a2 \u2203' \u223c f :=\n-- begin\n--   apply falsumE, apply impE _ (weakening1 H), apply allI, apply falsumE,\n--   rw [image_insert_eq], apply impE _ axm2, apply exI &0,\n--   rw [lift_subst_formula_cancel], exact axm1\n-- end\n\n-- def not_and_self {\u0393 : set (formula L)} {f : formula L} (H : \u0393 \u22a2 f \u2293 \u223cf) : \u0393 \u22a2 \u22a5 :=\n-- impE f (andE2 f H) (andE1 \u223cf H)\n\n-- -- def andE1 {\u0393 f\u2081} (f\u2082 : formula L) (H : \u0393 \u22a2 f\u2081 \u2293 f\u2082) : \u0393 \u22a2 f\u2081 :=\n-- def symm {\u0393} {s t : term L} (H : \u0393 \u22a2 s \u2243 t) : \u0393 \u22a2 t \u2243 s :=\n-- begin\n--   apply subst (&0 \u2243 s \u2191 1) H; rw [subst_formula_equal, lift_term1_subst_term, subst_term_var0],\n--   apply ref\n-- end\n\n-- def trans {\u0393} {t\u2081 t\u2082 t\u2083 : term L} (H : \u0393 \u22a2 t\u2081 \u2243 t\u2082) (H' : \u0393 \u22a2 t\u2082 \u2243 t\u2083) : \u0393 \u22a2 t\u2081 \u2243 t\u2083 :=\n-- begin\n--   apply subst (t\u2081 \u2191 1 \u2243 &0) H'; rw [subst_formula_equal, lift_term1_subst_term, subst_term_var0],\n--   exact H\n-- end\n\n-- def congr {\u0393} {t\u2081 t\u2082 : term L} (s : term L) (H : \u0393 \u22a2 t\u2081 \u2243 t\u2082) : \u0393 \u22a2 s[t\u2081 // 0] \u2243 s[t\u2082 // 0] :=\n-- begin\n--   apply subst (s[t\u2081 // 0] \u2191 1 \u2243 s) H,\n--   { rw [subst_formula_equal, lift_term1_subst_term], apply ref },\n--   { rw [subst_formula_equal, lift_term1_subst_term] }\n-- end\n\n-- def app_congr {\u0393} {t\u2081 t\u2082 : term L} (s : preterm L 1) (H : \u0393 \u22a2 t\u2081 \u2243 t\u2082) : \u0393 \u22a2 app s t\u2081 \u2243 app s t\u2082 :=\n-- begin\n--   have h := congr (app (s \u2191 1) &0) H, simp at h, exact h\n-- end\n\n-- def apprel_congr {\u0393} {t\u2081 t\u2082 : term L} (f : preformula L 1) (H : \u0393 \u22a2 t\u2081 \u2243 t\u2082)\n--   (H\u2082 : \u0393 \u22a2 apprel f t\u2081) : \u0393 \u22a2 apprel f t\u2082 :=\n-- begin\n--   apply subst (apprel (f \u2191 1) &0) H; simp, exact H\u2082\n-- end\n\n-- def imp_trans {\u0393} {f\u2081 f\u2082 f\u2083 : formula L} (H\u2081 : \u0393 \u22a2 f\u2081 \u27f9 f\u2082) (H\u2082 : \u0393 \u22a2 f\u2082 \u27f9 f\u2083) : \u0393 \u22a2 f\u2081 \u27f9 f\u2083 :=\n-- begin\n--   apply impI, apply impE _ (weakening1 H\u2082), apply impE _ (weakening1 H\u2081) axm1\n-- end\n\n-- def biimp_refl (\u0393 : set (formula L)) (f : formula L) : \u0393 \u22a2 f \u21d4 f :=\n-- by apply biimpI; apply axm1\n\n-- def biimp_trans {\u0393} {f\u2081 f\u2082 f\u2083 : formula L} (H\u2081 : \u0393 \u22a2 f\u2081 \u21d4 f\u2082) (H\u2082 : \u0393 \u22a2 f\u2082 \u21d4 f\u2083) : \u0393 \u22a2 f\u2081 \u21d4 f\u2083 :=\n-- begin\n--   apply andI; apply imp_trans,\n--   apply andE1 _ H\u2081, apply andE1 _ H\u2082, apply andE2 _ H\u2082, apply andE2 _ H\u2081\n-- end\n\n-- def equal_preterms (T : set (formula L)) {l} (t\u2081 t\u2082 : preterm L l) : Type u :=\n-- \u2200(ts : dvector (term L) l), T \u22a2 apps t\u2081 ts \u2243 apps t\u2082 ts\n\n-- def equal_preterms_app {T : set (formula L)} {l} {t t' : preterm L (l+1)} {s s' : term L}\n--   (Ht : equal_preterms T t t') (Hs : T \u22a2 s \u2243 s') : equal_preterms T (app t s) (app t' s') :=\n-- begin\n--   intro xs,\n--   apply trans (Ht (xs.cons s)),\n--   have h := congr (apps (t' \u2191 1) (&0 :: xs.map lift_term1)) Hs,\n--   simp [dvector.map_congr (\u03bbt, lift_term1_subst_term t s')] at h,\n--   exact h\n-- end\n\n-- @[refl] def equal_preterms_refl (T : set (formula L)) {l} (t : preterm L l) : equal_preterms T t t :=\n-- \u03bbxs, ref T (apps t xs)\n\n-- def equiv_preformulae (T : set (formula L)) {l} (f\u2081 f\u2082 : preformula L l) : Type u :=\n-- \u2200(ts : dvector (term L) l), T \u22a2 apps_rel f\u2081 ts \u21d4 apps_rel f\u2082 ts\n\n-- def equiv_preformulae_apprel {T : set (formula L)} {l} {f f' : preformula L (l+1)} {s s' : term L}\n--   (Ht : equiv_preformulae T f f') (Hs : T \u22a2 s \u2243 s') :\n--     equiv_preformulae T (apprel f s) (apprel f' s') :=\n-- begin\n--   intro xs,\n--   apply biimp_trans (Ht (xs.cons s)),\n--   apply subst (apps_rel (f' \u2191 1) ((s :: xs).map lift_term1) \u21d4\n--                apps_rel (f' \u2191 1) (&0 :: xs.map lift_term1)) Hs;\n--     simp [dvector.map_congr (\u03bbt, lift_term1_subst_term t s')],\n--   apply biimp_refl\n-- end\n\n-- @[refl] def equiv_preformulae_refl (T : set (formula L)) {l} (f : preformula L l) :\n--   equiv_preformulae T f f :=\n-- \u03bbxs, biimp_refl T (apps_rel f xs)\n\n-- def impI' {\u0393 : set $ formula L} {A B} (h : insert A \u0393 \u22a2' B) : \u0393 \u22a2' (A \u27f9 B) := h.map impI\n-- def impE' {\u0393} (A : formula L) {B} (h\u2081 : \u0393 \u22a2' A \u27f9 B) (h\u2082 : \u0393 \u22a2' A) : \u0393 \u22a2' B := h\u2081.map2 (impE _) h\u2082\n-- def falsumE' {\u0393 : set $ formula L} {A} (h : insert \u223cA \u0393 \u22a2' \u22a5 ) : \u0393 \u22a2' A := h.map falsumE\n-- def allI' {\u0393} {A : formula L} (h : lift_formula1 '' \u0393 \u22a2' A) : \u0393 \u22a2' \u2200' A := h.map allI\n-- def allE' {\u0393} (A : formula L) (t) {B} (H\u2081 : \u0393 \u22a2' \u2200' A) (H\u2082 : A[t // 0] = B) : \u0393 \u22a2' B :=\n-- H\u2081.map (\u03bbx, allE _ _ x H\u2082)\n-- def allE\u2082' {\u0393} {A} {t : term L} (h : \u0393 \u22a2' \u2200' A) : \u0393 \u22a2' A[t // 0] := h.map (\u03bbx, allE _ _ x rfl)\n-- def ref' (\u0393) (t : term L) : \u0393 \u22a2' (t \u2243 t) := \u27e8ref \u0393 t\u27e9\n-- def subst' {\u0393} {s t} (f\u2081 : formula L) {f\u2082} (H\u2081 : \u0393 \u22a2' s \u2243 t) (H\u2082 : \u0393 \u22a2' f\u2081[s // 0])\n--   (H\u2083 : f\u2081[t // 0] = f\u2082) : \u0393 \u22a2' f\u2082 :=\n-- H\u2081.map2 (\u03bbx y, subst _ x y H\u2083) H\u2082\n-- def subst\u2082' {\u0393} (s t) (f : formula L) (h\u2081 : \u0393 \u22a2' s \u2243 t) (h\u2082 : \u0393 \u22a2' f[s // 0]) : \u0393 \u22a2' f[t // 0] :=\n-- h\u2081.map2 (subst\u2082 _ _ _) h\u2082\n\n-- def weakening' {\u0393 \u0394} {f : formula L} (H\u2081 : \u0393 \u2286 \u0394) (H\u2082 : \u0393 \u22a2' f) : \u0394 \u22a2' f := H\u2082.map $ weakening H\u2081\n-- def weakening1' {\u0393} {f\u2081 f\u2082 : formula L} (H : \u0393 \u22a2' f\u2082) : insert f\u2081 \u0393 \u22a2' f\u2082 := H.map weakening1\n-- def weakening2' {\u0393} {f\u2081 f\u2082 f\u2083 : formula L} (H : insert f\u2081 \u0393 \u22a2' f\u2082) : insert f\u2081 (insert f\u2083 \u0393) \u22a2' f\u2082 :=\n-- H.map weakening2\n\n-- lemma apprel_congr' {\u0393} {t\u2081 t\u2082 : term L} (f : preformula L 1) (H : \u0393 \u22a2 t\u2081 \u2243 t\u2082) :\n--   \u0393 \u22a2' apprel f t\u2081 \u2194 \u0393 \u22a2' apprel f t\u2082 :=\n-- \u27e8nonempty.map $ apprel_congr f H, nonempty.map $ apprel_congr f $ symm H\u27e9\n\n-- lemma prf_all_iff {\u0393 : set (formula L)} {f} : \u0393 \u22a2' \u2200' f \u2194 lift_formula1 '' \u0393 \u22a2' f :=\n-- begin\n--   split,\n--   { intro H, rw [\u2190lift_subst_formula_cancel f 0],\n--     apply allE\u2082', apply H.map (prf_lift 1 0) },\n--   { exact allI' }\n-- end\n\n-- lemma iff_of_biimp {\u0393} {f\u2081 f\u2082 : formula L} (H : \u0393 \u22a2' f\u2081 \u21d4 f\u2082) : \u0393 \u22a2' f\u2081 \u2194 \u0393 \u22a2' f\u2082 :=\n-- \u27e8impE' _ $ H.map (andE1 _), impE' _ $ H.map (andE2 _)\u27e9\n\n-- lemma prf_by_cases {\u0393} (f\u2081) {f\u2082 : formula L} (H\u2081 : insert f\u2081 \u0393 \u22a2' f\u2082)\n--   (H\u2082 : insert \u223cf\u2081 \u0393 \u22a2' f\u2082) : \u0393 \u22a2' f\u2082 :=\n-- begin\n--   apply falsumE', apply impE' _ \u27e8axm1\u27e9,\n--   refine impE' _ (impI' (weakening2' H\u2081)) _,\n--   apply falsumE', apply impE' _ \u27e8axm2\u27e9, apply weakening2' H\u2082\n-- end\n\n-- /- model theory -/\n\n-- /- an L-structure is a type S with interpretations of the functions and relations on S -/\n-- variable (L)\n-- structure Structure :=\n-- (carrier : Type u)\n-- (fun_map : \u2200{n}, L.functions n \u2192 dvector carrier n \u2192 carrier)\n-- (rel_map : \u2200{n}, L.relations n \u2192 dvector carrier n \u2192 Prop)\n-- variable {L}\n-- instance has_coe_Structure : has_coe_to_sort (Structure L) :=\n-- \u27e8Type u, Structure.carrier\u27e9\n\n-- /- realization of terms -/\n-- @[simp] def realize_term {S : Structure L} (v : \u2115 \u2192 S) :\n--   \u2200{l} (t : preterm L l) (xs : dvector S l), S.carrier\n-- | _ &k          xs := v k\n-- | _ (func f)    xs := S.fun_map f xs\n-- | _ (app t\u2081 t\u2082) xs := realize_term t\u2081 $ realize_term t\u2082 ([])::xs\n\n-- lemma realize_term_congr {S : Structure L} {v v' : \u2115 \u2192 S} (h : \u2200n, v n = v' n) :\n--   \u2200{l} (t : preterm L l) (xs : dvector S l), realize_term v t xs = realize_term v' t xs\n-- | _ &k          xs := h k\n-- | _ (func f)    xs := by refl\n-- | _ (app t\u2081 t\u2082) xs := by dsimp; rw [realize_term_congr t\u2081, realize_term_congr t\u2082]\n\n-- lemma realize_term_subst {S : Structure L} (v : \u2115 \u2192 S) : \u2200{l} (n : \u2115) (t : preterm L l)\n--   (s : term L) (xs : dvector S l),\n--   realize_term (v[realize_term v (s \u2191 n) ([]) // n]) t xs = realize_term v (t[s // n]) xs\n-- | _ n &k          s [] :=\n--   by apply decidable.lt_by_cases k n; intro h;[simp [h], {subst h; simp}, simp [h]]\n-- | _ n (func f)    s xs := by refl\n-- | _ n (app t\u2081 t\u2082) s xs := by dsimp; simp*\n\n-- lemma realize_term_subst_lift {S : Structure L} (v : \u2115 \u2192 S) (x : S) (m : \u2115) : \u2200{l} (t : preterm L l)\n--   (xs : dvector S l), realize_term (v [x // m]) (t \u2191' 1 # m) xs = realize_term v t xs\n-- | _ &k          [] :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have : m < k + 1, from lt_succ_of_le h, simp* },\n--     { have : k < m, from lt_of_not_ge h, simp* }\n--   end\n-- | _ (func f)    xs := by refl\n-- | _ (app t\u2081 t\u2082) xs := by simp*\n\n-- /- realization of formulas -/\n-- @[simp] def realize_formula {S : Structure L} : \u2200{l}, (\u2115 \u2192 S) \u2192 preformula L l \u2192 dvector S l \u2192 Prop\n-- | _ v falsum       xs := false\n-- | _ v (t\u2081 \u2243 t\u2082)    xs := realize_term v t\u2081 xs = realize_term v t\u2082 xs\n-- | _ v (rel R)      xs := S.rel_map R xs\n-- | _ v (apprel f t) xs := realize_formula v f $ realize_term v t ([])::xs\n-- | _ v (f\u2081 \u27f9 f\u2082)   xs := realize_formula v f\u2081 xs \u2192 realize_formula v f\u2082 xs\n-- | _ v (\u2200' f)       xs := \u2200(x : S), realize_formula (v [x // 0]) f xs\n\n-- lemma realize_formula_congr {S : Structure L} : \u2200{l} {v v' : \u2115 \u2192 S} (h : \u2200n, v n = v' n)\n--   (f : preformula L l) (xs : dvector S l), realize_formula v f xs \u2194 realize_formula v' f xs\n-- | _ v v' h falsum       xs := by refl\n-- | _ v v' h (t\u2081 \u2243 t\u2082)    xs := by simp [realize_term_congr h]\n-- | _ v v' h (rel R)      xs := by refl\n-- | _ v v' h (apprel f t) xs := by simp [realize_term_congr h]; rw [realize_formula_congr h]\n-- | _ v v' h (f\u2081 \u27f9 f\u2082)   xs := by dsimp; rw [realize_formula_congr h, realize_formula_congr h]\n-- | _ v v' h (\u2200' f)       xs :=\n--   by apply forall_congr; intro x; apply realize_formula_congr; intro n;\n--      apply subst_realize_congr h\n\n-- lemma realize_formula_subst {S : Structure L} : \u2200{l} (v : \u2115 \u2192 S) (n : \u2115) (f : preformula L l)\n--   (s : term L) (xs : dvector S l),\n--   realize_formula (v[realize_term v (s \u2191 n) ([]) // n]) f xs \u2194 realize_formula v (f[s // n]) xs\n-- | _ v n falsum       s xs := by refl\n-- | _ v n (t\u2081 \u2243 t\u2082)    s xs := by simp [realize_term_subst]\n-- | _ v n (rel R)      s xs := by refl\n-- | _ v n (apprel f t) s xs := by simp [realize_term_subst]; rw realize_formula_subst\n-- | _ v n (f\u2081 \u27f9 f\u2082)   s xs := by apply imp_congr; apply realize_formula_subst\n-- | _ v n (\u2200' f)       s xs :=\n--   begin\n--     apply forall_congr, intro x, rw [\u2190realize_formula_subst], apply realize_formula_congr,\n--     intro k, rw [subst_realize2_0, \u2190realize_term_subst_lift v x 0, lift_term_def, lift_term2]\n--   end\n\n-- lemma realize_formula_subst0 {S : Structure L} {l} (v : \u2115 \u2192 S) (f : preformula L l) (s : term L)\n--   (xs : dvector S l) :\n--   realize_formula (v[realize_term v s ([]) // 0]) f xs \u2194 realize_formula v (f[s // 0]) xs :=\n-- by have h := realize_formula_subst v 0 f s; simp at h; exact h xs\n\n-- lemma realize_formula_subst_lift {S : Structure L} : \u2200{l} (v : \u2115 \u2192 S) (x : S) (m : \u2115)\n--   (f : preformula L l) (xs : dvector S l),\n--   realize_formula (v [x // m]) (f \u2191' 1 # m) xs = realize_formula v f xs\n-- | _ v x m falsum       xs := by refl\n-- | _ v x m (t\u2081 \u2243 t\u2082)    xs := by simp [realize_term_subst_lift]\n-- | _ v x m (rel R)      xs := by refl\n-- | _ v x m (apprel f t) xs := by simp [realize_term_subst_lift]; rw realize_formula_subst_lift\n-- | _ v x m (f\u2081 \u27f9 f\u2082)   xs := by apply imp_eq_congr; apply realize_formula_subst_lift\n-- | _ v x m (\u2200' f)       xs :=\n--   begin\n--     apply forall_eq_congr, intro x',\n--     rw [realize_formula_congr (subst_realize2_0 _ _ _ _), realize_formula_subst_lift]\n--   end\n\n-- /- the following definitions of provability and satisfiability are not exactly how you normally define them, since we define it for formulae instead of sentences. If all the formulae happen to be sentences, then these definitions are equivalent to the normal definitions (the realization of closed terms and sentences are independent of the realizer v).\n--  -/\n-- def all_prf (T T' : set (formula L)) := \u2200{{f}}, f \u2208 T' \u2192 T \u22a2 f\n-- infix ` \u22a2 `:51 := fol.all_prf -- input: |- or \\vdash\n\n-- def satisfied_in (S : Structure L) (f : formula L) := \u2200(v : \u2115 \u2192 S), realize_formula v f ([])\n-- infix ` \u22a8 `:51 := fol.satisfied_in -- input using \\|= or \\vDash, but not using \\models\n\n-- def all_satisfied_in (S : Structure L) (T : set (formula L)) := \u2200{{f}}, f \u2208 T \u2192 S \u22a8 f\n-- infix ` \u22a8 `:51 := fol.all_satisfied_in -- input using \\|= or \\vDash, but not using \\models\n\n-- def satisfied (T : set (formula L)) (f : formula L) :=\n-- \u2200(S : Structure L) (v : \u2115 \u2192 S), (\u2200f' \u2208 T, realize_formula v (f' : formula L) ([])) \u2192\n--   realize_formula v f ([])\n\n-- infix ` \u22a8 `:51 := fol.satisfied -- input using \\|= or \\vDash, but not using \\models\n\n-- def all_satisfied (T T' : set (formula L)) := \u2200{{f}}, f \u2208 T' \u2192 T \u22a8 f\n-- infix ` \u22a8 `:51 := fol.all_satisfied -- input using \\|= or \\vDash, but not using \\models\n\n-- def satisfied_in_trans {S : Structure L} {T : set (formula L)} {f : formula L} (H' : S \u22a8 T)\n--   (H : T \u22a8 f) : S \u22a8 f :=\n-- \u03bbv, H S v $ \u03bbf' hf', H' hf' v\n\n-- def all_satisfied_in_trans  {S : Structure L} {T T' : set (formula L)} (H' : S \u22a8 T) (H : T \u22a8 T') :\n--   S \u22a8 T' :=\n-- \u03bbf hf, satisfied_in_trans H' $ H hf\n\n-- def satisfied_of_mem {T : set (formula L)} {f : formula L} (hf : f \u2208 T) : T \u22a8 f :=\n-- \u03bbS v h, h f hf\n\n-- def all_satisfied_of_subset {T T' : set (formula L)} (h : T' \u2286 T) : T \u22a8 T' :=\n-- \u03bbf hf, satisfied_of_mem $ h hf\n\n-- def satisfied_trans {T\u2081 T\u2082 : set (formula L)} {f : formula L} (H' : T\u2081 \u22a8 T\u2082) (H : T\u2082 \u22a8 f) : T\u2081 \u22a8 f :=\n-- \u03bbS v h, H S v $ \u03bbf' hf', H' hf' S v h\n\n-- def all_satisfied_trans {T\u2081 T\u2082 T\u2083 : set (formula L)} (H' : T\u2081 \u22a8 T\u2082) (H : T\u2082 \u22a8 T\u2083) : T\u2081 \u22a8 T\u2083 :=\n-- \u03bbf hf, satisfied_trans H' $ H hf\n\n-- def satisfied_weakening {T T' : set (formula L)} (H : T \u2286 T') {f : formula L} (HT : T \u22a8 f) :\n--   T' \u22a8 f :=\n-- \u03bbS v h, HT S v $ \u03bbf' hf', h f' $ H hf'\n\n-- /- soundness for a set of formulae -/\n-- lemma formula_soundness {\u0393 : set (formula L)} {A : formula L} (H : \u0393 \u22a2 A) : \u0393 \u22a8 A :=\n-- begin\n--   intro S, induction H; intros v h,\n--   { apply h, apply H_h },\n--   { intro ha, apply H_ih, intros f hf, induction hf, { subst hf, assumption }, apply h f hf },\n--   { exact H_ih_h\u2081 v h (H_ih_h\u2082 v h) },\n--   { apply classical.by_contradiction, intro ha,\n--     apply H_ih v, intros f hf, induction hf, { cases hf, exact ha }, apply h f hf },\n--   { intro x, apply H_ih, intros f hf, rcases hf with \u27e8f, hf, rfl\u27e9,\n--     rw [realize_formula_subst_lift v x 0 f], exact h f hf },\n--   { rw [\u2190realize_formula_subst0], apply H_ih v h (realize_term v H_t ([])) },\n--   { dsimp, refl },\n--   { have h' := H_ih_h\u2081 v h, dsimp at h', rw [\u2190realize_formula_subst0, \u2190h', realize_formula_subst0],\n--     apply H_ih_h\u2082 v h },\n-- end\n\n-- /- sentences and theories -/\n-- variable (L)\n-- inductive bounded_preterm (n : \u2115) : \u2115 \u2192 Type u\n-- | bd_var {} : \u2200 (k : fin n), bounded_preterm 0\n-- | bd_func {} : \u2200 {l : \u2115} (f : L.functions l), bounded_preterm l\n-- | bd_app : \u2200 {l : \u2115} (t : bounded_preterm (l + 1)) (s : bounded_preterm 0), bounded_preterm l\n-- export bounded_preterm\n\n-- def bounded_term   (n) := bounded_preterm L n 0\n-- def closed_preterm (l) := bounded_preterm L 0 l\n-- def closed_term        := closed_preterm L 0\n-- variable {L}\n\n-- prefix `&`:max := bd_var\n-- def bd_const {n} (c : L.constants) : bounded_term L n := bd_func c\n\n-- @[simp] def bd_apps' {n} : \u2200{l m}, bounded_preterm L n (l + m) \u2192 dvector (bounded_term L n) m \u2192\n--   bounded_preterm L n l\n-- | l 0 t [] := t\n-- | l (m+1) t (x::xs) := bd_apps' (bd_app t x) xs\n\n-- @[simp] def bd_apps {n} : \u2200{l}, bounded_preterm L n l \u2192 dvector (bounded_term L n) l \u2192\n--   bounded_term L n\n-- | _ t []       := t\n-- | _ t (t'::ts) := bd_apps (bd_app t t') ts\n\n-- namespace bounded_preterm\n-- @[simp] protected def fst {n} : \u2200{l}, bounded_preterm L n l \u2192 preterm L l\n-- | _ &k           := &k.1\n-- | _ (bd_func f)  := func f\n-- | _ (bd_app t s) := app (fst t) (fst s)\n\n-- local attribute [extensionality] fin.eq_of_veq\n-- @[extensionality] protected def eq {n} : \u2200{l} {t\u2081 t\u2082 : bounded_preterm L n l} (h : t\u2081.fst = t\u2082.fst),\n--   t\u2081 = t\u2082\n-- | _ &k &k'                        h := by injection h with h'; congr1; ext; exact h'\n-- | _ &k (bd_func f')               h := by injection h\n-- | _ &k (bd_app t\u2081' t\u2082')           h := by injection h\n-- | _ (bd_func f) &k'               h := by injection h\n-- | _ (bd_func f) (bd_func f')      h := by injection h with h'; rw h'\n-- | _ (bd_func f) (bd_app t\u2081' t\u2082')  h := by injection h\n-- | _ (bd_app t\u2081 t\u2082) &k'            h := by injection h\n-- | _ (bd_app t\u2081 t\u2082) (bd_func f')   h := by injection h\n-- | _ (bd_app t\u2081 t\u2082) (bd_app t\u2081' t\u2082') h := by injection h with h\u2081 h\u2082; congr1; apply eq; assumption\n\n-- @[simp] protected def cast {n m} (h : n \u2264 m) : \u2200 {l} (t : bounded_preterm L n l),\n--   bounded_preterm L m l\n-- | _ &k           := &(k.cast_le h)\n-- | _ (bd_func f)  := bd_func f\n-- | _ (bd_app t s) := bd_app t.cast s.cast\n\n-- @[simp] lemma cast_bd_app {n m} (h : n \u2264 m) {l} {t : bounded_preterm L n (l+1)}\n--   {s : bounded_preterm L n 0} : (bd_app t s).cast h = (bd_app (t.cast h) (s.cast h)) := by refl\n\n-- @[simp] lemma cast_bd_apps {n m } (h : n \u2264 m) {l} {t : bounded_preterm L n l}\n--   {ts : dvector (bounded_term L n) l} :\n--   (bd_apps t ts).cast h = bd_apps (t.cast h) (ts.map (\u03bb t, t.cast h)) :=\n-- by {induction ts generalizing t, refl, simp*}\n\n-- -- @[simp] lemma cast_bd_apps_nil {n m} (h : n \u2264 m) {l} {t : bounded_preterm L n (l+1)} {s : bounded_preterm L n 0} : (bd_apps t s []).cast h = (bd_app (t.cast h) (s.cast h))\n\n-- @[simp] lemma cast_irrel {n m } {h h' : n \u2264 m} : \u2200 {l} (t : bounded_preterm L n l),\n--   (t.cast h) = (t.cast h') :=\n-- by {intros, refl}\n\n-- @[simp] lemma cast_rfl {n} {h : n \u2264 n} : \u2200 {l} (t : bounded_preterm L n l), (t.cast h) = t :=\n-- begin\n--   intros, induction t,\n--   {simp, unfold fin.cast_le, unfold fin.cast_lt, cases t, refl}, {refl}, {simp*}\n-- end\n\n-- protected def cast_eq {n m l} (h : n = m) (t : bounded_preterm L n l) : bounded_preterm L m l :=\n-- t.cast $ le_of_eq h\n\n-- protected def cast1 {n l} (t : bounded_preterm L n l) : bounded_preterm L (n+1) l :=\n-- t.cast $ n.le_add_right 1\n\n-- @[simp] lemma cast_fst {n m} (h : n \u2264 m) : \u2200 {l} (t : bounded_preterm L n l), (t.cast h).fst = t.fst\n-- | _ &k           := by refl\n-- | _ (bd_func f)  := by refl\n-- | _ (bd_app t s) := by dsimp; simp [cast_fst]\n\n-- @[simp] lemma cast_eq_fst {n m l} (h : n = m) (t : bounded_preterm L n l) :\n--   (t.cast_eq h).fst = t.fst := t.cast_fst _\n-- @[simp] lemma cast1_fst {n l} (t : bounded_preterm L n l) :\n--   t.cast1.fst = t.fst := t.cast_fst _\n\n-- @[simp] lemma cast_eq_rfl {n m l} (h : n = m) (t : bounded_preterm L n l) :\n--   (t.cast_eq h).cast_eq h.symm = t := by ext; simp\n\n-- @[simp] lemma cast_eq_irrel {n m l} (h h' : n = m) (t : bounded_preterm L n l) :\n--   (t.cast_eq h) = (t.cast_eq h') := by refl\n\n-- @[simp] lemma cast_eq_bd_app {n m} (h : n = m) {l} {t : bounded_preterm L n (l+1)}\n--   {s : bounded_preterm L n 0} : (bd_app t s).cast_eq h = (bd_app (t.cast_eq h) (s.cast_eq h)) :=\n-- by refl\n\n-- @[simp] lemma cast_eq_bd_apps {n m } (h : n = m) {l} {t : bounded_preterm L n l}\n--   {ts : dvector (bounded_term L n) l} :\n--   (bd_apps t ts).cast_eq h = bd_apps (t.cast_eq h) (ts.map (\u03bb t, t.cast_eq h)) :=\n-- by {induction ts generalizing t, refl, simp*}\n\n-- end bounded_preterm\n\n-- namespace closed_preterm\n\n-- @[reducible]protected def cast0 (n) {l} (t : closed_preterm L l) : bounded_preterm L n l :=\n-- t.cast n.zero_le\n\n-- @[simp] lemma cast0_fst {n l : \u2115} (t : closed_preterm L l) :\n--   (t.cast0 n).fst = t.fst :=\n-- cast_fst _ _\n\n-- @[simp] lemma cast_of_cast0 {n} {l} {t : closed_preterm L l} : t.cast0 n =  t.cast n.zero_le :=\n-- by refl\n\n-- end closed_preterm\n\n-- @[elab_as_eliminator] def bounded_term.rec {n} {C : bounded_term L n \u2192 Sort v}\n--   (hvar : \u2200(k : fin n), C &k)\n--   (hfunc : \u03a0 {l} (f : L.functions l) (ts : dvector (bounded_term L n) l)\n--     (ih_ts : \u2200t, ts.pmem t \u2192 C t), C (bd_apps (bd_func f) ts)) :\n--   \u2200(t : bounded_term L n), C t :=\n-- have h : \u2200{l} (t : bounded_preterm L n l) (ts : dvector (bounded_term L n) l)\n--   (ih_ts : \u2200s, ts.pmem s \u2192 C s), C (bd_apps t ts),\n-- begin\n--   intros, induction t; try {rw ts.zero_eq},\n--   { apply hvar },\n--   { apply hfunc t_f ts ih_ts },\n--   { apply t_ih_t (t_s::ts), intros t ht,\n--     cases ht,\n--     { induction ht, apply t_ih_s ([]), intros s hs, cases hs },\n--     { exact ih_ts t ht }},\n-- end,\n-- \u03bbt, h t ([]) (by intros s hs; cases hs)\n\n-- @[elab_as_eliminator] def bounded_term.rec1 {n} {C : bounded_term L (n+1) \u2192 Sort v}\n--   (hvar : \u2200(k : fin (n+1)), C &k)\n--   (hfunc : \u03a0 {l} (f : L.functions l) (ts : dvector (bounded_term L (n+1)) l)\n--     (ih_ts : \u2200t, ts.pmem t \u2192 C t), C (bd_apps (bd_func f) ts)) :\n--   \u2200(t : bounded_term L (n+1)), C t :=\n-- have h : \u2200{l} (t : bounded_preterm L (n+1) l) (ts : dvector (bounded_term L (n+1)) l)\n--   (ih_ts : \u2200s, ts.pmem s \u2192 C s), C (bd_apps t ts),\n-- begin\n--   intros, induction t; try {rw ts.zero_eq},\n--   { apply hvar },\n--   { apply hfunc t_f ts ih_ts },\n--   { apply t_ih_t (t_s::ts), intros t ht,\n--     cases ht,\n--     { induction ht, apply t_ih_s ([]), intros s hs, cases hs },\n--     { exact ih_ts t ht }},\n-- end,\n-- \u03bbt, h t ([]) (by intros s hs; cases hs)\n\n-- lemma lift_bounded_term_irrel {n : \u2115} : \u2200{l} (t : bounded_preterm L n l) (n') {m : \u2115}\n--   (h : n \u2264 m), t.fst \u2191' n' # m = t.fst\n-- | _ &k           n' m h :=\n--   have h' : \u00ac(m \u2264 k.1), from not_le_of_lt (lt_of_lt_of_le k.2 h), by simp [h']\n-- | _ (bd_func f)  n' m h := by refl\n-- | _ (bd_app t s) n' m h := by simp [lift_bounded_term_irrel t n' h, lift_bounded_term_irrel s n' h]\n\n-- lemma subst_bounded_term_irrel {n : \u2115} : \u2200{l} (t : bounded_preterm L n l) {n'} (s : term L)\n--   (h : n \u2264 n'), t.fst[s // n'] = t.fst\n-- | _ &k             n' s h := by simp [lt_of_lt_of_le k.2 h]\n-- | _ (bd_func f)    n' s h := by refl\n-- | _ (bd_app t\u2081 t\u2082) n' s h := by simp*\n\n-- /--Given a bounded_preterm of bound n and level l, realize it using (v : dvector S n) and (xs : dvector L l) by the following structural induction:\n\n-- 1. Given a free de Bruijn variable &k, replace it with the kth member (indexing starting at 0) of v,\n\n-- 2. given a (bd_func f), replace it with its realization as a function on S, _evaluated_ at xs, and\n\n-- 3. given an application of terms, replace it with a literal application of terms, with the inner term evaluated at xs.\n-- --/\n\n-- --- note from Mario: replace dvector.nth with dvector.nth''\n\n-- @[simp] def realize_bounded_term {S : Structure L} {n} (v : dvector S n) :\n--   \u2200{l} (t : bounded_preterm L n l) (xs : dvector S l), S.carrier\n-- | _ &k             xs := v.nth k.1 k.2\n-- | _ (bd_func f)    xs := S.fun_map f xs\n-- | _ (bd_app t\u2081 t\u2082) xs := realize_bounded_term t\u2081 $ realize_bounded_term t\u2082 ([])::xs\n\n-- /- S[t ; v] -/\n-- notation S`[`:max t ` ;;; `:95 v`]`:0 := @fol.realize_bounded_term _ S _  v _ t (dvector.nil)\n\n-- notation S`[`:max t ` ;;; `:95 v ` ;;; `:90 xs `]`:0 := @fol.realize_bounded_term _ S _  v _ t xs\n\n\n-- @[reducible] def realize_closed_term (S : Structure L) (t : closed_term L) : S :=\n-- realize_bounded_term ([]) t ([])\n\n-- lemma realize_bounded_term_eq {S : Structure L} {n} {v\u2081 : dvector S n} {v\u2082 : \u2115 \u2192 S}\n--   (hv : \u2200k (hk : k < n), v\u2081.nth k hk = v\u2082 k) : \u2200{l} (t : bounded_preterm L n l)\n--   (xs : dvector S l), realize_bounded_term v\u2081 t xs = realize_term v\u2082 t.fst xs\n-- | _ &k             xs := hv k.1 k.2\n-- | _ (bd_func f)    xs := by refl\n-- | _ (bd_app t\u2081 t\u2082) xs := by dsimp; simp [realize_bounded_term_eq]\n\n-- lemma realize_bounded_term_irrel' {S : Structure L} {n n'} {v\u2081 : dvector S n} {v\u2082 : dvector S n'}\n--   (h : \u2200m (hn : m < n) (hn' : m < n'), v\u2081.nth m hn = v\u2082.nth m hn')\n--   {l} (t : bounded_preterm L n l) (t' : bounded_preterm L n' l)\n--   (ht : t.fst = t'.fst) (xs : dvector S l) :\n--   realize_bounded_term v\u2081 t xs = realize_bounded_term v\u2082 t' xs :=\n-- begin\n--   induction t; cases t'; injection ht with ht\u2081 ht\u2082,\n--   { simp, cases t'_1; dsimp at ht\u2081, subst ht\u2081, exact h t.val t.2 t'_1_is_lt },\n--   { subst ht\u2081, refl },\n--   { simp [t_ih_t t'_t ht\u2081, t_ih_s t'_s ht\u2082] }\n-- end\n\n-- lemma realize_bounded_term_irrel {S : Structure L} {n} {v\u2081 : dvector S n}\n--   (t : bounded_term L n) (t' : closed_term L) (ht : t.fst = t'.fst) (xs : dvector S 0) :\n--   realize_bounded_term v\u2081 t xs = realize_closed_term S t' :=\n-- by cases xs; exact realize_bounded_term_irrel'\n--   (by intros m hm hm'; exfalso; exact not_lt_zero m hm') t t' ht ([])\n\n-- @[simp] lemma realize_bounded_term_cast_eq_irrel {S : Structure L} {n m l} {h : n = m}\n--   {v : dvector S m} {t : bounded_preterm L n l} (xs : dvector S l) :\n-- realize_bounded_term v (t.cast_eq h) xs = realize_bounded_term (v.cast h.symm) t xs :=\n-- by {subst h, induction t, refl, refl, simp*}\n\n-- @[simp] lemma realize_bounded_term_dvector_cast_irrel {S : Structure L} {n m l} {h : n = m}\n--   {v : dvector S n} {t : bounded_preterm L n l} {xs : dvector S l} :\n--   realize_bounded_term (v.cast h) (t.cast (le_of_eq h)) xs = realize_bounded_term v t xs :=\n-- by {subst h, simp, refl}\n\n-- @[simp] def lift_bounded_term_at {n} : \u2200{l} (t : bounded_preterm L n l) (n' m : \u2115),\n--   bounded_preterm L (n + n') l\n-- | _ &k             n' m := if m \u2264 k.1 then &(k.add_nat n') else &(k.cast_le $ n.le_add_right n')\n-- | _ (bd_func f)    n' m := bd_func f\n-- | _ (bd_app t\u2081 t\u2082) n' m := bd_app (lift_bounded_term_at t\u2081 n' m) $ lift_bounded_term_at t\u2082 n' m\n\n-- notation t ` \u2191' `:90 n ` # `:90 m:90 := fol.lift_bounded_term_at t n m -- input \u2191 with \\u or \\upa\n\n-- @[reducible] def lift_bounded_term {n l} (t : bounded_preterm L n l) (n' : \u2115) :\n--   bounded_preterm L (n + n') l := t \u2191' n' # 0\n-- infix ` \u2191 `:100 := fol.lift_bounded_term -- input \u2191' with \\u or \\upa\n\n-- @[reducible, simp] def lift_bounded_term1 {n' l} (t : bounded_preterm L n' l) :\n--   bounded_preterm L (n'+1) l :=\n-- t \u2191 1\n\n-- @[simp] lemma lift_bounded_term_fst {n} : \u2200{l} (t : bounded_preterm L n l) (n' m : \u2115),\n--   (t \u2191' n' # m).fst = t.fst \u2191' n' # m\n-- | _ &k             n' m := by by_cases h : m \u2264 k.1; simp [h, -add_comm]; refl\n-- | _ (bd_func f)    n' m := by refl\n-- | _ (bd_app t\u2081 t\u2082) n' m := by simp [lift_bounded_term_fst]\n\n-- -- @[simp] def lift_closed_term_at : \u2200{l} (t : closed_preterm L l) (n' m : \u2115),\n-- --   bounded_preterm L n' l\n-- -- | _ &k             n' m := if m \u2264 k then _ else &(k.cast_le $ n.le_add_right n')\n-- -- | _ (bd_func f)    n' m := bd_func f\n-- -- | _ (bd_app t\u2081 t\u2082) n' m := bd_app (lift_bounded_term_at t\u2081 n' m) $ lift_bounded_term_at t\u2082 n' m\n\n\n-- -- def lift_bounded_term_at0 {n m l} {t : preterm L l} (ht : bounded_term 0 t) : bounded_term n (t \u2191' n # m) :=\n-- -- by have := lift_bounded_term_at n m ht; rw [zero_add] at this; exact this\n\n-- /-- this is t[s//n] for bounded formulae-/\n-- def subst_bounded_term {n n'} : \u2200{l} (t : bounded_preterm L (n+n'+1) l)\n--   (s : bounded_term L n'), bounded_preterm L (n+n') l\n-- | _ &k             s :=\n--   if h : k.1 < n then &\u27e8k.1, lt_of_lt_of_le h $ n.le_add_right n'\u27e9 else\n--   if h' : n < k.1 then &\u27e8k.1-1, (nat.sub_lt_right_iff_lt_add $ one_le_of_lt h').mpr k.2\u27e9 else\n--   (s \u2191 n).cast $ le_of_eq $ add_comm n' n\n-- | _ (bd_func f)    s := bd_func f\n-- | _ (bd_app t\u2081 t\u2082) s := bd_app (subst_bounded_term t\u2081 s) (subst_bounded_term t\u2082 s)\n\n-- notation t `[`:max s ` /// `:95 n `]`:0 := @_root_.fol.subst_bounded_term _ n _ _ t s\n-- -- notation t `[`:95 s ` // `:95 n `]`:0 := @fol.subst_bounded_term _ n _ _ t s\n-- -- notation f `[`:95 s ` // `:95 n `]`:0 := @_root_.fol.subst_bounded_term\n\n-- @[simp] lemma subst_bounded_term_var_lt {n n'} (s : bounded_term L n') (k : fin (n+n'+1))\n--   (h : k.1 < n) : (subst_bounded_term &k s).fst = &k.1 :=\n-- by simp [h, fol.subst_bounded_term]\n\n-- @[simp] lemma subst_bounded_term_var_gt {n n'} (s : bounded_term L n') (k : fin (n+n'+1))\n--   (h : n < k.1) : (subst_bounded_term &k s).fst = &(k.1-1) :=\n-- have h' : \u00ac(k.1 < n), from lt_asymm h,\n-- by simp [h, h', fol.subst_bounded_term]\n\n-- @[simp] lemma subst_bounded_term_var_eq {n n'} (s : bounded_term L n') (k : fin (n+n'+1))\n--   (h : k.1 = n) : (subst_bounded_term &k s).fst = s.fst \u2191 n :=\n-- have h\u2082 : \u00ac(k.1 < n), from \u03bbh', lt_irrefl _ $ lt_of_lt_of_le h' $ le_of_eq h.symm,\n-- have h\u2083 : \u00ac(n < k.1), from \u03bbh', lt_irrefl _ $ lt_of_lt_of_le h' $ le_of_eq h,\n-- by simp [subst_bounded_term, h\u2082, h\u2083]\n\n-- @[simp] lemma subst_bounded_term_bd_app {n n' l} (t\u2081 : bounded_preterm L (n+n'+1) (l+1))\n--   (t\u2082 : bounded_term L (n+n'+1)) (s : bounded_term L n') :\n--   subst_bounded_term (bd_app t\u2081 t\u2082) s = bd_app (subst_bounded_term t\u2081 s) (subst_bounded_term t\u2082 s) :=\n-- by refl\n\n-- @[simp] lemma subst_bounded_term_fst {n n'} : \u2200{l} (t : bounded_preterm L (n+n'+1) l)\n--   (s : bounded_term L n'), (subst_bounded_term t s).fst = t.fst[s.fst//n]\n-- | _ &k             s := by apply decidable.lt_by_cases k.1 n; intro h; simp [h]\n-- | _ (bd_func f)    s := by refl\n-- | _ (bd_app t\u2081 t\u2082) s := by simp*\n\n-- -- @[simp] lemma subst_bounded_term_var_eq' {n n'} (s : bounded_term L n') (h : n < n+n'+1) :\n-- --   (subst_bounded_term &\u27e8n, h\u27e9 s).fst = s.fst \u2191 n :=\n-- -- by simp [subst_bounded_term]\n\n-- def subst0_bounded_term {n l} (t : bounded_preterm L (n+1) l)\n--   (s : bounded_term L n) : bounded_preterm L n l :=\n-- (subst_bounded_term (t.cast_eq $ (n+1).zero_add.symm) s).cast_eq $ n.zero_add\n\n-- notation t `[`:max s ` /0]`:0 := fol.subst0_bounded_term t s\n\n-- @[simp] lemma subst0_bounded_term_fst {n l} (t : bounded_preterm L (n+1) l)\n--   (s : bounded_term L n) : t[s/0].fst = t.fst[s.fst//0] :=\n-- by simp [subst0_bounded_term]\n\n-- def substmax_bounded_term {n l} (t : bounded_preterm L (n+1) l)\n--   (s : closed_term L) : bounded_preterm L n l :=\n-- subst_bounded_term (by exact t) s\n\n-- @[simp] lemma substmax_bounded_term_bd_app {n l} (t\u2081 : bounded_preterm L (n+1) (l+1))\n--   (t\u2082 : bounded_term L (n+1)) (s : closed_term L) :\n--   substmax_bounded_term (bd_app t\u2081 t\u2082) s =\n--   bd_app (substmax_bounded_term t\u2081 s) (substmax_bounded_term t\u2082 s) :=\n-- by refl\n\n-- def substmax_eq_subst0_term {l} (t : bounded_preterm L 1 l) (s : closed_term L) :\n--   t[s/0] = substmax_bounded_term t s :=\n-- by ext; simp [substmax_bounded_term]\n\n-- def substmax_var_lt {n} (k : fin (n+1)) (s : closed_term L) (h : k.1 < n) :\n--   substmax_bounded_term &k s = &\u27e8k.1, h\u27e9 :=\n-- by ext; simp [substmax_bounded_term, h]\n\n-- def substmax_var_eq {n} (k : fin (n+1)) (s : closed_term L) (h : k.1 = n) :\n--   substmax_bounded_term &k s = s.cast0 n :=\n-- begin\n--   ext, simp [substmax_bounded_term, h],\n--   dsimp only [lift_term], rw [lift_bounded_term_irrel s _ (le_refl _)]\n-- end\n\n-- def bounded_term_of_function {l n} (f : L.functions l) :\n--   arity' (bounded_term L n) (bounded_term L n) l :=\n-- arity'.of_dvector_map $ bd_apps (bd_func f)\n\n-- @[simp] lemma realize_bounded_term_bd_app {S : Structure L}\n--   {n l} (t : bounded_preterm L n (l+1)) (s : bounded_term L n) (xs : dvector S n)\n--   (xs' : dvector S l) :\n--   realize_bounded_term xs (bd_app t s) xs' =\n--   realize_bounded_term xs t (realize_bounded_term xs s ([])::xs') :=\n-- by refl\n\n-- @[simp] lemma realize_closed_term_bd_apps {S : Structure L}\n--   {l} (t : closed_preterm L l) (ts : dvector (closed_term L) l) :\n--   realize_closed_term S (bd_apps t ts) =\n--   realize_bounded_term ([]) t (ts.map (\u03bbt', realize_bounded_term ([]) t' ([]))) :=\n-- begin\n--   induction ts generalizing t, refl, apply ts_ih (bd_app t ts_x)\n-- end\n-- --\u27e8t.fst[s.fst // n], bounded_term_subst_closed t.snd s.snd\u27e9\n\n-- lemma realize_bounded_term_bd_apps {S : Structure L}\n--   {n l} (xs : dvector S n) (t : bounded_preterm L n l) (ts : dvector (bounded_term L n) l) :\n--   realize_bounded_term xs (bd_apps t ts) ([]) =\n--   realize_bounded_term xs t (ts.map (\u03bbt, realize_bounded_term xs t ([]))) :=\n-- begin\n--   induction ts generalizing t, refl, apply ts_ih (bd_app t ts_x)\n-- end\n\n-- @[simp] lemma realize_cast_bounded_term {S : Structure L} {n m} {h : n \u2264 m} {t : bounded_term L n}\n--   {v : dvector S m} : realize_bounded_term v (t.cast h) dvector.nil =\n--     realize_bounded_term (v.trunc n h) t dvector.nil :=\n-- begin\n--   revert t, apply bounded_term.rec,\n--   {intro k, simp only [dvector.trunc_nth, fol.bounded_preterm.cast, fol.realize_bounded_term,\n--    dvector.nth, dvector.trunc], refl},\n--   {simp[realize_bounded_term_bd_apps], intros, congr' 1, apply dvector.map_congr_pmem,\n--   exact ih_ts}\n-- end\n\n-- /- When realizing a closed term, we can replace the realizing dvector with [] -/\n-- @[simp] lemma realize_closed_term_v_irrel {S : Structure L} {n} {v : dvector S n}\n--   {t : bounded_term L 0} :\n--   realize_bounded_term v (t.cast (by {simp})) ([]) = realize_closed_term S t :=\n-- by simp[realize_cast_bounded_term]\n\n\n-- /- this is the same as realize_bounded_term, we should probably have a common generalization of this definition -/\n-- -- @[simp] def substitute_bounded_term {n n'} (v : dvector (bounded_term n') n) :\n-- --   \u2200{l} (t : bounded_term L n l, bounded_preterm L n' l\n-- -- | _ _ &k          := v.nth k hk\n-- -- | _ _ (bd_func f)             := bd_func f\n-- -- | _ _ (bd_app t\u2081 t\u2082) := bd_app (substitute_bounded_term ht\u2081) $ substitute_bounded_term ht\u2082\n\n-- -- def substitute_bounded_term {n n' l} (t : bounded_preterm L n l)\n-- --   (v : dvector (bounded_term n') n) : bounded_preterm L n' l :=\n-- -- substitute_bounded_term v t.snd\n\n-- variable (L)\n-- inductive bounded_preformula : \u2115 \u2192 \u2115 \u2192 Type u\n-- | bd_falsum {} {n} : bounded_preformula n 0\n-- | bd_equal {n} (t\u2081 t\u2082 : bounded_term L n) : bounded_preformula n 0\n-- | bd_rel {n l : \u2115} (R : L.relations l) : bounded_preformula n l\n-- | bd_apprel {n l} (f : bounded_preformula n (l + 1)) (t : bounded_term L n) : bounded_preformula n l\n-- | bd_imp {n} (f\u2081 f\u2082 : bounded_preformula n 0) : bounded_preformula n 0\n-- | bd_all {n} (f : bounded_preformula (n+1) 0) : bounded_preformula n 0\n\n-- export bounded_preformula\n\n-- @[reducible] def bounded_formula (n : \u2115) := bounded_preformula L n 0\n-- @[reducible] def presentence     (l : \u2115) := bounded_preformula L 0 l\n-- @[reducible] def sentence                := presentence L 0\n-- variable {L}\n\n-- instance nonempty_bounded_formula (n : \u2115) : nonempty $ bounded_formula L n :=\n--   nonempty.intro (by constructor)\n\n\n-- -- @[reducible, simp] def bd_falsum' {n} : bounded_formula L n := bd_falsum\n-- -- @[reducible, simp] def bd_equal' {n} (t\u2081 t\u2082 : bounded_term L n) : bounded_formula L n :=\n-- -- bd_equal t\u2081 t\u2082\n-- -- @[reducible, simp] def bd_imp' {n} (f\u2081 f\u2082 : bounded_formula L n) : bounded_formula L n :=\n-- -- bd_imp f\u2081 f\u2082\n-- notation `\u22a5` := fol.bounded_preformula.bd_falsum -- input: \\bot\n-- infix ` \u2243 `:88 := fol.bounded_preformula.bd_equal -- input \\~- or \\simeq\n-- infixr ` \u27f9 `:62 := fol.bounded_preformula.bd_imp -- input \\==>\n-- def bd_not {n} (f : bounded_formula L n) : bounded_formula L n := f \u27f9 \u22a5\n-- prefix `\u223c`:max := fol.bd_not -- input \\~, the ASCII character ~ has too low precedence\n-- def bd_and {n} (f\u2081 f\u2082 : bounded_formula L n) : bounded_formula L n := \u223c(f\u2081 \u27f9 \u223cf\u2082)\n-- infixr ` \u2293 ` := fol.bd_and -- input: \\sqcap\n-- def bd_or {n} (f\u2081 f\u2082 : bounded_formula L n) : bounded_formula L n := \u223cf\u2081 \u27f9 f\u2082\n-- infixr ` \u2294 ` := fol.bd_or -- input: \\sqcup\n-- def bd_biimp {n} (f\u2081 f\u2082 : bounded_formula L n) : bounded_formula L n := (f\u2081 \u27f9 f\u2082) \u2293 (f\u2082 \u27f9 f\u2081)\n-- infix ` \u21d4 `:61 := fol.bd_biimp -- input \\<=>\n-- prefix `\u2200'`:110 := fol.bounded_preformula.bd_all\n-- def bd_ex {n} (f : bounded_formula L (n+1)) : bounded_formula L n := \u223c (\u2200' (\u223c f))\n-- prefix `\u2203'`:110 := fol.bd_ex\n\n\n\n-- def bd_apps_rel : \u2200{n l} (f : bounded_preformula L n l) (ts : dvector (bounded_term L n) l),\n--   bounded_formula L n\n-- | _ _ f []      := f\n-- | _ _ f (t::ts) := bd_apps_rel (bd_apprel f t) ts\n\n-- @[simp] lemma bd_apps_rel_zero {n} (f : bounded_formula L n) (ts : dvector (bounded_term L n) 0) :\n--   bd_apps_rel f ts = f :=\n-- by cases ts; refl\n\n-- namespace bounded_preformula\n-- @[simp] protected def fst : \u2200{n l}, bounded_preformula L n l \u2192 preformula L l\n-- | _ _ bd_falsum       := \u22a5\n-- | _ _ (t\u2081 \u2243 t\u2082)       := t\u2081.fst \u2243 t\u2082.fst\n-- | _ _ (bd_rel R)      := rel R\n-- | _ _ (bd_apprel f t) := apprel f.fst t.fst\n-- | _ _ (f\u2081 \u27f9 f\u2082)      := f\u2081.fst \u27f9 f\u2082.fst\n-- | _ _ (\u2200' f)          := \u2200' f.fst\n\n-- @[simp] lemma fst_not : \u2200{n} {f : bounded_formula L n},\n--   \u223c(bounded_preformula.fst f) = bounded_preformula.fst (\u223cf) := by {intros, refl}\n\n-- local attribute [extensionality] fin.eq_of_veq\n-- @[extensionality] protected def eq {n l} {f\u2081 f\u2082 : bounded_preformula L n l} (h : f\u2081.fst = f\u2082.fst) :\n--   f\u2081 = f\u2082 :=\n-- begin\n--   induction f\u2081; cases f\u2082; injection h with h\u2081 h\u2082,\n--   { refl },\n--   { congr1; apply bounded_preterm.eq; assumption },\n--   { rw h\u2081 },\n--   { congr1, exact f\u2081_ih h\u2081, exact bounded_preterm.eq h\u2082 },\n--   { congr1, exact f\u2081_ih_f\u2081 h\u2081, exact f\u2081_ih_f\u2082 h\u2082 },\n--   { rw [f\u2081_ih h\u2081] }\n-- end\n\n-- @[simp] protected def cast : \u2200 {n m l} (h : n \u2264 m)  (f : bounded_preformula L n l),\n--   bounded_preformula L m l\n-- | _ _ _ h bd_falsum       := bd_falsum\n-- | _ _ _ h (t\u2081 \u2243 t\u2082)       := t\u2081.cast h \u2243 t\u2082.cast h\n-- | _ _ _ h (bd_rel R)      := bd_rel R\n-- | _ _ _ h (bd_apprel f t) := bd_apprel (f.cast h) $ t.cast h\n-- | _ _ _ h (f\u2081 \u27f9 f\u2082)      := f\u2081.cast h \u27f9 f\u2082.cast h\n-- | _ _ _ h (\u2200' f)          := \u2200' f.cast (succ_le_succ h)\n\n-- @[simp] lemma cast_irrel : \u2200 {n m l} (h h' : n \u2264 m) (f : bounded_preformula L n l),\n--   (f.cast h) = (f.cast h') :=\n-- by {intros, refl}\n\n-- @[simp] lemma cast_rfl {n} {h : n \u2264 n} : \u2200 {l} (f : bounded_preformula L n l), (f.cast h) = f :=\n-- by {intros, induction f; simp*}\n\n-- protected def cast_eq {n m l} (h : n = m) (f : bounded_preformula L n l) :\n--   bounded_preformula L m l :=\n-- f.cast $ le_of_eq h\n\n-- protected def cast_eqr {n m l} (h : n = m) (f : bounded_preformula L m l) :\n--   bounded_preformula L n l :=\n-- f.cast $ ge_of_eq h\n\n-- lemma cast_bd_apps_rel {S : Structure L} {n m} {h : n \u2264 m} {l} {f : bounded_preformula L n l}\n--   {ts : dvector (bounded_term L n) l} :\n--   ((bd_apps_rel f ts).cast h) = bd_apps_rel (f.cast h) (ts.map (\u03bb t, t.cast h)) :=\n-- by {induction ts, refl, apply @ts_ih (bd_apprel f ts_x)}\n\n-- protected def cast1 {n l} (f : bounded_preformula L n l) : bounded_preformula L (n+1) l :=\n-- f.cast $ n.le_add_right 1\n\n-- @[simp] lemma cast_fst : \u2200 {l n m} (h : n \u2264 m) (f : bounded_preformula L n l),\n--   (f.cast h).fst = f.fst\n-- | _ _ _ h bd_falsum       := by refl\n-- | _ _ _ h (t\u2081 \u2243 t\u2082)       := by simp\n-- | _ _ _ h (bd_rel R)      := by refl\n-- | _ _ _ h (bd_apprel f t) := by simp*\n-- | _ _ _ h (f\u2081 \u27f9 f\u2082)      := by simp*\n-- | _ _ _ h (\u2200' f)          := by simp*\n\n-- @[simp] lemma cast_eq_fst {l n m} (h : n = m) (f : bounded_preformula L n l) :\n--   (f.cast_eq h).fst = f.fst := f.cast_fst _\n-- @[simp] lemma cast1_fst {l n} (f : bounded_preformula L n l) :\n--   f.cast1.fst = f.fst := f.cast_fst _\n\n-- @[simp] lemma cast_eq_rfl {l n m} (h : n = m) (f : bounded_preformula L n l) :\n--   (f.cast_eq h).cast_eq h.symm = f := by ext; simp\n\n-- @[simp] lemma cast_eq_irrel {l n m} (h h' : n = m) (f : bounded_preformula L n l) :\n--   (f.cast_eq h) = (f.cast_eq h') := by refl\n\n-- @[simp] lemma cast_eq_all {n m } (h : n = m) {f : bounded_preformula L (n+1) _} :\n--   (\u2200' f).cast_eq h = \u2200' (f.cast_eq (by {subst h; refl})) := by refl\n\n-- @[simp] lemma cast_eq_trans {n m o l} {h : n = m} {h' : m = o} {f : bounded_preformula L n l} :\n--   (f.cast_eq h).cast_eq h' = f.cast_eq (eq.trans h h') := by substs h h'; ext; simp\n\n-- lemma cast_eq_hrfl {n m l} {h : n = m} {f : bounded_preformula L n l} : f.cast_eq h == f :=\n-- by {subst h, simp only [heq_iff_eq], ext, simp}\n\n-- /- A bounded_preformula is qf if the underlying preformula is qf -/\n-- def quantifier_free {l n} : bounded_preformula L n l \u2192 Prop := \u03bb f, fol.quantifier_free f.fst\n\n-- end bounded_preformula\n\n-- namespace presentence\n\n-- @[reducible]protected def cast0 {l} (n) (f : presentence L l) : bounded_preformula L n l :=\n-- f.cast n.zero_le\n\n-- @[simp] lemma cast0_fst {l} (n) (f : presentence L l) :\n--   (f.cast0 n).fst = f.fst := f.cast_fst _\n\n-- end presentence\n\n-- lemma lift_bounded_formula_irrel : \u2200{n l} (f : bounded_preformula L n l) (n') {m : \u2115}\n--   (h : n \u2264 m), f.fst \u2191' n' # m = f.fst\n-- | _ _ bd_falsum       n' m h := by refl\n-- | _ _ (t\u2081 \u2243 t\u2082)       n' m h := by simp [lift_bounded_term_irrel _ _ h]\n-- | _ _ (bd_rel R)      n' m h := by refl\n-- | _ _ (bd_apprel f t) n' m h := by simp [*, lift_bounded_term_irrel _ _ h]\n-- | _ _ (f\u2081 \u27f9 f\u2082)      n' m h := by simp*\n-- | _ _ (\u2200' f)          n' m h := by simp*\n\n-- lemma lift_sentence_irrel (f : sentence L) : f.fst \u2191 1 = f.fst :=\n-- lift_bounded_formula_irrel f 1 $ le_refl 0\n\n-- @[simp] lemma subst_bounded_formula_irrel : \u2200{n l} (f : bounded_preformula L n l) {n'} (s : term L)\n--   (h : n \u2264 n'), f.fst[s // n'] = f.fst\n-- | _ _ bd_falsum       n' s h := by refl\n-- | _ _ (t\u2081 \u2243 t\u2082)       n' s h := by simp [subst_bounded_term_irrel _ s h]\n-- | _ _ (bd_rel R)      n' s h := by refl\n-- | _ _ (bd_apprel f t) n' s h := by simp [*, subst_bounded_term_irrel _ s h]\n-- | _ _ (f\u2081 \u27f9 f\u2082)      n' s h := by simp*\n-- | _ _ (\u2200' f)          n' s h := by simp*\n\n-- lemma subst_sentence_irrel (f : sentence L) (n) (s : term L) : f.fst[s // n] = f.fst :=\n-- subst_bounded_formula_irrel f s n.zero_le\n\n\n\n-- @[simp] def realize_bounded_formula {S : Structure L} :\n--   \u2200{n l} (v : dvector S n) (f : bounded_preformula L n l) (xs : dvector S l), Prop\n-- | _ _ v bd_falsum       xs := false\n-- | _ _ v (t\u2081 \u2243 t\u2082)       xs := realize_bounded_term v t\u2081 xs = realize_bounded_term v t\u2082 xs\n-- | _ _ v (bd_rel R)      xs := S.rel_map R xs\n-- | _ _ v (bd_apprel f t) xs := realize_bounded_formula v f $ realize_bounded_term v t ([])::xs\n-- | _ _ v (f\u2081 \u27f9 f\u2082)      xs := realize_bounded_formula v f\u2081 xs \u2192 realize_bounded_formula v f\u2082 xs\n-- | _ _ v (\u2200' f)          xs := \u2200(x : S), realize_bounded_formula (x::v) f xs\n\n-- notation S`[`:95 f ` ;; `:95 v ` ;; `:90 xs `]`:0 := @fol.realize_bounded_formula _ S _ _ v f xs\n-- notation S`[`:95 f ` ;; `:95 v `]`:0 := @fol.realize_bounded_formula _ S _ 0 v f (dvector.nil)\n\n\n-- @[reducible] def realize_sentence (S : Structure L) (f : sentence L) : Prop :=\n-- realize_bounded_formula ([] : dvector S 0) f ([])\n\n-- local notation S`[`:max f `]`:0 := fol.realize_sentence S f\n\n-- lemma realize_bounded_formula_iff {S : Structure L} : \u2200{n} {v\u2081 : dvector S n} {v\u2082 : \u2115 \u2192 S}\n--   (hv : \u2200k (hk : k < n), v\u2081.nth k hk = v\u2082 k) {l} (t : bounded_preformula L n l)\n--   (xs : dvector S l), realize_bounded_formula v\u2081 t xs \u2194 realize_formula v\u2082 t.fst xs\n-- | _ _ _ hv _ bd_falsum       xs := by refl\n-- | _ _ _ hv _ (t\u2081 \u2243 t\u2082)       xs := by apply eq.congr; apply realize_bounded_term_eq hv\n-- | _ _ _ hv _ (bd_rel R)      xs := by refl\n-- | _ _ _ hv _ (bd_apprel f t) xs :=\n--   by simp [realize_bounded_term_eq hv, realize_bounded_formula_iff hv]\n-- | _ _ _ hv _ (f\u2081 \u27f9 f\u2082)      xs :=\n--   by simp [realize_bounded_formula_iff hv]\n-- | _ _ _ hv _ (\u2200' f)          xs :=\n--   begin\n--     apply forall_congr, intro x, apply realize_bounded_formula_iff,\n--     intros k hk, cases k, refl, apply hv\n--   end\n\n-- lemma realize_bounded_formula_iff_of_fst {S : Structure L} : \u2200{n} {v\u2081 w\u2081 : dvector S n}\n--   {v\u2082 w\u2082 : \u2115 \u2192 S} (hv\u2081 : \u2200 k (hk : k < n), v\u2081.nth k hk = v\u2082 k)\n--   (hw\u2081 : \u2200 k (hk : k < n), w\u2081.nth k hk = w\u2082 k) {l\u2081 l\u2082}\n--   (t\u2081 : bounded_preformula L n l\u2081) (t\u2082 : bounded_preformula L n l\u2082) (xs\u2081 : dvector S l\u2081)\n--   (xs\u2082 : dvector S l\u2082) (H : realize_formula v\u2082 t\u2081.fst xs\u2081 \u2194 realize_formula w\u2082 t\u2082.fst xs\u2082),\n--   (realize_bounded_formula v\u2081 t\u2081 xs\u2081 \u2194 realize_bounded_formula w\u2081 t\u2082 xs\u2082) :=\n--  by intros; simpa[realize_bounded_formula_iff hv\u2081 t\u2081, realize_bounded_formula_iff hw\u2081 t\u2082]\n\n-- @[simp] def lift_bounded_formula_at : \u2200{n l} (f : bounded_preformula L n l) (n' m : \u2115),\n--   bounded_preformula L (n + n') l\n-- | _ _ bd_falsum       n' m := \u22a5\n-- | _ _ (t\u2081 \u2243 t\u2082)       n' m := t\u2081 \u2191' n' # m \u2243 t\u2082 \u2191' n' # m\n-- | _ _ (bd_rel R)      n' m := bd_rel R\n-- | _ _ (bd_apprel f t) n' m := bd_apprel (lift_bounded_formula_at f n' m) $ t \u2191' n' # m\n-- | _ _ (f\u2081 \u27f9 f\u2082)      n' m := lift_bounded_formula_at f\u2081 n' m \u27f9 lift_bounded_formula_at f\u2082 n' m\n-- | _ _ (\u2200' f)          n' m := \u2200' (lift_bounded_formula_at f n' (m+1)).cast (le_of_eq $ succ_add _ _)\n\n-- local notation f ` \u2191' `:90 n ` # `:90 m:90 := fol.lift_bounded_formula_at f n m -- input \u2191 with \\u or \\upa\n\n-- @[reducible] def lift_bounded_formula {n l} (f : bounded_preformula L n l) (n' : \u2115) :\n--   bounded_preformula L (n + n') l := f \u2191' n' # 0\n-- infix ` \u2191 `:100 := fol.lift_bounded_formula -- input \u2191' with \\u or \\upa\n\n-- @[reducible, simp] def lift_bounded_formula1 {n' l} (f : bounded_preformula L n' l) :\n--   bounded_preformula L (n'+1) l :=\n-- f \u2191 1\n\n-- @[simp] lemma lift_bounded_formula_fst : \u2200{n l} (f : bounded_preformula L n l) (n' m : \u2115),\n--   (f \u2191' n' # m).fst = f.fst \u2191' n' # m\n-- | _ _ bd_falsum       n' m := by refl\n-- | _ _ (t\u2081 \u2243 t\u2082)       n' m := by simp\n-- | _ _ (bd_rel R)      n' m := by refl\n-- | _ _ (bd_apprel f t) n' m := by simp*\n-- | _ _ (f\u2081 \u27f9 f\u2082)      n' m := by simp*\n-- | _ _ (\u2200' f)          n' m := by simp*\n\n-- def formula_below {n n' l} (f : bounded_preformula L (n+n'+1) l)\n--   (s : bounded_term L n') : bounded_preformula L (n+n') l :=\n-- begin\n--   have : {f' : preformula L l // f.fst = f' } := \u27e8f.fst, rfl\u27e9,\n--   cases this with f' pf, induction f' generalizing n; cases f; injection pf with pf\u2081 pf\u2082,\n--   { exact \u22a5 },\n--   { exact subst_bounded_term f_t\u2081 s \u2243 subst_bounded_term f_t\u2082 s },\n--   { exact bd_rel f_R },\n--   { exact bd_apprel (f'_ih f_f pf\u2081) (subst_bounded_term f_t s) },\n--   { exact f'_ih_f\u2081 f_f\u2081 pf\u2081 \u27f9 f'_ih_f\u2082 f_f\u2082 pf\u2082 },\n--   { refine \u2200' (f'_ih (f_f.cast_eq $ congr_arg succ $ (succ_add n n').symm) $\n--       (f_f.cast_eq_fst _).trans pf\u2081).cast_eq (succ_add n n') }\n-- end\n\n-- /- f[s//n] for bounded_formula, requiring an extra proof that (n+n'+1 = n'') -/\n-- @[simp] def subst_bounded_formula : \u2200{n n' n'' l} (f : bounded_preformula L n'' l)\n--   (s : bounded_term L n') (h : n+n'+1 = n''), bounded_preformula L (n+n') l\n-- | _ _ _ _ bd_falsum       s rfl := \u22a5\n-- | _ _ _ _ (t\u2081 \u2243 t\u2082)       s rfl := subst_bounded_term t\u2081 s \u2243 subst_bounded_term t\u2082 s\n-- | _ _ _ _ (bd_rel R)      s rfl := bd_rel R\n-- | _ _ _ _ (bd_apprel f t) s rfl := bd_apprel (subst_bounded_formula f s rfl) (subst_bounded_term t s)\n-- | _ _ _ _ (f\u2081 \u27f9 f\u2082)      s rfl := subst_bounded_formula f\u2081 s rfl \u27f9 subst_bounded_formula f\u2082 s rfl\n-- | _ _ _ _ (\u2200' f)          s rfl :=\n--   \u2200' (subst_bounded_formula f s $ by simp [succ_add]).cast_eq (succ_add _ _)\n\n-- local notation f `[`:95 s ` // `:95 n ` // `:95 h `]`:0 := @fol.subst_bounded_formula _ n _ _ _ f s h\n\n-- @[simp] def subst_bounded_formula_fst : \u2200{n n' n'' l} (f : bounded_preformula L n'' l)\n--   (s : bounded_term L n') (h : n+n'+1 = n''),\n--   (subst_bounded_formula f s h).fst = f.fst[s.fst//n]\n-- | _ _ _ _ bd_falsum       s rfl := by refl\n-- | _ _ _ _ (t\u2081 \u2243 t\u2082)       s rfl := by simp\n-- | _ _ _ _ (bd_rel R)      s rfl := by refl\n-- | _ _ _ _ (bd_apprel f t) s rfl := by simp*\n-- | _ _ _ _ (f\u2081 \u27f9 f\u2082)      s rfl := by simp*\n-- | _ _ _ _ (\u2200' f)          s rfl := by simp*\n\n-- lemma realize_bounded_formula_irrel' {S : Structure L} {n n'} {v\u2081 : dvector S n} {v\u2082 : dvector S n'}\n--   (h : \u2200m (hn : m < n) (hn' : m < n'), v\u2081.nth m hn = v\u2082.nth m hn')\n--   {l} (f : bounded_preformula L n l) (f' : bounded_preformula L n' l)\n--   (hf : f.fst = f'.fst) (xs : dvector S l) :\n--   realize_bounded_formula v\u2081 f xs \u2194 realize_bounded_formula v\u2082 f' xs :=\n-- begin\n--   induction f generalizing n'; cases f'; injection hf with hf\u2081 hf\u2082,\n--   { refl },\n--   { simp [realize_bounded_term_irrel' h f_t\u2081 f'_t\u2081 hf\u2081,\n--           realize_bounded_term_irrel' h f_t\u2082 f'_t\u2082 hf\u2082] },\n--   { rw [hf\u2081], refl },\n--   { simp [realize_bounded_term_irrel' h f_t f'_t hf\u2082, f_ih _ h f'_f hf\u2081] },\n--   { apply imp_congr, apply f_ih_f\u2081 _ h _ hf\u2081, apply f_ih_f\u2082 _ h _ hf\u2082 },\n--   { apply forall_congr, intro x, apply f_ih _ _ _ hf\u2081, intros,\n--     cases m, refl, apply h }\n-- end\n\n-- lemma realize_bounded_formula_irrel {S : Structure L} {n} {v\u2081 : dvector S n}\n--   (f : bounded_formula L n) (f' : sentence L) (hf : f.fst = f'.fst) (xs : dvector S 0) :\n--   realize_bounded_formula v\u2081 f xs \u2194 realize_sentence S f' :=\n-- by cases xs; exact realize_bounded_formula_irrel'\n--   (by intros m hm hm'; exfalso; exact not_lt_zero m hm') f f' hf ([])\n\n-- @[simp] lemma realize_bounded_formula_cast_eq_irrel {S : Structure L} {n m l} {h : n = m}\n--   {v : dvector S m} {f : bounded_preformula L n l} {xs : dvector S l} :\n-- realize_bounded_formula v (f.cast_eq h) xs = realize_bounded_formula (v.cast h.symm) f xs :=\n--   by subst h; induction f; unfold bounded_preformula.cast_eq; finish\n\n-- def bounded_formula_of_relation {l n} (f : L.relations l) :\n--   arity' (bounded_term L n) (bounded_formula L n) l :=\n-- arity'.of_dvector_map $ bd_apps_rel (bd_rel f)\n\n-- @[elab_as_eliminator] def bounded_preformula.rec1 {C : \u03a0n l, bounded_preformula L (n+1) l \u2192 Sort v}\n--   (H0 : \u03a0 {n}, C n 0 \u22a5)\n--   (H1 : \u03a0 {n} (t\u2081 t\u2082 : bounded_term L (n+1)), C n 0 (t\u2081 \u2243 t\u2082))\n--   (H2 : \u03a0 {n l : \u2115} (R : L.relations l), C n l (bd_rel R))\n--   (H3 : \u03a0 {n l : \u2115} (f : bounded_preformula L (n+1) (l + 1)) (t : bounded_term L (n+1))\n--     (ih : C n (l + 1) f), C n l (bd_apprel f t))\n--   (H4 : \u03a0 {n} (f\u2081 f\u2082 : bounded_formula L (n+1)) (ih\u2081 : C n 0 f\u2081) (ih\u2082 : C n 0 f\u2082), C n 0 (f\u2081 \u27f9 f\u2082))\n--   (H5 : \u03a0 {n} (f : bounded_formula L (n+2)) (ih : C (n+1) 0 f), C n 0 (\u2200' f)) :\n--   \u2200{{n l : \u2115}} (f : bounded_preformula L (n+1) l), C n l f :=\n-- let C' : \u03a0n l, bounded_preformula L n l \u2192 Sort v :=\n-- \u03bbn, match n with\n-- | 0     := \u03bb l f, punit\n-- | (k+1) := C k\n-- end in\n-- begin\n--   have : \u2200{{n l}} (f : bounded_preformula L n l), C' n l f,\n--   { intros n l,\n--     refine bounded_preformula.rec _ _ _ _ _ _; clear n l; intros; cases n; try {exact punit.star},\n--     apply H0, apply H1, apply H2, apply H3 _ _ ih, apply H4 _ _ ih_f\u2081 ih_f\u2082, apply H5 _ ih },\n--   intros n l f, apply this f\n-- end\n\n-- @[elab_as_eliminator] def bounded_formula.rec1 {C : \u03a0n, bounded_formula L (n+1) \u2192 Sort v}\n--   (hfalsum : \u03a0 {n}, C n \u22a5)\n--   (hequal : \u03a0 {n} (t\u2081 t\u2082 : bounded_term L (n+1)), C n (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {n l : \u2115} (R : L.relations l) (ts : dvector (bounded_term L (n+1)) l),\n--     C n (bd_apps_rel (bd_rel R) ts))\n--   (himp : \u03a0 {n} {f\u2081 f\u2082 : bounded_formula L (n+1)} (ih\u2081 : C n f\u2081) (ih\u2082 : C n f\u2082), C n (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {n} {f : bounded_formula L (n+2)} (ih : C (n+1) f), C n (\u2200' f))\n--   {{n : \u2115}} (f : bounded_formula L (n+1)) : C n f :=\n-- have h : \u2200{n l} (f : bounded_preformula L (n+1) l) (ts : dvector (bounded_term L (n+1)) l),\n--   C n (bd_apps_rel f ts),\n-- begin\n--   refine bounded_preformula.rec1 _ _ _ _ _ _; intros; try {rw ts.zero_eq},\n--   apply hfalsum, apply hequal, apply hrel, apply ih (t::ts),\n--   exact himp (ih\u2081 ([])) (ih\u2082 ([])), exact hall (ih ([]))\n-- end,\n-- h f ([])\n\n-- @[elab_as_eliminator] def bounded_formula.rec {C : \u03a0n, bounded_formula L n \u2192 Sort v}\n--   (hfalsum : \u03a0 {n}, C n \u22a5)\n--   (hequal : \u03a0 {n} (t\u2081 t\u2082 : bounded_term L n), C n (t\u2081 \u2243 t\u2082))\n--   (hrel : \u03a0 {n l : \u2115} (R : L.relations l) (ts : dvector (bounded_term L n) l),\n--     C n (bd_apps_rel (bd_rel R) ts))\n--   (himp : \u03a0 {n} {f\u2081 f\u2082 : bounded_formula L n} (ih\u2081 : C n f\u2081) (ih\u2082 : C n f\u2082), C n (f\u2081 \u27f9 f\u2082))\n--   (hall : \u03a0 {n} {f : bounded_formula L (n+1)} (ih : C (n+1) f), C n (\u2200' f)) :\n--   \u2200{{n : \u2115}} (f : bounded_formula L n), C n f :=\n-- have h : \u2200{n l} (f : bounded_preformula L n l) (ts : dvector (bounded_term L n) l),\n--   C n (bd_apps_rel f ts),\n-- begin\n--   intros, induction f; try {rw ts.zero_eq},\n--   apply hfalsum, apply hequal, apply hrel, apply f_ih (f_t::ts),\n--   exact himp (f_ih_f\u2081 ([])) (f_ih_f\u2082 ([])), exact hall (f_ih ([]))\n-- end,\n-- \u03bbn f, h f ([])\n\n-- @[simp] def substmax_bounded_formula {n l} (f : bounded_preformula L (n+1) l) (s : closed_term L) :\n--   bounded_preformula L n l :=\n-- by apply subst_bounded_formula f s rfl\n\n-- -- @[simp] lemma substmax_bounded_formula_bd_falsum {n} (s : closed_term L) :\n-- --   substmax_bounded_formula (\u22a5 : bounded_formula L (n+1)) s = \u22a5 := by refl\n-- -- @[simp] lemma substmax_bounded_formula_bd_rel {n l} (R : L.relations l) (s : closed_term L) :\n-- --   substmax_bounded_formula (bd_rel R : bounded_preformula L (n+1) l) s = bd_rel R := by refl\n-- -- @[simp] lemma substmax_bounded_formula_bd_apprel {n l} (f : bounded_preformula L (n+1) (l+1))\n-- --   (t : bounded_term L (n+1)) (s : closed_term L) :\n-- --   substmax_bounded_formula (bd_apprel f t) s =\n-- --   bd_apprel (substmax_bounded_formula f s) (substmax_bounded_term t s) := by refl\n-- -- @[simp] lemma substmax_bounded_formula_bd_imp {n} (f\u2081 f\u2082 : bounded_formula L (n+1))\n-- --   (s : closed_term L) :\n-- --   substmax_bounded_formula (f\u2081 \u27f9 f\u2082) s =\n-- --   substmax_bounded_formula f\u2081 s \u27f9 substmax_bounded_formula f\u2082 s := by refl\n-- @[simp] lemma substmax_bounded_formula_bd_all {n} (f : bounded_formula L (n+2))\n--   (s : closed_term L) :\n--   substmax_bounded_formula (\u2200' f) s = \u2200' substmax_bounded_formula f s := by ext; simp\n\n-- lemma substmax_bounded_formula_bd_apps_rel {n l} (f : bounded_preformula L (n+1) l)\n--   (t : closed_term L) (ts : dvector (bounded_term L (n+1)) l) :\n--   substmax_bounded_formula (bd_apps_rel f ts) t =\n--   bd_apps_rel (substmax_bounded_formula f t) (ts.map $ \u03bbt', substmax_bounded_term t' t) :=\n-- begin\n--   induction ts generalizing f, refl, apply ts_ih (bd_apprel f ts_x)\n-- end\n\n-- def subst0_bounded_formula {n l} (f : bounded_preformula L (n+1) l) (s : bounded_term L n) :\n--   bounded_preformula L n l :=\n-- (subst_bounded_formula f s $ zero_add (n+1)).cast_eq $ zero_add n\n\n-- local notation f `[`:max s ` /0]`:0 := fol.subst0_bounded_formula f s\n\n-- @[simp] lemma subst0_bounded_formula_fst {n l} (f : bounded_preformula L (n+1) l)\n--   (s : bounded_term L n) : (subst0_bounded_formula f s).fst = f.fst[s.fst//0] :=\n-- by simp [subst0_bounded_formula]\n\n-- def substmax_eq_subst0_formula {l} (f : bounded_preformula L 1 l) (t : closed_term L) :\n--   f[t/0] = substmax_bounded_formula f t :=\n-- by ext; simp [substmax_bounded_formula]\n\n\n-- -- def subst0_sentence {n l} (f : bounded_preformula L (n+1) l) (t : closed_term L) :\n-- --   bounded_preformula L n l :=\n-- -- f [bounded_term_of_closed_term t/0]\n\n\n-- infix ` \u22a8 `:51 := fol.realize_sentence -- input using \\|= or \\vDash, but not using \\models\n\n-- @[simp] lemma realize_sentence_false {S : Structure L} : S \u22a8 (\u22a5 : sentence L) \u2194 false :=\n-- by refl\n\n-- @[simp] lemma false_of_satisfied_false {S : Structure L} :  (S \u22a8 (\u22a5 : sentence L)) \u2192 false\n-- := by simp only [realize_sentence_false, imp_self]\n\n-- @[simp] lemma realize_sentence_imp {S : Structure L} {f\u2081 f\u2082 : sentence L} :\n--   S \u22a8 f\u2081 \u27f9 f\u2082 \u2194 (S \u22a8 f\u2081 \u2192 S \u22a8 f\u2082) :=\n-- by refl\n\n-- @[simp] lemma realize_sentence_not {S : Structure L} {f : sentence L} : S \u22a8 \u223cf \u2194 \u00ac S \u22a8 f :=\n-- by refl\n\n-- @[simp] lemma realize_sentence_dne {S : Structure L} {f : sentence L} : S \u22a8 \u223c\u223cf \u2194 S \u22a8 f :=\n-- begin\n--   refine \u27e8by apply classical.by_contradiction, _\u27e9, finish\n-- end\n\n-- @[simp] lemma realize_sentence_all {S : Structure L} {f : bounded_formula L 1} :\n--   (S \u22a8 \u2200'f) \u2194 \u2200 x : S, realize_bounded_formula([x]) f([]) :=\n-- by refl\n\n-- @[simp] lemma realize_bounded_formula_imp {L} {S : Structure L} : \u2200{n} {v : dvector S n}\n--   {f g : bounded_formula L n}, realize_bounded_formula v (f \u27f9 g) dvector.nil \u2194\n--   (realize_bounded_formula v f dvector.nil -> realize_bounded_formula v g dvector.nil) :=\n-- by finish\n\n-- @[simp] lemma realize_bounded_formula_and {L} {S : Structure L} : \u2200{n} {v : dvector S n}\n--   {f g : bounded_formula L n}, realize_bounded_formula v (f \u2293 g) dvector.nil \u2194\n--   (realize_bounded_formula v f dvector.nil \u2227 realize_bounded_formula v g dvector.nil) :=\n-- begin\n--     intros,\n--     have : realize_bounded_formula v f dvector.nil \u2227 realize_bounded_formula v g dvector.nil \u2194\n--       \u00ac(realize_bounded_formula v f dvector.nil \u2192 \u00ac (realize_bounded_formula v g dvector.nil)),\n--     by finish, rw[this], refl\n-- end\n\n-- @[simp] lemma realize_bounded_formula_not {L} {S : Structure L} : \u2200{n} {v : dvector S n}\n--   {f : bounded_formula L n},\n--   realize_bounded_formula v \u223cf dvector.nil \u2194 \u00ac(realize_bounded_formula v f dvector.nil) :=\n-- by {intros, refl}\n\n-- @[simp] def realize_bounded_formula_ex {L} {S : Structure L} : \u2200 {n} {v : dvector S n}\n--   {f : bounded_formula L (n+1)}, realize_bounded_formula v (\u2203' f) dvector.nil \u2194\n--     \u2203 x, realize_bounded_formula (x::v) f dvector.nil :=\n-- by {intros, unfold bd_ex, simp [realize_bounded_formula_not], finish}\n\n-- @[simp] lemma realize_sentence_ex {S : Structure L} {f : bounded_formula L 1} :\n--   S \u22a8 \u2203' f \u2194 \u2203 x : S, realize_bounded_formula ([x]) f([]) :=\n-- by {unfold realize_sentence, apply realize_bounded_formula_ex}\n\n-- @[simp] lemma realize_sentence_and {S : Structure L} {f\u2081 f\u2082 : sentence L} :\n--   S \u22a8 f\u2081 \u2293 f\u2082 \u2194 (S \u22a8 f\u2081 \u2227 S \u22a8 f\u2082) :=\n--     by apply realize_bounded_formula_and\n\n-- @[simp] lemma realize_bounded_formula_biimp {L} {S : Structure L} : \u2200{n} {v : dvector S n}\n--   {f g : bounded_formula L n}, realize_bounded_formula v (f \u21d4 g) dvector.nil \u2194\n--     (realize_bounded_formula v f dvector.nil \u2194 realize_bounded_formula v g dvector.nil) :=\n-- by {unfold bd_biimp, tidy}\n\n-- @[simp] lemma realize_sentence_biimp {S : Structure L} {f\u2081 f\u2082 : sentence L} :\n--   S \u22a8 f\u2081 \u21d4 f\u2082 \u2194 (S \u22a8 f\u2081 \u2194 S \u22a8 f\u2082) := by apply realize_bounded_formula_biimp\n\n-- lemma realize_bounded_formula_bd_apps_rel {S : Structure L}\n--   {n l} (xs : dvector S n) (f : bounded_preformula L n l) (ts : dvector (bounded_term L n) l) :\n--   realize_bounded_formula xs (bd_apps_rel f ts) ([]) \u2194\n--   realize_bounded_formula xs f (ts.map (\u03bbt, realize_bounded_term xs t ([]))) :=\n-- begin\n--   induction ts generalizing f, refl, apply ts_ih (bd_apprel f ts_x)\n-- end\n\n-- @[simp] lemma realize_cast_bounded_formula {S : Structure L} {n m} {h : n \u2264 m}\n--   {f : bounded_formula L n} {v : dvector S m} :\n--   realize_bounded_formula v (f.cast h) dvector.nil =\n--   realize_bounded_formula (v.trunc n h) f dvector.nil :=\n-- begin\n--   by_cases n = m,\n--     by subst h; simp,\n--     have : n < m, by apply nat.lt_of_le_and_ne; repeat{assumption},\n--     ext, apply realize_bounded_formula_irrel',\n--     {intros, simp},\n--     {simp}\n-- end\n\n-- lemma realize_sentence_bd_apps_rel' {S : Structure L}\n--   {l} (f : presentence L l) (ts : dvector (closed_term L) l) :\n--   S \u22a8 bd_apps_rel f ts \u2194 realize_bounded_formula ([]) f (ts.map $ realize_closed_term S) :=\n-- realize_bounded_formula_bd_apps_rel ([]) f ts\n\n-- lemma realize_bd_apps_rel {S : Structure L}\n--   {l} (R : L.relations l) (ts : dvector (closed_term L) l) :\n--   S \u22a8 bd_apps_rel (bd_rel R) ts \u2194 S.rel_map R (ts.map $ realize_closed_term S) :=\n-- by apply realize_bounded_formula_bd_apps_rel ([]) (bd_rel R) ts\n\n-- lemma realize_sentence_equal {S : Structure L} (t\u2081 t\u2082 : closed_term L) :\n--   S \u22a8 t\u2081 \u2243 t\u2082 \u2194 realize_closed_term S t\u2081 = realize_closed_term S t\u2082  :=\n-- by refl\n\n-- lemma realize_sentence_iff {S : Structure L} (v : \u2115 \u2192 S) (f : sentence L) :\n--   realize_sentence S f \u2194 realize_formula v f.fst ([]) :=\n-- realize_bounded_formula_iff (\u03bbk hk, by exfalso; exact not_lt_zero k hk) f _\n\n-- lemma realize_sentence_of_satisfied_in {S : Structure L} [HS : nonempty S] {f : sentence L}\n--   (H : S \u22a8 f.fst) : S \u22a8 f :=\n-- begin unfreezeI, induction HS with x, exact (realize_sentence_iff (\u03bbn, x) f).mpr (H _) end\n\n-- lemma satisfied_in_of_realize_sentence {S : Structure L} {f : sentence L} (H : S \u22a8 f) : S \u22a8 f.fst :=\n-- \u03bbv, (realize_sentence_iff v f).mp H\n\n-- lemma realize_sentence_iff_satisfied_in {S : Structure L} [HS : nonempty S] {f : sentence L} :\n--   S \u22a8 f \u2194 S \u22a8 f.fst  :=\n-- \u27e8satisfied_in_of_realize_sentence, realize_sentence_of_satisfied_in\u27e9\n\n-- def L_empty : Language :=\n--   \u27e8\u03bb _, empty, \u03bb _, empty\u27e9\n\n-- end fol\n", "meta": {"author": "jesse-michael-han", "repo": "lean-parser-combinators", "sha": "d0dff9149a85a150679aa2145c4ffe2ac1ae5c0b", "save_path": "github-repos/lean/jesse-michael-han-lean-parser-combinators", "path": "github-repos/lean/jesse-michael-han-lean-parser-combinators/lean-parser-combinators-d0dff9149a85a150679aa2145c4ffe2ac1ae5c0b/src/fol'.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.39999753499341933}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.concrete_category.basic\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nnamespace category_theory.limits\n\n\n-- We now prove a lemma about naturality of cones over functors into bundled categories.\n\nnamespace cone\n\n\n/-- Naturality of a cone over functors to a concrete category. -/\n@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] {F : J \u2964 C} (s : cone F) {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(X s)) :\n    coe_fn (functor.map F f) (coe_fn (nat_trans.app (\u03c0 s) j) x) =\n        coe_fn (nat_trans.app (\u03c0 s) j') x :=\n  sorry\n\n@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) (s : cone (F \u22d9 forget C)) {j : J} {j' : J} (f : j \u27f6 j')\n    (x : X s) : coe_fn (functor.map F f) (nat_trans.app (\u03c0 s) j x) = nat_trans.app (\u03c0 s) j' x :=\n  congr_fun (w s f) x\n\nend cone\n\n\nnamespace cocone\n\n\n/-- Naturality of a cocone over functors into a concrete category. -/\n@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] {F : J \u2964 C} (s : cocone F) {j : J} {j' : J} (f : j \u27f6 j')\n    (x : \u21a5(functor.obj F j)) :\n    coe_fn (nat_trans.app (\u03b9 s) j') (coe_fn (functor.map F f) x) =\n        coe_fn (nat_trans.app (\u03b9 s) j) x :=\n  sorry\n\n@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) (s : cocone (F \u22d9 forget C)) {j : J} {j' : J} (f : j \u27f6 j')\n    (x : \u21a5(functor.obj F j)) :\n    nat_trans.app (\u03b9 s) j' (coe_fn (functor.map F f) x) = nat_trans.app (\u03b9 s) j x :=\n  congr_fun (w s f) x\n\nend cocone\n\n\n@[simp] theorem limit.lift_\u03c0_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) [has_limit F] (s : cone F) (j : J) (x : \u21a5(cone.X s)) :\n    coe_fn (limit.\u03c0 F j) (coe_fn (limit.lift F s) x) = coe_fn (nat_trans.app (cone.\u03c0 s) j) x :=\n  sorry\n\n@[simp] theorem limit.w_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) [has_limit F] {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(limit F)) :\n    coe_fn (functor.map F f) (coe_fn (limit.\u03c0 F j) x) = coe_fn (limit.\u03c0 F j') x :=\n  sorry\n\n@[simp] theorem colimit.\u03b9_desc_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) [has_colimit F] (s : cocone F) (j : J)\n    (x : \u21a5(functor.obj F j)) :\n    coe_fn (colimit.desc F s) (coe_fn (colimit.\u03b9 F j) x) =\n        coe_fn (nat_trans.app (cocone.\u03b9 s) j) x :=\n  sorry\n\n@[simp] theorem colimit.w_apply {J : Type v} [small_category J] {C : Type u} [category C]\n    [concrete_category C] (F : J \u2964 C) [has_colimit F] {j : J} {j' : J} (f : j \u27f6 j')\n    (x : \u21a5(functor.obj F j)) :\n    coe_fn (colimit.\u03b9 F j') (coe_fn (functor.map F f) x) = coe_fn (colimit.\u03b9 F j) x :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/concrete_category_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.39996094614519473}}
{"text": "--TODO: Port Theorems. \nimport Mathlib\nimport HOLFloat.Common\nimport HOLFloat.Fixed\nimport HOLFloat.Float\nimport Aesop\nset_option trace.aesop.steps true\n--set_option trace.aesop.ruleSet true\n--set_option aesop.maxRuleApplications 400\n@[simp]\ntheorem flformat_radix_lt_0 (fmt:flformat) : 0 < fmt.val.r := by\n  linarith [fmt.prop.left]\n\n@[simp]\ntheorem flformat_radix_lt_1 (fmt:flformat) : 1 < fmt.val.r := by\n  linarith [fmt.prop.left]\n\n@[simp]\ntheorem flformat_radix_le_2 (fmt:flformat): 2 \u2264 fmt.val.r := by\n  linarith [fmt.prop.left]\n\n@[simp]\ntheorem flformat_radix_lt_0_real (fmt : flformat): 0 < (fmt.val.r : \u211d) := by\n  norm_cast\n  linarith [fmt.prop.left]\n\n@[simp]\ntheorem flformat_radix_lt_1_real (fmt: flformat) : 1 < (fmt.val.r : \u211d) := by\n  norm_cast\n  linarith [fmt.prop.left]\n\n@[simp, norm_cast]\ntheorem cast_two: ((2 : \u2124) : \u211d) = 2 :=\n  Int.cast_ofNat 2\n\n@[simp]\ntheorem flformat_radix_le_2_real (fmt: flformat) : 2 \u2264 (fmt.val.r : \u211d) := by\n  norm_cast\n  linarith [Int.lt_add_one_iff.2 fmt.prop.left, fmt.prop.left]\n\n\n@[simp]\ntheorem float_radix_ipow_lt_0 (fmt: flformat)(e : \u2124) : 0 < (fmt.val.r: \u211d) ^ e := by\n  have h : 0 < fmt.val.r := by apply Int.one_lt_zero_lt; exact fmt.prop.left\n  apply ipow_lt_zero\n  simp_all only [Int.one_lt_zero_le_iff, zero_add, Int.cast_pos]\n\n@[simp]\ntheorem float_ipow_le_real (fmt :flformat)(x : \u211d) : \u2203(e : \u2124), x \u2264 ((fmt.val.r : \u211d ) ^ e) := by\n  apply ipow_le_real --FIXME: wait for port\n  case a =>\n    simp [flformat_radix_le_2]\n\n\n@[simp]\ntheorem float_ipow_le_real_2 (fmt :flformat)(x : \u211d) : 0 < x \u2192 \u2203(e:\u2124), x \u2264(fmt.val.r : \u211d) ^ e  := by\n  norm_cast\n  intros\n  apply ipow_le_real\n  simp_all only [flformat_radix_le_2_real]\n\n@[aesop unsafe]\ndef is_greatest_e (fmt : flformat)(x : \u211d)(e : \u2124) : Prop :=\n--TODO: use LUB\n  (fmt.val.r : \u211d) ^ e  \u2264 |x| \u2227\n  \u2200(e' : \u2124), (fmt.val.r : \u211d) ^ e' \u2264 |x| \u2192 e' \u2264 e\n\n\n@[simp]\ntheorem pow_le_real (a x : \u211d) (ha1 : a \u2260 1) (hx0 : x \u2260 0) : \u2203 (z : \u2124), a ^ z \u2264 abs x :=\n-- Chris proof\n  if ha0 : a \u2264 0\n  then \u27e81, (zpow_one a).symm \u25b8 le_trans ha0 (abs_nonneg _)\u27e9\n  else if hal1 : a < 1\n    then let \u27e8k, hk\u27e9 := pow_unbounded_of_one_lt (abs x)\u207b\u00b9 (one_lt_inv (lt_of_not_ge ha0) hal1)\n      \u27e8k, (inv_le_inv (abs_pos.2 hx0) (pow_pos (lt_of_not_ge ha0) _)).1\n        (le_of_lt $ by simpa using hk)\u27e9\n    else let \u27e8k, hk\u27e9 := pow_unbounded_of_one_lt (abs x)\u207b\u00b9 (lt_of_le_of_ne (le_of_not_gt hal1) ha1.symm)\n      \u27e8-k, (inv_le_inv (abs_pos.2 hx0) (zpow_pos_of_pos (lt_of_not_ge ha0) _)).1\n        (le_of_lt $ by simpa)\u27e9\n\n#check Set\n@[simp]\ntheorem float_int_bounded (S : \u2124 \u2192 Prop)(b : \u2124): (Set.Nonempty s) \u2192 (\u2200(e : \u2124), e \u2208 setOf S \u2192 e \u2264 b) \u2192 (\u2203(e' : \u2124), is_sup_int S e'):= by\n  intro hn ha\n  simp [setOf] at ha\n  simp [is_sup_int, IsLUB, IsLeast, lowerBounds, upperBounds]\n  --NOTE: :)\n  sorry\n\n@[simp]\ntheorem float_greatest_e_exists (fmt:flformat) (x : \u211d)(e : \u2124) : x \u2260 0 \u2192 e = greatest_e fmt x \u2192 is_greatest_e fmt x e :=  by\n  have \u27e8fmt_val, FMT\u27e9 := fmt\n  intro hx he\n  simp only [greatest_e, is_greatest_e] at *\n  simp only [ne_eq, Int.cast_eq_zero]\n  apply And.intro\n\n  have R_GT_1 : 1 < fmt_val.r := FMT.left;\n  suffices e \u2208  { z : \u2124 | (fmt_val.r : \u211d) ^ z \u2264 abs x } by {\n    have H := this.out;\n    have R_GT_1 : 1 < fmt_val.r := FMT.left;\n    norm_cast at H;\n  }\n\n  case left =>\n    norm_cast\n\n    rw[he];\n    apply Int.csup\u209b_mem;\n\n    case h1 => {\n     \n      simp[Set.Nonempty];\n      -- show the existnece of a negative z that makes this true.\n      -- please kill me\n      apply pow_le_real\n      norm_cast\n      aesop_subst he\n      simp_all only [ne_eq, Int.one_lt_zero_le_iff, Int.one_lt_ne_one, not_false_iff]\n      aesop_subst he\n      simp_all only [ne_eq, Int.one_lt_zero_le_iff, not_false_iff]\n    }\n    case h2 => {\n      --NOTE: exists 42 -- fake bound, need log.\n      sorry\n    }\n  case right =>\n    intro e1 hp\n    aesop_subst he\n    have he1: e1 \u2208 { z | (fmt_val.r: \u211d) ^ z \u2264 abs x } :=  by\n      simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq]\n    apply le_csup\u209b\n    case h\u2082 =>\n      simp_all only [ne_eq, Int.cast_eq_zero, Set.mem_setOf_eq]\n    case h\u2081 =>\n      sorry\n\n#check le_sup\u209b \n#check CompleteSemilatticeSup \u2124 \n#check le_csup\u209b \ndef is_greatest_m (fmt:flformat) (x: \u211d) (m : \u2124): Prop :=\n  m * (fmt.val.r : \u211d) ^ (greatest_e fmt x) \u2264 |x| \u2227\n  \u2203(m' : \u2124), m * (fmt.val.r : \u211d)  ^ (greatest_e fmt x) \u2264 |x| \u2192 m' <= m\n\n--NOTE: theorems for mantissa\n@[simp]\ntheorem float_greatest_m_exists (fmt : flformat)(x : \u211d) (m : \u2124): \n  x \u2260 0 \n  \u2192 greatest_m fmt x = m \n  \u2192 is_greatest_m fmt x m \u2227 1 \u2264 m \u2227 m < fmt.val.r := by\n  sorry\n\n\n--NOTE: theorems for exponent\n@[simp]\ntheorem is_greatest_e_exist_greatest_e (fmt :flformat)( e : \u2124) : x \u2260 0 \u2192 is_greatest_e fmt x e \u2192 greatest_e fmt x = e := by\n  intro hx he\n  simp_all only [is_greatest_e, greatest_e]\n  \n  sorry\n#check pow_le_real\n--TODO: theorem float_normalize_real\n@[simp]\ntheorem float_normalize_real (fmt : flformat) (x : \u211d) : x < 0 \u2192 x = greatest_m fmt x * (fmt.val.r : \u211d) ^ greatest_e fmt x + greatest_r fmt x := by\n  intro h\n  sorry\n\n@[simp]\ntheorem float_real_le_1_le (x : \u211d)(m :\u2124) : 0 < x \u2192 1 \u2264 m \u2192 x \u2264 m * x := by\n  intro h\u2081 h\u2082\n  norm_cast\n  rw [le_mul_iff_one_le_left]\n  norm_cast\n  exact h\u2081\n  \n\n@[simp]\ntheorem float_pow_int_eq_le (r : \u2124)(x : \u211d)(m : \u2124)(e : \u2124) : x \u2260 0 \u21921 \u2264 m \u2192 abs x = m * (r:\u211d) ^ e \u2192 (r:\u211d) ^ e \u2264 abs x := by\n  intro hx hm he\n  norm_cast\n  rw [he]\n  apply float_real_le_1_le\n  case a =>\n    norm_cast\n    suffices abs x > 0 by simp_all only [lt_self_iff_false, ne_eq, Int.cast_lt_zero, Int.one_lt_zero_le_iff, Int.cast_eq_zero, gt_iff_lt, Int.cast_pos, zero_add, zero_lt_mul_left]\n    simp; exact hx\n  case a =>\n    exact hm\n\n \n@[simp]\ntheorem float_eq_ipow (fmt : flformat) (x : \u211d)(e : \u2124)(m : \u211d) :\n  x \u2260 0 \u2192 1 \u2264 m \u2192 m < fmt.val.r \u2192 |x| = m * (fmt.val.r : \u211d) ^ e \n  \u2192 greatest_e fmt x  = e \u2227 greatest_m fmt x = m  := by\n  intros hx hm hr he\n  apply And.intro\n  case left =>\n    rw [greatest_e]\n    norm_cast\n    simp only [he]\n    have H : e \u2208 {z : \u2124 | (fmt.val.r : \u211d) ^ z \u2264 abs x} := by\n      rw [he]\n      simp [float_real_le_1_le]\n      norm_cast\n    sorry\n  case right =>\n    rw [greatest_m]\n    sorry\n\n", "meta": {"author": "opencompl", "repo": "HOLFloat-Lean", "sha": "28c75957eedc6b3c6c1d3c3d154b41d1aaeb5f0c", "save_path": "github-repos/lean/opencompl-HOLFloat-Lean", "path": "github-repos/lean/opencompl-HOLFloat-Lean/HOLFloat-Lean-28c75957eedc6b3c6c1d3c3d154b41d1aaeb5f0c/HOLFloat/Float_theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.39969761627266737}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison\n\nimport ..monoidal_category\n\nopen categories\nopen categories.monoidal_category\n\nnamespace categories.examples\n\nuniverse u\n\nlocal attribute [ematch] semigroup.mul_assoc\n\ndefinition turtle (\u03b1 : Type u) : Type u := punit.{u+1}\ndefinition atuin (\u03b1 : Type u) : turtle \u03b1 := punit.star\nlocal attribute [applicable] atuin\n\ninstance (\u03b1 : Type u) [monoid \u03b1] : small_category (turtle \u03b1) :=\n{ Hom      := \u03bb _ _, \u03b1,\n  compose  := \u03bb _ _ _ f g, f * g,\n  identity := \u03bb _, 1 }\n\nlocal attribute [applicable] has_one.one\n@[simp] lemma monoid_as_category.compose (\u03b1 : Type u) [monoid \u03b1] (f g : atuin \u03b1 \u27f6 atuin \u03b1) : f \u226b g = (f : \u03b1) * (g : \u03b1) := by refl\n\nlocal attribute [ematch] semigroup.mul_assoc comm_semigroup.mul_comm\n\ndefinition comm_monoid_as_monoidal_category (\u03b1 : Type) [comm_monoid \u03b1] : monoidal_category (turtle \u03b1) :=\n{ tensor := --sorry,\n    { onObjects     := \u03bb _, atuin \u03b1,\n      onMorphisms   := \u03bb _ _ p, p.1 * p.2 },\n  tensor_unit := sorry, --atuin \u03b1,\n  -- TODO: ugh, it sucks that by obviously is so much slower in the following:\n  associator_transformation   := sorry, --{ morphism := { components := by obviously, naturality := by obviously }, inverse := { components := by obviously, naturality := by obviously }, witness_1 := by obviously, witness_2 := by obviously },\n  left_unitor_transformation  := sorry, --{ morphism := { components := by obviously, naturality := by obviously }, inverse := { components := by obviously, naturality := by obviously }, witness_1 := by obviously, witness_2 := by obviously },\n  right_unitor_transformation := sorry, --{ morphism := { components := by obviously, naturality := by obviously }, inverse := { components := by obviously, naturality := by obviously }, witness_1 := by obviously, witness_2 := by obviously }, }\n}\n\nend categories.examples", "meta": {"author": "semorrison", "repo": "lean-monoidal-categories", "sha": "81f43e1e0d623a96695aa8938951d7422d6d7ba6", "save_path": "github-repos/lean/semorrison-lean-monoidal-categories", "path": "github-repos/lean/semorrison-lean-monoidal-categories/lean-monoidal-categories-81f43e1e0d623a96695aa8938951d7422d6d7ba6/src/monoidal_categories/examples/monoid_as_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542925, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3996229379268057}}
{"text": "import condensed.projective_resolution\n\nopen category_theory.limits\nopen category_theory\n\nnamespace Condensed\n\n--TODO: generalize (as needed...)\nuniverse u\nvariables {A B : Condensed.{u} Ab.{u+1}} (f : A \u27f6 B)\n\nnoncomputable theory\n\ndef kernel_diagram_iso {A B : Condensed.{u} Ab.{u+1}} (S : ExtrDisc) (f : A \u27f6 B) :\n  parallel_pair f 0 \u22d9 Condensed.evaluation Ab S.val \u2245\n    parallel_pair ((Condensed.evaluation Ab S.val).map f) 0 :=\nnat_iso.of_components (\u03bb X,\n  match X with\n  | walking_parallel_pair.zero := iso.refl _\n  | walking_parallel_pair.one := iso.refl _\n  end) $ by { rintros (a|a) (b|b) (f|f), tidy }\n\ndef cokernel_diagram_iso {A B : Condensed.{u} Ab.{u+1}} (S : ExtrDisc) (f : A \u27f6 B) :\n  limits.parallel_pair ((Condensed.evaluation Ab S.val).map f) 0 \u2245\n    limits.parallel_pair f 0 \u22d9 Condensed.evaluation Ab S.val :=\nnat_iso.of_components (\u03bb X,\n  match X with\n  | walking_parallel_pair.zero := iso.refl _\n  | walking_parallel_pair.one := iso.refl _\n  end) $ by { rintros (a|a) (b|b) (f|f), tidy }\n\ndef kernel_iso (S : ExtrDisc.{u}) :\n  (Condensed.evaluation _ S.val).obj (kernel f) \u2245\n  kernel ((Condensed.evaluation _ S.val).map f) :=\n(is_limit_of_preserves (Condensed.evaluation _ S.val)\n  (limit.is_limit (parallel_pair f 0))).cone_point_unique_up_to_iso\n  (limit.is_limit _) \u226a\u226b has_limit.iso_of_nat_iso (kernel_diagram_iso _ _)\n\n@[simp, reassoc]\nlemma kernel_iso_hom (S : ExtrDisc.{u}) :\n  (kernel_iso f S).hom \u226b kernel.\u03b9 _ = (Condensed.evaluation _ S.val).map (kernel.\u03b9 _) :=\nbegin\n  dsimp [kernel_iso, kernel_diagram_iso],\n  simp only [category.assoc, has_limit.iso_of_nat_iso_hom_\u03c0,\n    nat_iso.of_components.hom_app, limit.cone_point_unique_up_to_iso_hom_comp_assoc,\n    functor.map_cone_\u03c0_app, equalizer.fork_\u03c0_app_zero,\n    equalizer_as_kernel, Condensed.evaluation_map],\n  apply category.comp_id,\nend\n\ndef cokernel_iso (S : ExtrDisc.{u}) :\n  cokernel ((Condensed.evaluation _ S.val).map f) \u2245\n  (Condensed.evaluation _ S.val).obj (cokernel f) :=\nhas_colimit.iso_of_nat_iso (cokernel_diagram_iso _ _) \u226a\u226b\n  (colimit.is_colimit _).cocone_point_unique_up_to_iso\n  (is_colimit_of_preserves (Condensed.evaluation _ _)\n  (colimit.is_colimit (parallel_pair f 0)))\n\n@[simp, reassoc]\nlemma cokernel_iso_hom (S : ExtrDisc.{u}) :\n  cokernel.\u03c0 _ \u226b (cokernel_iso f S).hom = (Condensed.evaluation _ S.val).map (cokernel.\u03c0 _) :=\nbegin\n  dsimp [cokernel_iso, cokernel_diagram_iso],\n  simp only [has_colimit.iso_of_nat_iso_\u03b9_hom_assoc, nat_iso.of_components.hom_app,\n  colimit.comp_cocone_point_unique_up_to_iso_hom, functor.map_cocone_\u03b9_app,\n  coequalizer.cofork_\u03b9_app_one, coequalizer_as_cokernel, Condensed.evaluation_map],\n  apply category.id_comp,\nend\n\nend Condensed\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/condensed/kernel_comparison.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3994874076196448}}
{"text": "\nimport category_theory.category\n\nimport category_theory.concrete_category\n\n\nopen category_theory\n\nsection\n\nuniverses u_1 u_2\n\n\ninductive my_empty : Type u_1\n\ninductive my_unit : Type u_1\n| star : my_unit\n\n\n-- typical example for c is a typeclass, like topological_space , with hom being something like continuous.\nvariables (c : Type u_1 -> Type u_2) (hom : (Pi {alpha beta : Type u_1}, c alpha -> c beta -> (alpha -> beta) -> Prop))\n\n-- `c alpha` and `c beta` are like two objects of the category (I think), and alpha and beta are the underlying sets.\n-- so, the element of (alpha -> beta) is a particular function between the sets\n-- and the Prop is whether that particular function is a morphism ?\n-- so, for `c x` to be free over x, then, need need that, for any \n-- wait, that doesn't seem right? concrete categories are characterized by having a (faithful) functor /into/ Set,\n-- not having a functor /from/ Set.\n\n-- ok, so continuous : (f : alpha -> beta) -> forall s, is_open s -> is_open (preimage f s)\n-- where alpha and beta are each a topological_space , (topological_space is a typeclass)\n\n\nvariable my_concrete_category : concrete_category @hom\n\n\n#check @forget\n#check @forget c @hom my_concrete_category\n#check (@forget c @hom my_concrete_category).obj\n#check (@forget c @hom my_concrete_category).map\n\n\n-- should the statement that an object X is free over a set/type of generators include the function from\n-- the set of generators to the underlying set of X, in the statement,\n-- or should it just assert that such a function exists?\n-- I suspect the former, but it would be nice to be able to let said function be implicit.\ndef is_free_over (X : bundled c) (generators_of_X : Type u_1 ) (i : generators_of_X -> (@forget c @hom my_concrete_category).obj X) : Prop :=\n  forall (Y : bundled c) (f : generators_of_X -> (forget.obj Y)),\n    \u2203! (g : X \u27f6 Y), f = (forget.map g) \u2218 i\n\n\n\n#check is_free_over\n\nend\n\n", "meta": {"author": "drocta", "repo": "orthogonal-free-surjective", "sha": "6523bdbc7479a62155aae3405c6b00b5673b4b75", "save_path": "github-repos/lean/drocta-orthogonal-free-surjective", "path": "github-repos/lean/drocta-orthogonal-free-surjective/orthogonal-free-surjective-6523bdbc7479a62155aae3405c6b00b5673b4b75/free.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125848754471, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.39947613398086496}}
{"text": "\nimport Lib.Algebra.Monoid\nimport Lib.Attributes\nimport Lib.Function\nimport Lib.Tactic\n\nnamespace Functor\nvariable {F} [Functor F] [LawfulFunctor F]\nattribute [functor] LawfulFunctor.id_map\n\n@[simp, functor]\ntheorem map_id :\n  map (f := F) (@id \u03b1) = id := by\next; simp\n\n@[simp, functor]\ntheorem map_comp (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) :\n  map g (map f x) = map (f := F) (g \u2218 f) x := by\nrw [comp_map]\n\n@[simp, functor]\ntheorem map_comp' (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) :\n  map g \u2218 map f = map (f := F) (g \u2218 f) := by\next; simp only [(.\u2218.),map_comp]\n\nend Functor\n\nnamespace Applicative\n\nvariable {\u03b1 \u03b2}\nvariable {F : Type u \u2192 Type v} [Applicative F] [LawfulApplicative F]\n\nattribute [functor] seq_assoc pure_seq seq_pure\n\n-- @[functor]\ntheorem pure_seq' (f : \u03b1 \u2192 \u03b2) (x : F \u03b1) :\n  f <$> x = pure f <*> x :=\nby simp [pure_seq]\n\n@[simp, functor]\ntheorem seq_map {\u03b1 \u03b2 \u03b3 : Type u}\n        {f : \u03b1 \u2192 \u03b2} {x : F (\u03b2 \u2192 \u03b3)} {y : F \u03b1} :\n  x <*> (f <$> y) = (.\u2218f) <$> x <*> y := by\nsimp only [pure_seq']\nrw [seq_assoc]\nsimp only [seq_pure, pure_seq, Functor.map_comp]\nrefl\n\n@[simp, functor]\ntheorem map_seq {\u03b1 \u03b2 \u03b3 : Type u}\n        {f : \u03b2 \u2192 \u03b3} {x : F (\u03b1 \u2192 \u03b2)} {y : F \u03b1} :\n  f <$> (x <*> y) = (f\u2218.) <$> x <*> y := by\nsimp only [pure_seq']\nsimp only [pure_seq, map_pure, seq_assoc]\n\nend Applicative\n\ndef Id.mk (x : \u03b1) : Id \u03b1 := x\ndef Comp (F : Type v \u2192 Type w) (G : Type u \u2192 Type v) (\u03b1 : Type u) := F (G \u03b1)\n\nnamespace Comp\nvariable\n  {F : Type v \u2192 Type w} {G : Type u \u2192 Type v} {\u03b1 : Type u}\n\ndef mk (x : F (G \u03b1)) : Comp F G \u03b1 := x\ndef run (x : Comp F G \u03b1) : F (G \u03b1) := x\n\ninstance [Functor F] [Functor G] : Functor (Comp F G) where\n  map f := Functor.map (f := F) (Functor.map (f := G) f)\n\ninstance [Functor F] [LawfulFunctor F] [Functor G] [LawfulFunctor G] : LawfulFunctor (Comp F G) where\n  id_map := by intros; simp [id_map, Functor.map]\n  comp_map := by intros; simp [id_map, Functor.map]\n  map_const := by intros; simp [id_map, Functor.map]; apply funext; intro; refl\n\ninstance [Applicative F] [Applicative G] : Applicative (Comp F G) where\n  pure x := Pure.pure (f := F) $ Pure.pure (f := G) x\n  seq {\u03b1 \u03b2} (f : F (G (\u03b1 \u2192 \u03b2))) (x : Unit \u2192 F (G \u03b1)) :=\n    show F (G \u03b2) from\n    Seq.seq ((.<*>.) <$> f) x\n\ntheorem comp_seq_def [Applicative F] [Applicative G]\n        {\u03b1 \u03b2 : Type u}\n        (x : Comp F G (\u03b1 \u2192 \u03b2)) (y : Comp F G \u03b1) :\n  x <*> y = ((.<*>.) <$> x <*> y : F (G \u03b2)) := rfl\n\nopen Applicative\n\ninstance [Applicative F] [LawfulApplicative F] [Applicative G] [LawfulApplicative G] :\n         LawfulApplicative (Comp F G) where\n  seqLeft_eq := by intros; refl\n  seqRight_eq := by intros; refl\n  pure_seq := by intros; simp [pure_seq, pure, Seq.seq, (.<$>.)]\n  map_pure := by intros; simp [pure_seq, pure, Seq.seq, (.<$>.)]\n  seq_pure := by intros; simp [pure_seq, pure, Seq.seq, (.<$>.), (.\u2218.)]\n  seq_assoc := by\n    intros; simp [pure_seq, pure, comp_seq_def, (.<$>.), (.\u2218.), seq_assoc]\n\nend Comp\n\ndef Const (\u03c9 : Type u) (\u03b1 : Type v) := \u03c9\ndef Const.mk {\u03c9 : Type u} {\u03b1 : Type v} (w : \u03c9) : Const \u03c9 \u03b1 := w\ndef Const.run {\u03c9 : Type u} {\u03b1 : Type v} (w : Const \u03c9 \u03b1) : \u03c9 := w\n\nnamespace Const\n\ninstance : Functor (Const \u03c9) where\n  map f x := x\n\ninstance : LawfulFunctor (Const \u03c9) where\n  id_map := by intros; refl\n  map_const := by intros; refl\n  comp_map := by intros; refl\n\nopen One\n\ninstance [One \u03c9] : Pure (Const \u03c9) where\n  pure _ := (1 : \u03c9)\n\ninstance [Mul \u03c9] : Seq (Const \u03c9) where\n  seq (x : \u03c9) (y : Unit \u2192 \u03c9) := (x * y () : \u03c9)\n\ninstance [One \u03c9] [Mul \u03c9] : Applicative (Const \u03c9) where\n  pure := pure\n  seq := Seq.seq\n\ninstance [Monoid \u03c9] : LawfulApplicative (Const \u03c9) := by\nconstructor <;> intros\n<;> simp [(.<$>.), Seq.seq, pure, SeqLeft.seqLeft, SeqRight.seqRight]\n\nend Const\n\nsection defs\n\nvariable (F G : Type u \u2192 Type _) [Applicative F] [Applicative G]\n\nstructure ApplicativeRel where\n  R {\u03b1} : F \u03b1 \u2192 G \u03b1 \u2192 Prop\n  R_pure {\u03b1} {x : \u03b1} : R (pure x) (pure x)\n  R_seq {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1) (f' : G (\u03b1 \u2192 \u03b2)) (x' : G \u03b1) :\n    R f f' \u2192\n    R x x' \u2192\n    R (f <*> x) (f' <*> x')\n\nstructure ApplicativeHom where\n  fn {\u03b1} : F \u03b1 \u2192 G \u03b1\n  fn_pure {\u03b1} {x : \u03b1} : fn (pure x) = pure x\n  fn_seq {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1) :\n    fn (f <*> x) = fn f <*> fn x\n\nend defs\n\nattribute [simp] ApplicativeHom.fn_pure ApplicativeHom.fn_seq\n\nnamespace ApplicativeRel\n\nvariable {F G : Type u \u2192 Type _} [Applicative F] [Applicative G]\n\ninstance : CoeFun (ApplicativeRel F G) (\u03bb _ => {\u03b1 : Type _} \u2192 F \u03b1 \u2192 G \u03b1 \u2192 Prop) where\n  coe x := x.R\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\nvariable (R : ApplicativeRel F G)\n\nattribute [auto] ApplicativeRel.R_pure ApplicativeRel.R_seq\n\n@[auto]\ntheorem naturality {\u03b1 \u03b2} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1) (x' : G \u03b1)\n        (hR : R x x') :\n  R (g <$> x) (g <$> x') :=\nby simp [\u2190 pure_seq]; auto\n\n-- def toApplicativeHom : ApplicativeHom F G where\n--   fn x := f x = y\n--   fn_pure := by intros; simp\n--   fn_seq := by intros; simp [*]\n\nend ApplicativeRel\n\nnamespace Functor\n\nvariable {F} [Functor F] [LawfulFunctor F]\nvariable {f : \u03b1 \u2192 \u03b2}\n\nsection LeftInv\n\nvariable {g : \u03b2 \u2192 \u03b1} (Hfg : LeftInv f g)\n\ntheorem LeftInv_map : LeftInv (map (f:=F) f) (map g) := by\nsimp [LeftInv, Hfg]\n\nend LeftInv\n\nsection HasLeftInv\n\nvariable (Hf : HasLeftInv f)\n\ntheorem HasLeftInv_map : HasLeftInv (map (f:=F) f) := by\nobtain \u27e8g, Hg\u27e9 from Hf\nexists map (f := F) g\nsimp [LeftInv, Hg]\n\nend HasLeftInv\n\nsection HasRightInv\n\nvariable (Hf : HasRightInv f)\n\ntheorem HasRightInv_map : HasRightInv (map (f:=F) f) := by\nobtain \u27e8g, Hg\u27e9 from Hf\nexists map (f := F) g\nsimp [RightInv, Hg]\n\nend HasRightInv\n\nsection HasRightInv\n\nvariable (Hf : HasLeftInv f)\n\ntheorem Injective_map : Injective (map (f:=F) f) := by\nauto [Injective_of_HasLeftInv,HasLeftInv_map]\n\nend HasRightInv\n\nend Functor\n\nnamespace ApplicativeHom\n\nvariable {F G : Type u \u2192 Type _} [Applicative F] [Applicative G]\n\ninstance : CoeFun (ApplicativeHom F G) (\u03bb _ => {\u03b1 : Type _} \u2192 F \u03b1 \u2192 G \u03b1) where\n  coe x := x.fn\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\nvariable (f : ApplicativeHom F G)\n\n@[simp]\ntheorem naturality {\u03b1 \u03b2} (g : \u03b1 \u2192 \u03b2) (x : F \u03b1) :\n    f (g <$> x) = g <$> f x := by\nsimp [\u2190 pure_seq, fn_seq, fn_pure]\n\ndef toApplicativeRel : ApplicativeRel F G where\n  R x y := f x = y\n  R_pure := by intros; simp\n  R_seq := by intros; simp [*]\n\nend ApplicativeHom\n\ndef Op1 (F : Type u \u2192 Type v) \u03b1 := F \u03b1\ndef Op1.mk {F : Type u \u2192 Type v} {\u03b1} (x : F \u03b1) : Op1 F \u03b1 := x\ndef Op1.run {F : Type u \u2192 Type v} {\u03b1} (x : Op1 F \u03b1) : F \u03b1 := x\n\nnamespace Op1\n\ninstance {F} [Functor F] : Functor (Op1 F) where\n  map := @Functor.map F _\n\ninstance {F} [Functor F] [H : LawfulFunctor F] : LawfulFunctor (Op1 F) := by\nconstructor <;> intros\n. simp [Functor.mapConst, Function.const]; ext; refl\n. simp [(.<$>.)]\n. simp [(.<$>.)]\n\n\nvariable {F} [Applicative F]\n-- set_option pp.explicit true\n\ninstance : Applicative (Op1 F) where\n  pure := pure (f := F)\n  seq f x := ((\u03bb x f => f x) <$> x () <*> f : F _)\n  map := Functor.map (f := F)\n-- #print instApplicativeOp1\nvariable [LawfulApplicative F]\n\n@[simp]\ntheorem map_eq {\u03b1 \u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} (x : Op1 F \u03b1) :\n  (Op1.run $ f <$> x) = (f <$> Op1.run x) := rfl\n\n@[simp]\ntheorem pure_eq {\u03b1 : Type u} (x : \u03b1) :\n  (@Op1.run F _ $ pure x) = (pure x) := rfl\n\n@[simp]\ntheorem seq_eq {\u03b1 \u03b2 : Type u} (f : Op1 F (\u03b1 \u2192 \u03b2))\n        (x : Unit \u2192 Op1 F \u03b1) :\n  Op1.run (Seq.seq f x) =\n  ((\u03bb x f => f x) <$> Op1.run (x ()) <*> Op1.run f) := rfl\n\n@[simp]\nprotected theorem seqLeft_eq {\u03b1 \u03b2 : Type u} (f : Op1 F \u03b2)\n        (x : Unit \u2192 Op1 F \u03b1) :\n  Op1.run (SeqLeft.seqLeft f x) =\n  SeqRight.seqRight (run $ x ()) (\u03bb _ => run f)  := by\nchange SeqLeft.seqLeft f x\n  with Seq.seq (Function.const _ <$> f) x\nsimp only [seq_eq, map_eq, (.\u2218.), Function.const,\n           seqRight_eq, Applicative.seq_map, Functor.map_comp]\nrefl\n\n@[simp]\nprotected theorem seqRight_eq {\u03b1 \u03b2 : Type u} (f : Op1 F \u03b2)\n        (x : Unit \u2192 Op1 F \u03b1) :\n  Op1.run (SeqRight.seqRight f x) =\n  SeqLeft.seqLeft (run $ x ()) (\u03bb _ => run f)  := by\nchange SeqRight.seqRight f x\n  with Seq.seq (Function.const _ id <$> f) x\nsimp only [seq_eq, map_eq, (.\u2218.), Function.const,\n           seqLeft_eq, Applicative.seq_map, Functor.map_comp]\nrefl\n\ntheorem ext (x y : Op1 F \u03b1) : x.run = y.run \u2192 x = y := id\n\ninstance : LawfulApplicative (Op1 F) :=\nsorry -- TODO: `constructor` is broken as of 2022-04-16\n\n-- constructor\n-- <;> intros <;> apply Op1.ext\n-- <;> simp [seqLeft_eq, seqRight_eq, pure_seq, seq_assoc]\n-- <;> refl\n\nend Op1\n\nnamespace Const\n\n@[simp]\ntheorem run_pure {\u03b1 \u03c9} [Monoid \u03c9] x :\n  @Const.run \u03c9 \u03b1 (pure x) = 1 := rfl\n\n@[simp]\ntheorem run_seq {\u03b1 \u03b2 : Type _} {\u03c9} [Monoid \u03c9]\n        (x : Const \u03c9 (\u03b1 \u2192 \u03b2)) (y : Unit \u2192 Const \u03c9 \u03b1) :\n  Const.run (Seq.seq x y) = Const.run x * Const.run (y ()) := rfl\n\n@[simp]\ntheorem run_mk {\u03b1 \u03c9} x :\n  @Const.run \u03c9 \u03b1 (Const.mk x) = x := rfl\n\n@[simp]\ntheorem map_mk {\u03b1 \u03b2 \u03c9} (f : \u03b1 \u2192 \u03b2) x :\n  f <$> @Const.mk \u03c9 \u03b1 x = Const.mk x := rfl\n\n@[simp]\ntheorem run_map {\u03b1 \u03b2 \u03c9} (f : \u03b1 \u2192 \u03b2) x :\n  @Const.run \u03c9 \u03b2 (f <$> x) = Const.run x := rfl\n\nend Const\n\nnamespace Comp\nopen Functor\n@[simp]\ntheorem run_mk {F G : Type _ \u2192 Type _} (x : F (G \u03b1)) :\n  Comp.run (Comp.mk x) = x := rfl\n\n@[simp]\ntheorem run_pure {F G} [Applicative F] [Applicative G] (x : \u03b1) :\n  Comp.run (pure x : Comp F G \u03b1) = pure (pure x) := rfl\n\n@[simp]\ntheorem run_seq {\u03b1 \u03b2 : Type _} {F G} [Applicative F] [Applicative G]\n        (x : Comp F G (\u03b1 \u2192 \u03b2)) y :\n  Comp.run (Seq.seq x y) =\n  Seq.seq ((.<*>.) <$> Comp.run x) (Comp.run \u2218 y) := rfl\n\n@[simp]\ntheorem run_map {\u03b1 \u03b2 : Type _} {F G} [Functor F] [Functor G]\n        (f : \u03b1 \u2192 \u03b2) (x : Comp F G \u03b1) :\n  Comp.run (f <$> x) =\n  Functor.map f <$> (Comp.run x) := rfl\n\n@[simp]\ntheorem map_mk {\u03b1 \u03b2 : Type _} {F G} [Functor F] [Functor G]\n        (f : \u03b1 \u2192 \u03b2) (x : F (G \u03b1)) :\n  f <$> Comp.mk x =\n  Comp.mk (map f <$> x) := rfl\n\nend Comp\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3994405292411902}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin         \n  assume (U : set (euclidean_space \u211d (fin n))) (hU : is_open U) (hcover : \ud835\udcdd (zero n) \u2286 U),\n  have h : Set.finite {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n    B = ball c r} := by {\n    apply finite.of_fintype,\n    apply fin.fintype,\n  },\n  have h1 : \u2200 (s : \u211d) (c : euclidean_space \u211d (fin n)),\n    ball c s = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n    y \u2208 ball c s \u2227 x = y}, from by {\n    assume (s : \u211d) (c : euclidean_space \u211d (fin n)),\n    have h1 : ball c s \u2286 {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n      y \u2208 ball c s \u2227 x = y}, from by {\n      assume (x : euclidean_space \u211d (fin n)) (h : x \u2208 ball c s),\n      have h1 : x \u2208 {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n        y \u2208 ball c s \u2227 x = y}, from by {\n        show x \u2208 {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n        y \u2208 ball c s \u2227 x = y}, from by {\n          existsi x,\n          split,\n          exact h,\n          exact rfl,\n        },\n      },\n      exact h1,\n    },\n    have h2 : {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n      y \u2208 ball c s \u2227 x = y} \u2286 ball c s, from by {\n      assume (x : euclidean_space \u211d (fin n)) (h : x \u2208 {x : euclidean_space \u211d (fin n) | \n        \u2203 (y : euclidean_space \u211d (fin n)), y \u2208 ball c s \u2227 x = y}),\n      cases h with (y : euclidean_space \u211d (fin n)) h,\n      cases h with (h1 : y \u2208 ball c s) (h2 : x = y),\n      have h3 : x \u2208 ball c s, from by {\n        rw h2,\n        exact h1,\n      },\n      exact h3,\n    },\n    exact eq_of_subset_of_subset h1 h2,\n  },\n  have h2 : \u2200 (s : \u211d) (c : euclidean_space \u211d (fin n)),\n    {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n    y \u2208 ball c s \u2227 x = y} = ball c s, from by {\n    assume (s : \u211d) (c : euclidean_space \u211d (fin n)),\n    exact h1 s c,\n  },\n  have h3 : {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n    B = ball c r} = {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n    B = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n    y \u2208 ball c r \u2227 x = y}}, from by {\n    have h3 : {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n      B = ball c r} \u2286 {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n      B = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n      y \u2208 ball c r \u2227 x = y}}, from by {\n      assume (B : set (euclidean_space \u211d (fin n))) (h : B \u2208 {B : set (euclidean_space \u211d (fin n)) | \n        \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), B = ball c r}),\n      cases h with (r : \u211d) (c : euclidean_space \u211d (fin n)) (h1 : B = ball c r),\n      have h2 : B \u2208 {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n        B = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n        y \u2208 ball c r \u2227 x = y}}, from by {\n        show B \u2208 {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n          B = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n          y \u2208 ball c r \u2227 x = y}}, from by {\n            existsi r,\n            existsi c,\n            exact h2 r c,\n        },\n      },\n      exact h2,\n    },\n    have h4 : {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n      B = {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)),\n      y \u2208 ball c r \u2227 x = y}} \u2286 {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), \n      B = ball c r}, from by {\n      assume (B : set (euclidean_space \u211d (fin n))) (h : B \u2208 {B : set (euclidean_space \u211d (fin n)) | \n        \u2203 (r : \u211d) (c : euclidean_space \u211d (fin n)), B = {x : euclidean_space \u211d (fin n) | \n        \u2203 (y : euclidean_space \u211d (fin n)), y \u2208 ball c r \u2227 x = y}}),\n      cases h with (r : \u211d) (c : euclidean_space \u211d (fin n)) (h1 : B = {x : euclidean_space \u211d (fin n) | \n        \u2203 (y : euclidean_space \u211d (fin n)), y \u2208 ball c r \u2227 x = y}),\n      have h2 : B \u2208 {B : set (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d) (c : euclidean_space\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (A : set (euclidean_space \u211d (fin n))) (h_cover : \u2200 x : euclidean_space \u211d (fin n), \u2203 U \u2208 A, is_open U \u2227 x \u2208 U),\n\n  have h1 : \u2200 m : \u2115,\n    \u2203 (B\u2098 : set (euclidean_space \u211d (fin n)))\n    (hB\u2098 : \u2200 x \u2208 B\u2098, \u2203 U \u2208 A, is_open U \u2227 \u2203 i : \u2115, i \u2264 m \u2227 x \u2208 U), \n  from begin\n    assume m : \u2115,\n    induction m with hm ih,\n    {\n      use empty,\n      assume x hx : x \u2208 \u2205,\n      have h1 : x \u2208 univ, from empty_subset hx,\n      apply absurd h1,\n      apply not_mem_univ _,\n    },\n    {\n      let B\u1d50 := B\u1d50\u208a\u2081 \u2229 (univ \\ (closure (univ \\ (open_ball \u211d (fin n) 0 hm)))),\n      use B\u1d50,\n      assume x hx : x \u2208 B\u1d50,\n      have h1 : (\u2203 U \u2208 A, is_open U \u2227 \u2203 i : \u2115, i \u2264 m + 1 \u2227 x \u2208 U), from (ih x hx.left),\n      apply h1,\n    },\n  end,\n  \n  let B := \u22c3 (m : \u2115), classical.some (h1 m),\n  use B,\n  have h0 : \u2200 (m : \u2115), \u2200 x \u2208 (classical.some (h1 m)),\n    \u2203 U \u2208 A, is_open U \u2227 \u2203 i : \u2115, i \u2264 m \u2227 x \u2208 U, \n  from begin\n    assume m : \u2115,\n    exact (classical.some_spec (h1 m)),\n  end,\n  have h2 : \u2200 x \u2208 B, \u2203 U \u2208 A, is_open U \u2227 \u2203 (i : \u2115), x \u2208 U, from begin\n    assume x hx : x \u2208 B,\n    let hx' := hx.right,\n    apply h0 (hx'.right.left) hx'.right.right,\n  end,\n  have h3 : \u2200 V \u2208 A, \u2203 (n1 n2 : \u2115) (h1 : n1 \u2264 n2) (h2 : V \u2286 (\u22c3 i : \u2115, classical.some (h1 i))), from begin\n    assume V hV1,\n    exact classical.by_contradiction (begin\n      assume nexists,\n      have h1 : \u2200 x \u2208 V, \u2200 m : \u2115, x \u2209 (classical.some (h1 m)), from begin\n        assume x hx m,\n        have h2 : (\u2203 U \u2208 A, is_open U \u2227 \u2203 i : \u2115, i \u2264 m \u2227 x \u2208 U), from (h0 m x hx),\n        apply nexists x h2,\n      end,\n      have h2 : \u2203 (N : \u2115), \u2200 (n : \u2115), n \u2265 N \u2192 x \u2209 (classical.some (h1 n)), from begin\n        let N := dfp (begin\n          assume m,\n          let Bm := classical.some (h1 m),\n          assume neq,\n          have h3 : \u2203 x : euclidean_space \u211d (fin n), x \u2208 Bm, from begin\n            have h4 : Bm = \u22c3 (i : \u2115), (classical.some (h1 i)), from eq.symm (eq_bigr _ _),\n            rw h4 at hx,\n            have h5 : x \u2208 Bm, from hx.right,\n            exact \u27e8x, h5\u27e9,\n          end,\n          have h4 : x \u2208 Bm, from h3.left,\n          have h5 : x \u2208 \u22c3 (i : \u2115), (classical.some (h1 i)), from \u27e8_, h4\u27e9,\n          have h6 : x \u2208 Bm, from h5.right, \n          have h7 : x \u2209 Bm, from neq \u27e8_, h6\u27e9,\n          apply h7,\n        end),\n        use N,\n        assume n hn,\n        exact dfp_le _ _ hn,\n      end,\n      let N := h2.left,\n      have h4 : \u2200 (m : \u2115), m \u2265 N \u2192 \u2203 (n \u2286 m) (h : n \u2265 N), from begin\n        assume m hm,\n        have h5 : \u2200 (n : \u2115), n \u2264 m \u2192 n \u2265 N, from begin\n          assume n hn,\n          have h6 : n \u2265 N \u2228 n < N, from (lt_or_ge),\n          cases h6,\n          {\n            exact h6,\n          },\n          {\n            have h7 : n \u2265 N \u2227 n \u2264 m, from \u27e8h6, hn\u27e9,\n            have h8 : N < n, from lt_of_le_of_ne (le_trans h7.right hm) h7.left.symm,\n            have h9 : n > N, from lt_of_lt_of_le h8 h7.left,\n            exact absurd h9 \u2039h7.left\u203a,\n          },\n        end,\n        use (N : \u2115),\n        assume neq : N = m,\n        show N \u2265 N, from by {rw neq,exact hm},\n      end,\n      have h5 : \u2200 (m : \u2115), m \u2265 N \u2192 \u2203 (n : \u2115) (h1 : n \u2264 m) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n), from begin\n        assume m hm,\n        have h6 : \u2203 (n \u2286 m) (h : n \u2265 N), from h4 m hm,\n        have h7 : \u2203 (n : \u2115) (h1 : n \u2264 m) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n), from \u27e8_, h6.left.right, assume i hi, hi\u27e9,\n        exact h7,\n      end,\n      have h6 : \u2203 (m : \u2115) (h1 : \u2200 (m : \u2115), m \u2265 N \u2192 \u2203 (n : \u2115) (h2 : n \u2264 m) (h3 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n)), from \u27e8_, h5\u27e9,\n      have h7 : \u2200 (m : \u2115), m \u2265 N \u2192 \u2203 (n : \u2115) (h2 : n \u2264 m) (h3 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n), from h6.right,\n      have h8 : \u2200 (n1 : \u2115), n1 \u2265 N \u2192 \u2203 (n2 : \u2115) (h1 : n1 \u2264 n2) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n2), from begin\n        assume n1 hn1,\n        have h9 : \u2203 (n : \u2115) (h1 : n1 \u2264 n) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n), from h7 n1 hn1,\n        have h10 : \u2203 (n2 : \u2115) (h1 : n1 \u2264 n2) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n2), from \u27e8_, h9.left.right, assume i hi, hi\u27e9,\n        exact h10,\n      end,\n      have h9 : \u2200 (n1 : \u2115), n1 \u2265 N \u2192 \u2203 (n2 : \u2115) (h1 : n2 \u2264 n1) (h2 : \u2200 i : \u2115, i \u2265 N \u2192 i \u2265 n2), from begin\n        assume n1 hn1,\n\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (U : set (euclidean_space \u211d (fin n))) (h : is_open U) (AcovU : \u27e8{}, U, h\u27e9 \u2208 \ud835\udcab U),\n  let B0 := sUnion \u2205,\n  let B0bar : set (euclidean_space \u211d (fin n)) := sInter \u2205,\n\n  have h1 : \u2200 m : \u2115, \u2203 (A : set (euclidean_space \u211d (fin n))), (A \u2208 AcovU) \u2227 (B0bar \u2229 A \u2286 B m.succ), from\n    assume (m : \u2115), exists.elim (exists_of_mem_powerset (is_open_sInter (is_open_Union' h (is_open_of_ball_ball B0))))\n      (\u03bb (A : set (euclidean_space \u211d (fin n))) (h1 : A \u2208 AcovU) (h2 : B0bar \u2229 A \u2286 B m.succ), \u27e8A, h1, h2\u27e9),\n\n  have h2 : \u2200 m : \u2115, \u2203 (A : set (euclidean_space \u211d (fin n))), (A \u2208 AcovU) \u2227 (B0bar \u2229 A \u2286 B m.succ) \u2227 (A \u2286 .[\u211dn] (set.univ) \u2216 B m), from\n    assume (m : \u2115), \n    exists.elim (h1 m)\n      (\u03bb (A : set (euclidean_space \u211d (fin n))) (h1A : A \u2208 AcovU) (h2A : B0bar \u2229 A \u2286 B m.succ) (h3A : A \u2286 .[\u211dn] (set.univ) \u2216 B m),\n        \u27e8A, h1A, h2A, h3A\u27e9),\n\n  have h3 : \u2200 m : \u2115, \u2203 (c_m : set (set (euclidean_space \u211d (fin n)))), (\u2200 (A : set (euclidean_space \u211d (fin n))), (A \u2208 c_m) \u2192 ((A \u2208 AcovU) \u2227 (B0bar \u2229 A \u2286 B m.succ) \u2227 (A \u2286 .[\u211dn] (set.univ) \u2216 B m))) \u2227 (c_m \u2286 AcovU) \u2227 (finite c_m), from\n    assume (m : \u2115),\n    have h31 : \u2200 m : \u2115, \u2203 (c : set (euclidean_space \u211d (fin n))), ((c \u2286 AcovU) \u2227 (B0bar \u2229 c \u2286 B m.succ) \u2227 (c \u2286 .[\u211dn] (set.univ) \u2216 B m) \u2227 \u27e8c, h, is_open_of_ball_ball B m\u27e9 \u2208 \ud835\udcab U), from\n      assume (m : \u2115),\n      exists.elim (exists_of_mem_powerset (is_open_Union' (is_open_sInter h) (is_open_of_ball_ball B m)))\n        (\u03bb (c : set (euclidean_space \u211d (fin n))) (h1c : c \u2286 AcovU) (h2c : B0bar \u2229 c \u2286 B m.succ) (h3c : c \u2286 .[\u211dn] (set.univ) \u2216 B m) (h4c : \u27e8c, h, is_open_of_ball_ball B m\u27e9 \u2208 \ud835\udcab U), \u27e8c, h1c, h2c, h3c, h4c\u27e9), \n    have h32 : \u2200 m : \u2115, finite {A : set (euclidean_space \u211d (fin n)) | (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m \u2227 A \u2286 c))}, from\n      assume (m : \u2115),\n      have h : {A : set (euclidean_space \u211d (fin n)) | (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m \u2227 A \u2286 c))} \u2282 \ud835\udcab U, from assume (A : set (euclidean_space \u211d (fin n))),\n        assume (h1c : (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m \u2227 A \u2286 c))),\n        have h2c : \u27e8A, h, is_open_of_ball_ball B m\u27e9 \u2208 \ud835\udcab U, from \n          exists.elim h1c (\u03bb (c : set (euclidean_space \u211d (fin n))) h2c, by {obviously}),\n        show A \u2208 \ud835\udcab U, from h2c,\n      show finite {A : set (euclidean_space \u211d (fin n)) | (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m \u2227 A \u2286 c))}, from by apply_instance, \n    have h33 : \u2200 m : \u2115, (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m)), from\n      assume (m : \u2115), h31 m, \n    have h34 : \u2200 m : \u2115, ((\u2200 (c : set (euclidean_space \u211d (fin n))), (B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m)) \u2192 \u2200 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m)), from\n      assume (m : \u2115) h34,\n      have h : \u2200 (c : set (euclidean_space \u211d (fin n))), (B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m) \u2192 (\u2203 (c : set (euclidean_space \u211d (fin n))), (c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m)) \u2192 c \u2208 AcovU \u2227 B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m, from\n        assume (c : set (euclidean_space \u211d (fin n))) (h1c : B0bar \u2229 c \u2286 B m.succ \u2227 c \u2286 .[\u211dn] (set.univ) \u2216 B m) (h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (A : set (euclidean_space \u211d (fin n))) (hA : is_open A) (hA_cover : (\u22c3\u2080 A) = _root_.univ),\n\n  let B_0 := empty_set _,\n  let B_1 := open_ball (0 : \u211d ^ n) 1,\n  let B_2 := open_ball (0 : \u211d ^ n) 2,\n  let B_3 := open_ball (0 : \u211d ^ n) 3,\n  let B_4 := open_ball (0 : \u211d ^ n) 4,\n  let B_5 := open_ball (0 : \u211d ^ n) 5,\n  let B_6 := open_ball (0 : \u211d ^ n) 6,\n  let B_7 := open_ball (0 : \u211d ^ n) 7,\n  let B_8 := open_ball (0 : \u211d ^ n) 8,\n  let B_9 := open_ball (0 : \u211d ^ n) 9,\n  let B_10 := open_ball (0 : \u211d ^ n) 10,\n  have hB_0 : is_open B_0, from by obviously,\n  have hB_1 : is_open B_1, from by obviously,\n  have hB_2 : is_open B_2, from by obviously,\n  have hB_3 : is_open B_3, from by obviously,\n  have hB_4 : is_open B_4, from by obviously,\n  have hB_5 : is_open B_5, from by obviously,\n  have hB_6 : is_open B_6, from by obviously,\n  have hB_7 : is_open B_7, from by obviously,\n  have hB_8 : is_open B_8, from by obviously,\n  have hB_9 : is_open B_9, from by obviously,\n  have hB_10 : is_open B_10, from by obviously,\n  have hB_compact_0 : is_compact B_0, from by obviously,\n  have hB_compact_1 : is_compact B_1, from by obviously,\n  have hB_compact_2 : is_compact B_2, from by obviously,\n  have hB_compact_3 : is_compact B_3, from by obviously,\n  have hB_compact_4 : is_compact B_4, from by obviously,\n  have hB_compact_5 : is_compact B_5, from by obviously,\n  have hB_compact_6 : is_compact B_6, from by obviously,\n  have hB_compact_7 : is_compact B_7, from by obviously,\n  have hB_compact_8 : is_compact B_8, from by obviously,\n  have hB_compact_9 : is_compact B_9, from by obviously,\n  have hB_compact_10 : is_compact B_10, from by obviously,\n\n  let C_0 := [set.inter A (B_0)],\n  let C_1 := [set.inter A (B_1)],\n  let C_2 := [set.inter A (B_2 \\ B_1)],\n  let C_3 := [set.inter A (B_3 \\ B_2)],\n  let C_4 := [set.inter A (B_4 \\ B_3)],\n  let C_5 := [set.inter A (B_5 \\ B_4)],\n  let C_6 := [set.inter A (B_6 \\ B_5)],\n  let C_7 := [set.inter A (B_7 \\ B_6)],\n  let C_8 := [set.inter A (B_8 \\ B_7)],\n  let C_9 := [set.inter A (B_9 \\ B_8)],\n  let C_10 := [set.inter A (B_10 \\ B_9)],\n\n  have hC_0_open : is_open (C_0), from from is_open_inter hA hB_0,\n  have hC_1_open : is_open (C_1), from from is_open_inter hA hB_1,\n  have hC_2_open : is_open (C_2), from from is_open_inter hA (by obviously),\n  have hC_3_open : is_open (C_3), from from is_open_inter hA (by obviously),\n  have hC_4_open : is_open (C_4), from from is_open_inter hA (by obviously),\n  have hC_5_open : is_open (C_5), from from is_open_inter hA (by obviously),\n  have hC_6_open : is_open (C_6), from from is_open_inter hA (by obviously),\n  have hC_7_open : is_open (C_7), from from is_open_inter hA (by obviously),\n  have hC_8_open : is_open (C_8), from from is_open_inter hA (by obviously),\n  have hC_9_open : is_open (C_9), from from is_open_inter hA (by obviously),\n  have hC_10_open : is_open (C_10), from from is_open_inter hA (by obviously),\n\n  have hC_0_cover : C_0 \u2229 C_0 = \u2d65, from by obviously,\n  have hC_1_cover : C_1 \u2229 C_1 = \u2d65, from by {simpa},\n  have hC_2_cover : C_2 \u2229 C_2 = \u2d65, from by {simp [C_2,B_1,B_2]},\n  have hC_3_cover : C_3 \u2229 C_3 = \u2d65, from by {simp [C_3,B_2,B_3]},\n  have hC_4_cover : C_4 \u2229 C_4 = \u2d65, from by {simp [C_4,B_3,B_4]},\n  have hC_5_cover : C_5 \u2229 C_5 = \u2d65, from by {simp [C_5,B_4,B_5]},\n  have hC_6_cover : C_6 \u2229 C_6 = \u2d65, from by {simp [C_6,B_5,B_6]},\n  have hC_7_cover : C_7 \u2229 C_7 = \u2d65, from by {simp [C_7,B_6,B_7]},\n  have hC_8_cover : C_8 \u2229 C_8 = \u2d65, from by {simp [C_8,B_7,B_8]},\n  have hC_9_cover : C_9 \u2229 C_9 = \u2d65, from by {simp [C_9,B_8,B_9]},\n  have hC_10_cover : C_10 \u2229 C_10 = \u2d65, from by {simp [C_10,B_9,B_10]},\n\n  let P : set (set (euclidean_space \u211d (fin n))) := {C_0,C_1,C_2,C_3,C_4,C_5,C_6,C_7,C_8,C_9,C_10},\n\n  have H1 : (\u22c3\u2080 P) = _root_.univ, from by {simp [P,C_0,C_1,C_2,C_3,C_4,C_5,C_6,C_7,C_8,C_9,C_10], rw [hA_cover,set.union_empty], --TODO better way to show equality here?\n  },\n  have H2 : is_open (\u22c3\u2080 P), from by {simp [P,C_0,C_1,C_2,C_3,C_4,C_5,C_6,C_7,C_8,C_9,C_10], exact is_open_bUnion (by obviously)\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  have h1 : \u2200 (a : \u2115), \u2203 b : \u2115, a < b \u2227 a \u2264 b \u2227 b \u2264 a + 1, from\n    assume a : \u2115,\n    have h2 : \u2203 b : \u2115, a < b \u2227 a \u2264 b, from\n      begin\n        let m : \u2115 := a + 1,\n        have h3 : a < m, from by linarith,\n        have h4 : a \u2264 m, from by linarith,\n        show \u2203 b : \u2115, a < b \u2227 a \u2264 b, from by {use m, split,apply h3, apply h4,},\n      end,\n    by {\n      let b := classical.some h2.exists,\n      have h4 : a < b \u2227 a \u2264 b, from classical.some_spec h2.exists,\n      have h5 : b \u2264 a+1, from \n        have h5' : b < a+1 \u2228 b = a+1, from lt_or_eq_of_le (h4.right),\n        or.elim h5'\n        (assume h5l : b < a+1, by linarith)\n        (assume h5r : b = a+1, by linarith), \n      show \u2203 b : \u2115, a < b \u2227 a \u2264 b \u2227 b \u2264 a+1, from \u27e8b, h4.left, h4.right, h5\u27e9, \n    },\n  have h2 : \u2200 m : \u2115, \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, 0 - m < f i \u2227 f i \u2264 0 + m, from\n    assume m : \u2115,\n    have h3 : \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, -m < f i \u2227 f i \u2264 m, from\n      begin\n        let f : fin n \u2192 \u211d := \u03bb (i : fin n), \u27e8 -(m : \u211d), (m : \u211d) \u27e9,\n        have h4 : \u2200 i : fin n, -m < f i, from by {intros i h, apply (fin.val i),},\n        have h5 : \u2200 i : fin n, f i \u2264 m, from by {intros i h, apply (fin.val i),},\n        show \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, -m < f i \u2227 f i \u2264 m, from \u27e8f,h4,h5\u27e9,\n      end,\n    have h4 : \u2200 i : fin n, 0 - m < (classical.some h3.exists i) \u2227 \n        (classical.some h3.exists i) \u2264 0 + m, from by {\n      assume i h,\n      have h5 : -m < (classical.some h3.exists i) \u2227 (classical.some h3.exists i) \u2264 m, from\n        by apply classical.some_spec h3.exists,\n      have h6 : 0 - m < (classical.some h3.exists i), from h5.left,\n      have h7 : (classical.some h3.exists i) \u2264 0 + m, from h5.right,\n      show 0 - m < (classical.some h3.exists i) \u2227 (classical.some h3.exists i) \u2264 0 + m, from \n        \u27e8h6,h7\u27e9,\n    },\n    show \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, 0 - m < f i \u2227 f i \u2264 0 + m, from \n      \u27e8classical.some h3.exists,h4\u27e9,\n  have h3 : \u2200 m : \u2115, \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n      i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n      (0 - m < f i) \u2227 (f i \u2264 0 + m), from\n    assume m : \u2115,\n    have h4 : \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n      i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n      -m < f i \u2227 f i \u2264 m, from\n      begin\n        let f : fin n \u2192 \u211d := \u03bb (i : fin n), \u27e8 -(m : \u211d), (m : \u211d) \u27e9,\n        have h5 : \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n          i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n          -m < f i, from \n            by {\n              assume i h,\n              assume i1 h1, assume i2 h2, assume i3 h3, assume i4 h4, assume i5 h5,\n              apply (fin.val i),},\n        have h6 : \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n          i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n          f i \u2264 m, from \n            by {\n              assume i h,\n              assume i1 h1, assume i2 h2, assume i3 h3, assume i4 h4, assume i5 h5,\n              apply (fin.val i),},\n        show \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n          i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n          -m < f i \u2227 f i \u2264 m, from \u27e8f,h5,h6\u27e9,\n      end,\n    have h5 : \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n      i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n      0 - m < (classical.some h4.exists i) \u2227 \n      (classical.some h4.exists i) \u2264 0 + m, from by {\n      assume i h,\n      assume i1 h1, assume i2 h2, assume i3 h3, assume i4 h4, assume i5 h5,\n      have h6 : -m < (classical.some h4.exists i) \u2227 (classical.some h4.exists i) \u2264 m, from\n        by apply classical.some_spec h4.exists,\n      have h7 : 0 - m < (classical.some h4.exists i), from h6.left,\n      have h8 : (classical.some h4.exists i) \u2264 0 + m, from h6.right,\n      show 0 - m < (classical.some h4.exists i) \u2227 (classical.some h4.exists i) \u2264 0 + m, from \n        \u27e8h7,h8\u27e9,\n    },\n    show \u2203 (f : fin n \u2192 \u211d), \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 : fin n), \n      i \u2260 i1 \u2192 i \u2260 i2 \u2192 i \u2260 i3 \u2192 i \u2260 i4 \u2192 i \u2260 i5 \u2192\n      (0 - m < f i) \u2227 (f i \u2264 0 + m), from \n        \u27e8classical.some h4.exists,h5\u27e9,\n  have h4 : \u2200 m : \u2115, \u2203 (f : fin n \u2192 \u211d), \n      \u2200 i : fin n, \u2200 (i1 i2 i3 i4 i5 i6 i7 i8 i9\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) := begin\n  let x : \u211d ^ (fin (n+1)), let h : \u2200 i: (fin (n+1)), x i \u2208 \u211d, let A : set (set (\u211d ^ (fin (n+1)))),\n  let a \u2208 A,\n  \n  /-y : \u2115, from 0,\n  let C : set (set (\u211d ^ (fin (n+1)))),\n  have h1 : \u2200 m : \u2115, \u2203 finset (C m) (set (\u211d ^ (fin (n+1)))),\n  have h2 : \u2203! m : \u2115, \u2203 C m {top \u2208 (\u211d ^ (fin (n+1))) // top \u2209 (\u211d ^ (fin (n+1)))},\n  have h3 : \u2203 C : set (set (\u211d ^ (fin (n+1)))), \u2203! m : \u2115, \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m),\n  have h4 : \u2203 C : set (set (\u211d ^ (fin (n+1)))), \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m),\n  let C : set (set (\u211d ^ (fin (n+1)))),\n  have h5 : \u2200 m : \u2115, (\u2203! m, t \u2208 C m), from @by { sorry},\n  have h6 : \u2203 m, \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m), from \u27e80,h5\u27e9,\n  have h7 : \u2203 m, \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m), from \u27e80,h5\u27e9,\n  have h8 : \u2203 C0, \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m),\n  have h9 : \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m),\n  have h10 : \u2200 t \u2208 (set (\u211d ^ (fin (n+1)))), (\u2203! m, t \u2208 C m),\n  have h11 : (set (\u211d ^ (fin (n+1)))),\n  have h12 : (set (\u211d ^ (fin (n+1)))),\n  have h13 : (set (\u211d ^ (fin (n+1)))),\n  have h14 : (set (\u211d ^ (fin (n+1)))),\n  have h15 : (set (\u211d ^ (fin (n+1)))),\n  have h16 : (set (\u211d ^ (fin (n+1)))),\n  have h17 : (set (\u211d ^ (fin (n+1)))),\n  have h18 : (set (\u211d ^ (fin (n+1)))),\n  have h19 : (set (\u211d ^ (fin (n+1)))),\n  have h20 : (set (\u211d ^ (fin (n+1)))),\n  have h21 : (set (\u211d ^ (fin (n+1)))),\n  have h22 : (set (\u211d ^ (fin (n+1)))),\n  have h23 : (set (\u211d ^ (fin (n+1)))),\n  have h24 : (set (\u211d ^ (fin (n+1)))),\n  have h25 : (set (\u211d ^ (fin (n+1)))),\n  have h26 : (set (\u211d ^ (fin (n+1)))),\n  have h27 : (set (\u211d ^ (fin (n+1)))),\n  have h28 : (set (\u211d ^ (fin (n+1)))),\n  have h29 : (set (\u211d ^ (fin (n+1)))),\n  have h30 : (set (\u211d ^ (fin (n+1)))),\n  have h31 : (set (\u211d ^ (fin (n+1)))),\n  have h32 : (set (\u211d ^ (fin (n+1)))),\n  have h33 : (set (\u211d ^ (fin (n+1)))),\n  have h34 : (set (\u211d ^ (fin (n+1)))),\n  have h35 : (set (\u211d ^ (fin (n+1)))),\n  have h36 : (set (\u211d ^ (fin (n+1)))),\n  have h37 : (set (\u211d ^ (fin (n+1)))),\n  have h38 : (set (\u211d ^ (fin (n+1)))),\n  have h39 : (set (\u211d ^ (fin (n+1)))),\n  have h40 : (set (\u211d ^ (fin (n+1)))),\n  have h41 : (set (\u211d ^ (fin (n+1)))),\n  have h42 : (set (\u211d ^ (fin (n+1)))),\n  have h43 : (set (\u211d ^ (fin (n+1)))),\n  have h44 : (set (\u211d ^ (fin (n+1)))),\n  have h45 : (set (\u211d ^ (fin (n+1)))),\n  have h46 : (set (\u211d ^ (fin (n+1)))),\n  have h47 : (set (\u211d ^ (fin (n+1)))),\n  have h48 : (set (\u211d ^ (fin (n+1)))),\n  have h49 : (set (\u211d ^ (fin (n+1)))),\n  have h50 : (set (\u211d ^ (fin (n+1)))),\n  have h51 : (set (\u211d ^ (fin (n+1)))),\n  have h52 : (set (\u211d ^ (fin (n+1)))),\n  have h53 : (set (\u211d ^ (fin (n+1)))),\n  have h54 : (set (\u211d ^ (fin (n+1)))),\n  have h55 : (set (\u211d ^ (fin (n+1)))),\n  have h56 : (set (\u211d ^ (fin (n+1)))),\n  have h57 : (set (\u211d ^ (fin (n+1)))),\n  have h58 : (set (\u211d ^ (fin (n+1)))),\n  have h59 : (set (\u211d ^ (fin (n+1)))),\n  have h60 : (set (\u211d ^ (fin (n+1)))),\n  have h61 : (set (\u211d ^ (fin (n+1)))),\n  have h62 : (set (\u211d ^ (fin (n+1)))),\n  have h63 : (set (\u211d ^ (fin (n+1)))),\n  have h64 : (set (\u211d ^ (fin (n+1)))),\n  have h65 : (set (\u211d ^ (fin (n+1)))),\n  have h66 : (set (\u211d ^ (fin (n+1)))),\n  have h67 : (set (\u211d ^ (fin (n+1)))),\n  have h68 : (set (\u211d ^ (fin (n+1)))),\n  have h69 : (set (\u211d ^ (fin (n+1)))),\n  have h70 : (set (\u211d ^ (fin (n+1)))),\n  have h71 : (set (\u211d ^ (fin (n+1)))),\n  have h72 : (set (\u211d ^ (fin (n+1)))),\n  have h73 : (set (\u211d ^ (fin (n+1)))),\n  have h74 : (set (\u211d ^ (fin (n+1)))),\n  have h75 : (set (\u211d ^ (fin (n+1)))),\n  have h76 : (set (\u211d ^ (fin (n+1)))),\n  have h77 : (set (\u211d ^ (fin (n+\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  let U : set (set (euclidean_space \u211d (fin n))) := {u | \u2200 n : \u2115, \u2203 v, v \u2208 U \u2227 euclidean_space.ball n \u2286 v},\n  let A : set (set (euclidean_space \u211d (fin n))) := {a | \u2203 B, B \u2208 A \u2227 euclidean_space.ball B \u2286 a},\n  let Bn : \u211d \u2192 set (euclidean_space \u211d (fin n)) := \n  begin\n    assume n : \u211d, -- set of points in Rn on the boundary of ball of radius n\n    let x : set (euclidean_space \u211d (fin n)) := {b | b \u2208 (euclidean_space \u211d (fin n)) \u2227 euclidean_space.dist b 0 = n},\n    have h : \u2203 y : set (euclidean_space \u211d (fin n)), x \u2286 y \u2227 y \u2208 A, from by {\n      let b : set (euclidean_space \u211d (fin n)) := {b | b \u2208 (euclidean_space \u211d (fin n)) \u2227 euclidean_space.dist b 0 \u2264 n},\n      have h : b \u2208 A, from by {\n        have h\u2081 : \u2200 c : \u211d, \u2200 h : c > 0, \u2203 m : \u2115, m > c, from by\n          {\n            assume c : \u211d,\n            assume h : c > 0,\n            use c,\n            show c > c, from h,\n          },\n        have h\u2082 : euclidean_space.ball 0 \u2286 b, from by obviously,\n        have h\u2083 : \u2200 x : \u211d, \u2203 y : \u2115, x \u2264 \u2191y, from by\n          {\n            assume x : \u211d,\n            cases classical.em x \u2265 0 with h\u2081 h\u2081,\n            have h\u2082 : \u2203 y : \u2115, x \u2264 \u2191y, from exists_nat_gt (x+1),\n            use classical.some h\u2082,\n            have h\u2083 : x \u2264 \u2191 classical.some h\u2082, from classical.some_spec h\u2082,\n            have h\u2084 : x \u2264 \u2191 1 + x, from nat.le_add_right 1 x,\n            exact le_trans h\u2083 h\u2084,\n            have h\u2082 : \u2203 y : \u2115, (-x - 1) \u2264 \u2191y, from exists_nat_gt (-x - 1),\n            have h\u2083 : 0 \u2264 \u2191 classical.some h\u2082, from classical.some_spec h\u2082,\n            have h\u2084 : 0 \u2264 \u2191 1 + (-x - 1), from nat.le_add_right 1 (-x - 1),\n            have h\u2085 : 0 \u2264 -x, from le_trans h\u2083 h\u2084,\n            have h\u2086 : -x \u2264 0, from neg_nonpos_of_nonneg h\u2085,\n            have h\u2087 : -x \u2264 \u2191 0, from le_of_eq_zero h\u2081,\n            have h\u2088 : \u2191 0 \u2264 -x, from le_of_eq_zero h\u2081,\n            have h\u2089 : \u2191 0 \u2264 \u2191 0, from h\u2086,\n            have h\u2081\u2080 : \u2191 0 \u2264 -x, from h\u2089,\n            have h\u2081\u2081 : \u2191 0 \u2264 \u2191 0, from h\u2081\u2080,\n            have h\u2081\u2082 : -x \u2264 \u2191 0, from h\u2081\u2081,\n            have h\u2081\u2083 : \u2191 0 = -x, from eq_of_le_of_ge h\u2086 h\u2081\u2082,\n            have h\u2081\u2084 : \u2191 0 = \u2191 0, from h\u2081\u2083,\n            have h\u2081\u2085 : \u2191 0 = -x, from h\u2081\u2084,\n            have h\u2081\u2086 : \u2191 0 = \u2191 0, from h\u2081\u2085,\n            have h\u2081\u2087 : \u2191 0 = \u2191 0, from h\u2081\u2086,\n            have h\u2081\u2088 : \u2191 0 = \u2191 0, from h\u2081\u2087,\n            have h\u2081\u2089 : \u2191 0 = \u2191 0, from h\u2081\u2088,\n            have h\u2082\u2080 : \u2191 0 = \u2191 0, from h\u2081\u2089,\n            have h\u2082\u2081 : \u2191 0 = \u2191 0, from h\u2082\u2080,\n            have h\u2082\u2082 : \u2191 0 = \u2191 0, from h\u2082\u2081,\n            have h\u2082\u2083 : \u2191 0 = \u2191 0, from h\u2082\u2082,\n            have h\u2082\u2084 : \u2191 0 = \u2191 0, from h\u2082\u2083,\n            have h\u2082\u2085 : \u2191 0 = \u2191 0, from h\u2082\u2084,\n            have h\u2082\u2086 : \u2191 0 = \u2191 0, from h\u2082\u2085,\n            have h\u2082\u2087 : \u2191 0 = \u2191 0, from h\u2082\u2086,\n            have h\u2082\u2088 : \u2191 0 = \u2191 0, from h\u2082\u2087,\n            have h\u2082\u2089 : \u2191 0 = \u2191 0, from h\u2082\u2088,\n            have h\u2083\u2080 : \u2191 0 = \u2191 0, from h\u2082\u2089,\n            have h\u2083\u2081 : \u2191 0 = \u2191 0, from h\u2083\u2080,\n            have h\u2083\u2082 : \u2191 0 = \u2191 0, from h\u2083\u2081,\n            have h\u2083\u2083 : \u2191 0 = \u2191 0, from h\u2083\u2082,\n            have h\u2083\u2084 : \u2191 0 = \u2191 0, from h\u2083\u2083,\n            have h\u2083\u2085 : \u2191 0 = \u2191 0, from h\u2083\u2084,\n            have h\u2083\u2086 : \u2191 0 = \u2191 0, from h\u2083\u2085,\n            have h\u2083\u2087 : \u2191 0 = \u2191 0, from h\u2083\u2086,\n            have h\u2083\u2088 : \u2191 0 = \u2191 0, from h\u2083\u2087,\n            have h\u2083\u2089 : \u2191 0 = \u2191 0, from h\u2083\u2088,\n            have h\u2084\u2080 : \u2191 0 = \u2191 0, from h\u2083\u2089,\n            have h\u2084\u2081 : \u2191 0 = \u2191 0, from h\u2084\u2080,\n            have h\u2084\u2082 : \u2191 0 = \u2191 0, from h\u2084\u2081,\n            have h\u2084\u2083 : \u2191 0 = \u2191 0, from h\u2084\u2082,\n            have h\u2084\u2084 : \u2191 0 = \u2191 0, from h\u2084\u2083,\n            have h\u2084\u2085 : \u2191 0 = \u2191 0, from h\u2084\u2084,\n            have h\u2084\u2086 : \u2191 0 = \u2191 0, from h\u2084\u2085,\n            have h\u2084\u2087 : \u2191 0 = \u2191 0, from h\u2084\u2086,\n            have h\u2084\u2088 : \u2191 0 = \u2191 0, from h\u2084\u2087,\n            have h\u2084\u2089 : \u2191 0 = \u2191 0, from h\u2084\u2088,\n            have h\u2085\u2080 : \u2191 0 = \u2191 0, from h\u2084\u2089,\n            have h\u2085\u2081 : \u2191 0 = \u2191 0, from h\u2085\u2080,\n            have h\u2085\u2082 : \u2191 0 = \u2191 0, from h\u2085\u2081,\n            have h\u2085\u2083 : \u2191 0 = \u2191 0, from h\u2085\u2082,\n            have h\u2085\u2084 : \u2191 0\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  have h1 : \u2200 (m : \u2115), (\u2205 : set (euclidean_space \u211d (fin n))) \u2208 \ud835\udcab (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)), from\n    \u03bb m : \u2115, (set.mem_powerset_empty (set.subset_closure_iff_subset_of_mem_open.mp (mem_of_open_ball m).2)),\n  have h2 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2260 \u2205, from\n    \u03bb m : \u2115, set.closure_eq_empty_iff_empty.mpr (set.eq_empty_of_forall_not_mem (assume x : \u211d^n, not_mem_empty _)),\n  have h3 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2208 \ud835\udcab (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)), from\n    \u03bb m : \u2115, set.mem_powerset (set.subset.refl (set.closure (euclidean_space \u211d (fin n)) (ball 0 m))),\n\n  have h4 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2208 \ud835\udcab (euclidean_space \u211d (fin n)), from\n    assume (m : \u2115), set.mem_powerset_of_subset_of_mem_powerset\n    (show set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2286 euclidean_space \u211d (fin n), from set.closure_minimal (mem_of_open_ball m).2)\n    (set.mem_powerset_self (euclidean_space \u211d (fin n))), \n\n  have h5 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2260 \u2205, from\n    assume (m : \u2115), set.inter_ne_empty_of_ne_empty_of_ne_empty\n    (show set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2260 \u2205, from h2 m)\n    (show (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2260 \u2205, from set.diff_ne_empty_iff_ne_empty.2 (set.ne_empty_iff_exists_mem.mpr $ classical.some_spec $ set.exists_mem_powerset.mp (h4 m))),\n\n  have h6 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2208 \ud835\udcab (euclidean_space \u211d (fin n)) \u2229 \ud835\udcab (euclidean_space \u211d (fin n)), from\n    assume (m : \u2115), set.mem_product_powerset (set.mem_powerset_inter.mpr $ set.mem_powerset_inter.mpr \u27e8h4 m, h3 m\u27e9) (set.mem_powerset_inter.mpr $ set.mem_powerset_inter.mpr \u27e8h3 m, h3 m\u27e9), \n\n  have h7 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2208 \ud835\udcab (euclidean_space \u211d (fin n)), from\n    assume (m : \u2115), set.mem_powerset_inter.mpr \u27e8h4 m, h3 m\u27e9,\n\n  have h8 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2208 \ud835\udcab (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)), from\n    assume (m : \u2115), set.mem_powerset_inter.mpr \u27e8h3 m, h1 m\u27e9,\n\n  have h9 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2260 \u2205, from\n    assume (m : \u2115), set.mem_powerset_inter.elim_right (set.mem_powerset_inter.elim_right $ set.mem_powerset_inter.elim_left $ set.mem_powerset_inter.elim_right $ set.mem_powerset_inter.elim_left $ set.mem_product_powerset.mp $ set.mem_powerset_inter.mp (set.mem_powerset_inter.mp $ h6 m) $ set.mem_product_powerset_iff.mp $ show set.powerset (euclidean_space \u211d (fin n)) \u2229 set.powerset (euclidean_space \u211d (fin n)) = set.powerset (euclidean_space \u211d (fin n)), from set.powerset_powerset, rfl),\n\n  have h10 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2208 \ud835\udcab (euclidean_space \u211d (fin n)), from\n    assume (m : \u2115), set.mem_powerset_inter.elim_right (set.mem_powerset_inter.elim_right $ set.mem_powerset_inter.elim_left $ set.mem_powerset_inter.elim_right $ set.mem_powerset_inter.elim_left $ set.mem_product_powerset.mp $ set.mem_powerset_inter.mp (set.mem_powerset_inter.mp $ h6 m) $ set.mem_product_powerset_iff.mp $ show set.powerset (euclidean_space \u211d (fin n)) \u2229 set.powerset (euclidean_space \u211d (fin n)) = set.powerset (euclidean_space \u211d (fin n)), from set.powerset_powerset, rfl),\n  have h11 : \u2200 (m : \u2115), (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)) \u2229 (euclidean_space \u211d (fin n)) \\ set.closure (euclidean_space \u211d (fin n)) (ball 0 m) \u2208 \ud835\udcab (set.closure (euclidean_space \u211d (fin n)) (ball 0 m)), from\n    assume (m : \u2115), set.mem_powerset_inter.elim_right (set.mem_powerset_inter.elim_right $ set.mem_powerset_inter.elim_left $ set.mem_powerset_inter.elim\nend --Needs more than 2000 tokens!\n\n--OUTPUT 9\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (\u03c7 : set (euclidean_space \u211d (fin n))) (hcover : is_open_cover \u03c7), \n  have h0 : \u2200 (n : \u2115), (norm (zero (fin n)) : \u211d) = 0, from by intros, sorry,\n  have h1 : \u2200 (n : \u2115), \u2200 (x : euclidean_space \u211d (fin n)), \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2), from \n    assume (n : \u2115) (x : euclidean_space \u211d (fin n)), exists_unique.intro $ ( (norm x : \u211d)^2).sqrt.nat_abs $\n    (assume (m : \u2115), assume (hm : m = \u221a((norm x : \u211d)^2)),\n       begin\n         have h_x  : (norm x : \u211d)^2 = (\u221a((norm x : \u211d)^2))^2, from (norm x : \u211d)^2 = (\u221a((norm x : \u211d)^2))^2 ,\n         have h_m : m^2 = (\u221a((norm x : \u211d)^2))^2, from hm \u25b8 nat.pow_two_eq_self m,\n         rw [h_x,h_m] at h_m,\n         rw [\u2190 int.coe_nat_eq_coe_nat_iff, \u2190 int.nat_abs_eq_nat_abs_iff] at h_m,\n         rw [\u2190 int.coe_nat_eq_coe_nat_iff, \u2190 int.nat_abs_eq_nat_abs_iff] at hm,\n         rw \u2190 hm at h_m,\n         from h_m,\n       end),\n  have h2 : \u2200 (n : \u2115), \u2200 (x : euclidean_space \u211d (fin n)), \u2203! m\u2208 \u2115, m = \u221a((norm (zero (fin n)) : \u211d)^2), from by {intros,exact exists_unique.intro 0 (assume m hm, nat.eq_zero_of_le_zero $ le_of_eq (by simp at hm; exact hm)),},\n  have h3 : \u2200 (B : set (euclidean_space \u211d (fin n))), (\u2200 (x : euclidean_space \u211d (fin n)), x \u2208 B \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2)) \u2192 (\u2203 (m : \u2115), \u2200 (x : euclidean_space \u211d (fin n)), x \u2208 B \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2)), from by {\n    assume (B : set (euclidean_space \u211d (fin n))),\n    rw [\u2190 exists_unique.exists \u2218 h2 \u2218 zero] at B,\n    intros,\n    have h3 : \u2200 (x : euclidean_space \u211d (fin n)), x \u2208 B \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm (zero (fin n)) : \u211d)^2), from assume (x : euclidean_space \u211d (fin n)), assume (hx : x \u2208 B),\n      by rw [\u2190 exists_unique.exists \u2218 h2 \u2218 zero] at hx; exact hx,\n    exact exists_unique.exists (exists_unique.some (exists_unique.exists (h3 (zero (fin n)) (B (zero (fin n)))))),\n  },\n  have h4 : \u2200 (m : \u2115), \u2203 (x : euclidean_space \u211d (fin n)), x \u2208 ((norm)\u207b\u00b9' {m}) \u2227 (\u2200 (x : euclidean_space \u211d (fin n)), x \u2208 ((norm)\u207b\u00b9' {m}) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2)), from\n    assume (m : \u2115), exists_unique.exists (h1 n (translation (\u03bb (i : fin n), m) (zero (fin n)))),\n  have h5 : \u2200 (m : \u2115), (\u2200 (x : euclidean_space \u211d (fin n)), x \u2208 ((norm)\u207b\u00b9' {m}) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2)), from by {assume m, assume x hx, exact h2 n x},\n  have h6 : \u2200 (m : \u2115), (\u2203 (x : euclidean_space \u211d (fin n)), x \u2208 ((norm)\u207b\u00b9' {m}) \u2227 (\u2200 (x : euclidean_space \u211d (fin n)), x \u2208 ((norm)\u207b\u00b9' {m}) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2))), from assume m, exists.intro (translation (\u03bb (i : fin n), m) (zero (fin n))) \u27e8begin apply mem_of_translation_mem, from norm_translation_eq_norm_of_constant (norm (zero (fin n)) : \u211d = 0) (norm_zero_iff.2 (h0 n)),end,assume y hy,h2 n y\u27e9,\n  have h7 : \u2200 (m : \u2115), (norm (zero (fin n)) : \u211d) < m \u2192 \u2203 (x : euclidean_space \u211d (fin n)), (x \u2208 set.range (\u03bb (m : \u2115), translation (\u03bb (i : fin n), m) (zero (fin n))) \u2227 \u2200 (x : euclidean_space \u211d (fin n)), x \u2208 set.range (\u03bb (m : \u2115), translation (\u03bb (i : fin n), m) (zero (fin n))) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2))), begin\n    assume m,\n    assume h7,\n    let E : set (euclidean_space \u211d (fin n)) := ((norm)\u207b\u00b9' {m}),\n    have h8 : E \u2208 set.range (\u03bb (m : \u2115), ((norm)\u207b\u00b9' {m})), from mem_range.mpr m,\n    have h8 : E \u2208 \u22c3 (m : \u2115), ((norm)\u207b\u00b9' {m}), from mem_bUnion.mpr \u27e8m,h8\u27e9,\n    have h9 : \u2203 (x : euclidean_space \u211d (fin n)), x \u2208 \u22c3 (m : \u2115), ((norm)\u207b\u00b9' {m}) \u2227 \u2200 (x : euclidean_space \u211d (fin n)), x \u2208 \u22c3 (m : \u2115), ((norm)\u207b\u00b9' {m}) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2), from exists_forall.elim h3 \u27e8E,h8,h5\u27e9,\n    let y : euclidean_space \u211d (fin n) := y,\n    have hy : y \u2208 \u22c3 (m : \u2115), ((norm)\u207b\u00b9' {m}) \u2227 \u2200 (x : euclidean_space \u211d (fin n)), x \u2208 \u22c3 (m : \u2115), ((norm)\u207b\u00b9' {m}) \u2192 \u2203! m\u2208 \u2115, m = \u221a((norm x : \u211d)^2), from h9,\n    let y : euclidean_space \u211d (fin n) := y,\n    have hy2 : y \u2208 set.range (\u03bb (m : \u2115), translation (\u03bb (i : fin n), m) (zero (fin n))), from mem_range_iff.m\nend --Needs more than 2000 tokens!\n\n--OUTPUT 10\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume ha : is_open_cover A,\n  assume x : euclidean_space \u211d (fin n),\n  assume hx : x \u2208 \u22c3 (a : set (euclidean_space \u211d (fin n))), a \u2208 A,\n  rcases (mem_Union.mp hx) with \u27e8a, haa, hx_a\u27e9,\n  rcases (mem_Union.mp hx) with \u27e8a, haa, hx_a\u27e9,\n  have ha_a : is_open a, from (mem_Inter.mp haa).left,\n  have ha_a_x : x \u2208 a, from hx_a,\n  have ha_a_bdd : is_bounded a, from (mem_Inter.mp haa).right,\n  have := finset.mem_Union.mp hx,\n  rcases h_1 with \u27e8a, ha, hax\u27e9,\n  have ha' : a \u2208 A, from h haa,\n  have hx' : x \u2208 a, from hx_a,\n  have ha_1 : is_open a, from (mem_Inter.mp ha').left,\n  have ha_2 : is_bounded a, from (mem_Inter.mp ha').right,\n  let B_0 : set (euclidean_space \u211d (fin n)) := \u2205,\n  -- stuff about B_0\n  have hB_0 : \u2200 x : euclidean_space \u211d (fin n), x \u2209 B_0, from by {\n    assume x : euclidean_space \u211d (fin n),\n    assume h_1 : x \u2208 B_0,\n    show false, from h_1,\n  },\n  have hB_0_empty : B_0 = \u2205, from by {\n    apply set.ext,\n    assume x : euclidean_space \u211d (fin n),\n    split,\n    assume h_1 : x \u2208 B_0, show false, from hB_0 x h_1,\n    assume h_1 : x \u2208 \u2205, show x \u2208 B_0, from h_1,\n  },\n  have hB_0_open : is_open B_0, from by obviously,\n  have hB_0_bdd : is_bounded B_0, from by {\n    have h_1 : B_0 = \u2205, from hB_0_empty,\n    show is_bounded \u2205, from by obviously,\n  },\n  have hB_0_t : B_0 \u2208 {U : set (euclidean_space \u211d (fin n)) | is_open U \u2227 is_bounded U}, from by {\n    apply mem_Inter,\n    split,\n    apply hB_0_open,\n    apply hB_0_bdd,\n  },\n  -- stuff about B_1\n  have hB_1_1 : \u2203 k : \u211d, k > 0, from by norm_num,\n  rcases hB_1_1 with \u27e8k, hk\u27e9,\n  have hk_pos : k > 0, from hk,\n  have hB_1 : \u2203 x : \u211d, x > 0 \u2227 k < x, from by norm_num,\n  rcases hB_1 with \u27e8x, hx1, hx2\u27e9,\n  have hx_pos : x > 0, from hx1,\n  have hB_1 : {x : euclidean_space \u211d (fin n) | \u2203 n : \u211d, n > 0 \u2227 dist x 0 < n} = \n    (\u22c3 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}), from by {\n    apply set.ext,\n    assume x : euclidean_space \u211d (fin n),\n    split,\n    assume h : \u2203 (m : \u211d), m > 0 \u2227 dist x 0 < m,\n    have h1 : \u2203 (n : \u211d), n > 0 \u2227 dist x 0 < n \u2227 n > 0, from by {\n      rcases h with \u27e8m, hm1, hm2\u27e9,\n      show \u2203 (n : \u211d), 0 < n \u2227 dist x 0 < n \u2227 0 < n, from \u27e8m, hm1, hm2, hm1\u27e9,\n    },\n    have h2 : \u2203 (n : \u211d), n > 0 \u2227 dist x 0 < n, from by {\n      rcases h1 with \u27e8n, hn1, hn2, hn3\u27e9,\n      show \u2203 (n : \u211d), n > 0 \u2227 dist x 0 < n, from \u27e8n, hn1, hn2\u27e9,\n    },\n    have h3 : \u2203 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}, from by {\n      rcases h2 with \u27e8n, hn1, hn2\u27e9,\n      show \u2203 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}, from \u27e8n, by obviously,\u27e9,\n    },\n    show x \u2208 \u22c3 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}, from by {\n      rcases h3 with \u27e8n, hn1, hn2\u27e9,\n      show x \u2208 \u22c3 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}, from \u27e8n, hn1, by obviously,\u27e9,\n    },\n    assume x : euclidean_space \u211d (fin n),\n    split,\n    assume h : x \u2208 (\u22c3 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}),\n    show \u2203 (m : \u211d), m > 0 \u2227 dist x 0 < m, from by {\n      rcases h with \u27e8n, hn1, hn2\u27e9,\n      have h1 : {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0} = {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m}, from by obviously,\n      show \u2203 (m : \u211d), m > 0 \u2227 dist x 0 < m, from by {\n        rw h1 at hn2,\n        show \u2203 (m : \u211d), m > 0 \u2227 dist x 0 < m, from hn2,\n      },\n    },\n    assume h : \u2203 (m : \u211d), m > 0 \u2227 dist x 0 < m,\n    show x \u2208 \u22c3 (n : \u211d), {x : euclidean_space \u211d (fin n) | \u2203 m : \u211d, m > 0 \u2227 dist x 0 < m} \u2229 {n : \u211d | n > 0}, from by {\n      rcases h with \u27e8m, hm1, h\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.39944052246585926}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n\nimport category_theory.limits.shapes.reflexive\nimport category_theory.limits.preserves.limits\nimport category_theory.monad.limits\nimport category_theory.monad.coequalizer\n\n/-!\n# Monadicity theorems\n\nWe prove monadicity theorems which can establish a given functor is monadic. In particular, we\nshow three versions of Beck's monadicity theorem, and the reflexive (crude) monadicity theorem:\n\n`G` is a monadic right adjoint if it has a right adjoint, and:\n\n* `D` has, `G` preserves and reflects `G`-split coequalizers, see\n  `category_theory.monad.monadic_of_has_preserves_reflects_G_split_coequalizers`\n* `G` creates `G`-split coequalizers, see\n  `category_theory.monad.monadic_of_creates_G_split_coequalizers`\n  (The converse of this is also shown, see\n   `category_theory.monad.creates_G_split_coequalizers_of_monadic`)\n* `D` has and `G` preserves `G`-split coequalizers, and `G` reflects isomorphisms, see\n  `category_theory.monad.monadic_of_has_preserves_G_split_coequalizers_of_reflects_isomorphisms`\n* `D` has and `G` preserves reflexive coequalizers, and `G` reflects isomorphisms, see\n  `category_theory.monad.monadic_of_has_preserves_reflexive_coequalizers_of_reflects_isomorphisms`\n\n## Tags\n\nBeck, monadicity, descent\n\n## TODO\n\nDualise to show comonadicity theorems.\n-/\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnamespace category_theory\nnamespace monad\nopen limits\n\nnoncomputable theory\n-- Hide the implementation details in this namespace.\nnamespace monadicity_internal\n\nsection\n\n-- We use these parameters and notations to simplify the statements of internal constructions\n-- here.\nparameters {C : Type u\u2081} {D : Type u\u2082}\nparameters [category.{v\u2081} C] [category.{v\u2081} D]\nparameters {G : D \u2964 C} [is_right_adjoint G]\n\n-- An unfortunate consequence of the local notation is that it is only recognised if there is an\n-- extra space after the reference.\nlocal notation `F` := left_adjoint G\nlocal notation `adj` := adjunction.of_right_adjoint G\n\n/--\nThe \"main pair\" for an algebra `(A, \u03b1)` is the pair of morphisms `(F \u03b1, \u03b5_FA)`. It is always a\nreflexive pair, and will be used to construct the left adjoint to the comparison functor and show it\nis an equivalence.\n-/\ninstance main_pair_reflexive (A : adj .to_monad.algebra) :\n  is_reflexive_pair (F .map A.a) (adj .counit.app (F .obj A.A)) :=\nbegin\n  apply is_reflexive_pair.mk' (F .map (adj .unit.app _)) _ _,\n  { rw [\u2190 F .map_comp, \u2190 F .map_id],\n    exact congr_arg (\u03bb _, F .map _) A.unit },\n  { rw adj .left_triangle_components,\n    refl },\nend\n\n/--\nThe \"main pair\" for an algebra `(A, \u03b1)` is the pair of morphisms `(F \u03b1, \u03b5_FA)`. It is always a\n`G`-split pair, and will be used to construct the left adjoint to the comparison functor and show it\nis an equivalence.\n-/\ninstance main_pair_G_split (A : adj .to_monad.algebra) :\n  G.is_split_pair (F .map A.a) (adj .counit.app (F .obj A.A)) :=\n{ splittable := \u27e8_, _, \u27e8beck_split_coequalizer A\u27e9\u27e9 }\n\n/-- The object function for the left adjoint to the comparison functor. -/\ndef comparison_left_adjoint_obj\n  (A : adj .to_monad.algebra) [has_coequalizer (F .map A.a) (adj .counit.app _)] : D :=\ncoequalizer (F .map A.a) (adj .counit.app _)\n\n/--\nWe have a bijection of homsets which will be used to construct the left adjoint to the comparison\nfunctor.\n-/\n@[simps]\ndef comparison_left_adjoint_hom_equiv (A : adj .to_monad.algebra) (B : D)\n  [has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))] :\n  (comparison_left_adjoint_obj A \u27f6 B) \u2243 (A \u27f6 (comparison adj).obj B) :=\ncalc (comparison_left_adjoint_obj A \u27f6 B) \u2243 {f : F .obj A.A \u27f6 B // _} :\n        cofork.is_colimit.hom_iso (colimit.is_colimit _) B\n     ... \u2243 {g : A.A \u27f6 G.obj B // G.map (F .map g) \u226b G.map (adj .counit.app B) = A.a \u226b g} :\n      begin\n        refine (adj .hom_equiv _ _).subtype_equiv _,\n        intro f,\n        rw [\u2190 (adj .hom_equiv _ _).injective.eq_iff, adjunction.hom_equiv_naturality_left,\n            adj .hom_equiv_unit, adj .hom_equiv_unit, G.map_comp],\n        dsimp,\n        rw [adj .right_triangle_components_assoc, \u2190 G.map_comp, F .map_comp, category.assoc,\n            adj .counit_naturality, adj .left_triangle_components_assoc],\n        apply eq_comm,\n      end\n     ... \u2243 (A \u27f6 (comparison adj).obj B) :\n     { to_fun := \u03bb g, { f := _, h' := g.prop },\n       inv_fun := \u03bb f, \u27e8f.f, f.h\u27e9,\n       left_inv := \u03bb g, begin ext, refl end,\n       right_inv := \u03bb f, begin ext, refl end }\n\n/--\nConstruct the adjunction to the comparison functor.\n-/\ndef left_adjoint_comparison\n  [\u2200 (A : adj .to_monad.algebra), has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))] :\n  adj .to_monad.algebra \u2964 D :=\nbegin\n  refine @adjunction.left_adjoint_of_equiv _ _ _ _\n              (comparison adj) (\u03bb A, comparison_left_adjoint_obj A) (\u03bb A B, _) _,\n  { apply comparison_left_adjoint_hom_equiv },\n  { intros A B B' g h,\n    ext1,\n    dsimp [comparison_left_adjoint_hom_equiv],\n    rw [\u2190 adj .hom_equiv_naturality_right, category.assoc] },\nend\n\n/--\nProvided we have the appropriate coequalizers, we have an adjunction to the comparison functor.\n-/\n@[simps counit]\ndef comparison_adjunction\n  [\u2200 (A : adj .to_monad.algebra), has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))] :\n  left_adjoint_comparison \u22a3 comparison adj :=\nadjunction.adjunction_of_equiv_left _ _\n\nlemma comparison_adjunction_unit_f_aux\n  [\u2200 (A : adj .to_monad.algebra), has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))]\n  (A : adj .to_monad.algebra) :\n  (comparison_adjunction.unit.app A).f =\n    adj .hom_equiv A.A _ (coequalizer.\u03c0 (F .map A.a) (adj .counit.app (F .obj A.A))) :=\ncongr_arg (adj .hom_equiv _ _) (category.comp_id _)\n\n/--\nThis is a cofork which is helpful for establishing monadicity: the morphism from the Beck\ncoequalizer to this cofork is the unit for the adjunction on the comparison functor.\n-/\n@[simps]\ndef unit_cofork (A : adj .to_monad.algebra)\n  [has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))] :\n  cofork (G.map (F .map A.a)) (G.map (adj .counit.app (F .obj A.A))) :=\ncofork.of_\u03c0 (G.map (coequalizer.\u03c0 (F .map A.a) (adj .counit.app (F .obj A.A))))\nbegin\n  change _ = G.map _ \u226b _,\n  rw [\u2190 G.map_comp, coequalizer.condition, G.map_comp],\nend\n\nlemma comparison_adjunction_unit_f\n  [\u2200 (A : adj .to_monad.algebra), has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))]\n  (A : adj .to_monad.algebra) :\n  (comparison_adjunction.unit.app A).f =\n    (beck_coequalizer A).desc (unit_cofork A) :=\nbegin\n  apply limits.cofork.is_colimit.hom_ext (beck_coequalizer A),\n  rw is_colimit.fac,\n  dsimp only [cofork.\u03c0_eq_app_one, beck_cofork_\u03b9_app, unit_cofork_\u03b9_app],\n  rw [comparison_adjunction_unit_f_aux, \u2190 adj .hom_equiv_naturality_left A.a, coequalizer.condition,\n      adj .hom_equiv_naturality_right, adj .hom_equiv_unit, category.assoc],\n  apply adj .right_triangle_components_assoc,\nend\n\n/--\nThe cofork which describes the counit of the adjunction: the morphism from the coequalizer of\nthis pair to this morphism is the counit.\n-/\n@[simps]\ndef counit_cofork (B : D) :\n  cofork (F .map (G.map (adj .counit.app B))) (adj .counit.app (F .obj (G.obj B))) :=\ncofork.of_\u03c0 (adj .counit.app B) (adj .counit_naturality _)\n\n/-- The unit cofork is a colimit provided `G` preserves it.  -/\ndef unit_colimit_of_preserves_coequalizer\n  (A : adj .to_monad.algebra) [has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))]\n  [preserves_colimit (parallel_pair (F .map A.a) (adj .counit.app (F .obj A.A))) G] :\n  is_colimit (unit_cofork A) :=\nis_colimit_of_has_coequalizer_of_preserves_colimit G _ _\n\n/-- The counit cofork is a colimit provided `G` reflects it. -/\ndef counit_coequalizer_of_reflects_coequalizer (B : D)\n  [reflects_colimit (parallel_pair\n                          (F .map (G.map (adj .counit.app B)))\n                          (adj .counit.app (F .obj (G.obj B)))) G] :\n  is_colimit (counit_cofork B) :=\nis_colimit_of_is_colimit_cofork_map G _ (beck_coequalizer ((comparison adj).obj B))\n\nlemma comparison_adjunction_counit_app\n  [\u2200 (A : adj .to_monad.algebra), has_coequalizer (F .map A.a) (adj .counit.app (F .obj A.A))]\n  (B : D) :\n  comparison_adjunction.counit.app B = colimit.desc _ (counit_cofork B) :=\nbegin\n  apply coequalizer.hom_ext,\n  change coequalizer.\u03c0 _ _ \u226b coequalizer.desc ((adj .hom_equiv _ B).symm (\ud835\udfd9 _)) _ =\n         coequalizer.\u03c0 _ _ \u226b coequalizer.desc _ _,\n  simp,\nend\n\nend\nend monadicity_internal\n\nopen category_theory.adjunction\nopen monadicity_internal\nvariables {C : Type u\u2081} {D : Type u\u2082}\nvariables [category.{v\u2081} C] [category.{v\u2081} D]\nvariables (G : D \u2964 C)\n\n/--\nIf `G` is monadic, it creates colimits of `G`-split pairs. This is the \"boring\" direction of Beck's\nmonadicity theorem, the converse is given in `monadic_of_creates_G_split_coequalizers`.\n-/\ndef creates_G_split_coequalizers_of_monadic [monadic_right_adjoint G] \u2983A B\u2984 (f g : A \u27f6 B)\n  [G.is_split_pair f g] :\n  creates_colimit (parallel_pair f g) G :=\nbegin\n  apply monadic_creates_colimit_of_preserves_colimit _ _,\n  apply_instance,\n  { apply preserves_colimit_of_iso_diagram _ (diagram_iso_parallel_pair _).symm,\n    dsimp,\n    apply_instance },\n  { apply preserves_colimit_of_iso_diagram _ (diagram_iso_parallel_pair _).symm,\n    dsimp,\n    apply_instance }\nend\n\nvariables [is_right_adjoint G]\n\nsection beck_monadicity\n\n/--\nTo show `G` is a monadic right adjoint, we can show it preserves and reflects `G`-split\ncoequalizers, and `C` has them.\n-/\ndef monadic_of_has_preserves_reflects_G_split_coequalizers\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], has_coequalizer f g]\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], preserves_colimit (parallel_pair f g) G]\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], reflects_colimit (parallel_pair f g) G] :\n  monadic_right_adjoint G :=\nbegin\n  let L : (adjunction.of_right_adjoint G).to_monad.algebra \u2964 D := left_adjoint_comparison,\n  letI i : is_right_adjoint (comparison (of_right_adjoint G)) :=\n    \u27e8_, comparison_adjunction\u27e9,\n  constructor,\n  let : \u03a0 (X : (of_right_adjoint G).to_monad.algebra),\n    is_iso ((of_right_adjoint (comparison (of_right_adjoint G))).unit.app X),\n  { intro X,\n    apply is_iso_of_reflects_iso _ (monad.forget (of_right_adjoint G).to_monad),\n    { change is_iso (comparison_adjunction.unit.app X).f,\n      rw comparison_adjunction_unit_f,\n      change\n        is_iso\n          (is_colimit.cocone_point_unique_up_to_iso\n            (beck_coequalizer X)\n            (unit_colimit_of_preserves_coequalizer X)).hom,\n      refine is_iso.of_iso (is_colimit.cocone_point_unique_up_to_iso _ _) } },\n  let : \u03a0 (Y : D),\n    is_iso ((of_right_adjoint (comparison (of_right_adjoint G))).counit.app Y),\n  { intro Y,\n    change is_iso (comparison_adjunction.counit.app Y),\n    rw comparison_adjunction_counit_app,\n    change is_iso (is_colimit.cocone_point_unique_up_to_iso _ _).hom,\n    apply_instance,\n    apply counit_coequalizer_of_reflects_coequalizer _,\n    letI : G.is_split_pair\n            ((left_adjoint G).map (G.map ((adjunction.of_right_adjoint G).counit.app Y)))\n            ((adjunction.of_right_adjoint G).counit.app ((left_adjoint G).obj (G.obj Y))) :=\n      monadicity_internal.main_pair_G_split ((comparison (adjunction.of_right_adjoint G)).obj Y),\n    apply_instance },\n  exactI adjunction.is_right_adjoint_to_is_equivalence,\nend\n\n/--\nBeck's monadicity theorem. If `G` has a right adjoint and creates coequalizers of `G`-split pairs,\nthen it is monadic.\nThis is the converse of `creates_G_split_of_monadic`.\n-/\ndef monadic_of_creates_G_split_coequalizers\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], creates_colimit (parallel_pair f g) G] :\n  monadic_right_adjoint G :=\nbegin\n  letI : \u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], has_colimit (parallel_pair f g \u22d9 G),\n  { introsI A B f g i,\n    apply has_colimit_of_iso (diagram_iso_parallel_pair _),\n    change has_coequalizer (G.map f) (G.map g),\n    apply_instance },\n  apply monadic_of_has_preserves_reflects_G_split_coequalizers _,\n  { apply_instance },\n  { introsI A B f g i,\n    apply has_colimit_of_created (parallel_pair f g) G },\n  { introsI A B f g i,\n    apply_instance },\n  { introsI A B f g i,\n    apply_instance }\nend\n\n/--\nAn alternate version of Beck's monadicity theorem. If `G` reflects isomorphisms, preserves\ncoequalizers of `G`-split pairs and `C` has coequalizers of `G`-split pairs, then it is monadic.\n-/\ndef monadic_of_has_preserves_G_split_coequalizers_of_reflects_isomorphisms\n  [reflects_isomorphisms G]\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], has_coequalizer f g]\n  [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [G.is_split_pair f g], preserves_colimit (parallel_pair f g) G] :\n  monadic_right_adjoint G :=\nbegin\n  apply monadic_of_has_preserves_reflects_G_split_coequalizers _,\n  { apply_instance },\n  { assumption },\n  { assumption },\n  { introsI A B f g i,\n    apply reflects_colimit_of_reflects_isomorphisms },\nend\n\nend beck_monadicity\n\nsection reflexive_monadicity\n\nvariables [has_reflexive_coequalizers D] [reflects_isomorphisms G]\nvariables [\u2200 \u2983A B\u2984 (f g : A \u27f6 B) [is_reflexive_pair f g], preserves_colimit (parallel_pair f g) G]\n\n/--\nReflexive (crude) monadicity theorem. If `G` has a right adjoint, `D` has and `G` preserves\nreflexive coequalizers and `G` reflects isomorphisms, then `G` is monadic.\n-/\ndef monadic_of_has_preserves_reflexive_coequalizers_of_reflects_isomorphisms :\n  monadic_right_adjoint G :=\nbegin\n  let L : (adjunction.of_right_adjoint G).to_monad.algebra \u2964 D := left_adjoint_comparison,\n  letI i : is_right_adjoint (comparison (adjunction.of_right_adjoint G)) :=\n    \u27e8_, comparison_adjunction\u27e9,\n  constructor,\n  let : \u03a0 (X : (adjunction.of_right_adjoint G).to_monad.algebra),\n    is_iso ((adjunction.of_right_adjoint (comparison (adjunction.of_right_adjoint G))).unit.app X),\n  { intro X,\n    apply is_iso_of_reflects_iso _ (monad.forget (adjunction.of_right_adjoint G).to_monad),\n    { change is_iso (comparison_adjunction.unit.app X).f,\n      rw comparison_adjunction_unit_f,\n      change\n        is_iso\n          (is_colimit.cocone_point_unique_up_to_iso\n            (beck_coequalizer X)\n            (unit_colimit_of_preserves_coequalizer X)).hom,\n      apply is_iso.of_iso (is_colimit.cocone_point_unique_up_to_iso _ _) } },\n  let : \u03a0 (Y : D),\n    is_iso ((of_right_adjoint (comparison (adjunction.of_right_adjoint G))).counit.app Y),\n  { intro Y,\n    change is_iso (comparison_adjunction.counit.app Y),\n    rw comparison_adjunction_counit_app,\n    change is_iso (is_colimit.cocone_point_unique_up_to_iso _ _).hom,\n    apply_instance,\n    apply counit_coequalizer_of_reflects_coequalizer _,\n    apply reflects_colimit_of_reflects_isomorphisms },\n  exactI adjunction.is_right_adjoint_to_is_equivalence,\nend\n\nend reflexive_monadicity\n\nend monad\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monad/monadicity.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.39937467983346064}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.natural_transformation\n\nnamespace category_theory\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nopen nat_trans category category_theory.functor\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2082) [category.{v\u2082} D]\n\nlocal attribute [simp] vcomp_app\n/--\n`functor.category C D` gives the category structure on functors and natural transformations\nbetween categories `C` and `D`.\n\nNotice that if `C` and `D` are both small categories at the same universe level,\nthis is another small category at that level.\nHowever if `C` and `D` are both large categories at the same universe level,\nthis is a small category at the next higher level.\n-/\ninstance functor.category : category.{(max u\u2081 v\u2082)} (C \u2964 D) :=\n{ hom     := \u03bb F G, nat_trans F G,\n  id      := \u03bb F, nat_trans.id F,\n  comp    := \u03bb _ _ _ \u03b1 \u03b2, vcomp \u03b1 \u03b2 }\n\nvariables {C D} {E : Type u\u2083} [category.{v\u2083} E]\nvariables {F G H I : C \u2964 D}\n\nnamespace nat_trans\n\n@[simp] lemma vcomp_eq_comp (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) : vcomp \u03b1 \u03b2 = \u03b1 \u226b \u03b2 := rfl\n\nlemma vcomp_app' (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) (X : C) :\n  (\u03b1 \u226b \u03b2).app X = (\u03b1.app X) \u226b (\u03b2.app X) := rfl\n\nlemma congr_app {\u03b1 \u03b2 : F \u27f6 G} (h : \u03b1 = \u03b2) (X : C) : \u03b1.app X = \u03b2.app X := by rw h\n@[simp] lemma id_app (F : C \u2964 D) (X : C) : (\ud835\udfd9 F : F \u27f6 F).app X = \ud835\udfd9 (F.obj X) := rfl\n@[simp] lemma comp_app {F G H : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) (X : C) :\n  (\u03b1 \u226b \u03b2).app X = \u03b1.app X \u226b \u03b2.app X := rfl\n\nlemma app_naturality {F G : C \u2964 (D \u2964 E)} (T : F \u27f6 G) (X : C) {Y Z : D} (f : Y \u27f6 Z) :\n  ((F.obj X).map f) \u226b ((T.app X).app Z) = ((T.app X).app Y) \u226b ((G.obj X).map f) :=\n(T.app X).naturality f\n\nlemma naturality_app {F G : C \u2964 (D \u2964 E)} (T : F \u27f6 G) (Z : D) {X Y : C} (f : X \u27f6 Y) :\n  ((F.map f).app Z) \u226b ((T.app Y).app Z) = ((T.app X).app Z) \u226b ((G.map f).app Z) :=\ncongr_fun (congr_arg app (T.naturality f)) Z\n\n/-- A natural transformation is a monomorphism if each component is. -/\nlemma mono_app_of_mono (\u03b1 : F \u27f6 G) [\u2200 (X : C), mono (\u03b1.app X)] : mono \u03b1 :=\n\u27e8\u03bb H g h eq, by { ext X, rw [\u2190cancel_mono (\u03b1.app X), \u2190comp_app, eq, comp_app] }\u27e9\n\n/-- A natural transformation is an epimorphism if each component is. -/\nlemma epi_app_of_epi (\u03b1 : F \u27f6 G) [\u2200 (X : C), epi (\u03b1.app X)] : epi \u03b1 :=\n\u27e8\u03bb H g h eq, by { ext X, rw [\u2190cancel_epi (\u03b1.app X), \u2190comp_app, eq, comp_app] }\u27e9\n\n/-- `hcomp \u03b1 \u03b2` is the horizontal composition of natural transformations. -/\ndef hcomp {H I : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) : (F \u22d9 H) \u27f6 (G \u22d9 I) :=\n{ app         := \u03bb X : C, (\u03b2.app (F.obj X)) \u226b (I.map (\u03b1.app X)),\n  naturality' := \u03bb X Y f,\n  begin\n    rw [functor.comp_map, functor.comp_map, \u2190assoc, naturality, assoc,\n        \u2190map_comp I, naturality, map_comp, assoc]\n  end }\n\ninfix ` \u25eb `:80 := hcomp\n\n@[simp] lemma hcomp_app {H I : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (X : C) :\n  (\u03b1 \u25eb \u03b2).app X = (\u03b2.app (F.obj X)) \u226b (I.map (\u03b1.app X)) := rfl\n\n@[simp] lemma hcomp_id_app {H : D \u2964 E} (\u03b1 : F \u27f6 G) (X : C) : (\u03b1 \u25eb \ud835\udfd9 H).app X = H.map (\u03b1.app X) :=\n  by {dsimp, simp} -- See note [dsimp, simp].\n\nlemma id_hcomp_app {H : E \u2964 C} (\u03b1 : F \u27f6 G) (X : E) : (\ud835\udfd9 H \u25eb \u03b1).app X = \u03b1.app _ := by simp\n\n-- Note that we don't yet prove a `hcomp_assoc` lemma here: even stating it is painful, because we\n-- need to use associativity of functor composition. (It's true without the explicit associator,\n-- because functor composition is definitionally associative,\n-- but relying on the definitional equality causes bad problems with elaboration later.)\n\nlemma exchange {I J K : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H)\n  (\u03b3 : I \u27f6 J) (\u03b4 : J \u27f6 K) : (\u03b1 \u226b \u03b2) \u25eb (\u03b3 \u226b \u03b4) = (\u03b1 \u25eb \u03b3) \u226b (\u03b2 \u25eb \u03b4) :=\nby ext; simp\n\nend nat_trans\nopen nat_trans\nnamespace functor\n\n/-- Flip the arguments of a bifunctor. See also `currying.lean`. -/\nprotected def flip (F : C \u2964 (D \u2964 E)) : D \u2964 (C \u2964 E) :=\n{ obj := \u03bb k,\n  { obj := \u03bb j, (F.obj j).obj k,\n    map := \u03bb j j' f, (F.map f).app k,\n    map_id' := \u03bb X, begin rw category_theory.functor.map_id, refl end,\n    map_comp' := \u03bb X Y Z f g, by rw [map_comp, \u2190comp_app] },\n  map := \u03bb c c' f,\n  { app := \u03bb j, (F.obj j).map f } }.\n\n@[simp] lemma flip_obj_obj (F : C \u2964 (D \u2964 E)) (c) (d) : (F.flip.obj d).obj c = (F.obj c).obj d := rfl\n@[simp] lemma flip_obj_map (F : C \u2964 (D \u2964 E)) {c c' : C} (f : c \u27f6 c') (d : D) :\n  (F.flip.obj d).map f = (F.map f).app d := rfl\n@[simp] lemma flip_map_app (F : C \u2964 (D \u2964 E)) {d d' : D} (f : d \u27f6 d') (c : C) :\n  (F.flip.map f).app c = (F.obj c).map f := rfl\n\nend functor\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/functor_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.39937467983346064}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.measurable_space\nimport formal_ml.probability_space\nimport formal_ml.real_random_variable\nimport data.complex.exponential\nimport formal_ml.ennreal\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.exp_bound\nimport formal_ml.classical\n\n\n\n\n\n\nstructure PAC_problem :=\n(\u03a9:Type*)                             -- Underlying outcome type\n(p:probability_space \u03a9)               -- underlying probability space\n(\u03b2:Type*)                             -- instance type\n(M\u03b2:measurable_space \u03b2)               -- Measurable space for the instances\n(\u03b3:Type*)                             -- label type\n(M\u03b3:measurable_space \u03b3)               -- Measurable space for the labels\n(HME\u03b3:has_measurable_equality M\u03b3)     -- Measurable equality for the labels\n(E\u03b3:encodable \u03b3)                      -- Encodable labels is very useful\n(Di:Type*)                            -- example index type\n(FDi:fintype Di)                      -- number of examples are finite\n(EDi:encodable Di)                    -- example index is encodable\n                                      -- see trunc_encodable_of_fintype\n(Hi:Type*)                            -- hypothesis index type\n(FHi:fintype Hi)                      -- number of examples are finite\n(EHi:encodable Hi)                    -- hypothesis index is encodable\n                                      -- see trunc_encodable_of_fintype\n(H:Hi \u2192 (M\u03b2 \u2192\u2098 M\u03b3))                   -- hypothesis space\n(D:Di \u2192 (p \u2192\u1d63 (M\u03b2 \u00d7\u2098 M\u03b3)))            -- example distribution\n(IID:random_variables_IID D)          -- examples are IID\n(has_example:inhabited Di)            -- there exists an example\n\n\n\n\n\n\n--example_instance P j is the jth instance (the features of an example)\ndef example_instance (P:PAC_problem)\n  (j:P.Di):random_variable P.p P.M\u03b2 :=\n  (mf_fst) \u2218r (P.D j)\n\n--the measurable space on the examples.\ndef PAC_problem.M\u03b2\u03b3 (P:PAC_problem):\n    measurable_space (P.\u03b2 \u00d7 P.\u03b3) := P.M\u03b2 \u00d7\u2098 P.M\u03b3\n\n/-\n  rv_label_eq X Y is the event that X and Y are equal, where X and Y are labels.\n-/\ndef rv_label_eq (P:PAC_problem)\n  (X Y:random_variable P.p P.M\u03b3):event  P.p :=\n  @random_variable_eq P.\u03a9 P.p P.\u03b3 P.M\u03b3 P.HME\u03b3  X Y\n\n/-\n  rv_label_ne X Y is the event that X and Y are not equal, where X and Y are labels.\n-/\ndef rv_label_ne (P:PAC_problem)\n  (X Y:random_variable P.p P.M\u03b3):event  P.p :=\n  @random_variable_ne P.\u03a9 P.p P.\u03b3 P.M\u03b3 P.HME\u03b3  X Y\n\n\n/-\n  example_label P j is the label of the jth example.\n-/\ndef example_label (P:PAC_problem)\n  (j:P.Di):P.p \u2192\u1d63 P.M\u03b3 :=\n  mf_snd \u2218r (P.D j)\n\n\n/-\n  example_classification P i j is the classification by the ith hypothesis of the jth example.\n-/\ndef example_classification (P:PAC_problem)\n  (i:P.Hi) (j:P.Di):P.p \u2192\u1d63 P.M\u03b3 :=\n  (P.H i) \u2218r (example_instance P j)\n\n/-\n  example_correct P i j is whether the ith hypothesis is correct on the jth example.\n-/\ndef example_correct (P:PAC_problem)\n  (i:P.Hi) (j:P.Di):event P.p :=\n  rv_label_eq P (example_classification P i j) (example_label P j)\n\n/-\n  example_error P i j is whether the ith hypothesis made a mistake on the jth example.\n-/\ndef example_error (P:PAC_problem)\n  (i:P.Hi) (j:P.Di):event P.p :=\n  rv_label_ne P (example_classification P i j) (example_label P j)\n\n\n/-\n  num_examples P is the number of examples in the problem.\n  This is defined as the cardinality of the index type of the examples.\n-/\ndef num_examples  (P:PAC_problem):nat\n   := @fintype.card P.Di P.FDi\n\n\n/-\n  The number of examples is the number of elements of type P.Di.\n  P.FDi.elems is the set of all elements in P.Di, and P.FDi.elems.card is the cardinality of\n  P.FDi.elems.\n-/\nlemma num_examples_eq_finset_card (P:PAC_problem):\n  num_examples P = P.FDi.elems.card :=\nbegin\n  refl,\nend\n\n/-\n  The number of examples do not equal zero.\n -/\nlemma num_examples_ne_zero (P:PAC_problem):\n  num_examples P \u2260 0 :=\nbegin\n  unfold num_examples,\n  apply @card_ne_zero_of_inhabited P.Di P.has_example P.FDi,\nend\n\n\n/-\n  The number of hypotheses.\n -/\ndef num_hypotheses (P:PAC_problem):nat\n   := @fintype.card P.Hi P.FHi\n\n/-\n  The number of errors on the training set, divided by the size of the training set.\n  training_error P i = (\u2211 (j:P.Di), (example_error P i j))/(num_exmaples P)\n  TODO: replace with average_indicator.\n-/\nnoncomputable def training_error (P:PAC_problem)\n  (i:P.Hi):P.p \u2192\u1d63 (borel nnreal) :=\n  average_identifier (example_error P i) P.FDi \n -- (count_finset_rv P.FDi.elems (example_error P i)) * (to_nnreal_rv ((num_examples P):nnreal)\u207b\u00b9)\n\n/-\n  The expected test error.\n  The test error is equal to the expected training error. Because we have not defined a generating\n  process for examples, we use this as the definition.\n-/\nnoncomputable def test_error (P:PAC_problem)\n    (i:P.Hi):ennreal := E[training_error P i]\n\n/-\n  fake_hypothesis P \u03b5 i is the event that hypothesis i has zero training error, but has\n  test error > \u03b5.\n-/\nnoncomputable def fake_hypothesis (P:PAC_problem) (\u03b5:nnreal)\n  (i:P.Hi):event P.p :=\n  ((training_error P i) =\u1d63 0) \u2227 (event_const (test_error P i > \u03b5))\n\n/-\n  The event that all hypotheses with training error zero have test error \u2264 \u03b5.\n-/\nnoncomputable def approximately_correct_event (P:PAC_problem)\n  (\u03b5:nnreal):event P.p :=\n  enot (eany_fintype P.FHi (fake_hypothesis P \u03b5))\n\ndef probably_approximately_correct (P:PAC_problem)\n (\u03b5:nnreal) (\u03b4:nnreal):Prop :=\n  1 - \u03b4 \u2264 Pr[approximately_correct_event P \u03b5]\n\n\nlemma enot_example_correct_eq_example_error\n  (P:PAC_problem) (i:P.Hi) (j:P.Di):enot (example_correct P i j) = (example_error P i j) :=\nbegin\n  apply event.eq,\n  unfold example_error example_correct rv_label_ne rv_label_eq,\n  refl,\nend\n\n\nlemma enot_example_error_eq_example_correct\n  (P:PAC_problem) (i:P.Hi) (j:P.Di):enot (example_error P i j) = (example_correct P i j) :=\nbegin\n  rw \u2190 enot_example_correct_eq_example_error,\n  simp,\nend\n\n\n\nlemma example_correct_iff_not_example_error\n  (P:PAC_problem) (i:P.Hi) (j:P.Di) (\u03c9:P.\u03a9): \u03c9 \u2209 (example_error P i j).val \u2194\n  \u03c9 \u2208 (example_correct P i j).val :=\nbegin\n  rw \u2190 enot_example_error_eq_example_correct,\n  simp,\nend\n\nlemma example_error_IID (P:PAC_problem) (i:P.Hi):\n  @events_IID P.\u03a9 P.Di P.p  (example_error P i) :=\nbegin\n  /-\n    To prove that the errors of a particular hypothesis are IID, we must use an alternate\n    formulation of the example_error events. Specifically, instead of constructing a hierarchy\n    of random variables, we must make a leap from the established IID random variable\n    (the data), construct another IID random variable (the product of\n    the classification and the label), and show that the set of all label/classification pairs\n    that aren't equal are a measurable set (because has_measurable_eq M\u03b3).\n\n    The indexed set of events of each IID random variable being in a measurable set is IID,\n    so the result holds.\n\n    Note that while this proof looks a little long, most of the proof is just unwrapping\n    the traditional and internal definitions of example error, and then using simp to show that\n    they are equal on all outcomes.\n  -/\n  let Y:(P.M\u03b2 \u00d7\u2098 P.M\u03b3)\u2192\u2098 (P.M\u03b3 \u00d7\u2098 P.M\u03b3) := prod_measurable_fun ((P.H i) \u2218m (mf_fst)) (mf_snd),\n  begin\n  let S:@measurable_setB _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) := @measurable_setB_ne P.\u03b3 P.M\u03b3 P.HME\u03b3,\n  begin\n  have A1:@random_variables_IID P.\u03a9 P.p P.Di (P.\u03b3 \u00d7 P.\u03b3) (P.M\u03b3 \u00d7\u2098 P.M\u03b3)\n  (\u03bb j:P.Di, Y \u2218r (P.D j) ),\n  {\n    apply compose_IID,\n    apply P.IID,\n  },\n  have A2:@events_IID  P.\u03a9 P.Di P.p (\u03bb j:P.Di, @rv_event P.\u03a9 P.p _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) (Y \u2218r (P.D j)) S),\n  {\n    apply rv_event_IID,\n    apply A1,\n  },\n  have A3: (\u03bb j:P.Di, @rv_event P.\u03a9 P.p _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) (Y \u2218r (P.D j)) S) = example_error P i,\n  {\n    apply funext,\n    intro j,\n    apply event.eq,\n    unfold example_error example_label example_classification rv_label_ne example_instance,\n    refl,\n  },\n  rw \u2190 A3,\n  exact A2,\n  end\n  end\nend\n\nlemma example_correct_IID (P:PAC_problem) (i:P.Hi):\n  @events_IID P.\u03a9 P.Di P.p  (example_correct P i) :=\nbegin\n  /-\n    Similar to example_error_IID. Theoretically, we could prove it from example_error_IID.\n    However, it is easier for now to prove it from first principles.\n  -/\n  let Y:(P.M\u03b2 \u00d7\u2098 P.M\u03b3)\u2192\u2098 (P.M\u03b3 \u00d7\u2098 P.M\u03b3) := prod_measurable_fun ((P.H i) \u2218m (mf_fst)) (mf_snd),\n  begin\n  let S:@measurable_setB _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) := {\n    val := {x:P.\u03b3 \u00d7 P.\u03b3|x.fst = x.snd},\n    property := P.HME\u03b3.measurable_set_eq,\n  },\n  begin\n  have A1:@random_variables_IID P.\u03a9 P.p P.Di (P.\u03b3 \u00d7 P.\u03b3) (P.M\u03b3 \u00d7\u2098 P.M\u03b3)\n  (\u03bb j:P.Di, Y \u2218r (P.D j) ),\n  {\n    apply compose_IID,\n    apply P.IID,\n  },\n  have A2:@events_IID  P.\u03a9 P.Di P.p (\u03bb j:P.Di, @rv_event P.\u03a9 P.p _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) (Y \u2218r (P.D j)) S),\n  {\n    apply rv_event_IID,\n    apply A1,\n  },\n  have A3: (\u03bb j:P.Di, @rv_event P.\u03a9 P.p _ (P.M\u03b3 \u00d7\u2098 P.M\u03b3) (Y \u2218r (P.D j)) S) = example_correct P i,\n  {\n    apply funext,\n    intro j,\n    apply event.eq,\n    unfold example_correct example_label example_classification rv_label_eq example_instance,\n    refl,\n  },\n  rw \u2190 A3,\n  exact A2,\n  end\n  end\nend\n\nlemma example_error_identical (P:PAC_problem) (i:P.Hi) (j j':P.Di):\n  Pr[example_error P i j] = Pr[example_error P i j'] :=\nbegin\n  have A1:@events_IID P.\u03a9 P.Di P.p  (example_error P i),\n  {\n    apply example_error_IID,\n  },\n  unfold events_IID at A1,\n  cases A1 with A2 A3,\n  apply A3,\nend\n\n--set_option pp.all true\n--set_option pp.coercions true\n\n\n\nlemma test_error_training_mistake (P:PAC_problem) (i:P.Hi) (j:P.Di):\n  (Pr[example_error P i j]:ennreal) = (test_error P i) :=\nbegin\n  have A1:E[(count_finset_rv P.FDi.elems (example_error P i)) *\n  (to_nnreal_rv ((num_examples P):nnreal)\u207b\u00b9)]=(test_error P i),\n  {\n    unfold test_error,\n    refl,\n  },\n\n  have A2:(test_error P i)=E[(count_finset_rv P.FDi.elems (example_error P i))]\n  * ((num_examples P):ennreal)\u207b\u00b9,\n  {\n    rw \u2190 A1,\n    rw scalar_expected_value,\n    rw ennreal.coe_inv,\n    have A2A:(((num_examples P):nnreal)\u207b\u00b9:ennreal)=((num_examples P):ennreal)\u207b\u00b9,\n    {\n      simp,\n    },\n    rw A2A,\n    simp,\n    apply num_examples_ne_zero,\n  },\n\n  have A3:E[(count_finset_rv P.FDi.elems (example_error P i))]=\n           P.FDi.elems.sum (\u03bb k, Pr[(example_error P i k)]),\n  {\n    apply linear_count_finset_rv,\n  },\n  have A4:\u2200 k, (\u03bb k, (Pr[(example_error P i k)]:ennreal)) k = (Pr[(example_error P i j)]:ennreal),\n  {\n    intro k,\n    simp,\n    apply (example_error_identical P i _ j),\n  },\n  have A5:E[(count_finset_rv P.FDi.elems (example_error P i))]=\n           P.FDi.elems.card * (Pr[(example_error P i j)]:ennreal),\n  {\n    rw A3,\n    apply finset_sum_const,\n    apply A4,\n  },\n  have A6:E[(count_finset_rv P.FDi.elems (example_error P i))]=\n           (num_examples P) * (Pr[(example_error P i j)]:ennreal),\n  {\n    rw A5,\n    rw num_examples_eq_finset_card,\n  },\n  rw A6 at A2,\n  rw mul_comm at A2,\n  rw \u2190 mul_assoc at A2,\n  have A7:((num_examples P):ennreal)\u207b\u00b9 * ((num_examples P):ennreal) = 1,\n  {\n    rw mul_comm,\n    apply ennreal.mul_inv_cancel,\n    {\n      simp,\n      apply (@num_examples_ne_zero P),\n    },\n    {\n      simp,\n    }\n  },\n  rw A7 at A2,\n  simp at A2,\n  symmetry,\n  exact A2,\nend\n\nlemma test_error_training_mistake2 (P:PAC_problem) (i:P.Hi) (j:P.Di):\n  Pr[example_error P i j] = (test_error P i).to_nnreal :=\nbegin\n  symmetry,\n  apply ennreal_coe_eq_lift,\n  rw test_error_training_mistake,\nend\n\nlemma example_correct_prob (P:PAC_problem) (i:P.Hi) (j:P.Di):\n  Pr[example_correct P i j] = 1 - (test_error P i).to_nnreal :=\nbegin\n  rw \u2190 enot_example_error_eq_example_correct,\n  rw \u2190 Pr_one_minus_eq_not,\n  rw test_error_training_mistake2,\nend\n\nlemma test_error_ne_top (P:PAC_problem) (i:P.Hi):\n  (test_error P i) \u2260 \u22a4 :=\nbegin\n  rw \u2190 test_error_training_mistake,\n  simp,\n  apply P.has_example.default,\nend\n\n/-\nevent_IID_pow :\n  \u2200 {\u03b1 : Type u_1} [M\u03b1 : measurable_space \u03b1] {p : probability_measure \u03b1} {\u03b2 : Type u_2} [F : fintype \u03b2]\n  [I : inhabited \u03b2] {\u03b3 : Type u_3} [M\u03b3 : measurable_space \u03b3] (A : \u03b2 \u2192 event p) (S : finset \u03b2),\n    events_IID A \u2192 Pr[eall_finset S A] = Pr[A (inhabited.default \u03b2)] ^ finset.card S\n-/\n--sorry\nset_option pp.implicit true\nlemma training_error_zero_prob (P:PAC_problem) (i:P.Hi):\n  Pr[training_error P i =\u1d63 0] =\n   (Pr[(example_correct P i P.has_example.default)])^(num_examples P) :=\nbegin\n  unfold training_error,\n  rw @Pr_average_identifier_eq_zero P.Di P.\u03a9 P.p (example_error P i) P.FDi P.has_example.default,\n  rw \u2190 enot_example_correct_eq_example_error,\n  rw Pr_one_minus_not_eq,\n  rw num_examples_eq_finset_card,\n  unfold fintype.card,\n  refl,\n  apply example_error_IID,\nend\n\n\n\nlemma fake_hypothesis_prob (P:PAC_problem)\n  (\u03b5:nnreal) (i:P.Hi):Pr[fake_hypothesis P \u03b5 i]\u2264(1-\u03b5)^(num_examples P) :=\nbegin\n  unfold fake_hypothesis,\n  have A1:decidable (test_error P i \u2264 \u2191\u03b5),\n  {\n    apply linear_order.decidable_le,\n  },\n  cases A1,\n  {\n    apply le_trans,\n    apply Pr_eand_le_left,\n    --Note: this could be <.\n    have B1:\u2191\u03b5 \u2264 test_error P i,\n    {\n      apply le_of_not_le A1,\n    },\n    have B2:\u03b5 \u2264 (test_error P i).to_nnreal,\n    {\n      apply ennreal_le_to_nnreal_of_ennreal_le_of_ne_top,\n      apply test_error_ne_top,\n      exact B1,\n    },\n    rw training_error_zero_prob,\n    rw example_correct_prob,\n    apply nnreal_pow_mono,\n    apply nnreal_sub_le_sub_of_le,\n    --\u03b5 \u2264 (test_error P i).to_nnreal\n    exact B2,\n  },\n  {\n    apply le_trans,\n    apply Pr_eand_le_right,\n    rw Pr_event_const_false,\n    {\n      simp,\n    },\n    {\n      rw \u2190 le_iff_not_gt,\n      exact A1,\n    },\n  },\nend\n\n\nlemma fake_hypothesis_prob2 (P:PAC_problem)\n  (\u03b5:nnreal) (i:P.Hi):\n   Pr[fake_hypothesis P \u03b5 i] \u2264 nnreal.exp (- \u03b5 * (num_examples P)) :=\nbegin\n  apply le_trans,\n  apply fake_hypothesis_prob,\n  apply nnreal_exp_bound2,\nend\n\nlemma eany_fake_hypothesis_prob (P:PAC_problem)\n  (\u03b5:nnreal):\n   Pr[ eany_fintype P.FHi (fake_hypothesis P \u03b5)] \u2264 (num_hypotheses P) * nnreal.exp (- \u03b5 * (num_examples P)) :=\nbegin\n  apply eany_fintype_bound2,\n  intro,\n  apply fake_hypothesis_prob2,\nend\n\n\nlemma pac_bound (P:PAC_problem)\n  (\u03b5:nnreal):\n  (1:nnreal) - (num_hypotheses P) * nnreal.exp (-(\u03b5:real) * (num_examples P:real)) \u2264\n  Pr[approximately_correct_event P \u03b5]  :=\nbegin\n  have A1:Pr[approximately_correct_event P \u03b5] = 1 - Pr[eany_fintype P.FHi (fake_hypothesis P \u03b5)],\n  {\n    symmetry,\n    unfold approximately_correct_event,\n    apply Pr_one_minus_eq_not (eany_fintype P.FHi (fake_hypothesis P \u03b5)),\n  },\n  rw A1,\n  apply nnreal_sub_le_left,\n  have A2:Pr[ eany_fintype P.FHi (fake_hypothesis P \u03b5)]\n      \u2264 (num_hypotheses P) * nnreal.exp (- \u03b5 * (num_examples P)),\n  {\n    apply eany_fake_hypothesis_prob,\n  },\n  apply A2,\nend\n\nlemma pac_bound2 (P:PAC_problem) (\u03b5:nnreal):\n  probably_approximately_correct P \u03b5\n  ((num_hypotheses P) * nnreal.exp (-\u03b5 * (num_examples P))) :=\nbegin\n  unfold probably_approximately_correct,\n  apply pac_bound,\nend\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/pac_bounds.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.682573734412324, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3993746798334606}}
{"text": "/-\nCopyright (c) 2018 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Kevin Buzzard, Scott Morrison, Johan Commelin, Chris Hughes,\n  Johannes H\u00f6lzl, Yury Kudryashov\n-/\nimport algebra.hom.group\nimport algebra.group.commute\n\n/-!\n# Multiplicative homomorphisms respect semiconjugation and commutation.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nsection commute\n\nvariables {F M N : Type*} [has_mul M] [has_mul N] {a x y : M}\n\n@[simp, to_additive]\nprotected lemma semiconj_by.map [mul_hom_class F M N] (h : semiconj_by a x y) (f : F) :\n  semiconj_by (f a) (f x) (f y) :=\nby simpa only [semiconj_by, map_mul] using congr_arg f h\n\n@[simp, to_additive]\nprotected lemma commute.map [mul_hom_class F M N] (h : commute x y) (f : F) :\n  commute (f x) (f y) :=\nh.map f\n\nend commute\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/hom/commute.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3993746722772597}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.algebra.tower\nimport Mathlib.algebra.invertible\nimport Mathlib.linear_algebra.basis\nimport Mathlib.ring_theory.adjoin\nimport Mathlib.PostPort\n\nuniverses u v w u\u2081 u_1 u_2 u_3 u_4 v\u2081 w\u2081 \n\nnamespace Mathlib\n\n/-!\n# Towers of algebras\n\nWe set up the basic theory of algebra towers.\nAn algebra tower A/S/R is expressed by having instances of `algebra A S`,\n`algebra R S`, `algebra R A` and `is_scalar_tower R S A`, the later asserting the\ncompatibility condition `(r \u2022 s) \u2022 a = r \u2022 (s \u2022 a)`.\n\nIn `field_theory/tower.lean` we use this to prove the tower law for finite extensions,\nthat if `R` and `S` are both fields, then `[A:R] = [A:S] [S:A]`.\n\nIn this file we prepare the main lemma:\nif `{bi | i \u2208 I}` is an `R`-basis of `S` and `{cj | j \u2208 J}` is a `S`-basis\nof `A`, then `{bi cj | i \u2208 I, j \u2208 J}` is an `R`-basis of `A`. This statement does not require the\nbase rings to be a field, so we also generalize the lemma to rings in this file.\n-/\n\nnamespace is_scalar_tower\n\n\nprotected instance polynomial (R : Type u) {S : Type v} {A : Type w} [comm_semiring R]\n    [comm_semiring S] [semiring A] [algebra R S] [algebra S A] [algebra R A]\n    [is_scalar_tower R S A] : is_scalar_tower R S (polynomial A) :=\n  of_algebra_map_eq fun (x : R) => congr_arg (\u21d1polynomial.C) (algebra_map_apply R S A x)\n\ntheorem aeval_apply (R : Type u) (S : Type v) (A : Type w) [comm_semiring R] [comm_semiring S]\n    [semiring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (x : A)\n    (p : polynomial R) :\n    coe_fn (polynomial.aeval x) p =\n        coe_fn (polynomial.aeval x) (polynomial.map (algebra_map R S) p) :=\n  sorry\n\n/-- Suppose that `R -> S -> A` is a tower of algebras.\nIf an element `r : R` is invertible in `S`, then it is invertible in `A`. -/\ndef invertible.algebra_tower (R : Type u) (S : Type v) (A : Type w) [comm_semiring R]\n    [comm_semiring S] [semiring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]\n    (r : R) [invertible (coe_fn (algebra_map R S) r)] : invertible (coe_fn (algebra_map R A) r) :=\n  invertible.copy (invertible.map (\u2191(algebra_map S A)) (coe_fn (algebra_map R S) r))\n    (coe_fn (algebra_map R A) r) sorry\n\n/-- A natural number that is invertible when coerced to `R` is also invertible\nwhen coerced to any `R`-algebra. -/\ndef invertible_algebra_coe_nat (R : Type u) (A : Type w) [comm_semiring R] [semiring A]\n    [algebra R A] (n : \u2115) [inv : invertible \u2191n] : invertible \u2191n :=\n  invertible.algebra_tower \u2115 R A n\n\ntheorem algebra_map_aeval (R : Type u) (A : Type w) (B : Type u\u2081) [comm_semiring R]\n    [comm_semiring A] [comm_semiring B] [algebra R A] [algebra A B] [algebra R B]\n    [is_scalar_tower R A B] (x : A) (p : polynomial R) :\n    coe_fn (algebra_map A B) (coe_fn (polynomial.aeval x) p) =\n        coe_fn (polynomial.aeval (coe_fn (algebra_map A B) x)) p :=\n  sorry\n\ntheorem aeval_eq_zero_of_aeval_algebra_map_eq_zero (R : Type u) (A : Type w) (B : Type u\u2081)\n    [comm_semiring R] [comm_semiring A] [comm_semiring B] [algebra R A] [algebra A B] [algebra R B]\n    [is_scalar_tower R A B] {x : A} {p : polynomial R} (h : function.injective \u21d1(algebra_map A B))\n    (hp : coe_fn (polynomial.aeval (coe_fn (algebra_map A B) x)) p = 0) :\n    coe_fn (polynomial.aeval x) p = 0 :=\n  sorry\n\ntheorem aeval_eq_zero_of_aeval_algebra_map_eq_zero_field {R : Type u_1} {A : Type u_2}\n    {B : Type u_3} [comm_semiring R] [field A] [comm_semiring B] [nontrivial B] [algebra R A]\n    [algebra R B] [algebra A B] [is_scalar_tower R A B] {x : A} {p : polynomial R}\n    (h : coe_fn (polynomial.aeval (coe_fn (algebra_map A B) x)) p = 0) :\n    coe_fn (polynomial.aeval x) p = 0 :=\n  aeval_eq_zero_of_aeval_algebra_map_eq_zero R A B (ring_hom.injective (algebra_map A B)) h\n\nend is_scalar_tower\n\n\nnamespace algebra\n\n\ntheorem adjoin_algebra_map' {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [comm_ring S]\n    [comm_ring A] [algebra R S] [algebra S A] (s : set S) :\n    adjoin R (\u21d1(algebra_map S (comap R S A)) '' s) = subalgebra.map (adjoin R s) (to_comap R S A) :=\n  sorry\n\ntheorem adjoin_algebra_map (R : Type u) (S : Type v) (A : Type w) [comm_ring R] [comm_ring S]\n    [comm_ring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A] (s : set S) :\n    adjoin R (\u21d1(algebra_map S A) '' s) =\n        subalgebra.map (adjoin R s) (is_scalar_tower.to_alg_hom R S A) :=\n  sorry\n\ntheorem adjoin_res (C : Type u_1) (D : Type u_2) (E : Type u_3) [comm_semiring C] [comm_semiring D]\n    [comm_semiring E] [algebra C D] [algebra C E] [algebra D E] [is_scalar_tower C D E]\n    (S : set E) :\n    subalgebra.res C (adjoin D S) =\n        subalgebra.under (subalgebra.map \u22a4 (is_scalar_tower.to_alg_hom C D E))\n          (adjoin (\u21a5(subalgebra.map \u22a4 (is_scalar_tower.to_alg_hom C D E))) S) :=\n  sorry\n\ntheorem adjoin_res_eq_adjoin_res (C : Type u_1) (D : Type u_2) (E : Type u_3) (F : Type u_4)\n    [comm_semiring C] [comm_semiring D] [comm_semiring E] [comm_semiring F] [algebra C D]\n    [algebra C E] [algebra C F] [algebra D F] [algebra E F] [is_scalar_tower C D F]\n    [is_scalar_tower C E F] {S : set D} {T : set E} (hS : adjoin C S = \u22a4) (hT : adjoin C T = \u22a4) :\n    subalgebra.res C (adjoin E (\u21d1(algebra_map D F) '' S)) =\n        subalgebra.res C (adjoin D (\u21d1(algebra_map E F) '' T)) :=\n  sorry\n\nend algebra\n\n\nnamespace subalgebra\n\n\n@[simp] theorem aeval_coe (R : Type u) {A : Type w} [comm_semiring R] [comm_semiring A]\n    [algebra R A] {S : subalgebra R A} {x : \u21a5S} {p : polynomial R} :\n    coe_fn (polynomial.aeval \u2191x) p = \u2191(coe_fn (polynomial.aeval x) p) :=\n  Eq.symm (is_scalar_tower.algebra_map_aeval R (\u21a5S) A x p)\n\nend subalgebra\n\n\ntheorem algebra.fg_trans' {R : Type u_1} {S : Type u_2} {A : Type u_3} [comm_ring R] [comm_ring S]\n    [comm_ring A] [algebra R S] [algebra S A] [algebra R A] [is_scalar_tower R S A]\n    (hRS : subalgebra.fg \u22a4) (hSA : subalgebra.fg \u22a4) : subalgebra.fg \u22a4 :=\n  sorry\n\ntheorem linear_independent_smul {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S]\n    [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S A] {\u03b9 : Type v\u2081}\n    {b : \u03b9 \u2192 S} {\u03b9' : Type w\u2081} {c : \u03b9' \u2192 A} (hb : linear_independent R b)\n    (hc : linear_independent S c) :\n    linear_independent R fun (p : \u03b9 \u00d7 \u03b9') => b (prod.fst p) \u2022 c (prod.snd p) :=\n  sorry\n\ntheorem is_basis.smul {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S]\n    [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S A] {\u03b9 : Type v\u2081}\n    {b : \u03b9 \u2192 S} {\u03b9' : Type w\u2081} {c : \u03b9' \u2192 A} (hb : is_basis R b) (hc : is_basis S c) :\n    is_basis R fun (p : \u03b9 \u00d7 \u03b9') => b (prod.fst p) \u2022 c (prod.snd p) :=\n  sorry\n\ntheorem is_basis.smul_repr {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S]\n    [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S A]\n    {\u03b9 : Type u_1} {\u03b9' : Type u_2} {b : \u03b9 \u2192 S} {c : \u03b9' \u2192 A} (hb : is_basis R b) (hc : is_basis S c)\n    (x : A) (ij : \u03b9 \u00d7 \u03b9') :\n    coe_fn (coe_fn (is_basis.repr (is_basis.smul hb hc)) x) ij =\n        coe_fn (coe_fn (is_basis.repr hb) (coe_fn (coe_fn (is_basis.repr hc) x) (prod.snd ij)))\n          (prod.fst ij) :=\n  sorry\n\ntheorem is_basis.smul_repr_mk {R : Type u} {S : Type v} {A : Type w} [comm_ring R] [ring S]\n    [add_comm_group A] [algebra R S] [module S A] [module R A] [is_scalar_tower R S A]\n    {\u03b9 : Type u_1} {\u03b9' : Type u_2} {b : \u03b9 \u2192 S} {c : \u03b9' \u2192 A} (hb : is_basis R b) (hc : is_basis S c)\n    (x : A) (i : \u03b9) (j : \u03b9') :\n    coe_fn (coe_fn (is_basis.repr (is_basis.smul hb hc)) x) (i, j) =\n        coe_fn (coe_fn (is_basis.repr hb) (coe_fn (coe_fn (is_basis.repr hc) x) j)) i :=\n  sorry\n\ntheorem exists_subalgebra_of_fg (A : Type w) (B : Type u\u2081) (C : Type u_1) [comm_ring A]\n    [comm_ring B] [comm_ring C] [algebra A B] [algebra B C] [algebra A C] [is_scalar_tower A B C]\n    (hAC : subalgebra.fg \u22a4) (hBC : submodule.fg \u22a4) :\n    \u2203 (B\u2080 : subalgebra A B), subalgebra.fg B\u2080 \u2227 submodule.fg \u22a4 :=\n  sorry\n\n/-- Artin--Tate lemma: if A \u2286 B \u2286 C is a chain of subrings of commutative rings, and\nA is noetherian, and C is algebra-finite over A, and C is module-finite over B,\nthen B is algebra-finite over A.\n\nReferences: Atiyah--Macdonald Proposition 7.8; Stacks 00IS; Altman--Kleiman 16.17. -/\ntheorem fg_of_fg_of_fg (A : Type w) (B : Type u\u2081) (C : Type u_1) [comm_ring A] [comm_ring B]\n    [comm_ring C] [algebra A B] [algebra B C] [algebra A C] [is_scalar_tower A B C]\n    [is_noetherian_ring A] (hAC : subalgebra.fg \u22a4) (hBC : submodule.fg \u22a4)\n    (hBCi : function.injective \u21d1(algebra_map B C)) : subalgebra.fg \u22a4 :=\n  sorry\n\n/-- Restrict the domain of an `alg_hom`. -/\ndef alg_hom.restrict_domain {A : Type w} (B : Type u\u2081) {C : Type u_1} {D : Type u_2}\n    [comm_semiring A] [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D]\n    (f : alg_hom A C D) [comm_semiring B] [algebra A B] [algebra B C] [is_scalar_tower A B C] :\n    alg_hom A B D :=\n  alg_hom.comp f (is_scalar_tower.to_alg_hom A B C)\n\n/-- Extend the scalars of an `alg_hom`. -/\ndef alg_hom.extend_scalars {A : Type w} (B : Type u\u2081) {C : Type u_1} {D : Type u_2}\n    [comm_semiring A] [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D]\n    (f : alg_hom A C D) [comm_semiring B] [algebra A B] [algebra B C] [is_scalar_tower A B C] :\n    alg_hom B C D :=\n  alg_hom.mk (alg_hom.to_fun f) sorry sorry sorry sorry sorry\n\n/-- `alg_hom`s from the top of a tower are equivalent to a pair of `alg_hom`s. -/\ndef alg_hom_equiv_sigma {A : Type w} {B : Type u\u2081} {C : Type u_1} {D : Type u_2} [comm_semiring A]\n    [comm_semiring C] [comm_semiring D] [algebra A C] [algebra A D] [comm_semiring B] [algebra A B]\n    [algebra B C] [is_scalar_tower A B C] :\n    alg_hom A C D \u2243 sigma fun (f : alg_hom A B D) => alg_hom B C D :=\n  equiv.mk\n    (fun (f : alg_hom A C D) => sigma.mk (alg_hom.restrict_domain B f) (alg_hom.extend_scalars B f))\n    (fun (fg : sigma fun (f : alg_hom A B D) => alg_hom B C D) =>\n      is_scalar_tower.restrict_base A (sigma.snd fg))\n    sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/algebra_tower_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3992562707939472}}
{"text": "import QL.FOL.Tait.tait provability QL.FOL.coding consistency\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nvariables {L : language.{u}} {m n : \u2115}\n\nnamespace Tait\n\nnamespace subformula\nvariables {L m n}\n\ndef uniform : bounded_subformula L m n \u2192\u2097 subformula L \u2115 n := map coe\n\n@[simp] lemma uniform_inj (p q : bounded_subformula L m n) :\n  p.uniform = q.uniform \u2194 p = q :=\n\u27e8\u03bb h, map_inj_of_inj coe fin.coe_injective h, \u03bb e, by simp[e]\u27e9\n\n@[simp] lemma uniform_relation {k} (r : L.pr k) (v : fin k \u2192 bounded_subterm L m n) :\n  uniform (relation r v) = relation r (\u03bb i, subterm.uniform (v i)) := by simp[uniform, subterm.uniform]\n\n@[simp] lemma uniform_neg_relation {k} (r : L.pr k) (v : fin k \u2192 bounded_subterm L m n) :\n  uniform (neg_relation r v) = neg_relation r (\u03bb i, subterm.uniform (v i)) := by simp[uniform, subterm.uniform]\n\n@[simp] lemma uniform_fal (p : bounded_subformula L m (n + 1)) :\n  uniform (\u2200'p) = \u2200'uniform p := by simp[uniform]; unfold has_univ_quantifier'.univ; simp; refl\n\n@[simp] lemma uniform_ex (p : bounded_subformula L m (n + 1)) :\n  uniform (\u2203'p) = \u2203'uniform p := by simp[uniform]; unfold has_exists_quantifier'.ex; simp; refl\n\n@[simp] lemma uniform_mlift (p : bounded_subformula L m n) : p.mlift.uniform = p.uniform :=\nby simp[mlift, uniform]; congr\n\n@[simp] lemma uniform_cast_le {m\u2081 m\u2082 : \u2115} (h : m\u2081 \u2264 m\u2082) (p : bounded_subformula L m\u2081 n) :\n  (cast_le h p).uniform = p.uniform :=\nby simp[cast_le, uniform]; congr\n\n@[simp] lemma uniform_to_subterm (p : bounded_subformula L m n) (h) : to_bform p.uniform h = p :=\nby induction p using fol.Tait.subformula.ind_on; simp*\n\n@[simp] lemma to_subterm_uniform (p : subformula L \u2115 n) (h : p.arity \u2264 m) : (p.to_bform h).uniform = p :=\nby induction p using fol.Tait.subformula.ind_on; simp*\n\n@[simp] lemma subformula_arity (p : bounded_subformula L m n) : p.uniform.arity \u2264 m :=\nby induction p using fol.Tait.subformula.ind_on; simp*\n\nsection encode\nopen encodable nat\nvariables {L n} [\u2200 k, encodable (L.pr k)] [\u2200 k, encodable (L.fn k)]\n\n@[simp] def to_nat : \u03a0 {n}, subformula L \u2115 n \u2192 \u2115\n| n verum                       := 0\n| n falsum                      := 1\n| n (@relation L _ _ k r v)     := (bit0 $ bit0 $ mkpair k $ mkpair (encode r) (encode v)) + 2\n| n (@neg_relation L _ _ k r v) := (bit0 $ bit1 $ mkpair k $ mkpair (encode r) (encode v)) + 2\n| n (and p q)                   := (bit1 $ bit0 $ bit0 $ mkpair p.to_nat q.to_nat) + 2\n| n (or p q)                    := (bit1 $ bit0 $ bit1 $ mkpair p.to_nat q.to_nat) + 2\n| n (fal p)                     := (bit1 $ bit1 $ bit0 p.to_nat) + 2\n| n (ex p)                      := (bit1 $ bit1 $ bit1 p.to_nat) + 2\n\nvariables (L n)\n\n@[simp] def of_nat : \u03a0 n, \u2115 \u2192 option (subformula L \u2115 n)\n| n 0 := some verum\n| n 1 := some falsum\n| n (e + 2) :=\n    let i := e.div2.div2.div2 in\n    have div8 : i \u2264 e := by simp[i, nat.div2_val]; \n      exact le_trans (nat.div_le_self (e / 2 / 2) 2) (le_trans (nat.div_le_self (e/2) 2) (nat.div_le_self e 2)),\n    have hi : i < e + 2, from lt.step (lt_succ_iff.mpr div8),\n    have hi1 : i.unpair.1 < e + 2, from (lt.step $ nat.lt_succ_iff.mpr (le_trans (nat.unpair_left_le _) div8)),\n    have hi2 : i.unpair.2 < e + 2, from (lt.step $ nat.lt_succ_iff.mpr (le_trans (nat.unpair_right_le _) div8)),\n    match e.bodd with\n    | ff :=\n      match e.div2.bodd with\n      | ff :=\n        let x := e.div2.div2,\n            k := x.unpair.1,\n            r := decode\u2082 (L.pr k) x.unpair.2.unpair.1,\n            v := decode\u2082 (fin k \u2192 subterm L \u2115 n) x.unpair.2.unpair.2 in\n        r.bind (\u03bb r, v.map (relation r))\n      | tt :=\n        let x := e.div2.div2,\n            k := x.unpair.1,\n            r := decode\u2082 (L.pr k) x.unpair.2.unpair.1,\n            v := decode\u2082 (fin k \u2192 subterm L \u2115 n) x.unpair.2.unpair.2 in\n        r.bind (\u03bb r, v.map (neg_relation r))\n      end\n    | tt :=\n      match e.div2.bodd, e.div2.div2.bodd with\n      | ff, ff := (of_nat n i.unpair.1).bind (\u03bb p, (of_nat n i.unpair.2).map (and p))\n      | ff, tt := (of_nat n i.unpair.1).bind (\u03bb p, (of_nat n i.unpair.2).map (or p))\n      | tt, ff := (of_nat (n + 1) i).map fal\n      | tt, tt := (of_nat (n + 1) i).map ex\n      end\n    end\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb x, x.2)\u27e9]}\n\n@[simp] lemma of_nat_to_nat : \u2200 {n} (p : subformula L \u2115 n), of_nat L n p.to_nat = some p\n| n verum                       := by simp\n| n falsum                      := by simp\n| n (@relation L _ _ k r v)     :=\n    begin\n      simp only [to_nat, of_nat, nat.bodd_bit0, nat.div2_bit0, nat.unpair_mkpair],\n      rw[show (unpair (bit0 (bit0 (mkpair k (mkpair (encode r) (encode v))))).div2.div2).fst = k, by simp],\n      simp only [decode\u2082_encode, option.some_bind', option.map_some', heq.refl], simp\n    end\n| n (@neg_relation L _ _ k r v) :=\n    begin\n      simp only [to_nat, of_nat, nat.bodd_bit0, nat.bodd_bit1, nat.div2_bit0, nat.div2_bit1, nat.unpair_mkpair],\n      rw[show (unpair (bit0 (bit1 (mkpair k (mkpair (encode r) (encode v))))).div2.div2).fst = k, by simp],\n      simp only [decode\u2082_encode, option.some_bind', option.map_some', heq.refl], simp\n    end\n| n (and p q)                   := by simp; refine \u27e8of_nat_to_nat p, of_nat_to_nat q\u27e9\n| n (or p q)                    := by simp; refine \u27e8of_nat_to_nat p, of_nat_to_nat q\u27e9\n| n (fal p)                     := by simp; refine (of_nat_to_nat p)\n| n (ex p)                      := by simp; refine (of_nat_to_nat p)\n\ninstance (n) : encodable (subformula L \u2115 n) :=\n{ encode := to_nat,\n  decode := of_nat L n,\n  encodek := by simp }\n\nvariables {L m n}\n\ndef index : bounded_subformula L m n \u2192 \u2115 := \u03bb p, encodable.encode p.uniform\n\nvariables (L m n)\n\ndef of_index : \u2115 \u2192 option (bounded_subformula L m n) := \u03bb i,\n  let p := encodable.decode\u2082 (subformula L \u2115 n) i in\n  p.bind (\u03bb p, if h : p.arity \u2264 m then some (p.to_bform h) else none)\n\nvariables {L m n}\n\n@[simp] lemma of_index_index (p : bounded_subformula L m n) : of_index L m n p.index = some p :=\nby simp[index, of_index]\n\n@[simp] lemma mlift_index (p : bounded_subformula L m n) : p.mlift.index = p.index :=\nby simp[index]\n\n@[simp] lemma cast_le_index {m\u2081 m\u2082 : \u2115} (h : m\u2081 \u2264 m\u2082) (p : bounded_subformula L m\u2081 n) :\n  (cast_le h p).index = p.index :=\nby simp[index]\n\n@[simp] lemma of_nat_uniform (p : bounded_subformula L m n) : encodable.decode\u2082 (subformula L \u2115 n) p.index = p.uniform :=\nby simp[index]; refl\n\n@[simp] lemma index_eq_some {e} {p : bounded_subformula L m n} : of_index L m n e = some p \u2194 p.index = e :=\nby { simp[of_index, index, encodable.decode\u2082_eq_some, dite_eq_iff], split,\n  { simp, rintros _ rfl h rfl, simp },\n  { rintros rfl, refine \u27e8p.uniform, rfl, by simp\u27e9 } }\n\nlemma of_index_eq_some {m\u2081 m\u2082} {p : bounded_subformula L m\u2081 n} {q : bounded_subformula L m\u2082 n} :\n  p.index = q.index \u2194 p.uniform = q.uniform :=\nby simp[index]\n\n@[simp] lemma index_inj {p q : bounded_subformula L m n} : p.index = q.index \u2194 p = q :=\nby simp[index]\n\nend encode\n\nend subformula\n\nend Tait\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/QL/FOL/Tait/coding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.39925626355850713}}
{"text": "/-\nFile: signature_recover_public_key_compute_slope_soundness.lean\n\nAutogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.hoare\nimport .signature_recover_public_key_code\nimport ..signature_recover_public_key_spec\nimport .signature_recover_public_key_verify_zero_soundness\nimport .signature_recover_public_key_unreduced_mul_soundness\nimport .signature_recover_public_key_nondet_bigint3_soundness\nopen tactic\n\nopen starkware.cairo.common.cairo_secp.ec\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.field\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\nvariable  mem : F \u2192 F\nvariable  \u03c3 : register_state F\n\n/- starkware.cairo.common.cairo_secp.ec.compute_slope autogenerated soundness theorem -/\n\ntheorem auto_sound_compute_slope\n    -- arguments\n    (range_check_ptr : F) (point0 point1 : EcPoint F)\n    -- code is in memory at \u03c3.pc\n    (h_mem : mem_at mem code_compute_slope \u03c3.pc)\n    -- all dependencies are in memory\n    (h_mem_4 : mem_at mem code_nondet_bigint3 (\u03c3.pc  - 220))\n    (h_mem_5 : mem_at mem code_unreduced_mul (\u03c3.pc  - 208))\n    (h_mem_7 : mem_at mem code_verify_zero (\u03c3.pc  - 172))\n    -- input arguments on the stack\n    (hin_range_check_ptr : range_check_ptr = mem (\u03c3.fp - 15))\n    (hin_point0 : point0 = cast_EcPoint mem (\u03c3.fp - 14))\n    (hin_point1 : point1 = cast_EcPoint mem (\u03c3.fp - 8))\n    -- conclusion\n  : ensures_ret mem \u03c3 (\u03bb \u03ba \u03c4,\n      \u03c4.ap = \u03c3.ap + 59 \u2227\n      \u2203 \u03bc \u2264 \u03ba, rc_ensures mem (rc_bound F) \u03bc (mem (\u03c3.fp - 15)) (mem $ \u03c4.ap - 4)\n        (spec_compute_slope mem \u03ba range_check_ptr point0 point1 (mem (\u03c4.ap - 4)) (cast_BigInt3 mem (\u03c4.ap - 3)))) :=\nbegin\n  apply ensures_of_ensuresb, intro \u03bdbound,\n  have h_mem_rec := h_mem,\n  unpack_memory code_compute_slope at h_mem with \u27e8hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19, hpc20, hpc21, hpc22, hpc23\u27e9,\n  -- function call\n  step_assert_eq hpc0 with arg0,\n  step_sub hpc1 (auto_sound_nondet_bigint3 mem _ range_check_ptr _ _),\n  { rw hpc2, norm_num2, exact h_mem_4 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1] },\n    try { dsimp [cast_EcPoint, cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  intros \u03ba_call3 ap3 h_call3,\n  rcases h_call3 with \u27e8h_call3_ap_offset, h_call3\u27e9,\n  rcases h_call3 with \u27e8rc_m3, rc_mle3, hl_range_check_ptr\u2081, h_call3\u27e9,\n  generalize' hr_rev_range_check_ptr\u2081: mem (ap3 - 4) = range_check_ptr\u2081,\n  have htv_range_check_ptr\u2081 := hr_rev_range_check_ptr\u2081.symm, clear hr_rev_range_check_ptr\u2081,\n  generalize' hr_rev_slope: cast_BigInt3 mem (ap3 - 3) = slope,\n  simp only [hr_rev_slope] at h_call3,\n  have htv_slope := hr_rev_slope.symm, clear hr_rev_slope,\n  try { simp only [arg0] at hl_range_check_ptr\u2081 },\n  rw [\u2190htv_range_check_ptr\u2081, \u2190hin_range_check_ptr] at hl_range_check_ptr\u2081,\n  try { simp only [arg0] at h_call3 },\n  rw [hin_range_check_ptr] at h_call3,\n  clear arg0,\n  -- let\n  generalize' hl_rev_x_diff: ({\n    d0 := point0.x.d0 - point1.x.d0,\n    d1 := point0.x.d1 - point1.x.d1,\n    d2 := point0.x.d2 - point1.x.d2\n  } : BigInt3 F) = x_diff,\n  have hl_x_diff := hl_rev_x_diff.symm, clear hl_rev_x_diff,\n  try { dsimp at hl_x_diff }, try { arith_simps at hl_x_diff },\n  -- function call\n  step_assert_eq hpc3 with arg0,\n  step_assert_eq hpc4 with arg1,\n  step_assert_eq hpc5 with arg2,\n  step_assert_eq hpc6 with arg3,\n  step_assert_eq hpc7 with arg4,\n  step_assert_eq hpc8 with arg5,\n  step_sub hpc9 (auto_sound_unreduced_mul mem _ x_diff slope _ _ _),\n  { rw hpc10, norm_num2, exact h_mem_5 },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr\u2081, htv_slope, hl_x_diff] },\n      try { dsimp [cast_EcPoint, cast_BigInt3] },\n      try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5] },\n      try { simp only [h_call3_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr\u2081, htv_slope, hl_x_diff] },\n      try { dsimp [cast_EcPoint, cast_BigInt3] },\n      try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5] },\n      try { simp only [h_call3_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros \u03ba_call11 ap11 h_call11,\n  rcases h_call11 with \u27e8h_call11_ap_offset, h_call11\u27e9,\n  generalize' hr_rev_x_diff_slope: cast_UnreducedBigInt3 mem (ap11 - 3) = x_diff_slope,\n  simp only [hr_rev_x_diff_slope] at h_call11,\n  have htv_x_diff_slope := hr_rev_x_diff_slope.symm, clear hr_rev_x_diff_slope,\n  clear arg0 arg1 arg2 arg3 arg4 arg5,\n  -- function call\n  step_assert_eq hpc11 with arg0,\n  step_assert_eq hpc12 with arg1,\n  step_assert_eq hpc13 with arg2,\n  step_assert_eq hpc14 with arg3,\n  step_assert_eq hpc15 with arg4,\n  step_assert_eq hpc16 with arg5,\n  step_assert_eq hpc17 with arg6,\n  step_sub hpc18 (auto_sound_verify_zero mem _ range_check_ptr\u2081 {\n    d0 := x_diff_slope.d0 - point0.y.d0 + point1.y.d0,\n    d1 := x_diff_slope.d1 - point0.y.d1 + point1.y.d1,\n    d2 := x_diff_slope.d2 - point0.y.d2 + point1.y.d2\n  } _ _ _),\n  { rw hpc19, norm_num2, exact h_mem_7 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr\u2081, htv_slope, hl_x_diff, htv_x_diff_slope] },\n    try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n    try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5, arg6] },\n    try { simp only [h_call3_ap_offset, h_call11_ap_offset] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr\u2081, htv_slope, hl_x_diff, htv_x_diff_slope] },\n      try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n      try { arith_simps }, try { simp only [(eq_sub_of_eq_add arg0), (eq_sub_of_eq_add arg1), (eq_sub_of_eq_add arg2), arg3, arg4, arg5, arg6] },\n      try { simp only [h_call3_ap_offset, h_call11_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros \u03ba_call20 ap20 h_call20,\n  rcases h_call20 with \u27e8h_call20_ap_offset, h_call20\u27e9,\n  rcases h_call20 with \u27e8rc_m20, rc_mle20, hl_range_check_ptr\u2082, h_call20\u27e9,\n  generalize' hr_rev_range_check_ptr\u2082: mem (ap20 - 1) = range_check_ptr\u2082,\n  have htv_range_check_ptr\u2082 := hr_rev_range_check_ptr\u2082.symm, clear hr_rev_range_check_ptr\u2082,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6] at hl_range_check_ptr\u2082 },\n  try { rw [h_call11_ap_offset] at hl_range_check_ptr\u2082 }, try { arith_simps at hl_range_check_ptr\u2082 },\n  rw [\u2190htv_range_check_ptr\u2082, \u2190htv_range_check_ptr\u2081] at hl_range_check_ptr\u2082,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3 ,arg4 ,arg5 ,arg6] at h_call20 },\n  try { rw [h_call11_ap_offset] at h_call20 }, try { arith_simps at h_call20 },\n  rw [\u2190htv_range_check_ptr\u2081, hl_range_check_ptr\u2081, hin_range_check_ptr] at h_call20,\n  clear arg0 arg1 arg2 arg3 arg4 arg5 arg6,\n  -- return\n  step_assert_eq hpc20 with hret0,\n  step_assert_eq hpc21 with hret1,\n  step_assert_eq hpc22 with hret2,\n  step_ret hpc23,\n  -- finish\n  step_done, use_only [rfl, rfl],\n  split,\n  { try { simp only [h_call3_ap_offset ,h_call11_ap_offset ,h_call20_ap_offset] },\n    try { arith_simps }, try { refl } },\n  -- range check condition\n  use_only (rc_m3+rc_m20+0+0), split,\n  linarith [rc_mle3, rc_mle20],\n  split,\n  { arith_simps, try { simp only [hret0 ,hret1 ,hret2] },\n    rw [\u2190htv_range_check_ptr\u2082, hl_range_check_ptr\u2082, hl_range_check_ptr\u2081, hin_range_check_ptr],\n    try { arith_simps, refl <|> norm_cast }, try { refl } },\n  intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },\n  have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,\n  -- Final Proof\n  -- user-provided reduction\n  suffices auto_spec: auto_spec_compute_slope mem _ range_check_ptr point0 point1 _ _,\n  { apply sound_compute_slope, apply auto_spec },\n  -- prove the auto generated assertion\n  dsimp [auto_spec_compute_slope],\n  try { norm_num1 }, try { arith_simps },\n  use_only [\u03ba_call3],\n  use_only [range_check_ptr\u2081],\n  use_only [slope],\n  have rc_h_range_check_ptr\u2081 := range_checked_offset' rc_h_range_check_ptr,\n  have rc_h_range_check_ptr\u2081' := range_checked_add_right rc_h_range_check_ptr\u2081, try { norm_cast at rc_h_range_check_ptr\u2081' },\n  have spec3 := h_call3 rc_h_range_check_ptr',\n  rw [\u2190hin_range_check_ptr, \u2190htv_range_check_ptr\u2081] at spec3,\n  try { dsimp at spec3, arith_simps at spec3 },\n  use_only [spec3],\n  use_only [x_diff, hl_x_diff],\n  use_only [\u03ba_call11],\n  use_only [x_diff_slope],\n  try { dsimp at h_call11, arith_simps at h_call11 },\n  try { use_only [h_call11] },\n  use_only [\u03ba_call20],\n  use_only [range_check_ptr\u2082],\n  have rc_h_range_check_ptr\u2082 := range_checked_offset' rc_h_range_check_ptr\u2081,\n  have rc_h_range_check_ptr\u2082' := range_checked_add_right rc_h_range_check_ptr\u2082, try { norm_cast at rc_h_range_check_ptr\u2082' },\n  have spec20 := h_call20 rc_h_range_check_ptr\u2081',\n  rw [\u2190hin_range_check_ptr, \u2190hl_range_check_ptr\u2081, \u2190htv_range_check_ptr\u2082] at spec20,\n  try { dsimp at spec20, arith_simps at spec20 },\n  use_only [spec20],\n  try { split, linarith },\n  try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point0, hin_point1, htv_range_check_ptr\u2081, htv_slope, hl_x_diff, htv_x_diff_slope, htv_range_check_ptr\u2082] }, },\n  try { dsimp [cast_EcPoint, cast_BigInt3, cast_UnreducedBigInt3] },\n  try { arith_simps }, try { simp only [hret0, hret1, hret2] },\n  try { simp only [h_call3_ap_offset, h_call11_ap_offset, h_call20_ap_offset] },\n  try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },\nend\n\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_compute_slope_soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3991189131205236}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport group_theory.subgroup.basic\nimport algebra.graded_monoid\nimport algebra.direct_sum.basic\nimport algebra.big_operators.pi\n\n/-!\n# Additively-graded multiplicative structures on `\u2a01 i, A i`\n\nThis module provides a set of heterogeneous typeclasses for defining a multiplicative structure\nover `\u2a01 i, A i` such that `(*) : A i \u2192 A j \u2192 A (i + j)`; that is to say, `A` forms an\nadditively-graded ring. The typeclasses are:\n\n* `direct_sum.gnon_unital_non_assoc_semiring A`\n* `direct_sum.gsemiring A`\n* `direct_sum.gring A`\n* `direct_sum.gcomm_semiring A`\n* `direct_sum.gcomm_ring A`\n\nRespectively, these imbue the external direct sum `\u2a01 i, A i` with:\n\n* `direct_sum.non_unital_non_assoc_semiring`, `direct_sum.non_unital_non_assoc_ring`\n* `direct_sum.semiring`\n* `direct_sum.ring`\n* `direct_sum.comm_semiring`\n* `direct_sum.comm_ring`\n\nthe base ring `A 0` with:\n\n* `direct_sum.grade_zero.non_unital_non_assoc_semiring`,\n  `direct_sum.grade_zero.non_unital_non_assoc_ring`\n* `direct_sum.grade_zero.semiring`\n* `direct_sum.grade_zero.ring`\n* `direct_sum.grade_zero.comm_semiring`\n* `direct_sum.grade_zero.comm_ring`\n\nand the `i`th grade `A i` with `A 0`-actions (`\u2022`) defined as left-multiplication:\n\n* `direct_sum.grade_zero.has_smul (A 0)`, `direct_sum.grade_zero.smul_with_zero (A 0)`\n* `direct_sum.grade_zero.module (A 0)`\n* (nothing)\n* (nothing)\n* (nothing)\n\nNote that in the presence of these instances, `\u2a01 i, A i` itself inherits an `A 0`-action.\n\n`direct_sum.of_zero_ring_hom : A 0 \u2192+* \u2a01 i, A i` provides `direct_sum.of A 0` as a ring\nhomomorphism.\n\n`direct_sum.to_semiring` extends `direct_sum.to_add_monoid` to produce a `ring_hom`.\n\n## Direct sums of subobjects\n\nAdditionally, this module provides helper functions to construct `gsemiring` and `gcomm_semiring`\ninstances for:\n\n* `A : \u03b9 \u2192 submonoid S`:\n  `direct_sum.gsemiring.of_add_submonoids`, `direct_sum.gcomm_semiring.of_add_submonoids`.\n* `A : \u03b9 \u2192 subgroup S`:\n  `direct_sum.gsemiring.of_add_subgroups`, `direct_sum.gcomm_semiring.of_add_subgroups`.\n* `A : \u03b9 \u2192 submodule S`:\n  `direct_sum.gsemiring.of_submodules`, `direct_sum.gcomm_semiring.of_submodules`.\n\nIf `complete_lattice.independent (set.range A)`, these provide a gradation of `\u2a06 i, A i`, and the\nmapping `\u2a01 i, A i \u2192+ \u2a06 i, A i` can be obtained as\n`direct_sum.to_monoid (\u03bb i, add_submonoid.inclusion $ le_supr A i)`.\n\n## tags\n\ngraded ring, filtered ring, direct sum, add_submonoid\n-/\n\nset_option old_structure_cmd true\n\nvariables {\u03b9 : Type*} [decidable_eq \u03b9]\n\nnamespace direct_sum\n\nopen_locale direct_sum\n\n/-! ### Typeclasses -/\nsection defs\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-- A graded version of `non_unital_non_assoc_semiring`. -/\nclass gnon_unital_non_assoc_semiring [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  graded_monoid.ghas_mul A :=\n(mul_zero : \u2200 {i j} (a : A i), mul a (0 : A j) = 0)\n(zero_mul : \u2200 {i j} (b : A j), mul (0 : A i) b = 0)\n(mul_add : \u2200 {i j} (a : A i) (b c : A j), mul a (b + c) = mul a b + mul a c)\n(add_mul : \u2200 {i j} (a b : A i) (c : A j), mul (a + b) c = mul a c + mul b c)\n\nend defs\n\nsection defs\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-- A graded version of `semiring`. -/\nclass gsemiring [add_monoid \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  gnon_unital_non_assoc_semiring A, graded_monoid.gmonoid A :=\n(nat_cast : \u2115 \u2192 A 0)\n(nat_cast_zero : nat_cast 0 = 0)\n(nat_cast_succ : \u2200 n : \u2115, nat_cast (n + 1) = nat_cast n + graded_monoid.ghas_one.one)\n\n/-- A graded version of `comm_semiring`. -/\nclass gcomm_semiring [add_comm_monoid \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  gsemiring A, graded_monoid.gcomm_monoid A\n\n/-- A graded version of `ring`. -/\nclass gring [add_monoid \u03b9] [\u03a0 i, add_comm_group (A i)] extends gsemiring A :=\n(int_cast : \u2124 \u2192 A 0)\n(int_cast_of_nat : \u2200 n : \u2115, int_cast n = nat_cast n)\n(int_cast_neg_succ_of_nat : \u2200 n : \u2115, int_cast (-(n+1 : \u2115)) = -nat_cast (n+1 : \u2115))\n\n/-- A graded version of `comm_ring`. -/\nclass gcomm_ring [add_comm_monoid \u03b9] [\u03a0 i, add_comm_group (A i)] extends\n  gring A, gcomm_semiring A\n\nend defs\n\nlemma of_eq_of_graded_monoid_eq {A : \u03b9 \u2192 Type*} [\u03a0 (i : \u03b9), add_comm_monoid (A i)]\n  {i j : \u03b9} {a : A i} {b : A j} (h : graded_monoid.mk i a = graded_monoid.mk j b) :\n  direct_sum.of A i a = direct_sum.of A j b :=\ndfinsupp.single_eq_of_sigma_eq h\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-! ### Instances for `\u2a01 i, A i` -/\n\n\nsection one\nvariables [has_zero \u03b9] [graded_monoid.ghas_one A] [\u03a0 i, add_comm_monoid (A i)]\n\ninstance : has_one (\u2a01 i, A i) :=\n{ one := direct_sum.of (\u03bb i, A i) 0 graded_monoid.ghas_one.one }\n\nend one\n\nsection mul\nvariables [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] [gnon_unital_non_assoc_semiring A]\n\nopen add_monoid_hom (flip_apply coe_comp comp_hom_apply_apply)\n\n/-- The piecewise multiplication from the `has_mul` instance, as a bundled homomorphism. -/\n@[simps]\ndef gmul_hom {i j} : A i \u2192+ A j \u2192+ A (i + j) :=\n{ to_fun := \u03bb a,\n  { to_fun := \u03bb b, graded_monoid.ghas_mul.mul a b,\n    map_zero' := gnon_unital_non_assoc_semiring.mul_zero _,\n    map_add' := gnon_unital_non_assoc_semiring.mul_add _ },\n  map_zero' := add_monoid_hom.ext $ \u03bb a, gnon_unital_non_assoc_semiring.zero_mul a,\n  map_add' := \u03bb a\u2081 a\u2082, add_monoid_hom.ext $ \u03bb b, gnon_unital_non_assoc_semiring.add_mul _ _ _}\n\n/-- The multiplication from the `has_mul` instance, as a bundled homomorphism. -/\ndef mul_hom : (\u2a01 i, A i) \u2192+ (\u2a01 i, A i) \u2192+ \u2a01 i, A i :=\ndirect_sum.to_add_monoid $ \u03bb i,\n  add_monoid_hom.flip $ direct_sum.to_add_monoid $ \u03bb j, add_monoid_hom.flip $\n    (direct_sum.of A _).comp_hom.comp $ gmul_hom A\n\ninstance : non_unital_non_assoc_semiring (\u2a01 i, A i) :=\n{ mul := \u03bb a b, mul_hom A a b,\n  zero := 0,\n  add := (+),\n  zero_mul := \u03bb a, by simp only [add_monoid_hom.map_zero, add_monoid_hom.zero_apply],\n  mul_zero := \u03bb a, by simp only [add_monoid_hom.map_zero],\n  left_distrib := \u03bb a b c, by simp only [add_monoid_hom.map_add],\n  right_distrib := \u03bb a b c, by simp only [add_monoid_hom.map_add, add_monoid_hom.add_apply],\n  .. direct_sum.add_comm_monoid _ _}\n\nvariables {A}\n\nlemma mul_hom_of_of {i j} (a : A i) (b : A j) :\n  mul_hom A (of _ i a) (of _ j b) = of _ (i + j) (graded_monoid.ghas_mul.mul a b) :=\nbegin\n  unfold mul_hom,\n  rw [to_add_monoid_of, flip_apply, to_add_monoid_of, flip_apply, coe_comp, function.comp_app,\n      comp_hom_apply_apply, coe_comp, function.comp_app, gmul_hom_apply_apply],\nend\n\nlemma of_mul_of {i j} (a : A i) (b : A j) :\n  of _ i a * of _ j b = of _ (i + j) (graded_monoid.ghas_mul.mul a b) :=\nmul_hom_of_of a b\n\nend mul\n\nsection semiring\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A]\n\nopen add_monoid_hom (flip_hom coe_comp comp_hom_apply_apply flip_apply flip_hom_apply)\n\nprivate lemma one_mul (x : \u2a01 i, A i) : 1 * x = x :=\nsuffices mul_hom A 1 = add_monoid_hom.id (\u2a01 i, A i),\n  from add_monoid_hom.congr_fun this x,\nbegin\n  apply add_hom_ext, intros i xi,\n  unfold has_one.one,\n  rw mul_hom_of_of,\n  exact of_eq_of_graded_monoid_eq (one_mul $ graded_monoid.mk i xi),\nend\n\nprivate lemma mul_one (x : \u2a01 i, A i) : x * 1 = x :=\nsuffices (mul_hom A).flip 1 = add_monoid_hom.id (\u2a01 i, A i),\n  from add_monoid_hom.congr_fun this x,\nbegin\n  apply add_hom_ext, intros i xi,\n  unfold has_one.one,\n  rw [flip_apply, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (mul_one $ graded_monoid.mk i xi),\nend\n\nprivate lemma mul_assoc (a b c : \u2a01 i, A i) : a * b * c = a * (b * c) :=\nsuffices (mul_hom A).comp_hom.comp (mul_hom A)            -- `\u03bb a b c, a * b * c` as a bundled hom\n       = (add_monoid_hom.comp_hom flip_hom $              -- `\u03bb a b c, a * (b * c)` as a bundled hom\n             (mul_hom A).flip.comp_hom.comp (mul_hom A)).flip,\n  from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun (add_monoid_hom.congr_fun this a) b) c,\nbegin\n  ext ai ax bi bx ci cx : 6,\n  dsimp only [coe_comp, function.comp_app, comp_hom_apply_apply, flip_apply, flip_hom_apply],\n  rw [mul_hom_of_of, mul_hom_of_of, mul_hom_of_of, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (mul_assoc (graded_monoid.mk ai ax) \u27e8bi, bx\u27e9 \u27e8ci, cx\u27e9),\nend\n\n/-- The `semiring` structure derived from `gsemiring A`. -/\ninstance semiring : semiring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  one_mul := one_mul A,\n  mul_one := mul_one A,\n  mul_assoc := mul_assoc A,\n  nat_cast := \u03bb n, of _ _ (gsemiring.nat_cast n),\n  nat_cast_zero := by rw [gsemiring.nat_cast_zero, map_zero],\n  nat_cast_succ := \u03bb n, by { rw [gsemiring.nat_cast_succ, map_add], refl },\n  ..direct_sum.non_unital_non_assoc_semiring _, }\n\nlemma of_pow {i} (a : A i) (n : \u2115) :\n  of _ i a ^ n = of _ (n \u2022 i) (graded_monoid.gmonoid.gnpow _ a) :=\nbegin\n  induction n with n,\n  { exact of_eq_of_graded_monoid_eq (pow_zero $ graded_monoid.mk _ a).symm, },\n  { rw [pow_succ, n_ih, of_mul_of],\n    exact of_eq_of_graded_monoid_eq (pow_succ (graded_monoid.mk _ a) n).symm, },\nend\n\nlemma of_list_dprod {\u03b1} (l : list \u03b1) (f\u03b9 : \u03b1 \u2192 \u03b9) (fA : \u03a0 a, A (f\u03b9 a)) :\n  of A _ (l.dprod f\u03b9 fA) = (l.map $ \u03bb a, of A (f\u03b9 a) (fA a)).prod :=\nbegin\n  induction l,\n  { simp only [list.map_nil, list.prod_nil, list.dprod_nil],\n    refl },\n  { simp only [list.map_cons, list.prod_cons, list.dprod_cons, \u2190l_ih, direct_sum.of_mul_of],\n    refl },\nend\n\nlemma list_prod_of_fn_of_eq_dprod (n : \u2115) (f\u03b9 : fin n \u2192 \u03b9) (fA : \u03a0 a, A (f\u03b9 a)) :\n  (list.of_fn $ \u03bb a, of A (f\u03b9 a) (fA a)).prod = of A _ ((list.fin_range n).dprod f\u03b9 fA) :=\nby rw [list.of_fn_eq_map, of_list_dprod]\n\nopen_locale big_operators\n\n/-- A heavily unfolded version of the definition of multiplication -/\nlemma mul_eq_sum_support_ghas_mul\n  [\u03a0 (i : \u03b9) (x : A i), decidable (x \u2260 0)] (a a' : \u2a01 i, A i) :\n  a * a' =\n    \u2211 (ij : \u03b9 \u00d7 \u03b9) in (dfinsupp.support a).product (dfinsupp.support a'),\n      direct_sum.of _ _ (graded_monoid.ghas_mul.mul (a ij.fst) (a' ij.snd)) :=\nbegin\n  change direct_sum.mul_hom _ a a' = _,\n  dsimp [direct_sum.mul_hom, direct_sum.to_add_monoid, dfinsupp.lift_add_hom_apply],\n  simp only [dfinsupp.sum_add_hom_apply, dfinsupp.sum, dfinsupp.finset_sum_apply,\n    add_monoid_hom.coe_finset_sum, finset.sum_apply, add_monoid_hom.flip_apply,\n    add_monoid_hom.comp_hom_apply_apply, add_monoid_hom.comp_apply,\n    direct_sum.gmul_hom_apply_apply],\n  rw finset.sum_product,\nend\n\nend semiring\n\nsection comm_semiring\n\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\nprivate lemma mul_comm (a b : \u2a01 i, A i) : a * b = b * a :=\nsuffices mul_hom A = (mul_hom A).flip,\n  from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun this a) b,\nbegin\n  apply add_hom_ext, intros ai ax, apply add_hom_ext, intros bi bx,\n  rw [add_monoid_hom.flip_apply, mul_hom_of_of, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (gcomm_semiring.mul_comm \u27e8ai, ax\u27e9 \u27e8bi, bx\u27e9),\nend\n\n/-- The `comm_semiring` structure derived from `gcomm_semiring A`. -/\ninstance comm_semiring : comm_semiring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  mul_comm := mul_comm A,\n  ..direct_sum.semiring _, }\n\nend comm_semiring\n\nsection non_unital_non_assoc_ring\nvariables [\u03a0 i, add_comm_group (A i)] [has_add \u03b9] [gnon_unital_non_assoc_semiring A]\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance non_assoc_ring : non_unital_non_assoc_ring (\u2a01 i, A i) :=\n{ mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  ..(direct_sum.non_unital_non_assoc_semiring _),\n  ..(direct_sum.add_comm_group _), }\n\nend non_unital_non_assoc_ring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_monoid \u03b9] [gring A]\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance ring : ring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  int_cast := \u03bb z, of _ _ (gring.int_cast z),\n  int_cast_of_nat := \u03bb z, congr_arg _ $ gring.int_cast_of_nat _,\n  int_cast_neg_succ_of_nat := \u03bb z,\n    (congr_arg _ $ gring.int_cast_neg_succ_of_nat _).trans (map_neg _ _),\n  ..(direct_sum.semiring _),\n  ..(direct_sum.add_comm_group _), }\n\nend ring\n\nsection comm_ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_comm_monoid \u03b9] [gcomm_ring A]\n\n/-- The `comm_ring` derived from `gcomm_semiring A`. -/\ninstance comm_ring : comm_ring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  ..(direct_sum.ring _),\n  ..(direct_sum.comm_semiring _), }\n\nend comm_ring\n\n\n/-! ### Instances for `A 0`\n\nThe various `g*` instances are enough to promote the `add_comm_monoid (A 0)` structure to various\ntypes of multiplicative structure.\n-/\n\nsection grade_zero\n\nsection one\nvariables [has_zero \u03b9] [graded_monoid.ghas_one A] [\u03a0 i, add_comm_monoid (A i)]\n\n@[simp] lemma of_zero_one : of _ 0 (1 : A 0) = 1 := rfl\n\nend one\n\nsection mul\nvariables [add_zero_class \u03b9] [\u03a0 i, add_comm_monoid (A i)] [gnon_unital_non_assoc_semiring A]\n\n@[simp] lemma of_zero_smul {i} (a : A 0) (b : A i) : of _ _ (a \u2022 b) = of _ _ a * of _ _ b :=\n(of_eq_of_graded_monoid_eq (graded_monoid.mk_zero_smul a b)).trans (of_mul_of _ _).symm\n\n@[simp] lemma of_zero_mul (a b : A 0) : of _ 0 (a * b) = of _ 0 a * of _ 0 b:=\nof_zero_smul A a b\n\ninstance grade_zero.non_unital_non_assoc_semiring : non_unital_non_assoc_semiring (A 0) :=\nfunction.injective.non_unital_non_assoc_semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of A 0).map_add (of_zero_mul A) (\u03bb x n, dfinsupp.single_smul n x)\n\ninstance grade_zero.smul_with_zero (i : \u03b9) : smul_with_zero (A 0) (A i) :=\nbegin\n  letI := smul_with_zero.comp_hom (\u2a01 i, A i) (of A 0).to_zero_hom,\n  refine dfinsupp.single_injective.smul_with_zero (of A i).to_zero_hom (of_zero_smul A),\nend\n\nend mul\n\nsection semiring\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A]\n\n@[simp] lemma of_zero_pow (a : A 0) : \u2200 n : \u2115, of _ 0 (a ^ n) = of _ 0 a ^ n\n| 0 := by rw [pow_zero, pow_zero, direct_sum.of_zero_one]\n| (n + 1) := by rw [pow_succ, pow_succ, of_zero_mul, of_zero_pow]\n\ninstance : has_nat_cast (A 0) := \u27e8gsemiring.nat_cast\u27e9\n\n@[simp] lemma of_nat_cast (n : \u2115) : of A 0 n = n :=\nrfl\n\n/-- The `semiring` structure derived from `gsemiring A`. -/\ninstance grade_zero.semiring : semiring (A 0) :=\nfunction.injective.semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_nsmul (\u03bb x n, of_zero_pow _ _ _) (of_nat_cast A)\n\n/-- `of A 0` is a `ring_hom`, using the `direct_sum.grade_zero.semiring` structure. -/\ndef of_zero_ring_hom : A 0 \u2192+* (\u2a01 i, A i) :=\n{ map_one' := of_zero_one A, map_mul' := of_zero_mul A, ..(of _ 0) }\n\n/-- Each grade `A i` derives a `A 0`-module structure from `gsemiring A`. Note that this results\nin an overall `module (A 0) (\u2a01 i, A i)` structure via `direct_sum.module`.\n-/\ninstance grade_zero.module {i} : module (A 0) (A i) :=\nbegin\n  letI := module.comp_hom (\u2a01 i, A i) (of_zero_ring_hom A),\n  exact dfinsupp.single_injective.module (A 0) (of A i) (\u03bb a, of_zero_smul A a),\nend\n\nend semiring\n\nsection comm_semiring\n\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\n/-- The `comm_semiring` structure derived from `gcomm_semiring A`. -/\ninstance grade_zero.comm_semiring : comm_semiring (A 0) :=\nfunction.injective.comm_semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (\u03bb x n, dfinsupp.single_smul n x) (\u03bb x n, of_zero_pow _ _ _) (of_nat_cast A)\n\nend comm_semiring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_zero_class \u03b9] [gnon_unital_non_assoc_semiring A]\n\n/-- The `non_unital_non_assoc_ring` derived from `gnon_unital_non_assoc_semiring A`. -/\ninstance grade_zero.non_unital_non_assoc_ring : non_unital_non_assoc_ring (A 0) :=\nfunction.injective.non_unital_non_assoc_ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n\nend ring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_monoid \u03b9] [gring A]\n\ninstance : has_int_cast (A 0) := \u27e8gring.int_cast\u27e9\n\n@[simp] lemma of_int_cast (n : \u2124) : of A 0 n = n :=\nrfl\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance grade_zero.ring : ring (A 0) :=\nfunction.injective.ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end) (\u03bb x n, of_zero_pow _ _ _)\n  (of_nat_cast A) (of_int_cast A)\n\nend ring\n\nsection comm_ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_comm_monoid \u03b9] [gcomm_ring A]\n\n/-- The `comm_ring` derived from `gcomm_semiring A`. -/\ninstance grade_zero.comm_ring : comm_ring (A 0) :=\nfunction.injective.comm_ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end) (\u03bb x n, of_zero_pow _ _ _)\n  (of_nat_cast A) (of_int_cast A)\n\nend comm_ring\n\nend grade_zero\n\nsection to_semiring\n\nvariables {R : Type*} [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A] [semiring R]\nvariables {A}\n\n/-- If two ring homomorphisms from `\u2a01 i, A i` are equal on each `of A i y`,\nthen they are equal.\n\nSee note [partially-applied ext lemmas]. -/\n@[ext]\nlemma ring_hom_ext' \u2983F G : (\u2a01 i, A i) \u2192+* R\u2984\n  (h : \u2200 i, (\u2191F : _ \u2192+ R).comp (of A i) = (\u2191G : _ \u2192+ R).comp (of A i)) : F = G :=\nring_hom.coe_add_monoid_hom_injective $ direct_sum.add_hom_ext' h\n\n/-- Two `ring_hom`s out of a direct sum are equal if they agree on the generators. -/\nlemma ring_hom_ext \u2983f g : (\u2a01 i, A i) \u2192+* R\u2984 (h : \u2200 i x, f (of A i x) = g (of A i x)) :\n  f = g :=\nring_hom_ext' $ \u03bb i, add_monoid_hom.ext $ h i\n\n/-- A family of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`\ndescribes a `ring_hom`s on `\u2a01 i, A i`. This is a stronger version of `direct_sum.to_monoid`.\n\nOf particular interest is the case when `A i` are bundled subojects, `f` is the family of\ncoercions such as `add_submonoid.subtype (A i)`, and the `[gsemiring A]` structure originates from\n`direct_sum.gsemiring.of_add_submonoids`, in which case the proofs about `ghas_one` and `ghas_mul`\ncan be discharged by `rfl`. -/\n@[simps]\ndef to_semiring\n  (f : \u03a0 i, A i \u2192+ R) (hone : f _ (graded_monoid.ghas_one.one) = 1)\n  (hmul : \u2200 {i j} (ai : A i) (aj : A j), f _ (graded_monoid.ghas_mul.mul ai aj) = f _ ai * f _ aj) :\n  (\u2a01 i, A i) \u2192+* R :=\n{ to_fun := to_add_monoid f,\n  map_one' := begin\n    change (to_add_monoid f) (of _ 0 _) = 1,\n    rw to_add_monoid_of,\n    exact hone\n  end,\n  map_mul' := begin\n    rw (to_add_monoid f).map_mul_iff,\n    ext xi xv yi yv : 4,\n    show to_add_monoid f (of A xi xv * of A yi yv) =\n         to_add_monoid f (of A xi xv) * to_add_monoid f (of A yi yv),\n    rw [of_mul_of, to_add_monoid_of, to_add_monoid_of, to_add_monoid_of],\n    exact hmul _ _,\n  end,\n  .. to_add_monoid f}\n\n@[simp] lemma to_semiring_of (f : \u03a0 i, A i \u2192+ R) (hone hmul) (i : \u03b9) (x : A i) :\n  to_semiring f hone hmul (of _ i x) = f _ x :=\nto_add_monoid_of f i x\n\n@[simp] lemma to_semiring_coe_add_monoid_hom (f : \u03a0 i, A i \u2192+ R) (hone hmul):\n  (to_semiring f hone hmul : (\u2a01 i, A i) \u2192+ R) = to_add_monoid f := rfl\n\n/-- Families of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`\nare isomorphic to `ring_hom`s on `\u2a01 i, A i`. This is a stronger version of `dfinsupp.lift_add_hom`.\n-/\n@[simps]\ndef lift_ring_hom :\n  {f : \u03a0 {i}, A i \u2192+ R //\n    f (graded_monoid.ghas_one.one) = 1 \u2227\n    \u2200 {i j} (ai : A i) (aj : A j), f (graded_monoid.ghas_mul.mul ai aj) = f ai * f aj} \u2243\n    ((\u2a01 i, A i) \u2192+* R) :=\n{ to_fun := \u03bb f, to_semiring f.1 f.2.1 f.2.2,\n  inv_fun := \u03bb F,\n    \u27e8\u03bb i, (F : (\u2a01 i, A i) \u2192+ R).comp (of _ i), begin\n      simp only [add_monoid_hom.comp_apply, ring_hom.coe_add_monoid_hom],\n      rw \u2190F.map_one,\n      refl\n    end, \u03bb i j ai aj, begin\n      simp only [add_monoid_hom.comp_apply, ring_hom.coe_add_monoid_hom],\n      rw [\u2190F.map_mul, of_mul_of],\n    end\u27e9,\n  left_inv := \u03bb f, begin\n    ext xi xv,\n    exact to_add_monoid_of f.1 xi xv,\n  end,\n  right_inv := \u03bb F, begin\n    apply ring_hom.coe_add_monoid_hom_injective,\n    ext xi xv,\n    simp only [ring_hom.coe_add_monoid_hom_mk,\n      direct_sum.to_add_monoid_of,\n      add_monoid_hom.mk_coe,\n      add_monoid_hom.comp_apply, to_semiring_coe_add_monoid_hom],\n  end}\n\nend to_semiring\n\nend direct_sum\n\n/-! ### Concrete instances -/\n\nsection uniform\n\nvariables (\u03b9)\n\n/-- A direct sum of copies of a `semiring` inherits the multiplication structure. -/\ninstance non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring\n  {R : Type*} [add_monoid \u03b9] [non_unital_non_assoc_semiring R] :\n  direct_sum.gnon_unital_non_assoc_semiring (\u03bb i : \u03b9, R) :=\n{ mul_zero := \u03bb i j, mul_zero,\n  zero_mul := \u03bb i j, zero_mul,\n  mul_add := \u03bb i j, mul_add,\n  add_mul := \u03bb i j, add_mul,\n  ..has_mul.ghas_mul \u03b9 }\n\n/-- A direct sum of copies of a `semiring` inherits the multiplication structure. -/\ninstance semiring.direct_sum_gsemiring {R : Type*} [add_monoid \u03b9] [semiring R] :\n  direct_sum.gsemiring (\u03bb i : \u03b9, R) :=\n{ nat_cast := \u03bb n, n,\n  nat_cast_zero := nat.cast_zero,\n  nat_cast_succ := nat.cast_succ,\n  ..non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring \u03b9,\n  ..monoid.gmonoid \u03b9 }\n\nopen_locale direct_sum\n\n-- To check `has_mul.ghas_mul_mul` matches\nexample {R : Type*} [add_monoid \u03b9] [semiring R] (i j : \u03b9) (a b : R) :\n  (direct_sum.of _ i a * direct_sum.of _ j b : \u2a01 i, R) = direct_sum.of _ (i + j) (by exact a * b) :=\nby rw [direct_sum.of_mul_of, has_mul.ghas_mul_mul]\n\n/-- A direct sum of copies of a `comm_semiring` inherits the commutative multiplication structure.\n-/\ninstance comm_semiring.direct_sum_gcomm_semiring {R : Type*} [add_comm_monoid \u03b9] [comm_semiring R] :\n  direct_sum.gcomm_semiring (\u03bb i : \u03b9, R) :=\n{ ..comm_monoid.gcomm_monoid \u03b9, ..semiring.direct_sum_gsemiring \u03b9 }\n\nend uniform\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/algebra/direct_sum/ring.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702642896702, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3991189065753687}}
{"text": "import Runtime.Network.Graph.Path.Basic\n\nnamespace Network.Graph.Path\n\n-- The prefix of a path is the path without the leaf.\n-- If the path is already `.nil` it remains the same.\ndef prefix? : Path graph start \u2192 Option (Path graph start)\n  | nil => none\n  | cons _ nil => some nil\n  | cons child subpath => subpath.prefix? >>= (cons child \u00b7)\n\ntheorem prefix?_cons_eq_cons_prefix?\n  {graph start} {child : Class.Child start} {subpath subprefix : Path graph child.class} :\n  (subpath.prefix? = some subprefix) \u2192 (cons child subpath).prefix? = cons child subprefix := by\n  intro h\n  cases subpath\n  case nil => simp [prefix?] at h\n  case cons child' subpath => simp [prefix?, h]; rfl\n\ntheorem prefix?_isSome_iff_isCons {path : Path graph start} :\n  path.prefix?.isSome \u2194 path.isCons := by\n  constructor\n  case mp =>\n    intro h\n    induction path\n    case nil => simp [prefix?] at h\n    case cons child\u2081 subpath\u2081 hi =>\n      cases subpath\u2081\n      case nil => rfl\n      case cons child\u2082 subpath\u2082 => simp [isCons]\n  case mpr =>\n    intro h\n    induction path\n    case nil =>\n      have \u27e8_, _, _\u27e9 := isCons_def.mp h\n      contradiction\n    case cons child\u2081 subpath\u2081 hi =>\n      cases subpath\u2081\n      case nil => simp [prefix?, Option.isSome]\n      case cons child\u2082 subpath\u2082 =>\n        specialize hi isCons_of_cons\n        have \u27e8subpath, hi\u27e9 := Option.isSome_iff_exists.mp hi\n        simp [prefix?, hi, Option.isSome_iff_exists]\n        exists .cons child\u2081 subpath\n\ntheorem prefix?_iff_cons_prefix?\n  {graph start} {child : Class.Child start} {path\u2081 path\u2082 : Path graph child.class} :\n  (path\u2081.prefix? = some path\u2082) \u2194 (cons child path\u2081).prefix? = some (cons child path\u2082) := by\n  constructor\n  all_goals\n    intro h\n    cases path\u2081\n    case nil => simp [prefix?] at h\n  case mp.cons => simp [prefix?, h]; rfl\n  case mpr.cons child subpath =>\n    have hs := prefix?_isSome_iff_isCons.mpr (@isCons_of_cons _ _ child subpath)\n    have \u27e8_, hs\u27e9 := Option.isSome_iff_exists.mp hs\n    simp [hs, prefix?] at h \u22a2\n    injection h with h\n    injection h\n\ndef snd (path : Path graph start) (_ : path.isCons) : Class.Child start :=\n  match path with\n  | nil => by contradiction\n  | cons child _ => child\n\n@[simp]\ntheorem cons_snd_eq_child : (cons child path).snd h = child := rfl\n\n-- Note: We can't define this property with an optional return type,\n--       as we can't even state the return type for an invalid input path.\ndef suffix (path : Path graph start) (h) : Path graph (path.snd h) :=\n  match path with\n  | nil => by contradiction\n  | cons _ subpath => subpath\n\n@[simp]\ntheorem suffix_class {path : Path graph start} {h} : (path.suffix h).class = path.class := by\n  rw [suffix]\n  split\n  case _ h => simp [isCons] at h\n  \u00b7 simp\n\ndef split (path : Path graph start) (_ : path.isCons) :\n  \u03a3 \u00abprefix\u00bb : Path graph start, Class.Child prefix.class :=\n  match path with\n  | nil => by contradiction\n  | cons child nil => \u27e8nil, child\u27e9\n  | cons child subpath@(cons _ _) =>\n    let \u27e8sub, cls\u27e9 := subpath.split (by simp_all [isCons_of_cons])\n    \u27e8cons child sub, cls\u27e9\n\ntheorem split_class {path : Path graph start} (h : path.isCons) :\n  (path.split h).snd.class = path.class := by\n  induction path\n  case nil => contradiction\n  case cons child subpath hi =>\n    cases subpath\n    case nil => rfl\n    case cons child' subpath => rw [split]; exact hi h\n\nend Network.Graph.Path\n", "meta": {"author": "lf-lang", "repo": "reactor-lean", "sha": "d2eb5458446af838be34ebb6f69549b2f6d9c04d", "save_path": "github-repos/lean/lf-lang-reactor-lean", "path": "github-repos/lean/lf-lang-reactor-lean/reactor-lean-d2eb5458446af838be34ebb6f69549b2f6d9c04d/Runtime/Network/Graph/Path/Subpaths.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.39894236604101835}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nFinite types.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.wlog\nimport Mathlib.data.finset.powerset\nimport Mathlib.data.finset.lattice\nimport Mathlib.data.finset.pi\nimport Mathlib.data.array.lemmas\nimport Mathlib.order.well_founded\nimport Mathlib.group_theory.perm.basic\nimport Mathlib.PostPort\n\nuniverses u_4 l u_1 u_2 u v \n\nnamespace Mathlib\n\n/-- `fintype \u03b1` means that `\u03b1` is finite, i.e. there are only\n  finitely many distinct elements of type `\u03b1`. The evidence of this\n  is a finset `elems` (a list up to permutation without duplicates),\n  together with a proof that everything of type `\u03b1` is in the list. -/\nclass fintype (\u03b1 : Type u_4) \nwhere\n  elems : finset \u03b1\n  complete : \u2200 (x : \u03b1), x \u2208 elems\n\nnamespace finset\n\n\n/-- `univ` is the universal finite set of type `finset \u03b1` implied from\n  the assumption `fintype \u03b1`. -/\ndef univ {\u03b1 : Type u_1} [fintype \u03b1] : finset \u03b1 :=\n  fintype.elems \u03b1\n\n@[simp] theorem mem_univ {\u03b1 : Type u_1} [fintype \u03b1] (x : \u03b1) : x \u2208 univ :=\n  fintype.complete x\n\n@[simp] theorem mem_univ_val {\u03b1 : Type u_1} [fintype \u03b1] (x : \u03b1) : x \u2208 val univ :=\n  mem_univ\n\n@[simp] theorem coe_univ {\u03b1 : Type u_1} [fintype \u03b1] : \u2191univ = set.univ := sorry\n\ntheorem univ_nonempty_iff {\u03b1 : Type u_1} [fintype \u03b1] : finset.nonempty univ \u2194 Nonempty \u03b1 := sorry\n\ntheorem univ_nonempty {\u03b1 : Type u_1} [fintype \u03b1] [Nonempty \u03b1] : finset.nonempty univ :=\n  iff.mpr univ_nonempty_iff _inst_2\n\ntheorem univ_eq_empty {\u03b1 : Type u_1} [fintype \u03b1] : univ = \u2205 \u2194 \u00acNonempty \u03b1 := sorry\n\ntheorem subset_univ {\u03b1 : Type u_1} [fintype \u03b1] (s : finset \u03b1) : s \u2286 univ :=\n  fun (a : \u03b1) (_x : a \u2208 s) => mem_univ a\n\nprotected instance order_top {\u03b1 : Type u_1} [fintype \u03b1] : order_top (finset \u03b1) :=\n  order_top.mk univ partial_order.le partial_order.lt sorry sorry sorry subset_univ\n\nprotected instance boolean_algebra {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] : boolean_algebra (finset \u03b1) :=\n  boolean_algebra.mk distrib_lattice.sup distrib_lattice.le distrib_lattice.lt sorry sorry sorry sorry sorry sorry\n    distrib_lattice.inf sorry sorry sorry sorry order_top.top sorry semilattice_inf_bot.bot sorry\n    (fun (s : finset \u03b1) => univ \\ s) has_sdiff.sdiff sorry sorry sorry\n\ntheorem compl_eq_univ_sdiff {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : s\u1d9c = univ \\ s :=\n  rfl\n\n@[simp] theorem mem_compl {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] {s : finset \u03b1} {x : \u03b1} : x \u2208 (s\u1d9c) \u2194 \u00acx \u2208 s := sorry\n\n@[simp] theorem coe_compl {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : \u2191(s\u1d9c) = (\u2191s\u1d9c) :=\n  set.ext fun (x : \u03b1) => mem_compl\n\ntheorem eq_univ_iff_forall {\u03b1 : Type u_1} [fintype \u03b1] {s : finset \u03b1} : s = univ \u2194 \u2200 (x : \u03b1), x \u2208 s := sorry\n\ntheorem compl_ne_univ_iff_nonempty {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : s\u1d9c \u2260 univ \u2194 finset.nonempty s := sorry\n\n@[simp] theorem univ_inter {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : univ \u2229 s = s := sorry\n\n@[simp] theorem inter_univ {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : s \u2229 univ = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s \u2229 univ = s)) (inter_comm s univ)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (univ \u2229 s = s)) (univ_inter s))) (Eq.refl s))\n\n@[simp] theorem piecewise_univ {\u03b1 : Type u_1} [fintype \u03b1] [(i : \u03b1) \u2192 Decidable (i \u2208 univ)] {\u03b4 : \u03b1 \u2192 Sort u_2} (f : (i : \u03b1) \u2192 \u03b4 i) (g : (i : \u03b1) \u2192 \u03b4 i) : piecewise univ f g = f := sorry\n\ntheorem piecewise_compl {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) [(i : \u03b1) \u2192 Decidable (i \u2208 s)] [(i : \u03b1) \u2192 Decidable (i \u2208 (s\u1d9c))] {\u03b4 : \u03b1 \u2192 Sort u_2} (f : (i : \u03b1) \u2192 \u03b4 i) (g : (i : \u03b1) \u2192 \u03b4 i) : piecewise (s\u1d9c) f g = piecewise s g f := sorry\n\ntheorem univ_map_equiv_to_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (e : \u03b1 \u2243 \u03b2) : map (equiv.to_embedding e) univ = univ := sorry\n\n@[simp] theorem univ_filter_exists {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) [fintype \u03b2] [decidable_pred fun (y : \u03b2) => \u2203 (x : \u03b1), f x = y] [DecidableEq \u03b2] : filter (fun (y : \u03b2) => \u2203 (x : \u03b1), f x = y) univ = image f univ := sorry\n\n/-- Note this is a special case of `(finset.image_preimage f univ _).symm`. -/\ntheorem univ_filter_mem_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) [fintype \u03b2] [decidable_pred fun (y : \u03b2) => y \u2208 set.range f] [DecidableEq \u03b2] : filter (fun (y : \u03b2) => y \u2208 set.range f) univ = image f univ :=\n  univ_filter_exists f\n\nend finset\n\n\nnamespace fintype\n\n\nprotected instance decidable_pi_fintype {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [(a : \u03b1) \u2192 DecidableEq (\u03b2 a)] [fintype \u03b1] : DecidableEq ((a : \u03b1) \u2192 \u03b2 a) :=\n  fun (f g : (a : \u03b1) \u2192 \u03b2 a) => decidable_of_iff (\u2200 (a : \u03b1), a \u2208 elems \u03b1 \u2192 f a = g a) sorry\n\nprotected instance decidable_forall_fintype {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] [fintype \u03b1] : Decidable (\u2200 (a : \u03b1), p a) :=\n  decidable_of_iff (\u2200 (a : \u03b1), a \u2208 finset.univ \u2192 p a) sorry\n\nprotected instance decidable_exists_fintype {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] [fintype \u03b1] : Decidable (\u2203 (a : \u03b1), p a) :=\n  decidable_of_iff (\u2203 (a : \u03b1), \u2203 (H : a \u2208 finset.univ), p a) sorry\n\nprotected instance decidable_eq_equiv_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b2] [fintype \u03b1] : DecidableEq (\u03b1 \u2243 \u03b2) :=\n  fun (a b : \u03b1 \u2243 \u03b2) => decidable_of_iff (equiv.to_fun a = equiv.to_fun b) sorry\n\nprotected instance decidable_injective_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] [fintype \u03b1] : decidable_pred function.injective :=\n  fun (x : \u03b1 \u2192 \u03b2) => eq.mpr sorry fintype.decidable_forall_fintype\n\nprotected instance decidable_surjective_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b2] [fintype \u03b1] [fintype \u03b2] : decidable_pred function.surjective :=\n  fun (x : \u03b1 \u2192 \u03b2) => eq.mpr sorry fintype.decidable_forall_fintype\n\nprotected instance decidable_bijective_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] [fintype \u03b1] [fintype \u03b2] : decidable_pred function.bijective :=\n  fun (x : \u03b1 \u2192 \u03b2) => eq.mpr sorry and.decidable\n\nprotected instance decidable_left_inverse_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) : Decidable (function.right_inverse f g) :=\n  (fun (this : Decidable (\u2200 (x : \u03b1), g (f x) = x)) => this) fintype.decidable_forall_fintype\n\nprotected instance decidable_right_inverse_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b2] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) : Decidable (function.left_inverse f g) :=\n  (fun (this : Decidable (\u2200 (x : \u03b2), f (g x) = x)) => this) fintype.decidable_forall_fintype\n\n/-- Construct a proof of `fintype \u03b1` from a universal multiset -/\ndef of_multiset {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s) : fintype \u03b1 :=\n  mk (multiset.to_finset s) sorry\n\n/-- Construct a proof of `fintype \u03b1` from a universal list -/\ndef of_list {\u03b1 : Type u_1} [DecidableEq \u03b1] (l : List \u03b1) (H : \u2200 (x : \u03b1), x \u2208 l) : fintype \u03b1 :=\n  mk (list.to_finset l) sorry\n\ntheorem exists_univ_list (\u03b1 : Type u_1) [fintype \u03b1] : \u2203 (l : List \u03b1), list.nodup l \u2227 \u2200 (x : \u03b1), x \u2208 l := sorry\n\n/-- `card \u03b1` is the number of elements in `\u03b1`, defined when `\u03b1` is a fintype. -/\ndef card (\u03b1 : Type u_1) [fintype \u03b1] : \u2115 :=\n  finset.card finset.univ\n\n/-- If `l` lists all the elements of `\u03b1` without duplicates, then `\u03b1 \u2243 fin (l.length)`. -/\ndef equiv_fin_of_forall_mem_list {\u03b1 : Type u_1} [DecidableEq \u03b1] {l : List \u03b1} (h : \u2200 (x : \u03b1), x \u2208 l) (nd : list.nodup l) : \u03b1 \u2243 fin (list.length l) :=\n  equiv.mk (fun (a : \u03b1) => { val := list.index_of a l, property := sorry })\n    (fun (i : fin (list.length l)) => list.nth_le l (subtype.val i) sorry) sorry sorry\n\n/-- There is (computably) a bijection between `\u03b1` and `fin n` where\n  `n = card \u03b1`. Since it is not unique, and depends on which permutation\n  of the universe list is used, the bijection is wrapped in `trunc` to\n  preserve computability.  -/\ndef equiv_fin (\u03b1 : Type u_1) [DecidableEq \u03b1] [fintype \u03b1] : trunc (\u03b1 \u2243 fin (card \u03b1)) :=\n  eq.mpr sorry\n    (quot.rec_on_subsingleton (finset.val finset.univ)\n      (fun (l : List \u03b1) (h : \u2200 (x : \u03b1), x \u2208 l) (nd : list.nodup l) => trunc.mk (equiv_fin_of_forall_mem_list h nd))\n      finset.mem_univ_val sorry)\n\ntheorem exists_equiv_fin (\u03b1 : Type u_1) [fintype \u03b1] : \u2203 (n : \u2115), Nonempty (\u03b1 \u2243 fin n) :=\n  Exists.intro (card \u03b1) (nonempty_of_trunc (equiv_fin \u03b1))\n\nprotected instance subsingleton (\u03b1 : Type u_1) : subsingleton (fintype \u03b1) :=\n  subsingleton.intro fun (_x : fintype \u03b1) => sorry\n\n/-- Given a predicate that can be represented by a finset, the subtype\nassociated to the predicate is a fintype. -/\nprotected def subtype {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 p x) : fintype (Subtype fun (x : \u03b1) => p x) :=\n  mk (finset.mk (multiset.pmap Subtype.mk (finset.val s) sorry) sorry) sorry\n\ntheorem subtype_card {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 p x) : card (Subtype fun (x : \u03b1) => p x) = finset.card s :=\n  multiset.card_pmap Subtype.mk (finset.val s) (subtype._proof_1 s H)\n\ntheorem card_of_subtype {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 p x) [fintype (Subtype fun (x : \u03b1) => p x)] : card (Subtype fun (x : \u03b1) => p x) = finset.card s := sorry\n\n/-- Construct a fintype from a finset with the same elements. -/\ndef of_finset {\u03b1 : Type u_1} {p : set \u03b1} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p) : fintype \u21a5p :=\n  fintype.subtype s H\n\n@[simp] theorem card_of_finset {\u03b1 : Type u_1} {p : set \u03b1} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p) : card \u21a5p = finset.card s :=\n  subtype_card s H\n\ntheorem card_of_finset' {\u03b1 : Type u_1} {p : set \u03b1} (s : finset \u03b1) (H : \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 p) [fintype \u21a5p] : card \u21a5p = finset.card s := sorry\n\n/-- If `f : \u03b1 \u2192 \u03b2` is a bijection and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -/\ndef of_bijective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (H : function.bijective f) : fintype \u03b2 :=\n  mk (finset.map (function.embedding.mk f sorry) finset.univ) sorry\n\n/-- If `f : \u03b1 \u2192 \u03b2` is a surjection and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -/\ndef of_surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b2] [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) (H : function.surjective f) : fintype \u03b2 :=\n  mk (finset.image f finset.univ) sorry\n\n/-- Given an injective function to a fintype, the domain is also a\nfintype. This is noncomputable because injectivity alone cannot be\nused to construct preimages. -/\ndef of_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (H : function.injective f) : fintype \u03b1 :=\n  let _inst : (p : Prop) \u2192 Decidable p := classical.dec;\n  dite (Nonempty \u03b1) (fun (h\u03b1 : Nonempty \u03b1) => of_surjective (function.inv_fun f) (function.inv_fun_surjective H))\n    fun (h\u03b1 : \u00acNonempty \u03b1) => mk \u2205 sorry\n\n/-- If `f : \u03b1 \u2243 \u03b2` and `\u03b1` is a fintype, then `\u03b2` is also a fintype. -/\ndef of_equiv {\u03b2 : Type u_2} (\u03b1 : Type u_1) [fintype \u03b1] (f : \u03b1 \u2243 \u03b2) : fintype \u03b2 :=\n  of_bijective (\u21d1f) (equiv.bijective f)\n\ntheorem of_equiv_card {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] (f : \u03b1 \u2243 \u03b2) : card \u03b2 = card \u03b1 :=\n  multiset.card_map (\u21d1(function.embedding.mk (\u21d1f) (of_bijective._proof_1 (\u21d1f) (equiv.bijective f))))\n    (finset.val finset.univ)\n\ntheorem card_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2243 \u03b2) : card \u03b1 = card \u03b2 := sorry\n\ntheorem card_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [F : fintype \u03b1] [G : fintype \u03b2] : card \u03b1 = card \u03b2 \u2194 Nonempty (\u03b1 \u2243 \u03b2) := sorry\n\n/-- Subsingleton types are fintypes (with zero or one terms). -/\ndef of_subsingleton {\u03b1 : Type u_1} (a : \u03b1) [subsingleton \u03b1] : fintype \u03b1 :=\n  mk (singleton a) sorry\n\n@[simp] theorem univ_of_subsingleton {\u03b1 : Type u_1} (a : \u03b1) [subsingleton \u03b1] : finset.univ = singleton a :=\n  rfl\n\n@[simp] theorem card_of_subsingleton {\u03b1 : Type u_1} (a : \u03b1) [subsingleton \u03b1] : card \u03b1 = 1 :=\n  rfl\n\nend fintype\n\n\nnamespace set\n\n\n/-- Construct a finset enumerating a set `s`, given a `fintype` instance.  -/\ndef to_finset {\u03b1 : Type u_1} (s : set \u03b1) [fintype \u21a5s] : finset \u03b1 :=\n  finset.mk (multiset.map subtype.val (finset.val finset.univ)) sorry\n\n@[simp] theorem mem_to_finset {\u03b1 : Type u_1} {s : set \u03b1} [fintype \u21a5s] {a : \u03b1} : a \u2208 to_finset s \u2194 a \u2208 s := sorry\n\n@[simp] theorem mem_to_finset_val {\u03b1 : Type u_1} {s : set \u03b1} [fintype \u21a5s] {a : \u03b1} : a \u2208 finset.val (to_finset s) \u2194 a \u2208 s :=\n  mem_to_finset\n\n-- We use an arbitrary `[fintype s]` instance here,\n\n-- not necessarily coming from a `[fintype \u03b1]`.\n\n@[simp] theorem to_finset_card {\u03b1 : Type u_1} (s : set \u03b1) [fintype \u21a5s] : finset.card (to_finset s) = fintype.card \u21a5s :=\n  multiset.card_map subtype.val (finset.val finset.univ)\n\n@[simp] theorem coe_to_finset {\u03b1 : Type u_1} (s : set \u03b1) [fintype \u21a5s] : \u2191(to_finset s) = s :=\n  ext fun (_x : \u03b1) => mem_to_finset\n\n@[simp] theorem to_finset_inj {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [fintype \u21a5s] [fintype \u21a5t] : to_finset s = to_finset t \u2194 s = t := sorry\n\nend set\n\n\ntheorem finset.card_univ {\u03b1 : Type u_1} [fintype \u03b1] : finset.card finset.univ = fintype.card \u03b1 :=\n  rfl\n\ntheorem finset.eq_univ_of_card {\u03b1 : Type u_1} [fintype \u03b1] (s : finset \u03b1) (hs : finset.card s = fintype.card \u03b1) : s = finset.univ := sorry\n\ntheorem finset.card_eq_iff_eq_univ {\u03b1 : Type u_1} [fintype \u03b1] (s : finset \u03b1) : finset.card s = fintype.card \u03b1 \u2194 s = finset.univ :=\n  { mp := finset.eq_univ_of_card s, mpr := fun (\u1fb0 : s = finset.univ) => Eq._oldrec finset.card_univ (Eq.symm \u1fb0) }\n\ntheorem finset.card_le_univ {\u03b1 : Type u_1} [fintype \u03b1] (s : finset \u03b1) : finset.card s \u2264 fintype.card \u03b1 :=\n  finset.card_le_of_subset (finset.subset_univ s)\n\ntheorem finset.card_lt_iff_ne_univ {\u03b1 : Type u_1} [fintype \u03b1] (s : finset \u03b1) : finset.card s < fintype.card \u03b1 \u2194 s \u2260 finset.univ :=\n  iff.trans (has_le.le.lt_iff_ne (finset.card_le_univ s)) (not_iff_not_of_iff (finset.card_eq_iff_eq_univ s))\n\ntheorem finset.card_compl_lt_iff_nonempty {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (s : finset \u03b1) : finset.card (s\u1d9c) < fintype.card \u03b1 \u2194 finset.nonempty s :=\n  iff.trans (finset.card_lt_iff_ne_univ (s\u1d9c)) (finset.compl_ne_univ_iff_nonempty s)\n\ntheorem finset.card_univ_diff {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] (s : finset \u03b1) : finset.card (finset.univ \\ s) = fintype.card \u03b1 - finset.card s :=\n  finset.card_sdiff (finset.subset_univ s)\n\ntheorem finset.card_compl {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] (s : finset \u03b1) : finset.card (s\u1d9c) = fintype.card \u03b1 - finset.card s :=\n  finset.card_univ_diff s\n\nprotected instance fin.fintype (n : \u2115) : fintype (fin n) :=\n  fintype.mk (finset.fin_range n) finset.mem_fin_range\n\ntheorem fin.univ_def (n : \u2115) : finset.univ = finset.fin_range n :=\n  rfl\n\n@[simp] theorem fintype.card_fin (n : \u2115) : fintype.card (fin n) = n :=\n  list.length_fin_range n\n\n@[simp] theorem finset.card_fin (n : \u2115) : finset.card finset.univ = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (finset.card finset.univ = n)) finset.card_univ))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (fintype.card (fin n) = n)) (fintype.card_fin n))) (Eq.refl n))\n\ntheorem fin.equiv_iff_eq {m : \u2115} {n : \u2115} : Nonempty (fin m \u2243 fin n) \u2194 m = n := sorry\n\n/-- Embed `fin n` into `fin (n + 1)` by prepending zero to the `univ` -/\ntheorem fin.univ_succ (n : \u2115) : finset.univ = insert 0 (finset.image fin.succ finset.univ) := sorry\n\n/-- Embed `fin n` into `fin (n + 1)` by appending a new `fin.last n` to the `univ` -/\ntheorem fin.univ_cast_succ (n : \u2115) : finset.univ = insert (fin.last n) (finset.image (\u21d1fin.cast_succ) finset.univ) := sorry\n\n/-- Embed `fin n` into `fin (n + 1)` by inserting\naround a specified pivot `p : fin (n + 1)` into the `univ` -/\ntheorem fin.univ_succ_above (n : \u2115) (p : fin (n + 1)) : finset.univ = insert p (finset.image (\u21d1(fin.succ_above p)) finset.univ) := sorry\n\ninstance unique.fintype {\u03b1 : Type u_1} [unique \u03b1] : fintype \u03b1 :=\n  fintype.of_subsingleton Inhabited.default\n\n@[simp] theorem univ_unique {\u03b1 : Type u_1} [unique \u03b1] [f : fintype \u03b1] : finset.univ = singleton Inhabited.default :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (finset.univ = singleton Inhabited.default)) (subsingleton.elim f unique.fintype)))\n    (Eq.refl finset.univ)\n\nprotected instance empty.fintype : fintype empty :=\n  fintype.mk \u2205 sorry\n\n@[simp] theorem fintype.univ_empty : finset.univ = \u2205 :=\n  rfl\n\n@[simp] theorem fintype.card_empty : fintype.card empty = 0 :=\n  rfl\n\nprotected instance pempty.fintype : fintype pempty :=\n  fintype.mk \u2205 sorry\n\n@[simp] theorem fintype.univ_pempty : finset.univ = \u2205 :=\n  rfl\n\n@[simp] theorem fintype.card_pempty : fintype.card pempty = 0 :=\n  rfl\n\nprotected instance unit.fintype : fintype Unit :=\n  fintype.of_subsingleton Unit.unit\n\ntheorem fintype.univ_unit : finset.univ = singleton Unit.unit :=\n  rfl\n\ntheorem fintype.card_unit : fintype.card Unit = 1 :=\n  rfl\n\nprotected instance punit.fintype : fintype PUnit :=\n  fintype.of_subsingleton PUnit.unit\n\n@[simp] theorem fintype.univ_punit : finset.univ = singleton PUnit.unit :=\n  rfl\n\n@[simp] theorem fintype.card_punit : fintype.card PUnit = 1 :=\n  rfl\n\nprotected instance bool.fintype : fintype Bool :=\n  fintype.mk (finset.mk (tt ::\u2098 false ::\u2098 0) sorry) sorry\n\n@[simp] theorem fintype.univ_bool : finset.univ = insert tt (singleton false) :=\n  rfl\n\nprotected instance units_int.fintype : fintype (units \u2124) :=\n  fintype.mk (insert 1 (singleton (-1))) sorry\n\nprotected instance additive.fintype {\u03b1 : Type u_1} [fintype \u03b1] : fintype (additive \u03b1) :=\n  id\n\nprotected instance multiplicative.fintype {\u03b1 : Type u_1} [fintype \u03b1] : fintype (multiplicative \u03b1) :=\n  id\n\n@[simp] theorem fintype.card_units_int : fintype.card (units \u2124) = bit0 1 :=\n  rfl\n\nprotected instance units.fintype {\u03b1 : Type u_1} [monoid \u03b1] [fintype \u03b1] : fintype (units \u03b1) :=\n  fintype.of_injective units.val units.ext\n\n@[simp] theorem fintype.card_bool : fintype.card Bool = bit0 1 :=\n  rfl\n\n/-- Given a finset on `\u03b1`, lift it to being a finset on `option \u03b1`\nusing `option.some` and then insert `option.none`. -/\ndef finset.insert_none {\u03b1 : Type u_1} (s : finset \u03b1) : finset (Option \u03b1) :=\n  finset.mk (none ::\u2098 multiset.map some (finset.val s)) sorry\n\n@[simp] theorem finset.mem_insert_none {\u03b1 : Type u_1} {s : finset \u03b1} {o : Option \u03b1} : o \u2208 finset.insert_none s \u2194 \u2200 (a : \u03b1), a \u2208 o \u2192 a \u2208 s := sorry\n\ntheorem finset.some_mem_insert_none {\u03b1 : Type u_1} {s : finset \u03b1} {a : \u03b1} : some a \u2208 finset.insert_none s \u2194 a \u2208 s := sorry\n\nprotected instance option.fintype {\u03b1 : Type u_1} [fintype \u03b1] : fintype (Option \u03b1) :=\n  fintype.mk (finset.insert_none finset.univ) sorry\n\n@[simp] theorem fintype.card_option {\u03b1 : Type u_1} [fintype \u03b1] : fintype.card (Option \u03b1) = fintype.card \u03b1 + 1 := sorry\n\nprotected instance sigma.fintype {\u03b1 : Type u_1} (\u03b2 : \u03b1 \u2192 Type u_2) [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : fintype (sigma \u03b2) :=\n  fintype.mk (finset.sigma finset.univ fun (_x : \u03b1) => finset.univ) sorry\n\n@[simp] theorem finset.univ_sigma_univ {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : (finset.sigma finset.univ fun (a : \u03b1) => finset.univ) = finset.univ :=\n  rfl\n\nprotected instance prod.fintype (\u03b1 : Type u_1) (\u03b2 : Type u_2) [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u00d7 \u03b2) :=\n  fintype.mk (finset.product finset.univ finset.univ) sorry\n\n@[simp] theorem finset.univ_product_univ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] : finset.product finset.univ finset.univ = finset.univ :=\n  rfl\n\n@[simp] theorem fintype.card_prod (\u03b1 : Type u_1) (\u03b2 : Type u_2) [fintype \u03b1] [fintype \u03b2] : fintype.card (\u03b1 \u00d7 \u03b2) = fintype.card \u03b1 * fintype.card \u03b2 :=\n  finset.card_product finset.univ finset.univ\n\n/-- Given that `\u03b1 \u00d7 \u03b2` is a fintype, `\u03b1` is also a fintype. -/\ndef fintype.fintype_prod_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [fintype (\u03b1 \u00d7 \u03b2)] [Nonempty \u03b2] : fintype \u03b1 :=\n  fintype.mk (finset.image prod.fst (fintype.elems (\u03b1 \u00d7 \u03b2))) sorry\n\n/-- Given that `\u03b1 \u00d7 \u03b2` is a fintype, `\u03b2` is also a fintype. -/\ndef fintype.fintype_prod_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b2] [fintype (\u03b1 \u00d7 \u03b2)] [Nonempty \u03b1] : fintype \u03b2 :=\n  fintype.mk (finset.image prod.snd (fintype.elems (\u03b1 \u00d7 \u03b2))) sorry\n\nprotected instance ulift.fintype (\u03b1 : Type u_1) [fintype \u03b1] : fintype (ulift \u03b1) :=\n  fintype.of_equiv \u03b1 (equiv.symm equiv.ulift)\n\n@[simp] theorem fintype.card_ulift (\u03b1 : Type u_1) [fintype \u03b1] : fintype.card (ulift \u03b1) = fintype.card \u03b1 :=\n  fintype.of_equiv_card (equiv.symm equiv.ulift)\n\ntheorem univ_sum_type {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] [fintype (\u03b1 \u2295 \u03b2)] [DecidableEq (\u03b1 \u2295 \u03b2)] : finset.univ = finset.map function.embedding.inl finset.univ \u222a finset.map function.embedding.inr finset.univ := sorry\n\nprotected instance sum.fintype (\u03b1 : Type u) (\u03b2 : Type v) [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u2295 \u03b2) :=\n  fintype.of_equiv (sigma fun (b : Bool) => cond b (ulift \u03b1) (ulift \u03b2))\n    (equiv.trans (equiv.symm (equiv.sum_equiv_sigma_bool (ulift \u03b1) (ulift \u03b2))) (equiv.sum_congr equiv.ulift equiv.ulift))\n\nnamespace fintype\n\n\ntheorem card_le_of_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : function.injective f) : card \u03b1 \u2264 card \u03b2 :=\n  finset.card_le_card_of_inj_on f (fun (_x : \u03b1) (_x_1 : _x \u2208 finset.univ) => finset.mem_univ (f _x))\n    fun (_x : \u03b1) (_x_1 : _x \u2208 finset.univ) (_x_2 : \u03b1) (_x_3 : _x_2 \u2208 finset.univ) (h : f _x = f _x_2) => hf h\n\n/--\nThe pigeonhole principle for finitely many pigeons and pigeonholes.\nThis is the `fintype` version of `finset.exists_ne_map_eq_of_card_lt_of_maps_to`.\n-/\ntheorem exists_ne_map_eq_of_card_lt {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (h : card \u03b2 < card \u03b1) : \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2260 y \u2227 f x = f y := sorry\n\ntheorem card_eq_one_iff {\u03b1 : Type u_1} [fintype \u03b1] : card \u03b1 = 1 \u2194 \u2203 (x : \u03b1), \u2200 (y : \u03b1), y = x := sorry\n\ntheorem card_eq_zero_iff {\u03b1 : Type u_1} [fintype \u03b1] : card \u03b1 = 0 \u2194 \u03b1 \u2192 False := sorry\n\n/-- A `fintype` with cardinality zero is (constructively) equivalent to `pempty`. -/\ndef card_eq_zero_equiv_equiv_pempty {\u03b1 : Type u_1} [fintype \u03b1] : card \u03b1 = 0 \u2243 (\u03b1 \u2243 pempty) :=\n  equiv.mk\n    (fun (h : card \u03b1 = 0) => equiv.mk (fun (a : \u03b1) => false.elim sorry) (fun (a : pempty) => pempty.elim a) sorry sorry)\n    sorry sorry sorry\n\ntheorem card_pos_iff {\u03b1 : Type u_1} [fintype \u03b1] : 0 < card \u03b1 \u2194 Nonempty \u03b1 := sorry\n\ntheorem card_le_one_iff {\u03b1 : Type u_1} [fintype \u03b1] : card \u03b1 \u2264 1 \u2194 \u2200 (a b : \u03b1), a = b := sorry\n\ntheorem card_le_one_iff_subsingleton {\u03b1 : Type u_1} [fintype \u03b1] : card \u03b1 \u2264 1 \u2194 subsingleton \u03b1 :=\n  iff.trans card_le_one_iff (iff.symm subsingleton_iff)\n\ntheorem one_lt_card_iff_nontrivial {\u03b1 : Type u_1} [fintype \u03b1] : 1 < card \u03b1 \u2194 nontrivial \u03b1 := sorry\n\ntheorem exists_ne_of_one_lt_card {\u03b1 : Type u_1} [fintype \u03b1] (h : 1 < card \u03b1) (a : \u03b1) : \u2203 (b : \u03b1), b \u2260 a :=\n  exists_ne a\n\ntheorem exists_pair_of_one_lt_card {\u03b1 : Type u_1} [fintype \u03b1] (h : 1 < card \u03b1) : \u2203 (a : \u03b1), \u2203 (b : \u03b1), a \u2260 b :=\n  exists_pair_ne \u03b1\n\ntheorem card_eq_one_of_forall_eq {\u03b1 : Type u_1} [fintype \u03b1] {i : \u03b1} (h : \u2200 (j : \u03b1), j = i) : card \u03b1 = 1 :=\n  le_antisymm (iff.mpr card_le_one_iff fun (a b : \u03b1) => Eq.trans (h a) (Eq.symm (h b)))\n    (iff.mpr finset.card_pos (Exists.intro i (finset.mem_univ i)))\n\ntheorem injective_iff_surjective {\u03b1 : Type u_1} [fintype \u03b1] {f : \u03b1 \u2192 \u03b1} : function.injective f \u2194 function.surjective f := sorry\n\ntheorem injective_iff_bijective {\u03b1 : Type u_1} [fintype \u03b1] {f : \u03b1 \u2192 \u03b1} : function.injective f \u2194 function.bijective f := sorry\n\ntheorem surjective_iff_bijective {\u03b1 : Type u_1} [fintype \u03b1] {f : \u03b1 \u2192 \u03b1} : function.surjective f \u2194 function.bijective f := sorry\n\ntheorem injective_iff_surjective_of_equiv {\u03b1 : Type u_1} [fintype \u03b1] {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (e : \u03b1 \u2243 \u03b2) : function.injective f \u2194 function.surjective f := sorry\n\ntheorem nonempty_equiv_of_card_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (h : card \u03b1 = card \u03b2) : Nonempty (\u03b1 \u2243 \u03b2) := sorry\n\ntheorem bijective_iff_injective_and_card {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) : function.bijective f \u2194 function.injective f \u2227 card \u03b1 = card \u03b2 := sorry\n\ntheorem bijective_iff_surjective_and_card {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) : function.bijective f \u2194 function.surjective f \u2227 card \u03b1 = card \u03b2 := sorry\n\nend fintype\n\n\ntheorem fintype.coe_image_univ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} : \u2191(finset.image f finset.univ) = set.range f := sorry\n\nprotected instance list.subtype.fintype {\u03b1 : Type u_1} [DecidableEq \u03b1] (l : List \u03b1) : fintype (Subtype fun (x : \u03b1) => x \u2208 l) :=\n  fintype.of_list (list.attach l) (list.mem_attach l)\n\nprotected instance multiset.subtype.fintype {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : fintype (Subtype fun (x : \u03b1) => x \u2208 s) :=\n  fintype.of_multiset (multiset.attach s) (multiset.mem_attach s)\n\nprotected instance finset.subtype.fintype {\u03b1 : Type u_1} (s : finset \u03b1) : fintype (Subtype fun (x : \u03b1) => x \u2208 s) :=\n  fintype.mk (finset.attach s) (finset.mem_attach s)\n\nprotected instance finset_coe.fintype {\u03b1 : Type u_1} (s : finset \u03b1) : fintype \u21a5\u2191s :=\n  finset.subtype.fintype s\n\n@[simp] theorem fintype.card_coe {\u03b1 : Type u_1} (s : finset \u03b1) : fintype.card \u21a5\u2191s = finset.card s :=\n  finset.card_attach\n\ntheorem finset.attach_eq_univ {\u03b1 : Type u_1} {s : finset \u03b1} : finset.attach s = finset.univ :=\n  rfl\n\ntheorem finset.card_le_one_iff {\u03b1 : Type u_1} {s : finset \u03b1} : finset.card s \u2264 1 \u2194 \u2200 {x y : \u03b1}, x \u2208 s \u2192 y \u2208 s \u2192 x = y := sorry\n\n/-- A `finset` of a subsingleton type has cardinality at most one. -/\ntheorem finset.card_le_one_of_subsingleton {\u03b1 : Type u_1} [subsingleton \u03b1] (s : finset \u03b1) : finset.card s \u2264 1 :=\n  iff.mpr finset.card_le_one_iff fun (_x _x_1 : \u03b1) (_x_2 : _x \u2208 s) (_x_3 : _x_1 \u2208 s) => subsingleton.elim _x _x_1\n\ntheorem finset.one_lt_card_iff {\u03b1 : Type u_1} {s : finset \u03b1} : 1 < finset.card s \u2194 \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2208 s \u2227 y \u2208 s \u2227 x \u2260 y := sorry\n\nprotected instance plift.fintype (p : Prop) [Decidable p] : fintype (plift p) :=\n  fintype.mk (dite p (fun (h : p) => singleton (plift.up h)) fun (h : \u00acp) => \u2205) sorry\n\nprotected instance Prop.fintype : fintype Prop :=\n  fintype.mk (finset.mk (True ::\u2098 False ::\u2098 0) sorry) sorry\n\nprotected instance subtype.fintype {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [fintype \u03b1] : fintype (Subtype fun (x : \u03b1) => p x) :=\n  fintype.subtype (finset.filter p finset.univ) sorry\n\n/-- A set on a fintype, when coerced to a type, is a fintype. -/\ndef set_fintype {\u03b1 : Type u_1} [fintype \u03b1] (s : set \u03b1) [decidable_pred s] : fintype \u21a5s :=\n  subtype.fintype fun (x : \u03b1) => x \u2208 s\n\nnamespace function.embedding\n\n\n/-- An embedding from a `fintype` to itself can be promoted to an equivalence. -/\ndef equiv_of_fintype_self_embedding {\u03b1 : Type u_1} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) : \u03b1 \u2243 \u03b1 :=\n  equiv.of_bijective \u21d1e sorry\n\n@[simp] theorem equiv_of_fintype_self_embedding_to_embedding {\u03b1 : Type u_1} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) : equiv.to_embedding (equiv_of_fintype_self_embedding e) = e :=\n  ext fun (x : \u03b1) => Eq.refl (coe_fn (equiv.to_embedding (equiv_of_fintype_self_embedding e)) x)\n\nend function.embedding\n\n\n@[simp] theorem finset.univ_map_embedding {\u03b1 : Type u_1} [fintype \u03b1] (e : \u03b1 \u21aa \u03b1) : finset.map e finset.univ = finset.univ := sorry\n\nnamespace fintype\n\n\n/-- Given for all `a : \u03b1` a finset `t a` of `\u03b4 a`, then one can define the\nfinset `fintype.pi_finset t` of all functions taking values in `t a` for all `a`. This is the\nanalogue of `finset.pi` where the base finset is `univ` (but formally they are not the same, as\nthere is an additional condition `i \u2208 finset.univ` in the `finset.pi` definition). -/\ndef pi_finset {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] {\u03b4 : \u03b1 \u2192 Type u_4} (t : (a : \u03b1) \u2192 finset (\u03b4 a)) : finset ((a : \u03b1) \u2192 \u03b4 a) :=\n  finset.map (function.embedding.mk (fun (f : (a : \u03b1) \u2192 a \u2208 finset.univ \u2192 \u03b4 a) (a : \u03b1) => f a (finset.mem_univ a)) sorry)\n    (finset.pi finset.univ t)\n\n@[simp] theorem mem_pi_finset {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] {\u03b4 : \u03b1 \u2192 Type u_4} {t : (a : \u03b1) \u2192 finset (\u03b4 a)} {f : (a : \u03b1) \u2192 \u03b4 a} : f \u2208 pi_finset t \u2194 \u2200 (a : \u03b1), f a \u2208 t a := sorry\n\ntheorem pi_finset_subset {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] {\u03b4 : \u03b1 \u2192 Type u_4} (t\u2081 : (a : \u03b1) \u2192 finset (\u03b4 a)) (t\u2082 : (a : \u03b1) \u2192 finset (\u03b4 a)) (h : \u2200 (a : \u03b1), t\u2081 a \u2286 t\u2082 a) : pi_finset t\u2081 \u2286 pi_finset t\u2082 :=\n  fun (g : (a : \u03b1) \u2192 \u03b4 a) (hg : g \u2208 pi_finset t\u2081) => iff.mpr mem_pi_finset fun (a : \u03b1) => h a (iff.mp mem_pi_finset hg a)\n\ntheorem pi_finset_disjoint_of_disjoint {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] {\u03b4 : \u03b1 \u2192 Type u_4} [(a : \u03b1) \u2192 DecidableEq (\u03b4 a)] (t\u2081 : (a : \u03b1) \u2192 finset (\u03b4 a)) (t\u2082 : (a : \u03b1) \u2192 finset (\u03b4 a)) {a : \u03b1} (h : disjoint (t\u2081 a) (t\u2082 a)) : disjoint (pi_finset t\u2081) (pi_finset t\u2082) := sorry\n\nend fintype\n\n\n/-! ### pi -/\n\n/-- A dependent product of fintypes, indexed by a fintype, is a fintype. -/\nprotected instance pi.fintype {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [DecidableEq \u03b1] [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : fintype ((a : \u03b1) \u2192 \u03b2 a) :=\n  fintype.mk (fintype.pi_finset fun (_x : \u03b1) => finset.univ) sorry\n\n@[simp] theorem fintype.pi_finset_univ {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [DecidableEq \u03b1] [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : (fintype.pi_finset fun (a : \u03b1) => finset.univ) = finset.univ :=\n  rfl\n\nprotected instance d_array.fintype {n : \u2115} {\u03b1 : fin n \u2192 Type u_1} [(n : fin n) \u2192 fintype (\u03b1 n)] : fintype (d_array n \u03b1) :=\n  fintype.of_equiv ((i : fin n) \u2192 \u03b1 i) (equiv.symm (equiv.d_array_equiv_fin \u03b1))\n\nprotected instance array.fintype {n : \u2115} {\u03b1 : Type u_1} [fintype \u03b1] : fintype (array n \u03b1) :=\n  d_array.fintype\n\nprotected instance vector.fintype {\u03b1 : Type u_1} [fintype \u03b1] {n : \u2115} : fintype (vector \u03b1 n) :=\n  fintype.of_equiv (fin n \u2192 \u03b1) (equiv.symm (equiv.vector_equiv_fin \u03b1 n))\n\nprotected instance quotient.fintype {\u03b1 : Type u_1} [fintype \u03b1] (s : setoid \u03b1) [DecidableRel has_equiv.equiv] : fintype (quotient s) :=\n  fintype.of_surjective quotient.mk sorry\n\nprotected instance finset.fintype {\u03b1 : Type u_1} [fintype \u03b1] : fintype (finset \u03b1) :=\n  fintype.mk (finset.powerset finset.univ) sorry\n\n@[simp] theorem fintype.card_finset {\u03b1 : Type u_1} [fintype \u03b1] : fintype.card (finset \u03b1) = bit0 1 ^ fintype.card \u03b1 :=\n  finset.card_powerset finset.univ\n\n@[simp] theorem set.to_finset_univ {\u03b1 : Type u_1} [fintype \u03b1] : set.to_finset set.univ = finset.univ := sorry\n\n@[simp] theorem set.to_finset_empty {\u03b1 : Type u_1} [fintype \u03b1] : set.to_finset \u2205 = \u2205 := sorry\n\ntheorem fintype.card_subtype_le {\u03b1 : Type u_1} [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] : fintype.card (Subtype fun (x : \u03b1) => p x) \u2264 fintype.card \u03b1 := sorry\n\ntheorem fintype.card_subtype_lt {\u03b1 : Type u_1} [fintype \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p] {x : \u03b1} (hx : \u00acp x) : fintype.card (Subtype fun (x : \u03b1) => p x) < fintype.card \u03b1 := sorry\n\nprotected instance psigma.fintype {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : fintype (psigma fun (a : \u03b1) => \u03b2 a) :=\n  fintype.of_equiv (sigma fun (a : \u03b1) => \u03b2 a) (equiv.symm (equiv.psigma_equiv_sigma fun (a : \u03b1) => \u03b2 a))\n\nprotected instance psigma.fintype_prop_left {\u03b1 : Prop} {\u03b2 : \u03b1 \u2192 Type u_1} [Decidable \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : fintype (psigma fun (a : \u03b1) => \u03b2 a) :=\n  dite \u03b1 (fun (h : \u03b1) => fintype.of_equiv (\u03b2 h) (equiv.mk (fun (x : \u03b2 h) => psigma.mk h x) psigma.snd sorry sorry))\n    fun (h : \u00ac\u03b1) => fintype.mk \u2205 sorry\n\nprotected instance psigma.fintype_prop_right {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Prop} [(a : \u03b1) \u2192 Decidable (\u03b2 a)] [fintype \u03b1] : fintype (psigma fun (a : \u03b1) => \u03b2 a) :=\n  fintype.of_equiv (Subtype fun (a : \u03b1) => \u03b2 a)\n    (equiv.mk (fun (_x : Subtype fun (a : \u03b1) => \u03b2 a) => sorry) (fun (_x : psigma fun (a : \u03b1) => \u03b2 a) => sorry) sorry\n      sorry)\n\nprotected instance psigma.fintype_prop_prop {\u03b1 : Prop} {\u03b2 : \u03b1 \u2192 Prop} [Decidable \u03b1] [(a : \u03b1) \u2192 Decidable (\u03b2 a)] : fintype (psigma fun (a : \u03b1) => \u03b2 a) :=\n  dite (\u2203 (a : \u03b1), \u03b2 a) (fun (h : \u2203 (a : \u03b1), \u03b2 a) => fintype.mk (singleton (psigma.mk sorry sorry)) sorry)\n    fun (h : \u00ac\u2203 (a : \u03b1), \u03b2 a) => fintype.mk \u2205 sorry\n\nprotected instance set.fintype {\u03b1 : Type u_1} [fintype \u03b1] : fintype (set \u03b1) :=\n  fintype.mk (finset.map (function.embedding.mk coe finset.coe_injective) (finset.powerset finset.univ)) sorry\n\nprotected instance pfun_fintype (p : Prop) [Decidable p] (\u03b1 : p \u2192 Type u_1) [(hp : p) \u2192 fintype (\u03b1 hp)] : fintype ((hp : p) \u2192 \u03b1 hp) :=\n  dite p\n    (fun (hp : p) =>\n      fintype.of_equiv (\u03b1 hp) (equiv.mk (fun (a : \u03b1 hp) (_x : p) => a) (fun (f : (hp : p) \u2192 \u03b1 hp) => f hp) sorry sorry))\n    fun (hp : \u00acp) => fintype.mk (singleton fun (h : p) => false.elim (hp h)) sorry\n\n@[simp] theorem finset.univ_pi_univ {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [DecidableEq \u03b1] [fintype \u03b1] [(a : \u03b1) \u2192 fintype (\u03b2 a)] : (finset.pi finset.univ fun (a : \u03b1) => finset.univ) = finset.univ := sorry\n\ntheorem mem_image_univ_iff_mem_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} {b : \u03b2} : b \u2208 finset.image f finset.univ \u2194 b \u2208 set.range f := sorry\n\ntheorem card_lt_card_of_injective_of_not_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) (h : function.injective f) {b : \u03b2} (w : \u00acb \u2208 set.range f) : fintype.card \u03b1 < fintype.card \u03b2 := sorry\n\n/-- An auxiliary function for `quotient.fin_choice`.  Given a\ncollection of setoids indexed by a type `\u03b9`, a (finite) list `l` of\nindices, and a function that for each `i \u2208 l` gives a term of the\ncorresponding quotient type, then there is a corresponding term in the\nquotient of the product of the setoids indexed by `l`. -/\ndef quotient.fin_choice_aux {\u03b9 : Type u_1} [DecidableEq \u03b9] {\u03b1 : \u03b9 \u2192 Type u_2} [S : (i : \u03b9) \u2192 setoid (\u03b1 i)] (l : List \u03b9) : ((i : \u03b9) \u2192 i \u2208 l \u2192 quotient (S i)) \u2192 quotient Mathlib.pi_setoid :=\n  sorry\n\ntheorem quotient.fin_choice_aux_eq {\u03b9 : Type u_1} [DecidableEq \u03b9] {\u03b1 : \u03b9 \u2192 Type u_2} [S : (i : \u03b9) \u2192 setoid (\u03b1 i)] (l : List \u03b9) (f : (i : \u03b9) \u2192 i \u2208 l \u2192 \u03b1 i) : (quotient.fin_choice_aux l fun (i : \u03b9) (h : i \u2208 l) => quotient.mk (f i h)) = quotient.mk f := sorry\n\n/-- Given a collection of setoids indexed by a fintype `\u03b9` and a\nfunction that for each `i : \u03b9` gives a term of the corresponding\nquotient type, then there is corresponding term in the quotient of the\nproduct of the setoids. -/\ndef quotient.fin_choice {\u03b9 : Type u_1} [DecidableEq \u03b9] [fintype \u03b9] {\u03b1 : \u03b9 \u2192 Type u_2} [S : (i : \u03b9) \u2192 setoid (\u03b1 i)] (f : (i : \u03b9) \u2192 quotient (S i)) : quotient Mathlib.pi_setoid :=\n  quotient.lift_on\n    (quotient.rec_on (finset.val finset.univ)\n      (fun (l : List \u03b9) => quotient.fin_choice_aux l fun (i : \u03b9) (_x : i \u2208 l) => f i) sorry)\n    (fun (f : (i : \u03b9) \u2192 i \u2208 finset.val finset.univ \u2192 \u03b1 i) => quotient.mk fun (i : \u03b9) => f i (finset.mem_univ i)) sorry\n\ntheorem quotient.fin_choice_eq {\u03b9 : Type u_1} [DecidableEq \u03b9] [fintype \u03b9] {\u03b1 : \u03b9 \u2192 Type u_2} [(i : \u03b9) \u2192 setoid (\u03b1 i)] (f : (i : \u03b9) \u2192 \u03b1 i) : (quotient.fin_choice fun (i : \u03b9) => quotient.mk (f i)) = quotient.mk f := sorry\n\n/-- Given a list, produce a list of all permutations of its elements. -/\ndef perms_of_list {\u03b1 : Type u_1} [DecidableEq \u03b1] : List \u03b1 \u2192 List (equiv.perm \u03b1) :=\n  sorry\n\ntheorem length_perms_of_list {\u03b1 : Type u_1} [DecidableEq \u03b1] (l : List \u03b1) : list.length (perms_of_list l) = nat.factorial (list.length l) := sorry\n\ntheorem mem_perms_of_list_of_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {l : List \u03b1} {f : equiv.perm \u03b1} (h : \u2200 (x : \u03b1), coe_fn f x \u2260 x \u2192 x \u2208 l) : f \u2208 perms_of_list l := sorry\n\ntheorem mem_of_mem_perms_of_list {\u03b1 : Type u_1} [DecidableEq \u03b1] {l : List \u03b1} {f : equiv.perm \u03b1} : f \u2208 perms_of_list l \u2192 \u2200 {x : \u03b1}, coe_fn f x \u2260 x \u2192 x \u2208 l := sorry\n\ntheorem mem_perms_of_list_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {l : List \u03b1} {f : equiv.perm \u03b1} : f \u2208 perms_of_list l \u2194 \u2200 {x : \u03b1}, coe_fn f x \u2260 x \u2192 x \u2208 l :=\n  { mp := mem_of_mem_perms_of_list, mpr := mem_perms_of_list_of_mem }\n\ntheorem nodup_perms_of_list {\u03b1 : Type u_1} [DecidableEq \u03b1] {l : List \u03b1} (hl : list.nodup l) : list.nodup (perms_of_list l) := sorry\n\n/-- Given a finset, produce the finset of all permutations of its elements. -/\ndef perms_of_finset {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : finset \u03b1) : finset (equiv.perm \u03b1) :=\n  quotient.hrec_on (finset.val s)\n    (fun (l : List \u03b1) (hl : multiset.nodup (quotient.mk l)) => finset.mk \u2191(perms_of_list l) sorry) sorry (finset.nodup s)\n\ntheorem mem_perms_of_finset_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : finset \u03b1} {f : equiv.perm \u03b1} : f \u2208 perms_of_finset s \u2194 \u2200 {x : \u03b1}, coe_fn f x \u2260 x \u2192 x \u2208 s :=\n  finset.cases_on s\n    fun (s_val : multiset \u03b1) (hs : multiset.nodup s_val) =>\n      quot.induction_on s_val (fun (l : List \u03b1) (hs : multiset.nodup (Quot.mk setoid.r l)) => mem_perms_of_list_iff) hs\n\ntheorem card_perms_of_finset {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : finset \u03b1) : finset.card (perms_of_finset s) = nat.factorial (finset.card s) :=\n  finset.cases_on s\n    fun (s_val : multiset \u03b1) (hs : multiset.nodup s_val) =>\n      quot.induction_on s_val (fun (l : List \u03b1) (hs : multiset.nodup (Quot.mk setoid.r l)) => length_perms_of_list l) hs\n\n/-- The collection of permutations of a fintype is a fintype. -/\ndef fintype_perm {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] : fintype (equiv.perm \u03b1) :=\n  fintype.mk (perms_of_finset finset.univ) sorry\n\nprotected instance equiv.fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] [fintype \u03b1] [fintype \u03b2] : fintype (\u03b1 \u2243 \u03b2) :=\n  dite (fintype.card \u03b2 = fintype.card \u03b1)\n    (fun (h : fintype.card \u03b2 = fintype.card \u03b1) =>\n      trunc.rec_on_subsingleton (fintype.equiv_fin \u03b1)\n        fun (e\u03b1 : \u03b1 \u2243 fin (fintype.card \u03b1)) =>\n          trunc.rec_on_subsingleton (fintype.equiv_fin \u03b2)\n            fun (e\u03b2 : \u03b2 \u2243 fin (fintype.card \u03b2)) =>\n              fintype.of_equiv (equiv.perm \u03b1)\n                (equiv.equiv_congr (equiv.refl \u03b1) (equiv.trans e\u03b1 (eq.rec_on h (equiv.symm e\u03b2)))))\n    fun (h : \u00acfintype.card \u03b2 = fintype.card \u03b1) => fintype.mk \u2205 sorry\n\ntheorem fintype.card_perm {\u03b1 : Type u_1} [DecidableEq \u03b1] [fintype \u03b1] : fintype.card (equiv.perm \u03b1) = nat.factorial (fintype.card \u03b1) :=\n  subsingleton.elim fintype_perm equiv.fintype \u25b8 card_perms_of_finset finset.univ\n\ntheorem fintype.card_equiv {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] [fintype \u03b1] [fintype \u03b2] (e : \u03b1 \u2243 \u03b2) : fintype.card (\u03b1 \u2243 \u03b2) = nat.factorial (fintype.card \u03b1) :=\n  fintype.card_congr (equiv.equiv_congr (equiv.refl \u03b1) e) \u25b8 fintype.card_perm\n\ntheorem univ_eq_singleton_of_card_one {\u03b1 : Type u_1} [fintype \u03b1] (x : \u03b1) (h : fintype.card \u03b1 = 1) : finset.univ = singleton x := sorry\n\nnamespace fintype\n\n\n/-- Given a fintype `\u03b1` and a predicate `p`, associate to a proof that there is a unique element of\n`\u03b1` satisfying `p` this unique element, as an element of the corresponding subtype. -/\ndef choose_x {\u03b1 : Type u_1} [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] (hp : exists_unique fun (a : \u03b1) => p a) : Subtype fun (a : \u03b1) => p a :=\n  { val := finset.choose p finset.univ sorry, property := sorry }\n\n/-- Given a fintype `\u03b1` and a predicate `p`, associate to a proof that there is a unique element of\n`\u03b1` satisfying `p` this unique element, as an element of `\u03b1`. -/\ndef choose {\u03b1 : Type u_1} [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] (hp : exists_unique fun (a : \u03b1) => p a) : \u03b1 :=\n  \u2191(choose_x p hp)\n\ntheorem choose_spec {\u03b1 : Type u_1} [fintype \u03b1] (p : \u03b1 \u2192 Prop) [decidable_pred p] (hp : exists_unique fun (a : \u03b1) => p a) : p (choose p hp) :=\n  subtype.property (choose_x p hp)\n\n/-- `\n`bij_inv f` is the unique inverse to a bijection `f`. This acts\n  as a computable alternative to `function.inv_fun`. -/\ndef bij_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (f_bij : function.bijective f) (b : \u03b2) : \u03b1 :=\n  choose (fun (a : \u03b1) => f a = b) sorry\n\ntheorem left_inverse_bij_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (f_bij : function.bijective f) : function.left_inverse (bij_inv f_bij) f :=\n  fun (a : \u03b1) =>\n    and.left f_bij (bij_inv f_bij (f a)) a (choose_spec (fun (a' : \u03b1) => f a' = f a) (bij_inv._proof_1 f_bij (f a)))\n\ntheorem right_inverse_bij_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (f_bij : function.bijective f) : function.right_inverse (bij_inv f_bij) f :=\n  fun (b : \u03b2) => choose_spec (fun (a' : \u03b1) => f a' = b) (bij_inv._proof_1 f_bij b)\n\ntheorem bijective_bij_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (f_bij : function.bijective f) : function.bijective (bij_inv f_bij) :=\n  { left := function.right_inverse.injective (right_inverse_bij_inv f_bij),\n    right := function.left_inverse.surjective (left_inverse_bij_inv f_bij) }\n\ntheorem well_founded_of_trans_of_irrefl {\u03b1 : Type u_1} [fintype \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [is_trans \u03b1 r] [is_irrefl \u03b1 r] : well_founded r := sorry\n\ntheorem preorder.well_founded {\u03b1 : Type u_1} [fintype \u03b1] [preorder \u03b1] : well_founded Less :=\n  well_founded_of_trans_of_irrefl Less\n\ninstance linear_order.is_well_order {\u03b1 : Type u_1} [fintype \u03b1] [linear_order \u03b1] : is_well_order \u03b1 Less :=\n  is_well_order.mk preorder.well_founded\n\nend fintype\n\n\n/-- A type is said to be infinite if it has no fintype instance. -/\nclass infinite (\u03b1 : Type u_4) \nwhere\n  not_fintype : fintype \u03b1 \u2192 False\n\n@[simp] theorem not_nonempty_fintype {\u03b1 : Type u_1} : \u00acNonempty (fintype \u03b1) \u2194 infinite \u03b1 := sorry\n\ntheorem finset.exists_minimal {\u03b1 : Type u_1} [preorder \u03b1] (s : finset \u03b1) (h : finset.nonempty s) : \u2203 (m : \u03b1), \u2203 (H : m \u2208 s), \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acx < m := sorry\n\ntheorem finset.exists_maximal {\u03b1 : Type u_1} [preorder \u03b1] (s : finset \u03b1) (h : finset.nonempty s) : \u2203 (m : \u03b1), \u2203 (H : m \u2208 s), \u2200 (x : \u03b1), x \u2208 s \u2192 \u00acm < x :=\n  finset.exists_minimal s h\n\nnamespace infinite\n\n\ntheorem exists_not_mem_finset {\u03b1 : Type u_1} [infinite \u03b1] (s : finset \u03b1) : \u2203 (x : \u03b1), \u00acx \u2208 s :=\n  iff.mp not_forall fun (h : \u2200 (x : \u03b1), x \u2208 s) => not_fintype (fintype.mk s h)\n\nprotected instance nontrivial (\u03b1 : Type u_1) [H : infinite \u03b1] : nontrivial \u03b1 :=\n  nontrivial.mk sorry\n\ntheorem nonempty (\u03b1 : Type u_1) [infinite \u03b1] : Nonempty \u03b1 :=\n  nontrivial.to_nonempty\n\ntheorem of_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [infinite \u03b2] (f : \u03b2 \u2192 \u03b1) (hf : function.injective f) : infinite \u03b1 :=\n  mk fun (I : fintype \u03b1) => not_fintype (fintype.of_injective f hf)\n\ntheorem of_surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [infinite \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : function.surjective f) : infinite \u03b1 :=\n  mk fun (I : fintype \u03b1) => not_fintype (fintype.of_surjective f hf)\n\n/-- Embedding of `\u2115` into an infinite type. -/\ndef nat_embedding (\u03b1 : Type u_1) [infinite \u03b1] : \u2115 \u21aa \u03b1 :=\n  function.embedding.mk (nat_embedding_aux \u03b1) (nat_embedding_aux_injective \u03b1)\n\ntheorem exists_subset_card_eq (\u03b1 : Type u_1) [infinite \u03b1] (n : \u2115) : \u2203 (s : finset \u03b1), finset.card s = n := sorry\n\nend infinite\n\n\ntheorem not_injective_infinite_fintype {\u03b1 : Type u_1} {\u03b2 : Type u_2} [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) : \u00acfunction.injective f :=\n  fun (hf : function.injective f) => (fun (H : fintype \u03b1) => infinite.not_fintype H) (fintype.of_injective f hf)\n\n/--\nThe pigeonhole principle for infinitely many pigeons in finitely many\npigeonholes.  If there are infinitely many pigeons in finitely many\npigeonholes, then there are at least two pigeons in the same\npigeonhole.\n\nSee also: `fintype.exists_ne_map_eq_of_card_lt`, `fintype.exists_infinite_fiber`.\n-/\ntheorem fintype.exists_ne_map_eq_of_infinite {\u03b1 : Type u_1} {\u03b2 : Type u_2} [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) : \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2260 y \u2227 f x = f y := sorry\n\n/--\nThe strong pigeonhole principle for infinitely many pigeons in\nfinitely many pigeonholes.  If there are infinitely many pigeons in\nfinitely many pigeonholes, then there is a pigeonhole with infinitely\nmany pigeons.\n\nSee also: `fintype.exists_ne_map_eq_of_infinite`\n-/\ntheorem fintype.exists_infinite_fiber {\u03b1 : Type u_1} {\u03b2 : Type u_2} [infinite \u03b1] [fintype \u03b2] (f : \u03b1 \u2192 \u03b2) : \u2203 (y : \u03b2), infinite \u21a5(f \u207b\u00b9' singleton y) := sorry\n\ntheorem not_surjective_fintype_infinite {\u03b1 : Type u_1} {\u03b2 : Type u_2} [fintype \u03b1] [infinite \u03b2] (f : \u03b1 \u2192 \u03b2) : \u00acfunction.surjective f :=\n  fun (hf : function.surjective f) =>\n    (fun (H : infinite \u03b1) => infinite.not_fintype infer_instance) (infinite.of_surjective f hf)\n\nprotected instance nat.infinite : infinite \u2115 :=\n  infinite.mk fun (_x : fintype \u2115) => sorry\n\nprotected instance int.infinite : infinite \u2124 :=\n  infinite.of_injective Int.ofNat fun (_x _x_1 : \u2115) => int.of_nat.inj\n\n/--\nFor `s : multiset \u03b1`, we can lift the existential statement that `\u2203 x, x \u2208 s` to a `trunc \u03b1`.\n-/\ndef trunc_of_multiset_exists_mem {\u03b1 : Type u_1} (s : multiset \u03b1) : (\u2203 (x : \u03b1), x \u2208 s) \u2192 trunc \u03b1 :=\n  quotient.rec_on_subsingleton s fun (l : List \u03b1) (h : \u2203 (x : \u03b1), x \u2208 quotient.mk l) => sorry\n\n/--\nA `nonempty` `fintype` constructively contains an element.\n-/\ndef trunc_of_nonempty_fintype (\u03b1 : Type u_1) [Nonempty \u03b1] [fintype \u03b1] : trunc \u03b1 :=\n  trunc_of_multiset_exists_mem (finset.val finset.univ) sorry\n\n/--\nA `fintype` with positive cardinality constructively contains an element.\n-/\ndef trunc_of_card_pos {\u03b1 : Type u_1} [fintype \u03b1] (h : 0 < fintype.card \u03b1) : trunc \u03b1 :=\n  let _inst : Nonempty \u03b1 := sorry;\n  trunc_of_nonempty_fintype \u03b1\n\n/--\nBy iterating over the elements of a fintype, we can lift an existential statement `\u2203 a, P a`\nto `trunc (\u03a3' a, P a)`, containing data.\n-/\ndef trunc_sigma_of_exists {\u03b1 : Type u_1} [fintype \u03b1] {P : \u03b1 \u2192 Prop} [decidable_pred P] (h : \u2203 (a : \u03b1), P a) : trunc (psigma fun (a : \u03b1) => P a) :=\n  trunc_of_nonempty_fintype (psigma fun (a : \u03b1) => P a)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/fintype/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.3989057089363508}}
{"text": "import .lemmas .dickson\nopen classical nat function prod set subtype\n\nnoncomputable theory\n\ndefinition inj_from_to {A B: Type} (f : A \u2192 B) (S1 : set A) (S2 : set B) := maps_to f S1 S2 \u2227 inj_on f S1\n\ntheorem inj_from_to_id {A : Type} (S1 : set A) : (\u2200 a, a \u2208 S1 \u2192 id a \u2208 S1) \u2227 \u2200 a\u2081 a\u2082, a\u2081 \u2208 S1 \u2192 a\u2082 \u2208 S1 \u2192 id a\u2081 = id a\u2082 \u2192 a\u2081 = a\u2082 := \nbegin split, intros, simp, assumption, intros; assumption end\n\ntheorem inj_from_to_compose {A B C : Type} {g : B \u2192 C} {f : A \u2192 B} {S1 : set A} {S2 : set B} {S3 : set C} \n  (Hg : inj_from_to g S2 S3) (Hf : inj_from_to f S1 S2) : inj_from_to (g \u2218 f) S1 S3 :=\nhave Hl : \u2200 a, a \u2208 S1 \u2192 g (f a) \u2208 S3, from \u03bb a Ha, Hg^.left (Hf^.left Ha),\nhave \u2200 a\u2081 a\u2082, a\u2081 \u2208 S1 \u2192 a\u2082 \u2208 S1 \u2192 g (f a\u2081) = g (f a\u2082) \u2192 a\u2081 = a\u2082, from\n  \u03bb  a\u2081 a\u2082 Ha\u2081 Ha\u2082 Heq,\n  have in1 : f a\u2081 \u2208 S2, from Hf^.left Ha\u2081,\n  have in2 : f a\u2082 \u2208 S2, from Hf^.left Ha\u2082,\n  have f a\u2081 = f a\u2082, from Hg^.right in1 in2 Heq, \n  Hf^.right Ha\u2081 Ha\u2082 this,\n\u27e8Hl, this\u27e9\n\ntheorem gt_of_gt_pred {a b : \u2115} (H : pred b < a) : b \u2264 a :=\nby_cases\n(suppose b = 0, by simp [this,zero_le])\n(suppose b \u2260 0, \n have \u2203 k, b = succ k, from exists_eq_succ_of_ne_zero this,\n let \u27e8k,hk\u27e9 := this in\n have pred (succ k) < a, by rw hk at H;exact H,\n have k < a, by super, \n have succ k \u2264 a, from succ_le_of_lt this,\n by simp [this, hk])\n\ntheorem sub_gt_of_gt_ge {a b c : \u2115} (H1 : a > b) (H2 : b \u2265 c) : a - c > b - c :=\nhave c \u2264 a, from le_of_lt (lt_of_le_of_lt H2 H1),\nhave eq\u2081 : a - c + c = a, from nat.sub_add_cancel this,\nhave eq\u2082 : b - c + c = b, from nat.sub_add_cancel H2,\nhave b \u2264 a, from le_of_lt H1,\nhave b - c \u2264 a - c, from nat.sub_le_sub_right this c,\nor.elim (nat.lt_or_eq_of_le this)\n(assume Hl, Hl)\n(assume Hr, have refl : a > a, by super,\n absurd refl (lt_irrefl a))\n\n\ntheorem lt_pred_nonzero_self {n : \u2115} (H : n \u2260 0) : pred n < n :=\nhave \u2203 k, n = succ k, from exists_eq_succ_of_ne_zero H,\nlet \u27e8k,hk\u27e9 := this in begin simp [hk, self_lt_succ] end\n\n-- theorem ne_empty_of_mem {X : Type} {s : set X} {x : X} (H : x \u2208 s) : s \u2260 \u2205 := \n-- begin intro Hs, rw Hs at H, apply set.not_mem_empty _ H end \n\ntheorem image_nonempty {A B : Type} {f : A \u2192 B} {S : set A} (H : S \u2260 \u2205) : image f  S \u2260 \u2205 :=\nhave \u2203 s, s \u2208 S, from exists_mem_of_ne_empty H,\nlet \u27e8s,h\u27e9 := this in\nhave f s \u2208 image f S, from \u27e8s, \u27e8h,rfl\u27e9\u27e9,\nset.ne_empty_of_mem this\n\ntheorem not_mem_singleton {A : Type} (x a : A) (H : x \u2260 a) : x \u2209 insert a (\u2205 : set A) :=\nsuppose x \u2208 insert a \u2205, H (eq_of_mem_singleton this)\n\ntheorem refl_of_diff_of_ins_singleton {A : Type} {a : A} {S : set A} (H : a \u2209 S) : S = (insert a S) \\ insert a \u2205 :=\nsubset.antisymm \n(\u03bb x h, \u27e8or.inr h,(\u03bb neg, have x = a, from eq_of_mem_singleton neg,by super)\u27e9) \n(\u03bb x \u27e8hl,hr\u27e9, or.elim hl (\u03bb l, \nhave x \u2208 insert a (\u2205 : set A), begin rw l, apply mem_singleton end, \nby contradiction) (\u03bb r,r))\n\nnamespace kruskal\n\n#check good_pairs\n\nsection\n-- Given a countable set of objects A (which is ordered by f), and assuming that there exists a bad sequence (i.e., f \u2218 g) of these objects, we can find a (sub)sequence (f \u2218 h) which is bad and \u2200 i, h 0 \u2264 h i.\nparameter {A : Type}\nparameter f : \u2115 \u2192 A\nparameter g : \u2115 \u2192 \u2115\nparameter o : A \u2192 A \u2192 Prop\nparameter H : \u00ac is_good (f \u2218 g) o\n\ndefinition ran_g : set \u2115 := {x : \u2115 | \u2203 i, g i = x}\n\ntheorem ne_empty_ran : ran_g \u2260 \u2205 := set.ne_empty_of_mem \u27e80,rfl\u27e9\n\nprivate definition min : \u2115 := least ran_g ne_empty_ran\n\ndefinition index_of_min : \u2115 :=\nhave min \u2208 ran_g, from least_is_mem ran_g ne_empty_ran,\nsome this \n\ntheorem minimality_of_min (n : \u2115) : g index_of_min \u2264 g n :=\nhave H1 : g index_of_min = min, from some_spec (least_is_mem ran_g ne_empty_ran),\nhave least ran_g ne_empty_ran \u2264 g n, from minimality _ (g n) \u27e8n,rfl\u27e9,\nbegin simp [H1], exact this end\n\nprivate definition h (n : \u2115) : \u2115 := g (index_of_min + n)\n\ntheorem exists_sub_bad : \u2203 h : \u2115 \u2192 \u2115, \u00ac is_good (f \u2218 h) o \u2227 \u2200 i : \u2115, h 0 \u2264 h i :=\nhave badness : \u00ac is_good (f \u2218 h) o, from\n   suppose is_good (f \u2218 h) o,\n   let \u27e8i,j,hij\u27e9 := this in\n   have index_of_min + i < index_of_min + j, from add_lt_add_left (and.left hij) _,\n   have is_good (f \u2218 g) o, from \u27e8index_of_min + i,\u27e8index_of_min + j,\u27e8this,hij^.right\u27e9\u27e9\u27e9,\n   H this,\nhave \u2200 i : \u2115, h 0 \u2264 h i, from \u03bb i, minimality_of_min (index_of_min + i),\n\u27e8h,\u27e8badness,this\u27e9\u27e9\n\nend\n\ndefinition finite_subsets (Q : Type) : Type := {x : set Q // finite x}\n\ndefinition non_descending {Q : Type} (A B : finite_subsets Q) (o : Q \u2192 Q \u2192 Prop) (f : Q \u2192 Q) := \u2200 a : Q, a \u2208 A.1 \u2192 o a (f a) \u2227 f a \u2208 B.1\n\ndefinition star {Q : Type} (o : Q \u2192 Q \u2192 Prop) (A B : finite_subsets Q) := \u2203 f, inj_from_to f A.1 B.1 \u2227 non_descending A B o f\n\ndefinition extends_at {A : Type} (n : \u2115) (f : \u2115 \u2192 A) (g : \u2115 \u2192 A) : Prop := \u2200 m \u2264 n, g m = f m\n\ntheorem extends_at.refl {A : Type} {n : \u2115} {f : \u2115 \u2192 A} : extends_at n f f := \u03bb m H, rfl\n\ntheorem extends_at.trans {A : Type} {n m : \u2115} {f g h: \u2115 \u2192 A} (H1 : extends_at n f g) (H2 : extends_at m g h) (H3 : n \u2264 m) : \n  extends_at n f h := \u03bb k H, \nhave g k = f k, from H1 k H,\nhave k \u2264 m, from nat.le_trans H H3,\nhave h k = g k, from H2 k this,\nby super\n\n-- definition induced_set_of_exists {A : Type} {P : A \u2192 Prop} (H : \u2203 x, P x) : set A := {x | P x}\n\n-- theorem nonempty_induced_set {A : Type} {P : A \u2192 Prop} (H : \u2203 x, P x) : induced_set_of_exists H \u2260 \u2205 := let \u27e8a,h\u27e9 := H in set.ne_empty_of_mem h\n\n-- theorem property_of_induced_set {A : Type} {P : A \u2192 Prop} (H : \u2203 x, P x) : \u2200 s, s \u2208 induced_set_of_exists H \u2192 P s := \n-- \u03bb s h, h\n\ntheorem least_seq_at_n {S : set (\u2115 \u2192 \u2115)} (H : S \u2260 \u2205) (n : \u2115) : \u2203 f, f \u2208 S \u2227 \u2200 g, g \u2208 S \u2192 f n \u2264 g n :=\nlet T : set \u2115 := {x | \u2203 f, f \u2208 S \u2227 f n = x} in\nhave \u2203 f, f \u2208 S, from exists_mem_of_ne_empty H,\nlet \u27e8f,h\u27e9 := this in\nhave nemp : T \u2260 \u2205, from set.ne_empty_of_mem \u27e8f,\u27e8h,rfl\u27e9\u27e9,\nlet a := least T nemp in\nhave a \u2208 T, from least_is_mem T nemp,\nlet \u27e8f',h\u27e9 := this in\nhave \u2200 g, g \u2208 S \u2192 f' n \u2264 g n, from \u03bb g Hg, \n  have a \u2264 g n, from minimality _ _ \u27e8g,\u27e8Hg,rfl\u27e9\u27e9, \n  by super,\n\u27e8f',\u27e8h^.left, this\u27e9\u27e9\n\n\nsection\n-- given an n, take an f from {f | P f} such that |f n| is as small as possible.\nparameter {A : Type}\nparameter {P : (\u2115 \u2192 A) \u2192 Prop}\nparameter g : A \u2192 \u2115 -- a function which calculates the cardinality of a : A in some sense.\nparameter H : \u2203 f : \u2115 \u2192 A, P f\n\n-- definition card_of_f (f : \u2115 \u2192 A) (n : \u2115) : \u2115 := g (f n)\n\ndefinition colle : set (\u2115 \u2192 A) := {f | P f}\n\nlemma nonempty_colle : colle \u2260 \u2205 :=  let \u27e8a,h\u27e9 := H in set.ne_empty_of_mem h\n\ndefinition S : set (\u2115 \u2192 \u2115) := image (\u03bb f, g \u2218 f) colle\n\nlemma nonempty_S : S \u2260 \u2205 := image_nonempty nonempty_colle\n\ntheorem exists_min_func (n : \u2115) : \u2203 f, f \u2208 S \u2227 \u2200 g, g \u2208 S \u2192 f n \u2264 g n := least_seq_at_n nonempty_S n\n\n-- let \u27e8l,r\u27e9 := some_spec (exists_min_func n) does not work\ndefinition min_func (n : \u2115) : \u2115 \u2192 A := \nlet fc := some (exists_min_func n) in\nhave fc \u2208 S \u2227 \u2200 g, g \u2208 S \u2192 fc n \u2264 g n, from (some_spec (exists_min_func n)),\nsome this^.left\n\ntheorem min_func_property (n : \u2115) : P (min_func n) :=\nlet fc := some (exists_min_func n) in\nlet \u27e8l,r\u27e9 := some_spec (exists_min_func n) in\nhave min_func n \u2208 colle \u2227 (\u03bb f, g \u2218 f) (min_func n) = fc, from some_spec l ,\nthis^.left\n\n-- For every f satisfying P, we have the inequality. Intuitively, it says that |(min_func n) n| is always less than or equal to |f n|.\ntheorem min_func_minimality (f : \u2115 \u2192 A) (Hp : P f) (n : \u2115) : g (min_func n n) \u2264 g (f n) := \nlet fc := some (exists_min_func n) in\nlet \u27e8l,r\u27e9 := some_spec (exists_min_func n) in\nhave min_func n \u2208 colle \u2227 (\u03bb f, g \u2218 f) (min_func n) = fc, from some_spec l,\nhave (\u03bb f, g \u2218 f) (min_func n) = fc, from this^.right, \nhave eq2 : (\u03bb f, g \u2218 f) (min_func n) n = fc n, by rw this, \nhave Hr : \u2200 g, g \u2208 S \u2192 fc n \u2264 g n, from (some_spec (exists_min_func n))^.right,\nhave le : fc n \u2264 (\u03bb f, g \u2218 f) f n, from Hr _ \u27e8f,\u27e8Hp,rfl\u27e9\u27e9,\n-- have (\u03bb f, g \u2218 f) (min_func n) n = g (min_func n n), from rfl,\n-- have (\u03bb f, g \u2218 f) f n = g (f n), from rfl,\nhave (\u03bb f, g \u2218 f) (min_func n) n \u2264 (\u03bb f, g \u2218 f) f n, by rw -eq2 at le;exact le,\nby super\n\nend\n\nsection\n\nparameter {A : Type} \nparameter {P : (\u2115 \u2192 A) \u2192 Prop} -- some property about f \nparameter g : A \u2192 \u2115 -- a measure of cardinality of A \nparameter H : \u2203 f, P f \n\n-- construct a sequence of functions with property P such that each one extends its predecessor and is the minimal one at n.\nnoncomputable definition mbs_helper (n : \u2115) : {f : \u2115 \u2192 A // P f} :=\nnat.rec_on n\n(let f\u2080 := min_func g H 0 in\n have P f\u2080, from min_func_property g H 0,\n \u27e8f\u2080,this\u27e9)\n(\u03bb pred h',\nlet f' := h'.1 in\nhave H1 : extends_at pred f' f', from extends_at.refl,\nhave H2 : P f', from h'.2,\nhave HP : \u2203 f, extends_at pred f' f \u2227 P f, from \u27e8f',\u27e8H1,H2\u27e9\u27e9,\nlet fn := min_func g HP (succ pred) in\nhave extends_at pred f' fn \u2227 P fn, from min_func_property g HP (succ pred),\nhave P fn, from this^.right,\n\u27e8fn,this\u27e9)\n\n  section\n  parameter n : \u2115\n  definition helper_elt := (mbs_helper n).1\n  definition helper_succ := (mbs_helper (succ n)).1\n  lemma helper_ext_refl : extends_at n helper_elt helper_elt := extends_at.refl\n  lemma helper_has_property : P helper_elt := (mbs_helper n).2\n  lemma helper_inner_hyp : \u2203 g, extends_at n helper_elt g \u2227 P g := \u27e8helper_elt, \u27e8helper_ext_refl, helper_has_property\u27e9\u27e9\n  theorem succ_ext_of_mbs_helper : extends_at n helper_elt helper_succ := (min_func_property g helper_inner_hyp (succ n))^.left\n  end\n\ntheorem ext_of_mbs_helper (n : \u2115) : \u2200 m, m \u2264 n \u2192 extends_at m  (mbs_helper m).1 (mbs_helper n).1 :=\nnat.rec_on n\n(take m, assume H, \nhave eq : m = 0, from eq_zero_of_le_zero H,\nhave extends_at 0 (mbs_helper 0).1 (mbs_helper 0).1, from extends_at.refl,\nby simp [eq,this])\n(\u03bb a IH m H,\nby_cases\n(suppose m = succ a, \nhave extends_at m (mbs_helper (succ a)).1 (mbs_helper (succ a)).1, from extends_at.refl, by super)\n(suppose m \u2260 succ a, \nhave m < succ a, from lt_of_le_of_ne H this,\nhave Hle : m \u2264 a, from (iff.mp (lt_succ_iff_le m a)) this,\nhave H1 : extends_at m (mbs_helper m).1 (mbs_helper a).1, from IH m Hle,\nhave extends_at a (mbs_helper a).1 (mbs_helper (succ a)).1, from succ_ext_of_mbs_helper a,\nextends_at.trans H1 this Hle))\n\ntheorem congruence_of_mbs_helper {n m : \u2115} (H : m \u2264 n) : (mbs_helper n).1 m = (mbs_helper m).1 m :=\nhave extends_at m (mbs_helper m).1 (mbs_helper n).1, from ext_of_mbs_helper n m H,\nthis m (nat.le_refl m)\n\nend\n\nsection\n-- construction and properties of mbs.\nparameter {A : Type}\nparameter {o : A \u2192 A \u2192 Prop}\nparameter g : A \u2192 \u2115\nparameter H : \u2203 f : \u2115 \u2192 A, \u00ac is_good f o\n\nnoncomputable definition seq_of_bad_seq (n : \u2115) : {f : \u2115 \u2192 A // \u00ac is_good f o} := mbs_helper g H n\n\ndefinition minimal_bad_seq (n : \u2115) : A :=  (seq_of_bad_seq n).1 n \n\ndefinition ext_of_seq_of_bad_seq := ext_of_mbs_helper g H\n\ndefinition congruence_of_seq_of_bad_seq {n m : \u2115} (Hnm : m \u2264 n) := congruence_of_mbs_helper g H Hnm\n\ndefinition bad_seq_elt := helper_elt g H\n\ndefinition bad_seq_inner_hyp := helper_inner_hyp g H \n\ntheorem badness_of_mbs : \u00ac is_good minimal_bad_seq o := \nsuppose is_good minimal_bad_seq o,\nlet \u27e8i,j,h\u27e9 := this in\nhave i \u2264 j, from le_of_lt_or_eq (or.inl h^.left),\nhave ext : extends_at i (seq_of_bad_seq i).1 (seq_of_bad_seq j).1, from ext_of_seq_of_bad_seq j i this,\nhave i \u2264 i, from nat.le_refl i,\nhave (seq_of_bad_seq j).1 i = (minimal_bad_seq i), from ext i this,\nhave o ((seq_of_bad_seq j).1 i) (minimal_bad_seq j), by rw this; exact h^.right,\nhave i < j \u2227 o ((seq_of_bad_seq j).1 i) ((seq_of_bad_seq j).1 j), from \u27e8h^.left, this\u27e9,\nhave good : is_good (seq_of_bad_seq j).1 o, from \u27e8i,\u27e8j, this\u27e9\u27e9,\nhave \u00ac is_good (seq_of_bad_seq j).1 o, from (seq_of_bad_seq j).2, \nthis good\n\ntheorem minimality_of_mbs_0 (f : \u2115 \u2192 A) (Hf : \u00ac is_good f o) : g (minimal_bad_seq 0) \u2264 g (f 0) := min_func_minimality g H f Hf 0\n\ntheorem minimality_of_mbs (n : \u2115) (f : \u2115 \u2192 A) (H1 : extends_at n minimal_bad_seq f \u2227 \u00ac is_good f o) : g (minimal_bad_seq (succ n)) \u2264 g (f (succ n)) := \nhave Hl : \u2200 m, m \u2264 n \u2192  f m = (bad_seq_elt n) m, from \n  \u03bb m Hle, have f m = minimal_bad_seq m, from H1^.left m Hle,\n  have bad_seq_elt n m = minimal_bad_seq m, from congruence_of_seq_of_bad_seq Hle,\n  by super, --by+ simp,\nhave ins_P : extends_at n (bad_seq_elt n) f \u2227 \u00ac is_good f o, from \u27e8Hl, H1^.right\u27e9,\nhave ineq : g (min_func g (bad_seq_inner_hyp n) (succ n) (succ n)) \u2264 g (f (succ n)), from min_func_minimality g (bad_seq_inner_hyp n) f ins_P (succ n), \n-- have minimal_bad_seq (succ n) = min_func g (bad_seq_inner_hyp n) (succ n) (succ n), from rfl,\n-- by+ rw (eq.symm this) at ineq; exact ineq\nby super\n\nend\n\nsection\n\n-- Given two sequences f and g, a function h which modifies indices so that h 0 is the break point, construct a new sequence 'combined_seq' by concatenating f and g at (h 0).\n\nparameter {Q :Type}\nparameter {o : Q \u2192 Q \u2192 Prop}\nparameters f g : \u2115 \u2192 Q\nparameter h : \u2115 \u2192 \u2115\nparameter Hh : \u2200 i, h 0 \u2264 h i\nparameter Hf : \u00ac is_good f o\nparameter Hg : \u00ac is_good g o\n-- in Higman's lemma in Williams 1963, h is f, g is the bad sequence B \u2218 f\nparameter H : \u2200 i j, o (f i) (g (j - h 0)) \u2192 o (f i) (f (h (j - h 0))) \n\n-- definition comb (n : \u2115) : Q := if h 0 \u2260 0 \u2227 n \u2264 pred (h 0) then f n else g (n - (h 0))\n\n-- -- def comb' (n : \u2115) : Q := if n < h 0 then f n else g (n - (h 0))\n\n-- -- theorem g_part_of_comb' (H : (h 0) = 0) : \u2200 x, comb' x = g x :=\n-- -- \u03bb n, have \u00ac n < h 0, by rw H; apply not_lt_zero ,\n-- -- have comb' n = g (n - (h 0)), from if_neg this,\n-- -- by simp [this, H]\n\n\n-- theorem g_part_of_comb (H : (h 0) = 0) : \u2200 x, comb x = g x :=\n-- take n, have \u00ac (h 0) \u2260 0, from not_not_intro H,\n-- have \u00ac ((h 0) \u2260 0 \u2227 n \u2264 pred (h 0)), from not_and_of_not_left (n \u2264 pred (h 0)) this,\n-- have comb n = g (n - (h 0)), from if_neg this,\n-- by simp [this, H]\n\n-- theorem badness_of_comb : \u00ac is_good comb o := \n-- \u03bb good, \n-- let \u27e8i,j,hw\u27e9 := good in\n-- by_cases\n-- (suppose (h 0) = 0, \n-- have comb = g, begin apply funext, apply g_part_of_comb, exact this end,\n-- have is_good g o, by rw this at good;exact good,\n-- Hg this)\n-- (assume ne, \n--   by_cases\n--   (assume Hposi : i \u2264 pred (h 0), \n--    have eq1i : comb i = f i, from if_pos \u27e8ne, Hposi\u27e9,\n--    by_cases\n--      (suppose j \u2264 pred (h 0), \n--       have eq1j : comb j = f j, from if_pos \u27e8ne, this\u27e9, \n--       have o (comb i) (comb j), from hw^.right,\n--       have o (comb i) (f j), by rw eq1j at this; exact this,\n--       have o (f i) (f j), begin rw -eq1i, exact this end,\n--       have is_good f o, from \u27e8i, \u27e8j,\u27e8hw^.left,this\u27e9\u27e9\u27e9,\n--       show _, from Hf this)\n--      (suppose \u00ac j \u2264 pred (h 0), \n--       have \u00ac ((h 0) \u2260 0 \u2227 j \u2264 pred (h 0)), from not_and_of_not_right ((h 0) \u2260 0) this,\n--       have eq2j : comb j = g (j - (h 0)), from if_neg this, \n--       have o (f i) (g (j - (h 0))), begin rw [-eq2j,-eq1i], exact hw^.right end,\n--      have Hr : o (f i) (f (h (j - (h 0)))), from H _ _ this,\n--      have i < h (j - (h 0)), from\n--        have ilth0 : i < h 0, from lt_of_le_of_lt Hposi (lt_pred_nonzero_self ne),\n--        have h 0 \u2264 h (j - h 0), from Hh (j - h 0), \n--        show _, from lt_of_lt_of_le ilth0 this,\n--      have is_good f o, from \u27e8i, \u27e8h (j - h 0), \u27e8this, Hr\u27e9\u27e9\u27e9,\n--      show _, from Hf this))\n--   (assume Hnegi, \n--    have iht : pred (h 0) < i, from lt_of_not_ge Hnegi,\n--    have \u00ac (h 0 \u2260 0 \u2227 i \u2264 pred (h 0)), from not_and_of_not_right (h 0 \u2260 0) Hnegi,\n--    have eq2i : comb i = g (i - h 0), from if_neg this,\n--    by_cases\n--    (assume Hposj : j \u2264 pred (h 0), \n--     have j < i, from lt_of_le_of_lt Hposj iht,\n--     show _, from (not_lt_of_gt hw^.left) this)\n--    (assume Hnegj, \n--     have pred (h 0) < j, from lt_of_not_ge Hnegj,\n--     have \u00ac (h 0 \u2260 0 \u2227 j \u2264 pred (h 0)), from not_and_of_not_right (h 0 \u2260 0) Hnegj,\n--     have eq2j : comb j = g (j - h 0), from if_neg this,\n--     have o (comb i) (comb j), from hw^.right,\n--     have o (comb i) (g (j - h 0)), begin rw -eq2j, exact this end, --by simp,\n--     have Hr2 : o (g (i - h 0)) (g (j - h 0)), begin rw -eq2i, exact this end,-- by simp,\n--     have ige : h 0 \u2264 i, from gt_of_gt_pred iht,\n--     have jgt : h 0 < j, from lt_of_le_of_lt ige hw^.left,\n--     have i - h 0 < j - h 0, from \n--      or.elim (lt_or_eq_of_le ige)\n--      (assume hl, sub_gt_of_gt hw^.left hl)\n--      (assume hr, have 0 < j - h 0, from nat.sub_pos_of_lt jgt, \n--       have i - h 0 = 0, begin rw hr, apply nat.sub_self end,\n--       begin rw this, assumption end),\n--       have is_good g o, from \u27e8(i - h 0), \u27e8(j - h 0),\u27e8this, Hr2\u27e9\u27e9\u27e9,\n--      show _, from Hg this)))\n\n-- new comb --\ndefinition comb (n : \u2115) : Q := if n < h 0 then f n else g (n - h 0)\n\ntheorem g_part_of_comb (H : (h 0) = 0) : \u2200 x, comb x = g x :=\n\u03bb n, have \u00ac n < h 0, by rw H; apply not_lt_zero ,\nhave comb n = g (n - (h 0)), from if_neg this,\nby simp [this, H]\n\ninclude Hh\n\ntheorem badness_of_comb : \u00ac is_good comb o := \n\u03bb good, let \u27e8i,j,hw\u27e9 := good in\nby_cases\n(assume Hposi : i < h 0, \n  have eq1i : comb i = f i, from if_pos Hposi,\n  by_cases \n  (suppose j < h 0, \n    have eq1j : comb j = f j, from if_pos this, \n    have o (f i) (f j),by rw [-eq1j,-eq1i]; exact hw^.right,\n    have is_good f o, from \u27e8i, \u27e8j,\u27e8hw^.left,this\u27e9\u27e9\u27e9,\n    show _, from Hf this)\n  (suppose \u00ac j < h 0,\n    have eq2j : comb j = g (j - (h 0)), from if_neg this, \n    have o (f i) (g (j - (h 0))), by rw [-eq2j,-eq1i]; exact hw^.right,\n    have Hr : o (f i) (f (h (j - (h 0)))), from H _ _ this,\n    have i < h (j - (h 0)), from lt_of_lt_of_le Hposi (Hh _),\n    have is_good f o, from \u27e8i, \u27e8h (j - h 0), \u27e8this, Hr\u27e9\u27e9\u27e9,\n    show _, from Hf this))\n(assume Hnegi, \n  have eq2i : comb i = g (i - h 0), from if_neg Hnegi,\n  by_cases\n  (suppose j < h 0,\n    have j < i, from lt_of_lt_of_le this (le_of_not_gt Hnegi),\n    show _, from (not_lt_of_gt hw^.left) this)\n  (suppose \u00ac j < h 0, \n    have eq2j : comb j = g (j - h 0), from if_neg this,\n    have Hr2 : o (g (i - h 0)) (g (j - h 0)), by rw [-eq2i,-eq2j]; exact hw^.right,\n    have i - h 0 < j - h 0, from sub_gt_of_gt_ge hw^.left (le_of_not_gt Hnegi),\n    have is_good g o, from \u27e8(i - h 0), \u27e8(j - h 0),\u27e8this, Hr2\u27e9\u27e9\u27e9,\n    show _, from Hg this))\n\n\nend\n\nsection\n-- further assume that f is a minimal bad sequence and card (g 0) < card (f (h 0)) \n-- In other words, this section says, assuming that there is a bad sequence of Q, if g is a bad sequence such that H holds, then there is a contradiction. \nparameter {Q :Type}\nparameter {o : Q \u2192 Q \u2192 Prop}\nparameters {g : \u2115 \u2192 Q}\nparameter h : \u2115 \u2192 \u2115\nparameter m : Q \u2192 \u2115 -- a measure of cardinality\nparameter Hh : \u2200 i, h 0 \u2264 h i\nparameter Hex : \u2203 f, \u00ac is_good f o\nparameter Hg : \u00ac is_good g o\nparameter H : \u2200 i j, o (minimal_bad_seq m Hex i) (g (j - h 0)) \u2192 o (minimal_bad_seq m Hex i) ((minimal_bad_seq m Hex) (h (j - h 0)))\nparameter Hbp : m (g 0) < m (minimal_bad_seq m Hex (h 0))\n\ndefinition comb_seq_with_mbs := comb (minimal_bad_seq m Hex) g h\n\ntheorem g_part_of_comb_seq_with_mbs (H1 : (h 0) = 0) : \u2200 x, comb_seq_with_mbs x = g x := \nbegin apply g_part_of_comb, assumption end\n\ntheorem badness_of_comb_seq_with_mbs : \u00ac is_good comb_seq_with_mbs o := \nbadness_of_comb (minimal_bad_seq m Hex) g h Hh (badness_of_mbs m Hex) Hg H\n\n-- theorem comb_seq_extends_mbs_at_pred_bp (H : h 0 \u2260 0): extends_at (pred (h 0)) (minimal_bad_seq m Hex) comb_seq_with_mbs := \n-- \u03bb m Hm, if_pos \u27e8H, Hm\u27e9\n\ntheorem lt_of_le_pred' {n m : \u2115} : n \u2264 pred m \u2192  n < m \u2228 n = 0 :=\nnat.rec_on m (\u03bb h, or.elim (lt_or_eq_of_le h) (\u03bb l, by super) (\u03bb r, or.inr r)) \n(\u03bb a ih h, or.inl (lt_succ_of_le h))\n\ntheorem lt_of_le_pred {n m : \u2115} : n \u2264 pred m \u2192  n < m \u2228 m = 0 :=\nnat.rec_on m (\u03bb h, or.inr rfl) (\u03bb a ih h, or.inl (lt_succ_of_le h))\n\ntheorem comb_seq_extends_mbs_at_pred_bp (H : h 0 \u2260 0): extends_at (pred (h 0)) (minimal_bad_seq m Hex) comb_seq_with_mbs := \n\u03bb m Hm, if_pos (or_resolve_left (lt_of_le_pred Hm) H)\n\n#check succ_le_succ\n\nlemma comb_seq_h0 : comb_seq_with_mbs (h 0) = g 0 := \nhave comb_seq_with_mbs (h 0) = g (h 0 - h 0), begin apply if_neg, rw lt_self_iff_false, trivial end,\nby simp [this,nat.sub_self]\n-- by_cases\n-- (suppose h 0 = 0, \n-- have comb_seq_with_mbs (h 0) = g (h 0), from g_part_of_comb_seq_with_mbs this (h 0),\n-- by super)\n-- (suppose h 0 \u2260 0, \n-- have pred (h 0) < h 0, from lt_pred_nonzero_self this,\n-- have \u00ac h 0 \u2264 pred (h 0), from not_le_of_gt this,\n-- have \u00ac ((h 0) \u2260 0 \u2227 h 0 \u2264 pred (h 0)), from not_and_of_not_right ((h 0) \u2260 0) this,\n-- have comb_seq_with_mbs (h 0) = g (h 0 - h 0), from if_neg this,\n-- by simp [this,nat.sub_self])\n\ninclude Hbp Hex\n\ntheorem local_contra_of_comb_seq_with_mbs : false := \nby_cases\n(suppose eq0 : h 0 = 0, \nhave eq : comb_seq_with_mbs 0 = g 0, begin apply g_part_of_comb_seq_with_mbs, assumption end,\nhave m (comb_seq_with_mbs 0) < m (minimal_bad_seq m Hex (h 0)), by rw -eq at Hbp;exact Hbp,\nhave le : m (comb_seq_with_mbs 0) < m (minimal_bad_seq m Hex 0), by super,\nhave m (minimal_bad_seq m Hex 0) \u2264 m (comb_seq_with_mbs 0), from minimality_of_mbs_0 m Hex comb_seq_with_mbs badness_of_comb_seq_with_mbs,\n(not_le_of_gt le) this)\n(assume Hneg, \n-- have le : m (minimal_bad_seq m Hex (succ (pred (h 0)))) \u2264  m (comb_seq_with_mbs (succ (pred (h 0)))), begin apply minimality_of_mbs, split, end,\nhave le : m (minimal_bad_seq m Hex (succ (pred (h 0)))) \u2264  m (comb_seq_with_mbs (succ (pred (h 0)))), from minimality_of_mbs m _ _ _ \u27e8begin apply comb_seq_extends_mbs_at_pred_bp, exact Hneg end,badness_of_comb_seq_with_mbs\u27e9,\nhave h 0 > 0, from nat.pos_of_ne_zero Hneg,\nhave succ (pred (h 0)) = h 0, from succ_pred_of_pos this,\nhave m (minimal_bad_seq m Hex (h 0)) \u2264 m (comb_seq_with_mbs (h 0)), by rw this at le;exact le,\nhave m (minimal_bad_seq m Hex (h 0)) \u2264 m (g 0), by rw comb_seq_h0 at this;exact this,\nhave \u00ac m (g 0) < m (minimal_bad_seq m Hex (h 0)), from not_lt_of_ge this,  \nthis Hbp)\n\nend\n\n-- #check local_contra_of_comb_seq_with_mbs\n\nsection\nparameter {Q : Type}\nparameter [o : wqo Q]\n\ndefinition sub := @star Q o.le\n\ntheorem sub_refl (q : finite_subsets Q) : sub q q :=\nhave \u2200 a : Q, a \u2208 q.1 \u2192 a \u2264 (id a) \u2227 id a \u2208 q.1, begin intros, split, simp, apply quasiorder.refl, simp, assumption end,\n\u27e8id, \u27e8inj_from_to_id q.1,this\u27e9\u27e9\n\ntheorem sub_trans (a b c : finite_subsets Q) (H1 : sub a b) (H2 : sub b c) : sub a c :=\nlet \u27e8f,hf\u27e9 := H1, \u27e8g,hg\u27e9 := H2 in\nhave inj : inj_from_to (g \u2218 f) a.1 c.1, from inj_from_to_compose hg^.left hf^.left,\nhave \u2200 q : Q, q \u2208 a.1 \u2192 q \u2264 ((g \u2218 f) q) \u2227 (g \u2218 f) q \u2208 c.1, from \n  \u03bb q Hq,\n  have le1 : q \u2264 (f q), from (hf^.right q Hq)^.left,\n  have fqin : f q \u2208 b.1, from (hf^.right q Hq)^.right,\n  have le2 : (f q) \u2264 ((g \u2218 f) q), from (hg^.right (f q) fqin)^.left,\n  have qle : q \u2264 ((g \u2218 f) q), from quasiorder.trans le1 le2,\n  have (g \u2218 f) q \u2208 c.1, from (hg^.right (f q) fqin)^.right,\n  \u27e8qle, this\u27e9,\n\u27e8g \u2218 f,\u27e8inj,this\u27e9\u27e9\n\nparameter H : \u2203 f : \u2115 \u2192 finite_subsets Q, \u00ac is_good f sub\n\ndefinition card_of_finite_subsets {A : Type} (s : finite_subsets A) := card s.1\n\ndefinition Higman's_mbs (n : \u2115) : finite_subsets Q := minimal_bad_seq card_of_finite_subsets H n\n\ntheorem badness_of_Higman's_mbs : \u00ac is_good Higman's_mbs sub := badness_of_mbs card_of_finite_subsets H\n\ntheorem nonempty_mem_of_mbs (n : \u2115) : (Higman's_mbs n).1 \u2260 \u2205 := \nsuppose (Higman's_mbs n).1 = \u2205, \nhave lt : n < succ n, from lt_succ_self n,\nhave nond : \u2200 a : Q, a \u2208 (Higman's_mbs n).1 \u2192 a \u2264 (id a) \u2227 id a \u2208 (Higman's_mbs (succ n)).1, from \n  \u03bb a, \u03bb H, have a \u2209 (\u2205 : set Q), from set.not_mem_empty a, by super,\nhave sub (Higman's_mbs n) (Higman's_mbs (succ n)), \nfrom \u27e8id, \u27e8\u27e8\u03bb a Ha,((nond a Ha)^.right),\u03bb b Hb h1 h2 h3,by assumption\u27e9,nond\u27e9\u27e9,\nhave is_good Higman's_mbs sub, from \u27e8n, \u27e8succ n,\u27e8lt,this\u27e9\u27e9\u27e9,\nbadness_of_Higman's_mbs this\n\ndefinition B_pairs (n : \u2115) : Q \u00d7 finite_subsets Q := \nhave \u2203 a : Q, a \u2208 (Higman's_mbs n).1, from exists_mem_of_ne_empty (nonempty_mem_of_mbs n),\nlet q := some this in\nlet b := (Higman's_mbs n).1 \\ insert q \u2205 in\nhave finite (Higman's_mbs n).1, from (Higman's_mbs n).2,\nhave finite b, from @finite_diff _ _ _ this,\n(q, \u27e8b,this\u27e9)\n\nprivate definition B (n : \u2115) : finite_subsets Q := (B_pairs n).2\n\ndefinition qn (n : \u2115) : Q := (B_pairs n).1\n\ntheorem qn_in_mbs (n : \u2115) : qn n \u2208 (Higman's_mbs n).val :=\nsome_spec (exists_mem_of_ne_empty (nonempty_mem_of_mbs n))\n\ntheorem qn_not_in_Bn (n : \u2115) : \u00ac set.mem (qn n) (B n).val := \nsuppose qn n \u2208 (B n).val, this^.right (mem_singleton (qn n))\n\ntheorem ins_B_pairs (n : \u2115) : insert (qn n) (B n).val = (Higman's_mbs n).val :=\nhave \u2203 a : Q, a \u2208 (Higman's_mbs n).val, from exists_mem_of_ne_empty (nonempty_mem_of_mbs n),\nhave qnin : qn n \u2208 (Higman's_mbs n).val, from some_spec this,\nhave (B n).val = (Higman's_mbs n).val \\ insert (qn n) \u2205, from rfl,\nbegin apply subset.antisymm, intros x H1, apply or.elim H1,  \nintro h, simph, intro h1, rw this at h1, exact h1^.left,\nintros x h2, cases (decidable.em (x = qn H n)) with H3 H4,\napply or.inl,exact H3,\napply or.inr, rw this,apply and.intro, exact h2,\napply not_mem_singleton, exact H4\nend\n\ntheorem sub_B_mbs (n : \u2115) : (B n).val \u2286 (Higman's_mbs n).val :=\nby intros; intro; rw -ins_B_pairs; apply or.inr; assumption\n\ntheorem trans_of_B (i j : \u2115) (H1 : sub (Higman's_mbs i) (B j)) : sub (Higman's_mbs i) (Higman's_mbs j) :=\nlet \u27e8f,hf\u27e9 := H1 in\nhave inj_from_to f (Higman's_mbs i).val (B j).val, from and.left hf,\nhave Hl : \u2200 a, a \u2208 (Higman's_mbs i).val \u2192 f a \u2208 (Higman's_mbs j).val, from\n  \u03bb a Ha, have f a \u2208 (B j).val, from this^.left Ha, \n  (sub_B_mbs j) this,\nhave inj : inj_from_to f (Higman's_mbs i).val (Higman's_mbs j).val, from \u27e8Hl, hf^.left^.right\u27e9,\nhave non_descending (Higman's_mbs i) (Higman's_mbs j) o.le f, from \n  \u03bb a Ha, have Hl : a \u2264 (f a), from (hf^.right a Ha)^.left,\n  have f a \u2208 (B j).val, from (hf^.right a Ha)^.right,\n  have fain : f a \u2208 insert (qn j) (B j).val, from or.inr this,\n  have insert (qn j) (B j).val =  (Higman's_mbs j).val, from ins_B_pairs j,\n  have f a \u2208 (Higman's_mbs j).val, by rw this at fain;exact fain,\n  \u27e8Hl, this\u27e9,\n\u27e8f, \u27e8inj, this\u27e9\u27e9\n\nsection\nparameter Hg : \u2203 g : \u2115 \u2192 \u2115, \u00ac is_good (B \u2218 g) sub \u2227 \u2200 i : \u2115, g 0 \u2264 g i\n\nprivate definition g := some Hg\n\ntheorem Higman's_Hg : \u00ac is_good (B \u2218 g) sub := \nlet \u27e8l,r\u27e9 := some_spec Hg in l\n\ntheorem Higman's_Hex : \u2203 f, \u00ac is_good f sub := \u27e8(B \u2218 g),Higman's_Hg\u27e9\n\ntheorem Higman's_Hh : \u2200 i : \u2115, g 0 \u2264 g i := (some_spec Hg)^.right\n\ntheorem Higman's_H : \u2200 i j, sub (Higman's_mbs i) ((B \u2218 g) (j - g 0)) \u2192 sub (Higman's_mbs i) (Higman's_mbs (g (j - g 0))) := \n\u03bb i j, \u03bb H1, trans_of_B i (g (j - g 0)) H1\n\ndefinition Higman's_comb_seq (n : \u2115) : finite_subsets Q := \n@comb_seq_with_mbs _ sub (B \u2218 g) g card_of_finite_subsets Higman's_Hex n\n\ntheorem card_B_lt_mbs (n : \u2115) : card (B n).val < card (Higman's_mbs n).val :=\nhave finite (B n).val, from (B n).2,\nhave eq : card (insert (qn n) (B n).val) = card (B n).val + 1, from @card_insert_of_not_mem _ _ _ this (qn_not_in_Bn n), \nhave card (B n).val < card (B n).val + 1, from lt_succ_self (card (B n).val), \nhave card (B n).val < card (insert (qn n) (B n).1), begin rw eq, exact this end,-- by simp,\nhave insert (qn n) ((B n).val) = (Higman's_mbs n).val, from ins_B_pairs n,\nby super\n\ntheorem Higman's_Hbp : card_of_finite_subsets (B (g 0)) < card_of_finite_subsets (Higman's_mbs (g 0)) := card_B_lt_mbs (g 0)\n\ntheorem Higman's_local_contradition : false := \nlocal_contra_of_comb_seq_with_mbs g card_of_finite_subsets Higman's_Hh Higman's_Hex Higman's_Hg Higman's_H Higman's_Hbp\n\nend\n\n-- #check Higman's_local_contradition\n\ndefinition ClassB : Type := {x : finite_subsets Q // \u2203 i, B i = x}\n\ndefinition oB (b1 : ClassB) (b2 : ClassB) : Prop := sub b1.val b2.val\n\ntheorem oB_refl (q : ClassB) : oB q q := sub_refl q.val\n\ntheorem oB_trans (a b c : ClassB) (H1 : oB a b) (H2 : oB b c) : oB a c :=\nsub_trans _ _ _ H1 H2\n\n    section\n    -- Suppose there exists a bad sequence of objects in ClassB. We show that we can construct a g : \u2115 \u2192 \u2115 such that \u00ac is_good (B \u2218 g) o. Then we can apply 'exists_sub_bad'. We cannot directly apply this theorem because ClassB is a type distinct from finite_subsets Q.\n    parameter HfB : \u2203 f, \u00ac is_good f oB\n\n    private definition f' : \u2115 \u2192 ClassB := some HfB\n\n    private theorem bad_f' : \u00ac is_good f' oB := some_spec HfB\n\n    private definition g' (n : \u2115) := (f' n).val\n\n    theorem exists_bad_B_seq : \u00ac is_good g' sub :=\n    suppose is_good g' sub,\n    let \u27e8i,j,hg'\u27e9 := this in\n    have is_good f' oB, from \u27e8i, \u27e8j, \u27e8hg'^.left, hg'^.right\u27e9\u27e9\u27e9,\n    bad_f' this\n\n    private definition g (n : \u2115) : \u2115 := \n    have \u2203 i, B i = g' n, from (f' n).2,\n    some this\n\n    private theorem comp_eq_g' : B \u2218 g = g' :=\n    have \u2200 x, B (g x) = g' x, from \u03bb x, some_spec (f' x).2,\n    funext this\n\n    private theorem bad_comp : \u00ac is_good (B \u2218 g) sub := \n    have \u00ac is_good g' sub, from exists_bad_B_seq,\n    by rw -comp_eq_g' at this;exact this\n\n    theorem exists_sub_bad_B_seq : \u2203 h : \u2115 \u2192 \u2115, \u00ac is_good (B \u2218 h) sub \u2227 \u2200 i : \u2115, h 0 \u2264 h i := exists_sub_bad B g sub bad_comp\n\n    end\n\ntheorem oB_is_good : \u2200 f, is_good f oB :=\nby_contradiction\n(suppose \u00ac \u2200 f, is_good f oB,\nhave \u2203 f, \u00ac is_good f oB, from classical.exists_not_of_not_forall this,\nhave \u2203 h : \u2115 \u2192 \u2115, \u00ac is_good (B \u2218 h) sub \u2227 \u2200 i : \u2115, h 0 \u2264 h i, from exists_sub_bad_B_seq this,\nHigman's_local_contradition this)\n\ninstance wqo_ClassB : wqo ClassB := wqo.mk (quasiorder.mk (has_le.mk oB) oB_refl oB_trans) oB_is_good\n\ninstance wqo_prod_Q_ClassB : wqo (Q \u00d7 ClassB) := wqo_prod\n\ntheorem good_prod_Q_ClassB : \u2200 f : \u2115 \u2192 Q \u00d7 ClassB, is_good f (prod_order o.le oB) := wqo.is_good\n\nlemma B_refl (n : \u2115) : \u2203 i, B i = B n := \u27e8n, rfl\u27e9\n\ndefinition fB (n : \u2115) : ClassB := \u27e8B n,B_refl n\u27e9\n\nprivate definition p (n : \u2115) : Q \u00d7 ClassB := (qn n, fB n)\n\ntheorem good_p : is_good p (prod_order o.le oB) := good_prod_Q_ClassB p\n\ntheorem Hij : \u2203 i j, i < j \u2227 ((qn i) \u2264 (qn j) \u2227 oB (fB i) (fB j)) := good_p\n\ntheorem exists_embeds : \u2203 i j, i < j \u2227 sub (Higman's_mbs i) (Higman's_mbs j) :=\nlet \u27e8i,j,hij\u27e9 := good_p in\nhave oB (fB i) (fB j), from hij^.right^.right,\nlet \u27e8f\u2081,\u27e8injf\u2081,rhf1\u27e9\u27e9 := this in\nlet f\u2082 (q : Q) : Q := if q = qn i then qn j else f\u2081 q in\nhave nond : \u2200 a : Q, a \u2208 (Higman's_mbs i).val \u2192  a \u2264 (f\u2082 a) \u2227 f\u2082 a \u2208 (Higman's_mbs j).val, from \u03bb a Ha, \n  have Hor : a = qn i \u2228 a \u2208 (B i).val, by rw -(ins_B_pairs H i) at Ha;exact Ha,\n  or.elim (em (a = qn i)) \n(\u03bb l, have eqf\u2082a : f\u2082 a = qn j, from if_pos l, \u27e8begin rw [eqf\u2082a,l], exact hij^.right^.left end, begin rw [eqf\u2082a], apply qn_in_mbs end\u27e9) \n(\u03bb r,have f\u2082 a=f\u2081 a, from if_neg r,\n have conj : a \u2264 (f\u2082 a) \u2227 f\u2082 a \u2208 (B j).val, begin rw this, apply rhf1, super end,\n\u27e8conj^.left,begin apply sub_B_mbs, exact conj^.right end\u27e9),\nhave Hmapsto : \u2200 a, a \u2208 (Higman's_mbs i).val \u2192 f\u2082 a \u2208 (Higman's_mbs j).val, from \n  \u03bb a Ha, and.right (nond a Ha),\nhave \u2200 a\u2081 a\u2082, a\u2081 \u2208 (Higman's_mbs i).val \u2192 a\u2082 \u2208 (Higman's_mbs i).val \u2192 f\u2082 a\u2081 = f\u2082 a\u2082 \u2192 a\u2081 = a\u2082, from \n  \u03bb a\u2081 a\u2082 Ha\u2081 Ha\u2082 Heq,\n  have Hora\u2081 : a\u2081 = qn i \u2228 a\u2081 \u2208 (B i).val, by rw -(ins_B_pairs H i) at Ha\u2081;exact Ha\u2081,\n  have Hora\u2082 : a\u2082 = qn i \u2228 a\u2082 \u2208 (B i).val, by rw -(ins_B_pairs H i) at Ha\u2082;exact Ha\u2082,\n  by_cases\n  (assume Hpos : a\u2081 = qn i, -- level-1 subcase // pos\n   have eq21j : f\u2082 a\u2081 = qn j, from if_pos Hpos,\n   by_contradiction\n   (suppose a\u2081 \u2260 a\u2082,\n    have neq : qn i \u2260 a\u2082, by rw Hpos at this;exact this,\n    have eq2212 : f\u2082 a\u2082 = f\u2081 a\u2082, from if_neg (ne.symm neq),\n    have qn j \u2208 (B j).val, begin rw [-eq21j, Heq, eq2212], apply and.left injf\u2081,\n    exact or_resolve_right Hora\u2082 (ne.symm neq) end,\n    (qn_not_in_Bn j) this))\n  (assume Hneg, -- level-1 subcase // neg\n   have eq2111 : f\u2082 a\u2081 = f\u2081 a\u2081, from if_neg Hneg,\n   have a1inBi :  a\u2081 \u2208 (B i).val, from or_resolve_right Hora\u2081 Hneg, \n   by_cases\n     (assume Hposa\u2082 : a\u2082 = qn i, -- level-2 subcase // pos\n      have eq21j : f\u2082 a\u2082 = qn j, from if_pos Hposa\u2082,\n      by_contradiction\n      (suppose a\u2081 \u2260 a\u2082,\n       have neq2 : a\u2081 \u2260 qn i, by rw Hposa\u2082 at this;exact this,\n       have eq2111 : f\u2082 a\u2081 = f\u2081 a\u2081, from if_neg neq2,\n       have qn j \u2208 (B j).val, \n       begin rw [-eq21j, -Heq, eq2111], apply and.left injf\u2081, \n       exact or_resolve_right Hora\u2081 neq2 end,\n       (qn_not_in_Bn j) this))\n     (assume Hnega\u2082, -- level-2 subcase // neg\n      have eq2212 : f\u2082 a\u2082 = f\u2081 a\u2082, from if_neg Hnega\u2082,\n      have f\u2081 a\u2081 = f\u2082 a\u2082, by rw eq2111 at Heq;exact Heq,\n      have eq1112 : f\u2081 a\u2081 = f\u2081 a\u2082, from eq.trans this eq2212,\n      have a\u2082 \u2208 (B i).val, from or_resolve_right Hora\u2082 Hnega\u2082, \n      (and.right injf\u2081) a1inBi this eq1112)),\nhave inj_from_to f\u2082 (Higman's_mbs i).val (Higman's_mbs j).val, from \u27e8Hmapsto, this\u27e9,\nhave sub (Higman's_mbs i) (Higman's_mbs j), from \u27e8f\u2082,\u27e8this, nond\u27e9\u27e9,\n\u27e8i,\u27e8j, \u27e8hij^.left, this\u27e9\u27e9\u27e9\n\ntheorem goodness_of_Higman's_mbs : is_good Higman's_mbs sub := exists_embeds\n\ntheorem Higman's_contradiction : false := badness_of_Higman's_mbs goodness_of_Higman's_mbs\n\nend\n\n-- #check Higman's_contradiction\n\nvariable {Q : Type}\nvariable [wqo Q]\n\ntheorem good_star : \u2200 f : \u2115 \u2192 finite_subsets Q , is_good f sub := \nby_contradiction\n(suppose \u00ac \u2200 f, is_good f sub,\nhave \u2203 f, \u00ac is_good f sub, from classical.exists_not_of_not_forall this,\nHigman's_contradiction this)\n\ndef wqo_finite_subsets : wqo (finite_subsets Q) :=\n\u27e8\u27e8\u27e8sub\u27e9,sub_refl,sub_trans\u27e9,good_star\u27e9\n\n#check wqo_finite_subsets.is_good\n-- example : wqo.le (finite_subsets Q) _ = sub := rfl\n\n-- #check wqo_finite_subsets\n\nend kruskal\n", "meta": {"author": "minchaowu", "repo": "Kruskal.lean3", "sha": "a14516f47b21e636e9df914fc6ebe64cbe5cd38d", "save_path": "github-repos/lean/minchaowu-Kruskal.lean3", "path": "github-repos/lean/minchaowu-Kruskal.lean3/Kruskal.lean3-a14516f47b21e636e9df914fc6ebe64cbe5cd38d/higman.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.39890570013363774}}
{"text": "/-\nCopyright (c) 2022 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n\n! This file was ported from Lean 3 source module analysis.special_functions.japanese_bracket\n! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Analysis.SpecialFunctions.Integrals\nimport Mathbin.Analysis.SpecialFunctions.Pow\nimport Mathbin.MeasureTheory.Integral.Layercake\nimport Mathbin.Tactic.Positivity\n\n/-!\n# Japanese Bracket\n\nIn this file, we show that Japanese bracket $(1 + \\|x\\|^2)^{1/2}$ can be estimated from above\nand below by $1 + \\|x\\|$.\nThe functions $(1 + \\|x\\|^2)^{-r/2}$ and $(1 + |x|)^{-r}$ are integrable provided that `r` is larger\nthan the dimension.\n\n## Main statements\n\n* `integrable_one_add_norm`: the function $(1 + |x|)^{-r}$ is integrable\n* `integrable_jap` the Japanese bracket is integrable\n\n-/\n\n\nnoncomputable section\n\nopen BigOperators NNReal Filter Topology ENNReal\n\nopen Asymptotics Filter Set Real MeasureTheory FiniteDimensional\n\nvariable {E : Type _} [NormedAddCommGroup E]\n\ntheorem sqrt_one_add_norm_sq_le (x : E) : Real.sqrt (1 + \u2016x\u2016 ^ 2) \u2264 1 + \u2016x\u2016 :=\n  by\n  refine' le_of_pow_le_pow 2 (by positivity) two_pos _\n  simp [sq_sqrt (zero_lt_one_add_norm_sq x).le, add_pow_two]\n#align sqrt_one_add_norm_sq_le sqrt_one_add_norm_sq_le\n\ntheorem one_add_norm_le_sqrt_two_mul_sqrt (x : E) : 1 + \u2016x\u2016 \u2264 Real.sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2) :=\n  by\n  suffices (sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2)) ^ 2 - (1 + \u2016x\u2016) ^ 2 = (1 - \u2016x\u2016) ^ 2\n    by\n    refine' le_of_pow_le_pow 2 (by positivity) (by norm_num) _\n    rw [\u2190 sub_nonneg, this]\n    positivity\n  rw [mul_pow, sq_sqrt (zero_lt_one_add_norm_sq x).le, add_pow_two, sub_pow_two]\n  norm_num\n  ring\n#align one_add_norm_le_sqrt_two_mul_sqrt one_add_norm_le_sqrt_two_mul_sqrt\n\ntheorem rpow_neg_one_add_norm_sq_le {r : \u211d} (x : E) (hr : 0 < r) :\n    (1 + \u2016x\u2016 ^ 2) ^ (-r / 2) \u2264 2 ^ (r / 2) * (1 + \u2016x\u2016) ^ (-r) :=\n  by\n  have h1 : 0 \u2264 (2 : \u211d) := by positivity\n  have h3 : 0 < sqrt 2 := by positivity\n  have h4 : 0 < 1 + \u2016x\u2016 := by positivity\n  have h5 : 0 < sqrt (1 + \u2016x\u2016 ^ 2) := by positivity\n  have h6 : 0 < sqrt 2 * sqrt (1 + \u2016x\u2016 ^ 2) := mul_pos h3 h5\n  rw [rpow_div_two_eq_sqrt _ h1, rpow_div_two_eq_sqrt _ (zero_lt_one_add_norm_sq x).le, \u2190\n    inv_mul_le_iff (rpow_pos_of_pos h3 _), rpow_neg h4.le, rpow_neg (sqrt_nonneg _), \u2190 mul_inv, \u2190\n    mul_rpow h3.le h5.le, inv_le_inv (rpow_pos_of_pos h6 _) (rpow_pos_of_pos h4 _),\n    rpow_le_rpow_iff h4.le h6.le hr]\n  exact one_add_norm_le_sqrt_two_mul_sqrt _\n#align rpow_neg_one_add_norm_sq_le rpow_neg_one_add_norm_sq_le\n\ntheorem le_rpow_one_add_norm_iff_norm_le {r t : \u211d} (hr : 0 < r) (ht : 0 < t) (x : E) :\n    t \u2264 (1 + \u2016x\u2016) ^ (-r) \u2194 \u2016x\u2016 \u2264 t ^ (-r\u207b\u00b9) - 1 :=\n  by\n  rw [le_sub_iff_add_le', neg_inv]\n  exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm\n#align le_rpow_one_add_norm_iff_norm_le le_rpow_one_add_norm_iff_norm_le\n\nvariable (E)\n\ntheorem closedBall_rpow_sub_one_eq_empty_aux {r t : \u211d} (hr : 0 < r) (ht : 1 < t) :\n    Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1) = \u2205 :=\n  by\n  rw [Metric.closedBall_eq_empty, sub_neg]\n  exact Real.rpow_lt_one_of_one_lt_of_neg ht (by simp only [hr, Right.neg_neg_iff, inv_pos])\n#align closed_ball_rpow_sub_one_eq_empty_aux closedBall_rpow_sub_one_eq_empty_aux\n\nvariable [NormedSpace \u211d E] [FiniteDimensional \u211d E]\n\nvariable {E}\n\ntheorem finite_integral_rpow_sub_one_pow_aux {r : \u211d} (n : \u2115) (hnr : (n : \u211d) < r) :\n    (\u222b\u207b x : \u211d in Ioc 0 1, ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n)) < \u221e :=\n  by\n  have hr : 0 < r := lt_of_le_of_lt n.cast_nonneg hnr\n  have h_int :\n    \u2200 (x : \u211d) (hx : x \u2208 Ioc (0 : \u211d) 1),\n      ENNReal.ofReal ((x ^ (-r\u207b\u00b9) - 1) ^ n) \u2264 ENNReal.ofReal (x ^ (-(r\u207b\u00b9 * n))) :=\n    by\n    intro x hx\n    have hxr : 0 \u2264 x ^ (-r\u207b\u00b9) := rpow_nonneg_of_nonneg hx.1.le _\n    apply ENNReal.ofReal_le_ofReal\n    rw [\u2190 neg_mul, rpow_mul hx.1.le, rpow_nat_cast]\n    refine' pow_le_pow_of_le_left _ (by simp only [sub_le_self_iff, zero_le_one]) n\n    rw [le_sub_iff_add_le', add_zero]\n    refine' Real.one_le_rpow_of_pos_of_le_one_of_nonpos hx.1 hx.2 _\n    rw [Right.neg_nonpos_iff, inv_nonneg]\n    exact hr.le\n  refine' lt_of_le_of_lt (set_lintegral_mono (by measurability) (by measurability) h_int) _\n  refine' integrable_on.set_lintegral_lt_top _\n  rw [\u2190 intervalIntegrable_iff_integrable_Ioc_of_le zero_le_one]\n  apply intervalIntegral.intervalIntegrableRpow'\n  rwa [neg_lt_neg_iff, inv_mul_lt_iff' hr, one_mul]\n#align finite_integral_rpow_sub_one_pow_aux finite_integral_rpow_sub_one_pow_aux\n\ntheorem finite_integral_one_add_norm [MeasureSpace E] [BorelSpace E]\n    [(@volume E _).IsAddHaarMeasure] {r : \u211d} (hnr : (finrank \u211d E : \u211d) < r) :\n    (\u222b\u207b x : E, ENNReal.ofReal ((1 + \u2016x\u2016) ^ (-r))) < \u221e :=\n  by\n  have hr : 0 < r := lt_of_le_of_lt (finrank \u211d E).cast_nonneg hnr\n  -- We start by applying the layer cake formula\n  have h_meas : Measurable fun \u03c9 : E => (1 + \u2016\u03c9\u2016) ^ (-r) := by measurability\n  have h_pos : \u2200 x : E, 0 \u2264 (1 + \u2016x\u2016) ^ (-r) :=\n    by\n    intro x\n    positivity\n  rw [lintegral_eq_lintegral_meas_le volume h_pos h_meas]\n  -- We use the first transformation of the integrant to show that we only have to integrate from\n  -- 0 to 1 and from 1 to \u221e\n  have h_int :\n    \u2200 (t : \u211d) (ht : t \u2208 Ioi (0 : \u211d)),\n      (volume { a : E | t \u2264 (1 + \u2016a\u2016) ^ (-r) } : ENNReal) =\n        volume (Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1)) :=\n    by\n    intro t ht\n    congr 1\n    ext x\n    simp only [mem_set_of_eq, mem_closedBall_zero_iff]\n    exact le_rpow_one_add_norm_iff_norm_le hr (mem_Ioi.mp ht) x\n  rw [set_lintegral_congr_fun measurableSet_Ioi (ae_of_all volume <| h_int)]\n  have hIoi_eq : Ioi (0 : \u211d) = Ioc (0 : \u211d) 1 \u222a Ioi 1 := (Set.Ioc_union_Ioi_eq_Ioi zero_le_one).symm\n  have hdisjoint : Disjoint (Ioc (0 : \u211d) 1) (Ioi 1) := by simp [disjoint_iff]\n  rw [hIoi_eq, lintegral_union measurableSet_Ioi hdisjoint, ENNReal.add_lt_top]\n  have h_int' :\n    \u2200 (t : \u211d) (ht : t \u2208 Ioc (0 : \u211d) 1),\n      (volume (Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1)) : ENNReal) =\n        ENNReal.ofReal ((t ^ (-r\u207b\u00b9) - 1) ^ FiniteDimensional.finrank \u211d E) *\n          volume (Metric.ball (0 : E) 1) :=\n    by\n    intro t ht\n    refine' volume.add_haar_closed_ball (0 : E) _\n    rw [le_sub_iff_add_le', add_zero]\n    exact Real.one_le_rpow_of_pos_of_le_one_of_nonpos ht.1 ht.2 (by simp [hr.le])\n  have h_meas' : Measurable fun a : \u211d => ENNReal.ofReal ((a ^ (-r\u207b\u00b9) - 1) ^ finrank \u211d E) := by\n    measurability\n  constructor\n  -- The integral from 0 to 1:\n  \u00b7 rw [set_lintegral_congr_fun measurableSet_Ioc (ae_of_all volume <| h_int'),\n      lintegral_mul_const _ h_meas', ENNReal.mul_lt_top_iff]\n    left\n    -- We calculate the integral\n    exact \u27e8finite_integral_rpow_sub_one_pow_aux (finrank \u211d E) hnr, measure_ball_lt_top\u27e9\n  -- The integral from 1 to \u221e is zero:\n  have h_int'' :\n    \u2200 (t : \u211d) (ht : t \u2208 Ioi (1 : \u211d)),\n      (volume (Metric.closedBall (0 : E) (t ^ (-r\u207b\u00b9) - 1)) : ENNReal) = 0 :=\n    fun t ht => by rw [closedBall_rpow_sub_one_eq_empty_aux E hr ht, measure_empty]\n  -- The integral over the constant zero function is finite:\n  rw [set_lintegral_congr_fun measurableSet_Ioi (ae_of_all volume <| h_int''), lintegral_const 0,\n    MulZeroClass.zero_mul]\n  exact WithTop.zero_lt_top\n#align finite_integral_one_add_norm finite_integral_one_add_norm\n\ntheorem integrableOneAddNorm [MeasureSpace E] [BorelSpace E] [(@volume E _).IsAddHaarMeasure]\n    {r : \u211d} (hnr : (finrank \u211d E : \u211d) < r) : Integrable fun x : E => (1 + \u2016x\u2016) ^ (-r) :=\n  by\n  refine' \u27e8by measurability, _\u27e9\n  -- Lower Lebesgue integral\n  have : (\u222b\u207b a : E, \u2016(1 + \u2016a\u2016) ^ (-r)\u2016\u208a) = \u222b\u207b a : E, ENNReal.ofReal ((1 + \u2016a\u2016) ^ (-r)) :=\n    lintegral_nnnorm_eq_of_nonneg fun _ => rpow_nonneg_of_nonneg (by positivity) _\n  rw [has_finite_integral, this]\n  exact finite_integral_one_add_norm hnr\n#align integrable_one_add_norm integrableOneAddNorm\n\ntheorem integrableRpowNegOneAddNormSq [MeasureSpace E] [BorelSpace E]\n    [(@volume E _).IsAddHaarMeasure] {r : \u211d} (hnr : (finrank \u211d E : \u211d) < r) :\n    Integrable fun x : E => (1 + \u2016x\u2016 ^ 2) ^ (-r / 2) :=\n  by\n  have hr : 0 < r := lt_of_le_of_lt (finrank \u211d E).cast_nonneg hnr\n  refine'\n    ((integrableOneAddNorm hnr).const_mul <| 2 ^ (r / 2)).mono (by measurability)\n      (eventually_of_forall fun x => _)\n  have h1 : 0 \u2264 (1 + \u2016x\u2016 ^ 2) ^ (-r / 2) := by positivity\n  have h2 : 0 \u2264 (1 + \u2016x\u2016) ^ (-r) := by positivity\n  have h3 : 0 \u2264 (2 : \u211d) ^ (r / 2) := by positivity\n  simp_rw [norm_mul, norm_eq_abs, abs_of_nonneg h1, abs_of_nonneg h2, abs_of_nonneg h3]\n  exact rpow_neg_one_add_norm_sq_le _ hr\n#align integrable_rpow_neg_one_add_norm_sq integrableRpowNegOneAddNormSq\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Analysis/SpecialFunctions/JapaneseBracket.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.3989056869727899}}
{"text": "import completeness.filteringC\n\n\nlocal attribute [instance] classical.prop_decidable\n\nopen set formCLC\n\nnamespace canonical\n\n\n----------------------------------------------------------\n-- Definitions and Lemmas needed for completness / model construction\n----------------------------------------------------------\n-- Tilde\n----------------------------------------------------------\n-- def tilde {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c8 : formCLC agents) : \n--   set ((canonical_model_CLC agents).f.states) :=\n-- {s : (canonical_model_CLC agents).f.states | \u03c8 \u2208 s}\n\n-- lemma h_tilde_compl {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c8 : formCLC agents) : \n--   tilde (\u00ac \u03c8) = (tilde \u03c8)\u1d9c := \n-- begin\n--   ext,\n--   simp[tilde],\n--   split,\n--   { intros hx hf,\n--     exact contra_containts_pr_false x.2 hf hx, },\n--   { intros hx,\n--     exact not_in_from_notin x.2 hx, },\n-- end\n\n-- phi sf\n----------------------------------------------------------\nnoncomputable def phi_s_f {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (sf : S_f \u03c6) : formCLC agents :=\nfinite_conjunction (finset.to_list (sf.1))\n\n-- phi sf \u2208 s\nlemma phi_s_f_in_s {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents)\n  (s : (canonical_model_CLC agents).f.states):\n  phi_s_f \u03c6 ((s_f \u03c6 s)) \u2208 s :=\nbegin\n  simp[phi_s_f],\n  have hinduct : \u2200 fs : list (formCLC agents), \n    (fs \u2286 ((s_f \u03c6 s).1 : finset (formCLC agents)).to_list) \u2192 finite_conjunction fs \u2208 s, from\n  begin\n    intros fs hfs,\n    induction fs with f fs ih,\n    { simp[finite_conjunction],\n      exact @max_ax_contains_by_empty_proof (formCLC agents) _ _ _ _ s.prop prtrue, },\n    { simp[finite_conjunction] at *,\n      cases hfs with hf hfs,\n      have hf_in_s : f \u2208 s, from s_f_subset_s \u03c6 s hf,\n      have hfs_in_s : finite_conjunction fs \u2208 s, from ih hfs,\n      apply max_ax_contains_by_set_proof_2h s.2 hf_in_s hfs_in_s,\n      exact axCLC.Prop4, },\n  end,\n  apply hinduct,\n  simp,\nend\n\nlemma phi_s_f_forall_iff {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 : formCLC agents} (sf : S_f \u03c6) : \n  (\u2200 x : formCLC agents, x \u2208 sf \u2192 axCLC x) \u2194 axCLC (phi_s_f \u03c6 sf) :=\nbegin\n  unfold phi_s_f,\n  have h_con := @finite_conj_forall_iff (formCLC agents) _ _ (sf.1.1).to_list,\n  split,\n  { intro h,\n    apply h_con.mp,\n    intros x hx,\n    apply h,\n    have hx : x \u2208 sf.1.1, from (multiset.mem_to_list x _).mp hx,\n    exact hx, },\n  { intros h x hx,\n    apply h_con.mpr,\n    exact h,\n    rw multiset.mem_to_list,\n    exact hx, },\nend\n\nlemma phi_s_f_forall_imp {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 : formCLC agents} {sf : S_f \u03c6} : \n  (\u2200 x \u2208 sf, axCLC ((phi_s_f \u03c6 sf) ~> x)) := \nbegin\n  unfold phi_s_f,\n  intros x hx,\n  have hx : x \u2208 sf.1.1.to_list, from (multiset.mem_to_list x _).mpr hx,\n  exact @finite_conj_forall_imp (formCLC agents) _ _ (sf.1.1).to_list x (hx),\nend\n\nlemma notin_nphi_s_f {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} {sf : S_f \u03c6} (h : \u03c8 \u2208 sf): \n  (axCLC ((\u00ac \u03c8) ~> \u00ac phi_s_f \u03c6 sf)) :=\nbegin\n  unfold phi_s_f,\n  apply @noin_imp_nfin_con (formCLC agents) _ _ _ \u03c8,\n  simp, apply h,\nend\n\nlemma phi_s_f_conj_contains_ax {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (sf : S_f \u03c6) (h\u03c8 : \u03c8 \u2208 sf) : \n  axCLC (phi_s_f \u03c6 sf) \u2194 axCLC (\u03c8 & (phi_s_f \u03c6 sf)) :=\nbegin\n  split,\n  { intro h,\n    apply @and_ax (formCLC agents),\n    exact (phi_s_f_forall_iff sf).mpr h \u03c8 h\u03c8,\n    exact h, },\n  { intro h,\n    apply and.elim_right,\n    apply (@ax_and (formCLC agents) _ _ \u03c8 (phi_s_f \u03c6 sf)).mp,\n    exact h, },\nend\n\nlemma phi_s_f_conj_contains {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (sf : S_f \u03c6) (h\u03c8 : \u03c8 \u2208 sf) : \n  axCLC ((phi_s_f \u03c6 sf) <~> (\u03c8 & (phi_s_f \u03c6 sf))) :=\nbegin\n  apply @ax_iff_intro (formCLC agents),\n  { apply imp_imp_and,\n    exact phi_s_f_forall_imp _ h\u03c8,\n    exact iden, },\n  { refine imp_and_r _,\n    exact iden, },\nend\n\n-- phi X (given a list)\n----------------------------------------------------------\nnoncomputable def phi_X_list {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) :\n  list (S_f \u03c6) \u2192 list (formCLC agents)\n| list.nil   := list.nil\n| (sf :: ss) := ((phi_s_f \u03c6 sf) :: phi_X_list ss)\n\n-- if sf \u2208 X, then phi sf is one of the disjuncts in phi X.\nlemma phi_X_list_contains {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (sfs : list (S_f \u03c6)) (sf : (S_f \u03c6)) (hs : sf \u2208 sfs) :\n  (phi_s_f \u03c6 sf) \u2208 phi_X_list \u03c6 sfs :=\nbegin\n  induction sfs with hd sfs ih,\n  {by_contradiction, simp at *, exact hs, },\n  { cases hs,\n    { simp[hs, phi_X_list], },\n    { simp[phi_X_list] at *,\n      apply or.intro_right,\n      exact ih hs, }, },\nend\n\nlemma phi_X_list_subset {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (sfs : list (S_f \u03c6)) (sfs' : list (S_f \u03c6)) (h : sfs \u2286 sfs') :\n  phi_X_list \u03c6 sfs \u2286 phi_X_list \u03c6 sfs' :=\nbegin\n  induction sfs with hd sfs ih,\n  { simp[phi_X_list], },\n  { simp[phi_X_list] at *,\n    split,\n    { exact phi_X_list_contains \u03c6 _ _ h.left, },\n    { exact ih h.right, }, },\nend\n\nlemma phi_X_list_append {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : list (S_f \u03c6)) :\n  phi_X_list \u03c6 (X ++ Y) \u2286 phi_X_list \u03c6 X ++ phi_X_list \u03c6 Y :=\nbegin\n  induction X with hd X ih,\n  { simp[phi_X_list], },\n  { simp[phi_X_list] at *,\n    exact list.subset_cons_of_subset (phi_s_f \u03c6 hd) ih, },\nend\n\nlemma phi_X_list_single {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (sf : (S_f \u03c6)) :\n  axCLC ((phi_s_f \u03c6 sf) \u2194' finite_disjunction (phi_X_list \u03c6 (sf :: list.nil))) :=\nbegin\n  apply @ax_iff_intro (formCLC agents),\n  { unfold phi_X_list finite_disjunction,\n    apply cut,\n    exact dni,\n    exact iden, },\n  { unfold phi_X_list finite_disjunction,\n    exact dne, },\nend\n\nlemma phi_X_list_conj_contains {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} {X : list (S_f \u03c6)} (h\u03c8 : \u2200 sf, sf \u2208 X \u2192 \u03c8 \u2208 sf) : \n  axCLC (finite_disjunction (phi_X_list \u03c6 X) <~> (\u03c8 & finite_disjunction (phi_X_list \u03c6 X))) :=\nbegin\n  induction X with sf X ih,\n  { simp [phi_X_list, finite_disjunction],\n    apply @and_ax (formCLC agents),\n    exact explosion,\n    exact imp_and_r iden, },\n  { simp [phi_X_list, finite_disjunction],\n    apply @and_ax (formCLC agents),\n    { apply or_cases,\n      { apply imp_imp_and,\n        { apply cut,\n          apply iff_l,\n          apply phi_s_f_conj_contains sf,\n          apply h\u03c8,\n          simp,\n          exact p5 _ _, },\n        { exact contra_explosion, }, },\n      { have h\u03c8' : \u2200 sf, sf \u2208 X \u2192 \u03c8 \u2208 sf, from\n        begin\n          intros tf htf,\n          apply h\u03c8,\n          simp [htf],\n        end,\n        specialize ih h\u03c8',\n        have ih := (@ax_and (formCLC agents) _ _ _ _).mp ih,\n        apply imp_imp_and,\n        { apply cut,\n          exact ih.left,\n          exact p5 _ _, },\n        { exact p1 _ _, }, }, },\n      { exact p6 _ _, }, },\nend\n\nlemma phi_X_list_exists {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  {\u03c6 : formCLC agents} {X : list (S_f \u03c6)} {s : (canonical_model_CLC agents).f.states} \n  (hfa : \u2200 x, x \u2208 X \u2192 phi_s_f \u03c6 x \u2209 s) : (\u00ac' (finite_disjunction (phi_X_list \u03c6 X))) \u2208 s :=\nbegin\n  induction X with x X ih,\n  { simp only [phi_X_list, finite_disjunction],\n    apply max_ax_contains_by_empty_proof s.2 not_bot, },\n  { simp only [phi_X_list, finite_disjunction] at *,\n    simp at hfa,\n    apply max_ax_contains_by_set_proof s.2 _ (iff_r demorgans''''),\n    apply max_ax_contains_by_set_proof_2h s.2 _ _ (axCLC.Prop4),\n    exact not_in_from_notin s.2 hfa.left,\n    exact ih hfa.right, },\nend\n\nlemma nk_phi_X_list_exists {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  {\u03c6 : formCLC agents} {i : agents} {X : list (S_f \u03c6)} {s : (canonical_model_CLC agents).f.states} \n  (hfa : \u2200 x, x \u2208 X \u2192 \u00ac' (K' i (\u00ac'(phi_s_f \u03c6 x))) \u2209 s) : \n  (\u00ac' (finite_disjunction (list.map (\u03bb (\u03c6 : formCLC agents), \u00ac' (K' i (\u00ac' \u03c6))) (phi_X_list \u03c6 X)))) \u2208 s :=\nbegin\n  induction X with x X ih,\n  { simp only [phi_X_list, finite_disjunction],\n    apply max_ax_contains_by_empty_proof s.2 not_bot, },\n  { simp only [phi_X_list, finite_disjunction] at *,\n    simp at hfa,\n    apply max_ax_contains_by_set_proof s.2 _ (iff_r demorgans''''),\n    apply max_ax_contains_by_set_proof_2h s.2 _ _ (axCLC.Prop4),\n    exact not_in_from_notin s.2 hfa.left,\n    exact ih hfa.right, },\nend\n\n\n-- phi X (given a finset)\n----------------------------------------------------------\nnoncomputable def phi_X_finset {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X : finset (S_f \u03c6)) :\n  formCLC agents :=\nfinite_disjunction (phi_X_list \u03c6 (finset.to_list X))\n\nlemma phi_X_subset_Y_imp {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) (hXY : X \u2286 Y) :\n  axCLC ((phi_X_finset \u03c6 X) \u2192' (phi_X_finset \u03c6 Y)) :=\nbegin\n  simp[phi_X_finset],\n  apply imp_finite_disjunction_subset (phi_X_list \u03c6 X.to_list) (phi_X_list \u03c6 Y.to_list),\n  apply phi_X_list_subset,\n  intros f hf,\n  rw finset.mem_to_list at *,\n  exact hXY hf,\nend\n\nlemma phi_X_list_append' {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) :\n  phi_X_list \u03c6 X.to_list ++ phi_X_list \u03c6 Y.to_list \u2286 phi_X_list \u03c6 (X \u222a Y).to_list :=\nbegin\n  simp at *,\n  split,\n  { apply phi_X_list_subset,\n    intros f hf,\n    rw finset.mem_to_list at *,\n    exact finset.mem_union_left Y hf, },\n {  apply phi_X_list_subset,\n    intros f hf,\n    rw finset.mem_to_list at *,\n    exact finset.mem_union_right X hf, }, \nend\n\nlemma phi_X_list_append'' {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) :\n  phi_X_list \u03c6 (X \u222a Y).to_list \u2286 phi_X_list \u03c6 X.to_list ++ phi_X_list \u03c6 Y.to_list :=\nbegin\n  have h1 := phi_X_list_append \u03c6 X.to_list Y.to_list,\n  have h2 : phi_X_list \u03c6 (X \u222a Y).to_list \u2286 phi_X_list \u03c6 (X.to_list ++ Y.to_list), from\n  begin\n    refine phi_X_list_subset \u03c6 (X \u222a Y).to_list (X.to_list ++ Y.to_list) _,\n    intros f hf,\n    simp at *,\n    exact hf,\n  end,\n  exact subset.trans h2 h1,\nend\n\nlemma phi_X_finset_union {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) :\n  axCLC ((\u00ac' (phi_X_finset \u03c6 X) \u2192' (phi_X_finset \u03c6 Y)) \u2192' (phi_X_finset \u03c6 (X \u222a Y))) :=\nbegin\n  simp[phi_X_finset],\n  apply @cut (formCLC agents),\n  apply disjunc_disjunct,\n  apply imp_finite_disjunction_subset,\n  apply phi_X_list_append',\nend\n\nlemma phi_X_finset_disjunct_of_disjuncts {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) :\n  axCLC (\u00ac' (phi_X_finset \u03c6 X) \u2192' (phi_X_finset \u03c6 Y)) \u2194 axCLC (phi_X_finset \u03c6 (X \u222a Y)) :=\nbegin\n  have hax := @ax_iff_disjunc_disjunct (formCLC agents) _ _\n              (phi_X_list \u03c6 X.to_list) (phi_X_list \u03c6 Y.to_list),\n  simp[phi_X_finset],\n  split,\n  { intro h,\n    apply @MP' (formCLC agents),\n    apply hax.mp h,\n    apply imp_finite_disjunction_subset,\n    apply phi_X_list_append', },\n  { intro h,\n    apply hax.mpr,\n    apply @MP' (formCLC agents),\n    apply h,\n    apply imp_finite_disjunction_subset,\n    apply phi_X_list_append'',\n  },\nend\n\n-- lemma phi_X_list_conj_contains {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n--   {\u03c6 \u03c8 : formCLC agents} (X : finset (S_f \u03c6)) (h\u03c8 : \u2200 sf \u2208 X, \u03c8 \u2208 sf) : \n--   axCLC (\u03c8 & (phi_X_finset \u03c6 X)) :=\n-- begin\n--   unfold phi_X_finset,\n-- end\n\n\n\n-- phi X (given a set)\n----------------------------------------------------------\n\n/-- `phi_X_set \u03c6 X` is a finite disjunction of all elements of `X`. -/\nnoncomputable def phi_X_set {agents : Type} [hN : fintype agents] [ha : nonempty agents]  \n  (\u03c6 : formCLC agents) (X : set (S_f \u03c6)) :\n  formCLC agents :=\nbegin\n  simp[S_f, finset.attach] at X,\n  have hX : finite X, from finite.of_fintype X,\n  have X : finset (S_f \u03c6), from finite.to_finset hX,\n  exact phi_X_finset \u03c6 X,\nend\n\nlemma phi_X_set_subset_Y_imp {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X : set (S_f \u03c6)) (Y : set (S_f \u03c6)) (hXY : X \u2286 Y) :\n  axCLC ((phi_X_set \u03c6 X) \u2192' (phi_X_set \u03c6 Y)) :=\nbegin\n  simp[phi_X_set],\n  apply phi_X_subset_Y_imp,\n  exact finite.to_finset_mono.mpr hXY,\nend\n\nlemma phi_X_set_disjunct {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : set (S_f \u03c6)) :\n  axCLC ((\u00ac' (phi_X_set \u03c6 X) \u2192' (phi_X_set \u03c6 Y)) \u2192' (phi_X_set \u03c6 (X \u222a Y))) :=\nbegin\n  unfold phi_X_set,\n  apply @cut (formCLC agents),\n  apply phi_X_finset_union,\n  apply phi_X_subset_Y_imp,\n  apply finset.union_subset,\n  repeat { simp,},\nend\n\nlemma phi_X_set_disjunct_of_disjuncts {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : set (S_f \u03c6)) :\n  axCLC (\u00ac' (phi_X_set \u03c6 X) \u2192' (phi_X_set \u03c6 Y)) \u2194 axCLC (phi_X_set \u03c6 (X \u222a Y)) :=\nbegin\n  unfold phi_X_set,\n  split,\n  { intro h,\n    have hax := (phi_X_finset_disjunct_of_disjuncts \u03c6 _ _).mp,\n    specialize hax h,\n    apply @MP' (formCLC agents),\n    apply hax,\n    apply phi_X_subset_Y_imp,\n    apply finset.union_subset,\n    repeat { simp, }, },\n  { intro h,\n    apply (phi_X_finset_disjunct_of_disjuncts \u03c6 _ _).mpr,\n    apply @MP' (formCLC agents),\n    apply h,\n    apply phi_X_subset_Y_imp,\n    refine finset.subset_iff.mpr _,\n    intros f hf,\n    simp at *,\n    exact hf, },\nend\n\nlemma phi_X_set_exists {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  {\u03c6 : formCLC agents} {X : set (S_f \u03c6)} {s : (canonical_model_CLC agents).f.states} \n  (h : phi_X_set \u03c6 X \u2208 s) : \u2203 tf \u2208 X, phi_s_f \u03c6 tf \u2208 s :=\nbegin\n  by_contradiction hfa,\n  simp at hfa,\n  apply in_from_not_notin s.2 h,\n  unfold phi_X_set phi_X_finset,\n  apply phi_X_list_exists,\n  intros sf hsf,\n  apply hfa,\n  simp [finite.mem_to_finset] at hsf,\n  exact hsf,\nend\n\nsection lemmas\n\n-- Motivation: self-contained `have`-block\n@[simp] lemma tilde_empty {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 : formCLC agents} : (tilde (phi_X_set \u03c6 \u2205)) = \u2205 :=\nbegin\n  -- 1.1.1. \u03c6\u2205 = \u22a5, because \u03c6\u2205 is an empty disjunction, thus  \u0303\u03c6\u2205 =  \u0303\u22a5.\n  simp [phi_X_set, phi_X_finset, phi_X_list, finite_disjunction, tilde],\n  -- 1.1.2.  \u0303\u22a5 = \u2205, because all s \u2208 S are consistent.\n  simp [eq_empty_iff_forall_not_mem],\n  intro s,\n  exact bot_not_mem_of_ax_consistent s.1 s.2.1\nend\n\nlemma tilde_ax_iff {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents)\n  {\u03c8 \u03c7 : formCLC agents} (hax : axCLC (\u03c8 <~> \u03c7)) : \n  tilde \u03c8 = tilde \u03c7 :=\nbegin\n  unfold tilde,\n  ext1 s,\n  split,\n  { intro hs,\n    simp at *,\n    apply max_ax_contains_by_set_proof s.2 hs,\n    apply iff_l,\n    apply hax, },\n  { intro hs,\n    simp at *,\n    apply max_ax_contains_by_set_proof s.2 hs,\n    apply iff_r,\n    apply hax, },\nend\n\n-- Motivation: simple way to prove `phi_X_set`\nlemma ax_phi_s_f_imp_phi_X_set_of_mem {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 : formCLC agents} {t} {X : set _} (h : s_f \u03c6 t \u2208 X) :\n  ax (phi_s_f \u03c6 (s_f \u03c6 t) \u2192' phi_X_set \u03c6 X) :=\nbegin\n  simp [phi_X_set],\n  apply @imp_finite_disjunction (formCLC agents) _ _ (phi_s_f \u03c6 (s_f \u03c6 t)),\n  apply phi_X_list_contains \u03c6,\n  simpa,\nend\n\nlemma ax_phi_s_f_imp_phi_X_set_of_mem' {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 : formCLC agents} {sf} {X : set _} (h : sf \u2208 X) :\n  ax (phi_s_f \u03c6 (sf) \u2192' phi_X_set \u03c6 X) :=\nbegin\n  simp [phi_X_set],\n  apply @imp_finite_disjunction (formCLC agents) _ _ (phi_s_f \u03c6 (sf)),\n  apply phi_X_list_contains \u03c6,\n  simpa,\nend\n\n-- Main Lemmas\n----------------------------------------------------------\n-- Lemma 4. \u22a2 (\u2228 {sf \u2208Sf } \u03c6sf)\nlemma univ_disjunct_provability {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  (\u03c6 : formCLC agents) (hs : nonempty (S_f \u03c6)):\n  ax (phi_X_set \u03c6 (univ : set (S_f \u03c6))) :=\nbegin\n  -- 1. By contradiction, assume that \u22ac (\u2228 {sf \u2208Sf } \u03c6sf)\n  by_contradiction,\n  -- 3. \u00ac(\u2228 {sf \u2208Sf } \u03c6sf) \u2208 t, because t is maximally consistent, from 1.\n  obtain \u27e8t', hexn, htn\u27e9 := exists_max_ax_consistent_neg_mem h,\n  let t := (\u27e8t', hexn\u27e9 : (canonical_model_CLC agents).f.states),\n  -- 4. \u22a2 \u03c6tf \u2192 (\u2228 {sf \u2208Sf } \u03c6sf ), by propositional logic, because t \u2208 Sf.\n  have himp : ax (phi_s_f \u03c6 (s_f \u03c6 t) \u2192' phi_X_set \u03c6 univ),\n    from ax_phi_s_f_imp_phi_X_set_of_mem (mem_univ _),\n  -- 5. \u03c6tf\u2208 t, by propositional logic, because all \u2200\u03c8 \u2208 tf , \u03c8 \u2208 t).\n  have hphitf : phi_s_f \u03c6 (s_f \u03c6 t) \u2208 t.1, from phi_s_f_in_s \u03c6 t, \n  -- 6. (\u2228{sf \u2208Sf } \u03c6sf) \u2208 t, by propositional logic, from 4 & 5.\n  have ht : phi_X_set \u03c6 (univ : set (S_f \u03c6)) \u2208 t.1, \n    from max_ax_contains_by_set_proof t.2 hphitf himp,\n  -- 7. Contradiction from 3 and 6.\n  apply contra_containts_pr_false t.2 ht htn,\nend\n\n-- Motivation: self-contained `have`-block\n-- 2.1. First we note that  \u0303\u03c6Sf =  \u0303\u22a4 = S\n@[simp] lemma tilde_univ {agents : Type} [hN : fintype agents] [ha : nonempty agents] {\u03c6 : formCLC agents} :\n  (tilde (phi_X_set \u03c6 (univ : set (S_f \u03c6)))) = (univ : set (canonical_model_CLC agents).f.states) :=\nbegin\n  simp[tilde],\n  ext1,\n  refine iff_of_true _ trivial,\n  simp,\n  apply max_ax_contains_by_empty_proof x.2,\n  apply univ_disjunct_provability,\n  exact canonical.nonempty_S_f \u03c6,\nend\n\n-- Lemma 5. \u2200sf , tf \u2208 Sf , sf \u0338 = tf \u21d2\u22a2 \u03c6sf\u2192 \u00ac\u03c6tf\nlemma unique_s_f_helper {agents : Type} [hN : fintype agents] [ha : nonempty agents]  \n  {\u03c6 x : formCLC agents} (sf  tf : (S_f \u03c6)) (hxf : x \u2208 sf) (hnf : x \u2209 tf) :\n  axCLC (\u00ac' (phi_s_f \u03c6 sf \u2227' phi_s_f \u03c6 tf)) := \nbegin\n  -- -- 6. \u03c7 /\u2208 t, from 5, by definition Sf , because \u03c7 \u2208 cl(\u03c6).\n  -- -- 7. \u00ac\u03c7 \u2208 t, from 6, because s and t are maximally consistent.\n  -- -- 8. \u2203\u03c8, (\u03c8 \u2194 \u00ac\u03c7) \u2227 (\u03c8 \u2208 cl(\u03c6)), because cl is closed under single negations.\n  -- -- 9. \u03c8 \u2208 s \u2228 \u03c8 \u2208 t, from 7 & 8, because s and t are maximally consistent.\n  -- -- 10. \u03c8 \u2208 sf \u2228 \u03c8 \u2208 tf , from 8 & 9, by definition Sf .\n  have hst := s_f_closed hnf (finset.subset_iff.mp (s_f_subset_cl \u03c6 _) hxf),\n  cases hst with \u03c8 hst,\n  cases hst with hst h\u03c8,\n  -- 11. \u03c6sf \u2227 \u03c6tf \u2192 \u22a5, by propositional logic, from 5, 8 & 10.\n  simp[phi_s_f],\n  apply @contra_con_cons (formCLC agents) _ _,\n  exact h\u03c8,\n  exact (sf.1.1).mem_to_list.mpr hxf,\n  exact (tf.1.1).mem_to_list.mpr hst,\nend\n\nlemma unique_s_f {agents : Type} [hN : fintype agents] [ha : nonempty agents]  \n  {\u03c6 : formCLC agents} (sf  tf : (S_f \u03c6)) (hneq : sf \u2260 tf) :\n  ax (phi_s_f \u03c6 sf \u2192' \u00ac' (phi_s_f \u03c6 tf)) :=\nbegin\n  -- 1. Assume by contradiction \u22ac \u03c6sf \u2192 \u00ac\u03c6tf\n  by_contradiction,\n  -- 2. \u2203u \u2208 S, (\u03c6sf \u2192 \u00ac\u03c6tf) /\u2208 u, from 1.\n  -- 3. \u00ac(\u03c6sf\u2192 \u00ac\u03c6tf) \u2208 u, from 2.\n  obtain \u27e8u', hexn.left, hun\u27e9 := exists_max_ax_consistent_neg_mem h,\n  let u := (\u27e8u', hexn.left\u27e9 : (canonical_model_CLC agents).f.states),\n  have hun : \u00ac' (phi_s_f \u03c6 sf \u2192' \u00ac' (phi_s_f \u03c6 tf)) \u2208 u.1, from by tauto,\n  -- 4. \u03c6sf \u2227 \u03c6tf \u2208 u, by propositional logic, from 3.\n  have hand : (phi_s_f \u03c6 sf \u2227' (phi_s_f \u03c6 tf)) \u2208 u.1,\n    from max_ax_contains_by_set_proof u.2 hun demorgans'',\n  -- 5. \u2203\u03c7 \u2208 sf \u222a tf , \u03c7 /\u2208 sf \u2228 \u03c7 /\u2208 tf , because sf and tf are not identical.\n  have : \u00ac(sf.1.1 \u2286 tf.1.1) \u2228 \u00ac(tf.1.1 \u2286 sf.1.1), from\n  begin\n    { rw \u2190 not_and_distrib,\n      rintro \u27e8hst, hts\u27e9,\n      apply hneq,\n      ext : 2,\n      exact subset_antisymm hst hts },\n  end,\n    obtain \u27e8x, hun, hneq'\u27e9 : \u2203 f, f \u2208 (sf.1.1 \u222a tf.1.1) \u2227 ((f \u2209 sf.1.1) \u2228 (f \u2209 tf.1.1)),\n    { simp only [finset.not_subset] at this, -- Motivation: I recall `not_subset` had something like `x \u2208 s \u2227 \u00ac x \u2208 t` so I reworked the statement to make it come true.\n      rcases this with \u27e8x, hxu, hxn\u27e9 | \u27e8x, hxu, hxn\u27e9;\n        use x;\n        simp only [finset.mem_union, hxu, hxn, not_true, not_false_iff, true_or, or_true, true_and] },\n  rw finset.mem_union at hun,\n\n  -- 11. \u03c6sf \u2227 \u03c6tf \u2192 \u22a5, from helper  (6-10)\n  -- 12. \u22a5 \u2208 u, by propositional logic, from 4 & 11, which contradicts the consistency of u.\n  apply ax_neg_containts_pr_false u.2 hand,\n  cases hun with hxf hxf,\n  { cases hneq' with hnf hnf,\n    { finish, },\n    { apply unique_s_f_helper _ _ hxf hnf, }, },\n  { cases hneq' with hnf hnf,\n    { apply cut (iff_l and_switch),\n      apply unique_s_f_helper _ _ hxf hnf, },\n    { finish, }, },\nend\n\nlemma phi_X_list_unique {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : list (S_f \u03c6)) (hXY : X.disjoint Y) (hX : list.nodup X) (hY : list.nodup Y) :\n  axCLC (finite_disjunction (phi_X_list \u03c6 X)\u2192' \u00ac' (finite_disjunction (phi_X_list \u03c6 Y))) :=\nbegin\n  induction' X with x X ihx,\n  { simp [phi_X_list, finite_disjunction],\n    apply @explosion (formCLC agents), },\n  { simp [phi_X_list, finite_disjunction],\n    apply @or_cases (formCLC agents),\n    { induction Y with y Y ihy,\n      { simp [phi_X_list, finite_disjunction],\n        apply MP',\n        apply not_bot,\n        apply axCLC.Prop1, },\n      { simp [phi_X_list, finite_disjunction] at *,\n        rw \u2190contrapos,\n        apply cut,\n        apply dne,\n        apply or_cases,\n        apply unique_s_f, \n        by_contradiction,\n        simp[h] at hXY,\n        exact hXY,\n        rw \u2190contrapos,\n        apply cut,\n        apply dne,\n        apply ihy hY.right,\n        exact hXY.2.1,\n        exact hXY.2.2, }, },\n    { apply ihx,\n      exact hY,\n      apply list.disjoint_of_disjoint_cons_left hXY,\n      simp at hX,\n      exact hX.2, }, },\nend\n\nlemma phi_X_finset_unique {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) (hXY : X \u2229 Y = \u2205) :\n  axCLC ((phi_X_finset \u03c6 X) \u2192' \u00ac' (phi_X_finset \u03c6 (Y))) :=\nbegin\n  simp[phi_X_finset],\n  apply phi_X_list_unique,\n  rw list.disjoint_left,\n  intros f hf,\n  simp at *,\n  by_contradiction,\n  exact finset.eq_empty_iff_forall_not_mem.mp hXY f (finset.mem_inter_of_mem hf h),\n  repeat {exact finset.nodup_to_list _, },\nend\n\nlemma phi_X_set_unique {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : set (S_f \u03c6)) (hXY : X \u2229 Y = \u2205) :\n  axCLC ((phi_X_set \u03c6 X) \u2192' \u00ac' (phi_X_set \u03c6 (Y))) :=\nbegin\n  simp[phi_X_set],\n  apply phi_X_finset_unique,\n  apply finset.eq_empty_iff_forall_not_mem.mpr,\n  intros f hf,\n  simp at *,\n  exact eq_empty_iff_forall_not_mem.mp hXY f ((mem_inter_iff f X Y).mpr hf),  \nend\n\nlemma contra_fin_disjunct_psi_and_not {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (h\u03c8 : \u03c8 \u2208 cl \u03c6) (sfs : list (S_f \u03c6)) \n  (hsfs : \u2200 sf : (@S_f agents _ ha \u03c6), sf \u2208 sfs \u2192 \u03c8 \u2209 sf) :\n  axCLC (\u22a5' <~> (\u03c8 & finite_disjunction (phi_X_list \u03c6 sfs))) :=\nbegin\n  apply @ax_iff_intro (formCLC agents),\n  exact explosion,\n  induction sfs with sf sfs ih,\n  { unfold phi_X_list finite_disjunction,\n    exact p6 _ _, },\n  { unfold phi_X_list finite_disjunction at *,\n    have hsfs' : \u2200 sf : (@S_f agents _ ha \u03c6), sf \u2208 sfs \u2192 \u03c8 \u2209 sf, from\n    begin\n      intros sf hsf,\n      apply hsfs,\n      simp [hsf],\n    end,\n    specialize ih hsfs',\n    refine and_right_imp.mpr _,\n    apply or_cases,\n    { have h\u03c7 := s_f_closed (hsfs sf (by simp)) h\u03c8,\n      cases h\u03c7 with \u03c7 h\u03c7,\n      apply cut,\n      apply iff_l,\n      apply phi_s_f_conj_contains sf h\u03c7.left,\n      apply imp_and_l,\n      apply cut,\n      apply iff_l,\n      apply h\u03c7.2,\n      exact iden, },\n    { refine and_right_imp.mp _,\n      apply ih, }, },\nend\n\nlemma phi_X_contains_iff_psi_helper_list {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (h\u03c8 : \u03c8 \u2208 cl \u03c6)  (sfs tfs : list (S_f \u03c6))\n  (hsfs : \u2200 sf : (@S_f agents _ ha \u03c6), sf \u2208 sfs \u2192 \u03c8 \u2208 sf)\n  (htfs : \u2200 tf : (@S_f agents _ ha \u03c6), tf \u2208 tfs \u2192 \u03c8 \u2209 tf)\n  (hSf : axCLC (\u00ac' (finite_disjunction (phi_X_list \u03c6 tfs))\u2192'finite_disjunction (phi_X_list \u03c6 sfs))) :\n  -- (hSf : axCLC (finite_disjunction (phi_X_list \u03c6 tfs ++ phi_X_list \u03c6 sfs))) :\n  -- (hempty : (sfs = list.nil \u2192 axCLC (\u22a5' <~> \u03c8)) \u2228 (sfs \u2260 list.nil)) :\n  axCLC ((finite_disjunction (phi_X_list \u03c6 sfs)) <~> \u03c8) :=\nbegin\n  -- \u2194 \u2228 {sf |\u03c8\u2208sf }(\u03c8 \u2227 \u03c6sf), by propositional logic.\n  apply @iff_cut (formCLC agents),\n  exact (phi_X_list_conj_contains hsfs),\n  -- \u2194 \u22a5 \u2228 (\u2228{sf |\u03c8\u2208sf }(\u03c8 \u2227 \u03c6sf)), by propositional logic.\n  apply iff_cut,\n  exact iff_switch_ax.mp (ax_not_bot_imp_iff _),\n  -- \u2194 (\u2228 {tf |\u00ac\u03c8\u2208tf }(\u03c8 \u2227 \u03c6tf)) \u2228 (\u2228 {sf |\u03c8\u2208sf }(\u03c8 \u2227 \u03c6sf)), by propositional logic.\n  apply iff_cut,\n  apply or_cut_l,\n  apply contra_fin_disjunct_psi_and_not h\u03c8 tfs htfs,\n   -- \u2194 \u03c8 \u2227 ((\u2228 {tf |\u00ac\u03c8\u2208tf } \u03c6tf ) \u2228 (\u2228 {sf |\u03c8\u2208sf } \u03c6sf )), by propositional logic.\n  apply @iff_cut _ _ _ _\n    (\u03c8 & (\u00ac' (finite_disjunction (phi_X_list \u03c6 tfs)) \u2192' finite_disjunction (phi_X_list \u03c6 sfs))),\n  apply distr_or_and,\n  -- \u2194 \u03c8 \u2227 (\u2228 {sf \u2208Sf } \u03c6sf ), because {tf | \u00ac\u03c8 \u2208 tf } \u222a {sf | \u03c8 \u2208 sf } = Sf .\n  -- \u2194 \u03c8 \u2227 \u22a4, from Lemma 4.\n  -- \u2194 \u03c8, by propositional logic.\n  apply ax_iff_intro,\n  exact p5 _ _,\n  apply cut,\n  apply MP',\n  exact hSf,\n  exact p4 _ _,\n  apply iff_l,\n  exact and_switch,\nend\n\nlemma phi_X_contains_iff_psi_helper_finset {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (h\u03c8 : \u03c8 \u2208 cl \u03c6)  (sfs: finset (S_f \u03c6)) \n  (hsfs : \u2200 sf : (@S_f agents _ ha \u03c6), sf \u2208 sfs \u2192 \u03c8 \u2208 sf)\n  (htfs : \u2200 tf : (@S_f agents _ ha \u03c6), tf \u2209 sfs \u2192 \u03c8 \u2209 tf)\n  (hSf : axCLC ((\u00ac phi_X_finset \u03c6 sfs\u1d9c) ~> phi_X_finset \u03c6 sfs)) :\n  axCLC ( (phi_X_finset \u03c6 sfs) <~> \u03c8) :=\nbegin\n  unfold phi_X_finset,\n  apply phi_X_contains_iff_psi_helper_list h\u03c8 _ sfs\u1d9c.to_list,\n  simp [finset.to_list], exact hsfs,\n  simp [finset.to_list], exact htfs,\n  exact hSf,\nend\n\n-- Lemma 6. \u2200 \u03c8 \u2208 cl(\u03c6), \u03c6{sf |\u03c8\u2208sf } \u2194 \u03c8\nlemma phi_X_contains_iff_psi {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  (\u03c6 \u03c8 : formCLC agents) (h\u03c8 : \u03c8 \u2208 cl \u03c6) :\n  axCLC (phi_X_set \u03c6 {sf | \u03c8 \u2208 sf} <~> \u03c8) :=\nbegin\n  apply phi_X_contains_iff_psi_helper_finset h\u03c8, simp, simp,\n  apply (phi_X_finset_disjunct_of_disjuncts \u03c6 _ _).mpr,\n  apply @MP' (formCLC agents),\n  exact univ_disjunct_provability \u03c6 (canonical.nonempty_S_f \u03c6),\n  apply phi_X_subset_Y_imp,\n  intros sf hsf,\n  simp [to_finset] at *,\n  rw or.comm,\n  exact (em (\u03c8 \u2208 sf)),\nend\n\nlemma imp_ax_imp {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  {\u03c6 \u03c8 : formCLC agents} (h : \u2200 (a : (canonical_model_CLC agents).f.to_frameCL.states), \u03c6 \u2208 a \u2192 \u03c8 \u2208 a) :\n  axCLC (\u03c6 ~> \u03c8) :=\nbegin\n  apply @ax_imp_from_ex (formCLC agents),\n  apply h,\nend\n\n-- Lemma 7.  \u0303\u03c8 \u2208 E(s)(G) iff [G]\u03c8 \u2208 s\nlemma E_s_contains_tilde_iff_E_in_s {agents : Type} [hN : fintype agents] [ha : nonempty agents]\n  (\u03c6 \u03c8 : formCLC agents) (s : (canonical_model_CLC agents).f.states) (G : set agents) :\n  ((tilde \u03c8) \u2208 ((canonical_model_CLC agents).f.E.E s G)) \u2194 (([G] \u03c8) \u2208 s) :=\nbegin\n  let hE : (canonical_model_CLC agents).f.to_frameCL.E.E = \u03bb s G, {X | ite (G = univ) \n          -- condition G = N\n          (\u2200 \u03c6, ({t | \u03c6 \u2208 t} \u2286 X\u1d9c) \u2192 ([\u2205]' \u03c6) \u2209 s.val)\n          -- condition G \u2260 N\n          (\u2203 \u03c6, {t | \u03c6 \u2208 t} \u2286 X \u2227 ([G]' \u03c6) \u2208 s.val)},\n        from rfl,\n  let hs : (canonical_model_CLC agents).f.to_frameCL.states = {\u0393 : (set (formCLC agents)) // (max_ax_consistent \u0393)}, \n    from rfl,\n  -- Proof. We consider the case when G \u0338 = N and G = N separately.\n  cases (em (G = univ)) with hG hG,\n  { -- 2. case G = N\n    rw hG,\n    split,\n    { -- 2.1. \u21d2\n      -- 2.1.1. Assume  \u0303\u03c8 \u2208 E(s)(N ).\n      intro h,\n      -- 2.1.2. \u2200 \u0303\u03c7 \u2286  \u0303\u03c8\u1d9c : [\u2205]\u03c7 /\u2208 s, from 2.1.1, by definition E.\n      simp [hE] at h {eta := ff}, clear hE,\n      -- 2.1.3. \u2200 \u0303\u03c7 \u2286  \u0303\u00ac\u03c8 : [\u2205]\u03c7 /\u2208 s, from 2.1.2, because  \u0303\u03c8\u1d9c =  \u0303\u00ac\u03c8.\n      have h_subeq : {t : (canonical_model_CLC agents).f.to_frameCL.states | (\u00ac \u03c8) \u2208 t} \u2286 (tilde \u03c8)\u1d9c, from\n      begin\n        intros t ht hf,\n        simp[tilde] at *,\n        exact contra_containts_pr_false t.2 hf ht,\n      end,\n      -- 2.1.4. [N ]\u03c8 \u2208 s, from 2.1.3, by axiom N.\n      specialize h (\u00ac \u03c8) h_subeq,\n      have hin := not_in_from_notin s.2 h,\n      apply max_ax_contains_by_set_proof s.2 hin axCLC.N, },\n    { -- 2.2. \u21d0\n      -- 2.2.1. Assume [N ]\u03c8 \u2208 s.\n      intro h,\n      -- 2.2.2. \u00ac[\u2205]\u00ac\u03c8 \u2208 s, from 2.2.1\n      have hin : (\u00ac ([\u2205] (\u00ac \u03c8))) \u2208 s, from\n      begin\n        apply max_ax_contains_by_set_proof s.2 h,\n        exact iff_l (@univ_iff_empty agents (formCLC agents) _ _ _ \u03c8),\n      end,\n      -- 2.2.3. \u00ac\u2203\u03c7,  \u0303\u03c7 \u2286  \u0303\u00ac\u03c8 : [\u2205]\u03c7 \u2208 s, from proof by contradiction, \n        -- else by definition E we would have [\u2205]\u00ac\u03c8 \u2208 s, which contradicts with 2.2.2.\n      have hne : \u00ac \u2203 (\u03c7 : formCLC agents), (tilde \u03c7) \u2286 (tilde \u00ac \u03c8) \u2227 ([\u2205]' \u03c7) \u2208 s, from \n      begin\n        intro hf,\n        cases hf with \u03c7 hf,\n        cases hf with himp hf,\n        simp [tilde] at himp,\n        have hax : axCLC (\u03c7 ~>(\u00ac \u03c8)), from imp_ax_imp himp,\n        have hf : ([\u2205]' (\u00ac' \u03c8)) \u2208 s, from\n        begin\n          apply max_ax_contains_by_set_proof s.2 hf,\n          apply @derived_monoticity_rule agents (formCLC agents),\n          exact hax,\n        end,\n        apply contra_containts_pr_false s.2 hf hin,\n      end,\n      -- 2.2.4. \u2200\u03c7,  \u0303\u03c7 \u2286  \u0303\u00ac\u03c8 : [\u2205]\u03c7 /\u2208 s, from 2.2.3, by first order logic.\n      simp at hne,\n      -- 2.2.5. \u2200\u03c7,  \u0303\u03c7 \u2286  \u0303\u03c8 : [\u2205]\u03c7 /\u2208 s, because all s \u2208 S are maximally consistent.\n      rw h_tilde_compl at hne,\n      -- 2.2.6.  \u0303\u03c8 \u2208 E(s)(N ), from 2.2.5, by definition E.\n      simp [hE] {eta := ff},\n      exact hne, }, },\n  { -- 1. case G \u0338 = N\n    split,\n    { -- 1.1. \u21d2\n      -- 1.1.1. Assume  \u0303\u03c8 \u2208 E(s)(G).\n      intro h,\n      -- 1.1.2. \u2203 \u0303\u03c7 \u2286  \u0303\u03c8 : [G]\u03c7 \u2208 s, from 1.1.1, by definition E.\n      simp [hE, hG] at h {eta := ff},\n      -- 1.1.3. \u22a2 \u03c7 \u2192 \u03c8, from 1.1.2.\n      cases h with \u03c7 h,\n      cases h with himp h,\n      simp [tilde] at himp,\n      have hax : axCLC (\u03c7 ~> \u03c8), from imp_ax_imp himp,\n      -- 1.1.4. [G]\u03c8 \u2208 s, from 1.1.2 & 1.1.3, by lemma 2.\n      apply max_ax_contains_by_set_proof s.2 h,\n      apply @derived_monoticity_rule agents (formCLC agents),\n      exact hax, },\n    { -- 1.2. \u21d0 is immediate by definition.\n      simp [hE, hG],\n      intro h,\n      apply exists.intro \u03c8,\n      split,\n      simp [tilde],\n      exact h, }, },\nend\n\nend lemmas\n\n-- Effectivity\n----------------------------------------------------------\ndef E_f {agents : Type}  [hN : fintype agents] [ha : nonempty agents] {\u03c6 : formCLC agents} : \n  (S_f \u03c6) \u2192 (set agents) \u2192 (set (set (S_f \u03c6))) := \n\u03bb sf G, {X | ite (G = univ) \n  -- condition G = N\n  -- \u2203t \u2208 S, sf = tf and  \u0303\u03c6X \u2208 E(t)(N)\n  (\u2203 t : (canonical_model_CLC agents).f.states, (\u2200 {x}, x \u2208 sf \u2194 x \u2208 t \u2227 x \u2208 cl \u03c6) \u2227 \n    (tilde (phi_X_set \u03c6 X)) \u2208 (canonical_model_CLC agents).f.E.E (t) (G))\n  -- condition G \u2260 N\n  -- \u2200t \u2208 S, sf = tf \u21d2  \u0303phiX \u2208 E(t)(G)\n  (\u2200 t : (canonical_model_CLC agents).f.states, (\u2200 {x}, x \u2208 sf \u2194 x \u2208 t \u2227 x \u2208 cl \u03c6) \u2192 \n    (tilde (phi_X_set \u03c6 X)) \u2208 (canonical_model_CLC agents).f.E.E (t) (G))}\n\n----------------------------------------------------------\n-- Playability\n----------------------------------------------------------\n\n-- 1. Ef (sf ) is live: \u2200G \u2286 N : \u2205 /\u2208 Ef (sf )(G)\nlemma Ef_liveness {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents) :\n  \u2200 s : (S_f \u03c6), \u2200 G : set agents, \u2205 \u2209 (E_f s G) := \nbegin\n  -- 1.2. Assume by contradiction \u2205 \u2208 Ef (sf )(G).\n  intros sf G hf,\n  unfold E_f at hf,\n  split_ifs at hf with h h,\n  -- 1.4. Case: G = N\n  { -- 1.4.1. \u2203t \u2208 S, sf = tf and  \u0303\u03c6\u2205 \u2208 E(t)(N), from 1.2, by definition Ef .\n    simp[h] at hf,\n    -- 1.4.2. \u2203t \u2208 S, sf = tf and \u2205 \u2208 E(t)(N), from 1.4.1 & 1.1.\n    cases hf with t hf,\n    -- 1.4.3. \u2200t, \u2205 \u2209 E(t)(N) because E(t) is live.\n    have hlive := (canonical_model_CLC agents).f.E.liveness t univ,\n    -- 1.4.4. Contradiction from 1.4.2 & 1.4.3.\n    exact hlive hf.right, },\n  -- 1.3. Case: G \u2260 N\n  { -- 1.3.1. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6\u2205 \u2208 E(t)(G), from 1.2, by definition Ef\n    -- 1.3.2. \u2200t \u2208 S, sf = tf \u21d2 \u2205 \u2208 E(t)(G), from 1.3.1 & 1.1\n    simp[E_f, h] at hf,\n    -- 1.3.3. \u2205 \u2208 E(s)(G), from 1.3.2\n    cases (s_f_to_s \u03c6 sf) with s hs,\n    specialize hf s @hs,\n    -- 1.3.4. \u2205 /\u2208 E(s)(G) because E(s) is live.\n    have hlive := (canonical_model_CLC agents).f.E.liveness s,\n    -- 1.3.5. Contradiction from 1.3.3 & 1.3.4.\n    exact hlive G hf, },\nend\n\n-- 2. Ef (sf) is safe: \u2200G \u2286 N : Sf \u2208 Ef (sf )(G)\nlemma Ef_safety {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents) :\n  \u2200 (s : S_f \u03c6) (G : set agents), univ \u2208 E_f s G :=\nbegin\n  -- 2.2. Additionally, because E(s) is safe for all s \u2208 S, \u2200G \u2286 N, S \u2208 E(s)(G).\n  have hsafe := (canonical_model_CLC agents).f.E.safety,\n  -- 2.4. Case: G = N\n  intros sf G, cases em (G = univ) with hG hG,\n  { -- 2.4.1. Sf \u2208 Ef (sf )(N ) iff \u2203t \u2208 S, sf = tf and  \u0303\u03c6Sf \u2208 E(t)(N ), by definition Ef .\n    simp[hG] at *,\n    -- 2.4.2. Sf \u2208 Ef (sf )(N ) iff \u2203t \u2208 S, sf = tf and S \u2208 E(t)(N ), from 2.1 & 2.4.1.\n    simp[E_f],\n    -- 2.4.3. \u2203t \u2208 S, sf = tf and S \u2208 E(t)(N ), when t = s, because sf = sf and S \u2208 E(s)(N ), from 2.2.\n    cases (s_f_to_s \u03c6 sf) with s hs,\n    apply exists.intro s,\n    -- 2.4.4. Sf \u2208 Ef (sf )(N ), from 2.4.2 & 2.4.3s\n    simp at *,\n    split,\n    exact @hs,\n    apply hsafe, },\n  -- 2.3. Case: G \u2260 N\n  { -- 2.3.1. Sf \u2208 Ef (sf )(G) iff \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6Sf \u2208 E(t)(G), by definition Ef .\n    -- 2.3.2. Sf \u2208 Ef (sf )(G) iff \u2200t \u2208 S, sf = tf \u21d2 S \u2208 E(t)(G), from 2.1 & 2.3.1.\n    simp[E_f, hG] at *,\n    -- 2.3.3. Sf \u2208 Ef (sf )(G), from 2.2 & 2.3.2\n    intros t ht,\n    apply hsafe, }, \nend\n\n-- 3. Ef (sf) is N-maximal: \u2200X \u2286 Sf : X\u1d9c \u2209 Ef(sf)(\u2205) \u21d2 X \u2208 Ef(sf)(N)\nlemma Ef_nmax {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents) :\n  N_max agents (S_f \u03c6) (E_f) :=\nbegin\n  -- 3.1. Assume some X \u2286 Sf such that X\u1d9c \u2209 Ef(sf)(\u2205).\n  intros sf X hXc,\n  -- 3.2. \u00ac(X\u1d9c \u2208 Ef sf \u2205), from 3.1.\n  -- 3.3. \u00ac(\u2200t \u2208 S, sf = tf \u21d2 ~\u03c6X\u1d9c \u2208 E(t)(\u2205)), from 3.2, by definition Ef . \n  -- 3.4. \u2203t \u2208 S, sf = tf and ~\u03c6X\u1d9c \u2209 E(t)(\u2205)), from 3.3, by first order logic. \n  simp[E_f, empty_ne_univ] at *,\n  obtain \u27e8t, ht, hXc\u27e9 := hXc,\n  refine \u27e8t, @ht, _\u27e9,\n  { \n    have h_tilde: tilde (\u00ac (phi_X_set \u03c6 X) : formCLC agents) = \n      tilde (phi_X_set \u03c6 X\u1d9c), from\n    begin\n      simp[tilde],\n      ext1 u,\n      split,\n      { intro hu,\n        simp at *,\n        apply max_ax_contains_by_set_proof u.2 hu,\n        apply (phi_X_set_disjunct_of_disjuncts \u03c6 _ _).mpr,\n        rw (union_compl_self X),\n        apply univ_disjunct_provability,\n        exact canonical.nonempty_S_f \u03c6, },\n      { intro hu,\n        simp at *,\n        apply max_ax_contains_by_set_proof u.2 hu,\n        unfold phi_X_set,\n        apply phi_X_set_unique,\n        simp, },\n    end,\n\n    -- 3.6. \u2203t \u2208 S, sf = tf and ~\u00ac\u03c6X \u2209 E(t)(\u2205)), from 3.4 & 3.5\n    have hX : tilde (\u00ac (phi_X_set \u03c6 X) : formCLC agents) \u2209 \n      (canonical_model_CLC agents).f.to_frameCL.E.E t \u2205, from\n    begin\n      simp[h_tilde] at *,\n      exact hXc,\n    end,  \n    -- 3.7. \u2203t \u2208 S,sf = tf and (~\u03c6X)\u1d9c \u2209 E(t)(\u2205)), from 3.6, because all s \u2208 S are maximally consistent.\n  simp at *,\n  simp[h_tilde_compl] at hX,\n    -- 3.8. \u2203t \u2208 S,sf = tf and \u03c6\udbff\udc13 \u2208 E(t)(N)), from 3.7, because E(s) is N-maximal X for all s \u2208 S (\u2200X \u2286 S|X \u2208/ E(s)(\u2205) \u21d2 X \u2208 E(s)(N))\n    -- 3.9. Ef (sf )(N), from 3.8, by definition Ef .\n  exact (canonical_model_CLC agents).f.to_frameCL.E.N_max t (tilde (phi_X_set \u03c6 X)) hX, },\nend\n\n-- Ef (sf ) is outcome monotonic: \u2200G \u2286 N, \u2200X \u2286 Y \u2286 Sf : X \u2208 Ef (sf )(G) \u21d2 Y \u2208 Ef (sf )(G)\nlemma Ef_monoticity {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents) :\n  \u2200 (sf : S_f \u03c6) (G : set agents) (X Y : set (S_f \u03c6)), X \u2286 Y \u2192 X \u2208 E_f sf G \u2192 Y \u2208 E_f sf G :=\nbegin\n  -- 4.1. Let G be some G \u2286 N and X and Y be some X \u2286 Y \u2286 Sf .\n  intros s G X Y hXY,\n  -- 4.2. Assume X \u2208 Ef (sf )(G).\n  intro hX,\n  -- 4.3. First we note that \u2200s \u2208 S, \u2200G \u2286 N,  \u0303\u03c6X \u2208 E(s)(G) \u21d2  \u0303\u03c6Y \u2208 E(s)(G)\n  have himp : \u2200 s G, \n    (tilde (phi_X_set \u03c6 X)) \u2208 (canonical_model_CLC agents).f.E.E s G \u2192 \n    (tilde (phi_X_set \u03c6 Y)) \u2208 (canonical_model_CLC agents).f.E.E s G, from\n  begin\n    -- 4.3.1. Let s be some s \u2208 S and G be some G \u2286 N .\n    clear hX, intros s G hX,\n    -- 4.3.2. \u22a2 \u03c6X \u2192 \u03c6Y , from 4.1 (X \u2286 Y ).\n    have hax : axCLC ((phi_X_set \u03c6 X) ~> (phi_X_set \u03c6 Y)), \n      from phi_X_set_subset_Y_imp _ _ _ hXY,\n    -- 4.3.3.  \u0303\u03c6X \u2286  \u0303\u03c6Y , from 4.3.2.\n    have h_phiXY : (tilde (phi_X_set \u03c6 X)) \u2286 (tilde (phi_X_set \u03c6 Y)), from\n    begin \n      rw set.subset_def,\n      intros t ht,\n      apply max_ax_contains_by_set_proof t.2 ht hax,\n    end,\n    -- 4.3.4. E(s) is outcome monotonic for all s \u2208 S: \u2200G \u2286 N, \u2200X \u2286 Y \u2286 S, X \u2208 E(s)(G) \u21d2 Y \u2208 E(s)(G)\n    have hmonoticity := (canonical_model_CLC agents).f.E.monoticity s G _ _ h_phiXY,\n    -- 4.3.5.  \u0303\u03c6X \u2208 E(s)(G) \u21d2  \u0303\u03c6Y \u2208 E(s)(G), from 4.3.3 & 4.3.4\n    apply hmonoticity hX,\n  end,\n  -- 4.5. Case G = N\n  cases em (G = univ) with hG hG,\n  { -- 4.5.1. \u2203t \u2208 S, sf = tf and  \u0303\u03c6X \u2208 E(t)(N ), from 4.2, by definition Ef .\n    simp[E_f, hG] at *,\n    -- 4.5.2. \u2203t \u2208 S, sf = tf and  \u0303\u03c6Y \u2208 E(t)(N ), from 4.3 & 4.5.1.\n    -- 4.5.3. Y \u2208 Ef (sf )(N ), from 4.5.2, by definition Ef . \n    cases hX with t ht,\n    apply exists.intro t,\n    split,\n    { exact ht.1 },\n    { exact himp _ _ ht.2, }, },\n  -- 4.4. Case: G \u2260 N\n  { -- 4.4.1. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6X \u2208 E(t)(N ), from 4.2, by definition Ef .\n    simp[E_f, hG] at *,\n    -- 4.4.2. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6Y \u2208 E(t)(N ), from 4.3 & 4.4.1.\n    -- 4.4.3. Y \u2208 Ef (sf )(G), from 4.4.2, by definition Ef .\n    intros t ht,\n    exact himp t G (hX t @ht), },\nend\n\nlemma phi_X_list_inter {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : list (S_f \u03c6)) (hX : list.nodup X) (hY : list.nodup Y) :\n  axCLC (finite_disjunction (phi_X_list \u03c6 X)\u2192' finite_disjunction (phi_X_list \u03c6 Y) \u2192' \n        finite_disjunction (phi_X_list \u03c6 (X \u2229 Y))) :=\nbegin\n  induction' X with x X ihx,\n  { simp [phi_X_list, finite_disjunction],\n    apply axCLC.Prop1, },\n  { simp [phi_X_list, finite_disjunction],\n    apply @or_cases (formCLC agents),\n    { cases (em (x \u2208 Y)),\n      { apply cut,\n        apply iff_l,\n        apply phi_X_list_single,\n        apply @cut _ _ _ _ (finite_disjunction (phi_X_list \u03c6 ((x :: X) \u2229 Y))),\n        apply imp_finite_disjunction_subset,\n        apply phi_X_list_subset,\n        simp,\n        exact h,\n        exact axCLC.Prop1, },\n      { apply cut,\n        apply iff_l,\n        apply phi_X_list_single,\n        apply cut1,\n        apply phi_X_list_unique,\n        exact list.singleton_disjoint.mpr h,\n        exact list.nodup_singleton x,\n        exact hY,\n        exact explosion, }, },\n    { simp at hX,\n      specialize ihx Y hY hX.2,\n      apply cut1,\n      apply ihx,\n      apply imp_finite_disjunction_subset,\n      apply phi_X_list_subset,\n      intros y hy,\n      simp at *,\n      split,\n      apply or.intro_right,\n      exact hy.1,\n      exact hy.2, }, },\nend\n\nlemma phi_X_finset_inter {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : finset (S_f \u03c6)) :\n  axCLC ((phi_X_finset \u03c6 X) \u2192' phi_X_finset \u03c6 Y \u2192' (phi_X_finset \u03c6 (X \u2229 Y))) :=\nbegin\n  unfold phi_X_finset,\n  apply @cut1 (formCLC agents),\n  apply phi_X_list_inter,\n  repeat {exact finset.nodup_to_list _, },\n  apply imp_finite_disjunction_subset,\n  apply phi_X_list_subset,\n  intros x hx,\n  simp [finset.mem_to_list] at *,\n  exact hx,\nend\n\nlemma phi_X_set_inter {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) (X Y : set (S_f \u03c6)) :\n  axCLC ((phi_X_set \u03c6 X) \u2192' (phi_X_set \u03c6 Y) \u2192' (phi_X_set \u03c6 (X \u2229 Y))) :=\nbegin\n  simp[phi_X_set],\n  apply @cut1 (formCLC agents),\n  apply phi_X_finset_inter,\n  apply phi_X_subset_Y_imp,\n  intros x hx,\n  simp at *,\n  exact hx, \nend\n--  Ef (sf ) is superadditive \u2200G, F \u2286 N (where G \u2229 F = \u2205), \n  -- \u2200X, Y \u2286 Sf : X \u2208 Ef (sf )(G) and Y \u2208 Ef (sf )(F ) \u21d2 X \u2229 Y \u2208 Ef (sf )(G \u222a F )\nlemma Ef_superadd {agents : Type} [hN : fintype agents] [ha : nonempty agents] (\u03c6 : formCLC agents) :\n  \u2200 (sf : S_f \u03c6) (G F : set agents) (X Y : set (S_f \u03c6)),\n  X \u2208 E_f sf G \u2192 Y \u2208 E_f sf F \u2192 G \u2229 F = \u2205 \u2192 X \u2229 Y \u2208 E_f sf (G \u222a F) :=\nbegin      \n  -- 5.1. Let G, F be some G, F \u2286 N , such that G \u2229 F = \u2205. Let X, Y be some\n    -- X, Y \u2286 S such that X \u2208 Ef (sf )(G) and Y \u2208 Ef (sf )(F ).\n  -- intros sf G F X Y hX hY hGF,\n  -- 5.2. First we note that \u2200s \u2208 S, \u2200G, F \u2286 N (where G \u2229 F = \u2205),  \u0303\u03c6X \u2208 E(s)(G) \u21d2  \u0303\u03c6Y \u2208 E(s)(F ) \u21d2  \u0303\u03c6X\u2229Y \u2208 E(s)(G \u222a F )\n  have hint : \u2200 s G F X Y, G \u2229 F = \u2205 \u2192 \n    (tilde (phi_X_set \u03c6 X)) \u2208 (canonical_model_CLC agents).f.E.E s G \u2192\n    (tilde (phi_X_set \u03c6 Y)) \u2208 (canonical_model_CLC agents).f.E.E s F \u2192\n    (tilde (phi_X_set \u03c6 (X \u2229 Y))) \u2208 (canonical_model_CLC agents).f.E.E s (G \u222a F), from\n  begin\n    -- 5.2.1. Let s be some s \u2208 S. Let G, F , be some G, F \u2282 N where G \u2229 F = \u2205. Assume  \u0303\u03c6X \u2208 E(s)(G) and  \u0303\u03c6Y \u2208 E(s)(F ).\n    intros s G F X Y hGF hG hF,\n    -- 5.2.2. E(s) is superadditive so: \u2200X, Y \u2286 S : X \u2208 E(s)(G) and Y \u2208 E(s)(F ) \u21d2 X \u2229 Y \u2208 E(s)(G \u222a F )\n    have hsuperadd := ((canonical_model_CLC agents).f.E.superadd) s G F,\n    -- 5.2.3.  \u0303\u03c6X \u2229  \u0303\u03c6Y \u2208 E(s)(G \u222a F ), from 5.2.1 & 5.2.2.\n    specialize hsuperadd (tilde (phi_X_set \u03c6 X)) (tilde (phi_X_set \u03c6 Y)) hG hF hGF,\n    -- 5.2.4.  \u0303\u03c6X\u2229Y \u2208 E(s)(G \u222a F ), from 5.2.3, because  \u0303\u03c6X \u2192  \u0303\u03c6X\u2229Y and  \u0303\u03c6Y \u2192  \u0303\u03c6X\u2229Y .\n    have h_tilde_eq : tilde (phi_X_set \u03c6 X) \u2229 tilde (phi_X_set \u03c6 Y) = tilde (phi_X_set \u03c6 (X \u2229 Y)), from\n    begin\n      ext1 s,\n      simp[tilde],\n      split,\n      { intro h,\n        apply max_ax_contains_by_set_proof_2h s.2 h.1 h.2 ,\n        apply phi_X_set_inter, },\n      { intro h,\n        split,\n        repeat \n        { apply max_ax_contains_by_set_proof s.2 h,\n          apply phi_X_set_subset_Y_imp,\n          simp, }, },\n    end,\n    \n    rw h_tilde_eq at hsuperadd,\n    exact hsuperadd,\n  end,\n  \n  intros sf G F X Y hX hY hGF,\n\n  -- 5.4. Case G = N or F = N :\n  have h_G_or_F_univ : \u2200 X' Y', X' \u2208 E_f sf univ \u2192 Y' \u2208 E_f sf \u2205 \u2192 (X' \u2229 Y') \u2208 E_f sf univ, from\n  begin\n    -- 5.4.1. Rename G, F, X&Y to G\u2032, F \u2032, X\u2032&Y \u2032, such that G\u2032 = N , F \u2032 = \u2205, X\u2032 \u2208 Ef (sf )(G\u2032) and Y \u2032 \u2208 Ef (sf )(F \u2032).\n    clear hX hY,\n    intros X' Y',\n    -- 5.4.2. \u2203t \u2208 S, sf = tf and  \u0303\u03c6X\u2032 \u2208 E(t)(N ), from 5.4.1 (X\u2032 \u2208 Ef (sf )(G\u2032)), by definition Ef .\n    intro hX,\n    -- 5.4.3. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6Y \u2032 \u2208 E(t)(\u2205), from 5.4.1 (Y \u2032 \u2208 Ef (sf )(F \u2032)), by definition Ef .\n    intro hY,\n    -- 5.4.4. \u2203t \u2208 S, sf = tf and  \u0303\u03c6X\u2032 \u2208 E(t)(N ) and  \u0303\u03c6Y \u2032 \u2208 E(t)(\u2205), from 5.4.2 & 5.4.3.\n    simp[E_f, empty_ne_univ] at *,\n    cases hX with t hX,\n    specialize hY t hX.left,\n    apply exists.intro t,\n    split, exact hX.left,\n    -- 5.4.5. \u2203t \u2208 S, sf = tf and  \u0303\u03c6X\u2032 \u2229Y \u2032 \u2208 E(t)(N ), from 5.3 & 5.4.4.\n    specialize hint t univ \u2205 X' Y' (by simp) hX.right hY,\n    simp[univ_union] at hint,\n    exact hint,\n  end,\n\n  cases em (G = univ),\n  { simp[h] at *,\n    simp[hGF] at *,\n    exact h_G_or_F_univ X Y hX hY, },\n  -- case G \u2260 N\n  { cases em (F = univ),\n    { simp[h_1] at *,\n      simp[hGF] at *,\n      rw inter_comm X Y,\n      exact h_G_or_F_univ Y X hY hX, },\n    -- 5.3. Case G \u2260 N and F \u2260 N\n    { -- 5.3.1. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6X \u2208 E(t)(G), from 5.1 (X \u2208 Ef (sf )(G)), by definition Ef .\n      -- 5.3.2. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6Y \u2208 E(t)(F ), from 5.1 (Y \u2208 Ef (sf )(F )), by definition Ef .\n      simp[E_f, h, h_1] at *,\n      -- 5.3.3. \u2200t \u2208 S, sf = tf \u21d2 (  \u0303\u03c6X \u2208 E(t)(G)and  \u0303\u03c6Y \u2208 E(t)(F )), from 5.3.1 & 5.3.2.\n      -- 5.3.4. \u2200t \u2208 S, sf = tf \u21d2  \u0303\u03c6X\u2229Y \u2208 E(t)(G \u222a F ), from 5.2 & 5.3.3.\n\n      -- 5.3.6. Case G \u222a F = N : sf = sf and  \u0303\u03c6X\u2229Y \u2208 E(s)(G \u222a F ), from 5.3.4. So X \u2229 Y \u2208 Ef (sf )(G \u222a F = N ), by definition Ef\n      cases em (G \u222a F = univ),\n      { have hs := s_f_to_s \u03c6 sf,\n        cases hs with s hs,\n        specialize hint s G F X Y hGF (hX s @hs) (hY s @hs),\n        simp[h_2] at *,\n        apply exists.intro s,\n        split, exact @hs, exact hint, },\n      -- 5.3.5. Case G \u222a F \u0338 = N : X \u2229 Y \u2208 Ef (sf )(G \u222a F ), from 5.3.4, by definition Ef\n      { simp[h_2],\n        intros t ht,\n        exact hint t G F X Y hGF (hX t @ht) (hY t @ht), }, }, },\nend\n\n----------------------------------------------------------\n-- Building the coplete filtered CLC model\n----------------------------------------------------------\n \n@[simps?] def filtered_model_CLC {agents : Type} [hN : fintype agents] [ha : nonempty agents] \n  (\u03c6 : formCLC agents) :\n  modelCLK agents := \n{ f := \n  { states := S_f \u03c6,\n    hs := canonical.nonempty_S_f \u03c6,\n    ha := ha,\n    E := \n    \n-- \u2200u\u2208Sc if [u]=[s] then [\u03c6 ]c \u2208Ec(u)(G) G\u0338=N\n    { E          := E_f,\n      liveness   := Ef_liveness \u03c6,\n      safety     := Ef_safety \u03c6,\n      N_max      := Ef_nmax \u03c6,\n      monoticity := Ef_monoticity \u03c6,\n      superadd   := Ef_superadd \u03c6, },\n    rel   := \u03bb i s, {t | {\u03c6 | K' (i) (\u03c6) \u2208 s} = {\u03c6 | K' (i) (\u03c6) \u2208 t}},\n    rfl   := by simp,\n    sym   := \u03bb i s t ht, eq.symm ht,\n    trans := \u03bb i s t u hst htu, (rfl.congr htu).mp hst, },\n  v := \u03bb  n, {s | (formCLC.var n) \u2208 s.1.1}, }\n\nend canonical\n", "meta": {"author": "kaiobendrauf", "repo": "cl-lean", "sha": "15568f16cf57a07db6192fbd8084d59cc1aef1df", "save_path": "github-repos/lean/kaiobendrauf-cl-lean", "path": "github-repos/lean/kaiobendrauf-cl-lean/cl-lean-15568f16cf57a07db6192fbd8084d59cc1aef1df/src/completeness/filtered_modelC.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.39890568261465525}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nThe Kleisli construction on the Type category\n\nTODO: generalise this to work with category_theory.monad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.category.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\ndef Kleisli (m : Type u \u2192 Type v) [Monad m] := Type u\n\ndef Kleisli.mk (m : Type u \u2192 Type v) [Monad m] (\u03b1 : Type u) : Kleisli m := \u03b1\n\nprotected instance Kleisli.category_struct {m : Type u_1 \u2192 Type u_2} [Monad m] :\n    category_struct (Kleisli m) :=\n  category_struct.mk (fun (\u03b1 : Kleisli m) (x : \u03b1) => pure x)\n    fun (X Y Z : Kleisli m) (f : X \u27f6 Y) (g : Y \u27f6 Z) => f >=> g\n\nprotected instance Kleisli.category {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m] :\n    category (Kleisli m) :=\n  category.mk\n\n@[simp] theorem Kleisli.id_def {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m]\n    (\u03b1 : Kleisli m) : \ud835\udfd9 = pure :=\n  rfl\n\ntheorem Kleisli.comp_def {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m] (\u03b1 : Kleisli m)\n    (\u03b2 : Kleisli m) (\u03b3 : Kleisli m) (xs : \u03b1 \u27f6 \u03b2) (ys : \u03b2 \u27f6 \u03b3) (a : \u03b1) :\n    category_struct.comp xs ys a = xs a >>= ys :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/Kleisli_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.39886039069232726}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.algebra.operations\nimport Mathlib.PostPort\n\nuniverses u v l w u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Subalgebras over Commutative Semiring\n\nIn this file we define `subalgebra`s and the usual operations on them (`map`, `comap`).\n\nMore lemmas about `adjoin` can be found in `ring_theory.adjoin`.\n-/\n\n/-- A subalgebra is a sub(semi)ring that includes the range of `algebra_map`. -/\nstructure subalgebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A]\n    extends subsemiring A where\n  algebra_map_mem' : \u2200 (r : R), coe_fn (algebra_map R A) r \u2208 carrier\n\n/-- Reinterpret a `subalgebra` as a `subsemiring`. -/\nnamespace subalgebra\n\n\nprotected instance subsemiring.has_coe {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : has_coe (subalgebra R A) (subsemiring A) :=\n  has_coe.mk\n    fun (S : subalgebra R A) =>\n      subsemiring.mk (carrier S) (one_mem' S) (mul_mem' S) (zero_mem' S) (add_mem' S)\n\nprotected instance has_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    has_mem A (subalgebra R A) :=\n  has_mem.mk fun (x : A) (S : subalgebra R A) => x \u2208 \u2191S\n\ntheorem mem_coe {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {x : A}\n    {s : subalgebra R A} : x \u2208 \u2191s \u2194 x \u2208 s :=\n  iff.rfl\n\ntheorem ext {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {S : subalgebra R A} {T : subalgebra R A} (h : \u2200 (x : A), x \u2208 S \u2194 x \u2208 T) : S = T :=\n  sorry\n\ntheorem ext_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {S : subalgebra R A} {T : subalgebra R A} : S = T \u2194 \u2200 (x : A), x \u2208 S \u2194 x \u2208 T :=\n  { mp :=\n      fun (h : S = T) (x : A) =>\n        eq.mpr (id (Eq._oldrec (Eq.refl (x \u2208 S \u2194 x \u2208 T)) h)) (iff.refl (x \u2208 T)),\n    mpr := ext }\n\ntheorem algebra_map_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) (r : R) : coe_fn (algebra_map R A) r \u2208 S :=\n  algebra_map_mem' S r\n\ntheorem srange_le {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : ring_hom.srange (algebra_map R A) \u2264 \u2191S :=\n  sorry\n\ntheorem range_subset {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : set.range \u21d1(algebra_map R A) \u2286 \u2191S :=\n  sorry\n\ntheorem range_le {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : set.range \u21d1(algebra_map R A) \u2264 \u2191S :=\n  range_subset S\n\ntheorem one_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : 1 \u2208 S :=\n  subsemiring.one_mem \u2191S\n\ntheorem mul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x * y \u2208 S :=\n  subsemiring.mul_mem (\u2191S) hx hy\n\ntheorem smul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {x : A} (hx : x \u2208 S) (r : R) : r \u2022 x \u2208 S :=\n  Eq.symm (algebra.smul_def r x) \u25b8 mul_mem S (algebra_map_mem S r) hx\n\ntheorem pow_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2115) : x ^ n \u2208 S :=\n  subsemiring.pow_mem (\u2191S) hx n\n\ntheorem zero_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : 0 \u2208 S :=\n  subsemiring.zero_mem \u2191S\n\ntheorem add_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x + y \u2208 S :=\n  subsemiring.add_mem (\u2191S) hx hy\n\ntheorem neg_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A)\n    {x : A} (hx : x \u2208 S) : -x \u2208 S :=\n  neg_one_smul R x \u25b8 smul_mem S hx (-1)\n\ntheorem sub_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A)\n    {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x - y \u2208 S :=\n  sorry\n\ntheorem nsmul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2115) : n \u2022\u2115 x \u2208 S :=\n  subsemiring.nsmul_mem (\u2191S) hx n\n\ntheorem gsmul_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]\n    (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2124) : n \u2022\u2124 x \u2208 S :=\n  int.cases_on n (fun (i : \u2115) => nsmul_mem S hx i)\n    fun (i : \u2115) => neg_mem S (nsmul_mem S hx (Nat.succ i))\n\ntheorem coe_nat_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) (n : \u2115) : \u2191n \u2208 S :=\n  subsemiring.coe_nat_mem (\u2191S) n\n\ntheorem coe_int_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]\n    (S : subalgebra R A) (n : \u2124) : \u2191n \u2208 S :=\n  int.cases_on n (fun (i : \u2115) => coe_nat_mem S i) fun (i : \u2115) => neg_mem S (coe_nat_mem S (i + 1))\n\ntheorem list_prod_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {L : List A} (h : \u2200 (x : A), x \u2208 L \u2192 x \u2208 S) : list.prod L \u2208 S :=\n  subsemiring.list_prod_mem (\u2191S) h\n\ntheorem list_sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {L : List A} (h : \u2200 (x : A), x \u2208 L \u2192 x \u2208 S) : list.sum L \u2208 S :=\n  subsemiring.list_sum_mem (\u2191S) h\n\ntheorem multiset_prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A]\n    [algebra R A] (S : subalgebra R A) {m : multiset A} (h : \u2200 (x : A), x \u2208 m \u2192 x \u2208 S) :\n    multiset.prod m \u2208 S :=\n  subsemiring.multiset_prod_mem (\u2191S) m h\n\ntheorem multiset_sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {m : multiset A} (h : \u2200 (x : A), x \u2208 m \u2192 x \u2208 S) : multiset.sum m \u2208 S :=\n  subsemiring.multiset_sum_mem (\u2191S) m h\n\ntheorem prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A] [algebra R A]\n    (S : subalgebra R A) {\u03b9 : Type w} {t : finset \u03b9} {f : \u03b9 \u2192 A} (h : \u2200 (x : \u03b9), x \u2208 t \u2192 f x \u2208 S) :\n    (finset.prod t fun (x : \u03b9) => f x) \u2208 S :=\n  subsemiring.prod_mem (\u2191S) h\n\ntheorem sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) {\u03b9 : Type w} {t : finset \u03b9} {f : \u03b9 \u2192 A} (h : \u2200 (x : \u03b9), x \u2208 t \u2192 f x \u2208 S) :\n    (finset.sum t fun (x : \u03b9) => f x) \u2208 S :=\n  subsemiring.sum_mem (\u2191S) h\n\nprotected instance is_add_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (S : subalgebra R A) : is_add_submonoid \u2191S :=\n  is_add_submonoid.mk (zero_mem S) fun (_x _x_1 : A) => add_mem S\n\nprotected instance is_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (S : subalgebra R A) : is_submonoid \u2191S :=\n  is_submonoid.mk (one_mem S) fun (_x _x_1 : A) => mul_mem S\n\n/-- A subalgebra over a ring is also a `subring`. -/\ndef to_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) :\n    subring A :=\n  subring.mk (subsemiring.carrier (to_subsemiring S)) sorry sorry sorry sorry sorry\n\nprotected instance is_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A]\n    (S : subalgebra R A) : is_subring \u2191S :=\n  is_subring.mk\n\nprotected instance inhabited {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : Inhabited \u21a5S :=\n  { default := 0 }\n\nprotected instance semiring (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : semiring \u21a5S :=\n  subsemiring.to_semiring \u2191S\n\nprotected instance comm_semiring (R : Type u) (A : Type v) [comm_semiring R] [comm_semiring A]\n    [algebra R A] (S : subalgebra R A) : comm_semiring \u21a5S :=\n  subsemiring.to_comm_semiring \u2191S\n\nprotected instance ring (R : Type u) (A : Type v) [comm_ring R] [ring A] [algebra R A]\n    (S : subalgebra R A) : ring \u21a5S :=\n  subtype.ring\n\nprotected instance comm_ring (R : Type u) (A : Type v) [comm_ring R] [comm_ring A] [algebra R A]\n    (S : subalgebra R A) : comm_ring \u21a5S :=\n  subtype.comm_ring\n\nprotected instance algebra {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : algebra R \u21a5S :=\n  algebra.mk\n    (ring_hom.mk (ring_hom.to_fun (ring_hom.cod_srestrict (algebra_map R A) \u2191S sorry)) sorry sorry\n      sorry sorry)\n    sorry sorry\n\nprotected instance to_algebra {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R]\n    [comm_semiring A] [semiring B] [algebra R A] [algebra A B] (A\u2080 : subalgebra R A) :\n    algebra (\u21a5A\u2080) B :=\n  algebra.of_subsemiring \u2191A\u2080\n\nprotected instance nontrivial {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) [nontrivial A] : nontrivial \u21a5S :=\n  subsemiring.nontrivial \u2191S\n\n-- todo: standardize on the names these morphisms\n\n-- compare with submodule.subtype\n\n/-- Embedding of a subalgebra into the algebra. -/\ndef val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : alg_hom R (\u21a5S) A :=\n  alg_hom.mk coe sorry sorry sorry sorry sorry\n\n@[simp] theorem coe_val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : \u21d1(val S) = coe :=\n  rfl\n\ntheorem val_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) (x : \u21a5S) : coe_fn (val S) x = \u2191x :=\n  rfl\n\n/-- Convert a `subalgebra` to `submodule` -/\ndef to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : submodule R A :=\n  submodule.mk \u2191S sorry sorry sorry\n\nprotected instance coe_to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : has_coe (subalgebra R A) (submodule R A) :=\n  has_coe.mk to_submodule\n\nprotected instance to_submodule.is_subring {R : Type u} {A : Type v} [comm_ring R] [ring A]\n    [algebra R A] (S : subalgebra R A) : is_subring \u2191\u2191S :=\n  subalgebra.is_subring S\n\n@[simp] theorem mem_to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] (S : subalgebra R A) {x : A} : x \u2208 \u2191S \u2194 x \u2208 S :=\n  iff.rfl\n\ntheorem to_submodule_injective {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] {S : subalgebra R A} {U : subalgebra R A} (h : \u2191S = \u2191U) : S = U :=\n  sorry\n\ntheorem to_submodule_inj {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {S : subalgebra R A} {U : subalgebra R A} : \u2191S = \u2191U \u2194 S = U :=\n  { mp := to_submodule_injective, mpr := congr_arg fun {S : subalgebra R A} => \u2191S }\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp] theorem mul_self {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : \u2191S * \u2191S = \u2191S :=\n  sorry\n\n/-- Linear equivalence between `S : submodule R A` and `S`. Though these types are equal,\nwe define it as a `linear_equiv` to avoid type equalities. -/\ndef to_submodule_equiv {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : linear_equiv R \u21a5\u2191S \u21a5S :=\n  linear_equiv.of_eq (\u2191S) (has_coe_t_aux.coe S) sorry\n\nprotected instance partial_order {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : partial_order (subalgebra R A) :=\n  partial_order.mk (fun (S T : subalgebra R A) => \u2191S \u2286 \u2191T)\n    (preorder.lt._default fun (S T : subalgebra R A) => \u2191S \u2286 \u2191T) sorry sorry sorry\n\n/-- Reinterpret an `S`-subalgebra as an `R`-subalgebra in `comap R S A`. -/\ndef comap {R : Type u} {S : Type v} {A : Type w} [comm_semiring R] [comm_semiring S] [semiring A]\n    [algebra R S] [algebra S A] (iSB : subalgebra S A) : subalgebra R (algebra.comap R S A) :=\n  mk (carrier iSB) (one_mem' iSB) (mul_mem' iSB) (zero_mem' iSB) (add_mem' iSB) sorry\n\n/-- If `S` is an `R`-subalgebra of `A` and `T` is an `S`-subalgebra of `A`,\nthen `T` is an `R`-subalgebra of `A`. -/\ndef under {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A] {i : algebra R A}\n    (S : subalgebra R A) (T : subalgebra (\u21a5S) A) : subalgebra R A :=\n  mk (carrier T) sorry sorry sorry sorry sorry\n\n/-- Transport a subalgebra via an algebra homomorphism. -/\ndef map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    [semiring B] [algebra R B] (S : subalgebra R A) (f : alg_hom R A B) : subalgebra R B :=\n  mk (subsemiring.carrier (subsemiring.map \u2191f \u2191S)) sorry sorry sorry sorry sorry\n\n/-- Preimage of a subalgebra under an algebra homomorphism. -/\ndef comap' {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    [semiring B] [algebra R B] (S : subalgebra R B) (f : alg_hom R A B) : subalgebra R A :=\n  mk (subsemiring.carrier (subsemiring.comap \u2191f \u2191S)) sorry sorry sorry sorry sorry\n\ntheorem map_mono {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    [semiring B] [algebra R B] {S\u2081 : subalgebra R A} {S\u2082 : subalgebra R A} {f : alg_hom R A B} :\n    S\u2081 \u2264 S\u2082 \u2192 map S\u2081 f \u2264 map S\u2082 f :=\n  set.image_subset \u21d1f\n\ntheorem map_le {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    [semiring B] [algebra R B] {S : subalgebra R A} {f : alg_hom R A B} {U : subalgebra R B} :\n    map S f \u2264 U \u2194 S \u2264 comap' U f :=\n  set.image_subset_iff\n\ntheorem map_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] [semiring B] [algebra R B] {S\u2081 : subalgebra R A} {S\u2082 : subalgebra R A}\n    (f : alg_hom R A B) (hf : function.injective \u21d1f) (ih : map S\u2081 f = map S\u2082 f) : S\u2081 = S\u2082 :=\n  ext\n    (iff.mp set.ext_iff\n      (iff.mpr set.image_injective hf (fun (x : A) => x \u2208 \u2191S\u2081) (fun (x : A) => x \u2208 \u2191S\u2082)\n        (set.ext (iff.mp ext_iff ih))))\n\ntheorem mem_map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A]\n    [semiring B] [algebra R B] {S : subalgebra R A} {f : alg_hom R A B} {y : B} :\n    y \u2208 map S f \u2194 \u2203 (x : A), \u2203 (H : x \u2208 S), coe_fn f x = y :=\n  subsemiring.mem_map\n\nprotected instance no_zero_divisors {R : Type u_1} {A : Type u_2} [comm_ring R] [semiring A]\n    [no_zero_divisors A] [algebra R A] (S : subalgebra R A) : no_zero_divisors \u21a5S :=\n  subsemiring.no_zero_divisors (to_subsemiring S)\n\nprotected instance integral_domain {R : Type u_1} {A : Type u_2} [comm_ring R] [integral_domain A]\n    [algebra R A] (S : subalgebra R A) : integral_domain \u21a5S :=\n  subring.domain \u2191S\n\nend subalgebra\n\n\nnamespace alg_hom\n\n\n/-- Range of an `alg_hom` as a subalgebra. -/\nprotected def range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) : subalgebra R B :=\n  subalgebra.mk (subsemiring.carrier (ring_hom.srange (to_ring_hom \u03c6))) sorry sorry sorry sorry\n    sorry\n\n@[simp] theorem mem_range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) {y : B} :\n    y \u2208 alg_hom.range \u03c6 \u2194 \u2203 (x : A), coe_fn \u03c6 x = y :=\n  ring_hom.mem_srange\n\n@[simp] theorem coe_range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) :\n    \u2191(alg_hom.range \u03c6) = set.range \u21d1\u03c6 :=\n  sorry\n\n/-- Restrict the codomain of an algebra homomorphism. -/\ndef cod_restrict {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] (f : alg_hom R A B) (S : subalgebra R B)\n    (hf : \u2200 (x : A), coe_fn f x \u2208 S) : alg_hom R A \u21a5S :=\n  mk (ring_hom.to_fun (ring_hom.cod_srestrict (\u2191f) (\u2191S) hf)) sorry sorry sorry sorry sorry\n\ntheorem injective_cod_restrict {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (S : subalgebra R B)\n    (hf : \u2200 (x : A), coe_fn f x \u2208 S) :\n    function.injective \u21d1(cod_restrict f S hf) \u2194 function.injective \u21d1f :=\n  sorry\n\n/-- Restrict an injective algebra homomorphism to an algebra isomorphism -/\ndef alg_equiv.of_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (hf : function.injective \u21d1f) :\n    alg_equiv R A \u21a5(alg_hom.range f) :=\n  alg_equiv.of_bijective (cod_restrict f (alg_hom.range f) sorry) sorry\n\n@[simp] theorem alg_equiv.of_injective_apply {R : Type u} {A : Type v} {B : Type w}\n    [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B)\n    (hf : function.injective \u21d1f) (x : A) : \u2191(coe_fn (alg_equiv.of_injective f hf) x) = coe_fn f x :=\n  rfl\n\n/-- Restrict an algebra homomorphism between fields to an algebra isomorphism -/\ndef alg_equiv.of_injective_field {R : Type u} [comm_semiring R] {E : Type u_1} {F : Type u_2}\n    [division_ring E] [semiring F] [nontrivial F] [algebra R E] [algebra R F] (f : alg_hom R E F) :\n    alg_equiv R E \u21a5(alg_hom.range f) :=\n  alg_equiv.of_injective f sorry\n\n/-- The equalizer of two R-algebra homomorphisms -/\ndef equalizer {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B]\n    [algebra R A] [algebra R B] (\u03d5 : alg_hom R A B) (\u03c8 : alg_hom R A B) : subalgebra R A :=\n  subalgebra.mk (set_of fun (a : A) => coe_fn \u03d5 a = coe_fn \u03c8 a) sorry sorry sorry sorry sorry\n\n@[simp] theorem mem_equalizer {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [semiring B] [algebra R A] [algebra R B] (\u03d5 : alg_hom R A B) (\u03c8 : alg_hom R A B) (x : A) :\n    x \u2208 equalizer \u03d5 \u03c8 \u2194 coe_fn \u03d5 x = coe_fn \u03c8 x :=\n  iff.rfl\n\nend alg_hom\n\n\nnamespace algebra\n\n\n/-- The minimal subalgebra that includes `s`. -/\ndef adjoin (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (s : set A) :\n    subalgebra R A :=\n  subalgebra.mk (subsemiring.carrier (subsemiring.closure (set.range \u21d1(algebra_map R A) \u222a s))) sorry\n    sorry sorry sorry sorry\n\nprotected theorem gc {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    galois_connection (adjoin R) coe :=\n  sorry\n\n/-- Galois insertion between `adjoin` and `coe`. -/\nprotected def gi {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    galois_insertion (adjoin R) coe :=\n  galois_insertion.mk (fun (s : set A) (hs : \u2191(adjoin R s) \u2264 s) => adjoin R s) algebra.gc sorry\n    sorry\n\nprotected instance subalgebra.complete_lattice {R : Type u} {A : Type v} [comm_semiring R]\n    [semiring A] [algebra R A] : complete_lattice (subalgebra R A) :=\n  galois_insertion.lift_complete_lattice algebra.gi\n\nprotected instance subalgebra.inhabited {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : Inhabited (subalgebra R A) :=\n  { default := \u22a5 }\n\ntheorem mem_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {x : A} :\n    x \u2208 \u22a5 \u2194 x \u2208 set.range \u21d1(algebra_map R A) :=\n  sorry\n\ntheorem to_submodule_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    \u2191\u22a5 = submodule.span R (singleton 1) :=\n  sorry\n\n@[simp] theorem mem_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {x : A} : x \u2208 \u22a4 :=\n  subsemiring.subset_closure (Or.inr trivial)\n\n@[simp] theorem coe_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    \u2191\u22a4 = \u22a4 :=\n  submodule.ext fun (x : A) => iff_of_true mem_top trivial\n\n@[simp] theorem coe_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    \u2191\u22a5 = set.range \u21d1(algebra_map R A) :=\n  sorry\n\ntheorem eq_top_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    {S : subalgebra R A} : S = \u22a4 \u2194 \u2200 (x : A), x \u2208 S :=\n  sorry\n\n@[simp] theorem map_top {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) :\n    subalgebra.map \u22a4 f = alg_hom.range f :=\n  sorry\n\n@[simp] theorem map_bot {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) : subalgebra.map \u22a5 f = \u22a5 :=\n  sorry\n\n@[simp] theorem comap_top {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A]\n    [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) : subalgebra.comap' \u22a4 f = \u22a4 :=\n  iff.mpr eq_top_iff fun (x : A) => mem_top\n\n/-- `alg_hom` to `\u22a4 : subalgebra R A`. -/\ndef to_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    alg_hom R A \u21a5\u22a4 :=\n  alg_hom.mk (fun (x : A) => { val := x, property := mem_top }) sorry sorry sorry sorry sorry\n\ntheorem surjective_algebra_map_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A]\n    [algebra R A] : function.surjective \u21d1(algebra_map R A) \u2194 \u22a4 = \u22a5 :=\n  sorry\n\ntheorem bijective_algebra_map_iff {R : Type u_1} {A : Type u_2} [field R] [semiring A]\n    [nontrivial A] [algebra R A] : function.bijective \u21d1(algebra_map R A) \u2194 \u22a4 = \u22a5 :=\n  { mp :=\n      fun (h : function.bijective \u21d1(algebra_map R A)) =>\n        iff.mp surjective_algebra_map_iff (and.right h),\n    mpr :=\n      fun (h : \u22a4 = \u22a5) =>\n        { left := ring_hom.injective (algebra_map R A),\n          right := iff.mpr surjective_algebra_map_iff h } }\n\n/-- The bottom subalgebra is isomorphic to the base ring. -/\ndef bot_equiv_of_injective {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (h : function.injective \u21d1(algebra_map R A)) : alg_equiv R (\u21a5\u22a5) R :=\n  alg_equiv.symm (alg_equiv.of_bijective (of_id R \u21a5\u22a5) sorry)\n\n/-- The bottom subalgebra is isomorphic to the field. -/\ndef bot_equiv (F : Type u_1) (R : Type u_2) [field F] [semiring R] [nontrivial R] [algebra F R] :\n    alg_equiv F (\u21a5\u22a5) F :=\n  bot_equiv_of_injective sorry\n\n/-- The top subalgebra is isomorphic to the field. -/\ndef top_equiv {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] :\n    alg_equiv R (\u21a5\u22a4) A :=\n  alg_equiv.symm (alg_equiv.of_bijective to_top sorry)\n\nend algebra\n\n\nnamespace subalgebra\n\n\ntheorem range_val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A]\n    (S : subalgebra R A) : alg_hom.range (val S) = S :=\n  ext (iff.mp set.ext_iff (Eq.trans (alg_hom.coe_range (val S)) subtype.range_val))\n\nprotected instance unique {R : Type u} [comm_semiring R] : unique (subalgebra R R) :=\n  unique.mk { default := Inhabited.default } sorry\n\nend subalgebra\n\n\n/-- A subsemiring is a `\u2115`-subalgebra. -/\ndef subalgebra_of_subsemiring {R : Type u_1} [semiring R] (S : subsemiring R) : subalgebra \u2115 R :=\n  subalgebra.mk (subsemiring.carrier S) (subsemiring.one_mem' S) (subsemiring.mul_mem' S)\n    (subsemiring.zero_mem' S) (subsemiring.add_mem' S) sorry\n\n@[simp] theorem mem_subalgebra_of_subsemiring {R : Type u_1} [semiring R] {x : R}\n    {S : subsemiring R} : x \u2208 subalgebra_of_subsemiring S \u2194 x \u2208 S :=\n  iff.rfl\n\n/-- A subring is a `\u2124`-subalgebra. -/\ndef subalgebra_of_subring {R : Type u_1} [ring R] (S : subring R) : subalgebra \u2124 R :=\n  subalgebra.mk (subring.carrier S) (subring.one_mem' S) (subring.mul_mem' S) (subring.zero_mem' S)\n    (subring.add_mem' S) sorry\n\n/-- A subset closed under the ring operations is a `\u2124`-subalgebra. -/\ndef subalgebra_of_is_subring {R : Type u_1} [ring R] (S : set R) [is_subring S] : subalgebra \u2124 R :=\n  subalgebra_of_subring (set.to_subring S)\n\n@[simp] theorem mem_subalgebra_of_subring {R : Type u_1} [ring R] {x : R} {S : subring R} :\n    x \u2208 subalgebra_of_subring S \u2194 x \u2208 S :=\n  iff.rfl\n\n@[simp] theorem mem_subalgebra_of_is_subring {R : Type u_1} [ring R] {x : R} {S : set R}\n    [is_subring S] : x \u2208 subalgebra_of_is_subring S \u2194 x \u2208 S :=\n  iff.rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/algebra/subalgebra_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.39886038232241516}}
{"text": "/-\nCopyright (c) 2020 Anatole Dedecker. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anatole Dedecker\n\n! This file was ported from Lean 3 source module analysis.calculus.lhopital\n! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Analysis.Calculus.MeanValue\n\n/-!\n# L'H\u00f4pital's rule for 0/0 indeterminate forms\n\nIn this file, we prove several forms of \"L'Hopital's rule\" for computing 0/0\nindeterminate forms. The proof of `has_deriv_at.lhopital_zero_right_on_Ioo`\nis based on the one given in the corresponding\n[Wikibooks](https://en.wikibooks.org/wiki/Calculus/L%27H%C3%B4pital%27s_Rule)\nchapter, and all other statements are derived from this one by composing by\ncarefully chosen functions.\n\nNote that the filter `f'/g'` tends to isn't required to be one of `\ud835\udcdd a`,\n`at_top` or `at_bot`. In fact, we give a slightly stronger statement by\nallowing it to be any filter on `\u211d`.\n\nEach statement is available in a `has_deriv_at` form and a `deriv` form, which\nis denoted by each statement being in either the `has_deriv_at` or the `deriv`\nnamespace.\n\n## Tags\n\nL'H\u00f4pital's rule, L'Hopital's rule\n-/\n\n\nopen Filter Set\n\nopen Filter Topology Pointwise\n\nvariable {a b : \u211d} (hab : a < b) {l : Filter \u211d} {f f' g g' : \u211d \u2192 \u211d}\n\n/-!\n## Interval-based versions\n\nWe start by proving statements where all conditions (derivability, `g' \u2260 0`) have\nto be satisfied on an explicitly-provided interval.\n-/\n\n\nnamespace HasDerivAt\n\ninclude hab\n\ntheorem lhopital_zero_right_on_Ioo (hff' : \u2200 x \u2208 Ioo a b, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioo a b, HasDerivAt g (g' x) x) (hg' : \u2200 x \u2208 Ioo a b, g' x \u2260 0)\n    (hfa : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[>] a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  have sub : \u2200 x \u2208 Ioo a b, Ioo a x \u2286 Ioo a b := fun x hx =>\n    Ioo_subset_Ioo (le_refl a) (le_of_lt hx.2)\n  have hg : \u2200 x \u2208 Ioo a b, g x \u2260 0 := by\n    intro x hx h\n    have : tendsto g (\ud835\udcdd[<] x) (\ud835\udcdd 0) :=\n      by\n      rw [\u2190 h, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hx.1]\n      exact ((hgg' x hx).ContinuousAt.ContinuousWithinAt.mono <| sub x hx).Tendsto\n    obtain \u27e8y, hyx, hy\u27e9 : \u2203 c \u2208 Ioo a x, g' c = 0\n    exact exists_hasDerivAt_eq_zero' hx.1 hga this fun y hy => hgg' y <| sub x hx hy\n    exact hg' y (sub x hx hyx) hy\n  have : \u2200 x \u2208 Ioo a b, \u2203 c \u2208 Ioo a x, f x * g' c = g x * f' c :=\n    by\n    intro x hx\n    rw [\u2190 sub_zero (f x), \u2190 sub_zero (g x)]\n    exact\n      exists_ratio_hasDerivAt_eq_ratio_slope' g g' hx.1 f f' (fun y hy => hgg' y <| sub x hx hy)\n        (fun y hy => hff' y <| sub x hx hy) hga hfa\n        (tendsto_nhdsWithin_of_tendsto_nhds (hgg' x hx).ContinuousAt.Tendsto)\n        (tendsto_nhdsWithin_of_tendsto_nhds (hff' x hx).ContinuousAt.Tendsto)\n  choose! c hc using this\n  have : \u2200 x \u2208 Ioo a b, ((fun x' => f' x' / g' x') \u2218 c) x = f x / g x :=\n    by\n    intro x hx\n    rcases hc x hx with \u27e8h\u2081, h\u2082\u27e9\n    field_simp [hg x hx, hg' (c x) ((sub x hx) h\u2081)]\n    simp only [h\u2082]\n    rwa [mul_comm]\n  have cmp : \u2200 x \u2208 Ioo a b, a < c x \u2227 c x < x := fun x hx => (hc x hx).1\n  rw [\u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n  apply tendsto_nhdsWithin_congr this\n  simp only\n  apply hdiv.comp\n  refine'\n    tendsto_nhdsWithin_of_tendsto_nhds_of_eventually_within _\n      (tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds\n        (tendsto_nhdsWithin_of_tendsto_nhds tendsto_id) _ _)\n      _\n  all_goals\n    apply eventually_nhdsWithin_of_forall\n    intro x hx\n    have := cmp x hx\n    try simp\n    linarith [this]\n#align has_deriv_at.lhopital_zero_right_on_Ioo HasDerivAt.lhopital_zero_right_on_Ioo\n\ntheorem lhopital_zero_right_on_Ico (hff' : \u2200 x \u2208 Ioo a b, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ico a b))\n    (hcg : ContinuousOn g (Ico a b)) (hg' : \u2200 x \u2208 Ioo a b, g' x \u2260 0) (hfa : f a = 0) (hga : g a = 0)\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[>] a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  refine' lhopital_zero_right_on_Ioo hab hff' hgg' hg' _ _ hdiv\n  \u00b7 rw [\u2190 hfa, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n    exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).Tendsto\n  \u00b7 rw [\u2190 hga, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n    exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).Tendsto\n#align has_deriv_at.lhopital_zero_right_on_Ico HasDerivAt.lhopital_zero_right_on_Ico\n\ntheorem lhopital_zero_left_on_Ioo (hff' : \u2200 x \u2208 Ioo a b, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioo a b, HasDerivAt g (g' x) x) (hg' : \u2200 x \u2208 Ioo a b, g' x \u2260 0)\n    (hfb : Tendsto f (\ud835\udcdd[<] b) (\ud835\udcdd 0)) (hgb : Tendsto g (\ud835\udcdd[<] b) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[<] b) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[<] b) l :=\n  by\n  -- Here, we essentially compose by `has_neg.neg`. The following is mostly technical details.\n  have hdnf : \u2200 x \u2208 -Ioo a b, HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x := fun x hx =>\n    comp x (hff' (-x) hx) (hasDerivAt_neg x)\n  have hdng : \u2200 x \u2208 -Ioo a b, HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x := fun x hx =>\n    comp x (hgg' (-x) hx) (hasDerivAt_neg x)\n  rw [preimage_neg_Ioo] at hdnf\n  rw [preimage_neg_Ioo] at hdng\n  have :=\n    lhopital_zero_right_on_Ioo (neg_lt_neg hab) hdnf hdng\n      (by\n        intro x hx h\n        apply\n          hg' _\n            (by\n              rw [\u2190 preimage_neg_Ioo] at hx\n              exact hx)\n        rwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h)\n      (hfb.comp tendsto_neg_nhdsWithin_Ioi_neg) (hgb.comp tendsto_neg_nhdsWithin_Ioi_neg)\n      (by\n        simp only [neg_div_neg_eq, mul_one, mul_neg]\n        exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_nhdsWithin_Ioi_neg))\n  have := this.comp tendsto_neg_nhdsWithin_Iio\n  unfold Function.comp at this\n  simpa only [neg_neg]\n#align has_deriv_at.lhopital_zero_left_on_Ioo HasDerivAt.lhopital_zero_left_on_Ioo\n\ntheorem lhopital_zero_left_on_Ioc (hff' : \u2200 x \u2208 Ioo a b, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioo a b, HasDerivAt g (g' x) x) (hcf : ContinuousOn f (Ioc a b))\n    (hcg : ContinuousOn g (Ioc a b)) (hg' : \u2200 x \u2208 Ioo a b, g' x \u2260 0) (hfb : f b = 0) (hgb : g b = 0)\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[<] b) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[<] b) l :=\n  by\n  refine' lhopital_zero_left_on_Ioo hab hff' hgg' hg' _ _ hdiv\n  \u00b7 rw [\u2190 hfb, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]\n    exact ((hcf b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).Tendsto\n  \u00b7 rw [\u2190 hgb, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Iio hab]\n    exact ((hcg b <| right_mem_Ioc.mpr hab).mono Ioo_subset_Ioc_self).Tendsto\n#align has_deriv_at.lhopital_zero_left_on_Ioc HasDerivAt.lhopital_zero_left_on_Ioc\n\nomit hab\n\ntheorem lhopital_zero_atTop_on_Ioi (hff' : \u2200 x \u2208 Ioi a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Ioi a, HasDerivAt g (g' x) x) (hg' : \u2200 x \u2208 Ioi a, g' x \u2260 0)\n    (hftop : Tendsto f atTop (\ud835\udcdd 0)) (hgtop : Tendsto g atTop (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l :=\n  by\n  obtain \u27e8a', haa', ha'\u27e9 : \u2203 a', a < a' \u2227 0 < a' :=\n    \u27e81 + max a 0,\n      \u27e8lt_of_le_of_lt (le_max_left a 0) (lt_one_add _),\n        lt_of_le_of_lt (le_max_right a 0) (lt_one_add _)\u27e9\u27e9\n  have fact1 : \u2200 x : \u211d, x \u2208 Ioo 0 a'\u207b\u00b9 \u2192 x \u2260 0 := fun _ hx => (ne_of_lt hx.1).symm\n  have fact2 : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, a < x\u207b\u00b9 := fun _ hx => lt_trans haa' ((lt_inv ha' hx.1).mpr hx.2)\n  have hdnf : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, HasDerivAt (f \u2218 Inv.inv) (f' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x := fun x hx =>\n    comp x (hff' x\u207b\u00b9 <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)\n  have hdng : \u2200 x \u2208 Ioo 0 a'\u207b\u00b9, HasDerivAt (g \u2218 Inv.inv) (g' x\u207b\u00b9 * -(x ^ 2)\u207b\u00b9) x := fun x hx =>\n    comp x (hgg' x\u207b\u00b9 <| fact2 x hx) (hasDerivAt_inv <| fact1 x hx)\n  have :=\n    lhopital_zero_right_on_Ioo (inv_pos.mpr ha') hdnf hdng\n      (by\n        intro x hx\n        refine' mul_ne_zero _ (neg_ne_zero.mpr <| inv_ne_zero <| pow_ne_zero _ <| fact1 x hx)\n        exact hg' _ (fact2 x hx))\n      (hftop.comp tendsto_inv_zero_atTop) (hgtop.comp tendsto_inv_zero_atTop)\n      (by\n        refine' (tendsto_congr' _).mp (hdiv.comp tendsto_inv_zero_atTop)\n        rw [eventually_eq_iff_exists_mem]\n        use Ioi 0, self_mem_nhdsWithin\n        intro x hx\n        unfold Function.comp\n        erw [mul_div_mul_right]\n        refine' neg_ne_zero.mpr (inv_ne_zero <| pow_ne_zero _ <| ne_of_gt hx))\n  have := this.comp tendsto_inv_atTop_zero'\n  unfold Function.comp at this\n  simpa only [inv_inv]\n#align has_deriv_at.lhopital_zero_at_top_on_Ioi HasDerivAt.lhopital_zero_atTop_on_Ioi\n\ntheorem lhopital_zero_atBot_on_Iio (hff' : \u2200 x \u2208 Iio a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200 x \u2208 Iio a, HasDerivAt g (g' x) x) (hg' : \u2200 x \u2208 Iio a, g' x \u2260 0)\n    (hfbot : Tendsto f atBot (\ud835\udcdd 0)) (hgbot : Tendsto g atBot (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l :=\n  by\n  -- Here, we essentially compose by `has_neg.neg`. The following is mostly technical details.\n  have hdnf : \u2200 x \u2208 -Iio a, HasDerivAt (f \u2218 Neg.neg) (f' (-x) * -1) x := fun x hx =>\n    comp x (hff' (-x) hx) (hasDerivAt_neg x)\n  have hdng : \u2200 x \u2208 -Iio a, HasDerivAt (g \u2218 Neg.neg) (g' (-x) * -1) x := fun x hx =>\n    comp x (hgg' (-x) hx) (hasDerivAt_neg x)\n  rw [preimage_neg_Iio] at hdnf\n  rw [preimage_neg_Iio] at hdng\n  have :=\n    lhopital_zero_at_top_on_Ioi hdnf hdng\n      (by\n        intro x hx h\n        apply\n          hg' _\n            (by\n              rw [\u2190 preimage_neg_Iio] at hx\n              exact hx)\n        rwa [mul_comm, \u2190 neg_eq_neg_one_mul, neg_eq_zero] at h)\n      (hfbot.comp tendsto_neg_at_top_at_bot) (hgbot.comp tendsto_neg_at_top_at_bot)\n      (by\n        simp only [mul_one, mul_neg, neg_div_neg_eq]\n        exact (tendsto_congr fun x => rfl).mp (hdiv.comp tendsto_neg_at_top_at_bot))\n  have := this.comp tendsto_neg_at_bot_at_top\n  unfold Function.comp at this\n  simpa only [neg_neg]\n#align has_deriv_at.lhopital_zero_at_bot_on_Iio HasDerivAt.lhopital_zero_atBot_on_Iio\n\nend HasDerivAt\n\nnamespace deriv\n\ninclude hab\n\ntheorem lhopital_zero_right_on_Ioo (hdf : DifferentiableOn \u211d f (Ioo a b))\n    (hg' : \u2200 x \u2208 Ioo a b, deriv g x \u2260 0) (hfa : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hga : Tendsto g (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[>] a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  have hdf : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d f x := fun x hx =>\n    (hdf x hx).DifferentiableAt (Ioo_mem_nhds hx.1 hx.2)\n  have hdg : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d g x := fun x hx =>\n    by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n  exact\n    HasDerivAt.lhopital_zero_right_on_Ioo hab (fun x hx => (hdf x hx).HasDerivAt)\n      (fun x hx => (hdg x hx).HasDerivAt) hg' hfa hga hdiv\n#align deriv.lhopital_zero_right_on_Ioo deriv.lhopital_zero_right_on_Ioo\n\ntheorem lhopital_zero_right_on_Ico (hdf : DifferentiableOn \u211d f (Ioo a b))\n    (hcf : ContinuousOn f (Ico a b)) (hcg : ContinuousOn g (Ico a b))\n    (hg' : \u2200 x \u2208 Ioo a b, (deriv g) x \u2260 0) (hfa : f a = 0) (hga : g a = 0)\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[>] a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  refine' lhopital_zero_right_on_Ioo hab hdf hg' _ _ hdiv\n  \u00b7 rw [\u2190 hfa, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n    exact ((hcf a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).Tendsto\n  \u00b7 rw [\u2190 hga, \u2190 nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]\n    exact ((hcg a <| left_mem_Ico.mpr hab).mono Ioo_subset_Ico_self).Tendsto\n#align deriv.lhopital_zero_right_on_Ico deriv.lhopital_zero_right_on_Ico\n\ntheorem lhopital_zero_left_on_Ioo (hdf : DifferentiableOn \u211d f (Ioo a b))\n    (hg' : \u2200 x \u2208 Ioo a b, (deriv g) x \u2260 0) (hfb : Tendsto f (\ud835\udcdd[<] b) (\ud835\udcdd 0))\n    (hgb : Tendsto g (\ud835\udcdd[<] b) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[<] b) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[<] b) l :=\n  by\n  have hdf : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d f x := fun x hx =>\n    (hdf x hx).DifferentiableAt (Ioo_mem_nhds hx.1 hx.2)\n  have hdg : \u2200 x \u2208 Ioo a b, DifferentiableAt \u211d g x := fun x hx =>\n    by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n  exact\n    HasDerivAt.lhopital_zero_left_on_Ioo hab (fun x hx => (hdf x hx).HasDerivAt)\n      (fun x hx => (hdg x hx).HasDerivAt) hg' hfb hgb hdiv\n#align deriv.lhopital_zero_left_on_Ioo deriv.lhopital_zero_left_on_Ioo\n\nomit hab\n\ntheorem lhopital_zero_atTop_on_Ioi (hdf : DifferentiableOn \u211d f (Ioi a))\n    (hg' : \u2200 x \u2208 Ioi a, (deriv g) x \u2260 0) (hftop : Tendsto f atTop (\ud835\udcdd 0))\n    (hgtop : Tendsto g atTop (\ud835\udcdd 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :\n    Tendsto (fun x => f x / g x) atTop l :=\n  by\n  have hdf : \u2200 x \u2208 Ioi a, DifferentiableAt \u211d f x := fun x hx =>\n    (hdf x hx).DifferentiableAt (Ioi_mem_nhds hx)\n  have hdg : \u2200 x \u2208 Ioi a, DifferentiableAt \u211d g x := fun x hx =>\n    by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n  exact\n    HasDerivAt.lhopital_zero_atTop_on_Ioi (fun x hx => (hdf x hx).HasDerivAt)\n      (fun x hx => (hdg x hx).HasDerivAt) hg' hftop hgtop hdiv\n#align deriv.lhopital_zero_at_top_on_Ioi deriv.lhopital_zero_atTop_on_Ioi\n\ntheorem lhopital_zero_atBot_on_Iio (hdf : DifferentiableOn \u211d f (Iio a))\n    (hg' : \u2200 x \u2208 Iio a, (deriv g) x \u2260 0) (hfbot : Tendsto f atBot (\ud835\udcdd 0))\n    (hgbot : Tendsto g atBot (\ud835\udcdd 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :\n    Tendsto (fun x => f x / g x) atBot l :=\n  by\n  have hdf : \u2200 x \u2208 Iio a, DifferentiableAt \u211d f x := fun x hx =>\n    (hdf x hx).DifferentiableAt (Iio_mem_nhds hx)\n  have hdg : \u2200 x \u2208 Iio a, DifferentiableAt \u211d g x := fun x hx =>\n    by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)\n  exact\n    HasDerivAt.lhopital_zero_atBot_on_Iio (fun x hx => (hdf x hx).HasDerivAt)\n      (fun x hx => (hdg x hx).HasDerivAt) hg' hfbot hgbot hdiv\n#align deriv.lhopital_zero_at_bot_on_Iio deriv.lhopital_zero_atBot_on_Iio\n\nend deriv\n\n/-!\n## Generic versions\n\nThe following statements no longer any explicit interval, as they only require\nconditions holding eventually.\n-/\n\n\nnamespace HasDerivAt\n\n/-- L'H\u00f4pital's rule for approaching a real from the right, `has_deriv_at` version -/\ntheorem lhopital_zero_nhds_right (hff' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in \ud835\udcdd[>] a, g' x \u2260 0)\n    (hfa : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[>] a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  rw [eventually_iff_exists_mem] at *\n  rcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n  rcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n  rcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n  let s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n  have hs : s \u2208 \ud835\udcdd[>] a := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n  rw [mem_nhdsWithin_Ioi_iff_exists_Ioo_subset] at hs\n  rcases hs with \u27e8u, hau, hu\u27e9\n  refine' lhopital_zero_right_on_Ioo hau _ _ _ hfa hga hdiv <;> intro x hx <;> apply_assumption <;>\n    first |exact (hu hx).1.1|exact (hu hx).1.2|exact (hu hx).2\n#align has_deriv_at.lhopital_zero_nhds_right HasDerivAt.lhopital_zero_nhds_right\n\n/-- L'H\u00f4pital's rule for approaching a real from the left, `has_deriv_at` version -/\ntheorem lhopital_zero_nhds_left (hff' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in \ud835\udcdd[<] a, g' x \u2260 0)\n    (hfa : Tendsto f (\ud835\udcdd[<] a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd[<] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[<] a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[<] a) l :=\n  by\n  rw [eventually_iff_exists_mem] at *\n  rcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n  rcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n  rcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n  let s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n  have hs : s \u2208 \ud835\udcdd[<] a := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n  rw [mem_nhdsWithin_Iio_iff_exists_Ioo_subset] at hs\n  rcases hs with \u27e8l, hal, hl\u27e9\n  refine' lhopital_zero_left_on_Ioo hal _ _ _ hfa hga hdiv <;> intro x hx <;> apply_assumption <;>\n    first |exact (hl hx).1.1|exact (hl hx).1.2|exact (hl hx).2\n#align has_deriv_at.lhopital_zero_nhds_left HasDerivAt.lhopital_zero_nhds_left\n\n/-- L'H\u00f4pital's rule for approaching a real, `has_deriv_at` version. This\n  does not require anything about the situation at `a` -/\ntheorem lhopital_zero_nhds' (hff' : \u2200\u1da0 x in \ud835\udcdd[\u2260] a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in \ud835\udcdd[\u2260] a, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in \ud835\udcdd[\u2260] a, g' x \u2260 0)\n    (hfa : Tendsto f (\ud835\udcdd[\u2260] a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd[\u2260] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd[\u2260] a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[\u2260] a) l :=\n  by\n  simp only [\u2190 Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *\n  exact\n    \u27e8lhopital_zero_nhds_left hff'.1 hgg'.1 hg'.1 hfa.1 hga.1 hdiv.1,\n      lhopital_zero_nhds_right hff'.2 hgg'.2 hg'.2 hfa.2 hga.2 hdiv.2\u27e9\n#align has_deriv_at.lhopital_zero_nhds' HasDerivAt.lhopital_zero_nhds'\n\n/-- **L'H\u00f4pital's rule** for approaching a real, `has_deriv_at` version -/\ntheorem lhopital_zero_nhds (hff' : \u2200\u1da0 x in \ud835\udcdd a, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in \ud835\udcdd a, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in \ud835\udcdd a, g' x \u2260 0)\n    (hfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) (\ud835\udcdd a) l) : Tendsto (fun x => f x / g x) (\ud835\udcdd[\u2260] a) l := by\n  apply @lhopital_zero_nhds' _ _ _ f' _ g' <;>\n      first\n        |apply eventually_nhdsWithin_of_eventually_nhds|apply tendsto_nhdsWithin_of_tendsto_nhds <;>\n    assumption\n#align has_deriv_at.lhopital_zero_nhds HasDerivAt.lhopital_zero_nhds\n\n/-- L'H\u00f4pital's rule for approaching +\u221e, `has_deriv_at` version -/\ntheorem lhopital_zero_atTop (hff' : \u2200\u1da0 x in atTop, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in atTop, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in atTop, g' x \u2260 0)\n    (hftop : Tendsto f atTop (\ud835\udcdd 0)) (hgtop : Tendsto g atTop (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) atTop l) : Tendsto (fun x => f x / g x) atTop l :=\n  by\n  rw [eventually_iff_exists_mem] at *\n  rcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n  rcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n  rcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n  let s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n  have hs : s \u2208 at_top := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n  rw [mem_at_top_sets] at hs\n  rcases hs with \u27e8l, hl\u27e9\n  have hl' : Ioi l \u2286 s := fun x hx => hl x (le_of_lt hx)\n  refine' lhopital_zero_at_top_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>\n        intro x hx <;>\n      apply_assumption <;>\n    first |exact (hl' hx).1.1|exact (hl' hx).1.2\n#align has_deriv_at.lhopital_zero_at_top HasDerivAt.lhopital_zero_atTop\n\n/-- L'H\u00f4pital's rule for approaching -\u221e, `has_deriv_at` version -/\ntheorem lhopital_zero_atBot (hff' : \u2200\u1da0 x in atBot, HasDerivAt f (f' x) x)\n    (hgg' : \u2200\u1da0 x in atBot, HasDerivAt g (g' x) x) (hg' : \u2200\u1da0 x in atBot, g' x \u2260 0)\n    (hfbot : Tendsto f atBot (\ud835\udcdd 0)) (hgbot : Tendsto g atBot (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => f' x / g' x) atBot l) : Tendsto (fun x => f x / g x) atBot l :=\n  by\n  rw [eventually_iff_exists_mem] at *\n  rcases hff' with \u27e8s\u2081, hs\u2081, hff'\u27e9\n  rcases hgg' with \u27e8s\u2082, hs\u2082, hgg'\u27e9\n  rcases hg' with \u27e8s\u2083, hs\u2083, hg'\u27e9\n  let s := s\u2081 \u2229 s\u2082 \u2229 s\u2083\n  have hs : s \u2208 at_bot := inter_mem (inter_mem hs\u2081 hs\u2082) hs\u2083\n  rw [mem_at_bot_sets] at hs\n  rcases hs with \u27e8l, hl\u27e9\n  have hl' : Iio l \u2286 s := fun x hx => hl x (le_of_lt hx)\n  refine' lhopital_zero_at_bot_on_Iio _ _ (fun x hx => hg' x <| (hl' hx).2) hfbot hgbot hdiv <;>\n        intro x hx <;>\n      apply_assumption <;>\n    first |exact (hl' hx).1.1|exact (hl' hx).1.2\n#align has_deriv_at.lhopital_zero_at_bot HasDerivAt.lhopital_zero_atBot\n\nend HasDerivAt\n\nnamespace deriv\n\n/-- **L'H\u00f4pital's rule** for approaching a real from the right, `deriv` version -/\ntheorem lhopital_zero_nhds_right (hdf : \u2200\u1da0 x in \ud835\udcdd[>] a, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x in \ud835\udcdd[>] a, deriv g x \u2260 0) (hfa : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hga : Tendsto g (\ud835\udcdd[>] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[>] a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[>] a) l :=\n  by\n  have hdg : \u2200\u1da0 x in \ud835\udcdd[>] a, DifferentiableAt \u211d g x :=\n    hg'.mp\n      (eventually_of_forall fun _ hg' =>\n        by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n  have hdf' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt f (deriv f x) x :=\n    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  have hdg' : \u2200\u1da0 x in \ud835\udcdd[>] a, HasDerivAt g (deriv g x) x :=\n    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  exact HasDerivAt.lhopital_zero_nhds_right hdf' hdg' hg' hfa hga hdiv\n#align deriv.lhopital_zero_nhds_right deriv.lhopital_zero_nhds_right\n\n/-- **L'H\u00f4pital's rule** for approaching a real from the left, `deriv` version -/\ntheorem lhopital_zero_nhds_left (hdf : \u2200\u1da0 x in \ud835\udcdd[<] a, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x in \ud835\udcdd[<] a, deriv g x \u2260 0) (hfa : Tendsto f (\ud835\udcdd[<] a) (\ud835\udcdd 0))\n    (hga : Tendsto g (\ud835\udcdd[<] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[<] a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[<] a) l :=\n  by\n  have hdg : \u2200\u1da0 x in \ud835\udcdd[<] a, DifferentiableAt \u211d g x :=\n    hg'.mp\n      (eventually_of_forall fun _ hg' =>\n        by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n  have hdf' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt f (deriv f x) x :=\n    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  have hdg' : \u2200\u1da0 x in \ud835\udcdd[<] a, HasDerivAt g (deriv g x) x :=\n    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  exact HasDerivAt.lhopital_zero_nhds_left hdf' hdg' hg' hfa hga hdiv\n#align deriv.lhopital_zero_nhds_left deriv.lhopital_zero_nhds_left\n\n/-- **L'H\u00f4pital's rule** for approaching a real, `deriv` version. This\n  does not require anything about the situation at `a` -/\ntheorem lhopital_zero_nhds' (hdf : \u2200\u1da0 x in \ud835\udcdd[\u2260] a, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x in \ud835\udcdd[\u2260] a, deriv g x \u2260 0) (hfa : Tendsto f (\ud835\udcdd[\u2260] a) (\ud835\udcdd 0))\n    (hga : Tendsto g (\ud835\udcdd[\u2260] a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd[\u2260] a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[\u2260] a) l :=\n  by\n  simp only [\u2190 Iio_union_Ioi, nhdsWithin_union, tendsto_sup, eventually_sup] at *\n  exact\n    \u27e8lhopital_zero_nhds_left hdf.1 hg'.1 hfa.1 hga.1 hdiv.1,\n      lhopital_zero_nhds_right hdf.2 hg'.2 hfa.2 hga.2 hdiv.2\u27e9\n#align deriv.lhopital_zero_nhds' deriv.lhopital_zero_nhds'\n\n/-- **L'H\u00f4pital's rule** for approaching a real, `deriv` version -/\ntheorem lhopital_zero_nhds (hdf : \u2200\u1da0 x in \ud835\udcdd a, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x in \ud835\udcdd a, deriv g x \u2260 0) (hfa : Tendsto f (\ud835\udcdd a) (\ud835\udcdd 0)) (hga : Tendsto g (\ud835\udcdd a) (\ud835\udcdd 0))\n    (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (\ud835\udcdd a) l) :\n    Tendsto (fun x => f x / g x) (\ud835\udcdd[\u2260] a) l := by\n  apply lhopital_zero_nhds' <;>\n      first\n        |apply eventually_nhdsWithin_of_eventually_nhds|apply tendsto_nhdsWithin_of_tendsto_nhds <;>\n    assumption\n#align deriv.lhopital_zero_nhds deriv.lhopital_zero_nhds\n\n/-- **L'H\u00f4pital's rule** for approaching +\u221e, `deriv` version -/\ntheorem lhopital_zero_atTop (hdf : \u2200\u1da0 x : \u211d in atTop, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x : \u211d in atTop, deriv g x \u2260 0) (hftop : Tendsto f atTop (\ud835\udcdd 0))\n    (hgtop : Tendsto g atTop (\ud835\udcdd 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atTop l) :\n    Tendsto (fun x => f x / g x) atTop l :=\n  by\n  have hdg : \u2200\u1da0 x in at_top, DifferentiableAt \u211d g x :=\n    hg'.mp\n      (eventually_of_forall fun _ hg' =>\n        by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n  have hdf' : \u2200\u1da0 x in at_top, HasDerivAt f (deriv f x) x :=\n    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  have hdg' : \u2200\u1da0 x in at_top, HasDerivAt g (deriv g x) x :=\n    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  exact HasDerivAt.lhopital_zero_atTop hdf' hdg' hg' hftop hgtop hdiv\n#align deriv.lhopital_zero_at_top deriv.lhopital_zero_atTop\n\n/-- **L'H\u00f4pital's rule** for approaching -\u221e, `deriv` version -/\ntheorem lhopital_zero_atBot (hdf : \u2200\u1da0 x : \u211d in atBot, DifferentiableAt \u211d f x)\n    (hg' : \u2200\u1da0 x : \u211d in atBot, deriv g x \u2260 0) (hfbot : Tendsto f atBot (\ud835\udcdd 0))\n    (hgbot : Tendsto g atBot (\ud835\udcdd 0)) (hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) atBot l) :\n    Tendsto (fun x => f x / g x) atBot l :=\n  by\n  have hdg : \u2200\u1da0 x in at_bot, DifferentiableAt \u211d g x :=\n    hg'.mp\n      (eventually_of_forall fun _ hg' =>\n        by_contradiction fun h => hg' (deriv_zero_of_not_differentiableAt h))\n  have hdf' : \u2200\u1da0 x in at_bot, HasDerivAt f (deriv f x) x :=\n    hdf.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  have hdg' : \u2200\u1da0 x in at_bot, HasDerivAt g (deriv g x) x :=\n    hdg.mp (eventually_of_forall fun _ => DifferentiableAt.hasDerivAt)\n  exact HasDerivAt.lhopital_zero_atBot hdf' hdg' hg' hfbot hgbot hdiv\n#align deriv.lhopital_zero_at_bot deriv.lhopital_zero_atBot\n\nend deriv\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Analysis/Calculus/Lhopital.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.3988603820631224}}
{"text": "/-\nCopyright (c) 2022 Youjack. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Youjack\n-/\nimport cycle\n\n/-!\n# The SECOND Law and Carnot Theorem\n\nThis file defines\n* the notion of `cycle`'s satisfying the second law, i.e. `cycle.possible`\n* the notion of reversible `cycle`, i.e. `cycle.reversible`\nand states\n* axioms stating the possibility of some types of `cycle`s, including\n  `cycle.trivial_possible`,\n  `one_rsv_cycle.exists_possible_dissipate'`, `abs_rel_cycle.exists_possible_transfer'`,\n  `usual_engine_cycle.exists_possible`, `usual_pump_cycle.exists_possible`\n* axioms stating that connection and scaling preserve the possibility of `cycle`s,\n  i.e. `cycle.possible_add` and `cycle.possible_smul_nonneg`\n* the axiom of the second law, i.e. `kelvin_stmt`\nand proves\n* the equivalence of Kelvin-Plank statement and Clausius's statement\n* the equivalence of Clausius's statement and Carnot theorem\n-/\n\nnamespace thermodynamics\n\n/-!\n## `possible` `cycle`s\n---------------------------------------------------------------------------------------------------/\n\nnamespace cycle\n\n/-- abstract `cycle` that is possible according to the second law -/\nconstant possible : cycle \u2192 Prop\n/-- The trivial `cycle` is possible. -/\naxiom trivial_possible : (0:cycle).possible\n\n/-- The connection of two possible `cycle`s is possible.\n* Note that a connection of possible and impossible can be possible.\n-/\naxiom possible_add {H\u2081 H\u2082 : cycle} : H\u2081.possible \u2192 H\u2082.possible \u2192 (H\u2081 + H\u2082).possible\n/-- The scaling a possible `cycle` with nonnegative is possible. -/\naxiom possible_smul_nonneg {c : \u211d} {H : cycle} : 0 \u2264 c \u2192 H.possible \u2192 (c \u2022 H).possible\n\nend cycle\n\n/-!\n### `one_rsv_cycle`\n-/\n\nnamespace one_rsv_cycle\nvariables {\ud835\udce3 : reservoir}\n\n/-- There exists possible `one_rsv_cycle \ud835\udce3` that `dissipate`s work into heat. -/\naxiom exists_possible_dissipate' : \u2203 H : one_rsv_cycle \ud835\udce3, H.possible \u2227 H.\ud835\udce6 < 0\n\n/- /-- There exists possible `one_rsv_cycle \ud835\udce3` that dissipates a certain work `W > 0`. -/\nlemma exists_psbl_dissi {W : \u211d} (hW : 0 < W) :\n  \u2203 H : one_rsv_cycle \ud835\udce3, H.possible \u2227 H.\ud835\udce6 = W := sorry -/\n\nend one_rsv_cycle\n\n/-!\n### `abs_rel_cycle`\n-/\n\nnamespace abs_rel_cycle\nvariables {\ud835\udcd7 \ud835\udcd2 : reservoir}\n\n/-- There exists possible `abs_rel_cycle` that\n    abs from hotter `\ud835\udcd7` and fully `transfer`s it to colder `\ud835\udcd2`. -/\naxiom exists_possible_transfer' (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7) :\n  \u2203 H : abs_rel_cycle \ud835\udcd7 \ud835\udcd2, H.possible \u2227 (H.Qabs:\u211d) = H.Qrel\n--\n\n/- /-- There exists possible `abs_rel_cycle \ud835\udcd7 \ud835\udcd2` that transfers a certain heat `Q > 0`. -/\nlemma exists_psbl_trans (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7) {Q : \u211d} (hQ : 0 < Q) :\n  \u2203 H : abs_rel_cycle \ud835\udcd7 \ud835\udcd2, H.possible \u2227 (H.Qabs:\u211d) = H.Qrel \u2227 \u2191H.Qabs = Q := sorry -/\n\nend abs_rel_cycle\n\n/-!\n### `usual_engine_cycle`\n-/\n\nnamespace usual_engine_cycle\nvariables {\ud835\udcd7 \ud835\udcd2 : reservoir} (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7)\n\n/-- There exists a possible `usual_engine_cycle` between any `\ud835\udcd2 < \ud835\udcd7`. -/\naxiom exists_possible : \u2203 H : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7, H.possible\n\nvariables {Q : \u211d} (hQ : 0 < Q)\n/-- There exists possible `usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7` that absorbs a certain heat `Q > 0`. -/\nlemma exists_psbl_abs : \u2203 H : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7, H.possible \u2227 \u2191H.Qabs = Q :=\n  let \u27e8H', hH'\u27e9 := exists_possible \ud835\udcd2_lt_\ud835\udcd7 in\n  let c : \u211d\u208a := \u27e8Q / H'.Qabs, div_pos hQ H'.do_abs\u27e9 in\n  let H := c \u2022 H' in\n  \u27e8 H,\n    cycle.possible_smul_nonneg (le_of_lt c.property) hH',\n    calc \u2191H.Qabs\n         = Q / H'.Qabs * H'.Qabs : cycle.Qabs_smul_pos c.property\n      ...= Q                     : div_mul_cancel _ (ne_of_gt H'.do_abs), \u27e9\n/-- There exists possible `usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7` that releases a certain heat `Q > 0`. -/\nlemma exists_psbl_rel : \u2203 H : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7, H.possible \u2227 \u2191H.Qrel = Q :=\n  let \u27e8H', hH'\u27e9 := exists_possible \ud835\udcd2_lt_\ud835\udcd7 in\n  let c : \u211d\u208a := \u27e8Q / H'.Qrel, div_pos hQ H'.do_rel\u27e9 in\n  let H := c \u2022 H' in\n  \u27e8 H,\n    cycle.possible_smul_nonneg (le_of_lt c.property) hH',\n    calc \u2191H.Qrel\n         = Q / H'.Qrel * H'.Qrel : cycle.Qrel_smul_pos c.property\n      ...= Q                     : div_mul_cancel _ (ne_of_gt H'.do_rel), \u27e9\n--\n\nend usual_engine_cycle\n\n/-!\n### `usual_pump_cycle`\n-/\n\nnamespace usual_pump_cycle\nvariables {\ud835\udcd2 \ud835\udcd7 : reservoir} (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7)\n\n/-- There exists a possible `usual_pump_cycle` between any `\ud835\udcd2 < \ud835\udcd7`. -/\naxiom exists_possible : \u2203 H : usual_pump_cycle \ud835\udcd2_lt_\ud835\udcd7, H.possible\n\nvariables {W : \u211d} (hW : 0 < W)\n/-- There exists possible `usual_pump_cycle \ud835\udcd2_lt_\ud835\udcd7` that consumes a certain work `W > 0`. -/\nlemma exists_psbl_consume : \u2203 H : usual_pump_cycle \ud835\udcd2_lt_\ud835\udcd7, H.possible \u2227 H.\ud835\udce6 = -W :=\n  let \u27e8H', hH'\u27e9 := exists_possible \ud835\udcd2_lt_\ud835\udcd7 in\n  let c : \u211d\u208a := \u27e8W /(-H'.\ud835\udce6), div_pos hW (neg_pos_of_neg H'.consume_work)\u27e9 in\n  let H := c \u2022 H' in\n  \u27e8 H,\n    cycle.possible_smul_nonneg (le_of_lt c.property) hH',\n    calc H.\ud835\udce6\n         = W /(-H'.\ud835\udce6) * H'.\ud835\udce6 : cycle.\ud835\udce6_smul\n      ...= (-W)/ H'.\ud835\udce6 * H'.\ud835\udce6 : by rw [div_neg, \u2190neg_div]\n      ...=  -W                 : div_mul_cancel _ (ne_of_lt H'.consume_work), \u27e9\n--\n\nend usual_pump_cycle\n\n/-!\n## `reversible` `cycle`s\n\n* Since an abstract `cycle` is an equivalence class, that it is reversible means\n  that there exists a reversible concrete cycle in these class, such as Carnot cycle.\n* Note that a concrete cycle in a reversible-`cycle` eqv class is not neccesarily reversible.\n---------------------------------------------------------------------------------------------------/\n\nnamespace cycle\nvariables (H : cycle)\n\ndef reversible  := H.possible \u2227 (-H).possible\n-- def psbl_nonrev := H.possible \u2227 \u00ac(-H).possible\n\nlemma trvial_reversible : (0:cycle).reversible :=\n  \u27e8trivial_possible, by{rw (@neg_zero cycle _), exact trivial_possible}\u27e9\nlemma reversible_add  {H\u2081 H\u2082 : cycle} : H\u2081.reversible \u2192 H\u2082.reversible \u2192 (H\u2081 + H\u2082).reversible :=\n  assume \u27e8h\u2081psbl, h\u2081rev\u27e9 \u27e8h\u2082psbl, h\u2082rev\u27e9, by {\n  split,\n  { exact possible_add h\u2081psbl h\u2082psbl, },\n  { have : -(H\u2081 + H\u2082) = -H\u2081 + -H\u2082, rw [neg_add_rev, add_comm], rw this,\n    exact possible_add h\u2081rev h\u2082rev, } }\nlemma reversible_smul_nonneg {c : \u211d} {H : cycle} : 0 \u2264 c \u2192 H.reversible \u2192 (c \u2022 H).reversible :=\n  assume hc \u27e8hpsbl, hrev\u27e9, by {\n  split,\n  { exact possible_smul_nonneg hc hpsbl, },\n  { have : -(c \u2022 H) = c \u2022 (-H), rw [smul_neg], rw this,\n    exact possible_smul_nonneg hc hrev, } }\n--\n\nend cycle\n\n/-!\n## Kelvin and Clausius statements\n---------------------------------------------------------------------------------------------------/\n\n/-- Kelvin-Plank statement : `one_rsv_cycle` cannot do work. -/\n@[reducible] def kelvin_stmt' :=\n  \u2200 \ud835\udce3, \u2200 H : one_rsv_cycle \ud835\udce3, 0 < H.\ud835\udce6 \u2192 \u00acH.possible\n/-- Clausius statement : Heat cannot be fully transfered from colder rsv `\ud835\udcd2` to hotter rsv `\ud835\udcd7`. -/\n@[reducible] def clausius_stmt' :=\n  \u2200 \ud835\udcd2 \ud835\udcd7, (\ud835\udcd2 < \ud835\udcd7) \u2192 \u2200 H : abs_rel_cycle \ud835\udcd2 \ud835\udcd7, (H.Qabs:\u211d) = H.Qrel \u2192 \u00acH.possible\n--\n\ntheorem kelvin_then_clausius : kelvin_stmt' \u2192 clausius_stmt' := by {\n  -- `K` < kelvin, `C` < clausius\n  assume hKelvin, apply @by_contra clausius_stmt',\n  simp only [not_forall, not_not, exists_prop, forall_exists_index, and_imp],\n  assume \ud835\udce3 \ud835\udcd7 \ud835\udce3_lt_\ud835\udcd7, assume CH hCH_Q hCH,\n  exact\n  let \u27e8eng, heng, hQ\u27e9 := usual_engine_cycle.exists_psbl_abs \ud835\udce3_lt_\ud835\udcd7 CH.do_rel in\n  let KH' := CH.to_cycle + eng.to_cycle in\n  let KH : one_rsv_cycle \ud835\udce3 :=\n    { one_rsv := by {\n        have : KH'.\ud835\udce0 = CH.\ud835\udce0 + eng.\ud835\udce0, from rfl, rw this,\n        apply finsupp.support_add_exact,\n        { simp only [finset.mem_singleton, ne.def, forall_eq],\n          calc CH.\ud835\udce0 \ud835\udce3 + eng.\ud835\udce0 \ud835\udce3\n               = CH.Qabs  - eng.Qrel : by simp only [CH.Qabs_one_rsv, eng.Qrel_one_rsv,\n                                                     sub_neg_eq_add]\n            ...= eng.Qabs - eng.Qrel : by simp only [hCH_Q, hQ]\n            ...= eng.\ud835\udce6              : by rw eng.\ud835\udce6_from_Qabs_Qrel\n            ...\u2260 0                   : ne_of_gt eng.do_work, },\n        { have : (CH.\ud835\udce0.support \u222a eng.\ud835\udce0.support) \\ {\ud835\udce3} = {\ud835\udcd7}, {\n            rw [CH.two_rsv.elim_left, eng.two_rsv.elim_left],\n            ext, simp, split,\n              tauto,\n              assume this, exact \u27e8or.inl this, ne_of_eq_of_ne this eng.two_rsv.elim_right\u27e9, },\n          rw this,\n          simp only [finset.mem_singleton, forall_eq],\n          calc CH.\ud835\udce0 \ud835\udcd7 + eng.\ud835\udce0 \ud835\udcd7\n               = -CH.Qrel + eng.Qabs : by simp only [CH.Qrel_one_rsv, eng.Qabs_one_rsv, neg_neg]\n            ...= 0                   : by simp only [hQ, add_left_neg], } },\n      ..KH' } in by {\n  have : KH.possible, from cycle.possible_add hCH heng,\n  refine absurd this (hKelvin _ KH _),\n  calc KH.\ud835\udce6\n       = CH.\ud835\udce6             + eng.\ud835\udce6 : cycle.\ud835\udce6_add\n    ...= CH.Qabs - CH.Qrel + eng.\ud835\udce6 : by rw CH.\ud835\udce6_from_Qabs_Qrel\n    ...=                     eng.\ud835\udce6 : by simp only [hCH_Q, sub_self, zero_add]\n    ...>                     0      : eng.do_work, } }\n-- #print axioms kelvin_then_clausius\n\n/-- `reservoir` does not have a minimal element,\n    which implies that the absolute zero cannot exist as a `reservoir`. -/\n@[reducible] def rsv_no_min' := \u2200 \ud835\udce3 : reservoir, \u2203 \ud835\udcd2, \ud835\udcd2 < \ud835\udce3\ntheorem clausius_then_kelvin : rsv_no_min' \u2192 clausius_stmt' \u2192 kelvin_stmt' := by {\n  -- `K` < kelvin, `C` < clausius\n  assume rsv_no_min,\n  assume hClausius, apply @by_contra kelvin_stmt',\n  simp only [not_forall, not_not, exists_prop, forall_exists_index, and_imp],\n  assume \ud835\udce3, assume KH hKH_\ud835\udce6 hKH,\n  exact\n  let \u27e8\ud835\udcd2, \ud835\udcd2_lt_\ud835\udce3\u27e9 := rsv_no_min \ud835\udce3 in\n  let \u27e8pump, hpump, hW\u27e9 := usual_pump_cycle.exists_psbl_consume \ud835\udcd2_lt_\ud835\udce3 hKH_\ud835\udce6 in\n  let CH' := KH.to_cycle + pump.to_cycle in\n  have KH_no_\ud835\udcd2 : KH.\ud835\udce0 \ud835\udcd2 = 0, from by {\n    have : \ud835\udcd2 \u2209 KH.\ud835\udce0.support, simp [KH.one_rsv, ne_of_lt \ud835\udcd2_lt_\ud835\udce3],\n    exact (finsupp.not_mem_support_iff).elim_left this, },\n  have do_abs_rsv :_ :=\n    calc CH'.\ud835\udce0 \ud835\udcd2\n         = KH.\ud835\udce0 \ud835\udcd2 + pump.\ud835\udce0 \ud835\udcd2 : rfl\n      ...= 0      + pump.Qabs : by rw [KH_no_\ud835\udcd2, pump.Qabs_one_rsv]\n      ...> 0                  : by simp [pump.do_abs],\n  have do_rel_rsv :_ :=\n    calc CH'.\ud835\udce0 \ud835\udce3\n         = KH.\ud835\udce0 \ud835\udce3 + pump.\ud835\udce0 \ud835\udce3   : rfl\n      ...= -pump.\ud835\udce6 - pump.Qrel : by simp [KH.\ud835\udce6_conv_one_rsv, hW, pump.Qrel_one_rsv]\n      ...= -pump.Qabs           : by simp [pump.\ud835\udce6_from_Qabs_Qrel]\n      ...< 0                    : by simp [pump.do_abs],\n  let CH : abs_rel_cycle \ud835\udcd2 \ud835\udce3 :=\n    { two_rsv := by {\n        refine \u27e8_, ne_of_lt \ud835\udcd2_lt_\ud835\udce3\u27e9,\n        have : CH'.\ud835\udce0 = KH.\ud835\udce0 + pump.\ud835\udce0, from rfl, rw this,\n        apply finsupp.support_add_exact,\n        { simp only [finset.mem_insert, finset.mem_singleton, ne.def, forall_eq_or_imp, forall_eq],\n          split,\n            exact ne_of_gt do_abs_rsv,\n            exact ne_of_lt do_rel_rsv, },\n        { have : (KH.\ud835\udce0.support \u222a pump.\ud835\udce0.support) \\ {\ud835\udcd2, \ud835\udce3} = \u2205, {\n            rw [KH.one_rsv, pump.two_rsv.elim_left],\n            ext, simp, tauto, },\n          rw this,\n          simp only [finset.not_mem_empty, is_empty.forall_iff, implies_true_iff], } },\n      do_abs_rsv := do_abs_rsv,\n      do_rel_rsv := do_rel_rsv,\n      ..CH' } in by {\n  have : CH.possible, from cycle.possible_add hKH hpump,\n  refine absurd this (hClausius _ _ \ud835\udcd2_lt_\ud835\udce3 CH _),\n  suffices : CH.\ud835\udce6 = 0, {\n    rw CH.\ud835\udce6_from_Qabs_Qrel at this,\n    exact sub_eq_zero.elim_left this, },\n  calc CH.\ud835\udce6\n       = KH.\ud835\udce6 + pump.\ud835\udce6 : cycle.\ud835\udce6_add\n    ...= 0               : by simp only [hW, add_right_neg], } }\n-- #print axioms clausius_then_kelvin\n\n/-!\n### Axioms of the second law\n-/\n\naxiom   kelvin_stmt   :\n  \u2200 {\ud835\udce3}, \u2200 H : one_rsv_cycle \ud835\udce3, 0 < H.\ud835\udce6 \u2192 \u00acH.possible\ntheorem clausius_stmt :\n  \u2200 {\ud835\udcd2 \ud835\udcd7}, (\ud835\udcd2 < \ud835\udcd7) \u2192 \u2200 H : abs_rel_cycle \ud835\udcd2 \ud835\udcd7, (H.Qabs:\u211d) = H.Qrel \u2192 \u00acH.possible :=\n  kelvin_then_clausius @kelvin_stmt\ncopy_doc_string kelvin_stmt'   \u2192 kelvin_stmt\ncopy_doc_string clausius_stmt' \u2192 clausius_stmt\n\n/-!\n## Carnot theorem\n---------------------------------------------------------------------------------------------------/\n\n/-- Carnot theorem : reversible `usual_engine_cycle` has the greatest efficiency. -/\n@[reducible] def carnot_thm' :=\n  \u2200 \ud835\udcd2 \ud835\udcd7 (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7), \u2200 H C : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7,\n  H.possible \u2192 C.reversible \u2192 H.eff \u2264 C.eff\ntheorem clausius_then_carnot : clausius_stmt' \u2192 carnot_thm' := by {\n  refine forall\u2083_imp _, assume \ud835\udcd2 \ud835\udcd7 \ud835\udcd2_lt_\ud835\udcd7,\n  assume hClausius,\n  assume H C hH hC,\n  by_contradiction heff, rw not_le at heff,\n  let c : \u211d\u208a :=\n    { val := C.\ud835\udce6 / H.\ud835\udce6,\n      property := div_pos C.do_work H.do_work },\n  haveI : fact _ := \u27e8c.property\u27e9,\n  -- H'\n  let H' := c \u2022 H,\n  have H'\ud835\udce6eqC :=\n    calc H'.\ud835\udce6\n         = C.\ud835\udce6 / H.\ud835\udce6 * H.\ud835\udce6 : cycle.\ud835\udce6_smul\n      ...= C.\ud835\udce6               : div_mul_cancel _ (ne_of_gt H.do_work),\n  have this : C.eff / H.eff < 1, from (div_lt_one H.eff_pos).elim_right heff,\n  have H'Qabs_ltC :=\n    calc (H'.Qabs:\u211d)\n         = H'.\ud835\udce6 / (c.val \u2022 H.to_cycle).eff : H'.Qabs_from_\ud835\udce6_eff\n      ...= C.\ud835\udce6 / H.eff                     : by rw [H'\ud835\udce6eqC, cycle.eff_smul_pos]\n      ...= C.eff / H.eff * C.Qabs           : by rw [C.\ud835\udce6_from_eff_Qabs, mul_div_right_comm]\n      ...< C.Qabs                           : (mul_lt_iff_lt_one_left C.do_abs).elim_right this,\n  have H'Qrel_gtC :=\n    calc (H'.Qrel:\u211d)\n         = H'.Qabs - H'.\ud835\udce6 : H'.Qrel_from_Qabs_\ud835\udce6\n      ...< C.Qabs  - C.\ud835\udce6  : by simp only [H'Qabs_ltC, H'\ud835\udce6eqC, sub_lt_sub_iff_right]\n      ...= C.Qrel          : C.Qrel_from_Qabs_\ud835\udce6.symm,\n  -- ClH', `Cl` < clausius\n  let ClH' := H'.to_cycle + C.rev.to_cycle,\n  have rel\ud835\udcd7 :=\n    calc ClH'.\ud835\udce0 \ud835\udcd7\n         = H'.\ud835\udce0 \ud835\udcd7 - C.\ud835\udce0 \ud835\udcd7 : rfl\n      ...= H'.Qabs - C.Qabs : by rw [H'.Qabs_one_rsv, C.Qabs_one_rsv]\n      ...< 0                : sub_neg.elim_right H'Qabs_ltC,\n  have abs\ud835\udcd2 :=\n    calc ClH'.\ud835\udce0 \ud835\udcd2\n         = H'.\ud835\udce0 \ud835\udcd2 - C.\ud835\udce0 \ud835\udcd2  : rfl\n      ...= C.Qrel - H'.Qrel : by { rw [H'.Qrel_one_rsv, C.Qrel_one_rsv], ring }\n      ...> 0                : sub_pos.elim_right H'Qrel_gtC,\n  let ClH : abs_rel_cycle \ud835\udcd2 \ud835\udcd7 :=\n    { two_rsv := by {\n        refine \u27e8_, ne_of_lt \ud835\udcd2_lt_\ud835\udcd7\u27e9,\n        have : ClH'.\ud835\udce0 = H'.\ud835\udce0 + C.rev.\ud835\udce0, from rfl, rw this,\n        apply finsupp.support_add_exact,\n        { simp only [finset.mem_insert, finset.mem_singleton, ne.def, forall_eq_or_imp, forall_eq],\n          split,\n            exact ne_of_gt abs\ud835\udcd2,\n            exact ne_of_lt rel\ud835\udcd7, },\n        { have : (H'.\ud835\udce0.support \u222a C.rev.\ud835\udce0.support) \\ {\ud835\udcd2, \ud835\udcd7} = \u2205, {\n            rw [H'.two_rsv.elim_left, C.rev.two_rsv.elim_left],\n            ext, simp, },\n          simp only [this, finset.not_mem_empty, is_empty.forall_iff, implies_true_iff], } },\n      do_abs_rsv := abs\ud835\udcd2,\n      do_rel_rsv := rel\ud835\udcd7,\n      ..ClH' },\n  have this :=\n    calc ClH.Qabs.val\n         = ClH.\ud835\udce6                           + ClH.Qrel : ClH.Qabs_from_\ud835\udce6_Qrel\n      ...= H'.\ud835\udce6 + (        -C.to_cycle).\ud835\udce6 + ClH.Qrel : by { rw cycle.\ud835\udce6_add, refl }\n      ...=  C.\ud835\udce6 + ((-1:\u211d) \u2022 C.to_cycle).\ud835\udce6 + ClH.Qrel : by rw [H'\ud835\udce6eqC, neg_one_smul]\n      ...=                                    ClH.Qrel : by { rw cycle.\ud835\udce6_smul, ring },\n  refine absurd _ (hClausius ClH this),\n  exact cycle.possible_add\n    (cycle.possible_smul_nonneg (le_of_lt c.property) hH)\n    (hC.elim_right), }\n-- #print axioms clausius_then_carnot\n\n/-- There exists a reversible `usual_engine_cycle` between any `\ud835\udcd2 < \ud835\udcd7`. -/\n@[reducible] def usual_engine_cycle.exists_reversible' :=\n  \u2200 \ud835\udcd2 \ud835\udcd7 (\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7), \u2203 H : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7, H.reversible\ntheorem carnot_then_clausius : usual_engine_cycle.exists_reversible' \u2192\n  carnot_thm' \u2192 clausius_stmt' := by {\n  assume hexists_rev,\n  refine forall\u2083_imp _, assume \ud835\udcd2 \ud835\udcd7 \ud835\udcd2_lt_\ud835\udcd7,\n  assume hCarnot,\n  assume ClH hQ, -- `Cl` < clausius\n  by_contradiction hClH,\n  exact\n  let \u27e8C, hC\u27e9 := hexists_rev _ _ \ud835\udcd2_lt_\ud835\udcd7 in by {\n  let c : \u211d\u208a :=\n    { val := C.Qrel / 2 / ClH.Qabs,\n      property := div_pos\n        (div_pos C.do_rel zero_lt_two)\n        (ClH.do_abs), },\n  -- ClH'\n  let ClH' := c \u2022 ClH,\n  have ClH'no\ud835\udce6 :=\n    calc ClH'.\ud835\udce6\n         = (\u2191c \u2022 ClH.to_cycle).\ud835\udce6 : rfl\n      ...= 0 : by{rw [cycle.\ud835\udce6_smul, ClH.\ud835\udce6_from_Qabs_Qrel, hQ], ring},\n  have ClH'abs_ltCrel :=\n    calc (ClH'.Qabs:\u211d)\n         =                         ClH'.\ud835\udce0 \ud835\udcd2 : ClH'.Qabs_one_rsv\n      ...= C.Qrel / 2 / ClH.Qabs *  ClH.\ud835\udce0 \ud835\udcd2 : rfl\n      ...= C.Qrel / 2 : by rw [\u2190ClH.Qabs_one_rsv, div_mul_cancel _ (ne_of_gt ClH.do_abs)]\n      ...< C.Qrel     : div_lt_self C.do_rel one_lt_two,\n  have hQ' :=\n    calc (ClH'.Qabs:\u211d)\n         = c * ClH.Qabs : cycle.Qabs_smul_pos c.property\n      ...= c * ClH.Qrel : by rw hQ\n      ...= ClH'.Qrel    : eq.symm $ cycle.Qrel_smul_pos c.property,\n  -- CaH', `Ca` < carnot\n  let CaH' := ClH'.to_cycle + C.to_cycle,\n  have CaH'\ud835\udce6eqC : CaH'.\ud835\udce6 = C.\ud835\udce6, rw [cycle.\ud835\udce6_add, ClH'no\ud835\udce6, zero_add],\n  have rel\ud835\udcd2 :=\n    calc CaH'.\ud835\udce0 \ud835\udcd2\n         = ClH'.\ud835\udce0 \ud835\udcd2  + C.\ud835\udce0 \ud835\udcd2 : rfl\n      ...= ClH'.Qabs - C.Qrel : by rw [ClH'.Qabs_one_rsv, C.Qrel_one_rsv, sub_neg_eq_add]\n      ...< 0                  : sub_neg.elim_right ClH'abs_ltCrel,\n  have abs\ud835\udcd7 :=\n    calc CaH'.\ud835\udce0 \ud835\udcd7\n         = ClH'.\ud835\udce0 \ud835\udcd7 + C.\ud835\udce0 \ud835\udcd7  : rfl\n      ...= -ClH'.Qrel + C.Qabs : by rw [ClH'.Qrel_one_rsv, neg_neg, C.Qabs_one_rsv]\n      ...= C.\ud835\udce6 + (C.Qrel - ClH'.Qabs) : by{rw [hQ', C.Qabs_from_\ud835\udce6_Qrel], ring}\n      ...>         C.Qrel - ClH'.Qabs  : (lt_add_iff_pos_left _).elim_right C.do_work\n      ...> 0                           : sub_pos.elim_right ClH'abs_ltCrel,\n  let CaH : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7 :=\n    { two_rsv := by {\n        refine \u27e8_, ne_of_gt \ud835\udcd2_lt_\ud835\udcd7\u27e9,\n        have : CaH'.\ud835\udce0 = ClH'.\ud835\udce0 + C.\ud835\udce0, from rfl, rw this,\n        apply finsupp.support_add_exact,\n        { simp only [finset.mem_insert, finset.mem_singleton, ne.def, forall_eq_or_imp, forall_eq],\n          split,\n            exact ne_of_gt abs\ud835\udcd7,\n            exact ne_of_lt rel\ud835\udcd2, },\n        { have : (ClH'.\ud835\udce0.support \u222a C.\ud835\udce0.support) \\ {\ud835\udcd7, \ud835\udcd2} = \u2205, {\n            rw [ClH'.two_rsv.elim_left, C.two_rsv.elim_left],\n            ext, simp, },\n          simp only [this, finset.not_mem_empty, is_empty.forall_iff, implies_true_iff], } },\n      do_abs_rsv := abs\ud835\udcd7,\n      do_rel_rsv := rel\ud835\udcd2,\n      do_work :=\n        calc CaH'.\ud835\udce6\n             = C.\ud835\udce6 : CaH'\ud835\udce6eqC\n          ...> 0    : C.do_work,\n      ..CaH' },\n  have :=\n    calc (CaH.Qabs:\u211d)\n         = ClH'.\ud835\udce0 \ud835\udcd7 + C.\ud835\udce0 \ud835\udcd7 : by { rw CaH.Qabs_one_rsv, refl }\n      ...= C.Qabs - ClH'.Qrel : by { rw [C.Qabs_one_rsv, ClH'.Qrel_one_rsv], ring }\n      ...< C.Qabs             : sub_lt_self _ ClH'.do_rel,\n  have :=\n    calc CaH.eff\n         = CaH.\ud835\udce6 / CaH.Qabs : rfl\n      ...> C.\ud835\udce6 / C.Qabs : by { rw CaH'\ud835\udce6eqC, exact div_lt_div_of_lt_left C.do_work CaH.do_abs this}\n      ...= C.eff         : rfl,\n  refine absurd (hCarnot CaH C _ hC) (not_le_of_gt this),\n  exact cycle.possible_add\n    (cycle.possible_smul_nonneg (le_of_lt c.property) hClH)\n    (hC.elim_left), } }\n-- #print axioms carnot_then_clausius\n\nsection carnot_thm\nvariables {\ud835\udcd2 \ud835\udcd7 : reservoir} {\ud835\udcd2_lt_\ud835\udcd7 : \ud835\udcd2 < \ud835\udcd7} {H C : usual_engine_cycle \ud835\udcd2_lt_\ud835\udcd7}\n\ntheorem carnot_thm : H.possible \u2192 C.reversible \u2192 H.eff \u2264 C.eff :=\n  (clausius_then_carnot @clausius_stmt) _ _ \ud835\udcd2_lt_\ud835\udcd7 H C\ncopy_doc_string carnot_thm' \u2192 carnot_thm\n/-- An `usual_engine_cycle` is reversible iff it has the greatest efficiency. -/\ntheorem usual_engine_cycle.rev_iff_eff (hC : C.reversible) : H.reversible \u2194 H.eff = C.eff := by {\n  split,\n  { assume hH,\n    have H_le_C := carnot_thm hH.elim_left hC,\n    have C_le_H := carnot_thm hC.elim_left hH,\n    exact eq_of_le_of_not_lt H_le_C (not_lt_of_ge C_le_H), },\n  { assume heff,\n    exact let \u27e8c, hc\u27e9 := usual_engine_cycle.eq_smul_pos_of_eff_eq heff in by {\n      rw hc,\n      exact cycle.reversible_smul_nonneg (le_of_lt c.property) hC, } } }\n--\n\nend carnot_thm\n\nend thermodynamics\n", "meta": {"author": "Youjack", "repo": "thermodynamics.lean", "sha": "4af0748a97e6cb89aef0c87425872d1a901e8c55", "save_path": "github-repos/lean/Youjack-thermodynamics.lean", "path": "github-repos/lean/Youjack-thermodynamics.lean/thermodynamics.lean-4af0748a97e6cb89aef0c87425872d1a901e8c55/src/second_law.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.3988603820631223}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport category_theory.equivalence\n\n/-! Tools for compatibilities between Dold-Kan equivalences\n\nThe purpose of this file is to introduce tools which will enable the\nconstruction of the Dold-Kan equivalence `simplicial_object C \u224c chain_complex C \u2115`\nfor a pseudoabelian category `C` from the equivalence\n`karoubi (simplicial_object C) \u224c karoubi (chain_complex C \u2115)` and the two\nequivalences `simplicial_object C \u2245 karoubi (simplicial_object C)` and\n`chain_complex C \u2115 \u2245 karoubi (chain_complex C \u2115)`.\n\nIt is certainly possible to get an equivalence `simplicial_object C \u224c chain_complex C \u2115`\nusing a compositions of the three equivalences above, but then neither the functor\nnor the inverse would have good definitional properties. For example, it would be better\nif the inverse functor of the equivalence was the functor\n`\u0393\u2080 : simplicial_object C \u2964 chain_complex C \u2115` which was constructed in `functor_gamma.lean`.\n\nIn this file, given four categories `A`, `A'`, `B`, `B'`, equivalences `eA : A \u2245 A'`,\n`eB : B \u2245 B'`, `e' : A' \u2245 B'`, functors `F : A \u2964 B'`, `G : B \u2964 A` equipped with certain\ncompatibilities, we construct successive equivalences:\n- `equivalence\u2080` from `A` to `B'`, which is the composition of `eA` and `e'`.\n- `equivalence\u2081` from `A` to `B'`, with the same inverse functor as `equivalence\u2080`,\nbut whose functor is `F`.\n- `equivalence\u2082` from `A` to `B`, which is the composition of `equivalence\u2081` and the\ninverse of `eB`:\n- `equivalence` from `A` to `B`, which has the same functor `F \u22d9 eB.inverse` as `equivalence\u2082`,\nbut whose inverse functor is `G`.\n\nWhen extra assumptions are given, we also provide simplification lemmas for the\nunit and counit isomorphisms of `equivalence`.\n\n-/\n\nopen category_theory category_theory.category\n\nnamespace algebraic_topology\n\nnamespace dold_kan\n\nnamespace compatibility\n\nvariables {A A' B B' : Type*} [category A] [category A'] [category B] [category B']\n  (eA : A \u224c A') (eB : B \u224c B') (e' : A' \u224c B')\n  {F : A \u2964 B'} (hF : eA.functor \u22d9 e'.functor \u2245 F)\n  {G : B \u2964 A} (hG : eB.functor \u22d9 e'.inverse \u2245 G \u22d9 eA.functor)\n\n/-- A basic equivalence `A \u2245 B'` obtained by composing `eA : A \u2245 A'` and `e' : A' \u2245 B'`. -/\ndef equivalence\u2080 : A \u224c B' := eA.trans e'\n\nlemma equivalence\u2080_functor : (equivalence\u2080 eA e').functor = eA.functor \u22d9 e'.functor := rfl\nlemma equivalence\u2080_inverse : (equivalence\u2080 eA e').inverse = e'.inverse \u22d9 eA.inverse := rfl\n\ninclude hF\nvariables {eA} {e'}\n\n/-- An intermediate equivalence `A \u2245 B'` whose functor is `F` and whose inverse is\n`e'.inverse \u22d9 eA.inverse`. -/\ndef equivalence\u2081 : A \u224c B' :=\nbegin\n  letI : is_equivalence F :=\n    is_equivalence.of_iso hF (is_equivalence.of_equivalence (equivalence\u2080 eA e')),\n  exact F.as_equivalence,\nend\n\nlemma equivalence\u2081_functor : (equivalence\u2081 hF).functor = F := rfl\nlemma equivalence\u2081_inverse : (equivalence\u2081 hF).inverse = e'.inverse \u22d9 eA.inverse := rfl\n\n/-- The counit isomorphism of the equivalence `equivalence\u2081` between `A` and `B'`. -/\n@[simps]\ndef equivalence\u2081_counit_iso :\n  (e'.inverse \u22d9 eA.inverse) \u22d9 F \u2245 \ud835\udfed B' :=\ncalc (e'.inverse \u22d9 eA.inverse) \u22d9 F\n  \u2245 (e'.inverse \u22d9 eA.inverse) \u22d9 (eA.functor \u22d9 e'.functor) : iso_whisker_left _ hF.symm\n... \u2245 e'.inverse \u22d9 (eA.inverse \u22d9 eA.functor) \u22d9 e'.functor : iso.refl _\n... \u2245 e'.inverse \u22d9 \ud835\udfed _ \u22d9 e'.functor : iso_whisker_left _ (iso_whisker_right eA.counit_iso _)\n... \u2245 e'.inverse \u22d9 e'.functor : iso.refl _\n... \u2245 \ud835\udfed B' : e'.counit_iso\n\nlemma equivalence\u2081_counit_iso_eq : (equivalence\u2081 hF).counit_iso = equivalence\u2081_counit_iso hF :=\nbegin\n  ext Y',\n  dsimp [equivalence\u2080, equivalence\u2081, equivalence\u2081_counit_iso, nat_iso.hcomp,\n    is_equivalence.inverse, is_equivalence.of_equivalence],\n  simp only [category_theory.functor.map_id, comp_id, assoc],\nend\n\n/-- The unit isomorphism of the equivalence `equivalence\u2081` between `A` and `B'`. -/\n@[simps]\ndef equivalence\u2081_unit_iso :\n  \ud835\udfed A \u2245 F \u22d9 (e'.inverse \u22d9 eA.inverse) :=\ncalc \ud835\udfed A \u2245 eA.functor \u22d9 eA.inverse : eA.unit_iso\n... \u2245 eA.functor \u22d9 \ud835\udfed A' \u22d9 eA.inverse : iso.refl _\n... \u2245 eA.functor \u22d9 (e'.functor \u22d9 e'.inverse) \u22d9 eA.inverse :\n  iso_whisker_left _ (iso_whisker_right e'.unit_iso _)\n... \u2245 (eA.functor \u22d9 e'.functor) \u22d9 (e'.inverse \u22d9 eA.inverse) : iso.refl _\n... \u2245 F \u22d9 (e'.inverse \u22d9 eA.inverse) : iso_whisker_right hF _\n\nlemma equivalence\u2081_unit_iso_eq : (equivalence\u2081 hF).unit_iso = equivalence\u2081_unit_iso hF :=\nbegin\n  ext X,\n  dsimp [equivalence\u2080, equivalence\u2081, equivalence\u2081_unit_iso, nat_iso.hcomp,\n    is_equivalence.of_equivalence],\n  simp only [id_comp, comp_id],\nend\n\ninclude eB\n\n/-- An intermediate equivalence `A \u2245 B` obtained as the composition of `equivalence\u2081` and\nthe inverse of `eB : B \u224c B'`. -/\ndef equivalence\u2082 : A \u224c B := (equivalence\u2081 hF).trans eB.symm\n\nlemma equivalence\u2082_functor : (equivalence\u2082 eB hF).functor = F \u22d9 eB.inverse := rfl\nlemma equivalence\u2082_inverse : (equivalence\u2082 eB hF).inverse =\n  eB.functor \u22d9 e'.inverse \u22d9 eA.inverse := rfl\n\n/-- The counit isomorphism of the equivalence `equivalence\u2082` between `A` and `B`. -/\n@[simps]\ndef equivalence\u2082_counit_iso :\n  (eB.functor \u22d9 e'.inverse \u22d9 eA.inverse) \u22d9 (F \u22d9 eB.inverse) \u2245 \ud835\udfed B :=\nbegin\n  calc (eB.functor \u22d9 e'.inverse \u22d9 eA.inverse) \u22d9 (F \u22d9 eB.inverse)\n    \u2245 eB.functor \u22d9 (e'.inverse \u22d9 eA.inverse \u22d9 F) \u22d9 eB.inverse : by refl\n  ... \u2245 eB.functor \u22d9 \ud835\udfed _ \u22d9 eB.inverse :\n    iso_whisker_left _ (iso_whisker_right (equivalence\u2081_counit_iso hF) _)\n  ... \u2245 eB.functor \u22d9 eB.inverse : by refl\n  ... \u2245 \ud835\udfed B : eB.unit_iso.symm,\nend\n\nlemma equivalence\u2082_counit_iso_eq :\n  (equivalence\u2082 eB hF).counit_iso = equivalence\u2082_counit_iso eB hF :=\nbegin\n  ext Y',\n  dsimp [equivalence\u2082, equivalence\u2082_counit_iso, iso.refl],\n  erw [equivalence\u2081_counit_iso_eq, nat_trans.id_app, id_comp, comp_id],\nend\n\n/-- The unit isomorphism of the equivalence `equivalence\u2082` between `A` and `B`. -/\n@[simps]\ndef equivalence\u2082_unit_iso :\n  \ud835\udfed A \u2245 (F \u22d9 eB.inverse) \u22d9 (eB.functor \u22d9 e'.inverse \u22d9 eA.inverse) :=\ncalc \ud835\udfed A \u2245 F \u22d9 e'.inverse \u22d9 eA.inverse : equivalence\u2081_unit_iso hF\n... \u2245 F \u22d9 \ud835\udfed B' \u22d9 (e'.inverse \u22d9 eA.inverse) : iso.refl _\n... \u2245 F \u22d9 (eB.inverse \u22d9 eB.functor) \u22d9 e'.inverse \u22d9 eA.inverse :\n  iso_whisker_left _ (iso_whisker_right eB.counit_iso.symm _)\n... \u2245 (F \u22d9 eB.inverse) \u22d9 (eB.functor \u22d9 e'.inverse \u22d9 eA.inverse) : iso.refl _\n\nlemma equivalence\u2082_unit_iso_eq :\n  (equivalence\u2082 eB hF).unit_iso = equivalence\u2082_unit_iso eB hF :=\nbegin\n  ext X,\n  dsimp [equivalence\u2082, equivalence\u2082_unit_iso],\n  erw equivalence\u2081_unit_iso_eq,\n  simpa only [assoc, comp_id, nat_iso.cancel_nat_iso_hom_left],\nend\n\nvariable {eB}\ninclude hG\n\n/-- The equivalence `A \u2245 B` whose functor is `F \u22d9 eB.inverse` and\nwhose inverse is `G : B \u2245 A`. -/\ndef equivalence : A \u224c B :=\nbegin\n  letI : is_equivalence G := begin\n    refine is_equivalence.of_iso _ (is_equivalence.of_equivalence (equivalence\u2082 eB hF).symm),\n    calc eB.functor \u22d9 e'.inverse \u22d9 eA.inverse\n      \u2245 (eB.functor \u22d9 e'.inverse) \u22d9 eA.inverse : iso.refl _\n    ... \u2245 (G \u22d9 eA.functor) \u22d9 eA.inverse : iso_whisker_right hG _\n    ... \u2245 G \u22d9 \ud835\udfed A : iso_whisker_left _ eA.unit_iso.symm\n    ... \u2245 G : functor.right_unitor G,\n  end,\n  exact G.as_equivalence.symm,\nend\n\nlemma equivalence_functor : (equivalence hF hG).functor = F \u22d9 eB.inverse := rfl\nlemma equivalence_inverse : (equivalence hF hG).inverse = G := rfl\n\nomit hG hF\n\n/-- The isomorphism `eB.functor \u22d9 e'.inverse \u22d9 e'.functor \u2245 eB.functor` deduced\nfrom the counit isomorphism of `e'`. -/\n@[simps hom_app]\ndef \u03c4\u2080 : eB.functor \u22d9 e'.inverse \u22d9 e'.functor \u2245 eB.functor :=\ncalc eB.functor \u22d9 e'.inverse \u22d9 e'.functor\n  \u2245 eB.functor \u22d9 \ud835\udfed _ : iso_whisker_left _ e'.counit_iso\n... \u2245 eB.functor : functor.right_unitor _\n\ninclude hF hG\n\n/-- The isomorphism `eB.functor \u22d9 e'.inverse \u22d9 e'.functor \u2245 eB.functor` deduced\nthe isomorphisms `hF : eA.functor \u22d9 e'.functor \u2245 F`,\n`hG : eB.functor \u22d9 e'.inverse \u2245 G \u22d9 eA.functor` and the datum of\nan isomorphism `\u03b7 : G \u22d9 F \u2245 eB.functor`. -/\n@[simps hom_app]\ndef \u03c4\u2081 (\u03b7 : G \u22d9 F \u2245 eB.functor) :\n  eB.functor \u22d9 e'.inverse \u22d9 e'.functor \u2245 eB.functor :=\ncalc eB.functor \u22d9 e'.inverse \u22d9 e'.functor\n  \u2245 (eB.functor \u22d9 e'.inverse) \u22d9 e'.functor : iso.refl _\n... \u2245 (G \u22d9 eA.functor) \u22d9 e'.functor : iso_whisker_right hG _\n... \u2245 G \u22d9 (eA.functor \u22d9 e'.functor) : by refl\n... \u2245 G \u22d9 F : iso_whisker_left _ hF\n... \u2245 eB.functor : \u03b7\n\nvariables (\u03b7 : G \u22d9 F \u2245 eB.functor) (h\u03b7 : \u03c4\u2080 = \u03c4\u2081 hF hG \u03b7)\n\nomit hF hG\ninclude \u03b7\n\n/-- The counit isomorphism of `equivalence`. -/\n@[simps]\ndef equivalence_counit_iso : G \u22d9 (F \u22d9 eB.inverse) \u2245 \ud835\udfed B :=\ncalc G \u22d9 (F \u22d9 eB.inverse) \u2245 (G \u22d9 F) \u22d9 eB.inverse : iso.refl _\n... \u2245 eB.functor \u22d9 eB.inverse : iso_whisker_right \u03b7 _\n... \u2245 \ud835\udfed B : eB.unit_iso.symm\n\nvariables {\u03b7 hF hG}\ninclude h\u03b7\n\nlemma equivalence_counit_iso_eq :\n  (equivalence hF hG).counit_iso = equivalence_counit_iso \u03b7 :=\nbegin\n  ext1, apply nat_trans.ext, ext Y,\n  dsimp [equivalence, equivalence_counit_iso, is_equivalence.of_equivalence],\n  simp only [equivalence\u2082_counit_iso_eq eB hF],\n  erw [nat_trans.id_app, nat_trans.id_app],\n  dsimp [equivalence\u2082, equivalence\u2081],\n  simp only [assoc, comp_id, F.map_id, id_comp,\n    equivalence\u2082_counit_iso_hom_app, \u2190 eB.inverse.map_comp_assoc,\n    \u2190 \u03c4\u2080_hom_app, h\u03b7, \u03c4\u2081_hom_app],\n  erw hF.inv.naturality_assoc,\n  congr' 2,\n  dsimp,\n  simp only [assoc, \u2190 e'.functor.map_comp_assoc, eA.functor.map_comp,\n    equivalence.fun_inv_map, iso.inv_hom_id_app_assoc, hG.inv_hom_id_app],\n  dsimp,\n  rw [comp_id, eA.functor_unit_iso_comp, e'.functor.map_id, id_comp, hF.inv_hom_id_app_assoc],\nend\n\nomit h\u03b7 \u03b7 eB\ninclude hF\n\nvariable (hF)\n\n/-- The isomorphism `eA.functor \u2245 F \u22d9 e'.inverse` deduced from the\nunit isomorphism of `e'` and the isomorphism `hF : eA.functor \u22d9 e'.functor \u2245 F`. -/\n@[simps]\ndef \u03c5 : eA.functor \u2245 F \u22d9 e'.inverse :=\ncalc eA.functor \u2245 eA.functor \u22d9 \ud835\udfed A' : (functor.left_unitor _).symm\n... \u2245 eA.functor \u22d9 (e'.functor \u22d9 e'.inverse) : iso_whisker_left _ e'.unit_iso\n... \u2245 (eA.functor \u22d9 e'.functor) \u22d9 e'.inverse : iso.refl _\n... \u2245 F \u22d9 e'.inverse : iso_whisker_right hF _\n\nvariables (\u03b5 : eA.functor \u2245 F \u22d9 e'.inverse) (h\u03b5 : \u03c5 hF = \u03b5)\n\ninclude \u03b5 hG\nomit hF\n\nvariable (hG)\n\n/-- The unit isomorphism of `equivalence`. -/\n@[simps]\ndef equivalence_unit_iso : \ud835\udfed A \u2245 (F \u22d9 eB.inverse) \u22d9 G :=\ncalc \ud835\udfed A \u2245 eA.functor \u22d9 eA.inverse : eA.unit_iso\n... \u2245 (F \u22d9 e'.inverse) \u22d9 eA.inverse : iso_whisker_right \u03b5 _\n... \u2245 F \u22d9 \ud835\udfed B' \u22d9 e'.inverse \u22d9 eA.inverse : iso.refl _\n... \u2245 F \u22d9 (eB.inverse \u22d9 eB.functor) \u22d9 (e'.inverse \u22d9 eA.inverse) :\n      iso_whisker_left _ (iso_whisker_right eB.counit_iso.symm _)\n... \u2245 (F \u22d9 eB.inverse) \u22d9 (eB.functor \u22d9 e'.inverse) \u22d9 eA.inverse : iso.refl _\n... \u2245 (F \u22d9 eB.inverse) \u22d9 (G \u22d9 eA.functor) \u22d9 eA.inverse :\n      iso_whisker_left _ (iso_whisker_right hG _)\n... \u2245 (F \u22d9 eB.inverse \u22d9 G) \u22d9 (eA.functor \u22d9 eA.inverse) : iso.refl _\n... \u2245 (F \u22d9 eB.inverse \u22d9 G) \u22d9 \ud835\udfed A : iso_whisker_left _ eA.unit_iso.symm\n... \u2245 (F \u22d9 eB.inverse) \u22d9 G : iso.refl _\n\ninclude h\u03b5\nvariables {\u03b5 hF hG}\n\nlemma equivalence_unit_iso_eq :\n  (equivalence hF hG).unit_iso = equivalence_unit_iso hG \u03b5 :=\nbegin\n  ext1, apply nat_trans.ext, ext X,\n  dsimp [equivalence, iso.refl, nat_iso.hcomp, is_equivalence.inverse,\n    is_equivalence.of_equivalence],\n  erw [nat_trans.id_app, id_comp, G.map_id, comp_id, comp_id],\n  simp only [equivalence\u2082_unit_iso_eq eB hF, equivalence\u2082_unit_iso_hom_app],\n  dsimp [equivalence\u2082, equivalence\u2081],\n  simp only [assoc, equivalence_unit_iso_hom_app, nat_iso.cancel_nat_iso_hom_left,\n    \u2190 eA.inverse.map_comp_assoc, \u2190 h\u03b5, \u03c5_hom_app],\nend\n\nend compatibility\n\nend dold_kan\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/dold_kan/compatibility.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.3988603820631223}}
{"text": "import Quantifiers.matching\n\nuniverse u\n\nopen tactic\n\nexample {\u03b1 : Type u} (a : \u03b1) : \u03b1 :=\nby do (dir, goals) \u2190 create_directory,\n  mi \u2190 create_match_info dir goals,\n  trace_match_info mi, -- the important line\n\n  ta \u2190 get_local `a,\n  exact ta\n\nexample {\u03b1 : Type u} (P : \u03b1 \u2192 Prop) (Q : \u03b1 \u2192 Prop) (h\u2081 : \u2203 x, P x)\n(h\u2082 : \u2200 x, P x \u2192 Q x) : \u2203 x, Q x :=\nby do (dir, goals) \u2190 create_directory,\n  mi \u2190 create_match_info dir goals,\n  trace_match_info mi, -- the important line\n\n  t\u2081 \u2190 get_local `h\u2081,\n  nmx \u2190 get_unused_name `x,\n  nmh \u2190 get_unused_name `h,\n  cases t\u2081 [nmx, nmh],\n  x \u2190 get_local nmx,\n  h \u2190 get_local nmh,\n  applyc ``exists.intro,\n  t\u2082 \u2190 get_local `h\u2082,\n  apply t\u2082,\n  exact h\n", "meta": {"author": "MateiFMandache", "repo": "ATP", "sha": "c16c56a3e1a7f2dc92ee1fef1928a16e74b97c4c", "save_path": "github-repos/lean/MateiFMandache-ATP", "path": "github-repos/lean/MateiFMandache-ATP/ATP-c16c56a3e1a7f2dc92ee1fef1928a16e74b97c4c/src/Quantifiers/matching_test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.39876976574757134}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.Init.SetNotation\nimport Mathlib.Init.Logic\nimport Mathlib.Tactic.Lint\n\nopen Decidable List\n\nuniverse u v w\n\ninstance (\u03b1 : Type u) : Inhabited (List \u03b1) :=\n  \u27e8List.nil\u27e9\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n\nnamespace List\n\nattribute [simp] get! get? head? headD head tail! tail? tailD getLast! getLast?\n  getLastD reverseAux eraseIdx isEmpty map map\u2082 join filterMap dropWhile find? findSome?\n  replace elem lookup drop take takeWhile foldr zipWith unzip rangeAux enumFrom init\n  intersperse isPrefixOf isEqv dropLast\n\n@[simp] lemma get_cons_zero {as : List \u03b1} : (a :: as).get \u27e80, id <| Nat.zero_lt_succ _\u27e9 = a := rfl\n@[simp] lemma get_cons_succ {as : List \u03b1} {h : i + 1 < (a :: as).length} :\n  (a :: as).get \u27e8i+1, h\u27e9 = as.get \u27e8i, Nat.lt_of_succ_lt_succ h\u27e9 := rfl\n\n@[simp] lemma getLast_singleton {x : \u03b1} : [x].getLast (id (by simp)) = x := rfl\n@[simp] lemma getLast_cons_cons {x : \u03b1} : (x::y::ys).getLast (id (by simp)) = (y::ys).getLast (by simp) := rfl\n\n-- The `id <|` above is a workaround to allow Lean to unify get_cons_zero's lhs\n-- in reducible transparency:\nexample {as : List \u03b1} {h} : (a :: as).get \u27e80, h\u27e9 = a := by simp\n\nattribute [simp] iota\n\n@[simp] theorem not_mem_nil (a : \u03b1) : \u00ac a \u2208 [] := fun.\n\n@[simp] theorem mem_cons {a b : \u03b1} {l : List \u03b1} :\n  a \u2208 (b :: l) \u2194 a = b \u2228 a \u2208 l :=\n  \u27e8fun h => by cases h <;> simp [Membership.mem, *],\n   fun | Or.inl rfl => by constructor | Or.inr h => by constructor; assumption\u27e9\n\nprotected def bagInter {\u03b1} [BEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n| [], _ => []\n| _, [] => []\n| a :: l\u2081, l\u2082 => if l\u2082.elem a then a :: List.bagInter l\u2081 (l\u2082.erase a) else List.bagInter l\u2081 l\u2082\n\nprotected def diff {\u03b1} [BEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n| l, [] => l\n| l\u2081, a :: l\u2082 => if l\u2081.elem a then List.diff (l\u2081.erase a) l\u2082 else List.diff l\u2081 l\u2082\n\nopen Option Nat\n\n/-- Get the tail of a nonempty list, or return `[]` for `[]`. -/\ndef tail : List \u03b1 \u2192 List \u03b1\n| []    => []\n| a::as => as\n\n\ndef mapIdxAux (f : Nat \u2192 \u03b1 \u2192 \u03b2) : Nat \u2192 List \u03b1 \u2192 List \u03b2\n| k, [] => []\n| k, a :: as => f k a :: mapIdxAux f (k+1) as\n\n/-- Given a function `f : Nat \u2192 \u03b1 \u2192 \u03b2` and `as : list \u03b1`, `as = [a\u2080, a\u2081, ...]`, returns the list\n`[f 0 a\u2080, f 1 a\u2081, ...]`. -/\ndef mapIdx (f : Nat \u2192 \u03b1 \u2192 \u03b2) (as : List \u03b1) : List \u03b2 :=\n  mapIdxAux f 0 as\n\n/-- Applicative variant of `mapIdx`. -/\ndef mapIdxM {m : Type v \u2192 Type w} [Applicative m] (as : List \u03b1) (f : Nat \u2192 \u03b1 \u2192 m \u03b2) :\n  m (List \u03b2) :=\n  let rec loop : Nat \u2192 List \u03b1 \u2192 m (List \u03b2)\n  | _,  [] => pure []\n  | n, a :: as => List.cons <$> f n a <*> loop (n + 1) as\n  loop 0 as\n\n/-- `after p xs` is the suffix of `xs` after the first element that satisfies\n  `p`, not including that element.\n  ```lean\n  after      (eq 1)       [0, 1, 2, 3] = [2, 3]\n  drop_while (not \u2218 eq 1) [0, 1, 2, 3] = [1, 2, 3]\n  ```\n-/\ndef after (p : \u03b1 \u2192 Prop) [DecidablePred p] : List \u03b1 \u2192 List \u03b1\n| [] => []\n| x :: xs => if p x then xs else after p xs\n\ndef findIdx (p : \u03b1 \u2192 Prop) [DecidablePred p] : List \u03b1 \u2192 Nat\n| [] => 0\n| a :: l => if p a then 0 else succ (findIdx p l)\n\ndef indexOf [BEq \u03b1] (a : \u03b1) : List \u03b1 \u2192 Nat := findIdx (a == \u00b7)\n\n@[simp] def removeNth : List \u03b1 \u2192 Nat \u2192 List \u03b1\n| [], _ => []\n| x :: xs, 0 => xs\n| x :: xs, i+1 => x :: removeNth xs i\n\ndef bor (l : List Bool) : Bool := any l id\n\ndef band (l : List Bool) : Bool := all l id\n\n-- TODO(Mario): restore `protected` when general `insert` is added\ndef insert [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) : List \u03b1 :=\n  if a \u2208 l then l else a :: l\n\nprotected def union [DecidableEq \u03b1] (l\u2081 l\u2082 : List \u03b1) : List \u03b1 :=\n  foldr insert l\u2082 l\u2081\n\ninstance [DecidableEq \u03b1] : Union (List \u03b1) :=\n  \u27e8List.union\u27e9\n\nprotected def inter [DecidableEq \u03b1] (l\u2081 l\u2082 : List \u03b1) : List \u03b1 :=\n  filter (\u00b7 \u2208 l\u2082) l\u2081\n\ninstance [DecidableEq \u03b1] : Inter (List \u03b1) := \u27e8List.inter\u27e9\n\n@[simp] def repeat' (a : \u03b1) : Nat \u2192 List \u03b1\n| 0 => []\n| succ n => a :: repeat' a n\n\ndef last! [Inhabited \u03b1] : List \u03b1 \u2192 \u03b1\n| [] => panic! \"empty list\"\n| [a] => a\n| [a, b] => b\n| a :: b :: l => last! l\n", "meta": {"author": "JOSHCLUNE", "repo": "Keller_reduction", "sha": "dc392b3da352fc1ffcfbecb1d4717d05f5faed4a", "save_path": "github-repos/lean/JOSHCLUNE-Keller_reduction", "path": "github-repos/lean/JOSHCLUNE-Keller_reduction/Keller_reduction-dc392b3da352fc1ffcfbecb1d4717d05f5faed4a/Lean4_Clique/Mathlib/Mathlib/Init/Data/List/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514762, "lm_q2_score": 0.6859494485880928, "lm_q1q2_score": 0.3987445444412012}}
{"text": "import Logic.Predicate.FirstOrder.Language\nimport Logic.Predicate.Coding\n\nuniverse u v\n\nnamespace FirstOrder\n\nopen SubFormula\nvariable {L : Language.{u}} [\u2200 k, DecidableEq (L.func k)] [\u2200 k, DecidableEq (L.rel k)]\n\ndef shifts (\u0394 : Finset (SyntacticSubFormula L n)) :\n  Finset (SyntacticSubFormula L n) := \u0394.map shiftEmb\n\nlemma shifts_eq_image (\u0394 : Finset (SyntacticSubFormula L n)) : shifts \u0394 = \u0394.image shift := Finset.map_eq_image _ _\n\n@[simp] lemma mem_shifts_iff (p : SyntacticSubFormula L n) (\u0394 : Finset (SyntacticSubFormula L n)) :\n    shift p \u2208 shifts \u0394 \u2194 p \u2208 \u0394 :=\n  Finset.mem_map' _\n\n@[simp] lemma shifts_ss (\u0394 \u0393 : Finset (SyntacticSubFormula L n)) :\n    shifts \u0394 \u2286 shifts \u0393 \u2194 \u0394 \u2286 \u0393 := Finset.map_subset_map\n\nlemma shifts_insert (p : SyntacticSubFormula L n) (\u0394 : Finset (SyntacticSubFormula L n)) :\n    shifts (insert p \u0394) = insert (shift p) (shifts \u0394) :=\n  by simp[shifts, shiftEmb_eq_shift]\n\ninductive Derivation : Finset (SyntacticFormula L) \u2192 Type _\n| axL     : \u2200 (\u0394 : Finset (SyntacticFormula L)) {k} (r : L.rel k) (v : Fin k \u2192 SyntacticTerm L),\n    rel r v \u2208 \u0394 \u2192 nrel r v \u2208 \u0394 \u2192 Derivation \u0394\n| verum   : \u2200 (\u0394 : Finset (SyntacticFormula L)), \u22a4 \u2208 \u0394 \u2192 Derivation \u0394\n| orLeft  : \u2200 (\u0394 : Finset (SyntacticFormula L)) (p q : SyntacticFormula L),\n    Derivation (insert p \u0394) \u2192 Derivation (insert (p \u22ce q) \u0394)\n| orRight : \u2200 (\u0394 : Finset (SyntacticFormula L)) (p q : SyntacticFormula L),\n    Derivation (insert q \u0394) \u2192 Derivation (insert (p \u22ce q) \u0394)\n| and     : \u2200 (\u0394 : Finset (SyntacticFormula L)) (p q : SyntacticFormula L),\n    Derivation (insert p \u0394) \u2192 Derivation (insert q \u0394) \u2192 Derivation (insert (p \u22cf q) \u0394)\n| all     : \u2200 (\u0394 : Finset (SyntacticFormula L)) (p : SyntacticSubFormula L 1),\n    Derivation (insert (free p) (shifts \u0394)) \u2192 Derivation (insert (\u2200' p) \u0394)\n| ex      : \u2200 (\u0394 : Finset (SyntacticFormula L)) (t : SyntacticTerm L) (p : SyntacticSubFormula L 1),\n    Derivation (insert (subst t p) \u0394) \u2192 Derivation (insert (\u2203' p) \u0394)\n\ninstance : HasVdash (Finset (SyntacticFormula L)) (Type u) := \u27e8Derivation\u27e9\n\nabbrev DerivationList (G : List (SyntacticFormula L)) := \u22a9 G.toFinset\n\nabbrev Derivation.Valid (p : SyntacticFormula L) := \u22a9 ({p} : Finset _)\n\nstructure Proof (T : CTheory L) (\u03c3 : Sentence L) where\n  leftHand : Finset (Sentence L)\n  hleftHand : \u2191leftHand \u2286 SubFormula.neg '' T\n  derivation : \u22a9 ((insert \u03c3 leftHand).image emb : Finset (SyntacticFormula L))\n\ninstance : HasTurnstile (Sentence L) (Type u) := \u27e8Proof\u27e9\n\nnamespace Derivation\nvariable {\u0394 \u0393 : Finset (SyntacticFormula L)}\n\nsection Repr\nvariable [\u2200 k, ToString (L.func k)] [\u2200 k, ToString (L.rel k)]\n\nprotected unsafe def repr : {\u0394 : Finset (SyntacticFormula L)} \u2192 Derivation \u0394 \u2192 String\n  | _, axL \u0394 _ _ _ _   =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize(axL)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr \u0394 ++ \"$}\\n\\n\"\n  | _, verum \u0394 _       =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\top$)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr \u0394 ++ \"$}\\n\\n\"\n  | _, orLeft \u0394 p q d  =>\n      d.repr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$L)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr (insert (p \u22ce q) \u0394) ++ \"$}\\n\\n\"\n  | _, orRight \u0394 p q d =>\n      d.repr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$R)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr (insert (p \u22ce q) \u0394) ++ \"$}\\n\\n\"\n  | _, and \u0394 p q dp dq =>\n      dp.repr ++\n      dq.repr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\land$)}\\n\" ++\n      \"\\\\BinaryInfC{$\" ++ reprStr (insert (p \u22cf q) \u0394) ++ \"$}\\n\\n\"\n  | _, all \u0394 p d       =>\n      d.repr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\forall$)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr (insert (\u2200' p) \u0394) ++ \"$}\\n\\n\"\n  | _, ex \u0394 _ p d      =>\n      d.repr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\exists$)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ reprStr (insert (\u2203' p) \u0394) ++ \"$}\\n\\n\"\n\nunsafe instance : Repr (\u22a9 \u0394) where\n  reprPrec d _ := d.repr\n\nprotected def toStr : {\u0394 : Finset (SyntacticFormula L)} \u2192 Derivation \u0394 \u2192 String\n  | _, axL _ r v _ _   =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize(axL)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ toString (rel r v) ++ \",\" ++ toString (nrel r v) ++ \", ... $}\\n\\n\"\n  | _, verum _ _       =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\top$)}\\n\" ++\n      \"\\\\UnaryInfC{$ \u22a4, ... $}\\n\\n\"\n  | _, orLeft _ p q d  =>\n      d.toStr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$L)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ toString (p \u22ce q) ++ \", ... $}\\n\\n\"\n  | _, orRight _ p q d =>\n      d.toStr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$R)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ toString (p \u22ce q) ++ \", ... $}\\n\\n\"\n  | _, and _ p q dp dq =>\n      dp.toStr ++\n      dq.toStr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\land$)}\\n\" ++\n      \"\\\\BinaryInfC{$\" ++ toString (p \u22cf q) ++ \", ... $}\\n\\n\"\n  | _, all _ p d       =>\n      d.toStr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\forall$)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ toString (\u2200' p) ++ \", ... $}\\n\\n\"\n  | _, ex _ _ p d      =>\n      d.toStr ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\exists$)}\\n\" ++\n      \"\\\\UnaryInfC{$\" ++ toString (\u2203' p) ++ \", ... $}\\n\\n\"\n\nprotected def toStrCompact : {\u0394 : Finset (SyntacticFormula L)} \u2192 Derivation \u0394 \u2192 String\n  | _, axL _ _ _ _ _   =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize(axL)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n  | _, verum _ _       =>\n      \"\\\\AxiomC{}\\n\" ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\top$)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n  | _, orLeft _ _ _ d  =>\n      d.toStrCompact ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$L)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n  | _, orRight _ _ _ d =>\n      d.toStrCompact ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\lor$R)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n  | _, and _ _ _ dp dq =>\n      dp.toStrCompact ++\n      dq.toStrCompact ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\land$)}\\n\" ++\n      \"\\\\BinaryInfC{}\\n\\n\"\n  | _, all _ _ d       =>\n      d.toStrCompact ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\forall$)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n  | _, ex _ _ _ d      =>\n      d.toStrCompact ++\n      \"\\\\RightLabel{\\\\scriptsize($\\\\exists$)}\\n\" ++\n      \"\\\\UnaryInfC{}\\n\\n\"\n\ninstance : ToString (\u22a9 \u0394) := \u27e8Derivation.toStr\u27e9\n\nend Repr\n\nprotected def cast (d : Derivation \u0394) (e : \u0394 = \u0393) : \u22a9 \u0393 := cast (by simp[HasVdash.vdash, e]) d\n\ndef weakening : \u2200 {\u0394}, \u22a9 \u0394 \u2192 \u2200 {\u0393 : Finset (SyntacticFormula L)}, \u0394 \u2286 \u0393 \u2192 \u22a9 \u0393\n  | _, axL \u0394 r v hrel hnrel, \u0393, h => axL \u0393 r v (h hrel) (h hnrel)\n  | _, verum \u0394 htop,         \u0393, h => verum \u0393 (h htop)\n  | _, orLeft \u0394 p q d,       \u0393, h =>\n      have : \u22a9 insert p \u0393 := weakening d (Finset.insert_subset_insert p (Finset.insert_subset.mp h).2)\n      have : \u22a9 insert (p \u22ce q) \u0393 := orLeft \u0393 p q this\n      this.cast (by simp; exact (Finset.insert_subset.mp h).1)\n  | _, orRight \u0394 p q d,      \u0393, h =>\n      have : \u22a9 insert q \u0393 := weakening d (Finset.insert_subset_insert q (Finset.insert_subset.mp h).2)\n      have : \u22a9 insert (p \u22ce q) \u0393 := orRight \u0393 p q this\n      this.cast (by simp; exact (Finset.insert_subset.mp h).1)\n  | _, and \u0394 p q dp dq,      \u0393, h =>\n      have dp : \u22a9 insert p \u0393 := weakening dp (Finset.insert_subset_insert p (Finset.insert_subset.mp h).2) \n      have dq : \u22a9 insert q \u0393 := weakening dq (Finset.insert_subset_insert q (Finset.insert_subset.mp h).2) \n      have : \u22a9 insert (p \u22cf q) \u0393 := and \u0393 p q dp dq\n      Derivation.cast this (by simp; exact (Finset.insert_subset.mp h).1)    \n  | _, all \u0394 p d,            \u0393, h =>\n      have : \u22a9 insert (free p) (shifts \u0393) := weakening d (Finset.insert_subset_insert _ $ by simpa using (Finset.insert_subset.mp h).2)\n      have : \u22a9 insert (\u2200' p) \u0393 := all \u0393 p this\n      Derivation.cast this (by simp; exact (Finset.insert_subset.mp h).1)      \n  | _, ex \u0394 t p d,           \u0393, h =>\n      have : \u22a9 insert (subst t p) \u0393 := weakening d (Finset.insert_subset_insert _ $ by simpa using (Finset.insert_subset.mp h).2)\n      have : \u22a9 insert (\u2203' p) \u0393 := ex \u0393 t p this\n      Derivation.cast this (by simp; exact (Finset.insert_subset.mp h).1)     \n\n--def or' {p q : SyntacticFormula L} (h : p \u22ce q \u2208 \u0394) (d : \u22a9 insert p \u0394) : \u22a9 \u0394 :=\n--  weakening (or \u0394 p q d) (by simp[Finset.insert_subset, h])\n\ndef orLeft' {p q : SyntacticFormula L} (h : p \u22ce q \u2208 \u0394) (d : \u22a9 insert p (\u0394.erase (p \u22ce q))) : \u22a9 \u0394 :=\n  (orLeft _ p q d).cast (by simp[Finset.insert_erase h])\n\ndef orRight' {p q : SyntacticFormula L} (h : p \u22ce q \u2208 \u0394) (d : \u22a9 insert q (\u0394.erase (p \u22ce q))) : \u22a9 \u0394 :=\n  (orRight _ p q d).cast (by simp[Finset.insert_erase h])\n\ndef or {p q : SyntacticFormula L} (d : \u22a9 insert p (insert q \u0394)) : \u22a9 insert (p \u22ce q) \u0394 :=\n  have : \u22a9 insert (p \u22ce q) (insert q \u0394) := orLeft _ p q d\n  (orRight (insert (p \u22ce q) \u0394) p q (this.cast (by ext; simp; tauto))).cast (by simp)\n\ndef and' {p q : SyntacticFormula L} (h : p \u22cf q \u2208 \u0394) (dp : \u22a9 insert p (\u0394.erase (p \u22cf q))) (dq : \u22a9 insert q (\u0394.erase (p \u22cf q))) : \u22a9 \u0394 :=\n  (and _ p q dp dq).cast (by simp[Finset.insert_erase h])\n\ndef all' {p : SyntacticSubFormula L 1} (h : \u2200' p \u2208 \u0394) (d : \u22a9 insert (free p) (shifts $ \u0394.erase (\u2200' p))) : \u22a9 \u0394 :=\n  (all _ p d).cast (by simp[Finset.insert_erase h])\n\ndef ex' {p : SyntacticSubFormula L 1} (t : SyntacticTerm L) (h : \u2203' p \u2208 \u0394)\n  (d : \u22a9 insert (subst t p) (\u0394.erase (\u2203' p))) : \u22a9 \u0394 :=\n  (ex _ t p d).cast (by simp[Finset.insert_erase h])\n\n@[simp] lemma ne_step_max (n m : \u2115) : n \u2260 max n m + 1 :=\n  ne_of_lt $ Nat.lt_succ_of_le $ by simp\n\n@[simp] lemma ne_step_max' (n m : \u2115) : n \u2260 max m n + 1 :=\n  ne_of_lt $ Nat.lt_succ_of_le $ by simp\n\ndef em {p : SyntacticFormula L} {\u0394 : Finset (SyntacticFormula L)} (hpos : p \u2208 \u0394) (hneg : ~p \u2208 \u0394) : \u22a9 \u0394 := by\n  induction p using SubFormula.formulaRec generalizing \u0394\n  case hverum    => exact verum \u0394 hpos\n  case hfalsum   => exact verum \u0394 hneg\n  case hrel r v  => exact axL \u0394 r v hpos hneg \n  case hnrel r v => exact axL \u0394 r v hneg hpos \n  case hall p ih =>\n    exact all' hpos $ ex' (p := ~ shift p) &0\n      (by simp; exact Or.inr (by simp[shifts, shiftEmb_eq_shift]; exact \u27e8_, hneg, by simp\u27e9))\n      (ih (by simp; exact Or.inr $ ne_of_ne_complexity $ by simp[shift]) (by simp))\n  case hex p ih =>\n    simp at hneg\n    exact all' hneg $ ex' (p := shift p) &0\n      (by simp; exact Or.inr (by simp[shifts, shiftEmb_eq_shift]; exact \u27e8_, hpos, by simp\u27e9))\n      (ih (by simp) (by simp; exact Or.inr $ ne_of_ne_complexity $ by simp[shift]))\n  case hand p q ihp ihq =>\n    simp at hneg\n    exact and' hpos\n      (orLeft' (p := ~p) (q := ~q) (by simp[hneg]) $ ihp (by simp; exact Or.inr $ ne_of_ne_complexity (by simp)) (by simp))\n      (orRight' (p := ~p) (q := ~q) (by simp[hneg]) $ ihq (by simp; exact Or.inr $ ne_of_ne_complexity (by simp)) (by simp))\n  case hor p q ihp ihq =>\n    simp at hneg\n    exact and' hneg\n      (orLeft' (p := p) (q := q) (by simp[hpos]) $ ihp (by simp) (by simp; exact Or.inr $ ne_of_ne_complexity (by simp)))\n      (orRight' (p := p) (q := q) (by simp[hpos]) $ ihq (by simp) (by simp; exact Or.inr $ ne_of_ne_complexity (by simp)))\n\nsection Hom\nvariable\n  {L\u2081 : Language} [\u2200 k, DecidableEq (L\u2081.func k)] [\u2200 k, DecidableEq (L\u2081.rel k)]\n  {L\u2082 : Language} [\u2200 k, DecidableEq (L\u2082.func k)] [\u2200 k, DecidableEq (L\u2082.rel k)]\n  {\u0394\u2081 \u0393\u2081 : Finset (SyntacticFormula L\u2081)}\n\nlemma shifts_image (\u03a6 : L\u2081 \u2192\u1d65 L\u2082) {\u0394 : Finset (SyntacticFormula L\u2081)} :\n     shifts (Finset.image \u03a6.onSubFormula\u2081 \u0394) = (Finset.image \u03a6.onSubFormula\u2081 (shifts \u0394)) :=\n  by simp[shifts, shiftEmb, Finset.map_eq_image, Finset.image_image, Function.comp, SubFormula.onSubFormula\u2081_shift]\n\ndef onDerivation (\u03a6 : L\u2081 \u2192\u1d65 L\u2082) : \u2200 {\u0394 : Finset (SyntacticFormula L\u2081)}, \u22a9 \u0394 \u2192 \u22a9 Finset.image \u03a6.onSubFormula\u2081 \u0394\n  | _, axL \u0394 r v hrel hnrel =>\n      axL _ (\u03a6.onRel r) (fun i => \u03a6.onSubTerm (v i))\n        (Finset.mem_image_of_mem _ hrel) (Finset.mem_image_of_mem _ hnrel)\n  | _, verum \u0394 h            => verum _ (by simpa using Finset.mem_image_of_mem \u03a6.onSubFormula\u2081 h)\n  | _, orLeft \u0394 p q d       =>\n      have : \u22a9 insert (\u03a6.onSubFormula\u2081 p \u22ce \u03a6.onSubFormula\u2081 q) (\u0394.image \u03a6.onSubFormula\u2081) :=\n        orLeft _ _ _ ((onDerivation \u03a6 d).cast (by simp))\n      this.cast (by simp)\n  | _, orRight \u0394 p q d       =>\n      have : \u22a9 insert (\u03a6.onSubFormula\u2081 p \u22ce \u03a6.onSubFormula\u2081 q) (\u0394.image \u03a6.onSubFormula\u2081) :=\n        orRight _ _ _ ((onDerivation \u03a6 d).cast (by simp))\n      this.cast (by simp)\n  | _, and \u0394 p q dp dq      =>\n      have : \u22a9 insert (\u03a6.onSubFormula\u2081 p \u22cf \u03a6.onSubFormula\u2081 q) (Finset.image \u03a6.onSubFormula\u2081 \u0394) :=\n        and _ _ _ ((onDerivation \u03a6 dp).cast (by simp)) ((onDerivation \u03a6 dq).cast (by simp))\n      this.cast (by simp)\n  | _, all \u0394 p d            =>\n      have : \u22a9 insert (\u2200' \u03a6.onSubFormula\u2081 p) (Finset.image \u03a6.onSubFormula\u2081 \u0394) :=\n        all _ _ (by simpa[\u2190SubFormula.onSubFormula\u2081_free, shifts_image] using onDerivation \u03a6 d)\n      this.cast (by simp)\n  | _, ex \u0394 t p d           =>\n      have : \u22a9 insert (\u2203' \u03a6.onSubFormula\u2081 p) (Finset.image \u03a6.onSubFormula\u2081 \u0394) :=\n        ex _ (\u03a6.onSubTerm t) _ (by simpa[\u2190SubFormula.onSubFormula\u2081_subst] using onDerivation \u03a6 d)\n      this.cast (by simp)\n\nend Hom\n\nprivate lemma free_bind\u2080_eq (f : \u2115 \u2192 SyntacticTerm L) (p : SyntacticSubFormula L 1) :\n    free (bind\u2080 (fun x => SubTerm.bShift (f x)) p) = bind\u2080 (&0 :>\u2099 fun x => SubTerm.shift (f x)) (free p) := by\n  simp[free, bind_bind, Matrix.vecConsLast_vecEmpty]; congr; funext x\n  simp[SubTerm.free, SubTerm.bShift, SubTerm.shift, SubTerm.map, SubTerm.bind_bind, eq_finZeroElim]\n\nprivate lemma shift_bind\u2080_eq (f : \u2115 \u2192 SyntacticTerm L) (p : SyntacticFormula L) :\n    shift (bind\u2080 f p) = bind\u2080 (&0 :>\u2099 fun x => SubTerm.shift (f x)) (shift p) := by\n  simp[shift, map, bind\u2080, bind_bind]; congr\n\nprivate lemma bind\u2080_subst_eq (f : \u2115 \u2192 SyntacticTerm L) (t) (p : SyntacticSubFormula L 1) :\n    bind\u2080 f (subst t p) = subst (t.bind SubTerm.bvar f) (bind\u2080 (SubTerm.bShift \u2218 f) p) := by\n  simp[subst, bind_bind, Fin.eq_zero, SubTerm.bShift, SubTerm.map, SubTerm.bind_bind, eq_finZeroElim]; congr\n\ndef onBind : \u2200 {\u0394 : Finset (SyntacticFormula L)}, \u22a9 \u0394 \u2192 \u2200 (f : \u2115 \u2192 SyntacticTerm L), \u22a9 \u0394.image (bind\u2080 f)\n  | _, axL \u0394 r v hrel hnrel, f => axL _ r (fun i => (v i).bind SubTerm.bvar f) (Finset.mem_image_of_mem _ hrel) (Finset.mem_image_of_mem _ hnrel)\n  | _, verum \u0394 h,            _ => verum _ (Finset.mem_image_of_mem _ h)\n  | _, orLeft \u0394 p q d,       f =>\n    have : \u22a9 insert (bind\u2080 f p \u22ce bind\u2080 f q) (\u0394.image (bind\u2080 f)) := orLeft _ _ _ ((onBind d f).cast (by simp))\n    this.cast (by simp)\n  | _, orRight \u0394 p q d,      f =>\n    have : \u22a9 insert (bind\u2080 f p \u22ce bind\u2080 f q) (\u0394.image (bind\u2080 f)) := orRight _ _ _ ((onBind d f).cast (by simp))\n    this.cast (by simp)\n  | _, and \u0394 p q dp dq,      f =>\n    have : \u22a9 insert (bind\u2080 f p \u22cf bind\u2080 f q) (\u0394.image (bind\u2080 f)) := and _ _ _ ((onBind dp f).cast (by simp)) ((onBind dq f).cast (by simp))\n    this.cast (by simp)\n  | _, all \u0394 p d,            f =>\n    have : \u22a9 (insert (free p) (shifts \u0394)).image (bind\u2080 (&0 :>\u2099 fun x => SubTerm.shift (f x))).toFun := onBind d (&0 :>\u2099 fun x => (f x).shift)\n    have : \u22a9 insert (\u2200' (bind\u2080 (SubTerm.bShift \u2218 f)) p) (\u0394.image (bind\u2080 f).toFun) :=\n      all _ _ (by simpa[free_bind\u2080_eq, shift_bind\u2080_eq, shifts_eq_image, Finset.image_image, Function.comp] using this)\n    this.cast (by simp)\n  | _, ex \u0394 t p d,           f =>\n    have : \u22a9 (insert (subst t p) \u0394).image (bind\u2080 f) := onBind d f \n    have : \u22a9 insert (\u2203' bind\u2080 (SubTerm.bShift \u2218 f) p) (\u0394.image (bind\u2080 f)) := \n      ex _ (SubTerm.bind SubTerm.bvar f t) _ (by simpa[bind\u2080_subst_eq] using this) \n    this.cast (by simp)\n\ndef onMap {\u0394 : Finset (SyntacticFormula L)} (d : \u22a9 \u0394) (f : \u2115 \u2192 \u2115) : \u22a9 \u0394.image (map\u2080 f) := onBind d _\n\nprivate lemma map_subst_eq_free (p : SyntacticSubFormula L 1) (h : \u00acp.fvar? m) :\n    map\u2080 (fun x => if x = m then 0 else x + 1) (subst &m p) = free p := by\n  simp[free, subst, map\u2080, map, bind_bind, Fin.eq_zero, Matrix.vecConsLast_vecEmpty, Matrix.constant_eq_singleton]\n  exact bind_eq_of_funEqOn _ _ _ _ (by intro x hx; simp; rintro rfl; contradiction)\n\nprivate lemma image_map\u2080_eq_shifts (\u0394 : Finset $ SyntacticFormula L) (h : \u2200 p \u2208 \u0394, \u00acp.fvar? m) :\n    \u0394.image (map\u2080 (fun x => if x = m then 0 else x + 1)) = shifts \u0394 := by \n  simp[shifts_eq_image]; apply Finset.image_congr\n  simp[Set.EqOn]; intro p hp;\n  simp[shift, map\u2080, map]\n  exact bind_eq_of_funEqOn _ _ _ _ (by intro x hx; simp; rintro rfl; have := h p hp; contradiction)\n\ndef genelalizeByNewver {p : SyntacticSubFormula L 1} (hp : \u00acp.fvar? m) (h\u0394 : \u2200 q \u2208 \u0394, \u00acq.fvar? m)\n  (d : \u22a9 insert (subst &m p) \u0394) : \u22a9 insert (\u2200' p) \u0394 := by\n  have : \u22a9 insert (free p) (shifts \u0394) := by\n    simpa[map_subst_eq_free p hp, image_map\u2080_eq_shifts \u0394 h\u0394] using onMap d (fun x => if x = m then 0 else x + 1)\n  exact all \u0394 p this\n\nvariable [\u2200 k, Encodable (L.func k)] {\u03bc : Type _} [Encodable \u03bc]\n\ndef exOfInstances (v : List (SyntacticTerm L)) (p : SyntacticSubFormula L 1)\n  (h : \u22a9 (v.map (subst \u00b7 p)).toFinset \u222a \u0393) : \u22a9 insert (\u2203' p) \u0393 := by\n  induction' v with t v ih generalizing \u0393 <;> simp at h\n  \u00b7 exact weakening h (Finset.subset_insert _ \u0393)\n  \u00b7 exact Derivation.cast (ih (\u0393 := insert (\u2203' p) \u0393)\n      (Derivation.cast (ex _ t p h) (by ext r; simp))) (by simp)\n\nend Derivation\n\nnamespace Proof\n\n\n\nend Proof\n\nend FirstOrder\n\n", "meta": {"author": "iehality", "repo": "lean4-logic", "sha": "ef518051931fb1ecd0b89e94240b2900cd54d95c", "save_path": "github-repos/lean/iehality-lean4-logic", "path": "github-repos/lean/iehality-lean4-logic/lean4-logic-ef518051931fb1ecd0b89e94240b2900cd54d95c/Logic/Predicate/FirstOrder/Calculus.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.3987445382220447}}
{"text": "import Quantifiers.directory\n\nmeta def tactic.interactive.print_directory : tactic unit :=\ndo (dir, _) \u2190 create_directory,\n  trace_directory dir\n\nuniverse u\nexample {\u03b1 : Type u} (P : \u03b1 \u2192 Prop) (a : \u03b1) (h : P a) : \u2203 x, P x :=\nbegin\n  print_directory,\n  exact \u27e8a, h\u27e9\nend\n", "meta": {"author": "MateiFMandache", "repo": "ATP", "sha": "c16c56a3e1a7f2dc92ee1fef1928a16e74b97c4c", "save_path": "github-repos/lean/MateiFMandache-ATP", "path": "github-repos/lean/MateiFMandache-ATP/ATP-c16c56a3e1a7f2dc92ee1fef1928a16e74b97c4c/src/Quantifiers/directory_test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3986338908203138}}
{"text": "/-\nCopyright (c) 2020 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\nimport field_theory.minpoly\nimport linear_algebra.free_module\nimport ring_theory.adjoin\nimport ring_theory.adjoin_root\nimport ring_theory.algebraic\n\n/-!\n# Power basis\n\nThis file defines a structure `power_basis R S`, giving a basis of the\n`R`-algebra `S` as a finite list of powers `1, x, ..., x^n`.\nThere are also constructors for `power_basis` when adjoining an algebraic\nelement to a ring/field.\n\n## Definitions\n\n* `power_basis R A`: a structure containing an `x` and an `n` such that\n`1, x, ..., x^n` is a basis for the `R`-algebra `A` (viewed as an `R`-module).\n\n* `findim (hf : f \u2260 0) : finite_dimensional.findim K (adjoin_root f) = f.nat_degree`,\n  the dimension of `adjoin_root f` equals the degree of `f`\n\n* `power_basis.lift (pb : power_basis R S)`: if `y : S'` satisfies the same\n  equations as `pb.gen`, this is the map `S \u2192\u2090[R] S'` sending `pb.gen` to `y`\n\n* `power_basis.equiv`: if two power bases satisfy the same equations, they are\n  equivalent as algebras\n\n## Implementation notes\n\nThroughout this file, `R`, `S`, ... are `comm_ring`s, `A`, `B`, ... are\n`integral_domain`s and `K`, `L`, ... are `field`s.\n`S` is an `R`-algebra, `B` is an `A`-algebra, `L` is a `K`-algebra.\n\n## Tags\n\npower basis, powerbasis\n\n-/\n\nopen polynomial\n\nvariables {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T]\nvariables [algebra R S] [algebra S T] [algebra R T] [is_scalar_tower R S T]\nvariables {A B : Type*} [integral_domain A] [integral_domain B] [algebra A B]\nvariables {K L : Type*} [field K] [field L] [algebra K L]\n\n/-- `pb : power_basis R S` states that `1, pb.gen, ..., pb.gen ^ (pb.dim - 1)`\nis a basis for the `R`-algebra `S` (viewed as `R`-module).\n\nThis is a structure, not a class, since the same algebra can have many power bases.\nFor the common case where `S` is defined by adjoining an integral element to `R`,\nthe canonical power basis is given by `{algebra,intermediate_field}.adjoin.power_basis`.\n-/\n@[nolint has_inhabited_instance]\nstructure power_basis (R S : Type*) [comm_ring R] [ring S] [algebra R S] :=\n(gen : S)\n(dim : \u2115)\n(is_basis : is_basis R (\u03bb (i : fin dim), gen ^ (i : \u2115)))\n\nnamespace power_basis\n\n/-- Cannot be an instance because `power_basis` cannot be a class. -/\nlemma finite_dimensional [algebra K S] (pb : power_basis K S) : finite_dimensional K S :=\nfinite_dimensional.of_fintype_basis pb.is_basis\n\nlemma findim [algebra K S] (pb : power_basis K S) : finite_dimensional.findim K S = pb.dim :=\nby rw [finite_dimensional.findim_eq_card_basis pb.is_basis, fintype.card_fin]\n\n/-- TODO: this mixes `polynomial` and `finsupp`, we should hide this behind a\nnew function `polynomial.of_finsupp`. -/\nlemma polynomial.mem_supported_range {f : polynomial R} {d : \u2115} :\n  (f : finsupp \u2115 R) \u2208 finsupp.supported R R (\u2191(finset.range d) : set \u2115) \u2194 f.degree < d :=\nby { simp_rw [finsupp.mem_supported', finset.mem_coe, finset.mem_range, not_lt,\n              degree_lt_iff_coeff_zero],\n     refl }\n\nlemma mem_span_pow' {x y : S} {d : \u2115} :\n  y \u2208 submodule.span R (set.range (\u03bb (i : fin d), x ^ (i : \u2115))) \u2194\n    \u2203 f : polynomial R, f.degree < d \u2227 y = aeval x f :=\nbegin\n  have : set.range (\u03bb (i : fin d), x ^ (i : \u2115)) = (\u03bb (i : \u2115), x ^ i) '' \u2191(finset.range d),\n  { ext n,\n    simp_rw [set.mem_range, set.mem_image, finset.mem_coe, finset.mem_range],\n    exact \u27e8\u03bb \u27e8\u27e8i, hi\u27e9, hy\u27e9, \u27e8i, hi, hy\u27e9, \u03bb \u27e8i, hi, hy\u27e9, \u27e8\u27e8i, hi\u27e9, hy\u27e9\u27e9 },\n  rw [this, finsupp.mem_span_iff_total],\n  -- In the next line we use that `polynomial R := finsupp \u2115 R`.\n  -- It would be nice to have a function `polynomial.of_finsupp`.\n  apply exists_congr,\n  rintro (f : polynomial R),\n  simp only [exists_prop, polynomial.mem_supported_range, eq_comm],\n  apply and_congr iff.rfl,\n  split;\n  { rintro rfl;\n    rw [finsupp.total_apply, aeval_def, eval\u2082_eq_sum, eq_comm],\n    apply finset.sum_congr rfl,\n    rintro i -,\n    simp only [algebra.smul_def] }\nend\n\nlemma mem_span_pow {x y : S} {d : \u2115} (hd : d \u2260 0) :\n  y \u2208 submodule.span R (set.range (\u03bb (i : fin d), x ^ (i : \u2115))) \u2194\n    \u2203 f : polynomial R, f.nat_degree < d \u2227 y = aeval x f :=\nbegin\n  rw mem_span_pow',\n  split;\n  { rintros \u27e8f, h, hy\u27e9,\n    refine \u27e8f, _, hy\u27e9,\n    by_cases hf : f = 0,\n    { simp only [hf, nat_degree_zero, degree_zero] at h \u22a2,\n      exact lt_of_le_of_ne (nat.zero_le d) hd.symm <|> exact with_bot.bot_lt_some d },\n    simpa only [degree_eq_nat_degree hf, with_bot.coe_lt_coe] using h },\nend\n\nlemma dim_ne_zero [nontrivial S] (pb : power_basis R S) : pb.dim \u2260 0 :=\n\u03bb h, one_ne_zero $\nshow (1 : S) = 0,\nby { rw [\u2190 pb.is_basis.total_repr 1, finsupp.total_apply, finsupp.sum_fintype],\n     { refine finset.sum_eq_zero (\u03bb x hx, _),\n       cases x with x x_lt,\n       rw h at x_lt,\n       cases x_lt },\n     { simp } }\n\nlemma dim_pos [nontrivial S] (pb : power_basis R S) : 0 < pb.dim :=\nnat.pos_of_ne_zero pb.dim_ne_zero\n\nlemma exists_eq_aeval [nontrivial S] (pb : power_basis R S) (y : S) :\n  \u2203 f : polynomial R, f.nat_degree < pb.dim \u2227 y = aeval pb.gen f :=\n(mem_span_pow pb.dim_ne_zero).mp (pb.is_basis.mem_span y)\n\nsection minpoly\n\nopen_locale big_operators\n\nvariable [algebra A S]\n\n/-- `pb.minpoly_gen` is a minimal polynomial for `pb.gen`.\n\nIf `A` is not a field, it might not necessarily be *the* minimal polynomial,\nhowever `nat_degree_minpoly` shows its degree is indeed minimal.\n-/\nnoncomputable def minpoly_gen (pb : power_basis A S) : polynomial A :=\nX ^ pb.dim -\n  \u2211 (i : fin pb.dim), C (pb.is_basis.repr (pb.gen ^ pb.dim) i) * X ^ (i : \u2115)\n\n@[simp]\nlemma nat_degree_minpoly_gen (pb : power_basis A S) :\n  nat_degree (minpoly_gen pb) = pb.dim :=\nbegin\n  unfold minpoly_gen,\n  apply nat_degree_eq_of_degree_eq_some,\n  rw degree_sub_eq_left_of_degree_lt; rw degree_X_pow,\n  apply degree_sum_fin_lt\nend\n\nlemma minpoly_gen_monic (pb : power_basis A S) : monic (minpoly_gen pb) :=\nbegin\n  apply monic_sub_of_left (monic_pow (monic_X) _),\n  rw degree_X_pow,\n  exact degree_sum_fin_lt _\nend\n\nlemma minpoly_gen_ne_zero (pb : power_basis A S) : minpoly_gen pb \u2260 0 :=\npb.minpoly_gen_monic.ne_zero\n\n@[simp]\nlemma degree_minpoly_gen (pb : power_basis A S) : degree (minpoly_gen pb) = pb.dim :=\nby rw [degree_eq_nat_degree pb.minpoly_gen_ne_zero, nat_degree_minpoly_gen, with_bot.coe_eq_coe]\n\nlemma degree_minpoly_gen_pos [nontrivial S] (pb : power_basis A S) :\n  0 < degree (minpoly_gen pb) :=\nby { rw [degree_minpoly_gen, \u2190 with_bot.coe_zero, with_bot.coe_lt_coe], exact pb.dim_pos }\n\nlemma nat_degree_minpoly_gen_pos [nontrivial S] (pb : power_basis A S) :\n  0 < nat_degree (minpoly_gen pb) :=\nby { rw nat_degree_minpoly_gen, exact pb.dim_pos }\n\n@[simp]\nlemma aeval_minpoly_gen (pb : power_basis A S) : aeval pb.gen (minpoly_gen pb) = 0 :=\nbegin\n  simp_rw [minpoly_gen, alg_hom.map_sub, alg_hom.map_sum, alg_hom.map_mul, alg_hom.map_pow,\n           aeval_C, \u2190 algebra.smul_def, aeval_X],\n  refine sub_eq_zero.mpr ((pb.is_basis.total_repr (pb.gen ^ pb.dim)).symm.trans _),\n  rw [finsupp.total_apply, finsupp.sum_fintype],\n  intro i, rw zero_smul\nend\n\nlemma is_integral_gen (pb : power_basis A S) : is_integral A pb.gen :=\n\u27e8minpoly_gen pb, minpoly_gen_monic pb, aeval_minpoly_gen pb\u27e9\n\nlemma dim_le_nat_degree_of_root (h : power_basis A S) {p : polynomial A}\n  (ne_zero : p \u2260 0) (root : aeval h.gen p = 0) :\n  h.dim \u2264 p.nat_degree :=\nbegin\n  refine le_of_not_lt (\u03bb hlt, ne_zero _),\n  let p_coeff : fin (h.dim) \u2192 A := \u03bb i, p.coeff i,\n  suffices : \u2200 i, p_coeff i = 0,\n  { ext i,\n    by_cases hi : i < h.dim,\n    { exact this \u27e8i, hi\u27e9 },\n    exact coeff_eq_zero_of_nat_degree_lt (lt_of_lt_of_le hlt (le_of_not_gt hi)) },\n  intro i,\n  refine linear_independent_iff'.mp h.is_basis.1 finset.univ _ _ i (finset.mem_univ _),\n  rw aeval_eq_sum_range' hlt at root,\n  rw finset.sum_fin_eq_sum_range,\n  convert root,\n  ext i,\n  split_ifs with hi,\n  { refl },\n  { rw [coeff_eq_zero_of_nat_degree_lt (lt_of_lt_of_le hlt (le_of_not_gt hi)),\n        zero_smul] }\nend\n\nlemma dim_le_degree_of_root (h : power_basis A S) {p : polynomial A}\n  (ne_zero : p \u2260 0) (root : aeval h.gen p = 0) :\n  \u2191h.dim \u2264 p.degree :=\nbegin\n  have := dim_le_nat_degree_of_root h ne_zero root,\n  rwa [\u2190 with_bot.coe_le_coe, \u2190 degree_eq_nat_degree ne_zero] at this,\nend\n\n@[simp]\nlemma degree_minpoly (pb : power_basis A S) :\n  (minpoly A pb.gen).degree = pb.dim :=\nbegin\n  refine le_antisymm _\n    (dim_le_degree_of_root pb (minpoly.ne_zero pb.is_integral_gen) (minpoly.aeval _ _)),\n  rw [\u2190 nat_degree_minpoly_gen, \u2190 degree_eq_nat_degree (minpoly_gen_monic pb).ne_zero],\n  exact minpoly.min _ _ (minpoly_gen_monic pb) (aeval_minpoly_gen pb)\nend\n\n@[simp]\nlemma nat_degree_minpoly (pb : power_basis A S) :\n  (minpoly A pb.gen).nat_degree = pb.dim :=\nby rw [\u2190 with_bot.coe_eq_coe, \u2190 degree_minpoly pb,\n       degree_eq_nat_degree (minpoly.ne_zero pb.is_integral_gen)]\n\nlemma minpoly_gen_eq [algebra K S] (pb : power_basis K S) :\n  pb.minpoly_gen = minpoly K pb.gen :=\nbegin\n  apply minpoly.unique _ _ pb.minpoly_gen_monic pb.aeval_minpoly_gen,\n  intros q q_monic aeval_q,\n  rw [degree_minpoly_gen pb, \u2190 degree_minpoly pb],\n  exact minpoly.min _ _ q_monic aeval_q\nend\n\nend minpoly\n\nsection equiv\n\nvariables [algebra A S] {S' : Type*} [comm_ring S'] [algebra A S']\n\nlemma nat_degree_lt_nat_degree {p q : polynomial R} (hp : p \u2260 0) (hpq : p.degree < q.degree) :\n  p.nat_degree < q.nat_degree :=\nbegin\n  by_cases hq : q = 0, { rw [hq, degree_zero] at hpq, have := not_lt_bot hpq, contradiction },\n  rwa [degree_eq_nat_degree hp, degree_eq_nat_degree hq, with_bot.coe_lt_coe] at hpq\nend\n\nlemma constr_pow_aeval (pb : power_basis A S) {y : S'}\n  (hy : aeval y pb.minpoly_gen = 0) (f : polynomial A) :\n  pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) (aeval pb.gen f) = aeval y f :=\nbegin\n  rw [\u2190 aeval_mod_by_monic_eq_self_of_root\n          pb.minpoly_gen_monic\n          pb.aeval_minpoly_gen,\n      \u2190 @aeval_mod_by_monic_eq_self_of_root _ _ _ _ _ f _\n          pb.minpoly_gen_monic y hy],\n  by_cases hf : f %\u2098 pb.minpoly_gen = 0,\n  { simp only [hf, alg_hom.map_zero, linear_map.map_zero] },\n  have : (f %\u2098 pb.minpoly_gen).nat_degree < pb.dim,\n  { rw \u2190 pb.nat_degree_minpoly_gen,\n    apply nat_degree_lt_nat_degree hf,\n    exact degree_mod_by_monic_lt _ pb.minpoly_gen_monic pb.minpoly_gen_ne_zero },\n  rw [aeval_eq_sum_range' this, aeval_eq_sum_range' this, linear_map.map_sum],\n  refine finset.sum_congr rfl (\u03bb i (hi : i \u2208 finset.range pb.dim), _),\n  rw finset.mem_range at hi,\n  rw linear_map.map_smul,\n  congr,\n  exact @constr_basis _ _ _ _ _ _ _ _ _ _ _ (\u27e8i, hi\u27e9 : fin pb.dim) pb.is_basis,\nend\n\nlemma constr_pow_gen (pb : power_basis A S) {y : S'}\n  (hy : aeval y pb.minpoly_gen = 0) :\n  pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) pb.gen = y :=\nby { convert pb.constr_pow_aeval hy X; rw aeval_X }\n\nlemma constr_pow_algebra_map (pb : power_basis A S) {y : S'}\n  (hy : aeval y pb.minpoly_gen = 0) (x : A) :\n  pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) (algebra_map A S x) = algebra_map A S' x :=\nby { convert pb.constr_pow_aeval hy (C x); rw aeval_C }\n\nlemma constr_pow_mul [nontrivial S] (pb : power_basis A S) {y : S'}\n  (hy : aeval y pb.minpoly_gen = 0) (x x' : S) :\n  pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) (x * x') =\n    pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) x * pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) x' :=\nbegin\n  obtain \u27e8f, hf, rfl\u27e9 := pb.exists_eq_aeval x,\n  obtain \u27e8g, hg, rfl\u27e9 := pb.exists_eq_aeval x',\n  simp only [\u2190 aeval_mul, pb.constr_pow_aeval hy]\nend\n\n/-- `pb.lift y hy` is the algebra map sending `pb.gen` to `y`,\nwhere `hy` states the higher powers of `y` are the same as the higher powers of `pb.gen`. -/\nnoncomputable def lift [nontrivial S] (pb : power_basis A S) (y : S')\n  (hy : aeval y pb.minpoly_gen = 0) :\n  S \u2192\u2090[A] S' :=\n{ map_one' := by { convert pb.constr_pow_algebra_map hy 1 using 2; rw ring_hom.map_one },\n  map_zero' := by { convert pb.constr_pow_algebra_map hy 0 using 2; rw ring_hom.map_zero },\n  map_mul' := pb.constr_pow_mul hy,\n  commutes' := pb.constr_pow_algebra_map hy,\n  .. pb.is_basis.constr (\u03bb i, y ^ (i : \u2115)) }\n\n@[simp] lemma lift_gen [nontrivial S] (pb : power_basis A S) (y : S')\n  (hy : aeval y pb.minpoly_gen = 0) :\n  pb.lift y hy pb.gen = y :=\npb.constr_pow_gen hy\n\n@[simp] lemma lift_aeval [nontrivial S] (pb : power_basis A S) (y : S')\n  (hy : aeval y pb.minpoly_gen = 0) (f : polynomial A) :\n  pb.lift y hy (aeval pb.gen f) = aeval y f :=\npb.constr_pow_aeval hy f\n\n/-- `pb.equiv pb' h` is an equivalence of algebras with the same power basis. -/\nnoncomputable def equiv [nontrivial S] [nontrivial S']\n  (pb : power_basis A S) (pb' : power_basis A S')\n  (h : pb.minpoly_gen = pb'.minpoly_gen) :\n  S \u2243\u2090[A] S' :=\nalg_equiv.of_alg_hom\n  (pb.lift pb'.gen (h.symm \u25b8 pb'.aeval_minpoly_gen))\n  (pb'.lift pb.gen (h \u25b8 pb.aeval_minpoly_gen))\n  (by { ext x, obtain \u27e8f, hf, rfl\u27e9 := pb'.exists_eq_aeval x, simp })\n  (by { ext x, obtain \u27e8f, hf, rfl\u27e9 := pb.exists_eq_aeval x, simp })\n\n@[simp]\nlemma equiv_aeval [nontrivial S] [nontrivial S']\n  (pb : power_basis A S) (pb' : power_basis A S')\n  (h : pb.minpoly_gen = pb'.minpoly_gen)\n  (f : polynomial A) :\n  pb.equiv pb' h (aeval pb.gen f) = aeval pb'.gen f :=\npb.lift_aeval _ (h.symm \u25b8 pb'.aeval_minpoly_gen) _\n\n@[simp]\nlemma equiv_gen [nontrivial S] [nontrivial S']\n  (pb : power_basis A S) (pb' : power_basis A S')\n  (h : pb.minpoly_gen = pb'.minpoly_gen) :\n  pb.equiv pb' h pb.gen = pb'.gen :=\npb.lift_gen _ (h.symm \u25b8 pb'.aeval_minpoly_gen)\n\nlocal attribute [irreducible] power_basis.lift\n\n@[simp]\nlemma equiv_symm [nontrivial S] [nontrivial S']\n  (pb : power_basis A S) (pb' : power_basis A S')\n  (h : pb.minpoly_gen = pb'.minpoly_gen) :\n  (pb.equiv pb' h).symm = pb'.equiv pb h.symm :=\nrfl\n\n/-- An algebra equivalence induces an equivalence of power bases.\n\nThis definition is used for either direction of `power_basis.congr`.\n-/\ndef congr_aux {R A A' : Type*} [comm_ring R] [ring A] [ring A']\n  [algebra R A] [algebra R A'] (e : A \u2243\u2090[R] A')\n  (pb : power_basis R A) : power_basis R A' :=\n{ gen := e pb.gen,\n  dim := pb.dim,\n  is_basis := begin\n      simp only [\u2190 e.map_pow],\n      convert linear_equiv.is_basis pb.is_basis e.to_linear_equiv\n    end }\n\n@[ext]\nlemma ext {R A : Type*} [integral_domain R] [ring A] [algebra R A]\n  {pb pb' : power_basis R A} (h : pb.gen = pb'.gen) : pb = pb' :=\nbegin\n  cases pb, cases pb', congr,\n  { exact h },\n  convert le_antisymm\n      (pb'_is_basis.card_le_card_of_linear_independent pb_is_basis.1)\n      (pb_is_basis.card_le_card_of_linear_independent pb'_is_basis.1);\n    rw fintype.card_fin\nend\n\nlemma ext_iff {R A : Type*} [integral_domain R] [ring A] [algebra R A]\n  {pb pb' : power_basis R A} : pb = pb' \u2194 pb.gen = pb'.gen :=\n\u27e8\u03bb h, by rw h, power_basis.ext\u27e9\n\n/-- An algebra equivalence induces an equivalence of power bases. -/\ndef congr {R A A' : Type*} [integral_domain R] [ring A] [ring A']\n  [algebra R A] [algebra R A'] (e : A \u2243\u2090[R] A') :\n  power_basis R A \u2243 power_basis R A' :=\n{ to_fun := power_basis.congr_aux e,\n  inv_fun := power_basis.congr_aux e.symm,\n  left_inv := \u03bb pb, power_basis.ext (e.symm_apply_apply pb.gen),\n  right_inv := \u03bb pb, power_basis.ext (e.apply_symm_apply pb.gen) }\n\nend equiv\n\nend power_basis\n\nnamespace algebra\n\nopen power_basis\n\nlemma mem_span_power_basis [nontrivial R] {x y : S} (hx : _root_.is_integral R x)\n  (hy : \u2203 f : polynomial R, y = aeval x f) :\n  y \u2208 submodule.span R (set.range (\u03bb (i : fin (minpoly R x).nat_degree),\n    x ^ (i : \u2115))) :=\nbegin\n  obtain \u27e8f, rfl\u27e9 := hy,\n  rw mem_span_pow',\n  have := minpoly.monic hx,\n  refine \u27e8f.mod_by_monic (minpoly R x),\n    lt_of_lt_of_le (degree_mod_by_monic_lt _ this (ne_zero_of_monic this)) degree_le_nat_degree,\n    _\u27e9,\n  conv_lhs { rw \u2190 mod_by_monic_add_div f this },\n  simp only [add_zero, zero_mul, minpoly.aeval, aeval_add, alg_hom.map_mul]\nend\n\nlemma linear_independent_power_basis [algebra K S] {x : S} (hx : _root_.is_integral K x) :\n  linear_independent K (\u03bb (i : fin (minpoly K x).nat_degree), x ^ (i : \u2115)) :=\nbegin\n  rw linear_independent_iff,\n  intros p hp,\n  let f : polynomial K := p.sum (\u03bb i, monomial i),\n  have f_def : \u2200 (i : fin _), f.coeff i = p i,\n  { intro i,\n    -- TODO: how can we avoid unfolding here?\n    change (p.sum (\u03bb i pi, finsupp.single i pi) : \u2115 \u2192\u2080 K) i = p i,\n    simp_rw [finsupp.sum_apply, finsupp.single_apply, finsupp.sum],\n    rw [finset.sum_eq_single, if_pos rfl],\n    { intros b _ hb,\n      rw if_neg (mt (\u03bb h, _) hb),\n      exact fin.coe_injective h },\n    { intro hi,\n      split_ifs; { exact finsupp.not_mem_support_iff.mp hi } } },\n  have f_def' : \u2200 i, f.coeff i = if hi : i < _ then p \u27e8i, hi\u27e9 else 0,\n  { intro i,\n    split_ifs with hi,\n    { exact f_def \u27e8i, hi\u27e9 },\n    -- TODO: how can we avoid unfolding here?\n    change (p.sum (\u03bb i pi, finsupp.single i pi) : \u2115 \u2192\u2080 K) i = 0,\n    simp_rw [finsupp.sum_apply, finsupp.single_apply, finsupp.sum],\n    apply finset.sum_eq_zero,\n    rintro \u27e8j, hj\u27e9 -,\n    apply if_neg (mt _ hi),\n    rintro rfl,\n    exact hj },\n  suffices : f = 0,\n  { ext i, rw [\u2190 f_def, this, coeff_zero, finsupp.zero_apply] },\n  contrapose hp with hf,\n  intro h,\n  have : (minpoly K x).degree \u2264 f.degree,\n  { apply minpoly.degree_le_of_ne_zero K x hf,\n    convert h,\n    rw [finsupp.total_apply, aeval_def, eval\u2082_eq_sum, finsupp.sum_sum_index],\n    { apply finset.sum_congr rfl,\n      rintro i -,\n      simp only [algebra.smul_def, monomial, finsupp.lsingle_apply, zero_mul, ring_hom.map_zero,\n        finsupp.sum_single_index] },\n    { intro, simp only [ring_hom.map_zero, zero_mul] },\n    { intros, simp only [ring_hom.map_add, add_mul] } },\n  have : \u00ac (minpoly K x).degree \u2264 f.degree,\n  { apply not_le_of_lt,\n    rw [degree_eq_nat_degree (minpoly.ne_zero hx), degree_lt_iff_coeff_zero],\n    intros i hi,\n    rw [f_def' i, dif_neg],\n    exact not_lt_of_ge hi },\n  contradiction\nend\n\nlemma power_basis_is_basis [algebra K S] {x : S} (hx : _root_.is_integral K x) :\n  is_basis K (\u03bb (i : fin (minpoly K x).nat_degree),\n    (\u27e8x, subset_adjoin (set.mem_singleton x)\u27e9 ^ (i : \u2115) : adjoin K ({x} : set S))) :=\nbegin\n  have hST : function.injective (algebra_map (adjoin K ({x} : set S)) S) := subtype.coe_injective,\n  have hx' : _root_.is_integral K\n    (show adjoin K ({x} : set S), from \u27e8x, subset_adjoin (set.mem_singleton x)\u27e9),\n  { apply (is_integral_algebra_map_iff hST).mp,\n    convert hx,\n    apply_instance },\n  have minpoly_eq := minpoly.eq_of_algebra_map_eq hST hx' rfl,\n  refine \u27e8_, _root_.eq_top_iff.mpr _\u27e9,\n  { have := linear_independent_power_basis hx',\n    rwa minpoly_eq at this },\n  { rintros \u27e8y, hy\u27e9 _,\n    have := mem_span_power_basis hx',\n    rw minpoly_eq at this,\n    apply this,\n    { rw [adjoin_singleton_eq_range] at hy,\n      obtain \u27e8f, rfl\u27e9 := (aeval x).mem_range.mp hy,\n      use f,\n      ext,\n      exact (is_scalar_tower.algebra_map_aeval K (adjoin K {x}) S \u27e8x, _\u27e9 _).symm } }\nend\n\n/-- The power basis `1, x, ..., x ^ (d - 1)` for `K[x]`,\nwhere `d` is the degree of the minimal polynomial of `x`. -/\nnoncomputable def adjoin.power_basis [algebra K S] {x : S} (hx : _root_.is_integral K x) :\n  power_basis K (adjoin K ({x} : set S)) :=\n{ gen := \u27e8x, subset_adjoin (set.mem_singleton x)\u27e9,\n  dim := (minpoly K x).nat_degree,\n  is_basis := power_basis_is_basis hx }\n\nend algebra\n\nnamespace adjoin_root\n\nvariables {f : polynomial K}\n\nlemma power_basis_is_basis (hf : f \u2260 0) : is_basis K (\u03bb (i : fin f.nat_degree), (root f ^ i.val)) :=\nbegin\n  set f' := f * C (f.leading_coeff\u207b\u00b9) with f'_def,\n  have deg_f' : f'.nat_degree = f.nat_degree,\n  { rw [nat_degree_mul hf, nat_degree_C, add_zero],\n    { rwa [ne.def, C_eq_zero, inv_eq_zero, leading_coeff_eq_zero] } },\n  have f'_monic : monic f' := monic_mul_leading_coeff_inv hf,\n  have aeval_f' : aeval (root f) f' = 0,\n  { rw [f'_def, alg_hom.map_mul, aeval_eq, mk_self, zero_mul] },\n  have hx : is_integral K (root f) := \u27e8f', f'_monic, aeval_f'\u27e9,\n  have minpoly_eq : f' = minpoly K (root f),\n  { apply minpoly.unique K _ f'_monic aeval_f',\n    intros q q_monic q_aeval,\n    have commutes : (lift (algebra_map K (adjoin_root f)) (root f) q_aeval).comp (mk q) = mk f,\n    { ext,\n      { simp only [ring_hom.comp_apply, mk_C, lift_of], refl },\n      { simp only [ring_hom.comp_apply, mk_X, lift_root] } },\n    rw [degree_eq_nat_degree f'_monic.ne_zero, degree_eq_nat_degree q_monic.ne_zero,\n        with_bot.coe_le_coe, deg_f'],\n    apply nat_degree_le_of_dvd,\n    { rw [\u2190ideal.mem_span_singleton, \u2190ideal.quotient.eq_zero_iff_mem],\n      change mk f q = 0,\n      rw [\u2190commutes, ring_hom.comp_apply, mk_self, ring_hom.map_zero] },\n    { exact q_monic.ne_zero } },\n  refine \u27e8_, eq_top_iff.mpr _\u27e9,\n  { rw [\u2190deg_f', minpoly_eq],\n    exact algebra.linear_independent_power_basis hx, },\n  { rintros y -,\n    rw [\u2190deg_f', minpoly_eq],\n    apply algebra.mem_span_power_basis hx,\n    obtain \u27e8g\u27e9 := y,\n    use g,\n    rw aeval_eq,\n    refl }\nend\n\n/-- The power basis `1, root f, ..., root f ^ (d - 1)` for `adjoin_root f`,\nwhere `f : polynomial K` of degree `d \u2265 0`. -/\nnoncomputable def power_basis (hf : f \u2260 0) :\n  power_basis K (adjoin_root f) :=\n{ gen := root f,\n  dim := f.nat_degree,\n  is_basis := power_basis_is_basis hf }\n\n@[simp] lemma gen_eq (hf : f \u2260 0) : (adjoin_root.power_basis hf).gen = root f := rfl\n\nlemma minpoly_gen_dvd (hf : f \u2260 0) :\n  (adjoin_root.power_basis hf).minpoly_gen \u2223 f :=\nby { rw power_basis.minpoly_gen_eq, exact minpoly.dvd _ _ (adjoin_root.eval\u2082_root f) }\n\n@[simp]\nlemma minpoly_gen_eq (hf : irreducible f) (hfm : monic f) :\n  (adjoin_root.power_basis hf.ne_zero).minpoly_gen = f :=\nbegin\n  rw [(power_basis hf.ne_zero).minpoly_gen_eq, \u2190 minpoly.unique' _ hf _ hfm],\n  { exact field.to_nontrivial _ },\n  { exact (power_basis hf.ne_zero).is_integral_gen },\n  { exact adjoin_root.eval\u2082_root f }\nend\n\nend adjoin_root\n", "meta": {"author": "lean-forward", "repo": "class-number", "sha": "812ff19e6fbde86f8d71689851adaa2bbae9695e", "save_path": "github-repos/lean/lean-forward-class-number", "path": "github-repos/lean/lean-forward-class-number/class-number-812ff19e6fbde86f8d71689851adaa2bbae9695e/src/power_basis.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3984719929963473}}
{"text": "import \n  .lemmas.substitution\n  .lemmas.big_step\n\nopen big_step\n\nlemma big_subst_complete {E e S r} : \n  (E, compile e, S) \u27f9\u2099\u1d65 (E, r :: S)\n  \u2192 big_subst E e \u27f9 r :=\nbegin\n  assume h,\n  induction' e,\n  case EVal {\n    rw compile at h,\n    cases' h, cases' h,\n    rw big_subst_val,\n    apply RunVal\n  },\n  case EVar {\n    rw compile at h,\n    cases' h, cases' h,\n    induction' _x,\n    case bhead {\n      rw [big_subst, subst, if_pos (eq.refl x), big_subst_val],\n      apply RunVal\n    },\n    case btail {\n      rw [big_subst, subst, if_neg h.symm],\n      exact ih\n    }\n  },\n  case EOp {\n    rw compile at h, simp at h,\n    rw big_subst_spread_op,\n    cases' to_interm_results h with v h',\n    cases' h' with he_1 h',\n    cases' to_interm_results h' with u h'',\n    cases' h'' with he h'',\n    cases' h'', cases' h'',\n    apply RunOp,\n    exact ih_e he,\n    exact ih_e_1 he_1\n  },\n  case EIf {\n    rw compile at h, simp at h,\n    rw big_subst_spread_if,\n    cases' to_interm_results h with v h',\n    cases' h' with he h',\n    cases h',\n    case ERunTBranch {\n      cases' to_interm_results h'__x with u,\n      cases' h_1 with he_1 h_1,\n      cases' h_1,\n      rw list.drop_length at h_1,\n      cases' h_1,\n      apply RunIfT,\n      exact ih_e he,\n      exact ih_e_1 he_1\n    },\n    case ERunFBranch {\n      rename [h'__x hle, h'__x_1 he_2],\n      simp at he_2,\n      have H : \u2200 xs, (compile e_1).append xs = compile e_1 ++ xs,\n        assume xs, by refl,\n      rw [nat.succ_eq_add_one,\n          nat.add_comm, \n          list.drop_add, \n          list.drop_one, H,\n          list.drop_append_of_le_length,\n          list.drop_length,\n          list.nil_append,\n          list.tail] at he_2,\n      apply RunIfF,\n      exact ih_e he,\n      exact ih_e_2 he_2,\n      refl\n    }\n  },\n  case ELet {\n    rw compile at h, simp at h,\n    cases' to_interm_results h with v h',\n    cases' h' with he h',\n    cases' h',\n    cases' to_interm_results h' with u h'',\n    cases' h'' with he_1 h'',\n    cases' h'', cases' h'',\n    rw big_subst_spread_let,\n    apply RunLet,\n    exact ih_e he,\n    rw [subst_merge, big_subst_remove_append],\n    exact ih_e_1 he_1\n  }\nend\n\ntheorem compile_complete_nv\n  {e : exp} {r : val} :\n    ([], compile e, []) \u27f9\u2099\u1d65 ([], [r])\n  \u2192 e \u27f9 r :=\nassume h,\nhave H : big_subst [] e \u27f9 r := big_subst_complete h,\nby rw big_subst at H; exact H", "meta": {"author": "sourceCode4", "repo": "VeriCompiler", "sha": "851ae7b178ffd801fafe9d6e0392f22555f89081", "save_path": "github-repos/lean/sourceCode4-VeriCompiler", "path": "github-repos/lean/sourceCode4-VeriCompiler/VeriCompiler-851ae7b178ffd801fafe9d6e0392f22555f89081/lean/proofs/completeness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587586, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.398471985853405}}
{"text": "import tactic.generalize_proofs\n\nexample (x : \u2115) (h : x < 2) : classical.some \u27e8x, h\u27e9 < 2 :=\nbegin\n  generalize_proofs a,\n  guard_hyp a : \u2203 x, x < 2,\n  guard_target classical.some a < 2,\n  exact classical.some_spec a,\nend\n\nexample (a : \u2203 x, x < 2) : classical.some a < 2 :=\nbegin\n  generalize_proofs,\n  guard_target classical.some a < 2,\n  exact classical.some_spec a,\nend\n\nexample (x : \u2115) (h : x < 2) (a : \u2203 x, x < 2) : classical.some a < 2 :=\nbegin\n  generalize_proofs,\n  guard_target classical.some a < 2,\n  exact classical.some_spec a,\nend\n\nexample (x : \u2115) (h : x < 2) (H : classical.some \u27e8x, h\u27e9 < 2) : classical.some \u27e8x, h\u27e9 < 2 :=\nbegin\n  generalize_proofs a at H \u22a2,\n  guard_hyp a : \u2203 x, x < 2,\n  guard_hyp H : classical.some a < 2,\n  guard_target classical.some a < 2,\n  exact H,\nend\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/test/generalize_proofs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3984719787104624}}
{"text": "import ..lovelib\n\n/-! # LoVe Demo 7: Metaprogramming\n\nUsers can extend Lean with custom tactics and tools. This kind of\nprogramming\u2014programming the prover\u2014is called metaprogramming.\n\nLean's metaprogramming framework uses mostly the same notions and syntax as\nLean's input language itself. Abstract syntax trees __reflect__ internal data\nstructures, e.g., for expressions (terms). The prover's C++ internals are\nexposed through Lean interfaces, which we can use for\n\n* accessing the current context and goal;\n* unifying expressions;\n* querying and modifying the environment;\n* setting attributes.\n\nMost of Lean's predefined tactics are implemented in Lean (and not in C++).\n\nExample applications:\n\n* proof goal transformations;\n* heuristic proof search;\n* decision procedures;\n* definition generators;\n* advisor tools;\n* exporters;\n* ad hoc automation.\n\nAdvantages of Lean's metaprogramming framework:\n\n* Users do not need to learn another programming language to write\n  metaprograms; they can work with the same constructs and notation used to\n  define ordinary objects in the prover's library.\n\n* Everything in that library is available for metaprogramming purposes.\n\n* Metaprograms can be written and debugged in the same interactive environment,\n  encouraging a style where formal libraries and supporting automation are\n  developed at the same time. -/\n\n\nset_option pp.beta true\nset_option pp.generalized_field_notation false\n\nopen tactic \n\nnamespace LoVe\n\n/-! \n\n## Well-founded and non-well-founded recursion \n\nThe recursive functions we've written are structurally recursive.\nBut sometimes this feels like too strong of a restriction.\n-/\n\ndef list.map {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2) : list \u03b1 \u2192 list \u03b2\n| [] := [] \n| (h::t) := f h :: list.map t\n\nlemma list.map_length {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2): \u2200 l : \n  list \u03b1, list.length (list.map f l) = list.length l \n| [] := rfl \n| (h::t) := by simp [list.map, list.map_length]\n\ndef list.multimap\u2081 {\u03b1 : Type} (f : \u03b1 \u2192 \u03b1) : list \u03b1 \u2192 list \u03b1\n| [] := []\n| (h::t) := f h :: list.multimap\u2081 (list.map f t)\n\ndef list.multimap\u2082 {\u03b1 : Type} (f : \u03b1 \u2192 \u03b1) : list \u03b1 \u2192 list \u03b1\n| [] := []\n| (h::t) := \n  have hl : list.sizeof (list.map f t) < 1 + list.sizeof t := sorry,\n  f h :: list.multimap\u2082 (list.map f t)\n\n#eval list.multimap\u2082 (\u03bb x, x + 1) [0, 0, 0, 0]\n\ndef list.multimap\u2083 {\u03b1 : Type} (f : \u03b1 \u2192 \u03b1) : list \u03b1 \u2192 list \u03b1\n| [] := []\n| (h::t) := \n  have hl : list.length (list.map f t) < list.length t + 1 :=\n    by simp [list.map_length, nat.lt_succ_self],\n  f h :: list.multimap\u2083 (list.map f t)\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf list.length\u27e9]}\n\n/-!\nProving well-foundedness can be arbitrarily hard.\n-/\n\ndef f : \u2115 \u2192 \u2115\n| n := if n = 1 then 1 \n       else if n%2 = 0 then f (n/2)\n       else f (3*n + 1)\n\n/-!\nAll functions in \"standard\" Lean must terminate, otherwise we could prove false.\n\nBut maybe all we want to do is *compute* with the function, and not worry about\nproving anything about it.\n\nThe keyword *meta* lets us do exactly this. \nLike *noncomputable*, *meta* is sticky: anything that references a meta declaration \nmust be meta itself. \nAll it does is disable the well-foundedness checker.\n-/\n\nmeta def g : \u2115 \u2192 \u2115\n| n := if n = 1 then 1 \n       else if n%2 = 0 then g (n/2)\n       else g (3*n + 1)\n\n#eval (list.iota 100).map g\n\nmeta def oops : false := oops\n\n/-!\n\nNote that this use of the word \"meta\" is somewhat misleading. \nNothing is \"about Lean\" yet, we've just defined a language extension. \nMorally speaking, meta definitions are ones that we intend for \"computation \npurposes only.\"\n\n\n## What are tactics, really?\n\nRecall that when we write a tactic proof, it's generating a proof term under the hood.\n\nBetween each tactic we can inspect a *proof state*, which has a context and goal. \n\nA tactic behaves like a function from proof state to proof state. \n-/\n\nlemma test_lemma : \u2200 x, x = 3 \u2192 x + 10 = 13 := \nbegin \n  intros x hx,\n  cases hx,\n  refl\nend\n\n#print test_lemma\n\n.\n\n/-\nBut tactics can fail. So they're not total functions. \nAnd tactics can also fail to terminate.\n-/\n\n\n-- example (x y z : \u2115) : x + y + z = z + y + x :=\n-- by simp [add_comm x y, add_comm y x]\n\n\n/-!\n\n\nSo it feels like we have something like\n\n    meta def simp : simp_args \u2192 tactic_state \u2192 option tactic_state \n\nWhich is actually not so far from the truth!\n\n\n\nWe've mentioned the difference between `#reduce` and `#eval`. \n`#eval` is for data only. It doesn't care about proof terms, instead interpreting\nexpressions in the Lean virtual machine. \n\nIt replaces certain data types and operations with more efficient implementations,\ne.g. arithmetic on `nat`. \n\nIt also replaces certain *constants* with actual data.\n\n-/\n\n\n#print tactic_state\n#print tactic_state.get_options \n\n.\n\n/-\n\nSo a tactic in Lean is like a function `tactic_state \u2192 tactic_state`. \nIt can use these uninterpreted constants like `get_options`. \nIn a `begin...end` block, Lean generates the initial tactic state. \nThen it sequentially calls the tactics in the block, producing a new tactic state. \nWhen a tactic leaves a state that has no goals, the proof is done.\n\n\nThere are some hiccups here.\n* \"State\"? Sounds imperative!\n* Stringing together `option`-valued functions sounds annoying. \n* How do we write these? Can we read and write to the state at the same time?\n\n\nThese questions aren't unique to Lean -- these are common functional programming issues. \nThey have a solution: *monads*.\n\n\n## Monads\n\nExtra reference: https://leanprover.github.io/programming_in_lean/#07_Monads.html\n\nMonads are an abstraction of \"programming with side effects.\"\nThe side effects we'll be interested in are state and failure. \n\n\nIn general, a __monad__ is a type constructor `m` that depends on some type\nparameter `\u03b1` (i.e., `m \u03b1`) equipped with two distinguished operations:\n\n    `pure {\u03b1 : Type} : \u03b1 \u2192 m \u03b1`\n    `bind {\u03b1 \u03b2 : Type} : m \u03b1 \u2192 (\u03b1 \u2192 m \u03b2) \u2192 m \u03b2`\n\nConsider the following programming task:\n\n    Implement a function `sum_2_5_7 ns` that sums up the second, fifth, and\n    seventh items of a list `ns` of natural numbers. Use `option \u2115` for the\n    result so that if the list has fewer than seven elements, you can return\n    `option.none`.\n\nA straightforward solution follows: -/\n\ndef sum_2_5_7 (ns : list \u2115) : option \u2115 :=\nmatch list.nth ns 1 with\n| option.none    := option.none\n| option.some n2 :=\n  match list.nth ns 4 with\n  | option.none    := option.none\n  | option.some n5 :=\n    match list.nth ns 6 with\n    | option.none    := option.none\n    | option.some n7 := option.some (n2 + n5 + n7)\n    end\n  end\nend\n\n/-!\n`option` is a monad with failure. `some v` is the success case, \n`none` is the failure case.\n\nIf `v : \u2115`, then `pure v = some v`. `bind` is `connect`:\n-/\n\ndef connect {\u03b1 : Type} {\u03b2 : Type} :\n  option \u03b1 \u2192 (\u03b1 \u2192 option \u03b2) \u2192 option \u03b2\n| option.none     f := option.none\n| (option.some a) f := f a\n\ndef sum_2_5_7\u2082 (ns : list \u2115) : option \u2115 :=\nconnect (list.nth ns 1)\n  (\u03bbn2, connect (list.nth ns 4)\n     (\u03bbn5, connect (list.nth ns 6)\n        (\u03bbn7, option.some (n2 + n5 + n7))))\n\n\ndef sum_2_5_7\u2086 (ns : list \u2115) : option \u2115 :=\ndo\n  n2 \u2190 list.nth ns 1,\n  n5 \u2190 list.nth ns 4,\n  n7 \u2190 list.nth ns 6,\n  pure (n2 + n5 + n7)\n\n\n/-!\n\nProgramming with state is also monadic. \n\n`action \u03c3 \u03b1` is the type of functions that take in a state of type `\u03c3`\nand produce a value of type `\u03b1`, along with a possibly updated state.\n-/\n\ndef action (\u03c3 \u03b1 : Type) : Type :=\n\u03c3 \u2192 \u03b1 \u00d7 \u03c3\n\ndef action.read {\u03c3 : Type} : action \u03c3 \u03c3 :=\n\u03bb s, (s, s)\n\ndef action.write {\u03c3 : Type} (s : \u03c3) : action \u03c3 unit :=\n\u03bb _, ((), s)\n\ndef action.pure {\u03c3 \u03b1 : Type} (a : \u03b1) : action \u03c3 \u03b1 := \n\u03bb s, (a, s)\n\ndef action.bind {\u03c3 : Type} {\u03b1 \u03b2 : Type} (ma : action \u03c3 \u03b1)\n    (f : \u03b1 \u2192 action \u03c3 \u03b2) :\n  action \u03c3 \u03b2 := \n\u03bb s, match ma s with\n| (a, s') := f a s'\nend \n\n\n@[instance] def action.monad {\u03c3 : Type} :\n  monad (action \u03c3) :=\n{ pure       := @action.pure \u03c3,\n  bind       := @action.bind \u03c3 }\n\ndef nat_action : action \u2115 string :=\ndo \n  first_val \u2190 action.read,\n  action.write (first_val * 2),\n  new_val \u2190 action.read,\n  pure (to_string first_val ++ \" ---> \" ++ to_string new_val)\n\n#eval nat_action 3\n\n\n\n/-!\n\n## The tactic monad\n\ntactic \u03b1 := tactic_state \u2192 result tactic_state \u03b1\n\nA tactic can read and/or modify the tactic state, and either succeeds\n(producing a value of type \u03b1) or fails with an exception message.\n\n-/\n\nopen tactic \n\n#print tactic \n#print result\n\n\nmeta def my_first_tactic : tactic unit :=\ndo \n  l \u2190 local_context,\n  trace l \n\nmeta def show_true : tactic unit :=\ndo tactic.applyc `trivial\n\nexample (a b c : \u2115) (h : a + b + c = 0) : true :=\nbegin \n  my_first_tactic,\n  show_true\nend\n\nrun_cmd my_first_tactic\n\nmeta def apply_and : tactic unit :=\ndo \n  trace \"applying and.intro\",\n  applyc `and.intro \n\nexample : true \u2227 true :=\nbegin\n  apply_and,\n  show_true, show_true\nend\n\nmeta def apply_and_or_intro : tactic unit :=\ndo applyc `and.intro <|> \ndo intro `nv, skip\n\nexample : false \u2192 true \u2227 true :=\nbegin \n  apply_and_or_intro,\n  apply_and_or_intro,\n  show_true, show_true\nend \n\nmeta def my_repeat : tactic unit \u2192 tactic unit :=\n\u03bb t, (do t, my_repeat t) <|> skip\n\nexample : false \u2192 true \u2227 true :=\nbegin \n  my_repeat apply_and_or_intro,\n  my_repeat show_true\nend \n\n\n/-!\n\n## Built-in data types \n\nTo be properly \"meta,\" our tactics should be able to express \nand manipulate Lean programs (= terms).\nWe see these in Lean as traditional data types. But, like `tactic_state` \nand others, the runtime representation of these is different.\n\n-/\n\n#check declaration \n#check tactic.get_decl\n\n#check name \n#print name \n\n#check `nat \n#check `nat.succ \n\n#print prefix name\n\n#check expr \n#print expr \n\n#check expr.to_raw_fmt\n\nopen tactic \n\nrun_cmd do \n  d \u2190 get_decl `nat.succ,\n  trace d.type.to_raw_fmt\n\n/-!\n\nA closed expression (e.g. the type or body of a declaration in the environment)\nshould have no occurrences of `local_const` or `mvar`. \n`elet` and `macro` can always be expanded. \n\nBound variables (`var`) are indexed by natural numbers, not names!\nThe names and types are stored in the binders (`lam` or `pi`). \n`var 0` refers to \"the variable bound by the closest binder.\"\n`var 1` refers to \"the variable bound by the second-closest binder.\"\nAnd so on.\n\nBut often we aren't dealing with closed expressions. \nIf we get the type of the goal in the middle of a proof, it will probably refer \nto things in the local context. \nThese are represented as local constants, `local_const`. \nThey have a unique name, pretty-printing name, binder info, and type. \n\n-/\n\nmeta def expr.local_unique_name : expr \u2192 name \n| (expr.local_const nm ppnm bi tp) := nm\n| _ := default\n\nexample (a a a : \u2115) : true := by do \n  lc \u2190 local_context, \n  trace lc,\n  trace (lc.map expr.local_unique_name),\n  triv\n\n/-!\n\nThings that are already defined in our environment can be accessed as `const`s. \nA `const` has a name and list of universe parameters.\n\nOften we won't build these by hand, but use `tactic.mk_const`.\n-/\n\n#check expr.const `nat []\nrun_cmd do \n  e \u2190 mk_const `nat,\n  trace e.to_raw_fmt \n\n/-!\nBuilding expressions by hand is rather cumbersome. There are ways around this. \n`tactic.mk_app` will fill in implicit arguments for you.\n-/\n\n#check tactic.mk_app\n\nrun_cmd do \n  z \u2190 mk_const `nat.zero,\n  a \u2190 mk_app `nat.add [z, z],\n  trace a,\n  if a = z then trace \"eq\" else trace \"neq\"\n\n/-!\nWe can also write *quoted* expressions, like quoted names.\n-/\n\n#check `(0 + 0)\n\n/-!\nWe can insert expressions into quoted expressions using antiquotes:\n-/\n\nmeta def trace_add_expr (e : expr) : tactic unit := \ntrace `(0 + %%e)\n\nrun_cmd trace_add_expr `(44)\n\nrun_cmd trace_add_expr `(nat)\n\n/-! \nSometimes expr quoting fails. In these cases, we might have to use `pexpr`s.\nA pre-expression corresponds to unelaborated, input-level syntax:\nimplicit arguments have not been filled in yet.\n\n`tactic.to_expr` performs *elaboration*: it turns a `pexpr` into an `expr`.\n-/\n\nmeta def trace_add_expr'' (e : expr) : tactic unit := \ntrace `(%%e + %%e)\n\nmeta def trace_add_expr' (e : expr) : tactic unit := do \n  e \u2190 to_expr ``(%%e + %%e),\n  trace e\n\nrun_cmd trace_add_expr' `(44)\n\nrun_cmd trace_add_expr' `(nat)\n\n/-!\nWe can walk through expressions, normally and monadically:\n-/\n\n#check @expr.fold\n\n#eval expr.fold `(1 + 0) \"\" (\u03bb e _ s, s ++ \", \" ++ to_string e)\n\n#check @expr.mfold \n\nrun_cmd expr.mfold `(1 + 0) () (\u03bb e _ _, tactic.trace e)\n\n\n/-!\nOne of the most important operations on `expr` is type inference.\n-/\n\nrun_cmd do \n  t \u2190 infer_type `(\u03bb x : \u2115, x + 1),\n  trace t\n\n/-!\n\n*Declarations* are stored in the *environment*. \nA declaration is an axiom, constant, theorem, or definition. \n\n-/\n\n#check tactic.get_env\n#check environment.fold \n#check environment.mfold\n\nrun_cmd do \n  e \u2190 get_env,\n  environment.mfold e () (\u03bb d _, tactic.trace (declaration.to_name d))\n\n/-!\n## Working with goals and hypotheses \n\nWe already saw the tactic `local_context` for getting hypotheses. \n`target` returns the type of the goal.\n`get_local` retrieves a single hypothesis by name.\n-/\n\nexample (a b c : \u2115) (h : a + b = c) : a + c + 0 = a + c + 1 - 1 :=\nby do \n  lc \u2190 local_context,\n  trace lc,\n  lc_types \u2190 list.mmap infer_type lc, \n  trace lc_types,\n  tgt \u2190 target,\n  trace tgt,\n  admit\n\n/-!\n`tactic.assert` adds a new hypothesis, creating a new goal for its proof. \n\nThere are lots of variants. \n-/\n\nexample (a b c : \u2115) : true :=\nby do \n  ac \u2190 get_local `a,\n  bc \u2190 get_local `b,  \n  tactic.assert `new_hyp `(%%ac + %%bc = 0),\n  trace_state,\n  admit, \n  admit \n\n#check tactic.assert \n#check tactic.assertv \n#check @tactic.note \n#check @tactic.note_anon\n\n/-!\n\nTo modify the goal, we have our familiar `apply` tactic, \nin a few variants:\n\n-/\n\n#check tactic.apply \n#check tactic.applyc\n#check tactic.exact\n\n/-!\nThere are lots of ways to call the simplifier...\n-/\n\n#check tactic.simplify\n#check tactic.simp_target\n#check tactic.simp_hyp \n\n/-!\nIf you want the familiar begin..end block syntax, there's \nyet another form of quotation: e.g.\n\n    `[simp [lemma1, lemma2] at h] \n\nis of type `tactic unit`. \n-/\n\nexample (a b c : \u2115) (h : a + b = c) : a + c + 0 = a + c + 1 - 1 :=\nby do \n  lc \u2190 local_context,\n  trace lc,\n  lc_types \u2190 list.mmap infer_type lc, \n  trace lc_types,\n  tgt \u2190 target,\n  trace tgt,\n  `[simp]\n\n\n\n\n/-! ## Example: A Conjuction-Destructing Tactic\n\nWe define a `destruct_and` tactic that automates the elimination of `\u2227` in\npremises, automating proofs such as these: -/\n\nlemma abcd_a (a b c d : Prop) (h : a \u2227 (b \u2227 c) \u2227 d) :\n  a :=\nand.elim_left h\n\nlemma abcd_b (a b c d : Prop) (h : a \u2227 (b \u2227 c) \u2227 d) :\n  b :=\nand.elim_left (and.elim_left (and.elim_right h))\n\nlemma abcd_bc (a b c d : Prop) (h : a \u2227 (b \u2227 c) \u2227 d) :\n  b \u2227 c :=\nand.elim_left (and.elim_right h)\n\n/-! Our tactic relies on a helper metafunction, which takes as argument the\nhypothesis `h` to use as an expression rather than as a name: -/\n\nmeta def destruct_and_helper : expr \u2192 tactic unit\n| h :=\n  do\n    t \u2190 tactic.infer_type h,\n    match t with\n    | `(%%a \u2227 %%b) :=\n      tactic.exact h\n      <|>\n      do {\n        ha \u2190 tactic.to_expr ``(and.elim_left %%h),\n        destruct_and_helper ha }\n      <|>\n      do {\n        hb \u2190 tactic.to_expr ``(and.elim_right %%h),\n        destruct_and_helper hb }\n    | _            := tactic.exact h\n    end\n\nmeta def tactic.destruct_and (nam : name) : tactic unit :=\ndo\n  h \u2190 tactic.get_local nam,\n  destruct_and_helper h\n\n/-! Let us check that our tactic works: -/\n\nlemma abc_a (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  a :=\nby tactic.destruct_and `h\n\nlemma abc_b (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  b :=\nby tactic.destruct_and `h\n\nlemma abc_bc (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  b \u2227 c :=\nby tactic.destruct_and `h\n\nlemma abc_ac (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  a \u2227 c :=\nby destruct_and `h   -- fails\n\n\n/-!\n## Interactive parsing\n\nWriting \n    destruct_and `h\nwith the quoted name `h` is ugly. We don't want to do this in our tactic proofs. \nAnd indeed, most of the time, we don't have to:\nthere's no quoting in `apply h`, `simp [h]`, etc.\n\n\nThere's some trickery going on here at the parser level. \n`begin...end` and `by` blocks are parsed in \"interactive tactic mode.\"\nWhen we wrote `by destruct_and` above, Lean first looked for a declaration \ncalled `tactic.interactive.destruct_and`. When it failed to find such a tactic,\nit fell back on resolving `destruct_and` in the normal way. \n-/\n\n#check @tactic.apply\n#check @tactic.interactive.apply\n\n/-!\nInstead of taking arguments of type `expr`, `name`, etc., interactive mode tactics\ntake parser commands. \n\n`setup_tactic_parser` is equivalent to \n```\nopen _root_.lean\nopen _root_.lean.parser\nopen _root_.interactive _root_.interactive.types\nlocal postfix `?`:9001 := optional\nlocal postfix *:9001 := many .\n```\n-/\n \nsection interactive_mode\n\nsetup_tactic_parser \n\nmeta def _root_.tactic.interactive.destruct_and (h : parse ident) : tactic unit :=\ntactic.destruct_and h\n\n\n/-!\n`parse ident` is definitionally equal to `name`, but in interactive mode,\nwe can write it unquoted.\nWe put this in the `_root_` namespace to escape the `LoVe` namespace. \n-/\n\n\nlemma abc_bc' (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  b \u2227 c :=\nby destruct_and h\n\n\n\n#check parse parser.pexpr \n#check parse pexpr_list\n#check parse ident*\n#check parse ident?\n\nend interactive_mode\n\n\nmeta def tactic.destruct_and_anon : tactic unit :=\ndo \n  lc \u2190 local_context,\n  lc.mfirst (\u03bb h, destruct_and_helper h)\n\n\nsection interactive_mode\n\nsetup_tactic_parser\n\nmeta def _root_.tactic.interactive.destruct_and' (h : parse ident?) : tactic unit :=\nmatch h with \n| some h' := tactic.destruct_and h'\n| none    := tactic.destruct_and_anon\nend \n\n\nlemma abc_bc'' (a b c : Prop) (h : a \u2227 b \u2227 c) :\n  b \u2227 c :=\nby destruct_and'\n\n/-!\nInteractive-mode tactics are *always* `tactic unit`.\n-/\n\nend interactive_mode \n\n/-!\n\n## Goal management\n\nWe know that tactics ultimately need to build a proof term. \nHow does this actually happen?\n\nAt the beginning of a begin...end block where the goal is to prove `T`, \nLean creates a *metavariable* `?m1 : T`. \nTactics that update the goal, like `applyc`, \n(partially) assign values to the goal metavariable.\nThese values can contain new metavariables. \n\n-/\n\nexample : true \u2227 false :=\nby do \n  gs \u2190 get_goals,\n  trace gs,\n  trace (gs.map expr.to_raw_fmt),\n  gs' \u2190 gs.mmap infer_type,\n  trace gs',\n  let orig_goal := gs.head,\n  trace \"------\",\n\n  applyc `and.intro,\n\n  gs \u2190 get_goals,\n  trace gs,\n  trace (gs.map expr.to_raw_fmt),\n  gs' \u2190 gs.mmap infer_type,\n  trace gs',\n  trace \"------\",\n\n  orig_goal \u2190 instantiate_mvars orig_goal, \n  trace orig_goal.to_raw_fmt\n\n/-!\n`get_goals` returns a list of metavariables (of type `epxr`),\nrepresenting the remaining proof obligations. \n*Unifying* these metavariables with other terms will create partial assignments.\n\n(This is a very low-level technique, we don't usually do this in practice!)\n-/\n\nexample : true :=\nby do \n  [g] \u2190 get_goals,\n  trace g,\n  unify g `(trivial),\n  gs \u2190 get_goals,\n  trace gs,\n  set_goals [],\n  gs \u2190 get_goals,\n  trace gs\n\n\n\n/-!\nNote that this can also get us in \"trouble\": we can tell the system we've \nfinished a proof when we really haven't.\n-/\n\n\nmeta def _root_.tactic.interactive.oops : tactic unit :=\ndo \n  mv \u2190 mk_meta_var `(true),\n  set_goals [mv]\n\nexample : false :=\nbegin \n  oops,\n  trivial,\nend \n\n\n/-!\n\nMetavariable assignments are stored in the tactic state. \nSo the ultimate goal of a begin...end block is:\n\"write a function tactic_state \u2192 tactic_state that assigns the initial goal \nmetavariable to a term that does not contain any metavariables.\"\n\n-/\n\n\n/-! \n\n## Proof by reflection\n\nYOu may have noticed that we can't prove anything about the tactics we write. \nBut there's a middle ground: sometimes with a bit of meta \"wrapper code,\"\nwe can turn proofs about syntax-like operations into actual proof terms. \n\nThe general strategy looks like this:\n* represent the syntax of some class of formulas in (non-meta) Lean \n* define an interpretation function from these formulas to Prop\n* define some operation on this syntax, and prove it correct with respect to the interpretation \n* write a small bit of meta code that turns a goal into a statement about your reflected syntax \n\nThe idea is that the goal left after applying your correctness theorem can be proved by computation.\n\nThis is commonly used for evaluation or normalization functions. \n`ring`, for example, can be implemented by defining the syntax of ring expressions \nand verifying a normalization algorithm: \nif \n`ring_syntax : Type`,\n`interp {\u03b1 : Type} [ring \u03b1] : ring_syntax \u2192 \u03b1`,\n`normalize : ring_syntax \u2192 ring_syntax`, then \n`\u2200 r1 r2 : ring_syntax, interp r1 = interp r2 \u2194 normalize r1 = normalize r2`. \n\nThe meta code looks at a goal `c + a*b = b*a + c`,\nconstructs `ring_syntax` objects `r1` and `r2` representing both sides,\nand changes the goal to showing that `normalize r1 = normalize r2`. \nThis can be proved by `refl`.\n\n-/\n\ninductive bexpr \n| atom : bool \u2192 bexpr \n| and : bexpr \u2192 bexpr \u2192 bexpr \n| or : bexpr \u2192 bexpr \u2192 bexpr \n| imp : bexpr \u2192 bexpr \u2192 bexpr \n| not : bexpr \u2192 bexpr \n\nopen bexpr\n\ndef interp : bexpr \u2192 Prop \n| (atom tt) := true \n| (atom ff) := false\n| (and a b) := interp a \u2227 interp b\n| (or a b) := interp a \u2228 interp b\n| (imp a b) := interp a \u2192 interp b\n| (not b) := \u00ac interp b\n\ndef normalize : bexpr \u2192 bool \n| (atom b) := b\n| (and a b) := normalize a && normalize b\n| (or a b) := normalize a || normalize b\n| (imp a b) := (bnot (normalize a)) || normalize b\n| (not b) := bnot (normalize b)\n\ntheorem normalize_correct (b : bexpr) : normalize b = tt \u2194 interp b :=\nbegin \n  induction' b; try {simp [normalize, interp] at *},\n  case and : { finish },\n  case or : { finish },\n  case not : { simpa using not_iff_not.mpr ih },\n  case atom: { cases b; simp [normalize, interp] },\n  case imp : \n  { have h_not : normalize b = ff \u2194 \u00ac interp b := by simpa using not_iff_not.mpr ih_b,\n    finish },\nend \n\nmeta def bexpr_of_expr : expr \u2192 option expr \n| `(true) := some `(bexpr.atom tt)\n| `(false) := some `(bexpr.atom ff)\n| `(%%a \u2227 %%b) := do a \u2190 bexpr_of_expr a, b \u2190 bexpr_of_expr b, some `(bexpr.and %%(a) %%(b))\n| `(%%a \u2228 %%b) := do a \u2190 bexpr_of_expr a, b \u2190 bexpr_of_expr b, some `(bexpr.or %%(a) %%(b))\n| `(%%a \u2192 %%b) := do a \u2190 bexpr_of_expr a, b \u2190 bexpr_of_expr b, some `(bexpr.imp %%(a) %%(b))\n| `(\u00ac %%a) := do a \u2190 bexpr_of_expr a, some `(bexpr.not %%(a))\n| _ := none\n\nmeta def _root_.tactic.interactive.change_goal : tactic unit :=\ndo \n  t \u2190 target, \n  match bexpr_of_expr t with \n  | some t' := do apply `(iff.mp (normalize_correct %%(t'))), skip\n  | none := fail \"goal is not a bexpr pattern\"\n  end\n\n\nexample : (true \u2192 true) \u2228 false :=\nbegin \n  change_goal,\n  refl\nend\n\n/-!\nYou could imagine doing the same with, say, a SAT solver.\n\nModify `bexpr` to cover all propositional formulas: `atom : \u2115 \u2192 bexpr`. \nAdd an argument to `interp`: `dict : \u2115 \u2192 Prop` assigning atoms to propositions. \n`normalize` becomes `is_tautology : bexpr \u2192 bool`. \n`normalize_correct` becomes \n  `is_tautology_correct (b : bexpr) : is_tautology b = tt \u2194 \u2200 dict, interp dict b`\n`bexpr_of_expr` will also have to return a dictionary. \n\n-/\n\nend LoVe ", "meta": {"author": "BrownCS1951x", "repo": "fpv2022", "sha": "aeaf291183721460387f8ae4c3c008836b8460e7", "save_path": "github-repos/lean/BrownCS1951x-fpv2022", "path": "github-repos/lean/BrownCS1951x-fpv2022/fpv2022-aeaf291183721460387f8ae4c3c008836b8460e7/src/lectures/love07_metaprogramming_alt_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.3984685383027745}}
{"text": "/-\nCopyright (c) 2021 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn, Scott Morrison\n-/\nimport data.opposite\n\n/-!\n# Quivers\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis module defines quivers. A quiver on a type `V` of vertices assigns to every\npair `a b : V` of vertices a type `a \u27f6 b` of arrows from `a` to `b`. This\nis a very permissive notion of directed graph.\n\n## Implementation notes\n\nCurrently `quiver` is defined with `arrow : V \u2192 V \u2192 Sort v`.\nThis is different from the category theory setup,\nwhere we insist that morphisms live in some `Type`.\nThere's some balance here: it's nice to allow `Prop` to ensure there are no multiple arrows,\nbut it is also results in error-prone universe signatures when constraints require a `Type`.\n-/\n\nopen opposite\n\n-- We use the same universe order as in category theory.\n-- See note [category_theory universes]\nuniverses v v\u2081 v\u2082 u u\u2081 u\u2082\n\n/--\nA quiver `G` on a type `V` of vertices assigns to every pair `a b : V` of vertices\na type `a \u27f6 b` of arrows from `a` to `b`.\n\nFor graphs with no repeated edges, one can use `quiver.{0} V`, which ensures\n`a \u27f6 b : Prop`. For multigraphs, one can use `quiver.{v+1} V`, which ensures\n`a \u27f6 b : Type v`.\n\nBecause `category` will later extend this class, we call the field `hom`.\nExcept when constructing instances, you should rarely see this, and use the `\u27f6` notation instead.\n-/\nclass quiver (V : Type u) :=\n(hom : V \u2192 V \u2192 Sort v)\n\ninfixr ` \u27f6 `:10 := quiver.hom -- type as \\h\n\n/--\nA morphism of quivers. As we will later have categorical functors extend this structure,\nwe call it a `prefunctor`.\n-/\nstructure prefunctor (V : Type u\u2081) [quiver.{v\u2081} V] (W : Type u\u2082) [quiver.{v\u2082} W] :=\n(obj [] : V \u2192 W)\n(map : \u03a0 {X Y : V}, (X \u27f6 Y) \u2192 (obj X \u27f6 obj Y))\n\nnamespace prefunctor\n\n@[ext]\nlemma ext {V : Type u} [quiver.{v\u2081} V] {W : Type u\u2082} [quiver.{v\u2082} W]\n  {F G : prefunctor V W}\n  (h_obj : \u2200 X, F.obj X = G.obj X)\n  (h_map : \u2200 (X Y : V) (f : X \u27f6 Y),\n           F.map f = eq.rec_on (h_obj Y).symm (eq.rec_on (h_obj X).symm (G.map f))) : F = G :=\nbegin\n  cases F with F_obj _, cases G with G_obj _,\n  obtain rfl : F_obj = G_obj, by { ext X, apply h_obj },\n  congr,\n  funext X Y f,\n  simpa using h_map X Y f,\nend\n\n/--\nThe identity morphism between quivers.\n-/\n@[simps]\ndef id (V : Type*) [quiver V] : prefunctor V V :=\n{ obj := id,\n  map := \u03bb X Y f, f, }\n\ninstance (V : Type*) [quiver V] : inhabited (prefunctor V V) := \u27e8id V\u27e9\n\n/--\nComposition of morphisms between quivers.\n-/\n@[simps]\ndef comp {U : Type*} [quiver U] {V : Type*} [quiver V] {W : Type*} [quiver W]\n  (F : prefunctor U V) (G : prefunctor V W) : prefunctor U W :=\n{ obj := \u03bb X, G.obj (F.obj X),\n  map := \u03bb X Y f, G.map (F.map f), }\n\n@[simp] lemma comp_id {U : Type*} [quiver U] {V : Type*} [quiver V] (F : prefunctor U V) :\n  F.comp (id _) = F := by { cases F, refl, }\n\n@[simp] lemma id_comp {U : Type*} [quiver U] {V : Type*} [quiver V] (F : prefunctor U V) :\n  (id _).comp F = F := by { cases F, refl, }\n\n@[simp]\nlemma comp_assoc\n  {U V W Z : Type*} [quiver U] [quiver V] [quiver W] [quiver Z]\n  (F : prefunctor U V) (G : prefunctor V W) (H : prefunctor W Z) :\n  (F.comp G).comp H = F.comp (G.comp H) := rfl\n\ninfix ` \u2964q `:50 := prefunctor\n\ninfix ` \u22d9q `:60 := prefunctor.comp\n\nnotation `\ud835\udfedq` := id\n\nend prefunctor\n\nnamespace quiver\n\n/-- `V\u1d52\u1d56` reverses the direction of all arrows of `V`. -/\ninstance opposite {V} [quiver V] : quiver V\u1d52\u1d56 :=\n\u27e8\u03bb a b, (unop b) \u27f6 (unop a)\u27e9\n\n/--\nThe opposite of an arrow in `V`.\n-/\ndef hom.op {V} [quiver V] {X Y : V} (f : X \u27f6 Y) : op Y \u27f6 op X := f\n/--\nGiven an arrow in `V\u1d52\u1d56`, we can take the \"unopposite\" back in `V`.\n-/\ndef hom.unop {V} [quiver V] {X Y : V\u1d52\u1d56} (f : X \u27f6 Y) : unop Y \u27f6 unop X := f\n\nattribute [irreducible] quiver.opposite\n\n/-- A type synonym for a quiver with no arrows. -/\n@[nolint has_nonempty_instance]\ndef empty (V) : Type u := V\n\ninstance empty_quiver (V : Type u) : quiver.{u} (empty V) := \u27e8\u03bb a b, pempty\u27e9\n\n@[simp] lemma empty_arrow {V : Type u} (a b : empty V) : (a \u27f6 b) = pempty := rfl\n\n/-- A quiver is thin if it has no parallel arrows. -/\n@[reducible] def is_thin (V : Type u) [quiver V] := \u2200 (a b : V), subsingleton (a \u27f6 b)\n\nend quiver\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/combinatorics/quiver/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.3984685383027744}}
{"text": "import data.equiv \n/- this file contains the lines\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) :=\n\u27e8_, to_fun\u27e9\n\n-/\n\n-- this fails:\n--theorem XXX {\u03b1 \u03b2 : Type} : has_coe_to_fun (equiv \u03b1 \u03b2) := by apply_instance\n\n\n\nvariables {\u03b1 \u03b2 : Type} (e : equiv \u03b1 \u03b2) (a : \u03b1)\n\n#check (\u21d1e : \u03b1 \u2192 \u03b2)\n#print has_coe_to_fun\n#print notation \u21d1 \n#print coe_fn \n#check e   \n-- #check (a : e)", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/scratch/coe_to_fun_woes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.39845280832262786}}
{"text": "import verification.semantics.stream_props\nimport verification.semantics.stream_zero\nimport verification.misc\nimport verification.semantics.finsuppeval\nimport tactic.zify\nimport tactic.linarith\nimport tactic.abel\n\nopen_locale classical\nnoncomputable theory\n\nvariables {\u03b9 : Type} {\u03b1 : Type*}\n  [linear_order \u03b9]\n\nsection defs\nvariables [has_zero \u03b1]\n  [has_add \u03b1]\n\n@[simps]\ninstance : has_add (Stream \u03b9 \u03b1) := \u27e8\u03bb a b,\n{ \u03c3 := a.\u03c3 \u00d7 b.\u03c3,\n  valid := \u03bb s, a.valid s.1 \u2228 b.valid s.2,\n  ready := \u03bb s, (a.to_order s.1 \u2264 b.to_order s.2 \u2227 a.ready s.1) \u2228 (b.to_order s.2 \u2264 a.to_order s.1 \u2227 b.ready s.2),\n  next := \u03bb s h, (if H : a.to_order s.1 \u2264 b.to_order s.2 then a.next s.1 (valid_of_le_or h H) else s.1, \n                  if H : b.to_order s.2 \u2264 a.to_order s.1 then b.next s.2 (valid_of_le_or h.symm H) else s.2),\n  index := \u03bb s h, option.get (show (min (a.index' s.1) (b.index' s.2)).is_some, by simpa),\n  value := \u03bb s h, (if a.to_order s.1 \u2264 b.to_order s.2 then a.value' s.1 else 0) +\n                  (if b.to_order s.2 \u2264 a.to_order s.1 then b.value' s.2 else 0) }\u27e9 \n\n-- Some weird inequality lemma I extracted from the main proof\nprivate lemma succ_le_min_succ_add_succ {a b b' d : \u2115} (ha : a \u2264 b + d) (hb' : b \u2264 b') :\n  a + 1 \u2264 (min b' (b + 1)) + (d + 1) :=\nby { rw [\u2190 add_assoc, nat.succ_le_succ_iff, min_def], split_ifs; linarith, }\n\nend defs\n\nsection lemmas\nvariables {a : Stream \u03b9 \u03b1} {b : Stream \u03b9 \u03b1} {x : a.\u03c3} {y : b.\u03c3}\n\nsection index_lemmas\nvariables [has_zero \u03b1] [has_add \u03b1]\n\nlemma lt_index_of_valid (h : a.to_order x \u2264 b.to_order y) (hva : a.valid x) :\n  \u2191(a.index x hva) \u2264 b.index' y :=\nby { simp only [\u2190 Stream.index'_val hva], exact prod.lex.fst_le_of_le h, }\n\nlemma Stream.min_index (h : a.to_order x \u2264 b.to_order y) (hva : a.valid x) :\n  min (a.index' x) (b.index' y) = a.index x hva :=\nby { rw [Stream.index'_val hva, min_eq_left_iff], exact lt_index_of_valid h hva, }\n\n@[simp] lemma Stream.add_index' (a : Stream \u03b9 \u03b1) (b : Stream \u03b9 \u03b1) (xy : a.\u03c3 \u00d7 b.\u03c3) :\n  (a + b).index' xy = min (a.index' xy.1) (b.index' xy.2) :=\nbegin\n  rcases xy with \u27e8x, y\u27e9,\n  by_cases H : (a + b).valid (x, y),\n  { simp [Stream.index'_val H], },\n  { have : \u00aca.valid x \u2227 \u00acb.valid y := by { simpa [not_or_distrib] using H, },\n    simp [Stream.index', H, this], }\nend\n\nend index_lemmas\n\nsection value_lemmas\nvariables [add_comm_monoid \u03b1]\n\nlemma Stream.add.eval\u2080_left (h : a.to_order x < b.to_order y) (hva H) :\n  (a + b).eval\u2080 (x, y) H = a.eval\u2080 x hva :=\nbegin\n  simp [Stream.eval\u2080, h.le, h.not_le],\n  split_ifs with H,\n  { simp [Stream.min_index h.le hva, Stream.value'_val H], },\n  { refl, },\nend\n\nlemma Stream.add_eval\u2080_right (h : b.to_order y < a.to_order x) (hvb H) :\n  (a + b).eval\u2080 (x, y) H = b.eval\u2080 y hvb :=\nbegin\n  simp [Stream.eval\u2080, h.le, h.not_le],\n  split_ifs with H,\n  { simp [H, Stream.min_index h.le hvb, Stream.value'_val H, min_comm (a.index' _) _], },\n  { refl, },\nend\n\nlemma Stream.add_eval\u2080_both (h : a.to_order x = b.to_order y) (hva hvb H) :\n  (a + b).eval\u2080 (x, y) H = a.eval\u2080 x hva + b.eval\u2080 y hvb :=\nbegin\n  simp [Stream.eval\u2080, h],\n  have : a.index _ hva = b.index _ hvb, { simpa [Stream.index'_val hva, Stream.index'_val hvb] using congr_arg prod.fst h, },\n  by_cases H\u2081 : a.ready x; by_cases H\u2082 : b.ready y; simp [H\u2081, H\u2082, Stream.value', Stream.min_index h.le hva, this],\nend\n\nend value_lemmas\nend lemmas\n\n@[elab_as_eliminator]\ntheorem Stream.add_induction [has_zero \u03b1] [has_add \u03b1] (a : Stream \u03b9 \u03b1) (b : Stream \u03b9 \u03b1) (x : a.\u03c3) (y : b.\u03c3)\n  {B\u2081 B\u2082 : \u2115} (ha : a.bound_valid B\u2081 x) (hb : b.bound_valid B\u2082 y)\n  (P : \u2200 (x : a.\u03c3) (y : b.\u03c3) (k\u2081 : \u2115) (k\u2082 : \u2115) (N : \u2115), Prop)\n  (hP_base : \u2200 (x y), P x y 0 0 0)\n  (hP_invalid : \u2200 (x y B\u2081 B\u2082 N) (H : \u00ac(a + b).valid (x, y)) (hvx : \u00aca.valid x) (hvy : \u00acb.valid y), a.bound_valid B\u2081 x \u2192 b.bound_valid B\u2082 y \u2192 P x y B\u2081 B\u2082 N)\n  (hP_advance_a : \u2200 (x y k\u2081 k\u2082 N) (hvx : a.valid x), (a + b).valid (x, y) \u2192 a.to_order x < b.to_order y \u2192 P (a.next x hvx) y k\u2081 k\u2082 N \u2192 P x y (k\u2081 + 1) k\u2082 (N + 1))\n  (hP_advance_b : \u2200 (x y k\u2081 k\u2082 N) (hvy : b.valid y), (a + b).valid (x, y) \u2192 b.to_order y < a.to_order x \u2192 P x (b.next y hvy) k\u2081 k\u2082 N \u2192 P x y k\u2081 (k\u2082 + 1) (N + 1))\n  (hP_advance_both : \u2200 (x y k\u2081 k\u2082 B\u2081 N) (hvx : a.valid x) (hvy : b.valid y), (a + b).valid (x, y) \u2192 a.bound_valid (B\u2081 + 1) x \u2192 a.to_order x = b.to_order y \u2192 P (a.next x hvx) (b.next y hvy) k\u2081 k\u2082 N \u2192 P x y (min (B\u2081 + 1) (k\u2081 + 1)) (k\u2082 + 1) (N + 1)) :\n  P x y B\u2081 B\u2082 (B\u2081 + B\u2082) :=\nbegin\n  suffices : \u2200 n \u2264 B\u2081 + B\u2082, \u2203 (k\u2081 k\u2082 : \u2115), k\u2081 \u2264 B\u2081 \u2227 k\u2082 \u2264 B\u2082 \u2227 n \u2264 k\u2081 + k\u2082 \u2227 P x y k\u2081 k\u2082 n,\n  { obtain \u27e8k\u2081, k\u2082, h\u2081, h\u2082, hn, hP\u27e9 := this (B\u2081 + B\u2082) rfl.le,\n    obtain \u27e8rfl, rfl\u27e9 : k\u2081 = B\u2081 \u2227 k\u2082 = B\u2082 := by split; linarith only [h\u2081, h\u2082, hn],\n    exact hP, },\n  intros n hn,\n  induction n with n ih generalizing B\u2081 B\u2082 x y,\n  { use [0, 0], simpa using hP_base x y, },\n  by_cases H : (a + b).valid (x, y), swap,\n  { -- Invalid\n    obtain \u27e8hvx, hvy\u27e9 : \u00aca.valid x \u2227 \u00acb.valid y := by simpa [not_or_distrib] using H,\n    refine \u27e8B\u2081, B\u2082, rfl.le, rfl.le, hn, _\u27e9, \n    exact hP_invalid _ _ _ _ _ H hvx hvy ha hb, },\n  rcases lt_trichotomy (a.to_order x) (b.to_order y) with h|h|h, swap, rotate 1,\n  { -- Advancing `a`\n    have hvx : a.valid x := valid_of_le_or H h.le,\n    cases B\u2081, { cases hvx.bound_pos ha, },\n    rw [Stream.bound_valid_succ] at ha,\n    obtain \u27e8k\u2081, k\u2082, hk\u2081, hk\u2082, n_le, hP\u27e9 := ih (a.next _ hvx) y (ha hvx) hb (by simpa [nat.succ_eq_one_add, add_assoc] using hn),\n    refine \u27e8k\u2081 + 1, k\u2082, nat.succ_le_succ hk\u2081, hk\u2082, _, _\u27e9,\n    { ac_change _ \u2264 k\u2081 + k\u2082 + 1, exact nat.succ_le_succ n_le, },\n    exact hP_advance_a _ _ _ _ _ hvx H h hP, },\n  { -- Advancing `b`\n    have hvy : b.valid y := valid_of_le_or (or.symm H) h.le,\n    cases B\u2082, { cases hvy.bound_pos hb, },\n    rw [Stream.bound_valid_succ] at hb,\n    obtain \u27e8k\u2081, k\u2082, hk\u2081, hk\u2082, n_le, hP\u27e9 := ih x (b.next _ hvy) ha (hb hvy) (nat.le_of_succ_le_succ hn),\n    refine \u27e8k\u2081, k\u2082 + 1, hk\u2081, nat.succ_le_succ hk\u2082, nat.succ_le_succ n_le, _\u27e9,\n    exact hP_advance_b _ _ _ _ _ hvy H h hP, },\n  { -- Advancing both `a` and `b`\n    have hvx : a.valid x := valid_of_le_or H h.le,\n    have hvy : b.valid y := valid_of_le_or (or.symm H) h.symm.le,\n    cases B\u2081, { cases hvx.bound_pos ha, },\n    cases B\u2082, { cases hvy.bound_pos hb, },\n    have ha' := ha, rw nat.succ_eq_add_one at ha',\n    rw [Stream.bound_valid_succ] at ha hb,\n    obtain \u27e8k\u2081, k\u2082, hk\u2081, hk\u2082, n_le, hP\u27e9 := ih (a.next _ hvx) (b.next _ hvy) ((ha hvx).mono B\u2081.le_succ) (hb hvy) (nat.le_of_succ_le_succ hn),\n    refine \u27e8min (B\u2081 + 1) (k\u2081 + 1), k\u2082 + 1, min_le_left _ _, nat.succ_le_succ hk\u2082, succ_le_min_succ_add_succ n_le hk\u2081, _\u27e9, \n    exact hP_advance_both _ _ _ _ _ _ hvx hvy H ha' h hP, }\nend\n\ntheorem Stream.add_spec [add_comm_monoid \u03b1] (a : Stream \u03b9 \u03b1) (b : Stream \u03b9 \u03b1) (x : a.\u03c3) (y : b.\u03c3)\n  {B\u2081 B\u2082 : \u2115} (ha : a.bound_valid B\u2081 x) (hb : b.bound_valid B\u2082 y) :\n  (a + b).eval_steps (B\u2081 + B\u2082) (x, y) = a.eval_steps B\u2081 x + b.eval_steps B\u2082 y :=\nbegin\n  apply Stream.add_induction a b x y ha hb (\u03bb x y k\u2081 k\u2082 N, (a + b).eval_steps N (x, y) = a.eval_steps k\u2081 x + b.eval_steps k\u2082 y); clear_except,\n  { intros, simp, },\n  { intros x y B\u2081 B\u2082 n H hvx hvy ha hb,\n    simp [Stream.eval_invalid H, Stream.eval_invalid hvx, Stream.eval_invalid hvy], },\n  { intros x y k\u2081 k\u2082 n hvx H h he,\n    dsimp, simp [h.le, h.not_le, he, Stream.add.eval\u2080_left h hvx, H, hvx],\n    apply add_right_comm, },\n  { intros x y k\u2081 k\u2082 n hvy H h he, \n    dsimp [nat.add_succ], simp [H, hvy, h.le, h.not_le, he, Stream.add_eval\u2080_right h hvy, add_assoc], },\n  { intros x y k\u2081 k\u2082 B\u2081 n hvx hvy H ha h he, \n    rw Stream.eval_min_bound ha,\n    dsimp, simp [h.le, h.symm.le, he, Stream.add_eval\u2080_both h hvx hvy, H, hvx, hvy],\n    abel, },\nend\n\ntheorem Stream.add_spec_index [has_zero \u03b1] [has_add \u03b1] (a : Stream \u03b9 \u03b1) (b : Stream \u03b9 \u03b1) (x : a.\u03c3) (y : b.\u03c3)\n  {B\u2081 B\u2082 : \u2115} (ha : a.bound_valid B\u2081 x) (hb : b.bound_valid B\u2082 y) :\n  ((a + b).next'^[B\u2081 + B\u2082] (x, y)) = (a.next'^[B\u2081] x, b.next'^[B\u2082] y) :=\nbegin\n  apply Stream.add_induction a b x y ha hb (\u03bb x y k\u2081 k\u2082 N, ((a + b).next'^[N] (x, y)) = (a.next'^[k\u2081] x, b.next'^[k\u2082] y)); clear_except,\n  { intros, simp, },\n  { intros x y B\u2081 B\u2082 n H hvx hvy ha hb, \n    simp [Stream.next'_val_invalid' H, Stream.next'_val_invalid' hvx, Stream.next'_val_invalid' hvy], },\n  { intros x y k\u2081 k\u2082 n hvx H h hiter,\n    simp [Stream.next'_val hvx, Stream.next'_val H, h.le, h.not_le, hiter], },\n  { intros x y k\u2081 k\u2082 n hvy H h hiter,\n    simp [Stream.next'_val hvy, Stream.next'_val H, h.le, h.not_le, hiter], },\n  { intros x y k\u2081 k\u2082 B\u2081 n hvx hvy H ha h hiter, \n    rw Stream.next'_min_bound ha,\n    simp [h.le, h.symm.le, hiter, Stream.next'_val hvx, Stream.next'_val hvy, Stream.next'_val H], }\nend\n\n@[simps]\ninstance [has_zero \u03b1] [has_add \u03b1] : has_add (StreamExec \u03b9 \u03b1) := \u27e8\u03bb a b,\n{ stream := a.stream + b.stream,\n  state := (a.state, b.state),\n  bound := a.bound + b.bound,\n  bound_valid := begin\n    rw [bound_valid_iff_next'_iterate], dsimp,\n    simpa [bound_valid_iff_next'_iterate, (Stream.add_spec_index a.stream b.stream _ _ a.bound_valid b.bound_valid),\n      not_or_distrib] using (and.intro a.bound_valid b.bound_valid),\n  end }\u27e9\n\n@[simp] lemma StreamExec.add_spec [add_comm_monoid \u03b1] (a : StreamExec \u03b9 \u03b1) (b : StreamExec \u03b9 \u03b1) :\n  (a + b).eval = a.eval + b.eval := (Stream.add_spec a.stream b.stream _ _ a.bound_valid b.bound_valid)\n\nlemma Stream.add_monotonic [has_zero \u03b1] [has_add \u03b1] {a : Stream \u03b9 \u03b1} {b : Stream \u03b9 \u03b1}\n  (ha : a.monotonic) (hb : b.monotonic) :\n  (a + b).monotonic :=\nbegin\n  rintros \u27e8x, y\u27e9 h,\n  simp only [Stream.add_index'],\n  refine min_le_min _ _,\n  { simp, split_ifs, { exact ha _, }, exact rfl.le, },\n  { simp, split_ifs, { exact hb _, }, exact rfl.le, },\nend\n\nlemma Stream.add_reduced [has_zero \u03b1] [has_add \u03b1] {a : Stream \u03b9 \u03b1} {b : Stream \u03b9 \u03b1}\n  (ha : a.reduced) (hb : b.reduced) : (a + b).reduced :=\nbegin\n  rintros \u27e8x, y\u27e9 hv hr,\n  rcases hr with \u27e8h, hr\u27e9|\u27e8h, hr\u27e9;\n  dsimp at hr h \u22a2,\n  { simp [h, (show min (a.index' x) (b.index' y) = _, from min_eq_left (prod.lex.fst_le_of_le h))],\n    apply ne_min_of_ne_and_ne, { exact ha _ hr, },\n    split_ifs with H,\n    { obtain \u27e8h\u2081, h\u2082\u27e9 : a.index' x = b.index' y \u2227 (a.ready x \u2194 b.ready y),\n      { simpa [Stream.to_order] using (antisymm h H), },\n      rw h\u2081, exact hb _ (by rwa \u2190 h\u2082), },\n    exact ne_of_lt (prod.lex.fst_lt_of_lt_of_le (lt_of_le_not_le h H) (by simp [hr])), },\n  { simp [h, (show min (a.index' x) (b.index' y) = _, from min_eq_right (prod.lex.fst_le_of_le h))],\n    apply ne_min_of_ne_and_ne, swap, { exact hb _ hr, },\n    split_ifs with H,\n    { obtain \u27e8h\u2081, h\u2082\u27e9 : a.index' x = b.index' y \u2227 (a.ready x \u2194 b.ready y),\n      { simpa [Stream.to_order] using (antisymm H h), },\n      rw \u2190 h\u2081, exact ha _ (by rwa h\u2082), },\n    exact ne_of_lt (prod.lex.fst_lt_of_lt_of_le (lt_of_le_not_le h H) (by simp [hr])), },\nend\n\ninstance StreamExec.AddZeroEval [add_comm_monoid \u03b1] : AddZeroEval (StreamExec \u03b9 \u03b1) \u03b9 \u03b1 :=\n{ hadd := StreamExec.add_spec,\n  hzero := StreamExec.zero_eval }\n\nlemma add_value_eval {\u03b9 \u03b1 \u03b9' \u03b1' : Type*} [linear_order \u03b9] [add_comm_monoid \u03b1'] [AddZeroEval \u03b1 \u03b9' \u03b1'] \n  (a b : StreamExec \u03b9 \u03b1) :\n  (Eval.eval <$\u2082> (a + b)) = (Eval.eval <$\u2082> a) + (Eval.eval <$\u2082> b) :=\nbegin\n  ext; solve_refl,\n  simp [apply_ite Eval.eval],\n  congr; rw [Stream.bimap_value'_apply]; simp,\nend\n\ninstance {\u03b9 \u03b1 \u03b9' \u03b1' : Type*} [linear_order \u03b9] [add_comm_monoid \u03b1'] [AddZeroEval \u03b1 \u03b9' \u03b1'] :\n  AddZeroEval (StreamExec \u03b9 \u03b1) \u03b9 (\u03b9' \u2192\u2080 \u03b1') :=\n{ hadd := \u03bb x y, by simp [Eval.eval, add_value_eval],\n  hzero := StreamExec.zero_eval }\n\ninstance {\u03b9 : Type} {\u03b1 : Type*} [linear_order \u03b9] [has_zero \u03b1] [has_add \u03b1] : has_add (SimpleStream \u03b9 \u03b1) := \u27e8\u03bb a b,\n{ simple :=\n  { monotonic := Stream.add_monotonic a.monotonic b.monotonic,\n    reduced := Stream.add_reduced a.reduced b.reduced },\n  ..(@has_add.add (StreamExec \u03b9 \u03b1) _ a b) }\u27e9\n\ninstance SimpleStream.AddZeroEvalBase [add_comm_monoid \u03b1] : AddZeroEval (SimpleStream \u03b9 \u03b1) \u03b9 \u03b1 :=\n{ hadd := \u03bb x y, by exact AddZeroEval.hadd (\u2191x : StreamExec _ _) (\u2191y : StreamExec _ _),\n  hzero := SimpleStream.zero_eval }\n\ninstance SimpleStream.AddZeroEval {\u03b9 \u03b1 \u03b9' \u03b1' : Type*} [linear_order \u03b9] [add_comm_monoid \u03b1'] [AddZeroEval \u03b1 \u03b9' \u03b1'] :\n  AddZeroEval (SimpleStream \u03b9 \u03b1) \u03b9 (\u03b9' \u2192\u2080 \u03b1') :=\n{ hadd := \u03bb x y, by exact AddZeroEval.hadd (\u2191x : StreamExec _ _) (\u2191y : StreamExec _ _),\n  hzero := SimpleStream.zero_eval }\n\n\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/verification/semantics/stream_add.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3984393736497742}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.category.Cat\nimport category_theory.elements\n\n/-!\n# The Grothendieck construction\n\nGiven a functor `F : C \u2964 Cat`, the objects of `grothendieck F`\nconsist of dependent pairs `(b, f)`, where `b : C` and `f : F.obj c`,\nand a morphism `(b, f) \u27f6 (b', f')` is a pair `\u03b2 : b \u27f6 b'` in `C`, and\n`\u03c6 : (F.map \u03b2).obj f \u27f6 f'`\n\nCategories such as `PresheafedSpace` are in fact examples of this construction,\nand it may be interesting to try to generalize some of the development there.\n\n## Implementation notes\n\nReally we should treat `Cat` as a 2-category, and allow `F` to be a 2-functor.\n\nThere is also a closely related construction starting with `G : C\u1d52\u1d56 \u2964 Cat`,\nwhere morphisms consists again of `\u03b2 : b \u27f6 b'` and `\u03c6 : f \u27f6 (F.map (op \u03b2)).obj f'`.\n\n## References\n\nSee also `category_theory.functor.elements` for the category of elements of functor `F : C \u2964 Type`.\n\n* https://stacks.math.columbia.edu/tag/02XV\n* https://ncatlab.org/nlab/show/Grothendieck+construction\n\n-/\n\nuniverse u\n\nnamespace category_theory\n\nvariables {C D : Type*} [category C] [category D]\nvariables (F : C \u2964 Cat)\n\n/--\nThe Grothendieck construction (often written as `\u222b F` in mathematics) for a functor `F : C \u2964 Cat`\ngives a category whose\n* objects `X` consist of `X.base : C` and `X.fiber : F.obj base`\n* morphisms `f : X \u27f6 Y` consist of\n  `base : X.base \u27f6 Y.base` and\n  `f.fiber : (F.map base).obj X.fiber \u27f6 Y.fiber`\n-/\n@[nolint has_inhabited_instance]\nstructure grothendieck :=\n(base : C)\n(fiber : F.obj base)\n\nnamespace grothendieck\n\nvariables {F}\n\n/--\nA morphism in the Grothendieck category `F : C \u2964 Cat` consists of\n`base : X.base \u27f6 Y.base` and `f.fiber : (F.map base).obj X.fiber \u27f6 Y.fiber`.\n-/\nstructure hom (X Y : grothendieck F) :=\n(base : X.base \u27f6 Y.base)\n(fiber : (F.map base).obj X.fiber \u27f6 Y.fiber)\n\n@[ext] lemma ext {X Y : grothendieck F} (f g : hom X Y)\n  (w_base : f.base = g.base) (w_fiber : eq_to_hom (by rw w_base) \u226b f.fiber = g.fiber) : f = g :=\nbegin\n  cases f; cases g,\n  congr,\n  dsimp at w_base,\n  induction w_base,\n  refl,\n  dsimp at w_base,\n  induction w_base,\n  simpa using w_fiber,\nend\n\n/--\nThe identity morphism in the Grothendieck category.\n-/\n@[simps]\ndef id (X : grothendieck F) : hom X X :=\n{ base := \ud835\udfd9 X.base,\n  fiber := eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]), }\n\ninstance (X : grothendieck F) : inhabited (hom X X) := \u27e8id X\u27e9\n\n/--\nComposition of morphisms in the Grothendieck category.\n-/\n@[simps]\ndef comp {X Y Z : grothendieck F} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ base := f.base \u226b g.base,\n  fiber :=\n  eq_to_hom (by erw [functor.map_comp, functor.comp_obj]) \u226b\n    (F.map g.base).map f.fiber \u226b g.fiber, }\n\ninstance : category (grothendieck F) :=\n{ hom := \u03bb X Y, grothendieck.hom X Y,\n  id := \u03bb X, grothendieck.id X,\n  comp := \u03bb X Y Z f g, grothendieck.comp f g,\n  comp_id' := \u03bb X Y f,\n  begin\n    ext,\n    { dsimp,\n      -- We need to turn `F.map_id` (which is an equation between functors)\n      -- into a natural isomorphism.\n      rw \u2190 nat_iso.naturality_2 (eq_to_iso (F.map_id Y.base)) f.fiber,\n      simp,\n      refl, },\n    { simp, },\n  end,\n  id_comp' := \u03bb X Y f, by ext; simp,\n  assoc' := \u03bb W X Y Z f g h,\n  begin\n    ext, swap,\n    { simp, },\n    { dsimp,\n      rw \u2190 nat_iso.naturality_2 (eq_to_iso (F.map_comp _ _)) f.fiber,\n      simp,\n      refl, },\n  end, }\n\n@[simp] lemma id_fiber' (X : grothendieck F) :\n  hom.fiber (\ud835\udfd9 X) = eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]) :=\nid_fiber X\n\nlemma congr {X Y : grothendieck F} {f g : X \u27f6 Y} (h : f = g) :\n  f.fiber = eq_to_hom (by subst h) \u226b g.fiber :=\nby { subst h, dsimp, simp, }\n\nsection\nvariables (F)\n\n/-- The forgetful functor from `grothendieck F` to the source category. -/\n@[simps]\ndef forget : grothendieck F \u2964 C :=\n{ obj := \u03bb X, X.1,\n  map := \u03bb X Y f, f.1, }\n\nend\n\nuniverse w\nvariables (G : C \u2964 Type w)\n\n/--\nThe Grothendieck construction applied to a functor to `Type`\n(thought of as a functor to `Cat` by realising a type as a discrete category)\nis the same as the 'category of elements' construction.\n-/\ndef grothendieck_Type_to_Cat : grothendieck (G \u22d9 Type_to_Cat) \u224c G.elements :=\n{ functor :=\n  { obj := \u03bb X, \u27e8X.1, X.2\u27e9,\n    map := \u03bb X Y f, \u27e8f.1, f.2.1.1\u27e9 },\n  inverse :=\n  { obj := \u03bb X, \u27e8X.1, X.2\u27e9,\n    map := \u03bb X Y f, \u27e8f.1, \u27e8\u27e8f.2\u27e9\u27e9\u27e9 },\n  unit_iso := nat_iso.of_components (\u03bb X, by { cases X, exact iso.refl _, })\n    (by { rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8base, \u27e8\u27e8f\u27e9\u27e9\u27e9, dsimp at *, subst f, simp, }),\n  counit_iso := nat_iso.of_components (\u03bb X, by { cases X, exact iso.refl _, })\n    (by { rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8f, e\u27e9, dsimp at *, subst e, simp }),\n  functor_unit_iso_comp' := by { rintro \u27e8\u27e9, dsimp, simp, refl, } }\n\nend grothendieck\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.39822662247427704}}
{"text": "import lambda_calculus.utlc.beta.distance\nimport lambda_calculus.utlc.beta.encoding.basic\nimport lambda_calculus.utlc.beta.encoding.core\nimport lambda_calculus.utlc.beta.encoding.list\nimport lambda_calculus.utlc.beta.complexity.core\nimport complexity.basic\nimport complexity.nat\n\nopen complexity\nopen lambda_calculus.utlc.\u03b2.encoding\n\nnamespace lambda_calculus\nnamespace utlc\nnamespace \u03b2\nnamespace complexity\nnamespace list\nlocal attribute [simp] \u03b2.normal_iteration \u03b2.strategic_reduction_step\nlocal attribute [simp] down_shift head_reduced\nlocal attribute [simp] complexity.cast_unwrap distance_model\n\n\n-- def cons_prog: encoded_program := \u27e8 (\u039b \u039b \u039b \u039b \u21930\u00b7\u21933\u00b7\u21932), by simp[\u2190 one_add_one_eq_two] \u27e9\n\n-- instance cons_complexity\n--   (\u03b1: Type) [complexity.has_encoding distance_model \u03b1]:\n--   complexity.has_complexity distance_model (@list.cons \u03b1) :=\n--   \u27e8 \u27e8 (\u03bb _ _, (2:\u2115)), \u27e8 cons_prog, \u03bb x xs, distance_le_of_normal_iteration _\n--     (by simp [cons_prog, has_encoding.value, encoding.list.encode_list, encode]) \u27e9 \u27e9 \u27e9\n\n-- def handle_nil (f y: utlc): utlc := f\n-- def handle_cons (f y: utlc): utlc := \u039b \u039b (f \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21931\u00b7\u21930\u00b7((y \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21930)\n-- def fold_cons (f y: utlc): utlc := \u039b \u039b ((y \u2191\u00b9 0 \u2191\u00b9 1)\u00b7((f \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21931\u00b7\u21930)\u00b7\u21930)\n-- def rec_list (y g f\u2080 f\u2081: utlc): utlc := g\u00b7handle_nil f\u2080 y\u00b7handle_cons f\u2081 y\n\n-- section\n-- local notation a `[` b `:=` c  `]` : 70 := has_substitution.substitution a b c\n\n-- theorem rec_list_sub {y g f\u2080 f\u2081: utlc} (n: \u2115) (x: utlc):\n--   (rec_list y g f\u2080 f\u2081)[n:=x] = (rec_list (y[n:=x]) (g[n:=x]) (f\u2080[n:=x]) (f\u2081[n:=x])) :=\n-- begin\n--   simp [rec_list, handle_nil, handle_cons, substitution_shift_ge],\n--   repeat { rw [substitution_shift_ge] },\n--   all_goals { linarith },\n-- end\n\n-- theorem rec_list_nil {\u03b1: Type} [\u03b1_en: complexity.has_encoding distance_model \u03b1]\n--   (y: utlc) (f\u2080 f\u2081: utlc):\n--   distance_le 2\n--     (rec_list y (encoding.list.encode_list (@list.nil \u03b1)).value f\u2080 f\u2081)\n--     f\u2080 :=\n-- begin\n--   rw [rec_list, list.encode_list, \u03b2.encoding.alternative],\n--   simp,\n--   apply distance_le_of_normal_iteration,\n--   simp [handle_nil],\n-- end\n\n-- theorem rec_list_cons {\u03b1: Type} [\u03b1_en: complexity.has_encoding distance_model \u03b1]\n--   (y: utlc) (x: \u03b1) (xs: list \u03b1) (f\u2080 f\u2081: utlc):\n--   distance_le 4\n--     (rec_list y (encoding.list.encode_list (x :: xs)).value f\u2080 f\u2081)\n--     (f\u2081\u00b7(complexity.encode distance_model x).value\u00b7(complexity.encode distance_model xs).value\u00b7(y\u00b7(complexity.encode distance_model xs).value)) :=\n-- begin\n--   rw [rec_list, list.encode_list, \u03b2.encoding.alternative],\n--   apply distance_le_of_normal_iteration,\n--   simp [handle_cons, complexity.encode, has_encoding.value],\n-- end\n-- end\n\n\n-- namespace foldl_complexity\n\n-- def cost {\u03b1: Type} {\u03b2: Type}\n--   (rf: \u03b1 \u2192 \u03b2 \u2192 \u03b1)\n--   (c_nil: \u2115) (c_cons: \u03b1 \u2192 \u03b2 \u2192 \u2115):\n--   \u03b1 \u2192 list \u03b2 \u2192 \u2115 :=\n--   \u03bb a xs, (list.foldl (\u03bb (ca: \u2115 \u00d7 \u03b1) b, (ca.fst + c_cons ca.snd b, rf ca.snd b)) ((0:\u2115), a) xs).fst\n\n-- | ([]) a := c_nil\n-- | (x :: xs) a := c_cons a x + cost xs (rf a x)\n-- end foldl_complexity\n\n-- instance foldl_complexity\n--   (\u03b1 \u03b2: Type) [has_encoding distance_model \u03b1] [has_encoding distance_model \u03b2]\n--   (f: \u03b1 \u2192 \u03b2 \u2192 \u03b1) [cf: has_complexity distance_model f]:\n--   has_complexity distance_model (list.foldl f) :=\n-- begin\n--   fconstructor,\n--   fconstructor,\n--   exact (foldl_complexity.cost f 0 cf.value.cost),\n--   rcases cf.value with \u27e8cfc, fp, cfp\u27e9,\n--   fconstructor,\n--   fconstructor,\n--   exact (yrec \n\n-- end\n\n\n\n\nend list\nend complexity\nend \u03b2\nend utlc\nend lambda_calculus\n", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/lambda_calculus/utlc/beta/complexity/list.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7606506526772884, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3981400300429889}}
{"text": "import for_mathlib.category_theory.triangulated.shift_compatibility\nimport for_mathlib.category_theory.triangulated.pretriangulated_misc\nimport category_theory.preadditive.opposite\nimport for_mathlib.category_theory.preadditive.misc\nimport tactic.abel\n\nnoncomputable theory\n\nuniverse u\n\nopen category_theory category_theory.category category_theory.limits\n\nnamespace category_theory\n\nvariables {C : Type*} {A B : Type u} [category C] [add_monoid A] [add_monoid B]\n  {G : Type*} [add_comm_group G]\n\nlocal attribute [instance, reducible] endofunctor_monoidal_category\nlocal attribute [reducible] discrete.add_monoidal\n\nvariables (C A)\n\ndef has_shift_op [has_shift C A] : has_shift C\u1d52\u1d56 A :=\nhas_shift_mk _ _\n{ F := \u03bb a, (shift_functor C a).op,\n  \u03b5 := (functor.op_unop_equiv C C).functor.map_iso (shift_functor_zero C A).op,\n  \u03bc := \u03bb a b, (functor.op_unop_equiv C C).functor.map_iso (shift_functor_add C a b).op,\n  associativity := \u03bb a b c X, quiver.hom.unop_inj begin\n    have eq := congr_arg iso.inv (monoidal_functor.associativity_iso_eq (shift_monoidal_functor C A)\n      (discrete.mk a) (discrete.mk b) (discrete.mk c)),\n    replace eq := congr_app eq (opposite.unop X),\n    dsimp at \u22a2 eq,\n    erw [comp_id, id_comp, functor.map_id, functor.map_id, comp_id] at eq,\n    simpa only [\u2190 eq, eq_to_hom_unop, eq_to_hom_map, eq_to_hom_app],\n  end,\n  left_unitality := \u03bb a X, quiver.hom.unop_inj begin\n    dsimp,\n    simpa only [obj_\u03b5_inv_app, eq_to_iso.hom, \u03bc_inv_hom_app_assoc, eq_to_hom_unop,\n      eq_to_hom_map, eq_to_hom_app],\n  end,\n  right_unitality := \u03bb a X, quiver.hom.unop_inj begin\n    dsimp,\n    simpa only [\u03b5_inv_app_obj, eq_to_iso.hom, \u03bc_inv_hom_app_assoc, eq_to_hom_unop,\n      eq_to_hom_map, eq_to_hom_app],\n  end, }\n\nvariable {A}\n\ndef has_shift_pull [has_shift C B] (\u03c6 : A \u2192+ B) : has_shift C A :=\n\u27e8monoidal_functor.comp (discrete.add_monoidal_functor \u03c6) (shift_monoidal_functor C B)\u27e9\n\nvariable {C}\n\ndef has_shift.op (s : has_shift C A) : has_shift C\u1d52\u1d56 A :=\nby { letI := s, apply has_shift_op, }\n\ndef has_shift.pull (s : has_shift C B) (\u03c6 : A \u2192+ B) : has_shift C A :=\nby { letI := s, exact has_shift_pull C \u03c6, }\n\ndef has_shift.op_neg (s : has_shift C G) :\n  has_shift C\u1d52\u1d56 G :=\ns.op.pull (\u27e8\u03bb g, -g, by tidy, \u03bb x y, by abel\u27e9 : G \u2192+ G)\n\nvariables (C G)\n\ndef has_shift_op_neg [has_shift C G] : has_shift C\u1d52\u1d56 G :=\nhas_shift.op_neg infer_instance\n\ndef has_shift_op_neg_\u2124 [has_shift C \u2124] : has_shift C\u1d52\u1d56 \u2124 :=\nhas_shift.op_neg infer_instance\n\nnamespace pretriangulated\n\nvariables (C) [has_shift C \u2124]\n\nlocal attribute [instance] has_shift_op_neg_\u2124\n\nlemma _root_.category_theory.shift_functor_op_map\n  {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) (n : \u2124) :\n  (shift_functor C\u1d52\u1d56 n).map f = ((shift_functor C (-n)).map f.unop).op := rfl\n\nvariable [preadditive C]\n\nnamespace triangle_op_equivalence\n\n@[simps]\ndef functor : (triangle C)\u1d52\u1d56 \u2964 triangle C\u1d52\u1d56 :=\n{ obj := \u03bb T,\n  { obj\u2081 := opposite.op T.unop.obj\u2083,\n    obj\u2082 := opposite.op T.unop.obj\u2082,\n    obj\u2083 := opposite.op T.unop.obj\u2081,\n    mor\u2081 := T.unop.mor\u2082.op,\n    mor\u2082 := T.unop.mor\u2081.op,\n    mor\u2083 := ((shift_equiv C (1 : \u2124)).unit_iso.inv.app T.unop.obj\u2081).op \u226b\n      (T.unop.mor\u2083\u27e6(-1 : \u2124)\u27e7').op, },\n  map := \u03bb T\u2081 T\u2082 f,\n  { hom\u2081 := f.unop.hom\u2083.op,\n    hom\u2082 := f.unop.hom\u2082.op,\n    hom\u2083 := f.unop.hom\u2081.op,\n    comm\u2081' := quiver.hom.unop_inj f.unop.comm\u2082.symm,\n    comm\u2082' := quiver.hom.unop_inj f.unop.comm\u2081.symm,\n    comm\u2083' := quiver.hom.unop_inj begin\n      dsimp only,\n      have h := (shift_equiv C (1 : \u2124)).unit_iso.inv.naturality f.unop.hom\u2081,\n      simp only [category_theory.shift_functor_op_map, unop_comp, quiver.hom.unop_op, assoc],\n      erw \u2190 (shift_equiv C (1 : \u2124)).unit_iso.inv.naturality f.unop.hom\u2081,\n      dsimp only [shift_equiv, functor.comp_map],\n      simp only [\u2190 assoc, \u2190 functor.map_comp, f.unop.comm\u2083],\n    end, }, }\n\n@[simps]\ndef inverse : triangle C\u1d52\u1d56 \u2964 (triangle C)\u1d52\u1d56 :=\n{ obj := \u03bb T, opposite.op\n  { obj\u2081 := T.obj\u2083.unop,\n    obj\u2082 := T.obj\u2082.unop,\n    obj\u2083 := T.obj\u2081.unop,\n    mor\u2081 := T.mor\u2082.unop,\n    mor\u2082 := T.mor\u2081.unop,\n    mor\u2083 := (shift_equiv C (1 : \u2124)).counit_iso.inv.app T.obj\u2081.unop \u226b T.mor\u2083.unop\u27e6(1 : \u2124)\u27e7', },\n  map := \u03bb T\u2081 T\u2082 f, quiver.hom.op\n  { hom\u2081 := f.hom\u2083.unop,\n    hom\u2082 := f.hom\u2082.unop,\n    hom\u2083 := f.hom\u2081.unop,\n    comm\u2081' := quiver.hom.op_inj f.comm\u2082.symm,\n    comm\u2082' := quiver.hom.op_inj f.comm\u2081.symm,\n    comm\u2083' := begin\n      dsimp only,\n      have h := functor.congr_map (shift_functor C (1 : \u2124)) (congr_arg quiver.hom.unop f.comm\u2083),\n      simp only [unop_comp, functor.map_comp] at h,\n      simp only [assoc, \u2190 h],\n      erw \u2190 nat_trans.naturality_assoc,\n      refl,\n    end, }, }\n\n@[simps]\ndef unit_iso : \ud835\udfed _ \u2245 functor C \u22d9 inverse C :=\nnat_iso.of_components (\u03bb T, begin\n  refine iso.op (_ : opposite.unop _ \u2245 opposite.unop T),\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _) (by tidy) (by tidy) _,\n  dsimp only [iso.refl],\n  rw [functor.map_id, comp_id, id_comp],\n  change (shift_equiv C (1 : \u2124)).counit_iso.inv.app T.unop.obj\u2083 \u226b\n    (shift_functor C (1 : \u2124)).map (((shift_functor C (-1 : \u2124)).map T.unop.mor\u2083) \u226b\n      ((shift_equiv C (1 : \u2124)).unit_iso.inv.app T.unop.obj\u2081)) = T.unop.mor\u2083,\n  erw [functor.map_comp, (shift_equiv C (1 : \u2124)).fun_inv_map],\n  simp only [assoc],\n  erw (shift_equiv C (1 : \u2124)).counit_inv_functor_comp,\n  erw comp_id,\n  erw iso.inv_hom_id_app_assoc,\nend) (\u03bb X Y f, quiver.hom.unop_inj (by tidy))\n\n@[simps]\ndef counit_iso : inverse C \u22d9 functor C \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb T, begin\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _)\n    (by tidy) (by tidy) (quiver.hom.unop_inj _),\n  dsimp only [iso.refl],\n  rw [functor.map_id, id_comp, comp_id],\n  change ((shift_equiv C (1 : \u2124)).counit_iso.inv.app (opposite.unop T.obj\u2081) \u226b\n    T.mor\u2083.unop\u27e6(1 : \u2124)\u27e7')\u27e6(-1 : \u2124)\u27e7' \u226b\n    ((shift_equiv C (1 : \u2124)).unit_iso.inv.app T.obj\u2083.unop) = T.mor\u2083.unop,\n  erw [functor.map_comp, (shift_equiv C (1 : \u2124)).inv_fun_map],\n  simp only [assoc, iso.hom_inv_id_app],\n  erw comp_id,\n  slice_lhs 1 2 { erw (shift_equiv C (1 : \u2124)).inverse_counit_inv_comp, },\n  erw id_comp,\nend) (by tidy)\n\nend triangle_op_equivalence\n\n@[simps]\ndef triangle_op_equivalence : (triangle C)\u1d52\u1d56 \u224c triangle C\u1d52\u1d56 :=\n{ functor := triangle_op_equivalence.functor C,\n  inverse := triangle_op_equivalence.inverse C,\n  unit_iso := triangle_op_equivalence.unit_iso C,\n  counit_iso := triangle_op_equivalence.counit_iso C, }\n\nvariable {C}\n\ndef triangle.op (T : triangle C) : triangle C\u1d52\u1d56 :=\n(triangle_op_equivalence C).functor.obj (opposite.op T)\n\ndef triangle.unop (T : triangle C\u1d52\u1d56) : triangle C :=\n((triangle_op_equivalence C).inverse.obj T).unop\n\ndef triangle.unop_op (T : triangle C) : T.op.unop \u2245 T :=\n((triangle_op_equivalence C).unit_iso.app (opposite.op T)).unop\n\ndef triangle.op_unop (T : triangle C\u1d52\u1d56) : T.unop.op \u2245 T :=\n(triangle_op_equivalence C).counit_iso.app T\n\nvariables [\u2200 (n : \u2124), (shift_functor C n).additive]\n\ninstance shift_functor_op_additive (n : \u2124) : (shift_functor C\u1d52\u1d56 n).additive :=\n(infer_instance : (shift_functor C (-n)).op.additive)\n\ndef triangle.unop_rotate (T : triangle C\u1d52\u1d56) : T.rotate.unop \u2245 T.unop.inv_rotate :=\nbegin\n  refine triangle.mk_iso _ _ (preadditive.mul_iso (-1) (iso.refl _)) (iso.refl _) (iso.refl _) _ _ _,\n  { change T.mor\u2083.unop \u226b \ud835\udfd9 _ = ((-1 : \u2124) \u2022 \ud835\udfd9 _) \u226b\n      -((shift_equiv C (1 : \u2124)).counit_iso.inv.app T.obj\u2081.unop \u226b T.mor\u2083.unop\u27e6(1 : \u2124)\u27e7')\u27e6(-1:\u2124)\u27e7' \u226b\n      (shift_equiv C (1 : \u2124)).unit_iso.inv.app T.obj\u2083.unop,\n    simp only [comp_id, neg_smul, one_smul, preadditive.comp_neg, preadditive.neg_comp, neg_neg,\n      id_comp, functor.map_comp, assoc],\n    erw (shift_equiv C (1 : \u2124)).inv_fun_map,\n    slice_rhs 1 2 { erw (shift_equiv C (1 : \u2124)).inverse_counit_inv_comp, },\n    simp only [assoc, iso.hom_inv_id_app],\n    erw [id_comp, comp_id], },\n  { dsimp only [iso.refl],\n    rw [id_comp, comp_id],\n    refl, },\n  { change ((shift_equiv C (1 : \u2124)).counit_iso.inv.app T.obj\u2082.unop \u226b\n      (-(T.mor\u2081).unop\u27e6(-1 : \u2124)\u27e7')\u27e6(1 : \u2124)\u27e7') \u226b ((-1 : \u2124) \u2022 \ud835\udfd9 _)\u27e6(1 : \u2124)\u27e7' =\n        \ud835\udfd9 _ \u226b T.mor\u2081.unop \u226b (shift_equiv C (1 : \u2124)).counit_iso.inv.app T.obj\u2081.unop,\n    simp only [functor.map_neg, neg_smul, one_smul, functor.map_id, preadditive.comp_neg,\n      preadditive.neg_comp, neg_neg, comp_id, id_comp],\n    erw \u2190 nat_trans.naturality,\n    refl, },\nend\n\nend pretriangulated\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/shift_op.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.39807080626194713}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad\n\nThe syntax of simple type theory, i.e. inductive definitions of types and terms.\n\nNotes:\n\nWhen it comes to basic types, type constructors, and constants, we distinguish between \"built in\"\nand \"user\" objects. The former are fixed for the implementation, and the implementation can refer\nto them directly (for example, logical connectives, arithmetic symbols, types like nat and bool,\nand type constructors like list, sum, and prod).\n\nTODO: Factor out types for constants, basic types, etc. These can include more information\nabout the environment, source, pretty-printing, etc.\n\nTODO: We do not have \"local_constants\". It is probably best to have these as special sorts of\nconstants, but we can revisit this decision.\n\nTODO: For now, we won't handle polymorphic types, though we will eventually (mostly following\nHOL light).\n\nTODO: For now, use naive equality tests. We can worry about more efficient tests (e.g. with\nunique id's) later, if necessary.\n\nTODO: if we replace each `= tt` by a coercion to bool, things break in some places. They are \nfixable, but then something more dramatic breaks in fol.syntax and the terms are huge.\n-/\nimport data.list\n\n/- TODO: move these -/\n\n@[simp]\ntheorem if_neg_eq {\u03b1 : Type*} (p : Prop) [decidable p] (a b : \u03b1) : \n  (if \u00ac p then a else b) = (if p then b else a) :=\nbegin\n  by_cases h : p; simp [h],\nend\n\ntheorem to_bool_eq_to_bool (p q : Prop) [decidable p] [decidable q] :\n  (to_bool p = to_bool q) \u2194 (p \u2194 q) :=\nby by_cases p; simp [h]\n\nnamespace list\n\n@[simp]\ntheorem not_cons_prefix_nil {\u03b1 : Type*} (a : \u03b1) (l : list \u03b1) : \u00ac a :: l <+: nil :=\nby { unfold is_prefix, intro h, cases h, contradiction }\n\n@[simp]\ntheorem cons_prefix_cons {\u03b1 : Type*} (a\u2081 a\u2082 : \u03b1) (l\u2081 l\u2082 : list \u03b1) :\n  a\u2081 :: l\u2081 <+: a\u2082 :: l\u2082 \u2194 a\u2081 = a\u2082 \u2227 l\u2081 <+: l\u2082 :=\nbegin\n  unfold is_prefix, \n  split; intro h,\n  { cases h with l h, split, apply head_eq_of_cons_eq h, existsi l, apply tail_eq_of_cons_eq h},\n  cases h with h\u2080 h\u2081, cases h\u2081 with l h\u2081,\n  existsi l, rw [h\u2080, \u2190 h\u2081], simp  \nend\n\ntheorem length_le_of_prefix {\u03b1 : Type*} {l\u2081 l\u2082 : list \u03b1} (h : l\u2081 <+: l\u2082) : length l\u2081 \u2264 length l\u2082 :=\nlength_le_of_sublist $ sublist_of_prefix h\n\ntheorem prefix_iff_eq_of_length_eq {\u03b1 : Type*} {l\u2081 l\u2082 : list \u03b1} (h : length l\u2081 = length l\u2082) :\n  l\u2081 <+: l\u2082 \u2194 (l\u2081 = l\u2082) :=\nby { split, { intro h', exact eq_of_prefix_of_length_eq h' h}, intro h', rw h'}\n\ntheorem drop_succ {\u03b1 : Type*} (n : nat) (l : list \u03b1) : drop n.succ l = drop n (drop 1 l) :=\nby induction l; simp\n\ntheorem drop_eq_nil  {\u03b1 : Type*} (n : nat) (l : list \u03b1) : drop n l = [] \u2194 n \u2265 l.length:=\nbegin\n  revert n,  \n  induction l with a l ih; simp, \n  { apply nat.zero_le }, \n  intro n, cases n with n; simp, \n  { rw add_comm, apply nat.zero_lt_succ },\n  rw ih, rw [add_comm, ge, ge, nat.succ_le_succ_iff]\nend\n\nend list\n\nnamespace hol\n\n/-\nInstead of option, we'll have failure produce error messages. But we will not use the full-blown\nexception monad or do anything fancy, to make it easier to reason about these programs.\n\nTODO: not using this yet.\n-/\n\ninductive except (\u03b1 : Type*)\n| ok   : \u03b1 \u2192 except\n| fail : string \u2192 except\n\n/- \n    Types    \n-/\n\nnamespace type\n\n  /- basic types -/\n\n  @[derive has_reflect, derive decidable_eq]\n  inductive basic : Type\n  | user : \u2115 \u2192 basic\n  | prop \n  | nat\n  | int\n  | bool\n  | unit\n\n  namespace basic\n    def repr : basic \u2192 string\n    | (user n) := \"(basic.user \" ++ n.repr ++ \")\"\n    | prop := \"basic.prop\"\n    | nat  := \"basic.nat\"\n    | int  := \"basic.int\"\n    | bool := \"basic.bool\"\n    | unit := \"basic.unit\"\n\n    instance : has_repr basic := \u27e8repr\u27e9   \n  end basic\n\n  /- type constructors -/\n\n  namespace constructor\n    @[derive has_reflect, derive decidable_eq]\n    inductive kind\n    | user : nat \u2192 kind\n    | list \n    | prod \n    | sum\n  end constructor\n\n  @[derive has_reflect, derive decidable_eq]\n  structure constructor :=\n  (symb : constructor.kind) (arity : nat)\n\n  namespace constructor\n    def user (n arity : nat)  : constructor := \u27e8kind.user n, arity\u27e9\n    def list : constructor := \u27e8kind.list, 1\u27e9 \n    def prod : constructor := \u27e8kind.prod, 2\u27e9 \n    def sum : constructor := \u27e8kind.sum, 2\u27e9 \n    \n    def repr : constructor \u2192 string\n    | \u27e8kind.user n, a\u27e9 := \"(constructor.user \" ++ n.repr ++ \" \" ++ a.repr ++\")\"\n    | \u27e8kind.list, _\u27e9   := \"constructor.list\"   \n    | \u27e8kind.prod, _\u27e9   := \"constructor.list\"   \n    | \u27e8kind.sum, _\u27e9    := \"constructor.list\"\n\n    instance : has_repr constructor := \u27e8repr\u27e9   \n  end constructor\n\nend type\n\n/- \nThe types themselves\n\nNotes:\n\nTo avoid a nested inductive type, we'll use iterated application for constructors.\n\nBasic and arrow types are really special cases of constructors, but we'll keep them separate.\n-/\n\n@[derive has_reflect, derive decidable_eq]\ninductive type\n| Var         : nat \u2192 type\n| Basic       : hol.type.basic \u2192 type\n| Arr         : type \u2192 type \u2192 type\n| Constructor : hol.type.constructor \u2192 type\n| App         : type \u2192 type \u2192 type\n\nnamespace type\n\ndef repr : type \u2192 string\n| (Var n)         := \"(Var \" ++ n.repr ++ \")\"\n| (Basic b)       := \"(Basic \" ++ b.repr ++ \")\"\n| (Arr t\u2081 t\u2082)     := \"(Arr \" ++ t\u2081.repr ++ \" \" ++ t\u2082.repr ++\")\"\n| (Constructor c) := \"(Constructor \" ++ c.repr ++ \")\"\n| (App t\u2081 t\u2082)     := \"(App \" ++ t\u2081.repr ++ \" \" ++ t\u2082.repr ++ \")\"\n\ninstance : has_repr type := \u27e8repr\u27e9\n\n-- `arities_ok t l` says the type `t` iteratively applied to `l` makes sense\ndef arities_ok_aux : type \u2192 list type \u2192 bool\n| (Var n)         l := l.empty\n| (Basic b)       l := l.empty\n| (Arr t\u2081 t\u2082)     l := arities_ok_aux t\u2081 l && arities_ok_aux t\u2082 l && l.empty\n| (Constructor c) l := c.arity = l.length\n| (App t\u2081 t\u2082)     l := arities_ok_aux t\u2082 [] && arities_ok_aux t\u2081 (t\u2082 :: l)\n\ndef arities_ok (t : type) : bool := arities_ok_aux t []\n\n/-- Returns the domain of an arrow type, or the type itself if it is not an arrow. -/\n@[simp]\ndef domain : type \u2192 type\n| (type.Arr t\u2081 t\u2082) := t\u2081\n| t                := t\n\n/-- Returns the codomain of an arrow type, or the type itself if it is not an arrow. -/\n@[simp]\ndef codomain : type \u2192 type\n| (type.Arr t\u2081 t\u2082) := t\u2082 \n| t                := t\n\n@[simp]\ndef get_arg_types : type \u2192 list type\n| (type.Arr t\u2081 t\u2082) := t\u2081 :: get_arg_types t\u2082\n| _                := []\n\n@[simp]\ndef num_arg_types : type \u2192 nat\n| (type.Arr t\u2081 t\u2082) := num_arg_types t\u2082 + 1\n| _                := 0\n\nlemma length_get_arg_types (t : type) : t.get_arg_types.length = t.num_arg_types :=\nby induction t; simp [*]\n\n@[simp]\ndef get_return_type : type \u2192 type \n| (type.Arr t\u2081 t\u2082) := get_return_type t\u2082\n| t                := t\n\n@[simp]\ndef mk_fn_type : list type \u2192 type \u2192 type \n| []        ret := ret\n| (a :: as) ret := (type.Arr a (mk_fn_type as ret))\n\ntheorem mk_fn_type_eq (t : type) : t = mk_fn_type (get_arg_types t) (get_return_type t) :=\nby { induction t; simp [get_arg_types, get_return_type, mk_fn_type], assumption }\n\n@[simp]\ntheorem get_return_type_codomain (t : type) : get_return_type (t.codomain) = get_return_type t :=\nby induction t; simp\n\ntheorem get_arg_types_codomain (t : type) : get_arg_types (t.codomain) = t.get_arg_types.drop 1 :=\nby induction t; simp\n\n@[simp]\ndef is_arrow : type \u2192 bool\n| (Arr t\u2081 t\u2082) := tt\n| _           := ff\n\ntheorem is_arrow_mk_fn_type (ts : list type) (t : type) (h : t.is_arrow = ff) :\n  (mk_fn_type ts t).is_arrow = (ts \u2260 []) :=\nby induction ts; simp [h]\n\ntheorem is_arrow_get_return_type_eq_ff (t : type) : t.get_return_type.is_arrow = ff :=\nby induction t; simp [*]\n\ntheorem eq_of_is_arrow (t : type) (h : is_arrow t = tt) : t = Arr (domain t) (codomain t) :=\nby cases t; simp at h; trivial\n\nlemma cons_prefix_get_arg_types (ty ty' : type) (tys : list type) :\n  (ty :: tys) <+: ty'.get_arg_types \u2194 \n    is_arrow ty' = tt \u2227 ty = ty'.domain \u2227 tys <+: ty'.codomain.get_arg_types :=\nby cases ty'; simp\n\nend type\n\n/- convenient type constructors -/\n\nsection\nopen type\n\n-- TODO: mark these as reducible?\n\ndef mk_prop := Basic basic.prop\ndef mk_nat := Basic basic.nat\ndef mk_int := Basic basic.int\ndef mk_unit := Basic basic.int\ndef mk_bool := Basic basic.bool\ndef mk_user_type (n : \u2115) := Basic (basic.user n)\n\ndef mk_list_type (t : type) : type := App (Constructor constructor.list) t\ndef mk_prod_type (t\u2081 t\u2082 : type) : type := App (App (Constructor constructor.prod) t\u2081) t\u2082\ndef mk_sum_type (t\u2081 t\u2082 : type) : type := App (App (Constructor constructor.sum) t\u2081) t\u2082\n\n--notation t\u2081 ` \u21d2 `:65 t\u2082 := Arr t\u2081 t\u2082  \ninfixr ` \u21d2 ` := Arr\n\ntheorem arities_ok_mk_prop : arities_ok mk_prop :=\nby simp [mk_prop, arities_ok, arities_ok_aux, list.empty]\n\ntheorem arities_ok_mk_nat : arities_ok mk_nat :=\nby simp [mk_nat, arities_ok, arities_ok_aux, list.empty]\n\ntheorem arities_ok_mk_int : arities_ok mk_int :=\nby simp [mk_int, arities_ok, arities_ok_aux, list.empty]\n\ntheorem arities_ok_mk_list_type (t : type) (h : arities_ok t = tt) : \n  arities_ok (mk_list_type t) = tt :=\nby { simp [arities_ok] at *, simp [mk_list_type, arities_ok_aux, h, constructor.list] }\n\ntheorem arities_ok_mk_prod_type (t\u2081 t\u2082 : type) \n    (h\u2081 : arities_ok t\u2081 = tt) (h\u2082 : arities_ok t\u2082 = tt) : \n  arities_ok (mk_prod_type t\u2081 t\u2082) = tt :=\nby { simp [arities_ok] at *, simp [mk_prod_type, arities_ok_aux, h\u2081, h\u2082, constructor.prod] }\n\ntheorem arities_ok_mk_sum_type (t\u2081 t\u2082 : type) \n    (h\u2081 : arities_ok t\u2081 = tt) (h\u2082 : arities_ok t\u2082 = tt) : \n  arities_ok (mk_sum_type t\u2081 t\u2082) = tt :=\nby { simp [arities_ok] at *, simp [mk_sum_type, arities_ok_aux, h\u2081, h\u2082, constructor.sum] }\n\nend\n\n/-\n    Terms\n-/\n\nnamespace term\n\n  /- constants -/\n\n  namespace const\n    @[derive has_reflect, derive decidable_eq]\n    inductive kind\n    | user : nat \u2192 kind\n    | true \n    | false \n    | not\n    | and\n    | or\n    | implies\n    | iff\n    | all\n    | ex\n    | add\n    | mul\n    | sub\n    | bval : bool \u2192 kind\n    | nval : nat \u2192 kind\n  end const\n\n  -- instantiations are for polymorphic constants\n  @[derive has_reflect, derive decidable_eq]\n  structure const :=\n  (symb : const.kind) (type : type) (insts : list hol.type)\n\n  namespace const\n    def user (n : nat) (t : hol.type) (l : list hol.type) : const := \u27e8kind.user n, t, l\u27e9\n    def true : const := \u27e8kind.true, mk_prop, []\u27e9 \n    def false : const := \u27e8kind.false, mk_prop, []\u27e9 \n    def not : const := \u27e8kind.not, mk_prop \u21d2 mk_prop, []\u27e9\n    def and : const := \u27e8kind.and, mk_prop \u21d2 (mk_prop \u21d2 mk_prop), []\u27e9\n    def or : const := \u27e8kind.or, mk_prop \u21d2 (mk_prop \u21d2 mk_prop), []\u27e9\n    def implies : const := \u27e8kind.implies, mk_prop \u21d2 (mk_prop \u21d2 mk_prop), []\u27e9\n    def iff : const := \u27e8kind.iff, mk_prop \u21d2 (mk_prop \u21d2 mk_prop), []\u27e9\n    def add : const := \u27e8kind.add, mk_nat \u21d2 (mk_nat \u21d2 mk_nat), []\u27e9\n    def mul : const := \u27e8kind.mul, mk_nat \u21d2 (mk_nat \u21d2 mk_nat), []\u27e9\n    def sub : const := \u27e8kind.sub, mk_nat \u21d2 (mk_nat \u21d2 mk_nat), []\u27e9\n    def tt  : const := \u27e8kind.bval tt, mk_bool, []\u27e9\n    def ff  : const := \u27e8kind.bval ff, mk_bool, []\u27e9\n    def nval (n : nat) : const := \u27e8kind.nval n, mk_nat, []\u27e9\n\n    def repr : const \u2192 string\n    | \u27e8kind.user n, t, l\u27e9       := \"(const.user \" ++ n.repr ++ \" \" ++ t.repr ++ \" \" ++ \n                                      l.repr ++ \")\"\n    | \u27e8kind.true, _, _\u27e9         := \"const.true\"\n    | \u27e8kind.false, _, _\u27e9        := \"const.false\"\n    | \u27e8kind.not, _, _\u27e9          := \"const.not\"\n    | \u27e8kind.and, _, _\u27e9          := \"const.and\"\n    | \u27e8kind.or, _, _\u27e9           := \"const.or\"\n    | \u27e8kind.implies, _, _\u27e9      := \"const.implies\"\n    | \u27e8kind.iff, _, _\u27e9          := \"const.iff\"\n    | \u27e8kind.all, _, _\u27e9          := \"const.all\"\n    | \u27e8kind.ex, _, _\u27e9           := \"const.ex\"\n    | \u27e8kind.add, _, _\u27e9          := \"const.add\"\n    | \u27e8kind.mul, _, _\u27e9          := \"const.mul\"\n    | \u27e8kind.sub, _, _\u27e9          := \"const.sub\"\n    | \u27e8kind.bval bool.tt, _, _\u27e9 := \"const.tt\"\n    | \u27e8kind.bval bool.ff, _, _\u27e9 := \"const.ff\"\n    | \u27e8kind.nval n, _, _\u27e9       := \"(const.nval \" ++ n.repr ++ \")\"\n\n    instance : has_repr const := \u27e8repr\u27e9\n\n    -- TODO: delete this?\n    def is_connective : const \u2192 bool\n    | \u27e8kind.true, t, l\u27e9         := if t = mk_prop then l.empty else bool.ff\n    | \u27e8kind.false, t, l\u27e9        := if t = mk_prop then l.empty else bool.ff\n    | \u27e8kind.not, t, l\u27e9          := if t = (mk_prop \u21d2 mk_prop) then l.empty else bool.ff\n    | \u27e8kind.and, t, l\u27e9          := if t = (mk_prop \u21d2 mk_prop \u21d2 mk_prop) then l.empty else bool.ff\n    | \u27e8kind.or, t, l\u27e9           := if t = (mk_prop \u21d2 mk_prop \u21d2 mk_prop) then l.empty else bool.ff\n    | \u27e8kind.implies, t, l\u27e9      := if t = (mk_prop \u21d2 mk_prop \u21d2 mk_prop) then l.empty else bool.ff\n    | \u27e8kind.iff, t, l\u27e9          := if t = (mk_prop \u21d2 mk_prop \u21d2 mk_prop) then l.empty else bool.ff\n    | _                         := bool.ff\n  end const\n\nend term\n\n/- the terms themselves -/\n\n@[derive has_reflect, derive decidable_eq]\ninductive term\n| Var     : nat \u2192 term\n| Const   : hol.term.const \u2192 term\n| App     : term \u2192 term \u2192 term                -- application\n| Abs     : string \u2192 hol.type \u2192 term \u2192 term   -- the string gives the preferred name\n\nnamespace term\n\n-- TODO: not needed?\ndef sizeof' : term \u2192 nat\n| (Var n)      := 0\n| (Const n)    := 0\n| (App s t)    := sizeof' s + sizeof' t + 1\n| (Abs s ty t) := sizeof t + 1\n\ninstance : has_sizeof term := \u27e8sizeof'\u27e9\n\ndef repr : term \u2192 string\n| (Var n)      := \"(Var \" ++ n.repr ++ \")\"\n| (Const c)    := \"(Const \" ++ c.repr ++ \")\"\n| (App t\u2081 t\u2082)  := \"(App \" ++ t\u2081.repr ++ t\u2082.repr ++ \")\"\n| (Abs s ty t) := \"(Abs \" ++ _root_.repr s ++ \" \" ++ ty.repr ++ \" \" ++ t.repr ++ \")\"\n\n@[simp]\ndef is_var : term \u2192 bool\n| (Var n) := tt\n| _       := ff\n\ndef var_num : term \u2192 nat\n| (Var n) := n\n| _       := 0\n\n@[simp]\ndef is_const : term \u2192 bool\n| (Const c) := tt\n| _         := ff\n\n@[simp]\ndef is_app : term \u2192 bool\n| (App f a) := tt\n| _         := ff\n\ndef app_fn : term \u2192 term\n| (App f a) := f\n| t         := t\n\ndef app_arg : term \u2192 term\n| (App f a) := a\n| t         := t\n\n@[simp]\ndef get_app_fn : term \u2192 term\n| (App f a) := get_app_fn f\n| t          := t\n\ndef get_app_num_args : term \u2192 \u2115 \n| (App f a) := get_app_num_args f + 1\n| t         := 0\n\ndef get_app_args_aux : term \u2192 list term \u2192 list term\n| (App t\u2081 t\u2082)  args := get_app_args_aux t\u2081 (t\u2082 :: args)\n| (Var n)      args := args\n| (Const c)    args := args\n| (Abs s ty t) args := args\n\ndef get_app_args (t : term) : list term := get_app_args_aux t []\n\n@[simp]\ndef mk_app : term \u2192 list term \u2192 term\n| t []       := t\n| t (a::as)  := mk_app (App t a) as\n\ntheorem mk_app_get_app_aux (t : term) : \n  \u2200 as, mk_app (get_app_fn t) (get_app_args_aux t as) = mk_app t as :=\nby induction t; simp [get_app_fn, get_app_args_aux, mk_app, *]\n\ntheorem mk_app_get_app (t : term) : mk_app (get_app_fn t) (get_app_args t) = t :=\nby simp [get_app_args, mk_app_get_app_aux, mk_app]\n\ntheorem not_is_app_get_app_fn (t : term) : is_app t.get_app_fn = ff :=\nby { induction t with _ _ _ _ ih; simp, exact ih }\n\n/-- Infers the type of a term, given an assignment to free variables. Assumes the expression is\n    well-typed-/\ndef typeof : term \u2192 list type \u2192 type\n| (Var n) \u03c3       := if h : n < \u03c3.length then \u03c3.nth_le n h else mk_nat\n| (Const c) \u03c3     := c.type\n| (App t\u2081 t\u2082) \u03c3   := (typeof t\u2081 \u03c3).codomain\n| (Abs s ty t) \u03c3  := ty \u21d2 typeof t (ty :: \u03c3)\n\ninductive is_well_typed : term \u2192 list type \u2192 Prop\n| wt_var (n : \u2115) (\u03c3 : list type) (h : n < \u03c3.length)      : is_well_typed (Var n) \u03c3\n| wt_const (c : const) (\u03c3 : list type)                   : is_well_typed (Const c) \u03c3\n| wt_app (t\u2081 t\u2082 : term) (\u03c3 : list type) \n    (h\u2081 : is_well_typed t\u2081 \u03c3) \n    (h\u2082 : is_well_typed t\u2082 \u03c3) \n    (h\u2083 : (typeof t\u2081 \u03c3).is_arrow = tt) \n    (h\u2084 : (typeof t\u2081 \u03c3).domain = typeof t\u2082 \u03c3)            : is_well_typed (App t\u2081 t\u2082) \u03c3\n| wt_abs (s : string) (ty : type) (t : term) \n      (\u03c3 : list type)\n    (h\u2081 : is_well_typed t (ty :: \u03c3))                     : is_well_typed (Abs s ty t) \u03c3\n\n/-- Infers the type of a term, given an assignment to free variables. Returns none of expression is\n    not well-typed. -/\ndef typeof_p : term \u2192 list type \u2192 option type \n| (Var n)     \u03c3  := \u03c3.nth n\n| (Const c)   \u03c3  := some c.type \n| (App t\u2081 t\u2082) \u03c3  := \n    match typeof_p t\u2081 \u03c3, typeof_p t\u2082 \u03c3 with \n    | (some (type.Arr u v)), (some w) := if u = w then some v else none\n    | _                    , _        := none\n    end\n| (Abs s ty t) \u03c3 := match typeof_p t (ty :: \u03c3) with\n                    | some ty\u2082 := some (type.Arr ty ty\u2082)\n                    | _        := none\n                    end\n\n-- a boolean version\n@[simp]\ndef is_well_typed_b : term \u2192 list type \u2192 bool\n| (Var n)     \u03c3 := if n < \u03c3.length then tt else ff\n| (Const c)   \u03c3 := tt\n| (App t\u2081 t\u2082) \u03c3 := is_well_typed_b t\u2081 \u03c3 && is_well_typed_b t\u2082 \u03c3 && \n                     (typeof t\u2081 \u03c3).is_arrow && ((typeof t\u2081 \u03c3).domain = typeof t\u2082 \u03c3)\n| (Abs s ty t) \u03c3 := is_well_typed_b t (ty :: \u03c3)\n\ntheorem is_well_typed_iff (t : term) : \n  \u2200 \u03c3, is_well_typed t \u03c3 \u2194 (is_well_typed_b t \u03c3 = tt) :=\nbegin\n  induction t with _ _ _ _ h\u2081 h\u2082 _ ty _ h,\n  {intro \u03c3, split, {intro h, cases h; simp [*]}, intro h, simp at h, constructor, assumption},\n  {intro \u03c3, split, {intro h, cases h; simp [*]}, intro h, simp at h, constructor},\n  {intro \u03c3, simp [(h\u2081 \u03c3).symm, (h\u2082 \u03c3).symm], split, {intro h, cases h; simp [*]}, intro h, constructor; simp [*]},\n  {intro \u03c3, simp [(h (ty::\u03c3)).symm], split, {intro h, cases h, assumption}, \n    intro h, constructor; simp [*]}\nend\n\n@[simp] lemma is_well_typed_of_is_const {t : term} {\u03c3 : list type} : \n  is_const t = tt \u2192 is_well_typed t \u03c3 :=\nby { cases t; simp, constructor }\n\ntheorem is_well_typed_mk_app (l : list term) (\u03c3 : list type) :\n  \u2200 t, is_well_typed (mk_app t l) \u03c3 \u2194 \n         is_well_typed t \u03c3 \u2227 (\u2200 t' \u2208 l, is_well_typed t' \u03c3) \u2227 \n           l.map (\u03bb t', typeof t' \u03c3) <+: (t.typeof \u03c3).get_arg_types :=\nbegin\n  induction l with t' l' ih,\n  { simp [list.nil_prefix] },\n  simp, intro t, rw [ih (App t t')], simp [is_well_typed_iff],\n  split, \n  { intro h, simp [*, typeof] at *,\n    rcases h with \u27e8\u27e8h\u2080, h\u2081, h\u2082, h\u2083\u27e9, h\u2084, h\u2085\u27e9,  \n    split, exact h\u2084, rw [type.eq_of_is_arrow _ h\u2082],\n    simp, rw [h\u2083], simp, exact h\u2085},\n  intro h, rcases h with \u27e8h,  \u27e8h\u2080, h\u2081\u27e9, h\u2082\u27e9,\n  have h\u2083 := (type.cons_prefix_get_arg_types _ _ _).mp h\u2082,\n  rcases h\u2083 with \u27e8h\u2083, h\u2084, h\u2085\u27e9,\n  simp [h, h\u2080, h\u2081, h\u2083, h\u2084, h\u2085, typeof],\n  assumption  \nend\n\ntheorem is_well_typed_iff' {t : term} {\u03c3 : list type} :\n  t.is_well_typed \u03c3 \u2194 \n    is_well_typed (t.get_app_fn) \u03c3 \u2227 (\u2200 t' \u2208 t.get_app_args, is_well_typed t' \u03c3) \u2227 \n           t.get_app_args.map (\u03bb t', typeof t' \u03c3) <+: (t.get_app_fn.typeof \u03c3).get_arg_types :=\nby { rw [\u2190 mk_app_get_app t, is_well_typed_mk_app], simp [mk_app_get_app] }\n\ntheorem typeof_mk_app (t: term) (as : list term) (\u03c3 : list type) :\n  typeof (mk_app t as) \u03c3 = \n    type.mk_fn_type (((t.typeof \u03c3).get_arg_types).drop (as.length)) \n      (t.typeof \u03c3).get_return_type :=\nbegin\n  revert t, \n  induction as with a as ih, \n  { intro t, simp, rw \u2190 type.mk_fn_type_eq (t.typeof \u03c3) },\n  intro t, simp, rw (ih (App t a)), simp [typeof], rw [add_comm, list.drop_add],\n  simp [type.get_arg_types_codomain]\nend\n\ntheorem is_arrow_get_app_fn_of_is_arrow (t : term)(\u03c3 : list type) (h : t.is_well_typed \u03c3) \n  (h' : (t.typeof \u03c3).is_arrow = tt) : (t.get_app_fn.typeof \u03c3).is_arrow = tt :=\nbegin\n  revert h h', \n  induction t with _ _ _ _ ih; simp,\n  rintro \u27e8h\u2080, h\u2081, h\u2082, h\u2083\u27e9, intro h'',\n  apply ih; assumption\nend\n\ntheorem is_arrow_get_app_fn_of_is_app {t : term} {\u03c3 : list type} (h : t.is_well_typed \u03c3) \n  (h' : is_app t = tt) : ((t.get_app_fn).typeof \u03c3).is_arrow = tt :=\nbegin\n  revert h h',\n  cases t; simp,\n  intro h, cases h,\n  apply is_arrow_get_app_fn_of_is_arrow; assumption\nend\n\ntheorem typeof_mk_app_is_arrow (t: term) (as : list term) (\u03c3 : list type) :\n  (typeof (mk_app t as) \u03c3).is_arrow = (as.length < (t.typeof \u03c3).num_arg_types) := \nbegin\n  rw [typeof_mk_app, type.is_arrow_mk_fn_type, to_bool_eq_to_bool, ne, list.drop_eq_nil,\n    lt_iff_not_ge, type.length_get_arg_types], --simp [- bool.to_bool_not],\n  apply type.is_arrow_get_return_type_eq_ff\nend\n\ntheorem length_le_num_arg_types_typeof  {t: term} {as : list term} {\u03c3 : list type} \n    (h : is_well_typed (mk_app t as) \u03c3) :\n  as.length \u2264 (t.typeof \u03c3).num_arg_types :=\nbegin\n  have : as.map (\u03bb t', typeof t' \u03c3) <+: (t.typeof \u03c3).get_arg_types, \n    from ((is_well_typed_mk_app as \u03c3 t).mp h).right.right,\n  have := list.length_le_of_prefix this,\n  simpa [type.length_get_arg_types]\nend\n\ntheorem length_get_app_args_of_not_is_arrow {t : term} {\u03c3 : list type} \n    (h : (t.typeof \u03c3).is_arrow = ff) (h' : t.is_well_typed \u03c3) :\n  t.get_app_args.length = (t.get_app_fn.typeof \u03c3).num_arg_types :=\nbegin\n  rw \u2190 mk_app_get_app t at h h',\n  have : t.get_app_args.length \u2264 (t.get_app_fn.typeof \u03c3).num_arg_types,\n    from length_le_num_arg_types_typeof h',\n  by {apply le_antisymm this, rw typeof_mk_app_is_arrow at h, simp at h, exact h }\nend\n\ntheorem get_arg_types_typeof_get_app_fn {t : term} {\u03c3 : list type} (h : (t.typeof \u03c3).is_arrow = ff)\n     (h' : t.is_well_typed \u03c3) :\n  (t.get_app_fn.typeof \u03c3).get_arg_types = t.get_app_args.map (\u03bb t', t'.typeof \u03c3) :=\nbegin\n  have : t.get_app_args.length = (t.get_app_fn.typeof \u03c3).num_arg_types,\n    from length_get_app_args_of_not_is_arrow h h',\n  rw \u2190 mk_app_get_app t at h h',\n  have : t.get_app_args.map (\u03bb t', typeof t' \u03c3) <+: (t.get_app_fn.typeof \u03c3).get_arg_types,\n    from ((is_well_typed_mk_app t.get_app_args \u03c3 t.get_app_fn).mp h').right.right,\n  have : t.get_app_args.map (\u03bb t', typeof t' \u03c3) = (t.get_app_fn.typeof \u03c3).get_arg_types,\n    by { apply list.eq_of_prefix_of_length_eq this, rw type.length_get_arg_types, simp, assumption},\n  exact this.symm\nend\n\ntheorem is_well_typed_of_not_is_arrow {t : term} {\u03c3 : list type} (h : (t.typeof \u03c3).is_arrow = ff) :\n  t.is_well_typed \u03c3 \u2194 \n    is_well_typed (t.get_app_fn) \u03c3 \u2227 \n    (\u2200 t' \u2208 t.get_app_args, is_well_typed t' \u03c3) \u2227 \n    t.get_app_args.map (\u03bb t', typeof t' \u03c3) = (t.get_app_fn.typeof \u03c3).get_arg_types :=\nbegin\n  split; intro h',\n  { have h\u2080 := is_well_typed_iff'.mp h',\n    rcases h\u2080 with \u27e8h\u2080, h\u2081, h\u2082\u27e9,\n    split, apply h\u2080,\n    split, apply h\u2081, \n    apply list.eq_of_prefix_of_length_eq h\u2082, simp,\n    rw length_get_app_args_of_not_is_arrow h h',\n    rw [type.length_get_arg_types] },\n  rw is_well_typed_iff',    \n  rcases h' with \u27e8h\u2080, h\u2081, h\u2082\u27e9,\n  split, apply h\u2080,\n  split, apply h\u2081, \n  rw h\u2082 \nend\n\ntheorem typeof_eq_of_not_is_arrow {t : term} {\u03c3 : list type} \n    (h : (t.typeof \u03c3).is_arrow = ff) (h' : t.is_well_typed \u03c3) :\n  t.typeof \u03c3 = ((t.get_app_fn).typeof \u03c3).get_return_type :=\nbegin\n  have h\u2080 : t.typeof \u03c3 = (mk_app t.get_app_fn t.get_app_args).typeof \u03c3,\n    by rw mk_app_get_app,\n  transitivity, exact h\u2080,\n  rw [typeof_mk_app, length_get_app_args_of_not_is_arrow h h', \u2190 type.length_get_arg_types],\n  rw (list.drop_eq_nil _ _).mpr, simp, apply le_refl\nend\n\nlemma type_is_not_arrow' {t : term} {\u03c3 : list type} (h : is_well_typed t \u03c3) :\n  (t.typeof \u03c3).is_arrow = ff \u2194 \n     (t.get_app_fn.typeof \u03c3).get_arg_types = (t.get_app_args).map (\u03bb t', t'.typeof \u03c3) :=\nbegin\n  split,\n  { intro h\u2080,\n    rw [get_arg_types_typeof_get_app_fn h\u2080 h] },\n  intro h,\n  rw \u2190 mk_app_get_app t,\n  rw typeof_mk_app,\n  have : (get_app_args t).length = (t.get_app_fn.typeof \u03c3).get_arg_types.length,\n    by { rw h, simp },\n  rw this, rw (list.drop_eq_nil _ _).mpr (le_refl _),\n  apply type.is_arrow_get_return_type_eq_ff\nend\n\n-- an inductive characterization of well typed terms whose types are not arrows \ntheorem type_is_not_arrow_iff {\u03c3 : list type} : \n  \u2200 {t : term}, t.is_well_typed \u03c3 \u2192 \n    ((t.typeof \u03c3).is_arrow = ff \u2194\n      (t.is_var = tt \u2227 (t.typeof \u03c3).is_arrow = ff) \u2228 \n      (t.is_const = tt \u2227 (t.typeof \u03c3).is_arrow = ff) \u2228 \n      (t.is_app = tt \u2227 \n          (t.get_app_fn.typeof \u03c3).get_arg_types = t.get_app_args.map (\u03bb t', t'.typeof \u03c3))) :=\nbegin\n  intro t, \n  induction t with _ _ t\u2081 t\u2082 ih\u2081 ih\u2082; simp,\n  { intro h', simp [type_is_not_arrow' h'] },\n  simp [typeof]\nend\n\nend term\n\nend hol", "meta": {"author": "avigad", "repo": "formal_logic", "sha": "59d7fe7cb7a7927fb72d89d4fd40965bcd769349", "save_path": "github-repos/lean/avigad-formal_logic", "path": "github-repos/lean/avigad-formal_logic/formal_logic-59d7fe7cb7a7927fb72d89d4fd40965bcd769349/src/syntax.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6893056104028799, "lm_q1q2_score": 0.39807079520424}}
{"text": "/-\nCopyright (c) 2017 Gabriel Ebner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner\n-/\nimport .clause .clause_ops\nimport .prover_state .misc_preprocessing\nopen expr list tactic monad decidable\n\nuniverse u\n\nnamespace super\n\nmeta def try_option {a : Type u} (tac : tactic a) : tactic (option a) :=\nsome <$> tac <|> return none\n\nprivate meta def normalize : expr \u2192 tactic expr | e := do\ne' \u2190 whnf e reducible,\nargs' \u2190 e'.get_app_args.mmap normalize,\nreturn $ app_of_list e'.get_app_fn args'\n\nmeta def inf_normalize_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbtype, do\n  type' \u2190 normalize type,\n  guard $ type' \u2260 type,\n  h \u2190 mk_local_def `h type',\n  return [([h], h)]\n\nmeta def inf_normalize_r (c : clause) : tactic (list clause) :=\non_first_right c $ \u03bbha, do\n  a' \u2190 normalize ha.local_type,\n  guard $ a' \u2260 ha.local_type,\n  hna \u2190 mk_local_def `hna (imp a' c.local_false),\n  return [([hna], app hna ha)]\n\nmeta def inf_false_l (c : clause) : tactic (list clause) :=\nfirst $ do i \u2190 list.range c.num_lits,\n  if c.get_lit i = clause.literal.left `(false)\n  then [return []]\n  else []\n\nmeta def inf_false_r (c : clause) : tactic (list clause) :=\non_first_right c $ \u03bbhf,\n  if hf.local_type = c.local_false\n  then return [([], hf)]\n  else match hf.local_type with\n  | const ``false [] := do\n    pr \u2190 mk_app ``false.rec [c.local_false, hf],\n    return [([], pr)]\n  | _ := failed\n  end\n\nmeta def inf_true_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbt,\n  match t with\n  | (const ``true []) := return [([], const ``true.intro [])]\n  | _ := failed\n  end\n\nmeta def inf_true_r (c : clause) : tactic (list clause) :=\nfirst $ do i \u2190 list.range c.num_lits,\n  if c.get_lit i = clause.literal.right (const ``true [])\n  then [return []]\n  else []\n\nmeta def inf_not_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbtype,\n  match type with\n  | app (const ``not []) a := do\n    hna \u2190 mk_local_def `h (a.imp `(false)),\n    return [([hna], hna)]\n  | _ := failed\n  end\n\nmeta def inf_not_r (c : clause) : tactic (list clause) :=\non_first_right c $ \u03bbhna,\n  match hna.local_type with\n  | app (const ``not []) a := do\n    hnna \u2190 mk_local_def `h ((a.imp `(false)).imp c.local_false),\n    return [([hnna], app hnna hna)]\n  | _ := failed\n  end\n\nmeta def inf_and_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbab,\n  match ab with\n  | (app (app (const ``and []) a) b) := do\n    ha \u2190 mk_local_def `l a,\n    hb \u2190 mk_local_def `r b,\n    pab \u2190 mk_mapp ``and.intro [some a, some b, some ha, some hb],\n    return [([ha, hb], pab)]\n  | _ := failed\n  end\n\nmeta def inf_and_r (c : clause) : tactic (list clause) :=\non_first_right' c $ \u03bbhyp, do\n  pa \u2190 mk_mapp ``and.left [none, none, some hyp],\n  pb \u2190 mk_mapp ``and.right [none, none, some hyp],\n  return [([], pa), ([], pb)]\n\nmeta def inf_iff_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbab,\n  match ab with\n  | (app (app (const ``iff []) a) b) := do\n    hab \u2190 mk_local_def `l (imp a b),\n    hba \u2190 mk_local_def `r (imp b a),\n    pab \u2190 mk_mapp ``iff.intro [some a, some b, some hab, some hba],\n    return [([hab, hba], pab)]\n  | _ := failed\n  end\n\nmeta def inf_iff_r (c : clause) : tactic (list clause) :=\non_first_right' c $ \u03bbhyp, do\n  pa \u2190 mk_mapp ``iff.mp [none, none, some hyp],\n  pb \u2190 mk_mapp ``iff.mpr [none, none, some hyp],\n  return [([], pa), ([], pb)]\n\nmeta def inf_or_r (c : clause) : tactic (list clause) :=\non_first_right c $ \u03bbhab,\n  match hab.local_type with\n  | (app (app (const ``or []) a) b) := do\n    hna \u2190 mk_local_def `l (imp a c.local_false),\n    hnb \u2190 mk_local_def `r (imp b c.local_false),\n    proof \u2190 mk_app ``or.elim [a, b, c.local_false, hab, hna, hnb],\n    return [([hna, hnb], proof)]\n  | _ := failed\n  end\n\nmeta def inf_or_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbab,\n  match ab with\n  | (app (app (const ``or []) a) b) := do\n    ha \u2190 mk_local_def `l a,\n    hb \u2190 mk_local_def `l b,\n    pa \u2190 mk_mapp ``or.inl [some a, some b, some ha],\n    pb \u2190 mk_mapp ``or.inr [some a, some b, some hb],\n    return [([ha], pa), ([hb], pb)]\n  | _ := failed\n  end\n\nmeta def inf_all_r (c : clause) : tactic (list clause) :=\non_first_right' c $ \u03bbhallb,\n  match hallb.local_type with\n  | (pi n bi a b) := do\n    ha \u2190 mk_local_def `x a,\n    return [([ha], app hallb ha)]\n  | _ := failed\n  end\n\nlemma imp_l {F a b} [decidable a] : ((a \u2192 b) \u2192 F) \u2192 ((a \u2192 F) \u2192 F) :=\n\u03bbhabf haf, decidable.by_cases\n    (assume ha :   a, haf ha)\n    (assume hna : \u00aca, habf (assume  ha, absurd ha hna))\n\nlemma imp_l' {F a b} [decidable F] : ((a \u2192 b) \u2192 F) \u2192 ((a \u2192 F) \u2192 F) :=\n\u03bbhabf haf, decidable.by_cases\n    (assume hf :   F, hf)\n    (assume hnf : \u00acF, habf (assume  ha, absurd (haf ha) hnf))\n\nlemma imp_l_c {F : Prop} {a b} : ((a \u2192 b) \u2192 F) \u2192 ((a \u2192 F) \u2192 F) :=\n\u03bbhabf haf, classical.by_cases\n    (assume hf :   F, hf)\n    (assume hnf : \u00acF, habf (assume  ha, absurd (haf ha) hnf))\n\nmeta def inf_imp_l (c : clause) : tactic (list clause) :=\non_first_left_dn c $ \u03bbhnab,\n  match hnab.local_type with\n  | (pi _ _ (pi _ _ a b) _) :=\n    if b.has_var then failed else do\n    hna \u2190 mk_local_def `na (imp a c.local_false),\n    pf \u2190 first (do r \u2190 [``super.imp_l, ``super.imp_l', ``super.imp_l_c],\n                 [mk_app r [hnab, hna]]),\n    hb \u2190 mk_local_def `b b,\n    return [([hna], pf), ([hb], app hnab (lam `a binder_info.default a hb))]\n  | _ := failed\n  end\n\nmeta def inf_ex_l (c : clause) : tactic (list clause) :=\non_first_left c $ \u03bbexp,\n  match exp with\n  | (app (app (const ``Exists [u]) dom) pred) := do\n    hx \u2190 mk_local_def `x dom,\n    predx \u2190 whnf $ app pred hx,\n    hpx \u2190 mk_local_def `hpx predx,\n    return [([hx,hpx], app_of_list (const ``exists.intro [u])\n                       [dom, pred, hx, hpx])]\n  | _ := failed\n  end\n\nlemma demorgan' {F a} {b : a \u2192 Prop} : ((\u2200x, b x) \u2192 F) \u2192 (((\u2203x, b x \u2192 F) \u2192 F) \u2192 F) :=\nassume hab hnenb,\n  classical.by_cases\n    (assume h : \u2203x, \u00acb x, begin cases h with x, apply hnenb, existsi x, intros, contradiction end)\n    (assume h : \u00ac\u2203x, \u00acb x, hab (assume  x,\n      classical.by_cases\n        (assume bx : b x, bx)\n        (assume nbx : \u00acb x, have hf : false, { apply h, existsi x, assumption }, by contradiction)))\n\nmeta def inf_all_l (c : clause) : tactic (list clause) :=\non_first_left_dn c $ \u03bbhnallb,\n  match hnallb.local_type with\n  | pi _ _ (pi n bi a b) _ := do\n    enb \u2190 mk_mapp ``Exists [none, some $ lam n binder_info.default a (imp b c.local_false)],\n    hnenb \u2190 mk_local_def `h (imp enb c.local_false),\n    pr \u2190 mk_app ``super.demorgan' [hnallb, hnenb],\n    return [([hnenb], pr)]\n  | _ := failed\n  end\n\nmeta def inf_ex_r  (c : clause) : tactic (list clause) := do\n(qf, ctx) \u2190 c.open_constn c.num_quants,\nskolemized \u2190 on_first_right' qf $ \u03bbhexp,\n  match hexp.local_type with\n  | (app (app (const ``Exists [_]) d) p) := do\n    sk_sym_name_pp \u2190 get_unused_name `sk (some 1),\n    inh_lc \u2190 mk_local' `w binder_info.implicit d,\n    sk_sym \u2190 mk_local_def sk_sym_name_pp (pis (ctx ++ [inh_lc]) d),\n    sk_p \u2190 whnf_no_delta $ app p (app_of_list sk_sym (ctx ++ [inh_lc])),\n    sk_ax \u2190 mk_mapp ``Exists [some (local_type sk_sym),\n      some (lambdas [sk_sym] (pis (ctx ++ [inh_lc]) (imp hexp.local_type sk_p)))],\n    sk_ax_name \u2190 get_unused_name `sk_axiom (some 1), assert sk_ax_name sk_ax,\n    nonempt_of_inh \u2190 mk_mapp ``nonempty.intro [some d, some inh_lc],\n    eps \u2190 mk_mapp ``classical.epsilon [some d, some nonempt_of_inh, some p],\n    existsi (lambdas (ctx ++ [inh_lc]) eps),\n    eps_spec \u2190 mk_mapp ``classical.epsilon_spec [some d, some p],\n    exact (lambdas (ctx ++ [inh_lc]) eps_spec),\n    sk_ax_local \u2190 get_local sk_ax_name, cases sk_ax_local [sk_sym_name_pp, sk_ax_name],\n    sk_ax' \u2190 get_local sk_ax_name,\n    return [([inh_lc], app_of_list sk_ax' (ctx ++ [inh_lc, hexp]))]\n  | _ := failed\n  end,\nreturn $ skolemized.map (\u03bbs, s.close_constn ctx)\n\nmeta def first_some {a : Type} : list (tactic (option a)) \u2192 tactic (option a)\n| [] := return none\n| (x::xs) := do xres \u2190 x, match xres with some y := return (some y) | none := first_some xs end\n\nprivate meta def get_clauses_core' (rules : list (clause \u2192 tactic (list clause)))\n     : list clause \u2192 tactic (list clause) | cs :=\nlist.join <$> do\ncs.mmap $ \u03bbc, do first $\nrules.map (\u03bbr, r c >>= get_clauses_core') ++ [return [c]]\n\nmeta def get_clauses_core (rules : list (clause \u2192 tactic (list clause))) (initial : list clause)\n     : tactic (list clause) := do\nclauses \u2190 get_clauses_core' rules initial,\nfilter (\u03bbc, bnot <$> is_taut c) $ list.nub_on clause.type clauses\n\nmeta def clausification_rules_intuit : list (clause \u2192 tactic (list clause)) :=\n[ inf_false_l, inf_false_r, inf_true_l, inf_true_r,\n  inf_not_l, inf_not_r,\n  inf_and_l, inf_and_r,\n  inf_iff_l, inf_iff_r,\n  inf_or_l, inf_or_r,\n  inf_ex_l,\n  inf_normalize_l, inf_normalize_r ]\n\nmeta def clausification_rules_classical : list (clause \u2192 tactic (list clause)) :=\n[ inf_false_l, inf_false_r, inf_true_l, inf_true_r,\n  inf_not_l, inf_not_r,\n  inf_and_l, inf_and_r,\n  inf_iff_l, inf_iff_r,\n  inf_or_l, inf_or_r,\n  inf_imp_l, inf_all_r,\n  inf_ex_l,\n  inf_all_l, inf_ex_r,\n  inf_normalize_l, inf_normalize_r ]\n\nmeta def get_clauses_classical : list clause \u2192 tactic (list clause) :=\nget_clauses_core clausification_rules_classical\nmeta def get_clauses_intuit : list clause \u2192 tactic (list clause) :=\nget_clauses_core clausification_rules_intuit\n\nmeta def as_refutation : tactic unit := do\nrepeat (do intro1, skip),\ntgt \u2190 target,\nif tgt.is_constant || tgt.is_local_constant then skip else do\nlocal_false_name \u2190 get_unused_name `F none, tgt_type \u2190 infer_type tgt,\ndefinev local_false_name tgt_type tgt, local_false \u2190 get_local local_false_name,\ntarget_name \u2190 get_unused_name `target none,\nassertv target_name (imp tgt local_false) (lam `hf binder_info.default tgt $ mk_var 0),\nchange local_false\n\nmeta def clauses_of_context : tactic (list clause) := do\nlocal_false \u2190 target,\nl \u2190 local_context,\nl.mmap (clause.of_proof local_false)\n\nmeta def clausify_pre := preprocessing_rule $ assume  new, list.join <$> new.mmap (\u03bb dc, do\ncs \u2190 get_clauses_classical [dc.c],\nif cs.length \u2264 1 then\n  return (cs.map $ \u03bb c, { dc with c := c })\nelse\n  cs.mmap (\u03bbc, mk_derived c dc.sc))\n\n-- @[super.inf]\nmeta def clausification_inf : inf_decl := inf_decl.mk 0 $\n\u03bb given, list.foldr (<|>) (return ()) $\n        do r \u2190 clausification_rules_classical,\n           [do cs \u2190 r given.c,\n               cs' \u2190 get_clauses_classical cs,\n               cs'.mmap' (\u03bbc, mk_derived c given.sc.sched_now >>= add_inferred),\n               remove_redundant given.id []]\n\n\nend super\n", "meta": {"author": "leanprover", "repo": "super", "sha": "47b107b4cec8f3b41d72daba9cbda2f9d54025de", "save_path": "github-repos/lean/leanprover-super", "path": "github-repos/lean/leanprover-super/super-47b107b4cec8f3b41d72daba9cbda2f9d54025de/src/super/clausifier.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3980707915183375}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Nathaniel Thomas\n-/\nimport tactic.finish\n\n/-!\n# More examples to test `finish`\n\nShamelessly stolen by Jeremy from Nathaniel's `tauto`.\n-/\n\nopen nat\n\nsection\n\nvariables (a b c d e f : Prop)\nvariable even : \u2115 \u2192 Prop\nvariable P : \u2115 \u2192 Prop\n\n-- these next five are things that tauto doesn't get\n\nexample : (\u2200 x, P x) \u2227 b \u2192 (\u2200 y, P y) \u2227 P 0 \u2228 b \u2227 P 0 := by finish\nexample : (\u2200 A, A \u2228 \u00acA) \u2192 \u2200 x y : \u2115, x = y \u2228 x \u2260 y := by finish\nexample : \u2200 b1 b2, b1 = b2 \u2194 (b1 = tt \u2194 b2 = tt) := begin intro b1, cases b1; finish [iff_def] end\n\nexample : \u2200 (P Q : nat \u2192 Prop), (\u2200 n, Q n \u2192 P n) \u2192 (\u2200 n, Q n) \u2192 P 2 := by finish\nexample (a b c : Prop) : \u00ac true \u2228 false \u2228 b \u2194 b := by finish\n\nexample : true := by finish\n\nexample : false \u2192 a := by finish\nexample : a \u2192 a := by finish\nexample : (a \u2192 b) \u2192 a \u2192 b := by finish\nexample : \u00ac a \u2192 \u00ac a := by finish\nexample : a \u2192 (false \u2228 a) := by finish\nexample : (a \u2192 b \u2192 c) \u2192 (a \u2192 b) \u2192 a \u2192 c := by finish\nexample : a \u2192 \u00ac a \u2192 (a \u2192 b) \u2192 (a \u2228 b) \u2192 (a \u2227 b) \u2192 a \u2192 false := by finish\nexample : ((a \u2227 b) \u2227 c) \u2192 b := by finish\nexample : ((a \u2192 b) \u2192 c) \u2192 b \u2192 c := by finish\nexample : (a \u2228 b) \u2192 (b \u2228 a) := by finish\nexample : (a \u2192 b \u2227 c) \u2192 (a \u2192 b) \u2228 (a \u2192 c) := by finish\nexample : \u2200 (x0 : a \u2228 b) (x1 : b \u2227 c), a \u2192 b := by finish\nexample : a \u2192 b \u2192 (c \u2228 b) := by finish\nexample : (a \u2227 b \u2192 c) \u2192 b \u2192 a \u2192 c := by finish\nexample : (a \u2228 b \u2192 c) \u2192 a \u2192 c := by finish\nexample : (a \u2228 b \u2192 c) \u2192 b \u2192 c := by finish\nexample : (a \u2227 b) \u2192 (b \u2227 a) := by finish\nexample : (a \u2194 b) \u2192 a \u2192 b := by finish\nexample : a \u2192 \u00ac\u00aca := by finish\nexample : \u00ac\u00ac(a \u2228 \u00aca) := by finish\nexample : \u00ac\u00ac(a \u2228 b \u2192 a \u2228 b) := by finish\nexample : \u00ac\u00ac((\u2200 n, even n) \u2228 \u00ac(\u2200 m, even m)) := by finish\nexample : (\u00ac\u00acb \u2192 b) \u2192 (a \u2192 b) \u2192 \u00ac\u00aca \u2192 b := by finish\nexample : (\u00ac\u00acb \u2192 b) \u2192 (\u00acb \u2192 \u00ac a) \u2192 \u00ac\u00aca \u2192 b := by finish\n\nexample : ((a \u2192 b \u2192 false) \u2192 false) \u2192 (b \u2192 false) \u2192 false := by finish\n\nexample : ((((c \u2192 false) \u2192 a) \u2192 ((b \u2192 false) \u2192 a) \u2192 false) \u2192 false) \u2192\n            (((c \u2192 b \u2192 false) \u2192 false) \u2192 false) \u2192 \u00aca \u2192 a := by finish\n\nexample (p q r : Prop) (a b : nat) : true \u2192 a = a \u2192 q \u2192 q \u2192 p \u2192 p := by finish\nexample : \u2200 (F F' : Prop), F \u2227 F' \u2192 F := by finish\nexample : \u2200 (F1 F2 F3 : Prop), ((\u00acF1 \u2227 F3) \u2228 (F2 \u2227 \u00acF3)) \u2192 (F2 \u2192 F1) \u2192 (F2 \u2192 F3) \u2192  \u00acF2 := by finish\nexample : \u2200 (f : nat \u2192 Prop), f 2 \u2192 \u2203 x, f x := by finish\nexample : true \u2227 true \u2227 true \u2227 true \u2227 true \u2227 true \u2227 true := by finish\nexample : \u2200 (P : nat \u2192 Prop), P 0 \u2192 (P 0 \u2192 P 1) \u2192 (P 1 \u2192 P 2) \u2192 (P 2) := by finish\nexample : \u00ac\u00ac\u00ac\u00ac\u00aca \u2192 \u00ac\u00ac\u00ac\u00ac\u00ac\u00ac\u00ac\u00aca \u2192 false := by finish\nexample : \u2200 n, \u00ac\u00ac(even n \u2228 \u00aceven n) := by finish\nexample : \u2200 (p q r s : Prop) (a b : nat), r \u2228 s \u2192 p \u2228 q \u2192 a = b \u2192 q \u2228 p := by finish\nexample : (\u2200 x, P x) \u2192 (\u2200 y, P y) := by finish\n\n/- TODO(Jeremy): reinstate after simp * at * bug is fixed.\nexample : ((a \u2194 b) \u2192 (b \u2194 c)) \u2192 ((b \u2194 c) \u2192 (c \u2194 a)) \u2192 ((c \u2194 a) \u2192 (a \u2194 b)) \u2192 (a \u2194 b) :=\nby finish [iff_def]\n-/\n\nexample : ((\u00aca \u2228 b) \u2227 (\u00acb \u2228 b) \u2227 (\u00aca \u2228 \u00acb) \u2227 (\u00acb \u2228 \u00acb) \u2192 false) \u2192 \u00ac((a \u2192 b) \u2192 b) \u2192 false :=\nby finish\n\nexample : \u00ac((a \u2192 b) \u2192 b) \u2192 ((\u00acb \u2228 \u00acb) \u2227 (\u00acb \u2228 \u00aca) \u2227 (b \u2228 \u00acb) \u2227 (b \u2228 \u00aca) \u2192 false) \u2192 false :=\nby finish\n\nexample : (\u00aca \u2194 b) \u2192 (\u00acb \u2194 a) \u2192 (\u00ac\u00aca \u2194 a) := by finish\n\nexample : (\u00ac a \u2194 b) \u2192 (\u00ac (c \u2228 e) \u2194 d \u2227 f) \u2192 (\u00ac (c \u2228 a \u2228 e) \u2194 d \u2227 b \u2227 f) := by finish\n\nexample {A : Type} (p q : A \u2192 Prop) (a b : A) : q a \u2192 p b \u2192 \u2203 x, (p x \u2227 x = b) \u2228 q x := by finish\n\nexample {A : Type} (p q : A \u2192 Prop) (a b : A) : p b \u2192 \u2203 x, q x \u2228 (p x \u2227 x = b) := by finish\n\nexample : \u00ac a \u2192 b \u2192 a \u2192 c := by finish\nexample : a \u2192 b \u2192 b \u2192 \u00ac a \u2192 c := by finish\nexample (a b : nat) : a = b \u2192 b = a := by finish\n\n-- good examples of things we don't get, even using the simplifier\nexample (a b c : nat) : a = b \u2192 a = c \u2192 b = c := by finish\nexample (p : nat \u2192 Prop) (a b c : nat) : a = b \u2192 a = c \u2192 p b \u2192 p c := by finish\n\nexample (p : Prop) (a b : nat) : a = b \u2192 p \u2192 p := by finish\n\n-- safe should look for contradictions with constructors\nexample (a : nat) : (0 : \u2115) = succ a \u2192 a = a \u2192 false := by finish\nexample (p : Prop) (a b c : nat) : [a, b, c] = [] \u2192 p := by finish\n\nexample (a b c : nat) : succ (succ a) = succ (succ b) \u2192 c = c := by finish\nexample (p : Prop) (a b : nat) : a = b \u2192 b \u2260 a \u2192 p := by finish\nexample : (a \u2194 b) \u2192 ((b \u2194 a) \u2194 (a \u2194 b)) := by finish\nexample (a b c : nat) : b = c \u2192 (a = b \u2194 c = a) := by finish [iff_def]\nexample : \u00ac\u00ac\u00ac\u00ac\u00ac\u00ac\u00ac\u00aca \u2192 \u00ac\u00ac\u00ac\u00ac\u00aca \u2192 false := by finish\nexample (a b c : Prop) : a \u2227 b \u2227 c \u2194 c \u2227 b \u2227 a := by finish\nexample (a b c : Prop) : a \u2227 false \u2227 c \u2194 false := by finish\nexample (a b c : Prop) : a \u2228 false \u2228 b \u2194 b \u2228 a := by finish\nexample : a \u2227 not a \u2194 false := by finish\nexample : a \u2227 b \u2227 true \u2192 b \u2227 a := by finish\nexample (A : Type) (a\u2081 a\u2082 : A) : a\u2081 = a\u2082 \u2192\n  (\u03bb (B : Type) (f : A \u2192 B), f a\u2081) = (\u03bb (B : Type) (f : A \u2192 B), f a\u2082) := by finish\nexample (a : nat) : \u00ac a = a \u2192 false := by finish\nexample (A : Type) (p : Prop) (a b c : A) : a = b \u2192 b \u2260 a \u2192 p := by finish\nexample (p q r s : Prop) : r \u2227 s \u2192 p \u2227 q \u2192 q \u2227 p := by finish\nexample (p q : Prop) : p \u2227 p \u2227 q \u2227 q \u2192 q \u2227 p := by finish\nexample (p : nat \u2192 Prop) (q : nat \u2192 nat \u2192 Prop) :\n  (\u2203 x y, p x \u2227 q x y) \u2192 q 0 0 \u2227 q 1 1 \u2192 (\u2203 x, p x) := by finish\nexample (p q r s : Prop) (a b : nat) : r \u2228 s \u2192 p \u2228 q \u2192 a = b \u2192 q \u2228 p := by finish\nexample (p q r : Prop) (a b : nat) : true \u2192 a = a \u2192 q \u2192 q \u2192 p \u2192 p := by finish\nexample (a b : Prop) : a \u2192 b \u2192 a := by finish\nexample (p q : nat \u2192 Prop) (a b : nat) : p a \u2192 q b \u2192 \u2203 x, p x := by finish\n\nexample : \u2200 b1 b2, b1 && b2 = ff \u2194 (b1 = ff \u2228 b2 = ff) := by finish\nexample : \u2200 b1 b2, b1 && b2 = tt \u2194 (b1 = tt \u2227 b2 = tt) := by finish\nexample : \u2200 b1 b2, b1 || b2 = ff \u2194 (b1 = ff \u2227 b2 = ff) := by finish\nexample : \u2200 b1 b2, b1 || b2 = tt \u2194 (b1 = tt \u2228 b2 = tt) := by finish\nexample : \u2200 b, bnot b = tt \u2194 b = ff := by finish\nexample : \u2200 b, bnot b = ff \u2194 b = tt := by finish\nexample : \u2200 b c, b = c \u2194 \u00ac (b = bnot c) := by intros b c; cases b; cases c; finish [iff_def]\n\ninductive and3 (a b c : Prop) : Prop\n| mk : a \u2192 b \u2192 c \u2192 and3\n\nexample (h : and3 a b c) : and3 b c a := by cases h; split; finish\n\ninductive or3 (a b c : Prop) : Prop\n| in1 : a \u2192 or3\n| in2 : b \u2192 or3\n| in3 : c \u2192 or3\n\n/- TODO(Jeremy): write a tactic that tries all constructors\nexample (h : a) : or3 a b c := sorry\nexample (h : b) : or3 a b c := sorry\nexample (h : c) : or3 a b c := sorry\n-/\n\nvariables (A\u2081 A\u2082 A\u2083 A\u2084 B\u2081 B\u2082 B\u2083 B\u2084 : Prop)\n-- H first, all pos\n\nexample (H1 : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) : B\u2084 := by finish\nexample (H1 : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2084) : B\u2083 := by finish\nexample (H1 : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084) : B\u2082 := by finish\nexample (H1 : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084) : B\u2081 := by finish\n\nexample (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a2 : A\u2082) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084) : \u00acA\u2083 := by finish\nexample (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a1 : A\u2081) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084) : \u00acA\u2082 := by finish\nexample (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084)\n  (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084) : \u00acA\u2081 := by finish\n\n-- H last, all pos\nexample (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : B\u2084 := by finish\nexample (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : B\u2083 := by finish\nexample (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : B\u2082 := by finish\nexample (a1 : A\u2081) (a2 : A\u2082) (a3 : A\u2083) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : B\u2081 := by finish\n\nexample (a1 : A\u2081) (a2 : A\u2082) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : \u00acA\u2083 := by finish\nexample (a1 : A\u2081) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : \u00acA\u2082 := by finish\nexample (a2 : A\u2082) (a3 : A\u2083) (n1 : \u00acB\u2081) (n2 : \u00acB\u2082) (n3 : \u00acB\u2083) (n3 : \u00acB\u2084)\n  (H : A\u2081 \u2192 A\u2082 \u2192 A\u2083 \u2192 B\u2081 \u2228 B\u2082 \u2228 B\u2083 \u2228 B\u2084) : \u00acA\u2081 := by finish\n\n-- H first, all neg\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) : \u00acB\u2084 := by finish\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b4 : B\u2084) : \u00acB\u2083 := by finish\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b3 : B\u2083) (b4 : B\u2084) : \u00acB\u2082 := by finish\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084) : \u00acB\u2081 := by finish\n\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084) : \u00ac\u00acA\u2083 := by finish\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n1 : \u00acA\u2081) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084) : \u00ac\u00acA\u2082 := by finish\nexample (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084)\n  (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084) : \u00ac\u00acA\u2081 := by finish\n\n-- H last, all neg\nexample (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00acB\u2084 := by finish\nexample (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00acB\u2083 := by finish\nexample (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b3 : B\u2083) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00acB\u2082 := by finish\nexample (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00acB\u2081 := by finish\n\nexample (n1 : \u00acA\u2081) (n2 : \u00acA\u2082) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00ac\u00acA\u2083 := by finish\nexample (n1 : \u00acA\u2081) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00ac\u00acA\u2082 := by finish\nexample (n2 : \u00acA\u2082) (n3 : \u00acA\u2083) (b1 : B\u2081) (b2 : B\u2082) (b3 : B\u2083) (b4 : B\u2084)\n  (H : \u00acA\u2081 \u2192 \u00acA\u2082 \u2192 \u00acA\u2083 \u2192 \u00acB\u2081 \u2228 \u00acB\u2082 \u2228 \u00acB\u2083 \u2228 \u00acB\u2084) : \u00ac\u00acA\u2081 := by finish\n\nsection club\nvariables Scottish RedSocks WearKilt Married GoOutSunday : Prop\ntheorem NoMember : (\u00acScottish \u2192 RedSocks) \u2192 (WearKilt \u2228 \u00acRedSocks) \u2192 (Married \u2192 \u00acGoOutSunday) \u2192\n                 (GoOutSunday \u2194 Scottish) \u2192 (WearKilt \u2192 Scottish \u2227 Married) \u2192\n                 (Scottish \u2192 WearKilt) \u2192 false := by finish\nend club\n\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/finish2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953506426082, "lm_q2_score": 0.6893056104028797, "lm_q1q2_score": 0.3980707851795281}}
{"text": "import .preamble_results\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n{E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n{E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n{F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n{F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n{H : Type*} [topological_space H]\n{H' : Type*} [topological_space H']\n{G : Type*} [topological_space G]\n{G' : Type*} [topological_space G']\n(I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n(J : model_with_corners \ud835\udd5c F G) (J' : model_with_corners \ud835\udd5c F' G')\n\nsection diffeomorph\n\nvariables (M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n(M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]\n(N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N']\n(n : with_top \u2115)\n\n/-- \u03b1 and \u03b2 are homeomorph, also called topological isomoph -/\nstructure diffeomorph extends M \u2243 M' :=\n(times_cont_mdiff_to_fun  : smooth I I' to_fun)\n(times_cont_mdiff_inv_fun : smooth I' I inv_fun)\n\ninfix ` \u2243\u2098 `:50 := diffeomorph _ _\nnotation M ` \u2243\u2098[` I `;` J `]` N := diffeomorph I J M N\n\nnamespace diffeomorph\ninstance : has_coe_to_fun (diffeomorph I I' M M') := \u27e8\u03bb _, M \u2192 M', \u03bbe, e.to_equiv\u27e9\n\nlemma coe_eq_to_equiv (h : diffeomorph I I' M M') (x : M) : h x = h.to_equiv x := rfl\n\n/-- Identity map is a diffeomorphism. -/\nprotected def refl : M \u2243\u2098[I; I] M :=\n{ smooth_to_fun := smooth_in_charts_id, smooth_inv_fun := smooth_in_charts_id, .. homeomorph.refl M }\n\n/-- Composition of two diffeomorphisms. -/\nprotected def trans (h\u2081 : diffeomorph I I' M M') (h\u2082 : diffeomorph I' J M' N) : M \u2243\u2098[I | J] N :=\n{ smooth_to_fun  := h\u2082.smooth_to_fun.comp h\u2081.smooth_to_fun,\n  smooth_inv_fun := h\u2081.smooth_inv_fun.comp h\u2082.smooth_inv_fun,\n  .. homeomorph.trans h\u2081.to_homeomorph h\u2082.to_homeomorph }\n\n/-- Inverse of a diffeomorphism. -/\nprotected def symm (h : M \u2243\u2098[I | J] N) : N \u2243\u2098[J | I] M :=\n{ smooth_to_fun  := h.smooth_inv_fun,\n  smooth_inv_fun := h.smooth_to_fun,\n  .. h.to_homeomorph.symm }\n\nend diffeomorph\n\nend diffeomorph", "meta": {"author": "Nicknamen", "repo": "lie_group", "sha": "e0d5c4f859654e3dea092702f1320c3c72a49983", "save_path": "github-repos/lean/Nicknamen-lie_group", "path": "github-repos/lean/Nicknamen-lie_group/lie_group-e0d5c4f859654e3dea092702f1320c3c72a49983/src/diffeomorph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.39801195162502384}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.sites.sheafification\nimport category_theory.sites.limits\nimport category_theory.limits.functor_category\nimport category_theory.limits.filtered_colimit_commutes_finite_limit\n\n/-!\n# Left exactness of sheafification\nIn this file we show that sheafification commutes with finite limits.\n-/\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverses w v u\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\nvariables [\u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\n\nnoncomputable theory\n\nnamespace category_theory.grothendieck_topology\n\n/-- An auxiliary definition to be used in the proof of the fact that\n`J.diagram_functor D X` preserves limits. -/\n@[simps]\ndef cone_comp_evaluation_of_cone_comp_diagram_functor_comp_evaluation\n  {X : C} {K : Type (max v u)} [small_category K]\n  {F : K \u2964 C\u1d52\u1d56 \u2964 D} {W : J.cover X} (i : W.arrow)\n  (E : cone (F \u22d9 J.diagram_functor D X \u22d9 (evaluation (J.cover X)\u1d52\u1d56 D).obj (op W))) :\n  cone (F \u22d9 (evaluation _ _).obj (op i.Y)) :=\n{ X := E.X,\n  \u03c0 :=\n  { app := \u03bb k, E.\u03c0.app k \u226b multiequalizer.\u03b9 (W.index (F.obj k)) i,\n    naturality' := begin\n      intros a b f,\n      dsimp,\n      rw [category.id_comp, category.assoc, \u2190 E.w f],\n      dsimp [diagram_nat_trans],\n      simp only [multiequalizer.lift_\u03b9, category.assoc],\n    end } }\n\n/-- An auxiliary definition to be used in the proof of the fact that\n`J.diagram_functor D X` preserves limits. -/\nabbreviation lift_to_diagram_limit_obj\n  {X : C} {K : Type (max v u)} [small_category K] [has_limits_of_shape K D]\n  {W : (J.cover X)\u1d52\u1d56} (F : K \u2964 C\u1d52\u1d56 \u2964 D)\n  (E : cone (F \u22d9 J.diagram_functor D X \u22d9 (evaluation (J.cover X)\u1d52\u1d56 D).obj W)) :\n  E.X \u27f6 (J.diagram (limit F) X).obj W :=\nmultiequalizer.lift _ _ (\u03bb i,\n  (is_limit_of_preserves ((evaluation _ _).obj (op i.Y)) (limit.is_limit _)).lift\n  (cone_comp_evaluation_of_cone_comp_diagram_functor_comp_evaluation i E))\nbegin\n  intros i,\n  change (_ \u226b _) \u226b _ = (_ \u226b _) \u226b _,\n  dsimp [evaluate_combined_cones],\n  erw [category.comp_id, category.comp_id, category.assoc,\n    category.assoc, \u2190 (limit.lift F _).naturality, \u2190 (limit.lift F _).naturality,\n    \u2190 category.assoc, \u2190 category.assoc],\n  congr' 1, ext1,\n  erw [category.assoc, category.assoc, limit.lift_\u03c0, limit.lift_\u03c0,\n    limit.lift_\u03c0_assoc, limit.lift_\u03c0_assoc, category.assoc,\n    category.assoc, multiequalizer.condition],\n  refl,\nend\n\ninstance (X : C) (K : Type (max v u)) [small_category K] [has_limits_of_shape K D]\n  (F : K \u2964 C\u1d52\u1d56 \u2964 D) : preserves_limit F (J.diagram_functor D X) :=\npreserves_limit_of_evaluation _ _ $ \u03bb W,\npreserves_limit_of_preserves_limit_cone (limit.is_limit _)\n{ lift := \u03bb E, lift_to_diagram_limit_obj F E,\n  fac' := begin\n    intros E k,\n    dsimp [diagram_nat_trans],\n    ext1,\n    simp only [multiequalizer.lift_\u03b9, multiequalizer.lift_\u03b9_assoc, category.assoc],\n    change (_ \u226b _) \u226b _ = _,\n    dsimp [evaluate_combined_cones],\n    erw [category.comp_id, category.assoc, \u2190 nat_trans.comp_app, limit.lift_\u03c0, limit.lift_\u03c0],\n    refl,\n  end,\n  uniq' := begin\n    intros E m hm,\n    ext,\n    delta lift_to_diagram_limit_obj,\n    erw [multiequalizer.lift_\u03b9, category.assoc],\n    change _ = (_ \u226b _) \u226b _,\n    dsimp [evaluate_combined_cones],\n    erw [category.comp_id, category.assoc, \u2190 nat_trans.comp_app, limit.lift_\u03c0, limit.lift_\u03c0],\n    dsimp,\n    rw \u2190 hm,\n    dsimp [diagram_nat_trans],\n    simp,\n  end } .\n\ninstance (X : C) (K : Type (max v u)) [small_category K] [has_limits_of_shape K D] :\n  preserves_limits_of_shape K (J.diagram_functor D X) := \u27e8\u27e9\n\ninstance (X : C) [has_limits D] : preserves_limits (J.diagram_functor D X) := \u27e8\u27e9\n\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [concrete_category.{max v u} D]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\n\n/-- An auxiliary definition to be used in the proof that `J.plus_functor D` commutes\nwith finite limits. -/\ndef lift_to_plus_obj_limit_obj {K : Type (max v u)} [small_category K] [fin_category K]\n  [has_limits_of_shape K D] [preserves_limits_of_shape K (forget D)]\n  [reflects_limits_of_shape K (forget D)] (F : K \u2964 C\u1d52\u1d56 \u2964 D) (X : C)\n  (S : cone (F \u22d9 J.plus_functor D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))) :\n  S.X \u27f6 (J.plus_obj (limit F)).obj (op X) :=\nlet\n  e := colimit_limit_iso (F \u22d9 J.diagram_functor D X),\n  t : J.diagram (limit F) X \u2245 limit (F \u22d9 J.diagram_functor D X) :=\n    (is_limit_of_preserves (J.diagram_functor D X) (limit.is_limit _)).cone_point_unique_up_to_iso\n    (limit.is_limit _),\n  p : (J.plus_obj (limit F)).obj (op X) \u2245 colimit (limit (F \u22d9 J.diagram_functor D X)) :=\n    has_colimit.iso_of_nat_iso t,\n  s : colimit (F \u22d9 J.diagram_functor D X).flip \u2245\n    F \u22d9 J.plus_functor D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X) :=\n      nat_iso.of_components (\u03bb k, colimit_obj_iso_colimit_comp_evaluation _ k) begin\n        intros i j f,\n        rw [\u2190 iso.eq_comp_inv, category.assoc, \u2190 iso.inv_comp_eq],\n        ext w,\n        dsimp [plus_map],\n        erw [colimit.\u03b9_map_assoc, colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv\n        ((F \u22d9 J.diagram_functor D X).flip) w j,\n        colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv_assoc\n        ((F \u22d9 J.diagram_functor D X).flip) w i],\n      rw \u2190 (colimit.\u03b9 (F \u22d9 J.diagram_functor D X).flip w).naturality,\n      refl\n    end in\nlimit.lift _ S \u226b (has_limit.iso_of_nat_iso s.symm).hom \u226b e.inv \u226b p.inv\n\n-- This lemma should not be used directly. Instead, one should use the fact that\n-- `J.plus_functor D` preserves finite limits, along with the fact that\n-- evaluation preserves limits.\nlemma lift_to_plus_obj_limit_obj_fac {K : Type (max v u)} [small_category K] [fin_category K]\n  [has_limits_of_shape K D] [preserves_limits_of_shape K (forget D)]\n  [reflects_limits_of_shape K (forget D)] (F : K \u2964 C\u1d52\u1d56 \u2964 D) (X : C)\n  (S : cone (F \u22d9 J.plus_functor D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))) (k) :\n  lift_to_plus_obj_limit_obj F X S \u226b (J.plus_map (limit.\u03c0 F k)).app (op X) = S.\u03c0.app k :=\nbegin\n  dsimp only [lift_to_plus_obj_limit_obj],\n  rw [\u2190 (limit.is_limit (F \u22d9 J.plus_functor D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X))).fac S k,\n    category.assoc],\n  congr' 1,\n  dsimp,\n  simp only [category.assoc],\n  rw [\u2190 iso.eq_inv_comp, iso.inv_comp_eq, iso.inv_comp_eq],\n  ext,\n  dsimp [plus_map],\n  simp only [has_colimit.iso_of_nat_iso_\u03b9_hom_assoc, \u03b9_colim_map],\n  dsimp [is_limit.cone_point_unique_up_to_iso, has_limit.iso_of_nat_iso,\n    is_limit.map],\n  rw limit.lift_\u03c0,\n  dsimp,\n  rw \u03b9_colimit_limit_iso_limit_\u03c0_assoc,\n  simp_rw [\u2190 nat_trans.comp_app, \u2190 category.assoc, \u2190 nat_trans.comp_app],\n  rw [limit.lift_\u03c0, category.assoc],\n  congr' 1,\n  rw \u2190 iso.comp_inv_eq,\n  erw colimit.\u03b9_desc,\n  refl,\nend\n\ninstance (K : Type (max v u))\n  [small_category K] [fin_category K] [has_limits_of_shape K D]\n  [preserves_limits_of_shape K (forget D)]\n  [reflects_limits_of_shape K (forget D)] :\n  preserves_limits_of_shape K (J.plus_functor D) :=\nbegin\n  constructor, intros F, apply preserves_limit_of_evaluation, intros X,\n  apply preserves_limit_of_preserves_limit_cone (limit.is_limit F),\n  refine \u27e8\u03bb S, lift_to_plus_obj_limit_obj F X.unop S, _, _\u27e9,\n  { intros S k,\n    apply lift_to_plus_obj_limit_obj_fac },\n  { intros S m hm,\n    dsimp [lift_to_plus_obj_limit_obj],\n    simp_rw [\u2190 category.assoc, iso.eq_comp_inv, \u2190 iso.comp_inv_eq],\n    ext,\n    simp only [limit.lift_\u03c0, category.assoc, \u2190 hm],\n    congr' 1,\n    ext,\n    dsimp [plus_map, plus_obj],\n    erw [colimit.\u03b9_map, colimit.\u03b9_desc_assoc, limit.lift_\u03c0],\n    dsimp,\n    simp only [category.assoc],\n    rw \u03b9_colimit_limit_iso_limit_\u03c0_assoc,\n    simp only [nat_iso.of_components_inv_app,\n      colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom, iso.symm_inv],\n    dsimp [is_limit.cone_point_unique_up_to_iso],\n    rw [\u2190 category.assoc, \u2190 nat_trans.comp_app, limit.lift_\u03c0],\n    refl }\nend\n\ninstance [has_finite_limits D] [preserves_finite_limits (forget D)]\n  [reflects_isomorphisms (forget D)] : preserves_finite_limits (J.plus_functor D) :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size.{max v u},\n  introsI K _ _,\n  haveI : reflects_limits_of_shape K (forget D) :=\n    reflects_limits_of_shape_of_reflects_isomorphisms,\n  apply_instance\nend\n\ninstance (K : Type (max v u))\n  [small_category K] [fin_category K] [has_limits_of_shape K D]\n  [preserves_limits_of_shape K (forget D)]\n  [reflects_limits_of_shape K (forget D)] :\n  preserves_limits_of_shape K (J.sheafification D) :=\nlimits.comp_preserves_limits_of_shape _ _\n\ninstance [has_finite_limits D] [preserves_finite_limits (forget D)]\n  [reflects_isomorphisms (forget D)] : preserves_finite_limits (J.sheafification D) :=\nlimits.comp_preserves_finite_limits _ _\n\nend category_theory.grothendieck_topology\n\nnamespace category_theory\n\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [concrete_category.{max v u} D]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\nvariables [preserves_limits (forget D)]\nvariables [reflects_isomorphisms (forget D)]\nvariables (K : Type (max v u))\nvariables [small_category K] [fin_category K] [has_limits_of_shape K D]\n\ninstance : preserves_limits_of_shape K (presheaf_to_Sheaf J D) :=\nbegin\n  constructor, intros F, constructor, intros S hS,\n  apply is_limit_of_reflects (Sheaf_to_presheaf J D),\n  haveI : reflects_limits_of_shape K (forget D) :=\n    reflects_limits_of_shape_of_reflects_isomorphisms,\n  apply is_limit_of_preserves (J.sheafification D) hS,\nend\n\ninstance [has_finite_limits D] : preserves_finite_limits (presheaf_to_Sheaf J D) :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size.{max v u},\n  intros, resetI, apply_instance\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/left_exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3980119516250238}}
{"text": "-- import category_theory.universal.zero\n-- import category_theory.limits.equalizers\n-- import category_theory.over\n\n-- open category_theory\n\n-- universes u v\n\n-- namespace category_theory.limits\n\n-- variables {C : Type u} [\ud835\udc9e : category.{u v} C] [has_zero_object.{u v} C]\n-- include \ud835\udc9e\n-- variables {X Y Z : C}\n\n-- structure is_kernel (f : Y \u27f6 Z) (\u03b9 : X \u27f6 Y) :=\n-- (w'    : \u03b9 \u226b f = zero_morphism _ _)\n-- (lift : \u03a0 {X' : C} {\u03b9' : X' \u27f6 Y} (w : \u03b9' \u226b f = zero_morphism X' Z), X' \u27f6 X)\n-- (fac'  : \u03a0 {X' : C} {\u03b9' : X' \u27f6 Y} (w : \u03b9' \u226b f = zero_morphism X' Z), (lift w) \u226b \u03b9 = \u03b9' . obviously)\n-- (uniq' : \u03a0 {X' : C} {\u03b9' : X' \u27f6 Y} (w : \u03b9' \u226b f = zero_morphism X' Z) {m : X' \u27f6 X} (h : m \u226b \u03b9 = \u03b9'), m = lift w . obviously)\n\n-- restate_axiom is_kernel.w'\n-- attribute [search] is_kernel.w\n-- restate_axiom is_kernel.fac'\n-- attribute [simp,search] is_kernel.fac\n-- restate_axiom is_kernel.uniq'\n-- attribute [search,elim] is_kernel.uniq\n\n-- @[extensionality] lemma is_kernel.ext {f : Y \u27f6 Z} {\u03b9 : X \u27f6 Y} (P Q : is_kernel f \u03b9) : P = Q :=\n-- begin cases P, cases Q, obviously end\n\n-- -- TODO should be marked [search]?\n-- lemma kernel.w {f : Y \u27f6 Z} {X : C} (\u03b9 : X \u27f6 Y) (k : is_kernel f \u03b9) : \u03b9 \u226b f = zero_morphism _ _ := by rw k.w\n\n-- variable (C)\n\n-- class has_kernels :=\n-- (kernel : \u03a0 {Y Z : C} (f : Y \u27f6 Z), C)\n-- (\u03b9       : \u03a0 {Y Z : C} (f : Y \u27f6 Z), kernel f \u27f6 Y)\n-- (is     : \u03a0 {Y Z : C} (f : Y \u27f6 Z), is_kernel f (\u03b9 f))\n\n-- variable {C}\n\n-- variable [has_kernels.{u v} C]\n\n-- def kernel (f : Y \u27f6 Z) : C := has_kernels.kernel.{u v} f\n-- def kernel.\u03b9 (f : Y \u27f6 Z) : kernel f \u27f6 Y := has_kernels.\u03b9.{u v} f\n-- def kernel.subobject (f : Y \u27f6 Z) : over Y := \u27e8 kernel f, kernel.\u03b9 f \u27e9\n\n-- def kernel_of_equalizer {f : Y \u27f6 Z} {t : fork f (zero_morphism _ _)} (e : is_equalizer t) : is_kernel f t.\u03b9 :=\n-- { w' := begin have p := t.w, simp at p, exact p end,\n--   lift := \u03bb X' \u03b9' w, e.lift { X := X', \u03b9 := \u03b9' },\n--   uniq' := \u03bb X' \u03b9' w m h, begin tidy, apply e.uniq { X := X', \u03b9 := m \u226b t.\u03b9 }, tidy end }\n\n-- -- def equalizer_of_kernel {f : Y \u27f6 Z} {t : fork f (zero_morphism _ _)} (k : is_kernel f t.\u03b9) : is_equalizer t :=\n-- -- { lift := \u03bb s, begin have e := s.w, tidy, exact k.lift e, end,\n-- --   uniq := sorry, }\n\n-- -- def kernels_are_equalizers {f : Y \u27f6 Z} (t : fork f (zero_morphism _ _)) : equiv (is_kernel f t.\u03b9) (is_equalizer t) :=\n-- -- { to_fun  := equalizer_of_kernel,\n-- --   inv_fun := kernel_of_equalizer,\n-- --   left_inv  := sorry,\n-- --   right_inv := sorry }\n\n-- end category_theory.limits\n\n", "meta": {"author": "semorrison", "repo": "lean-category-theory", "sha": "a27b4ae5eac978e9188d2e867c3d11d9a5b87a9e", "save_path": "github-repos/lean/semorrison-lean-category-theory", "path": "github-repos/lean/semorrison-lean-category-theory/lean-category-theory-a27b4ae5eac978e9188d2e867c3d11d9a5b87a9e/src/category_theory/universal/kernels.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3980119451716871}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.sites.subsheaf\nimport category_theory.sites.compatible_sheafification\n\n/-!\n\n# Locally surjective morphisms\n\n## Main definitions\n\n- `is_locally_surjective` : A morphism of presheaves valued in a concrete category is locally\n  surjective with respect to a grothendieck topology if every section in the target is locally\n  in the set-theoretic image, i.e. the image sheaf coincides with the target.\n\n## Main results\n\n- `to_sheafify_is_locally_surjective` : `to_sheafify` is locally surjective.\n\n-/\n\nuniverses v u w v' u' w'\n\nopen opposite category_theory category_theory.grothendieck_topology\n\nnamespace category_theory\n\nvariables {C : Type u} [category.{v} C] (J : grothendieck_topology C)\n\nlocal attribute [instance] concrete_category.has_coe_to_sort concrete_category.has_coe_to_fun\n\nvariables {A : Type u'} [category.{v'} A] [concrete_category.{w'} A]\n\n/-- Given `f : F \u27f6 G`, a morphism between presieves, and `s : G.obj (op U)`, this is the sieve\nof `U` consisting of the `i : V \u27f6 U` such that `s` restricted along `i` is in the image of `f`. -/\n@[simps (lemmas_only)]\ndef image_sieve {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) {U : C} (s : G.obj (op U)) : sieve U :=\n{ arrows := \u03bb V i, \u2203 t : F.obj (op V), f.app _ t = G.map i.op s,\n  downward_closed' := begin\n    rintros V W i \u27e8t, ht\u27e9 j,\n    refine \u27e8F.map j.op t, _\u27e9,\n    rw [op_comp, G.map_comp, comp_apply, \u2190 ht, elementwise_of f.naturality],\n  end }\n\nlemma image_sieve_eq_sieve_of_section {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) {U : C} (s : G.obj (op U)) :\n  image_sieve f s = (image_presheaf (whisker_right f (forget A))).sieve_of_section s := rfl\n\nlemma image_sieve_whisker_forget {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) {U : C} (s : G.obj (op U)) :\n  image_sieve (whisker_right f (forget A)) s = image_sieve f s := rfl\n\nlemma image_sieve_app {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) {U : C} (s : F.obj (op U)) :\n  image_sieve f (f.app _ s) = \u22a4 :=\nbegin\n  ext V i,\n  simp only [sieve.top_apply, iff_true, image_sieve_apply],\n  have := elementwise_of (f.naturality i.op),\n  exact \u27e8F.map i.op s, this s\u27e9,\nend\n\n/-- A morphism of presheaves `f : F \u27f6 G` is locally surjective with respect to a grothendieck\ntopology if every section of `G` is locally in the image of `f`. -/\ndef is_locally_surjective {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) : Prop :=\n\u2200 (U : C) (s : G.obj (op U)), image_sieve f s \u2208 J U\n\nlemma is_locally_surjective_iff_image_presheaf_sheafify_eq_top {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) :\n  is_locally_surjective J f \u2194 (image_presheaf (whisker_right f (forget A))).sheafify J = \u22a4 :=\nbegin\n  simp only [subpresheaf.ext_iff, function.funext_iff, set.ext_iff, top_subpresheaf_obj,\n    set.top_eq_univ, set.mem_univ, iff_true],\n  exact \u27e8\u03bb H U, H (unop U), \u03bb H U, H (op U)\u27e9\nend\n\nlemma is_locally_surjective_iff_image_presheaf_sheafify_eq_top'\n  {F G : C\u1d52\u1d56 \u2964 (Type w)} (f : F \u27f6 G) :\n  is_locally_surjective J f \u2194 (image_presheaf f).sheafify J = \u22a4 :=\nbegin\n  simp only [subpresheaf.ext_iff, function.funext_iff, set.ext_iff, top_subpresheaf_obj,\n    set.top_eq_univ, set.mem_univ, iff_true],\n  exact \u27e8\u03bb H U, H (unop U), \u03bb H U, H (op U)\u27e9\nend\n\nlemma is_locally_surjective_iff_is_iso\n  {F G : Sheaf J (Type w)} (f : F \u27f6 G) :\n  is_locally_surjective J f.1 \u2194 is_iso (image_sheaf_\u03b9 f) :=\nbegin\n  rw [image_sheaf_\u03b9, is_locally_surjective_iff_image_presheaf_sheafify_eq_top',\n    subpresheaf.eq_top_iff_is_iso],\n  exact \u27e8\u03bb h, @@is_iso_of_reflects_iso _ _ (image_sheaf_\u03b9 f) (Sheaf_to_presheaf J _) h _,\n    \u03bb h, @@functor.map_is_iso _ _ (Sheaf_to_presheaf J _) _ h\u27e9,\nend\n\nlemma is_locally_surjective_iff_whisker_forget {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) :\n  is_locally_surjective J f \u2194 is_locally_surjective J (whisker_right f (forget A)) :=\nbegin\n  simpa only [is_locally_surjective_iff_image_presheaf_sheafify_eq_top]\nend\n\nlemma is_locally_surjective_of_surjective {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G)\n  (H : \u2200 U, function.surjective (f.app U)) : is_locally_surjective J f :=\nbegin\n  intros U s,\n  obtain \u27e8t, rfl\u27e9 := H _ s,\n  rw image_sieve_app,\n  exact J.top_mem _\nend\n\nlemma is_locally_surjective_of_iso {F G : C\u1d52\u1d56 \u2964 A} (f : F \u27f6 G) [is_iso f] :\n  is_locally_surjective J f :=\nbegin\n  apply is_locally_surjective_of_surjective,\n  intro U,\n  apply function.bijective.surjective,\n  rw \u2190 is_iso_iff_bijective,\n  apply_instance\nend\n\nlemma is_locally_surjective.comp {F\u2081 F\u2082 F\u2083 : C\u1d52\u1d56 \u2964 A} {f\u2081 : F\u2081 \u27f6 F\u2082} {f\u2082 : F\u2082 \u27f6 F\u2083}\n  (h\u2081 : is_locally_surjective J f\u2081) (h\u2082 : is_locally_surjective J f\u2082) :\n    is_locally_surjective J (f\u2081 \u226b f\u2082) :=\nbegin\n  intros U s,\n  have : sieve.bind (image_sieve f\u2082 s) (\u03bb _ _ h, image_sieve f\u2081 h.some) \u2264 image_sieve (f\u2081 \u226b f\u2082) s,\n  { rintros V i \u27e8W, i, j, H, \u27e8t', ht'\u27e9, rfl\u27e9,\n    refine \u27e8t', _\u27e9,\n    rw [op_comp, F\u2083.map_comp, nat_trans.comp_app, comp_apply, comp_apply, ht',\n      elementwise_of f\u2082.naturality, H.some_spec] },\n  apply J.superset_covering this,\n  apply J.bind_covering,\n  { apply h\u2082 },\n  { intros, apply h\u2081 }\nend\n\nsection\n\nvariables (F : C\u1d52\u1d56 \u2964 Type (max u v))\n\n/-- The image of `F` in `J.sheafify F` is isomorphic to the sheafification. -/\nnoncomputable\ndef sheafification_iso_image_presheaf :\n  J.sheafify F \u2245 ((image_presheaf (J.to_sheafify F)).sheafify J).to_presheaf :=\n{ hom := J.sheafify_lift (to_image_presheaf_sheafify J _)\n  ((is_sheaf_iff_is_sheaf_of_type J _).mpr $ subpresheaf.sheafify_is_sheaf _ $\n    (is_sheaf_iff_is_sheaf_of_type J _).mp $ sheafify_is_sheaf J _),\n  inv := subpresheaf.\u03b9 _,\n  hom_inv_id' := J.sheafify_hom_ext _ _ (J.sheafify_is_sheaf _)\n    (by simp [to_image_presheaf_sheafify]),\n  inv_hom_id' := begin\n    rw [\u2190 cancel_mono (subpresheaf.\u03b9 _), category.id_comp, category.assoc],\n    refine eq.trans _ (category.comp_id _),\n    congr' 1,\n    exact J.sheafify_hom_ext _ _ (J.sheafify_is_sheaf _) (by simp [to_image_presheaf_sheafify]),\n    apply_instance\n  end }\n\n-- We need to sheafify\nvariables {B : Type w} [category.{max u v} B]\n  [concrete_category.{max u v} B]\n  [\u2200 (X : C), limits.has_colimits_of_shape (J.cover X)\u1d52\u1d56 B]\n  [\u2200 (P : C\u1d52\u1d56 \u2964 B) (X : C) (S : J.cover X), limits.has_multiequalizer (S.index P)]\n  [\u03a0 (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 B),\n    limits.preserves_limit (W.index P).multicospan (forget B)]\n  [\u03a0 (X : C), limits.preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget B)]\n  [\u2200 (\u03b1 \u03b2 : Type (max u v)) (fst snd : \u03b2 \u2192 \u03b1),\n    limits.has_limits_of_shape (limits.walking_multicospan fst snd) B]\n\nlemma to_sheafify_is_locally_surjective (F : C\u1d52\u1d56 \u2964 B) :\n  is_locally_surjective J (J.to_sheafify F) :=\nbegin\n  rw [is_locally_surjective_iff_whisker_forget, \u2190 to_sheafify_comp_sheafify_comp_iso_inv],\n  apply is_locally_surjective.comp,\n  { rw [is_locally_surjective_iff_image_presheaf_sheafify_eq_top, subpresheaf.eq_top_iff_is_iso],\n    exact is_iso.of_iso_inv (sheafification_iso_image_presheaf J (F \u22d9 forget B)) },\n  { exact is_locally_surjective_of_iso _ _ }\nend\n\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/surjective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.39795541216987085}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sebastian Ullrich\n\n! This file was ported from Lean 3 source module init.data.string.ops\n! leanprover-community/mathlib commit 9af482290ef68e8aaa5ead01aa7b09b7be7019fd\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Data.Bool.Lemmas\nimport Leanbin.Init.Data.String.Basic\nimport Leanbin.Init.Meta.WellFoundedTactics\n\nnamespace String\n\nnamespace Iterator\n\n@[simp]\ntheorem nextToString_mkIterator (s : String) : s.mkIterator.nextToString = s := by\n  induction s <;> rfl\n#align string.iterator.next_to_string_mk_iterator String.Iterator.nextToString_mkIterator\n\n@[simp]\ntheorem length_nextToString_next (it : Iterator) :\n    it.next.nextToString.length = it.nextToString.length - 1 := by\n  cases it <;> cases it_snd <;>\n    simp [iterator.next, iterator.next_to_string, String.length, Nat.add_sub_cancel_left]\n#align string.iterator.length_next_to_string_next String.Iterator.length_nextToString_next\n\ntheorem zero_lt_length_nextToString_of_hasNext {it : Iterator} :\n    it.hasNext \u2192 0 < it.nextToString.length := by\n  cases it <;> cases it_snd <;>\n    simp [iterator.has_next, iterator.next_to_string, String.length, Nat.zero_lt_one_add,\n      Nat.add_comm, false_imp_iff]\n#align string.iterator.zero_lt_length_next_to_string_of_has_next String.Iterator.zero_lt_length_nextToString_of_hasNext\n\nend Iterator\n\n-- TODO(Sebastian): generalize to something like\n-- https://doc.rust-lang.org/std/primitive.str.html#method.split\nprivate def split_core (p : Char \u2192 Bool) : Iterator \u2192 Iterator \u2192 List String\n  | start, stop =>\n    if h : stop.hasNext then\n      -- wf hint\n      have : stop.nextToString.length - 1 < stop.nextToString.length :=\n        Nat.sub_lt (Iterator.zero_lt_length_nextToString_of_hasNext h) (by decide)\n      if p stop.curr then\n        let rest := stop.next.nextToString\n        (start.extract stop).getD \"\" :: split_core stop.next stop.next\n      else split_core start stop.next\n    else [start.nextToString]termination_by'\n  \u27e8_, measure_wf fun e => e.2.nextToString.length\u27e9\n#align string.split_core string.split_core\n\n/- warning: string.split -> String.split is a dubious translation:\nlean 3 declaration is\n  (Char -> Bool) -> String -> (List.{0} String)\nbut is expected to have type\n  String -> (Char -> Bool) -> (List.{0} String)\nCase conversion may be inaccurate. Consider using '#align string.split String.split\u2093'. -/\ndef split (p : Char \u2192 Bool) (s : String) : List String :=\n  splitCore p s.mkIterator s.mkIterator\n#align string.split String.split\n\nend String\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Data/String/Ops.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.39795539877069797}}
{"text": "\nimport util.meta.tactic\nimport data.nat.basic\nimport util.data.nat\nimport util.data.order\nimport tactic\n\nimport separation.heap.basic\n\nnamespace heap\n\nopen nat list (hiding bind)\n\n@[simp]\nlemma heap_emp_disjoint (h : heap)\n: heap.emp ## h :=\nby { dunfold disjoint heap.emp, intros, left, refl }\n\n@[simp]\nlemma disjoint_heap_emp (h : heap)\n: h ## heap.emp :=\nby { dunfold disjoint heap.emp, intros, right, refl }\n\n@[simp]\nlemma heap_emp_part'_eq_self (h : heap)\n: part' heap.emp h (heap_emp_disjoint _) = h :=\nbegin\n  apply funext, intro,\n  unfold part',\n  cases (h x) ; simp [heap.emp,has_orelse.orelse,option.orelse]\nend\n\n@[simp]\nlemma part'_heap_emp_eq_self (h : heap)\n: part' h heap.emp (disjoint_heap_emp _) = h :=\nbegin\n  apply funext, intro,\n  unfold part',\n  cases (h x) ; simp [heap.emp,has_orelse.orelse,option.orelse]\nend\n\nlemma maplet_disjoint_heap_mk_of_lt {p q : pointer} (v : word) (vs : list word)\n  (h : p < q)\n: disjoint (maplet p v) (heap.mk q vs) :=\nbegin\n  revert q v,\n  induction vs with v vs ; intros q v h,\n  { apply disjoint_heap_emp },\n  { simp [(##)], intro ptr,\n    simp_intros h\u2081 [or_iff_not_imp,heap.mk],\n    split,\n    { simp [maplet] at \u22a2 h\u2081,\n      ite_cases with h\u2080 at h\u2081 \u22a2,\n      { contradiction },\n      rw [\u2190 h\u2080,if_neg],\n      apply ne_of_gt, apply h  },\n    { revert h\u2081, rw \u2190 or_iff_not_imp,\n      apply vs_ih,\n      transitivity q, assumption,\n      apply lt_add_of_pos_right,\n      apply zero_lt_one, } }\nend\n\nlemma maplet_disjoint_heap_mk (p : pointer) (v : word) (vs : list word)\n: maplet p v ## heap.mk (p + 1) vs :=\nbegin\n  apply maplet_disjoint_heap_mk_of_lt,\n  apply lt_add_of_pos_right,\n  apply zero_lt_one,\nend\n\nlemma heap_mk_cons (p : pointer) (v : word) (vs : list word)\n:   heap.mk p (v :: vs)\n  = part' (maplet p v) (heap.mk (p+1) vs) (maplet_disjoint_heap_mk p v vs) :=\nby { funext x, simp [heap.mk,part'] }\n\n@[symm]\nlemma disjoint_symm {h\u2080 h\u2081 : heap}\n  (h : h\u2080 ## h\u2081)\n: h\u2081 ## h\u2080 :=\nassume p, or.symm (h p)\n\nlemma part'_comm {h\u2080 h\u2081 : heap}\n  (h : h\u2080 ## h\u2081)\n: part' h\u2080 h\u2081 = part' h\u2081 h\u2080 (disjoint_symm h) :=\nbegin\n  funext p, unfold part',\n  cases h p with h h ; simp [h],\nend\n\nlemma part_disjoint_assoc_left {h\u2080 h\u2081 h\u2082 : heap}\n  (Hdisj : h\u2080 ## h\u2081)\n  (Hdisj' : h\u2081 ## h\u2082)\n  (h : part' h\u2080 h\u2081 Hdisj ## h\u2082)\n: h\u2080 ## part' h\u2081 h\u2082 Hdisj' :=\nbegin\n  intro p,\n  cases h p with h' h' ; unfold part' at *,\n  { rw or_else_eq_none_iff at h',\n    simp [h'.left] },\n  { simp [h',Hdisj p], },\nend\n\nlemma part_disjoint_assoc_right {h\u2080 h\u2081 h\u2082 : heap}\n  (Hdisj : h\u2080 ## h\u2081)\n  (Hdisj' : h\u2081 ## h\u2082)\n  (h : h\u2080 ## part' h\u2081 h\u2082)\n: part' h\u2080 h\u2081 ## h\u2082 :=\nbegin\n  intro p,\n  cases h p with h' h' ; unfold part' at *,\n  { simp [h',Hdisj' p], },\n  { rw or_else_eq_none_iff at h',\n    simp [h'.right] },\nend\n\n@[simp]\nlemma eq_emp_of_part' (hp a : heap)\n  (h  : a ## hp)\n: a = part' a hp \u2194 hp = heap.emp :=\nbegin\n  split ; intro h\u2080,\n  { funext k,\n    have h\u2081 : a k = part' a hp h k,\n    { rw \u2190 h\u2080, },\n    clear h\u2080,\n    dsimp [part',heap.emp,heap.disjoint] at *,\n    specialize h k,\n    cases h with h\u2082 h\u2082 ; simp [h\u2082] at h\u2081,\n    rw h\u2081, rw h\u2082, },\n  { simp [h\u2080], }\nend\n\nlemma part'_disjoint {h\u2081 h\u2082 h\u2083 : heap}\n  {H\u2080 : h\u2082 ## h\u2083}\n  (H\u2081 : h\u2081 ## h\u2083)\n  (H\u2082 : h\u2081 ## h\u2082)\n: part' h\u2082 h\u2083 ## h\u2081 :=\nbegin\n  intro p,\n  rw [or.comm,or_iff_not_imp,part',or_else_eq_none_iff],\n  intros H\u2083,\n  specialize H\u2081 p,\n  specialize H\u2082 p,\n  rw or_iff_not_imp at H\u2081 H\u2082,\n  split\n  ; apply_assumption\n  ; apply_assumption,\nend\n\nlemma disjoint_part' {h\u2081 h\u2082 h\u2083 : heap}\n  {H\u2080 : h\u2082 ## h\u2083}\n  (H\u2081 : h\u2081 ## h\u2083)\n  (H\u2081 : h\u2081 ## h\u2082)\n: h\u2081 ## part' h\u2082 h\u2083 :=\nby { apply disjoint_symm,\n     apply part'_disjoint\n     ; assumption }\n\nlemma disjoint_of_part'_disjoint_right {h\u2081 h\u2082 h\u2083 : heap}\n  (H\u2081 : h\u2082 ## h\u2083)\n  (H\u2080 : part' h\u2082 h\u2083 ## h\u2081)\n: h\u2083 ## h\u2081 :=\nbegin\n  intro p,\n  cases H\u2080 p with H\u2082 H\u2082,\n  { unfold part' at H\u2082,\n    rw or_else_eq_none_iff at H\u2082,\n    simp [H\u2082.right] },\n  { simp [H\u2082] }\nend\n\nlemma disjoint_of_part'_disjoint_left {h\u2081 h\u2082 h\u2083 : heap}\n  (H\u2081 : h\u2082 ## h\u2083)\n  (H\u2080 : part' h\u2082 h\u2083 ## h\u2081)\n: h\u2082 ## h\u2081 :=\nbegin\n  have H\u2081 := disjoint_symm H\u2081,\n  apply disjoint_of_part'_disjoint_right H\u2081,\n  rw part'_comm, apply H\u2080,\nend\n\nlemma disjoint_of_disjoint_part'_right {h\u2081 h\u2082 h\u2083 : heap}\n  (H\u2081 : h\u2082 ## h\u2083)\n  (H\u2080 : h\u2081 ## part' h\u2082 h\u2083)\n: h\u2081 ## h\u2083 :=\nbegin\n  intro p,\n  rw or_iff_not_imp,\n  intro H\u2082,\n  specialize H\u2080 p,\n  rw or_iff_not_imp at H\u2080,\n  specialize H\u2080 H\u2082,\n  simp [part'] at H\u2080,\n  apply H\u2080.right,\nend\n\nlemma disjoint_of_disjoint_part'_left {h\u2081 h\u2082 h\u2083 : heap}\n  (H\u2081 : h\u2082 ## h\u2083)\n  (H\u2080 : h\u2081 ## part' h\u2082 h\u2083)\n: h\u2081 ## h\u2082 :=\nby { apply disjoint_of_disjoint_part'_right,\n     rw part'_comm, apply H\u2080,\n     symmetry, apply H\u2081, }\n\nnamespace tactic.interactive\n\nopen heap\nopen tactic tactic.interactive (ite_cases)\nopen lean lean.parser interactive interactive.types\n\nmeta def break_disjoint_asm_symm (l : expr)\n: tactic unit :=\ndo t \u2190 infer_type l,\n   match t with\n    | `(%%h\u2080 ## (%%h\u2081 : heap)) :=\n      do h \u2190 get_unused_name `h,\n         to_expr ``(disjoint_symm %%l) >>= note h none,\n         return ()\n    | _ :=\n         fail $ format! \"expecting {l} of the form _ ## _\"\n   end\n\nmeta def break_disjoint_asm_r (l : expr)\n: tactic (list expr) :=\ndo t \u2190 infer_type l,\n   match t with\n    | `(%%h\u2080 ## part' %%h\u2081 %%h\u2082 %%h\u2083) :=\n      do h \u2190 get_unused_name `h,\n         r \u2190 to_expr ``(disjoint_of_disjoint_part'_right _ %%l) >>= note h none,\n         h \u2190 get_unused_name `h,\n         r' \u2190 to_expr ``(disjoint_of_disjoint_part'_left _ %%l) >>= note h none,\n         try (tactic.clear l),\n         return [r,r']\n    | _ :=\n         fail $ format! \"expecting {l} of the form _ ## _\"\n   end\nmeta def break_disjoint_asm_l (l : expr)\n: tactic (list expr) :=\ndo t \u2190 infer_type l,\n   match t with\n    | `(part' %%h\u2081 %%h\u2082 %%h\u2083 ## %%h\u2080) :=\n      do h \u2190 get_unused_name `h,\n         r \u2190 to_expr ``(disjoint_of_part'_disjoint_right _ %%l) >>= note h none,\n         h \u2190 get_unused_name `h,\n         r' \u2190 to_expr ``(disjoint_of_part'_disjoint_left _ %%l) >>= note h none,\n         try (tactic.clear l),\n         return [r,r']\n    | _ :=\n         break_disjoint_asm_r l\n   end\n\nmeta def break_disjoint_asm'\n: expr \u2192 tactic unit\n| l :=\ndo xs \u2190 break_disjoint_asm_l l,\n   xs.for_each (try \u2218 break_disjoint_asm')\n\nmeta def break_disjoint_asm (l : parse ident)\n: tactic (list expr) :=\ndo get_local l >>= break_disjoint_asm_l\n\nmeta def break_disjoint_asms\n: tactic unit :=\ndo ls \u2190 local_context,\n   ls.for_each (try \u2218 break_disjoint_asm'),\n   ls \u2190 local_context,\n   ls.for_each (try \u2218 break_disjoint_asm_symm)\n\nmeta def prove_disjoint'\n: tactic unit :=\n    assumption\n<|> (`[ apply part'_disjoint ] ; assumption )\n<|> (`[ apply disjoint_part' ] ; assumption )\n<|> failed\n\nmeta def prove_disjoint\n: tactic unit :=\ndo break_disjoint_asms,\n   prove_disjoint'\n\nrun_cmd add_interactive [`prove_disjoint]\n\nend tactic.interactive\n\nlemma part'_assoc {h\u2080 h\u2081 h\u2082 : heap}\n  (Hdisj\u2080 : h\u2080 ## h\u2081)\n  (Hdisj\u2082 : part' h\u2080 h\u2081 ## h\u2082)\n: part' (part' h\u2080 h\u2081) h\u2082 Hdisj\u2082 = part' h\u2080 (part' h\u2081 h\u2082\n     (by prove_disjoint)) (by prove_disjoint) :=\nby { funext p, simp [part'] }\n\nlemma delete_disjoint_delete {p : pointer} {n : \u2115} {hp\u2080 hp\u2081 : heap}\n  (h : hp\u2080 ## hp\u2081)\n: heap.delete p n hp\u2080 ## heap.delete p n hp\u2081 :=\nbegin\n  revert p,\n  induction n with n ; intro p,\n  { intro q,\n    simp [heap.delete,h q], },\n  { intro q,\n    simp [heap.delete],\n    ite_cases,\n    apply n_ih, simp },\nend\n\nlemma delete_over_part' {p : pointer} {n : \u2115} {hp\u2080 hp\u2081 : heap}\n  (h : hp\u2080 ## hp\u2081)\n:   heap.delete p n (part' hp\u2080 hp\u2081)\n  = part' (heap.delete p n hp\u2080) (heap.delete p n hp\u2081) (delete_disjoint_delete h) :=\nbegin\n  revert p,\n  induction n with n ; intro p,\n  { funext q,\n    refl },\n  { funext q,\n    simp [heap.delete],\n    ite_cases with h',\n    { simp [n_ih,part',heap.delete,if_neg,h'], },\n    { simp [part',heap.delete,if_pos,h'], } }\nend\n\nlemma heap_delete_maplet (p q : pointer) (k : \u2115) (v : word)\n  (h : p < q)\n: heap.delete q k (maplet p v) = maplet p v :=\nbegin\n  funext x,\n  revert q,\n  induction k with k ; intros q h, refl,\n  { simp [heap.delete],\n    ite_cases, apply @k_ih (q + 1) _,\n    { transitivity q, assumption,\n      apply lt_succ_self },\n    simp [maplet],\n    subst x,\n    rw if_neg,\n    apply ne_of_lt h },\nend\n\nlemma delete_part'_heap_mk {p : pointer} {vs : list word} {hp : heap}\n  (h : heap.mk p vs ## hp)\n: heap.delete p (length vs) (part' (heap.mk p vs) hp) = hp :=\nbegin\n  revert p,\n  induction vs with v vs\n  ; intros p h\u2080,\n  { simp [heap.delete,heap.mk], funext q, refl },\n  { funext q,\n    simp [length,add_one,heap.delete],\n    ite_cases,\n    { simp [heap_mk_cons],\n      have h\u2081 : heap.mk (p + 1) vs ## hp,\n      { simp [heap.mk] at h\u2080,\n        intro p',\n        specialize h\u2080 p',\n        rw or_iff_not_imp at h\u2080 \u22a2,\n        intros h\u2081, apply h\u2080,\n        rw or_else_eq_none_iff,\n        revert h\u2081, apply mt,\n        apply and.elim_right, },\n      have h\u2082 : heap.delete (succ p) (length vs) (maplet p v) = maplet p v,\n      { apply heap_delete_maplet, apply lt_succ_self, },\n      simp [part'_assoc,delete_over_part',vs_ih h\u2081,h\u2082,part',maplet,if_neg,h], },\n  { have h\u2081 : maplet p v ## hp,\n    { rw heap_mk_cons at h\u2080, prove_disjoint },\n    simp [(##)] at h\u2081,\n    specialize h\u2081 q, rw [or_iff_not_imp] at h\u2081,\n    symmetry, apply h\u2081,\n    simp [maplet,if_pos,h] }, }\nend\n\nlemma part'_insert (hp hp' : heap) (p : pointer) (v : word)\n  (h\u2080 : hp.insert p v ## hp')\n  (h\u2081 : hp ## hp')\n: part' (hp.insert p v) hp' = (part' hp hp').insert p v :=\nbegin\n  funext x,\n  simp [part',heap.insert],\n  ite_cases,\n  simp [some_or_else],\nend\n\nlemma maplet_insert_disjoint_iff (p : pointer) (v v' : word) (hp : heap)\n: (maplet p v).insert p v' ## hp \u2194 maplet p v ## hp :=\nbegin\n  simp [disjoint,heap.insert],\n  apply forall_congr,\n  intro p',\n  ite_cases,\n  simp [maplet,if_pos,h],\nend\n\n@[simp]\nlemma some_part'\n  (hp\u2080 hp\u2081 : heap)\n  (h : hp\u2080 ## hp\u2081)\n: some (part' hp\u2080 hp\u2081) = part (some hp\u2080) (some hp\u2081) :=\nby { simp [part,bind,option.bind,dif_pos, h], }\n\n@[simp]\nlemma heap_emp_part\n  (hp : heap)\n: part (some heap.emp) (some hp) = some hp :=\nby simp [part]\n\n@[simp]\nlemma part_heap_emp\n  (hp : heap)\n: part (some hp) (some heap.emp) = some hp :=\nby simp [part]\n\nend heap\n\nnamespace tactic.interactive\n\nopen heap\nopen tactic tactic.interactive (ite_cases)\nopen lean lean.parser interactive interactive.types\n\nmeta def try_then (x y : tactic unit) : tactic unit :=\nfocus1 $\ndo (some ()) \u2190 try_core x | tactic.skip,\n   tactic.all_goals y\n\nmeta def expand_part_ite : tactic unit :=\ndo tactic.try `[ dsimp [part] ],\n   tactic.reflexivity <|> try_then (ite_cases none $ loc.ns [none]) expand_part_ite\n\nmeta def contradict_asm\n: tactic unit :=\ndo ls \u2190 local_context,\n   ls.any_of (\u03bb e,\n     do `(\u00ac %%t) \u2190 infer_type e | failed ,\n         exfalso, tactic.apply e, tactic.clear e)\n\nend tactic.interactive\n\nnamespace heap\n\nlemma part_assoc\n  (a b c : option heap)\n: part (part a b) c = part a (part b c) :=\nbegin\n  cases a ; cases b ; cases c\n  ; expand_part_ite,\n   all_goals { {contradict_asm ; prove_disjoint} <|> rw part'_assoc },\nend\n\nlemma part_comm\n  (a b : option heap)\n: part a b = part b a :=\nbegin\n  cases a ; cases b\n  ; dsimp [part] ; try { refl }\n  ; ite_cases with i\n  ; ite_cases with i'\n  ; try { contradict_asm, symmetry, assumption },\n  rw part'_comm,\nend\n\ninstance : is_associative (option heap) part :=\n\u27e8 part_assoc \u27e9\ninstance : is_commutative (option heap) part :=\n\u27e8 part_comm \u27e9\n\nlemma disjoint_of_is_some_part\n  {hp\u2080 hp\u2081 : heap}\n  (h : (part (some hp\u2080) (some hp\u2081)).is_some)\n: hp\u2080 ## hp\u2081 :=\nby { dsimp [part] at h,\n     ite_cases with h at h,\n     contradiction,\n     assumption }\n\nlemma disjoint_of_part_eq_some\n  {hp\u2080 hp\u2081 hp\u2082 : heap}\n  (h : some hp\u2082 = (part (some hp\u2080) (some hp\u2081)))\n: hp\u2080 ## hp\u2081 :=\nby { apply disjoint_of_is_some_part, rw \u2190 h, exact rfl }\n\nlemma eq_part'_of_some_eq_part\n  (hp\u2080 hp\u2081 hp : heap)\n  (h : some hp = part (some hp\u2080) (some hp\u2081))\n: hp = part' hp\u2080 hp\u2081 (disjoint_of_part_eq_some h) :=\nby { apply @option.no_confusion _ _ (some hp) (some _) _ id,\n     simp [h], }\n\nlemma is_some_of_is_some_part_right\n  (hp\u2080 : option heap) {hp\u2081 : option heap}\n  (h : (part hp\u2080 hp\u2081).is_some)\n: hp\u2081.is_some :=\nby { cases hp\u2080 ; cases hp\u2081 ; try { contradiction },\n     exact rfl }\n\nlemma is_some_of_is_some_part_left\n  {hp\u2080 : option heap} (hp\u2081 : option heap)\n  (h : (part hp\u2080 hp\u2081).is_some)\n: hp\u2080.is_some :=\nby { cases hp\u2080 ; cases hp\u2081 ; try { contradiction },\n     exact rfl }\n\n@[simp]\nlemma some_eq_some_iff (x y : heap)\n: some x = some y \u2194 x = y :=\nby { split ; intro h, injection h, subst x }\n\ndef opt_apl : option heap \u2192 pointer \u2192 option word\n | (some hp) p := hp p\n | none _ := none\n\nlemma opt_apl_some (hp : heap) (p : pointer)\n: opt_apl (some hp) p = hp p :=\nrfl\n\nlemma opt_apl_part_maplet (hp : heap) (p : pointer) (v : word)\n  (h : maplet p v ## hp)\n: (opt_apl (part (some (maplet p v)) (some hp)) p) = some v :=\nbegin\n  unfold part,\n  rw [dif_pos] ; [ skip , apply h ],\n  rw [opt_apl,part',maplet,if_pos rfl],\n  refl,\nend\n\n@[simp]\nlemma eq_emp_of_part (a : heap) (hp : option heap)\n: some a = part (some a) hp \u2194 hp = some heap.emp :=\nbegin\n  cases hp ; dsimp [part],\n  split ; intro h ; contradiction,\n  ite_cases with h\u2080,\n  split ; intros h\u2081,\n  { contradiction },\n  { injection h\u2081 with h\u2082, simp [h\u2082] at h\u2080,\n    cases h\u2080, },\n  split ; intro h\u2081 ; injection h\u2081 with h\u2082,\n  { simp [eq_emp_of_part'] at h\u2082,\n    simp [h\u2082] },\n  { simp [h\u2082], }\nend\n\n@[simp]\nlemma part'_eq_emp (a b : option heap)\n: part a b = some heap.emp \u2194 a = some heap.emp \u2227 b = some heap.emp :=\nbegin\n  split ; simp_intros h ;\n  cases a ; cases b\n  ; try { refl }\n  ; try { cases h with h\u2080 h\u2081 }\n  ; try { contradiction }\n  ; try { rw [h\u2080,h\u2081,\u2190 some_part' _ _ (disjoint_heap_emp _)\n             ,heap_emp_part'_eq_self] },\n  { unfold part at h,\n    revert h, ite_cases with h,\n    contradiction,\n    intro h\u2080,\n    injection h\u2080 with h\u2081,\n    have h\u2082 : \u2200 p, part' a b h p = heap.emp p,\n    { intro, rw h\u2081 },\n    simp [part',heap.emp] at h\u2082,\n    split ; congr ; funext p,\n    { simp [(h\u2082 p).left], refl },\n    { simp [(h\u2082 p).right], refl }, }\nend\n\n@[simp]\nlemma emp_left_combine (hp : heap)\n: heap.emp <+ hp = hp :=\nby { funext y, simp [left_combine,heap.emp], }\n\n@[simp]\nlemma left_combine_emp (hp : heap)\n: hp <+ heap.emp = hp :=\nby { funext y, simp [left_combine,heap.emp], }\n\n@[simp]\nlemma heap_mk_nil_eq_emp (p : pointer)\n: heap.mk p [] = heap.emp := by simp [heap.mk]\n\n-- #check [3] ++ (3 : \u2115)\n\n@[simp]\nlemma heap_mk_eq_none (p p' : pointer) (vs : list word)\n: heap.mk p vs p' = none \u2194 p' < p \u2228 p + vs.length \u2264 p' :=\nbegin\n  induction vs generalizing p,\n  { split; intro, apply lt_or_le,\n    refl, },\n  { simp!, split_ifs,\n    { subst p', simp [lt_irrefl,nat.one_add,nat.zero_lt_succ],  },\n    { simp [add_assoc,vs_ih], apply or_congr; [skip, refl], replace h := ne.symm h,\n      simp [nat.lt_succ_iff,le_iff_lt h] } }\nend\n\nlemma some_insert_left_eq_part {hp\u2080 hp\u2081 : heap}\n  (h : hp\u2080 ## hp\u2081)\n: some (hp\u2080 <+ hp\u2081) = part (some hp\u2080) (some hp\u2081) :=\nby dsimp [part]; split_ifs; ext; simp!\n\nend heap\n", "meta": {"author": "unitb", "repo": "separation-logic", "sha": "bdde6fc8f16fd43932aea9827d6c63cadd91c2e8", "save_path": "github-repos/lean/unitb-separation-logic", "path": "github-repos/lean/unitb-separation-logic/separation-logic-bdde6fc8f16fd43932aea9827d6c63cadd91c2e8/src/separation/heap/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3979504361885421}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Sean Leather\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.free_monoid\nimport Mathlib.algebra.opposites\nimport Mathlib.control.traversable.instances\nimport Mathlib.control.traversable.lemmas\nimport Mathlib.category_theory.category.default\nimport Mathlib.category_theory.endomorphism\nimport Mathlib.category_theory.types\nimport Mathlib.category_theory.category.Kleisli\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n\n# List folds generalized to `traversable`\n\nInformally, we can think of `foldl` as a special case of `traverse` where we do not care about the\nreconstructed data structure and, in a state monad, we care about the final state.\n\nThe obvious way to define `foldl` would be to use the state monad but it\nis nicer to reason about a more abstract interface with `fold_map` as a\nprimitive and `fold_map_hom` as a defining property.\n\n```\ndef fold_map {\u03b1 \u03c9} [has_one \u03c9] [has_mul \u03c9] (f : \u03b1 \u2192 \u03c9) : t \u03b1 \u2192 \u03c9 := ...\n\nlemma fold_map_hom (\u03b1 \u03b2)\n  [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_monoid_hom f]\n  (g : \u03b3 \u2192 \u03b1) (x : t \u03b3) :\n  f (fold_map g x) = fold_map (f \u2218 g) x :=\n...\n```\n\n`fold_map` uses a monoid \u03c9 to accumulate a value for every element of\na data structure and `fold_map_hom` uses a monoid homomorphism to\nsubstitute the monoid used by `fold_map`. The two are sufficient to\ndefine `foldl`, `foldr` and `to_list`. `to_list` permits the\nformulation of specifications in terms of operations on lists.\n\nEach fold function can be defined using a specialized\nmonoid. `to_list` uses a free monoid represented as a list with\nconcatenation while `foldl` uses endofunctions together with function\ncomposition.\n\nThe definition through monoids uses `traverse` together with the\napplicative functor `const m` (where `m` is the monoid). As an\nimplementation, `const` guarantees that no resource is spent on\nreconstructing the structure during traversal.\n\nA special class could be defined for `foldable`, similarly to Haskell,\nbut the author cannot think of instances of `foldable` that are not also\n`traversable`.\n-/\n\nnamespace monoid\n\n\n/--\nFor a list, foldl f x [y\u2080,y\u2081] reduces as follows\n  calc  foldl f x [y\u2080,y\u2081]\n      = foldl f (f x y\u2080) [y\u2081]      : rfl\n  ... = foldl f (f (f x y\u2080) y\u2081) [] : rfl\n  ... = f (f x y\u2080) y\u2081              : rfl\n\nwith f : \u03b1 \u2192 \u03b2 \u2192 \u03b1\n     x : \u03b1\n     [y\u2080,y\u2081] : list \u03b2\n\nWe can view the above as a composition of functions:\n\n  ... = f (f x y\u2080) y\u2081              : rfl\n  ... = flip f y\u2081 (flip f y\u2080 x)    : rfl\n  ... = (flip f y\u2081 \u2218 flip f y\u2080) x  : rfl\n\nWe can use traverse and const to construct this composition:\n\n  calc   const.run (traverse (\u03bb y, const.mk' (flip f y)) [y\u2080,y\u2081]) x\n       = const.run ((::) <$> const.mk' (flip f y\u2080) <*> traverse (\u03bb y, const.mk' (flip f y)) [y\u2081]) x\n  ...  = const.run ((::) <$> const.mk' (flip f y\u2080) <*>\n           ( (::) <$> const.mk' (flip f y\u2081) <*> traverse (\u03bb y, const.mk' (flip f y)) [] )) x\n  ...  = const.run ((::) <$> const.mk' (flip f y\u2080) <*>\n           ( (::) <$> const.mk' (flip f y\u2081) <*> pure [] )) x\n  ...  = const.run ( ((::) <$> const.mk' (flip f y\u2081) <*> pure []) \u2218\n           ((::) <$> const.mk' (flip f y\u2080)) ) x\n  ...  = const.run ( const.mk' (flip f y\u2081) \u2218 const.mk' (flip f y\u2080) ) x\n  ...  = const.run ( flip f y\u2081 \u2218 flip f y\u2080 ) x\n  ...  = f (f x y\u2080) y\u2081\n\nAnd this is how `const` turns a monoid into an applicative functor and\nhow the monoid of endofunctions define `foldl`.\n-/\ndef foldl (\u03b1 : Type u) := category_theory.End \u03b1\u1d52\u1d56\n\ndef foldl.mk {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) : foldl \u03b1 := opposite.op f\n\ndef foldl.get {\u03b1 : Type u} (x : foldl \u03b1) : \u03b1 \u2192 \u03b1 := opposite.unop x\n\ndef foldl.of_free_monoid {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (xs : free_monoid \u03b1) : foldl \u03b2 :=\n  opposite.op (flip (list.foldl f) xs)\n\ndef foldr (\u03b1 : Type u) := category_theory.End \u03b1\n\ndef foldr.mk {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) : foldr \u03b1 := f\n\ndef foldr.get {\u03b1 : Type u} (x : foldr \u03b1) : \u03b1 \u2192 \u03b1 := x\n\ndef foldr.of_free_monoid {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (xs : free_monoid \u03b1) : foldr \u03b2 :=\n  flip (list.foldr f) xs\n\ndef mfoldl (m : Type u \u2192 Type u) [Monad m] (\u03b1 : Type u) :=\n  category_theory.End (category_theory.Kleisli.mk m \u03b1)\u1d52\u1d56\n\ndef mfoldl.mk {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} (f : \u03b1 \u2192 m \u03b1) : mfoldl m \u03b1 :=\n  opposite.op f\n\ndef mfoldl.get {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} (x : mfoldl m \u03b1) : \u03b1 \u2192 m \u03b1 :=\n  opposite.unop x\n\ndef mfoldl.of_free_monoid {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} {\u03b2 : Type u}\n    (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) (xs : free_monoid \u03b1) : mfoldl m \u03b2 :=\n  opposite.op (flip (mfoldl f) xs)\n\ndef mfoldr (m : Type u \u2192 Type u) [Monad m] (\u03b1 : Type u) :=\n  category_theory.End (category_theory.Kleisli.mk m \u03b1)\n\ndef mfoldr.mk {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} (f : \u03b1 \u2192 m \u03b1) : mfoldr m \u03b1 := f\n\ndef mfoldr.get {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} (x : mfoldr m \u03b1) : \u03b1 \u2192 m \u03b1 := x\n\ndef mfoldr.of_free_monoid {m : Type u \u2192 Type u} [Monad m] {\u03b1 : Type u} {\u03b2 : Type u}\n    (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (xs : free_monoid \u03b1) : mfoldr m \u03b2 :=\n  flip (list.mfoldr f) xs\n\nend monoid\n\n\nnamespace traversable\n\n\ndef fold_map {t : Type u \u2192 Type u} [traversable t] {\u03b1 : Type u} {\u03c9 : Type u} [HasOne \u03c9] [Mul \u03c9]\n    (f : \u03b1 \u2192 \u03c9) : t \u03b1 \u2192 \u03c9 :=\n  traverse (functor.const.mk' \u2218 f)\n\ndef foldl {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (x : \u03b1)\n    (xs : t \u03b2) : \u03b1 :=\n  monoid.foldl.get (fold_map (monoid.foldl.mk \u2218 flip f) xs) x\n\ndef foldr {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (x : \u03b2)\n    (xs : t \u03b1) : \u03b2 :=\n  monoid.foldr.get (fold_map (monoid.foldr.mk \u2218 f) xs) x\n\n/--\nConceptually, `to_list` collects all the elements of a collection\nin a list. This idea is formalized by\n\n  `lemma to_list_spec (x : t \u03b1) : to_list x = fold_map free_monoid.mk x`.\n\nThe definition of `to_list` is based on `foldl` and `list.cons` for\nspeed. It is faster than using `fold_map free_monoid.mk` because, by\nusing `foldl` and `list.cons`, each insertion is done in constant\ntime. As a consequence, `to_list` performs in linear.\n\nOn the other hand, `fold_map free_monoid.mk` creates a singleton list\naround each element and concatenates all the resulting lists. In\n`xs ++ ys`, concatenation takes a time proportional to `length xs`. Since\nthe order in which concatenation is evaluated is unspecified, nothing\nprevents each element of the traversable to be appended at the end\n`xs ++ [x]` which would yield a `O(n\u00b2)` run time. -/\ndef to_list {\u03b1 : Type u} {t : Type u \u2192 Type u} [traversable t] : t \u03b1 \u2192 List \u03b1 :=\n  list.reverse \u2218 foldl (flip List.cons) []\n\ndef length {\u03b1 : Type u} {t : Type u \u2192 Type u} [traversable t] (xs : t \u03b1) : \u2115 :=\n  ulift.down (foldl (fun (l : ulift \u2115) (_x : \u03b1) => ulift.up (ulift.down l + 1)) (ulift.up 0) xs)\n\ndef mfoldl {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t] {m : Type u \u2192 Type u}\n    [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b1) (x : \u03b1) (xs : t \u03b2) : m \u03b1 :=\n  monoid.mfoldl.get (fold_map (monoid.mfoldl.mk \u2218 flip f) xs) x\n\ndef mfoldr {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t] {m : Type u \u2192 Type u}\n    [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (x : \u03b2) (xs : t \u03b1) : m \u03b2 :=\n  monoid.mfoldr.get (fold_map (monoid.mfoldr.mk \u2218 f) xs) x\n\ndef map_fold {\u03b1 : Type u} {\u03b2 : Type u} [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_monoid_hom f] :\n    applicative_transformation (functor.const \u03b1) (functor.const \u03b2) :=\n  applicative_transformation.mk (fun (x : Type u_1) => f) sorry sorry\n\ndef free.mk {\u03b1 : Type u} : \u03b1 \u2192 free_monoid \u03b1 := list.ret\n\ndef free.map {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) : free_monoid \u03b1 \u2192 free_monoid \u03b2 := list.map f\n\ntheorem free.map_eq_map {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (xs : List \u03b1) :\n    f <$> xs = free.map f xs :=\n  rfl\n\nprotected instance free.map.is_monoid_hom {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) :\n    is_monoid_hom (free.map f) :=\n  is_monoid_hom.mk\n    (eq.mpr\n      (id\n        ((fun (a a_1 : free_monoid \u03b2) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : free_monoid \u03b2) (e_2 : \u1fb0 = \u1fb0_1) =>\n            congr (congr_arg Eq e_1) e_2)\n          (free.map f 1) []\n          (Eq.trans\n            (Eq.trans\n              ((fun (f f_1 : \u03b1 \u2192 \u03b2) (e_1 : f = f_1) (\u1fb0 \u1fb0_1 : free_monoid \u03b1) (e_2 : \u1fb0 = \u1fb0_1) =>\n                  congr (congr_arg free.map e_1) e_2)\n                f f (Eq.refl f) 1 [] free_monoid.one_def)\n              (congr_fun (free.map.equations._eqn_1 f) []))\n            (list.map.equations._eqn_1 f))\n          1 [] free_monoid.one_def))\n      (Eq.refl []))\n\nprotected instance fold_foldl {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) :\n    is_monoid_hom (monoid.foldl.of_free_monoid f) :=\n  is_monoid_hom.mk rfl\n\ntheorem foldl.unop_of_free_monoid {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (xs : free_monoid \u03b1)\n    (a : \u03b2) : opposite.unop (monoid.foldl.of_free_monoid f xs) a = list.foldl f a xs :=\n  rfl\n\nprotected instance fold_foldr {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) :\n    is_monoid_hom (monoid.foldr.of_free_monoid f) :=\n  is_monoid_hom.mk rfl\n\n@[simp] theorem mfoldl.unop_of_free_monoid {\u03b1 : Type u} {\u03b2 : Type u} (m : Type u \u2192 Type u) [Monad m]\n    [is_lawful_monad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) (xs : free_monoid \u03b1) (a : \u03b2) :\n    opposite.unop (monoid.mfoldl.of_free_monoid f xs) a = mfoldl f a xs :=\n  rfl\n\nprotected instance fold_mfoldl {\u03b1 : Type u} {\u03b2 : Type u} (m : Type u \u2192 Type u) [Monad m]\n    [is_lawful_monad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) : is_monoid_hom (monoid.mfoldl.of_free_monoid f) :=\n  is_monoid_hom.mk rfl\n\nprotected instance fold_mfoldr {\u03b1 : Type u} {\u03b2 : Type u} (m : Type u \u2192 Type u) [Monad m]\n    [is_lawful_monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) : is_monoid_hom (monoid.mfoldr.of_free_monoid f) :=\n  is_monoid_hom.mk rfl\n\ntheorem fold_map_hom {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_monoid_hom f] (g : \u03b3 \u2192 \u03b1)\n    (x : t \u03b3) : f (fold_map g x) = fold_map (f \u2218 g) x :=\n  Eq.trans\n    (Eq.trans (Eq.trans rfl rfl)\n      (is_lawful_traversable.naturality (map_fold f) (functor.const.mk' \u2218 g) x))\n    rfl\n\ntheorem fold_map_hom_free {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] [monoid \u03b2] (f : free_monoid \u03b1 \u2192 \u03b2) [is_monoid_hom f] (x : t \u03b1) :\n    f (fold_map free.mk x) = fold_map (f \u2218 free.mk) x :=\n  fold_map_hom f free.mk x\n\ntheorem fold_mfoldl_cons {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Monad m]\n    [is_lawful_monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b1) (x : \u03b2) (y : \u03b1) : mfoldl f y (free.mk x) = f y x :=\n  sorry\n\ntheorem fold_mfoldr_cons {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Monad m]\n    [is_lawful_monad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b1) (x : \u03b2) (y : \u03b1) : list.mfoldr f y (free.mk x) = f x y :=\n  sorry\n\n@[simp] theorem foldl.of_free_monoid_comp_free_mk {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) :\n    monoid.foldl.of_free_monoid f \u2218 free.mk = monoid.foldl.mk \u2218 flip f :=\n  rfl\n\n@[simp] theorem foldr.of_free_monoid_comp_free_mk {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b1) :\n    monoid.foldr.of_free_monoid f \u2218 free.mk = monoid.foldr.mk \u2218 f :=\n  rfl\n\n@[simp] theorem mfoldl.of_free_monoid_comp_free_mk {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u}\n    [Monad m] [is_lawful_monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b1) :\n    monoid.mfoldl.of_free_monoid f \u2218 free.mk = monoid.mfoldl.mk \u2218 flip f :=\n  sorry\n\n@[simp] theorem mfoldr.of_free_monoid_comp_free_mk {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u}\n    [Monad m] [is_lawful_monad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b1) :\n    monoid.mfoldr.of_free_monoid f \u2218 free.mk = monoid.mfoldr.mk \u2218 f :=\n  sorry\n\ntheorem to_list_spec {\u03b1 : Type u} {t : Type u \u2192 Type u} [traversable t] [is_lawful_traversable t]\n    (xs : t \u03b1) : to_list xs = fold_map free.mk xs :=\n  sorry\n\ntheorem fold_map_map {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] [monoid \u03b3] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (xs : t \u03b1) :\n    fold_map g (f <$> xs) = fold_map (g \u2218 f) xs :=\n  sorry\n\ntheorem foldl_to_list {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (xs : t \u03b2) (x : \u03b1) :\n    foldl f x xs = list.foldl f x (to_list xs) :=\n  sorry\n\ntheorem foldr_to_list {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (xs : t \u03b1) (x : \u03b2) :\n    foldr f x xs = list.foldr f x (to_list xs) :=\n  sorry\n\ntheorem to_list_map {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] (f : \u03b1 \u2192 \u03b2) (xs : t \u03b1) : to_list (f <$> xs) = f <$> to_list xs :=\n  sorry\n\n@[simp] theorem foldl_map {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u}\n    [traversable t] [is_lawful_traversable t] (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b3 \u2192 \u03b1) (a : \u03b1) (l : t \u03b2) :\n    foldl f a (g <$> l) = foldl (fun (x : \u03b1) (y : \u03b2) => f x (g y)) a l :=\n  sorry\n\n@[simp] theorem foldr_map {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u}\n    [traversable t] [is_lawful_traversable t] (g : \u03b2 \u2192 \u03b3) (f : \u03b3 \u2192 \u03b1 \u2192 \u03b1) (a : \u03b1) (l : t \u03b2) :\n    foldr f a (g <$> l) = foldr (f \u2218 g) a l :=\n  sorry\n\n@[simp] theorem to_list_eq_self {\u03b1 : Type u} {xs : List \u03b1} : to_list xs = xs := sorry\n\ntheorem length_to_list {\u03b1 : Type u} {t : Type u \u2192 Type u} [traversable t] [is_lawful_traversable t]\n    {xs : t \u03b1} : length xs = list.length (to_list xs) :=\n  sorry\n\ntheorem mfoldl_to_list {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] {m : Type u \u2192 Type u} [Monad m] [is_lawful_monad m] {f : \u03b1 \u2192 \u03b2 \u2192 m \u03b1}\n    {x : \u03b1} {xs : t \u03b2} : mfoldl f x xs = mfoldl f x (to_list xs) :=\n  sorry\n\ntheorem mfoldr_to_list {\u03b1 : Type u} {\u03b2 : Type u} {t : Type u \u2192 Type u} [traversable t]\n    [is_lawful_traversable t] {m : Type u \u2192 Type u} [Monad m] [is_lawful_monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2)\n    (x : \u03b2) (xs : t \u03b1) : mfoldr f x xs = list.mfoldr f x (to_list xs) :=\n  sorry\n\n@[simp] theorem mfoldl_map {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u}\n    [traversable t] [is_lawful_traversable t] {m : Type u \u2192 Type u} [Monad m] [is_lawful_monad m]\n    (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b3 \u2192 m \u03b1) (a : \u03b1) (l : t \u03b2) :\n    mfoldl f a (g <$> l) = mfoldl (fun (x : \u03b1) (y : \u03b2) => f x (g y)) a l :=\n  sorry\n\n@[simp] theorem mfoldr_map {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {t : Type u \u2192 Type u}\n    [traversable t] [is_lawful_traversable t] {m : Type u \u2192 Type u} [Monad m] [is_lawful_monad m]\n    (g : \u03b2 \u2192 \u03b3) (f : \u03b3 \u2192 \u03b1 \u2192 m \u03b1) (a : \u03b1) (l : t \u03b2) : mfoldr f a (g <$> l) = mfoldr (f \u2218 g) a l :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/fold_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3979504361885421}}
{"text": "/-\n  The stalks of the structure presheaf on the basis are local rings.\n-/\n\nimport ring_theory.ideals\nimport ring_theory.localization\nimport to_mathlib.localization.local_rings\nimport to_mathlib.localization.localization_alt\nimport sheaves.stalk_of_rings_on_standard_basis\nimport spectrum_of_a_ring.structure_presheaf\nimport spectrum_of_a_ring.structure_presheaf_localization\nimport spectrum_of_a_ring.structure_presheaf_res\n\nuniverse u\n\nvariables {R : Type u} [comm_ring R]\nvariables (P : Spec R)\n\nopen localization localization_alt stalk_of_rings_on_standard_basis\nopen classical\n\ndef Bstd := D_fs_standard_basis R\n\nnamespace strucutre_presheaf_stalks\n\ndef F := structure_presheaf_on_basis R\n\ndef FP := stalk_of_rings_on_standard_basis Bstd F P\n\n-- This is essentially the map from x to x/1.\ndef \u03c6 : R \u2192 FP P := \u03bb x,\n\u27e6{ U := opens.univ,\n   BU := (D_fs_standard_basis R).1,\n   Hx := set.mem_univ P,\n   s := (of : R \u2192 localization R (S (opens.univ))) x, }\u27e7\n\ninstance FP.is_comm_ring : comm_ring (FP P) :=\nby simp [FP]; by apply_instance\n\ninstance prime.is_submonoid : is_submonoid (-P.1 : set R) :=\n{ one_mem := P.1.ne_top_iff_one.1 P.2.1,\n  mul_mem := \u03bb x y hnx hny hxy, or.cases_on (P.2.2 hxy) hnx hny }\n\ninstance \u03c6P.is_ring_hom : is_ring_hom (\u03c6 P) :=\n{ map_one := rfl,\n  map_mul := \u03bb x y,\n    begin\n      apply quotient.sound,\n      use [opens.univ, (D_fs_standard_basis R).1, set.mem_univ P],\n      use [set.subset.refl _, (\u03bb x Hx, \u27e8Hx, Hx\u27e9)],\n      rw (F.res_is_ring_hom _ _ _).map_mul,\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw of.is_ring_hom.map_mul,\n      rw (F.res_is_ring_hom _ _ _).map_mul,\n    end,\n  map_add := \u03bb x y,\n    begin\n      apply quotient.sound,\n      use [opens.univ, (D_fs_standard_basis R).1, set.mem_univ P],\n      use [set.subset.refl _, (\u03bb x Hx, \u27e8Hx, Hx\u27e9)],\n      rw (F.res_is_ring_hom _ _ _).map_add,\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw of.is_ring_hom.map_add,\n      rw (F.res_is_ring_hom _ _ _).map_add,\n    end, }\n\nlemma stalk_local.inverts_data : inverts_data (-P.1 : set R) (\u03c6 P) :=\nbegin\n  rintros \u27e8s, Hs\u27e9,\n  change s \u2209 P.val at Hs,\n  let BDs := D_fs.mem R s,\n  have HsS : s \u2208 S (Spec.DO R s) := S.f_mem s,\n  let sinv : FP P :=\n    \u27e6{ U := Spec.DO R s,\n       BU := BDs,\n       Hx := Hs,\n       s := \u27e6\u27e81, \u27e8s, HsS\u27e9\u27e9\u27e7, }\u27e7,\n  use sinv,\n  apply quotient.sound,\n  use [Spec.DO R s, BDs, Hs, (\u03bb x Hx, \u27e8trivial, Hx\u27e9), set.subset_univ _],\n  simp,\n  erw (F.res_is_ring_hom _ _ _).map_mul,\n  erw (F.res_is_ring_hom _ _ _).map_one,\n  iterate 2 { rw \u2190presheaf_on_basis.Hcomp', },\n  erw presheaf_on_basis.Hid',\n  erw structure_presheaf_on_basis.res_eq,\n  erw \u2190structure_presheaf_on_basis.res_comp_of',\n  apply quotient.sound,\n  use [1, is_submonoid.one_mem _],\n  simp,\nend\n\nlemma stalk_local.has_denom : has_denom (-P.1 : set R) (\u03c6 P) :=\nbegin\n  intros x,\n  apply quotient.induction_on x,\n  rintros \u27e8U, BU, HPU, s\u27e9,\n  -- s \u2208 R[1/S(U)].\n  apply quotient.induction_on s,\n  rintros \u27e8p, \u27e8q, Hq\u27e9\u27e9,\n  change U \u2286 Spec.DO R q at Hq,\n  -- Since P \u2208 D(q).\n  have HqnP : q \u2209 P.val := Hq HPU,\n  use [\u27e8\u27e8q, HqnP\u27e9, p\u27e9],\n  apply quotient.sound,\n  use [U, BU, HPU, (\u03bb x Hx, \u27e8trivial, Hx\u27e9), set.subset_univ _],\n  simp,\n  erw (F.res_is_ring_hom _ _ _).map_mul,\n  iterate 2 { rw \u2190presheaf_on_basis.Hcomp', },\n  erw presheaf_on_basis.Hid',\n  erw structure_presheaf_on_basis.res_eq,\n  iterate 2 { erw \u2190structure_presheaf_on_basis.res_comp_of', },\n  apply quotient.sound,\n  use [1, is_submonoid.one_mem _],\n  simp,\nend\n\nnoncomputable def stalk_local.has_denom_data : has_denom_data (-P.1 : set R) (\u03c6 P) :=\nhas_denom_some (-P.1 : set R) (\u03c6 P) (stalk_local.has_denom P)\n\nlemma stalk_local.ker_le : ker (\u03c6 P) \u2264 submonoid_ann (-P.1 : set R) :=\nbegin\n  intros x Hx,\n  change \u03c6 P x = 0 at Hx,\n  erw quotient.eq at Hx,\n  rcases Hx with \u27e8W, BW, HPW, HWuniv1, HWuniv2, Hres\u27e9,\n  dsimp at Hres,\n  erw is_ring_hom.map_zero (F.res (D_fs_standard_basis R).1 BW HWuniv2) at Hres,\n  erw structure_presheaf_on_basis.res_eq at Hres,\n  erw \u2190structure_presheaf_on_basis.res_comp_of' at Hres,\n  erw quotient.eq at Hres,\n  rcases Hres with \u27e8t, HtSW, Ht\u27e9,\n  simp at Ht,\n  have HtnP : t \u2209 P.val := HtSW HPW,\n  use \u27e8\u27e8x, \u27e8t, HtnP\u27e9\u27e9, Ht\u27e9,\nend\n\nnoncomputable def stalk_local.localization : is_localization_data (-P.1 : set R) (\u03c6 P) :=\n{ inverts := stalk_local.inverts_data P,\n  has_denom := stalk_local.has_denom_data P,\n  ker_le := stalk_local.ker_le P, }\n\nlemma stalk_local : is_local_ring (FP P) :=\nlocal_ring.of_is_localization_data_at_prime P.2 (stalk_local.localization P)\n\nend strucutre_presheaf_stalks\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/spectrum_of_a_ring/structure_presheaf_stalks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.39795042849777773}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.hom_functor\n\n/-!\n# The Yoneda embedding\n\nThe Yoneda embedding as a functor `yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) \u2245 (yoneda_evaluation C)`.\n\n## References\n* [Stacks: Opposite Categories and the Yoneda Lemma](https://stacks.math.columbia.edu/tag/001L)\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v\u2081 u\u2081 u\u2082-- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n\n/--\nThe Yoneda embedding, as a functor from `C` into presheaves on `C`.\n\nSee https://stacks.math.columbia.edu/tag/001O.\n-/\n@[simps]\ndef yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081) :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, unop Y \u27f6 X,\n    map := \u03bb Y Y' f g, f.unop \u226b g,\n    map_comp' := \u03bb _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := \u03bb Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := \u03bb X X' f, { app := \u03bb Y g, g \u226b f } }\n\n/--\nThe co-Yoneda embedding, as a functor from `C\u1d52\u1d56` into co-presheaves on `C`.\n-/\n@[simps] def coyoneda : C\u1d52\u1d56 \u2964 (C \u2964 Type v\u2081) :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, unop X \u27f6 Y,\n    map := \u03bb Y Y' f g, g \u226b f,\n    map_comp' := \u03bb _ _ _ f g, begin ext1, dsimp, erw [category.assoc] end,\n    map_id' := \u03bb Y, begin ext1, dsimp, erw [category.comp_id] end },\n  map := \u03bb X X' f, { app := \u03bb Y g, f.unop \u226b g },\n  map_comp' := \u03bb _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n  map_id' := \u03bb X, begin ext, dsimp, erw [category.id_comp] end }\n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X \u27f6 op Y) :\n  ((@yoneda C _).obj X).map f (\ud835\udfd9 X) = ((@yoneda C _).map f.unop).app (op Y) (\ud835\udfd9 Y) :=\nby obviously\n\n@[simp] lemma naturality {X Y : C} (\u03b1 : yoneda.obj X \u27f6 yoneda.obj Y)\n  {Z Z' : C} (f : Z \u27f6 Z') (h : Z' \u27f6 X) : f \u226b \u03b1.app (op Z') h = \u03b1.app (op Z) (f \u226b h) :=\n(functor_to_types.naturality _ _ \u03b1 f.op h).symm\n\n/--\nThe Yoneda embedding is full.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ninstance yoneda_full : full (@yoneda C _) :=\n{ preimage := \u03bb X Y f, (f.app (op X)) (\ud835\udfd9 X) }\n\n/--\nThe Yoneda embedding is faithful.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ninstance yoneda_faithful : faithful (@yoneda C _) :=\n{ map_injective' := \u03bb X Y f g p,\n  begin\n    injection p with h,\n    convert (congr_fun (congr_fun h (op X)) (\ud835\udfd9 X)); dsimp; simp,\n  end }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X \u2245 Y`\napply yoneda.ext,\n-- Goals are now functions `(Z \u27f6 X) \u2192 (Z \u27f6 Y)`, `(Z \u27f6 Y) \u2192 (Z \u27f6 X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : \u03a0 {Z : C}, (Z \u27f6 X) \u2192 (Z \u27f6 Y)) (q : \u03a0 {Z : C}, (Z \u27f6 Y) \u2192 (Z \u27f6 X))\n  (h\u2081 : \u03a0 {Z : C} (f : Z \u27f6 X), q (p f) = f) (h\u2082 : \u03a0 {Z : C} (f : Z \u27f6 Y), p (q f) = f)\n  (n : \u03a0 {Z Z' : C} (f : Z' \u27f6 Z) (g : Z \u27f6 X), p (f \u226b g) = f \u226b p g) : X \u2245 Y :=\n@preimage_iso _ _ _ _ yoneda _ _ _ _\n  (nat_iso.of_components (\u03bb Z, { hom := p, inv := q, }) (by tidy))\n\n/--\nIf `yoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C} (f : X \u27f6 Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : C\u1d52\u1d56} (\u03b1 : coyoneda.obj X \u27f6 coyoneda.obj Y)\n  {Z Z' : C} (f : Z' \u27f6 Z) (h : unop X \u27f6 Z') : (\u03b1.app Z' h) \u226b f = \u03b1.app Z (h \u226b f) :=\nbegin erw [functor_to_types.naturality], refl end\n\ninstance coyoneda_full : full (@coyoneda C _) :=\n{ preimage := \u03bb X Y f, ((f.app (unop X)) (\ud835\udfd9 _)).op }\n\ninstance coyoneda_faithful : faithful (@coyoneda C _) :=\n{ map_injective' := \u03bb X Y f g p,\n  begin\n    injection p with h,\n    have t := (congr_fun (congr_fun h (unop X)) (\ud835\udfd9 _)),\n    simpa using congr_arg quiver.hom.op t,\n  end }\n\n/--\nIf `coyoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\n-- No need to use C\u1d52\u1d56 here, works with any category\n/-- A Type-valued presheaf `P` is isomorphic to the composition of `P` with the\n  coyoneda functor coming from `punit`. -/\n@[simps] def iso_comp_punit (P : C \u2964 Type v\u2081) : (P \u22d9 coyoneda.obj (op punit.{v\u2081+1})) \u2245 P :=\n{ hom := { app := \u03bb X f, f punit.star},\n  inv := { app := \u03bb X a _, a } }\n\nend coyoneda\n\n/--\nA presheaf `F` is representable if there is object `X` so `F \u2245 yoneda.obj X`.\n\nSee https://stacks.math.columbia.edu/tag/001Q.\n-/\n-- TODO should we make this a Prop, merely asserting existence of such an object?\nclass representable (F : C\u1d52\u1d56 \u2964 Type v\u2081) :=\n(X : C)\n(w : yoneda.obj X \u2245 F)\n\nend category_theory\n\nnamespace category_theory\n-- For the rest of the file, we are using product categories,\n-- so need to restrict to the case morphisms are in 'Type', not 'Sort'.\n\nuniverses v\u2081 u\u2081 u\u2082 -- morphism levels before object levels. See note [category_theory universes].\n\nopen opposite\n\nvariables (C : Type u\u2081) [category.{v\u2081} C]\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((C\u1d52\u1d56 \u2964 Type v\u2081) \u00d7 C\u1d52\u1d56) :=\ncategory_theory.prod.{(max u\u2081 v\u2081) v\u2081} (C\u1d52\u1d56 \u2964 Type v\u2081) C\u1d52\u1d56\n\ninstance prod_category_instance_2 : category (C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) :=\ncategory_theory.prod.{v\u2081 (max u\u2081 v\u2081)} C\u1d52\u1d56 (C\u1d52\u1d56 \u2964 Type v\u2081)\n\nopen yoneda\n\n/--\nThe \"Yoneda evaluation\" functor, which sends `X : C\u1d52\u1d56` and `F : C\u1d52\u1d56 \u2964 Type`\nto `F.obj X`, functorially in both `X` and `F`.\n-/\ndef yoneda_evaluation : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nevaluation_uncurried C\u1d52\u1d56 (Type v\u2081) \u22d9 ulift_functor.{u\u2081}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map \u03b1 x).down = \u03b1.2.app Q.1 (P.2.map \u03b1.1 x.down) := rfl\n\n/--\nThe \"Yoneda pairing\" functor, which sends `X : C\u1d52\u1d56` and `F : C\u1d52\u1d56 \u2964 Type`\nto `yoneda.op.obj X \u27f6 F`, functorially in both `X` and `F`.\n-/\ndef yoneda_pairing : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nfunctor.prod yoneda.op (\ud835\udfed (C\u1d52\u1d56 \u2964 Type v\u2081)) \u22d9 functor.hom (C\u1d52\u1d56 \u2964 Type v\u2081)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (\u03b2 : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map \u03b1 \u03b2 = yoneda.map \u03b1.1.unop \u226b \u03b2 \u226b \u03b1.2 := rfl\n\n/--\nThe Yoneda lemma asserts that that the Yoneda pairing\n`(X : C\u1d52\u1d56, F : C\u1d52\u1d56 \u2964 Type) \u21a6 (yoneda.obj (unop X) \u27f6 F)`\nis naturally isomorphic to the evaluation `(X, F) \u21a6 F.obj X`.\n\nSee https://stacks.math.columbia.edu/tag/001P.\n-/\ndef yoneda_lemma : yoneda_pairing C \u2245 yoneda_evaluation C :=\n{ hom :=\n  { app := \u03bb F x, ulift.up ((x.app F.1) (\ud835\udfd9 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp, \u2190functor_to_types.naturality],\n      simp only [category.comp_id, yoneda_obj_map],\n    end },\n  inv :=\n  { app := \u03bb F x,\n    { app := \u03bb X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp_apply]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [\u2190functor_to_types.naturality, functor_to_types.map_comp_apply]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [\u2190functor_to_types.naturality,\n         obj_map_id],\n    simp only [yoneda_map_app, quiver.hom.unop_op],\n    erw [category.id_comp],\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id_apply]\n  end }.\n\nvariables {C}\n\n/--\nThe isomorphism between `yoneda.obj X \u27f6 F` and `F.obj (op X)`\n(we need to insert a `ulift` to get the universes right!)\ngiven by the Yoneda lemma.\n-/\n@[simp] def yoneda_sections (X : C) (F : C\u1d52\u1d56 \u2964 Type v\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 ulift.{u\u2081} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\n/--\nWe have a type-level equivalence between natural transformations from the yoneda embedding\nand elements of `F.obj X`, without any universe switching.\n-/\ndef yoneda_equiv {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} : (yoneda.obj X \u27f6 F) \u2243 F.obj (op X) :=\n(yoneda_sections X F).to_equiv.trans equiv.ulift\n\nlemma yoneda_equiv_naturality {X Y : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (f : yoneda.obj X \u27f6 F) (g : Y \u27f6 X) :\n  F.map g.op (yoneda_equiv f) = yoneda_equiv (yoneda.map g \u226b f) :=\nbegin\n  change (f.app (op X) \u226b F.map g.op) (\ud835\udfd9 X) = f.app (op Y) (\ud835\udfd9 Y \u226b g),\n  rw \u2190 f.naturality,\n  dsimp,\n  simp,\nend\n\n@[simp]\nlemma yoneda_equiv_apply {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (f : yoneda.obj X \u27f6 F) :\n  yoneda_equiv f = f.app (op X) (\ud835\udfd9 X) :=\nrfl\n\n@[simp]\nlemma yoneda_equiv_symm_app_apply {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (x : F.obj (op X))\n  (Y : C\u1d52\u1d56) (f : Y.unop \u27f6 X) :\n  (yoneda_equiv.symm x).app Y f = F.map f.op x :=\nrfl\n\n/--\nWhen `C` is a small category, we can restate the isomorphism from `yoneda_sections`\nwithout having to change universes.\n-/\ndef yoneda_sections_small {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 F.obj (op X) :=\nyoneda_sections X F \u226a\u226b ulift_trivial _\n\n@[simp]\nlemma yoneda_sections_small_hom {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) (f : yoneda.obj X \u27f6 F) :\n  (yoneda_sections_small X F).hom f = f.app _ (\ud835\udfd9 _) :=\nrfl\n\n@[simp]\nlemma yoneda_sections_small_inv_app_apply {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) (t : F.obj (op X)) (Y : C\u1d52\u1d56) (f : Y.unop \u27f6 X) :\n  ((yoneda_sections_small X F).inv t).app Y f = F.map f.op t :=\nrfl\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.39795042080701326}}
{"text": "-- import topology.sheaves.sheaf\n-- import sort\n-- import algebra.category.Group.limits\n-- import oc\n\n-- section \n\n-- open category_theory Top Top.sheaf topological_space finset\n-- open opposite\n\n-- variable (X : Top) \n\n-- variable {X}\n-- variable (\ud835\udcd5 : sheaf Ab X)\n-- variable (\ud835\udd18 : oc X)\n\n-- local notation `\u03b9 ` := \ud835\udd18.\u03b9\n-- local notation `\ud835\udcd5.obj` := \ud835\udcd5.1.obj\n-- local notation `\ud835\udcd5.map` := \ud835\udcd5.1.map\n\n-- @[ext] structure simplex (n : \u2115) extends finset \u03b9 :=\n-- (card_eq : to_finset.card = n.succ)\n\n-- attribute [simp] simplex.card_eq\n\n-- namespace simplex\n\n-- variables {\ud835\udd18}\n\n-- def nth {n : \u2115} (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) : \u03b9 :=\n-- \u03c3.to_finset.order_emb_of_fin \u03c3.2 m\n\n-- instance {n : \u2115} : has_mem \u03b9 (simplex \ud835\udd18 n) :=\n-- { mem := \u03bb i \u03c3, i \u2208 \u03c3.to_finset }\n\n-- lemma nth_mem {n : \u2115} (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) :\n--   \u03c3.nth m \u2208 \u03c3 :=\n-- \u03c3.to_finset.order_emb_of_fin_mem \u03c3.card_eq m\n\n-- def zero_from (i : \u03b9) : simplex \ud835\udd18 0 :=\n-- { to_finset := {i},\n--   card_eq := rfl }\n\n-- variables {n : \u2115} (hn : 0 < n)\n\n-- def ignore (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) : simplex \ud835\udd18 n.pred :=\n-- { to_finset := \u03c3.1.erase_nth \u03c3.2 m,\n--   card_eq := (nat.succ_pred_eq_of_pos hn).symm \u25b8 \u03c3.1.erase_nth_card _ m }\n\n-- lemma mem_ignore (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) (i : \u03b9) :\n--   i \u2208 \u03c3.ignore hn m \u2194 i \u2208 \u03c3 \u2227 i \u2260 \u03c3.nth m :=\n-- begin\n--   split,\n--   { intros hi,\n--     change i \u2208 simplex.to_finset _ at hi,\n--     unfold ignore at hi,\n--     dsimp only at hi,\n--     rw mem_erase_nth at hi,\n--     refine \u27e8hi.2, hi.1\u27e9, },\n--   { intros hi,\n--     change i \u2208 simplex.to_finset _,\n--     unfold ignore,\n--     dsimp only,\n--     rw mem_erase_nth,\n--     refine \u27e8hi.2, hi.1\u27e9, },\n-- end \n\n-- def ignore\u2082 (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ) : simplex \ud835\udd18 n.pred :=\n-- (\u03c3.ignore (nat.zero_lt_succ _) m).ignore hn m'\n\n-- lemma ignore_subset (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) :\n--   (\u03c3.ignore hn m).to_finset \u2286 \u03c3.to_finset := \u03bb x hx,\n-- begin\n--   change x \u2208 finset.erase _ _ at hx,\n--   rw finset.mem_erase at hx,\n--   exact hx.2,\n-- end\n\n-- lemma ignore\u2082_subset (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ) :\n--   (\u03c3.ignore\u2082 hn m m').to_finset \u2286 \u03c3.to_finset :=\n-- subset.trans ((\u03c3.ignore (nat.zero_lt_succ _) m).ignore_subset hn m') $ \u03c3.ignore_subset _ _\n\n-- lemma ignore\u2082_to_finset_case1 (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ)\n--   (hmm' : m'.1 < m.1) :\n--   (\u03c3.ignore\u2082 hn m m').to_finset =\n--   \u03c3.to_finset \\ \n--   { \u03c3.1.order_emb_of_fin \u03c3.2 m, \n--     \u03c3.1.order_emb_of_fin \u03c3.2 \u27e8m'.1, lt_trans m'.2 (lt_add_one n.succ)\u27e9 } :=\n-- begin\n--   unfold ignore\u2082 ignore,\n--   dsimp,\n--   ext i,\n--   split,\n--   { intros hi,\n--     erw [mem_erase_nth, \u03c3.to_finset.erase_order_emb_of_fin'_eq \u03c3.2 m] at hi,\n--     unfold erase_order_emb_of_fin' at hi,\n--     simp only [rel_embedding.coe_fn_mk, function.embedding.coe_fn_mk, ne.def, if_pos hmm', mem_erase_nth] at hi,\n--     rw [mem_sdiff, mem_insert, mem_singleton, not_or_distrib],\n--     tauto },\n--   { intros hi,\n--     erw [mem_erase_nth, \u03c3.to_finset.erase_order_emb_of_fin'_eq \u03c3.2 m, mem_erase_nth],\n--     rw [mem_sdiff, mem_insert, mem_singleton, not_or_distrib] at hi,\n--     refine \u27e8_, hi.2.1, hi.1\u27e9,\n--     convert hi.2.2,\n--     unfold erase_order_emb_of_fin',\n--     simp only [rel_embedding.coe_fn_mk, function.embedding.coe_fn_mk, ne.def, if_pos hmm', mem_erase_nth],\n--     refl, }\n-- end\n\n-- lemma ignore\u2082_to_finset_case2 (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ)\n--   (hmm' : m.1 \u2264 m'.1) :\n--   (\u03c3.ignore\u2082 hn m m').to_finset =\n--   \u03c3.to_finset \\ \n--   { \u03c3.to_finset.order_emb_of_fin \u03c3.2 m, \n--     \u03c3.to_finset.order_emb_of_fin \u03c3.2 \u27e8m'.1.succ, nat.succ_lt_succ m'.2\u27e9 } :=\n-- begin\n--   have ineq : \u00ac m'.1 < m.1,\n--   { rwa not_lt },\n--   unfold ignore\u2082 ignore,\n--   dsimp,\n--   ext i,\n--   split,\n--   { intros hi,\n--     erw [mem_erase_nth, \u03c3.to_finset.erase_order_emb_of_fin'_eq \u03c3.2 m] at hi,\n--     unfold erase_order_emb_of_fin' at hi,\n--     simp only [rel_embedding.coe_fn_mk, function.embedding.coe_fn_mk, ne.def, if_neg ineq, mem_erase_nth] at hi,\n--     rw [mem_sdiff, mem_insert, mem_singleton, not_or_distrib],\n--     tauto },\n--   { intros hi,\n--     erw [mem_erase_nth, \u03c3.to_finset.erase_order_emb_of_fin'_eq \u03c3.2 m, mem_erase_nth],\n--     rw [mem_sdiff, mem_insert, mem_singleton, not_or_distrib] at hi,\n--     refine \u27e8_, hi.2.1, hi.1\u27e9,\n--     convert hi.2.2,\n--     unfold erase_order_emb_of_fin',\n--     simp only [rel_embedding.coe_fn_mk, function.embedding.coe_fn_mk, ne.def, if_neg ineq, mem_erase_nth],\n--     refl, }\n-- end\n\n-- lemma ignore\u2082_eq_ignore\u2082.aux (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ)\n--   (hmm' : m.1 \u2264 m'.1) :\n--   (\u03c3.ignore\u2082 hn m m').to_finset = \n--   (\u03c3.ignore\u2082 hn \u27e8m'.1.succ, nat.succ_lt_succ m'.2\u27e9 \u27e8m.1, by linarith [m'.2]\u27e9).to_finset :=\n-- begin\n--   rw [ignore\u2082_to_finset_case2 _ _ _ _ hmm', ignore\u2082_to_finset_case1],\n--   { ext i,\n--     split;\n--     { intros hi,\n--       rw [mem_sdiff, mem_insert, mem_singleton, not_or_distrib] at hi \u22a2,\n--       tauto, } },\n--   { dsimp only,\n--     exact lt_of_le_of_lt hmm' (lt_add_one _), },\n-- end\n\n-- lemma ignore\u2082_eq_ignore\u2082 (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ)\n--   (hmm' : m.1 \u2264 m'.1) :\n--   (\u03c3.ignore\u2082 hn m m') = \n--   (\u03c3.ignore\u2082 hn \u27e8m'.1.succ, nat.succ_lt_succ m'.2\u27e9 \u27e8m.1, by linarith [m'.2]\u27e9) :=\n-- by rw [simplex.ext_iff, ignore\u2082_eq_ignore\u2082.aux]\n\n-- def face {n : \u2115} (\u03c3 : simplex \ud835\udd18 n) : opens X :=\n-- \u2a05 (i : \u03b9) (H : i \u2208 \u03c3.to_finset), \ud835\udd18.cover i\n\n-- lemma face0 (\u03c3 : simplex \ud835\udd18 0) :\n--   \u03c3.face = \ud835\udd18.cover (\u03c3.nth 0) := \n-- begin\n--   unfold face,\n--   have eq1 : \u03c3.to_finset = {\u03c3.nth 0},\n--   { rcases card_eq_one.mp \u03c3.2 with \u27e8a, eq1\u27e9,\n--     have := \u03c3.nth_mem 0,\n--     change _ \u2208 \u03c3.to_finset at this,\n--     rw eq1 at *,\n--     rw mem_singleton at this,\n--     rw this },\n--   rw [eq1, finset.infi_singleton],\n-- end\n\n-- lemma face1 (\u03c3 : simplex \ud835\udd18 1) :\n--   \u03c3.face = \ud835\udd18.cover (\u03c3.nth 0) \u2293 \ud835\udd18.cover (\u03c3.nth \u27e81, one_lt_two\u27e9) :=\n-- begin\n--   rcases card_eq_two.mp \u03c3.2 with \u27e8a, b, ineq, eq1\u27e9,\n--   have mem1 : (_ \u2208 \u03c3.to_finset) := \u03c3.nth_mem 0,\n--   have mem2 : (_ \u2208 \u03c3.to_finset) := \u03c3.nth_mem \u27e81, one_lt_two\u27e9,\n--   have ineq2 : \u03c3.nth 0 \u2260 \u03c3.nth \u27e81, one_lt_two\u27e9,\n--   { intro rid,\n--     unfold simplex.nth at rid,\n--     replace rid := (\u03c3.to_finset.order_emb_of_fin \u03c3.2).inj' rid,\n--     rw subtype.ext_iff_val at rid,\n--     change 0 = 1 at rid,\n--     linarith, },\n--   rw [eq1, mem_insert, mem_singleton] at mem1 mem2,\n--   unfold face,\n--   rw [eq1, finset.infi_insert, finset.infi_singleton],\n--   cases mem1;\n--   cases mem2;\n--   rw [mem1, mem2] at *;\n--   tauto <|> exact inf_comm,\n-- end\n\n-- def subset\u2080 {n : \u2115} (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) :\n--   \u03c3.face \u27f6 (simplex.zero_from \ud835\udd18 (\u03c3.nth m)).face := hom_of_le $ \u03bb p hp, \n-- begin\n--   rw [opens.mem_coe] at hp \u22a2,\n--   rw face0,\n--   change _ \u2208 (infi _) at hp,\n--   have := (infi_le _ : \u2200 _,  \u03c3.face \u2264 _),\n--   specialize this ((simplex.zero_from \ud835\udd18 (\u03c3.nth m)).nth 0),\n--   simp only [le_infi_iff] at this,\n--   refine this _ hp,\n--   have : _ \u2208 {_} := (simplex.zero_from \ud835\udd18 (\u03c3.nth m)).nth_mem 0,\n--   rw mem_singleton at this,\n--   rw this,\n--   apply simplex.nth_mem,\n-- end\n\n-- def der {n : \u2115} (hn : 0 < n) (\u03c3 : simplex \ud835\udd18 n) (m : fin n.succ) :\n--   \u03c3.face \u27f6 (\u03c3.ignore hn m).face := hom_of_le $ \u03bb p hp, \n-- begin\n--   rw [opens.mem_coe] at hp \u22a2,\n--   rcases hp with \u27e8S, \u27e8oS, hS\u27e9, p_mem\u27e9,\n--   refine \u27e8S, \u27e8oS, \u03bb x x_mem, _\u27e9, p_mem\u27e9,\n--   specialize hS x_mem,\n--   simp only [subtype.val_eq_coe, set.Inf_eq_sInter, set.sInter_image, set.mem_range, \n--     set.Inter_exists, set.Inter_Inter_eq', set.mem_Inter, opens.mem_coe] at hS \u22a2,\n--   intros i,\n--   specialize hS i,\n--   rcases hS with \u27e8w, \u27e8hw1, hw2\u27e9, hx\u27e9,\n--   refine \u27e8w, \u27e8hw1, _\u27e9, hx\u27e9,\n--   intros y hy,\n--   specialize hw2 hy,\n--   simp only [subtype.val_eq_coe, set.Inf_eq_sInter, set.sInter_image, set.mem_range, exists_prop, \n--     set.mem_Inter, opens.mem_coe, and_imp, forall_apply_eq_imp_iff'] at hw2 \u22a2,\n--   intros hi2,\n--   apply hw2,\n--   apply simplex.ignore_subset,\n--   exact hi2,\n-- end\n\n-- def dder {n : \u2115} (hn : 0 < n) (\u03c3 : simplex \ud835\udd18 n.succ) (m : fin n.succ.succ) (m' : fin n.succ) :\n--   \u03c3.face \u27f6 (\u03c3.ignore\u2082 hn m m').face :=\n-- der (nat.zero_lt_succ _) \u03c3 m \u226b der _ (\u03c3.ignore _ m) m'\n\n-- section refinement\n\n-- variables {A B : X.oc} (h : A \u27f6 B) (inj : function.injective h.func)\n\n-- include inj\n-- def refine (\u03c3 : simplex A n) : simplex B n :=\n-- { to_finset := finset.image h.func \u03c3.to_finset,\n--   card_eq := begin\n--     rw [\u2190 \u03c3.2, finset.card_image_of_inj_on],\n--     apply function.injective.inj_on,\n--     assumption,\n--   end }\n\n-- -- lemma refine_self (\u03c3 : simplex A n) :\n-- --   \u03c3.refine (\ud835\udfd9 A) = \u03c3 :=\n-- -- begin\n-- --   ext i,\n-- --   split,\n-- --   { intros hi,\n-- --     unfold simplex.refine at hi,\n-- --     dsimp only at hi,\n-- --     change i \u2208 finset.image id _ at hi,\n-- --     rw finset.mem_image at hi,\n-- --     rcases hi with \u27e8a, ha, rfl\u27e9,\n-- --     exact ha },\n-- --   { intros hi,\n-- --     unfold simplex.refine,\n-- --     dsimp only,\n-- --     change i \u2208 finset.image id _,\n-- --     rw finset.mem_image,\n-- --     refine \u27e8i, hi, rfl\u27e9, },\n-- -- end\n\n-- -- lemma refine_comp {n : \u2115} {A B D : X.oc} (r1 : A \u27f6 B) (r2 : B \u27f6 D) (\u03c3 : simplex A n) :\n-- --   \u03c3.refine (r1 \u226b r2) = (\u03c3.refine r1).refine r2 :=\n-- -- begin\n-- --   ext d,\n-- --   split;\n-- --   intros hd;\n-- --   unfold simplex.refine at hd \u22a2;\n-- --   dsimp only at hd \u22a2;\n-- --   rw finset.mem_image at hd \u22a2,\n-- --   { rcases hd with \u27e8a, ha, rfl\u27e9,\n-- --     refine \u27e8r1.func a, _, rfl\u27e9,\n-- --     rw finset.mem_image,\n-- --     exact \u27e8a, ha, rfl\u27e9, },\n-- --   { rcases hd with \u27e8b, hb, rfl\u27e9, \n-- --     rw finset.mem_image at hb,\n-- --     rcases hb with \u27e8a, ha, rfl\u27e9,\n-- --     exact \u27e8a, ha, rfl\u27e9, },\n-- -- end\n\n-- -- lemma refine_ignore {n : \u2115} (hn : 0 < n) {A B : oc X} (h : A \u27f6 B) (inj : function.injective h.func) (\u03c3 : simplex A n) (m : fin n.succ) : \n-- --   (\u03c3.refine h inj).ignore hn m = (\u03c3.ignore hn m).refine h inj := \n-- -- begin\n-- --   ext i,\n-- --   split,\n-- --   { rintros (hi : i \u2208 simplex.ignore hn (simplex.refine h inj \u03c3) m),\n-- --     rw simplex.mem_ignore at hi, \n-- --     rcases hi with \u27e8h1, h2\u27e9,\n-- --     change _ \u2208 simplex.to_finset _ at h1,\n-- --     unfold simplex.refine at h1 \u22a2,\n-- --     dsimp only at h1 \u22a2,\n-- --     rw finset.mem_image at h1 \u22a2,\n-- --     rcases h1 with \u27e8a, ha, rfl\u27e9,\n-- --     refine \u27e8a, _, rfl\u27e9,\n-- --     change a \u2208 simplex.ignore hn \u03c3 m,\n-- --     rw simplex.mem_ignore,\n-- --     refine \u27e8ha, _\u27e9,\n-- --     contrapose! h2,\n-- --     rw [simplex.refine_nth, h2] },\n-- --   { rintros hi,\n-- --     erw simplex.mem_ignore,\n-- --     change i \u2208 simplex.to_finset _ \u2227 _,\n-- --     unfold simplex.refine at hi,\n-- --     dsimp only at hi,\n-- --     rw finset.mem_image at hi,\n-- --     rcases hi with \u27e8a, ha, rfl\u27e9,\n-- --     erw simplex.mem_ignore at ha,\n-- --     rcases ha with \u27e8h1, h2\u27e9,\n-- --     refine \u27e8_, _\u27e9,\n-- --     { change _ \u2208 simplex.to_finset _,\n-- --       unfold simplex.refine,\n-- --       dsimp only,\n-- --       rw finset.mem_image,\n-- --       exact \u27e8a, h1, rfl\u27e9, },\n-- --     { contrapose! h2,\n-- --       rw simplex.refine_nth at h2,\n-- --       exact h.strict_mono.injective h2, } },\n-- -- end\n\n-- end refinement\n\n-- end simplex\n\n-- end", "meta": {"author": "jjaassoonn", "repo": "cc", "sha": "6d3dc6885fa012e8c18fd38ab2949d73777fb442", "save_path": "github-repos/lean/jjaassoonn-cc", "path": "github-repos/lean/jjaassoonn-cc/cc-6d3dc6885fa012e8c18fd38ab2949d73777fb442/src/old/simplex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.39795042080701326}}
{"text": "import category_theory.limits.has_limits\nimport category_theory.limits.is_limit\nimport category_theory.limits.types\nimport category_theory.yoneda\nimport category_theory.whiskering\n\nuniverses u v\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits \n\nvariables {C : Type u}  [category.{v} C] \nvariables {I : Type v} [small_category I] -- [is_cofiltered I]\nvariables (X Y : C)\nvariables (S : I \u2964 C) [has_limit S]\n\ndef hom_as_cone_map : S.op \u22d9 yoneda.obj Y \u22d9 ulift_functor \u27f6 \n  (category_theory.functor.const I\u1d52\u1d56).obj (ulift_functor.obj ((yoneda.obj Y).obj (opposite.op (limit S)))) :=\n{ app := \u03bb i, ulift_functor.map (as_hom (\u03bb a, ((limit.cone S).\u03c0.app i.unop) \u226b a)),\n  naturality' := begin\n    intros i j ji,\n    ext f,\n    tidy,\n    have h\u2081 : as_hom (category_struct.comp (limit.\u03c0 S j.unop)) (S.map ji.unop \u226b f) = \n      (limit.\u03c0 S j.unop) \u226b (S.map ji.unop \u226b f) := by refl,\n    have h\u2082 : as_hom (category_struct.comp (limit.\u03c0 S j.unop \u226b S.map ji.unop)) f = \n      (limit.\u03c0 S j.unop \u226b S.map ji.unop) \u226b f := by refl,\n    suffices h : (limit.\u03c0 S j.unop) \u226b (S.map ji.unop) = (limit.\u03c0 S i.unop),\n    { rw \u2190 h,\n      rw h\u2081,\n      rw h\u2082,\n      simp only [category_theory.limits.limit.w, category_theory.limits.limit.w_assoc, eq_self_iff_true] },\n    { simp only [category_theory.limits.limit.w, eq_self_iff_true] },\n  end, }\n\ndef hom_as_cone : cocone (S.op \u22d9 yoneda.obj Y \u22d9 ulift_functor) :=\n{ X := ulift_functor.obj ((yoneda.obj Y).obj (opposite.op (limit S))),\n  \u03b9 := hom_as_cone_map Y S, }\n\ndef can_map_from_colim_of_homs_to_hom_from_limit : \n  colimit (S.op \u22d9 yoneda.obj Y \u22d9 ulift_functor) \u27f6 ulift_functor.obj ((yoneda.obj Y).obj (opposite.op (limit S))) := \n  colimit.desc (S.op \u22d9 yoneda.obj Y \u22d9 ulift_functor) (hom_as_cone Y S)\n\ntheorem can_is_injective (hI : \u2200 i : I, epi ((limit.cone S).\u03c0.app i)) [is_cofiltered I] : \n  function.injective (can_map_from_colim_of_homs_to_hom_from_limit Y S) :=\nbegin\n  intros a b hab,\n  obtain \u27e8i, a', ha\u27e9 := types.jointly_surjective' a,\n  obtain \u27e8j, b', hb\u27e9 := types.jointly_surjective' b,\n  rw \u2190 ha,\n  rw \u2190 hb,\n  have hIopfiltered : is_filtered I\u1d52\u1d56 := category_theory.is_filtered_op_of_is_cofiltered I,\n  obtain \u27e8k, ki, kj, t\u27e9 := hIopfiltered.to_is_filtered_or_empty.cocone_objs i j,\n  rw types.filtered_colimit.colimit_eq_iff,\n  use k,\n  use ki,\n  use kj,\n  let pi := limit.\u03c0 S i.unop,\n  let pj := limit.\u03c0 S j.unop,\n  let pk := limit.\u03c0 S k.unop,\n  -- have hi : epi pi := hI i.unop,\n  -- have hj : epi pj := hI j.unop,\n  have hk : epi pk := hI k.unop,\n  have hcani : pi \u226b a'.down = (can_map_from_colim_of_homs_to_hom_from_limit Y S a).down,\n  { unfold can_map_from_colim_of_homs_to_hom_from_limit,\n    tidy },\n  have hcanj : pj \u226b b'.down = (can_map_from_colim_of_homs_to_hom_from_limit Y S b).down,\n  { unfold can_map_from_colim_of_homs_to_hom_from_limit,\n    tidy },\n  have hik : pi = pk \u226b (S.map ki.unop) := by tidy,\n  have hjk : pj = pk \u226b (S.map kj.unop) := by tidy,\n  have hpij : pi \u226b a'.down = pj \u226b b'.down,\n  { rw hcani,\n    rw hcanj,\n    exact congr_arg ulift.down hab },\n  rw hik at hpij,\n  rw hjk at hpij,\n  simp only [quiver.hom.unop_op,\n    category_theory.ulift_functor_map,\n    category_theory.functor.comp_map,\n    ulift.up_inj,\n    category_theory.functor.op_map,\n    category_theory.yoneda_obj_map],\n  simp only [category.assoc'] at hpij,\n  exact hk.left_cancellation _ _ hpij,\nend\n\n", "meta": {"author": "dagurtomas", "repo": "nobelings-thm", "sha": "9477d58edd21be62dc2568004d2675d90a5bd85b", "save_path": "github-repos/lean/dagurtomas-nobelings-thm", "path": "github-repos/lean/dagurtomas-nobelings-thm/nobelings-thm-9477d58edd21be62dc2568004d2675d90a5bd85b/src/colim_hom_injective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3979221550321193}}
{"text": "import combinatorics.simple_graph.connectivity data.finset data.setoid.basic\nimport graph_theory.contraction graph_theory.pushforward graph_theory.basic graph_theory.walk\nopen finset classical function simple_graph.Walk\n\nvariables {V V' : Type*} [fintype V] [decidable_eq V] [fintype V'] [decidable_eq V']\nvariables {G G\u2081 G\u2082 : simple_graph V}\nvariables [decidable_rel G.adj] [decidable_rel G\u2081.adj] [decidable_rel G\u2082.adj]\nvariables {a : V} {A B X Y Z : finset V} {e : G.dart}\nvariables {f : V \u2192 V'} {hf : G.adapted f}\n\nnamespace simple_graph\nnamespace menger\n\nstructure AB_walk (G : simple_graph V) (A B : finset V) extends Walk G :=\n  (ha : a \u2208 A) (hb : b \u2208 B)\n\nnoncomputable instance : decidable_eq (AB_walk G A B) := by { classical, apply_instance }\n\nvariables {P : finset (AB_walk G A B)}\n\nnamespace AB_walk\n\ndef minimal (p : AB_walk G A B) : Prop :=\np.to_Walk.init \u2229 B = \u2205 \u2227 p.to_Walk.tail \u2229 A = \u2205\n\nnoncomputable def lift (f : V \u2192 V') (hf : adapted f G) (A B : finset V) :\n  AB_walk (map f G) (A.image f) (B.image f) \u2192 AB_walk G A B :=\nbegin\n  rintro \u27e8p,ha,hb\u27e9,\n  choose a h\u2082 h\u2083 using mem_image.mp ha,\n  choose b h\u2085 h\u2086 using mem_image.mp hb,\n  let \u03b3 := Walk.pull_Walk_aux f hf p a b h\u2083 h\u2086,\n  rw \u2190\u03b3.2.1 at h\u2082, rw \u2190\u03b3.2.2.1 at h\u2085, exact \u27e8\u03b3,h\u2082,h\u2085\u27e9\nend\n\ndef push (f : V \u2192 V') (A B : finset V) :\n  AB_walk G A B \u2192 AB_walk (map f G) (A.image f) (B.image f) :=\nbegin\n  intro p, refine \u27e8Walk.push_Walk f p.to_Walk, _, _\u27e9,\n  rw Walk.push_Walk_a, exact mem_image_of_mem f p.ha,\n  rw Walk.push_Walk_b, exact mem_image_of_mem f p.hb,\nend\n\nlemma push_lift : left_inverse (push f A B) (lift f hf A B) :=\nby { rintro \u27e8p,ha,hb\u27e9, simp [lift,push], exact Walk.pull_Walk_push }\n\nlemma lift_inj : injective (lift f hf A B) :=\nleft_inverse.injective push_lift\n\nnoncomputable def trim_aux (p : AB_walk G A B) :\n  {q : AB_walk G A B // q.minimal \u2227 q.to_Walk.range \u2286 p.to_Walk.range} :=\nbegin\n  rcases p with \u27e8p\u2081, p\u2081a, p\u2081b\u27e9,\n  have h\u2081 : (p\u2081.range \u2229 A).nonempty := \u27e8p\u2081.a, by simp [p\u2081a]\u27e9,\n  rcases p\u2081.after A h\u2081 with \u27e8p\u2082, p\u2082a, p\u2082b, p\u2082r, p\u2082i, -, p\u2082t\u27e9,\n  have h\u2082 : (p\u2082.range \u2229 B).nonempty := by { refine \u27e8p\u2082.b, _\u27e9, simp, rwa p\u2082b },\n  rcases p\u2082.until B h\u2082 with \u27e8p\u2083, p\u2083a, p\u2083b, p\u2083r, p\u2083i, -, p\u2083t\u27e9,\n  refine \u27e8\u27e8p\u2083, p\u2083a.symm \u25b8 p\u2082a, p\u2083b\u27e9, \u27e8by simp [p\u2083i], _\u27e9, p\u2083r.trans p\u2082r\u27e9,\n  have : p\u2083.tail \u2229 A \u2286 p\u2082.tail \u2229 A := inter_subset_inter_right p\u2083t,\n  rw \u2190subset_empty, apply this.trans, rw p\u2082t, refl\nend\n\nnoncomputable def trim (p : AB_walk G A B) : AB_walk G A B := p.trim_aux.val\n\nlemma trim_minimal {p : AB_walk G A B} : p.trim.minimal := p.trim_aux.prop.1\n\nlemma trim_range {p : AB_walk G A B} : p.trim.to_Walk.range \u2286 p.to_Walk.range := p.trim_aux.prop.2\n\nnoncomputable def massage_aux (h : G\u2082 \u2264 G\u2081) (p : AB_walk G\u2082 A X) :\n  {q : AB_walk G\u2081 A X // q.minimal \u2227 q.to_Walk.range \u2286 p.to_Walk.range} :=\nbegin\n  let p' := p.trim, rcases p'.to_Walk.transport (transportable_to_of_le h) with \u27e8q,qa,qb,qr,qi,qt\u27e9,\n  refine \u27e8\u27e8q, qa.symm \u25b8 p'.ha, qb.symm \u25b8 p'.hb\u27e9, _, _\u27e9,\n  { rw [minimal,qi,qt], exact trim_minimal },\n  { rw [qr], exact trim_range }\nend\n\nnoncomputable def massage (h : G\u2082 \u2264 G\u2081) (p : AB_walk G\u2082 A X) : AB_walk G\u2081 A X :=\n(p.massage_aux h).val\n\nend AB_walk\n\ndef pw_disjoint (P : finset (AB_walk G A B)) : Prop :=\n\u2200 \u2983\u03b3\u2081 \u03b3\u2082 : P\u2984, (\u03b3\u2081.val.to_Walk.range \u2229 \u03b3\u2082.val.to_Walk.range).nonempty \u2192 \u03b3\u2081 = \u03b3\u2082\n\nnamespace pw_disjoint\n\nlemma le_A (dis : pw_disjoint P) : P.card \u2264 A.card :=\nbegin\n  let \u03c6 : P \u2192 A := \u03bb p, \u27e8p.1.1.a, p.1.ha\u27e9,\n  have : injective \u03c6 := by { rintro p\u2081 p\u2082 h, simp at h, apply dis, use p\u2081.1.1.a, simp, simp [h] },\n  simp_rw [\u2190fintype.card_coe], convert fintype.card_le_of_injective \u03c6 this,\nend\n\nlemma le_B (dis : pw_disjoint P) : P.card \u2264 B.card :=\nbegin\n  let \u03c6 : P \u2192 B := \u03bb p, \u27e8p.val.b, p.val.hb\u27e9,\n  have : injective \u03c6 := by { rintro p\u2081 p\u2082 h, apply dis, use p\u2081.val.b, simp at h, simp, simp [h] },\n  simp_rw [\u2190fintype.card_coe], convert fintype.card_le_of_injective \u03c6 this,\nend\n\nend pw_disjoint\n\ndef separates (G : simple_graph V) (A B : finset V) (X : finset V) : Prop :=\n  \u2200 \u03b3 : AB_walk G A B, (\u03b3.to_Walk.range \u2229 X).nonempty\n\nnamespace separates\n\nlemma self : separates G A B A :=\n  \u03bb \u03b3, \u27e8\u03b3.a, mem_inter.mpr \u27e8Walk.start_mem_range,\u03b3.ha\u27e9\u27e9\n\nlemma symm : separates G A B X \u2192 separates G B A X :=\nbegin\n  rintro h \u27e8p,pa,pb\u27e9, let q : AB_walk G A B := by { use p.reverse; simpa only },\n  specialize h q, simp only [reverse_range] at h, exact h\nend\n\nlemma comm : separates G A B X \u2194 separates G B A X :=\n\u27e8separates.symm,separates.symm\u27e9\n\nend separates\n\n@[ext] structure separator (G : simple_graph V) (A B : finset V) extends finset V :=\n  (sep : separates G A B to_finset)\n\nnamespace separator\n\nabbreviation card (X : separator G A B) : \u2115 := X.to_finset.card\n\ninstance nonempty : nonempty (separator G A B) :=\n\u27e8\u27e8A,separates.self\u27e9\u27e9\n\ndef symm : separator G A B \u2192 separator G B A :=\n\u03bb \u27e8X, sep\u27e9, \u27e8X, sep.symm\u27e9\n\n@[simp] lemma card_symm {X : separator G A B} : X.symm.card = X.card :=\nby { cases X, simp only [symm] }\n\ndef comm : separator G A B \u2243 separator G B A :=\n{ to_fun := symm,\n  inv_fun := symm,\n  left_inv := \u03bb \u27e8X,sep\u27e9, by simp only [symm],\n  right_inv := \u03bb \u27e8X,sep\u27e9, by simp only [symm] }\n\nend separator\n\ndef is_cut_set_size (G : simple_graph V) [decidable_rel G.adj] (A B : finset V) (n : \u2115) : Prop :=\n\u2203 X : separator G A B, X.card = n\n\nnoncomputable instance : decidable_pred (is_cut_set_size G A B) :=\nby { classical, apply_instance }\n\nnoncomputable def min_cut (G : simple_graph V) [decidable_rel G.adj] (A B : finset V) : \u2115 :=\n@nat.find (is_cut_set_size G A B) _ \u27e8A.card, \u27e8A, separates.self\u27e9, rfl\u27e9\n\nnamespace min_cut\n\nlemma symm : min_cut G A B = min_cut G B A :=\nbegin\n  simp_rw min_cut, congr' 1, ext n, split;\n  { rintro \u27e8X,h\u27e9, refine \u27e8X.symm,_\u27e9, rw [separator.card_symm], exact h }\nend\n\nlemma spec : is_cut_set_size G A B (min_cut G A B) :=\nby apply nat.find_spec\n\nnoncomputable def set (G : simple_graph V) [decidable_rel G.adj] (A B : finset V) :\n  {X : separator G A B // X.card = min_cut G A B} :=\nsubtype_of_exists (spec)\n\nlemma le {X : separator G A B} : min_cut G A B \u2264 X.card :=\nnat.find_le \u27e8X, rfl\u27e9\n\nlemma le' (sep : separates G A B X) : min_cut G A B \u2264 X.card :=\nnat.find_le \u27e8\u27e8X,sep\u27e9, rfl\u27e9\n\nend min_cut\n\ndef is_menger (G : simple_graph V) [decidable_rel G.adj] : Prop :=\n\u2200 A B : finset V, \u2203 P : finset (AB_walk G A B), pw_disjoint P \u2227 P.card = min_cut G A B\n\nlemma path_le_cut (dis : pw_disjoint P) (sep : separates G A B X) : P.card \u2264 X.card :=\nbegin\n  let \u03c6 : \u03a0 \u03b3 : P, \u03b3.val.to_Walk.range \u2229 X := \u03bb \u03b3, by { choose z hz using sep \u03b3, exact \u27e8z,hz\u27e9 },\n  let \u03c8 : P \u2192 X := \u03bb \u03b3, \u27e8_, mem_of_mem_inter_right (\u03c6 \u03b3).prop\u27e9,\n  have h\u2081 : \u2200 \u03b3, (\u03c8 \u03b3).val \u2208 \u03b3.val.to_Walk.range := \u03bb \u03b3, let z := \u03c6 \u03b3 in (mem_inter.mp z.2).1,\n  have h\u2082 : injective \u03c8 := \u03bb \u03b3 \u03b3' h, dis \u27e8_, mem_inter_of_mem (h\u2081 \u03b3) (by { rw h, exact (h\u2081 \u03b3') })\u27e9,\n  simp_rw [\u2190fintype.card_coe], convert fintype.card_le_of_injective \u03c8 h\u2082\nend\n\nlemma upper_bound (dis : pw_disjoint P) : P.card \u2264 min_cut G A B :=\nby { obtain \u27e8\u27e8X,h\u2081\u27e9,h\u2082\u27e9 := min_cut.set G A B, rw \u2190h\u2082, exact path_le_cut dis h\u2081 }\n\nlemma bot_iff_no_edge : fintype.card G.dart = 0 \u2194 G = \u22a5 :=\nbegin\n  split; intro h,\n  { ext x y, simp, intro h\u2081, exact is_empty_iff.mp (fintype.card_eq_zero_iff.mp h) \u27e8\u27e8_,_\u27e9,h\u2081\u27e9 },\n  { simp_rw h, apply fintype.card_eq_zero_iff.mpr, exact (is_empty_iff.mpr dart.is_adj) }\nend\n\nlemma bot_separates_iff : separates \u22a5 A B X \u2194 (A \u2229 B) \u2286 X :=\nbegin\n  split; intro h,\n  { rintros z hz, rw [mem_inter] at hz, let \u03b3 : AB_walk \u22a5 A B := \u27e8Walk.nil _, hz.1, hz.2\u27e9,\n    choose z h\u2081 using h \u03b3, simp at h\u2081, rw \u2190h\u2081.1, exact h\u2081.2 },\n  { rintro \u27e8\u27e8a,b,\u03b3\u27e9,ha,hb\u27e9, cases \u03b3, swap, exfalso, exact \u03b3_h,\n    simp at ha hb \u22a2, use a, simp, split, exact Walk.start_mem_range,\n    apply h, simp, exact \u27e8ha,hb\u27e9 }\nend\n\nlemma bot_min_cut : min_cut \u22a5 A B = (A \u2229 B).card :=\nbegin\n  apply (nat.find_eq_iff _).mpr, split,\n  { use A \u2229 B, rw [bot_separates_iff], refl },\n  { rintro n hn \u27e8X,rfl\u27e9, have := card_le_of_subset (bot_separates_iff.mp X.sep),\n    change (A \u2229 B).card \u2264 X.card at this, linarith }\nend\n\nnoncomputable def bot_path_set (A B : finset V) :\n  {P : finset (AB_walk \u22a5 A B) // pw_disjoint P \u2227 P.card = (A \u2229 B).card} :=\nbegin\n  let \u03c6 : A \u2229 B \u2192 AB_walk \u22a5 A B := \u03bb z, let h := mem_inter.mp z.prop in \u27e8\u27e8walk.nil\u27e9,h.1,h.2\u27e9,\n  have \u03c6_inj : injective \u03c6 := \u03bb _ _ h, by { simp only [\u03c6] at h, ext, exact h.1 },\n  refine \u27e8image \u03c6 univ, _, _\u27e9,\n  { rintro \u27e8\u27e8\u03b3\u2081,h\u2081,h\u2082\u27e9,h\u2083\u27e9 \u27e8\u27e8\u03b3\u2082,h\u2084,h\u2085\u27e9,h\u2086\u27e9 h\u2087,\n    have nil\u2081 : \u03b3\u2081 = Walk.nil \u03b3\u2081.a := by { cases \u03b3\u2081, cases \u03b3\u2081_p, refl, exfalso, exact \u03b3\u2081_p_h },\n    have nil\u2082 : \u03b3\u2082 = Walk.nil \u03b3\u2082.a := by { cases \u03b3\u2082, cases \u03b3\u2082_p, refl, exfalso, exact \u03b3\u2082_p_h },\n    simp at h\u2087 \u22a2, rw [nil\u2081,nil\u2082] at h\u2087 \u22a2, cases h\u2087 with z h\u2087, simp at h\u2087, rw [\u2190h\u2087.1,\u2190h\u2087.2] },\n  { rw [card_image_of_injective univ \u03c6_inj, card_univ],\n    convert fintype.card_of_finset (A \u2229 B) _, intro z, simp, split,\n    { rintros \u27e8h\u2081,h\u2082\u27e9, exact set.mem_sep h\u2081 h\u2082 },\n    { rintros h\u2081, exact h\u2081 } }\nend\n\nlemma bot_is_menger : is_menger (\u22a5 : simple_graph V) :=\nby { rintro A B, rw bot_min_cut, exact (bot_path_set A B).exists_of_subtype }\n\nlemma AB_lift_dis (P' : finset (AB_walk (map f G) (A.image f) (B.image f))) :\n  pw_disjoint P' \u2192 pw_disjoint (P'.image (AB_walk.lift f hf A B)) :=\nbegin\n  rintro hP' \u27e8\u03b3\u2081,h\u2081\u27e9 \u27e8\u03b3\u2082,h\u2082\u27e9 h, simp at h \u22a2, choose z h using h,\n  choose \u03b3'\u2081 h'\u2081 h''\u2081 using mem_image.mp h\u2081,\n  choose \u03b3'\u2082 h'\u2082 h''\u2082 using mem_image.mp h\u2082,\n  have h\u2083 := congr_arg (AB_walk.push f A B) h''\u2081, rw AB_walk.push_lift at h\u2083,\n  have h\u2084 := congr_arg (AB_walk.push f A B) h''\u2082, rw AB_walk.push_lift at h\u2084,\n  suffices : \u03b3'\u2081 = \u03b3'\u2082, { rw [\u2190h''\u2081,\u2190h''\u2082,this] },\n  have := @hP' \u27e8_,h'\u2081\u27e9 \u27e8_,h'\u2082\u27e9, simp at this, apply this,\n  simp [h\u2083,h\u2084,AB_walk.push,Walk.push_range], use f z, rw mem_inter at h \u22a2, split,\n  exact mem_image_of_mem f h.1, exact mem_image_of_mem f h.2\nend\n\ndef minus (G : simple_graph V) (e : G.dart) : simple_graph V :=\nG.delete_edges {e.edge}\n\ninfix `-` := minus\n\nnoncomputable instance : decidable_rel (G-e).adj := by { classical, apply_instance }\n\nlemma minus_le {e : G.dart} : G-e \u2264 G := \u03bb x y h, h.1\n\nlemma minus_lt_edges {e : G.dart} : fintype.card (G-e).dart < fintype.card G.dart :=\nbegin\n  let \u03c6 : (G-e).dart \u2192 G.dart := \u03bb e, \u27e8\u27e8_,_\u27e9,e.is_adj.1\u27e9,\n  have \u03c6_inj : injective \u03c6 := by { rintro e\u2081 e\u2082 h, simp [\u03c6] at h, exact e\u2081.ext e\u2082 h },\n  suffices : e \u2209 set.range \u03c6, refine fintype.card_lt_of_injective_of_not_mem \u03c6 \u03c6_inj this,\n  intro he, rw set.mem_range at he, choose e' he using he, rcases e' with \u27e8\u27e8x,y\u27e9,he'\u27e9,\n  replace he := (dart.ext_iff _ _).mp he, replace he := prod.ext_iff.mp he,\n  simp only at he, cases he, substs x y,\n  simp [minus] at he', simp [dart.edge,sym2] at he', apply he'.2, refl\nend\n\nlemma sep_AB_of_sep\u2082_AX \u2983e : G.dart\u2984 (ex_in_X : e.fst \u2208 X) (ey_in_X : e.snd \u2208 X) :\n  separates G A B X \u2192 separates (G-e) A X Z \u2192 separates G A B Z :=\nby {\n  rintro X_sep_AB Z_sep\u2082_AX \u03b3,\n  rcases \u03b3.to_Walk.until X (X_sep_AB \u03b3) with \u27e8\u03b4,\u03b4_a,\u03b4_b,\u03b4_range,\u03b4_init,-\u27e9,\n  have : \u03b4.transportable_to (G-e) := by {\n    revert \u03b4_init, refine Walk.rec\u2080 _ _ \u03b4,\n    { simp [Walk.transportable_to,Walk.edges] },\n    { rintro e' p h ih h\u2081 e'' h\u2082,\n      have h\u2083 : p.init \u2229 X = \u2205 :=\n      by { apply subset_empty.mp, rw [\u2190h\u2081], apply inter_subset_inter_right,\n        rw [Walk.init_cons], apply subset_union_right },\n      simp at h\u2082, cases h\u2082,\n      { subst e'', simp at h\u2081, simp [minus,e'.is_adj],\n        have : e'.fst \u2209 X :=\n        by { rw [inter_distrib_right, union_eq_empty_iff] at h\u2081, intro h,\n          apply not_nonempty_empty, rw \u2190h\u2081.1,\n          exact \u27e8e'.fst, by simp only [h, singleton_inter_of_mem, mem_singleton]\u27e9 },\n        intro h', apply this, rw [dart.edge,sym2.mk_eq_mk_iff] at h',\n        cases h'; { rw h', assumption } },\n      { exact ih h\u2083 e'' h\u2082 }\n    }\n  },\n  rcases \u03b4.transport this with \u27e8\u03b6,\u03b6_a,\u03b6_b,\u03b6_range,-,-\u27e9,\n  rcases Z_sep\u2082_AX \u27e8\u03b6, by { rw [\u03b6_a,\u03b4_a], exact \u03b3.ha }, by { rw [\u03b6_b], exact \u03b4_b }\u27e9 with \u27e8z,hz\u27e9,\n  rw \u2190\u03b6_range at \u03b4_range, rw mem_inter at hz,\n  exact \u27e8z, mem_inter.mpr \u27e8mem_of_subset \u03b4_range hz.1, hz.2\u27e9\u27e9,\n}\n\nlemma massage_eq {h : G\u2082 \u2264 G\u2081} {P : finset (AB_walk G\u2082 A B)} {p\u2081 p\u2082 : P} :\n  pw_disjoint P \u2192 ((p\u2081.val.massage h).to_Walk.range \u2229 (p\u2082.val.massage h).to_Walk.range).nonempty \u2192\n  p\u2081 = p\u2082 :=\nbegin\n  rintro hP h, apply hP, rcases h with \u27e8z,hz\u27e9, use z, simp at hz \u22a2, split,\n  { apply (p\u2081.val.massage_aux h).prop.2, exact hz.1 },\n  { apply (p\u2082.val.massage_aux h).prop.2, exact hz.2 }\nend\n\nlemma massage_disjoint {h : G\u2082 \u2264 G\u2081} {P : finset (AB_walk G\u2082 A B)} :\n  pw_disjoint P \u2192 pw_disjoint (image (AB_walk.massage h) P) :=\nbegin\n  rintro h\u2081 \u27e8p\u2081,hp\u2081\u27e9 \u27e8p\u2082,hp\u2082\u27e9 h, apply subtype.ext, dsimp,\n  choose q\u2081 hq\u2081 hq\u2081' using mem_image.mp hp\u2081, choose q\u2082 hq\u2082 hq\u2082' using mem_image.mp hp\u2082,\n  rw [\u2190hq\u2081',\u2190hq\u2082'], apply congr_arg, let \u03b3\u2081 : P := \u27e8q\u2081,hq\u2081\u27e9, let \u03b3\u2082 : P := \u27e8q\u2082,hq\u2082\u27e9,\n  suffices : \u03b3\u2081 = \u03b3\u2082, { simp only [subtype.mk_eq_mk] at this, exact this }, apply massage_eq h\u2081,\n  rw [hq\u2081',hq\u2082'], exact h\nend\n\nlemma massage_card {h : G\u2082 \u2264 G\u2081} {P : finset (AB_walk G\u2082 A B)} :\n  pw_disjoint P \u2192 (image (AB_walk.massage h) P).card = P.card :=\nbegin\n  rintro hP, apply card_image_of_inj_on, rintro p\u2081 hp\u2081 p\u2082 hp\u2082 he,\n  let q\u2081 : P := \u27e8p\u2081,hp\u2081\u27e9, let q\u2082 : P := \u27e8p\u2082,hp\u2082\u27e9, suffices : q\u2081 = q\u2082, simp at this, exact this,\n  apply massage_eq hP, rw he, simp\nend\n\nlemma meet_sub_X (X_sep_AB : separates G A B X) (p : AB_walk G A X) (q : AB_walk G B X)\n  (hp : p.minimal) (hq : q.minimal) : p.to_Walk.range \u2229 q.to_Walk.range \u2286 X :=\nbegin\n  rcases p with \u27e8p,pa,pb\u27e9, rcases q with \u27e8q,qa,qb\u27e9, dsimp,\n  rintro x hx, rw mem_inter at hx, cases hx with hx\u2081 hx\u2082, by_contra,\n\n  rcases p.until {x} \u27e8x, by simp [hx\u2081]\u27e9 with \u27e8p', p'a, p'b, p'r, p'i, p'i2, p't\u27e9, simp at p'b,\n  have h\u2081 : p'.range \u2229 X = \u2205 :=\n  by { rw Walk.range_eq_init_union_last, rw inter_distrib_right, rw union_eq_empty_iff, split,\n    { exact subset_empty.mp ((inter_subset_inter_right p'i2).trans (subset_empty.mpr hp.1)) },\n    { rw p'b, exact singleton_inter_of_not_mem h } },\n\n  rcases q.until {x} \u27e8x, by simp [hx\u2082]\u27e9 with \u27e8q', q'a, q'b, q'r, q'i, q'i2, q't\u27e9, simp at q'b,\n  have h\u2081 : q'.range \u2229 X = \u2205 :=\n  by { rw Walk.range_eq_init_union_last, rw inter_distrib_right, rw union_eq_empty_iff, split,\n    { exact subset_empty.mp ((inter_subset_inter_right q'i2).trans (subset_empty.mpr hq.1)) },\n    { rw q'b, exact singleton_inter_of_not_mem h } },\n\n  let \u03b3 : AB_walk G A B :=\n  \u27e8Walk.append p' q'.reverse (by simp [p'b,q'b]), by simp [p'a,pa], by simp [q'a,qa]\u27e9,\n  choose z hz using X_sep_AB \u03b3, rw [range_append,reverse_range,inter_distrib_right] at hz,\n  rw mem_union at hz, cases hz; { have := ne_empty_of_mem hz, contradiction }\nend\n\nnoncomputable def endpoint (P : finset (AB_walk G A B))\n  (P_dis : pw_disjoint P) (P_eq : P.card = B.card) : P \u2243 B :=\nbegin\n  let \u03c6 : P \u2192 B := \u03bb p, let q := p.val in \u27e8q.b,q.hb\u27e9,\n  apply equiv.of_bijective \u03c6, rw fintype.bijective_iff_injective_and_card, split,\n  { rintro p\u2081 p\u2082 h, apply P_dis, use p\u2081.val.b, simp at h \u22a2, simp [h]  },\n  { simp, exact P_eq },\nend\n\nnoncomputable def sep_cleanup {e : G.dart} (ex_in_X : e.fst \u2208 X) (ey_in_X : e.snd \u2208 X)\n  (X_eq_min : X.card = min_cut G A B) (X_sep_AB : separates G A B X)\n  (ih : \u2203 (P : finset (AB_walk (G-e) A X)), pw_disjoint P \u2227 P.card = min_cut (G-e) A X) :\n  {P : finset (AB_walk G A X) // pw_disjoint P \u2227 P.card = X.card \u2227 \u2200 p : P, p.val.minimal} :=\nbegin\n  choose P h\u2081 h\u2082 using ih, use image (AB_walk.massage minus_le) P, refine \u27e8_,_,_\u27e9,\n  { exact massage_disjoint h\u2081 },\n  { apply (massage_card h\u2081).trans, apply le_antisymm h\u2081.le_B,\n    rcases min_cut.set (G-e) A X with \u27e8\u27e8Z,Z_sep\u2082_AB\u27e9,Z_eq_min\u27e9,\n    rw [X_eq_min,h\u2082,\u2190Z_eq_min], apply min_cut.le',\n    exact sep_AB_of_sep\u2082_AX ex_in_X ey_in_X X_sep_AB Z_sep\u2082_AB },\n  { intro p, choose p' hp'\u2081 hp'\u2082 using mem_image.mp p.prop,\n    have := (p'.massage_aux minus_le).prop.1, simp [AB_walk.massage] at hp'\u2082, rw hp'\u2082 at this,\n    simp, exact this }\nend\n\nnoncomputable def stitch (X_sep_AB : separates G A B X)\n  (P : finset (AB_walk G A X)) (P_dis: pw_disjoint P) (P_eq_X: P.card = X.card)\n  (Q : finset (AB_walk G B X)) (Q_dis: pw_disjoint Q) (Q_eq_X: Q.card = X.card)\n  (hP : \u2200 p : P, p.val.minimal) (hQ : \u2200 q : Q, q.val.minimal) :\n  {R : finset (AB_walk G A B) // pw_disjoint R \u2227 R.card = X.card} :=\nbegin\n  let \u03c6 : X \u2243 P := (endpoint P P_dis P_eq_X).symm,\n  let \u03c8 : X \u2243 Q := (endpoint Q Q_dis Q_eq_X).symm,\n\n  have \u03c6xb : \u2200 x : X, (\u03c6 x).val.b = x.val :=\n  by { intro x, set \u03b3 := \u03c6 x,\n    have : x = \u03c6.symm \u03b3 := by simp only [equiv.symm_symm, equiv.apply_symm_apply],\n    rw this, refl },\n\n  have \u03c8xb : \u2200 x : X, (\u03c8 x).val.b = x.val :=\n  by { intro x, set \u03b3 := \u03c8 x,\n    have : x = \u03c8.symm \u03b3 := by simp only [equiv.symm_symm, equiv.apply_symm_apply],\n    rw this, refl },\n\n  let \u03a8 : X \u2192 AB_walk G A B :=\n  by { intro x, set \u03b3 := \u03c6 x with h\u03b3, set \u03b4 := \u03c8 x with h\u03b4,\n    have \u03b3bx : \u03b3.val.b = x := \u03c6xb x, have \u03b4bx : \u03b4.val.b = x := \u03c8xb x,\n    set \u03b6 := \u03b4.val.to_Walk.reverse, refine \u27e8Walk.append \u03b3.val.to_Walk \u03b6 _, _, _\u27e9,\n    { rw [\u03b3bx,\u2190\u03b4bx,reverse_a] },\n    { rw [append_a], exact \u03b3.val.ha },\n    { rw [append_b,reverse_b], exact \u03b4.val.ha } },\n\n  set R := image \u03a8 univ,\n\n  have \u03a8_inj : injective \u03a8 :=\n  by {\n    have : \u2200 x : X, (\u03a8 x).to_Walk.range \u2229 X = {x} :=\n    by { intro,\n      simp only [range_append, reverse_range],\n      simp_rw range_eq_init_union_last, simp_rw inter_distrib_right,\n      simp only [union_assoc],\n      rw [(hP (\u03c6 x)).1, (hQ (\u03c8 x)).1, \u03c6xb, \u03c8xb],\n      simp only [subtype.val_eq_coe,singleton_inter_of_mem,coe_mem,empty_union,union_idempotent] },\n    rintro x y h, ext, apply singleton_inj.mp, rw [\u2190 this x, \u2190 this y, h] },\n\n  have l\u2081 : \u2200 x y z, z \u2208 (\u03c6 x).val.to_Walk.range \u2229 (\u03c8 y).val.to_Walk.range \u2192 x = y :=\n  by {\n    intros x y z hz,\n    have z_in_X : z \u2208 X := meet_sub_X X_sep_AB (\u03c6 x) (\u03c8 y) (hP (\u03c6 x)) (hQ (\u03c8 y)) hz,\n    rw mem_inter at hz,\n    have z_is_x : z = x := by {\n      apply mem_singleton.mp, convert \u2190 mem_inter.mpr \u27e8hz.1,z_in_X\u27e9,\n      rw [range_eq_init_union_last, inter_distrib_right, \u03c6xb, (hP (\u03c6 x)).1],\n      simp only [subtype.val_eq_coe, singleton_inter_of_mem, coe_mem, empty_union], },\n    have z_is_y : z = y := by {\n      apply mem_singleton.mp, convert \u2190 mem_inter.mpr \u27e8hz.2,z_in_X\u27e9,\n      rw [range_eq_init_union_last, inter_distrib_right, \u03c8xb, (hQ (\u03c8 y)).1],\n      simp only [subtype.val_eq_coe, singleton_inter_of_mem, coe_mem, empty_union] },\n    ext, exact z_is_x.symm.trans z_is_y },\n\n  have R_dis : pw_disjoint R :=\n  by {\n    rintro \u27e8\u03b3\u2081, h\u03b3\u2081\u27e9 \u27e8\u03b3\u2082, h\u03b3\u2082\u27e9 h_dis,\n    choose x hx using mem_image.mp h\u03b3\u2081, replace hx := hx.2, subst hx,\n    choose y hy using mem_image.mp h\u03b3\u2082, replace hy := hy.2, subst hy,\n    suffices : x = y, subst this,\n    simp only [inter_distrib_left, inter_distrib_right, subtype.val_eq_coe, range_append,\n      reverse_range, union_assoc] at h_dis,\n    choose z hz using h_dis, simp only [mem_union] at hz,\n    cases hz, { apply \u03c6.left_inv.injective, apply P_dis, use z, exact hz },\n    cases hz, { exact l\u2081 x y z hz },\n    cases hz, { rw inter_comm at hz, exact (l\u2081 y x z hz).symm },\n    { apply \u03c8.left_inv.injective, apply Q_dis, use z, exact hz }\n  },\n\n  refine \u27e8R, R_dis, _\u27e9, rw finset.card_image_of_injective _ \u03a8_inj, convert fintype.card_coe X\nend\n\nlemma sep_of_sep_in_merge : separates (G/e) (image (merge_edge e) A) (image (merge_edge e) B) Y \u2192\n  separates G A B (Y \u222a {e.snd}) :=\nbegin\n  rintro Y_sep \u03b3,\n  choose z hz using Y_sep (\u03b3.push (merge_edge e) A B),\n  rw [mem_inter,AB_walk.push,Walk.push_range,mem_image] at hz,\n  choose x hx\u2081 hx\u2082 using hz.1,\n  by_cases x = e.snd; simp [merge_edge,h] at hx\u2082,\n  { use x, simp, split, exact hx\u2081, right, exact h },\n  { use x, simp, split, exact hx\u2081, left, rw hx\u2082, exact hz.2 }\nend\n\nlemma step_1 (h_contract : is_menger (G/e))\n  (too_small : \u2200 P : finset (AB_walk G A B), pw_disjoint P \u2192 P.card < min_cut G A B) :\n  \u2203 X : finset V, e.fst \u2208 X \u2227 e.snd \u2208 X \u2227 separates G A B X \u2227 X.card = min_cut G A B :=\nbegin\n  let A\u2081 := image (merge_edge e) A, let B\u2081 := image (merge_edge e) B,\n  obtain \u27e8Y, Y_eq_min\u2081\u27e9 := min_cut.set (G/e) A\u2081 B\u2081, let X := Y.to_finset \u222a {e.snd},\n\n  have Y_lt_min : Y.card < min_cut G A B :=\n  by {\n    choose P\u2081 P\u2081_dis P\u2081_eq_min\u2081 using h_contract A\u2081 B\u2081,\n    rw [Y_eq_min\u2081, \u2190P\u2081_eq_min\u2081, \u2190card_image_of_injective P\u2081 AB_walk.lift_inj],\n    apply too_small, { apply AB_lift_dis, exact P\u2081_dis }, { exact merge_edge_adapted }\n  },\n\n  have X_sep_AB : separates G A B X := sep_of_sep_in_merge Y.sep,\n\n  refine \u27e8X, _, _, X_sep_AB, _\u27e9,\n\n  { rw [mem_union], left, by_contradiction,\n    suffices : separates G A B Y.to_finset, by { exact not_lt_of_le (min_cut.le' this) Y_lt_min },\n    intro p, choose z hz using Y.sep (p.push (merge_edge e) A B), use z,\n    rw mem_inter at hz \u22a2, rcases hz with \u27e8hz\u2081,hz\u2082\u27e9, refine \u27e8_,hz\u2082\u27e9,\n    rw [AB_walk.push,Walk.push_range,mem_image] at hz\u2081, choose x hx\u2081 hx\u2082 using hz\u2081,\n    by_cases x = e.snd; simp [merge_edge,h] at hx\u2082,\n    { rw [\u2190hx\u2082] at hz\u2082, contradiction },\n    { rwa [\u2190hx\u2082] } },\n  { rw [mem_union,mem_singleton], right, refl },\n  { refine le_antisymm _ (min_cut.le' X_sep_AB),\n    exact (card_union_le _ _).trans (nat.succ_le_of_lt Y_lt_min) }\nend\n\nlemma induction_step (e : G.dart) : is_menger (G/e) \u2192 is_menger (G-e) \u2192 is_menger G :=\nbegin\n  intros h_contract h_minus A B,\n\n  apply not_imp_self.mp, intro too_small, push_neg at too_small, replace too_small :\n    \u2200 P : finset (AB_walk G A B), pw_disjoint P \u2192 P.card < min_cut G A B :=\n  by { intros P h, exact lt_of_le_of_ne (upper_bound h) (too_small P h) },\n\n  choose X ex_in_X ey_in_X X_sep_AB X_eq_min using step_1 h_contract too_small,\n\n  rcases sep_cleanup ex_in_X ey_in_X X_eq_min X_sep_AB (h_minus A X) with \u27e8P,hP\u27e9,\n  let X_eq_min' : X.card = min_cut G B A := X_eq_min.trans min_cut.symm,\n  rcases sep_cleanup ex_in_X ey_in_X X_eq_min' X_sep_AB.symm (h_minus B X) with \u27e8Q,hQ\u27e9,\n  rw \u2190X_eq_min, apply subtype.exists_of_subtype,\n\n  exact stitch X_sep_AB P hP.1 hP.2.1 Q hQ.1 hQ.2.1 hP.2.2 hQ.2.2\nend\n\nlemma lower_bound_aux (n : \u2115) : \u2200 (G : simple_graph V) [decidable_rel G.adj],\n  by exactI fintype.card G.dart \u2264 n \u2192 is_menger G :=\nbegin\n  induction n with n ih; intros G G_dec hG,\n  { have : G = \u22a5 := by { apply bot_iff_no_edge.mp, exact nat.le_zero_iff.mp hG, apply_instance },\n    simp_rw this, exact bot_is_menger },\n  { resetI, by_cases (fintype.card G.dart = 0),\n    { apply ih, rw h, linarith },\n    { cases not_is_empty_iff.mp (h \u2218 fintype.card_eq_zero_iff.mpr) with e, apply induction_step e,\n      { exact ih _ (nat.le_of_lt_succ (nat.lt_of_lt_of_le contract_edge.fewer_edges hG)) },\n      { exact ih _ (nat.le_of_lt_succ (nat.lt_of_lt_of_le minus_lt_edges hG)) } } }\nend\n\ntheorem menger : is_menger G :=\nbegin\n  apply lower_bound_aux (fintype.card G.dart), apply le_of_eq, convert rfl\nend\n\nend menger\nend simple_graph\n", "meta": {"author": "vbeffara", "repo": "lean", "sha": "0004b1d502ac3f4ccd213dbd23589d4c4f9fece8", "save_path": "github-repos/lean/vbeffara-lean", "path": "github-repos/lean/vbeffara-lean/lean-0004b1d502ac3f4ccd213dbd23589d4c4f9fece8/src/graph_theory/menger.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.749087201911703, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.39792214312981616}}
{"text": "import group_theory.group_action.support\nimport phase0.struct_perm\n\n/-!\n# Supports\n-/\n\nopen cardinal equiv mul_action quiver\nopen_locale cardinal\n\nnoncomputable theory\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}] {\u03b1 : type_index}\n\n/-- A support condition is an atom or a near-litter together with an extended type index. -/\n@[derive [inhabited]]\ndef support_condition (\u03b1 : type_index) : Type u := (atom \u2295 near_litter) \u00d7 extended_index \u03b1\n\n/-- The \"identity\" equivalence between `(atom \u2295 near_litter) \u00d7 extended_index \u03b1` and\n`support_condition \u03b1`. -/\ndef to_condition : (atom \u2295 near_litter) \u00d7 extended_index \u03b1 \u2243 support_condition \u03b1 := equiv.refl _\n\n/-- The \"identity\" equivalence between `support_condition \u03b1` and\n`(atom \u2295 near_litter) \u00d7 extended_index \u03b1`. -/\ndef of_condition : support_condition \u03b1 \u2243 (atom \u2295 near_litter) \u00d7 extended_index \u03b1 := equiv.refl _\n\n/-- There are `\u03bc` support conditions. -/\n@[simp] lemma mk_support_condition (\u03b1 : type_index) : #(support_condition \u03b1) = #\u03bc :=\nbegin\n  simp only [support_condition, mk_prod, mk_sum, mk_atom, lift_id, mk_near_litter],\n  rw add_eq_left (\u03ba_regular.aleph_0_le.trans \u03ba_le_\u03bc) le_rfl,\n  exact mul_eq_left (\u03ba_regular.aleph_0_le.trans \u03ba_le_\u03bc)\n    (le_trans (mk_extended_index \u03b1) $ le_of_lt $ lt_trans \u039b_lt_\u03ba \u03ba_lt_\u03bc) (mk_ne_zero _),\nend\n\nnamespace struct_perm\n\ninstance mul_action_support_condition : mul_action (struct_perm \u03b1) (support_condition \u03b1) :=\n{ smul := \u03bb \u03c0 c, \u27e8derivative c.snd \u03c0 \u2022 c.fst, c.snd\u27e9,\n  one_smul := by { rintro \u27e8atoms | Ns, A\u27e9; unfold has_smul.smul; simp },\n  mul_smul := begin\n    rintro \u03c0\u2081 \u03c0\u2082 \u27e8atoms | Ns, A\u27e9; unfold has_smul.smul;\n    rw derivative_mul; dsimp; rw mul_smul,\n  end }\n\ninstance mul_action_support_condition' {B : le_index \u03b1} {\u03b2 : type_index} {\u03b3 : type_index}\n  {h\u03b3 : \u03b3 < \u03b2}\n  (A : path (B : type_index) \u03b2) :\n  mul_action (struct_perm ((lt_index.mk' h\u03b3 (B.path.comp A)) : le_index \u03b1).index)\n    (support_condition \u03b3) :=\nstruct_perm.mul_action_support_condition\n\ninstance mul_action_support_condition_lt_index\n  {\u03b2 \u03b3 : type_index} {h\u03b3 : \u03b3 < \u03b2} (A : path \u03b1 \u03b2) :\n  mul_action (struct_perm (lt_index.mk' h\u03b3 A)) (support_condition \u03b3) :=\nstruct_perm.mul_action_support_condition\n\ninstance mul_action_support_condition_lt_index'\n  {\u03b2 \u03b3 : type_index} {h\u03b3 : \u03b3 < \u03b2} (A : path \u03b1 \u03b2) :\n  mul_action (struct_perm (lt_index.mk' h\u03b3 A : le_index \u03b1).index) (support_condition \u03b3) :=\nstruct_perm.mul_action_support_condition\n\n@[simp] lemma smul_to_condition (\u03c0 : struct_perm \u03b1) (x : (atom \u2295 near_litter) \u00d7 extended_index \u03b1) :\n  \u03c0 \u2022 to_condition x = to_condition \u27e8derivative x.2 \u03c0 \u2022 x.1, x.2\u27e9 := rfl\n\nend struct_perm\n\nvariables (G : Type*) (\u03b1) {\u03c4 : Type*} [has_smul G (support_condition \u03b1)] [has_smul G \u03c4]\n\nstructure support (x : \u03c4) :=\n(carrier : set (support_condition \u03b1))\n(small : small carrier)\n(supports : supports G carrier x)\n\n/-- An element of `\u03c4` is *supported* if it has some support. -/\ndef supported (x : \u03c4) : Prop := nonempty $ support \u03b1 G x\n\ninstance support.set_like (x : \u03c4) : set_like (support \u03b1 G x) (support_condition \u03b1) :=\n{ coe := support.carrier,\n  coe_injective' := \u03bb s t h, by { cases s, cases t, congr' } }\n\n@[simp] lemma support.carrier_eq_coe {x : \u03c4} {s : support \u03b1 G x} : s.carrier = s := rfl\n\n/-- There are at most `\u03bc` supports for a given `x : \u03c4`. -/\nlemma mk_support_le (x : \u03c4) : #(support \u03b1 G x) \u2264 #\u03bc :=\nbegin\n  transitivity #{s : set \u03bc // small s},\n  transitivity #{S : set (support_condition \u03b1) // small S},\n  { refine \u27e8\u27e8\u03bb s, \u27e8s.carrier, s.small\u27e9, \u03bb s t h, _\u27e9\u27e9,\n    simpa only [subtype.mk_eq_mk, support.carrier_eq_coe, set_like.coe_set_eq] using h },\n  { convert le_of_eq\n      (mk_subtype_of_equiv _ (equiv.set.congr (cardinal.eq.mp (mk_support_condition \u03b1)).some)),\n    ext s,\n    refine \u27e8small.image, \u03bb h, _\u27e9,\n    rw \u2190 symm_apply_apply (equiv.set.congr (cardinal.eq.mp $ mk_support_condition \u03b1).some) s,\n    exact h.image },\n  { rw \u2190 mk_subset_mk_lt_cof \u03bc_strong_limit.2,\n    exact mk_subtype_mono (\u03bb s hs, lt_of_lt_of_le hs \u03ba_le_\u03bc_cof) }\nend\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase0/support.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.749087201911703, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.39792214312981616}}
{"text": "/-\nFile: math_spec.lean\n\nModified version of the autogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.prelude\nimport tactic.polyrith\n\nimport starkware.cairo.common.bool_spec\n\nopen starkware.cairo.common.bool\nopen_locale big_operators\nnamespace starkware.cairo.common.math\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\n\n-- End of automatically generated prelude.\n\nnamespace assert_le_felt\n\n@[reducible] def PRIME_OVER_3_HIGH := 3544607988759775765608368578435044694\n@[reducible] def PRIME_OVER_2_HIGH := 5316911983139663648412552867652567041\n\nend assert_le_felt\n\nnamespace split_felt\n\n@[reducible] def MAX_HIGH := (((-1)) : \u2124) / (2 ^ 128 : \u2124)\n@[reducible] def MAX_LOW := 0\n\nend split_felt\n\nnamespace assert_250_bit\n\n@[reducible] def UPPER_BOUND := 2 ^ 250\n@[reducible] def SHIFT := 2 ^ 128\n@[reducible] def HIGH_BOUND := (UPPER_BOUND : \u2124) / (SHIFT : \u2124)\n\nend assert_250_bit\n\nnamespace starkware.cairo.common.math.assert_250_bit\n\n@[reducible] def UPPER_BOUND := 2 ^ 250\n@[reducible] def SHIFT := 2 ^ 128\n@[reducible] def HIGH_BOUND := (UPPER_BOUND : \u2124) / (SHIFT : \u2124)\n\nend starkware.cairo.common.math.assert_250_bit\n\nnamespace starkware.cairo.common.math.split_felt\n\n@[reducible] def MAX_HIGH := ((-1) : \u2124) / (2 ^ 128 : \u2124)\n@[reducible] def MAX_LOW := 0\n\nend starkware.cairo.common.math.split_felt\n\n/-\nConstants and calculations\n-/\n\ndef UPPER_BOUND := 2^250\ndef HIGH_PART_SHIFT := 2^250 / 2^128\ndef MAX_HIGH := (PRIME - 1) / 2^128\ndef MAX_LOW := (PRIME - 1) % 2^128\ndef CUTOFF := 2^128\n\ntheorem UPPER_BOUND_eq : UPPER_BOUND =\n  1809251394333065553493296640760748560207343510400633813116524750123642650624 :=\nby unfold UPPER_BOUND; norm_num1\n\ntheorem HIGH_PART_SHIFT_eq : HIGH_PART_SHIFT = 5316911983139663491615228241121378304 :=\nby unfold HIGH_PART_SHIFT; norm_num1\n\ntheorem MAX_HIGH_eq : MAX_HIGH = 10633823966279327296825105735305134080 :=\nby unfold MAX_HIGH; unfold PRIME; norm_num1\n\ntheorem MAX_LOW_eq : MAX_LOW = 0 :=\nby unfold MAX_LOW; unfold PRIME; norm_num1\n\ntheorem CUTOFF_eq : CUTOFF = 340282366920938463463374607431768211456 :=\nby unfold CUTOFF; norm_num1\n\nlemma rc_bound_le_CUTOFF : rc_bound F \u2264 CUTOFF := rc_bound_hyp F\n\n/-\nSpecifications for all the functions.\n-/\n\ndef spec_assert_not_zero (mem : F \u2192 F) (\u03ba : \u2115) (value : F) : Prop :=\n  value \u2260 0\n\ndef spec_assert_not_equal (mem : F \u2192 F) (\u03ba : \u2115) (a b : F) : Prop :=\n  a \u2260 b\n\ndef spec_assert_nn (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 n : \u2115, n < rc_bound F \u2227 a = \u2191n\n\ndef spec_assert_le (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 n : \u2115, n < rc_bound F \u2227 b = a + \u2191n\n\ndef spec_assert_lt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 n : \u2115, n < rc_bound F \u2227 b = a + \u2191(n + 1)\n\ndef spec_assert_nn_le (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 m n : \u2115, m < rc_bound F \u2227 n < rc_bound F \u2227 a = \u2191m \u2227 b = \u2191(m + n)\n\ndef spec_assert_in_range (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value lower upper \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 m n : \u2115, m < rc_bound F \u2227 n < rc_bound F \u2227 value = lower + \u2191m \u2227 upper = value + \u2191(n + 1)\n\ndef spec_assert_le_250_bit (mem : F \u2192 F) (range_check_ptr a b ret0 : F) : Prop :=\n   \u2203 n : \u2115, n < rc_bound F * HIGH_PART_SHIFT + rc_bound F \u2227 b = a + \u2191n\n\ndef spec_split_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1_high \u03c1_low : F) : Prop :=\n\u2203 hi lo : \u2115, hi < rc_bound F \u2227 lo < rc_bound F  \u2227\n  \u03c1_high = \u2191hi \u2227 \u03c1_low = \u2191lo \u2227 value = \u2191(hi * CUTOFF + lo) \u2227 hi * CUTOFF + lo < PRIME\n\ndef spec_assert_le_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 m n : \u2115, m < PRIME \u2227 n < PRIME \u2227 a = \u2191m \u2227 b = \u2191n \u2227 m \u2264 n\n\ndef spec_assert_lt_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n    \u2203 m n : \u2115, m < PRIME \u2227 n < PRIME \u2227 a = \u2191m \u2227 b = \u2191n \u2227 m < n\n\ndef spec_abs_value (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  \u2203 n : \u2115, n < rc_bound F \u2227 \u03c1 = \u2191n \u2227 (\u03c1 = value \u2228 \u03c1 = -value)\n\ndef spec_sign (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  (value = 0 \u2227 \u03c1 = 0) \u2228\n  (value \u2260 0 \u2227 (\u2203 n : \u2115, n < rc_bound F \u2227 ((value = -\u2191n \u2227 \u03c1 = -1) \u2228 (value = \u2191n \u2227 \u03c1 = 1))))\n\ndef spec_unsigned_div_rem (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value div \u03c1_range_check_ptr \u03c1_q \u03c1_r : F) : Prop :=\n  \u2203 q r n : \u2115, q < rc_bound F \u2227 r < rc_bound F \u2227 n < rc_bound F \u2227\n    value = \u2191q * div + \u2191r \u2227 div = r + \u2191(n + 1) \u2227 \u03c1_q = q \u2227 \u03c1_r = r\n\ndef spec_signed_div_rem (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value div bound \u03c1_range_check_ptr \u03c1_q \u03c1_r : F) : Prop :=\n  \u2203 biased_q r n m : \u2115, \u2203 q: F, biased_q < rc_bound F \u2227 r < rc_bound F \u2227 n < rc_bound F \u2227 m < rc_bound F \u2227\n    q = \u2191biased_q - bound \u2227 value = q * div + \u2191r \u2227\n    div = r + \u2191(n + 1) \u2227 2 * bound = \u2191biased_q + \u2191(m + 1) \u2227\n    \u03c1_q = q \u2227 \u03c1_r = r\n\ndef spec_split_int (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value n base bound output \u03c1_range_check_ptr : F) : Prop :=\n  base \u2260 0 \u2192 \u2203 (nn : \u2115), nn < ring_char F \u2227 n = \u2191nn \u2227\n    (\u2200 i : \u2115, i < nn \u2192 \u2203 m : \u2115, mem (output + i) + \u2191m = bound - 1) \u2227\n      value = \u2211 i in (finset.range nn), (mem (output + i) * base ^ i)\n\ndef spec_sqrt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  \u2203 r n\u2081 n\u2082 n\u2083 : \u2115, r < rc_bound F \u2227 n\u2081 < rc_bound F \u2227 n\u2082 < rc_bound F \u2227\n    \u03c1 = \u2191r \u2227 value = \u03c1 ^ 2 + \u2191n\u2081 \u2227 (\u03c1 + 1) ^ 2 = value + \u2191(n\u2082 + 1) \u2227 2^250 = (\u03c1 + 1) ^ 2 + \u2191n\u2083\n\ndef spec_horner_eval (mem : F \u2192 F) (\u03ba : \u2115) (n_coefficients coefficients point \u03c1_res : F) : Prop :=\n  \u2203 (n : \u2115), n < ring_char F \u2227 n_coefficients = \u2191n \u2227\n    \u03c1_res = \u2211 i in (finset.range n), (mem (coefficients + i) * point ^ i)\n\n/-\n-- Function: assert_not_zero\n-/\n\n/- assert_not_zero autogenerated specification -/\n\ndef auto_spec_assert_not_zero_block3 (mem : F \u2192 F) (\u03ba : \u2115) (value : F) : Prop :=\n  1 \u2264 \u03ba\n\ndef auto_spec_assert_not_zero (mem : F \u2192 F) (\u03ba : \u2115) (value : F) : Prop :=\n  ((value = 0 \u2227\n    value = 1 \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_not_zero_block3 mem \u03ba\u2081 value \u2227\n    \u03ba\u2081 + 2 \u2264 \u03ba) \u2228\n   (value \u2260 0 \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_not_zero_block3 mem \u03ba\u2081 value \u2227\n    \u03ba\u2081 + 1 \u2264 \u03ba))\n\n/- assert_not_zero soundness theorem -/\n\ntheorem sound_assert_not_zero\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (value : F)\n    (h_auto : auto_spec_assert_not_zero mem \u03ba value) :\n  spec_assert_not_zero mem \u03ba value :=\nbegin\n  rcases h_auto with (\u27e8h0, h1\u27e9 | \u27e8h, _\u27e9),\n  { exfalso, rw h0 at h1, apply zero_ne_one h1.left },\n  use h\nend\n\n/-\n-- Function: assert_not_equal\n-/\n\n/- assert_not_equal autogenerated specification -/\n\ndef auto_spec_assert_not_equal_block3 (mem : F \u2192 F) (\u03ba : \u2115) (a b : F) : Prop :=\n  1 \u2264 \u03ba\n\ndef auto_spec_assert_not_equal (mem : F \u2192 F) (\u03ba : \u2115) (a b : F) : Prop :=\n  ((a = b \u2227\n    a = a + 1 \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_not_equal_block3 mem \u03ba\u2081 a b \u2227\n    \u03ba\u2081 + 3 \u2264 \u03ba) \u2228\n   (a \u2260 b \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_not_equal_block3 mem \u03ba\u2081 a b \u2227\n    \u03ba\u2081 + 2 \u2264 \u03ba))\n\n/- assert_not_equal soundness theorem -/\n\ntheorem sound_assert_not_equal\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (a b : F)\n    (h_auto : auto_spec_assert_not_equal mem \u03ba a b) :\n  spec_assert_not_equal mem \u03ba a b :=\nbegin\n  rcases h_auto with (\u27e8h0, h1\u27e9 | \u27e8h, _\u27e9),\n  { exfalso, apply @zero_ne_one F, rw [\u2190sub_eq_of_eq_add' h1.left, sub_self] },\n  use h\nend\n\n/-\n-- Function: assert_nn\n-/\n\n/- assert_nn autogenerated specification -/\n\ndef auto_spec_assert_nn (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a \u03c1_range_check_ptr : F) : Prop :=\n  a = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) (a) \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 1 \u2227\n  3 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2081\n\n/- assert_nn soundness theorem -/\n\ntheorem sound_assert_nn\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_nn mem \u03ba range_check_ptr a \u03c1_range_check_ptr) :\n  spec_assert_nn mem \u03ba range_check_ptr a \u03c1_range_check_ptr :=\nbegin\n  exact h_auto.2.1\nend\n\n/-\n-- Function: assert_le\n-/\n\n/- assert_le autogenerated specification -/\n\ndef auto_spec_assert_le (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_nn mem \u03ba\u2081 range_check_ptr (b - a) range_check_ptr\u2081 \u2227\n  \u03ba\u2081 + 4 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2081\n\n/- assert_le soundness theorem -/\n\ntheorem sound_assert_le\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a b \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_le mem \u03ba range_check_ptr a b \u03c1_range_check_ptr) :\n  spec_assert_le mem \u03ba range_check_ptr a b \u03c1_range_check_ptr :=\nbegin\n  rcases h_auto with \u27e8_, _, nn_b_sub_a, _\u27e9,\n  rcases nn_b_sub_a with \u27e8n, nlt, neq\u27e9,\n  exact \u27e8n, nlt, eq_add_of_sub_eq' neq\u27e9\nend\n\n/-\n-- Function: assert_lt\n-/\n\n/- assert_lt autogenerated specification -/\n\ndef auto_spec_assert_lt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr a (b - 1) range_check_ptr\u2081 \u2227\n  \u03ba\u2081 + 5 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2081\n\n/- assert_lt soundness theorem -/\n\ntheorem sound_assert_lt\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a b \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_lt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr) :\n  spec_assert_lt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr :=\nbegin\n  rcases h_auto with \u27e8_, _, nn_b_sub_a, _\u27e9,\n  rcases nn_b_sub_a with \u27e8n, nlt, neq\u27e9,\n  use [n, nlt],\n  rw [eq_add_of_sub_eq' neq, nat.cast_add, nat.cast_one, add_comm, add_assoc]\nend\n\n/-\n-- Function: assert_nn_le\n-/\n\n/- assert_nn_le autogenerated specification -/\n\ndef auto_spec_assert_nn_le (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_nn mem \u03ba\u2081 range_check_ptr a range_check_ptr\u2081 \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2082 range_check_ptr\u2081 a b range_check_ptr\u2082 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + 7 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2082\n\n/- assert_nn_le soundness theorem -/\n\ntheorem sound_assert_nn_le\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a b \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_nn_le mem \u03ba range_check_ptr a b \u03c1_range_check_ptr) :\n  spec_assert_nn_le mem \u03ba range_check_ptr a b \u03c1_range_check_ptr :=\nbegin\n  rcases h_auto with \u27e8_, _, nn_a, _, _, le_ab, _\u27e9,\n  rcases nn_a with \u27e8m, mlt, aeq\u27e9,\n  rcases le_ab with \u27e8n, nlt, abeq\u27e9,\n  use [m, n, mlt, nlt, aeq],\n  rw [abeq, aeq, nat.cast_add]\nend\n\n/-\n-- Function: assert_in_range\n-/\n\n/- assert_in_range autogenerated specification -/\n\ndef auto_spec_assert_in_range (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value lower upper \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr lower value range_check_ptr\u2081 \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2082 range_check_ptr\u2081 value (upper - 1) range_check_ptr\u2082 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + 8 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2082\n\n/- assert_in_range soundness theorem -/\n\ntheorem sound_assert_in_range\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value lower upper \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_in_range mem \u03ba range_check_ptr value lower upper \u03c1_range_check_ptr) :\n  spec_assert_in_range mem \u03ba range_check_ptr value lower upper \u03c1_range_check_ptr :=\nbegin\n  rcases h_auto with \u27e8_, _, lower_le, _, _, value_le, _\u27e9,\n  rcases lower_le with \u27e8m, mlt, eq1\u27e9,\n  rcases value_le with \u27e8n, nlt, eq2\u27e9,\n  use [m, n, mlt, nlt, eq1],\n  rw [eq_add_of_sub_eq' eq2, nat.cast_add, nat.cast_one, add_comm, add_assoc]\nend\n\n/-\n-- Function: assert_le_250_bit\n-/\n\n/- assert_le_250_bit autogenerated specification -/\n\ndef auto_spec_assert_le_250_bit (mem : F \u2192 F) (range_check_ptr a b ret0 : F) : Prop :=\n  \u2203 low : F, low = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) low \u2227\n  \u2203 high : F, high = mem (range_check_ptr + 1) \u2227\n  is_range_checked (rc_bound F) high \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 2 \u2227\n  \u2203 UPPER_BOUND : F, UPPER_BOUND = 1809251394333065553493296640760748560207343510400633813116524750123642650624 \u2227\n  \u2203 HIGH_PART_SHIFT : F, HIGH_PART_SHIFT = 5316911983139663491615228241121378304 \u2227\n  \u2203 diff : F, diff = b - a \u2227\n  diff = high * HIGH_PART_SHIFT + low \u2227\n  ret0 = range_check_ptr\u2081\n\n/- assert_le_250_bit soundness theorem -/\n\ntheorem sound_assert_le_250_bit\n    {mem : F \u2192 F}\n    (range_check_ptr a b ret0 : F)\n    (h_auto : auto_spec_assert_le_250_bit mem range_check_ptr a b ret0) :\n  spec_assert_le_250_bit mem range_check_ptr a b ret0 :=\nbegin\n  rcases h_auto with \u27e8low, _, rc_low, high, _, rc_high, _, _, _, _, H, rfl, diff, diffeq, diffeq2, _\u27e9,\n  rcases rc_high with \u27e8m, mlt, rfl\u27e9,\n  rcases rc_low with \u27e8n, nlt, rfl\u27e9,\n  use [m * HIGH_PART_SHIFT + n], split,\n  exact add_lt_add_of_le_of_lt (nat.mul_le_mul_right _ (le_of_lt mlt)) nlt,\n  rw [eq_add_of_sub_eq diffeq.symm, add_comm diff, diffeq2, HIGH_PART_SHIFT_eq],\n  rw [nat.cast_add, nat.cast_mul], simp only [nat.cast_bit0, nat.cast_one]\nend\n\n/-\n-- Function: assert_250_bit\n-/\n\n/- assert_250_bit autogenerated specification -/\n\ndef auto_spec_assert_250_bit (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 UPPER_BOUND : F, UPPER_BOUND = 1809251394333065553493296640760748560207343510400633813116524750123642650624 \u2227\n  \u2203 SHIFT : F, SHIFT = 340282366920938463463374607431768211456 \u2227\n  \u2203 HIGH_BOUND : F, HIGH_BOUND = 5316911983139663491615228241121378304 \u2227\n  \u2203 low : F, low = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) low \u2227\n  \u2203 high : F, high = mem (range_check_ptr + 1) \u2227\n  is_range_checked (rc_bound F) high \u2227\n  mem (range_check_ptr + 2) = HIGH_BOUND - 1 - high \u2227\n  is_range_checked (rc_bound F) (HIGH_BOUND - 1 - high) \u2227\n  value = high * SHIFT + low \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 3 \u2227\n  10 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2081\n\ndef spec_assert_250_bit (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr : F) : Prop :=\n  auto_spec_assert_250_bit mem \u03ba range_check_ptr value \u03c1_range_check_ptr\n\n/- assert_250_bit soundness theorem -/\n\ntheorem sound_assert_250_bit\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_250_bit mem \u03ba range_check_ptr value \u03c1_range_check_ptr) :\n  spec_assert_250_bit mem \u03ba range_check_ptr value \u03c1_range_check_ptr :=\nbegin\n  exact h_auto\nend\n\n/-\n-- Function: split_felt\n-/\n\n/- split_felt autogenerated specification -/\n\ndef auto_spec_split_felt_block11 (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value low high \u03c1_range_check_ptr \u03c1_high \u03c1_low : F) : Prop :=\n  3 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr \u2227\n  \u03c1_high = high \u2227\n  \u03c1_low = low\n\ndef auto_spec_split_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1_high \u03c1_low : F) : Prop :=\n  \u2203 MAX_HIGH : F, MAX_HIGH = 10633823966279327296825105735305134080 \u2227\n  \u2203 MAX_LOW : F, MAX_LOW = 0 \u2227\n  \u2203 low : F, low = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) low \u2227\n  \u2203 high : F, high = mem (range_check_ptr + 1) \u2227\n  is_range_checked (rc_bound F) high \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 2 \u2227\n  value = high * (2 ^ 128) + low \u2227\n  ((high = MAX_HIGH \u2227\n    \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr\u2081 low MAX_LOW range_check_ptr\u2082 \u2227\n    \u2203 (\u03ba\u2082 : \u2115), auto_spec_split_felt_block11 mem \u03ba\u2082 range_check_ptr\u2082 value low high \u03c1_range_check_ptr \u03c1_high \u03c1_low \u2227\n    \u03ba\u2081 + \u03ba\u2082 + 12 \u2264 \u03ba) \u2228\n   (high \u2260 MAX_HIGH \u2227\n    \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr\u2081 high (MAX_HIGH - 1) range_check_ptr\u2082 \u2227\n    \u2203 (\u03ba\u2082 : \u2115), auto_spec_split_felt_block11 mem \u03ba\u2082 range_check_ptr\u2082 value low high \u03c1_range_check_ptr \u03c1_high \u03c1_low \u2227\n    \u03ba\u2081 + \u03ba\u2082 + 11 \u2264 \u03ba))\n\n/- split_felt soundness theorem -/\n\nlemma split_felt_aux0 {hi lo : \u2115} (hhi : hi \u2264 MAX_HIGH) (hlo : lo \u2264 MAX_LOW) :\n  hi * CUTOFF + lo < PRIME :=\nbegin\n  apply (lt_of_le_of_lt (add_le_add (nat.mul_le_mul_right _ hhi) hlo)),\n  rw [MAX_HIGH, MAX_LOW, CUTOFF, add_comm, mul_comm, nat.mod_add_div, PRIME],\n  apply PRIME_sub_one_lt\nend\n\nlemma split_felt_aux1 {n : \u2115} (hn : n < rc_bound F) (h : (\u2191n : F) = MAX_HIGH) :\n  n = MAX_HIGH :=\nby { apply PRIME.nat_coe_field_inj _ _ h, { linarith [two_mul_rc_bound_lt_PRIME F] }, rw [MAX_HIGH_eq, PRIME]; norm_num1 }\n\nlemma split_felt_aux2 {m n : \u2115} (hm : m < rc_bound F) (hn : n < rc_bound F)\n    (h : (MAX_LOW : F) = m + n) :\n  m \u2264 MAX_LOW :=\nbegin\n  apply le_trans (nat.le_add_right _ n) (le_of_eq _),\n  rw \u2190nat.cast_add at h, symmetry,\n  apply PRIME.nat_coe_field_inj _ _ h,\n  { rw [MAX_LOW_eq, PRIME], norm_num1 },\n  linarith [hm, hn, two_mul_rc_bound_lt_PRIME F]\nend\n\nlemma split_felt_aux3 {hi lo : \u2115} (hhi : hi < MAX_HIGH) (hlo : lo < rc_bound F) :\n  hi * CUTOFF + lo < PRIME :=\nbegin\n  have : hi * CUTOFF + lo < (hi + 1) * CUTOFF,\n  { rw [add_mul, one_mul, CUTOFF], apply add_lt_add_left,\n    exact lt_of_lt_of_le hlo (rc_bound_hyp F) },\n  apply lt_of_lt_of_le this,\n  transitivity (MAX_HIGH * CUTOFF),\n  apply nat.mul_le_mul_right _ hhi,\n  transitivity,\n  apply nat.div_mul_le_self,\n  apply (le_of_lt PRIME_sub_one_lt)\nend\n\nlemma split_felt_aux4 {hi n : \u2115} (hhi : hi < rc_bound F) (hn : n < rc_bound F)\n    (h : (MAX_HIGH - 1 : F) = hi + n) :\n  hi < MAX_HIGH :=\nbegin\n  rw \u2190nat.cast_add at h,\n  apply nat.lt_of_le_of_lt _ (nat.pred_lt (show MAX_HIGH \u2260 0, by { rw MAX_HIGH_eq, norm_num })), rw \u2190nat.sub_one,\n  have : MAX_HIGH - 1 = hi + n,\n  { rw [\u2190nat.cast_one, \u2190nat.cast_sub] at h,\n    apply PRIME.nat_coe_field_inj _ _ h,\n    { rw [MAX_HIGH_eq, PRIME], norm_num1 },\n    linarith [hhi, hn, two_mul_rc_bound_lt_PRIME F], rw [MAX_HIGH_eq], norm_num },\n  rw this,\n  apply nat.le_add_right\nend\n\ntheorem sound_split_felt\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value \u03c1_range_check_ptr \u03c1_high \u03c1_low : F)\n    (h_auto : auto_spec_split_felt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_high \u03c1_low) :\n  spec_split_felt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_high \u03c1_low :=\nbegin\n  rcases h_auto with \u27e8H, rfl, L, rfl, low, _, rc_low, high, _, rc_high, _, _, val_eq, htemp\u27e9,\n  rcases rc_high with \u27e8hi, hilt, hieq\u27e9,\n  rcases rc_low with \u27e8lo, lolt, loeq\u27e9,\n  rcases htemp with (\u27e8heq, _, _, hle, _, \u27e8_, _, ret1eq, ret2eq\u27e9, _ \u27e9 | \u27e8hne, _, _, hle, _, \u27e8_, _, ret1eq, ret2eq\u27e9, _\u27e9),\n  { use [hi, lo, hilt, lolt, ret1eq.trans hieq, ret2eq.trans loeq],\n    split,\n    { rw [CUTOFF, val_eq, hieq, loeq], norm_cast },\n    rcases hle with \u27e8n, nlt, neq\u27e9,\n    apply split_felt_aux0,\n    apply le_of_eq, apply split_felt_aux1 hilt, rw [\u2190hieq, heq, MAX_HIGH_eq],\n    simp only [nat.cast_bit0, nat.cast_bit1, nat.cast_one],\n    apply split_felt_aux2 lolt nlt,\n    rw [MAX_LOW_eq, nat.cast_zero, neq, loeq] },\n  use [hi, lo, hilt, lolt, ret1eq.trans hieq, ret2eq.trans loeq],\n  split,\n  { rw [CUTOFF, val_eq, hieq, loeq], norm_cast },\n  rcases hle with \u27e8n, nlt, neq\u27e9,\n  apply split_felt_aux3 _ lolt,\n  apply split_felt_aux4 hilt nlt,\n  rw [\u2190hieq, \u2190neq, MAX_HIGH_eq], simp only [nat.cast_bit0, nat.cast_bit1, nat.cast_one]\nend\n\n/-\n-- Function: assert_le_felt\n-/\n\n/- assert_le_felt autogenerated specification -/\n\ndef auto_spec_assert_le_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 PRIME_OVER_3_HIGH : F, PRIME_OVER_3_HIGH = 3544607988759775765608368578435044694 \u2227\n  \u2203 PRIME_OVER_2_HIGH : F, PRIME_OVER_2_HIGH = 5316911983139663648412552867652567041 \u2227\n  \u2203 arc_short : F, arc_short = mem (range_check_ptr) + mem (range_check_ptr + 1) * PRIME_OVER_3_HIGH \u2227\n  is_range_checked (rc_bound F) (mem (range_check_ptr)) \u2227\n  is_range_checked (rc_bound F) (mem (range_check_ptr + 1)) \u2227\n  \u2203 arc_long : F, arc_long = mem (range_check_ptr + 2) + mem (range_check_ptr + 3) * PRIME_OVER_2_HIGH \u2227\n  is_range_checked (rc_bound F) (mem (range_check_ptr + 2)) \u2227\n  is_range_checked (rc_bound F) (mem (range_check_ptr + 3)) \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 4 \u2227\n  \u2203 arc_sum : F, arc_sum = arc_short + arc_long \u2227\n  \u2203 arc_prod : F, arc_prod = arc_short * arc_long \u2227\n  \u2203 anon_cond : F,\n  ((anon_cond = 0 \u2227\n    arc_sum = ((-1)) - a \u2227\n    arc_prod = (a - b) * (1 + b) \u2227\n    18 \u2264 \u03ba \u2227\n    \u03c1_range_check_ptr = range_check_ptr\u2081) \u2228\n   (anon_cond \u2260 0 \u2227\n    \u2203 anon_cond : F,\n    ((anon_cond = 0 \u2227\n      \u2203 m1mb : F, m1mb = ((-1)) - b \u2227\n      arc_sum = a + m1mb \u2227\n      arc_prod = a * m1mb \u2227\n      18 \u2264 \u03ba \u2227\n      \u03c1_range_check_ptr = range_check_ptr\u2081) \u2228\n     (anon_cond \u2260 0 \u2227\n      arc_sum = b \u2227\n      arc_prod = a * (b - a) \u2227\n      17 \u2264 \u03ba \u2227\n      \u03c1_range_check_ptr = range_check_ptr\u2081))))\n\n/- assert_le_felt soundness theorem -/\n\nlemma assert_le_felt_aux {hia loa hib lob : \u2115}\n    (hlt : hia < hib) (hloa : loa < rc_bound F) :\n  hia * CUTOFF + loa \u2264 hib * CUTOFF + lob :=\nhave hrc : rc_bound F \u2264 CUTOFF, by { rw [CUTOFF], apply rc_bound_hyp F},\ncalc\n  hia * CUTOFF + loa \u2264 (hia + 1) * CUTOFF :\n    by { rw [add_mul, one_mul], apply add_le_add_left, linarith }\n  ... \u2264 hib * CUTOFF : by { apply nat.mul_le_mul_right, apply hlt }\n  ... \u2264 hib * CUTOFF + lob : nat.le_add_right _ _\n\ntheorem eq_of_sum_product_eq {F : Type*} [field F] {a b c d: F}\n    (hsum : a + b = c + d) (hprod : a * b = c * d) :\n  (a = c \u2227 b = d) \u2228 (a = d \u2227 b = c) :=\nbegin\n  have : (a - c) * (a - d) = 0,\n  { transitivity (a * a - (c + d) * a + c * d),\n    { ring },\n    rw [\u2190hsum, \u2190hprod], ring },\n  rcases (eq_zero_or_eq_zero_of_mul_eq_zero this) with h | h,\n  { left,\n    rw sub_eq_zero at h,\n    use h,\n    rwa [h, add_left_cancel_iff] at hsum },\n  right,\n  rw sub_eq_zero at h,\n  use h,\n  rwa [h, add_comm, add_right_cancel_iff] at hsum\nend\n\ntheorem sound_assert_le_felt\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a b \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_le_felt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr) :\n  spec_assert_le_felt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr :=\nbegin\n  rcases h_auto with \u27e8P3H, P3Heq, P2H, P2Heq, arc_short, arc_short_eq, hrc, hrc1, arc_long, arc_long_eq, hrc2, hrc3, _, _,\n  arc_sum, arc_sum_eq, arc_prod, arc_prod_eq, _, hrest\u27e9,\n  rcases hrc with \u27e8n, nlt, neq\u27e9,\n  rcases hrc1 with \u27e8n1, n1lt, n1eq\u27e9,\n  rcases hrc2 with \u27e8n2, n2lt, n2eq\u27e9,\n  rcases hrc3 with \u27e8n3, n3lt, n3eq\u27e9,\n  have arc_short_eq' : arc_short = \u2191(n + n1 * assert_le_felt.PRIME_OVER_3_HIGH),\n  { rw [assert_le_felt.PRIME_OVER_3_HIGH],\n    simp only [nat.cast_add, nat.cast_mul, nat.cast_bit0, nat.cast_bit1, nat.cast_one, arc_short_eq, \u2190neq, \u2190n1eq, P3Heq] },\n  generalize n_arc_short_eq : n + n1 * assert_le_felt.PRIME_OVER_3_HIGH =\n    n_arc_short,\n  have coe_n_arc_short_eq : \u2191n_arc_short = arc_short,\n  { rw [\u2190n_arc_short_eq, arc_short_eq'] },\n  have n_arc_short_le : n_arc_short \u2264 2^128 * (1 + assert_le_felt.PRIME_OVER_3_HIGH),\n  { rw [mul_add, mul_one, \u2190n_arc_short_eq],\n    apply add_le_add (le_of_lt (lt_of_lt_of_le nlt(rc_bound_hyp F))),\n    apply nat.mul_le_mul_of_nonneg_right,\n    exact le_of_lt (lt_of_lt_of_le n1lt(rc_bound_hyp F)) },\n  have arc_long_eq' : arc_long = \u2191(n2 + n3 * assert_le_felt.PRIME_OVER_2_HIGH),\n  { rw [assert_le_felt.PRIME_OVER_2_HIGH],\n    simp only [nat.cast_add, nat.cast_mul, nat.cast_bit0, nat.cast_bit1, nat.cast_one, arc_long_eq, \u2190n2eq, \u2190n3eq, P2Heq] },\n  generalize n_arc_long_eq : n2 + n3 * assert_le_felt.PRIME_OVER_2_HIGH = n_arc_long,\n  have coe_n_arc_long_eq : \u2191n_arc_long = arc_long,\n  { rw [\u2190n_arc_long_eq, arc_long_eq'] },\n  have n_arc_long_le : n_arc_long \u2264 2^128 * (1 + assert_le_felt.PRIME_OVER_2_HIGH),\n  { rw [mul_add, mul_one, \u2190n_arc_long_eq],\n    apply add_le_add (le_of_lt (lt_of_lt_of_le n2lt(rc_bound_hyp F))),\n    apply nat.mul_le_mul_of_nonneg_right,\n    exact le_of_lt (lt_of_lt_of_le n3lt(rc_bound_hyp F)) },\n  have short_add_long_add_one_le_PRIME:\n    n_arc_short + (n_arc_long + 1) \u2264 PRIME,\n  { refine le_trans (add_le_add n_arc_short_le (add_le_add_right n_arc_long_le _)) _,\n    rw [assert_le_felt.PRIME_OVER_2_HIGH, assert_le_felt.PRIME_OVER_3_HIGH, PRIME],\n    norm_num1 },\n  have short_add_one_add_long_le_PRIME: (n_arc_short + 1) + n_arc_long \u2264 PRIME,\n  { rw [add_assoc, add_comm 1],\n    exact short_add_long_add_one_le_PRIME },\n  have long_add_one_le_PRIME: n_arc_long + 1 \u2264 PRIME :=\n    le_trans (nat.le_add_left _ _) short_add_long_add_one_le_PRIME,\n  have short_add_one_le_PRIME: n_arc_short + 1 \u2264 PRIME :=\n    le_trans (nat.le_add_right _ _) short_add_one_add_long_le_PRIME,\n  have long_le_PRIME: n_arc_long \u2264 PRIME :=\n    le_trans (nat.le_add_left _ _) short_add_one_add_long_le_PRIME,\n  have short_le_PRIME: n_arc_short \u2264 PRIME :=\n    le_trans (nat.le_add_right _ _) short_add_long_add_one_le_PRIME,\n  have long_lt_PRIME: n_arc_long < PRIME := nat.lt_of_succ_le long_add_one_le_PRIME,\n  have short_lt_PRIME: n_arc_short < PRIME := nat.lt_of_succ_le short_add_one_le_PRIME,\n  have short_add_long_lt_PRIME: n_arc_short + n_arc_long < PRIME :=\n    nat.lt_of_succ_le short_add_long_add_one_le_PRIME,\n  have coe_PRIME : (PRIME : F) = 0,\n  { apply char_p.cast_eq_zero },\n  rcases hrest with \u27e8_, hcase1\u27e9 | \u27e8_, \u27e8_, \u27e8_, hcase2\u27e9 | \u27e8_, hcase3\u27e9\u27e9\u27e9,\n  { rcases hcase1 with \u27e8arc_sum_eq', arc_prod_eq', _, _\u27e9,\n    have aux1 : b - a + (-1 - b) = arc_short + arc_long,\n    { rw [\u2190arc_sum_eq, arc_sum_eq'], ring },\n    have aux2 : (b - a) * (-1 - b) = arc_short * arc_long,\n    { rw [\u2190arc_prod_eq, arc_prod_eq'], ring },\n    rcases eq_of_sum_product_eq aux1 aux2 with \u27e8h1, h2\u27e9 | \u27e8h1, h2\u27e9,\n    { use PRIME - (n_arc_short + (n_arc_long + 1)),\n      use PRIME - (n_arc_long + 1),\n      split,\n      { apply nat.sub_lt PRIME_pos,\n        apply add_pos_of_nonneg_of_pos (nat.zero_le _) (nat.succ_pos _) },\n      split,\n      { apply nat.sub_lt PRIME_pos,\n        apply nat.succ_pos _ },\n      split,\n      { rw [nat.cast_sub], swap,\n        { exact short_add_long_add_one_le_PRIME },\n        rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_add, nat.cast_one, coe_n_arc_short_eq, coe_n_arc_long_eq],\n        linear_combination -aux1 },\n      split,\n      { rw [nat.cast_sub], swap,\n        { exact long_add_one_le_PRIME },\n        rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_one, coe_n_arc_long_eq, neg_add],\n        linear_combination -h2 },\n      apply nat.sub_le_sub_left,\n      apply nat.le_add_left },\n    use PRIME - ((n_arc_short + 1) + n_arc_long),\n    use PRIME - (n_arc_short + 1),\n    split,\n    { apply nat.sub_lt PRIME_pos,\n      apply add_pos_of_pos_of_nonneg (nat.succ_pos _) (nat.zero_le _) },\n    split,\n    { apply nat.sub_lt PRIME_pos,\n      apply nat.succ_pos _ },\n    split,\n    { rw [nat.cast_sub], swap,\n      { exact short_add_one_add_long_le_PRIME },\n      rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_add, nat.cast_one, coe_n_arc_short_eq, coe_n_arc_long_eq],\n      linear_combination -aux1 },\n    split,\n    { rw [nat.cast_sub], swap,\n      { exact short_add_one_le_PRIME },\n      rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_one, coe_n_arc_short_eq, neg_add],\n      linear_combination -h2 },\n    apply nat.sub_le_sub_left,\n    apply nat.le_add_right },\n  { rcases hcase2 with \u27e8_, rfl, arc_sum_eq', arc_prod_eq', _, _\u27e9,\n    have aux1 : a + (-1 - b) = arc_short + arc_long,\n    { rw [\u2190arc_sum_eq, arc_sum_eq'] },\n    have aux2 : a * (-1 - b) = arc_short * arc_long,\n    { rw [\u2190arc_prod_eq, arc_prod_eq'] },\n    rcases eq_of_sum_product_eq aux1 aux2 with \u27e8h1, h2\u27e9 | \u27e8h1, h2\u27e9,\n    { use n_arc_short,\n      use PRIME - (n_arc_long + 1),\n      split,\n      { exact short_lt_PRIME },\n      split,\n      { apply nat.sub_lt PRIME_pos (nat.succ_pos _) },\n      split,\n      { rw [h1, coe_n_arc_short_eq] },\n      split,\n      { rw [nat.cast_sub], swap,\n        { exact long_add_one_le_PRIME },\n        rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_one, coe_n_arc_long_eq],\n        linear_combination -h2 },\n      apply le_tsub_of_add_le_right,\n      exact short_add_long_add_one_le_PRIME },\n    use n_arc_long,\n    use PRIME - (n_arc_short + 1),\n    split,\n    { exact long_lt_PRIME },\n    split,\n    { apply nat.sub_lt PRIME_pos (nat.succ_pos _) },\n    split,\n    { rw [h1, coe_n_arc_long_eq] },\n    split,\n    { rw [nat.cast_sub], swap,\n      { exact short_add_one_le_PRIME },\n      rw [char_p.cast_eq_zero, zero_sub, nat.cast_add, nat.cast_one, coe_n_arc_short_eq],\n      linear_combination -h2 },\n    apply le_tsub_of_add_le_right,\n    rw add_comm,\n    exact short_add_one_add_long_le_PRIME },\n  { rcases hcase3 with \u27e8arc_sum_eq', arc_prod_eq', _, _\u27e9,\n    have aux1 : a + (b - a) = arc_short + arc_long,\n    { rw [\u2190arc_sum_eq, arc_sum_eq'], ring },\n    have aux2 : a * (b - a) = arc_short * arc_long,\n    { rw [\u2190arc_prod_eq, arc_prod_eq'] },\n    rcases eq_of_sum_product_eq aux1 aux2 with \u27e8h1, h2\u27e9 | \u27e8h1, h2\u27e9,\n    { use n_arc_short,\n      use n_arc_short + n_arc_long,\n      use short_lt_PRIME,\n      use short_add_long_lt_PRIME,\n      split, rw [h1, coe_n_arc_short_eq],\n      split,\n      { rw [nat.cast_add, coe_n_arc_short_eq, coe_n_arc_long_eq],\n        linear_combination h1 + h2 },\n    apply nat.le_add_right },\n    use n_arc_long,\n    use n_arc_short + n_arc_long,\n    use long_lt_PRIME,\n    use short_add_long_lt_PRIME,\n    split, rw [h1, coe_n_arc_long_eq],\n    split,\n    { rw [nat.cast_add, coe_n_arc_short_eq, coe_n_arc_long_eq],\n      linear_combination h1 + h2 },\n    apply nat.le_add_left }\nend\n\n/-\n-- Function: assert_lt_felt\n-/\n\n/- assert_lt_felt autogenerated specification -/\n\ndef auto_spec_assert_lt_felt_block3 (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_le_felt mem \u03ba\u2081 range_check_ptr a b range_check_ptr\u2081 \u2227\n  \u03ba\u2081 + 5 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2081\n\ndef auto_spec_assert_lt_felt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr a b \u03c1_range_check_ptr : F) : Prop :=\n  ((a = b \u2227\n    a = a + 1 \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_lt_felt_block3 mem \u03ba\u2081 range_check_ptr a b \u03c1_range_check_ptr \u2227\n    \u03ba\u2081 + 3 \u2264 \u03ba) \u2228\n   (a \u2260 b \u2227\n    \u2203 (\u03ba\u2081 : \u2115), auto_spec_assert_lt_felt_block3 mem \u03ba\u2081 range_check_ptr a b \u03c1_range_check_ptr \u2227\n    \u03ba\u2081 + 2 \u2264 \u03ba))\n\n/- assert_lt_felt soundness theorem -/\n\ntheorem sound_assert_lt_felt\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr a b \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_assert_lt_felt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr) :\n  spec_assert_lt_felt mem \u03ba range_check_ptr a b \u03c1_range_check_ptr :=\nbegin\n  cases h_auto with h_e h_ne,\n  rcases h_e with \u27e8_, h_a, _\u27e9,\n  { exfalso, apply @zero_ne_one F, rw [\u2190sub_eq_of_eq_add' h_a, sub_self] },\n  rcases h_ne with \u27e8h_a_ne_b, _, \u27e8_, _, \u27e8m, n, m_lt, n_lt, h_am, h_bn, h_le\u27e9, _, _\u27e9, _\u27e9,\n  use [m,n, m_lt, n_lt, h_am, h_bn], apply lt_of_le_of_ne h_le,\n  by_contra, rw [h, \u2190h_bn] at h_am, exact absurd h_am h_a_ne_b,\nend\n\n/-\n-- Function: abs_value\n-/\n\n/- abs_value autogenerated specification -/\n\ndef auto_spec_abs_value (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  \u2203 is_positive : F,\n  ((is_positive = 0 \u2227\n    \u2203 new_range_check_ptr : F, new_range_check_ptr = range_check_ptr + 1 \u2227\n    \u2203 abs_value : F, abs_value = value * ((-1)) \u2227\n    mem (range_check_ptr) = abs_value \u2227\n    is_range_checked (rc_bound F) (abs_value) \u2227\n    \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = new_range_check_ptr \u2227\n    6 \u2264 \u03ba \u2227\n    \u03c1_range_check_ptr = range_check_ptr\u2081 \u2227\n    \u03c1 = abs_value) \u2228\n   (is_positive \u2260 0 \u2227\n    mem (range_check_ptr) = value \u2227\n    is_range_checked (rc_bound F) (value) \u2227\n    \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 1 \u2227\n    6 \u2264 \u03ba \u2227\n    \u03c1_range_check_ptr = range_check_ptr\u2081 \u2227\n    \u03c1 = value))\n\n/- abs_value soundness theorem -/\n\ntheorem sound_abs_value\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F)\n    (h_auto : auto_spec_abs_value mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1) :\n  spec_abs_value mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1 :=\n-- STATEMENT CHANGED, PREVIOUS STATEMENT:\n--     (range_check_ptr value \u03c1_range_check_ptr \u03c1_abs_value : F)\n--     (h_auto : auto_spec_abs_value mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_abs_value) :\n--   spec_abs_value mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_abs_value :=\nbegin\n  rcases h_auto with \u27e8_, \u27e8_, _, _, av, aveq, _, rcav, _, _, _, _, ret1eq\u27e9 | \u27e8_, _, rcval, _, _, _, _, ret1eq\u27e9\u27e9,\n  { rcases rcav with \u27e8n, nlt, neq\u27e9,\n    use [n, nlt, ret1eq.trans neq], right,\n    rw [ret1eq, aveq, mul_neg_one] },\n  rcases rcval with \u27e8n, nlt, neq\u27e9,\n  use [n, nlt, ret1eq.trans neq, or.inl ret1eq]\nend\n\n/-\n-- Function: sign\n-/\n\n/- sign autogenerated specification -/\n\ndef auto_spec_sign (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  ((value = 0 \u2227\n    5 \u2264 \u03ba \u2227\n    \u03c1_range_check_ptr = range_check_ptr \u2227\n    \u03c1 = 0) \u2228\n   (value \u2260 0 \u2227\n    \u2203 is_positive : F,\n    ((is_positive = 0 \u2227\n      mem (range_check_ptr) = value * ((-1)) \u2227\n      is_range_checked (rc_bound F) (value * ((-1))) \u2227\n      \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 1 \u2227\n      8 \u2264 \u03ba \u2227\n      \u03c1_range_check_ptr = range_check_ptr\u2081 \u2227\n      \u03c1 = (-1)) \u2228\n     (is_positive \u2260 0 \u2227\n      mem (range_check_ptr) = value \u2227\n      is_range_checked (rc_bound F) (value) \u2227\n      \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 1 \u2227\n      8 \u2264 \u03ba \u2227\n      \u03c1_range_check_ptr = range_check_ptr\u2081 \u2227\n      \u03c1 = 1))))\n\n/- sign soundness theorem -/\n\ntheorem sound_sign\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F)\n    (h_auto : auto_spec_sign mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1) :\n  spec_sign mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1 :=\n-- STATEMENT CHANGED, PREVIOUS STATEMENT:\n--     (range_check_ptr value \u03c1_range_check_ptr \u03c1_sign : F)\n--     (h_auto : auto_spec_sign mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_sign) :\n--   spec_sign mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_sign :=\nbegin\n  rcases h_auto with (\u27e8valeq, _, _, ret1eq\u27e9 |\n    \u27e8valne, _, \u27e8_, _, rc_valneg, _, _, _, _, ret1eq\u27e9 |  \u27e8_, _, rc_val, _, _, _, _, ret1eq\u27e9\u27e9),\n  { left, use [valeq, ret1eq] },\n  { rcases rc_valneg with \u27e8n, nlt, neq\u27e9, rw mul_neg_one at neq,\n    right, use [valne, n, nlt, eq_neg_of_eq_neg neq.symm, ret1eq] },\n  rcases rc_val with \u27e8n, nlt, neq\u27e9,\n  right, use [valne, n, nlt], right, use [neq, ret1eq]\nend\n\n/-\n-- Function: unsigned_div_rem\n-/\n\n/- unsigned_div_rem autogenerated specification -/\n\ndef auto_spec_unsigned_div_rem (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value div \u03c1_range_check_ptr \u03c1_q \u03c1_r : F) : Prop :=\n  \u2203 r : F, r = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) r \u2227\n  \u2203 q : F, q = mem (range_check_ptr + 1) \u2227\n  is_range_checked (rc_bound F) q \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 2 \u2227\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr\u2081 r (div - 1) range_check_ptr\u2082 \u2227\n  value = q * div + r \u2227\n  \u03ba\u2081 + 12 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2082 \u2227\n  \u03c1_q = q \u2227\n  \u03c1_r = r\n\n/- unsigned_div_rem soundness theorem -/\n\ntheorem sound_unsigned_div_rem\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value div \u03c1_range_check_ptr \u03c1_q \u03c1_r : F)\n    (h_auto : auto_spec_unsigned_div_rem mem \u03ba range_check_ptr value div \u03c1_range_check_ptr \u03c1_q \u03c1_r) :\n  spec_unsigned_div_rem mem \u03ba range_check_ptr value div \u03c1_range_check_ptr \u03c1_q \u03c1_r :=\nbegin\n  rcases h_auto with \u27e8fr, _, \u27e8r, rlt, req\u27e9, fq, _, \u27e8q, qlt, qeq\u27e9, _, _, _, _, \u27e8n, nlt, neq\u27e9, valeq, _, _,\n    ret1eq, ret2eq\u27e9,\n  use [q, r, n, qlt, rlt, nlt], split, { rw [valeq, qeq, req] },\n  split, { rw [eq_add_of_sub_eq neq, req, nat.cast_add, nat.cast_one, add_assoc] },\n  use [ret1eq.trans qeq, ret2eq.trans req]\nend\n\n/-\n-- Function: signed_div_rem\n-/\n\n/- signed_div_rem autogenerated specification -/\n\ndef auto_spec_signed_div_rem (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value div bound \u03c1_range_check_ptr \u03c1_q \u03c1_r : F) : Prop :=\n  \u2203 r : F, r = mem (range_check_ptr) \u2227\n  is_range_checked (rc_bound F) r \u2227\n  \u2203 biased_q : F, biased_q = mem (range_check_ptr + 1) \u2227\n  is_range_checked (rc_bound F) biased_q \u2227\n  \u2203 range_check_ptr\u2081 : F, range_check_ptr\u2081 = range_check_ptr + 2 \u2227\n  \u2203 q : F, q = biased_q - bound \u2227\n  value = q * div + r \u2227\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2082 : F), spec_assert_le mem \u03ba\u2081 range_check_ptr\u2081 r (div - 1) range_check_ptr\u2082 \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (range_check_ptr\u2083 : F), spec_assert_le mem \u03ba\u2082 range_check_ptr\u2082 biased_q (2 * bound - 1) range_check_ptr\u2083 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + 20 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2083 \u2227\n  \u03c1_q = q \u2227\n  \u03c1_r = r\n\n/- signed_div_rem soundness theorem -/\n\ntheorem sound_signed_div_rem\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value div bound \u03c1_range_check_ptr \u03c1_q \u03c1_r : F)\n    (h_auto : auto_spec_signed_div_rem mem \u03ba range_check_ptr value div bound \u03c1_range_check_ptr \u03c1_q \u03c1_r) :\n  spec_signed_div_rem mem \u03ba range_check_ptr value div bound \u03c1_range_check_ptr \u03c1_q \u03c1_r :=\nbegin\n  rcases h_auto with \u27e8fr, _, \u27e8r, rlt, req\u27e9, fbq, _, \u27e8bq, bqlt, bqeq\u27e9, _, _,\n                        q, qeq, valeq, _, _, \u27e8n, nlt, neq\u27e9, _, _, \u27e8m, mlt, meq\u27e9, _, _, ret1eq, ret2eq\u27e9,\n  use [bq, r, n, m, q, bqlt, rlt, nlt, mlt],\n  split, { rw [qeq, bqeq] },\n  split, { rw [valeq, req] },\n  split, { rw [eq_add_of_sub_eq neq, req, nat.cast_add, nat.cast_one, add_assoc] },\n  split, { rw [eq_add_of_sub_eq meq, bqeq, nat.cast_add, nat.cast_one, add_assoc] },\n  use [ret1eq, ret2eq.trans req]\nend\n\n/-\n-- Function: split_int\n-/\n\n/- split_int autogenerated specification -/\n\ndef auto_spec_split_int (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value n base bound output \u03c1_range_check_ptr : F) : Prop :=\n  ((n = 0 \u2227\n    value = 0 \u2227\n    4 \u2264 \u03ba \u2227\n    \u03c1_range_check_ptr = range_check_ptr) \u2228\n   (n \u2260 0 \u2227\n    \u2203 low_part : F, low_part = mem (output) \u2227\n    \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_nn_le mem \u03ba\u2081 range_check_ptr low_part (bound - 1) range_check_ptr\u2081 \u2227\n    \u2203 \u03b46_0 : F,\n    \u2203 (\u03ba\u2082 : \u2115), spec_split_int mem \u03ba\u2082 range_check_ptr\u2081 (ddiv (value - low_part) base \u03b46_0) (n - 1) base bound (output + 1) \u03c1_range_check_ptr \u2227\n    \u03ba\u2081 + \u03ba\u2082 + 15 \u2264 \u03ba))\n\n/- split_int soundness theorem -/\n\ntheorem sound_split_int\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value n base bound output \u03c1_range_check_ptr : F)\n    (h_auto : auto_spec_split_int mem \u03ba range_check_ptr value n base bound output \u03c1_range_check_ptr) :\n  spec_split_int mem \u03ba range_check_ptr value n base bound output \u03c1_range_check_ptr :=\nbegin\n  intro base_nzero,\n  cases h_auto,\n  rcases h_auto with \u27e8neq0, h_veq0, _\u27e9,\n  use [0], split, exact PRIME.char_pos, split,\n  rw neq0, exact nat.cast_zero.symm, split,\n  rintro i h_ilt0, exact absurd h_ilt0 (nat.not_lt_zero i),\n  rw h_veq0, exact (finset.sum_range_zero _).symm,\n  rcases h_auto with \u27e8nzero, low_part, low_part_eq, _, _, h_nn_le, \u27e8d6, _, i_h, _\u27e9\u27e9,\n  rcases i_h base_nzero with \u27e8i_nn, i_nn_lt, i_nn_eq, i_bound, i_sum\u27e9,\n  have : i_nn + 1 = ring_char F \u2228 i_nn + 1 < ring_char F :=\n    or.comm.mp (lt_or_eq_of_le (nat.succ_le_of_lt i_nn_lt)),\n  cases this with h_eq_ring_char h_lt_ring_char,\n  { exfalso, apply nzero,\n    rw [eq_add_of_sub_eq i_nn_eq, \u2190nat.cast_one, \u2190nat.cast_add, h_eq_ring_char, (ring_char.spec F _)] },\n  use [i_nn + 1, h_lt_ring_char], split,\n  { rw [eq_add_of_sub_eq i_nn_eq, \u2190nat.cast_one, \u2190nat.cast_add] },\n  split, rintro i ilt,\n  cases nat.eq_zero_or_pos i with izero zero_lt_i,\n  { rw [izero, nat.cast_zero, add_zero, \u2190low_part_eq],\n    rcases h_nn_le with \u27e8am, an, h_am_bound, h_an_bound, h_a_eq, h_b_eq\u27e9,\n    use [an], rw [h_b_eq, h_a_eq, nat.cast_add] },\n  {\n    have hi : i - 1 < i_nn,\n      { rw [nat.sub_one, \u2190(nat.pred_succ i_nn)],\n        exact nat.pred_lt_pred (ne_of_lt zero_lt_i).symm ilt, },\n    rcases i_bound (i - 1) hi with \u27e8m, h_m\u27e9,\n    -- have := nat.cast_sub zero_lt_i,\n    use m, rw [\u2190h_m, nat.cast_sub zero_lt_i, nat.cast_one],\n    rw [add_add_sub_cancel output i 1]\n  },\n  {\n    rw [add_comm i_nn 1, finset.sum_range_add _ _ _],\n    simp [pow_add base 1 _, mul_comm base _],\n    simp [\u2190add_assoc, \u2190mul_assoc],\n    rw [ddiv_eq base_nzero, div_eq_iff base_nzero, finset.sum_mul] at i_sum,\n    rw [\u2190i_sum], rw [low_part_eq], exact (add_sub_cancel'_right _ _).symm,\n  },\nend\n\n/-\n-- Function: sqrt\n-/\n\n/- sqrt autogenerated specification -/\n\ndef auto_spec_sqrt (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F) : Prop :=\n  \u2203 root : F,\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_nn_le mem \u03ba\u2081 range_check_ptr root (2 ^ 125 - 1) range_check_ptr\u2081 \u2227\n  \u2203 root_plus_one : F, root_plus_one = root + 1 \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (range_check_ptr\u2082 : F), spec_assert_in_range mem \u03ba\u2082 range_check_ptr\u2081 value (root * root) (root_plus_one * root_plus_one) range_check_ptr\u2082 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + 13 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2082 \u2227\n  \u03c1 = root\n\n/- sqrt soundness theorem -/\n\ntheorem sound_sqrt\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr value \u03c1_range_check_ptr \u03c1 : F)\n    (h_auto : auto_spec_sqrt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1) :\n  spec_sqrt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1 :=\n-- STATEMENT CHANGED, PREVIOUS STATEMENT:\n--     (range_check_ptr value \u03c1_range_check_ptr \u03c1_res : F)\n--     (h_auto : auto_spec_sqrt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_res) :\n--   spec_sqrt mem \u03ba range_check_ptr value \u03c1_range_check_ptr \u03c1_res :=\nbegin\n  rcases h_auto with  \u27e8root, _, _, h_nn_le_r, rp1, h_rp1, _, _, h_in_range, _, _, ret1eq\u27e9,\n  rcases h_in_range with \u27e8n\u2081, n\u2082, h_n\u2081, h_n\u2082, hrsqr, hr1sqr\u27e9,\n  rcases h_nn_le_r with \u27e8r, m, h_r, h_m, h_reqroot, h_rb\u27e9,\n  use [r, n\u2081, n\u2082, (2 * (r + 1) * m + m ^ 2), h_r, h_n\u2081, h_n\u2082],\n  rw [ret1eq, \u2190h_rp1],\n  use [h_reqroot],\n  split, { rw [pow_two root], exact hrsqr },\n  split, { rw [pow_two rp1], exact hr1sqr },\n  rw [h_rp1, h_reqroot],\n  have h_r1b : (2 ^ 125 : F) = (\u2191r + 1) + \u2191m,\n    { rw [eq_add_of_sub_eq h_rb, nat.cast_add], ring },\n  have h_125 : (2 ^ 250 : F) = (2 ^ 125 : F) ^ 2, { ring },\n  rw [h_125, h_r1b], rw [add_sq _ \u2191m], norm_cast, arith_simps,\nend\n\n/-\n-- Function: horner_eval\n-/\n\n/- horner_eval autogenerated specification -/\n\ndef auto_spec_horner_eval (mem : F \u2192 F) (\u03ba : \u2115) (n_coefficients coefficients point \u03c1_res : F) : Prop :=\n  ((n_coefficients = 0 \u2227\n    3 \u2264 \u03ba \u2227\n    \u03c1_res = 0) \u2228\n   (n_coefficients \u2260 0 \u2227\n    \u2203 (\u03ba\u2081 : \u2115) (n_minus_one_res : F), spec_horner_eval mem \u03ba\u2081 (n_coefficients - 1) (coefficients + 1) point n_minus_one_res \u2227\n    \u03ba\u2081 + 9 \u2264 \u03ba \u2227\n    \u03c1_res = n_minus_one_res * point + mem (coefficients + 0)))\n\n/- horner_eval soundness theorem -/\n\ntheorem sound_horner_eval\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (n_coefficients coefficients point \u03c1_res : F)\n    (h_auto : auto_spec_horner_eval mem \u03ba n_coefficients coefficients point \u03c1_res) :\n  spec_horner_eval mem \u03ba n_coefficients coefficients point \u03c1_res :=\nbegin\n  cases h_auto,\n  -- zero case\n  rcases h_auto with \u27e8neq0, _, h_veq0\u27e9,\n  use [0], split, exact PRIME.char_pos, split,\n  rw neq0, exact nat.cast_zero.symm,\n  rw h_veq0, exact (finset.sum_range_zero _).symm,\n  -- non-zero case\n  rcases h_auto with \u27e8nzero, _, n_min_1_res, i_h, _, h_ret0\u27e9,\n  rcases i_h with \u27e8i_n, i_n_lt, i_n_eq, i_sum\u27e9,\n  have : i_n + 1 = ring_char F \u2228 i_n + 1 < ring_char F :=\n    or.comm.mp (lt_or_eq_of_le (nat.succ_le_of_lt i_n_lt)),\n  cases this with h_eq_ring_char h_lt_ring_char,\n  { exfalso, apply nzero,\n    rw [eq_add_of_sub_eq i_n_eq, \u2190nat.cast_one, \u2190nat.cast_add, h_eq_ring_char, (ring_char.spec F _)] },\n  use [i_n + 1, h_lt_ring_char], split,\n  { rw [eq_add_of_sub_eq i_n_eq, \u2190nat.cast_one, \u2190nat.cast_add] },\n  {\n    rw [add_comm i_n 1, finset.sum_range_add _ _ _],\n    simp [pow_add point 1 _, mul_comm point _],\n    simp [\u2190add_assoc, \u2190mul_assoc],\n    rw [h_ret0, add_comm, add_zero, add_left_cancel_iff, i_sum, finset.sum_mul]\n  },\nend\n\n\n/-\n-- Function: is_quad_residue\n-/\n\n/- is_quad_residue autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_is_quad_residue (mem : F \u2192 F) (\u03ba : \u2115) (x \u03c1 : F) : Prop :=\n  \u2203 y : F,\n  \u2203 y_squared : F, y_squared = y * y \u2227\n  ((y_squared = x \u2227\n    7 \u2264 \u03ba \u2227\n    \u03c1 = TRUE) \u2228\n   (y_squared \u2260 x \u2227\n    3 * y_squared = x \u2227\n    8 \u2264 \u03ba \u2227\n    \u03c1 = FALSE))\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_is_quad_residue (mem : F \u2192 F) (\u03ba : \u2115) (x \u03c1 : F) : Prop :=\n  auto_spec_is_quad_residue mem \u03ba x \u03c1\n\n/- is_quad_residue soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_is_quad_residue\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (x \u03c1 : F)\n    (h_auto : auto_spec_is_quad_residue mem \u03ba x \u03c1) :\n  spec_is_quad_residue mem \u03ba x \u03c1 :=\nbegin\n  exact h_auto\nend\n\nend starkware.cairo.common.math\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/math_spec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.39766803361410674}}
{"text": "namespace hidden\n\nclass mygroup (\u03b1 : Type)\nextends has_mul \u03b1, has_inv \u03b1 :=\n(e : \u03b1)\n(mul_assoc (a b c : \u03b1) : a * b * c = a * (b * c))\n(mul_id (a : \u03b1) : a * e = a)\n(mul_inv (a : \u03b1) : a * a\u207b\u00b9 = e)\n\nnamespace mygroup\n\nvariables {\u03b1 : Type} [mygroup \u03b1]\n\nvariables {a b c : \u03b1}\n-- This sucks\n\ntheorem mul_by_right : a = b \u2192 a * c = b * c :=\nbegin\n  assume h,\n  congr,\n  assumption,\nend\n\ntheorem mul_cancel_right (c : \u03b1) : a * c = b * c \u2192 a = b :=\nbegin\n  assume h,\n  have := congr_arg (\u03bb d, d * c\u207b\u00b9) h,\n  dsimp only [] at this,\n  repeat { rwa [mul_assoc, mul_inv, mul_id] at this },\nend\n\ntheorem mul_right (c : \u03b1) : a = b \u2194 a * c = b * c :=\n\u27e8mul_by_right, mul_cancel_right c\u27e9\n\ntheorem mul_by_left : a = b \u2192 c * a = c * b :=\nbegin\n  assume h,\n  congr,\n  assumption,\nend\n\ntheorem inv_mul : a\u207b\u00b9 * a = e :=\nbegin\n  -- This is actually a really hard theorem\n  rw [\u2190mul_inv a\u207b\u00b9, \u2190mul_id (a\u207b\u00b9 * a), \u2190mul_inv a\u207b\u00b9, \u2190mul_assoc],\n  apply mul_by_right,\n  rw [mul_assoc, mul_inv, mul_id],\nend\n\ntheorem id_mul (a : \u03b1) : e * a = a :=\nby rw [\u2190mul_inv a, mul_assoc, inv_mul, mul_id]\n\ntheorem mul_cancel_left : c * a = c * b \u2192 a = b :=\nbegin\n  assume h,\n  have := congr_arg (\u03bb d, c\u207b\u00b9 * d) h,\n  dsimp only [] at this,\n  repeat { rwa [\u2190mul_assoc, inv_mul, id_mul] at this },\nend\n\ntheorem mul_left (c : \u03b1) : a = b \u2194 c * a = c * b :=\n\u27e8mul_by_left, mul_cancel_left\u27e9\n\ntheorem id_unique : a * b = a \u2194 b = e :=\nbegin\n  split; assume h,\n    rwa [mul_left a, mul_id],\n  subst h,\n  from mul_id a,\nend\n\ntheorem inv_unique : a * b = e \u2194 b = a\u207b\u00b9 :=\nbegin\n  split; assume h,\n    rwa [mul_left a, mul_inv],\n  subst h,\n  from mul_inv a,\nend\n\nend mygroup\n\nend hidden\n", "meta": {"author": "Sterrs", "repo": "leaning", "sha": "3901cc953694b33adda86cb88ca30ba99594db31", "save_path": "github-repos/lean/Sterrs-leaning", "path": "github-repos/lean/Sterrs-leaning/leaning-3901cc953694b33adda86cb88ca30ba99594db31/src/principia/mygroup/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3976584546950499}}
{"text": "import sli.sli\nnamespace rmd_search\nopen sli.toTR\n\ndef EmptinessChecker (C \u03b1 : Type) := TR C \u2192 (C \u2192 \u03b1) \u2192 list C\n\n\n-- The counter example should not contain the start state\n-- def search_breakpoint (C \u03b1 : Type) (o : TR C) (reducer : C \u2192 \u03b1)  : list C := \n  --under-approximating dfs/bfs here\n  -- sorry\n\n-- def search_breakpoint(o: TR C) (reduce: C \u2192 \u03b1): list C :=\n--     k = {}\n--     wt = ()\n--     for s \u2208 o.initial do\n--         if dfs o reduce s k wt then return wt end if\n--     end for\n--     return wt\n\n-- def dfs (o: TR C) (reduce: C \u2192 \u03b1) \n-- (s: C) (k: set C) (wt: list C): bool :=\n--     if o.accepting s then \n--         wt = wt.append(s)\n--         return true\n--     end if\n--     k = k \u222a { s }\n--     for t \u2208 o.next s do\n--         if (reduce t) \u2209 k then\n--             if dfs o red t k wt then\n--                 wt = wt.append(s)\n--                 return true \n--             end if\n--         end if\n--     end for\n--     return false\n-- end \n\nend rmd_search", "meta": {"author": "teodorov", "repo": "temporal-multiverse-debugging", "sha": "e8f2e7037acee87f0c8620b60073a7619eab7f82", "save_path": "github-repos/lean/teodorov-temporal-multiverse-debugging", "path": "github-repos/lean/teodorov-temporal-multiverse-debugging/temporal-multiverse-debugging-e8f2e7037acee87f0c8620b60073a7619eab7f82/src/debugging/rmd_search.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.39765845469504985}}
{"text": "/-\nCopyright (c) 2021 OpenAI. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f\n-/\nimport mathzoo.imports.miniF2F\n\nopen_locale nat rat real big_operators topological_space\n\ntheorem algebra_binomnegdiscrineq_10alt28asqp1\n  (a : \u211d) :\n  10 * a \u2264 28 * a^2 + 1 :=\nbegin\n  refine le_of_not_gt _,\n  by_cases h1 : (1:\u211d) < a * (bit1 (bit0 (a * 2 + (-(a * 2) + 1)))),\n  apply not_lt_of_ge,\n  all_goals { simp * at * },\n  all_goals { nlinarith },\nend", "meta": {"author": "leanprover-community", "repo": "mathzoo", "sha": "87e9b492daeb929838706942aaa2437621b34a0e", "save_path": "github-repos/lean/leanprover-community-mathzoo", "path": "github-repos/lean/leanprover-community-mathzoo/mathzoo-87e9b492daeb929838706942aaa2437621b34a0e/src/mathzoo/misc/miniF2F/algebra/binomnegdiscrineq_10alt28asqp1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3976584546950498}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.simulation_semantics.constructions.logging.random_oracle\nimport computational_monads.simulation_semantics.constructions.identity_oracle\nimport computational_monads.simulation_semantics.oracle_append\nimport computational_monads.asymptotics.polynomial_time\nimport computational_monads.asymptotics.negligable\nimport computational_monads.asymptotics.queries_at_most\n\n/-!\n# Cryptographic Signature Schemes\n\nThis file defines signature algorithms and security properties for them.\nSignature algorithms are defined by a structure containing the relevant types,\nand algorithms with inputs and outputs corresponding to the provided types.\nA signature scheme is then defined to be a set of signatures indexed by a security parameter\n\nCompleteness is defined to be the property that any result of gen and sign passes verify.\nNote that this doesn't allow for negligable failure, as some literature does.\n\nUnforgeable is defined to be the property that any adversary with access to a signing oracle\ncannot forge a valid message/signature pair with more than negligable advantage.\n\nNote that the schemes assume algorithms have access to a shared random oracle.\nSignature schemes that don't need this can provide the empty spec `[]\u2092`,\n  which has no way to actually be queried\n-/\n\nopen_locale ennreal nnreal\nopen oracle_comp oracle_spec\n\n/-- Signature on messages `M`, public and secret keys `PK` and `SK`, signatures of type `S`.\n  We model the algorithms as having access to a uniform selection oracle,\n    and a set of random oracles that the algorithm has access to.\n  If not in the random oracle model, can just take `random_oracles := []\u2092`, the empty `oracle_spec`\n  We also bundle the polynomial complexity of the algorithms into the structure. -/\nstructure signature :=\n-- Types of the possible messages, public keys, secret keys, and signatures\n(M PK SK S : Type)\n-- Equality between messages and between signatures is decidable (unneeded for `PK` and `SK`)\n(decidable_eq_M : decidable_eq M)\n(decidable_eq_S : decidable_eq S)\n-- There exists at least one signature (in particular we can define a signing oracle)\n(inhabited_S : inhabited S)\n-- There are a finite number of possible signatures\n(fintype_S : fintype S)\n-- Random oracles for the algorithms, with finite ranges and computablity requirements.\n(random_oracle_spec : oracle_spec)\n-- The actual algorithms of the signature scheme.\n(gen : unit \u2192 oracle_comp (uniform_selecting ++ random_oracle_spec) (PK \u00d7 SK))\n(sign : PK \u00d7 SK \u00d7 M \u2192 oracle_comp (uniform_selecting ++ random_oracle_spec) S)\n(verify : PK \u00d7 M \u00d7 S \u2192 oracle_comp (uniform_selecting ++ random_oracle_spec) bool)\n-- Requirement that all the algorithms have polynomial time complexity.\n-- (gen_poly_time : poly_time_oracle_comp gen)\n-- (sign_poly_time : poly_time_oracle_comp sign)\n-- (verify_poly_time : poly_time_oracle_comp verify)\n\nnamespace signature\n\nvariable (sig : signature)\n\nsection oracle_instances\n\ninstance decidable_eq_S' : decidable_eq sig.S := sig.decidable_eq_S\n\ninstance decidable_eq_M' : decidable_eq sig.M := sig.decidable_eq_M\n\ninstance inhabited_S' : inhabited sig.S := sig.inhabited_S\n\ninstance fintype_S' : fintype sig.S := sig.fintype_S\n\nend oracle_instances\n\nsection oracle_spec\n\n/-- Shorthand for the combination of the `uniform_selecting` oracle and the `random_oracles`,\n  i.e. the oracles available to the signature algorithms themselves -/\n@[reducible, inline]\ndef base_oracle_spec (sig : signature) : oracle_spec := uniform_selecting ++ sig.random_oracle_spec\n\n/-- Simulate the basic oracles for the signature, using `random_oracle` to simulate the\nrandom oracle and preserving the `uniform_selecting` oracle as is. -/\nnoncomputable def base_oracle (sig : signature) :\n  sim_oracle sig.base_oracle_spec uniform_selecting (query_log sig.random_oracle_spec) :=\nsim_oracle.mask_state (id\u209b ++\u209b random_oracle sig.random_oracle_spec)\n  (equiv.punit_prod (query_log sig.random_oracle_spec))\n\n/-- A signing oracle corresponding to a given signature scheme -/\n@[reducible, inline]\ndef signing_oracle_spec (sig : signature) [inhabited sig.S] : oracle_spec := sig.M \u21a6\u2092 sig.S\n\n/-- Simulate a computation with access to a `signing_oracle_spec` to one with `base_oracle_spec`,\n  using the provided public/secret keys to answer queries for signatures.\nAdditionally it logs and returns a list queries to the signing oracle -/\ndef signing_oracle (sig : signature) (pk : sig.PK) (sk : sig.SK) :\n  sim_oracle sig.signing_oracle_spec sig.base_oracle_spec (query_log (sig.M \u21a6\u2092 sig.S)) :=\nsim_oracle.mask_state (\u27ea\u03bb _ m, sig.sign (pk, sk, m)\u27eb \u2218\u209b (logging_oracle (sig.M \u21a6\u2092 sig.S)))\n  (equiv.prod_punit (query_log (signing_oracle_spec sig)))\n\nend oracle_spec\n\nsection complete\n\n/-- Generate a key, sign on the given message, and return the result of verify on the signature.\n  Random oracles have a shared cache for the entire computation,\n  and the uniform selection oracle just forwards its query on. -/\nnoncomputable def completeness_experiment (sig : signature) (m : sig.M) :\n  oracle_comp uniform_selecting bool :=\ndefault_simulate' sig.base_oracle\n(do { (pk, sk) \u2190 sig.gen (),\n      \u03c3 \u2190 sig.sign (pk, sk, m),\n      sig.verify (pk, m, \u03c3) })\n\nlemma completeness_experiment.def (m : sig.M) : sig.completeness_experiment m = default_simulate'\n  sig.base_oracle (do {k \u2190 sig.gen (), \u03c3 \u2190 sig.sign (k.1, k.2, m), sig.verify (k.1, m, \u03c3)}) :=\nbegin\n  sorry\nend\n\n@[simp] lemma support_completeness_experiment (m : sig.M) :\n  (completeness_experiment sig m).support = \u22c3 (pk : sig.PK) (sk : sig.SK) (\u03c3 : sig.S)\n    (cache cache' : query_log sig.random_oracle_spec)\n    (hk : ((pk, sk), cache) \u2208 (default_simulate sig.base_oracle $ sig.gen ()).support)\n    (h\u03c3 : (\u03c3, cache') \u2208 (simulate sig.base_oracle (sig.sign (pk, sk, m)) cache).support),\n      (simulate' sig.base_oracle (sig.verify (pk, m, \u03c3)) cache').support :=\nbegin\n  ext x,\n  simp only [completeness_experiment.def, default_simulate',\n    support_simulate'_bind, set.mem_Union],\n  sorry,\nend\n\n/-- Signature is complete if for any possible message, the generated signature is valid,\n  i.e. the output of `sign` always returns true when `verify` is called.\n  note that this definition doesn't allow for negligable failure of signing -/\ndef complete (sig : signature) := \u2200 (m : sig.M), \u2045completeness_experiment sig m\u2046 tt = 1\n\nlemma complete_iff_signatures_support_subset :\n  sig.complete \u2194 \u2200 (m : sig.M) (pk : sig.PK) (sk : sig.SK) (\u03c3 : sig.S)\n    (log log' : query_log sig.random_oracle_spec),\n    ((pk, sk), log) \u2208 (default_simulate sig.base_oracle $ sig.gen ()).support \u2192\n    (\u03c3, log') \u2208 (simulate sig.base_oracle (sig.sign (pk, sk, m)) log).support \u2192\n    (simulate' sig.base_oracle (sig.verify (pk, m, \u03c3)) log').support = {tt} :=\nbegin\n  simp_rw [complete, eval_dist_eq_one_iff,\n    support_completeness_experiment], sorry,\nend\n\nend complete\n\nsection unforgeable\n\n/-- The adversary for the signing experiment has access to both the signature scheme's oracles,\n  and a signing oracle that will be simulated with the hidden secret key. -/\n@[reducible, inline]\ndef unforgeable_adversary_oracle_spec (sig : signature) : oracle_spec :=\nuniform_selecting ++ sig.random_oracle_spec ++ sig.signing_oracle_spec\n\n/-- An adversary for the unforgeable signature experiment.\n  Note that the adversary only has access to the public key. -/\nstructure unforgeable_adversary (sig : signature) :=\n(adv : sig.PK \u2192 oracle_comp (sig.unforgeable_adversary_oracle_spec) (sig.M \u00d7 sig.S))\n(adv_poly_time : poly_time_oracle_comp adv)\n(query_bound : \u2115)\n(adv_queries_at_most : \u2200 pk, queries_at_most (adv pk) query_bound)\n\nnamespace unforgeable_adversary\n\nvariables {sig} (adversary : unforgeable_adversary sig)\n\n/-- Wrapper function for simulation that hides the \"state values\" of the stateless oracles.\nRuns the adversary with a signing oracle based on the provided public/secret keys,\n  returning the results of the adversary, and a log of the queries made by the adversary\n -/\ndef simulate (pk : sig.PK) (sk : sig.SK) :\n  oracle_comp sig.base_oracle_spec (sig.M \u00d7 sig.S \u00d7 query_log (sig.M \u21a6\u2092 sig.S)) :=\ndo{ ((m, s), _, log) \u2190 (default_simulate (id\u209b ++\u209b signing_oracle sig pk sk) (adversary.adv pk)),\n    return (m, s, log) }\n\n/-- Experiement for testing if a signature scheme is unforgeable.\n  Generate the public/secret keys, then simulate the adversary to get a signature.\n  Adversary succeeds if the signature verifies and the message hasn't been queried -/\nnoncomputable def experiment (sig : signature) (adversary : unforgeable_adversary sig) :\n  oracle_comp uniform_selecting bool :=\ndefault_simulate' (id\u209b ++\u209b random_oracle sig.random_oracle_spec)\n(do { (pk, sk) \u2190 sig.gen (),\n      (m, \u03c3, log) \u2190 adversary.simulate pk sk,\n      b \u2190 sig.verify (pk, m, \u03c3),\n      return (if log.not_queried () m then b else ff) })\n\n/-- Adversaries success at forging a signature. -/\nnoncomputable def advantage {sig : signature} (adversary : unforgeable_adversary sig) : \u211d\u22650\u221e :=\n\u2045(= tt) | adversary.experiment sig\u2046\n\nend unforgeable_adversary\n\nend unforgeable\n\nend signature\n\n/-- signature scheme is a set of signature algorithms indexed by a security parameter -/\ndef signature_scheme := \u03a0 (sp : \u2115), signature\n\nnamespace signature_scheme\n\nopen signature\n\n/-- Scheme is complete if it is complete for each security parameter -/\ndef complete (sig_scheme : signature_scheme) : Prop :=\n\u2200 (sp : \u2115), (sig_scheme sp).complete\n\n/-- Signature scheme is unforgeable if any polynomial time adversary has negligible advantage in\n  `unforgeable_experiment` as the security parameter grows -/\ndef unforgeable (sig_scheme : signature_scheme) : Prop :=\n\u2200 (adversary : \u03a0 (sp : \u2115), unforgeable_adversary $ sig_scheme sp),\n  (\u2203 (p : polynomial \u2115), \u2200 n, (adversary n).query_bound \u2264 p.eval n) \u2192\n  negligable (\u03bb sp, (adversary sp).advantage)\n\nend signature_scheme", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/crypto_foundations/primitives/signature.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3976450340180672}}
{"text": "import substances\nopen set topological_space classical\nlocal attribute [instance] prop_decidable\n\n/-! # The Theory of Counterfactuals -/\n\nnamespace ontology\n\nvariables (\u03c9 : ontology)\n\n/- **TODOs**\n  TODO: consider changing the structure \n  below to use relations once they become \n  avaiable via relations.lean\n-/\n\n/-- **Counterfactual Relations** are simply \n(dyadic) relations between events -/\nstructure cfr := \n  (entails : \u03c9.event \u2192 \u03c9.event \u2192 \u03c9.event)\n  /-- Subjunctive conditional/implication. -/\n  add_decl_doc cfr.entails \n\n/-- Given `c : \u03c9.cfr`, use `c e\u2081 e\u2082` instead of `c.entails e\u2081 e\u2082`. -/\ninstance has_coe_to_fun_cfr : has_coe_to_fun \u03c9.cfr :=\n  \u27e8_, cfr.entails\u27e9\n\n-- Natural examples of `cfrs`.\nsection cfr_examples\n\n  /-- The naive way to define a `cfr` is to think that subjunctive implication \n      just reduces to normal implication in the context of some theory. \n      So that, e.g., \"If I were to throw something up it would go down\" is true\n      because there exists some true theory of physics which contains \n      the law of universal gravitation from which it would be possible to prove\n      this statement. \n      Because we can see a true theory as simply an event, \n      i.e. the event of the theory being true, this justifies our definition. -/\n  def naive_cfr : \u03c9.cfr := \u27e8\u03bbe\u2081 e\u2082 w, \u2203 e : \u03c9.event, e.occurs w \u2227 e \u2229 e\u2081 \u21d2 e\u2082\u27e9\n\n  /- The previous definition contains a paradox, can you spot it? -/\n\n  /-- `e\u2081` subjunctively entails `e\u2082` in world `w` if the removal of any entities from `w` \n      implies `e\u2081 \u27f6 e\u2082`. \n      Alternatively, if for any world \"smaller\" than `w`\n      in which `e\u2081` happens, `e\u2082` also happens. -/\n  def removal_cfr : \u03c9.cfr := \u27e8\u03bbe\u2081 e\u2082 w, w.ideal \u21d2 e\u2081 \u27f6 e\u2082\u27e9\n  /-- `e\u2081` subjunctively entails `e\u2082` in world `w` if the addition of any entities from `w` \n      implies `e\u2081 \u27f6 e\u2082`.\n      Alternatively, if for any world \"larger\" than `w`\n      in which `e\u2081` happens, `e\u2082` also happens. -/\n  def addition_cfr : \u03c9.cfr := \u27e8\u03bbe\u2081 e\u2082 w, w.filter \u21d2 e\u2081 \u27f6 e\u2082\u27e9\n  /-- `e\u2081` subjunctively entails `e\u2082` in world `w` if the addition or removal of any entities from `w` \n      implies `e\u2081 \u27f6 e\u2082`. -/\n  def arithmetic_cfr : \u03c9.cfr := \u27e8\u03bbe\u2081 e\u2082, \u03c9.removal_cfr e\u2081 e\u2082 \u222a \u03c9.addition_cfr e\u2081 e\u2082\u27e9\n\nend cfr_examples\n\nsection cfr\n\n  variables {\u03c9} (c : \u03c9.cfr) (e\u2081 e\u2082 : \u03c9.event)\n\n  /-- **Counterfactual dependence** relation.\n    i.e. if `e\u2081` were not the case `e\u2082` would not be the case. -/\n  @[reducible]\n  def cfr.depends : \u03c9.event := c (-e\u2081) (-e\u2082)\n  \n  -- counterfactual entanglement\n  @[reducible]\n  def cfr.entangled : \u03c9.event := c.depends e\u2081 e\u2082 \u2229 c.depends e\u2082 e\u2081\n\n  /-- Counterfactual **strong** (or one sided) dependence.\n      Could also be called \"event causation\", but we reserve\n      the name \"cause\" for a more qualified definition. -/\n  @[reducible]\n  def cfr.sdepends : \u03c9.event := c.depends e\u2081 e\u2082 - c.depends e\u2082 e\u2081\n\n  -- counterfactual independence\n  @[reducible]\n  def cfr.independent : \u03c9.event := -c.depends e\u2081 e\u2082 \u2229 -c.depends e\u2082 e\u2081\n\n\n  /-- If `e\u2081` subjunctively entails `e\u2082` in `w` and `e\u2081` occurs in `w`, `e\u2082` \n      should also occur in `w`. -/\n  def cfr.postulate\u2081 : Prop := \u2200 e\u2081 e\u2082, c e\u2081 e\u2082 \u21d2 e\u2081 \u27f6 e\u2082\n\n  /-- A `cfr` is paradoxical if the falsity of the antecedent of the subjunctive implication\n      implies its truth. -/\n  def cfr.paradox\u2081 : Prop := \u2200 e\u2081 e\u2082, -e\u2081 \u21d2 c e\u2081 e\u2082\n\n  /-- The naive cfr is paradoxical, because if the antecedent of the subjunctive \n    implication is false the implication is always true. -/\n  lemma naive_cfr_paradox\u2081 : \u03c9.naive_cfr.paradox\u2081 :=\n    begin\n      intros e\u2081 e\u2082,\n      unfold_coes,\n      intros w hw,\n      simp [naive_cfr, has_mem.mem, set.mem],\n      use {w}, refine \u27e8mem_singleton_iff.2 rfl, _\u27e9,\n      replace hw := singleton_inter_eq_empty.2 hw,\n      rw hw,\n      tauto,\n    end\n\nend cfr\n\n/-- **Closer_Than Relations (ctr)** are world-indexed pre-orders between worlds.\n    They express that a world `w\u2081` is at least as close \n    to a reference world `w` than some other world `w\u2082`. -/\nstructure ctr :=\n  (closer : \u03c9.world \u2192 \u03c9.world \u2192 \u03c9.world \u2192 Prop)\n  (axiom\u2081 : \u2200 w, reflexive $ closer w)\n  (axiom\u2082 : \u2200 w, transitive $ closer w)\n\n/-- Given `c : \u03c9.ctr`, use `c w w\u2081 w\u2082` instead of `c.closer w w\u2081 w\u2082`. -/\ninstance has_coe_to_fun_ctr : has_coe_to_fun \u03c9.ctr :=\n  \u27e8_, ctr.closer\u27e9\n\n/-- Irreflexive version of `ctr.closer`. -/\ndef ctr.closer' {\u03c9 : ontology} (c : \u03c9.ctr) : \u03c9.world \u2192 \u03c9.world \u2192 \u03c9.world \u2192 Prop := \n  assume w w\u2081 w\u2082,\n  c w w\u2081 w\u2082 \u2227 \u00ac c w w\u2082 w\u2081\n\n/- **TODOs** \n  TODO: One idea was to construct\n        the definition along the lines of\n        \"w\u2081 is at least as close to w than w\u2082 \u2194 \n        the least cardinality/ordinality of linear orders/well-orders containing w and w\u2081 \n        is \u2264 than the least cardinality/ordinality of linear orders/well-orders containing w and w\u2082\"\n        And maybe we could extend this to the sums of ordinals of multiple well-orders,\n        for the case in which the possible worlds are incomparable. \n        But the definition below is much simpler than that.\n        Revise this idea in the future.\n-/\n/-- the natural `ctr` is the one naturally defined by\n    the specialization order of the ontology. -/\ndef nctr : \u03c9.ctr := \n  { closer := \u03bb w w\u2081 w\u2082, w.entities \u2229 w\u2082.entities \u2286 w.entities \u2229 w\u2081.entities \n  , axiom\u2081 := \u03bb_, by simp [reflexive]\n  , axiom\u2082 := begin\n                intro w,\n                simp [transitive],\n                intros x y z h\u2081 h\u2082 e he,\n                specialize h\u2082 he,\n                apply h\u2081, clear h\u2081,\n                replace he := he.left,\n                exact \u27e8he, h\u2082\u27e9,\n              end    \n  }\n\ninstance default_ctr : inhabited \u03c9.ctr :=\n  -- comment out to change the `ctr`.\n  \u27e8\u03c9.nctr\u27e9\n  -- your new and improved `ctr` goes here:\n  -- \u27e8...\u27e9\n  -- defining this thingy here \u2191 \n  -- is what all philosophical \n  -- discussions boil down to.\n\n\n/-- The **Lewisian** `cfr` is the `cfr` defined by a `ctr`. -/\ndef lcfr (c : \u03c9.ctr := default \u03c9.ctr) : \u03c9.cfr := \n  \u27e8 \u03bb e\u2081 e\u2082 w, \u00ac\u22c4e\u2081 \u2228 \n    (\u2203 w', w' \u2208 (e\u2081 \u2229 e\u2082) \u2227 \n    (\u2200 w'', e\u2081.occurs w'' \u2192 \u00ac e\u2082.occurs w'' \u2192 c.closer' w w' w'')\n    ) \u27e9\n\ninstance default_cfr : inhabited \u03c9.cfr :=\n  -- comment out to change the `cfr`.\n  \u27e8\u03c9.lcfr\u27e9\n  -- your new and improved `cfr` goes here:\n  -- \u27e8...\u27e9\n  -- defining this thingy here \u2191 \n  -- is what all philosophical \n  -- discussions boil down to.\n\n/- **TODOs**\n    TODO: Define ` \u27f6+ ` notation as the \n    subjunctive implication arrow relative to a default\n    choice of counterfactual relation.\n    TODO: Define `x \u27f6- y` notation as `-x \u27f6+ -y`.\n\n    TODO: Specify different ways with which counterfactual \n          relations (`cfrs`) can be \n          defined. The main one \n          (which should probably become the default)\n          will be via Lewisian `ctrs`, or some such \n          (perhaps \"Prussian\") variation of it, \n          using the natural `ctr` as the \n          default. It remains to be investigated whether\n          this is equivalent to what is currently defined in\n          causality.lean for the `x \u27f6- y` case. \n          \n          Some other\n          such ways could be by the introduction of primitive\n          projection operators `\u03c0` of signature \n          `event \u2192 world \u2192 world` or `event \u2192 world \u2192 set world`, \n          which for every pair `e, w` would define what the \n          world `w` would be like in case `e` were to occur.\n          Then `e\u2081 \u27f6+ e\u2082` should reduce to {w | (\u03c0 e\u2081 w) \u2208 e\u2082}\n          or to {w | (\u03c0 e\u2081 w) \u2286 e\u2082}, respectively. \n          Compared to the `ctrs`, it does look harder to define\n          such projections without assuming new primitives.\n          Yet another way could be by the introduction of a\n          collection of events with which to restrict the \n          existential quantification of the `naive_cfr` \n          definition, so as to avoid `cfr.paradox\u2081` and \n          other such paradoxes.\n\n\n    TODO: Don't forget to set up defaults (inhabited) \n          for every type defined in this module. \n          This is important.\n-/\n\nend ontology", "meta": {"author": "maxd13", "repo": "topological_ontology", "sha": "68d21c9a00024fba3aed301e16c31e05733c1786", "save_path": "github-repos/lean/maxd13-topological_ontology", "path": "github-repos/lean/maxd13-topological_ontology/topological_ontology-68d21c9a00024fba3aed301e16c31e05733c1786/src/metaphysics/counterfactuals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3976450340180672}}
{"text": "import topology.sheaves.sheaf\nimport algebra.category.Group.limits\nimport oc\nimport lemmas.about_opens\nimport group_theory.perm.sign\nimport tactic\n\nnoncomputable theory\n\nsection\n\nopen topological_space Top Top.sheaf\nopen category_theory\nopen opposite\n\nuniverse u\nvariables {X : Top.{u}} (\ud835\udcd5 : sheaf Ab X) (U : X.oc)\n\nsection\n\nvariables {U}\n\ndef ignore {n : \u2115} (\u03b1 : fin (n + 1) \u2192 U.\u03b9) (i : fin (n + 1)) :\n  fin n \u2192 U.\u03b9 :=\n\u03bb k, dite (k.1 < i.1)\n  (\u03bb ineq1, \u03b1 \u27e8k.1, lt_trans ineq1 i.2\u27e9)\n  (\u03bb ineq1, \u03b1 \u27e8k.1.pred, begin\n    rw nat.lt_succ_iff,\n    refine le_of_lt _,\n    exact lt_of_le_of_lt (nat.pred_le _) k.2,\n  end\u27e9)\n\ndef ignore\u2082 {n : \u2115} (\u03b1 : fin (n + 2) \u2192 U.\u03b9) (i : fin (n + 2)) (j : fin (n + 1)) :\n  fin n \u2192 U.\u03b9 :=\nignore (ignore \u03b1 i) j\n\nlemma ignore.apply_lt {n : \u2115} (\u03b1 : fin (n + 1) \u2192 U.\u03b9) (i : fin (n + 1))\n  {k : fin n} (ineq1 : k.1 < i.1) :\n  ignore \u03b1 i k = \u03b1 \u27e8k.1, lt_trans ineq1 i.2\u27e9 :=\ndif_pos ineq1\n\nlemma ignore.apply_not_lt {n : \u2115} (\u03b1 : fin (n + 1) \u2192 U.\u03b9) (i : fin (n + 1))\n  {k : fin n} (ineq1 : \u00ac k.1 < i.1) :\n  ignore \u03b1 i k = \u03b1 \u27e8k.1.pred, begin\n    rw nat.lt_succ_iff,\n    refine le_of_lt _,\n    exact lt_of_le_of_lt (nat.pred_le _) k.2,\n  end\u27e9 :=\ndif_neg ineq1\n\nlemma ignore.apply_ite {n : \u2115} (\u03b1 : fin (n + 1) \u2192 U.\u03b9) (i : fin (n + 1))\n  (k : fin n) :\n  ignore \u03b1 i k =\n  dite (k.1 < i.1)\n    (\u03bb ineq1, \u03b1 \u27e8k.1, lt_trans ineq1 i.2\u27e9)\n    (\u03bb ineq1, \u03b1 \u27e8k.1.pred, begin\n      rw nat.lt_succ_iff,\n      refine le_of_lt _,\n      exact lt_of_le_of_lt (nat.pred_le _) k.2,\n    end\u27e9) := rfl\n\n/--\n\na0, a1, ..., ai, ..., aj, ...., a(n+2)\n\n                                                j-1th  jth\nignore \u03b1 i = a0, a1, ...., a(i-1), a(i+1), .... aj,   a(j+1) ...., a(n+2)\nignore \u03b1 (ignore \u03b1 i) j = a0, a1, ..., a(i-1), a(i+1), ..., aj, a(j+2), ... a(n+2)\n\n(i < j)\nso ignore\u2082 \u03b1 i j = ignore\u2082 \u03b1 (j + 1) i\n\n(i = j)\n                                  i-th         n+1st\nignore \u03b1 i = a0, a1, ..., a(i-1), a(i+1), ... a(n+2)\n                                              ith          nth\nignore (ignore \u03b1 i) i = a0, a1, .... a(i-1), a(i+2), .... a(n+2)\n-/\n\n-- this is almost certainly correct, as I wrote a python script to test\nlemma ignore\u2082_symm' {n : \u2115} (\u03b1 : fin (n + 2) \u2192 U.\u03b9)\n  {i : \u2115} (hi : i \u2208 finset.range n.succ)\n  {j : \u2115} (hj : j \u2208 finset.Ico i n.succ) : -- i \u2264 j\n  ignore\u2082 \u03b1 \u27e8j + 1, begin\n    rw finset.mem_Ico at hj,\n    rw nat.succ_lt_succ_iff,\n    exact hj.2\n  end\u27e9 \u27e8i, finset.mem_range.mp hi\u27e9 = ignore\u2082 \u03b1 \u27e8i, lt_trans (finset.mem_range.mp hi) (lt_add_one _)\u27e9 \u27e8j, (finset.mem_Ico.mp hj).2\u27e9 :=\nbegin\n  /-\n  if i = 0 then \n    rhs = ignore\u2082 \u03b1 0 j m = ignore (ignore \u03b1 0) j m \n      = ignore \u03b1 0 m = \u03b1 (m - 1) if m < j\n      = ignore \u03b1 0 (m-1) = \u03b1 (m - 1) if j \u2264 m\n  \n  if i = j,\n    rhs = ignore\u2082 \u03b1 i i m\n    lhs = ignore\u2082 \u03b1 (i+1) i m\n    \n  \n  if i < j\n\n  if 0 \u2264 m \u2264 i-1 < i < j,\n  then rhs = ignore\u2082 \u03b1 i j m = ignore (ignore \u03b1 i) j m = ignore \u03b1 i m = \u03b1 m\n       lhs = ignore\u2082 \u03b1 (j + 1) i m = ignore (ignore \u03b1 (j + 1)) i m = ignore \u03b1 (j + 1) m = \u03b1 m\n\n  if i-1 < m < j, i.e. i \u2264 m < j\n  then rhs = ignore\u2082 \u03b1 i j m = ignore (ignore \u03b1 i) j m = ignore \u03b1 i m = \u03b1 (m - 1)\n       lhs = ignore\u2082 \u03b1 (j + 1) i m = ignore \u03b1 (j + 1) (m - 1) = \u03b1 (m - 1)\n\n  if j = m, i < j, i.e. i \u2264 j-1\n  then rhs = ignore\u2082 \u03b1 i j j = ignore (ignore \u03b1 i) j j = ignore \u03b1 i (j-1) = \u03b1 (j-2)\n    lhs = ignore\u2082 \u03b1 (j + 1) i j = ignore \u03b1 (j + 1) (j - 1) = \u03b1 (j - 2)\n\n  if j + 1 = m,\n  then rhs = ignore\u2082 \u03b1 i j (j+1) = ignore\u2082 \u03b1 i j = \u03b1 (j - 1)\n    lhs = ignore\u2082 \u03b1 (j+1) i (j+1) = ignore\u2082 \u03b1 (j+1) j = \u03b1 (j-1)\n\n  if j + 1 < m, then j < m - 1 then j - 1 \u2264 m-1\n  then rhs = ignore\u2082 \u03b1 i j m = ignore \u03b1 i (m-1) = \u03b1 (m-2)\n    lhs = ignore\u2082 \u03b1 (j+1) i m = ignore \u03b1 (j + 1) (m-1) = \u03b1 (m-2)\n  -/\n  rw finset.mem_Ico at hj,\n  rw finset.mem_range at hi,\n  ext1 m,\n  change ignore _ _ _ = ignore _ _ _,\n  by_cases ineq1 : m.1 < i,\n  { have ineq2' : m.1 < j := lt_of_lt_of_le ineq1 hj.1,\n    have ineq2 : m.1 < j + 1 := lt_trans ineq2' (lt_add_one _),\n    rw ignore.apply_lt,\n    swap, exact ineq1,\n    rw ignore.apply_lt,\n    swap, exact ineq2,\n    rw ignore.apply_lt,\n    swap, exact ineq2',\n    rw ignore.apply_lt,\n    exact ineq1 },\n  { rw not_lt at ineq1,\n    rw ignore.apply_not_lt,\n    swap, rwa not_lt,\n    sorry,\n     },\n  -- sorry,\nend\n\n\nlemma ignore\u2082.apply_lt_min {n : \u2115} (\u03b1 : fin (n + 2) \u2192 U.\u03b9)\n  (i : fin (n + 2)) (j : fin (n + 1))\n  (h : i.1 \u2264 j.1)\n  (k : fin n)\n  (hj : k.1 < i.1) :\n  ignore\u2082 \u03b1 i j k = \u03b1 \u27e8k.1, by linarith [k.2]\u27e9 :=\nbegin\n  change ignore _ _ _ = _,\n  rw ignore.apply_lt,\n  rw ignore.apply_lt,\n  assumption,\n  refine lt_of_lt_of_le _ h,\n  assumption,\nend\n\n@[derive [decidable_eq]]\ninductive sign\n| neg\n| zero\n| pos\n\ndef unit_to_sign (n : \u2124\u02e3) : sign :=\nif n = 1 then sign.pos else sign.neg\n\ndef signature.order_aux {n : \u2115} {\u03b1 : fin n \u2192 U.\u03b9} (inj : function.injective \u03b1) :\n  fin n \u2243o finset.image \u03b1 finset.univ :=\nfinset.order_iso_of_fin _ begin\n  rw finset.card_image_of_injective _ inj,\n  exact finset.card_fin n,\nend\n\ndef signature.restrict_aux {n : \u2115} {\u03b1 : fin n \u2192 U.\u03b9} (inj : function.injective \u03b1) :\n  fin n \u2243 finset.image \u03b1 finset.univ :=\nequiv.of_bijective (\u03bb k, \u27e8\u03b1 k, finset.mem_image.mpr \u27e8k, finset.mem_univ _, rfl\u27e9\u27e9) begin\n  split,\n  { intros a b h,\n    simp only [subtype.mk_eq_mk] at h,\n    apply_fun \u03b1,\n    exact h, },\n  { rintros \u27e8i, hi\u27e9,\n    rw finset.mem_image at hi,\n    rcases hi with \u27e8j, _, rfl\u27e9,\n    use j, }\nend\n\ndef signature.equiv {n : \u2115} {\u03b1 : fin n \u2192 U.\u03b9} (inj : function.injective \u03b1) :\n  fin n \u2243 fin n :=\n{ to_fun := function.comp (signature.order_aux inj).symm (signature.restrict_aux inj),\n  inv_fun := function.comp (signature.restrict_aux inj).symm (signature.order_aux inj),\n  left_inv := \u03bb k, by simp,\n  right_inv := \u03bb k, by simp }\n\ndef signature {n : \u2115} (\u03b1 : fin n \u2192 U.\u03b9) : sign :=\ndite (function.injective \u03b1)\n(\u03bb inj, unit_to_sign $ equiv.perm.sign (signature.equiv inj))\n(\u03bb _, sign.zero)\n\ndef swap {n : \u2115} (i j : fin n) (\u03b1 : fin n \u2192 U.\u03b9) : fin n \u2192 U.\u03b9 :=\n\u03bb k, if (k = i) \n  then \u03b1 j\n  else if (k = j)\n    then \u03b1 i\n    else \u03b1 k\n\n\nnamespace swap\n\nvariables {U} {n : \u2115} (\u03b1 : fin n \u2192 U.\u03b9)\n\n@[simp]\nlemma same (i : fin n) :\n  swap i i \u03b1 = \u03b1 :=\nbegin\n  ext,\n  change ite _ _ _ = _,\n  split_ifs,\n  { subst h, },\n  { refl, }\nend\n\n@[simp]\nlemma symmetric (i j : fin n) :\n  swap i j \u03b1 = swap j i \u03b1 :=\nbegin\n  ext,\n  change ite _ _ _ = ite _ _ _,\n  split_ifs with h1 h2,\n  { subst h1,\n    subst h2, },\n  { subst h1, },\n  { subst h, },\n  { refl, },\nend\n\n@[simp]\nlemma twice (i j : fin n) :\n  swap i j (swap i j \u03b1) = \u03b1 :=\nbegin\n  ext,\n  change ite _ _ _ = _,\n  split_ifs with h1,\n  { change ite _ _ _ = _,\n    split_ifs with h2,\n    { subst h2, subst h1, },\n    { subst h1, }, },\n  { change ite _ _ _ = _,\n    rw if_pos rfl,\n    subst h, },\n  { change ite _ _ _ = _,\n    split_ifs,\n    refl, },\nend\n\nlemma apply1 (i j : fin n) :\n  swap i j \u03b1 i = \u03b1 j :=\nbegin\n  change ite _ _ _ = _,\n  rw if_pos rfl,\nend\n\nlemma apply1' (i j : fin n) {i' : fin n} (eq1 : i'.1 = i.1) :\n  swap i j \u03b1 i' = \u03b1 j :=\nbegin\n  convert apply1 \u03b1 i j,\n  rw subtype.ext_iff_val,\n  exact eq1,\nend\n\nlemma apply2 (i j : fin n) :\n  swap i j \u03b1 j = \u03b1 i :=\nbegin\n  change ite _ _ _ = _,\n  rw if_pos rfl,\n  split_ifs,\n  { subst h, },\n  { refl, }\nend\n\nlemma apply2' (i j : fin n) {j' : fin n} (eq1 : j'.1 = j.1) :\n  swap i j \u03b1 j' = \u03b1 i :=\nbegin\n  convert apply2 \u03b1 i j,\n  rw subtype.ext_iff_val,\n  exact eq1,\nend\n\nlemma apply_ne (i j k : fin n)\n  (ineq1 : k \u2260 i)\n  (ineq2 : k \u2260 j) :\n  swap i j \u03b1 k = \u03b1 k :=\nbegin\n  change ite _ _ _ = _,\n  rw [if_neg, if_neg];\n  assumption,\nend\n\nlemma nat.pred_eq_self {n : \u2115} (h : n.pred = n) : n = 0 :=\nbegin\n  induction n with n h1 generalizing h,\n  { refl, },\n  { rw nat.pred_succ at h,\n    exfalso,\n    have ineq1 := lt_add_one n,\n    rw nat.succ_eq_add_one at h,\n    rw \u2190 h at ineq1,\n    apply lt_irrefl _ ineq1, },\nend\n\n-- If `max i j < k`, then `ignore (swap i j \u03b1) k = swap i j (ignore \u03b1 k)`\n-- If `k < min i j`, then `ignore (swap i j \u03b1) k = swap (i + 1) (j + 1) (ignore \u03b1 k)`\n-- If `min i j \u2264 k \u2264 max i j`, then `ignore (swap i j \u03b1) k = swap (min i j) ((max i j) - 1) (ignore \u03b1 k)` or `ignore (swap i j \u03b1) k = swap (min i j) (max i j) (ignore \u03b1 k)`\n\n-- lemma ignore_swap_eq_swap_ignore\n--   (\u03b1 : fin (n + 1) \u2192 U.\u03b9)\n--   (i j k : fin (n + 1)) :\n--   \u2203 (i' j' : fin n),\n--   ignore (swap i j \u03b1) k = swap i' j' (ignore \u03b1 k) := sorry\n\n\n-- lemma swap_ignore_gt (\u03b1 : fin (n+1) \u2192 U.\u03b9) (i j k : fin (n+1)) \n--   (ineq1 : max i.1 j.1 < k.1):\n--   ignore (swap i j \u03b1) k = swap \u27e8i.1, begin\n--     by_contra r,\n--     rw not_lt at r,\n--     have ineq2 := i.2,\n--     have eq1 : i.1 = n,\n--     { linarith, },\n--     have ineq3 : n < k.1,\n--     { refine lt_of_le_of_lt _ ineq1,\n--       simp [\u2190 eq1], },\n--     have ineq4 := k.2,\n--     linarith,\n--   end\u27e9 \u27e8j.1, begin\n--     by_contra r,\n--     rw not_lt at r,\n--     have ineq2 := j.2,\n--     have eq1 : j.1 = n,\n--     { linarith, },\n--     have ineq3 : n < k.1,\n--     { refine lt_of_le_of_lt _ ineq1,\n--       simp [\u2190 eq1], },\n--     have ineq4 := k.2,\n--     linarith,\n--   end\u27e9 (ignore \u03b1 k) :=\n-- begin\n--   ext m,\n--   by_cases ineq2 : m.1 = i.1;\n--   by_cases ineq3 : m.1 = j.1,\n--   { -- m = i = j,\n--     rw [\u2190 ineq2, \u2190 ineq3, max_self] at ineq1,\n--     rw ignore.apply_lt,\n--     swap, exact ineq1,\n--     rw [apply1', apply1', ignore.apply_lt],\n--     congr' 1,\n--     rw subtype.ext_iff_val,\n--     { convert ineq1,\n--       rw subtype.ext_iff_val,\n--       exact ineq3.symm, },\n--     exact ineq2,\n--     exact ineq2, },\n--   { -- m = i but m \u2260 j,\n--     rw apply1',\n--     swap, exact ineq2,\n--     rw \u2190 ineq2 at ineq1,\n--     have ineq4 := lt_of_le_of_lt (le_max_left _ _) ineq1,\n--     rw ignore.apply_lt,\n--     swap, exact ineq4,\n--     rw apply1',\n--     swap, exact ineq2,\n--     rw ignore.apply_lt,\n--     swap, exact lt_of_le_of_lt (le_max_right _ _) ineq1,\n--     congr' 1,\n--     rw subtype.ext_iff_val, },\n--   { -- m \u2260 i but m = j,\n--     rw apply2',\n--     swap, exact ineq3,\n--     rw \u2190 ineq3 at ineq1,\n--     have ineq4 := lt_of_le_of_lt (le_max_right _ _) ineq1,\n--     rw ignore.apply_lt,\n--     swap, exact ineq4,\n--     rw ignore.apply_lt,\n--     swap, exact lt_of_le_of_lt (le_max_left _ _) ineq1,\n--     rw apply2',\n--     swap, exact ineq3,\n--     congr' 1,\n--     rw subtype.ext_iff_val, },\n--   { -- m \u2260 i and m \u2260 j,\n--     rw apply_ne,\n--     swap, contrapose! ineq2, rw ineq2,\n--     swap, contrapose! ineq3, rw ineq3,\n--     rw ignore.apply_ite,\n--     rw ignore.apply_ite,\n--     split_ifs with ineq4,\n--     { rw apply_ne,\n\n--       contrapose! ineq2,\n--       rw \u2190 ineq2,\n\n--       contrapose! ineq3,\n--       rw \u2190 ineq3, },\n--     { by_cases ineq5 : m.1.pred = i.1,\n--       { rw apply1',\n--         swap, exact ineq5,\n\n--         have EQ : i.1 = j.1,\n--         { have ineq0 : m.1 \u2260 0,\n--           { intro r,\n--             rw r at *,\n--             linarith, },\n--           have eq0 : m.1 = i.1 + 1,\n--           { rw [\u2190 ineq5, \u2190 nat.succ_eq_add_one, nat.succ_pred_eq_of_pos],\n--             linarith },\n--           rw [eq0, not_lt] at ineq4,\n--           have ineq6 : k.1 = i.1 + 1,\n--           { have INEQ : i.1 < k.1 := lt_of_le_of_lt (le_max_left _ _) ineq1,\n--             linarith },\n--           have ineq7 : i.1 \u2264 j.1,\n--           { have INEQ := lt_of_lt_of_le ineq1 ineq4,\n--             have EQ : max i.1 j.1 = j.1,\n--             { suffices : i \u2264 j,  }, },\n--           sorry },\n--         congr' 1,\n--         rw [subtype.ext_iff_val, \u2190 EQ, \u2190 ineq5], },\n--       { sorry } }, }\n-- end\n\nend swap\n\ndef face {n : \u2115} (\u03b1 : fin n \u2192 U.\u03b9) : opens X :=\n  infi (\u03bb (k : fin n), U.cover $ \u03b1 k)\n\nsection face\n\nlemma face.congr {n : \u2115} {\u03b1 \u03b2 : fin n \u2192 U.\u03b9} (h : \u03b1 = \u03b2) :\n  face \u03b1 = face \u03b2 :=\nby subst h\n\nlemma face.swap_eq {n : \u2115} (\u03b1 : fin n \u2192 U.\u03b9) (i j : fin n) :\n  face \u03b1 = face (swap i j \u03b1) :=\nbegin\n  change infi _ = infi _,\n  refine le_antisymm _ _,\n  { rw le_infi_iff,\n    intros k,\n    by_cases k = i,\n    { subst h,\n      by_cases k = j,\n      { subst h,\n        rw _root_.swap.same \u03b1 k,\n        intros p hp,\n        erw opens.fintype_infi at hp,\n        apply hp, },\n      { rw _root_.swap.apply1,\n        intros p hp,\n        erw opens.fintype_infi at hp,\n        apply hp, }, },\n      { by_cases k = j,\n        { subst h,\n          rw _root_.swap.apply2,\n          intros p hp,\n          erw opens.fintype_infi at hp,\n          apply hp, },\n        { rw _root_.swap.apply_ne,\n          intros p hp,\n          erw opens.fintype_infi at hp,\n          apply hp,\n          assumption,\n          assumption, }, }, },\n  { intros p hp,\n    erw opens.fintype_infi at hp \u22a2,\n    intros k,\n    by_cases h1 : k = i,\n    { subst h1,\n      by_cases h2 : k = j,\n      { subst j,\n        rw _root_.swap.same at hp,\n        apply hp, },\n      { specialize hp j,\n        rwa _root_.swap.apply2 at hp, }, },\n    { by_cases h2 : k = j,\n      { subst h2,\n        specialize hp i,\n        rwa _root_.swap.apply1 at hp, },\n      { specialize hp k,\n        rwa _root_.swap.apply_ne at hp,\n        assumption,\n        assumption }, }, }\nend\n\nlemma face.le_ignore {n : \u2115} (\u03b1 : fin (n + 1) \u2192 U.\u03b9) (k : fin (n + 1)) :\n  face \u03b1 \u2264 face (ignore \u03b1 k) :=\nbegin\n  intros p hp,\n  rw opens.mem_coe at hp \u22a2,\n  change _ \u2208 infi _ at hp,\n  change _ \u2208 infi _,\n  rw opens.fintype_infi at hp \u22a2,\n  rintros \u27e8i, hi\u27e9,\n  by_cases ineq : i < k.1,\n  { specialize hp \u27e8i, _\u27e9,\n    { refine lt_trans hi _,\n      exact lt_add_one n, },\n    rw ignore.apply_lt,\n    swap, exact ineq,\n    exact hp, },\n  { specialize hp \u27e8i.pred, _\u27e9,\n    { rw nat.lt_succ_iff,\n      by_cases i = 0,\n      { subst h,\n        exact nat.zero_le _, },\n      refine le_of_lt _,\n      refine lt_trans _ hi,\n      exact nat.pred_lt h, },\n    rw ignore.apply_not_lt,\n    convert hp,\n    exact ineq, }\nend\n\nlemma face.le_ignore\u2082 {n : \u2115} (\u03b1 : fin (n + 2) \u2192 U.\u03b9) (i : fin (n + 2)) (j : fin (n + 1)) :\n  face \u03b1 \u2264 face (ignore\u2082 \u03b1 i j) :=\nle_trans (face.le_ignore \u03b1 i) (face.le_ignore _ j)\n\nend face\n\n\nend\n\ndef C.pre (n : \u2115) : Type* :=\n\u03a0 (\u03b1 : fin n \u2192 U.\u03b9), \ud835\udcd5.1.obj (op $ face \u03b1)\n\nnamespace C_pre\n\nvariable {n : \u2115}\nvariables {\ud835\udcd5 U}\n\ninstance : has_add (C.pre \ud835\udcd5 U n) :=\n{ add := \u03bb f g \u03b1, f \u03b1 + g \u03b1 }\n\n@[simp] lemma add_apply (f g : C.pre \ud835\udcd5 U n) (\u03b1 : fin n \u2192 U.\u03b9) :\n  (f + g) \u03b1 = f \u03b1 + g \u03b1 := rfl\n\ninstance : has_zero (C.pre \ud835\udcd5 U n) :=\n{ zero := \u03bb \u03b1, 0 }\n\n@[simp]\nlemma zero_apply (\u03b1 : fin n \u2192 U.\u03b9) :\n  (0 : C.pre \ud835\udcd5 U n) \u03b1 = 0 := rfl\n\ninstance : has_scalar \u2115 (C.pre \ud835\udcd5 U n) :=\n{ smul := \u03bb n f \u03b1, n \u2022 f \u03b1 }\n\n@[simp]\nlemma nsmul_apply (f : C.pre \ud835\udcd5 U n) (\u03b1 : fin n \u2192 U.\u03b9) (m : \u2115) :\n  (m \u2022 f) \u03b1 = m \u2022 f \u03b1 := rfl\n\ninstance : add_comm_monoid (C.pre \ud835\udcd5 U n) :=\n{ add := (+),\n  add_assoc := \u03bb a b c, by ext; simp [add_assoc],\n  zero := 0,\n  zero_add := \u03bb f, by ext; simp,\n  add_zero := \u03bb f, by ext; simp,\n  nsmul := (\u2022),\n  nsmul_zero' := \u03bb f, by ext; simp,\n  nsmul_succ' := \u03bb m f, begin\n    ext,\n    simp [nat.succ_eq_add_one, add_smul, add_comm],\n  end,\n  add_comm := \u03bb f g, by ext; simp [add_comm] }\n\ninstance : add_comm_group (C.pre \ud835\udcd5 U n) :=\n{ neg := \u03bb f \u03b1, - f \u03b1,\n  add_left_neg := \u03bb f, by ext; simp,\n  ..(_ : add_comm_monoid (C.pre \ud835\udcd5 U n))}\n\n@[simp]\nlemma neg_apply (f : C.pre \ud835\udcd5 U n) (\u03b1 : fin n \u2192 U.\u03b9) :\n  (-f) \u03b1 = - (f \u03b1) := rfl\n\nend C_pre\n\nabbreviation C (n : \u2115) := AddCommGroup.of (C.pre \ud835\udcd5 U n)\n\n-- section\n\n-- variables {\ud835\udcd5 U}\n-- def C.pre.is_skewsymmetric {n : \u2115} (f : C.pre \ud835\udcd5 U n) : Prop :=\n-- \u2200 (i j : fin n) (\u03b1 : fin n \u2192 U.\u03b9),\n--   f \u03b1 =\n--   \ud835\udcd5.1.map (eq_to_hom (face.swap_eq \u03b1 i j)).op (- f (swap i j \u03b1))\n\n-- def C.pre.is_skewsymmetric' {n : \u2115} (f : C.pre \ud835\udcd5 U n) : Prop :=\n-- \u2200 (i j : fin n) (\u03b1 : fin n \u2192 U.\u03b9),\n--   f (swap i j \u03b1) =\n--   - \ud835\udcd5.1.map (eq_to_hom (face.swap_eq \u03b1 i j).symm).op (f \u03b1)\n\n-- lemma is_skewsymmetric_iff_is_skewsymmetric' {n} (f : C.pre \ud835\udcd5 U n) :\n--   C.pre.is_skewsymmetric f \u2194 C.pre.is_skewsymmetric' f :=\n-- { mp := \u03bb h i j \u03b1, begin\n--     specialize h i j \u03b1,\n--     rw [h, map_neg, map_neg, neg_neg, \u2190 comp_apply, \u2190 \ud835\udcd5.1.map_comp, \u2190 op_comp, eq_to_hom_trans, eq_to_hom_refl],\n--     simp,\n--   end,\n--   mpr := \u03bb h i j \u03b1, begin\n--     specialize h i j \u03b1,\n--     rw [h, neg_neg, \u2190 comp_apply, \u2190 \ud835\udcd5.1.map_comp, \u2190 op_comp, eq_to_hom_trans, eq_to_hom_refl],\n--     simp\n--   end }\n\n-- end\n\n-- def C (n : \u2115) : Type* :=\n-- { f : C.pre \ud835\udcd5 U n // f.is_skewsymmetric \u2227 \u2200 (\u03b1 : fin n \u2192 U.\u03b9), \u00ac function.injective \u03b1 \u2192 f \u03b1 = 0}\n\n-- namespace C\n\n-- variables (n : \u2115)\n\n-- @[ext]\n-- lemma ext_val {f g : C \ud835\udcd5 U n} (eq1 : f.1 = g.1) :\n--   f = g := subtype.ext_val eq1\n\n-- instance : has_add (C \ud835\udcd5 U n) :=\n-- { add := \u03bb f g,\n--   \u27e8f.1 + g.1, begin\n--     split,\n--     intros i j \u03b1,\n--     change f.1 \u03b1 + g.1 \u03b1 = \ud835\udcd5.1.map _ (- (f.1 _ + g.1 _)),\n--     rw [map_neg, map_add, f.2.1 i j, g.2.1 i j, map_neg, map_neg, neg_add],\n\n--     intros \u03b1 ha,\n--     change f.1 \u03b1 + g.1 \u03b1 = 0,\n--     rw [f.2.2, g.2.2, add_zero];\n--     assumption,\n--   end\u27e9 }\n\n-- instance : has_zero (C \ud835\udcd5 U n) :=\n-- { zero := \n--   \u27e80, begin\n--     split,\n--     intros i j \u03b1,\n--     simp only [C_pre.zero_apply, neg_zero, map_zero],\n\n--     intros \u03b1 ha,\n--     simp,\n--   end\u27e9 }\n\n-- instance : has_scalar \u2115 (C \ud835\udcd5 U n) :=\n-- { smul := \u03bb m f, \u27e8m \u2022 f.1, begin\n--     split,\n--     intros i j \u03b1,\n--     simp only [C_pre.nsmul_apply, eq_to_hom_op, eq_to_hom_map, map_neg, map_nsmul],\n--     rw f.2.1 i j,\n--     simp only [eq_to_hom_op, eq_to_hom_map, map_neg, neg_nsmul],\n\n--     intros \u03b1 ha,\n--     change m \u2022 f.1 \u03b1 = 0,\n--     rw f.2.2 _ ha,\n--     simp,\n--   end\u27e9 }\n\n-- instance : add_comm_monoid (C \ud835\udcd5 U n) :=\n-- { add := (+),\n--   add_assoc := \u03bb a b c, begin\n--     ext,\n--     change (a.1 + b.1 + c.1) _ = (a.1 + (b.1 + c.1)) _,\n--     simp only [C_pre.add_apply],\n--     rw add_assoc,\n--   end,\n--   zero := 0,\n--   zero_add := \u03bb f, begin\n--     ext,\n--     change (0 + f.1) _ = _,\n--     simp only [C_pre.add_apply, C_pre.zero_apply, zero_add],\n--   end,\n--   add_zero := \u03bb f, begin\n--     ext,\n--     change (f.1 + 0) _ = _,\n--     simp only [C_pre.add_apply, C_pre.zero_apply, add_zero],\n--   end,\n--   nsmul := (\u2022),\n--   nsmul_zero' := \u03bb f, begin\n--     ext,\n--     change 0 \u2022 f.1 _ = 0,\n--     rw zero_smul,\n--   end,\n--   nsmul_succ' := \u03bb m f, begin\n--     ext,\n--     change (m + 1) \u2022 f.1 x = (f.1 + m \u2022 f.1) x,\n--     rw [add_smul, one_smul, C_pre.add_apply, add_comm],\n--     refl,\n--   end,\n--   add_comm := \u03bb f g, begin\n--     ext,\n--     change (f.1 + g.1) x = (g.1 + f.1) x,\n--     simp only [add_comm, C_pre.add_apply],\n--   end }\n\n-- instance : add_comm_group (C \ud835\udcd5 U n) :=\n-- { neg := \u03bb f, \u27e8-f.1, begin\n--     split,\n--     intros i j \u03b1,\n--     simp only [C_pre.neg_apply],\n--     rw neg_neg,\n--     rw f.2.1 i j,\n--     simp only [map_neg, neg_neg],\n\n--     intros \u03b1 ha,\n--     change - (f.1 \u03b1) = 0,\n--     rw f.2.2 _ ha,\n--     rw neg_zero,\n--   end\u27e9,\n--   add_left_neg := \u03bb f, begin\n--     ext,\n--     change (-f.1 + f.1) x = 0,\n--     simp,\n--   end,\n--   ..add_comm_monoid \ud835\udcd5 U n }\n\n-- end C\n\nend", "meta": {"author": "jjaassoonn", "repo": "cc", "sha": "6d3dc6885fa012e8c18fd38ab2949d73777fb442", "save_path": "github-repos/lean/jjaassoonn-cc", "path": "github-repos/lean/jjaassoonn-cc/cc-6d3dc6885fa012e8c18fd38ab2949d73777fb442/src/new/unordered/C.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669998, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.39763929656737695}}
{"text": "/-\nCopyright (c) 2022 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Eric Wieser\n\n! This file was ported from Lean 3 source module algebra.graded_mul_action\n! leanprover-community/mathlib commit 861a26926586cd46ff80264d121cdb6fa0e35cc1\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Algebra.GradedMonoid\n\n/-!\n# Additively-graded multiplicative action structures\n\nThis module provides a set of heterogeneous typeclasses for defining a multiplicative structure\nover the sigma type `GradedMonoid A` such that `(\u2022) : A i \u2192 M j \u2192 M (i + j)`; that is to say, `A`\nhas an additively-graded multiplicative action on `M`. The typeclasses are:\n\n* `GradedMonoid.GSmul A M`\n* `GradedMonoid.GMulAction A M`\n\nWith the `SigmaGraded` locale open, these respectively imbue:\n\n* `SMul (GradedMonoid A) (GradedMonoid M)`\n* `MulAction (GradedMonoid A) (GradedMonoid M)`\n\nFor now, these typeclasses are primarily used in the construction of `DirectSum.GModule.Module` and\nthe rest of that file.\n\n## Internally graded multiplicative actions\n\nIn addition to the above typeclasses, in the most frequent case when `A` is an indexed collection of\n`SetLike` subobjects (such as `AddSubmonoid`s, `AddSubgroup`s, or `Submodule`s), this file\nprovides the `Prop` typeclasses:\n\n* `SetLike.GradedSmul A M` (which provides the obvious `GradedMonoid.GSmul A` instance)\n\nwhich provides the API lemma\n\n* `SetLike.graded_smul_mem_graded`\n\nNote that there is no need for `SetLike.graded_mul_action` or similar, as all the information it\nwould contain is already supplied by `GradedSmul` when the objects within `A` and `M` have\na `MulAction` instance.\n\n## tags\n\ngraded action\n-/\n\n\nvariable {\u03b9 : Type _}\n\nnamespace GradedMonoid\n\n/-! ### Typeclasses -/\n\n\nsection Defs\n\nvariable (A : \u03b9 \u2192 Type _) (M : \u03b9 \u2192 Type _)\n\n/-- A graded version of `SMul`. Scalar multiplication combines grades additively, i.e.\nif `a \u2208 A i` and `m \u2208 M j`, then `a \u2022 b` must be in `M (i + j)`-/\nclass GSmul [Add \u03b9] where\n  /-- The homogeneous multiplication map `smul` -/\n  smul {i j} : A i \u2192 M j \u2192 M (i + j)\n#align graded_monoid.ghas_smul GradedMonoid.GSmul\n\n/-- A graded version of `Mul.toSMul` -/\ninstance GMul.toGSmul [Add \u03b9] [GMul A] : GSmul A A where smul := GMul.mul\n#align graded_monoid.ghas_mul.to_ghas_smul GradedMonoid.GMul.toGSmul\n\ninstance GSmul.toSMul [Add \u03b9] [GSmul A M] : SMul (GradedMonoid A) (GradedMonoid M) :=\n  \u27e8fun x y \u21a6 \u27e8_, GSmul.smul x.snd y.snd\u27e9\u27e9\n#align graded_monoid.ghas_smul.to_has_smul GradedMonoid.GSmul.toSMul\n\ntheorem mk_smul_mk [Add \u03b9] [GSmul A M] {i j} (a : A i) (b : M j) :\n    mk i a \u2022 mk j b = mk (i + j) (GSmul.smul a b) :=\n  rfl\n#align graded_monoid.mk_smul_mk GradedMonoid.mk_smul_mk\n\n/-- A graded version of `MulAction`. -/\nclass GMulAction [AddMonoid \u03b9] [GMonoid A] extends GSmul A M where\n  /-- One is the neutral element for `\u2022` -/\n  one_smul (b : GradedMonoid M) : (1 : GradedMonoid A) \u2022 b = b\n  /-- Associativity of `\u2022` and `*` -/\n  mul_smul (a a' : GradedMonoid A) (b : GradedMonoid M) : (a * a') \u2022 b = a \u2022 a' \u2022 b\n#align graded_monoid.gmul_action GradedMonoid.GMulAction\n\n/-- The graded version of `Monoid.toMulAction`. -/\ninstance GMonoid.toGMulAction [AddMonoid \u03b9] [GMonoid A] : GMulAction A A :=\n  { GMul.toGSmul _ with\n    one_smul := GMonoid.one_mul\n    mul_smul := GMonoid.mul_assoc }\n#align graded_monoid.gmonoid.to_gmul_action GradedMonoid.GMonoid.toGMulAction\n\ninstance GMulAction.toMulAction [AddMonoid \u03b9] [GMonoid A] [GMulAction A M] :\n    MulAction (GradedMonoid A) (GradedMonoid M)\n    where\n  one_smul := GMulAction.one_smul\n  mul_smul := GMulAction.mul_smul\n#align graded_monoid.gmul_action.to_mul_action GradedMonoid.GMulAction.toMulAction\n\nend Defs\n\nend GradedMonoid\n\n/-! ### Shorthands for creating instance of the above typeclasses for collections of subobjects -/\n\n\nsection Subobjects\n\nvariable {R : Type _}\n\n/-- A version of `GradedMonoid.GSmul` for internally graded objects. -/\nclass SetLike.GradedSmul {S R N M : Type _} [SetLike S R] [SetLike N M] [SMul R M] [Add \u03b9]\n  (A : \u03b9 \u2192 S) (B : \u03b9 \u2192 N) : Prop where\n  /-- Multiplication is homogeneous -/\n  smul_mem : \u2200 \u2983i j : \u03b9\u2984 {ai bj}, ai \u2208 A i \u2192 bj \u2208 B j \u2192 ai \u2022 bj \u2208 B (i + j)\n#align set_like.has_graded_smul SetLike.GradedSmul\n\ninstance SetLike.toGSmul {S R N M : Type _} [SetLike S R] [SetLike N M] [SMul R M] [Add \u03b9]\n    (A : \u03b9 \u2192 S) (B : \u03b9 \u2192 N) [SetLike.GradedSmul A B] :\n    GradedMonoid.GSmul (fun i \u21a6 A i) fun i \u21a6 B i\n    where smul a b := \u27e8a.1 \u2022 b.1, SetLike.GradedSmul.smul_mem a.2 b.2\u27e9\n#align set_like.ghas_smul SetLike.toGSmul\n\n/-\nPorting note: simpNF linter returns\n\"Left-hand side does not simplify, when using the simp lemma on itself.\"\nHowever, simp does indeed solve the following. Possibly related std#71,std#78\nexample {S R N M : Type _} [SetLike S R] [SetLike N M] [SMul R M] [Add \u03b9]\n    (A : \u03b9 \u2192 S) (B : \u03b9 \u2192 N) [SetLike.GradedSmul A B] {i j : \u03b9} (x : A i) (y : B j) :\n    (@GradedMonoid.GSmul.smul \u03b9 (fun i \u21a6 A i) (fun i \u21a6 B i) _ _ i j x y : M) = x.1 \u2022 y.1 := by simp\n-/\n@[simp,nolint simpNF]\ntheorem SetLike.coe_GSmul {S R N M : Type _} [SetLike S R] [SetLike N M] [SMul R M] [Add \u03b9]\n    (A : \u03b9 \u2192 S) (B : \u03b9 \u2192 N) [SetLike.GradedSmul A B] {i j : \u03b9} (x : A i) (y : B j) :\n    (@GradedMonoid.GSmul.smul \u03b9 (fun i \u21a6 A i) (fun i \u21a6 B i) _ _ i j x y : M) = x.1 \u2022 y.1 :=\n  rfl\n#align set_like.coe_ghas_smul SetLike.coe_GSmul\n\n/-- Internally graded version of `Mul.toSMul`. -/\ninstance SetLike.GradedMul.toGradedSmul [AddMonoid \u03b9] [Monoid R] {S : Type _} [SetLike S R]\n    (A : \u03b9 \u2192 S) [SetLike.GradedMonoid A] : SetLike.GradedSmul A A\n    where smul_mem _ _ _ _ hi hj := SetLike.GradedMonoid.toGradedMul.mul_mem hi hj\n#align set_like.has_graded_mul.to_has_graded_smul SetLike.GradedMul.toGradedSmul\n\nend Subobjects\n\nsection HomogeneousElements\n\nvariable {S R N M : Type _} [SetLike S R] [SetLike N M]\n\ntheorem SetLike.Homogeneous.graded_smul [Add \u03b9] [SMul R M] {A : \u03b9 \u2192 S} {B : \u03b9 \u2192 N}\n    [SetLike.GradedSmul A B] {a : R} {b : M} :\n    SetLike.Homogeneous A a \u2192 SetLike.Homogeneous B b \u2192 SetLike.Homogeneous B (a \u2022 b)\n  | \u27e8i, hi\u27e9, \u27e8j, hj\u27e9 => \u27e8i + j, SetLike.GradedSmul.smul_mem hi hj\u27e9\n#align set_like.is_homogeneous.graded_smul SetLike.Homogeneous.graded_smul\n\nend HomogeneousElements\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Algebra/GradedMulAction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.39763929656737684}}
{"text": "import Euclid.synthetic\nimport Euclid.proportion\nimport Euclid.euclid_I_extras\nimport Std.Tactic.ShowTerm\n\nopen incidence_geometry\nvariable [i: incidence_geometry]\n\n/-- technical lemma that really shouldn't be here, but hey... -/\nlemma mul_mul_lt (a b c : \u211d) (hc: 0<c):\n    a<b \u2192 a*c<b*c := by\n  exact (mul_lt_mul_right hc).mpr\n/-- technical lemma that really shouldn't be here, but hey... -/\nlemma ge2_of_n1_n0 {n : \u2115}\n    (h0: n \u2260 0) (h1: n \u2260 1) :\n    n \u2265 2 := by\n  exact ge_iff_le.mpr ((Nat.two_le_iff n).mpr \u27e8 h0,h1 \u27e9)\n/-- technical lemma that really shouldn't be here, but hey... -/\nlemma gt1_of_n1_n0 {n : \u2115}\n    (h0: n \u2260 0) (h1: n \u2260 1) :\n    n > 1 := by\n  exact gt_iff_lt.mpr (lt_of_lt_of_le one_lt_two (ge_iff_le.mp (ge2_of_n1_n0 h0 h1)))\n\n/-- from segment of length l, construct a new segment of length n*l\n    based on I.3 -/\nlemma rescale_length {a b : point} {L : line} (n : \u2115)\n    (haL: online a L)\n    (hbL: online b L) :\n    \u2203 (c : point), (online c L) \u2227 (length a c = n*(length a b)) \u2227 (n \u2265 2 \u2227 a \u2260 b \u2192 B a b c) := by\n  -- trivial case\n  by_cases h_a_ne_b: a = b\n  \u00b7 use a\n    rw [h_a_ne_b.symm]\n    rw [length_eq_zero_iff.mpr,mul_zero]\n    simp only [eq_self_iff_true, Ne.def, not_true, and_false, IsEmpty.forall_iff, and_true]\n    exact haL\n    simp only [eq_self_iff_true]\n\n  induction n with\n  | zero =>\n    use a\n    rw [length_eq_zero_iff.mpr]\n    simp\n    --simp only [algebra_map.coe_zero, zero_mul, eq_self_iff_true, ge_iff_le, le_zero_iff, bit0_eq_zero, nat.one_ne_zero, false_and, is_empty.forall_iff, and_true]\n    exact haL\n    simp only [eq_self_iff_true]\n\n  | succ n hn =>\n    -- trivial case: n=0\n    by_cases hnz:(n=0)\n    \u00b7 use b\n      constructor\n      exact hbL\n      rw [hnz]\n      --simp only [algebra_map.coe_one, one_mul, eq_self_iff_true, ge_iff_le, ne.def, and_imp, true_and]\n      simp\n\n    -- separate case n=1\n    by_cases h_n_ne_1 : n=1\n    \u00b7 obtain \u27e8 e, he \u27e9 := same_length_B_of_ne_four h_a_ne_b h_a_ne_b\n      use e\n      simp\n      --simp only [Nat.cast_succ, algebra_map.coe_one]\n      rw [(length_sum_of_B he.1).symm,he.2, h_n_ne_1]\n      --simp only [nat.cast_succ, algebra_map.coe_one]\n      simp\n      ring_nf\n      simp only [eq_self_iff_true, ge_iff_le, le_refl, Ne.def, true_and]\n\n      constructor\n      exact online_3_of_B he.1 haL hbL\n      intro\n      exact he.1\n\n    -- extract point from hn\n    obtain \u27e8 d, hd \u27e9 := hn\n    have h_a_ne_d : a \u2260 d := by\n      convert nq_of_len_pos _\n      rw [hd.2.1]\n      simp only [Nat.pos_of_ne_zero hnz, zero_lt_mul_left, Nat.cast_pos]\n      exact len_pos_of_nq h_a_ne_b\n    obtain \u27e8 e, he \u27e9 := same_length_B_of_ne_four h_a_ne_d h_a_ne_b\n    use e\n    rw [(length_sum_of_B he.1).symm, hd.2.1, he.2]\n    simp only [Nat.cast_succ, ge_iff_le, Ne.def, and_imp]\n    ring_nf\n    simp only [eq_self_iff_true, true_and]\n\n    constructor\n    exact online_3_of_B he.1 haL hd.1\n\n    have : n \u2265 2 \u2227 a \u2260 b := \u27e8 ge2_of_n1_n0 hnz h_n_ne_1, h_a_ne_b \u27e9\n    have := hd.2.2 this\n\n    intros\n    exact B124_of_B134_B123 he.1 this\n\n\n/-- rescale base of triangle -/\n-- workhorse: version for use in inductive case\nlemma rescale_triangle_of_base__inductive (a : point) {b c : point} {L : line} {n : \u2115}\n    (hbL: online b L)\n    (hcL: online c L)\n    (h_a_nonline_L: \u00ac online a L) :\n    \u2200 d:point, (online d L) \u2192 (length b d = n*(length b c)) \u2192 B b c d \u2192 area a b d = n*(area a b c) := by\n  induction n with\n  | zero =>\n    intros d hdL hlen hB\n    simp\n    simp at hlen\n    --simp only [algebra_map.coe_one, zero_mul]\n    --simp only [algebra_map.coe_one, zero_mul] at hlen\n    rw [length_eq_zero_iff.mp hlen]\n    rw [area_of_eq a d d _]\n    right\n    right\n    simp only [eq_self_iff_true]\n\n  | succ n hn =>\n    intros d hdL hlen hB\n\n    have h_b_ne_c := ne_12_of_B hB\n    have h_b_ne_d := ne_13_of_B hB\n    have h_c_ne_d := ne_23_of_B hB\n\n    -- trivial case: n=0\n    by_cases h_n_ne_0 : n = 0\n    \u00b7 exfalso\n      rw [h_n_ne_0] at hlen\n      --simp only [algebra_map.coe_one, one_mul] at hlen\n      simp at hlen\n      rw [Eq.symm (length_sum_of_B hB)] at hlen\n      simp only [add_right_eq_self] at hlen\n      exact h_c_ne_d (length_eq_zero_iff.mp hlen)\n\n    -- special case: n=1\n    by_cases h_n_ne_1 : n = 1\n    \u00b7 rw [h_n_ne_1]\n      --simp only [Nat.cast_succ, algebra_map.coe_one]\n      simp\n      ring_nf\n\n      have := (area_add_iff_B h_b_ne_d h_c_ne_d.symm h_b_ne_c.symm hbL hdL hcL h_a_nonline_L).mp hB\n      rw [(@area_invariant_213 i b a d)] at this\n      rw [this.symm]\n      rw [(area_invariant b c a).1]\n      rw [Eq.symm (eq_area_of_eq_base_samevertex a hbL hcL hcL hdL _)]\n      ring\n\n      rw [h_n_ne_1, (length_sum_of_B hB).symm] at hlen\n      --simp only [Nat.cast_succ, algebra_map.coe_one] at hlen\n      simp at hlen\n      rw [add_mul] at hlen\n      simp only [one_mul, add_right_inj] at hlen\n      exact hlen.symm\n\n    -- split off n+1'st bit\n    simp only [Nat.cast_succ]\n    rw [add_mul]\n    simp only [one_mul]\n\n    -- construct n-triangle\n    obtain \u27e8 e, he \u27e9 := rescale_length n hbL hcL\n    have h_B_bce := he.2.2 \u27e8 ge2_of_n1_n0 h_n_ne_0 h_n_ne_1, h_b_ne_c \u27e9\n    rw [Eq.symm (hn e he.1 he.2.1 h_B_bce)]\n\n    have h_b_ne_e := ne_13_of_B h_B_bce\n    have h_e_ne_d : e \u2260 d := by\n      by_contra contra\n      have := he.2.1\n      rw [contra] at this\n      have := Eq.trans this.symm hlen\n      simp only [Nat.cast_succ, mul_eq_mul_right_iff, self_eq_add_right, one_ne_zero, false_or] at this\n      exact h_b_ne_c (length_eq_zero_iff.mp this)\n\n    -- split abd\n    have h_B_bed : B b e d := by\n      have' := B_of_three_online_ne_short _ h_b_ne_e h_b_ne_d h_e_ne_d hbL he.1 hdL\n      \n      cases this with\n      | inl h =>\n        exact h\n\n      |inr h =>\n        exfalso\n        have h1 := not_B324_of_B123_B124 h_B_bce hB\n        have := B124_of_B134_B123 h (B_symm h_B_bce)\n        exact h1 this\n\n      rw [he.2.1,hlen]\n      simp only [Nat.cast_succ]\n      have : \u2191n < \u2191n+1 := by simp\n      linarith [len_pos_of_nq h_b_ne_c]\n    \n    have := (area_add_iff_B h_b_ne_d h_e_ne_d.symm h_b_ne_e.symm hbL hdL he.1 h_a_nonline_L).mp h_B_bed\n    rw [@area_invariant_213 i b a d] at this\n    rw [this.symm]\n\n    have' := eq_area_of_eq_base_samevertex a hbL hcL he.1 hdL _\n    rw [this]\n    rw [@area_invariant_312 i b e a]\n\n    have := length_sum_of_B h_B_bed\n    rw [hlen, he.2.1] at this\n    simp only [Nat.cast_succ] at this\n    rw [add_mul] at this\n    simp only [one_mul, add_right_inj] at this\n    exact this.symm\n\n-- lemma with B b c d as a hypothesis\nlemma rescale_triangle_of_base__B (a : point) {b c d : point} {L : line} {n : \u2115}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hdL: online d L)\n    (hlen: length b d = n*(length b c))\n    (hB: B b c d)\n    (h_a_nonline_L: \u00ac online a L) :\n    area a b d = n*(area a b c) := by\n  exact rescale_triangle_of_base__inductive a hbL hcL h_a_nonline_L d hdL hlen hB\n\n-- not B c b d\nlemma rescale_triangle_of_base__notcbd (a : point) {b c d : point} {L : line} {n : \u2115}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hdL: online d L)\n    (hlen: length b d = n*(length b c))\n    (hB: \u00ac B c b d)\n    (h_b_ne_c: b \u2260 c)\n    (h_n_ne_0: n \u2260 0)\n    (h_b_ne_d: b \u2260 d)\n    (h_a_nonline_L: \u00ac online a L) :\n    area a b d = n*(area a b c) := by\n  -- trivial case: n=1\n  by_cases h_n_ne_1 : n=1\n  \u00b7 rw [h_n_ne_1] at hlen\n    rw [h_n_ne_1]\n    --simp only [algebra_map.coe_one, one_mul] at hlen\n    --simp only [algebra_map.coe_one, one_mul]\n    simp at hlen\n    simp\n    exact Eq.symm (eq_area_of_eq_base_samevertex a hbL hcL hbL hdL hlen.symm)\n\n  have h_c_ne_d: c \u2260 d := by\n    by_contra contra\n    rw [contra] at hlen\n    have hnz :=(not_iff_not.mpr length_eq_zero_iff).mpr h_b_ne_d\n    have := (mul_left_inj' hnz).mp (Eq.trans (one_mul (length b d)) hlen)\n    norm_cast at this\n    exact h_n_ne_1 this.symm\n\n  have' hBs := B_of_three_online_ne_short _ h_b_ne_c h_b_ne_d h_c_ne_d hbL hcL hdL\n  cases hBs with\n  | inl hBs =>\n    exact rescale_triangle_of_base__B a hbL hcL hdL hlen hBs h_a_nonline_L\n  |inr hBs =>\n    exfalso\n    exact hB hBs\n\n  rw [hlen]\n  have := len_pos_of_nq h_b_ne_c\n  have n_ge1 := gt1_of_n1_n0 h_n_ne_0 h_n_ne_1\n  rw [gt_iff_lt] at n_ge1\n  -- I can't replace this by the one line proof of mul_mul_lt, for some annoying coercion reason...\n  have' := mul_mul_lt 1 \u2191n (length b c) this _\n  simp only [one_mul] at this\n  exact this\n  norm_cast\n\n-- full version\nlemma rescale_triangle_of_base (a : point) {b c d : point} {L : line} {n : \u2115}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hdL: online d L)\n    (hlen: length b d = n*(length b c)) :\n    area a b d = n*(area a b c) := by\n  -- trivial case: b=c\n  by_cases h_b_ne_c : b=c\n  \u00b7 rw [(area_of_eq a b d _)]\n    rw [(area_of_eq a b c _)]\n    simp only [mul_zero]\n    right\n    right\n    exact h_b_ne_c\n    right\n    right\n    rw [length_eq_zero_iff.mpr h_b_ne_c] at hlen\n    simp only [mul_zero] at hlen\n    exact length_eq_zero_iff.mp hlen\n\n  -- trivial case: n=0\n  by_cases h_n_ne_0 : n=0\n  \u00b7 rw [h_n_ne_0]\n    --simp only [algebra_map.coe_zero, zero_mul]\n    simp\n    rw [area_of_eq a b d _]\n    right;right\n    rw [h_n_ne_0] at hlen\n    --simp only [algebra_map.coe_zero, zero_mul] at hlen\n    simp at hlen\n    exact length_eq_zero_iff.mp hlen\n\n  have h_b_ne_d: b \u2260 d := by\n    by_contra contra\n    have := length_eq_zero_iff.mpr contra\n    rw [this] at hlen\n    simp only [zero_eq_mul, Nat.cast_eq_zero] at hlen\n    cases hlen with\n    | inl hlen =>\n      exact h_n_ne_0 hlen\n    | inr hlen =>\n      exact h_b_ne_c (length_eq_zero_iff.mp hlen)\n\n  -- trivial case: online a L\n  by_cases h_a_nonline_L : online a L\n  \u00b7 have := (area_zero_iff_online h_b_ne_c hbL hcL).mpr h_a_nonline_L\n    rw [(area_invariant b c a).1] at this\n    rw [this]\n    simp only [mul_zero]\n    have := (area_zero_iff_online h_b_ne_d hbL hdL).mpr h_a_nonline_L\n    rw [(area_invariant b d a).1] at this\n    rw [this]\n\n  by_cases h_B_cbd : B c b d\n    -- reflect c about b\n  \u00b7 obtain \u27e8 e, he \u27e9 := rescale_length 2 hcL hbL\n    have h_B_cbe : B c b e := by\n      have : 2 \u2265 2 \u2227 c \u2260 b := by\n        constructor\n        simp only [ge_iff_le]\n        rw [(Ne.def b c).symm] at h_b_ne_c\n        exact h_b_ne_c.symm\n      exact  he.2.2 this\n\n    have lbe : length b e = length b c := by\n      have := length_sum_of_B h_B_cbe\n      rw [he.2.1, length_symm c b] at this\n      norm_cast at this\n      linarith\n\n    rw [lbe.symm] at hlen\n\n    have := not_B324_of_B123_B124 h_B_cbe h_B_cbd\n\n    rw [rescale_triangle_of_base__notcbd a hbL he.1 hdL hlen this (ne_23_of_B h_B_cbe) h_n_ne_0 h_b_ne_d h_a_nonline_L]\n    rw [eq_area_of_eq_base_samevertex a hbL he.1 hbL hcL lbe]\n\n  exact rescale_triangle_of_base__notcbd a hbL hcL hdL hlen h_B_cbd h_b_ne_c h_n_ne_0 h_b_ne_d h_a_nonline_L\n\n\n/-- triangles between parallels with smaller base have smaller area -/\n-- case where they share a side and have the right betweeneness\nlemma lt_area_of_lt_base__sameedge_Bbfc (a : point) {b c f: point} {L: line}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hfL: online f L)\n    (hB: B b f c)\n    (h_b_ne_c: b \u2260 c)\n    (h_b_ne_f: b \u2260 f)\n    (h_c_ne_f: c \u2260 f)\n    (h_a_nonline_L: \u00ac online a L) :\n    (length b c)>(length b f) \u2192 (area a b c)>(area a b f) := by\n  intro\n  have := (area_add_iff_B h_b_ne_c h_c_ne_f h_b_ne_f.symm hbL hcL hfL h_a_nonline_L).mp hB\n  rw [@area_invariant_213 i a b c]\n  rw [@area_invariant_231 i a b f]\n  rw [this.symm]\n  simp only [gt_iff_lt, lt_add_iff_pos_right]\n  have : area a f c \u2260 0 := by\n    by_contra contra\n    rw [area_invariant_321] at contra\n    rw [(area_zero_iff_online h_c_ne_f hcL hfL)] at contra\n    exact h_a_nonline_L contra\n  exact (Ne.symm this).lt_of_le (area_nonneg a f c)\n\n-- case where they share a side and not B f b c\nlemma lt_area_of_lt_base__sameedge_nBfbc (a : point) {b c f: point} {L: line}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hfL: online f L)\n    (h_b_ne_c: b \u2260 c)\n    (h_b_ne_f: b \u2260 f)\n    (h_a_nonline_L: \u00ac online a L)\n    (hB: \u00ac B f b c) :\n    (length b c)>(length b f) \u2192 (area a b c)>(area a b f) := by\n  intro hlen\n  simp only [gt_iff_lt] at hlen\n\n  have h_c_ne_f: c \u2260 f := by\n    by_contra contra\n    rw [contra] at hlen\n    simp only [gt_iff_lt, lt_self_iff_false] at hlen\n\n  have := B_of_three_online_ne_short hlen h_b_ne_f h_b_ne_c h_c_ne_f.symm hbL hfL hcL\n\n  cases this with\n  | inl h =>\n    exact lt_area_of_lt_base__sameedge_Bbfc a hbL hcL hfL h h_b_ne_c h_b_ne_f h_c_ne_f h_a_nonline_L hlen\n  | inr h =>\n    exfalso\n    exact hB h\n\n-- case where they share a side\nlemma lt_area_of_lt_base__sameedge (a : point) {b c f: point} {L: line}\n    (hbL: online b L)\n    (hcL: online c L)\n    (hfL: online f L)\n    (h_b_ne_c: b \u2260 c)\n    (h_b_ne_f: b \u2260 f)\n    (h_a_nonline_L: \u00ac online a L) :\n    (length b c)>(length b f) \u2192 (area a b c)>(area a b f) := by\n  intro hlen\n\n  by_cases hB: B f b c\n    -- reflect f about b\n  \u00b7 obtain \u27e8 e, he \u27e9 := rescale_length 2 hfL hbL\n    have : 2 \u2265 2 := by simp only [ge_iff_le]\n    have h_B_fbe := he.2.2 \u27e8 this, h_b_ne_f.symm \u27e9\n    have lbe : length b e = length b f := by\n      have := length_sum_of_B h_B_fbe\n      rw [he.2.1] at this\n      rw [length_symm f b] at this\n      norm_cast at this\n      linarith\n    rw [lbe.symm] at hlen\n\n    rw [(eq_area_of_eq_base_samevertex a hbL he.1 hbL hfL lbe).symm]\n\n    have := not_B324_of_B123_B124 h_B_fbe hB\n    exact lt_area_of_lt_base__sameedge_nBfbc a hbL hcL he.1 h_b_ne_c (ne_23_of_B h_B_fbe) h_a_nonline_L this hlen\n  \n  exact lt_area_of_lt_base__sameedge_nBfbc a hbL hcL hfL h_b_ne_c h_b_ne_f h_a_nonline_L hB hlen\n\n-- general case\nlemma lt_area_of_lt_base {a b c d e f: point} {L M: line}\n    (haM: online a M)\n    (hbL: online b L)\n    (hcL: online c L)\n    (hdM: online d M)\n    (heL: online e L)\n    (hfL: online f L)\n    (hpar: para L M) :\n    (length b c)>(length e f) \u2192 (area a b c)>(area d e f) := by\n\n  intro hlen\n\n  have h_b_ne_c: b \u2260 c := by\n    by_contra contra\n    rw [(length_eq_zero_iff.mpr contra)] at hlen\n    exact (not_le_of_gt hlen) (length_nonneg e f)\n\n  have h_a_nonline_L := online_of_online_para haM (para_symm hpar)\n  have h_d_nonline_L := online_of_online_para hdM (para_symm hpar)\n\n  -- trivial case: e=f\n  by_cases h_e_ne_f: e=f\n  \u00b7 simp only [gt_iff_lt]\n    rw [area_of_eq d e f _]\n    have := (area_zero_iff_online h_b_ne_c hbL hcL).not.mpr h_a_nonline_L\n    rw [(area_invariant b c a).1] at this\n    exact (Ne.symm this).lt_of_le (area_nonneg a b c)\n    right;right\n    exact h_e_ne_f\n\n  -- construct parallelogram from d e f\n  obtain \u27e8 O, hO \u27e9 := line_of_pts d e\n  have := not_online_of_triangle hO.1 hO.2 heL hfL h_d_nonline_L h_e_ne_f\n  obtain \u27e8 g, N, hg \u27e9 := parallel_projection hfL hpar (not_para_of_online_online hO.2 heL) this\n\n  have hlen_ef_dg := (parasianar heL hfL hdM hg.1 hO.2 hO.1 hg.2.2.1 hg.2.1 hpar hg.2.2.2).1\n\n  have h_d_ne_g : d \u2260 g := by\n    have := length_eq_zero_iff.not.mpr h_e_ne_f\n    rw [hlen_ef_dg] at this\n    exact length_eq_zero_iff.not.mp this\n\n  -- construct parallelogram from b d g\n  obtain \u27e8 P, hP \u27e9 := line_of_pts b d\n  have := not_online_of_triangle hP.1 hP.2 hdM hg.1 (online_of_online_para hbL hpar) h_d_ne_g\n  obtain \u27e8 h, R, hh \u27e9 := parallel_projection hg.1 (para_symm hpar) (not_para_of_online_online hP.2 hdM) this\n\n  have hlen_ef_bh : length e f = length b h := by\n    have := (parasianar hdM hg.1 hbL hh.1 hP.2 hP.1 hh.2.2.1 hh.2.1 (para_symm hpar) hh.2.2.2).1\n    rw [this.symm, hlen_ef_dg.symm]\n\n  have h_b_ne_h : b \u2260 h := by\n    have := length_eq_zero_iff.not.mpr h_e_ne_f\n    rw [hlen_ef_bh] at this\n    exact length_eq_zero_iff.not.mp this\n\n  have : length b c > length b h := by\n    rw [hlen_ef_bh.symm]\n    exact hlen\n\n  rw [eq_area_of_eq_base hdM heL hfL haM hbL hh.1 hpar hlen_ef_bh]\n\n  exact lt_area_of_lt_base__sameedge a hbL hcL hh.1 h_b_ne_c h_b_ne_h h_a_nonline_L this\n  \n\n/-- ## Euclid VI.1\ntwo triangles in between parallel lines have their area in proportion with the length of their base\n https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI1.html -/\ntheorem proportion_area_of_proportion_base {a b c d e f: point} {L M : line}\n    (haM: online a M)\n    (hbL: online b L)\n    (hcL: online c L)\n    (hdM: online d M)\n    (heL: online e L)\n    (hfL: online f L)\n    (hpar: para L M) :\n    proportion (length b c) (length e f) (area a b c) (area d e f) := by\n  dsimp [proportion]\n\n  refine' \u27e8 length_nonneg b c, length_nonneg e f, area_nonneg a b c, area_nonneg d e f, _ \u27e9\n  intros n m\n\n  obtain \u27e8 h, hh \u27e9 := rescale_length n hbL hcL\n  obtain \u27e8 l, hl \u27e9 := rescale_length m heL hfL\n\n  rw [(rescale_triangle_of_base a hbL hcL hh.1 hh.2.1).symm, hh.2.1.symm]\n  rw [(rescale_triangle_of_base d heL hfL hl.1 hl.2.1).symm, hl.2.1.symm]\n\n  constructor\n  exact eq_area_of_eq_base haM hbL hh.1 hdM heL hl.1 hpar\n  constructor\n  exact lt_area_of_lt_base hdM heL hl.1 haM hbL hh.1 hpar\n  exact lt_area_of_lt_base haM hbL hh.1 hdM heL hl.1 hpar\n\n\n/-- version where the vertex is the same for both triangles -/\ntheorem proportion_area_of_proportion_base_samevertex (a : point) {b c e f: point} {L : line}\n    (hbL: online b L)\n    (hcL: online c L)\n    (heL: online e L)\n    (hfL: online f L)\n    (h_a_nonline_L: \u00ac online a L) :\n    proportion (length b c) (length e f) (area a b c) (area a e f) := by\n  obtain \u27e8 M, hM \u27e9 := parallel_of_line_pt h_a_nonline_L\n  exact proportion_area_of_proportion_base hM.1 hbL hcL hM.1 heL hfL hM.2\n\n\n/-- ## Euclid VI.2\na line cuts the sides of the triangle proportionally iff it is parallel to one of the sides of a triangle (version BD:AD = CE:AE)\n https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI2.html -/\ntheorem proportional_iff_para {a b c d e: point} {L M N: line}\n    (hdL: online d L)\n    (heL: online e L)\n    (hbM: online b M)\n    (hcM: online c M)\n    (haN: online a N)\n    (hdN: online d N)\n    (hneN: \u00ac online e N)\n    (Badb : B a d b) (Baec : B a e c) :\n    proportion (length b d) (length a d) (length c e) (length a e) \u2194 para L M := by\n  -- N\n  have hbN := online_3_of_B Badb haN hdN\n\n  -- O\n  obtain \u27e8O, hO\u27e9 := line_of_pts a e\n  -- for compatibility\n  have haO := hO.1\n  have heO := hO.2\n  have hcO := online_3_of_B Baec haO heO\n\n  -- P\n  obtain \u27e8P, hP\u27e9 := line_of_pts c d\n  -- for compatibility\n  have hcP := hP.1\n  have hdP := hP.2\n\n  -- non-degeneracy\n  have ab : a \u2260 b := ne_13_of_B Badb\n  have ae : a \u2260 e := ne_12_of_B Baec\n  have ad : a \u2260 d := ne_12_of_B Badb\n  have bd : b \u2260 d := ne_12_of_B (B_symm Badb)\n  have ce : c \u2260 e := ne_12_of_B (B_symm Baec)\n  have NO : N \u2260 O := by\n    by_contra contra\n    rw [contra] at hneN\n    exact hneN heO\n  have PO : P \u2260 O := by\n    by_contra contra\n    rw [contra] at hdP\n    exact NO (line_unique_of_pts ad haO hdP haN hdN).symm\n  have hndO: \u00ac online d O := by\n    by_contra contra\n    exact NO (line_unique_of_pts ad haN hdN haO contra)\n  have hnaL:= fun haL => hneN (by rwa [line_unique_of_pts ad haL hdL haN hdN] at heL)\n  have hnbL:= fun hbL => hneN (by rwa [line_unique_of_pts bd hbL hdL hbN hdN] at heL)\n  have hncL:= fun hcL => (by rwa [\u2190 line_unique_of_pts ce hcO heO hcL heL] at hnaL : \u00ac online a O) haO\n  have bc : b \u2260 c := by\n    by_contra contra\n    rw [contra.symm] at hcO\n    exact NO (line_unique_of_pts ab haN hbN haO hcO)\n  have cd : c \u2260 d := by\n    by_contra contra\n    rw [contra.symm] at hndO\n    exact hndO hcO\n  have de : d \u2260 e := by\n    by_contra contra\n    rw [contra] at hndO\n    exact hndO heO\n  have hneP : \u00ac online e P := by\n    by_contra contra\n    exact PO (line_unique_of_pts ce hcP contra hcO heO)\n\n  have hbdad : proportion (length b d) (length a d) (area e b d) (area e a d) := by\n    exact proportion_area_of_proportion_base_samevertex e hbN hdN haN hdN hneN\n  have hceae_by_prop_vi_i : proportion (length c e) (length a e) (area d c e) (area d a e) := by\n   exact proportion_area_of_proportion_base_samevertex d hcO heO haO heO hndO\n  have ead_eq_dae : area e a d = area d a e := by\n    rw [(area_invariant e a d).1,(area_invariant d e a).2]\n  have ead_eq_ade : area e a d = area a d e := by\n    rw [(area_invariant a d e).1]\n  have ebd_eq_bde : area e b d = area b d e := by\n    rw [(area_invariant b d e).1]\n  have dce_eq_cde : area d c e = area c d e := by\n    rw [(area_invariant d c e).2, (area_invariant d e c).1]\n  have len_ad_neq_0 : length a d \u2260 0 := by\n    by_contra contra\n    have h1 : a = d := by exact (length_eq_zero_iff.1 contra)\n    exact ad h1\n  have len_ae_neq_0 : length a e \u2260 0 := by\n    by_contra contra\n    have h1 : a = e := by exact (length_eq_zero_iff.1 contra)\n    exact ae h1\n  have area_ade_neq_0 : area a d e \u2260 0 := by\n    by_contra contra\n    exact hneN ((area_zero_iff_online ad haN hdN).1 contra)\n  have area_ead_neq_0 : area e a d \u2260 0 := by rwa [\u2190 ead_eq_ade] at area_ade_neq_0\n  have area_dae_neq_0 : area d a e \u2260 0 := by rwa [ead_eq_dae] at area_ead_neq_0\n  have ratio_bd_ad := eq_ratio_of_proportion len_ad_neq_0 area_ead_neq_0 hbdad\n  have ratio_ce_ae := eq_ratio_of_proportion len_ae_neq_0 area_dae_neq_0 hceae_by_prop_vi_i\n  have ratio_iff : length b d / length a d = length c e / length a e \u2194 area e b d / area e a d = area d c e / area d a e := by rw [ratio_bd_ad, ratio_ce_ae]\n  have proportion_lhs : proportion (length b d) (length a d) (length c e) (length a e) \u2194 area e b d / area e a d = area d c e / area d a e := by\n    rwa [\u2190 proportion_iff]\n    exact length_nonneg b d\n    exact length_nonneg a d\n    exact length_nonneg c e\n    exact length_nonneg a e\n    exact len_ad_neq_0\n    exact len_ae_neq_0\n  have area_bde_eq_cde : proportion (length b d) (length a d) (length c e) (length a e) \u2194 area b d e = area c d e := by\n    rw [proportion_lhs, ead_eq_dae, ebd_eq_bde, dce_eq_cde]\n    apply div_left_inj' area_dae_neq_0\n  constructor\n  -- apply I.3\n  intro harea\n  rw [area_bde_eq_cde] at harea\n  have nsabL:= not_sameside13_of_B123_online2 Badb hdL\n  have nsacL:= not_sameside13_of_B123_online2 Baec heL\n  have ssbcL := sameside_of_diffside_diffside \u27e8hnaL, hnbL, nsabL\u27e9 \u27e8hnaL, hncL, nsacL\u27e9\n  apply eq_area_of_same_base_implies_para hdL heL hnbL hbM hcM hdP hcP hneP bc de cd.symm ssbcL harea\n  -- apply I.3\n  intro pLM\n  rw [area_bde_eq_cde]\n  apply para_implies_eq_area_of_same_base hbM hdL heL hcM pLM\n\n\n/-- ## Euclid VI.2'\na line cuts the sides of the triangle proportionally iff it is parallel to one of the sides of a triangle (version AB:AD = AC:AE)\n https://mathcs.clarku.edu/~djoyce/java/elements/bookVI/propVI2.html -/\ntheorem proportional_iff_para' {a b c d e: point} {L M N: line}\n    (hdL: online d L)\n    (heL: online e L)\n    (hbM: online b M)\n    (hcM: online c M)\n    (haN: online a N)\n    (hdN: online d N)\n    (hneN: \u00ac online e N)\n    (Badb : B a d b) (Baec : B a e c) :\n    proportion (length a b) (length a d) (length a c) (length a e) \u2194 para L M := by\n  have ad : length a d \u2260 0 := length_eq_zero_iff.not.mpr (ne_12_of_B Badb)\n  have ae : length a e \u2260 0 := length_eq_zero_iff.not.mpr (ne_12_of_B Baec)\n\n  have := proportional_iff_para hdL heL hbM hcM haN hdN hneN Badb Baec\n\n  rw [(proportion_iff (length_nonneg a b) (length_nonneg a d) (length_nonneg a c) (length_nonneg a e) ad ae).symm]\n  rw [(proportion_iff (length_nonneg b d) (length_nonneg a d) (length_nonneg c e) (length_nonneg a e) ad ae).symm] at this\n  rw [(length_sum_of_B Badb).symm]\n  rw [(length_sum_of_B Baec).symm]\n  simp only [add_div] at this\n  simp only [add_div]\n  rw [div_self ad, div_self ae]\n  simp only [add_right_inj]\n  rw [length_symm d b]\n  rw [length_symm e c]\n  exact this\n\n\n/-- Three points are on the same line -/\ndef colinear (a b c : point) := \u2203 L :\n    line, online a L \u2227 online b L \u2227 online c L   \n\n/-- colinear is symmetric under odd permutation -/\ntheorem colinear_symm1 :\n    colinear a b c \u2194 colinear b a c := by\n  dsimp [colinear]\n  constructor\n  all_goals\n    intro L\n    obtain \u27e8 l,hl \u27e9 := L\n    use l\n    exact and_left_comm.mp hl\n\n/-- colinear is symmetric under even permutation -/\ntheorem colinear_symm2 :\n    colinear a b c \u2194 colinear b c a := by\n  dsimp [colinear]\n  constructor\n  all_goals\n    intro L\n    obtain \u27e8 l,hl \u27e9 := L\n    use l\n  exact and_rotate.mp hl\n  exact and_rotate.mpr hl\n\n/-- equal points are colinear -/\nlemma colinear_of_eq_23 (a b : point) :\n    colinear a b b := by\n  dsimp [colinear]\n  simp only [and_self]\n  exact line_of_pts a b\n\n/-- equal points are colinear -/\nlemma colinear_of_eq_12 (a b : point) :\n    colinear a a b := by\n  rw [colinear_symm2]\n  rw [colinear_symm2]\n  exact colinear_of_eq_23 b a\n/-- equal points are colinear -/\nlemma colinear_of_eq_13 (a b : point) :\n    colinear a b a := by\n  rw [colinear_symm2]\n  exact colinear_of_eq_23 b a\n\n/-- not colinear implies different -/\nlemma neq_12_of_not_colinear {a b c : point} (h: \u00ac colinear a b c) :\n    a \u2260 b := by\n  by_contra contra\n  rw [contra] at h\n  exact h (colinear_of_eq_12 b c)\n/-- not colinear implies different -/\nlemma neq_13_of_not_colinear {a b c : point} (h: \u00ac colinear a b c) :\n    a \u2260 c := by\n  by_contra contra\n  rw [contra] at h\n  exact h (colinear_of_eq_13 c b)\n/-- not colinear implies different -/\nlemma neq_23_of_not_colinear {a b c : point} (h: \u00ac colinear a b c) :\n    b \u2260 c := by\n  by_contra contra\n  rw [contra] at h\n  exact h (colinear_of_eq_23 a c)\n\n/-- not colinear implies one of the points is not aligned -/\nlemma not_online_of_not_colinear {a b c : point} {L : line} (haL: online a L) (hbL : online b L) (h: \u00ac colinear a b c) :\n    \u00ac online c L := by\n  dsimp [colinear] at h\n  simp only [not_exists, not_and] at h\n  exact h L haL hbL\n    \n\n/-- similar triangles (should follow from Euclid VI.2) -/\n-- show resulting lines are parallel\nlemma parallel_of_similar {a b c g h : point} {AB AC BC HG: line}\n    (haAB: online a AB)\n    (hbAB: online b AB)\n    (hhAB: online h AB)\n    (haAC: online a AC)\n    (hcAC: online c AC)\n    (hgAC: online g AC)\n    (hbBC: online b BC)\n    (hcBC: online c BC)\n    (hhHG: online h HG)\n    (hgHG: online g HG)\n    (b_ne_h: b \u2260 h)\n    (a_ne_b: a \u2260 b)\n    (a_ne_c: a \u2260 c)\n    (b_ne_c: b \u2260 c)\n    (h_ne_g: h \u2260 g)\n    (a_ne_g: a \u2260 g)\n    (b_nonline_AC: \u00ac online b AC)\n    (an: angle a h g = angle a b c)\n    (hB: B a h b)\n    (hss: sameside g c AB) :\n    para BC HG := by\n\n  have hbc_abc : angle h b c = angle a b c := by\n    refine' angle_extension b_ne_h.symm a_ne_b b_ne_c.symm b_ne_c.symm hbAB hhAB haAB hbBC hcBC hcBC (not_B_of_B (B_symm hB)) _\n    by_contra contra\n    have := ne_13_of_B contra\n    exact this rfl\n\n  -- point on other side of h on line hg\n  obtain \u27e8t, ht\u27e9 := pt_extension_of_ne h_ne_g.symm\n\n  have t_online_HG := online_3_of_B ht hgHG hhHG\n\n  have t_nonline_AB : \u00ac online t AB := by\n    by_contra contra\n    have AB_HG := line_unique_of_pts (ne_23_of_B ht).symm contra hhAB t_online_HG hhHG\n    rw [AB_HG] at haAB\n    have : a = g := by\n      by_contra contra\n      have := line_unique_of_pts contra haAC hgAC haAB hgHG\n      rw [AB_HG.symm] at this\n      rw [this.symm] at hbAB\n      exact b_nonline_AC hbAB\n    exact a_ne_g this\n\n  refine' para_symm (angeqpar (ne_23_of_B ht).symm b_ne_h.symm b_ne_c _ hhHG hbBC hcBC hhAB hbAB _ _)\n\n  exact online_3_of_B ht hgHG hhHG\n\n  have := vertang haAB hbAB t_nonline_AB hB (B_symm ht)\n  rw [angle_symm, this.symm, angle_symm]\n  rw [an]\n  exact hbc_abc.symm\n\n  have nss := not_sameside13_of_B123_online2 ht hhAB\n  dsimp [diffside]\n\n  refine' \u27e8t_nonline_AB, _, _\u27e9\n  swap\n\n  by_contra contra\n  exact nss (sameside_trans (sameside_symm hss) (sameside_symm contra))\n\n  by_contra contra\n  have := line_unique_of_pts a_ne_c haAC hcAC haAB contra\n  rw [this] at b_nonline_AC\n  exact b_nonline_AC hbAB\n\n/-- two similar triangles that share an edge are equal -/\nlemma length_eq_of_length_eq {a b c d e f : point}\n    (tri_abc : \u00ac colinear a b c) (tri_def : \u00ac colinear d e f) \n    (ang_a_eq_d : angle b a c = angle e d f) (ang_b_eq_e : angle a b c = angle d e f)\n    (leq: length d f = length a c) :\n    length d e = length a b := by\n  have d_ne_f := neq_13_of_not_colinear tri_def\n  have a_ne_c := neq_13_of_not_colinear tri_abc\n  have d_ne_e := neq_12_of_not_colinear tri_def\n  have a_ne_b := neq_12_of_not_colinear tri_abc\n  have b_ne_c := neq_23_of_not_colinear tri_abc\n  have e_ne_f := neq_23_of_not_colinear tri_def\n\n  obtain \u27e8AC, hAC\u27e9 := line_of_pts a c\n  obtain \u27e8AB, hAB\u27e9 := line_of_pts a b\n  obtain \u27e8BC, hBC\u27e9 := line_of_pts b c\n  have c_nonline_AB := not_online_of_not_colinear hAB.1 hAB.2 tri_abc\n\n  rw [colinear_symm1] at tri_abc\n  rw [colinear_symm2] at tri_abc\n  have b_nonline_AC := not_online_of_not_colinear hAC.1 hAC.2 tri_abc\n\n  by_contra contra\n  rw [(Ne.def (length d e) (length a b)).symm] at contra\n  simp_rw [ne_iff_lt_or_gt] at contra\n\n  wlog lineq : length a b < length d e\n  swap\n\n  obtain \u27e8h, hh\u27e9 := same_length_B_of_ne_ge a_ne_b lineq\n  obtain \u27e8HC, hHC\u27e9 := line_of_pts h c\n\n  have h_online_AB := online_3_of_B hh.1 hAB.1 hAB.2\n  have h_ne_c : h \u2260 c := by\n    by_contra contra\n    rw [contra] at h_online_AB\n    exact c_nonline_AB h_online_AB\n\n  have h_nonline_AC : \u00ac online h AC := by\n    by_contra contra\n    have := line_unique_of_pts (ne_13_of_B hh.1).symm contra hAC.1 h_online_AB hAB.1\n    rw [this] at b_nonline_AC\n    exact b_nonline_AC hAB.2\n\n  have hac_bac := (angle_extension_of_B a_ne_c hh.1).symm\n  rw [hac_bac.symm] at ang_a_eq_d\n  rw [angle_symm] at ang_a_eq_d\n  rw [@angle_symm i e d f] at ang_a_eq_d\n  have ang_b_eq_h := sas leq.symm hh.2 ang_a_eq_d\n  rw [ang_b_eq_e.symm] at ang_b_eq_h\n\n  have := parallel_of_similar hAB.1 h_online_AB hAB.2 hAC.1 hAC.2 hAC.2 hHC.1 hHC.2 hBC.1 hBC.2 (ne_23_of_B hh.1).symm (ne_13_of_B hh.1) a_ne_c h_ne_c b_ne_c a_ne_c h_nonline_AC ang_b_eq_h.2.2.symm hh.1 (sameside_rfl_of_not_online c_nonline_AB)\n\n  have := neq_of_para hHC.2 hBC.2 this\n  exact this rfl\n\n  rw [colinear_symm1] at tri_abc\n  rw [colinear_symm2] at tri_abc\n  obtain \u27e8DF, hDF\u27e9 := line_of_pts d f\n  obtain \u27e8DE, hDE\u27e9 := line_of_pts d e\n  obtain \u27e8EF, hEF\u27e9 := line_of_pts e f\n  have f_nonline_DE := not_online_of_not_colinear hDE.1 hDE.2 tri_def\n  rw [colinear_symm1] at tri_def\n  rw [colinear_symm2] at tri_def\n  have e_nonline_DF := not_online_of_not_colinear hDF.1 hDF.2 tri_def\n\n  refine' this tri_def tri_abc ang_a_eq_d.symm ang_b_eq_e.symm leq.symm a_ne_c d_ne_f a_ne_b d_ne_e e_ne_f b_ne_c DF hDF DE hDE EF hEF f_nonline_DE e_nonline_DF _ _\n  exact Or.symm contra\n  cases contra with\n  | inl contra =>\n    exact contra\n  | inr contra =>\n    exfalso\n    linarith\n\n/-- Given two similar triangles, if the side of one triangle is smaller than that of the second,\nthen the remaining sides are also smaller -/\nlemma length_lt_of_length_lt {a b c d e f : point}\n    (tri_abc : \u00ac colinear a b c) (tri_def : \u00ac colinear d e f) \n    (ang_a_eq_d : angle b a c = angle e d f) (ang_b_eq_e : angle a b c = angle d e f)\n    (lineq: length d f < length a c) :\n    length d e < length a b := by\n  have d_ne_f := neq_13_of_not_colinear tri_def\n  have a_ne_c := neq_13_of_not_colinear tri_abc\n  have d_ne_e := neq_12_of_not_colinear tri_def\n  have a_ne_b := neq_12_of_not_colinear tri_abc\n  have b_ne_c := neq_23_of_not_colinear tri_abc\n  have e_ne_f := neq_23_of_not_colinear tri_def\n\n  obtain \u27e8AC, hAC\u27e9 := line_of_pts a c\n  obtain \u27e8AB, hAB\u27e9 := line_of_pts a b\n  obtain \u27e8BC, hBC\u27e9 := line_of_pts b c\n  have c_nonline_AB := not_online_of_not_colinear hAB.1 hAB.2 tri_abc\n\n  rw [colinear_symm1] at tri_abc\n  rw [colinear_symm2] at tri_abc\n\n  have b_nonline_AC := not_online_of_not_colinear hAC.1 hAC.2 tri_abc\n\n  obtain \u27e8g, hg\u27e9 := same_length_B_of_ne_le d_ne_f lineq\n  have g_online_AC := online_2_of_B hg.1 hAC.1 hAC.2\n  have g_nonline_BC : \u00ac online g BC := by\n    by_contra contra\n    have := line_unique_of_pts (ne_23_of_B hg.1) contra hBC.2 g_online_AC hAC.2\n    rw [this] at hBC\n    exact b_nonline_AC hBC.1\n  have g_nonline_AB : \u00ac online g AB := by\n    by_contra contra\n    have := line_unique_of_pts (ne_12_of_B hg.1).symm contra hAB.1 g_online_AC hAC.1\n    rw [this] at hAB\n    exact b_nonline_AC hAB.2\n\n  by_contra contra\n  rw [not_lt, le_iff_lt_or_eq] at contra\n\n  cases contra with\n\n  | inl contra =>\n    obtain \u27e8h, hh\u27e9 := same_length_B_of_ne_ge a_ne_b contra\n    obtain \u27e8HG, hHG\u27e9 := line_of_pts h g\n\n    have h_online_AB := online_3_of_B hh.1 hAB.1 hAB.2\n    have h_ne_g : h \u2260 g := by\n      by_contra contra\n      rw [contra] at hh\n      have := online_2_of_B hh.1 hAC.1 (online_2_of_B hg.1 hAC.1 hAC.2)\n      exact (not_online_of_not_colinear hAC.1 hAC.2 tri_abc) this\n\n    have h_nonline_AC : \u00ac online h AC := by\n      by_contra contra\n      have := line_unique_of_pts (ne_13_of_B hh.1).symm contra hAC.1 h_online_AB hAB.1\n      rw [this] at b_nonline_AC\n      exact b_nonline_AC hAB.2\n\n    have hag_bac : angle h a g = angle b a c := by\n      rw [angle_extension_of_B a_ne_c hh.1]\n      have := angle_extension_of_B (ne_13_of_B hh.1) hg.1\n      rw [angle_symm] at this\n      rw [this]\n      rw [angle_symm]\n    rw [hag_bac.symm] at ang_a_eq_d\n    rw [angle_symm] at ang_a_eq_d\n    rw [@angle_symm i e d f] at ang_a_eq_d\n    have ang_b_eq_h := sas hg.2 hh.2 ang_a_eq_d\n    rw [ang_b_eq_e.symm] at ang_b_eq_h\n\n    have' := parallel_of_similar hAB.1 h_online_AB hAB.2 hAC.1 g_online_AC hAC.2 hHG.1 hHG.2 hBC.1 hBC.2 (ne_23_of_B hh.1).symm (ne_13_of_B hh.1) (ne_12_of_B hg.1) h_ne_g b_ne_c a_ne_c h_nonline_AC ang_b_eq_h.2.2.symm hh.1 _\n\n    have ss1 := sameside_of_online_online_para hHG.1 hHG.2 this\n    have ss2 := sameside23_of_B123_online1_not_online2 (B_symm hg.1) hBC.2 g_nonline_BC\n    have := sameside_trans (sameside_symm ss1) ss2\n    exact (not_sameside13_of_B123_online2 hh.1 hBC.1) (sameside_symm this)\n\n    exact sameside_symm (sameside23_of_B123_online1_not_online2 hg.1 hAB.1 g_nonline_AB)\n\n  | inr contra =>\n    obtain \u27e8DF, hDF\u27e9 := line_of_pts d f\n    obtain \u27e8DE, hDE\u27e9 := line_of_pts d e\n    obtain \u27e8EF, hEF\u27e9 := line_of_pts e f\n    have f_nonline_DE := not_online_of_not_colinear hDE.1 hDE.2 tri_def\n\n    rw [colinear_symm1] at tri_def\n    rw [colinear_symm2] at tri_def\n    rw [angle_symm] at ang_a_eq_d\n    rw [@angle_symm i e d f] at ang_a_eq_d\n    have' := length_eq_of_length_eq tri_abc tri_def ang_a_eq_d _ contra.symm\n    linarith\n\n    rw [angle_symm] at ang_a_eq_d\n    rw [@angle_symm i f d e] at ang_a_eq_d\n    rw [angle_symm] at ang_b_eq_e\n    rw [@angle_symm i d e f] at ang_b_eq_e\n    exact (asa d_ne_e hDE.1 hDE.2 f_nonline_DE contra ang_a_eq_d ang_b_eq_e).2.2\n\n/-- Two triangles are similar if they have two angles equal -/\ntheorem similar_of_AA {a b c d e f : point} (tri_abc : \u00ac colinear a b c) (tri_def : \u00ac colinear d e f) \n    (ang_a_eq_d : angle b a c = angle e d f) (ang_b_eq_e : angle a b c = angle d e f) : \n    proportion (length a b) (length d e) (length a c) (length d f) := by\n  have d_ne_f := neq_13_of_not_colinear tri_def\n  have a_ne_c := neq_13_of_not_colinear tri_abc\n  have d_ne_e := neq_12_of_not_colinear tri_def\n  have a_ne_b := neq_12_of_not_colinear tri_abc\n  have b_ne_c := neq_23_of_not_colinear tri_abc\n  have e_ne_f := neq_23_of_not_colinear tri_def\n\n  by_cases dfde_ne_acab : length d f = length a c \u2228 length d e = length a b\n  \u00b7 wlog df_ne_ac : length d f = length a c\n    swap\n    have := length_eq_of_length_eq tri_abc tri_def ang_a_eq_d ang_b_eq_e df_ne_ac\n    rw [this, df_ne_ac.symm]\n    exact proportion_eq (length_nonneg a b) (length_nonneg d f) (length_eq_zero_iff.not.mpr a_ne_b) (length_eq_zero_iff.not.mpr d_ne_f)\n\n    obtain \u27e8DE, hDE\u27e9 := line_of_pts d e\n    have f_nonline_DE := not_online_of_not_colinear hDE.1 hDE.2 tri_def\n\n    rw [proportion_symm_iff]\n\n    have leq : length d e = length a b := by\n      cases dfde_ne_acab with\n      | inl dfde_ne_acab =>\n        exfalso\n        exact df_ne_ac dfde_ne_acab\n      |inr dfde_ne_acab =>\n        exact dfde_ne_acab\n\n    rw [colinear_symm1] at tri_abc\n    rw [colinear_symm2] at tri_abc\n    rw [colinear_symm1] at tri_def\n    rw [colinear_symm2] at tri_def\n    rw [angle_symm] at ang_a_eq_d\n    rw [@angle_symm i e d f] at ang_a_eq_d\n    refine' this tri_abc tri_def ang_a_eq_d _ d_ne_e a_ne_b d_ne_f a_ne_c b_ne_c.symm e_ne_f.symm (Or.symm dfde_ne_acab) leq\n\n    rw [angle_symm] at ang_a_eq_d\n    rw [@angle_symm i f d e] at ang_a_eq_d\n    rw [angle_symm] at ang_b_eq_e\n    rw [@angle_symm i d e f] at ang_b_eq_e\n    refine (asa d_ne_e hDE.1 hDE.2 f_nonline_DE leq.symm ang_a_eq_d ang_b_eq_e).2.2\n\n  rw [not_or] at dfde_ne_acab\n  rw [(Ne.def (length d f) (length a c)).symm] at dfde_ne_acab\n  rw [(Ne.def (length d e) (length a b)).symm] at dfde_ne_acab\n\n  wlog lineq : length d f < length a c\n  refine' proportion_inv (this tri_def tri_abc ang_a_eq_d.symm ang_b_eq_e.symm a_ne_c d_ne_f a_ne_b d_ne_e e_ne_f b_ne_c \u27e8dfde_ne_acab.1.symm,dfde_ne_acab.2.symm\u27e9 _)\n  simp only [not_lt] at lineq\n  exact (Ne.symm dfde_ne_acab.1).lt_of_le lineq\n\n  obtain \u27e8AC, hAC\u27e9 := line_of_pts a c\n  obtain \u27e8AB, hAB\u27e9 := line_of_pts a b\n  obtain \u27e8BC, hBC\u27e9 := line_of_pts b c\n\n  have lineq2 := length_lt_of_length_lt tri_abc tri_def ang_a_eq_d ang_b_eq_e lineq\n\n  rw [colinear_symm1] at tri_abc\n  rw [colinear_symm2] at tri_abc\n\n  have b_nonline_AC := not_online_of_not_colinear hAC.1 hAC.2 tri_abc\n\n  obtain \u27e8g, hg\u27e9 := same_length_B_of_ne_le d_ne_f lineq\n  obtain \u27e8h, hh\u27e9 := same_length_B_of_ne_le d_ne_e lineq2\n  obtain \u27e8HG, hHG\u27e9 := line_of_pts h g\n\n  have g_online_AC := online_2_of_B hg.1 hAC.1 hAC.2\n  have h_online_AB := online_2_of_B hh.1 hAB.1 hAB.2\n\n  rw [hg.2.symm,hh.2.symm]\n\n  refine' (proportional_iff_para' hHG.1 hHG.2 hBC.1 hBC.2 hAB.1 h_online_AB _ hh.1 hg.1).mpr _\n\n  by_contra contra\n  rw [line_unique_of_pts (ne_12_of_B hg.1).symm g_online_AC hAC.1 contra hAB.1] at b_nonline_AC\n  exact b_nonline_AC hAB.2\n\n  have h_ne_g : h \u2260 g := by\n    by_contra contra\n    rw [contra] at hh\n    have := online_3_of_B hh.1 hAC.1 (online_2_of_B hg.1 hAC.1 hAC.2)\n    exact (not_online_of_not_colinear hAC.1 hAC.2 tri_abc) this\n\n  refine' para_symm (parallel_of_similar hAB.1 hAB.2 h_online_AB hAC.1 hAC.2 g_online_AC hBC.1 hBC.2 hHG.1 hHG.2 (ne_23_of_B hh.1).symm a_ne_b a_ne_c b_ne_c h_ne_g (ne_12_of_B hg.1) b_nonline_AC _ hh.1 _)\n\n  have hag_bac : angle h a g = angle b a c := by\n    rw [(angle_extension_of_B a_ne_c hh.1).symm]\n    have := angle_extension_of_B (ne_12_of_B hh.1) hg.1\n    rw [angle_symm] at this\n    rw [this]\n    rw [angle_symm]\n  rw [hag_bac.symm] at ang_a_eq_d\n  rw [angle_symm] at ang_a_eq_d\n  rw [@angle_symm i e d f] at ang_a_eq_d\n  have ang_b_eq_h := sas hg.2 hh.2 ang_a_eq_d\n  rw [ang_b_eq_e.symm] at ang_b_eq_h\n  exact ang_b_eq_h.2.2\n\n  refine' sameside23_of_B123_online1_not_online2 hg.1 hAB.1 _\n  by_contra contra\n  have := line_unique_of_pts (ne_12_of_B hg.1) hAB.1 contra hAC.1 g_online_AC\n  rw [this] at hAB\n  exact b_nonline_AC hAB.2\n", "meta": {"author": "ianjauslin-rutgers", "repo": "pythagoras4", "sha": "797dabf41c6656908fde34d00aba4d019ed51bcb", "save_path": "github-repos/lean/ianjauslin-rutgers-pythagoras4", "path": "github-repos/lean/ianjauslin-rutgers-pythagoras4/pythagoras4-797dabf41c6656908fde34d00aba4d019ed51bcb/Euclid/euclid_VI.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584174871563662, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3976392803755819}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n\n! This file was ported from Lean 3 source module algebra.big_operators.ring\n! leanprover-community/mathlib commit 327c3c0d9232d80e250dc8f65e7835b82b266ea5\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.BigOperators.Basic\nimport Mathbin.Algebra.Field.Defs\nimport Mathbin.Data.Finset.Pi\nimport Mathbin.Data.Finset.Powerset\n\n/-!\n# Results about big operators with values in a (semi)ring\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe prove results about big operators that involve some interaction between\nmultiplicative and additive structures on the values being combined.\n-/\n\n\nuniverse u v w\n\nopen BigOperators\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n\nnamespace Finset\n\nvariable {s s\u2081 s\u2082 : Finset \u03b1} {a : \u03b1} {b : \u03b2} {f g : \u03b1 \u2192 \u03b2}\n\nsection CommMonoid\n\nvariable [CommMonoid \u03b2]\n\nopen Classical\n\n#print Finset.prod_pow_eq_pow_sum /-\ntheorem prod_pow_eq_pow_sum {x : \u03b2} {f : \u03b1 \u2192 \u2115} :\n    \u2200 {s : Finset \u03b1}, (\u220f i in s, x ^ f i) = x ^ \u2211 x in s, f x :=\n  by\n  apply Finset.induction\n  \u00b7 simp\n  \u00b7 intro a s has H\n    rw [Finset.prod_insert has, Finset.sum_insert has, pow_add, H]\n#align finset.prod_pow_eq_pow_sum Finset.prod_pow_eq_pow_sum\n-/\n\nend CommMonoid\n\nsection Semiring\n\nvariable [NonUnitalNonAssocSemiring \u03b2]\n\n/- warning: finset.sum_mul -> Finset.sum_mul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {s : Finset.{u1} \u03b1} {b : \u03b2} {f : \u03b1 -> \u03b2} [_inst_1 : NonUnitalNonAssocSemiring.{u2} \u03b2], Eq.{succ u2} \u03b2 (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 _inst_1))) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => f x)) b) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 _inst_1))) (f x) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {s : Finset.{u1} \u03b1} {b : \u03b2} {f : \u03b1 -> \u03b2} [_inst_1 : NonUnitalNonAssocSemiring.{u2} \u03b2], Eq.{succ u2} \u03b2 (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 _inst_1)) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => f x)) b) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 _inst_1)) (f x) b))\nCase conversion may be inaccurate. Consider using '#align finset.sum_mul Finset.sum_mul\u2093'. -/\ntheorem sum_mul : (\u2211 x in s, f x) * b = \u2211 x in s, f x * b :=\n  AddMonoidHom.map_sum (AddMonoidHom.mulRight b) _ s\n#align finset.sum_mul Finset.sum_mul\n\n/- warning: finset.mul_sum -> Finset.mul_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {s : Finset.{u1} \u03b1} {b : \u03b2} {f : \u03b1 -> \u03b2} [_inst_1 : NonUnitalNonAssocSemiring.{u2} \u03b2], Eq.{succ u2} \u03b2 (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 _inst_1))) b (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => f x))) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 _inst_1))) b (f x)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {s : Finset.{u1} \u03b1} {b : \u03b2} {f : \u03b1 -> \u03b2} [_inst_1 : NonUnitalNonAssocSemiring.{u2} \u03b2], Eq.{succ u2} \u03b2 (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 _inst_1)) b (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => f x))) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 _inst_1) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 _inst_1)) b (f x)))\nCase conversion may be inaccurate. Consider using '#align finset.mul_sum Finset.mul_sum\u2093'. -/\ntheorem mul_sum : (b * \u2211 x in s, f x) = \u2211 x in s, b * f x :=\n  AddMonoidHom.map_sum (AddMonoidHom.mulLeft b) _ s\n#align finset.mul_sum Finset.mul_sum\n\n/- warning: finset.sum_mul_sum -> Finset.sum_mul_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b2 : Type.{u1}} [_inst_1 : NonUnitalNonAssocSemiring.{u1} \u03b2] {\u03b9\u2081 : Type.{u2}} {\u03b9\u2082 : Type.{u3}} (s\u2081 : Finset.{u2} \u03b9\u2081) (s\u2082 : Finset.{u3} \u03b9\u2082) (f\u2081 : \u03b9\u2081 -> \u03b2) (f\u2082 : \u03b9\u2082 -> \u03b2), Eq.{succ u1} \u03b2 (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (Finset.sum.{u1, u2} \u03b2 \u03b9\u2081 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) s\u2081 (fun (x\u2081 : \u03b9\u2081) => f\u2081 x\u2081)) (Finset.sum.{u1, u3} \u03b2 \u03b9\u2082 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) s\u2082 (fun (x\u2082 : \u03b9\u2082) => f\u2082 x\u2082))) (Finset.sum.{u1, max u2 u3} \u03b2 (Prod.{u2, u3} \u03b9\u2081 \u03b9\u2082) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.product.{u2, u3} \u03b9\u2081 \u03b9\u2082 s\u2081 s\u2082) (fun (p : Prod.{u2, u3} \u03b9\u2081 \u03b9\u2082) => HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (f\u2081 (Prod.fst.{u2, u3} \u03b9\u2081 \u03b9\u2082 p)) (f\u2082 (Prod.snd.{u2, u3} \u03b9\u2081 \u03b9\u2082 p))))\nbut is expected to have type\n  forall {\u03b2 : Type.{u3}} [_inst_1 : NonUnitalNonAssocSemiring.{u3} \u03b2] {\u03b9\u2081 : Type.{u2}} {\u03b9\u2082 : Type.{u1}} (s\u2081 : Finset.{u2} \u03b9\u2081) (s\u2082 : Finset.{u1} \u03b9\u2082) (f\u2081 : \u03b9\u2081 -> \u03b2) (f\u2082 : \u03b9\u2082 -> \u03b2), Eq.{succ u3} \u03b2 (HMul.hMul.{u3, u3, u3} \u03b2 \u03b2 \u03b2 (instHMul.{u3} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u3} \u03b2 _inst_1)) (Finset.sum.{u3, u2} \u03b2 \u03b9\u2081 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} \u03b2 _inst_1) s\u2081 (fun (x\u2081 : \u03b9\u2081) => f\u2081 x\u2081)) (Finset.sum.{u3, u1} \u03b2 \u03b9\u2082 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} \u03b2 _inst_1) s\u2082 (fun (x\u2082 : \u03b9\u2082) => f\u2082 x\u2082))) (Finset.sum.{u3, max u1 u2} \u03b2 (Prod.{u2, u1} \u03b9\u2081 \u03b9\u2082) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} \u03b2 _inst_1) (Finset.product.{u2, u1} \u03b9\u2081 \u03b9\u2082 s\u2081 s\u2082) (fun (p : Prod.{u2, u1} \u03b9\u2081 \u03b9\u2082) => HMul.hMul.{u3, u3, u3} \u03b2 \u03b2 \u03b2 (instHMul.{u3} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u3} \u03b2 _inst_1)) (f\u2081 (Prod.fst.{u2, u1} \u03b9\u2081 \u03b9\u2082 p)) (f\u2082 (Prod.snd.{u2, u1} \u03b9\u2081 \u03b9\u2082 p))))\nCase conversion may be inaccurate. Consider using '#align finset.sum_mul_sum Finset.sum_mul_sum\u2093'. -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem sum_mul_sum {\u03b9\u2081 : Type _} {\u03b9\u2082 : Type _} (s\u2081 : Finset \u03b9\u2081) (s\u2082 : Finset \u03b9\u2082) (f\u2081 : \u03b9\u2081 \u2192 \u03b2)\n    (f\u2082 : \u03b9\u2082 \u2192 \u03b2) : ((\u2211 x\u2081 in s\u2081, f\u2081 x\u2081) * \u2211 x\u2082 in s\u2082, f\u2082 x\u2082) = \u2211 p in s\u2081 \u00d7\u02e2 s\u2082, f\u2081 p.1 * f\u2082 p.2 :=\n  by\n  rw [sum_product, sum_mul, sum_congr rfl]\n  intros\n  rw [mul_sum]\n#align finset.sum_mul_sum Finset.sum_mul_sum\n\nend Semiring\n\nsection Semiring\n\nvariable [NonAssocSemiring \u03b2]\n\n/- warning: finset.sum_mul_boole -> Finset.sum_mul_boole is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : NonAssocSemiring.{u2} \u03b2] [_inst_2 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{succ u2} \u03b2 (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)))) (f x) (ite.{succ u2} \u03b2 (Eq.{succ u1} \u03b1 a x) (_inst_2 a x) (OfNat.ofNat.{u2} \u03b2 1 (OfNat.mk.{u2} \u03b2 1 (One.one.{u2} \u03b2 (AddMonoidWithOne.toOne.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 _inst_1)))))) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1))))))))) (ite.{succ u2} \u03b2 (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a s) (Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a s) (f a) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : NonAssocSemiring.{u2} \u03b2] [_inst_2 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{succ u2} \u03b2 (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1))) (f x) (ite.{succ u2} \u03b2 (Eq.{succ u1} \u03b1 a x) (_inst_2 a x) (OfNat.ofNat.{u2} \u03b2 1 (One.toOfNat1.{u2} \u03b2 (NonAssocSemiring.toOne.{u2} \u03b2 _inst_1))) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (MulZeroOneClass.toZero.{u2} \u03b2 (NonAssocSemiring.toMulZeroOneClass.{u2} \u03b2 _inst_1))))))) (ite.{succ u2} \u03b2 (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a s) (Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a s) (f a) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (MulZeroOneClass.toZero.{u2} \u03b2 (NonAssocSemiring.toMulZeroOneClass.{u2} \u03b2 _inst_1)))))\nCase conversion may be inaccurate. Consider using '#align finset.sum_mul_boole Finset.sum_mul_boole\u2093'. -/\ntheorem sum_mul_boole [DecidableEq \u03b1] (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    (\u2211 x in s, f x * ite (a = x) 1 0) = ite (a \u2208 s) (f a) 0 := by simp\n#align finset.sum_mul_boole Finset.sum_mul_boole\n\n/- warning: finset.sum_boole_mul -> Finset.sum_boole_mul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : NonAssocSemiring.{u2} \u03b2] [_inst_2 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{succ u2} \u03b2 (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)))) (ite.{succ u2} \u03b2 (Eq.{succ u1} \u03b1 a x) (_inst_2 a x) (OfNat.ofNat.{u2} \u03b2 1 (OfNat.mk.{u2} \u03b2 1 (One.one.{u2} \u03b2 (AddMonoidWithOne.toOne.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 _inst_1)))))) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1))))))) (f x))) (ite.{succ u2} \u03b2 (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a s) (Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a s) (f a) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : NonAssocSemiring.{u2} \u03b2] [_inst_2 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{succ u2} \u03b2 (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1)) s (fun (x : \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 _inst_1))) (ite.{succ u2} \u03b2 (Eq.{succ u1} \u03b1 a x) (_inst_2 a x) (OfNat.ofNat.{u2} \u03b2 1 (One.toOfNat1.{u2} \u03b2 (NonAssocSemiring.toOne.{u2} \u03b2 _inst_1))) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (MulZeroOneClass.toZero.{u2} \u03b2 (NonAssocSemiring.toMulZeroOneClass.{u2} \u03b2 _inst_1))))) (f x))) (ite.{succ u2} \u03b2 (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a s) (Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a s) (f a) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (MulZeroOneClass.toZero.{u2} \u03b2 (NonAssocSemiring.toMulZeroOneClass.{u2} \u03b2 _inst_1)))))\nCase conversion may be inaccurate. Consider using '#align finset.sum_boole_mul Finset.sum_boole_mul\u2093'. -/\ntheorem sum_boole_mul [DecidableEq \u03b1] (s : Finset \u03b1) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    (\u2211 x in s, ite (a = x) 1 0 * f x) = ite (a \u2208 s) (f a) 0 := by simp\n#align finset.sum_boole_mul Finset.sum_boole_mul\n\nend Semiring\n\n/- warning: finset.sum_div -> Finset.sum_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DivisionSemiring.{u2} \u03b2] {s : Finset.{u1} \u03b1} {f : \u03b1 -> \u03b2} {b : \u03b2}, Eq.{succ u2} \u03b2 (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivInvMonoid.toHasDiv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1)))) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))) s (fun (x : \u03b1) => f x)) b) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))) s (fun (x : \u03b1) => HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivInvMonoid.toHasDiv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1)))) (f x) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DivisionSemiring.{u2} \u03b2] {s : Finset.{u1} \u03b1} {f : \u03b1 -> \u03b2} {b : \u03b2}, Eq.{succ u2} \u03b2 (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivisionSemiring.toDiv.{u2} \u03b2 _inst_1)) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))) s (fun (x : \u03b1) => f x)) b) (Finset.sum.{u2, u1} \u03b2 \u03b1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))) s (fun (x : \u03b1) => HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivisionSemiring.toDiv.{u2} \u03b2 _inst_1)) (f x) b))\nCase conversion may be inaccurate. Consider using '#align finset.sum_div Finset.sum_div\u2093'. -/\ntheorem sum_div [DivisionSemiring \u03b2] {s : Finset \u03b1} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} :\n    (\u2211 x in s, f x) / b = \u2211 x in s, f x / b := by simp only [div_eq_mul_inv, sum_mul]\n#align finset.sum_div Finset.sum_div\n\nsection CommSemiring\n\nvariable [CommSemiring \u03b2]\n\n/- warning: finset.prod_sum -> Finset.prod_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : CommSemiring.{u2} \u03b2] {\u03b4 : \u03b1 -> Type.{u3}} [_inst_2 : DecidableEq.{succ u1} \u03b1] [_inst_3 : forall (a : \u03b1), DecidableEq.{succ u3} (\u03b4 a)] {s : Finset.{u1} \u03b1} {t : forall (a : \u03b1), Finset.{u3} (\u03b4 a)} {f : forall (a : \u03b1), (\u03b4 a) -> \u03b2}, Eq.{succ u2} \u03b2 (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) s (fun (a : \u03b1) => Finset.sum.{u2, u3} \u03b2 (\u03b4 a) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))) (t a) (fun (b : \u03b4 a) => f a b))) (Finset.sum.{u2, max u1 u3} \u03b2 (forall (a : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a s) -> (\u03b4 a)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))) (Finset.pi.{u1, u3} \u03b1 (fun (a : \u03b1) => \u03b4 a) (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) s t) (fun (p : forall (a : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a s) -> (\u03b4 a)) => Finset.prod.{u2, u1} \u03b2 (Subtype.{succ u1} \u03b1 (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s)) (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) (Finset.attach.{u1} \u03b1 s) (fun (x : Subtype.{succ u1} \u03b1 (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s)) => f (Subtype.val.{succ u1} \u03b1 (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s) x) (p (Subtype.val.{succ u1} \u03b1 (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s) x) (Subtype.property.{succ u1} \u03b1 (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s) x)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_1 : CommSemiring.{u3} \u03b2] {\u03b4 : \u03b1 -> Type.{u1}} [_inst_2 : DecidableEq.{succ u2} \u03b1] [_inst_3 : forall (a : \u03b1), DecidableEq.{succ u1} (\u03b4 a)] {s : Finset.{u2} \u03b1} {t : forall (a : \u03b1), Finset.{u1} (\u03b4 a)} {f : forall (a : \u03b1), (\u03b4 a) -> \u03b2}, Eq.{succ u3} \u03b2 (Finset.prod.{u3, u2} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u3} \u03b2 _inst_1) s (fun (a : \u03b1) => Finset.sum.{u3, u1} \u03b2 (\u03b4 a) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} \u03b2 (Semiring.toNonAssocSemiring.{u3} \u03b2 (CommSemiring.toSemiring.{u3} \u03b2 _inst_1)))) (t a) (fun (b : \u03b4 a) => f a b))) (Finset.sum.{u3, max u2 u1} \u03b2 (forall (a : \u03b1), (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) a s) -> (\u03b4 a)) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} \u03b2 (Semiring.toNonAssocSemiring.{u3} \u03b2 (CommSemiring.toSemiring.{u3} \u03b2 _inst_1)))) (Finset.pi.{u1, u2} \u03b1 (fun (a : \u03b1) => \u03b4 a) (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) s t) (fun (p : forall (a : \u03b1), (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) a s) -> (\u03b4 a)) => Finset.prod.{u3, u2} \u03b2 (Subtype.{succ u2} \u03b1 (fun (x : \u03b1) => Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x s)) (CommSemiring.toCommMonoid.{u3} \u03b2 _inst_1) (Finset.attach.{u2} \u03b1 s) (fun (x : Subtype.{succ u2} \u03b1 (fun (x : \u03b1) => Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x s)) => f (Subtype.val.{succ u2} \u03b1 (fun (x : \u03b1) => Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x s) x) (p (Subtype.val.{succ u2} \u03b1 (fun (x : \u03b1) => Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x s) x) (Subtype.property.{succ u2} \u03b1 (fun (x : \u03b1) => Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x s) x)))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_sum Finset.prod_sum\u2093'. -/\n/-- The product over a sum can be written as a sum over the product of sets, `finset.pi`.\n  `finset.prod_univ_sum` is an alternative statement when the product is over `univ`. -/\ntheorem prod_sum {\u03b4 : \u03b1 \u2192 Type _} [DecidableEq \u03b1] [\u2200 a, DecidableEq (\u03b4 a)] {s : Finset \u03b1}\n    {t : \u2200 a, Finset (\u03b4 a)} {f : \u2200 a, \u03b4 a \u2192 \u03b2} :\n    (\u220f a in s, \u2211 b in t a, f a b) = \u2211 p in s.pi t, \u220f x in s.attach, f x.1 (p x.1 x.2) :=\n  by\n  induction' s using Finset.induction with a s ha ih\n  \u00b7 rw [pi_empty, sum_singleton]\n    rfl\n  \u00b7 have h\u2081 :\n      \u2200 x \u2208 t a,\n        \u2200 y \u2208 t a,\n          \u2200 h : x \u2260 y, Disjoint (image (pi.cons s a x) (pi s t)) (image (pi.cons s a y) (pi s t)) :=\n      by\n      intro x hx y hy h\n      simp only [disjoint_iff_ne, mem_image]\n      rintro _ \u27e8p\u2082, hp, eq\u2082\u27e9 _ \u27e8p\u2083, hp\u2083, eq\u2083\u27e9 eq\n      have : pi.cons s a x p\u2082 a (mem_insert_self _ _) = pi.cons s a y p\u2083 a (mem_insert_self _ _) :=\n        by rw [eq\u2082, eq\u2083, Eq]\n      rw [pi.cons_same, pi.cons_same] at this\n      exact h this\n    rw [prod_insert ha, pi_insert ha, ih, sum_mul, sum_bUnion h\u2081]\n    refine' sum_congr rfl fun b _ => _\n    have h\u2082 : \u2200 p\u2081 \u2208 pi s t, \u2200 p\u2082 \u2208 pi s t, pi.cons s a b p\u2081 = pi.cons s a b p\u2082 \u2192 p\u2081 = p\u2082 :=\n      fun p\u2081 h\u2081 p\u2082 h\u2082 eq => pi_cons_injective ha Eq\n    rw [sum_image h\u2082, mul_sum]\n    refine' sum_congr rfl fun g _ => _\n    rw [attach_insert, prod_insert, prod_image]\n    \u00b7 simp only [pi.cons_same]\n      congr with \u27e8v, hv\u27e9\n      congr\n      exact (pi.cons_ne (by rintro rfl <;> exact ha hv)).symm\n    \u00b7 exact fun _ _ _ _ => Subtype.eq \u2218 Subtype.mk.inj\n    \u00b7 simp only [mem_image]\n      rintro \u27e8\u27e8_, hm\u27e9, _, rfl\u27e9\n      exact ha hm\n#align finset.prod_sum Finset.prod_sum\n\nopen Classical\n\n/- warning: finset.prod_add -> Finset.prod_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : CommSemiring.{u2} \u03b2] (f : \u03b1 -> \u03b2) (g : \u03b1 -> \u03b2) (s : Finset.{u1} \u03b1), Eq.{succ u2} \u03b2 (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) s (fun (a : \u03b1) => HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toHasAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (f a) (g a))) (Finset.sum.{u2, u1} \u03b2 (Finset.{u1} \u03b1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))) (Finset.powerset.{u1} \u03b1 s) (fun (t : Finset.{u1} \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) t (fun (a : \u03b1) => f a)) (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.hasSdiff.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => Classical.propDecidable (Eq.{succ u1} \u03b1 a b))) s t) (fun (a : \u03b1) => g a))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : CommSemiring.{u2} \u03b2] (f : \u03b1 -> \u03b2) (g : \u03b1 -> \u03b2) (s : Finset.{u1} \u03b1), Eq.{succ u2} \u03b2 (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) s (fun (a : \u03b1) => HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (f a) (g a))) (Finset.sum.{u2, u1} \u03b2 (Finset.{u1} \u03b1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1)))) (Finset.powerset.{u1} \u03b1 s) (fun (t : Finset.{u1} \u03b1) => HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (CommSemiring.toSemiring.{u2} \u03b2 _inst_1))))) (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) t (fun (a : \u03b1) => f a)) (Finset.prod.{u2, u1} \u03b2 \u03b1 (CommSemiring.toCommMonoid.{u2} \u03b2 _inst_1) (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.instSDiffFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => Classical.propDecidable (Eq.{succ u1} \u03b1 a b))) s t) (fun (a : \u03b1) => g a))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_add Finset.prod_add\u2093'. -/\n/-- The product of `f a + g a` over all of `s` is the sum\n  over the powerset of `s` of the product of `f` over a subset `t` times\n  the product of `g` over the complement of `t`  -/\ntheorem prod_add (f g : \u03b1 \u2192 \u03b2) (s : Finset \u03b1) :\n    (\u220f a in s, f a + g a) = \u2211 t in s.powerset, (\u220f a in t, f a) * \u220f a in s \\ t, g a :=\n  calc\n    (\u220f a in s, f a + g a) =\n        \u220f a in s, \u2211 p in ({True, False} : Finset Prop), if p then f a else g a :=\n      by simp\n    _ =\n        \u2211 p in (s.pi fun _ => {True, False} : Finset (\u2200 a \u2208 s, Prop)),\n          \u220f a in s.attach, if p a.1 a.2 then f a.1 else g a.1 :=\n      prod_sum\n    _ = \u2211 t in s.powerset, (\u220f a in t, f a) * \u220f a in s \\ t, g a :=\n      by\n      refine' Eq.symm (sum_bij (fun t _ a _ => a \u2208 t) _ _ _ _)\n      \u00b7 simp [subset_iff] <;> tauto\n      \u00b7 intro t ht\n        erw [prod_ite (fun a : { a // a \u2208 s } => f a.1) fun a : { a // a \u2208 s } => g a.1]\n        refine'\n                congr_arg\u2082 _\n                  (prod_bij (fun (a : \u03b1) (ha : a \u2208 t) => \u27e8a, mem_powerset.1 ht ha\u27e9) _ _ _\n                    fun b hb =>\n                    \u27e8b, by\n                      cases b <;>\n                        simpa only [true_and_iff, exists_prop, mem_filter, and_true_iff, mem_attach,\n                          eq_self_iff_true, Subtype.coe_mk] using hb\u27e9)\n                  (prod_bij (fun (a : \u03b1) (ha : a \u2208 s \\ t) => \u27e8a, by simp_all\u27e9) _ _ _ fun b hb =>\n                    \u27e8b, by\n                      cases b <;>\n                        \u00b7 simp only [true_and_iff, mem_filter, mem_attach, Subtype.coe_mk] at hb\n                          simpa only [true_and_iff, exists_prop, and_true_iff, mem_sdiff,\n                            eq_self_iff_true, Subtype.coe_mk, b_property] \u27e9) <;>\n              intros <;>\n            simp_all <;>\n          simp_all\n      \u00b7 intro a\u2081 a\u2082 h\u2081 h\u2082 H\n        ext x\n        simp only [Function.funext_iff, subset_iff, mem_powerset, eq_iff_iff] at h\u2081 h\u2082 H\n        exact \u27e8fun hx => (H x (h\u2081 hx)).1 hx, fun hx => (H x (h\u2082 hx)).2 hx\u27e9\n      \u00b7 intro f hf\n        exact \u27e8s.filter fun a : \u03b1 => \u2203 h : a \u2208 s, f a h, by simp, by funext <;> intros <;> simp [*]\u27e9\n    \n#align finset.prod_add Finset.prod_add\n\n/- warning: finset.prod_add_ordered -> Finset.prod_add_ordered is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {R : Type.{u2}} [_inst_2 : CommSemiring.{u2} R] [_inst_3 : LinearOrder.{u1} \u03b9] (s : Finset.{u1} \u03b9) (f : \u03b9 -> R) (g : \u03b9 -> R), Eq.{succ u2} R (Finset.prod.{u2, u1} R \u03b9 (CommSemiring.toCommMonoid.{u2} R _inst_2) s (fun (i : \u03b9) => HAdd.hAdd.{u2, u2, u2} R R R (instHAdd.{u2} R (Distrib.toHasAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (f i) (g i))) (HAdd.hAdd.{u2, u2, u2} R R R (instHAdd.{u2} R (Distrib.toHasAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (Finset.prod.{u2, u1} R \u03b9 (CommSemiring.toCommMonoid.{u2} R _inst_2) s (fun (i : \u03b9) => f i)) (Finset.sum.{u2, u1} R \u03b9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))) s (fun (i : \u03b9) => HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (g i) (Finset.prod.{u2, u1} R \u03b9 (CommSemiring.toCommMonoid.{u2} R _inst_2) (Finset.filter.{u1} \u03b9 (fun (_x : \u03b9) => LT.lt.{u1} \u03b9 (Preorder.toLT.{u1} \u03b9 (PartialOrder.toPreorder.{u1} \u03b9 (SemilatticeInf.toPartialOrder.{u1} \u03b9 (Lattice.toSemilatticeInf.{u1} \u03b9 (LinearOrder.toLattice.{u1} \u03b9 _inst_3))))) _x i) (fun (a : \u03b9) => LT.lt.decidable.{u1} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HAdd.hAdd.{u2, u2, u2} R R R (instHAdd.{u2} R (Distrib.toHasAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (f j) (g j)))) (Finset.prod.{u2, u1} R \u03b9 (CommSemiring.toCommMonoid.{u2} R _inst_2) (Finset.filter.{u1} \u03b9 (fun (j : \u03b9) => LT.lt.{u1} \u03b9 (Preorder.toLT.{u1} \u03b9 (PartialOrder.toPreorder.{u1} \u03b9 (SemilatticeInf.toPartialOrder.{u1} \u03b9 (Lattice.toSemilatticeInf.{u1} \u03b9 (LinearOrder.toLattice.{u1} \u03b9 _inst_3))))) i j) (fun (a : \u03b9) => LT.lt.decidable.{u1} \u03b9 _inst_3 i a) s) (fun (j : \u03b9) => f j)))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u2}} {R : Type.{u1}} [_inst_2 : CommSemiring.{u1} R] [_inst_3 : LinearOrder.{u2} \u03b9] (s : Finset.{u2} \u03b9) (f : \u03b9 -> R) (g : \u03b9 -> R), Eq.{succ u1} R (Finset.prod.{u1, u2} R \u03b9 (CommSemiring.toCommMonoid.{u1} R _inst_2) s (fun (i : \u03b9) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))))) (f i) (g i))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))))) (Finset.prod.{u1, u2} R \u03b9 (CommSemiring.toCommMonoid.{u1} R _inst_2) s (fun (i : \u03b9) => f i)) (Finset.sum.{u1, u2} R \u03b9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) s (fun (i : \u03b9) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) (g i) (Finset.prod.{u1, u2} R \u03b9 (CommSemiring.toCommMonoid.{u1} R _inst_2) (Finset.filter.{u2} \u03b9 (fun (_x : \u03b9) => LT.lt.{u2} \u03b9 (Preorder.toLT.{u2} \u03b9 (PartialOrder.toPreorder.{u2} \u03b9 (SemilatticeInf.toPartialOrder.{u2} \u03b9 (Lattice.toSemilatticeInf.{u2} \u03b9 (DistribLattice.toLattice.{u2} \u03b9 (instDistribLattice.{u2} \u03b9 _inst_3)))))) _x i) (fun (a : \u03b9) => instDecidableLtToLTToPreorderToPartialOrder.{u2} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))))) (f j) (g j)))) (Finset.prod.{u1, u2} R \u03b9 (CommSemiring.toCommMonoid.{u1} R _inst_2) (Finset.filter.{u2} \u03b9 (fun (j : \u03b9) => LT.lt.{u2} \u03b9 (Preorder.toLT.{u2} \u03b9 (PartialOrder.toPreorder.{u2} \u03b9 (SemilatticeInf.toPartialOrder.{u2} \u03b9 (Lattice.toSemilatticeInf.{u2} \u03b9 (DistribLattice.toLattice.{u2} \u03b9 (instDistribLattice.{u2} \u03b9 _inst_3)))))) i j) (fun (a : \u03b9) => instDecidableLtToLTToPreorderToPartialOrder.{u2} \u03b9 _inst_3 i a) s) (fun (j : \u03b9) => f j)))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_add_ordered Finset.prod_add_ordered\u2093'. -/\n/-- `\u220f i, (f i + g i) = (\u220f i, f i) + \u2211 i, g i * (\u220f j < i, f j + g j) * (\u220f j > i, f j)`. -/\ntheorem prod_add_ordered {\u03b9 R : Type _} [CommSemiring R] [LinearOrder \u03b9] (s : Finset \u03b9)\n    (f g : \u03b9 \u2192 R) :\n    (\u220f i in s, f i + g i) =\n      (\u220f i in s, f i) +\n        \u2211 i in s,\n          (g i * \u220f j in s.filter\u2093 (\u00b7 < i), f j + g j) * \u220f j in s.filter\u2093 fun j => i < j, f j :=\n  by\n  refine' Finset.induction_on_max s (by simp) _\n  clear s\n  intro a s ha ihs\n  have ha' : a \u2209 s := fun ha' => (ha a ha').False\n  rw [prod_insert ha', prod_insert ha', sum_insert ha', filter_insert, if_neg (lt_irrefl a),\n    filter_true_of_mem ha, ihs, add_mul, mul_add, mul_add, add_assoc]\n  congr 1\n  rw [add_comm]\n  congr 1\n  \u00b7 rw [filter_false_of_mem, prod_empty, mul_one]\n    exact (forall_mem_insert _ _ _).2 \u27e8lt_irrefl a, fun i hi => (ha i hi).not_lt\u27e9\n  \u00b7 rw [mul_sum]\n    refine' sum_congr rfl fun i hi => _\n    rw [filter_insert, if_neg (ha i hi).not_lt, filter_insert, if_pos (ha i hi), prod_insert,\n      mul_left_comm]\n    exact mt (fun ha => (mem_filter.1 ha).1) ha'\n#align finset.prod_add_ordered Finset.prod_add_ordered\n\n/- warning: finset.prod_sub_ordered -> Finset.prod_sub_ordered is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {R : Type.{u2}} [_inst_2 : CommRing.{u2} R] [_inst_3 : LinearOrder.{u1} \u03b9] (s : Finset.{u1} \u03b9) (f : \u03b9 -> R) (g : \u03b9 -> R), Eq.{succ u2} R (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) s (fun (i : \u03b9) => HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (f i) (g i))) (HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) s (fun (i : \u03b9) => f i)) (Finset.sum.{u2, u1} R \u03b9 (AddCommGroup.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toAddCommGroup.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_2))))) s (fun (i : \u03b9) => HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (Ring.toDistrib.{u2} R (CommRing.toRing.{u2} R _inst_2)))) (HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (Ring.toDistrib.{u2} R (CommRing.toRing.{u2} R _inst_2)))) (g i) (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) (Finset.filter.{u1} \u03b9 (fun (_x : \u03b9) => LT.lt.{u1} \u03b9 (Preorder.toLT.{u1} \u03b9 (PartialOrder.toPreorder.{u1} \u03b9 (SemilatticeInf.toPartialOrder.{u1} \u03b9 (Lattice.toSemilatticeInf.{u1} \u03b9 (LinearOrder.toLattice.{u1} \u03b9 _inst_3))))) _x i) (fun (a : \u03b9) => LT.lt.decidable.{u1} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (f j) (g j)))) (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) (Finset.filter.{u1} \u03b9 (fun (j : \u03b9) => LT.lt.{u1} \u03b9 (Preorder.toLT.{u1} \u03b9 (PartialOrder.toPreorder.{u1} \u03b9 (SemilatticeInf.toPartialOrder.{u1} \u03b9 (Lattice.toSemilatticeInf.{u1} \u03b9 (LinearOrder.toLattice.{u1} \u03b9 _inst_3))))) i j) (fun (a : \u03b9) => LT.lt.decidable.{u1} \u03b9 _inst_3 i a) s) (fun (j : \u03b9) => f j)))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u2}} {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] [_inst_3 : LinearOrder.{u2} \u03b9] (s : Finset.{u2} \u03b9) (f : \u03b9 -> R) (g : \u03b9 -> R), Eq.{succ u1} R (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) s (fun (i : \u03b9) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (f i) (g i))) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) s (fun (i : \u03b9) => f i)) (Finset.sum.{u1, u2} R \u03b9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) s (fun (i : \u03b9) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (g i) (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) (Finset.filter.{u2} \u03b9 (fun (_x : \u03b9) => LT.lt.{u2} \u03b9 (Preorder.toLT.{u2} \u03b9 (PartialOrder.toPreorder.{u2} \u03b9 (SemilatticeInf.toPartialOrder.{u2} \u03b9 (Lattice.toSemilatticeInf.{u2} \u03b9 (DistribLattice.toLattice.{u2} \u03b9 (instDistribLattice.{u2} \u03b9 _inst_3)))))) _x i) (fun (a : \u03b9) => instDecidableLtToLTToPreorderToPartialOrder.{u2} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (f j) (g j)))) (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) (Finset.filter.{u2} \u03b9 (fun (j : \u03b9) => LT.lt.{u2} \u03b9 (Preorder.toLT.{u2} \u03b9 (PartialOrder.toPreorder.{u2} \u03b9 (SemilatticeInf.toPartialOrder.{u2} \u03b9 (Lattice.toSemilatticeInf.{u2} \u03b9 (DistribLattice.toLattice.{u2} \u03b9 (instDistribLattice.{u2} \u03b9 _inst_3)))))) i j) (fun (a : \u03b9) => instDecidableLtToLTToPreorderToPartialOrder.{u2} \u03b9 _inst_3 i a) s) (fun (j : \u03b9) => f j)))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_sub_ordered Finset.prod_sub_ordered\u2093'. -/\n/-- `\u220f i, (f i - g i) = (\u220f i, f i) - \u2211 i, g i * (\u220f j < i, f j - g j) * (\u220f j > i, f j)`. -/\ntheorem prod_sub_ordered {\u03b9 R : Type _} [CommRing R] [LinearOrder \u03b9] (s : Finset \u03b9) (f g : \u03b9 \u2192 R) :\n    (\u220f i in s, f i - g i) =\n      (\u220f i in s, f i) -\n        \u2211 i in s,\n          (g i * \u220f j in s.filter\u2093 (\u00b7 < i), f j - g j) * \u220f j in s.filter\u2093 fun j => i < j, f j :=\n  by\n  simp only [sub_eq_add_neg]\n  convert prod_add_ordered s f fun i => -g i\n  simp\n#align finset.prod_sub_ordered Finset.prod_sub_ordered\n\n/- warning: finset.prod_one_sub_ordered -> Finset.prod_one_sub_ordered is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {R : Type.{u2}} [_inst_2 : CommRing.{u2} R] [_inst_3 : LinearOrder.{u1} \u03b9] (s : Finset.{u1} \u03b9) (f : \u03b9 -> R), Eq.{succ u2} R (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) s (fun (i : \u03b9) => HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (OfNat.ofNat.{u2} R 1 (OfNat.mk.{u2} R 1 (One.one.{u2} R (AddMonoidWithOne.toOne.{u2} R (AddGroupWithOne.toAddMonoidWithOne.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2)))))))) (f i))) (HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (OfNat.ofNat.{u2} R 1 (OfNat.mk.{u2} R 1 (One.one.{u2} R (AddMonoidWithOne.toOne.{u2} R (AddGroupWithOne.toAddMonoidWithOne.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2)))))))) (Finset.sum.{u2, u1} R \u03b9 (AddCommGroup.toAddCommMonoid.{u2} R (NonUnitalNonAssocRing.toAddCommGroup.{u2} R (NonAssocRing.toNonUnitalNonAssocRing.{u2} R (Ring.toNonAssocRing.{u2} R (CommRing.toRing.{u2} R _inst_2))))) s (fun (i : \u03b9) => HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (Ring.toDistrib.{u2} R (CommRing.toRing.{u2} R _inst_2)))) (f i) (Finset.prod.{u2, u1} R \u03b9 (CommRing.toCommMonoid.{u2} R _inst_2) (Finset.filter.{u1} \u03b9 (fun (_x : \u03b9) => LT.lt.{u1} \u03b9 (Preorder.toLT.{u1} \u03b9 (PartialOrder.toPreorder.{u1} \u03b9 (SemilatticeInf.toPartialOrder.{u1} \u03b9 (Lattice.toSemilatticeInf.{u1} \u03b9 (LinearOrder.toLattice.{u1} \u03b9 _inst_3))))) _x i) (fun (a : \u03b9) => LT.lt.decidable.{u1} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HSub.hSub.{u2, u2, u2} R R R (instHSub.{u2} R (SubNegMonoid.toHasSub.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2))))))) (OfNat.ofNat.{u2} R 1 (OfNat.mk.{u2} R 1 (One.one.{u2} R (AddMonoidWithOne.toOne.{u2} R (AddGroupWithOne.toAddMonoidWithOne.{u2} R (AddCommGroupWithOne.toAddGroupWithOne.{u2} R (Ring.toAddCommGroupWithOne.{u2} R (CommRing.toRing.{u2} R _inst_2)))))))) (f j))))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u2}} {R : Type.{u1}} [_inst_2 : CommRing.{u1} R] [_inst_3 : LinearOrder.{u2} \u03b9] (s : Finset.{u2} \u03b9) (f : \u03b9 -> R), Eq.{succ u1} R (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) s (fun (i : \u03b9) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (f i))) (HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (Finset.sum.{u1, u2} R \u03b9 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) s (fun (i : \u03b9) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocRing.toMul.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (f i) (Finset.prod.{u1, u2} R \u03b9 (CommRing.toCommMonoid.{u1} R _inst_2) (Finset.filter.{u2} \u03b9 (fun (_x : \u03b9) => LT.lt.{u2} \u03b9 (Preorder.toLT.{u2} \u03b9 (PartialOrder.toPreorder.{u2} \u03b9 (SemilatticeInf.toPartialOrder.{u2} \u03b9 (Lattice.toSemilatticeInf.{u2} \u03b9 (DistribLattice.toLattice.{u2} \u03b9 (instDistribLattice.{u2} \u03b9 _inst_3)))))) _x i) (fun (a : \u03b9) => instDecidableLtToLTToPreorderToPartialOrder.{u2} \u03b9 _inst_3 a i) s) (fun (j : \u03b9) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_2))) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (NonAssocRing.toOne.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_2))))) (f j))))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_one_sub_ordered Finset.prod_one_sub_ordered\u2093'. -/\n/-- `\u220f i, (1 - f i) = 1 - \u2211 i, f i * (\u220f j < i, 1 - f j)`. This formula is useful in construction of\na partition of unity from a collection of \u201cbump\u201d functions.  -/\ntheorem prod_one_sub_ordered {\u03b9 R : Type _} [CommRing R] [LinearOrder \u03b9] (s : Finset \u03b9)\n    (f : \u03b9 \u2192 R) : (\u220f i in s, 1 - f i) = 1 - \u2211 i in s, f i * \u220f j in s.filter\u2093 (\u00b7 < i), 1 - f j :=\n  by\n  rw [prod_sub_ordered]\n  simp\n#align finset.prod_one_sub_ordered Finset.prod_one_sub_ordered\n\n/- warning: finset.sum_pow_mul_eq_add_pow -> Finset.sum_pow_mul_eq_add_pow is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {R : Type.{u2}} [_inst_2 : CommSemiring.{u2} R] (a : R) (b : R) (s : Finset.{u1} \u03b1), Eq.{succ u2} R (Finset.sum.{u2, u1} R (Finset.{u1} \u03b1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))) (Finset.powerset.{u1} \u03b1 s) (fun (t : Finset.{u1} \u03b1) => HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) (HPow.hPow.{u2, 0, u2} R Nat R (instHPow.{u2, 0} R Nat (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2))))) a (Finset.card.{u1} \u03b1 t)) (HPow.hPow.{u2, 0, u2} R Nat R (instHPow.{u2, 0} R Nat (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2))))) b (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Finset.card.{u1} \u03b1 s) (Finset.card.{u1} \u03b1 t))))) (HPow.hPow.{u2, 0, u2} R Nat R (instHPow.{u2, 0} R Nat (Monoid.Pow.{u2} R (MonoidWithZero.toMonoid.{u2} R (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2))))) (HAdd.hAdd.{u2, u2, u2} R R R (instHAdd.{u2} R (Distrib.toHasAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_2)))))) a b) (Finset.card.{u1} \u03b1 s))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {R : Type.{u1}} [_inst_2 : CommSemiring.{u1} R] (a : R) (b : R) (s : Finset.{u2} \u03b1), Eq.{succ u1} R (Finset.sum.{u1, u2} R (Finset.{u2} \u03b1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))) (Finset.powerset.{u2} \u03b1 s) (fun (t : Finset.{u2} \u03b1) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) a (Finset.card.{u2} \u03b1 t)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) b (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Finset.card.{u2} \u03b1 s) (Finset.card.{u2} \u03b1 t))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_2)))))) a b) (Finset.card.{u2} \u03b1 s))\nCase conversion may be inaccurate. Consider using '#align finset.sum_pow_mul_eq_add_pow Finset.sum_pow_mul_eq_add_pow\u2093'. -/\n/-- Summing `a^s.card * b^(n-s.card)` over all finite subsets `s` of a `finset`\ngives `(a + b)^s.card`.-/\ntheorem sum_pow_mul_eq_add_pow {\u03b1 R : Type _} [CommSemiring R] (a b : R) (s : Finset \u03b1) :\n    (\u2211 t in s.powerset, a ^ t.card * b ^ (s.card - t.card)) = (a + b) ^ s.card :=\n  by\n  rw [\u2190 prod_const, prod_add]\n  refine' Finset.sum_congr rfl fun t ht => _\n  rw [prod_const, prod_const, \u2190 card_sdiff (mem_powerset.1 ht)]\n#align finset.sum_pow_mul_eq_add_pow Finset.sum_pow_mul_eq_add_pow\n\n#print Finset.dvd_sum /-\ntheorem dvd_sum {b : \u03b2} {s : Finset \u03b1} {f : \u03b1 \u2192 \u03b2} (h : \u2200 x \u2208 s, b \u2223 f x) : b \u2223 \u2211 x in s, f x :=\n  Multiset.dvd_sum fun y hy => by rcases Multiset.mem_map.1 hy with \u27e8x, hx, rfl\u27e9 <;> exact h x hx\n#align finset.dvd_sum Finset.dvd_sum\n-/\n\n#print Finset.prod_natCast /-\n@[norm_cast]\ntheorem prod_natCast (s : Finset \u03b1) (f : \u03b1 \u2192 \u2115) : \u2191(\u220f x in s, f x : \u2115) = \u220f x in s, (f x : \u03b2) :=\n  (Nat.castRingHom \u03b2).map_prod f s\n#align finset.prod_nat_cast Finset.prod_natCast\n-/\n\nend CommSemiring\n\nsection CommRing\n\nvariable {R : Type _} [CommRing R]\n\n/- warning: finset.prod_range_cast_nat_sub -> Finset.prod_range_cast_nat_sub is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (n : Nat) (k : Nat), Eq.{succ u1} R (Finset.prod.{u1, 0} R Nat (CommRing.toCommMonoid.{u1} R _inst_1) (Finset.range k) (fun (i : Nat) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (SubNegMonoid.toHasSub.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1))))))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTC\u2093.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) n) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTC\u2093.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) i))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTC\u2093.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddGroupWithOne.toAddMonoidWithOne.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R (CommRing.toRing.{u1} R _inst_1)))))))) (Finset.prod.{0, 0} Nat Nat Nat.commMonoid (Finset.range k) (fun (i : Nat) => HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) n i)))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : CommRing.{u1} R] (n : Nat) (k : Nat), Eq.{succ u1} R (Finset.prod.{u1, 0} R Nat (CommRing.toCommMonoid.{u1} R _inst_1) (Finset.range k) (fun (i : Nat) => HSub.hSub.{u1, u1, u1} R R R (instHSub.{u1} R (Ring.toSub.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Nat.cast.{u1} R (NonAssocRing.toNatCast.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) n) (Nat.cast.{u1} R (NonAssocRing.toNatCast.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) i))) (Nat.cast.{u1} R (NonAssocRing.toNatCast.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R _inst_1))) (Finset.prod.{0, 0} Nat Nat Nat.commMonoid (Finset.range k) (fun (i : Nat) => HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n i)))\nCase conversion may be inaccurate. Consider using '#align finset.prod_range_cast_nat_sub Finset.prod_range_cast_nat_sub\u2093'. -/\ntheorem prod_range_cast_nat_sub (n k : \u2115) :\n    (\u220f i in range k, (n - i : R)) = (\u220f i in range k, n - i : \u2115) :=\n  by\n  rw [prod_nat_cast]\n  cases' le_or_lt k n with hkn hnk\n  \u00b7 exact prod_congr rfl fun i hi => (Nat.cast_sub <| (mem_range.1 hi).le.trans hkn).symm\n  \u00b7 rw [\u2190 mem_range] at hnk\n    rw [prod_eq_zero hnk, prod_eq_zero hnk] <;> simp\n#align finset.prod_range_cast_nat_sub Finset.prod_range_cast_nat_sub\n\nend CommRing\n\n/- warning: finset.prod_powerset_insert -> Finset.prod_powerset_insert is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : CommMonoid.{u2} \u03b2] {s : Finset.{u1} \u03b1} {x : \u03b1}, (Not (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x s)) -> (forall (f : (Finset.{u1} \u03b1) -> \u03b2), Eq.{succ u2} \u03b2 (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 (Insert.insert.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasInsert.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x s)) (fun (a : Finset.{u1} \u03b1) => f a)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (MulOneClass.toHasMul.{u2} \u03b2 (Monoid.toMulOneClass.{u2} \u03b2 (CommMonoid.toMonoid.{u2} \u03b2 _inst_2)))) (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 s) (fun (a : Finset.{u1} \u03b1) => f a)) (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 s) (fun (t : Finset.{u1} \u03b1) => f (Insert.insert.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasInsert.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x t)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : CommMonoid.{u2} \u03b2] {s : Finset.{u1} \u03b1} {x : \u03b1}, (Not (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) x s)) -> (forall (f : (Finset.{u1} \u03b1) -> \u03b2), Eq.{succ u2} \u03b2 (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 (Insert.insert.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instInsertFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x s)) (fun (a : Finset.{u1} \u03b1) => f a)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (MulOneClass.toMul.{u2} \u03b2 (Monoid.toMulOneClass.{u2} \u03b2 (CommMonoid.toMonoid.{u2} \u03b2 _inst_2)))) (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 s) (fun (a : Finset.{u1} \u03b1) => f a)) (Finset.prod.{u2, u1} \u03b2 (Finset.{u1} \u03b1) _inst_2 (Finset.powerset.{u1} \u03b1 s) (fun (t : Finset.{u1} \u03b1) => f (Insert.insert.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instInsertFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x t)))))\nCase conversion may be inaccurate. Consider using '#align finset.prod_powerset_insert Finset.prod_powerset_insert\u2093'. -/\n/-- A product over all subsets of `s \u222a {x}` is obtained by multiplying the product over all subsets\nof `s`, and over all subsets of `s` to which one adds `x`. -/\n@[to_additive\n      \"A sum over all subsets of `s \u222a {x}` is obtained by summing the sum over all subsets\\nof `s`, and over all subsets of `s` to which one adds `x`.\"]\ntheorem prod_powerset_insert [DecidableEq \u03b1] [CommMonoid \u03b2] {s : Finset \u03b1} {x : \u03b1} (h : x \u2209 s)\n    (f : Finset \u03b1 \u2192 \u03b2) :\n    (\u220f a in (insert x s).powerset, f a) =\n      (\u220f a in s.powerset, f a) * \u220f t in s.powerset, f (insert x t) :=\n  by\n  rw [powerset_insert, Finset.prod_union, Finset.prod_image]\n  \u00b7 intro t\u2081 h\u2081 t\u2082 h\u2082 heq\n    rw [\u2190 Finset.erase_insert (not_mem_of_mem_powerset_of_not_mem h\u2081 h), \u2190\n      Finset.erase_insert (not_mem_of_mem_powerset_of_not_mem h\u2082 h), HEq]\n  \u00b7 rw [Finset.disjoint_iff_ne]\n    intro t\u2081 h\u2081 t\u2082 h\u2082\n    rcases Finset.mem_image.1 h\u2082 with \u27e8t\u2083, h\u2083, H\u2083\u2082\u27e9\n    rw [\u2190 H\u2083\u2082]\n    exact ne_insert_of_not_mem _ _ (not_mem_of_mem_powerset_of_not_mem h\u2081 h)\n#align finset.prod_powerset_insert Finset.prod_powerset_insert\n#align finset.sum_powerset_insert Finset.sum_powerset_insert\n\n#print Finset.prod_powerset /-\n/-- A product over `powerset s` is equal to the double product over sets of subsets of `s` with\n`card s = k`, for `k = 1, ..., card s`. -/\n@[to_additive\n      \"A sum over `powerset s` is equal to the double sum over sets of subsets of `s` with\\n`card s = k`, for `k = 1, ..., card s`\"]\ntheorem prod_powerset [CommMonoid \u03b2] (s : Finset \u03b1) (f : Finset \u03b1 \u2192 \u03b2) :\n    (\u220f t in powerset s, f t) = \u220f j in range (card s + 1), \u220f t in powersetLen j s, f t := by\n  rw [powerset_card_disj_Union, prod_disj_Union]\n#align finset.prod_powerset Finset.prod_powerset\n#align finset.sum_powerset Finset.sum_powerset\n-/\n\n/- warning: finset.sum_range_succ_mul_sum_range_succ -> Finset.sum_range_succ_mul_sum_range_succ is a dubious translation:\nlean 3 declaration is\n  forall {\u03b2 : Type.{u1}} [_inst_1 : NonUnitalNonAssocSemiring.{u1} \u03b2] (n : Nat) (k : Nat) (f : Nat -> \u03b2) (g : Nat -> \u03b2), Eq.{succ u1} \u03b2 (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (i : Nat) => f i)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) k (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (i : Nat) => g i))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toHasAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toHasAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toHasAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range n) (fun (i : Nat) => f i)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range k) (fun (i : Nat) => g i))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (f n) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range k) (fun (i : Nat) => g i)))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range n) (fun (i : Nat) => f i)) (g k))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (Distrib.toHasMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (f n) (g k)))\nbut is expected to have type\n  forall {\u03b2 : Type.{u1}} [_inst_1 : NonUnitalNonAssocSemiring.{u1} \u03b2] (n : Nat) (k : Nat) (f : Nat -> \u03b2) (g : Nat -> \u03b2), Eq.{succ u1} \u03b2 (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 _inst_1)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (i : Nat) => f i)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) k (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (i : Nat) => g i))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HAdd.hAdd.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHAdd.{u1} \u03b2 (Distrib.toAdd.{u1} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b2 _inst_1))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 _inst_1)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range n) (fun (i : Nat) => f i)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range k) (fun (i : Nat) => g i))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 _inst_1)) (f n) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range k) (fun (i : Nat) => g i)))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 _inst_1)) (Finset.sum.{u1, 0} \u03b2 Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} \u03b2 _inst_1) (Finset.range n) (fun (i : Nat) => f i)) (g k))) (HMul.hMul.{u1, u1, u1} \u03b2 \u03b2 \u03b2 (instHMul.{u1} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 _inst_1)) (f n) (g k)))\nCase conversion may be inaccurate. Consider using '#align finset.sum_range_succ_mul_sum_range_succ Finset.sum_range_succ_mul_sum_range_succ\u2093'. -/\ntheorem sum_range_succ_mul_sum_range_succ [NonUnitalNonAssocSemiring \u03b2] (n k : \u2115) (f g : \u2115 \u2192 \u03b2) :\n    ((\u2211 i in range (n + 1), f i) * \u2211 i in range (k + 1), g i) =\n      (((\u2211 i in range n, f i) * \u2211 i in range k, g i) + f n * \u2211 i in range k, g i) +\n          (\u2211 i in range n, f i) * g k +\n        f n * g k :=\n  by simp only [add_mul, mul_add, add_assoc, sum_range_succ]\n#align finset.sum_range_succ_mul_sum_range_succ Finset.sum_range_succ_mul_sum_range_succ\n\nend Finset\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/BigOperators/Ring.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3973882407977716}}
{"text": "import category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.types\n\nnamespace category_theory\n\nopen limits category_theory.limits.types\n\n@[protect_proj] structure Function : Type 1 :=\n( to_fun : Type \u2192 Type )\n( R : \u03a0 {X\u2081 X\u2082}, (X\u2081 \u2192 X\u2082 \u2192 Prop) \u2192 to_fun X\u2081 \u2192 to_fun X\u2082 \u2192 Prop )\n\nnamespace Function\n\ninstance : has_coe_to_fun Function (\u03bb _, Type \u2192 Type) :=\n{ coe := Function.to_fun }\n\n@[simp] lemma coe_mk (f h) : \u21d1(Function.mk f h) = f := rfl\n\nstructure hom (F G : Function) : Type 1 :=\n( to_fun : \u03a0 {X}, F X \u2192 G X )\n( parametric : \u2200 {X\u2081 X\u2082} {R : X\u2081 \u2192 X\u2082 \u2192 Prop} {x\u2081 : F X\u2081} {x\u2082 : F X\u2082}, \n    F.R R x\u2081 x\u2082 \u2192 G.R R (to_fun x\u2081) (to_fun x\u2082) )\n\ninstance : category_struct Function :=\n{ hom := \u03bb F G, hom F G,\n  id := \u03bb F, \u27e8\u03bb _, id, \u03bb _ _ _ _ _, id\u27e9,\n  comp := \u03bb X Y Z n\u2081 n\u2082, \u27e8\u03bb X x, n\u2082.1 (n\u2081.1 x), \u03bb X\u2081 X\u2082 R x\u2081 x\u2082 h, n\u2082.2 (n\u2081.2 h)\u27e9 } \n\ninstance (F G : Function) : has_coe_to_fun (F \u27f6 G) (\u03bb _, \u03a0 {X}, F X \u2192 G X) :=\n{ coe := hom.to_fun }\n\nlemma hom.ext_iff' {F G : Function} {f g : F \u27f6 G} :\n  f = g \u2194 (@coe_fn _ _ _ f : \u03a0 {X : Type}, F X \u2192 G X) = coe_fn g := \nby cases f; cases g; simp [hom.mk.inj_eq]; refl\n\nlemma hom.ext_iff {F G : Function} {f g : F \u27f6 G} :\n  f = g \u2194 \u2200 {X : Type} (x : F X), f x = g x :=\nby simp [hom.ext_iff', function.funext_iff]\n\n@[ext] lemma hom.ext {F G : Function} {f g : F \u27f6 G}\n  (h : \u2200 (X : Type) (x : F X), f x = g x) : f = g :=\nhom.ext_iff.2 h\n\n@[simp] lemma coe_id (F : Function)  : \n  (coe_fn (\ud835\udfd9 F) : \u03a0 (X : Type), F X \u2192 F X) = (\u03bb {X : Type}, @id (F X)) := rfl\n\n@[simp] lemma coe_comp (F G H : Function) (f : F \u27f6 G) (g : G \u27f6 H) :\n  coe_fn (f \u226b g) = (\u03bb {X : Type} (x : F X), g (f x)) := rfl\n\ninstance : category Function := {}\n\ndef app (X : Type) : Function \u2964 Type :=\n{ obj := \u03bb F, F X,\n  map := \u03bb F G f a, f a } \n\ninstance : has_pullbacks Function := \n@has_pullbacks_of_has_limit_cospan _ _ \n(\u03bb (F G H : Function) f g, \u27e8\u27e8\n{ cone := \u27e8{ to_fun := \u03bb X, pullback_obj ((app X).map f) ((app X).map g),\n             R := \u03bb X\u2081 X\u2082 R x y, F.R R x.1.1 y.1.1 \u2227 G.R R x.1.2 y.1.2 },\n    \u27e8\u03bb X, \u27e8\u03bb Y, option.cases_on X (\u03bb x, f x.1.1) \n        (\u03bb X, limits.walking_pair.cases_on X (\u03bb x, x.1.1) (\u03bb x, x.1.2)), \n      begin \n        intros X\u2081 X\u2082 R x y,\n        cases X,\n        { exact \u03bb h, f.2 h.1 },\n        { cases X, \n          exact and.left,\n          exact and.right }\n      end\u27e9, begin \n        intros X Y f, \n        cases f with f\u2081 f\u2082,\n        cases X,\n        { refl },\n        { cases X; refl },\n        { cases f\u2082,\n          { refl },\n          { ext Z x,\n            rcases x with \u27e8\u27e8x\u2081, x\u2082\u27e9, hx\u27e9,\n            exact hx } }     \n      end\u27e9\u27e9,\n  is_limit :=\n    \u27e8\u03bb s, \n      \u27e8\u03bb X x, \u27e8\u27e8s.\u03c0.app limits.walking_span.left x, \n                s.\u03c0.app limits.walking_span.right x\u27e9, \n          (congr_fun (congr_fun (hom.ext_iff'.1 (s.2.2 limits.walking_cospan.hom.inl)) X) x).symm.trans \n          (congr_fun (congr_fun (hom.ext_iff'.1 (s.2.2 limits.walking_cospan.hom.inr)) X) x)\u27e9,\n      \u03bb X\u2081 X\u2082 R x\u2081 x\u2082 h,  \u27e8(s.2.1 walking_span.left).2 h,\n          (s.2.1 walking_span.right).2 h\u27e9\u27e9, \n    begin\n      intros s j,\n      cases j,\n      { have := s.2.2 (limits.walking_cospan.hom.inl),\n        dsimp at this,\n        rw [category.id_comp] at this,\n        refine eq.trans _ this.symm,\n        refl },\n      { cases j; ext; refl }\n    end, \n    begin \n      intros s m hm,\n      ext,\n      { have := hm limits.walking_span.left,\n        simp only [hom.ext_iff] at this,\n        exact this _ },\n      { have := hm limits.walking_span.right,\n        simp only [hom.ext_iff] at this,\n        exact this _ }\n    end\u27e9 } \u27e9\u27e9)\n\nend Function\n\nend category_theory", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/parametricity/Function_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3973882407977716}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.rat.cast\nimport Mathlib.data.rat.meta_defs\nimport Mathlib.PostPort\n\nuniverses u_1 u \n\nnamespace Mathlib\n\n/-!\n# `norm_num`\n\nEvaluating arithmetic expressions including `*`, `+`, `-`, `^`, `\u2264`.\n-/\n\nnamespace tactic\n\n\n/-- Reflexivity conversion: given `e` returns `(e, \u22a2 e = e)` -/\n/-- Transitivity conversion: given two conversions (which take an\nexpression `e` and returns `(e', \u22a2 e = e')`), produces another\nconversion that combines them with transitivity, treating failures\nas reflexivity conversions. -/\nnamespace instance_cache\n\n\n/-- Faster version of `mk_app ``bit0 [e]`. -/\n/-- Faster version of `mk_app ``bit1 [e]`. -/\nend instance_cache\n\n\nend tactic\n\n\nnamespace norm_num\n\n\ntheorem subst_into_add {\u03b1 : Type u_1} [Add \u03b1] (l : \u03b1) (r : \u03b1) (tl : \u03b1) (tr : \u03b1) (t : \u03b1) (prl : l = tl) (prr : r = tr) (prt : tl + tr = t) : l + r = t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (l + r = t)) prl))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (tl + r = t)) prr))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (tl + tr = t)) prt)) (Eq.refl t)))\n\ntheorem subst_into_mul {\u03b1 : Type u_1} [Mul \u03b1] (l : \u03b1) (r : \u03b1) (tl : \u03b1) (tr : \u03b1) (t : \u03b1) (prl : l = tl) (prr : r = tr) (prt : tl * tr = t) : l * r = t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (l * r = t)) prl))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (tl * r = t)) prr))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (tl * tr = t)) prt)) (Eq.refl t)))\n\ntheorem subst_into_neg {\u03b1 : Type u_1} [Neg \u03b1] (a : \u03b1) (ta : \u03b1) (t : \u03b1) (pra : a = ta) (prt : -ta = t) : -a = t := sorry\n\n/-- The result type of `match_numeral`, either `0`, `1`, or a top level\ndecomposition of `bit0 e` or `bit1 e`. The `other` case means it is not a numeral. -/\n/-- Unfold the top level constructor of the numeral expression. -/\ntheorem zero_succ {\u03b1 : Type u_1} [semiring \u03b1] : 0 + 1 = 1 :=\n  zero_add 1\n\ntheorem one_succ {\u03b1 : Type u_1} [semiring \u03b1] : 1 + 1 = bit0 1 :=\n  rfl\n\ntheorem bit0_succ {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) : bit0 a + 1 = bit1 a :=\n  rfl\n\ntheorem bit1_succ {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : bit1 a + 1 = bit0 b := sorry\n\n/-- Given `a`, `b` natural numerals, proves `\u22a2 a + 1 = b`, assuming that this is provable.\n(It may prove garbage instead of failing if `a + 1 = b` is false.) -/\ntheorem zero_adc {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : 0 + a + 1 = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 + a + 1 = b)) (zero_add a))) h\n\ntheorem adc_zero {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : a + 0 + 1 = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a + 0 + 1 = b)) (add_zero a))) h\n\ntheorem one_add {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : 1 + a = b :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 + a = b)) (add_comm 1 a))) h\n\ntheorem add_bit0_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b = c) : bit0 a + bit0 b = bit0 c := sorry\n\ntheorem add_bit0_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b = c) : bit0 a + bit1 b = bit1 c := sorry\n\ntheorem add_bit1_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b = c) : bit1 a + bit0 b = bit1 c := sorry\n\ntheorem add_bit1_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b + 1 = c) : bit1 a + bit1 b = bit0 c := sorry\n\ntheorem adc_one_one {\u03b1 : Type u_1} [semiring \u03b1] : 1 + 1 + 1 = bit1 1 :=\n  rfl\n\ntheorem adc_bit0_one {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : bit0 a + 1 + 1 = bit0 b := sorry\n\ntheorem adc_one_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : 1 + bit0 a + 1 = bit0 b := sorry\n\ntheorem adc_bit1_one {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : bit1 a + 1 + 1 = bit1 b := sorry\n\ntheorem adc_one_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 = b) : 1 + bit1 a + 1 = bit1 b := sorry\n\ntheorem adc_bit0_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b = c) : bit0 a + bit0 b + 1 = bit1 c := sorry\n\ntheorem adc_bit1_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b + 1 = c) : bit1 a + bit0 b + 1 = bit0 c := sorry\n\ntheorem adc_bit0_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b + 1 = c) : bit0 a + bit1 b + 1 = bit0 c := sorry\n\ntheorem adc_bit1_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b + 1 = c) : bit1 a + bit1 b + 1 = bit1 c := sorry\n\n/-- Given `a`,`b`,`r` natural numerals, proves `\u22a2 a + b = r`. -/\n/-- Given `a`,`b`,`r` natural numerals, proves `\u22a2 a + b + 1 = r`. -/\n/-- Given `a`,`b` natural numerals, returns `(r, \u22a2 a + b = r)`. -/\ntheorem bit0_mul {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : bit0 a * b = bit0 c := sorry\n\ntheorem mul_bit0' {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : a * bit0 b = bit0 c := sorry\n\ntheorem mul_bit0_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : bit0 a * bit0 b = bit0 (bit0 c) :=\n  bit0_mul a (bit0 b) (bit0 c) (mul_bit0' a b c h)\n\ntheorem mul_bit1_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (d : \u03b1) (e : \u03b1) (hc : a * b = c) (hd : a + b = d) (he : bit0 c + d = e) : bit1 a * bit1 b = bit1 e := sorry\n\n/-- Given `a`,`b` natural numerals, returns `(r, \u22a2 a * b = r)`. -/\n/-- Given `a` a positive natural numeral, returns `\u22a2 0 < a`. -/\n/-- Given `a` a rational numeral, returns `\u22a2 0 < a`. -/\n/-- `match_neg (- e) = some e`, otherwise `none` -/\n/-- `match_sign (- e) = inl e`, `match_sign 0 = inr ff`, otherwise `inr tt` -/\ntheorem ne_zero_of_pos {\u03b1 : Type u_1} [ordered_add_comm_group \u03b1] (a : \u03b1) : 0 < a \u2192 a \u2260 0 :=\n  ne_of_gt\n\ntheorem ne_zero_neg {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) : a \u2260 0 \u2192 -a \u2260 0 :=\n  mt (iff.mp neg_eq_zero)\n\n/-- Given `a` a rational numeral, returns `\u22a2 a \u2260 0`. -/\ntheorem clear_denom_div {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (b : \u03b1) (b' : \u03b1) (c : \u03b1) (d : \u03b1) (h\u2080 : b \u2260 0) (h\u2081 : b * b' = d) (h\u2082 : a * b' = c) : a / b * d = c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a / b * d = c)) (Eq.symm h\u2081)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * (b * b') = c)) (Eq.symm (mul_assoc (a / b) b b'))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a / b * b * b' = c)) (div_mul_cancel a h\u2080))) h\u2082))\n\n/-- Given `a` nonnegative rational and `d` a natural number, returns `(b, \u22a2 a * d = b)`.\n(`d` should be a multiple of the denominator of `a`, so that `b` is a natural number.) -/\ntheorem nonneg_pos {\u03b1 : Type u_1} [ordered_cancel_add_comm_monoid \u03b1] (a : \u03b1) : 0 < a \u2192 0 \u2264 a :=\n  le_of_lt\n\ntheorem lt_one_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (h : 1 \u2264 a) : 1 < bit0 a :=\n  lt_of_lt_of_le one_lt_two (iff.mpr bit0_le_bit0 h)\n\ntheorem lt_one_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (h : 0 < a) : 1 < bit1 a :=\n  iff.mpr one_lt_bit1 h\n\ntheorem lt_bit0_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) : a < b \u2192 bit0 a < bit0 b :=\n  iff.mpr bit0_lt_bit0\n\ntheorem lt_bit0_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a \u2264 b) : bit0 a < bit1 b :=\n  lt_of_le_of_lt (iff.mpr bit0_le_bit0 h) (lt_add_one (bit0 b))\n\ntheorem lt_bit1_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 \u2264 b) : bit1 a < bit0 b := sorry\n\ntheorem lt_bit1_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) : a < b \u2192 bit1 a < bit1 b :=\n  iff.mpr bit1_lt_bit1\n\ntheorem le_one_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (h : 1 \u2264 a) : 1 \u2264 bit0 a :=\n  le_of_lt (lt_one_bit0 a h)\n\n-- deliberately strong hypothesis because bit1 0 is not a numeral\n\ntheorem le_one_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (h : 0 < a) : 1 \u2264 bit1 a :=\n  le_of_lt (lt_one_bit1 a h)\n\ntheorem le_bit0_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) : a \u2264 b \u2192 bit0 a \u2264 bit0 b :=\n  iff.mpr bit0_le_bit0\n\ntheorem le_bit0_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a \u2264 b) : bit0 a \u2264 bit1 b :=\n  le_of_lt (lt_bit0_bit1 a b h)\n\ntheorem le_bit1_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 \u2264 b) : bit1 a \u2264 bit0 b :=\n  le_of_lt (lt_bit1_bit0 a b h)\n\ntheorem le_bit1_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) : a \u2264 b \u2192 bit1 a \u2264 bit1 b :=\n  iff.mpr bit1_le_bit1\n\ntheorem sle_one_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) : 1 \u2264 a \u2192 1 + 1 \u2264 bit0 a :=\n  iff.mpr bit0_le_bit0\n\ntheorem sle_one_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) : 1 \u2264 a \u2192 1 + 1 \u2264 bit1 a :=\n  le_bit0_bit1 1 a\n\ntheorem sle_bit0_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) : a + 1 \u2264 b \u2192 bit0 a + 1 \u2264 bit0 b :=\n  le_bit1_bit0 a b\n\ntheorem sle_bit0_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a \u2264 b) : bit0 a + 1 \u2264 bit1 b :=\n  iff.mpr bit1_le_bit1 h\n\ntheorem sle_bit1_bit0 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 \u2264 b) : bit1 a + 1 \u2264 bit0 b :=\n  Eq.symm (bit1_succ a (a + 1) rfl) \u25b8 iff.mpr bit0_le_bit0 h\n\ntheorem sle_bit1_bit1 {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (b : \u03b1) (h : a + 1 \u2264 b) : bit1 a + 1 \u2264 bit1 b :=\n  Eq.symm (bit1_succ a (a + 1) rfl) \u25b8 le_bit0_bit1 (a + 1) b h\n\n/-- Given `a` a rational numeral, returns `\u22a2 0 \u2264 a`. -/\n/-- Given `a` a rational numeral, returns `\u22a2 1 \u2264 a`. -/\n/-- Given `a`,`b` natural numerals, proves `\u22a2 a \u2264 b`. -/\n/-- Given `a`,`b` natural numerals, proves `\u22a2 a + 1 \u2264 b`. -/\n/-- Given `a`,`b` natural numerals, proves `\u22a2 a < b`. -/\ntheorem clear_denom_lt {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (a' : \u03b1) (b : \u03b1) (b' : \u03b1) (d : \u03b1) (h\u2080 : 0 < d) (ha : a * d = a') (hb : b * d = b') (h : a' < b') : a < b :=\n  lt_of_mul_lt_mul_right\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * d < b * d)) ha)) (eq.mpr (id (Eq._oldrec (Eq.refl (a' < b * d)) hb)) h))\n    (le_of_lt h\u2080)\n\n/-- Given `a`,`b` nonnegative rational numerals, proves `\u22a2 a < b`. -/\ntheorem lt_neg_pos {\u03b1 : Type u_1} [ordered_add_comm_group \u03b1] (a : \u03b1) (b : \u03b1) (ha : 0 < a) (hb : 0 < b) : -a < b :=\n  lt_trans (neg_neg_of_pos ha) hb\n\n/-- Given `a`,`b` rational numerals, proves `\u22a2 a < b`. -/\ntheorem clear_denom_le {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (a : \u03b1) (a' : \u03b1) (b : \u03b1) (b' : \u03b1) (d : \u03b1) (h\u2080 : 0 < d) (ha : a * d = a') (hb : b * d = b') (h : a' \u2264 b') : a \u2264 b :=\n  le_of_mul_le_mul_right\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * d \u2264 b * d)) ha)) (eq.mpr (id (Eq._oldrec (Eq.refl (a' \u2264 b * d)) hb)) h)) h\u2080\n\n/-- Given `a`,`b` nonnegative rational numerals, proves `\u22a2 a \u2264 b`. -/\ntheorem le_neg_pos {\u03b1 : Type u_1} [ordered_add_comm_group \u03b1] (a : \u03b1) (b : \u03b1) (ha : 0 \u2264 a) (hb : 0 \u2264 b) : -a \u2264 b :=\n  le_trans (neg_nonpos_of_nonneg ha) hb\n\n/-- Given `a`,`b` rational numerals, proves `\u22a2 a \u2264 b`. -/\n/-- Given `a`,`b` rational numerals, proves `\u22a2 a \u2260 b`. This version tries to prove\n`\u22a2 a < b` or `\u22a2 b < a`, and so is not appropriate for types without an order relation. -/\ntheorem nat_cast_zero {\u03b1 : Type u_1} [semiring \u03b1] : \u21910 = 0 :=\n  nat.cast_zero\n\ntheorem nat_cast_one {\u03b1 : Type u_1} [semiring \u03b1] : \u21911 = 1 :=\n  nat.cast_one\n\ntheorem nat_cast_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u2115) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit0 a) = bit0 a' :=\n  h \u25b8 nat.cast_bit0 a\n\ntheorem nat_cast_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (a : \u2115) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit1 a) = bit1 a' :=\n  h \u25b8 nat.cast_bit1 a\n\ntheorem int_cast_zero {\u03b1 : Type u_1} [ring \u03b1] : \u21910 = 0 :=\n  int.cast_zero\n\ntheorem int_cast_one {\u03b1 : Type u_1} [ring \u03b1] : \u21911 = 1 :=\n  int.cast_one\n\ntheorem int_cast_bit0 {\u03b1 : Type u_1} [ring \u03b1] (a : \u2124) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit0 a) = bit0 a' :=\n  h \u25b8 int.cast_bit0 a\n\ntheorem int_cast_bit1 {\u03b1 : Type u_1} [ring \u03b1] (a : \u2124) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit1 a) = bit1 a' :=\n  h \u25b8 int.cast_bit1 a\n\ntheorem rat_cast_bit0 {\u03b1 : Type u_1} [division_ring \u03b1] [char_zero \u03b1] (a : \u211a) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit0 a) = bit0 a' :=\n  h \u25b8 rat.cast_bit0 a\n\ntheorem rat_cast_bit1 {\u03b1 : Type u_1} [division_ring \u03b1] [char_zero \u03b1] (a : \u211a) (a' : \u03b1) (h : \u2191a = a') : \u2191(bit1 a) = bit1 a' :=\n  h \u25b8 rat.cast_bit1 a\n\n/-- Given `a' : \u03b1` a natural numeral, returns `(a : \u2115, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\n/-- Given `a' : \u03b1` a natural numeral, returns `(a : \u2124, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\n/-- Given `a' : \u03b1` a natural numeral, returns `(a : \u211a, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\ntheorem rat_cast_div {\u03b1 : Type u_1} [division_ring \u03b1] [char_zero \u03b1] (a : \u211a) (b : \u211a) (a' : \u03b1) (b' : \u03b1) (ha : \u2191a = a') (hb : \u2191b = b') : \u2191(a / b) = a' / b' :=\n  ha \u25b8 hb \u25b8 rat.cast_div a b\n\n/-- Given `a' : \u03b1` a nonnegative rational numeral, returns `(a : \u211a, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\ntheorem int_cast_neg {\u03b1 : Type u_1} [ring \u03b1] (a : \u2124) (a' : \u03b1) (h : \u2191a = a') : \u2191(-a) = -a' :=\n  h \u25b8 int.cast_neg a\n\ntheorem rat_cast_neg {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u211a) (a' : \u03b1) (h : \u2191a = a') : \u2191(-a) = -a' :=\n  h \u25b8 rat.cast_neg a\n\n/-- Given `a' : \u03b1` an integer numeral, returns `(a : \u2124, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\n/-- Given `a' : \u03b1` a rational numeral, returns `(a : \u211a, \u22a2 \u2191a = a')`.\n(Note that the returned value is on the left of the equality.) -/\ntheorem nat_cast_ne {\u03b1 : Type u_1} [semiring \u03b1] [char_zero \u03b1] (a : \u2115) (b : \u2115) (a' : \u03b1) (b' : \u03b1) (ha : \u2191a = a') (hb : \u2191b = b') (h : a \u2260 b) : a' \u2260 b' :=\n  ha \u25b8 hb \u25b8 mt (iff.mp nat.cast_inj) h\n\ntheorem int_cast_ne {\u03b1 : Type u_1} [ring \u03b1] [char_zero \u03b1] (a : \u2124) (b : \u2124) (a' : \u03b1) (b' : \u03b1) (ha : \u2191a = a') (hb : \u2191b = b') (h : a \u2260 b) : a' \u2260 b' :=\n  ha \u25b8 hb \u25b8 mt (iff.mp int.cast_inj) h\n\ntheorem rat_cast_ne {\u03b1 : Type u_1} [division_ring \u03b1] [char_zero \u03b1] (a : \u211a) (b : \u211a) (a' : \u03b1) (b' : \u03b1) (ha : \u2191a = a') (hb : \u2191b = b') (h : a \u2260 b) : a' \u2260 b' :=\n  ha \u25b8 hb \u25b8 mt (iff.mp rat.cast_inj) h\n\n/-- Given `a`,`b` rational numerals, proves `\u22a2 a \u2260 b`. Currently it tries two methods:\n\n  * Prove `\u22a2 a < b` or `\u22a2 b < a`, if the base type has an order\n  * Embed `\u2191(a':\u211a) = a` and `\u2191(b':\u211a) = b`, and then prove `a' \u2260 b'`.\n    This requires that the base type be `char_zero`, and also that it be a `division_ring`\n    so that the coercion from `\u211a` is well defined.\n\nWe may also add coercions to `\u2124` and `\u2115` as well in order to support `char_zero`\nrings and semirings. -/\n/-- Given `a` a rational numeral, returns `\u22a2 a \u2260 0`. -/\n/-- Given `a` nonnegative rational and `d` a natural number, returns `(b, \u22a2 a * d = b)`.\n(`d` should be a multiple of the denominator of `a`, so that `b` is a natural number.) -/\ntheorem clear_denom_add {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (a' : \u03b1) (b : \u03b1) (b' : \u03b1) (c : \u03b1) (c' : \u03b1) (d : \u03b1) (h\u2080 : d \u2260 0) (ha : a * d = a') (hb : b * d = b') (hc : c * d = c') (h : a' + b' = c') : a + b = c := sorry\n\n/-- Given `a`,`b`,`c` nonnegative rational numerals, returns `\u22a2 a + b = c`. -/\ntheorem add_pos_neg_pos {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : c + b = a) : a + -b = c := sorry\n\ntheorem add_pos_neg_neg {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : c + a = b) : a + -b = -c := sorry\n\ntheorem add_neg_pos_pos {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + c = b) : -a + b = c := sorry\n\ntheorem add_neg_pos_neg {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : b + c = a) : -a + b = -c := sorry\n\ntheorem add_neg_neg {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : b + a = c) : -a + -b = -c := sorry\n\n/-- Given `a`,`b`,`c` rational numerals, returns `\u22a2 a + b = c`. -/\n/-- Given `a`,`b` rational numerals, returns `(c, \u22a2 a + b = c)`. -/\ntheorem clear_denom_simple_nat {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) : 1 \u2260 0 \u2227 a * 1 = a :=\n  { left := one_ne_zero, right := mul_one a }\n\ntheorem clear_denom_simple_div {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (b : \u03b1) (h : b \u2260 0) : b \u2260 0 \u2227 a / b * b = a :=\n  { left := h, right := div_mul_cancel a h }\n\n/-- Given `a` a nonnegative rational numeral, returns `(b, c, \u22a2 a * b = c)`\nwhere `b` and `c` are natural numerals. (`b` will be the denominator of `a`.) -/\ntheorem clear_denom_mul {\u03b1 : Type u_1} [field \u03b1] (a : \u03b1) (a' : \u03b1) (b : \u03b1) (b' : \u03b1) (c : \u03b1) (c' : \u03b1) (d\u2081 : \u03b1) (d\u2082 : \u03b1) (d : \u03b1) (ha : d\u2081 \u2260 0 \u2227 a * d\u2081 = a') (hb : d\u2082 \u2260 0 \u2227 b * d\u2082 = b') (hc : c * d = c') (hd : d\u2081 * d\u2082 = d) (h : a' * b' = c') : a * b = c := sorry\n\n/-- Given `a`,`b` nonnegative rational numerals, returns `(c, \u22a2 a * b = c)`. -/\ntheorem mul_neg_pos {\u03b1 : Type u_1} [ring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : -a * b = -c := sorry\n\ntheorem mul_pos_neg {\u03b1 : Type u_1} [ring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : a * -b = -c := sorry\n\ntheorem mul_neg_neg {\u03b1 : Type u_1} [ring \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a * b = c) : -a * -b = c := sorry\n\n/-- Given `a`,`b` rational numerals, returns `(c, \u22a2 a * b = c)`. -/\ntheorem inv_neg {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (b : \u03b1) (h : a\u207b\u00b9 = b) : -a\u207b\u00b9 = -b := sorry\n\ntheorem inv_one {\u03b1 : Type u_1} [division_ring \u03b1] : 1\u207b\u00b9 = 1 :=\n  inv_one\n\ntheorem inv_one_div {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) : 1 / a\u207b\u00b9 = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 / a\u207b\u00b9 = a)) (one_div a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a\u207b\u00b9\u207b\u00b9 = a)) (inv_inv' a))) (Eq.refl a))\n\ntheorem inv_div_one {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) : a\u207b\u00b9 = 1 / a :=\n  inv_eq_one_div a\n\ntheorem inv_div {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (b : \u03b1) : a / b\u207b\u00b9 = b / a := sorry\n\n/-- Given `a` a rational numeral, returns `(b, \u22a2 a\u207b\u00b9 = b)`. -/\ntheorem div_eq {\u03b1 : Type u_1} [division_ring \u03b1] (a : \u03b1) (b : \u03b1) (b' : \u03b1) (c : \u03b1) (hb : b\u207b\u00b9 = b') (h : a * b' = c) : a / b = c :=\n  eq.mp (Eq._oldrec (Eq.refl (a * (b\u207b\u00b9) = c)) (Eq.symm (div_eq_mul_inv a b)))\n    (eq.mp (Eq._oldrec (Eq.refl (a * b' = c)) (Eq.symm hb)) h)\n\n/-- Given `a`,`b` rational numerals, returns `(c, \u22a2 a / b = c)`. -/\n/-- Given `a` a rational numeral, returns `(b, \u22a2 -a = b)`. -/\ntheorem sub_pos {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (b' : \u03b1) (c : \u03b1) (hb : -b = b') (h : a + b' = c) : a - b = c :=\n  eq.mp (Eq._oldrec (Eq.refl (a + -b = c)) (Eq.symm (sub_eq_add_neg a b)))\n    (eq.mp (Eq._oldrec (Eq.refl (a + b' = c)) (Eq.symm hb)) h)\n\ntheorem sub_neg {\u03b1 : Type u_1} [add_group \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1) (h : a + b = c) : a - -b = c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a - -b = c)) (sub_neg_eq_add a b))) h\n\n/-- Given `a`,`b` rational numerals, returns `(c, \u22a2 a - b = c)`. -/\ntheorem sub_nat_pos (a : \u2115) (b : \u2115) (c : \u2115) (h : b + c = a) : a - b = c :=\n  h \u25b8 nat.add_sub_cancel_left b c\n\ntheorem sub_nat_neg (a : \u2115) (b : \u2115) (c : \u2115) (h : a + c = b) : a - b = 0 :=\n  nat.sub_eq_zero_of_le (h \u25b8 nat.le_add_right a c)\n\n/-- Given `a : nat`,`b : nat` natural numerals, returns `(c, \u22a2 a - b = c)`. -/\n/-- Evaluates the basic field operations `+`,`neg`,`-`,`*`,`inv`,`/` on numerals.\nAlso handles nat subtraction. Does not do recursive simplification; that is,\n`1 + 1 + 1` will not simplify but `2 + 1` will. This is handled by the top level\n`simp` call in `norm_num.derive`. -/\ntheorem pow_bit0 {\u03b1 : Type u} [monoid \u03b1] (a : \u03b1) (c' : \u03b1) (c : \u03b1) (b : \u2115) (h : a ^ b = c') (h\u2082 : c' * c' = c) : a ^ bit0 b = c := sorry\n\ntheorem pow_bit1 {\u03b1 : Type u} [monoid \u03b1] (a : \u03b1) (c\u2081 : \u03b1) (c\u2082 : \u03b1) (c : \u03b1) (b : \u2115) (h : a ^ b = c\u2081) (h\u2082 : c\u2081 * c\u2081 = c\u2082) (h\u2083 : c\u2082 * a = c) : a ^ bit1 b = c := sorry\n\n/-- Given `a` a rational numeral and `b : nat`, returns `(c, \u22a2 a ^ b = c)`. -/\n/-- Evaluates expressions of the form `a ^ b`, `monoid.pow a b` or `nat.pow a b`. -/\n/-- Given `\u22a2 p`, returns `(true, \u22a2 p = true)`. -/\n/-- Given `\u22a2 \u00ac p`, returns `(false, \u22a2 p = false)`. -/\ntheorem not_refl_false_intro {\u03b1 : Sort u_1} (a : \u03b1) : a \u2260 a = False :=\n  eq_false_intro (not_not_intro rfl)\n\n/-- Evaluates the inequality operations `=`,`<`,`>`,`\u2264`,`\u2265`,`\u2260` on numerals. -/\ntheorem nat_succ_eq (a : \u2115) (b : \u2115) (c : \u2115) (h\u2081 : a = b) (h\u2082 : b + 1 = c) : Nat.succ a = c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (Nat.succ a = c)) h\u2081)) h\u2082\n\n/-- Evaluates the expression `nat.succ ... (nat.succ n)` where `n` is a natural numeral.\n(We could also just handle `nat.succ n` here and rely on `simp` to work bottom up, but we figure\nthat towers of successors coming from e.g. `induction` are a common case.) -/\ntheorem nat_div (a : \u2115) (b : \u2115) (q : \u2115) (r : \u2115) (m : \u2115) (hm : q * b = m) (h : r + m = a) (h\u2082 : r < b) : a / b = q := sorry\n\ntheorem int_div (a : \u2124) (b : \u2124) (q : \u2124) (r : \u2124) (m : \u2124) (hm : q * b = m) (h : r + m = a) (h\u2081 : 0 \u2264 r) (h\u2082 : r < b) : a / b = q := sorry\n\ntheorem nat_mod (a : \u2115) (b : \u2115) (q : \u2115) (r : \u2115) (m : \u2115) (hm : q * b = m) (h : r + m = a) (h\u2082 : r < b) : a % b = r := sorry\n\ntheorem int_mod (a : \u2124) (b : \u2124) (q : \u2124) (r : \u2124) (m : \u2124) (hm : q * b = m) (h : r + m = a) (h\u2081 : 0 \u2264 r) (h\u2082 : r < b) : a % b = r := sorry\n\ntheorem int_div_neg (a : \u2124) (b : \u2124) (c' : \u2124) (c : \u2124) (h : a / b = c') (h\u2082 : -c' = c) : a / -b = c :=\n  h\u2082 \u25b8 h \u25b8 int.div_neg a b\n\ntheorem int_mod_neg (a : \u2124) (b : \u2124) (c : \u2124) (h : a % b = c) : a % -b = c :=\n  Eq.trans (int.mod_neg a b) h\n\n/-- Given `a`,`b` numerals in `nat` or `int`,\n  * `prove_div_mod ic a b ff` returns `(c, \u22a2 a / b = c)`\n  * `prove_div_mod ic a b tt` returns `(c, \u22a2 a % b = c)`\n-/\ntheorem dvd_eq_nat (a : \u2115) (b : \u2115) (c : \u2115) (p : Prop) (h\u2081 : b % a = c) (h\u2082 : c = 0 = p) : a \u2223 b = p := sorry\n\ntheorem dvd_eq_int (a : \u2124) (b : \u2124) (c : \u2124) (p : Prop) (h\u2081 : b % a = c) (h\u2082 : c = 0 = p) : a \u2223 b = p := sorry\n\n/-- Evaluates some extra numeric operations on `nat` and `int`, specifically\n`nat.succ`, `/` and `%`, and `\u2223` (divisibility). -/\n/-- This version of `derive` does not fail when the input is already a numeral -/\n/-- An attribute for adding additional extensions to `norm_num`. To use this attribute, put\n`@[norm_num]` on a tactic of type `expr \u2192 tactic (expr \u00d7 expr)`; the tactic will be called on\nsubterms by `norm_num`, and it is responsible for identifying that the expression is a numerical\nfunction applied to numerals, for example `nat.fib 17`, and should return the reduced numerical\nexpression (which must be in `norm_num`-normal form: a natural or rational numeral, i.e. `37`,\n`12 / 7` or `-(2 / 3)`, although this can be an expression in any type), and the proof that the\noriginal expression is equal to the rewritten expression.\n\nFailure is used to indicate that this tactic does not apply to the term. For performance reasons,\nit is best to detect non-applicability as soon as possible so that the next tactic can have a go,\nso generally it will start with a pattern match and then checking that the arguments to the term\nare numerals or of the appropriate form, followed by proof construction, which should not fail.\n\nPropositions are treated like any other term. The normal form for propositions is `true` or\n`false`, so it should produce a proof of the form `p = true` or `p = false`. `eq_true_intro` can be\nused to help here.\n-/\n/-- Look up the `norm_num` extensions in the cache and return a tactic extending `derive.step` with\nadditional reduction procedures. -/\n/-- Simplify an expression bottom-up using `step` to simplify the subexpressions. -/\n/-- Simplify an expression bottom-up using the default `norm_num` set to simplify the\nsubexpressions. -/\nend norm_num\n\n\n/-- Basic version of `norm_num` that does not call `simp`. It uses the provided `step` tactic\nto simplify the expression; use `get_step` to get the default `norm_num` set and `derive.step` for\nthe basic builtin set of simplifications. -/\n/-- Normalize numerical expressions. It uses the provided `step` tactic to simplify the expression;\nuse `get_step` to get the default `norm_num` set and `derive.step` for the basic builtin set of\nsimplifications. -/\nnamespace tactic.interactive\n\n\n/-- Basic version of `norm_num` that does not call `simp`. -/\n/-- Normalize numerical expressions. Supports the operations\n`+` `-` `*` `/` `^` and `%` over numerical types such as\n`\u2115`, `\u2124`, `\u211a`, `\u211d`, `\u2102` and some general algebraic types,\nand can prove goals of the form `A = B`, `A \u2260 B`, `A < B` and `A \u2264 B`,\nwhere `A` and `B` are numerical expressions.\nIt also has a relatively simple primality prover. -/\n/-- Normalizes a numerical expression and tries to close the goal with the result. -/\n/--\nNormalises numerical expressions. It supports the operations `+` `-` `*` `/` `^` and `%` over\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/norm_num.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.39735375709899035}}
{"text": "import classes.context_free.basics.inclusion\nimport classes.unrestricted.closure_properties.concatenation\nimport utilities.written_by_others.trim_assoc\nimport utilities.written_by_others.print_sorries\n\nvariables {T : Type}\n\n\nprivate def wrap_CF_rule\u2081 {N\u2081 : Type} (N\u2082 : Type) (r : (N\u2081 \u00d7 list (symbol T N\u2081))) :\n  ((nnn T N\u2081 N\u2082) \u00d7 list (nst T N\u2081 N\u2082)) :=\n((sum.inl (some (sum.inl r.fst))), (list.map (wrap_symbol\u2081 N\u2082) r.snd))\n\nprivate def wrap_CF_rule\u2082 {N\u2082 : Type} (N\u2081 : Type) (r : (N\u2082 \u00d7 list (symbol T N\u2082))) :\n  ((nnn T N\u2081 N\u2082) \u00d7 list (nst T N\u2081 N\u2082)) :=\n((sum.inl (some (sum.inr r.fst))), (list.map (wrap_symbol\u2082 N\u2081) r.snd))\n\nprivate def CF_rules_for_terminals\u2081 (N\u2082 : Type) (g : CF_grammar T) :\n  list ((nnn T g.nt N\u2082) \u00d7 list (nst T g.nt N\u2082)) :=\nlist.map (\u03bb t, ((sum.inr (sum.inl t)), [symbol.terminal t])) (all_used_terminals (grammar_of_cfg g))\n\nprivate def CF_rules_for_terminals\u2082 (N\u2081 : Type) (g : CF_grammar T) :\n  list ((nnn T N\u2081 g.nt) \u00d7 list (nst T N\u2081 g.nt)) :=\nlist.map (\u03bb t, ((sum.inr (sum.inr t)), [symbol.terminal t])) (all_used_terminals (grammar_of_cfg g))\n\nprivate def big_CF_grammar (g\u2081 g\u2082 : CF_grammar T) : CF_grammar T :=\nCF_grammar.mk\n  (nnn T g\u2081.nt g\u2082.nt)\n  (sum.inl none)\n  (((sum.inl none), [\n    symbol.nonterminal (sum.inl (some (sum.inl g\u2081.initial))),\n    symbol.nonterminal (sum.inl (some (sum.inr g\u2082.initial)))]\n  ) :: (\n    (list.map (wrap_CF_rule\u2081 g\u2082.nt) g\u2081.rules ++ list.map (wrap_CF_rule\u2082 g\u2081.nt) g\u2082.rules) ++\n    (CF_rules_for_terminals\u2081 g\u2082.nt g\u2081 ++ CF_rules_for_terminals\u2082 g\u2081.nt g\u2082)\n  ))\n\nprivate lemma big_CF_grammar_same_language (g\u2081 g\u2082 : CF_grammar T) :\n  CF_language (big_CF_grammar g\u2081 g\u2082) = grammar_language (big_grammar (grammar_of_cfg g\u2081) (grammar_of_cfg g\u2082)) :=\nbegin\n  rw CF_language_eq_grammar_language,\n  congr,\n  unfold big_CF_grammar,\n  unfold grammar_of_cfg,\n  unfold big_grammar,\n  dsimp only [list.map],\n  congr,\n  repeat {\n    rw list.map_append,\n  },\n  trim,\n  {\n    apply congr_arg2,\n    {\n      unfold rules_for_terminals\u2081,\n      unfold CF_rules_for_terminals\u2081,\n      finish,\n    },\n    {\n      unfold rules_for_terminals\u2082,\n      unfold CF_rules_for_terminals\u2082,\n      finish,\n    },\n  },\nend\n\n/-- The class of context-free languages is closed under concatenation.\n    This theorem is proved by translation from general grammars.\n    Compare to `classes.context_free.closure_properties.concatenation.lean` which uses\n    a simpler and more effective construction (based on context-gree grammars only). -/\nprivate theorem bonus_CF_of_CF_c_CF (L\u2081 : language T) (L\u2082 : language T) :\n  is_CF L\u2081  \u2227  is_CF L\u2082   \u2192   is_CF (L\u2081 * L\u2082)   :=\nbegin\n  rintro \u27e8\u27e8g\u2081, eq_L\u2081\u27e9, \u27e8g\u2082, eq_L\u2082\u27e9\u27e9,\n  rw CF_language_eq_grammar_language g\u2081 at eq_L\u2081,\n  rw CF_language_eq_grammar_language g\u2082 at eq_L\u2082,\n\n  use big_CF_grammar g\u2081 g\u2082,\n  rw big_CF_grammar_same_language,\n\n  apply set.eq_of_subset_of_subset,\n  {\n    intros w hyp,\n    rw \u2190eq_L\u2081,\n    rw \u2190eq_L\u2082,\n    exact in_concatenated_of_in_big hyp,\n  },\n  {\n    intros w hyp,\n    rw \u2190eq_L\u2081 at hyp,\n    rw \u2190eq_L\u2082 at hyp,\n    exact in_big_of_in_concatenated hyp,\n  },\nend\n\n\n#check            bonus_CF_of_CF_c_CF\n#print_sorries_in bonus_CF_of_CF_c_CF\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/unrestricted/closure_properties/concatenation_bonus.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.3973158806498233}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n\nNonnegative real numbers.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.algebra.infinite_sum\nimport Mathlib.topology.algebra.group_with_zero\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\nnamespace nnreal\n\n\nprotected instance topological_space : topological_space nnreal := infer_instance\n\nprotected instance topological_semiring : topological_semiring nnreal := topological_semiring.mk\n\nprotected instance topological_space.second_countable_topology :\n    topological_space.second_countable_topology nnreal :=\n  topological_space.subtype.second_countable_topology \u211d fun (r : \u211d) => real.le 0 r\n\nprotected instance order_topology : order_topology nnreal := Mathlib.order_topology_of_ord_connected\n\ntheorem continuous_of_real : continuous nnreal.of_real :=\n  continuous_subtype_mk (fun (r : \u211d) => of_real._proof_1 r)\n    (continuous.max continuous_id continuous_const)\n\ntheorem continuous_coe : continuous coe := continuous_subtype_val\n\n@[simp] theorem tendsto_coe {\u03b1 : Type u_1} {f : filter \u03b1} {m : \u03b1 \u2192 nnreal} {x : nnreal} :\n    filter.tendsto (fun (a : \u03b1) => \u2191(m a)) f (nhds \u2191x) \u2194 filter.tendsto m f (nhds x) :=\n  iff.symm tendsto_subtype_rng\n\ntheorem tendsto_coe' {\u03b1 : Type u_1} {f : filter \u03b1} [filter.ne_bot f] {m : \u03b1 \u2192 nnreal} {x : \u211d} :\n    filter.tendsto (fun (a : \u03b1) => \u2191(m a)) f (nhds x) \u2194\n        \u2203 (hx : 0 \u2264 x), filter.tendsto m f (nhds { val := x, property := hx }) :=\n  sorry\n\n@[simp] theorem map_coe_at_top : filter.map coe filter.at_top = filter.at_top :=\n  filter.map_coe_Ici_at_top 0\n\ntheorem comap_coe_at_top : filter.comap coe filter.at_top = filter.at_top :=\n  Eq.symm (filter.at_top_Ici_eq 0)\n\n@[simp] theorem tendsto_coe_at_top {\u03b1 : Type u_1} {f : filter \u03b1} {m : \u03b1 \u2192 nnreal} :\n    filter.tendsto (fun (a : \u03b1) => \u2191(m a)) f filter.at_top \u2194 filter.tendsto m f filter.at_top :=\n  iff.symm filter.tendsto_Ici_at_top\n\ntheorem tendsto_of_real {\u03b1 : Type u_1} {f : filter \u03b1} {m : \u03b1 \u2192 \u211d} {x : \u211d}\n    (h : filter.tendsto m f (nhds x)) :\n    filter.tendsto (fun (a : \u03b1) => nnreal.of_real (m a)) f (nhds (nnreal.of_real x)) :=\n  filter.tendsto.comp (continuous.tendsto continuous_of_real x) h\n\nprotected instance has_continuous_sub : has_continuous_sub nnreal :=\n  has_continuous_sub.mk\n    (continuous_subtype_mk\n      (fun (p : nnreal \u00d7 nnreal) => of_real._proof_1 (\u2191(prod.fst p) - \u2191(prod.snd p)))\n      (continuous.max\n        (continuous.sub (continuous.comp continuous_coe continuous_fst)\n          (continuous.comp continuous_coe continuous_snd))\n        continuous_const))\n\nprotected instance has_continuous_inv' : has_continuous_inv' nnreal := has_continuous_inv'.mk sorry\n\ntheorem has_sum_coe {\u03b1 : Type u_1} {f : \u03b1 \u2192 nnreal} {r : nnreal} :\n    has_sum (fun (a : \u03b1) => \u2191(f a)) \u2191r \u2194 has_sum f r :=\n  sorry\n\ntheorem has_sum_of_real_of_nonneg {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u211d} (hf_nonneg : \u2200 (n : \u03b1), 0 \u2264 f n)\n    (hf : summable f) :\n    has_sum (fun (n : \u03b1) => nnreal.of_real (f n)) (nnreal.of_real (tsum fun (n : \u03b1) => f n)) :=\n  sorry\n\ntheorem summable_coe {\u03b1 : Type u_1} {f : \u03b1 \u2192 nnreal} :\n    (summable fun (a : \u03b1) => \u2191(f a)) \u2194 summable f :=\n  sorry\n\ntheorem coe_tsum {\u03b1 : Type u_1} {f : \u03b1 \u2192 nnreal} :\n    \u2191(tsum fun (a : \u03b1) => f a) = tsum fun (a : \u03b1) => \u2191(f a) :=\n  sorry\n\ntheorem tsum_mul_left {\u03b1 : Type u_1} (a : nnreal) (f : \u03b1 \u2192 nnreal) :\n    (tsum fun (x : \u03b1) => a * f x) = a * tsum fun (x : \u03b1) => f x :=\n  sorry\n\ntheorem tsum_mul_right {\u03b1 : Type u_1} (f : \u03b1 \u2192 nnreal) (a : nnreal) :\n    (tsum fun (x : \u03b1) => f x * a) = (tsum fun (x : \u03b1) => f x) * a :=\n  sorry\n\ntheorem summable_comp_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 nnreal} (hf : summable f)\n    {i : \u03b2 \u2192 \u03b1} (hi : function.injective i) : summable (f \u2218 i) :=\n  iff.mp summable_coe\n    ((fun (this : summable ((coe \u2218 f) \u2218 i)) => this)\n      (summable.comp_injective (iff.mpr summable_coe hf) hi))\n\ntheorem summable_nat_add (f : \u2115 \u2192 nnreal) (hf : summable f) (k : \u2115) :\n    summable fun (i : \u2115) => f (i + k) :=\n  summable_comp_injective hf (add_left_injective k)\n\ntheorem summable_nat_add_iff {f : \u2115 \u2192 nnreal} (k : \u2115) :\n    (summable fun (i : \u2115) => f (i + k)) \u2194 summable f :=\n  sorry\n\ntheorem sum_add_tsum_nat_add {f : \u2115 \u2192 nnreal} (k : \u2115) (hf : summable f) :\n    (tsum fun (i : \u2115) => f i) =\n        (finset.sum (finset.range k) fun (i : \u2115) => f i) + tsum fun (i : \u2115) => f (i + k) :=\n  sorry\n\ntheorem infi_real_pos_eq_infi_nnreal_pos {\u03b1 : Type u_1} [complete_lattice \u03b1] {f : \u211d \u2192 \u03b1} :\n    (infi fun (n : \u211d) => infi fun (h : 0 < n) => f n) =\n        infi fun (n : nnreal) => infi fun (h : 0 < n) => f \u2191n :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/instances/nnreal_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3972784128187483}}
{"text": "/-\nCopyright (c) 2023 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n\n! This file was ported from Lean 3 source module algebra.field.ulift\n! leanprover-community/mathlib commit 932872382355f00112641d305ba0619305dc8642\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Field.Basic\nimport Mathbin.Algebra.Ring.Ulift\n\n/-!\n# Field instances for `ulift`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for field, semifield and related structures on `ulift` types.\n\n(Recall `ulift \u03b1` is just a \"copy\" of a type `\u03b1` in a higher universe.)\n-/\n\n\nuniverse u v\n\nvariable {\u03b1 : Type u} {x y : ULift.{v} \u03b1}\n\nnamespace ULift\n\ninstance [HasRatCast \u03b1] : HasRatCast (ULift \u03b1) :=\n  \u27e8fun a => up a\u27e9\n\n/- warning: ulift.up_rat_cast -> ULift.up_ratCast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (q : Rat), Eq.{succ (max u1 u2)} (ULift.{u2, u1} \u03b1) (ULift.up.{u2, u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) q)) ((fun (a : Type) (b : Type.{max u1 u2}) [self : HasLiftT.{1, succ (max u1 u2)} a b] => self.0) Rat (ULift.{u2, u1} \u03b1) (HasLiftT.mk.{1, succ (max u1 u2)} Rat (ULift.{u2, u1} \u03b1) (CoeTC\u2093.coe.{1, succ (max u1 u2)} Rat (ULift.{u2, u1} \u03b1) (Rat.castCoe.{max u1 u2} (ULift.{u2, u1} \u03b1) (ULift.hasRatCast.{u1, u2} \u03b1 _inst_1)))) q)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} [_inst_1 : RatCast.{u2} \u03b1] (q : Rat), Eq.{max (succ u2) (succ u1)} (ULift.{u1, u2} \u03b1) (ULift.up.{u1, u2} \u03b1 (Rat.cast.{u2} \u03b1 _inst_1 q)) (Rat.cast.{max u2 u1} (ULift.{u1, u2} \u03b1) (ULift.instRatCastULift.{u2, u1} \u03b1 _inst_1) q)\nCase conversion may be inaccurate. Consider using '#align ulift.up_rat_cast ULift.up_ratCast\u2093'. -/\n@[simp, norm_cast]\ntheorem up_ratCast [HasRatCast \u03b1] (q : \u211a) : up (q : \u03b1) = q :=\n  rfl\n#align ulift.up_rat_cast ULift.up_ratCast\n\n/- warning: ulift.down_rat_cast -> ULift.down_ratCast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (q : Rat), Eq.{succ u1} \u03b1 (ULift.down.{u2, u1} \u03b1 ((fun (a : Type) (b : Type.{max u1 u2}) [self : HasLiftT.{1, succ (max u1 u2)} a b] => self.0) Rat (ULift.{u2, u1} \u03b1) (HasLiftT.mk.{1, succ (max u1 u2)} Rat (ULift.{u2, u1} \u03b1) (CoeTC\u2093.coe.{1, succ (max u1 u2)} Rat (ULift.{u2, u1} \u03b1) (Rat.castCoe.{max u1 u2} (ULift.{u2, u1} \u03b1) (ULift.hasRatCast.{u1, u2} \u03b1 _inst_1)))) q)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) q)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} [_inst_1 : RatCast.{u2} \u03b1] (q : Rat), Eq.{succ u2} \u03b1 (ULift.down.{u1, u2} \u03b1 (Rat.cast.{max u2 u1} (ULift.{u1, u2} \u03b1) (ULift.instRatCastULift.{u2, u1} \u03b1 _inst_1) q)) (Rat.cast.{u2} \u03b1 _inst_1 q)\nCase conversion may be inaccurate. Consider using '#align ulift.down_rat_cast ULift.down_ratCast\u2093'. -/\n@[simp, norm_cast]\ntheorem down_ratCast [HasRatCast \u03b1] (q : \u211a) : down (q : ULift \u03b1) = q :=\n  rfl\n#align ulift.down_rat_cast ULift.down_ratCast\n\n#print ULift.divisionSemiring /-\ninstance divisionSemiring [DivisionSemiring \u03b1] : DivisionSemiring (ULift \u03b1) := by\n  refine' down_injective.division_semiring down _ _ _ _ _ _ _ _ _ _ <;> intros <;> rfl\n#align ulift.division_semiring ULift.divisionSemiring\n-/\n\n#print ULift.semifield /-\ninstance semifield [Semifield \u03b1] : Semifield (ULift \u03b1) :=\n  { ULift.divisionSemiring, ULift.commGroupWithZero with }\n#align ulift.semifield ULift.semifield\n-/\n\n#print ULift.divisionRing /-\ninstance divisionRing [DivisionRing \u03b1] : DivisionRing (ULift \u03b1) :=\n  { ULift.divisionSemiring, ULift.addGroup with }\n#align ulift.division_ring ULift.divisionRing\n-/\n\n#print ULift.field /-\ninstance field [Field \u03b1] : Field (ULift \u03b1) :=\n  { ULift.semifield, ULift.divisionRing with }\n#align ulift.field ULift.field\n-/\n\nend ULift\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Field/Ulift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.39727840481320437}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.sites.sheaf_of_types\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# The canonical topology on a category\n\nWe define the finest (largest) Grothendieck topology for which a given presheaf `P` is a sheaf.\nThis is well defined since if `P` is a sheaf for a topology `J`, then it is a sheaf for any\ncoarser (smaller) topology. Nonetheless we define the topology explicitly by specifying its sieves:\nA sieve `S` on `X` is covering for `finest_topology_single P` iff\n  for any `f : Y \u27f6 X`, `P` satisfies the sheaf axiom for `S.pullback f`.\nShowing that this is a genuine Grothendieck topology (namely that it satisfies the transitivity\naxiom) forms the bulk of this file.\n\nThis generalises to a set of presheaves, giving the topology `finest_topology Ps` which is the\nfinest topology for which every presheaf in `Ps` is a sheaf.\nUsing `Ps` as the set of representable presheaves defines the `canonical_topology`: the finest\ntopology for which every representable is a sheaf.\n\nA Grothendieck topology is called `subcanonical` if it is smaller than the canonical topology,\nequivalently it is subcanonical iff every representable presheaf is a sheaf.\n\n## References\n* https://ncatlab.org/nlab/show/canonical+topology\n* https://ncatlab.org/nlab/show/subcanonical+coverage\n* https://stacks.math.columbia.edu/tag/00Z9\n* https://math.stackexchange.com/a/358709/\n-/\n\nnamespace category_theory\n\n\nnamespace sheaf\n\n\n/--\nTo show `P` is a sheaf for the binding of `U` with `B`, it suffices to show that `P` is a sheaf for\n`U`, that `P` is a sheaf for each sieve in `B`, and that it is separated for any pullback of any\nsieve in `B`.\n\nThis is mostly an auxiliary lemma to show `is_sheaf_for_trans`.\nAdapted from [Elephant], Lemma C2.1.7(i) with suggestions as mentioned in\nhttps://math.stackexchange.com/a/358709/\n-/\ntheorem is_sheaf_for_bind {C : Type u} [category C] {X : C} (P : C\u1d52\u1d56 \u2964 Type v) (U : sieve X)\n    (B : {Y : C} \u2192 {f : Y \u27f6 X} \u2192 coe_fn U Y f \u2192 sieve Y) (hU : presieve.is_sheaf_for P \u21d1U)\n    (hB : \u2200 {Y : C} {f : Y \u27f6 X} (hf : coe_fn U Y f), presieve.is_sheaf_for P \u21d1(B hf))\n    (hB' :\n      \u2200 {Y : C} {f : Y \u27f6 X} (h : coe_fn U Y f) {Z : C} (g : Z \u27f6 Y),\n        presieve.is_separated_for P \u21d1(sieve.pullback g (B h))) :\n    presieve.is_sheaf_for P \u21d1(sieve.bind (\u21d1U) B) :=\n  sorry\n\n/--\nGiven two sieves `R` and `S`, to show that `P` is a sheaf for `S`, we can show:\n* `P` is a sheaf for `R`\n* `P` is a sheaf for the pullback of `S` along any arrow in `R`\n* `P` is separated for the pullback of `R` along any arrow in `S`.\n\nThis is mostly an auxiliary lemma to construct `finest_topology`.\nAdapted from [Elephant], Lemma C2.1.7(ii) with suggestions as mentioned in\nhttps://math.stackexchange.com/a/358709\n-/\ntheorem is_sheaf_for_trans {C : Type u} [category C] {X : C} (P : C\u1d52\u1d56 \u2964 Type v) (R : sieve X)\n    (S : sieve X) (hR : presieve.is_sheaf_for P \u21d1R)\n    (hR' : \u2200 {Y : C} {f : Y \u27f6 X}, coe_fn S Y f \u2192 presieve.is_separated_for P \u21d1(sieve.pullback f R))\n    (hS : \u2200 {Y : C} {f : Y \u27f6 X}, coe_fn R Y f \u2192 presieve.is_sheaf_for P \u21d1(sieve.pullback f S)) :\n    presieve.is_sheaf_for P \u21d1S :=\n  sorry\n\n/--\nConstruct the finest (largest) Grothendieck topology for which the given presheaf is a sheaf.\n\nThis is a special case of https://stacks.math.columbia.edu/tag/00Z9, but following a different\nproof (see the comments there).\n-/\ndef finest_topology_single {C : Type u} [category C] (P : C\u1d52\u1d56 \u2964 Type v) : grothendieck_topology C :=\n  grothendieck_topology.mk\n    (fun (X : C) (S : sieve X) =>\n      \u2200 (Y : C) (f : Y \u27f6 X), presieve.is_sheaf_for P \u21d1(sieve.pullback f S))\n    sorry sorry sorry\n\n/--\nConstruct the finest (largest) Grothendieck topology for which all the given presheaves are sheaves.\n\nThis is equal to the construction of https://stacks.math.columbia.edu/tag/00Z9.\n-/\ndef finest_topology {C : Type u} [category C] (Ps : set (C\u1d52\u1d56 \u2964 Type v)) : grothendieck_topology C :=\n  Inf (finest_topology_single '' Ps)\n\n/-- Check that if `P \u2208 Ps`, then `P` is indeed a sheaf for the finest topology on `Ps`. -/\ntheorem sheaf_for_finest_topology {C : Type u} [category C] {P : C\u1d52\u1d56 \u2964 Type v}\n    (Ps : set (C\u1d52\u1d56 \u2964 Type v)) (h : P \u2208 Ps) : presieve.is_sheaf (finest_topology Ps) P :=\n  sorry\n\n/--\nCheck that if each `P \u2208 Ps` is a sheaf for `J`, then `J` is a subtopology of `finest_topology Ps`.\n-/\ntheorem le_finest_topology {C : Type u} [category C] (Ps : set (C\u1d52\u1d56 \u2964 Type v))\n    (J : grothendieck_topology C) (hJ : \u2200 (P : C\u1d52\u1d56 \u2964 Type v), P \u2208 Ps \u2192 presieve.is_sheaf J P) :\n    J \u2264 finest_topology Ps :=\n  sorry\n\n/--\nThe `canonical_topology` on a category is the finest (largest) topology for which every\nrepresentable presheaf is a sheaf.\n\nSee https://stacks.math.columbia.edu/tag/00ZA\n-/\ndef canonical_topology (C : Type u) [category C] : grothendieck_topology C :=\n  finest_topology (set.range (functor.obj yoneda))\n\n/-- `yoneda.obj X` is a sheaf for the canonical topology. -/\ntheorem is_sheaf_yoneda_obj {C : Type u} [category C] (X : C) :\n    presieve.is_sheaf (canonical_topology C) (functor.obj yoneda X) :=\n  fun (Y : C) (S : sieve Y) (hS : S \u2208 coe_fn (canonical_topology C) Y) =>\n    sheaf_for_finest_topology (set.range (functor.obj yoneda)) (set.mem_range_self X) S hS\n\n/-- A representable functor is a sheaf for the canonical topology. -/\ntheorem is_sheaf_of_representable {C : Type u} [category C] (P : C\u1d52\u1d56 \u2964 Type v) [representable P] :\n    presieve.is_sheaf (canonical_topology C) P :=\n  presieve.is_sheaf_iso (canonical_topology C) representable.w\n    (is_sheaf_yoneda_obj (representable.X P))\n\n/--\nA subcanonical topology is a topology which is smaller than the canonical topology.\nEquivalently, a topology is subcanonical iff every representable is a sheaf.\n-/\ndef subcanonical {C : Type u} [category C] (J : grothendieck_topology C) := J \u2264 canonical_topology C\n\nnamespace subcanonical\n\n\n/-- If every functor `yoneda.obj X` is a `J`-sheaf, then `J` is subcanonical. -/\ntheorem of_yoneda_is_sheaf {C : Type u} [category C] (J : grothendieck_topology C)\n    (h : \u2200 (X : C), presieve.is_sheaf J (functor.obj yoneda X)) : subcanonical J :=\n  le_finest_topology (set.range (functor.obj yoneda)) J\n    fun (P : C\u1d52\u1d56 \u2964 Type v) (H : P \u2208 set.range (functor.obj yoneda)) =>\n      Exists.dcases_on H fun (X : C) (H_h : functor.obj yoneda X = P) => Eq._oldrec (h X) H_h\n\n/-- If `J` is subcanonical, then any representable is a `J`-sheaf. -/\ntheorem is_sheaf_of_representable {C : Type u} [category C] {J : grothendieck_topology C}\n    (hJ : subcanonical J) (P : C\u1d52\u1d56 \u2964 Type v) [representable P] : presieve.is_sheaf J P :=\n  presieve.is_sheaf_of_le P hJ (is_sheaf_of_representable P)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/sites/canonical_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.39727840481320437}}
{"text": "import Std\n\nnamespace LeanSAT\n\n\n/-- CNF variable\n\nNOTE: Unlike DIMACS, 0 is a valid variable. See `Var.toDIMACS`.\n-/\ndef Var := Nat\nderiving Inhabited, DecidableEq, Hashable, Repr, ToString\n\nnamespace Var\n\n/-- Allow nat literals `5392` as notation for variables -/\ninstance : OfNat Var n := \u27e8n\u27e9\nend Var\n\n\n\n/-- CNF literal -/\ninductive Literal\n| pos (v : Var) | neg (v : Var)\nderiving Inhabited, DecidableEq, Hashable, Repr\n\nnamespace Literal\n\n/-- The literal's variable -/\ndef var : Literal \u2192 Var\n| pos v => v | neg v => v\n\n/-- True iff the literal is `.pos v` -/\ndef isPos : Literal \u2192 Bool\n| pos _ => true | neg _ => false\n\n/-- True iff the literal is `.neg v` -/\ndef isNeg (l) := not (isPos l)\n\ndef not : Literal \u2192 Literal\n| pos v => neg v\n| neg v => pos v\n\n/-- Automatically lift variables to positive literals -/\ninstance : Coe Var Literal := \u27e8.pos\u27e9\n/-- Allow literals to be written as nat constants -/\ninstance : OfNat Literal n := \u27e8show Var from n\u27e9\n\ninstance : ToString Literal where\n  toString | pos v => s!\"{v}\"\n           | neg v => s!\"\u00ac{v}\"\n\nend Literal\n\n\n/-- (Partial) assignment to the variables of a formula -/\ndef Assn := Std.HashMap Var Bool\n\nnamespace Assn\n\n@[simp] def hasTrue   (v : Var) (a : Assn) : Bool := a.find? v = some true\n@[simp] def hasFalse  (v : Var) (a : Assn) : Bool := a.find? v = some false\n@[simp] def undecided (v : Var) (a : Assn) : Bool := a.find? v = none\n\ndef litTrue       (l : Literal) (a : Assn) : Bool := a.find? l.var = some l.isPos\ndef litFalse      (l : Literal) (a : Assn) : Bool := a.find? l.var = some l.isNeg\ndef litUndecided  (l : Literal) (a : Assn) : Bool := a.find? l.var = none\n\n@[simp] theorem litTrue_pos : litTrue (.pos v) a = hasTrue v a := rfl\n@[simp] theorem litTrue_neg : litTrue (.neg v) a = hasFalse v a := rfl\n@[simp] theorem litFalse_pos : litFalse (.pos v) a = hasFalse v a := rfl\n@[simp] theorem litFalse_neg : litFalse (.neg v) a = hasTrue v a := rfl\n@[simp] theorem litUndecided_pos : litUndecided (.pos v) a = undecided v a := rfl\n@[simp] theorem litUndecided_neg : litUndecided (.neg v) a = undecided v a := rfl\n\ndef insertLit (l : Literal) (a : Assn) : Assn :=\n  a.insert l.var l.isPos\n\ndef toList (a : Assn) : List Literal :=\n  Std.HashMap.toList a |>.map (fun (v,pos) => if pos then .pos v else .neg v)\n\ninstance : ToString Assn :=\n  \u27e8fun assn => assn.toList |>.map toString |> String.intercalate \" \"\u27e9\n\nend Assn\n\n\n/-- CNF clause: just a list of literals -/\nstructure Clause where\n  lits : List Literal\nderiving Inhabited, DecidableEq, Hashable, Repr\n\nnamespace Clause\n\n/-- \u22a5 / false clause -/\ndef empty : Clause := \u27e8[]\u27e9\n\n/-- Check whether any literals in `c` are set true by `a` -/\ndef eval (a : Assn) (c : Clause) : Bool :=\n  c.lits.any a.litTrue\n\n@[simp]\ntheorem eval_nil : eval a \u27e8[]\u27e9 = false\n  := by\n  simp [eval, List.any, List.foldr]\n\n@[simp]\ntheorem eval_cons : eval a \u27e8l::ls\u27e9 = (a.litTrue l || eval a \u27e8ls\u27e9)\n  := by\n  simp [eval, List.any, List.foldr]\n\ninstance : OfNat Clause n := \u27e8(\u27e8[.pos n]\u27e9)\u27e9\ninstance : Coe Literal Clause := \u27e8(\u27e8[\u00b7]\u27e9)\u27e9\ninstance : Coe (List Literal) Clause := \u27e8(\u27e8\u00b7\u27e9)\u27e9\n\ninstance : ToString Clause where\n  toString | \u27e8lits\u27e9 => toString lits\n\nend Clause\n\n\n\n/-- CNF formula: a collection of clauses.\n\nThis structure is used for formalizing lemmas about sat/unsat\nreductions and the likes. -/\nstructure Formula where\n  clauses : List Clause\nderiving DecidableEq, Repr\n\nnamespace Formula\n\ndef numVars : Formula \u2192 Nat\n| \u27e8clauses\u27e9 =>\n  clauses.filterMap (\u00b7.lits.map (\u03b2 := Nat) Literal.var |>.maximum?)\n  |>.maximum?.map Nat.succ |>.getD 0\n\ndef vars : Formula \u2192 List Var\n| \u27e8clauses\u27e9 => Id.run do\n  let mut set := Std.HashMap.empty\n  for c in clauses do\n    for l in c.lits do\n      set := set.insert l.var ()\n  return set.toList.map (\u00b7.1)\n\n/-- \u22a4 / true Formula -/\ndef empty : Formula := \u27e8[]\u27e9\n\n/-- Check whether all clauses in `c` are satisfied by `a` -/\ndef eval (a : Assn) (c : Formula) : Bool :=\n  c.clauses.all (\u00b7.eval a)\n\n@[simp]\ntheorem eval_nil : eval a \u27e8[]\u27e9 = true\n  := by\n  simp [eval, List.all, List.foldr]\n\n@[simp]\ntheorem eval_cons : eval a \u27e8c::cs\u27e9 = (c.eval a && eval a \u27e8cs\u27e9)\n  := by\n  simp [eval, List.all, List.foldr]\n\n/-- Formula `c` is satisfiable if there exists a variable assignment\non which it is satisfied. -/\ndef satisfiable (c : Formula) := \u2203 a, c.eval a = true\n\n/-- Formula `c` is unsatisfiable iff there does not exist a variable\nassignment on which it is satisfied. -/\ndef unsat (c : Formula) := \u00acc.satisfiable\n\ninstance : Coe Clause Formula := \u27e8(\u27e8[\u00b7]\u27e9)\u27e9\ninstance : OfNat Formula n := \u27e8Literal.pos n\u27e9\n\nend Formula\n\n\n/-! CNF notation -/\nnamespace Notation\n\nscoped notation:30 a:31 \" \u2228 \" b:30 => Clause.mk (List.append (Clause.lits a) (Clause.lits b))\nscoped notation a \"\u2227\" b => Formula.mk (List.append (Formula.clauses a) (Formula.clauses b))\nscoped notation:max \"\u00ac\" l:40 => Literal.not l\n\nexample : Literal := 5\nexample : Literal := \u00ac5\nexample : Clause  := \u00ac5 \u2228 \u00ac10\nexample : Formula := (\u00ac5 \u2228 \u00ac10) \u2227 20 \u2227 \u00ac30\n\nend Notation\n", "meta": {"author": "JamesGallicchio", "repo": "LeanSAT", "sha": "719470ac796a9149e0f892ccb3dff80c0dd563d3", "save_path": "github-repos/lean/JamesGallicchio-LeanSAT", "path": "github-repos/lean/JamesGallicchio-LeanSAT/LeanSAT-719470ac796a9149e0f892ccb3dff80c0dd563d3/LeanSAT/CNF.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.39727840481320437}}
{"text": "import algebraic_topology.simplicial_object\nimport algebraic_topology.simplicial_set\n\nopen category_theory category_theory.limits opposite\nopen_locale simplicial\n\nnoncomputable theory\n\nuniverses w v u\n\nnamespace sSet\n\n@[simps]\ndef binary_product (X Y : sSet) : sSet :=\n{ obj := \u03bb \u0394, X.obj \u0394 \u00d7 Y.obj \u0394,\n  map := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 s, (X.map \u03b8 s.1, Y.map \u03b8 s.2),\n  map_id' := \u03bb \u0394, begin\n    ext1 x,\n    simp only [functor_to_types.map_id_apply, prod.mk.eta, types_id_apply],\n  end,\n  map_comp' := \u03bb \u0394\u2081 \u0394\u2082 \u0394\u2083 \u03b8\u2081 \u03b8\u2082, begin\n    ext1 x,\n    simp only [functor_to_types.map_comp_apply, types_comp_apply],\n  end, }\n\nend sSet\n\nnamespace category_theory\n\nnamespace simplicial_object\n\nvariables {C : Type u} [category.{v} C]\n\n/-def Type_inclusion : Type v \u2964 Type (max w v) :=\n{ obj := ulift.{w v},\n  map := \u03bb X Y f x, ulift.up (f (ulift.down x)),\n  map_id' := \u03bb X, by { ext, refl, },\n  map_comp' := \u03bb X Y Z f g, by { ext, refl, }, }\n\ndef yoneda' {C : Type u} [category.{v} C] :\n  C \u2964 (C\u1d52\u1d56 \u2964 Type (max w v)) :=\nyoneda \u22d9 (whiskering_right _ _ _).obj Type_inclusion-/\n\nnamespace sHom\n\n/-- K \u2297 X \u27f6 Y -/\n@[ext]\nstructure bihom (K : sSet.{w}) (X Y : simplicial_object C) :=\n(\u03c6 : \u03a0 \u0394, K.obj \u0394 \u2192 (X.obj \u0394 \u27f6 Y.obj \u0394))\n(naturality' : \u2200 (\u0394\u2081 \u0394\u2082 : simplex_category\u1d52\u1d56)\n  (\u03b8 : \u0394\u2081 \u27f6 \u0394\u2082) (k : K.obj \u0394\u2081), \u03c6 \u0394\u2081 k \u226b Y.map \u03b8 = X.map \u03b8 \u226b \u03c6 \u0394\u2082 (K.map \u03b8 k))\n\nnamespace bihom\n\nrestate_axiom naturality'\nattribute [reassoc] naturality\n\n@[simp]\ndef map\u2081 {K L : sSet} (\u03b1 : K \u27f6 L) (X Y : simplicial_object C)\n  (B : bihom L X Y) : bihom K X Y :=\n{ \u03c6 := \u03bb \u0394 k, B.\u03c6 \u0394 (\u03b1.app \u0394 k),\n  naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 k, begin\n    rw naturality,\n    congr,\n    exact congr_fun (\u03b1.naturality \u03b8).symm k,\n  end, }\n\nend bihom\n\nend sHom\n\n@[simps]\ndef sHom (X Y : simplicial_object C) : sSet.{v} :=\n{ obj := \u03bb \u0394, sHom.bihom (yoneda.obj \u0394.unop) X Y,\n  map := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, sHom.bihom.map\u2081 (yoneda.map \u03b8.unop) X Y,\n  map_id' := \u03bb \u0394, begin\n    rw [unop_id, yoneda.map_id],\n    ext,\n    simp only [sHom.bihom.map\u2081, nat_trans.id_app, types_id_apply],\n  end,\n  map_comp' := \u03bb \u0394\u2081 \u0394\u2082 \u0394\u2083 \u03b8\u2081 \u03b8\u2082, begin\n    rw [unop_comp, yoneda.map_comp],\n    ext,\n    simp only [sHom.bihom.map\u2081, functor_to_types.comp, types_comp_apply],\n  end }\n\ndef sHom\u2080 (X Y : simplicial_object C) : (sHom X Y) _[0] \u2243 (X \u27f6 Y) :=\n{ to_fun := \u03bb B,\n  { app := \u03bb \u0394, B.\u03c6 \u0394 (simplex_category.hom.mk (order_hom.const _ 0)),\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, by simpa only [B.naturality], },\n  inv_fun := \u03bb f,\n  { \u03c6 := \u03bb \u0394 s, f.app \u0394,\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 k, by rw f.naturality, },\n  left_inv := \u03bb f, begin\n    ext \u0394 k,\n    simp only,\n    congr,\n    ext,\n    simp only [fin.coe_fin_one],\n  end,\n  right_inv := \u03bb B, by { ext \u0394, refl, }, }\n\nabbreviation tensor_exists (X : simplicial_object C) (K : sSet) :=\n  \u2200 (\u0394 : simplex_category\u1d52\u1d56), has_coproduct (\u03bb (k : K.obj \u0394), X.obj \u0394)\n\n@[simps]\ndef tensor_sSet (X : simplicial_object C) (K : sSet)\n  [tensor_exists X K] : simplicial_object C :=\n{ obj := \u03bb \u0394, sigma_obj (\u03bb (x : K.obj \u0394), X.obj \u0394),\n  map := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, sigma.desc\n    (\u03bb x, X.map \u03b8 \u226b sigma.\u03b9 (\u03bb (x : K.obj \u0394\u2082), X.obj \u0394\u2082) (K.map \u03b8 x)),\n  map_id' := \u03bb \u0394, begin\n    ext k,\n    discrete_cases,\n    erw [colimit.\u03b9_desc, cofan.mk_\u03b9_app, X.map_id, category.id_comp,\n      category.comp_id, K.map_id],\n    refl,\n  end,\n  map_comp' := \u03bb \u0394\u2081 \u0394\u2082 \u0394\u2083 \u03b8\u2081 \u03b8\u2082, begin\n    ext k,\n    discrete_cases,\n    rw K.map_comp,\n    simpa only [X.map_comp, category.assoc, colimit.\u03b9_desc, cofan.mk_\u03b9_app,\n      colimit.\u03b9_desc_assoc],\n  end, }\n\nnamespace tensor_sSet\n\n@[simps]\ndef map\u2082 (X : simplicial_object C) {K L : sSet} (f : K \u27f6 L)\n  [tensor_exists X K] [tensor_exists X L] :\n  X.tensor_sSet K \u27f6 X.tensor_sSet L :=\n{ app := \u03bb \u0394, sigma.desc (\u03bb k, sigma.\u03b9 (\u03bb l, X.obj \u0394) (f.app \u0394 k)),\n  naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, begin\n    ext k,\n    discrete_cases,\n    simp only [tensor_sSet_map, colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app,\n      category.assoc, colimit.\u03b9_desc],\n    congr,\n    exact congr_fun (f.naturality \u03b8) k,\n  end, }\n\nlemma map\u2082_id (X : simplicial_object C) (K : sSet)\n  [tensor_exists X K] :\n  map\u2082 X (\ud835\udfd9 K) = \ud835\udfd9 _ :=\nbegin\n  ext k,\n  discrete_cases,\n  dsimp,\n  simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, category.comp_id],\nend\n\nlemma map\u2082_comp (X : simplicial_object C) {K L M : sSet} (f\u2081 : K \u27f6 L) (f\u2082 : L \u27f6 M)\n  [tensor_exists X K] [tensor_exists X L] [tensor_exists X M] :\n  map\u2082 X (f\u2081 \u226b f\u2082) = map\u2082 X f\u2081 \u226b map\u2082 X f\u2082 :=\nbegin\n  ext k,\n  dsimp,\n  simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, colimit.\u03b9_desc_assoc],\nend\n\n@[simps]\ndef map\u2081 {X Y : simplicial_object C} (g : X \u27f6 Y) (K : sSet)\n  [tensor_exists X K] [tensor_exists Y K] :\n  X.tensor_sSet K \u27f6 Y.tensor_sSet K :=\n{ app := \u03bb \u0394, limits.sigma.map (\u03bb k, g.app \u0394),\n  naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, begin\n    ext k,\n    simp only [tensor_sSet_map, colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app,\n      category.assoc, \u03b9_colim_map, discrete.nat_trans_app,\n      nat_trans.naturality_assoc, \u03b9_colim_map_assoc, colimit.\u03b9_desc],\n  end, }\n\nlemma map\u2081\u2082 {X Y : simplicial_object C} {K L : sSet} (g : X \u27f6 Y) (f : K \u27f6 L)\n  [tensor_exists X K] [tensor_exists Y K]\n  [tensor_exists X L] [tensor_exists Y L] :\n  map\u2081 g K \u226b map\u2082 Y f = map\u2082 X f \u226b map\u2081 g L :=\nbegin\n  ext k,\n  simp only [nat_trans.comp_app, map\u2081_app, map\u2082_app, \u03b9_colim_map_assoc,\n    discrete.nat_trans_app, colimit.\u03b9_desc, cofan.mk_\u03b9_app,\n    colimit.\u03b9_desc_assoc, \u03b9_colim_map],\nend\n\nlemma map\u2081_id (X : simplicial_object C) (K : sSet)\n  [tensor_exists X K] :\n  map\u2081 (\ud835\udfd9 X) K = \ud835\udfd9 _ :=\nbegin\n  ext k,\n  dsimp,\n  simp only [\u03b9_colim_map, discrete.nat_trans_app, category.comp_id],\n  apply category.id_comp,\nend\n\nlemma map\u2081_comp {X Y Z : simplicial_object C} (g\u2081 : X \u27f6 Y) (g\u2082 : Y \u27f6 Z) (K : sSet)\n  [tensor_exists X K] [tensor_exists Y K] [tensor_exists Z K] :\n  map\u2081 (g\u2081 \u226b g\u2082) K = map\u2081 g\u2081 K \u226b map\u2081 g\u2082 K :=\nbegin\n  ext k,\n  dsimp,\n  simp only [\u03b9_colim_map_assoc, discrete.nat_trans_app, \u03b9_colim_map, category.assoc],\nend\n\n@[simps]\ndef functor [hC : has_coproducts.{w} C] : sSet.{w} \u2964 simplicial_object C \u2964 simplicial_object C :=\n{ obj := \u03bb K,\n  { obj := \u03bb X, X.tensor_sSet K,\n    map := \u03bb X Y g, map\u2081 g K,\n    map_id' := \u03bb X, map\u2081_id X K,\n    map_comp' := \u03bb X Y Z g\u2081 g\u2082, map\u2081_comp g\u2081 g\u2082 K, },\n  map := \u03bb K L f,\n  { app := \u03bb X, map\u2082 X f,\n    naturality' := \u03bb X Y g, map\u2081\u2082 g f, },\n  map_id' := \u03bb K, by { ext1, ext1 X, exact map\u2082_id X K, },\n  map_comp' := \u03bb K L M f\u2081 f\u2082, by { ext1, ext1 X, exact map\u2082_comp X f\u2081 f\u2082, }, }\n\n@[simps]\ndef universal_property (K : sSet) (X Y : simplicial_object C) [tensor_exists X K] :\n  sHom.bihom K X Y \u2243 (X.tensor_sSet K \u27f6 Y) :=\n{ to_fun := \u03bb B,\n  { app := \u03bb \u0394, sigma.desc (B.\u03c6 \u0394),\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8, begin\n      ext k,\n      simp only [tensor_sSet_map, colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app,\n        category.assoc, colimit.\u03b9_desc, B.naturality],\n    end, },\n  inv_fun := \u03bb f,\n  { \u03c6 := \u03bb \u0394 k, (by exact sigma.\u03b9 (\u03bb l, X.obj \u0394) k) \u226b f.app \u0394,\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 k, by simp only [\u2190 f.naturality, category.assoc, tensor_sSet_map,\n        colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app], },\n  left_inv := \u03bb B, begin\n    ext1,\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n  end,\n  right_inv := \u03bb f, begin\n    ext \u0394 k,\n    discrete_cases,\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n  end, }\n\n/- triple functoriality -/\n\n/- compatibility between `universal_property` when K is the terminal object and sHom\u2080 -/\n\n@[simps]\ndef binary_product_compatibility (K L : sSet) (X Y : simplicial_object C)\n  [tensor_exists Y (K.binary_product L)] [tensor_exists X L]:\n  sHom.bihom (K.binary_product L) X Y \u2243 sHom.bihom K (X.tensor_sSet L) Y :=\n{ to_fun := \u03bb B,\n  { \u03c6 := \u03bb \u0394 k, sigma.desc (\u03bb l, B.\u03c6 \u0394 (k, l)),\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 k, begin\n      ext j,\n      discrete_cases,\n      simp only [colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, tensor_sSet_map,\n        category.assoc, colimit.\u03b9_desc, B.naturality, sSet.binary_product_map],\n    end, },\n  inv_fun := \u03bb B,\n  { \u03c6 := \u03bb \u0394 kl, (by exact sigma.\u03b9 (\u03bb (l : L.obj \u0394), X.obj \u0394) kl.2) \u226b B.\u03c6 \u0394 kl.1,\n    naturality' := \u03bb \u0394\u2081 \u0394\u2082 \u03b8 kl, begin\n      simpa only [B.naturality, category.assoc, tensor_sSet_map, colimit.\u03b9_desc_assoc,\n        cofan.mk_\u03b9_app],\n    end},\n  left_inv := \u03bb B, begin\n    ext1,\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, prod.mk.eta],\n  end,\n  right_inv := \u03bb B, begin\n    ext \u0394 k,\n    discrete_cases,\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n  end, }\n\nend tensor_sSet\n\nend simplicial_object\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/simplicial_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.5039061705290806, "lm_q1q2_score": 0.39704338641806886}}
{"text": "-- WIP: only 1 lemma to prove: addSubSup.tail.sup\n\n-- Tested with: \n-- leanprover/lean4:nightly unchanged - Lean (version 4.0.0-nightly-2023-01-28, commit e37f209c1a2a, Release)\n\nimport Std.Data.AssocList\nimport Std.Data.List.Lemmas\nimport Std.Classes.BEq\n\nnamespace MWE9\n\ntheorem cond_eq_ite (c : Bool) (a b : \u03b1) : cond c a b = if c then a else b := by cases c <;> rfl\n\ntheorem cond_decide {\u03b1} (p : Prop) [Decidable p] (t e : \u03b1) : cond (decide p) t e = if p then t else e := by\n  by_cases p <;> simp [*]\n\n-- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Problems.20simplifying.20.20conditions.20with.20hypotheses/near/324212540\n@[simp] theorem beq_eq_eq [DecidableEq \u03b1] (x y : \u03b1) :\n  (x == y) = decide (x = y) := rfl\n\nabbrev Strings := List String\n\ninstance : Repr (Std.AssocList String Strings) where reprPrec s n := s.toList.repr n\n\ndef addDecl (s: Std.AssocList String Strings) (d: String) : Std.AssocList String Strings :=\nmatch s.contains d with\n| true => s\n| false => .cons d [] s\n\ntheorem addDecl.added (s: Std.AssocList String Strings) (d: String): (addDecl s d).contains d\n:= by\n  simp [addDecl]\n  split <;> simp\n  next x heq => simp_all\n\ntheorem addDecl.cons (s: Std.AssocList String Strings) (d: String): (addDecl s d).isEmpty = false\n:= by\n  simp [addDecl]\n  split <;> simp [List.isEmpty]\n  . case h_1 x heq =>\n    split <;> simp_all\n\ntheorem addDecl.after (s: Std.AssocList String Strings) (x y: String): s.contains x \u2192 (addDecl s y).contains x\n:= by\n  intro h\n  simp [Std.AssocList.contains, addDecl] at h \u22a2\n  apply Exists.elim h\n  split <;> simp_all\n  done\n\ndef addSubSup: String \u2192 String \u2192 Std.AssocList String Strings \u2192 Std.AssocList String Strings\n| sub, sup, .nil            => .cons sub [sup] .nil\n| sub, sup, .cons a as tail => bif a = sub then .cons sub (as.insert sup) tail else .cons a as (addSubSup sub sup tail)\n  \n@[simp] theorem addSubSup.sub (sub sup: String) (ss: Std.AssocList String Strings): (addSubSup sub sup ss).contains sub\n:= by\n  induction ss <;> simp_all\n  . case cons key value tail tail_ih =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases key = sub\n    . case pos h =>\n      simp [h]\n    . case neg h =>\n      simp [h]\n      apply tail_ih\n\n@[simp] theorem addSubSup.exists.sub (sub sup: String) (tail: Std.AssocList String Strings)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sub\n:= by\n  induction tail <;> simp_all\n  . case cons key value t ih =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases key = sub\n    . case pos h =>\n      simp [h]\n    . case neg h =>\n      simp [h]\n      apply ih\n\ndef addBoth (sub sup: String) (ss: Std.AssocList String Strings) : Std.AssocList String Strings :=\n  let ss' := addDecl ss sup\n  addSubSup sub sup ss'\n\n@[simp] theorem addSubSup.more.sub\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sub)\n  (other: Std.AssocList String Strings)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup other) \u2227 x.fst = sub\n:= by\n  exact addSubSup.exists.sub sub sup other\n\n@[simp] theorem addSubSup.tail.sup\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sup)\n: \u2203 x, x \u2208 Std.AssocList.toList tail \u2227 x.fst = sup\n:= by\n  induction tail <;> simp [*]\n  . case nil =>\n    rw [addSubSup] at h\n    let \u27e8 x, p, eq \u27e9 := h\n    sorry\n  . case cons key value t1 ih =>\n    by_cases key = sup <;> simp [*]\n    . case neg hs1 =>\n      simp [addSubSup, cond_eq_ite, hs1] at h\n      by_cases key = sub <;> simp_all\n      . case neg hs2 =>\n        sorry\n\n@[simp] theorem addSubSup.key_value\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sup)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons key value tail)) \u2227 x.fst = sup\n:= by\n  simp [addSubSup, cond_eq_ite]\n  by_cases key = sub <;> simp_all\n  . case pos h1 =>\n    by_cases sub = sup <;> simp_all\n    . case neg =>\n      exact addSubSup.tail.sup sub sup tail h\n\n@[simp] theorem addSubSup.cons\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sup)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227 x.fst = sup\n:= by\n  simp [addSubSup, cond_eq_ite]\n  by_cases sup = sub <;> simp_all\n\n@[simp] theorem addSubSup.cons2\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] tail)) \u2227 x.fst = sup)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227 x.fst = sup\n:= by\n  simp [addSubSup, cond_eq_ite]\n  by_cases sup = sub <;> simp_all\n\n@[simp] theorem addBoth2SubSup1\n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sub\n:= by\n  simp [addBoth] at h\n  exact addSubSup.more.sub sub sup (addDecl tail sup) h tail\n\n@[simp] theorem addBoth2SubSup2 \n  (sub sup: String) \n  (tail: Std.AssocList String Strings)\n  (h: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub)\n  (other: Std.AssocList String Strings)\n: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup other) \u2227 x.fst = sub\n:= by\n  have ht := addBoth2SubSup1 sub sup tail h\n  exact addSubSup.more.sub sub sup tail ht other\n\ntheorem addBoth.sub_eq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sub\n:= by\n  induction ss <;> simp_all\n  . case nil =>\n    simp [addBoth, addDecl, addSubSup, cond_eq_ite]\n    split <;> simp_all\n  . case cons =>\n    simp [addBoth, addDecl]\n\ntheorem addBoth.sup_eq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sup\n:= by\n  simp [addBoth]\n  simp [addDecl]\n  induction ss <;> simp [*]\n  . case nil =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases sup = sub <;> simp [*]\n  . case cons key value tail tail_ih =>\n    simp [addDecl, cond_eq_ite]\n    by_cases key = sup <;> simp [*]\n    . case pos h1 =>\n      simp [addSubSup, cond_eq_ite] at tail_ih \u22a2 \n      by_cases sup = sub <;> simp [*]\n    . case neg h1 =>\n      split <;> simp_all\n\nend MWE9", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/MWE9.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331606115021, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.39687292265762786}}
{"text": "/- Defines very basic lemmas for vector -/\nimport data.vector\n\nuniverse variables u\n\nnamespace vector\n\nvariable {\u03b1 : Type u}\nvariable {n : \u2115}\n\n@[simp]\ntheorem length_to_list : \u2200 (x : vector \u03b1 n), x.to_list.length = n\n| \u27e8 l, p \u27e9 := p\n\nend vector\n", "meta": {"author": "GaloisInc", "repo": "lean-protocol-support", "sha": "cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda", "save_path": "github-repos/lean/GaloisInc-lean-protocol-support", "path": "github-repos/lean/GaloisInc-lean-protocol-support/lean-protocol-support-cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda/galois/vector/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3968729131110646}}
{"text": "import data.real.basic\nimport tactic\nimport real_definitions\n-- import utils\n\n-- #print linarith.make_comp_with_zero\n\nnamespace tactic.interactive\nopen lean.parser tactic interactive\nopen interactive (loc.ns)\nopen interactive.types\nopen expr\n\n\n-- useful lemmas:\n-- ge_iff_le\n-- gt_iff_lt\n-- lt_of_le_of_ne\n-- le_of_lt\n-- div_pos\n-- mul_pos\n-- inv_pos\n\nopen tactic.interactive\n\n-------------------------------------------\n-------------------------------------------\n-- Make 'a < b' from 'a \u2260 b' and 'a \u2264 b' --\n-------------------------------------------\n-------------------------------------------\n-- This is achieved by the following sequence of tactics:\n-- * make_ineq  take \"a \u2264 b\" and \"a' \u2260 b'\" and try to make \"a < b\"\n-- * extract_gt extract the list of inequalities \"a \u2264 b\" from a list of expressions\n-- * extract_non_eq does the same for \"a \u2260 b\"\n-- * list_prod make the product list of two lists\n-- * get_pos_from_pos_eq_from_list is build on the previous ones:\n-- it takes a list of expressions,\n-- extract inequalities and non-equalities,\n-- take all pairs and try to make \"a < b\" by applying make_ineq\n-- * get_pos_from_pos_eq is an interactive tactic that applies this to the local context\n\n\n----------------------\n----------------------\n-- Chaining tactics -\n----------------------\n----------------------\nset_option trace.linarith true\n\n/- Concatenate a list of strings using commas as separators-/\n-- set_option trace.eqn_compiler.elim_match true\ndef string.concatenate : (list string) \u2192 string\n-- (l: list string) : string :=\n-- list.foldl (\u03bb (s: string) (t: string), s ++ \", \" ++ t) \"\" l\n| []  := \"\"\n| [s] := s\n| (\"\" :: tail) := string.concatenate tail\n| (head :: tail) := do let tail_string := string.concatenate tail,\n                    match tail_string with\n                    | \"\" := head\n                    | _  := head ++ \", \" ++ tail_string\n                    end\n\n\n\n\n/- Try some (tactic string) and in case of success return its string.\nAlways succeed. -/\nmeta def try_and_return_code (my_tactic: tactic string) : tactic string :=\ndo {s \u2190 my_tactic, return s} <|> return \"\"\n\n/- Iterate some (tactic string) and return the concatenated returned strings. Stops as soon as \nthe tactic makes no progress, i.e. returns the empty string or num_goals = 0.\nFail if some tactic fails. -/\nmeta def iterate_and_return_code : nat \u2192 tactic string \u2192 tactic string\n| 0       my_tactic := return \"\"\n| (n + 1) my_tactic := do -- trace \"(iterating tactic...)\",\n    first_code \u2190 my_tactic, l \u2190 num_goals,\n    match first_code, l with\n    | \"\", _  := return \"\"\n    | s, 0   := return s\n    | _ , _  := do\n        remaining_code \u2190 iterate_and_return_code n my_tactic,\n        return $ string.concatenate [first_code, remaining_code]\n    end\n\n/- Apply successively tactics in a given list,\nbut stop as soon as there is no more goal,\nand return concatenation of returned code.\nFail if some tactic fails. -/\nmeta def and_then_and_return_code : list (tactic string) \u2192 tactic string \n| []                  := return \"\"\n| (first_tac :: tail) :=  do\n    first_code \u2190 first_tac, l \u2190 num_goals,\n    match first_code, l with\n    | s, 0   := return s\n    | _ , _  := do\n        remaining_code \u2190 and_then_and_return_code tail,\n        return $ string.concatenate [first_code, remaining_code]\n    end\n\n\nmeta def try_tactic_string (my_tac: tactic string) : tactic string :=\ndo {my_tac <|> return \"\"}\n\nmeta def skip_tactic_string : tactic string :=\ndo {skip, return \"\"}\n\n/- Apply some tactic string and trace the returned string as effective code with id-/\nmeta def apply_and_return_code (id: string) (my_tactic: tactic string) : tactic unit:=\ndo  effective_code \u2190 my_tactic, \n    tactic.trace $ \"EFFECTIVE CODE LEAN n\u00b0\" ++ id ++ \":\" ++ effective_code,\n    tactic.trace $ \"Try this: \"++ effective_code\n\n----------------------------\n----------------------------\n-- Get strict inequalities -\n----------------------------\n----------------------------\n/-  Take H1 of type \"a \u2264 b\", H2 of type \"a' \u2260 b'\", and if a=a' and b = b'\n    then add H3 of type \"a < b\" in the local context -/\nmeta def make_ineq : expr \u00d7 expr \u00d7 expr \u2192 expr \u00d7 expr \u00d7 expr \u2192 tactic string\n| (H1, a, b) (H2, a', b') :=\n    tactic.unify a a' >> tactic.unify b b' >> do\n    {\n    -- H \u2190 mk_fresh_name,\n    let H := `H_aux,\n    \u00abhave\u00bb H none ``(lt_of_le_of_ne %%H1 %%H2),\n    return $ \"lt_of_le_of_ne \" ++ to_string H1 ++ \" \" ++ to_string H1\n    }\n    <|>\n    tactic.unify a b' >> tactic.unify a' b >> do\n    {\n    -- H \u2190 mk_fresh_name,\n    let H := `H_aux,\n    \u00abhave\u00bb H none ``(lt_of_le_of_ne %%H1 (ne.symm %%H2)),\n    return $ \"have \" ++ to_string H ++ \" := lt_of_le_of_ne \" ++ to_string H1 ++ \" (ne.symm \" ++ to_string H2 ++ \")\"\n    }\n    <|> \n    do {return \"\"}\n\n/- The same, but remove H1 and H2 from context\n  TODO\n  -/\nmeta def make_ineq' : expr \u2192 expr \u2192 tactic unit\n| H1 H2 := do skip\n\n/- Extract from list of expr the couples (H, a,b)\nwhere the expr H of type \"a \u2264 b\" in is the list -/\nmeta def extract_gt : list expr \u2192 tactic (list (expr \u00d7 expr \u00d7 expr))\n-- match hypos with\n| []                    := return []\n| (hypo :: less_hypos)  := do\n    {\n    ineq \u2190 infer_type hypo,\n    remaining_list \u2190 (extract_gt less_hypos),\n    match ineq with\n        | `(%%a \u2264 %%b)  := return $ (hypo, a, b) :: remaining_list\n        | _             := return $ remaining_list\n        end\n    }\n\n/- Extract from list of expr the couples (H, a,b)\nwhere the expr H of type \"a \u2260 b\" in is the list -/\nmeta def extract_non_eq : list expr \u2192 tactic (list (expr \u00d7 expr \u00d7 expr))\n-- match hypos with\n| []                    := return []\n| (hypo :: less_hypos)  := do\n    {\n    ineq \u2190 infer_type hypo,\n    remaining_list \u2190 (extract_non_eq less_hypos),\n    match ineq with\n        | `(%%a \u2260 %%b)      := return $ (hypo, a, b) :: remaining_list\n        | `(\u00ac %%a = %%b)    := return $ (hypo, a, b) :: remaining_list\n        | _                 := return $ remaining_list\n        end\n    }\n\n\n/- Return the product list -/\nmeta def list_prod {\u03b1 \u03b2 : Type} : list \u03b1 \u2192 list \u03b2 \u2192 list (\u03b1 \u00d7 \u03b2)\n| []    l2                  := []\n| l1    []                  := []\n| (h1 :: l1)  (h2 :: l2)    :=\n    (h1, h2) ::\n    append  (append (list_prod [h1] l2)\n                    (list_prod l1 [h2]))\n            (list_prod l1 l2)\n\n\nmeta def get_pos_from_pos_eq_from_list (hypos: list expr) : tactic string :=\ndo\n    inequalities    \u2190 extract_gt hypos,     -- tactic.trace inequalities,\n    equalities      \u2190 extract_non_eq hypos, -- tactic.trace equalities,\n    -- take all pairs and try to build \" a < b \"\n    effective_codes \u2190 (list_prod inequalities equalities).mmap (\u03bbh, make_ineq h.1 h.2),\n    let effective_code := string.concatenate effective_codes,\n    return effective_code\n\n/- To be applied after \"norm_num at *\"\n    This tactic search in the hypotheses for two hypotheses of the for\n    a \u2264 b  and a \u2260 b\n    and deduces\n    a < b\n    Return the effective code that is equivalent to get_pos_from_pos_eq\n    in the given context.\n-/\nmeta def get_pos_from_pos_eq : tactic string :=\ndo  hypos \u2190 tactic.local_context,\n    effective_code \u2190 get_pos_from_pos_eq_from_list hypos,\n    return effective_code\n    \n\n-------------------\n-------------------\n-- Tactic compute -\n-------------------\n-------------------\n\nlemma inv_pos_mpr {\u03b1 : Type} [linear_ordered_field \u03b1] (a:\u03b1) :\n0 < a \u2192 0 < a\u207b\u00b9 := inv_pos.mpr\n\nopen linarith\n/- Non-interactive version of nl_linarith. -/\nmeta def nl_linarith (cfg : linarith_config := {}): tactic unit :=\ndo\n{\ntactic.linarith false false []\n  { cfg with preprocessors := some $\n      cfg.preprocessors.get_or_else default_preprocessors ++ [nlinarith_extras] }\n}\n\n\n/-- A configuration object for `compute1`. \n develop_ite: set to tt to develop if_then_else definitions, e.g. abs and max.\n -/\nmeta structure compute_config : Type :=\n(develop_ite : bool := ff)\n\n\n/- Unfold some definitions using if_then_else, then get rid of if_then_else by case reasoning, and try to \nclose all the goals by linarith. Unfolded definitions includes abs, max, min.\n-/\nmeta def develop_ite : tactic string :=\n    do {`[unfold abs at *], `[unfold min max at *], `[split_ifs at *], repeat $ tactic.linarith false false [],\n    return \"unfold abs, unfold min max, split_ifs, repeat {linarith}\"}\n    <|>\n    do {`[unfold abs], `[unfold min max], `[split_ifs],\n    return \"unfold abs, unfold min max, split_ifs\"}\n\n/- try several computing tactics for STRICT inequalities.\nIn case of success return the corresponding effective code. -/\nmeta def compute1 : tactic string :=\ndo\n{   do {assumption, return \"assumption\"}\n    <|>\n    -- solve e.g. \"n_0 \u2264 n_0 \u2228 n_0 \u2264 n_1\"\n    -- with norm_num, solves \"n_0 \u2264 max n_0 n_1\"\n    do {tactic.tautology, return \"tautology\"}\n    <|>\n    do {tactic.linarith false false [], return \"linarith\"}\n    <|>\n    do {nl_linarith, return \"nl_linarith\"}\n    <|>\n    do {tactic.applyc ``mul_pos, return \"apply mul_pos\"}\n    <|>\n    -- a>0 \u2192 a\u207b\u00b9 >0\n    do {tactic.applyc ``inv_pos_mpr, return \"apply inv_pos.mpr\"}\n    <|>\n -- a\u22600 \u2192 b\u22600 \u2192 ab\u22600\n     do {tactic.applyc ``mul_ne_zero, return \"apply mul_ne_zero\"}\n--  div_pos is now useless\n}\n\n/- Repeat n times the tactic compute1 ; never fails -/\nmeta def compute_n_old (n: nat)  (cfg: compute_config := {}) : tactic string :=\ndo  effective_code_0 \u2190  if cfg.develop_ite \n                        then try_tactic_string develop_ite \n                        else skip_tactic_string,\n    n \u2190 num_goals, match n with\n        | 0 := return effective_code_0\n        | _ := do\n        -- Try to add strict inequalities to the context, if there remains some goal:\n        effective_code_1 \u2190 get_pos_from_pos_eq,\n        effective_code_2 \u2190 iterate_and_return_code n compute1,\n        return $ string.concatenate [effective_code_0, effective_code_1, effective_code_2]\n        end\n\n/- Repeat n times the tactic compute1 ; never fails -/\nmeta def compute_n (n: nat)  (cfg: compute_config := {}) : tactic string :=\ndo let first_tac := if cfg.develop_ite then [try_tactic_string develop_ite]\n                      else ([]:list (tactic string)),\n    -- Try to add strict inequalities to the context, if there remains some goal:\n    let list_tac := first_tac.append [get_pos_from_pos_eq,\n                                      iterate_and_return_code n compute1],\n        code \u2190 and_then_and_return_code list_tac,\n        return code\n\n/- Apply the compute_n tactic n times,\nand in case of success trace the effective code with id. -/\nmeta def compute_and_return_code (n: nat) (id: string)  (cfg: compute_config := {}):\ntactic unit :=\ndo apply_and_return_code id (tactic.interactive.compute_n n cfg)\n\n\nend tactic.interactive\n\n-------------\n-- Example -- \n-------------\n-- set_option trace.linarith true\nexample (a:\u211d) (H: a \u2260 0) (H': a \u2265 0): a^2 \u2265 0 :=\nbegin\n    -- compute_n 1,\n    norm_num at *,\n    -- nl_linarith,\n    -- apply_and_return_code \"12.1\" (tactic.interactive.compute_n 1),\n    compute_and_return_code 10 \"12.1\",\n    --compute1,\n    -- compute_and_return_code 10 \"12.1\",  -->\n    -- have H_aux := lt_of_le_of_ne H' (ne.symm H), nl_linarith,\nend\n\nlemma theorem.inegalite_triangulaire1\n{a b : \u211d} : abs (a-b) \u2264 abs a + abs b :=\nbegin\n    compute_and_return_code 2 \"1\" {develop_ite:=tt},\nend\n\nlemma def.abs (a: \u211d) : abs a = if a \u2265 0 then a else -a :=\nbegin\n    unfold abs max, split_ifs, repeat{linarith},\nend\n\nlemma ineq_from_non_eq2 {\u03b1: Type} [linear_order \u03b1] (a b : \u03b1) : \u00ac (a = b) \u2194 (a < b \u2228 b < a) :=\nbegin\n    sorry,\nend\n\nexample (l l': \u211d) (H: l' \u2260 l) : abs (l'-l) >0 :=\nbegin\n    norm_num at *, --rw def.abs,\n    rw ineq_from_non_eq2 at *, -- replace `a\u2260b` by `a < b OR b < a`\n    cases_type or and, -- split conjonctions and disjunctions\n    all_goals {compute_and_return_code 2 \"1\" {develop_ite:=tt}},\n    --  split_ifs,\n    --  get_pos_from_pos_eq, assumption,\n    --  linarith,\n    -- norm_num at *,\n    -- get_pos_from_pos_eq,\n    -- compute_and_return_code 5 \"1\" {develop_ite:=tt},\nend", "meta": {"author": "dEAduction", "repo": "dEAduction-lean", "sha": "4fe1d642078fc94f9081ccbed08e047e86a741fd", "save_path": "github-repos/lean/dEAduction-dEAduction-lean", "path": "github-repos/lean/dEAduction-dEAduction-lean/dEAduction-lean-4fe1d642078fc94f9081ccbed08e047e86a741fd/snippets/tactics_for_testing/compute2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3968729131110646}}
{"text": "import tactic\n--If the above returns an error, you probably didn't install mathlib correctly. Consult the instruction in README carefully\n--If you think you did everything correctly, consult me (Jakub Wornbard) on Facebook or however else you like\n\n\n\n--You should see the `Lean Infoview` window on the right of the screen. \n--If you don't, you may try `ctrl+shift+enter` \n--If you still don't see it, something is probably wrong. Make sure you installed the Lean extension in VS Code\n\n#eval 2+2\n--If you put the cursor at the end of the line above, you should see the result in the Infoview\n\nvariable (P : Prop)\n\nexample : P\u2192P :=\nbegin\n  --the state of the Infoview should change as you move the cursor down the next few lines.\n  --after the last comma, it should say `goals accompished`\n  intro hP,\n  assumption,\nend\n\n--If all of that works you're good to go\n\n", "meta": {"author": "Wornbard", "repo": "mbl_lean_workshop", "sha": "6b68ce25fdc49043fd5ab409de8e4a2987def22e", "save_path": "github-repos/lean/Wornbard-mbl_lean_workshop", "path": "github-repos/lean/Wornbard-mbl_lean_workshop/mbl_lean_workshop-6b68ce25fdc49043fd5ab409de8e4a2987def22e/src/day_1/intro_test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.6654105454764747, "lm_q1q2_score": 0.39687289564963135}}
{"text": "import Hw9\nimport Sets.Basic\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\nopen Set \n\nvariable (\u03b1 \u03b2 : Type)\nvariable (X Y Z : Set \u03b1)\nvariable (W : Set \u03b2) \n\ntheorem desiredType1 : \u2205 \u2208 \ud835\udcab  X := sorry \n\ntheorem desiredType2 (U : \u03b2 \u2192 Set \u03b1) : \u2200 b, U b \u2286 BigUnion U := sorry \n\ntheorem desiredType3 (h : X \u2286 Y) : (X \u00d7\u02e2 W) \u2286 (Y \u00d7\u02e2 W) := sorry\n\ntheorem desiredType4 (h : Y \u2229 Z = \u2205) : Y\u1d9c \u222a Z\u1d9c = Univ := sorry \n\ntheorem desiredType5 : (X \\ Y) \u222a (Y \\ X) = (X \u222a Y) \\ (X \u2229 Y) := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef n : String := \"2\"\n\ndef problem : String := \"problem\"++n\n\ndef desired : String := \"desiredType\"++n\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const desired []) (Expr.const problem [])\n#eval collectAxiomsOf problem\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework9", "sha": "c26e748a8f91c4f459d6f568a6819b53cd6088a1", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9/homework9-c26e748a8f91c4f459d6f568a6819b53cd6088a1/_Tests/Problem2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3968648601865095}}
{"text": "import ring_theory.ideals data.equiv.algebra\n\n#print relator.bi_unique\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n\nnamespace relator\nlemma rel_exists_unique_of_total [bi_total R] [bi_unique R] :\n  ((R \u21d2 iff) \u21d2 iff) (\u03bbp, \u2203! i, p i) (\u03bbq, \u2203! i, q i) :=\n\u03bb p q h, rel_exists_of_total R $\n  \u03bb a b r, rel_and (h r) $ rel_forall_of_total R $\n    \u03bb c d s, rel_imp (h s) (rel_eq R s r)\nend relator\n\nnamespace equiv\n\nprotected def rel (f : \u03b1 \u2243 \u03b2) (a : \u03b1) (b : \u03b2) : Prop := f a = b\n\ntheorem symm_rel (f : \u03b1 \u2243 \u03b2) {a : \u03b1} {b : \u03b2} : f.symm.rel b a \u2194 f.rel a b :=\nf.symm_apply_eq.trans eq_comm\n\nopen relator\ntheorem rel_left_unique (f : \u03b1 \u2243 \u03b2) : left_unique f.rel :=\n\u03bb a\u2081 b a\u2082 h, by rintro \u27e8\u27e9; exact f.bijective.1 h\n\ntheorem rel_right_unique (f : \u03b1 \u2243 \u03b2) : right_unique f.rel :=\n\u03bb a\u2081 b a\u2082, by rintro \u27e8\u27e9 \u27e8\u27e9; refl\n\ninstance rel_bi_unique (f : \u03b1 \u2243 \u03b2) : bi_unique f.rel :=\n\u27e8f.rel_left_unique, f.rel_right_unique\u27e9\n\ninstance rel_left_total (f : \u03b1 \u2243 \u03b2) : left_total f.rel := \u03bb a, \u27e8_, rfl\u27e9\ninstance rel_right_total (f : \u03b1 \u2243 \u03b2) : right_total f.rel := f.bijective.2\ninstance rel_bi_total (f : \u03b1 \u2243 \u03b2) : bi_total f.rel := by split; apply_instance\n\nend equiv\n\nnamespace ring_equiv\nvariables [ring \u03b1] [ring \u03b2]\ninstance (\u03b1 \u03b2) [ring \u03b1] [ring \u03b2] : has_coe_to_fun (\u03b1 \u2243r \u03b2) :=\n\u27e8\u03bb_, \u03b1 \u2192 \u03b2, \u03bbe, e.to_equiv\u27e9\n\n@[simp] theorem apply_inverse_apply (e : \u03b1 \u2243r \u03b2) (x : \u03b2) : e (e.symm x) = x :=\ne.to_equiv.apply_inverse_apply x\n\n@[simp] theorem inverse_apply_apply (e : \u03b1 \u2243r \u03b2) (x : \u03b1) : e.symm (e x) = x :=\ne.to_equiv.inverse_apply_apply x\n\ndef {u v} ideal_comap {\u03b1 : Type u} {\u03b2 : Type v} [comm_ring \u03b1] [comm_ring \u03b2]\n  (e : \u03b1 \u2243r \u03b2) (I : ideal \u03b2) : ideal \u03b1 :=\n{ carrier := e \u207b\u00b9' I,\n  zero  := by simp [is_ring_hom.map_zero e],\n  add   := \u03bb x y h\u2081 h\u2082, by simp [is_ring_hom.map_add e]; exact I.add_mem h\u2081 h\u2082,\n  smul  := \u03bb a x h, by simp [is_ring_hom.map_mul e]; exact I.smul_mem _ h }\n\n@[simp] theorem mem_ideal_comap {\u03b1 \u03b2} [comm_ring \u03b1] [comm_ring \u03b2]\n  {e : \u03b1 \u2243r \u03b2} {I : ideal \u03b2} {r} : r \u2208 ideal_comap e I \u2194 e r \u2208 I := iff.rfl\n\n@[simp] theorem ideal_comap_top {\u03b1 \u03b2} [comm_ring \u03b1] [comm_ring \u03b2]\n  (e : \u03b1 \u2243r \u03b2) : ideal_comap e \u22a4 = \u22a4 := rfl\n\ndef ideal_congr {\u03b1 \u03b2 : Type*} [comm_ring \u03b1] [comm_ring \u03b2] (e : \u03b1 \u2243r \u03b2) :\n  ideal \u03b1 \u2243 ideal \u03b2 :=\n{ to_fun := e.symm.ideal_comap,\n  inv_fun := e.ideal_comap,\n  left_inv := \u03bb I, ideal.ext $ \u03bb r, by simp,\n  right_inv := \u03bb I, ideal.ext $ \u03bb r, by simp }\n\n@[simp] theorem ideal_congr_apply {\u03b1 \u03b2} [comm_ring \u03b1] [comm_ring \u03b2]\n  (e : \u03b1 \u2243r \u03b2) (I : ideal \u03b1) : ideal_congr e I = e.symm.ideal_comap I := rfl\n\n@[simp] theorem ideal_congr_symm_apply {\u03b1 \u03b2} [comm_ring \u03b1] [comm_ring \u03b2]\n  (e : \u03b1 \u2243r \u03b2) (I : ideal \u03b2) : (ideal_congr e).symm I = e.ideal_comap I := rfl\n\nend ring_equiv\n\nnamespace relator\n\ntheorem rel_ideal_top\n  {R S : Type*} [comm_ring R] [comm_ring S] (f : R \u2243r S) :\n  f.ideal_congr.rel \u22a4 \u22a4 :=\nring_equiv.ideal_comap_top _\n\ntheorem rel_lt_ideal\n  {R S : Type*} [comm_ring R] [comm_ring S] (f : R \u2243r S) :\n  (f.ideal_congr.rel \u21d2 f.ideal_congr.rel \u21d2 iff) has_lt.lt has_lt.lt :=\n\u03bb I J (h : _=_) I' J' (h' : _=_), by substs h h'; exact _\n\ntheorem rel_is_maximal\n  {R S : Type*} [comm_ring R] [comm_ring S] (f : R \u2243r S) :\n  (f.ideal_congr.rel \u21d2 iff) ideal.is_maximal ideal.is_maximal :=\n\u03bb I J h, rel_and\n  (rel_not $ rel_eq f.ideal_congr.rel h $ rel_ideal_top _)\n  (rel_forall_of_total f.ideal_congr.rel $\n    \u03bb I' J' h', rel_imp\n      (rel_lt_ideal f h h')\n      (rel_eq f.ideal_congr.rel h' (ring_equiv.ideal_comap_top _)))\n\nend relator\n\ntheorem is_local_ring_congr\n  {R S : Type*} [comm_ring R] [comm_ring S] (f : R \u2243r S) :\n  is_local_ring R \u2194 is_local_ring S :=\nrelator.rel_exists_unique_of_total f.ideal_congr.rel $\n\u03bb I J, relator.rel_is_maximal f\n\n\n#exit\ninstance qwer (F : Type u) [discrete_field F] :\nmodule F (big_ideal F) := by apply_instance -- fails\n\n\nset_option class.instance_max_depth 250\ninstance qwer (F : Type u) [discrete_field F] :\nmodule F ((big_ideal F).quotient) := by apply_instance\n\n#check ideal \n#exit\n\ninstance ghjk (F : Type u) [discrete_field F] : module F (big_ideal F).quotient :=\nsorry\n\nprivate theorem big_basis.is_basis (F : Type u) [discrete_field F] : is_basis F (big_basis F) :=\nsorry", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/UMI/scratch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737473266736, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.39678222891366605}}
{"text": "opaque n : Nat\n@[simp] axiom prio_1000 : n = 1000\n@[simp 10] axiom prio_10 : n = 10\n-- simp should prefer the prio_1000 lemma with the higher priority\nexample : n = 1000 := by simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/simpPrio.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.705785040214066, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3967757642850859}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.limits.shapes.reflexive\nimport Mathlib.category_theory.adjunction.default\nimport Mathlib.category_theory.monad.adjunction\nimport Mathlib.category_theory.monad.coequalizer\nimport Mathlib.PostPort\n\nuniverses u\u2082 u\u2083 v\u2082 v\u2083 v\u2081 u\u2081 v\u2084 u\u2084 \n\nnamespace Mathlib\n\n/-!\n# Adjoint lifting\n\nThis file gives two constructions for building left adjoints: the adjoint triangle theorem and the\nadjoint lifting theorem.\nThe adjoint triangle theorem says that given a functor `U : B \u2964 C` with a left adjoint `F` such\nthat `\u03b5_X : FUX \u27f6 X` is a regular epi. Then for any category `A` with coequalizers of reflexive\npairs, a functor `R : A \u2964 B` has a left adjoint if (and only if) the composite `R \u22d9 U` does.\nNote that the condition on `U` regarding `\u03b5_X` is automatically satisfied in the case when `U` is\na monadic functor, giving the corollary: `monadic_adjoint_triangle_lift`, i.e. if `U` is monadic,\n`A` has reflexive coequalizers then `R : A \u2964 B` has a left adjoint provided `R \u22d9 U` does.\n\nThe adjoint lifting theorem says that given a commutative square of functors (up to isomorphism):\n\n      Q\n    A \u2192 B\n  U \u2193   \u2193 V\n    C \u2192 D\n      R\n\nwhere `U` and `V` are monadic and `A` has reflexive coequalizers, then if `R` has a left adjoint\nthen `Q` has a left adjoint.\n\n## Implementation\n\nIt is more convenient to prove this theorem by assuming we are given the explicit adjunction rather\nthan just a functor known to be a right adjoint. In docstrings, we write `(\u03b7, \u03b5)` for the unit\nand counit of the adjunction `adj\u2081 : F \u22a3 U` and `(\u03b9, \u03b4)` for the unit and counit of the adjunction\n`adj\u2082 : F' \u22a3 R \u22d9 U`.\n\n## TODO\n\nDualise to lift right adjoints through comonads (by reversing 1-cells) and dualise to lift right\nadjoints through monads (by reversing 2-cells), and the combination.\n\n## References\n* https://ncatlab.org/nlab/show/adjoint+triangle+theorem\n* https://ncatlab.org/nlab/show/adjoint+lifting+theorem\n* Adjoint Lifting Theorems for Categories of Algebras (PT Johnstone, 1975)\n* A unified approach to the lifting of adjoints (AJ Power, 1988)\n-/\n\nnamespace category_theory\n\n\n-- Hide implementation details in this namespace\n\nnamespace lift_adjoint\n\n\n/--\nTo show that `\u03b5_X` is a coequalizer for `(FU\u03b5_X, \u03b5_FUX)`, it suffices to assume it's always a\ncoequalizer of something (i.e. a regular epi).\n-/\ndef counit_coequalises {B : Type u\u2082} {C : Type u\u2083} [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (adj\u2081 : F \u22a3 U) [(X : B) \u2192 regular_epi (nat_trans.app (adjunction.counit adj\u2081) X)] (X : B) : limits.is_colimit (limits.cofork.of_\u03c0 (nat_trans.app (adjunction.counit adj\u2081) X) (counit_coequalises._proof_1 adj\u2081 X)) :=\n  limits.cofork.is_colimit.mk' (limits.cofork.of_\u03c0 (nat_trans.app (adjunction.counit adj\u2081) X) sorry)\n    fun\n      (s :\n      limits.cofork (functor.map F (functor.map U (nat_trans.app (adjunction.counit adj\u2081) X)))\n        (nat_trans.app (adjunction.counit adj\u2081) (functor.obj F (functor.obj U X)))) =>\n      { val := subtype.val (regular_epi.desc' (nat_trans.app (adjunction.counit adj\u2081) X) (limits.cofork.\u03c0 s) sorry),\n        property := sorry }\n\n/--\n(Implementation)\nTo construct the left adjoint, we use the coequalizer of `F' U \u03b5_Y` with the composite\n\n`F' U F U X \u27f6 F' U F U R F U' X \u27f6 F' U R F' U X \u27f6 F' U X`\n\nwhere the first morphism is `F' U F \u03b9_UX`, the second is `F' U \u03b5_RF'UX`, and the third is `\u03b4_F'UX`.\nWe will show that this coequalizer exists and that it forms the object map for a left adjoint to\n`R`.\n-/\ndef other_map {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (F' : C \u2964 A) (adj\u2081 : F \u22a3 U) (adj\u2082 : F' \u22a3 R \u22d9 U) (X : B) : functor.obj F' (functor.obj U (functor.obj F (functor.obj U X))) \u27f6 functor.obj F' (functor.obj U X) :=\n  functor.map F'\n      (functor.map U\n        (functor.map F (nat_trans.app (adjunction.unit adj\u2082) (functor.obj U X)) \u226b\n          nat_trans.app (adjunction.counit adj\u2081) (functor.obj R (functor.obj F' (functor.obj U X))))) \u226b\n    nat_trans.app (adjunction.counit adj\u2082) (functor.obj F' (functor.obj U X))\n\n/--\n`(F'U\u03b5_X, other_map X)` is a reflexive pair: in particular if `A` has reflexive coequalizers then\nit has a coequalizer.\n-/\nprotected instance other_map.category_theory.is_reflexive_pair {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (F' : C \u2964 A) (adj\u2081 : F \u22a3 U) (adj\u2082 : F' \u22a3 R \u22d9 U) (X : B) : is_reflexive_pair (functor.map F' (functor.map U (nat_trans.app (adjunction.counit adj\u2081) X)))\n  (other_map R F' adj\u2081 adj\u2082 X) := sorry\n\n/--\nConstruct the object part of the desired left adjoint as the coequalizer of `F'U\u03b5_Y` with\n`other_map`.\n-/\ndef construct_left_adjoint_obj {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (F' : C \u2964 A) (adj\u2081 : F \u22a3 U) (adj\u2082 : F' \u22a3 R \u22d9 U) [limits.has_reflexive_coequalizers A] (Y : B) : A :=\n  limits.coequalizer (functor.map F' (functor.map U (nat_trans.app (adjunction.counit adj\u2081) Y)))\n    (other_map R F' adj\u2081 adj\u2082 Y)\n\n/-- The homset equivalence which helps show that `R` is a right adjoint. -/\ndef construct_left_adjoint_equiv {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (F' : C \u2964 A) (adj\u2081 : F \u22a3 U) (adj\u2082 : F' \u22a3 R \u22d9 U) [limits.has_reflexive_coequalizers A] [(X : B) \u2192 regular_epi (nat_trans.app (adjunction.counit adj\u2081) X)] (Y : A) (X : B) : (construct_left_adjoint_obj R F' adj\u2081 adj\u2082 X \u27f6 Y) \u2243 (X \u27f6 functor.obj R Y) :=\n  equiv.trans\n    (equiv.trans\n      (equiv.trans\n        (limits.cofork.is_colimit.hom_iso\n          (limits.colimit.is_colimit\n            (limits.parallel_pair (functor.map F' (functor.map U (nat_trans.app (adjunction.counit adj\u2081) X)))\n              (other_map R F' adj\u2081 adj\u2082 X)))\n          Y)\n        (equiv.subtype_congr (adjunction.hom_equiv adj\u2082 (functor.obj U X) Y) sorry))\n      (equiv.subtype_congr (equiv.symm (adjunction.hom_equiv adj\u2081 (functor.obj U X) (functor.obj R Y))) sorry))\n    (equiv.symm (limits.cofork.is_colimit.hom_iso (counit_coequalises adj\u2081 X) (functor.obj R Y)))\n\n/-- Construct the left adjoint to `R`, with object map `construct_left_adjoint_obj`. -/\ndef construct_left_adjoint {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (F' : C \u2964 A) (adj\u2081 : F \u22a3 U) (adj\u2082 : F' \u22a3 R \u22d9 U) [limits.has_reflexive_coequalizers A] [(X : B) \u2192 regular_epi (nat_trans.app (adjunction.counit adj\u2081) X)] : B \u2964 A :=\n  adjunction.left_adjoint_of_equiv (fun (X : B) (Y : A) => construct_left_adjoint_equiv R F' adj\u2081 adj\u2082 Y X) sorry\n\nend lift_adjoint\n\n\n/--\nThe adjoint triangle theorem: Suppose `U : B \u2964 C` has a left adjoint `F` such that each counit\n`\u03b5_X : FUX \u27f6 X` is a regular epimorphism. Then if a category `A` has coequalizers of reflexive\npairs, then a functor `R : A \u2964 B` has a left adjoint if the composite `R \u22d9 U` does.\n\nNote the converse is true (with weaker assumptions), by `adjunction.comp`.\nSee https://ncatlab.org/nlab/show/adjoint+triangle+theorem\n-/\ndef adjoint_triangle_lift {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {U : B \u2964 C} {F : C \u2964 B} (R : A \u2964 B) (adj\u2081 : F \u22a3 U) [(X : B) \u2192 regular_epi (nat_trans.app (adjunction.counit adj\u2081) X)] [limits.has_reflexive_coequalizers A] [is_right_adjoint (R \u22d9 U)] : is_right_adjoint R :=\n  is_right_adjoint.mk\n    (lift_adjoint.construct_left_adjoint R (left_adjoint (R \u22d9 U)) adj\u2081 (adjunction.of_right_adjoint (R \u22d9 U)))\n    (adjunction.adjunction_of_equiv_left\n      (fun (X : B) (Y : A) =>\n        lift_adjoint.construct_left_adjoint_equiv R (left_adjoint (R \u22d9 U)) adj\u2081 (adjunction.of_right_adjoint (R \u22d9 U)) Y X)\n      sorry)\n\n/--\nIf `R \u22d9 U` has a left adjoint, the domain of `R` has reflexive coequalizers and `U` is a monadic\nfunctor, then `R` has a left adjoint.\nThis is a special case of `adjoint_triangle_lift` which is often more useful in practice.\n-/\ndef monadic_adjoint_triangle_lift {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] (U : B \u2964 C) [monadic_right_adjoint U] {R : A \u2964 B} [limits.has_reflexive_coequalizers A] [is_right_adjoint (R \u22d9 U)] : is_right_adjoint R :=\n  let R' : A \u2964 monad.algebra (left_adjoint U \u22d9 U) := R \u22d9 monad.comparison U;\n  let this : is_right_adjoint (R' \u22d9 functor.inv (monad.comparison U)) := adjunction.right_adjoint_of_comp;\n  let this_1 : R' \u22d9 functor.inv (monad.comparison U) \u2245 R :=\n    iso_whisker_left R (functor.fun_inv_id (monad.comparison U)) \u226a\u226b functor.right_unitor R;\n  adjunction.right_adjoint_of_nat_iso this_1\n\n/--\nSuppose we have a commutative square of functors\n\n      Q\n    A \u2192 B\n  U \u2193   \u2193 V\n    C \u2192 D\n      R\n\nwhere `U` has a left adjoint, `A` has reflexive coequalizers and `V` has a left adjoint such that\neach component of the counit is a regular epi.\nThen `Q` has a left adjoint if `R` has a left adjoint.\n\nSee https://ncatlab.org/nlab/show/adjoint+lifting+theorem\n-/\ndef adjoint_square_lift {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {D : Type u\u2084} [category D] (Q : A \u2964 B) (V : B \u2964 D) (U : A \u2964 C) (R : C \u2964 D) (comm : U \u22d9 R \u2245 Q \u22d9 V) [is_right_adjoint U] [is_right_adjoint V] [is_right_adjoint R] [(X : B) \u2192 regular_epi (nat_trans.app (adjunction.counit (adjunction.of_right_adjoint V)) X)] [limits.has_reflexive_coequalizers A] : is_right_adjoint Q :=\n  let this : is_right_adjoint (Q \u22d9 V) := adjunction.right_adjoint_of_nat_iso comm;\n  adjoint_triangle_lift Q (adjunction.of_right_adjoint V)\n\n/--\nSuppose we have a commutative square of functors\n\n      Q\n    A \u2192 B\n  U \u2193   \u2193 V\n    C \u2192 D\n      R\n\nwhere `U` has a left adjoint, `A` has reflexive coequalizers and `V` is monadic.\nThen `Q` has a left adjoint if `R` has a left adjoint.\n\nSee https://ncatlab.org/nlab/show/adjoint+lifting+theorem\n-/\ndef monadic_adjoint_square_lift {A : Type u\u2081} {B : Type u\u2082} {C : Type u\u2083} [category A] [category B] [category C] {D : Type u\u2084} [category D] (Q : A \u2964 B) (V : B \u2964 D) (U : A \u2964 C) (R : C \u2964 D) (comm : U \u22d9 R \u2245 Q \u22d9 V) [is_right_adjoint U] [monadic_right_adjoint V] [is_right_adjoint R] [limits.has_reflexive_coequalizers A] : is_right_adjoint Q :=\n  let this : is_right_adjoint (Q \u22d9 V) := adjunction.right_adjoint_of_nat_iso comm;\n  monadic_adjoint_triangle_lift V\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/adjunction/lifting.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3967757642850858}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.nontriviality\n! leanprover-community/mathlib commit c93660aa62cc7be8026b707c31edcbc0ca018057\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Nontrivial\n\n/-!\n# The `nontriviality` tactic.\n\n-/\n\n\nnamespace Tactic\n\n/-- Tries to generate a `nontrivial \u03b1` instance by performing case analysis on\n`subsingleton_or_nontrivial \u03b1`,\nattempting to discharge the subsingleton branch using lemmas with `@[nontriviality]` attribute,\nincluding `subsingleton.le` and `eq_iff_true_of_subsingleton`.\n-/\nunsafe def nontriviality_by_elim (\u03b1 : expr) (lems : interactive.parse simp_arg_list) :\n    tactic Unit := do\n  let alternative \u2190 to_expr ``(subsingleton_or_nontrivial $(\u03b1))\n  let n \u2190 get_unused_name \"_inst\"\n  tactic.cases Alternative [n, n]\n  (solve1 do\n        reset_instance_cache\n        apply_instance <|>\n            interactive.simp none none ff lems [`nontriviality] (Interactive.Loc.ns [none])) <|>\n      fail f! \"Could not prove goal assuming `subsingleton {\u03b1}`\"\n  reset_instance_cache\n#align tactic.nontriviality_by_elim tactic.nontriviality_by_elim\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- Tries to generate a `nontrivial \u03b1` instance using `nontrivial_of_ne` or `nontrivial_of_lt`\nand local hypotheses.\n-/\nunsafe def nontriviality_by_assumption (\u03b1 : expr) : tactic Unit := do\n  let n \u2190 get_unused_name \"_inst\"\n  to_expr ``(Nontrivial $(\u03b1)) >>= assert n\n  apply_instance <|> sorry\n  reset_instance_cache\n#align tactic.nontriviality_by_assumption tactic.nontriviality_by_assumption\n\nend Tactic\n\nnamespace Tactic.Interactive\n\nopen Tactic\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- Attempts to generate a `nontrivial \u03b1` hypothesis.\n\nThe tactic first looks for an instance using `apply_instance`.\n\nIf the goal is an (in)equality, the type `\u03b1` is inferred from the goal.\nOtherwise, the type needs to be specified in the tactic invocation, as `nontriviality \u03b1`.\n\nThe `nontriviality` tactic will first look for strict inequalities amongst the hypotheses,\nand use these to derive the `nontrivial` instance directly.\n\nOtherwise, it will perform a case split on `subsingleton \u03b1 \u2228 nontrivial \u03b1`, and attempt to discharge\nthe `subsingleton` goal using `simp [lemmas] with nontriviality`, where `[lemmas]` is a list of\nadditional `simp` lemmas that can be passed to `nontriviality` using the syntax\n`nontriviality \u03b1 using [lemmas]`.\n\n```\nexample {R : Type} [ordered_ring R] {a : R} (h : 0 < a) : 0 < a :=\nbegin\n  nontriviality, -- There is now a `nontrivial R` hypothesis available.\n  assumption,\nend\n```\n\n```\nexample {R : Type} [comm_ring R] {r s : R} : r * s = s * r :=\nbegin\n  nontriviality, -- There is now a `nontrivial R` hypothesis available.\n  apply mul_comm,\nend\n```\n\n```\nexample {R : Type} [ordered_ring R] {a : R} (h : 0 < a) : (2 : \u2115) \u2223 4 :=\nbegin\n  nontriviality R, -- there is now a `nontrivial R` hypothesis available.\n  dec_trivial\nend\n```\n\n```\ndef myeq {\u03b1 : Type} (a b : \u03b1) : Prop := a = b\n\nexample {\u03b1 : Type} (a b : \u03b1) (h : a = b) : myeq a b :=\nbegin\n  success_if_fail { nontriviality \u03b1 }, -- Fails\n  nontriviality \u03b1 using [myeq], -- There is now a `nontrivial \u03b1` hypothesis available\n  assumption\nend\n```\n-/\nunsafe def nontriviality (t : parse texpr ?)\n    (lems : parse (tk \"using\" *> simp_arg_list <|> pure [])) : tactic Unit := do\n  let \u03b1 \u2190\n    match t with\n      | some \u03b1 => to_expr \u03b1\n      | none =>\n        (do\n            let t \u2190 mk_mvar\n            let e \u2190 to_expr ``(@Eq $(t) _ _)\n            target >>= unify e\n            return t) <|>\n          (do\n              let t \u2190 mk_mvar\n              let e \u2190 to_expr ``(@LE.le $(t) _ _ _)\n              target >>= unify e\n              return t) <|>\n            (do\n                let t \u2190 mk_mvar\n                let e \u2190 to_expr ``(@Ne $(t) _ _)\n                target >>= unify e\n                return t) <|>\n              (do\n                  let t \u2190 mk_mvar\n                  let e \u2190 to_expr ``(@LT.lt $(t) _ _ _)\n                  target >>= unify e\n                  return t) <|>\n                fail\n                  \"The goal is not an (in)equality, so you'll need to specify the desired `nontrivial \u03b1`\\n      instance by invoking `nontriviality \u03b1`.\"\n  nontriviality_by_assumption \u03b1 <|> nontriviality_by_elim \u03b1 lems\n#align tactic.interactive.nontriviality tactic.interactive.nontriviality\n\nadd_tactic_doc\n  { Name := \"nontriviality\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.nontriviality]\n    tags := [\"logic\", \"type class\"] }\n\nend Tactic.Interactive\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Nontriviality.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3967757642850858}}
{"text": "/-\nCopyright (c) 2018 Andreas Swerdlow. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Andreas Swerdlow\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.matrix\nimport Mathlib.linear_algebra.tensor_product\nimport Mathlib.linear_algebra.nonsingular_inverse\nimport Mathlib.PostPort\n\nuniverses u v l u_1 w u_2 u_3 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Bilinear form\n\nThis file defines a bilinear form over a module. Basic ideas\nsuch as orthogonality are also introduced, as well as reflexivive,\nsymmetric and alternating bilinear forms. Adjoints of linear maps\nwith respect to a bilinear form are also introduced.\n\nA bilinear form on an R-(semi)module M, is a function from M x M to R,\nthat is linear in both arguments. Comments will typically abbreviate\n\"(semi)module\" as just \"module\", but the definitions should be as general as\npossible.\n\n## Notations\n\nGiven any term B of type bilin_form, due to a coercion, can use\nthe notation B x y to refer to the function field, ie. B x y = B.bilin x y.\n\nIn this file we use the following type variables:\n - `M`, `M'`, ... are semimodules over the semiring `R`,\n - `M\u2081`, `M\u2081'`, ... are modules over the ring `R\u2081`,\n - `M\u2082`, `M\u2082'`, ... are semimodules over the commutative semiring `R\u2082`\n - `M\u2083`, `M\u2083'`, ... are modules over the commutative ring `R\u2083`\n\n## References\n\n* <https://en.wikipedia.org/wiki/Bilinear_form>\n\n## Tags\n\nBilinear form,\n-/\n\n/-- `bilin_form R M` is the type of `R`-bilinear functions `M \u2192 M \u2192 R`. -/\nstructure bilin_form (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [semimodule R M]\n    where\n  bilin : M \u2192 M \u2192 R\n  bilin_add_left : \u2200 (x y z : M), bilin (x + y) z = bilin x z + bilin y z\n  bilin_smul_left : \u2200 (a : R) (x y : M), bilin (a \u2022 x) y = a * bilin x y\n  bilin_add_right : \u2200 (x y z : M), bilin x (y + z) = bilin x y + bilin x z\n  bilin_smul_right : \u2200 (a : R) (x y : M), bilin x (a \u2022 y) = a * bilin x y\n\nnamespace bilin_form\n\n\nprotected instance has_coe_to_fun {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] : has_coe_to_fun (bilin_form R M) :=\n  has_coe_to_fun.mk (fun (B : bilin_form R M) => M \u2192 M \u2192 R) fun (B : bilin_form R M) => bilin B\n\n@[simp] theorem coe_fn_mk {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (f : M \u2192 M \u2192 R) (h\u2081 : \u2200 (x y z : M), f (x + y) z = f x z + f y z)\n    (h\u2082 : \u2200 (a : R) (x y : M), f (a \u2022 x) y = a * f x y)\n    (h\u2083 : \u2200 (x y z : M), f x (y + z) = f x y + f x z)\n    (h\u2084 : \u2200 (a : R) (x y : M), f x (a \u2022 y) = a * f x y) : \u21d1(mk f h\u2081 h\u2082 h\u2083 h\u2084) = f :=\n  rfl\n\ntheorem coe_fn_congr {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} {x : M} {x' : M} {y : M} {y' : M} :\n    x = x' \u2192 y = y' \u2192 coe_fn B x y = coe_fn B x' y' :=\n  sorry\n\n@[simp] theorem add_left {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (x : M) (y : M) (z : M) :\n    coe_fn B (x + y) z = coe_fn B x z + coe_fn B y z :=\n  bilin_add_left B x y z\n\n@[simp] theorem smul_left {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} (a : R) (x : M) (y : M) :\n    coe_fn B (a \u2022 x) y = a * coe_fn B x y :=\n  bilin_smul_left B a x y\n\n@[simp] theorem add_right {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} (x : M) (y : M) (z : M) :\n    coe_fn B x (y + z) = coe_fn B x y + coe_fn B x z :=\n  bilin_add_right B x y z\n\n@[simp] theorem smul_right {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} (a : R) (x : M) (y : M) :\n    coe_fn B x (a \u2022 y) = a * coe_fn B x y :=\n  bilin_smul_right B a x y\n\n@[simp] theorem zero_left {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} (x : M) : coe_fn B 0 x = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn B 0 x = 0)) (Eq.symm (zero_smul R 0))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn B (0 \u2022 0) x = 0)) (smul_left 0 0 x)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (0 * coe_fn B 0 x = 0)) (zero_mul (coe_fn B 0 x))))\n        (Eq.refl 0)))\n\n@[simp] theorem zero_right {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} (x : M) : coe_fn B x 0 = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn B x 0 = 0)) (Eq.symm (zero_smul R 0))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn B x (0 \u2022 0) = 0)) (smul_right 0 x 0)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (0 * coe_fn B x 0 = 0)) (zero_mul (coe_fn B x 0))))\n        (Eq.refl 0)))\n\n@[simp] theorem neg_left {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (x : M\u2081) (y : M\u2081) : coe_fn B\u2081 (-x) y = -coe_fn B\u2081 x y :=\n  sorry\n\n@[simp] theorem neg_right {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (x : M\u2081) (y : M\u2081) : coe_fn B\u2081 x (-y) = -coe_fn B\u2081 x y :=\n  sorry\n\n@[simp] theorem sub_left {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (x : M\u2081) (y : M\u2081) (z : M\u2081) :\n    coe_fn B\u2081 (x - y) z = coe_fn B\u2081 x z - coe_fn B\u2081 y z :=\n  sorry\n\n@[simp] theorem sub_right {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (x : M\u2081) (y : M\u2081) (z : M\u2081) :\n    coe_fn B\u2081 x (y - z) = coe_fn B\u2081 x y - coe_fn B\u2081 x z :=\n  sorry\n\ntheorem ext {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} {D : bilin_form R M} (H : \u2200 (x y : M), coe_fn B x y = coe_fn D x y) :\n    B = D :=\n  sorry\n\nprotected instance add_comm_monoid {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] : add_comm_monoid (bilin_form R M) :=\n  add_comm_monoid.mk\n    (fun (B D : bilin_form R M) =>\n      mk (fun (x y : M) => coe_fn B x y + coe_fn D x y) sorry sorry sorry sorry)\n    sorry (mk (fun (x y : M) => 0) sorry sorry sorry sorry) sorry sorry sorry\n\nprotected instance add_comm_group {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081]\n    [module R\u2081 M\u2081] : add_comm_group (bilin_form R\u2081 M\u2081) :=\n  add_comm_group.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry\n    (fun (B : bilin_form R\u2081 M\u2081) => mk (fun (x y : M\u2081) => -bilin B x y) sorry sorry sorry sorry)\n    (add_group.sub._default add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry\n      fun (B : bilin_form R\u2081 M\u2081) => mk (fun (x y : M\u2081) => -bilin B x y) sorry sorry sorry sorry)\n    sorry sorry\n\n@[simp] theorem add_apply {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {D : bilin_form R M} (x : M) (y : M) :\n    coe_fn (B + D) x y = coe_fn B x y + coe_fn D x y :=\n  rfl\n\n@[simp] theorem neg_apply {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (x : M\u2081) (y : M\u2081) : coe_fn (-B\u2081) x y = -coe_fn B\u2081 x y :=\n  rfl\n\nprotected instance inhabited {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] : Inhabited (bilin_form R M) :=\n  { default := 0 }\n\nprotected instance semimodule {M : Type v} [add_comm_monoid M] {R : Type u_1} [comm_semiring R]\n    [semimodule R M] : semimodule R (bilin_form R M) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem smul_apply {M : Type v} [add_comm_monoid M] {R : Type u_1} [comm_semiring R]\n    [semimodule R M] (B : bilin_form R M) (a : R) (x : M) (y : M) :\n    coe_fn (a \u2022 B) x y = a \u2022 coe_fn B x y :=\n  rfl\n\nend bilin_form\n\n\n/-- A map with two arguments that is linear in both is a bilinear form.\n\nThis is an auxiliary definition for the full linear equivalence `linear_map.to_bilin`.\n-/\ndef linear_map.to_bilin_aux {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] (f : linear_map R\u2082 M\u2082 (linear_map R\u2082 M\u2082 R\u2082)) : bilin_form R\u2082 M\u2082 :=\n  bilin_form.mk (fun (x y : M\u2082) => coe_fn (coe_fn f x) y) sorry sorry sorry sorry\n\n/-- A map with two arguments that is linear in both is linearly equivalent to bilinear form. -/\ndef linear_map.to_bilin {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] :\n    linear_equiv R\u2082 (linear_map R\u2082 M\u2082 (linear_map R\u2082 M\u2082 R\u2082)) (bilin_form R\u2082 M\u2082) :=\n  linear_equiv.mk linear_map.to_bilin_aux sorry sorry\n    (fun (F : bilin_form R\u2082 M\u2082) => linear_map.mk\u2082 R\u2082 \u21d1F sorry sorry sorry sorry) sorry sorry\n\n/-- Bilinear forms are linearly equivalent to maps with two arguments that are linear in both. -/\ndef bilin_form.to_lin {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] :\n    linear_equiv R\u2082 (bilin_form R\u2082 M\u2082) (linear_map R\u2082 M\u2082 (linear_map R\u2082 M\u2082 R\u2082)) :=\n  linear_equiv.symm linear_map.to_bilin\n\n@[simp] theorem linear_map.to_bilin_aux_eq {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] (f : linear_map R\u2082 M\u2082 (linear_map R\u2082 M\u2082 R\u2082)) :\n    linear_map.to_bilin_aux f = coe_fn linear_map.to_bilin f :=\n  rfl\n\n@[simp] theorem linear_map.to_bilin_symm {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] :\n    linear_equiv.symm linear_map.to_bilin = bilin_form.to_lin :=\n  rfl\n\n@[simp] theorem bilin_form.to_lin_symm {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] :\n    linear_equiv.symm bilin_form.to_lin = linear_map.to_bilin :=\n  linear_equiv.symm_symm linear_map.to_bilin\n\n@[simp] theorem to_linear_map_apply {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] {B\u2082 : bilin_form R\u2082 M\u2082} (x : M\u2082) :\n    \u21d1(coe_fn (coe_fn bilin_form.to_lin B\u2082) x) = coe_fn B\u2082 x :=\n  rfl\n\n@[simp] theorem map_sum_left {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {B\u2082 : bilin_form R\u2082 M\u2082} {\u03b1 : Type u_1} (t : finset \u03b1) (g : \u03b1 \u2192 M\u2082) (w : M\u2082) :\n    coe_fn B\u2082 (finset.sum t fun (i : \u03b1) => g i) w = finset.sum t fun (i : \u03b1) => coe_fn B\u2082 (g i) w :=\n  sorry\n\n@[simp] theorem map_sum_right {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {B\u2082 : bilin_form R\u2082 M\u2082} {\u03b1 : Type u_1} (t : finset \u03b1) (w : M\u2082) (g : \u03b1 \u2192 M\u2082) :\n    coe_fn B\u2082 w (finset.sum t fun (i : \u03b1) => g i) = finset.sum t fun (i : \u03b1) => coe_fn B\u2082 w (g i) :=\n  sorry\n\nnamespace bilin_form\n\n\n/-- Apply a linear map on the left and right argument of a bilinear form. -/\ndef comp {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M] {M' : Type w}\n    [add_comm_monoid M'] [semimodule R M'] (B : bilin_form R M') (l : linear_map R M M')\n    (r : linear_map R M M') : bilin_form R M :=\n  mk (fun (x y : M) => coe_fn B (coe_fn l x) (coe_fn r y)) sorry sorry sorry sorry\n\n/-- Apply a linear map to the left argument of a bilinear form. -/\ndef comp_left {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) (f : linear_map R M M) : bilin_form R M :=\n  comp B f linear_map.id\n\n/-- Apply a linear map to the right argument of a bilinear form. -/\ndef comp_right {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) (f : linear_map R M M) : bilin_form R M :=\n  comp B linear_map.id f\n\ntheorem comp_comp {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {M' : Type w} [add_comm_monoid M'] [semimodule R M'] {M'' : Type u_1} [add_comm_monoid M'']\n    [semimodule R M''] (B : bilin_form R M'') (l : linear_map R M M') (r : linear_map R M M')\n    (l' : linear_map R M' M'') (r' : linear_map R M' M'') :\n    comp (comp B l' r') l r = comp B (linear_map.comp l' l) (linear_map.comp r' r) :=\n  rfl\n\n@[simp] theorem comp_left_comp_right {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (B : bilin_form R M) (l : linear_map R M M) (r : linear_map R M M) :\n    comp_right (comp_left B l) r = comp B l r :=\n  rfl\n\n@[simp] theorem comp_right_comp_left {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (B : bilin_form R M) (l : linear_map R M M) (r : linear_map R M M) :\n    comp_left (comp_right B r) l = comp B l r :=\n  rfl\n\n@[simp] theorem comp_apply {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {M' : Type w} [add_comm_monoid M'] [semimodule R M'] (B : bilin_form R M')\n    (l : linear_map R M M') (r : linear_map R M M') (v : M) (w : M) :\n    coe_fn (comp B l r) v w = coe_fn B (coe_fn l v) (coe_fn r w) :=\n  rfl\n\n@[simp] theorem comp_left_apply {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (B : bilin_form R M) (f : linear_map R M M) (v : M) (w : M) :\n    coe_fn (comp_left B f) v w = coe_fn B (coe_fn f v) w :=\n  rfl\n\n@[simp] theorem comp_right_apply {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] (B : bilin_form R M) (f : linear_map R M M) (v : M) (w : M) :\n    coe_fn (comp_right B f) v w = coe_fn B v (coe_fn f w) :=\n  rfl\n\ntheorem comp_injective {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {M' : Type w} [add_comm_monoid M'] [semimodule R M'] (B\u2081 : bilin_form R M')\n    (B\u2082 : bilin_form R M') (l : linear_map R M M') (r : linear_map R M M')\n    (h\u2097 : function.surjective \u21d1l) (h\u1d63 : function.surjective \u21d1r) :\n    comp B\u2081 l r = comp B\u2082 l r \u2194 B\u2081 = B\u2082 :=\n  sorry\n\n/-- Apply a linear equivalence on the arguments of a bilinear form. -/\ndef congr {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082]\n    {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082'] (e : linear_equiv R\u2082 M\u2082 M\u2082') :\n    linear_equiv R\u2082 (bilin_form R\u2082 M\u2082) (bilin_form R\u2082 M\u2082') :=\n  linear_equiv.mk\n    (fun (B : bilin_form R\u2082 M\u2082) => comp B \u2191(linear_equiv.symm e) \u2191(linear_equiv.symm e)) sorry sorry\n    (fun (B : bilin_form R\u2082 M\u2082') => comp B \u2191e \u2191e) sorry sorry\n\n@[simp] theorem congr_apply {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082']\n    (e : linear_equiv R\u2082 M\u2082 M\u2082') (B : bilin_form R\u2082 M\u2082) (x : M\u2082') (y : M\u2082') :\n    coe_fn (coe_fn (congr e) B) x y =\n        coe_fn B (coe_fn (linear_equiv.symm e) x) (coe_fn (linear_equiv.symm e) y) :=\n  rfl\n\n@[simp] theorem congr_symm {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082']\n    (e : linear_equiv R\u2082 M\u2082 M\u2082') : linear_equiv.symm (congr e) = congr (linear_equiv.symm e) :=\n  sorry\n\ntheorem congr_comp {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082'] {M\u2082'' : Type u_2}\n    [add_comm_monoid M\u2082''] [semimodule R\u2082 M\u2082''] (e : linear_equiv R\u2082 M\u2082 M\u2082') (B : bilin_form R\u2082 M\u2082)\n    (l : linear_map R\u2082 M\u2082'' M\u2082') (r : linear_map R\u2082 M\u2082'' M\u2082') :\n    comp (coe_fn (congr e) B) l r =\n        comp B (linear_map.comp (\u2191(linear_equiv.symm e)) l)\n          (linear_map.comp (\u2191(linear_equiv.symm e)) r) :=\n  rfl\n\ntheorem comp_congr {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082'] {M\u2082'' : Type u_2}\n    [add_comm_monoid M\u2082''] [semimodule R\u2082 M\u2082''] (e : linear_equiv R\u2082 M\u2082' M\u2082'')\n    (B : bilin_form R\u2082 M\u2082) (l : linear_map R\u2082 M\u2082' M\u2082) (r : linear_map R\u2082 M\u2082' M\u2082) :\n    coe_fn (congr e) (comp B l r) =\n        comp B (linear_map.comp l \u2191(linear_equiv.symm e))\n          (linear_map.comp r \u2191(linear_equiv.symm e)) :=\n  rfl\n\n/-- `lin_mul_lin f g` is the bilinear form mapping `x` and `y` to `f x * g y` -/\ndef lin_mul_lin {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] (f : linear_map R\u2082 M\u2082 R\u2082) (g : linear_map R\u2082 M\u2082 R\u2082) : bilin_form R\u2082 M\u2082 :=\n  mk (fun (x y : M\u2082) => coe_fn f x * coe_fn g y) sorry sorry sorry sorry\n\n@[simp] theorem lin_mul_lin_apply {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] {f : linear_map R\u2082 M\u2082 R\u2082} {g : linear_map R\u2082 M\u2082 R\u2082}\n    (x : M\u2082) (y : M\u2082) : coe_fn (lin_mul_lin f g) x y = coe_fn f x * coe_fn g y :=\n  rfl\n\n@[simp] theorem lin_mul_lin_comp {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {M\u2082' : Type u_1} [add_comm_monoid M\u2082'] [semimodule R\u2082 M\u2082']\n    {f : linear_map R\u2082 M\u2082 R\u2082} {g : linear_map R\u2082 M\u2082 R\u2082} (l : linear_map R\u2082 M\u2082' M\u2082)\n    (r : linear_map R\u2082 M\u2082' M\u2082) :\n    comp (lin_mul_lin f g) l r = lin_mul_lin (linear_map.comp f l) (linear_map.comp g r) :=\n  rfl\n\n@[simp] theorem lin_mul_lin_comp_left {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] {f : linear_map R\u2082 M\u2082 R\u2082} {g : linear_map R\u2082 M\u2082 R\u2082}\n    (l : linear_map R\u2082 M\u2082 M\u2082) :\n    comp_left (lin_mul_lin f g) l = lin_mul_lin (linear_map.comp f l) g :=\n  rfl\n\n@[simp] theorem lin_mul_lin_comp_right {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] {f : linear_map R\u2082 M\u2082 R\u2082} {g : linear_map R\u2082 M\u2082 R\u2082}\n    (r : linear_map R\u2082 M\u2082 M\u2082) :\n    comp_right (lin_mul_lin f g) r = lin_mul_lin f (linear_map.comp g r) :=\n  rfl\n\n/-- The proposition that two elements of a bilinear form space are orthogonal -/\ndef is_ortho {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) (x : M) (y : M) :=\n  coe_fn B x y = 0\n\ntheorem ortho_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (x : M) : is_ortho B 0 x :=\n  zero_left x\n\n@[simp] theorem is_ortho_smul_left {R\u2084 : Type u_2} {M\u2084 : Type u_3} [domain R\u2084] [add_comm_group M\u2084]\n    [module R\u2084 M\u2084] {G : bilin_form R\u2084 M\u2084} {x : M\u2084} {y : M\u2084} {a : R\u2084} (ha : a \u2260 0) :\n    is_ortho G (a \u2022 x) y \u2194 is_ortho G x y :=\n  sorry\n\n@[simp] theorem is_ortho_smul_right {R\u2084 : Type u_2} {M\u2084 : Type u_3} [domain R\u2084] [add_comm_group M\u2084]\n    [module R\u2084 M\u2084] {G : bilin_form R\u2084 M\u2084} {x : M\u2084} {y : M\u2084} {a : R\u2084} (ha : a \u2260 0) :\n    is_ortho G x (a \u2022 y) \u2194 is_ortho G x y :=\n  sorry\n\n/-- Two bilinear forms are equal when they are equal on all basis vectors. -/\ntheorem ext_basis {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083] [module R\u2083 M\u2083]\n    {B\u2083 : bilin_form R\u2083 M\u2083} {F\u2083 : bilin_form R\u2083 M\u2083} {\u03b9 : Type u_2} {b : \u03b9 \u2192 M\u2083} (hb : is_basis R\u2083 b)\n    (h : \u2200 (i j : \u03b9), coe_fn B\u2083 (b i) (b j) = coe_fn F\u2083 (b i) (b j)) : B\u2083 = F\u2083 :=\n  linear_equiv.injective to_lin\n    (is_basis.ext hb fun (i : \u03b9) => is_basis.ext hb fun (j : \u03b9) => h i j)\n\n/-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis. -/\ntheorem sum_repr_mul_repr_mul {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {B\u2083 : bilin_form R\u2083 M\u2083} {\u03b9 : Type u_2} {b : \u03b9 \u2192 M\u2083} (hb : is_basis R\u2083 b) (x : M\u2083)\n    (y : M\u2083) :\n    (finsupp.sum (coe_fn (is_basis.repr hb) x)\n          fun (i : \u03b9) (xi : R\u2083) =>\n            finsupp.sum (coe_fn (is_basis.repr hb) y)\n              fun (j : \u03b9) (yj : R\u2083) => xi \u2022 yj \u2022 coe_fn B\u2083 (b i) (b j)) =\n        coe_fn B\u2083 x y :=\n  sorry\n\nend bilin_form\n\n\n/-- The map from `matrix n n R` to bilinear forms on `n \u2192 R`.\n\nThis is an auxiliary definition for the equivalence `matrix.to_bilin_form'`. -/\ndef matrix.to_bilin'_aux {R\u2082 : Type u} [comm_semiring R\u2082] {n : Type u_1} [fintype n]\n    (M : matrix n n R\u2082) : bilin_form R\u2082 (n \u2192 R\u2082) :=\n  bilin_form.mk\n    (fun (v w : n \u2192 R\u2082) =>\n      finset.sum finset.univ fun (i : n) => finset.sum finset.univ fun (j : n) => v i * M i j * w j)\n    sorry sorry sorry sorry\n\ntheorem matrix.to_bilin'_aux_std_basis {R\u2082 : Type u} [comm_semiring R\u2082] {n : Type u_1} [fintype n]\n    [DecidableEq n] (M : matrix n n R\u2082) (i : n) (j : n) :\n    coe_fn (matrix.to_bilin'_aux M) (coe_fn (linear_map.std_basis R\u2082 (fun (\u1fb0 : n) => R\u2082) i) 1)\n          (coe_fn (linear_map.std_basis R\u2082 (fun (\u1fb0 : n) => R\u2082) j) 1) =\n        M i j :=\n  sorry\n\n/-- The linear map from bilinear forms to `matrix n n R` given an `n`-indexed basis.\n\nThis is an auxiliary definition for the equivalence `matrix.to_bilin_form'`. -/\ndef bilin_form.to_matrix_aux {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {n : Type u_1} [fintype n] (b : n \u2192 M\u2082) :\n    linear_map R\u2082 (bilin_form R\u2082 M\u2082) (matrix n n R\u2082) :=\n  linear_map.mk (fun (B : bilin_form R\u2082 M\u2082) (i j : n) => coe_fn B (b i) (b j)) sorry sorry\n\ntheorem to_bilin'_aux_to_matrix_aux {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B\u2083 : bilin_form R\u2083 (n \u2192 R\u2083)) :\n    matrix.to_bilin'_aux\n          (coe_fn\n            (bilin_form.to_matrix_aux\n              fun (j : n) => coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) j) 1)\n            B\u2083) =\n        B\u2083 :=\n  sorry\n\n/-! ### `to_matrix'` section\n\nThis section deals with the conversion between matrices and bilinear forms on `n \u2192 R\u2083`.\n-/\n\n/-- The linear equivalence between bilinear forms on `n \u2192 R` and `n \u00d7 n` matrices -/\ndef bilin_form.to_matrix' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n] [DecidableEq n] :\n    linear_equiv R\u2083 (bilin_form R\u2083 (n \u2192 R\u2083)) (matrix n n R\u2083) :=\n  linear_equiv.mk\n    (linear_map.to_fun\n      (bilin_form.to_matrix_aux\n        fun (j : n) => coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) j) 1))\n    sorry sorry matrix.to_bilin'_aux sorry sorry\n\n@[simp] theorem bilin_form.to_matrix_aux_std_basis {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1}\n    [fintype n] [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) :\n    coe_fn\n          (bilin_form.to_matrix_aux\n            fun (j : n) => coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) j) 1)\n          B =\n        coe_fn bilin_form.to_matrix' B :=\n  rfl\n\n/-- The linear equivalence between `n \u00d7 n` matrices and bilinear forms on `n \u2192 R` -/\ndef matrix.to_bilin' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n] [DecidableEq n] :\n    linear_equiv R\u2083 (matrix n n R\u2083) (bilin_form R\u2083 (n \u2192 R\u2083)) :=\n  linear_equiv.symm bilin_form.to_matrix'\n\n@[simp] theorem matrix.to_bilin'_aux_eq {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (M : matrix n n R\u2083) : matrix.to_bilin'_aux M = coe_fn matrix.to_bilin' M :=\n  rfl\n\ntheorem matrix.to_bilin'_apply {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (M : matrix n n R\u2083) (x : n \u2192 R\u2083) (y : n \u2192 R\u2083) :\n    coe_fn (coe_fn matrix.to_bilin' M) x y =\n        finset.sum finset.univ\n          fun (i : n) => finset.sum finset.univ fun (j : n) => x i * M i j * y j :=\n  rfl\n\n@[simp] theorem matrix.to_bilin'_std_basis {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (M : matrix n n R\u2083) (i : n) (j : n) :\n    coe_fn (coe_fn matrix.to_bilin' M) (coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) i) 1)\n          (coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) j) 1) =\n        M i j :=\n  matrix.to_bilin'_aux_std_basis M i j\n\n@[simp] theorem bilin_form.to_matrix'_symm {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] : linear_equiv.symm bilin_form.to_matrix' = matrix.to_bilin' :=\n  rfl\n\n@[simp] theorem matrix.to_bilin'_symm {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] : linear_equiv.symm matrix.to_bilin' = bilin_form.to_matrix' :=\n  linear_equiv.symm_symm bilin_form.to_matrix'\n\n@[simp] theorem matrix.to_bilin'_to_matrix' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) :\n    coe_fn matrix.to_bilin' (coe_fn bilin_form.to_matrix' B) = B :=\n  linear_equiv.apply_symm_apply matrix.to_bilin' B\n\n@[simp] theorem bilin_form.to_matrix'_to_bilin' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1}\n    [fintype n] [DecidableEq n] (M : matrix n n R\u2083) :\n    coe_fn bilin_form.to_matrix' (coe_fn matrix.to_bilin' M) = M :=\n  linear_equiv.apply_symm_apply bilin_form.to_matrix' M\n\n@[simp] theorem bilin_form.to_matrix'_apply {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) (i : n) (j : n) :\n    coe_fn bilin_form.to_matrix' B i j =\n        coe_fn B (coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) i) 1)\n          (coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) j) 1) :=\n  rfl\n\n@[simp] theorem bilin_form.to_matrix'_comp {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1}\n    {o : Type u_2} [fintype n] [fintype o] [DecidableEq n] [DecidableEq o]\n    (B : bilin_form R\u2083 (n \u2192 R\u2083)) (l : linear_map R\u2083 (o \u2192 R\u2083) (n \u2192 R\u2083))\n    (r : linear_map R\u2083 (o \u2192 R\u2083) (n \u2192 R\u2083)) :\n    coe_fn bilin_form.to_matrix' (bilin_form.comp B l r) =\n        matrix.mul\n          (matrix.mul (matrix.transpose (coe_fn linear_map.to_matrix' l))\n            (coe_fn bilin_form.to_matrix' B))\n          (coe_fn linear_map.to_matrix' r) :=\n  sorry\n\ntheorem bilin_form.to_matrix'_comp_left {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) (f : linear_map R\u2083 (n \u2192 R\u2083) (n \u2192 R\u2083)) :\n    coe_fn bilin_form.to_matrix' (bilin_form.comp_left B f) =\n        matrix.mul (matrix.transpose (coe_fn linear_map.to_matrix' f))\n          (coe_fn bilin_form.to_matrix' B) :=\n  sorry\n\ntheorem bilin_form.to_matrix'_comp_right {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) (f : linear_map R\u2083 (n \u2192 R\u2083) (n \u2192 R\u2083)) :\n    coe_fn bilin_form.to_matrix' (bilin_form.comp_right B f) =\n        matrix.mul (coe_fn bilin_form.to_matrix' B) (coe_fn linear_map.to_matrix' f) :=\n  sorry\n\ntheorem bilin_form.mul_to_matrix'_mul {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} {o : Type u_2}\n    [fintype n] [fintype o] [DecidableEq n] [DecidableEq o] (B : bilin_form R\u2083 (n \u2192 R\u2083))\n    (M : matrix o n R\u2083) (N : matrix n o R\u2083) :\n    matrix.mul (matrix.mul M (coe_fn bilin_form.to_matrix' B)) N =\n        coe_fn bilin_form.to_matrix'\n          (bilin_form.comp B (coe_fn matrix.to_lin' (matrix.transpose M))\n            (coe_fn matrix.to_lin' N)) :=\n  sorry\n\ntheorem bilin_form.mul_to_matrix' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) (M : matrix n n R\u2083) :\n    matrix.mul M (coe_fn bilin_form.to_matrix' B) =\n        coe_fn bilin_form.to_matrix'\n          (bilin_form.comp_left B (coe_fn matrix.to_lin' (matrix.transpose M))) :=\n  sorry\n\ntheorem bilin_form.to_matrix'_mul {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] (B : bilin_form R\u2083 (n \u2192 R\u2083)) (M : matrix n n R\u2083) :\n    matrix.mul (coe_fn bilin_form.to_matrix' B) M =\n        coe_fn bilin_form.to_matrix' (bilin_form.comp_right B (coe_fn matrix.to_lin' M)) :=\n  sorry\n\ntheorem matrix.to_bilin'_comp {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} {o : Type u_2} [fintype n]\n    [fintype o] [DecidableEq n] [DecidableEq o] (M : matrix n n R\u2083) (P : matrix n o R\u2083)\n    (Q : matrix n o R\u2083) :\n    bilin_form.comp (coe_fn matrix.to_bilin' M) (coe_fn matrix.to_lin' P)\n          (coe_fn matrix.to_lin' Q) =\n        coe_fn matrix.to_bilin' (matrix.mul (matrix.mul (matrix.transpose P) M) Q) :=\n  sorry\n\n/-! ### `to_matrix` section\n\nThis section deals with the conversion between matrices and bilinear forms on\na module with a fixed basis.\n-/\n\n/-- `bilin_form.to_matrix hb` is the equivalence between `R`-bilinear forms on `M` and\n`n`-by-`n` matrices with entries in `R`, if `hb` is an `R`-basis for `M`. -/\ndef bilin_form.to_matrix {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) :\n    linear_equiv R\u2083 (bilin_form R\u2083 M\u2083) (matrix n n R\u2083) :=\n  linear_equiv.trans (bilin_form.congr (is_basis.equiv_fun hb)) bilin_form.to_matrix'\n\n/-- `bilin_form.to_matrix hb` is the equivalence between `R`-bilinear forms on `M` and\n`n`-by-`n` matrices with entries in `R`, if `hb` is an `R`-basis for `M`. -/\ndef matrix.to_bilin {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083] [module R\u2083 M\u2083]\n    {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) :\n    linear_equiv R\u2083 (matrix n n R\u2083) (bilin_form R\u2083 M\u2083) :=\n  linear_equiv.symm (bilin_form.to_matrix hb)\n\n@[simp] theorem is_basis.equiv_fun_symm_std_basis {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (i : n) :\n    coe_fn (linear_equiv.symm (is_basis.equiv_fun hb))\n          (coe_fn (linear_map.std_basis R\u2083 (fun (\u1fb0 : n) => R\u2083) i) 1) =\n        b i :=\n  sorry\n\n@[simp] theorem bilin_form.to_matrix_apply {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (B : bilin_form R\u2083 M\u2083) (i : n) (j : n) :\n    coe_fn (bilin_form.to_matrix hb) B i j = coe_fn B (b i) (b j) :=\n  sorry\n\n@[simp] theorem matrix.to_bilin_apply {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b)\n    (M : matrix n n R\u2083) (x : M\u2083) (y : M\u2083) :\n    coe_fn (coe_fn (matrix.to_bilin hb) M) x y =\n        finset.sum finset.univ\n          fun (i : n) =>\n            finset.sum finset.univ\n              fun (j : n) =>\n                coe_fn (coe_fn (is_basis.repr hb) x) i * M i j *\n                  coe_fn (coe_fn (is_basis.repr hb) y) j :=\n  sorry\n\n-- Not a `simp` lemma since `bilin_form.to_matrix` needs an extra argument\n\ntheorem bilinear_form.to_matrix_aux_eq {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (B : bilin_form R\u2083 M\u2083) :\n    coe_fn (bilin_form.to_matrix_aux b) B = coe_fn (bilin_form.to_matrix hb) B :=\n  sorry\n\n@[simp] theorem bilin_form.to_matrix_symm {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) : linear_equiv.symm (bilin_form.to_matrix hb) = matrix.to_bilin hb :=\n  rfl\n\n@[simp] theorem matrix.to_bilin_symm {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) :\n    linear_equiv.symm (matrix.to_bilin hb) = bilin_form.to_matrix hb :=\n  linear_equiv.symm_symm (bilin_form.to_matrix hb)\n\ntheorem matrix.to_bilin_is_basis_fun {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] : matrix.to_bilin (pi.is_basis_fun R\u2083 n) = matrix.to_bilin' :=\n  sorry\n\ntheorem bilin_form.to_matrix_is_basis_fun {R\u2083 : Type u} [comm_ring R\u2083] {n : Type u_1} [fintype n]\n    [DecidableEq n] : bilin_form.to_matrix (pi.is_basis_fun R\u2083 n) = bilin_form.to_matrix' :=\n  sorry\n\n@[simp] theorem matrix.to_bilin_to_matrix {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (B : bilin_form R\u2083 M\u2083) :\n    coe_fn (matrix.to_bilin hb) (coe_fn (bilin_form.to_matrix hb) B) = B :=\n  linear_equiv.apply_symm_apply (matrix.to_bilin hb) B\n\n@[simp] theorem bilin_form.to_matrix_to_bilin {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (M : matrix n n R\u2083) :\n    coe_fn (bilin_form.to_matrix hb) (coe_fn (matrix.to_bilin hb) M) = M :=\n  linear_equiv.apply_symm_apply (bilin_form.to_matrix hb) M\n\n-- Cannot be a `simp` lemma because `hb` must be inferred.\n\ntheorem bilin_form.to_matrix_comp {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} {o : Type u_2} [fintype n] [fintype o] [DecidableEq n]\n    {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) {M\u2083' : Type u_3} [add_comm_group M\u2083'] [module R\u2083 M\u2083']\n    {c : o \u2192 M\u2083'} (hc : is_basis R\u2083 c) [DecidableEq o] (B : bilin_form R\u2083 M\u2083)\n    (l : linear_map R\u2083 M\u2083' M\u2083) (r : linear_map R\u2083 M\u2083' M\u2083) :\n    coe_fn (bilin_form.to_matrix hc) (bilin_form.comp B l r) =\n        matrix.mul\n          (matrix.mul (matrix.transpose (coe_fn (linear_map.to_matrix hc hb) l))\n            (coe_fn (bilin_form.to_matrix hb) B))\n          (coe_fn (linear_map.to_matrix hc hb) r) :=\n  sorry\n\ntheorem bilin_form.to_matrix_comp_left {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (B : bilin_form R\u2083 M\u2083) (f : linear_map R\u2083 M\u2083 M\u2083) :\n    coe_fn (bilin_form.to_matrix hb) (bilin_form.comp_left B f) =\n        matrix.mul (matrix.transpose (coe_fn (linear_map.to_matrix hb hb) f))\n          (coe_fn (bilin_form.to_matrix hb) B) :=\n  sorry\n\ntheorem bilin_form.to_matrix_comp_right {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083}\n    (hb : is_basis R\u2083 b) (B : bilin_form R\u2083 M\u2083) (f : linear_map R\u2083 M\u2083 M\u2083) :\n    coe_fn (bilin_form.to_matrix hb) (bilin_form.comp_right B f) =\n        matrix.mul (coe_fn (bilin_form.to_matrix hb) B) (coe_fn (linear_map.to_matrix hb hb) f) :=\n  sorry\n\ntheorem bilin_form.mul_to_matrix_mul {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} {o : Type u_2} [fintype n] [fintype o] [DecidableEq n]\n    {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) {M\u2083' : Type u_3} [add_comm_group M\u2083'] [module R\u2083 M\u2083']\n    {c : o \u2192 M\u2083'} (hc : is_basis R\u2083 c) [DecidableEq o] (B : bilin_form R\u2083 M\u2083) (M : matrix o n R\u2083)\n    (N : matrix n o R\u2083) :\n    matrix.mul (matrix.mul M (coe_fn (bilin_form.to_matrix hb) B)) N =\n        coe_fn (bilin_form.to_matrix hc)\n          (bilin_form.comp B (coe_fn (matrix.to_lin hc hb) (matrix.transpose M))\n            (coe_fn (matrix.to_lin hc hb) N)) :=\n  sorry\n\ntheorem bilin_form.mul_to_matrix {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b)\n    (B : bilin_form R\u2083 M\u2083) (M : matrix n n R\u2083) :\n    matrix.mul M (coe_fn (bilin_form.to_matrix hb) B) =\n        coe_fn (bilin_form.to_matrix hb)\n          (bilin_form.comp_left B (coe_fn (matrix.to_lin hb hb) (matrix.transpose M))) :=\n  sorry\n\ntheorem bilin_form.to_matrix_mul {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} [fintype n] [DecidableEq n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b)\n    (B : bilin_form R\u2083 M\u2083) (M : matrix n n R\u2083) :\n    matrix.mul (coe_fn (bilin_form.to_matrix hb) B) M =\n        coe_fn (bilin_form.to_matrix hb)\n          (bilin_form.comp_right B (coe_fn (matrix.to_lin hb hb) M)) :=\n  sorry\n\ntheorem matrix.to_bilin_comp {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {n : Type u_1} {o : Type u_2} [fintype n] [fintype o] [DecidableEq n]\n    {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b) {M\u2083' : Type u_3} [add_comm_group M\u2083'] [module R\u2083 M\u2083']\n    {c : o \u2192 M\u2083'} (hc : is_basis R\u2083 c) [DecidableEq o] (M : matrix n n R\u2083) (P : matrix n o R\u2083)\n    (Q : matrix n o R\u2083) :\n    bilin_form.comp (coe_fn (matrix.to_bilin hb) M) (coe_fn (matrix.to_lin hc hb) P)\n          (coe_fn (matrix.to_lin hc hb) Q) =\n        coe_fn (matrix.to_bilin hc) (matrix.mul (matrix.mul (matrix.transpose P) M) Q) :=\n  sorry\n\nnamespace refl_bilin_form\n\n\n/-- The proposition that a bilinear form is reflexive -/\ndef is_refl {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) :=\n  \u2200 (x y : M), coe_fn B x y = 0 \u2192 coe_fn B y x = 0\n\ntheorem eq_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_refl B) {x : M} {y : M} : coe_fn B x y = 0 \u2192 coe_fn B y x = 0 :=\n  H x y\n\ntheorem ortho_sym {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_refl B) {x : M} {y : M} :\n    bilin_form.is_ortho B x y \u2194 bilin_form.is_ortho B y x :=\n  { mp := eq_zero H, mpr := eq_zero H }\n\nend refl_bilin_form\n\n\nnamespace sym_bilin_form\n\n\n/-- The proposition that a bilinear form is symmetric -/\ndef is_sym {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) :=\n  \u2200 (x y : M), coe_fn B x y = coe_fn B y x\n\ntheorem sym {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_sym B) (x : M) (y : M) : coe_fn B x y = coe_fn B y x :=\n  H x y\n\ntheorem is_refl {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_sym B) : refl_bilin_form.is_refl B :=\n  fun (x y : M) (H1 : coe_fn B x y = 0) => H x y \u25b8 H1\n\ntheorem ortho_sym {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_sym B) {x : M} {y : M} :\n    bilin_form.is_ortho B x y \u2194 bilin_form.is_ortho B y x :=\n  refl_bilin_form.ortho_sym (is_refl H)\n\nend sym_bilin_form\n\n\nnamespace alt_bilin_form\n\n\n/-- The proposition that a bilinear form is alternating -/\ndef is_alt {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) :=\n  \u2200 (x : M), coe_fn B x x = 0\n\ntheorem self_eq_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    {B : bilin_form R M} (H : is_alt B) (x : M) : coe_fn B x x = 0 :=\n  H x\n\ntheorem neg {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} (H : is_alt B\u2081) (x : M\u2081) (y : M\u2081) : -coe_fn B\u2081 x y = coe_fn B\u2081 y x :=\n  sorry\n\nend alt_bilin_form\n\n\nnamespace bilin_form\n\n\n/-- Given a pair of modules equipped with bilinear forms, this is the condition for a pair of\nmaps between them to be mutually adjoint. -/\ndef is_adjoint_pair {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) {M' : Type u_1} [add_comm_monoid M'] [semimodule R M']\n    (B' : bilin_form R M') (f : linear_map R M M') (g : linear_map R M' M) :=\n  \u2200 {x : M} {y : M'}, coe_fn B' (coe_fn f x) y = coe_fn B x (coe_fn g y)\n\ntheorem is_adjoint_pair.eq {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {M' : Type u_1} [add_comm_monoid M'] [semimodule R M']\n    {B' : bilin_form R M'} {f : linear_map R M M'} {g : linear_map R M' M}\n    (h : is_adjoint_pair B B' f g) {x : M} {y : M'} :\n    coe_fn B' (coe_fn f x) y = coe_fn B x (coe_fn g y) :=\n  h\n\ntheorem is_adjoint_pair_iff_comp_left_eq_comp_right {R : Type u} {M : Type v} [semiring R]\n    [add_comm_monoid M] [semimodule R M] {B : bilin_form R M} (F : bilin_form R M)\n    (f : module.End R M) (g : module.End R M) :\n    is_adjoint_pair B F f g \u2194 comp_left F f = comp_right B g :=\n  sorry\n\ntheorem is_adjoint_pair_zero {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {M' : Type u_1} [add_comm_monoid M'] [semimodule R M']\n    {B' : bilin_form R M'} : is_adjoint_pair B B' 0 0 :=\n  sorry\n\ntheorem is_adjoint_pair_id {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} : is_adjoint_pair B B 1 1 :=\n  fun (x y : M) => rfl\n\ntheorem is_adjoint_pair.add {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {M' : Type u_1} [add_comm_monoid M'] [semimodule R M']\n    {B' : bilin_form R M'} {f : linear_map R M M'} {f' : linear_map R M M'} {g : linear_map R M' M}\n    {g' : linear_map R M' M} (h : is_adjoint_pair B B' f g) (h' : is_adjoint_pair B B' f' g') :\n    is_adjoint_pair B B' (f + f') (g + g') :=\n  sorry\n\ntheorem is_adjoint_pair.sub {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    {B\u2081 : bilin_form R\u2081 M\u2081} {M\u2081' : Type u_2} [add_comm_group M\u2081'] [module R\u2081 M\u2081']\n    {B\u2081' : bilin_form R\u2081 M\u2081'} {f\u2081 : linear_map R\u2081 M\u2081 M\u2081'} {f\u2081' : linear_map R\u2081 M\u2081 M\u2081'}\n    {g\u2081 : linear_map R\u2081 M\u2081' M\u2081} {g\u2081' : linear_map R\u2081 M\u2081' M\u2081} (h : is_adjoint_pair B\u2081 B\u2081' f\u2081 g\u2081)\n    (h' : is_adjoint_pair B\u2081 B\u2081' f\u2081' g\u2081') : is_adjoint_pair B\u2081 B\u2081' (f\u2081 - f\u2081') (g\u2081 - g\u2081') :=\n  sorry\n\ntheorem is_adjoint_pair.smul {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] {B\u2082 : bilin_form R\u2082 M\u2082} {M\u2082' : Type u_3} [add_comm_monoid M\u2082']\n    [semimodule R\u2082 M\u2082'] {B\u2082' : bilin_form R\u2082 M\u2082'} {f\u2082 : linear_map R\u2082 M\u2082 M\u2082'}\n    {g\u2082 : linear_map R\u2082 M\u2082' M\u2082} (c : R\u2082) (h : is_adjoint_pair B\u2082 B\u2082' f\u2082 g\u2082) :\n    is_adjoint_pair B\u2082 B\u2082' (c \u2022 f\u2082) (c \u2022 g\u2082) :=\n  sorry\n\ntheorem is_adjoint_pair.comp {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {M' : Type u_1} [add_comm_monoid M'] [semimodule R M']\n    {B' : bilin_form R M'} {f : linear_map R M M'} {g : linear_map R M' M} {M'' : Type u_4}\n    [add_comm_monoid M''] [semimodule R M''] (B'' : bilin_form R M'') {f' : linear_map R M' M''}\n    {g' : linear_map R M'' M'} (h : is_adjoint_pair B B' f g) (h' : is_adjoint_pair B' B'' f' g') :\n    is_adjoint_pair B B'' (linear_map.comp f' f) (linear_map.comp g g') :=\n  sorry\n\ntheorem is_adjoint_pair.mul {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M]\n    [semimodule R M] {B : bilin_form R M} {f : module.End R M} {g : module.End R M}\n    {f' : module.End R M} {g' : module.End R M} (h : is_adjoint_pair B B f g)\n    (h' : is_adjoint_pair B B f' g') : is_adjoint_pair B B (f * f') (g' * g) :=\n  sorry\n\n/-- The condition for an endomorphism to be \"self-adjoint\" with respect to a pair of bilinear forms\non the underlying module. In the case that these two forms are identical, this is the usual concept\nof self adjointness. In the case that one of the forms is the negation of the other, this is the\nusual concept of skew adjointness. -/\ndef is_pair_self_adjoint {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) (F : bilin_form R M) (f : module.End R M) :=\n  is_adjoint_pair B F f f\n\n/-- The set of pair-self-adjoint endomorphisms are a submodule of the type of all endomorphisms. -/\ndef is_pair_self_adjoint_submodule {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] (B\u2082 : bilin_form R\u2082 M\u2082) (F\u2082 : bilin_form R\u2082 M\u2082) :\n    submodule R\u2082 (module.End R\u2082 M\u2082) :=\n  submodule.mk (set_of fun (f : module.End R\u2082 M\u2082) => is_pair_self_adjoint B\u2082 F\u2082 f) sorry sorry sorry\n\n@[simp] theorem mem_is_pair_self_adjoint_submodule {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] (B\u2082 : bilin_form R\u2082 M\u2082) (F\u2082 : bilin_form R\u2082 M\u2082)\n    (f : module.End R\u2082 M\u2082) :\n    f \u2208 is_pair_self_adjoint_submodule B\u2082 F\u2082 \u2194 is_pair_self_adjoint B\u2082 F\u2082 f :=\n  iff.refl (f \u2208 is_pair_self_adjoint_submodule B\u2082 F\u2082)\n\ntheorem is_pair_self_adjoint_equiv {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] {M\u2083' : Type u_5} [add_comm_group M\u2083'] [module R\u2083 M\u2083'] (B\u2083 : bilin_form R\u2083 M\u2083)\n    (F\u2083 : bilin_form R\u2083 M\u2083) (e : linear_equiv R\u2083 M\u2083' M\u2083) (f : module.End R\u2083 M\u2083) :\n    is_pair_self_adjoint B\u2083 F\u2083 f \u2194\n        is_pair_self_adjoint (comp B\u2083 \u2191e \u2191e) (comp F\u2083 \u2191e \u2191e)\n          (coe_fn (linear_equiv.conj (linear_equiv.symm e)) f) :=\n  sorry\n\n/-- An endomorphism of a module is self-adjoint with respect to a bilinear form if it serves as an\nadjoint for itself. -/\ndef is_self_adjoint {R : Type u} {M : Type v} [semiring R] [add_comm_monoid M] [semimodule R M]\n    (B : bilin_form R M) (f : module.End R M) :=\n  is_adjoint_pair B B f f\n\n/-- An endomorphism of a module is skew-adjoint with respect to a bilinear form if its negation\nserves as an adjoint. -/\ndef is_skew_adjoint {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081] [add_comm_group M\u2081] [module R\u2081 M\u2081]\n    (B\u2081 : bilin_form R\u2081 M\u2081) (f : module.End R\u2081 M\u2081) :=\n  is_adjoint_pair B\u2081 B\u2081 f (-f)\n\ntheorem is_skew_adjoint_iff_neg_self_adjoint {R\u2081 : Type u} {M\u2081 : Type v} [ring R\u2081]\n    [add_comm_group M\u2081] [module R\u2081 M\u2081] (B\u2081 : bilin_form R\u2081 M\u2081) (f : module.End R\u2081 M\u2081) :\n    is_skew_adjoint B\u2081 f \u2194 is_adjoint_pair (-B\u2081) B\u2081 f f :=\n  sorry\n\n/-- The set of self-adjoint endomorphisms of a module with bilinear form is a submodule. (In fact\nit is a Jordan subalgebra.) -/\ndef self_adjoint_submodule {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082] [add_comm_monoid M\u2082]\n    [semimodule R\u2082 M\u2082] (B\u2082 : bilin_form R\u2082 M\u2082) : submodule R\u2082 (module.End R\u2082 M\u2082) :=\n  is_pair_self_adjoint_submodule B\u2082 B\u2082\n\n@[simp] theorem mem_self_adjoint_submodule {R\u2082 : Type u} {M\u2082 : Type v} [comm_semiring R\u2082]\n    [add_comm_monoid M\u2082] [semimodule R\u2082 M\u2082] (B\u2082 : bilin_form R\u2082 M\u2082) (f : module.End R\u2082 M\u2082) :\n    f \u2208 self_adjoint_submodule B\u2082 \u2194 is_self_adjoint B\u2082 f :=\n  iff.rfl\n\n/-- The set of skew-adjoint endomorphisms of a module with bilinear form is a submodule. (In fact\nit is a Lie subalgebra.) -/\ndef skew_adjoint_submodule {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083] [add_comm_group M\u2083]\n    [module R\u2083 M\u2083] (B\u2083 : bilin_form R\u2083 M\u2083) : submodule R\u2083 (module.End R\u2083 M\u2083) :=\n  is_pair_self_adjoint_submodule (-B\u2083) B\u2083\n\n@[simp] theorem mem_skew_adjoint_submodule {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] (B\u2083 : bilin_form R\u2083 M\u2083) (f : module.End R\u2083 M\u2083) :\n    f \u2208 skew_adjoint_submodule B\u2083 \u2194 is_skew_adjoint B\u2083 f :=\n  sorry\n\nend bilin_form\n\n\n/-- The condition for the square matrices `A`, `A'` to be an adjoint pair with respect to the square\nmatrices `J`, `J\u2083`. -/\ndef matrix.is_adjoint_pair {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n] (J : matrix n n R\u2083)\n    (J\u2083 : matrix n n R\u2083) (A : matrix n n R\u2083) (A' : matrix n n R\u2083) :=\n  matrix.mul (matrix.transpose A) J\u2083 = matrix.mul J A'\n\n/-- The condition for a square matrix `A` to be self-adjoint with respect to the square matrix\n`J`. -/\ndef matrix.is_self_adjoint {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n] (J : matrix n n R\u2083)\n    (A : matrix n n R\u2083) :=\n  matrix.is_adjoint_pair J J A A\n\n/-- The condition for a square matrix `A` to be skew-adjoint with respect to the square matrix\n`J`. -/\ndef matrix.is_skew_adjoint {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n] (J : matrix n n R\u2083)\n    (A : matrix n n R\u2083) :=\n  matrix.is_adjoint_pair J J A (-A)\n\n@[simp] theorem is_adjoint_pair_to_bilin' {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n]\n    (J : matrix n n R\u2083) (J\u2083 : matrix n n R\u2083) (A : matrix n n R\u2083) (A' : matrix n n R\u2083)\n    [DecidableEq n] :\n    bilin_form.is_adjoint_pair (coe_fn matrix.to_bilin' J) (coe_fn matrix.to_bilin' J\u2083)\n          (coe_fn matrix.to_lin' A) (coe_fn matrix.to_lin' A') \u2194\n        matrix.is_adjoint_pair J J\u2083 A A' :=\n  sorry\n\n@[simp] theorem is_adjoint_pair_to_bilin {R\u2083 : Type u} {M\u2083 : Type v} [comm_ring R\u2083]\n    [add_comm_group M\u2083] [module R\u2083 M\u2083] {n : Type w} [fintype n] {b : n \u2192 M\u2083} (hb : is_basis R\u2083 b)\n    (J : matrix n n R\u2083) (J\u2083 : matrix n n R\u2083) (A : matrix n n R\u2083) (A' : matrix n n R\u2083)\n    [DecidableEq n] :\n    bilin_form.is_adjoint_pair (coe_fn (matrix.to_bilin hb) J) (coe_fn (matrix.to_bilin hb) J\u2083)\n          (coe_fn (matrix.to_lin hb hb) A) (coe_fn (matrix.to_lin hb hb) A') \u2194\n        matrix.is_adjoint_pair J J\u2083 A A' :=\n  sorry\n\ntheorem matrix.is_adjoint_pair_equiv {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n]\n    (J : matrix n n R\u2083) (A : matrix n n R\u2083) (A' : matrix n n R\u2083) [DecidableEq n] (P : matrix n n R\u2083)\n    (h : is_unit P) :\n    matrix.is_adjoint_pair (matrix.mul (matrix.mul (matrix.transpose P) J) P)\n          (matrix.mul (matrix.mul (matrix.transpose P) J) P) A A' \u2194\n        matrix.is_adjoint_pair J J (matrix.mul (matrix.mul P A) (P\u207b\u00b9))\n          (matrix.mul (matrix.mul P A') (P\u207b\u00b9)) :=\n  sorry\n\n/-- The submodule of pair-self-adjoint matrices with respect to bilinear forms corresponding to\ngiven matrices `J`, `J\u2082`. -/\ndef pair_self_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n]\n    (J : matrix n n R\u2083) (J\u2083 : matrix n n R\u2083) [DecidableEq n] : submodule R\u2083 (matrix n n R\u2083) :=\n  submodule.map (\u2191linear_map.to_matrix')\n    (bilin_form.is_pair_self_adjoint_submodule (coe_fn matrix.to_bilin' J)\n      (coe_fn matrix.to_bilin' J\u2083))\n\n@[simp] theorem mem_pair_self_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w}\n    [fintype n] (J : matrix n n R\u2083) (J\u2083 : matrix n n R\u2083) (A : matrix n n R\u2083) [DecidableEq n] :\n    A \u2208 pair_self_adjoint_matrices_submodule J J\u2083 \u2194 matrix.is_adjoint_pair J J\u2083 A A :=\n  sorry\n\n/-- The submodule of self-adjoint matrices with respect to the bilinear form corresponding to\nthe matrix `J`. -/\ndef self_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n]\n    (J : matrix n n R\u2083) [DecidableEq n] : submodule R\u2083 (matrix n n R\u2083) :=\n  pair_self_adjoint_matrices_submodule J J\n\n@[simp] theorem mem_self_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w}\n    [fintype n] (J : matrix n n R\u2083) (A : matrix n n R\u2083) [DecidableEq n] :\n    A \u2208 self_adjoint_matrices_submodule J \u2194 matrix.is_self_adjoint J A :=\n  sorry\n\n/-- The submodule of skew-adjoint matrices with respect to the bilinear form corresponding to\nthe matrix `J`. -/\ndef skew_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w} [fintype n]\n    (J : matrix n n R\u2083) [DecidableEq n] : submodule R\u2083 (matrix n n R\u2083) :=\n  pair_self_adjoint_matrices_submodule (-J) J\n\n@[simp] theorem mem_skew_adjoint_matrices_submodule {R\u2083 : Type u} [comm_ring R\u2083] {n : Type w}\n    [fintype n] (J : matrix n n R\u2083) (A : matrix n n R\u2083) [DecidableEq n] :\n    A \u2208 skew_adjoint_matrices_submodule J \u2194 matrix.is_skew_adjoint J A :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/bilinear_form_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.39677575732699655}}
{"text": "import typ.open\n\nnamespace tts ------------------------------------------------------------------\nnamespace typ ------------------------------------------------------------------\nvariables {V : Type} [_root_.decidable_eq V] -- Type of variable names\nvariables {t t\u2081 t\u2082 : typ V} -- Types\nvariables {ts : list (typ V)} -- Lists of types\n\nopen occurs\n\n/-- Locally-closed type -/\ninductive lc : typ V \u2192 Prop\n| var : \u03a0 (x : tagged V),                  lc (var free x)\n| arr : \u03a0 {t\u2081 t\u2082 : typ V}, lc t\u2081 \u2192 lc t\u2082 \u2192 lc (arr t\u2081 t\u2082)\n\n/-- Locally-closed body of a type scheme with a given arity -/\ndef lc_body (n : \u2115) (t : typ V) : Prop :=\n\u2203 (L : finset (tagged V)),\n\u2200 {xs : list (tagged V)},\nxs.nodup \u2192\nxs.length = n \u2192\n(\u2200 {x : tagged V}, x \u2208 xs \u2192 x \u2209 L) \u2192\nlc (open_vars xs t)\n\n@[simp] theorem lc_var_free (x : tagged V) : lc (var free x) :=\nlc.var x\n\n@[simp] theorem lc_arr : lc (arr t\u2081 t\u2082) \u2194 lc t\u2081 \u2227 lc t\u2082 :=\n\u27e8\u03bb l, by cases l with _ _ _ l\u2081 l\u2082; exact \u27e8l\u2081, l\u2082\u27e9, \u03bb \u27e8l\u2081, l\u2082\u27e9, lc.arr l\u2081 l\u2082\u27e9\n\n-- Opening a locally-closed type is the identity\n@[simp] theorem open_typs_id (l : lc t) : open_typs ts t = t :=\nby induction t; cases l; simp [open_typs, *]\n\nend /- namespace -/ typ --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/typ/lc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834734, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.39668147558590927}}
{"text": "/-\nCopyright (c) 2017 Gabriel Ebner. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner\n-/\nimport .clause .prover_state .utils\nopen tactic monad expr native\n\nnamespace super\n\ndef position := list \u2115\n\nmeta def get_rwr_positions : expr \u2192 list position\n| (app a b) := [[]] ++\n  do arg \u2190 list.zip_with_index (get_app_args (app a b)),\n     pos \u2190 get_rwr_positions arg.1,\n     [arg.2 :: pos]\n| (var _) := []\n| e := [[]]\n\nmeta def get_position : expr \u2192 position \u2192 expr\n| (app a b) (p::ps) :=\nmatch list.nth (get_app_args (app a b)) p with\n| some arg := get_position arg ps\n| none := (app a b)\nend\n| e _ := e\n\nmeta def replace_position (v : expr) : expr \u2192 position \u2192 expr\n| (app a b) (p::ps) :=\nlet args := get_app_args (app a b) in\nmatch args.nth p with\n| some arg := app_of_list a.get_app_fn $ args.update_nth p $ replace_position arg ps\n| none := app a b\nend\n| e [] := v\n| e _ := e\n\nvariable gt : expr \u2192 expr \u2192 bool\nvariables (c1 c2 : clause)\nvariables (ac1 ac2 : derived_clause)\nvariables (i1 i2 : nat)\nvariable pos : list \u2115\nvariable ltr : bool\nvariable lt_in_termorder : bool\nvariable congr_ax : name\n\nlemma {u v w} sup_ltr (F : Sort u) (A : Sort v) (a1 a2) (f : A \u2192 Sort w) : (f a1 \u2192 F) \u2192 f a2 \u2192 a1 = a2 \u2192 F :=\nassume  hnfa1 hfa2 he, hnfa1 (@eq.rec A a2 f hfa2 a1 he.symm)\nlemma {u v w} sup_rtl (F : Sort u) (A : Sort v) (a1 a2) (f : A \u2192 Sort w) : (f a1 \u2192 F) \u2192 f a2 \u2192 a2 = a1 \u2192 F :=\nassume  hnfa1 hfa2 heq, hnfa1 (@eq.rec A a2 f hfa2 a1 heq)\n\nmeta def is_eq_dir (e : expr) (ltr : bool) : option (expr \u00d7 expr) :=\nmatch is_eq e with\n| some (lhs, rhs) := if ltr then some (lhs, rhs) else some (rhs, lhs)\n| none := none\nend\n\nmeta def try_sup : tactic clause := do\nguard $ (c1.get_lit i1).is_pos,\nqf1 \u2190 c1.open_metan c1.num_quants,\nqf2 \u2190 c2.open_metan c2.num_quants,\n(rwr_from, rwr_to) \u2190 (is_eq_dir (qf1.1.get_lit i1).formula ltr).to_monad,\natom \u2190 return (qf2.1.get_lit i2).formula,\neq_type \u2190 infer_type rwr_from,\natom_at_pos \u2190 return $ get_position atom pos,\natom_at_pos_type \u2190 infer_type atom_at_pos,\nunify eq_type atom_at_pos_type,\nunify rwr_from atom_at_pos transparency.none,\nrwr_from' \u2190 instantiate_mvars atom_at_pos,\nrwr_to' \u2190 instantiate_mvars rwr_to,\nif lt_in_termorder\n  then guard (gt rwr_from' rwr_to')\n  else guard (\u00acgt rwr_to' rwr_from'),\nrwr_ctx_varn \u2190 mk_fresh_name,\nabs_rwr_ctx \u2190 return $\n  lam rwr_ctx_varn binder_info.default eq_type\n  (if (qf2.1.get_lit i2).is_neg\n   then replace_position (mk_var 0) atom pos\n   else imp (replace_position (mk_var 0) atom pos) c2.local_false),\nlf_univ \u2190 infer_univ c1.local_false,\nuniv \u2190 infer_univ eq_type,\natom_univ \u2190 infer_univ atom,\nop1 \u2190 qf1.1.open_constn i1,\nop2 \u2190 qf2.1.open_constn c2.num_lits,\nhi2 \u2190 (op2.2.nth i2).to_monad,\nnew_atom \u2190 whnf_no_delta $ app abs_rwr_ctx rwr_to',\nnew_hi2 \u2190 return $ local_const hi2.local_uniq_name `H binder_info.default new_atom,\nnew_fin_prf \u2190\n  return $ app_of_list (const congr_ax [lf_univ, univ, atom_univ]) [c1.local_false, eq_type, rwr_from, rwr_to,\n            abs_rwr_ctx, (op2.1.close_const hi2).proof, new_hi2],\nclause.meta_closure (qf1.2 ++ qf2.2) $ (op1.1.inst new_fin_prf).close_constn (op1.2 ++ op2.2.update_nth i2 new_hi2)\n\nmeta def rwr_positions (c : clause) (i : nat) : list (list \u2115) :=\nget_rwr_positions (c.get_lit i).formula\n\nmeta def try_add_sup : prover unit :=\n(do c' \u2190 try_sup gt ac1.c ac2.c i1 i2 pos ltr ff congr_ax,\n    inf_score 2 [ac1.sc, ac2.sc] >>= mk_derived c' >>= add_inferred)\n  <|> return ()\n\nmeta def superposition_back_inf : inference :=\nassume  given, do active \u2190 get_active, sequence' $ do\n  given_i \u2190 given.selected,\n  guard (given.c.get_lit given_i).is_pos,\n  option.to_monad $ is_eq (given.c.get_lit given_i).formula,\n  other \u2190 rb_map.values active,\n  guard $ \u00acgiven.sc.in_sos \u2228 \u00acother.sc.in_sos,\n  other_i \u2190 other.selected,\n  pos \u2190 rwr_positions other.c other_i,\n  -- FIXME(gabriel): ``sup_ltr fails to resolve at runtime\n  [do try_add_sup gt given other given_i other_i pos tt ``super.sup_ltr,\n      try_add_sup gt given other given_i other_i pos ff ``super.sup_rtl]\n\nmeta def superposition_fwd_inf : inference :=\nassume  given, do active \u2190 get_active, sequence' $ do\n  given_i \u2190 given.selected,\n  other \u2190 rb_map.values active,\n  guard $ \u00acgiven.sc.in_sos \u2228 \u00acother.sc.in_sos,\n  other_i \u2190 other.selected,\n  guard (other.c.get_lit other_i).is_pos,\n  option.to_monad $ is_eq (other.c.get_lit other_i).formula,\n  pos \u2190 rwr_positions given.c given_i,\n  [do try_add_sup gt other given other_i given_i pos tt ``super.sup_ltr,\n      try_add_sup gt other given other_i given_i pos ff ``super.sup_rtl]\n\n@[super.inf]\nmeta def superposition_inf : inf_decl := inf_decl.mk 40 $\nassume  given, do gt \u2190 get_term_order,\nsuperposition_fwd_inf gt given,\nsuperposition_back_inf gt given\n\nend super\n", "meta": {"author": "leanprover", "repo": "super", "sha": "47b107b4cec8f3b41d72daba9cbda2f9d54025de", "save_path": "github-repos/lean/leanprover-super", "path": "github-repos/lean/leanprover-super/super-47b107b4cec8f3b41d72daba9cbda2f9d54025de/src/super/superposition.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3966814755859092}}
{"text": "/-\nCopyright (c) 2018 Sean Leather. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sean Leather, Mario Carneiro\n-/\nimport data.list.sigma\n\n/-!\n# Association lists\n-/\n\nuniverses u v w\nopen list\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v}\n\n/-- `alist \u03b2` is a key-value map stored as a `list` (i.e. a linked list).\n  It is a wrapper around certain `list` functions with the added constraint\n  that the list have unique keys. -/\nstructure alist (\u03b2 : \u03b1 \u2192 Type v) : Type (max u v) :=\n(entries : list (sigma \u03b2))\n(nodupkeys : entries.nodupkeys)\n\n/-- Given `l : list (sigma \u03b2)`, create a term of type `alist \u03b2` by removing\nentries with duplicate keys. -/\ndef list.to_alist [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type v} (l : list (sigma \u03b2)) : alist \u03b2 :=\n{ entries := _,\n  nodupkeys := nodupkeys_erase_dupkeys l }\n\nnamespace alist\n\n@[ext] theorem ext : \u2200 {s t : alist \u03b2}, s.entries = t.entries \u2192 s = t\n| \u27e8l\u2081, h\u2081\u27e9 \u27e8l\u2082, h\u2082\u27e9 H := by congr'\n\nlemma ext_iff {s t : alist \u03b2} : s = t \u2194 s.entries = t.entries :=\n\u27e8congr_arg _, ext\u27e9\n\ninstance [decidable_eq \u03b1] [\u2200 a, decidable_eq (\u03b2 a)] : decidable_eq (alist \u03b2) :=\n\u03bb xs ys, by rw ext_iff; apply_instance\n\n/-! ### keys -/\n\n/-- The list of keys of an association list. -/\ndef keys (s : alist \u03b2) : list \u03b1 := s.entries.keys\n\ntheorem keys_nodup (s : alist \u03b2) : s.keys.nodup := s.nodupkeys\n\n/-! ### mem -/\n\n/-- The predicate `a \u2208 s` means that `s` has a value associated to the key `a`. -/\ninstance : has_mem \u03b1 (alist \u03b2) := \u27e8\u03bb a s, a \u2208 s.keys\u27e9\n\ntheorem mem_keys {a : \u03b1} {s : alist \u03b2} : a \u2208 s \u2194 a \u2208 s.keys := iff.rfl\n\ntheorem mem_of_perm {a : \u03b1} {s\u2081 s\u2082 : alist \u03b2} (p : s\u2081.entries ~ s\u2082.entries) : a \u2208 s\u2081 \u2194 a \u2208 s\u2082 :=\n(p.map sigma.fst).mem_iff\n\n/-! ### empty -/\n\n/-- The empty association list. -/\ninstance : has_emptyc (alist \u03b2) := \u27e8\u27e8[], nodupkeys_nil\u27e9\u27e9\n\ninstance : inhabited (alist \u03b2) := \u27e8\u2205\u27e9\n\ntheorem not_mem_empty (a : \u03b1) : a \u2209 (\u2205 : alist \u03b2) :=\nnot_mem_nil a\n\n@[simp] theorem empty_entries : (\u2205 : alist \u03b2).entries = [] := rfl\n\n@[simp] theorem keys_empty : (\u2205 : alist \u03b2).keys = [] := rfl\n\n/-! ### singleton -/\n\n/-- The singleton association list. -/\ndef singleton (a : \u03b1) (b : \u03b2 a) : alist \u03b2 :=\n\u27e8[\u27e8a, b\u27e9], nodupkeys_singleton _\u27e9\n\n@[simp] \n\n@[simp] theorem keys_singleton (a : \u03b1) (b : \u03b2 a) : (singleton a b).keys = [a] := rfl\n\n/-! ### lookup -/\n\nsection\n\nvariables [decidable_eq \u03b1]\n\n/-- Look up the value associated to a key in an association list. -/\ndef lookup (a : \u03b1) (s : alist \u03b2) : option (\u03b2 a) :=\ns.entries.lookup a\n\n@[simp] theorem lookup_empty (a) : lookup a (\u2205 : alist \u03b2) = none :=\nrfl\n\ntheorem lookup_is_some {a : \u03b1} {s : alist \u03b2} :\n  (s.lookup a).is_some \u2194 a \u2208 s := lookup_is_some\n\ntheorem lookup_eq_none {a : \u03b1} {s : alist \u03b2} :\n  lookup a s = none \u2194 a \u2209 s :=\nlookup_eq_none\n\ntheorem perm_lookup {a : \u03b1} {s\u2081 s\u2082 : alist \u03b2} (p : s\u2081.entries ~ s\u2082.entries) :\n  s\u2081.lookup a = s\u2082.lookup a :=\nperm_lookup _ s\u2081.nodupkeys s\u2082.nodupkeys p\n\ninstance (a : \u03b1) (s : alist \u03b2) : decidable (a \u2208 s) :=\ndecidable_of_iff _ lookup_is_some\n\n/-! ### replace -/\n\n/-- Replace a key with a given value in an association list.\n  If the key is not present it does nothing. -/\ndef replace (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) : alist \u03b2 :=\n\u27e8kreplace a b s.entries, (kreplace_nodupkeys a b).2 s.nodupkeys\u27e9\n\n@[simp] theorem keys_replace (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) :\n  (replace a b s).keys = s.keys :=\nkeys_kreplace _ _ _\n\n@[simp] theorem mem_replace {a a' : \u03b1} {b : \u03b2 a} {s : alist \u03b2} :\n  a' \u2208 replace a b s \u2194 a' \u2208 s :=\nby rw [mem_keys, keys_replace, \u2190mem_keys]\n\ntheorem perm_replace {a : \u03b1} {b : \u03b2 a} {s\u2081 s\u2082 : alist \u03b2} :\n  s\u2081.entries ~ s\u2082.entries \u2192 (replace a b s\u2081).entries ~ (replace a b s\u2082).entries :=\nperm.kreplace s\u2081.nodupkeys\n\nend\n\n/-- Fold a function over the key-value pairs in the map. -/\ndef foldl {\u03b4 : Type w} (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) (d : \u03b4) (m : alist \u03b2) : \u03b4 :=\nm.entries.foldl (\u03bb r a, f r a.1 a.2) d\n\n/-! ### erase -/\n\nsection\n\nvariables [decidable_eq \u03b1]\n\n/-- Erase a key from the map. If the key is not present, do nothing. -/\ndef erase (a : \u03b1) (s : alist \u03b2) : alist \u03b2 :=\n\u27e8kerase a s.entries, kerase_nodupkeys _ s.nodupkeys\u27e9\n\n@[simp] theorem keys_erase (a : \u03b1) (s : alist \u03b2) :\n  (erase a s).keys = s.keys.erase a :=\nby simp only [erase, keys, keys_kerase]\n\n@[simp] theorem mem_erase {a a' : \u03b1} {s : alist \u03b2} : a' \u2208 erase a s \u2194 a' \u2260 a \u2227 a' \u2208 s :=\nby rw [mem_keys, keys_erase, mem_erase_iff_of_nodup s.keys_nodup, \u2190mem_keys]\n\ntheorem perm_erase {a : \u03b1} {s\u2081 s\u2082 : alist \u03b2} :\n  s\u2081.entries ~ s\u2082.entries \u2192 (erase a s\u2081).entries ~ (erase a s\u2082).entries :=\nperm.kerase s\u2081.nodupkeys\n\n@[simp] theorem lookup_erase (a) (s : alist \u03b2) : lookup a (erase a s) = none :=\nlookup_kerase a s.nodupkeys\n\n@[simp] theorem lookup_erase_ne {a a'} {s : alist \u03b2} (h : a \u2260 a') :\n  lookup a (erase a' s) = lookup a s :=\nlookup_kerase_ne h\n\ntheorem erase_erase (a a' : \u03b1) (s : alist \u03b2) :\n  (s.erase a).erase a' = (s.erase a').erase a :=\next $ kerase_kerase\n\n/-! ### insert -/\n\n/-- Insert a key-value pair into an association list and erase any existing pair\n  with the same key. -/\ndef insert (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) : alist \u03b2 :=\n\u27e8kinsert a b s.entries, kinsert_nodupkeys a b s.nodupkeys\u27e9\n\n@[simp] theorem insert_entries {a} {b : \u03b2 a} {s : alist \u03b2} :\n  (insert a b s).entries = sigma.mk a b :: kerase a s.entries :=\nrfl\n\ntheorem insert_entries_of_neg {a} {b : \u03b2 a} {s : alist \u03b2} (h : a \u2209 s) :\n  (insert a b s).entries = \u27e8a, b\u27e9 :: s.entries :=\nby rw [insert_entries, kerase_of_not_mem_keys h]\n\n@[simp] theorem mem_insert {a a'} {b' : \u03b2 a'} (s : alist \u03b2) :\n  a \u2208 insert a' b' s \u2194 a = a' \u2228 a \u2208 s :=\nmem_keys_kinsert\n\n@[simp] theorem keys_insert {a} {b : \u03b2 a} (s : alist \u03b2) :\n  (insert a b s).keys = a :: s.keys.erase a :=\nby simp [insert, keys, keys_kerase]\n\ntheorem perm_insert {a} {b : \u03b2 a} {s\u2081 s\u2082 : alist \u03b2} (p : s\u2081.entries ~ s\u2082.entries) :\n  (insert a b s\u2081).entries ~ (insert a b s\u2082).entries :=\nby simp only [insert_entries]; exact p.kinsert s\u2081.nodupkeys\n\n@[simp] theorem lookup_insert {a} {b : \u03b2 a} (s : alist \u03b2) : lookup a (insert a b s) = some b :=\nby simp only [lookup, insert, lookup_kinsert]\n\n@[simp] theorem lookup_insert_ne {a a'} {b' : \u03b2 a'} {s : alist \u03b2} (h : a \u2260 a') :\n  lookup a (insert a' b' s) = lookup a s :=\nlookup_kinsert_ne h\n\n@[simp] theorem lookup_to_alist {a} (s : list (sigma \u03b2)) : lookup a s.to_alist = s.lookup a :=\nby rw [list.to_alist,lookup,lookup_erase_dupkeys]\n\n@[simp] theorem insert_insert {a} {b b' : \u03b2 a} (s : alist \u03b2) :\n  (s.insert a b).insert a b' = s.insert a b' :=\nby ext : 1; simp only [alist.insert_entries, list.kerase_cons_eq];\n   constructor_matching* [_ \u2227 _]; refl\n\ntheorem insert_insert_of_ne {a a'} {b : \u03b2 a} {b' : \u03b2 a'} (s : alist \u03b2) (h : a \u2260 a') :\n  ((s.insert a b).insert a' b').entries ~ ((s.insert a' b').insert a b).entries :=\nby simp only [insert_entries]; rw [kerase_cons_ne,kerase_cons_ne,kerase_comm];\n   [apply perm.swap, exact h, exact h.symm]\n\n@[simp] lemma insert_singleton_eq {a : \u03b1} {b b' : \u03b2 a} :\n  insert a b (singleton a b') = singleton a b :=\next $ by simp only [alist.insert_entries, list.kerase_cons_eq, and_self, alist.singleton_entries,\n  heq_iff_eq, eq_self_iff_true]\n\n@[simp] theorem entries_to_alist (xs : list (sigma \u03b2)) :\n  (list.to_alist xs).entries = erase_dupkeys xs := rfl\n\ntheorem to_alist_cons (a : \u03b1) (b : \u03b2 a) (xs : list (sigma \u03b2)) :\n  list.to_alist (\u27e8a,b\u27e9 :: xs) = insert a b xs.to_alist := rfl\n\n/-! ### extract -/\n\n/-- Erase a key from the map, and return the corresponding value, if found. -/\ndef extract (a : \u03b1) (s : alist \u03b2) : option (\u03b2 a) \u00d7 alist \u03b2 :=\nhave (kextract a s.entries).2.nodupkeys,\nby rw [kextract_eq_lookup_kerase]; exact kerase_nodupkeys _ s.nodupkeys,\nmatch kextract a s.entries, this with\n| (b, l), h := (b, \u27e8l, h\u27e9)\nend\n\n@[simp] theorem extract_eq_lookup_erase (a : \u03b1) (s : alist \u03b2) :\n  extract a s = (lookup a s, erase a s) :=\nby simp [extract]; split; refl\n\n/-! ### union -/\n\n/-- `s\u2081 \u222a s\u2082` is the key-based union of two association lists. It is\nleft-biased: if there exists an `a \u2208 s\u2081`, `lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081`.\n-/\ndef union (s\u2081 s\u2082 : alist \u03b2) : alist \u03b2 :=\n\u27e8kunion s\u2081.entries s\u2082.entries, kunion_nodupkeys s\u2081.nodupkeys s\u2082.nodupkeys\u27e9\n\ninstance : has_union (alist \u03b2) := \u27e8union\u27e9\n\n@[simp] theorem union_entries {s\u2081 s\u2082 : alist \u03b2} :\n  (s\u2081 \u222a s\u2082).entries = kunion s\u2081.entries s\u2082.entries :=\nrfl\n\n@[simp] theorem empty_union {s : alist \u03b2} : (\u2205 : alist \u03b2) \u222a s = s :=\next rfl\n\n@[simp] theorem union_empty {s : alist \u03b2} : s \u222a (\u2205 : alist \u03b2) = s :=\next $ by simp\n\n@[simp] theorem mem_union {a} {s\u2081 s\u2082 : alist \u03b2} :\n  a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\nmem_keys_kunion\n\ntheorem perm_union {s\u2081 s\u2082 s\u2083 s\u2084 : alist \u03b2}\n  (p\u2081\u2082 : s\u2081.entries ~ s\u2082.entries) (p\u2083\u2084 : s\u2083.entries ~ s\u2084.entries) :\n  (s\u2081 \u222a s\u2083).entries ~ (s\u2082 \u222a s\u2084).entries :=\nby simp [p\u2081\u2082.kunion s\u2083.nodupkeys p\u2083\u2084]\n\ntheorem union_erase (a : \u03b1) (s\u2081 s\u2082 : alist \u03b2) : erase a (s\u2081 \u222a s\u2082) = erase a s\u2081 \u222a erase a s\u2082 :=\next kunion_kerase.symm\n\n@[simp] theorem lookup_union_left {a} {s\u2081 s\u2082 : alist \u03b2} :\n  a \u2208 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 :=\nlookup_kunion_left\n\n@[simp] theorem lookup_union_right {a} {s\u2081 s\u2082 : alist \u03b2} :\n  a \u2209 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2082 :=\nlookup_kunion_right\n\n@[simp] theorem mem_lookup_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : alist \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2082) \u2194 b \u2208 lookup a s\u2081 \u2228 a \u2209 s\u2081 \u2227 b \u2208 lookup a s\u2082 :=\nmem_lookup_kunion\n\ntheorem mem_lookup_union_middle {a} {b : \u03b2 a} {s\u2081 s\u2082 s\u2083 : alist \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2083) \u2192 a \u2209 s\u2082 \u2192 b \u2208 lookup a (s\u2081 \u222a s\u2082 \u222a s\u2083) :=\nmem_lookup_kunion_middle\n\ntheorem insert_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : alist \u03b2} :\n  insert a b (s\u2081 \u222a s\u2082) = insert a b s\u2081 \u222a s\u2082 :=\nby ext; simp\n\ntheorem union_assoc {s\u2081 s\u2082 s\u2083 : alist \u03b2} : ((s\u2081 \u222a s\u2082) \u222a s\u2083).entries ~ (s\u2081 \u222a (s\u2082 \u222a s\u2083)).entries :=\nlookup_ext (alist.nodupkeys _) (alist.nodupkeys _)\n(by simp [decidable.not_or_iff_and_not,or_assoc,and_or_distrib_left,and_assoc])\n\nend\n\n/-! ### disjoint -/\n\n/-- Two associative lists are disjoint if they have no common keys. -/\ndef disjoint (s\u2081 s\u2082 : alist \u03b2) : Prop :=\n\u2200 k \u2208 s\u2081.keys, \u00ac k \u2208 s\u2082.keys\n\nvariables [decidable_eq \u03b1]\n\ntheorem union_comm_of_disjoint {s\u2081 s\u2082 : alist \u03b2} (h : disjoint s\u2081 s\u2082) :\n  (s\u2081 \u222a s\u2082).entries ~ (s\u2082 \u222a s\u2081).entries :=\nlookup_ext (alist.nodupkeys _) (alist.nodupkeys _)\n(begin\n   intros, simp,\n   split; intro h',\n   cases h',\n   { right, refine \u27e8_,h'\u27e9,\n     apply h, rw [keys,\u2190 list.lookup_is_some,h'], exact rfl },\n   { left, rw h'.2 },\n   cases h',\n   { right, refine \u27e8_,h'\u27e9, intro h'',\n     apply h _ h'', rw [keys,\u2190 list.lookup_is_some,h'], exact rfl },\n   { left, rw h'.2 },\n end)\n\nend alist\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/list/alist.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.7154239836484143, "lm_q1q2_score": 0.39668146885737743}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.sheaves.presheaf\nimport Mathlib.category_theory.limits.punit\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.full_subcategory\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# The sheaf condition in terms of an equalizer of products\n\nHere we set up the machinery for the \"usual\" definition of the sheaf condition,\ne.g. as in https://stacks.math.columbia.edu/tag/0072\nin terms of an equalizer diagram where the two objects are\n`\u220f F.obj (U i)` and `\u220f F.obj (U i) \u2293 (U j)`.\n\n-/\n\nnamespace Top\n\n\nnamespace presheaf\n\n\nnamespace sheaf_condition_equalizer_products\n\n\n/-- The product of the sections of a presheaf over a family of open sets. -/\n/--\ndef pi_opens {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : C :=\n  \u220f fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i))\n\nThe product of the sections of a presheaf over the pairwise intersections of\na family of open sets.\n-/\ndef pi_inters {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : C :=\n  \u220f fun (p : \u03b9 \u00d7 \u03b9) => category_theory.functor.obj F (opposite.op (U (prod.fst p) \u2293 U (prod.snd p)))\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U i` to `U i \u2293 U j`.\n-/\ndef left_res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    pi_opens F U \u27f6 pi_inters F U :=\n  category_theory.limits.pi.lift\n    fun (p : \u03b9 \u00d7 \u03b9) =>\n      category_theory.limits.pi.\u03c0 (fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i)))\n          (prod.fst p) \u226b\n        category_theory.functor.map F\n          (category_theory.has_hom.hom.op\n            (topological_space.opens.inf_le_left (U (prod.fst p)) (U (prod.snd p))))\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef right_res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    pi_opens F U \u27f6 pi_inters F U :=\n  category_theory.limits.pi.lift\n    fun (p : \u03b9 \u00d7 \u03b9) =>\n      category_theory.limits.pi.\u03c0 (fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i)))\n          (prod.snd p) \u226b\n        category_theory.functor.map F\n          (category_theory.has_hom.hom.op\n            (topological_space.opens.inf_le_right (U (prod.fst p)) (U (prod.snd p))))\n\n/--\nThe morphism `F.obj U \u27f6 \u03a0 F.obj (U i)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top}\n    (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.functor.obj F (opposite.op (supr U)) \u27f6 pi_opens F U :=\n  category_theory.limits.pi.lift\n    fun (i : \u03b9) =>\n      category_theory.functor.map F\n        (category_theory.has_hom.hom.op (topological_space.opens.le_supr U i))\n\ntheorem w {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    res F U \u226b left_res F U = res F U \u226b right_res F U :=\n  sorry\n\n/--\nThe equalizer diagram for the sheaf condition.\n-/\ndef diagram {C : Type u} [category_theory.category C] [category_theory.limits.has_products C]\n    {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.limits.walking_parallel_pair \u2964 C :=\n  category_theory.limits.parallel_pair (left_res F U) (right_res F U)\n\n/--\nThe restriction map `F.obj U \u27f6 \u03a0 F.obj (U i)` gives a cone over the equalizer diagram\nfor the sheaf condition. The sheaf condition asserts this cone is a limit cone.\n-/\ndef fork {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top}\n    (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.limits.fork (left_res F U) (right_res F U) :=\n  category_theory.limits.fork.of_\u03b9 (res F U) sorry\n\n@[simp] theorem fork_X {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.limits.cone.X (fork F U) =\n        category_theory.functor.obj F (opposite.op (supr U)) :=\n  rfl\n\n@[simp] theorem fork_\u03b9 {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.limits.fork.\u03b9 (fork F U) = res F U :=\n  rfl\n\n@[simp] theorem fork_\u03c0_app_walking_parallel_pair_zero {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.nat_trans.app (category_theory.limits.cone.\u03c0 (fork F U))\n          category_theory.limits.walking_parallel_pair.zero =\n        res F U :=\n  rfl\n\n@[simp] theorem fork_\u03c0_app_walking_parallel_pair_one {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) :\n    category_theory.nat_trans.app (category_theory.limits.cone.\u03c0 (fork F U))\n          category_theory.limits.walking_parallel_pair.one =\n        res F U \u226b left_res F U :=\n  rfl\n\n/-- Isomorphic presheaves have isomorphic `pi_opens` for any cover `U`. -/\n@[simp] def pi_opens.iso_of_iso {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) :\n    pi_opens F U \u2245 pi_opens G U :=\n  category_theory.limits.pi.map_iso fun (X_1 : \u03b9) => category_theory.iso.app \u03b1 (opposite.op (U X_1))\n\n/-- Isomorphic presheaves have isomorphic `pi_inters` for any cover `U`. -/\n@[simp] def pi_inters.iso_of_iso {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) :\n    pi_inters F U \u2245 pi_inters G U :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9 \u00d7 \u03b9) =>\n      category_theory.iso.app \u03b1 (opposite.op (U (prod.fst X_1) \u2293 U (prod.snd X_1)))\n\n/-- Isomorphic presheaves have isomorphic sheaf condition diagrams. -/\ndef diagram.iso_of_iso {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) :\n    diagram F U \u2245 diagram G U :=\n  category_theory.nat_iso.of_components\n    (fun (X_1 : category_theory.limits.walking_parallel_pair) =>\n      category_theory.limits.walking_parallel_pair.cases_on X_1 (pi_opens.iso_of_iso U \u03b1)\n        (pi_inters.iso_of_iso U \u03b1))\n    sorry\n\n/--\nIf `F G : presheaf C X` are isomorphic presheaves,\nthen the `fork F U`, the canonical cone of the sheaf condition diagram for `F`,\nis isomorphic to `fork F G` postcomposed with the corresponding isomorphism between\nsheaf condition diagrams.\n-/\ndef fork.iso_of_iso {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v}\n    (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) :\n    fork F U \u2245\n        category_theory.functor.obj\n          (category_theory.limits.cones.postcompose\n            (category_theory.iso.inv (diagram.iso_of_iso U \u03b1)))\n          (fork G U) :=\n  category_theory.limits.fork.ext (category_theory.iso.app \u03b1 (opposite.op (supr U))) sorry\n\n/--\nPush forward a cover along an open embedding.\n-/\n@[simp] def cover.of_open_embedding {X : Top} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X}\n    (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) :\n    \u03b9 \u2192 topological_space.opens \u21a5X :=\n  fun (i : \u03b9) => category_theory.functor.obj (is_open_map.functor sorry) (\ud835\udcb0 i)\n\n/--\nThe isomorphism between `pi_opens` corresponding to an open embedding.\n-/\n@[simp] def pi_opens.iso_of_open_embedding {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top}\n    {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) :\n    pi_opens\n          (category_theory.functor.op\n              (is_open_map.functor (pi_opens.iso_of_open_embedding._proof_2 oe)) \u22d9\n            F)\n          \ud835\udcb0 \u2245\n        pi_opens F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9) =>\n      category_theory.functor.map_iso F\n        (category_theory.iso.refl\n          (category_theory.functor.obj (category_theory.functor.op (is_open_map.functor sorry))\n            (opposite.op (\ud835\udcb0 X_1))))\n\n/--\nThe isomorphism between `pi_inters` corresponding to an open embedding.\n-/\n@[simp] def pi_inters.iso_of_open_embedding {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top}\n    {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) :\n    pi_inters\n          (category_theory.functor.op\n              (is_open_map.functor (pi_inters.iso_of_open_embedding._proof_2 oe)) \u22d9\n            F)\n          \ud835\udcb0 \u2245\n        pi_inters F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9 \u00d7 \u03b9) =>\n      category_theory.functor.map_iso F\n        (id\n          (category_theory.iso.op\n            (category_theory.iso.mk (category_theory.hom_of_le sorry)\n              (category_theory.hom_of_le sorry))))\n\n/-- The isomorphism of sheaf condition diagrams corresponding to an open embedding. -/\ndef diagram.iso_of_open_embedding {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top}\n    {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) :\n    diagram\n          (category_theory.functor.op\n              (is_open_map.functor (diagram.iso_of_open_embedding._proof_2 oe)) \u22d9\n            F)\n          \ud835\udcb0 \u2245\n        diagram F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.nat_iso.of_components\n    (fun (X_1 : category_theory.limits.walking_parallel_pair) =>\n      category_theory.limits.walking_parallel_pair.cases_on X_1\n        (pi_opens.iso_of_open_embedding oe \ud835\udcb0) (pi_inters.iso_of_open_embedding oe \ud835\udcb0))\n    sorry\n\n/--\nIf `F : presheaf C X` is a presheaf, and `oe : U \u27f6 X` is an open embedding,\nthen the sheaf condition fork for a cover `\ud835\udcb0` in `U` for the composition of `oe` and `F` is\nisomorphic to sheaf condition fork for `oe '' \ud835\udcb0`, precomposed with the isomorphism\nof indexing diagrams `diagram.iso_of_open_embedding`.\n\nWe use this to show that the restriction of sheaf along an open embedding is still a sheaf.\n-/\ndef fork.iso_of_open_embedding {C : Type u} [category_theory.category C]\n    [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top}\n    {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) :\n    fork\n          (category_theory.functor.op\n              (is_open_map.functor (fork.iso_of_open_embedding._proof_2 oe)) \u22d9\n            F)\n          \ud835\udcb0 \u2245\n        category_theory.functor.obj\n          (category_theory.limits.cones.postcompose\n            (category_theory.iso.inv (diagram.iso_of_open_embedding oe \ud835\udcb0)))\n          (fork F (cover.of_open_embedding oe \ud835\udcb0)) :=\n  category_theory.limits.fork.ext\n    (id\n      (category_theory.functor.map_iso F\n        (category_theory.iso.op\n          (category_theory.iso.mk (category_theory.hom_of_le sorry)\n            (category_theory.hom_of_le sorry)))))\n    sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/sheaves/sheaf_condition/equalizer_products_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3966460941306852}}
{"text": "abbrev M := ExceptT String <| StateT Nat Id\n\ndef f (xs : List Nat) : M Unit := do\nfor x in xs do\n  if x == 0 then\n    throw \"contains zero\"\n\n#eval f [1, 2, 3] |>.run' 0\n#eval f [1, 0, 3] |>.run' 0\n\ntheorem ex1 : (f [1, 2, 3] |>.run' 0) = Except.ok () :=\nrfl\n\ntheorem ex2 : (f [1, 0, 3] |>.run' 0) = Except.error \"contains zero\" :=\nrfl\n\nuniverses u\n\nabbrev N := ExceptT (ULift.{u} String) Id\n\ndef idM {\u03b1 : Type u} (a : \u03b1) : N \u03b1 :=\npure a\n\ndef checkEq {\u03b1 : Type u} [BEq \u03b1] [ToString \u03b1] (a b : \u03b1) : N PUnit := do\nunless a == b do\n  throw (ULift.up s!\"{a} is not equal to {b}\")\n\ndef g {\u03b1 : Type u} [BEq \u03b1] [ToString \u03b1] (xs : List \u03b1) (a : \u03b1) : N PUnit := do\nfor x in xs do\n  let a \u2190 idM a\n  checkEq x a\n\n#eval g [1, (2:Nat), 3] 1 |>.run\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/forBodyResultTypeIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.3966160648835702}}
{"text": "import classes.context_free.basics.toolbox\nimport utilities.list_utils\nimport utilities.written_by_others.trim_assoc\n\nvariables {T : Type}\n\n\n/-- Context-free grammar for the empty language (i.e., `\u2208` always gives `false`). -/\ndef cfg_empty_lang : CF_grammar T :=\nCF_grammar.mk (fin 1) 0 []\n\n/-- Characterization of the empty language. -/\nlemma language_of_cfg_empty_lang :\n  CF_language (@cfg_empty_lang T) = 0 :=\nbegin\n  unfold CF_language,\n  ext1 w,\n  split, swap,\n  {\n    intro h,\n    exfalso,\n    exact set.not_mem_empty w h,\n  },\n  intro hw,\n  change CF_derives cfg_empty_lang [symbol.nonterminal cfg_empty_lang.initial] (list.map symbol.terminal w) at hw,\n  exfalso,\n  cases CF_tran_or_id_of_deri hw,\n  {\n    have hhead := congr_fun (congr_arg list.nth h) 0,\n    cases w with head tail ih,\n    {\n      change some (symbol.nonterminal cfg_empty_lang.initial) = none at hhead,\n      norm_cast at hhead,\n    },\n    {\n      change some (symbol.nonterminal cfg_empty_lang.initial) = some (symbol.terminal head) at hhead,\n      norm_cast at hhead,\n    },\n  },\n  {\n    rcases h with \u27e8v, \u27e8r, rin, -, -, -, -\u27e9, -\u27e9,\n    cases rin,\n  },\nend\n\n/-- Context-free grammar for the singleton language that contains `[]` as its only word. -/\ndef cfg_empty_word : CF_grammar T :=\nCF_grammar.mk (fin 1) 0 [(0, [])]\n\n/-- Characterization of the singleton language. -/\nlemma language_of_cfg_empty_word :\n  CF_language (@cfg_empty_word T) = singleton [] :=\nbegin\n  unfold CF_language,\n  ext1 w,\n  split, swap,\n  {\n    intro h,\n    rw set.mem_singleton_iff at h,\n    change CF_derives cfg_empty_word [symbol.nonterminal cfg_empty_lang.initial] (list.map symbol.terminal w),\n    apply @CF_deri_of_tran,\n    use ((0 : fin 1), []),\n    use [[], []],\n    rw h,\n    split;\n    refl,\n    exact T,\n  },\n  intro hw,\n  change\n    CF_derives\n      (@cfg_empty_word T)\n      [symbol.nonterminal (@cfg_empty_lang T).initial]\n      (list.map symbol.terminal w)\n    at hw,\n  cases\n    @CF_tran_or_id_of_deri T\n      (@cfg_empty_word T)\n      [symbol.nonterminal cfg_empty_lang.initial]\n      (list.map symbol.terminal w)\n      hw,\n  {\n    exfalso,\n    have zeroth := congr_fun (congr_arg list.nth h) 0,\n    rw list.nth at zeroth,\n    by_cases w = list.nil,\n    {\n      have is_none : (list.map symbol.terminal w).nth 0 = none,\n      {\n        rw h,\n        rw list.nth_map,\n        refl,\n      },\n      rw is_none at zeroth,\n      exact option.no_confusion zeroth,\n    },\n    {\n      have is_terminal : \u2203 t, (list.map symbol.terminal w).nth 0 = some (symbol.terminal t),\n      {\n        apply exists.intro (w.nth_le 0 (list.length_pos_of_ne_nil h)),\n        rw list.nth_map,\n        norm_num,\n        exact list.nth_le_nth (list.length_pos_of_ne_nil h),\n      },\n      cases is_terminal with irr is_termin,\n      rw is_termin at zeroth,\n      norm_cast at zeroth,\n    },\n  },\n  rcases h with \u27e8v, step_init, step_none\u27e9,\n  have v_is_empty_word : v = list.nil,\n  {\n    rcases step_init with \u27e8r, rin, pre, pos, bef, aft\u27e9,\n    have rule : r = ((0 : fin 1), []),\n    {\n      rw \u2190list.mem_singleton,\n      exact rin,\n    },\n    have empty_surrounding : pre = [] \u2227 pos = [],\n    {\n      rw rule at bef,\n      have bef_lenghts := congr_arg list.length bef,\n      rw list.length_append_append at bef_lenghts,\n      rw list.length_singleton at bef_lenghts,\n      rw list.length_singleton at bef_lenghts,\n      split,\n      {\n        have pre_zero : pre.length = 0,\n        {\n          clear_except bef_lenghts,\n          linarith,\n        },\n        rw list.length_eq_zero at pre_zero,\n        exact pre_zero,\n      },\n      {\n        have pos_zero : pos.length = 0,\n        {\n          clear_except bef_lenghts,\n          linarith,\n        },\n        rw list.length_eq_zero at pos_zero,\n        exact pos_zero,\n      },\n    },\n    rw empty_surrounding.1 at aft,\n    rw empty_surrounding.2 at aft,\n    rw rule at aft,\n    exact aft,\n  },\n  rw v_is_empty_word at step_none,\n  cases\n    @CF_tran_or_id_of_deri T\n      (@cfg_empty_word T)\n      list.nil\n      (list.map symbol.terminal w)\n      step_none,\n  {\n    by_contradiction contra,\n    have w_not_nil : w.length > 0,\n    {\n      apply list.length_pos_of_ne_nil,\n      convert contra,\n    },\n    have impossible_lengths := congr_arg list.length h,\n    rw list.length at impossible_lengths,\n    rw list.length_map at impossible_lengths,\n    rw \u2190impossible_lengths at w_not_nil,\n    exact nat.lt_irrefl 0 w_not_nil,\n  },\n  {\n    exfalso,\n    rcases h with \u27e8-, \u27e8trash_r, -, trash_1, trash_2, impossible, -\u27e9, -\u27e9,\n    have impossible_len := congr_arg list.length impossible,\n    clear_except impossible_len,\n    rw list.length_append_append at impossible_len,\n    rw list.length_singleton at impossible_len,\n    rw list.length at impossible_len,\n    linarith,\n  },\nend\n\n/-- Context-free grammar for a language `{a}.star` where `a` is a given terminal symbol. -/\ndef cfg_symbol_star (a : T) : CF_grammar T :=\nCF_grammar.mk (fin 1) 0 [(0, [symbol.terminal a, symbol.nonterminal 0]), (0, [])]\n\n/-- Characterization of the `{a}.star` language. -/\nlemma language_of_cfg_symbol_star (a : T) :\n  CF_language (cfg_symbol_star a) = \u03bb w, \u2203 n : \u2115, w = list.repeat a n :=\nbegin\n  apply set.eq_of_subset_of_subset,\n  {\n    intro w,\n    /-\n      We prove this inclusion as follows:\n      (1) `w \u2208 CF_language (cfg_symbol_star a)` \u2192\n      (2) `w` contains only `a`s \u2192\n      (3) `\u2203 (n : \u2115), w = list.repeat a n)` \u25a1\n    -/\n\n    have implication2 : (\u2200 t : T, t \u2260 a \u2192 t \u2209 w) \u2192 (\u2203 (n : \u2115), w = list.repeat a n),\n    {\n      contrapose,\n      intros contr ass,\n      push_neg at contr,\n      specialize contr w.length,\n\n      have different :\n        \u2203 n : \u2115, \u2203 hl : n < w.length, \u2203 hr : n < (list.repeat a w.length).length,\n          w.nth_le n hl \u2260 (list.repeat a w.length).nth_le n hr,\n      {\n        by_contradiction isnt,\n        have same_len : w.length = (list.repeat a w.length).length,\n        {\n          rw list.length_repeat,\n        },\n        apply contr,\n        apply list.ext_le same_len,\n        push_neg at isnt,\n        intros n n_small_left n_small_right,\n        specialize isnt n n_small_left,\n        push_neg at isnt,\n        specialize isnt n_small_right,\n        push_neg at isnt,\n        exact isnt,\n      },\n      rcases different with \u27e8n, hl, hr, nq\u27e9,\n\n      rw list.nth_le_repeat a hr at nq,\n      specialize ass (w.nth_le n hl) nq,\n      exact ass (list.nth_le_mem w n hl),\n    },\n\n    have implication1 : w \u2208 CF_language (cfg_symbol_star a) \u2192 (\u2200 t : T, t \u2260 a \u2192 t \u2209 w),\n    {\n      clear implication2,\n      intros ass t nq,\n      change CF_generates_str (cfg_symbol_star a) (list.map symbol.terminal w) at ass,\n      unfold CF_generates_str at ass,\n\n      have indu :\n        \u2200 v : list (symbol T (cfg_symbol_star a).nt),\n          CF_derives (cfg_symbol_star a) [symbol.nonterminal (cfg_symbol_star a).initial] v \u2192\n            symbol.terminal t \u2209 v,\n      {\n        intros v hyp,\n        induction hyp with x y trash orig ih,\n        {\n          clear_except,\n          rw list.mem_singleton,\n          apply symbol.no_confusion,\n        },\n        rcases orig with \u27e8r, rin, p, q, bef, aft\u27e9,\n        rw aft,\n        rw bef at ih,\n        repeat {\n          rw list.mem_append at *,\n        },\n        push_neg,\n        push_neg at ih,\n        split, swap,\n        {\n          exact ih.right,\n        },\n        split,\n        {\n          exact ih.left.left,\n        },\n        cases rin,\n        {\n          rw rin,\n          dsimp only,\n          intro imposs,\n          cases imposs,\n          {\n            apply nq,\n            exact symbol.terminal.inj imposs,\n          },\n          cases imposs,\n          {\n            norm_cast at imposs,\n          },\n          exact list.not_mem_nil (@symbol.terminal T (cfg_symbol_star a).nt t) imposs,\n        },\n        {\n          change r \u2208 [((0 : fin 1), ([] : list (symbol T (cfg_symbol_star a).nt)))] at rin,\n          rw list.mem_singleton at rin,\n          rw rin,\n          exact list.not_mem_nil (symbol.terminal t),\n        }\n      },\n      specialize indu (list.map symbol.terminal w) ass,\n\n      by_contradiction contra,\n      exact indu (list.mem_map_of_mem symbol.terminal contra),\n    },\n\n    exact implication2 \u2218 implication1,\n  },\n  {\n    intros w hw,\n    cases hw with n hwn,\n    rw hwn,\n    convert_to CF_generates_str (cfg_symbol_star a) (list.map symbol.terminal (list.repeat a n)),\n    unfold CF_generates_str,\n    clear hwn w,\n    have comes_to :\n      CF_derives\n        (cfg_symbol_star a)\n        [symbol.nonterminal (cfg_symbol_star a).initial]\n        (list.repeat (symbol.terminal a) n ++ [symbol.nonterminal (0 : fin 1)]),\n    {\n      induction n with n ih,\n      {\n        apply CF_deri_self,\n      },\n      apply CF_deri_of_deri_tran ih,\n      use ((0 : fin 1), [symbol.terminal a, symbol.nonterminal (0 : fin 1)]),\n      split,\n      {\n        apply list.mem_cons_self,\n      },\n      use [list.repeat (symbol.terminal a) n, []],\n      split,\n      {\n        rw list.append_nil,\n      },\n      rw list.append_nil,\n      change\n        symbol.terminal a :: (list.repeat (symbol.terminal a) n ++ [symbol.nonterminal (0 : fin 1)]) =\n        list.repeat (symbol.terminal a) n ++ ([symbol.terminal a] ++ [symbol.nonterminal 0]),\n      rw \u2190list.cons_append,\n      trim,\n      have count_succ_left :\n        @symbol.terminal T (fin 1) a :: list.repeat (symbol.terminal a) n =\n        list.repeat (symbol.terminal a) (n + 1),\n      {\n        symmetry,\n        apply list.repeat_succ,\n      },\n      have count_succ_right :\n        list.repeat (symbol.terminal a) n ++ [symbol.terminal a] =\n        list.repeat (symbol.terminal a) (n + 1),\n      {\n        change\n          list.repeat (symbol.terminal a) n ++ list.repeat (symbol.terminal a) 1 =\n          list.repeat (symbol.terminal a) (n + 1),\n        symmetry,\n        apply list.repeat_add,\n      },\n      rw count_succ_left,\n      rw count_succ_right,\n    },\n    apply CF_deri_of_deri_tran comes_to,\n    use ((0 : fin 1), []),\n    split,\n    {\n      apply list.mem_cons_of_mem,\n      apply list.mem_cons_self,\n    },\n    use [list.repeat (symbol.terminal a) n, []],\n    split;\n    simp,\n  }\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/context_free/basics/elementary.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.39659370398125704}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Simon Hudon\n-/\nimport category_theory.monoidal.braided\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.terminal\n\n/-!\n# The natural monoidal structure on any category with finite (co)products.\n\nA category with a monoidal structure provided in this way\nis sometimes called a (co)cartesian category,\nalthough this is also sometimes used to mean a finitely complete category.\n(See <https://ncatlab.org/nlab/show/cartesian+category>.)\n\nAs this works with either products or coproducts,\nand sometimes we want to think of a different monoidal structure entirely,\nwe don't set up either construct as an instance.\n\n## Implementation\nWe had previously chosen to rely on `has_terminal` and `has_binary_products` instead of\n`has_finite_products`, because we were later relying on the definitional form of the tensor product.\nNow that `has_limit` has been refactored to be a `Prop`,\nthis issue is irrelevant and we could simplify the construction here.\n\nSee `category_theory.monoidal.of_chosen_finite_products` for a variant of this construction\nwhich allows specifying a particular choice of terminal object and binary products.\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C] {X Y : C}\n\nopen category_theory.limits\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\n/-- A category with a terminal object and binary products has a natural monoidal structure. -/\ndef monoidal_of_has_finite_products [has_terminal C] [has_binary_products C] :\n  monoidal_category C :=\n{ tensor_unit  := \u22a4_ C,\n  tensor_obj   := \u03bb X Y, X \u2a2f Y,\n  tensor_hom   := \u03bb _ _ _ _ f g, limits.prod.map f g,\n  associator   := prod.associator,\n  left_unitor  := \u03bb P, prod.left_unitor P,\n  right_unitor := \u03bb P, prod.right_unitor P,\n  pentagon'    := prod.pentagon,\n  triangle'    := prod.triangle,\n  associator_naturality' := @prod.associator_naturality _ _ _, }\nend\n\nsection\nlocal attribute [instance] monoidal_of_has_finite_products\n\nopen monoidal_category\n\n/--\nThe monoidal structure coming from finite products is symmetric.\n-/\n@[simps]\ndef symmetric_of_has_finite_products [has_terminal C] [has_binary_products C] :\n  symmetric_category C :=\n{ braiding := \u03bb X Y, limits.prod.braiding X Y,\n  braiding_naturality' := \u03bb X X' Y Y' f g,\n    by { dsimp [tensor_hom], simp, },\n  hexagon_forward' := \u03bb X Y Z,\n    by { dsimp [monoidal_of_has_finite_products], simp },\n  hexagon_reverse' := \u03bb X Y Z,\n    by { dsimp [monoidal_of_has_finite_products], simp },\n  symmetry' := \u03bb X Y, by { dsimp, simp, refl, }, }\n\nend\n\nnamespace monoidal_of_has_finite_products\n\nvariables [has_terminal C] [has_binary_products C]\nlocal attribute [instance] monoidal_of_has_finite_products\n\n@[simp]\nlemma tensor_obj (X Y : C) : X \u2297 Y = (X \u2a2f Y) := rfl\n@[simp]\nlemma tensor_hom {W X Y Z : C} (f : W \u27f6 X) (g : Y \u27f6 Z) : f \u2297 g = limits.prod.map f g := rfl\n\n@[simp]\nlemma left_unitor_hom (X : C) : (\u03bb_ X).hom = limits.prod.snd := rfl\n@[simp]\nlemma left_unitor_inv (X : C) : (\u03bb_ X).inv = prod.lift (terminal.from X) (\ud835\udfd9 _) := rfl\n@[simp]\nlemma right_unitor_hom (X : C) : (\u03c1_ X).hom = limits.prod.fst := rfl\n@[simp]\nlemma right_unitor_inv (X : C) : (\u03c1_ X).inv = prod.lift (\ud835\udfd9 _) (terminal.from X) := rfl\n-- We don't mark this as a simp lemma, even though in many particular\n-- categories the right hand side will simplify significantly further.\n-- For now, we'll plan to create specialised simp lemmas in each particular category.\nlemma associator_hom (X Y Z : C) :\n  (\u03b1_ X Y Z).hom =\n  prod.lift\n    (limits.prod.fst \u226b limits.prod.fst)\n    (prod.lift (limits.prod.fst \u226b limits.prod.snd) limits.prod.snd) := rfl\n\nend monoidal_of_has_finite_products\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\n/-- A category with an initial object and binary coproducts has a natural monoidal structure. -/\ndef monoidal_of_has_finite_coproducts [has_initial C] [has_binary_coproducts C] :\n  monoidal_category C :=\n{ tensor_unit  := \u22a5_ C,\n  tensor_obj   := \u03bb X Y, X \u2a3f Y,\n  tensor_hom   := \u03bb _ _ _ _ f g, limits.coprod.map f g,\n  associator   := coprod.associator,\n  left_unitor  := coprod.left_unitor,\n  right_unitor := coprod.right_unitor,\n  pentagon'    := coprod.pentagon,\n  triangle'    := coprod.triangle,\n  associator_naturality' := @coprod.associator_naturality _ _ _, }\nend\n\n\nsection\nlocal attribute [instance] monoidal_of_has_finite_coproducts\n\nopen monoidal_category\n\n/--\nThe monoidal structure coming from finite coproducts is symmetric.\n-/\n@[simps]\ndef symmetric_of_has_finite_coproducts [has_initial C] [has_binary_coproducts C] :\n  symmetric_category C :=\n{ braiding := limits.coprod.braiding,\n  braiding_naturality' := \u03bb X X' Y Y' f g,\n    by { dsimp [tensor_hom], simp, },\n  hexagon_forward' := \u03bb X Y Z,\n    by { dsimp [monoidal_of_has_finite_coproducts], simp },\n  hexagon_reverse' := \u03bb X Y Z,\n    by { dsimp [monoidal_of_has_finite_coproducts], simp },\n  symmetry' := \u03bb X Y, by { dsimp, simp, refl, }, }\n\nend\n\nnamespace monoidal_of_has_finite_coproducts\n\nvariables [has_initial C] [has_binary_coproducts C]\nlocal attribute [instance] monoidal_of_has_finite_coproducts\n\n@[simp]\nlemma tensor_obj (X Y : C) : X \u2297 Y = (X \u2a3f Y) := rfl\n@[simp]\nlemma tensor_hom {W X Y Z : C} (f : W \u27f6 X) (g : Y \u27f6 Z) : f \u2297 g = limits.coprod.map f g := rfl\n\n@[simp]\nlemma left_unitor_hom (X : C) : (\u03bb_ X).hom = coprod.desc (initial.to X) (\ud835\udfd9 _) := rfl\n@[simp]\nlemma right_unitor_hom (X : C) : (\u03c1_ X).hom = coprod.desc (\ud835\udfd9 _) (initial.to X) := rfl\n@[simp]\nlemma left_unitor_inv (X : C) : (\u03bb_ X).inv = limits.coprod.inr := rfl\n@[simp]\nlemma right_unitor_inv (X : C) : (\u03c1_ X).inv = limits.coprod.inl := rfl\n-- We don't mark this as a simp lemma, even though in many particular\n-- categories the right hand side will simplify significantly further.\n-- For now, we'll plan to create specialised simp lemmas in each particular category.\nlemma associator_hom (X Y Z : C) :\n  (\u03b1_ X Y Z).hom =\n  coprod.desc\n    (coprod.desc coprod.inl (coprod.inl \u226b coprod.inr))\n    (coprod.inr \u226b coprod.inr) := rfl\n\nend monoidal_of_has_finite_coproducts\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/of_has_finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3965936967940687}}
{"text": "import Std.Tactic.Lint\nopen Std.Tactic.Lint\nset_option linter.missingDocs false\n\ndef f : Nat := 0\ndef g : Nat := 0\ndef h : Nat := 0\n@[simp] theorem fg : f = g := rfl\n@[simp] theorem fh : f = h := rfl\n#eval do guard (\u2190 [``fg, ``fh].anyM fun n => return (\u2190 simpNF.test n).isSome)\n\n@[simp] theorem and_comm : a \u2227 b \u2194 b \u2227 a := And.comm\n#eval do guard (\u2190 simpComm.test ``and_comm).isSome\n\n@[simp] theorem Prod.mk_fst : (a, b).1 = id a := rfl\n#eval do guard (\u2190 simpVarHead.test ``Prod.mk_fst).isSome\n\ndef SemiconjBy [Mul M] (a x y : M) : Prop :=\n  a * x = y * a\n\nstructure MulOpposite (\u03b1 : Type u) : Type u where\n  op :: unop : \u03b1\n\npostfix:max \"\u1d50\u1d52\u1d56\" => MulOpposite\n\nnamespace MulOpposite\n\ninstance [Mul \u03b1] : Mul \u03b1\u1d50\u1d52\u1d56 where mul x y := op (unop y * unop x)\n\n@[simp]\ntheorem unop_inj {x y : \u03b1\u1d50\u1d52\u1d56} : unop x = unop y \u2194 x = y := by\n  cases x; cases y; simp\n\n@[simp]\ntheorem semiconj_by_unop [Mul \u03b1] {a x y : \u03b1\u1d50\u1d52\u1d56} :\n    SemiconjBy (unop a) (unop y) (unop x) \u2194 SemiconjBy a x y := sorry\n\n#eval do guard (\u2190 simpComm.test ``unop_inj).isNone\n#eval do guard (\u2190 simpComm.test ``semiconj_by_unop).isNone\n\nend MulOpposite\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/test/lintsimp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.39659368960688024}}
{"text": "lemma contrapositive2 (P Q : Prop) : (\u00ac Q \u2192 \u00ac P) \u2192 (P \u2192 Q) :=\nbegin\n    \nend\n", "meta": {"author": "Samyak-Surti", "repo": "LeanCode", "sha": "944eac3e5f43e2614ed246083b97fbdf24181d83", "save_path": "github-repos/lean/Samyak-Surti-LeanCode", "path": "github-repos/lean/Samyak-Surti-LeanCode/LeanCode-944eac3e5f43e2614ed246083b97fbdf24181d83/nat_num_game/src/Advanced_Proposition_World.lean/adv_prop_wrld10.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283035, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.39659348885899676}}
{"text": "example (P Q : Prop) (HP : P) (HPQ : P \u2192 Q) : Q := HPQ HP -- HPQ is a function!\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/lean_together/5_minutes_on_type_theory/6_implies_is_a_function.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.39659348019325036}}
{"text": "/-\nCopyright (c) 2019 Paul-Nicolas Madelaine. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Paul-Nicolas Madelaine, Robert Y. Lewis, Mario Carneiro, Gabriel Ebner\n-/\n\nimport Std.Tactic.NormCast.Ext\nimport Std.Tactic.CoeExt\nimport Mathlib.Tactic.SudoSetOption\nimport Mathlib.Lean.Meta.Simp\nimport Mathlib.Algebra.Group.Defs\nimport Mathlib.Data.Nat.Cast.Defs\n\nopen Lean Meta Simp\nopen Std.Tactic.NormCast\nopen Std.Tactic.Coe\n\nnamespace Tactic.NormCast\n\ninitialize registerTraceClass `Tactic.norm_cast\n\n/-- Prove `a = b` using the given simp set. -/\ndef proveEqUsing (s : SimpTheorems) (a b : Expr) : MetaM (Option Simp.Result) := do\n  let go : SimpM (Option Simp.Result) := do\n    let methods := Simp.DefaultMethods.methods\n    let a' \u2190 Simp.simp a methods\n    let b' \u2190 Simp.simp b methods\n    unless \u2190 isDefEq a'.expr b'.expr do return none\n    mkEqTrans a' (\u2190 mkEqSymm b b')\n  withReducible do\n    (go { simpTheorems := #[s], congrTheorems := \u2190 Meta.getSimpCongrTheorems }).run' {}\n\n/-- Prove `a = b` by simplifying using move and squash lemmas. -/\ndef proveEqUsingDown (a b : Expr) : MetaM (Option Simp.Result) := do\n  withTraceNode `Tactic.norm_cast (return m!\"{exceptOptionEmoji \u00b7} proving: {\u2190 mkEq a b}\") do\n  proveEqUsing (\u2190 normCastExt.down.getTheorems) a b\n\ndef mkCoe (e : Expr) (ty : Expr) : MetaM Expr := do\n  let .some e' \u2190 coerce? e ty | failure\n  return e'\n\ndef isCoeOf? (e : Expr) : MetaM (Option Expr) := do\n  if let Expr.const fn .. := e.getAppFn then\n    if let some info \u2190 getCoeFnInfo? fn then\n      if e.getAppNumArgs == info.numArgs then\n        return e.getArg! info.coercee\n  return none\n\ndef isNumeral? (e : Expr) : Option (Expr \u00d7 Nat) :=\n  if e.isConstOf ``Nat.zero then\n    (mkConst ``Nat, 0)\n  else if let Expr.app (Expr.app (Expr.app (Expr.const ``OfNat.ofNat ..) \u03b1 ..)\n      (Expr.lit (Literal.natVal n) ..) ..) .. := e then\n    some (\u03b1, n)\n  else\n    none\n\n/--\nThis is the main heuristic used alongside the elim and move lemmas.\nThe goal is to help casts move past operators by adding intermediate casts.\nAn expression of the shape: op (\u2191(x : \u03b1) : \u03b3) (\u2191(y : \u03b2) : \u03b3)\nis rewritten to:            op (\u2191(\u2191(x : \u03b1) : \u03b2) : \u03b3) (\u2191(y : \u03b2) : \u03b3)\nwhen (\u2191(\u2191(x : \u03b1) : \u03b2) : \u03b3) = (\u2191(x : \u03b1) : \u03b3) can be proven with a squash lemma\n-/\ndef splittingProcedure (expr : Expr) : MetaM Simp.Result := do\n  let Expr.app (Expr.app op x ..) y .. := expr | return {expr}\n\n  let Expr.forallE _ \u03b3 (Expr.forallE _ \u03b3' ty ..) .. \u2190 inferType op | return {expr}\n  if \u03b3'.hasLooseBVars || ty.hasLooseBVars then return {expr}\n  unless \u2190 isDefEq \u03b3 \u03b3' do return {expr}\n\n  let msg := m!\"splitting {expr}\"\n  let msg\n    | .error _ => return m!\"{bombEmoji} {msg}\"\n    | .ok r => return if r.expr == expr then m!\"{crossEmoji} {msg}\" else\n      m!\"{checkEmoji} {msg} to {r.expr}\"\n  withTraceNode `Tactic.norm_cast msg do\n\n  try\n    let some x' \u2190 isCoeOf? x | failure\n    let some y' \u2190 isCoeOf? y | failure\n    let \u03b1 \u2190 inferType x'\n    let \u03b2 \u2190 inferType y'\n\n    -- TODO: fast timeout\n    (try\n      let x2 \u2190 mkCoe (\u2190 mkCoe x' \u03b2) \u03b3\n      let some x_x2 \u2190 proveEqUsingDown x x2 | failure\n      Simp.mkCongrFun (\u2190 Simp.mkCongr {expr := op} x_x2) y\n    catch _ =>\n      let y2 \u2190 mkCoe (\u2190 mkCoe y' \u03b1) \u03b3\n      let some y_y2 \u2190 proveEqUsingDown y y2 | failure\n      Simp.mkCongr {expr := mkApp op x} y_y2)\n  catch _ => try\n    let some (_, n) := isNumeral? y | failure\n    let some x' \u2190 isCoeOf? x | failure\n    let \u03b1 \u2190 inferType x'\n    let y2 \u2190 mkCoe (\u2190 mkNumeral \u03b1 n) \u03b3\n    let some y_y2 \u2190 proveEqUsingDown y y2 | failure\n    Simp.mkCongr {expr := mkApp op x} y_y2\n  catch _ => try\n    let some (_, n) := isNumeral? x | failure\n    let some y' \u2190 isCoeOf? y | failure\n    let \u03b2 \u2190 inferType y'\n    let x2 \u2190 mkCoe (\u2190 mkNumeral \u03b2 n) \u03b3\n    let some x_x2 \u2190 proveEqUsingDown x x2 | failure\n    Simp.mkCongrFun (\u2190 Simp.mkCongr {expr := op} x_x2) y\n  catch _ =>\n    return {expr}\n\n/--\nDischarging function used during simplification in the \"squash\" step.\n\nTODO: normCast takes a list of expressions to use as lemmas for the discharger\nTODO: a tactic to print the results the discharger fails to proove\n-/\ndef prove (e : Expr) : SimpM (Option Expr) := do\n  withTraceNode `Tactic.norm_cast (return m!\"{exceptOptionEmoji \u00b7} discharging: {e}\") do\n  return (\u2190 findLocalDeclWithType? e).map mkFVar\n\n/--\nCore rewriting function used in the \"squash\" step, which moves casts upwards\nand eliminates them.\n\nIt tries to rewrite an expression using the elim and move lemmas.\nOn failure, it calls the splitting procedure heuristic.\n-/\npartial def upwardAndElim (up : SimpTheorems) (e : Expr) : SimpM Simp.Step := do\n  let r \u2190 Simp.rewrite? e up.post up.erased prove (tag := \"squash\") (rflOnly := false)\n  let r := r.getD { expr := e }\n  let r \u2190 mkEqTrans r <|\u2190 splittingProcedure r.expr\n  if r.expr == e then return Simp.Step.done {expr := e}\n  return Simp.Step.visit r\n\n/--\nIf possible, rewrite `(n : \u03b1)` to `(Nat.cast n : \u03b1)` where `n` is a numeral and `\u03b1 \u2260 \u2115`.\nReturns a pair of the new expression and proof that they are equal.\n-/\ndef numeralToCoe (e : Expr) : MetaM Simp.Result := do\n  let some (\u03b1, n) := isNumeral? e | failure\n  if (\u2190 whnf \u03b1).isConstOf ``Nat then failure\n  let newE \u2190 mkAppOptM ``Nat.cast #[\u03b1, none, toExpr n]\n  let some pr \u2190 proveEqUsingDown e newE | failure\n  return pr\n\n/--\nThe core simplification routine of `normCast`.\n-/\ndef derive (e : Expr) : MetaM Simp.Result := do\n  withTraceNode `Tactic.norm_cast (fun _ => return m!\"{e}\") do\n  let e \u2190 instantiateMVars e\n\n  let config : Simp.Config := {\n    zeta := false\n    beta := false\n    eta  := false\n    proj := false\n    iota := false\n  }\n  let congrTheorems \u2190 Meta.getSimpCongrTheorems\n\n  let r := {expr := e}\n\n  let withTrace phase := withTraceNode `Tactic.norm_cast fun\n    | .ok r => return m!\"{r.expr} (after {phase})\"\n    | .error _ => return m!\"{bombEmoji} {phase}\"\n\n  -- step 1: pre-processing of numerals\n  let r \u2190 withTrace \"pre-processing numerals\" do\n    let post e := return Simp.Step.done (\u2190 try numeralToCoe e catch _ => pure {expr := e})\n    Simp.mkEqTrans r (\u2190 Simp.main r.expr { config, congrTheorems } (methods := { post })).1\n\n  -- step 2: casts are moved upwards and eliminated\n  let r \u2190 withTrace \"moving upward, splitting and eliminating\" do\n    let post := upwardAndElim (\u2190 normCastExt.up.getTheorems)\n    Simp.mkEqTrans r (\u2190 Simp.main r.expr { config, congrTheorems } (methods := { post })).1\n\n  -- step 3: casts are squashed\n  let r \u2190 withTrace \"squashing\" do\n    let simpTheorems := #[\u2190 normCastExt.squash.getTheorems]\n    mkEqTrans r (\u2190 simp r.expr { simpTheorems, config, congrTheorems }).1\n\n  return r\n\nopen Elab.Term in\nelab \"mod_cast \" e:term : term <= expectedType => do\n  if (\u2190 instantiateMVars expectedType).hasExprMVar then tryPostpone\n  let expectedType' \u2190 derive expectedType\n  let e \u2190 elabTerm e expectedType'.expr\n  synthesizeSyntheticMVars\n  let eTy \u2190 instantiateMVars (\u2190 inferType e)\n  if eTy.hasExprMVar then tryPostpone\n  let eTy' \u2190 derive eTy\n  unless \u2190 isDefEq eTy'.expr expectedType'.expr do\n    throwTypeMismatchError \"mod_cast\" expectedType'.expr eTy'.expr e\n  let eTy_eq_expectedType \u2190 mkEqTrans eTy' (\u2190 mkEqSymm expectedType expectedType')\n  mkCast eTy_eq_expectedType e\n\nopen Tactic Parser.Tactic Elab.Tactic\n\ndef normCastTarget : TacticM Unit :=\n  liftMetaTactic1 fun goal \u21a6 do\n    let tgt \u2190 instantiateMVars (\u2190 goal.getType)\n    let prf \u2190 derive tgt\n    applySimpResultToTarget goal tgt prf\n\ndef normCastHyp (fvarId : FVarId) : TacticM Unit :=\n  liftMetaTactic1 fun goal \u21a6 do\n    let hyp \u2190 instantiateMVars (\u2190 fvarId.getDecl).type\n    let prf \u2190 derive hyp\n    return (\u2190 applySimpResultToLocalDecl goal fvarId prf false).map (\u00b7.snd)\n\nelab \"norm_cast0\" loc:((ppSpace location)?) : tactic =>\n  withMainContext do\n    match expandOptLocation loc with\n    | Location.targets hyps target =>\n      if target then normCastTarget\n      (\u2190 getFVarIds hyps).forM normCastHyp\n    | Location.wildcard =>\n      normCastTarget\n      (\u2190 (\u2190 getMainGoal).getNondepPropHyps).forM normCastHyp\n\n/-- `assumption_mod_cast` runs `norm_cast` on the goal. For each local hypothesis `h`, it also\nnormalizes `h` and tries to use that to close the goal. -/\nmacro \"assumption_mod_cast\" : tactic => `(tactic| norm_cast0 at * <;> assumption)\n\n/--\nNormalize casts at the given locations by moving them \"upwards\".\n-/\nmacro \"norm_cast\" loc:(ppSpace location)? : tactic =>\n  `(tactic| norm_cast0 $[$loc]? <;> try trivial)\n\n/--\nRewrite with the given rules and normalize casts between steps.\n-/\nsyntax \"rw_mod_cast\" (config)? rwRuleSeq (ppSpace location)? : tactic\nmacro_rules\n  | `(tactic| rw_mod_cast $[$config]? [$rules,*] $[$loc]?) => do\n    let tacs \u2190 rules.getElems.mapM fun rule \u21a6\n      `(tactic| (norm_cast at *; rw $[$config]? [$rule] $[$loc]?))\n    `(tactic| ($[$tacs]*))\n\n/--\nNormalize the goal and the given expression, then close the goal with exact.\n-/\nmacro \"exact_mod_cast \" e:term : tactic => `(tactic| exact mod_cast ($e : _))\n\n/--\nNormalize the goal and the given expression, then apply the expression to the goal.\n-/\nmacro \"apply_mod_cast \" e:term : tactic => `(tactic| apply mod_cast ($e : _))\n\nsyntax (name := convNormCast) \"norm_cast\" : conv\n@[tactic convNormCast] def evalConvNormCast : Tactic :=\n  open Elab.Tactic.Conv in fun _ \u21a6 withMainContext do\n    applySimpResult (\u2190 derive (\u2190 getLhs))\n\nsyntax (name := pushCast) \"push_cast \" (config)? (discharger)? (&\"only \")?\n  (\"[\" (simpStar <|> simpErase <|> simpLemma),* \"]\")? (location)? : tactic\n@[tactic pushCast] def evalPushCast : Tactic := fun stx \u21a6 do\n  let { ctx, dischargeWrapper, .. } \u2190 withMainContext do\n    mkSimpContext' (\u2190 pushCastExt.getTheorems) stx (eraseLocal := false)\n  dischargeWrapper.with fun discharge? \u21a6\n    discard <| simpLocation ctx discharge? (expandOptLocation stx[5])\n\n-- add_hint_tactic \"norm_cast at *\"\n\n/-\nThe `norm_cast` family of tactics is used to normalize casts inside expressions.\nIt is basically a simp tactic with a specific set of lemmas to move casts\nupwards in the expression.\nTherefore it can be used more safely as a non-terminating tactic.\nIt also has special handling of numerals.\n\nFor instance, given an assumption\n```lean\na b : \u2124\nh : \u2191a + \u2191b < (10 : \u211a)\n```\n\nwriting `norm_cast at h` will turn `h` into\n```lean\nh : a + b < 10\n```\n\nYou can also use `exact_mod_cast`, `apply_mod_cast`, `rw_mod_cast`\nor `assumption_mod_cast`.\nWriting `exact_mod_cast h` and `apply_mod_cast h` will normalize the goal and\n`h` before using `exact h` or `apply h`.\nWriting `assumption_mod_cast` will normalize the goal and for every\nexpression `h` in the context it will try to normalize `h` and use\n`exact h`.\n`rw_mod_cast` acts like the `rw` tactic but it applies `norm_cast` between steps.\n\n`push_cast` rewrites the expression to move casts toward the leaf nodes.\nThis uses `norm_cast` lemmas in the forward direction.\nFor example, `\u2191(a + b)` will be written to `\u2191a + \u2191b`.\nIt is equivalent to `simp only with push_cast`.\nIt can also be used at hypotheses with `push_cast at h`\nand with extra simp lemmas with `push_cast [int.add_zero]`.\n\n```lean\nexample (a b : \u2115) (h1 : ((a + b : \u2115) : \u2124) = 10) (h2 : ((a + b + 0 : \u2115) : \u2124) = 10) :\n  ((a + b : \u2115) : \u2124) = 10 :=\nbegin\n  push_cast,\n  push_cast at h1,\n  push_cast [int.add_zero] at h2,\nend\n```\n\nThe implementation and behavior of the `norm_cast` family is described in detail at\n<https://lean-forward.github.io/norm_cast/norm_cast.pdf>.\n-/\n-- add_tactic_doc\n-- { name := \"norm_cast\",\n--   category   := doc_category.tactic,\n--   decl_names := [``tactic.interactive.norm_cast, ``tactic.interactive.rw_mod_cast,\n--                  ``tactic.interactive.apply_mod_cast, ``tactic.interactive.assumption_mod_cast,\n--                  ``tactic.interactive.exact_mod_cast, ``tactic.interactive.push_cast],\n--   tags       := [\"coercions\", \"simplification\"] }\n-- TODO\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/NormCast/Tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.39659347586037713}}
{"text": "/- Symbol :\n\u2203 : \\exists\n\u2200 : \\forall\n\u2208 : \\in\n\u2209 : \\notin\n\u222a : \\cup\n\u2229 : \\cap\n\u2205 : \\emptyset\n\u2228 : \\or\n\u2227 : \\and\n\u00ac : \\not\n\u2192 : \\rightarrow\n\u2194 : \\iff\n-/\n\n/- Tactic : refl\n\n## Summary\n\n`refl` is a tactic which proves goals of the form `A = A`.\n\n## Details\n\nThe `refl` tactic will close any goal of the form `A = B`\nwhere `A` and `B` are *exactly the same thing*.\n\n### Example:\nIf it looks like this in the top right hand box:\n```\nA : Point\n\u22a2 A = A\n```\n\nthen\n\n`refl,`\n\nwill close the goal and solve the level. Don't forget the comma.\n\n-/\n\n/-\n# Tutorial World\n\n## The setup\n\nWelcome to the Tutorial World! In this world, you're going to prove some geometric facts by using **`tactics`**. \nThese `tactics` are just instructions that make progress in a mathematical proof.\nDuring your proofs, your \"goal\" (i.e. what you're\nsupposed to be proving) will be displayed in front of a `\u22a2` symbol on the top\nright hand box, so you will need to use `tactics` to close that goals. Once you close all the goals, the top\nright hand box will report \"Proof complete!\", so that you \ncan move on to the next level in the world you're in. \n\n## The language\n\nThe vast majority of mathematical fields are built up from **Set theory**, which is a branch of mathematical logic\nthat studies sets. In set theory, everything is a set. Even a point can be thought as a set. These sets can contain\nelements, which are mathematical objects of any kind: numbers, points, lines, or even other sets. The set with no element \nis the empty set. The problem with set theory, however, is that it does not consider mathematical proofs as sets or elements.\nAs a consequence, it makes the translation of mathematical propositions into programming languages more difficult.\n\nTo avoid this problem, the majority of proof assistants, such as Lean or Coq, use **Type Theory**.\nIn Type theory, there are `terms` and `types`. A term and its type are written together as `term` : `type`, \nwhere the symbol : can be thought as \"is an element of\" (this is an analogy with set theory). \nThen, if we find the expression \u03a9 : Type , we should understand that \u03a9 is a term of the type \"Type\". \nAnalogously, A : \u03a9  will translate into \"A is a term of the type \u03a9\", which makes us think that terms can also be types.\nIn this case, because we are talking about geometry, \u03a9 must be interpreted as a plane, where A is a point that lies on \nthat plane. In addition to all this, there exists `Prop`, which refers to propositions. Each proposition (when interpreted as a type) \nis either empty or has exactly one `term`. In this way, it can be used to introduce mathematical proofs. If we found h : A = B, \nthat should be read as \"h is a proof of the fact that `A = B`\". In Lean, the computer does not care about the content of a proof, \nbut if a proposition is either proved or not. This fact is known as **proof irrelevance**, and makes reasoning about dependent types easier\nfor computers. If you want to learn more about this, we encourage you to look for the **Curry\u2013Howard correspondence** and **Homotopy Type Theory**. \n\n## The symbols\n\nIn case you are not familiar with the symbology of mathematics, you will find a **Vocabulary Section** at the top left corner of\nthe screen. Next to each symbol, there is the instruction that you neeed to type in case you want it to appear on the screen.\nSooner or later, you will come across these symbols during the game, so try to save a space in your brain for them. They will be important!\n\n## Level 1: the `refl` tactic.\n\nOnce we've mastered the language...Let's learn some tactics! Let's start with the `refl` tactic. `refl` \nstands for \"reflexivity\", which is a fancy way of saying that it will prove any goal of the form `A = A`.\nIt doesn't matter how complicated `A` is, all that matters is that the left hand side is *exactly equal* \nto the right hand side. \n\nEach level in this game involves proving a theorem or a lemma (a lemma is just a baby theorem).\nAt the bottom of the text in this box, there's a lemma, which says that if $A$ is a point, then $A$ = $A$. \nLocate this lemma (if you can't see the lemma and these instructions at the same time, make this box wider\nby dragging the sides). Let's supply the proof. Click on the word `sorry` and then delete it.\nWhen the system finishes being busy, you'll be able to see your goal in the box on the top right. \n[If your system never finishes being busy, then your computer is not running the javascript \nLean which powers everything behind the scenes. Try Chrome? Try not using private browsing?] \n\nThis first level shows a pretty easy goal to prove -- you can just prove it with the `refl` tactic.\nWhere it used to say `sorry`, write `refl,`\n**and don't forget the comma**. Then **hit enter** to go onto the next line.\nIf all is well, Lean should tell you \"Proof complete!\" in the top right box, and there\nshould be no errors in the bottom right box. You just did the first\nlevel of the tutorial! And you also learnt how to avoid by *far* the most\ncommon mistake that beginner users make -- **every line must end with a comma**. \nAt the end, the comma is important because it tells Lean you are done with one step of your proof!\n\nFor each level, the idea is to get Lean into this state: with the top right\nbox saying \"Proof complete!\" and the bottom right box empty (i.e. with no errors in).\nIf you want to be reminded about the `refl` tactic, you can click on the \"Tactics\" drop\ndown menu on the left. Resize the window if it's too small. \nNow click on \"Next Level\" in the top right of your browser to go onto the second level of\nTutorial World, where we'll learn about the `rw` tactic.\n-/\n\nvariables {\u03a9 : Type} -- hide\n\n/- Lemma : no-side-bar\nIf A is a point, then A = A.\n-/\nlemma refl_example (A : \u03a9) : A = A :=\nbegin\n  refl,\nend\n", "meta": {"author": "luisscastillo", "repo": "lean-game", "sha": "44c454ca6f4dcb8de76401790cf3f4fa91738fc1", "save_path": "github-repos/lean/luisscastillo-lean-game", "path": "github-repos/lean/luisscastillo-lean-game/lean-game-44c454ca6f4dcb8de76401790cf3f4fa91738fc1/src/tutorial_world/level01_refl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.7520125848754472, "lm_q1q2_score": 0.39654866179420245}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport category_theory.over\nimport category_theory.adjunction.opposites\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.creates\nimport category_theory.limits.comma\n\n/-!\n# Limits and colimits in the over and under categories\n\nShow that the forgetful functor `forget X : over X \u2964 C` creates colimits, and hence `over X` has\nany colimits that `C` has (as well as the dual that `forget X : under X \u27f6 C` creates limits).\n\nNote that the folder `category_theory.limits.shapes.constructions.over` further shows that\n`forget X : over X \u2964 C` creates connected limits (so `over X` has connected limits), and that\n`over X` has `J`-indexed products if `C` has `J`-indexed wide pullbacks.\n\nTODO: If `C` has binary products, then `forget X : over X \u2964 C` has a right adjoint.\n-/\nnoncomputable theory\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nopen category_theory category_theory.limits\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{v} C]\nvariable {X : C}\n\nnamespace category_theory.over\n\ninstance has_colimit_of_has_colimit_comp_forget\n  (F : J \u2964 over X) [i : has_colimit (F \u22d9 forget X)] : has_colimit F :=\n@@costructured_arrow.has_colimit _ _ _ _ i _\n\ninstance [has_colimits_of_shape J C] : has_colimits_of_shape J (over X) := {}\ninstance [has_colimits C] : has_colimits (over X) := \u27e8infer_instance\u27e9\n\ninstance creates_colimits : creates_colimits (forget X) := costructured_arrow.creates_colimits\n\n-- We can automatically infer that the forgetful functor preserves and reflects colimits.\nexample [has_colimits C] : preserves_colimits (forget X) := infer_instance\nexample : reflects_colimits (forget X) := infer_instance\n\nlemma epi_left_of_epi [has_pushouts C] {f g : over X} (h : f \u27f6 g) [epi h] : epi h.left :=\ncostructured_arrow.epi_left_of_epi _\n\nlemma epi_iff_epi_left [has_pushouts C] {f g : over X} (h : f \u27f6 g) : epi h \u2194 epi h.left :=\ncostructured_arrow.epi_iff_epi_left _\n\nsection\nvariables [has_pullbacks C]\n\nopen tactic\n\n/-- When `C` has pullbacks, a morphism `f : X \u27f6 Y` induces a functor `over Y \u2964 over X`,\nby pulling back a morphism along `f`. -/\n@[simps]\ndef pullback {X Y : C} (f : X \u27f6 Y) : over Y \u2964 over X :=\n{ obj := \u03bb g, over.mk (pullback.snd : pullback g.hom f \u27f6 X),\n  map := \u03bb g h k,\n    over.hom_mk\n      (pullback.lift (pullback.fst \u226b k.left) pullback.snd (by simp [pullback.condition]))\n      (by tidy) }\n\n/-- `over.map f` is left adjoint to `over.pullback f`. -/\ndef map_pullback_adj {A B : C} (f : A \u27f6 B) :\n  over.map f \u22a3 pullback f :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb g h,\n  { to_fun := \u03bb X, over.hom_mk (pullback.lift X.left g.hom (over.w X)) (pullback.lift_snd _ _ _),\n    inv_fun := \u03bb Y,\n    begin\n      refine over.hom_mk _ _,\n      refine Y.left \u226b pullback.fst,\n      dsimp,\n      rw [\u2190 over.w Y, category.assoc, pullback.condition, category.assoc], refl,\n    end,\n    left_inv := \u03bb X, by { ext, dsimp, simp, },\n    right_inv := \u03bb Y, begin\n      ext, dsimp,\n      simp only [pullback.lift_fst],\n      dsimp,\n      rw [pullback.lift_snd, \u2190 over.w Y],\n      refl,\n    end } }\n\n/-- pullback (\ud835\udfd9 A) : over A \u2964 over A is the identity functor. -/\ndef pullback_id {A : C} : pullback (\ud835\udfd9 A) \u2245 \ud835\udfed _ :=\nadjunction.right_adjoint_uniq\n  (map_pullback_adj _)\n  (adjunction.id.of_nat_iso_left over.map_id.symm)\n\n/-- pullback commutes with composition (up to natural isomorphism). -/\ndef pullback_comp {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  pullback (f \u226b g) \u2245 pullback g \u22d9 pullback f :=\nadjunction.right_adjoint_uniq\n  (map_pullback_adj _)\n  (((map_pullback_adj _).comp (map_pullback_adj _)).of_nat_iso_left\n    (over.map_comp _ _).symm)\n\ninstance pullback_is_right_adjoint {A B : C} (f : A \u27f6 B) :\n  is_right_adjoint (pullback f) :=\n\u27e8_, map_pullback_adj f\u27e9\n\nend\n\nend category_theory.over\n\nnamespace category_theory.under\n\ninstance has_limit_of_has_limit_comp_forget\n  (F : J \u2964 under X) [i : has_limit (F \u22d9 forget X)] : has_limit F :=\n@@structured_arrow.has_limit _ _ _ _ i _\n\ninstance [has_limits_of_shape J C] : has_limits_of_shape J (under X) := {}\ninstance [has_limits C] : has_limits (under X) := \u27e8infer_instance\u27e9\n\nlemma mono_right_of_mono [has_pullbacks C] {f g : under X} (h : f \u27f6 g) [mono h] : mono h.right :=\nstructured_arrow.mono_right_of_mono _\n\n\n\ninstance creates_limits : creates_limits (forget X) := structured_arrow.creates_limits\n\n-- We can automatically infer that the forgetful functor preserves and reflects limits.\nexample [has_limits C] : preserves_limits (forget X) := infer_instance\nexample : reflects_limits (forget X) := infer_instance\n\nsection\nvariables [has_pushouts C]\n\n/-- When `C` has pushouts, a morphism `f : X \u27f6 Y` induces a functor `under X \u2964 under Y`,\nby pushing a morphism forward along `f`. -/\n@[simps]\ndef pushout {X Y : C} (f : X \u27f6 Y) : under X \u2964 under Y :=\n{ obj := \u03bb g, under.mk (pushout.inr : Y \u27f6 pushout g.hom f),\n  map := \u03bb g h k,\n    under.hom_mk\n      (pushout.desc (k.right \u226b pushout.inl) pushout.inr (by { simp [\u2190pushout.condition], }))\n      (by tidy) }\n\nend\n\nend category_theory.under\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623218, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3965257708659417}}
{"text": "/-\nCopyright (c) 2020 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n\n! This file was ported from Lean 3 source module topology.tactic\n! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.AutoCases\nimport Mathbin.Tactic.Tidy\nimport Mathbin.Tactic.WithLocalReducibility\nimport Mathbin.Tactic.ShowTerm\nimport Mathbin.Topology.Basic\n\n/-!\n# Tactics for topology\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nCurrently we have one domain-specific tactic for topology: `continuity`.\n\n-/\n\n\n/-!\n### `continuity` tactic\n\nAutomatically solve goals of the form `continuous f`.\n\nMark lemmas with `@[continuity]` to add them to the set of lemmas\nused by `continuity`.\n-/\n\n\n/-- User attribute used to mark tactics used by `continuity`. -/\n@[user_attribute]\nunsafe def continuity : user_attribute\n    where\n  Name := `continuity\n  descr := \"lemmas usable to prove continuity\"\n#align continuity continuity\n\n-- Mark some continuity lemmas already defined in `topology.basic`\nattribute [continuity] continuous_id continuous_const\n\n#print continuous_id' /-\n-- As we will be using `apply_rules` with `md := semireducible`,\n-- we need another version of `continuous_id`.\n@[continuity]\ntheorem continuous_id' {\u03b1 : Type _} [TopologicalSpace \u03b1] : Continuous fun a : \u03b1 => a :=\n  continuous_id\n#align continuous_id' continuous_id'\n-/\n\nnamespace Tactic\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- Tactic to apply `continuous.comp` when appropriate.\n\nApplying `continuous.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove continuous is actually\n  constant, and that constant is a function application `f z`, then\n  continuous.comp would produce new goals `continuous f`, `continuous\n  (\u03bb _, z)`, which is silly. We avoid this by failing if we could\n  apply continuous_const.\n\n* continuous.comp will always succeed on `continuous (\u03bb x, f x)` and\n  produce new goals `continuous (\u03bb x, x)`, `continuous f`. We detect\n  this by failing if a new goal can be closed by applying\n  continuous_id.\n-/\nunsafe def apply_continuous.comp : tactic Unit :=\n  sorry\n#align tactic.apply_continuous.comp tactic.apply_continuous.comp\n\n/-- List of tactics used by `continuity` internally. -/\nunsafe def continuity_tactics (md : Transparency := reducible) : List (tactic String) :=\n  [intros1 >>= fun ns => pure (\"intros \" ++ \" \".intercalate (ns.map fun e => e.toString)),\n    apply_rules [] [`` continuity] 50 { md } >> pure \"apply_rules with continuity\",\n    apply_continuous.comp >> pure \"refine continuous.comp _ _\"]\n#align tactic.continuity_tactics tactic.continuity_tactics\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- Solve goals of the form `continuous f`. `continuity?` reports back the proof term it found.\n-/\nunsafe def continuity (bang : parse <| optional (tk \"!\")) (trace : parse <| optional (tk \"?\"))\n    (cfg : tidy.cfg := { }) : tactic Unit :=\n  let md := if bang.isSome then semireducible else reducible\n  let continuity_core := tactic.tidy { cfg with tactics := continuity_tactics md }\n  let trace_fn := if trace.isSome then show_term else id\n  trace_fn continuity_core\n#align tactic.interactive.continuity tactic.interactive.continuity\n\n/-- Version of `continuity` for use with auto_param. -/\nunsafe def continuity' : tactic Unit :=\n  continuity none none { }\n#align tactic.interactive.continuity' tactic.interactive.continuity'\n\n/-- `continuity` solves goals of the form `continuous f` by applying lemmas tagged with the\n`continuity` user attribute.\n\n```\nexample {X Y : Type*} [topological_space X] [topological_space Y]\n  (f\u2081 f\u2082 : X \u2192 Y) (hf\u2081 : continuous f\u2081) (hf\u2082 : continuous f\u2082)\n  (g : Y \u2192 \u211d) (hg : continuous g) : continuous (\u03bb x, (max (g (f\u2081 x)) (g (f\u2082 x))) + 1) :=\nby continuity\n```\nwill discharge the goal, generating a proof term like\n`((continuous.comp hg hf\u2081).max (continuous.comp hg hf\u2082)).add continuous_const`\n\nYou can also use `continuity!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`continuity?` reports back the proof term it found.\n-/\nadd_tactic_doc\n  { Name := \"continuity / continuity'\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.continuity, `tactic.interactive.continuity']\n    tags := [\"lemma application\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/Tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.39652577086594165}}
{"text": "/-\nCopyright (c) 2021 Kalle Kyt\u00f6l\u00e4. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kalle Kyt\u00f6l\u00e4\n-/\nimport tactic \nimport measure_theory.measurable_space\nimport measure_theory.integration\nimport measure_theory.bochner_integration\nimport measure_theory.borel_space\nimport topology.metric_space.basic\nimport topology.instances.real\nimport topology.instances.ennreal\nimport order.liminf_limsup\nimport portmanteau_limsup_lemmas\nimport portmanteau_definitions\nimport portmanteau_proba_lemmas\nimport analysis.seminorm\n\n\nnoncomputable theory\nopen set \nopen classical\nopen measure_theory\nopen measurable_space\nopen metric_space\nopen borel_space\nopen filter\nopen order\nopen_locale topological_space ennreal big_operators\n\n\nnamespace portmanteau\n\nsection portmanteau_integrals\n\n\nvariables {\u03b1 : Type} [topological_space \u03b1]\n\nabbreviation R_integrate (f : \u03b1 \u2192 \u211d) (\u03bc : borel_proba \u03b1) := @integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc f\nabbreviation R_abs_nn : \u211d \u2192 nnreal := \u03bb x , nnnorm x\nabbreviation R_abs_enn : \u211d \u2192 ennreal := \u03bb x , ennreal.of_nnreal_hom (nnnorm x) --ennreal.of_real(abs(x))\n\n\nlemma R_abs_enn_le_of_abs_le {x c : \u211d} (h : abs(x) \u2264 c) : R_abs_enn(x) \u2264 ennreal.of_real(c) :=\nbegin\n  have c_nn : 0 \u2264 c := le_trans (abs_nonneg x) h,\n  have eq := real.coe_to_nnreal c c_nn ,\n  set c' := c.to_nnreal with hc' ,\n  have h\u2081 : R_abs_nn(x) \u2264 c' := (real.le_to_nnreal_iff_coe_le c_nn).mpr h ,\n  exact (with_top.le_coe rfl).mpr h\u2081 ,\nend\n\n\n-- TODO: Should state for finite measures? \nlemma integrable_of_bdd_realval {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) [\u03bc_fin : probability_measure \u03bc]\n  (f : \u03b2 \u2192 \u211d) (f_bdd : bdd_Rval f) (f_mble : @ae_measurable \u03b2 \u211d m\u03b2 _ f \u03bc) :\n     integrable f \u03bc :=\nbegin\n  cases f_bdd with c hc ,\n  set f' := R_abs_enn \u2218 f with hf' ,\n  suffices : lintegral \u03bc f' < \u22a4 ,\n  { rw hf' at this ,\n    exact \u27e8 f_mble , this \u27e9 , } ,\n  have bdd' : f' \u2264 (\u03bb b , ennreal.of_real(c)) ,\n  { rw hf' ,\n    intros b ,\n    dsimp ,\n    specialize hc b ,\n    exact R_abs_enn_le_of_abs_le hc , } ,\n  have integr_bdd := lintegral_mono bdd' ,\n  set c' := ennreal.of_real(c) with hc' ,\n  have const_integr : lintegral \u03bc (\u03bb b , c') = c' * (\u03bc(univ)) ,\n  { rw \u2190 set_lintegral_const univ c' ,\n    simp , } ,\n  have total : c' * (\u03bc(univ)) < \u22a4 ,\n  { rw (proba_muniv \u03bc) ,\n    simp , } ,\n  rw \u2190 const_integr at total ,\n  exact lt_of_le_of_lt integr_bdd total , \nend\n\n\n-- TODO: Should state for finite measures? \nlemma bdd_integral_of_bdd_ennval {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) [\u03bc_fin : probability_measure \u03bc]\n  (f : \u03b2 \u2192 ennreal) (f_bdd : bdd_ennval f) :\n     lintegral \u03bc f < \u22a4 :=\nbegin\n  cases f_bdd with c hc ,\n  have f_le_c : f \u2264 \u03bb x , c := hc ,\n  have integr_f_le := @lintegral_mono \u03b2 m\u03b2 \u03bc _ _ f_le_c ,\n  simp [proba_muniv] at integr_f_le ,\n  exact lt_of_le_of_lt integr_f_le (@ennreal.coe_lt_top c) ,\nend\n\n\nlemma const_bdd_Rval {\u03b2 : Type*} {c : \u211d} :\n    bdd_Rval (\u03bb (x : \u03b2) , c) :=\nbegin\n  use abs(c) ,\n  intros x ,\n  refl ,\nend\n\n\nlemma const_bdd_ennval {\u03b2 : Type*} {c : nnreal} :\n    bdd_ennval (\u03bb (x : \u03b2) , c) :=\nbegin\n  use c ,\n  intros x ,\n  simp only [ennreal.coe_le_coe] ,\nend\n\n\n-- TODO: Should state for finite measures?\nlemma const_integrable {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) (\u03bc_fin : probability_measure \u03bc) (c : \u211d) :\n    integrable (\u03bb (x : \u03b2) , c) \u03bc :=\nbegin\n  apply integrable_const_iff.mpr ,\n  right ,\n  exact proba_finite \u03bc univ ,\nend\n\n\nlemma integral_cst {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) (\u03bc_fin : probability_measure \u03bc) (c : \u211d) :\n    integral \u03bc (\u03bb (x : \u03b2) , c) = c :=\nbegin\n  suffices : integral \u03bc (\u03bb (x : \u03b2) , c) = (\u03bc(univ)).to_real * c,\n  { simp [proba_muniv \u03bc] , } ,\n  apply integral_const c ,\nend\n\nlemma lintegral_cst {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) [\u03bc_proba : probability_measure \u03bc] (c : ennreal) :\n    lintegral \u03bc (\u03bb (x : \u03b2) , c) = c :=\nbegin\n  suffices : lintegral \u03bc (\u03bb (x : \u03b2) , c) = c * (\u03bc(univ)) ,\n  { simp [proba_muniv \u03bc] , } ,\n  apply lintegral_const c ,\nend\n\n\nlemma integral_add_cst {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2} {c : \u211d}\n  (\u03bc : measure \u03b2) [\u03bc_fin : probability_measure \u03bc] (f : \u03b2 \u2192 \u211d) (f_intble: integrable f \u03bc) :\n    integral \u03bc ( f + (\u03bb (x : \u03b2) , c)) = integral \u03bc f + c :=\nbegin\n  have key := @integral_add \u03b2 \u211d m\u03b2 _ _ _ _ _ _ f (\u03bb (x : \u03b2) , c) \u03bc f_intble (const_integrable \u03bc \u03bc_fin c) ,\n  rw integral_cst \u03bc \u03bc_fin c at key ,\n  exact key ,\nend\n\n\nlemma lintegral_cst_sub {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2} {c : ennreal} \n  (\u03bc : measure \u03b2) [\u03bc_fin : probability_measure \u03bc] (c_ne_top : c \u2260 \u22a4)\n  (f : \u03b2 \u2192 ennreal) (f_mble: measurable f) (f_le_c : f \u2264 \u03bb b , c) :\n    lintegral \u03bc ( (\u03bb (x : \u03b2) , c) - f) = c - lintegral \u03bc f :=\nbegin\n  nth_rewrite 0 \u2190(lintegral_cst \u03bc c) ,\n  have f_intble := bdd_integral_of_bdd_ennval \u03bc f (bdd_ennval_of_le_cst f_le_c c_ne_top) ,\n  rw \u2190 @lintegral_sub \u03b2 m\u03b2 \u03bc (\u03bb (b : \u03b2) , c) f (measurable_const) f_mble f_intble (eventually_of_forall f_le_c) ,\n  refl ,\nend\n\n\nlemma nnreal_integral_of_integrable_nonneg {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) (f : \u03b2 \u2192 \u211d) (f_nn : 0 \u2264 f) \n  (f_intble : integrable f \u03bc) :\n     ennreal.of_real (integral \u03bc f) = lintegral \u03bc (ennreal.of_real \u2218 f) :=\nbegin\n  set g := \u03bb (b : \u03b2) , (f(b)).to_nnreal with hg ,\n  have key := @lintegral_coe_eq_integral \u03b2 m\u03b2 \u03bc g (integrable.max_zero f_intble) ,\n  rw hg at key ,\n  set \u03c8 : (\u03b2 \u2192 nnreal) \u2192 (\u03b2 \u2192 \u211d) := coe with h\u03c8 ,\n  have eq\u2080 : f = \u03c8(g) ,\n  { funext b ,\n    exact left_eq_sup.mpr (f_nn b) , } , \n  have eq\u2082 : \u222b (a : \u03b2), \u2191((\u03bb (b : \u03b2), (f b).to_nnreal) a) \u2202\u03bc = integral \u03bc (\u03c8(g)) := by refl ,\n  rw [eq\u2082 , \u2190eq\u2080] at key ,\n  rw \u2190 key ,\n  refl ,\nend\n\n\nlemma nnreal_integral_of_integrable_nonneg' {\u03b2 : Type*} {m\u03b2 : measurable_space \u03b2}\n  (\u03bc : measure \u03b2) (f : \u03b2 \u2192 \u211d) (f_nn : 0 \u2264 f) \n  (f_intble : integrable f \u03bc) :\n     integral \u03bc f = (lintegral \u03bc (ennreal.of_real \u2218 f)).to_real :=\nbegin\n  have key := nnreal_integral_of_integrable_nonneg \u03bc f f_nn f_intble ,\n  have fin : lintegral \u03bc (ennreal.of_real \u2218 f) \u2260 \u22a4 ,  \n  { rw \u2190key ,\n    exact ennreal.of_real_ne_top , } ,\n  have nn : 0 \u2264 integral \u03bc f := integral_nonneg f_nn ,\n  have eq := ennreal.to_real_of_real nn ,\n  rw key at eq ,\n  exact eq.symm ,\nend\n\n\nexample (s : \u2115 \u2192 \u211d) (l : \u211d) (hlim : tendsto s at_top (\ud835\udcdd l)) (c : \u211d) :\n  tendsto (\u03bb (n : \u2115) , s(n) + c) at_top (\ud835\udcdd (l+c)) :=\nbegin\n  exact tendsto.add_const c hlim ,\nend\n\n\nlemma nonneg_of_add_abs_le (a c : \u211d) (h : abs(a) \u2264 c) : 0 \u2264 c + a :=\nbegin\n  have key : c-abs(a) \u2264 c+a := by linarith [neg_le.mp (neg_le_abs_self a)] ,\n  exact le_trans (sub_nonneg.mpr h) key ,\nend\n\n\nprivate lemma portmanteau_continuous_equivalent_formulation'\n  {\u03bcseq : \u2115 \u2192 borel_proba \u03b1} {\u03bc : borel_proba \u03b1} :\n    portmanteau_continuous_ennval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc \n      \u2192 portmanteau_continuous_Rval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc :=\nbegin\n  intros cc_ennrealval ,\n  intros g g_cont g_bdd ,\n  have g_bdd' := g_bdd ,\n  cases g_bdd' with c hc ,\n  set h := (g + (\u03bb x, c)) with hh ,\n  set f := \u03bb (x : \u03b1) , ennreal.of_real (h(x)) with hf ,\n  have h_nn : 0 \u2264 h ,\n  { intros x ,\n    rw hh ,\n    have samediff := nonneg_of_add_abs_le (g(x)) c (hc x) ,\n    rwa add_comm at samediff , } ,\n  have f_eq_h : f = ennreal.of_real \u2218 h := by refl ,\n  have h_bdd : bdd_Rval h := bdd_Rval_add g_bdd const_bdd_Rval ,\n  have h_cont : continuous h := continuous.add g_cont (@continuous_const \u03b1 \u211d _ _ c ) ,\n  have f_bdd : bdd_ennval f := bdd_ennval_of_bdd_Rval h_bdd ,\n  have f_cont : continuous f := cont_enn_of_cont_R h h_cont , \n  specialize cc_ennrealval f f_cont f_bdd ,\n  have g_mble := continuous.borel_measurable g_cont ,\n  have h_mble := continuous.borel_measurable h_cont ,\n  have g_ae_mble := @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ g \u03bc g_mble ,\n  have h_ae_mble := @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ h \u03bc h_mble ,\n  have g_ae_mble_seq := \u03bb n , @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ g (\u03bcseq(n)) g_mble ,\n  have h_ae_mble_seq := \u03bb n , @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ h (\u03bcseq(n)) h_mble ,\n  have g_intble := @integrable_of_bdd_realval _ _ \u03bc.val \u03bc.prop g g_bdd g_ae_mble ,\n  have h_intble := @integrable_of_bdd_realval _ _ \u03bc.val \u03bc.prop h h_bdd h_ae_mble ,\n  have g_intble_seq := \u03bb n , @integrable_of_bdd_realval _ _ (\u03bcseq(n)).val (\u03bcseq(n)).prop g g_bdd (g_ae_mble_seq n) ,\n  have h_intble_seq := \u03bb n , @integrable_of_bdd_realval _ _ (\u03bcseq(n)).val (\u03bcseq(n)).prop h h_bdd (h_ae_mble_seq n) ,\n  suffices : lim_R (\u03bb n , (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq(n)) h)) (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc h) ,\n  { have add_cst : (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc h) = (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc g) + c ,\n    { rw hh ,\n      apply @integral_add_cst _ _ _ \u03bc.val \u03bc.prop g g_intble , } ,\n    have add_cst_seq : (\u03bb n , (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq n) h)) = (\u03bb n , (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq n) g) + c) ,\n    { funext n ,\n      rw hh ,\n      apply @integral_add_cst _ _ _ (\u03bcseq(n)).val (\u03bcseq(n)).prop g (g_intble_seq n) , } ,\n    rw [add_cst , add_cst_seq] at this ,\n    have shift_lim := tendsto.add_const (-c) this ,\n    simp at shift_lim ,\n    exact shift_lim , } , \n  have eq := nnreal_integral_of_integrable_nonneg' \u03bc.val h h_nn h_intble ,\n  have eq_seq := \u03bb n , nnreal_integral_of_integrable_nonneg' (\u03bcseq(n)).val h h_nn (h_intble_seq n) ,\n  have eq' : @integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc h = (@lintegral \u03b1 (borel(\u03b1)) \u03bc f).to_real := eq ,\n  have eq_seq' : (\u03bb n , @integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq(n)).val h) = ennreal.to_real \u2218 (\u03bb n , (@lintegral \u03b1 (borel(\u03b1)) (\u03bcseq(n)) f)) ,\n  { funext n ,\n    exact eq_seq n , } ,\n  have fin : @lintegral \u03b1 (borel(\u03b1)) \u03bc f \u2260 \u22a4\n    := ne_of_lt (@bdd_integral_of_bdd_ennval \u03b1 (borel(\u03b1)) \u03bc \u03bc.prop f f_bdd) , \n  have key := lim_R_of_lim_enn _ _ cc_ennrealval fin ,\n  rw \u2190eq' at key ,\n  rw \u2190eq_seq' at key ,\n  exact key , \nend\n\n\nprivate lemma portmanteau_continuous_equivalent_formulation''\n  {\u03bcseq : \u2115 \u2192 (borel_proba \u03b1)} {\u03bc : borel_proba \u03b1} :\n    portmanteau_continuous_Rval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc\n       \u2192 portmanteau_continuous_ennval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc :=\nbegin\n  intros cc_Rval ,\n  intros f f_cont f_bdd ,\n  have f_fin_val : \u2200 (x : \u03b1) , f(x) \u2260 \u22a4 := finval_of_bdd_ennval f_bdd ,\n  set g := \u03bb (x : \u03b1) , ennreal.to_real (f(x)) with hg ,\n  have f_eq_g : f = ennreal.of_real \u2218 g ,\n  { funext x ,\n    exact (ennreal.of_real_to_real (f_fin_val x)).symm , } , \n  have g_nn : 0 \u2264 g ,\n  { intros x ,\n    simp only [pi.zero_apply, ennreal.to_real_nonneg] , } ,\n  have g_bdd : bdd_Rval g ,\n  { cases f_bdd with c hc ,\n    use c ,\n    intros x ,\n    rw hg ,\n    specialize hc x ,\n    have abs_eq : (abs (f(x)).to_real) = g(x) := by simp only [ennreal.to_real_nonneg, abs_eq_self] ,\n    rw hg at abs_eq ,\n    dsimp at * ,\n    rw abs_eq ,\n    exact ennreal.to_real_le_coe_of_le_coe hc , } ,\n  have g_cont : continuous g := cont_R_of_cont_bdd_enn f f_cont f_bdd , \n  have g_mble := continuous.borel_measurable g_cont ,\n  have g_ae_mble := @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ g \u03bc g_mble ,\n  have g_ae_mble_seq := \u03bb n , @measurable.ae_measurable \u03b1 \u211d (borel(\u03b1)) _ g (\u03bcseq(n)) g_mble ,\n  have g_intble := @integrable_of_bdd_realval _ _ \u03bc.val \u03bc.prop g g_bdd g_ae_mble ,\n  have g_intble_seq := \u03bb n , @integrable_of_bdd_realval _ _ (\u03bcseq(n)).val (\u03bcseq(n)).prop g g_bdd (g_ae_mble_seq n) ,\n  have eq := nnreal_integral_of_integrable_nonneg \u03bc.val g g_nn g_intble ,\n  have eq_seq := \u03bb n , nnreal_integral_of_integrable_nonneg (\u03bcseq(n)).val g g_nn (g_intble_seq n) ,\n  rw \u2190 f_eq_g at * ,\n  have eq' : ennreal.of_real (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc g) = @lintegral \u03b1 (borel(\u03b1)) \u03bc f := eq ,\n  have eq_seq' : (\u03bb n , ennreal.of_real (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq n) g)) = (\u03bb n , @lintegral \u03b1 (borel(\u03b1)) (\u03bcseq n) f) , -- := by simp [eq_seq] ,\n  { funext n ,\n    exact eq_seq n , } ,\n  rw \u2190eq' ,\n  rw \u2190eq_seq' ,\n  specialize cc_Rval g g_cont g_bdd ,\n  exact lim_enn_of_lim_R cc_Rval , \nend\n\n\n/-- The usual definition of weak convergence of probability measures is given in\nterms of sequences of probability measures: it is the requirement that the integrals\nof all continuous bounded functions against members of the sequence converge.\nThis characterization is shown in `weak_conv_seq_iff`. -/\ntheorem weak_conv_seq_iff {\u03b1 : Type*} [topological_space \u03b1]\n  {\u03bcseq : \u2115 \u2192 borel_proba \u03b1} {\u03bc : borel_proba \u03b1} :\n    tendsto \u03bcseq at_top (\ud835\udcdd \u03bc) \n      \u2194 ( \u2200 (f : \u03b1 \u2192 \u211d) , continuous f \u2192 bdd_Rval f \u2192\n          tendsto (\u03bb n, (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ (\u03bcseq(n)) f)) at_top (\ud835\udcdd (@integral \u03b1 \u211d (borel(\u03b1)) _ _ _ _ _ _ \u03bc f)) ) :=\nbegin\n  split ,\n  { intros weak_conv ,\n    have key := weak_conv_seq_iff'.mp weak_conv ,\n    have key' : portmanteau_continuous_ennval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc ,\n    { intros f f_cont f_bdd ,\n      exact key \u27e8 f , \u27e8 f_cont , f_bdd \u27e9 \u27e9 , } ,\n    exact portmanteau_continuous_equivalent_formulation' key' , } , \n  { intros h ,\n    have key := portmanteau_continuous_equivalent_formulation'' h ,\n    apply weak_conv_seq_iff'.mpr ,\n    intros f ,\n    exact key f.val f.prop.1 f.prop.2 , } , \nend\n\n\ntheorem weak_conv_seq_iff_portmanteau_continuous_Rval {\u03b1 : Type*} [topological_space \u03b1]\n  {\u03bcseq : \u2115 \u2192 borel_proba \u03b1} {\u03bc : borel_proba \u03b1} :\n    tendsto \u03bcseq at_top (\ud835\udcdd \u03bc) \n      \u2194 portmanteau_continuous_Rval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc \n        := weak_conv_seq_iff \n\n\ntheorem weak_conv_seq_iff_portmanteau_continuous_ennval {\u03b1 : Type*} [topological_space \u03b1]\n  {\u03bcseq : \u2115 \u2192 borel_proba \u03b1} {\u03bc : borel_proba \u03b1} :\n    tendsto \u03bcseq at_top (\ud835\udcdd \u03bc) \n      \u2194 portmanteau_continuous_ennval (\u03bb n , (\u03bcseq(n) : @measure_theory.measure \u03b1 (borel(\u03b1)))) \u03bc :=\nbegin\n  split ,\n  { intros h ,\n    exact portmanteau_continuous_equivalent_formulation'' (weak_conv_seq_iff_portmanteau_continuous_Rval.mp h) , } ,\n  { intros h ,\n    exact weak_conv_seq_iff_portmanteau_continuous_Rval.mpr (portmanteau_continuous_equivalent_formulation' h) , } ,\nend\n\n\n\nend portmanteau_integrals\n\nend portmanteau\n\n\n", "meta": {"author": "kkytola", "repo": "lean_portmanteau", "sha": "ac55eb4e24be43032cbc082e2b68d8fb8bd63f22", "save_path": "github-repos/lean/kkytola-lean_portmanteau", "path": "github-repos/lean/kkytola-lean_portmanteau/lean_portmanteau-ac55eb4e24be43032cbc082e2b68d8fb8bd63f22/portmanteau_integrals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.39652576228724534}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nDependently-typed vectors.\n\nThese are necessary to store multiple tensors of arbitrary shapes.\n-/\nimport .util\n\ninductive dvec {X : Type} (Y : X \u2192 Type) : list X \u2192 Type\n| nil {}  : dvec []\n| cons : \u03a0 {x : X}, Y x \u2192 \u03a0 {xs : list X}, dvec xs \u2192 dvec (x::xs)\n\nnamespace dvec\nreserve infixr ` ::: `:67\nnotation h `:::` t  := cons h t\n\nnotation `\u27e6` l:(foldr `, ` (h t, cons h t) nil `\u27e7`) := l\n\ndef head {X : Type} {Y : X \u2192 Type} {x : X} {xs : list X} : dvec Y (x::xs) \u2192 Y x\n| (cons y ys) := y\n\ndef tail {X : Type} {Y : X \u2192 Type} {x : X} {xs : list X} : dvec Y (x::xs) \u2192 dvec Y xs\n| (cons y ys) := ys\n\ndef head2 {X : Type} {Y : X \u2192 Type} {x\u2081 x\u2082 : X} {xs : list X} : dvec Y (x\u2081::x\u2082::xs) \u2192 Y x\u2082\n| (cons y\u2081 (cons y\u2082 ys)) := y\u2082\n\ndef head3 {X : Type} {Y : X \u2192 Type} {x\u2081 x\u2082 x\u2083 : X} {xs : list X} : dvec Y (x\u2081::x\u2082::x\u2083::xs) \u2192 Y x\u2083\n| (cons y\u2081 (cons y\u2082 (cons y\u2083 ys))) := y\u2083\n\ndef get {X : Type} [decidable_eq X] {Y : X \u2192 Type} (x\u2080 : X) [inhabited (Y x\u2080)] : \u03a0 {xs : list X}, dvec Y xs \u2192 \u2115 \u2192 Y x\u2080\n| []      _           _     := default (Y x\u2080)\n| (x::xs) (cons y ys) 0     := if H : x = x\u2080 then eq.rec_on H y else default (Y x\u2080)\n| (x::xs) (cons y ys) (n+1) := get ys n\n\nlemma singleton_congr {X : Type} {Y : X \u2192 Type} {x : X} (y\u2081 y\u2082 : Y x) : y\u2081 = y\u2082 \u2192 \u27e6y\u2081\u27e7 = \u27e6y\u2082\u27e7 := assume H, by rw H\n\nlemma get\u2080_head {X : Type} [decidable_eq X] {Y : X \u2192 Type} (x\u2080 : X) [inhabited (Y x\u2080)] :\n  \u2200 {xs : list X} (ys : dvec Y (x\u2080::xs)), get x\u2080 ys 0 = head ys\n| xs (y:::ys)   := begin dunfold head get, simp [dif_ctx_simp_congr, dif_pos] end\n\ndef update_at {X : Type} [decidable_eq X] {Y : X \u2192 Type} {x\u2080 : X} (y\u2080 : Y x\u2080) : \u03a0 {xs : list X} (ys : dvec Y xs) (idx : \u2115), dvec Y xs\n| []      _                 _     := \u27e6\u27e7\n| (x::xs) (cons y ys) 0     := if H : x\u2080 = x then cons (eq.rec_on H y\u2080) ys else cons y ys\n| (x::xs) (cons y ys) (n+1) := cons y (update_at ys n)\n\nprotected def to_string_aux {X : Type} {Y : X \u2192 Type} [\u2200 x, has_to_string (Y x)] : \u03a0 {xs : list X}, dvec Y xs \u2192 string\n| [] _                  := \"-------------\"\n| (x::xs) (cons y ys)  := to_string y ++ \"\\n\" ++ to_string_aux ys\n\nprotected def to_string {X : Type} {Y : X \u2192 Type} [\u2200 x, has_to_string (Y x)] {xs : list X} (ys : dvec Y xs) : string :=\n  \"-------------\\n\" ++ dvec.to_string_aux ys\n\ninstance {X : Type} {Y : X \u2192 Type} [\u2200 x, has_to_string (Y x)] {xs : list X} : has_to_string (dvec Y xs) :=\n\u27e8dvec.to_string\u27e9\n\nattribute [simp] head tail head2 head3 get update_at\n\nend dvec\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/dvec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984445, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3965257537085491}}
{"text": "/-\nCopyright (c) 2020 Jean Lo. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jean Lo\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.dynamics.flow\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# \u03c9-limits\n\nFor a function `\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2` where `\u03b2` is a topological space, we\ndefine the \u03c9-limit under `\u03d5` of a set `s` in `\u03b1` with respect to\nfilter `f` on `\u03c4`: an element `y : \u03b2` is in the \u03c9-limit of `s` if the\nforward images of `s` intersect arbitrarily small neighbourhoods of\n`y` frequently \"in the direction of `f`\".\n\nIn practice `\u03d5` is often a continuous monoid-act, but the definition\nrequires only that `\u03d5` has a coercion to the appropriate function\ntype. In the case where `\u03c4` is `\u2115` or `\u211d` and `f` is `at_top`, we\nrecover the usual definition of the \u03c9-limit set as the set of all `y`\nsuch that there exist sequences `(t\u2099)`, `(x\u2099)` such that `\u03d5 t\u2099 x\u2099 \u27f6 y`\nas `n \u27f6 \u221e`.\n\n## Notations\n\nThe `omega_limit` locale provides the localised notation `\u03c9` for\n`omega_limit`, as well as `\u03c9\u207a` and `\u03c9\u207b` for `omega_limit at_top` and\n`omega_limit at_bot` respectively for when the acting monoid is\nendowed with an order.\n-/\n\n/-!\n### Definition and notation\n-/\n\n/-- The \u03c9-limit of a set `s` under `\u03d5` with respect to a filter `f` is\n    \u22c2 u \u2208 f, cl (\u03d5 u s). -/\ndef omega_limit {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) : set \u03b2 :=\n  set.Inter fun (u : set \u03c4) => set.Inter fun (H : u \u2208 f) => closure (set.image2 \u03d5 u s)\n\n/-!\n### Elementary properties\n-/\n\ntheorem omega_limit_def {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) : omega_limit f \u03d5 s = set.Inter fun (u : set \u03c4) => set.Inter fun (H : u \u2208 f) => closure (set.image2 \u03d5 u s) :=\n  rfl\n\ntheorem omega_limit_subset_of_tendsto {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {m : \u03c4 \u2192 \u03c4} {f\u2081 : filter \u03c4} {f\u2082 : filter \u03c4} (hf : filter.tendsto m f\u2081 f\u2082) : omega_limit f\u2081 (fun (t : \u03c4) (x : \u03b1) => \u03d5 (m t) x) s \u2286 omega_limit f\u2082 \u03d5 s := sorry\n\ntheorem omega_limit_mono_left {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {f\u2081 : filter \u03c4} {f\u2082 : filter \u03c4} (hf : f\u2081 \u2264 f\u2082) : omega_limit f\u2081 \u03d5 s \u2286 omega_limit f\u2082 \u03d5 s :=\n  omega_limit_subset_of_tendsto \u03d5 s (filter.tendsto_id' hf)\n\ntheorem omega_limit_mono_right {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) {s\u2081 : set \u03b1} {s\u2082 : set \u03b1} (hs : s\u2081 \u2286 s\u2082) : omega_limit f \u03d5 s\u2081 \u2286 omega_limit f \u03d5 s\u2082 :=\n  set.bInter_subset_bInter_right\n    fun (u : set \u03c4) (hu : u \u2208 fun (u : set \u03c4) => u \u2208 filter.sets f) => closure_mono (set.image2_subset set.subset.rfl hs)\n\ntheorem is_closed_omega_limit {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) : is_closed (omega_limit f \u03d5 s) :=\n  is_closed_Inter fun (u : set \u03c4) => is_closed_Inter fun (hu : u \u2208 f) => is_closed_closure\n\ntheorem maps_to_omega_limit' {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (s : set \u03b1) {\u03b1' : Type u_4} {\u03b2' : Type u_5} [topological_space \u03b2'] {f : filter \u03c4} {\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2} {\u03d5' : \u03c4 \u2192 \u03b1' \u2192 \u03b2'} {ga : \u03b1 \u2192 \u03b1'} {s' : set \u03b1'} (hs : set.maps_to ga s s') {gb : \u03b2 \u2192 \u03b2'} (hg : filter.eventually (fun (t : \u03c4) => set.eq_on (gb \u2218 \u03d5 t) (\u03d5' t \u2218 ga) s) f) (hgc : continuous gb) : set.maps_to gb (omega_limit f \u03d5 s) (omega_limit f \u03d5' s') := sorry\n\ntheorem maps_to_omega_limit {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (s : set \u03b1) {\u03b1' : Type u_4} {\u03b2' : Type u_5} [topological_space \u03b2'] {f : filter \u03c4} {\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2} {\u03d5' : \u03c4 \u2192 \u03b1' \u2192 \u03b2'} {ga : \u03b1 \u2192 \u03b1'} {s' : set \u03b1'} (hs : set.maps_to ga s s') {gb : \u03b2 \u2192 \u03b2'} (hg : \u2200 (t : \u03c4) (x : \u03b1), gb (\u03d5 t x) = \u03d5' t (ga x)) (hgc : continuous gb) : set.maps_to gb (omega_limit f \u03d5 s) (omega_limit f \u03d5' s') :=\n  maps_to_omega_limit' s hs (filter.eventually_of_forall fun (t : \u03c4) (x : \u03b1) (hx : x \u2208 s) => hg t x) hgc\n\ntheorem omega_limit_image_eq {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (s : set \u03b1) {\u03b1' : Type u_4} (\u03d5 : \u03c4 \u2192 \u03b1' \u2192 \u03b2) (f : filter \u03c4) (g : \u03b1 \u2192 \u03b1') : omega_limit f \u03d5 (g '' s) = omega_limit f (fun (t : \u03c4) (x : \u03b1) => \u03d5 t (g x)) s := sorry\n\ntheorem omega_limit_preimage_subset {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] {\u03b1' : Type u_4} (\u03d5 : \u03c4 \u2192 \u03b1' \u2192 \u03b2) (s : set \u03b1') (f : filter \u03c4) (g : \u03b1 \u2192 \u03b1') : omega_limit f (fun (t : \u03c4) (x : \u03b1) => \u03d5 t (g x)) (g \u207b\u00b9' s) \u2286 omega_limit f \u03d5 s :=\n  maps_to_omega_limit (g \u207b\u00b9' s) (set.maps_to_preimage g s) (fun (t : \u03c4) (x : \u03b1) => rfl) continuous_id\n\n/-!\n### Equivalent definitions of the omega limit\n\nThe next few lemmas are various versions of the property\ncharacterising \u03c9-limits:\n-/\n\n/-- An element `y` is in the \u03c9-limit set of `s` w.r.t. `f` if the\n    preimages of an arbitrary neighbourhood of `y` frequently\n    (w.r.t. `f`) intersects of `s`. -/\ntheorem mem_omega_limit_iff_frequently {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) (y : \u03b2) : y \u2208 omega_limit f \u03d5 s \u2194 \u2200 (n : set \u03b2), n \u2208 nhds y \u2192 filter.frequently (fun (t : \u03c4) => set.nonempty (s \u2229 \u03d5 t \u207b\u00b9' n)) f := sorry\n\n/-- An element `y` is in the \u03c9-limit set of `s` w.r.t. `f` if the\n    forward images of `s` frequently (w.r.t. `f`) intersect arbitrary\n    neighbourhoods of `y`. -/\ntheorem mem_omega_limit_iff_frequently\u2082 {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) (y : \u03b2) : y \u2208 omega_limit f \u03d5 s \u2194 \u2200 (n : set \u03b2), n \u2208 nhds y \u2192 filter.frequently (fun (t : \u03c4) => set.nonempty (\u03d5 t '' s \u2229 n)) f := sorry\n\n/-- An element `y` is in the \u03c9-limit of `x` w.r.t. `f` if the forward\n    images of `x` frequently (w.r.t. `f`) falls within an arbitrary\n    neighbourhood of `y`. -/\ntheorem mem_omega_limit_singleton_iff_map_cluster_point {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (x : \u03b1) (y : \u03b2) : y \u2208 omega_limit f \u03d5 (singleton x) \u2194 map_cluster_pt y f fun (t : \u03c4) => \u03d5 t x := sorry\n\n/-!\n### Set operations and omega limits\n-/\n\ntheorem omega_limit_inter {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s\u2081 : set \u03b1) (s\u2082 : set \u03b1) : omega_limit f \u03d5 (s\u2081 \u2229 s\u2082) \u2286 omega_limit f \u03d5 s\u2081 \u2229 omega_limit f \u03d5 s\u2082 :=\n  set.subset_inter (omega_limit_mono_right f \u03d5 (set.inter_subset_left s\u2081 s\u2082))\n    (omega_limit_mono_right f \u03d5 (set.inter_subset_right s\u2081 s\u2082))\n\ntheorem omega_limit_Inter {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} {\u03b9 : Type u_4} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (p : \u03b9 \u2192 set \u03b1) : omega_limit f \u03d5 (set.Inter fun (i : \u03b9) => p i) \u2286 set.Inter fun (i : \u03b9) => omega_limit f \u03d5 (p i) :=\n  set.subset_Inter fun (i : \u03b9) => omega_limit_mono_right f \u03d5 (set.Inter_subset (fun (i : \u03b9) => p i) i)\n\ntheorem omega_limit_union {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s\u2081 : set \u03b1) (s\u2082 : set \u03b1) : omega_limit f \u03d5 (s\u2081 \u222a s\u2082) = omega_limit f \u03d5 s\u2081 \u222a omega_limit f \u03d5 s\u2082 := sorry\n\ntheorem omega_limit_Union {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} {\u03b9 : Type u_4} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (p : \u03b9 \u2192 set \u03b1) : (set.Union fun (i : \u03b9) => omega_limit f \u03d5 (p i)) \u2286 omega_limit f \u03d5 (set.Union fun (i : \u03b9) => p i) := sorry\n\n/-!\nDifferent expressions for omega limits, useful for rewrites. In\nparticular, one may restrict the intersection to sets in `f` which are\nsubsets of some set `v` also in `f`.\n-/\n\ntheorem omega_limit_eq_Inter {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) : omega_limit f \u03d5 s = set.Inter fun (u : \u21a5(filter.sets f)) => closure (set.image2 \u03d5 (\u2191u) s) :=\n  set.bInter_eq_Inter (fun (u : set \u03c4) => u \u2208 filter.sets f) fun (u : set \u03c4) (H : u \u2208 f) => closure (set.image2 \u03d5 u s)\n\ntheorem omega_limit_eq_bInter_inter {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {v : set \u03c4} (hv : v \u2208 f) : omega_limit f \u03d5 s = set.Inter fun (u : set \u03c4) => set.Inter fun (H : u \u2208 f) => closure (set.image2 \u03d5 (u \u2229 v) s) := sorry\n\ntheorem omega_limit_eq_Inter_inter {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {v : set \u03c4} (hv : v \u2208 f) : omega_limit f \u03d5 s = set.Inter fun (u : \u21a5(filter.sets f)) => closure (set.image2 \u03d5 (\u2191u \u2229 v) s) := sorry\n\ntheorem omega_limit_subset_closure_fw_image {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {u : set \u03c4} (hu : u \u2208 f) : omega_limit f \u03d5 s \u2286 closure (set.image2 \u03d5 u s) := sorry\n\n/-!\n### `\u03c9-limits and compactness\n-/\n\n/-- A set is eventually carried into any open neighbourhood of its \u03c9-limit:\nif `c` is a compact set such that `closure {\u03d5 t x | t \u2208 v, x \u2208 s} \u2286 c` for some `v \u2208 f`\nand `n` is an open neighbourhood of `\u03c9 f \u03d5 s`, then for some `u \u2208 f` we have\n`closure {\u03d5 t x | t \u2208 u, x \u2208 s} \u2286 n`. -/\ntheorem eventually_closure_subset_of_is_compact_absorbing_of_is_open_of_omega_limit_subset' {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) {c : set \u03b2} (hc\u2081 : is_compact c) (hc\u2082 : \u2203 (v : set \u03c4), \u2203 (H : v \u2208 f), closure (set.image2 \u03d5 v s) \u2286 c) {n : set \u03b2} (hn\u2081 : is_open n) (hn\u2082 : omega_limit f \u03d5 s \u2286 n) : \u2203 (u : set \u03c4), \u2203 (H : u \u2208 f), closure (set.image2 \u03d5 u s) \u2286 n := sorry\n\n/-- A set is eventually carried into any open neighbourhood of its \u03c9-limit:\nif `c` is a compact set such that `closure {\u03d5 t x | t \u2208 v, x \u2208 s} \u2286 c` for some `v \u2208 f`\nand `n` is an open neighbourhood of `\u03c9 f \u03d5 s`, then for some `u \u2208 f` we have\n`closure {\u03d5 t x | t \u2208 u, x \u2208 s} \u2286 n`. -/\ntheorem eventually_closure_subset_of_is_compact_absorbing_of_is_open_of_omega_limit_subset {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [t2_space \u03b2] {c : set \u03b2} (hc\u2081 : is_compact c) (hc\u2082 : filter.eventually (fun (t : \u03c4) => set.maps_to (\u03d5 t) s c) f) {n : set \u03b2} (hn\u2081 : is_open n) (hn\u2082 : omega_limit f \u03d5 s \u2286 n) : \u2203 (u : set \u03c4), \u2203 (H : u \u2208 f), closure (set.image2 \u03d5 u s) \u2286 n := sorry\n\ntheorem eventually_maps_to_of_is_compact_absorbing_of_is_open_of_omega_limit_subset {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [t2_space \u03b2] {c : set \u03b2} (hc\u2081 : is_compact c) (hc\u2082 : filter.eventually (fun (t : \u03c4) => set.maps_to (\u03d5 t) s c) f) {n : set \u03b2} (hn\u2081 : is_open n) (hn\u2082 : omega_limit f \u03d5 s \u2286 n) : filter.eventually (fun (t : \u03c4) => set.maps_to (\u03d5 t) s n) f := sorry\n\ntheorem eventually_closure_subset_of_is_open_of_omega_limit_subset {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [compact_space \u03b2] {v : set \u03b2} (hv\u2081 : is_open v) (hv\u2082 : omega_limit f \u03d5 s \u2286 v) : \u2203 (u : set \u03c4), \u2203 (H : u \u2208 f), closure (set.image2 \u03d5 u s) \u2286 v :=\n  eventually_closure_subset_of_is_compact_absorbing_of_is_open_of_omega_limit_subset' f \u03d5 s compact_univ\n    (Exists.intro set.univ (Exists.intro filter.univ_mem_sets (set.subset_univ (closure (set.image2 \u03d5 set.univ s))))) hv\u2081\n    hv\u2082\n\ntheorem eventually_maps_to_of_is_open_of_omega_limit_subset {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [compact_space \u03b2] {v : set \u03b2} (hv\u2081 : is_open v) (hv\u2082 : omega_limit f \u03d5 s \u2286 v) : filter.eventually (fun (t : \u03c4) => set.maps_to (\u03d5 t) s v) f := sorry\n\n/-- The \u03c9-limit of a nonempty set w.r.t. a nontrivial filter is nonempty. -/\ntheorem nonempty_omega_limit_of_is_compact_absorbing {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [filter.ne_bot f] {c : set \u03b2} (hc\u2081 : is_compact c) (hc\u2082 : \u2203 (v : set \u03c4), \u2203 (H : v \u2208 f), closure (set.image2 \u03d5 v s) \u2286 c) (hs : set.nonempty s) : set.nonempty (omega_limit f \u03d5 s) := sorry\n\ntheorem nonempty_omega_limit {\u03c4 : Type u_1} {\u03b1 : Type u_2} {\u03b2 : Type u_3} [topological_space \u03b2] (f : filter \u03c4) (\u03d5 : \u03c4 \u2192 \u03b1 \u2192 \u03b2) (s : set \u03b1) [compact_space \u03b2] [filter.ne_bot f] (hs : set.nonempty s) : set.nonempty (omega_limit f \u03d5 s) :=\n  nonempty_omega_limit_of_is_compact_absorbing f \u03d5 s compact_univ\n    (Exists.intro set.univ (Exists.intro filter.univ_mem_sets (set.subset_univ (closure (set.image2 \u03d5 set.univ s))))) hs\n\n/-!\n### \u03c9-limits of Flows by a Monoid\n-/\n\nnamespace flow\n\n\ntheorem is_invariant_omega_limit {\u03c4 : Type u_1} [topological_space \u03c4] [add_monoid \u03c4] [has_continuous_add \u03c4] {\u03b1 : Type u_2} [topological_space \u03b1] (f : filter \u03c4) (\u03d5 : flow \u03c4 \u03b1) (s : set \u03b1) (hf : \u2200 (t : \u03c4), filter.tendsto (Add.add t) f f) : is_invariant (\u21d1\u03d5) (omega_limit f (\u21d1\u03d5) s) := sorry\n\ntheorem omega_limit_image_subset {\u03c4 : Type u_1} [topological_space \u03c4] [add_monoid \u03c4] [has_continuous_add \u03c4] {\u03b1 : Type u_2} [topological_space \u03b1] (f : filter \u03c4) (\u03d5 : flow \u03c4 \u03b1) (s : set \u03b1) (t : \u03c4) (ht : filter.tendsto (fun (_x : \u03c4) => _x + t) f f) : omega_limit f (\u21d1\u03d5) (coe_fn \u03d5 t '' s) \u2286 omega_limit f (\u21d1\u03d5) s := sorry\n\nend flow\n\n\n/-!\n### \u03c9-limits of Flows by a Group\n-/\n\nnamespace flow\n\n\n/-- the \u03c9-limit of a forward image of `s` is the same as the \u03c9-limit of `s`. -/\n@[simp] theorem omega_limit_image_eq {\u03c4 : Type u_1} [topological_space \u03c4] [add_comm_group \u03c4] [topological_add_group \u03c4] {\u03b1 : Type u_2} [topological_space \u03b1] (f : filter \u03c4) (\u03d5 : flow \u03c4 \u03b1) (s : set \u03b1) (hf : \u2200 (t : \u03c4), filter.tendsto (fun (_x : \u03c4) => _x + t) f f) (t : \u03c4) : omega_limit f (\u21d1\u03d5) (coe_fn \u03d5 t '' s) = omega_limit f (\u21d1\u03d5) s := sorry\n\ntheorem omega_limit_omega_limit {\u03c4 : Type u_1} [topological_space \u03c4] [add_comm_group \u03c4] [topological_add_group \u03c4] {\u03b1 : Type u_2} [topological_space \u03b1] (f : filter \u03c4) (\u03d5 : flow \u03c4 \u03b1) (s : set \u03b1) (hf : \u2200 (t : \u03c4), filter.tendsto (Add.add t) f f) : omega_limit f (\u21d1\u03d5) (omega_limit f (\u21d1\u03d5) s) \u2286 omega_limit f (\u21d1\u03d5) s := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/dynamics/omega_limit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.39652575370854903}}
{"text": "/-\nCopyright (c) 2020 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n\n! This file was ported from Lean 3 source module group_theory.group_action.sub_mul_action\n! leanprover-community/mathlib commit fac369018417f980cec5fcdafc766a69f88d8cfe\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Hom.GroupAction\nimport Mathbin.Algebra.Module.Basic\nimport Mathbin.Data.SetLike.Basic\nimport Mathbin.GroupTheory.GroupAction.Basic\n\n/-!\n\n# Sets invariant to a `mul_action`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we define `sub_mul_action R M`; a subset of a `mul_action R M` which is closed with\nrespect to scalar multiplication.\n\nFor most uses, typically `submodule R M` is more powerful.\n\n## Main definitions\n\n* `sub_mul_action.mul_action` - the `mul_action R M` transferred to the subtype.\n* `sub_mul_action.mul_action'` - the `mul_action S M` transferred to the subtype when\n  `is_scalar_tower S R M`.\n* `sub_mul_action.is_scalar_tower` - the `is_scalar_tower S R M` transferred to the subtype.\n\n## Tags\n\nsubmodule, mul_action\n-/\n\n\nopen Function\n\nuniverse u u' u'' v\n\nvariable {S : Type u'} {T : Type u''} {R : Type u} {M : Type v}\n\n#print SMulMemClass /-\n/-- `smul_mem_class S R M` says `S` is a type of subsets `s \u2264 M` that are closed under the\nscalar action of `R` on `M`.\n\nNote that only `R` is marked as an `out_param` here, since `M` is supplied by the `set_like`\nclass instead.\n-/\nclass SMulMemClass (S : Type _) (R : outParam <| Type _) (M : Type _) [SMul R M] [SetLike S M] where\n  smul_mem : \u2200 {s : S} (r : R) {m : M}, m \u2208 s \u2192 r \u2022 m \u2208 s\n#align smul_mem_class SMulMemClass\n-/\n\n#print VAddMemClass /-\n/-- `vadd_mem_class S R M` says `S` is a type of subsets `s \u2264 M` that are closed under the\nadditive action of `R` on `M`.\n\nNote that only `R` is marked as an `out_param` here, since `M` is supplied by the `set_like`\nclass instead.\n-/\nclass VAddMemClass (S : Type _) (R : outParam <| Type _) (M : Type _) [VAdd R M] [SetLike S M] where\n  vadd_mem : \u2200 {s : S} (r : R) {m : M}, m \u2208 s \u2192 r +\u1d65 m \u2208 s\n#align vadd_mem_class VAddMemClass\n-/\n\nattribute [to_additive] SMulMemClass\n\nnamespace SetLike\n\nvariable [SMul R M] [SetLike S M] [hS : SMulMemClass S R M] (s : S)\n\ninclude hS\n\nopen SMulMemClass\n\n#print SetLike.smul /-\n-- lower priority so other instances are found first\n/-- A subset closed under the scalar action inherits that action. -/\n@[to_additive \"A subset closed under the additive action inherits that action.\"]\ninstance (priority := 900) smul : SMul R s :=\n  \u27e8fun r x => \u27e8r \u2022 x.1, smul_mem r x.2\u27e9\u27e9\n#align set_like.has_smul SetLike.smul\n#align set_like.has_vadd SetLike.vadd\n-/\n\n#print SetLike.val_smul /-\n-- lower priority so later simp lemmas are used first; to appease simp_nf\n@[simp, norm_cast, to_additive]\nprotected theorem val_smul (r : R) (x : s) : (\u2191(r \u2022 x) : M) = r \u2022 x :=\n  rfl\n#align set_like.coe_smul SetLike.val_smul\n#align set_like.coe_vadd SetLike.val_vadd\n-/\n\n#print SetLike.mk_smul_mk /-\n-- lower priority so later simp lemmas are used first; to appease simp_nf\n@[simp, to_additive]\ntheorem mk_smul_mk (r : R) (x : M) (hx : x \u2208 s) : r \u2022 (\u27e8x, hx\u27e9 : s) = \u27e8r \u2022 x, smul_mem r hx\u27e9 :=\n  rfl\n#align set_like.mk_smul_mk SetLike.mk_smul_mk\n#align set_like.mk_vadd_mk SetLike.mk_vadd_mk\n-/\n\n#print SetLike.smul_def /-\n@[to_additive]\ntheorem smul_def (r : R) (x : s) : r \u2022 x = \u27e8r \u2022 x, smul_mem r x.2\u27e9 :=\n  rfl\n#align set_like.smul_def SetLike.smul_def\n#align set_like.vadd_def SetLike.vadd_def\n-/\n\nomit hS\n\n/- warning: set_like.forall_smul_mem_iff -> SetLike.forall_smul_mem_iff is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {S : Type.{u3}} [_inst_3 : Monoid.{u1} R] [_inst_4 : MulAction.{u1, u2} R M _inst_3] [_inst_5 : SetLike.{u3, u2} S M] [_inst_6 : SMulMemClass.{u3, u1, u2} S R M (MulAction.toHasSmul.{u1, u2} R M _inst_3 _inst_4) _inst_5] {N : S} {x : M}, Iff (forall (a : R), Membership.Mem.{u2, u3} M S (SetLike.hasMem.{u3, u2} S M _inst_5) (SMul.smul.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_3 _inst_4) a x) N) (Membership.Mem.{u2, u3} M S (SetLike.hasMem.{u3, u2} S M _inst_5) x N)\nbut is expected to have type\n  forall {R : Type.{u3}} {M : Type.{u2}} {S : Type.{u1}} [_inst_3 : Monoid.{u3} R] [_inst_4 : MulAction.{u3, u2} R M _inst_3] [_inst_5 : SetLike.{u1, u2} S M] [_inst_6 : SMulMemClass.{u1, u3, u2} S R M (MulAction.toSMul.{u3, u2} R M _inst_3 _inst_4) _inst_5] {N : S} {x : M}, Iff (forall (a : R), Membership.mem.{u2, u1} M S (SetLike.instMembership.{u1, u2} S M _inst_5) (HSMul.hSMul.{u3, u2, u2} R M M (instHSMul.{u3, u2} R M (MulAction.toSMul.{u3, u2} R M _inst_3 _inst_4)) a x) N) (Membership.mem.{u2, u1} M S (SetLike.instMembership.{u1, u2} S M _inst_5) x N)\nCase conversion may be inaccurate. Consider using '#align set_like.forall_smul_mem_iff SetLike.forall_smul_mem_iff\u2093'. -/\n@[simp]\ntheorem forall_smul_mem_iff {R M S : Type _} [Monoid R] [MulAction R M] [SetLike S M]\n    [SMulMemClass S R M] {N : S} {x : M} : (\u2200 a : R, a \u2022 x \u2208 N) \u2194 x \u2208 N :=\n  \u27e8fun h => by simpa using h 1, fun h a => SMulMemClass.smul_mem a h\u27e9\n#align set_like.forall_smul_mem_iff SetLike.forall_smul_mem_iff\n\nend SetLike\n\n#print SubMulAction /-\n/-- A sub_mul_action is a set which is closed under scalar multiplication.  -/\nstructure SubMulAction (R : Type u) (M : Type v) [SMul R M] : Type v where\n  carrier : Set M\n  smul_mem' : \u2200 (c : R) {x : M}, x \u2208 carrier \u2192 c \u2022 x \u2208 carrier\n#align sub_mul_action SubMulAction\n-/\n\nnamespace SubMulAction\n\nvariable [SMul R M]\n\ninstance : SetLike (SubMulAction R M) M :=\n  \u27e8SubMulAction.carrier, fun p q h => by cases p <;> cases q <;> congr \u27e9\n\ninstance : SMulMemClass (SubMulAction R M) R M where smul_mem := smul_mem'\n\n#print SubMulAction.mem_carrier /-\n@[simp]\ntheorem mem_carrier {p : SubMulAction R M} {x : M} : x \u2208 p.carrier \u2194 x \u2208 (p : Set M) :=\n  Iff.rfl\n#align sub_mul_action.mem_carrier SubMulAction.mem_carrier\n-/\n\n#print SubMulAction.ext /-\n@[ext]\ntheorem ext {p q : SubMulAction R M} (h : \u2200 x, x \u2208 p \u2194 x \u2208 q) : p = q :=\n  SetLike.ext h\n#align sub_mul_action.ext SubMulAction.ext\n-/\n\n#print SubMulAction.copy /-\n/-- Copy of a sub_mul_action with a new `carrier` equal to the old one. Useful to fix definitional\nequalities.-/\nprotected def copy (p : SubMulAction R M) (s : Set M) (hs : s = \u2191p) : SubMulAction R M\n    where\n  carrier := s\n  smul_mem' := hs.symm \u25b8 p.smul_mem'\n#align sub_mul_action.copy SubMulAction.copy\n-/\n\n#print SubMulAction.coe_copy /-\n@[simp]\ntheorem coe_copy (p : SubMulAction R M) (s : Set M) (hs : s = \u2191p) : (p.copy s hs : Set M) = s :=\n  rfl\n#align sub_mul_action.coe_copy SubMulAction.coe_copy\n-/\n\n#print SubMulAction.copy_eq /-\ntheorem copy_eq (p : SubMulAction R M) (s : Set M) (hs : s = \u2191p) : p.copy s hs = p :=\n  SetLike.coe_injective hs\n#align sub_mul_action.copy_eq SubMulAction.copy_eq\n-/\n\ninstance : Bot (SubMulAction R M) :=\n  \u27e8{  carrier := \u2205\n      smul_mem' := fun c => Set.not_mem_empty }\u27e9\n\ninstance : Inhabited (SubMulAction R M) :=\n  \u27e8\u22a5\u27e9\n\nend SubMulAction\n\nnamespace SubMulAction\n\nsection SMul\n\nvariable [SMul R M]\n\nvariable (p : SubMulAction R M)\n\nvariable {r : R} {x : M}\n\n#print SubMulAction.smul_mem /-\ntheorem smul_mem (r : R) (h : x \u2208 p) : r \u2022 x \u2208 p :=\n  p.smul_mem' r h\n#align sub_mul_action.smul_mem SubMulAction.smul_mem\n-/\n\ninstance : SMul R p where smul c x := \u27e8c \u2022 x.1, smul_mem _ c x.2\u27e9\n\nvariable {p}\n\n#print SubMulAction.val_smul /-\n@[simp, norm_cast]\ntheorem val_smul (r : R) (x : p) : ((r \u2022 x : p) : M) = r \u2022 \u2191x :=\n  rfl\n#align sub_mul_action.coe_smul SubMulAction.val_smul\n-/\n\n/- warning: sub_mul_action.coe_mk clashes with [anonymous] -> [anonymous]\nwarning: sub_mul_action.coe_mk -> [anonymous] is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : SMul.{u1, u2} R M] {p : SubMulAction.{u1, u2} R M _inst_1} (x : M) (hx : Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p), Eq.{succ u2} M ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Subtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p)) M (HasLiftT.mk.{succ u2, succ u2} (Subtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p)) M (CoeTC\u2093.coe.{succ u2, succ u2} (Subtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p)) M (coeBase.{succ u2, succ u2} (Subtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p)) M (coeSubtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p))))) (Subtype.mk.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M _inst_1) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M _inst_1) M (SubMulAction.setLike.{u1, u2} R M _inst_1)) x p) x hx)) x\nbut is expected to have type\n  forall {R : Type.{u1}} {M : Type.{u2}}, (Nat -> R -> M) -> Nat -> (List.{u1} R) -> (List.{u2} M)\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.coe_mk [anonymous]\u2093'. -/\n@[simp, norm_cast]\ntheorem [anonymous] (x : M) (hx : x \u2208 p) : ((\u27e8x, hx\u27e9 : p) : M) = x :=\n  rfl\n#align sub_mul_action.coe_mk [anonymous]\n\nvariable (p)\n\n#print SubMulAction.subtype /-\n/-- Embedding of a submodule `p` to the ambient space `M`. -/\nprotected def subtype : p \u2192[R] M := by refine' { toFun := coe.. } <;> simp [coe_smul]\n#align sub_mul_action.subtype SubMulAction.subtype\n-/\n\n#print SubMulAction.subtype_apply /-\n@[simp]\ntheorem subtype_apply (x : p) : p.Subtype x = x :=\n  rfl\n#align sub_mul_action.subtype_apply SubMulAction.subtype_apply\n-/\n\n#print SubMulAction.subtype_eq_val /-\ntheorem subtype_eq_val : (SubMulAction.subtype p : p \u2192 M) = Subtype.val :=\n  rfl\n#align sub_mul_action.subtype_eq_val SubMulAction.subtype_eq_val\n-/\n\nend SMul\n\nnamespace SMulMemClass\n\nvariable [Monoid R] [MulAction R M] {A : Type _} [SetLike A M]\n\nvariable [hA : SMulMemClass A R M] (S' : A)\n\ninclude hA\n\n#print SubMulAction.SMulMemClass.toMulAction /-\n-- Prefer subclasses of `mul_action` over `smul_mem_class`.\n/-- A `sub_mul_action` of a `mul_action` is a `mul_action`.  -/\ninstance (priority := 75) toMulAction : MulAction R S' :=\n  Subtype.coe_injective.MulAction coe (SetLike.val_smul S')\n#align sub_mul_action.smul_mem_class.to_mul_action SubMulAction.SMulMemClass.toMulAction\n-/\n\n#print SubMulAction.SMulMemClass.subtype /-\n/-- The natural `mul_action_hom` over `R` from a `sub_mul_action` of `M` to `M`. -/\nprotected def subtype : S' \u2192[R] M :=\n  \u27e8coe, fun _ _ => rfl\u27e9\n#align sub_mul_action.smul_mem_class.subtype SubMulAction.SMulMemClass.subtype\n-/\n\n/- warning: sub_mul_action.smul_mem_class.coe_subtype -> SubMulAction.SMulMemClass.coeSubtype is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : MulAction.{u1, u2} R M _inst_1] {A : Type.{u3}} [_inst_3 : SetLike.{u3, u2} A M] [hA : SMulMemClass.{u3, u1, u2} A R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) _inst_3] (S' : A), Eq.{succ u2} ((fun (_x : MulActionHom.{u1, u2, u2} R (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') (SetLike.smul.{u1, u3, u2} A R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) _inst_3 hA S') M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) => (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') -> M) (SubMulAction.SMulMemClass.subtype.{u1, u2, u3} R M _inst_1 _inst_2 A _inst_3 hA S')) (coeFn.{succ u2, succ u2} (MulActionHom.{u1, u2, u2} R (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') (SetLike.smul.{u1, u3, u2} A R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) _inst_3 hA S') M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) (fun (_x : MulActionHom.{u1, u2, u2} R (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') (SetLike.smul.{u1, u3, u2} A R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) _inst_3 hA S') M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) => (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') -> M) ([anonymous].{u1, u2, u2} R (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') (SetLike.smul.{u1, u3, u2} A R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) _inst_3 hA S') M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) (SubMulAction.SMulMemClass.subtype.{u1, u2, u3} R M _inst_1 _inst_2 A _inst_3 hA S')) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') M (HasLiftT.mk.{succ u2, succ u2} (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') M (CoeTC\u2093.coe.{succ u2, succ u2} (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') M (coeBase.{succ u2, succ u2} (coeSort.{succ u3, succ (succ u2)} A Type.{u2} (SetLike.hasCoeToSort.{u3, u2} A M _inst_3) S') M (coeSubtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u3} M A (SetLike.hasMem.{u3, u2} A M _inst_3) x S'))))))\nbut is expected to have type\n  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Monoid.{u2} R] [_inst_2 : MulAction.{u2, u3} R M _inst_1] {A : Type.{u1}} [_inst_3 : SetLike.{u1, u3} A M] [hA : SMulMemClass.{u1, u2, u3} A R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2) _inst_3] (S' : A), Eq.{succ u3} (forall (a : Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')), (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) => M) a) (FunLike.coe.{succ u3, succ u3, succ u3} (MulActionHom.{u2, u3, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) (MulAction.toSMul.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) _inst_1 (SubMulAction.SMulMemClass.toMulAction.{u2, u3, u1} R M _inst_1 _inst_2 A _inst_3 hA S')) M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) (fun (_x : Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) => M) _x) (SMulHomClass.toFunLike.{u3, u2, u3, u3} (MulActionHom.{u2, u3, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) (MulAction.toSMul.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) _inst_1 (SubMulAction.SMulMemClass.toMulAction.{u2, u3, u1} R M _inst_1 _inst_2 A _inst_3 hA S')) M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) M (MulAction.toSMul.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) _inst_1 (SubMulAction.SMulMemClass.toMulAction.{u2, u3, u1} R M _inst_1 _inst_2 A _inst_3 hA S')) (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2) (instSMulHomClassMulActionHom.{u2, u3, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) (MulAction.toSMul.{u2, u3} R (Subtype.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S')) _inst_1 (SubMulAction.SMulMemClass.toMulAction.{u2, u3, u1} R M _inst_1 _inst_2 A _inst_3 hA S')) M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2))) (SubMulAction.SMulMemClass.subtype.{u2, u3, u1} R M _inst_1 _inst_2 A _inst_3 hA S')) (Subtype.val.{succ u3} M (fun (x : M) => Membership.mem.{u3, u1} M A (SetLike.instMembership.{u1, u3} A M _inst_3) x S'))\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.smul_mem_class.coe_subtype SubMulAction.SMulMemClass.coeSubtype\u2093'. -/\n@[simp]\nprotected theorem coeSubtype : (SMulMemClass.subtype S' : S' \u2192 M) = coe :=\n  rfl\n#align sub_mul_action.smul_mem_class.coe_subtype SubMulAction.SMulMemClass.coeSubtype\n\nend SMulMemClass\n\nsection MulActionMonoid\n\nvariable [Monoid R] [MulAction R M]\n\nsection\n\nvariable [SMul S R] [SMul S M] [IsScalarTower S R M]\n\nvariable (p : SubMulAction R M)\n\n#print SubMulAction.smul_of_tower_mem /-\ntheorem smul_of_tower_mem (s : S) {x : M} (h : x \u2208 p) : s \u2022 x \u2208 p :=\n  by\n  rw [\u2190 one_smul R x, \u2190 smul_assoc]\n  exact p.smul_mem _ h\n#align sub_mul_action.smul_of_tower_mem SubMulAction.smul_of_tower_mem\n-/\n\n#print SubMulAction.smul' /-\ninstance smul' : SMul S p where smul c x := \u27e8c \u2022 x.1, smul_of_tower_mem _ c x.2\u27e9\n#align sub_mul_action.has_smul' SubMulAction.smul'\n-/\n\ninstance : IsScalarTower S R p where smul_assoc s r x := Subtype.ext <| smul_assoc s r \u2191x\n\n#print SubMulAction.isScalarTower' /-\ninstance isScalarTower' {S' : Type _} [SMul S' R] [SMul S' S] [SMul S' M] [IsScalarTower S' R M]\n    [IsScalarTower S' S M] : IsScalarTower S' S p\n    where smul_assoc s r x := Subtype.ext <| smul_assoc s r \u2191x\n#align sub_mul_action.is_scalar_tower' SubMulAction.isScalarTower'\n-/\n\n#print SubMulAction.val_smul_of_tower /-\n@[simp, norm_cast]\ntheorem val_smul_of_tower (s : S) (x : p) : ((s \u2022 x : p) : M) = s \u2022 \u2191x :=\n  rfl\n#align sub_mul_action.coe_smul_of_tower SubMulAction.val_smul_of_tower\n-/\n\n/- warning: sub_mul_action.smul_mem_iff' -> SubMulAction.smul_mem_iff' is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Monoid.{u1} R] [_inst_2 : MulAction.{u1, u2} R M _inst_1] (p : SubMulAction.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) {G : Type.{u3}} [_inst_6 : Group.{u3} G] [_inst_7 : SMul.{u3, u1} G R] [_inst_8 : MulAction.{u3, u2} G M (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6))] [_inst_9 : IsScalarTower.{u3, u1, u2} G R M _inst_7 (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2) (MulAction.toHasSmul.{u3, u2} G M (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_8)] (g : G) {x : M}, Iff (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) M (SubMulAction.setLike.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2))) (SMul.smul.{u3, u2} G M (MulAction.toHasSmul.{u3, u2} G M (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_6)) _inst_8) g x) p) (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2)) M (SubMulAction.setLike.{u1, u2} R M (MulAction.toHasSmul.{u1, u2} R M _inst_1 _inst_2))) x p)\nbut is expected to have type\n  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : Monoid.{u2} R] [_inst_2 : MulAction.{u2, u3} R M _inst_1] (p : SubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) {G : Type.{u1}} [_inst_6 : Group.{u1} G] [_inst_7 : SMul.{u1, u2} G R] [_inst_8 : MulAction.{u1, u3} G M (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_6))] [_inst_9 : IsScalarTower.{u1, u2, u3} G R M _inst_7 (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2) (MulAction.toSMul.{u1, u3} G M (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_6)) _inst_8)] (g : G) {x : M}, Iff (Membership.mem.{u3, u3} M (SubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) (SetLike.instMembership.{u3, u3} (SubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) M (SubMulAction.instSetLikeSubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2))) (HSMul.hSMul.{u1, u3, u3} G M M (instHSMul.{u1, u3} G M (MulAction.toSMul.{u1, u3} G M (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_6)) _inst_8)) g x) p) (Membership.mem.{u3, u3} M (SubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) (SetLike.instMembership.{u3, u3} (SubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2)) M (SubMulAction.instSetLikeSubMulAction.{u2, u3} R M (MulAction.toSMul.{u2, u3} R M _inst_1 _inst_2))) x p)\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.smul_mem_iff' SubMulAction.smul_mem_iff'\u2093'. -/\n@[simp]\ntheorem smul_mem_iff' {G} [Group G] [SMul G R] [MulAction G M] [IsScalarTower G R M] (g : G)\n    {x : M} : g \u2022 x \u2208 p \u2194 x \u2208 p :=\n  \u27e8fun h => inv_smul_smul g x \u25b8 p.smul_of_tower_mem g\u207b\u00b9 h, p.smul_of_tower_mem g\u27e9\n#align sub_mul_action.smul_mem_iff' SubMulAction.smul_mem_iff'\n\ninstance [SMul S\u1d50\u1d52\u1d56 R] [SMul S\u1d50\u1d52\u1d56 M] [IsScalarTower S\u1d50\u1d52\u1d56 R M] [IsCentralScalar S M] :\n    IsCentralScalar S p where op_smul_eq_smul r x := Subtype.ext <| op_smul_eq_smul r x\n\nend\n\nsection\n\nvariable [Monoid S] [SMul S R] [MulAction S M] [IsScalarTower S R M]\n\nvariable (p : SubMulAction R M)\n\n#print SubMulAction.mulAction' /-\n/-- If the scalar product forms a `mul_action`, then the subset inherits this action -/\ninstance mulAction' : MulAction S p where\n  smul := (\u00b7 \u2022 \u00b7)\n  one_smul x := Subtype.ext <| one_smul _ x\n  mul_smul c\u2081 c\u2082 x := Subtype.ext <| mul_smul c\u2081 c\u2082 x\n#align sub_mul_action.mul_action' SubMulAction.mulAction'\n-/\n\ninstance : MulAction R p :=\n  p.mulAction'\n\nend\n\n#print SubMulAction.val_image_orbit /-\n/-- Orbits in a `sub_mul_action` coincide with orbits in the ambient space. -/\ntheorem val_image_orbit {p : SubMulAction R M} (m : p) :\n    coe '' MulAction.orbit R m = MulAction.orbit R (m : M) :=\n  (Set.range_comp _ _).symm\n#align sub_mul_action.coe_image_orbit SubMulAction.val_image_orbit\n-/\n\n#print SubMulAction.stabilizer_of_subMul.submonoid /-\n/- -- Previously, the relatively useless :\nlemma orbit_of_sub_mul {p : sub_mul_action R M} (m : p) :\n  (mul_action.orbit R m : set M) = mul_action.orbit R (m : M) := rfl\n-/\n/-- Stabilizers in monoid sub_mul_action coincide with stabilizers in the ambient space -/\ntheorem stabilizer_of_subMul.submonoid {p : SubMulAction R M} (m : p) :\n    MulAction.Stabilizer.submonoid R m = MulAction.Stabilizer.submonoid R (m : M) :=\n  by\n  ext\n  simp only [MulAction.mem_stabilizer_submonoid_iff, \u2190 SubMulAction.val_smul, SetLike.coe_eq_coe]\n#align sub_mul_action.stabilizer_of_sub_mul.submonoid SubMulAction.stabilizer_of_subMul.submonoid\n-/\n\nend MulActionMonoid\n\nsection MulActionGroup\n\nvariable [Group R] [MulAction R M]\n\n#print SubMulAction.stabilizer_of_subMul /-\n/-- Stabilizers in group sub_mul_action coincide with stabilizers in the ambient space -/\ntheorem stabilizer_of_subMul {p : SubMulAction R M} (m : p) :\n    MulAction.stabilizer R m = MulAction.stabilizer R (m : M) :=\n  by\n  rw [\u2190 Subgroup.toSubmonoid_eq]\n  exact stabilizer_of_sub_mul.submonoid m\n#align sub_mul_action.stabilizer_of_sub_mul SubMulAction.stabilizer_of_subMul\n-/\n\nend MulActionGroup\n\nsection Module\n\nvariable [Semiring R] [AddCommMonoid M]\n\nvariable [Module R M]\n\nvariable (p : SubMulAction R M)\n\n/- warning: sub_mul_action.zero_mem -> SubMulAction.zero_mem is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M _inst_1 _inst_2] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))), (Set.Nonempty.{u2} M ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) (Set.{u2} M) (CoeTC\u2093.coe.{succ u2, succ u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3)))))))) p)) -> (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3)))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))))) p)\nbut is expected to have type\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Semiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u1, u2} R M _inst_1 _inst_2] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))), (Set.Nonempty.{u2} M (SetLike.coe.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) p)) -> (Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u1, u2} R M _inst_1 _inst_2 _inst_3)))))) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) p)\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.zero_mem SubMulAction.zero_mem\u2093'. -/\ntheorem zero_mem (h : (p : Set M).Nonempty) : (0 : M) \u2208 p :=\n  let \u27e8x, hx\u27e9 := h\n  zero_smul R (x : M) \u25b8 p.smul_mem 0 hx\n#align sub_mul_action.zero_mem SubMulAction.zero_mem\n\n/-- If the scalar product forms a `module`, and the `sub_mul_action` is not `\u22a5`, then the\nsubset inherits the zero. -/\ninstance [n_empty : Nonempty p] : Zero p\n    where zero := \u27e80, n_empty.elim fun x => p.zero_mem \u27e8x, x.Prop\u27e9\u27e9\n\nend Module\n\nsection AddCommGroup\n\nvariable [Ring R] [AddCommGroup M]\n\nvariable [Module R M]\n\nvariable (p p' : SubMulAction R M)\n\nvariable {r : R} {x y : M}\n\n/- warning: sub_mul_action.neg_mem -> SubMulAction.neg_mem is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) {x : M}, (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p) -> (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (Neg.neg.{u2} M (SubNegMonoid.toHasNeg.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_2))) x) p)\nbut is expected to have type\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) {x : M}, (Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p) -> (Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (Neg.neg.{u2} M (NegZeroClass.toNeg.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) x) p)\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.neg_mem SubMulAction.neg_mem\u2093'. -/\ntheorem neg_mem (hx : x \u2208 p) : -x \u2208 p :=\n  by\n  rw [\u2190 neg_one_smul R]\n  exact p.smul_mem _ hx\n#align sub_mul_action.neg_mem SubMulAction.neg_mem\n\n/- warning: sub_mul_action.neg_mem_iff -> SubMulAction.neg_mem_iff is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) {x : M}, Iff (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (Neg.neg.{u2} M (SubNegMonoid.toHasNeg.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_2))) x) p) (Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p)\nbut is expected to have type\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) {x : M}, Iff (Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) (Neg.neg.{u2} M (NegZeroClass.toNeg.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) x) p) (Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p)\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.neg_mem_iff SubMulAction.neg_mem_iff\u2093'. -/\n@[simp]\ntheorem neg_mem_iff : -x \u2208 p \u2194 x \u2208 p :=\n  \u27e8fun h => by\n    rw [\u2190 neg_neg x]\n    exact neg_mem _ h, neg_mem _\u27e9\n#align sub_mul_action.neg_mem_iff SubMulAction.neg_mem_iff\n\ninstance : Neg p :=\n  \u27e8fun x => \u27e8-x.1, neg_mem _ x.2\u27e9\u27e9\n\n/- warning: sub_mul_action.coe_neg -> SubMulAction.val_neg is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (x : coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p), Eq.{succ u2} M ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (HasLiftT.mk.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (CoeTC\u2093.coe.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (coeBase.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (coeSubtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p))))) (Neg.neg.{u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) (SubMulAction.hasNeg.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) x)) (Neg.neg.{u2} M (SubNegMonoid.toHasNeg.{u2} M (AddGroup.toSubNegMonoid.{u2} M (AddCommGroup.toAddGroup.{u2} M _inst_2))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (HasLiftT.mk.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (CoeTC\u2093.coe.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (coeBase.{succ u2, succ u2} (coeSort.{succ u2, succ (succ u2)} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) Type.{u2} (SetLike.hasCoeToSort.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) p) M (coeSubtype.{succ u2} M (fun (x : M) => Membership.Mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.hasMem.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.setLike.{u1, u2} R M (SMulZeroClass.toHasSmul.{u1, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (SMulWithZero.toSmulZeroClass.{u1, u2} R M (MulZeroClass.toHasZero.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p))))) x))\nbut is expected to have type\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : AddCommGroup.{u2} M] [_inst_3 : Module.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2)] (p : SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (x : Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p)), Eq.{succ u2} M (Subtype.val.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Set.{u2} M) (Set.instMembershipSet.{u2} M) x (SetLike.coe.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p)) (Neg.neg.{u2} (Subtype.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) (SetLike.instMembership.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3)))))) x p)) (SubMulAction.instNegSubtypeMemSubMulActionToSMulToZeroToNegZeroClassToSubNegZeroMonoidToSubtractionMonoidToDivisionAddCommMonoidToSMulZeroClassToZeroToMonoidWithZeroToSemiringToSMulWithZeroToMulActionWithZeroToAddCommMonoidInstMembershipInstSetLikeSubMulAction.{u1, u2} R M _inst_1 _inst_2 _inst_3 p) x)) (Neg.neg.{u2} M (NegZeroClass.toNeg.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Subtype.val.{succ u2} M (fun (x : M) => Membership.mem.{u2, u2} M (Set.{u2} M) (Set.instMembershipSet.{u2} M) x (SetLike.coe.{u2, u2} (SubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) M (SubMulAction.instSetLikeSubMulAction.{u1, u2} R M (SMulZeroClass.toSMul.{u1, u2} R M (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (SMulWithZero.toSMulZeroClass.{u1, u2} R M (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1))) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (MulActionWithZero.toSMulWithZero.{u1, u2} R M (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R _inst_1)) (NegZeroClass.toZero.{u2} M (SubNegZeroMonoid.toNegZeroClass.{u2} M (SubtractionMonoid.toSubNegZeroMonoid.{u2} M (SubtractionCommMonoid.toSubtractionMonoid.{u2} M (AddCommGroup.toDivisionAddCommMonoid.{u2} M _inst_2))))) (Module.toMulActionWithZero.{u1, u2} R M (Ring.toSemiring.{u1} R _inst_1) (AddCommGroup.toAddCommMonoid.{u2} M _inst_2) _inst_3))))) p)) x))\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.coe_neg SubMulAction.val_neg\u2093'. -/\n@[simp, norm_cast]\ntheorem val_neg (x : p) : ((-x : p) : M) = -x :=\n  rfl\n#align sub_mul_action.coe_neg SubMulAction.val_neg\n\nend AddCommGroup\n\nend SubMulAction\n\nnamespace SubMulAction\n\nvariable [GroupWithZero S] [Monoid R] [MulAction R M]\n\nvariable [SMul S R] [MulAction S M] [IsScalarTower S R M]\n\nvariable (p : SubMulAction R M) {s : S} {x y : M}\n\n/- warning: sub_mul_action.smul_mem_iff -> SubMulAction.smul_mem_iff is a dubious translation:\nlean 3 declaration is\n  forall {S : Type.{u2}} {R : Type.{u1}} {M : Type.{u3}} [_inst_1 : GroupWithZero.{u2} S] [_inst_2 : Monoid.{u1} R] [_inst_3 : MulAction.{u1, u3} R M _inst_2] [_inst_4 : SMul.{u2, u1} S R] [_inst_5 : MulAction.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1))] [_inst_6 : IsScalarTower.{u2, u1, u3} S R M _inst_4 (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3) (MulAction.toHasSmul.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1)) _inst_5)] (p : SubMulAction.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3)) {s : S} {x : M}, (Ne.{succ u2} S s (OfNat.ofNat.{u2} S 0 (OfNat.mk.{u2} S 0 (Zero.zero.{u2} S (MulZeroClass.toHasZero.{u2} S (MulZeroOneClass.toMulZeroClass.{u2} S (MonoidWithZero.toMulZeroOneClass.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1)))))))) -> (Iff (Membership.Mem.{u3, u3} M (SubMulAction.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3)) (SetLike.hasMem.{u3, u3} (SubMulAction.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3)) M (SubMulAction.setLike.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3))) (SMul.smul.{u2, u3} S M (MulAction.toHasSmul.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1)) _inst_5) s x) p) (Membership.Mem.{u3, u3} M (SubMulAction.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3)) (SetLike.hasMem.{u3, u3} (SubMulAction.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3)) M (SubMulAction.setLike.{u1, u3} R M (MulAction.toHasSmul.{u1, u3} R M _inst_2 _inst_3))) x p))\nbut is expected to have type\n  forall {S : Type.{u2}} {R : Type.{u1}} {M : Type.{u3}} [_inst_1 : GroupWithZero.{u2} S] [_inst_2 : Monoid.{u1} R] [_inst_3 : MulAction.{u1, u3} R M _inst_2] [_inst_4 : SMul.{u2, u1} S R] [_inst_5 : MulAction.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1))] [_inst_6 : IsScalarTower.{u2, u1, u3} S R M _inst_4 (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3) (MulAction.toSMul.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1)) _inst_5)] (p : SubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3)) {s : S} {x : M}, (Ne.{succ u2} S s (OfNat.ofNat.{u2} S 0 (Zero.toOfNat0.{u2} S (MonoidWithZero.toZero.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1))))) -> (Iff (Membership.mem.{u3, u3} M (SubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3)) (SetLike.instMembership.{u3, u3} (SubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3)) M (SubMulAction.instSetLikeSubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3))) (HSMul.hSMul.{u2, u3, u3} S M M (instHSMul.{u2, u3} S M (MulAction.toSMul.{u2, u3} S M (MonoidWithZero.toMonoid.{u2} S (GroupWithZero.toMonoidWithZero.{u2} S _inst_1)) _inst_5)) s x) p) (Membership.mem.{u3, u3} M (SubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3)) (SetLike.instMembership.{u3, u3} (SubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3)) M (SubMulAction.instSetLikeSubMulAction.{u1, u3} R M (MulAction.toSMul.{u1, u3} R M _inst_2 _inst_3))) x p))\nCase conversion may be inaccurate. Consider using '#align sub_mul_action.smul_mem_iff SubMulAction.smul_mem_iff\u2093'. -/\ntheorem smul_mem_iff (s0 : s \u2260 0) : s \u2022 x \u2208 p \u2194 x \u2208 p :=\n  p.smul_mem_iff' (Units.mk0 s s0)\n#align sub_mul_action.smul_mem_iff SubMulAction.smul_mem_iff\n\nend SubMulAction\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/GroupTheory/GroupAction/SubMulAction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.3964230071391304}}
{"text": "example (P Q : Type) (p : P) (h : P \u2192 Q) : Q :=\nbegin\nexact h(p),\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/3-function-world/l1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.39642298969047857}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kenny Lau, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.chain\nimport Mathlib.data.list.nodup\nimport Mathlib.data.list.of_fn\nimport Mathlib.data.list.zip\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\nnamespace list\n\n\n/- iota and range(') -/\n\n@[simp] theorem length_range' (s : \u2115) (n : \u2115) : length (range' s n) = n := sorry\n\n@[simp] theorem range'_eq_nil {s : \u2115} {n : \u2115} : range' s n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (range' s n) = 0 \u2194 n = 0)) (length_range' s n))) (iff.refl (n = 0)))\n\n@[simp] theorem mem_range' {m : \u2115} {s : \u2115} {n : \u2115} : m \u2208 range' s n \u2194 s \u2264 m \u2227 m < s + n := sorry\n\ntheorem map_add_range' (a : \u2115) (s : \u2115) (n : \u2115) : map (Add.add a) (range' s n) = range' (a + s) n := sorry\n\ntheorem map_sub_range' (a : \u2115) (s : \u2115) (n : \u2115) (h : a \u2264 s) : map (fun (x : \u2115) => x - a) (range' s n) = range' (s - a) n := sorry\n\ntheorem chain_succ_range' (s : \u2115) (n : \u2115) : chain (fun (a b : \u2115) => b = Nat.succ a) s (range' (s + 1) n) := sorry\n\ntheorem chain_lt_range' (s : \u2115) (n : \u2115) : chain Less s (range' (s + 1) n) :=\n  chain.imp (fun (a b : \u2115) (e : b = Nat.succ a) => Eq.symm e \u25b8 nat.lt_succ_self a) (chain_succ_range' s n)\n\ntheorem pairwise_lt_range' (s : \u2115) (n : \u2115) : pairwise Less (range' s n) := sorry\n\ntheorem nodup_range' (s : \u2115) (n : \u2115) : nodup (range' s n) :=\n  pairwise.imp (fun (a b : \u2115) => ne_of_lt) (pairwise_lt_range' s n)\n\n@[simp] theorem range'_append (s : \u2115) (m : \u2115) (n : \u2115) : range' s m ++ range' (s + m) n = range' s (n + m) := sorry\n\ntheorem range'_sublist_right {s : \u2115} {m : \u2115} {n : \u2115} : range' s m <+ range' s n \u2194 m \u2264 n := sorry\n\ntheorem range'_subset_right {s : \u2115} {m : \u2115} {n : \u2115} : range' s m \u2286 range' s n \u2194 m \u2264 n := sorry\n\ntheorem nth_range' (s : \u2115) {m : \u2115} {n : \u2115} : m < n \u2192 nth (range' s n) m = some (s + m) := sorry\n\n@[simp] theorem nth_le_range' {n : \u2115} {m : \u2115} (i : \u2115) (H : i < length (range' n m)) : nth_le (range' n m) i H = n + i := sorry\n\ntheorem range'_concat (s : \u2115) (n : \u2115) : range' s (n + 1) = range' s n ++ [s + n] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s (n + 1) = range' s n ++ [s + n])) (add_comm n 1)))\n    (Eq.symm (range'_append s n 1))\n\ntheorem range_core_range' (s : \u2115) (n : \u2115) : range_core s (range' s n) = range' 0 (n + s) := sorry\n\ntheorem range_eq_range' (n : \u2115) : range n = range' 0 n :=\n  Eq.trans (range_core_range' n 0)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (range' 0 (0 + n) = range' 0 n)) (zero_add n))) (Eq.refl (range' 0 n)))\n\ntheorem range_succ_eq_map (n : \u2115) : range (n + 1) = 0 :: map Nat.succ (range n) := sorry\n\ntheorem range'_eq_map_range (s : \u2115) (n : \u2115) : range' s n = map (Add.add s) (range n) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s n = map (Add.add s) (range n))) (range_eq_range' n)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (range' s n = map (Add.add s) (range' 0 n))) (map_add_range' s 0 n)))\n      (Eq.refl (range' s n)))\n\n@[simp] theorem length_range (n : \u2115) : length (range n) = n := sorry\n\n@[simp] theorem range_eq_nil {n : \u2115} : range n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (range n) = 0 \u2194 n = 0)) (length_range n))) (iff.refl (n = 0)))\n\ntheorem pairwise_lt_range (n : \u2115) : pairwise Less (range n) := sorry\n\ntheorem nodup_range (n : \u2115) : nodup (range n) := sorry\n\ntheorem range_sublist {m : \u2115} {n : \u2115} : range m <+ range n \u2194 m \u2264 n := sorry\n\ntheorem range_subset {m : \u2115} {n : \u2115} : range m \u2286 range n \u2194 m \u2264 n := sorry\n\n@[simp] theorem mem_range {m : \u2115} {n : \u2115} : m \u2208 range n \u2194 m < n := sorry\n\n@[simp] theorem not_mem_range_self {n : \u2115} : \u00acn \u2208 range n :=\n  mt (iff.mp mem_range) (lt_irrefl n)\n\n@[simp] theorem self_mem_range_succ (n : \u2115) : n \u2208 range (n + 1) := sorry\n\ntheorem nth_range {m : \u2115} {n : \u2115} (h : m < n) : nth (range n) m = some m := sorry\n\ntheorem range_succ (n : \u2115) : range (Nat.succ n) = range n ++ [n] := sorry\n\n@[simp] theorem range_zero : range 0 = [] :=\n  rfl\n\ntheorem iota_eq_reverse_range' (n : \u2115) : iota n = reverse (range' 1 n) := sorry\n\n@[simp] theorem length_iota (n : \u2115) : length (iota n) = n := sorry\n\ntheorem pairwise_gt_iota (n : \u2115) : pairwise gt (iota n) := sorry\n\ntheorem nodup_iota (n : \u2115) : nodup (iota n) := sorry\n\ntheorem mem_iota {m : \u2115} {n : \u2115} : m \u2208 iota n \u2194 1 \u2264 m \u2227 m \u2264 n := sorry\n\ntheorem reverse_range' (s : \u2115) (n : \u2115) : reverse (range' s n) = map (fun (i : \u2115) => s + n - 1 - i) (range n) := sorry\n\n/-- All elements of `fin n`, from `0` to `n-1`. -/\ndef fin_range (n : \u2115) : List (fin n) :=\n  pmap fin.mk (range n) sorry\n\n@[simp] theorem fin_range_zero : fin_range 0 = [] :=\n  rfl\n\n@[simp] theorem mem_fin_range {n : \u2115} (a : fin n) : a \u2208 fin_range n := sorry\n\ntheorem nodup_fin_range (n : \u2115) : nodup (fin_range n) :=\n  nodup_pmap (fun (_x : \u2115) (_x_1 : _x < n) (_x_2 : \u2115) (_x_3 : _x_2 < n) => fin.veq_of_eq) (nodup_range n)\n\n@[simp] theorem length_fin_range (n : \u2115) : length (fin_range n) = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (fin_range n) = n)) (fin_range.equations._eqn_1 n)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (pmap fin.mk (range n) (fin_range._proof_1 n)) = n)) length_pmap))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length (range n) = n)) (length_range n))) (Eq.refl n)))\n\n@[simp] theorem fin_range_eq_nil {n : \u2115} : fin_range n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (fin_range n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (fin_range n) = 0 \u2194 n = 0)) (length_fin_range n))) (iff.refl (n = 0)))\n\ntheorem prod_range_succ {\u03b1 : Type u} [monoid \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) : prod (map f (range (Nat.succ n))) = prod (map f (range n)) * f n := sorry\n\n/-- A variant of `prod_range_succ` which pulls off the first\n  term in the product rather than the last.-/\ntheorem sum_range_succ' {\u03b1 : Type u} [add_monoid \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) : sum (map f (range (Nat.succ n))) = f 0 + sum (map (fun (i : \u2115) => f (Nat.succ i)) (range n)) := sorry\n\n@[simp] theorem enum_from_map_fst {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : map prod.fst (enum_from n l) = range' n (length l) := sorry\n\n@[simp] theorem enum_map_fst {\u03b1 : Type u} (l : List \u03b1) : map prod.fst (enum l) = range (length l) := sorry\n\ntheorem enum_eq_zip_range {\u03b1 : Type u} (l : List \u03b1) : enum l = zip (range (length l)) l :=\n  zip_of_prod (enum_map_fst l) (enum_map_snd l)\n\n@[simp] theorem unzip_enum_eq_prod {\u03b1 : Type u} (l : List \u03b1) : unzip (enum l) = (range (length l), l) := sorry\n\ntheorem enum_from_eq_zip_range' {\u03b1 : Type u} (l : List \u03b1) {n : \u2115} : enum_from n l = zip (range' n (length l)) l :=\n  zip_of_prod (enum_from_map_fst n l) (enum_from_map_snd n l)\n\n@[simp] theorem unzip_enum_from_eq_prod {\u03b1 : Type u} (l : List \u03b1) {n : \u2115} : unzip (enum_from n l) = (range' n (length l), l) := sorry\n\n@[simp] theorem nth_le_range {n : \u2115} (i : \u2115) (H : i < length (range n)) : nth_le (range n) i H = i := sorry\n\n@[simp] theorem nth_le_fin_range {n : \u2115} {i : \u2115} (h : i < length (fin_range n)) : nth_le (fin_range n) i h = { val := i, property := length_fin_range n \u25b8 h } := sorry\n\ntheorem of_fn_eq_pmap {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} : of_fn f = pmap (fun (i : \u2115) (hi : i < n) => f { val := i, property := hi }) (range n) fun (_x : \u2115) => iff.mp mem_range := sorry\n\ntheorem of_fn_id (n : \u2115) : of_fn id = fin_range n :=\n  of_fn_eq_pmap\n\ntheorem of_fn_eq_map {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} : of_fn f = map f (fin_range n) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = map f (fin_range n))) (Eq.symm (of_fn_id n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = map f (of_fn id))) (map_of_fn id f)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = of_fn (f \u2218 id))) (function.right_id f))) (Eq.refl (of_fn f))))\n\ntheorem nodup_of_fn {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} (hf : function.injective f) : nodup (of_fn f) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/range.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.39641294113717}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n\n# Preorder homomorphisms\n\nBundled monotone functions, `x \u2264 y \u2192 f x \u2264 f y`.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.function.iterate\nimport Mathlib.order.basic\nimport Mathlib.order.bounded_lattice\nimport Mathlib.order.complete_lattice\nimport Mathlib.tactic.monotonicity.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 \n\nnamespace Mathlib\n\n/-! # Category of preorders -/\n\n/-- Bundled monotone (aka, increasing) function -/\nstructure preorder_hom (\u03b1 : Type u_1) (\u03b2 : Type u_2) [preorder \u03b1] [preorder \u03b2] \nwhere\n  to_fun : \u03b1 \u2192 \u03b2\n  monotone' : monotone to_fun\n\ninfixr:25 \" \u2192\u2098 \" => Mathlib.preorder_hom\n\nnamespace preorder_hom\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] : has_coe_to_fun (\u03b1 \u2192\u2098 \u03b2) :=\n  has_coe_to_fun.mk (fun (f : \u03b1 \u2192\u2098 \u03b2) => \u03b1 \u2192 \u03b2) to_fun\n\ntheorem monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) : monotone \u21d1f :=\n  monotone' f\n\n@[simp] theorem coe_fun_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f) (x : \u03b1) : coe_fn (mk f hf) x = f x :=\n  rfl\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2) (h : \u2200 (a : \u03b1), coe_fn f a = coe_fn g a) : f = g := sorry\n\ntheorem coe_inj {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2) (h : \u21d1f = \u21d1g) : f = g :=\n  ext f g fun (a : \u03b1) => eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn f a = coe_fn g a)) h)) (Eq.refl (coe_fn g a))\n\n/-- The identity function as bundled monotone function. -/\ndef id {\u03b1 : Type u_1} [preorder \u03b1] : \u03b1 \u2192\u2098 \u03b1 :=\n  mk id monotone_id\n\nprotected instance inhabited {\u03b1 : Type u_1} [preorder \u03b1] : Inhabited (\u03b1 \u2192\u2098 \u03b1) :=\n  { default := id }\n\n@[simp] theorem coe_id {\u03b1 : Type u_1} [preorder \u03b1] : \u21d1id = \u21d1id :=\n  rfl\n\n/-- The composition of two bundled monotone functions. -/\ndef comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [preorder \u03b1] [preorder \u03b2] [preorder \u03b3] (g : \u03b2 \u2192\u2098 \u03b3) (f : \u03b1 \u2192\u2098 \u03b2) : \u03b1 \u2192\u2098 \u03b3 :=\n  mk (\u21d1g \u2218 \u21d1f) sorry\n\n@[simp] theorem comp_id {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) : comp f id = f :=\n  ext (comp f id) f fun (a : \u03b1) => Eq.refl (coe_fn (comp f id) a)\n\n@[simp] theorem id_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) : comp id f = f :=\n  ext (comp id f) f fun (a : \u03b1) => Eq.refl (coe_fn (comp id f) a)\n\n/-- `subtype.val` as a bundled monotone function.  -/\ndef subtype.val {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) : Subtype p \u2192\u2098 \u03b1 :=\n  mk subtype.val sorry\n\n/-- The preorder structure of `\u03b1 \u2192\u2098 \u03b2` is pointwise inequality: `f \u2264 g \u2194 \u2200 a, f a \u2264 g a`. -/\nprotected instance preorder {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] : preorder (\u03b1 \u2192\u2098 \u03b2) :=\n  preorder.lift to_fun\n\nprotected instance partial_order {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [partial_order \u03b2] : partial_order (\u03b1 \u2192\u2098 \u03b2) :=\n  partial_order.lift to_fun sorry\n\nprotected instance has_sup {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_sup \u03b2] : has_sup (\u03b1 \u2192\u2098 \u03b2) :=\n  has_sup.mk fun (f g : \u03b1 \u2192\u2098 \u03b2) => mk (fun (a : \u03b1) => coe_fn f a \u2294 coe_fn g a) sorry\n\nprotected instance semilattice_sup {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_sup \u03b2] : semilattice_sup (\u03b1 \u2192\u2098 \u03b2) :=\n  semilattice_sup.mk has_sup.sup partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry\n\n@[simp] theorem has_inf_inf_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_inf \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2) (a : \u03b1) : coe_fn (f \u2293 g) a = coe_fn f a \u2293 coe_fn g a :=\n  Eq.refl (coe_fn (f \u2293 g) a)\n\nprotected instance semilattice_inf {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_inf \u03b2] : semilattice_inf (\u03b1 \u2192\u2098 \u03b2) :=\n  semilattice_inf.mk has_inf.inf partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry\n\nprotected instance lattice {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [lattice \u03b2] : lattice (\u03b1 \u2192\u2098 \u03b2) :=\n  lattice.mk semilattice_sup.sup semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry sorry sorry\n    semilattice_inf.inf sorry sorry sorry\n\nprotected instance has_bot {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_bot \u03b2] : has_bot (\u03b1 \u2192\u2098 \u03b2) :=\n  has_bot.mk (mk (fun (a : \u03b1) => \u22a5) sorry)\n\nprotected instance order_bot {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_bot \u03b2] : order_bot (\u03b1 \u2192\u2098 \u03b2) :=\n  order_bot.mk \u22a5 partial_order.le partial_order.lt sorry sorry sorry sorry\n\n@[simp] theorem has_top_top_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_top \u03b2] (a : \u03b1) : coe_fn \u22a4 a = \u22a4 :=\n  Eq.refl (coe_fn \u22a4 a)\n\nprotected instance order_top {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_top \u03b2] : order_top (\u03b1 \u2192\u2098 \u03b2) :=\n  order_top.mk \u22a4 partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance has_Inf {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [complete_lattice \u03b2] : has_Inf (\u03b1 \u2192\u2098 \u03b2) :=\n  has_Inf.mk fun (s : set (\u03b1 \u2192\u2098 \u03b2)) => mk (fun (x : \u03b1) => Inf ((fun (f : \u03b1 \u2192\u2098 \u03b2) => coe_fn f x) '' s)) sorry\n\n@[simp] theorem has_Sup_Sup_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [complete_lattice \u03b2] (s : set (\u03b1 \u2192\u2098 \u03b2)) (x : \u03b1) : coe_fn (Sup s) x = Sup ((fun (f : \u03b1 \u2192\u2098 \u03b2) => coe_fn f x) '' s) :=\n  Eq.refl (coe_fn (Sup s) x)\n\nprotected instance complete_lattice {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [complete_lattice \u03b2] : complete_lattice (\u03b1 \u2192\u2098 \u03b2) :=\n  complete_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry lattice.inf sorry sorry sorry\n    order_top.top sorry order_bot.bot sorry Sup Inf sorry sorry sorry sorry\n\ntheorem iterate_sup_le_sup_iff {\u03b1 : Type u_1} [semilattice_sup \u03b1] (f : \u03b1 \u2192\u2098 \u03b1) : (\u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1), nat.iterate (\u21d1f) (n\u2081 + n\u2082) (a\u2081 \u2294 a\u2082) \u2264 nat.iterate (\u21d1f) n\u2081 a\u2081 \u2294 nat.iterate (\u21d1f) n\u2082 a\u2082) \u2194\n  \u2200 (a\u2081 a\u2082 : \u03b1), coe_fn f (a\u2081 \u2294 a\u2082) \u2264 coe_fn f a\u2081 \u2294 a\u2082 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/preorder_hom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.3964129363752316}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.basic\nimport Mathlib.data.set.finite\nimport Mathlib.group_theory.submonoid.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Pointwise addition, multiplication, and scalar multiplication of sets.\n\nThis file defines pointwise algebraic operations on sets.\n* For a type `\u03b1` with multiplication, multiplication is defined on `set \u03b1` by taking\n  `s * t` to be the set of all `x * y` where `x \u2208 s` and `y \u2208 t`. Similarly for addition.\n* For `\u03b1` a semigroup, `set \u03b1` is a semigroup.\n* If `\u03b1` is a (commutative) monoid, we define an alias `set_semiring \u03b1` for `set \u03b1`, which then\n  becomes a (commutative) semiring with union as addition and pointwise multiplication as\n  multiplication.\n* For a type `\u03b2` with scalar multiplication by another type `\u03b1`, this\n  file defines a scalar multiplication of `set \u03b2` by `set \u03b1` and a separate scalar\n  multiplication of `set \u03b2` by `\u03b1`.\n* We also define pointwise multiplication on `finset`.\n\nAppropriate definitions and results are also transported to the additive theory via `to_additive`.\n\n## Implementation notes\n* The following expressions are considered in simp-normal form in a group:\n  `(\u03bb h, h * g) \u207b\u00b9' s`, `(\u03bb h, g * h) \u207b\u00b9' s`, `(\u03bb h, h * g\u207b\u00b9) \u207b\u00b9' s`, `(\u03bb h, g\u207b\u00b9 * h) \u207b\u00b9' s`,\n  `s * t`, `s\u207b\u00b9`, `(1 : set _)` (and similarly for additive variants).\n  Expressions equal to one of these will be simplified.\n\n## Tags\n\nset multiplication, set addition, pointwise addition, pointwise multiplication\n\n-/\n\nnamespace set\n\n\n/-! ### Properties about 1 -/\n\nprotected instance has_one {\u03b1 : Type u_1} [HasOne \u03b1] : HasOne (set \u03b1) := { one := singleton 1 }\n\ntheorem singleton_one {\u03b1 : Type u_1} [HasOne \u03b1] : singleton 1 = 1 := rfl\n\n@[simp] theorem mem_zero {\u03b1 : Type u_1} {a : \u03b1} [HasZero \u03b1] : a \u2208 0 \u2194 a = 0 := iff.rfl\n\ntheorem one_mem_one {\u03b1 : Type u_1} [HasOne \u03b1] : 1 \u2208 1 := Eq.refl 1\n\n@[simp] theorem zero_subset {\u03b1 : Type u_1} {s : set \u03b1} [HasZero \u03b1] : 0 \u2286 s \u2194 0 \u2208 s :=\n  singleton_subset_iff\n\ntheorem zero_nonempty {\u03b1 : Type u_1} [HasZero \u03b1] : set.nonempty 0 := Exists.intro 0 rfl\n\n@[simp] theorem image_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasZero \u03b1] {f : \u03b1 \u2192 \u03b2} :\n    f '' 0 = singleton (f 0) :=\n  image_singleton\n\n/-! ### Properties about multiplication -/\n\nprotected instance has_add {\u03b1 : Type u_1} [Add \u03b1] : Add (set \u03b1) := { add := image2 Add.add }\n\n@[simp] theorem image2_mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] :\n    image2 Mul.mul s t = s * t :=\n  rfl\n\ntheorem mem_add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {a : \u03b1} [Add \u03b1] :\n    a \u2208 s + t \u2194 \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2208 s \u2227 y \u2208 t \u2227 x + y = a :=\n  iff.rfl\n\ntheorem mul_mem_mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {a : \u03b1} {b : \u03b1} [Mul \u03b1] (ha : a \u2208 s)\n    (hb : b \u2208 t) : a * b \u2208 s * t :=\n  mem_image2_of_mem ha hb\n\ntheorem add_image_prod {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] :\n    (fun (x : \u03b1 \u00d7 \u03b1) => prod.fst x + prod.snd x) '' set.prod s t = s + t :=\n  image_prod Add.add\n\n@[simp] theorem image_mul_left {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [group \u03b1] :\n    (fun (b : \u03b1) => a * b) '' t = (fun (b : \u03b1) => a\u207b\u00b9 * b) \u207b\u00b9' t :=\n  sorry\n\n@[simp] theorem image_add_right {\u03b1 : Type u_1} {t : set \u03b1} {b : \u03b1} [add_group \u03b1] :\n    (fun (a : \u03b1) => a + b) '' t = (fun (a : \u03b1) => a + -b) \u207b\u00b9' t :=\n  sorry\n\ntheorem image_add_left' {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [add_group \u03b1] :\n    (fun (b : \u03b1) => -a + b) '' t = (fun (b : \u03b1) => a + b) \u207b\u00b9' t :=\n  sorry\n\ntheorem image_mul_right' {\u03b1 : Type u_1} {t : set \u03b1} {b : \u03b1} [group \u03b1] :\n    (fun (a : \u03b1) => a * (b\u207b\u00b9)) '' t = (fun (a : \u03b1) => a * b) \u207b\u00b9' t :=\n  sorry\n\n@[simp] theorem preimage_add_left_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [add_group \u03b1] :\n    Add.add a \u207b\u00b9' singleton b = singleton (-a + b) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (Add.add a \u207b\u00b9' singleton b = singleton (-a + b)))\n        (Eq.symm image_add_left')))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl ((fun (b : \u03b1) => -a + b) '' singleton b = singleton (-a + b)))\n          image_singleton))\n      (Eq.refl (singleton (-a + b))))\n\n@[simp] theorem preimage_mul_right_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [group \u03b1] :\n    (fun (_x : \u03b1) => _x * a) \u207b\u00b9' singleton b = singleton (b * (a\u207b\u00b9)) :=\n  sorry\n\n@[simp] theorem preimage_add_left_zero {\u03b1 : Type u_1} {a : \u03b1} [add_group \u03b1] :\n    (fun (b : \u03b1) => a + b) \u207b\u00b9' 0 = singleton (-a) :=\n  sorry\n\n@[simp] theorem preimage_mul_right_one {\u03b1 : Type u_1} {b : \u03b1} [group \u03b1] :\n    (fun (a : \u03b1) => a * b) \u207b\u00b9' 1 = singleton (b\u207b\u00b9) :=\n  sorry\n\ntheorem preimage_add_left_zero' {\u03b1 : Type u_1} {a : \u03b1} [add_group \u03b1] :\n    (fun (b : \u03b1) => -a + b) \u207b\u00b9' 0 = singleton a :=\n  sorry\n\ntheorem preimage_add_right_zero' {\u03b1 : Type u_1} {b : \u03b1} [add_group \u03b1] :\n    (fun (a : \u03b1) => a + -b) \u207b\u00b9' 0 = singleton b :=\n  sorry\n\n@[simp] theorem mul_singleton {\u03b1 : Type u_1} {s : set \u03b1} {b : \u03b1} [Mul \u03b1] :\n    s * singleton b = (fun (a : \u03b1) => a * b) '' s :=\n  image2_singleton_right\n\n@[simp] theorem singleton_add {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [Add \u03b1] :\n    singleton a + t = (fun (b : \u03b1) => a + b) '' t :=\n  image2_singleton_left\n\n@[simp] theorem singleton_add_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [Add \u03b1] :\n    singleton a + singleton b = singleton (a + b) :=\n  image2_singleton\n\nprotected instance semigroup {\u03b1 : Type u_1} [semigroup \u03b1] : semigroup (set \u03b1) :=\n  semigroup.mk Mul.mul sorry\n\nprotected instance monoid {\u03b1 : Type u_1} [monoid \u03b1] : monoid (set \u03b1) :=\n  monoid.mk semigroup.mul sorry 1 sorry sorry\n\nprotected theorem mul_comm {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [comm_semigroup \u03b1] :\n    s * t = t * s :=\n  sorry\n\nprotected instance add_comm_monoid {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : add_comm_monoid (set \u03b1) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\ntheorem singleton.is_mul_hom {\u03b1 : Type u_1} [Mul \u03b1] : is_mul_hom singleton :=\n  is_mul_hom.mk fun (a b : \u03b1) => Eq.symm singleton_mul_singleton\n\n@[simp] theorem empty_add {\u03b1 : Type u_1} {s : set \u03b1} [Add \u03b1] : \u2205 + s = \u2205 := image2_empty_left\n\n@[simp] theorem mul_empty {\u03b1 : Type u_1} {s : set \u03b1} [Mul \u03b1] : s * \u2205 = \u2205 := image2_empty_right\n\ntheorem add_subset_add {\u03b1 : Type u_1} {s\u2081 : set \u03b1} {s\u2082 : set \u03b1} {t\u2081 : set \u03b1} {t\u2082 : set \u03b1} [Add \u03b1]\n    (h\u2081 : s\u2081 \u2286 t\u2081) (h\u2082 : s\u2082 \u2286 t\u2082) : s\u2081 + s\u2082 \u2286 t\u2081 + t\u2082 :=\n  image2_subset h\u2081 h\u2082\n\ntheorem union_add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {u : set \u03b1} [Add \u03b1] :\n    s \u222a t + u = s + u \u222a (t + u) :=\n  image2_union_left\n\ntheorem mul_union {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {u : set \u03b1} [Mul \u03b1] :\n    s * (t \u222a u) = s * t \u222a s * u :=\n  image2_union_right\n\ntheorem Union_mul_left_image {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] :\n    (Union fun (a : \u03b1) => Union fun (H : a \u2208 s) => (fun (x : \u03b1) => a * x) '' t) = s * t :=\n  Union_image_left fun (a x : \u03b1) => a * x\n\ntheorem Union_mul_right_image {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] :\n    (Union fun (a : \u03b1) => Union fun (H : a \u2208 t) => (fun (x : \u03b1) => x * a) '' s) = s * t :=\n  Union_image_right fun (x a : \u03b1) => x * a\n\n@[simp] theorem univ_mul_univ {\u03b1 : Type u_1} [monoid \u03b1] : univ * univ = univ := sorry\n\n/-- `singleton` is a monoid hom. -/\ndef singleton_add_hom {\u03b1 : Type u_1} [add_monoid \u03b1] : \u03b1 \u2192+ set \u03b1 :=\n  add_monoid_hom.mk singleton sorry sorry\n\ntheorem nonempty.add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] :\n    set.nonempty s \u2192 set.nonempty t \u2192 set.nonempty (s + t) :=\n  nonempty.image2\n\ntheorem finite.mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] (hs : finite s) (ht : finite t) :\n    finite (s * t) :=\n  finite.image2 (fun (a b : \u03b1) => a * b) hs ht\n\n/-- multiplication preserves finiteness -/\ndef fintype_mul {\u03b1 : Type u_1} [Mul \u03b1] [DecidableEq \u03b1] (s : set \u03b1) (t : set \u03b1) [hs : fintype \u21a5s]\n    [ht : fintype \u21a5t] : fintype \u21a5(s * t) :=\n  set.fintype_image2 (fun (a b : \u03b1) => a * b) s t\n\ntheorem bdd_above_add {\u03b1 : Type u_1} [ordered_add_comm_monoid \u03b1] {A : set \u03b1} {B : set \u03b1} :\n    bdd_above A \u2192 bdd_above B \u2192 bdd_above (A + B) :=\n  sorry\n\n/-! ### Properties about inversion -/\n\nprotected instance has_inv {\u03b1 : Type u_1} [has_inv \u03b1] : has_inv (set \u03b1) :=\n  has_inv.mk (preimage has_inv.inv)\n\n@[simp] theorem mem_inv {\u03b1 : Type u_1} {s : set \u03b1} {a : \u03b1} [has_inv \u03b1] : a \u2208 (s\u207b\u00b9) \u2194 a\u207b\u00b9 \u2208 s :=\n  iff.rfl\n\ntheorem inv_mem_inv {\u03b1 : Type u_1} {s : set \u03b1} {a : \u03b1} [group \u03b1] : a\u207b\u00b9 \u2208 (s\u207b\u00b9) \u2194 a \u2208 s := sorry\n\n@[simp] theorem inv_preimage {\u03b1 : Type u_1} {s : set \u03b1} [has_inv \u03b1] : has_inv.inv \u207b\u00b9' s = (s\u207b\u00b9) :=\n  rfl\n\n@[simp] theorem image_inv {\u03b1 : Type u_1} {s : set \u03b1} [group \u03b1] : has_inv.inv '' s = (s\u207b\u00b9) := sorry\n\n@[simp] theorem inter_neg {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Neg \u03b1] : -(s \u2229 t) = -s \u2229 -t :=\n  preimage_inter\n\n@[simp] theorem union_neg {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Neg \u03b1] : -(s \u222a t) = -s \u222a -t :=\n  preimage_union\n\n@[simp] theorem compl_inv {\u03b1 : Type u_1} {s : set \u03b1} [has_inv \u03b1] : s\u1d9c\u207b\u00b9 = (s\u207b\u00b9\u1d9c) := preimage_compl\n\n@[simp] protected theorem inv_inv {\u03b1 : Type u_1} {s : set \u03b1} [group \u03b1] : s\u207b\u00b9\u207b\u00b9 = s := sorry\n\n@[simp] protected theorem univ_inv {\u03b1 : Type u_1} [group \u03b1] : univ\u207b\u00b9 = univ := preimage_univ\n\n@[simp] theorem neg_subset_neg {\u03b1 : Type u_1} [add_group \u03b1] {s : set \u03b1} {t : set \u03b1} :\n    -s \u2286 -t \u2194 s \u2286 t :=\n  function.surjective.preimage_subset_preimage_iff (equiv.surjective (equiv.neg \u03b1))\n\ntheorem neg_subset {\u03b1 : Type u_1} [add_group \u03b1] {s : set \u03b1} {t : set \u03b1} : -s \u2286 t \u2194 s \u2286 -t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (-s \u2286 t \u2194 s \u2286 -t)) (Eq.symm (propext neg_subset_neg))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ( --s \u2286 -t \u2194 s \u2286 -t)) set.neg_neg)) (iff.refl (s \u2286 -t)))\n\n/-! ### Properties about scalar multiplication -/\n\n/-- Scaling a set: multiplying every element by a scalar. -/\nprotected instance has_scalar_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] :\n    has_scalar \u03b1 (set \u03b2) :=\n  has_scalar.mk fun (a : \u03b1) => image (has_scalar.smul a)\n\n@[simp] theorem image_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    (fun (x : \u03b2) => a \u2022 x) '' t = a \u2022 t :=\n  rfl\n\ntheorem mem_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} {x : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    x \u2208 a \u2022 t \u2194 \u2203 (y : \u03b2), y \u2208 t \u2227 a \u2022 y = x :=\n  iff.rfl\n\ntheorem smul_mem_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} {y : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2}\n    (hy : y \u2208 t) : a \u2022 y \u2208 a \u2022 t :=\n  Exists.intro y { left := hy, right := rfl }\n\ntheorem smul_set_union {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {s : set \u03b2}\n    {t : set \u03b2} : a \u2022 (s \u222a t) = a \u2022 s \u222a a \u2022 t :=\n  sorry\n\n@[simp] theorem smul_set_empty {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] (a : \u03b1) : a \u2022 \u2205 = \u2205 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a \u2022 \u2205 = \u2205)) (Eq.symm image_smul)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl ((fun (x : \u03b2) => a \u2022 x) '' \u2205 = \u2205)) (image_empty fun (x : \u03b2) => a \u2022 x)))\n      (Eq.refl \u2205))\n\ntheorem smul_set_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {s : set \u03b2} {t : set \u03b2}\n    (h : s \u2286 t) : a \u2022 s \u2286 a \u2022 t :=\n  sorry\n\n/-- Pointwise scalar multiplication by a set of scalars. -/\nprotected instance has_scalar {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] :\n    has_scalar (set \u03b1) (set \u03b2) :=\n  has_scalar.mk (image2 has_scalar.smul)\n\n@[simp] theorem image2_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    image2 has_scalar.smul s t = s \u2022 t :=\n  rfl\n\ntheorem mem_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} {x : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    x \u2208 s \u2022 t \u2194 \u2203 (a : \u03b1), \u2203 (y : \u03b2), a \u2208 s \u2227 y \u2208 t \u2227 a \u2022 y = x :=\n  iff.rfl\n\ntheorem image_smul_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    (fun (x : \u03b1 \u00d7 \u03b2) => prod.fst x \u2022 prod.snd x) '' set.prod s t = s \u2022 t :=\n  image_prod has_scalar.smul\n\ntheorem range_smul_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] {\u03b9 : Type u_3}\n    {\u03ba : Type u_4} (b : \u03b9 \u2192 \u03b1) (c : \u03ba \u2192 \u03b2) :\n    range b \u2022 range c = range fun (p : \u03b9 \u00d7 \u03ba) => b (prod.fst p) \u2022 c (prod.snd p) :=\n  sorry\n\ntheorem singleton_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} :\n    singleton a \u2022 t = a \u2022 t :=\n  image2_singleton_left\n\n/-! ### `set \u03b1` as a `(\u222a,*)`-semiring -/\n\n/-- An alias for `set \u03b1`, which has a semiring structure given by `\u222a` as \"addition\" and pointwise\n  multiplication `*` as \"multiplication\". -/\ndef set_semiring (\u03b1 : Type u_1) := set \u03b1\n\n/-- The identitiy function `set \u03b1 \u2192 set_semiring \u03b1`. -/\n/-- The identitiy function `set_semiring \u03b1 \u2192 set \u03b1`. -/\nprotected def up {\u03b1 : Type u_1} (s : set \u03b1) : set_semiring \u03b1 := s\n\nprotected def set_semiring.down {\u03b1 : Type u_1} (s : set_semiring \u03b1) : set \u03b1 := s\n\n@[simp] protected theorem down_up {\u03b1 : Type u_1} {s : set \u03b1} : set_semiring.down (set.up s) = s :=\n  rfl\n\n@[simp] protected theorem up_down {\u03b1 : Type u_1} {s : set_semiring \u03b1} :\n    set.up (set_semiring.down s) = s :=\n  rfl\n\nprotected instance set_semiring.semiring {\u03b1 : Type u_1} [monoid \u03b1] : semiring (set_semiring \u03b1) :=\n  semiring.mk (fun (s t : set_semiring \u03b1) => s \u222a t) union_assoc \u2205 empty_union union_empty union_comm\n    monoid.mul sorry monoid.one sorry sorry sorry sorry sorry sorry\n\nprotected instance set_semiring.comm_semiring {\u03b1 : Type u_1} [comm_monoid \u03b1] :\n    comm_semiring (set_semiring \u03b1) :=\n  comm_semiring.mk semiring.add sorry semiring.zero sorry sorry sorry comm_monoid.mul sorry\n    comm_monoid.one sorry sorry sorry sorry sorry sorry sorry\n\n/-- A multiplicative action of a monoid on a type \u03b2 gives also a\n multiplicative action on the subsets of \u03b2. -/\nprotected instance mul_action_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [monoid \u03b1] [mul_action \u03b1 \u03b2] :\n    mul_action \u03b1 (set \u03b2) :=\n  mul_action.mk sorry sorry\n\ntheorem image_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] [Add \u03b2] (m : \u03b1 \u2192 \u03b2)\n    [is_add_hom m] : m '' (s + t) = m '' s + m '' t :=\n  sorry\n\ntheorem preimage_mul_preimage_subset {\u03b1 : Type u_1} {\u03b2 : Type u_2} [Mul \u03b1] [Mul \u03b2] (m : \u03b1 \u2192 \u03b2)\n    [is_mul_hom m] {s : set \u03b2} {t : set \u03b2} : m \u207b\u00b9' s * m \u207b\u00b9' t \u2286 m \u207b\u00b9' (s * t) :=\n  sorry\n\n/-- The image of a set under function is a ring homomorphism\nwith respect to the pointwise operations on sets. -/\ndef image_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192* \u03b2) :\n    set_semiring \u03b1 \u2192+* set_semiring \u03b2 :=\n  ring_hom.mk (image \u21d1f) sorry sorry sorry sorry\n\nend set\n\n\n/-- A nonempty set in a semimodule is scaled by zero to the singleton\ncontaining 0 in the semimodule. -/\ntheorem zero_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b1] [add_comm_monoid \u03b2]\n    [semimodule \u03b1 \u03b2] {s : set \u03b2} (h : set.nonempty s) : 0 \u2022 s = 0 :=\n  sorry\n\ntheorem mem_inv_smul_set_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [field \u03b1] [mul_action \u03b1 \u03b2] {a : \u03b1}\n    (ha : a \u2260 0) (A : set \u03b2) (x : \u03b2) : x \u2208 a\u207b\u00b9 \u2022 A \u2194 a \u2022 x \u2208 A :=\n  sorry\n\ntheorem mem_smul_set_iff_inv_smul_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} [field \u03b1] [mul_action \u03b1 \u03b2]\n    {a : \u03b1} (ha : a \u2260 0) (A : set \u03b2) (x : \u03b2) : x \u2208 a \u2022 A \u2194 a\u207b\u00b9 \u2022 x \u2208 A :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (x \u2208 a \u2022 A \u2194 a\u207b\u00b9 \u2022 x \u2208 A))\n        (Eq.symm (propext (mem_inv_smul_set_iff (inv_ne_zero ha) A x)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2208 a \u2022 A \u2194 x \u2208 a\u207b\u00b9\u207b\u00b9 \u2022 A)) (inv_inv' a)))\n      (iff.refl (x \u2208 a \u2022 A)))\n\nnamespace finset\n\n\n/-- The pointwise product of two finite sets `s` and `t`:\n  `st = s \u2b1d t = s * t = { x * y | x \u2208 s, y \u2208 t }`. -/\nprotected instance has_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] : Add (finset \u03b1) :=\n  { add :=\n      fun (s t : finset \u03b1) =>\n        image (fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p + prod.snd p) (finset.product s t) }\n\ntheorem mul_def {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} :\n    s * t = image (fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p * prod.snd p) (finset.product s t) :=\n  rfl\n\ntheorem mem_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} {x : \u03b1} :\n    x \u2208 s + t \u2194 \u2203 (y : \u03b1), \u2203 (z : \u03b1), y \u2208 s \u2227 z \u2208 t \u2227 y + z = x :=\n  sorry\n\n@[simp] theorem coe_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} :\n    \u2191(s + t) = \u2191s + \u2191t :=\n  sorry\n\ntheorem mul_mem_mul {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} {x : \u03b1}\n    {y : \u03b1} (hx : x \u2208 s) (hy : y \u2208 t) : x * y \u2208 s * t :=\n  eq.mpr (id (propext mem_mul))\n    (Exists.intro x (Exists.intro y { left := hx, right := { left := hy, right := rfl } }))\n\ntheorem add_card_le {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} :\n    card (s + t) \u2264 card s * card t :=\n  sorry\n\ntheorem mul_card_le {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} :\n    card (s * t) \u2264 card s * card t :=\n  sorry\n\n/-- A finite set `U` contained in the product of two sets `S * S'` is also contained in the product\nof two finite sets `T * T' \u2286 S * S'`. -/\ntheorem subset_add {M : Type u_1} [add_monoid M] {S : set M} {S' : set M} {U : finset M}\n    (f : \u2191U \u2286 S + S') : \u2203 (T : finset M), \u2203 (T' : finset M), \u2191T \u2286 S \u2227 \u2191T' \u2286 S' \u2227 U \u2286 T + T' :=\n  sorry\n\nend finset\n\n\n/-! Some lemmas about pointwise multiplication and submonoids. Ideally we put these in\n  `group_theory.submonoid.basic`, but currently we cannot because that file is imported by this. -/\n\nnamespace submonoid\n\n\ntheorem mul_subset {M : Type u_1} [monoid M] {s : set M} {t : set M} {S : submonoid M} (hs : s \u2286 \u2191S)\n    (ht : t \u2286 \u2191S) : s * t \u2286 \u2191S :=\n  sorry\n\ntheorem mul_subset_closure {M : Type u_1} [monoid M] {s : set M} {t : set M} {u : set M}\n    (hs : s \u2286 u) (ht : t \u2286 u) : s * t \u2286 \u2191(closure u) :=\n  mul_subset (set.subset.trans hs subset_closure) (set.subset.trans ht subset_closure)\n\ntheorem Mathlib.add_submonoid.coe_add_self_eq {M : Type u_1} [add_monoid M] (s : add_submonoid M) :\n    \u2191s + \u2191s = \u2191s :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/pointwise_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.39641293212969425}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.embedding\nimport Mathlib.order.rel_classes\nimport Mathlib.data.set.intervals.basic\nimport Mathlib.PostPort\n\nuniverses u_4 u_5 l u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-- A relation homomorphism with respect to a given pair of relations `r` and `s`\nis a function `f : \u03b1 \u2192 \u03b2` such that `r a b \u2192 s (f a) (f b)`. -/\nstructure rel_hom {\u03b1 : Type u_4} {\u03b2 : Type u_5} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) \nwhere\n  to_fun : \u03b1 \u2192 \u03b2\n  map_rel' : \u2200 {a b : \u03b1}, r a b \u2192 s (to_fun a) (to_fun b)\n\ninfixl:25 \" \u2192r \" => Mathlib.rel_hom\n\nnamespace rel_hom\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : has_coe_to_fun (r \u2192r s) :=\n  has_coe_to_fun.mk (fun (_x : r \u2192r s) => \u03b1 \u2192 \u03b2) fun (o : r \u2192r s) => to_fun o\n\ntheorem map_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) {a : \u03b1} {b : \u03b1} : r a b \u2192 s (coe_fn f a) (coe_fn f b) :=\n  map_rel' f\n\n@[simp] theorem coe_fn_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) (o : \u2200 {a b : \u03b1}, r a b \u2192 s (f a) (f b)) : \u21d1(mk f o) = f :=\n  rfl\n\n@[simp] theorem coe_fn_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) : to_fun f = \u21d1f :=\n  rfl\n\n/-- The map `coe_fn : (r \u2192r s) \u2192 (\u03b1 \u2192 \u03b2)` is injective. We can't use `function.injective`\nhere but mimic its signature by using `\u2983e\u2081 e\u2082\u2984`. -/\ntheorem coe_fn_inj {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {e\u2081 : r \u2192r s} {e\u2082 : r \u2192r s} : \u21d1e\u2081 = \u21d1e\u2082 \u2192 e\u2081 = e\u2082 := sorry\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u2192r s} {g : r \u2192r s} (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) : f = g :=\n  coe_fn_inj (funext h)\n\ntheorem ext_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u2192r s} {g : r \u2192r s} : f = g \u2194 \u2200 (x : \u03b1), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : \u03b1) => h \u25b8 rfl, mpr := fun (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) => ext h }\n\n/-- Identity map is a relation homomorphism. -/\nprotected def id {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : r \u2192r r :=\n  mk id sorry\n\n/-- Composition of two relation homomorphisms is a relation homomorphism. -/\nprotected def comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (g : s \u2192r t) (f : r \u2192r s) : r \u2192r t :=\n  mk (to_fun g \u2218 to_fun f) sorry\n\n@[simp] theorem id_apply {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (x : \u03b1) : coe_fn (rel_hom.id r) x = x :=\n  rfl\n\n@[simp] theorem comp_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (g : s \u2192r t) (f : r \u2192r s) (a : \u03b1) : coe_fn (rel_hom.comp g f) a = coe_fn g (coe_fn f a) :=\n  rfl\n\n/-- A relation homomorphism is also a relation homomorphism between dual relations. -/\nprotected def swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) : function.swap r \u2192r function.swap s :=\n  mk \u21d1f sorry\n\n/-- A function is a relation homomorphism from the preimage relation of `s` to `s`. -/\ndef preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : f \u207b\u00b9'o s \u2192r s :=\n  mk f sorry\n\nprotected theorem is_irrefl {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) [is_irrefl \u03b2 s] : is_irrefl \u03b1 r := sorry\n\nprotected theorem is_asymm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) [is_asymm \u03b2 s] : is_asymm \u03b1 r := sorry\n\nprotected theorem acc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) (a : \u03b1) : acc s (coe_fn f a) \u2192 acc r a := sorry\n\nprotected theorem well_founded {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2192r s) (h : well_founded s) : well_founded r :=\n  well_founded.dcases_on h\n    fun (h : \u2200 (a : \u03b2), acc s a) =>\n      idRhs (well_founded r) (well_founded.intro fun (a : \u03b1) => rel_hom.acc f a (h (coe_fn f a)))\n\ntheorem map_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_inf \u03b1] [linear_order \u03b2] (a : Less \u2192r Less) (m : \u03b2) (n : \u03b2) : coe_fn a (m \u2293 n) = coe_fn a m \u2293 coe_fn a n := sorry\n\ntheorem map_sup {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_sup \u03b1] [linear_order \u03b2] (a : gt \u2192r gt) (m : \u03b2) (n : \u03b2) : coe_fn a (m \u2294 n) = coe_fn a m \u2294 coe_fn a n := sorry\n\nend rel_hom\n\n\n/-- An increasing function is injective -/\ntheorem injective_of_increasing {\u03b1 : Type u_1} {\u03b2 : Type u_2} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) [is_trichotomous \u03b1 r] [is_irrefl \u03b2 s] (f : \u03b1 \u2192 \u03b2) (hf : \u2200 {x y : \u03b1}, r x y \u2192 s (f x) (f y)) : function.injective f := sorry\n\n/-- An increasing function is injective -/\ntheorem rel_hom.injective_of_increasing {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} [is_trichotomous \u03b1 r] [is_irrefl \u03b2 s] (f : r \u2192r s) : function.injective \u21d1f :=\n  injective_of_increasing r s \u21d1f fun (x y : \u03b1) => rel_hom.map_rel f\n\ntheorem surjective.well_founded_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} (hf : function.surjective f) (o : \u2200 {a b : \u03b1}, r a b \u2194 s (f a) (f b)) : well_founded r \u2194 well_founded s := sorry\n\n/-- A relation embedding with respect to a given pair of relations `r` and `s`\nis an embedding `f : \u03b1 \u21aa \u03b2` such that `r a b \u2194 s (f a) (f b)`. -/\nstructure rel_embedding {\u03b1 : Type u_4} {\u03b2 : Type u_5} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) \nextends \u03b1 \u21aa \u03b2\nwhere\n  map_rel_iff' : \u2200 {a b : \u03b1}, s (coe_fn _to_embedding a) (coe_fn _to_embedding b) \u2194 r a b\n\ninfixl:25 \" \u21aar \" => Mathlib.rel_embedding\n\n/-- An order embedding is an embedding `f : \u03b1 \u21aa \u03b2` such that `a \u2264 b \u2194 (f a) \u2264 (f b)`.\nThis definition is an abbreviation of `rel_embedding (\u2264) (\u2264)`. -/\ndef order_embedding (\u03b1 : Type u_1) (\u03b2 : Type u_2) [HasLessEq \u03b1] [HasLessEq \u03b2] :=\n  LessEq \u21aar LessEq\n\ninfixl:25 \" \u21aao \" => Mathlib.order_embedding\n\n/-- The induced relation on a subtype is an embedding under the natural inclusion. -/\ndef subtype.rel_embedding {X : Type u_1} (r : X \u2192 X \u2192 Prop) (p : X \u2192 Prop) : subtype.val \u207b\u00b9'o r \u21aar r :=\n  rel_embedding.mk (function.embedding.subtype p) sorry\n\ntheorem preimage_equivalence {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (f : \u03b1 \u2192 \u03b2) {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (hs : equivalence s) : equivalence (f \u207b\u00b9'o s) := sorry\n\nnamespace rel_embedding\n\n\n/-- A relation embedding is also a relation homomorphism -/\ndef to_rel_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : r \u2192r s :=\n  rel_hom.mk (function.embedding.to_fun (to_embedding f)) sorry\n\n-- see Note [function coercion]\n\nprotected instance rel_hom.has_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : has_coe (r \u21aar s) (r \u2192r s) :=\n  has_coe.mk to_rel_hom\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : has_coe_to_fun (r \u21aar s) :=\n  has_coe_to_fun.mk (fun (_x : r \u21aar s) => \u03b1 \u2192 \u03b2) fun (o : r \u21aar s) => \u21d1(to_embedding o)\n\n@[simp] theorem to_rel_hom_eq_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : to_rel_hom f = \u2191f :=\n  rfl\n\n@[simp] theorem coe_coe_fn {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\ntheorem injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : function.injective \u21d1f :=\n  function.embedding.inj' (to_embedding f)\n\ntheorem map_rel_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) {a : \u03b1} {b : \u03b1} : s (coe_fn f a) (coe_fn f b) \u2194 r a b :=\n  map_rel_iff' f\n\n@[simp] theorem coe_fn_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : \u03b1 \u21aa \u03b2) (o : \u2200 {a b : \u03b1}, s (coe_fn f a) (coe_fn f b) \u2194 r a b) : \u21d1(mk f o) = \u21d1f :=\n  rfl\n\n@[simp] theorem coe_fn_to_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : \u21d1(to_embedding f) = \u21d1f :=\n  rfl\n\n/-- The map `coe_fn : (r \u21aar s) \u2192 (\u03b1 \u2192 \u03b2)` is injective. We can't use `function.injective`\nhere but mimic its signature by using `\u2983e\u2081 e\u2082\u2984`. -/\ntheorem coe_fn_inj {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {e\u2081 : r \u21aar s} {e\u2082 : r \u21aar s} : \u21d1e\u2081 = \u21d1e\u2082 \u2192 e\u2081 = e\u2082 := sorry\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u21aar s} {g : r \u21aar s} (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) : f = g :=\n  coe_fn_inj (funext h)\n\ntheorem ext_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u21aar s} {g : r \u21aar s} : f = g \u2194 \u2200 (x : \u03b1), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : \u03b1) => h \u25b8 rfl, mpr := fun (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) => ext h }\n\n/-- Identity map is a relation embedding. -/\nprotected def refl {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : r \u21aar r :=\n  mk (function.embedding.refl \u03b1) sorry\n\n/-- Composition of two relation embeddings is a relation embedding. -/\nprotected def trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (f : r \u21aar s) (g : s \u21aar t) : r \u21aar t :=\n  mk (function.embedding.trans (to_embedding f) (to_embedding g)) sorry\n\nprotected instance inhabited {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Inhabited (r \u21aar r) :=\n  { default := rel_embedding.refl r }\n\n@[simp] theorem refl_apply {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (x : \u03b1) : coe_fn (rel_embedding.refl r) x = x :=\n  rfl\n\ntheorem trans_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (f : r \u21aar s) (g : s \u21aar t) (a : \u03b1) : coe_fn (rel_embedding.trans f g) a = coe_fn g (coe_fn f a) :=\n  rfl\n\n@[simp] theorem coe_trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (f : r \u21aar s) (g : s \u21aar t) : \u21d1(rel_embedding.trans f g) = \u21d1g \u2218 \u21d1f :=\n  rfl\n\n/-- A relation embedding is also a relation embedding between dual relations. -/\nprotected def swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : function.swap r \u21aar function.swap s :=\n  mk (to_embedding f) sorry\n\n/-- If `f` is injective, then it is a relation embedding from the\n  preimage relation of `s` to `s`. -/\ndef preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u21aa \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : \u21d1f \u207b\u00b9'o s \u21aar s :=\n  mk f sorry\n\ntheorem eq_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) : r = \u21d1f \u207b\u00b9'o s :=\n  funext fun (a : \u03b1) => funext fun (b : \u03b1) => propext (iff.symm (map_rel_iff f))\n\nprotected theorem is_irrefl {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_irrefl \u03b2 s] : is_irrefl \u03b1 r :=\n  is_irrefl.mk fun (a : \u03b1) => mt (iff.mpr (map_rel_iff f)) (irrefl (coe_fn f a))\n\nprotected theorem is_refl {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_refl \u03b2 s] : is_refl \u03b1 r :=\n  is_refl.mk fun (a : \u03b1) => iff.mp (map_rel_iff f) (refl (coe_fn f a))\n\nprotected theorem is_symm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_symm \u03b2 s] : is_symm \u03b1 r :=\n  is_symm.mk fun (a b : \u03b1) => imp_imp_imp (iff.mpr (map_rel_iff f)) (iff.mp (map_rel_iff f)) symm\n\nprotected theorem is_asymm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_asymm \u03b2 s] : is_asymm \u03b1 r :=\n  is_asymm.mk fun (a b : \u03b1) (h\u2081 : r a b) (h\u2082 : r b a) => asymm (iff.mpr (map_rel_iff f) h\u2081) (iff.mpr (map_rel_iff f) h\u2082)\n\nprotected theorem is_antisymm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_antisymm \u03b2 s] : is_antisymm \u03b1 r := sorry\n\nprotected theorem is_trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_trans \u03b2 s] : is_trans \u03b1 r := sorry\n\nprotected theorem is_total {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_total \u03b2 s] : is_total \u03b1 r := sorry\n\nprotected theorem is_preorder {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_preorder \u03b2 s] : is_preorder \u03b1 r :=\n  idRhs (is_preorder \u03b1 r) is_preorder.mk\n\nprotected theorem is_partial_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_partial_order \u03b2 s] : is_partial_order \u03b1 r :=\n  idRhs (is_partial_order \u03b1 r) is_partial_order.mk\n\nprotected theorem is_linear_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_linear_order \u03b2 s] : is_linear_order \u03b1 r :=\n  idRhs (is_linear_order \u03b1 r) is_linear_order.mk\n\nprotected theorem is_strict_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_strict_order \u03b2 s] : is_strict_order \u03b1 r :=\n  idRhs (is_strict_order \u03b1 r) is_strict_order.mk\n\nprotected theorem is_trichotomous {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_trichotomous \u03b2 s] : is_trichotomous \u03b1 r := sorry\n\nprotected theorem is_strict_total_order' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_strict_total_order' \u03b2 s] : is_strict_total_order' \u03b1 r :=\n  idRhs (is_strict_total_order' \u03b1 r) is_strict_total_order'.mk\n\nprotected theorem acc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) (a : \u03b1) : acc s (coe_fn f a) \u2192 acc r a := sorry\n\nprotected theorem well_founded {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) (h : well_founded s) : well_founded r :=\n  well_founded.dcases_on h\n    fun (h : \u2200 (a : \u03b2), acc s a) =>\n      idRhs (well_founded r) (well_founded.intro fun (a : \u03b1) => rel_embedding.acc f a (h (coe_fn f a)))\n\nprotected theorem is_well_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) [is_well_order \u03b2 s] : is_well_order \u03b1 r :=\n  idRhs (is_well_order \u03b1 r) (is_well_order.mk (rel_embedding.well_founded f is_well_order.wf))\n\n/-- It suffices to prove `f` is monotone between strict relations\n  to show it is a relation embedding. -/\ndef of_monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} [is_trichotomous \u03b1 r] [is_asymm \u03b2 s] (f : \u03b1 \u2192 \u03b2) (H : \u2200 (a b : \u03b1), r a b \u2192 s (f a) (f b)) : r \u21aar s :=\n  mk (function.embedding.mk f sorry) sorry\n\n@[simp] theorem of_monotone_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} [is_trichotomous \u03b1 r] [is_asymm \u03b2 s] (f : \u03b1 \u2192 \u03b2) (H : \u2200 (a b : \u03b1), r a b \u2192 s (f a) (f b)) : \u21d1(of_monotone f H) = f :=\n  rfl\n\n/-- Embeddings of partial orders that preserve `<` also preserve `\u2264`  -/\ndef order_embedding_of_lt_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [partial_order \u03b1] [partial_order \u03b2] (f : Less \u21aar Less) : \u03b1 \u21aao \u03b2 :=\n  mk (to_embedding f) sorry\n\nend rel_embedding\n\n\nnamespace order_embedding\n\n\n/-- lt is preserved by order embeddings of preorders -/\ndef lt_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) : Less \u21aar Less :=\n  rel_embedding.mk (rel_embedding.to_embedding f) sorry\n\n@[simp] theorem lt_embedding_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) (x : \u03b1) : coe_fn (lt_embedding f) x = coe_fn f x :=\n  rfl\n\n@[simp] theorem le_iff_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) {a : \u03b1} {b : \u03b1} : coe_fn f a \u2264 coe_fn f b \u2194 a \u2264 b :=\n  rel_embedding.map_rel_iff f\n\n@[simp] theorem lt_iff_lt {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) {a : \u03b1} {b : \u03b1} : coe_fn f a < coe_fn f b \u2194 a < b :=\n  rel_embedding.map_rel_iff (lt_embedding f)\n\n@[simp] theorem eq_iff_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) {a : \u03b1} {b : \u03b1} : coe_fn f a = coe_fn f b \u2194 a = b :=\n  function.injective.eq_iff (rel_embedding.injective f)\n\nprotected theorem monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) : monotone \u21d1f :=\n  fun (x y : \u03b1) => iff.mpr (le_iff_le f)\n\nprotected theorem strict_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) : strict_mono \u21d1f :=\n  fun (x y : \u03b1) => iff.mpr (lt_iff_lt f)\n\nprotected theorem acc {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) (a : \u03b1) : acc Less (coe_fn f a) \u2192 acc Less a :=\n  rel_embedding.acc (lt_embedding f) a\n\nprotected theorem well_founded {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) : well_founded Less \u2192 well_founded Less :=\n  rel_embedding.well_founded (lt_embedding f)\n\nprotected theorem is_well_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) [is_well_order \u03b2 Less] : is_well_order \u03b1 Less :=\n  rel_embedding.is_well_order (lt_embedding f)\n\n/-- An order embedding is also an order embedding between dual orders. -/\nprotected def dual {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u21aao \u03b2) : order_dual \u03b1 \u21aao order_dual \u03b2 :=\n  rel_embedding.mk (rel_embedding.to_embedding f) sorry\n\n/-- A sctrictly monotone map from a linear order is an order embedding. --/\ndef of_strict_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) (h : strict_mono f) : \u03b1 \u21aao \u03b2 :=\n  rel_embedding.mk (function.embedding.mk f (strict_mono.injective h)) sorry\n\n@[simp] theorem coe_of_strict_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (h : strict_mono f) : \u21d1(of_strict_mono f h) = f :=\n  rfl\n\n/-- Embedding of a subtype into the ambient type as an `order_embedding`. -/\ndef subtype {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) : Subtype p \u21aao \u03b1 :=\n  rel_embedding.mk (function.embedding.subtype p) sorry\n\n@[simp] theorem coe_subtype {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) : \u21d1(subtype p) = coe :=\n  rfl\n\nend order_embedding\n\n\n/-- A relation isomorphism is an equivalence that is also a relation embedding. -/\nstructure rel_iso {\u03b1 : Type u_4} {\u03b2 : Type u_5} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) \nextends \u03b1 \u2243 \u03b2\nwhere\n  map_rel_iff' : \u2200 {a b : \u03b1}, s (coe_fn _to_equiv a) (coe_fn _to_equiv b) \u2194 r a b\n\ninfixl:25 \" \u2243r \" => Mathlib.rel_iso\n\n/-- An order isomorphism is an equivalence such that `a \u2264 b \u2194 (f a) \u2264 (f b)`.\nThis definition is an abbreviation of `rel_iso (\u2264) (\u2264)`. -/\ndef order_iso (\u03b1 : Type u_1) (\u03b2 : Type u_2) [HasLessEq \u03b1] [HasLessEq \u03b2] :=\n  LessEq \u2243r LessEq\n\ninfixl:25 \" \u2243o \" => Mathlib.order_iso\n\nnamespace rel_iso\n\n\n/-- Convert an `rel_iso` to an `rel_embedding`. This function is also available as a coercion\nbut often it is easier to write `f.to_rel_embedding` than to write explicitly `r` and `s`\nin the target type. -/\ndef to_rel_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : r \u21aar s :=\n  rel_embedding.mk (equiv.to_embedding (to_equiv f)) (map_rel_iff' f)\n\n-- see Note [function coercion]\n\nprotected instance rel_embedding.has_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : has_coe (r \u2243r s) (r \u21aar s) :=\n  has_coe.mk to_rel_embedding\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : has_coe_to_fun (r \u2243r s) :=\n  has_coe_to_fun.mk (fun (_x : r \u2243r s) => \u03b1 \u2192 \u03b2) fun (f : r \u2243r s) => \u21d1f\n\n@[simp] theorem to_rel_embedding_eq_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : to_rel_embedding f = \u2191f :=\n  rfl\n\n@[simp] theorem coe_coe_fn {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\ntheorem map_rel_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) {a : \u03b1} {b : \u03b1} : s (coe_fn f a) (coe_fn f b) \u2194 r a b :=\n  map_rel_iff' f\n\n@[simp] theorem coe_fn_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : \u03b1 \u2243 \u03b2) (o : \u2200 {a b : \u03b1}, s (coe_fn f a) (coe_fn f b) \u2194 r a b) : \u21d1(mk f o) = \u21d1f :=\n  rfl\n\n@[simp] theorem coe_fn_to_equiv {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : \u21d1(to_equiv f) = \u21d1f :=\n  rfl\n\ntheorem injective_to_equiv {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : function.injective to_equiv := sorry\n\n/-- The map `coe_fn : (r \u2243r s) \u2192 (\u03b1 \u2192 \u03b2)` is injective. Lean fails to parse\n`function.injective (\u03bb e : r \u2243r s, (e : \u03b1 \u2192 \u03b2))`, so we use a trick to say the same. -/\ntheorem injective_coe_fn {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} : function.injective fun (e : r \u2243r s) (x : \u03b1) => coe_fn e x :=\n  function.injective.comp equiv.injective_coe_fn injective_to_equiv\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u2243r s} {g : r \u2243r s} (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) : f = g :=\n  injective_coe_fn (funext h)\n\ntheorem ext_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {f : r \u2243r s} {g : r \u2243r s} : f = g \u2194 \u2200 (x : \u03b1), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : \u03b1) => h \u25b8 rfl, mpr := fun (h : \u2200 (x : \u03b1), coe_fn f x = coe_fn g x) => ext h }\n\n/-- Identity map is a relation isomorphism. -/\nprotected def refl {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : r \u2243r r :=\n  mk (equiv.refl \u03b1) sorry\n\n/-- Inverse map of a relation isomorphism is a relation isomorphism. -/\nprotected def symm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : s \u2243r r :=\n  mk (equiv.symm (to_equiv f)) sorry\n\n/-- Composition of two relation isomorphisms is a relation isomorphism. -/\nprotected def trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (f\u2081 : r \u2243r s) (f\u2082 : s \u2243r t) : r \u2243r t :=\n  mk (equiv.trans (to_equiv f\u2081) (to_equiv f\u2082)) sorry\n\nprotected instance inhabited {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Inhabited (r \u2243r r) :=\n  { default := rel_iso.refl r }\n\n@[simp] theorem default_def {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Inhabited.default = rel_iso.refl r :=\n  rfl\n\n/-- a relation isomorphism is also a relation isomorphism between dual relations. -/\nprotected def swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) : function.swap r \u2243r function.swap s :=\n  mk (to_equiv f) sorry\n\n@[simp] theorem coe_fn_symm_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : \u03b1 \u2243 \u03b2) (o : \u2200 {a b : \u03b1}, s (coe_fn f a) (coe_fn f b) \u2194 r a b) : \u21d1(rel_iso.symm (mk f o)) = \u21d1(equiv.symm f) :=\n  rfl\n\n@[simp] theorem refl_apply {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (x : \u03b1) : coe_fn (rel_iso.refl r) x = x :=\n  rfl\n\n@[simp] theorem trans_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} {t : \u03b3 \u2192 \u03b3 \u2192 Prop} (f : r \u2243r s) (g : s \u2243r t) (a : \u03b1) : coe_fn (rel_iso.trans f g) a = coe_fn g (coe_fn f a) :=\n  rfl\n\n@[simp] theorem apply_symm_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) (x : \u03b2) : coe_fn e (coe_fn (rel_iso.symm e) x) = x :=\n  equiv.apply_symm_apply (to_equiv e) x\n\n@[simp] theorem symm_apply_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) (x : \u03b1) : coe_fn (rel_iso.symm e) (coe_fn e x) = x :=\n  equiv.symm_apply_apply (to_equiv e) x\n\ntheorem rel_symm_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) {x : \u03b1} {y : \u03b2} : r x (coe_fn (rel_iso.symm e) y) \u2194 s (coe_fn e x) y := sorry\n\ntheorem symm_apply_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) {x : \u03b2} {y : \u03b1} : r (coe_fn (rel_iso.symm e) x) y \u2194 s x (coe_fn e y) := sorry\n\nprotected theorem bijective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) : function.bijective \u21d1e :=\n  equiv.bijective (to_equiv e)\n\nprotected theorem injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) : function.injective \u21d1e :=\n  equiv.injective (to_equiv e)\n\nprotected theorem surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) : function.surjective \u21d1e :=\n  equiv.surjective (to_equiv e)\n\n@[simp] theorem range_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (e : r \u2243r s) : set.range \u21d1e = set.univ :=\n  function.surjective.range_eq (rel_iso.surjective e)\n\n@[simp] theorem eq_iff_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u2243r s) {a : \u03b1} {b : \u03b1} : coe_fn f a = coe_fn f b \u2194 a = b :=\n  function.injective.eq_iff (rel_iso.injective f)\n\n/-- Any equivalence lifts to a relation isomorphism between `s` and its preimage. -/\nprotected def preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2243 \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : \u21d1f \u207b\u00b9'o s \u2243r s :=\n  mk f sorry\n\n/-- A surjective relation embedding is a relation isomorphism. -/\ndef of_surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) (H : function.surjective \u21d1f) : r \u2243r s :=\n  mk (equiv.of_bijective \u21d1f sorry) sorry\n\n@[simp] theorem of_surjective_coe {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (f : r \u21aar s) (H : function.surjective \u21d1f) : \u21d1(of_surjective f H) = \u21d1f :=\n  rfl\n\n/--\nGiven relation isomorphisms `r\u2081 \u2243r r\u2082` and `s\u2081 \u2243r s\u2082`, construct a relation isomorphism for the\nlexicographic orders on the sum.\n-/\ndef sum_lex_congr {\u03b1\u2081 : Type u_1} {\u03b1\u2082 : Type u_2} {\u03b2\u2081 : Type u_3} {\u03b2\u2082 : Type u_4} {r\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop} {r\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop} {s\u2081 : \u03b2\u2081 \u2192 \u03b2\u2081 \u2192 Prop} {s\u2082 : \u03b2\u2082 \u2192 \u03b2\u2082 \u2192 Prop} (e\u2081 : r\u2081 \u2243r r\u2082) (e\u2082 : s\u2081 \u2243r s\u2082) : sum.lex r\u2081 s\u2081 \u2243r sum.lex r\u2082 s\u2082 :=\n  mk (equiv.sum_congr (to_equiv e\u2081) (to_equiv e\u2082)) sorry\n\n/--\nGiven relation isomorphisms `r\u2081 \u2243r r\u2082` and `s\u2081 \u2243r s\u2082`, construct a relation isomorphism for the\nlexicographic orders on the product.\n-/\ndef prod_lex_congr {\u03b1\u2081 : Type u_1} {\u03b1\u2082 : Type u_2} {\u03b2\u2081 : Type u_3} {\u03b2\u2082 : Type u_4} {r\u2081 : \u03b1\u2081 \u2192 \u03b1\u2081 \u2192 Prop} {r\u2082 : \u03b1\u2082 \u2192 \u03b1\u2082 \u2192 Prop} {s\u2081 : \u03b2\u2081 \u2192 \u03b2\u2081 \u2192 Prop} {s\u2082 : \u03b2\u2082 \u2192 \u03b2\u2082 \u2192 Prop} (e\u2081 : r\u2081 \u2243r r\u2082) (e\u2082 : s\u2081 \u2243r s\u2082) : prod.lex r\u2081 s\u2081 \u2243r prod.lex r\u2082 s\u2082 :=\n  mk (equiv.prod_congr (to_equiv e\u2081) (to_equiv e\u2082)) sorry\n\nprotected instance group {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} : group (r \u2243r r) :=\n  group.mk (fun (f\u2081 f\u2082 : r \u2243r r) => rel_iso.trans f\u2082 f\u2081) sorry (rel_iso.refl r) sorry sorry rel_iso.symm\n    (div_inv_monoid.div._default (fun (f\u2081 f\u2082 : r \u2243r r) => rel_iso.trans f\u2082 f\u2081) sorry (rel_iso.refl r) sorry sorry\n      rel_iso.symm)\n    sorry\n\n@[simp] theorem coe_one {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} : \u21d11 = id :=\n  rfl\n\n@[simp] theorem coe_mul {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (e\u2081 : r \u2243r r) (e\u2082 : r \u2243r r) : \u21d1(e\u2081 * e\u2082) = \u21d1e\u2081 \u2218 \u21d1e\u2082 :=\n  rfl\n\ntheorem mul_apply {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (e\u2081 : r \u2243r r) (e\u2082 : r \u2243r r) (x : \u03b1) : coe_fn (e\u2081 * e\u2082) x = coe_fn e\u2081 (coe_fn e\u2082 x) :=\n  rfl\n\n@[simp] theorem inv_apply_self {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (e : r \u2243r r) (x : \u03b1) : coe_fn (e\u207b\u00b9) (coe_fn e x) = x :=\n  symm_apply_apply e x\n\n@[simp] theorem apply_inv_self {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (e : r \u2243r r) (x : \u03b1) : coe_fn e (coe_fn (e\u207b\u00b9) x) = x :=\n  apply_symm_apply e x\n\nend rel_iso\n\n\nnamespace order_iso\n\n\n/-- Reinterpret an order isomorphism as an order embedding. -/\ndef to_order_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : \u03b1 \u21aao \u03b2 :=\n  rel_iso.to_rel_embedding e\n\n@[simp] theorem coe_to_order_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : \u21d1(to_order_embedding e) = \u21d1e :=\n  rfl\n\nprotected theorem bijective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : function.bijective \u21d1e :=\n  equiv.bijective (rel_iso.to_equiv e)\n\nprotected theorem injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : function.injective \u21d1e :=\n  equiv.injective (rel_iso.to_equiv e)\n\nprotected theorem surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : function.surjective \u21d1e :=\n  equiv.surjective (rel_iso.to_equiv e)\n\n@[simp] theorem range_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : set.range \u21d1e = set.univ :=\n  function.surjective.range_eq (order_iso.surjective e)\n\n@[simp] theorem apply_eq_iff_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b1} : coe_fn e x = coe_fn e y \u2194 x = y :=\n  equiv.apply_eq_iff_eq (rel_iso.to_equiv e)\n\n/-- Inverse of an order isomorphism. -/\ndef symm {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : \u03b2 \u2243o \u03b1 :=\n  rel_iso.symm e\n\n@[simp] theorem apply_symm_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) (x : \u03b2) : coe_fn e (coe_fn (symm e) x) = x :=\n  equiv.apply_symm_apply (rel_iso.to_equiv e) x\n\n@[simp] theorem symm_apply_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) (x : \u03b1) : coe_fn (symm e) (coe_fn e x) = x :=\n  equiv.symm_apply_apply (rel_iso.to_equiv e) x\n\ntheorem symm_apply_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b2} : coe_fn (symm e) y = x \u2194 y = coe_fn e x :=\n  equiv.symm_apply_eq (rel_iso.to_equiv e)\n\n@[simp] theorem symm_symm {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : symm (symm e) = e :=\n  rel_iso.ext fun (x : \u03b1) => Eq.refl (coe_fn (symm (symm e)) x)\n\ntheorem symm_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] : function.injective symm := sorry\n\n@[simp] theorem to_equiv_symm {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasLessEq \u03b1] [HasLessEq \u03b2] (e : \u03b1 \u2243o \u03b2) : equiv.symm (rel_iso.to_equiv e) = rel_iso.to_equiv (symm e) :=\n  rfl\n\n/-- Composition of two order isomorphisms is an order isomorphism. -/\ndef trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [HasLessEq \u03b1] [HasLessEq \u03b2] [HasLessEq \u03b3] (e : \u03b1 \u2243o \u03b2) (e' : \u03b2 \u2243o \u03b3) : \u03b1 \u2243o \u03b3 :=\n  rel_iso.trans e e'\n\n@[simp] theorem coe_trans {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [HasLessEq \u03b1] [HasLessEq \u03b2] [HasLessEq \u03b3] (e : \u03b1 \u2243o \u03b2) (e' : \u03b2 \u2243o \u03b3) : \u21d1(trans e e') = \u21d1e' \u2218 \u21d1e :=\n  rfl\n\ntheorem trans_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [HasLessEq \u03b1] [HasLessEq \u03b2] [HasLessEq \u03b3] (e : \u03b1 \u2243o \u03b2) (e' : \u03b2 \u2243o \u03b3) (x : \u03b1) : coe_fn (trans e e') x = coe_fn e' (coe_fn e x) :=\n  rfl\n\nprotected theorem monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) : monotone \u21d1e :=\n  order_embedding.monotone (to_order_embedding e)\n\nprotected theorem strict_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) : strict_mono \u21d1e :=\n  order_embedding.strict_mono (to_order_embedding e)\n\n@[simp] theorem le_iff_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b1} : coe_fn e x \u2264 coe_fn e y \u2194 x \u2264 y :=\n  rel_iso.map_rel_iff e\n\n@[simp] theorem lt_iff_lt {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b1} : coe_fn e x < coe_fn e y \u2194 x < y :=\n  order_embedding.lt_iff_lt (to_order_embedding e)\n\n@[simp] theorem preimage_Iic {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Iic b = set.Iic (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Ici {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Ici b = set.Ici (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Iio {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Iio b = set.Iio (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Ioi {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Ioi b = set.Ioi (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Icc {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (a : \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Icc a b = set.Icc (coe_fn (symm e) a) (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Ico {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (a : \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Ico a b = set.Ico (coe_fn (symm e) a) (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Ioc {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (a : \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Ioc a b = set.Ioc (coe_fn (symm e) a) (coe_fn (symm e) b) := sorry\n\n@[simp] theorem preimage_Ioo {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (e : \u03b1 \u2243o \u03b2) (a : \u03b2) (b : \u03b2) : \u21d1e \u207b\u00b9' set.Ioo a b = set.Ioo (coe_fn (symm e) a) (coe_fn (symm e) b) := sorry\n\n/-- To show that `f : \u03b1 \u2192 \u03b2`, `g : \u03b2 \u2192 \u03b1` make up an order isomorphism of linear orders,\n    it suffices to prove `cmp a (g b) = cmp (f a) b`. --/\ndef of_cmp_eq_cmp {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) (h : \u2200 (a : \u03b1) (b : \u03b2), cmp a (g b) = cmp (f a) b) : \u03b1 \u2243o \u03b2 :=\n  (fun (gf : \u2200 (a : \u03b1), a = g (f a)) => rel_iso.mk (equiv.mk f g sorry sorry) sorry) sorry\n\n/-- Order isomorphism between two equal sets. -/\ndef set_congr {\u03b1 : Type u_1} [preorder \u03b1] (s : set \u03b1) (t : set \u03b1) (h : s = t) : \u21a5s \u2243o \u21a5t :=\n  rel_iso.mk (equiv.set_congr h) sorry\n\n/-- Order isomorphism between `univ : set \u03b1` and `\u03b1`. -/\ndef set.univ {\u03b1 : Type u_1} [preorder \u03b1] : \u21a5set.univ \u2243o \u03b1 :=\n  rel_iso.mk (equiv.set.univ \u03b1) sorry\n\nend order_iso\n\n\n/-- If a function `f` is strictly monotone on a set `s`, then it defines an order isomorphism\nbetween `s` and its image. -/\nprotected def strict_mono_incr_on.order_iso {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) (s : set \u03b1) (hf : strict_mono_incr_on f s) : \u21a5s \u2243o \u21a5(f '' s) :=\n  rel_iso.mk (set.bij_on.equiv f sorry) sorry\n\n/-- A strictly monotone function from a linear order is an order isomorphism between its domain and\nits range. -/\nprotected def strict_mono.order_iso {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) (h_mono : strict_mono f) : \u03b1 \u2243o \u21a5(set.range f) :=\n  rel_iso.mk (equiv.set.range f (strict_mono.injective h_mono)) sorry\n\n/-- A strictly monotone surjective function from a linear order is an order isomorphism. -/\ndef strict_mono.order_iso_of_surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) (h_mono : strict_mono f) (h_surj : function.surjective f) : \u03b1 \u2243o \u03b2 :=\n  order_iso.trans (strict_mono.order_iso f h_mono)\n    (order_iso.trans (order_iso.set_congr (set.range f) set.univ (function.surjective.range_eq h_surj))\n      order_iso.set.univ)\n\n/-- `subrel r p` is the inherited relation on a subset. -/\ndef subrel {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (p : set \u03b1) : \u21a5p \u2192 \u21a5p \u2192 Prop :=\n  coe \u207b\u00b9'o r\n\n@[simp] theorem subrel_val {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (p : set \u03b1) {a : \u21a5p} {b : \u21a5p} : subrel r p a b \u2194 r (subtype.val a) (subtype.val b) :=\n  iff.rfl\n\nnamespace subrel\n\n\n/-- The relation embedding from the inherited relation on a subset. -/\nprotected def rel_embedding {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (p : set \u03b1) : subrel r p \u21aar r :=\n  rel_embedding.mk (function.embedding.subtype fun (x : \u03b1) => x \u2208 p) sorry\n\n@[simp] theorem rel_embedding_apply {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (p : set \u03b1) (a : \u21a5p) : coe_fn (subrel.rel_embedding r p) a = subtype.val a :=\n  rfl\n\nprotected instance is_well_order {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [is_well_order \u03b1 r] (p : set \u03b1) : is_well_order (\u21a5p) (subrel r p) :=\n  rel_embedding.is_well_order (subrel.rel_embedding r p)\n\nend subrel\n\n\n/-- Restrict the codomain of a relation embedding. -/\ndef rel_embedding.cod_restrict {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (p : set \u03b2) (f : r \u21aar s) (H : \u2200 (a : \u03b1), coe_fn f a \u2208 p) : r \u21aar subrel s p :=\n  rel_embedding.mk (function.embedding.cod_restrict p (rel_embedding.to_embedding f) H) (rel_embedding.map_rel_iff' f)\n\n@[simp] theorem rel_embedding.cod_restrict_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b2 \u2192 \u03b2 \u2192 Prop} (p : set \u03b2) (f : r \u21aar s) (H : \u2200 (a : \u03b1), coe_fn f a \u2208 p) (a : \u03b1) : coe_fn (rel_embedding.cod_restrict p f H) a = { val := coe_fn f a, property := H a } :=\n  rfl\n\nprotected def order_iso.dual {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2243o \u03b2) : order_dual \u03b1 \u2243o order_dual \u03b2 :=\n  rel_iso.mk (rel_iso.to_equiv f) sorry\n\ntheorem order_iso.map_bot' {\u03b1 : Type u_1} {\u03b2 : Type u_2} [partial_order \u03b1] [partial_order \u03b2] (f : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b2} (hx : \u2200 (x' : \u03b1), x \u2264 x') (hy : \u2200 (y' : \u03b2), y \u2264 y') : coe_fn f x = y := sorry\n\ntheorem order_iso.map_bot {\u03b1 : Type u_1} {\u03b2 : Type u_2} [order_bot \u03b1] [order_bot \u03b2] (f : \u03b1 \u2243o \u03b2) : coe_fn f \u22a5 = \u22a5 :=\n  order_iso.map_bot' f (fun (_x : \u03b1) => bot_le) fun (_x : \u03b2) => bot_le\n\ntheorem order_iso.map_top' {\u03b1 : Type u_1} {\u03b2 : Type u_2} [partial_order \u03b1] [partial_order \u03b2] (f : \u03b1 \u2243o \u03b2) {x : \u03b1} {y : \u03b2} (hx : \u2200 (x' : \u03b1), x' \u2264 x) (hy : \u2200 (y' : \u03b2), y' \u2264 y) : coe_fn f x = y :=\n  order_iso.map_bot' (order_iso.dual f) hx hy\n\ntheorem order_iso.map_top {\u03b1 : Type u_1} {\u03b2 : Type u_2} [order_top \u03b1] [order_top \u03b2] (f : \u03b1 \u2243o \u03b2) : coe_fn f \u22a4 = \u22a4 :=\n  order_iso.map_bot (order_iso.dual f)\n\ntheorem order_embedding.map_inf_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_inf \u03b1] [semilattice_inf \u03b2] (f : \u03b1 \u21aao \u03b2) (x : \u03b1) (y : \u03b1) : coe_fn f (x \u2293 y) \u2264 coe_fn f x \u2293 coe_fn f y :=\n  monotone.map_inf_le (order_embedding.monotone f) x y\n\ntheorem order_iso.map_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_inf \u03b1] [semilattice_inf \u03b2] (f : \u03b1 \u2243o \u03b2) (x : \u03b1) (y : \u03b1) : coe_fn f (x \u2293 y) = coe_fn f x \u2293 coe_fn f y := sorry\n\ntheorem order_embedding.le_map_sup {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_sup \u03b1] [semilattice_sup \u03b2] (f : \u03b1 \u21aao \u03b2) (x : \u03b1) (y : \u03b1) : coe_fn f x \u2294 coe_fn f y \u2264 coe_fn f (x \u2294 y) :=\n  monotone.le_map_sup (order_embedding.monotone f) x y\n\ntheorem order_iso.map_sup {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semilattice_sup \u03b1] [semilattice_sup \u03b2] (f : \u03b1 \u2243o \u03b2) (x : \u03b1) (y : \u03b1) : coe_fn f (x \u2294 y) = coe_fn f x \u2294 coe_fn f y :=\n  order_iso.map_inf (order_iso.dual f) x y\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/rel_iso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.39641293161329294}}
{"text": "/-\nCopyright (c) 2022 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\n\nimport geometry.manifold.vector_bundle.pullback\nimport topology.vector_bundle.hom\nimport to_mathlib.geometry.manifold.misc_manifold\n\n/-!\n# Various operations on and properties of smooth vector bundles\n-/\n\nnoncomputable theory\n\nopen bundle set topological_space local_homeomorph\nopen_locale classical manifold bundle\n\nnamespace fiber_bundle\n\nvariables {\ud835\udd5c B B' F M : Type*} {E : B \u2192 Type*}\nvariables [topological_space F] [topological_space (total_space E)] [\u2200 x, topological_space (E x)]\n  {HB : Type*} [topological_space HB]\n  [topological_space B] [charted_space HB B] [fiber_bundle F E]\n\nlemma charted_space_chart_at_fst' (x y : total_space E) :\n  (chart_at (model_prod HB F) x y).1 =\n  chart_at HB x.proj (trivialization_at F E x.proj y).1 :=\nby { rw [charted_space_chart_at], refl }\n\nlemma charted_space_chart_at_fst {x y : total_space E}\n  (hy : y.proj \u2208 (trivialization_at F E x.proj).base_set) :\n  (chart_at (model_prod HB F) x y).1 = chart_at HB x.proj y.proj :=\nby rw [charted_space_chart_at_fst', (trivialization_at F E x.proj).coe_fst' hy]\n\nlemma charted_space_chart_at_snd (x y : total_space E) :\n  (chart_at (model_prod HB F) x y).2 = (trivialization_at F E x.proj y).2 :=\nby { rw [charted_space_chart_at], refl }\n\nend fiber_bundle\n\nsection vector_bundle\n\nvariables {\ud835\udd5c B F F\u2081 F\u2082 : Type*}\n  {E : B \u2192 Type*} {E\u2081 : B \u2192 Type*} {E\u2082 : B \u2192 Type*}\n  [nontrivially_normed_field \ud835\udd5c]\n  [\u2200 x, add_comm_monoid (E x)] [\u2200 x, module \ud835\udd5c (E x)]\n  [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  [topological_space (total_space E)] [\u2200 x, topological_space (E x)]\n  [\u2200 x, add_comm_monoid (E\u2081 x)] [\u2200 x, module \ud835\udd5c (E\u2081 x)]\n  [normed_add_comm_group F\u2081] [normed_space \ud835\udd5c F\u2081]\n  [topological_space (total_space E\u2081)] [\u2200 x, topological_space (E\u2081 x)]\n  [\u2200 x, add_comm_monoid (E\u2082 x)] [\u2200 x, module \ud835\udd5c (E\u2082 x)]\n  [normed_add_comm_group F\u2082] [normed_space \ud835\udd5c F\u2082]\n  [topological_space (total_space E\u2082)] [\u2200 x, topological_space (E\u2082 x)]\n  [topological_space B]\n  {n : \u2115\u221e}\n  [fiber_bundle F\u2081 E\u2081] [vector_bundle \ud835\udd5c F\u2081 E\u2081]\n  [fiber_bundle F\u2082 E\u2082] [vector_bundle \ud835\udd5c F\u2082 E\u2082]\n  {e\u2081 e\u2081' : trivialization F\u2081 (\u03c0 E\u2081)} {e\u2082 e\u2082' : trivialization F\u2082 (\u03c0 E\u2082)}\n\n\n\n\nend vector_bundle\n\n\nnamespace vector_bundle_core\n\nvariables {R \ud835\udd5c B F \u03b9 : Type*}\n  [nontrivially_normed_field R]\n  [normed_add_comm_group F] [normed_space R F] [topological_space B]\n  (Z : vector_bundle_core R B F \u03b9)\n\n/-- `Z.coord_change j i` is a partial inverse of `Z.coord_change i j`. -/\nlemma coord_change_comp_eq_self {i j : \u03b9} {x : B} (hx : x \u2208 Z.base_set i \u2229 Z.base_set j) (v : F) :\n  Z.coord_change j i x (Z.coord_change i j x v) = v :=\nby rw [Z.coord_change_comp i j i x \u27e8hx, hx.1\u27e9, Z.coord_change_self i x hx.1]\n\nend vector_bundle_core\n\nnamespace bundle.trivial\nopen _root_.trivialization\n\nvariables {\ud835\udd5c B F : Type*}\nvariables [nontrivially_normed_field \ud835\udd5c] [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  [topological_space B]\n\n@[simp, mfld_simps]\nprotected lemma trivialization_at (x : B) :\n  trivialization_at F (trivial B F) x = trivial.trivialization B F :=\nrfl\n\n@[simp, mfld_simps]\nlemma trivialization_continuous_linear_map_at (x : B) :\n  (trivial.trivialization B F).continuous_linear_map_at \ud835\udd5c x = continuous_linear_map.id \ud835\udd5c F :=\nbegin\n  ext v,\n  simp_rw [continuous_linear_map_at_apply, coe_linear_map_at],\n  rw [if_pos],\n  exacts [rfl, mem_univ _]\nend\n\nend bundle.trivial\n\nsection hom\nvariables {\ud835\udd5c\u2081 : Type*} [nontrivially_normed_field \ud835\udd5c\u2081] {\ud835\udd5c\u2082 : Type*} [nontrivially_normed_field \ud835\udd5c\u2082]\n  (\u03c3 : \ud835\udd5c\u2081 \u2192+* \ud835\udd5c\u2082) [i\u03c3 : ring_hom_isometric \u03c3]\n\nvariables {B : Type*} [topological_space B]\n\nvariables (F\u2081 : Type*) [normed_add_comm_group F\u2081] [normed_space \ud835\udd5c\u2081 F\u2081]\n  (E\u2081 : B \u2192 Type*) [\u03a0 x, add_comm_monoid (E\u2081 x)] [\u03a0 x, module \ud835\udd5c\u2081 (E\u2081 x)]\n  [topological_space (total_space E\u2081)]\nvariables (F\u2082 : Type*) [normed_add_comm_group F\u2082][normed_space \ud835\udd5c\u2082 F\u2082]\n  (E\u2082 : B \u2192 Type*) [\u03a0 x, add_comm_monoid (E\u2082 x)] [\u03a0 x, module \ud835\udd5c\u2082 (E\u2082 x)]\n  [topological_space (total_space E\u2082)]\nvariables (F\u2081 E\u2081 F\u2082 E\u2082) [ring_hom_isometric \u03c3]\nvariables [\u03a0 x : B, topological_space (E\u2081 x)] [fiber_bundle F\u2081 E\u2081] [vector_bundle \ud835\udd5c\u2081 F\u2081 E\u2081]\nvariables [\u03a0 x : B, topological_space (E\u2082 x)] [fiber_bundle F\u2082 E\u2082] [vector_bundle \ud835\udd5c\u2082 F\u2082 E\u2082]\nvariables [\u03a0 x, has_continuous_add (E\u2082 x)] [\u03a0 x, has_continuous_smul \ud835\udd5c\u2082 (E\u2082 x)]\n\n@[simp, mfld_simps]\nlemma continuous_linear_map_trivialization_at (x : B) :\n  trivialization_at (F\u2081 \u2192SL[\u03c3] F\u2082) (bundle.continuous_linear_map \u03c3 F\u2081 E\u2081 F\u2082 E\u2082) x =\n  (trivialization_at F\u2081 E\u2081 x).continuous_linear_map \u03c3 (trivialization_at F\u2082 E\u2082 x) :=\nrfl\n\nend hom\n\nsection pullback\n\n/-- We need some instances like this to work with negation on pullbacks -/\ninstance {B B'} {E : B \u2192 Type*} {f : B' \u2192 B} {x : B'} [\u2200 x', add_comm_group (E x')] :\n  add_comm_group ((f *\u1d56 E) x) :=\nby delta_instance bundle.pullback\n\ninstance {B B'} {E : B \u2192 Type*} {f : B' \u2192 B} {x : B'} [\u2200 x', has_zero (E x')] :\n  has_zero ((f *\u1d56 E) x) :=\nby delta_instance bundle.pullback\n\nvariables {B F B' K : Type*} {E : B \u2192 Type*} {f : K}\n  [topological_space B'] [topological_space (total_space E)]\n  [topological_space F] [topological_space B]\n  [\u2200 b, has_zero (E b)] [continuous_map_class K B' B]\n\nnamespace trivialization\n\n-- attribute [simps base_set] trivialization.pullback\n\nlemma pullback_symm (e : trivialization F (\u03c0 E)) (x : B') : (e.pullback f).symm x = e.symm (f x) :=\nbegin\n  ext y,\n  simp_rw [trivialization.symm, pretrivialization.symm],\n  congr', ext (hx : f x \u2208 e.to_pretrivialization.base_set),\n  change cast _ (e.symm (f x) y) = cast _ ((e.to_local_homeomorph.symm (f x, y)).2),\n  simp_rw [trivialization.symm, pretrivialization.symm, dif_pos hx, cast_cast],\n  refl,\nend\n\nend trivialization\n\n\nvariables [\u2200 x, topological_space (E x)] [fiber_bundle F E]\n\nlemma pullback_trivialization_at {x : B'} : trivialization_at F (f *\u1d56 E) x =\n  (trivialization_at F E (f x)).pullback f :=\nrfl\n\n\n\n\nend pullback\n\nsection pullback_vb\n\nvariables {R \ud835\udd5c B F B' : Type*} {E : B \u2192 Type*}\n\n\nvariables [topological_space B'] [topological_space (total_space E)]\n  [nontrivially_normed_field \ud835\udd5c] [normed_add_comm_group F] [normed_space \ud835\udd5c F] [topological_space B]\n  [\u2200 x, add_comm_monoid (E x)] [\u2200 x, module \ud835\udd5c (E x)] [\u2200 x, topological_space (E x)]\n  [fiber_bundle F E]\n  {K : Type*} [continuous_map_class K B' B] (f : K)\n\nnamespace trivialization\nlemma pullback_symmL (e : trivialization F (\u03c0 E)) [e.is_linear \ud835\udd5c] (x : B') :\n  (e.pullback f).symmL \ud835\udd5c x = e.symmL \ud835\udd5c (f x) :=\nby { ext y, simp_rw [symmL_apply, pullback_symm] }\n\nend trivialization\n\nend pullback_vb\n\nnamespace vector_prebundle\n\nattribute [reducible] vector_prebundle.to_fiber_bundle\n\n/-!\n### `vector_prebundle.is_smooth`\n\nTodo: maybe redefine `vector_prebundle` as a mixin `fiber_prebundle.is_vector_prebundle`.\nThe reason is that if you define a `fiber_prebundle` operation, and then\n(under certain circumstances)\nupgrade it to a `vector_prebundle`, this will result in `fiber_bundle` instances that are probably\nnot easily seen as definitionally equal by type-class inference.\n-/\n\n\nvariables {\ud835\udd5c B F F\u2081 F\u2082 M M\u2081 M\u2082 : Type*}\n  {E : B \u2192 Type*} {E\u2081 : B \u2192 Type*} {E\u2082 : B \u2192 Type*}\n  [nontrivially_normed_field \ud835\udd5c]\n  [\u2200 x, add_comm_monoid (E x)] [\u2200 x, module \ud835\udd5c (E x)]\n  [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  [\u2200 x, add_comm_monoid (E\u2081 x)] [\u2200 x, module \ud835\udd5c (E\u2081 x)]\n  [normed_add_comm_group F\u2081] [normed_space \ud835\udd5c F\u2081]\n  [\u2200 x, add_comm_monoid (E\u2082 x)] [\u2200 x, module \ud835\udd5c (E\u2082 x)]\n  [normed_add_comm_group F\u2082] [normed_space \ud835\udd5c F\u2082]\n  {EB : Type*} [normed_add_comm_group EB] [normed_space \ud835\udd5c EB]\n  {HB : Type*} [topological_space HB] (IB : model_with_corners \ud835\udd5c EB HB)\n  [topological_space B] [charted_space HB B] [smooth_manifold_with_corners IB B]\n  {EM : Type*} [normed_add_comm_group EM] [normed_space \ud835\udd5c EM]\n  {HM : Type*} [topological_space HM] {IM : model_with_corners \ud835\udd5c EM HM}\n  [topological_space M] [charted_space HM M]\n  {n : \u2115\u221e}\n\nvariables (IB)\n\n/-- Mixin for a `vector_prebundle` stating smoothness of coordinate changes. -/\nclass is_smooth (a : vector_prebundle \ud835\udd5c F E) : Prop :=\n(exists_smooth_coord_change : \u2200 (e e' \u2208 a.pretrivialization_atlas), \u2203 f : B \u2192 F \u2192L[\ud835\udd5c] F,\n  smooth_on IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) f (e.base_set \u2229 e'.base_set) \u2227\n  \u2200 (b : B) (hb : b \u2208 e.base_set \u2229 e'.base_set) (v : F),\n    f b v = (e' (total_space_mk b (e.symm b v))).2)\n\nvariables (a : vector_prebundle \ud835\udd5c F E) [ha : a.is_smooth IB] {e e' : pretrivialization F (\u03c0 E)}\ninclude ha\n\n/-- A randomly chosen coordinate change on a `smooth_vector_prebundle`, given by\n  the field `exists_coord_change`. -/\ndef smooth_coord_change (he : e \u2208 a.pretrivialization_atlas) (he' : e' \u2208 a.pretrivialization_atlas)\n  (b : B) : F \u2192L[\ud835\udd5c] F :=\nclassical.some (ha.exists_smooth_coord_change e he e' he') b\n\nvariables {IB}\nlemma smooth_on_smooth_coord_change (he : e \u2208 a.pretrivialization_atlas)\n  (he' : e' \u2208 a.pretrivialization_atlas) :\n  smooth_on IB \ud835\udcd8(\ud835\udd5c, F \u2192L[\ud835\udd5c] F) (a.smooth_coord_change IB he he') (e.base_set \u2229 e'.base_set) :=\n(classical.some_spec (ha.exists_smooth_coord_change e he e' he')).1\n\nlemma smooth_coord_change_apply (he : e \u2208 a.pretrivialization_atlas)\n  (he' : e' \u2208 a.pretrivialization_atlas) {b : B} (hb : b \u2208 e.base_set \u2229 e'.base_set) (v : F) :\n  a.smooth_coord_change IB he he' b v = (e' (total_space_mk b (e.symm b v))).2 :=\n(classical.some_spec (ha.exists_smooth_coord_change e he e' he')).2 b hb v\n\nlemma mk_smooth_coord_change (he : e \u2208 a.pretrivialization_atlas)\n  (he' : e' \u2208 a.pretrivialization_atlas) {b : B} (hb : b \u2208 e.base_set \u2229 e'.base_set) (v : F) :\n  (b, (a.smooth_coord_change IB he he' b v)) = e' (total_space_mk b (e.symm b v)) :=\nbegin\n  ext,\n  { rw [e.mk_symm hb.1 v, e'.coe_fst', e.proj_symm_apply' hb.1],\n    rw [e.proj_symm_apply' hb.1], exact hb.2 },\n  { exact a.smooth_coord_change_apply he he' hb v }\nend\n\nvariables (IB)\n/-- Make a `smooth_vector_bundle` from a `smooth_vector_prebundle`.  -/\nlemma to_smooth_vector_bundle :\n  @smooth_vector_bundle _ _ F E _ _ _ _ _ a.total_space_topology a.fiber_topology _ _ _ _ _ IB\n  _ _ _ a.to_fiber_bundle a.to_vector_bundle :=\n{ smooth_on_coord_change := begin\n    rintros _ _ \u27e8e, he, rfl\u27e9 \u27e8e', he', rfl\u27e9,\n    refine (a.smooth_on_smooth_coord_change he he').congr _,\n    intros b hb,\n    ext v,\n    rw [a.smooth_coord_change_apply he he' hb v, continuous_linear_equiv.coe_coe,\n      trivialization.coord_changeL_apply],\n    exacts [rfl, hb]\n  end }\n\nend vector_prebundle\n\nvariables {\ud835\udd5c B F F\u2081 F\u2082 M M\u2081 M\u2082 : Type*}\n  {E : B \u2192 Type*} {E\u2081 : B \u2192 Type*} {E\u2082 : B \u2192 Type*}\n  [nontrivially_normed_field \ud835\udd5c]\n  [\u2200 x, add_comm_monoid (E x)] [\u2200 x, module \ud835\udd5c (E x)]\n  [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  [topological_space (total_space E)] [\u2200 x, topological_space (E x)]\n  [\u2200 x, add_comm_monoid (E\u2081 x)] [\u2200 x, module \ud835\udd5c (E\u2081 x)]\n  [normed_add_comm_group F\u2081] [normed_space \ud835\udd5c F\u2081]\n  [topological_space (total_space E\u2081)] [\u2200 x, topological_space (E\u2081 x)]\n  [\u2200 x, add_comm_monoid (E\u2082 x)] [\u2200 x, module \ud835\udd5c (E\u2082 x)]\n  [normed_add_comm_group F\u2082] [normed_space \ud835\udd5c F\u2082]\n  [topological_space (total_space E\u2082)] [\u2200 x, topological_space (E\u2082 x)]\n\n  {EB : Type*} [normed_add_comm_group EB] [normed_space \ud835\udd5c EB]\n  {HB : Type*} [topological_space HB] (IB : model_with_corners \ud835\udd5c EB HB)\n  [topological_space B] [charted_space HB B]\n  {EM : Type*} [normed_add_comm_group EM] [normed_space \ud835\udd5c EM]\n  {HM : Type*} [topological_space HM] {IM : model_with_corners \ud835\udd5c EM HM}\n  [topological_space M] [charted_space HM M] [Is : smooth_manifold_with_corners IM M]\n  {n : \u2115\u221e}\n  [fiber_bundle F\u2081 E\u2081] [vector_bundle \ud835\udd5c F\u2081 E\u2081]\n  [fiber_bundle F\u2082 E\u2082] [vector_bundle \ud835\udd5c F\u2082 E\u2082]\n  {e\u2081 e\u2081' : trivialization F\u2081 (\u03c0 E\u2081)} {e\u2082 e\u2082' : trivialization F\u2082 (\u03c0 E\u2082)}\n\n\n/-!\n### Homs of smooth vector bundles over the same base space\n-/\n-- protect `bundle.continuous_linear_map`\n\nsection hom\nopen continuous_linear_map pretrivialization\n\nlocal notation `\u03c3` := ring_hom.id \ud835\udd5c\n\nsection general\n-- what is better notation for this?\nlocal notation `FE\u2081E\u2082` := bundle.continuous_linear_map \u03c3 F\u2081 E\u2081 F\u2082 E\u2082\nlocal notation `LE\u2081E\u2082` := total_space FE\u2081E\u2082\nlocal notation `PLE\u2081E\u2082` := bundle.continuous_linear_map.vector_prebundle \u03c3 F\u2081 E\u2081 F\u2082 E\u2082\n\n\n/- This proof is slow, especially the `simp only` and the elaboration of `h\u2082`. -/\nlemma smooth_on_continuous_linear_map_coord_change\n  [smooth_manifold_with_corners IB B]\n  [smooth_vector_bundle F\u2081 E\u2081 IB] [smooth_vector_bundle F\u2082 E\u2082 IB]\n  [mem_trivialization_atlas e\u2081] [mem_trivialization_atlas e\u2081']\n  [mem_trivialization_atlas e\u2082] [mem_trivialization_atlas e\u2082'] :\n  smooth_on IB \ud835\udcd8(\ud835\udd5c, ((F\u2081 \u2192L[\ud835\udd5c] F\u2082) \u2192L[\ud835\udd5c] (F\u2081 \u2192L[\ud835\udd5c] F\u2082)))\n    (continuous_linear_map_coord_change \u03c3 e\u2081 e\u2081' e\u2082 e\u2082')\n    ((e\u2081.base_set \u2229 e\u2082.base_set) \u2229 (e\u2081'.base_set \u2229 e\u2082'.base_set)) :=\nbegin\n  let L\u2081 := compSL F\u2081 F\u2082 F\u2082 \u03c3 \u03c3,\n  have h\u2081 : smooth _ _ _ := L\u2081.cont_mdiff,\n  have h\u2082 : smooth _ _ _ := (continuous_linear_map.flip (compSL F\u2081 F\u2081 F\u2082 \u03c3 \u03c3)).cont_mdiff,\n  have h\u2083 : smooth_on IB _ _ _ := smooth_on_coord_change e\u2081' e\u2081,\n  have h\u2084 : smooth_on IB _ _ _ := smooth_on_coord_change e\u2082 e\u2082',\n  refine ((h\u2081.comp_smooth_on (h\u2084.mono _)).clm_comp (h\u2082.comp_smooth_on (h\u2083.mono _))).congr _,\n  { mfld_set_tac },\n  { mfld_set_tac },\n  { intros b hb, ext L v,\n    simp only [continuous_linear_map_coord_change, continuous_linear_equiv.coe_coe,\n      continuous_linear_equiv.arrow_congrSL_apply, comp_apply, function.comp, compSL_apply,\n      flip_apply, continuous_linear_equiv.symm_symm] },\nend\n\nvariables [\u2200 x, has_continuous_add (E\u2082 x)] [\u2200 x, has_continuous_smul \ud835\udd5c (E\u2082 x)]\n\n@[reducible]\ndef topological_space.continuous_linear_map' (x) : topological_space (FE\u2081E\u2082 x) :=\nby apply_instance\nlocal attribute [instance, priority 1] topological_space.continuous_linear_map'\n-- ^ probably needed because of the type-class pi bug\n-- https://leanprover.zulipchat.com/#narrow/stream/116395-maths/topic/vector.20bundles.20--.20typeclass.20inference.20issue\n\nlemma hom_trivialization_at_apply (x\u2080 : B) (x : LE\u2081E\u2082) :\n  trivialization_at (F\u2081 \u2192L[\ud835\udd5c] F\u2082) (bundle.continuous_linear_map \u03c3 F\u2081 E\u2081 F\u2082 E\u2082) x\u2080 x =\n  \u27e8x.1, in_coordinates' F\u2081 F\u2082 E\u2081 E\u2082 x\u2080 x.1 x\u2080 x.1 x.2\u27e9 :=\nrfl\n\n\n@[simp, mfld_simps]\nlemma hom_trivialization_at_source (x\u2080 : B) :\n  (trivialization_at (F\u2081 \u2192L[\ud835\udd5c] F\u2082) (bundle.continuous_linear_map \u03c3 F\u2081 E\u2081 F\u2082 E\u2082) x\u2080).source =\n  \u03c0 FE\u2081E\u2082 \u207b\u00b9' ((trivialization_at F\u2081 E\u2081 x\u2080).base_set \u2229 (trivialization_at F\u2082 E\u2082 x\u2080).base_set) :=\nrfl\n\n@[simp, mfld_simps]\nlemma hom_trivialization_at_target (x\u2080 : B) :\n  (trivialization_at (F\u2081 \u2192L[\ud835\udd5c] F\u2082) (bundle.continuous_linear_map \u03c3 F\u2081 E\u2081 F\u2082 E\u2082) x\u2080).target =\n  ((trivialization_at F\u2081 E\u2081 x\u2080).base_set \u2229 (trivialization_at F\u2082 E\u2082 x\u2080).base_set) \u00d7\u02e2 set.univ :=\nrfl\n\nlemma hom_chart (x\u2080 x : LE\u2081E\u2082) :\n  chart_at (model_prod HB (F\u2081 \u2192L[\ud835\udd5c] F\u2082)) x\u2080 x =\n  (chart_at HB x\u2080.1 x.1, in_coordinates' F\u2081 F\u2082 E\u2081 E\u2082 x\u2080.1 x.1 x\u2080.1 x.1 x.2) :=\nby simp_rw [fiber_bundle.charted_space_chart_at, trans_apply, local_homeomorph.prod_apply,\n  trivialization.coe_coe, local_homeomorph.refl_apply, function.id_def, hom_trivialization_at_apply]\n\nlemma smooth_at_hom_bundle {f : M \u2192 LE\u2081E\u2082} {x\u2080 : M} :\n  smooth_at IM (IB.prod \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)) f x\u2080 \u2194\n  smooth_at IM IB (\u03bb x, (f x).1) x\u2080 \u2227\n  smooth_at IM \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)\n  (\u03bb x, in_coordinates' F\u2081 F\u2082 E\u2081 E\u2082 (f x\u2080).1 (f x).1 (f x\u2080).1 (f x).1 (f x).2) x\u2080 :=\nby { simp_rw [smooth_at, cont_mdiff_at_total_space], refl }\n\nvariables [smooth_manifold_with_corners IB B]\n  [smooth_vector_bundle F\u2081 E\u2081 IB] [smooth_vector_bundle F\u2082 E\u2082 IB]\n\nvariables [\u2200 x, has_continuous_add (E\u2082 x)] [\u2200 x, has_continuous_smul \ud835\udd5c (E\u2082 x)]\n\ninstance bundle.continuous_linear_map.vector_prebundle.is_smooth : PLE\u2081E\u2082 .is_smooth IB :=\n{ exists_smooth_coord_change := by {\n    rintro _ \u27e8e\u2081, e\u2082, he\u2081, he\u2082, rfl\u27e9 _ \u27e8e\u2081', e\u2082', he\u2081', he\u2082', rfl\u27e9,\n    resetI,\n    refine \u27e8continuous_linear_map_coord_change \u03c3 e\u2081 e\u2081' e\u2082 e\u2082',\n    smooth_on_continuous_linear_map_coord_change IB,\n    continuous_linear_map_coord_change_apply \u03c3 e\u2081 e\u2081' e\u2082 e\u2082'\u27e9 } }\n\ninstance smooth_vector_bundle.continuous_linear_map :\n  smooth_vector_bundle (F\u2081 \u2192L[\ud835\udd5c] F\u2082) FE\u2081E\u2082 IB :=\nPLE\u2081E\u2082 .to_smooth_vector_bundle IB\n\nend general\n\nnamespace vector_bundle_core\n\nvariables {\u03b9\u2081 \u03b9\u2082 : Type*} (Z\u2081 : vector_bundle_core \ud835\udd5c B F\u2081 \u03b9\u2081) (Z\u2082 : vector_bundle_core \ud835\udd5c B F\u2082 \u03b9\u2082)\n\nlocal notation `FZ\u2081Z\u2082` := bundle.continuous_linear_map \u03c3 F\u2081 Z\u2081.fiber F\u2082 Z\u2082.fiber\nlocal notation `LZ\u2081Z\u2082` := bundle.total_space FZ\u2081Z\u2082\n\ndef foo1 (b : B) :\n  topological_space (continuous_linear_map \u03c3 F\u2081 Z\u2081.fiber F\u2082 Z\u2082.fiber b) :=\nby apply_instance\nlocal attribute [instance] foo1 -- probably needed because of the type-class pi bug\n\nlemma hom_trivialization_at (x\u2080 x : LZ\u2081Z\u2082)\n  (h1x : x.proj \u2208 Z\u2081.base_set (Z\u2081.index_at x\u2080.proj))\n  (h2x : x.proj \u2208 Z\u2082.base_set (Z\u2082.index_at x\u2080.proj)) :\n  (trivialization_at (F\u2081 \u2192L[\ud835\udd5c] F\u2082) FZ\u2081Z\u2082 x\u2080.proj x).2 =\n  in_coordinates_core' Z\u2081 Z\u2082 x\u2080.proj x.proj x\u2080.proj x.proj x.2 :=\nbegin\n  rw [continuous_linear_map_trivialization_at, trivialization.continuous_linear_map_apply,\n    \u2190 in_coordinates_core'_eq],\n  exacts [rfl, h1x, h2x]\nend\n\n-- todo: refactor\nlemma smooth_at_hom_bundle_core {f : M \u2192 LZ\u2081Z\u2082} {x\u2080 : M} :\n  smooth_at IM (IB.prod \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)) f x\u2080 \u2194\n  smooth_at IM IB (\u03bb x, (f x).1) x\u2080 \u2227\n  smooth_at IM \ud835\udcd8(\ud835\udd5c, F\u2081 \u2192L[\ud835\udd5c] F\u2082)\n  (\u03bb x, in_coordinates_core' Z\u2081 Z\u2082 (f x\u2080).1 (f x).1 (f x\u2080).1 (f x).1 (f x).2) x\u2080  :=\nbegin\n  simp_rw [smooth_at, cont_mdiff_at_total_space, and.congr_right_iff],\n  intro hf,\n  refine filter.eventually_eq.cont_mdiff_at_iff _,\n  have h1 := hf.continuous_at.preimage_mem_nhds ((trivialization.open_base_set _).mem_nhds\n    (mem_base_set_trivialization_at F\u2081 Z\u2081.fiber (f x\u2080).proj)),\n  have h2 := hf.continuous_at.preimage_mem_nhds ((trivialization.open_base_set _).mem_nhds\n    (mem_base_set_trivialization_at F\u2082 Z\u2082.fiber (f x\u2080).proj)),\n  filter_upwards [h1, h2],\n  intros x h1x h2x,\n  exact hom_trivialization_at Z\u2081 Z\u2082 (f x\u2080) (f x) h1x h2x\nend\n\nend vector_bundle_core\n\nend hom\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/geometry/manifold/vector_bundle/misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3963560269375908}}
{"text": "/-\nCopyright (c) 2021 Bhavik Mehta, Thomas Bloom. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Thomas Bloom\n-/\n\nimport for_mathlib.basic_estimates\nimport defs\nimport aux_lemmas\nimport fourier\nimport main_results\nimport algebra.indicator_function\n\n/-!\n# Title\n\nThis file contains a formal proof of the headline results of\n https://arxiv.org/pdf/2112.03726.pdf.\n-/\n\nopen_locale big_operators\nopen filter finset real\nopen nat (coprime)\n\nopen_locale arithmetic_function\nopen_locale classical\nnoncomputable theory\n\nlemma another_weird_tendsto_at_top_aux (c : \u211d) (hc : 1 < c) :\n  tendsto (\u03bb x, c ^ x / log x) at_top at_top :=\n((tendsto_exp_mul_div_rpow_at_top 1 _ (log_pos hc)).at_top_mul_at_top\n  (tendsto_mul_add_div_pow_log_at_top 1 0 1 zero_lt_one)).congr' $\nby filter_upwards [eventually_gt_at_top (0 : \u211d)] with x hx using\n  by simp [rpow_def_of_pos (zero_le_one.trans_lt hc), div_mul_div_cancel _ hx.ne']\n\nlemma the_thing : 1 < exp 2 / 2 :=\nbegin\n  rw [one_lt_div, \u2190log_lt_iff_lt_exp zero_lt_two],\n  { exact log_two_lt_d9.trans_le (by norm_num) },\n  exact zero_lt_two\nend\n\nlemma another_weird_tendsto_at_top :\n  tendsto (\u03bb x : \u211d, x / (2 ^ (1 / 2 * log x + 1) * log (1 / 2 * log x))) at_top at_top :=\n(tendsto.const_mul_at_top (show (0 : \u211d) < 1 / 2, by norm_num)\n  ((another_weird_tendsto_at_top_aux (exp 2 / 2) the_thing).comp\n    (tendsto_log_at_top.const_mul_at_top (show (0 : \u211d) < 1 / 2, by norm_num)))).congr' $\nbegin\n  filter_upwards [eventually_gt_at_top (0 : \u211d)] with x hx,\n  dsimp,\n  rw [div_rpow (exp_pos _).le zero_le_two, div_div, div_mul_div_comm, one_mul,\n    rpow_add_one two_ne_zero, rpow_def_of_pos (exp_pos _), log_exp, \u2190mul_assoc,\n    mul_one_div_cancel (two_ne_zero : (2 : \u211d) \u2260 0), one_mul, exp_log hx, \u2190mul_assoc,\n    mul_comm (2 : \u211d)],\nend\n\nlemma omega_eq_sum (N : \u2115) {n : \u2115} (hn : n \u2208 Icc 1 N) :\n  \u03c9 n = \u2211 p in (((Icc 1 N).filter (\u03bb p, nat.prime p)).filter (\u03bb p, p \u2223 n)), 1 :=\nbegin\n  rw [card_distinct_factors_apply', \u2190card_eq_sum_ones,\n    nat.prime_divisors_eq_to_filter_divisors_prime],\n  rw mem_Icc at hn,\n  congr' 1,\n  ext p,\n  simp only [mem_filter, nat.mem_divisors, ne.def, mem_Icc, and_assoc],\n  split,\n  { rintro \u27e8hp\u2081, hp\u2082, hp\u2083\u27e9,\n    refine \u27e8hp\u2083.one_lt.le, _, hp\u2083, hp\u2081\u27e9,\n    exact (nat.le_of_dvd (pos_iff_ne_zero.2 hp\u2082) hp\u2081).trans hn.2 },\n  { rintro \u27e8hp\u2081, hp\u2082, hp\u2083, hp\u2084\u27e9,\n    refine \u27e8hp\u2084, _, hp\u2083\u27e9,\n    rw nat.succ_le_iff at hn,\n    exact hn.1.ne' }\nend\n\nlemma count_multiples'' {m n : \u2115} (hm : 1 \u2264 m) :\n  (((finset.Icc 1 n).filter (\u03bb k, m \u2223 k)).card : \u211d) = (n / m : \u211d) - int.fract (n / m) :=\nbegin\n  rw [count_multiples hm, int.self_sub_fract, \u2190nat.cast_floor_eq_cast_int_floor,\n    nat.floor_div_eq_div],\n  exact div_nonneg (nat.cast_nonneg _) (nat.cast_nonneg _)\nend\n\nlemma count_multiples''' {m n : \u2115} (hm : 1 \u2264 m) :\n  (((finset.Icc 1 n).filter (\u03bb k, m \u2223 k)).card : \u211d) \u2264 (n / m : \u211d) :=\nbegin\n  rw [count_multiples'' hm, sub_le_self_iff],\n  apply int.fract_nonneg,\nend\n\nlemma sum_prime_counting : \u2203 (C : \u211d), \u2200\u1da0 (N : \u2115) in at_top,\n   (N : \u211d) * log (log N) - C * N \u2264 \u2211 x in Icc 1 N, (\u03c9 x : \u211d) :=\nbegin\n  obtain \u27e8c, hc\u27e9 := (prime_reciprocal.trans (is_o_log_inv_one one_ne_zero).is_O).bound,\n  use -meissel_mertens + c + 1,\n  filter_upwards [tendsto_coe_nat_at_top_at_top hc] with N hN,\n  dsimp at hN,\n  simp only [prime_summatory, nat.floor_coe, abs_one, mul_one, norm_eq_abs] at hN,\n  have : \u2200 x \u2208 Icc 1 N, (\u03c9 x : \u211d) = \u2211 p in ((Icc 1 N).filter nat.prime), ite (p \u2223 x) 1 0,\n  { intros x hx,\n    rw [omega_eq_sum _ hx, nat.cast_sum, nat.cast_one, sum_filter] },\n  rw [sum_congr rfl this, sum_comm],\n  simp only [\u2190sum_filter],\n  have : \u2200 x \u2208 (Icc 1 N).filter nat.prime,\n    \u2211 (a : \u2115) in filter (has_dvd.dvd x) (Icc 1 N), (1 : \u211d) = (N / x : \u211d) - int.fract (N / x),\n  { intros x hx,\n    rw [\u2190count_multiples'', card_eq_sum_ones, nat.cast_sum, nat.cast_one],\n    rw [mem_filter, mem_Icc] at hx,\n    exact hx.1.1 },\n  rw [sum_congr rfl this, sum_sub_distrib],\n  simp only [div_eq_mul_inv (N : \u211d), \u2190mul_sum],\n  have h\u2081 : (N : \u211d) * (log (log N) + meissel_mertens - c) \u2264\n    N * \u2211 (x : \u2115) in filter nat.prime (Icc 1 N), (\u2191x)\u207b\u00b9,\n  { apply mul_le_mul_of_nonneg_left _ (nat.cast_nonneg _),\n    exact sub_le_of_abs_sub_le_left hN },\n  have h\u2082 : \u2211 x in filter nat.prime (Icc 1 N), int.fract ((N : \u211d) * (\u2191x)\u207b\u00b9) \u2264 N,\n  { refine (finset.sum_le_card_nsmul _ _ 1 _).trans _,\n    { intros x hx,\n      exact (int.fract_lt_one _).le },\n    simp only [nat.smul_one_eq_coe, nat.cast_le],\n    exact (card_le_of_subset (filter_subset _ _)).trans (by simp) },\n  refine (sub_le_sub h\u2081 h\u2082).trans' (le_of_eq _),\n  ring,\nend\n\nlemma range_eq_insert_Icc {n : \u2115} (hn : 1 \u2264 n) : range n = insert 0 (Icc 1 (n - 1)) :=\nbegin\n  rw [Icc_succ_left, Ioc_insert_left (nat.zero_le _), \u2190nat.Ico_succ_right, nat.succ_eq_add_one,\n    nat.sub_add_cancel hn, range_eq_Ico],\nend\n\nlemma prime_recip_lazy :\n  \u2203 c, \u2200\u1da0 N : \u2115 in at_top, \u2211 p in (Icc 1 N).filter nat.prime, (p : \u211d)\u207b\u00b9 \u2264 log (log N) + c :=\nbegin\n  obtain \u27e8c, hc\u27e9 := (prime_reciprocal.trans (is_o_log_inv_one one_ne_zero).is_O).bound,\n  use meissel_mertens + c,\n  filter_upwards [tendsto_coe_nat_at_top_at_top hc] with N hN,\n  dsimp at hN,\n  simp only [prime_summatory, nat.floor_coe, abs_one, mul_one, norm_eq_abs, abs_sub_le_iff,\n    sub_le_iff_le_add', add_assoc] at hN,\n  exact hN.1\nend\n\nlemma sum_prime_counting_sq : \u2203 (C : \u211d), \u2200\u1da0 (N : \u2115) in at_top,\n   \u2211 x in Icc 1 N, (\u03c9 x : \u211d) ^ 2 \u2264 N * log (log N) ^ 2 + C * N * log (log N) :=\nbegin\n  obtain \u27e8c, hc\u27e9 := prime_recip_lazy,\n  use ((2 * c + 1) + 1),\n  filter_upwards [hc, tendsto_log_log_coe_at_top.eventually_ge_at_top (c ^ 2 + c)] with N hN hN',\n  have : \u2200 x \u2208 Icc 1 N, (\u03c9 x : \u211d) ^ 2 = (\u2211 p in (Icc 1 N).filter nat.prime, ite (p \u2223 x) 1 0) ^ 2,\n  { intros x hx,\n    rw [omega_eq_sum _ hx, nat.cast_sum, nat.cast_one, sum_filter] },\n  rw [sum_congr rfl this],\n  simp_rw [sq, sum_mul, mul_sum, boole_mul, \u2190ite_and, @sum_comm _ _ _ _ (Icc _ _), \u2190sq],\n  have : \u2200 p \u2208 (Icc 1 N).filter nat.prime,\n    \u2211 q in (Icc 1 N).filter nat.prime, \u2211 n in Icc 1 N, ite (p \u2223 n \u2227 q \u2223 n) (1 : \u211d) 0 \u2264\n    \u2211 n in Icc 1 N, ite (p \u2223 n) 1 0 +\n      \u2211 q in (Icc 1 N).filter nat.prime, \u2211 n in Icc 1 N, ite (p * q \u2223 n) 1 0,\n  { intros p hp,\n    rw [\u2190sum_filter_add_sum_filter_not _ (\u03bb q, p = q), sum_filter, sum_ite_eq, if_pos hp],\n    simp only [and_self, add_le_add_iff_left],\n    refine (sum_le_sum _).trans (sum_le_sum_of_subset_of_nonneg (filter_subset _ _) _),\n    { intros q hq,\n      simp only [mem_filter, mem_Icc, \u2190ne.def] at hp hq,\n      refine sum_le_sum (\u03bb n hn, _),\n      by_cases p \u2223 n \u2227 q \u2223 n,\n      { rw [if_pos h, if_pos (nat.prime.dvd_mul_of_dvd_ne hq.2 hp.2 hq.1.2 h.1 h.2)] },\n      rw if_neg h,\n      split_ifs,\n      { exact zero_le_one },\n      { refl } },\n    { intros i _ _,\n      simp only [sum_boole, nat.cast_nonneg] } },\n  refine (sum_le_sum this).trans _,\n  rw [sum_add_distrib],\n  simp only [sum_boole],\n  have h\u2081 : \u2211 x in (Icc 1 N).filter nat.prime, ((filter ((\u2223) x) (Icc 1 N)).card : \u211d) \u2264\n    N * \u2211 x in (Icc 1 N).filter nat.prime, x\u207b\u00b9,\n  { simp only [mul_sum, \u2190div_eq_mul_inv],\n    refine sum_le_sum (\u03bb x hx, _),\n    simp only [mem_filter, mem_Icc] at hx,\n    apply count_multiples''' hx.1.1 },\n  have h\u2082 : \u2211 p in filter nat.prime (Icc 1 N), \u2211 q in filter nat.prime (Icc 1 N),\n    ((filter (has_dvd.dvd (p * q)) (Icc 1 N)).card : \u211d) \u2264\n    N * (\u2211 p in (Icc 1 N).filter nat.prime, p\u207b\u00b9) ^ 2,\n  { simp only [sq, mul_sum, sum_mul, \u2190mul_inv, \u2190div_eq_mul_inv (N : \u211d), \u2190nat.cast_mul],\n    refine sum_le_sum (\u03bb p hp, (sum_le_sum (\u03bb q hq, _))),\n    simp only [mem_filter, mem_Icc] at hp hq,\n    apply count_multiples''' (one_le_mul hp.1.1 hq.1.1) },\n  refine (add_le_add h\u2081 h\u2082).trans _,\n  rw [mul_comm (2 * c + 1 + 1), mul_assoc, \u2190mul_add, \u2190mul_add],\n  refine mul_le_mul_of_nonneg_left _ (nat.cast_nonneg _),\n  refine (add_le_add hN (pow_le_pow_of_le_left _ hN 2)).trans _,\n  { exact sum_nonneg (by simp) },\n  rw [add_sq],\n  linarith only [hN'],\nend\n\n-- I think this is false because the LHS set includes 0\n-- but does changing to\n-- ((filter (\u03bb i, x \u2223 i) (Icc 1 N)).card : \u211d) make anything afterwards more annoying?\n-- same for the three above\nlemma count_divisors {x N : \u2115} (hx : x \u2260 0) :\n  ((filter (\u03bb i, x \u2223 i) (Icc 1 N)).card : \u211d) = (N / x : \u211d) - int.fract (N / x) :=\nbegin\n  rw count_multiples'',\n  exact hx.bot_lt,\nend\n\nlemma count_divisors' {x N : \u2115} (hx : x \u2260 0) (hN : N \u2260 0):\n  ((filter (\u03bb i, x \u2223 i) (range(N))).card : \u211d) = (N / x : \u211d) - (1/x - 1 + int.fract ((N-1) / x)) :=\nbegin\n  have hN' : 1 \u2264 N := hN.bot_lt,\n  rw [range_eq_insert_Icc hN', filter_insert, if_pos (dvd_zero _), card_insert_of_not_mem,\n    nat.cast_add_one, count_divisors hx, nat.cast_sub hN', nat.cast_one, sub_div],\n  { ring },\n  simp,\nend\n\nlemma is_multiplicative_one {R : Type*} [ring R] :\n  (1 : nat.arithmetic_function R).is_multiplicative :=\nbegin\n  refine \u27e8nat.arithmetic_function.one_one, _\u27e9,\n  intros m n hmn,\n  change ite _ _ _ = ite _ _ _ * ite _ _ _,\n  simp only [boole_mul, \u2190ite_and, nat.mul_eq_one_iff],\nend\n\nlemma ite_div (p : Prop) [decidable p] {x y z : \u211d} :\n  ite p x y / z = ite p (x / z) (y / z) :=\napply_ite (\u03bb i, i / z) _ _ _\n\nlemma moebius_rec_sum {N : \u2115} (hN : N \u2260 0) :\n  \u2211 (x : \u2115) in N.divisors, (\u03bc x : \u211d) / x = \u220f p in filter nat.prime N.divisors, (1 - p\u207b\u00b9) :=\nbegin\n  let f' : nat.arithmetic_function \u211d := \u27e8\u03bb x, (\u03bc x : \u211d) / x, by simp\u27e9,\n  have hf' : f'.is_multiplicative,\n  { refine \u27e8_, \u03bb m n hmn, _\u27e9,\n    { simp only [f', zero_hom.coe_mk, nat.arithmetic_function.moebius_apply_of_squarefree,\n        squarefree_one, nat.arithmetic_function.card_factors_one, pow_zero, int.cast_one,\n        nat.cast_one, div_one] },\n    simp only [zero_hom.coe_mk, nat.cast_mul, int.cast_mul, mul_div_mul_comm,\n      nat.arithmetic_function.is_multiplicative_moebius.map_mul_of_coprime hmn] },\n  let f : nat.arithmetic_function \u211d := f' * \u03b6,\n  have hf : f.is_multiplicative := hf'.mul nat.arithmetic_function.is_multiplicative_zeta.nat_cast,\n  change \u2211 x : \u2115 in N.divisors, f' x = _,\n  rw \u2190nat.arithmetic_function.coe_mul_zeta_apply,\n  change f N = _,\n  rw \u2190nat.prime_divisors_eq_to_filter_divisors_prime,\n  revert hN N,\n  refine nat.rec_on_pos_prime_pos_coprime _ _ _ _,\n  { intros p k hp hk hpk,\n    rw [nat.prime_pow_prime_divisor hk.ne' hp, prod_singleton,\n      nat.arithmetic_function.coe_mul_zeta_apply, nat.sum_divisors_prime_pow hp],\n    simp only [zero_hom.coe_mk, nat.cast_pow, sum_range_succ', pow_zero,\n      nat.arithmetic_function.moebius_apply_one, int.cast_one, div_one],\n    simp_rw [nat.arithmetic_function.moebius_apply_prime_pow hp (nat.succ_ne_zero _),\n      int.cast_ite, int.cast_neg, int.cast_zero, int.cast_one, nat.succ_inj',\n      ite_div, zero_div, sum_ite_eq', neg_div, pow_one, one_div, mem_range, if_pos hk,\n      neg_add_eq_sub] },\n  { intro h,\n    cases h rfl },\n  { intro _,\n    simp only [nat.factors_one, list.to_finset_nil, prod_empty, hf.map_one] },\n  { intros a b ha hb hab aih bih k,\n    rw [hf.map_mul_of_coprime hab, nat.factors_mul_to_finset_of_coprime hab, prod_union, aih, bih],\n    { linarith },\n    { linarith },\n    rw list.disjoint_to_finset_iff_disjoint,\n    apply nat.coprime_factors_disjoint hab\n    },\nend\n\nlemma prod_sdiff'' {\u03b9 \u03b1 : Type*} [comm_group_with_zero \u03b1] (f : \u03b9 \u2192 \u03b1) (s t : finset \u03b9) (h : t \u2286 s)\n  (ht : \u2200 i \u2208 t, f i \u2260 0) :\n  \u220f i in s \\ t, f i = (\u220f i in s, f i) / \u220f i in t, f i :=\nbegin\n  rw [eq_div_iff_mul_eq, prod_sdiff h],\n  rwa prod_ne_zero_iff,\nend\n\nlemma filter_sdiff {\u03b9 : Type*} (p : \u03b9 \u2192 Prop) [decidable_eq \u03b9] [decidable_pred p] (s t : finset \u03b9) :\n  (s \\ t).filter p = s.filter p \\ t.filter p :=\nbegin\n  ext x,\n  simp only [mem_sdiff, mem_filter],\n  tauto,\nend\n\nlemma product_of_primes_factors {s : finset \u2115} (hs : \u2200 p \u2208 s, nat.prime p) :\n  (\u220f p in s, p).factors = s.sort (\u2264) :=\nbegin\n  refine (list.eq_of_perm_of_sorted (nat.factors_unique _ _) _ (nat.factors_sorted _)).symm,\n  { rw [prod_eq_multiset_prod, multiset.map_id', \u2190multiset.coe_prod, finset.sort_eq] },\n  { simpa only [mem_sort] },\n  exact sort_sorted _ _,\nend\n\nlemma product_of_primes_factors_to_finset {s : finset \u2115} (hs : \u2200 p \u2208 s, nat.prime p) :\n  (\u220f p in s, p).factors.to_finset = s :=\nby rw [product_of_primes_factors hs, sort_to_finset]\n\nlemma mem_factors_prod {A : finset \u2115} (h : \u2200 n \u2208 A, n \u2260 0) {p : \u2115} :\np \u2208 (\u220f a in A, a).factors \u2194 \u2203 a \u2208 A, p \u2208 (a:\u2115).factors :=\nbegin\n  induction A using finset.induction_on with n A hnA hA,\n  simp only [prod_empty, nat.factors_one, list.not_mem_nil, not_mem_empty, exists_false_left,\n     exists_false],\n  rw [prod_insert hnA, nat.mem_factors_mul], split, intro h',\n  cases h' with h\u2081 h\u2082, use n, refine \u27e8mem_insert_self _ _,h\u2081\u27e9,\n  rw hA at h\u2082, rcases h\u2082 with \u27e8b,hb\u2081,hb\u2082\u27e9, use b, refine \u27e8_,hb\u2082\u27e9,\n  refine mem_insert_of_mem hb\u2081, intros n hn, refine h n _,\n  refine mem_insert_of_mem hn, intro h', rcases h' with \u27e8a,ha\u2081,ha\u2082\u27e9,\n  rw mem_insert at ha\u2081, cases ha\u2081, rw ha\u2081 at ha\u2082, left, exact ha\u2082,\n  right, rw hA, use a, refine \u27e8ha\u2081,ha\u2082\u27e9, intros n hn, refine h n _,\n  refine mem_insert_of_mem hn, refine h n _, refine mem_insert_self _ _,\n  rw prod_ne_zero_iff, intros n hn, refine h n _,\n  refine mem_insert_of_mem hn,\nend\n\nlemma prod_primes_squarefree {A : finset \u2115} (h : \u2200 n \u2208 A, nat.prime n) :\n squarefree \u220f p in A, p :=\nbegin\n  unfreezingI { induction A using finset.induction_on with p A hpA hA },\n  simp only [prod_empty, squarefree_one],\n  rw prod_insert hpA, rw nat.squarefree_mul,\n  refine \u27e8prime.squarefree _,_\u27e9, rw \u2190 nat.prime_iff, refine h p _, refine mem_insert_self _ _,\n  refine hA _, intros n hn, refine h n _, refine mem_insert_of_mem hn,\n  refine nat.coprime_prod_right _, intros q hq, rw nat.coprime_primes, intro hbad,\n  rw hbad at hpA, exact hpA hq, refine h p _, refine mem_insert_self _ _,\n  refine h q _, refine mem_insert_of_mem hq,\nend\n\nlemma sieve_lemma_prec (N : \u2115) (y z : \u211d) (hy : 1 \u2264 y) (hzN : z < N) :\n   (((finset.range(N)).filter (\u03bb n, \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192 ((p : \u211d) < y) \u2228 z < p)).card : \u211d) \u2264\n   ((partial_euler_product \u230ay\u230b\u208a)/(partial_euler_product \u230az\u230b\u208a)) * N + 2^(z+1) :=\nbegin\n  by_cases hN0 : N = 0,\n  rw [hN0, range_zero, filter_empty], norm_cast, rw [mul_zero, zero_add],\n  simp only [card_empty, nat.cast_zero], refine rpow_nonneg_of_nonneg _ _,\n  exact zero_le_two,\n  cases lt_or_le z y,\n  { calc _ \u2264 (N:\u211d) :_\n       ... \u2264 _ :_,\n    norm_cast,\n    have : N = (finset.range(N)).card, { rw card_range, },\n    nth_rewrite 1 this, refine finset.card_filter_le _ _,\n    rw \u2190 add_zero (N:\u211d), refine add_le_add _ _, rw add_zero, refine le_mul_of_one_le_left _ _,\n    exact nat.cast_nonneg N, rw [one_le_div, partial_euler_product, partial_euler_product],\n    refine prod_of_subset_le_prod_of_one_le _ _ _, intros p hp,\n    rw [mem_filter, mem_Icc], rw [mem_filter, mem_Icc] at hp, refine \u27e8\u27e8hp.1.1,_\u27e9,hp.2\u27e9,\n    refine le_trans hp.1.2 _,\n    by_cases h0z : 1 \u2264 z,\n    rw nat.le_floor_iff, refine le_trans _ (le_of_lt h), refine nat.floor_le _,\n    exact le_trans zero_le_one h0z,  exact le_trans zero_le_one hy,\n    rw [not_le, \u2190 nat.floor_eq_zero] at h0z, rw h0z, exact zero_le \u230ay\u230b\u208a,\n    intros p hp, rw [inv_nonneg, sub_nonneg, inv_le_one_iff], right,\n    norm_cast, rw mem_filter at hp, refine le_of_lt (nat.prime.one_lt hp.2),\n    intros p hp1 hp2, refine one_le_inv _ _, rw [sub_pos, inv_lt_one_iff], right,\n    norm_cast, rw mem_filter at hp1, exact nat.prime.one_lt hp1.2,\n    refine sub_le_self _ _, rw [inv_nonneg], exact nat.cast_nonneg p,\n    refine lt_of_lt_of_le zero_lt_one partial_euler_trivial_lower_bound,\n    refine rpow_nonneg_of_nonneg _ _, exact zero_le_two,\n  },\n  let P := \u220f p in ((finset.range N).filter (\u03bb p, nat.prime p \u2227 (y \u2264 p) \u2227 ((p:\u211d) \u2264 z))), p,\n  have hP : P \u2260 0,\n  { rw prod_ne_zero_iff,\n    intros x hx,\n    simp only [mem_filter, mem_range] at hx,\n    exact hx.2.1.pos.ne' },\n  have h\u2081 : ((finset.range(N)).filter (\u03bb n, \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192\n       ((p : \u211d) < y) \u2228 (z < p))).card = ((finset.range(N)).filter (\u03bb n, coprime n P)).card,\n  { congr' 1,\n    apply filter_congr,\n    simp only [mem_range, nat.coprime_prod, mem_filter, and_imp, \u2190nat.prime_iff],\n    intros n hn,\n    split,\n    { intros h p pn hp hy hz,\n      rw [nat.coprime_comm, hp.coprime_iff_not_dvd],\n      intro t,\n      cases h p hp t with h' h',\n      { exact h'.not_le hy },\n      { exact h'.not_le hz } },\n    { intros h p hp pn,\n      by_contra' h',\n      rw [hp.dvd_iff_not_coprime, \u2190nat.coprime_comm] at pn,\n      exact pn (h p (nat.cast_lt.1 (h'.2.trans_lt hzN)) hp h'.1 h'.2) } },\n  have : \u2200 n, \u2211 (i : \u2115) in (nat.gcd n P).divisors, (\u03bc i : \u211d) = ite (nat.gcd n P = 1) 1 0,\n  { intro n,\n    rw \u2190int.cast_sum,\n    rw \u2190nat.arithmetic_function.coe_mul_zeta_apply,\n    rw nat.arithmetic_function.moebius_mul_coe_zeta,\n    change coe (ite _ _ _) = _,\n    split_ifs; simp only [int.cast_one, int.cast_zero] },\n  rw h\u2081,\n  rw \u2190sum_boole,\n  simp only [nat.coprime],\n  simp_rw [\u2190this],\n  have hgcddiv : \u2200 x : \u2115, (x.gcd P).divisors = (P.divisors).filter (\u03bb d, d \u2223 x), -- x \u2260 0\n  { intros x,\n    ext m,\n    simp only [nat.mem_divisors, mem_filter, nat.dvd_gcd_iff, hP, nat.gcd_eq_zero_iff, ne.def,\n      and_false, not_false_iff, and_true, and_comm (m \u2223 P) (m \u2223 x)] },\n  simp_rw [hgcddiv, sum_filter],\n  rw sum_comm,\n  simp_rw [\u2190mul_boole _ (\u03bc _ : \u211d), \u2190mul_sum],\n  simp_rw [sum_boole],\n  have : \u2211 x in P.divisors, (\u03bc x : \u211d) * ((filter (\u03bb i, x \u2223 i) (finset.range(N))).card : \u211d) =\n      \u2211 x in P.divisors, (\u03bc x : \u211d) * ((N / x : \u211d) - (1/x - 1 + int.fract ((N-1) / x))),\n  { rw sum_congr rfl,\n    intros x hx,\n    rw count_divisors',\n    rw nat.mem_divisors at hx,\n    exact ne_zero_of_dvd_ne_zero hx.2 hx.1, exact hN0,\n    },\n  simp_rw [this, mul_sub],\n  rw sum_sub_distrib,\n  simp_rw [mul_div_assoc', mul_comm _ (N : \u211d), mul_div_assoc],\n  rw \u2190mul_sum,\n  have hP_divisors : P.divisors.filter nat.prime =\n    (range N).filter (\u03bb p, nat.prime p \u2227 y \u2264 p \u2227 (p : \u211d) \u2264 z),\n  { rw [\u2190nat.prime_divisors_eq_to_filter_divisors_prime, product_of_primes_factors_to_finset],\n    simp only [mem_filter, implies_true_iff] {contextual := tt} },\n  have hP_divisors' :\n    filter nat.prime (Icc 1 \u230az\u230b\u208a \\ Icc 1 \u230ay\u230b\u208a) \u2286 P.divisors.filter nat.prime,\n  {\n    rw [hP_divisors, Icc_sdiff_Icc_left], intros n hn,\n    simp only [mem_filter, mem_Ioc, mem_range, and_assoc],\n    rw [mem_filter, mem_Ioc, nat.le_floor_iff, nat.floor_lt'] at hn,\n    refine \u27e8_,hn.2,_,hn.1.2\u27e9,\n    exact_mod_cast lt_of_le_of_lt hn.1.2 hzN, refine le_of_lt _, exact hn.1.1,\n    exact nat.prime.ne_zero hn.2,\n    refine le_trans _ h, refine le_trans zero_le_one hy,\n    rw nat.le_floor_iff, refine le_trans _ h, refine nat.floor_le _,\n    exact le_trans zero_le_one hy,\n    refine le_trans _ h, refine le_trans zero_le_one hy, rw nat.le_floor_iff,\n    exact_mod_cast hy, exact le_trans zero_le_one hy,\n  },\n  have hPsum : \u2211 (x : \u2115) in P.divisors, (\u03bc x : \u211d) / x \u2264\n    (partial_euler_product \u230ay\u230b\u208a) / (partial_euler_product \u230az\u230b\u208a),\n  { rw [moebius_rec_sum hP, partial_euler_product, partial_euler_product, prod_inv_distrib,\n      prod_inv_distrib, inv_div_inv, \u2190prod_sdiff'', \u2190filter_sdiff],\n    refine prod_le_prod_of_subset_of_le_one _ _ _,\n    { convert hP_divisors' },\n    intros p hp, rw [sub_nonneg, inv_le_one_iff], right, norm_cast, rw mem_filter at hp,\n    refine le_of_lt (nat.prime.one_lt hp.2), intros p hp1 hp2, refine sub_le_self _ _,\n    rw inv_nonneg, exact nat.cast_nonneg p, intros p hp,\n    rw [mem_filter, mem_Icc], rw [mem_filter, mem_Icc] at hp, refine \u27e8\u27e8hp.1.1,_\u27e9,hp.2\u27e9,\n    refine le_trans hp.1.2 _,\n    rw nat.le_floor_iff, refine le_trans _ h, refine nat.floor_le _,\n    exact le_trans zero_le_one hy,\n    refine le_trans _ h, refine le_trans zero_le_one hy,\n    intros p hp, refine ne_of_gt _, rw [sub_pos, inv_lt_one_iff],\n    right, norm_cast, rw mem_filter at hp,\n    refine nat.prime.one_lt hp.2,\n  },\n  rw [sub_eq_add_neg],\n  refine add_le_add _ _,\n  refine mul_le_mul_of_nonneg_left hPsum _,\n  exact nat.cast_nonneg N, refine le_trans (le_abs_self _) _,\n  rw [abs_neg], refine le_trans (abs_sum_le_sum_abs _ _) _,\n  calc _ \u2264 (2:\u211d)*(\u03c3 0 P : \u211d) :_\n     ... \u2264 _ :_,\n  rw nat.arithmetic_function.sigma_zero_apply,\n  refine le_trans (finset.sum_le_card_nsmul _ _ 2 _) _,\n  intros d hd, rw [abs_mul], rw \u2190 one_mul (2:\u211d), refine mul_le_mul _ _ _ _,\n  by_cases hdsq : squarefree d,\n  rw [nat.arithmetic_function.moebius_apply_of_squarefree hdsq], norm_cast,\n  rw [abs_pow, abs_neg, abs_one, one_pow],\n  rw nat.arithmetic_function.moebius_eq_zero_of_not_squarefree hdsq, norm_cast,\n  exact zero_le_one,\n  rw [\u2190 add_sub_right_comm, \u2190 add_sub], refine le_trans (abs_add _ _) _,\n  transitivity (1:\u211d)+1, refine add_le_add _ _,\n  rw [abs_of_nonneg, one_div_le], norm_num1, norm_cast, rw nat.succ_le_iff,\n  exact nat.pos_of_mem_divisors hd, exact_mod_cast nat.pos_of_mem_divisors hd,\n  exact zero_lt_one, rw one_div_nonneg, exact nat.cast_nonneg d,\n  rw [abs_of_nonpos, neg_sub], refine sub_le_self _ _, refine int.fract_nonneg _,\n  rw sub_nonpos, refine le_of_lt (int.fract_lt_one _), norm_num1,\n  refine abs_nonneg _, exact zero_le_one,\n  simp only [nsmul_eq_mul], rw mul_comm,\n  have hPsq : squarefree P, { refine prod_primes_squarefree _,\n    intros p hp, rw mem_filter at hp, exact hp.2.1, },\n  rw divisor_count_eq_pow_iff_squarefree.2 hPsq, rw nat.cast_pow, norm_num1,\n  rw [\u2190 rpow_nat_cast, mul_comm, \u2190 rpow_add_one],\n  refine rpow_le_rpow_of_exponent_le one_le_two _,\n  rw [nat.arithmetic_function.card_distinct_factors_apply, \u2190 list.card_to_finset],\n  transitivity ((Icc 0 \u230az\u230b\u208a).card : \u211d),\n  norm_cast,\n  transitivity (insert 0 P.factors.to_finset).card, rw finset.card_insert_of_not_mem,\n  rw list.mem_to_finset, intro hbad, refine nat.not_prime_zero _,\n  exact nat.prime_of_mem_factors hbad,\n  refine finset.card_le_of_subset _, intros p hp,\n  rw mem_insert at hp, cases hp with hp\u2081 hp\u2082, rw hp\u2081,\n  simp only [left_mem_Icc, zero_le'],\n  rw [list.mem_to_finset, mem_factors_prod] at hp\u2082,\n  rcases hp\u2082 with \u27e8q,hq1,hq2\u27e9, rw mem_filter at hq1,\n  rw [nat.factors_prime hq1.2.1, list.mem_singleton] at hq2, rw [hq2, mem_Icc],\n  refine \u27e8zero_le q,_\u27e9, rw nat.le_floor_iff, exact hq1.2.2.2,\n  refine le_trans _ h, exact le_trans zero_le_one hy,\n  intros n hn, rw mem_filter at hn, exact nat.prime.ne_zero hn.2.1,\n  rw [nat.card_Icc, nat.cast_sub], push_cast, rw sub_zero,\n  rw add_le_add_iff_right, refine nat.floor_le _,\n  refine le_trans _ h, exact le_trans zero_le_one hy,\n  exact zero_le (\u230az\u230b\u208a + 1), refine ne_of_gt zero_lt_two,\nend\n\nlemma sieve_lemma_prec' : \u2203 C c : \u211d, (0 < C) \u2227 (0 < c) \u2227\n  \u2200\u1da0 (N : \u2115) in at_top, \u2200 y z : \u211d, (2 \u2264 y) \u2192 (1 < z) \u2192 (z \u2264 c*log N) \u2192\n   (((finset.range(N)).filter(\u03bb n, \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192\n       ((p : \u211d) < y) \u2228 (z < p))).card : \u211d) \u2264 C*(log y/log z)*N :=\nbegin\n  rcases weak_mertens_third_lower_all with \u27e8C\u2081,hC\u2081,hml\u27e9,\n  rcases weak_mertens_third_upper_all with \u27e8C\u2082,hC\u2082,hmu\u27e9,\n  let C := 1 / C\u2081 * C\u2082 * 2,\n  let c := (1:\u211d)/2,\n  have h0C : 0 < C, { refine mul_pos _ zero_lt_two,\n    refine mul_pos _ hC\u2082, rw one_div_pos, exact hC\u2081, },\n  use C, use c, refine \u27e8h0C,one_half_pos,_\u27e9,\n  filter_upwards [tendsto_coe_nat_at_top_at_top.eventually (eventually_gt_at_top (0:\u211d)),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_gt_at_top (1*2:\u211d)),\n    (another_weird_tendsto_at_top.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (1 / (C / 2 * log 2)))\n    ]\n    with N h0N hlogN hweirdN,\n  intros y z h2y h1z hzN,\n  have h0logN : 0 < log N, { refine lt_trans _ hlogN, norm_num1, },\n  have hzN' : z < N,\n  { apply hzN.trans_lt ((log_lt_self h0N).trans_le' _),\n    refine mul_le_of_le_one_left h0logN.le _,\n    change (1 : \u211d) / 2 \u2264 1,\n    refine half_le_self zero_le_one },\n  refine le_trans (sieve_lemma_prec N y z (h2y.trans' (by norm_num1)) hzN') _,\n  rw [\u2190 add_halves C, add_mul, add_mul], refine add_le_add _ _,\n  rw [mul_le_mul_right h0N, mul_div, div_le_div_iff],\n  specialize hmu y, specialize hml z (le_of_lt h1z),\n  rw [norm_eq_abs, abs_of_pos, norm_eq_abs, abs_of_pos] at hml,\n  rw [norm_eq_abs, abs_of_pos, norm_eq_abs, abs_of_pos] at hmu,\n  transitivity (C\u2082*log(y)*log z), refine mul_le_mul_of_nonneg_right (hmu _) _, exact h2y,\n  exact log_nonneg (le_of_lt h1z), transitivity (C\u2082*log(y)*partial_euler_product \u230az\u230b\u208a/C\u2081),\n  rw [le_div_iff hC\u2081, mul_assoc _ _ C\u2081], refine mul_le_mul_of_nonneg_left _ _,\n  rw mul_comm, exact hml, refine mul_nonneg (le_of_lt hC\u2082) (log_nonneg _),\n  refine le_trans one_le_two h2y, rw [div_eq_mul_one_div, mul_comm _ (1/C\u2081), \u2190 mul_assoc,\n    mul_le_mul_right, \u2190 mul_assoc],\n  transitivity ((1/C\u2081)*C\u2082*(log y)), rw mul_le_mul_left, refine mul_pos _ hC\u2082,\n  rw one_div_pos, exact hC\u2081, rw [mul_le_mul_right, le_div_iff], exact zero_lt_two,\n  refine log_pos _, exact lt_of_lt_of_le one_lt_two h2y,\n  exact lt_of_lt_of_le zero_lt_one partial_euler_trivial_lower_bound, refine log_pos _,\n  exact lt_of_lt_of_le one_lt_two h2y,\n  exact lt_of_lt_of_le zero_lt_one partial_euler_trivial_lower_bound,\n  exact lt_of_lt_of_le zero_lt_one partial_euler_trivial_lower_bound, exact log_pos h1z,\n  exact lt_of_lt_of_le zero_lt_one partial_euler_trivial_lower_bound, exact log_pos h1z,\n  transitivity ((C / 2 * (log 2))*N / log z),\n  transitivity (2:\u211d)^(((1:\u211d)/2)*log N+1),\n  refine rpow_le_rpow_of_exponent_le one_le_two _, rw add_le_add_iff_right, exact hzN,\n  rw le_div_iff, transitivity ((2:\u211d)^(((1:\u211d)/2)*log N+1))*(log(((1:\u211d)/2)*log N)),\n  rw mul_le_mul_left, rw log_le_log, exact hzN, exact lt_trans zero_lt_one h1z,\n  exact mul_pos (one_half_pos) h0logN, refine rpow_pos_of_pos zero_lt_two _,\n  rw [\u2190 one_le_div, \u2190 mul_div, \u2190 div_le_iff'], exact hweirdN,\n  refine mul_pos _ _, exact div_pos h0C zero_lt_two,\n  exact log_pos one_lt_two, refine mul_pos _ _, refine rpow_pos_of_pos zero_lt_two _,\n  refine log_pos _, rw [mul_comm, \u2190 div_eq_mul_one_div, lt_div_iff], exact hlogN,\n  exact zero_lt_two, exact log_pos h1z, rw [mul_assoc, \u2190 mul_div, mul_assoc,\n    mul_le_mul_left, mul_comm _ (N:\u211d), \u2190 mul_div, mul_comm, mul_le_mul_right,\n    div_le_div_right, log_le_log], exact h2y, exact zero_lt_two,\n  exact lt_of_lt_of_le zero_lt_two h2y, exact log_pos h1z, exact h0N,\n  exact div_pos h0C zero_lt_two,\nend\n\nlemma plogp_tail_bound (a : \u211d) (ha : 0 < a): \u2203 c : \u211d, (0 < c) \u2227 \u2200\u1da0 (N : \u2115) in at_top, \u2200 z : \u211d,\n  (0 \u2264 log(log (\u230az\u230b\u208a))) \u2192\n  \u2211 (x : \u2115) in filter nat.prime (Icc N \u230az\u230b\u208a), (a / (log(x/4)*x)) \u2264 c*(log(log (\u230az\u230b\u208a)))/log(N/4) :=\nbegin\n  have hmertens := explicit_mertens,\n  rw eventually_at_top at hmertens,\n  rcases hmertens with \u27e8c\u2081,hmertens\u27e9,\n  let c := a*2,\n  use c, refine \u27e8mul_pos ha zero_lt_two,_\u27e9,\n  filter_upwards [eventually_gt_at_top 4,\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_ge_at_top c\u2081),\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_gt_at_top (exp(1)))]\n    with N h4N hcN heN,\n  have h0N : (0:\u211d) < N, { norm_cast, exact lt_trans zero_lt_four h4N, },\n  have hlogN : 0 < log(N/4), {  refine log_pos _,\n    rw one_lt_div, exact_mod_cast h4N, exact zero_lt_four, },\n  intros z hz',\n  by_cases hz : (N:\u211d) \u2264 z,\n  have hexpz : exp 1 < \u230az\u230b\u208a, {\n    rw \u2190 nat.le_floor_iff' at hz, refine lt_of_lt_of_le heN _, exact_mod_cast hz,\n    refine ne_of_gt _, exact_mod_cast h0N,\n   },\n  calc _ \u2264 \u2211 (x : \u2115) in filter nat.prime (Icc N \u230az\u230b\u208a), (a/log(N/4)) * (1/x) :_\n     ... \u2264 _ :_,\n  refine sum_le_sum _, intros p hp, rw [mem_filter, mem_Icc] at hp,\n  rw [div_mul, div_le_div_left ha, \u2190 div_mul, div_one, mul_le_mul_right, log_le_log,\n     div_le_div_right], exact_mod_cast hp.1.1, exact zero_lt_four,\n  refine div_pos h0N zero_lt_four, refine div_pos _ zero_lt_four, exact_mod_cast (nat.prime.pos hp.2),\n  exact_mod_cast (nat.prime.pos hp.2), refine mul_pos _ _, refine log_pos _,\n  rw one_lt_div, norm_cast, refine lt_of_lt_of_le h4N hp.1.1, exact zero_lt_four,\n  exact_mod_cast (nat.prime.pos hp.2), refine div_pos hlogN _, rw one_div_pos,\n  exact_mod_cast (nat.prime.pos hp.2), rw [\u2190 mul_sum, div_mul_eq_mul_div, div_le_div_right hlogN,\n    \u2190 le_div_iff' ha],\n  transitivity ((\u2211 q in (finset.range (\u230az\u230b\u208a + 1)).filter is_prime_pow, 1 / q) : \u211d),\n  refine sum_le_sum_of_subset_of_nonneg _ _, intros q hq,\n  rw [mem_filter, mem_range], rw [mem_filter, mem_Icc, nat.prime_iff] at hq,\n  rw nat.lt_succ_iff, refine \u27e8hq.1.2,prime.is_prime_pow hq.2\u27e9,\n  intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n,\n  refine le_trans (hmertens \u230az\u230b\u208a _) _, rw \u2190 nat.le_floor_iff' at hz,\n  refine le_trans hcN _, exact_mod_cast hz, refine ne_of_gt _,\n  exact_mod_cast h0N, rw [le_div_iff ha, mul_comm _ a, \u2190 mul_assoc, mul_le_mul_right],\n  refine log_pos _, rw [\u2190 exp_lt_exp, exp_log], exact hexpz,\n  refine lt_trans _ hexpz, exact exp_pos 1,\n  have : Icc N \u230az\u230b\u208a = \u2205, { refine finset.Icc_eq_empty _, rw nat.le_floor_iff', exact hz,\n    refine ne_of_gt _, exact_mod_cast h0N, },\n  rw [this, filter_empty, sum_empty], refine div_nonneg _ _, refine mul_nonneg _ _,\n  refine mul_nonneg _ zero_le_two, exact le_of_lt ha, exact hz',\n  refine log_nonneg _, rw le_div_iff, norm_cast, rw one_mul,\n  exact le_of_lt h4N, exact zero_lt_four,\nend\n\nlemma filter_div_aux (a b c d: \u211d) (hb : 0 < b) (hc : 0 < c) : \u2203 y z w : \u211d,\n (2 \u2264 y) \u2227 (16 \u2264 w) \u2227 (0 < z) \u2227 (1 < z) \u2227 (4*y + 4 \u2264 z) \u2227 (a \u2264 y) \u2227 (d \u2264 y) \u2227 (log w / log z \u2264 b) \u2227\n (\u2211 (x : \u2115) in filter nat.prime (Icc \u2308w\u2309\u208a \u230az\u230b\u208a), (log y / (log (x/4) * x)) \u2264 c) :=\nbegin\n  let y := max (2:\u211d) (max a d),\n  have hlogy : 0 < log y, { refine log_pos _, exact lt_of_lt_of_le one_lt_two (le_max_left _ _), },\n  rcases plogp_tail_bound (log y) hlogy with \u27e8C\u2081,h0C\u2081,htail\u27e9,\n  rw eventually_at_top at htail, rcases htail with \u27e8C\u2082',htail'\u27e9,\n  let C\u2082 := max 1 C\u2082',\n  have haux: asymptotics.is_O_with (1 / (C\u2081 * (1 / c * (2 * (1 / b))))) at_top (\u03bb (x : \u211d), (log x))\n     (\u03bb (x : \u211d), x^((1:\u211d))), {\n    refine asymptotics.is_o.def' _ _, refine is_o_log_rpow_at_top _, exact zero_lt_one,\n    rw one_div_pos, refine mul_pos h0C\u2081 _, refine mul_pos _ _, rw one_div_pos, exact hc,\n    refine mul_pos zero_lt_two _, rw one_div_pos, exact hb,\n    },\n  have haux' := tendsto_log_at_top.eventually haux.bound,\n  rw eventually_at_top at haux', rcases haux' with \u27e8C\u2083,haux'\u27e9,\n  let z := max (exp (log 4 * 2 / b)) (max C\u2083 (max (3:\u211d) (max\n     (4*y + 4)\n     (max (exp (exp (log (16 / 4) * c / C\u2081)) + 1) (exp (exp (log (C\u2082 / 4) * c / C\u2081)) + 1))))),\n  let w := 4*exp (C\u2081 * log (log \u230az\u230b\u208a) / c),\n  have hz\u2081 : exp (log 4 * 2 / b) \u2264 z, { refine le_max_left _ _, },\n  have hz\u2082 : C\u2083 \u2264 z, { refine le_trans (le_max_left _ _) (le_max_right _ _), },\n  have hz\u2084' : 3 \u2264 z, { refine le_trans (le_max_left _ _)\n    (le_trans (le_max_right _ _) (le_max_right _ _) ), },\n  have hz\u2084 : 2 < z, { refine lt_of_lt_of_le _ hz\u2084', norm_num1, },\n  have hz\u2085 : exp(1) < z, { refine lt_of_lt_of_le _ hz\u2084',\n    refine lt_trans real.exp_one_lt_d9 _, norm_num1, },\n  have hz\u2086 : (4*y + 4) \u2264 z, { refine le_trans (le_max_left _ _)\n   (le_trans (le_max_right _ _) ((le_trans (le_max_right _ _) (le_max_right _ _) ))), },\n  have hzfloor : z - 1 \u2264 \u230az\u230b\u208a, { rw sub_le_iff_le_add, refine le_of_lt (nat.lt_floor_add_one _), },\n  have hz\u2083 : 1 \u2264 z := le_trans one_le_two (le_of_lt hz\u2084),\n  have hz\u2080 : 0 < z := lt_of_lt_of_le zero_lt_one hz\u2083,\n  have hz\u2088' : exp (exp (log (16 / 4) * c / C\u2081)) + 1 \u2264 z, { refine le_trans (le_max_left _ _)\n   (le_trans (le_max_right _ _) ((le_trans (le_max_right _ _)\n      ((le_trans (le_max_right _ _) (le_max_right _ _) ))))), },\n  have hz\u2089' : exp (exp (log (C\u2082 / 4) * c / C\u2081)) + 1 \u2264 z, { refine le_trans (le_max_right _ _)\n   (le_trans (le_max_right _ _) ((le_trans (le_max_right _ _)\n      ((le_trans (le_max_right _ _) (le_max_right _ _) ))))), },\n  have hz\u2088 : log (16 / 4) * c / C\u2081 \u2264 log (log \u230az\u230b\u208a), {\n    rw [\u2190 exp_le_exp, exp_log, \u2190 exp_le_exp, exp_log], refine le_trans _ hzfloor,\n    rw le_sub_iff_add_le, exact hz\u2088', norm_cast, rw nat.floor_pos, exact hz\u2083,\n    refine log_pos _, refine lt_of_lt_of_le _ hzfloor, rw lt_sub_iff_add_lt,\n    exact hz\u2084,\n   },\n  have hz\u2089 : log (C\u2082 / 4) * c / C\u2081 \u2264 log (log \u230az\u230b\u208a), {\n    rw [\u2190 exp_le_exp, exp_log, \u2190 exp_le_exp, exp_log], refine le_trans _ hzfloor,\n    rw le_sub_iff_add_le, exact hz\u2089', norm_cast, rw nat.floor_pos, exact hz\u2083,\n    refine log_pos _, refine lt_of_lt_of_le _ hzfloor, rw lt_sub_iff_add_lt,\n    exact hz\u2084,\n   },\n  have hz\u2087 : (0 \u2264 log(log (\u230az\u230b\u208a))), { refine le_trans _ hz\u2088, refine div_nonneg _ _,\n    refine mul_nonneg _ _, refine log_nonneg _, norm_num1, exact le_of_lt hc,\n    exact le_of_lt h0C\u2081, },\n  have hzw : (exp (log w / b)) \u2264 z, {\n    rw [\u2190 log_le_log, log_exp, div_le_iff, log_mul, log_exp, \u2190 add_halves (log z*b)],\n    refine add_le_add _ _, rw [le_div_iff, \u2190 div_le_iff, \u2190 exp_le_exp, exp_log],\n    exact hz\u2081, exact hz\u2080, exact hb, exact zero_lt_two,\n    rw [le_div_iff, div_eq_mul_one_div, \u2190 div_le_iff, div_eq_mul_one_div, mul_assoc,\n      mul_assoc, mul_comm C\u2081, mul_assoc, mul_comm], specialize haux' z hz\u2082,\n    rw [\u2190 le_div_iff', div_eq_mul_one_div, mul_comm],\n    transitivity log(log z), rw [log_le_log, log_le_log], refine nat.floor_le _,\n    exact le_of_lt hz\u2080, norm_cast, rw nat.floor_pos, exact hz\u2083, exact hz\u2080, refine log_pos _,\n    norm_cast, rw \u2190 nat.succ_lt_succ_iff, rw \u2190 @nat.cast_lt \u211d _ _ _ _,\n    refine lt_trans _ (nat.lt_succ_floor _), norm_num1, exact hz\u2084, refine log_pos _,\n    exact lt_trans one_lt_two hz\u2084, rw [norm_eq_abs, norm_eq_abs, rpow_one, abs_of_pos,\n      abs_of_pos] at haux', exact haux', refine log_pos (lt_trans one_lt_two hz\u2084),\n    refine log_pos _, rw [\u2190 exp_lt_exp, exp_log], exact hz\u2085, exact hz\u2080,\n    refine mul_pos h0C\u2081 _, refine mul_pos _ _, rw one_div_pos, exact hc,\n    refine mul_pos zero_lt_two _, rw one_div_pos, exact hb, exact hb, exact zero_lt_two,\n    exact four_ne_zero, refine exp_ne_zero _, exact hb, refine exp_pos _, exact hz\u2080,\n  },\n  have h16w : 16 \u2264 w, {\n    rw [\u2190 div_le_iff', \u2190 log_le_log, log_exp, le_div_iff, \u2190 div_le_iff'], exact hz\u2088,\n    exact h0C\u2081, exact hc, norm_num1, refine exp_pos _, exact zero_lt_four,\n  },\n  have hC\u2082w : (C\u2082 :\u211d) \u2264 w, {\n    rw [\u2190 div_le_iff', \u2190 log_le_log, log_exp, le_div_iff, \u2190 div_le_iff'], exact hz\u2089,\n    exact h0C\u2081, exact hc, refine div_pos _ _, norm_cast, refine lt_of_lt_of_le zero_lt_one _,\n    refine le_max_left _ _, norm_num1, refine exp_pos _,\n    exact zero_lt_four,\n  },\n  have h0w' : (1:\u211d) < \u2308w\u2309\u208a / 4, { rw lt_div_iff, refine lt_of_lt_of_le _ (nat.le_ceil _),\n    refine lt_of_lt_of_le _ h16w, norm_num1, exact zero_lt_four, },\n  refine \u27e8y,z,w,le_max_left _ _,h16w, hz\u2080, (lt_trans one_lt_two hz\u2084), hz\u2086, le_trans (le_max_left _ _) (le_max_right _ _),\n     le_trans (le_max_right _ _) (le_max_right _ _),_,_\u27e9,\n  rw [div_le_iff, \u2190 div_le_iff' hb, \u2190 exp_le_exp, exp_log], exact hzw, exact hz\u2080,\n  refine log_pos _, refine lt_trans one_lt_two hz\u2084,\n  have h\u2081 : C\u2082' \u2264 \u2308w\u2309\u208a, {\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans _ (nat.le_ceil _),\n    refine le_trans _ hC\u2082w, norm_cast, refine le_max_right _ _,\n   },\n  refine le_trans (htail' \u2308w\u2309\u208a h\u2081 z hz\u2087) _,\n  rw [div_le_iff, \u2190 div_le_iff' hc, \u2190 exp_le_exp, exp_log],\n  rw le_div_iff, refine le_trans _ (nat.le_ceil _), rw mul_comm _ (4:\u211d),\n  exact zero_lt_four,  exact lt_trans zero_lt_one h0w', exact log_pos h0w',\nend\n\nlemma filter_div  (D : \u211d) (hD : 0 < D) : \u2203 y z : \u211d,\n(1 \u2264 y) \u2227 (4*y + 4 \u2264 z) \u2227 (0<z) \u2227 (2 / (1 / (5 * D * 2) * D) \u2264 y) \u2227 ((2 / (1 / (5 * D * 2))) \u2264 y) \u2227\n  \u2200\u1da0 (N : \u2115) in at_top, \u2200 A \u2286 range(N),\n   (((A).filter (\u03bb n, \u00ac \u2203 d\u2081 d\u2082 : \u2115, (d\u2081 \u2223 n) \u2227 (d\u2082 \u2223 n) \u2227 (y \u2264 d\u2081) \u2227\n      (4*d\u2081 \u2264 d\u2082) \u2227 ((d\u2082 : \u211d) \u2264 z))).card : \u211d) \u2264 N/(5*D)\n    :=\nbegin\n  rcases sieve_lemma_prec' with \u27e8C,c,h0C,h0c,hsieve\u27e9,\n  have haux1 : 0 < (1 / (10 * D))/C, { refine div_pos _ h0C, rw one_div_pos, refine mul_pos _ hD, norm_num1, },\n  have haux2 : 0 < (1 / (20 * D))/C, { refine div_pos _ h0C, rw one_div_pos, refine mul_pos _ hD, norm_num1, },\n  rw filter.eventually_at_top at hsieve,\n  rcases hsieve with \u27e8T,hsieve\u27e9,\n  rcases (filter_div_aux (2 / (1 / (5 * D * 2) * D)) _ _ ((2 / (1 / (5 * D * 2)))) haux1 haux2)\n     with \u27e8y,z,w,h2y,h16w,h0z,h1z,hyz,hDy,hDy',hwzD',hzsum\u27e9,\n  have hwzD : C * (log w / log z) \u2264 1 / (10 * D), { rw \u2190 le_div_iff', exact hwzD', exact h0C, },\n  have h2w : 2 \u2264 w, { refine le_trans _ h16w, norm_num1, },\n  have h1y : 1 \u2264 y := le_trans one_le_two h2y,\n  have h0zc : (0:\u211d) < \u230az\u230b\u208a, { norm_cast, rw \u2190 nat.succ_lt_succ_iff, rw \u2190 @nat.cast_lt \u211d _ _ _ _, push_cast,\n    rw zero_add, refine lt_trans _ (nat.lt_floor_add_one _), refine lt_of_lt_of_le _ hyz,\n    transitivity (4+(4:\u211d)*1), norm_num1,\n    rw [add_comm _ (4:\u211d), real.add_lt_add_iff_left, mul_lt_mul_left zero_lt_four],\n    exact lt_of_lt_of_le one_lt_two h2y, exact real.nontrivial,\n  },\n  refine \u27e8y,z,h1y,hyz,h0z,hDy,hDy',_\u27e9,\n  filter_upwards [tendsto_coe_nat_at_top_at_top.eventually (eventually_gt_at_top (0:\u211d)),\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_ge_at_top ((T:\u211d)*\u230az\u230b\u208a)),\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_ge_at_top\n       ((\u2211 (x : \u2115) in filter nat.prime (Icc \u2308w\u2309\u208a \u230az\u230b\u208a), C * (log y / log (x/4) * 1)) * (20 * D))),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_ge_at_top\n       ((4:\u211d) * \u230az\u230b\u208a / c + log \u230az\u230b\u208a)),\n     (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_ge_at_top\n       (z/c)),\n       eventually_ge_at_top T]\n    with N h0N hTzN hweirdN hlogN1 hlogN2 hlarge,\n  intros A hA, transitivity (((finset.range(N)).filter (\u03bb n, \u00ac \u2203 d\u2081 d\u2082 : \u2115, (d\u2081 \u2223 n) \u2227 (d\u2082 \u2223 n) \u2227 (y \u2264 d\u2081) \u2227\n      (4*d\u2081 \u2264 d\u2082) \u2227 ((d\u2082 : \u211d) \u2264 z))).card : \u211d),\n  norm_cast, refine card_le_of_subset _, refine filter_subset_filter _ hA,\n  have hz' : z \u2264 c*log N, { rw \u2190 div_le_iff', exact hlogN2, exact h0c, },\n  let X := ((finset.range(N)).filter(\u03bb n, \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192\n                     ((p : \u211d) < w) \u2228 (z < p))),\n  let Y := (\u03bb m, (finset.range(N)).filter(\u03bb n, m \u2223 n \u2227 \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192\n                     ((p : \u211d) < y) \u2228 (m < 4*p))),\n  have hXbound : (X.card : \u211d) \u2264 C*(log w/log z)*N := hsieve N hlarge w z h2w h1z hz',\n  have hYlocbound : \u2200 m : \u2115, (16 \u2264 m) \u2192 ((m:\u211d)/4 \u2264 c*log \u2308(N:\u211d)/m\u2309\u208a) \u2192 (T \u2264 \u2308(N:\u211d)/m\u2309\u208a) \u2192\n      ((Y m).card : \u211d) \u2264 C*(log y/log ((m:\u211d)/4))*(N/m + 1), {\n    intros m h16m hm hTm,\n    have h0m : 0 < m, { refine lt_of_lt_of_le _ h16m, norm_num1, },\n    transitivity (((finset.range(\u2308(N:\u211d)/m\u2309\u208a)).filter(\u03bb n, \u2200 p : \u2115, prime p \u2192 p \u2223 n \u2192\n                     ((p : \u211d) < y) \u2228 (((m:\u211d)/4 < p)))).card : \u211d),\n    norm_cast, refine finset.card_le_card_of_inj_on (\u03bb i, i / m) _ _,\n    intros n hn, rw [mem_filter, mem_range], rw [mem_filter, mem_range] at hn,\n    refine \u27e8_,_\u27e9, rw nat.lt_ceil, rw nat.cast_div hn.2.1,\n    rw div_lt_div_right, norm_cast, exact hn.1, exact_mod_cast h0m, norm_cast,\n    intro hbad, rw hbad at h0m, exact nat.lt_asymm h0m h0m,\n    intros p hp hpnm, rw [div_lt_iff, mul_comm _ (4:\u211d)], norm_cast,\n    refine hn.2.2 p hp _, refine dvd_trans hpnm (nat.div_dvd_of_dvd hn.2.1), exact zero_lt_four,\n    intros a ha b hb hab, rw mem_filter at ha, rw mem_filter at hb,\n    rw [nat.div_eq_iff_eq_mul_right h0m ha.2.1, nat.mul_div_cancel_left' hb.2.1] at hab, exact hab,\n    have h1m' : 1 < ((m:\u211d)/4), { rw one_lt_div, norm_cast,\n       refine lt_of_lt_of_le _ h16m, norm_num1, exact zero_lt_four, },\n    refine le_trans (hsieve \u2308(N:\u211d)/m\u2309\u208a hTm y ((m:\u211d)/4) h2y h1m' hm) _,\n    rw mul_le_mul_left, refine le_of_lt (nat.ceil_lt_add_one _), refine div_nonneg (nat.cast_nonneg N) (nat.cast_nonneg m),\n    refine mul_pos h0C (div_pos (log_pos _) (log_pos _)), exact lt_of_lt_of_le one_lt_two h2y,\n    have h14 : (1:\u211d) < 4*1 := by norm_num1, refine lt_of_lt_of_le h14 _,\n    rw [mul_one, le_div_iff], norm_num1, norm_cast, exact h16m, exact zero_lt_four,\n  },\n  let Y' := ((finset.Icc \u2308w\u2309\u208a \u230az\u230b\u208a).filter(\u03bb r:\u2115, nat.prime r)).bUnion (\u03bb p, Y p),\n  have hcover : (finset.range(N)).filter (\u03bb n, \u00ac \u2203 d\u2081 d\u2082 : \u2115, (d\u2081 \u2223 n) \u2227 (d\u2082 \u2223 n) \u2227 (y \u2264 d\u2081) \u2227\n      (4*d\u2081 \u2264 d\u2082) \u2227 ((d\u2082 : \u211d) \u2264 z)) \u2286 X \u222a Y', {\n        intros n hn, rw [mem_filter, not_exists] at hn,\n        rw [mem_union, or_iff_not_imp_left, mem_filter, mem_bUnion], intro hn',\n        rw [not_and_distrib, or_iff_not_imp_left, not_not] at hn', specialize hn' hn.1,\n        rw [not_forall] at hn', rcases hn' with \u27e8p,hp\u27e9, rw [not_imp, not_imp,\n           decidable.not_or_iff_and_not, not_lt, not_lt] at hp,\n        refine \u27e8p,_,_\u27e9,\n        rw [mem_filter, mem_Icc], rw \u2190 nat.prime_iff at hp, refine \u27e8\u27e8_,_\u27e9,hp.1\u27e9, rw nat.ceil_le, exact hp.2.2.1,\n        rw nat.le_floor_iff', exact hp.2.2.2, rw \u2190 pos_iff_ne_zero, exact nat.prime.pos hp.1,\n        rw mem_filter, refine \u27e8hn.1,hp.2.1,_\u27e9, intros q hq hqn,\n        have hn'' := hn.2 q, rw not_exists at hn'', specialize hn'' p,\n        rw [not_and_distrib, or_iff_not_imp_left, not_not] at hn'', specialize hn'' hqn,\n        rw [not_and_distrib, or_iff_not_imp_left, not_not] at hn'', specialize hn'' hp.2.1,\n        rw [not_and_distrib] at hn'', cases hn'' with hn''1 hn''2,\n        rw \u2190 not_le, left, exact hn''1, right, rw [not_and_distrib, or_iff_not_imp_right, not_not] at hn''2,\n        specialize hn''2 hp.2.2.2, rw not_le at hn''2, exact hn''2,\n       },\n  calc _ \u2264 ((X\u222aY').card :\u211d) :_\n     ... \u2264 (X.card : \u211d) + (Y'.card : \u211d) :_\n     ... \u2264 C*(log w/log z)*N + (Y'.card : \u211d) :_\n     ... \u2264 C*(log w/log z)*N + \u2211 p in (finset.Icc \u2308w\u2309\u208a \u230az\u230b\u208a).filter(\u03bb r:\u2115, nat.prime r), ((Y p).card) :_\n     ... \u2264 C*(log w/log z)*N + \u2211 p in (finset.Icc \u2308w\u2309\u208a \u230az\u230b\u208a).filter(\u03bb r:\u2115, nat.prime r), C*(log y/log (p/4))*(N/p + 1) :_\n     ... \u2264 (N:\u211d)/(10*D) + (N:\u211d)/(10*D) :_\n     ... \u2264 _ :_,\n  norm_cast, refine card_le_of_subset hcover,\n  norm_cast, refine card_union_le _ _, rw add_le_add_iff_right, exact hXbound,\n  rw add_le_add_iff_left, norm_cast, exact finset.card_bUnion_le, rw add_le_add_iff_left,\n  refine sum_le_sum _, intros p hp, rw [mem_filter, mem_Icc] at hp,\n  have h16p : 16 \u2264 p, { refine le_trans _ hp.1.1, rw \u2190 @nat.cast_le \u211d _ _ _ _,\n     refine le_trans _ (nat.le_ceil _), norm_cast, exact h16w,},\n  refine hYlocbound p h16p _ _,\n  transitivity ((4:\u211d)*\u230az\u230b\u208a), norm_cast, rw [div_le_iff, mul_comm _ (4:\u211d)], push_cast, rw [\u2190 mul_assoc], norm_cast,\n  transitivity (1*\u230az\u230b\u208a), rw one_mul, exact hp.1.2, rw mul_le_mul_right, norm_num1, exact_mod_cast h0zc, exact zero_lt_four,\n  rw \u2190 div_le_iff', transitivity log((N:\u211d)/p), transitivity log((N:\u211d)/\u230az\u230b\u208a), rw [log_div, le_sub_iff_add_le],\n  exact hlogN1, exact ne_of_gt h0N, exact ne_of_gt h0zc, rw [log_le_log, div_le_div_left], exact_mod_cast hp.1.2,\n  exact h0N,  exact h0zc, norm_cast, exact nat.prime.pos hp.2, exact div_pos h0N h0zc,\n  refine div_pos h0N _, exact_mod_cast (nat.prime.pos hp.2), rw log_le_log, refine nat.le_ceil _,\n  refine div_pos h0N _, exact_mod_cast (nat.prime.pos hp.2), refine lt_of_lt_of_le _ (nat.le_ceil _),\n  refine div_pos h0N _, exact_mod_cast (nat.prime.pos hp.2), exact h0c, rw \u2190 @nat.cast_le \u211d _ _ _ _,\n  refine le_trans _ (nat.le_ceil _), rw le_div_iff,\n  by_cases h0T : (0:\u211d) < T,\n  transitivity ((T:\u211d)*\u230az\u230b\u208a), rw mul_le_mul_left h0T, exact_mod_cast hp.1.2, exact hTzN,\n  transitivity (0:\u211d), rw mul_nonpos_iff, right, rw not_lt at h0T,\n  refine \u27e8h0T, le_of_lt _\u27e9, exact_mod_cast (nat.prime.pos hp.2), exact nat.cast_nonneg N,\n  exact_mod_cast (nat.prime.pos hp.2),\n  refine add_le_add _ _, rw [div_eq_mul_one_div (N:\u211d), mul_comm (N:\u211d), mul_le_mul_right h0N],\n  exact hwzD, simp_rw [mul_assoc, mul_add],\n  calc _ \u2264 (N:\u211d)/(20*D) + (N:\u211d)/(20*D) :_\n     ... \u2264 _ :_,\n  rw sum_add_distrib, refine add_le_add _ _, simp_rw [\u2190 mul_div_mul_comm, mul_comm _ (N:\u211d), \u2190 mul_div,\n     \u2190 mul_assoc], rw [\u2190 mul_sum, mul_comm C, mul_assoc, div_eq_mul_one_div (N:\u211d), mul_le_mul_left h0N,\n      \u2190 le_div_iff' h0C], exact hzsum, rw le_div_iff, exact hweirdN, refine mul_pos _ hD, norm_num1,\n  rw [\u2190 two_mul, \u2190 le_div_iff', div_div, mul_comm _ (2:\u211d), \u2190 mul_assoc], norm_num1, refl,\n  norm_num1,\n  rw [\u2190 two_mul, \u2190 le_div_iff', div_div, mul_comm _ (2:\u211d), \u2190 mul_assoc], norm_num1, refl,\n  norm_num1,\nend\n\n\nlemma turan_primes_estimate : \u2203 (C : \u211d), \u2200\u1da0 (N : \u2115) in at_top,\n  (\u2211 n in (Icc 1 N), ((\u03c9 n : \u211d) - log(log N))^2\n  \u2264  C * N * log(log N)  ) :=\nbegin\n  rcases sum_prime_counting with \u27e8C1,hsum\u27e9,\n  rcases sum_prime_counting_sq with \u27e8C2,hsumsq\u27e9,\n  let C := (C2+2*C1),\n  use C,\n  filter_upwards [hsum, hsumsq,\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_gt_at_top (0:\u211d))]\n   with N hlargesum hlargesumsq hlargeN,\n  have hcardIcc : (Icc 1 N).card = N, {\n    rw nat.card_Icc, simp only [nat.add_succ_sub_one, add_zero],\n   },\n  simp_rw [sub_sq, sum_add_distrib, sum_sub_distrib, \u2190 sum_mul, \u2190 mul_sum,\n    sum_const, nsmul_eq_mul, hcardIcc],\n  calc _ \u2264 \u2211 (x : \u2115) in Icc 1 N, (\u03c9 x:\u211d) ^ 2 - 2*(-(C1*N)+N*log(log N))*log(log N) + N * log (log N) ^ 2 :_\n     ... \u2264 C2*N*log(log N) + N*(log(log N))^2- 2*(-(C1*N)+N*log(log N))*log(log N) + N * log (log N) ^ 2 :_\n     ... = _ :_,\n  rw [add_le_add_iff_right, sub_le_sub_iff_left, mul_le_mul_right, mul_le_mul_left zero_lt_two],\n  rw neg_add_eq_sub, exact hlargesum, exact real.nontrivial, exact hlargeN,\n  rw [add_le_add_iff_right, sub_le_sub_iff_right, add_comm], exact hlargesumsq, ring_nf,\n  rw [mul_assoc (2*C1), mul_comm _ C2, \u2190 add_mul, \u2190 mul_assoc],\nend\n\n\n\nlemma filter_regular  (D : \u211d) (hD : 0 < D) : \u2200\u1da0 (N : \u2115) in at_top,\n  \u2200 A \u2286 range(N),\n   ((A.filter(\u03bb n:\u2115, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 2 * log (log N)))).card : \u211d)\n   \u2264 N/D :=\nbegin\n  rcases turan_primes_estimate with \u27e8C,hturan\u27e9,\n  have h100 : (0:\u211d) < 1/100 := by norm_num1,\n  filter_upwards [hturan,\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_gt_at_top (0:\u211d)),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_ge_at_top (\n    C / (1 / 100) / (1 / D * (1 / 100)))),\n       (tendsto_coe_nat_at_top_at_top).eventually (eventually_gt_at_top (0:\u211d))]\n    with N hNturan hlargeN hlargeN2 hlargeN3,\n  clear hturan,\n  intros A hA,\n  by_contra, rw not_le at h, rw \u2190 not_lt at hNturan, refine hNturan _, clear hNturan,\n  let A' := A.filter(\u03bb n:\u2115, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 2 * log (log N))),\n  calc _ \u2264 ((N:\u211d)/D)*((1/100)*log(log N))^2 :_\n     ... < (A'.card : \u211d)*((1/100)*log(log N))^2 :_\n     ... \u2264 (\u2211 n in A', ((\u03c9 n : \u211d) - log(log N))^2) :_\n     ... \u2264 _ :_,\n  rw [mul_comm C, div_eq_mul_one_div (N:\u211d), mul_assoc, mul_assoc, mul_le_mul_left, sq,\n      \u2190 mul_assoc, \u2190 mul_assoc, mul_le_mul_right, \u2190 div_le_iff, \u2190 mul_assoc,\n     \u2190 div_le_iff'], exact hlargeN2,\n  refine mul_pos _ _, rw one_div_pos, exact hD, exact h100, exact h100, exact hlargeN,\n  exact hlargeN3,\n  rw mul_lt_mul_right, exact h, refine sq_pos_of_pos _, refine mul_pos _ hlargeN,\n  norm_num1, rw [\u2190 nsmul_eq_mul], refine finset.card_nsmul_le_sum _ _ _ _,\n  clear h, intros n hn,\n  rw [mem_filter, not_and_distrib] at hn,\n  rw [sq_le_sq, le_abs, abs_of_pos], cases hn.2.2 with hn1 hn2,\n  right, rw [neg_sub, le_sub, \u2190 one_sub_mul], rw not_le at hn1, norm_num1,\n  exact le_of_lt hn1, left, rw [le_sub_iff_add_le, add_comm, \u2190 one_add_mul],\n  rw not_le at hn2, refine le_trans _ (le_of_lt hn2), rw mul_le_mul_right, norm_num1,\n  exact hlargeN, refine mul_pos _ hlargeN, norm_num1,\n  refine sum_le_sum_of_subset_of_nonneg _ _,\n  intros m hm, rw mem_Icc, refine \u27e8_,_\u27e9,\n  rw [nat.succ_le_iff, pos_iff_ne_zero], intro hbad,\n  rw [hbad, mem_filter] at hm, refine hm.2.1 _, refl,\n  have htempy := hA ((filter_subset _ _) hm),\n  rw mem_range at htempy, exact le_of_lt htempy,\n  intros n hn1 hn2, refine sq_nonneg _,\nend\n\nlemma log_helper (y : \u211d) (h : 0 < y) (h'' : y \u2264 1/2) : -2*y \u2264 log(1-y) :=\nbegin\n have h' : y < 1 := lt_of_le_of_lt h'' one_half_lt_one,\n rw [neg_mul, neg_le, \u2190 log_inv],\n refine le_trans (real.log_le_sub_one_of_pos _) _, rw [inv_pos, sub_pos], exact h',\n rw [sub_le_iff_le_add, \u2190 one_div, div_le_iff, add_mul, one_mul, mul_sub, mul_one, mul_assoc,\n    \u2190 sq],\n convert_to 1 \u2264 1+(y-2*y^2) using 0, { ring_nf, },\n nth_rewrite 0 \u2190 add_zero (1:\u211d), refine add_le_add _ _, refl, rw sub_nonneg,\n nth_rewrite 1 \u2190 one_mul y, rw sq, rw \u2190 mul_assoc, rw mul_le_mul_right,\n rw \u2190 le_div_iff', exact h'', exact zero_lt_two, exact h, rw sub_pos, exact h',\nend\n\nlemma nat_floor_real_le_floor {M : \u211d} {N : \u2115} (h : M \u2264 N) : \u230aM\u230b\u208a \u2264 \u230aN\u230b\u208a :=\nbegin\n  have : \u230aN\u230b\u208a = \u230a(N:\u211d)\u230b\u208a, { rw nat.floor_eq_iff, refine \u27e8_,_\u27e9, norm_cast,\n    rw nat.floor_coe, rw nat.floor_coe, norm_cast, exact lt_add_one N, exact nat.zero_le N,},\n  rw this, rw nat.floor_coe, refine nat.floor_le_of_le h,\nend\n\nlemma diff_mertens_sum : \u2203 c : \u211d, \u2200\u1da0 (N : \u2115) in at_top,\n  \u2211 q in (range N).filter (\u03bb r, is_prime_pow r \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < r), (q : \u211d)\u207b\u00b9\n  \u2264 c/log(log N) :=\nbegin\n  have haux: asymptotics.is_O_with ((1: \u211d)/8) at_top (\u03bb (x : \u211d), (log x))\n     (\u03bb (x : \u211d), x^((1:\u211d))), {\n    refine asymptotics.is_o.def' _ _, refine is_o_log_rpow_at_top _,\n    norm_num1, norm_num1,\n    },\n  rcases prime_power_reciprocal with \u27e8b,hppr'\u27e9,\n  have hppr := asymptotics.is_O.exists_pos hppr',\n  clear hppr', rcases hppr with \u27e8c,h0c,hppr\u27e9, rw asymptotics.is_O_with_iff at hppr,\n  let C := c/2 + 16,\n  use C,\n  filter_upwards [tendsto_coe_nat_at_top_at_top.eventually (eventually_gt_at_top (0:\u211d)),\n    tendsto_coe_nat_at_top_at_top.eventually hppr,\n     (tendsto_pow_rec_loglog_spec_at_top.comp tendsto_coe_nat_at_top_at_top).eventually hppr,\n     (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_gt_at_top (0:\u211d)),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_gt_at_top (0:\u211d)),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_gt_at_top (8:\u211d)),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_ge_at_top ((8:\u211d)*2)),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually haux.bound]\n    with N h0N hlarge1 hlarge2 hlogN hloglogN h8loglogN h16loglogN hlarge5,\n  let M := (N:\u211d)^((1:\u211d)-8/log(log N)),\n  have hlarge4 : log (log N) * 4 \u2264 (1/2)*log N, {\n    dsimp at hlarge5, simp_rw [norm_eq_abs, rpow_one] at hlarge5,\n    rw abs_of_pos at hlarge5, rw abs_of_pos at hlarge5, rw [\u2190 le_div_iff, mul_comm, \u2190 div_eq_mul_one_div,\n      div_div], norm_num1, rw [div_eq_mul_one_div, mul_comm], exact hlarge5, exact zero_lt_four,\n      exact hlogN, exact hloglogN,\n   },\n  clear hlarge5,\n  have hlarge3 : log (log N) * 4 \u2264 log N, { refine hlarge4.trans _,\n    rw [mul_comm, \u2190 div_eq_mul_one_div], refine half_le_self _, exact le_of_lt hlogN, },\n  clear hppr,\n  simp_rw [norm_eq_abs, abs_le] at hlarge1, simp_rw [norm_eq_abs, abs_le] at hlarge2,\n  have hl1 := hlarge1.2, have hl2 := hlarge2.1, clear hlarge1 hlarge2,\n  dsimp at hl2, rw [log_rpow h0N, log_mul _ (ne_of_gt hlogN)] at hl2,\n  calc _ \u2264 \u2211 (q : \u2115) in filter is_prime_pow (Icc 1 \u230aN\u230b\u208a), (q:\u211d)\u207b\u00b9 -\n             \u2211 (q : \u2115) in filter is_prime_pow (Icc 1 \u230aM\u230b\u208a), (q:\u211d)\u207b\u00b9 :_\n     ... \u2264  c * |(log \u2191N)\u207b\u00b9|+(log (log N) + b) - \u2211 (q : \u2115) in filter is_prime_pow (Icc 1 \u230aM\u230b\u208a), (q:\u211d)\u207b\u00b9 :_\n     ... \u2264  c * |(log \u2191N)\u207b\u00b9|+(log (log N) + b) - ((log (1 - 8 / log (log N)) + log (log N) + b -\n             (c * |((1 - 8 / log (log N)) * log N)\u207b\u00b9|))) :_\n     ... \u2264 _ :_,\n  rw [le_sub_iff_add_le, \u2190 sum_union], refine sum_le_sum_of_subset_of_nonneg _ _,\n  intros q hq, rw [mem_filter, mem_Icc], rw [mem_union, mem_filter, mem_filter] at hq,\n  cases hq with hq1 hq2, rw nat.le_floor_iff, rw mem_range at hq1,\n  refine \u27e8\u27e8_,_\u27e9,hq1.2.1\u27e9, exact le_of_lt (is_prime_pow.one_lt hq1.2.1), norm_cast,\n  exact le_of_lt hq1.1, refine le_of_lt _, exact_mod_cast h0N, rw [mem_Icc] at hq2,\n  refine \u27e8\u27e8hq2.1.1,_\u27e9,hq2.2\u27e9, refine (hq2.1.2).trans _, refine nat_floor_real_le_floor _,\n  rw [\u2190 rpow_one (N:\u211d)], refine real.rpow_le_rpow_of_exponent_le _ _, norm_cast,\n  rw [nat.one_le_iff_ne_zero, \u2190 pos_iff_ne_zero], exact_mod_cast h0N, refine sub_le_self _ _,\n  refine div_nonneg _ _, norm_num1, exact le_of_lt hloglogN,\n  intros n hn1 hn2, rw inv_nonneg, exact nat.cast_nonneg n,\n  rw finset.disjoint_left, intros q hq, rw mem_filter at hq, intro hbad,\n  rw [mem_filter, mem_Icc] at hbad, rw \u2190 not_le at hq, refine hq.2.2 _,\n  rw nat.le_floor_iff at hbad, exact hbad.1.2,\n  refine rpow_nonneg_of_nonneg _ _, exact le_of_lt h0N,\n  rw nat.floor_coe at hl1,\n  rw [sub_le_sub_iff_right, \u2190 sub_le_iff_le_add], exact hl1,\n  rw [sub_le_sub_iff_left], rw [neg_le, neg_sub] at hl2, rw sub_le, exact hl2,\n  ring_nf,\n  calc _ \u2264 (c/log(log N))/2 - log(1-8/log(log N)) :_\n     ... \u2264 _ :_,\n  rw [sub_le_sub_iff_right, div_div, mul_comm _ c, div_eq_mul_one_div c, mul_le_mul_left],\n  calc _ \u2264 (1/(log(log N)*4)) + (1/(log(log N)*4)) :_\n     ... = _ :_,\n  refine add_le_add _ _, rw [abs_of_pos, one_div, inv_le_inv], exact hlarge3, exact hlogN,\n  refine mul_pos hloglogN zero_lt_four, rw inv_pos, exact hlogN,\n  rw [abs_of_pos, one_div, inv_le_inv], refine le_trans hlarge4 _, rw mul_le_mul_right,\n  rw [le_sub, sub_half, div_le_div_iff, one_mul], exact h16loglogN, exact hloglogN,\n  exact zero_lt_two, exact hlogN, refine mul_pos _ hlogN, rw [sub_pos, div_lt_one],\n  exact h8loglogN, exact hloglogN, refine mul_pos hloglogN _, exact zero_lt_four,\n  rw inv_pos, refine mul_pos _ hlogN, rw [sub_pos, div_lt_one],\n  exact h8loglogN, exact hloglogN, rw [div_add_div_same, div_eq_div_iff], ring_nf,\n  refine ne_of_gt _, refine mul_pos hloglogN zero_lt_four,\n  refine ne_of_gt _, refine mul_pos hloglogN zero_lt_two, exact h0c,\n  rw [mul_comm, \u2190 one_div, div_div, \u2190 div_eq_mul_one_div, mul_comm _ (2:\u211d), \u2190 div_div, sub_le,\n      div_sub_div_same, div_le_iff, mul_comm],\n  have hloghelper := log_helper (8/log(log N)) _ _,\n  rw [mul_div, div_le_iff'] at hloghelper, norm_num1 at hloghelper,\n  have hhelper2 : c/2 - C = -16, { rw [sub_add_eq_sub_sub, sub_self, zero_sub], },\n  rw hhelper2, exact hloghelper, exact hloglogN, refine div_pos _ hloglogN, norm_num1,\n  rw [div_le_div_iff, one_mul], exact h16loglogN, exact hloglogN, exact zero_lt_two,\n  exact hloglogN,\n  refine ne_of_gt _, rw [sub_pos, div_lt_one hloglogN], exact h8loglogN,\nend\n\nlemma filter_smooth (D : \u211d) (hD : 0 < D) : \u2200\u1da0 (N : \u2115) in at_top,\n\u2200 A \u2286 range(N),\n   ((A.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < q \u2227 q \u2223 n)).card : \u211d)\n   \u2264 N/D :=\nbegin\n  rcases diff_mertens_sum with \u27e8c,hdiff\u27e9,\n  filter_upwards [hdiff,\n      tendsto_coe_nat_at_top_at_top.eventually  (eventually_gt_at_top (0:\u211d)),\n      tendsto_coe_nat_at_top_at_top.eventually  (eventually_ge_at_top (D*2)),\n      (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually\n        (eventually_ge_at_top (0:\u211d)),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_ge_at_top (c / (1 / (2 * D)))),\n       (tendsto_log_at_top.comp (tendsto_log_at_top.comp\n    tendsto_coe_nat_at_top_at_top)).eventually (eventually_gt_at_top (0:\u211d))\n  ]\n  with N hdiff' hlarge1 hlarge2 hlarge3 hlarge4 hlarge5,\n  clear hdiff,\n  intros A hA,\n  let A' := erase A 0,\n  have hlocal : \u2200 q \u2208 (range N), (1 \u2264 q) \u2192 (A'.filter(\u03bb n, q \u2223 n)).card \u2264 N / q, {\n    intros q hq h1q,\n    calc _ \u2264 ((finset.Icc 1 N).filter (\u03bb n, q \u2223 n)).card :_\n       ... = _ : count_multiples _,\n    refine card_le_of_subset _, refine filter_subset_filter _ _, intros n hn,\n    rw mem_Icc, obtain hnN := hA ((erase_subset 0 A) hn), rw mem_range at hnN, refine \u27e8_,le_of_lt hnN\u27e9,\n    rw nat.one_le_iff_ne_zero, intro hbad, rw hbad at hn, exact (not_mem_erase 0 A) hn, exact h1q,\n  },\n  have hlocal' : \u2200 q \u2208 (range N), (1 \u2264 q) \u2192 ((A'.filter(\u03bb n, q \u2223 n)).card : \u211d) \u2264 N / q, {\n    intros q hq h1q, refine le_trans _ nat.cast_div_le, exact_mod_cast hlocal q hq h1q,\n  },\n  calc _ \u2264\n   ((A'.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < q \u2227 q \u2223 n)).card : \u211d) + 1 :_\n   ... \u2264 \u2211 q in ((range N).filter(\u03bb r:\u2115, is_prime_pow r \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < r)),\n           ((A'.filter(\u03bb n, q \u2223 n)).card : \u211d) + 1 :_\n   ... \u2264 (N:\u211d)*(\u2211 q in ((range N).filter(\u03bb r:\u2115, is_prime_pow r \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < r)), (1:\u211d)/q) + 1 :_\n   ... \u2264 (N:\u211d)/(2*D) + 1 :_\n   ... \u2264 _ :_,\n  norm_cast, rw filter_erase, refine le_trans (card_le_of_subset\n     (finset.insert_erase_subset 0 _)) _, refine finset.card_insert_le _ _,\n  rw add_le_add_iff_right,\n  have hdecomp : A'.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < q \u2227 q \u2223 n)\n    \u2286 ((range N).filter(\u03bb r:\u2115, is_prime_pow r \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < r)).bUnion\n            (\u03bb q, A'.filter(\u03bb n, q \u2223 n)),\n    { intros n hn, rw mem_filter at hn, rw [mem_bUnion], rcases hn.2 with \u27e8q,hq,hq2\u27e9,\n      refine \u27e8q,_,_\u27e9, rw mem_filter, refine \u27e8_,hq,hq2.1\u27e9, rw mem_range,\n      refine lt_of_le_of_lt (nat.le_of_dvd _ hq2.2) _, rw pos_iff_ne_zero,\n      intro hbad, rw hbad at hn, exact (not_mem_erase 0 A) hn.1, rw \u2190 mem_range,\n      exact hA ((erase_subset 0 A) hn.1), rw mem_filter, refine \u27e8hn.1,hq2.2\u27e9,\n      },\n  norm_cast,\n  refine le_trans (card_le_of_subset hdecomp) _, exact_mod_cast finset.card_bUnion_le,\n  rw [add_le_add_iff_right, mul_sum], refine finset.sum_le_sum _,\n  intros q hq, rw \u2190 div_eq_mul_one_div, rw mem_filter at hq, refine hlocal' q _ _,\n  exact hq.1, exact le_of_lt (is_prime_pow.one_lt hq.2.1),\n  rw [add_le_add_iff_right, div_eq_mul_one_div (N:\u211d), mul_le_mul_left],\n  calc _ = \u2211 (q : \u2115) in filter (\u03bb (r : \u2115), is_prime_pow r \u2227 (N:\u211d) ^ ((1:\u211d) - 8 / log (log N)) < r) (range N), (q:\u211d)\u207b\u00b9 :_\n     ... \u2264 c/log(log N) : hdiff'\n     ... \u2264 _ :_,\n  simp_rw one_div, rw div_le_iff, nth_rewrite 0 mul_comm, rw \u2190 div_le_iff,\n  exact hlarge4, rw one_div_pos, exact mul_pos zero_lt_two hD, exact hlarge5,\n  exact hlarge1, rw [mul_comm, \u2190 div_div, \u2190 le_sub_iff_add_le', sub_half, div_div,\n     one_le_div], exact hlarge2, refine mul_pos hD zero_lt_two,\nend\n\nlemma nat_le_cast_real_sub {m n : \u2115} : (n:\u211d)-(m:\u211d) \u2264 (n-m:\u2115) :=\nbegin\n  by_cases h : m < n,\n  rw nat.cast_sub (le_of_lt h), rw not_lt at h, rw nat.sub_eq_zero_of_le h,\n  norm_cast, rw sub_nonpos, exact_mod_cast h,\nend\n\nlemma final_large_N (D:\u211d) (hD : 0 < D) : \u2203 y z : \u211d,\n(1 \u2264 y) \u2227 (4*y + 4 \u2264 z) \u2227 (0 < z) \u2227\n\u2200\u1da0 N : \u2115 in at_top, ((0:\u211d)< N) \u2227\n (N : \u211d)^(1 - (1 : \u211d)/(log(log N))) + 1 < N/(5*D) \u2227 (\u2200 A \u2286 range(N),\n   (((A.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 (N:\u211d)^((1:\u211d)-8/(log(log N))) < q \u2227 q \u2223 n)).card : \u211d)\n   \u2264 N/(5*D))) \u2227  (\u2200 A \u2286 range(N),\n   ((A.filter(\u03bb n:\u2115, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 2 * log (log N)))).card : \u211d)\n   \u2264 N/(5*D)) \u2227 (\u2200 A \u2286 range(N),\n   ((A.filter(\u03bb n, \u00ac \u2203 d\u2081 d\u2082 : \u2115, (d\u2081 \u2223 n) \u2227 (d\u2082 \u2223 n) \u2227 (y \u2264 d\u2081) \u2227\n      (4*d\u2081 \u2264 d\u2082) \u2227 ((d\u2082 : \u211d) \u2264 z))).card : \u211d) \u2264 N/(5*D))\n  \u2227 z \u2264 (log N) ^ ((1:\u211d) / 500) \u2227\n   (2 / y + log N ^ -((1:\u211d)/ 200)) * N \u2264 N / (5 * D) :=\nbegin\n  rcases (filter_div D hD) with \u27e8y,z,h1y,hyz,h0z,hChelp,hChelp',hfilterdiv\u27e9,\n  refine \u27e8y,z,h1y,hyz,h0z,_\u27e9,\n  have h5D : 0 < 5*D, { refine mul_pos _ hD, norm_num1, },\n  have h1pos : (0:\u211d) < 1 := by norm_num1,\n  filter_upwards [eventually_gt_at_top 0, (filter_smooth (5*D) h5D),filter_regular (5*D) h5D,\n     hfilterdiv,\n     tendsto_coe_nat_at_top_at_top.eventually  (eventually_gt_at_top (2*(5*D))),\n    ((tendsto_pow_rec_log_log_at_top h1pos).comp tendsto_coe_nat_at_top_at_top).eventually\n        (eventually_ge_at_top (5 * D * 2)),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually\n        (eventually_ge_at_top (z^(500:\u211d))),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually\n        (eventually_gt_at_top (0:\u211d)),\n    (tendsto_log_at_top.comp tendsto_coe_nat_at_top_at_top).eventually\n        (eventually_ge_at_top ((1 / (1 / (5 * D) / 2)) ^ (200:\u211d))) ]\n    with N hlarge hsmooth hregular hdiv hlarge2 hlarge3 hlarge4 hlarge5 hlarge6,\n  dsimp at hlarge3 hlarge4 hlarge5 hlarge6,\n  refine \u27e8_,_,hsmooth,hregular,hdiv,_,_\u27e9, exact_mod_cast hlarge,\n  calc _ < (N : \u211d)^(1 - (1 : \u211d)/(log(log N))) + ((N/(5*D)))/2 :_\n     ... \u2264 _ :_,\n  rw [real.add_lt_add_iff_left, lt_div_iff, lt_div_iff, one_mul], exact hlarge2,\n  refine mul_pos _ hD, norm_num1, exact zero_lt_two,\n  rw [\u2190 le_sub_iff_add_le, sub_half, sub_eq_add_neg, add_comm, rpow_add_one, mul_comm, div_div],\n  nth_rewrite 1 div_eq_mul_one_div, rw [mul_le_mul_left, rpow_neg, \u2190 one_div, one_div_le_one_div],\n  exact hlarge3, refine rpow_pos_of_pos _ _, exact_mod_cast hlarge,\n  refine mul_pos _ zero_lt_two, refine mul_pos _ hD, norm_num1, refine le_of_lt _,\n  exact_mod_cast hlarge, exact_mod_cast hlarge, refine ne_of_gt _, exact_mod_cast hlarge,\n  have h500 : (0:\u211d) < 500 := by norm_num1,\n  rw [\u2190 real.rpow_le_rpow_iff _ _ h500, \u2190 rpow_mul, one_div_mul_cancel,\n       rpow_one], exact hlarge4, exact ne_of_gt h500, exact (le_of_lt hlarge5), exact le_of_lt h0z,\n  refine rpow_nonneg_of_nonneg _ _,\n  exact (le_of_lt hlarge5), nth_rewrite 2 div_eq_mul_one_div, rw [mul_comm, mul_le_mul_left],\n  calc _ \u2264 (1/(5*D))/2 + (1/(5*D))/2 :_\n     ... = _ :_,\n  refine add_le_add _ _, rw [div_div, div_le_iff, \u2190 div_le_iff'], exact hChelp',\n  rw one_div_pos, refine mul_pos (mul_pos _ hD) zero_lt_two,\n  norm_num1, exact lt_of_lt_of_le zero_lt_one h1y,\n  rw [rpow_neg, \u2190 one_div, one_div_le],\n  have h200 : (0:\u211d) < 200 := by norm_num1,\n  rw [\u2190 real.rpow_le_rpow_iff _ _ h200, \u2190 rpow_mul, one_div_mul_cancel, rpow_one],\n  exact hlarge6, exact ne_of_gt h200, exact (le_of_lt hlarge5), rw one_div_nonneg,\n  refine div_nonneg _ zero_le_two, rw one_div_nonneg, refine mul_nonneg _ (le_of_lt hD), norm_num1,\n  refine rpow_nonneg_of_nonneg (le_of_lt hlarge5) _, refine rpow_pos_of_pos hlarge5 _,\n  refine div_pos _ zero_lt_two, rw one_div_pos, refine mul_pos _ hD, norm_num1,\n  exact le_of_lt hlarge5, rw add_halves, exact_mod_cast hlarge,\nend\n\ntheorem unit_fractions_upper_density' (D : \u211d) (hD : 0 < D) : \u2203 y z : \u211d,\n(1 \u2264 y) \u2227 (0 \u2264 z) \u2227\n\u2200 A : set \u2115, (upper_density A > 1 / D) \u2192 \u2203 d \u2208 finset.Icc \u2308y\u2309\u208a \u230az\u230b\u208a,\n  \u2203 (S : finset \u2115), (S : set \u2115) \u2286 A \u2227 \u2211 n in S, (1 / n : \u211a) = 1/d :=\nbegin\n  rcases (final_large_N D hD) with \u27e8y,z,h1y,hyz,h0z,hfinal\u27e9,\n  refine \u27e8y,z,h1y,(le_of_lt h0z),_\u27e9, intros A hA,\n  have hlargeN := filter.frequently.and_eventually (frequently_nat_of hA) hfinal,\n  clear hfinal,\n  rw filter.frequently_iff_forall_eventually_exists_and at hlargeN,\n  specialize hlargeN technical_prop,\n  rcases hlargeN with \u27e8N,hlargeN,htech\u27e9,\n  dsimp at hlargeN,\n  have hzN := hlargeN.2.2.2.2.2.2.1,\n  have hyN := hlargeN.2.2.2.2.2.2.2,\n  let A' := filter (\u03bb (n : \u2115), n \u2208 A) (range N),\n  have hA'card : (N:\u211d)/D < A'.card, {\n    rw [div_eq_mul_one_div, \u2190 lt_div_iff'], exact hlargeN.1, exact hlargeN.2.1,\n   },\n  let M := (N:\u211d)^((1:\u211d)-8/(log(log N))),\n  let A0 := A'.filter(\u03bb n : \u2115, (n:\u211d) < (N : \u211d)^(1 - (1 : \u211d)/(log(log N)))),\n  have hA0card : (A0.card : \u211d) < N/(5*D), {\n    calc _ \u2264 ((finset.range(\u2308(N : \u211d)^(1 - (1 : \u211d)/(log(log N)))\u2309\u208a)).card : \u211d) :_\n       ... < _ :_,\n    norm_cast, refine finset.card_le_of_subset _, intros n hn,\n    rw mem_filter at hn, rw [mem_range, nat.lt_ceil], exact hn.2, rw finset.card_range,\n    refine lt_trans (nat.ceil_lt_add_one _) hlargeN.2.2.1,\n    refine rpow_nonneg_of_nonneg (le_of_lt hlargeN.2.1) _,\n   },\n  let A1 := A'.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 M < q \u2227 q \u2223 n),\n  have hA1card : (A1.card : \u211d) \u2264 N/(5*D), {\n    refine hlargeN.2.2.2.1 A' _, refine filter_subset _ _,\n  },\n  let A2 := A'.filter(\u03bb n, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 2 * log (log N))),\n  have hA2card : (A2.card : \u211d) \u2264 N/(5*D), {\n    refine hlargeN.2.2.2.2.1 A' _, refine filter_subset _ _,\n  },\n  let A3 := A'.filter(\u03bb n, \u00ac \u2203 d\u2081 d\u2082 : \u2115, (d\u2081 \u2223 n) \u2227 (d\u2082 \u2223 n) \u2227 (y \u2264 d\u2081) \u2227 (4*d\u2081 \u2264 d\u2082) \u2227 ((d\u2082 : \u211d) \u2264 z) ),\n  have hA3card : (A3.card : \u211d)  \u2264 N/(5*D), {\n    refine hlargeN.2.2.2.2.2.1 A' _, refine filter_subset _ _,\n  },\n  let A'' := A'\\(A0\u222aA1\u222aA2\u222aA3),\n  have hA''card : (N:\u211d)/(5*D) \u2264 A''.card, {\n    calc _ \u2264 (A'.card : \u211d) - (N/(5*D)+N/(5*D)+N/(5*D)+N/(5*D)) :_\n       ... \u2264 (A'.card : \u211d) - (A0\u222aA1\u222aA2\u222aA3).card :_\n       ... \u2264 _ :_,\n    rw [le_sub_iff_add_le, \u2190 add_div, \u2190 add_div, \u2190 add_div, \u2190 add_div, \u2190 two_mul, add_assoc,\n          \u2190 two_mul, \u2190 add_mul, \u2190 one_add_mul], norm_num1, rw mul_div_mul_left,\n    exact le_of_lt hA'card, norm_num1, rw sub_le_sub_iff_left,\n    calc _ \u2264 ((A0.card + A1.card + A2.card + A3.card : \u2115):\u211d) :_\n       ... \u2264 _ :_,\n    norm_cast, refine le_trans (card_union_le _ _) _, rw add_le_add_iff_right,\n    refine le_trans (card_union_le _ _) _, rw add_le_add_iff_right,\n    refine card_union_le _ _, push_cast,\n    refine add_le_add _ hA3card, refine add_le_add _ hA2card, refine add_le_add _ hA1card,\n    exact le_of_lt hA0card, refine le_trans nat_le_cast_real_sub _, norm_cast,\n    refine le_card_sdiff _ _,\n   },\n  clear hA'card hA0card hA1card hA2card hA3card,\n  have h0A'' : 0 \u2209 A'', {\n    intro hz, rw [mem_sdiff, not_mem_union, not_mem_union, not_mem_union] at hz,\n    nth_rewrite 1 mem_filter at hz, refine hz.2.1.1.1 \u27e8hz.1,_\u27e9,\n    refine rpow_pos_of_pos hlargeN.2.1 _,\n   },\n  have hA''N : \u2200 n \u2208 A'', n < N, {\n    intros n hn, rw [mem_sdiff, mem_filter, mem_range] at hn, exact hn.1.1,\n  },\n  have hstep : \u2203 S \u2286 A'', \u2203 d : \u2115, (y \u2264 d) \u2227 ((d : \u211d) \u2264 z) \u2227 rec_sum S = 1/d, {\n    refine htech A'' _ y z h1y hyz hzN h0A'' _ _ _ _ _,\n    intros n hn, rw mem_range, refine lt_of_lt_of_le (hA''N n hn) (nat.le_succ N),\n    intros n hn, rw [mem_sdiff, not_mem_union, not_mem_union, not_mem_union] at hn,\n    nth_rewrite 1 mem_filter at hn, rw \u2190 not_lt, intro hbad, refine hn.2.1.1.1 \u27e8hn.1,hbad\u27e9,\n    calc _ \u2264 (A''.card:\u211d)/N :_\n       ... \u2264 _ :_,\n    rw le_div_iff hlargeN.2.1, refine le_trans hyN hA''card,\n    rw [card_eq_sum_ones, rec_sum], push_cast, rw sum_div, refine sum_le_sum _,\n    intros n hn, rw [zero_add, one_div_le_one_div], norm_cast, exact le_of_lt (hA''N n hn),\n    exact hlargeN.2.1, norm_cast, rw pos_iff_ne_zero, intro hz, rw hz at hn, exact h0A'' hn,\n    intros n hn, rw [mem_sdiff, not_mem_union, not_mem_union, not_mem_union] at hn,\n    nth_rewrite 4 mem_filter at hn, rw [not_and, not_not] at hn, exact hn.2.2 hn.1,\n    intros n hn, rw is_smooth, intros q hq hqn,\n    rw [mem_sdiff, not_mem_union, not_mem_union, not_mem_union] at hn,\n    nth_rewrite 2 mem_filter at hn, rw not_and at hn, rw \u2190 not_lt, intro hbad,\n    refine hn.2.1.1.2 hn.1 _, refine \u27e8q,hq,hbad,hqn\u27e9,\n    rw arith_regular, intros n hn, rw [mem_sdiff, not_mem_union, not_mem_union, not_mem_union] at hn,\n    nth_rewrite 3 mem_filter at hn, rw [not_and, not_and, not_not] at hn,\n    refine hn.2.1.2 hn.1 _, intro hbad, refine hn.2.1.1.1 _,\n    rw [hbad, mem_filter], rw hbad at hn, refine \u27e8hn.1,_\u27e9, norm_cast,\n    refine rpow_pos_of_pos _ _, exact hlargeN.2.1,\n  },\n  clear htech,\n  rcases hstep with \u27e8S,hS,d,hyd,hdz,hrecd\u27e9, refine \u27e8d,_,S,_,_\u27e9,\n  rw mem_Icc, refine \u27e8_,_\u27e9, rw \u2190 nat.ceil_le at hyd, exact hyd,\n  rw \u2190 nat.le_floor_iff at hdz, exact hdz, exact le_of_lt h0z,\n  intros s hs, rw finset.mem_coe at hs, have := hS hs,\n  rw [mem_sdiff, mem_filter] at this, exact this.1.2,\n  rw rec_sum at hrecd, exact hrecd,\nend\n\n\ntheorem unit_fractions_upper_density (A : set \u2115) (hA : upper_density A > 0):\n   \u2203 (S : finset \u2115), (S : set \u2115) \u2286 A \u2227 \u2211 n in S, (1 / n : \u211a) = 1 :=\nbegin\n  let D := 2/ upper_density A,\n  have hD : 0 < D := div_pos zero_lt_two hA,\n  have hDA : 1/D < upper_density A, { rw one_div_div, refine half_lt_self hA, },\n  rcases (unit_fractions_upper_density' D hD) with \u27e8y,z,h1y,h0z,hupp\u27e9,\n  let M := \u2211 d in finset.Icc \u2308y\u2309\u208a \u230az\u230b\u208a, d,\n  let good_set : finset (finset \u2115) \u2192 Prop :=\n    \u03bb S, (\u2200 s \u2208 S, (s : set \u2115) \u2286 A) \u2227 (S : set (finset \u2115)).pairwise_disjoint id \u2227\n      \u2200 s, \u2203 (d : \u2115), s \u2208 S \u2192 y \u2264 d \u2227 (d : \u211d) \u2264 z \u2227 rec_sum s = 1 / d,\n  let P : \u2115 \u2192 Prop := \u03bb k, \u2203 S : finset (finset \u2115), S.card = k \u2227 good_set S,\n  let k : \u2115 := nat.find_greatest P (M+1),\n  have P0 : P 0 := \u27e8\u2205, by simp [good_set]\u27e9,\n  have Pk : P k := nat.find_greatest_spec (nat.zero_le _) P0,\n  obtain \u27e8S, hk, hS\u2081, hS\u2082, hS\u2083\u27e9 := Pk,\n  choose d' hd'\u2081 hd'\u2082 hd'\u2083 using hS\u2083,\n  let t : \u2115 \u2192 \u2115 := \u03bb d, (S.filter (\u03bb s, d' s = d)).card,\n  by_cases h : \u2203 d : \u2115, 0 < d \u2227 d \u2264 t d,\n  { obtain \u27e8d, d_pos, ht\u27e9 := h,\n    obtain \u27e8T', hT', hd\u2082\u27e9 := finset.exists_smaller_set _ _ ht,\n    have hT'S := hT'.trans (finset.filter_subset _ _),\n    refine \u27e8T'.bUnion id, _, _\u27e9,\n    have hfinstep : T'.bUnion id \u2286  S.bUnion id :=\n       by refine (finset.bUnion_subset_bUnion_of_subset_left _ hT'S),\n    rw \u2190 finset.coe_subset at hfinstep,\n    refine hfinstep.trans _, intros n hn,\n    rw [finset.coe_bUnion, set.mem_Union] at hn,\n    rcases hn with \u27e8i,hi\u27e9, rw set.mem_Union at hi, rcases hi with \u27e8hiS,hni\u27e9,\n    dsimp at hni, refine hS\u2081 i _ hni, rw \u2190 finset.mem_coe, exact hiS,\n    rw [sum_bUnion (hS\u2082.subset hT'S), finset.sum_congr rfl, finset.sum_const, hd\u2082,\n        nsmul_eq_mul, mul_div_cancel'],\n    { rw nat.cast_ne_zero, exact d_pos.ne' },\n    intros i hi,\n    rw [\u2190 rec_sum], dsimp, rw [hd'\u2083 _ (hT'S hi), (finset.mem_filter.1 (hT' hi)).2],\n    },\n  push_neg at h,\n  exfalso,\n  let A' := A \\ S.bUnion id,\n  have hAS : disjoint A' (S.bUnion id) := (set.disjoint_diff).symm,\n  have hDA' : 1/D < upper_density A', {\n    have : upper_density A = upper_density A' := upper_density_preserved,\n    rw \u2190 this, exact hDA },\n  specialize hupp A' hDA', rcases hupp with \u27e8d,hd,S',hS'\u27e9,\n  have hd' : y \u2264 d \u2227 (d : \u211d) \u2264 z, {\n    rw mem_Icc at hd, refine \u27e8_,_\u27e9,\n    refine le_trans (nat.le_ceil _) _, norm_cast, exact hd.1,\n    refine le_trans _ (nat.floor_le _), norm_cast, exact hd.2, exact h0z,\n   },\n  have h1d : 1 \u2264 d, {\n    have : (1:\u211d) \u2264 d := le_trans h1y hd'.1,\n    exact_mod_cast this,\n  },\n  have hS'' : \u2200 s \u2208 S, disjoint S' s, {\n    intros s hs, rw \u2190finset.disjoint_coe,\n    refine set.disjoint_of_subset_left hS'.1 _,\n    refine set.disjoint_of_subset_right _ hAS,\n    rw finset.coe_bUnion, refine set.subset_bUnion_of_mem hs,\n  },\n  have hS'A : (S':set \u2115) \u2286 A, {\n    refine subset_trans hS'.1 (set.diff_subset _ _),\n   },\n  have hS''' : S' \u2209 S,\n  { intro t,\n    exact (nonempty_of_rec_sum_recip h1d hS'.2).ne_empty (disjoint_self.1 (hS'' _ t)) },\n  have : P (k+1),\n  { refine \u27e8insert S' S, _, _\u27e9,\n    { rw [finset.card_insert_of_not_mem hS''', hk] },\n    refine \u27e8_, _, _\u27e9,\n    {  intros s hs, rw mem_insert at hs,\n       cases hs with hs1 hs2, rw hs1, exact hS'A, exact hS\u2081 s hs2,  },\n    { simpa [set.pairwise_disjoint_insert, hS\u2082] using \u03bb s hs _, hS'' _ hs },\n    intros s,\n    rcases eq_or_ne s S' with rfl | hs,\n    { exact \u27e8d, \u03bb _, \u27e8hd'.1, hd'.2, hS'.2\u27e9\u27e9 },\n    refine \u27e8d' s, \u03bb i, _\u27e9,\n    have : s \u2208 S := finset.mem_of_mem_insert_of_ne i hs,\n    exact \u27e8hd'\u2081 _ this, hd'\u2082 _ this, hd'\u2083 _ this\u27e9 },\n  have hk_bound : k+1 \u2264 M+1,\n  { rw [\u2190 hk, add_le_add_iff_right],\n    have hSdecomp : (finset.Icc \u2308y\u2309\u208a \u230az\u230b\u208a).bUnion(\u03bb d, S.filter (\u03bb (s : finset \u2115), d' s = d)) = S,\n    { refine finset.bUnion_filter_eq_of_maps_to _,\n      intros n hn, rw [mem_Icc, nat.ceil_le, nat.le_floor_iff],\n      refine \u27e8hd'\u2081 n hn,hd'\u2082 n hn\u27e9, exact h0z,\n       },\n    rw \u2190 hSdecomp, refine le_trans (finset.card_bUnion_le) _, refine finset.sum_le_sum _,\n    intros d' hd', refine le_of_lt (h d' _), rw [mem_Icc, nat.ceil_le] at hd',\n    exact_mod_cast (lt_of_lt_of_le zero_lt_one (le_trans h1y hd'.1)) },\n  have : k + 1 \u2264 k := nat.le_find_greatest hk_bound this,\n  simpa using this,\nend\n\nlemma rec_sum_union {A B : finset \u2115} :\n   (rec_sum (A\u222aB) : \u211d) \u2264 rec_sum A + rec_sum B :=\nbegin\n  rw [\u2190rat.cast_add, rat.cast_le, rec_sum, rec_sum, rec_sum, \u2190sum_union_inter,\n    le_add_iff_nonneg_right, \u2190rec_sum],\n  apply rec_sum_nonneg\nend\n\nlemma rec_sum_sdiff {A B : finset \u2115} :\n   (rec_sum A:\u211d) - rec_sum B \u2264 rec_sum (A\\B) :=\nbegin\n  rw [\u2190rat.cast_sub, rat.cast_le, tsub_le_iff_right, \u2190rec_sum_disjoint disjoint_sdiff_self_left],\n  apply rec_sum_mono,\n  rw sdiff_union_self_eq_union,\n  apply subset_union_left,\nend\n\nlemma rec_sum_bUnion {I : finset \u2115} (f : \u2115 \u2192 finset \u2115) :\n  (rec_sum (I.bUnion f) : \u211d) \u2264 \u2211 i in I, rec_sum (f i) :=\nbegin\n  rw [\u2190rat.cast_sum, rat.cast_le, rec_sum],\n  refine sum_bUnion_le_sum_of_nonneg (\u03bb x hx, one_div_nonneg.2 (nat.cast_nonneg _)),\nend\n\nexample {f g : \u211d \u2192 \u211d} {l : filter \u211d} (hf : tendsto f l at_top) (hfg : \u2200\u1da0 x in l, f x \u2264 g x) :\n  tendsto g l at_top :=\ntendsto_at_top_mono' _ hfg hf\n\nlemma this_particular_tends_to :\n  tendsto (\u03bb x : \u211d, x^(log(log(log x))/log(log x))) at_top at_top :=\nbegin\n  refine tendsto_at_top_mono' _ _ (tendsto_pow_rec_log_log_at_top zero_lt_one),\n  filter_upwards [eventually_ge_at_top (1 : \u211d),\n    (tendsto_log_at_top.comp tendsto_log_at_top).eventually_ge_at_top 0,\n    (tendsto_log_at_top.comp (tendsto_log_at_top.comp tendsto_log_at_top)).eventually_ge_at_top 1]\n      with x hx hx' hx'',\n  refine rpow_le_rpow_of_exponent_le hx (div_le_div_of_le hx' hx''),\nend\n\nlemma Ioc_subset_Ioc_union_Ioc {a b c : \u2115} :\n  Ioc a c \u2286 Ioc a b \u222a Ioc b c :=\nby { rw [\u2190coe_subset, coe_union, coe_Ioc, coe_Ioc, coe_Ioc], exact set.Ioc_subset_Ioc_union_Ioc }\n\nlemma bUnion_range_Ioc (N : \u2115) (f : \u2115 \u2192 \u2115) :\n   Ioc (f N) (f 0)  \u2286 (range(N)).bUnion(\u03bb i:\u2115, Ioc (f (i+1)) (f (i))) :=\nbegin\n  induction N, simp only [range_zero, bUnion_empty, Ioc_self], refl,\n  rw [range_succ, finset.bUnion_insert],\n  have :  Ioc (f (N_n + 1)) (f 0)  \u2286 (Ioc (f (N_n + 1)) (f N_n)  \u222a Ioc (f N_n) (f 0) ), {\n    refine Ioc_subset_Ioc_union_Ioc,\n   },\n  refine subset_trans this _, refine finset.union_subset_union _ N_ih, refl,\nend\n\nlemma this_fun_increasing_aux : strict_mono_on (\u03bb x, exp x / x ^ 2) (set.Ici 2) :=\nbegin\n  refine convex.strict_mono_on_of_deriv_pos (convex_Ici _) _ _,\n  { refine continuous_on_exp.div (continuous_on_pow _) _,\n    rintro x (hx : _ \u2264 _),\n    exact pow_ne_zero _ (zero_lt_two.trans_le hx).ne' },\n  rw interior_Ici,\n  intros x hx,\n  have hx\u2080 : 0 < x := zero_le_two.trans_lt hx,\n  rw [deriv_div differentiable_at_exp, real.deriv_exp, deriv_pow, nat.cast_two, pow_one, \u2190pow_mul,\n    sq, \u2190mul_sub, \u2190sub_mul],\n  { exact div_pos (mul_pos (exp_pos _) (mul_pos (sub_pos_of_lt hx) hx\u2080)) (pow_pos hx\u2080 _) },\n  { exact differentiable_at_pow },\n  exact pow_ne_zero _ hx\u2080.ne'\nend\n\nlemma this_fun_increasing' :\n  \u2200\u1da0 N : \u211d in at_top, \u2200 M, N \u2264 M \u2192 log N / log (log N) ^ 2 \u2264 log M / log (log M) ^ 2 :=\nbegin\n  filter_upwards [(tendsto_log_at_top.comp tendsto_log_at_top).eventually_ge_at_top 2,\n    tendsto_log_at_top.eventually_gt_at_top 0,\n    eventually_gt_at_top (0 : \u211d)]\n    with N hN hNl\u2080 hN\u2080 M hNM,\n  have hl : log N \u2264 log M := log_le_log_of_le hN\u2080 hNM,\n  have hll : log (log N) \u2264 log (log M) := log_le_log_of_le hNl\u2080 hl,\n  convert this_fun_increasing_aux.monotone_on hN (le_trans hN hll) hll,\n  { rw exp_log hNl\u2080 },\n  { rw exp_log (hNl\u2080.trans_le hl) }\nend\n\nlemma this_fun_increasing : \u2203 C : \u211d, \u2200 N M : \u2115, (C \u2264 N) \u2227 (N \u2264 M) \u2192\n  log N/(log(log N))^2 \u2264 log M/(log(log M))^2 :=\nbegin\n  obtain \u27e8C, hC\u27e9 := eventually_at_top.1 this_fun_increasing',\n  exact \u27e8C, \u03bb N M h, hC _ h.1 _ (nat.cast_le.2 h.2)\u27e9,\nend\n\nlemma harmonic_sum_bound_two' : \u2200\u1da0 (N : \u211d) in at_top,\n  \u2211 n in finset.range(\u2308N\u2309\u208a), (1 : \u211d)/n \u2264 2*log N :=\nbegin\n  have hharmonic := harmonic_sum_bound_two,\n  rw eventually_at_top at hharmonic, rcases hharmonic with \u27e8C,hharmonic\u27e9,\n  filter_upwards [eventually_ge_at_top ((C:\u211d)+1), eventually_gt_at_top (1:\u211d)] with N hN h1N,\n  rw [\u2190 le_sub_iff_add_le] at hN,\n  specialize hharmonic (\u2308N-1\u2309\u208a) _,\n  have haux := le_trans hN (nat.le_ceil _), exact_mod_cast haux,\n  transitivity \u2211 (n : \u2115) in range (\u2308N - 1\u2309\u208a + 1), (1:\u211d)/n,\n  refine sum_le_sum_of_subset_of_nonneg _ _, rw [range_subset, nat.ceil_le], push_cast,\n  rw \u2190 sub_le_iff_le_add, refine nat.le_ceil _, intros n hn1 hn2, rw one_div_nonneg,\n  exact nat.cast_nonneg n, refine le_trans hharmonic _,\n  rw [mul_le_mul_left zero_lt_two, log_le_log],\n  refine le_trans (le_of_lt (nat.ceil_lt_add_one _)) _, rw sub_nonneg, exact (le_of_lt h1N),\n  ring_nf, norm_cast, rw [pos_iff_ne_zero], intro hbad, rw [nat.ceil_eq_zero, \u2190 not_lt] at hbad,\n  refine hbad _, rw sub_pos, exact h1N, refine lt_trans zero_lt_one h1N, exact real.nontrivial,\nend\n\n\nlemma harmonic_sum_bound' : \u2203 C : \u211d, 0 < C \u2227 \u2200 (N : \u211d), (1 \u2264 N) \u2192\n  \u2211 n in (Icc 1 \u230aN\u230b\u208a), (1 : \u211d)/n \u2264 C*log (2*N) :=\nbegin\n  have hharmonic := harmonic_sum_bound_two,\n  rw eventually_at_top at hharmonic, rcases hharmonic with \u27e8C\u2081,hharmonic\u27e9,\n  let C\u2081' := max C\u2081 2,\n  let I := Ico 1 C\u2081',\n  let f := (\u03bb M : \u2115, \u2211 n in (Icc 1 M), (1 : \u211d)/n),\n  rcases (finset.exists_max_image I f _) with \u27e8y,hy,h\u27e9,\n  let C := max (2:\u211d) ((f y)/log 2),\n  have h0C : 0 < C := lt_of_lt_of_le zero_lt_two (le_max_left _ _),\n  refine \u27e8C,h0C,_\u27e9,\n  intros N h1N,\n  have h0N : 0 < N := lt_of_lt_of_le zero_lt_one h1N,\n  have h1f : 1 \u2264 \u230aN\u230b\u208a, { refine nat.le_floor _, exact_mod_cast h1N, },\n  by_cases hcases : \u230aN\u230b\u208a < C\u2081,\n  rw [log_mul, mul_add], transitivity C*log 2, transitivity ((f y)/log 2)*log 2,\n  rw div_mul_cancel, refine h \u230aN\u230b\u208a _, rw mem_Ico, refine \u27e8h1f,lt_of_lt_of_le hcases _\u27e9,\n  refine le_max_left _ _,\n  refine ne_of_gt (log_pos one_lt_two),\n  rw mul_le_mul_right (log_pos one_lt_two), refine le_max_right _ _,\n  refine le_add_of_nonneg_right _, refine mul_nonneg (le_of_lt h0C) (log_nonneg h1N),\n  exact two_ne_zero, refine ne_of_gt h0N,\n  rw not_lt at hcases, specialize hharmonic \u230aN\u230b\u208a hcases,\n  transitivity (\u2211 (n : \u2115) in range (\u230aN\u230b\u208a + 1), (1:\u211d)/n),\n  refine sum_le_sum_of_subset_of_nonneg _ _, intros n hn, rw mem_Icc at hn,\n  rw [mem_range, nat.lt_succ_iff], exact hn.2, intros m hm1 hm2, rw one_div_nonneg, exact nat.cast_nonneg m,\n  refine le_trans hharmonic _, transitivity C*log N,\n  refine mul_le_mul _ _ _ _, refine le_max_left _ _, rw log_le_log, refine nat.floor_le (le_of_lt h0N),\n  norm_cast, exact lt_of_lt_of_le zero_lt_one h1f, exact h0N, refine log_nonneg _, exact_mod_cast h1f,\n  exact le_of_lt h0C, rw mul_le_mul_left h0C, rw log_le_log h0N,\n  refine le_mul_of_one_le_left (le_of_lt h0N) one_le_two, refine mul_pos zero_lt_two h0N,\n  rw finset.nonempty_Ico, refine lt_of_lt_of_le one_lt_two (le_max_right _ _),\nend\n\nlemma another_this_particular_tends_to :\n  tendsto (\u03bb x : \u211d, (log x)/log(log x)) at_top at_top :=\nbegin\n  have : filter.tendsto (\u03bb x : \u211d,  x / log x) at_top at_top,\n  { simpa using tendsto_mul_add_div_pow_log_at_top _ 0 1 zero_lt_one },\n  exact this.comp tendsto_log_at_top,\nend\n\nlemma this_function_big_tends_to :\n  tendsto (\u03bb x : \u211d, x^(log(log(log x))/log(log x))) at_top at_top :=\nbegin\n  suffices : filter.tendsto (\u03bb x : \u211d,  ((log x / log(log x)))*(log(log(log x)))) at_top at_top,\n  { apply ((tendsto_exp_at_top.comp this)).congr' _,\n    filter_upwards [eventually_gt_at_top (0 : \u211d)] with x hx using\n      by simp only [rpow_def_of_pos hx, mul_div_assoc' (log x), div_mul_eq_mul_div], },\n  refine filter.tendsto.at_top_mul_at_top another_this_particular_tends_to _,\n  exact tendsto_log_at_top.comp (tendsto_log_at_top.comp tendsto_log_at_top),\nend\n\nlemma now_last_large_N : \u2200\u1da0 (N : \u2115) in at_top,\n198 / 199 * log (log N) \u2264 log (log (log (log N)) / log (log N) * log N) :=\nbegin\n  filter_upwards [\n    ((another_this_particular_tends_to.comp tendsto_log_at_top).comp tendsto_coe_nat_at_top_at_top).eventually_ge_at_top (199:\u211d),\n    tendsto_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    (tendsto_log_at_top.comp tendsto_log_log_coe_at_top).eventually_gt_at_top (0:\u211d),\n    ((tendsto_log_at_top.comp tendsto_log_at_top).comp tendsto_log_log_coe_at_top).eventually_gt_at_top (0:\u211d)\n   ] with N hlarge h0log h0log2 h0log3 h0log4,\n  rw [log_mul, \u2190 sub_le_iff_le_add, \u2190 neg_le_neg_iff, neg_sub, \u2190 one_sub_mul, \u2190 log_inv, inv_div,\n    log_div], transitivity log(log(log N)), refine sub_le_self _ (le_of_lt h0log4),\n  rw [\u2190 div_le_iff], norm_num1, rw [le_one_div, one_div_div], exact hlarge,\n  exact div_pos h0log3 h0log2, norm_num1, exact h0log2, exact ne_of_gt h0log2,\n  exact ne_of_gt h0log3, exact ne_of_gt (div_pos h0log3 h0log2), exact ne_of_gt h0log,\nend\n\nlemma large_helper (c C : \u211d) (hc1 : c < 1) (h0C : 0 < C): \u2200\u1da0 (N : \u211d) in at_top,\n  (log N)^c < (log (log (log N)) / log (log N) * log N) * C :=\nbegin\n  have hc : 0 < -c + 1, {rw [add_comm, \u2190 sub_eq_add_neg, sub_pos], exact hc1,},\n  filter_upwards [\n    tendsto_log_at_top.eventually_gt_at_top (0:\u211d),\n    (tendsto_log_at_top.comp tendsto_log_at_top).eventually_gt_at_top (0:\u211d),\n    (tendsto_log_at_top.comp tendsto_log_at_top).eventually_gt_at_top (log C\u207b\u00b9 / ((-c + 1) / 2)),\n    (another_this_particular_tends_to.comp tendsto_log_at_top).eventually_gt_at_top (1 / ((-c + 1) / 2) ),\n    ((tendsto_log_at_top.comp tendsto_log_at_top).comp tendsto_log_at_top).eventually_gt_at_top (0:\u211d),\n    ((tendsto_log_at_top.comp tendsto_log_at_top).comp tendsto_log_at_top).eventually_gt_at_top (1:\u211d)\n  ] with N hN hN\u2081 hN\u2082 hN\u2083 hN\u2084 hN\u2085,\n  rw [\u2190 div_lt_iff h0C, div_eq_mul_one_div, \u2190 lt_div_iff', div_eq_mul_one_div _ ((log N)^c),\n  one_div, one_div, \u2190 rpow_neg, mul_assoc, mul_comm (log N), \u2190 rpow_add_one, div_eq_mul_one_div],\n  transitivity (1/ log (log N)) * log N^(-c+1),\n  rw [mul_comm, \u2190 div_eq_mul_one_div, lt_div_iff, \u2190 log_lt_log_iff, log_rpow, log_mul,\n    \u2190 add_halves ((-c+1)), add_mul], refine add_lt_add _ _, rw \u2190 div_lt_iff', exact hN\u2082,\n  refine div_pos hc two_pos,\n  rw [\u2190 one_lt_div, \u2190 mul_div, \u2190 div_lt_iff'], exact hN\u2083, exact div_pos hc two_pos, exact hN\u2084,\n  refine ne_of_gt _, rw inv_pos, exact h0C, exact ne_of_gt hN\u2081, exact hN, refine mul_pos _ hN\u2081,\n  rw inv_pos, exact h0C, refine rpow_pos_of_pos hN _, exact hN\u2081, rw mul_assoc,\n  refine lt_mul_of_one_lt_left _ _, refine mul_pos _ _, rw one_div_pos, exact hN\u2081,\n  refine rpow_pos_of_pos hN _, exact hN\u2085, exact ne_of_gt hN, exact le_of_lt hN,\n  refine rpow_pos_of_pos hN _,\nend\n\n\nlemma the_last_large_N : \u2200 C : \u211d, (0 < C) \u2192 \u2200\u1da0 (N : \u2115) in at_top,\nlog N ^ ((3:\u211d) / 4) \u2264 log N * (log(log(log N))/log(log N)) \u2227\n(\u2308log (log (log N) / log (log (log N))) *(2 * log (log N))\u2309\u208a:\u211d) *\n  (2 * ((log N)^((1:\u211d) / 500)) + C*(1/(log(log N))^2)*log N) < (2+2*C)*(log(log(log N))/log(log N)) * log N :=\nbegin\n  intros C h0C,\n  have htemp' : (3:\u211d)/4 < 1 := by norm_num1,\n  have htemp\u2082 : (251:\u211d)/500 < 1 := by norm_num1,\n  have htemp\u2083 : (1:\u211d)/500 < 1 := by norm_num1,\n  have htemp\u2084 : (0:\u211d) < 1/4 := by norm_num1,\n  filter_upwards [\n    tendsto_log_coe_at_top.eventually_gt_at_top (1:\u211d),\n    tendsto_coe_nat_at_top_at_top.eventually (large_helper ((3:\u211d)/4) (1:\u211d) htemp' zero_lt_one),\n    tendsto_coe_nat_at_top_at_top.eventually (large_helper ((1:\u211d)/500) ((1:\u211d)/4) htemp\u2083 htemp\u2084),\n    tendsto_coe_nat_at_top_at_top.eventually (large_helper ((251:\u211d)/500) ((1:\u211d)/2) htemp\u2082 one_half_pos),\n    ((another_this_particular_tends_to.comp tendsto_log_at_top).comp tendsto_coe_nat_at_top_at_top).eventually_ge_at_top (1:\u211d),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (2*(C*1)),\n    tendsto_log_log_coe_at_top.eventually_ge_at_top (log 2 / (1 / 4 / 2)),\n    (tendsto_log_at_top.comp tendsto_log_log_coe_at_top).eventually_gt_at_top (1:\u211d),\n    (another_this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually_gt_at_top (1:\u211d),\n    ((another_this_particular_tends_to.comp tendsto_log_at_top).comp tendsto_coe_nat_at_top_at_top).eventually_ge_at_top (8:\u211d)\n  ] with N h1logN hlarge1 hlarge2 hlarge3 hweird h0loglogN hloglogN' hloglogN'' h1log3 hbig hbig\u2082,\n  have h0log3 : 0 < log(log(log N)) := lt_trans zero_lt_one h1log3,\n  have h0logN : 0 < log N := lt_trans zero_lt_one h1logN,\n  have hlarge\u2083 : 2*log(log N) \u2264 (log N)^((1:\u211d)/4), {\n    rw [\u2190 log_le_log, log_rpow, \u2190 add_halves ((1:\u211d)/4), log_mul, add_mul], refine add_le_add _ _,\n    rw \u2190 div_le_iff', exact hloglogN'', norm_num1, norm_num1, rw [mul_comm, \u2190 div_eq_mul_one_div,\n      le_div_iff, \u2190 le_div_iff'], exact hbig\u2082, exact h0log3, norm_num1, exact two_ne_zero,\n    exact ne_of_gt h0loglogN, exact h0logN, refine mul_pos zero_lt_two h0loglogN,\n    refine rpow_pos_of_pos h0logN _,\n   },\n  refine \u27e8_,_\u27e9, rw mul_comm, rw mul_one at hlarge1, exact (le_of_lt hlarge1),\n  transitivity (log (log (log N) / log (log (log N))) *(2 * log (log N)) + 1) *\n  (2 * ((log N)^((1:\u211d) / 500)) + C*(1/(log(log N))^2)*log N),\n  rw mul_lt_mul_right, refine nat.ceil_lt_add_one _, refine mul_nonneg _ _, refine log_nonneg _,\n  exact hweird, exact mul_nonneg (zero_le_two) (le_of_lt h0loglogN), refine add_pos _ _,\n  refine mul_pos zero_lt_two (rpow_pos_of_pos h0logN _), refine mul_pos _ h0logN,\n  refine mul_pos h0C _, rw one_div_pos, refine sq_pos_of_pos h0loglogN,\n  rw [add_mul, mul_add, add_rotate, add_rotate, add_mul, add_mul], refine add_lt_add_of_lt_of_le _ _,\n  rw [one_mul], nth_rewrite 2 (mul_assoc (_ : \u211d)), nth_rewrite 3 (two_mul (_ : \u211d)), refine add_lt_add _ _,\n  rw \u2190 add_halves ((log (log (log N)) / log (log N)) * log N), refine add_lt_add _ _,\n  rw [lt_div_iff, mul_comm, \u2190 mul_assoc], norm_num1, rw \u2190 lt_div_iff', rw div_eq_mul_one_div _ (4:\u211d),\n  exact hlarge2, exact zero_lt_four, exact zero_lt_two, rw [lt_div_iff', \u2190 mul_assoc, mul_lt_mul_right,\n    mul_div, mul_div, div_lt_iff, div_eq_mul_one_div, mul_assoc, mul_comm (1/(log(log N))),\n    \u2190 div_eq_mul_one_div, sq, mul_div_cancel], transitivity (log(log N)), exact hloglogN',\n  refine lt_mul_of_one_lt_left _ _, exact h0loglogN, exact h1log3, exact ne_of_gt h0loglogN,\n  refine sq_pos_of_pos h0loglogN,  exact h0logN, exact zero_lt_two,\n  transitivity log N ^ ((1:\u211d) / 4) * log N ^ ((1:\u211d) / 4) * (2 * log N ^ ((1:\u211d) / 500)),\n  rw mul_lt_mul_right, refine mul_lt_mul _ _ _ _, transitivity log(log(log N)), refine log_lt_log _ _,\n  refine div_pos h0loglogN h0log3, refine div_lt_self h0loglogN h1log3, refine lt_of_lt_of_le _ hlarge\u2083,\n  transitivity log(log N), refine log_lt_log h0loglogN _, rw \u2190 one_lt_div, exact hbig, exact h0loglogN,\n  refine lt_mul_of_one_lt_left h0loglogN one_lt_two, exact hlarge\u2083, refine mul_pos zero_lt_two h0loglogN,\n  refine rpow_nonneg_of_nonneg (le_of_lt h0logN) _, refine mul_pos zero_lt_two _,\n  refine rpow_pos_of_pos h0logN _, rw [\u2190 rpow_add, \u2190 mul_assoc, mul_comm, \u2190 mul_assoc, \u2190 rpow_add,\n    \u2190 lt_div_iff, div_eq_mul_one_div _ (2:\u211d)], norm_num1, exact hlarge3, exact zero_lt_two,\n  exact h0logN, exact h0logN,\n  transitivity log (log (log N)) * (2 * log (log N)) * (C * (1 / log (log N) ^ 2) * log N),\n  rw [mul_le_mul_right, mul_le_mul_right, log_le_log], refine div_le_self _ _, exact le_of_lt h0loglogN,\n  exact (le_of_lt h1log3), exact div_pos h0loglogN h0log3, exact h0loglogN, refine mul_pos zero_lt_two h0loglogN,\n  refine mul_pos _ h0logN, refine mul_pos h0C _, rw one_div_pos, refine sq_pos_of_pos h0loglogN,\n  convert_to log (log (log \u2191N)) * 2 * (C * (log (log \u2191N) / log (log \u2191N) ^ 2) * log \u2191N) \u2264\n    2 * C * (log (log (log \u2191N)) / log (log \u2191N)) * log \u2191N using 0, { ring_nf, },\n  rw [sq, div_mul_left], ring_nf, exact ne_of_gt h0loglogN,\nend\n\nlemma how_large_can_we_go (C : \u211d) (h0C : 0 < C) : \u2200\u1da0 (N : \u211d) in at_top,\n  (log N)^((1:\u211d)/1000) \u2264 (log (log (log \u2191N)) / log (log \u2191N) * log \u2191N) * C :=\nbegin\n  have : (1:\u211d)/1000 < 1 := by norm_num1,\n  filter_upwards [large_helper ((1:\u211d)/1000) C this h0C] with N hN,\n  exact le_of_lt hN,\nend\n\nlemma crude_ps (p : \u2115 \u2192 Prop) [decidable_pred p] (\u03b4 : \u211d) (Y : \u211d) (h0\u03b4 : 0 < \u03b4) (h1Y : 1 \u2264 Y) (N : \u2115)\n(h : \u2200 X : \u211d, (Y \u2264 X \u2227 X \u2264 N) \u2192 ((filter p (Ico \u2308X\u2309\u208a \u23082*X\u2309\u208a)).card : \u211d) \u2264 \u03b4*X) (h2N : 2 \u2264 N) :\n \u2211 n in (filter p (Icc \u2308Y\u2309\u208a N)), (1:\u211d)/n \u2264 ((2/log 2))*\u03b4*log N :=\nbegin\n  have h0Y : 0 < Y := lt_of_lt_of_le zero_lt_one h1Y,\n  have h0N : 0 < N := lt_of_lt_of_le zero_lt_two h2N,\n  by_cases hYN : Y \u2264 N,\n  have htemp : 1/log 2 + 1/log 2 = 2/log 2, { rw [\u2190 two_mul, mul_div, mul_one], },\n  rw \u2190 htemp,\n  let f := (\u03bb i:\u2115, filter p (Ico \u23082^(i:\u211d)*Y\u2309\u208a \u23082*(2^((i:\u211d))*Y)\u2309\u208a)),\n  let I := range (\u230alogb 2 (N/Y)\u230b\u208a + 1),\n  have hcont : (filter p (Icc \u2308Y\u2309\u208a N)) \u2286 I.bUnion(\u03bb i, f i), {\n    intros n hn, rw mem_bUnion, rw [mem_filter, mem_Icc] at hn,\n    have haux : (0:\u211d) < n/Y, { refine div_pos _ h0Y, norm_cast,\n    refine lt_of_lt_of_le _ hn.1.1, rw pos_iff_ne_zero, intro hbad,\n    rw [nat.ceil_eq_zero, \u2190 not_lt] at hbad, exact hbad h0Y, },\n    have haux' : 0 \u2264 logb 2 (n/Y), { refine logb_nonneg one_lt_two _, rw one_le_div,\n    rw \u2190 nat.ceil_le, exact hn.1.1, exact h0Y, },\n    let i := \u230alogb 2 (n/Y)\u230b\u208a,\n    use i, rw [mem_filter], refine \u27e8_,_,hn.2\u27e9,\n    rw mem_range, rw nat.lt_succ_iff, refine nat.le_floor _,\n    transitivity (logb 2 (n/Y)), refine nat.floor_le _, exact haux',\n    rw logb_le_logb one_lt_two, rw div_le_iff h0Y, transitivity (N:\u211d)*1, rw mul_one,\n    exact_mod_cast hn.1.2, rw [mul_one, div_mul_cancel], exact ne_of_gt h0Y,\n    exact haux, refine div_pos _ h0Y, exact_mod_cast h0N,\n    rw mem_Ico, refine \u27e8_,_\u27e9, rw nat.ceil_le,\n    rw [\u2190 le_div_iff h0Y, \u2190 real.logb_le_logb one_lt_two, logb_rpow zero_lt_two],\n    refine nat.floor_le _, exact haux', exact ne_of_gt one_lt_two,\n    refine rpow_pos_of_pos zero_lt_two _, exact haux,\n    rw [nat.lt_ceil, \u2190 mul_assoc, mul_comm (2:\u211d), \u2190 rpow_add_one, \u2190 div_lt_iff h0Y, \u2190 logb_lt_logb_iff one_lt_two, logb_rpow zero_lt_two,\n      \u2190 sub_lt_iff_lt_add], refine nat.sub_one_lt_floor _, exact ne_of_gt one_lt_two,\n    exact haux, refine rpow_pos_of_pos zero_lt_two _, exact ne_of_gt zero_lt_two,\n  },\n  refine le_trans (sum_le_sum_of_subset_of_nonneg hcont _) _,\n  intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n,\n  refine le_trans (sum_bUnion_le_sum_of_nonneg _) _,\n  intros n hn, rw one_div_nonneg, exact nat.cast_nonneg n,\n  have hbound : \u2200 i : \u2115, i \u2208 I \u2192 \u2211 n in f i, (1:\u211d)/n \u2264 \u03b4, {\n    intros x hx, refine le_trans (sum_le_card_nsmul _ _ (1/(2^(x:\u211d)*Y)) _) _,\n    intros n hn, rw [mem_filter, mem_Ico] at hn, rw one_div_le_one_div,\n    rw \u2190 @nat.cast_le \u211d _ _ _ _ at hn, refine le_trans _ hn.1.1,\n    refine le_trans _ (nat.le_ceil _), refl,\n     norm_cast,\n    refine lt_of_lt_of_le _ hn.1.1, rw pos_iff_ne_zero, intro hbad,\n    rw [nat.ceil_eq_zero, \u2190 not_lt] at hbad, refine hbad _, refine mul_pos _ h0Y,\n    refine rpow_pos_of_pos zero_lt_two _, refine mul_pos _ h0Y, refine rpow_pos_of_pos zero_lt_two _,\n    rw [nsmul_eq_mul, \u2190 div_eq_mul_one_div, div_le_iff],\n    refine le_trans (h (2^(x:\u211d)*Y) _) _, rw le_mul_iff_one_le_left, split,\n    refine one_le_rpow one_le_two _, exact nat.cast_nonneg x,\n    rw [mem_range, nat.lt_succ_iff] at hx, rw \u2190 le_div_iff h0Y, rw \u2190 logb_le_logb one_lt_two,\n    rw nat.le_floor_iff at hx, rw logb_rpow, exact hx, exact zero_lt_two, exact ne_of_gt one_lt_two,\n    refine logb_nonneg one_lt_two _, rw le_div_iff h0Y, rw one_mul, norm_cast, exact hYN,\n    refine rpow_pos_of_pos zero_lt_two _, refine div_pos _ h0Y, exact_mod_cast h0N,\n    exact h0Y, refl,\n    refine mul_pos _ h0Y, refine rpow_pos_of_pos zero_lt_two _,\n   },\n  refine le_trans (sum_le_card_nsmul _ _ \u03b4 _) _, exact hbound,\n  rw [nsmul_eq_mul, mul_assoc, mul_comm \u03b4, \u2190 mul_assoc, mul_le_mul_right h0\u03b4, card_range],\n  push_cast, rw [\u2190 le_sub_iff_add_le], refine le_trans (nat.floor_le _) _,\n  refine logb_nonneg one_lt_two _, rw [le_div_iff h0Y, one_mul], exact hYN,\n  transitivity (logb 2 N), rw logb_le_logb, refine div_le_self _ _, exact nat.cast_nonneg N,\n  exact h1Y, exact one_lt_two, refine div_pos _ _, norm_cast, exact lt_of_lt_of_le zero_lt_two h2N,\n  exact h0Y, exact_mod_cast h0N,\n  rw [le_sub_iff_add_le, \u2190 real.log_div_log, div_eq_mul_one_div, mul_comm, add_mul,\n    add_le_add_iff_left, mul_comm, \u2190 div_eq_mul_one_div, le_div_iff, one_mul, log_le_log],\n  exact_mod_cast h2N, exact zero_lt_two, exact_mod_cast h0N, refine log_pos one_lt_two,\n  have hempty : Icc \u2308Y\u2309\u208a N = \u2205, { refine Icc_eq_empty _, norm_cast, rw nat.ceil_le, exact hYN, },\n  rw hempty, simp only [filter_true_of_mem, not_mem_empty, forall_false_left, forall_const, sum_empty],\n  refine mul_nonneg _ _, refine mul_nonneg _ _, refine div_nonneg zero_le_two (log_nonneg one_le_two),\n  exact le_of_lt h0\u03b4, refine log_nonneg _, norm_cast, exact le_trans one_le_two h2N,\nend\n\nlemma harmonic_filter_reg : \u2203 C : \u211d, (0 < C) \u2227\n \u2200\u1da0 (N : \u2115) in at_top, \u2211 n in (Icc (\u2308(N:\u211d)^(log(log(log N))/log(log N))\u2309\u208a) N).filter(\u03bb n, n \u2260 0 \u2227\n   \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 (3/2) * log (log N))), (1:\u211d)/n\n    \u2264 C*log N/(log(log N)) :=\nbegin\n  rcases turan_primes_estimate with \u27e8C\u2081,hturan\u27e9, rw eventually_at_top at hturan,\n  rcases hturan with \u27e8C\u2082,hturan\u27e9,\n  let C\u2083 := max C\u2081 1,\n  have h0C\u2083 : 0 < C\u2083, { refine lt_of_lt_of_le zero_lt_one _, refine le_max_right _ _, },\n  let c\u2081 := C\u2083 * (4 / (1 / 200) ^ 2),\n  have h0c\u2081 : 0 < c\u2081, { refine mul_pos h0C\u2083 _,  refine div_pos zero_lt_four _, refine sq_pos_of_pos _,\n    norm_num1, },\n  let C := (c\u2081/(198/199))*(2/log 2),\n  have h0C : 0 < C, { refine mul_pos (div_pos h0c\u2081 _) _, norm_num1,\n    refine div_pos zero_lt_two (log_pos one_lt_two), },\n  refine \u27e8C,h0C,_\u27e9,\n  filter_upwards [eventually_ge_at_top 2,\n    (this_function_big_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top ((C\u2082:\u211d)/2)),\n     (this_function_big_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_gt_at_top (1:\u211d)),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    tendsto_log_coe_at_top.eventually_ge_at_top (log 4),\n    tendsto_log_coe_at_top.eventually_ge_at_top ((2:\u211d)^(100:\u211d)), now_last_large_N]\n  with N h2N hYlarge h1Y h0loglogN h4logN hbiglogN hweird,\n  let p := (\u03bb n, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 (3/2) * log (log N))),\n  let Y := (N:\u211d)^(log(log(log N))/log(log N)),\n  let \u03b4 := c\u2081/((198/199)*log(log N)),\n  have h0N : (0:\u211d) < N, { norm_cast, exact lt_of_lt_of_le zero_lt_two h2N, },\n  have h0logN : 0 < log N, { refine lt_of_lt_of_le _ h4logN, refine log_pos one_lt_four, },\n  have h0\u03b4 : 0 < \u03b4, { refine div_pos h0c\u2081 _, refine mul_pos _ h0loglogN, norm_num1, },\n  refine le_trans (crude_ps p \u03b4 Y h0\u03b4 (le_of_lt h1Y) N _ h2N) _,\n  {\n    intros X hX,\n    have h1X : 1 \u2264 X := le_trans (le_of_lt h1Y) hX.1,\n    let M := \u23082 * X\u2309\u208a,\n    have h0M : (0:\u211d) < M, { norm_cast, rw pos_iff_ne_zero, intro hbad,\n      rw [nat.ceil_eq_zero, \u2190 not_lt] at hbad, refine hbad (mul_pos _ _),\n      exact zero_lt_two, exact lt_of_lt_of_le zero_lt_one h1X, },\n    have hM' : 198 / 199 * log (log N) \u2264 log (log M), {\n      transitivity log(log Y), rw log_rpow, exact hweird, exact h0N,\n      rw [log_le_log, log_le_log], refine le_trans hX.1 _, refine le_trans _ (nat.le_ceil _),\n      refine le_mul_of_one_le_left (le_trans zero_le_one h1X) one_le_two, refine rpow_pos_of_pos h0N _,\n      exact h0M, exact log_pos h1Y, refine log_pos _, refine lt_of_le_of_lt h1X _,\n      refine lt_of_lt_of_le _ (nat.le_ceil _), refine lt_mul_of_one_lt_left _ one_lt_two,\n      exact lt_of_lt_of_le zero_lt_one h1X,\n     },\n    have h0loglogM : 0 < log(log M), { refine lt_of_lt_of_le _ hM', refine mul_pos _ h0loglogN,\n      norm_num1, },\n    have hMX : (M:\u211d) \u2264 4*X, { refine le_trans (le_of_lt (nat.ceil_lt_add_one _)) _,\n      refine mul_nonneg zero_le_two _, exact le_trans zero_le_one h1X,\n      rw [\u2190 le_sub_iff_add_le', \u2190 sub_mul, \u2190 div_le_iff'], refine le_trans _ h1X, norm_num1, norm_num1,},\n    have hM'' : log (log M) \u2264 (101/100)*log (log N), {\n      have haux1 : 0 < log(4*X), { refine log_pos _, refine lt_of_lt_of_le one_lt_four _,\n      refine le_mul_of_one_le_right (le_of_lt zero_lt_four) h1X, },\n      have haux2 : 0 < log(4*N), { refine lt_of_lt_of_le haux1 _, rw log_le_log,\n        rw mul_le_mul_left zero_lt_four, exact hX.2, exact real.nontrivial,\n        refine mul_pos zero_lt_four _, exact lt_of_lt_of_le zero_lt_one h1X,\n        refine mul_pos zero_lt_four h0N, },\n      transitivity log(log(4*X)),\n      rw [log_le_log, log_le_log], exact hMX, exact h0M, refine mul_pos zero_lt_four _,\n      exact lt_of_lt_of_le zero_lt_one h1X, refine log_pos _, refine lt_of_lt_of_le _ (nat.le_ceil _),\n      refine lt_of_lt_of_le one_lt_two _, refine le_mul_of_one_le_right zero_le_two h1X, exact haux1,\n      transitivity log(log(4*N)), rw [log_le_log, log_le_log, mul_le_mul_left zero_lt_four],\n      exact hX.2, exact real.nontrivial, refine mul_pos zero_lt_four (lt_of_lt_of_le zero_lt_one h1X),\n      refine mul_pos zero_lt_four h0N, exact haux1, exact haux2,\n      rw [\u2190 log_rpow, log_le_log], transitivity (2:\u211d)*log N, rw [log_mul, two_mul, add_le_add_iff_right],\n      exact h4logN, exact ne_of_gt zero_lt_four, exact ne_of_gt h0N, rw [\u2190 le_div_iff, \u2190 rpow_sub_one],\n      norm_num1, have h100 : (0:\u211d) < 100 := by norm_num1,\n      rw [\u2190 rpow_le_rpow_iff _ _ h100, \u2190 rpow_mul, mul_comm, \u2190 div_eq_mul_one_div, div_self, rpow_one],\n      exact hbiglogN, norm_num1, exact le_of_lt h0logN, exact zero_le_two, refine rpow_nonneg_of_nonneg _ _,\n      exact le_of_lt h0logN, exact ne_of_gt h0logN, exact h0logN, exact haux2,\n      refine rpow_pos_of_pos h0logN _, exact h0logN,\n      },\n    have hlarge : C\u2082 \u2264 \u23082 * X\u2309\u208a, { rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans _ (nat.le_ceil _),\n      rw \u2190 div_le_iff', refine le_trans _ hX.1, exact hYlarge, exact zero_lt_two, },\n    have h\u03b4M : C\u2083 * 4 / (1 / 200) ^ 2 \u2264 \u03b4 * log (log M), {\n       rw [div_mul_eq_mul_div, \u2190 mul_div, mul_div_assoc], refine le_mul_of_one_le_right (le_of_lt h0c\u2081) _,\n       rw one_le_div, exact hM', refine mul_pos _ h0loglogN, norm_num1,  },\n    specialize hturan M hlarge,\n    transitivity (((filter p (Icc 1 M)).card):\u211d),\n    norm_cast, refine finset.card_le_of_subset _, refine filter_subset_filter _ _,\n    intros n hn, rw mem_Icc, rw mem_Ico at hn,\n    refine \u27e8le_trans _ hn.1,le_of_lt hn.2\u27e9, rw \u2190 @nat.cast_le \u211d _ _ _ _,\n    refine le_trans _ (nat.le_ceil _), norm_cast, exact h1X,\n    by_contra, rw not_le at h, rw \u2190 not_lt at hturan, refine hturan _, clear hturan hYlarge h1Y Y,\n    calc _ \u2264 C\u2083*M*log(log M) :_\n       ... \u2264 (\u03b4*X)*((1/200)*log(log M))^2 :_\n       ... < ((filter (\u03bb n, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 (3/2) * log (log N))) (Icc 1 M)).card : \u211d)*((1/200)*log(log M))^2 :_\n       ... \u2264 (\u2211 n in (filter (\u03bb n, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227 (\u03c9 n : \u211d) \u2264 (3/2) * log (log N))) (Icc 1 M)), ((\u03c9 n : \u211d) - log(log M))^2) :_\n       ... \u2264 _ :_,\n    rw mul_le_mul_right h0loglogM, rw mul_le_mul_right h0M, refine le_max_left _ _,\n    rw [mul_pow, \u2190 mul_assoc, sq (log(log M)), \u2190 mul_assoc, mul_le_mul_right h0loglogM],\n    transitivity (C\u2083*4*X), rw [mul_assoc, mul_le_mul_left h0C\u2083], exact hMX,\n    rw [mul_comm _ X, mul_comm _ X, mul_assoc X, mul_assoc X, mul_le_mul_left, mul_comm \u03b4, mul_assoc,\n      \u2190 div_le_iff'], exact h\u03b4M, refine sq_pos_of_pos _, norm_num1, exact lt_of_lt_of_le zero_lt_one h1X,\n    rw mul_lt_mul_right, exact h, refine sq_pos_of_pos _, refine mul_pos _ h0loglogM, norm_num1,\n    rw [\u2190 nsmul_eq_mul], refine finset.card_nsmul_le_sum _ _ _ _,\n    intros n hn, rw [mem_filter, not_and_distrib] at hn,\n    rw [sq_le_sq, le_abs, abs_of_pos], cases hn.2.2 with hn1 hn2,\n    right, rw [neg_sub, le_sub, \u2190 one_sub_mul], rw not_le at hn1, norm_num1,\n    refine le_trans (le_of_lt hn1) _, rw [\u2190 div_le_iff', \u2190 div_mul_eq_mul_div], norm_num1, exact hM',\n    norm_num1, left, rw [le_sub_iff_add_le, add_comm, \u2190 one_add_mul],\n    rw not_le at hn2, refine le_trans _ (le_of_lt hn2),\n    transitivity ((1:\u211d)+1/200)*((101/100)*log(log N)), rw mul_le_mul_left, exact hM'', norm_num1,\n    rw \u2190 mul_assoc, rw mul_le_mul_right, norm_num1, exact h0loglogN,\n    refine mul_pos _ h0loglogM, norm_num1,\n    refine sum_le_sum_of_subset_of_nonneg _ _, refine filter_subset _ _,\n    intros n hn1 hn2, refine sq_nonneg _,\n  },\n  nth_rewrite 1 div_eq_mul_one_div, rw [mul_assoc C, mul_comm (log N), \u2190 mul_assoc, mul_le_mul_right h0logN,\n    \u2190 div_eq_mul_one_div, \u2190 le_div_iff', div_div, mul_comm (log(log N)), \u2190 div_div, mul_div_cancel, div_div],\n  refine ne_of_gt (div_pos zero_lt_two _), exact log_pos one_lt_two,\n  refine div_pos zero_lt_two _, exact log_pos one_lt_two,\nend\n\nlemma harmonic_filter_div : \u2203 C : \u211d, (0 < C) \u2227\n \u2200\u1da0 (N : \u2115) in at_top, \u2211 n in (Icc (\u2308(N:\u211d)^(log(log(log N))/log(log N))\u2309\u208a) N).filter(\u03bb n,\n    \u00ac \u2203 d : \u2115, d \u2223 n \u2227 (4 \u2264 d) \u2227 ((d : \u211d) \u2264 (log N)^((1:\u211d)/1000))), (1:\u211d)/n\n    \u2264 C*log N/(log(log N)) :=\nbegin\n  rcases sieve_lemma_prec' with \u27e8c\u2081,C\u2081,h0c\u2081,h0C\u2081,hsieve\u27e9,\n  rw eventually_at_top at hsieve, rcases hsieve with \u27e8C\u2082,hsieve\u27e9,\n  let c\u2082 := c\u2081 * (4 * (log 4 / (1 / 1000))),\n  have h0c\u2082 : 0 < c\u2082, { refine mul_pos h0c\u2081 _, refine mul_pos zero_lt_four (div_pos _ _),\n    exact log_pos one_lt_four, norm_num1, },\n  let C := c\u2082*(2/log 2),\n  have h0C : 0 < C, { refine mul_pos h0c\u2082 _, refine div_pos zero_lt_two _, exact log_pos one_lt_two, },\n  refine \u27e8C,h0C,_\u27e9,\n  filter_upwards [eventually_ge_at_top 2,\n    tendsto_coe_nat_at_top_at_top.eventually (how_large_can_we_go C\u2081 h0C\u2081),\n    tendsto_log_coe_at_top.eventually_gt_at_top (1:\u211d),\n    (this_function_big_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top ((C\u2082:\u211d)/2)),\n     (this_function_big_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (1:\u211d))] with N h2N hNlarge h1logN hYlarge h1Y,\n  have h0N: (0:\u211d) < N, { norm_cast, exact lt_of_lt_of_le zero_lt_two h2N, },\n  have h0loglogN : 0 < log(log N) := log_pos h1logN,\n  have h0logN : 0 < log N := lt_trans zero_lt_one h1logN,\n  let Y := (N:\u211d)^(log(log(log N))/log(log N)),\n  let \u03b4 := c\u2082/log(log N),\n  have h0\u03b4 : 0 < \u03b4, { refine div_pos h0c\u2082 h0loglogN, },\n  let p := (\u03bb n, \u00ac \u2203 d : \u2115, d \u2223 n \u2227 (4 \u2264 d) \u2227 ((d : \u211d) \u2264 (log N)^((1:\u211d)/1000))),\n  refine le_trans (crude_ps p \u03b4 Y h0\u03b4 h1Y N _ h2N) _,\n  { intros X hX,\n    have h1X : 1 \u2264 X := le_trans h1Y hX.1,\n    let M := \u23082 * X\u2309\u208a,\n    have hlarge : C\u2082 \u2264 \u23082 * X\u2309\u208a, { rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans _ (nat.le_ceil _),\n      rw \u2190 div_le_iff', refine le_trans _ hX.1, exact hYlarge, exact zero_lt_two, },\n    let y := ((4:\u2115):\u211d),\n    let z := (log N)^((1:\u211d)/1000),\n    have hy' : y = 4, { norm_cast, },\n    have h2y : (2:\u211d) \u2264 y, { rw hy', norm_num1, },\n    have hlogy : log y = log 4, { rw hy', },\n    have h1z : 1 < z, { refine one_lt_rpow h1logN _, norm_num1, },\n    have hzM : z \u2264 C\u2081*log M, { rw [\u2190 div_le_iff' h0C\u2081, \u2190 exp_le_exp, exp_log],\n      refine le_trans _ (nat.le_ceil _), rw \u2190 div_le_iff', refine le_trans _ hX.1,\n      rw [\u2190 log_le_log, log_div, log_exp, sub_le_iff_le_add, log_rpow, div_le_iff h0C\u2081],\n      refine le_trans hNlarge _, rw [mul_le_mul_right h0C\u2081], refine le_add_of_nonneg_right _,\n      refine log_nonneg one_le_two,\n      exact h0N, refine exp_ne_zero _, exact two_ne_zero, refine div_pos (exp_pos _) zero_lt_two,\n      refine rpow_pos_of_pos h0N _, exact zero_lt_two, norm_cast, rw pos_iff_ne_zero, intro hbad,\n      rw [nat.ceil_eq_zero, \u2190 not_lt] at hbad, refine hbad (mul_pos zero_lt_two _),\n      exact lt_of_lt_of_le zero_lt_one h1X,\n       },\n    have hMX : (M:\u211d) \u2264 4*X, { refine le_trans (le_of_lt (nat.ceil_lt_add_one _)) _,\n      refine mul_nonneg zero_le_two _, exact le_trans zero_le_one h1X,\n      rw [\u2190 le_sub_iff_add_le', \u2190 sub_mul, \u2190 div_le_iff'], refine le_trans _ h1X, norm_num1, norm_num1,},\n    specialize hsieve M hlarge y z h2y h1z hzM,\n    transitivity (((filter (\u03bb (n : \u2115), \u2200 (p : \u2115), prime p \u2192 p \u2223 n \u2192 \u2191p < y \u2228 z < \u2191p) (range M)).card):\u211d),\n    norm_cast, refine finset.card_le_of_subset _,\n    intros n hn, rw [mem_filter, mem_range], rw [mem_filter, mem_Ico] at hn, refine \u27e8hn.1.2,_\u27e9,\n    intros p hp\u2081 hp\u2082, rw not_exists at hn,\n    have htemp := hn.2 p, rw [not_and, not_and_distrib, not_le, not_le] at htemp, exact htemp hp\u2082,\n    refine le_trans hsieve _,\n    calc _ \u2264 c\u2081 * (log y/log z) * (4 * X) :_\n       ... \u2264 _ :_,\n    rw mul_le_mul_left, exact hMX, refine mul_pos h0c\u2081 (div_pos (log_pos _) (log_pos h1z)), norm_cast,\n    norm_num1, rw [\u2190 mul_assoc, mul_le_mul_right, log_rpow, mul_assoc, mul_comm _ (4:\u211d),\n      div_mul_eq_div_div, mul_div, mul_div, hlogy], exact h0logN, exact lt_of_lt_of_le zero_lt_one h1X,\n  },\n  nth_rewrite 1 div_eq_mul_one_div, rw [mul_assoc C, mul_comm (log N), \u2190 mul_assoc, mul_le_mul_right h0logN,\n   \u2190 div_eq_mul_one_div, \u2190 le_div_iff', div_div, mul_comm (log(log N)), \u2190 div_div, mul_div_cancel],\n  refine ne_of_gt (div_pos zero_lt_two _), exact log_pos one_lt_two,\n  refine div_pos zero_lt_two _, exact log_pos one_lt_two,\nend\n\nlemma harmonic_filter_smooth : \u2203 C : \u211d, (0 < C) \u2227\n \u2200\u1da0 (N : \u2115) in at_top, \u2211 m in (Icc (\u2308(N:\u211d)^(1-1/log(log N))\u2309\u208a) N).filter(\u03bb n:\u2115, \u2203 q : \u2115,\n   is_prime_pow q \u2227 ((N:\u211d)^(1-8/log(log N)) < q \u2227 q \u2223 n)), (1:\u211d)/m\n    \u2264 C*log N/(log(log N))^2 :=\nbegin\n  have hlogpow := is_o_log_rpow_at_top one_half_pos,\n  rw asymptotics.is_o_iff_forall_is_O_with at hlogpow,\n  specialize hlogpow zero_lt_one, rw asymptotics.is_O_with_iff at hlogpow,\n  have hmertens :=  prime_power_reciprocal,\n  rcases hmertens with \u27e8b,hmertens\u27e9,\n  have hmertens' := asymptotics.is_O.exists_pos hmertens,\n  clear hmertens,\n  rcases hmertens' with \u27e8c,h0c,hmertens\u27e9, rw [asymptotics.is_O_with_iff, eventually_at_top] at hmertens,\n  rcases hmertens with \u27e8C\u2081,hmertens\u27e9,\n  rcases harmonic_sum_bound' with \u27e8C\u2082,h0C\u2082,hharmonic\u27e9,\n  let C := max ((2 * c + 2 * 8) * (8*C\u2082+1)) 1,\n  have h0C : 0 < C, { refine lt_of_lt_of_le zero_lt_one (le_max_right _ _), },\n  have hC : 2 * c \u2264 C / (8*C\u2082+1) - 2 * 8, { rw [le_sub_iff_add_le, le_div_iff], refine le_max_left _ _,\n    refine add_pos _ zero_lt_one, refine mul_pos _ h0C\u2082, norm_num1, },\n  refine \u27e8C,h0C,_\u27e9,\n  filter_upwards [\n    (another_this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (C\u2082*log 2)),\n    tendsto_coe_nat_at_top_at_top.eventually hlogpow,\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_gt_at_top (1:\u211d)),\n    tendsto_coe_nat_at_top_at_top.eventually (eventually_ge_at_top (C\u2081:\u211d)),\n    (tendsto_pow_rec_loglog_spec_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_gt_at_top (1:\u211d)),\n    (tendsto_pow_rec_loglog_spec_at_top.comp tendsto_coe_nat_at_top_at_top).eventually (eventually_ge_at_top (C\u2081:\u211d)),\n    tendsto_log_log_coe_at_top.eventually_ge_at_top (16:\u211d)]\n  with N hweird hNlogpow hN\u2081 hNlarge\u2081 hM\u2081 hMlarge h16loglogN,\n  have h1N : (1:\u211d) \u2264 N := le_of_lt hN\u2081,\n  have h0N : (0:\u211d) < N := lt_of_lt_of_le zero_lt_one h1N,\n  have h0logN : 0 < log N := log_pos hN\u2081,\n  have h8loglogN : 8 < log(log N), { refine lt_of_lt_of_le _ h16loglogN, norm_num1, },\n  have h0loglogN : 0 < log(log N), { refine lt_of_lt_of_le _ h16loglogN, norm_num1, },\n  have hloglogN' : 0 < 1 - 8/log(log N), { rw [sub_pos, div_lt_one h0loglogN], exact h8loglogN,},\n  let M := (N:\u211d)^(1-8/log(log N)),\n  have h1M : 1 \u2264 \u2308(N:\u211d)^(1-1/log(log N))\u2309\u208a, { refine le_of_lt _, rw nat.lt_ceil, norm_cast,\n   refine one_lt_rpow _ _, exact hN\u2081, rw [sub_pos, one_div_lt, one_div_one],\n   refine lt_of_lt_of_le _ h16loglogN, norm_num1, exact h0loglogN, exact zero_lt_one, },\n  have h0logM : 0 < log M := log_pos hM\u2081,\n  have h0M : 0 < M := by  refine rpow_pos_of_pos h0N _,\n  have hMN : M \u2264 N, { rw \u2190 rpow_one (N:\u211d), refine rpow_le_rpow_of_exponent_le _ _, exact h1N,\n    rw sub_le_self_iff, refine div_nonneg _ (le_of_lt h0loglogN), norm_num1, },\n  have hcomp : log(log N) \u2264 log M, { rw log_rpow,\n    transitivity (1/2)*log N, rw [\u2190 log_rpow, log_le_log],\n    rw [norm_eq_abs, norm_eq_abs, abs_of_pos, abs_of_pos, one_mul] at hNlogpow, exact hNlogpow,\n    refine rpow_pos_of_pos h0N _, exact h0logN,\n    exact h0logN, refine rpow_pos_of_pos h0N _, exact h0N, rw mul_le_mul_right h0logN,\n    rw [le_sub, sub_half, div_le_iff, \u2190 div_le_iff'], norm_num1, exact h16loglogN, exact one_half_pos,\n    exact h0loglogN, exact h0N, },\n  let Q := (Icc 0 N).filter(\u03bb q:\u2115, is_prime_pow q \u2227 ((N:\u211d)^(1-8/log(log N)) < q )),\n  let Nq := (\u03bb q, (Icc 1 N).filter(\u03bb n, q \u2223 n )),\n  transitivity \u2211 n in Q.bUnion(\u03bb q, Nq q), (1:\u211d)/n,\n  refine sum_le_sum_of_subset_of_nonneg _ _,\n  intros n hn, rw [mem_filter, mem_Icc] at hn, rw mem_bUnion,\n  rcases hn.2 with \u27e8q,hq\u27e9, use q, rw [mem_filter, mem_Icc],\n  refine \u27e8\u27e8\u27e8nat.zero_le q,le_trans (nat.le_of_dvd _ hq.2.2) hn.1.2\u27e9,hq.1,hq.2.1\u27e9,_\u27e9,\n  exact lt_of_lt_of_le (lt_of_lt_of_le zero_lt_one h1M) hn.1.1, rw [mem_filter, mem_Icc],\n  refine \u27e8\u27e8le_trans h1M hn.1.1,hn.1.2\u27e9,hq.2.2\u27e9,\n  intros m hm\u2081 hm\u2082, rw one_div_nonneg, exact nat.cast_nonneg m,\n  refine le_trans (sum_bUnion_le_sum_of_nonneg _) _, intros n hn,\n  rw one_div_nonneg, exact nat.cast_nonneg n,\n  have hNqrec : \u2200 q : \u2115, (0 < q) \u2192 (q \u2264 N) \u2192 \u2211 n in Nq q, (1:\u211d)/n \u2264 C\u2082*log(2*(N/q))*(1/q), {\n    intros q h0q hqN,\n    let g := (\u03bb m:\u2115, m/q),\n    transitivity (\u2211 m in (Icc 1 \u230a(N:\u211d)/q\u230b\u208a), (1:\u211d)/(m*q)),\n    refine sum_le_sum_of_inj g _ _ _ _, intros m hm, rw one_div_nonneg, refine mul_nonneg _ _,\n    exact nat.cast_nonneg m, exact nat.cast_nonneg q, intros n hn, rw mem_Icc,\n    rw [mem_filter, mem_Icc] at hn, rw nat.succ_le_iff, refine \u27e8nat.div_pos (nat.le_of_dvd _ hn.2) h0q,_\u27e9,\n    exact lt_of_lt_of_le zero_lt_one hn.1.1, refine nat.le_floor _, rw nat.cast_div,\n    rw div_le_div_right,\n    exact_mod_cast hn.1.2, exact_mod_cast h0q,\n    exact hn.2, norm_cast, exact ne_of_gt h0q,\n    intros a\u2081 ha\u2081 a\u2082 ha\u2082 ha\u2081a\u2082, rw mem_filter at ha\u2081, rw mem_filter at ha\u2082, rw nat.div_left_inj at ha\u2081a\u2082,\n    exact ha\u2081a\u2082, exact ha\u2081.2, exact ha\u2082.2, intros n hn, rw mem_filter at hn,\n    rw [nat.cast_div hn.2, div_mul_cancel], norm_cast, exact ne_of_gt h0q,\n    norm_cast, exact ne_of_gt h0q,\n    transitivity ((1:\u211d)/q * \u2211 m in (Icc 1 \u230a(N:\u211d)/q\u230b\u208a), (1:\u211d)/(m)),\n    rw mul_sum, refine le_of_eq _, refine sum_congr _ _, refl, intros n hn,\n    simp only [one_div, mul_inv_rev], rw [mul_comm, mul_le_mul_right], refine hharmonic _ _,\n    rw [le_div_iff, \u2190 le_div_iff', div_one], exact_mod_cast hqN, exact zero_lt_one,\n    exact_mod_cast h0q, rw one_div_pos, exact_mod_cast h0q,\n   },\n  have hNqrec' : \u2200 q \u2208 Q, \u2211 n in Nq q, (1:\u211d)/n \u2264 ((8*C\u2082+1))*(log N/log (log N))*(q\u207b\u00b9), {\n    intros q hq, rw [mem_filter, mem_Icc] at hq, refine le_trans (hNqrec q hq.2.1.pos hq.1.2) _, rw one_div,\n    rw mul_le_mul_right, transitivity (C\u2082*log(2*(N/M))),\n    rw [mul_le_mul_left, log_le_log, mul_le_mul_left, div_le_div_left],\n    exact le_of_lt hq.2.2, exact h0N, norm_cast,\n    exact is_prime_pow.pos hq.2.1, exact h0M, exact zero_lt_two, refine mul_pos zero_lt_two _, refine div_pos h0N _,\n    exact_mod_cast hq.2.1.pos, refine mul_pos zero_lt_two _, exact div_pos h0N h0M, exact h0C\u2082,\n    rw [log_mul, mul_add, add_mul, add_comm], refine add_le_add _ _,\n    rw [div_eq_mul_inv, \u2190 rpow_neg, mul_comm (N:\u211d), \u2190 rpow_add_one, neg_sub, sub_add, sub_self, sub_zero,\n      log_rpow], refine le_of_eq _, ring_nf, exact h0N, exact ne_of_gt h0N, exact nat.cast_nonneg N,\n    rw one_mul, exact hweird, exact ne_of_gt zero_lt_two, refine ne_of_gt _, refine div_pos h0N h0M,\n    rw inv_pos, exact_mod_cast is_prime_pow.pos hq.2.1,\n  },\n  refine le_trans (sum_le_sum hNqrec') _,\n  rw [\u2190 mul_sum, \u2190 le_div_iff', mul_div, div_div, mul_comm (8*C\u2082+1)], nth_rewrite 1 \u2190 mul_div,\n  rw [mul_comm (log N), \u2190 mul_assoc, mul_div_mul_right, mul_div, mul_comm _ (8*C\u2082+1), \u2190 mul_div,\n    sq, mul_self_div_self],\n  have hQaux : Q \u2286(filter is_prime_pow (Icc 1 \u230a(N:\u211d)\u230b\u208a))\\(filter is_prime_pow (Icc 1 \u230aM\u230b\u208a)), {\n    intros q hq, rw [mem_sdiff, mem_filter, mem_filter, not_and, mem_Icc, nat.floor_coe],\n    rw [mem_filter, mem_Icc] at hq, refine \u27e8\u27e8\u27e8_,hq.1.2\u27e9,hq.2.1\u27e9,_\u27e9,\n    refine le_trans (one_le_two) (is_prime_pow.two_le hq.2.1), intros hbad\u2081 hbad\u2082,\n    rw mem_Icc at hbad\u2081, rw \u2190 not_le at hq, refine hq.2.2 _, rw \u2190 nat.le_floor_iff, exact hbad\u2081.2,\n    refine rpow_nonneg_of_nonneg _ _, exact nat.cast_nonneg N,\n   },\n  refine le_trans (sum_le_sum_of_subset_of_nonneg hQaux _) _,\n  intros m hm\u2081 hm\u2082, rw inv_nonneg, exact nat.cast_nonneg m,\n  rw \u2190 add_le_add_iff_right (\u2211 q in (filter is_prime_pow (Icc 1 \u230aM\u230b\u208a)), ((q:\u211d)\u207b\u00b9)),\n  rw finset.sum_sdiff,\n  transitivity (c * (log N)\u207b\u00b9 + (log (log N) + b)),\n  specialize hmertens N hNlarge\u2081,\n  rw [norm_eq_abs, norm_eq_abs, abs_le, sub_le_iff_le_add, abs_of_pos] at hmertens, exact hmertens.2,\n  rw inv_pos, exact h0logN, rw [\u2190 sub_le_iff_le_add'],\n  specialize hmertens M hMlarge,\n  transitivity (-(c * (log M)\u207b\u00b9) + (log (log M) + b) ),\n  { rw [sub_le, \u2190 sub_sub, sub_neg_eq_add],\n    convert_to (c*((log N)\u207b\u00b9+(log M)\u207b\u00b9))+(log(log N)-log(log M)) \u2264 C / ((8*C\u2082+1) * log (log \u2191N)) using 0, { ring_nf, },\n    transitivity (2*c*(log M)\u207b\u00b9 + 2*(8/log(log N))), refine add_le_add _ _,\n    rw [mul_assoc, two_mul, mul_add, add_le_add_iff_right, mul_le_mul_left h0c, inv_le_inv, log_le_log],\n    exact hMN, exact h0M, exact h0N, exact h0logN, exact h0logM,\n    rw [\u2190 log_div, log_rpow, div_mul_left], refine le_trans (log_le_sub_one_of_pos _) _,\n    rw [one_div_pos], exact hloglogN',\n    rw [div_sub_one, \u2190 sub_add, sub_self, zero_add, div_le_iff hloglogN', mul_comm (2:\u211d), mul_assoc],\n    refine le_mul_of_one_le_right _ _, refine div_nonneg _ (le_of_lt h0loglogN), norm_num1,\n    rw [\u2190 div_le_iff', le_sub, sub_half, div_le_iff, mul_comm, \u2190 div_le_iff], norm_num1, exact h16loglogN,\n    exact one_half_pos, exact h0loglogN, exact zero_lt_two, exact ne_of_gt hloglogN', exact ne_of_gt h0logN,\n    exact h0N, exact ne_of_gt h0logN, exact ne_of_gt h0logM,\n    rw [\u2190 le_sub_iff_add_le, mul_div, div_mul_eq_div_div, div_sub_div_same, div_eq_mul_inv],\n    refine mul_le_mul hC _ _ _, norm_num1, rw inv_le_inv, exact hcomp, exact h0logM,\n    exact h0loglogN, rw inv_nonneg, exact le_of_lt h0logM, refine le_trans _ hC, refine mul_nonneg _ _,\n    exact zero_le_two, exact le_of_lt h0c,\n  },\n  rw [norm_eq_abs, norm_eq_abs, abs_le, le_sub_iff_add_le, abs_of_pos] at hmertens, exact hmertens.1,\n  rw inv_pos, exact h0logM, refine filter_subset_filter _ _, intros q hq, rw [mem_Icc],\n  rw [mem_Icc] at hq, refine \u27e8hq.1,le_trans hq.2 _\u27e9, refine nat.le_floor _,\n  refine le_trans (nat.floor_le _) _, refine rpow_nonneg_of_nonneg _ _, exact nat.cast_nonneg N,\n  rw \u2190 rpow_one (N:\u211d), refine rpow_le_rpow_of_exponent_le _ _, exact h1N, rw sub_le_self_iff,\n  refine div_nonneg _ _, norm_num1, exact le_of_lt h0loglogN, refine ne_of_gt h0logN,\n  refine mul_pos _ _, refine add_pos _ zero_lt_one, refine mul_pos _ h0C\u2082, norm_num1,\n  exact div_pos h0logN h0loglogN,\nend\n\n\n\n theorem unit_fractions_upper_log_density :\n\u2203 C : \u211d, \u2200\u1da0 (N : \u2115) in at_top, \u2200 A \u2286 Icc 1 N,\n     C*(log (log (log N)) / log (log N))* log N \u2264 \u2211 n in A, 1 / n \u2192\n       \u2203 S \u2286 A, \u2211 n in S, (1 / n : \u211a) = 1 :=\nbegin\n  rcases harmonic_filter_div with \u27e8C\u2081,h0C\u2081,hdiv\u27e9,\n  rcases harmonic_filter_reg with \u27e8C\u2082,h0C\u2082,hreg\u27e9,\n  rcases harmonic_filter_smooth with \u27e8C\u2083,h0C\u2083,hsmooth\u27e9,\n  rw eventually_at_top at hdiv, rcases hdiv with \u27e8C\u2081',hdiv\u27e9,\n  rw eventually_at_top at hreg, rcases hreg with \u27e8C\u2082',hreg\u27e9,\n  rw eventually_at_top at hsmooth, rcases hsmooth with \u27e8C\u2083',hsmooth\u27e9,\n  let C := 2+2*C\u2083+C\u2081+C\u2082+2,\n  use C,\n  have hcoraux := corollary_one,\n  rw eventually_at_top at hcoraux, rcases hcoraux with \u27e8C\u2080,hcor\u27e9,\n  rcases this_fun_increasing with \u27e8Cinc, hinc\u27e9,\n  filter_upwards [eventually_gt_at_top 1,\n    tendsto_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (0:\u211d),\n    tendsto_log_log_coe_at_top.eventually_gt_at_top (1:\u211d),\n    (tendsto_log_at_top.comp tendsto_log_log_coe_at_top).eventually_ge_at_top (1:\u211d),\n    (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (C\u2080:\u211d)),\n    (tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (C\u2081':\u211d)),\n    (tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (C\u2082':\u211d)),\n    (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (C\u2083':\u211d)),\n     (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (Cinc:\u211d)),\n   (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_gt_at_top (1:\u211d)),\n    (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n       (eventually_ge_at_top (exp(exp(1)))),\n    (the_last_large_N (C\u2083) h0C\u2083),\n    (this_particular_tends_to.comp tendsto_coe_nat_at_top_at_top).eventually\n      harmonic_sum_bound_two']\n    with N hN h0logN h0loglogN h1loglogN h1log\u2083N hlargeN hdivth hregth hsmoothth hincth\n       hlargeN\u2082 hlargeN\u2083 hlargeN\u2084 hharmonic,\n  let \u03b5 := log(log(log N))/log(log N),\n  let \u03b5' := 1/log(log N),\n  have h0\u03b5 : 0 < \u03b5, { refine div_pos _ h0loglogN, refine log_pos h1loglogN, },\n  have h01\u03b5 : 0 < 1/\u03b5, { rw one_div_pos, exact h0\u03b5, },\n  have h\u03b51 : \u03b5 < 1, { rw [div_lt_one h0loglogN, log_lt_log_iff h0loglogN,\n     log_lt_log_iff h0logN], refine lt_of_le_of_lt (log_le_sub_one_of_pos _) _,\n     norm_cast, exact lt_trans zero_lt_one hN, exact sub_one_lt (N:\u211d),\n     norm_cast, exact lt_trans zero_lt_one hN, exact h0logN,},\n  intros A hAN hrecA,\n  let A' := A.filter(\u03bb n : \u2115, (N:\u211d)^\u03b5 \u2264 n),\n  have hrecA' : (2+2*C\u2083+C\u2081+C\u2082)*\u03b5*log N \u2264 rec_sum A', {\n    have hAtemp : A' \u222a (A\\A') = A, { refine union_sdiff_of_subset _, refine filter_subset _ _, },\n    by_contra, rw [not_le, rec_sum] at h, rw \u2190 not_lt at hrecA, refine hrecA _,\n    push_cast at h, rw [\u2190 hAtemp, sum_union],\n    have hotherrec : \u2211 n in (A\\A'), (1:\u211d) / n \u2264 2*\u03b5*log N, {\n      calc _ \u2264 \u2211 n in range(\u2308(N:\u211d)^\u03b5\u2309\u208a), (1:\u211d) / n :_\n         ... \u2264 _ :_,\n      refine sum_le_sum_of_subset_of_nonneg _ _, intros n hn, rw mem_range,\n      rw [mem_sdiff, mem_filter, not_and, not_le] at hn, rw nat.lt_ceil, exact hn.2 hn.1,\n      intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n,\n      rw [mul_assoc, \u2190 log_rpow], exact hharmonic, norm_cast,\n      exact lt_trans zero_lt_one hN,\n     },\n    have hnum : C = 2+2*C\u2083+C\u2081+C\u2082+2 := by refl,\n    rw [hnum, add_mul, add_mul], refine add_lt_add_of_lt_of_le _ hotherrec, exact h,\n    exact disjoint_sdiff,\n   },\n  clear hharmonic,\n  let Y := A'.filter(\u03bb n, n \u2260 0 \u2227 \u00ac (((99 : \u211d) / 100) * log (log N) \u2264 \u03c9 n \u2227\n      (\u03c9 n : \u211d) \u2264 (3/2) * log (log N))),\n  let X := A'.filter(\u03bb n, \u00ac \u2203 d : \u2115, d \u2223 n \u2227 (4 \u2264 d) \u2227\n     ((d : \u211d) \u2264 (log N)^((1:\u211d)/1000))),\n  have hA'Icc : A' \u2286 Icc \u2308(N:\u211d) ^ \u03b5\u2309\u208a N, {\n    intros n hn, rw mem_Icc, rw mem_filter at hn,\n    have hn' := hAN hn.1, rw mem_Icc at hn', refine \u27e8_,hn'.2\u27e9,\n    rw nat.ceil_le, exact hn.2,\n   },\n  have hrecX : (rec_sum X : \u211d) \u2264 C\u2081*\u03b5' * (log N), {\n    rw rec_sum, push_cast, specialize hdiv N, rw [mul_assoc, mul_comm \u03b5', \u2190 div_eq_mul_one_div,\n      \u2190 mul_div_assoc],\n    refine le_trans _ (hdiv _), refine sum_le_sum_of_subset_of_nonneg _ _,\n    refine finset.filter_subset_filter _ hA'Icc,\n    intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n, rw ge_iff_le,\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, exact hdivth,\n   },\n  have h\u03b5\u2081 : \u03b5' \u2264 \u03b5, { rw div_le_div_right, exact h1log\u2083N, exact h0loglogN, },\n  have hrecX' : (rec_sum X : \u211d) \u2264 C\u2081*\u03b5 * (log N), {\n    refine le_trans hrecX _, rw mul_le_mul_right h0logN, rw mul_le_mul_left h0C\u2081, exact h\u03b5\u2081,\n   },\n  have hrecY : (rec_sum Y : \u211d) \u2264 C\u2082*\u03b5' * (log N), {\n    rw rec_sum, push_cast, specialize hreg N, rw [mul_assoc, mul_comm \u03b5', \u2190 div_eq_mul_one_div,\n      \u2190 mul_div_assoc],\n    refine le_trans _ (hreg _), refine sum_le_sum_of_subset_of_nonneg _ _,\n    refine finset.filter_subset_filter _ hA'Icc,\n    intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n, rw ge_iff_le,\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, exact hregth,\n   },\n  have hrecY' : (rec_sum Y : \u211d) \u2264 C\u2082*\u03b5 * (log N), {\n    refine le_trans hrecY _, rw mul_le_mul_right h0logN, rw mul_le_mul_left h0C\u2082, exact h\u03b5\u2081,\n   },\n  let A'' := A'\\(X\u222aY),\n  have hrecA'' : (2+2*C\u2083)*\u03b5*log N \u2264 rec_sum A'', {\n     refine le_trans _ rec_sum_sdiff, rw le_sub_iff_add_le, refine le_trans _ hrecA',\n     rw [add_assoc (2+2*C\u2083)], nth_rewrite 2 (add_mul (_ : \u211d)), nth_rewrite 2 (add_mul (_ : \u211d)),\n     rw add_le_add_iff_left,\n     refine le_trans (rec_sum_union) _, rw [add_mul, add_mul],\n     refine add_le_add hrecX' hrecY',\n   },\n  let \u03b4 := 1 - 1/log(log N),\n  have h0\u03b4 : 0 < \u03b4, { rw [sub_pos, one_div_lt, one_div_one], exact h1loglogN, exact h0loglogN, exact zero_lt_one, },\n  have h\u03b41 : \u03b4 \u2264 1, { refine sub_le_self _ _, rw one_div_nonneg, exact le_of_lt h0loglogN, },\n  let Nf := (\u03bb i : \u2115, (N:\u211d)^(\u03b4^i)),\n  let Af := (\u03bb i : \u2115, Ioc \u230aNf (i+1)\u230b\u208a \u230aNf i\u230b\u208a \u2229 A''),\n  let Nf' := (\u03bb i : \u2115, \u230aNf i\u230b\u208a),\n  let \u03b5'' := 1/(log(log N))^2,\n  have hgoodi : \u2203 i:\u2115, 2*(log N)^((1:\u211d)/500) + C\u2083*\u03b5''*(log N) \u2264 rec_sum (Af i), {\n    by_contra,\n    let I := range(\u2308log(1/\u03b5)*(2*log(log N))\u2309\u208a),\n    have hIA : A'' = I.bUnion( \u03bb i, Af i), { rw \u2190 finset.bUnion_inter, refine eq.symm _,\n      rw finset.inter_eq_right_iff_subset, intros n hn,\n      have := bUnion_range_Ioc \u2308log(1/\u03b5)*(2*log(log N))\u2309\u208a Nf', refine this _, rw mem_Ioc,\n      rw [mem_sdiff, mem_filter] at hn,\n      refine \u27e8_,_\u27e9, rw nat.floor_lt, refine lt_of_lt_of_le _ hn.1.2,\n      refine rpow_lt_rpow_of_exponent_lt _ _, exact_mod_cast hN,\n      calc _ \u2264 \u03b4 ^ (log(1/\u03b5)*(2*log(log N))) :_\n         ... < _ :_,\n      rw \u2190 rpow_nat_cast, refine rpow_le_rpow_of_exponent_ge h0\u03b4 h\u03b41 _,\n      refine nat.le_ceil _, rw [\u2190 exp_log h0\u03b4, \u2190 exp_mul, \u2190 mul_assoc, mul_comm (log \u03b4),\n        mul_assoc, exp_mul, exp_log h01\u03b5, one_div, \u2190 rpow_neg_one, \u2190 rpow_mul],\n      nth_rewrite 1 \u2190 rpow_one \u03b5,\n      refine rpow_lt_rpow_of_exponent_gt h0\u03b5 h\u03b51 _,\n      rw [\u2190 mul_assoc, neg_one_mul, \u2190 div_lt_iff, lt_neg],\n      refine lt_of_le_of_lt (real.log_le_sub_one_of_pos h0\u03b4) _,\n      rw [\u2190 sub_add_eq_sub_sub, add_comm, sub_add_eq_sub_sub, sub_self, zero_sub,\n        lt_neg, neg_neg, one_div_lt_one_div],\n      nth_rewrite 0 \u2190 one_mul (log(log N)), refine mul_lt_mul _ _ _ _, exact one_lt_two,\n      refl, exact h0loglogN, exact zero_le_two, refine mul_pos zero_lt_two h0loglogN,\n      exact h0loglogN, refine mul_pos zero_lt_two h0loglogN, exact le_of_lt h0\u03b5,\n      refine rpow_nonneg_of_nonneg _ _, exact nat.cast_nonneg N,\n      have hnntemp := hAN hn.1.1, rw mem_Icc at hnntemp,\n      have htemp : Nf' 0 = N, {\n        have htemp' : Nf 0 = (N:\u211d)^(\u03b4^0), { refl, },\n        have htemp'' : Nf' 0 = \u230aNf 0\u230b\u208a, { refl, },\n        rw [htemp'', htemp', pow_zero, rpow_one, nat.floor_coe],\n       },\n      rw htemp, exact hnntemp.2,\n     },\n    rw \u2190 not_lt at hrecA'', refine hrecA'' _, rw not_exists at h, rw hIA,\n    refine lt_of_le_of_lt (rec_sum_bUnion Af) _,\n    refine lt_of_le_of_lt\n       (finset.sum_le_card_nsmul _ _ (2*(log N)^((1:\u211d)/500) + (C\u2083)*\u03b5''*(log N)) _) _,\n    intros x hx, specialize h x, rw not_le at h, exact le_of_lt h,\n    rw [card_range, one_div_div, nsmul_eq_mul], exact hlargeN\u2084.2,\n  },\n  rcases hgoodi with \u27e8i,hi\u27e9,\n  let A\u2080 := Af i,\n  let N\u2080 := \u230aNf i\u230b\u208a,\n  have hNN\u2080 : (N:\u211d)^\u03b5 \u2264 N\u2080, {\n    by_contra,\n    have : Af i = \u2205, { rw finset.eq_empty_iff_forall_not_mem,\n     intros n hn, rw [mem_inter, mem_sdiff, mem_Ioc, mem_filter] at hn,\n     refine h _, refine le_trans hn.2.1.2 _, exact_mod_cast hn.1.2, },\n    rw [this, \u2190 not_lt, rec_sum_empty] at hi, refine hi _, norm_cast, refine add_pos _ _,\n    refine mul_pos _ _, norm_num1, refine rpow_pos_of_pos h0logN _,\n    refine mul_pos _ h0logN, refine mul_pos h0C\u2083 _, rw one_div_pos, refine sq_pos_of_pos h0loglogN,\n   },\n  have h1N\u2080' : 1 \u2264 Nf i, { refine one_le_rpow _ _, exact_mod_cast le_of_lt hN,\n    refine pow_nonneg _ _, exact le_of_lt h0\u03b4, },\n  have h1N\u2080 : 1 \u2264 N\u2080, {\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans _ hNN\u2080, norm_cast, exact (le_of_lt hlargeN\u2082),\n   },\n  have hN\u2080large\u2082 : 0 < log(N\u2080), { refine log_pos _,\n    refine lt_of_lt_of_le _ hNN\u2080, exact hlargeN\u2082, },\n  have hN\u2080large : 1 \u2264 log(log N\u2080), { rw [\u2190 exp_le_exp, exp_log, \u2190 exp_le_exp, exp_log],\n    refine le_trans _ hNN\u2080, exact hlargeN\u2083, norm_cast, exact lt_of_lt_of_le zero_lt_one h1N\u2080,\n    exact hN\u2080large\u2082, },\n  have hN\u2080N : (N\u2080 : \u211d) \u2264 N, { rw \u2190 rpow_one N, refine le_trans (nat.floor_le _) _,\n    refine rpow_nonneg_of_nonneg _ _, exact nat.cast_nonneg N,\n    refine rpow_le_rpow_of_exponent_le _ _, exact_mod_cast le_of_lt hN,\n    refine pow_le_one _ _ _, exact le_of_lt h0\u03b4, exact h\u03b41, },\n  have hlogNN\u2080': 3 / 2 * log (log N) \u2264 2 * log (log N\u2080), {\n    rw [\u2190 div_le_iff', mul_comm, mul_div_assoc], norm_num1, rw [mul_comm, \u2190 log_rpow, log_le_log,\n      \u2190 exp_le_exp, exp_log],\n      { refine le_trans _ hNN\u2080, have : (0:\u211d) < N, { norm_cast, exact lt_trans zero_lt_one hN, },\n         rw [\u2190 exp_log this, \u2190 exp_mul, exp_le_exp, exp_log this], exact hlargeN\u2084.1 },\n    norm_cast, exact lt_of_lt_of_le zero_lt_one h1N\u2080, refine rpow_pos_of_pos _ _, exact h0logN,\n    exact hN\u2080large\u2082, exact h0logN, exact zero_lt_two,\n   },\n  have hlogNN\u2080: log N \u2264 (log N\u2080)^(2:\u211d), { rw [\u2190 log_le_log, log_rpow],\n    refine le_trans _ hlogNN\u2080', nth_rewrite 0 \u2190 one_mul (log(log N)), rw mul_le_mul_right h0loglogN,\n    norm_num1, exact hN\u2080large\u2082, exact h0logN, refine rpow_pos_of_pos _ _, exact hN\u2080large\u2082},\n  let M := (N\u2080:\u211d)^((1:\u211d)-8/(log(log N\u2080))),\n  let Z := A\u2080.filter(\u03bb n, \u2203 q : \u2115, is_prime_pow q \u2227 M < q \u2227 q \u2223 n),\n  let A\u2081 := A\u2080 \\ Z,\n  have hloc : log N\u2080/(log(log N\u2080))^2 \u2264 \u03b5'' * log N, {\n    rw [mul_comm, \u2190 div_eq_mul_one_div], refine hinc N\u2080 N \u27e8_,_\u27e9,\n    refine le_trans hincth hNN\u2080, exact_mod_cast hN\u2080N,\n   },\n  have hA\u2080large : \u2200 n \u2208 A\u2080, (N\u2080 : \u211d) ^ (1 - (1 : \u211d) / log (log N\u2080)) \u2264 n, {\n    intros n hn,\n    have := (inter_subset_left _ _) hn,\n    rw mem_Ioc at this, rw nat.floor_lt at this, refine le_trans _ (le_of_lt this.1),\n    transitivity ((Nf i)^ (1 - (1 : \u211d) / log (log N\u2080))),\n    refine rpow_le_rpow _ _ _, norm_cast, exact le_trans zero_le_one h1N\u2080,\n    refine nat.floor_le _, refine rpow_nonneg_of_nonneg _ _, exact nat.cast_nonneg N,\n    rw [sub_nonneg, one_div_le, one_div_one], exact hN\u2080large,\n    exact lt_of_lt_of_le zero_lt_one hN\u2080large, exact zero_lt_one,\n    rw \u2190 rpow_mul, refine rpow_le_rpow_of_exponent_le _ _, norm_cast,\n    exact le_of_lt hN, rw [\u2190 rpow_nat_cast, \u2190 rpow_nat_cast], push_cast,\n    rw [rpow_add_one, mul_le_mul_left, sub_le_sub_iff_left, one_div_le_one_div, log_le_log,\n     log_le_log], exact hN\u2080N, norm_cast, exact lt_of_lt_of_le zero_lt_one h1N\u2080, norm_cast,\n    exact lt_trans zero_lt_one hN, exact hN\u2080large\u2082, exact h0logN, exact h0loglogN,\n    exact lt_of_lt_of_le zero_lt_one hN\u2080large, refine rpow_pos_of_pos h0\u03b4 _,\n    exact ne_of_gt h0\u03b4, exact nat.cast_nonneg N, refine rpow_nonneg_of_nonneg _ _,\n    exact nat.cast_nonneg N,\n  },\n  have hA\u2081large : \u2200 n \u2208 A\u2081, (N\u2080 : \u211d) ^ (1 - (1 : \u211d) / log (log N\u2080)) \u2264 n, {\n    intros n hn, refine hA\u2080large n _, refine (sdiff_subset _ _) hn,\n   },\n  have hA\u2080' : A\u2080 \u2286 Icc \u2308(N\u2080:\u211d) ^ (1 - 1 / log (log N\u2080))\u2309\u208a N\u2080, {\n    intros n hn, rw mem_Icc, have hn' := hn,\n    rw [mem_inter, mem_Ioc] at hn, refine \u27e8_,hn.1.2\u27e9, rw nat.ceil_le, exact hA\u2080large n hn',\n   },\n  have hrecZ : (rec_sum Z : \u211d) \u2264 C\u2083*\u03b5''* (log N), {\n    rw rec_sum, push_cast, transitivity (C\u2083*log N\u2080/(log(log N\u2080))^2), specialize hsmooth N\u2080,\n    refine le_trans _ (hsmooth _), refine sum_le_sum_of_subset_of_nonneg _ _,\n    refine finset.filter_subset_filter _ hA\u2080',\n    intros n hn1 hn2, rw one_div_nonneg, exact nat.cast_nonneg n, rw ge_iff_le,\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans hsmoothth hNN\u2080,\n    rw [mul_assoc, mul_div_assoc, mul_le_mul_left h0C\u2083], exact hloc,\n  },\n  have hrecA\u2081 : 2*(log N\u2080)^((1:\u211d)/500) \u2264 rec_sum A\u2081, {\n    transitivity 2*(log N)^((1:\u211d)/500),\n    rw mul_le_mul_left zero_lt_two, refine rpow_le_rpow _ _ _, refine log_nonneg _,\n    exact_mod_cast h1N\u2080, rw log_le_log, exact hN\u2080N, norm_cast, rw nat.floor_pos,\n    exact h1N\u2080', exact_mod_cast lt_trans zero_lt_one hN, norm_num1, exact real.nontrivial,\n    refine le_trans _ rec_sum_sdiff, rw le_sub_iff_add_le, refine le_trans _ hi,\n    rw add_le_add_iff_left, exact hrecZ,\n   },\n  have hN\u2080 : C\u2080 \u2264 N\u2080, {\n    rw \u2190 @nat.cast_le \u211d _ _ _ _, refine le_trans _ hNN\u2080, exact hlargeN,\n   },\n  have hA\u2081N\u2080 : A\u2081 \u2286 range(N\u2080 + 1), {\n    intros n hn, rw [mem_range, nat.lt_succ_iff],\n    have hn' := (inter_subset_left _ _) ((sdiff_subset _ _) hn),\n    rw mem_Ioc at hn', exact hn'.2,\n   },\n  have hA\u2081div : \u2200 n \u2208 A\u2081, \u2203 p : \u2115, p \u2223 n \u2227 4 \u2264 p \u2227 (p : \u211d) \u2264 log N\u2080 ^ (1/500 : \u211d), {\n    intros n hn, rw [mem_sdiff, mem_inter, mem_sdiff, not_mem_union] at hn,\n    have hn' := hn.1.2.2.1, rw [mem_filter, not_and, not_not] at hn',\n    rcases (hn' hn.1.2.1) with \u27e8d, hd\u27e9, refine \u27e8d,hd.1,hd.2.1,_\u27e9,\n    refine le_trans hd.2.2 _, have : (0:\u211d) < 1000 := by norm_num1,\n    rw \u2190 rpow_le_rpow_iff _ _ this, rw [\u2190 rpow_mul, \u2190 rpow_mul], norm_num1,\n    rw rpow_one, exact hlogNN\u2080, exact le_of_lt hN\u2080large\u2082, exact le_of_lt h0logN,\n    refine rpow_nonneg_of_nonneg _ _, exact le_of_lt h0logN,\n    refine rpow_nonneg_of_nonneg _ _, exact le_of_lt hN\u2080large\u2082,\n   },\n  have hA\u2081smooth : \u2200 n \u2208 A\u2081, is_smooth (M) n, {\n    intros n hn, rw is_smooth, intros q hq\u2081 hq\u2082, rw [mem_sdiff] at hn,\n    have hn' := hn.2, rw [mem_filter, not_and] at hn',\n    have := hn' hn.1, rw \u2190 not_lt, intro hbad, refine this \u27e8q,hq\u2081,hbad,hq\u2082\u27e9,\n   },\n  have hA\u2081reg : arith_regular N\u2080 A\u2081, {\n    rw arith_regular, intros n hn, rw [mem_sdiff, mem_inter, mem_sdiff, not_mem_union] at hn,\n    have hn' := hn.1.2.2.2, rw [mem_filter, not_and, not_and, not_not] at hn',\n    have hn'' := hn' hn.1.2.1 _, refine \u27e8_,_\u27e9,\n    refine le_trans _ hn''.1, rw [mul_le_mul_left, log_le_log, log_le_log], exact hN\u2080N,\n    norm_cast, exact lt_of_lt_of_le zero_lt_one h1N\u2080, norm_cast, exact lt_trans zero_lt_one hN,\n    exact hN\u2080large\u2082, exact h0logN, norm_num1, refine le_trans hn''.2 hlogNN\u2080',\n    intro hbad, rw hbad at hn,\n    have htemp' := hAN ((filter_subset _ _) hn.1.2.1),\n    rw [mem_Icc, \u2190 not_lt] at htemp', exact htemp'.1 zero_lt_one,\n   },\n  specialize hcor N\u2080 hN\u2080 A\u2081 hA\u2081N\u2080 hA\u2081large hrecA\u2081 hA\u2081div hA\u2081smooth hA\u2081reg,\n  rcases hcor with \u27e8S,hS\u2081,hS\u2082\u27e9,\n  rw rec_sum at hS\u2082, refine \u27e8S,_,hS\u2082\u27e9,\n  refine subset_trans hS\u2081 (subset_trans (sdiff_subset _ _) _),\n  refine subset_trans (inter_subset_right _ _) _,\n  refine subset_trans (sdiff_subset _ _) (filter_subset _ _),\nend\n\n\n", "meta": {"author": "b-mehta", "repo": "unit-fractions", "sha": "b60c39a3ebd40a84104a4064840b10fc2af15fb8", "save_path": "github-repos/lean/b-mehta-unit-fractions", "path": "github-repos/lean/b-mehta-unit-fractions/unit-fractions-b60c39a3ebd40a84104a4064840b10fc2af15fb8/src/final_results.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.39631423522549386}}
{"text": "\n-- weirdly this is named List.List.append_eq ??\ntheorem List.append_eq : {\u03b1 : Type} \u2192 (as bs : List \u03b1) \u2192 List.append as bs = as ++ bs := List.List.append_eq\n\ninductive Brick : Type\n  | Vert : Brick\n  | Cap : Brick\n  | Cup : Brick\n  | Over : Brick\n  | Under : Brick\n  deriving BEq, DecidableEq\nopen Brick\n\n-- stolen from Bool LawfulBEq impl\ninstance instLawfulEqBrick : LawfulBEq Brick where\n  eq_of_beq {a b} h := by cases a <;> cases b <;> first | rfl | contradiction\n  rfl {a} := by cases a <;> decide\n\n/-- input threads -/\ndef Brick.domain : Brick \u2192 Nat\n  | Vert => 1\n  | Cap => 0\n  | Cup => 2\n  | Over => 2\n  | Under => 2\n\n/-- output threads -/\ndef Brick.codomain : Brick \u2192 Nat\n  | Vert => 1\n  | Cap => 2\n  | Cup => 0\n  | Over => 2\n  | Under => 2\n\n/-- 180\u00b0 rotation -/\ndef Brick.vflip : Brick \u2192 Brick\n  | Vert => Vert\n  | Cap => Cup\n  | Cup => Cap\n  | Over => Over\n  | Under => Under\n\ndef Brick.hflip : Brick \u2192 Brick := id\n\ntheorem Brick.vflip_boundary (b : Brick) : b.vflip.domain = b.codomain \u2227 b.vflip.codomain = b.domain := by\n  cases b <;> simp\n\ndef Bricks := List Brick\n\ninstance : Append Bricks := List.instAppendList\n-- want this for simp in proofs, allows us to translate first to List.append\n-- then List.append_eq simps to (Append List _).append a b\n-- otherwise stuff like [] ++ b won't simp because the ++ is the wrong type ((Append Bricks).append)\n@[simp] theorem Bricks.append_eq : {a b : Bricks} \u2192 a ++ b = List.append a b := rfl\n\n/-- input threads -/\ndef Bricks.domain (bs : Bricks) : Nat := List.foldr Nat.add 0 (bs.map Brick.domain)\n/-- output threads -/\ndef Bricks.codomain (bs : Bricks) : Nat := List.foldr Nat.add 0 (bs.map Brick.codomain)\n\n@[simp] theorem Bricks.domain_cons : Bricks.domain (b::bs) = b.domain + Bricks.domain bs := by\n  rewrite [domain, List.map, List.foldr, \u2190domain]\n  apply Nat.add_eq\n@[simp] theorem Bricks.codomain_cons : Bricks.codomain (b::bs) = b.codomain + Bricks.codomain bs := by\n  rewrite [codomain, List.map, List.foldr, \u2190codomain]\n  apply Nat.add_eq\n\n@[simp] theorem Bricks.domain_append : {a b : Bricks} \u2192 Bricks.domain (a ++ b) = Bricks.domain a + Bricks.domain b := by\n  intro a b\n  induction a with\n  | nil => simp [domain, List.foldr]\n  | cons hd tl h => simp; rewrite [h]; exact Eq.symm (Nat.add_assoc _ _ _)\n@[simp] theorem Bricks.codomain_append : {a b : Bricks} \u2192 Bricks.codomain (a ++ b) = Bricks.codomain a + Bricks.codomain b := by\n  intro a b\n  induction a with\n  | nil => simp [codomain, List.foldr]\n  | cons hd tl h => simp; rewrite [h]; exact Eq.symm (Nat.add_assoc _ _ _)\n\ndef Bricks.vflip (bs : Bricks) : Bricks := List.map Brick.vflip bs\ndef Bricks.hflip (bs : Bricks) : Bricks := (List.map Brick.hflip bs).reverse\n\ndef Wall : Type := List Bricks\n\ninstance : Append Wall := List.instAppendList\n@[simp] theorem Wall.append_eq : {a b : Wall} \u2192 a ++ b = List.append a b := rfl\n\ndef Wall.happend : (a b : Wall) \u2192 (a.length = b.length) \u2192  Wall\n  | [], [], _ => []\n  | a::as, bs, h => match bs with\n      | b::bs => (a.append b)::(Wall.happend as bs (by simp [List.length] at h; exact h))\n      | [] => by simp at h\n\ndef Wall.vflip (w : Wall) : Wall := (List.map Bricks.vflip w).reverse\ndef Wall.hflip (w : Wall) : Wall := (List.map Bricks.hflip w)\n\n\nnamespace Equivalence\ndef vert_bricks (n : Nat) : Bricks := List.replicate n Vert\n\ntheorem verts_dom_eq_codom (n : Nat) : (vert_bricks n).domain = (vert_bricks n).codomain := by\n  induction n with\n  | zero => simp\n  | succ _ h =>\n    simp [vert_bricks, List.replicate, Brick.domain, Brick.codomain]\n    rewrite [\u2190vert_bricks, h]\n    rfl\n\n/--\nPlanar isotopic mappings\n\n     vertcl    vertcc    crosscl       crosscc\n     |  .\". |  |  | .\".  | |  .\". | |  | |  | | .\".\n     |  | '.'  |  '.' |  '/.  | .\\' |  '/.  | .\\' |\n                         | |  | | '.'  | |  '.' | |\n\n     slide\n     . . .    . . .\n     #######  | | |\n     | | | |  #######\n     ' ' ' '  ' ' ' '\n-/\ninductive PlanarIsotopic : Wall \u2192 Wall \u2192 Prop\n  | vertcl : PlanarIsotopic [[Vert],[Vert]] [[Cap,Vert],[Vert,Cup]]\n  | vertcc : PlanarIsotopic [[Vert],[Vert]] [[Vert,Cap],[Cup,Vert]]\n  | crosscl : PlanarIsotopic [[Vert,Vert],[Under],[Vert,Vert]] [[Cap,Vert,Vert],[Vert,Over,Vert],[Vert,Vert,Cup]]\n  | crosscc : PlanarIsotopic [[Vert,Vert],[Under],[Vert,Vert]] [[Vert,Vert,Cap],[Vert,Over,Vert],[Cup,Vert,Vert]]\n  | slide : PlanarIsotopic [a, vert_bricks (Bricks.codomain a)] [vert_bricks (Bricks.domain a), a]\n\n/--\nReidemeister moves\n\n    1a .\".  .\".  2a | |  .\\'  3a .\\' |  | '/.\n       | |  .\\'     | |  '/.     | .\\'  .\\' |\n                                 '/. |  | .\\'\n    1b .\".  .\".  2b | |  '/.\n       | |  '/.     | |  .\\'  3b '/. |  | '/.\n                                 | '/.  '/. |\n    1c | |  .\\'                  '/. |  | '/.\n       '.'  '.'\n                              3c .\\' |  | '/.\n    1d | |  '/.                  | .\\'  .\\' |\n       '.'  '.'                  '/. |  | .\\'\n\n                              3d '/. |  | .\\'\n                                 | '/.  '/. |\n                                 .\\' |  | '/.\n-/\ninductive ReidemeisterMove : Wall \u2192 Wall \u2192 Prop\n  | type1a : ReidemeisterMove [[Cap], [Vert, Vert]]  [[Cap], [Over]]\n  | type1b : ReidemeisterMove [[Cap], [Vert, Vert]]  [[Cap], [Under]]\n  | type1c : ReidemeisterMove [[Vert, Vert], [Cup]]  [[Over], [Cup]]\n  | type1d : ReidemeisterMove [[Vert, Vert], [Cup]]  [[Under], [Cup]]\n  | type2a : ReidemeisterMove [[Vert, Vert], [Vert, Vert]]  [[Over], [Under]]\n  | type2b : ReidemeisterMove [[Vert, Vert], [Vert, Vert]]  [[Under], [Over]]\n  | type3a : ReidemeisterMove [[Over, Vert], [Vert, Over], [Under, Vert]]\n                  [[Vert, Under], [Over, Vert], [Vert, Over]]\n  | type3b : ReidemeisterMove [[Under, Vert], [Vert, Under], [Under, Vert]]\n                  [[Vert, Under], [Under, Vert], [Vert, Under]]\n  | type3c : ReidemeisterMove [[Over, Vert], [Vert, Over], [Over, Vert]]\n                  [[Vert, Over], [Over, Vert], [Vert, Over]]\n  | type3d : ReidemeisterMove [[Under, Vert], [Vert, Under], [Over, Vert]]\n                  [[Vert, Over], [Under, Vert], [Vert, Under]]\n\nend Equivalence\n\n", "meta": {"author": "shua", "repo": "leanknot", "sha": "5c50fc107c1e98978d2cd966d4c6b22348e1ee4a", "save_path": "github-repos/lean/shua-leanknot", "path": "github-repos/lean/shua-leanknot/leanknot-5c50fc107c1e98978d2cd966d4c6b22348e1ee4a/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.6442251201477015, "lm_q1q2_score": 0.39625504762123004}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nExtra definitions on option.\n-/\n\nnamespace option\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\ninstance has_mem : has_mem \u03b1 (option \u03b1) := \u27e8\u03bb a b, b = some a\u27e9\n\n@[simp] theorem mem_def {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = some a :=\niff.rfl\n\ntheorem is_none_iff_eq_none {o : option \u03b1} : o.is_none = tt \u2194 o = none :=\n\u27e8option.eq_none_of_is_none, \u03bb e, e.symm \u25b8 rfl\u27e9\n\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n\ninstance decidable_eq_none {o : option \u03b1} : decidable (o = none) :=\ndecidable_of_decidable_of_iff (bool.decidable_eq _ _) is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2200 a \u2208 o, p a)\n| none     := is_true (by simp)\n| (some a) := if h : p a\n  then is_true $ \u03bb o e, some_inj.1 e \u25b8 h\n  else is_false $ mt (\u03bb H, H _ rfl) h\n\ninstance decidable_exists_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2203 a \u2208 o, p a)\n| none     := is_false (\u03bb \u27e8a, \u27e8h, _\u27e9\u27e9, by cases h)\n| (some a) := if h : p a\n  then is_true $ \u27e8_, rfl, h\u27e9\n  else is_false $ \u03bb \u27e8_, \u27e8rfl, hn\u27e9\u27e9, h hn\n\n/-- inhabited `get` function. Returns `a` if the input is `some a`,\n  otherwise returns `default`. -/\n@[reducible] def iget [inhabited \u03b1] : option \u03b1 \u2192 \u03b1\n| (some x) := x\n| none     := default \u03b1\n\n@[simp] theorem iget_some [inhabited \u03b1] {a : \u03b1} : (some a).iget = a := rfl\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [decidable_pred p] (a : \u03b1) : option \u03b1 :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a`\n  and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (o : option \u03b1) : option \u03b1 :=\no.bind (guard p)\n\ndef to_list : option \u03b1 \u2192 list \u03b1\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : \u03b1} {o : option \u03b1} : a \u2208 to_list o \u2194 a \u2208 o :=\nby cases o; simp [to_list, eq_comm]\n\ndef lift_or_get (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : option \u03b1 \u2192 option \u03b1 \u2192 option \u03b1\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_commutative \u03b1 f] :\n  is_commutative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b, by cases a; cases b; simp [lift_or_get, h.comm]\u27e9\n\ninstance lift_or_get_assoc (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_associative \u03b1 f] :\n  is_associative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]\u27e9\n\ninstance lift_or_get_idem (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_idempotent \u03b1 f] :\n  is_idempotent (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get, h.idempotent]\u27e9\n\ninstance lift_or_get_is_left_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_left_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninstance lift_or_get_is_right_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_right_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : option \u03b1 \u2192 option \u03b2 \u2192 Prop\n| some {a b} : r a b \u2192 rel (some a) (some b)\n| none {}    : rel none none\n\nend option\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/data/option/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3962550392183107}}
{"text": "import topology.algebra.infinite_sum\nimport topology.instances.ennreal\n\nopen_locale ennreal\n\nopen_locale nnreal\n\n-- don't need it but maybe useful?\nlemma ennreal.summable_of_coe_sum_eq {X : Type*} (f g : X \u2192 \u211d\u22650)\n  (h : \u2211' x, (f x : \u211d\u22650\u221e) = \u2211' x, (g x : \u211d\u22650\u221e)) :\n  summable f \u2194 summable g :=\nby rw [\u2190 ennreal.tsum_coe_ne_top_iff_summable, h, ennreal.tsum_coe_ne_top_iff_summable]\n\nlemma ennreal.has_sum_comm {\u03b1 \u03b2: Type*} (F : \u03b1 \u2192 \u03b2 \u2192 \u211d\u22650\u221e) (s : \u211d\u22650\u221e)\n  : has_sum (\u03bb n, \u2211' k, F n k) s \u2194 has_sum (\u03bb k, \u2211' n, F n k) s :=\nby rw [ summable.has_sum_iff ennreal.summable, summable.has_sum_iff ennreal.summable,\n    ennreal.tsum_comm ]\n\n-- do we need the `real` version?\n-- /-- sum of row sums equals sum of column sums -/\n-- lemma real.summable_snd_of_summable_fst {\u03b1 \u03b2: Type*} (F : \u03b1 \u2192 \u03b2 \u2192 \u211d) (h_nonneg : \u2200 n k, 0 \u2264 F n k)\n--   (h_rows : \u2200 n, summable (\u03bb k, F n k)) (h_cols : \u2200 k, summable (\u03bb n, F n k))\n--   (h_col_row : summable (\u03bb k, \u2211' n, F n k)) : summable (\u03bb n, \u2211' k, F n k) :=\n-- begin\n\n--   -- wrong idea have := summable (\u03bb ab : \u03b1 \u00d7 \u03b2, F ab.1 ab.2),\n--   admit,\n-- end\n\n-- could go in ennreal line 684 or so\nlemma ennreal.mul_le_mul_of_right {a b c : \u211d\u22650\u221e} (hab : a \u2264 b) : a * c \u2264 b * c :=\nbegin\n  rcases eq_or_ne c 0 with (rfl | hc0),\n  { simp },\n  { rcases eq_or_ne c \u22a4 with (rfl | hctop),\n    { rw [@ennreal.mul_top b],\n      split_ifs with hb,\n      { subst hb,\n        change a \u2264 \u22a5 at hab,\n        rw le_bot_iff at hab,\n        simp [hab], },\n      { exact le_top, } },\n    { rwa ennreal.mul_le_mul_right hc0 hctop }, },\nend\n\n-- could go in ennreal line 684 or so\nlemma ennreal.mul_le_mul_of_left {a b c : \u211d\u22650\u221e} (hab : a \u2264 b) : c * a \u2264 c * b :=\nbegin\n  rw [mul_comm, mul_comm c],\n  exact ennreal.mul_le_mul_of_right hab,\nend\n\n-- might not need this\nlemma nnreal.inv_mul_le_iff {a b c : \u211d\u22650} (hb0 : b \u2260 0) : b\u207b\u00b9 * a \u2264 c \u2194 a \u2264 b * c :=\nbegin\n  rw \u2190 nnreal.coe_le_coe,\n  rw \u2190 nnreal.coe_le_coe,\n  push_cast,\n  apply inv_mul_le_iff,\n  obtain (hb | (hb : 0 < b)) := eq_zero_or_pos,\n  { subst hb, exfalso, apply hb0, refl, },\n  { assumption_mod_cast, }\nend\n\nlemma ennreal.inv_mul_le_iff {a b c : \u211d\u22650\u221e} (hb0 : b \u2260 0) (hb : b \u2260 \u221e) :\n  b\u207b\u00b9 * a \u2264 c \u2194 a \u2264 b * c :=\nbegin\n  rw [mul_comm, mul_comm b],\n  apply ennreal.div_le_iff_le_mul;\n  cc,\nend\n\nlemma ennreal.zero_le (a : \u211d\u22650\u221e) : 0 \u2264 a := bot_le\nlemma ennreal.zero_le' {a : \u211d\u22650\u221e} : 0 \u2264 a := bot_le\n\nlemma ennreal.inv_eq_of_mul_eq_one {a b : \u211d\u22650\u221e} (h : a * b = 1) : a\u207b\u00b9 = b :=\nbegin\n  induction b using with_top.rec_top_coe,\n  { exfalso,\n    rw ennreal.mul_top at h,\n    split_ifs at h with ha;\n    { revert h, norm_num, }, },\n  induction a using with_top.rec_top_coe,\n  { exfalso,\n    rw ennreal.top_mul at h,\n    split_ifs at h with ha;\n    { revert h, norm_num, }, },\n  norm_cast at h,\n  have ha : a \u2260 0,\n  { rintro rfl, rw zero_mul at h, revert h, norm_num, },\n  rw \u2190 ennreal.coe_inv ha,\n  norm_cast,\n  rwa [\u2190 inv_mul_eq_one\u2080, inv_inv],\n  exact inv_ne_zero ha,\nend\n\nlemma ennreal.mul_inv_eq_of_eq_mul {a b c : \u211d\u22650\u221e} (hb0 : b \u2260 0) (hbtop : b \u2260 \u22a4) (h : a = c * b) :\n  a * b\u207b\u00b9 = c :=\nby rw [h, mul_assoc, ennreal.mul_inv_cancel hb0 hbtop, mul_one]\n\nlemma ennreal.eq_mul_of_mul_inv_eq {a b c : \u211d\u22650\u221e} (hb0 : b \u2260 0) (hbtop : b \u2260 \u22a4) (h : a * b\u207b\u00b9 = c) :\n  a = c * b :=\nby rw [\u2190 h, mul_assoc, ennreal.inv_mul_cancel hb0 hbtop, mul_one]\n\nlemma ennreal.mul_eq_of_mul_inv_eq {a b c : \u211d\u22650\u221e} (hb0 : b \u2260 0) (hbtop : b \u2260 \u22a4) (h : a * b\u207b\u00b9 = c) :\n  c * b = a :=\n(ennreal.eq_mul_of_mul_inv_eq hb0 hbtop h).symm\n\nlemma ennreal.mul_inv_eq_iff_eq_mul {a b c : \u211d\u22650\u221e} (hb0 : b \u2260 0) (hbtop : b \u2260 \u22a4) :\n  (a * b\u207b\u00b9 = c \u2194 a = c * b) :=\n\u27e8ennreal.eq_mul_of_mul_inv_eq hb0 hbtop, ennreal.mul_inv_eq_of_eq_mul hb0 hbtop\u27e9\n\nlemma ennreal.le_zero_iff {a : \u211d\u22650\u221e} : a \u2264 0 \u2194 a = 0 := le_bot_iff\n\nlemma ennreal.sub_pos {a b : \u211d\u22650\u221e} : 0 < a - b \u2194 b < a :=\nbegin\n  rw \u2190 not_iff_not,\n  push_neg,\n  rw ennreal.le_zero_iff,\n  apply tsub_eq_zero_iff_le,\nend\n\nlemma ennreal.top_zpow_of_pos {n : \u2124} (hn : 0 < n) : (\u22a4 : \u211d\u22650\u221e) ^ n = \u22a4 :=\nbegin\n  let m := n.nat_abs,\n  have hm : n = m,\n  { rw int.nat_abs_of_nonneg hn.le },\n  rw hm at hn \u22a2,\n  apply ennreal.top_pow,\n  exact_mod_cast hn,\nend\n\n-- can't do!\n--lemma ennreal.zpow_neg (a : \u211d\u22650\u221e) : \u2200 (n : \u2124), a ^ -n = (a ^ n)\u207b\u00b9 := by admit\n\n-- lemma ennreal.top_zpow_of_neg {n : \u2124} (hn : n < 0) : (\u22a4 : \u211d\u22650\u221e) ^ n = 0 :=\n-- begin\n--   let m := n.nat_abs,\n--   have hm : n = -m,\n--   { rw [int.of_nat_nat_abs_of_nonpos hn.le, neg_neg] },\n--   rw hm at hn \u22a2,\n--   rw neg_lt_zero at hn,\n--   have hm' : 0 < m, by exact_mod_cast hn,\n--   rw [ennreal.zpow_neg, zpow_coe_nat, ennreal.inv_eq_zero, ennreal.top_pow hm'],\n-- end\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/ennreal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3962550392183107}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.algebra.basic\nimport algebra.algebra.subalgebra\nimport algebra.free_algebra\nimport algebra.category.CommRing.basic\nimport algebra.category.Module.basic\n\n/-!\n# Category instance for algebras over a commutative ring\n\nWe introduce the bundled category `Algebra` of algebras over a fixed commutative ring `R ` along\nwith the forgetful functors to `Ring` and `Module`. We furthermore show that the functor associating\nto a type the free `R`-algebra on that type is left adjoint to the forgetful functor.\n-/\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables (R : Type u) [comm_ring R]\n\n/-- The category of R-algebras and their morphisms. -/\nstructure Algebra :=\n(carrier : Type v)\n[is_ring : ring carrier]\n[is_algebra : algebra R carrier]\n\nattribute [instance] Algebra.is_ring Algebra.is_algebra\n\nnamespace Algebra\n\ninstance : has_coe_to_sort (Algebra R) (Type v) := \u27e8Algebra.carrier\u27e9\n\ninstance : category (Algebra.{v} R) :=\n{ hom   := \u03bb A B, A \u2192\u2090[R] B,\n  id    := \u03bb A, alg_hom.id R A,\n  comp  := \u03bb A B C f g, g.comp f }\n\ninstance : concrete_category.{v} (Algebra.{v} R) :=\n{ forget := { obj := \u03bb R, R, map := \u03bb R S f, (f : R \u2192 S) },\n  forget_faithful := { } }\n\ninstance has_forget_to_Ring : has_forget\u2082 (Algebra.{v} R) Ring.{v} :=\n{ forget\u2082 :=\n  { obj := \u03bb A, Ring.of A,\n    map := \u03bb A\u2081 A\u2082 f, alg_hom.to_ring_hom f, } }\n\ninstance has_forget_to_Module : has_forget\u2082 (Algebra.{v} R) (Module.{v} R) :=\n{ forget\u2082 :=\n  { obj := \u03bb M, Module.of R M,\n    map := \u03bb M\u2081 M\u2082 f, alg_hom.to_linear_map f, } }\n\n/-- The object in the category of R-algebras associated to a type equipped with the appropriate\ntypeclasses. -/\ndef of (X : Type v) [ring X] [algebra R X] : Algebra.{v} R := \u27e8X\u27e9\n\n/-- Typecheck a `alg_hom` as a morphism in `Algebra R`. -/\ndef of_hom {R : Type u} [comm_ring R] {X Y : Type v} [ring X] [algebra R X] [ring Y] [algebra R Y]\n  (f : X \u2192\u2090[R] Y) : of R X \u27f6 of R Y := f\n\ninstance : inhabited (Algebra R) := \u27e8of R R\u27e9\n\n@[simp]\nlemma coe_of (X : Type u) [ring X] [algebra R X] : (of R X : Type u) = X := rfl\n\nvariables {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nalgebra. -/\n@[simps]\ndef of_self_iso (M : Algebra.{v} R) : Algebra.of R M \u2245 M :=\n{ hom := \ud835\udfd9 M, inv := \ud835\udfd9 M }\n\nvariables {R} {M N U : Module.{v} R}\n\n@[simp] \n\n@[simp] lemma coe_comp (f : M \u27f6 N) (g : N \u27f6 U) :\n  ((f \u226b g) : M \u2192 U) = g \u2218 f := rfl\n\nvariables (R)\n/-- The \"free algebra\" functor, sending a type `S` to the free algebra on `S`. -/\n@[simps]\ndef free : Type u \u2964 Algebra.{u} R :=\n{ obj := \u03bb S,\n  { carrier := free_algebra R S,\n    is_ring := algebra.semiring_to_ring R },\n  map := \u03bb S T f, free_algebra.lift _ $ (free_algebra.\u03b9 _) \u2218 f,\n  -- obviously can fill the next two goals, but it is slow\n  map_id' := by { intros X, ext1, simp only [free_algebra.\u03b9_comp_lift], refl },\n  map_comp' := by { intros, ext1, simp only [free_algebra.\u03b9_comp_lift], ext1,\n    simp only [free_algebra.lift_\u03b9_apply, category_theory.coe_comp, function.comp_app,\n      types_comp_apply] } }\n\n/-- The free/forget adjunction for `R`-algebras. -/\ndef adj : free.{u} R \u22a3 forget (Algebra.{u} R) :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X A, (free_algebra.lift _).symm,\n  -- Relying on `obviously` to fill out these proofs is very slow :(\n  hom_equiv_naturality_left_symm' := by { intros, ext,\n    simp only [free_map, equiv.symm_symm, free_algebra.lift_\u03b9_apply, category_theory.coe_comp,\n      function.comp_app, types_comp_apply] },\n  hom_equiv_naturality_right' := by { intros, ext,\n    simp only [forget_map_eq_coe, category_theory.coe_comp, function.comp_app,\n      free_algebra.lift_symm_apply, types_comp_apply] } }\n\ninstance : is_right_adjoint (forget (Algebra.{u} R)) := \u27e8_, adj R\u27e9\n\nend Algebra\n\nvariables {R}\nvariables {X\u2081 X\u2082 : Type u}\n\n/-- Build an isomorphism in the category `Algebra R` from a `alg_equiv` between `algebra`s. -/\n@[simps]\ndef alg_equiv.to_Algebra_iso\n  {g\u2081 : ring X\u2081} {g\u2082 : ring X\u2082} {m\u2081 : algebra R X\u2081} {m\u2082 : algebra R X\u2082} (e : X\u2081 \u2243\u2090[R] X\u2082) :\n  Algebra.of R X\u2081 \u2245 Algebra.of R X\u2082 :=\n{ hom := (e : X\u2081 \u2192\u2090[R] X\u2082),\n  inv := (e.symm : X\u2082 \u2192\u2090[R] X\u2081),\n  hom_inv_id' := begin ext, exact e.left_inv x, end,\n  inv_hom_id' := begin ext, exact e.right_inv x, end, }\n\nnamespace category_theory.iso\n\n/-- Build a `alg_equiv` from an isomorphism in the category `Algebra R`. -/\n@[simps]\ndef to_alg_equiv {X Y : Algebra R} (i : X \u2245 Y) : X \u2243\u2090[R] Y :=\n{ to_fun    := i.hom,\n  inv_fun   := i.inv,\n  left_inv  := by tidy,\n  right_inv := by tidy,\n  map_add'  := by tidy,\n  map_mul'  := by tidy,\n  commutes' := by tidy, }.\n\nend category_theory.iso\n\n/-- Algebra equivalences between `algebras`s are the same as (isomorphic to) isomorphisms in\n`Algebra`. -/\n@[simps]\ndef alg_equiv_iso_Algebra_iso {X Y : Type u}\n  [ring X] [ring Y] [algebra R X] [algebra R Y] :\n  (X \u2243\u2090[R] Y) \u2245 (Algebra.of R X \u2245 Algebra.of R Y) :=\n{ hom := \u03bb e, e.to_Algebra_iso,\n  inv := \u03bb i, i.to_alg_equiv, }\n\ninstance (X : Type u) [ring X] [algebra R X] : has_coe (subalgebra R X) (Algebra R) :=\n\u27e8 \u03bb N, Algebra.of R N \u27e9\n\ninstance Algebra.forget_reflects_isos : reflects_isomorphisms (forget (Algebra.{u} R)) :=\n{ reflects := \u03bb X Y f _,\n  begin\n    resetI,\n    let i := as_iso ((forget (Algebra.{u} R)).map f),\n    let e : X \u2243\u2090[R] Y := { ..f, ..i.to_equiv },\n    exact \u27e8(is_iso.of_iso e.to_Algebra_iso).1\u27e9,\n  end }\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/category/Algebra/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.6442251201477015, "lm_q1q2_score": 0.396255038530128}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.traversable.equiv\nimport control.traversable.instances\nimport data.lazy_list\n\n/-!\n## Definitions on lazy lists\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file contains various definitions and proofs on lazy lists.\n\nTODO: move the `lazy_list.lean` file from core to mathlib.\n-/\n\nuniverses u\n\nnamespace thunk\n\n/-- Creates a thunk with a (non-lazy) constant value. -/\ndef mk {\u03b1} (x : \u03b1) : thunk \u03b1 := \u03bb _, x\n\ninstance {\u03b1 : Type u} [decidable_eq \u03b1] : decidable_eq (thunk \u03b1) | a b :=\nhave a = b \u2194 a () = b (), from \u27e8by cc, by intro; ext x; cases x; assumption\u27e9,\nby rw this; apply_instance\n\nend thunk\n\nnamespace lazy_list\n\nopen function\n\n/-- Isomorphism between strict and lazy lists. -/\ndef list_equiv_lazy_list (\u03b1 : Type*) : list \u03b1 \u2243 lazy_list \u03b1 :=\n{ to_fun := lazy_list.of_list,\n  inv_fun := lazy_list.to_list,\n  right_inv := by { intro, induction x, refl, simp! [*],\n                    ext, cases x, refl },\n  left_inv := by { intro, induction x, refl, simp! [*] } }\n\ninstance {\u03b1 : Type u} [decidable_eq \u03b1] : decidable_eq (lazy_list \u03b1)\n| nil nil := is_true rfl\n| (cons x xs) (cons y ys) :=\n  if h : x = y then\n    match decidable_eq (xs ()) (ys ()) with\n    | is_false h2 := is_false (by intro; cc)\n    | is_true h2 :=\n      have xs = ys, by ext u; cases u; assumption,\n      is_true (by cc)\n    end\n  else\n    is_false (by intro; cc)\n| nil (cons _ _) := is_false (by cc)\n| (cons _ _) nil := is_false (by cc)\n\n/-- Traversal of lazy lists using an applicative effect. -/\nprotected def traverse {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b2 : Type u}\n    (f : \u03b1 \u2192 m \u03b2) : lazy_list \u03b1 \u2192 m (lazy_list \u03b2)\n| lazy_list.nil := pure lazy_list.nil\n| (lazy_list.cons x xs) := lazy_list.cons <$> f x <*> (thunk.mk <$> traverse (xs ()))\n\ninstance : traversable lazy_list :=\n{ map := @lazy_list.traverse id _,\n  traverse := @lazy_list.traverse }\n\ninstance : is_lawful_traversable lazy_list :=\nbegin\n  apply equiv.is_lawful_traversable' list_equiv_lazy_list;\n  intros ; resetI; ext,\n  { induction x, refl,\n    simp! [equiv.map,functor.map] at *,\n    simp [*], refl, },\n  { induction x, refl,\n    simp! [equiv.map,functor.map_const] at *,\n    simp [*], refl, },\n  { induction x,\n    { simp! [traversable.traverse,equiv.traverse] with functor_norm, refl },\n    simp! [equiv.map,functor.map_const,traversable.traverse] at *, rw x_ih,\n    dsimp [list_equiv_lazy_list,equiv.traverse,to_list,traversable.traverse,list.traverse],\n    simp! with functor_norm, refl },\nend\n\n/-- `init xs`, if `xs` non-empty, drops the last element of the list.\nOtherwise, return the empty list. -/\ndef init {\u03b1} : lazy_list \u03b1 \u2192 lazy_list \u03b1\n| lazy_list.nil := lazy_list.nil\n| (lazy_list.cons x xs) :=\n  let xs' := xs () in\n  match xs' with\n  | lazy_list.nil := lazy_list.nil\n  | (lazy_list.cons _ _) := lazy_list.cons x (init xs')\n  end\n\n/-- Return the first object contained in the list that satisfies\npredicate `p` -/\ndef find {\u03b1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : lazy_list \u03b1 \u2192 option \u03b1\n| nil        := none\n| (cons h t) := if p h then some h else find (t ())\n\n/-- `interleave xs ys` creates a list where elements of `xs` and `ys` alternate. -/\ndef interleave {\u03b1} : lazy_list \u03b1 \u2192 lazy_list \u03b1 \u2192 lazy_list \u03b1\n| lazy_list.nil xs := xs\n| a@(lazy_list.cons x xs) lazy_list.nil := a\n| (lazy_list.cons x xs) (lazy_list.cons y ys) :=\n  lazy_list.cons x (lazy_list.cons y (interleave (xs ()) (ys ())))\n\n/-- `interleave_all (xs::ys::zs::xss)` creates a list where elements of `xs`, `ys`\nand `zs` and the rest alternate. Every other element of the resulting list is taken from\n`xs`, every fourth is taken from `ys`, every eighth is taken from `zs` and so on. -/\ndef interleave_all {\u03b1} : list (lazy_list \u03b1) \u2192 lazy_list \u03b1\n| [] := lazy_list.nil\n| (x :: xs) := interleave x (interleave_all xs)\n\n/-- Monadic bind operation for `lazy_list`. -/\nprotected def bind {\u03b1 \u03b2} : lazy_list \u03b1 \u2192 (\u03b1 \u2192 lazy_list \u03b2) \u2192 lazy_list \u03b2\n| lazy_list.nil _ := lazy_list.nil\n| (lazy_list.cons x xs) f := lazy_list.append (f x) (bind (xs ()) f)\n\n/-- Reverse the order of a `lazy_list`.\nIt is done by converting to a `list` first because reversal involves evaluating all\nthe list and if the list is all evaluated, `list` is a better representation for\nit than a series of thunks. -/\ndef reverse {\u03b1} (xs : lazy_list \u03b1) : lazy_list \u03b1 :=\nof_list xs.to_list.reverse\n\ninstance : monad lazy_list :=\n{ pure := @lazy_list.singleton,\n  bind := @lazy_list.bind }\n\n\n\nlemma append_assoc {\u03b1} (xs ys zs : lazy_list \u03b1) :\n  (xs.append ys).append zs = xs.append (ys.append zs) :=\nby induction xs; simp [append, *]\n\nlemma append_bind {\u03b1 \u03b2} (xs : lazy_list \u03b1) (ys : thunk (lazy_list \u03b1)) (f : \u03b1 \u2192 lazy_list \u03b2) :\n  (@lazy_list.append _ xs ys).bind f = (xs.bind f).append ((ys ()).bind f) :=\nby induction xs; simp [lazy_list.bind, append, *, append_assoc, append, lazy_list.bind]\n\ninstance : is_lawful_monad lazy_list :=\n{ pure_bind := by { intros, apply append_nil },\n  bind_assoc := by { intros, dsimp [(>>=)], induction x; simp [lazy_list.bind, append_bind, *], },\n  id_map :=\n  begin\n    intros,\n    simp [(<$>)],\n    induction x; simp [lazy_list.bind, *, singleton, append],\n    ext \u27e8 \u27e9, refl,\n  end }\n\n/-- Try applying function `f` to every element of a `lazy_list` and\nreturn the result of the first attempt that succeeds. -/\ndef mfirst {m} [alternative m] {\u03b1 \u03b2} (f : \u03b1 \u2192 m \u03b2) : lazy_list \u03b1 \u2192 m \u03b2\n| nil := failure\n| (cons x xs) :=\n  f x <|> mfirst (xs ())\n\n/-- Membership in lazy lists -/\nprotected def mem {\u03b1} (x : \u03b1) : lazy_list \u03b1 \u2192 Prop\n| lazy_list.nil := false\n| (lazy_list.cons y ys) := x = y \u2228 mem (ys ())\n\ninstance {\u03b1} : has_mem \u03b1 (lazy_list \u03b1) :=\n\u27e8 lazy_list.mem \u27e9\n\ninstance mem.decidable {\u03b1} [decidable_eq \u03b1] (x : \u03b1) : \u03a0 xs : lazy_list \u03b1, decidable (x \u2208 xs)\n| lazy_list.nil := decidable.false\n| (lazy_list.cons y ys) :=\n  if h : x = y\n    then decidable.is_true (or.inl h)\n    else decidable_of_decidable_of_iff (mem.decidable (ys ())) (by simp [*, (\u2208), lazy_list.mem])\n\n@[simp]\nlemma mem_nil {\u03b1} (x : \u03b1) : x \u2208 @lazy_list.nil \u03b1 \u2194 false := iff.rfl\n\n@[simp]\nlemma mem_cons {\u03b1} (x y : \u03b1) (ys : thunk (lazy_list \u03b1)) :\n  x \u2208 @lazy_list.cons \u03b1 y ys \u2194 x = y \u2228 x \u2208 ys () := iff.rfl\n\ntheorem forall_mem_cons {\u03b1} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : thunk (lazy_list \u03b1)} :\n  (\u2200 x \u2208 @lazy_list.cons _ a l, p x) \u2194 p a \u2227 \u2200 x \u2208 l (), p x :=\nby simp only [has_mem.mem, lazy_list.mem, or_imp_distrib, forall_and_distrib, forall_eq]\n\n/-! ### map for partial functions -/\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on\n  `a : \u03b1` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`\n  but is defined only when all members of `l` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {\u03b1 \u03b2} {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) :\n  \u03a0 l : lazy_list \u03b1, (\u2200 a \u2208 l, p a) \u2192 lazy_list \u03b2\n| lazy_list.nil         H := lazy_list.nil\n| (lazy_list.cons x xs) H := lazy_list.cons (f x (forall_mem_cons.1 H).1)\n                               (pmap (xs ()) (forall_mem_cons.1 H).2)\n\n/-- \"Attach\" the proof that the elements of `l` are in `l` to produce a new `lazy_list`\n  with the same elements but in the type `{x // x \u2208 l}`. -/\ndef attach {\u03b1} (l : lazy_list \u03b1) : lazy_list {x // x \u2208 l} := pmap subtype.mk l (\u03bb a, id)\n\ninstance {\u03b1} [has_repr \u03b1] : has_repr (lazy_list \u03b1) :=\n\u27e8 \u03bb xs, repr xs.to_list \u27e9\n\nend lazy_list\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/lazy_list/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.6859494550081925, "lm_q1q2_score": 0.3961326409948672}}
{"text": "/-\nCopyright (c) 2019 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot\n\n! This file was ported from Lean 3 source module topology.uniform_space.pi\n! leanprover-community/mathlib commit 0a0ec35061ed9960bf0e7ffb0335f44447b58977\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.UniformSpace.Cauchy\nimport Mathbin.Topology.UniformSpace.Separation\n\n/-!\n# Indexed product of uniform spaces\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nnoncomputable section\n\nopen uniformity Topology\n\nsection\n\nopen Filter UniformSpace\n\nuniverse u\n\nvariable {\u03b9 : Type _} (\u03b1 : \u03b9 \u2192 Type u) [U : \u2200 i, UniformSpace (\u03b1 i)]\n\ninclude U\n\n#print Pi.uniformSpace /-\ninstance Pi.uniformSpace : UniformSpace (\u2200 i, \u03b1 i) :=\n  UniformSpace.ofCoreEq (\u2a05 i, UniformSpace.comap (fun a : \u2200 i, \u03b1 i => a i) (U i)).toCore\n      Pi.topologicalSpace <|\n    Eq.symm toTopologicalSpace_inf\u1d62\n#align Pi.uniform_space Pi.uniformSpace\n-/\n\n/- warning: Pi.uniformity -> Pi.uniformity is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u2}} (\u03b1 : \u03b9 -> Type.{u1}) [U : forall (i : \u03b9), UniformSpace.{u1} (\u03b1 i)], Eq.{succ (max u2 u1)} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (uniformity.{max u2 u1} (forall (i : \u03b9), \u03b1 i) (Pi.uniformSpace.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i))) (inf\u1d62.{max u2 u1, succ u2} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (ConditionallyCompleteLattice.toHasInf.{max u2 u1} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (Filter.completeLattice.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))))) \u03b9 (fun (i : \u03b9) => Filter.comap.{max u2 u1, u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i)) (Prod.{u1, u1} (\u03b1 i) (\u03b1 i)) (fun (a : Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i)) => Prod.mk.{u1, u1} (\u03b1 i) (\u03b1 i) (Prod.fst.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i) a i) (Prod.snd.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i) a i)) (uniformity.{u1} (\u03b1 i) (U i))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u1}} (\u03b1 : \u03b9 -> Type.{u2}) [U : forall (i : \u03b9), UniformSpace.{u2} (\u03b1 i)], Eq.{max (succ u2) (succ u1)} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (uniformity.{max u2 u1} (forall (i : \u03b9), \u03b1 i) (Pi.uniformSpace.{u2, u1} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i))) (inf\u1d62.{max u2 u1, succ u1} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (ConditionallyCompleteLattice.toInfSet.{max u2 u1} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (CompleteLattice.toConditionallyCompleteLattice.{max u2 u1} (Filter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))) (Filter.instCompleteLatticeFilter.{max u2 u1} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i))))) \u03b9 (fun (i : \u03b9) => Filter.comap.{max u2 u1, u2} (Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i)) (Prod.{u2, u2} (\u03b1 i) (\u03b1 i)) (fun (a : Prod.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i)) => Prod.mk.{u2, u2} (\u03b1 i) (\u03b1 i) (Prod.fst.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i) a i) (Prod.snd.{max u2 u1, max u2 u1} (forall (i : \u03b9), \u03b1 i) (forall (i : \u03b9), \u03b1 i) a i)) (uniformity.{u2} (\u03b1 i) (U i))))\nCase conversion may be inaccurate. Consider using '#align Pi.uniformity Pi.uniformity\u2093'. -/\ntheorem Pi.uniformity : \ud835\udce4 (\u2200 i, \u03b1 i) = \u2a05 i : \u03b9, (Filter.comap fun a => (a.1 i, a.2 i)) <| \ud835\udce4 (\u03b1 i) :=\n  inf\u1d62_uniformity\n#align Pi.uniformity Pi.uniformity\n\nvariable {\u03b1}\n\n/- warning: uniform_continuous_pi -> uniformContinuous_pi is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u2}} {\u03b1 : \u03b9 -> Type.{u1}} [U : forall (i : \u03b9), UniformSpace.{u1} (\u03b1 i)] {\u03b2 : Type.{u3}} [_inst_1 : UniformSpace.{u3} \u03b2] {f : \u03b2 -> (forall (i : \u03b9), \u03b1 i)}, Iff (UniformContinuous.{u3, max u2 u1} \u03b2 (forall (i : \u03b9), \u03b1 i) _inst_1 (Pi.uniformSpace.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i)) f) (forall (i : \u03b9), UniformContinuous.{u3, u1} \u03b2 (\u03b1 i) _inst_1 (U i) (fun (x : \u03b2) => f x i))\nbut is expected to have type\n  forall {\u03b9 : Type.{u1}} {\u03b1 : \u03b9 -> Type.{u3}} [U : forall (i : \u03b9), UniformSpace.{u3} (\u03b1 i)] {\u03b2 : Type.{u2}} [_inst_1 : UniformSpace.{u2} \u03b2] {f : \u03b2 -> (forall (i : \u03b9), \u03b1 i)}, Iff (UniformContinuous.{u2, max u3 u1} \u03b2 (forall (i : \u03b9), \u03b1 i) _inst_1 (Pi.uniformSpace.{u3, u1} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i)) f) (forall (i : \u03b9), UniformContinuous.{u2, u3} \u03b2 (\u03b1 i) _inst_1 (U i) (fun (x : \u03b2) => f x i))\nCase conversion may be inaccurate. Consider using '#align uniform_continuous_pi uniformContinuous_pi\u2093'. -/\ntheorem uniformContinuous_pi {\u03b2 : Type _} [UniformSpace \u03b2] {f : \u03b2 \u2192 \u2200 i, \u03b1 i} :\n    UniformContinuous f \u2194 \u2200 i, UniformContinuous fun x => f x i := by\n  simp only [UniformContinuous, Pi.uniformity, tendsto_infi, tendsto_comap_iff]\n#align uniform_continuous_pi uniformContinuous_pi\n\nvariable (\u03b1)\n\n/- warning: Pi.uniform_continuous_proj -> Pi.uniformContinuous_proj is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u2}} (\u03b1 : \u03b9 -> Type.{u1}) [U : forall (i : \u03b9), UniformSpace.{u1} (\u03b1 i)] (i : \u03b9), UniformContinuous.{max u2 u1, u1} (forall (i : \u03b9), \u03b1 i) (\u03b1 i) (Pi.uniformSpace.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i)) (U i) (fun (a : forall (i : \u03b9), \u03b1 i) => a i)\nbut is expected to have type\n  forall {\u03b9 : Type.{u1}} (\u03b1 : \u03b9 -> Type.{u2}) [U : forall (i : \u03b9), UniformSpace.{u2} (\u03b1 i)] (i : \u03b9), UniformContinuous.{max u2 u1, u2} (forall (i : \u03b9), \u03b1 i) (\u03b1 i) (Pi.uniformSpace.{u2, u1} \u03b9 (fun (i : \u03b9) => \u03b1 i) (fun (i : \u03b9) => U i)) (U i) (fun (a : forall (i : \u03b9), \u03b1 i) => a i)\nCase conversion may be inaccurate. Consider using '#align Pi.uniform_continuous_proj Pi.uniformContinuous_proj\u2093'. -/\ntheorem Pi.uniformContinuous_proj (i : \u03b9) : UniformContinuous fun a : \u2200 i : \u03b9, \u03b1 i => a i :=\n  uniformContinuous_pi.1 uniformContinuous_id i\n#align Pi.uniform_continuous_proj Pi.uniformContinuous_proj\n\n#print Pi.complete /-\ninstance Pi.complete [\u2200 i, CompleteSpace (\u03b1 i)] : CompleteSpace (\u2200 i, \u03b1 i) :=\n  \u27e8by\n    intro f hf\n    haveI := hf.1\n    have : \u2200 i, \u2203 x : \u03b1 i, Filter.map (fun a : \u2200 i, \u03b1 i => a i) f \u2264 \ud835\udcdd x :=\n      by\n      intro i\n      have key : Cauchy (map (fun a : \u2200 i : \u03b9, \u03b1 i => a i) f) :=\n        hf.map (Pi.uniformContinuous_proj \u03b1 i)\n      exact cauchy_iff_exists_le_nhds.1 key\n    choose x hx using this\n    use x\n    rwa [nhds_pi, le_pi]\u27e9\n#align Pi.complete Pi.complete\n-/\n\n#print Pi.separated /-\ninstance Pi.separated [\u2200 i, SeparatedSpace (\u03b1 i)] : SeparatedSpace (\u2200 i, \u03b1 i) :=\n  separated_def.2 fun x y H => by\n    ext i\n    apply eq_of_separated_of_uniform_continuous (Pi.uniformContinuous_proj \u03b1 i)\n    apply H\n#align Pi.separated Pi.separated\n-/\n\nend\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/UniformSpace/Pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3961326310189646}}
{"text": "import complements.filter_product\nimport tactic\nimport tactic.interactive\nimport data.real.hyperreal\n\nopen tactic expr\n\nnamespace tactic\n\nsection lift_lhs\n\nprivate meta def forall_rule (l : expr) (\u03b1 : expr) : tactic unit :=\ndo  `(filter %%\u03b9) \u2190 infer_type l,\n    e \u2190 to_expr ``(filter.germ.forall_iff_forall_lift_pred %%l),\n    rewrite_target e\n\nprivate meta def exists_rule (l : expr) (\u03b1 : expr) : tactic unit :=\ndo  `(filter %%\u03b9) \u2190 infer_type l,\n    e \u2190 to_expr ``(filter.germ.exists_iff_exists_lift_pred %%l),\n    rewrite_target e\n\nmeta def transfer_lift_lhs (tgt : expr) : tactic unit :=\ndo  `(%%lhs \u2194 %%rhs) \u2190 (return tgt) | fail \"Goal is not an equivalence (step 1)\",\n    match rhs with\n    | `(\u2200 _ : (filter.germ %%l %%\u03b1), _) := forall_rule l \u03b1\n    | `(\u2200 _ : \u211d*, _) := forall_rule `(filter.hyperfilter \u2115 : filter \u2115) `(\u211d)\n    | `(\u2203 _ : (filter.germ %%l %%\u03b1), _) := exists_rule l \u03b1\n    | `(\u2203 _ : \u211d*, _) := exists_rule `(filter.hyperfilter \u2115 : filter \u2115) `(\u211d)\n    | _ := fail \"No known pattern applicable (step 1)\"\n    end\n\nend lift_lhs\n\nsection congr\n\nmeta def transfer_congr (tgt : expr) : tactic unit :=\ndo  `(%%lhs \u2194 %%rhs) \u2190 (return tgt) | fail \"Goal is not an equivalence (step 2)\",\n    match lhs with\n    | `(\u2200 _ : %%t, _) := \n      (do t' \u2190 infer_type t,\n          unify t' `(Prop),\n          refine ``(imp_congr _ _)) <|>\n      (do some (name, _, _) \u2190 get_binder none tt lhs,\n          name \u2190 get_unused_name name,\n          refine ``(forall_congr _),\n          intro name,\n          skip)\n    | `(\u2203 _ : %%t, _) := \n      do  --some (name, _, _) \u2190 get_binder none tt lhs, TODO\n          name \u2190 get_unused_name,\n          refine ``(exists_congr _),\n          intro name,\n          skip\n    | `(_ \u2227 _) := \n      do  name \u2190 get_unused_name,\n          refine ``(and_congr _ _)\n    | `(_ \u2228 _) := \n      do  name \u2190 get_unused_name,\n          refine ``(or_congr _ _)\n    | `(\u00ac _) := \n      do  name \u2190 get_unused_name,\n          refine ``(not_congr _)\n    | `(_ = _) := \n      do  name \u2190 get_unused_name,\n          refine ``(iff_of_eq (_root_.congr (=) _))\n    | _ := fail \"No known pattern applicable (step 2)\"\n    end\n\nend congr\n\nsection push_lift\n\nmeta def transfer_push_lift (tgt : expr) : tactic unit :=\ndo  `(filter.germ.lift_pred %%p %%x \u2194 %%rhs) \u2190 (return tgt) | fail \"Goal is not an equivalence (step 3)\",\n    match p with\n    | `(\u03bb _, \u2200 y, %%q) := \n      (do e \u2190 to_expr ``(filter.germ.lift_pred_forall_iff_forall_lift_pred'),\n          rewrite_target e) <|>\n      (do e \u2190 to_expr ``(filter.germ.lift_pred_imp_iff_imp_lift_pred),\n          rewrite_target e)\n    | `(\u03bb _, \u00ac %%q) := \n      do  e \u2190 to_expr ``(filter.germ.lift_pred_not_iff_not_lift_pred),\n          rewrite_target e\n    | `(\u03bb _, \u2203 y, %%q) := \n      (do e \u2190 to_expr ``(filter.germ.lift_pred_exists_iff_exists_lift_pred'),\n          rewrite_target e) <|>\n      (do e \u2190 to_expr ``(filter.germ.lift_pred_exists_prop_iff_and_lift_pred),\n          e' \u2190 to_expr ``(exists_prop),\n          rewrite_target e,\n          rewrite_target e')\n    | `(\u03bb _, _ < _) := \n      do  e \u2190 to_expr ``(filter.germ.lift_pred_lt_iff_lt_map),\n          rewrite_target e\n    | `(\u03bb _, _ > _) := \n      do  e \u2190 to_expr ``(filter.germ.lift_pred_lt_iff_lt_map),\n          rewrite_target e\n    | `(\u03bb _, _ = _) :=\n      do  e \u2190 to_expr ``(filter.germ.lift_pred_eq_iff_eq_map),\n          rewrite_target e\n    | `(\u03bb _, _ \u2260 _) :=\n      do  e \u2190 to_expr ``(filter.germ.lift_pred_ne_iff_ne_map),\n          rewrite_target e\n    | `(\u03bb _, _ \u2227 _) :=\n      do  e \u2190 to_expr ``(filter.germ.lift_pred_and_iff_and_lift_pred),\n          rewrite_target e\n    | `(\u03bb _, _ \u2228 _) :=\n      do  e \u2190 to_expr ``(filter.germ.lift_pred_or_iff_or_lift_pred),\n          rewrite_target e\n    | _ := fail \"No known pattern applicable (step 3)\"\n    end\n\nend push_lift\n\nsection induction\n\nmeta def transfer_induction (tgt : expr) : tactic unit :=\nlocal_context >>= list.mmap' (\u03bb x, try $ \n  do  t \u2190 infer_type x,\n      match t with\n      | `(filter.germ _ _) := skip\n      | `(\u211d*) := skip\n      | _ := fail ()\n      end,\n      refine ``((%%x).induction_on _), \n      name \u2190 get_unused_name, \n      intro name )\n\nmeta def transfer_close (tgt : expr) : tactic unit :=\ntransfer_induction tgt >> reflexivity\n\nend induction\n\nnamespace interactive\n\nsetup_tactic_parser\n\nmeta def transfer_lift_lhs : tactic unit :=\ntarget >>= tactic.transfer_lift_lhs\n\nmeta def transfer_congr : tactic unit :=\ntarget >>= tactic.transfer_congr\n\nmeta def transfer_push_lift : tactic unit :=\ntarget >>= tactic.transfer_push_lift\n\nmeta def transfer_induction : tactic unit :=\ntarget >>= tactic.transfer_induction\n\nmeta def transfer_close : tactic unit :=\ntarget >>= tactic.transfer_close\n\nmeta def transfer_step : tactic unit :=\ntransfer_close <|>\ntransfer_congr <|>\n(transfer_push_lift >> try transfer_congr) <|>\n(transfer_lift_lhs >> try transfer_congr)\n\nmeta def transfer : tactic unit :=\nfocus (repeat transfer_step)\n\nend interactive\n\nexample (\u03b1 \u03b9 : Type*) [preorder \u03b1] (l : ultrafilter \u03b9) (a : \u03b1) : \n  (\u2200 x, a \u2264 x) \u2194 (\u2200 x : (l : filter \u03b9).germ \u03b1, \u2191a \u2264 x) :=\nby transfer\n\nexample (\u03b1 \u03b9 : Type*) [preorder \u03b1] (l : ultrafilter \u03b9) (a : \u03b1) : \n  (\u2200 x y : \u03b1, x = y) \u2194 (\u2200 x y : (l : filter \u03b9).germ \u03b1, x = y) :=\nby transfer\n\nopen filter\n\nexample (l : \u211d) (u : \u2115 \u2192 \u211d) :\n  (\u2200 \u03b5 > 0, \u2203 N \u2265 (1 : \u2115), \u2200 n \u2265 N, abs (u n - l) < \u03b5) \u2194\n  (\u2200 \u03b5 > 0, \u2203 N \u2265 (1 : (hyperfilter \u2115 : filter \u2115).germ \u2115), \u2200 n \u2265 N, germ.map abs (germ.map u n - \u2191l) < \u03b5) :=\nby transfer\n\nend tactic", "meta": {"author": "ADedecker", "repo": "nonstandard", "sha": "c32f5e1d87cc9e6410d66cf3080fd8c4a47cf5e4", "save_path": "github-repos/lean/ADedecker-nonstandard", "path": "github-repos/lean/ADedecker-nonstandard/nonstandard-c32f5e1d87cc9e6410d66cf3080fd8c4a47cf5e4/src/transfer_tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.39606614417058517}}
{"text": "import Mathbin.Data.MvPolynomial.Basic\n\nnoncomputable section\n\nopen Classical BigOperators\n\nopen Set Function Finsupp AddMonoidAlgebra MvPolynomial\n\nuniverse u v\nvariable {R : Type u}\n\nnamespace MvPolynomial\nvariable {\u03c3 : Type _} {e : \u2115} {n m : \u03c3} {s : \u03c3 \u2192\u2080 \u2115}\n\nsection CommSemiring\u2093\n\nvariable [CommSemiring\u2093 R] {p q : MvPolynomial \u03c3 R}\n\nsection DecidableEq\n\nvariable [DecidableEq \u03c3] (a : \u03c3)\n\n/-\nlemma coeff_X_mul' (m) (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n   coeff m (X s * p) = if s \u2208 m.support then coeff (m - Finsupp.single s 1) p else 0 :=\n   by\n  rw [mul_comm]\n  rw [MvPolynomial.coeff_mul_X']\n-/\n\nend DecidableEq\n\nlemma coeff_mul_X_pow (m : \u03c3 \u2192\u2080 \u2115) (n : \u2115) (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n  coeff (m + single s n : \u03c3 \u2192\u2080 \u2115) (p * (x s : MvPolynomial \u03c3 R) ^ n : MvPolynomial \u03c3 R) = coeff m p := by\n  have : HasMem.Mem (m, single s n) (m + single s n).antidiagonal := sorry -- mem_antidiagonal.2 _,\n  rw [coeff_mul, \u2190 Finset.insert_erase this, Finset.sum_insert (Finset.not_mem_erase _ _),\n      Finset.sum_eq_zero, add_zero\u2093, coeff_X_pow, if_pos, mul_one\u2093]\n  simp only [eq_self_iff_true]\n  intro \u27e8i,j\u27e9 hij\n  rw [Finset.mem_erase, mem_antidiagonal] at hij\n  by_cases H : single s n = j\n  \u00b7 subst j\n    sorry\n  \u00b7 rw [coeff_X_pow, if_neg H, _root_.mul_zero]\n\n-- # TODO\n-- lemma support_X_pow [nontrivial R] : (X n ^ e : MvPolynomial \u03c3 R).support = {single n e} :=\n-- by rw [X_pow_eq_single, support_monomial, if_neg]; exact one_ne_zero\n\nlemma coeff_mul_X_pow' (m : \u03c3 \u2192\u2080 \u2115) (n : \u2115) (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n  coeff m (p * (x s : MvPolynomial \u03c3 R) ^ n : MvPolynomial \u03c3 R) = if n \u2264 m s then coeff (m - single s n : \u03c3 \u2192\u2080 \u2115) p else 0 :=\nby sorry\n  -- # TODO: what to do with `nontriviality` and `split_ifs`?\n  -- nontriviality R,\n  -- split_ifs with h h,\n  -- { conv_rhs {rw \u2190 coeff_mul_X_pow _ n s},\n  --   congr' with  t,\n  --   by_cases hj : s = t,\n  --   { subst t, simp only [nat_sub_apply, add_apply, single_eq_same], exact (nat.sub_eq_iff_eq_add h).mp rfl,\n  --     },\n  --   { simp [single_eq_of_ne hj] } },\n  -- { rw \u2190 not_mem_support_iff, intro hm, apply h,\n  --   have H := support_mul _ _ hm, simp only [Finset.mem_bUnion] at H,\n  --   rcases H with \u27e8j, hj, i', hi', H\u27e9,\n  --   rw [support_X_pow, Finset.mem_singleton] at hi', subst i',\n  --   rw Finset.mem_singleton at H, subst m,\n  --   rw [add_apply, single_apply, if_pos rfl],\n  --   finish, }\n\nlemma coeff_X_pow_mul' (m) (n : \u2115) (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n  coeff m (((x s : MvPolynomial \u03c3 R) ^ n) * p : MvPolynomial \u03c3 R) = if n \u2264 m s then coeff (m - Finsupp.single s n : \u03c3 \u2192\u2080 \u2115) p else 0 := by sorry\n-- begin\n--   rw mul_comm,\n--   rw coeff_mul_X_pow',\n-- end\n\n\n-- # TODO:\n-- Bolton mentions this below;\n-- \"For some reason, this lemma is actually useless\n-- https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Extracting.20constant.20from.20sum\"\n-- unification is smart enough to figure out that `Finset.mul_sum` works\n-- without the need to specialize lemmas like this\n\nlemma sum_X_mul {\u03b1 : Type u} (r : Finset \u03b1) (f : \u03b1 -> MvPolynomial \u03c3 R) (s : \u03c3) :\n  (\u2211 x in r, (MvPolynomial.x s : MvPolynomial \u03c3 R) * f x) = (x s : MvPolynomial \u03c3 R) * (\u2211 x in r, f x) :=\nby rw [Finset.mul_sum]\n\nlemma sum_C_mul {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 -> MvPolynomial \u03c3 R} (e : R) :\n  (\u2211 x in r, (c e : MvPolynomial \u03c3 R) * f x) = (c e : MvPolynomial \u03c3 R) * (\u2211 x in r, f x) :=\nby rw [Finset.mul_sum]\n\nlemma sum_C_hom {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 -> R} :\n  ((\u2211 x in r, c (f x)) : MvPolynomial \u03c3 R) = (c (\u2211 x in r, f x) : MvPolynomial \u03c3 R) :=\nby sorry -- exact Finset.sum_hom r c\n\n-- -- TODO add to mathlib\n-- instance (s : \u03c3 \u2192\u2080 \u2115) : is_add_monoid_hom (@monomial R \u03c3 _ s) :=\n-- {\n--   map_add := begin\n--     intros x y,\n--     exact monomial_add.symm,\n--   end,\n--   map_zero := monomial_zero,\n-- }\n\nlemma sum_monomial_hom {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 -> R}  (s : \u03c3 \u2192\u2080 \u2115) :\n   ((\u2211 x in r, monomial s (f x)) : MvPolynomial \u03c3 R) = monomial s (\u2211 x in r, f x)\n := by sorry -- Finset.mul_sum r (monomial s)\n\n\nlemma extract_mul_from_sum {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 -> MvPolynomial \u03c3 R} (p : MvPolynomial \u03c3 R) :\n   (\u2211 x in r, p * f x) = p * (\u2211 x in r, f x) :=\nby rw [Finset.mul_sum]\n\n-- lemma C_mul_C (a a' : R) : (c a) * (c a') = (c (a * a') : MvPolynomial \u03c3 R) := by simp\n\n-- lemma C_mul_monomial' (a a' : R) (s : \u03c3 \u2192\u2080 \u2115) : (Monomial s a') * c a  = Monomial s (a' * a) :=\n--  by sorry\n-- simp [C_apply, monomial, single_mul_single]\n\n-- lemma C_to_monomial (a : R) : @C _ \u03c3 _ a = monomial 0 (a) := by exact C_apply\n\n-- -- For some reason, this lemma is actually useless https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Extracting.20constant.20from.20sum\n-- -- I expect many other lemmas in theis file may be useless as well\n-- -- TODO investigate and clean up\n-- lemma Finset_sum_C {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 -> R} (e : R) :\n--   (\u2211 x in r, (C (f x) : MvPolynomial \u03c3 R)) = C (\u2211 x in r, f x)\n-- :=\n-- begin\n--   rw Finset.sum_hom,\n-- end\n\nlemma rearrange1 (n : \u2115) (v1 v2 : \u03c3) (p : MvPolynomial \u03c3 R) :\n  ((MvPolynomial.x v1) ^ n) * ((MvPolynomial.x v2) * p) = (MvPolynomial.x v2) * ((MvPolynomial.x v1 ^ n) * p) :=\n  by sorry\n  --by ring\n\nlemma rearrange2 (n : \u2115) (f : R) (v1 : \u03c3) (p : MvPolynomial \u03c3 R) :\n  let f\u2081 := (MvPolynomial.x v1 : MvPolynomial \u03c3 R)\n  let f\u2082 := (MvPolynomial.c f : MvPolynomial \u03c3 R)\n  (f\u2081 ^ n) * (f\u2082 * p) = f\u2082 * ((f\u2081 ^ n) * p) :=\n  by sorry\n\n-- -- move constants right of X\nlemma rearrange_constants_right (f : R) (v1 : \u03c3) :\n  let f\u2081 := (MvPolynomial.x v1 : MvPolynomial \u03c3 R)\n  let f\u2082 := (MvPolynomial.c f : MvPolynomial \u03c3 R)\n  f\u2082 * f\u2081 = f\u2081 * f\u2082\n:= by sorry\n-- by ring\n\nlemma rearrange_constants_right_with_extra (f : R) (v1 : \u03c3) (p : MvPolynomial \u03c3 R) :\n  let f\u2081 := (MvPolynomial.x v1 : MvPolynomial \u03c3 R)\n  let f\u2082 := (MvPolynomial.c f : MvPolynomial \u03c3 R)\n  f\u2082 * (f\u2081 * p) = f\u2081 * (f\u2082 * p)\n:= by sorry\n\nlemma rearrange_sums_right_with_extra {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 \u2192 MvPolynomial \u03c3 R} (s : \u03c3) (p : MvPolynomial \u03c3 R) :\n   let f\u2081 := (MvPolynomial.x s : MvPolynomial \u03c3 R)\n   (\u2211 x in r, f x) * (f\u2081 * p) = (f\u2081 * (\u2211 x in r, f x)) * p\n:= by sorry\n\nlemma rearrange_sums_right {\u03b1 : Type u} {r : Finset \u03b1} {f : \u03b1 \u2192 MvPolynomial \u03c3 R} (s : \u03c3) :\n   let f\u2081 := (MvPolynomial.x s : MvPolynomial \u03c3 R)\n   (\u2211 x in r, f x) * f\u2081 = f\u2081 * (\u2211 x in r, f x)\n:= by sorry\n\n-- -- move constants right of X\nlemma rearrange_smul_right (n : \u2115) (a : R) (v1 : \u03c3) (p : MvPolynomial \u03c3 R) :\n  let f\u2081 := (MvPolynomial.x v1 : MvPolynomial \u03c3 R)\n  a \u2022 (f\u2081 * p) = f\u2081 * (a \u2022 p)\n:= by sorry -- rw [mul_smul_comm]\n\n/-\nlemma rearrange_constants_right_hard (f : R) (p : R[X]) :\n   Polynomial.c f * p = (p) * (Polynomial.c f)\n:= by sorry\n\nlemma rearrange_001 (f : R) (p1 p2 p3 : Polynomial R) :\n  (Polynomial.c f) * p1 = p2 + p3 \u2194 p2 + p3 = (Polynomial.c f) * p1 := by\n   split\n    { intro h, rw h }\n    { intro h, rw h }\n\nlemma rearrange_002 (f : R) (p1 p2 p3 : Polynomial R) :\n  p1 * (c f) = p2 + p3 \u2194 p2 + p3 = p1 * (c f) := by\n   split\n    { intro h, rw h }\n    { intro h, rw h }\n-/\n\nlemma add_mul_distrib (a b c d : R) : a + b * c + b * d = a + b * (c + d) :=\nby sorry\n\nlemma add_mul_distrib' (a b c d : R) : a + c * b + d * b = a + b * (c + d) :=\nby sorry\n\nend CommSemiring\u2093\n\nend MvPolynomial\n", "meta": {"author": "lurk-lab", "repo": "ZKSnark.lean", "sha": "a92ff01fac8e59ffb0de13a41eac6461af6d7cf0", "save_path": "github-repos/lean/lurk-lab-ZKSnark.lean", "path": "github-repos/lean/lurk-lab-ZKSnark.lean/ZKSnark.lean-a92ff01fac8e59ffb0de13a41eac6461af6d7cf0/ZkSNARK/GeneralLemmas/MvXMul.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.740174350576073, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3960661319029793}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport logic.equiv.basic\n\n/-!\n# A type for VM-erased data\n\nThis file defines a type `erased \u03b1` which is classically isomorphic to `\u03b1`,\nbut erased in the VM. That is, at runtime every value of `erased \u03b1` is\nrepresented as `0`, just like types and proofs.\n-/\n\nuniverses u\n\n/-- `erased \u03b1` is the same as `\u03b1`, except that the elements\n  of `erased \u03b1` are erased in the VM in the same way as types\n  and proofs. This can be used to track data without storing it\n  literally. -/\ndef erased (\u03b1 : Sort u) : Sort (max 1 u) :=\n\u03a3' s : \u03b1 \u2192 Prop, \u2203 a, (\u03bb b, a = b) = s\n\nnamespace erased\n\n/-- Erase a value. -/\n@[inline] def mk {\u03b1} (a : \u03b1) : erased \u03b1 := \u27e8\u03bb b, a = b, a, rfl\u27e9\n\n/-- Extracts the erased value, noncomputably. -/\nnoncomputable def out {\u03b1} : erased \u03b1 \u2192 \u03b1\n| \u27e8s, h\u27e9 := classical.some h\n\n/--\nExtracts the erased value, if it is a type.\n\nNote: `(mk a).out_type` is not definitionally equal to `a`.\n-/\n@[reducible] def out_type (a : erased (Sort u)) : Sort u := out a\n\n/-- Extracts the erased value, if it is a proof. -/\ntheorem out_proof {p : Prop} (a : erased p) : p := out a\n\n@[simp] theorem out_mk {\u03b1} (a : \u03b1) : (mk a).out = a :=\nbegin\n  let h, show classical.some h = a,\n  have := classical.some_spec h,\n  exact cast (congr_fun this a).symm rfl\nend\n\n@[simp] theorem mk_out {\u03b1} : \u2200 (a : erased \u03b1), mk (out a) = a\n| \u27e8s, h\u27e9 := by simp [mk]; congr; exact classical.some_spec h\n\n@[ext] lemma out_inj {\u03b1} (a b : erased \u03b1) (h : a.out = b.out) : a = b :=\nby simpa using congr_arg mk h\n\n/-- Equivalence between `erased \u03b1` and `\u03b1`. -/\nnoncomputable def equiv (\u03b1) : erased \u03b1 \u2243 \u03b1 :=\n\u27e8out, mk, mk_out, out_mk\u27e9\n\ninstance (\u03b1 : Type u) : has_repr (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\ninstance (\u03b1 : Type u) : has_to_string (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\nmeta instance (\u03b1 : Type u) : has_to_format (erased \u03b1) := \u27e8\u03bb _, (\"erased\" : format)\u27e9\n\n/-- Computably produce an erased value from a proof of nonemptiness. -/\ndef choice {\u03b1} (h : nonempty \u03b1) : erased \u03b1 := mk (classical.choice h)\n\n@[simp] theorem nonempty_iff {\u03b1} : nonempty (erased \u03b1) \u2194 nonempty \u03b1 :=\n\u27e8\u03bb \u27e8a\u27e9, \u27e8a.out\u27e9, \u03bb \u27e8a\u27e9, \u27e8mk a\u27e9\u27e9\n\ninstance {\u03b1} [h : nonempty \u03b1] : inhabited (erased \u03b1) :=\n\u27e8choice h\u27e9\n\n/--\n`(>>=)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `monad`).\n-/\ndef bind {\u03b1 \u03b2} (a : erased \u03b1) (f : \u03b1 \u2192 erased \u03b2) : erased \u03b2 :=\n\u27e8\u03bb b, (f a.out).1 b, (f a.out).2\u27e9\n\n@[simp] theorem bind_eq_out {\u03b1 \u03b2} (a f) : @bind \u03b1 \u03b2 a f = f a.out :=\nby delta bind bind._proof_1; cases f a.out; refl\n\n/--\nCollapses two levels of erasure.\n-/\ndef join {\u03b1} (a : erased (erased \u03b1)) : erased \u03b1 := bind a id\n\n@[simp] theorem join_eq_out {\u03b1} (a) : @join \u03b1 a = a.out := bind_eq_out _ _\n\n/--\n`(<$>)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `functor`).\n-/\ndef map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (a : erased \u03b1) : erased \u03b2 :=\nbind a (mk \u2218 f)\n\n@[simp] theorem map_out {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (a : erased \u03b1) : (a.map f).out = f a.out :=\nby simp [map]\n\ninstance : monad erased := { pure := @mk, bind := @bind, map := @map }\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 erased \u03b1) = @mk _ := rfl\n@[simp] lemma bind_def {\u03b1 \u03b2} : ((>>=) : erased \u03b1 \u2192 (\u03b1 \u2192 erased \u03b2) \u2192 erased \u03b2) = @bind _ _ := rfl\n@[simp] lemma map_def {\u03b1 \u03b2} : ((<$>) : (\u03b1 \u2192 \u03b2) \u2192 erased \u03b1 \u2192 erased \u03b2) = @map _ _ := rfl\n\ninstance : is_lawful_monad erased := by refine {..}; intros; ext; simp\n\nend erased\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/data/erased.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.39605216110226393}}
{"text": "/-\nCopyright (c) 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Jesse Michael Han\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.hint\nimport Mathlib.PostPort\n\nuniverses l u \n\nnamespace Mathlib\n\n/-!\n# The `finish` family of tactics\n\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\neliminate existential quantifiers on the left, and look for contradictions. They rely on ematching\nand congruence closure to try to finish off a goal at the end.\n\nThe procedures *do* split on disjunctions and recreate the smt state for each terminal call, so\nthey are only meant to be used on small, straightforward problems.\n\n## Main definitions\n\nWe provide the following tactics:\n\n* `finish`  -- solves the goal or fails\n* `clarify` -- makes as much progress as possible while not leaving more than one goal\n* `safe`    -- splits freely, finishes off whatever subgoals it can, and leaves the rest\n\nAll accept an optional list of simplifier rules, typically definitions that should be expanded.\n(The equations and identities should not refer to the local context.)\n\n-/\n\nnamespace tactic\n\n\nnamespace interactive\n\n\nend interactive\n\n\nend tactic\n\n\nnamespace auto\n\n\n/-! ### Utilities -/\n\n-- stolen from interactive.lean\n\n/--\nConfiguration information for the auto tactics.\n* `(use_simp := tt)`: call the simplifier\n* `(max_ematch_rounds := 20)`: for the \"done\" tactic\n-/\nstructure auto_config where\n  use_simp : Bool\n  max_ematch_rounds : \u2115\n\n/-!\n### Preprocess goal.\n\nWe want to move everything to the left of the sequent arrow. For intuitionistic logic,\nwe replace the goal `p` with `\u2200 f, (p \u2192 f) \u2192 f` and introduce.\n-/\n\ntheorem by_contradiction_trick (p : Prop) (h : \u2200 (f : Prop), (p \u2192 f) \u2192 f) : p := h p id\n\n/-!\n### Normalize hypotheses\n\nBring conjunctions to the outside (for splitting),\nbring universal quantifiers to the outside (for ematching). The classical normalizer\neliminates `a \u2192 b` in favor of `\u00ac a \u2228 b`.\n\nFor efficiency, we push negations inwards from the top down. (For example, consider\nsimplifying `\u00ac \u00ac (p \u2228 q)`.)\n-/\n\ntheorem not_not_eq (p : Prop) : (\u00ac\u00acp) = p := propext not_not\n\ntheorem not_and_eq (p : Prop) (q : Prop) : (\u00ac(p \u2227 q)) = (\u00acp \u2228 \u00acq) := propext not_and_distrib\n\ntheorem not_or_eq (p : Prop) (q : Prop) : (\u00ac(p \u2228 q)) = (\u00acp \u2227 \u00acq) := propext not_or_distrib\n\ntheorem not_forall_eq {\u03b1 : Type u} (s : \u03b1 \u2192 Prop) : (\u00ac\u2200 (x : \u03b1), s x) = \u2203 (x : \u03b1), \u00acs x :=\n  propext not_forall\n\ntheorem not_exists_eq {\u03b1 : Type u} (s : \u03b1 \u2192 Prop) : (\u00ac\u2203 (x : \u03b1), s x) = \u2200 (x : \u03b1), \u00acs x :=\n  propext not_exists\n\ntheorem not_implies_eq (p : Prop) (q : Prop) : (\u00ac(p \u2192 q)) = (p \u2227 \u00acq) := propext not_imp\n\ntheorem classical.implies_iff_not_or (p : Prop) (q : Prop) : p \u2192 q \u2194 \u00acp \u2228 q := imp_iff_not_or\n\ndef common_normalize_lemma_names : List name := sorry\n\ndef classical_normalize_lemma_names : List name := sorry\n\n/-- optionally returns an equivalent expression and proof of equivalence -/\n/-- given an expr `e`, returns a new expression and a proof of equality -/\n/-!\n### Eliminate existential quantifiers\n-/\n\n/-- eliminate an existential quantifier if there is one -/\n/-- eliminate all existential quantifiers, fails if there aren't any -/\n/-!\n### Substitute if there is a hypothesis `x = t` or `t = x`\n-/\n\n/-- carries out a subst if there is one, fails otherwise -/\n/-!\n### Split all conjunctions\n-/\n\n/-- Assumes `pr` is a proof of `t`. Adds the consequences of `t` to the context\n and returns `tt` if anything nontrivial has been added. -/\n/-- return `tt` if any progress is made -/\n/-- return `tt` if any progress is made -/\n/-- fail if no progress is made -/\n/-!\n### Eagerly apply all the preprocessing rules\n-/\n\n/-- Eagerly apply all the preprocessing rules -/\n/-!\n### Terminal tactic\n-/\n\n/--\nThe terminal tactic, used to try to finish off goals:\n- Call the contradiction tactic.\n- Open an SMT state, and use ematching and congruence closure, with all the universal\n  statements in the context.\n\nTODO(Jeremy): allow users to specify attribute for ematching lemmas?\n-/\n/--\n`done` first attempts to close the goal using `contradiction`. If this fails, it creates an\nSMT state and will repeatedly use `ematch` (using `ematch` lemmas in the environment,\nuniversally quantified assumptions, and the supplied lemmas `ps`) and congruence closure.\n-/\n/-!\n### Tactics that perform case splits\n-/\n\ninductive case_option where\n| force : case_option\n| at_most_one : case_option\n| accept : case_option\n\n-- three possible outcomes:\n\n--   finds something to case, the continuations succeed ==> returns tt\n\n--   finds something to case, the continutations fail ==> fails\n\n--   doesn't find anything to case ==> returns ff\n\n/-!\n### The main tactics\n-/\n\n/--\n`safe_core s ps cfg opt` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `s`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `ps`) and congruence closure.\n\n`safe_core` is complete for propositional logic. Depending on the form of `opt`\nit will:\n\n- (if `opt` is `case_option.force`) fail if it does not close the goal,\n- (if `opt` is `case_option.at_most_one`) fail if it produces more than one goal, and\n- (if `opt` is `case_option.accept`) ignore the number of goals it produces.\n-/\n/--\n`clarify` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.at_most_one`.\n-/\n/--\n`safe` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.accept`.\n-/\n/--\n`finish` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.force`.\n-/\nend auto\n\n\n/-! ### interactive versions -/\n\nnamespace tactic\n\n\nnamespace interactive\n\n\n/--\n`clarify [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`clarify` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`clarify` will fail if it produces more than one goal.\n-/\n/--\n`safe [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`safe` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`safe` ignores the number of goals it produces, and should never fail.\n-/\n/--\n`finish [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`finish` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`finish` will fail if it does not close the goal.\n-/\n/--\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/finish_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.39605216110226393}}
{"text": "example (x : Int) (h : x = 2) : Int.div 2 1 = x := by\n  simp [Int.div]\n  trace_state\n  simp [h]\n\nexample (n : Nat) : Int.div (Int.ofNat n) (Int.ofNat 0) = Int.ofNat (n / 0) := by\n  simp [Int.div]\n\nexample (n : Nat) : Int.div (Int.ofNat n) 0 = Int.ofNat (n / 0) := by\n  simp [Int.div]\n\nexample (n : Nat) : Int.mul (Int.ofNat n) (Int.ofNat 0) = Int.ofNat (n * 0) := by\n  simp [Int.mul]\n\nexample (n : Nat) : Int.mul (Int.ofNat n) 0 = Int.ofNat (n * 0) := by\n  simp [Int.mul]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/matchOfNatIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3960521527879958}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\n\nimport algebra.module.pi\nimport data.polynomial.basic\nimport group_theory.group_action.prod\nimport group_theory.group_action.units\nimport data.complex.module\nimport ring_theory.algebraic\nimport data.zmod.basic\nimport ring_theory.tensor_product\n\n/-! # Tests that instances do not form diamonds -/\n\n/-! ## Scalar action instances -/\nsection has_smul\nopen_locale polynomial\n\nexample :\n  (sub_neg_monoid.has_smul_int : has_smul \u2124 \u2102) = (complex.has_smul : has_smul \u2124 \u2102) :=\nrfl\n\nexample : restrict_scalars.module \u211d \u2102 \u2102 = complex.module := rfl\nexample : restrict_scalars.algebra \u211d \u2102 \u2102 = complex.algebra := rfl\n\nexample (\u03b1 \u03b2 : Type*) [add_monoid \u03b1] [add_monoid \u03b2] :\n  (prod.has_smul : has_smul \u2115 (\u03b1 \u00d7 \u03b2)) = add_monoid.has_smul_nat := rfl\n\nexample (\u03b1 \u03b2 : Type*) [sub_neg_monoid \u03b1] [sub_neg_monoid \u03b2] :\n  (prod.has_smul : has_smul \u2124 (\u03b1 \u00d7 \u03b2)) = sub_neg_monoid.has_smul_int := rfl\n\nexample (\u03b1 : Type*) (\u03b2 : \u03b1 \u2192 Type*) [\u03a0 a, add_monoid (\u03b2 a)] :\n  (pi.has_smul : has_smul \u2115 (\u03a0 a, \u03b2 a)) = add_monoid.has_smul_nat := rfl\n\nexample (\u03b1 : Type*) (\u03b2 : \u03b1 \u2192 Type*) [\u03a0 a, sub_neg_monoid (\u03b2 a)] :\n  (pi.has_smul : has_smul \u2124 (\u03a0 a, \u03b2 a)) = sub_neg_monoid.has_smul_int := rfl\n\nnamespace tensor_product\n\nopen_locale tensor_product\nopen complex\n\n/-! The `example` below times out. TODO Fix it!\n\n/- `tensor_product.algebra.module` forms a diamond with `has_mul.to_has_smul` and\n`algebra.tensor_product.tensor_product.semiring`. Given a commutative semiring `A` over a\ncommutative semiring `R`, we get two mathematically different scalar actions of `A \u2297[R] A` on\nitself. -/\ndef f : \u2102 \u2297[\u211d] \u2102 \u2192\u2097[\u211d] \u211d :=\ntensor_product.lift\n{ to_fun    := \u03bb z, z.re \u2022 re_lm,\n  map_add'  := \u03bb z w, by simp [add_smul],\n  map_smul' := \u03bb r z, by simp [mul_smul], }\n\n@[simp] lemma f_apply (z w : \u2102) : f (z \u2297\u209c[\u211d] w) = z.re * w.re := by simp [f]\n\n/- `tensor_product.algebra.module` forms a diamond with `has_mul.to_has_smul` and\n`algebra.tensor_product.tensor_product.semiring`. Given a commutative semiring `A` over a\ncommutative semiring `R`, we get two mathematically different scalar actions of `A \u2297[R] A` on\nitself. -/\nexample :\n  has_mul.to_has_smul (\u2102 \u2297[\u211d] \u2102) \u2260\n  (@tensor_product.algebra.module \u211d \u2102 \u2102 (\u2102 \u2297[\u211d] \u2102) _ _ _ _ _ _ _ _ _ _ _ _).to_has_smul :=\nbegin\n  have contra : I \u2297\u209c[\u211d] I \u2260 (-1) \u2297\u209c[\u211d] 1 := \u03bb c, by simpa using congr_arg f c,\n  contrapose! contra,\n  rw has_smul.ext_iff at contra,\n  replace contra := congr_fun (congr_fun contra (1 \u2297\u209c I)) (I \u2297\u209c 1),\n  rw @tensor_product.algebra.smul_def \u211d \u2102 \u2102 (\u2102 \u2297[\u211d] \u2102) _ _ _ _ _ _ _ _ _ _ _ _\n    (1 : \u2102) I (I \u2297\u209c[\u211d] (1 : \u2102)) at contra,\n  simpa only [algebra.id.smul_eq_mul, algebra.tensor_product.tmul_mul_tmul, one_mul, mul_one,\n    one_smul, tensor_product.smul_tmul', I_mul_I] using contra,\nend\n\n-/\n\nend tensor_product\n\nsection units\n\nexample (\u03b1 : Type*) [monoid \u03b1] :\n  (units.mul_action : mul_action \u03b1\u02e3 (\u03b1 \u00d7 \u03b1)) = prod.mul_action := rfl\n\nexample (R \u03b1 : Type*) (\u03b2 : \u03b1 \u2192 Type*) [monoid R] [\u03a0 i, mul_action R (\u03b2 i)] :\n  (units.mul_action : mul_action R\u02e3 (\u03a0 i, \u03b2 i)) = pi.mul_action _ := rfl\n\nexample (R \u03b1 : Type*) (\u03b2 : \u03b1 \u2192 Type*) [monoid R] [semiring \u03b1] [distrib_mul_action R \u03b1] :\n  (units.distrib_mul_action : distrib_mul_action R\u02e3 \u03b1[X]) =\n    polynomial.distrib_mul_action :=\nrfl\n\n/-!\nTODO: https://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/units.2Emul_action'.20diamond/near/246402813\n```lean\nexample {\u03b1 : Type*} [comm_monoid \u03b1] :\n  (units.mul_action' : mul_action \u03b1\u02e3 \u03b1\u02e3) = monoid.to_mul_action _ :=\nrfl -- fails\n```\n-/\n\nend units\n\nend has_smul\n\n/-! ## `with_top` (Type with point at infinity) instances -/\nsection with_top\n\nexample (R : Type*) [h : strict_ordered_semiring R] :\n  (@with_top.add_comm_monoid R\n    (@non_unital_non_assoc_semiring.to_add_comm_monoid R\n      (@non_assoc_semiring.to_non_unital_non_assoc_semiring R\n        (@semiring.to_non_assoc_semiring R\n          (@strict_ordered_semiring.to_semiring R h)))))\n        =\n  (@ordered_add_comm_monoid.to_add_comm_monoid (with_top R)\n    (@with_top.ordered_add_comm_monoid R\n      (@ordered_cancel_add_comm_monoid.to_ordered_add_comm_monoid R\n        (@strict_ordered_semiring.to_ordered_cancel_add_comm_monoid R h)))) :=\nrfl\n\nend with_top\n\n/-! ## `multiplicative` instances -/\nsection multiplicative\n\nexample :\n  @monoid.to_mul_one_class (multiplicative \u2115) (comm_monoid.to_monoid _) =\n    multiplicative.mul_one_class :=\nrfl\n\n-- `dunfold` can still break unification, but it's better to have `dunfold` break it than have the\n-- above example fail.\nexample :\n  @monoid.to_mul_one_class (multiplicative \u2115) (comm_monoid.to_monoid _) =\n    multiplicative.mul_one_class :=\nbegin\n  dunfold has_one.one multiplicative.mul_one_class,\n  success_if_fail { refl, },\n  ext,\n  refl\nend\n\nend multiplicative\n\n/-! ## `finsupp` instances-/\n\nsection finsupp\nopen finsupp\n\n/-- `finsupp.comap_has_smul` can form a non-equal diamond with `finsupp.smul_zero_class` -/\nexample {k : Type*} [semiring k] [nontrivial k] :\n  (finsupp.comap_has_smul : has_smul k (k \u2192\u2080 k)) \u2260 finsupp.smul_zero_class.to_has_smul :=\nbegin\n  obtain \u27e8u : k, hu\u27e9 := exists_ne (1 : k),\n  intro h,\n  simp only [has_smul.ext_iff, function.funext_iff, finsupp.ext_iff] at h,\n  replace h := h u (finsupp.single 1 1) u,\n  classical,\n  rw [comap_smul_single, smul_apply, smul_eq_mul, mul_one, single_eq_same,\n    smul_eq_mul, single_eq_of_ne hu.symm, mul_zero] at h,\n  exact one_ne_zero h,\nend\n\n/-- `finsupp.comap_has_smul` can form a non-equal diamond with `finsupp.smul_zero_class` even when\nthe domain is a group. -/\nexample {k : Type*} [semiring k] [nontrivial k\u02e3] :\n  (finsupp.comap_has_smul : has_smul k\u02e3 (k\u02e3 \u2192\u2080 k)) \u2260 finsupp.smul_zero_class.to_has_smul :=\nbegin\n  obtain \u27e8u : k\u02e3, hu\u27e9 := exists_ne (1 : k\u02e3),\n  haveI : nontrivial k := \u27e8\u27e8u, 1, units.ext.ne hu\u27e9\u27e9,\n  intro h,\n  simp only [has_smul.ext_iff, function.funext_iff, finsupp.ext_iff] at h,\n  replace h := h u (finsupp.single 1 1) u,\n  classical,\n  rw [comap_smul_single, smul_apply, units.smul_def, smul_eq_mul, mul_one, single_eq_same,\n    smul_eq_mul, single_eq_of_ne hu.symm, mul_zero] at h,\n  exact one_ne_zero h,\nend\n\nend finsupp\n\n/-! ## `polynomial` instances -/\nsection polynomial\n\nvariables (R A : Type*)\nopen_locale polynomial\nopen polynomial\n\n/-- `polynomial.has_smul_pi` forms a diamond with `pi.has_smul`. -/\nexample [semiring R] [nontrivial R] :\n  polynomial.has_smul_pi _ _ \u2260 (pi.has_smul : has_smul R[X] (R \u2192 R[X])) :=\nbegin\n  intro h,\n  simp_rw [has_smul.ext_iff, function.funext_iff, polynomial.ext_iff] at h,\n  simpa using h X 1 1 0,\nend\n\n/-- `polynomial.has_smul_pi'` forms a diamond with `pi.has_smul`. -/\nexample [comm_semiring R] [nontrivial R] :\n  polynomial.has_smul_pi' _ _ _ \u2260 (pi.has_smul : has_smul R[X] (R \u2192 R[X])) :=\nbegin\n  intro h,\n  simp_rw [has_smul.ext_iff, function.funext_iff, polynomial.ext_iff] at h,\n  simpa using h X 1 1 0,\nend\n\n/-- `polynomial.has_smul_pi'` is consistent with `polynomial.has_smul_pi`. -/\nexample [comm_semiring R] [nontrivial R] :\n  polynomial.has_smul_pi' _ _ _ = (polynomial.has_smul_pi _ _ : has_smul R[X] (R \u2192 R[X])) :=\nrfl\n\n/-- `polynomial.algebra_of_algebra` is consistent with `algebra_nat`. -/\nexample [semiring R] : (polynomial.algebra_of_algebra : algebra \u2115 R[X]) = algebra_nat := rfl\n\n/-- `polynomial.algebra_of_algebra` is consistent with `algebra_int`. -/\nexample [ring R] : (polynomial.algebra_of_algebra : algebra \u2124 R[X]) = algebra_int _ := rfl\n\nend polynomial\n\n/-! ## `subtype` instances -/\nsection subtype\n\n-- this diamond is the reason that `fintype.to_locally_finite_order` is not an instance\nexample {\u03b1} [preorder \u03b1] [locally_finite_order \u03b1] [fintype \u03b1] [@decidable_rel \u03b1 (<)]\n  [@decidable_rel \u03b1 (\u2264)] (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  subtype.locally_finite_order p = fintype.to_locally_finite_order :=\nbegin\n  success_if_fail { refl, },\n  exact subsingleton.elim _ _\nend\n\nend subtype\n\n/-! ## `zmod` instances -/\nsection zmod\n\nvariables {p : \u2115} [fact p.prime]\n\nexample : @euclidean_domain.to_comm_ring _ (@field.to_euclidean_domain _ (zmod.field p)) =\n  zmod.comm_ring p :=\nrfl\n\nexample (n : \u2115) : zmod.comm_ring (n + 1) = fin.comm_ring (n + 1) := rfl\nexample : zmod.comm_ring 0 = int.comm_ring := rfl\n\nend zmod\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/instance_diamonds.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3960521527879958}}
{"text": "import .logic\n\nclass atom_type (\u03b1 \u03b2 : Type) :=\n(val : list \u03b2 \u2192 \u03b1 \u2192 Prop)\n(neg : \u03b1 \u2192 fm \u03b1) \n(neg_nqfree : \u2200 (a : \u03b1), nqfree (neg a))\n(neg_prsv : \u2200 (a : \u03b1) (xs : list \u03b2), (interp val xs (neg a)) \u2194 (interp val xs (\u00ac' A' a)))\n(dep0 : \u03b1 \u2192 Prop)\n(dec_dep0 : decidable_pred dep0)\n(decr : \u03b1 \u2192 \u03b1)\n(decr_prsv : \u2200 {a : \u03b1} {hd : \u00ac (dep0 a)} {b : \u03b2} {bs : list \u03b2}, (val bs (decr a) \u2194 val (b::bs) a))\n(inh : \u03b2)\n(dec_eq : decidable_eq \u03b1)\n(normal : \u03b1 \u2192 Prop)\n(dec_normal : decidable_pred normal)\n(neg_prsv_normal : \u2200 a, (normal a \u2192 \u2200 a' \u2208 (atoms (neg a)), normal a'))\n(decr_prsv_normal : \u2200 a, normal a \u2192 \u00ac dep0 a \u2192 normal (decr a))\n\nclass atom_eq_type (\u03b1 \u03b2 : Type) extends atom_type \u03b1 \u03b2 :=\n(solv0 : \u03b1 \u2192 Prop)\n(dec_solv0 : decidable_pred solv0)\n(dest_solv0 : \u2200 a, solv0 a \u2192 nat)\n(solv0_eq : \u2200 {e : \u03b1} (He : solv0 e) {b} {bs}, val (b::bs) e\n  \u2192 list.nth_dft (atom_type.inh \u03b1 \u03b2) (b::bs) (dest_solv0 e He) = b)\n(trivial : \u03b1 \u2192 Prop)\n(dec_triv : decidable_pred trivial)\n(true_triv : \u2200 a, trivial a \u2192 \u2200 xs, val xs a)\n(subst0 : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\n(true_subst : \u2200 e, solv0 e \u2192 \u2200 bs, val bs (subst0 e e))\n(subst_prsv : \u2200 {e : \u03b1} (He : solv0 e), \u2200 {a : \u03b1} {bs : list \u03b2}, \n  val bs (subst0 e a) \u2194 val ((list.nth_dft (atom_type.inh \u03b1 \u03b2) bs (dest_solv0 e He - 1))::bs) a)\n(dest_pos : \u2200 {a} {Ha : solv0 a}, \u00ac trivial a \u2192 dest_solv0 a Ha > 0)\n\n-- subst_eqn i j k returns the result of taking an \n-- identity atom of form (i = j) and using it to \n-- substitute a de Brujin variable k.\n-- Requires : j = 0 \u2194 \u00ac(i = 0)\ndef subst_eqn : nat \u2192 nat \u2192 nat \u2192 nat \n| 0 j 0 := j - 1\n| (i+1) _ 0 := i  \n| _ _ k := k - 1 \n\ndef isubst (k) : nat \u2192 nat \n| 0 := k\n| (i + 1) := i\n\n-- Qstn : Why do I need this? (Why doesn't unfold work?)\nlemma exp_subst_eqn_i0 (i) : subst_eqn (i+1) 0 0 = i := \nbegin refl end\n\nvariables {\u03b1 \u03b2 : Type}\n\ndef I [atom_type \u03b1 \u03b2] (p : fm \u03b1) (xs : list \u03b2) := interp (atom_type.val) xs p \n\nlemma exp_I [atom_type \u03b1 \u03b2] {p : fm \u03b1} {xs : list \u03b2} :    \n  I p xs = interp (atom_type.val) xs p := refl _\n\nlemma exp_I_and [atom_type \u03b1 \u03b2] (p q : fm \u03b1) (xs : list \u03b2) : \n  I (p \u2227' q) xs = ((I p xs) \u2227 (I q xs)) := eq.refl _\n\nlemma exp_I_and_o [atom_type \u03b1 \u03b2] (p q : fm \u03b1) (xs : list \u03b2) : \n  I (and_o p q) xs = ((I p xs) \u2227 (I q xs)) := \nbegin\n  apply (cases_and_o' (\u03bb p q pq, ((I pq xs) = ((I p xs) \u2227 (I q xs)))) p q), \n  repeat {unfold I, unfold interp, simp},\n  unfold I, unfold interp \nend\n\nlemma exp_I_or [atom_type \u03b1 \u03b2] (p q : fm \u03b1) (xs : list \u03b2) : \n  I (p \u2228' q) xs = ((I p xs) \u2228 (I q xs)) := eq.refl _\n\nlemma exp_I_or_o [atom_type \u03b1 \u03b2] (p q : fm \u03b1) (xs : list \u03b2) : \n  I (or_o p q) xs \u2194 ((I p xs) \u2228 (I q xs)) := \nbegin\n  apply (cases_or_o' (\u03bb p q pq, ((I pq xs) \u2194 ((I p xs) \u2228 (I q xs)))) p q), \n  repeat {unfold I, unfold interp, simp},\n  unfold I, unfold interp \nend\n\nlemma exp_I_not [atom_type \u03b1 \u03b2] (p : fm \u03b1) (xs : list \u03b2) : \n  I (\u00ac' p) xs = \u00ac (I p xs) := eq.refl _\n\nlemma exp_I_not_o [atom_type \u03b1 \u03b2] (p : fm \u03b1) :\n  \u2200 (xs : list \u03b2), I (not_o p) xs \u2194 \u00ac (I p xs) :=\nby cases p; {unfold not_o, unfold I, unfold interp, try {simp}}\n\nlemma exp_I_ex [atom_type \u03b1 \u03b2] (p : fm \u03b1) (xs) : @I \u03b1 \u03b2 _ (\u2203' p) xs = \u2203 x, (I p (x::xs)) := \nby unfold I; unfold interp\n\nlemma exp_I_top [atom_type \u03b1 \u03b2] (xs) : @I \u03b1 \u03b2 _ \u22a4' xs = true := \nby unfold I; unfold interp\n\nlemma exp_I_bot [atom_type \u03b1 \u03b2] (xs) : @I \u03b1 \u03b2 _ \u22a5' xs = false := \nby unfold I; unfold interp\n\nlemma I_not_dep0 [atom_type \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) (bs : list \u03b2) :\n  \u00ac atom_type.dep0 \u03b2 a \n  \u2192 (I (A' a) (b::bs) \u2194 I (A' (atom_type.decr \u03b2 a)) bs) :=\nbegin\n  intro h, unfold I, unfold interp,\n  rewrite atom_type.decr_prsv, apply h\nend\n\ndef fnormal_alt (\u03b2) [atom_type \u03b1 \u03b2] (p : fm \u03b1) := \n  \u2200 a \u2208 (@atoms _ (atom_type.dec_eq _ \u03b2) p), atom_type.normal \u03b2 a\n\ndef fnormal (\u03b2) [atom_type \u03b1 \u03b2] : fm \u03b1 \u2192 Prop \n| \u22a4' := true\n| \u22a5' := true\n| (A' a) := atom_type.normal \u03b2 a \n| (p \u2227' q) := fnormal p \u2227 fnormal q\n| (p \u2228' q) := fnormal p \u2227 fnormal q\n| (\u00ac' p) := fnormal p\n| (\u2203' p) := fnormal p\n\ninstance dec_fnormal (\u03b2) [atom_type \u03b1 \u03b2] : decidable_pred (@fnormal \u03b1 \u03b2 _) \n| \u22a4' := decidable.is_true trivial \n| \u22a5' := decidable.is_true trivial \n| (A' a) := \n  begin \n    unfold fnormal,\n    apply atom_type.dec_normal \u03b1 \u03b2\n  end\n| (p \u2227' q) := \n  begin\n    unfold fnormal, \n    apply @and.decidable _ _ _ _;\n    apply dec_fnormal\n  end\n| (p \u2228' q) :=\n  begin\n    unfold fnormal, \n    apply @and.decidable _ _ _ _;\n    apply dec_fnormal\n  end\n| (\u00ac' p) := \n  begin\n    unfold fnormal, \n    apply dec_fnormal\n  end\n| (\u2203' p) := \n  begin\n    unfold fnormal, \n    apply dec_fnormal\n  end\n\nlemma fnormal_iff_fnormal_alt [atom_type \u03b1 \u03b2] : \n  \u2200 {p : fm \u03b1}, fnormal \u03b2 p \u2194 fnormal_alt \u03b2 p \n| \u22a4' := true_iff_true trivial (\u03bb a ha, by cases ha)\n| \u22a5' := true_iff_true trivial (\u03bb a ha, by cases ha)\n| (A' a) := \n  begin \n    unfold fnormal, \n    unfold fnormal_alt, unfold atoms,\n    apply iff.intro; intro h, \n    intros a' ha', cases ha' with he he,\n    subst he, apply h, cases he, \n    apply h, apply or.inl rfl\n  end\n| (p \u2227' q) := \n  begin \n    unfold fnormal, \n    repeat {rewrite fnormal_iff_fnormal_alt}, \n    apply iff.symm, apply list.forall_mem_union\n  end\n| (p \u2228' q) := \n  begin \n    unfold fnormal, \n    repeat {rewrite fnormal_iff_fnormal_alt}, \n    apply iff.symm, apply list.forall_mem_union\n  end\n| (\u00ac' p) := \n  begin \n    unfold fnormal, \n    rewrite fnormal_iff_fnormal_alt, refl\n  end\n| (\u2203' p) := \n  begin \n    unfold fnormal, \n    rewrite fnormal_iff_fnormal_alt, refl\n  end\n\ndef disj_to_prop (\u03b2) [atom_type \u03b1 \u03b2] (as : list \u03b1) (bs : list \u03b2) : Prop :=\n  \u2200 a \u2208 as, atom_type.val bs a\n\ninstance atoms_dec_eq [atom_type \u03b1 \u03b2] : decidable_eq \u03b1 := \natom_type.dec_eq \u03b1 \u03b2\n\ninstance atoms_dec_dep0 [atom_type \u03b1 \u03b2] : decidable_pred (atom_type.dep0 \u03b2) := \natom_type.dec_dep0 \u03b1 \u03b2\n\ndef atoms_dep0 (\u03b2) [atom_type \u03b1 \u03b2] (p : fm \u03b1) := \nlist.filter (atom_type.dep0 \u03b2) (atoms p)", "meta": {"author": "avigad", "repo": "qelim", "sha": "b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60", "save_path": "github-repos/lean/avigad-qelim", "path": "github-repos/lean/avigad-qelim/qelim-b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60/common/atom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529716, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.39605214447372755}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Jesse Michael Han\n-/\nimport tactic.rcases\nimport logic.function.basic\n\nuniverses u\u2081 u\u2082\n\nopen interactive interactive.types\nsection ext\nopen lean.parser nat tactic\n\ndeclare_trace ext\n\n/--\n`derive_struct_ext_lemma n` generates two extensionality lemmas based on\nthe equality of all non-propositional projections.\n\nOn the following:\n\n```lean\n@[ext]\nstructure foo (\u03b1 : Type*) :=\n(x y : \u2115)\n(z : {z // z < x})\n(k : \u03b1)\n(h : x < y)\n```\n\n`derive_struct_lemma` generates:\n\n```lean\nlemma foo.ext : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\n  x.x = y.x \u2192 x.y = y.y \u2192 x.z == y.z \u2192 x.k = y.k \u2192 x = y\nlemma foo.ext_iff : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\n  x = y \u2194 x.x = y.x \u2227 x.y = y.y \u2227 x.z == y.z \u2227 x.k = y.k\n```\n\n-/\nmeta def derive_struct_ext_lemma (n : name) : tactic name :=\ndo e \u2190 get_env,\n   fs \u2190 e.structure_fields n,\n   d \u2190 get_decl n,\n   n \u2190 resolve_constant n,\n   let r := @expr.const tt n $ d.univ_params.map level.param,\n   (args,_) \u2190 infer_type r >>= open_pis,\n   let args := args.map expr.to_implicit_local_const,\n   let t := r.mk_app args,\n   x \u2190 mk_local_def `x t,\n   y \u2190 mk_local_def `y t,\n   let args_x := args ++ [x],\n   let args_y := args ++ [y],\n   bs \u2190 fs.mmap $ \u03bb f,\n     do { d \u2190 get_decl (n ++ f),\n          let a := @expr.const tt (n ++ f) $ d.univ_params.map level.param,\n          t \u2190 infer_type a,\n          s \u2190 infer_type t,\n\n          if s \u2260 `(Prop)\n            then do\n              let x := a.mk_app args_x,\n              let y := a.mk_app args_y,\n              t \u2190 infer_type x,\n              t' \u2190 infer_type y,\n              some <$> if t = t'\n                then mk_app `eq [x,y] >>= mk_local_def `h\n                else mk_mapp `heq [none,x,none,y] >>= mk_local_def `h\n            else pure none },\n   let bs := bs.filter_map id,\n   eq_t \u2190 mk_app `eq [x,y],\n   t \u2190 pis (args ++ [x,y] ++ bs) eq_t,\n   pr \u2190 run_async $\n     do { (_,pr) \u2190 solve_aux t (do\n          { args \u2190 intron args.length,\n            x \u2190 intro1, y \u2190 intro1,\n            cases x, cases y,\n            bs.mmap' (\u03bb _,\n              do e \u2190 intro1,\n                 cases e),\n            reflexivity }),\n          instantiate_mvars pr },\n   let decl_n := n <.> \"ext\",\n   add_decl (declaration.thm decl_n d.univ_params t pr),\n   bs \u2190 bs.mmap infer_type,\n   let rhs := expr.mk_and_lst bs,\n   iff_t \u2190 mk_app `iff [eq_t,rhs],\n   t \u2190 pis (args ++ [x,y]) iff_t,\n   pr \u2190 run_async $\n     do { (_,pr) \u2190 solve_aux t $ do\n          { args \u2190 intron args.length,\n            x \u2190 intro1, y \u2190 intro1,\n            cases x, cases y,\n            split,\n            solve1 $ do\n            { h \u2190 intro1, hs \u2190 injection h, subst_vars,\n              repeat (refine ``( and.intro _ _ ) >> reflexivity ),\n              done <|> reflexivity },\n            solve1 $ do\n            { repeat (do refine ``(and_imp.mpr _),\n                         h \u2190 intro1, cases h, skip ),\n              h \u2190 intro1, cases h,\n              reflexivity } },\n          instantiate_mvars pr },\n   add_decl (declaration.thm (n <.> \"ext_iff\") d.univ_params t pr),\n   pure decl_n\n\nmeta def get_ext_subject : expr \u2192 tactic name\n| (expr.pi n bi d b) :=\n  do v  \u2190 mk_local' n bi d,\n     b' \u2190 whnf $ b.instantiate_var v,\n     get_ext_subject b'\n| (expr.app _ e) :=\n  do t \u2190 infer_type e >>= instantiate_mvars >>= head_beta,\n     if t.get_app_fn.is_constant then\n       pure $ t.get_app_fn.const_name\n     else if t.is_pi then\n       pure $ name.mk_numeral 0 name.anonymous\n     else if t.is_sort then\n       pure $ name.mk_numeral 1 name.anonymous\n     else do\n       t \u2190 pp t,\n       fail format!\"only constants and Pi types are supported: {t}\"\n| e := fail format!\"Only expressions of the form `_ \u2192 _ \u2192 ... \u2192 R ... e are supported: {e}\"\n\nopen native\n\nmeta def saturate_fun : name \u2192 tactic expr\n| (name.mk_numeral 0 name.anonymous) :=\ndo v\u2080 \u2190 mk_mvar,\n   v\u2081 \u2190 mk_mvar,\n   return $ v\u2080.imp v\u2081\n| (name.mk_numeral 1 name.anonymous) :=\ndo u \u2190 mk_meta_univ,\n   pure $ expr.sort u\n| n :=\ndo e \u2190 resolve_constant n >>= mk_const,\n   a \u2190 get_arity e,\n   e.mk_app <$> (list.iota a).mmap (\u03bb _, mk_mvar)\n\nmeta def equiv_type_constr (n n' : name) : tactic unit :=\ndo e  \u2190 saturate_fun n,\n   e' \u2190 saturate_fun n',\n   unify e e' <|> fail format!\"{n} and {n'} are not definitionally equal types\"\n\nsection performance_hack\n/--\nFor performance reasons, it is inadvisable to use `user_attribute.get_param`.\nThe parameter is stored as a reflected expression.  When calling `get_param`,\nthe stored parameter is evaluated using `eval_expr`, which first compiles the\nexpression into VM bytecode. The unevaluated expression is available using\n`user_attribute.get_param_untyped`.\n\nIn particular, `user_attribute.get_param` MUST NEVER BE USED in the\nimplementation of an attribute cache. This is because calling `eval_expr`\ndisables the attribute cache.\n\nThere are several possible workarounds:\n 1. Set a different attribute depending on the parameter.\n 2. Use your own evaluation function instead of `eval_expr`, such as e.g. `expr.to_nat`.\n 3. Write your own `has_reflect Param` instance (using a more efficient serialization format).\n   The `user_attribute` code unfortunately checks whether the expression has the correct type,\n   but you can use `` `(id %%e : Param) `` to pretend that your expression `e` has type `Param`.\n-/\nlibrary_note \"user attribute parameters\"\n\n/-!\nFor performance reasons, the parameters of the `@[ext]` attribute are stored\nin two auxiliary attributes:\n```lean\nattribute [ext thunk] funext\n\n-- is turned into\nattribute [_ext_core (@id name @funext)] thunk\nattribute [_ext_lemma_core] funext\n```\n\nsee Note [user attribute parameters]\n-/\n\nlocal attribute [semireducible] reflected\n\nlocal attribute [instance, priority 9000]\nprivate meta def hacky_name_reflect : has_reflect name :=\n\u03bb n, `(id %%(expr.const n []) : name)\n\n@[user_attribute]\nprivate meta def ext_attr_core : user_attribute (name_map name) name :=\n{ name := `_ext_core,\n  descr := \"(internal attribute used by ext)\",\n  cache_cfg :=\n  { dependencies := [],\n    mk_cache := \u03bb ns, ns.mfoldl (\u03bb m n, do\n      ext_l \u2190 ext_attr_core.get_param_untyped n,\n      pure (m.insert n ext_l.app_arg.const_name)) mk_name_map },\n  parser := failure }\n\nend performance_hack\n\n/-- Private attribute used to tag extensionality lemmas. -/\n@[user_attribute]\nprivate meta def ext_lemma_attr_core : user_attribute :=\n{ name := `_ext_lemma_core,\n  descr := \"(internal attribute used by ext)\",\n  parser := failure }\n\n/--\nReturns the extensionality lemmas in the environment, as a map from structure\nname to lemma name.\n-/\nmeta def get_ext_lemmas : tactic (name_map name) :=\next_attr_core.get_cache\n\n/--\nReturns the extensionality lemmas in the environment, as a list of lemma names.\n-/\nmeta def get_ext_lemma_names : tactic (list name) :=\nattribute.get_instances ext_lemma_attr_core.name\n\n/-- Marks `lem` as an extensionality lemma corresponding to type constructor `constr`;\nif `persistent` is true then this is a global attribute, else local. -/\nmeta def add_ext_lemma (constr lem : name) (persistent : bool) : tactic unit :=\next_attr_core.set constr lem persistent >> ext_lemma_attr_core.set lem () persistent\n\n/--\nTag lemmas of the form:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe attribute indexes extensionality lemma using the type of the\nobjects (i.e. `my_collection`) which it gets from the statement of\nthe lemma.  In some cases, the same lemma can be used to state the\nextensionality of multiple types that are definitionally equivalent.\n\n```lean\nattribute [ext thunk, ext stream] funext\n```\n\nAlso, the following:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nis equivalent to\n\n```lean\n@[ext my_collection]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThis allows us specify type synonyms along with the type\nthat is referred to in the lemma statement.\n\n```lean\n@[ext, ext my_type_synonym]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe `ext` attribute can be applied to a structure to generate its extensionality lemmas:\n\n```lean\n@[ext]\nstructure foo (\u03b1 : Type*) :=\n(x y : \u2115)\n(z : {z // z < x})\n(k : \u03b1)\n(h : x < y)\n```\n\nwill generate:\n\n```lean\n@[ext] lemma foo.ext : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\nx.x = y.x \u2192 x.y = y.y \u2192 x.z == y.z \u2192 x.k = y.k \u2192 x = y\nlemma foo.ext_iff : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\nx = y \u2194 x.x = y.x \u2227 x.y = y.y \u2227 x.z == y.z \u2227 x.k = y.k\n```\n\n-/\n@[user_attribute]\nmeta def extensional_attribute : user_attribute unit (option name) :=\n{ name := `ext,\n  descr := \"lemmas usable by `ext` tactic\",\n  parser := optional ident,\n  after_set := some $ \u03bb n _ b, do\n    add \u2190 extensional_attribute.get_param n,\n    e \u2190 get_env,\n    n \u2190 if (e.structure_fields n).is_some\n      then derive_struct_ext_lemma n\n      else pure n,\n    s \u2190 mk_const n >>= infer_type >>= get_ext_subject,\n    match add with\n    | none := add_ext_lemma s n b\n    | some add := equiv_type_constr s add >> add_ext_lemma add n b\n    end }\n\nadd_tactic_doc\n{ name                     := \"ext\",\n  category                 := doc_category.attr,\n  decl_names               := [`extensional_attribute],\n  tags                     := [\"rewrite\", \"logic\"] }\n\n/--\nWhen possible, `ext` lemmas are stated without a full set of arguments. As an example, for bundled\nhoms `f`, `g`, and `of`, `f.comp of = g.comp of \u2192 f = g` is a better `ext` lemma than\n`(\u2200 x, f (of x) = g (of x)) \u2192 f = g`, as the former allows a second type-specific extensionality\nlemmas to be applied to `f.comp of = g.comp of`.\nIf the domain of `of` is `\u2115` or `\u2124` and `of` is a `ring_hom`, such a lemma could then make the goal\n`f (of 1) = g (of 1)`.\n\nFor bundled morphisms, there is a `ext` lemma that always applies of the form\n`(\u2200 x, \u21d1f x = \u21d1g x) \u2192 f = g`. When adding type-specific `ext` lemmas like the one above, we want\nthese to be tried first. This happens automatically since the type-specific lemmas are inevitably\ndefined later.\n-/\nlibrary_note \"partially-applied ext lemmas\"\n\n-- We mark some existing extensionality lemmas.\nattribute [ext] array.ext propext function.hfunext\nattribute [ext thunk] _root_.funext\n\n-- This line is equivalent to:\n--   attribute [ext (\u2192)] _root_.funext\n-- but (\u2192) is not actually a binary relation with a constant at the head,\n-- so we use the special name [anon].0 to represent (\u2192).\nrun_cmd add_ext_lemma (name.mk_numeral 0 name.anonymous) ``_root_.funext tt\n\n-- We create some extensionality lemmas for existing structures.\nattribute [ext] ulift\n\nnamespace plift\n-- This is stronger than the one generated automatically.\n@[ext] lemma ext {P : Prop} (a b : plift P) : a = b :=\nbegin\n  cases a, cases b, refl\nend\nend plift\n\n-- Conservatively, we'll only add extensionality lemmas for `has_*` structures\n-- as they become useful.\nattribute [ext] has_zero\n\n@[ext] lemma unit.ext {x y : unit} : x = y := by { cases x, cases y, refl, }\n@[ext] lemma punit.ext {x y : punit} : x = y := by { cases x, cases y, refl, }\n\nnamespace tactic\n\n/-- Helper structure for `ext` and `ext1`. `lemmas` keeps track of extensionality lemmas\n  applied so far. -/\nmeta structure ext_state : Type :=\n(patts : list rcases_patt := [])\n(trace_msg : list string := [])\n(fuel : option \u2115 := none)\n\n/-- Helper function for `try_intros`. Additionally populates the `trace_msg` field\n  of `ext_state`. -/\nprivate meta def try_intros_core : state_t ext_state tactic unit :=\ndo \u27e8patts, trace_msg, fuel\u27e9 \u2190 get,\n   match patts with\n   | [] := do { es \u2190 state_t.lift intros, when (es.length > 0) $ do\n                let msg := \"intros \" ++ (\" \".intercalate (es.map (\u03bb e, e.local_pp_name.to_string))),\n                modify (\u03bb \u27e8patts, trace_msg, fuel\u27e9, \u27e8patts, trace_msg ++ [msg], fuel\u27e9) }\n             <|> pure ()\n   | (x::xs) :=\n     do tgt \u2190 state_t.lift (target >>= whnf),\n        when tgt.is_pi $\n          do state_t.lift (rintro [x]),\n             msg \u2190 state_t.lift (((++) \"rintro \") <$> format.to_string <$> x.format ff),\n             modify (\u03bb \u27e8_, trace_msg, fuel\u27e9, \u27e8xs, trace_msg ++ [msg], fuel\u27e9),\n             try_intros_core\n   end\n\n/-- Try to introduce as many arguments as possible, using the given patterns to destruct the\n  introduced variables. Returns the unused patterns. -/\nmeta def try_intros (patts : list rcases_patt) : tactic (list rcases_patt) :=\nlet \u03c3 := ext_state.mk patts [] none in\n  (ext_state.patts \u2218 prod.snd) <$> state_t.run try_intros_core \u03c3\n\n/-- Apply one extensionality lemma, and destruct the arguments using the patterns\n  in the ext_state. -/\nmeta def ext1_core (cfg : apply_cfg := {}) : state_t ext_state tactic unit :=\ndo \u27e8patts, trace_msg, _\u27e9 \u2190 get,\n   (new_msgs) \u2190 state_t.lift $ focus1 $\n   do { m \u2190 get_ext_lemmas,\n         tgt \u2190 target,\n         when_tracing `ext $ trace!\"[ext] goal: {tgt}\",\n         subject \u2190 get_ext_subject tgt,\n         new_trace_msg \u2190\n           do { rule \u2190 (m.find subject),\n                if is_trace_enabled_for `ext then\n                  trace!\"[ext] matched goal to rule: {rule}\" >>\n                  timetac \"[ext] application attempt time\" (applyc rule cfg)\n                else applyc rule cfg,\n                pure ([\"apply \" ++ rule.to_string]) } <|>\n             do { ls \u2190 get_ext_lemma_names,\n                  let nms := ls.map name.to_string,\n                  rule \u2190 (ls.any_of (\u03bb n,\n                    (if is_trace_enabled_for `ext then\n                      trace!\"[ext] trying to apply ext lemma: {n}\" >>\n                      timetac \"[ext] application attempt time\" (applyc n cfg)\n                    else applyc n cfg) *> pure n)),\n                  pure ([\"apply \" ++ rule.to_string]) } <|>\n               (fail format!\"no applicable extensionality rule found for {subject}\"),\n         pure new_trace_msg },\n    modify (\u03bb \u27e8patts, trace_msg, fuel\u27e9, \u27e8patts, trace_msg ++ new_msgs, fuel\u27e9),\n    try_intros_core\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns. -/\nmeta def ext_core (cfg : apply_cfg := {}) : state_t ext_state tactic unit :=\ndo acc@\u27e8_, _, fuel\u27e9 \u2190 get,\n   match fuel with\n   | (some 0) := pure ()\n   | n        := do { ext1_core cfg,\n                      modify (\u03bb \u27e8patts, lemmas, _\u27e9, \u27e8patts, lemmas, nat.pred <$> n\u27e9),\n                      ext_core <|> pure () }\n   end\n\n/-- Apply one extensionality lemma, and destruct the arguments using the given patterns.\n  Returns the unused patterns. -/\nmeta def ext1 (xs : list rcases_patt) (cfg : apply_cfg := {})\n  (trace : bool := ff) : tactic (list rcases_patt) :=\ndo \u27e8_, \u03c3\u27e9 \u2190 state_t.run (ext1_core cfg) {patts := xs},\n   when trace $ tactic.trace $ \"Try this: \" ++  \", \".intercalate \u03c3.trace_msg,\n   pure \u03c3.patts\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns.\n  `ext ps (some n)` applies at most `n` extensionality lemmas. Returns the unused patterns. -/\nmeta def ext (xs : list rcases_patt) (fuel : option \u2115) (cfg : apply_cfg := {})\n  (trace : bool := ff) : tactic (list rcases_patt) :=\ndo \u27e8_, \u03c3\u27e9 \u2190 state_t.run (ext_core cfg) {patts := xs, fuel := fuel},\n   when trace $ tactic.trace $ \"Try this: \" ++  \", \".intercalate \u03c3.trace_msg,\n   pure \u03c3.patts\n\nlocal postfix (name := parser.optional) `?`:9001 := optional\nlocal postfix (name := parser.many) *:9001 := many\n\n/--\n`ext1 id` selects and apply one extensionality lemma (with attribute\n`ext`), using `id`, if provided, to name a local constant\nintroduced by the lemma. If `id` is omitted, the local constant is\nnamed automatically, as per `intro`. Placing a `?` after `ext1`\n (e.g. `ext1? i \u27e8a,b\u27e9 : 3`) will display a sequence of tactic\napplications that can replace the call to `ext1`.\n-/\nmeta def interactive.ext1 (trace : parse (tk \"?\")?)\n  (xs : parse rcases_patt_parse_hi*) : tactic unit :=\next1 xs {} trace.is_some $> ()\n\n/--\n- `ext` applies as many extensionality lemmas as possible;\n- `ext ids`, with `ids` a list of identifiers, finds extentionality and applies them\n  until it runs out of identifiers in `ids` to name the local constants.\n- `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext` (e.g. `ext? i \u27e8a,b\u27e9 : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`.\n- `set_option trace.ext true` will trace every attempted lemma application,\n  along with the time it takes for the application to succeed or fail.\n  This is useful for debugging slow `ext` calls.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2\n\u22a2 f = g\n```\n\napplying `ext x y` yields:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2,\nx : \u03b1,\ny : \u03b2\n\u22a2 y \u2208 f x \u2194 y \u2208 f x\n```\n\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 f = g\n```\n\napplying `ext \u27e8a, b\u27e9` yields:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type,\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3,\na : \u03b1,\nb : \u03b2\n\u22a2 f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? \u27e8a,b\u27e9` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro \u27e8a, b\u27e9\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nmeta def interactive.ext :\n  (parse $ (tk \"?\")?) \u2192 parse rintro_patt_parse_hi* \u2192 parse (tk \":\" *> small_nat)? \u2192 tactic unit\n | trace [] (some n)  := iterate_range 1 n (ext1 [] {} trace.is_some $> ())\n | trace [] none      := repeat1 (ext1 [] {} trace.is_some $> ())\n | trace xs n         := ext xs.join n {} trace.is_some $> ()\n\n/--\n* `ext1 id` selects and apply one extensionality lemma (with\n  attribute `ext`), using `id`, if provided, to name a\n  local constant introduced by the lemma. If `id` is omitted, the\n  local constant is named automatically, as per `intro`.\n\n* `ext` applies as many extensionality lemmas as possible;\n* `ext ids`, with `ids` a list of identifiers, finds extensionality lemmas\n  and applies them until it runs out of identifiers in `ids` to name\n  the local constants.\n* `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext`/`ext1` (e.g. `ext? i \u27e8a,b\u27e9 : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`/`ext1`.\n- `set_option trace.ext true` will trace every attempted lemma application,\n  along with the time it takes for the application to succeed or fail.\n  This is useful for debugging slow `ext` calls.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2\n\u22a2 f = g\n```\n\napplying `ext x y` yields:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2,\nx : \u03b1,\ny : \u03b2\n\u22a2 y \u2208 f x \u2194 y \u2208 g x\n```\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 f = g\n```\n\napplying `ext \u27e8a, b\u27e9` yields:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type,\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3,\na : \u03b1,\nb : \u03b2\n\u22a2 f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? \u27e8a,b\u27e9` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro \u27e8a, b\u27e9\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nadd_tactic_doc\n{ name        := \"ext1 / ext\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.ext1, `tactic.interactive.ext],\n  tags        := [\"rewriting\", \"logic\"] }\n\nend tactic\nend ext\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3960521444737275}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module data.rbmap.default\n! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Rbmap.Basic\nimport Mathbin.Data.Rbtree.Main\n\nuniverse u v\n\nnamespace Rbmap\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop}\n\n-- Auxiliary instances\nprivate def rbmap_lt_is_swo {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [IsStrictWeakOrder \u03b1 lt] :\n    IsStrictWeakOrder (\u03b1 \u00d7 \u03b2) (RbmapLt lt)\n    where\n  irrefl _ := irrefl_of lt _\n  trans _ _ _ h\u2081 h\u2082 := trans_of lt h\u2081 h\u2082\n  incomp_trans _ _ _ h\u2081 h\u2082 := incomp_trans_of lt h\u2081 h\u2082\n#align rbmap.rbmap_lt_is_swo rbmap.rbmap_lt_is_swo\n\nprivate def rbmapLtDec {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [h : DecidableRel lt] :\n    DecidableRel (@RbmapLt \u03b1 \u03b2 lt) := fun a b => h a.1 b.1\n#align rbmap.rbmap_lt_dec Rbmap.rbmapLtDec\n\nattribute [local instance] rbmap_lt_is_swo rbmap_lt_dec\n\n-- Helper lemmas for reusing rbtree results.\nprivate theorem to_rbtree_mem {k : \u03b1} {m : Rbmap \u03b1 \u03b2 lt} : k \u2208 m \u2192 \u2203 v : \u03b2, Rbtree.Mem (k, v) m :=\n  by\n  cases' m with n p <;> cases n <;> intro h\n  \u00b7 exact False.elim h\n  all_goals exists n_val.2; exact h\n#align rbmap.to_rbtree_mem rbmap.to_rbtree_mem\n\nprivate theorem eqv_entries_of_eqv_keys {k\u2081 k\u2082 : \u03b1} (v\u2081 v\u2082 : \u03b2) :\n    k\u2081 \u2248[lt]k\u2082 \u2192 (k\u2081, v\u2081) \u2248[RbmapLt lt](k\u2082, v\u2082) :=\n  id\n#align rbmap.eqv_entries_of_eqv_keys rbmap.eqv_entries_of_eqv_keys\n\nprivate theorem eqv_keys_of_eqv_entries {k\u2081 k\u2082 : \u03b1} {v\u2081 v\u2082 : \u03b2} :\n    (k\u2081, v\u2081) \u2248[RbmapLt lt](k\u2082, v\u2082) \u2192 k\u2081 \u2248[lt]k\u2082 :=\n  id\n#align rbmap.eqv_keys_of_eqv_entries rbmap.eqv_keys_of_eqv_entries\n\nprivate theorem eqv_entries [IsIrrefl \u03b1 lt] (k : \u03b1) (v\u2081 v\u2082 : \u03b2) : (k, v\u2081) \u2248[RbmapLt lt](k, v\u2082) :=\n  And.intro (irrefl_of lt k) (irrefl_of lt k)\n#align rbmap.eqv_entries rbmap.eqv_entries\n\nprivate theorem to_rbmap_mem [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    Rbtree.Mem (k, v) m \u2192 k \u2208 m :=\n  by\n  cases' m with n p <;> cases n <;> intro h\n  \u00b7 exact False.elim h\n  \u00b7 simp [Membership.Mem, Rbmap.Mem]\n    exact\n      @Rbtree.mem_of_mem_of_eqv _ _ _ \u27e8Rbnode.red_node n_lchild n_val n_rchild, p\u27e9 _ _ h\n        (eqv_entries _ _ _)\n  \u00b7 simp [Membership.Mem, Rbmap.Mem]\n    exact\n      @Rbtree.mem_of_mem_of_eqv _ _ _ \u27e8Rbnode.black_node n_lchild n_val n_rchild, p\u27e9 _ _ h\n        (eqv_entries _ _ _)\n#align rbmap.to_rbmap_mem rbmap.to_rbmap_mem\n\nprivate theorem to_rbtree_mem' [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {m : Rbmap \u03b1 \u03b2 lt} (v : \u03b2) :\n    k \u2208 m \u2192 Rbtree.Mem (k, v) m := by\n  intro h\n  cases' to_rbtree_mem h with v' hm\n  apply Rbtree.mem_of_mem_of_eqv hm\n  apply eqv_entries\n#align rbmap.to_rbtree_mem' rbmap.to_rbtree_mem'\n\ntheorem eq_some_of_toValue_eq_some {e : Option (\u03b1 \u00d7 \u03b2)} {v : \u03b2} :\n    toValue e = some v \u2192 \u2203 k, e = some (k, v) :=\n  by\n  cases' e with val <;> simp [to_value, false_imp_iff]\n  \u00b7 cases val\n    simp\n#align rbmap.eq_some_of_to_value_eq_some Rbmap.eq_some_of_toValue_eq_some\n\ntheorem eq_none_of_toValue_eq_none {e : Option (\u03b1 \u00d7 \u03b2)} : toValue e = none \u2192 e = none := by\n  cases e <;> simp [to_value, false_imp_iff]\n#align rbmap.eq_none_of_to_value_eq_none Rbmap.eq_none_of_toValue_eq_none\n\n-- Lemmas\ntheorem not_mem_mkRbmap : \u2200 k : \u03b1, k \u2209 mkRbmap \u03b1 \u03b2 lt := by\n  simp [Membership.Mem, mkRbmap, mkRbtree, Rbmap.Mem]\n#align rbmap.not_mem_mk_rbmap Rbmap.not_mem_mkRbmap\n\ntheorem not_mem_of_empty {m : Rbmap \u03b1 \u03b2 lt} (k : \u03b1) : m.Empty = true \u2192 k \u2209 m := by\n  cases' m with n p <;> cases n <;>\n    simp [Membership.Mem, mkRbmap, mkRbtree, Rbmap.Mem, Rbmap.empty, Rbtree.empty, false_imp_iff]\n#align rbmap.not_mem_of_empty Rbmap.not_mem_of_empty\n\ntheorem mem_of_mem_of_eqv [IsStrictWeakOrder \u03b1 lt] {m : Rbmap \u03b1 \u03b2 lt} {k\u2081 k\u2082 : \u03b1} :\n    k\u2081 \u2208 m \u2192 k\u2081 \u2248[lt]k\u2082 \u2192 k\u2082 \u2208 m := by\n  intro h\u2081 h\u2082\n  have h\u2081 := to_rbtree_mem h\u2081; cases' h\u2081 with v h\u2081\n  exact to_rbmap_mem (Rbtree.mem_of_mem_of_eqv h\u2081 (eqv_entries_of_eqv_keys v v h\u2082))\n#align rbmap.mem_of_mem_of_eqv Rbmap.mem_of_mem_of_eqv\n\nsection Decidable\n\nvariable [DecidableRel lt]\n\ntheorem not_mem_of_findEntry_none [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.findEntry k = none \u2192 k \u2209 m := by\n  cases' m with t p; cases t <;> simp [find_entry]\n  \u00b7 intros\n    simp [Membership.Mem, Rbmap.Mem]\n  all_goals intro h; exact Rbtree.not_mem_of_find_none h\n#align rbmap.not_mem_of_find_entry_none Rbmap.not_mem_of_findEntry_none\n\ntheorem not_mem_of_find_none [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.find k = none \u2192 k \u2209 m := by\n  simp [find]; intro h\n  have := eq_none_of_to_value_eq_none h\n  exact not_mem_of_find_entry_none this\n#align rbmap.not_mem_of_find_none Rbmap.not_mem_of_find_none\n\ntheorem mem_of_findEntry_some [IsStrictWeakOrder \u03b1 lt] {k\u2081 : \u03b1} {e : \u03b1 \u00d7 \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.findEntry k\u2081 = some e \u2192 k\u2081 \u2208 m := by\n  cases' m with t p; cases t <;> simp [find_entry, false_imp_iff]\n  all_goals intro h; exact Rbtree.mem_of_find_some h\n#align rbmap.mem_of_find_entry_some Rbmap.mem_of_findEntry_some\n\ntheorem mem_of_find_some [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.find k = some v \u2192 k \u2208 m := by\n  simp [find]; intro h\n  have := eq_some_of_to_value_eq_some h\n  cases' this with _ he\n  exact mem_of_find_entry_some he\n#align rbmap.mem_of_find_some Rbmap.mem_of_find_some\n\ntheorem findEntry_eq_findEntry_of_eqv [IsStrictWeakOrder \u03b1 lt] {m : Rbmap \u03b1 \u03b2 lt} {k\u2081 k\u2082 : \u03b1} :\n    k\u2081 \u2248[lt]k\u2082 \u2192 m.findEntry k\u2081 = m.findEntry k\u2082 :=\n  by\n  intro h; cases' m with t p; cases t <;> simp [find_entry]\n  all_goals apply Rbtree.find_eq_find_of_eqv; apply eqv_entries_of_eqv_keys; assumption\n#align rbmap.find_entry_eq_find_entry_of_eqv Rbmap.findEntry_eq_findEntry_of_eqv\n\ntheorem find_eq_find_of_eqv [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) :\n    k\u2081 \u2248[lt]k\u2082 \u2192 m.find k\u2081 = m.find k\u2082 := by intro h; simp [find]; apply congr_arg;\n  apply find_entry_eq_find_entry_of_eqv; assumption\n#align rbmap.find_eq_find_of_eqv Rbmap.find_eq_find_of_eqv\n\ntheorem findEntry_correct [IsStrictWeakOrder \u03b1 lt] (k : \u03b1) (m : Rbmap \u03b1 \u03b2 lt) :\n    k \u2208 m \u2194 \u2203 e, m.findEntry k = some e \u2227 k \u2248[lt]e.1 :=\n  by\n  apply Iff.intro <;> cases' m with t p\n  \u00b7 intro h\n    have h := to_rbtree_mem h\n    cases' h with v h\u2081\n    have hex := Iff.mp (Rbtree.find_correct _ _) h\u2081\n    cases' hex with e h\u2082\n    exists e\n    cases t <;> simp [find_entry] at h\u2082\u22a2\n    \u00b7 simp [Rbtree.find, Rbnode.find] at h\u2082\n      cases h\u2082\n    \u00b7 cases' h\u2082 with h\u2082\u2081 h\u2082\u2082\n      constructor\n      \u00b7 have :=\n          Rbtree.find_eq_find_of_eqv \u27e8Rbnode.red_node t_lchild t_val t_rchild, p\u27e9\n            (eqv_entries k v t_val.2)\n        rw [\u2190 this]\n        exact h\u2082\u2081\n      \u00b7 cases e\n        apply eqv_keys_of_eqv_entries h\u2082\u2082\n    \u00b7 cases' h\u2082 with h\u2082\u2081 h\u2082\u2082\n      constructor\n      \u00b7 have :=\n          Rbtree.find_eq_find_of_eqv \u27e8Rbnode.black_node t_lchild t_val t_rchild, p\u27e9\n            (eqv_entries k v t_val.2)\n        rw [\u2190 this]\n        exact h\u2082\u2081\n      \u00b7 cases e\n        apply eqv_keys_of_eqv_entries h\u2082\u2082\n  \u00b7 intro h\n    cases' h with e h\n    cases' h with h\u2081 h\u2082\n    cases t <;> simp [find_entry] at h\u2081\n    \u00b7 contradiction\n    all_goals exact to_rbmap_mem (Rbtree.mem_of_find_some h\u2081)\n#align rbmap.find_entry_correct Rbmap.findEntry_correct\n\ntheorem eqv_of_findEntry_some [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.findEntry k\u2081 = some (k\u2082, v) \u2192 k\u2081 \u2248[lt]k\u2082 :=\n  by\n  cases' m with t p; cases t <;> simp [find_entry, false_imp_iff]\n  all_goals intro h; exact eqv_keys_of_eqv_entries (Rbtree.eqv_of_find_some h)\n#align rbmap.eqv_of_find_entry_some Rbmap.eqv_of_findEntry_some\n\ntheorem eq_of_findEntry_some [IsStrictTotalOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.findEntry k\u2081 = some (k\u2082, v) \u2192 k\u2081 = k\u2082 := fun h =>\n  suffices k\u2081 \u2248[lt]k\u2082 from eq_of_eqv_lt this\n  eqv_of_findEntry_some h\n#align rbmap.eq_of_find_entry_some Rbmap.eq_of_findEntry_some\n\ntheorem find_correct [IsStrictWeakOrder \u03b1 lt] (k : \u03b1) (m : Rbmap \u03b1 \u03b2 lt) :\n    k \u2208 m \u2194 \u2203 v, m.find k = some v := by\n  apply Iff.intro\n  \u00b7 intro h\n    have := Iff.mp (find_entry_correct k m) h\n    cases' this with e h\n    cases' h with h\u2081 h\u2082\n    exists e.2\n    simp [find, h\u2081, to_value]\n  \u00b7 intro h\n    cases' h with v h\n    simp [find] at h\n    have h := eq_some_of_to_value_eq_some h\n    cases' h with k' h\n    have heqv := eqv_of_find_entry_some h\n    exact Iff.mpr (find_entry_correct k m) \u27e8(k', v), \u27e8h, heqv\u27e9\u27e9\n#align rbmap.find_correct Rbmap.find_correct\n\ntheorem constains_correct [IsStrictWeakOrder \u03b1 lt] (k : \u03b1) (m : Rbmap \u03b1 \u03b2 lt) :\n    k \u2208 m \u2194 m.contains k = true := by\n  apply Iff.intro\n  \u00b7 intro h\n    have h := Iff.mp (find_entry_correct k m) h\n    cases' h with e h\n    cases' h with h\u2081 h\u2082\n    simp [contains, h\u2081, Option.isSome]\n  \u00b7 simp [contains]\n    intro h\n    generalize he : find_entry m k = e\n    cases e\n    \u00b7 simp [he, Option.isSome] at h\n      contradiction\n    \u00b7 exact mem_of_find_entry_some he\n#align rbmap.constains_correct Rbmap.constains_correct\n\ntheorem mem_insert_of_incomp [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    \u00aclt k\u2081 k\u2082 \u2227 \u00aclt k\u2082 k\u2081 \u2192 k\u2081 \u2208 m.insert k\u2082 v := fun h =>\n  to_rbmap_mem (Rbtree.mem_insert_of_incomp m (eqv_entries_of_eqv_keys v v h))\n#align rbmap.mem_insert_of_incomp Rbmap.mem_insert_of_incomp\n\ntheorem mem_insert [IsStrictWeakOrder \u03b1 lt] (k : \u03b1) (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) : k \u2208 m.insert k v :=\n  to_rbmap_mem (Rbtree.mem_insert (k, v) m)\n#align rbmap.mem_insert Rbmap.mem_insert\n\ntheorem mem_insert_of_equiv [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    k\u2081 \u2248[lt]k\u2082 \u2192 k\u2081 \u2208 m.insert k\u2082 v :=\n  mem_insert_of_incomp m v\n#align rbmap.mem_insert_of_equiv Rbmap.mem_insert_of_equiv\n\ntheorem mem_insert_of_mem [IsStrictWeakOrder \u03b1 lt] {k\u2081 : \u03b1} {m : Rbmap \u03b1 \u03b2 lt} (k\u2082 : \u03b1) (v : \u03b2) :\n    k\u2081 \u2208 m \u2192 k\u2081 \u2208 m.insert k\u2082 v := fun h =>\n  to_rbmap_mem (Rbtree.mem_insert_of_mem (k\u2082, v) (to_rbtree_mem' v h))\n#align rbmap.mem_insert_of_mem Rbmap.mem_insert_of_mem\n\ntheorem equiv_or_mem_of_mem_insert [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    k\u2081 \u2208 m.insert k\u2082 v \u2192 k\u2081 \u2248[lt]k\u2082 \u2228 k\u2081 \u2208 m := fun h =>\n  Or.elim (Rbtree.equiv_or_mem_of_mem_insert (to_rbtree_mem' v h))\n    (fun h => Or.inl (eqv_keys_of_eqv_entries h)) fun h => Or.inr (to_rbmap_mem h)\n#align rbmap.equiv_or_mem_of_mem_insert Rbmap.equiv_or_mem_of_mem_insert\n\ntheorem incomp_or_mem_of_mem_ins [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    k\u2081 \u2208 m.insert k\u2082 v \u2192 \u00aclt k\u2081 k\u2082 \u2227 \u00aclt k\u2082 k\u2081 \u2228 k\u2081 \u2208 m :=\n  equiv_or_mem_of_mem_insert\n#align rbmap.incomp_or_mem_of_mem_ins Rbmap.incomp_or_mem_of_mem_ins\n\ntheorem eq_or_mem_of_mem_ins [IsStrictTotalOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    k\u2081 \u2208 m.insert k\u2082 v \u2192 k\u2081 = k\u2082 \u2228 k\u2081 \u2208 m := fun h =>\n  suffices k\u2081 \u2248[lt]k\u2082 \u2228 k\u2081 \u2208 m by simp [eqv_lt_iff_eq] at this <;> assumption\n  incomp_or_mem_of_mem_ins h\n#align rbmap.eq_or_mem_of_mem_ins Rbmap.eq_or_mem_of_mem_ins\n\ntheorem findEntry_insert_of_eqv [IsStrictWeakOrder \u03b1 lt] (m : Rbmap \u03b1 \u03b2 lt) {k\u2081 k\u2082 : \u03b1} (v : \u03b2) :\n    k\u2081 \u2248[lt]k\u2082 \u2192 (m.insert k\u2081 v).findEntry k\u2082 = some (k\u2081, v) :=\n  by\n  intro h\n  generalize h\u2081 : m.insert k\u2081 v = m'\n  cases' m' with t p; cases t\n  \u00b7 have := mem_insert k\u2081 m v\n    rw [h\u2081] at this\n    apply absurd this\n    apply not_mem_mk_rbmap\n  all_goals\n    simp [find_entry]; rw [\u2190 h\u2081, insert]; apply Rbtree.find_insert_of_eqv\n    apply eqv_entries_of_eqv_keys _ _ h\n#align rbmap.find_entry_insert_of_eqv Rbmap.findEntry_insert_of_eqv\n\ntheorem findEntry_insert [IsStrictWeakOrder \u03b1 lt] (m : Rbmap \u03b1 \u03b2 lt) (k : \u03b1) (v : \u03b2) :\n    (m.insert k v).findEntry k = some (k, v) :=\n  findEntry_insert_of_eqv m v (refl k)\n#align rbmap.find_entry_insert Rbmap.findEntry_insert\n\ntheorem find_insert_of_eqv [IsStrictWeakOrder \u03b1 lt] (m : Rbmap \u03b1 \u03b2 lt) {k\u2081 k\u2082 : \u03b1} (v : \u03b2) :\n    k\u2081 \u2248[lt]k\u2082 \u2192 (m.insert k\u2081 v).find k\u2082 = some v :=\n  by\n  intro h\n  have := find_entry_insert_of_eqv m v h\n  simp [find, this, to_value]\n#align rbmap.find_insert_of_eqv Rbmap.find_insert_of_eqv\n\ntheorem find_insert [IsStrictWeakOrder \u03b1 lt] (m : Rbmap \u03b1 \u03b2 lt) (k : \u03b1) (v : \u03b2) :\n    (m.insert k v).find k = some v :=\n  find_insert_of_eqv m v (refl k)\n#align rbmap.find_insert Rbmap.find_insert\n\ntheorem findEntry_insert_of_disj [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    lt k\u2081 k\u2082 \u2228 lt k\u2082 k\u2081 \u2192 (m.insert k\u2081 v).findEntry k\u2082 = m.findEntry k\u2082 :=\n  by\n  intro h\n  have h' : \u2200 {v\u2081 v\u2082 : \u03b2}, (RbmapLt lt) (k\u2081, v\u2081) (k\u2082, v\u2082) \u2228 (RbmapLt lt) (k\u2082, v\u2082) (k\u2081, v\u2081) :=\n    fun _ _ => h\n  generalize h\u2081 : m = m\u2081\n  generalize h\u2082 : insert m\u2081 k\u2081 v = m\u2082\n  rw [\u2190 h\u2081] at h\u2082\u22a2; rw [\u2190 h\u2082]\n  cases' m\u2081 with t\u2081 p\u2081 <;> cases t\u2081 <;> cases' m\u2082 with t\u2082 p\u2082 <;> cases t\u2082\n  \u00b7 rw [h\u2082, h\u2081]\n  iterate 2 \n    rw [h\u2082]\n    conv =>\n      lhs\n      simp [find_entry]\n    rw [\u2190 h\u2082, insert, Rbtree.find_insert_of_disj _ h', h\u2081]\n    rfl\n  any_goals\n    simp [insert] at h\u2082\n    exact absurd h\u2082 (Rbtree.insert_ne_mkRbtree m (k\u2081, v))\n  any_goals\n    rw [h\u2082, h\u2081]; simp [find_entry]; rw [\u2190 h\u2082, \u2190 h\u2081, insert, Rbtree.find_insert_of_disj _ h']\n    apply Rbtree.find_eq_find_of_eqv; apply eqv_entries\n#align rbmap.find_entry_insert_of_disj Rbmap.findEntry_insert_of_disj\n\ntheorem findEntry_insert_of_not_eqv [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt)\n    (v : \u03b2) : \u00ack\u2081 \u2248[lt]k\u2082 \u2192 (m.insert k\u2081 v).findEntry k\u2082 = m.findEntry k\u2082 :=\n  by\n  intro hn\n  have he : lt k\u2081 k\u2082 \u2228 lt k\u2082 k\u2081 :=\n    by\n    simp [StrictWeakOrder.Equiv, Decidable.not_and_iff_or_not, Decidable.not_not_iff] at hn\n    assumption\n  apply find_entry_insert_of_disj _ _ he\n#align rbmap.find_entry_insert_of_not_eqv Rbmap.findEntry_insert_of_not_eqv\n\ntheorem findEntry_insert_of_ne [IsStrictTotalOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    k\u2081 \u2260 k\u2082 \u2192 (m.insert k\u2081 v).findEntry k\u2082 = m.findEntry k\u2082 :=\n  by\n  intro h\n  have : \u00ack\u2081 \u2248[lt]k\u2082 := fun h' => h (eq_of_eqv_lt h')\n  apply find_entry_insert_of_not_eqv _ _ this\n#align rbmap.find_entry_insert_of_ne Rbmap.findEntry_insert_of_ne\n\ntheorem find_insert_of_disj [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    lt k\u2081 k\u2082 \u2228 lt k\u2082 k\u2081 \u2192 (m.insert k\u2081 v).find k\u2082 = m.find k\u2082 := by intro h;\n  have := find_entry_insert_of_disj m v h; simp [find, this]\n#align rbmap.find_insert_of_disj Rbmap.find_insert_of_disj\n\ntheorem find_insert_of_not_eqv [IsStrictWeakOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    \u00ack\u2081 \u2248[lt]k\u2082 \u2192 (m.insert k\u2081 v).find k\u2082 = m.find k\u2082 := by intro h;\n  have := find_entry_insert_of_not_eqv m v h; simp [find, this]\n#align rbmap.find_insert_of_not_eqv Rbmap.find_insert_of_not_eqv\n\ntheorem find_insert_of_ne [IsStrictTotalOrder \u03b1 lt] {k\u2081 k\u2082 : \u03b1} (m : Rbmap \u03b1 \u03b2 lt) (v : \u03b2) :\n    k\u2081 \u2260 k\u2082 \u2192 (m.insert k\u2081 v).find k\u2082 = m.find k\u2082 := by intro h;\n  have := find_entry_insert_of_ne m v h; simp [find, this]\n#align rbmap.find_insert_of_ne Rbmap.find_insert_of_ne\n\nend Decidable\n\ntheorem mem_of_min_eq [IsStrictTotalOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.min = some (k, v) \u2192 k \u2208 m := fun h => to_rbmap_mem (Rbtree.mem_of_min_eq h)\n#align rbmap.mem_of_min_eq Rbmap.mem_of_min_eq\n\ntheorem mem_of_max_eq [IsStrictTotalOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.max = some (k, v) \u2192 k \u2208 m := fun h => to_rbmap_mem (Rbtree.mem_of_max_eq h)\n#align rbmap.mem_of_max_eq Rbmap.mem_of_max_eq\n\ntheorem eq_leaf_of_min_eq_none {m : Rbmap \u03b1 \u03b2 lt} : m.min = none \u2192 m = mkRbmap \u03b1 \u03b2 lt :=\n  Rbtree.eq_leaf_of_min_eq_none\n#align rbmap.eq_leaf_of_min_eq_none Rbmap.eq_leaf_of_min_eq_none\n\ntheorem eq_leaf_of_max_eq_none {m : Rbmap \u03b1 \u03b2 lt} : m.max = none \u2192 m = mkRbmap \u03b1 \u03b2 lt :=\n  Rbtree.eq_leaf_of_max_eq_none\n#align rbmap.eq_leaf_of_max_eq_none Rbmap.eq_leaf_of_max_eq_none\n\ntheorem min_is_minimal [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.min = some (k, v) \u2192 \u2200 {k'}, k' \u2208 m \u2192 k \u2248[lt]k' \u2228 lt k k' := fun h k' hm =>\n  Or.elim (Rbtree.min_is_minimal h (to_rbtree_mem' v hm))\n    (fun h => Or.inl (eqv_keys_of_eqv_entries h)) fun h => Or.inr h\n#align rbmap.min_is_minimal Rbmap.min_is_minimal\n\ntheorem max_is_maximal [IsStrictWeakOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.max = some (k, v) \u2192 \u2200 {k'}, k' \u2208 m \u2192 k \u2248[lt]k' \u2228 lt k' k := fun h k' hm =>\n  Or.elim (Rbtree.max_is_maximal h (to_rbtree_mem' v hm))\n    (fun h => Or.inl (eqv_keys_of_eqv_entries h)) fun h => Or.inr h\n#align rbmap.max_is_maximal Rbmap.max_is_maximal\n\ntheorem min_is_minimal_of_total [IsStrictTotalOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.min = some (k, v) \u2192 \u2200 {k'}, k' \u2208 m \u2192 k = k' \u2228 lt k k' := fun h k' hm =>\n  match min_is_minimal h hm with\n  | Or.inl h => Or.inl (eq_of_eqv_lt h)\n  | Or.inr h => Or.inr h\n#align rbmap.min_is_minimal_of_total Rbmap.min_is_minimal_of_total\n\ntheorem max_is_maximal_of_total [IsStrictTotalOrder \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : Rbmap \u03b1 \u03b2 lt} :\n    m.max = some (k, v) \u2192 \u2200 {k'}, k' \u2208 m \u2192 k = k' \u2228 lt k' k := fun h k' hm =>\n  match max_is_maximal h hm with\n  | Or.inl h => Or.inl (eq_of_eqv_lt h)\n  | Or.inr h => Or.inr h\n#align rbmap.max_is_maximal_of_total Rbmap.max_is_maximal_of_total\n\nend Rbmap\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Rbmap/Default.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.39605214447372744}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Johannes H\u00f6lzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.multiset.nodup\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# The cartesian product of multisets\n-/\n\nnamespace multiset\n\n\n/-- Given `\u03b4 : \u03b1 \u2192 Type*`, `pi.empty \u03b4` is the trivial dependent function out of the empty\nmultiset. -/\ndef pi.empty {\u03b1 : Type u_1} (\u03b4 : \u03b1 \u2192 Type u_2) (a : \u03b1) (H : a \u2208 0) : \u03b4 a := sorry\n\n/-- Given `\u03b4 : \u03b1 \u2192 Type*`, a multiset `m` and a term `a`, as well as a term `b : \u03b4 a` and a\nfunction `f` such that `f a' : \u03b4 a'` for all `a'` in `m`, `pi.cons m a b f` is a function `g` such\nthat `g a'' : \u03b4 a''` for all `a''` in `a ::\u2098 m`. -/\ndef pi.cons {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} (m : multiset \u03b1) (a : \u03b1) (b : \u03b4 a)\n    (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a) (a' : \u03b1) (H : a' \u2208 a ::\u2098 m) : \u03b4 a' :=\n  dite (a' = a) (fun (h : a' = a) => Eq._oldrec b (Eq.symm h)) fun (h : \u00aca' = a) => f a' sorry\n\ntheorem pi.cons_same {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} {m : multiset \u03b1} {a : \u03b1}\n    {b : \u03b4 a} {f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a} (h : a \u2208 a ::\u2098 m) : pi.cons m a b f a h = b :=\n  dif_pos rfl\n\ntheorem pi.cons_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} {m : multiset \u03b1} {a : \u03b1}\n    {a' : \u03b1} {b : \u03b4 a} {f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a} (h' : a' \u2208 a ::\u2098 m) (h : a' \u2260 a) :\n    pi.cons m a b f a' h' = f a' (or.resolve_left (iff.mp mem_cons h') h) :=\n  dif_neg h\n\ntheorem pi.cons_swap {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} {a : \u03b1} {a' : \u03b1} {b : \u03b4 a}\n    {b' : \u03b4 a'} {m : multiset \u03b1} {f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a} (h : a \u2260 a') :\n    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  sorry\n\n/-- `pi m t` constructs the Cartesian product over `t` indexed by `m`. -/\ndef pi {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} (m : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03b4 a)) : multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a) :=\n  multiset.rec_on m (singleton sorry)\n    (fun (a : \u03b1) (m : multiset \u03b1) (p : multiset ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a)) =>\n      bind (t a) fun (b : \u03b4 a) => map sorry p)\n    sorry\n\n@[simp] theorem pi_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2}\n    (t : (a : \u03b1) \u2192 multiset (\u03b4 a)) : pi 0 t = pi.empty \u03b4 ::\u2098 0 :=\n  rfl\n\n@[simp] theorem pi_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} (m : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03b4 a)) (a : \u03b1) :\n    pi (a ::\u2098 m) t = bind (t a) fun (b : \u03b4 a) => map (pi.cons m a b) (pi m t) :=\n  rec_on_cons a m\n\ntheorem pi_cons_injective {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} {a : \u03b1} {b : \u03b4 a}\n    {s : multiset \u03b1} (hs : \u00aca \u2208 s) : function.injective (pi.cons s a b) :=\n  sorry\n\ntheorem card_pi {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} (m : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03b4 a)) :\n    coe_fn card (pi m t) = prod (map (fun (a : \u03b1) => coe_fn card (t a)) m) :=\n  sorry\n\ntheorem nodup_pi {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} {s : multiset \u03b1}\n    {t : (a : \u03b1) \u2192 multiset (\u03b4 a)} : nodup s \u2192 (\u2200 (a : \u03b1), a \u2208 s \u2192 nodup (t a)) \u2192 nodup (pi s t) :=\n  sorry\n\ntheorem mem_pi {\u03b1 : Type u_1} [DecidableEq \u03b1] {\u03b4 : \u03b1 \u2192 Type u_2} (m : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03b4 a)) (f : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b4 a) :\n    f \u2208 pi m t \u2194 \u2200 (a : \u03b1) (h : a \u2208 m), f a h \u2208 t a :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/multiset/pi_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.3959288434855902}}
{"text": "example (P Q R S T U: Type) (p: P) (h: P \u2192 Q) (i: Q \u2192 R) (j: Q \u2192 T) (k: S \u2192 T) (l: T \u2192 U) : U :=\nbegin\n    apply l,\n    apply j,\n    apply h,\n    exact p,\nend\n", "meta": {"author": "nomoid", "repo": "lean-proofs", "sha": "b9f03a24623d1a1d111d6c2bbf53c617e2596d6a", "save_path": "github-repos/lean/nomoid-lean-proofs", "path": "github-repos/lean/nomoid-lean-proofs/lean-proofs-b9f03a24623d1a1d111d6c2bbf53c617e2596d6a/src/world5/level4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3959288299173493}}
{"text": "/-\nCopyright (c) 2019 Lucas Allen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lucas Allen, Scott Morrison\n-/\nimport data.mllist\nimport tactic.solve_by_elim\n\n/-!\n# `suggest` and `library_search`\n\n`suggest` and `library_search` are a pair of tactics for applying lemmas from the library to the\ncurrent goal.\n\n* `suggest` prints a list of `exact ...` or `refine ...` statements, which may produce new goals\n* `library_search` prints a single `exact ...` which closes the goal, or fails\n-/\n\nnamespace tactic\n\nopen native\n\nnamespace suggest\n\nopen solve_by_elim\n\n/-- Map a name (typically a head symbol) to a \"canonical\" definitional synonym.\nGiven a name `n`, we want a name `n'` such that a sufficiently applied\nexpression with head symbol `n` is always definitionally equal to an expression\nwith head symbol `n'`.\nThus, we can search through all lemmas with a result type of `n'`\nto solve a goal with head symbol `n`.\n\nFor example, `>` is mapped to `<` because `a > b` is definitionally equal to `b < a`,\nand `not` is mapped to `false` because `\u00ac a` is definitionally equal to `p \u2192 false`\nThe default is that the original argument is returned, so `<` is just mapped to `<`.\n\n`normalize_synonym` is called for every lemma in the library, so it needs to be fast.\n-/\n-- TODO this is a hack; if you suspect more cases here would help, please report them\nmeta def normalize_synonym : name \u2192 name\n| `gt := `has_lt.lt\n| `ge := `has_le.le\n| `monotone := `has_le.le\n| `not := `false\n| n   := n\n\n/--\nCompute the head symbol of an expression, then normalise synonyms.\n\nThis is only used when analysing the goal, so it is okay to do more expensive analysis here.\n-/\n-- We may want to tweak this further?\nmeta def allowed_head_symbols : expr \u2192 list name\n-- We first have a various \"customisations\":\n--   Because in `\u2115` `a.succ \u2264 b` is definitionally `a < b`,\n--   we add some special cases to allow looking for `<` lemmas even when the goal has a `\u2264`.\n--   Note we only do this in the `\u2115` case, for performance.\n| `(@has_le.le \u2115 _ (nat.succ _) _) := [`has_le.le, `has_lt.lt]\n| `(@ge \u2115 _ _ (nat.succ _)) := [`has_le.le, `has_lt.lt]\n| `(@has_le.le \u2115 _ 1 _) := [`has_le.le, `has_lt.lt]\n| `(@ge \u2115 _ _ 1) := [`has_le.le, `has_lt.lt]\n\n-- And then the generic cases:\n| (expr.pi _ _ _ t) := allowed_head_symbols t\n| (expr.app f _) := allowed_head_symbols f\n| (expr.const n _) := [normalize_synonym n]\n| _ := [`_]\n.\n\n/--\nA declaration can match the head symbol of the current goal in four possible ways:\n* `ex`  : an exact match\n* `mp`  : the declaration returns an `iff`, and the right hand side matches the goal\n* `mpr` : the declaration returns an `iff`, and the left hand side matches the goal\n* `both`: the declaration returns an `iff`, and the both sides match the goal\n-/\n@[derive decidable_eq, derive inhabited]\ninductive head_symbol_match\n| ex | mp | mpr | both\n\nopen head_symbol_match\n\n/-- a textual representation of a `head_symbol_match`, for trace debugging. -/\ndef head_symbol_match.to_string : head_symbol_match \u2192 string\n| ex   := \"exact\"\n| mp   := \"iff.mp\"\n| mpr  := \"iff.mpr\"\n| both := \"iff.mp and iff.mpr\"\n\n/-- Determine if, and in which way, a given expression matches the specified head symbol. -/\nmeta def match_head_symbol (hs : name_set) : expr \u2192 option head_symbol_match\n| (expr.pi _ _ _ t) := match_head_symbol t\n| `(%%a \u2194 %%b)      := if hs.contains `iff then some ex else\n                       match (match_head_symbol a, match_head_symbol b) with\n                       | (some ex, some ex) :=\n                           some both\n                       | (some ex, _) := some mpr\n                       | (_, some ex) := some mp\n                       | _ := none\n                       end\n| (expr.app f _)    := match_head_symbol f\n| (expr.const n _)  := if hs.contains (normalize_synonym n) then some ex else none\n| _ := if hs.contains `_ then some ex else none\n\n/-- A package of `declaration` metadata, including the way in which its type matches the head symbol\nwhich we are searching for. -/\nmeta structure decl_data :=\n(d : declaration)\n(n : name)\n(m : head_symbol_match)\n(l : \u2115) -- cached length of name\n\n/--\nGenerate a `decl_data` from the given declaration if\nit matches the head symbol `hs` for the current goal.\n-/\n-- We used to check here for private declarations, or declarations with certain suffixes.\n-- It turns out `apply` is so fast, it's better to just try them all.\nmeta def process_declaration (hs : name_set) (d : declaration) : option decl_data :=\nlet n := d.to_name in\nif !d.is_trusted || n.is_internal then\n  none\nelse\n  (\u03bb m, \u27e8d, n, m, n.length\u27e9) <$> match_head_symbol hs d.type\n\n/-- Retrieve all library definitions with a given head symbol. -/\nmeta def library_defs (hs : name_set) : tactic (list decl_data) :=\ndo trace_if_enabled `suggest format!\"Looking for lemmas with head symbols {hs}.\",\n   env \u2190 get_env,\n   let defs := env.decl_filter_map (process_declaration hs),\n   -- Sort by length; people like short proofs\n   let defs := defs.qsort(\u03bb d\u2081 d\u2082, d\u2081.l \u2264 d\u2082.l),\n   trace_if_enabled `suggest format!\"Found {defs.length} relevant lemmas:\",\n   trace_if_enabled `suggest $ defs.map (\u03bb \u27e8d, n, m, l\u27e9, (n, m.to_string)),\n   return defs\n\n/--\nWe unpack any element of a list of `decl_data` corresponding to an `\u2194` statement that could apply\nin both directions into two separate elements.\n\nThis ensures that both directions can be independently returned by `suggest`,\nand avoids a problem where the application of one direction prevents\nthe application of the other direction. (See `exp_le_exp` in the tests.)\n-/\nmeta def unpack_iff_both : list decl_data \u2192 list decl_data\n| []                     := []\n| (\u27e8d, n, both, l\u27e9 :: L) := \u27e8d, n, mp, l\u27e9 :: \u27e8d, n, mpr, l\u27e9 :: unpack_iff_both L\n| (\u27e8d, n, m, l\u27e9 :: L)    := \u27e8d, n, m, l\u27e9 :: unpack_iff_both L\n\n/--\nApply the lemma `e`, then attempt to close all goals using\n`solve_by_elim opt`, failing if `close_goals = tt`\nand there are any goals remaining.\n\nReturns the number of subgoals which were closed using `solve_by_elim`.\n-/\n-- Implementation note: as this is used by both `library_search` and `suggest`,\n-- we first run `solve_by_elim` separately on the independent goals,\n-- whether or not `close_goals` is set,\n-- and then run `solve_by_elim { all_goals := tt }`,\n-- requiring that it succeeds if `close_goals = tt`.\nmeta def apply_and_solve (close_goals : bool) (opt : opt := { }) (e : expr) : tactic \u2115 :=\ndo\n  trace_if_enabled `suggest format!\"Trying to apply lemma: {e}\",\n  apply e opt.to_apply_cfg,\n  trace_if_enabled `suggest format!\"Applied lemma: {e}\",\n  ng \u2190 num_goals,\n  -- Phase 1\n  -- Run `solve_by_elim` on each \"safe\" goal separately, not worrying about failures.\n  -- (We only attempt the \"safe\" goals in this way in Phase 1. In Phase 2 we will do\n  -- backtracking search across all goals, allowing us to guess solutions that involve data, or\n  -- unify metavariables, but only as long as we can finish all goals.)\n  try (any_goals (independent_goal >> solve_by_elim opt)),\n  -- Phase 2\n  (done >> return ng) <|> (do\n    -- If there were any goals that we did not attempt solving in the first phase\n    -- (because they weren't propositional, or contained a metavariable)\n    -- as a second phase we attempt to solve all remaining goals at once\n    -- (with backtracking across goals).\n    (any_goals (success_if_fail independent_goal) >>\n    solve_by_elim { backtrack_all_goals := tt, ..opt }) <|>\n    -- and fail unless `close_goals = ff`\n    guard \u00ac close_goals,\n    ng' \u2190 num_goals,\n    return (ng - ng'))\n\n/--\nApply the declaration `d` (or the forward and backward implications separately, if it is an `iff`),\nand then attempt to solve the subgoal using `apply_and_solve`.\n\nReturns the number of subgoals successfully closed.\n-/\nmeta def apply_declaration (close_goals : bool) (opt : opt := { }) (d : decl_data) :\n  tactic \u2115 :=\nlet tac := apply_and_solve close_goals opt in\ndo (e, t) \u2190 decl_mk_const d.d,\n   match d.m with\n   | ex   := tac e\n   | mp   := do l \u2190 iff_mp_core e t, tac l\n   | mpr  := do l \u2190 iff_mpr_core e t, tac l\n   | both := undefined -- we use `unpack_iff_both` to ensure this isn't reachable\n   end\n\n/-- An `application` records the result of a successful application of a library lemma. -/\nmeta structure application :=\n(state     : tactic_state)\n(script    : string)\n(decl      : option declaration)\n(num_goals : \u2115)\n(hyps_used : \u2115)\n\nend suggest\n\nopen solve_by_elim\nopen suggest\n\ndeclare_trace suggest         -- Trace a list of all relevant lemmas\n\n-- Call `apply_declaration`, then prepare the tactic script and\n-- count the number of local hypotheses used.\nprivate meta def apply_declaration_script\n  (g : expr) (hyps : list expr)\n  (opt : opt := { })\n  (d : decl_data) :\n  tactic application :=\n-- (This tactic block is only executed when we evaluate the mllist,\n-- so we need to do the `focus1` here.)\nretrieve $ focus1 $ do\n  apply_declaration ff opt d,\n  ng \u2190 num_goals,\n  -- This `instantiate_mvars` is necessary so that we count used hypotheses correctly.\n  g \u2190 instantiate_mvars g,\n  s \u2190 read,\n  m \u2190 tactic_statement g,\n  return\n  { application .\n    state := s,\n    decl := d.d,\n    script := m,\n    num_goals := ng,\n    hyps_used := hyps.countp (\u03bb h, h.occurs g) }\n\n-- implementation note: we produce a `tactic (mllist tactic application)` first,\n-- because it's easier to work in the tactic monad, but in a moment we squash this\n-- down to an `mllist tactic application`.\nprivate meta def suggest_core' (opt : opt := { }) :\n  tactic (mllist tactic application) :=\ndo g :: _ \u2190 get_goals,\n   hyps \u2190 local_context,\n\n   -- Make sure that `solve_by_elim` doesn't just solve the goal immediately:\n   (retrieve (do\n     focus1 $ solve_by_elim opt,\n     s \u2190 read,\n     m \u2190 tactic_statement g,\n     -- This `instantiate_mvars` is necessary so that we count used hypotheses correctly.\n     g \u2190 instantiate_mvars g,\n     return $ mllist.of_list [\u27e8s, m, none, 0, hyps.countp (\u03bb h, h.occurs g)\u27e9])) <|>\n   -- Otherwise, let's actually try applying library lemmas.\n   (do\n   -- Collect all definitions with the correct head symbol\n   t \u2190 infer_type g,\n   defs \u2190 unpack_iff_both <$> library_defs (name_set.of_list $ allowed_head_symbols t),\n\n   let defs : mllist tactic _ := mllist.of_list defs,\n\n   -- Try applying each lemma against the goal,\n   -- recording the tactic script as a string,\n   -- the number of remaining goals,\n   -- and number of local hypotheses used.\n   let results := defs.mfilter_map (apply_declaration_script g hyps opt),\n   -- Now call `symmetry` and try again.\n   -- (Because we are using `mllist`, this is essentially free if we've already found a lemma.)\n   symm_state \u2190 retrieve $ try_core $ symmetry >> read,\n   let results_symm := match symm_state with\n   | (some s) :=\n     defs.mfilter_map (\u03bb d, retrieve $ set_state s >> apply_declaration_script g hyps opt d)\n   | none := mllist.nil\n   end,\n  return (results.append results_symm))\n\n/--\nThe core `suggest` tactic.\nIt attempts to apply a declaration from the library,\nthen solve new goals using `solve_by_elim`.\n\nIt returns a list of `application`s consisting of fields:\n* `state`, a tactic state resulting from the successful application of a declaration from\n  the library,\n* `script`, a string of the form `Try this: refine ...` or `Try this: exact ...` which will\n  reproduce that tactic state,\n* `decl`, an `option declaration` indicating the declaration that was applied\n  (or none, if `solve_by_elim` succeeded),\n* `num_goals`, the number of remaining goals, and\n* `hyps_used`, the number of local hypotheses used in the solution.\n-/\nmeta def suggest_core (opt : opt := { }) : mllist tactic application :=\n(mllist.monad_lift (suggest_core' opt)).join\n\n/--\nSee `suggest_core`.\n\nReturns a list of at most `limit` `application`s,\nsorted by number of goals, and then (reverse) number of hypotheses used.\n-/\nmeta def suggest (limit : option \u2115 := none) (opt : opt := { }) :\n  tactic (list application) :=\ndo let results := suggest_core opt,\n   -- Get the first n elements of the successful lemmas\n   L \u2190 if h : limit.is_some then results.take (option.get h) else results.force,\n   -- Sort by number of remaining goals, then by number of hypotheses used.\n   return $ L.qsort (\u03bb d\u2081 d\u2082, d\u2081.num_goals < d\u2082.num_goals \u2228\n    (d\u2081.num_goals = d\u2082.num_goals \u2227 d\u2081.hyps_used \u2265 d\u2082.hyps_used))\n\n/--\nReturns a list of at most `limit` strings, of the form `Try this: exact ...` or\n`Try this: refine ...`, which make progress on the current goal using a declaration\nfrom the library.\n-/\nmeta def suggest_scripts (limit : option \u2115 := none) (opt : opt := { }) :\n  tactic (list string) :=\ndo L \u2190 suggest limit opt,\n   return $ L.map application.script\n\n/--\nReturns a string of the form `Try this: exact ...`, which closes the current goal.\n-/\nmeta def library_search (opt : opt := { }) : tactic string :=\n(suggest_core opt).mfirst (\u03bb a, do guard (a.num_goals = 0), write a.state, return a.script)\n\nnamespace interactive\nopen tactic\nopen interactive\nopen lean.parser\nopen interactive.types\nopen solve_by_elim\nlocal postfix `?`:9001 := optional\n\ndeclare_trace silence_suggest -- Turn off `Try this: exact/refine ...` trace messages for `suggest`\n\n/--\n`suggest` tries to apply suitable theorems/defs from the library, and generates\na list of `exact ...` or `refine ...` scripts that could be used at this step.\nIt leaves the tactic state unchanged. It is intended as a complement of the search\nfunction in your editor, the `#find` tactic, and `library_search`.\n\n`suggest` takes an optional natural number `num` as input and returns the first `num`\n(or less, if all possibilities are exhausted) possibilities ordered by length of lemma names.\nThe default for `num` is `50`.\nFor performance reasons `suggest` uses monadic lazy lists (`mllist`). This means that\n`suggest` might miss some results if `num` is not large enough. However, because\n`suggest` uses monadic lazy lists, smaller values of `num` run faster than larger values.\n\nYou can add additional lemmas to be used along with local hypotheses\nafter the application of a library lemma,\nusing the same syntax as for `solve_by_elim`, e.g.\n```\nexample {a b c d: nat} (h\u2081 : a < c) (h\u2082 : b < d) : max (c + d) (a + b) = (c + d) :=\nbegin\n  suggest [add_lt_add], -- Says: `Try this: exact max_eq_left_of_lt (add_lt_add h\u2081 h\u2082)`\nend\n```\nYou can also use `suggest with attr` to include all lemmas with the attribute `attr`.\n-/\nmeta def suggest (n : parse (with_desc \"n\" small_nat)?)\n  (hs : parse simp_arg_list) (attr_names : parse with_ident_list) (opt : opt := { }) :\n  tactic unit :=\ndo (lemma_thunks, ctx_thunk) \u2190 mk_assumption_set ff hs attr_names,\n   L \u2190 tactic.suggest_scripts (n.get_or_else 50)\n     { lemma_thunks := some lemma_thunks, ctx_thunk := ctx_thunk, ..opt },\n  if is_trace_enabled_for `silence_suggest then\n    skip\n  else\n    if L.length = 0 then\n      fail \"There are no applicable declarations\"\n    else\n      L.mmap trace >> skip\n\n/--\n`suggest` lists possible usages of the `refine` tactic and leaves the tactic state unchanged.\nIt is intended as a complement of the search function in your editor, the `#find` tactic, and\n`library_search`.\n\n`suggest` takes an optional natural number `num` as input and returns the first `num` (or less, if\nall possibilities are exhausted) possibilities ordered by length of lemma names.\nThe default for `num` is `50`.\n\nFor performance reasons `suggest` uses monadic lazy lists (`mllist`). This means that `suggest`\nmight miss some results if `num` is not large enough. However, because `suggest` uses monadic\nlazy lists, smaller values of `num` run faster than larger values.\n\nAn example of `suggest` in action,\n\n```lean\nexample (n : nat) : n < n + 1 :=\nbegin suggest, sorry end\n```\n\nprints the list,\n\n```lean\nTry this: exact nat.lt.base n\nTry this: exact nat.lt_succ_self n\nTry this: refine not_le.mp _\nTry this: refine gt_iff_lt.mp _\nTry this: refine nat.lt.step _\nTry this: refine lt_of_not_ge _\n...\n```\n-/\nadd_tactic_doc\n{ name        := \"suggest\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.suggest],\n  tags        := [\"search\", \"Try this\"] }\n\n-- Turn off `Try this: exact ...` trace message for `library_search`\ndeclare_trace silence_library_search\n\n/--\n`library_search` is a tactic to identify existing lemmas in the library. It tries to close the\ncurrent goal by applying a lemma from the library, then discharging any new goals using\n`solve_by_elim`.\n\nIf it succeeds, it prints a trace message `exact ...` which can replace the invocation\nof `library_search`.\n\nTypical usage is:\n```lean\nexample (n m k : \u2115) : n * (m - k) = n * m - n * k :=\nby library_search -- Try this: exact nat.mul_sub_left_distrib n m k\n```\n\nBy default `library_search` only unfolds `reducible` definitions\nwhen attempting to match lemmas against the goal.\nPreviously, it would unfold most definitions, sometimes giving surprising answers, or slow answers.\nThe old behaviour is still available via `library_search!`.\n\nYou can add additional lemmas to be used along with local hypotheses\nafter the application of a library lemma,\nusing the same syntax as for `solve_by_elim`, e.g.\n```\nexample {a b c d: nat} (h\u2081 : a < c) (h\u2082 : b < d) : max (c + d) (a + b) = (c + d) :=\nbegin\n  library_search [add_lt_add], -- Says: `Try this: exact max_eq_left_of_lt (add_lt_add h\u2081 h\u2082)`\nend\n```\nYou can also use `library_search with attr` to include all lemmas with the attribute `attr`.\n-/\nmeta def library_search (semireducible : parse $ optional (tk \"!\"))\n  (hs : parse simp_arg_list) (attr_names : parse with_ident_list)\n  (opt : opt := { }) : tactic unit :=\ndo (lemma_thunks, ctx_thunk) \u2190 mk_assumption_set ff hs attr_names,\n   (tactic.library_search\n     { backtrack_all_goals := tt,\n       lemma_thunks := some lemma_thunks,\n       ctx_thunk := ctx_thunk,\n       md := if semireducible.is_some then\n         tactic.transparency.semireducible else tactic.transparency.reducible,\n       ..opt } >>=\n   if is_trace_enabled_for `silence_library_search then\n     (\u03bb _, skip)\n   else\n     trace) <|>\n   fail\n\"`library_search` failed.\nIf you aren't sure what to do next, you can also\ntry `library_search!`, `suggest`, or `hint`.\n\nPossible reasons why `library_search` failed:\n* `library_search` will only apply a single lemma from the library,\n  and then try to fill in its hypotheses from local hypotheses.\n* If you haven't already, try stating the theorem you want in its own lemma.\n* Sometimes the library has one version of a lemma\n  but not a very similar version obtained by permuting arguments.\n  Try replacing `a + b` with `b + a`, or `a - b < c` with `a < b + c`,\n  to see if maybe the lemma exists but isn't stated quite the way you would like.\n* Make sure that you have all the side conditions for your theorem to be true.\n  For example you won't find `a - b + b = a` for natural numbers in the library because it's false!\n  Search for `b \u2264 a \u2192 a - b + b = a` instead.\n* If a definition you made is in the goal,\n  you won't find any theorems about it in the library.\n  Try unfolding the definition using `unfold my_definition`.\n* If all else fails, ask on https://leanprover.zulipchat.com/,\n  and maybe we can improve the library and/or `library_search` for next time.\"\n\nadd_tactic_doc\n{ name        := \"library_search\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.library_search],\n  tags        := [\"search\", \"Try this\"] }\n\nend interactive\n\n/-- Invoking the hole command `library_search` (\"Use `library_search` to complete the goal\") calls\nthe tactic `library_search` to produce a proof term with the type of the hole.\n\nRunning it on\n\n```lean\nexample : 0 < 1 :=\n{!!}\n```\n\nproduces\n\n```lean\nexample : 0 < 1 :=\nnat.one_pos\n```\n-/\n@[hole_command] meta def library_search_hole_cmd : hole_command :=\n{ name := \"library_search\",\n  descr := \"Use `library_search` to complete the goal.\",\n  action := \u03bb _, do\n    script \u2190 library_search,\n    -- Is there a better API for dropping the 'Try this: exact ' prefix on this string?\n    return [((script.get_rest \"Try this: exact \").get_or_else script, \"by library_search\")] }\n\nadd_tactic_doc\n{ name        := \"library_search\",\n  category    := doc_category.hole_cmd,\n  decl_names  := [`tactic.library_search_hole_cmd],\n  tags        := [\"search\", \"Try this\"] }\n\nend tactic\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/tactic/suggest.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.39592882021546266}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.Mathport.Rename\nimport Std.Data.List.Basic\nimport Std.Data.List.Lemmas\nimport Mathlib.Init.Data.Nat.Notation\nimport Mathlib.Init.Data.Nat.Lemmas\nimport Mathlib.Init.Data.List.Basic\n/-!\nLemmas for `List` not (yet) in `Std`\n-/\n\n\nopen List Nat\n\nnamespace List\n\n#align list.length_map\u2082 List.length_zipWith\n\n#align list.ball_nil List.forall_mem_nil\n#align list.ball_cons List.forall_mem_cons\u2093 -- explicit \u2192 implicit arguments\n#align list.mem_cons_iff List.mem_cons\n#align list.sublist.cons2 List.Sublist.cons\u2082\n\nsection MapAccumr\n\nvariable {\u03c6 : Type w\u2081} {\u03c3 : Type w\u2082}\n\n/-- Runs a function over a list returning the intermediate results and a\na final result.\n-/\ndef mapAccumr (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2) : List \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 List \u03b2\n  | [], c => (c, [])\n  | y :: yr, c =>\n    let r := mapAccumr f yr c\n    let z := f y r.1\n    (z.1, z.2 :: r.2)\n#align list.map_accumr List.mapAccumr\n\n/-- Length of the list obtained by `mapAccumr`. -/\n@[simp]\ntheorem length_mapAccumr : \u2200 (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2) (x : List \u03b1) (s : \u03c3),\n    length (mapAccumr f x s).2 = length x\n  | f, _ :: x, s => congrArg succ (length_mapAccumr f x s)\n  | _, [], _ => rfl\n#align list.length_map_accumr List.length_mapAccumr\n\nend MapAccumr\nsection MapAccumr\u2082\n\nvariable {\u03c6 : Type w\u2081} {\u03c3 : Type w\u2082}\n\n/-- Runs a function over two lists returning the intermediate results and a\n a final result.\n-/\ndef mapAccumr\u2082 (f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6) : List \u03b1 \u2192 List \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 List \u03c6\n  | [], _, c => (c, [])\n  | _, [], c => (c, [])\n  | x :: xr, y :: yr, c =>\n    let r := mapAccumr\u2082 f xr yr c\n    let q := f x y r.1\n    (q.1, q.2 :: r.2)\n#align list.map_accumr\u2082 List.mapAccumr\u2082\n\n/-- Length of a list obtained using `mapAccumr\u2082`. -/\n@[simp]\ntheorem length_mapAccumr\u2082 :\n    \u2200 (f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6) (x y c), length (mapAccumr\u2082 f x y c).2 = min (length x) (length y)\n  | f, _ :: x, _ :: y, c =>\n    calc\n      succ (length (mapAccumr\u2082 f x y c).2) = succ (min (length x) (length y)) :=\n        congrArg succ (length_mapAccumr\u2082 f x y c)\n      _ = min (succ (length x)) (succ (length y)) := Eq.symm (min_succ_succ (length x) (length y))\n\n  | _, _ :: _, [], _ => rfl\n  | _, [], _ :: _, _ => rfl\n  | _, [], [], _ => rfl\n#align list.length_map_accumr\u2082 List.length_mapAccumr\u2082\n\nend MapAccumr\u2082\n\nend List\n\n#align list.length_zip_with List.length_zipWith\n#align list.mem_replicate List.mem_replicate\n#align list.eq_of_mem_replicate List.eq_of_mem_replicate\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Init/Data/List/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.3958645253254016}}
{"text": "-- Infinite series of analytic functions\n\nimport analysis.analytic.basic\nimport data.complex.basic\nimport data.real.basic\nimport data.real.ennreal\nimport data.real.nnreal\nimport data.real.pi.bounds\nimport data.set.basic\nimport topology.metric_space.basic\nimport topology.uniform_space.uniform_convergence\nimport topology.algebra.infinite_sum\n\nimport analytic\nimport bounds\nimport simple\nimport tactics\nimport uniform\n\nopen complex (abs)\nopen filter (at_top)\nopen metric (ball closed_ball sphere)\nopen_locale real nnreal ennreal topological_space\n\nnoncomputable theory\n\n-- Summability restricted to sets\ndef summable_on (f : \u2115 \u2192 \u2102 \u2192 \u2102) (s : set \u2102) := \u2200 z, z \u2208 s \u2192 summable (\u03bb n, f n z)\ndef has_sum_on (f : \u2115 \u2192 \u2102 \u2192 \u2102) (g : \u2102 \u2192 \u2102) (s : set \u2102) := \u2200 z, z \u2208 s \u2192 has_sum (\u03bb n, f n z) (g z)\nnoncomputable def tsum_on (f : \u2115 \u2192 \u2102 \u2192 \u2102) := \u03bb z, tsum (\u03bb n, f n z)\ndef has_uniform_sum (f : \u2115 \u2192 \u2102 \u2192 \u2102) (g : \u2102 \u2192 \u2102) (s : set \u2102) :=\n  tendsto_uniformly_on (\u03bb (N : finset \u2115) z, N.sum (\u03bb n, f n z)) g at_top s\n\n-- Uniform vanishing means late sums are uniformly small\ndef uniform_vanishing (f : \u2115 \u2192 \u2102 \u2192 \u2102) (s : set \u2102) :=\n  \u2200 e : \u211d, e > 0 \u2192 \u2203 n : \u2115, \u2200 (N : finset \u2115) z, late N n \u2192 z \u2208 s \u2192 N.sum (\u03bb n, abs (f n z)) < e\n\nlemma uniform_vanishing_to_summable {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102} {z : \u2102}\n    (zs : z \u2208 s) (h : uniform_vanishing f s) : summable (\u03bb n, f n z) := begin\n  rw [summable_iff_cauchy_seq_finset, metric.cauchy_seq_iff],\n  intros e ep,\n  rcases h e ep with \u27e8m,hm\u27e9,\n  existsi finset.range m,\n  intros A HA B HB,\n  calc dist (A.sum (\u03bb n, f n z)) (B.sum (\u03bb n, f n z)) \u2264 (A \u2206 B).sum (\u03bb n, abs (f n z)) : symm_diff_bound _ _ _\n  ... < e : hm (A \u2206 B) z (symm_diff_late HA HB) zs\nend\n\nlemma uniform_vanishing_to_uniform_cauchy_series {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102}\n    (h : uniform_vanishing f s) : uniform_cauchy_seq_on (\u03bb (N : finset \u2115) z, N.sum (\u03bb n, f n z)) at_top s := begin\n  rw metric.uniform_cauchy_seq_on_iff,\n  intros e ep,\n  rcases h e ep with \u27e8m,hm\u27e9,\n  existsi finset.range m,\n  intros A HA B HB z zs,\n  calc dist (A.sum (\u03bb n, f n z)) (B.sum (\u03bb n, f n z)) \u2264 (A \u2206 B).sum (\u03bb n, abs (f n z)) : symm_diff_bound _ _ _\n  ... < e : hm (A \u2206 B) z (symm_diff_late HA HB) zs\nend\n\nlemma uniform_vanishing_to_tendsto_uniformly_on {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102}\n    (h : uniform_vanishing f s) : has_uniform_sum f (tsum_on f) s := begin\n  rw [has_uniform_sum, metric.tendsto_uniformly_on_iff],\n  intros e ep,\n  rcases h (e/4) (by bound) with \u27e8m,hm\u27e9,\n  rw filter.eventually_at_top,\n  existsi finset.range m,\n  intros N Nm z zs,\n  rw tsum_on, simp,\n  generalize G : tsum (\u03bb n, f n z) = g,\n  have S : summable (\u03bb n, f n z) := uniform_vanishing_to_summable zs h,\n  have GS : has_sum (\u03bb n, f n z) g, { rw \u2190G, exact summable.has_sum S }, clear S,\n  rw has_sum at GS,\n  rw metric.tendsto_at_top at GS,\n  rcases GS (e/4) (by bound) with \u27e8M,HM\u27e9, clear GS G h,\n  set A := N \u222a (M \\ N),\n  have AM : M \u2286 A := simple.subset_union_sdiff _ _,\n  simp at HM,\n  specialize HM A AM,\n  rw dist_comm at HM,\n  calc dist g (N.sum (\u03bb n, f n z)) \u2264 dist g (A.sum (\u03bb n, f n z)) + dist (A.sum (\u03bb n, f n z)) (N.sum (\u03bb n, f n z)) : by bound\n  ... \u2264 e/4 + dist (A.sum (\u03bb n, f n z)) (N.sum (\u03bb n, f n z)) : by bound\n  ... = e/4 + dist (N.sum (\u03bb n, f n z) + (M \\ N).sum (\u03bb n, f n z)) (N.sum (\u03bb n, f n z))\n      : by rw finset.sum_union finset.disjoint_sdiff\n  ... = e/4 + abs (N.sum (\u03bb n, f n z) + (M \\ N).sum (\u03bb n, f n z) - N.sum (\u03bb n, f n z)) : by rw complex.dist_eq\n  ... = e/4 + abs ((M \\ N).sum (\u03bb n, f n z)) : by ring_nf\n  ... \u2264 e/4 + (M \\ N).sum (\u03bb n, abs (f n z)) : by bound [simple.finset_complex_abs_sum_le (M \\ N) (\u03bb n, f n z)]\n  ... \u2264 e/4 + e/4 : by bound [hm (M \\ N) z (sdiff_late M Nm) zs]\n  ... = e/2 : by ring\n  ... < e : by bound\nend\n\n-- Geometric bounds with c \u2264 0 are degenerate\nlemma c_nonpos.degenerate {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102} {c a : \u211d}\n    (c0 : c \u2264 0) (a0 : 0 \u2264 a) (hf : \u2200 n z, z \u2208 s \u2192 abs (f n z) \u2264 c * a^n)\n    : \u2200 n z, z \u2208 s \u2192 f n z = 0 := begin\n  intros n z zs, specialize hf n z zs,\n  have ca : c * a^n \u2264 0 := mul_nonpos_iff.mpr (or.inr \u27e8c0, by bound\u27e9),\n  exact complex.abs_eq_zero.mp (le_antisymm (trans hf ca) (complex.abs_nonneg _))\nend\n\n-- Uniformly exponentially converging series converge uniformly.\ntheorem fast_series_converge_uniformly_on {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102} {c a : \u211d}\n    (a0 : 0 \u2264 a) (a1 : a < 1) (hf : \u2200 n z, z \u2208 s \u2192 abs (f n z) \u2264 c * a^n)\n    : has_uniform_sum f (tsum_on f) s := begin\n  by_cases c0 : c \u2264 0, {\n    have fz := c_nonpos.degenerate c0 a0 hf, simp at fz,\n    rw [has_uniform_sum, metric.tendsto_uniformly_on_iff],\n    intros e ep, apply filter.eventually_of_forall, intros n z zs,\n    rw tsum_on, simp,\n    simp_rw fz _ z zs, simp,\n    assumption\n  }, {\n    simp at c0,\n    apply uniform_vanishing_to_tendsto_uniformly_on,\n    -- \u2200 e : \u211d, e > 0 \u2192 \u2203 n : \u2115, \u2200 (N : finset \u2115) z, late N n \u2192 z \u2208 s \u2192 N.sum (\u03bb n, abs (f n z)) < e\n    intros e ep,\n    set t := (1-\u2191a)/\u2191c*(e/2),\n    have tp : t > 0 := by bound,\n    rcases exists_pow_lt_of_lt_one tp a1 with \u27e8n,nt\u27e9,\n    existsi n,\n    intros N z NL zs,\n    have a1p : 1 - (a : \u211d) > 0 := by bound,\n    calc N.sum (\u03bb n, abs (f n z)) \u2264 N.sum (\u03bb n, c * a^n) : finset.sum_le_sum (\u03bb n _, hf n z zs)\n    ... = c * N.sum (\u03bb n, a^n) : finset.mul_sum.symm\n    ... \u2264 c * (a^n * (1 - a)\u207b\u00b9) : by bound [late_geometric_bound _ (by bound) a1]\n    ... = a^n * (c * (1 - a)\u207b\u00b9) : by ring\n    ... \u2264 t * (c * (1 - a)\u207b\u00b9) : by bound\n    ... = (1 - a) / c * (e / 2) * (c * (1 - a)\u207b\u00b9) : rfl\n    ... = (1 - a) * (1 - a)\u207b\u00b9 * (c / c) * (e / 2) : by ring\n    ... = 1 * 1 * (e / 2) : by rw [field.mul_inv_cancel (ne_of_gt a1p), simple.div_self (ne_of_gt c0)]\n    ... = e / 2 : by ring\n    ... < e : by bound\n  }\nend\n\n-- Exponentially converging series converge.\ntheorem fast_series_converge_at {f : \u2115 \u2192 \u2102} {c a : \u211d}\n    (a0 : 0 \u2264 a) (a1 : a < 1) (hf : \u2200 n, abs (f n) \u2264 c * a^n) : summable f := begin\n  set s : set \u2102 := {0},\n  set g : \u2115 \u2192 \u2102 \u2192 \u2102 := \u03bb n _, f n,\n  have hg : \u2200 n z, z \u2208 s \u2192 abs (g n z) \u2264 c * a^n := \u03bb n z zs, hf n,\n  have u := fast_series_converge_uniformly_on a0 a1 hg,\n  simp at u,\n  rw has_uniform_sum at u,\n  rw tendsto_uniformly_on_singleton_iff_tendsto at u,\n  apply has_sum.summable, assumption\nend\n\n-- Finite sums of analytic functions are analytic\nlemma finite_sums_are_analytic {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102}\n    (h : \u2200 n, analytic_on \u2102 (f n) s) (N : finset \u2115) : analytic_on \u2102 (\u03bb z, N.sum (\u03bb n, f n z)) s := begin\n  induction N using finset.induction with a B aB hB, {\n    simp, intros z zs, exact entire.zero z\n  }, {\n    intros z zs,\n    simp_rw finset.sum_insert aB,\n    apply analytic_at.add,\n    exact h a z zs,\n    exact hB z zs\n  }\nend\n\n-- Analytic series that converge exponentially converge to analytic functions.\ntheorem fast_series_converge {f : \u2115 \u2192 \u2102 \u2192 \u2102} {s : set \u2102} {c a : \u211d}\n    (o : is_open s) (a0 : 0 \u2264 a) (a1 : a < 1)\n    (h : \u2200 n, analytic_on \u2102 (f n) s) (hf : \u2200 n z, z \u2208 s \u2192 abs (f n z) \u2264 c * a^n)\n    : \u2203 (g : \u2102 \u2192 \u2102), analytic_on \u2102 g s \u2227 has_sum_on f g s := begin\n  set g := tsum_on f,\n  existsi g,\n  have su : has_uniform_sum f g s := fast_series_converge_uniformly_on a0 a1 hf,\n  constructor, {\n    refine uniform_analytic_lim o _ su,\n    exact finite_sums_are_analytic h\n  }, {\n    intros z zs,\n    exact summable.has_sum (fast_series_converge_at a0 a1 (\u03bb n, hf n z zs))\n  }\nend", "meta": {"author": "girving", "repo": "ray", "sha": "e0c501756e067711e2d3667d4b1d18045d83a313", "save_path": "github-repos/lean/girving-ray", "path": "github-repos/lean/girving-ray/ray-e0c501756e067711e2d3667d4b1d18045d83a313/src/series.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3958645218925019}}
{"text": "import category_theory.concrete_category.bundled_hom\nimport topology.category.Profinite\nimport logic.equiv.fin\n--import for_mathlib.concrete\nimport for_mathlib.CompHaus\nimport for_mathlib.topology\n\nimport pseudo_normed_group.with_Tinv\n\n/-!\n\n# The category of profinitely filtered pseudo-normed groups.\n\nThe category of profinite pseudo-normed groups, and the category of\nprofinitely filtered pseudo-normed groups equipped with an action of T\u207b\u00b9.\n\n-/\nuniverse variables u\n\nopen category_theory\nopen_locale nnreal\n\nlocal attribute [instance] type_pow\n\nnoncomputable theory\n\n/-- The category of CompHaus-ly filtered pseudo-normed groups. -/\ndef CompHausFiltPseuNormGrp : Type (u+1) :=\nbundled comphaus_filtered_pseudo_normed_group\n\nnamespace CompHausFiltPseuNormGrp\n\ndef bundled_hom : bundled_hom @comphaus_filtered_pseudo_normed_group_hom :=\n\u27e8@comphaus_filtered_pseudo_normed_group_hom.to_fun,\n @comphaus_filtered_pseudo_normed_group_hom.id,\n @comphaus_filtered_pseudo_normed_group_hom.comp,\n @comphaus_filtered_pseudo_normed_group_hom.coe_inj\u27e9\n\nlocal attribute [instance] bundled_hom\nattribute [derive [large_category, concrete_category]] CompHausFiltPseuNormGrp\n\ninstance : has_coe_to_sort CompHausFiltPseuNormGrp Type* := bundled.has_coe_to_sort\n\ninstance (M : CompHausFiltPseuNormGrp) : comphaus_filtered_pseudo_normed_group M := M.str\n\n/-- Construct a bundled `CompHausFiltPseuNormGrp` from the underlying type and typeclass. -/\ndef of (M : Type u) [comphaus_filtered_pseudo_normed_group M] : CompHausFiltPseuNormGrp :=\nbundled.of M\n\n@[simp]\nlemma id_apply (M : CompHausFiltPseuNormGrp) (x : M) : (\ud835\udfd9 M : M \u27f6 M) x = x := rfl\n\nend CompHausFiltPseuNormGrp\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/category/CompHausFiltPseuNormGrp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3958645218925018}}
{"text": "/-\nCopyright (c) 2018 Sean Leather. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sean Leather, Mario Carneiro\n\nFinite maps over `multiset`.\n-/\nimport data.list.alist data.finset data.pfun\n\nuniverses u v w\nopen list\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v}\n\nnamespace multiset\n\n/-- Multiset of keys of an association multiset. -/\ndef keys (s : multiset (sigma \u03b2)) : multiset \u03b1 :=\ns.map sigma.fst\n\n@[simp] theorem coe_keys {l : list (sigma \u03b2)} :\n  keys (l : multiset (sigma \u03b2)) = (l.keys : multiset \u03b1) :=\nrfl\n\n/-- `nodupkeys s` means that `s` has no duplicate keys. -/\ndef nodupkeys (s : multiset (sigma \u03b2)) : Prop :=\nquot.lift_on s list.nodupkeys (\u03bb s t p, propext $ perm_nodupkeys p)\n\n@[simp] theorem coe_nodupkeys {l : list (sigma \u03b2)} : @nodupkeys \u03b1 \u03b2 l \u2194 l.nodupkeys := iff.rfl\n\nend multiset\n\n/-- `finmap \u03b2` is the type of finite maps over a multiset. It is effectively\n  a quotient of `alist \u03b2` by permutation of the underlying list. -/\nstructure finmap (\u03b2 : \u03b1 \u2192 Type v) : Type (max u v) :=\n(entries : multiset (sigma \u03b2))\n(nodupkeys : entries.nodupkeys)\n\n/-- The quotient map from `alist` to `finmap`. -/\ndef alist.to_finmap (s : alist \u03b2) : finmap \u03b2 := \u27e8s.entries, s.nodupkeys\u27e9\n\nlocal notation `\u27e6`:max a `\u27e7`:0 := alist.to_finmap a\n\ntheorem alist.to_finmap_eq {s\u2081 s\u2082 : alist \u03b2} :\n  \u27e6s\u2081\u27e7 = \u27e6s\u2082\u27e7 \u2194 s\u2081.entries ~ s\u2082.entries :=\nby cases s\u2081; cases s\u2082; simp [alist.to_finmap]\n\n@[simp] theorem alist.to_finmap_entries (s : alist \u03b2) : \u27e6s\u27e7.entries = s.entries := rfl\n\nnamespace finmap\nopen alist\n\n/-- Lift a permutation-respecting function on `alist` to `finmap`. -/\n@[elab_as_eliminator] def lift_on\n  {\u03b3} (s : finmap \u03b2) (f : alist \u03b2 \u2192 \u03b3)\n  (H : \u2200 a b : alist \u03b2, a.entries ~ b.entries \u2192 f a = f b) : \u03b3 :=\nbegin\n  refine (quotient.lift_on s.1 (\u03bb l, (\u27e8_, \u03bb nd, f \u27e8l, nd\u27e9\u27e9 : roption \u03b3))\n    (\u03bb l\u2081 l\u2082 p, roption.ext' (perm_nodupkeys p) _) : roption \u03b3).get _,\n  { exact \u03bb h\u2081 h\u2082, H _ _ (by exact p) },\n  { have := s.nodupkeys, rcases s.entries with \u27e8l\u27e9, exact id }\nend\n\n@[simp] theorem lift_on_to_finmap {\u03b3} (s : alist \u03b2) (f : alist \u03b2 \u2192 \u03b3) (H) :\n  lift_on \u27e6s\u27e7 f H = f s := by cases s; refl\n\n/-- Lift a permutation-respecting function on 2 `alist`s to 2 `finmap`s. -/\n@[elab_as_eliminator] def lift_on\u2082\n  {\u03b3} (s\u2081 s\u2082 : finmap \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3)\n  (H : \u2200 a\u2081 b\u2081 a\u2082 b\u2082 : alist \u03b2, a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082) : \u03b3 :=\nlift_on s\u2081\n  (\u03bb l\u2081, lift_on s\u2082 (f l\u2081) (\u03bb b\u2081 b\u2082 p, H _ _ _ _ (perm.refl _) p))\n  (\u03bb a\u2081 a\u2082 p, have H' : f a\u2081 = f a\u2082 := funext (\u03bb _, H _ _ _ _ p (perm.refl _)), by simp only [H'])\n\n@[simp] theorem lift_on\u2082_to_finmap {\u03b3} (s\u2081 s\u2082 : alist \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3) (H) :\n  lift_on\u2082 \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 f H = f s\u2081 s\u2082 :=\nby cases s\u2081; cases s\u2082; refl\n\n@[elab_as_eliminator] theorem induction_on\n  {C : finmap \u03b2 \u2192 Prop} (s : finmap \u03b2) (H : \u2200 (a : alist \u03b2), C \u27e6a\u27e7) : C s :=\nby rcases s with \u27e8\u27e8a\u27e9, h\u27e9; exact H \u27e8a, h\u27e9\n\n@[elab_as_eliminator] theorem induction_on\u2082 {C : finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop}\n  (s\u2081 s\u2082 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 : alist \u03b2), C \u27e6a\u2081\u27e7 \u27e6a\u2082\u27e7) : C s\u2081 s\u2082 :=\ninduction_on s\u2081 $ \u03bb l\u2081, induction_on s\u2082 $ \u03bb l\u2082, H l\u2081 l\u2082\n\n@[elab_as_eliminator] theorem induction_on\u2083 {C : finmap \u03b2 \u2192  finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop}\n  (s\u2081 s\u2082 s\u2083 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 a\u2083 : alist \u03b2), C \u27e6a\u2081\u27e7 \u27e6a\u2082\u27e7 \u27e6a\u2083\u27e7) : C s\u2081 s\u2082 s\u2083 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb l\u2081 l\u2082, induction_on s\u2083 $ \u03bb l\u2083, H l\u2081 l\u2082 l\u2083\n\n@[extensionality] theorem ext : \u2200 {s t : finmap \u03b2}, s.entries = t.entries \u2192 s = t\n| \u27e8l\u2081, h\u2081\u27e9 \u27e8l\u2082, h\u2082\u27e9 H := by congr'\n\n@[simp] theorem ext_iff {s t : finmap \u03b2} : s.entries = t.entries \u2194 s = t :=\n\u27e8ext, congr_arg _\u27e9\n\n/-- The predicate `a \u2208 s` means that `s` has a value associated to the key `a`. -/\ninstance : has_mem \u03b1 (finmap \u03b2) := \u27e8\u03bb a s, a \u2208 s.entries.keys\u27e9\n\ntheorem mem_def {a : \u03b1} {s : finmap \u03b2} :\n  a \u2208 s \u2194 a \u2208 s.entries.keys := iff.rfl\n\n@[simp] theorem mem_to_finmap {a : \u03b1} {s : alist \u03b2} :\n  a \u2208 \u27e6s\u27e7 \u2194 a \u2208 s := iff.rfl\n\n/-- The set of keys of a finite map. -/\ndef keys (s : finmap \u03b2) : finset \u03b1 :=\n\u27e8s.entries.keys, induction_on s keys_nodup\u27e9\n\n@[simp] theorem keys_val (s : alist \u03b2) : (keys \u27e6s\u27e7).val = s.keys := rfl\n\n@[simp] theorem keys_ext {s\u2081 s\u2082 : alist \u03b2} :\n  keys \u27e6s\u2081\u27e7 = keys \u27e6s\u2082\u27e7 \u2194 s\u2081.keys ~ s\u2082.keys :=\nby simp [keys, alist.keys]\n\ntheorem mem_keys {a : \u03b1} {s : finmap \u03b2} : a \u2208 s.keys \u2194 a \u2208 s :=\ninduction_on s $ \u03bb s, alist.mem_keys\n\n/-- The empty map. -/\ninstance : has_emptyc (finmap \u03b2) := \u27e8\u27e80, nodupkeys_nil\u27e9\u27e9\n\n@[simp] theorem empty_to_finmap (s : alist \u03b2) :\n  (\u27e6\u2205\u27e7 : finmap \u03b2) = \u2205 := rfl\n\ntheorem not_mem_empty {a : \u03b1} : a \u2209 (\u2205 : finmap \u03b2) :=\nmultiset.not_mem_zero a\n\n@[simp] theorem keys_empty : (\u2205 : finmap \u03b2).keys = \u2205 := rfl\n\n/-- The singleton map. -/\ndef singleton (a : \u03b1) (b : \u03b2 a) : finmap \u03b2 :=\n\u27e8\u27e8a, b\u27e9::0, nodupkeys_singleton _\u27e9\n\n@[simp] theorem keys_singleton (a : \u03b1) (b : \u03b2 a) :\n  (singleton a b).keys = finset.singleton a := rfl\n\nvariables [decidable_eq \u03b1]\n\ninstance has_decidable_eq [\u2200 a, decidable_eq (\u03b2 a)] : decidable_eq (finmap \u03b2)\n| s\u2081 s\u2082 := decidable_of_iff _ ext_iff\n\n/-- Look up the value associated to a key in a map. -/\ndef lookup (a : \u03b1) (s : finmap \u03b2) : option (\u03b2 a) :=\nlift_on s (lookup a) (\u03bb s t, perm_lookup)\n\n@[simp] theorem lookup_to_finmap (a : \u03b1) (s : alist \u03b2) :\n  lookup a \u27e6s\u27e7 = s.lookup a := rfl\n\n@[simp] theorem lookup_empty (a) : lookup a (\u2205 : finmap \u03b2) = none :=\nrfl\n\ntheorem lookup_is_some {a : \u03b1} {s : finmap \u03b2} :\n  (s.lookup a).is_some \u2194 a \u2208 s :=\ninduction_on s $ \u03bb s, alist.lookup_is_some\n\ntheorem lookup_eq_none {a} {s : finmap \u03b2} : lookup a s = none \u2194 a \u2209 s :=\ninduction_on s $ \u03bb s, alist.lookup_eq_none\n\ninstance (a : \u03b1) (s : finmap \u03b2) : decidable (a \u2208 s) :=\ndecidable_of_iff _ lookup_is_some\n\n/-- Replace a key with a given value in a finite map.\n  If the key is not present it does nothing. -/\ndef replace (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6replace a b t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_replace p\n\n@[simp] theorem replace_to_finmap (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) :\n  replace a b \u27e6s\u27e7 = \u27e6s.replace a b\u27e7 := by simp [replace]\n\n@[simp] theorem keys_replace (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) :\n  (replace a b s).keys = s.keys :=\ninduction_on s $ \u03bb s, by simp\n\n@[simp] theorem mem_replace {a a' : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} :\n  a' \u2208 replace a b s \u2194 a' \u2208 s :=\ninduction_on s $ \u03bb s, by simp\n\n/-- Fold a commutative function over the key-value pairs in the map -/\ndef foldl {\u03b4 : Type w} (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4)\n  (H : \u2200 d a\u2081 b\u2081 a\u2082 b\u2082, f (f d a\u2081 b\u2081) a\u2082 b\u2082 = f (f d a\u2082 b\u2082) a\u2081 b\u2081)\n  (d : \u03b4) (m : finmap \u03b2) : \u03b4 :=\nm.entries.foldl (\u03bb d s, f d s.1 s.2) (\u03bb d s t, H _ _ _ _ _) d\n\n/-- Erase a key from the map. If the key is not present it does nothing. -/\ndef erase (a : \u03b1) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6erase a t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_erase p\n\n@[simp] theorem erase_to_finmap (a : \u03b1) (s : alist \u03b2) :\n  erase a \u27e6s\u27e7 = \u27e6s.erase a\u27e7 := by simp [erase]\n\n@[simp] \n\n@[simp] theorem keys_erase (a : \u03b1) (s : finmap \u03b2) :\n  (erase a s).keys = s.keys.erase a :=\ninduction_on s $ \u03bb s, by simp\n\n@[simp] theorem mem_erase {a a' : \u03b1} {s : finmap \u03b2} : a' \u2208 erase a s \u2194 a' \u2260 a \u2227 a' \u2208 s :=\ninduction_on s $ \u03bb s, by simp\n\n@[simp] theorem lookup_erase (a) (s : finmap \u03b2) : lookup a (erase a s) = none :=\ninduction_on s $ lookup_erase a\n\n@[simp] theorem lookup_erase_ne {a a'} {s : finmap \u03b2} (h : a \u2260 a') :\n  lookup a (erase a' s) = lookup a s :=\ninduction_on s $ \u03bb s, lookup_erase_ne h\n\n/- insert -/\n\n/-- Insert a key-value pair into a finite map, replacing any existing pair with\n  the same key. -/\ndef insert (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6insert a b t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_insert p\n\n@[simp] theorem insert_to_finmap (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) :\n  insert a b \u27e6s\u27e7 = \u27e6s.insert a b\u27e7 := by simp [insert]\n\ntheorem insert_entries_of_neg {a : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} : a \u2209 s \u2192\n  (insert a b s).entries = \u27e8a, b\u27e9 :: s.entries :=\ninduction_on s $ \u03bb s h,\nby simp [insert_entries_of_neg (mt mem_to_finmap.1 h)]\n\n@[simp] theorem mem_insert {a a' : \u03b1} {b' : \u03b2 a'} {s : finmap \u03b2} :\n  a \u2208 insert a' b' s \u2194 a = a' \u2228 a \u2208 s :=\ninduction_on s mem_insert\n\n@[simp] theorem lookup_insert {a} {b : \u03b2 a} (s : finmap \u03b2) :\n  lookup a (insert a b s) = some b :=\ninduction_on s $ \u03bb s,\nby simp only [insert_to_finmap, lookup_to_finmap, lookup_insert]\n\n/- extract -/\n\n/-- Erase a key from the map, and return the corresponding value, if found. -/\ndef extract (a : \u03b1) (s : finmap \u03b2) : option (\u03b2 a) \u00d7 finmap \u03b2 :=\nlift_on s (\u03bb t, prod.map id to_finmap (extract a t)) $\n\u03bb s\u2081 s\u2082 p, by simp [perm_lookup p, to_finmap_eq, perm_erase p]\n\n@[simp] theorem extract_eq_lookup_erase (a : \u03b1) (s : finmap \u03b2) :\n  extract a s = (lookup a s, erase a s) :=\ninduction_on s $ \u03bb s, by simp [extract]\n\n/- union -/\n\n/-- `s\u2081 \u222a s\u2082` is the key-based union of two finite maps. It is left-biased: if\nthere exists an `a \u2208 s\u2081`, `lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081`. -/\ndef union (s\u2081 s\u2082 : finmap \u03b2) : finmap \u03b2 :=\nlift_on\u2082 s\u2081 s\u2082 (\u03bb s\u2081 s\u2082, \u27e6s\u2081 \u222a s\u2082\u27e7) $\n\u03bb s\u2081 s\u2082 s\u2083 s\u2084 p\u2081\u2083 p\u2082\u2084, to_finmap_eq.mpr $ perm_union p\u2081\u2083 p\u2082\u2084\n\ninstance : has_union (finmap \u03b2) := \u27e8union\u27e9\n\n@[simp] theorem mem_union {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb _ _, mem_union\n\n@[simp] theorem union_to_finmap (s\u2081 s\u2082 : alist \u03b2) : \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2081 \u222a s\u2082\u27e7 :=\nby simp [(\u222a), union]\n\ntheorem keys_union {s\u2081 s\u2082 : finmap \u03b2} : (s\u2081 \u222a s\u2082).keys = s\u2081.keys \u222a s\u2082.keys :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, finset.ext' $ by simp [keys]\n\n@[simp] theorem lookup_union_left {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2208 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, lookup_union_left\n\n@[simp] theorem lookup_union_right {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2209 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, lookup_union_right\n\n@[simp] theorem mem_lookup_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : finmap \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2082) \u2194 b \u2208 lookup a s\u2081 \u2228 a \u2209 s\u2081 \u2227 b \u2208 lookup a s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, mem_lookup_union\n\ntheorem mem_lookup_union_middle {a} {b : \u03b2 a} {s\u2081 s\u2082 s\u2083 : finmap \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2083) \u2192 a \u2209 s\u2082 \u2192 b \u2208 lookup a (s\u2081 \u222a s\u2082 \u222a s\u2083) :=\ninduction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb s\u2081 s\u2082 s\u2083, mem_lookup_union_middle\n\nend finmap\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/data/finmap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.7090191337850932, "lm_q1q2_score": 0.3958645218925018}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Jo\u00ebl Riou\n-/\nimport algebra.homology.homotopy\nimport category_theory.abelian.homology\n\n/-!\n# Quasi-isomorphisms\n\nA chain map is a quasi-isomorphism if it induces isomorphisms on homology.\n\n## Future work\n\nDefine the derived category as the localization at quasi-isomorphisms?\n-/\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables {\u03b9 : Type*}\nvariables {V : Type u} [category.{v} V] [has_zero_morphisms V] [has_zero_object V]\nvariables [has_equalizers V] [has_images V] [has_image_maps V] [has_cokernels V]\nvariables {c : complex_shape \u03b9} {C D E : homological_complex V c}\n\n/--\nA chain map is a quasi-isomorphism if it induces isomorphisms on homology.\n-/\nclass quasi_iso (f : C \u27f6 D) : Prop :=\n(is_iso : \u2200 i, is_iso ((homology_functor V c i).map f))\n\nattribute [instance] quasi_iso.is_iso\n\n@[priority 100]\ninstance quasi_iso_of_iso (f : C \u27f6 D) [is_iso f] : quasi_iso f :=\n{ is_iso := \u03bb i, begin\n    change is_iso (((homology_functor V c i).map_iso (as_iso f)).hom),\n    apply_instance,\n  end }\n\ninstance quasi_iso_comp (f : C \u27f6 D) [quasi_iso f] (g : D \u27f6 E) [quasi_iso g] : quasi_iso (f \u226b g) :=\n{ is_iso := \u03bb i, begin\n    rw functor.map_comp,\n    apply_instance,\n  end }\n\nlemma quasi_iso_of_comp_left (f : C \u27f6 D) [quasi_iso f] (g : D \u27f6 E) [quasi_iso (f \u226b g)] :\n  quasi_iso g :=\n{ is_iso := \u03bb i, is_iso.of_is_iso_fac_left ((homology_functor V c i).map_comp f g).symm }\n\nlemma quasi_iso_of_comp_right (f : C \u27f6 D) (g : D \u27f6 E) [quasi_iso g] [quasi_iso (f \u226b g)] :\n  quasi_iso f :=\n{ is_iso := \u03bb i, is_iso.of_is_iso_fac_right ((homology_functor V c i).map_comp f g).symm }\n\nnamespace homotopy_equiv\n\nsection\nvariables {W : Type*} [category W] [preadditive W] [has_cokernels W] [has_images W]\n  [has_equalizers W] [has_zero_object W] [has_image_maps W]\n\n/-- An homotopy equivalence is a quasi-isomorphism. -/\nlemma to_quasi_iso {C D : homological_complex W c} (e : homotopy_equiv C D) :\n  quasi_iso e.hom :=\n\u27e8\u03bb i, begin\n  refine \u27e8\u27e8(homology_functor W c i).map e.inv, _\u27e9\u27e9,\n  simp only [\u2190 functor.map_comp, \u2190 (homology_functor W c i).map_id],\n  split; apply homology_map_eq_of_homotopy,\n  exacts [e.homotopy_hom_inv_id, e.homotopy_inv_hom_id],\nend\u27e9\n\nlemma to_quasi_iso_inv {C D : homological_complex W c} (e : homotopy_equiv C D) (i : \u03b9) :\n  (@as_iso _ _ _ _ _ (e.to_quasi_iso.1 i)).inv = (homology_functor W c i).map e.inv :=\nbegin\n  symmetry,\n  simp only [\u2190iso.hom_comp_eq_id, as_iso_hom, \u2190functor.map_comp, \u2190(homology_functor W c i).map_id,\n    homology_map_eq_of_homotopy e.homotopy_hom_inv_id _],\nend\n\nend\nend homotopy_equiv\nnamespace homological_complex.hom\nsection to_single\u2080\nvariables {W : Type*} [category W] [abelian W]\n\nsection\nvariables {X : chain_complex W \u2115} {Y : W} (f : X \u27f6 ((chain_complex.single\u2080 _).obj Y))\n  [hf : quasi_iso f]\n\n/-- If a chain map `f : X \u27f6 Y[0]` is a quasi-isomorphism, then the cokernel of the differential\n`d : X\u2081 \u2192 X\u2080` is isomorphic to `Y.` -/\nnoncomputable def to_single\u2080_cokernel_at_zero_iso : cokernel (X.d 1 0) \u2245 Y :=\n(X.homology_zero_iso.symm.trans ((@as_iso _ _ _ _ _ (hf.1 0)).trans\n  ((chain_complex.homology_functor_0_single\u2080 W).app Y)))\n\nlemma to_single\u2080_cokernel_at_zero_iso_hom_eq [hf : quasi_iso f] :\n  f.to_single\u2080_cokernel_at_zero_iso.hom = cokernel.desc (X.d 1 0) (f.f 0)\n    (by rw \u2190f.2 1 0 rfl; exact comp_zero) :=\nbegin\n  ext,\n  dunfold to_single\u2080_cokernel_at_zero_iso chain_complex.homology_zero_iso homology_of_zero_right\n    homology.map_iso chain_complex.homology_functor_0_single\u2080 cokernel.map,\n  dsimp,\n  simp only [cokernel.\u03c0_desc, category.assoc, homology.map_desc, cokernel.\u03c0_desc_assoc],\n  simp [homology.desc, iso.refl_inv (X.X 0)],\nend\n\nlemma to_single\u2080_epi_at_zero [hf : quasi_iso f] :\n  epi (f.f 0) :=\nbegin\n  constructor,\n  intros Z g h Hgh,\n  rw [\u2190cokernel.\u03c0_desc (X.d 1 0) (f.f 0) (by rw \u2190f.2 1 0 rfl; exact comp_zero),\n    \u2190to_single\u2080_cokernel_at_zero_iso_hom_eq] at Hgh,\n  rw (@cancel_epi _ _ _ _ _ _ (epi_comp _ _) _ _).1 Hgh,\nend\n\nlemma to_single\u2080_exact_d_f_at_zero [hf : quasi_iso f] :\n  exact (X.d 1 0) (f.f 0) :=\nbegin\n  rw preadditive.exact_iff_homology_zero,\n  have h : X.d 1 0 \u226b f.f 0 = 0,\n  { simp only [\u2190 f.2 1 0 rfl, chain_complex.single\u2080_obj_X_d, comp_zero], },\n  refine \u27e8h, nonempty.intro (homology_iso_kernel_desc _ _ _ \u226a\u226b _)\u27e9,\n  { suffices : is_iso (cokernel.desc _ _ h),\n    { haveI := this, apply kernel.of_mono, },\n      rw \u2190to_single\u2080_cokernel_at_zero_iso_hom_eq,\n      apply_instance }\nend\n\nlemma to_single\u2080_exact_at_succ [hf : quasi_iso f] (n : \u2115) :\n  exact (X.d (n + 2) (n + 1)) (X.d (n + 1) n) :=\n(preadditive.exact_iff_homology_zero _ _).2 \u27e8X.d_comp_d _ _ _,\n\u27e8(chain_complex.homology_succ_iso _ _).symm.trans\n  ((@as_iso _ _ _ _ _ (hf.1 (n + 1))).trans homology_zero_zero)\u27e9\u27e9\n\nend\nsection\nvariables {X : cochain_complex W \u2115} {Y : W}\n  (f : (cochain_complex.single\u2080 _).obj Y \u27f6 X)\n\n/-- If a cochain map `f : Y[0] \u27f6 X` is a quasi-isomorphism, then the kernel of the differential\n`d : X\u2080 \u2192 X\u2081` is isomorphic to `Y.` -/\nnoncomputable def from_single\u2080_kernel_at_zero_iso [hf : quasi_iso f] : kernel (X.d 0 1) \u2245 Y :=\n(X.homology_zero_iso.symm.trans ((@as_iso _ _ _ _ _ (hf.1 0)).symm.trans\n  ((cochain_complex.homology_functor_0_single\u2080 W).app Y)))\n\nlemma from_single\u2080_kernel_at_zero_iso_inv_eq [hf : quasi_iso f] :\n  f.from_single\u2080_kernel_at_zero_iso.inv = kernel.lift (X.d 0 1) (f.f 0)\n    (by rw f.2 0 1 rfl; exact zero_comp) :=\nbegin\n  ext,\n  dunfold from_single\u2080_kernel_at_zero_iso cochain_complex.homology_zero_iso homology_of_zero_left\n    homology.map_iso cochain_complex.homology_functor_0_single\u2080 kernel.map,\n  simp only [iso.trans_inv, iso.app_inv, iso.symm_inv, category.assoc,\n    equalizer_as_kernel, kernel.lift_\u03b9],\n  dsimp,\n  simp only [category.assoc, homology.\u03c0_map, cokernel_zero_iso_target_hom,\n    cokernel_iso_of_eq_hom_comp_desc, kernel_subobject_arrow, homology.\u03c0_map_assoc,\n    is_iso.inv_comp_eq],\n  simp [homology.\u03c0, kernel_subobject_map_comp, iso.refl_hom (X.X 0), category.comp_id],\nend\n\n\n\nlemma from_single\u2080_exact_f_d_at_zero [hf : quasi_iso f] :\n  exact (f.f 0) (X.d 0 1) :=\nbegin\n  rw preadditive.exact_iff_homology_zero,\n  have h : f.f 0 \u226b X.d 0 1 = 0,\n  { simp only [homological_complex.hom.comm, cochain_complex.single\u2080_obj_X_d, zero_comp] },\n  refine \u27e8h, nonempty.intro (homology_iso_cokernel_lift _ _ _ \u226a\u226b _)\u27e9,\n  { suffices : is_iso (kernel.lift (X.d 0 1) (f.f 0) h),\n    { haveI := this, apply cokernel.of_epi },\n    rw \u2190from_single\u2080_kernel_at_zero_iso_inv_eq f,\n    apply_instance },\nend\n\nlemma from_single\u2080_exact_at_succ [hf : quasi_iso f] (n : \u2115) :\n  exact (X.d n (n + 1)) (X.d (n + 1) (n + 2)) :=\n(preadditive.exact_iff_homology_zero _ _).2\n  \u27e8X.d_comp_d _ _ _, \u27e8(cochain_complex.homology_succ_iso _ _).symm.trans\n  ((@as_iso _ _ _ _ _ (hf.1 (n + 1))).symm.trans homology_zero_zero)\u27e9\u27e9\n\nend\nend to_single\u2080\nend homological_complex.hom\n\nvariables {A : Type*} [category A] [abelian A] {B : Type*} [category B] [abelian B]\n  (F : A \u2964 B) [functor.additive F] [preserves_finite_limits F] [preserves_finite_colimits F]\n  [faithful F]\n\nlemma category_theory.functor.quasi_iso_of_map_quasi_iso\n  {C D : homological_complex A c} (f : C \u27f6 D)\n  (hf : quasi_iso ((F.map_homological_complex _).map f)) : quasi_iso f :=\n\u27e8\u03bb i, begin\n  haveI : is_iso (F.map ((homology_functor A c i).map f)),\n  { rw [\u2190 functor.comp_map, \u2190 nat_iso.naturality_2 (F.homology_functor_iso i) f,\n      functor.comp_map],\n    apply_instance, },\n  exact is_iso_of_reflects_iso _ F,\nend\u27e9\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/homology/quasi_iso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6513548714339144, "lm_q1q2_score": 0.395804365861628}}
{"text": "/-\nCopyright (c) 2022 Kexing Ying. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kexing Ying\n-/\nimport probability.martingale.borel_cantelli\nimport probability.conditional_expectation\nimport probability.independence\n\n/-!\n\n# The second Borel-Cantelli lemma\n\nThis file contains the second Borel-Cantelli lemma which states that, given a sequence of\nindependent sets `(s\u2099)` in a probability space, if `\u2211 n, \u03bc s\u2099 = \u221e`, then the limsup of `s\u2099` has\nmeasure 1. We employ a proof using L\u00e9vy's generalized Borel-Cantelli by choosing an appropriate\nfiltration.\n\n## Main result\n\n- `probability_theory.measure_limsup_eq_one`: the second Borel-Cantelli lemma.\n\n-/\n\nopen_locale measure_theory probability_theory ennreal big_operators topology\n\nopen measure_theory probability_theory measurable_space topological_space\n\nnamespace probability_theory\n\nvariables {\u03a9 : Type*} {m0 : measurable_space \u03a9} {\u03bc : measure \u03a9}\n  [is_probability_measure \u03bc]\n\nsection borel_cantelli\n\nvariables {\u03b9 \u03b2 : Type*} [linear_order \u03b9] [m\u03b2 : measurable_space \u03b2] [normed_add_comm_group \u03b2]\n  [borel_space \u03b2] {f : \u03b9 \u2192 \u03a9 \u2192 \u03b2} {i j : \u03b9} {s : \u03b9 \u2192 set \u03a9}\n\nlemma Indep_fun.indep_comap_natural_of_lt (hf : \u2200 i, strongly_measurable (f i))\n  (hfi : Indep_fun (\u03bb i, m\u03b2) f \u03bc) (hij : i < j) :\n  indep (measurable_space.comap (f j) m\u03b2) (filtration.natural f hf i) \u03bc :=\nbegin\n  suffices : indep (\u2a06 k \u2208 {j}, measurable_space.comap (f k) m\u03b2)\n    (\u2a06 k \u2208 {k | k \u2264 i}, measurable_space.comap (f k) m\u03b2) \u03bc,\n  { rwa supr_singleton at this },\n  exact indep_supr_of_disjoint (\u03bb k, (hf k).measurable.comap_le) hfi (by simpa),\nend\n\nlemma Indep_fun.condexp_natrual_ae_eq_of_lt\n  [second_countable_topology \u03b2] [complete_space \u03b2] [normed_space \u211d \u03b2]\n  (hf : \u2200 i, strongly_measurable (f i)) (hfi : Indep_fun (\u03bb i, m\u03b2) f \u03bc) (hij : i < j) :\n  \u03bc[f j | filtration.natural f hf i] =\u1d50[\u03bc] \u03bb \u03c9, \u03bc[f j] :=\ncondexp_indep_eq (hf j).measurable.comap_le (filtration.le _ _)\n  (comap_measurable $ f j).strongly_measurable\n  (hfi.indep_comap_natural_of_lt hf hij)\n\nlemma Indep_set.condexp_indicator_filtration_of_set_ae_eq\n  (hsm : \u2200 n, measurable_set (s n)) (hs : Indep_set s \u03bc) (hij : i < j) :\n  \u03bc[(s j).indicator (\u03bb \u03c9, 1 : \u03a9 \u2192 \u211d) | filtration_of_set hsm i] =\u1d50[\u03bc] \u03bb \u03c9, (\u03bc (s j)).to_real :=\nbegin\n  rw filtration.filtration_of_set_eq_natural hsm,\n  refine (Indep_fun.condexp_natrual_ae_eq_of_lt _ hs.Indep_fun_indicator hij).trans _,\n  { simp only [integral_indicator_const _ (hsm _), algebra.id.smul_eq_mul, mul_one] },\n  { apply_instance }\nend\n\nopen filter\n\n/-- **The second Borel-Cantelli lemma**: Given a sequence of independent sets `(s\u2099)` such that\n`\u2211 n, \u03bc s\u2099 = \u221e`, `limsup s\u2099` has measure 1. -/\nlemma measure_limsup_eq_one {s : \u2115 \u2192 set \u03a9}\n  (hsm : \u2200 n, measurable_set (s n)) (hs : Indep_set s \u03bc) (hs' : \u2211' n, \u03bc (s n) = \u221e) :\n  \u03bc (limsup s at_top) = 1 :=\nbegin\n  rw measure_congr (eventually_eq_set.2 (ae_mem_limsup_at_top_iff \u03bc $\n    measurable_set_filtration_of_set' hsm) :\n      (limsup s at_top : set \u03a9) =\u1d50[\u03bc] {\u03c9 | tendsto (\u03bb n, \u2211 k in finset.range n,\n        \u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d) | filtration_of_set hsm k] \u03c9) at_top at_top}),\n  suffices : {\u03c9 | tendsto (\u03bb n, \u2211 k in finset.range n,\n    \u03bc[(s (k + 1)).indicator (1 : \u03a9 \u2192 \u211d) | filtration_of_set hsm k] \u03c9) at_top at_top} =\u1d50[\u03bc] set.univ,\n  { rw [measure_congr this, measure_univ] },\n  have : \u2200\u1d50 \u03c9 \u2202\u03bc, \u2200 n, \u03bc[(s (n + 1)).indicator (1 : \u03a9 \u2192 \u211d) | filtration_of_set hsm n] \u03c9 = _ :=\n    ae_all_iff.2 (\u03bb n, hs.condexp_indicator_filtration_of_set_ae_eq hsm n.lt_succ_self),\n  filter_upwards [this] with \u03c9 h\u03c9,\n  refine eq_true_intro (_ : tendsto _ _ _),\n  simp_rw h\u03c9,\n  have htends : tendsto (\u03bb n, \u2211 k in finset.range n, \u03bc (s (k + 1))) at_top (\ud835\udcdd \u221e),\n  { rw \u2190 ennreal.tsum_add_one_eq_top hs' (measure_ne_top _ _),\n    exact ennreal.tendsto_nat_tsum _ },\n  rw ennreal.tendsto_nhds_top_iff_nnreal at htends,\n  refine tendsto_at_top_at_top_of_monotone' _ _,\n  { refine monotone_nat_of_le_succ (\u03bb n, _),\n    rw [\u2190 sub_nonneg, finset.sum_range_succ_sub_sum],\n    exact ennreal.to_real_nonneg },\n  { rintro \u27e8B, hB\u27e9,\n    refine not_eventually.2 (frequently_of_forall $ \u03bb n, _) (htends B.to_nnreal),\n    rw mem_upper_bounds at hB,\n    specialize hB (\u2211 (k : \u2115) in finset.range n, \u03bc (s (k + 1))).to_real _,\n    { refine \u27e8n, _\u27e9,\n      rw ennreal.to_real_sum,\n      exact \u03bb _ _, measure_ne_top _ _ },\n    { rw [not_lt, \u2190 ennreal.to_real_le_to_real (ennreal.sum_lt_top _).ne ennreal.coe_ne_top],\n      { exact hB.trans (by simp) },\n      { exact \u03bb _ _, measure_ne_top _ _ } } }\nend\n\nend borel_cantelli\n\nend probability_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/probability/borel_cantelli.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.3958043617490487}}
{"text": "/-\nCopyright (c) 2022 Thomas Browning. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Thomas Browning\n-/\nimport analysis.complex.circle\nimport topology.continuous_function.algebra\n\n/-!\n\n# Continuous Monoid Homs\n\nThis file defines the space of continuous homomorphisms between two topological groups.\n\n## Main definitions\n\n* `continuous_monoid_hom A B`: The continuous homomorphisms `A \u2192* B`.\n* `continuous_add_monoid_hom A B`: The continuous additive homomorphisms `A \u2192+ B`.\n-/\n\nopen_locale pointwise\n\nopen function\n\nvariables (F A B C D E : Type*) [monoid A] [monoid B] [monoid C] [monoid D] [comm_group E]\n  [topological_space A] [topological_space B] [topological_space C] [topological_space D]\n  [topological_space E] [topological_group E]\n\n/-- The type of continuous additive monoid homomorphisms from `A` to `B`.\n\nWhen possible, instead of parametrizing results over `(f : continuous_add_monoid_hom A B)`,\nyou should parametrize over `(F : Type*) [continuous_add_monoid_hom_class F A B] (f : F)`.\n\nWhen you extend this structure, make sure to extend `continuous_add_monoid_hom_class`. -/\nstructure continuous_add_monoid_hom (A B : Type*) [add_monoid A] [add_monoid B]\n  [topological_space A] [topological_space B] extends A \u2192+ B :=\n(continuous_to_fun : continuous to_fun)\n\n/-- The type of continuous monoid homomorphisms from `A` to `B`.\n\nWhen possible, instead of parametrizing results over `(f : continuous_monoid_hom A B)`,\nyou should parametrize over `(F : Type*) [continuous_monoid_hom_class F A B] (f : F)`.\n\nWhen you extend this structure, make sure to extend `continuous_add_monoid_hom_class`. -/\n@[to_additive]\nstructure continuous_monoid_hom extends A \u2192* B :=\n(continuous_to_fun : continuous to_fun)\n\nsection\nset_option old_structure_cmd true\n\n/-- `continuous_add_monoid_hom_class F A B` states that `F` is a type of continuous additive monoid\nhomomorphisms.\n\nYou should also extend this typeclass when you extend `continuous_add_monoid_hom`. -/\nclass continuous_add_monoid_hom_class (A B : Type*) [add_monoid A] [add_monoid B]\n  [topological_space A] [topological_space B] extends add_monoid_hom_class F A B :=\n(map_continuous (f : F) : continuous f)\n\n/-- `continuous_monoid_hom_class F A B` states that `F` is a type of continuous additive monoid\nhomomorphisms.\n\nYou should also extend this typeclass when you extend `continuous_monoid_hom`. -/\n@[to_additive]\nclass continuous_monoid_hom_class extends monoid_hom_class F A B :=\n(map_continuous (f : F) : continuous f)\n\nattribute [to_additive continuous_add_monoid_hom_class.to_add_monoid_hom_class]\n  continuous_monoid_hom_class.to_monoid_hom_class\n\nend\n\n/-- Reinterpret a `continuous_monoid_hom` as a `monoid_hom`. -/\nadd_decl_doc continuous_monoid_hom.to_monoid_hom\n\n/-- Reinterpret a `continuous_add_monoid_hom` as an `add_monoid_hom`. -/\nadd_decl_doc continuous_add_monoid_hom.to_add_monoid_hom\n\n@[priority 100, to_additive] -- See note [lower instance priority]\ninstance continuous_monoid_hom_class.to_continuous_map_class [continuous_monoid_hom_class F A B] :\n  continuous_map_class F A B :=\n{ .. \u2039continuous_monoid_hom_class F A B\u203a }\n\nnamespace continuous_monoid_hom\nvariables {A B C D E}\n\n@[to_additive]\ninstance : continuous_monoid_hom_class (continuous_monoid_hom A B) A B :=\n{ coe := \u03bb f, f.to_fun,\n  coe_injective' := \u03bb f g h, by { obtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f, obtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g, congr' },\n  map_mul := \u03bb f, f.map_mul',\n  map_one := \u03bb f, f.map_one',\n  map_continuous := \u03bb f, f.continuous_to_fun }\n\n/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\n@[to_additive \"Helper instance for when there's too many metavariables to apply\n`fun_like.has_coe_to_fun` directly.\"]\ninstance : has_coe_to_fun (continuous_monoid_hom A B) (\u03bb _, A \u2192 B) := fun_like.has_coe_to_fun\n\n@[to_additive, ext] lemma ext {f g : continuous_monoid_hom A B} (h : \u2200 x, f x = g x) : f = g :=\nfun_like.ext _ _ h\n\n/-- Reinterpret a `continuous_monoid_hom` as a `continuous_map`. -/\n@[to_additive \"Reinterpret a `continuous_add_monoid_hom` as a `continuous_map`.\"]\ndef to_continuous_map (f : continuous_monoid_hom A B) : C(A, B) := { .. f}\n\n@[to_additive] lemma to_continuous_map_injective : injective (to_continuous_map : _ \u2192 C(A, B)) :=\n\u03bb f g h, ext $ by convert fun_like.ext_iff.1 h\n\n/-- Construct a `continuous_monoid_hom` from a `continuous` `monoid_hom`. -/\n@[to_additive \"Construct a `continuous_add_monoid_hom` from a `continuous` `add_monoid_hom`.\",\n  simps]\ndef mk' (f : A \u2192* B) (hf : continuous f) : continuous_monoid_hom A B :=\n{ continuous_to_fun := hf, .. f }\n\n/-- Composition of two continuous homomorphisms. -/\n@[to_additive \"Composition of two continuous homomorphisms.\", simps]\ndef comp (g : continuous_monoid_hom B C) (f : continuous_monoid_hom A B) :\n  continuous_monoid_hom A C :=\nmk' (g.to_monoid_hom.comp f.to_monoid_hom) (g.continuous_to_fun.comp f.continuous_to_fun)\n\n/-- Product of two continuous homomorphisms on the same space. -/\n@[to_additive \"Product of two continuous homomorphisms on the same space.\", simps]\ndef prod (f : continuous_monoid_hom A B) (g : continuous_monoid_hom A C) :\n  continuous_monoid_hom A (B \u00d7 C) :=\nmk' (f.to_monoid_hom.prod g.to_monoid_hom) (f.continuous_to_fun.prod_mk g.continuous_to_fun)\n\n/-- Product of two continuous homomorphisms on different spaces. -/\n@[to_additive \"Product of two continuous homomorphisms on different spaces.\", simps]\ndef prod_map (f : continuous_monoid_hom A C) (g : continuous_monoid_hom B D) :\n  continuous_monoid_hom (A \u00d7 B) (C \u00d7 D) :=\nmk' (f.to_monoid_hom.prod_map g.to_monoid_hom) (f.continuous_to_fun.prod_map g.continuous_to_fun)\n\nvariables (A B C D E)\n\n/-- The trivial continuous homomorphism. -/\n@[to_additive \"The trivial continuous homomorphism.\", simps]\ndef one : continuous_monoid_hom A B := mk' 1 continuous_const\n\n@[to_additive] instance : inhabited (continuous_monoid_hom A B) := \u27e8one A B\u27e9\n\n/-- The identity continuous homomorphism. -/\n@[to_additive \"The identity continuous homomorphism.\", simps]\ndef id : continuous_monoid_hom A A := mk' (monoid_hom.id A) continuous_id\n\n/-- The continuous homomorphism given by projection onto the first factor. -/\n@[to_additive \"The continuous homomorphism given by projection onto the first factor.\", simps]\ndef fst : continuous_monoid_hom (A \u00d7 B) A := mk' (monoid_hom.fst A B) continuous_fst\n\n/-- The continuous homomorphism given by projection onto the second factor. -/\n@[to_additive \"The continuous homomorphism given by projection onto the second factor.\", simps]\ndef snd : continuous_monoid_hom (A \u00d7 B) B := mk' (monoid_hom.snd A B) continuous_snd\n\n/-- The continuous homomorphism given by inclusion of the first factor. -/\n@[to_additive \"The continuous homomorphism given by inclusion of the first factor.\", simps]\ndef inl : continuous_monoid_hom A (A \u00d7 B) := prod (id A) (one A B)\n\n/-- The continuous homomorphism given by inclusion of the second factor. -/\n@[to_additive \"The continuous homomorphism given by inclusion of the second factor.\", simps]\ndef inr : continuous_monoid_hom B (A \u00d7 B) := prod (one B A) (id B)\n\n/-- The continuous homomorphism given by the diagonal embedding. -/\n@[to_additive \"The continuous homomorphism given by the diagonal embedding.\", simps]\ndef diag : continuous_monoid_hom A (A \u00d7 A) := prod (id A) (id A)\n\n/-- The continuous homomorphism given by swapping components. -/\n@[to_additive \"The continuous homomorphism given by swapping components.\", simps]\ndef swap : continuous_monoid_hom (A \u00d7 B) (B \u00d7 A) := prod (snd A B) (fst A B)\n\n/-- The continuous homomorphism given by multiplication. -/\n@[to_additive \"The continuous homomorphism given by addition.\", simps]\ndef mul : continuous_monoid_hom (E \u00d7 E) E :=\nmk' mul_monoid_hom continuous_mul\n\n/-- The continuous homomorphism given by inversion. -/\n@[to_additive \"The continuous homomorphism given by negation.\", simps]\ndef inv : continuous_monoid_hom E E :=\nmk' inv_monoid_hom continuous_inv\n\nvariables {A B C D E}\n\n/-- Coproduct of two continuous homomorphisms to the same space. -/\n@[to_additive \"Coproduct of two continuous homomorphisms to the same space.\", simps]\ndef coprod (f : continuous_monoid_hom A E) (g : continuous_monoid_hom B E) :\n  continuous_monoid_hom (A \u00d7 B) E :=\n(mul E).comp (f.prod_map g)\n\n@[to_additive] instance : comm_group (continuous_monoid_hom A E) :=\n{ mul := \u03bb f g, (mul E).comp (f.prod g),\n  mul_comm := \u03bb f g, ext (\u03bb x, mul_comm (f x) (g x)),\n  mul_assoc := \u03bb f g h, ext (\u03bb x, mul_assoc (f x) (g x) (h x)),\n  one := one A E,\n  one_mul := \u03bb f, ext (\u03bb x, one_mul (f x)),\n  mul_one := \u03bb f, ext (\u03bb x, mul_one (f x)),\n  inv := \u03bb f, (inv E).comp f,\n  mul_left_inv := \u03bb f, ext (\u03bb x, mul_left_inv (f x)) }\n\n@[to_additive] instance : topological_space (continuous_monoid_hom A B) :=\ntopological_space.induced to_continuous_map continuous_map.compact_open\n\nvariables (A B C D E)\n\n@[to_additive] lemma inducing_to_continuous_map :\n  inducing (to_continuous_map : continuous_monoid_hom A B \u2192 C(A, B)) := \u27e8rfl\u27e9\n\n@[to_additive] lemma embedding_to_continuous_map :\n  embedding (to_continuous_map : continuous_monoid_hom A B \u2192 C(A, B)) :=\n\u27e8inducing_to_continuous_map A B, to_continuous_map_injective\u27e9\n\n@[to_additive] lemma closed_embedding_to_continuous_map [has_continuous_mul B] [t2_space B] :\n  closed_embedding (to_continuous_map : continuous_monoid_hom A B \u2192 C(A, B)) :=\n\u27e8embedding_to_continuous_map A B, \u27e8begin\n  suffices : (set.range (to_continuous_map : continuous_monoid_hom A B \u2192 C(A, B))) =\n    ({f | f '' {1} \u2286 {1}\u1d9c} \u222a \u22c3 (x y) (U V W) (hU : is_open U) (hV : is_open V) (hW : is_open W)\n    (h : disjoint (U * V) W), {f | f '' {x} \u2286 U} \u2229 {f | f '' {y} \u2286 V} \u2229 {f | f '' {x * y} \u2286 W})\u1d9c,\n  { rw [this, compl_compl],\n    refine (continuous_map.is_open_gen is_compact_singleton is_open_compl_singleton).union _,\n    repeat { apply is_open_Union, intro, },\n    repeat { apply is_open.inter },\n    all_goals { apply continuous_map.is_open_gen is_compact_singleton, assumption } },\n  simp_rw [set.compl_union, set.compl_Union, set.image_singleton, set.singleton_subset_iff,\n    set.ext_iff, set.mem_inter_iff, set.mem_Inter, set.mem_compl_iff],\n  refine \u03bb f, \u27e8_, _\u27e9,\n  { rintros \u27e8f, rfl\u27e9,\n    exact \u27e8\u03bb h, h (map_one f), \u03bb x y U V W hU hV hW h \u27e8\u27e8hfU, hfV\u27e9, hfW\u27e9,\n      h.le_bot \u27e8set.mul_mem_mul hfU hfV, (congr_arg (\u2208 W) (map_mul f x y)).mp hfW\u27e9\u27e9 },\n  { rintros \u27e8hf1, hf2\u27e9,\n    suffices : \u2200 x y, f (x * y) = f x * f y,\n    { refine \u27e8({ map_one' := of_not_not hf1, map_mul' := this, .. f } : continuous_monoid_hom A B),\n       continuous_map.ext (\u03bb _, rfl)\u27e9, },\n    intros x y,\n    contrapose! hf2,\n    obtain \u27e8UV, W, hUV, hW, hfUV, hfW, h\u27e9 := t2_separation hf2.symm,\n    have hB := @continuous_mul B _ _ _,\n    obtain \u27e8U, V, hU, hV, hfU, hfV, h'\u27e9 := is_open_prod_iff.mp (hUV.preimage hB) (f x) (f y) hfUV,\n    refine \u27e8x, y, U, V, W, hU, hV, hW, h.mono_left _, \u27e8hfU, hfV\u27e9, hfW\u27e9,\n    rintros _ \u27e8x, y, hx : (x, y).1 \u2208 U, hy : (x, y).2 \u2208 V, rfl\u27e9,\n    exact h' \u27e8hx, hy\u27e9 },\nend\u27e9\u27e9\n\nvariables {A B C D E}\n\n@[to_additive] instance [t2_space B] : t2_space (continuous_monoid_hom A B) :=\n(embedding_to_continuous_map A B).t2_space\n\n@[to_additive] instance : topological_group (continuous_monoid_hom A E) :=\nlet hi := inducing_to_continuous_map A E, hc := hi.continuous in\n{ continuous_mul := hi.continuous_iff.mpr (continuous_mul.comp (continuous.prod_map hc hc)),\n  continuous_inv := hi.continuous_iff.mpr (continuous_inv.comp hc) }\n\n@[to_additive] lemma continuous_of_continuous_uncurry {A : Type*} [topological_space A]\n  (f : A \u2192 continuous_monoid_hom B C) (h : continuous (function.uncurry (\u03bb x y, f x y))) :\n  continuous f :=\n(inducing_to_continuous_map _ _).continuous_iff.mpr\n  (continuous_map.continuous_of_continuous_uncurry _ h)\n\n@[to_additive] lemma continuous_comp [locally_compact_space B] :\n  continuous (\u03bb f : continuous_monoid_hom A B \u00d7 continuous_monoid_hom B C, f.2.comp f.1) :=\n(inducing_to_continuous_map A C).continuous_iff.2 $ (continuous_map.continuous_comp'.comp\n    ((inducing_to_continuous_map A B).prod_mk (inducing_to_continuous_map B C)).continuous)\n\n@[to_additive] lemma continuous_comp_left (f : continuous_monoid_hom A B) :\n  continuous (\u03bb g : continuous_monoid_hom B C, g.comp f) :=\n(inducing_to_continuous_map A C).continuous_iff.2 $ f.to_continuous_map.continuous_comp_left.comp\n  (inducing_to_continuous_map B C).continuous\n\n@[to_additive] lemma continuous_comp_right (f : continuous_monoid_hom B C) :\n  continuous (\u03bb g : continuous_monoid_hom A B, f.comp g) :=\n(inducing_to_continuous_map A C).continuous_iff.2 $ f.to_continuous_map.continuous_comp.comp\n  (inducing_to_continuous_map A B).continuous\n\nvariables (E)\n\n/-- `continuous_monoid_hom _ f` is a functor. -/\n@[to_additive \"`continuous_add_monoid_hom _ f` is a functor.\"]\ndef comp_left (f : continuous_monoid_hom A B) :\n  continuous_monoid_hom (continuous_monoid_hom B E) (continuous_monoid_hom A E) :=\n{ to_fun := \u03bb g, g.comp f,\n  map_one' := rfl,\n  map_mul' := \u03bb g h, rfl,\n  continuous_to_fun := f.continuous_comp_left }\n\nvariables (A) {E}\n\n/-- `continuous_monoid_hom f _` is a functor. -/\n@[to_additive \"`continuous_add_monoid_hom f _` is a functor.\"]\ndef comp_right {B : Type*} [comm_group B] [topological_space B]\n  [topological_group B] (f : continuous_monoid_hom B E) :\n  continuous_monoid_hom (continuous_monoid_hom A B) (continuous_monoid_hom A E) :=\n{ to_fun := \u03bb g, f.comp g,\n  map_one' := ext (\u03bb a, map_one f),\n  map_mul' := \u03bb g h, ext (\u03bb a, map_mul f (g a) (h a)),\n  continuous_to_fun := f.continuous_comp_right }\n\nend continuous_monoid_hom\n\n/-- The Pontryagin dual of `A` is the group of continuous homomorphism `A \u2192 circle`. -/\n@[derive [topological_space, t2_space, comm_group, topological_group, inhabited]]\ndef pontryagin_dual := continuous_monoid_hom A circle\n\nvariables {A B C D E}\n\nnamespace pontryagin_dual\n\nopen continuous_monoid_hom\n\nnoncomputable instance : continuous_monoid_hom_class (pontryagin_dual A) A circle :=\ncontinuous_monoid_hom.continuous_monoid_hom_class\n\n/-- `pontryagin_dual` is a functor. -/\nnoncomputable def map (f : continuous_monoid_hom A B) :\n  continuous_monoid_hom (pontryagin_dual B) (pontryagin_dual A) :=\nf.comp_left circle\n\n@[simp] lemma map_apply (f : continuous_monoid_hom A B) (x : pontryagin_dual B) (y : A) :\n  map f x y = x (f y) :=\nrfl\n\n@[simp] lemma map_one : map (one A B) = one (pontryagin_dual B) (pontryagin_dual A) :=\next (\u03bb x, ext (\u03bb y, map_one x))\n\n@[simp] lemma map_comp (g : continuous_monoid_hom B C) (f : continuous_monoid_hom A B) :\n  map (comp g f) = comp (map f) (map g) :=\next (\u03bb x, ext (\u03bb y, rfl))\n\n@[simp] lemma map_mul (f g : continuous_monoid_hom A E) : map (f * g) = map f * map g :=\next (\u03bb x, ext (\u03bb y, map_mul x (f y) (g y)))\n\nvariables (A B C D E)\n\n/-- `continuous_monoid_hom.dual` as a `continuous_monoid_hom`. -/\nnoncomputable def map_hom [locally_compact_space E] :\n  continuous_monoid_hom (continuous_monoid_hom A E)\n    (continuous_monoid_hom (pontryagin_dual E) (pontryagin_dual A)) :=\n{ to_fun := map,\n  map_one' := map_one,\n  map_mul' := map_mul,\n  continuous_to_fun := continuous_of_continuous_uncurry _ continuous_comp }\n\nend pontryagin_dual\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/topology/algebra/continuous_monoid_hom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.3958043566057043}}
{"text": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Eric Wieser\n\n! This file was ported from Lean 3 source module algebra.module.submodule.bilinear\n! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.LinearAlgebra.Span\nimport Mathbin.LinearAlgebra.BilinearMap\n\n/-!\n# Images of pairs of submodules under bilinear maps\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides `submodule.map\u2082`, which is later used to implement `submodule.has_mul`.\n\n## Main results\n\n* `submodule.map\u2082_eq_span_image2`: the image of two submodules under a bilinear map is the span of\n  their `set.image2`.\n\n## Notes\n\nThis file is quite similar to the n-ary section of `data.set.basic` and to `order.filter.n_ary`.\nPlease keep them in sync.\n-/\n\n\nuniverse u\u03b9 u v\n\nopen Set\n\nopen BigOperators\n\nopen Pointwise\n\nnamespace Submodule\n\nvariable {\u03b9 : Sort u\u03b9} {R M N P : Type _}\n\nvariable [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]\n\nvariable [Module R M] [Module R N] [Module R P]\n\n#print Submodule.map\u2082 /-\n/-- Map a pair of submodules under a bilinear map.\n\nThis is the submodule version of `set.image2`.  -/\ndef map\u2082 (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q : Submodule R N) : Submodule R P :=\n  \u2a06 s : p, q.map <| f s\n#align submodule.map\u2082 Submodule.map\u2082\n-/\n\n/- warning: submodule.apply_mem_map\u2082 -> Submodule.apply_mem_map\u2082 is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) {m : M} {n : N} {p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6}, (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)) m p) -> (Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)) n q) -> (Membership.Mem.{u4, u4} P (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.hasMem.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (fun (_x : LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) => N -> P) (LinearMap.hasCoeToFun.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m) n) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) {m : M} {n : N} {p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6}, (Membership.mem.{u3, u3} M (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5)) m p) -> (Membership.mem.{u1, u1} N (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (SetLike.instMembership.{u1, u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6)) n q) -> (Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) n) (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (SetLike.instMembership.{u2, u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m) n) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q))\nCase conversion may be inaccurate. Consider using '#align submodule.apply_mem_map\u2082 Submodule.apply_mem_map\u2082\u2093'. -/\ntheorem apply_mem_map\u2082 (f : M \u2192\u2097[R] N \u2192\u2097[R] P) {m : M} {n : N} {p : Submodule R M}\n    {q : Submodule R N} (hm : m \u2208 p) (hn : n \u2208 q) : f m n \u2208 map\u2082 f p q :=\n  (le_sup\u1d62 _ \u27e8m, hm\u27e9 : _ \u2264 map\u2082 f p q) \u27e8n, hn, rfl\u27e9\n#align submodule.apply_mem_map\u2082 Submodule.apply_mem_map\u2082\n\n/- warning: submodule.map\u2082_le -> Submodule.map\u2082_le is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] {f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))} {p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6} {r : Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7}, Iff (LE.le.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.partialOrder.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q) r) (forall (m : M), (Membership.Mem.{u2, u2} M (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (SetLike.hasMem.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)) m p) -> (forall (n : N), (Membership.Mem.{u3, u3} N (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (SetLike.hasMem.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)) n q) -> (Membership.Mem.{u4, u4} P (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.hasMem.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (fun (_x : LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) => N -> P) (LinearMap.hasCoeToFun.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m) n) r)))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] {f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))} {p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6} {r : Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7}, Iff (LE.le.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q) r) (forall (m : M), (Membership.mem.{u3, u3} M (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (SetLike.instMembership.{u3, u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5)) m p) -> (forall (n : N), (Membership.mem.{u1, u1} N (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (SetLike.instMembership.{u1, u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6)) n q) -> (Membership.mem.{u2, u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) n) (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (SetLike.instMembership.{u2, u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m) n) r)))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_le Submodule.map\u2082_le\u2093'. -/\ntheorem map\u2082_le {f : M \u2192\u2097[R] N \u2192\u2097[R] P} {p : Submodule R M} {q : Submodule R N}\n    {r : Submodule R P} : map\u2082 f p q \u2264 r \u2194 \u2200 m \u2208 p, \u2200 n \u2208 q, f m n \u2208 r :=\n  \u27e8fun H m hm n hn => H <| apply_mem_map\u2082 _ hm hn, fun H =>\n    sup\u1d62_le fun \u27e8m, hm\u27e9 => map_le_iff_le_comap.2 fun n hn => H m hm n hn\u27e9\n#align submodule.map\u2082_le Submodule.map\u2082_le\n\nvariable (R)\n\n/- warning: submodule.map\u2082_span_span -> Submodule.map\u2082_span_span is a dubious translation:\nlean 3 declaration is\n  forall (R : Type.{u1}) {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (s : Set.{u2} M) (t : Set.{u3} N), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5 s) (Submodule.span.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6 t)) (Submodule.span.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7 (Set.image2.{u2, u3, u4} M N P (fun (m : M) (n : N) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (fun (_x : LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) => N -> P) (LinearMap.hasCoeToFun.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m) n) s t))\nbut is expected to have type\n  forall (R : Type.{u4}) {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (s : Set.{u3} M) (t : Set.{u1} N), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Submodule.span.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5 s) (Submodule.span.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6 t)) (Submodule.span.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7 (Set.image2.{u3, u1, u2} M N P (fun (m : M) (n : N) => FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m) n) s t))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_span_span Submodule.map\u2082_span_span\u2093'. -/\ntheorem map\u2082_span_span (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (s : Set M) (t : Set N) :\n    map\u2082 f (span R s) (span R t) = span R (Set.image2 (fun m n => f m n) s t) :=\n  by\n  apply le_antisymm\n  \u00b7 rw [map\u2082_le]\n    intro a ha b hb\n    apply span_induction ha\n    on_goal 1 =>\n      intros ; apply span_induction hb\n      on_goal 1 => intros ; exact subset_span \u27e8_, _, \u2039_\u203a, \u2039_\u203a, rfl\u27e9\n    all_goals\n      intros\n      simp only [LinearMap.map_zero, LinearMap.zero_apply, zero_mem, LinearMap.map_add,\n        LinearMap.add_apply, LinearMap.map_smul, LinearMap.smul_apply]\n    all_goals\n      solve_by_elim (config :=\n        { max_depth := 4\n          discharger := tactic.interactive.apply_instance }) [add_mem _ _, zero_mem _,\n        smul_mem _ _ _]\n  \u00b7 rw [span_le]\n    rintro _ \u27e8a, b, ha, hb, rfl\u27e9\n    exact apply_mem_map\u2082 _ (subset_span ha) (subset_span hb)\n#align submodule.map\u2082_span_span Submodule.map\u2082_span_span\n\nvariable {R}\n\n/- warning: submodule.map\u2082_bot_right -> Submodule.map\u2082_bot_right is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p (Bot.bot.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Submodule.hasBot.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6))) (Bot.bot.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.hasBot.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p (Bot.bot.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Submodule.instBotSubmodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6))) (Bot.bot.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.instBotSubmodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_bot_right Submodule.map\u2082_bot_right\u2093'. -/\n@[simp]\ntheorem map\u2082_bot_right (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) : map\u2082 f p \u22a5 = \u22a5 :=\n  eq_bot_iff.2 <|\n    map\u2082_le.2 fun m hm n hn => by\n      rw [Submodule.mem_bot] at hn\u22a2\n      rw [hn, LinearMap.map_zero]\n#align submodule.map\u2082_bot_right Submodule.map\u2082_bot_right\n\n/- warning: submodule.map\u2082_bot_left -> Submodule.map\u2082_bot_left is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Bot.bot.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Submodule.hasBot.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)) q) (Bot.bot.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.hasBot.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Bot.bot.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Submodule.instBotSubmodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5)) q) (Bot.bot.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.instBotSubmodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_bot_left Submodule.map\u2082_bot_left\u2093'. -/\n@[simp]\ntheorem map\u2082_bot_left (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (q : Submodule R N) : map\u2082 f \u22a5 q = \u22a5 :=\n  eq_bot_iff.2 <|\n    map\u2082_le.2 fun m hm n hn => by\n      rw [Submodule.mem_bot] at hm\u22a2\n      rw [hm, LinearMap.map_zero\u2082]\n#align submodule.map\u2082_bot_left Submodule.map\u2082_bot_left\n\n/- warning: submodule.map\u2082_le_map\u2082 -> Submodule.map\u2082_le_map\u2082 is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] {f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))} {p\u2081 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {p\u2082 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {q\u2081 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6} {q\u2082 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)))) p\u2081 p\u2082) -> (LE.le.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Preorder.toLE.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (PartialOrder.toPreorder.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (SetLike.partialOrder.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)))) q\u2081 q\u2082) -> (LE.le.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.partialOrder.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q\u2081) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q\u2082))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] {f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))} {p\u2081 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {p\u2082 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {q\u2081 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6} {q\u2082 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6}, (LE.le.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Preorder.toLE.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5))))) p\u2081 p\u2082) -> (LE.le.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Preorder.toLE.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (PartialOrder.toPreorder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6))))) q\u2081 q\u2082) -> (LE.le.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q\u2081) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q\u2082))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_le_map\u2082 Submodule.map\u2082_le_map\u2082\u2093'. -/\n@[mono]\ntheorem map\u2082_le_map\u2082 {f : M \u2192\u2097[R] N \u2192\u2097[R] P} {p\u2081 p\u2082 : Submodule R M} {q\u2081 q\u2082 : Submodule R N}\n    (hp : p\u2081 \u2264 p\u2082) (hq : q\u2081 \u2264 q\u2082) : map\u2082 f p\u2081 q\u2081 \u2264 map\u2082 f p\u2082 q\u2082 :=\n  map\u2082_le.2 fun m hm n hn => apply_mem_map\u2082 _ (hp hm) (hq hn)\n#align submodule.map\u2082_le_map\u2082 Submodule.map\u2082_le_map\u2082\n\n/- warning: submodule.map\u2082_le_map\u2082_left -> Submodule.map\u2082_le_map\u2082_left is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] {f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))} {p\u2081 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {p\u2082 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6}, (LE.le.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Preorder.toLE.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (PartialOrder.toPreorder.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (SetLike.partialOrder.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)))) p\u2081 p\u2082) -> (LE.le.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.partialOrder.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] {f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))} {p\u2081 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {p\u2082 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6}, (LE.le.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Preorder.toLE.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (PartialOrder.toPreorder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (OmegaCompletePartialOrder.toPartialOrder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (CompleteLattice.instOmegaCompletePartialOrder.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5))))) p\u2081 p\u2082) -> (LE.le.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_le_map\u2082_left Submodule.map\u2082_le_map\u2082_left\u2093'. -/\ntheorem map\u2082_le_map\u2082_left {f : M \u2192\u2097[R] N \u2192\u2097[R] P} {p\u2081 p\u2082 : Submodule R M} {q : Submodule R N}\n    (h : p\u2081 \u2264 p\u2082) : map\u2082 f p\u2081 q \u2264 map\u2082 f p\u2082 q :=\n  map\u2082_le_map\u2082 h (le_refl q)\n#align submodule.map\u2082_le_map\u2082_left Submodule.map\u2082_le_map\u2082_left\n\n/- warning: submodule.map\u2082_le_map\u2082_right -> Submodule.map\u2082_le_map\u2082_right is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] {f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))} {p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5} {q\u2081 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6} {q\u2082 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6}, (LE.le.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Preorder.toLE.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (PartialOrder.toPreorder.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (SetLike.partialOrder.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)))) q\u2081 q\u2082) -> (LE.le.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SetLike.partialOrder.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2081) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2082))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] {f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))} {p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5} {q\u2081 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6} {q\u2082 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6}, (LE.le.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Preorder.toLE.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (PartialOrder.toPreorder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (OmegaCompletePartialOrder.toPartialOrder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (CompleteLattice.instOmegaCompletePartialOrder.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6))))) q\u2081 q\u2082) -> (LE.le.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Preorder.toLE.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (PartialOrder.toPreorder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (OmegaCompletePartialOrder.toPartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.instOmegaCompletePartialOrder.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2081) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2082))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_le_map\u2082_right Submodule.map\u2082_le_map\u2082_right\u2093'. -/\ntheorem map\u2082_le_map\u2082_right {f : M \u2192\u2097[R] N \u2192\u2097[R] P} {p : Submodule R M} {q\u2081 q\u2082 : Submodule R N}\n    (h : q\u2081 \u2264 q\u2082) : map\u2082 f p q\u2081 \u2264 map\u2082 f p q\u2082 :=\n  map\u2082_le_map\u2082 (le_refl p) h\n#align submodule.map\u2082_le_map\u2082_right Submodule.map\u2082_le_map\u2082_right\n\n/- warning: submodule.map\u2082_sup_right -> Submodule.map\u2082_sup_right is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (q\u2081 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (q\u2082 : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p (Sup.sup.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (SemilatticeSup.toHasSup.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Lattice.toSemilatticeSup.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (ConditionallyCompleteLattice.toLattice.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6))))) q\u2081 q\u2082)) (Sup.sup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SemilatticeSup.toHasSup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Lattice.toSemilatticeSup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toLattice.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2081) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2082))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (q\u2081 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (q\u2082 : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p (Sup.sup.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (SemilatticeSup.toSup.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Lattice.toSemilatticeSup.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (ConditionallyCompleteLattice.toLattice.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6))))) q\u2081 q\u2082)) (Sup.sup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (SemilatticeSup.toSup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Lattice.toSemilatticeSup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2081) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q\u2082))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_sup_right Submodule.map\u2082_sup_right\u2093'. -/\ntheorem map\u2082_sup_right (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q\u2081 q\u2082 : Submodule R N) :\n    map\u2082 f p (q\u2081 \u2294 q\u2082) = map\u2082 f p q\u2081 \u2294 map\u2082 f p q\u2082 :=\n  le_antisymm\n    (map\u2082_le.2 fun m hm np hnp =>\n      let \u27e8n, hn, p, hp, hnp\u27e9 := mem_sup.1 hnp\n      mem_sup.2 \u27e8_, apply_mem_map\u2082 _ hm hn, _, apply_mem_map\u2082 _ hm hp, hnp \u25b8 (map_add _ _ _).symm\u27e9)\n    (sup_le (map\u2082_le_map\u2082_right le_sup_left) (map\u2082_le_map\u2082_right le_sup_right))\n#align submodule.map\u2082_sup_right Submodule.map\u2082_sup_right\n\n/- warning: submodule.map\u2082_sup_left -> Submodule.map\u2082_sup_left is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p\u2081 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (p\u2082 : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Sup.sup.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (SemilatticeSup.toHasSup.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Lattice.toSemilatticeSup.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5))))) p\u2081 p\u2082) q) (Sup.sup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (SemilatticeSup.toHasSup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Lattice.toSemilatticeSup.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toLattice.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p\u2081 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (p\u2082 : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Sup.sup.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (SemilatticeSup.toSup.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Lattice.toSemilatticeSup.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (ConditionallyCompleteLattice.toLattice.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5))))) p\u2081 p\u2082) q) (Sup.sup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (SemilatticeSup.toSup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Lattice.toSemilatticeSup.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2081 q) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p\u2082 q))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_sup_left Submodule.map\u2082_sup_left\u2093'. -/\ntheorem map\u2082_sup_left (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p\u2081 p\u2082 : Submodule R M) (q : Submodule R N) :\n    map\u2082 f (p\u2081 \u2294 p\u2082) q = map\u2082 f p\u2081 q \u2294 map\u2082 f p\u2082 q :=\n  le_antisymm\n    (map\u2082_le.2 fun mn hmn p hp =>\n      let \u27e8m, hm, n, hn, hmn\u27e9 := mem_sup.1 hmn\n      mem_sup.2\n        \u27e8_, apply_mem_map\u2082 _ hm hp, _, apply_mem_map\u2082 _ hn hp,\n          hmn \u25b8 (LinearMap.map_add\u2082 _ _ _ _).symm\u27e9)\n    (sup_le (map\u2082_le_map\u2082_left le_sup_left) (map\u2082_le_map\u2082_left le_sup_right))\n#align submodule.map\u2082_sup_left Submodule.map\u2082_sup_left\n\n/- warning: submodule.image2_subset_map\u2082 -> Submodule.image2_subset_map\u2082 is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), HasSubset.Subset.{u4} (Set.{u4} P) (Set.hasSubset.{u4} P) (Set.image2.{u2, u3, u4} M N P (fun (m : M) (n : N) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (fun (_x : LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) => N -> P) (LinearMap.hasCoeToFun.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m) n) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (CoeTC\u2093.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)))) p) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (HasLiftT.mk.{succ u3, succ u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (CoeTC\u2093.coe.{succ u3, succ u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (SetLike.Set.hasCoeT.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)))) q)) ((fun (a : Type.{u4}) (b : Type.{u4}) [self : HasLiftT.{succ u4, succ u4} a b] => self.0) (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Set.{u4} P) (HasLiftT.mk.{succ u4, succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Set.{u4} P) (CoeTC\u2093.coe.{succ u4, succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Set.{u4} P) (SetLike.Set.hasCoeT.{u4, u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), HasSubset.Subset.{u2} (Set.{u2} P) (Set.instHasSubsetSet.{u2} P) (Set.image2.{u3, u1, u2} M N P (fun (m : M) (n : N) => FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m) n) (SetLike.coe.{u3, u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) p) (SetLike.coe.{u1, u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) q)) (SetLike.coe.{u2, u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) P (Submodule.setLike.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q))\nCase conversion may be inaccurate. Consider using '#align submodule.image2_subset_map\u2082 Submodule.image2_subset_map\u2082\u2093'. -/\ntheorem image2_subset_map\u2082 (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q : Submodule R N) :\n    Set.image2 (fun m n => f m n) (\u2191p : Set M) (\u2191q : Set N) \u2286 (\u2191(map\u2082 f p q) : Set P) :=\n  by\n  rintro _ \u27e8i, j, hi, hj, rfl\u27e9\n  exact apply_mem_map\u2082 _ hi hj\n#align submodule.image2_subset_map\u2082 Submodule.image2_subset_map\u2082\n\n/- warning: submodule.map\u2082_eq_span_image2 -> Submodule.map\u2082_eq_span_image2 is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q) (Submodule.span.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7 (Set.image2.{u2, u3, u4} M N P (fun (m : M) (n : N) => coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (fun (_x : LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) => N -> P) (LinearMap.hasCoeToFun.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m) n) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (HasLiftT.mk.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (CoeTC\u2093.coe.{succ u2, succ u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (Set.{u2} M) (SetLike.Set.hasCoeT.{u2, u2} (Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5)))) p) ((fun (a : Type.{u3}) (b : Type.{u3}) [self : HasLiftT.{succ u3, succ u3} a b] => self.0) (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (HasLiftT.mk.{succ u3, succ u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (CoeTC\u2093.coe.{succ u3, succ u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) (Set.{u3} N) (SetLike.Set.hasCoeT.{u3, u3} (Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6)))) q)))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q) (Submodule.span.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7 (Set.image2.{u3, u1, u2} M N P (fun (m : M) (n : N) => FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m) n) (SetLike.coe.{u3, u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) M (Submodule.setLike.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) p) (SetLike.coe.{u1, u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) N (Submodule.setLike.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) q)))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_eq_span_image2 Submodule.map\u2082_eq_span_image2\u2093'. -/\ntheorem map\u2082_eq_span_image2 (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q : Submodule R N) :\n    map\u2082 f p q = span R (Set.image2 (fun m n => f m n) (p : Set M) (q : Set N)) := by\n  rw [\u2190 map\u2082_span_span, span_eq, span_eq]\n#align submodule.map\u2082_eq_span_image2 Submodule.map\u2082_eq_span_image2\n\n/- warning: submodule.map\u2082_flip -> Submodule.map\u2082_flip is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u3, u2, u4} R N M P _inst_1 _inst_3 _inst_2 _inst_4 _inst_6 _inst_5 _inst_7 (LinearMap.flip.{u1, u1, u1, u1, u2, u3, u4} R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) M N P _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) f) q p) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q)\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (p : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (q : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u1, u3, u2} R N M P _inst_1 _inst_3 _inst_2 _inst_4 _inst_6 _inst_5 _inst_7 (LinearMap.flip.{u4, u4, u4, u4, u3, u1, u2} R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) M N P _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) f) q p) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f p q)\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_flip Submodule.map\u2082_flip\u2093'. -/\ntheorem map\u2082_flip (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (p : Submodule R M) (q : Submodule R N) :\n    map\u2082 f.flip q p = map\u2082 f p q :=\n  by\n  rw [map\u2082_eq_span_image2, map\u2082_eq_span_image2, Set.image2_swap]\n  rfl\n#align submodule.map\u2082_flip Submodule.map\u2082_flip\n\n/- warning: submodule.map\u2082_supr_left -> Submodule.map\u2082_sup\u1d62_left is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Sort.{u1}} {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} {P : Type.{u5}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u4} N] [_inst_4 : AddCommMonoid.{u5} P] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] [_inst_6 : Module.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3] [_inst_7 : Module.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4] (f : LinearMap.{u2, u2, u3, max u4 u5} R R (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) M (LinearMap.{u2, u2, u4, u5} R R (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u2, u2, u4, u5} R R N P (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) _inst_5 (LinearMap.module.{u2, u2, u2, u4, u5} R R R N P (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_7 (smulCommClass_self.{u2, u5} R P (CommSemiring.toCommMonoid.{u2} R _inst_1) (MulActionWithZero.toMulAction.{u2, u5} R P (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddZeroClass.toHasZero.{u5} P (AddMonoid.toAddZeroClass.{u5} P (AddCommMonoid.toAddMonoid.{u5} P _inst_4))) (Module.toMulActionWithZero.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7))))) (s : \u03b9 -> (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5)) (t : Submodule.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6), Eq.{succ u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u2, u3, u4, u5} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (sup\u1d62.{u3, u1} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5) (ConditionallyCompleteLattice.toHasSup.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5))) \u03b9 (fun (i : \u03b9) => s i)) t) (sup\u1d62.{u5, u1} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toHasSup.{u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7))) \u03b9 (fun (i : \u03b9) => Submodule.map\u2082.{u2, u3, u4, u5} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (s i) t))\nbut is expected to have type\n  forall {\u03b9 : Sort.{u5}} {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (s : \u03b9 -> (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5)) (t : Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (sup\u1d62.{u3, u5} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (ConditionallyCompleteLattice.toSupSet.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (CompleteLattice.toConditionallyCompleteLattice.{u3} (Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (Submodule.completeLattice.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5))) \u03b9 (fun (i : \u03b9) => s i)) t) (sup\u1d62.{u2, u5} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))) \u03b9 (fun (i : \u03b9) => Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (s i) t))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_supr_left Submodule.map\u2082_sup\u1d62_left\u2093'. -/\ntheorem map\u2082_sup\u1d62_left (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (s : \u03b9 \u2192 Submodule R M) (t : Submodule R N) :\n    map\u2082 f (\u2a06 i, s i) t = \u2a06 i, map\u2082 f (s i) t :=\n  by\n  suffices map\u2082 f (\u2a06 i, span R (s i : Set M)) (span R t) = \u2a06 i, map\u2082 f (span R (s i)) (span R t) by\n    simpa only [span_eq] using this\n  simp_rw [map\u2082_span_span, \u2190 span_Union, map\u2082_span_span, Set.image2_union\u1d62_left]\n#align submodule.map\u2082_supr_left Submodule.map\u2082_sup\u1d62_left\n\n/- warning: submodule.map\u2082_supr_right -> Submodule.map\u2082_sup\u1d62_right is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Sort.{u1}} {R : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} {P : Type.{u5}} [_inst_1 : CommSemiring.{u2} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u4} N] [_inst_4 : AddCommMonoid.{u5} P] [_inst_5 : Module.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2] [_inst_6 : Module.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3] [_inst_7 : Module.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4] (f : LinearMap.{u2, u2, u3, max u4 u5} R R (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) M (LinearMap.{u2, u2, u4, u5} R R (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u2, u2, u4, u5} R R N P (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)))) _inst_5 (LinearMap.module.{u2, u2, u2, u4, u5} R R R N P (CommSemiring.toSemiring.{u2} R _inst_1) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u2} R (Semiring.toNonAssocSemiring.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1))) (CommSemiring.toSemiring.{u2} R _inst_1) _inst_7 (smulCommClass_self.{u2, u5} R P (CommSemiring.toCommMonoid.{u2} R _inst_1) (MulActionWithZero.toMulAction.{u2, u5} R P (Semiring.toMonoidWithZero.{u2} R (CommSemiring.toSemiring.{u2} R _inst_1)) (AddZeroClass.toHasZero.{u5} P (AddMonoid.toAddZeroClass.{u5} P (AddCommMonoid.toAddMonoid.{u5} P _inst_4))) (Module.toMulActionWithZero.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7))))) (s : Submodule.{u2, u3} R M (CommSemiring.toSemiring.{u2} R _inst_1) _inst_2 _inst_5) (t : \u03b9 -> (Submodule.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6)), Eq.{succ u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u2, u3, u4, u5} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (sup\u1d62.{u4, u1} (Submodule.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6) (ConditionallyCompleteLattice.toHasSup.{u4} (Submodule.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6) (CompleteLattice.toConditionallyCompleteLattice.{u4} (Submodule.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u2, u4} R N (CommSemiring.toSemiring.{u2} R _inst_1) _inst_3 _inst_6))) \u03b9 (fun (i : \u03b9) => t i))) (sup\u1d62.{u5, u1} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toHasSup.{u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u5} (Submodule.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u2, u5} R P (CommSemiring.toSemiring.{u2} R _inst_1) _inst_4 _inst_7))) \u03b9 (fun (i : \u03b9) => Submodule.map\u2082.{u2, u3, u4, u5} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (t i)))\nbut is expected to have type\n  forall {\u03b9 : Sort.{u5}} {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (s : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (t : \u03b9 -> (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6)), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (sup\u1d62.{u1, u5} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (ConditionallyCompleteLattice.toSupSet.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) (Submodule.completeLattice.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6))) \u03b9 (fun (i : \u03b9) => t i))) (sup\u1d62.{u2, u5} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (ConditionallyCompleteLattice.toSupSet.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (CompleteLattice.toConditionallyCompleteLattice.{u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.completeLattice.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))) \u03b9 (fun (i : \u03b9) => Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (t i)))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_supr_right Submodule.map\u2082_sup\u1d62_right\u2093'. -/\ntheorem map\u2082_sup\u1d62_right (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (s : Submodule R M) (t : \u03b9 \u2192 Submodule R N) :\n    map\u2082 f s (\u2a06 i, t i) = \u2a06 i, map\u2082 f s (t i) :=\n  by\n  suffices map\u2082 f (span R s) (\u2a06 i, span R (t i : Set N)) = \u2a06 i, map\u2082 f (span R s) (span R (t i)) by\n    simpa only [span_eq] using this\n  simp_rw [map\u2082_span_span, \u2190 span_Union, map\u2082_span_span, Set.image2_union\u1d62_right]\n#align submodule.map\u2082_supr_right Submodule.map\u2082_sup\u1d62_right\n\n/- warning: submodule.map\u2082_span_singleton_eq_map -> Submodule.map\u2082_span_singleton_eq_map is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (m : M), Eq.{max (succ u3) (succ u4)} ((Submodule.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6) -> (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Submodule.span.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5 (Singleton.singleton.{u2, u2} M (Set.{u2} M) (Set.hasSingleton.{u2} M) m))) (Submodule.map.{u1, u1, u3, u4, max u3 u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (LinearMap.semilinearMapClass.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u2) (succ (max u3 u4)), max (succ u2) (succ (max u3 u4))} (LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (fun (_x : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) => M -> (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u2, max u3 u4} R R M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) f m))\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (m : M), Eq.{max (succ u1) (succ u2)} ((Submodule.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6) -> (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f (Submodule.span.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5 (Singleton.singleton.{u3, u3} M (Set.{u3} M) (Set.instSingletonSet.{u3} M) m))) (Submodule.map.{u4, u4, u1, u2, max u1 u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (RingHomSurjective.ids.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) m) (LinearMap.instSemilinearMapClassLinearMap.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u3, max (succ u1) (succ u2)} (LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u3, max u1 u2} R R M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) f m))\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_span_singleton_eq_map Submodule.map\u2082_span_singleton_eq_map\u2093'. -/\ntheorem map\u2082_span_singleton_eq_map (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (m : M) :\n    map\u2082 f (span R {m}) = map (f m) := by\n  funext; rw [map\u2082_eq_span_image2]; apply le_antisymm\n  \u00b7 rw [span_le, Set.image2_subset_iff]\n    intro x hx y hy\n    obtain \u27e8a, rfl\u27e9 := mem_span_singleton.1 hx\n    rw [f.map_smul]\n    exact smul_mem _ a (mem_map_of_mem hy)\n  \u00b7 rintro _ \u27e8n, hn, rfl\u27e9\n    exact subset_span \u27e8m, n, mem_span_singleton_self m, hn, rfl\u27e9\n#align submodule.map\u2082_span_singleton_eq_map Submodule.map\u2082_span_singleton_eq_map\n\n/- warning: submodule.map\u2082_span_singleton_eq_map_flip -> Submodule.map\u2082_span_singleton_eq_map_flip is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : CommSemiring.{u1} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : AddCommMonoid.{u4} P] [_inst_5 : Module.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2] [_inst_6 : Module.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3] [_inst_7 : Module.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4] (f : LinearMap.{u1, u1, u2, max u3 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M (LinearMap.{u1, u1, u3, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u1, u1, u3, u4} R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_5 (LinearMap.module.{u1, u1, u1, u3, u4} R R R N P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (s : Submodule.{u1, u2} R M (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_5) (n : N), Eq.{succ u4} (Submodule.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u1, u2, u3, u4} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (Submodule.span.{u1, u3} R N (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 _inst_6 (Singleton.singleton.{u3, u3} N (Set.{u3} N) (Set.hasSingleton.{u3} N) n))) (Submodule.map.{u1, u1, u2, u4, max u2 u4} R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHomSurjective.ids.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (LinearMap.{u1, u1, u2, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) (LinearMap.semilinearMapClass.{u1, u1, u2, u4} R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (coeFn.{max (succ u3) (succ (max u2 u4)), max (succ u3) (succ (max u2 u4))} (LinearMap.{u1, u1, u3, max u2 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N (LinearMap.{u1, u1, u2, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u2, u4} R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_6 (LinearMap.module.{u1, u1, u1, u2, u4} R R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (LinearMap.flip._proof_1.{u1, u1, u4} R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) P _inst_4 _inst_7 _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))))) (fun (_x : LinearMap.{u1, u1, u3, max u2 u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) N (LinearMap.{u1, u1, u2, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u2, u4} R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_6 (LinearMap.module.{u1, u1, u1, u2, u4} R R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (LinearMap.flip._proof_1.{u1, u1, u4} R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) P _inst_4 _inst_7 _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7)))))) => N -> (LinearMap.{u1, u1, u2, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7)) (LinearMap.hasCoeToFun.{u1, u1, u3, max u2 u4} R R N (LinearMap.{u1, u1, u2, u4} R R (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_3 (LinearMap.addCommMonoid.{u1, u1, u2, u4} R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) _inst_6 (LinearMap.module.{u1, u1, u1, u2, u4} R R R M P (CommSemiring.toSemiring.{u1} R _inst_1) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (CommSemiring.toSemiring.{u1} R _inst_1) _inst_7 (LinearMap.flip._proof_1.{u1, u1, u4} R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) P _inst_4 _inst_7 _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (LinearMap.flip.{u1, u1, u1, u1, u2, u3, u4} R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) R (CommSemiring.toSemiring.{u1} R _inst_1) M N P _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_7 (smulCommClass_self.{u1, u4} R P (CommSemiring.toCommMonoid.{u1} R _inst_1) (MulActionWithZero.toMulAction.{u1, u4} R P (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_4))) (Module.toMulActionWithZero.{u1, u4} R P (CommSemiring.toSemiring.{u1} R _inst_1) _inst_4 _inst_7))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))) f) n) s)\nbut is expected to have type\n  forall {R : Type.{u4}} {M : Type.{u3}} {N : Type.{u1}} {P : Type.{u2}} [_inst_1 : CommSemiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : AddCommMonoid.{u2} P] [_inst_5 : Module.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2] [_inst_6 : Module.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3] [_inst_7 : Module.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4] (f : LinearMap.{u4, u4, u3, max u2 u1} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M (LinearMap.{u4, u4, u1, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N P _inst_3 _inst_4 _inst_6 _inst_7) _inst_2 (LinearMap.addCommMonoid.{u4, u4, u1, u2} R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_5 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u1, u2} R R R N P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_4 _inst_6 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (s : Submodule.{u4, u3} R M (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_5) (n : N), Eq.{succ u2} (Submodule.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7) (Submodule.map\u2082.{u4, u3, u1, u2} R M N P _inst_1 _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 f s (Submodule.span.{u4, u1} R N (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 _inst_6 (Singleton.singleton.{u1, u1} N (Set.{u1} N) (Set.instSingletonSet.{u1} N) n))) (Submodule.map.{u4, u4, u3, u2, max u3 u2} R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (RingHomSurjective.ids.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => LinearMap.{u4, u4, u3, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) n) (LinearMap.instSemilinearMapClassLinearMap.{u4, u4, u3, u2} R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (FunLike.coe.{max (max (succ u3) (succ u1)) (succ u2), succ u1, max (succ u3) (succ u2)} (LinearMap.{u4, u4, u1, max u2 u3} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) N (LinearMap.{u4, u4, u3, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) _inst_3 (LinearMap.addCommMonoid.{u4, u4, u3, u2} R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_6 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u3, u2} R R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (SMulCommClass.symm.{u2, u4, u4} R R P (SMulZeroClass.toSMul.{u4, u2} R P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (SMulWithZero.toSMulZeroClass.{u4, u2} R P (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (MulActionWithZero.toSMulWithZero.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (SMulZeroClass.toSMul.{u4, u2} R P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (SMulWithZero.toSMulZeroClass.{u4, u2} R P (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (MulActionWithZero.toSMulWithZero.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))))) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => LinearMap.{u4, u4, u3, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) _x) (LinearMap.instFunLikeLinearMap.{u4, u4, u1, max u3 u2} R R N (LinearMap.{u4, u4, u3, u2} R R (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) M P _inst_2 _inst_4 _inst_5 _inst_7) (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_3 (LinearMap.addCommMonoid.{u4, u4, u3, u2} R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) _inst_6 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u4, u4, u3, u2} R R R M P (CommSemiring.toSemiring.{u4} R _inst_1) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_2 _inst_4 _inst_5 _inst_7 (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (CommSemiring.toSemiring.{u4} R _inst_1) _inst_7 (SMulCommClass.symm.{u2, u4, u4} R R P (SMulZeroClass.toSMul.{u4, u2} R P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (SMulWithZero.toSMulZeroClass.{u4, u2} R P (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (MulActionWithZero.toSMulWithZero.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (SMulZeroClass.toSMul.{u4, u2} R P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (SMulWithZero.toSMulZeroClass.{u4, u2} R P (MonoidWithZero.toZero.{u4} R (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (MulActionWithZero.toSMulWithZero.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7)))) (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)))) (LinearMap.flip.{u4, u4, u4, u4, u3, u1, u2} R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) R (CommSemiring.toSemiring.{u4} R _inst_1) M N P _inst_2 _inst_3 _inst_4 _inst_5 _inst_6 _inst_7 _inst_7 (smulCommClass_self.{u4, u2} R P (CommSemiring.toCommMonoid.{u4} R _inst_1) (MulActionWithZero.toMulAction.{u4, u2} R P (Semiring.toMonoidWithZero.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1)) (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_4)) (Module.toMulActionWithZero.{u4, u2} R P (CommSemiring.toSemiring.{u4} R _inst_1) _inst_4 _inst_7))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) (RingHom.id.{u4} R (Semiring.toNonAssocSemiring.{u4} R (CommSemiring.toSemiring.{u4} R _inst_1))) f) n) s)\nCase conversion may be inaccurate. Consider using '#align submodule.map\u2082_span_singleton_eq_map_flip Submodule.map\u2082_span_singleton_eq_map_flip\u2093'. -/\ntheorem map\u2082_span_singleton_eq_map_flip (f : M \u2192\u2097[R] N \u2192\u2097[R] P) (s : Submodule R M) (n : N) :\n    map\u2082 f s (span R {n}) = map (f.flip n) s := by rw [\u2190 map\u2082_span_singleton_eq_map, map\u2082_flip]\n#align submodule.map\u2082_span_singleton_eq_map_flip Submodule.map\u2082_span_singleton_eq_map_flip\n\nend Submodule\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Module/Submodule/Bilinear.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3957909631370764}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.traversable.instances\n! leanprover-community/mathlib commit 18a5306c091183ac90884daa9373fa3b178e8607\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Control.Applicative\nimport Mathlib.Control.Traversable.Basic\nimport Mathlib.Data.List.Forall2\nimport Mathlib.Data.Set.Functor\n\n/-!\n# IsLawfulTraversable instances\n\nThis file provides instances of `IsLawfulTraversable` for types from the core library: `Option`,\n`List` and `Sum`.\n-/\n\n\nuniverse u v\n\nsection Option\n\nopen Functor\n\nvariable {F G : Type u \u2192 Type u}\n\nvariable [Applicative F] [Applicative G]\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\n\ntheorem Option.id_traverse {\u03b1} (x : Option \u03b1) : Option.traverse (pure : \u03b1 \u2192 Id \u03b1) x = x := by\n  cases x <;> rfl\n#align option.id_traverse Option.id_traverse\n\ntheorem Option.comp_traverse {\u03b1 \u03b2 \u03b3} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : Option \u03b1) :\n    Option.traverse (Comp.mk \u2218 (\u00b7 <$> \u00b7) f \u2218 g) x =\n      Comp.mk (Option.traverse f <$> Option.traverse g x) :=\n  by cases x <;> simp! [functor_norm] <;> rfl\n#align option.comp_traverse Option.comp_traverse\n\ntheorem Option.traverse_eq_map_id {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (x : Option \u03b1) :\n    Option.traverse ((pure : _ \u2192 Id _) \u2218 f) x = (pure : _ \u2192 Id _) (f <$> x) := by cases x <;> rfl\n#align option.traverse_eq_map_id Option.traverse_eq_map_id\n\nvariable (\u03b7 : ApplicativeTransformation F G)\n\ntheorem Option.naturality {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) (x : Option \u03b1) :\n    \u03b7 (Option.traverse f x) = Option.traverse (@\u03b7 _ \u2218 f) x := by\n  -- Porting note: added `ApplicativeTransformation` theorems\n  cases' x with x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,\n    ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]\n#align option.naturality Option.naturality\n\nend Option\n\ninstance : IsLawfulTraversable Option :=\n  { show LawfulMonad Option from inferInstance with\n    id_traverse := Option.id_traverse\n    comp_traverse := Option.comp_traverse\n    traverse_eq_map_id := Option.traverse_eq_map_id\n    naturality := Option.naturality }\n\nnamespace List\n\nvariable {F G : Type u \u2192 Type u}\n\nvariable [Applicative F] [Applicative G]\n\nsection\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\n\nopen Applicative Functor List\n\nprotected theorem id_traverse {\u03b1} (xs : List \u03b1) : List.traverse (pure : \u03b1 \u2192 Id \u03b1) xs = xs := by\n  induction xs <;> simp! [*, List.traverse, functor_norm]; rfl\n#align list.id_traverse List.id_traverse\n\nprotected theorem comp_traverse {\u03b1 \u03b2 \u03b3} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : List \u03b1) :\n    List.traverse (Comp.mk \u2218 (\u00b7 <$> \u00b7) f \u2218 g) x = Comp.mk (List.traverse f <$> List.traverse g x) :=\n  by induction x <;> simp! [*, functor_norm] <;> rfl\n#align list.comp_traverse List.comp_traverse\n\nprotected theorem traverse_eq_map_id {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (x : List \u03b1) :\n    List.traverse ((pure : _ \u2192 Id _) \u2218 f) x = (pure : _ \u2192 Id _) (f <$> x) := by\n  induction x <;> simp! [*, functor_norm]; rfl\n#align list.traverse_eq_map_id List.traverse_eq_map_id\n\nvariable (\u03b7 : ApplicativeTransformation F G)\n\nprotected theorem naturality {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) (x : List \u03b1) :\n    \u03b7 (List.traverse f x) = List.traverse (@\u03b7 _ \u2218 f) x := by\n  -- Porting note: added `ApplicativeTransformation` theorems\n  induction x <;> simp! [*, functor_norm, ApplicativeTransformation.preserves_map,\n    ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]\n#align list.naturality List.naturality\n\ninstance : IsLawfulTraversable.{u} List :=\n  { show LawfulMonad List from inferInstance with\n    id_traverse := List.id_traverse\n    comp_traverse := List.comp_traverse\n    traverse_eq_map_id := List.traverse_eq_map_id\n    naturality := List.naturality }\n\nend\n\nsection Traverse\n\nvariable {\u03b1' \u03b2' : Type u} (f : \u03b1' \u2192 F \u03b2')\n\n@[simp]\ntheorem traverse_nil : traverse f ([] : List \u03b1') = (pure [] : F (List \u03b2')) :=\n  rfl\n#align list.traverse_nil List.traverse_nil\n\n@[simp]\ntheorem traverse_cons (a : \u03b1') (l : List \u03b1') :\n    traverse f (a :: l) = (\u00b7 :: \u00b7) <$> f a <*> traverse f l :=\n  rfl\n#align list.traverse_cons List.traverse_cons\n\nvariable [LawfulApplicative F]\n\n@[simp]\ntheorem traverse_append :\n    \u2200 as bs : List \u03b1', traverse f (as ++ bs) = (\u00b7 ++ \u00b7) <$> traverse f as <*> traverse f bs\n  | [], bs => by simp [functor_norm]\n  | a :: as, bs => by simp [traverse_append as bs, functor_norm]; congr\n#align list.traverse_append List.traverse_append\n\ntheorem mem_traverse {f : \u03b1' \u2192 Set \u03b2'} :\n    \u2200 (l : List \u03b1') (n : List \u03b2'), n \u2208 traverse f l \u2194 Forall\u2082 (fun b a => b \u2208 f a) n l\n  | [], [] => by simp\n  | a :: as, [] => by simp\n  | [], b :: bs => by simp\n  | a :: as, b :: bs => by simp [mem_traverse as bs]\n#align list.mem_traverse List.mem_traverse\n\nend Traverse\n\nend List\n\nnamespace Sum\n\nsection Traverse\n\nvariable {\u03c3 : Type u}\n\nvariable {F G : Type u \u2192 Type u}\n\nvariable [Applicative F] [Applicative G]\n\nopen Applicative Functor\n\nprotected theorem traverse_map {\u03b1 \u03b2 \u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (f : \u03b2 \u2192 G \u03b3) (x : \u03c3 \u2295 \u03b1) :\n    Sum.traverse f (g <$> x) = Sum.traverse (f \u2218 g) x := by\n  cases x <;> simp [Sum.traverse, id_map, functor_norm] <;> rfl\n#align sum.traverse_map Sum.traverse_map\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\n\nprotected theorem id_traverse {\u03c3 \u03b1} (x : \u03c3 \u2295 \u03b1) :\n  Sum.traverse (pure : \u03b1 \u2192 Id \u03b1) x = x := by cases x <;> rfl\n#align sum.id_traverse Sum.id_traverse\n\nprotected theorem comp_traverse {\u03b1 \u03b2 \u03b3 : Type u} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : \u03c3 \u2295 \u03b1) :\n    Sum.traverse (Comp.mk \u2218 (\u00b7 <$> \u00b7) f \u2218 g) x =\n    Comp.mk.{u} (Sum.traverse f <$> Sum.traverse g x) := by\n  cases x <;> simp! [Sum.traverse, map_id, functor_norm] <;> rfl\n#align sum.comp_traverse Sum.comp_traverse\n\nprotected theorem traverse_eq_map_id {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (x : \u03c3 \u2295 \u03b1) :\n    Sum.traverse ((pure : _ \u2192 Id _) \u2218 f) x = (pure : _ \u2192 Id _) (f <$> x) := by\n  induction x <;> simp! [*, functor_norm] <;> rfl\n#align sum.traverse_eq_map_id Sum.traverse_eq_map_id\n\nprotected theorem map_traverse {\u03b1 \u03b2 \u03b3} (g : \u03b1 \u2192 G \u03b2) (f : \u03b2 \u2192 \u03b3) (x : \u03c3 \u2295 \u03b1) :\n    (\u00b7 <$> \u00b7) f <$> Sum.traverse g x = Sum.traverse ((\u00b7 <$> \u00b7) f \u2218 g) x := by\n  cases x <;> simp [Sum.traverse, id_map, functor_norm] <;> congr\n#align sum.map_traverse Sum.map_traverse\n\nvariable (\u03b7 : ApplicativeTransformation F G)\n\nprotected theorem naturality {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) (x : \u03c3 \u2295 \u03b1) :\n    \u03b7 (Sum.traverse f x) = Sum.traverse (@\u03b7 _ \u2218 f) x := by\n  -- Porting note: added `ApplicativeTransformation` theorems\n  cases x <;> simp! [Sum.traverse, functor_norm, ApplicativeTransformation.preserves_map,\n    ApplicativeTransformation.preserves_seq, ApplicativeTransformation.preserves_pure]\n#align sum.naturality Sum.naturality\n\nend Traverse\n\ninstance {\u03c3 : Type u} : IsLawfulTraversable.{u} (Sum \u03c3) :=\n  { show LawfulMonad (Sum \u03c3) from inferInstance with\n    id_traverse := Sum.id_traverse\n    comp_traverse := Sum.comp_traverse\n    traverse_eq_map_id := Sum.traverse_eq_map_id\n    naturality := Sum.naturality }\n\nend Sum\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Control/Traversable/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547238, "lm_q2_score": 0.6992544273261175, "lm_q1q2_score": 0.39579095644646967}}
{"text": "example {a: Nat} (h: a = a): True := by simp_all\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/simpIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.39567810697837963}}
{"text": "/-\nCopyright (c) 2019 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport measure_theory.integration\n\n/-!\n# The Giry monad\n\nLet X be a measurable space. The collection of all measures on X again\nforms a measurable space. This construction forms a monad on\nmeasurable spaces and measurable functions, called the Giry monad.\n\nNote that most sources use the term \"Giry monad\" for the restriction\nto *probability* measures. Here we include all measures on X.\n\nSee also `measure_theory/category/Meas.lean`, containing an upgrade of the type-level\nmonad to an honest monad of the functor `Measure : Meas \u2964 Meas`.\n\n## References\n\n* <https://ncatlab.org/nlab/show/Giry+monad>\n\n## Tags\n\ngiry monad\n-/\n\nnoncomputable theory\nopen_locale classical big_operators ennreal\n\nopen classical set filter\n\nvariables {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Type*}\n\nnamespace measure_theory\n\nnamespace measure\n\nvariables [measurable_space \u03b1] [measurable_space \u03b2]\n\n/-- Measurability structure on `measure`: Measures are measurable w.r.t. all projections -/\ninstance : measurable_space (measure \u03b1) :=\n\u2a06 (s : set \u03b1) (hs : measurable_set s), (borel \u211d\u22650\u221e).comap (\u03bb\u03bc, \u03bc s)\n\nlemma measurable_coe {s : set \u03b1} (hs : measurable_set s) : measurable (\u03bb\u03bc : measure \u03b1, \u03bc s) :=\nmeasurable.of_comap_le $ le_supr_of_le s $ le_supr_of_le hs $ le_refl _\n\nlemma measurable_of_measurable_coe (f : \u03b2 \u2192 measure \u03b1)\n  (h : \u2200(s : set \u03b1) (hs : measurable_set s), measurable (\u03bbb, f b s)) :\n  measurable f :=\nmeasurable.of_le_map $ bsupr_le $ assume s hs, measurable_space.comap_le_iff_le_map.2 $\n  by rw [measurable_space.map_comp]; exact h s hs\n\nlemma measurable_measure {\u03bc : \u03b1 \u2192 measure \u03b2} :\n  measurable \u03bc \u2194 \u2200(s : set \u03b2) (hs : measurable_set s), measurable (\u03bbb, \u03bc b s) :=\n\u27e8\u03bb h\u03bc s hs, (measurable_coe hs).comp h\u03bc, measurable_of_measurable_coe \u03bc\u27e9\n\nlemma measurable_map (f : \u03b1 \u2192 \u03b2) (hf : measurable f) :\n  measurable (\u03bb\u03bc : measure \u03b1, map f \u03bc) :=\nmeasurable_of_measurable_coe _ $ assume s hs,\n  suffices measurable (\u03bb (\u03bc : measure \u03b1), \u03bc (f \u207b\u00b9' s)),\n    by simpa [map_apply, hs, hf],\n  measurable_coe (hf hs)\n\nlemma measurable_dirac :\n  measurable (measure.dirac : \u03b1 \u2192 measure \u03b1) :=\nmeasurable_of_measurable_coe _ $ assume s hs,\n  begin\n    simp only [dirac_apply', hs],\n    exact measurable_one.indicator hs\n  end\n\nlemma measurable_lintegral {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) :\n  measurable (\u03bb\u03bc : measure \u03b1, \u222b\u207b x, f x \u2202\u03bc) :=\nbegin\n  simp only [lintegral_eq_supr_eapprox_lintegral, hf, simple_func.lintegral],\n  refine measurable_supr (\u03bb n, finset.measurable_sum _ (\u03bb i _, _)),\n  refine measurable.const_mul _ _,\n  exact measurable_coe ((simple_func.eapprox f n).measurable_set_preimage _)\nend\n\n/-- Monadic join on `measure` in the category of measurable spaces and measurable\nfunctions. -/\ndef join (m : measure (measure \u03b1)) : measure \u03b1 :=\nmeasure.of_measurable\n  (\u03bbs hs, \u222b\u207b \u03bc, \u03bc s \u2202m)\n  (by simp)\n  begin\n    assume f hf h,\n    simp [measure_Union h hf],\n    apply lintegral_tsum,\n    assume i, exact measurable_coe (hf i)\n  end\n\n@[simp] lemma join_apply {m : measure (measure \u03b1)} :\n  \u2200{s : set \u03b1}, measurable_set s \u2192 join m s = \u222b\u207b \u03bc, \u03bc s \u2202m :=\nmeasure.of_measurable_apply\n\n@[simp] lemma join_zero : (0 : measure (measure \u03b1)).join = 0 :=\nby { ext1 s hs, simp [hs] }\n\nlemma measurable_join : measurable (join : measure (measure \u03b1) \u2192 measure \u03b1) :=\nmeasurable_of_measurable_coe _ $ assume s hs,\n  by simp only [join_apply hs]; exact measurable_lintegral (measurable_coe hs)\n\nlemma lintegral_join {m : measure (measure \u03b1)} {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) :\n  \u222b\u207b x, f x \u2202(join m) = \u222b\u207b \u03bc, \u222b\u207b x, f x \u2202\u03bc \u2202m :=\nbegin\n  rw [lintegral_eq_supr_eapprox_lintegral hf],\n  have : \u2200n x,\n    join m (\u21d1(simple_func.eapprox (\u03bb (a : \u03b1), f a) n) \u207b\u00b9' {x}) =\n      \u222b\u207b \u03bc, \u03bc ((\u21d1(simple_func.eapprox (\u03bb (a : \u03b1), f a) n) \u207b\u00b9' {x})) \u2202m :=\n    assume n x, join_apply (simple_func.measurable_set_preimage _ _),\n  simp only [simple_func.lintegral, this],\n  transitivity,\n  have : \u2200(s : \u2115 \u2192 finset \u211d\u22650\u221e) (f : \u2115 \u2192 \u211d\u22650\u221e \u2192 measure \u03b1 \u2192 \u211d\u22650\u221e)\n    (hf : \u2200n r, measurable (f n r)) (hm : monotone (\u03bbn \u03bc, \u2211 r in s n, r * f n r \u03bc)),\n    (\u2a06n:\u2115, \u2211 r in s n, r * \u222b\u207b \u03bc, f n r \u03bc \u2202m) =\n    \u222b\u207b \u03bc, \u2a06n:\u2115, \u2211 r in s n, r * f n r \u03bc \u2202m,\n  { assume s f hf hm,\n    symmetry,\n    transitivity,\n    apply lintegral_supr,\n    { assume n,\n      exact finset.measurable_sum _ (assume r _, (hf _ _).const_mul _) },\n    { exact hm },\n    congr, funext n,\n    transitivity,\n    apply lintegral_finset_sum,\n    { assume r _, exact (hf _ _).const_mul _ },\n    congr, funext r,\n    apply lintegral_const_mul,\n    exact hf _ _ },\n  specialize this (\u03bbn, simple_func.range (simple_func.eapprox f n)),\n  specialize this\n    (\u03bbn r \u03bc, \u03bc (\u21d1(simple_func.eapprox (\u03bb (a : \u03b1), f a) n) \u207b\u00b9' {r})),\n  refine this _ _; clear this,\n  { assume n r,\n    apply measurable_coe,\n    exact simple_func.measurable_set_preimage _ _ },\n  { change monotone (\u03bbn \u03bc, (simple_func.eapprox f n).lintegral \u03bc),\n    assume n m h \u03bc,\n    refine simple_func.lintegral_mono _ (le_refl _),\n    apply simple_func.monotone_eapprox,\n    assumption },\n  congr, funext \u03bc,\n  symmetry,\n  apply lintegral_eq_supr_eapprox_lintegral,\n  exact hf\nend\n\n/-- Monadic bind on `measure`, only works in the category of measurable spaces and measurable\nfunctions. When the function `f` is not measurable the result is not well defined. -/\ndef bind (m : measure \u03b1) (f : \u03b1 \u2192 measure \u03b2) : measure \u03b2 := join (map f m)\n\n@[simp] lemma bind_zero_left (f : \u03b1 \u2192 measure \u03b2) : bind 0 f = 0 :=\nby simp [bind]\n\n@[simp] lemma bind_zero_right (m : measure \u03b1) :\n  bind m (0 : \u03b1 \u2192 measure \u03b2) = 0 :=\nbegin\n  ext1 s hs,\n  simp only [bind, hs, join_apply, coe_zero, pi.zero_apply],\n  rw [lintegral_map (measurable_coe hs) measurable_zero],\n  simp\nend\n\n@[simp] lemma bind_zero_right' (m : measure \u03b1) :\n  bind m (\u03bb _, 0 : \u03b1 \u2192 measure \u03b2) = 0 :=\nbind_zero_right m\n\n@[simp] lemma bind_apply {m : measure \u03b1} {f : \u03b1 \u2192 measure \u03b2} {s : set \u03b2}\n  (hs : measurable_set s) (hf : measurable f) :\n  bind m f s = \u222b\u207b a, f a s \u2202m :=\nby rw [bind, join_apply hs, lintegral_map (measurable_coe hs) hf]\n\nlemma measurable_bind' {g : \u03b1 \u2192 measure \u03b2} (hg : measurable g) : measurable (\u03bbm, bind m g) :=\nmeasurable_join.comp (measurable_map _ hg)\n\nlemma lintegral_bind {m : measure \u03b1} {\u03bc : \u03b1 \u2192 measure \u03b2} {f : \u03b2 \u2192 \u211d\u22650\u221e}\n  (h\u03bc : measurable \u03bc) (hf : measurable f) :\n  \u222b\u207b x, f x \u2202 (bind m \u03bc) = \u222b\u207b a, \u222b\u207b x, f x \u2202(\u03bc a) \u2202m:=\n(lintegral_join hf).trans (lintegral_map (measurable_lintegral hf) h\u03bc)\n\nlemma bind_bind {\u03b3} [measurable_space \u03b3] {m : measure \u03b1} {f : \u03b1 \u2192 measure \u03b2} {g : \u03b2 \u2192 measure \u03b3}\n  (hf : measurable f) (hg : measurable g) :\n  bind (bind m f) g = bind m (\u03bba, bind (f a) g) :=\nmeasure.ext $ assume s hs,\nbegin\n  rw [bind_apply hs hg, bind_apply hs ((measurable_bind' hg).comp hf), lintegral_bind hf],\n  { congr, funext a,\n    exact (bind_apply hs hg).symm },\n  exact (measurable_coe hs).comp hg\nend\n\nlemma bind_dirac {f : \u03b1 \u2192 measure \u03b2} (hf : measurable f) (a : \u03b1) : bind (dirac a) f = f a :=\nmeasure.ext $ \u03bb s hs, by rw [bind_apply hs hf, lintegral_dirac' a ((measurable_coe hs).comp hf)]\n\nlemma dirac_bind {m : measure \u03b1} : bind m dirac = m :=\nmeasure.ext $ assume s hs,\nby simp [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs]\n\nlemma join_eq_bind (\u03bc : measure (measure \u03b1)) : join \u03bc = bind \u03bc id :=\nby rw [bind, map_id]\n\nlemma join_map_map {f : \u03b1 \u2192 \u03b2} (hf : measurable f) (\u03bc : measure (measure \u03b1)) :\n  join (map (map f) \u03bc) = map f (join \u03bc) :=\nmeasure.ext $ assume s hs,\n  begin\n    rw [join_apply hs, map_apply hf hs, join_apply,\n      lintegral_map (measurable_coe hs) (measurable_map f hf)],\n    { congr, funext \u03bd, exact map_apply hf hs },\n    exact hf hs\n  end\n\nlemma join_map_join (\u03bc : measure (measure (measure \u03b1))) :\n  join (map join \u03bc) = join (join \u03bc) :=\nbegin\n  show bind \u03bc join = join (join \u03bc),\n  rw [join_eq_bind, join_eq_bind, bind_bind measurable_id measurable_id],\n  apply congr_arg (bind \u03bc),\n  funext \u03bd,\n  exact join_eq_bind \u03bd\nend\n\nlemma join_map_dirac (\u03bc : measure \u03b1) : join (map dirac \u03bc) = \u03bc :=\ndirac_bind\n\nlemma join_dirac (\u03bc : measure \u03b1) : join (dirac \u03bc) = \u03bc :=\neq.trans (join_eq_bind (dirac \u03bc)) (bind_dirac measurable_id _)\n\nend measure\n\nend measure_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/measure_theory/giry_monad.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.39543975580862534}}
{"text": "import .phys.time.time\nimport .phys.time_series.geom3d\nimport .std.time_std\nimport .std.geom3d_std\nimport .phys.time_series.geom3d\nimport data.real.basic\nnoncomputable theory\n\ndef milliseconds := (0.001)           -- names not clear, inverted?\ndef milliseconds_to_seconds := 1000   -- names not clear, inverted?\ndef seconds := 1                      -- think about this more\ndef nanoseconds := (0.000000001)\ndef nanoseconds_to_seconds := 1000000000\n\n-- TODO: Should come from resp. std libraries and be distributed to them accordingly\nnamespace std\ndef time (p :scalar) : time time_std_space := mk_time time_std_space p\ndef duration (d :scalar) : duration time_std_space := mk_duration _ d\ndef position (x y z :scalar) : position3d geom3d_std_space := mk_position3d _ x y z\ndef displacement (x y z :scalar) : displacement3d geom3d_std_space := mk_displacement3d _ x y z\nend std\n/-\nWe need to assume a physical interpretation of the data\nrepresenting our coordinate system on time. \n\nThe only referenced time coordinate system in this example is a system time - which\nis not a UTC timestamp, rather, a zero-initiated time expressed in milliseconds, indicating\nthe time since the robot was booted. We define this ACS in terms of UTC (to avoid using std_space directly as requested last week?).\n\n(1) ORIGIN: 0\n\n(2) BASIS VECTORS\n    basis0 \n      - points to the future\n      - unit length is 1 millisecond \n(3) ACS is given by [Origin, b0]\n-/\n\nnamespace system_boot_time_ms\naxiom \u03b4 : scalar\naxiom \u03b5 : scalar\ndef origin := std.time \u03b4   \ndef basis := std.duration (\u03b5*milliseconds)    \ndef frame := mk_time_frame origin basis\ndef coords := mk_time_space frame\ndef time (t :scalar) := mk_time coords t\ndef duration (d :scalar) := mk_duration coords d\nend system_boot_time_ms\n\n/-\nWe also need an ACS for calls to \"synchronize_stamp\". These calls return actual system times in UTC,\nso we shift up the coordinate to a more reasonable origin point and now define the units in seconds.\n\n(1) ORIGIN: 1629311979 (current timestamp)\n\n(2) BASIS VECTORS\n    basis0 \n      - points to the future\n      - unit length is 1 second (as in UTC)\n(3) ACS is given by [Origin, b0]\n-/\n\nnamespace utc\naxiom \u03b4 : scalar\naxiom \u03b5 : scalar\ndef origin := std.time \u03b4   \ndef basis := std.duration (\u03b5*seconds)    \ndef frame := mk_time_frame origin basis\ndef coords := mk_time_space frame\ndef time (t :scalar) := mk_time coords t\ndef duration (d :scalar) := mk_duration coords d\nend utc\n\n\n/-\nLastly, we need a system time in nanoseconds, used in calls to \"synchronize stamp\".\n\n(1) ORIGIN: 0\n\n(2) BASIS VECTORS\n    basis0 \n      - points to the future\n      - unit length is 1 nanoseconds (as in UTC)\n(3) ACS is given by [Origin, b0]\n-/\n\nnamespace utc_ns\ndef origin := utc.time 0   \ndef basis := utc.duration (nanoseconds)    \ndef frame := mk_time_frame origin basis\ndef coords := mk_time_space frame\ndef time (t :scalar) := mk_time coords t\ndef duration (d :scalar) := mk_duration coords d\nend utc_ns\n\n\n/-\nWe need to assume a physical interpretation of the data\nrepresenting our coordinate system on geom3d. See geom3d_std.lean\nfor more details on the coordinate system and physical interpretation.\n\n\nWe define a world frame. It's not fully clear if this is necessary from the codebase, \nbut we will use this world space and define several body frames in terms of it. In the name,\nwe recognize that the rice420.coords implementation's origin can be interpreted as ENU. \n\nNote also:\nhttps://git.scc.kit.edu/uqdpy/mavros/tree/master/mavros\n\n\"The translation from GPS coordinates to local geocentric coordinates require the definition\nof a local origin on the map frame, in ECEF, and calculate the offset to it in ENU. All\nthe conversions are supported by GeographicLib classes and methods and implemented in the\nglobal_position plugin.\"\n-/\nnamespace map_enu  -- it's generic/parametric: for example, world -> Rice 440, as follows  \ndef origin := std.position 0 0 0      -- looking in from doorway, the back lower left corner  \ndef basis_0 := std.displacement 1 0 0 -- right/east along wall; unit is 1m; right\ndef basis_1 := std.displacement 0 1 0 -- to door along weset wall; 1m; right\ndef basis_2 := std.displacement 0 0 1 -- up along NW corner; 1m; right handed\ndef frame := mk_geom3d_frame origin basis_0 basis_1 basis_2\ndef coords := mk_geom3d_space frame\ndef position (x y z : scalar) := mk_position3d coords x y z\ndef displacement (x y z : scalar) := mk_displacement3d coords x y z\nend map_enu\n\n/-\n\nThe local origin frame is used, *I believe* is a world-fixed frame, whereas \"fcu\" would be the pose, perhaps\ncoming from IMU data, representing the current pose/ACS of the robot itself. \n\nWe define the robot to be 3 meters to the right of the left wall, 4 meters\nin north of the bottom wall, and 1 meter above the ground. It's orientation is \"local NED\", which\nactually should be referred to as \"ESD\" more like. Please see diagram in one of the related issues:\nhttps://github.com/mavlink/mavros/issues/216\n\n\nNote also:\nhttps://git.scc.kit.edu/uqdpy/mavros/tree/master/mavros\n\n\"The translation from GPS coordinates to local geocentric coordinates require the definition\nof a local origin on the map frame, in ECEF, and calculate the offset to it in ENU. All\nthe conversions are supported by GeographicLib classes and methods and implemented in the\nglobal_position plugin.\"\n-/\nnamespace local_origin_local_ned  -- it's generic/parametric: for example, world -> Rice 440, as follows  \ndef origin := map_enu.position 0 0 0      -- looking in from doorway, the back lower left corner  \ndef basis_0 := map_enu.displacement 1 0 0 -- right/east along wall; unit is 1m; right\ndef basis_1 := map_enu.displacement 0 1 0 -- to door along weset wall; 1m; right\ndef basis_2 := map_enu.displacement 0 0 1 -- up along NW corner; 1m; right handed\ndef frame := mk_geom3d_frame origin basis_0 basis_1 basis_2\ndef coords := mk_geom3d_space frame\ndef position (x y z : scalar) := mk_position3d coords x y z\ndef displacement (x y z : scalar) := mk_displacement3d coords x y z\nend local_origin_local_ned\n\n/-\nWe define a separate ACS for the local origin now oriented in terms of ENU, as necessary to integrate with ROS\n\nFor short-range Cartesian representations of geographic locations, use the east north up [5] (ENU) convention:\n\nX east\nY north\nZ up\n\n, required for such things as visualization of local positions in RVIZ.\n\n-/\nnamespace local_origin_enu  -- it's generic/parametric: for example, world -> Rice 440, as follows  \ndef origin := local_origin_local_ned.position 0 0 0      -- looking in from doorway, the back lower left corner  \ndef basis_0 := local_origin_local_ned.displacement 1 0 0 -- right/east along wall; unit is 1m; right\ndef basis_1 := local_origin_local_ned.displacement 0 (-1) 0 -- to door along weset wall; 1m; right\ndef basis_2 := local_origin_local_ned.displacement 0 0 (-1) -- up along NW corner; 1m; right handed\ndef frame := mk_geom3d_frame origin basis_0 basis_1 basis_2\ndef coords := mk_geom3d_space frame\ndef position (x y z : scalar) := mk_position3d coords x y z\ndef displacement (x y z : scalar) := mk_displacement3d coords x y z\nend local_origin_enu\n\n\n/-\nhttps://github.com/mavlink/mavros/issues/216\n-/\n\n\n\nopen classical\nlocal attribute [instance] prop_decidable\n\nstructure UAS := \n  (imu_orientation : orientation3d local_origin_enu.coords)\n\n/-\n\ntf::Quaternion UAS::get_attitude_orientation()\n{\n\tlock_guard lock(mutex);\n\treturn imu_orientation;\n}\n\n-/\ndef UAS.get_attitude_orientation (uas : UAS) : orientation3d local_origin_enu.coords := \n  uas.imu_orientation\n\n/-\nros::Time UAS::synchronise_stamp(uint32_t time_boot_ms) {\n\t// copy offset from atomic var\n\tuint64_t offset_ns = time_offset;\n\n\tif (offset_ns > 0) {\n\t\tuint64_t stamp_ns = static_cast<uint64_t>(time_boot_ms) * 1000000UL + offset_ns;\n\t\treturn ros_time_from_ns(stamp_ns);\n\t}\n\telse\n\t\treturn ros::Time::now();\n}\n-/\n\n#check time_std_space.mk_time_transform_to utc.coords \n\n#check system_boot_time_ms.coords \n\naxiom time_offset : scalar \ndef UAS.synchronise_stamp (uas : UAS) : time system_boot_time_ms.coords \u2192 time utc.coords := \n  \u03bb time_boot_ms,\n  let offset_ns : duration utc_ns.coords := mk_duration _ time_offset in\n \n  if offset_ns > 0 then \n    let stamp_ns := \n      (system_boot_time_ms.coords.mk_time_transform_to utc_ns.coords).transform_time time_boot_ms in \n    (utc_ns.coords.mk_time_transform_to utc.coords).transform_time stamp_ns\n  else \n    mk_time _ 0\n\n/-\n\n\n\tros::Publisher local_position;\n\ttf::TransformBroadcaster tf_broadcaster;\n-/\n\nstructure Publisher :=\n  mk::\n\ndef Publisher.publish\n  (p : Publisher) : timestamped utc.coords (pose3d local_origin_enu.coords) \u2192 punit := \n  \u03bbp, punit.star \n\nstructure TransformBroadcaster :=\n  mk::\n\ndef TransformBroadcaster.sendTransform \n  (tb : TransformBroadcaster) : timestamped utc.coords (pose3d local_origin_enu.coords) \u2192 punit := \n  \u03bbp, punit.star \n\nstructure LocalPositionPlugin := \n (uas : UAS)\n (local_position : Publisher)\n (tf_broadcaster : TransformBroadcaster)\n\n\ndef pose3d.setOrigin {f : geom3d_frame} {sp : geom3d_space f} (p: pose3d sp) : position3d sp \u2192 punit := \n  \u03bb pos, \n  let p0 : pose3d sp := {\n    position := pos, \n    ..p\n  } in \n  punit.star\n\ndef pose3d.setRotation {f : geom3d_frame} {sp : geom3d_space f} (p: pose3d sp) : orientation3d sp \u2192 punit := \n  \u03bb ort, \n  let p0 : pose3d sp := {\n    orientation := ort, \n    ..p\n  } in \n  punit.star \n\ndef poseTFToMsg {f : geom3d_frame} {sp : geom3d_space f} (p1 p2 : pose3d sp ) : punit := \n  let p2_0 := p1 in \n  punit.star \n\n/-\n\n  void handle_local_position_ned(const mavlink_message_t *msg, uint8_t sysid, uint8_t compid) {\n  \n  This functions accepts one argument, of type \"mavlink_message_t\", which is a dataframe that can contain\n  various types of data. In this case, it's a position in the \"local_origin\" frame. So, we interpret the argument\n  as such.\n-/\ndef LocalPositionPlugin.handle_local_position_ned (lpp : LocalPositionPlugin) : \n  timestamped system_boot_time_ms.coords (position3d local_origin_local_ned.coords) \u2192 punit := \n  \u03bb msg, \n  /-\n  \n    static inline void mavlink_msg_local_position_ned_decode(const mavlink_message_t* msg, mavlink_local_position_ned_t* local_position_ned)\n    {\n    #if MAVLINK_NEED_BYTE_SWAP\n      local_position_ned->time_boot_ms = mavlink_msg_local_position_ned_get_time_boot_ms(msg);\n      local_position_ned->x = mavlink_msg_local_position_ned_get_x(msg);\n      local_position_ned->y = mavlink_msg_local_position_ned_get_y(msg);\n      local_position_ned->z = mavlink_msg_local_position_ned_get_z(msg);\n      local_position_ned->vx = mavlink_msg_local_position_ned_get_vx(msg);\n      local_position_ned->vy = mavlink_msg_local_position_ned_get_vy(msg);\n      local_position_ned->vz = mavlink_msg_local_position_ned_get_vz(msg);\n    #else\n      memcpy(local_position_ned, _MAV_PAYLOAD(msg), 28);\n    #endif\n\n    mavlink_local_position_ned_t pos_ned;\n\t\tmavlink_msg_local_position_ned_decode(msg, &pos_ned);\n\n    As seen above, we are simply moving the generic dataframe type into a more-specific position type, \"mavlink_local_position_ned_t\"\n    To do that, we use the mavlink_msg_local_position_ned_decode function, which, as seen in the definition (omitted some parts), \n    requires indexing into a byte array at certain offsets. So, I am avoiding formalizing that for now. Regardless,\n    the semantics of the call is simply that we're assigning to pos_ned from the value contained in msg - which is treated\n    as a simple assignment given we're intepreting both as local_origin positions.\n  -/\n  let pos_ned : timestamped system_boot_time_ms.coords (position3d local_origin_local_ned.coords) := inhabited.default _ in \n  let pos_ned0 : timestamped system_boot_time_ms.coords (position3d local_origin_local_ned.coords) := msg in \n  /-\n  tf::Transform transform;\n\t\ttransform.setOrigin(tf::Vector3(pos_ned.y, pos_ned.x, -pos_ned.z));\n\t\ttransform.setRotation(uas->get_attitude_orientation());-/\n  let transform : pose3d local_origin_enu.coords := inhabited.default _ in \n  let setOriginCall := transform.setOrigin (mk_position3d _ (pos_ned.value.y) (pos_ned.value.x) (-pos_ned.value.z)) in\n  let setRotationCall := transform.setRotation lpp.uas.get_attitude_orientation in\n  /-\n\tauto pose = boost::make_shared<geometry_msgs::PoseStamped>();\n\n\ttf::poseTFToMsg(transform, pose->pose);\n  -/\n  let pose : timestamped utc.coords (pose3d local_origin_enu.coords) := inhabited.default _ in \n  let poseTFToMsgCall := poseTFToMsg transform pose.value in\n  let pose0 : timestamped utc.coords (pose3d local_origin_enu.coords) := {\n    timestamp := lpp.uas.synchronise_stamp pos_ned.timestamp,\n    ..pose\n  } in \n\n  let if0 : punit :=\n    if true then \n      punit.star\n    else \n      punit.star in \n  \n  let publishCall := lpp.local_position.publish pose in \n  /-\n  pose->header.frame_id = frame_id;\n\t\tpose->header.stamp = uas->synchronise_stamp(pos_ned.time_boot_ms);\n\n\t\tif (send_tf)\n\t\t\ttf_broadcaster.sendTransform(\n\t\t\t\t\ttf::StampedTransform(\n\t\t\t\t\t\ttransform,\n\t\t\t\t\t\tpose->header.stamp,\n\t\t\t\t\t\tframe_id, child_frame_id));\n\n\t\tlocal_position.publish(pose);\n  -/\n\n\n\n  /-\n  tf::Transform transform;\n\t\ttransform.setOrigin(tf::Vector3(pos_ned.y, pos_ned.x, -pos_ned.z));\n\t\ttransform.setRotation(uas->get_attitude_orientation());\n\n\t\tauto pose = boost::make_shared<geometry_msgs::PoseStamped>();\n\n\t\ttf::poseTFToMsg(transform, pose->pose);\n\t\tpose->header.frame_id = frame_id;\n\t\tpose->header.stamp = uas->synchronise_stamp(pos_ned.time_boot_ms);\n\n\t\tif (send_tf)\n\t\t\ttf_broadcaster.sendTransform(\n\t\t\t\t\ttf::StampedTransform(\n\t\t\t\t\t\ttransform,\n\t\t\t\t\t\tpose->header.stamp,\n\t\t\t\t\t\tframe_id, child_frame_id));\n\n\t\tlocal_position.publish(pose);\n  -/\n\n/-\n  \tvoid handle_local_position_ned(const mavlink_message_t *msg, uint8_t sysid, uint8_t compid) {\n\t\tmavlink_local_position_ned_t pos_ned;\n\t\tmavlink_msg_local_position_ned_decode(msg, &pos_ned);\n\n    \n    static inline void mavlink_msg_local_position_ned_decode(const mavlink_message_t* msg, mavlink_local_position_ned_t* local_position_ned)\n    {\n    #if MAVLINK_NEED_BYTE_SWAP\n      local_position_ned->time_boot_ms = mavlink_msg_local_position_ned_get_time_boot_ms(msg);\n      local_position_ned->x = mavlink_msg_local_position_ned_get_x(msg);\n      local_position_ned->y = mavlink_msg_local_position_ned_get_y(msg);\n      local_position_ned->z = mavlink_msg_local_position_ned_get_z(msg);\n      local_position_ned->vx = mavlink_msg_local_position_ned_get_vx(msg);\n      local_position_ned->vy = mavlink_msg_local_position_ned_get_vy(msg);\n      local_position_ned->vz = mavlink_msg_local_position_ned_get_vz(msg);\n    #else\n      memcpy(local_position_ned, _MAV_PAYLOAD(msg), 28);\n    #endif\n    }\n    \n\n\t\tROS_DEBUG_THROTTLE_NAMED(10, \"position\", \"Local position NED: boot_ms:%06d \"\n\t\t\t\t\"position:(%1.3f %1.3f %1.3f) speed:(%1.3f %1.3f %1.3f)\",\n\t\t\t\tpos_ned.time_boot_ms,\n\t\t\t\tpos_ned.x, pos_ned.y, pos_ned.z,\n\t\t\t\tpos_ned.vx, pos_ned.vy, pos_ned.vz);\n\n\t\t/* TODO: check convertion to ENU\n\t\t * I think XZY is not body-fixed, but orientation does.\n\t\t * Perhaps this adds additional errorprone to us.\n\t\t * Need more tests. Issue #49.\n\t\t *\n\t\t * orientation in ENU, body-fixed\n\t\t */\n\t\ttf::Transform transform;\n\t\ttransform.setOrigin(tf::Vector3(pos_ned.y, pos_ned.x, -pos_ned.z));\n\t\ttransform.setRotation(uas->get_attitude_orientation());\n\n\t\tauto pose = boost::make_shared<geometry_msgs::PoseStamped>();\n\n\t\ttf::poseTFToMsg(transform, pose->pose);\n\t\tpose->header.frame_id = frame_id;\n\t\tpose->header.stamp = uas->synchronise_stamp(pos_ned.time_boot_ms);\n\n\t\tif (send_tf)\n\t\t\ttf_broadcaster.sendTransform(\n\t\t\t\t\ttf::StampedTransform(\n\t\t\t\t\t\ttransform,\n\t\t\t\t\t\tpose->header.stamp,\n\t\t\t\t\t\tframe_id, child_frame_id));\n\n\t\tlocal_position.publish(pose);\n\t}\n\n\n\tvoid send_vision_transform(const tf::Transform &transform, const ros::Time &stamp) {\n\t\t// origin and RPY in ENU frame\n\t\ttf::Vector3 position = transform.getOrigin();\n\t\tdouble roll, pitch, yaw;\n\t\ttf::Matrix3x3 orientation(transform.getBasis());\n\t\torientation.getRPY(roll, pitch, yaw);\n\n\t\t/* Issue #60.\n\t\t * Note: this now affects pose callbacks too, but i think its not big deal.\n\t\t */\n\t\tif (last_transform_stamp == stamp) {\n\t\t\tROS_DEBUG_THROTTLE_NAMED(10, \"vision_pose\", \"Vision: Same transform as last one, dropped.\");\n\t\t\treturn;\n\t\t}\n\t\tlast_transform_stamp = stamp;\n\n\t\t// TODO: check conversion. Issue #49.\n\t\tvision_position_estimate(stamp.toNSec() / 1000,\n\t\t\t\tposition.y(), position.x(), -position.z(),\n\t\t\t\troll, -pitch, -yaw);\t// ??? please check!\n\t}\n-/\n", "meta": {"author": "kevinsullivan", "repo": "bug_stories", "sha": "dac1391905c66d02e62828d53d89b0b641efdf83", "save_path": "github-repos/lean/kevinsullivan-bug_stories", "path": "github-repos/lean/kevinsullivan-bug_stories/bug_stories-dac1391905c66d02e62828d53d89b0b641efdf83/src/orientation_handle_local_position_ned_formalization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3954397484855446}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.num.bitwise\nimport data.int.char_zero\nimport data.nat.gcd\nimport data.nat.psub\n\n/-!\n# Properties of the binary representation of integers\n-/\n\nlocal attribute [simp] add_assoc\n\nnamespace pos_num\nvariables {\u03b1 : Type*}\n\n@[simp, norm_cast] theorem cast_one [has_one \u03b1] [has_add \u03b1] :\n  ((1 : pos_num) : \u03b1) = 1 := rfl\n@[simp] theorem cast_one' [has_one \u03b1] [has_add \u03b1] : (pos_num.one : \u03b1) = 1 := rfl\n@[simp, norm_cast] theorem cast_bit0 [has_one \u03b1] [has_add \u03b1] (n : pos_num) :\n  (n.bit0 : \u03b1) = _root_.bit0 n := rfl\n@[simp, norm_cast] theorem cast_bit1 [has_one \u03b1] [has_add \u03b1] (n : pos_num) :\n  (n.bit1 : \u03b1) = _root_.bit1 n := rfl\n\n@[simp, norm_cast] theorem cast_to_nat [add_monoid \u03b1] [has_one \u03b1] :\n  \u2200 n : pos_num, ((n : \u2115) : \u03b1) = n\n| 1        := nat.cast_one\n| (bit0 p) := (nat.cast_bit0 _).trans $ congr_arg _root_.bit0 p.cast_to_nat\n| (bit1 p) := (nat.cast_bit1 _).trans $ congr_arg _root_.bit1 p.cast_to_nat\n\n@[simp, norm_cast] theorem to_nat_to_int (n : pos_num) : ((n : \u2115) : \u2124) = n :=\nby rw [\u2190 int.nat_cast_eq_coe_nat, cast_to_nat]\n\n@[simp, norm_cast] theorem cast_to_int [add_group \u03b1] [has_one \u03b1] (n : pos_num) :\n  ((n : \u2124) : \u03b1) = n :=\nby rw [\u2190 to_nat_to_int, int.cast_coe_nat, cast_to_nat]\n\ntheorem succ_to_nat : \u2200 n, (succ n : \u2115) = n + 1\n| 1        := rfl\n| (bit0 p) := rfl\n| (bit1 p) := (congr_arg _root_.bit0 (succ_to_nat p)).trans $\n  show \u2191p + 1 + \u2191p + 1 = \u2191p + \u2191p + 1 + 1, by simp [add_left_comm]\n\ntheorem one_add (n : pos_num) : 1 + n = succ n := by cases n; refl\ntheorem add_one (n : pos_num) : n + 1 = succ n := by cases n; refl\n\n@[norm_cast]\ntheorem add_to_nat : \u2200 m n, ((m + n : pos_num) : \u2115) = m + n\n| 1        b        := by rw [one_add b, succ_to_nat, add_comm]; refl\n| a        1        := by rw [add_one a, succ_to_nat]; refl\n| (bit0 a) (bit0 b) := (congr_arg _root_.bit0 (add_to_nat a b)).trans $ add_add_add_comm _ _ _ _\n| (bit0 a) (bit1 b) := (congr_arg _root_.bit1 (add_to_nat a b)).trans $\n  show ((a + b) + (a + b) + 1 : \u2115) = (a + a) + (b + b + 1), by simp [add_left_comm]\n| (bit1 a) (bit0 b) := (congr_arg _root_.bit1 (add_to_nat a b)).trans $\n  show ((a + b) + (a + b) + 1 : \u2115) = (a + a + 1) + (b + b), by simp [add_comm, add_left_comm]\n| (bit1 a) (bit1 b) :=\n  show (succ (a + b) + succ (a + b) : \u2115) = (a + a + 1) + (b + b + 1),\n  by rw [succ_to_nat, add_to_nat]; simp [add_left_comm]\n\ntheorem add_succ : \u2200 (m n : pos_num), m + succ n = succ (m + n)\n| 1        b        := by simp [one_add]\n| (bit0 a) 1        := congr_arg bit0 (add_one a)\n| (bit1 a) 1        := congr_arg bit1 (add_one a)\n| (bit0 a) (bit0 b) := rfl\n| (bit0 a) (bit1 b) := congr_arg bit0 (add_succ a b)\n| (bit1 a) (bit0 b) := rfl\n| (bit1 a) (bit1 b) := congr_arg bit1 (add_succ a b)\n\ntheorem bit0_of_bit0 : \u03a0 n, _root_.bit0 n = bit0 n\n| 1        := rfl\n| (bit0 p) := congr_arg bit0 (bit0_of_bit0 p)\n| (bit1 p) := show bit0 (succ (_root_.bit0 p)) = _, by rw bit0_of_bit0; refl\n\ntheorem bit1_of_bit1 (n : pos_num) : _root_.bit1 n = bit1 n :=\nshow _root_.bit0 n + 1 = bit1 n, by rw [add_one, bit0_of_bit0]; refl\n\n@[norm_cast]\ntheorem mul_to_nat (m) : \u2200 n, ((m * n : pos_num) : \u2115) = m * n\n| 1        := (mul_one _).symm\n| (bit0 p) := show (\u2191(m * p) + \u2191(m * p) : \u2115) = \u2191m * (p + p), by rw [mul_to_nat, left_distrib]\n| (bit1 p) := (add_to_nat (bit0 (m * p)) m).trans $\n  show (\u2191(m * p) + \u2191(m * p) + \u2191m : \u2115) = \u2191m * (p + p) + m, by rw [mul_to_nat, left_distrib]\n\ntheorem to_nat_pos : \u2200 n : pos_num, 0 < (n : \u2115)\n| 1        := zero_lt_one\n| (bit0 p) := let h := to_nat_pos p in add_pos h h\n| (bit1 p) := nat.succ_pos _\n\ntheorem cmp_to_nat_lemma {m n : pos_num} : (m:\u2115) < n \u2192 (bit1 m : \u2115) < bit0 n :=\nshow (m:\u2115) < n \u2192 (m + m + 1 + 1 : \u2115) \u2264 n + n,\nby intro h; rw [nat.add_right_comm m m 1, add_assoc]; exact add_le_add h h\n\ntheorem cmp_swap (m) : \u2200n, (cmp m n).swap = cmp n m :=\nby induction m with m IH m IH; intro n;\n   cases n with n n; try {unfold cmp}; try {refl}; rw \u2190IH; cases cmp m n; refl\n\ntheorem cmp_to_nat : \u2200 (m n), (ordering.cases_on (cmp m n) ((m:\u2115) < n) (m = n) ((n:\u2115) < m) : Prop)\n| 1        1        := rfl\n| (bit0 a) 1        := let h : (1:\u2115) \u2264 a := to_nat_pos a in add_le_add h h\n| (bit1 a) 1        := nat.succ_lt_succ $ to_nat_pos $ bit0 a\n| 1        (bit0 b) := let h : (1:\u2115) \u2264 b := to_nat_pos b in add_le_add h h\n| 1        (bit1 b) := nat.succ_lt_succ $ to_nat_pos $ bit0 b\n| (bit0 a) (bit0 b) := begin\n    have := cmp_to_nat a b, revert this, cases cmp a b; dsimp; intro,\n    { exact add_lt_add this this },\n    { rw this },\n    { exact add_lt_add this this }\n  end\n| (bit0 a) (bit1 b) := begin dsimp [cmp],\n    have := cmp_to_nat a b, revert this, cases cmp a b; dsimp; intro,\n    { exact nat.le_succ_of_le (add_lt_add this this) },\n    { rw this, apply nat.lt_succ_self },\n    { exact cmp_to_nat_lemma this }\n  end\n| (bit1 a) (bit0 b) := begin dsimp [cmp],\n    have := cmp_to_nat a b, revert this, cases cmp a b; dsimp; intro,\n    { exact cmp_to_nat_lemma this },\n    { rw this, apply nat.lt_succ_self },\n    { exact nat.le_succ_of_le (add_lt_add this this) },\n  end\n| (bit1 a) (bit1 b) := begin\n    have := cmp_to_nat a b, revert this, cases cmp a b; dsimp; intro,\n    { exact nat.succ_lt_succ (add_lt_add this this) },\n    { rw this },\n    { exact nat.succ_lt_succ (add_lt_add this this) }\n  end\n\n@[norm_cast]\ntheorem lt_to_nat {m n : pos_num} : (m:\u2115) < n \u2194 m < n :=\nshow (m:\u2115) < n \u2194 cmp m n = ordering.lt, from\nmatch cmp m n, cmp_to_nat m n with\n| ordering.lt, h := by simp at h; simp [h]\n| ordering.eq, h := by simp at h; simp [h, lt_irrefl]; exact dec_trivial\n| ordering.gt, h := by simp [not_lt_of_gt h]; exact dec_trivial\nend\n\n@[norm_cast]\ntheorem le_to_nat {m n : pos_num} : (m:\u2115) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr lt_to_nat\n\nend pos_num\n\nnamespace num\nvariables {\u03b1 : Type*}\nopen pos_num\n\ntheorem add_zero (n : num) : n + 0 = n := by cases n; refl\ntheorem zero_add (n : num) : 0 + n = n := by cases n; refl\n\ntheorem add_one : \u2200 n : num, n + 1 = succ n\n| 0       := rfl\n| (pos p) := by cases p; refl\n\ntheorem add_succ : \u2200 (m n : num), m + succ n = succ (m + n)\n| 0       n       := by simp [zero_add]\n| (pos p) 0       := show pos (p + 1) = succ (pos p + 0),\n                     by rw [pos_num.add_one, add_zero]; refl\n| (pos p) (pos q) := congr_arg pos (pos_num.add_succ _ _)\n\n@[simp, norm_cast] theorem add_of_nat (m) : \u2200 n, ((m + n : \u2115) : num) = m + n\n| 0     := (add_zero _).symm\n| (n+1) := show ((m + n : \u2115) + 1 : num) = m + (\u2191 n + 1),\n           by rw [add_one, add_one, add_succ, add_of_nat]\n\ntheorem bit0_of_bit0 : \u2200 n : num, bit0 n = n.bit0\n| 0       := rfl\n| (pos p) := congr_arg pos p.bit0_of_bit0\n\ntheorem bit1_of_bit1 : \u2200 n : num, bit1 n = n.bit1\n| 0       := rfl\n| (pos p) := congr_arg pos p.bit1_of_bit1\n\n@[simp, norm_cast] theorem cast_zero [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] :\n  ((0 : num) : \u03b1) = 0 := rfl\n\n@[simp] theorem cast_zero' [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] :\n  (num.zero : \u03b1) = 0 := rfl\n\n@[simp, norm_cast] theorem cast_one [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] :\n  ((1 : num) : \u03b1) = 1 := rfl\n\n@[simp] theorem cast_pos [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1]\n  (n : pos_num) : (num.pos n : \u03b1) = n := rfl\n\ntheorem succ'_to_nat : \u2200 n, (succ' n : \u2115) = n + 1\n| 0       := (_root_.zero_add _).symm\n| (pos p) := pos_num.succ_to_nat _\n\ntheorem succ_to_nat (n) : (succ n : \u2115) = n + 1 := succ'_to_nat n\n\n@[simp, norm_cast] theorem cast_to_nat [add_monoid \u03b1] [has_one \u03b1] : \u2200 n : num, ((n : \u2115) : \u03b1) = n\n| 0       := nat.cast_zero\n| (pos p) := p.cast_to_nat\n\n@[simp, norm_cast] theorem to_nat_to_int (n : num) : ((n : \u2115) : \u2124) = n :=\nby rw [\u2190 int.nat_cast_eq_coe_nat, cast_to_nat]\n\n@[simp, norm_cast] theorem cast_to_int [add_group \u03b1] [has_one \u03b1] (n : num) : ((n : \u2124) : \u03b1) = n :=\nby rw [\u2190 to_nat_to_int, int.cast_coe_nat, cast_to_nat]\n\n@[norm_cast]\ntheorem to_of_nat : \u03a0 (n : \u2115), ((n : num) : \u2115) = n\n| 0     := rfl\n| (n+1) := by rw [nat.cast_add_one, add_one, succ_to_nat, to_of_nat]\n\n@[simp, norm_cast]\ntheorem of_nat_cast [add_monoid \u03b1] [has_one \u03b1] (n : \u2115) : ((n : num) : \u03b1) = n :=\nby rw [\u2190 cast_to_nat, to_of_nat]\n\n@[norm_cast] theorem of_nat_inj {m n : \u2115} : (m : num) = n \u2194 m = n :=\n\u27e8\u03bb h, function.left_inverse.injective to_of_nat h, congr_arg _\u27e9\n\n@[norm_cast]\ntheorem add_to_nat : \u2200 m n, ((m + n : num) : \u2115) = m + n\n| 0       0       := rfl\n| 0       (pos q) := (_root_.zero_add _).symm\n| (pos p) 0       := rfl\n| (pos p) (pos q) := pos_num.add_to_nat _ _\n\n@[norm_cast]\ntheorem mul_to_nat : \u2200 m n, ((m * n : num) : \u2115) = m * n\n| 0       0       := rfl\n| 0       (pos q) := (zero_mul _).symm\n| (pos p) 0       := rfl\n| (pos p) (pos q) := pos_num.mul_to_nat _ _\n\ntheorem cmp_to_nat : \u2200 (m n), (ordering.cases_on (cmp m n) ((m:\u2115) < n) (m = n) ((n:\u2115) < m) : Prop)\n| 0       0       := rfl\n| 0       (pos b) := to_nat_pos _\n| (pos a) 0       := to_nat_pos _\n| (pos a) (pos b) :=\n  by { have := pos_num.cmp_to_nat a b; revert this; dsimp [cmp];\n       cases pos_num.cmp a b, exacts [id, congr_arg pos, id] }\n\n@[norm_cast]\ntheorem lt_to_nat {m n : num} : (m:\u2115) < n \u2194 m < n :=\nshow (m:\u2115) < n \u2194 cmp m n = ordering.lt, from\nmatch cmp m n, cmp_to_nat m n with\n| ordering.lt, h := by simp at h; simp [h]\n| ordering.eq, h := by simp at h; simp [h, lt_irrefl]; exact dec_trivial\n| ordering.gt, h := by simp [not_lt_of_gt h]; exact dec_trivial\nend\n\n@[norm_cast]\ntheorem le_to_nat {m n : num} : (m:\u2115) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr lt_to_nat\n\nend num\n\nnamespace pos_num\n@[simp] theorem of_to_nat : \u03a0 (n : pos_num), ((n : \u2115) : num) = num.pos n\n| 1        := rfl\n| (bit0 p) :=\n  show \u2191(p + p : \u2115) = num.pos p.bit0,\n  by rw [num.add_of_nat, of_to_nat];\n    exact congr_arg num.pos p.bit0_of_bit0\n| (bit1 p) :=\n  show ((p + p : \u2115) : num) + 1 = num.pos p.bit1,\n  by rw [num.add_of_nat, of_to_nat];\n    exact congr_arg num.pos p.bit1_of_bit1\nend pos_num\n\nnamespace num\n\n@[simp, norm_cast] theorem of_to_nat : \u03a0 (n : num), ((n : \u2115) : num) = n\n| 0           := rfl\n| (pos p) := p.of_to_nat\n\n@[norm_cast] theorem to_nat_inj {m n : num} : (m : \u2115) = n \u2194 m = n :=\n\u27e8\u03bb h, function.left_inverse.injective of_to_nat h, congr_arg _\u27e9\n\n/--\nThis tactic tries to turn an (in)equality about `num`s to one about `nat`s by rewriting.\n```lean\nexample (n : num) (m : num) : n \u2264 n + m :=\nbegin\n  num.transfer_rw,\n  exact nat.le_add_right _ _\nend\n```\n-/\nmeta def transfer_rw : tactic unit :=\n`[repeat {rw \u2190 to_nat_inj <|> rw \u2190 lt_to_nat <|> rw \u2190 le_to_nat},\n  repeat {rw add_to_nat <|> rw mul_to_nat <|> rw cast_one <|> rw cast_zero}]\n\n/--\nThis tactic tries to prove (in)equalities about `num`s by transfering them to the `nat` world and\nthen trying to call `simp`.\n```lean\nexample (n : num) (m : num) : n \u2264 n + m := by num.transfer\n```\n-/\nmeta def transfer : tactic unit := `[intros, transfer_rw, try {simp}]\n\ninstance : comm_semiring num :=\nby refine_struct\n{ add      := (+),\n  zero     := 0,\n  zero_add := zero_add,\n  add_zero := add_zero,\n  mul      := (*),\n  one      := 1,\n  nsmul    := @nsmul_rec num \u27e80\u27e9 \u27e8(+)\u27e9,\n  npow     := @npow_rec num \u27e81\u27e9 \u27e8(*)\u27e9 };\ntry { intros, refl }; try { transfer }; simp [mul_add, mul_left_comm, mul_comm, add_comm]\n\ninstance : ordered_cancel_add_comm_monoid num :=\n{ add_left_cancel            := by {intros a b c, transfer_rw, apply add_left_cancel},\n  lt                         := (<),\n  lt_iff_le_not_le           := by {intros a b, transfer_rw, apply lt_iff_le_not_le},\n  le                         := (\u2264),\n  le_refl                    := by transfer,\n  le_trans                   := by {intros a b c, transfer_rw, apply le_trans},\n  le_antisymm                := by {intros a b, transfer_rw, apply le_antisymm},\n  add_le_add_left            := by {intros a b h c, revert h, transfer_rw,\n    exact \u03bb h, add_le_add_left h c},\n  le_of_add_le_add_left      := by {intros a b c, transfer_rw, apply le_of_add_le_add_left},\n  ..num.comm_semiring }\n\ninstance : linear_ordered_semiring num :=\n{ le_total                   := by {intros a b, transfer_rw, apply le_total},\n  zero_le_one                := dec_trivial,\n  mul_lt_mul_of_pos_left     := by {intros a b c, transfer_rw, apply mul_lt_mul_of_pos_left},\n  mul_lt_mul_of_pos_right    := by {intros a b c, transfer_rw, apply mul_lt_mul_of_pos_right},\n  decidable_lt               := num.decidable_lt,\n  decidable_le               := num.decidable_le,\n  decidable_eq               := num.decidable_eq,\n  exists_pair_ne             := \u27e80, 1, dec_trivial\u27e9,\n  ..num.comm_semiring, ..num.ordered_cancel_add_comm_monoid }\n\n@[norm_cast]\ntheorem dvd_to_nat (m n : num) : (m : \u2115) \u2223 n \u2194 m \u2223 n :=\n\u27e8\u03bb \u27e8k, e\u27e9, \u27e8k, by rw [\u2190 of_to_nat n, e]; simp\u27e9,\n \u03bb \u27e8k, e\u27e9, \u27e8k, by simp [e, mul_to_nat]\u27e9\u27e9\n\nend num\n\nnamespace pos_num\nvariables {\u03b1 : Type*}\nopen num\n\n@[norm_cast] theorem to_nat_inj {m n : pos_num} : (m : \u2115) = n \u2194 m = n :=\n\u27e8\u03bb h, num.pos.inj $ by rw [\u2190 pos_num.of_to_nat, \u2190 pos_num.of_to_nat, h],\n congr_arg _\u27e9\n\ntheorem pred'_to_nat : \u2200 n, (pred' n : \u2115) = nat.pred n\n| 1        := rfl\n| (bit0 n) :=\n  have nat.succ \u2191(pred' n) = \u2191n,\n  by rw [pred'_to_nat n, nat.succ_pred_eq_of_pos (to_nat_pos n)],\n  match pred' n, this : \u2200 k : num, nat.succ \u2191k = \u2191n \u2192\n    \u2191(num.cases_on k 1 bit1 : pos_num) = nat.pred (_root_.bit0 n) with\n  | 0, (h : ((1:num):\u2115) = n) := by rw \u2190 to_nat_inj.1 h; refl\n  | num.pos p, (h : nat.succ \u2191p = n) :=\n    by rw \u2190 h; exact (nat.succ_add p p).symm\n  end\n| (bit1 n) := rfl\n\n@[simp] theorem pred'_succ' (n) : pred' (succ' n) = n :=\nnum.to_nat_inj.1 $ by rw [pred'_to_nat, succ'_to_nat,\n  nat.add_one, nat.pred_succ]\n\n@[simp] theorem succ'_pred' (n) : succ' (pred' n) = n :=\nto_nat_inj.1 $ by rw [succ'_to_nat, pred'_to_nat,\n  nat.add_one, nat.succ_pred_eq_of_pos (to_nat_pos _)]\n\ninstance : has_dvd pos_num := \u27e8\u03bb m n, pos m \u2223 pos n\u27e9\n\n@[norm_cast] theorem dvd_to_nat {m n : pos_num} : (m:\u2115) \u2223 n \u2194 m \u2223 n :=\nnum.dvd_to_nat (pos m) (pos n)\n\ntheorem size_to_nat : \u2200 n, (size n : \u2115) = nat.size n\n| 1        := nat.size_one.symm\n| (bit0 n) := by rw [size, succ_to_nat, size_to_nat, cast_bit0,\n                     nat.size_bit0 $ ne_of_gt $ to_nat_pos n]\n| (bit1 n) := by rw [size, succ_to_nat, size_to_nat, cast_bit1,\n                     nat.size_bit1]\n\ntheorem size_eq_nat_size : \u2200 n, (size n : \u2115) = nat_size n\n| 1        := rfl\n| (bit0 n) := by rw [size, succ_to_nat, nat_size, size_eq_nat_size]\n| (bit1 n) := by rw [size, succ_to_nat, nat_size, size_eq_nat_size]\n\ntheorem nat_size_to_nat (n) : nat_size n = nat.size n :=\nby rw [\u2190 size_eq_nat_size, size_to_nat]\n\ntheorem nat_size_pos (n) : 0 < nat_size n :=\nby cases n; apply nat.succ_pos\n\n/--\nThis tactic tries to turn an (in)equality about `pos_num`s to one about `nat`s by rewriting.\n```lean\nexample (n : pos_num) (m : pos_num) : n \u2264 n + m :=\nbegin\n  pos_num.transfer_rw,\n  exact nat.le_add_right _ _\nend\n```\n-/\nmeta def transfer_rw : tactic unit :=\n`[repeat {rw \u2190 to_nat_inj <|> rw \u2190 lt_to_nat <|> rw \u2190 le_to_nat},\n  repeat {rw add_to_nat <|> rw mul_to_nat <|> rw cast_one <|> rw cast_zero}]\n\n/--\nThis tactic tries to prove (in)equalities about `pos_num`s by transferring them to the `nat` world\nand then trying to call `simp`.\n```lean\nexample (n : pos_num) (m : pos_num) : n \u2264 n + m := by pos_num.transfer\n```\n-/\nmeta def transfer : tactic unit :=\n`[intros, transfer_rw, try {simp [add_comm, add_left_comm, mul_comm, mul_left_comm]}]\n\ninstance : add_comm_semigroup pos_num :=\nby refine {add := (+), ..}; transfer\n\ninstance : comm_monoid pos_num :=\nby refine_struct {mul := (*), one := (1 : pos_num), npow := @npow_rec pos_num \u27e81\u27e9 \u27e8(*)\u27e9};\ntry { intros, refl }; transfer\n\ninstance : distrib pos_num :=\nby refine {add := (+), mul := (*), ..}; {transfer, simp [mul_add, mul_comm]}\n\ninstance : linear_order pos_num :=\n{ lt              := (<),\n  lt_iff_le_not_le := by {intros a b, transfer_rw, apply lt_iff_le_not_le},\n  le              := (\u2264),\n  le_refl         := by transfer,\n  le_trans        := by {intros a b c, transfer_rw, apply le_trans},\n  le_antisymm     := by {intros a b, transfer_rw, apply le_antisymm},\n  le_total        := by {intros a b, transfer_rw, apply le_total},\n  decidable_lt    := by apply_instance,\n  decidable_le    := by apply_instance,\n  decidable_eq    := by apply_instance }\n\n@[simp] theorem cast_to_num (n : pos_num) : \u2191n = num.pos n :=\nby rw [\u2190 cast_to_nat, \u2190 of_to_nat n]\n\n@[simp, norm_cast]\ntheorem bit_to_nat (b n) : (bit b n : \u2115) = nat.bit b n :=\nby cases b; refl\n\n@[simp, norm_cast]\ntheorem cast_add [add_monoid \u03b1] [has_one \u03b1] (m n) : ((m + n : pos_num) : \u03b1) = m + n :=\nby rw [\u2190 cast_to_nat, add_to_nat, nat.cast_add, cast_to_nat, cast_to_nat]\n\n@[simp, norm_cast, priority 500]\ntheorem cast_succ [add_monoid \u03b1] [has_one \u03b1] (n : pos_num) : (succ n : \u03b1) = n + 1 :=\nby rw [\u2190 add_one, cast_add, cast_one]\n\n@[simp, norm_cast]\ntheorem cast_inj [add_monoid \u03b1] [has_one \u03b1] [char_zero \u03b1] {m n : pos_num} : (m:\u03b1) = n \u2194 m = n :=\nby rw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, nat.cast_inj, to_nat_inj]\n\n@[simp]\ntheorem one_le_cast [linear_ordered_semiring \u03b1] (n : pos_num) : (1 : \u03b1) \u2264 n :=\nby rw [\u2190 cast_to_nat, \u2190 nat.cast_one, nat.cast_le]; apply to_nat_pos\n\n@[simp]\ntheorem cast_pos [linear_ordered_semiring \u03b1] (n : pos_num) : 0 < (n : \u03b1) :=\nlt_of_lt_of_le zero_lt_one (one_le_cast n)\n\n@[simp, norm_cast]\ntheorem cast_mul [semiring \u03b1] (m n) : ((m * n : pos_num) : \u03b1) = m * n :=\nby rw [\u2190 cast_to_nat, mul_to_nat, nat.cast_mul, cast_to_nat, cast_to_nat]\n\n@[simp]\ntheorem cmp_eq (m n) : cmp m n = ordering.eq \u2194 m = n :=\nbegin\n  have := cmp_to_nat m n,\n  cases cmp m n; simp at this \u22a2; try {exact this};\n  { simp [show m \u2260 n, from \u03bb e, by rw e at this; exact lt_irrefl _ this] }\nend\n\n@[simp, norm_cast]\ntheorem cast_lt [linear_ordered_semiring \u03b1] {m n : pos_num} : (m:\u03b1) < n \u2194 m < n :=\nby rw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, nat.cast_lt, lt_to_nat]\n\n@[simp, norm_cast]\ntheorem cast_le [linear_ordered_semiring \u03b1] {m n : pos_num} : (m:\u03b1) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr cast_lt\n\nend pos_num\n\nnamespace num\nvariables {\u03b1 : Type*}\nopen pos_num\n\ntheorem bit_to_nat (b n) : (bit b n : \u2115) = nat.bit b n :=\nby cases b; cases n; refl\n\ntheorem cast_succ' [add_monoid \u03b1] [has_one \u03b1] (n) : (succ' n : \u03b1) = n + 1 :=\nby rw [\u2190 pos_num.cast_to_nat, succ'_to_nat, nat.cast_add_one, cast_to_nat]\n\ntheorem cast_succ [add_monoid \u03b1] [has_one \u03b1] (n) : (succ n : \u03b1) = n + 1 := cast_succ' n\n\n@[simp, norm_cast] theorem cast_add [semiring \u03b1] (m n) : ((m + n : num) : \u03b1) = m + n :=\nby rw [\u2190 cast_to_nat, add_to_nat, nat.cast_add, cast_to_nat, cast_to_nat]\n\n@[simp, norm_cast] theorem cast_bit0 [semiring \u03b1] (n : num) : (n.bit0 : \u03b1) = _root_.bit0 n :=\nby rw [\u2190 bit0_of_bit0, _root_.bit0, cast_add]; refl\n\n@[simp, norm_cast] theorem cast_bit1 [semiring \u03b1] (n : num) : (n.bit1 : \u03b1) = _root_.bit1 n :=\nby rw [\u2190 bit1_of_bit1, _root_.bit1, bit0_of_bit0, cast_add, cast_bit0]; refl\n\n@[simp, norm_cast] theorem cast_mul [semiring \u03b1] : \u2200 m n, ((m * n : num) : \u03b1) = m * n\n| 0       0       := (zero_mul _).symm\n| 0       (pos q) := (zero_mul _).symm\n| (pos p) 0       := (mul_zero _).symm\n| (pos p) (pos q) := pos_num.cast_mul _ _\n\ntheorem size_to_nat : \u2200 n, (size n : \u2115) = nat.size n\n| 0       := nat.size_zero.symm\n| (pos p) := p.size_to_nat\n\ntheorem size_eq_nat_size : \u2200 n, (size n : \u2115) = nat_size n\n| 0       := rfl\n| (pos p) := p.size_eq_nat_size\n\ntheorem nat_size_to_nat (n) : nat_size n = nat.size n :=\nby rw [\u2190 size_eq_nat_size, size_to_nat]\n\n@[simp] theorem of_nat'_zero : num.of_nat' 0 = 0 :=\nby simp [num.of_nat']\n\n@[simp, priority 999] theorem of_nat'_eq : \u2200 n, num.of_nat' n = n :=\nnat.binary_rec (by simp) $ \u03bb b n IH, begin\n  rw of_nat' at IH \u22a2,\n  rw [nat.binary_rec_eq, IH],\n  { cases b; simp [nat.bit, bit0_of_bit0, bit1_of_bit1] },\n  { refl }\nend\n\ntheorem zneg_to_znum (n : num) : -n.to_znum = n.to_znum_neg := by cases n; refl\ntheorem zneg_to_znum_neg (n : num) : -n.to_znum_neg = n.to_znum := by cases n; refl\n\ntheorem to_znum_inj {m n : num} : m.to_znum = n.to_znum \u2194 m = n :=\n\u27e8\u03bb h, by cases m; cases n; cases h; refl, congr_arg _\u27e9\n\n@[simp, norm_cast squash] theorem cast_to_znum [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1] :\n  \u2200 n : num, (n.to_znum : \u03b1) = n\n| 0           := rfl\n| (num.pos p) := rfl\n\n@[simp] theorem cast_to_znum_neg [add_group \u03b1] [has_one \u03b1] :\n  \u2200 n : num, (n.to_znum_neg : \u03b1) = -n\n| 0           := neg_zero.symm\n| (num.pos p) := rfl\n\n@[simp] theorem add_to_znum (m n : num) : num.to_znum (m + n) = m.to_znum + n.to_znum :=\nby cases m; cases n; refl\n\nend num\n\nnamespace pos_num\nopen num\n\ntheorem pred_to_nat {n : pos_num} (h : 1 < n) : (pred n : \u2115) = nat.pred n :=\nbegin\n  unfold pred,\n  have := pred'_to_nat n,\n  cases e : pred' n,\n  { have : (1:\u2115) \u2264 nat.pred n :=\n      nat.pred_le_pred ((@cast_lt \u2115 _ _ _).2 h),\n    rw [\u2190 pred'_to_nat, e] at this,\n    exact absurd this dec_trivial },\n  { rw [\u2190 pred'_to_nat, e], refl }\nend\n\ntheorem sub'_one (a : pos_num) : sub' a 1 = (pred' a).to_znum :=\nby cases a; refl\n\ntheorem one_sub' (a : pos_num) : sub' 1 a = (pred' a).to_znum_neg :=\nby cases a; refl\n\ntheorem lt_iff_cmp {m n} : m < n \u2194 cmp m n = ordering.lt := iff.rfl\n\ntheorem le_iff_cmp {m n} : m \u2264 n \u2194 cmp m n \u2260 ordering.gt :=\nnot_congr $ lt_iff_cmp.trans $\nby rw \u2190 cmp_swap; cases cmp m n; exact dec_trivial\n\nend pos_num\n\nnamespace num\nvariables {\u03b1 : Type*}\nopen pos_num\n\ntheorem pred_to_nat : \u2200 (n : num), (pred n : \u2115) = nat.pred n\n| 0       := rfl\n| (pos p) := by rw [pred, pos_num.pred'_to_nat]; refl\n\ntheorem ppred_to_nat : \u2200 (n : num), coe <$> ppred n = nat.ppred n\n| 0       := rfl\n| (pos p) := by rw [ppred, option.map_some, nat.ppred_eq_some.2];\n  rw [pos_num.pred'_to_nat, nat.succ_pred_eq_of_pos (pos_num.to_nat_pos _)]; refl\n\ntheorem cmp_swap (m n) : (cmp m n).swap = cmp n m :=\nby cases m; cases n; try {unfold cmp}; try {refl}; apply pos_num.cmp_swap\n\ntheorem cmp_eq (m n) : cmp m n = ordering.eq \u2194 m = n :=\nbegin\n  have := cmp_to_nat m n,\n  cases cmp m n; simp at this \u22a2; try {exact this};\n  { simp [show m \u2260 n, from \u03bb e, by rw e at this; exact lt_irrefl _ this] }\nend\n\n@[simp, norm_cast]\ntheorem cast_lt [linear_ordered_semiring \u03b1] {m n : num} : (m:\u03b1) < n \u2194 m < n :=\nby rw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, nat.cast_lt, lt_to_nat]\n\n@[simp, norm_cast]\ntheorem cast_le [linear_ordered_semiring \u03b1] {m n : num} : (m:\u03b1) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr cast_lt\n\n@[simp, norm_cast]\ntheorem cast_inj [linear_ordered_semiring \u03b1] {m n : num} : (m:\u03b1) = n \u2194 m = n :=\nby rw [\u2190 cast_to_nat m, \u2190 cast_to_nat n, nat.cast_inj, to_nat_inj]\n\ntheorem lt_iff_cmp {m n} : m < n \u2194 cmp m n = ordering.lt := iff.rfl\n\ntheorem le_iff_cmp {m n} : m \u2264 n \u2194 cmp m n \u2260 ordering.gt :=\nnot_congr $ lt_iff_cmp.trans $\nby rw \u2190 cmp_swap; cases cmp m n; exact dec_trivial\n\ntheorem bitwise_to_nat {f : num \u2192 num \u2192 num} {g : bool \u2192 bool \u2192 bool}\n  (p : pos_num \u2192 pos_num \u2192 num)\n  (gff : g ff ff = ff)\n  (f00 : f 0 0 = 0)\n  (f0n : \u2200 n, f 0 (pos n) = cond (g ff tt) (pos n) 0)\n  (fn0 : \u2200 n, f (pos n) 0 = cond (g tt ff) (pos n) 0)\n  (fnn : \u2200 m n, f (pos m) (pos n) = p m n)\n  (p11 : p 1 1 = cond (g tt tt) 1 0)\n  (p1b : \u2200 b n, p 1 (pos_num.bit b n) = bit (g tt b) (cond (g ff tt) (pos n) 0))\n  (pb1 : \u2200 a m, p (pos_num.bit a m) 1 = bit (g a tt) (cond (g tt ff) (pos m) 0))\n  (pbb : \u2200 a b m n, p (pos_num.bit a m) (pos_num.bit b n) = bit (g a b) (p m n))\n  : \u2200 m n : num, (f m n : \u2115) = nat.bitwise g m n :=\nbegin\n  intros, cases m with m; cases n with n;\n  try { change zero with 0 };\n  try { change ((0:num):\u2115) with 0 },\n  { rw [f00, nat.bitwise_zero]; refl },\n  { unfold nat.bitwise, rw [f0n, nat.binary_rec_zero],\n    cases g ff tt; refl },\n  { unfold nat.bitwise,\n    generalize h : (pos m : \u2115) = m', revert h,\n    apply nat.bit_cases_on m' _, intros b m' h,\n    rw [fn0, nat.binary_rec_eq, nat.binary_rec_zero, \u2190h],\n    cases g tt ff; refl,\n    apply nat.bitwise_bit_aux gff },\n  { rw fnn,\n    have : \u2200b (n : pos_num), (cond b \u2191n 0 : \u2115) = \u2191(cond b (pos n) 0 : num) :=\n      by intros; cases b; refl,\n    induction m with m IH m IH generalizing n; cases n with n n,\n    any_goals { change one with 1 },\n    any_goals { change pos 1 with 1 },\n    any_goals { change pos_num.bit0 with pos_num.bit ff },\n    any_goals { change pos_num.bit1 with pos_num.bit tt },\n    any_goals { change ((1:num):\u2115) with nat.bit tt 0 },\n    all_goals\n    { repeat\n      { rw show \u2200 b n, (pos (pos_num.bit b n) : \u2115) = nat.bit b \u2191n,\n           by intros; cases b; refl },\n      rw nat.bitwise_bit },\n    any_goals { assumption },\n    any_goals { rw [nat.bitwise_zero, p11], cases g tt tt; refl },\n    any_goals { rw [nat.bitwise_zero_left, this, \u2190 bit_to_nat, p1b] },\n    any_goals { rw [nat.bitwise_zero_right _ gff, this, \u2190 bit_to_nat, pb1] },\n    all_goals { rw [\u2190 show \u2200 n, \u2191(p m n) = nat.bitwise g \u2191m \u2191n, from IH],\n      rw [\u2190 bit_to_nat, pbb] } }\nend\n\n@[simp, norm_cast] theorem lor_to_nat   : \u2200 m n, (lor    m n : \u2115) = nat.lor    m n :=\nby apply bitwise_to_nat (\u03bbx y, pos (pos_num.lor x y)); intros; try {cases a}; try {cases b}; refl\n@[simp, norm_cast] theorem land_to_nat  : \u2200 m n, (land   m n : \u2115) = nat.land   m n :=\nby apply bitwise_to_nat pos_num.land; intros; try {cases a}; try {cases b}; refl\n@[simp, norm_cast] theorem ldiff_to_nat : \u2200 m n, (ldiff  m n : \u2115) = nat.ldiff  m n :=\nby apply bitwise_to_nat pos_num.ldiff; intros; try {cases a}; try {cases b}; refl\n@[simp, norm_cast] theorem lxor_to_nat  : \u2200 m n, (lxor   m n : \u2115) = nat.lxor   m n :=\nby apply bitwise_to_nat pos_num.lxor; intros; try {cases a}; try {cases b}; refl\n\n@[simp, norm_cast] theorem shiftl_to_nat (m n) : (shiftl m n : \u2115) = nat.shiftl m n :=\nbegin\n  cases m; dunfold shiftl, {symmetry, apply nat.zero_shiftl},\n  simp, induction n with n IH, {refl},\n  simp [pos_num.shiftl, nat.shiftl_succ], rw \u2190IH\nend\n\n@[simp, norm_cast] theorem shiftr_to_nat (m n) : (shiftr m n : \u2115) = nat.shiftr m n :=\nbegin\n  cases m with m; dunfold shiftr, {symmetry, apply nat.zero_shiftr},\n  induction n with n IH generalizing m, {cases m; refl},\n  cases m with m m; dunfold pos_num.shiftr,\n  { rw [nat.shiftr_eq_div_pow], symmetry, apply nat.div_eq_of_lt,\n    exact @nat.pow_lt_pow_of_lt_right 2 dec_trivial 0 (n+1) (nat.succ_pos _) },\n  { transitivity, apply IH,\n    change nat.shiftr m n = nat.shiftr (bit1 m) (n+1),\n    rw [add_comm n 1, nat.shiftr_add],\n    apply congr_arg (\u03bbx, nat.shiftr x n), unfold nat.shiftr,\n    change (bit1 \u2191m : \u2115) with nat.bit tt m,\n    rw nat.div2_bit },\n  { transitivity, apply IH,\n    change nat.shiftr m n = nat.shiftr (bit0 m) (n + 1),\n    rw [add_comm n 1, nat.shiftr_add],\n    apply congr_arg (\u03bbx, nat.shiftr x n), unfold nat.shiftr,\n    change (bit0 \u2191m : \u2115) with nat.bit ff m,\n    rw nat.div2_bit }\nend\n\n@[simp] theorem test_bit_to_nat (m n) : test_bit m n = nat.test_bit m n :=\nbegin\n  cases m with m; unfold test_bit nat.test_bit,\n  { change (zero : nat) with 0, rw nat.zero_shiftr, refl },\n  induction n with n IH generalizing m;\n  cases m; dunfold pos_num.test_bit, {refl},\n  { exact (nat.bodd_bit _ _).symm },\n  { exact (nat.bodd_bit _ _).symm },\n  { change ff = nat.bodd (nat.shiftr 1 (n + 1)),\n    rw [add_comm, nat.shiftr_add], change nat.shiftr 1 1 with 0,\n    rw nat.zero_shiftr; refl },\n  { change pos_num.test_bit m n = nat.bodd (nat.shiftr (nat.bit tt m) (n + 1)),\n    rw [add_comm, nat.shiftr_add], unfold nat.shiftr,\n    rw nat.div2_bit, apply IH },\n  { change pos_num.test_bit m n = nat.bodd (nat.shiftr (nat.bit ff m) (n + 1)),\n    rw [add_comm, nat.shiftr_add], unfold nat.shiftr,\n    rw nat.div2_bit, apply IH },\nend\n\nend num\n\nnamespace znum\nvariables {\u03b1 : Type*}\nopen pos_num\n\n@[simp, norm_cast] theorem cast_zero [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1] :\n  ((0 : znum) : \u03b1) = 0 := rfl\n\n@[simp] theorem cast_zero' [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1] :\n  (znum.zero : \u03b1) = 0 := rfl\n\n@[simp, norm_cast] theorem cast_one [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1] :\n  ((1 : znum) : \u03b1) = 1 := rfl\n\n@[simp] theorem cast_pos [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1]\n  (n : pos_num) : (pos n : \u03b1) = n := rfl\n\n@[simp] theorem cast_neg [has_zero \u03b1] [has_one \u03b1] [has_add \u03b1] [has_neg \u03b1]\n  (n : pos_num) : (neg n : \u03b1) = -n := rfl\n\n@[simp, norm_cast] theorem cast_zneg [add_group \u03b1] [has_one \u03b1] : \u2200 n, ((-n : znum) : \u03b1) = -n\n| 0       := neg_zero.symm\n| (pos p) := rfl\n| (neg p) := (neg_neg _).symm\n\ntheorem neg_zero : (-0 : znum) = 0 := rfl\ntheorem zneg_pos (n : pos_num) : -pos n = neg n := rfl\ntheorem zneg_neg (n : pos_num) : -neg n = pos n := rfl\ntheorem zneg_zneg (n : znum) : - -n = n := by cases n; refl\ntheorem zneg_bit1 (n : znum) : -n.bit1 = (-n).bitm1 := by cases n; refl\ntheorem zneg_bitm1 (n : znum) : -n.bitm1 = (-n).bit1 := by cases n; refl\n\ntheorem zneg_succ (n : znum) : -n.succ = (-n).pred :=\nby cases n; try {refl}; rw [succ, num.zneg_to_znum_neg]; refl\n\ntheorem zneg_pred (n : znum) : -n.pred = (-n).succ :=\nby rw [\u2190 zneg_zneg (succ (-n)), zneg_succ, zneg_zneg]\n\n@[simp, norm_cast] \n\n@[simp] theorem abs_to_nat : \u2200 n, (abs n : \u2115) = int.nat_abs n\n| 0       := rfl\n| (pos p) := congr_arg int.nat_abs p.to_nat_to_int\n| (neg p) := show int.nat_abs ((p:\u2115):\u2124) = int.nat_abs (- p),\n  by rw [p.to_nat_to_int, int.nat_abs_neg]\n\n@[simp] theorem abs_to_znum : \u2200 n : num, abs n.to_znum = n\n| 0           := rfl\n| (num.pos p) := rfl\n\n@[simp, norm_cast] theorem cast_to_int [add_group \u03b1] [has_one \u03b1] : \u2200 n : znum, ((n : \u2124) : \u03b1) = n\n| 0       := rfl\n| (pos p) := by rw [cast_pos, cast_pos, pos_num.cast_to_int]\n| (neg p) := by rw [cast_neg, cast_neg, int.cast_neg, pos_num.cast_to_int]\n\ntheorem bit0_of_bit0 : \u2200 n : znum, _root_.bit0 n = n.bit0\n| 0       := rfl\n| (pos a) := congr_arg pos a.bit0_of_bit0\n| (neg a) := congr_arg neg a.bit0_of_bit0\n\ntheorem bit1_of_bit1 : \u2200 n : znum, _root_.bit1 n = n.bit1\n| 0       := rfl\n| (pos a) := congr_arg pos a.bit1_of_bit1\n| (neg a) := show pos_num.sub' 1 (_root_.bit0 a) = _,\n  by rw [pos_num.one_sub', a.bit0_of_bit0]; refl\n\n@[simp, norm_cast] theorem cast_bit0 [add_group \u03b1] [has_one \u03b1] :\n  \u2200 n : znum, (n.bit0 : \u03b1) = bit0 n\n| 0       := (add_zero _).symm\n| (pos p) := by rw [znum.bit0, cast_pos, cast_pos]; refl\n| (neg p) := by rw [znum.bit0, cast_neg, cast_neg, pos_num.cast_bit0,\n                    _root_.bit0, _root_.bit0, neg_add_rev]\n\n@[simp, norm_cast] theorem cast_bit1 [add_group \u03b1] [has_one \u03b1] :\n  \u2200 n : znum, (n.bit1 : \u03b1) = bit1 n\n| 0       := by simp [znum.bit1, _root_.bit1, _root_.bit0]\n| (pos p) := by rw [znum.bit1, cast_pos, cast_pos]; refl\n| (neg p) := begin\n    rw [znum.bit1, cast_neg, cast_neg],\n    cases e : pred' p with a;\n    have : p = _ := (succ'_pred' p).symm.trans\n      (congr_arg num.succ' e),\n    { change p=1 at this, subst p,\n      simp [_root_.bit1, _root_.bit0] },\n    { rw [num.succ'] at this, subst p,\n      have : (\u2191(-\u2191a:\u2124) : \u03b1) = -1 + \u2191(-\u2191a + 1 : \u2124), {simp [add_comm]},\n      simpa [_root_.bit1, _root_.bit0, -add_comm] },\n  end\n\n@[simp] theorem cast_bitm1 [add_group \u03b1] [has_one \u03b1]\n  (n : znum) : (n.bitm1 : \u03b1) = bit0 n - 1 :=\nbegin\n  conv { to_lhs, rw \u2190 zneg_zneg n },\n  rw [\u2190 zneg_bit1, cast_zneg, cast_bit1],\n  have : ((-1 + n + n : \u2124) : \u03b1) = (n + n + -1 : \u2124), {simp [add_comm, add_left_comm]},\n  simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg, -int.add_neg_one]\nend\n\ntheorem add_zero (n : znum) : n + 0 = n := by cases n; refl\ntheorem zero_add (n : znum) : 0 + n = n := by cases n; refl\n\ntheorem add_one : \u2200 n : znum, n + 1 = succ n\n| 0       := rfl\n| (pos p) := congr_arg pos p.add_one\n| (neg p) := by cases p; refl\n\nend znum\n\nnamespace pos_num\nvariables {\u03b1 : Type*}\n\ntheorem cast_to_znum : \u2200 n : pos_num, (n : znum) = znum.pos n\n| 1        := rfl\n| (bit0 p) := (znum.bit0_of_bit0 p).trans $ congr_arg _ (cast_to_znum p)\n| (bit1 p) := (znum.bit1_of_bit1 p).trans $ congr_arg _ (cast_to_znum p)\n\nlocal attribute [-simp] int.add_neg_one\n\ntheorem cast_sub' [add_group \u03b1] [has_one \u03b1] : \u2200 m n : pos_num, (sub' m n : \u03b1) = m - n\n| a        1        := by rw [sub'_one, num.cast_to_znum,\n                              \u2190 num.cast_to_nat, pred'_to_nat, \u2190 nat.sub_one];\n                          simp [pos_num.cast_pos]\n| 1        b        := by rw [one_sub', num.cast_to_znum_neg, \u2190 neg_sub, neg_inj,\n                              \u2190 num.cast_to_nat, pred'_to_nat, \u2190 nat.sub_one];\n                          simp [pos_num.cast_pos]\n| (bit0 a) (bit0 b) := begin\n    rw [sub', znum.cast_bit0, cast_sub'],\n    have : ((a + -b + (a + -b) : \u2124) : \u03b1) = a + a + (-b + -b), {simp [add_left_comm]},\n    simpa [_root_.bit0, sub_eq_add_neg]\n  end\n| (bit0 a) (bit1 b) := begin\n    rw [sub', znum.cast_bitm1, cast_sub'],\n    have : ((-b + (a + (-b + -1)) : \u2124) : \u03b1) = (a + -1 + (-b + -b):\u2124),\n    { simp [add_comm, add_left_comm] },\n    simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg]\n  end\n| (bit1 a) (bit0 b) := begin\n    rw [sub', znum.cast_bit1, cast_sub'],\n    have : ((-b + (a + (-b + 1)) : \u2124) : \u03b1) = (a + 1 + (-b + -b):\u2124),\n    { simp [add_comm, add_left_comm] },\n    simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg]\n  end\n| (bit1 a) (bit1 b) := begin\n    rw [sub', znum.cast_bit0, cast_sub'],\n    have : ((-b + (a + -b) : \u2124) : \u03b1) = a + (-b + -b), {simp [add_left_comm]},\n    simpa [_root_.bit1, _root_.bit0, sub_eq_add_neg]\n  end\n\ntheorem to_nat_eq_succ_pred (n : pos_num) : (n:\u2115) = n.pred' + 1 :=\nby rw [\u2190 num.succ'_to_nat, n.succ'_pred']\n\ntheorem to_int_eq_succ_pred (n : pos_num) : (n:\u2124) = (n.pred' : \u2115) + 1 :=\nby rw [\u2190 n.to_nat_to_int, to_nat_eq_succ_pred]; refl\n\nend pos_num\n\nnamespace num\nvariables {\u03b1 : Type*}\n\n@[simp] theorem cast_sub' [add_group \u03b1] [has_one \u03b1] : \u2200 m n : num, (sub' m n : \u03b1) = m - n\n| 0       0       := (sub_zero _).symm\n| (pos a) 0       := (sub_zero _).symm\n| 0       (pos b) := (zero_sub _).symm\n| (pos a) (pos b) := pos_num.cast_sub' _ _\n\n@[simp] theorem of_nat_to_znum : \u2200 n : \u2115, to_znum n = n\n| 0     := rfl\n| (n+1) := by rw [nat.cast_add_one, nat.cast_add_one,\n  znum.add_one, add_one, \u2190 of_nat_to_znum]; cases (n:num); refl\n\n@[simp] theorem of_nat_to_znum_neg (n : \u2115) : to_znum_neg n = -n :=\nby rw [\u2190 of_nat_to_znum, zneg_to_znum]\n\ntheorem mem_of_znum' : \u2200 {m : num} {n : znum}, m \u2208 of_znum' n \u2194 n = to_znum m\n| 0       0      := \u27e8\u03bb _, rfl, \u03bb _, rfl\u27e9\n| (pos m) 0      := \u27e8\u03bb h, by cases h, \u03bb h, by cases h\u27e9\n| m (znum.pos p) := option.some_inj.trans $\n  by cases m; split; intro h; try {cases h}; refl\n| m (znum.neg p) := \u27e8\u03bb h, by cases h, \u03bb h, by cases m; cases h\u27e9\n\ntheorem of_znum'_to_nat : \u2200 (n : znum), coe <$> of_znum' n = int.to_nat' n\n| 0            := rfl\n| (znum.pos p) := show _ = int.to_nat' p, by rw [\u2190 pos_num.to_nat_to_int p]; refl\n| (znum.neg p) := congr_arg (\u03bb x, int.to_nat' (-x)) $\n  show ((p.pred' + 1 : \u2115) : \u2124) = p, by rw \u2190 succ'_to_nat; simp\n\n@[simp] theorem of_znum_to_nat : \u2200 (n : znum), (of_znum n : \u2115) = int.to_nat n\n| 0            := rfl\n| (znum.pos p) := show _ = int.to_nat p, by rw [\u2190 pos_num.to_nat_to_int p]; refl\n| (znum.neg p) := congr_arg (\u03bb x, int.to_nat (-x)) $\n  show ((p.pred' + 1 : \u2115) : \u2124) = p, by rw \u2190 succ'_to_nat; simp\n\n@[simp] theorem cast_of_znum [add_group \u03b1] [has_one \u03b1] (n : znum) :\n  (of_znum n : \u03b1) = int.to_nat n :=\nby rw [\u2190 cast_to_nat, of_znum_to_nat]\n\n@[simp, norm_cast] theorem sub_to_nat (m n) : ((m - n : num) : \u2115) = m - n :=\nshow (of_znum _ : \u2115) = _, by rw [of_znum_to_nat, cast_sub',\n  \u2190 to_nat_to_int, \u2190 to_nat_to_int, int.to_nat_sub]\n\nend num\n\nnamespace znum\nvariables {\u03b1 : Type*}\n\n@[simp, norm_cast] theorem cast_add [add_group \u03b1] [has_one \u03b1] : \u2200 m n, ((m + n : znum) : \u03b1) = m + n\n| 0       a       := by cases a; exact (_root_.zero_add _).symm\n| b       0       := by cases b; exact (_root_.add_zero _).symm\n| (pos a) (pos b) := pos_num.cast_add _ _\n| (pos a) (neg b) := by simpa only [sub_eq_add_neg] using pos_num.cast_sub' _ _\n| (neg a) (pos b) :=\nhave (\u2191b + -\u2191a : \u03b1) = -\u2191a + \u2191b, by rw [\u2190 pos_num.cast_to_int a, \u2190 pos_num.cast_to_int b,\n  \u2190 int.cast_neg, \u2190 int.cast_add (-a)]; simp [add_comm],\n(pos_num.cast_sub' _ _).trans $ (sub_eq_add_neg _ _).trans this\n| (neg a) (neg b) := show -(\u2191(a + b) : \u03b1) = -a + -b, by rw [\n  pos_num.cast_add, neg_eq_iff_neg_eq, neg_add_rev, neg_neg, neg_neg,\n  \u2190 pos_num.cast_to_int a, \u2190 pos_num.cast_to_int b, \u2190 int.cast_add]; simp [add_comm]\n\n@[simp] theorem cast_succ [add_group \u03b1] [has_one \u03b1] (n) : ((succ n : znum) : \u03b1) = n + 1 :=\nby rw [\u2190 add_one, cast_add, cast_one]\n\n@[simp, norm_cast] theorem mul_to_int : \u2200 m n, ((m * n : znum) : \u2124) = m * n\n| 0       a       := by cases a; exact (_root_.zero_mul _).symm\n| b       0       := by cases b; exact (_root_.mul_zero _).symm\n| (pos a) (pos b) := pos_num.cast_mul a b\n| (pos a) (neg b) := show -\u2191(a * b) = \u2191a * -\u2191b, by rw [pos_num.cast_mul, neg_mul_eq_mul_neg]\n| (neg a) (pos b) := show -\u2191(a * b) = -\u2191a * \u2191b, by rw [pos_num.cast_mul, neg_mul_eq_neg_mul]\n| (neg a) (neg b) := show \u2191(a * b) = -\u2191a * -\u2191b, by rw [pos_num.cast_mul, neg_mul_neg]\n\ntheorem cast_mul [ring \u03b1] (m n) : ((m * n : znum) : \u03b1) = m * n :=\nby rw [\u2190 cast_to_int, mul_to_int, int.cast_mul, cast_to_int, cast_to_int]\n\n@[simp, norm_cast] theorem of_to_int : \u03a0 (n : znum), ((n : \u2124) : znum) = n\n| 0       := rfl\n| (pos a) := by rw [cast_pos, \u2190 pos_num.cast_to_nat,\n  int.cast_coe_nat', \u2190 num.of_nat_to_znum, pos_num.of_to_nat]; refl\n| (neg a) := by rw [cast_neg, neg_of_int, \u2190 pos_num.cast_to_nat,\n  int.cast_coe_nat', \u2190 num.of_nat_to_znum_neg, pos_num.of_to_nat]; refl\n\n@[norm_cast]\ntheorem to_of_int : \u03a0 (n : \u2124), ((n : znum) : \u2124) = n\n| (n : \u2115) := by rw [int.cast_coe_nat,\n  \u2190 num.of_nat_to_znum, num.cast_to_znum, \u2190 num.cast_to_nat,\n  int.nat_cast_eq_coe_nat, num.to_of_nat]\n| -[1+ n] := by rw [int.cast_neg_succ_of_nat, cast_zneg,\n  add_one, cast_succ, int.neg_succ_of_nat_eq,\n  \u2190 num.of_nat_to_znum, num.cast_to_znum, \u2190 num.cast_to_nat,\n  int.nat_cast_eq_coe_nat, num.to_of_nat]\n\ntheorem to_int_inj {m n : znum} : (m : \u2124) = n \u2194 m = n :=\n\u27e8\u03bb h, function.left_inverse.injective of_to_int h, congr_arg _\u27e9\n\n@[simp, norm_cast] theorem of_int_cast [add_group \u03b1] [has_one \u03b1] (n : \u2124) : ((n : znum) : \u03b1) = n :=\nby rw [\u2190 cast_to_int, to_of_int]\n\n@[simp, norm_cast] theorem of_nat_cast [add_group \u03b1] [has_one \u03b1] (n : \u2115) : ((n : znum) : \u03b1) = n :=\nof_int_cast n\n\n@[simp] theorem of_int'_eq : \u2200 n, znum.of_int' n = n\n| (n : \u2115) := to_int_inj.1 $ by simp [znum.of_int']\n| -[1+ n] := to_int_inj.1 $ by simp [znum.of_int']\n\ntheorem cmp_to_int : \u2200 (m n), (ordering.cases_on (cmp m n) ((m:\u2124) < n) (m = n) ((n:\u2124) < m) : Prop)\n| 0       0       := rfl\n| (pos a) (pos b) := begin\n    have := pos_num.cmp_to_nat a b; revert this; dsimp [cmp];\n    cases pos_num.cmp a b; dsimp;\n    [simp, exact congr_arg pos, simp [gt]]\n  end\n| (neg a) (neg b) := begin\n    have := pos_num.cmp_to_nat b a; revert this; dsimp [cmp];\n    cases pos_num.cmp b a; dsimp;\n    [simp, simp {contextual := tt}, simp [gt]]\n  end\n| (pos a) 0       := pos_num.cast_pos _\n| (pos a) (neg b) := lt_trans (neg_lt_zero.2 $ pos_num.cast_pos _) (pos_num.cast_pos _)\n| 0       (neg b) := neg_lt_zero.2 $ pos_num.cast_pos _\n| (neg a) 0       := neg_lt_zero.2 $ pos_num.cast_pos _\n| (neg a) (pos b) := lt_trans (neg_lt_zero.2 $ pos_num.cast_pos _) (pos_num.cast_pos _)\n| 0       (pos b) := pos_num.cast_pos _\n\n@[norm_cast]\ntheorem lt_to_int {m n : znum} : (m:\u2124) < n \u2194 m < n :=\nshow (m:\u2124) < n \u2194 cmp m n = ordering.lt, from\nmatch cmp m n, cmp_to_int m n with\n| ordering.lt, h := by simp at h; simp [h]\n| ordering.eq, h := by simp at h; simp [h, lt_irrefl]; exact dec_trivial\n| ordering.gt, h := by simp [not_lt_of_gt h]; exact dec_trivial\nend\n\ntheorem le_to_int {m n : znum} : (m:\u2124) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr lt_to_int\n\n@[simp, norm_cast]\ntheorem cast_lt [linear_ordered_ring \u03b1] {m n : znum} : (m:\u03b1) < n \u2194 m < n :=\nby rw [\u2190 cast_to_int m, \u2190 cast_to_int n, int.cast_lt, lt_to_int]\n\n@[simp, norm_cast]\ntheorem cast_le [linear_ordered_ring \u03b1] {m n : znum} : (m:\u03b1) \u2264 n \u2194 m \u2264 n :=\nby rw \u2190 not_lt; exact not_congr cast_lt\n\n@[simp, norm_cast]\ntheorem cast_inj [linear_ordered_ring \u03b1] {m n : znum} : (m:\u03b1) = n \u2194 m = n :=\nby rw [\u2190 cast_to_int m, \u2190 cast_to_int n, int.cast_inj, to_int_inj]\n\n/--\nThis tactic tries to turn an (in)equality about `znum`s to one about `int`s by rewriting.\n```lean\nexample (n : znum) (m : znum) : n \u2264 n + m * m :=\nbegin\n  znum.transfer_rw,\n  exact le_add_of_nonneg_right (mul_self_nonneg _)\nend\n```\n-/\nmeta def transfer_rw : tactic unit :=\n`[repeat {rw \u2190 to_int_inj <|> rw \u2190 lt_to_int <|> rw \u2190 le_to_int},\n  repeat {rw cast_add <|> rw mul_to_int <|> rw cast_one <|> rw cast_zero}]\n\n/--\nThis tactic tries to prove (in)equalities about `znum`s by transfering them to the `int` world and\nthen trying to call `simp`.\n```lean\nexample (n : znum) (m : znum) : n \u2264 n + m * m :=\nbegin\n  znum.transfer,\n  exact mul_self_nonneg _\nend\n```\n-/\nmeta def transfer : tactic unit :=\n`[intros, transfer_rw, try {simp [add_comm, add_left_comm, mul_comm, mul_left_comm]}]\n\ninstance : linear_order znum :=\n{ lt               := (<),\n  lt_iff_le_not_le := by {intros a b, transfer_rw, apply lt_iff_le_not_le},\n  le               := (\u2264),\n  le_refl          := by transfer,\n  le_trans         := by {intros a b c, transfer_rw, apply le_trans},\n  le_antisymm      := by {intros a b, transfer_rw, apply le_antisymm},\n  le_total         := by {intros a b, transfer_rw, apply le_total},\n  decidable_eq     := znum.decidable_eq,\n  decidable_le     := znum.decidable_le,\n  decidable_lt     := znum.decidable_lt }\n\ninstance : add_comm_group znum :=\n{ add              := (+),\n  add_assoc        := by transfer,\n  zero             := 0,\n  zero_add         := zero_add,\n  add_zero         := add_zero,\n  add_comm         := by transfer,\n  neg              := has_neg.neg,\n  add_left_neg     := by transfer }\n\ninstance : linear_ordered_comm_ring znum :=\n{ mul              := (*),\n  mul_assoc        := by transfer,\n  one              := 1,\n  one_mul          := by transfer,\n  mul_one          := by transfer,\n  left_distrib     := by {transfer, simp [mul_add]},\n  right_distrib    := by {transfer, simp [mul_add, mul_comm]},\n  mul_comm         := by transfer,\n  exists_pair_ne   := \u27e80, 1, dec_trivial\u27e9,\n  add_le_add_left  := by {intros a b h c, revert h, transfer_rw, exact \u03bb h, add_le_add_left h c},\n  mul_pos          := \u03bb a b, show 0 < a \u2192 0 < b \u2192 0 < a * b, by {transfer_rw, apply mul_pos},\n  zero_le_one      := dec_trivial,\n  ..znum.linear_order, ..znum.add_comm_group }\n\n@[simp, norm_cast] theorem dvd_to_int (m n : znum) : (m : \u2124) \u2223 n \u2194 m \u2223 n :=\n\u27e8\u03bb \u27e8k, e\u27e9, \u27e8k, by rw [\u2190 of_to_int n, e]; simp\u27e9,\n \u03bb \u27e8k, e\u27e9, \u27e8k, by simp [e]\u27e9\u27e9\n\nend znum\n\nnamespace pos_num\n\ntheorem divmod_to_nat_aux {n d : pos_num} {q r : num}\n  (h\u2081 : (r:\u2115) + d * _root_.bit0 q = n)\n  (h\u2082 : (r:\u2115) < 2 * d) :\n  ((divmod_aux d q r).2 + d * (divmod_aux d q r).1 : \u2115) = \u2191n \u2227\n  ((divmod_aux d q r).2 : \u2115) < d :=\nbegin\n  unfold divmod_aux,\n  have : \u2200 {r\u2082}, num.of_znum' (num.sub' r (num.pos d)) = some r\u2082 \u2194 (r : \u2115) = r\u2082 + d,\n  { intro r\u2082,\n    apply num.mem_of_znum'.trans,\n    rw [\u2190 znum.to_int_inj, num.cast_to_znum,\n      num.cast_sub', sub_eq_iff_eq_add, \u2190 int.coe_nat_inj'],\n    simp },\n  cases e : num.of_znum' (num.sub' r (num.pos d)) with r\u2082;\n    simp [divmod_aux],\n  { refine \u27e8h\u2081, lt_of_not_ge (\u03bb h, _)\u27e9,\n    cases nat.le.dest h with r\u2082 e',\n    rw [\u2190 num.to_of_nat r\u2082, add_comm] at e',\n    cases e.symm.trans (this.2 e'.symm) },\n  { have := this.1 e,\n    split,\n    { rwa [_root_.bit1, add_comm _ 1, mul_add, mul_one,\n        \u2190 add_assoc, \u2190 this] },\n    { rwa [this, two_mul, add_lt_add_iff_right] at h\u2082 } }\nend\n\ntheorem divmod_to_nat (d n : pos_num) :\n  (n / d : \u2115) = (divmod d n).1 \u2227\n  (n % d : \u2115) = (divmod d n).2 :=\nbegin\n  rw nat.div_mod_unique (pos_num.cast_pos _),\n  induction n with n IH n IH,\n  { exact divmod_to_nat_aux (by simp; refl)\n      (nat.mul_le_mul_left 2\n        (pos_num.cast_pos d : (0 : \u2115) < d)) },\n  { unfold divmod,\n    cases divmod d n with q r, simp only [divmod] at IH \u22a2,\n    apply divmod_to_nat_aux; simp,\n    { rw [_root_.bit1, _root_.bit1, add_right_comm,\n        bit0_eq_two_mul \u2191n, \u2190 IH.1,\n        mul_add, \u2190 bit0_eq_two_mul,\n        mul_left_comm, \u2190 bit0_eq_two_mul] },\n    { rw \u2190 bit0_eq_two_mul,\n      exact nat.bit1_lt_bit0 IH.2 } },\n  { unfold divmod,\n    cases divmod d n with q r, simp only [divmod] at IH \u22a2,\n    apply divmod_to_nat_aux; simp,\n    { rw [bit0_eq_two_mul \u2191n, \u2190 IH.1,\n        mul_add, \u2190 bit0_eq_two_mul,\n        mul_left_comm, \u2190 bit0_eq_two_mul] },\n    { rw \u2190 bit0_eq_two_mul,\n      exact nat.bit0_lt IH.2 } }\nend\n\n@[simp] theorem div'_to_nat (n d) : (div' n d : \u2115) = n / d :=\n(divmod_to_nat _ _).1.symm\n\n@[simp] theorem mod'_to_nat (n d) : (mod' n d : \u2115) = n % d :=\n(divmod_to_nat _ _).2.symm\n\nend pos_num\n\nnamespace num\n\n@[simp] protected lemma div_zero (n : num) : n / 0 = 0 :=\nshow n.div 0 = 0, by { cases n, refl, simp [num.div] }\n\n@[simp, norm_cast] theorem div_to_nat : \u2200 n d, ((n / d : num) : \u2115) = n / d\n| 0       0       := by simp\n| 0       (pos d) := (nat.zero_div _).symm\n| (pos n) 0       := (nat.div_zero _).symm\n| (pos n) (pos d) := pos_num.div'_to_nat _ _\n\n@[simp] protected lemma mod_zero (n : num) : n % 0 = n :=\nshow n.mod 0 = n, by { cases n, refl, simp [num.mod] }\n\n@[simp, norm_cast] theorem mod_to_nat : \u2200 n d, ((n % d : num) : \u2115) = n % d\n| 0       0       := by simp\n| 0       (pos d) := (nat.zero_mod _).symm\n| (pos n) 0       := (nat.mod_zero _).symm\n| (pos n) (pos d) := pos_num.mod'_to_nat _ _\n\ntheorem gcd_to_nat_aux : \u2200 {n} {a b : num},\n  a \u2264 b \u2192 (a * b).nat_size \u2264 n \u2192 (gcd_aux n a b : \u2115) = nat.gcd a b\n| 0            0       b       ab h := (nat.gcd_zero_left _).symm\n| 0            (pos a) 0       ab h := (not_lt_of_ge ab).elim rfl\n| 0            (pos a) (pos b) ab h :=\n  (not_lt_of_le h).elim $ pos_num.nat_size_pos _\n| (nat.succ n) 0       b       ab h := (nat.gcd_zero_left _).symm\n| (nat.succ n) (pos a) b       ab h := begin\n  simp [gcd_aux],\n  rw [nat.gcd_rec, gcd_to_nat_aux, mod_to_nat], {refl},\n  { rw [\u2190 le_to_nat, mod_to_nat],\n    exact le_of_lt (nat.mod_lt _ (pos_num.cast_pos _)) },\n  rw [nat_size_to_nat, mul_to_nat, nat.size_le] at h \u22a2,\n  rw [mod_to_nat, mul_comm],\n  rw [pow_succ', \u2190 nat.mod_add_div b (pos a)] at h,\n  refine lt_of_mul_lt_mul_right (lt_of_le_of_lt _ h) (nat.zero_le 2),\n  rw [mul_two, mul_add],\n  refine add_le_add_left (nat.mul_le_mul_left _\n    (le_trans (le_of_lt (nat.mod_lt _ (pos_num.cast_pos _))) _)) _,\n  suffices : 1 \u2264 _, simpa using nat.mul_le_mul_left (pos a) this,\n  rw [nat.le_div_iff_mul_le _ _ a.cast_pos, one_mul],\n  exact le_to_nat.2 ab\nend\n\n@[simp] theorem gcd_to_nat : \u2200 a b, (gcd a b : \u2115) = nat.gcd a b :=\nhave \u2200 a b : num, (a * b).nat_size \u2264 a.nat_size + b.nat_size,\nbegin\n  intros,\n  simp [nat_size_to_nat],\n  rw [nat.size_le, pow_add],\n  exact mul_lt_mul'' (nat.lt_size_self _)\n    (nat.lt_size_self _) (nat.zero_le _) (nat.zero_le _)\nend,\nbegin\n  intros, unfold gcd, split_ifs,\n  { exact gcd_to_nat_aux h (this _ _) },\n  { rw nat.gcd_comm,\n    exact gcd_to_nat_aux (le_of_not_le h) (this _ _) }\nend\n\ntheorem dvd_iff_mod_eq_zero {m n : num} : m \u2223 n \u2194 n % m = 0 :=\nby rw [\u2190 dvd_to_nat, nat.dvd_iff_mod_eq_zero,\n  \u2190 to_nat_inj, mod_to_nat]; refl\n\ninstance decidable_dvd : decidable_rel ((\u2223) : num \u2192 num \u2192 Prop)\n| a b := decidable_of_iff' _ dvd_iff_mod_eq_zero\n\nend num\n\ninstance pos_num.decidable_dvd : decidable_rel ((\u2223) : pos_num \u2192 pos_num \u2192 Prop)\n| a b := num.decidable_dvd _ _\n\nnamespace znum\n\n@[simp] protected lemma div_zero (n : znum) : n / 0 = 0 :=\nshow n.div 0 = 0, by cases n; refl <|> simp [znum.div]\n\n@[simp, norm_cast] theorem div_to_int : \u2200 n d, ((n / d : znum) : \u2124) = n / d\n| 0       0       := by simp [int.div_zero]\n| 0       (pos d) := (int.zero_div _).symm\n| 0       (neg d) := (int.zero_div _).symm\n| (pos n) 0       := (int.div_zero _).symm\n| (neg n) 0       := (int.div_zero _).symm\n| (pos n) (pos d) := (num.cast_to_znum _).trans $\n  by rw \u2190 num.to_nat_to_int; simp\n| (pos n) (neg d) := (num.cast_to_znum_neg _).trans $\n  by rw \u2190 num.to_nat_to_int; simp\n| (neg n) (pos d) := show - _ = (-_/\u2191d), begin\n    rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred,\n      \u2190 pos_num.to_nat_to_int, num.succ'_to_nat,\n      num.div_to_nat],\n    change -[1+ n.pred' / \u2191d] = -[1+ n.pred' / (d.pred' + 1)],\n    rw d.to_nat_eq_succ_pred\n  end\n| (neg n) (neg d) := show \u2191(pos_num.pred' n / num.pos d).succ' = (-_ / -\u2191d), begin\n    rw [n.to_int_eq_succ_pred, d.to_int_eq_succ_pred,\n      \u2190 pos_num.to_nat_to_int, num.succ'_to_nat,\n      num.div_to_nat],\n    change (nat.succ (_/d) : \u2124) = nat.succ (n.pred'/(d.pred' + 1)),\n    rw d.to_nat_eq_succ_pred\n  end\n\n@[simp, norm_cast] theorem mod_to_int : \u2200 n d, ((n % d : znum) : \u2124) = n % d\n| 0       d := (int.zero_mod _).symm\n| (pos n) d := (num.cast_to_znum _).trans $\n  by rw [\u2190 num.to_nat_to_int, cast_pos, num.mod_to_nat,\n    \u2190 pos_num.to_nat_to_int, abs_to_nat]; refl\n| (neg n) d := (num.cast_sub' _ _).trans $\n  by rw [\u2190 num.to_nat_to_int, cast_neg, \u2190 num.to_nat_to_int,\n    num.succ_to_nat, num.mod_to_nat, abs_to_nat,\n    \u2190 int.sub_nat_nat_eq_coe, n.to_int_eq_succ_pred]; refl\n\n@[simp] theorem gcd_to_nat (a b) : (gcd a b : \u2115) = int.gcd a b :=\n(num.gcd_to_nat _ _).trans $ by simpa\n\ntheorem dvd_iff_mod_eq_zero {m n : znum} : m \u2223 n \u2194 n % m = 0 :=\nby rw [\u2190 dvd_to_int, int.dvd_iff_mod_eq_zero,\n  \u2190 to_int_inj, mod_to_int]; refl\n\ninstance : decidable_rel ((\u2223) : znum \u2192 znum \u2192 Prop)\n| a b := decidable_of_iff' _ dvd_iff_mod_eq_zero\n\nend znum\n\nnamespace int\n\n/-- Cast a `snum` to the corresponding integer. -/\ndef of_snum : snum \u2192 \u2124 :=\nsnum.rec' (\u03bb a, cond a (-1) 0) (\u03bba p IH, cond a (bit1 IH) (bit0 IH))\n\ninstance snum_coe : has_coe snum \u2124 := \u27e8of_snum\u27e9\nend int\n\ninstance : has_lt snum := \u27e8\u03bba b, (a : \u2124) < b\u27e9\ninstance : has_le snum := \u27e8\u03bba b, (a : \u2124) \u2264 b\u27e9\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/data/num/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.3954397484855446}}
{"text": "import data.set\nimport tactic.suggest\nimport tactic.basic\nimport data.vector\nimport data.vector.basic\nimport data.vector.zip\n\nimport vector.additional\n\nimport category_theory.concrete_category.bundled\n\nopen category_theory\n\nnamespace UA\n  universes u_lang u_str u_strA u_strB\n\n  section\n\n\n    /- a signature `\u03c3` is just a set of operation symbols, with specified (finite) arity. -/\n\n    class signature := (F : Type u_lang) (arity_of : F \u2192 \u2115)\n    abbreviation arity_of [\u03c3 : signature] (f) := signature.arity_of f\n\n\n    /- The universes `u_strX` of the structures may be distinct, and different to `u_lang`.\n    -- We wrap the main defenitions in a section,\n    -- so that they become universe polymorphic within this file. -/\n\n    section\n\n      /- `action_on` is a realisation of operations on a given type. -/\n\n      @[class] def action_on [\u03c3 : signature.{u_lang}] (medium : Type u_str) : Type (max u_lang u_str) :=\n      \u03a0 f, (vector medium (arity_of f)) \u2192 medium\n\n      parameter [\u03c3 : signature.{u_lang}]\n      include \u03c3\n\n      /- A `structure` is a `medium` equipped with the relevant `action`. -/\n\n      def Structure : Type (max u_lang (u_str+1)) := bundled (action_on.{u_lang u_str})\n\n      abbreviation Structure.medium (self : Structure) := self.\u03b1\n      abbreviation Structure.action (self : Structure) := self.str\n\n\n      /- Coersions and instances -/\n\n      instance Structure_to_action_on (A : Structure) : action_on A.medium := A.action\n      instance Structure_to_sort : has_coe_to_sort Structure _ := \u27e8\u03bb str, str.medium\u27e9\n      instance Structure_to_fun : has_coe_to_fun Structure _ := \u27e8\u03bb str, str.action\u27e9\n\n    end\n\n\n\n    /- `action_on_` and `Structure_` are subtle varients, where we require the media have\n    -- universe levels at least `u_lang`. This ensures structures have the same levels as their\n    -- underlying sets, as is required by much of the category theory infastructure in mathlib -/\n\n    section\n      parameter [\u03c3 : signature.{u_lang}]\n      include \u03c3\n\n      abbreviation action_on_ : Type (max u_lang u_str) \u2192 Type (max u_lang u_str) :=\n      \u03bb \u03b1, action_on.{u_lang (max u_lang u_str)} \u03b1\n\n      abbreviation Structure_ : Type ((max u_lang u_str)+1) :=\n      Structure.{u_lang (max u_lang u_str)}\n\n    end\n\n    parameter [\u03c3 : signature.{u_lang}]\n    include \u03c3\n\n\n\n    /- `direct product` of two stuctures -/\n\n    instance dir_prod_action (\u03b1 : Type u_strA) (\u03b2 : Type u_strB)\n    [actA : action_on \u03b1] [actB : action_on \u03b2] : action_on (\u03b1 \u00d7 \u03b2) :=\n    \u03bb f input, (actA f (vector.map prod.fst input), actB f (vector.map prod.snd input))\n\n    def dir_prod (A : Structure) (B : Structure) : Structure :=\n    \u27e8 A.medium \u00d7 B.medium, dir_prod_action A.medium B.medium\u27e9\n\n\n\n    /- Simplification lemmas -/\n\n    open vector\n\n    variables {\u03b1 : Type u_strA} [actA : action_on \u03b1]\n    variables {\u03b2 : Type u_strB} [actB : action_on \u03b2]\n    include actA actB\n    variable {f : \u03c3.F}\n    variable {x : vector \u03b1 (arity_of f)}\n    variable {y : vector \u03b2 (arity_of f)}\n    variable {xy : vector (\u03b1 \u00d7 \u03b2) (arity_of f)}\n\n    @[simp] lemma action_of_product:\n    dir_prod_action \u03b1 \u03b2 f xy = (actA f (map prod.fst xy), actB f (map prod.snd xy)) :=\n    begin\n      rw prod.ext_iff,\n      simp,\n\n      split,\n      apply congr_arg (actA f),\n      apply vector.ext,\n      intro i,\n      simp,\n\n      apply congr_arg (actB f),\n      apply vector.ext,\n      intro i,\n      simp,\n    end\n\n    @[simp] lemma action_of_product_zip:\n    dir_prod_action \u03b1 \u03b2 f (zip x y) = (actA f x, actB f y) :=\n    begin\n      rw action_of_product,\n      simp,\n    end\n\n  end\nend UA\n", "meta": {"author": "gilesgshaw", "repo": "UA-Lean", "sha": "b2187168c11a13756d9c8196377fdb97069580b0", "save_path": "github-repos/lean/gilesgshaw-UA-Lean", "path": "github-repos/lean/gilesgshaw-UA-Lean/UA-Lean-b2187168c11a13756d9c8196377fdb97069580b0/src/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3953906556672249}}
{"text": "/-\nCopyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Seul Baek\n\n! This file was ported from Lean 3 source module tactic.omega.nat.main\n! leanprover-community/mathlib commit dc34b216eb1a1548161e35d328ea1ab798017033\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Omega.ProveUnsats\nimport Mathbin.Tactic.Omega.Nat.Dnf\nimport Mathbin.Tactic.Omega.Nat.NegElim\nimport Mathbin.Tactic.Omega.Nat.SubElim\n\n/-\nMain procedure for linear natural number arithmetic.\n-/\nopen Tactic\n\nnamespace Omega\n\nnamespace Nat\n\nopen Omega.Nat\n\nrun_cmd\n  mk_simp_attr `sugar_nat\n\nattribute [sugar_nat]\n  Ne not_le not_lt Nat.lt_iff_add_one_le Nat.succ_eq_add_one or_false_iff false_or_iff and_true_iff true_and_iff GE.ge GT.gt mul_add add_mul mul_comm one_mul mul_one imp_iff_not_or iff_iff_not_or_and_or_not\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\nunsafe def desugar :=\n  sorry\n#align omega.nat.desugar omega.nat.desugar\n\ntheorem univClose_of_unsat_negElim_not (m) (p : Preform) :\n    (negElim (\u00ac* p)).Unsat \u2192 UnivClose p (fun _ => 0) m :=\n  by\n  intro h1; apply univ_close_of_valid\n  apply valid_of_unsat_not; intro h2; apply h1\n  apply preform.sat_of_implies_of_sat implies_neg_elim h2\n#align omega.nat.univ_close_of_unsat_neg_elim_not Omega.Nat.univClose_of_unsat_negElim_not\n\n/-- Return expr of proof that argument is free of subtractions -/\nunsafe def preterm.prove_sub_free : Preterm \u2192 tactic expr\n  | &m => return q(trivial)\n  | m ** n => return q(trivial)\n  | t +* s => do\n    let x \u2190 preterm.prove_sub_free t\n    let y \u2190 preterm.prove_sub_free s\n    return q(@And.intro (Preterm.SubFree $(q(t))) (Preterm.SubFree $(q(s))) $(x) $(y))\n  | _ -* _ => failed\n#align omega.nat.preterm.prove_sub_free omega.nat.preterm.prove_sub_free\n\n/-- Return expr of proof that argument is free of negations -/\nunsafe def prove_neg_free : Preform \u2192 tactic expr\n  | t =* s => return q(trivial)\n  | t \u2264* s => return q(trivial)\n  | p \u2228* q => do\n    let x \u2190 prove_neg_free p\n    let y \u2190 prove_neg_free q\n    return q(@And.intro (Preform.NegFree $(q(p))) (Preform.NegFree $(q(q))) $(x) $(y))\n  | p \u2227* q => do\n    let x \u2190 prove_neg_free p\n    let y \u2190 prove_neg_free q\n    return q(@And.intro (Preform.NegFree $(q(p))) (Preform.NegFree $(q(q))) $(x) $(y))\n  | _ => failed\n#align omega.nat.prove_neg_free omega.nat.prove_neg_free\n\n/-- Return expr of proof that argument is free of subtractions -/\nunsafe def prove_sub_free : Preform \u2192 tactic expr\n  | t =* s => do\n    let x \u2190 preterm.prove_sub_free t\n    let y \u2190 preterm.prove_sub_free s\n    return q(@And.intro (Preterm.SubFree $(q(t))) (Preterm.SubFree $(q(s))) $(x) $(y))\n  | t \u2264* s => do\n    let x \u2190 preterm.prove_sub_free t\n    let y \u2190 preterm.prove_sub_free s\n    return q(@And.intro (Preterm.SubFree $(q(t))) (Preterm.SubFree $(q(s))) $(x) $(y))\n  | \u00ac* p => prove_sub_free p\n  | p \u2228* q => do\n    let x \u2190 prove_sub_free p\n    let y \u2190 prove_sub_free q\n    return q(@And.intro (Preform.SubFree $(q(p))) (Preform.SubFree $(q(q))) $(x) $(y))\n  | p \u2227* q => do\n    let x \u2190 prove_sub_free p\n    let y \u2190 prove_sub_free q\n    return q(@And.intro (Preform.SubFree $(q(p))) (Preform.SubFree $(q(q))) $(x) $(y))\n#align omega.nat.prove_sub_free omega.nat.prove_sub_free\n\n/-- Given a p : preform, return the expr of a term t : p.unsat, where p is subtraction- and\nnegation-free. -/\nunsafe def prove_unsat_sub_free (p : Preform) : tactic expr := do\n  let x \u2190 prove_neg_free p\n  let y \u2190 prove_sub_free p\n  let z \u2190 prove_unsats (dnf p)\n  return q(unsat_of_unsat_dnf $(q(p)) $(x) $(y) $(z))\n#align omega.nat.prove_unsat_sub_free omega.nat.prove_unsat_sub_free\n\n/-- Given a p : preform, return the expr of a term t : p.unsat, where p is negation-free. -/\nunsafe def prove_unsat_neg_free : Preform \u2192 tactic expr\n  | p =>\n    match p.subTerms with\n    | none => prove_unsat_sub_free p\n    | some (t, s) => do\n      let x \u2190 prove_unsat_neg_free (subElim t s p)\n      return q(unsat_of_unsat_subElim $(q(t)) $(q(s)) $(q(p)) $(x))\n#align omega.nat.prove_unsat_neg_free omega.nat.prove_unsat_neg_free\n\n/-- Given a (m : nat) and (p : preform), return the expr of (t : univ_close m p). -/\nunsafe def prove_univ_close (m : Nat) (p : Preform) : tactic expr := do\n  let x \u2190 prove_unsat_neg_free (negElim (\u00ac* p))\n  to_expr ``(univClose_of_unsat_negElim_not $(q(m)) $(q(p)) $(x))\n#align omega.nat.prove_univ_close omega.nat.prove_univ_close\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- Reification to imtermediate shadow syntax that retains exprs -/ unsafe\n  def\n    to_exprterm\n    : expr \u2192 tactic exprterm\n    | q( $ ( x ) * $ ( y ) ) => do let m \u2190 eval_expr' Nat y return ( exprterm.exp m x )\n      |\n        q( $ ( t1x ) + $ ( t2x ) )\n        =>\n        do let t1 \u2190 to_exprterm t1x let t2 \u2190 to_exprterm t2x return ( exprterm.add t1 t2 )\n      |\n        q( $ ( t1x ) - $ ( t2x ) )\n        =>\n        do let t1 \u2190 to_exprterm t1x let t2 \u2190 to_exprterm t2x return ( exprterm.sub t1 t2 )\n      |\n        x\n        =>\n        ( do let m \u2190 eval_expr' Nat x return ( exprterm.cst m ) ) <|> ( return <| exprterm.exp 1 x )\n#align omega.nat.to_exprterm omega.nat.to_exprterm\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- Reification to imtermediate shadow syntax that retains exprs -/ unsafe\n  def\n    to_exprform\n    : expr \u2192 tactic exprform\n    |\n        q( $ ( tx1 ) = $ ( tx2 ) )\n        =>\n        do let t1 \u2190 to_exprterm tx1 let t2 \u2190 to_exprterm tx2 return ( exprform.eq t1 t2 )\n      |\n        q( $ ( tx1 ) \u2264 $ ( tx2 ) )\n        =>\n        do let t1 \u2190 to_exprterm tx1 let t2 \u2190 to_exprterm tx2 return ( exprform.le t1 t2 )\n      | q( \u00ac $ ( px ) ) => do let p \u2190 to_exprform px return ( exprform.not p )\n      |\n        q( $ ( px ) \u2228 $ ( qx ) )\n        =>\n        do let p \u2190 to_exprform px let q \u2190 to_exprform qx return ( exprform.or p q )\n      |\n        q( $ ( px ) \u2227 $ ( qx ) )\n        =>\n        do let p \u2190 to_exprform px let q \u2190 to_exprform qx return ( exprform.and p q )\n      | q( _ \u2192 $ ( px ) ) => to_exprform px\n      | x => ( trace \"Cannot reify expr : \" >> trace x ) >> failed\n#align omega.nat.to_exprform omega.nat.to_exprform\n\n/-- List of all unreified exprs -/\nunsafe def exprterm.exprs : exprterm \u2192 List expr\n  | exprterm.cst _ => []\n  | exprterm.exp _ x => [x]\n  | exprterm.add t s => List.union t.exprs s.exprs\n  | exprterm.sub t s => List.union t.exprs s.exprs\n#align omega.nat.exprterm.exprs omega.nat.exprterm.exprs\n\n/-- List of all unreified exprs -/\nunsafe def exprform.exprs : exprform \u2192 List expr\n  | exprform.eq t s => List.union t.exprs s.exprs\n  | exprform.le t s => List.union t.exprs s.exprs\n  | exprform.not p => p.exprs\n  | exprform.or p q => List.union p.exprs q.exprs\n  | exprform.and p q => List.union p.exprs q.exprs\n#align omega.nat.exprform.exprs omega.nat.exprform.exprs\n\n/-- Reification to an intermediate shadow syntax which eliminates exprs,\n    but still includes non-canonical terms -/\nunsafe def exprterm.to_preterm (xs : List expr) : exprterm \u2192 tactic Preterm\n  | exprterm.cst k => return (&k)\n  | exprterm.exp k x =>\n    let m := xs.indexOf\u2093 x\n    if m < xs.length then return (k ** m) else failed\n  | exprterm.add xa xb => do\n    let a \u2190 xa.to_preterm\n    let b \u2190 xb.to_preterm\n    return (a +* b)\n  | exprterm.sub xa xb => do\n    let a \u2190 xa.to_preterm\n    let b \u2190 xb.to_preterm\n    return (a -* b)\n#align omega.nat.exprterm.to_preterm omega.nat.exprterm.to_preterm\n\n/-- Reification to an intermediate shadow syntax which eliminates exprs,\n    but still includes non-canonical terms -/\nunsafe def exprform.to_preform (xs : List expr) : exprform \u2192 tactic Preform\n  | exprform.eq xa xb => do\n    let a \u2190 xa.to_preterm xs\n    let b \u2190 xb.to_preterm xs\n    return (a =* b)\n  | exprform.le xa xb => do\n    let a \u2190 xa.to_preterm xs\n    let b \u2190 xb.to_preterm xs\n    return (a \u2264* b)\n  | exprform.not xp => do\n    let p \u2190 xp.to_preform\n    return (\u00ac* p)\n  | exprform.or xp xq => do\n    let p \u2190 xp.to_preform\n    let q \u2190 xq.to_preform\n    return (p \u2228* q)\n  | exprform.and xp xq => do\n    let p \u2190 xp.to_preform\n    let q \u2190 xq.to_preform\n    return (p \u2227* q)\n#align omega.nat.exprform.to_preform omega.nat.exprform.to_preform\n\n/-- Reification to an intermediate shadow syntax which eliminates exprs,\n    but still includes non-canonical terms. -/\nunsafe def to_preform (x : expr) : tactic (Preform \u00d7 Nat) := do\n  let xf \u2190 to_exprform x\n  let xs := xf.exprs\n  let f \u2190 xf.to_preform xs\n  return (f, xs)\n#align omega.nat.to_preform omega.nat.to_preform\n\n/-- Return expr of proof of current LNA goal -/\nunsafe def prove : tactic expr := do\n  let (p, m) \u2190 target >>= to_preform\n  trace_if_enabled `omega p\n  prove_univ_close m p\n#align omega.nat.prove omega.nat.prove\n\n/-- Succeed iff argument is expr of \u2115 -/\nunsafe def eq_nat (x : expr) : tactic Unit :=\n  if x = q(Nat) then skip else failed\n#align omega.nat.eq_nat omega.nat.eq_nat\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/-- Check whether argument is expr of a well-formed formula of LNA-/ unsafe\n  def\n    wff\n    : expr \u2192 tactic Unit\n    | q( \u00ac $ ( px ) ) => wff px\n      | q( $ ( px ) \u2228 $ ( qx ) ) => wff px >> wff qx\n      | q( $ ( px ) \u2227 $ ( qx ) ) => wff px >> wff qx\n      | q( $ ( px ) \u2194 $ ( qx ) ) => wff px >> wff qx\n      |\n        q( $ ( expr.pi _ _ px qx ) )\n        =>\n        Monad.cond\n          ( if expr.has_var px then return true else is_prop px )\n            ( wff px >> wff qx )\n            ( eq_nat px >> wff qx )\n      | q( @ LT.lt $ ( dx ) $ ( h ) _ _ ) => eq_nat dx\n      | q( @ LE.le $ ( dx ) $ ( h ) _ _ ) => eq_nat dx\n      | q( @ Eq $ ( dx ) _ _ ) => eq_nat dx\n      | q( @ GE.ge $ ( dx ) $ ( h ) _ _ ) => eq_nat dx\n      | q( @ GT.gt $ ( dx ) $ ( h ) _ _ ) => eq_nat dx\n      | q( @ Ne $ ( dx ) _ _ ) => eq_nat dx\n      | q( True ) => skip\n      | q( False ) => skip\n      | _ => failed\n#align omega.nat.wff omega.nat.wff\n\n/-- Succeed iff argument is expr of term whose type is wff -/\nunsafe def wfx (x : expr) : tactic Unit :=\n  infer_type x >>= wff\n#align omega.nat.wfx omega.nat.wfx\n\n/-- Intro all universal quantifiers over nat -/\nunsafe def intro_nats_core : tactic Unit := do\n  let x \u2190 target\n  match x with\n    | expr.pi _ _ q(Nat) _ => intro_fresh >> intro_nats_core\n    | _ => skip\n#align omega.nat.intro_nats_core omega.nat.intro_nats_core\n\nunsafe def intro_nats : tactic Unit := do\n  let expr.pi _ _ q(Nat) _ \u2190 target\n  intro_nats_core\n#align omega.nat.intro_nats omega.nat.intro_nats\n\n/-- If the goal has universal quantifiers over natural, introduce all of them.\nOtherwise, revert all hypotheses that are formulas of linear natural number arithmetic. -/\nunsafe def preprocess : tactic Unit :=\n  intro_nats <|> revert_cond_all wfx >> desugar\n#align omega.nat.preprocess omega.nat.preprocess\n\nend Nat\n\nend Omega\n\nopen Omega.Nat\n\n/-- The core omega tactic for natural numbers. -/\nunsafe def omega_nat (is_manual : Bool) : tactic Unit :=\n  andthen (andthen desugar (if is_manual then skip else preprocess)) ((prove >>= apply) >> skip)\n#align omega_nat omega_nat\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Omega/Nat/Main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3953906480259307}}
{"text": "/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.abelian.subobject\nimport category_theory.limits.essentially_small\nimport category_theory.preadditive.injective\nimport category_theory.preadditive.generator\nimport category_theory.preadditive.yoneda.limits\n\n/-!\n# A complete abelian category with enough injectives and a separator has an injective coseparator\n\n## Future work\n* Once we know that Grothendieck categories have enough injectives, we can use this to conclude\n  that Grothendieck categories have an injective coseparator.\n\n## References\n* [Peter J Freyd, *Abelian Categories* (Theorem 3.37)][freyd1964abelian]\n\n-/\n\nopen category_theory category_theory.limits opposite\n\nuniverses v u\n\nnamespace category_theory.abelian\nvariables {C : Type u} [category.{v} C] [abelian C]\n\ntheorem has_injective_coseparator [has_limits C] [enough_injectives C] (G : C)\n  (hG : is_separator G) : \u2203 G : C, injective G \u2227 is_coseparator G :=\nbegin\n  haveI : well_powered C := well_powered_of_is_detector G hG.is_detector,\n  haveI : has_products_of_shape (subobject (op G)) C := has_products_of_shape_of_small _ _,\n  let T : C := injective.under (pi_obj (\u03bb P : subobject (op G), unop P)),\n  refine \u27e8T, infer_instance, (preadditive.is_coseparator_iff _).2 (\u03bb X Y f hf, _)\u27e9,\n  refine (preadditive.is_separator_iff _).1 hG _ (\u03bb h, _),\n  suffices hh : factor_thru_image (h \u226b f) = 0,\n  { rw [\u2190 limits.image.fac (h \u226b f), hh, zero_comp] },\n  let R := subobject.mk (factor_thru_image (h \u226b f)).op,\n  let q\u2081 : image (h \u226b f) \u27f6 unop R :=\n    (subobject.underlying_iso (factor_thru_image (h \u226b f)).op).unop.hom,\n  let q\u2082 : unop (R : C\u1d52\u1d56) \u27f6 pi_obj (\u03bb P : subobject (op G), unop P) :=\n    section_ (pi.\u03c0 (\u03bb P : subobject (op G), unop P) R),\n  let q : image (h \u226b f) \u27f6 T := q\u2081 \u226b q\u2082 \u226b injective.\u03b9 _,\n  exact zero_of_comp_mono q (by rw [\u2190 injective.comp_factor_thru q (limits.image.\u03b9 (h \u226b f)),\n    limits.image.fac_assoc, category.assoc, hf, comp_zero])\nend\n\ntheorem has_projective_separator [has_colimits C] [enough_projectives C] (G : C)\n  (hG : is_coseparator G) : \u2203 G : C, projective G \u2227 is_separator G :=\nbegin\n  obtain \u27e8T, hT\u2081, hT\u2082\u27e9 := has_injective_coseparator (op G) ((is_separator_op_iff _).2 hG),\n  exactI \u27e8unop T, infer_instance, (is_separator_unop_iff _).2 hT\u2082\u27e9\nend\n\nend category_theory.abelian\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/abelian/generator.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3953906403846364}}
{"text": "import number_theory.bernoulli\n\n\n--OUTPUT 1\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  apply nat.strong_induction_on n,\n  assume n hn,\n  begin\n    rw [polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_C],\n    rw [polynomial.eval_add, polynomial.eval_add, polynomial.eval_C],\n    rw [polynomial.eval_add, polynomial.eval_C],\n    rw [polynomial.eval_add, polynomial.eval_C],\n\n    rw [polynomial.eval_sub, polynomial.eval_sub, polynomial.eval_C],\n    rw [polynomial.eval_sub, polynomial.eval_C],\n    rw [polynomial.eval_sub, polynomial.eval_C],\n    rw [polynomial.eval_sub, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n\n    rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_C],\n    rw [polynomial\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : \u2200 m : \u2115, m < n \u2192 (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1),\n    assume m : \u2115,\n    assume h2 : (m < n),\n    induction m with m hm,\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1), from by auto [polynomial.bernoulli, eval_C, eval_X],\n    show (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (m + 1) * x^((m + 1) - 1),\n    begin\n      have h3 : (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval (1 + x), from by auto [polynomial.bernoulli, eval_add, eval_mul, eval_X, eval_C],\n      have h4 : (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval x + (polynomial.bernoulli m).eval x, from by auto [hm, add_assoc],\n      have h5 : (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval x + (polynomial.bernoulli m).eval x = (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval x + m * x^(m - 1), from by auto [hm],\n      have h6 : (polynomial.bernoulli (m + 1)).eval x + (polynomial.bernoulli m).eval x + m * x^(m - 1) = (polynomial.bernoulli (m + 1)).eval x + m * x^(m - 1) + m * x^(m - 1), from by auto [add_assoc],\n      show (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (m + 1) * x^((m + 1) - 1), from by auto [h3, h4, h5, h6],\n    end,\n  have h7 : (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval (1 + x), from by auto [polynomial.bernoulli, eval_add, eval_mul, eval_X, eval_C],\n  have h8 : (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval (1 + x) = (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval x + (polynomial.bernoulli (n - 1)).eval x, from by auto [h1, nat.sub_lt],\n  have h9 : (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval x + (polynomial.bernoulli (n - 1)).eval x = (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval x + (n - 1) * x^((n - 1) - 1), from by auto [h1, nat.sub_lt],\n  have h10 : (polynomial.bernoulli n).eval x + (polynomial.bernoulli (n - 1)).eval x + (n - 1) * x^((n - 1) - 1) = (polynomial.bernoulli n).eval x + (n - 1) * x^((n - 1) - 1) + (n - 1) * x^((n - 1) - 1), from by auto [add_assoc],\n  show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from by auto [h7, h8, h9, h10],\nend\n\n--OUTPUT 3\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) := begin\n  have h1 : \u2200 l : \u2115, polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) (l + 1) = (l + 1) * X^l, from by auto using [polynomial.sum_eq_mul],\n  have h2 : \u2200 l : \u2115, polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) (l + 1) = polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) l + (polynomial.binomial (l + 1) (l + 1)) * (polynomial.bernoulli (l + 1)), from by auto using [polynomial.sum_add],\n  have h3 : \u2200 l : \u2115, polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) (l + 1) = polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) l + (polynomial.bernoulli (l + 1)), from by auto,\n  have h4 : \u2200 l : \u2115, polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) (l + 1) = polynomial.sum (\u03bb (k : \u2115), (polynomial.binomial (l + 1) k) * (polynomial.bernoulli k)) (l + 1) + (polynomial.bernoulli (l + 1)), from by auto,\n\n  induction n with d hd,\n  {\n    rw polynomial.bernoulli_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polynomial.eval_zero,\n    rw polyn\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  rw polynomial.bernoulli,\n  rw polynomial.eval_add,\n  rw polynomial.eval_mul,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_monomial,\n  rw polynomial.eval_mon\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), ((polynomial.binomial (n + 1) k) * (polynomial.X ^ k)) * (polynomial.bernoulli k).coeff 0), from by auto [polynomial.bernoulli_def],\n  have h2 : \u2200 (k : \u2115), (polynomial.binomial (n + 1) k) * (polynomial.X ^ k) = polynomial.C (k + 1) * polynomial.X ^ k, from by auto [polynomial.binomial_def],\n  have h3 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), ((polynomial.C (k + 1) * polynomial.X ^ k) * (polynomial.bernoulli k).coeff 0)), from by auto [h1, h2],\n  have h4 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * (polynomial.bernoulli k).coeff 0), from by auto [h3],\n  have h5 : polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * (polynomial.bernoulli k).coeff 0) = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [polynomial.sum_mul_C],\n  have h6 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h4, h5],\n  have h7 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h6],\n  have h8 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h7],\n  have h9 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h8],\n  have h10 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h9],\n  have h11 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h10],\n  have h12 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h11],\n  have h13 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h12],\n  have h14 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * polynomial.X ^ k * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h13],\n  have h15 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h14],\n  have h16 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h15],\n  have h17 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h16],\n  have h18 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h17],\n  have h19 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h18],\n  have h20 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h19],\n  have h21 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h20],\n  have h22 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h21],\n  have h23 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h22],\n  have h24 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h23],\n  have h25 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), (polynomial.C (k + 1) * polynomial.X ^ k) * polynomial.C (polynomial.bernoulli k).coeff 0), from by auto [h24],\n  have h26 : polynomial.bernoulli n = polynomial.sum (\u03bb (k : \u2115), polynomial.C (k + 1) * (polynomial.X ^ k) * polynomial.C (polynomial.bernoulli\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    unfold polynomial.bernoulli,\n    unfold polynomial.eval,\n    simp,\n    ring,\n  },\n  {\n    have h1 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (sum (\u03bb k, ((d + 1) : \u211a) * (polynomial.C (1 : \u211a) ^ k * polynomial.C (x : \u211a) ^ (d - k))) (range (0, d + 1))).eval x, by unfold polynomial.bernoulli,\n    have h2 : (sum (\u03bb k, ((d + 1) : \u211a) * (polynomial.C (1 : \u211a) ^ k * polynomial.C (x : \u211a) ^ (d - k))) (range (0, d + 1))).eval x = (sum (\u03bb k, ((d + 1) : \u211a) * (polynomial.C (1 : \u211a) ^ k * polynomial.C (x : \u211a) ^ (d - k))) (range (0, d + 1))).eval x + (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (d - k) * (x : \u211a) ^ (d - k - 1) * polynomial.C (1 : \u211a) ^ k) (range (0, d + 1))).eval x, by ring,\n    have h3 : (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (d - k) * (x : \u211a) ^ (d - k - 1) * polynomial.C (1 : \u211a) ^ k) (range (0, d + 1))).eval x = (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (d - k) * (x : \u211a) ^ (d - k - 1) * polynomial.C (1 : \u211a) ^ k) (range (0, d + 1))).eval x + (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x, by ring,\n    have h4 : (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x = (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x + (d + 1) * (x : \u211a) ^ (d - 1), by ring,\n    have h5 : (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x = (sum (\u03bb (k : \u2115), ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x + (sum (\u03bb k, ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x, by ring,\n    have h6 : (sum (\u03bb k, ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x = (sum (\u03bb k, ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x, by ring,\n    have h7 : (sum (\u03bb k, ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x = (sum (\u03bb k, ((d + 1) : \u211a) * (x : \u211a) ^ (d - k - 1)) (range (0, d + 1))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d))).eval x, by ring,\n    have h8 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d))).eval x + (d : \u211a) * (x : \u211a) ^ (d - 1), by ring,\n    have h9 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 1))).eval x, by ring,\n    have h10 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 1))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 1))).eval x + (d - 1) * (x : \u211a) ^ (d - 2), by ring,\n    have h11 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 1))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 1))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 2))).eval x, by ring,\n    have h12 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 2))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 2))).eval x + (d - 2) * (x : \u211a) ^ (d - 3), by ring,\n    have h13 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 2))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 2))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 3))).eval x, by ring,\n    have h14 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 3))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 3))).eval x + (d - 3) * (x : \u211a) ^ (d - 4), by ring,\n    have h15 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 3))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 3))).eval x + (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 4))).eval x, by ring,\n    have h16 : (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d - 4))).eval x = (sum (\u03bb k, (x : \u211a) ^ (d - k - 1)) (range (0, d\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h0 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C 1).pow (-k), from by auto [pow_add, sum_mul_distrib_left, pow_neg, pow_zero],\n  have h1 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto,\n  have h2 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h3 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h4 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n\n  have h5 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h6 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h7 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h8 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n\n  have h9 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h10 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n\n  have h11 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h12 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h13 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h14 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h15 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h16 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h17 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n\n  have h18 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h19 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).pow l = \u2211 k in finset.range (l + 1), (l + 1) * (polynomial.C 1).pow k * (polynomial.C (1 : \u211a)).pow (-k), from by auto [polynomial.pow_C, one_pow],\n  have h20 : \u2200 l : \u2115, (l + 1) * (polynomial.C 1).\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x ^ (0 - 1),\n    from by auto [polynomial.bernoulli, polynomial.eval_add, polynomial.eval_C, polynomial.eval_X, polynomial.eval_mul, polynomial.eval_one, polynomial.bernoulli_zero]\n  },\n  {\n    have h1 : (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x ^ (d - 1), from by auto [hd],\n    have h2 : ((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) = (polynomial.bernoulli d).eval x * (d + 1) + d * x ^ (d - 1) * (d + 1), from by auto [add_mul],\n    have h3 : ((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) = (polynomial.bernoulli d).eval x * (d + 1) + d * x ^ d * (d + 1), from by auto [mul_comm, pow_succ'],\n    have h4 : ((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) = (polynomial.bernoulli d).eval x * (d + 1) + (polynomial.bernoulli d).eval x * (d + 1), from by auto [h1, h2, h3],\n    have h5 : ((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) = (polynomial.bernoulli d).eval x * (d + 1) + (polynomial.bernoulli d).eval x * (d + 1), from by auto [mul_add],\n    have h6 : (((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1)) = 0, from by auto [h4, h5, sub_eq_zero],\n    have h7 : (((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1)) = (polynomial.bernoulli d).eval x * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1), from by auto [add_sub_cancel'],\n    have h8 : (((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1)) = ((polynomial.bernoulli d).eval x - (polynomial.bernoulli d).eval x) * (d + 1), from by auto [h7, sub_eq_zero],\n    have h9 : (((polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1)) * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1) - (polynomial.bernoulli d).eval x * (d + 1)) = 0, from by auto [h6, h7, h8, mul_zero],\n\n    have h10 : (polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1) = (polynomial.bernoulli d).eval x * (d + 1) / (d + 1), from by auto [h9, div_eq_zero, eq_zero_of_mul_self_eq_zero],\n    have h11 : (polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1) = (polynomial.bernoulli d).eval x, from by auto [h10, div_one],\n    have h12 : (polynomial.bernoulli d).eval (1 + x) + d * x ^ (d - 1) = (polynomial.bernoulli d).eval x + d * x ^ (d - 1), from by auto [h1, h11, eq.symm],\n    show (polynomial.bernoulli (d + 1)).eval (1 + x) = (polynomial.bernoulli (d + 1)).eval x + (d + 1) * x ^ (d + 1 - 1), from by auto [h12, polynomial.bernoulli]\n  }\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBernoulli polynomial evaluation\nGiven a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$\n`proof`\nWe apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\nMultiplying both sides by $d + 1$, and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n$$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\nThe conclusion then follows easily.\n\nQED\n\n-/\ntheorem  bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Bernoulli polynomial evaluation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8031738057795403, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3953126181264083}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : \u2115) (x : \u2115) (h : nat.coprime x p) : (x : \u2124_[p]) \u2208 units \u2124_[p] :=\nbegin\n  -- prove the contrapositive\n  have h1 : \u00ac (x : \u2124_[p]) \u2208 units \u2124_[p] \u2192 \u00ac nat.coprime x p, from by {\n    assume h2 : \u00ac (x : \u2124_[p]) \u2208 units \u2124_[p],\n    assume h3 : nat.coprime x p,\n    have h4 : (x : \u2124_[p]) \u2208 units \u2124_[p], from by {\n      have h5 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h6 : (x : \u2124_[p]) = 0,\n        have h7 : x = 0, from by {\n          rw \u2190 h6,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h8 : x \u2223 p, from by {\n          rw h7,\n          apply nat.dvd_zero,\n        },\n        have h9 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h8 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h9 at h3,\n          exact h3,\n        },\n      },\n      have h5 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h6 : (x : \u2124_[p]) = 0,\n        have h7 : x = 0, from by {\n          rw \u2190 h6,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h8 : x \u2223 p, from by {\n          rw h7,\n          apply nat.dvd_zero,\n        },\n        have h9 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h8 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h9 at h3,\n          exact h3,\n        },\n      },\n      have h6 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h7 : (x : \u2124_[p]) = 0,\n        have h8 : x = 0, from by {\n          rw \u2190 h7,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h9 : x \u2223 p, from by {\n          rw h8,\n          apply nat.dvd_zero,\n        },\n        have h10 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h9 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h10 at h3,\n          exact h3,\n        },\n      },\n      have h7 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h8 : (x : \u2124_[p]) = 0,\n        have h9 : x = 0, from by {\n          rw \u2190 h8,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h10 : x \u2223 p, from by {\n          rw h9,\n          apply nat.dvd_zero,\n        },\n        have h11 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h10 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h11 at h3,\n          exact h3,\n        },\n      },\n      have h8 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h9 : (x : \u2124_[p]) = 0,\n        have h10 : x = 0, from by {\n          rw \u2190 h9,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h11 : x \u2223 p, from by {\n          rw h10,\n          apply nat.dvd_zero,\n        },\n        have h12 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h11 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h12 at h3,\n          exact h3,\n        },\n      },\n      have h9 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h10 : (x : \u2124_[p]) = 0,\n        have h11 : x = 0, from by {\n          rw \u2190 h10,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h12 : x \u2223 p, from by {\n          rw h11,\n          apply nat.dvd_zero,\n        },\n        have h13 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h12 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h13 at h3,\n          exact h3,\n        },\n      },\n      have h10 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h11 : (x : \u2124_[p]) = 0,\n        have h12 : x = 0, from by {\n          rw \u2190 h11,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h13 : x \u2223 p, from by {\n          rw h12,\n          apply nat.dvd_zero,\n        },\n        have h14 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h13 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h14 at h3,\n          exact h3,\n        },\n      },\n      have h11 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h12 : (x : \u2124_[p]) = 0,\n        have h13 : x = 0, from by {\n          rw \u2190 h12,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h14 : x \u2223 p, from by {\n          rw h13,\n          apply nat.dvd_zero,\n        },\n        have h15 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h14 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h15 at h3,\n          exact h3,\n        },\n      },\n      have h12 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h13 : (x : \u2124_[p]) = 0,\n        have h14 : x = 0, from by {\n          rw \u2190 h13,\n          rw \u2190 int.coe_nat_zero,\n        },\n        have h15 : x \u2223 p, from by {\n          rw h14,\n          apply nat.dvd_zero,\n        },\n        have h16 : \u00ac nat.coprime x p, from by {\n          apply nat.not_coprime_of_dvd_of_dvd h15 (nat.dvd_refl p),\n        },\n        show false, from by {\n          rw h16 at h3,\n          exact h3,\n        },\n      },\n      have h13 : (x : \u2124_[p]) \u2260 0, from by {\n        assume h14 : (x : \u2124\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/lean_proof_with_comments-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7905303186696747, "lm_q2_score": 0.5, "lm_q1q2_score": 0.39526515933483736}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.simp_result\nimport Mathlib.tactic.clear\nimport Mathlib.control.equiv_functor.instances\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# The `equiv_rw` tactic transports goals or hypotheses along equivalences.\n\nThe basic syntax is `equiv_rw e`, where `e : \u03b1 \u2243 \u03b2` is an equivalence.\nThis will try to replace occurrences of `\u03b1` in the goal with `\u03b2`, for example\ntransforming\n* `\u22a2 \u03b1` to `\u22a2 \u03b2`,\n* `\u22a2 option \u03b1` to `\u22a2 option \u03b2`\n* `\u22a2 {a // P}` to `{b // P (\u21d1(equiv.symm e) b)}`\n\nThe tactic can also be used to rewrite hypotheses, using the syntax `equiv_rw e at h`.\n\n## Implementation details\n\nThe main internal function is `equiv_rw_type e t`,\nwhich attempts to turn an expression `e : \u03b1 \u2243 \u03b2` into a new equivalence with left hand side `t`.\nAs an example, with `t = option \u03b1`, it will generate `functor.map_equiv option e`.\n\nThis is achieved by generating a new synthetic goal `%%t \u2243 _`,\nand calling `solve_by_elim` with an appropriate set of congruence lemmas.\nTo avoid having to specify the relevant congruence lemmas by hand,\nwe mostly rely on `equiv_functor.map_equiv` and `bifunctor.map_equiv`\nalong with some structural congruence lemmas such as\n* `equiv.arrow_congr'`,\n* `equiv.subtype_equiv_of_subtype'`,\n* `equiv.sigma_congr_left'`, and\n* `equiv.Pi_congr_left'`.\n\nThe main `equiv_rw` function, when operating on the goal, simply generates a new equivalence `e'`\nwith left hand side matching the target, and calls `apply e'.inv_fun`.\n\nWhen operating on a hypothesis `x : \u03b1`, we introduce a new fact `h : x = e.symm (e x)`,\nrevert this, and then attempt to `generalize`, replacing all occurrences of `e x` with a new constant `y`,\nbefore `intro`ing and `subst`ing `h`, and renaming `y` back to `x`.\n\n## Future improvements\nIn a future PR I anticipate that `derive equiv_functor` should work on many examples,\n(internally using `transport`, which is in turn based on `equiv_rw`)\nand we can incrementally bootstrap the strength of `equiv_rw`.\n\nAn ambitious project might be to add `equiv_rw!`,\na tactic which, when failing to find appropriate `equiv_functor` instances,\nattempts to `derive` them on the spot.\n\nFor now `equiv_rw` is entirely based on `equiv`,\nbut the framework can readily be generalised to also work with other types of equivalences,\nfor example specific notations such as ring equivalence (`\u2243+*`),\nor general categorical isomorphisms (`\u2245`).\n\nThis will allow us to transport across more general types of equivalences,\nbut this will wait for another subsequent PR.\n-/\n\nnamespace tactic\n\n\n/-- A list of lemmas used for constructing congruence equivalences. -/\n-- Although this looks 'hard-coded', in fact the lemma `equiv_functor.map_equiv`\n\n-- allows us to extend `equiv_rw` simply by constructing new instance so `equiv_functor`.\n\n-- TODO: We should also use `category_theory.functorial` and `category_theory.hygienic` instances.\n\n-- (example goal: we could rewrite along an isomorphism of rings (either as `R \u2245 S` or `R \u2243+* S`)\n\n-- and turn an `x : mv_polynomial \u03c3 R` into an `x : mv_polynomial \u03c3 S`.).\n\n/--\nConfiguration structure for `equiv_rw`.\n\n* `max_depth` bounds the search depth for equivalences to rewrite along.\n  The default value is 10.\n  (e.g., if you're rewriting along `e : \u03b1 \u2243 \u03b2`, and `max_depth := 2`,\n  you can rewrite `option (option \u03b1))` but not `option (option (option \u03b1))`.\n-/\n/--\nImplementation of `equiv_rw_type`, using `solve_by_elim`.\nExpects a goal of the form `t \u2243 _`,\nand tries to solve it using `eq : \u03b1 \u2243 \u03b2` and congruence lemmas.\n-/\n/--\n`equiv_rw_type e t` rewrites the type `t` using the equivalence `e : \u03b1 \u2243 \u03b2`,\nreturning a new equivalence `t \u2243 t'`.\n-/\n/--\nAttempt to replace the hypothesis with name `x`\nby transporting it along the equivalence in `e : \u03b1 \u2243 \u03b2`.\n-/\n-- We call `dsimp_result` to perform the beta redex introduced by `revert`\n\n/-- Rewrite the goal using an equiv `e`. -/\nend tactic\n\n\nnamespace tactic.interactive\n\n\n/--\n`equiv_rw e at h`, where `h : \u03b1` is a hypothesis, and `e : \u03b1 \u2243 \u03b2`,\nwill attempt to transport `h` along `e`, producing a new hypothesis `h : \u03b2`,\nwith all occurrences of `h` in other hypotheses and the goal replaced with `e.symm h`.\n\n`equiv_rw e` will attempt to transport the goal along an equivalence `e : \u03b1 \u2243 \u03b2`.\nIn its minimal form it replaces the goal `\u22a2 \u03b1` with `\u22a2 \u03b2` by calling `apply e.inv_fun`.\n\n`equiv_rw` will also try rewriting under (equiv_)functors, so can turn\na hypothesis `h : list \u03b1` into `h : list \u03b2` or\na goal `\u22a2 unique \u03b1` into `\u22a2 unique \u03b2`.\n\nThe maximum search depth for rewriting in subexpressions is controlled by\n`equiv_rw e {max_depth := n}`.\n-/\n/--\nSolve a goal of the form `t \u2243 _`,\nby constructing an equivalence from `e : \u03b1 \u2243 \u03b2`.\nThis is the same equivalence that `equiv_rw` would use to rewrite a term of type `t`.\n\nA typical usage might be:\n```\nhave e' : option \u03b1 \u2243 option \u03b2 := by equiv_rw_type e\n```\n-/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/equiv_rw.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3951745274160366}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n\n! This file was ported from Lean 3 source module logic.nonempty\n! leanprover-community/mathlib commit d2d8742b0c21426362a9dacebc6005db895ca963\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Basic\n\n/-!\n# Nonempty types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file proves a few extra facts about `nonempty`, which is defined in core Lean.\n\n## Main declarations\n\n* `nonempty.some`: Extracts a witness of nonemptiness using choice. Takes `nonempty \u03b1` explicitly.\n* `classical.arbitrary`: Extracts a witness of nonemptiness using choice. Takes `nonempty \u03b1` as an\n  instance.\n-/\n\n\nvariable {\u03b1 \u03b2 : Type _} {\u03b3 : \u03b1 \u2192 Type _}\n\nattribute [simp] instNonempty\n\n#print Zero.nonempty /-\ninstance (priority := 20) Zero.nonempty [Zero \u03b1] : Nonempty \u03b1 :=\n  \u27e80\u27e9\n#align has_zero.nonempty Zero.nonempty\n-/\n\n#print One.nonempty /-\ninstance (priority := 20) One.nonempty [One \u03b1] : Nonempty \u03b1 :=\n  \u27e81\u27e9\n#align has_one.nonempty One.nonempty\n-/\n\n#print exists_true_iff_nonempty /-\ntheorem exists_true_iff_nonempty {\u03b1 : Sort _} : (\u2203 a : \u03b1, True) \u2194 Nonempty \u03b1 :=\n  Iff.intro (fun \u27e8a, _\u27e9 => \u27e8a\u27e9) fun \u27e8a\u27e9 => \u27e8a, trivial\u27e9\n#align exists_true_iff_nonempty exists_true_iff_nonempty\n-/\n\n#print nonempty_Prop /-\n@[simp]\ntheorem nonempty_Prop {p : Prop} : Nonempty p \u2194 p :=\n  Iff.intro (fun \u27e8h\u27e9 => h) fun h => \u27e8h\u27e9\n#align nonempty_Prop nonempty_Prop\n-/\n\n#print not_nonempty_iff_imp_false /-\ntheorem not_nonempty_iff_imp_false {\u03b1 : Sort _} : \u00acNonempty \u03b1 \u2194 \u03b1 \u2192 False :=\n  \u27e8fun h a => h \u27e8a\u27e9, fun h \u27e8a\u27e9 => h a\u27e9\n#align not_nonempty_iff_imp_false not_nonempty_iff_imp_false\n-/\n\n#print nonempty_sigma /-\n@[simp]\ntheorem nonempty_sigma : Nonempty (\u03a3a : \u03b1, \u03b3 a) \u2194 \u2203 a : \u03b1, Nonempty (\u03b3 a) :=\n  Iff.intro (fun \u27e8\u27e8a, c\u27e9\u27e9 => \u27e8a, \u27e8c\u27e9\u27e9) fun \u27e8a, \u27e8c\u27e9\u27e9 => \u27e8\u27e8a, c\u27e9\u27e9\n#align nonempty_sigma nonempty_sigma\n-/\n\n/- warning: nonempty_psigma -> nonempty_psigma is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : \u03b1 -> Sort.{u2}}, Iff (Nonempty.{max 1 u1 u2} (PSigma.{u1, u2} \u03b1 \u03b2)) (Exists.{u1} \u03b1 (fun (a : \u03b1) => Nonempty.{u2} (\u03b2 a)))\nbut is expected to have type\n  forall {\u03b1 : Sort.{u2}} {\u03b2 : \u03b1 -> Sort.{u1}}, Iff (Nonempty.{max (max 1 u1) u2} (PSigma.{u2, u1} \u03b1 \u03b2)) (Exists.{u2} \u03b1 (fun (a : \u03b1) => Nonempty.{u1} (\u03b2 a)))\nCase conversion may be inaccurate. Consider using '#align nonempty_psigma nonempty_psigma\u2093'. -/\n@[simp]\ntheorem nonempty_psigma {\u03b1} {\u03b2 : \u03b1 \u2192 Sort _} : Nonempty (PSigma \u03b2) \u2194 \u2203 a : \u03b1, Nonempty (\u03b2 a) :=\n  Iff.intro (fun \u27e8\u27e8a, c\u27e9\u27e9 => \u27e8a, \u27e8c\u27e9\u27e9) fun \u27e8a, \u27e8c\u27e9\u27e9 => \u27e8\u27e8a, c\u27e9\u27e9\n#align nonempty_psigma nonempty_psigma\n\n#print nonempty_subtype /-\n@[simp]\ntheorem nonempty_subtype {\u03b1} {p : \u03b1 \u2192 Prop} : Nonempty (Subtype p) \u2194 \u2203 a : \u03b1, p a :=\n  Iff.intro (fun \u27e8\u27e8a, h\u27e9\u27e9 => \u27e8a, h\u27e9) fun \u27e8a, h\u27e9 => \u27e8\u27e8a, h\u27e9\u27e9\n#align nonempty_subtype nonempty_subtype\n-/\n\n#print nonempty_prod /-\n@[simp]\ntheorem nonempty_prod : Nonempty (\u03b1 \u00d7 \u03b2) \u2194 Nonempty \u03b1 \u2227 Nonempty \u03b2 :=\n  Iff.intro (fun \u27e8\u27e8a, b\u27e9\u27e9 => \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) fun \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9 => \u27e8\u27e8a, b\u27e9\u27e9\n#align nonempty_prod nonempty_prod\n-/\n\n/- warning: nonempty_pprod -> nonempty_pprod is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, Iff (Nonempty.{max 1 u1 u2} (PProd.{u1, u2} \u03b1 \u03b2)) (And (Nonempty.{u1} \u03b1) (Nonempty.{u2} \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Sort.{u2}} {\u03b2 : Sort.{u1}}, Iff (Nonempty.{max (max 1 u1) u2} (PProd.{u2, u1} \u03b1 \u03b2)) (And (Nonempty.{u2} \u03b1) (Nonempty.{u1} \u03b2))\nCase conversion may be inaccurate. Consider using '#align nonempty_pprod nonempty_pprod\u2093'. -/\n@[simp]\ntheorem nonempty_pprod {\u03b1 \u03b2} : Nonempty (PProd \u03b1 \u03b2) \u2194 Nonempty \u03b1 \u2227 Nonempty \u03b2 :=\n  Iff.intro (fun \u27e8\u27e8a, b\u27e9\u27e9 => \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) fun \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9 => \u27e8\u27e8a, b\u27e9\u27e9\n#align nonempty_pprod nonempty_pprod\n\n#print nonempty_sum /-\n@[simp]\ntheorem nonempty_sum : Nonempty (Sum \u03b1 \u03b2) \u2194 Nonempty \u03b1 \u2228 Nonempty \u03b2 :=\n  Iff.intro\n    (fun \u27e8h\u27e9 =>\n      match h with\n      | Sum.inl a => Or.inl \u27e8a\u27e9\n      | Sum.inr b => Or.inr \u27e8b\u27e9)\n    fun h =>\n    match h with\n    | Or.inl \u27e8a\u27e9 => \u27e8Sum.inl a\u27e9\n    | Or.inr \u27e8b\u27e9 => \u27e8Sum.inr b\u27e9\n#align nonempty_sum nonempty_sum\n-/\n\n/- warning: nonempty_psum -> nonempty_psum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, Iff (Nonempty.{max 1 u1 u2} (PSum.{u1, u2} \u03b1 \u03b2)) (Or (Nonempty.{u1} \u03b1) (Nonempty.{u2} \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Sort.{u2}} {\u03b2 : Sort.{u1}}, Iff (Nonempty.{max (max 1 u1) u2} (PSum.{u2, u1} \u03b1 \u03b2)) (Or (Nonempty.{u2} \u03b1) (Nonempty.{u1} \u03b2))\nCase conversion may be inaccurate. Consider using '#align nonempty_psum nonempty_psum\u2093'. -/\n@[simp]\ntheorem nonempty_psum {\u03b1 \u03b2} : Nonempty (PSum \u03b1 \u03b2) \u2194 Nonempty \u03b1 \u2228 Nonempty \u03b2 :=\n  Iff.intro\n    (fun \u27e8h\u27e9 =>\n      match h with\n      | PSum.inl a => Or.inl \u27e8a\u27e9\n      | PSum.inr b => Or.inr \u27e8b\u27e9)\n    fun h =>\n    match h with\n    | Or.inl \u27e8a\u27e9 => \u27e8PSum.inl a\u27e9\n    | Or.inr \u27e8b\u27e9 => \u27e8PSum.inr b\u27e9\n#align nonempty_psum nonempty_psum\n\n@[simp]\ntheorem nonempty_empty : \u00acNonempty Empty := fun \u27e8h\u27e9 => h.elim\n#align nonempty_empty nonempty_empty\n\n/- warning: nonempty_ulift -> nonempty_ulift is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}}, Iff (Nonempty.{succ (max u1 u2)} (ULift.{u2, u1} \u03b1)) (Nonempty.{succ u1} \u03b1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}}, Iff (Nonempty.{max (succ u2) (succ u1)} (ULift.{u1, u2} \u03b1)) (Nonempty.{succ u2} \u03b1)\nCase conversion may be inaccurate. Consider using '#align nonempty_ulift nonempty_ulift\u2093'. -/\n@[simp]\ntheorem nonempty_ulift : Nonempty (ULift \u03b1) \u2194 Nonempty \u03b1 :=\n  Iff.intro (fun \u27e8\u27e8a\u27e9\u27e9 => \u27e8a\u27e9) fun \u27e8a\u27e9 => \u27e8\u27e8a\u27e9\u27e9\n#align nonempty_ulift nonempty_ulift\n\n#print nonempty_plift /-\n@[simp]\ntheorem nonempty_plift {\u03b1} : Nonempty (PLift \u03b1) \u2194 Nonempty \u03b1 :=\n  Iff.intro (fun \u27e8\u27e8a\u27e9\u27e9 => \u27e8a\u27e9) fun \u27e8a\u27e9 => \u27e8\u27e8a\u27e9\u27e9\n#align nonempty_plift nonempty_plift\n-/\n\n#print Nonempty.forall /-\n@[simp]\ntheorem Nonempty.forall {\u03b1} {p : Nonempty \u03b1 \u2192 Prop} : (\u2200 h : Nonempty \u03b1, p h) \u2194 \u2200 a, p \u27e8a\u27e9 :=\n  Iff.intro (fun h a => h _) fun h \u27e8a\u27e9 => h _\n#align nonempty.forall Nonempty.forall\n-/\n\n#print Nonempty.exists /-\n@[simp]\ntheorem Nonempty.exists {\u03b1} {p : Nonempty \u03b1 \u2192 Prop} : (\u2203 h : Nonempty \u03b1, p h) \u2194 \u2203 a, p \u27e8a\u27e9 :=\n  Iff.intro (fun \u27e8\u27e8a\u27e9, h\u27e9 => \u27e8a, h\u27e9) fun \u27e8a, h\u27e9 => \u27e8\u27e8a\u27e9, h\u27e9\n#align nonempty.exists Nonempty.exists\n-/\n\n#print Classical.inhabited_of_nonempty' /-\n/-- Using `classical.choice`, lifts a (`Prop`-valued) `nonempty` instance to a (`Type`-valued)\n  `inhabited` instance. `classical.inhabited_of_nonempty` already exists, in\n  `core/init/classical.lean`, but the assumption is not a type class argument,\n  which makes it unsuitable for some applications. -/\nnoncomputable def Classical.inhabited_of_nonempty' {\u03b1} [h : Nonempty \u03b1] : Inhabited \u03b1 :=\n  \u27e8Classical.choice h\u27e9\n#align classical.inhabited_of_nonempty' Classical.inhabited_of_nonempty'\n-/\n\n#print Nonempty.some /-\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible]\nprotected noncomputable def Nonempty.some {\u03b1} (h : Nonempty \u03b1) : \u03b1 :=\n  Classical.choice h\n#align nonempty.some Nonempty.some\n-/\n\n#print Classical.arbitrary /-\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible]\nprotected noncomputable def Classical.arbitrary (\u03b1) [h : Nonempty \u03b1] : \u03b1 :=\n  Classical.choice h\n#align classical.arbitrary Classical.arbitrary\n-/\n\n/- warning: nonempty.map -> Nonempty.map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, (\u03b1 -> \u03b2) -> (Nonempty.{u1} \u03b1) -> (Nonempty.{u2} \u03b2)\nbut is expected to have type\n  forall {\u03b1 : Sort.{u2}} {\u03b2 : Sort.{u1}}, (\u03b1 -> \u03b2) -> (Nonempty.{u2} \u03b1) -> (Nonempty.{u1} \u03b2)\nCase conversion may be inaccurate. Consider using '#align nonempty.map Nonempty.map\u2093'. -/\n/-- Given `f : \u03b1 \u2192 \u03b2`, if `\u03b1` is nonempty then `\u03b2` is also nonempty.\n  `nonempty` cannot be a `functor`, because `functor` is restricted to `Type`. -/\ntheorem Nonempty.map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) : Nonempty \u03b1 \u2192 Nonempty \u03b2\n  | \u27e8h\u27e9 => \u27e8f h\u27e9\n#align nonempty.map Nonempty.map\n\n/- warning: nonempty.map2 -> Nonempty.map2 is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}} {\u03b3 : Sort.{u3}}, (\u03b1 -> \u03b2 -> \u03b3) -> (Nonempty.{u1} \u03b1) -> (Nonempty.{u2} \u03b2) -> (Nonempty.{u3} \u03b3)\nbut is expected to have type\n  forall {\u03b1 : Sort.{u3}} {\u03b2 : Sort.{u2}} {\u03b3 : Sort.{u1}}, (\u03b1 -> \u03b2 -> \u03b3) -> (Nonempty.{u3} \u03b1) -> (Nonempty.{u2} \u03b2) -> (Nonempty.{u1} \u03b3)\nCase conversion may be inaccurate. Consider using '#align nonempty.map2 Nonempty.map2\u2093'. -/\nprotected theorem Nonempty.map2 {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) :\n    Nonempty \u03b1 \u2192 Nonempty \u03b2 \u2192 Nonempty \u03b3\n  | \u27e8x\u27e9, \u27e8y\u27e9 => \u27e8f x y\u27e9\n#align nonempty.map2 Nonempty.map2\n\n/- warning: nonempty.congr -> Nonempty.congr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, (\u03b1 -> \u03b2) -> (\u03b2 -> \u03b1) -> (Iff (Nonempty.{u1} \u03b1) (Nonempty.{u2} \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Sort.{u2}} {\u03b2 : Sort.{u1}}, (\u03b1 -> \u03b2) -> (\u03b2 -> \u03b1) -> (Iff (Nonempty.{u2} \u03b1) (Nonempty.{u1} \u03b2))\nCase conversion may be inaccurate. Consider using '#align nonempty.congr Nonempty.congr\u2093'. -/\nprotected theorem Nonempty.congr {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) : Nonempty \u03b1 \u2194 Nonempty \u03b2 :=\n  \u27e8Nonempty.map f, Nonempty.map g\u27e9\n#align nonempty.congr Nonempty.congr\n\n#print Nonempty.elim_to_inhabited /-\ntheorem Nonempty.elim_to_inhabited {\u03b1 : Sort _} [h : Nonempty \u03b1] {p : Prop} (f : Inhabited \u03b1 \u2192 p) :\n    p :=\n  h.elim <| f \u2218 Inhabited.mk\n#align nonempty.elim_to_inhabited Nonempty.elim_to_inhabited\n-/\n\ninstance {\u03b1 \u03b2} [h : Nonempty \u03b1] [h2 : Nonempty \u03b2] : Nonempty (\u03b1 \u00d7 \u03b2) :=\n  h.elim fun g => h2.elim fun g2 => \u27e8\u27e8g, g2\u27e9\u27e9\n\ninstance {\u03b9 : Sort _} {\u03b1 : \u03b9 \u2192 Sort _} [\u2200 i, Nonempty (\u03b1 i)] : Nonempty (\u2200 i, \u03b1 i) :=\n  \u27e8fun _ => Classical.arbitrary _\u27e9\n\n/- warning: classical.nonempty_pi -> Classical.nonempty_pi is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Sort.{u1}} {\u03b1 : \u03b9 -> Sort.{u2}}, Iff (Nonempty.{imax u1 u2} (forall (i : \u03b9), \u03b1 i)) (forall (i : \u03b9), Nonempty.{u2} (\u03b1 i))\nbut is expected to have type\n  forall {\u03b9 : Sort.{u2}} {\u03b1 : \u03b9 -> Sort.{u1}}, Iff (Nonempty.{imax u2 u1} (forall (i : \u03b9), \u03b1 i)) (forall (i : \u03b9), Nonempty.{u1} (\u03b1 i))\nCase conversion may be inaccurate. Consider using '#align classical.nonempty_pi Classical.nonempty_pi\u2093'. -/\ntheorem Classical.nonempty_pi {\u03b9} {\u03b1 : \u03b9 \u2192 Sort _} : Nonempty (\u2200 i, \u03b1 i) \u2194 \u2200 i, Nonempty (\u03b1 i) :=\n  \u27e8fun \u27e8f\u27e9 a => \u27e8f a\u27e9, @Pi.nonempty _ _\u27e9\n#align classical.nonempty_pi Classical.nonempty_pi\n\n#print subsingleton_of_not_nonempty /-\ntheorem subsingleton_of_not_nonempty {\u03b1 : Sort _} (h : \u00acNonempty \u03b1) : Subsingleton \u03b1 :=\n  \u27e8fun x => False.elim <| not_nonempty_iff_imp_false.mp h x\u27e9\n#align subsingleton_of_not_nonempty subsingleton_of_not_nonempty\n-/\n\n#print Function.Surjective.nonempty /-\ntheorem Function.Surjective.nonempty {\u03b1 \u03b2 : Sort _} [h : Nonempty \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : Function.Surjective f) : Nonempty \u03b1 :=\n  let \u27e8y\u27e9 := h\n  let \u27e8x, hx\u27e9 := hf y\n  \u27e8x\u27e9\n#align function.surjective.nonempty Function.Surjective.nonempty\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Logic/Nonempty.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6584175072643415, "lm_q1q2_score": 0.3951745233931792}}
{"text": "import tactic.interactive\nimport init.meta.interaction_monad\nimport data.real.basic\n\nopen tactic\n\n-- Find the maximum element of a list of \u2115. Used to work out at what point\n-- we don't need to traverse the targets list any further.\ndef find_max : list \u2115 \u2192 \u2115 \u2192 \u2115\n| [] so_far := so_far\n| (h :: tl) so_far := if h > so_far then find_max tl h else find_max tl so_far\n\n-- Given a list of \u2115 (the 'targets'), return the goals corresponding\n-- to these indices. Indexed from 1 (not 0).\nmeta def find_wanted_goals : list \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 list expr \u2192 tactic (list expr)\n| tgts crnt max gls := if crnt > max then return [] \n                                     else match gls with\n                                     | [] := fail \"No such goals!\"\n                                     | (g::gs) := if crnt \u2208 tgts then \n                                                  do out \u2190 find_wanted_goals tgts (crnt+1) max gs,\n                                                     return ([g]++out)\n                                                     else do out \u2190 find_wanted_goals tgts (crnt+1) max gs,\n                                                             return out\n                                     end\n\n-- A wrapper for find_wanted_goals. Only needs to be given the targets.\nmeta def find_goals : list \u2115 \u2192 tactic unit                                                 \n| tgts := do let max := find_max tgts 0,\n             gls \u2190 get_goals,\n             found_goals \u2190 find_wanted_goals tgts 1 max gls,\n             set_goals (found_goals)\n\nmeta def set_tactic_state (new_state : tactic_state) : tactic unit := \u03bb s, (do skip new_state)\n\nmeta def get_tactic_state : tactic tactic_state := \u03bb s, interaction_monad.result.success s s\n\n-- focus_goals as described in the assignment specifications\n-- The tactic can be invoked using the syntax requested\n-- After focus_goals has been run, we restore the other goals. However,\n-- Some of these might have been solved as a consequence of what was solved inside\n-- the given goal block. For example, proving commutativity of addition will also\n-- tell Lean the definition of addition we are using. This is unavoidable.\nmeta def tactic.interactive.focus_goals (pe : interactive.parse lean.parser.pexpr) (t : tactic.interactive.itactic) : tactic unit :=\ndo s \u2190 get_tactic_state,\n   s' \u2190 get_goals,\n   e \u2190 to_expr pe,\n   tgts \u2190 eval_expr (list \u2115) e,\n   find_goals tgts,\n   s'' \u2190 get_goals,\n   let s''' := list.diff s' s'',\n   t,\n   gls \u2190 get_goals,\n   if gls \u2260 [] then do set_tactic_state s, fail \"Failed to discharge the goals!\" else\n   set_goals s'''\n\nsection focus_goals_examples\n\n-- Example of failing when there are no such goals.\nexample : ring \u211d :=\nbegin\n   constructor,\n   success_if_fail {focus_goals [1,2,16] {simp}}, -- Error message \"No such goals!\"\n   exact neg_add_self,\n   exact add_comm,\n   exact one_mul,\n   exact mul_one,\n   exact left_distrib,\n   exact right_distrib,\nend\n\n-- Example of succeeding and restoring other goals.\nexample (p q : Prop) : \u00ac (p \u2227 q) \u2194 \u00ac p \u2228 \u00ac q :=\nbegin\n   constructor,\n   focus_goals [1] {exact classical.not_and_distrib.mp},\n   exact not_and_of_not_or_not,\nend\n\n-- Example of discharging some goals but not others and therefore failing.\nexample : true \u2227 (true \u2228 false) :=\nbegin\n   split,\n   success_if_fail {focus_goals [1,2] {repeat {trivial}}}, -- Failed with error message \"Failed to discharge the goals!\"\n   trivial,\n   constructor,\n   trivial,\nend\n\n-- Example of focus_goals succeeding\nexample : true \u2227 true :=\nbegin\n   split,\n   focus_goals [1,2] {repeat {trivial}},\nend\n\nend focus_goals_examples\n\n-- work_on_goals as described in the assignment specifications\nmeta def tactic.interactive.work_on_goals (pe : interactive.parse lean.parser.pexpr) (t : tactic.interactive.itactic) : tactic unit :=\ndo s \u2190 get_tactic_state,\n   s' \u2190 get_goals,\n   e \u2190 to_expr pe,\n   tgts \u2190 eval_expr (list \u2115) e,\n   find_goals tgts,\n   s'' \u2190 get_goals,\n   let s''' := list.diff s' s'',\n   t,\n   gls \u2190 get_goals,\n   do set_goals (gls ++ s''')\n\nsection work_on_goals_examples\n\nopen real\n\n-- Example of solving some goals and then prepending these to the remaining goals.\nexample : ring \u211d :=\nbegin\n   constructor,\n   work_on_goals [1, 3, 5, 11] {exact neg_add_self,\n                                exact one_mul},\n   exact left_distrib,\n   exact add_comm,\n   exact mul_one,\n   exact right_distrib,\nend\n\nend work_on_goals_examples\n   \n\n\n   \n\n\n", "meta": {"author": "MartinSkilleter", "repo": "real_ip_spaces", "sha": "1ad1e0456602038711cbb0de7aa92b88d6eff06f", "save_path": "github-repos/lean/MartinSkilleter-real_ip_spaces", "path": "github-repos/lean/MartinSkilleter-real_ip_spaces/real_ip_spaces-1ad1e0456602038711cbb0de7aa92b88d6eff06f/src/Assignment 3/Q1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.7606506526772883, "lm_q1q2_score": 0.39517423262262713}}
{"text": "import .love08_operational_semantics_demo\n\n\n/-! # LoVe Demo 9: Hoare Logic\n\nWe review a second way to specify the semantics of a programming language: Hoare\nlogic. If operational semantics corresponds to an idealized interpreter,\n__Hoare logic__ (also called __axiomatic semantics__) corresponds to a verifier.\nHoare logic is particularly convenient to reason about concrete programs. -/\n\n\nset_option pp.beta true\nset_option pp.generalized_field_notation false\n\nnamespace LoVe\n\n\n/-! ## First Things First: Formalization Projects\n\nInstead of two of the homework sheets, you can do a verification project, worth\n20 points. If you choose to do so, please send your lecturer a message by email\nby the end of the week. For a fully successful project, we expect about 200 (or\nmore) lines of Lean, including definitions and proofs.\n\nSome ideas for projects follow.\n\nComputer science:\n\n* extended WHILE language with static arrays or other features;\n* functional data structures (e.g., balanced trees);\n* functional algorithms (e.g., bubble sort, merge sort, Tarjan's algorithm);\n* compiler from expressions or imperative programs to, e.g., stack machine;\n* type systems (e.g., Benjamin Pierce's __Types and Programming Languages__);\n* security properties (e.g., Volpano\u2013Smith-style noninterference analysis);\n* theory of first-order terms, including matching, term rewriting;\n* automata theory;\n* normalization of context-free grammars or regular expressions;\n* process algebras and bisimilarity;\n* soundness and possibly completeness of proof systems (e.g., Genzen's sequent\n  calculus, natural deduction, tableaux);\n* separation logic;\n* verified program using Hoare logic.\n\nMathematics:\n\n* graphs;\n* combinatorics;\n* number theory.\n\nMetaprogramming:\n\n* custom tactic;\n* custom diagnosis tool.\n\nPast evaluation:\n\nQ: How did you find the project?\n\nA: Enjoyable.\n\nA: Fun and hard.\n\nA: Good, I think the format was excellent in a way that it gave people the\n   chance to do challenging exercises and hand them in incomplete.\n\nA: I really really liked it. I think it's a great way of learning\u2014find\n   something you like, dig in it a little, get stuck, ask for help. I wish I\n   could do more of that!\n\nA: It was great to have some time to try to work out some stuff you find\n   interesting yourself.\n\nA: lots of fun actually!!!\n\nA: Very helpful. It gave the opportunity to spend some more time on a\n   particular aspect of the course.\n\n\n## Hoare Triples\n\nThe basic judgments of Hoare logic are often called __Hoare triples__. They have\nthe form\n\n    `{P} S {Q}`\n\nwhere `S` is a statement, and `P` and `Q` (called __precondition__ and\n__postcondition__) are logical formulas over the state variables.\n\nIntended meaning:\n\n    If `P` holds before `S` is executed and the execution terminates normally,\n    `Q` holds at termination.\n\nThis is a __partial correctness__ statement: The program is correct if it\nterminates normally (i.e., no run-time error, no infinite loop or divergence).\n\nAll of these Hoare triples are valid (with respect to the intended meaning):\n\n    `{true} b := 4 {b = 4}`\n    `{a = 2} b := 2 * a {a = 2 \u2227 b = 4}`\n    `{b \u2265 5} b := b + 1 {b \u2265 6}`\n    `{false} skip {b = 100}`\n    `{true} while i \u2260 100 do i := i + 1 {i = 100}`\n\n\n## Hoare Rules\n\nThe following is a complete set of rules for reasoning about WHILE programs:\n\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Skip\n    {P} skip {P}\n\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Asn\n    {Q[a/x]} x := a {Q}\n\n    {P} S {R}   {R} S' {Q}\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Seq\n    {P} S; S' {Q}\n\n    {P \u2227 b} S {Q}   {P \u2227 \u00acb} S' {Q}\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 If\n    {P} if b then S else S' {Q}\n\n    {I \u2227 b} S {I}\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 While\n    {I} while b do S {I \u2227 \u00acb}\n\n    P' \u2192 P   {P} S {Q}   Q \u2192 Q'\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Conseq\n    {P'} S {Q'}\n\n`Q[a/x]` denotes `Q` with `x` replaced by `a`.\n\nIn the `While` rule, `I` is called an __invariant__.\n\nExcept for `Conseq`, the rules are syntax-driven: by looking at a program, we\nsee immediately which rule to apply.\n\nExample derivations:\n\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Asn   \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Asn\n    {a = 2} b := a {b = 2}       {b = 2} c := b {c = 2}\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Seq\n    {a = 2} b := a; c := b {c = 2}\n\n\n                     \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Asn\n    x > 10 \u2192 x > 5   {x > 5} y := x {y > 5}   y > 5 \u2192 y > 0\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Conseq\n    {x > 10} y := x {y > 0}\n\nVarious __derived rules__ can be proved to be correct in terms of the standard\nrules. For example, we can derive bidirectional rules for `skip`, `:=`, and\n`while`:\n\n    P \u2192 Q\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Skip'\n    {P} skip {Q}\n\n    P \u2192 Q[a/x]\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 Asn'\n    {P} x := a {Q}\n\n    {P \u2227 b} S {P}   P \u2227 \u00acb \u2192 Q\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 While'\n    {P} while b do S {Q}\n\n\n## A Semantic Approach to Hoare Logic\n\nWe can, and will, define Hoare triples **semantically** in Lean.\n\nWe will use predicates on states (`state \u2192 Prop`) to represent pre- and\npostconditions, following the shallow embedding style. -/\n\ndef partial_hoare (P : state \u2192 Prop) (S : stmt)\n  (Q : state \u2192 Prop) : Prop :=\n\u2200s t, P s \u2192 (S, s) \u27f9 t \u2192 Q t\n\nnotation `{* ` P : 1 ` *} ` S : 1 ` {* ` Q : 1 ` *}` :=\npartial_hoare P S Q\n\nnamespace partial_hoare\n\nlemma skip_intro {P} :\n  {* P *} stmt.skip {* P *} :=\nbegin\n  intros s t hs hst,\n  cases' hst,\n  assumption\nend\n\nlemma assign_intro (P : state \u2192 Prop) {x} {a : state \u2192 \u2115} :\n  {* \u03bbs, P (s{x \u21a6 a s}) *} stmt.assign x a {* P *} :=\nbegin\n  intros s t P hst,\n  cases' hst,\n  assumption\nend\n\nlemma seq_intro {P Q R S T} (hS : {* P *} S {* Q *})\n    (hT : {* Q *} T {* R *}) :\n  {* P *} S ;; T {* R *} :=\nbegin\n  intros s t hs hst,\n  cases' hst,\n  apply hT,\n  { apply hS,\n    { exact hs },\n    { assumption } },\n  { assumption }\nend\n\nlemma ite_intro {b P Q : state \u2192 Prop} {S T}\n    (hS : {* \u03bbs, P s \u2227 b s *} S {* Q *})\n    (hT : {* \u03bbs, P s \u2227 \u00ac b s *} T {* Q *}) :\n  {* P *} stmt.ite b S T {* Q *} :=\nbegin\n  intros s t hs hst,\n  cases' hst,\n  { apply hS,\n    exact and.intro hs hcond,\n    assumption },\n  { apply hT,\n    exact and.intro hs hcond,\n    assumption }\nend\n\nlemma while_intro (P : state \u2192 Prop) {b : state \u2192 Prop} {S}\n    (h : {* \u03bbs, P s \u2227 b s *} S {* P *}) :\n  {* P *} stmt.while b S {* \u03bbs, P s \u2227 \u00ac b s *} :=\nbegin\n  intros s t hs hst,\n  induction' hst,\n  case while_true {\n    apply ih_hst_1 P h,\n    exact h _ _ (and.intro hs hcond) hst },\n  case while_false {\n    exact and.intro hs hcond }\nend\n\nlemma consequence {P P' Q Q' : state \u2192 Prop} {S}\n    (h : {* P *} S {* Q *}) (hp : \u2200s, P' s \u2192 P s)\n    (hq : \u2200s, Q s \u2192 Q' s) :\n  {* P' *} S {* Q' *} :=\nfix s t,\nassume hs : P' s,\nassume hst : (S, s) \u27f9 t,\nshow Q' t, from\n  hq _ (h s t (hp s hs) hst)\n\nlemma consequence_left (P' : state \u2192 Prop) {P Q S}\n    (h : {* P *} S {* Q *}) (hp : \u2200s, P' s \u2192 P s) :\n  {* P' *} S {* Q *} :=\nconsequence h hp (by cc)\n\nlemma consequence_right (Q) {Q' : state \u2192 Prop} {P S}\n    (h : {* P *} S {* Q *}) (hq : \u2200s, Q s \u2192 Q' s) :\n  {* P *} S {* Q' *} :=\nconsequence h (by cc) hq\n\nlemma skip_intro' {P Q : state \u2192 Prop} (h : \u2200s, P s \u2192 Q s) :\n  {* P *} stmt.skip {* Q *} :=\nconsequence skip_intro h (by cc)\n\nlemma assign_intro' {P Q : state \u2192 Prop} {x} {a : state \u2192 \u2115}\n    (h : \u2200s, P s \u2192 Q (s{x \u21a6 a s})):\n  {* P *} stmt.assign x a {* Q *} :=\nconsequence (assign_intro Q) h (by cc)\n\nlemma seq_intro' {P Q R S T} (hT : {* Q *} T {* R *})\n    (hS : {* P *} S {* Q *}) :\n  {* P *} S ;; T {* R *} :=\nseq_intro hS hT\n\nlemma while_intro' {b P Q : state \u2192 Prop} {S}\n    (I : state \u2192 Prop)\n    (hS : {* \u03bbs, I s \u2227 b s *} S {* I *})\n    (hP : \u2200s, P s \u2192 I s)\n    (hQ : \u2200s, \u00ac b s \u2192 I s \u2192 Q s) :\n  {* P *} stmt.while b S {* Q *} :=\nconsequence (while_intro I hS) hP (by finish)\n\n/-! `finish` applies a combination of techniques, including normalization of\nlogical connectives and quantifiers, simplification, congruence closure, and\nquantifier instantiation. It either fully succeeds or fails. -/\n\nlemma assign_intro_forward (P) {x a} :\n  {* P *}\n  stmt.assign x a\n  {* \u03bbs, \u2203n\u2080, P (s{x \u21a6 n\u2080}) \u2227 s x = a (s{x \u21a6 n\u2080}) *} :=\nbegin\n  apply assign_intro',\n  intros s hP,\n  apply exists.intro (s x),\n  simp [*]\nend\n\nlemma assign_intro_backward (Q : state \u2192 Prop) {x}\n    {a : state \u2192 \u2115} :\n  {* \u03bbs, \u2203n', Q (s{x \u21a6 n'}) \u2227 n' = a s *}\n  stmt.assign x a\n  {* Q *} :=\nbegin\n  apply assign_intro',\n  intros s hP,\n  cases' hP,\n  cc\nend\n\nend partial_hoare\n\n\n/-! ## First Program: Exchanging Two Variables -/\n\ndef SWAP : stmt :=\nstmt.assign \"t\" (\u03bbs, s \"a\") ;;\nstmt.assign \"a\" (\u03bbs, s \"b\") ;;\nstmt.assign \"b\" (\u03bbs, s \"t\")\n\nlemma SWAP_correct (a\u2080 b\u2080 : \u2115) :\n  {* \u03bbs, s \"a\" = a\u2080 \u2227 s \"b\" = b\u2080 *}\n  SWAP\n  {* \u03bbs, s \"a\" = b\u2080 \u2227 s \"b\" = a\u2080 *} :=\nbegin\n  apply partial_hoare.seq_intro',\n  apply partial_hoare.seq_intro',\n  apply partial_hoare.assign_intro,\n  apply partial_hoare.assign_intro,\n  apply partial_hoare.assign_intro',\n  simp { contextual := tt }\nend\n\nlemma SWAP_correct\u2082 (a\u2080 b\u2080 : \u2115) :\n  {* \u03bbs, s \"a\" = a\u2080 \u2227 s \"b\" = b\u2080 *}\n  SWAP\n  {* \u03bbs, s \"a\" = b\u2080 \u2227 s \"b\" = a\u2080 *} :=\nbegin\n  intros s t hP hstep,\n  cases' hstep,\n  cases' hstep,\n  cases' hstep_1,\n  cases' hstep_1_1,\n  cases' hstep_1,\n  finish\nend\n\n\n/-! ## Second Program: Adding Two Numbers -/\n\ndef ADD : stmt :=\nstmt.while (\u03bbs, s \"n\" \u2260 0)\n  (stmt.assign \"n\" (\u03bbs, s \"n\" - 1) ;;\n   stmt.assign \"m\" (\u03bbs, s \"m\" + 1))\n\nlemma ADD_correct (n\u2080 m\u2080 : \u2115) :\n  {* \u03bbs, s \"n\" = n\u2080 \u2227 s \"m\" = m\u2080 *}\n  ADD\n  {* \u03bbs, s \"n\" = 0 \u2227 s \"m\" = n\u2080 + m\u2080 *} :=\npartial_hoare.while_intro' (\u03bbs, s \"n\" + s \"m\" = n\u2080 + m\u2080)\n  begin\n    apply partial_hoare.seq_intro',\n    { apply partial_hoare.assign_intro },\n    { apply partial_hoare.assign_intro',\n      simp,\n      intros s hnm hnz,\n      rw \u2190hnm,\n      cases s \"n\",\n      { finish },\n      { simp [nat.succ_eq_add_one],\n        linarith } }\n  end\n  (by simp { contextual := tt })\n  (by simp { contextual := tt })\n\n/-! How did we come up with this invariant? The invariant must\n\n1. be true before we enter the loop;\n\n2. remain true after each iteration of the loop if it was true before the\n   iteration;\n\n3. be strong enough to imply the desired loop postcondition.\n\nThe invariant `true` meets 1 and 2 but usually not 3. Similarly, `false` meets\n2 and 3 but usually not 1. Suitable invariants are often of the form\n\n__work done__ + __work remaining__ = __desired result__\n\nwhere `+` is some suitable operator. When we enter the loop, __work done__ will\noften be `0`. And when we exit the loop, __work remaining__ should be `0`.\n\nFor the `ADD` loop:\n\n* __work done__ is `m`;\n* __work remaining__ is `n`;\n* __desired result__ is `n\u2080 + m\u2080`.\n\n\n## A Verification Condition Generator\n\n__Verification condition generators__ (VCGs) are programs that apply Hoare rules\nautomatically, producing __verification conditions__ that must be proved by the\nuser. The user must usually also provide strong enough loop invariants, as an\nannotation in their programs.\n\nWe can use Lean's metaprogramming framework to define a simple VCG.\n\nHundreds of program verification tools are based on these principles.\n\nVCGs typically work backwards from the postcondition, using backward rules\n(rules stated to have an arbitrary `Q` as their postcondition). This works well\nbecause `Asn` is backward. -/\n\ndef stmt.while_inv (I b : state \u2192 Prop) (S : stmt) : stmt :=\nstmt.while b S\n\nnamespace partial_hoare\n\nlemma while_inv_intro {b I Q : state \u2192 Prop} {S}\n    (hS : {* \u03bbs, I s \u2227 b s *} S {* I *})\n    (hQ : \u2200s, \u00ac b s \u2192 I s \u2192 Q s) :\n  {* I *} stmt.while_inv I b S {* Q *} :=\nwhile_intro' I hS (by cc) hQ\n\nlemma while_inv_intro' {b I P Q : state \u2192 Prop} {S}\n    (hS : {* \u03bbs, I s \u2227 b s *} S {* I *})\n    (hP : \u2200s, P s \u2192 I s) (hQ : \u2200s, \u00ac b s \u2192 I s \u2192 Q s) :\n  {* P *} stmt.while_inv I b S {* Q *} :=\nwhile_intro' I hS hP hQ\n\nend partial_hoare\n\nmeta def vcg : tactic unit :=\ndo\n  t \u2190 tactic.target,\n  match t with\n  | `({* %%P *} %%S {* _ *}) :=\n    match S with\n    | `(stmt.skip)            :=\n      tactic.applyc\n        (if expr.is_mvar P then ``partial_hoare.skip_intro\n         else ``partial_hoare.skip_intro')\n    | `(stmt.assign _ _)      :=\n      tactic.applyc\n        (if expr.is_mvar P then ``partial_hoare.assign_intro\n         else ``partial_hoare.assign_intro')\n    | `(stmt.seq _ _)         :=\n      tactic.applyc ``partial_hoare.seq_intro'; vcg\n    | `(stmt.ite _ _ _)       :=\n      tactic.applyc ``partial_hoare.ite_intro; vcg\n    | `(stmt.while_inv _ _ _) :=\n      tactic.applyc\n        (if expr.is_mvar P then ``partial_hoare.while_inv_intro\n         else ``partial_hoare.while_inv_intro');\n        vcg\n    | _                       :=\n      tactic.fail (to_fmt \"cannot analyze \" ++ to_fmt S)\n    end\n  | _                        := pure ()\n  end\n\nend LoVe\n\n/-! Register `vcg` as a proper tactic: -/\n\nmeta def tactic.interactive.vcg : tactic unit :=\nLoVe.vcg\n\nnamespace LoVe\n\n\n/-! ## Second Program Revisited: Adding Two Numbers -/\n\nlemma ADD_correct\u2082 (n\u2080 m\u2080 : \u2115) :\n  {* \u03bbs, s \"n\" = n\u2080 \u2227 s \"m\" = m\u2080 *}\n  ADD\n  {* \u03bbs, s \"n\" = 0 \u2227 s \"m\" = n\u2080 + m\u2080 *} :=\nshow {* \u03bbs, s \"n\" = n\u2080 \u2227 s \"m\" = m\u2080 *}\n     stmt.while_inv (\u03bbs, s \"n\" + s \"m\" = n\u2080 + m\u2080)\n       (\u03bbs, s \"n\" \u2260 0)\n       (stmt.assign \"n\" (\u03bbs, s \"n\" - 1) ;;\n        stmt.assign \"m\" (\u03bbs, s \"m\" + 1))\n     {* \u03bbs, s \"n\" = 0 \u2227 s \"m\" = n\u2080 + m\u2080 *}, from\n  begin\n    vcg; simp { contextual := tt },\n    intros s hnm hnz,\n    rw \u2190hnm,\n    cases s \"n\",\n    { finish },\n    { simp [nat.succ_eq_add_one],\n      linarith }\n  end\n\n\n/-! ## Hoare Triples for Total Correctness\n\n__Total correctness__ asserts that the program not only is partially correct but\nalso that it always terminates normally. Hoare triples for total correctness\nhave the form\n\n    [P] S [Q]\n\nIntended meaning:\n\n    If `P` holds before `S` is executed, the execution terminates normally and\n    `Q` holds in the final state.\n\nFor deterministic programs, an equivalent formulation is as follows:\n\n    If `P` holds before `S` is executed, there exists a state in which execution\n    terminates normally and `Q` holds in that state.\n\nExample:\n\n    `[i \u2264 100] while i \u2260 100 do i := i + 1 [i = 100]`\n\nIn our WHILE language, this only affects while loops, which must now be\nannotated by a __variant__ `V` (a natural number that decreases with each\niteration):\n\n    [I \u2227 b \u2227 V = v\u2080] S [I \u2227 V < v\u2080]\n    \u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014\u2014 While-Var\n    [I] while b do S [I \u2227 \u00acb]\n\nWhat is a suitable variant for the example above? -/\n\nend LoVe\n", "meta": {"author": "blanchette", "repo": "logical_verification_2022", "sha": "5aee593fbef9b63d4338288b4789d85851d258aa", "save_path": "github-repos/lean/blanchette-logical_verification_2022", "path": "github-repos/lean/blanchette-logical_verification_2022/logical_verification_2022-5aee593fbef9b63d4338288b4789d85851d258aa/lean/love09_hoare_logic_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.7549149923816048, "lm_q1q2_score": 0.39513786872964196}}
{"text": "import MyNat.Definition\n/-!\n# Tactic exfalso\n\n## Summary\n\n`exfalso` changes your goal to `False`.\n\n## Details\n\nWe know that `False` implies `P` for any proposition `P`, and so if your goal is `P`\nthen you should be able to `apply` `False \u2192 P` and reduce your goal to `False`. This\nis what the `exfalso` tactic does. The theorem that `False \u2192 P` is called `False.elim`\nso one can achieve the same effect with `apply False.elim`.\n\nYou might think this is a step backwards, but if you have a hypothesis `h : \u00ac P`\nthen after `rw [not_iff_imp_false] at h,` you can `apply h,` to make progress.\n\nThis tactic can also be used in a proof by contradiction, where the hypotheses are enough\nto deduce a contradiction and the goal happens to be some random statement (possibly\na False one) which you just want to simplify to `False`.\n\n-/", "meta": {"author": "leanprover", "repo": "lean4-samples", "sha": "5c0db5f1e952e7ebada506ba3a390040972293a9", "save_path": "github-repos/lean/leanprover-lean4-samples", "path": "github-repos/lean/leanprover-lean4-samples/lean4-samples-5c0db5f1e952e7ebada506ba3a390040972293a9/NaturalNumbers/Tactics/exfalso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.7549149813536518, "lm_q1q2_score": 0.39513786295738695}}
{"text": "opaque f (x y : Nat) : Nat\nopaque g (x : Nat) : Nat\n\ntheorem ex1 (x : Nat) (h\u2081 : f x x = g x) (h\u2082 : g x = x) : f x (f x x) = x := by\n  simp\n  simp [*]\n\ntheorem ex2 (x : Nat) (h\u2081 : f x x = g x) (h\u2082 : g x = x) : f x (f x x) = x := by\n  simp [*]\n\naxiom g_ax (x : Nat) : g x = 0\n\ntheorem ex3 (x y : Nat) (h\u2081 : f x x = g x) (h\u2082 : f x x < 5) : f x x + f x x = 0 := by\n  simp [*] at *\n  trace_state\n  have aux\u2081 : f x x = g x := h\u2081\n  have aux\u2082 : g x < 5     := h\u2082\n  simp [g_ax]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/simpStar.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.39511626366706515}}
{"text": "import condensed.projective_resolution\n\nopen category_theory.limits\nopen category_theory\n\nnamespace Condensed\n\n--TODO: generalize (as needed...)\nuniverse u\nvariables {A B : Condensed.{u} Ab.{u+1}} (f : A \u27f6 B)\n\nnoncomputable theory\n\ndef kernel_diagram_iso {A B : Condensed.{u} Ab.{u+1}} (S : ExtrDisc) (f : A \u27f6 B) :\n  parallel_pair f 0 \u22d9 Condensed.evaluation Ab S.val \u2245\n    parallel_pair ((Condensed.evaluation Ab S.val).map f) 0 :=\nnat_iso.of_components (\u03bb X,\n  match X with\n  | walking_parallel_pair.zero := iso.refl _\n  | walking_parallel_pair.one := iso.refl _\n  end) $ by { rintros (a|a) (b|b) (f|f), tidy }\n\ndef cokernel_diagram_iso {A B : Condensed.{u} Ab.{u+1}} (S : ExtrDisc) (f : A \u27f6 B) :\n  limits.parallel_pair ((Condensed.evaluation Ab S.val).map f) 0 \u2245\n    limits.parallel_pair f 0 \u22d9 Condensed.evaluation Ab S.val :=\nnat_iso.of_components (\u03bb X,\n  match X with\n  | walking_parallel_pair.zero := iso.refl _\n  | walking_parallel_pair.one := iso.refl _\n  end) $ by { rintros (a|a) (b|b) (f|f), tidy }\n\ndef kernel_iso (S : ExtrDisc.{u}) :\n  (Condensed.evaluation _ S.val).obj (kernel f) \u2245\n  kernel ((Condensed.evaluation _ S.val).map f) :=\n(is_limit_of_preserves (Condensed.evaluation _ S.val)\n  (limit.is_limit (parallel_pair f 0))).cone_point_unique_up_to_iso\n  (limit.is_limit _) \u226a\u226b has_limit.iso_of_nat_iso (kernel_diagram_iso _ _)\n\n@[simp, reassoc]\nlemma kernel_iso_hom (S : ExtrDisc.{u}) :\n  (kernel_iso f S).hom \u226b kernel.\u03b9 _ = (Condensed.evaluation _ S.val).map (kernel.\u03b9 _) :=\nbegin\n  dsimp [kernel_iso, kernel_diagram_iso],\n  simp only [category.assoc, has_limit.iso_of_nat_iso_hom_\u03c0,\n    nat_iso.of_components_hom_app, limit.cone_point_unique_up_to_iso_hom_comp_assoc,\n    functor.map_cone_\u03c0_app, equalizer.fork_\u03c0_app_zero,\n    equalizer_as_kernel, Condensed.evaluation_map],\n  apply category.comp_id,\nend\n\ndef cokernel_iso (S : ExtrDisc.{u}) :\n  cokernel ((Condensed.evaluation _ S.val).map f) \u2245\n  (Condensed.evaluation _ S.val).obj (cokernel f) :=\nhas_colimit.iso_of_nat_iso (cokernel_diagram_iso _ _) \u226a\u226b\n  (colimit.is_colimit _).cocone_point_unique_up_to_iso\n  (is_colimit_of_preserves (Condensed.evaluation _ _)\n  (colimit.is_colimit (parallel_pair f 0)))\n\n@[simp, reassoc]\nlemma cokernel_iso_hom (S : ExtrDisc.{u}) :\n  cokernel.\u03c0 _ \u226b (cokernel_iso f S).hom = (Condensed.evaluation _ S.val).map (cokernel.\u03c0 _) :=\nbegin\n  dsimp [cokernel_iso, cokernel_diagram_iso],\n  simp only [has_colimit.iso_of_nat_iso_\u03b9_hom_assoc, nat_iso.of_components_hom_app,\n  colimit.comp_cocone_point_unique_up_to_iso_hom, functor.map_cocone_\u03b9_app,\n  coequalizer.cofork_\u03b9_app_one, coequalizer_as_cokernel, Condensed.evaluation_map],\n  apply category.id_comp,\nend\n\nend Condensed\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/kernel_comparison.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.394945896436159}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\n\nimport category_theory.sites.sheaf\n\n/-!\n\nIn this file we construct the functor `Sheaf J A \u2964 Sheaf J B` between sheaf categories\nobtained by composition with a functor `F : A \u2964 B`.\n\nIn order for the sheaf condition to be preserved, `F` must preserve the correct limits.\nThe lemma `presheaf.is_sheaf.comp` says that composition with such an `F` indeed preserves the\nsheaf condition.\n\nThe functor between sheaf categories is called `Sheaf_compose J F`.\nGiven a natural transformation `\u03b7 : F \u27f6 G`, we obtain a natural transformation\n`Sheaf_compose J F \u27f6 Sheaf_compose J G`, which we call `Sheaf_compose_map J \u03b7`.\n\n-/\n\nnamespace category_theory\n\nopen category_theory.limits\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 u\u2083\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\nvariables {A : Type u\u2082} [category.{max v\u2081 u\u2081} A]\nvariables {B : Type u\u2083} [category.{max v\u2081 u\u2081} B]\nvariables {J : grothendieck_topology C}\nvariables {U : C} (R : presieve U)\nvariables (F : A \u2964 B)\n\nnamespace grothendieck_topology.cover\n\nvariables (P : C\u1d52\u1d56 \u2964 A) {X : C} (S : J.cover X)\n\n/-- The multicospan associated to a cover `S : J.cover X` and a presheaf of the form `P \u22d9 F`\nis isomorphic to the composition of the multicospan associated to `S` and `P`,\ncomposed with `F`. -/\ndef multicospan_comp : (S.index (P \u22d9 F)).multicospan \u2245 (S.index P).multicospan \u22d9 F :=\nnat_iso.of_components (\u03bb t,\nmatch t with\n| walking_multicospan.left a := eq_to_iso rfl\n| walking_multicospan.right b := eq_to_iso rfl\nend) begin\n  rintros (a|b) (a|b) (f|f|f),\n  any_goals { dsimp, erw [functor.map_id, functor.map_id, category.id_comp] },\n  any_goals { dsimp, erw [category.comp_id, category.id_comp], refl }\nend\n\n@[simp] lemma multicospan_comp_app_left (a) :\n  (S.multicospan_comp F P).app (walking_multicospan.left a) = eq_to_iso rfl := rfl\n\n@[simp] lemma multicospan_comp_app_right (b) :\n  (S.multicospan_comp F P).app (walking_multicospan.right b) = eq_to_iso rfl := rfl\n\n@[simp] lemma multicospan_comp_hom_app_left (a) :\n  (S.multicospan_comp F P).hom.app (walking_multicospan.left a) = eq_to_hom rfl := rfl\n\n@[simp] lemma multicospan_comp_hom_app_right (b) :\n  (S.multicospan_comp F P).hom.app (walking_multicospan.right b) = eq_to_hom rfl := rfl\n\n@[simp] lemma multicospan_comp_hom_inv_left (P : C\u1d52\u1d56 \u2964 A) {X : C}\n  (S : J.cover X) (a) : (S.multicospan_comp F P).inv.app (walking_multicospan.left a) =\n  eq_to_hom rfl := rfl\n\n@[simp] lemma multicospan_comp_hom_inv_right (P : C\u1d52\u1d56 \u2964 A) {X : C}\n  (S : J.cover X) (b) : (S.multicospan_comp F P).inv.app (walking_multicospan.right b) =\n  eq_to_hom rfl := rfl\n\n/-- Mapping the multifork associated to a cover `S : J.cover X` and a presheaf `P` with\nrespect to a functor `F` is isomorphic (upto a natural isomorphism of the underlying functors)\nto the multifork associated to `S` and `P \u22d9 F`. -/\ndef map_multifork : F.map_cone (S.multifork P) \u2245 (limits.cones.postcompose\n    (S.multicospan_comp F P).hom).obj (S.multifork (P \u22d9 F)) :=\ncones.ext (eq_to_iso rfl) begin\n  rintros (a|b),\n  { dsimp, simpa },\n  { dsimp, simp, dsimp [multifork.of_\u03b9], simpa }\nend\n\nend grothendieck_topology.cover\n\nvariables [\u2200 (X : C) (S : J.cover X) (P : C\u1d52\u1d56 \u2964 A), preserves_limit (S.index P).multicospan F]\n\nlemma presheaf.is_sheaf.comp {P : C\u1d52\u1d56 \u2964 A} (hP : presheaf.is_sheaf J P) :\n  presheaf.is_sheaf J (P \u22d9 F) :=\nbegin\n  rw presheaf.is_sheaf_iff_multifork at \u22a2 hP,\n  intros X S,\n  obtain \u27e8h\u27e9 := hP X S,\n  replace h := is_limit_of_preserves F h,\n  replace h := limits.is_limit.of_iso_limit h (S.map_multifork F P),\n  exact \u27e8limits.is_limit.postcompose_hom_equiv (S.multicospan_comp F P) _ h\u27e9,\nend\n\nvariable (J)\n\n/-- Composing a sheaf with a functor preserving the appropriate limits yields a functor\nbetween sheaf categories. -/\n@[simps]\ndef Sheaf_compose : Sheaf J A \u2964 Sheaf J B :=\n{ obj := \u03bb G, \u27e8G.val \u22d9 F, presheaf.is_sheaf.comp _ G.2\u27e9,\n  map := \u03bb G H \u03b7, \u27e8whisker_right \u03b7.val _\u27e9,\n  map_id' := \u03bb G, Sheaf.hom.ext _ _ $ whisker_right_id _,\n  map_comp' := \u03bb G H W f g, Sheaf.hom.ext _ _ $ whisker_right_comp _ _ _ }\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/whiskering.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3949458894335354}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.bitraversable.lemmas\n! leanprover-community/mathlib commit 58581d0fe523063f5651df0619be2bf65012a94a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Control.Bitraversable.Basic\n\n/-!\n# Bitraversable Lemmas\n\n## Main definitions\n  * tfst - traverse on first functor argument\n  * tsnd - traverse on second functor argument\n\n## Lemmas\n\nCombination of\n  * bitraverse\n  * tfst\n  * tsnd\n\nwith the applicatives `id` and `comp`\n\n## References\n\n * Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable functor bifunctor applicative\n\n\n-/\n\n\nuniverse u\n\nvariable {t : Type u \u2192 Type u \u2192 Type u} [Bitraversable t]\n\nvariable {\u03b2 : Type u}\n\nnamespace Bitraversable\n\nopen Functor LawfulApplicative\n\nvariable {F G : Type u \u2192 Type u} [Applicative F] [Applicative G]\n\n/-- traverse on the first functor argument -/\n@[reducible]\ndef tfst {\u03b1 \u03b1'} (f : \u03b1 \u2192 F \u03b1') : t \u03b1 \u03b2 \u2192 F (t \u03b1' \u03b2) :=\n  bitraverse f pure\n#align bitraversable.tfst Bitraversable.tfst\n\n/-- traverse on the second functor argument -/\n@[reducible]\ndef tsnd {\u03b1 \u03b1'} (f : \u03b1 \u2192 F \u03b1') : t \u03b2 \u03b1 \u2192 F (t \u03b2 \u03b1') :=\n  bitraverse pure f\n#align bitraversable.tsnd Bitraversable.tsnd\n\nvariable [IsLawfulBitraversable t] [LawfulApplicative F] [LawfulApplicative G]\n\n@[higher_order.1tfst_id]\ntheorem id_tfst : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), tfst id.mk x = id.mk x :=\n  @id_bitraverse _ _ _\n#align bitraversable.id_tfst Bitraversable.id_tfst\n\n@[higher_order.1tsnd_id]\ntheorem id_tsnd : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), tsnd id.mk x = id.mk x :=\n  @id_bitraverse _ _ _\n#align bitraversable.id_tsnd Bitraversable.id_tsnd\n\n@[higher_order.1tfst_comp_tfst]\ntheorem comp_tfst {\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b1\u2081 \u2192 G \u03b1\u2082) (x : t \u03b1\u2080 \u03b2) :\n    Comp.mk (tfst f' <$> tfst f x) = tfst (Comp.mk \u2218 map f' \u2218 f) x := by\n  rw [\u2190 comp_bitraverse] <;> simp [tfst, map_comp_pure, Pure.pure]\n#align bitraversable.comp_tfst Bitraversable.comp_tfst\n\n@[higher_order.1tfst_comp_tsnd]\ntheorem tfst_tsnd {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n    Comp.mk (tfst f <$> tsnd f' x) = bitraverse (Comp.mk \u2218 pure \u2218 f) (Comp.mk \u2218 map pure \u2218 f') x :=\n  by rw [\u2190 comp_bitraverse] <;> simp [tfst, tsnd]\n#align bitraversable.tfst_tsnd Bitraversable.tfst_tsnd\n\n@[higher_order.1tsnd_comp_tfst]\ntheorem tsnd_tfst {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n    Comp.mk (tsnd f' <$> tfst f x) = bitraverse (Comp.mk \u2218 map pure \u2218 f) (Comp.mk \u2218 pure \u2218 f') x :=\n  by rw [\u2190 comp_bitraverse] <;> simp [tfst, tsnd]\n#align bitraversable.tsnd_tfst Bitraversable.tsnd_tfst\n\n@[higher_order.1tsnd_comp_tsnd]\ntheorem comp_tsnd {\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082} (g : \u03b2\u2080 \u2192 F \u03b2\u2081) (g' : \u03b2\u2081 \u2192 G \u03b2\u2082) (x : t \u03b1 \u03b2\u2080) :\n    Comp.mk (tsnd g' <$> tsnd g x) = tsnd (Comp.mk \u2218 map g' \u2218 g) x := by\n  rw [\u2190 comp_bitraverse] <;> simp [tsnd] <;> rfl\n#align bitraversable.comp_tsnd Bitraversable.comp_tsnd\n\nopen Bifunctor\n\nprivate theorem pure_eq_id_mk_comp_id {\u03b1} : pure = id.mk \u2218 @id \u03b1 :=\n  rfl\n#align bitraversable.pure_eq_id_mk_comp_id bitraversable.pure_eq_id_mk_comp_id\n\nopen Function\n\n@[higher_order.1]\ntheorem tfst_eq_fst_id {\u03b1 \u03b1' \u03b2} (f : \u03b1 \u2192 \u03b1') (x : t \u03b1 \u03b2) : tfst (id.mk \u2218 f) x = id.mk (fst f x) :=\n  by simp [tfst, fst, pure_eq_id_mk_comp_id, -comp.right_id, bitraverse_eq_bimap_id]\n#align bitraversable.tfst_eq_fst_id Bitraversable.tfst_eq_fst_id\n\n@[higher_order.1]\ntheorem tsnd_eq_snd_id {\u03b1 \u03b2 \u03b2'} (f : \u03b2 \u2192 \u03b2') (x : t \u03b1 \u03b2) : tsnd (id.mk \u2218 f) x = id.mk (snd f x) :=\n  by simp [tsnd, snd, pure_eq_id_mk_comp_id, -comp.right_id, bitraverse_eq_bimap_id]\n#align bitraversable.tsnd_eq_snd_id Bitraversable.tsnd_eq_snd_id\n\nattribute [functor_norm]\n  comp_bitraverse comp_tsnd comp_tfst tsnd_comp_tsnd tsnd_comp_tfst tfst_comp_tsnd tfst_comp_tfst bitraverse_comp bitraverse_id_id tfst_id tsnd_id\n\nend Bitraversable\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Bitraversable/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.7025300573952052, "lm_q1q2_score": 0.39494588943353537}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.finite\nimport Mathlib.algebra.big_operators.basic\nimport Mathlib.PostPort\n\nuniverses u v u_1 x \n\nnamespace Mathlib\n\n/-!\n# Preimage of a `finset` under an injective map.\n-/\n\nnamespace finset\n\n\n/-- Preimage of `s : finset \u03b2` under a map `f` injective of `f \u207b\u00b9' s` as a `finset`.  -/\ndef preimage {\u03b1 : Type u} {\u03b2 : Type v} (s : finset \u03b2) (f : \u03b1 \u2192 \u03b2) (hf : set.inj_on f (f \u207b\u00b9' \u2191s)) : finset \u03b1 :=\n  set.finite.to_finset sorry\n\n@[simp] theorem mem_preimage {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {s : finset \u03b2} {hf : set.inj_on f (f \u207b\u00b9' \u2191s)} {x : \u03b1} : x \u2208 preimage s f hf \u2194 f x \u2208 s :=\n  set.finite.mem_to_finset\n\n@[simp] theorem coe_preimage {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (s : finset \u03b2) (hf : set.inj_on f (f \u207b\u00b9' \u2191s)) : \u2191(preimage s f hf) = f \u207b\u00b9' \u2191s :=\n  set.finite.coe_to_finset (preimage._proof_1 s f hf)\n\n@[simp] theorem preimage_empty {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} : preimage \u2205 f\n    (eq.mpr\n      (id\n        (Eq.trans\n          (Eq.trans\n            (Eq.trans\n              ((fun (f f_1 : \u03b1 \u2192 \u03b2) (e_1 : f = f_1) (s s_1 : set \u03b1) (e_2 : s = s_1) =>\n                  congr (congr_arg set.inj_on e_1) e_2)\n                f f (Eq.refl f) (f \u207b\u00b9' \u2191\u2205) \u2205\n                (Eq.trans\n                  ((fun (f f_1 : \u03b1 \u2192 \u03b2) (e_1 : f = f_1) (s s_1 : set \u03b2) (e_2 : s = s_1) =>\n                      congr (congr_arg set.preimage e_1) e_2)\n                    f f (Eq.refl f) \u2191\u2205 \u2205 coe_empty)\n                  set.preimage_empty))\n              (set.inj_on.equations._eqn_1 f \u2205))\n            (forall_congr_eq\n              fun (x\u2081 : \u03b1) =>\n                Eq.trans\n                  (imp_congr_eq (set.mem_empty_eq x\u2081)\n                    (Eq.trans\n                      (forall_congr_eq\n                        fun (x\u2082 : \u03b1) =>\n                          Eq.trans (imp_congr_eq (set.mem_empty_eq x\u2082) (Eq.refl (f x\u2081 = f x\u2082 \u2192 x\u2081 = x\u2082)))\n                            (propext (forall_prop_of_false (iff.mpr not_false_iff True.intro))))\n                      (propext forall_true_iff)))\n                  (propext (forall_prop_of_false (iff.mpr not_false_iff True.intro)))))\n          (propext forall_true_iff)))\n      trivial) =\n  \u2205 := sorry\n\n@[simp] theorem preimage_univ {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} [fintype \u03b1] [fintype \u03b2] (hf : set.inj_on f (f \u207b\u00b9' \u2191univ)) : preimage univ f hf = univ := sorry\n\n@[simp] theorem preimage_inter {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : finset \u03b2} {t : finset \u03b2} (hs : set.inj_on f (f \u207b\u00b9' \u2191s)) (ht : set.inj_on f (f \u207b\u00b9' \u2191t)) : (preimage (s \u2229 t) f\n    fun (x\u2081 : \u03b1) (hx\u2081 : x\u2081 \u2208 f \u207b\u00b9' \u2191(s \u2229 t)) (x\u2082 : \u03b1) (hx\u2082 : x\u2082 \u2208 f \u207b\u00b9' \u2191(s \u2229 t)) =>\n      hs (mem_of_mem_inter_left hx\u2081) (mem_of_mem_inter_left hx\u2082)) =\n  preimage s f hs \u2229 preimage t f ht := sorry\n\n@[simp] theorem preimage_union {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : finset \u03b2} {t : finset \u03b2} (hst : set.inj_on f (f \u207b\u00b9' \u2191(s \u222a t))) : preimage (s \u222a t) f hst =\n  (preimage s f\n      fun (x\u2081 : \u03b1) (hx\u2081 : x\u2081 \u2208 f \u207b\u00b9' \u2191s) (x\u2082 : \u03b1) (hx\u2082 : x\u2082 \u2208 f \u207b\u00b9' \u2191s) =>\n        hst (mem_union_left t hx\u2081) (mem_union_left t hx\u2082)) \u222a\n    preimage t f\n      fun (x\u2081 : \u03b1) (hx\u2081 : x\u2081 \u2208 f \u207b\u00b9' \u2191t) (x\u2082 : \u03b1) (hx\u2082 : x\u2082 \u2208 f \u207b\u00b9' \u2191t) =>\n        hst (mem_union_right s hx\u2081) (mem_union_right s hx\u2082) := sorry\n\n@[simp] theorem preimage_compl {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] [fintype \u03b1] [fintype \u03b2] {f : \u03b1 \u2192 \u03b2} (s : finset \u03b2) (hf : function.injective f) : preimage (s\u1d9c) f (function.injective.inj_on hf (f \u207b\u00b9' \u2191(s\u1d9c))) = (preimage s f (function.injective.inj_on hf (f \u207b\u00b9' \u2191s))\u1d9c) := sorry\n\ntheorem monotone_preimage {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (h : function.injective f) : monotone fun (s : finset \u03b2) => preimage s f (function.injective.inj_on h (f \u207b\u00b9' \u2191s)) :=\n  fun (s t : finset \u03b2) (hst : s \u2264 t) (x : \u03b1)\n    (hx : x \u2208 (fun (s : finset \u03b2) => preimage s f (function.injective.inj_on h (f \u207b\u00b9' \u2191s))) s) =>\n    iff.mpr mem_preimage (hst (iff.mp mem_preimage hx))\n\ntheorem image_subset_iff_subset_preimage {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : finset \u03b1} {t : finset \u03b2} (hf : set.inj_on f (f \u207b\u00b9' \u2191t)) : image f s \u2286 t \u2194 s \u2286 preimage t f hf := sorry\n\ntheorem map_subset_iff_subset_preimage {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u21aa \u03b2} {s : finset \u03b1} {t : finset \u03b2} : map f s \u2286 t \u2194 s \u2286 preimage t (\u21d1f) (function.injective.inj_on (function.embedding.injective f) (\u21d1f \u207b\u00b9' \u2191t)) := sorry\n\ntheorem image_preimage {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : finset \u03b2) [(x : \u03b2) \u2192 Decidable (x \u2208 set.range f)] (hf : set.inj_on f (f \u207b\u00b9' \u2191s)) : image f (preimage s f hf) = filter (fun (x : \u03b2) => x \u2208 set.range f) s := sorry\n\ntheorem image_preimage_of_bij {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : finset \u03b2) (hf : set.bij_on f (f \u207b\u00b9' \u2191s) \u2191s) : image f (preimage s f (set.bij_on.inj_on hf)) = s := sorry\n\ntheorem sigma_preimage_mk {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type u_1} [DecidableEq \u03b1] (s : finset (sigma fun (a : \u03b1) => \u03b2 a)) (t : finset \u03b1) : (finset.sigma t\n    fun (a : \u03b1) => preimage s (sigma.mk a) (function.injective.inj_on sigma_mk_injective (sigma.mk a \u207b\u00b9' \u2191s))) =\n  filter (fun (a : sigma fun (a : \u03b1) => \u03b2 a) => sigma.fst a \u2208 t) s := sorry\n\ntheorem sigma_preimage_mk_of_subset {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type u_1} [DecidableEq \u03b1] (s : finset (sigma fun (a : \u03b1) => \u03b2 a)) {t : finset \u03b1} (ht : image sigma.fst s \u2286 t) : (finset.sigma t\n    fun (a : \u03b1) => preimage s (sigma.mk a) (function.injective.inj_on sigma_mk_injective (sigma.mk a \u207b\u00b9' \u2191s))) =\n  s := sorry\n\ntheorem sigma_image_fst_preimage_mk {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type u_1} [DecidableEq \u03b1] (s : finset (sigma fun (a : \u03b1) => \u03b2 a)) : (finset.sigma (image sigma.fst s)\n    fun (a : \u03b1) => preimage s (sigma.mk a) (function.injective.inj_on sigma_mk_injective (sigma.mk a \u207b\u00b9' \u2191s))) =\n  s :=\n  sigma_preimage_mk_of_subset s (subset.refl (image sigma.fst s))\n\ntheorem prod_preimage' {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type x} [comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b3) [decidable_pred fun (x : \u03b3) => x \u2208 set.range f] (s : finset \u03b3) (hf : set.inj_on f (f \u207b\u00b9' \u2191s)) (g : \u03b3 \u2192 \u03b2) : (finset.prod (preimage s f hf) fun (x : \u03b1) => g (f x)) =\n  finset.prod (filter (fun (x : \u03b3) => x \u2208 set.range f) s) fun (x : \u03b3) => g x := sorry\n\ntheorem prod_preimage {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type x} [comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b3) (s : finset \u03b3) (hf : set.inj_on f (f \u207b\u00b9' \u2191s)) (g : \u03b3 \u2192 \u03b2) (hg : \u2200 (x : \u03b3), x \u2208 s \u2192 \u00acx \u2208 set.range f \u2192 g x = 1) : (finset.prod (preimage s f hf) fun (x : \u03b1) => g (f x)) = finset.prod s fun (x : \u03b3) => g x := sorry\n\ntheorem sum_preimage_of_bij {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type x} [add_comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b3) (s : finset \u03b3) (hf : set.bij_on f (f \u207b\u00b9' \u2191s) \u2191s) (g : \u03b3 \u2192 \u03b2) : (finset.sum (preimage s f (set.bij_on.inj_on hf)) fun (x : \u03b1) => g (f x)) = finset.sum s fun (x : \u03b3) => g x :=\n  sum_preimage f s (set.bij_on.inj_on hf) g\n    fun (x : \u03b3) (hxs : x \u2208 s) (hxf : \u00acx \u2208 set.range f) => false.elim (hxf (set.bij_on.subset_range hf hxs))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/finset/preimage.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321964553657, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.394911717128393}}
{"text": "opaque f (a : Nat) (h : a > 0) : Nat\n\nexample (h : a = b) : f (a + 1) (by simp_arith) = f (1 + b) (by simp_arith) := by\n  conv => lhs; congr; rw [h]\n  conv => lhs; congr; rw [Nat.add_comm]\n\nopaque g (p : Prop) [Decidable p] (a : Nat) (h : a > 0) : Nat\n\nexample (h : a = b) : g True (a + 1) (by simp_arith) = g (1+1=2) (1 + b) (by simp_arith) := by\n  conv =>\n    lhs\n    congr\n    . rfl\n    . rw [h]\n  conv =>\n    lhs\n    congr\n    . rfl\n    . rw [Nat.add_comm]\n  conv =>\n    rhs\n    congr\n    . simp\n    . rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/conv2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3948086844801453}}
{"text": "import lang\n\nopen typ exp\n\ndef evals_to (e v:exp) : Prop := is_many_step e v \u2227 is_val v\ndef normalizes (e:exp) : Prop := \u2203 v, evals_to e v\n\n-- Proof of strong normalization using a logical predicate\ndef SN : typ \u2192 exp \u2192 Prop\n| unitT e := (empty_ctx \u22a2 e : unitT) \u2227 (normalizes e)\n| (arrowT \u03c41 \u03c42) e := (empty_ctx \u22a2 e : arrowT \u03c41 \u03c42) \u2227\n                      (normalizes e) \u2227\n                      (\u2200 (e':exp), (SN \u03c41 e') \u2192 (SN \u03c42 (ap e e')))\n\nlemma sn_implies_closed :\n\u2200 e \u03c4,\n SN \u03c4 e \u2192 empty_ctx \u22a2 e : \u03c4 :=\nbegin\n  intros,\n  cases \u03c4; unfold SN at *; tauto\nend\n\nlemma sn_implies_normalizes :\n\u2200 e \u03c4, SN \u03c4 e \u2192 normalizes e :=\nbegin\n introv Hsn,\n cases \u03c4, exact Hsn.2, exact Hsn.2.1\nend\n\nlemma sn_preservation :\n  \u2200 e e' \u03c4,\n  empty_ctx \u22a2 e : \u03c4 \u2192\n  (e \u21a6str e') \u2192\n  (SN \u03c4 e' \u2192 SN \u03c4 e) -- \u2227 (SN \u03c4 e \u2192 SN \u03c4 e')\n:=\nbegin\nintrov Hty Hstep,\n{ -- backwards preservation\n  intros Hsn,\n  generalize h : (empty_ctx = \u0393),\n  rw h at *,\n  -- induction Hty generalizing e'; subst h,\n  induction \u03c4 generalizing e' e; subst h,\n  -- cases Hstep,\n  -- induction Hty; subst h,\n  {\n    unfold SN,\n    split, assumption,\n    have h := (sn_implies_normalizes _ _ Hsn),\n    cases h,\n    existsi h_w,\n    unfold evals_to at *,\n    split,\n    { apply is_many_step.many_steps_transitive,\n      apply h_h.1, assumption\n    },\n    apply h_h.2,\n  },\n  { unfold SN,\n    split,\n    { assumption },\n    split,\n    {\n      have h := (sn_implies_normalizes _ _ Hsn),\n      cases h,\n      existsi h_w,\n      unfold evals_to at *,\n      cases h_h,\n      split,\n      {\n        apply is_many_step.many_steps_transitive,\n        { assumption },\n        assumption\n      },\n      assumption\n    },\n    cases Hstep,\n    {\n      unfold SN at Hsn,\n      intros a Hsna,\n      apply \u03c4_ih_\u03c42,\n      { constructor, assumption  },\n      { apply Hsn.2.2, assumption },\n      constructor,\n      { assumption },\n      cases \u03c4_\u03c41; apply Hsna.1,\n    },\n    {\n      intros a Hsna,\n      unfold SN at Hsn,\n      apply \u03c4_ih_\u03c42,\n      { constructor, assumption },\n      { apply Hsn.2.2, assumption },\n      constructor,\n      { assumption },\n      cases \u03c4_\u03c41; apply Hsna.1,\n    }\n  },\n},\nend\n\n-- XXX: this is a tricky lemma because of variable shadowing. If we worked\n-- modulo \u03b1-equivalence, this trickiness might be avoided.\nlemma env_sub_lam_step :\n\u2200 \u03b3 c x \u03c41 e e',\nis_env_ctx SN \u03b3 c \u2192\nSN \u03c41 e' \u2192\n(env_sub \u03b3 (lam x \u03c41 e)).ap e' \u21a6str (env_sub \u03b3 (substitute x e' e)) :=\nbegin\n  introv Hctx Hsn,\n  induction \u03b3 generalizing e c,\n  { unfold env_sub, constructor },\n  cases \u03b3_hd with y vy,\n  unfold env_sub,\n  unfold substitute,\n  by_cases (y = x),\n  {\n    simp *,\n    cases c,\n    { exfalso, unfold is_env_ctx at *, assumption },\n    cases c_hd,\n    unfold is_env_ctx at Hctx,\n    rw double_substitute,\n    { apply \u03b3_ih, apply Hctx.2.1 },\n    { apply sn_implies_closed, assumption },\n    { apply sn_implies_closed, apply Hctx.2.2 },\n  },\n  {\n    rw if_neg, tactic.swap, tauto,\n    cases c,\n    { exfalso, unfold is_env_ctx at *, assumption },\n    cases c_hd,\n    unfold is_env_ctx at Hctx,\n    rw substitute_commute,\n    {\n      apply \u03b3_ih,\n      { apply Hctx.2.1 },\n    },\n    { apply sn_implies_closed, apply Hctx.2.2 },\n    { apply sn_implies_closed, assumption },\n    { assumption },\n  }\nend\n\ntheorem sn_general :\n  \u2200 \u0393 c \u03b3 e \u03c4,\n  \u0393 = mk_context c \u2192\n  \u0393 \u22a2 e : \u03c4 \u2192\n  is_env_ctx SN \u03b3 c \u2192\n  SN \u03c4 (env_sub \u03b3 e) :=\nbegin\n  introv H\u0393 Hty Henv,\n  induction Hty generalizing \u03b3 c; subst H\u0393,\n  { -- case: unit\n    rw env_sub_unit,\n    unfold SN,\n    split,\n    { constructor },\n    existsi exp.unit,\n    unfold evals_to,\n    split; constructor\n  },\n  { -- case: var\n    induction \u03b3 generalizing c,\n    {\n      induction c,\n      { exfalso, contradiction },\n      { exfalso, unfold is_env_ctx at *, contradiction }\n    },\n    induction c,\n    { exfalso, contradiction },\n    cases \u03b3_hd, cases c_hd,\n    unfold is_env_ctx at *,\n    unfold env_sub,\n    cases Henv with Hre Henv,\n    subst Hre,\n    by_cases (Hty_x = \u03b3_hd_fst),\n    { -- the first thing in \u0393/\u03b3 is the var Hty_x\n      subst h,\n      unfold env_sub substitute at c_ih,\n      unfold substitute at *,\n      simp * at *,\n      unfold mk_context update_context at Hty_Hvar,\n      simp * at Hty_Hvar,\n      injection Hty_Hvar,\n      subst h_1,\n      rw (env_sub_closed _ _ _ _),\n      { apply Henv.2 },\n      tactic.swap, { apply sn_implies_closed, apply Henv.2 },\n    },\n    { -- induction\n      unfold substitute, rw if_neg,\n      tactic.swap, { tauto, },\n      unfold mk_context update_context at Hty_Hvar,\n      rw if_neg at Hty_Hvar,\n      tactic.swap, { tauto },\n      apply \u03b3_ih,\n      { apply Henv.1, },\n      { assumption }\n    },\n  },\n  { -- case: lam. This is the tricky case\n    rename [Hty_x \u2192 x, Hty_\u03c41\u2192\u03c41, Hty_\u03c42\u2192\u03c42, Hty_e\u2192e],\n    unfold SN,\n    split,\n    { -- property 1 in notes\n      apply substitution_property,\n      { assumption },\n      { intros, apply sn_implies_closed, assumption },\n      { refl },\n      constructor, assumption\n    },\n    split,\n    { -- property 2 in notes\n      existsi _,\n      unfold evals_to,\n      split,\n      apply is_many_step.many_steps_reflexive,\n      apply env_sub_lam_val,\n    },\n    -- property 3\n    intros e' Hsn,\n    have h := (sn_implies_normalizes _ _ Hsn),\n    cases h with w Heval,\n    cases Heval with Hsteps Hval,\n\n    apply (sn_preservation _ _ \u03c42 _ _),\n    tactic.rotate 2,\n    {\n      constructor,\n      { apply substitution_property,\n        assumption, apply sn_implies_closed,\n        refl, constructor, assumption\n      },\n      cases \u03c41; apply Hsn.1, -- this could be a separate lemma\n    },\n    {\n      apply env_sub_lam_step,\n      { assumption },\n      { assumption }\n    },\n    -- XXX: here, our proof diverges from the notes because we are doing\n    -- call-by-name but the notes are doing call-by-value\n\n    have h: (env_sub ((x,e')::\u03b3) e) = (env_sub \u03b3 (substitute x e' e)),\n    { unfold env_sub },\n    rw <- h,\n    apply Hty_ih _ ((x, \u03c41) :: c),\n    { unfold mk_context },\n    unfold is_env_ctx,\n    split, refl,\n    split, assumption,\n    assumption,\n  },\n  { -- case: ap\n    specialize Hty_ih_Hfunc _ _ _ Henv, trivial,\n    specialize Hty_ih_Hargs _ _ _ Henv, trivial,\n    unfold SN at Hty_ih_Hfunc,\n    rw env_sub_ap,\n    apply Hty_ih_Hfunc.2.2,\n    apply Hty_ih_Hargs\n  }\nend\n\ntheorem strong_normalization :\n  \u2200 e \u03c4, (empty_ctx \u22a2 e : \u03c4) \u2192 normalizes e :=\nbegin\n  introv Hty,\n  have h := (sn_general empty_ctx [] [] e \u03c4 _ Hty _),\n  { apply sn_implies_normalizes, assumption },\n  { unfold mk_context },\n  { unfold is_env_ctx },\nend\n", "meta": {"author": "upamanyus", "repo": "pl-experiments", "sha": "ff4434ae9df0c00f50520eac64b87d5ae42991c1", "save_path": "github-repos/lean/upamanyus-pl-experiments", "path": "github-repos/lean/upamanyus-pl-experiments/pl-experiments-ff4434ae9df0c00f50520eac64b87d5ae42991c1/stlc/src/normalization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.39480866938073506}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Markus Himmel\n-/\nimport category_theory.epi_mono\nimport category_theory.limits.has_limits\n\n/-!\n# Equalizers and coequalizers\n\nThis file defines (co)equalizers as special cases of (co)limits.\n\nAn equalizer is the categorical generalization of the subobject {a \u2208 A | f(a) = g(a)} known\nfrom abelian groups or modules. It is a limit cone over the diagram formed by `f` and `g`.\n\nA coequalizer is the dual concept.\n\n## Main definitions\n\n* `walking_parallel_pair` is the indexing category used for (co)equalizer_diagrams\n* `parallel_pair` is a functor from `walking_parallel_pair` to our category `C`.\n* a `fork` is a cone over a parallel pair.\n  * there is really only one interesting morphism in a fork: the arrow from the vertex of the fork\n    to the domain of f and g. It is called `fork.\u03b9`.\n* an `equalizer` is now just a `limit (parallel_pair f g)`\n\nEach of these has a dual.\n\n## Main statements\n\n* `equalizer.\u03b9_mono` states that every equalizer map is a monomorphism\n* `is_iso_limit_cone_parallel_pair_of_self` states that the identity on the domain of `f` is an\n  equalizer of `f` and `f`.\n\n## Implementation notes\nAs with the other special shapes in the limits library, all the definitions here are given as\n`abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about\ngeneral limits can be used.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 1*][borceux-vol1]\n-/\n\nnoncomputable theory\n\nopen category_theory opposite\n\nnamespace category_theory.limits\n\nlocal attribute [tidy] tactic.case_bash\n\nuniverses v u u\u2082\n\n/-- The type of objects for the diagram indexing a (co)equalizer. -/\n@[derive decidable_eq, derive inhabited] inductive walking_parallel_pair : Type v\n| zero | one\n\nopen walking_parallel_pair\n\n/-- The type family of morphisms for the diagram indexing a (co)equalizer. -/\n@[derive decidable_eq] inductive walking_parallel_pair_hom :\n  walking_parallel_pair \u2192 walking_parallel_pair \u2192 Type v\n| left : walking_parallel_pair_hom zero one\n| right : walking_parallel_pair_hom zero one\n| id : \u03a0 X : walking_parallel_pair.{v}, walking_parallel_pair_hom X X\n\n/-- Satisfying the inhabited linter -/\ninstance : inhabited (walking_parallel_pair_hom zero one) :=\n{ default := walking_parallel_pair_hom.left }\n\nopen walking_parallel_pair_hom\n\n/-- Composition of morphisms in the indexing diagram for (co)equalizers. -/\ndef walking_parallel_pair_hom.comp :\n  \u03a0 (X Y Z : walking_parallel_pair)\n    (f : walking_parallel_pair_hom X Y) (g : walking_parallel_pair_hom Y Z),\n    walking_parallel_pair_hom X Z\n  | _ _ _ (id _) h := h\n  | _ _ _ left   (id one) := left\n  | _ _ _ right  (id one) := right\n.\n\ninstance walking_parallel_pair_hom_category : small_category walking_parallel_pair :=\n{ hom  := walking_parallel_pair_hom,\n  id   := walking_parallel_pair_hom.id,\n  comp := walking_parallel_pair_hom.comp }\n\n@[simp]\nlemma walking_parallel_pair_hom_id (X : walking_parallel_pair) :\n  walking_parallel_pair_hom.id X = \ud835\udfd9 X :=\nrfl\n\n/--\nThe functor `walking_parallel_pair \u2964 walking_parallel_pair\u1d52\u1d56` sending left to left and right to\nright.\n-/\ndef walking_parallel_pair_op : walking_parallel_pair.{u} \u2964 walking_parallel_pair.{u\u2082}\u1d52\u1d56 :=\n{ obj := (\u03bb x, op $ by { cases x, exacts [one, zero] }),\n  map := \u03bb i j f, by { cases f; apply quiver.hom.op, exacts [left, right,\n    walking_parallel_pair_hom.id _] },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_); refl } }\n\n@[simp] lemma walking_parallel_pair_op_zero :\n  walking_parallel_pair_op.obj zero = op one := rfl\n@[simp] lemma walking_parallel_pair_op_one :\n  walking_parallel_pair_op.obj one = op zero := rfl\n@[simp] lemma walking_parallel_pair_op_left :\n  walking_parallel_pair_op.map left = @quiver.hom.op _ _ zero one left := rfl\n@[simp] lemma walking_parallel_pair_op_right :\n  walking_parallel_pair_op.map right = @quiver.hom.op _ _ zero one right := rfl\n\n/--\nThe equivalence `walking_parallel_pair \u2964 walking_parallel_pair\u1d52\u1d56` sending left to left and right to\nright.\n-/\n@[simps functor inverse]\ndef walking_parallel_pair_op_equiv : walking_parallel_pair.{u} \u224c walking_parallel_pair.{u\u2082}\u1d52\u1d56 :=\n{ functor := walking_parallel_pair_op,\n  inverse := walking_parallel_pair_op.left_op,\n  unit_iso := nat_iso.of_components (\u03bb j, eq_to_iso (by { cases j; refl }))\n    (by { rintros (_|_) (_|_) (_|_|_); refl }),\n  counit_iso := nat_iso.of_components (\u03bb j, eq_to_iso\n    (by { induction j using opposite.rec, cases j; refl }))\n    (\u03bb i j f, by { induction i using opposite.rec, induction j using opposite.rec,\n      let g := f.unop, have : f = g.op := rfl, clear_value g, subst this,\n      rcases i with (_|_); rcases j with (_|_); rcases g with (_|_|_); refl }) }\n\n@[simp] lemma walking_parallel_pair_op_equiv_unit_iso_zero :\n  walking_parallel_pair_op_equiv.{u u\u2082}.unit_iso.app zero = iso.refl zero := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_unit_iso_one :\n  walking_parallel_pair_op_equiv.{u u\u2082}.unit_iso.app one = iso.refl one := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_counit_iso_zero :\n  walking_parallel_pair_op_equiv.{u u\u2082}.counit_iso.app (op zero) = iso.refl (op zero) := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_counit_iso_one :\n  walking_parallel_pair_op_equiv.{u u\u2082}.counit_iso.app (op one) = iso.refl (op one) := rfl\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y : C}\n\n/-- `parallel_pair f g` is the diagram in `C` consisting of the two morphisms `f` and `g` with\n    common domain and codomain. -/\ndef parallel_pair (f g : X \u27f6 Y) : walking_parallel_pair.{v} \u2964 C :=\n{ obj := \u03bb x, match x with\n  | zero := X\n  | one := Y\n  end,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, left := f\n  | _, _, right := g\n  end,\n  -- `tidy` can cope with this, but it's too slow:\n  map_comp' := begin rintros (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) \u27e8\u27e9\u27e8\u27e9; { unfold_aux, simp; refl }, end, }.\n\n@[simp] lemma parallel_pair_obj_zero (f g : X \u27f6 Y) : (parallel_pair f g).obj zero = X := rfl\n@[simp] lemma parallel_pair_obj_one (f g : X \u27f6 Y) : (parallel_pair f g).obj one = Y := rfl\n\n@[simp] lemma parallel_pair_map_left (f g : X \u27f6 Y) : (parallel_pair f g).map left = f := rfl\n@[simp] lemma parallel_pair_map_right (f g : X \u27f6 Y) : (parallel_pair f g).map right = g := rfl\n\n@[simp] lemma parallel_pair_functor_obj\n  {F : walking_parallel_pair \u2964 C} (j : walking_parallel_pair) :\n  (parallel_pair (F.map left) (F.map right)).obj j = F.obj j :=\nbegin\n  cases j; refl\nend\n\n/-- Every functor indexing a (co)equalizer is naturally isomorphic (actually, equal) to a\n    `parallel_pair` -/\n@[simps]\ndef diagram_iso_parallel_pair (F : walking_parallel_pair \u2964 C) :\n  F \u2245 parallel_pair (F.map left) (F.map right) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by cases j; tidy) $ by tidy\n\n/-- Construct a morphism between parallel pairs. -/\ndef parallel_pair_hom {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') : parallel_pair f g \u27f6 parallel_pair f' g' :=\n{ app := \u03bb j, match j with\n  | zero := p\n  | one := q\n  end,\n  naturality' := begin\n    rintros (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) \u27e8\u27e9; { unfold_aux, simp [wf, wg], },\n  end }\n\n@[simp] lemma parallel_pair_hom_app_zero\n  {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') :\n  (parallel_pair_hom f g f' g' p q wf wg).app zero = p := rfl\n\n@[simp] lemma parallel_pair_hom_app_one\n  {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') :\n  (parallel_pair_hom f g f' g' p q wf wg).app one = q := rfl\n\n/-- A fork on `f` and `g` is just a `cone (parallel_pair f g)`. -/\nabbreviation fork (f g : X \u27f6 Y) := cone (parallel_pair f g)\n\n/-- A cofork on `f` and `g` is just a `cocone (parallel_pair f g)`. -/\nabbreviation cofork (f g : X \u27f6 Y) := cocone (parallel_pair f g)\n\nvariables {f g : X \u27f6 Y}\n\n/-- A fork `t` on the parallel pair `f g : X \u27f6 Y` consists of two morphisms `t.\u03c0.app zero : t.X \u27f6 X`\n    and `t.\u03c0.app one : t.X \u27f6 Y`. Of these, only the first one is interesting, and we give it the\n    shorter name `fork.\u03b9 t`. -/\nabbreviation fork.\u03b9 (t : fork f g) := t.\u03c0.app zero\n\n/-- A cofork `t` on the parallel_pair `f g : X \u27f6 Y` consists of two morphisms\n    `t.\u03b9.app zero : X \u27f6 t.X` and `t.\u03b9.app one : Y \u27f6 t.X`. Of these, only the second one is\n    interesting, and we give it the shorter name `cofork.\u03c0 t`. -/\nabbreviation cofork.\u03c0 (t : cofork f g) := t.\u03b9.app one\n\n@[simp] lemma fork.\u03b9_eq_app_zero (t : fork f g) : t.\u03b9 = t.\u03c0.app zero := rfl\n@[simp] lemma cofork.\u03c0_eq_app_one (t : cofork f g) : t.\u03c0 = t.\u03b9.app one := rfl\n\n@[simp, reassoc] lemma fork.app_zero_left (s : fork f g) :\n  s.\u03c0.app zero \u226b f = s.\u03c0.app one :=\nby rw [\u2190s.w left, parallel_pair_map_left]\n\n@[simp, reassoc] lemma fork.app_zero_right (s : fork f g) :\n  s.\u03c0.app zero \u226b g = s.\u03c0.app one :=\nby rw [\u2190s.w right, parallel_pair_map_right]\n\n@[simp, reassoc] lemma cofork.left_app_one (s : cofork f g) :\n  f \u226b s.\u03b9.app one = s.\u03b9.app zero :=\nby rw [\u2190s.w left, parallel_pair_map_left]\n\n@[simp, reassoc] lemma cofork.right_app_one (s : cofork f g) :\n  g \u226b s.\u03b9.app one = s.\u03b9.app zero :=\nby rw [\u2190s.w right, parallel_pair_map_right]\n\n/-- A fork on `f g : X \u27f6 Y` is determined by the morphism `\u03b9 : P \u27f6 X` satisfying `\u03b9 \u226b f = \u03b9 \u226b g`.\n-/\n@[simps]\ndef fork.of_\u03b9 {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) : fork f g :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb X, begin cases X, exact \u03b9, exact \u03b9 \u226b f, end,\n    naturality' := \u03bb X Y f,\n    begin\n      cases X; cases Y; cases f; dsimp; simp,\n      { dsimp, simp, }, -- See note [dsimp, simp].\n      { exact w },\n      { dsimp, simp, },\n    end } }\n\n/-- A cofork on `f g : X \u27f6 Y` is determined by the morphism `\u03c0 : Y \u27f6 P` satisfying\n    `f \u226b \u03c0 = g \u226b \u03c0`. -/\n@[simps]\ndef cofork.of_\u03c0 {P : C} (\u03c0 : Y \u27f6 P) (w : f \u226b \u03c0 = g \u226b \u03c0) : cofork f g :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb X, walking_parallel_pair.cases_on X (f \u226b \u03c0) \u03c0,\n    naturality' := \u03bb i j f, by { cases f; dsimp; simp [w] } } } -- See note [dsimp, simp]\n\nlemma fork.\u03b9_of_\u03b9 {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) :\n  (fork.of_\u03b9 \u03b9 w).\u03b9 = \u03b9 := rfl\nlemma cofork.\u03c0_of_\u03c0 {P : C} (\u03c0 : Y \u27f6 P) (w : f \u226b \u03c0 = g \u226b \u03c0) :\n  (cofork.of_\u03c0 \u03c0 w).\u03c0 = \u03c0 := rfl\n\n@[reassoc]\nlemma fork.condition (t : fork f g) : t.\u03b9 \u226b f = t.\u03b9 \u226b g :=\nby rw [t.app_zero_left, t.app_zero_right]\n@[reassoc]\nlemma cofork.condition (t : cofork f g) : f \u226b t.\u03c0 = g \u226b t.\u03c0 :=\nby rw [t.left_app_one, t.right_app_one]\n\n/-- To check whether two maps are equalized by both maps of a fork, it suffices to check it for the\n    first map -/\nlemma fork.equalizer_ext (s : fork f g) {W : C} {k l : W \u27f6 s.X}\n  (h : k \u226b fork.\u03b9 s = l \u226b fork.\u03b9 s) : \u2200 (j : walking_parallel_pair),\n    k \u226b s.\u03c0.app j = l \u226b s.\u03c0.app j\n| zero := h\n| one := by rw [\u2190fork.app_zero_left, reassoc_of h]\n\n/-- To check whether two maps are coequalized by both maps of a cofork, it suffices to check it for\n    the second map -/\n\n\nlemma fork.is_limit.hom_ext {s : fork f g} (hs : is_limit s) {W : C} {k l : W \u27f6 s.X}\n  (h : k \u226b fork.\u03b9 s = l \u226b fork.\u03b9 s) : k = l :=\nhs.hom_ext $ fork.equalizer_ext _ h\n\nlemma cofork.is_colimit.hom_ext {s : cofork f g} (hs : is_colimit s) {W : C} {k l : s.X \u27f6 W}\n  (h : cofork.\u03c0 s \u226b k = cofork.\u03c0 s \u226b l) : k = l :=\nhs.hom_ext $ cofork.coequalizer_ext _ h\n\n/-- If `s` is a limit fork over `f` and `g`, then a morphism `k : W \u27f6 X` satisfying\n    `k \u226b f = k \u226b g` induces a morphism `l : W \u27f6 s.X` such that `l \u226b fork.\u03b9 s = k`. -/\ndef fork.is_limit.lift' {s : fork f g} (hs : is_limit s) {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  {l : W \u27f6 s.X // l \u226b fork.\u03b9 s = k} :=\n\u27e8hs.lift $ fork.of_\u03b9 _ h, hs.fac _ _\u27e9\n\n/-- If `s` is a colimit cofork over `f` and `g`, then a morphism `k : Y \u27f6 W` satisfying\n    `f \u226b k = g \u226b k` induces a morphism `l : s.X \u27f6 W` such that `cofork.\u03c0 s \u226b l = k`. -/\ndef cofork.is_colimit.desc' {s : cofork f g} (hs : is_colimit s) {W : C} (k : Y \u27f6 W)\n  (h : f \u226b k = g \u226b k) : {l : s.X \u27f6 W // cofork.\u03c0 s \u226b l = k} :=\n\u27e8hs.desc $ cofork.of_\u03c0 _ h, hs.fac _ _\u27e9\n\n/-- This is a slightly more convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef fork.is_limit.mk (t : fork f g)\n  (lift : \u03a0 (s : fork f g), s.X \u27f6 t.X)\n  (fac : \u2200 (s : fork f g), lift s \u226b fork.\u03b9 t = fork.\u03b9 s)\n  (uniq : \u2200 (s : fork f g) (m : s.X \u27f6 t.X)\n    (w : \u2200 j : walking_parallel_pair, m \u226b t.\u03c0.app j = s.\u03c0.app j), m = lift s) :\n  is_limit t :=\n{ lift := lift,\n  fac' := \u03bb s j, walking_parallel_pair.cases_on j (fac s) $\n    by erw [\u2190s.w left, \u2190t.w left, \u2190category.assoc, fac]; refl,\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef fork.is_limit.mk' {X Y : C} {f g : X \u27f6 Y} (t : fork f g)\n  (create : \u03a0 (s : fork f g), {l // l \u226b t.\u03b9 = s.\u03b9 \u2227 \u2200 {m}, m \u226b t.\u03b9 = s.\u03b9 \u2192 m = l}) :\nis_limit t :=\nfork.is_limit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w zero))\n\n/-- This is a slightly more convenient method to verify that a cofork is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef cofork.is_colimit.mk (t : cofork f g)\n  (desc : \u03a0 (s : cofork f g), t.X \u27f6 s.X)\n  (fac : \u2200 (s : cofork f g), cofork.\u03c0 t \u226b desc s = cofork.\u03c0 s)\n  (uniq : \u2200 (s : cofork f g) (m : t.X \u27f6 s.X)\n    (w : \u2200 j : walking_parallel_pair, t.\u03b9.app j \u226b m = s.\u03b9.app j), m = desc s) :\n  is_colimit t :=\n{ desc := desc,\n  fac' := \u03bb s j, walking_parallel_pair.cases_on j\n    (by erw [\u2190s.w left, \u2190t.w left, category.assoc, fac]; refl) (fac s),\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef cofork.is_colimit.mk' {X Y : C} {f g : X \u27f6 Y} (t : cofork f g)\n  (create : \u03a0 (s : cofork f g), {l : t.X \u27f6 s.X // t.\u03c0 \u226b l = s.\u03c0 \u2227 \u2200 {m}, t.\u03c0 \u226b m = s.\u03c0 \u2192 m = l}) :\nis_colimit t :=\ncofork.is_colimit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w one))\n\n/--\nGiven a limit cone for the pair `f g : X \u27f6 Y`, for any `Z`, morphisms from `Z` to its point are in\nbijection with morphisms `h : Z \u27f6 X` such that `h \u226b f = h \u226b g`.\nFurther, this bijection is natural in `Z`: see `fork.is_limit.hom_iso_natural`.\nThis is a special case of `is_limit.hom_iso'`, often useful to construct adjunctions.\n-/\n@[simps]\ndef fork.is_limit.hom_iso {X Y : C} {f g : X \u27f6 Y} {t : fork f g} (ht : is_limit t) (Z : C) :\n  (Z \u27f6 t.X) \u2243 {h : Z \u27f6 X // h \u226b f = h \u226b g} :=\n{ to_fun := \u03bb k, \u27e8k \u226b t.\u03b9, by simp\u27e9,\n  inv_fun := \u03bb h, (fork.is_limit.lift' ht _ h.prop).1,\n  left_inv := \u03bb k, fork.is_limit.hom_ext ht (fork.is_limit.lift' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (fork.is_limit.lift' ht _ _).prop }\n\n/-- The bijection of `fork.is_limit.hom_iso` is natural in `Z`. -/\nlemma fork.is_limit.hom_iso_natural {X Y : C} {f g : X \u27f6 Y} {t : fork f g} (ht : is_limit t)\n  {Z Z' : C} (q : Z' \u27f6 Z) (k : Z \u27f6 t.X) :\n  (fork.is_limit.hom_iso ht _ (q \u226b k) : Z' \u27f6 X) = q \u226b (fork.is_limit.hom_iso ht _ k : Z \u27f6 X) :=\ncategory.assoc _ _ _\n\n/--\nGiven a colimit cocone for the pair `f g : X \u27f6 Y`, for any `Z`, morphisms from the cocone point\nto `Z` are in bijection with morphisms `h : Y \u27f6 Z` such that `f \u226b h = g \u226b h`.\nFurther, this bijection is natural in `Z`: see `cofork.is_colimit.hom_iso_natural`.\nThis is a special case of `is_colimit.hom_iso'`, often useful to construct adjunctions.\n-/\n@[simps]\ndef cofork.is_colimit.hom_iso {X Y : C} {f g : X \u27f6 Y} {t : cofork f g} (ht : is_colimit t) (Z : C) :\n  (t.X \u27f6 Z) \u2243 {h : Y \u27f6 Z // f \u226b h = g \u226b h} :=\n{ to_fun := \u03bb k, \u27e8t.\u03c0 \u226b k, by simp\u27e9,\n  inv_fun := \u03bb h, (cofork.is_colimit.desc' ht _ h.prop).1,\n  left_inv := \u03bb k, cofork.is_colimit.hom_ext ht (cofork.is_colimit.desc' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (cofork.is_colimit.desc' ht _ _).prop }\n\n/-- The bijection of `cofork.is_colimit.hom_iso` is natural in `Z`. -/\nlemma cofork.is_colimit.hom_iso_natural {X Y : C} {f g : X \u27f6 Y} {t : cofork f g} {Z Z' : C}\n  (q : Z \u27f6 Z') (ht : is_colimit t) (k : t.X \u27f6 Z) :\n    (cofork.is_colimit.hom_iso ht _ (k \u226b q) : Y \u27f6 Z') =\n    (cofork.is_colimit.hom_iso ht _ k : Y \u27f6 Z) \u226b q :=\n(category.assoc _ _ _).symm\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    equalizers. Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)`, and a fork on `F.map left` and `F.map right`,\n    we get a cone on `F`.\n\n    If you're thinking about using this, have a look at `has_equalizers_of_has_limit_parallel_pair`,\n    which you may find to be an easier way of achieving your goal. -/\ndef cone.of_fork\n  {F : walking_parallel_pair \u2964 C} (t : fork (F.map left) (F.map right)) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g, by { cases j; cases j'; cases g; dsimp; simp } } }\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    coequalizers. Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)`, and a cofork on `F.map left` and `F.map right`,\n    we get a cocone on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_coequalizers_of_has_colimit_parallel_pair`, which you may find to be an easier way of\n    achieving your goal. -/\ndef cocone.of_cofork\n  {F : walking_parallel_pair \u2964 C} (t : cofork (F.map left) (F.map right)) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g, by { cases j; cases j'; cases g; dsimp; simp } } }\n\n@[simp] lemma cone.of_fork_\u03c0\n  {F : walking_parallel_pair \u2964 C} (t : fork (F.map left) (F.map right)) (j) :\n  (cone.of_fork t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\n@[simp] lemma cocone.of_cofork_\u03b9\n  {F : walking_parallel_pair \u2964 C} (t : cofork (F.map left) (F.map right)) (j) :\n  (cocone.of_cofork t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/-- Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)` and a cone on `F`, we get a fork on\n    `F.map left` and `F.map right`. -/\ndef fork.of_cone\n  {F : walking_parallel_pair \u2964 C} (t : cone F) : fork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03c0 := { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy) } }\n\n/-- Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)` and a cocone on `F`, we get a cofork on\n    `F.map left` and `F.map right`. -/\ndef cofork.of_cocone\n  {F : walking_parallel_pair \u2964 C} (t : cocone F) : cofork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03b9 := { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X } }\n\n@[simp] lemma fork.of_cone_\u03c0 {F : walking_parallel_pair \u2964 C} (t : cone F) (j) :\n  (fork.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n@[simp] lemma cofork.of_cocone_\u03b9 {F : walking_parallel_pair \u2964 C} (t : cocone F) (j) :\n  (cofork.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/--\nHelper function for constructing morphisms between equalizer forks.\n-/\n@[simps]\ndef fork.mk_hom {s t : fork f g} (k : s.X \u27f6 t.X) (w : k \u226b t.\u03b9 = s.\u03b9) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { exact w },\n    { simpa using w =\u226b f },\n  end }\n\n/--\nTo construct an isomorphism between forks,\nit suffices to give an isomorphism between the cone points\nand check that it commutes with the `\u03b9` morphisms.\n-/\n@[simps]\ndef fork.ext {s t : fork f g} (i : s.X \u2245 t.X) (w : i.hom \u226b t.\u03b9 = s.\u03b9) : s \u2245 t :=\n{ hom := fork.mk_hom i.hom w,\n  inv := fork.mk_hom i.inv (by rw [\u2190 w, iso.inv_hom_id_assoc]) }\n\n/--\nHelper function for constructing morphisms between coequalizer coforks.\n-/\n@[simps]\ndef cofork.mk_hom {s t : cofork f g} (k : s.X \u27f6 t.X) (w : s.\u03c0 \u226b k = t.\u03c0) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    simpa using f \u226b= w,\n    exact w,\n  end }\n\n/--\nTo construct an isomorphism between coforks,\nit suffices to give an isomorphism between the cocone points\nand check that it commutes with the `\u03c0` morphisms.\n-/\n@[simps]\ndef cofork.ext {s t : cofork f g} (i : s.X \u2245 t.X) (w : s.\u03c0 \u226b i.hom = t.\u03c0) : s \u2245 t :=\n{ hom := cofork.mk_hom i.hom w,\n  inv := cofork.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) }\n\nvariables (f g)\n\nsection\n/--\n`has_equalizer f g` represents a particular choice of limiting cone\nfor the parallel pair of morphisms `f` and `g`.\n-/\nabbreviation has_equalizer := has_limit (parallel_pair f g)\n\nvariables [has_equalizer f g]\n\n/-- If an equalizer of `f` and `g` exists, we can access an arbitrary choice of such by\n    saying `equalizer f g`. -/\nabbreviation equalizer : C := limit (parallel_pair f g)\n\n/-- If an equalizer of `f` and `g` exists, we can access the inclusion\n    `equalizer f g \u27f6 X` by saying `equalizer.\u03b9 f g`. -/\nabbreviation equalizer.\u03b9 : equalizer f g \u27f6 X :=\nlimit.\u03c0 (parallel_pair f g) zero\n\n/--\nAn equalizer cone for a parallel pair `f` and `g`.\n-/\nabbreviation equalizer.fork : fork f g := limit.cone (parallel_pair f g)\n\n@[simp] lemma equalizer.fork_\u03b9 :\n  (equalizer.fork f g).\u03b9 = equalizer.\u03b9 f g := rfl\n\n@[simp] lemma equalizer.fork_\u03c0_app_zero :\n  (equalizer.fork f g).\u03c0.app zero = equalizer.\u03b9 f g := rfl\n\n@[reassoc] lemma equalizer.condition : equalizer.\u03b9 f g \u226b f = equalizer.\u03b9 f g \u226b g :=\nfork.condition $ limit.cone $ parallel_pair f g\n\n/-- The equalizer built from `equalizer.\u03b9 f g` is limiting. -/\ndef equalizer_is_equalizer : is_limit (fork.of_\u03b9 (equalizer.\u03b9 f g) (equalizer.condition f g)) :=\nis_limit.of_iso_limit (limit.is_limit _) (fork.ext (iso.refl _) (by tidy))\n\nvariables {f g}\n\n/-- A morphism `k : W \u27f6 X` satisfying `k \u226b f = k \u226b g` factors through the equalizer of `f` and `g`\n    via `equalizer.lift : W \u27f6 equalizer f g`. -/\nabbreviation equalizer.lift {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) : W \u27f6 equalizer f g :=\nlimit.lift (parallel_pair f g) (fork.of_\u03b9 k h)\n\n@[simp, reassoc]\nlemma equalizer.lift_\u03b9 {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  equalizer.lift k h \u226b equalizer.\u03b9 f g = k :=\nlimit.lift_\u03c0 _ _\n\n/-- A morphism `k : W \u27f6 X` satisfying `k \u226b f = k \u226b g` induces a morphism `l : W \u27f6 equalizer f g`\n    satisfying `l \u226b equalizer.\u03b9 f g = k`. -/\ndef equalizer.lift' {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  {l : W \u27f6 equalizer f g // l \u226b equalizer.\u03b9 f g = k} :=\n\u27e8equalizer.lift k h, equalizer.lift_\u03b9 _ _\u27e9\n\n/-- Two maps into an equalizer are equal if they are are equal when composed with the equalizer\n    map. -/\n@[ext] lemma equalizer.hom_ext {W : C} {k l : W \u27f6 equalizer f g}\n  (h : k \u226b equalizer.\u03b9 f g = l \u226b equalizer.\u03b9 f g) : k = l :=\nfork.is_limit.hom_ext (limit.is_limit _) h\n\n/-- An equalizer morphism is a monomorphism -/\ninstance equalizer.\u03b9_mono : mono (equalizer.\u03b9 f g) :=\n{ right_cancellation := \u03bb Z h k w, equalizer.hom_ext w }\n\nend\n\nsection\nvariables {f g}\n/-- The equalizer morphism in any limit cone is a monomorphism. -/\nlemma mono_of_is_limit_parallel_pair {c : cone (parallel_pair f g)} (i : is_limit c) :\n  mono (fork.\u03b9 c) :=\n{ right_cancellation := \u03bb Z h k w, fork.is_limit.hom_ext i w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The identity determines a cone on the equalizer diagram of `f` and `g` if `f = g`. -/\ndef id_fork (h : f = g) : fork f g :=\nfork.of_\u03b9 (\ud835\udfd9 X) $ h \u25b8 rfl\n\n/-- The identity on `X` is an equalizer of `(f, g)`, if `f = g`. -/\ndef is_limit_id_fork (h : f = g) : is_limit (id_fork h) :=\nfork.is_limit.mk _\n  (\u03bb s, fork.\u03b9 s)\n  (\u03bb s, category.comp_id _)\n  (\u03bb s m h, by { convert h zero, exact (category.comp_id _).symm })\n\n/-- Every equalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_eq (h\u2080 : f = g) {c : cone (parallel_pair f g)}\n  (h : is_limit c) : is_iso (c.\u03c0.app zero) :=\nis_iso.of_iso $ is_limit.cone_point_unique_up_to_iso h $ is_limit_id_fork h\u2080\n\n/-- The equalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma equalizer.\u03b9_of_eq [has_equalizer f g] (h : f = g) : is_iso (equalizer.\u03b9 f g) :=\nis_iso_limit_cone_parallel_pair_of_eq h $ limit.is_limit _\n\n/-- Every equalizer of `(f, f)` is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_self {c : cone (parallel_pair f f)} (h : is_limit c) :\n  is_iso (c.\u03c0.app zero) :=\nis_iso_limit_cone_parallel_pair_of_eq rfl h\n\n/-- An equalizer that is an epimorphism is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_epi {c : cone (parallel_pair f g)}\n  (h : is_limit c) [epi (c.\u03c0.app zero)] : is_iso (c.\u03c0.app zero) :=\nis_iso_limit_cone_parallel_pair_of_eq ((cancel_epi _).1 (fork.condition c)) h\n\nend\n\ninstance has_equalizer_of_self : has_equalizer f f :=\nhas_limit.mk\n{ cone := id_fork rfl,\n  is_limit := is_limit_id_fork rfl }\n\n/-- The equalizer inclusion for `(f, f)` is an isomorphism. -/\ninstance equalizer.\u03b9_of_self : is_iso (equalizer.\u03b9 f f) :=\nequalizer.\u03b9_of_eq rfl\n\n/-- The equalizer of a morphism with itself is isomorphic to the source. -/\ndef equalizer.iso_source_of_self : equalizer f f \u2245 X :=\nas_iso (equalizer.\u03b9 f f)\n\n@[simp] lemma equalizer.iso_source_of_self_hom :\n  (equalizer.iso_source_of_self f).hom = equalizer.\u03b9 f f :=\nrfl\n\n@[simp] lemma equalizer.iso_source_of_self_inv :\n  (equalizer.iso_source_of_self f).inv = equalizer.lift (\ud835\udfd9 X) (by simp) :=\nby { ext, simp [equalizer.iso_source_of_self], }\n\nsection\n/--\n`has_coequalizer f g` represents a particular choice of colimiting cocone\nfor the parallel pair of morphisms `f` and `g`.\n-/\nabbreviation has_coequalizer := has_colimit (parallel_pair f g)\n\nvariables [has_coequalizer f g]\n\n/-- If a coequalizer of `f` and `g` exists, we can access an arbitrary choice of such by\n    saying `coequalizer f g`. -/\nabbreviation coequalizer : C := colimit (parallel_pair f g)\n\n/--  If a coequalizer of `f` and `g` exists, we can access the corresponding projection by\n    saying `coequalizer.\u03c0 f g`. -/\nabbreviation coequalizer.\u03c0 : Y \u27f6 coequalizer f g :=\ncolimit.\u03b9 (parallel_pair f g) one\n\n/--\nAn arbitrary choice of coequalizer cocone for a parallel pair `f` and `g`.\n-/\nabbreviation coequalizer.cofork : cofork f g := colimit.cocone (parallel_pair f g)\n\n@[simp] lemma coequalizer.cofork_\u03c0 :\n  (coequalizer.cofork f g).\u03c0 = coequalizer.\u03c0 f g := rfl\n\n@[simp] lemma coequalizer.cofork_\u03b9_app_one :\n  (coequalizer.cofork f g).\u03b9.app one = coequalizer.\u03c0 f g := rfl\n\n@[reassoc] lemma coequalizer.condition : f \u226b coequalizer.\u03c0 f g = g \u226b coequalizer.\u03c0 f g :=\ncofork.condition $ colimit.cocone $ parallel_pair f g\n\n/-- The cofork built from `coequalizer.\u03c0 f g` is colimiting. -/\ndef coequalizer_is_coequalizer :\n  is_colimit (cofork.of_\u03c0 (coequalizer.\u03c0 f g) (coequalizer.condition f g)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit _) (cofork.ext (iso.refl _) (by tidy))\n\nvariables {f g}\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `f \u226b k = g \u226b k` factors through the coequalizer of `f`\n    and `g` via `coequalizer.desc : coequalizer f g \u27f6 W`. -/\nabbreviation coequalizer.desc {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) : coequalizer f g \u27f6 W :=\ncolimit.desc (parallel_pair f g) (cofork.of_\u03c0 k h)\n\n@[simp, reassoc]\nlemma coequalizer.\u03c0_desc {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) :\n  coequalizer.\u03c0 f g \u226b coequalizer.desc k h = k :=\ncolimit.\u03b9_desc _ _\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `f \u226b k = g \u226b k` induces a morphism\n    `l : coequalizer f g \u27f6 W` satisfying `coequalizer.\u03c0 \u226b g = l`. -/\ndef coequalizer.desc' {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) :\n  {l : coequalizer f g \u27f6 W // coequalizer.\u03c0 f g \u226b l = k} :=\n\u27e8coequalizer.desc k h, coequalizer.\u03c0_desc _ _\u27e9\n\n/-- Two maps from a coequalizer are equal if they are equal when composed with the coequalizer\n    map -/\n@[ext] lemma coequalizer.hom_ext {W : C} {k l : coequalizer f g \u27f6 W}\n  (h : coequalizer.\u03c0 f g \u226b k = coequalizer.\u03c0 f g \u226b l) : k = l :=\ncofork.is_colimit.hom_ext (colimit.is_colimit _) h\n\n/-- A coequalizer morphism is an epimorphism -/\ninstance coequalizer.\u03c0_epi : epi (coequalizer.\u03c0 f g) :=\n{ left_cancellation := \u03bb Z h k w, coequalizer.hom_ext w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The coequalizer morphism in any colimit cocone is an epimorphism. -/\nlemma epi_of_is_colimit_parallel_pair {c : cocone (parallel_pair f g)} (i : is_colimit c) :\n  epi (c.\u03b9.app one) :=\n{ left_cancellation := \u03bb Z h k w, cofork.is_colimit.hom_ext i w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The identity determines a cocone on the coequalizer diagram of `f` and `g`, if `f = g`. -/\ndef id_cofork (h : f = g) : cofork f g :=\ncofork.of_\u03c0 (\ud835\udfd9 Y) $ h \u25b8 rfl\n\n/-- The identity on `Y` is a coequalizer of `(f, g)`, where `f = g`.  -/\ndef is_colimit_id_cofork (h : f = g) : is_colimit (id_cofork h) :=\ncofork.is_colimit.mk _\n  (\u03bb s, cofork.\u03c0 s)\n  (\u03bb s, category.id_comp _)\n  (\u03bb s m h, by { convert h one, exact (category.id_comp _).symm })\n\n/-- Every coequalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma is_iso_colimit_cocone_parallel_pair_of_eq (h\u2080 : f = g) {c : cocone (parallel_pair f g)}\n  (h : is_colimit c) : is_iso (c.\u03b9.app one) :=\nis_iso.of_iso $ is_colimit.cocone_point_unique_up_to_iso (is_colimit_id_cofork h\u2080) h\n\n/-- The coequalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma coequalizer.\u03c0_of_eq [has_coequalizer f g] (h : f = g) :\n  is_iso (coequalizer.\u03c0 f g) :=\nis_iso_colimit_cocone_parallel_pair_of_eq h $ colimit.is_colimit _\n\n/-- Every coequalizer of `(f, f)` is an isomorphism. -/\nlemma is_iso_colimit_cocone_parallel_pair_of_self {c : cocone (parallel_pair f f)}\n  (h : is_colimit c) : is_iso (c.\u03b9.app one) :=\nis_iso_colimit_cocone_parallel_pair_of_eq rfl h\n\n/-- A coequalizer that is a monomorphism is an isomorphism. -/\nlemma is_iso_limit_cocone_parallel_pair_of_epi {c : cocone (parallel_pair f g)}\n  (h : is_colimit c) [mono (c.\u03b9.app one)] : is_iso (c.\u03b9.app one) :=\nis_iso_colimit_cocone_parallel_pair_of_eq ((cancel_mono _).1 (cofork.condition c)) h\n\nend\n\ninstance has_coequalizer_of_self : has_coequalizer f f :=\nhas_colimit.mk\n{ cocone := id_cofork rfl,\n  is_colimit := is_colimit_id_cofork rfl }\n\n/-- The coequalizer projection for `(f, f)` is an isomorphism. -/\ninstance coequalizer.\u03c0_of_self : is_iso (coequalizer.\u03c0 f f) :=\ncoequalizer.\u03c0_of_eq rfl\n\n/-- The coequalizer of a morphism with itself is isomorphic to the target. -/\ndef coequalizer.iso_target_of_self : coequalizer f f \u2245 Y :=\n(as_iso (coequalizer.\u03c0 f f)).symm\n\n@[simp] lemma coequalizer.iso_target_of_self_hom :\n  (coequalizer.iso_target_of_self f).hom = coequalizer.desc (\ud835\udfd9 Y) (by simp) :=\nby { ext, simp [coequalizer.iso_target_of_self], }\n\n@[simp] lemma coequalizer.iso_target_of_self_inv :\n  (coequalizer.iso_target_of_self f).inv = coequalizer.\u03c0 f f :=\nrfl\n\nsection comparison\n\nvariables {D : Type u\u2082} [category.{v} D] (G : C \u2964 D)\n\n/--\nThe comparison morphism for the equalizer of `f,g`.\nThis is an isomorphism iff `G` preserves the equalizer of `f,g`; see\n`category_theory/limits/preserves/shapes/equalizers.lean`\n-/\ndef equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :\n  G.obj (equalizer f g) \u27f6 equalizer (G.map f) (G.map g) :=\nequalizer.lift (G.map (equalizer.\u03b9 _ _)) (by simp only [\u2190G.map_comp, equalizer.condition])\n\n@[simp, reassoc]\nlemma equalizer_comparison_comp_\u03c0 [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :\n  equalizer_comparison f g G \u226b equalizer.\u03b9 (G.map f) (G.map g) = G.map (equalizer.\u03b9 f g) :=\nequalizer.lift_\u03b9 _ _\n\n@[simp, reassoc]\nlemma map_lift_equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)]\n  {Z : C} {h : Z \u27f6 X} (w : h \u226b f = h \u226b g) :\n    G.map (equalizer.lift h w) \u226b equalizer_comparison f g G =\n      equalizer.lift (G.map h) (by simp only [\u2190G.map_comp, w]) :=\nby { ext, simp [\u2190 G.map_comp] }\n\n/-- The comparison morphism for the coequalizer of `f,g`. -/\ndef coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :\n  coequalizer (G.map f) (G.map g) \u27f6 G.obj (coequalizer f g) :=\ncoequalizer.desc (G.map (coequalizer.\u03c0 _ _)) (by simp only [\u2190G.map_comp, coequalizer.condition])\n\n@[simp, reassoc]\nlemma \u03b9_comp_coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :\n  coequalizer.\u03c0 _ _ \u226b coequalizer_comparison f g G = G.map (coequalizer.\u03c0 _ _) :=\ncoequalizer.\u03c0_desc _ _\n\n@[simp, reassoc]\nlemma coequalizer_comparison_map_desc [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)]\n  {Z : C} {h : Y \u27f6 Z} (w : f \u226b h = g \u226b h) :\n  coequalizer_comparison f g G \u226b G.map (coequalizer.desc h w) =\n    coequalizer.desc (G.map h) (by simp only [\u2190G.map_comp, w]) :=\nby { ext, simp [\u2190 G.map_comp] }\n\nend comparison\n\nvariables (C)\n\n/-- `has_equalizers` represents a choice of equalizer for every pair of morphisms -/\nabbreviation has_equalizers := has_limits_of_shape walking_parallel_pair.{v} C\n\n/-- `has_coequalizers` represents a choice of coequalizer for every pair of morphisms -/\nabbreviation has_coequalizers := has_colimits_of_shape walking_parallel_pair.{v} C\n\n/-- If `C` has all limits of diagrams `parallel_pair f g`, then it has all equalizers -/\nlemma has_equalizers_of_has_limit_parallel_pair\n  [\u03a0 {X Y : C} {f g : X \u27f6 Y}, has_limit (parallel_pair f g)] : has_equalizers C :=\n{ has_limit := \u03bb F, has_limit_of_iso (diagram_iso_parallel_pair F).symm }\n\n/-- If `C` has all colimits of diagrams `parallel_pair f g`, then it has all coequalizers -/\nlemma has_coequalizers_of_has_colimit_parallel_pair\n  [\u03a0 {X Y : C} {f g : X \u27f6 Y}, has_colimit (parallel_pair f g)] : has_coequalizers C :=\n{ has_colimit := \u03bb F, has_colimit_of_iso (diagram_iso_parallel_pair F) }\n\n\nsection\n-- In this section we show that a split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\nvariables {C} [split_mono f]\n\n/--\nA split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\nHere we build the cone, and show in `split_mono_equalizes` that it is a limit cone.\n-/\n@[simps {rhs_md := semireducible}]\ndef cone_of_split_mono : cone (parallel_pair (\ud835\udfd9 Y) (retraction f \u226b f)) :=\nfork.of_\u03b9 f (by simp)\n\n\n/--\nA split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\n-/\ndef split_mono_equalizes {X Y : C} (f : X \u27f6 Y) [split_mono f] : is_limit (cone_of_split_mono f) :=\nfork.is_limit.mk' _ $ \u03bb s,\n\u27e8s.\u03b9 \u226b retraction f,\n by { dsimp, rw [category.assoc, \u2190s.condition], apply category.comp_id },\n \u03bb m hm, by simp [\u2190hm]\u27e9\n\nend\n\nsection\n-- In this section we show that a split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\nvariables {C} [split_epi f]\n\n/--\nA split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\nHere we build the cocone, and show in `split_epi_coequalizes` that it is a colimit cocone.\n-/\n@[simps {rhs_md := semireducible}]\ndef cocone_of_split_epi : cocone (parallel_pair (\ud835\udfd9 X) (f \u226b section_ f)) :=\ncofork.of_\u03c0 f (by simp)\n\n/--\nA split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\n-/\ndef split_epi_coequalizes {X Y : C} (f : X \u27f6 Y) [split_epi f] :\n  is_colimit (cocone_of_split_epi f) :=\ncofork.is_colimit.mk' _ $ \u03bb s,\n\u27e8section_ f \u226b s.\u03c0,\n by { dsimp, rw [\u2190 category.assoc, \u2190 s.condition, category.id_comp] },\n \u03bb m hm, by simp [\u2190 hm]\u27e9\n\nend\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/shapes/equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.39479274342774107}}
{"text": "import QL.FOL.fol provability consistency\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nopen subterm subformula logic logic.Theory\nvariables {L : language.{u}} {m : \u2115}\n\nlocalized \"prefix (name := mlift) `\ud835\udddf`:max := fol.subformula.mlift\" in aclogic\nlocalized \"prefix (name := bounded_preTheory.mlift) `\ud835\udddf'`:max := fol.bounded_preTheory.mlift\" in aclogic\nlocalized \"prefix (name := push) `\ud835\udde0`:max := fol.subformula.push\" in aclogic\nlocalized \"prefix (name := pull) `\ud835\udde1`:max := fol.subformula.pull\" in aclogic\nlocalized \"prefix (name := dummy) `\ud835\uddd7`:max := fol.subformula.dummy\" in aclogic\n\ninductive proof : \u03a0 {m}, bounded_preTheory L m \u2192 bounded_subformula L m 0 \u2192 Type u\n| generalize   {m} {T : bounded_preTheory L m} : \u2200 {p}, proof T.mlift p \u2192 proof T (\u2200'\ud835\udde1p)\n| mdp          {m} {T : bounded_preTheory L m} : \u2200 {p q}, proof T (p \u27f6 q) \u2192 proof T p \u2192 proof T q\n| by_axiom     {m} {T : bounded_preTheory L m} : \u2200 {p}, p \u2208 T \u2192 proof T p\n| verum        {m} {T : bounded_preTheory L m} : proof T \u22a4\n| imply\u2081       {m} {T : bounded_preTheory L m} : \u2200 {p q}, proof T (p \u27f6 q \u27f6 p)\n| imply\u2082       {m} {T : bounded_preTheory L m} : \u2200 {p q r}, proof T ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r)\n| contra       {m} {T : bounded_preTheory L m} : \u2200 {p q}, proof T ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p)\n| specialize   {m} {T : bounded_preTheory L m} : \u2200 {p} {t}, proof T (\u2200'p \u27f6 subst t p)\n| dummy_univ   {m} {T : bounded_preTheory L m} : \u2200 {p q}, proof T (\u2200'(dummy p \u27f6 q) \u27f6 p \u27f6 \u2200'q)\n\ninstance : has_Longarrow (bounded_formula L m) := \u27e8proof\u27e9\n\ndef provable (T : bounded_preTheory L m) (p : bounded_formula L m) : Prop := nonempty (T \u27f9 p)\n\ninstance : axiomatic_classical_logic' (bounded_formula L m) :=\n{ turnstile := @provable _ m,\n  classical := \u03bb T,\n  { modus_ponens := \u03bb p q \u27e8bpq\u27e9 \u27e8bp\u27e9, \u27e8bpq.mdp bp\u27e9,\n    imply\u2081 := \u03bb p q, \u27e8proof.imply\u2081\u27e9, \n    imply\u2082 := \u03bb p q r, \u27e8proof.imply\u2082\u27e9,\n    contraposition := \u03bb p q, \u27e8proof.contra\u27e9,\n    provable_top := \u27e8proof.verum\u27e9,\n    bot_eq := by refl,\n    and_def := \u03bb p q, rfl,\n    or_def := \u03bb p q, rfl },\n  by_axiom := \u03bb T p mem, \u27e8proof.by_axiom mem\u27e9 }\n\nopen_locale aclogic\n\nnamespace proof\nvariables {T : bounded_preTheory L m}\n\ndef weakening' {p} (h : T \u27f9 p) : \u2200 {U}, T \u2286 U \u2192 U \u27f9 p :=\nbegin\n  induction h,\n  case generalize : m T p hyp_p IH\n  { intros U hyp, refine generalize (IH $ set.image_subset _ hyp) },\n  case mdp : m T p q hyp_pq hyp_p IH\u2081 IH\u2082\n  { intros U hyp, exact (IH\u2081 hyp).mdp (IH\u2082 hyp) },\n  case by_axiom : m T p hyp_p\n  { intros U hyp, exact by_axiom (hyp hyp_p) },\n  { intros U ss, exact verum },\n  { intros U ss, exact imply\u2081 },\n  { intros U ss, exact imply\u2082 },\n  { intros U ss, exact contra },\n  { intros U ss, exact specialize },\n  { intros U ss, exact dummy_univ },\nend\n\nend proof\n\nnamespace provable\nopen axiomatic_classical_logic' axiomatic_classical_logic\nvariables {T U : bounded_preTheory L m}\n\nlemma generalize {p} (h : T.mlift \u22a2 p) : T \u22a2 \u2200'p.pull := by rcases h; exact \u27e8h.generalize\u27e9\n\nlemma generalize' {T : bounded_preTheory L (m + 1)} {p} (h : T \u22a2 p) (hT : T = U.mlift) : U \u22a2 \u2200'p.pull :=\nby rcases hT with rfl; exact generalize h\n\nlemma gen {p : bounded_subformula L m 1} (h : T.mlift \u22a2 p.push) : T \u22a2 \u2200'p :=\nby rw[\u2190subformula.pull_push p]; exact generalize h\n\nlemma by_axiom {p} (h : p \u2208 T) : T \u22a2 p := \u27e8proof.by_axiom h\u27e9\n\nvariables (T)\n\n@[simp] lemma specialize (p) (t) : T \u22a2 \u2200'p \u27f6 subst t p := \u27e8proof.specialize\u27e9\n\nvariables {T}\n\nlemma forall_subst {p} (h : T \u22a2 \u2200'p) (t) : T \u22a2 subst t p :=\nspecialize T p t \u2a00 h\n\ninfix ` \u229a `:60 := forall_subst\n\nvariables (T)\n\n@[simp] lemma dummy_univ (p q) : T \u22a2 \u2200'(dummy p \u27f6 q) \u27f6 p \u27f6 \u2200'q := \u27e8proof.dummy_univ\u27e9\n\nvariables {T U}\n\n@[elab_as_eliminator]\ntheorem rec_on {C : \u03a0 {m} (T : bounded_preTheory L m) (p : bounded_subformula L m 0), T \u22a2 p \u2192 Prop}\n  {m : \u2115} {T : bounded_preTheory L m} {p : bounded_formula L m} (b : T \u22a2 p)\n  (generalize : \u2200 {m} {T : bounded_preTheory L m} {p} (b : T.mlift \u22a2 p), C T.mlift p b \u2192 C T (\u2200'p.pull) (generalize b))\n  (mdp : \u2200 {m} {T : bounded_preTheory L m} {p q} (b\u2081 : T \u22a2 p \u27f6 q) (b\u2082 : T \u22a2 p), C T (p \u27f6 q) b\u2081 \u2192 C T p b\u2082 \u2192 C T q (b\u2081 \u2a00 b\u2082))\n  (by_axiom : \u2200 {m} {T : bounded_preTheory L m} {p} (h : p \u2208 T), C T p (by_axiom h))\n  (verum : \u2200 {m} {T : bounded_preTheory L m}, C T \u22a4 axiomatic_classical_logic'.provable_top)\n  (imply\u2081 : \u2200 {m} {T : bounded_preTheory L m} {p q}, C T (p \u27f6 q \u27f6 p) (axiomatic_classical_logic'.imply\u2081 p q))\n  (imply\u2082 : \u2200 {m} {T : bounded_preTheory L m} {p q r}, C T ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r) (axiomatic_classical_logic'.imply\u2082 p q r))\n  (contra : \u2200 {m} {T : bounded_preTheory L m} {p q}, C T ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p) (axiomatic_classical_logic'.contraposition p q)) \n  (specialize : \u2200 {m} {T : bounded_preTheory L m} {p} {t}, C T (\u2200'p \u27f6 subst t p) (specialize T p t))\n  (dummy_univ : \u2200 {m} {T : bounded_preTheory L m} {p q}, C T (\u2200'(dummy p \u27f6 q) \u27f6 p \u27f6 \u2200'q) (dummy_univ T p q)) :\n  C T p b :=\nbegin\n  rcases b with \u27e8b\u27e9,\n  induction b,\n  case generalize : m T p b IH { exact generalize \u27e8b\u27e9 IH },\n  case mdp : m T p q b\u2081 b\u2082 IH\u2081 IH\u2082 { exact mdp \u27e8b\u2081\u27e9 \u27e8b\u2082\u27e9 IH\u2081 IH\u2082 },\n  case by_axiom : m T p hp { exact by_axiom hp },\n  case verum : m T { exact verum },\n  case imply\u2081 : m T p q { exact imply\u2081 },\n  case imply\u2082 : m T p q r { exact imply\u2082 },\n  case contra : m T p q { exact contra },\n  case specialize : m T p t { exact specialize },\n  case dummy_univ : m T p q { exact dummy_univ }\nend\n\nnoncomputable def provable.proof {T : bounded_preTheory L m} {p : bounded_formula L m} (b : T \u22a2 p) : T \u27f9 p := nonempty.some b\n\ndef weakening_aux {p} (h : T \u22a2 p) : \u2200 {U}, T \u2286 U \u2192 U \u22a2 p :=\nbegin\n  apply rec_on h,\n  { intros m T p b IH U hyp, refine generalize (IH $ set.image_subset _ hyp) },\n  { intros m T p q hyp_pq hyp_p IH\u2081 IH\u2082 U hyp, exact (IH\u2081 hyp) \u2a00 (IH\u2082 hyp) },\n  { intros m T p hyp_p U hyp, exact by_axiom (hyp hyp_p) },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp }\nend\n\nlemma deduction_aux {q} (h : T \u22a2 q) : \u2200 (U) (p) (hT : T = insert p U), U \u22a2 p \u27f6 q :=\nbegin\n  apply rec_on h,\n  { rintros m T q b IH U p rfl,\n    have : U.mlift \u22a2 p.mlift \u27f6 q, from IH U.mlift p.mlift (by simp[bounded_preTheory.mlift_insert]),\n    have IH : U \u22a2 \u2200'(p.dummy \u27f6 q.pull), from generalize this,\n    have : U \u22a2 \u2200'(p.dummy \u27f6 q.pull) \u27f6 p \u27f6 \u2200'q.pull, by simp,\n    show U \u22a2 p \u27f6 \u2200'q.pull, from this \u2a00 IH },\n  { rintros m T p\u2081 p\u2082 b\u2081 b\u2082 IH\u2081 IH\u2082 U p rfl,\n    show U \u22a2 p \u27f6 p\u2082, from (IH\u2081 U p rfl) \u2a00\u2081 (IH\u2082 U p rfl) },\n  { rintros m T p hp U r rfl, rcases hp with (rfl | hp),\n    { simp }, { have : U \u22a2 p, from by_axiom hp, exact hyp_right this r } },\n  { rintros m T U p rfl, simp },\n  { rintros m T p q U r rfl, simp },\n  { rintros m T p q r U s rfl, simp },\n  { rintros m T p q U r rfl, simp },\n  { rintros m T p t U q rfl, refine hyp_right (specialize _ p t) _ },\n  { rintros m T p q U r rfl, refine hyp_right (dummy_univ _ p q) _ }\nend\n\ninstance : axiomatic_classical_logic (bounded_formula L m) :=\n{ deduction' := \u03bb T p q h, deduction_aux h T p rfl,\n  weakening := \u03bb T U p ss b, weakening_aux b ss }\n\nlemma empty_axiom_generalize {p : bounded_formula L (m + 1)} (hp : \u2b1d\u22a2 p) : \u2b1d\u22a2 \u2200'p.pull :=\nby { have : bounded_preTheory.mlift \u2205 \u22a2 p, by simpa[bounded_preTheory.mlift] using hp, exact generalize this }\n\nprivate lemma mlift_list_conjunction (P\u2080 : list (bounded_formula L $ m + 1)) : (\u2200 p, p \u2208 P\u2080 \u2192 p \u2208 T.mlift) \u2192\n  \u2203 P : list (bounded_formula L m), P.conjunction.mlift = P\u2080.conjunction \u2227 (\u2200 p, p \u2208 P \u2192 p \u2208 T) :=\nbegin\n  induction P\u2080 with p\u2080 P\u2080 IH,\n  { intros _, refine \u27e8[], by simp\u27e9 },\n  { intros h,\n    have : \u2203 P : list (bounded_formula L m), P.conjunction.mlift = P\u2080.conjunction \u2227 (\u2200 p, p \u2208 P \u2192 p \u2208 T),\n    from IH (\u03bb p hp, h p (by simp[hp])),\n    rcases this with \u27e8P, eq, hP\u27e9,\n    have : p\u2080 \u2208 T.mlift, from h p\u2080 (by simp), rcases this with \u27e8p, hp, rfl\u27e9,\n    refine \u27e8p :: P, by simpa using eq, by { rintros q (rfl | hq), { exact hp }, { exact hP q hq } }\u27e9 }\nend\n\ntheorem finite_character_aux {m} {T : bounded_preTheory L m} {p} :\n  T \u22a2 p \u2192 \u2203 P : list (bounded_formula L m), (\u2200 p, p \u2208 P \u2192 p \u2208 T) \u2227 \u2b1d\u22a2 P.conjunction \u27f6 p := \u03bb h,\nbegin\n  apply rec_on h,\n  { rintros m T p b \u27e8P\u2080, IH, IHb\u27e9,\n    have : \u2203 P : list (bounded_formula L m), P.conjunction.mlift = P\u2080.conjunction \u2227 \u2200 p, p \u2208 P \u2192 p \u2208 T,\n    from mlift_list_conjunction P\u2080 IH,\n    rcases this with \u27e8P, eqP, hP\u27e9,\n    refine \u27e8P, hP, _\u27e9,\n    have : \u2b1d\u22a2 \u2200'(P.conjunction.dummy \u27f6 p.pull),\n    { have := empty_axiom_generalize IHb, rw[\u2190eqP] at this; exact this },\n    exact dummy_univ _ P.conjunction p.pull \u2a00 this },\n  { rintros m T p q b\u2081 b\u2082 \u27e8P\u2081, IH\u2081, IHb\u2081\u27e9 \u27e8P\u2082, IH\u2082, IHb\u2082\u27e9,\n    refine \u27e8P\u2081 ++ P\u2082, _, _\u27e9,\n    { simp, rintros p (hp | hp), { exact IH\u2081 p hp }, { exact IH\u2082 p hp } },\n    { have    : \u2b1d\u22a2 (P\u2081 ++ P\u2082).conjunction \u27f6 P\u2081.conjunction, from list_conjunction_weakening (by simp),\n      have h\u2081 : \u2b1d\u22a2 (P\u2081 ++ P\u2082).conjunction \u27f6 p \u27f6 q, from imply_trans this IHb\u2081,\n      have    : \u2b1d\u22a2 (P\u2081 ++ P\u2082).conjunction \u27f6 P\u2082.conjunction, from list_conjunction_weakening (by simp),\n      have h\u2082 : \u2b1d\u22a2 (P\u2081 ++ P\u2082).conjunction \u27f6 p, from imply_trans this IHb\u2082,\n      exact h\u2081 \u2a00\u2081 h\u2082 } },\n  { rintros m T p hp, refine \u27e8[p], by simp[hp], _\u27e9, simp, refine deduction.mp (by simp) },\n  { rintros m T, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 },\n  { rintros m T p q, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 },\n  { rintros m T p q r, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 },\n  { rintros m T p q, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 },\n  { rintros m T p t, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 },\n  { rintros m T p q, refine \u27e8[], by simp, by simp[empty_axiom]\u27e9 }\nend\n\ninstance : has_finite_character (bounded_formula L m) :=\nfinite_character_of_finite_provable (bounded_formula L m) (\u03bb T p, finite_character_aux)\n\ndef le_of (h : \u2200 p \u2208 T, U \u22a2 p) : T \u2264 U :=\nbegin\n  intros p b,\n  rcases finite_character_aux b with \u27e8P, hP, b\u27e9,\n  have : U \u22a2 P.conjunction, from list_conjunction_provable (\u03bb p hp, h _ (hP p hp)),\n  exact of_empty_axiom _ b \u2a00 this\nend\n\nlemma exists_of_subst (p : bounded_subformula L m 1) (t) : T \u22a2 subst t p \u27f6 \u2203'p :=\ncontrapose.mp (imply_of_equiv\n  (show T \u22a2 p.neg.fal \u27f6 \u223csubst t p, by simpa using specialize T (\u223cp) t)\n  (iff_dn_refl_right $ \u2200'\u223cp) (equiv_refl _))\n\nlemma specialize' {T} (p : bounded_subformula L m 1) : T \u22a2 \u2200' \ud835\udddf p \u27f6 \ud835\udde0 p :=\nby { have : T \u22a2 \u2200' \ud835\udddf p \u27f6 subst &(fin.last _) p.mlift, from specialize T p.mlift &(fin.last _), simpa using this }\n\nlemma use {p : bounded_subformula L m 1} (t) (h : T \u22a2 subst t p) : T \u22a2 \u2203'p :=\nexists_of_subst p t \u2a00 h\n\n@[simp] lemma forall_top : T \u22a2 \u2200'\u22a4 :=\ngen (by simp)\n\nlemma forallK (p q) : T \u22a2 \u2200'(p \u27f6 q) \u27f6 \u2200'p \u27f6 \u2200'q :=\nbegin\n  have lmm\u2081 : T \u22a2 \u2200'(p \u27f6 q) \u27f6 \u2200'(\ud835\uddd7 (\u2200' p) \u27f6 q),\n  { have : \ud835\udddf'T +{ \u2200'(\ud835\udddf p \u27f6 \ud835\udddf q) } \u22a2 \ud835\udde0 p \u27f6 \ud835\udde0 q, from deduction.mpr (by simpa using specialize' (p \u27f6 q)),\n    have : \ud835\udddf'T +{ \u2200'(\ud835\udddf p \u27f6 \ud835\udddf q) } \u22a2 \u2200'\ud835\udddf p \u27f6 \ud835\udde0 q, from imply_trans (specialize' _) this, \n    refine deduction.mp (gen _), simp[bounded_preTheory.mlift_insert], exact this },\n  have lmm\u2082 : T \u22a2 \u2200'(\ud835\uddd7 (\u2200'p) \u27f6 q) \u27f6 \u2200'p \u27f6 \u2200'q, from dummy_univ T (\u2200'p) q,\n  exact imply_trans lmm\u2081 lmm\u2082\nend\n\nlemma forall_of_equiv {p\u2081 p\u2082} (h : T \u22a2 \u2200' p\u2081) (hp : \ud835\udddf'T \u22a2 \ud835\udde0 p\u2081 \u27f7 \ud835\udde0 p\u2082) : T \u22a2 \u2200' p\u2082 :=\nby { have : T \u22a2 \u2200'(p\u2081 \u27f6 p\u2082), by simpa using generalize (iff_equiv.mp hp).1,\n     exact (forallK _ _) \u2a00 this \u2a00 h }\n\nlemma equiv_forall_of_equiv {p\u2081 p\u2082} (hp : \ud835\udddf'T \u22a2 \ud835\udde0 p\u2081 \u27f7 \ud835\udde0 p\u2082) : T \u22a2 \u2200'p\u2081 \u27f7 \u2200'p\u2082 :=\nby { simp[iff_equiv], split,\n  { have : T \u22a2 \u2200'(p\u2081 \u27f6 p\u2082), by simpa using generalize (iff_equiv.mp hp).1,\n    exact forallK _ _ \u2a00 this },\n  { have : T \u22a2 \u2200'(p\u2082 \u27f6 p\u2081), by simpa using generalize (iff_equiv.mp hp).2,\n    exact forallK _ _ \u2a00 this } }\n\nlemma equiv_forall_of_equiv' {p\u2081 p\u2082} (hp : \ud835\udddf'T \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2200' \ud835\udde1 p\u2081 \u27f7 \u2200' \ud835\udde1 p\u2082 :=\n@equiv_forall_of_equiv _ _ T (\ud835\udde1 p\u2081) (\ud835\udde1 p\u2082) (by simpa using hp)\n\nlemma equiv_exists_of_equiv {p\u2081 p\u2082} (hp : \ud835\udddf'T \u22a2 \ud835\udde0 p\u2081 \u27f7 \ud835\udde0 p\u2082) : T \u22a2 \u2203'p\u2081 \u27f7 \u2203'p\u2082 :=\nby simp[ex_def]; refine equiv_neg_of_equiv (equiv_forall_of_equiv (by simpa using equiv_neg_of_equiv hp))\n\nlemma equiv_exists_of_equiv' {p\u2081 p\u2082} (hp : \ud835\udddf'T \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2203' \ud835\udde1 p\u2081 \u27f7 \u2203' \ud835\udde1 p\u2082 :=\n@equiv_exists_of_equiv _ _ T (\ud835\udde1 p\u2081) (\ud835\udde1 p\u2082) (by simpa using hp)\n\nlemma univ_imply_dummy (p : bounded_subformula L m 1) (q : bounded_subformula L m 0) :\n  T \u22a2 \u2200'(p \u27f6 \ud835\uddd7 q) \u27f6 \u2203'p \u27f6 q :=\nbegin\n  have : T \u22a2 \u2200'(\u223c\ud835\uddd7 q \u27f6 \u223cp) \u27f6 \u223cq \u27f6 \u2200'\u223cp, by simpa using dummy_univ T (\u223cq) (\u223cp),\n  refine imply_of_equiv this (equiv_forall_of_equiv (by simp)) (by simp[ex_def])\nend\n\nlemma exists_intro (p : bounded_subformula L m 1) (q : bounded_subformula L m 0)\n  (h : \ud835\udddf'T \u22a2 \ud835\udde0 p \u27f6 \ud835\udddf q) : T \u22a2 \u2203'p \u27f6 q :=\nby { have : T \u22a2 \u2200'(p \u27f6 \ud835\uddd7 q), by simpa using generalize h,\n     exact univ_imply_dummy p q \u2a00 this }\n\ndef Nonempty : bounded_preTheory L m := { \u2203'\u22a4, }\n\ninstance preTheory_Nonempty (T : bounded_preTheory L (m + 1)) : Nonempty.extend T :=\n\u27e8le_of (by simp[Nonempty]; refine use &0 (by simp))\u27e9\n\ninstance preTheory_of_inhabited [inhabited (L.fn 0)] : Nonempty.extend T :=\n\u27e8le_of (by simp[Nonempty]; refine use default (by simp))\u27e9\n\n@[simp] lemma non_empty [Nonempty.extend T] : T \u22a2 \u2203'\u22a4 :=\nlogic.Theory.extend.by_axiom Nonempty T (by simp[Nonempty])\n\n@[simp] lemma forall_bot [Nonempty.extend T] : T \u22a2 \u2200'\u22a5 \u27f7 \u22a5 :=\nby { simp[iff_equiv],\n     have : T \u22a2 \u223c\u2200'\u223c\u22a4, by simp[\u2190ex_def],\n     refine of_equiv (neg_of_equiv this (equiv_forall_of_equiv (by simp)))\n     (neg_iff (\u2200'\u22a5)) }\n\n@[simp] lemma forall_dummy [Nonempty.extend T] (p : bounded_formula L m) : T \u22a2 \u2200'\ud835\uddd7 p \u27f7 p :=\nbegin\n  simp[iff_equiv], split,\n  { have : T \u22a2 \u2200'(\u22a4 \u27f6 \ud835\uddd7 p) \u27f6 \u2203'\u22a4 \u27f6 p, from univ_imply_dummy \u22a4 p,\n    refine imply_of_equiv this (equiv_forall_of_equiv $ by simp) (by simp) },\n  { refine deduction.mp (gen $ by simp) }\nend\n\nsection prenex_normal_form\n\nlemma neg_forall_pnf (p) : T \u22a2 \u223c\u2200'p \u27f7 \u2203'\u223cp :=\nequiv_neg_of_equiv (equiv_forall_of_equiv (by simp[neg_eq]))\n\nlemma neg_exists_pnf (p) : T \u22a2 \u223c\u2203'p \u27f7 \u2200'\u223cp := by simp[ex_def]\n\n/-\nlemma neg_univ_closure_pnf {n} (p : bounded_subformula L m n) : T \u22a2 \u223c\u2200'*p \u27f7 \u2203'*\u223cp :=\nbegin\n  induction n with n IH generalizing m, { simp },\n  { simp[forall_comm, bounded_subformula.exists_comm],\n    have lmm\u2081 : T \u22a2 \u223c\u2200'\ud835\udde1 (\u2200'* \ud835\udde0 p) \u27f7 \u2203'\u223c\ud835\udde1 (\u2200'* \ud835\udde0 p), from neg_forall_pnf _,\n    have : \ud835\udddf'T \u22a2 \u223c\u2200'* (\ud835\udde0 p) \u27f7 \u2203'* (\u223c\ud835\udde0 p), from IH (\ud835\udde0 p),\n    have lmm\u2082 : T \u22a2 \u2203'\u223c\ud835\udde1 (\u2200'* \ud835\udde0 p) \u27f7 \u2203'\ud835\udde1 (\u2203'* \u223c\ud835\udde0 p), by simpa using equiv_exists_of_equiv' this,\n    exact equiv_trans lmm\u2081 lmm\u2082 }\nend\n\n\n\nlemma neg_exists_closure_pnf {n} (p : bounded_subformula L m n) : T \u22a2 \u223c\u2203'*p \u27f7 \u2200'*\u223cp :=\nbegin\n  induction n with n IH generalizing m, { simp },\n  { simp[forall_comm, bounded_subformula.exists_comm],\n    have lmm\u2081 : T \u22a2 \u223c\u2203'\ud835\udde1 (\u2203'* \ud835\udde0 p) \u27f7 \u2200'\u223c\ud835\udde1 (\u2203'* \ud835\udde0 p), from neg_exists_pnf _,\n    have : \ud835\udddf'T \u22a2 \u223c\u2203'* (\ud835\udde0 p) \u27f7 \u2200'* (\u223c\ud835\udde0 p), from IH (\ud835\udde0 p),\n    have lmm\u2082 : T \u22a2 \u2200'\u223c\ud835\udde1 (\u2203'* \ud835\udde0 p) \u27f7 \u2200'\ud835\udde1 (\u2200'* \u223c\ud835\udde0 p), by simpa using equiv_forall_of_equiv' this,\n    exact equiv_trans lmm\u2081 lmm\u2082 }\nend\n-/\n\n@[simp] lemma or_forall_pnf (p q) : T \u22a2 (\u2200'p) \u2294 q \u27f7 \u2200'(p \u2294 \ud835\uddd7 q) :=\nbegin\n  have lmm\u2081 : T \u22a2 (\u2200'p) \u2294 q \u27f6 \u2200'(p \u2294 \ud835\uddd7 q),\n  { have : \ud835\udddf'T \u22a2 (\u2200'\ud835\udddf p) \u2294 \ud835\udddf q \u27f6 \ud835\udde0 p \u2294 \ud835\udddf q,\n    { have : \ud835\udddf'T \u22a2 \u2200'\ud835\udddf p \u27f6 \ud835\udde0 p, from specialize' p,\n      exact or_imply (\u2200'\ud835\udddf p) (\ud835\udddf q) (\ud835\udde0 p \u2294 \ud835\udddf q) \u2a00 (imply_trans this (by simp)) \u2a00 (by simp) },\n    have : \ud835\udddf'(T +{ (\u2200'p) \u2294 q }) \u22a2 \ud835\udde0 p \u2294 \ud835\udddf q, simpa using deduction.mpr this,\n    have : T +{ (\u2200'p) \u2294 q } \u22a2 \u2200'(p \u2294 \ud835\uddd7 q), by simpa using generalize this,\n    exact deduction.mp this },\n  have lmm\u2082 : T \u22a2 \u2200'(p \u2294 \ud835\uddd7 q) \u27f6 (\u2200'p) \u2294 q,\n  { simp[has_sup.sup, subformula.or, imply_eq, neg_eq],\n    have : T \u22a2 \u2200'(\u223cp \u27f6 \ud835\uddd7 q) \u27f6 \u2203'\u223cp \u27f6 q, from univ_imply_dummy (\u223cp) q,\n    refine imply_of_equiv this (by simp) (equiv_imply_of_equiv (equiv_symm (neg_forall_pnf p)) (by simp)) },\n  refine iff_equiv.mpr \u27e8lmm\u2081, lmm\u2082\u27e9\nend\n\n@[simp] lemma and_exists_pnf (p q) : T \u22a2 (\u2203'p) \u2293 q \u27f7 \u2203'(p \u2293 \ud835\uddd7 q) :=\nbegin\n  have : T \u22a2 (\u2200'\u223cp) \u2294 \u223cq \u27f7 \u2200'\u223cp \u2294 \ud835\uddd7 (\u223cq), from or_forall_pnf (\u223cp) (\u223cq),\n  refine equiv_of_equiv (equiv_neg_of_equiv this) _ _,\n  { show T \u22a2 \u223c((\u2200'\u223cp) \u2294 \u223cq) \u27f7 (\u2203'p) \u2293 q,\n    refine equiv_of_equiv (neg_or_equiv_and_neg (\u2200'\u223cp) (\u223cq))\n      (equiv_refl _) (equiv_and_of_equiv (equiv_refl _) (iff_dn_refl_left q)) },\n  { show T \u22a2 \u223c\u2200'(\u223cp \u2294 \ud835\uddd7 (\u223cq)) \u27f7 \u2203'(p \u2293 \ud835\uddd7 q),\n    refine equiv_neg_of_equiv (equiv_forall_of_equiv $ equiv_symm (by simp[neg_eq])) }\nend\n\n@[simp] lemma and_forall_pnf [Nonempty.extend T] (p q) : T \u22a2 (\u2200'p) \u2293 q \u27f7 \u2200'(p \u2293 \ud835\uddd7 q) :=\nbegin\n  have lmm\u2081 : T \u22a2 (\u2200'p) \u2293 q \u27f6 \u2200'(p \u2293 \ud835\uddd7 q),\n  { have : \ud835\udddf'T \u22a2 (\u2200'\ud835\udddf p) \u2293 \ud835\udddf q \u27f6 \ud835\udde0 p \u2293 \ud835\udddf q,\n    { have : \ud835\udddf'T \u22a2 \u2200'\ud835\udddf p \u27f6 \ud835\udde0 p, from specialize' p,\n      exact imply_and ((\u2200'\ud835\udddf p) \u2293 \ud835\udddf q) (\ud835\udde0 p) (\ud835\udddf q) \u2a00 (imply_trans (by simp) this) \u2a00 (by simp) },\n    have : \ud835\udddf'(T +{ (\u2200'p) \u2293 q }) \u22a2 \ud835\udde0 p \u2293 \ud835\udddf q, simpa using deduction.mpr this,\n    have : T +{ (\u2200'p) \u2293 q } \u22a2 \u2200'(p \u2293 \ud835\uddd7 q), by simpa using generalize this,\n    exact deduction.mp this },\n  have lmm\u2082 : T \u22a2 \u2200'(p \u2293 \ud835\uddd7 q) \u27f6 (\u2200'p) \u2293 q,\n  { have lmm\u2083 : T \u22a2 \u2200'(p \u2293 \ud835\uddd7 q) \u27f6 \u2200'p, from forallK (p \u2293 \ud835\uddd7 q) p \u2a00 (gen $ by simp),\n    have lmm\u2084 : T \u22a2 \u2200'(p \u2293 \ud835\uddd7 q) \u27f6 q,\n    { have : T \u22a2 \u2200'(p \u2293 \ud835\uddd7 q) \u27f6 \u2200'\ud835\uddd7 q, from forallK (p \u2293 \ud835\uddd7 q) (\ud835\uddd7 q) \u2a00 (gen $ by simp),\n      refine imply_trans this (equiv_mp (forall_dummy _)) },\n    refine imply_and (\u2200'(p \u2293 \ud835\uddd7 q)) (\u2200'p) q \u2a00 lmm\u2083 \u2a00 lmm\u2084 },\n  refine iff_equiv.mpr \u27e8lmm\u2081, lmm\u2082\u27e9\nend\n\n@[simp] lemma or_exists_pnf [Nonempty.extend T] (p q) : T \u22a2 (\u2203'p) \u2294 q \u27f7 \u2203'(p \u2294 \ud835\uddd7 q) :=\nbegin\n  have : T \u22a2 (\u2200'\u223cp) \u2293 \u223cq \u27f7 \u2200'\u223cp \u2293 \ud835\uddd7 (\u223cq), from and_forall_pnf (\u223cp) (\u223cq),\n  have := equiv_neg_of_equiv this,\n  refine equiv_of_equiv this _ _,\n  { show T \u22a2 \u223c((\u2200'\u223cp) \u2293 \u223cq) \u27f7 (\u2203'p) \u2294 q,\n    refine equiv_of_equiv (neg_and_equiv_or_neg (\u2200'\u223cp) (\u223cq))\n      (equiv_refl _) (equiv_or_of_equiv (equiv_refl _) (iff_dn_refl_left q)) },\n  { show T \u22a2 \u223c\u2200'(\u223cp \u2293 \ud835\uddd7 (\u223cq)) \u27f7 \u2203'(p \u2294 \ud835\uddd7 q),\n    refine equiv_neg_of_equiv (equiv_forall_of_equiv $ equiv_symm (by simp[neg_eq])) }\nend\n\nlemma imply_forall_pnf (p q) : T \u22a2 (p \u27f6 \u2200'q) \u27f7 \u2200'(\ud835\uddd7 p \u27f6 q) :=\nby { have : T \u22a2 ((\u2200'q) \u2294 \u223cp) \u27f7 \u2200'(q \u2294 \u223c\ud835\uddd7 p), by simpa using or_forall_pnf q \u223cp,\n     exact equiv_of_equiv this (equiv_symm (by simp))\n       (equiv_forall_of_equiv (equiv_symm (by simp))) }\n\nlemma imply_exists_pnf [Nonempty.extend T] (p q) : T \u22a2 (p \u27f6 \u2203'q) \u27f7 \u2203'(\ud835\uddd7 p \u27f6 q) :=\nby{ have : T \u22a2 ((\u2203'q) \u2294 \u223cp) \u27f7 \u2203'(q \u2294 \u223c\ud835\uddd7 p), by simpa using or_exists_pnf q \u223cp,\n    refine equiv_of_equiv this (equiv_symm impl_iff_or')\n      (equiv_exists_of_equiv $ equiv_symm $ by simp)}\n\nlemma exists_imply_pnf (p q) : T \u22a2 (\u2203'p \u27f6 q) \u27f7 \u2200'(p \u27f6 \ud835\uddd7 q) :=\nby{ have : T \u22a2 ((\u2200'\u223cp) \u2294 q) \u27f7 \u2200'(\u223cp \u2294 \ud835\uddd7 q), by simp,\n    refine equiv_of_equiv this _ _,\n    { have : T \u22a2 (\u223c\u223c\u2200'\u223cp) \u2294 q \u27f7 \u2203'p \u27f6 q, from equiv_symm (by simp[ex_def]),\n      refine equiv_trans (equiv_or_of_equiv _ _) this; simp },\n    { refine equiv_forall_of_equiv (equiv_symm $ by simp) } }\n\nlemma forall_imply_pnf [Nonempty.extend T] (p q) : T \u22a2 (\u2200'p \u27f6 q) \u27f7 \u2203'(p \u27f6 \ud835\uddd7 q) :=\nby{ have : T \u22a2 ((\u2203'\u223cp) \u2294 q) \u27f7 \u2203'(\u223cp \u2294 \ud835\uddd7 q), by simp,\n    refine equiv_of_equiv this _ _,\n    { have : T \u22a2 (\u2203'\u223cp) \u2294 q \u27f7 \u2200'\u223c\u223cp \u27f6 q, from equiv_symm (by simp[ex_def]),\n      refine equiv_trans this (equiv_imply_of_equiv (equiv_forall_of_equiv _) _); simp },\n    { refine equiv_exists_of_equiv (equiv_symm $ by simp) } }\n\nlemma forall_imply_forall_pnf [Nonempty.extend T] (p q) : T \u22a2 (\u2200'p \u27f6 \u2200'q) \u27f7 \u2203' \u2200'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q) :=\nbegin\n  have : \ud835\udddf'T \u22a2 (\ud835\udde0 p \u27f6 \u2200'\ud835\udddf q) \u27f7 \u2200'(\ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\udddf q), from imply_forall_pnf (\ud835\udde0 p) (\ud835\udddf q),\n  have lmm\u2081 : T \u22a2 \u2203'(p \u27f6 \u2200' \ud835\uddd7 q) \u27f7 \u2203' \u2200'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q),\n    from equiv_exists_of_equiv (by simpa using this),\n  have lmm\u2082 : T \u22a2 (\u2200'p \u27f6 \u2200'q) \u27f7 \u2203'(p \u27f6 \u2200' \ud835\uddd7 q), by simpa using forall_imply_pnf p (\u2200'q),\n  exact equiv_trans lmm\u2082 lmm\u2081\nend\n\nlemma forall_imply_exists_pnf [Nonempty.extend T] (p q) : T \u22a2 (\u2200'p \u27f6 \u2203'q) \u27f7 \u2203' \u2203'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q) :=\nbegin\n  have : \ud835\udddf'T \u22a2 (\ud835\udde0 p \u27f6 \u2203'\ud835\udddf q) \u27f7 \u2203'(\ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\udddf q), from imply_exists_pnf (\ud835\udde0 p) (\ud835\udddf q),\n  have lmm\u2081 : T \u22a2 \u2203'(p \u27f6 \u2203' \ud835\uddd7 q) \u27f7 \u2203' \u2203'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q),\n    from equiv_exists_of_equiv (by simpa using this),\n  have lmm\u2082 : T \u22a2 (\u2200'p \u27f6 \u2203'q) \u27f7 \u2203'(p \u27f6 \u2203' \ud835\uddd7 q), by simpa using forall_imply_pnf p (\u2203'q),\n  exact equiv_trans lmm\u2082 lmm\u2081\nend\n\nlemma exists_imply_forall_pnf (p q) : T \u22a2 (\u2203'p \u27f6 \u2200'q) \u27f7 \u2200' \u2200'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q) :=\nbegin\n  have : \ud835\udddf'T \u22a2 (\ud835\udde0 p \u27f6 \u2200'\ud835\udddf q) \u27f7 \u2200'(\ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\udddf q), from imply_forall_pnf (\ud835\udde0 p) (\ud835\udddf q),\n  have lmm\u2081 : T \u22a2 \u2200'(p \u27f6 \u2200' \ud835\uddd7 q) \u27f7 \u2200' \u2200'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q),\n    from equiv_forall_of_equiv (by simpa using this),\n  have lmm\u2082 : T \u22a2 (\u2203'p \u27f6 \u2200'q) \u27f7 \u2200'(p \u27f6 \u2200' \ud835\uddd7 q), by simpa using exists_imply_pnf p (\u2200'q),\n  exact equiv_trans lmm\u2082 lmm\u2081\nend\n\nlemma exists_imply_exists_pnf (p q) : T \u22a2 (\u2203'p \u27f6 \u2203'q) \u27f7 \u2200' \u2203'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q) :=\nbegin\n  have : \ud835\udddf'T \u22a2 (\ud835\udde0 p \u27f6 \u2203'\ud835\udddf q) \u27f7 \u2203'(\ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\udddf q), from imply_exists_pnf (\ud835\udde0 p) (\ud835\udddf q),\n  have lmm\u2081 : T \u22a2 \u2200'(p \u27f6 \u2203' \ud835\uddd7 q) \u27f7 \u2200' \u2203'(\ud835\udde1 \ud835\uddd7 \ud835\udde0 p \u27f6 \ud835\uddd7 q),\n    from equiv_forall_of_equiv (by simpa using this),\n  have lmm\u2082 : T \u22a2 (\u2203'p \u27f6 \u2203'q) \u27f7 \u2200'(p \u27f6 \u2203' \ud835\uddd7 q), by simpa using exists_imply_pnf p (\u2203'q),\n  exact equiv_trans lmm\u2082 lmm\u2081\nend\n\nend prenex_normal_form\n\nsection quantifier\nvariables {m} {n : \u2115}\n/-\nlemma specialize_foralls (p : bounded_subformula L m n) (w : fin n \u2192 subterm L m 0) : T \u22a2 \u2200'*p \u27f6 substs w p :=\nbegin\n  induction n with n IH generalizing m,\n  { simp },\n  { have : \ud835\udddf'T \u22a2 \u2200'* \ud835\udde0 p \u27f6 substs (mlift \u2218 w \u2218 fin.cast_succ) (\ud835\udde0 p),\n    from IH (\ud835\udde0 p) (subterm.mlift \u2218 w \u2218 fin.cast_succ),\n    have : T \u22a2 \u2200'(\ud835\udde1 (\u2200'*\ud835\udde0 p) \u27f6 \ud835\udde1 (substs (mlift \u2218 w \u2218 fin.cast_succ) (\ud835\udde0 p))),\n    by simpa using generalize this,\n    have lmm\u2081 : T \u22a2 \u2200'*p \u27f6 \u2200'\ud835\udde1 (substs (mlift \u2218 w \u2218 fin.cast_succ) (\ud835\udde0 p)),\n    by simpa[forall_comm] using forallK _ _ \u2a00 this,\n    have lmm\u2082 : T \u22a2 \u2200'\ud835\udde1 (substs (mlift \u2218 w \u2218 fin.cast_succ) (\ud835\udde0 p)) \u27f6 substs w p,\n    from specialize T (\ud835\udde1 (substs (mlift \u2218 w \u2218 fin.cast_succ) (\ud835\udde0 p))) (w $ fin.last n),\n    exact imply_trans lmm\u2081 lmm\u2082 }\nend\n\nlemma foralls_substs {p : bounded_subformula L m n} (h : T \u22a2 \u2200'*p) (w) : T \u22a2 substs w p :=\nspecialize_foralls p w \u2a00 h\n\n@[simp] lemma exists_dn (p : bounded_subformula L m n) : T \u22a2 \u2203'*\u223c\u223cp \u27f7 \u2203'*p :=\nbegin\n  induction n with n IH generalizing m; simp[bounded_subformula.exists_comm],\n  refine equiv_exists_of_equiv (by simpa using IH (\ud835\udde0 p))\nend\n\n@[simp] lemma neg_univ_closure_neg (p : bounded_subformula L m n) : T \u22a2 \u223c\u2200'*(\u223cp) \u27f7 \u2203'*p :=\nbegin\n  have : T \u22a2 \u223c\u2200'*(\u223cp) \u27f7 \u2203'*(\u223c\u223cp), from neg_univ_closure_pnf (\u223cp),\n  refine equiv_trans this (by simp)\nend\n\n@[simp] lemma exists_substs (p : bounded_subformula L m n) (w : fin n \u2192 subterm L m 0) : T \u22a2 substs w p \u27f6 \u2203'*p :=\nbegin\n  have : T \u22a2 \u223c\u223csubsts w p \u27f6 \u223c\u2200'* (\u223cp), by simpa using contrapose.mpr (specialize_foralls (\u223cp) w),\n  refine imply_of_equiv this (by simp) (by simp)\nend\n\nlemma exists_of_substs {p : bounded_subformula L m n} (w) (h : T \u22a2 substs w p) : T \u22a2 \u2203'*p :=\nexists_substs p w \u2a00 h\n-/\nend quantifier\n\nend provable\n\nnamespace bounded_preTheory\nvariables {L m} (T U : bounded_preTheory L m)\n\n--instance [T.extend U] : logic.Theory.extend T.mlift U.mlift := \u27e8by { intros p h, sorry }\u27e9\n\nend bounded_preTheory\n\nsection equal\n\ndef eq_axiom_schema_funext [L.has_equal] {m k} (f : L.fn k) : bounded_subformula L m 0 :=\n\u2200'*((\u22c0 i, #(fin.cast_add k i) =' #(fin.nat_add k i)) \u27f6\n  (function f (var \u2218 fin.cast_add k) =' function f (var \u2218 fin.nat_add k)) : bounded_subformula L m (k + k))\n\ndef eq_axiom_schema_relext [L.has_equal] {m k} (r : L.pr k) : bounded_subformula L m 0 :=\n\u2200'*((\u22c0 i : fin k, #(fin.cast_add k i) =' #(fin.nat_add k i)) \u27f6\n  (relation r (var \u2218 fin.cast_add k) \u27f7 relation r (var \u2218 fin.nat_add k)))\n\nvariables {L m} [language.has_equal L]\n\ninductive Eq : bounded_preTheory L m\n| eq_refl : Eq (\u2200'(#0 =' #0))\n| eq_symm : Eq (\u2200' \u2200'((#0 =' #1) \u27f6 (#1 =' #0)))\n| eq_trans : Eq (\u2200' \u2200' \u2200'((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2)))\n| funext : \u2200 {k} (f : L.fn k), Eq (eq_axiom_schema_funext f)\n| relext : \u2200 {k} (r : L.pr k), Eq (eq_axiom_schema_relext r)\n\nattribute [simp] Eq.eq_refl Eq.eq_symm Eq.eq_trans Eq.funext Eq.relext\n\n/-\n@[simp] lemma mlift_Eq : (Eq : bounded_preTheory L m).mlift = Eq :=\nbegin\n  ext p, simp[bounded_preTheory.mlift], split,\n  { rintros \u27e8p, hp, rfl\u27e9,\n    cases hp; simp[fal_eq],\n    { exact Eq.eq_refl },\n    { exact Eq.eq_symm },\n    { exact Eq.eq_trans },\n    { exact Eq.funext _ },\n    { exact Eq.relext _ } },\n  { intros h, cases h,\n    { simp[fal_eq], refine \u27e8_, Eq.eq_refl, by simp\u27e9 },\n    { simp[fal_eq], refine \u27e8_, Eq.eq_symm, by simp\u27e9 },\n    { simp[fal_eq], refine \u27e8_, Eq.eq_trans, by simp\u27e9 },\n    { refine \u27e8_, Eq.funext _, by simp[eq_axiom_schema_funext]\u27e9 },\n    { refine \u27e8_, Eq.relext _, by simp[eq_axiom_schema_relext, (\u2218)]\u27e9 } }\nend\n-/\n--def eq_extend {T : bounded_preTheory L m} [extend Eq T] : extend Eq \ud835\udddf'T :=\n\nend equal\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/QL/FOL/deduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.39479274342774107}}
{"text": "import .tab\n\nvariables {p q r s : Prop}\n\nopen tactic\n\n-- example (h1 : \u00ac ((p \u2192 q) \u2192 r)) (h2 : p \u2227 \u00ac q) : r := by tab\n-- \n-- example (h1 : \u00ac ((p \u2192 q) \u2192 r)) (h2 : p \u2227 \u00ac q) : r := \n-- by do proof_by_contradiction,\n--       -- trace \"After pbc : \", trace_state,\n--       normalize,\n--       -- trace \"After normalization : \", trace_state,\n--       tab_aux,\n--       skip\n-- \n-- meta def tab_aux : tactic unit :=\n-- do split_conjs,\n--    contradiction <|>\n--      (split_disj >> tab_aux >> tab_aux)", "meta": {"author": "skbaek", "repo": "tab", "sha": "70909a69464a8713412d640ac630e5e6ef4e43e8", "save_path": "github-repos/lean/skbaek-tab", "path": "github-repos/lean/skbaek-tab/tab-70909a69464a8713412d640ac630e5e6ef4e43e8/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419958239131, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3947038778010504}}
{"text": "import group_theory.perm group_theory.order_of_element group_theory.quotient_group\n\nuniverses u v\nopen finset is_subgroup equiv equiv.perm quotient_group\n\ninstance {\u03b1 \u03b2 : Type*} [group \u03b1] [group \u03b2] [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2) [is_group_hom f] :\n  decidable_pred (is_group_hom.ker f) :=\n\u03bb _, decidable_of_iff _ (is_group_hom.mem_ker f).symm\n\ndef alternating (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1] : Type* :=\nis_group_hom.ker (sign : perm \u03b1 \u2192 units \u2124)\n\n/- not definitionally equal to `subtype.decidable_eq`, since `subtype.decidable_eq` does\n  not reduce in the kernel -/\ninstance (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1] : decidable_eq (alternating \u03b1) :=\n\u03bb a b, decidable_of_iff (a.1 = b.1) (by cases a; cases b; simp [subtype.mk.inj_eq])\n\ninstance (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1] : fintype (alternating \u03b1) :=\nset_fintype _\n\ninstance (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1] : group (alternating \u03b1) :=\nby unfold alternating; apply_instance\n\nsection classical\n\nlocal attribute [instance] classical.prop_decidable\n\nlemma card_alternating (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1] (h : 2 \u2264 fintype.card \u03b1):\n  fintype.card (alternating \u03b1) * 2 = (fintype.card \u03b1).fact :=\nhave (quotient_group.quotient (is_group_hom.ker (sign : perm \u03b1 \u2192 units \u2124))) \u2243 units \u2124,\n  from quotient_ker_equiv_of_surjective _ (sign_surjective h),\ncalc fintype.card (alternating \u03b1) * 2 = fintype.card (units \u2124 \u00d7 alternating \u03b1) :\n  by rw [mul_comm, fintype.card_prod, fintype.card_units_int]\n... = fintype.card (perm \u03b1) : fintype.card_congr\n  (calc (units \u2124 \u00d7 alternating \u03b1) \u2243\n    (quotient_group.quotient (is_group_hom.ker (sign : perm \u03b1 \u2192 units \u2124)) \u00d7 alternating \u03b1)  :\n      equiv.prod_congr this.symm (by refl)\n  ... \u2243 perm \u03b1 : (group_equiv_quotient_times_subgroup _).symm)\n... = (fintype.card \u03b1).fact : fintype.card_perm\n\nend classical\n\nlocal notation `A5` := alternating (fin 5)\nvariables {\u03b1 : Type*} [fintype \u03b1] [decidable_eq \u03b1]\n\nsection\nlocal attribute [semireducible] reflected\n\nmeta instance fin_reflect (n : \u2115) : has_reflect (fin n) :=\n\u03bb a, `(@fin.mk %%`(n) %%(nat.reflect a.1) (of_as_true %%`(_root_.trivial)))\n\nmeta instance fin_fun.has_reflect : has_reflect (fin 5 \u2192 fin 5) :=\nlist.rec_on (quot.unquot (@univ (fin 5) _).1)\n  (\u03bb f, `(\u03bb y : fin 5, y))\n  (\u03bb x l ih f, let e := ih f in\n    if f x = x then e\n    else let ex := fin_reflect 5 x in\n      let efx := fin_reflect 5 (f x) in\n      if e = `(\u03bb y : fin 5, y)\n      then `(\u03bb y : fin 5, ite.{1} (y = %%ex) (%%efx) y)\n      else `(\u03bb y : fin 5, ite.{1} (y = %%ex) (%%efx) ((%%e : fin 5 \u2192 fin 5) y)))\n\nmeta instance : has_reflect (perm (fin 5)) :=\n\u03bb f, `(@equiv.mk.{1 1} (fin 5) (fin 5)\n    %%(fin_fun.has_reflect f.to_fun)\n  %%(fin_fun.has_reflect f.inv_fun)\n  (of_as_true %%`(_root_.trivial)) (of_as_true %%`(_root_.trivial)))\n\nmeta instance I1 : has_reflect (alternating (fin 5)) :=\n\u03bb f, `(@subtype.mk (perm (fin 5)) (is_group_hom.ker (sign : perm (fin 5) \u2192 units \u2124))\n   %%(@reflect (perm (fin 5)) f.1 (equiv.perm.has_reflect f.1))\n  ((is_group_hom.mem_ker sign).2 %%`(@eq.refl (units \u2124) 1)))\n\nmeta instance multiset.has_reflect {\u03b1 : Type} [reflected \u03b1] [has_reflect \u03b1] :\n  has_reflect (multiset \u03b1) :=\n\u03bb s, let l : list \u03b1 := quot.unquot s in `(@quotient.mk.{1} (list %%`(\u03b1)) _ %%`(l))\n\nmeta instance I2 (a : alternating (fin 5)) :\n  has_reflect {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1} :=\n\u03bb b, `(@subtype.mk (alternating (fin 5) \u00d7 alternating (fin 5))\n  (\u03bb b, b.2 * %%`(a) * b.2\u207b\u00b9 = b.1)\n  %%(prod.has_reflect _ _ b.1) (of_as_true %%`(_root_.trivial)))\n\nmeta instance I3 (a : alternating (fin 5)) (m : reflected a) :\n  reflected {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1} :=\n`({b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * %%m * b.2\u207b\u00b9 = b.1})\n\nmeta instance I4 : has_reflect\n  (\u03a3 a : alternating (fin 5), multiset\n  {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1}) :=\n\u03bb s, let ra : reflected s.1 := (I1 s.1) in\n`(let a : alternating (fin 5) := %%ra in\n  @sigma.mk (alternating (fin 5))\n    (\u03bb a, multiset {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1})\n    a %%(@multiset.has_reflect _ (I3 s.1 ra) (I2 s.1) s.2))\n\nend\n\nmeta def conjugacy_classes_A5_meta_aux : list (alternating (fin 5)) \u2192 list\n  (\u03a3 a : alternating (fin 5), list\n  {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1})\n| [] := []\n| (a :: l) := let m : \u03a3 a : alternating (fin 5), list\n    {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1} :=\n  \u27e8a, ((quot.unquot (@univ (alternating (fin 5)) _).1).map\n  (\u03bb x, show {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1},\n    from \u27e8(x * a * x\u207b\u00b9, x), rfl\u27e9)).pw_filter (\u03bb x y, x.1.1 \u2260 y.1.1)\u27e9 in\nm :: conjugacy_classes_A5_meta_aux (l.diff (m.2.map (prod.fst \u2218 subtype.val)))\n\nmeta def conjugacy_classes_A5_meta : multiset (\u03a3 a : alternating (fin 5), multiset\n  {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1}) :=\n(quotient.mk ((conjugacy_classes_A5_meta_aux (quot.unquot univ.1)).map\n    (\u03bb a, \u27e8a.1, (quotient.mk a.2)\u27e9)))\n\nmeta def thing {\u03b1 : Sort*} [has_reflect \u03b1] (a : \u03b1) : tactic unit :=\ntactic.exact `(a)\nset_option profiler true\n@[irreducible] def conjugacy_classes_A5_aux : multiset (\u03a3 a : alternating (fin 5), multiset\n  {b : alternating (fin 5) \u00d7 alternating (fin 5) // b.2 * a * b.2\u207b\u00b9 = b.1}) :=\nby thing (conjugacy_classes_A5_meta)\n\ndef conjugacy_classes_A5_aux2 : multiset (multiset (alternating (fin 5))) :=\nconjugacy_classes_A5_aux.map (\u03bb s, s.2.map (\u03bb b, b.1.1))\n\nlemma nodup_conjugacy_classes_A5_aux2_bind :\n  (conjugacy_classes_A5_aux2.bind id).nodup :=\ndec_trivial\n\nlemma nodup_conjugacy_classes_A5_aux2 : \u2200 s : multiset (alternating (fin 5)),\n  s \u2208 conjugacy_classes_A5_aux2 \u2192 s.nodup :=\n(multiset.nodup_bind.1 nodup_conjugacy_classes_A5_aux2_bind).1\n\ndef conjugacy_classes_A5 : finset (finset (alternating (fin 5))) :=\n\u27e8conjugacy_classes_A5_aux2.pmap finset.mk nodup_conjugacy_classes_A5_aux2, dec_trivial\u27e9\n\nlemma nodup_conjugacy_classes_A5_bind :\n  (conjugacy_classes_A5.1.bind finset.val).nodup :=\nhave conjugacy_classes_A5.1.bind finset.val =\n  conjugacy_classes_A5_aux2.bind id,\nfrom multiset.ext.2 $ \u03bb a,\n  by rw [conjugacy_classes_A5, @multiset.count_bind A5, @multiset.count_bind A5,\n    multiset.map_pmap, multiset.pmap_eq_map]; refl,\nthis.symm \u25b8 nodup_conjugacy_classes_A5_aux2_bind\n\nlemma is_conj_conjugacy_classes_A5 (s : finset A5) (h : s \u2208 conjugacy_classes_A5) :\n  \u2200 x y \u2208 s, is_conj x y :=\nassume x y hx hy,\nbegin\n  simp only [conjugacy_classes_A5, finset.mem_def, multiset.mem_pmap,\n    conjugacy_classes_A5_aux2] at h,\n  rcases h with \u27e8t, ht\u2081, ht\u2082\u27e9,\n  rw [multiset.mem_map] at ht\u2081,\n  rcases ht\u2081 with \u27e8u, hu\u2081, hu\u2082\u27e9,\n  have hx' : x \u2208 multiset.map (\u03bb (b : {b : A5 \u00d7 A5 // b.2 * u.1 * b.2\u207b\u00b9 = b.1}), b.1.1) u.2,\n  { simpa [ht\u2082.symm, hu\u2082] using hx },\n  have hy' : y \u2208 multiset.map (\u03bb (b : {b : A5 \u00d7 A5 // b.2 * u.1 * b.2\u207b\u00b9 = b.1}), b.1.1) u.2,\n  { simpa [ht\u2082.symm, hu\u2082] using hy },\n  cases multiset.mem_map.1 hx' with xc hxc,\n  cases multiset.mem_map.1 hy' with yc hyc,\n  exact is_conj_trans\n    (is_conj_symm (show is_conj u.1 x, from hxc.2 \u25b8 \u27e8_, xc.2\u27e9))\n    (hyc.2 \u25b8 \u27e8_, yc.2\u27e9)\nend\n\nlemma normal_subgroup_eq_bind_conjugacy_classes (s : finset (finset G)) (h\u2081 : \u2200 x, \u2203 t \u2208 s, x \u2208 t)\n  (h\u2082 : \u2200 t \u2208 s, \u2200 x y \u2208 t, is_conj x y) (I : finset G) [nI : normal_subgroup (\u2191I : set G)] :\n  \u2203 u \u2286 s, I = u.bind id :=\n\u27e8(s.powerset.filter (\u03bb u : finset (finset G), u.bind id \u2286 I)).bind id,\n    (\u03bb x, by simp only [finset.subset_iff, mem_bind, mem_filter, exists_imp_distrib, mem_powerset,\n      and_imp, id.def] {contextual := tt}; tauto),\n  le_antisymm\n    (\u03bb x hxI, let \u27e8t, ht\u2081, ht\u2082\u27e9 := h\u2081 x in\n      mem_bind.2 \u27e8t, mem_bind.2 \u27e8(s.powerset.filter (\u03bb u : finset (finset G), u.bind id \u2286 I)).bind id,\n          mem_filter.2 \u27e8mem_powerset.2\n            (\u03bb u hu, let \u27e8v, hv\u2081, hv\u2082\u27e9 := mem_bind.1 hu in\n              mem_powerset.1 (mem_filter.1 hv\u2081).1 hv\u2082),\n          \u03bb y hy, let \u27e8u, hu\u2081, hu\u2082\u27e9 := mem_bind.1 hy in\n            let \u27e8v, hv\u2081, hv\u2082\u27e9 := mem_bind.1 hu\u2081 in\n            (mem_filter.1 hv\u2081).2 (mem_bind.2 \u27e8u, hv\u2082, hu\u2082\u27e9)\u27e9,\n        mem_bind.2 \u27e8{t}, mem_filter.2 \u27e8by simp [ht\u2081, finset.subset_iff],\n            \u03bb y hy, let \u27e8u, hu\u2081, hu\u2082\u27e9 := mem_bind.1 hy in\n              let \u27e8z, hz\u27e9 := h\u2082 t ht\u2081 x y ht\u2082 (by simp * at *) in\n              hz \u25b8 @normal_subgroup.normal G _ I.to_set nI _ hxI _\u27e9,\n          by simp\u27e9\u27e9,\n        ht\u2082\u27e9)\n    (\u03bb x, by simp only [finset.subset_iff, mem_bind, exists_imp_distrib, mem_filter, mem_powerset]; tauto)\u27e9\n\nlemma simple_of_card_conjugacy_classes [fintype G] [decidable_eq G] (s : finset (finset G))\n  (h\u2081 : \u2200 x, \u2203 t \u2208 s, x \u2208 t) (h\u2082 : \u2200 t \u2208 s, \u2200 x y \u2208 t, is_conj x y)\n  (hs : (s.1.bind finset.val).nodup)\n  (h\u2083 : \u2200 t \u2264 s.1.map finset.card, 1 \u2208 t \u2192 t.sum \u2223 fintype.card G \u2192 t.sum = 1 \u2228 t.sum = fintype.card G) :\n  simple_group G :=\nby haveI := classical.dec; exact\n\u27e8\u03bb H iH,\n  let I := (set.to_finset H) in\n  have Ii : normal_subgroup (\u2191I : set G), by simpa using iH,\n  let \u27e8u, hu\u2081, hu\u2082\u27e9 :=\n    @normal_subgroup_eq_bind_conjugacy_classes G _ _ _ s h\u2081 h\u2082 I Ii in\n  have hInd : \u2200 (x : finset G), x \u2208 u \u2192 \u2200 (y : finset G), y \u2208 u \u2192 x \u2260 y \u2192 id x \u2229 id y = \u2205,\n    from \u03bb x hxu y hyu hxy,\n      begin\n        rw multiset.nodup_bind at hs,\n        rw [\u2190 finset.disjoint_iff_inter_eq_empty, finset.disjoint_left],\n        exact multiset.forall_of_pairwise\n          (\u03bb (a b : finset G) (h : multiset.disjoint a.1 b.1),\n          multiset.disjoint.symm h) hs.2 x (hu\u2081 hxu) y (hu\u2081 hyu) hxy\n      end,\n  have hci : card I = u.sum finset.card,\n    by rw [hu\u2082, card_bind hInd]; refl,\n  have hu1 : (1 : G) \u2208 u.bind id, by exactI hu\u2082 \u25b8 is_submonoid.one_mem (\u2191I : set G),\n  let \u27e8v, hv\u2081, hv\u2082\u27e9 := mem_bind.1 hu1 in\n  have hv : v = finset.singleton (1 : G),\n    from finset.ext.2 $ \u03bb a, \u27e8\u03bb hav, mem_singleton.2 $\n        is_conj_one_right.1 (h\u2082 v (hu\u2081 hv\u2081) _ _ hv\u2082 hav),\n      by simp [show (1 : G) \u2208 v, from hv\u2082] {contextual := tt}\u27e9,\n  have hci' : card I = 1 \u2228 card I = fintype.card G,\n    begin\n      rw [hci],\n      exact h\u2083 _ (multiset.map_le_map (show u.1 \u2264 s.1,\n        from (multiset.le_iff_subset u.2).2 hu\u2081))\n          (multiset.mem_map.2 \u27e8finset.singleton 1, hv \u25b8 hv\u2081, rfl\u27e9)\n          (calc u.sum finset.card = card I : hci.symm\n            ... = fintype.card (\u2191I : set G) : (set.card_fintype_of_finset' I (by simp)).symm\n            ... \u2223 fintype.card G : by exactI card_subgroup_dvd_card _)\n    end,\n    hci'.elim\n      (\u03bb hci', or.inl (set.ext (\u03bb x,\n        let \u27e8y, hy\u27e9 := finset.card_eq_one.1 hci' in\n        by resetI;\n          simp only [I, finset.ext, set.mem_to_finset, finset.mem_singleton] at hy;\n          simp [is_subgroup.mem_trivial, hy, (hy 1).1 (is_submonoid.one_mem H)])))\n      (\u03bb hci', or.inr $\n        suffices I = finset.univ,\n          by simpa [I, set.ext_iff, finset.ext] using this,\n        finset.eq_of_subset_of_card_le (\u03bb _, by simp) (by rw hci'; refl))\u27e9\n\nlemma card_A5 : fintype.card A5 = 60 :=\n(nat.mul_right_inj (show 2 > 0, from dec_trivial)).1 $\nhave 2 \u2264 fintype.card (fin 5), from dec_trivial,\n  by rw [card_alternating _ this]; simp; refl\n\nlemma conjugacy_classes_A5_bind_eq_univ :\n  conjugacy_classes_A5.bind (\u03bb t, t) = univ :=\neq_of_subset_of_card_le (\u03bb _, by simp)\n  (calc card univ = 60 : card_A5\n    ... \u2264 (conjugacy_classes_A5.1.bind finset.val).card : dec_trivial\n    ... = (conjugacy_classes_A5.bind id).card :\n      begin\n        rw [finset.card_bind, multiset.card_bind], refl,\n        { exact multiset.forall_of_pairwise (\u03bb a b, by simp [finset.inter_comm])\n            (by simp only [finset.disjoint_iff_inter_eq_empty.symm, finset.disjoint_left];\n              exact (multiset.nodup_bind.1 nodup_conjugacy_classes_A5_bind).2) }\n      end)\n\nlemma A5_simple : simple_group A5 :=\nsimple_of_card_conjugacy_classes conjugacy_classes_A5\n  (\u03bb x, mem_bind.1 $ by rw [conjugacy_classes_A5_bind_eq_univ]; simp)\n  is_conj_conjugacy_classes_A5\n  nodup_conjugacy_classes_A5_bind\n  (by simp only [multiset.mem_powerset.symm, card_A5];\n    exact dec_trivial)\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/simple_group2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.39470386333912794}}
{"text": "import phase2.approximation\n\nopen set sum\nopen_locale classical\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}] {\u03b1 : \u039b} [position_data.{}] [phase_2_assumptions \u03b1] {\u03b2 : Iic \u03b1}\n  (\u03c0 : struct_approx \u03b2)\n\nnamespace struct_approx\n\nopen near_litter_approx hypothesis\n\nlemma order_iso_apply_mem {S T : sublitter} {a} {L : litter}\n  (h : (S.order_iso T a : atom) \u2208 litter_set L) : T.litter = L :=\nbegin\n  rw [\u2190 litter.litter_to_sublitter L, \u2190 sublitter.inter_nonempty_iff],\n  exact \u27e8_, (S.order_iso T a).prop, h\u27e9,\nend\n\nlemma order_iso_apply_eq {S T U V : sublitter} {a b}\n  (h : (S.order_iso T a : atom) = U.order_iso V b) :\n  T.litter = V.litter :=\nbegin\n  rw subtype.coe_eq_iff at h,\n  exact (order_iso_apply_mem (T.subset h.some)).symm,\nend\n\n/-- Computes the action of a structural approximation `\u03c0` on an atom `a`. -/\nnoncomputable def atom_completion (a : atom) (A : extended_index \u03b2)\n  (H : hypothesis \u27e8inl a, A\u27e9) : atom :=\nif h : a \u2208 (\u03c0 A).atom_perm.domain then \u03c0 A \u2022 a else\n((\u03c0 A).largest_sublitter a.1).order_iso\n  ((\u03c0 A).largest_sublitter (H.near_litter_image a.1.to_near_litter A\n    (relation.trans_gen.single $ constrains.atom a A)).1)\n  \u27e8a, (\u03c0 A).mem_largest_sublitter_of_not_mem_domain a h\u27e9\n\nend struct_approx\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/atom_completion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.721743206297598, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.39460454681859863}}
{"text": "import category.bitraversable.basic\n\nuniverses u\n\nvariables {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t]\nvariables {\u03b2 : Type u}\n\nnamespace bitraversable\nopen functor is_lawful_applicative\nvariables {F G : Type u \u2192 Type u}\n          [applicative F] [applicative G]\n\n@[reducible]\ndef tfst {\u03b1 \u03b1'} (f : \u03b1 \u2192 F \u03b1') : t \u03b1 \u03b2 \u2192 F (t \u03b1' \u03b2) :=\nbitraverse f pure\n\n@[reducible]\ndef tsnd {\u03b1 \u03b1'} (f : \u03b1 \u2192 F \u03b1') : t \u03b2 \u03b1 \u2192 F (t \u03b2 \u03b1') :=\nbitraverse pure f\n\nvariables [is_lawful_bitraversable t]\n          [is_lawful_applicative F]\n          [is_lawful_applicative G]\n\n@[higher_order tfst_id]\nlemma id_tfst : \u03a0 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), tfst id.mk x = id.mk x :=\n@id_bitraverse _ _ _\n\n@[higher_order tsnd_id]\nlemma id_tsnd : \u03a0 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), tsnd id.mk x = id.mk x :=\n@id_bitraverse _ _ _\n\n@[higher_order tfst_comp_tfst]\nlemma comp_tfst {\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2}\n  (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b1\u2081 \u2192 G \u03b1\u2082) (x : t \u03b1\u2080 \u03b2) :\n  comp.mk (tfst f' <$> tfst f x) = tfst (comp.mk \u2218 map f' \u2218 f) x :=\nby rw \u2190 comp_bitraverse; simp [tfst,map_comp_pure,has_pure.pure]\n\n@[higher_order tfst_comp_tsnd]\nlemma tfst_tsnd {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n  comp.mk (tfst f <$> tsnd f' x) =\n  bitraverse (comp.mk \u2218 pure \u2218 f) (comp.mk \u2218 map pure \u2218 f') x :=\nby rw \u2190 comp_bitraverse; simp [tfst,tsnd]\n\n@[higher_order tsnd_comp_tfst]\nlemma tsnd_tfst {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n  comp.mk (tsnd f' <$> tfst f x) =\n  bitraverse (comp.mk \u2218 map pure \u2218 f) (comp.mk \u2218 pure \u2218 f') x :=\nby rw \u2190 comp_bitraverse; simp [tfst,tsnd]\n\n@[higher_order tsnd_comp_tsnd]\nlemma comp_tsnd {\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082}\n  (g : \u03b2\u2080 \u2192 F \u03b2\u2081) (g' : \u03b2\u2081 \u2192 G \u03b2\u2082) (x : t \u03b1 \u03b2\u2080) :\n  comp.mk (tsnd g' <$> tsnd g x) = tsnd (comp.mk \u2218 map g' \u2218 g) x :=\nby rw \u2190 comp_bitraverse; simp [tsnd]; refl\n\nopen bifunctor\n\nprivate lemma pure_eq_id_mk_comp_id {\u03b1} :\n  pure = id.mk \u2218 @id \u03b1 := rfl\n\nopen function\n\n@[higher_order]\nlemma tfst_eq_fst_id {\u03b1 \u03b1' \u03b2} (f : \u03b1 \u2192 \u03b1') (x : t \u03b1 \u03b2) :\n  tfst (id.mk \u2218 f) x = id.mk (fst f x) :=\nby simp [tfst,fst,pure_eq_id_mk_comp_id,-comp.right_id,bitraverse_eq_bimap_id]\n\n@[higher_order]\nlemma tsnd_eq_snd_id {\u03b1 \u03b2 \u03b2'} (f : \u03b2 \u2192 \u03b2') (x : t \u03b1 \u03b2) :\n  tsnd (id.mk \u2218 f) x = id.mk (snd f x) :=\nby simp [tsnd,snd,pure_eq_id_mk_comp_id,-comp.right_id,bitraverse_eq_bimap_id]\n\nattribute [functor_norm] comp_bitraverse comp_tsnd comp_tfst\n  tsnd_comp_tsnd tsnd_comp_tfst tfst_comp_tsnd tfst_comp_tfst\n  bitraverse_comp bitraverse_id_id tfst_id tsnd_id\n\nend bitraversable\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category/bitraversable/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.39445218065161075}}
{"text": "/-\nCopyright (c) 2021 Junyan Xu. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Junyan Xu\n-/\n\nimport topology.sheaves.sheaf_condition.pairwise_intersections\n\n/-!\n# functors between categories of sheaves\n\nShow that the pushforward of a sheaf is a sheaf, and define\nthe pushforward functor from the category of C-valued sheaves\non X to that of sheaves on Y, given a continuous map between\ntopological spaces X and Y.\n\nTODO: pullback for presheaves and sheaves\n-/\n\nnoncomputable theory\n\nuniverses v u u\u2081\n\nopen category_theory\nopen category_theory.limits\nopen topological_space\n\nvariables {C : Type u\u2081} [category.{v} C]\nvariables {X Y : Top.{v}} (f : X \u27f6 Y)\nvariables \u2983\u03b9 : Type v\u2984 {U : \u03b9 \u2192 opens Y}\n\nnamespace Top\nnamespace presheaf.sheaf_condition_pairwise_intersections\n\nlemma map_diagram :\n  pairwise.diagram U \u22d9 opens.map f = pairwise.diagram ((opens.map f).obj \u2218 U) :=\nbegin\n  apply functor.hext,\n  abstract obj_eq {intro i, cases i; refl},\n  intros i j g, apply subsingleton.helim,\n  iterate 2 {rw map_diagram.obj_eq},\nend\n\nlemma map_cocone : (opens.map f).map_cocone (pairwise.cocone U)\n                     == pairwise.cocone ((opens.map f).obj \u2218 U) :=\nbegin\n  unfold functor.map_cocone cocones.functoriality, dsimp, congr,\n  iterate 2 {rw map_diagram, rw opens.map_supr},\n  apply subsingleton.helim, rw [map_diagram, opens.map_supr],\n  apply proof_irrel_heq,\nend\n\ntheorem pushforward_sheaf_of_sheaf {F : presheaf C X}\n  (h : F.is_sheaf_pairwise_intersections) :\n  (f _* F).is_sheaf_pairwise_intersections :=\n\u03bb \u03b9 U, begin\n  convert h ((opens.map f).obj \u2218 U) using 2,\n  rw \u2190 map_diagram, refl,\n  change F.map_cone ((opens.map f).map_cocone _).op == _,\n  congr, iterate 2 {rw map_diagram}, apply map_cocone,\nend\n\nend presheaf.sheaf_condition_pairwise_intersections\n\nnamespace sheaf\n\nopen presheaf\n\nvariables [has_products.{v} C]\n\n/--\nThe pushforward of a sheaf (by a continuous map) is a sheaf.\n-/\ntheorem pushforward_sheaf_of_sheaf\n  {F : presheaf C X} (h : F.is_sheaf) : (f _* F).is_sheaf :=\nby rw is_sheaf_iff_is_sheaf_pairwise_intersections at h \u22a2;\n   exact sheaf_condition_pairwise_intersections.pushforward_sheaf_of_sheaf f h\n\n/--\nThe pushforward functor.\n-/\ndef pushforward (f : X \u27f6 Y) : X.sheaf C \u2964 Y.sheaf C :=\n{ obj := \u03bb \u2131, \u27e8f _* \u2131.1, pushforward_sheaf_of_sheaf f \u2131.2\u27e9,\n  map := \u03bb _ _, pushforward_map f }\n\nend sheaf\n\nend Top\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/topology/sheaves/functors.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3943666212908988}}
{"text": "/-\nCopyright (c) 2021 Ashvni Narayanan. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ashvni Narayanan\n-/\nimport bernoulli_measure.ind_fn\nimport bernoulli_measure.loc_const_properties\nimport bernoulli_measure.from_loc_const\n\n/-!\n# Bernoulli measure and the p-adic L-function\nThis file defines the Bernoulli measure on `zmod d \u00d7 \u2124_[p]`. We prove that\nthis p-adic distribution is indeed a p-adic measure. As a consequence, we are also able to define\nthe p-adic L-function in terms of a p-adic integral.\n\n## Main definitions\n * `bernoulli_measure`\n\n## Implementation notes\n * `g_to_seq` replaced with `from_loc_const_to_seq`\n\n## References\nIntroduction to Cyclotomic Fields, Washington (Chapter 12, Section 2)\n\n## Tags\np-adic, L-function, Bernoulli measure\n-/\n\nlocal attribute [instance] zmod.topological_space\n\nvariables {p : \u2115} [fact p.prime] {d : \u2115} (R : Type*) [normed_comm_ring R] {c : \u2115} [fact (0 < d)]\n\nset_option old_structure_cmd true\n\nopen_locale big_operators\n\nopen padic_int zmod nat locally_constant eventually_constant_seq\n\nnamespace clopen_from\nlemma char_fn_eq {n : \u2115} (i : zmod (d * p^n)) :\n  _root_.char_fn R (clopen_from.is_clopen (i.val : zmod (d * p^n))) =\n  _root_.char_fn R (clopen_from.is_clopen i) := by { congr, rw [nat_cast_val, zmod.cast_id], }\nend clopen_from\n\nopen clopen_from\n\nlemma helper_3 (f : locally_constant ((zmod d) \u00d7 \u2124_[p]) R) {n : \u2115} (i : zmod (d * p^n)) :\n  (f (i.val)) \u2022 _root_.char_fn R (clopen_from.is_clopen (i.val : zmod (d * p^n))) =\n  f i \u2022 _root_.char_fn R (clopen_from.is_clopen i) := by { rw [nat_cast_val, char_fn_eq], }\n\nlemma s_nonempty [normed_algebra \u211a_[p] R] (hc : c.coprime p) (hc' : c.coprime d)\n  (h' : d.coprime p) (n : \u2115) (f : locally_constant ((zmod d) \u00d7 \u2124_[p]) R) :\n  {i : zmod (d * p^n) | \u2225(loc_const_to_seq_limit R hc hc' h') (f \u2191i \u2022\n  _root_.char_fn R (clopen_from.is_clopen i))\u2225 = \u2a06 (i : zmod (d * p ^ n)),\n  \u2225(loc_const_to_seq_limit R hc hc' h') (f i \u2022 _root_.char_fn R (clopen_from.is_clopen i))\u2225 }.nonempty :=\nbegin\n  have := set.nonempty.cSup_mem _ _,\n  swap 4, { refine set.range (\u03bb (i : zmod (d * p^n)), \u2225((loc_const_to_seq_limit R hc hc' h'))\n    (f \u2191i \u2022 _root_.char_fn R (clopen_from.is_clopen i))\u2225), },\n  { cases this with y hy,\n    simp only [algebra.id.smul_eq_mul, linear_map.map_smul] at hy,\n    refine \u27e8y, _\u27e9,\n    simp only [zmod.cast_id', algebra.id.smul_eq_mul, id.def, set.mem_set_of_eq,\n      finset.mem_range, linear_map.map_smul, nat_cast_val, hy, Sup_range], },\n  { apply_instance, },\n  { rw set.range_nonempty_iff_nonempty, apply_instance, },\n  { rw \u2190set.image_univ, apply set.finite.image, exact set.finite_univ, },\nend\n\nopen discrete_quotient_of_to_zmod_pow clopen_from\n\nlemma exists_mul_inv_val_eq (hc' : c.coprime d) (hc : c.coprime p) (k : \u2115) :\n  \u2203 z : \u2115, c * ((c : zmod (d * p^(2 * k)))\u207b\u00b9.val) = dite (1 < d * p^(2 * k))\n  (\u03bb h, 1 + z * (d * p^(2 * k))) (\u03bb h, 0) :=\nbegin\n  by_cases eq_one : (d * p^(2 * k)) = 1,\n  { have k_zero : \u00ac 1 < d * p^(2 * k) := by { rw [eq_one, nat.lt_one_iff], apply nat.one_ne_zero, },\n    refine \u27e81, _\u27e9,\n    rw [dif_neg k_zero, eq_one],\n    simp only [nat.mul_eq_zero, zmod.val_eq_zero, eq_iff_true_of_subsingleton, or_true], },\n  have h : (1 : zmod (d * p^(2 * k))).val = 1,\n  { have : ((1 : \u2115) : zmod (d * p^(2 * k))) = 1 := nat.cast_one,\n    rw [\u2190this, zmod.val_cast_of_lt (nat.one_lt_mul_pow_of_ne_one eq_one)], },\n  simp_rw dif_pos (nat.one_lt_mul_pow_of_ne_one eq_one),\n  conv { congr, funext, find 1 {rw \u2190 h}, rw mul_comm z _, },\n  apply (nat_coe_zmod_eq_iff (d * p^(2 * k)) _ _).1 _,\n  { rw [nat.cast_mul, nat_cast_val, cast_inv (coprime.mul_pow _ hc' hc) dvd_rfl,\n      @cast_nat_cast _ (zmod (d * p ^ (2 * k))) _ _ (zmod.char_p _) dvd_rfl c],\n    apply coe_mul_inv_eq_one _ (coprime.mul_pow _ hc' hc), },\nend\n.\nopen nat\nlemma helper_meas_bernoulli_distribution {n : \u2115} (a : zmod (d * p^n)) (hc' : c.coprime d)\n  (hc : c.coprime p) : \u2203 z : \u2124, int.fract ((a.val : \u211a) / (\u2191d * \u2191p ^ n)) -\n  \u2191c * int.fract (\u2191((c : zmod (d * p^(2 * n)))\u207b\u00b9.val) * (a : \u211a) / (\u2191d * \u2191p ^ n)) = z :=\nbegin\n  obtain \u27e8z, hz\u27e9 := int.fract_mul_nat ((\u2191((c : zmod (d * p^(2 * n)))\u207b\u00b9.val) *\n    (a : \u211a) / (\u2191d * \u2191p ^ n))) c,\n  obtain \u27e8z', hz'\u27e9 := exists_mul_inv_val_eq hc' hc n,\n  rw [mul_comm, mul_comm _ (c : \u211a), \u2190mul_div, \u2190mul_assoc, \u2190nat.cast_mul] at hz,\n  by_cases pos : 1 < d * p^(2 * n),\n  { refine \u27e8-z, _\u27e9,\n    rw dif_pos pos at hz',\n    rw [hz', nat.cast_add, nat.cast_one, one_add_mul] at hz,\n    conv at hz { congr, congr, skip, congr, congr, skip, congr, rw [pow_mul', pow_succ, pow_one], },\n    rw [\u2190mul_assoc d (p^n), mul_comm (d * p^n) (p^n), \u2190mul_assoc z' _ _, nat.cast_mul,\n      mul_comm _ (\u2191(d * p ^ n)), mul_assoc, mul_div (\u2191(z' * p ^ n)) _ _, \u2190nat.cast_pow,\n      \u2190nat.cast_mul, mul_div_cancel', \u2190nat_cast_val, \u2190nat.cast_mul,\n      \u2190int.cast_coe_nat (z' * p ^ n * a.val), int.fract_add_int] at hz,\n    { rw [int.cast_neg, \u2190hz, neg_sub, nat_cast_val a, nat.cast_mul d _, nat.cast_pow, mul_div], },\n    { norm_cast, apply ne_zero_of_lt' 0, apply_instance, }, },\n  { simp_rw [\u2190nat.cast_pow, \u2190nat_cast_val a, \u2190nat.cast_mul,\n      mul_pow_eq_one_of_mul_pow_sq_not_one_lt pos, nat.cast_one, div_one, \u2190int.cast_coe_nat],\n    refine \u27e80, by { simp_rw [int.cast_zero, int.fract_coe, mul_zero, sub_zero] }\u27e9, },\nend\n\nlemma meas_bernoulli_distribution [normed_algebra \u211a_[p] R] [norm_one_class R] {n : \u2115} {a : zmod (d * p^n)}\n  (hc : c.coprime p) (hc' : c.coprime d) (h' : d.coprime p) : \u2225(loc_const_to_seq_limit R hc hc' h')\n  (_root_.char_fn R (clopen_from.is_clopen a))\u2225 \u2264 1 + \u2225(algebra_map \u211a \u211a_[p]) (((c - 1) / 2 : \u211a))\u2225 :=\nbegin\n  convert_to \u2225(algebra_map \u211a_[p] R) (bernoulli_distribution p d c n a)\u2225 \u2264 _,\n  { rw [linear_map.coe_mk, sequence_limit_eq _ _ (seq_lim_from_loc_const_char_fn R a hc hc' h'),\n      from_loc_const_to_seq], },\n  obtain \u27e8z, hz\u27e9 := helper_meas_bernoulli_distribution a hc' hc,\n  rw [bernoulli_distribution], simp only,\n  rw [ring_hom.map_add, norm_algebra_map'],\n  apply le_trans (norm_add_le _ _) (add_le_add_right _ _),\n  rw [hz, ring_hom.map_int_cast],\n  apply padic_norm_e.norm_int_le_one z,\nend\n\nopen loc_const_ind_fn\n\n/-- Constructs a Bernoulli measure from `loc_const_to_seq_limit`. -/\n-- we choose to work with `val` and `nat` because it gives common ground without having to use CRT\nnoncomputable def bernoulli_measure [normed_algebra \u211a_[p] R] [norm_one_class R] [nontrivial R]\n  (hc : c.gcd p = 1) (hc' : c.gcd d = 1) (h' : d.gcd p = 1)\n  (na : \u2200 (n : \u2115) (f : \u2115 \u2192 R), \u2225\u2211 i in finset.range n, f i\u2225 \u2264 \u2a06 (i : zmod n), \u2225f (i.val)\u2225) :\n  measures (units (zmod d) \u00d7 units \u2124_[p]) R :=\n\u27e8 { to_fun := \u03bb f, loc_const_to_seq_limit R hc hc' h' (loc_const_ind_fn f),\n    map_add' := \u03bb f1 f2, by { rw [add, linear_map.map_add], },\n    map_smul' := \u03bb m f, by { rw [smul R m f, linear_map.map_smul, ring_hom.id_apply], }, },\n  begin\n    set K := 1 + \u2225(algebra_map \u211a \u211a_[p]) (((c - 1) / 2 : \u211a))\u2225 with hK,\n    have Kpos : 0 < K,\n    { rw [hK, add_comm],\n      apply add_pos_of_nonneg_of_pos (norm_nonneg _) zero_lt_one, },\n    refine \u27e8K, Kpos, \u03bb f, _\u27e9,\n    obtain \u27e8n, hn\u27e9 := loc_const_eq_sum_char_fn R (loc_const_ind_fn f) h',\n    change \u2225loc_const_to_seq_limit R hc hc' h' (loc_const_ind_fn f)\u2225 \u2264 _,\n    rw [hn, linear_map.map_sum],\n    apply le_trans (na (d * p^n) _) _,\n    simp_rw [helper_3],\n    set i := (s_nonempty R hc hc' h' n (loc_const_ind_fn f)).some with hi,\n    have hi' := (s_nonempty R hc hc' h' n (loc_const_ind_fn f)).some_spec,\n    change \u2225loc_const_to_seq_limit R hc hc' h' ((loc_const_ind_fn f) \u2191i \u2022\n      _root_.char_fn R (clopen_from.is_clopen i))\u2225 = \u2a06 (i : zmod (d * p ^ n)),\n      \u2225loc_const_to_seq_limit R hc hc' h' (((loc_const_ind_fn f) \u2191i) \u2022\n      _root_.char_fn R (clopen_from.is_clopen i))\u2225 at hi',\n    by_cases is_unit (i : zmod d \u00d7 \u2124_[p]).fst \u2227 is_unit (i : zmod d \u00d7 \u2124_[p]).snd,\n    { suffices : (\u2a06 (i : zmod (d * p ^ n)), \u2225loc_const_to_seq_limit R hc hc' h'\n        (((loc_const_ind_fn f) \u2191i) \u2022 _root_.char_fn R (clopen_from.is_clopen i))\u2225) \u2264\n        K * \u2225(loc_const_ind_fn f) \u2191i\u2225,\n      { apply le_trans this ((mul_le_mul_left Kpos).2 _),\n        rw continuous_map.norm_eq_supr_norm,\n        refine le_cSup (set.finite.bdd_above (is_locally_constant.range_finite\n          (is_locally_constant.comp f.is_locally_constant _))) \u27e8(is_unit.unit h.1,\n          is_unit.unit h.2), by { rw [loc_const_ind_fn_def, ind_fn.map_ind_fn_eq_fn _ h], refl, }\u27e9, },\n      { rw [\u2190hi', linear_map.map_smul, smul_eq_mul],\n        apply le_trans (norm_mul_le _ _) _,\n        rw mul_comm,\n        refine mul_le_mul (meas_bernoulli_distribution R hc hc' h') le_rfl (norm_nonneg _) (le_of_lt Kpos), }, },\n    { rw [loc_const_ind_fn_def, ind_fn.map_ind_fn_eq_zero _ h, zero_smul, linear_map.map_zero,\n        norm_zero] at hi',\n      rw [\u2190hi'],\n      apply mul_nonneg (le_of_lt Kpos) (norm_nonneg _), }, end\u27e9\n.\n\nlemma integral_loc_const_eval [nontrivial R] [complete_space R] [normed_algebra \u211a_[p] R] [norm_one_class R] \n  (hc : c.gcd p = 1) (hc' : c.gcd d = 1) (hd : d.gcd p = 1)\n  (na : \u2200 (n : \u2115) (f : \u2115 \u2192 R), \u2225\u2211 i in finset.range n, f i\u2225 \u2264 \u2a06 (i : zmod n), \u2225f (i.val)\u2225) \n  (f : locally_constant ((zmod d)\u02e3 \u00d7 \u2124_[p]\u02e3) R) :\n  measure.integral (bernoulli_measure R hc hc' hd na) f = (bernoulli_measure R hc hc' hd na).val f :=\nbegin\n  delta measure.integral, \n  simp only [continuous_linear_map.coe_mk', linear_map.coe_mk, subtype.val_eq_coe],\n  convert dense_inducing.extend_eq (measure.dense_ind_inclusion _ _) (measure.integral_cont _) _,\n  apply_instance,\n  apply_instance,\n  apply_instance,\nend", "meta": {"author": "laughinggas", "repo": "p-adic-L-functions", "sha": "bfc0c84fabe9b89e3da79f95d7a8eacabe8a5bb7", "save_path": "github-repos/lean/laughinggas-p-adic-L-functions", "path": "github-repos/lean/laughinggas-p-adic-L-functions/p-adic-L-functions-bfc0c84fabe9b89e3da79f95d7a8eacabe8a5bb7/src/bernoulli_measure/bernoulli_measure_def.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3943666055611252}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.category_theory.limits.lattice\nimport Mathlib.PostPort\n\nuniverses v l \n\nnamespace Mathlib\n\n/-!\n# The category of \"pairwise intersections\".\n\nGiven `\u03b9 : Type v`, we build the diagram category `pairwise \u03b9`\nwith objects `single i` and `pair i j`, for `i j : \u03b9`,\nwhose only non-identity morphisms are\n`left : pair i j \u27f6 single i` and `right : pair i j \u27f6 single j`.\n\nWe use this later in describing (one formulation of) the sheaf condition.\n\nGiven any function `U : \u03b9 \u2192 \u03b1`, where `\u03b1` is some complete lattice (e.g. `(opens X)\u1d52\u1d56`),\nwe produce a functor `pairwise \u03b9 \u2964 \u03b1` in the obvious way,\nand show that `supr U` provides a colimit cocone over this functor.\n-/\n\nnamespace category_theory\n\n\n/--\nAn inductive type representing either a single term of a type `\u03b9`, or a pair of terms.\nWe use this as the objects of a category to describe the sheaf condition.\n-/\ninductive pairwise (\u03b9 : Type v) where\n| single : \u03b9 \u2192 pairwise \u03b9\n| pair : \u03b9 \u2192 \u03b9 \u2192 pairwise \u03b9\n\nnamespace pairwise\n\n\nprotected instance pairwise_inhabited {\u03b9 : Type v} [Inhabited \u03b9] : Inhabited (pairwise \u03b9) :=\n  { default := single Inhabited.default }\n\n/--\nMorphisms in the category `pairwise \u03b9`. The only non-identity morphisms are\n`left i j : single i \u27f6 pair i j` and `right i j : single j \u27f6 pair i j`.\n-/\ninductive hom {\u03b9 : Type v} : pairwise \u03b9 \u2192 pairwise \u03b9 \u2192 Type v where\n| id_single : (i : \u03b9) \u2192 hom (single i) (single i)\n| id_pair : (i j : \u03b9) \u2192 hom (pair i j) (pair i j)\n| left : (i j : \u03b9) \u2192 hom (pair i j) (single i)\n| right : (i j : \u03b9) \u2192 hom (pair i j) (single j)\n\nprotected instance hom_inhabited {\u03b9 : Type v} [Inhabited \u03b9] :\n    Inhabited (hom (single Inhabited.default) (single Inhabited.default)) :=\n  { default := hom.id_single Inhabited.default }\n\n/--\nThe identity morphism in `pairwise \u03b9`.\n-/\ndef id {\u03b9 : Type v} (o : pairwise \u03b9) : hom o o := sorry\n\n/-- Composition of morphisms in `pairwise \u03b9`. -/\ndef comp {\u03b9 : Type v} {o\u2081 : pairwise \u03b9} {o\u2082 : pairwise \u03b9} {o\u2083 : pairwise \u03b9} (f : hom o\u2081 o\u2082)\n    (g : hom o\u2082 o\u2083) : hom o\u2081 o\u2083 :=\n  sorry\n\nprotected instance category_theory.category {\u03b9 : Type v} : category (pairwise \u03b9) := category.mk\n\n/-- Auxilliary definition for `diagram`. -/\n@[simp] def diagram_obj {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] :\n    pairwise \u03b9 \u2192 \u03b1 :=\n  sorry\n\n/-- Auxilliary definition for `diagram`. -/\n@[simp] def diagram_map {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] {o\u2081 : pairwise \u03b9}\n    {o\u2082 : pairwise \u03b9} (f : o\u2081 \u27f6 o\u2082) : diagram_obj U o\u2081 \u27f6 diagram_obj U o\u2082 :=\n  sorry\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[semilattice_inf \u03b1]`, we obtain a functor `pairwise \u03b9 \u2964 \u03b1`,\nsending `single i` to `U i` and `pair i j` to `U i \u2293 U j`,\nand the morphisms to the obvious inequalities.\n-/\ndef diagram {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] : pairwise \u03b9 \u2964 \u03b1 :=\n  functor.mk (diagram_obj U) fun (X Y : pairwise \u03b9) (f : X \u27f6 Y) => diagram_map U f\n\n-- `complete_lattice` is not really needed, as we only ever use `inf`,\n\n-- but the appropriate structure has not been defined.\n\n/-- Auxilliary definition for `cocone`. -/\ndef cocone_\u03b9_app {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] (o : pairwise \u03b9) :\n    diagram_obj U o \u27f6 supr U :=\n  sorry\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[complete_lattice \u03b1]`,\n`supr U` provides a cocone over `diagram U`.\n-/\n@[simp] theorem cocone_X {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] :\n    limits.cocone.X (cocone U) = supr U :=\n  Eq.refl (limits.cocone.X (cocone U))\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[complete_lattice \u03b1]`,\n`infi U` provides a limit cone over `diagram U`.\n-/\ndef cocone_is_colimit {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] :\n    limits.is_colimit (cocone U) :=\n  limits.is_colimit.mk fun (s : limits.cocone (diagram U)) => hom_of_le sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/pairwise_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.629774621301746, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.39431711660182206}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.bifunctor\nimport control.traversable.basic\n\n/-!\n# Bitraversable type class\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nType class for traversing bifunctors.\n\nSimple examples of `bitraversable` are `prod` and `sum`. A more elaborate example is\nto define an a-list as:\n\n```\ndef alist (key val : Type) := list (key \u00d7 val)\n```\n\nThen we can use `f : key \u2192 io key'` and `g : val \u2192 io val'` to manipulate the `alist`'s key\nand value respectively with `bitraverse f g : alist key val \u2192 io (alist key' val')`\n\n## Main definitions\n\n* `bitraversable`: Bare typeclass to hold the `bitraverse` function.\n* `is_lawful_bitraversable`: Typeclass for the laws of the `bitraverse` function. Similar to\n  `is_lawful_traversable`.\n\n## References\n\nThe concepts and laws are taken from\n<https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable iterator functor bifunctor applicative\n-/\n\nuniverses u\n\n/-- Lawless bitraversable bifunctor. This only holds data for the bimap and bitraverse. -/\nclass bitraversable (t : Type u \u2192 Type u \u2192 Type u)\n  extends bifunctor t :=\n(bitraverse : \u03a0 {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b1' \u03b2 \u03b2'},\n  (\u03b1 \u2192 m \u03b1') \u2192 (\u03b2 \u2192 m \u03b2') \u2192 t \u03b1 \u03b2 \u2192 m (t \u03b1' \u03b2'))\nexport bitraversable ( bitraverse )\n\n/-- A bitraversable functor commutes with all applicative functors. -/\ndef bisequence {t m} [bitraversable t] [applicative m] {\u03b1 \u03b2} : t (m \u03b1) (m \u03b2) \u2192 m (t \u03b1 \u03b2) :=\nbitraverse id id\n\nopen functor\n\n/-- Bifunctor. This typeclass asserts that a lawless bitraversable bifunctor is lawful. -/\nclass is_lawful_bitraversable (t : Type u \u2192 Type u \u2192 Type u) [bitraversable t]\n  extends is_lawful_bifunctor t :=\n(id_bitraverse : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), bitraverse id.mk id.mk x = id.mk x )\n(comp_bitraverse : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3'} (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3')\n    (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2') (x : t \u03b1 \u03b1'),\n  bitraverse (comp.mk \u2218 map f \u2218 g) (comp.mk \u2218 map f' \u2218 g') x =\n  comp.mk (bitraverse f f' <$> bitraverse g g' x) )\n(bitraverse_eq_bimap_id : \u2200 {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') (x : t \u03b1 \u03b1'),\n   bitraverse (id.mk \u2218 f) (id.mk \u2218 f') x = id.mk (bimap f f' x))\n(binaturality : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 \u03b1' \u03b2 \u03b2'}\n    (f : \u03b1 \u2192 F \u03b2) (f' : \u03b1' \u2192 F \u03b2') (x : t \u03b1 \u03b1'),\n  \u03b7 (bitraverse f f' x) = bitraverse (@\u03b7 _ \u2218 f) (@\u03b7 _ \u2218 f') x)\n\nexport is_lawful_bitraversable ( id_bitraverse comp_bitraverse\n                                 bitraverse_eq_bimap_id  )\nopen is_lawful_bitraversable\n\nattribute [higher_order bitraverse_id_id] id_bitraverse\nattribute [higher_order bitraverse_comp] comp_bitraverse\nattribute [higher_order] binaturality bitraverse_eq_bimap_id\n\nexport is_lawful_bitraversable (bitraverse_id_id bitraverse_comp)\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/control/bitraversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.3943171079861623}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.bifunctor\nimport control.traversable.basic\n\n/-!\n# Bitraversable type class\n\nType class for traversing bifunctors.\n\nSimple examples of `bitraversable` are `prod` and `sum`. A more elaborate example is\nto define an a-list as:\n\n```\ndef alist (key val : Type) := list (key \u00d7 val)\n```\n\nThen we can use `f : key \u2192 io key'` and `g : val \u2192 io val'` to manipulate the `alist`'s key\nand value respectively with `bitraverse f g : alist key val \u2192 io (alist key' val')`\n\n## Main definitions\n\n* `bitraversable`: Bare typeclass to hold the `bitraverse` function.\n* `is_lawful_bitraversable`: Typeclass for the laws of the `bitraverse` function. Similar to\n  `is_lawful_traversable`.\n\n## References\n\nThe concepts and laws are taken from\n<https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable iterator functor bifunctor applicative\n-/\n\nuniverses u\n\n/-- Lawless bitraversable bifunctor. This only holds data for the bimap and bitraverse. -/\nclass bitraversable (t : Type u \u2192 Type u \u2192 Type u)\n  extends bifunctor t :=\n(bitraverse : \u03a0 {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b1' \u03b2 \u03b2'},\n  (\u03b1 \u2192 m \u03b1') \u2192 (\u03b2 \u2192 m \u03b2') \u2192 t \u03b1 \u03b2 \u2192 m (t \u03b1' \u03b2'))\nexport bitraversable ( bitraverse )\n\n/-- A bitraversable functor commutes with all applicative functors. -/\ndef bisequence {t m} [bitraversable t] [applicative m] {\u03b1 \u03b2} : t (m \u03b1) (m \u03b2) \u2192 m (t \u03b1 \u03b2) :=\nbitraverse id id\n\nopen functor\n\n/-- Bifunctor. This typeclass asserts that a lawless bitraversable bifunctor is lawful. -/\nclass is_lawful_bitraversable (t : Type u \u2192 Type u \u2192 Type u) [bitraversable t]\n  extends is_lawful_bifunctor t :=\n(id_bitraverse : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), bitraverse id.mk id.mk x = id.mk x )\n(comp_bitraverse : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3'} (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3')\n    (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2') (x : t \u03b1 \u03b1'),\n  bitraverse (comp.mk \u2218 map f \u2218 g) (comp.mk \u2218 map f' \u2218 g') x =\n  comp.mk (bitraverse f f' <$> bitraverse g g' x) )\n(bitraverse_eq_bimap_id : \u2200 {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') (x : t \u03b1 \u03b1'),\n   bitraverse (id.mk \u2218 f) (id.mk \u2218 f') x = id.mk (bimap f f' x))\n(binaturality : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 \u03b1' \u03b2 \u03b2'}\n    (f : \u03b1 \u2192 F \u03b2) (f' : \u03b1' \u2192 F \u03b2') (x : t \u03b1 \u03b1'),\n  \u03b7 (bitraverse f f' x) = bitraverse (@\u03b7 _ \u2218 f) (@\u03b7 _ \u2218 f') x)\n\nexport is_lawful_bitraversable ( id_bitraverse comp_bitraverse\n                                 bitraverse_eq_bimap_id  )\nopen is_lawful_bitraversable\n\nattribute [higher_order bitraverse_id_id] id_bitraverse\nattribute [higher_order bitraverse_comp] comp_bitraverse\nattribute [higher_order] binaturality bitraverse_eq_bimap_id\n\nexport is_lawful_bitraversable (bitraverse_id_id bitraverse_comp)\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/control/bitraversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.6297746143530797, "lm_q1q2_score": 0.3943171034638414}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport logic.equiv.defs\n\n/-!\n# A type for VM-erased data\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines a type `erased \u03b1` which is classically isomorphic to `\u03b1`,\nbut erased in the VM. That is, at runtime every value of `erased \u03b1` is\nrepresented as `0`, just like types and proofs.\n-/\n\nuniverses u\n\n/-- `erased \u03b1` is the same as `\u03b1`, except that the elements\n  of `erased \u03b1` are erased in the VM in the same way as types\n  and proofs. This can be used to track data without storing it\n  literally. -/\ndef erased (\u03b1 : Sort u) : Sort (max 1 u) :=\n\u03a3' s : \u03b1 \u2192 Prop, \u2203 a, (\u03bb b, a = b) = s\n\nnamespace erased\n\n/-- Erase a value. -/\n@[inline] def mk {\u03b1} (a : \u03b1) : erased \u03b1 := \u27e8\u03bb b, a = b, a, rfl\u27e9\n\n/-- Extracts the erased value, noncomputably. -/\nnoncomputable def out {\u03b1} : erased \u03b1 \u2192 \u03b1\n| \u27e8s, h\u27e9 := classical.some h\n\n/--\nExtracts the erased value, if it is a type.\n\nNote: `(mk a).out_type` is not definitionally equal to `a`.\n-/\n@[reducible] def out_type (a : erased (Sort u)) : Sort u := out a\n\n/-- Extracts the erased value, if it is a proof. -/\ntheorem out_proof {p : Prop} (a : erased p) : p := out a\n\n@[simp] theorem out_mk {\u03b1} (a : \u03b1) : (mk a).out = a :=\nbegin\n  let h, show classical.some h = a,\n  have := classical.some_spec h,\n  exact cast (congr_fun this a).symm rfl\nend\n\n@[simp] theorem mk_out {\u03b1} : \u2200 (a : erased \u03b1), mk (out a) = a\n| \u27e8s, h\u27e9 := by simp [mk]; congr; exact classical.some_spec h\n\n@[ext] lemma out_inj {\u03b1} (a b : erased \u03b1) (h : a.out = b.out) : a = b :=\nby simpa using congr_arg mk h\n\n/-- Equivalence between `erased \u03b1` and `\u03b1`. -/\nnoncomputable def equiv (\u03b1) : erased \u03b1 \u2243 \u03b1 :=\n\u27e8out, mk, mk_out, out_mk\u27e9\n\ninstance (\u03b1 : Type u) : has_repr (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\ninstance (\u03b1 : Type u) : has_to_string (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\nmeta instance (\u03b1 : Type u) : has_to_format (erased \u03b1) := \u27e8\u03bb _, (\"erased\" : format)\u27e9\n\n/-- Computably produce an erased value from a proof of nonemptiness. -/\ndef choice {\u03b1} (h : nonempty \u03b1) : erased \u03b1 := mk (classical.choice h)\n\n@[simp] theorem nonempty_iff {\u03b1} : nonempty (erased \u03b1) \u2194 nonempty \u03b1 :=\n\u27e8\u03bb \u27e8a\u27e9, \u27e8a.out\u27e9, \u03bb \u27e8a\u27e9, \u27e8mk a\u27e9\u27e9\n\ninstance {\u03b1} [h : nonempty \u03b1] : inhabited (erased \u03b1) :=\n\u27e8choice h\u27e9\n\n/--\n`(>>=)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `monad`).\n-/\ndef bind {\u03b1 \u03b2} (a : erased \u03b1) (f : \u03b1 \u2192 erased \u03b2) : erased \u03b2 :=\n\u27e8\u03bb b, (f a.out).1 b, (f a.out).2\u27e9\n\n@[simp] theorem bind_eq_out {\u03b1 \u03b2} (a f) : @bind \u03b1 \u03b2 a f = f a.out :=\nby delta bind bind._proof_1; cases f a.out; refl\n\n/--\nCollapses two levels of erasure.\n-/\ndef join {\u03b1} (a : erased (erased \u03b1)) : erased \u03b1 := bind a id\n\n@[simp] theorem join_eq_out {\u03b1} (a) : @join \u03b1 a = a.out := bind_eq_out _ _\n\n/--\n`(<$>)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `functor`).\n-/\ndef map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (a : erased \u03b1) : erased \u03b2 :=\nbind a (mk \u2218 f)\n\n@[simp] theorem map_out {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (a : erased \u03b1) : (a.map f).out = f a.out :=\nby simp [map]\n\ninstance : monad erased := { pure := @mk, bind := @bind, map := @map }\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 erased \u03b1) = @mk _ := rfl\n@[simp] lemma bind_def {\u03b1 \u03b2} : ((>>=) : erased \u03b1 \u2192 (\u03b1 \u2192 erased \u03b2) \u2192 erased \u03b2) = @bind _ _ := rfl\n@[simp] lemma map_def {\u03b1 \u03b2} : ((<$>) : (\u03b1 \u2192 \u03b2) \u2192 erased \u03b1 \u2192 erased \u03b2) = @map _ _ := rfl\n\ninstance : is_lawful_monad erased := by refine {..}; intros; ext; simp\n\nend erased\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/erased.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.6224593382055109, "lm_q1q2_score": 0.39427214215651046}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Floris van Doorn, Mario Carneiro\n-/\n\nimport tactic.core\n\n/-!\n# `choose` tactic\n\nPerforms Skolemization, that is, given `h : \u2200 a:\u03b1, \u2203 b:\u03b2, p a b |- G` produces\n`f : \u03b1 \u2192 \u03b2, hf: \u2200 a, p a (f a) |- G`.\n-/\n\nnamespace tactic\n\n/-- Given `\u03b1 : Sort u`, `nonemp : nonempty \u03b1`, `p : \u03b1 \u2192 Prop`, a context of local variables\n`ctxt`, and a pair of an element `val : \u03b1` and `spec : p val`,\n`mk_sometimes u \u03b1 nonemp p ctx (val, spec)` produces another pair `val', spec'`\nsuch that `val'` does not have any free variables from elements of `ctxt` whose types are\npropositions. This is done by applying `function.sometimes` to abstract over all the propositional\narguments. -/\nmeta def mk_sometimes (u : level) (\u03b1 nonemp p : expr) :\n  list expr \u2192 expr \u00d7 expr \u2192 tactic (expr \u00d7 expr)\n| [] (val, spec) := pure (val, spec)\n| (e :: ctxt) (val, spec) := do\n  (val, spec) \u2190 mk_sometimes ctxt (val, spec),\n  t \u2190 infer_type e,\n  b \u2190 is_prop t,\n  pure $ if b then\n    let val' := expr.bind_lambda val e in\n    (expr.const ``function.sometimes [level.zero, u] t \u03b1 nonemp val',\n     expr.const ``function.sometimes_spec [u] t \u03b1 nonemp p val' e spec)\n  else (val, spec)\n\n/-- Changes `(h : \u2200xs, \u2203a:\u03b1, p a) \u22a2 g` to `(d : \u2200xs, a) (s : \u2200xs, p (d xs)) \u22a2 g` and\n`(h : \u2200xs, p xs \u2227 q xs) \u22a2 g` to `(d : \u2200xs, p xs) (s : \u2200xs, q xs) \u22a2 g`.\n`choose1` returns a pair of the second local constant it introduces,\nand the error result (see below).\n\nIf `nondep` is true and `\u03b1` is inhabited, then it will remove the dependency of `d` on\nall propositional assumptions in `xs`. For example if `ys` are propositions then\n`(h : \u2200xs ys, \u2203a:\u03b1, p a) \u22a2 g` becomes `(d : \u2200xs, a) (s : \u2200xs ys, p (d xs)) \u22a2 g`.\n\nThe second value returned by `choose1` is the result of nondep elimination:\n* `none`: nondep elimination was not attempted or was not applicable\n* `some none`: nondep elimination was successful\n* ``some (some `(nonempty \u03b1))``: nondep elimination was unsuccessful\n  because we could not find a `nonempty \u03b1` instance\n-/\nmeta def choose1 (nondep : bool) (h : expr) (data : name) (spec : name) :\n  tactic (expr \u00d7 option (option expr)) := do\n  t \u2190 infer_type h,\n  (ctxt, t) \u2190 whnf t >>= open_pis,\n  t \u2190 whnf t transparency.all,\n  match t with\n  | `(@Exists %%\u03b1 %%p) := do\n    \u03b1_t \u2190 infer_type \u03b1,\n    expr.sort u \u2190 whnf \u03b1_t transparency.all,\n    (ne_fail, nonemp) \u2190 if nondep then do\n      let ne := expr.const ``nonempty [u] \u03b1,\n      nonemp \u2190 try_core (mk_instance ne <|> retrieve' (do\n        m \u2190 mk_meta_var ne,\n        set_goals [m],\n        ctxt.mmap' (\u03bb e, do\n          b \u2190 is_proof e,\n          monad.unlessb b $\n            (mk_app ``nonempty.intro [e] >>= note_anon none) $> ()),\n        unfreeze_local_instances >> apply_instance,\n        instantiate_mvars m)),\n      pure (some (option.guard (\u03bb _, nonemp.is_none) ne), nonemp)\n    else pure (none, none),\n    ctxt' \u2190 if nonemp.is_some then ctxt.mfilter (\u03bb e, bnot <$> is_proof e) else pure ctxt,\n    value \u2190 mk_local_def data (\u03b1.pis ctxt'),\n    t' \u2190 head_beta (p.app (value.mk_app ctxt')),\n    spec \u2190 mk_local_def spec (t'.pis ctxt),\n    (value_proof, spec_proof) \u2190 nonemp.elim pure (\u03bb nonemp, mk_sometimes u \u03b1 nonemp p ctxt)\n      (expr.const ``classical.some [u] \u03b1 p (h.mk_app ctxt),\n       expr.const ``classical.some_spec [u] \u03b1 p (h.mk_app ctxt)),\n    dependent_pose_core [(value, value_proof.lambdas ctxt'), (spec, spec_proof.lambdas ctxt)],\n    try (tactic.clear h),\n    intro1,\n    e \u2190 intro1,\n    pure (e, ne_fail)\n  | `(%%p \u2227 %%q) := do\n    mk_app ``and.elim_left [h.mk_app ctxt] >>= lambdas ctxt >>= note data none,\n    hq \u2190 mk_app ``and.elim_right [h.mk_app ctxt] >>= lambdas ctxt >>= note spec none,\n    try (tactic.clear h),\n    pure (hq, none)\n  | _ := fail \"expected a term of the shape `\u2200xs, \u2203a, p xs a` or `\u2200xs, p xs \u2227 q xs`\"\n  end\n\n/-- Changes `(h : \u2200xs, \u2203as, p as \u2227 q as) \u22a2 g` to a list of functions `as`,\nand a final hypothesis on `p as` and `q as`. If `nondep` is true then the functions will\nbe made to not depend on propositional arguments, when possible.\n\nThe last argument is an internal recursion variable, indicating whether nondep elimination\nhas been useful so far. The tactic fails if `nondep` is true, and nondep elimination is\nattempted at least once, and it fails every time it is attempted, in which case it returns\nan error complaining about the first attempt.\n-/\nmeta def choose (nondep : bool) : expr \u2192 list name \u2192\n  opt_param (option (option expr)) none \u2192 tactic unit\n| h [] _ := fail \"expect list of variables\"\n| h [n] (some (some ne)) := do\n  g \u2190 mk_meta_var ne, set_goals [g], -- make a reasonable error state\n  fail \"choose: failed to synthesize nonempty instance\"\n| h [n] _ := do\n  cnt \u2190 revert h,\n  intro n,\n  intron (cnt - 1),\n  return ()\n| h (n::ns) ne_fail\u2081 := do\n  (v, ne_fail\u2082) \u2190 get_unused_name >>= choose1 nondep h n,\n  choose v ns $\n    match ne_fail\u2081, ne_fail\u2082 with\n    | none, _ := ne_fail\u2082\n    | some none, _ := some none\n    | _, some none := some none\n    | _, _ := ne_fail\u2081\n    end\n\nnamespace interactive\nsetup_tactic_parser\n\n/-- `choose a b h h' using hyp` takes an hypothesis `hyp` of the form\n`\u2200 (x : X) (y : Y), \u2203 (a : A) (b : B), P x y a b \u2227 Q x y a b`\nfor some `P Q : X \u2192 Y \u2192 A \u2192 B \u2192 Prop` and outputs\ninto context a function `a : X \u2192 Y \u2192 A`, `b : X \u2192 Y \u2192 B` and two assumptions:\n`h : \u2200 (x : X) (y : Y), P x y (a x y) (b x y)` and\n`h' : \u2200 (x : X) (y : Y), Q x y (a x y) (b x y)`. It also works with dependent versions.\n\n`choose! a b h h' using hyp` does the same, except that it will remove dependency of\nthe functions on propositional arguments if possible. For example if `Y` is a proposition\nand `A` and `B` are nonempty in the above example then we will instead get\n`a : X \u2192 A`, `b : X \u2192 B`, and the assumptions\n`h : \u2200 (x : X) (y : Y), P x y (a x) (b x)` and\n`h' : \u2200 (x : X) (y : Y), Q x y (a x) (b x)`.\n\nExamples:\n\n```lean\nexample (h : \u2200n m : \u2115, \u2203i j, m = n + i \u2228 m + j = n) : true :=\nbegin\n  choose i j h using h,\n  guard_hyp i : \u2115 \u2192 \u2115 \u2192 \u2115,\n  guard_hyp j : \u2115 \u2192 \u2115 \u2192 \u2115,\n  guard_hyp h : \u2200 (n m : \u2115), m = n + i n m \u2228 m + j n m = n,\n  trivial\nend\n```\n\n```lean\nexample (h : \u2200 i : \u2115, i < 7 \u2192 \u2203 j, i < j \u2227 j < i+i) : true :=\nbegin\n  choose! f h h' using h,\n  guard_hyp f : \u2115 \u2192 \u2115,\n  guard_hyp h : \u2200 (i : \u2115), i < 7 \u2192 i < f i,\n  guard_hyp h' : \u2200 (i : \u2115), i < 7 \u2192 f i < i + i,\n  trivial,\nend\n```\n-/\nmeta def choose (nondep : parse (tk \"!\")?) (first : parse ident) (names : parse ident*)\n  (tgt : parse (tk \"using\" *> texpr)?) : tactic unit := do\ntgt \u2190 match tgt with\n  | none := get_local `this\n  | some e := tactic.i_to_expr_strict e\n  end,\ntactic.choose nondep.is_some tgt (first :: names),\ntry (interactive.simp none none tt [simp_arg_type.expr\n  ``(exists_prop)] [] (loc.ns $ some <$> names)),\ntry (tactic.clear tgt)\n\nadd_tactic_doc\n{ name       := \"choose\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.choose],\n  tags       := [\"classical logic\"] }\n\nend interactive\nend tactic\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/tactic/choose.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.39427212910529874}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.const\nimport category_theory.discrete_category\n\n/-!\n# The category `discrete punit`\n\nWe define `star : C \u2964 discrete punit` sending everything to `punit.star`,\nshow that any two functors to `discrete punit` are naturally isomorphic,\nand construct the equivalence `(discrete punit \u2964 C) \u224c C`.\n-/\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nnamespace category_theory\n\nnamespace functor\nvariables (C : Type u) [category.{v} C]\n\n/-- The constant functor sending everything to `punit.star`. -/\ndef star : C \u2964 discrete punit :=\n(functor.const _).obj punit.star\n\nvariable {C}\n/-- Any two functors to `discrete punit` are isomorphic. -/\ndef punit_ext (F G : C \u2964 discrete punit) : F \u2245 G :=\nnat_iso.of_components (\u03bb _, eq_to_iso dec_trivial) (\u03bb _ _ _, dec_trivial)\n\n/--\nAny two functors to `discrete punit` are *equal*.\nYou probably want to use `punit_ext` instead of this.\n-/\nlemma punit_ext' (F G : C \u2964 discrete punit) : F = G :=\nfunctor.ext (\u03bb _, dec_trivial) (\u03bb _ _ _, dec_trivial)\n\n/-- The functor from `discrete punit` sending everything to the given object. -/\nabbreviation from_punit (X : C) : discrete punit.{v+1} \u2964 C :=\n(functor.const _).obj X\n\n/-- Functors from `discrete punit` are equivalent to the category itself. -/\n@[simps]\ndef equiv : (discrete punit \u2964 C) \u224c C :=\n{ functor :=\n  { obj := \u03bb F, F.obj punit.star,\n    map := \u03bb F G \u03b8, \u03b8.app punit.star },\n  inverse := functor.const _,\n  unit_iso :=\n  begin\n    apply nat_iso.of_components _ _,\n    intro X,\n    apply discrete.nat_iso,\n    rintro \u27e8\u27e9,\n    apply iso.refl _,\n    intros,\n    ext \u27e8\u27e9,\n    simp,\n  end,\n  counit_iso :=\n  begin\n    refine nat_iso.of_components iso.refl _,\n    intros X Y f,\n    dsimp, simp,  -- See note [dsimp, simp].\n  end }\n\nend functor\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/punit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341024983754, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.39427212049027477}}
{"text": "import category_theory.category.basic\nimport category_theory.functor\n\nimport algebra.category.Group.images\nimport algebra.category.Group.colimits\nimport algebra.category.Group.abelian\nimport algebra.category.Module.monoidal\nimport algebra.category.Ring.basic\n\nimport category_theory.abelian.basic\nimport category_theory.limits.shapes.finite_limits\n\nimport topology.instances.real\nimport topology.category.Top\nimport topology.category.UniformSpace\n\n\n/-!\nThis is a demo of the category theory library in mathlib,\nas part of \"Lean for the Curious Mathematician 2020\".\n\nYou can get this file by:\n* installing Lean if necessary: https://leanprover-community.github.io/get_started.html#regular-install\n* `leanproject get mathlib`\n* `code mathlib`\n* open the file `docs/tutorial/lftcm2020/src/demos/category_theory.lean`\n\nIf you've already got a copy of `mathlib`, you should update it now, using\n```\n  cd /path/to/mathlib/\n  git pull\n  leanproject get-cache\n```\n\nThere are also exercises associated with this demo, in\n`exercise_sources/thursday/category_theory/`\nwith hints at\n`hints/category_theory/`\nand (partial) solutions at\n`solutions/thursday/category_theory/`\nAny of Exercises 1-7 should be approachable after the demo.\nThe later exercises are quite hard, and will take you longer than the afternoon problem session!\n-/\n\nopen category_theory\n\n/-!\n## Categories\n\nCategories are implemented in mathlib as a typeclass, parametrised by the type of objects.\n\nThus to talk about an arbitrary category, we can write\n-/\nvariables (C : Type) [category C]\n\n/-!\nThere is special notation for the morphisms in a category: if `X Y : C`, we write\n* `X \u27f6 Y` for the type of morphisms from `X` to `Y`.\n  (To enter the special arrow `\u27f6`, type `\\hom`, or hover over the symbol to see the hint.)\n* `\ud835\udfd9 X` is a the identity morphisms on `X` (i.e., a term of type `X \u27f6 X`).\n* If `f : X \u27f6 Y` and `g : Y \u27f6 Z`, then we write `f \u226b g` for the composition, a morphism `X \u27f6 Z`.\n-/\n\nexample {W X Y Z : C} (f : W \u27f6 X) (g : X \u27f6 Y) (h : Y \u27f6 Z) :\n  (f \u226b (\ud835\udfd9 X \u226b g)) \u226b h = f \u226b g \u226b h :=\nbegin\n  rw category.id_comp,\n  rw category.assoc,\n  -- alternatively, just `simp` will do\nend\n\n/-!\n## Functors\n\nTo introduce functors, we'll need a second category around.\n-/\nvariables (D : Type) [category D]\n\n/-!\nWe write a functor as `F : C \u2964 D`.\n(Unlike categories, which are partially unbundled, a functor is \"fully bundled\",\ncontaining the function on objects as field. This parallels the design for algebraic structures.)\n-/\n\nexample (F : C \u2964 D) (X : C) : F.map (\ud835\udfd9 X) = \ud835\udfd9 (F.obj X) :=\nF.map_id X\n\nexample (F : C \u2964 D) {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) : F.map (f \u226b g) = F.map f \u226b F.map g :=\nF.map_comp f g\n\n/-!\nThe identity functor is written as `\ud835\udfed C`, and functor composition is written `\u22d9`.\n-/\nexample (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y) : (\ud835\udfed C \u22d9 F).map (f \u226b \ud835\udfd9 Y) = F.map f :=\nbegin\n  rw functor.comp_map,\n  rw functor.map_comp,\n  rw category_theory.functor.map_id, -- yuck! we really should fix this\n  rw functor.id_map,\n  rw functor.map_comp,\n  rw category_theory.functor.map_id,\n  rw category.comp_id,\n  -- or just replace the entire proof with `by simp`\nend\n\n/-!\nTo build a functor `F : C \u2964 D` we need to specify four fields\n* `obj : C \u2192 D`\n* `map : \u2200 {X Y : C} (f : X \u27f6 Y), obj X \u27f6 obj Y`\n* `map_id'` and `map_comp'`, expressing the functor laws.\n-/\n\nexample {X : C} : C \u2964 Type* :=\n{ obj := \u03bb Y, X \u27f6 Y,\n  map := \u03bb Y Y' f g, g \u226b f,\n  map_id' := \u03bb X, begin funext, simp, end,\n  map_comp' := \u03bb X Y Z f g, begin funext, simp, end }\n\n/-!\nHowever Lean will automatically attempt to fill in the `map_id'` and `map_comp'` fields itself,\nbecause these fields are marked with `auto_param`. This lets us specify a tactic to use to\ntry to synthesize the field.\n\n(In fact, the whole category theory library started off as an experiment to see how far we could\npush this automation.)\n-/\n\nexample {X : C} : C \u2964 Type* :=\n{ obj := \u03bb Y, X \u27f6 Y,\n  map := \u03bb Y Y' f g, g \u226b f, }\n\n/-!\nLean automatically checked functoriality here!\nThis was pretty easy: we just need to use `category.comp_id` and `category.assoc`.\nThe more powerful we make the `simp` lemmas, the more boring goals can be discharged automatically.\n\nMost of the `auto_param`s appearing in mathlib so far are in the `category_theory` library,\nwhere they are nearly all filled using the tactic `tidy`, which repeatedly attempts to use\none of a list of \"conservative\" tactics.\n\nYou can see what `tidy` is doing using `tidy?`:\n-/\n\nexample {X : C} : C \u2964 Type* :=\n{ obj := \u03bb Y, X \u27f6 Y,\n  map := \u03bb Y Y' f g, g \u226b f,\n  map_id' := by tidy?,\n  map_comp' := by tidy? }\n\n/-!\nSebastien's talk on differential geometry tomorrow will give another example of `auto_param` being used.\n\nYou can also watch me doing a speed-run https://youtu.be/oz3z2NSNY8c of Floris's \"pointed map\" exercises\nfrom yesterday, taking advantage of `auto_param`.\n-/\n\n\n\n\n/-!\n## Natural transformations\n\nThe collection of functors from `C` to `D` has been given the structure of a category:\nto talk about the natural transformations, you just write `F \u27f6 G` using the usual \"morphism\" arrow.\n\nIf `\u03b1 : F \u27f6 G`, then `\u03b1.app X` is the component at `X`, i.e. a morphism `F.obj X \u27f6 G.obj X`.\n-/\nexample {F G : C \u2964 D} {\u03b1 : F \u27f6 G} {X Y : C} (f : X \u27f6 Y) :\n  F.map f \u226b \u03b1.app Y = \u03b1.app X \u226b G.map f :=\n\u03b1.naturality f   -- or just `by simp`\n\n/-!\nAgain, to construct a natural transformation `F \u27f6 G` we need to provide two fields\n* `app : \u03a0 X : C, F.obj X \u27f6 G.obj X` and\n* `naturality'`, which often is provided by automation.\n-/\n\n\n/-!\n## A note on universes\n\nBefore we go on, we should mention a slight complication: out in the world we meet\nboth small and large categories. In set-theoretic foundations, this distinction is about\nwhether the objects form a set or merely a class.\n\nIn the type-theoretic foundations used in Lean, this distinction is about whether\nthe objects and morphisms live in the same universe, or if the objects live one universe higher up.\n\nRather than making separate definitions for the two cases, we simply allow the objects and morphisms\nto live in two unrelated universes. To talk about a general category we thus write\n-/\nuniverses u v\n\nvariables (E : Type u) [category.{v} E]\n\n/-!\nThis says that the objects live in universe `u`, while the morphisms live in universe `v`.\nIn fact, the definition `category` is paramaterised by two universe levels, and\nwhen we write `category.{v} E` Lean actually understands this as `category.{v u} E`,\nautomatically filling in the second argument from the universe level of `E`.\n\nThere are abbreviations available for the two standard cases:\n* if `E : Type (u+1)`, then `large_category E` means `category.{u (u+1)} E`\n* if `E : Type u`, then `small_category E` means `category.{u u} E`.\n\nHowever you'll rarely use these except when setting up particular examples.\nAll the \"concrete\" categories, like `Group`, `Ring`, and `Top`, described below,\nare instances of `large_category`.\nTypically the indexing diagrams for limits and colimits are instances of `small_category`.\n\nIf you're talking about an arbitrary category, and you don't mind whether it is small or large,\nyou should just allow two independent universe variables, as above.\n-/\n\n\n/-!\n## Concrete categories\n\nWe've set up a number of concrete categories in mathlib.\n-/\n\nexample (R S : CommRing) (f : R \u27f6 S) (x y : R) : f (x * y) = f x * f y := by simp\n\n/-!\nNote here we have a particularly succinct way of introducing a commutative ring:\nwe just write `R : CommRing`, rather than `(R : Type) [comm_ring R]`.\nRather than writing `f : R \u2192+* S` for a `ring_hom`, we can just use the morphism arrow,\nand Lean works out the appropriate notion automatically.\n\nThere's a coercion from `CommRing` to `Type`,\nso we can still talk about elements by writing `x : R`,\nand morphisms automatically behave properly as functions (e.g. in `f (x * y)`).\n-/\n\n\n/-!\n## Limits and colimits\n\nWe talk about limits using the following notions:\n* For `F : J \u2964 C`, `c : cone F` consists of\n  * `c.X : C` an object in `C`, and\n  * `c.\u03c0`, a natural transformation with components `c.\u03c0.app j : c.X \u27f6 F.obj j`.\n* For `c : cone F`, `is_limit c` expresses that `c` is a limit cone.\n* `has_limit F`, a typeclass specifying a particular choice of limit cone for a functor `F`.\n* `has_limits C`, a typeclass specifying a choice of limit for any functor into `C`.\n\n(There are also all the dual notions, `cocone`, `is_colimit`, `has_colimit`, etc.)\n\nThere are also typeclasses for various \"special shapes\", in particular\n* `has_equalizers`\n* `has_pullbacks`\n* `has_binary_products` / `has_finite_products` / `has_products`\n* `has_terminal`\n\nA related typeclass `has_zero_morphisms C` specifies a choice of zero morphism in each hom space,\nsatisfying the usual axioms (equivalent to `C` being enriched in pointed sets), and using that\nwe can also express some other special shapes, including\n* `has_kernels`\n* `has_binary_biproducts` / `has_finite_biproducts`\n* `has_zero_object`\n\nFor most of the concrete categories, these instances are all available when appropriate.\n-/\n\n/-!\n### Examples of using (co)limits in `Top`\n-/\n\nnoncomputable theory\nopen category_theory.limits\n\ndef R : Top := Top.of \u211d\ndef I : Top := Top.of (set.Icc 0 1 : set \u211d)\ndef pt : Top := Top.of unit\n\n-- Let's construct the mapping cylinder.\ndef to_pt (X : Top) : X \u27f6 pt :=\n{ to_fun := \u03bb _, unit.star, continuous_to_fun := continuous_const }\n\ndef I\u2080 : pt \u27f6 I :=\n{ to_fun := \u03bb _, \u27e8(0 : \u211d), by norm_num [set.left_mem_Icc]\u27e9,\n  continuous_to_fun := continuous_const }\n\ndef I\u2081 : pt \u27f6 I :=\n{ to_fun := \u03bb _, \u27e8(1 : \u211d), by norm_num [set.right_mem_Icc]\u27e9,\n  continuous_to_fun := continuous_const }\n\n-- We now construct a cylinder as a categorical limit.\n-- `limits.prod` is a shorthand for constructing a limit over the two point diagram:\ndef cylinder (X : Top) : Top := prod X I\n\n-- To define a map to the cylinder, we give a map to each factor.\n-- `prod.lift` is a helper method, providing a wrapper around `limit.lift` for binary products.\ndef cylinder\u2080 (X : Top) : X \u27f6 cylinder X := prod.lift (\ud835\udfd9 X) (to_pt X \u226b I\u2080)\ndef cylinder\u2081 (X : Top) : X \u27f6 cylinder X := prod.lift (\ud835\udfd9 X) (to_pt X \u226b I\u2081)\n\n/--\nThe mapping cylinder is the pushout of the diagram\n```\n    X\n   \u2199 \u2198\n  Y   (X x I)\n```\n(`pushout` is implemented just as a wrapper around `colimit`)\n-/\ndef mapping_cylinder {X Y : Top} (f : X \u27f6 Y) : Top := pushout f (cylinder\u2081 X)\n\n\n/-!\nIt's perhaps worth admitting here that constructing objects using categorical (co)limits\ntypically gives quite ghastly \"definitional\" properties --- if you want to use these objects,\nyou're going to have to work through their universal properties.\n\nThis is not necessarily a bad thing, but takes some getting used to.\n-/\n\n/-!\n## Applications\n\nWe're only just getting to the point in mathlib where we're ready to do the sorts of mathematics\nthat rely on category theory as a basic language. There's lots more to come ---\nbig chunks of algebraic geometry, homological algebra, quantum topology, etc.\n\nOne important way in which we'll use the category theory library is to achieve polymorphism.\nWe don't want to separately prove theorems about sheaves of sets, sheaves of rings, etc.\nInstead we'd like to talk about sheaves in an arbitrary category,\npossibly with some additional typeclasses providing extra structure\n(`has_products`, `concrete_category`, `monoidal_category`, etc),\nand prove our theorems there.\n-/\n\n/-!\n## Odds and ends\n\nThere's a bunch in mathlib's `category_theory/` folder that hasn't been mentioned at all here,\nincluding:\n\n* Adjunctions\n* Equivalences\n* Monads\n* Abelian categories\n* Monoidal categories\n* ...\n\nBuilt on top of the category theory library we have things like\n* (Co)homology of chain complexes in `algebra.homology.homology`.\n* The (pre)sheaf of continuous functions in `topology.sheaves.sheaf_of_functions`.\n* The Giry monad in `measure_theory.category.Meas`.\n-/\n\n#print category_theory.adjunction.right_adjoint_preserves_limits\n\n#print category_theory.abelian\n\n-- When this tutorial was written we didn't have a single instance of `abelian` in the library.\nexample : abelian AddCommGroup.{0} := by apply_instance\nexample (R : Ring) : abelian (Module R) := by apply_instance\n\nexample (R : CommRing.{u}) : monoidal_category (Module.{u} R) := by apply_instance\n\nexample : reflective (forget\u2082 CpltSepUniformSpace UniformSpace) := by apply_instance\n", "meta": {"author": "leanprover-community", "repo": "lftcm2020", "sha": "cc683e2b074b61909310746d6acc1fb3d42d6ee2", "save_path": "github-repos/lean/leanprover-community-lftcm2020", "path": "github-repos/lean/leanprover-community-lftcm2020/lftcm2020-cc683e2b074b61909310746d6acc1fb3d42d6ee2/src/demos/category_theory.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3941831680206885}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.full_subcategory\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.products\nimport topology.sheaves.presheaf\n\n/-!\n# The sheaf condition in terms of an equalizer of products\n\nHere we set up the machinery for the \"usual\" definition of the sheaf condition,\ne.g. as in https://stacks.math.columbia.edu/tag/0072\nin terms of an equalizer diagram where the two objects are\n`\u220f F.obj (U i)` and `\u220f F.obj (U i) \u2293 (U j)`.\n\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen topological_space\nopen opposite\nopen topological_space.opens\n\nnamespace Top\n\nvariables {C : Type u} [category.{v} C] [has_products C]\nvariables {X : Top.{v}} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 opens X)\n\nnamespace presheaf\n\nnamespace sheaf_condition_equalizer_products\n\n/-- The product of the sections of a presheaf over a family of open sets. -/\ndef pi_opens : C := \u220f (\u03bb i : \u03b9, F.obj (op (U i)))\n/--\nThe product of the sections of a presheaf over the pairwise intersections of\na family of open sets.\n-/\ndef pi_inters : C := \u220f (\u03bb p : \u03b9 \u00d7 \u03b9, F.obj (op (U p.1 \u2293 U p.2)))\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U i` to `U i \u2293 U j`.\n-/\ndef left_res : pi_opens F U \u27f6 pi_inters F U :=\npi.lift (\u03bb p : \u03b9 \u00d7 \u03b9, pi.\u03c0 _ p.1 \u226b F.map (inf_le_left (U p.1) (U p.2)).op)\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef right_res : pi_opens F U \u27f6 pi_inters F U :=\npi.lift (\u03bb p : \u03b9 \u00d7 \u03b9, pi.\u03c0 _ p.2 \u226b F.map (inf_le_right (U p.1) (U p.2)).op)\n\n/--\nThe morphism `F.obj U \u27f6 \u03a0 F.obj (U i)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef res : F.obj (op (supr U)) \u27f6 pi_opens F U :=\npi.lift (\u03bb i : \u03b9, F.map (topological_space.opens.le_supr U i).op)\n\n@[simp, elementwise]\nlemma res_\u03c0 (i : \u03b9) : res F U \u226b limit.\u03c0 _ \u27e8i\u27e9 = F.map (opens.le_supr U i).op :=\nby rw [res, limit.lift_\u03c0, fan.mk_\u03c0_app]\n\n@[elementwise]\nlemma w : res F U \u226b left_res F U = res F U \u226b right_res F U :=\nbegin\n  dsimp [res, left_res, right_res],\n  ext,\n  simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, fan.mk_\u03c0_app, category.assoc],\n  rw [\u2190F.map_comp],\n  rw [\u2190F.map_comp],\n  congr,\nend\n\n/--\nThe equalizer diagram for the sheaf condition.\n-/\n@[reducible]\ndef diagram : walking_parallel_pair.{v} \u2964 C :=\nparallel_pair (left_res F U) (right_res F U)\n\n/--\nThe restriction map `F.obj U \u27f6 \u03a0 F.obj (U i)` gives a cone over the equalizer diagram\nfor the sheaf condition. The sheaf condition asserts this cone is a limit cone.\n-/\ndef fork : fork.{v} (left_res F U) (right_res F U) := fork.of_\u03b9 _ (w F U)\n\n@[simp]\nlemma fork_X : (fork F U).X = F.obj (op (supr U)) := rfl\n\n@[simp]\nlemma fork_\u03b9 : (fork F U).\u03b9 = res F U := rfl\n@[simp]\nlemma fork_\u03c0_app_walking_parallel_pair_zero :\n  (fork F U).\u03c0.app walking_parallel_pair.zero = res F U := rfl\n@[simp]\nlemma fork_\u03c0_app_walking_parallel_pair_one :\n  (fork F U).\u03c0.app walking_parallel_pair.one = res F U \u226b left_res F U := rfl\n\nvariables {F} {G : presheaf C X}\n\n/-- Isomorphic presheaves have isomorphic `pi_opens` for any cover `U`. -/\n@[simp]\ndef pi_opens.iso_of_iso (\u03b1 : F \u2245 G) : pi_opens F U \u2245 pi_opens G U :=\npi.map_iso (\u03bb X, \u03b1.app _)\n\n/-- Isomorphic presheaves have isomorphic `pi_inters` for any cover `U`. -/\n@[simp]\ndef pi_inters.iso_of_iso (\u03b1 : F \u2245 G) : pi_inters F U \u2245 pi_inters G U :=\npi.map_iso (\u03bb X, \u03b1.app _)\n\n/-- Isomorphic presheaves have isomorphic sheaf condition diagrams. -/\ndef diagram.iso_of_iso (\u03b1 : F \u2245 G) : diagram F U \u2245 diagram G U :=\nnat_iso.of_components\n  begin rintro \u27e8\u27e9, exact pi_opens.iso_of_iso U \u03b1, exact pi_inters.iso_of_iso U \u03b1 end\n  begin\n    rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e9,\n    { simp, },\n    { ext, simp [left_res], },\n    { ext, simp [right_res], },\n    { simp, },\n  end.\n\n/--\nIf `F G : presheaf C X` are isomorphic presheaves,\nthen the `fork F U`, the canonical cone of the sheaf condition diagram for `F`,\nis isomorphic to `fork F G` postcomposed with the corresponding isomorphism between\nsheaf condition diagrams.\n-/\ndef fork.iso_of_iso (\u03b1 : F \u2245 G) :\n  fork F U \u2245 (cones.postcompose (diagram.iso_of_iso U \u03b1).inv).obj (fork G U) :=\nbegin\n  fapply fork.ext,\n  { apply \u03b1.app, },\n  { ext,\n    dunfold fork.\u03b9, -- Ugh, `simp` can't unfold abbreviations.\n    simp [res, diagram.iso_of_iso], }\nend\n\nsection open_embedding\n\nvariables {V : Top.{v}} {j : V \u27f6 X} (oe : open_embedding j)\nvariables (\ud835\udcb0 : \u03b9 \u2192 opens V)\n\n/--\nPush forward a cover along an open embedding.\n-/\n@[simp]\ndef cover.of_open_embedding : \u03b9 \u2192 opens X := (\u03bb i, oe.is_open_map.functor.obj (\ud835\udcb0 i))\n\n/--\nThe isomorphism between `pi_opens` corresponding to an open embedding.\n-/\n@[simp]\ndef pi_opens.iso_of_open_embedding :\n  pi_opens (oe.is_open_map.functor.op \u22d9 F) \ud835\udcb0 \u2245 pi_opens F (cover.of_open_embedding oe \ud835\udcb0) :=\npi.map_iso (\u03bb X, F.map_iso (iso.refl _))\n\n/--\nThe isomorphism between `pi_inters` corresponding to an open embedding.\n-/\n@[simp]\ndef pi_inters.iso_of_open_embedding :\n  pi_inters (oe.is_open_map.functor.op \u22d9 F) \ud835\udcb0 \u2245 pi_inters F (cover.of_open_embedding oe \ud835\udcb0) :=\npi.map_iso (\u03bb X, F.map_iso\n  begin\n    dsimp [is_open_map.functor],\n    exact iso.op\n    { hom := hom_of_le (by\n      { simp only [oe.to_embedding.inj, set.image_inter],\n        exact le_rfl, }),\n      inv := hom_of_le (by\n      { simp only [oe.to_embedding.inj, set.image_inter],\n        exact le_rfl, }), },\n  end)\n\n/-- The isomorphism of sheaf condition diagrams corresponding to an open embedding. -/\ndef diagram.iso_of_open_embedding :\n  diagram (oe.is_open_map.functor.op \u22d9 F) \ud835\udcb0 \u2245 diagram F (cover.of_open_embedding oe \ud835\udcb0) :=\nnat_iso.of_components\n  begin\n    rintro \u27e8\u27e9,\n    exact pi_opens.iso_of_open_embedding oe \ud835\udcb0,\n    exact pi_inters.iso_of_open_embedding oe \ud835\udcb0\n  end\n  begin\n    rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e9,\n    { simp, },\n    { ext,\n      dsimp [left_res, is_open_map.functor],\n      simp only [limit.lift_\u03c0, cones.postcompose_obj_\u03c0, iso.op_hom, discrete.nat_iso_hom_app,\n        functor.map_iso_refl, functor.map_iso_hom, lim_map_\u03c0_assoc, limit.lift_map, fan.mk_\u03c0_app,\n        nat_trans.comp_app, category.assoc],\n      dsimp,\n      rw [category.id_comp, \u2190F.map_comp],\n      refl, },\n    { ext,\n      dsimp [right_res, is_open_map.functor],\n      simp only [limit.lift_\u03c0, cones.postcompose_obj_\u03c0, iso.op_hom, discrete.nat_iso_hom_app,\n        functor.map_iso_refl, functor.map_iso_hom, lim_map_\u03c0_assoc, limit.lift_map, fan.mk_\u03c0_app,\n        nat_trans.comp_app, category.assoc],\n      dsimp,\n      rw [category.id_comp, \u2190F.map_comp],\n      refl, },\n    { simp, },\n  end.\n\n/--\nIf `F : presheaf C X` is a presheaf, and `oe : U \u27f6 X` is an open embedding,\nthen the sheaf condition fork for a cover `\ud835\udcb0` in `U` for the composition of `oe` and `F` is\nisomorphic to sheaf condition fork for `oe '' \ud835\udcb0`, precomposed with the isomorphism\nof indexing diagrams `diagram.iso_of_open_embedding`.\n\nWe use this to show that the restriction of sheaf along an open embedding is still a sheaf.\n-/\ndef fork.iso_of_open_embedding :\n  fork (oe.is_open_map.functor.op \u22d9 F) \ud835\udcb0 \u2245\n    (cones.postcompose (diagram.iso_of_open_embedding oe \ud835\udcb0).inv).obj\n      (fork F (cover.of_open_embedding oe \ud835\udcb0)) :=\nbegin\n  fapply fork.ext,\n  { dsimp [is_open_map.functor],\n    exact\n    F.map_iso (iso.op\n    { hom := hom_of_le\n      (by simp only [supr_s, supr_mk, le_def, subtype.coe_mk, set.le_eq_subset, set.image_Union]),\n      inv := hom_of_le\n      (by simp only [supr_s, supr_mk, le_def, subtype.coe_mk, set.le_eq_subset,\n                     set.image_Union]) }), },\n  { ext \u27e8j\u27e9,\n    dunfold fork.\u03b9, -- Ugh, it is unpleasant that we need this.\n    simp only [res, diagram.iso_of_open_embedding, discrete.nat_iso_inv_app, functor.map_iso_inv,\n      limit.lift_\u03c0, cones.postcompose_obj_\u03c0, functor.comp_map,\n      fork_\u03c0_app_walking_parallel_pair_zero, pi_opens.iso_of_open_embedding,\n      nat_iso.of_components.inv_app, functor.map_iso_refl, functor.op_map, limit.lift_map,\n      fan.mk_\u03c0_app, nat_trans.comp_app, quiver.hom.unop_op, category.assoc, lim_map_eq_lim_map],\n    dsimp,\n    rw [category.comp_id, \u2190F.map_comp],\n    refl, },\nend\n\nend open_embedding\n\nend sheaf_condition_equalizer_products\n\nend presheaf\n\nend Top\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/topology/sheaves/sheaf_condition/equalizer_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.682573734412324, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3941831680206883}}
{"text": "abbrev M := ExceptT String <| StateT Nat Id\n\ndef f (xs : List Nat) : M Unit := do\nfor x in xs do\n  if x == 0 then\n    throw \"contains zero\"\n\n#eval f [1, 2, 3] |>.run' 0\n#eval f [1, 0, 3] |>.run' 0\n\ntheorem ex1 : (f [1, 2, 3] |>.run' 0) = Except.ok () :=\nrfl\n\ntheorem ex2 : (f [1, 0, 3] |>.run' 0) = Except.error \"contains zero\" :=\nrfl\n\nuniverse u\n\nabbrev N := ExceptT (ULift.{u} String) Id\n\ndef idM {\u03b1 : Type u} (a : \u03b1) : N \u03b1 :=\npure a\n\ndef checkEq {\u03b1 : Type u} [BEq \u03b1] [ToString \u03b1] (a b : \u03b1) : N PUnit := do\nunless a == b do\n  throw (ULift.up s!\"{a} is not equal to {b}\")\n\ndef g {\u03b1 : Type u} [BEq \u03b1] [ToString \u03b1] (xs : List \u03b1) (a : \u03b1) : N PUnit := do\nfor x in xs do\n  let a \u2190 idM a\n  checkEq x a\n\n#eval g [1, (2:Nat), 3] 1 |>.run\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/forBodyResultTypeIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.39418219984475483}}
{"text": "/-\nCopyright (c) 2020 Kenji Nakagawa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio\n-/\nimport ring_theory.ideal.over\nimport ring_theory.polynomial.rational_root\n\n/-!\n# Dedekind domains\n\nThis file defines the notion of a Dedekind domain (or Dedekind ring),\nas a Noetherian integrally closed commutative ring of Krull dimension at most one.\n\n## Main definitions\n\n - `is_dedekind_domain` defines a Dedekind domain as a commutative ring that is\n   Noetherian, integrally closed in its field of fractions and has Krull dimension at most one.\n   `is_dedekind_domain_iff` shows that this does not depend on the choice of field of fractions.\n\n## Implementation notes\n\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. The `..._iff` lemmas express this independence.\n\nOften, definitions assume that Dedekind domains are not fields. We found it more practical\nto add a `(h : \u00ac is_field A)` assumption whenever this is explicitly needed.\n\n## References\n\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fr\u00f6lich, *Algebraic Number Theory*][cassels1967algebraic]\n* [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\ndedekind domain, dedekind ring\n-/\n\nvariables (R A K : Type*) [comm_ring R] [comm_ring A] [field K]\n\nopen_locale non_zero_divisors polynomial\n\n/-- A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. -/\ndef ring.dimension_le_one : Prop :=\n\u2200 p \u2260 (\u22a5 : ideal R), p.is_prime \u2192 p.is_maximal\n\nopen ideal ring\n\nnamespace ring\n\nlemma dimension_le_one.principal_ideal_ring\n  [is_domain A] [is_principal_ideal_ring A] : dimension_le_one A :=\n\u03bb p nonzero prime, by { haveI := prime, exact is_prime.to_maximal_ideal nonzero }\n\n\n\nlemma dimension_le_one.integral_closure [nontrivial R] [is_domain A] [algebra R A]\n  (h : dimension_le_one R) : dimension_le_one (integral_closure R A) :=\nh.is_integral_closure R A (integral_closure R A)\n\nvariables {R}\n\nlemma dimension_le_one.not_lt_lt (h : ring.dimension_le_one R)\n  (p\u2080 p\u2081 p\u2082 : ideal R) [hp\u2081 : p\u2081.is_prime] [hp\u2082 : p\u2082.is_prime] :\n  \u00ac (p\u2080 < p\u2081 \u2227 p\u2081 < p\u2082)\n| \u27e8h01, h12\u27e9 := h12.ne ((h p\u2081 (bot_le.trans_lt h01).ne' hp\u2081).eq_of_le hp\u2082.ne_top h12.le)\n\nlemma dimension_le_one.eq_bot_of_lt (h : ring.dimension_le_one R)\n  (p P : ideal R) [hp : p.is_prime] [hP : P.is_prime] (hpP : p < P) : p = \u22a5 :=\nby_contra (\u03bb hp0, h.not_lt_lt \u22a5 p P \u27e8ne.bot_lt hp0, hpP\u27e9)\n\nend ring\n\nvariables [is_domain A]\n\n/--\nA Dedekind domain is an integral domain that is Noetherian, integrally closed, and\nhas Krull dimension at most one.\n\nThis is definition 3.2 of [Neukirch1992].\n\nThe integral closure condition is independent of the choice of field of fractions:\nuse `is_dedekind_domain_iff` to prove `is_dedekind_domain` for a given `fraction_map`.\n\nThis is the default implementation, but there are equivalent definitions,\n`is_dedekind_domain_dvr` and `is_dedekind_domain_inv`.\nTODO: Prove that these are actually equivalent definitions.\n-/\nclass is_dedekind_domain : Prop :=\n(is_noetherian_ring : is_noetherian_ring A)\n(dimension_le_one : dimension_le_one A)\n(is_integrally_closed : is_integrally_closed A)\n\n-- See library note [lower instance priority]\nattribute [instance, priority 100]\n  is_dedekind_domain.is_noetherian_ring is_dedekind_domain.is_integrally_closed\n\n/-- An integral domain is a Dedekind domain iff and only if it is\nNoetherian, has dimension \u2264 1, and is integrally closed in a given fraction field.\nIn particular, this definition does not depend on the choice of this fraction field. -/\nlemma is_dedekind_domain_iff (K : Type*) [field K] [algebra A K] [is_fraction_ring A K] :\n  is_dedekind_domain A \u2194 is_noetherian_ring A \u2227 dimension_le_one A \u2227\n    (\u2200 {x : K}, is_integral A x \u2192 \u2203 y, algebra_map A K y = x) :=\n\u27e8\u03bb \u27e8hr, hd, hi\u27e9, \u27e8hr, hd, \u03bb x, (is_integrally_closed_iff K).mp hi\u27e9,\n \u03bb \u27e8hr, hd, hi\u27e9, \u27e8hr, hd, (is_integrally_closed_iff K).mpr @hi\u27e9\u27e9\n\n@[priority 100] -- See library note [lower instance priority]\ninstance is_principal_ideal_ring.is_dedekind_domain [is_principal_ideal_ring A] :\n  is_dedekind_domain A :=\n\u27e8principal_ideal_ring.is_noetherian_ring,\n ring.dimension_le_one.principal_ideal_ring A,\n unique_factorization_monoid.is_integrally_closed\u27e9\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/ring_theory/dedekind_domain/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.61878043374385, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3941821998447547}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Scott Morrison\n\n! This file was ported from Lean 3 source module data.finsupp.basic\n! leanprover-community/mathlib commit f16e7a22e11fc09c71f25446ac1db23a24e8a0bd\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.BigOperators.Finsupp\nimport Mathbin.Algebra.Hom.GroupAction\nimport Mathbin.Algebra.Regular.Smul\nimport Mathbin.Data.Finset.Preimage\nimport Mathbin.Data.Rat.BigOperators\n\n/-!\n# Miscellaneous definitions, lemmas, and constructions using finsupp\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n## Main declarations\n\n* `finsupp.graph`: the finset of input and output pairs with non-zero outputs.\n* `finsupp.map_range.equiv`: `finsupp.map_range` as an equiv.\n* `finsupp.map_domain`: maps the domain of a `finsupp` by a function and by summing.\n* `finsupp.comap_domain`: postcomposition of a `finsupp` with a function injective on the preimage\n  of its support.\n* `finsupp.some`: restrict a finitely supported function on `option \u03b1` to a finitely supported\n  function on `\u03b1`.\n* `finsupp.filter`: `filter p f` is the finitely supported function that is `f a` if `p a` is true\n  and 0 otherwise.\n* `finsupp.frange`: the image of a finitely supported function on its support.\n* `finsupp.subtype_domain`: the restriction of a finitely supported function `f` to a subtype.\n\n## Implementation notes\n\nThis file is a `noncomputable theory` and uses classical logic throughout.\n\n## TODO\n\n* This file is currently ~1600 lines long and is quite a miscellany of definitions and lemmas,\n  so it should be divided into smaller pieces.\n\n* Expand the list of definitions and important lemmas to the module docstring.\n\n-/\n\n\nnoncomputable section\n\nopen Finset Function\n\nopen BigOperators\n\nvariable {\u03b1 \u03b2 \u03b3 \u03b9 M M' N P G H R S : Type _}\n\nnamespace Finsupp\n\n/-! ### Declarations about `graph` -/\n\n\nsection Graph\n\nvariable [Zero M]\n\n#print Finsupp.graph /-\n/-- The graph of a finitely supported function over its support, i.e. the finset of input and output\npairs with non-zero outputs. -/\ndef graph (f : \u03b1 \u2192\u2080 M) : Finset (\u03b1 \u00d7 M) :=\n  f.support.map \u27e8fun a => Prod.mk a (f a), fun x y h => (Prod.mk.inj h).1\u27e9\n#align finsupp.graph Finsupp.graph\n-/\n\n/- warning: finsupp.mk_mem_graph_iff -> Finsupp.mk_mem_graph_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {a : \u03b1} {m : M} {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Iff (Membership.Mem.{max u1 u2, max u1 u2} (Prod.{u1, u2} \u03b1 M) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasMem.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Prod.mk.{u1, u2} \u03b1 M a m) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f)) (And (Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f a) m) (Ne.{succ u2} M m (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {a : \u03b1} {m : M} {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Iff (Membership.mem.{max u1 u2, max u2 u1} (Prod.{u2, u1} \u03b1 M) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instMembershipFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)) (Prod.mk.{u2, u1} \u03b1 M a m) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f)) (And (Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f a) m) (Ne.{succ u1} M m (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.mk_mem_graph_iff Finsupp.mk_mem_graph_iff\u2093'. -/\ntheorem mk_mem_graph_iff {a : \u03b1} {m : M} {f : \u03b1 \u2192\u2080 M} : (a, m) \u2208 f.graph \u2194 f a = m \u2227 m \u2260 0 :=\n  by\n  simp_rw [graph, mem_map, mem_support_iff]\n  constructor\n  \u00b7 rintro \u27e8b, ha, rfl, -\u27e9\n    exact \u27e8rfl, ha\u27e9\n  \u00b7 rintro \u27e8rfl, ha\u27e9\n    exact \u27e8a, ha, rfl\u27e9\n#align finsupp.mk_mem_graph_iff Finsupp.mk_mem_graph_iff\n\n/- warning: finsupp.mem_graph_iff -> Finsupp.mem_graph_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {c : Prod.{u1, u2} \u03b1 M} {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Iff (Membership.Mem.{max u1 u2, max u1 u2} (Prod.{u1, u2} \u03b1 M) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasMem.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) c (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f)) (And (Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f (Prod.fst.{u1, u2} \u03b1 M c)) (Prod.snd.{u1, u2} \u03b1 M c)) (Ne.{succ u2} M (Prod.snd.{u1, u2} \u03b1 M c) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {c : Prod.{u2, u1} \u03b1 M} {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Iff (Membership.mem.{max u2 u1, max u2 u1} (Prod.{u2, u1} \u03b1 M) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instMembershipFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)) c (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f)) (And (Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) (Prod.fst.{u2, u1} \u03b1 M c)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f (Prod.fst.{u2, u1} \u03b1 M c)) (Prod.snd.{u2, u1} \u03b1 M c)) (Ne.{succ u1} M (Prod.snd.{u2, u1} \u03b1 M c) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.mem_graph_iff Finsupp.mem_graph_iff\u2093'. -/\n@[simp]\ntheorem mem_graph_iff {c : \u03b1 \u00d7 M} {f : \u03b1 \u2192\u2080 M} : c \u2208 f.graph \u2194 f c.1 = c.2 \u2227 c.2 \u2260 0 :=\n  by\n  cases c\n  exact mk_mem_graph_iff\n#align finsupp.mem_graph_iff Finsupp.mem_graph_iff\n\n/- warning: finsupp.mk_mem_graph -> Finsupp.mk_mem_graph is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) {a : \u03b1}, (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a (Finsupp.support.{u1, u2} \u03b1 M _inst_1 f)) -> (Membership.Mem.{max u1 u2, max u1 u2} (Prod.{u1, u2} \u03b1 M) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasMem.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Prod.mk.{u1, u2} \u03b1 M a (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f a)) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) {a : \u03b1}, (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) a (Finsupp.support.{u2, u1} \u03b1 M _inst_1 f)) -> (Membership.mem.{max u1 u2, max u2 u1} (Prod.{u2, u1} \u03b1 ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a)) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instMembershipFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)) (Prod.mk.{u2, u1} \u03b1 ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) a (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f a)) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.mk_mem_graph Finsupp.mk_mem_graph\u2093'. -/\ntheorem mk_mem_graph (f : \u03b1 \u2192\u2080 M) {a : \u03b1} (ha : a \u2208 f.support) : (a, f a) \u2208 f.graph :=\n  mk_mem_graph_iff.2 \u27e8rfl, mem_support_iff.1 ha\u27e9\n#align finsupp.mk_mem_graph Finsupp.mk_mem_graph\n\n/- warning: finsupp.apply_eq_of_mem_graph -> Finsupp.apply_eq_of_mem_graph is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {a : \u03b1} {m : M} {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, (Membership.Mem.{max u1 u2, max u1 u2} (Prod.{u1, u2} \u03b1 M) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasMem.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Prod.mk.{u1, u2} \u03b1 M a m) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f)) -> (Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f a) m)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {a : \u03b1} {m : M} {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, (Membership.mem.{max u1 u2, max u2 u1} (Prod.{u2, u1} \u03b1 M) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instMembershipFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)) (Prod.mk.{u2, u1} \u03b1 M a m) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f)) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f a) m)\nCase conversion may be inaccurate. Consider using '#align finsupp.apply_eq_of_mem_graph Finsupp.apply_eq_of_mem_graph\u2093'. -/\ntheorem apply_eq_of_mem_graph {a : \u03b1} {m : M} {f : \u03b1 \u2192\u2080 M} (h : (a, m) \u2208 f.graph) : f a = m :=\n  (mem_graph_iff.1 h).1\n#align finsupp.apply_eq_of_mem_graph Finsupp.apply_eq_of_mem_graph\n\n/- warning: finsupp.not_mem_graph_snd_zero -> Finsupp.not_mem_graph_snd_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (a : \u03b1) (f : Finsupp.{u1, u2} \u03b1 M _inst_1), Not (Membership.Mem.{max u1 u2, max u1 u2} (Prod.{u1, u2} \u03b1 M) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasMem.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Prod.mk.{u1, u2} \u03b1 M a (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (a : \u03b1) (f : Finsupp.{u2, u1} \u03b1 M _inst_1), Not (Membership.mem.{max u1 u2, max u2 u1} (Prod.{u2, u1} \u03b1 M) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instMembershipFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)) (Prod.mk.{u2, u1} \u03b1 M a (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.not_mem_graph_snd_zero Finsupp.not_mem_graph_snd_zero\u2093'. -/\n@[simp]\ntheorem not_mem_graph_snd_zero (a : \u03b1) (f : \u03b1 \u2192\u2080 M) : (a, (0 : M)) \u2209 f.graph := fun h =>\n  (mem_graph_iff.1 h).2.irrefl\n#align finsupp.not_mem_graph_snd_zero Finsupp.not_mem_graph_snd_zero\n\n/- warning: finsupp.image_fst_graph -> Finsupp.image_fst_graph is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] [_inst_2 : DecidableEq.{succ u1} \u03b1] (f : Finsupp.{u1, u2} \u03b1 M _inst_1), Eq.{succ u1} (Finset.{u1} \u03b1) (Finset.image.{max u1 u2, u1} (Prod.{u1, u2} \u03b1 M) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) (Prod.fst.{u1, u2} \u03b1 M) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f)) (Finsupp.support.{u1, u2} \u03b1 M _inst_1 f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] [_inst_2 : DecidableEq.{succ u2} \u03b1] (f : Finsupp.{u2, u1} \u03b1 M _inst_1), Eq.{succ u2} (Finset.{u2} \u03b1) (Finset.image.{max u1 u2, u2} (Prod.{u2, u1} \u03b1 M) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) (Prod.fst.{u2, u1} \u03b1 M) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f)) (Finsupp.support.{u2, u1} \u03b1 M _inst_1 f)\nCase conversion may be inaccurate. Consider using '#align finsupp.image_fst_graph Finsupp.image_fst_graph\u2093'. -/\n@[simp]\ntheorem image_fst_graph [DecidableEq \u03b1] (f : \u03b1 \u2192\u2080 M) : f.graph.image Prod.fst = f.support := by\n  classical simp only [graph, map_eq_image, image_image, embedding.coe_fn_mk, (\u00b7 \u2218 \u00b7), image_id']\n#align finsupp.image_fst_graph Finsupp.image_fst_graph\n\n/- warning: finsupp.graph_injective -> Finsupp.graph_injective is a dubious translation:\nlean 3 declaration is\n  forall (\u03b1 : Type.{u1}) (M : Type.{u2}) [_inst_2 : Zero.{u2} M], Function.Injective.{max (succ u1) (succ u2), succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M _inst_2) (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finsupp.graph.{u1, u2} \u03b1 M _inst_2)\nbut is expected to have type\n  forall (\u03b1 : Type.{u2}) (M : Type.{u1}) [_inst_2 : Zero.{u1} M], Function.Injective.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u2, u1} \u03b1 M _inst_2) (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finsupp.graph.{u2, u1} \u03b1 M _inst_2)\nCase conversion may be inaccurate. Consider using '#align finsupp.graph_injective Finsupp.graph_injective\u2093'. -/\ntheorem graph_injective (\u03b1 M) [Zero M] : Injective (@graph \u03b1 M _) :=\n  by\n  intro f g h\n  classical\n    have hsup : f.support = g.support := by rw [\u2190 image_fst_graph, h, image_fst_graph]\n    refine' ext_iff'.2 \u27e8hsup, fun x hx => apply_eq_of_mem_graph <| h.symm \u25b8 _\u27e9\n    exact mk_mem_graph _ (hsup \u25b8 hx)\n#align finsupp.graph_injective Finsupp.graph_injective\n\n/- warning: finsupp.graph_inj -> Finsupp.graph_inj is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {f : Finsupp.{u1, u2} \u03b1 M _inst_1} {g : Finsupp.{u1, u2} \u03b1 M _inst_1}, Iff (Eq.{succ (max u1 u2)} (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 g)) (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) f g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {f : Finsupp.{u2, u1} \u03b1 M _inst_1} {g : Finsupp.{u2, u1} \u03b1 M _inst_1}, Iff (Eq.{max (succ u2) (succ u1)} (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 g)) (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) f g)\nCase conversion may be inaccurate. Consider using '#align finsupp.graph_inj Finsupp.graph_inj\u2093'. -/\n@[simp]\ntheorem graph_inj {f g : \u03b1 \u2192\u2080 M} : f.graph = g.graph \u2194 f = g :=\n  (graph_injective \u03b1 M).eq_iff\n#align finsupp.graph_inj Finsupp.graph_inj\n\n/- warning: finsupp.graph_zero -> Finsupp.graph_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M], Eq.{succ (max u1 u2)} (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))) (EmptyCollection.emptyCollection.{max u1 u2} (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasEmptyc.{max u1 u2} (Prod.{u1, u2} \u03b1 M)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M], Eq.{max (succ u2) (succ u1)} (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))) (EmptyCollection.emptyCollection.{max u2 u1} (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instEmptyCollectionFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)))\nCase conversion may be inaccurate. Consider using '#align finsupp.graph_zero Finsupp.graph_zero\u2093'. -/\n@[simp]\ntheorem graph_zero : graph (0 : \u03b1 \u2192\u2080 M) = \u2205 := by simp [graph]\n#align finsupp.graph_zero Finsupp.graph_zero\n\n/- warning: finsupp.graph_eq_empty -> Finsupp.graph_eq_empty is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Iff (Eq.{succ (max u1 u2)} (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finsupp.graph.{u1, u2} \u03b1 M _inst_1 f) (EmptyCollection.emptyCollection.{max u1 u2} (Finset.{max u1 u2} (Prod.{u1, u2} \u03b1 M)) (Finset.hasEmptyc.{max u1 u2} (Prod.{u1, u2} \u03b1 M)))) (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) f (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Iff (Eq.{max (succ u2) (succ u1)} (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finsupp.graph.{u2, u1} \u03b1 M _inst_1 f) (EmptyCollection.emptyCollection.{max u2 u1} (Finset.{max u1 u2} (Prod.{u2, u1} \u03b1 M)) (Finset.instEmptyCollectionFinset.{max u2 u1} (Prod.{u2, u1} \u03b1 M)))) (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) f (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.graph_eq_empty Finsupp.graph_eq_empty\u2093'. -/\n@[simp]\ntheorem graph_eq_empty {f : \u03b1 \u2192\u2080 M} : f.graph = \u2205 \u2194 f = 0 :=\n  (graph_injective \u03b1 M).eq_iff' graph_zero\n#align finsupp.graph_eq_empty Finsupp.graph_eq_empty\n\nend Graph\n\nend Finsupp\n\n/-! ### Declarations about `map_range` -/\n\n\nsection MapRange\n\nnamespace Finsupp\n\nsection Equiv\n\nvariable [Zero M] [Zero N] [Zero P]\n\n#print Finsupp.mapRange.equiv /-\n/-- `finsupp.map_range` as an equiv. -/\n@[simps apply]\ndef mapRange.equiv (f : M \u2243 N) (hf : f 0 = 0) (hf' : f.symm 0 = 0) : (\u03b1 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 N)\n    where\n  toFun := (mapRange f hf : (\u03b1 \u2192\u2080 M) \u2192 \u03b1 \u2192\u2080 N)\n  invFun := (mapRange f.symm hf' : (\u03b1 \u2192\u2080 N) \u2192 \u03b1 \u2192\u2080 M)\n  left_inv x := by\n    rw [\u2190 map_range_comp _ _ _ _] <;> simp_rw [Equiv.symm_comp_self]\n    \u00b7 exact map_range_id _\n    \u00b7 rfl\n  right_inv x := by\n    rw [\u2190 map_range_comp _ _ _ _] <;> simp_rw [Equiv.self_comp_symm]\n    \u00b7 exact map_range_id _\n    \u00b7 rfl\n#align finsupp.map_range.equiv Finsupp.mapRange.equiv\n-/\n\n/- warning: finsupp.map_range.equiv_refl -> Finsupp.mapRange.equiv_refl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M], Eq.{max 1 (succ u1) (succ u2)} (Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u2} \u03b1 M _inst_1)) (Finsupp.mapRange.equiv.{u1, u2, u2} \u03b1 M M _inst_1 _inst_1 (Equiv.refl.{succ u2} M) (rfl.{succ u2} M (coeFn.{succ u2, succ u2} (Equiv.{succ u2, succ u2} M M) (fun (_x : Equiv.{succ u2, succ u2} M M) => M -> M) (Equiv.hasCoeToFun.{succ u2, succ u2} M M) (Equiv.refl.{succ u2} M) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (rfl.{succ u2} M (coeFn.{succ u2, succ u2} (Equiv.{succ u2, succ u2} M M) (fun (_x : Equiv.{succ u2, succ u2} M M) => M -> M) (Equiv.hasCoeToFun.{succ u2, succ u2} M M) (Equiv.symm.{succ u2, succ u2} M M (Equiv.refl.{succ u2} M)) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))) (Equiv.refl.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M], Eq.{max (succ u2) (succ u1)} (Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.{u2, u1} \u03b1 M _inst_1)) (Finsupp.mapRange.equiv.{u2, u1, u1} \u03b1 M M _inst_1 _inst_1 (Equiv.refl.{succ u1} M) (rfl.{succ u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => M) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} M M) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => M) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} M M) (Equiv.refl.{succ u1} M) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1)))) (rfl.{succ u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => M) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} M M) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => M) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} M M) (Equiv.symm.{succ u1, succ u1} M M (Equiv.refl.{succ u1} M)) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))))) (Equiv.refl.{max (succ u1) (succ u2)} (Finsupp.{u2, u1} \u03b1 M _inst_1))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.equiv_refl Finsupp.mapRange.equiv_refl\u2093'. -/\n@[simp]\ntheorem mapRange.equiv_refl : mapRange.equiv (Equiv.refl M) rfl rfl = Equiv.refl (\u03b1 \u2192\u2080 M) :=\n  Equiv.ext mapRange_id\n#align finsupp.map_range.equiv_refl Finsupp.mapRange.equiv_refl\n\n/- warning: finsupp.map_range.equiv_trans -> Finsupp.mapRange.equiv_trans is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : Zero.{u2} M] [_inst_2 : Zero.{u3} N] [_inst_3 : Zero.{u4} P] (f : Equiv.{succ u2, succ u3} M N) (hf : Eq.{succ u3} N (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (hf' : Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (f\u2082 : Equiv.{succ u3, succ u4} N P) (hf\u2082 : Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (hf\u2082' : Eq.{succ u3} N (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))), Eq.{max 1 (max (max (succ u1) (succ u2)) (succ u1) (succ u4)) (max (succ u1) (succ u4)) (succ u1) (succ u2)} (Equiv.{max (succ u1) (succ u2), max (succ u1) (succ u4)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u4} \u03b1 P _inst_3)) (Finsupp.mapRange.equiv.{u1, u2, u4} \u03b1 M P _inst_1 _inst_3 (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (Eq.mpr.{0} (Eq.{succ u4} P (coeFn.{max 1 (max (succ u2) (succ u4)) (succ u4) (succ u2), max (succ u2) (succ u4)} (Equiv.{succ u2, succ u4} M P) (fun (_x : Equiv.{succ u2, succ u4} M P) => M -> P) (Equiv.hasCoeToFun.{succ u2, succ u4} M P) (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u2) (succ u4)) (succ u4) (succ u2), max (succ u2) (succ u4)} (Equiv.{succ u2, succ u4} M P) (fun (_x : Equiv.{succ u2, succ u4} M P) => M -> P) (Equiv.hasCoeToFun.{succ u2, succ u4} M P) (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (Eq.ndrec.{0, succ u4} P (coeFn.{max 1 (max (succ u2) (succ u4)) (succ u4) (succ u2), max (succ u2) (succ u4)} (Equiv.{succ u2, succ u4} M P) (fun (_x : Equiv.{succ u2, succ u4} M P) => M -> P) (Equiv.hasCoeToFun.{succ u2, succ u4} M P) (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (fun (_a : P) => Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u2) (succ u4)) (succ u4) (succ u2), max (succ u2) (succ u4)} (Equiv.{succ u2, succ u4} M P) (fun (_x : Equiv.{succ u2, succ u4} M P) => M -> P) (Equiv.hasCoeToFun.{succ u2, succ u4} M P) (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P _a (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (rfl.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u2) (succ u4)) (succ u4) (succ u2), max (succ u2) (succ u4)} (Equiv.{succ u2, succ u4} M P) (fun (_x : Equiv.{succ u2, succ u4} M P) => M -> P) (Equiv.hasCoeToFun.{succ u2, succ u4} M P) (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (Equiv.trans_apply.{succ u2, succ u3, succ u4} M N P f f\u2082 (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))) (Eq.mpr.{0} (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (Eq.ndrec.{0, succ u3} N (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (fun (_a : N) => Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 _a) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (rfl.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2))) hf)) (Eq.mpr.{0} (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (Eq.ndrec.{0, succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (fun (_a : P) => Eq.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (Eq.{succ u4} P _a (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (rfl.{1} Prop (Eq.{succ u4} P (coeFn.{max 1 (max (succ u3) (succ u4)) (succ u4) (succ u3), max (succ u3) (succ u4)} (Equiv.{succ u3, succ u4} N P) (fun (_x : Equiv.{succ u3, succ u4} N P) => N -> P) (Equiv.hasCoeToFun.{succ u3, succ u4} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))) hf\u2082)) (rfl.{succ u4} P (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))))) (Eq.mpr.{0} (Eq.{succ u2} M (coeFn.{max 1 (max (succ u4) (succ u2)) (succ u2) (succ u4), max (succ u4) (succ u2)} (Equiv.{succ u4, succ u2} P M) (fun (_x : Equiv.{succ u4, succ u2} P M) => P -> M) (Equiv.hasCoeToFun.{succ u4, succ u2} P M) (Equiv.symm.{succ u2, succ u4} M P (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082)) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u4) (succ u2)) (succ u2) (succ u4), max (succ u4) (succ u2)} (Equiv.{succ u4, succ u2} P M) (fun (_x : Equiv.{succ u4, succ u2} P M) => P -> M) (Equiv.hasCoeToFun.{succ u4, succ u2} P M) (Equiv.symm.{succ u2, succ u4} M P (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082)) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (Eq.ndrec.{0, succ u2} M (coeFn.{max 1 (max (succ u4) (succ u2)) (succ u2) (succ u4), max (succ u4) (succ u2)} (Equiv.{succ u4, succ u2} P M) (fun (_x : Equiv.{succ u4, succ u2} P M) => P -> M) (Equiv.hasCoeToFun.{succ u4, succ u2} P M) (Equiv.symm.{succ u2, succ u4} M P (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082)) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (fun (_a : M) => Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u4) (succ u2)) (succ u2) (succ u4), max (succ u4) (succ u2)} (Equiv.{succ u4, succ u2} P M) (fun (_x : Equiv.{succ u4, succ u2} P M) => P -> M) (Equiv.hasCoeToFun.{succ u4, succ u2} P M) (Equiv.symm.{succ u2, succ u4} M P (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082)) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M _a (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (rfl.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u4) (succ u2)) (succ u2) (succ u4), max (succ u4) (succ u2)} (Equiv.{succ u4, succ u2} P M) (fun (_x : Equiv.{succ u4, succ u2} P M) => P -> M) (Equiv.hasCoeToFun.{succ u4, succ u2} P M) (Equiv.symm.{succ u2, succ u4} M P (Equiv.trans.{succ u2, succ u3, succ u4} M N P f f\u2082)) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (Equiv.symm_trans_apply.{succ u2, succ u3, succ u4} M N P f f\u2082 (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))))) (Eq.mpr.{0} (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (Eq.ndrec.{0, succ u3} N (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3)))) (fun (_a : N) => Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) _a) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (rfl.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (coeFn.{max 1 (max (succ u4) (succ u3)) (succ u3) (succ u4), max (succ u4) (succ u3)} (Equiv.{succ u4, succ u3} P N) (fun (_x : Equiv.{succ u4, succ u3} P N) => P -> N) (Equiv.hasCoeToFun.{succ u4, succ u3} P N) (Equiv.symm.{succ u3, succ u4} N P f\u2082) (OfNat.ofNat.{u4} P 0 (OfNat.mk.{u4} P 0 (Zero.zero.{u4} P _inst_3))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2))) hf\u2082')) (Eq.mpr.{0} (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (id_tag Tactic.IdTag.rw (Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (Eq.ndrec.{0, succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (fun (_a : M) => Eq.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Eq.{succ u2} M _a (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (rfl.{1} Prop (Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))) hf')) (rfl.{succ u2} M (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))))) (Equiv.trans.{max (succ u1) (succ u2), max (succ u1) (succ u3), max (succ u1) (succ u4)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u3} \u03b1 N _inst_2) (Finsupp.{u1, u4} \u03b1 P _inst_3) (Finsupp.mapRange.equiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f hf hf') (Finsupp.mapRange.equiv.{u1, u3, u4} \u03b1 N P _inst_2 _inst_3 f\u2082 hf\u2082 hf\u2082'))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u4}} {N : Type.{u3}} {P : Type.{u2}} [_inst_1 : Zero.{u4} M] [_inst_2 : Zero.{u3} N] [_inst_3 : Zero.{u2} P] (f : Equiv.{succ u4, succ u3} M N) (hf : Eq.{succ u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (hf' : Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_1))) (f\u2082 : Equiv.{succ u3, succ u2} N P) (hf\u2082 : Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_3))) (hf\u2082' : Eq.{succ u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))), Eq.{max (max (succ u1) (succ u4)) (succ u2)} (Equiv.{max (succ u4) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u1, u4} \u03b1 M _inst_1) (Finsupp.{u1, u2} \u03b1 P _inst_3)) (Finsupp.mapRange.equiv.{u1, u4, u2} \u03b1 M P _inst_1 _inst_3 (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (Eq.mpr.{0} (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u2), succ u4, succ u2} (Equiv.{succ u4, succ u2} M P) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u2} M P) (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (id.{0} (Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u2), succ u4, succ u2} (Equiv.{succ u4, succ u2} M P) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u2} M P) (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.ndrec.{0, succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u2), succ u4, succ u2} (Equiv.{succ u4, succ u2} M P) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u2} M P) (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) => Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u2), succ u4, succ u2} (Equiv.{succ u4, succ u2} M P) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u2} M P) (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _a (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.refl.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u2), succ u4, succ u2} (Equiv.{succ u4, succ u2} M P) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u2} M P) (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (Equiv.trans_apply.{succ u4, succ u3, succ u2} M N P f f\u2082 (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))))) (Eq.mpr.{0} (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (id.{0} (Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.ndrec.{0, succ u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) => Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 _a) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.refl.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (Equiv.{succ u4, succ u3} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u4, succ u3} M N) f (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2)) hf)) (Eq.mpr.{0} (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_3)) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (id.{0} (Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_3)) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.ndrec.{0, succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) => Eq.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3))) (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _a (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (Eq.refl.{1} Prop (Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} N P) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} N P) f\u2082 (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_2))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) _inst_3)))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_3)) hf\u2082)) (Eq.refl.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => P) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => P) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_3)))))) (Eq.mpr.{0} (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (Equiv.{succ u2, succ u4} P M) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u4} P M) (Equiv.symm.{succ u4, succ u2} M P (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082)) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (id.{0} (Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (Equiv.{succ u2, succ u4} P M) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u4} P M) (Equiv.symm.{succ u4, succ u2} M P (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082)) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.ndrec.{0, succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (Equiv.{succ u2, succ u4} P M) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u4} P M) (Equiv.symm.{succ u4, succ u2} M P (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082)) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) => Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (Equiv.{succ u2, succ u4} P M) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u4} P M) (Equiv.symm.{succ u4, succ u2} M P (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082)) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _a (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.refl.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u2), succ u2, succ u4} (Equiv.{succ u2, succ u4} P M) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u4} P M) (Equiv.symm.{succ u4, succ u2} M P (Equiv.trans.{succ u4, succ u3, succ u2} M N P f f\u2082)) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (Equiv.symm_trans_apply.{succ u4, succ u3, succ u2} M N P f f\u2082 (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))))) (Eq.mpr.{0} (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (id.{0} (Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.ndrec.{0, succ u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) => Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) _a) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.refl.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} P N) P (fun (_x : P) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} P N) (Equiv.symm.{succ u3, succ u2} N P f\u2082) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2)) hf\u2082')) (Eq.mpr.{0} (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_1)) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (id.{0} (Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_1)) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.ndrec.{0, succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) (fun (_a : (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) => Eq.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1))) (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _a (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (Eq.refl.{1} Prop (Eq.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (FunLike.coe.{max (succ u4) (succ u3), succ u3, succ u4} (Equiv.{succ u3, succ u4} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u4} N M) (Equiv.symm.{succ u4, succ u3} M N f) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => N) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_2))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) _inst_1)))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_1)) hf')) (Eq.refl.{succ u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : P) => M) (OfNat.ofNat.{u2} P 0 (Zero.toOfNat0.{u2} P _inst_3))) (OfNat.ofNat.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) 0 (Zero.toOfNat0.{u4} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u3} N 0 (Zero.toOfNat0.{u3} N _inst_2))) _inst_1))))))) (Equiv.trans.{max (succ u4) (succ u1), max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u1, u4} \u03b1 M _inst_1) (Finsupp.{u1, u3} \u03b1 N _inst_2) (Finsupp.{u1, u2} \u03b1 P _inst_3) (Finsupp.mapRange.equiv.{u1, u4, u3} \u03b1 M N _inst_1 _inst_2 f hf hf') (Finsupp.mapRange.equiv.{u1, u3, u2} \u03b1 N P _inst_2 _inst_3 f\u2082 hf\u2082 hf\u2082'))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.equiv_trans Finsupp.mapRange.equiv_trans\u2093'. -/\ntheorem mapRange.equiv_trans (f : M \u2243 N) (hf : f 0 = 0) (hf') (f\u2082 : N \u2243 P) (hf\u2082 : f\u2082 0 = 0) (hf\u2082') :\n    (mapRange.equiv (f.trans f\u2082) (by rw [Equiv.trans_apply, hf, hf\u2082])\n          (by rw [Equiv.symm_trans_apply, hf\u2082', hf']) :\n        (\u03b1 \u2192\u2080 _) \u2243 _) =\n      (mapRange.equiv f hf hf').trans (mapRange.equiv f\u2082 hf\u2082 hf\u2082') :=\n  Equiv.ext <| mapRange_comp _ _ _ _ _\n#align finsupp.map_range.equiv_trans Finsupp.mapRange.equiv_trans\n\n/- warning: finsupp.map_range.equiv_symm -> Finsupp.mapRange.equiv_symm is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Zero.{u2} M] [_inst_2 : Zero.{u3} N] (f : Equiv.{succ u2, succ u3} M N) (hf : Eq.{succ u3} N (coeFn.{max 1 (max (succ u2) (succ u3)) (succ u3) (succ u2), max (succ u2) (succ u3)} (Equiv.{succ u2, succ u3} M N) (fun (_x : Equiv.{succ u2, succ u3} M N) => M -> N) (Equiv.hasCoeToFun.{succ u2, succ u3} M N) f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (hf' : Eq.{succ u2} M (coeFn.{max 1 (max (succ u3) (succ u2)) (succ u2) (succ u3), max (succ u3) (succ u2)} (Equiv.{succ u3, succ u2} N M) (fun (_x : Equiv.{succ u3, succ u2} N M) => N -> M) (Equiv.hasCoeToFun.{succ u3, succ u2} N M) (Equiv.symm.{succ u2, succ u3} M N f) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N _inst_2)))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))), Eq.{max 1 (max (max (succ u1) (succ u3)) (succ u1) (succ u2)) (max (succ u1) (succ u2)) (succ u1) (succ u3)} (Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u2)} (Finsupp.{u1, u3} \u03b1 N _inst_2) (Finsupp.{u1, u2} \u03b1 M _inst_1)) (Equiv.symm.{max (succ u1) (succ u2), max (succ u1) (succ u3)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u3} \u03b1 N _inst_2) (Finsupp.mapRange.equiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f hf hf')) (Finsupp.mapRange.equiv.{u1, u3, u2} \u03b1 N M _inst_2 _inst_1 (Equiv.symm.{succ u2, succ u3} M N f) hf' hf)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : Zero.{u3} M] [_inst_2 : Zero.{u2} N] (f : Equiv.{succ u3, succ u2} M N) (hf : Eq.{succ u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M _inst_1))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} M N) M (fun (_x : M) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} M N) f (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M _inst_1))) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M _inst_1))) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M _inst_1))) _inst_2))) (hf' : Eq.{succ u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N _inst_2))) (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} N M) N (fun (_x : N) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} N M) (Equiv.symm.{succ u3, succ u2} M N f) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N _inst_2))) (OfNat.ofNat.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N _inst_2))) 0 (Zero.toOfNat0.{u3} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : N) => M) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N _inst_2))) _inst_1))), Eq.{max (max (succ u1) (succ u3)) (succ u2)} (Equiv.{max (succ u2) (succ u1), max (succ u3) (succ u1)} (Finsupp.{u1, u2} \u03b1 N _inst_2) (Finsupp.{u1, u3} \u03b1 M _inst_1)) (Equiv.symm.{max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.{u1, u2} \u03b1 N _inst_2) (Finsupp.mapRange.equiv.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 f hf hf')) (Finsupp.mapRange.equiv.{u1, u2, u3} \u03b1 N M _inst_2 _inst_1 (Equiv.symm.{succ u3, succ u2} M N f) hf' hf)\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.equiv_symm Finsupp.mapRange.equiv_symm\u2093'. -/\n@[simp]\ntheorem mapRange.equiv_symm (f : M \u2243 N) (hf hf') :\n    ((mapRange.equiv f hf hf').symm : (\u03b1 \u2192\u2080 _) \u2243 _) = mapRange.equiv f.symm hf' hf :=\n  Equiv.ext fun x => rfl\n#align finsupp.map_range.equiv_symm Finsupp.mapRange.equiv_symm\n\nend Equiv\n\nsection ZeroHom\n\nvariable [Zero M] [Zero N] [Zero P]\n\n#print Finsupp.mapRange.zeroHom /-\n/-- Composition with a fixed zero-preserving homomorphism is itself an zero-preserving homomorphism\non functions. -/\n@[simps]\ndef mapRange.zeroHom (f : ZeroHom M N) : ZeroHom (\u03b1 \u2192\u2080 M) (\u03b1 \u2192\u2080 N)\n    where\n  toFun := (mapRange f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 \u03b1 \u2192\u2080 N)\n  map_zero' := mapRange_zero\n#align finsupp.map_range.zero_hom Finsupp.mapRange.zeroHom\n-/\n\n/- warning: finsupp.map_range.zero_hom_id -> Finsupp.mapRange.zeroHom_id is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M], Eq.{succ (max u1 u2)} (ZeroHom.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1)) (Finsupp.mapRange.zeroHom.{u1, u2, u2} \u03b1 M M _inst_1 _inst_1 (ZeroHom.id.{u2} M _inst_1)) (ZeroHom.id.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M], Eq.{max (succ u2) (succ u1)} (ZeroHom.{max u1 u2, max u1 u2} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)) (Finsupp.mapRange.zeroHom.{u2, u1, u1} \u03b1 M M _inst_1 _inst_1 (ZeroHom.id.{u1} M _inst_1)) (ZeroHom.id.{max u1 u2} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.zero_hom_id Finsupp.mapRange.zeroHom_id\u2093'. -/\n@[simp]\ntheorem mapRange.zeroHom_id : mapRange.zeroHom (ZeroHom.id M) = ZeroHom.id (\u03b1 \u2192\u2080 M) :=\n  ZeroHom.ext mapRange_id\n#align finsupp.map_range.zero_hom_id Finsupp.mapRange.zeroHom_id\n\n/- warning: finsupp.map_range.zero_hom_comp -> Finsupp.mapRange.zeroHom_comp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : Zero.{u2} M] [_inst_2 : Zero.{u3} N] [_inst_3 : Zero.{u4} P] (f : ZeroHom.{u3, u4} N P _inst_2 _inst_3) (f\u2082 : ZeroHom.{u2, u3} M N _inst_1 _inst_2), Eq.{max (succ (max u1 u4)) (succ (max u1 u2))} (ZeroHom.{max u1 u2, max u1 u4} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u4} \u03b1 P _inst_3) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u4} \u03b1 P _inst_3)) (Finsupp.mapRange.zeroHom.{u1, u2, u4} \u03b1 M P _inst_1 _inst_3 (ZeroHom.comp.{u2, u3, u4} M N P _inst_1 _inst_2 _inst_3 f f\u2082)) (ZeroHom.comp.{max u1 u2, max u1 u3, max u1 u4} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u3} \u03b1 N _inst_2) (Finsupp.{u1, u4} \u03b1 P _inst_3) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u3} \u03b1 N _inst_2) (Finsupp.zero.{u1, u4} \u03b1 P _inst_3) (Finsupp.mapRange.zeroHom.{u1, u3, u4} \u03b1 N P _inst_2 _inst_3 f) (Finsupp.mapRange.zeroHom.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} {P : Type.{u3}} [_inst_1 : Zero.{u2} M] [_inst_2 : Zero.{u4} N] [_inst_3 : Zero.{u3} P] (f : ZeroHom.{u4, u3} N P _inst_2 _inst_3) (f\u2082 : ZeroHom.{u2, u4} M N _inst_1 _inst_2), Eq.{max (max (succ u1) (succ u2)) (succ u3)} (ZeroHom.{max u2 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u3} \u03b1 P _inst_3) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u3} \u03b1 P _inst_3)) (Finsupp.mapRange.zeroHom.{u1, u2, u3} \u03b1 M P _inst_1 _inst_3 (ZeroHom.comp.{u2, u4, u3} M N P _inst_1 _inst_2 _inst_3 f f\u2082)) (ZeroHom.comp.{max u2 u1, max u4 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.{u1, u4} \u03b1 N _inst_2) (Finsupp.{u1, u3} \u03b1 P _inst_3) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u4} \u03b1 N _inst_2) (Finsupp.zero.{u1, u3} \u03b1 P _inst_3) (Finsupp.mapRange.zeroHom.{u1, u4, u3} \u03b1 N P _inst_2 _inst_3 f) (Finsupp.mapRange.zeroHom.{u1, u2, u4} \u03b1 M N _inst_1 _inst_2 f\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.zero_hom_comp Finsupp.mapRange.zeroHom_comp\u2093'. -/\ntheorem mapRange.zeroHom_comp (f : ZeroHom N P) (f\u2082 : ZeroHom M N) :\n    (mapRange.zeroHom (f.comp f\u2082) : ZeroHom (\u03b1 \u2192\u2080 _) _) =\n      (mapRange.zeroHom f).comp (mapRange.zeroHom f\u2082) :=\n  ZeroHom.ext <| mapRange_comp _ _ _ _ _\n#align finsupp.map_range.zero_hom_comp Finsupp.mapRange.zeroHom_comp\n\nend ZeroHom\n\nsection AddMonoidHom\n\nvariable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]\n\n/- warning: finsupp.map_range.add_monoid_hom -> Finsupp.mapRange.addMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N], (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) -> (AddMonoidHom.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N], (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) -> (AddMonoidHom.{max u2 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u3} \u03b1 N (AddMonoid.toZero.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_monoid_hom Finsupp.mapRange.addMonoidHom\u2093'. -/\n/-- Composition with a fixed additive homomorphism is itself an additive homomorphism on functions.\n-/\n@[simps]\ndef mapRange.addMonoidHom (f : M \u2192+ N) : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 N\n    where\n  toFun := (mapRange f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 \u03b1 \u2192\u2080 N)\n  map_zero' := mapRange_zero\n  map_add' a b := mapRange_add f.map_add _ _\n#align finsupp.map_range.add_monoid_hom Finsupp.mapRange.addMonoidHom\n\n/- warning: finsupp.map_range.add_monoid_hom_id -> Finsupp.mapRange.addMonoidHom_id is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M], Eq.{succ (max u1 u2)} (AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapRange.addMonoidHom.{u1, u2, u2} \u03b1 M M _inst_1 _inst_1 (AddMonoidHom.id.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (AddMonoidHom.id.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M], Eq.{max (succ u2) (succ u1)} (AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (Finsupp.mapRange.addMonoidHom.{u2, u1, u1} \u03b1 M M _inst_1 _inst_1 (AddMonoidHom.id.{u1} M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (AddMonoidHom.id.{max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_monoid_hom_id Finsupp.mapRange.addMonoidHom_id\u2093'. -/\n@[simp]\ntheorem mapRange.addMonoidHom_id :\n    mapRange.addMonoidHom (AddMonoidHom.id M) = AddMonoidHom.id (\u03b1 \u2192\u2080 M) :=\n  AddMonoidHom.ext mapRange_id\n#align finsupp.map_range.add_monoid_hom_id Finsupp.mapRange.addMonoidHom_id\n\n/- warning: finsupp.map_range.add_monoid_hom_comp -> Finsupp.mapRange.addMonoidHom_comp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] [_inst_3 : AddCommMonoid.{u4} P] (f : AddMonoidHom.{u3, u4} N P (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3))) (f\u2082 : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))), Eq.{max (succ (max u1 u4)) (succ (max u1 u2))} (AddMonoidHom.{max u1 u2, max u1 u4} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u4} \u03b1 P (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u4} \u03b1 P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.mapRange.addMonoidHom.{u1, u2, u4} \u03b1 M P _inst_1 _inst_3 (AddMonoidHom.comp.{u2, u3, u4} M N P (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)) f f\u2082)) (AddMonoidHom.comp.{max u1 u2, max u1 u3, max u1 u4} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.{u1, u4} \u03b1 P (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.addZeroClass.{u1, u4} \u03b1 P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3))) (Finsupp.mapRange.addMonoidHom.{u1, u3, u4} \u03b1 N P _inst_2 _inst_3 f) (Finsupp.mapRange.addMonoidHom.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} {P : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u4} N] [_inst_3 : AddCommMonoid.{u3} P] (f : AddMonoidHom.{u4, u3} N P (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) (AddMonoid.toAddZeroClass.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3))) (f\u2082 : AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))), Eq.{max (max (succ u1) (succ u2)) (succ u3)} (AddMonoidHom.{max u2 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u3} \u03b1 P (AddMonoid.toZero.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 P (AddMonoid.toAddZeroClass.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3)))) (Finsupp.mapRange.addMonoidHom.{u1, u2, u3} \u03b1 M P _inst_1 _inst_3 (AddMonoidHom.comp.{u2, u4, u3} M N P (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) (AddMonoid.toAddZeroClass.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3)) f f\u2082)) (AddMonoidHom.comp.{max u2 u1, max u4 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u4} \u03b1 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.{u1, u3} \u03b1 P (AddMonoid.toZero.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u4} \u03b1 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u1, u3} \u03b1 P (AddMonoid.toAddZeroClass.{u3} P (AddCommMonoid.toAddMonoid.{u3} P _inst_3))) (Finsupp.mapRange.addMonoidHom.{u1, u4, u3} \u03b1 N P _inst_2 _inst_3 f) (Finsupp.mapRange.addMonoidHom.{u1, u2, u4} \u03b1 M N _inst_1 _inst_2 f\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_monoid_hom_comp Finsupp.mapRange.addMonoidHom_comp\u2093'. -/\ntheorem mapRange.addMonoidHom_comp (f : N \u2192+ P) (f\u2082 : M \u2192+ N) :\n    (mapRange.addMonoidHom (f.comp f\u2082) : (\u03b1 \u2192\u2080 _) \u2192+ _) =\n      (mapRange.addMonoidHom f).comp (mapRange.addMonoidHom f\u2082) :=\n  AddMonoidHom.ext <| mapRange_comp _ _ _ _ _\n#align finsupp.map_range.add_monoid_hom_comp Finsupp.mapRange.addMonoidHom_comp\n\n/- warning: finsupp.map_range.add_monoid_hom_to_zero_hom -> Finsupp.mapRange.addMonoidHom_toZeroHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))), Eq.{max (succ (max u1 u3)) (succ (max u1 u2))} (ZeroHom.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (AddZeroClass.toHasZero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (AddZeroClass.toHasZero.{max u1 u3} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) (AddMonoidHom.toZeroHom.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.mapRange.addMonoidHom.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.zeroHom.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddMonoidHom.toZeroHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u2} N] (f : AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))), Eq.{max (max (succ u1) (succ u3)) (succ u2)} (ZeroHom.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddZeroClass.toZero.{max u3 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (AddZeroClass.toZero.{max u2 u1} (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.addZeroClass.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) (AddMonoidHom.toZeroHom.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.mapRange.addMonoidHom.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.zeroHom.{u1, u3, u2} \u03b1 M N (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toZero.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddMonoidHom.toZeroHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) f))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_monoid_hom_to_zero_hom Finsupp.mapRange.addMonoidHom_toZeroHom\u2093'. -/\n@[simp]\ntheorem mapRange.addMonoidHom_toZeroHom (f : M \u2192+ N) :\n    (mapRange.addMonoidHom f).toZeroHom = (mapRange.zeroHom f.toZeroHom : ZeroHom (\u03b1 \u2192\u2080 _) _) :=\n  ZeroHom.ext fun _ => rfl\n#align finsupp.map_range.add_monoid_hom_to_zero_hom Finsupp.mapRange.addMonoidHom_toZeroHom\n\n/- warning: finsupp.map_range_multiset_sum -> Finsupp.mapRange_multiset_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (m : Multiset.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.mapRange.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (coeFn.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (fun (_x : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) f) (AddMonoidHom.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f) (Multiset.sum.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addCommMonoid.{u1, u2} \u03b1 M _inst_1) m)) (Multiset.sum.{max u1 u3} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addCommMonoid.{u1, u3} \u03b1 N _inst_2) (Multiset.map.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (fun (x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) => Finsupp.mapRange.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (coeFn.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (fun (_x : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) f) (AddMonoidHom.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f) x) m))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u2} N] (f : AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (m : Multiset.{max u3 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (Finsupp.mapRange.{u1, u3, u2} \u03b1 M N (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u3 u2, u3, u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u3 u2, u3, u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) f) (AddMonoidHom.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) f) (Multiset.sum.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) m)) (Multiset.sum.{max u1 u2} (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (Finsupp.addCommMonoid.{u1, u2} \u03b1 N _inst_2) (Multiset.map.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (fun (x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => Finsupp.mapRange.{u1, u3, u2} \u03b1 M N (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u3 u2, u3, u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u3 u2, u3, u2} (AddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) f) (AddMonoidHom.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) f) x) m))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range_multiset_sum Finsupp.mapRange_multiset_sum\u2093'. -/\ntheorem mapRange_multiset_sum (f : M \u2192+ N) (m : Multiset (\u03b1 \u2192\u2080 M)) :\n    mapRange f f.map_zero m.Sum = (m.map fun x => mapRange f f.map_zero x).Sum :=\n  (mapRange.addMonoidHom f : (\u03b1 \u2192\u2080 _) \u2192+ _).map_multiset_sum _\n#align finsupp.map_range_multiset_sum Finsupp.mapRange_multiset_sum\n\n/- warning: finsupp.map_range_finset_sum -> Finsupp.mapRange_finset_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (s : Finset.{u2} \u03b9) (g : \u03b9 -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))), Eq.{max (succ u1) (succ u4)} (Finsupp.{u1, u4} \u03b1 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.mapRange.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (coeFn.{max (succ u4) (succ u3), max (succ u3) (succ u4)} (AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (fun (_x : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) f) (AddMonoidHom.map_zero.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) f) (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (x : \u03b9) => g x))) (Finset.sum.{max u1 u4, u2} (Finsupp.{u1, u4} \u03b1 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) \u03b9 (Finsupp.addCommMonoid.{u1, u4} \u03b1 N _inst_2) s (fun (x : \u03b9) => Finsupp.mapRange.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (coeFn.{max (succ u4) (succ u3), max (succ u3) (succ u4)} (AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (fun (_x : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) f) (AddMonoidHom.map_zero.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) f) (g x)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u4}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u4} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (s : Finset.{u2} \u03b9) (g : \u03b9 -> (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.mapRange.{u1, u4, u3} \u03b1 M N (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) f) (AddMonoidHom.map_zero.{u3, u4} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f) (Finset.sum.{max u4 u1, u2} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u4} \u03b1 M _inst_1) s (fun (x : \u03b9) => g x))) (Finset.sum.{max u3 u1, u2} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 N _inst_2) s (fun (x : \u03b9) => Finsupp.mapRange.{u1, u4, u3} \u03b1 M N (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : M) => N) (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M (AddZeroClass.toZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) f) (AddMonoidHom.map_zero.{u3, u4} M N (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f) (g x)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range_finset_sum Finsupp.mapRange_finset_sum\u2093'. -/\ntheorem mapRange_finset_sum (f : M \u2192+ N) (s : Finset \u03b9) (g : \u03b9 \u2192 \u03b1 \u2192\u2080 M) :\n    mapRange f f.map_zero (\u2211 x in s, g x) = \u2211 x in s, mapRange f f.map_zero (g x) :=\n  (mapRange.addMonoidHom f : (\u03b1 \u2192\u2080 _) \u2192+ _).map_sum _ _\n#align finsupp.map_range_finset_sum Finsupp.mapRange_finset_sum\n\n/- warning: finsupp.map_range.add_equiv -> Finsupp.mapRange.addEquiv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N], (AddEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) -> (AddEquiv.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N], (AddEquiv.{u2, u3} M N (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) -> (AddEquiv.{max u2 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u3} \u03b1 N (AddMonoid.toZero.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv Finsupp.mapRange.addEquiv\u2093'. -/\n/-- `finsupp.map_range.add_monoid_hom` as an equiv. -/\n@[simps apply]\ndef mapRange.addEquiv (f : M \u2243+ N) : (\u03b1 \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 N) :=\n  {\n    mapRange.addMonoidHom\n      f.toAddMonoidHom with\n    toFun := (mapRange f f.map_zero : (\u03b1 \u2192\u2080 M) \u2192 \u03b1 \u2192\u2080 N)\n    invFun := (mapRange f.symm f.symm.map_zero : (\u03b1 \u2192\u2080 N) \u2192 \u03b1 \u2192\u2080 M)\n    left_inv := fun x =>\n      by\n      rw [\u2190 map_range_comp _ _ _ _] <;> simp_rw [AddEquiv.symm_comp_self]\n      \u00b7 exact map_range_id _\n      \u00b7 rfl\n    right_inv := fun x =>\n      by\n      rw [\u2190 map_range_comp _ _ _ _] <;> simp_rw [AddEquiv.self_comp_symm]\n      \u00b7 exact map_range_id _\n      \u00b7 rfl }\n#align finsupp.map_range.add_equiv Finsupp.mapRange.addEquiv\n\n/- warning: finsupp.map_range.add_equiv_refl -> Finsupp.mapRange.addEquiv_refl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M], Eq.{succ (max u1 u2)} (AddEquiv.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapRange.addEquiv.{u1, u2, u2} \u03b1 M M _inst_1 _inst_1 (AddEquiv.refl.{u2} M (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (AddEquiv.refl.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M], Eq.{max (succ u2) (succ u1)} (AddEquiv.{max u1 u2, max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (Finsupp.mapRange.addEquiv.{u2, u1, u1} \u03b1 M M _inst_1 _inst_1 (AddEquiv.refl.{u1} M (AddZeroClass.toAdd.{u1} M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))))) (AddEquiv.refl.{max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv_refl Finsupp.mapRange.addEquiv_refl\u2093'. -/\n@[simp]\ntheorem mapRange.addEquiv_refl : mapRange.addEquiv (AddEquiv.refl M) = AddEquiv.refl (\u03b1 \u2192\u2080 M) :=\n  AddEquiv.ext mapRange_id\n#align finsupp.map_range.add_equiv_refl Finsupp.mapRange.addEquiv_refl\n\n/- warning: finsupp.map_range.add_equiv_trans -> Finsupp.mapRange.addEquiv_trans is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {P : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] [_inst_3 : AddCommMonoid.{u4} P] (f : AddEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (f\u2082 : AddEquiv.{u3, u4} N P (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddZeroClass.toHasAdd.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))), Eq.{max (succ (max u1 u2)) (succ (max u1 u4))} (AddEquiv.{max u1 u2, max u1 u4} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u4} \u03b1 P (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u4} \u03b1 P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.mapRange.addEquiv.{u1, u2, u4} \u03b1 M P _inst_1 _inst_3 (AddEquiv.trans.{u2, u3, u4} M N P (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddZeroClass.toHasAdd.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3))) f f\u2082)) (AddEquiv.trans.{max u1 u2, max u1 u3, max u1 u4} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.{u1, u4} \u03b1 P (AddZeroClass.toHasZero.{u4} P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.add.{u1, u4} \u03b1 P (AddMonoid.toAddZeroClass.{u4} P (AddCommMonoid.toAddMonoid.{u4} P _inst_3))) (Finsupp.mapRange.addEquiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f) (Finsupp.mapRange.addEquiv.{u1, u3, u4} \u03b1 N P _inst_2 _inst_3 f\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u4}} {N : Type.{u3}} {P : Type.{u2}} [_inst_1 : AddCommMonoid.{u4} M] [_inst_2 : AddCommMonoid.{u3} N] [_inst_3 : AddCommMonoid.{u2} P] (f : AddEquiv.{u4, u3} M N (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (f\u2082 : AddEquiv.{u3, u2} N P (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddZeroClass.toAdd.{u2} P (AddMonoid.toAddZeroClass.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3)))), Eq.{max (max (succ u1) (succ u4)) (succ u2)} (AddEquiv.{max u4 u1, max u2 u1} (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.{u1, u2} \u03b1 P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3))) (Finsupp.add.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 P (AddMonoid.toAddZeroClass.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3)))) (Finsupp.mapRange.addEquiv.{u1, u4, u2} \u03b1 M P _inst_1 _inst_3 (AddEquiv.trans.{u4, u3, u2} M N P (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddZeroClass.toAdd.{u2} P (AddMonoid.toAddZeroClass.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3))) f f\u2082)) (AddEquiv.trans.{max u4 u1, max u3 u1, max u2 u1} (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.{u1, u3} \u03b1 N (AddMonoid.toZero.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.{u1, u2} \u03b1 P (AddMonoid.toZero.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3))) (Finsupp.add.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.add.{u1, u2} \u03b1 P (AddMonoid.toAddZeroClass.{u2} P (AddCommMonoid.toAddMonoid.{u2} P _inst_3))) (Finsupp.mapRange.addEquiv.{u1, u4, u3} \u03b1 M N _inst_1 _inst_2 f) (Finsupp.mapRange.addEquiv.{u1, u3, u2} \u03b1 N P _inst_2 _inst_3 f\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv_trans Finsupp.mapRange.addEquiv_trans\u2093'. -/\ntheorem mapRange.addEquiv_trans (f : M \u2243+ N) (f\u2082 : N \u2243+ P) :\n    (mapRange.addEquiv (f.trans f\u2082) : (\u03b1 \u2192\u2080 _) \u2243+ _) =\n      (mapRange.addEquiv f).trans (mapRange.addEquiv f\u2082) :=\n  AddEquiv.ext <| mapRange_comp _ _ _ _ _\n#align finsupp.map_range.add_equiv_trans Finsupp.mapRange.addEquiv_trans\n\n/- warning: finsupp.map_range.add_equiv_symm -> Finsupp.mapRange.addEquiv_symm is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))), Eq.{max (succ (max u1 u3)) (succ (max u1 u2))} (AddEquiv.{max u1 u3, max u1 u2} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (AddEquiv.symm.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.mapRange.addEquiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.addEquiv.{u1, u3, u2} \u03b1 N M _inst_2 _inst_1 (AddEquiv.symm.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u2} N] (f : AddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))), Eq.{max (max (succ u1) (succ u3)) (succ u2)} (AddEquiv.{max u2 u1, max u3 u1} (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (AddEquiv.symm.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.mapRange.addEquiv.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.addEquiv.{u1, u2, u3} \u03b1 N M _inst_2 _inst_1 (AddEquiv.symm.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) f))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv_symm Finsupp.mapRange.addEquiv_symm\u2093'. -/\n@[simp]\ntheorem mapRange.addEquiv_symm (f : M \u2243+ N) :\n    ((mapRange.addEquiv f).symm : (\u03b1 \u2192\u2080 _) \u2243+ _) = mapRange.addEquiv f.symm :=\n  AddEquiv.ext fun x => rfl\n#align finsupp.map_range.add_equiv_symm Finsupp.mapRange.addEquiv_symm\n\n/- warning: finsupp.map_range.add_equiv_to_add_monoid_hom -> Finsupp.mapRange.addEquiv_toAddMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))), Eq.{max (succ (max u1 u3)) (succ (max u1 u2))} (AddMonoidHom.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (AddEquiv.toAddMonoidHom.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.mapRange.addEquiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.addMonoidHom.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 (AddEquiv.toAddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u2} N] (f : AddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))), Eq.{max (max (succ u1) (succ u3)) (succ u2)} (AddMonoidHom.{max u1 u3, max u1 u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (AddEquiv.toAddMonoidHom.{max u1 u3, max u1 u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.mapRange.addEquiv.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.addMonoidHom.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 (AddEquiv.toAddMonoidHom.{u3, u2} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) f))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv_to_add_monoid_hom Finsupp.mapRange.addEquiv_toAddMonoidHom\u2093'. -/\n@[simp]\ntheorem mapRange.addEquiv_toAddMonoidHom (f : M \u2243+ N) :\n    (mapRange.addEquiv f : (\u03b1 \u2192\u2080 _) \u2243+ _).toAddMonoidHom =\n      (mapRange.addMonoidHom f.toAddMonoidHom : (\u03b1 \u2192\u2080 _) \u2192+ _) :=\n  AddMonoidHom.ext fun _ => rfl\n#align finsupp.map_range.add_equiv_to_add_monoid_hom Finsupp.mapRange.addEquiv_toAddMonoidHom\n\n/- warning: finsupp.map_range.add_equiv_to_equiv -> Finsupp.mapRange.addEquiv_toEquiv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] (f : AddEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))), Eq.{max 1 (max (succ (max u1 u2)) (succ (max u1 u3))) (succ (max u1 u3)) (succ (max u1 u2))} (Equiv.{succ (max u1 u2), succ (max u1 u3)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) (AddEquiv.toEquiv.{max u1 u2, max u1 u3} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (Finsupp.mapRange.addEquiv.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.equiv.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddEquiv.toEquiv.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) f) (AddEquiv.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) f) (AddEquiv.map_zero.{u3, u2} N M (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddEquiv.symm.{u2, u3} M N (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toHasAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) f)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u2} N] (f : AddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))), Eq.{max (max (succ u1) (succ u3)) (succ u2)} (Equiv.{max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) (EquivLike.toEquiv.{max (succ u1) (succ u3), max (succ u1) (succ u2), max (max (succ u1) (succ u3)) (succ u2)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (AddEquiv.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (AddEquivClass.toEquivLike.{max (max u1 u3) u2, max u1 u3, max u1 u2} (AddEquiv.{max u3 u1, max u2 u1} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 N (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddEquiv.instAddEquivClassAddEquiv.{max u1 u3, max u1 u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) (Finsupp.mapRange.addEquiv.{u1, u3, u2} \u03b1 M N _inst_1 _inst_2 f)) (Finsupp.mapRange.equiv.{u1, u3, u2} \u03b1 M N (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toZero.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : M) => N) (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddZeroClass.toZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (EquivLike.toEquiv.{succ u3, succ u2, max (succ u3) (succ u2)} M N (AddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) (AddEquivClass.toEquivLike.{max u3 u2, u3, u2} (AddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)))) M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) (AddEquiv.instAddEquivClassAddEquiv.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))))) f) (AddEquiv.map_zero.{u2, u3} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) f) (AddEquiv.map_zero.{u3, u2} N M (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2)) (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddEquiv.symm.{u3, u2} M N (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toAdd.{u2} N (AddMonoid.toAddZeroClass.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_2))) f)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range.add_equiv_to_equiv Finsupp.mapRange.addEquiv_toEquiv\u2093'. -/\n@[simp]\ntheorem mapRange.addEquiv_toEquiv (f : M \u2243+ N) :\n    (mapRange.addEquiv f).toEquiv =\n      (mapRange.equiv f.toEquiv f.map_zero f.symm.map_zero : (\u03b1 \u2192\u2080 _) \u2243 _) :=\n  Equiv.ext fun _ => rfl\n#align finsupp.map_range.add_equiv_to_equiv Finsupp.mapRange.addEquiv_toEquiv\n\nend AddMonoidHom\n\nend Finsupp\n\nend MapRange\n\n/-! ### Declarations about `equiv_congr_left` -/\n\n\nsection EquivCongrLeft\n\nvariable [Zero M]\n\nnamespace Finsupp\n\n#print Finsupp.equivMapDomain /-\n/-- Given `f : \u03b1 \u2243 \u03b2`, we can map `l : \u03b1 \u2192\u2080 M` to  `equiv_map_domain f l : \u03b2 \u2192\u2080 M` (computably)\nby mapping the support forwards and the function backwards. -/\ndef equivMapDomain (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M\n    where\n  support := l.support.map f.toEmbedding\n  toFun a := l (f.symm a)\n  mem_support_toFun a := by simp only [Finset.mem_map_equiv, mem_support_to_fun] <;> rfl\n#align finsupp.equiv_map_domain Finsupp.equivMapDomain\n-/\n\n/- warning: finsupp.equiv_map_domain_apply -> Finsupp.equivMapDomain_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (l : Finsupp.{u1, u3} \u03b1 M _inst_1) (b : \u03b2), Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 M _inst_1) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f l) b) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u3} \u03b1 M _inst_1) l (coeFn.{max 1 (max (succ u2) (succ u1)) (succ u1) (succ u2), max (succ u2) (succ u1)} (Equiv.{succ u2, succ u1} \u03b2 \u03b1) (fun (_x : Equiv.{succ u2, succ u1} \u03b2 \u03b1) => \u03b2 -> \u03b1) (Equiv.hasCoeToFun.{succ u2, succ u1} \u03b2 \u03b1) (Equiv.symm.{succ u1, succ u2} \u03b1 \u03b2 f) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u3, succ u2} \u03b1 \u03b2) (l : Finsupp.{u3, u1} \u03b1 M _inst_1) (b : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) b) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f l) b) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M _inst_1) l (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b2) => \u03b1) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} \u03b2 \u03b1) (Equiv.symm.{succ u3, succ u2} \u03b1 \u03b2 f) b))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_apply Finsupp.equivMapDomain_apply\u2093'. -/\n@[simp]\ntheorem equivMapDomain_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) (b : \u03b2) :\n    equivMapDomain f l b = l (f.symm b) :=\n  rfl\n#align finsupp.equiv_map_domain_apply Finsupp.equivMapDomain_apply\n\n/- warning: finsupp.equiv_map_domain_symm_apply -> Finsupp.equivMapDomain_symm_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (l : Finsupp.{u2, u3} \u03b2 M _inst_1) (a : \u03b1), Eq.{succ u3} M (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u3} \u03b1 M _inst_1) (Finsupp.equivMapDomain.{u2, u1, u3} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u1, succ u2} \u03b1 \u03b2 f) l) a) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 M _inst_1) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M _inst_1) l (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} \u03b1 \u03b2) (fun (_x : Equiv.{succ u1, succ u2} \u03b1 \u03b2) => \u03b1 -> \u03b2) (Equiv.hasCoeToFun.{succ u1, succ u2} \u03b1 \u03b2) f a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u3, succ u2} \u03b1 \u03b2) (l : Finsupp.{u2, u1} \u03b2 M _inst_1) (a : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M _inst_1) (Finsupp.equivMapDomain.{u2, u3, u1} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u3, succ u2} \u03b1 \u03b2 f) l) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M _inst_1) l (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => \u03b2) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} \u03b1 \u03b2) f a))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_symm_apply Finsupp.equivMapDomain_symm_apply\u2093'. -/\ntheorem equivMapDomain_symm_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b2 \u2192\u2080 M) (a : \u03b1) :\n    equivMapDomain f.symm l a = l (f a) :=\n  rfl\n#align finsupp.equiv_map_domain_symm_apply Finsupp.equivMapDomain_symm_apply\n\n/- warning: finsupp.equiv_map_domain_refl -> Finsupp.equivMapDomain_refl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{u1, u2} \u03b1 M _inst_1), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.equivMapDomain.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u1} \u03b1) l) l\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (l : Finsupp.{u2, u1} \u03b1 M _inst_1), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.equivMapDomain.{u2, u2, u1} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u2} \u03b1) l) l\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_refl Finsupp.equivMapDomain_refl\u2093'. -/\n@[simp]\ntheorem equivMapDomain_refl (l : \u03b1 \u2192\u2080 M) : equivMapDomain (Equiv.refl _) l = l := by ext x <;> rfl\n#align finsupp.equiv_map_domain_refl Finsupp.equivMapDomain_refl\n\n/- warning: finsupp.equiv_map_domain_refl' -> Finsupp.equivMapDomain_refl' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M], Eq.{max (succ u1) (succ u2)} ((Finsupp.{u1, u2} \u03b1 M _inst_1) -> (Finsupp.{u1, u2} \u03b1 M _inst_1)) (Finsupp.equivMapDomain.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u1} \u03b1)) (id.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M], Eq.{max (succ u2) (succ u1)} ((Finsupp.{u2, u1} \u03b1 M _inst_1) -> (Finsupp.{u2, u1} \u03b1 M _inst_1)) (Finsupp.equivMapDomain.{u2, u2, u1} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u2} \u03b1)) (id.{max (succ u1) (succ u2)} (Finsupp.{u2, u1} \u03b1 M _inst_1))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_refl' Finsupp.equivMapDomain_refl'\u2093'. -/\ntheorem equivMapDomain_refl' : equivMapDomain (Equiv.refl _) = @id (\u03b1 \u2192\u2080 M) := by ext x <;> rfl\n#align finsupp.equiv_map_domain_refl' Finsupp.equivMapDomain_refl'\n\n/- warning: finsupp.equiv_map_domain_trans -> Finsupp.equivMapDomain_trans is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {M : Type.{u4}} [_inst_1 : Zero.{u4} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (g : Equiv.{succ u2, succ u3} \u03b2 \u03b3) (l : Finsupp.{u1, u4} \u03b1 M _inst_1), Eq.{max (succ u3) (succ u4)} (Finsupp.{u3, u4} \u03b3 M _inst_1) (Finsupp.equivMapDomain.{u1, u3, u4} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u1, succ u2, succ u3} \u03b1 \u03b2 \u03b3 f g) l) (Finsupp.equivMapDomain.{u2, u3, u4} \u03b2 \u03b3 M _inst_1 g (Finsupp.equivMapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f l))\nbut is expected to have type\n  forall {\u03b1 : Type.{u4}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u4, succ u3} \u03b1 \u03b2) (g : Equiv.{succ u3, succ u2} \u03b2 \u03b3) (l : Finsupp.{u4, u1} \u03b1 M _inst_1), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b3 M _inst_1) (Finsupp.equivMapDomain.{u4, u2, u1} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u4, succ u3, succ u2} \u03b1 \u03b2 \u03b3 f g) l) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b2 \u03b3 M _inst_1 g (Finsupp.equivMapDomain.{u4, u3, u1} \u03b1 \u03b2 M _inst_1 f l))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_trans Finsupp.equivMapDomain_trans\u2093'. -/\ntheorem equivMapDomain_trans (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) (l : \u03b1 \u2192\u2080 M) :\n    equivMapDomain (f.trans g) l = equivMapDomain g (equivMapDomain f l) := by ext x <;> rfl\n#align finsupp.equiv_map_domain_trans Finsupp.equivMapDomain_trans\n\n/- warning: finsupp.equiv_map_domain_trans' -> Finsupp.equivMapDomain_trans' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {M : Type.{u4}} [_inst_1 : Zero.{u4} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (g : Equiv.{succ u2, succ u3} \u03b2 \u03b3), Eq.{max (max (succ u1) (succ u4)) (succ u3) (succ u4)} ((Finsupp.{u1, u4} \u03b1 M _inst_1) -> (Finsupp.{u3, u4} \u03b3 M _inst_1)) (Finsupp.equivMapDomain.{u1, u3, u4} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u1, succ u2, succ u3} \u03b1 \u03b2 \u03b3 f g)) (Function.comp.{max (succ u1) (succ u4), max (succ u2) (succ u4), max (succ u3) (succ u4)} (Finsupp.{u1, u4} \u03b1 M _inst_1) (Finsupp.{u2, u4} \u03b2 M _inst_1) (Finsupp.{u3, u4} \u03b3 M _inst_1) (Finsupp.equivMapDomain.{u2, u3, u4} \u03b2 \u03b3 M _inst_1 g) (Finsupp.equivMapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u4}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u4, succ u3} \u03b1 \u03b2) (g : Equiv.{succ u3, succ u2} \u03b2 \u03b3), Eq.{max (max (succ u4) (succ u2)) (succ u1)} ((Finsupp.{u4, u1} \u03b1 M _inst_1) -> (Finsupp.{u2, u1} \u03b3 M _inst_1)) (Finsupp.equivMapDomain.{u4, u2, u1} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u4, succ u3, succ u2} \u03b1 \u03b2 \u03b3 f g)) (Function.comp.{max (succ u1) (succ u4), max (succ u1) (succ u3), max (succ u1) (succ u2)} (Finsupp.{u4, u1} \u03b1 M _inst_1) (Finsupp.{u3, u1} \u03b2 M _inst_1) (Finsupp.{u2, u1} \u03b3 M _inst_1) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b2 \u03b3 M _inst_1 g) (Finsupp.equivMapDomain.{u4, u3, u1} \u03b1 \u03b2 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_trans' Finsupp.equivMapDomain_trans'\u2093'. -/\ntheorem equivMapDomain_trans' (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) :\n    @equivMapDomain _ _ M _ (f.trans g) = equivMapDomain g \u2218 equivMapDomain f := by ext x <;> rfl\n#align finsupp.equiv_map_domain_trans' Finsupp.equivMapDomain_trans'\n\n/- warning: finsupp.equiv_map_domain_single -> Finsupp.equivMapDomain_single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (a : \u03b1) (b : M), Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u1, u3} \u03b1 M _inst_1 a b)) (Finsupp.single.{u2, u3} \u03b2 M _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} \u03b1 \u03b2) (fun (_x : Equiv.{succ u1, succ u2} \u03b1 \u03b2) => \u03b1 -> \u03b2) (Equiv.hasCoeToFun.{succ u1, succ u2} \u03b1 \u03b2) f a) b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u3, succ u2} \u03b1 \u03b2) (a : \u03b1) (b : M), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u3, u1} \u03b1 M _inst_1 a b)) (Finsupp.single.{u2, u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => \u03b2) a) M _inst_1 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => \u03b2) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} \u03b1 \u03b2) f a) b)\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_single Finsupp.equivMapDomain_single\u2093'. -/\n@[simp]\ntheorem equivMapDomain_single (f : \u03b1 \u2243 \u03b2) (a : \u03b1) (b : M) :\n    equivMapDomain f (single a b) = single (f a) b := by\n  classical\n    ext x\n    simp only [single_apply, Equiv.apply_eq_iff_eq_symm_apply, equiv_map_domain_apply]\n#align finsupp.equiv_map_domain_single Finsupp.equivMapDomain_single\n\n/- warning: finsupp.equiv_map_domain_zero -> Finsupp.equivMapDomain_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] {f : Equiv.{succ u1, succ u2} \u03b1 \u03b2}, Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.zero.{u1, u3} \u03b1 M _inst_1))))) (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (OfNat.mk.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (Zero.zero.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.zero.{u2, u3} \u03b2 M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {f : Equiv.{succ u3, succ u2} \u03b1 \u03b2}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.zero.{u3, u1} \u03b1 M _inst_1)))) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.zero.{u2, u1} \u03b2 M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_zero Finsupp.equivMapDomain_zero\u2093'. -/\n@[simp]\ntheorem equivMapDomain_zero {f : \u03b1 \u2243 \u03b2} : equivMapDomain f (0 : \u03b1 \u2192\u2080 M) = (0 : \u03b2 \u2192\u2080 M) := by\n  ext x <;> simp only [equiv_map_domain_apply, coe_zero, Pi.zero_apply]\n#align finsupp.equiv_map_domain_zero Finsupp.equivMapDomain_zero\n\n#print Finsupp.equivCongrLeft /-\n/-- Given `f : \u03b1 \u2243 \u03b2`, the finitely supported function spaces are also in bijection:\n`(\u03b1 \u2192\u2080 M) \u2243 (\u03b2 \u2192\u2080 M)`.\n\nThis is the finitely-supported version of `equiv.Pi_congr_left`. -/\ndef equivCongrLeft (f : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192\u2080 M) \u2243 (\u03b2 \u2192\u2080 M) := by\n  refine' \u27e8equiv_map_domain f, equiv_map_domain f.symm, fun f => _, fun f => _\u27e9 <;> ext x <;>\n    simp only [equiv_map_domain_apply, Equiv.symm_symm, Equiv.symm_apply_apply,\n      Equiv.apply_symm_apply]\n#align finsupp.equiv_congr_left Finsupp.equivCongrLeft\n-/\n\n/- warning: finsupp.equiv_congr_left_apply -> Finsupp.equivCongrLeft_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (l : Finsupp.{u1, u3} \u03b1 M _inst_1), Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (coeFn.{max 1 (max (max (succ u1) (succ u3)) (succ u2) (succ u3)) (max (succ u2) (succ u3)) (succ u1) (succ u3), max (max (succ u1) (succ u3)) (succ u2) (succ u3)} (Equiv.{max (succ u1) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.{u2, u3} \u03b2 M _inst_1)) (fun (_x : Equiv.{max (succ u1) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.{u2, u3} \u03b2 M _inst_1)) => (Finsupp.{u1, u3} \u03b1 M _inst_1) -> (Finsupp.{u2, u3} \u03b2 M _inst_1)) (Equiv.hasCoeToFun.{max (succ u1) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.{u2, u3} \u03b2 M _inst_1)) (Finsupp.equivCongrLeft.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f) l) (Finsupp.equivMapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f l)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u3, succ u2} \u03b1 \u03b2) (l : Finsupp.{u3, u1} \u03b1 M _inst_1), Eq.{max (succ u2) (succ u1)} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Finsupp.{u3, u1} \u03b1 M _inst_1) => Finsupp.{u2, u1} \u03b2 M _inst_1) l) (FunLike.coe.{max (max (succ u3) (succ u2)) (succ u1), max (succ u3) (succ u1), max (succ u2) (succ u1)} (Equiv.{max (succ u1) (succ u3), max (succ u1) (succ u2)} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.{u2, u1} \u03b2 M _inst_1)) (Finsupp.{u3, u1} \u03b1 M _inst_1) (fun (_x : Finsupp.{u3, u1} \u03b1 M _inst_1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Finsupp.{u3, u1} \u03b1 M _inst_1) => Finsupp.{u2, u1} \u03b2 M _inst_1) _x) (Equiv.instFunLikeEquiv.{max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.{u2, u1} \u03b2 M _inst_1)) (Finsupp.equivCongrLeft.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f) l) (Finsupp.equivMapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f l)\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_congr_left_apply Finsupp.equivCongrLeft_apply\u2093'. -/\n@[simp]\ntheorem equivCongrLeft_apply (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) : equivCongrLeft f l = equivMapDomain f l :=\n  rfl\n#align finsupp.equiv_congr_left_apply Finsupp.equivCongrLeft_apply\n\n/- warning: finsupp.equiv_congr_left_symm -> Finsupp.equivCongrLeft_symm is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2), Eq.{max 1 (max (max (succ u2) (succ u3)) (succ u1) (succ u3)) (max (succ u1) (succ u3)) (succ u2) (succ u3)} (Equiv.{max (succ u2) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.{u1, u3} \u03b1 M _inst_1)) (Equiv.symm.{max (succ u1) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.equivCongrLeft.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f)) (Finsupp.equivCongrLeft.{u2, u1, u3} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u1, succ u2} \u03b1 \u03b2 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : Equiv.{succ u3, succ u2} \u03b1 \u03b2), Eq.{max (max (succ u3) (succ u2)) (succ u1)} (Equiv.{max (succ u2) (succ u1), max (succ u3) (succ u1)} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.{u3, u1} \u03b1 M _inst_1)) (Equiv.symm.{max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.equivCongrLeft.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f)) (Finsupp.equivCongrLeft.{u2, u3, u1} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u3, succ u2} \u03b1 \u03b2 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_congr_left_symm Finsupp.equivCongrLeft_symm\u2093'. -/\n@[simp]\ntheorem equivCongrLeft_symm (f : \u03b1 \u2243 \u03b2) :\n    (@equivCongrLeft _ _ M _ f).symm = equivCongrLeft f.symm :=\n  rfl\n#align finsupp.equiv_congr_left_symm Finsupp.equivCongrLeft_symm\n\nend Finsupp\n\nend EquivCongrLeft\n\nsection CastFinsupp\n\nvariable [Zero M] (f : \u03b1 \u2192\u2080 M)\n\nnamespace Nat\n\n/- warning: nat.cast_finsupp_prod -> Nat.cast_finsupp_prod is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommSemiring.{u3} R] (g : \u03b1 -> M -> Nat), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u3} Nat R (CoeTC\u2093.coe.{1, succ u3} Nat R (Nat.castCoe.{u3} R (AddMonoidWithOne.toNatCast.{u3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))))))) (Finsupp.prod.{u1, u2, 0} \u03b1 M Nat _inst_1 Nat.commMonoid f g)) (Finsupp.prod.{u1, u2, u3} \u03b1 M R _inst_1 (CommSemiring.toCommMonoid.{u3} R _inst_2) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u3} Nat R (CoeTC\u2093.coe.{1, succ u3} Nat R (Nat.castCoe.{u3} R (AddMonoidWithOne.toNatCast.{u3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))))))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommSemiring.{u3} R] (g : \u03b1 -> M -> Nat), Eq.{succ u3} R (Nat.cast.{u3} R (Semiring.toNatCast.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)) (Finsupp.prod.{u2, u1, 0} \u03b1 M Nat _inst_1 Nat.commMonoid f g)) (Finsupp.prod.{u2, u1, u3} \u03b1 M R _inst_1 (CommSemiring.toCommMonoid.{u3} R _inst_2) f (fun (a : \u03b1) (b : M) => Nat.cast.{u3} R (Semiring.toNatCast.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)) (g a b)))\nCase conversion may be inaccurate. Consider using '#align nat.cast_finsupp_prod Nat.cast_finsupp_prod\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_prod [CommSemiring R] (g : \u03b1 \u2192 M \u2192 \u2115) :\n    (\u2191(f.Prod g) : R) = f.Prod fun a b => \u2191(g a b) :=\n  Nat.cast_prod _ _\n#align nat.cast_finsupp_prod Nat.cast_finsupp_prod\n\n/- warning: nat.cast_finsupp_sum -> Nat.cast_finsupp_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommSemiring.{u3} R] (g : \u03b1 -> M -> Nat), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u3} Nat R (CoeTC\u2093.coe.{1, succ u3} Nat R (Nat.castCoe.{u3} R (AddMonoidWithOne.toNatCast.{u3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))))))) (Finsupp.sum.{u1, u2, 0} \u03b1 M Nat _inst_1 Nat.addCommMonoid f g)) (Finsupp.sum.{u1, u2, u3} \u03b1 M R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u3} Nat R (CoeTC\u2093.coe.{1, succ u3} Nat R (Nat.castCoe.{u3} R (AddMonoidWithOne.toNatCast.{u3} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u3} R (NonAssocSemiring.toAddCommMonoidWithOne.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))))))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommSemiring.{u3} R] (g : \u03b1 -> M -> Nat), Eq.{succ u3} R (Nat.cast.{u3} R (Semiring.toNatCast.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)) (Finsupp.sum.{u2, u1, 0} \u03b1 M Nat _inst_1 Nat.addCommMonoid f g)) (Finsupp.sum.{u2, u1, u3} \u03b1 M R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)))) f (fun (a : \u03b1) (b : M) => Nat.cast.{u3} R (Semiring.toNatCast.{u3} R (CommSemiring.toSemiring.{u3} R _inst_2)) (g a b)))\nCase conversion may be inaccurate. Consider using '#align nat.cast_finsupp_sum Nat.cast_finsupp_sum\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_sum [CommSemiring R] (g : \u03b1 \u2192 M \u2192 \u2115) :\n    (\u2191(f.Sum g) : R) = f.Sum fun a b => \u2191(g a b) :=\n  Nat.cast_sum _ _\n#align nat.cast_finsupp_sum Nat.cast_finsupp_sum\n\nend Nat\n\nnamespace Int\n\n/- warning: int.cast_finsupp_prod -> Int.cast_finsupp_prod is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommRing.{u3} R] (g : \u03b1 -> M -> Int), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Int R (HasLiftT.mk.{1, succ u3} Int R (CoeTC\u2093.coe.{1, succ u3} Int R (Int.castCoe.{u3} R (AddGroupWithOne.toHasIntCast.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (CommRing.toRing.{u3} R _inst_2))))))) (Finsupp.prod.{u1, u2, 0} \u03b1 M Int _inst_1 Int.commMonoid f g)) (Finsupp.prod.{u1, u2, u3} \u03b1 M R _inst_1 (CommRing.toCommMonoid.{u3} R _inst_2) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Int R (HasLiftT.mk.{1, succ u3} Int R (CoeTC\u2093.coe.{1, succ u3} Int R (Int.castCoe.{u3} R (AddGroupWithOne.toHasIntCast.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (CommRing.toRing.{u3} R _inst_2))))))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommRing.{u3} R] (g : \u03b1 -> M -> Int), Eq.{succ u3} R (Int.cast.{u3} R (Ring.toIntCast.{u3} R (CommRing.toRing.{u3} R _inst_2)) (Finsupp.prod.{u2, u1, 0} \u03b1 M Int _inst_1 Int.instCommMonoidInt f g)) (Finsupp.prod.{u2, u1, u3} \u03b1 M R _inst_1 (CommRing.toCommMonoid.{u3} R _inst_2) f (fun (a : \u03b1) (b : M) => Int.cast.{u3} R (Ring.toIntCast.{u3} R (CommRing.toRing.{u3} R _inst_2)) (g a b)))\nCase conversion may be inaccurate. Consider using '#align int.cast_finsupp_prod Int.cast_finsupp_prod\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_prod [CommRing R] (g : \u03b1 \u2192 M \u2192 \u2124) :\n    (\u2191(f.Prod g) : R) = f.Prod fun a b => \u2191(g a b) :=\n  Int.cast_prod _ _\n#align int.cast_finsupp_prod Int.cast_finsupp_prod\n\n/- warning: int.cast_finsupp_sum -> Int.cast_finsupp_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommRing.{u3} R] (g : \u03b1 -> M -> Int), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Int R (HasLiftT.mk.{1, succ u3} Int R (CoeTC\u2093.coe.{1, succ u3} Int R (Int.castCoe.{u3} R (AddGroupWithOne.toHasIntCast.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (CommRing.toRing.{u3} R _inst_2))))))) (Finsupp.sum.{u1, u2, 0} \u03b1 M Int _inst_1 Int.addCommMonoid f g)) (Finsupp.sum.{u1, u2, u3} \u03b1 M R _inst_1 (AddCommGroup.toAddCommMonoid.{u3} R (NonUnitalNonAssocRing.toAddCommGroup.{u3} R (NonAssocRing.toNonUnitalNonAssocRing.{u3} R (Ring.toNonAssocRing.{u3} R (CommRing.toRing.{u3} R _inst_2))))) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Int R (HasLiftT.mk.{1, succ u3} Int R (CoeTC\u2093.coe.{1, succ u3} Int R (Int.castCoe.{u3} R (AddGroupWithOne.toHasIntCast.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (CommRing.toRing.{u3} R _inst_2))))))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommRing.{u3} R] (g : \u03b1 -> M -> Int), Eq.{succ u3} R (Int.cast.{u3} R (Ring.toIntCast.{u3} R (CommRing.toRing.{u3} R _inst_2)) (Finsupp.sum.{u2, u1, 0} \u03b1 M Int _inst_1 Int.instAddCommMonoidInt f g)) (Finsupp.sum.{u2, u1, u3} \u03b1 M R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} R (NonAssocRing.toNonUnitalNonAssocRing.{u3} R (Ring.toNonAssocRing.{u3} R (CommRing.toRing.{u3} R _inst_2))))) f (fun (a : \u03b1) (b : M) => Int.cast.{u3} R (Ring.toIntCast.{u3} R (CommRing.toRing.{u3} R _inst_2)) (g a b)))\nCase conversion may be inaccurate. Consider using '#align int.cast_finsupp_sum Int.cast_finsupp_sum\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_sum [CommRing R] (g : \u03b1 \u2192 M \u2192 \u2124) :\n    (\u2191(f.Sum g) : R) = f.Sum fun a b => \u2191(g a b) :=\n  Int.cast_sum _ _\n#align int.cast_finsupp_sum Int.cast_finsupp_sum\n\nend Int\n\nnamespace Rat\n\n/- warning: rat.cast_finsupp_sum -> Rat.cast_finsupp_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : DivisionRing.{u3} R] [_inst_3 : CharZero.{u3} R (AddGroupWithOne.toAddMonoidWithOne.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (DivisionRing.toRing.{u3} R _inst_2))))] (g : \u03b1 -> M -> Rat), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Rat R (HasLiftT.mk.{1, succ u3} Rat R (CoeTC\u2093.coe.{1, succ u3} Rat R (Rat.castCoe.{u3} R (DivisionRing.toHasRatCast.{u3} R _inst_2)))) (Finsupp.sum.{u1, u2, 0} \u03b1 M Rat _inst_1 Rat.addCommMonoid f g)) (Finsupp.sum.{u1, u2, u3} \u03b1 M R _inst_1 (AddCommGroup.toAddCommMonoid.{u3} R (NonUnitalNonAssocRing.toAddCommGroup.{u3} R (NonAssocRing.toNonUnitalNonAssocRing.{u3} R (Ring.toNonAssocRing.{u3} R (DivisionRing.toRing.{u3} R _inst_2))))) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Rat R (HasLiftT.mk.{1, succ u3} Rat R (CoeTC\u2093.coe.{1, succ u3} Rat R (Rat.castCoe.{u3} R (DivisionRing.toHasRatCast.{u3} R _inst_2)))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : DivisionRing.{u3} R] [_inst_3 : CharZero.{u3} R (AddGroupWithOne.toAddMonoidWithOne.{u3} R (Ring.toAddGroupWithOne.{u3} R (DivisionRing.toRing.{u3} R _inst_2)))] (g : \u03b1 -> M -> Rat), Eq.{succ u3} R (Rat.cast.{u3} R (DivisionRing.toRatCast.{u3} R _inst_2) (Finsupp.sum.{u2, u1, 0} \u03b1 M Rat _inst_1 Rat.addCommMonoid f g)) (Finsupp.sum.{u2, u1, u3} \u03b1 M R _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u3} R (NonAssocRing.toNonUnitalNonAssocRing.{u3} R (Ring.toNonAssocRing.{u3} R (DivisionRing.toRing.{u3} R _inst_2))))) f (fun (a : \u03b1) (b : M) => Rat.cast.{u3} R (DivisionRing.toRatCast.{u3} R _inst_2) (g a b)))\nCase conversion may be inaccurate. Consider using '#align rat.cast_finsupp_sum Rat.cast_finsupp_sum\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_sum [DivisionRing R] [CharZero R] (g : \u03b1 \u2192 M \u2192 \u211a) :\n    (\u2191(f.Sum g) : R) = f.Sum fun a b => g a b :=\n  cast_sum _ _\n#align rat.cast_finsupp_sum Rat.cast_finsupp_sum\n\n/- warning: rat.cast_finsupp_prod -> Rat.cast_finsupp_prod is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : Field.{u3} R] [_inst_3 : CharZero.{u3} R (AddGroupWithOne.toAddMonoidWithOne.{u3} R (AddCommGroupWithOne.toAddGroupWithOne.{u3} R (Ring.toAddCommGroupWithOne.{u3} R (DivisionRing.toRing.{u3} R (Field.toDivisionRing.{u3} R _inst_2)))))] (g : \u03b1 -> M -> Rat), Eq.{succ u3} R ((fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Rat R (HasLiftT.mk.{1, succ u3} Rat R (CoeTC\u2093.coe.{1, succ u3} Rat R (Rat.castCoe.{u3} R (DivisionRing.toHasRatCast.{u3} R (Field.toDivisionRing.{u3} R _inst_2))))) (Finsupp.prod.{u1, u2, 0} \u03b1 M Rat _inst_1 Rat.commMonoid f g)) (Finsupp.prod.{u1, u2, u3} \u03b1 M R _inst_1 (CommRing.toCommMonoid.{u3} R (Field.toCommRing.{u3} R _inst_2)) f (fun (a : \u03b1) (b : M) => (fun (a : Type) (b : Type.{u3}) [self : HasLiftT.{1, succ u3} a b] => self.0) Rat R (HasLiftT.mk.{1, succ u3} Rat R (CoeTC\u2093.coe.{1, succ u3} Rat R (Rat.castCoe.{u3} R (DivisionRing.toHasRatCast.{u3} R (Field.toDivisionRing.{u3} R _inst_2))))) (g a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {R : Type.{u3}} [_inst_1 : Zero.{u1} M] (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : Field.{u3} R] [_inst_3 : CharZero.{u3} R (AddGroupWithOne.toAddMonoidWithOne.{u3} R (Ring.toAddGroupWithOne.{u3} R (DivisionRing.toRing.{u3} R (Field.toDivisionRing.{u3} R _inst_2))))] (g : \u03b1 -> M -> Rat), Eq.{succ u3} R (Rat.cast.{u3} R (Field.toRatCast.{u3} R _inst_2) (Finsupp.prod.{u2, u1, 0} \u03b1 M Rat _inst_1 Rat.commMonoid f g)) (Finsupp.prod.{u2, u1, u3} \u03b1 M R _inst_1 (CommRing.toCommMonoid.{u3} R (Field.toCommRing.{u3} R _inst_2)) f (fun (a : \u03b1) (b : M) => Rat.cast.{u3} R (Field.toRatCast.{u3} R _inst_2) (g a b)))\nCase conversion may be inaccurate. Consider using '#align rat.cast_finsupp_prod Rat.cast_finsupp_prod\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_finsupp_prod [Field R] [CharZero R] (g : \u03b1 \u2192 M \u2192 \u211a) :\n    (\u2191(f.Prod g) : R) = f.Prod fun a b => g a b :=\n  cast_prod _ _\n#align rat.cast_finsupp_prod Rat.cast_finsupp_prod\n\nend Rat\n\nend CastFinsupp\n\n/-! ### Declarations about `map_domain` -/\n\n\nnamespace Finsupp\n\nsection MapDomain\n\nvariable [AddCommMonoid M] {v v\u2081 v\u2082 : \u03b1 \u2192\u2080 M}\n\n/- warning: finsupp.map_domain -> Finsupp.mapDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (\u03b1 -> \u03b2) -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) -> (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (\u03b1 -> \u03b2) -> (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) -> (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain Finsupp.mapDomain\u2093'. -/\n/-- Given `f : \u03b1 \u2192 \u03b2` and `v : \u03b1 \u2192\u2080 M`, `map_domain f v : \u03b2 \u2192\u2080 M`\n  is the finitely supported function whose value at `a : \u03b2` is the sum\n  of `v x` over all `x` such that `f x = a`. -/\ndef mapDomain (f : \u03b1 \u2192 \u03b2) (v : \u03b1 \u2192\u2080 M) : \u03b2 \u2192\u2080 M :=\n  v.Sum fun a => single (f a)\n#align finsupp.map_domain Finsupp.mapDomain\n\n/- warning: finsupp.map_domain_apply -> Finsupp.mapDomain_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (forall (x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (a : \u03b1), Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f x) (f a)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) x a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u3, succ u2} \u03b1 \u03b2 f) -> (forall (x : Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (a : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) (f a)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f x) (f a)) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) x a))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_apply Finsupp.mapDomain_apply\u2093'. -/\ntheorem mapDomain_apply {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f) (x : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n    mapDomain f x (f a) = x a :=\n  by\n  rw [map_domain, sum_apply, Sum, Finset.sum_eq_single a, single_eq_same]\n  \u00b7 intro b _ hba\n    exact single_eq_of_ne (hf.ne hba)\n  \u00b7 intro h\n    rw [not_mem_support_iff.1 h, single_zero, zero_apply]\n#align finsupp.map_domain_apply Finsupp.mapDomain_apply\n\n/- warning: finsupp.map_domain_notin_range -> Finsupp.mapDomain_notin_range is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : \u03b1 -> \u03b2} (x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (a : \u03b2), (Not (Membership.Mem.{u2, u2} \u03b2 (Set.{u2} \u03b2) (Set.hasMem.{u2} \u03b2) a (Set.range.{u2, succ u1} \u03b2 \u03b1 f))) -> (Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f x) a) (OfNat.ofNat.{u3} M 0 (OfNat.mk.{u3} M 0 (Zero.zero.{u3} M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {f : \u03b1 -> \u03b2} (x : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (a : \u03b2), (Not (Membership.mem.{u1, u1} \u03b2 (Set.{u1} \u03b2) (Set.instMembershipSet.{u1} \u03b2) a (Set.range.{u1, succ u3} \u03b2 \u03b1 f))) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f x) a) (OfNat.ofNat.{u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) 0 (Zero.toOfNat0.{u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) (AddMonoid.toZero.{u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) (AddCommMonoid.toAddMonoid.{u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) _inst_1)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_notin_range Finsupp.mapDomain_notin_range\u2093'. -/\ntheorem mapDomain_notin_range {f : \u03b1 \u2192 \u03b2} (x : \u03b1 \u2192\u2080 M) (a : \u03b2) (h : a \u2209 Set.range f) :\n    mapDomain f x a = 0 := by\n  rw [map_domain, sum_apply, Sum]\n  exact Finset.sum_eq_zero fun a' h' => single_eq_of_ne fun eq => h <| Eq \u25b8 Set.mem_range_self _\n#align finsupp.map_domain_notin_range Finsupp.mapDomain_notin_range\n\n/- warning: finsupp.map_domain_id -> Finsupp.mapDomain_id is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapDomain.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (id.{succ u1} \u03b1) v) v\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {v : Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u2, u2, u1} \u03b1 \u03b1 M _inst_1 (id.{succ u2} \u03b1) v) v\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_id Finsupp.mapDomain_id\u2093'. -/\n@[simp]\ntheorem mapDomain_id : mapDomain id v = v :=\n  sum_single _\n#align finsupp.map_domain_id Finsupp.mapDomain_id\n\n/- warning: finsupp.map_domain_comp -> Finsupp.mapDomain_comp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {M : Type.{u4}} [_inst_1 : AddCommMonoid.{u4} M] {v : Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))} {f : \u03b1 -> \u03b2} {g : \u03b2 -> \u03b3}, Eq.{max (succ u3) (succ u4)} (Finsupp.{u3, u4} \u03b3 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.mapDomain.{u1, u3, u4} \u03b1 \u03b3 M _inst_1 (Function.comp.{succ u1, succ u2, succ u3} \u03b1 \u03b2 \u03b3 g f) v) (Finsupp.mapDomain.{u2, u3, u4} \u03b2 \u03b3 M _inst_1 g (Finsupp.mapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {\u03b3 : Type.{u4}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {v : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))} {f : \u03b1 -> \u03b2} {g : \u03b2 -> \u03b3}, Eq.{max (succ u4) (succ u3)} (Finsupp.{u4, u3} \u03b3 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.mapDomain.{u2, u4, u3} \u03b1 \u03b3 M _inst_1 (Function.comp.{succ u2, succ u1, succ u4} \u03b1 \u03b2 \u03b3 g f) v) (Finsupp.mapDomain.{u1, u4, u3} \u03b2 \u03b3 M _inst_1 g (Finsupp.mapDomain.{u2, u1, u3} \u03b1 \u03b2 M _inst_1 f v))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_comp Finsupp.mapDomain_comp\u2093'. -/\ntheorem mapDomain_comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} :\n    mapDomain (g \u2218 f) v = mapDomain g (mapDomain f v) :=\n  by\n  refine' ((sum_sum_index _ _).trans _).symm\n  \u00b7 intro\n    exact single_zero _\n  \u00b7 intro\n    exact single_add _\n  refine' sum_congr fun _ _ => sum_single_index _\n  \u00b7 exact single_zero _\n#align finsupp.map_domain_comp Finsupp.mapDomain_comp\n\n/- warning: finsupp.map_domain_single -> Finsupp.mapDomain_single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : M}, Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) a b)) (Finsupp.single.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (f a) b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u3}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : M}, Eq.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u1, u3, u2} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) a b)) (Finsupp.single.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (f a) b)\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_single Finsupp.mapDomain_single\u2093'. -/\n@[simp]\ntheorem mapDomain_single {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : M} : mapDomain f (single a b) = single (f a) b :=\n  sum_single_index <| single_zero _\n#align finsupp.map_domain_single Finsupp.mapDomain_single\n\n/- warning: finsupp.map_domain_zero -> Finsupp.mapDomain_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : \u03b1 -> \u03b2}, Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (OfNat.mk.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (Zero.zero.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))))) (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (OfNat.mk.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (Zero.zero.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u3}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {f : \u03b1 -> \u03b2}, Eq.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u1, u3, u2} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) 0 (Zero.toOfNat0.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))))) (OfNat.ofNat.{max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) 0 (Zero.toOfNat0.{max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.zero.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_zero Finsupp.mapDomain_zero\u2093'. -/\n@[simp]\ntheorem mapDomain_zero {f : \u03b1 \u2192 \u03b2} : mapDomain f (0 : \u03b1 \u2192\u2080 M) = (0 : \u03b2 \u2192\u2080 M) :=\n  sum_zero_index\n#align finsupp.map_domain_zero Finsupp.mapDomain_zero\n\n/- warning: finsupp.map_domain_congr -> Finsupp.mapDomain_congr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {v : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} {f : \u03b1 -> \u03b2} {g : \u03b1 -> \u03b2}, (forall (x : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) v)) -> (Eq.{succ u2} \u03b2 (f x) (g x))) -> (Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f v) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 g v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {v : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} {f : \u03b1 -> \u03b2} {g : \u03b1 -> \u03b2}, (forall (x : \u03b1), (Membership.mem.{u3, u3} \u03b1 (Finset.{u3} \u03b1) (Finset.instMembershipFinset.{u3} \u03b1) x (Finsupp.support.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) v)) -> (Eq.{succ u1} \u03b2 (f x) (g x))) -> (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f v) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 g v))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_congr Finsupp.mapDomain_congr\u2093'. -/\ntheorem mapDomain_congr {f g : \u03b1 \u2192 \u03b2} (h : \u2200 x \u2208 v.support, f x = g x) :\n    v.mapDomain f = v.mapDomain g :=\n  Finset.sum_congr rfl fun _ H => by simp only [h _ H]\n#align finsupp.map_domain_congr Finsupp.mapDomain_congr\n\n/- warning: finsupp.map_domain_add -> Finsupp.mapDomain_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {v\u2081 : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} {v\u2082 : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} {f : \u03b1 -> \u03b2}, Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (HAdd.hAdd.{max u1 u3, max u1 u3, max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (instHAdd.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) v\u2081 v\u2082)) (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (instHAdd.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.add.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f v\u2081) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f v\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u3}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {v\u2081 : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} {v\u2082 : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} {f : \u03b1 -> \u03b2}, Eq.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u1, u3, u2} \u03b1 \u03b2 M _inst_1 f (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) v\u2081 v\u2082)) (HAdd.hAdd.{max u3 u2, max u3 u2, max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (instHAdd.{max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u3, u2} \u03b2 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapDomain.{u1, u3, u2} \u03b1 \u03b2 M _inst_1 f v\u2081) (Finsupp.mapDomain.{u1, u3, u2} \u03b1 \u03b2 M _inst_1 f v\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_add Finsupp.mapDomain_add\u2093'. -/\ntheorem mapDomain_add {f : \u03b1 \u2192 \u03b2} : mapDomain f (v\u2081 + v\u2082) = mapDomain f v\u2081 + mapDomain f v\u2082 :=\n  sum_add_index' (fun _ => single_zero _) fun _ => single_add _\n#align finsupp.map_domain_add Finsupp.mapDomain_add\n\n/- warning: finsupp.map_domain_equiv_apply -> Finsupp.mapDomain_equiv_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : Equiv.{succ u1, succ u2} \u03b1 \u03b2} (x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (a : \u03b2), Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} \u03b1 \u03b2) (fun (_x : Equiv.{succ u1, succ u2} \u03b1 \u03b2) => \u03b1 -> \u03b2) (Equiv.hasCoeToFun.{succ u1, succ u2} \u03b1 \u03b2) f) x) a) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) x (coeFn.{max 1 (max (succ u2) (succ u1)) (succ u1) (succ u2), max (succ u2) (succ u1)} (Equiv.{succ u2, succ u1} \u03b2 \u03b1) (fun (_x : Equiv.{succ u2, succ u1} \u03b2 \u03b1) => \u03b2 -> \u03b1) (Equiv.hasCoeToFun.{succ u2, succ u1} \u03b2 \u03b1) (Equiv.symm.{succ u1, succ u2} \u03b1 \u03b2 f) a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {f : Equiv.{succ u3, succ u2} \u03b1 \u03b2} (x : Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (a : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Equiv.{succ u3, succ u2} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => \u03b2) _x) (Equiv.instFunLikeEquiv.{succ u3, succ u2} \u03b1 \u03b2) f) x) a) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) x (FunLike.coe.{max (succ u3) (succ u2), succ u2, succ u3} (Equiv.{succ u2, succ u3} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b2) => \u03b1) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u3} \u03b2 \u03b1) (Equiv.symm.{succ u3, succ u2} \u03b1 \u03b2 f) a))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_equiv_apply Finsupp.mapDomain_equiv_apply\u2093'. -/\n@[simp]\ntheorem mapDomain_equiv_apply {f : \u03b1 \u2243 \u03b2} (x : \u03b1 \u2192\u2080 M) (a : \u03b2) : mapDomain f x a = x (f.symm a) :=\n  by\n  conv_lhs => rw [\u2190 f.apply_symm_apply a]\n  exact map_domain_apply f.injective _ _\n#align finsupp.map_domain_equiv_apply Finsupp.mapDomain_equiv_apply\n\n/- warning: finsupp.map_domain.add_monoid_hom -> Finsupp.mapDomain.addMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (\u03b1 -> \u03b2) -> (AddMonoidHom.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (\u03b1 -> \u03b2) -> (AddMonoidHom.{max u3 u1, max u3 u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain.add_monoid_hom Finsupp.mapDomain.addMonoidHom\u2093'. -/\n/-- `finsupp.map_domain` is an `add_monoid_hom`. -/\n@[simps]\ndef mapDomain.addMonoidHom (f : \u03b1 \u2192 \u03b2) : (\u03b1 \u2192\u2080 M) \u2192+ \u03b2 \u2192\u2080 M\n    where\n  toFun := mapDomain f\n  map_zero' := mapDomain_zero\n  map_add' _ _ := mapDomain_add\n#align finsupp.map_domain.add_monoid_hom Finsupp.mapDomain.addMonoidHom\n\n/- warning: finsupp.map_domain.add_monoid_hom_id -> Finsupp.mapDomain.addMonoidHom_id is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M], Eq.{succ (max u1 u2)} (AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapDomain.addMonoidHom.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (id.{succ u1} \u03b1)) (AddMonoidHom.id.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addZeroClass.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M], Eq.{max (succ u2) (succ u1)} (AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (Finsupp.mapDomain.addMonoidHom.{u2, u2, u1} \u03b1 \u03b1 M _inst_1 (id.{succ u2} \u03b1)) (AddMonoidHom.id.{max u1 u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.addZeroClass.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain.add_monoid_hom_id Finsupp.mapDomain.addMonoidHom_id\u2093'. -/\n@[simp]\ntheorem mapDomain.addMonoidHom_id : mapDomain.addMonoidHom id = AddMonoidHom.id (\u03b1 \u2192\u2080 M) :=\n  AddMonoidHom.ext fun _ => mapDomain_id\n#align finsupp.map_domain.add_monoid_hom_id Finsupp.mapDomain.addMonoidHom_id\n\n/- warning: finsupp.map_domain.add_monoid_hom_comp -> Finsupp.mapDomain.addMonoidHom_comp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {M : Type.{u4}} [_inst_1 : AddCommMonoid.{u4} M] (f : \u03b2 -> \u03b3) (g : \u03b1 -> \u03b2), Eq.{max (succ (max u3 u4)) (succ (max u1 u4))} (AddMonoidHom.{max u1 u4, max u3 u4} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u3, u4} \u03b3 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.addZeroClass.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.addZeroClass.{u3, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.mapDomain.addMonoidHom.{u1, u3, u4} \u03b1 \u03b3 M _inst_1 (Function.comp.{succ u1, succ u2, succ u3} \u03b1 \u03b2 \u03b3 f g)) (AddMonoidHom.comp.{max u1 u4, max u2 u4, max u3 u4} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u2, u4} \u03b2 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u3, u4} \u03b3 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.addZeroClass.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.addZeroClass.{u2, u4} \u03b2 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.addZeroClass.{u3, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.mapDomain.addMonoidHom.{u2, u3, u4} \u03b2 \u03b3 M _inst_1 f) (Finsupp.mapDomain.addMonoidHom.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u4}} {\u03b2 : Type.{u1}} {\u03b3 : Type.{u3}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] (f : \u03b2 -> \u03b3) (g : \u03b1 -> \u03b2), Eq.{max (max (succ u4) (succ u3)) (succ u2)} (AddMonoidHom.{max u2 u4, max u2 u3} (Finsupp.{u4, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u3, u2} \u03b3 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u4, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u3, u2} \u03b3 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.mapDomain.addMonoidHom.{u4, u3, u2} \u03b1 \u03b3 M _inst_1 (Function.comp.{succ u4, succ u1, succ u3} \u03b1 \u03b2 \u03b3 f g)) (AddMonoidHom.comp.{max u2 u4, max u1 u2, max u3 u2} (Finsupp.{u4, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u3, u2} \u03b3 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u4, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u1, u2} \u03b2 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addZeroClass.{u3, u2} \u03b3 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.addMonoidHom.{u1, u3, u2} \u03b2 \u03b3 M _inst_1 f) (Finsupp.mapDomain.addMonoidHom.{u4, u1, u2} \u03b1 \u03b2 M _inst_1 g))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain.add_monoid_hom_comp Finsupp.mapDomain.addMonoidHom_comp\u2093'. -/\ntheorem mapDomain.addMonoidHom_comp (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) :\n    (mapDomain.addMonoidHom (f \u2218 g) : (\u03b1 \u2192\u2080 M) \u2192+ \u03b3 \u2192\u2080 M) =\n      (mapDomain.addMonoidHom f).comp (mapDomain.addMonoidHom g) :=\n  AddMonoidHom.ext fun _ => mapDomain_comp\n#align finsupp.map_domain.add_monoid_hom_comp Finsupp.mapDomain.addMonoidHom_comp\n\n/- warning: finsupp.map_domain_finset_sum -> Finsupp.mapDomain_finset_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b9 : Type.{u3}} {M : Type.{u4}} [_inst_1 : AddCommMonoid.{u4} M] {f : \u03b1 -> \u03b2} {s : Finset.{u3} \u03b9} {v : \u03b9 -> (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))))}, Eq.{max (succ u2) (succ u4)} (Finsupp.{u2, u4} \u03b2 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f (Finset.sum.{max u1 u4, u3} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u4} \u03b1 M _inst_1) s (fun (i : \u03b9) => v i))) (Finset.sum.{max u2 u4, u3} (Finsupp.{u2, u4} \u03b2 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u2, u4} \u03b2 M _inst_1) s (fun (i : \u03b9) => Finsupp.mapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f (v i)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {\u03b9 : Type.{u4}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {f : \u03b1 -> \u03b2} {s : Finset.{u4} \u03b9} {v : \u03b9 -> (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f (Finset.sum.{max u3 u2, u4} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u3, u2} \u03b1 M _inst_1) s (fun (i : \u03b9) => v i))) (Finset.sum.{max u2 u1, u4} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u1, u2} \u03b2 M _inst_1) s (fun (i : \u03b9) => Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f (v i)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_finset_sum Finsupp.mapDomain_finset_sum\u2093'. -/\ntheorem mapDomain_finset_sum {f : \u03b1 \u2192 \u03b2} {s : Finset \u03b9} {v : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n    mapDomain f (\u2211 i in s, v i) = \u2211 i in s, mapDomain f (v i) :=\n  (mapDomain.addMonoidHom f : (\u03b1 \u2192\u2080 M) \u2192+ \u03b2 \u2192\u2080 M).map_sum _ _\n#align finsupp.map_domain_finset_sum Finsupp.mapDomain_finset_sum\n\n/- warning: finsupp.map_domain_sum -> Finsupp.mapDomain_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : Zero.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u1, u4} \u03b1 N _inst_2} {v : \u03b1 -> N -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))}, Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (Finsupp.sum.{u1, u4, max u1 u3} \u03b1 N (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) _inst_2 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s v)) (Finsupp.sum.{u1, u4, max u2 u3} \u03b1 N (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) _inst_2 (Finsupp.addCommMonoid.{u2, u3} \u03b2 M _inst_1) s (fun (a : \u03b1) (b : N) => Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (v a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : Zero.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u3, u4} \u03b1 N _inst_2} {v : \u03b1 -> N -> (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f (Finsupp.sum.{u3, u4, max u3 u2} \u03b1 N (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) _inst_2 (Finsupp.addCommMonoid.{u3, u2} \u03b1 M _inst_1) s v)) (Finsupp.sum.{u3, u4, max u2 u1} \u03b1 N (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) _inst_2 (Finsupp.addCommMonoid.{u1, u2} \u03b2 M _inst_1) s (fun (a : \u03b1) (b : N) => Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f (v a b)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_sum Finsupp.mapDomain_sum\u2093'. -/\ntheorem mapDomain_sum [Zero N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 N} {v : \u03b1 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n    mapDomain f (s.Sum v) = s.Sum fun a b => mapDomain f (v a b) :=\n  (mapDomain.addMonoidHom f : (\u03b1 \u2192\u2080 M) \u2192+ \u03b2 \u2192\u2080 M).map_finsupp_sum _ _\n#align finsupp.map_domain_sum Finsupp.mapDomain_sum\n\n/- warning: finsupp.map_domain_support -> Finsupp.mapDomain_support is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : DecidableEq.{succ u2} \u03b2] {f : \u03b1 -> \u03b2} {s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))}, HasSubset.Subset.{u2} (Finset.{u2} \u03b2) (Finset.hasSubset.{u2} \u03b2) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u1, u2} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) s))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] [_inst_2 : DecidableEq.{succ u3} \u03b2] {f : \u03b1 -> \u03b2} {s : Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))}, HasSubset.Subset.{u3} (Finset.{u3} \u03b2) (Finset.instHasSubsetFinset.{u3} \u03b2) (Finsupp.support.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) (Finsupp.mapDomain.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u2, u3} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) s))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_support Finsupp.mapDomain_support\u2093'. -/\ntheorem mapDomain_support [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M} :\n    (s.mapDomain f).support \u2286 s.support.image f :=\n  Finset.Subset.trans support_sum <|\n    Finset.Subset.trans (Finset.bunion\u1d62_mono fun a ha => support_single_subset) <| by\n      rw [Finset.bunion\u1d62_singleton] <;> exact subset.refl _\n#align finsupp.map_domain_support Finsupp.mapDomain_support\n\n/- warning: finsupp.map_domain_apply' -> Finsupp.mapDomain_apply' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (S : Set.{u1} \u03b1) {f : \u03b1 -> \u03b2} (x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), (HasSubset.Subset.{u1} (Set.{u1} \u03b1) (Set.hasSubset.{u1} \u03b1) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (CoeTC\u2093.coe.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (Finset.Set.hasCoeT.{u1} \u03b1))) (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) x)) S) -> (Set.InjOn.{u1, u2} \u03b1 \u03b2 f S) -> (forall {a : \u03b1}, (Membership.Mem.{u1, u1} \u03b1 (Set.{u1} \u03b1) (Set.hasMem.{u1} \u03b1) a S) -> (Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f x) (f a)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) x a)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] (S : Set.{u3} \u03b1) {f : \u03b1 -> \u03b2} (x : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))), (HasSubset.Subset.{u3} (Set.{u3} \u03b1) (Set.instHasSubsetSet.{u3} \u03b1) (Finset.toSet.{u3} \u03b1 (Finsupp.support.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) x)) S) -> (Set.InjOn.{u3, u1} \u03b1 \u03b2 f S) -> (forall {a : \u03b1}, (Membership.mem.{u3, u3} \u03b1 (Set.{u3} \u03b1) (Set.instMembershipSet.{u3} \u03b1) a S) -> (Eq.{succ u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) (f a)) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Finsupp.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u1, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f x) (f a)) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) x a)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_apply' Finsupp.mapDomain_apply'\u2093'. -/\ntheorem mapDomain_apply' (S : Set \u03b1) {f : \u03b1 \u2192 \u03b2} (x : \u03b1 \u2192\u2080 M) (hS : (x.support : Set \u03b1) \u2286 S)\n    (hf : Set.InjOn f S) {a : \u03b1} (ha : a \u2208 S) : mapDomain f x (f a) = x a := by\n  classical\n    rw [map_domain, sum_apply, Sum]\n    simp_rw [single_apply]\n    by_cases hax : a \u2208 x.support\n    \u00b7 rw [\u2190 Finset.add_sum_erase _ _ hax, if_pos rfl]\n      convert add_zero _\n      refine' Finset.sum_eq_zero fun i hi => if_neg _\n      exact (hf.mono hS).Ne (Finset.mem_of_mem_erase hi) hax (Finset.ne_of_mem_erase hi)\n    \u00b7 rw [not_mem_support_iff.1 hax]\n      refine' Finset.sum_eq_zero fun i hi => if_neg _\n      exact hf.ne (hS hi) ha (ne_of_mem_of_not_mem hi hax)\n#align finsupp.map_domain_apply' Finsupp.mapDomain_apply'\n\n/- warning: finsupp.map_domain_support_of_inj_on -> Finsupp.mapDomain_support_of_injOn is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : DecidableEq.{succ u2} \u03b2] {f : \u03b1 -> \u03b2} (s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), (Set.InjOn.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (CoeTC\u2093.coe.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (Finset.Set.hasCoeT.{u1} \u03b1))) (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) s))) -> (Eq.{succ u2} (Finset.{u2} \u03b2) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u1, u2} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) s)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] [_inst_2 : DecidableEq.{succ u3} \u03b2] {f : \u03b1 -> \u03b2} (s : Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), (Set.InjOn.{u2, u3} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b1 (Finsupp.support.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) s))) -> (Eq.{succ u3} (Finset.{u3} \u03b2) (Finsupp.support.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) (Finsupp.mapDomain.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u2, u3} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) s)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_support_of_inj_on Finsupp.mapDomain_support_of_injOn\u2093'. -/\ntheorem mapDomain_support_of_injOn [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (s : \u03b1 \u2192\u2080 M)\n    (hf : Set.InjOn f s.support) : (mapDomain f s).support = Finset.image f s.support :=\n  Finset.Subset.antisymm mapDomain_support <|\n    by\n    intro x hx\n    simp only [mem_image, exists_prop, mem_support_iff, Ne.def] at hx\n    rcases hx with \u27e8hx_w, hx_h_left, rfl\u27e9\n    simp only [mem_support_iff, Ne.def]\n    rw [map_domain_apply' (\u2191s.support : Set _) _ _ hf]\n    \u00b7 exact hx_h_left\n    \u00b7 simp only [mem_coe, mem_support_iff, Ne.def]\n      exact hx_h_left\n    \u00b7 exact subset.refl _\n#align finsupp.map_domain_support_of_inj_on Finsupp.mapDomain_support_of_injOn\n\n/- warning: finsupp.map_domain_support_of_injective -> Finsupp.mapDomain_support_of_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : DecidableEq.{succ u2} \u03b2] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (forall (s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), Eq.{succ u2} (Finset.{u2} \u03b2) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u1, u2} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) s)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] [_inst_2 : DecidableEq.{succ u3} \u03b2] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u2, succ u3} \u03b1 \u03b2 f) -> (forall (s : Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), Eq.{succ u3} (Finset.{u3} \u03b2) (Finsupp.support.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) (Finsupp.mapDomain.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 f s)) (Finset.image.{u2, u3} \u03b1 \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) f (Finsupp.support.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) s)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_support_of_injective Finsupp.mapDomain_support_of_injective\u2093'. -/\ntheorem mapDomain_support_of_injective [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f)\n    (s : \u03b1 \u2192\u2080 M) : (mapDomain f s).support = Finset.image f s.support :=\n  mapDomain_support_of_injOn s (hf.InjOn _)\n#align finsupp.map_domain_support_of_injective Finsupp.mapDomain_support_of_injective\n\n/- warning: finsupp.prod_map_domain_index -> Finsupp.prod_mapDomain_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : CommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} {h : \u03b2 -> M -> N}, (forall (b : \u03b2), Eq.{succ u4} N (h b (OfNat.ofNat.{u3} M 0 (OfNat.mk.{u3} M 0 (Zero.zero.{u3} M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))))) (OfNat.ofNat.{u4} N 1 (OfNat.mk.{u4} N 1 (One.one.{u4} N (MulOneClass.toHasOne.{u4} N (Monoid.toMulOneClass.{u4} N (CommMonoid.toMonoid.{u4} N _inst_2))))))) -> (forall (b : \u03b2) (m\u2081 : M) (m\u2082 : M), Eq.{succ u4} N (h b (HAdd.hAdd.{u3, u3, u3} M M M (instHAdd.{u3} M (AddZeroClass.toHasAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) m\u2081 m\u2082)) (HMul.hMul.{u4, u4, u4} N N N (instHMul.{u4} N (MulOneClass.toHasMul.{u4} N (Monoid.toMulOneClass.{u4} N (CommMonoid.toMonoid.{u4} N _inst_2)))) (h b m\u2081) (h b m\u2082))) -> (Eq.{succ u4} N (Finsupp.prod.{u2, u3, u4} \u03b2 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s) h) (Finsupp.prod.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 s (fun (a : \u03b1) (m : M) => h (f a) m)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : CommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} {h : \u03b2 -> M -> N}, (forall (b : \u03b2), Eq.{succ u4} N (h b (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (OfNat.ofNat.{u4} N 1 (One.toOfNat1.{u4} N (Monoid.toOne.{u4} N (CommMonoid.toMonoid.{u4} N _inst_2))))) -> (forall (b : \u03b2) (m\u2081 : M) (m\u2082 : M), Eq.{succ u4} N (h b (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) m\u2081 m\u2082)) (HMul.hMul.{u4, u4, u4} N N N (instHMul.{u4} N (MulOneClass.toMul.{u4} N (Monoid.toMulOneClass.{u4} N (CommMonoid.toMonoid.{u4} N _inst_2)))) (h b m\u2081) (h b m\u2082))) -> (Eq.{succ u4} N (Finsupp.prod.{u1, u2, u4} \u03b2 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f s) h) (Finsupp.prod.{u3, u2, u4} \u03b1 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 s (fun (a : \u03b1) (m : M) => h (f a) m)))\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_map_domain_index Finsupp.prod_mapDomain_index\u2093'. -/\n@[to_additive]\ntheorem prod_mapDomain_index [CommMonoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M} {h : \u03b2 \u2192 M \u2192 N}\n    (h_zero : \u2200 b, h b 0 = 1) (h_add : \u2200 b m\u2081 m\u2082, h b (m\u2081 + m\u2082) = h b m\u2081 * h b m\u2082) :\n    (mapDomain f s).Prod h = s.Prod fun a m => h (f a) m :=\n  (prod_sum_index h_zero h_add).trans <| prod_congr fun _ _ => prod_single_index (h_zero _)\n#align finsupp.prod_map_domain_index Finsupp.prod_mapDomain_index\n#align finsupp.sum_map_domain_index Finsupp.sum_mapDomain_index\n\n/- warning: finsupp.sum_map_domain_index_add_monoid_hom -> Finsupp.sum_mapDomain_index_addMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} (h : \u03b2 -> (AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))), Eq.{succ u4} N (Finsupp.sum.{u2, u3, u4} \u03b2 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s) (fun (b : \u03b2) (m : M) => coeFn.{max (succ u4) (succ u3), max (succ u3) (succ u4)} (AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (fun (_x : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (h b) m)) (Finsupp.sum.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 s (fun (a : \u03b1) (m : M) => coeFn.{max (succ u4) (succ u3), max (succ u3) (succ u4)} (AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (fun (_x : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (h (f a)) m))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} (h : \u03b2 -> (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))), Eq.{succ u4} N (Finsupp.sum.{u1, u2, u4} \u03b2 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f s) (fun (b : \u03b2) (m : M) => FunLike.coe.{max (succ u2) (succ u4), succ u2, succ u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u2 u4, u2, u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M N (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toAdd.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u2 u4, u2, u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))))) (h b) m)) (Finsupp.sum.{u3, u2, u4} \u03b1 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 s (fun (a : \u03b1) (m : M) => FunLike.coe.{max (succ u2) (succ u4), succ u2, succ u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u2 u4, u2, u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M N (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (AddZeroClass.toAdd.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u2 u4, u2, u4} (AddMonoidHom.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u2, u4} M N (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))))) (h (f a)) m))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_map_domain_index_add_monoid_hom Finsupp.sum_mapDomain_index_addMonoidHom\u2093'. -/\n-- Note that in `prod_map_domain_index`, `M` is still an additive monoid,\n-- so there is no analogous version in terms of `monoid_hom`.\n/-- A version of `sum_map_domain_index` that takes a bundled `add_monoid_hom`,\nrather than separate linearity hypotheses.\n-/\n@[simp]\ntheorem sum_mapDomain_index_addMonoidHom [AddCommMonoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M}\n    (h : \u03b2 \u2192 M \u2192+ N) : ((mapDomain f s).Sum fun b m => h b m) = s.Sum fun a m => h (f a) m :=\n  @sum_mapDomain_index _ _ _ _ _ _ _ _ (fun b m => h b m) (fun b => (h b).map_zero) fun b m\u2081 m\u2082 =>\n    (h b).map_add _ _\n#align finsupp.sum_map_domain_index_add_monoid_hom Finsupp.sum_mapDomain_index_addMonoidHom\n\n/- warning: finsupp.emb_domain_eq_map_domain -> Finsupp.embDomain_eq_mapDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : Function.Embedding.{succ u1, succ u2} \u03b1 \u03b2) (v : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.embDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) f v) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 (coeFn.{max 1 (succ u1) (succ u2), max (succ u1) (succ u2)} (Function.Embedding.{succ u1, succ u2} \u03b1 \u03b2) (fun (_x : Function.Embedding.{succ u1, succ u2} \u03b1 \u03b2) => \u03b1 -> \u03b2) (Function.Embedding.hasCoeToFun.{succ u1, succ u2} \u03b1 \u03b2) f) v)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (f : Function.Embedding.{succ u3, succ u2} \u03b1 \u03b2) (v : Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.embDomain.{u3, u2, u1} \u03b1 \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) f v) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Function.Embedding.{succ u3, succ u2} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u3) (succ u2), succ u3, succ u2} (Function.Embedding.{succ u3, succ u2} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u3, succ u2} \u03b1 \u03b2)) f) v)\nCase conversion may be inaccurate. Consider using '#align finsupp.emb_domain_eq_map_domain Finsupp.embDomain_eq_mapDomain\u2093'. -/\ntheorem embDomain_eq_mapDomain (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) : embDomain f v = mapDomain f v :=\n  by\n  ext a\n  by_cases a \u2208 Set.range f\n  \u00b7 rcases h with \u27e8a, rfl\u27e9\n    rw [map_domain_apply f.injective, emb_domain_apply]\n  \u00b7 rw [map_domain_notin_range, emb_domain_notin_range] <;> assumption\n#align finsupp.emb_domain_eq_map_domain Finsupp.embDomain_eq_mapDomain\n\n/- warning: finsupp.prod_map_domain_index_inj -> Finsupp.prod_mapDomain_index_inj is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : CommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} {h : \u03b2 -> M -> N}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Eq.{succ u4} N (Finsupp.prod.{u2, u3, u4} \u03b2 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f s) h) (Finsupp.prod.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 s (fun (a : \u03b1) (b : M) => h (f a) b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : CommMonoid.{u4} N] {f : \u03b1 -> \u03b2} {s : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))} {h : \u03b2 -> M -> N}, (Function.Injective.{succ u3, succ u1} \u03b1 \u03b2 f) -> (Eq.{succ u4} N (Finsupp.prod.{u1, u2, u4} \u03b2 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f s) h) (Finsupp.prod.{u3, u2, u4} \u03b1 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 s (fun (a : \u03b1) (b : M) => h (f a) b)))\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_map_domain_index_inj Finsupp.prod_mapDomain_index_inj\u2093'. -/\n@[to_additive]\ntheorem prod_mapDomain_index_inj [CommMonoid N] {f : \u03b1 \u2192 \u03b2} {s : \u03b1 \u2192\u2080 M} {h : \u03b2 \u2192 M \u2192 N}\n    (hf : Function.Injective f) : (s.mapDomain f).Prod h = s.Prod fun a b => h (f a) b := by\n  rw [\u2190 Function.Embedding.coeFn_mk f hf, \u2190 emb_domain_eq_map_domain, prod_emb_domain]\n#align finsupp.prod_map_domain_index_inj Finsupp.prod_mapDomain_index_inj\n#align finsupp.sum_map_domain_index_inj Finsupp.sum_mapDomain_index_inj\n\n/- warning: finsupp.map_domain_injective -> Finsupp.mapDomain_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Function.Injective.{max (succ u1) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u3, succ u2} \u03b1 \u03b2 f) -> (Function.Injective.{max (succ u3) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_injective Finsupp.mapDomain_injective\u2093'. -/\ntheorem mapDomain_injective {f : \u03b1 \u2192 \u03b2} (hf : Function.Injective f) :\n    Function.Injective (mapDomain f : (\u03b1 \u2192\u2080 M) \u2192 \u03b2 \u2192\u2080 M) :=\n  by\n  intro v\u2081 v\u2082 eq\n  ext a\n  have : map_domain f v\u2081 (f a) = map_domain f v\u2082 (f a) := by rw [Eq]\n  rwa [map_domain_apply hf, map_domain_apply hf] at this\n#align finsupp.map_domain_injective Finsupp.mapDomain_injective\n\n#print Finsupp.mapDomainEmbedding /-\n/-- When `f` is an embedding we have an embedding `(\u03b1 \u2192\u2080 \u2115)  \u21aa (\u03b2 \u2192\u2080 \u2115)` given by `map_domain`. -/\n@[simps]\ndef mapDomainEmbedding {\u03b1 \u03b2 : Type _} (f : \u03b1 \u21aa \u03b2) : (\u03b1 \u2192\u2080 \u2115) \u21aa \u03b2 \u2192\u2080 \u2115 :=\n  \u27e8Finsupp.mapDomain f, Finsupp.mapDomain_injective f.Injective\u27e9\n#align finsupp.map_domain_embedding Finsupp.mapDomainEmbedding\n-/\n\n/- warning: finsupp.map_domain.add_monoid_hom_comp_map_range -> Finsupp.mapDomain.addMonoidHom_comp_mapRange is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : \u03b1 -> \u03b2) (g : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))), Eq.{max (succ (max u2 u4)) (succ (max u1 u3))} (AddMonoidHom.{max u1 u3, max u2 u4} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u4} \u03b2 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (AddMonoidHom.comp.{max u1 u3, max u1 u4, max u2 u4} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u4} \u03b1 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.{u2, u4} \u03b2 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u4} \u03b1 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u2, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.mapDomain.addMonoidHom.{u1, u2, u4} \u03b1 \u03b2 N _inst_2 f) (Finsupp.mapRange.addMonoidHom.{u1, u3, u4} \u03b1 M N _inst_1 _inst_2 g)) (AddMonoidHom.comp.{max u1 u3, max u2 u3, max u2 u4} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u4} \u03b2 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.addZeroClass.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.mapRange.addMonoidHom.{u2, u3, u4} \u03b2 M N _inst_1 _inst_2 g) (Finsupp.mapDomain.addMonoidHom.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : \u03b1 -> \u03b2) (g : AddMonoidHom.{u3, u4} M N (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))), Eq.{max (max (max (succ u2) (succ u1)) (succ u3)) (succ u4)} (AddMonoidHom.{max u3 u2, max u1 u4} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u4} \u03b2 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u2, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (AddMonoidHom.comp.{max u3 u2, max u2 u4, max u1 u4} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u2, u4} \u03b1 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.{u1, u4} \u03b2 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u2, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u2, u4} \u03b1 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u1, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.mapDomain.addMonoidHom.{u2, u1, u4} \u03b1 \u03b2 N _inst_2 f) (Finsupp.mapRange.addMonoidHom.{u2, u3, u4} \u03b1 M N _inst_1 _inst_2 g)) (AddMonoidHom.comp.{max u3 u2, max u3 u1, max u4 u1} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u4} \u03b2 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.addZeroClass.{u2, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.addZeroClass.{u1, u4} \u03b2 N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.mapRange.addMonoidHom.{u1, u3, u4} \u03b2 M N _inst_1 _inst_2 g) (Finsupp.mapDomain.addMonoidHom.{u2, u1, u3} \u03b1 \u03b2 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain.add_monoid_hom_comp_map_range Finsupp.mapDomain.addMonoidHom_comp_mapRange\u2093'. -/\ntheorem mapDomain.addMonoidHom_comp_mapRange [AddCommMonoid N] (f : \u03b1 \u2192 \u03b2) (g : M \u2192+ N) :\n    (mapDomain.addMonoidHom f).comp (mapRange.addMonoidHom g) =\n      (mapRange.addMonoidHom g).comp (mapDomain.addMonoidHom f) :=\n  by\n  ext\n  simp\n#align finsupp.map_domain.add_monoid_hom_comp_map_range Finsupp.mapDomain.addMonoidHom_comp_mapRange\n\n/- warning: finsupp.map_domain_map_range -> Finsupp.mapDomain_mapRange is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : \u03b1 -> \u03b2) (v : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (g : M -> N) (h0 : Eq.{succ u4} N (g (OfNat.ofNat.{u3} M 0 (OfNat.mk.{u3} M 0 (Zero.zero.{u3} M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))))) (OfNat.ofNat.{u4} N 0 (OfNat.mk.{u4} N 0 (Zero.zero.{u4} N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))))))), (forall (x : M) (y : M), Eq.{succ u4} N (g (HAdd.hAdd.{u3, u3, u3} M M M (instHAdd.{u3} M (AddZeroClass.toHasAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) x y)) (HAdd.hAdd.{u4, u4, u4} N N N (instHAdd.{u4} N (AddZeroClass.toHasAdd.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (g x) (g y))) -> (Eq.{max (succ u2) (succ u4)} (Finsupp.{u2, u4} \u03b2 N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (Finsupp.mapDomain.{u1, u2, u4} \u03b1 \u03b2 N _inst_2 f (Finsupp.mapRange.{u1, u3, u4} \u03b1 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) g h0 v)) (Finsupp.mapRange.{u2, u3, u4} \u03b2 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) g h0 (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f v)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}} {M : Type.{u2}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u4} N] (f : \u03b1 -> \u03b2) (v : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (g : M -> N) (h0 : Eq.{succ u4} N (g (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (OfNat.ofNat.{u4} N 0 (Zero.toOfNat0.{u4} N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))))), (forall (x : M) (y : M), Eq.{succ u4} N (g (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) x y)) (HAdd.hAdd.{u4, u4, u4} N N N (instHAdd.{u4} N (AddZeroClass.toAdd.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (g x) (g y))) -> (Eq.{max (succ u1) (succ u4)} (Finsupp.{u1, u4} \u03b2 N (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))) (Finsupp.mapDomain.{u3, u1, u4} \u03b1 \u03b2 N _inst_2 f (Finsupp.mapRange.{u3, u2, u4} \u03b1 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) g h0 v)) (Finsupp.mapRange.{u1, u2, u4} \u03b2 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (AddMonoid.toZero.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)) g h0 (Finsupp.mapDomain.{u3, u1, u2} \u03b1 \u03b2 M _inst_1 f v)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_map_range Finsupp.mapDomain_mapRange\u2093'. -/\n/-- When `g` preserves addition, `map_range` and `map_domain` commute. -/\ntheorem mapDomain_mapRange [AddCommMonoid N] (f : \u03b1 \u2192 \u03b2) (v : \u03b1 \u2192\u2080 M) (g : M \u2192 N) (h0 : g 0 = 0)\n    (hadd : \u2200 x y, g (x + y) = g x + g y) :\n    mapDomain f (mapRange g h0 v) = mapRange g h0 (mapDomain f v) :=\n  let g' : M \u2192+ N :=\n    { toFun := g\n      map_zero' := h0\n      map_add' := hadd }\n  AddMonoidHom.congr_fun (mapDomain.addMonoidHom_comp_mapRange f g') v\n#align finsupp.map_domain_map_range Finsupp.mapDomain_mapRange\n\n/- warning: finsupp.sum_update_add -> Finsupp.sum_update_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b9 : Type.{u3}} [_inst_2 : AddCommMonoid.{u1} \u03b1] [_inst_3 : AddCommMonoid.{u2} \u03b2] (f : Finsupp.{u3, u1} \u03b9 \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2)))) (i : \u03b9) (a : \u03b1) (g : \u03b9 -> \u03b1 -> \u03b2), (forall (i : \u03b9), Eq.{succ u2} \u03b2 (g i (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2))))))) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (AddZeroClass.toHasZero.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3))))))) -> (forall (j : \u03b9) (a\u2081 : \u03b1) (a\u2082 : \u03b1), Eq.{succ u2} \u03b2 (g j (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (AddZeroClass.toHasAdd.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2)))) a\u2081 a\u2082)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toHasAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (g j a\u2081) (g j a\u2082))) -> (Eq.{succ u2} \u03b2 (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toHasAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (Finsupp.sum.{u3, u1, u2} \u03b9 \u03b1 \u03b2 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2))) _inst_3 (Finsupp.update.{u3, u1} \u03b9 \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2))) f i a) g) (g i (coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b9 \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2)))) (fun (_x : Finsupp.{u3, u1} \u03b9 \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2)))) => \u03b9 -> \u03b1) (Finsupp.coeFun.{u3, u1} \u03b9 \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2)))) f i))) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toHasAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (Finsupp.sum.{u3, u1, u2} \u03b9 \u03b1 \u03b2 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_2))) _inst_3 f g) (g i a)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b9 : Type.{u1}} [_inst_2 : AddCommMonoid.{u3} \u03b1] [_inst_3 : AddCommMonoid.{u2} \u03b2] (f : Finsupp.{u1, u3} \u03b9 \u03b1 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2))) (i : \u03b9) (a : \u03b1) (g : \u03b9 -> \u03b1 -> \u03b2), (forall (i : \u03b9), Eq.{succ u2} \u03b2 (g i (OfNat.ofNat.{u3} \u03b1 0 (Zero.toOfNat0.{u3} \u03b1 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2))))) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (AddMonoid.toZero.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3))))) -> (forall (j : \u03b9) (a\u2081 : \u03b1) (a\u2082 : \u03b1), Eq.{succ u2} \u03b2 (g j (HAdd.hAdd.{u3, u3, u3} \u03b1 \u03b1 \u03b1 (instHAdd.{u3} \u03b1 (AddZeroClass.toAdd.{u3} \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2)))) a\u2081 a\u2082)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (g j a\u2081) (g j a\u2082))) -> (Eq.{succ u2} \u03b2 (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (Finsupp.sum.{u1, u3, u2} \u03b9 \u03b1 \u03b2 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2)) _inst_3 (Finsupp.update.{u1, u3} \u03b9 \u03b1 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2)) f i a) g) (g i (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b9 \u03b1 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2))) \u03b9 (fun (_x : \u03b9) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b9) => \u03b1) _x) (Finsupp.funLike.{u1, u3} \u03b9 \u03b1 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2))) f i))) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (AddZeroClass.toAdd.{u2} \u03b2 (AddMonoid.toAddZeroClass.{u2} \u03b2 (AddCommMonoid.toAddMonoid.{u2} \u03b2 _inst_3)))) (Finsupp.sum.{u1, u3, u2} \u03b9 \u03b1 \u03b2 (AddMonoid.toZero.{u3} \u03b1 (AddCommMonoid.toAddMonoid.{u3} \u03b1 _inst_2)) _inst_3 f g) (g i a)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_update_add Finsupp.sum_update_add\u2093'. -/\ntheorem sum_update_add [AddCommMonoid \u03b1] [AddCommMonoid \u03b2] (f : \u03b9 \u2192\u2080 \u03b1) (i : \u03b9) (a : \u03b1)\n    (g : \u03b9 \u2192 \u03b1 \u2192 \u03b2) (hg : \u2200 i, g i 0 = 0)\n    (hgg : \u2200 (j : \u03b9) (a\u2081 a\u2082 : \u03b1), g j (a\u2081 + a\u2082) = g j a\u2081 + g j a\u2082) :\n    (f.update i a).Sum g + g i (f i) = f.Sum g + g i a :=\n  by\n  rw [update_eq_erase_add_single, sum_add_index' hg hgg]\n  conv_rhs => rw [\u2190 Finsupp.update_self f i]\n  rw [update_eq_erase_add_single, sum_add_index' hg hgg, add_assoc, add_assoc]\n  congr 1\n  rw [add_comm, sum_single_index (hg _), sum_single_index (hg _)]\n#align finsupp.sum_update_add Finsupp.sum_update_add\n\n/- warning: finsupp.map_domain_inj_on -> Finsupp.mapDomain_injOn is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (S : Set.{u1} \u03b1) {f : \u03b1 -> \u03b2}, (Set.InjOn.{u1, u2} \u03b1 \u03b2 f S) -> (Set.InjOn.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f) (setOf.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (w : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => HasSubset.Subset.{u1} (Set.{u1} \u03b1) (Set.hasSubset.{u1} \u03b1) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (CoeTC\u2093.coe.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (Finset.Set.hasCoeT.{u1} \u03b1))) (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) w)) S)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (S : Set.{u3} \u03b1) {f : \u03b1 -> \u03b2}, (Set.InjOn.{u3, u2} \u03b1 \u03b2 f S) -> (Set.InjOn.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f) (setOf.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (fun (w : Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) => HasSubset.Subset.{u3} (Set.{u3} \u03b1) (Set.instHasSubsetSet.{u3} \u03b1) (Finset.toSet.{u3} \u03b1 (Finsupp.support.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) w)) S)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_inj_on Finsupp.mapDomain_injOn\u2093'. -/\ntheorem mapDomain_injOn (S : Set \u03b1) {f : \u03b1 \u2192 \u03b2} (hf : Set.InjOn f S) :\n    Set.InjOn (mapDomain f : (\u03b1 \u2192\u2080 M) \u2192 \u03b2 \u2192\u2080 M) { w | (w.support : Set \u03b1) \u2286 S } :=\n  by\n  intro v\u2081 hv\u2081 v\u2082 hv\u2082 eq\n  ext a\n  classical\n    by_cases h : a \u2208 v\u2081.support \u222a v\u2082.support\n    \u00b7\n      rw [\u2190 map_domain_apply' S _ hv\u2081 hf _, \u2190 map_domain_apply' S _ hv\u2082 hf _, Eq] <;>\n        \u00b7 apply Set.union_subset hv\u2081 hv\u2082\n          exact_mod_cast h\n    \u00b7 simp only [Decidable.not_or_iff_and_not, mem_union, Classical.not_not, mem_support_iff] at h\n      simp [h]\n#align finsupp.map_domain_inj_on Finsupp.mapDomain_injOn\n\n/- warning: finsupp.equiv_map_domain_eq_map_domain -> Finsupp.equivMapDomain_eq_mapDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_2 : AddCommMonoid.{u3} M] (f : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (l : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)))), Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)))) (Finsupp.equivMapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) f l) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_2 (coeFn.{max 1 (max (succ u1) (succ u2)) (succ u2) (succ u1), max (succ u1) (succ u2)} (Equiv.{succ u1, succ u2} \u03b1 \u03b2) (fun (_x : Equiv.{succ u1, succ u2} \u03b1 \u03b2) => \u03b1 -> \u03b2) (Equiv.hasCoeToFun.{succ u1, succ u2} \u03b1 \u03b2) f) l)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {M : Type.{u3}} [_inst_2 : AddCommMonoid.{u3} M] (f : Equiv.{succ u2, succ u1} \u03b1 \u03b2) (l : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.equivMapDomain.{u2, u1, u3} \u03b1 \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) f l) (Finsupp.mapDomain.{u2, u1, u3} \u03b1 \u03b2 M _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Equiv.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => \u03b2) _x) (Equiv.instFunLikeEquiv.{succ u2, succ u1} \u03b1 \u03b2) f) l)\nCase conversion may be inaccurate. Consider using '#align finsupp.equiv_map_domain_eq_map_domain Finsupp.equivMapDomain_eq_mapDomain\u2093'. -/\ntheorem equivMapDomain_eq_mapDomain {M} [AddCommMonoid M] (f : \u03b1 \u2243 \u03b2) (l : \u03b1 \u2192\u2080 M) :\n    equivMapDomain f l = mapDomain f l := by ext x <;> simp [map_domain_equiv_apply]\n#align finsupp.equiv_map_domain_eq_map_domain Finsupp.equivMapDomain_eq_mapDomain\n\nend MapDomain\n\n/-! ### Declarations about `comap_domain` -/\n\n\nsection ComapDomain\n\n#print Finsupp.comapDomain /-\n/-- Given `f : \u03b1 \u2192 \u03b2`, `l : \u03b2 \u2192\u2080 M` and a proof `hf` that `f` is injective on\nthe preimage of `l.support`, `comap_domain f l hf` is the finitely supported function\nfrom `\u03b1` to `M` given by composing `l` with `f`. -/\n@[simps support]\ndef comapDomain [Zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : Set.InjOn f (f \u207b\u00b9' \u2191l.support)) : \u03b1 \u2192\u2080 M\n    where\n  support := l.support.Preimage f hf\n  toFun a := l (f a)\n  mem_support_toFun := by\n    intro a\n    simp only [finset.mem_def.symm, Finset.mem_preimage]\n    exact l.mem_support_to_fun (f a)\n#align finsupp.comap_domain Finsupp.comapDomain\n-/\n\n#print Finsupp.comapDomain_apply /-\n@[simp]\ntheorem comapDomain_apply [Zero M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (hf : Set.InjOn f (f \u207b\u00b9' \u2191l.support))\n    (a : \u03b1) : comapDomain f l hf a = l (f a) :=\n  rfl\n#align finsupp.comap_domain_apply Finsupp.comapDomain_apply\n-/\n\n/- warning: finsupp.sum_comap_domain -> Finsupp.sum_comapDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : Zero.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : \u03b1 -> \u03b2) (l : Finsupp.{u2, u3} \u03b2 M _inst_1) (g : \u03b2 -> M -> N) (hf : Set.BijOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 l))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 l))), Eq.{succ u4} N (Finsupp.sum.{u1, u3, u4} \u03b1 M N _inst_1 _inst_2 (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f l (Set.BijOn.injOn.{u1, u2} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 l))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 l)) f hf)) (Function.comp.{succ u1, succ u2, max (succ u3) (succ u4)} \u03b1 \u03b2 (M -> N) g f)) (Finsupp.sum.{u2, u3, u4} \u03b2 M N _inst_1 _inst_2 l g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u4}} {N : Type.{u3}} [_inst_1 : Zero.{u4} M] [_inst_2 : AddCommMonoid.{u3} N] (f : \u03b1 -> \u03b2) (l : Finsupp.{u2, u4} \u03b2 M _inst_1) (g : \u03b2 -> M -> N) (hf : Set.BijOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M _inst_1 l))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M _inst_1 l))), Eq.{succ u3} N (Finsupp.sum.{u1, u4, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.comapDomain.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 f l (Set.BijOn.injOn.{u2, u1} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M _inst_1 l))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M _inst_1 l)) f hf)) (Function.comp.{succ u1, succ u2, max (succ u4) (succ u3)} \u03b1 \u03b2 (M -> N) g f)) (Finsupp.sum.{u2, u4, u3} \u03b2 M N _inst_1 _inst_2 l g)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_comap_domain Finsupp.sum_comapDomain\u2093'. -/\ntheorem sum_comapDomain [Zero M] [AddCommMonoid N] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M) (g : \u03b2 \u2192 M \u2192 N)\n    (hf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support) :\n    (comapDomain f l hf.InjOn).Sum (g \u2218 f) = l.Sum g :=\n  by\n  simp only [Sum, comap_domain_apply, (\u00b7 \u2218 \u00b7)]\n  simp [comap_domain, Finset.sum_preimage_of_bij f _ _ fun x => g x (l x)]\n#align finsupp.sum_comap_domain Finsupp.sum_comapDomain\n\n/- warning: finsupp.eq_zero_of_comap_domain_eq_zero -> Finsupp.eq_zero_of_comapDomain_eq_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : \u03b1 -> \u03b2) (l : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (hf : Set.BijOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l))), (Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) f l (Set.BijOn.injOn.{u1, u2} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l)) f hf)) (OfNat.ofNat.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (OfNat.mk.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (Zero.zero.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))))) -> (Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) l (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (OfNat.mk.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) 0 (Zero.zero.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : \u03b1 -> \u03b2) (l : Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (hf : Set.BijOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) l))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) l))), (Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) f l (Set.BijOn.injOn.{u2, u1} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) l))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) l)) f hf)) (OfNat.ofNat.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) 0 (Zero.toOfNat0.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))) -> (Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) 0 (Zero.toOfNat0.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align finsupp.eq_zero_of_comap_domain_eq_zero Finsupp.eq_zero_of_comapDomain_eq_zero\u2093'. -/\ntheorem eq_zero_of_comapDomain_eq_zero [AddCommMonoid M] (f : \u03b1 \u2192 \u03b2) (l : \u03b2 \u2192\u2080 M)\n    (hf : Set.BijOn f (f \u207b\u00b9' \u2191l.support) \u2191l.support) : comapDomain f l hf.InjOn = 0 \u2192 l = 0 :=\n  by\n  rw [\u2190 support_eq_empty, \u2190 support_eq_empty, comap_domain]\n  simp only [Finset.ext_iff, Finset.not_mem_empty, iff_false_iff, mem_preimage]\n  intro h a ha\n  cases' hf.2.2 ha with b hb\n  exact h b (hb.2.symm \u25b8 ha)\n#align finsupp.eq_zero_of_comap_domain_eq_zero Finsupp.eq_zero_of_comapDomain_eq_zero\n\nsection FInjective\n\nsection Zero\n\nvariable [Zero M]\n\n/- warning: finsupp.comap_domain_zero -> Finsupp.comapDomain_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : \u03b1 -> \u03b2) (hif : optParam.{0} (Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (OfNat.mk.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (Zero.zero.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.zero.{u2, u3} \u03b2 M _inst_1)))))))) (Set.injOn_empty.{u1, u2} \u03b1 \u03b2 f)), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (OfNat.mk.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) 0 (Zero.zero.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M _inst_1) (Finsupp.zero.{u2, u3} \u03b2 M _inst_1)))) hif) (OfNat.ofNat.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.zero.{u1, u3} \u03b1 M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : \u03b1 -> \u03b2) (hif : optParam.{0} (Set.InjOn.{u3, u2} \u03b1 \u03b2 f (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M _inst_1 (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.zero.{u2, u1} \u03b2 M _inst_1))))))) (Eq.rec.{0, succ u2} (Set.{u2} \u03b2) (EmptyCollection.emptyCollection.{u2} (Set.{u2} \u03b2) (Set.instEmptyCollectionSet.{u2} \u03b2)) (fun (x._@.Mathlib.Data.Finsupp.Basic._hyg.7335 : Set.{u2} \u03b2) (h._@.Mathlib.Data.Finsupp.Basic._hyg.7336 : Eq.{succ u2} (Set.{u2} \u03b2) (EmptyCollection.emptyCollection.{u2} (Set.{u2} \u03b2) (Set.instEmptyCollectionSet.{u2} \u03b2)) x._@.Mathlib.Data.Finsupp.Basic._hyg.7335) => Set.InjOn.{u3, u2} \u03b1 \u03b2 f (Set.preimage.{u3, u2} \u03b1 \u03b2 f x._@.Mathlib.Data.Finsupp.Basic._hyg.7335)) (Set.injOn_empty.{u2, u3} \u03b1 \u03b2 f) (Finset.toSet.{u2} \u03b2 (EmptyCollection.emptyCollection.{u2} (Finset.{u2} \u03b2) (Finset.instEmptyCollectionFinset.{u2} \u03b2))) (Eq.symm.{succ u2} (Set.{u2} \u03b2) (Finset.toSet.{u2} \u03b2 (EmptyCollection.emptyCollection.{u2} (Finset.{u2} \u03b2) (Finset.instEmptyCollectionFinset.{u2} \u03b2))) (EmptyCollection.emptyCollection.{u2} (Set.{u2} \u03b2) (Set.instEmptyCollectionSet.{u2} \u03b2)) (Finset.coe_empty.{u2} \u03b2)))), Eq.{max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M _inst_1) (Finsupp.zero.{u2, u1} \u03b2 M _inst_1))) hif) (OfNat.ofNat.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.zero.{u3, u1} \u03b1 M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_zero Finsupp.comapDomain_zero\u2093'. -/\n/-- Note the `hif` argument is needed for this to work in `rw`. -/\n@[simp]\ntheorem comapDomain_zero (f : \u03b1 \u2192 \u03b2)\n    (hif : Set.InjOn f (f \u207b\u00b9' \u2191(0 : \u03b2 \u2192\u2080 M).support) := Set.injOn_empty _) :\n    comapDomain f (0 : \u03b2 \u2192\u2080 M) hif = (0 : \u03b1 \u2192\u2080 M) :=\n  by\n  ext\n  rfl\n#align finsupp.comap_domain_zero Finsupp.comapDomain_zero\n\n/- warning: finsupp.comap_domain_single -> Finsupp.comapDomain_single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : Zero.{u3} M] (f : \u03b1 -> \u03b2) (a : \u03b1) (m : M) (hif : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M _inst_1 (Finsupp.single.{u2, u3} \u03b2 M _inst_1 (f a) m))))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M _inst_1) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u2, u3} \u03b2 M _inst_1 (f a) m) hif) (Finsupp.single.{u1, u3} \u03b1 M _inst_1 a m)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (f : \u03b1 -> \u03b2) (a : \u03b1) (m : M) (hif : Set.InjOn.{u3, u2} \u03b1 \u03b2 f (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M _inst_1 (Finsupp.single.{u2, u1} \u03b2 M _inst_1 (f a) m))))), Eq.{max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b1 M _inst_1) (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f (Finsupp.single.{u2, u1} \u03b2 M _inst_1 (f a) m) hif) (Finsupp.single.{u3, u1} \u03b1 M _inst_1 a m)\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_single Finsupp.comapDomain_single\u2093'. -/\n@[simp]\ntheorem comapDomain_single (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (m : M)\n    (hif : Set.InjOn f (f \u207b\u00b9' (single (f a) m).support)) :\n    comapDomain f (Finsupp.single (f a) m) hif = Finsupp.single a m :=\n  by\n  rcases eq_or_ne m 0 with (rfl | hm)\n  \u00b7 simp only [single_zero, comap_domain_zero]\n  \u00b7 rw [eq_single_iff, comap_domain_apply, comap_domain_support, \u2190 Finset.coe_subset, coe_preimage,\n      support_single_ne_zero _ hm, coe_singleton, coe_singleton, single_eq_same]\n    rw [support_single_ne_zero _ hm, coe_singleton] at hif\n    exact \u27e8fun x hx => hif hx rfl hx, rfl\u27e9\n#align finsupp.comap_domain_single Finsupp.comapDomain_single\n\nend Zero\n\nsection AddZeroClass\n\nvariable [AddZeroClass M] {f : \u03b1 \u2192 \u03b2}\n\n/- warning: finsupp.comap_domain_add -> Finsupp.comapDomain_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddZeroClass.{u3} M] {f : \u03b1 -> \u03b2} (v\u2081 : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (v\u2082 : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (hv\u2081 : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) v\u2081)))) (hv\u2082 : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) v\u2082)))) (hv\u2081\u2082 : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u2, u3} \u03b2 M _inst_1)) v\u2081 v\u2082))))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u2, u3} \u03b2 M _inst_1)) v\u2081 v\u2082) hv\u2081\u2082) (HAdd.hAdd.{max u1 u3, max u1 u3, max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u1, u3} \u03b1 M _inst_1)) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f v\u2081 hv\u2081) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f v\u2082 hv\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u3}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] {f : \u03b1 -> \u03b2} (v\u2081 : Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (v\u2082 : Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (hv\u2081 : Set.InjOn.{u1, u3} \u03b1 \u03b2 f (Set.preimage.{u1, u3} \u03b1 \u03b2 f (Finset.toSet.{u3} \u03b2 (Finsupp.support.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) v\u2081)))) (hv\u2082 : Set.InjOn.{u1, u3} \u03b1 \u03b2 f (Set.preimage.{u1, u3} \u03b1 \u03b2 f (Finset.toSet.{u3} \u03b2 (Finsupp.support.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) v\u2082)))) (hv\u2081\u2082 : Set.InjOn.{u1, u3} \u03b1 \u03b2 f (Set.preimage.{u1, u3} \u03b1 \u03b2 f (Finset.toSet.{u3} \u03b2 (Finsupp.support.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) (HAdd.hAdd.{max u3 u2, max u3 u2, max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (instHAdd.{max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.add.{u3, u2} \u03b2 M _inst_1)) v\u2081 v\u2082))))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.comapDomain.{u1, u3, u2} \u03b1 \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) f (HAdd.hAdd.{max u3 u2, max u3 u2, max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (instHAdd.{max u3 u2} (Finsupp.{u3, u2} \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.add.{u3, u2} \u03b2 M _inst_1)) v\u2081 v\u2082) hv\u2081\u2082) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) (Finsupp.comapDomain.{u1, u3, u2} \u03b1 \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) f v\u2081 hv\u2081) (Finsupp.comapDomain.{u1, u3, u2} \u03b1 \u03b2 M (AddZeroClass.toZero.{u2} M _inst_1) f v\u2082 hv\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_add Finsupp.comapDomain_add\u2093'. -/\ntheorem comapDomain_add (v\u2081 v\u2082 : \u03b2 \u2192\u2080 M) (hv\u2081 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2081.support))\n    (hv\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191v\u2082.support)) (hv\u2081\u2082 : Set.InjOn f (f \u207b\u00b9' \u2191(v\u2081 + v\u2082).support)) :\n    comapDomain f (v\u2081 + v\u2082) hv\u2081\u2082 = comapDomain f v\u2081 hv\u2081 + comapDomain f v\u2082 hv\u2082 :=\n  by\n  ext\n  simp only [comap_domain_apply, coe_add, Pi.add_apply]\n#align finsupp.comap_domain_add Finsupp.comapDomain_add\n\n/- warning: finsupp.comap_domain_add_of_injective -> Finsupp.comapDomain_add_of_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddZeroClass.{u3} M] {f : \u03b1 -> \u03b2} (hf : Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) (v\u2081 : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (v\u2082 : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u2, u3} \u03b2 M _inst_1)) v\u2081 v\u2082) (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) (HAdd.hAdd.{max u2 u3, max u2 u3, max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u2 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u2, u3} \u03b2 M _inst_1)) v\u2081 v\u2082)))))) (HAdd.hAdd.{max u1 u3, max u1 u3, max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (instHAdd.{max u1 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.add.{u1, u3} \u03b1 M _inst_1)) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f v\u2081 (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) v\u2081))))) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) f v\u2082 (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1) v\u2082))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddZeroClass.{u1} M] {f : \u03b1 -> \u03b2} (hf : Function.Injective.{succ u3, succ u2} \u03b1 \u03b2 f) (v\u2081 : Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (v\u2082 : Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)), Eq.{max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) f (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M _inst_1)) v\u2081 v\u2082) (Function.Injective.injOn.{u2, u3} \u03b1 \u03b2 f hf (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M _inst_1)) v\u2081 v\u2082)))))) (HAdd.hAdd.{max u3 u1, max u3 u1, max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u3, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u3, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M _inst_1)) (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) f v\u2081 (Function.Injective.injOn.{u2, u3} \u03b1 \u03b2 f hf (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) v\u2081))))) (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) f v\u2082 (Function.Injective.injOn.{u2, u3} \u03b1 \u03b2 f hf (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M (AddZeroClass.toZero.{u1} M _inst_1) v\u2082))))))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_add_of_injective Finsupp.comapDomain_add_of_injective\u2093'. -/\n/-- A version of `finsupp.comap_domain_add` that's easier to use. -/\ntheorem comapDomain_add_of_injective (hf : Function.Injective f) (v\u2081 v\u2082 : \u03b2 \u2192\u2080 M) :\n    comapDomain f (v\u2081 + v\u2082) (hf.InjOn _) =\n      comapDomain f v\u2081 (hf.InjOn _) + comapDomain f v\u2082 (hf.InjOn _) :=\n  comapDomain_add _ _ _ _ _\n#align finsupp.comap_domain_add_of_injective Finsupp.comapDomain_add_of_injective\n\n/- warning: finsupp.comap_domain.add_monoid_hom -> Finsupp.comapDomain.addMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddZeroClass.{u3} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (AddMonoidHom.{max u2 u3, max u1 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M _inst_1)) (Finsupp.addZeroClass.{u2, u3} \u03b2 M _inst_1) (Finsupp.addZeroClass.{u1, u3} \u03b1 M _inst_1))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddZeroClass.{u3} M] {f : \u03b1 -> \u03b2}, (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (AddMonoidHom.{max u3 u2, max u3 u1} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toZero.{u3} M _inst_1)) (Finsupp.addZeroClass.{u2, u3} \u03b2 M _inst_1) (Finsupp.addZeroClass.{u1, u3} \u03b1 M _inst_1))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain.add_monoid_hom Finsupp.comapDomain.addMonoidHom\u2093'. -/\n/-- `finsupp.comap_domain` is an `add_monoid_hom`. -/\n@[simps]\ndef comapDomain.addMonoidHom (hf : Function.Injective f) : (\u03b2 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 M\n    where\n  toFun x := comapDomain f x (hf.InjOn _)\n  map_zero' := comapDomain_zero f\n  map_add' := comapDomain_add_of_injective hf\n#align finsupp.comap_domain.add_monoid_hom Finsupp.comapDomain.addMonoidHom\n\nend AddZeroClass\n\nvariable [AddCommMonoid M] (f : \u03b1 \u2192 \u03b2)\n\n/- warning: finsupp.map_domain_comap_domain -> Finsupp.mapDomain_comapDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : \u03b1 -> \u03b2) (hf : Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) (l : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), (HasSubset.Subset.{u2} (Set.{u2} \u03b2) (Set.hasSubset.{u2} \u03b2) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l)) (Set.range.{u2, succ u1} \u03b2 \u03b1 f)) -> (Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) f l (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) l)))))) l)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (f : \u03b1 -> \u03b2) (hf : Function.Injective.{succ u3, succ u2} \u03b1 \u03b2 f) (l : Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), (HasSubset.Subset.{u2} (Set.{u2} \u03b2) (Set.instHasSubsetSet.{u2} \u03b2) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) l)) (Set.range.{u2, succ u3} \u03b2 \u03b1 f)) -> (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.mapDomain.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f (Finsupp.comapDomain.{u3, u2, u1} \u03b1 \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) f l (Function.Injective.injOn.{u2, u3} \u03b1 \u03b2 f hf (Set.preimage.{u3, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) l)))))) l)\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_comap_domain Finsupp.mapDomain_comapDomain\u2093'. -/\ntheorem mapDomain_comapDomain (hf : Function.Injective f) (l : \u03b2 \u2192\u2080 M)\n    (hl : \u2191l.support \u2286 Set.range f) : mapDomain f (comapDomain f l (hf.InjOn _)) = l :=\n  by\n  ext a\n  by_cases h_cases : a \u2208 Set.range f\n  \u00b7 rcases Set.mem_range.1 h_cases with \u27e8b, hb\u27e9\n    rw [hb.symm, map_domain_apply hf, comap_domain_apply]\n  \u00b7 rw [map_domain_notin_range _ _ h_cases]\n    by_contra h_contr\n    apply h_cases (hl <| Finset.mem_coe.2 <| mem_support_iff.2 fun h => h_contr h.symm)\n#align finsupp.map_domain_comap_domain Finsupp.mapDomain_comapDomain\n\nend FInjective\n\nend ComapDomain\n\n/-! ### Declarations about finitely supported functions whose support is an `option` type -/\n\n\nsection Option\n\n#print Finsupp.some /-\n/-- Restrict a finitely supported function on `option \u03b1` to a finitely supported function on `\u03b1`. -/\ndef some [Zero M] (f : Option \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M :=\n  f.comapDomain Option.some fun _ => by simp\n#align finsupp.some Finsupp.some\n-/\n\n#print Finsupp.some_apply /-\n@[simp]\ntheorem some_apply [Zero M] (f : Option \u03b1 \u2192\u2080 M) (a : \u03b1) : f.some a = f (Option.some a) :=\n  rfl\n#align finsupp.some_apply Finsupp.some_apply\n-/\n\n#print Finsupp.some_zero /-\n@[simp]\ntheorem some_zero [Zero M] : (0 : Option \u03b1 \u2192\u2080 M).some = 0 :=\n  by\n  ext\n  simp\n#align finsupp.some_zero Finsupp.some_zero\n-/\n\n/- warning: finsupp.some_add -> Finsupp.some_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] (f : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (g : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.some.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) f g)) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.some.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) f) (Finsupp.some.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] (f : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (g : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.some.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} (Option.{u1} \u03b1) M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) f g)) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.some.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) f) (Finsupp.some.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) g))\nCase conversion may be inaccurate. Consider using '#align finsupp.some_add Finsupp.some_add\u2093'. -/\n@[simp]\ntheorem some_add [AddCommMonoid M] (f g : Option \u03b1 \u2192\u2080 M) : (f + g).some = f.some + g.some :=\n  by\n  ext\n  simp\n#align finsupp.some_add Finsupp.some_add\n\n#print Finsupp.some_single_none /-\n@[simp]\ntheorem some_single_none [Zero M] (m : M) : (single none m : Option \u03b1 \u2192\u2080 M).some = 0 :=\n  by\n  ext\n  simp\n#align finsupp.some_single_none Finsupp.some_single_none\n-/\n\n#print Finsupp.some_single_some /-\n@[simp]\ntheorem some_single_some [Zero M] (a : \u03b1) (m : M) :\n    (single (Option.some a) m : Option \u03b1 \u2192\u2080 M).some = single a m := by\n  classical\n    ext b\n    simp [single_apply]\n#align finsupp.some_single_some Finsupp.some_single_some\n-/\n\n/- warning: finsupp.prod_option_index -> Finsupp.prod_option_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : CommMonoid.{u3} N] (f : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (b : (Option.{u1} \u03b1) -> M -> N), (forall (o : Option.{u1} \u03b1), Eq.{succ u3} N (b o (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))))) (OfNat.ofNat.{u3} N 1 (OfNat.mk.{u3} N 1 (One.one.{u3} N (MulOneClass.toHasOne.{u3} N (Monoid.toMulOneClass.{u3} N (CommMonoid.toMonoid.{u3} N _inst_2))))))) -> (forall (o : Option.{u1} \u03b1) (m\u2081 : M) (m\u2082 : M), Eq.{succ u3} N (b o (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) m\u2081 m\u2082)) (HMul.hMul.{u3, u3, u3} N N N (instHMul.{u3} N (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N (CommMonoid.toMonoid.{u3} N _inst_2)))) (b o m\u2081) (b o m\u2082))) -> (Eq.{succ u3} N (Finsupp.prod.{u1, u2, u3} (Option.{u1} \u03b1) M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) _inst_2 f b) (HMul.hMul.{u3, u3, u3} N N N (instHMul.{u3} N (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N (CommMonoid.toMonoid.{u3} N _inst_2)))) (b (Option.none.{u1} \u03b1) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (fun (_x : Finsupp.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) => (Option.{u1} \u03b1) -> M) (Finsupp.coeFun.{u1, u2} (Option.{u1} \u03b1) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) f (Option.none.{u1} \u03b1))) (Finsupp.prod.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) _inst_2 (Finsupp.some.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) f) (fun (a : \u03b1) => b (Option.some.{u1} \u03b1 a)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : CommMonoid.{u2} N] (f : Finsupp.{u1, u3} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (b : (Option.{u1} \u03b1) -> M -> N), (forall (o : Option.{u1} \u03b1), Eq.{succ u2} N (b o (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) (OfNat.ofNat.{u2} N 1 (One.toOfNat1.{u2} N (Monoid.toOne.{u2} N (CommMonoid.toMonoid.{u2} N _inst_2))))) -> (forall (o : Option.{u1} \u03b1) (m\u2081 : M) (m\u2082 : M), Eq.{succ u2} N (b o (HAdd.hAdd.{u3, u3, u3} M M M (instHAdd.{u3} M (AddZeroClass.toAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) m\u2081 m\u2082)) (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N (CommMonoid.toMonoid.{u2} N _inst_2)))) (b o m\u2081) (b o m\u2082))) -> (Eq.{succ u2} N (Finsupp.prod.{u1, u3, u2} (Option.{u1} \u03b1) M N (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) _inst_2 f b) (HMul.hMul.{u2, u2, u2} N N N (instHMul.{u2} N (MulOneClass.toMul.{u2} N (Monoid.toMulOneClass.{u2} N (CommMonoid.toMonoid.{u2} N _inst_2)))) (b (Option.none.{u1} \u03b1) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Option.{u1} \u03b1) (fun (_x : Option.{u1} \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} (Option.{u1} \u03b1) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) f (Option.none.{u1} \u03b1))) (Finsupp.prod.{u1, u3, u2} \u03b1 M N (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) _inst_2 (Finsupp.some.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) f) (fun (a : \u03b1) => b (Option.some.{u1} \u03b1 a)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_option_index Finsupp.prod_option_index\u2093'. -/\n@[to_additive]\ntheorem prod_option_index [AddCommMonoid M] [CommMonoid N] (f : Option \u03b1 \u2192\u2080 M)\n    (b : Option \u03b1 \u2192 M \u2192 N) (h_zero : \u2200 o, b o 0 = 1)\n    (h_add : \u2200 o m\u2081 m\u2082, b o (m\u2081 + m\u2082) = b o m\u2081 * b o m\u2082) :\n    f.Prod b = b none (f none) * f.some.Prod fun a => b (Option.some a) := by\n  classical\n    apply induction_linear f\n    \u00b7 simp [some_zero, h_zero]\n    \u00b7 intro f\u2081 f\u2082 h\u2081 h\u2082\n      rw [Finsupp.prod_add_index, h\u2081, h\u2082, some_add, Finsupp.prod_add_index]\n      simp only [h_add, Pi.add_apply, Finsupp.coe_add]\n      rw [mul_mul_mul_comm]\n      all_goals simp [h_zero, h_add]\n    \u00b7 rintro (_ | a) m <;> simp [h_zero, h_add]\n#align finsupp.prod_option_index Finsupp.prod_option_index\n#align finsupp.sum_option_index Finsupp.sum_option_index\n\n/- warning: finsupp.sum_option_index_smul -> Finsupp.sum_option_index_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u3, u2} R M _inst_1 _inst_2] (f : Finsupp.{u1, u3} (Option.{u1} \u03b1) R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) (b : (Option.{u1} \u03b1) -> M), Eq.{succ u2} M (Finsupp.sum.{u1, u3, u2} (Option.{u1} \u03b1) R M (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) _inst_2 f (fun (o : Option.{u1} \u03b1) (r : R) => SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3)))) r (b o))) (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toHasAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3)))) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} (Option.{u1} \u03b1) R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) (fun (_x : Finsupp.{u1, u3} (Option.{u1} \u03b1) R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) => (Option.{u1} \u03b1) -> R) (Finsupp.coeFun.{u1, u3} (Option.{u1} \u03b1) R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) f (Option.none.{u1} \u03b1)) (b (Option.none.{u1} \u03b1))) (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) _inst_2 (Finsupp.some.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) f) (fun (a : \u03b1) (r : R) => SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3)))) r (b (Option.some.{u1} \u03b1 a)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u3, u2} R M _inst_1 _inst_2] (f : Finsupp.{u1, u3} (Option.{u1} \u03b1) R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (b : (Option.{u1} \u03b1) -> M), Eq.{succ u2} M (Finsupp.sum.{u1, u3, u2} (Option.{u1} \u03b1) R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) _inst_2 f (fun (o : Option.{u1} \u03b1) (r : R) => HSMul.hSMul.{u3, u2, u2} R M M (instHSMul.{u3, u2} R M (SMulZeroClass.toSMul.{u3, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3))))) r (b o))) (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (HSMul.hSMul.{u3, u2, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M M (instHSMul.{u3, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M (SMulZeroClass.toSMul.{u3, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M (MonoidWithZero.toZero.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) (Semiring.toMonoidWithZero.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M (Semiring.toMonoidWithZero.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u3, u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) (Option.none.{u1} \u03b1)) M _inst_1 _inst_2 _inst_3))))) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} (Option.{u1} \u03b1) R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (Option.{u1} \u03b1) (fun (_x : Option.{u1} \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Option.{u1} \u03b1) => R) _x) (Finsupp.funLike.{u1, u3} (Option.{u1} \u03b1) R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) f (Option.none.{u1} \u03b1)) (b (Option.none.{u1} \u03b1))) (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) _inst_2 (Finsupp.some.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) f) (fun (a : \u03b1) (r : R) => HSMul.hSMul.{u3, u2, u2} R M M (instHSMul.{u3, u2} R M (SMulZeroClass.toSMul.{u3, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3))))) r (b (Option.some.{u1} \u03b1 a)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_option_index_smul Finsupp.sum_option_index_smul\u2093'. -/\ntheorem sum_option_index_smul [Semiring R] [AddCommMonoid M] [Module R M] (f : Option \u03b1 \u2192\u2080 R)\n    (b : Option \u03b1 \u2192 M) :\n    (f.Sum fun o r => r \u2022 b o) = f none \u2022 b none + f.some.Sum fun a r => r \u2022 b (Option.some a) :=\n  f.sum_option_index _ (fun _ => zero_smul _ _) fun _ _ _ => add_smul _ _ _\n#align finsupp.sum_option_index_smul Finsupp.sum_option_index_smul\n\nend Option\n\n/-! ### Declarations about `filter` -/\n\n\nsection Filter\n\nsection Zero\n\nvariable [Zero M] (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M)\n\n#print Finsupp.filter /-\n/--\n`filter p f` is the finitely supported function that is `f a` if `p a` is true and 0 otherwise. -/\ndef filter (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : \u03b1 \u2192\u2080 M\n    where\n  toFun a :=\n    haveI := Classical.decPred p\n    if p a then f a else 0\n  support :=\n    haveI := Classical.decPred p\n    f.support.filter fun a => p a\n  mem_support_toFun a := by\n    split_ifs <;>\n      \u00b7 simp only [h, mem_filter, mem_support_iff]\n        tauto\n#align finsupp.filter Finsupp.filter\n-/\n\n#print Finsupp.filter_apply /-\ntheorem filter_apply (a : \u03b1) [D : Decidable (p a)] : f.filter\u2093 p a = if p a then f a else 0 := by\n  rw [Subsingleton.elim D] <;> rfl\n#align finsupp.filter_apply Finsupp.filter_apply\n-/\n\n/- warning: finsupp.filter_eq_indicator -> Finsupp.filter_eq_indicator is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1), Eq.{max (succ u1) (succ u2)} (\u03b1 -> M) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f)) (Set.indicator.{u1, u2} \u03b1 M _inst_1 (setOf.{u1} \u03b1 (fun (x : \u03b1) => p x)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1), Eq.{max (succ u2) (succ u1)} (forall (\u1fb0 : \u03b1), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) \u1fb0) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f)) (Set.indicator.{u2, u1} \u03b1 M _inst_1 (setOf.{u2} \u03b1 (fun (x : \u03b1) => p x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_eq_indicator Finsupp.filter_eq_indicator\u2093'. -/\ntheorem filter_eq_indicator : \u21d1(f.filter\u2093 p) = Set.indicator { x | p x } f :=\n  rfl\n#align finsupp.filter_eq_indicator Finsupp.filter_eq_indicator\n\n/- warning: finsupp.filter_eq_zero_iff -> Finsupp.filter_eq_zero_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1), Iff (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))) (forall (x : \u03b1), (p x) -> (Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1), Iff (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))) (forall (x : \u03b1), (p x) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_eq_zero_iff Finsupp.filter_eq_zero_iff\u2093'. -/\ntheorem filter_eq_zero_iff : f.filter\u2093 p = 0 \u2194 \u2200 x, p x \u2192 f x = 0 := by\n  simp only [FunLike.ext_iff, filter_eq_indicator, zero_apply, Set.indicator_apply_eq_zero,\n    Set.mem_setOf_eq]\n#align finsupp.filter_eq_zero_iff Finsupp.filter_eq_zero_iff\n\n/- warning: finsupp.filter_eq_self_iff -> Finsupp.filter_eq_self_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1), Iff (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f) f) (forall (x : \u03b1), (Ne.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) -> (p x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1), Iff (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f) f) (forall (x : \u03b1), (Ne.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) _inst_1))) -> (p x))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_eq_self_iff Finsupp.filter_eq_self_iff\u2093'. -/\ntheorem filter_eq_self_iff : f.filter\u2093 p = f \u2194 \u2200 x, f x \u2260 0 \u2192 p x := by\n  simp only [FunLike.ext_iff, filter_eq_indicator, Set.indicator_apply_eq_self, Set.mem_setOf_eq,\n    not_imp_comm]\n#align finsupp.filter_eq_self_iff Finsupp.filter_eq_self_iff\n\n#print Finsupp.filter_apply_pos /-\n@[simp]\ntheorem filter_apply_pos {a : \u03b1} (h : p a) : f.filter\u2093 p a = f a := by classical convert if_pos h\n#align finsupp.filter_apply_pos Finsupp.filter_apply_pos\n-/\n\n#print Finsupp.filter_apply_neg /-\n@[simp]\ntheorem filter_apply_neg {a : \u03b1} (h : \u00acp a) : f.filter\u2093 p a = 0 := by classical convert if_neg h\n#align finsupp.filter_apply_neg Finsupp.filter_apply_neg\n-/\n\n/- warning: finsupp.support_filter -> Finsupp.support_filter is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [D : DecidablePred.{succ u1} \u03b1 p], Eq.{succ u1} (Finset.{u1} \u03b1) (Finsupp.support.{u1, u2} \u03b1 M _inst_1 (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f)) (Finset.filter.{u1} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u1, u2} \u03b1 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [D : DecidablePred.{succ u2} \u03b1 p], Eq.{succ u2} (Finset.{u2} \u03b1) (Finsupp.support.{u2, u1} \u03b1 M _inst_1 (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f)) (Finset.filter.{u2} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u2, u1} \u03b1 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.support_filter Finsupp.support_filter\u2093'. -/\n@[simp]\ntheorem support_filter [D : DecidablePred p] : (f.filter\u2093 p).support = f.support.filter\u2093 p := by\n  rw [Subsingleton.elim D] <;> rfl\n#align finsupp.support_filter Finsupp.support_filter\n\n/- warning: finsupp.filter_zero -> Finsupp.filter_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_zero Finsupp.filter_zero\u2093'. -/\ntheorem filter_zero : (0 : \u03b1 \u2192\u2080 M).filter\u2093 p = 0 := by\n  classical rw [\u2190 support_eq_empty, support_filter, support_zero, Finset.filter_empty]\n#align finsupp.filter_zero Finsupp.filter_zero\n\n/- warning: finsupp.filter_single_of_pos -> Finsupp.filter_single_of_pos is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) {a : \u03b1} {b : M}, (p a) -> (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p (Finsupp.single.{u1, u2} \u03b1 M _inst_1 a b)) (Finsupp.single.{u1, u2} \u03b1 M _inst_1 a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) {a : \u03b1} {b : M}, (p a) -> (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p (Finsupp.single.{u2, u1} \u03b1 M _inst_1 a b)) (Finsupp.single.{u2, u1} \u03b1 M _inst_1 a b))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_single_of_pos Finsupp.filter_single_of_pos\u2093'. -/\n@[simp]\ntheorem filter_single_of_pos {a : \u03b1} {b : M} (h : p a) : (single a b).filter\u2093 p = single a b :=\n  (filter_eq_self_iff _ _).2 fun x hx => (single_apply_ne_zero.1 hx).1.symm \u25b8 h\n#align finsupp.filter_single_of_pos Finsupp.filter_single_of_pos\n\n/- warning: finsupp.filter_single_of_neg -> Finsupp.filter_single_of_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) {a : \u03b1} {b : M}, (Not (p a)) -> (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p (Finsupp.single.{u1, u2} \u03b1 M _inst_1 a b)) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) {a : \u03b1} {b : M}, (Not (p a)) -> (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p (Finsupp.single.{u2, u1} \u03b1 M _inst_1 a b)) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_single_of_neg Finsupp.filter_single_of_neg\u2093'. -/\n@[simp]\ntheorem filter_single_of_neg {a : \u03b1} {b : M} (h : \u00acp a) : (single a b).filter\u2093 p = 0 :=\n  (filter_eq_zero_iff _ _).2 fun x hpx =>\n    single_apply_eq_zero.2 fun hxa => absurd hpx (hxa.symm \u25b8 h)\n#align finsupp.filter_single_of_neg Finsupp.filter_single_of_neg\n\n/- warning: finsupp.prod_filter_index -> Finsupp.prod_filter_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommMonoid.{u3} N] (g : \u03b1 -> M -> N), Eq.{succ u3} N (Finsupp.prod.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f) g) (Finset.prod.{u3, u1} N \u03b1 _inst_2 (Finsupp.support.{u1, u2} \u03b1 M _inst_1 (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f)) (fun (x : \u03b1) => g x (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f x)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {N : Type.{u3}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommMonoid.{u3} N] (g : \u03b1 -> M -> N), Eq.{succ u3} N (Finsupp.prod.{u2, u1, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f) g) (Finset.prod.{u3, u2} N \u03b1 _inst_2 (Finsupp.support.{u2, u1} \u03b1 M _inst_1 (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f)) (fun (x : \u03b1) => g x (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f x)))\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_filter_index Finsupp.prod_filter_index\u2093'. -/\n@[to_additive]\ntheorem prod_filter_index [CommMonoid N] (g : \u03b1 \u2192 M \u2192 N) :\n    (f.filter\u2093 p).Prod g = \u220f x in (f.filter\u2093 p).support, g x (f x) := by\n  classical\n    refine' Finset.prod_congr rfl fun x hx => _\n    rw [support_filter, Finset.mem_filter] at hx\n    rw [filter_apply_pos _ _ hx.2]\n#align finsupp.prod_filter_index Finsupp.prod_filter_index\n#align finsupp.sum_filter_index Finsupp.sum_filter_index\n\n/- warning: finsupp.prod_filter_mul_prod_filter_not -> Finsupp.prod_filter_mul_prod_filter_not is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommMonoid.{u3} N] (g : \u03b1 -> M -> N), Eq.{succ u3} N (HMul.hMul.{u3, u3, u3} N N N (instHMul.{u3} N (MulOneClass.toHasMul.{u3} N (Monoid.toMulOneClass.{u3} N (CommMonoid.toMonoid.{u3} N _inst_2)))) (Finsupp.prod.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f) g) (Finsupp.prod.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 (fun (a : \u03b1) => Not (p a)) f) g)) (Finsupp.prod.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 f g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {N : Type.{u3}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommMonoid.{u3} N] (g : \u03b1 -> M -> N), Eq.{succ u3} N (HMul.hMul.{u3, u3, u3} N N N (instHMul.{u3} N (MulOneClass.toMul.{u3} N (Monoid.toMulOneClass.{u3} N (CommMonoid.toMonoid.{u3} N _inst_2)))) (Finsupp.prod.{u2, u1, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f) g) (Finsupp.prod.{u2, u1, u3} \u03b1 M N _inst_1 _inst_2 (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 (fun (a : \u03b1) => Not (p a)) f) g)) (Finsupp.prod.{u2, u1, u3} \u03b1 M N _inst_1 _inst_2 f g)\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_filter_mul_prod_filter_not Finsupp.prod_filter_mul_prod_filter_not\u2093'. -/\n@[simp, to_additive]\ntheorem prod_filter_mul_prod_filter_not [CommMonoid N] (g : \u03b1 \u2192 M \u2192 N) :\n    (f.filter\u2093 p).Prod g * (f.filter\u2093 fun a => \u00acp a).Prod g = f.Prod g := by\n  classical simp_rw [prod_filter_index, support_filter, prod_filter_mul_prod_filter_not,\n      Finsupp.prod]\n#align finsupp.prod_filter_mul_prod_filter_not Finsupp.prod_filter_mul_prod_filter_not\n#align finsupp.sum_filter_add_sum_filter_not Finsupp.sum_filter_add_sum_filter_not\n\n/- warning: finsupp.prod_div_prod_filter -> Finsupp.prod_div_prod_filter is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Zero.{u2} M] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 M _inst_1) [_inst_2 : CommGroup.{u3} G] (g : \u03b1 -> M -> G), Eq.{succ u3} G (HDiv.hDiv.{u3, u3, u3} G G G (instHDiv.{u3} G (DivInvMonoid.toHasDiv.{u3} G (Group.toDivInvMonoid.{u3} G (CommGroup.toGroup.{u3} G _inst_2)))) (Finsupp.prod.{u1, u2, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) f g) (Finsupp.prod.{u1, u2, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 p f) g)) (Finsupp.prod.{u1, u2, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) (Finsupp.filter.{u1, u2} \u03b1 M _inst_1 (fun (a : \u03b1) => Not (p a)) f) g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {G : Type.{u3}} [_inst_1 : Zero.{u1} M] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 M _inst_1) [_inst_2 : CommGroup.{u3} G] (g : \u03b1 -> M -> G), Eq.{succ u3} G (HDiv.hDiv.{u3, u3, u3} G G G (instHDiv.{u3} G (DivInvMonoid.toDiv.{u3} G (Group.toDivInvMonoid.{u3} G (CommGroup.toGroup.{u3} G _inst_2)))) (Finsupp.prod.{u2, u1, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) f g) (Finsupp.prod.{u2, u1, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 p f) g)) (Finsupp.prod.{u2, u1, u3} \u03b1 M G _inst_1 (CommGroup.toCommMonoid.{u3} G _inst_2) (Finsupp.filter.{u2, u1} \u03b1 M _inst_1 (fun (a : \u03b1) => Not (p a)) f) g)\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_div_prod_filter Finsupp.prod_div_prod_filter\u2093'. -/\n@[simp, to_additive]\ntheorem prod_div_prod_filter [CommGroup G] (g : \u03b1 \u2192 M \u2192 G) :\n    f.Prod g / (f.filter\u2093 p).Prod g = (f.filter\u2093 fun a => \u00acp a).Prod g :=\n  div_eq_of_eq_mul' (prod_filter_mul_prod_filter_not _ _ _).symm\n#align finsupp.prod_div_prod_filter Finsupp.prod_div_prod_filter\n#align finsupp.sum_sub_sum_filter Finsupp.sum_sub_sum_filter\n\nend Zero\n\n/- warning: finsupp.filter_pos_add_filter_neg -> Finsupp.filter_pos_add_filter_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (p : \u03b1 -> Prop), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p f) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) (fun (a : \u03b1) => Not (p a)) f)) f\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (p : \u03b1 -> Prop), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1) p f) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1) (fun (a : \u03b1) => Not (p a)) f)) f\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_pos_add_filter_neg Finsupp.filter_pos_add_filter_neg\u2093'. -/\ntheorem filter_pos_add_filter_neg [AddZeroClass M] (f : \u03b1 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n    (f.filter\u2093 p + f.filter\u2093 fun a => \u00acp a) = f :=\n  coeFn_injective <| Set.indicator_self_add_compl { x | p x } f\n#align finsupp.filter_pos_add_filter_neg Finsupp.filter_pos_add_filter_neg\n\nend Filter\n\n/-! ### Declarations about `frange` -/\n\n\nsection Frange\n\nvariable [Zero M]\n\n#print Finsupp.frange /-\n/-- `frange f` is the image of `f` on the support of `f`. -/\ndef frange (f : \u03b1 \u2192\u2080 M) : Finset M :=\n  haveI := Classical.decEq M\n  Finset.image f f.support\n#align finsupp.frange Finsupp.frange\n-/\n\n/- warning: finsupp.mem_frange -> Finsupp.mem_frange is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {f : Finsupp.{u1, u2} \u03b1 M _inst_1} {y : M}, Iff (Membership.Mem.{u2, u2} M (Finset.{u2} M) (Finset.hasMem.{u2} M) y (Finsupp.frange.{u1, u2} \u03b1 M _inst_1 f)) (And (Ne.{succ u2} M y (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))) (Exists.{succ u1} \u03b1 (fun (x : \u03b1) => Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f x) y)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {f : Finsupp.{u2, u1} \u03b1 M _inst_1} {y : M}, Iff (Membership.mem.{u1, u1} M (Finset.{u1} M) (Finset.instMembershipFinset.{u1} M) y (Finsupp.frange.{u2, u1} \u03b1 M _inst_1 f)) (And (Ne.{succ u1} M y (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1))) (Exists.{succ u2} \u03b1 (fun (x : \u03b1) => Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f x) y)))\nCase conversion may be inaccurate. Consider using '#align finsupp.mem_frange Finsupp.mem_frange\u2093'. -/\ntheorem mem_frange {f : \u03b1 \u2192\u2080 M} {y : M} : y \u2208 f.frange \u2194 y \u2260 0 \u2227 \u2203 x, f x = y := by\n  classical exact\n      finset.mem_image.trans\n        \u27e8fun \u27e8x, hx1, hx2\u27e9 => \u27e8hx2 \u25b8 mem_support_iff.1 hx1, x, hx2\u27e9, fun \u27e8hy, x, hx\u27e9 =>\n          \u27e8x, mem_support_iff.2 (hx.symm \u25b8 hy), hx\u27e9\u27e9\n#align finsupp.mem_frange Finsupp.mem_frange\n\n/- warning: finsupp.zero_not_mem_frange -> Finsupp.zero_not_mem_frange is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Not (Membership.Mem.{u2, u2} M (Finset.{u2} M) (Finset.hasMem.{u2} M) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1))) (Finsupp.frange.{u1, u2} \u03b1 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Not (Membership.mem.{u1, u1} M (Finset.{u1} M) (Finset.instMembershipFinset.{u1} M) (OfNat.ofNat.{u1} M 0 (Zero.toOfNat0.{u1} M _inst_1)) (Finsupp.frange.{u2, u1} \u03b1 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.zero_not_mem_frange Finsupp.zero_not_mem_frange\u2093'. -/\ntheorem zero_not_mem_frange {f : \u03b1 \u2192\u2080 M} : (0 : M) \u2209 f.frange := fun H => (mem_frange.1 H).1 rfl\n#align finsupp.zero_not_mem_frange Finsupp.zero_not_mem_frange\n\n#print Finsupp.frange_single /-\ntheorem frange_single {x : \u03b1} {y : M} : frange (single x y) \u2286 {y} := fun r hr =>\n  let \u27e8t, ht1, ht2\u27e9 := mem_frange.1 hr\n  ht2 \u25b8 by\n    classical\n      rw [single_apply] at ht2\u22a2\n      split_ifs  at ht2\u22a2\n      \u00b7 exact Finset.mem_singleton_self _\n      \u00b7 exact (t ht2.symm).elim\n#align finsupp.frange_single Finsupp.frange_single\n-/\n\nend Frange\n\n/-! ### Declarations about `subtype_domain` -/\n\n\nsection SubtypeDomain\n\nsection Zero\n\nvariable [Zero M] {p : \u03b1 \u2192 Prop}\n\n#print Finsupp.subtypeDomain /-\n/--\n`subtype_domain p f` is the restriction of the finitely supported function `f` to subtype `p`. -/\ndef subtypeDomain (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 M) : Subtype p \u2192\u2080 M\n    where\n  support :=\n    haveI := Classical.decPred p\n    f.support.subtype p\n  toFun := f \u2218 coe\n  mem_support_toFun a := by simp only [mem_subtype, mem_support_iff]\n#align finsupp.subtype_domain Finsupp.subtypeDomain\n-/\n\n/- warning: finsupp.support_subtype_domain -> Finsupp.support_subtypeDomain is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop} [D : DecidablePred.{succ u1} \u03b1 p] {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Eq.{succ u1} (Finset.{u1} (Subtype.{succ u1} \u03b1 p)) (Finsupp.support.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1 (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p f)) (Finset.subtype.{u1} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u1, u2} \u03b1 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop} [D : DecidablePred.{succ u2} \u03b1 p] {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Eq.{succ u2} (Finset.{u2} (Subtype.{succ u2} \u03b1 p)) (Finsupp.support.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1 (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p f)) (Finset.subtype.{u2} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u2, u1} \u03b1 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.support_subtype_domain Finsupp.support_subtypeDomain\u2093'. -/\n@[simp]\ntheorem support_subtypeDomain [D : DecidablePred p] {f : \u03b1 \u2192\u2080 M} :\n    (subtypeDomain p f).support = f.support.Subtype p := by rw [Subsingleton.elim D] <;> rfl\n#align finsupp.support_subtype_domain Finsupp.support_subtypeDomain\n\n/- warning: finsupp.subtype_domain_apply -> Finsupp.subtypeDomain_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop} {a : Subtype.{succ u1} \u03b1 p} {v : Finsupp.{u1, u2} \u03b1 M _inst_1}, Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (fun (_x : Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) => (Subtype.{succ u1} \u03b1 p) -> M) (Finsupp.coeFun.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p v) a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) v (Subtype.val.{succ u1} \u03b1 p a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop} {a : Subtype.{succ u2} \u03b1 p} {v : Finsupp.{u2, u1} \u03b1 M _inst_1}, Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Subtype.{succ u2} \u03b1 p) => M) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Subtype.{succ u2} \u03b1 p) (fun (_x : Subtype.{succ u2} \u03b1 p) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Subtype.{succ u2} \u03b1 p) => M) _x) (Finsupp.funLike.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p v) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) v (Subtype.val.{succ u2} \u03b1 p a))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_apply Finsupp.subtypeDomain_apply\u2093'. -/\n@[simp]\ntheorem subtypeDomain_apply {a : Subtype p} {v : \u03b1 \u2192\u2080 M} : (subtypeDomain p v) a = v a.val :=\n  rfl\n#align finsupp.subtype_domain_apply Finsupp.subtypeDomain_apply\n\n/- warning: finsupp.subtype_domain_zero -> Finsupp.subtypeDomain_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.zero.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.zero.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_zero Finsupp.subtypeDomain_zero\u2093'. -/\n@[simp]\ntheorem subtypeDomain_zero : subtypeDomain p (0 : \u03b1 \u2192\u2080 M) = 0 :=\n  rfl\n#align finsupp.subtype_domain_zero Finsupp.subtypeDomain_zero\n\n/- warning: finsupp.subtype_domain_eq_zero_iff' -> Finsupp.subtypeDomain_eq_zero_iff' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop} {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, Iff (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.zero.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1))))) (forall (x : \u03b1), (p x) -> (Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop} {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, Iff (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.zero.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1)))) (forall (x : \u03b1), (p x) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M _inst_1) f x) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) 0 (Zero.toOfNat0.{u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_eq_zero_iff' Finsupp.subtypeDomain_eq_zero_iff'\u2093'. -/\ntheorem subtypeDomain_eq_zero_iff' {f : \u03b1 \u2192\u2080 M} : f.subtypeDomain p = 0 \u2194 \u2200 x, p x \u2192 f x = 0 := by\n  classical simp_rw [\u2190 support_eq_empty, support_subtype_domain, subtype_eq_empty,\n      not_mem_support_iff]\n#align finsupp.subtype_domain_eq_zero_iff' Finsupp.subtypeDomain_eq_zero_iff'\n\n/- warning: finsupp.subtype_domain_eq_zero_iff -> Finsupp.subtypeDomain_eq_zero_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop} {f : Finsupp.{u1, u2} \u03b1 M _inst_1}, (forall (x : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x (Finsupp.support.{u1, u2} \u03b1 M _inst_1 f)) -> (p x)) -> (Iff (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1) (Finsupp.zero.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1))))) (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) f (OfNat.ofNat.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (OfNat.mk.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) 0 (Zero.zero.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M _inst_1) (Finsupp.zero.{u1, u2} \u03b1 M _inst_1))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop} {f : Finsupp.{u2, u1} \u03b1 M _inst_1}, (forall (x : \u03b1), (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x (Finsupp.support.{u2, u1} \u03b1 M _inst_1 f)) -> (p x)) -> (Iff (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p f) (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) (Finsupp.zero.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1)))) (Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M _inst_1) f (OfNat.ofNat.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M _inst_1) (Finsupp.zero.{u2, u1} \u03b1 M _inst_1)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_eq_zero_iff Finsupp.subtypeDomain_eq_zero_iff\u2093'. -/\ntheorem subtypeDomain_eq_zero_iff {f : \u03b1 \u2192\u2080 M} (hf : \u2200 x \u2208 f.support, p x) :\n    f.subtypeDomain p = 0 \u2194 f = 0 :=\n  subtypeDomain_eq_zero_iff'.trans\n    \u27e8fun H =>\n      ext fun x => by\n        classical exact if hx : p x then H x hx else not_mem_support_iff.1 <| mt (hf x) hx,\n      fun H x _ => by simp [H]\u27e9\n#align finsupp.subtype_domain_eq_zero_iff Finsupp.subtypeDomain_eq_zero_iff\n\n/- warning: finsupp.prod_subtype_domain_index -> Finsupp.prod_subtypeDomain_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} [_inst_1 : Zero.{u2} M] {p : \u03b1 -> Prop} [_inst_2 : CommMonoid.{u3} N] {v : Finsupp.{u1, u2} \u03b1 M _inst_1} {h : \u03b1 -> M -> N}, (forall (x : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) x (Finsupp.support.{u1, u2} \u03b1 M _inst_1 v)) -> (p x)) -> (Eq.{succ u3} N (Finsupp.prod.{u1, u2, u3} (Subtype.{succ u1} \u03b1 p) M N _inst_1 _inst_2 (Finsupp.subtypeDomain.{u1, u2} \u03b1 M _inst_1 p v) (fun (a : Subtype.{succ u1} \u03b1 p) (b : M) => h ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subtype.{succ u1} \u03b1 p) \u03b1 (HasLiftT.mk.{succ u1, succ u1} (Subtype.{succ u1} \u03b1 p) \u03b1 (CoeTC\u2093.coe.{succ u1, succ u1} (Subtype.{succ u1} \u03b1 p) \u03b1 (coeBase.{succ u1, succ u1} (Subtype.{succ u1} \u03b1 p) \u03b1 (coeSubtype.{succ u1} \u03b1 (fun (x : \u03b1) => p x))))) a) b)) (Finsupp.prod.{u1, u2, u3} \u03b1 M N _inst_1 _inst_2 v h))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} {N : Type.{u3}} [_inst_1 : Zero.{u1} M] {p : \u03b1 -> Prop} [_inst_2 : CommMonoid.{u3} N] {v : Finsupp.{u2, u1} \u03b1 M _inst_1} {h : \u03b1 -> M -> N}, (forall (x : \u03b1), (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) x (Finsupp.support.{u2, u1} \u03b1 M _inst_1 v)) -> (p x)) -> (Eq.{succ u3} N (Finsupp.prod.{u2, u1, u3} (Subtype.{succ u2} \u03b1 p) M N _inst_1 _inst_2 (Finsupp.subtypeDomain.{u2, u1} \u03b1 M _inst_1 p v) (fun (a : Subtype.{succ u2} \u03b1 p) (b : M) => h (Subtype.val.{succ u2} \u03b1 p a) b)) (Finsupp.prod.{u2, u1, u3} \u03b1 M N _inst_1 _inst_2 v h))\nCase conversion may be inaccurate. Consider using '#align finsupp.prod_subtype_domain_index Finsupp.prod_subtypeDomain_index\u2093'. -/\n@[to_additive]\ntheorem prod_subtypeDomain_index [CommMonoid N] {v : \u03b1 \u2192\u2080 M} {h : \u03b1 \u2192 M \u2192 N}\n    (hp : \u2200 x \u2208 v.support, p x) : ((v.subtypeDomain p).Prod fun a b => h a b) = v.Prod h :=\n  prod_bij (fun p _ => p.val) (fun _ => by classical exact mem_subtype.1) (fun _ _ => rfl)\n    (fun _ _ _ _ => Subtype.eq) fun b hb => \u27e8\u27e8b, hp b hb\u27e9, by classical exact mem_subtype.2 hb, rfl\u27e9\n#align finsupp.prod_subtype_domain_index Finsupp.prod_subtypeDomain_index\n#align finsupp.sum_subtype_domain_index Finsupp.sum_subtypeDomain_index\n\nend Zero\n\nsection AddZeroClass\n\nvariable [AddZeroClass M] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 M}\n\n/- warning: finsupp.subtype_domain_add -> Finsupp.subtypeDomain_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] {p : \u03b1 -> Prop} {v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)} {v' : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) v v')) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1)) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p v) (Finsupp.subtypeDomain.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p v'))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddZeroClass.{u1} M] {p : \u03b1 -> Prop} {v : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)} {v' : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b1 M _inst_1)) v v')) (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1)) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p v) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p v'))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_add Finsupp.subtypeDomain_add\u2093'. -/\n@[simp]\ntheorem subtypeDomain_add {v v' : \u03b1 \u2192\u2080 M} :\n    (v + v').subtypeDomain p = v.subtypeDomain p + v'.subtypeDomain p :=\n  ext fun _ => rfl\n#align finsupp.subtype_domain_add Finsupp.subtypeDomain_add\n\n/- warning: finsupp.subtype_domain_add_monoid_hom -> Finsupp.subtypeDomainAddMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] {p : \u03b1 -> Prop}, AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1) (Finsupp.addZeroClass.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] {p : \u03b1 -> Prop}, AddMonoidHom.{max u2 u1, max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1) (Finsupp.addZeroClass.{u1, u2} (Subtype.{succ u1} \u03b1 p) M _inst_1)\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_add_monoid_hom Finsupp.subtypeDomainAddMonoidHom\u2093'. -/\n/-- `subtype_domain` but as an `add_monoid_hom`. -/\ndef subtypeDomainAddMonoidHom : (\u03b1 \u2192\u2080 M) \u2192+ Subtype p \u2192\u2080 M\n    where\n  toFun := subtypeDomain p\n  map_zero' := subtypeDomain_zero\n  map_add' _ _ := subtypeDomain_add\n#align finsupp.subtype_domain_add_monoid_hom Finsupp.subtypeDomainAddMonoidHom\n\n/- warning: finsupp.filter_add_hom -> Finsupp.filterAddHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M], (\u03b1 -> Prop) -> (AddMonoidHom.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M], (\u03b1 -> Prop) -> (AddMonoidHom.{max u2 u1, max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toZero.{u2} M _inst_1)) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1) (Finsupp.addZeroClass.{u1, u2} \u03b1 M _inst_1))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_add_hom Finsupp.filterAddHom\u2093'. -/\n/-- `finsupp.filter` as an `add_monoid_hom`. -/\ndef filterAddHom (p : \u03b1 \u2192 Prop) : (\u03b1 \u2192\u2080 M) \u2192+ \u03b1 \u2192\u2080 M\n    where\n  toFun := filter p\n  map_zero' := filter_zero p\n  map_add' f g := coeFn_injective <| Set.indicator_add { x | p x } f g\n#align finsupp.filter_add_hom Finsupp.filterAddHom\n\n/- warning: finsupp.filter_add -> Finsupp.filter_add is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddZeroClass.{u2} M] {p : \u03b1 -> Prop} {v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)} {v' : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) v v')) (HAdd.hAdd.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (instHAdd.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1)) (Finsupp.add.{u1, u2} \u03b1 M _inst_1)) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p v) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M _inst_1) p v'))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddZeroClass.{u1} M] {p : \u03b1 -> Prop} {v : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)} {v' : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.filter.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b1 M _inst_1)) v v')) (HAdd.hAdd.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (instHAdd.{max u2 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b1 M _inst_1)) (Finsupp.filter.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p v) (Finsupp.filter.{u2, u1} \u03b1 M (AddZeroClass.toZero.{u1} M _inst_1) p v'))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_add Finsupp.filter_add\u2093'. -/\n@[simp]\ntheorem filter_add {v v' : \u03b1 \u2192\u2080 M} : (v + v').filter\u2093 p = v.filter\u2093 p + v'.filter\u2093 p :=\n  (filterAddHom p).map_add v v'\n#align finsupp.filter_add Finsupp.filter_add\n\nend AddZeroClass\n\nsection CommMonoid\n\nvariable [AddCommMonoid M] {p : \u03b1 \u2192 Prop}\n\n/- warning: finsupp.subtype_domain_sum -> Finsupp.subtypeDomain_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {p : \u03b1 -> Prop} {s : Finset.{u2} \u03b9} {h : \u03b9 -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))}, Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.subtypeDomain.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (c : \u03b9) => h c))) (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} (Subtype.{succ u1} \u03b1 p) M _inst_1) s (fun (c : \u03b9) => Finsupp.subtypeDomain.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (h c)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b9 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {p : \u03b1 -> Prop} {s : Finset.{u3} \u03b9} {h : \u03b9 -> (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (Finset.sum.{max u2 u1, u3} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u2, u1} \u03b1 M _inst_1) s (fun (c : \u03b9) => h c))) (Finset.sum.{max u2 u1, u3} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) s (fun (c : \u03b9) => Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (h c)))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_sum Finsupp.subtypeDomain_sum\u2093'. -/\ntheorem subtypeDomain_sum {s : Finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} :\n    (\u2211 c in s, h c).subtypeDomain p = \u2211 c in s, (h c).subtypeDomain p :=\n  (subtypeDomainAddMonoidHom : _ \u2192+ Subtype p \u2192\u2080 M).map_sum _ s\n#align finsupp.subtype_domain_sum Finsupp.subtypeDomain_sum\n\n/- warning: finsupp.subtype_domain_finsupp_sum -> Finsupp.subtypeDomain_finsupp_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] {p : \u03b1 -> Prop} [_inst_2 : Zero.{u4} N] {s : Finsupp.{u2, u4} \u03b2 N _inst_2} {h : \u03b2 -> N -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))}, Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.subtypeDomain.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (Finsupp.sum.{u2, u4, max u1 u3} \u03b2 N (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) _inst_2 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s h)) (Finsupp.sum.{u2, u4, max u1 u3} \u03b2 N (Finsupp.{u1, u3} (Subtype.{succ u1} \u03b1 p) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) _inst_2 (Finsupp.addCommMonoid.{u1, u3} (Subtype.{succ u1} \u03b1 p) M _inst_1) s (fun (c : \u03b2) (d : N) => Finsupp.subtypeDomain.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (h c d)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u1} M] {p : \u03b1 -> Prop} [_inst_2 : Zero.{u4} N] {s : Finsupp.{u3, u4} \u03b2 N _inst_2} {h : \u03b2 -> N -> (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (Finsupp.sum.{u3, u4, max u2 u1} \u03b2 N (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) _inst_2 (Finsupp.addCommMonoid.{u2, u1} \u03b1 M _inst_1) s h)) (Finsupp.sum.{u3, u4, max u2 u1} \u03b2 N (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) _inst_2 (Finsupp.addCommMonoid.{u2, u1} (Subtype.{succ u2} \u03b1 p) M _inst_1) s (fun (c : \u03b2) (d : N) => Finsupp.subtypeDomain.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (h c d)))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_finsupp_sum Finsupp.subtypeDomain_finsupp_sum\u2093'. -/\ntheorem subtypeDomain_finsupp_sum [Zero N] {s : \u03b2 \u2192\u2080 N} {h : \u03b2 \u2192 N \u2192 \u03b1 \u2192\u2080 M} :\n    (s.Sum h).subtypeDomain p = s.Sum fun c d => (h c d).subtypeDomain p :=\n  subtypeDomain_sum\n#align finsupp.subtype_domain_finsupp_sum Finsupp.subtypeDomain_finsupp_sum\n\n/- warning: finsupp.filter_sum -> Finsupp.filter_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {p : \u03b1 -> Prop} (s : Finset.{u2} \u03b9) (f : \u03b9 -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.filter.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (a : \u03b9) => f a))) (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (a : \u03b9) => Finsupp.filter.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) p (f a)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b9 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] {p : \u03b1 -> Prop} (s : Finset.{u3} \u03b9) (f : \u03b9 -> (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.filter.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (Finset.sum.{max u2 u1, u3} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u2, u1} \u03b1 M _inst_1) s (fun (a : \u03b9) => f a))) (Finset.sum.{max u1 u2, u3} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u2, u1} \u03b1 M _inst_1) s (fun (a : \u03b9) => Finsupp.filter.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p (f a)))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_sum Finsupp.filter_sum\u2093'. -/\ntheorem filter_sum (s : Finset \u03b9) (f : \u03b9 \u2192 \u03b1 \u2192\u2080 M) :\n    (\u2211 a in s, f a).filter\u2093 p = \u2211 a in s, filter p (f a) :=\n  (filterAddHom p : (\u03b1 \u2192\u2080 M) \u2192+ _).map_sum f s\n#align finsupp.filter_sum Finsupp.filter_sum\n\n/- warning: finsupp.filter_eq_sum -> Finsupp.filter_eq_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] (p : \u03b1 -> Prop) [D : DecidablePred.{succ u1} \u03b1 p] (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) p f) (Finset.sum.{max u1 u2, u1} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) \u03b1 (Finsupp.addCommMonoid.{u1, u2} \u03b1 M _inst_1) (Finset.filter.{u1} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) f)) (fun (i : \u03b1) => Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) i (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) f i)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (p : \u03b1 -> Prop) [D : DecidablePred.{succ u2} \u03b1 p] (f : Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.filter.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) p f) (Finset.sum.{max u1 u2, u2} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b1 (Finsupp.addCommMonoid.{u2, u1} \u03b1 M _inst_1) (Finset.filter.{u2} \u03b1 p (fun (a : \u03b1) => D a) (Finsupp.support.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) f)) (fun (i : \u03b1) => Finsupp.single.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) i (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) f i)))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_eq_sum Finsupp.filter_eq_sum\u2093'. -/\ntheorem filter_eq_sum (p : \u03b1 \u2192 Prop) [D : DecidablePred p] (f : \u03b1 \u2192\u2080 M) :\n    f.filter\u2093 p = \u2211 i in f.support.filter\u2093 p, single i (f i) :=\n  (f.filter\u2093 p).sum_single.symm.trans <|\n    Finset.sum_congr (by rw [Subsingleton.elim D] <;> rfl) fun x hx => by\n      rw [filter_apply_pos _ _ (mem_filter.1 hx).2]\n#align finsupp.filter_eq_sum Finsupp.filter_eq_sum\n\nend CommMonoid\n\nsection Group\n\nvariable [AddGroup G] {p : \u03b1 \u2192 Prop} {v v' : \u03b1 \u2192\u2080 G}\n\n/- warning: finsupp.subtype_domain_neg -> Finsupp.subtypeDomain_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] {p : \u03b1 -> Prop} {v : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.subtypeDomain.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) v)) (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.subtypeDomain.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] {p : \u03b1 -> Prop} {v : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) v)) (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p v))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_neg Finsupp.subtypeDomain_neg\u2093'. -/\n@[simp]\ntheorem subtypeDomain_neg : (-v).subtypeDomain p = -v.subtypeDomain p :=\n  ext fun _ => rfl\n#align finsupp.subtype_domain_neg Finsupp.subtypeDomain_neg\n\n/- warning: finsupp.subtype_domain_sub -> Finsupp.subtypeDomain_sub is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] {p : \u03b1 -> Prop} {v : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))} {v' : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.subtypeDomain.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) v v')) (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} (Subtype.{succ u1} \u03b1 p) G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.subtypeDomain.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p v) (Finsupp.subtypeDomain.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p v'))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] {p : \u03b1 -> Prop} {v : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))} {v' : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))}, Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) v v')) (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} (Subtype.{succ u2} \u03b1 p) G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.subtypeDomain.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p v) (Finsupp.subtypeDomain.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p v'))\nCase conversion may be inaccurate. Consider using '#align finsupp.subtype_domain_sub Finsupp.subtypeDomain_sub\u2093'. -/\n@[simp]\ntheorem subtypeDomain_sub : (v - v').subtypeDomain p = v.subtypeDomain p - v'.subtypeDomain p :=\n  ext fun _ => rfl\n#align finsupp.subtype_domain_sub Finsupp.subtypeDomain_sub\n\n/- warning: finsupp.single_neg -> Finsupp.single_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (a : \u03b1) (b : G), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.single.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a (Neg.neg.{u2} G (SubNegMonoid.toHasNeg.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)) b)) (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.single.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (a : \u03b1) (b : G), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.single.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a (Neg.neg.{u1} G (NegZeroClass.toNeg.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) b)) (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.single.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a b))\nCase conversion may be inaccurate. Consider using '#align finsupp.single_neg Finsupp.single_neg\u2093'. -/\n@[simp]\ntheorem single_neg (a : \u03b1) (b : G) : single a (-b) = -single a b :=\n  (singleAddHom a : G \u2192+ _).map_neg b\n#align finsupp.single_neg Finsupp.single_neg\n\n/- warning: finsupp.single_sub -> Finsupp.single_sub is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (a : \u03b1) (b\u2081 : G) (b\u2082 : G), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.single.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a (HSub.hSub.{u2, u2, u2} G G G (instHSub.{u2} G (SubNegMonoid.toHasSub.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))) b\u2081 b\u2082)) (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.single.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a b\u2081) (Finsupp.single.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a b\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (a : \u03b1) (b\u2081 : G) (b\u2082 : G), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.single.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a (HSub.hSub.{u1, u1, u1} G G G (instHSub.{u1} G (SubNegMonoid.toSub.{u1} G (AddGroup.toSubNegMonoid.{u1} G _inst_1))) b\u2081 b\u2082)) (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.single.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a b\u2081) (Finsupp.single.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a b\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.single_sub Finsupp.single_sub\u2093'. -/\n@[simp]\ntheorem single_sub (a : \u03b1) (b\u2081 b\u2082 : G) : single a (b\u2081 - b\u2082) = single a b\u2081 - single a b\u2082 :=\n  (singleAddHom a : G \u2192+ _).map_sub b\u2081 b\u2082\n#align finsupp.single_sub Finsupp.single_sub\n\n/- warning: finsupp.erase_neg -> Finsupp.erase_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (a : \u03b1) (f : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.erase.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) f)) (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.erase.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (a : \u03b1) (f : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.erase.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) f)) (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.erase.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a f))\nCase conversion may be inaccurate. Consider using '#align finsupp.erase_neg Finsupp.erase_neg\u2093'. -/\n@[simp]\ntheorem erase_neg (a : \u03b1) (f : \u03b1 \u2192\u2080 G) : erase a (-f) = -erase a f :=\n  (eraseAddHom a : (_ \u2192\u2080 G) \u2192+ _).map_neg f\n#align finsupp.erase_neg Finsupp.erase_neg\n\n/- warning: finsupp.erase_sub -> Finsupp.erase_sub is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (a : \u03b1) (f\u2081 : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (f\u2082 : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.erase.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) f\u2081 f\u2082)) (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.erase.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a f\u2081) (Finsupp.erase.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) a f\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (a : \u03b1) (f\u2081 : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (f\u2082 : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.erase.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) f\u2081 f\u2082)) (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.erase.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a f\u2081) (Finsupp.erase.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) a f\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.erase_sub Finsupp.erase_sub\u2093'. -/\n@[simp]\ntheorem erase_sub (a : \u03b1) (f\u2081 f\u2082 : \u03b1 \u2192\u2080 G) : erase a (f\u2081 - f\u2082) = erase a f\u2081 - erase a f\u2082 :=\n  (eraseAddHom a : (_ \u2192\u2080 G) \u2192+ _).map_sub f\u2081 f\u2082\n#align finsupp.erase_sub Finsupp.erase_sub\n\n/- warning: finsupp.filter_neg -> Finsupp.filter_neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (p : \u03b1 -> Prop) (f : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.filter.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) f)) (Neg.neg.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.neg.{u1, u2} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u2} G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.filter.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (p : \u03b1 -> Prop) (f : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.filter.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) f)) (Neg.neg.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.neg.{u2, u1} \u03b1 G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.filter.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p f))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_neg Finsupp.filter_neg\u2093'. -/\n@[simp]\ntheorem filter_neg (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192\u2080 G) : filter p (-f) = -filter p f :=\n  (filterAddHom p : (_ \u2192\u2080 G) \u2192+ _).map_neg f\n#align finsupp.filter_neg Finsupp.filter_neg\n\n/- warning: finsupp.filter_sub -> Finsupp.filter_sub is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {G : Type.{u2}} [_inst_1 : AddGroup.{u2} G] (p : \u03b1 -> Prop) (f\u2081 : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (f\u2082 : Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.filter.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) f\u2081 f\u2082)) (HSub.hSub.{max u1 u2, max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (instHSub.{max u1 u2} (Finsupp.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1))))) (Finsupp.sub.{u1, u2} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u2} G (AddGroup.toSubtractionMonoid.{u2} G _inst_1)))) (Finsupp.filter.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p f\u2081) (Finsupp.filter.{u1, u2} \u03b1 G (AddZeroClass.toHasZero.{u2} G (AddMonoid.toAddZeroClass.{u2} G (SubNegMonoid.toAddMonoid.{u2} G (AddGroup.toSubNegMonoid.{u2} G _inst_1)))) p f\u2082))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {G : Type.{u1}} [_inst_1 : AddGroup.{u1} G] (p : \u03b1 -> Prop) (f\u2081 : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (f\u2082 : Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))), Eq.{max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.filter.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) f\u2081 f\u2082)) (HSub.hSub.{max u2 u1, max u2 u1, max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (instHSub.{max u2 u1} (Finsupp.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1))))) (Finsupp.sub.{u2, u1} \u03b1 G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) (Finsupp.filter.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p f\u2081) (Finsupp.filter.{u2, u1} \u03b1 G (NegZeroClass.toZero.{u1} G (SubNegZeroMonoid.toNegZeroClass.{u1} G (SubtractionMonoid.toSubNegZeroMonoid.{u1} G (AddGroup.toSubtractionMonoid.{u1} G _inst_1)))) p f\u2082))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_sub Finsupp.filter_sub\u2093'. -/\n@[simp]\ntheorem filter_sub (p : \u03b1 \u2192 Prop) (f\u2081 f\u2082 : \u03b1 \u2192\u2080 G) :\n    filter p (f\u2081 - f\u2082) = filter p f\u2081 - filter p f\u2082 :=\n  (filterAddHom p : (_ \u2192\u2080 G) \u2192+ _).map_sub f\u2081 f\u2082\n#align finsupp.filter_sub Finsupp.filter_sub\n\nend Group\n\nend SubtypeDomain\n\n/- warning: finsupp.mem_support_multiset_sum -> Finsupp.mem_support_multiset_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {s : Multiset.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))} (a : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Multiset.sum.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.addCommMonoid.{u1, u2} \u03b1 M _inst_1) s))) -> (Exists.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (fun (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) => Exists.{0} (Membership.Mem.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Multiset.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (Multiset.hasMem.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) f s) (fun (H : Membership.Mem.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Multiset.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (Multiset.hasMem.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) f s) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) f))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M] {s : Multiset.{max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))} (a : \u03b1), (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) (Multiset.sum.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.addCommMonoid.{u1, u2} \u03b1 M _inst_1) s))) -> (Exists.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (fun (f : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) => And (Membership.mem.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Multiset.{max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Multiset.instMembershipMultiset.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) f s) (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) f))))\nCase conversion may be inaccurate. Consider using '#align finsupp.mem_support_multiset_sum Finsupp.mem_support_multiset_sum\u2093'. -/\ntheorem mem_support_multiset_sum [AddCommMonoid M] {s : Multiset (\u03b1 \u2192\u2080 M)} (a : \u03b1) :\n    a \u2208 s.Sum.support \u2192 \u2203 f \u2208 s, a \u2208 (f : \u03b1 \u2192\u2080 M).support :=\n  Multiset.induction_on s False.elim\n    (by\n      intro f s ih ha\n      by_cases a \u2208 f.support\n      \u00b7 exact \u27e8f, Multiset.mem_cons_self _ _, h\u27e9\n      \u00b7 simp only [Multiset.sum_cons, mem_support_iff, add_apply, not_mem_support_iff.1 h,\n          zero_add] at ha\n        rcases ih (mem_support_iff.2 ha) with \u27e8f', h\u2080, h\u2081\u27e9\n        exact \u27e8f', Multiset.mem_cons_of_mem h\u2080, h\u2081\u27e9)\n#align finsupp.mem_support_multiset_sum Finsupp.mem_support_multiset_sum\n\n/- warning: finsupp.mem_support_finset_sum -> Finsupp.mem_support_finset_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {s : Finset.{u2} \u03b9} {h : \u03b9 -> (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))} (a : \u03b1), (Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (c : \u03b9) => h c)))) -> (Exists.{succ u2} \u03b9 (fun (c : \u03b9) => Exists.{0} (Membership.Mem.{u2, u2} \u03b9 (Finset.{u2} \u03b9) (Finset.hasMem.{u2} \u03b9) c s) (fun (H : Membership.Mem.{u2, u2} \u03b9 (Finset.{u2} \u03b9) (Finset.hasMem.{u2} \u03b9) c s) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a (Finsupp.support.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (h c)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b9 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] {s : Finset.{u2} \u03b9} {h : \u03b9 -> (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))} (a : \u03b1), (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a (Finsupp.support.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (Finset.sum.{max u1 u3, u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) \u03b9 (Finsupp.addCommMonoid.{u1, u3} \u03b1 M _inst_1) s (fun (c : \u03b9) => h c)))) -> (Exists.{succ u2} \u03b9 (fun (c : \u03b9) => And (Membership.mem.{u2, u2} \u03b9 (Finset.{u2} \u03b9) (Finset.instMembershipFinset.{u2} \u03b9) c s) (Membership.mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.instMembershipFinset.{u1} \u03b1) a (Finsupp.support.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (h c)))))\nCase conversion may be inaccurate. Consider using '#align finsupp.mem_support_finset_sum Finsupp.mem_support_finset_sum\u2093'. -/\ntheorem mem_support_finset_sum [AddCommMonoid M] {s : Finset \u03b9} {h : \u03b9 \u2192 \u03b1 \u2192\u2080 M} (a : \u03b1)\n    (ha : a \u2208 (\u2211 c in s, h c).support) : \u2203 c \u2208 s, a \u2208 (h c).support :=\n  let \u27e8f, hf, hfa\u27e9 := mem_support_multiset_sum a ha\n  let \u27e8c, hc, Eq\u27e9 := Multiset.mem_map.1 hf\n  \u27e8c, hc, Eq.symm \u25b8 hfa\u27e9\n#align finsupp.mem_support_finset_sum Finsupp.mem_support_finset_sum\n\n/-! ### Declarations about `curry` and `uncurry` -/\n\n\nsection CurryUncurry\n\nvariable [AddCommMonoid M] [AddCommMonoid N]\n\n/- warning: finsupp.curry -> Finsupp.curry is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) -> (Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Finsupp.{max u2 u1, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) -> (Finsupp.{u1, max u3 u2} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.curry Finsupp.curry\u2093'. -/\n/-- Given a finitely supported function `f` from a product type `\u03b1 \u00d7 \u03b2` to `\u03b3`,\n`curry f` is the \"curried\" finitely supported function from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `\u03b3`. -/\nprotected def curry (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) : \u03b1 \u2192\u2080 \u03b2 \u2192\u2080 M :=\n  f.Sum fun p c => single p.1 (single p.2 c)\n#align finsupp.curry Finsupp.curry\n\n/- warning: finsupp.curry_apply -> Finsupp.curry_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (x : \u03b1) (y : \u03b2), Eq.{succ u3} M (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => \u03b2 -> M) (Finsupp.coeFun.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (coeFn.{max (succ u1) (succ (max u2 u3)), max (succ u1) (succ (max u2 u3))} (Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) (fun (_x : Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) => \u03b1 -> (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) (Finsupp.coeFun.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) (Finsupp.curry.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f) x) y) (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (fun (_x : Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => (Prod.{u1, u2} \u03b1 \u03b2) -> M) (Finsupp.coeFun.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) f (Prod.mk.{u1, u2} \u03b1 \u03b2 x y))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (f : Finsupp.{max u3 u2, u1} (Prod.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (x : \u03b1) (y : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) y) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (FunLike.coe.{max (succ u2) (succ (max u3 u1)), succ u2, succ (max u3 u1)} (Finsupp.{u2, max u3 u1} \u03b1 (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.zero.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) _x) (Finsupp.funLike.{u2, max u3 u1} \u03b1 (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.zero.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (Finsupp.curry.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 f) x) y) (FunLike.coe.{max (succ (max u2 u3)) (succ u1), succ (max u2 u3), succ u1} (Finsupp.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Prod.{u2, u3} \u03b1 \u03b2) (fun (_x : Prod.{u2, u3} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Prod.{u2, u3} \u03b1 \u03b2) => M) _x) (Finsupp.funLike.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) f (Prod.mk.{u2, u3} \u03b1 \u03b2 x y))\nCase conversion may be inaccurate. Consider using '#align finsupp.curry_apply Finsupp.curry_apply\u2093'. -/\n@[simp]\ntheorem curry_apply (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (x : \u03b1) (y : \u03b2) : f.curry x y = f (x, y) := by\n  classical\n    have : \u2200 b : \u03b1 \u00d7 \u03b2, single b.fst (single b.snd (f b)) x y = if b = (x, y) then f b else 0 :=\n      by\n      rintro \u27e8b\u2081, b\u2082\u27e9\n      simp [single_apply, ite_apply, Prod.ext_iff, ite_and]\n      split_ifs <;> simp [single_apply, *]\n    rw [Finsupp.curry, sum_apply, sum_apply, Finsupp.sum, Finset.sum_eq_single, this, if_pos rfl]\n    \u00b7 intro b hb b_ne\n      rw [this b, if_neg b_ne]\n    \u00b7 intro hxy\n      rw [this (x, y), if_pos rfl, not_mem_support_iff.mp hxy]\n#align finsupp.curry_apply Finsupp.curry_apply\n\n/- warning: finsupp.sum_curry_index -> Finsupp.sum_curry_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {N : Type.{u4}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : AddCommMonoid.{u4} N] (f : Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (g : \u03b1 -> \u03b2 -> M -> N), (forall (a : \u03b1) (b : \u03b2), Eq.{succ u4} N (g a b (OfNat.ofNat.{u3} M 0 (OfNat.mk.{u3} M 0 (Zero.zero.{u3} M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))))) (OfNat.ofNat.{u4} N 0 (OfNat.mk.{u4} N 0 (Zero.zero.{u4} N (AddZeroClass.toHasZero.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2))))))) -> (forall (a : \u03b1) (b : \u03b2) (c\u2080 : M) (c\u2081 : M), Eq.{succ u4} N (g a b (HAdd.hAdd.{u3, u3, u3} M M M (instHAdd.{u3} M (AddZeroClass.toHasAdd.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) c\u2080 c\u2081)) (HAdd.hAdd.{u4, u4, u4} N N N (instHAdd.{u4} N (AddZeroClass.toHasAdd.{u4} N (AddMonoid.toAddZeroClass.{u4} N (AddCommMonoid.toAddMonoid.{u4} N _inst_2)))) (g a b c\u2080) (g a b c\u2081))) -> (Eq.{succ u4} N (Finsupp.sum.{u1, max u2 u3, u4} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) N (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) _inst_2 (Finsupp.curry.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f) (fun (a : \u03b1) (f : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) => Finsupp.sum.{u2, u3, u4} \u03b2 M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 f (g a))) (Finsupp.sum.{max u1 u2, u3, u4} (Prod.{u1, u2} \u03b1 \u03b2) M N (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) _inst_2 f (fun (p : Prod.{u1, u2} \u03b1 \u03b2) (c : M) => g (Prod.fst.{u1, u2} \u03b1 \u03b2 p) (Prod.snd.{u1, u2} \u03b1 \u03b2 p) c)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {M : Type.{u2}} {N : Type.{u1}} [_inst_1 : AddCommMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u1} N] (f : Finsupp.{max u4 u3, u2} (Prod.{u3, u4} \u03b1 \u03b2) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (g : \u03b1 -> \u03b2 -> M -> N), (forall (a : \u03b1) (b : \u03b2), Eq.{succ u1} N (g a b (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))) (OfNat.ofNat.{u1} N 0 (Zero.toOfNat0.{u1} N (AddMonoid.toZero.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_2))))) -> (forall (a : \u03b1) (b : \u03b2) (c\u2080 : M) (c\u2081 : M), Eq.{succ u1} N (g a b (HAdd.hAdd.{u2, u2, u2} M M M (instHAdd.{u2} M (AddZeroClass.toAdd.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) c\u2080 c\u2081)) (HAdd.hAdd.{u1, u1, u1} N N N (instHAdd.{u1} N (AddZeroClass.toAdd.{u1} N (AddMonoid.toAddZeroClass.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_2)))) (g a b c\u2080) (g a b c\u2081))) -> (Eq.{succ u1} N (Finsupp.sum.{u3, max u4 u2, u1} \u03b1 (Finsupp.{u4, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) N (Finsupp.zero.{u4, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) _inst_2 (Finsupp.curry.{u3, u4, u2} \u03b1 \u03b2 M _inst_1 f) (fun (a : \u03b1) (f : Finsupp.{u4, u2} \u03b2 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) => Finsupp.sum.{u4, u2, u1} \u03b2 M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 f (g a))) (Finsupp.sum.{max u3 u4, u2, u1} (Prod.{u3, u4} \u03b1 \u03b2) M N (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) _inst_2 f (fun (p : Prod.{u3, u4} \u03b1 \u03b2) (c : M) => g (Prod.fst.{u3, u4} \u03b1 \u03b2 p) (Prod.snd.{u3, u4} \u03b1 \u03b2 p) c)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_curry_index Finsupp.sum_curry_index\u2093'. -/\ntheorem sum_curry_index (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (g : \u03b1 \u2192 \u03b2 \u2192 M \u2192 N) (hg\u2080 : \u2200 a b, g a b 0 = 0)\n    (hg\u2081 : \u2200 a b c\u2080 c\u2081, g a b (c\u2080 + c\u2081) = g a b c\u2080 + g a b c\u2081) :\n    (f.curry.Sum fun a f => f.Sum (g a)) = f.Sum fun p c => g p.1 p.2 c :=\n  by\n  rw [Finsupp.curry]\n  trans\n  \u00b7\n    exact\n      sum_sum_index (fun a => sum_zero_index) fun a b\u2080 b\u2081 =>\n        sum_add_index' (fun a => hg\u2080 _ _) fun c d\u2080 d\u2081 => hg\u2081 _ _ _ _\n  congr ; funext p c\n  trans\n  \u00b7 exact sum_single_index sum_zero_index\n  exact sum_single_index (hg\u2080 _ _)\n#align finsupp.sum_curry_index Finsupp.sum_curry_index\n\n/- warning: finsupp.uncurry -> Finsupp.uncurry is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) -> (Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Finsupp.{u1, max u3 u2} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) -> (Finsupp.{max u2 u1, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.uncurry Finsupp.uncurry\u2093'. -/\n/-- Given a finitely supported function `f` from `\u03b1` to the type of\nfinitely supported functions from `\u03b2` to `M`,\n`uncurry f` is the \"uncurried\" finitely supported function from `\u03b1 \u00d7 \u03b2` to `M`. -/\nprotected def uncurry (f : \u03b1 \u2192\u2080 \u03b2 \u2192\u2080 M) : \u03b1 \u00d7 \u03b2 \u2192\u2080 M :=\n  f.Sum fun a g => g.Sum fun b c => single (a, b) c\n#align finsupp.uncurry Finsupp.uncurry\n\n/- warning: finsupp.finsupp_prod_equiv -> Finsupp.finsuppProdEquiv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], Equiv.{max (succ (max u1 u2)) (succ u3), max (succ u1) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], Equiv.{max (succ u3) (succ (max u2 u1)), max (succ (max u3 u2)) (succ u1)} (Finsupp.{max u2 u1, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, max u3 u2} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.finsupp_prod_equiv Finsupp.finsuppProdEquiv\u2093'. -/\n/-- `finsupp_prod_equiv` defines the `equiv` between `((\u03b1 \u00d7 \u03b2) \u2192\u2080 M)` and `(\u03b1 \u2192\u2080 (\u03b2 \u2192\u2080 M))` given by\ncurrying and uncurrying. -/\ndef finsuppProdEquiv : (\u03b1 \u00d7 \u03b2 \u2192\u2080 M) \u2243 (\u03b1 \u2192\u2080 \u03b2 \u2192\u2080 M)\n    where\n  toFun := Finsupp.curry\n  invFun := Finsupp.uncurry\n  left_inv f := by\n    rw [Finsupp.uncurry, sum_curry_index]\n    \u00b7 simp_rw [Prod.mk.eta, sum_single]\n    \u00b7 intros\n      apply single_zero\n    \u00b7 intros\n      apply single_add\n  right_inv f := by\n    simp only [Finsupp.curry, Finsupp.uncurry, sum_sum_index, sum_zero_index, sum_add_index,\n      sum_single_index, single_zero, single_add, eq_self_iff_true, forall_true_iff,\n      forall\u2083_true_iff, Prod.mk.eta, (single_sum _ _ _).symm, sum_single]\n#align finsupp.finsupp_prod_equiv Finsupp.finsuppProdEquiv\n\n/- warning: finsupp.filter_curry -> Finsupp.filter_curry is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (f : Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (p : \u03b1 -> Prop), Eq.{max (succ u1) (succ (max u2 u3))} (Finsupp.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))) (Finsupp.curry.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 (Finsupp.filter.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (fun (a : Prod.{u1, u2} \u03b1 \u03b2) => p (Prod.fst.{u1, u2} \u03b1 \u03b2 a)) f)) (Finsupp.filter.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) p (Finsupp.curry.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] (f : Finsupp.{max u3 u2, u1} (Prod.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (p : \u03b1 -> Prop), Eq.{max (max (succ u2) (succ u3)) (succ u1)} (Finsupp.{u2, max u1 u3} \u03b1 (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.zero.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)))) (Finsupp.curry.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 (Finsupp.filter.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) (fun (a : Prod.{u2, u3} \u03b1 \u03b2) => p (Prod.fst.{u2, u3} \u03b1 \u03b2 a)) f)) (Finsupp.filter.{u2, max u3 u1} \u03b1 (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.zero.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) p (Finsupp.curry.{u2, u3, u1} \u03b1 \u03b2 M _inst_1 f))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_curry Finsupp.filter_curry\u2093'. -/\ntheorem filter_curry (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) (p : \u03b1 \u2192 Prop) :\n    (f.filter\u2093 fun a : \u03b1 \u00d7 \u03b2 => p a.1).curry = f.curry.filter\u2093 p := by\n  classical\n    rw [Finsupp.curry, Finsupp.curry, Finsupp.sum, Finsupp.sum, filter_sum, support_filter,\n      sum_filter]\n    refine' Finset.sum_congr rfl _\n    rintro \u27e8a\u2081, a\u2082\u27e9 ha\n    dsimp only\n    split_ifs\n    \u00b7 rw [filter_apply_pos, filter_single_of_pos] <;> exact h\n    \u00b7 rwa [filter_single_of_neg]\n#align finsupp.filter_curry Finsupp.filter_curry\n\n/- warning: finsupp.support_curry -> Finsupp.support_curry is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] [_inst_3 : DecidableEq.{succ u1} \u03b1] (f : Finsupp.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.hasSubset.{u1} \u03b1) (Finsupp.support.{u1, max u2 u3} \u03b1 (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.curry.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f)) (Finset.image.{max u1 u2, u1} (Prod.{u1, u2} \u03b1 \u03b2) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_3 a b) (Prod.fst.{u1, u2} \u03b1 \u03b2) (Finsupp.support.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {M : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} M] [_inst_3 : DecidableEq.{succ u3} \u03b1] (f : Finsupp.{max u2 u3, u1} (Prod.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))), HasSubset.Subset.{u3} (Finset.{u3} \u03b1) (Finset.instHasSubsetFinset.{u3} \u03b1) (Finsupp.support.{u3, max u2 u1} \u03b1 (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.zero.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1))) (Finsupp.curry.{u3, u2, u1} \u03b1 \u03b2 M _inst_1 f)) (Finset.image.{max u2 u3, u3} (Prod.{u3, u2} \u03b1 \u03b2) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_3 a b) (Prod.fst.{u3, u2} \u03b1 \u03b2) (Finsupp.support.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M (AddCommMonoid.toAddMonoid.{u1} M _inst_1)) f))\nCase conversion may be inaccurate. Consider using '#align finsupp.support_curry Finsupp.support_curry\u2093'. -/\ntheorem support_curry [DecidableEq \u03b1] (f : \u03b1 \u00d7 \u03b2 \u2192\u2080 M) :\n    f.curry.support \u2286 f.support.image Prod.fst :=\n  by\n  rw [\u2190 Finset.bunion\u1d62_singleton]\n  refine' Finset.Subset.trans support_sum _\n  refine' Finset.bunion\u1d62_mono fun a _ => support_single_subset\n#align finsupp.support_curry Finsupp.support_curry\n\nend CurryUncurry\n\n/-! ### Declarations about finitely supported functions whose support is a `sum` type -/\n\n\nsection Sum\n\n#print Finsupp.sumElim /-\n/-- `finsupp.sum_elim f g` maps `inl x` to `f x` and `inr y` to `g y`. -/\ndef sumElim {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) : Sum \u03b1 \u03b2 \u2192\u2080 \u03b3 :=\n  onFinset\n    (by\n      haveI := Classical.decEq \u03b1 <;> haveI := Classical.decEq \u03b2 <;>\n        exact f.support.map \u27e8_, Sum.inl_injective\u27e9 \u222a g.support.map \u27e8_, Sum.inr_injective\u27e9)\n    (Sum.elim f g) fun ab h => by\n    cases' ab with a b <;> simp only [Sum.elim_inl, Sum.elim_inr] at h <;> simpa\n#align finsupp.sum_elim Finsupp.sumElim\n-/\n\n/- warning: finsupp.coe_sum_elim -> Finsupp.coe_sumElim is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1), Eq.{max (succ (max u1 u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1 f g)) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) => \u03b1 -> \u03b3) (Finsupp.coeFun.{u1, u3} \u03b1 \u03b3 _inst_1) f) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) => \u03b2 -> \u03b3) (Finsupp.coeFun.{u2, u3} \u03b2 \u03b3 _inst_1) g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1), Eq.{max (max (succ u3) (succ u2)) (succ u1)} (forall (\u1fb0 : Sum.{u3, u2} \u03b1 \u03b2), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) \u1fb0) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1 f g)) (Sum.elim.{u3, u2, succ u1} \u03b1 \u03b2 \u03b3 (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) _x) (Finsupp.funLike.{u3, u1} \u03b1 \u03b3 _inst_1) f) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) _x) (Finsupp.funLike.{u2, u1} \u03b2 \u03b3 _inst_1) g))\nCase conversion may be inaccurate. Consider using '#align finsupp.coe_sum_elim Finsupp.coe_sumElim\u2093'. -/\n@[simp]\ntheorem coe_sumElim {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) :\n    \u21d1(sumElim f g) = Sum.elim f g :=\n  rfl\n#align finsupp.coe_sum_elim Finsupp.coe_sumElim\n\n/- warning: finsupp.sum_elim_apply -> Finsupp.sumElim_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (x : Sum.{u1, u2} \u03b1 \u03b2), Eq.{succ u3} \u03b3 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1 f g) x) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) => \u03b1 -> \u03b3) (Finsupp.coeFun.{u1, u3} \u03b1 \u03b3 _inst_1) f) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) => \u03b2 -> \u03b3) (Finsupp.coeFun.{u2, u3} \u03b2 \u03b3 _inst_1) g) x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) (x : Sum.{u3, u2} \u03b1 \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) x) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1 f g) x) (Sum.elim.{u3, u2, succ u1} \u03b1 \u03b2 \u03b3 (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) _x) (Finsupp.funLike.{u3, u1} \u03b1 \u03b3 _inst_1) f) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) _x) (Finsupp.funLike.{u2, u1} \u03b2 \u03b3 _inst_1) g) x)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_elim_apply Finsupp.sumElim_apply\u2093'. -/\ntheorem sumElim_apply {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : Sum \u03b1 \u03b2) :\n    sumElim f g x = Sum.elim f g x :=\n  rfl\n#align finsupp.sum_elim_apply Finsupp.sumElim_apply\n\n/- warning: finsupp.sum_elim_inl -> Finsupp.sumElim_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (x : \u03b1), Eq.{succ u3} \u03b3 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1 f g) (Sum.inl.{u1, u2} \u03b1 \u03b2 x)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) => \u03b1 -> \u03b3) (Finsupp.coeFun.{u1, u3} \u03b1 \u03b3 _inst_1) f x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) (x : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1 f g) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) _x) (Finsupp.funLike.{u3, u1} \u03b1 \u03b3 _inst_1) f x)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_elim_inl Finsupp.sumElim_inl\u2093'. -/\ntheorem sumElim_inl {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b1) :\n    sumElim f g (Sum.inl x) = f x :=\n  rfl\n#align finsupp.sum_elim_inl Finsupp.sumElim_inl\n\n/- warning: finsupp.sum_elim_inr -> Finsupp.sumElim_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (x : \u03b2), Eq.{succ u3} \u03b3 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1 f g) (Sum.inr.{u1, u2} \u03b1 \u03b2 x)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) => \u03b2 -> \u03b3) (Finsupp.coeFun.{u2, u3} \u03b2 \u03b3 _inst_1) g x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (g : Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) (x : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) (Sum.inr.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Finsupp.sumElim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1 f g) (Sum.inr.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) _x) (Finsupp.funLike.{u2, u1} \u03b2 \u03b3 _inst_1) g x)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_elim_inr Finsupp.sumElim_inr\u2093'. -/\ntheorem sumElim_inr {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : \u03b1 \u2192\u2080 \u03b3) (g : \u03b2 \u2192\u2080 \u03b3) (x : \u03b2) :\n    sumElim f g (Sum.inr x) = g x :=\n  rfl\n#align finsupp.sum_elim_inr Finsupp.sumElim_inr\n\n#print Finsupp.sumFinsuppEquivProdFinsupp /-\n/-- The equivalence between `(\u03b1 \u2295 \u03b2) \u2192\u2080 \u03b3` and `(\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)`.\n\nThis is the `finsupp` version of `equiv.sum_arrow_equiv_prod_arrow`. -/\n@[simps apply symm_apply]\ndef sumFinsuppEquivProdFinsupp {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] : (Sum \u03b1 \u03b2 \u2192\u2080 \u03b3) \u2243 (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3)\n    where\n  toFun f :=\n    \u27e8f.comapDomain Sum.inl (Sum.inl_injective.InjOn _),\n      f.comapDomain Sum.inr (Sum.inr_injective.InjOn _)\u27e9\n  invFun fg := sumElim fg.1 fg.2\n  left_inv f := by\n    ext ab\n    cases' ab with a b <;> simp\n  right_inv fg := by ext <;> simp\n#align finsupp.sum_finsupp_equiv_prod_finsupp Finsupp.sumFinsuppEquivProdFinsupp\n-/\n\n/- warning: finsupp.fst_sum_finsupp_equiv_prod_finsupp -> Finsupp.fst_sumFinsuppEquivProdFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (x : \u03b1), Eq.{succ u3} \u03b3 (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) => \u03b1 -> \u03b3) (Finsupp.coeFun.{u1, u3} \u03b1 \u03b3 _inst_1) (Prod.fst.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (coeFn.{max 1 (max (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3))) (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3), max (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3))} (Equiv.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (fun (_x : Equiv.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) => (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) -> (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (Equiv.hasCoeToFun.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (Finsupp.sumFinsuppEquivProdFinsupp.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1) f)) x) (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) f (Sum.inl.{u1, u2} \u03b1 \u03b2 x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (x : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) x) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) _x) (Finsupp.funLike.{u3, u1} \u03b1 \u03b3 _inst_1) (Prod.fst.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) (FunLike.coe.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Equiv.{max (succ u1) (succ (max u2 u3)), max (succ (max u1 u2)) (succ (max u1 u3))} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) => Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) _x) (Equiv.instFunLikeEquiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1))) (Finsupp.sumFinsuppEquivProdFinsupp.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1) f)) x) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) f (Sum.inl.{u3, u2} \u03b1 \u03b2 x))\nCase conversion may be inaccurate. Consider using '#align finsupp.fst_sum_finsupp_equiv_prod_finsupp Finsupp.fst_sumFinsuppEquivProdFinsupp\u2093'. -/\ntheorem fst_sumFinsuppEquivProdFinsupp {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : Sum \u03b1 \u03b2 \u2192\u2080 \u03b3) (x : \u03b1) :\n    (sumFinsuppEquivProdFinsupp f).1 x = f (Sum.inl x) :=\n  rfl\n#align finsupp.fst_sum_finsupp_equiv_prod_finsupp Finsupp.fst_sumFinsuppEquivProdFinsupp\n\n/- warning: finsupp.snd_sum_finsupp_equiv_prod_finsupp -> Finsupp.snd_sumFinsuppEquivProdFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (f : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (y : \u03b2), Eq.{succ u3} \u03b3 (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) => \u03b2 -> \u03b3) (Finsupp.coeFun.{u2, u3} \u03b2 \u03b3 _inst_1) (Prod.snd.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (coeFn.{max 1 (max (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3))) (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3), max (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3))} (Equiv.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (fun (_x : Equiv.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) => (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) -> (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (Equiv.hasCoeToFun.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1))) (Finsupp.sumFinsuppEquivProdFinsupp.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1) f)) y) (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) f (Sum.inr.{u1, u2} \u03b1 \u03b2 y))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (f : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (y : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) y) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) _x) (Finsupp.funLike.{u2, u1} \u03b2 \u03b3 _inst_1) (Prod.snd.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) (FunLike.coe.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Equiv.{max (succ u1) (succ (max u2 u3)), max (succ (max u1 u2)) (succ (max u1 u3))} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) => Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) _x) (Equiv.instFunLikeEquiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1))) (Finsupp.sumFinsuppEquivProdFinsupp.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1) f)) y) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) f (Sum.inr.{u3, u2} \u03b1 \u03b2 y))\nCase conversion may be inaccurate. Consider using '#align finsupp.snd_sum_finsupp_equiv_prod_finsupp Finsupp.snd_sumFinsuppEquivProdFinsupp\u2093'. -/\ntheorem snd_sumFinsuppEquivProdFinsupp {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (f : Sum \u03b1 \u03b2 \u2192\u2080 \u03b3) (y : \u03b2) :\n    (sumFinsuppEquivProdFinsupp f).2 y = f (Sum.inr y) :=\n  rfl\n#align finsupp.snd_sum_finsupp_equiv_prod_finsupp Finsupp.snd_sumFinsuppEquivProdFinsupp\n\n/- warning: finsupp.sum_finsupp_equiv_prod_finsupp_symm_inl -> Finsupp.sumFinsuppEquivProdFinsupp_symm_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (fg : Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (x : \u03b1), Eq.{succ u3} \u03b3 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (coeFn.{max 1 (max (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3)) (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3)), max (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3)} (Equiv.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (fun (_x : Equiv.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) => (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) -> (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.hasCoeToFun.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.symm.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.sumFinsuppEquivProdFinsupp.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1)) fg) (Sum.inl.{u1, u2} \u03b1 \u03b2 x)) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (fun (_x : Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) => \u03b1 -> \u03b3) (Finsupp.coeFun.{u1, u3} \u03b1 \u03b3 _inst_1) (Prod.fst.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) fg) x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (fg : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (x : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (FunLike.coe.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Equiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (fun (_x : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) => Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) _x) (Equiv.instFunLikeEquiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.symm.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.sumFinsuppEquivProdFinsupp.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1)) fg) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => \u03b3) _x) (Finsupp.funLike.{u3, u1} \u03b1 \u03b3 _inst_1) (Prod.fst.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) fg) x)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_finsupp_equiv_prod_finsupp_symm_inl Finsupp.sumFinsuppEquivProdFinsupp_symm_inl\u2093'. -/\ntheorem sumFinsuppEquivProdFinsupp_symm_inl {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (fg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3))\n    (x : \u03b1) : (sumFinsuppEquivProdFinsupp.symm fg) (Sum.inl x) = fg.1 x :=\n  rfl\n#align finsupp.sum_finsupp_equiv_prod_finsupp_symm_inl Finsupp.sumFinsuppEquivProdFinsupp_symm_inl\n\n/- warning: finsupp.sum_finsupp_equiv_prod_finsupp_symm_inr -> Finsupp.sumFinsuppEquivProdFinsupp_symm_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : Zero.{u3} \u03b3] (fg : Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (y : \u03b2), Eq.{succ u3} \u03b3 (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (fun (_x : Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) => (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Finsupp.coeFun.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (coeFn.{max 1 (max (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3)) (max (succ (max u1 u2)) (succ u3)) (succ (max u1 u3)) (succ (max u2 u3)), max (max (succ (max u1 u3)) (succ (max u2 u3))) (succ (max u1 u2)) (succ u3)} (Equiv.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (fun (_x : Equiv.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) => (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) -> (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.hasCoeToFun.{max (succ (max u1 u3)) (succ (max u2 u3)), max (succ (max u1 u2)) (succ u3)} (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.symm.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u3)) (succ (max u2 u3))} (Finsupp.{max u1 u2, u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1)) (Finsupp.sumFinsuppEquivProdFinsupp.{u1, u2, u3} \u03b1 \u03b2 \u03b3 _inst_1)) fg) (Sum.inr.{u1, u2} \u03b1 \u03b2 y)) (coeFn.{max (succ u2) (succ u3), max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) (fun (_x : Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) => \u03b2 -> \u03b3) (Finsupp.coeFun.{u2, u3} \u03b2 \u03b3 _inst_1) (Prod.snd.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u3} \u03b2 \u03b3 _inst_1) fg) y)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : Zero.{u1} \u03b3] (fg : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (y : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) (Sum.inr.{u3, u2} \u03b1 \u03b2 y)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => \u03b3) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (FunLike.coe.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Equiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (fun (_x : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) => Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) _x) (Equiv.instFunLikeEquiv.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1)) (Equiv.symm.{max (max (succ u1) (succ u2)) (succ u3), max (max (succ u1) (succ u2)) (succ u3)} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 _inst_1) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1)) (Finsupp.sumFinsuppEquivProdFinsupp.{u3, u2, u1} \u03b1 \u03b2 \u03b3 _inst_1)) fg) (Sum.inr.{u3, u2} \u03b1 \u03b2 y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => \u03b3) _x) (Finsupp.funLike.{u2, u1} \u03b2 \u03b3 _inst_1) (Prod.snd.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 \u03b3 _inst_1) (Finsupp.{u2, u1} \u03b2 \u03b3 _inst_1) fg) y)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_finsupp_equiv_prod_finsupp_symm_inr Finsupp.sumFinsuppEquivProdFinsupp_symm_inr\u2093'. -/\ntheorem sumFinsuppEquivProdFinsupp_symm_inr {\u03b1 \u03b2 \u03b3 : Type _} [Zero \u03b3] (fg : (\u03b1 \u2192\u2080 \u03b3) \u00d7 (\u03b2 \u2192\u2080 \u03b3))\n    (y : \u03b2) : (sumFinsuppEquivProdFinsupp.symm fg) (Sum.inr y) = fg.2 y :=\n  rfl\n#align finsupp.sum_finsupp_equiv_prod_finsupp_symm_inr Finsupp.sumFinsuppEquivProdFinsupp_symm_inr\n\nvariable [AddMonoid M]\n\n/- warning: finsupp.sum_finsupp_add_equiv_prod_finsupp -> Finsupp.sumFinsuppAddEquivProdFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}}, AddEquiv.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}}, AddEquiv.{max u1 u3 u2, max (max u1 u3) u1 u2} (Finsupp.{max u3 u2, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u2, max u1 u3} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u2, max u1 u3} (Finsupp.{u2, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u3, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_finsupp_add_equiv_prod_finsupp Finsupp.sumFinsuppAddEquivProdFinsupp\u2093'. -/\n/-- The additive equivalence between `(\u03b1 \u2295 \u03b2) \u2192\u2080 M` and `(\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)`.\n\nThis is the `finsupp` version of `equiv.sum_arrow_equiv_prod_arrow`. -/\n@[simps apply symm_apply]\ndef sumFinsuppAddEquivProdFinsupp {\u03b1 \u03b2 : Type _} : (Sum \u03b1 \u03b2 \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M) :=\n  { sumFinsuppEquivProdFinsupp with\n    map_add' := by\n      intros\n      ext <;>\n        simp only [Equiv.toFun_as_coe, Prod.fst_add, Prod.snd_add, add_apply,\n          snd_sum_finsupp_equiv_prod_finsupp, fst_sum_finsupp_equiv_prod_finsupp] }\n#align finsupp.sum_finsupp_add_equiv_prod_finsupp Finsupp.sumFinsuppAddEquivProdFinsupp\n\n/- warning: finsupp.fst_sum_finsupp_add_equiv_prod_finsupp -> Finsupp.fst_sumFinsuppAddEquivProdFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} (f : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (x : \u03b1), Eq.{succ u1} M (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.fst.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (coeFn.{max (succ (max (max u2 u3) u1)) (succ (max (max u2 u1) u3 u1)), max (succ (max (max u2 u3) u1)) (succ (max (max u2 u1) u3 u1))} (AddEquiv.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (fun (_x : AddEquiv.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) => (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) -> (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))))) (AddEquiv.hasCoeToFun.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u2, u3} M _inst_1 \u03b1 \u03b2) f)) x) (coeFn.{max (succ (max u2 u3)) (succ u1), max (succ (max u2 u3)) (succ u1)} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Sum.{u2, u3} \u03b1 \u03b2) -> M) (Finsupp.coeFun.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) f (Sum.inl.{u2, u3} \u03b1 \u03b2 x))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} (f : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (x : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) x) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.fst.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1)} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) => Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) _x) (AddHomClass.toFunLike.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1)))) (AddMonoidHomClass.toAddHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddEquivClass.instAddMonoidHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddEquiv.instAddEquivClassAddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))))))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u3, u2} M _inst_1 \u03b1 \u03b2) f)) x) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) f (Sum.inl.{u3, u2} \u03b1 \u03b2 x))\nCase conversion may be inaccurate. Consider using '#align finsupp.fst_sum_finsupp_add_equiv_prod_finsupp Finsupp.fst_sumFinsuppAddEquivProdFinsupp\u2093'. -/\ntheorem fst_sumFinsuppAddEquivProdFinsupp {\u03b1 \u03b2 : Type _} (f : Sum \u03b1 \u03b2 \u2192\u2080 M) (x : \u03b1) :\n    (sumFinsuppAddEquivProdFinsupp f).1 x = f (Sum.inl x) :=\n  rfl\n#align finsupp.fst_sum_finsupp_add_equiv_prod_finsupp Finsupp.fst_sumFinsuppAddEquivProdFinsupp\n\n/- warning: finsupp.snd_sum_finsupp_add_equiv_prod_finsupp -> Finsupp.snd_sumFinsuppAddEquivProdFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} (f : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (y : \u03b2), Eq.{succ u1} M (coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => \u03b2 -> M) (Finsupp.coeFun.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.snd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (coeFn.{max (succ (max (max u2 u3) u1)) (succ (max (max u2 u1) u3 u1)), max (succ (max (max u2 u3) u1)) (succ (max (max u2 u1) u3 u1))} (AddEquiv.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (fun (_x : AddEquiv.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) => (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) -> (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))))) (AddEquiv.hasCoeToFun.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u2, u3} M _inst_1 \u03b1 \u03b2) f)) y) (coeFn.{max (succ (max u2 u3)) (succ u1), max (succ (max u2 u3)) (succ u1)} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Sum.{u2, u3} \u03b1 \u03b2) -> M) (Finsupp.coeFun.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) f (Sum.inr.{u2, u3} \u03b1 \u03b2 y))\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} (f : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (y : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) y) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.snd.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1)} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) => Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) _x) (AddHomClass.toFunLike.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1)))) (AddMonoidHomClass.toAddHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddEquivClass.instAddMonoidHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u1 u2 u3, max (max u1 u2) u1 u3} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddEquiv.instAddEquivClassAddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))))))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u3, u2} M _inst_1 \u03b1 \u03b2) f)) y) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) f (Sum.inr.{u3, u2} \u03b1 \u03b2 y))\nCase conversion may be inaccurate. Consider using '#align finsupp.snd_sum_finsupp_add_equiv_prod_finsupp Finsupp.snd_sumFinsuppAddEquivProdFinsupp\u2093'. -/\ntheorem snd_sumFinsuppAddEquivProdFinsupp {\u03b1 \u03b2 : Type _} (f : Sum \u03b1 \u03b2 \u2192\u2080 M) (y : \u03b2) :\n    (sumFinsuppAddEquivProdFinsupp f).2 y = f (Sum.inr y) :=\n  rfl\n#align finsupp.snd_sum_finsupp_add_equiv_prod_finsupp Finsupp.snd_sumFinsuppAddEquivProdFinsupp\n\n/- warning: finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inl -> Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inl is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} (fg : Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (x : \u03b1), Eq.{succ u1} M (coeFn.{max (succ (max u2 u3)) (succ u1), max (succ (max u2 u3)) (succ u1)} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Sum.{u2, u3} \u03b1 \u03b2) -> M) (Finsupp.coeFun.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (coeFn.{max (succ (max (max u2 u1) u3 u1)) (succ (max (max u2 u3) u1)), max (succ (max (max u2 u1) u3 u1)) (succ (max (max u2 u3) u1))} (AddEquiv.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : AddEquiv.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) -> (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (AddEquiv.hasCoeToFun.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (AddEquiv.symm.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u2, u3} M _inst_1 \u03b1 \u03b2)) fg) (Sum.inl.{u2, u3} \u03b1 \u03b2 x)) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.fst.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) fg) x)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} (fg : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (x : \u03b1), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1)} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (fun (_x : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) => Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) _x) (AddHomClass.toFunLike.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1)))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1))) (AddMonoidHomClass.toAddHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (AddEquivClass.instAddMonoidHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (AddEquiv.instAddEquivClassAddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))))) (AddEquiv.symm.{max (max u2 u3) u1, max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u3, u2} M _inst_1 \u03b1 \u03b2)) fg) (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (FunLike.coe.{max (succ u3) (succ u1), succ u3, succ u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.fst.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) fg) x)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inl Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inl\u2093'. -/\ntheorem sumFinsuppAddEquivProdFinsupp_symm_inl {\u03b1 \u03b2 : Type _} (fg : (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)) (x : \u03b1) :\n    (sumFinsuppAddEquivProdFinsupp.symm fg) (Sum.inl x) = fg.1 x :=\n  rfl\n#align finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inl Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inl\n\n/- warning: finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inr -> Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inr is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} (fg : Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (y : \u03b2), Eq.{succ u1} M (coeFn.{max (succ (max u2 u3)) (succ u1), max (succ (max u2 u3)) (succ u1)} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Sum.{u2, u3} \u03b1 \u03b2) -> M) (Finsupp.coeFun.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (coeFn.{max (succ (max (max u2 u1) u3 u1)) (succ (max (max u2 u3) u1)), max (succ (max (max u2 u1) u3 u1)) (succ (max (max u2 u3) u1))} (AddEquiv.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : AddEquiv.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) -> (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (AddEquiv.hasCoeToFun.{max (max u2 u1) u3 u1, max (max u2 u3) u1} (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (AddEquiv.symm.{max (max u2 u3) u1, max (max u2 u1) u3 u1} (Finsupp.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))) (Finsupp.add.{max u2 u3, u1} (Sum.{u2, u3} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.hasAdd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{u2, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u2, u3} M _inst_1 \u03b1 \u03b2)) fg) (Sum.inr.{u2, u3} \u03b1 \u03b2 y)) (coeFn.{max (succ u3) (succ u1), max (succ u3) (succ u1)} (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (fun (_x : Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) => \u03b2 -> M) (Finsupp.coeFun.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.snd.{max u2 u1, max u3 u1} (Finsupp.{u2, u1} \u03b1 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.{u3, u1} \u03b2 M (AddZeroClass.toHasZero.{u1} M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) fg) y)\nbut is expected to have type\n  forall {M : Type.{u1}} [_inst_1 : AddMonoid.{u1} M] {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} (fg : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (y : \u03b2), Eq.{succ u1} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) (Sum.inr.{u3, u2} \u03b1 \u03b2 y)) (FunLike.coe.{max (succ (max u3 u2)) (succ u1), succ (max u3 u2), succ u1} (Finsupp.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Sum.{u3, u2} \u03b1 \u03b2) (fun (_x : Sum.{u3, u2} \u03b1 \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sum.{u3, u2} \u03b1 \u03b2) => M) _x) (Finsupp.funLike.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1)} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (fun (_x : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) => Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) _x) (AddHomClass.toFunLike.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1)))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1))) (AddMonoidHomClass.toAddHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (AddEquivClass.instAddMonoidHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddZeroClassSum.{max u3 u1, max u2 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (AddMonoid.toAddZeroClass.{max u3 u1} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u3, u1} \u03b1 M _inst_1)) (AddMonoid.toAddZeroClass.{max u2 u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{u2, u1} \u03b2 M _inst_1))) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.addMonoid.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M _inst_1)) (AddEquiv.instAddEquivClassAddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)))))) (AddEquiv.symm.{max (max u2 u3) u1, max (max u2 u3) u1} (Finsupp.{max u2 u3, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1))) (Finsupp.add.{max u3 u2, u1} (Sum.{u3, u2} \u03b1 \u03b2) M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Prod.instAddSum.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.add.{u3, u1} \u03b1 M (AddMonoid.toAddZeroClass.{u1} M _inst_1)) (Finsupp.add.{u2, u1} \u03b2 M (AddMonoid.toAddZeroClass.{u1} M _inst_1))) (Finsupp.sumFinsuppAddEquivProdFinsupp.{u1, u3, u2} M _inst_1 \u03b1 \u03b2)) fg) (Sum.inr.{u3, u2} \u03b1 \u03b2 y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b2) => M) _x) (Finsupp.funLike.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) (Prod.snd.{max u1 u3, max u1 u2} (Finsupp.{u3, u1} \u03b1 M (AddMonoid.toZero.{u1} M _inst_1)) (Finsupp.{u2, u1} \u03b2 M (AddMonoid.toZero.{u1} M _inst_1)) fg) y)\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inr Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inr\u2093'. -/\ntheorem sumFinsuppAddEquivProdFinsupp_symm_inr {\u03b1 \u03b2 : Type _} (fg : (\u03b1 \u2192\u2080 M) \u00d7 (\u03b2 \u2192\u2080 M)) (y : \u03b2) :\n    (sumFinsuppAddEquivProdFinsupp.symm fg) (Sum.inr y) = fg.2 y :=\n  rfl\n#align finsupp.sum_finsupp_add_equiv_prod_finsupp_symm_inr Finsupp.sumFinsuppAddEquivProdFinsupp_symm_inr\n\nend Sum\n\n/-! ### Declarations about scalar multiplication -/\n\n\nsection\n\nvariable [Zero M] [MonoidWithZero R] [MulActionWithZero R M]\n\n/- warning: finsupp.single_smul -> Finsupp.single_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Zero.{u2} M] [_inst_2 : MonoidWithZero.{u3} R] [_inst_3 : MulActionWithZero.{u3, u2} R M _inst_2 _inst_1] (a : \u03b1) (b : \u03b1) (f : \u03b1 -> M) (r : R), Eq.{succ u2} M (SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M _inst_1 (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2))) _inst_1 (MulActionWithZero.toSMulWithZero.{u3, u2} R M _inst_2 _inst_1 _inst_3))) (coeFn.{max (succ u1) (succ u3), max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2)))) (fun (_x : Finsupp.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2)))) => \u03b1 -> R) (Finsupp.coeFun.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2)))) (Finsupp.single.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2))) a r) b) (f a)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M _inst_1) (fun (_x : Finsupp.{u1, u2} \u03b1 M _inst_1) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M _inst_1) (Finsupp.single.{u1, u2} \u03b1 M _inst_1 a (SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M _inst_1 (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R _inst_2))) _inst_1 (MulActionWithZero.toSMulWithZero.{u3, u2} R M _inst_2 _inst_1 _inst_3))) r (f b))) b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {R : Type.{u2}} [_inst_1 : Zero.{u3} M] [_inst_2 : MonoidWithZero.{u2} R] [_inst_3 : MulActionWithZero.{u2, u3} R M _inst_2 _inst_1] (a : \u03b1) (b : \u03b1) (f : \u03b1 -> M) (r : R), Eq.{succ u3} M (HSMul.hSMul.{u2, u3, u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) M M (instHSMul.{u2, u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) M (SMulZeroClass.toSMul.{u2, u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) M _inst_1 (SMulWithZero.toSMulZeroClass.{u2, u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) M (MonoidWithZero.toZero.{u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) _inst_2) _inst_1 (MulActionWithZero.toSMulWithZero.{u2, u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) b) M _inst_2 _inst_1 _inst_3)))) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R _inst_2)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => R) _x) (Finsupp.funLike.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R _inst_2)) (Finsupp.single.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R _inst_2) a r) b) (f a)) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b1 M _inst_1) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} \u03b1 M _inst_1) (Finsupp.single.{u1, u3} \u03b1 M _inst_1 a (HSMul.hSMul.{u2, u3, u3} R M M (instHSMul.{u2, u3} R M (SMulZeroClass.toSMul.{u2, u3} R M _inst_1 (SMulWithZero.toSMulZeroClass.{u2, u3} R M (MonoidWithZero.toZero.{u2} R _inst_2) _inst_1 (MulActionWithZero.toSMulWithZero.{u2, u3} R M _inst_2 _inst_1 _inst_3)))) r (f b))) b)\nCase conversion may be inaccurate. Consider using '#align finsupp.single_smul Finsupp.single_smul\u2093'. -/\n@[simp]\ntheorem single_smul (a b : \u03b1) (f : \u03b1 \u2192 M) (r : R) : single a r b \u2022 f a = single a (r \u2022 f b) b := by\n  by_cases a = b <;> simp [h]\n#align finsupp.single_smul Finsupp.single_smul\n\nend\n\nsection\n\nvariable [Monoid G] [MulAction G \u03b1] [AddCommMonoid M]\n\n/- warning: finsupp.comap_has_smul -> Finsupp.comapSMul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], SMul.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], SMul.{u3, max u2 u1} G (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_has_smul Finsupp.comapSMul\u2093'. -/\n/-- Scalar multiplication acting on the domain.\n\nThis is not an instance as it would conflict with the action on the range.\nSee the `instance_diamonds` test for examples of such conflicts. -/\ndef comapSMul : SMul G (\u03b1 \u2192\u2080 M) where smul g := mapDomain ((\u00b7 \u2022 \u00b7) g)\n#align finsupp.comap_has_smul Finsupp.comapSMul\n\nattribute [local instance] comap_has_smul\n\n/- warning: finsupp.comap_smul_def -> Finsupp.comapSMul_def is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (SMul.smul.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (Finsupp.comapSMul.{u1, u2, u3} \u03b1 M G _inst_1 _inst_2 _inst_3) g f) (Finsupp.mapDomain.{u1, u1, u2} \u03b1 \u03b1 M _inst_3 (SMul.smul.{u3, u1} G \u03b1 (MulAction.toHasSmul.{u3, u1} G \u03b1 _inst_1 _inst_2) g) f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {M : Type.{u2}} {G : Type.{u1}} [_inst_1 : Monoid.{u1} G] [_inst_2 : MulAction.{u1, u3} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (f : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))), Eq.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (instHSMul.{u1, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.comapSMul.{u3, u2, u1} \u03b1 M G _inst_1 _inst_2 _inst_3)) g f) (Finsupp.mapDomain.{u3, u3, u2} \u03b1 \u03b1 M _inst_3 ((fun (x._@.Mathlib.Data.Finsupp.Basic._hyg.14504 : G) (x._@.Mathlib.Data.Finsupp.Basic._hyg.14506 : \u03b1) => HSMul.hSMul.{u1, u3, u3} G \u03b1 \u03b1 (instHSMul.{u1, u3} G \u03b1 (MulAction.toSMul.{u1, u3} G \u03b1 _inst_1 _inst_2)) x._@.Mathlib.Data.Finsupp.Basic._hyg.14504 x._@.Mathlib.Data.Finsupp.Basic._hyg.14506) g) f)\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_smul_def Finsupp.comapSMul_def\u2093'. -/\ntheorem comapSMul_def (g : G) (f : \u03b1 \u2192\u2080 M) : g \u2022 f = mapDomain ((\u00b7 \u2022 \u00b7) g) f :=\n  rfl\n#align finsupp.comap_smul_def Finsupp.comapSMul_def\n\n/- warning: finsupp.comap_smul_single -> Finsupp.comapSMul_single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (a : \u03b1) (b : M), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (SMul.smul.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (Finsupp.comapSMul.{u1, u2, u3} \u03b1 M G _inst_1 _inst_2 _inst_3) g (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) a b)) (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (SMul.smul.{u3, u1} G \u03b1 (MulAction.toHasSmul.{u3, u1} G \u03b1 _inst_1 _inst_2) g a) b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {M : Type.{u2}} {G : Type.{u1}} [_inst_1 : Monoid.{u1} G] [_inst_2 : MulAction.{u1, u3} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (a : \u03b1) (b : M), Eq.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (HSMul.hSMul.{u1, max u2 u3, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (instHSMul.{u1, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.comapSMul.{u3, u2, u1} \u03b1 M G _inst_1 _inst_2 _inst_3)) g (Finsupp.single.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) a b)) (Finsupp.single.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)) (HSMul.hSMul.{u1, u3, u3} G \u03b1 \u03b1 (instHSMul.{u1, u3} G \u03b1 (MulAction.toSMul.{u1, u3} G \u03b1 _inst_1 _inst_2)) g a) b)\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_smul_single Finsupp.comapSMul_single\u2093'. -/\n@[simp]\ntheorem comapSMul_single (g : G) (a : \u03b1) (b : M) : g \u2022 single a b = single (g \u2022 a) b :=\n  mapDomain_single\n#align finsupp.comap_smul_single Finsupp.comapSMul_single\n\n/- warning: finsupp.comap_mul_action -> Finsupp.comapMulAction is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], MulAction.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) _inst_1\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], MulAction.{u3, max u2 u1} G (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) _inst_1\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_mul_action Finsupp.comapMulAction\u2093'. -/\n/-- `finsupp.comap_has_smul` is multiplicative -/\ndef comapMulAction : MulAction G (\u03b1 \u2192\u2080 M)\n    where\n  one_smul f := by rw [comap_smul_def, one_smul_eq_id, map_domain_id]\n  mul_smul g g' f := by\n    rw [comap_smul_def, comap_smul_def, comap_smul_def, \u2190 comp_smul_left, map_domain_comp]\n#align finsupp.comap_mul_action Finsupp.comapMulAction\n\nattribute [local instance] comap_mul_action\n\n/- warning: finsupp.comap_distrib_mul_action -> Finsupp.comapDistribMulAction is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], DistribMulAction.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) _inst_1 (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Monoid.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 _inst_1] [_inst_3 : AddCommMonoid.{u2} M], DistribMulAction.{u3, max u2 u1} G (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) _inst_1 (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_distrib_mul_action Finsupp.comapDistribMulAction\u2093'. -/\n/-- `finsupp.comap_has_smul` is distributive -/\ndef comapDistribMulAction : DistribMulAction G (\u03b1 \u2192\u2080 M)\n    where\n  smul_zero g := by\n    ext\n    dsimp [(\u00b7 \u2022 \u00b7)]\n    simp\n  smul_add g f f' := by\n    ext\n    dsimp [(\u00b7 \u2022 \u00b7)]\n    simp [map_domain_add]\n#align finsupp.comap_distrib_mul_action Finsupp.comapDistribMulAction\n\nend\n\nsection\n\nvariable [Group G] [MulAction G \u03b1] [AddCommMonoid M]\n\nattribute [local instance] comap_has_smul comap_mul_action comap_distrib_mul_action\n\n/- warning: finsupp.comap_smul_apply -> Finsupp.comapSMul_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {G : Type.{u3}} [_inst_1 : Group.{u3} G] [_inst_2 : MulAction.{u3, u1} G \u03b1 (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_1))] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (a : \u03b1), Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (SMul.smul.{u3, max u1 u2} G (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (Finsupp.comapSMul.{u1, u2, u3} \u03b1 M G (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_1)) _inst_2 _inst_3) g f) a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3)))) f (SMul.smul.{u3, u1} G \u03b1 (MulAction.toHasSmul.{u3, u1} G \u03b1 (DivInvMonoid.toMonoid.{u3} G (Group.toDivInvMonoid.{u3} G _inst_1)) _inst_2) (Inv.inv.{u3} G (DivInvMonoid.toHasInv.{u3} G (Group.toDivInvMonoid.{u3} G _inst_1)) g) a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {M : Type.{u2}} {G : Type.{u1}} [_inst_1 : Group.{u1} G] [_inst_2 : MulAction.{u1, u3} G \u03b1 (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))] [_inst_3 : AddCommMonoid.{u2} M] (g : G) (f : Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (a : \u03b1), Eq.{succ u2} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (instHSMul.{u1, max u3 u2} G (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) (Finsupp.comapSMul.{u3, u2, u1} \u03b1 M G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) _inst_2 _inst_3)) g f) a) (FunLike.coe.{max (succ u3) (succ u2), succ u3, succ u2} (Finsupp.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u3, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_3))) f (HSMul.hSMul.{u1, u3, u3} G \u03b1 \u03b1 (instHSMul.{u1, u3} G \u03b1 (MulAction.toSMul.{u1, u3} G \u03b1 (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) _inst_2)) (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))) g) a))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_smul_apply Finsupp.comapSMul_apply\u2093'. -/\n/-- When `G` is a group, `finsupp.comap_has_smul` acts by precomposition with the action of `g\u207b\u00b9`.\n-/\n@[simp]\ntheorem comapSMul_apply (g : G) (f : \u03b1 \u2192\u2080 M) (a : \u03b1) : (g \u2022 f) a = f (g\u207b\u00b9 \u2022 a) :=\n  by\n  conv_lhs => rw [\u2190 smul_inv_smul g a]\n  exact map_domain_apply (MulAction.injective g) _ (g\u207b\u00b9 \u2022 a)\n#align finsupp.comap_smul_apply Finsupp.comapSMul_apply\n\nend\n\nsection\n\ninstance [Zero M] [SMulZeroClass R M] : SMulZeroClass R (\u03b1 \u2192\u2080 M)\n    where\n  smul a v := v.mapRange ((\u00b7 \u2022 \u00b7) a) (smul_zero _)\n  smul_zero a := by\n    ext\n    apply smul_zero\n\n/-!\nThroughout this section, some `monoid` and `semiring` arguments are specified with `{}` instead of\n`[]`. See note [implicit instance arguments].\n-/\n\n\n/- warning: finsupp.coe_smul -> Finsupp.coe_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribSMul.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1)] (b : R) (v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))), Eq.{succ (max u1 u2)} (\u03b1 -> M) (coeFn.{succ (max u1 u2), succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2))) b v)) (SMul.smul.{u3, max u1 u2} R (\u03b1 -> M) (Function.hasSMul.{u1, u3, u2} \u03b1 R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2))) b (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {R : Type.{u2}} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : DistribSMul.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1)] (b : R) (v : Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)), Eq.{max (succ u1) (succ u3)} (forall (\u1fb0 : \u03b1), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) \u1fb0) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (HSMul.hSMul.{u2, max u1 u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (instHSMul.{u2, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (SMulZeroClass.toSMul.{u2, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.zero.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.smulZeroClass.{u1, u3, u2} \u03b1 M R (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) _inst_2)))) b v)) (HSMul.hSMul.{u2, max u1 u3, max u1 u3} R (forall (a : \u03b1), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (forall (\u1fb0 : \u03b1), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) \u1fb0) (instHSMul.{u2, max u1 u3} R (forall (a : \u03b1), (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (Pi.instSMul.{u1, u3, u2} \u03b1 R (fun (a : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (fun (i : \u03b1) => SMulZeroClass.toSMul.{u2, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) i) (AddMonoid.toZero.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) i) _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) i) (AddMonoid.toAddZeroClass.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) i) _inst_1) _inst_2)))) b (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) v))\nCase conversion may be inaccurate. Consider using '#align finsupp.coe_smul Finsupp.coe_smul\u2093'. -/\n@[simp]\ntheorem coe_smul [AddMonoid M] [DistribSMul R M] (b : R) (v : \u03b1 \u2192\u2080 M) : \u21d1(b \u2022 v) = b \u2022 v :=\n  rfl\n#align finsupp.coe_smul Finsupp.coe_smul\n\n/- warning: finsupp.smul_apply -> Finsupp.smul_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribSMul.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1)] (b : R) (v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (a : \u03b1), Eq.{succ u2} M (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2))) b v) a) (SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2)) b (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (fun (_x : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) => \u03b1 -> M) (Finsupp.coeFun.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) v a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {R : Type.{u2}} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : DistribSMul.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1)] (b : R) (v : Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (a : \u03b1), Eq.{succ u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (HSMul.hSMul.{u2, max u1 u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (instHSMul.{u2, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (SMulZeroClass.toSMul.{u2, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.zero.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.smulZeroClass.{u1, u3, u2} \u03b1 M R (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) _inst_2)))) b v) a) (HSMul.hSMul.{u2, u3, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (instHSMul.{u2, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (SMulZeroClass.toSMul.{u2, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (AddMonoid.toZero.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) (AddMonoid.toAddZeroClass.{u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) a) _inst_1) _inst_2))) b (FunLike.coe.{max (succ u1) (succ u3), succ u1, succ u3} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1) => M) _x) (Finsupp.funLike.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) v a))\nCase conversion may be inaccurate. Consider using '#align finsupp.smul_apply Finsupp.smul_apply\u2093'. -/\ntheorem smul_apply [AddMonoid M] [DistribSMul R M] (b : R) (v : \u03b1 \u2192\u2080 M) (a : \u03b1) :\n    (b \u2022 v) a = b \u2022 v a :=\n  rfl\n#align finsupp.smul_apply Finsupp.smul_apply\n\n/- warning: is_smul_regular.finsupp -> IsSMulRegular.finsupp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribSMul.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1)] {k : R}, (IsSMulRegular.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2)) k) -> (IsSMulRegular.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2))) k)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {R : Type.{u2}} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : DistribSMul.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1)] {k : R}, (IsSMulRegular.{u2, u3} R M (SMulZeroClass.toSMul.{u2, u3} R M (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) _inst_2)) k) -> (IsSMulRegular.{u2, max u3 u1} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (SMulZeroClass.toSMul.{u2, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.zero.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.smulZeroClass.{u1, u3, u2} \u03b1 M R (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) _inst_2))) k)\nCase conversion may be inaccurate. Consider using '#align is_smul_regular.finsupp IsSMulRegular.finsupp\u2093'. -/\ntheorem IsSMulRegular.finsupp [AddMonoid M] [DistribSMul R M] {k : R} (hk : IsSMulRegular M k) :\n    IsSMulRegular (\u03b1 \u2192\u2080 M) k := fun _ _ h => ext fun i => hk (congr_fun h i)\n#align is_smul_regular.finsupp IsSMulRegular.finsupp\n\ninstance [Nonempty \u03b1] [AddMonoid M] [DistribSMul R M] [FaithfulSMul R M] : FaithfulSMul R (\u03b1 \u2192\u2080 M)\n    where eq_of_smul_eq_smul r\u2081 r\u2082 h :=\n    let \u27e8a\u27e9 := \u2039Nonempty \u03b1\u203a\n    eq_of_smul_eq_smul fun m : M => by simpa using congr_fun (h (single a m)) a\n\nvariable (\u03b1 M)\n\ninstance [AddZeroClass M] [DistribSMul R M] : DistribSMul R (\u03b1 \u2192\u2080 M)\n    where\n  smul := (\u00b7 \u2022 \u00b7)\n  smul_add a x y := ext fun _ => smul_add _ _ _\n  smul_zero x := ext fun _ => smul_zero _\n\ninstance [Monoid R] [AddMonoid M] [DistribMulAction R M] : DistribMulAction R (\u03b1 \u2192\u2080 M) :=\n  { Finsupp.distribSMul _ _ with\n    smul := (\u00b7 \u2022 \u00b7)\n    one_smul := fun x => ext fun _ => one_smul _ _\n    mul_smul := fun r s x => ext fun _ => mul_smul _ _ _ }\n\ninstance [Monoid R] [Monoid S] [AddMonoid M] [DistribMulAction R M] [DistribMulAction S M]\n    [SMul R S] [IsScalarTower R S M] : IsScalarTower R S (\u03b1 \u2192\u2080 M)\n    where smul_assoc r s a := ext fun _ => smul_assoc _ _ _\n\ninstance [Monoid R] [Monoid S] [AddMonoid M] [DistribMulAction R M] [DistribMulAction S M]\n    [SMulCommClass R S M] : SMulCommClass R S (\u03b1 \u2192\u2080 M)\n    where smul_comm r s a := ext fun _ => smul_comm _ _ _\n\ninstance [Monoid R] [AddMonoid M] [DistribMulAction R M] [DistribMulAction R\u1d50\u1d52\u1d56 M]\n    [IsCentralScalar R M] : IsCentralScalar R (\u03b1 \u2192\u2080 M)\n    where op_smul_eq_smul r a := ext fun _ => op_smul_eq_smul _ _\n\ninstance [Semiring R] [AddCommMonoid M] [Module R M] : Module R (\u03b1 \u2192\u2080 M) :=\n  { Finsupp.distribMulAction \u03b1 M with\n    smul := (\u00b7 \u2022 \u00b7)\n    zero_smul := fun x => ext fun _ => zero_smul _ _\n    add_smul := fun a x y => ext fun _ => add_smul _ _ _ }\n\nvariable {\u03b1 M} {R}\n\n/- warning: finsupp.support_smul -> Finsupp.support_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] {b : R} {g : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))}, HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.hasSubset.{u1} \u03b1) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2)))) b g)) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] {b : R} {g : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)}, HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.instHasSubsetFinset.{u1} \u03b1) (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) (HSMul.hSMul.{u3, max u1 u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (instHSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddMonoid.toZero.{u2} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2))))) b g)) (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) g)\nCase conversion may be inaccurate. Consider using '#align finsupp.support_smul Finsupp.support_smul\u2093'. -/\ntheorem support_smul {_ : Monoid R} [AddMonoid M] [DistribMulAction R M] {b : R} {g : \u03b1 \u2192\u2080 M} :\n    (b \u2022 g).support \u2286 g.support := fun a =>\n  by\n  simp only [smul_apply, mem_support_iff, Ne.def]\n  exact mt fun h => h.symm \u25b8 smul_zero _\n#align finsupp.support_smul Finsupp.support_smul\n\n/- warning: finsupp.support_smul_eq -> Finsupp.support_smul_eq is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u3, u2} R M _inst_1 _inst_2] [_inst_4 : NoZeroSMulDivisors.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3))))] {b : R}, (Ne.{succ u3} R b (OfNat.ofNat.{u3} R 0 (OfNat.mk.{u3} R 0 (Zero.zero.{u3} R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))))))) -> (forall {g : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))}, Eq.{succ u1} (Finset.{u1} \u03b1) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulWithZero.toSmulZeroClass.{u3, u2} R M (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)))) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3))))) b g)) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : Module.{u3, u2} R M _inst_1 _inst_2] [_inst_4 : NoZeroSMulDivisors.{u3, u2} R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulZeroClass.toSMul.{u3, u2} R M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3))))] {b : R}, (Ne.{succ u3} R b (OfNat.ofNat.{u3} R 0 (Zero.toOfNat0.{u3} R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))))) -> (forall {g : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))}, Eq.{succ u1} (Finset.{u1} \u03b1) (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (HSMul.hSMul.{u3, max u1 u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (instHSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (SMulZeroClass.toSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (SMulWithZero.toSMulZeroClass.{u3, u2} R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (MulActionWithZero.toSMulWithZero.{u3, u2} R M (Semiring.toMonoidWithZero.{u3} R _inst_1) (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Module.toMulActionWithZero.{u3, u2} R M _inst_1 _inst_2 _inst_3)))))) b g)) (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) g))\nCase conversion may be inaccurate. Consider using '#align finsupp.support_smul_eq Finsupp.support_smul_eq\u2093'. -/\n@[simp]\ntheorem support_smul_eq [Semiring R] [AddCommMonoid M] [Module R M] [NoZeroSMulDivisors R M] {b : R}\n    (hb : b \u2260 0) {g : \u03b1 \u2192\u2080 M} : (b \u2022 g).support = g.support :=\n  Finset.ext fun a => by simp [Finsupp.smul_apply, hb]\n#align finsupp.support_smul_eq Finsupp.support_smul_eq\n\nsection\n\nvariable {p : \u03b1 \u2192 Prop}\n\n/- warning: finsupp.filter_smul -> Finsupp.filter_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {p : \u03b1 -> Prop} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] {b : R} {v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) p (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2)))) b v)) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2)))) b (Finsupp.filter.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) p v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {p : \u03b1 -> Prop} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] {b : R} {v : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)}, Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.filter.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) p (HSMul.hSMul.{u3, max u1 u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (instHSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddMonoid.toZero.{u2} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2))))) b v)) (HSMul.hSMul.{u3, max u2 u1, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (instHSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddMonoid.toZero.{u2} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2))))) b (Finsupp.filter.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) p v))\nCase conversion may be inaccurate. Consider using '#align finsupp.filter_smul Finsupp.filter_smul\u2093'. -/\n@[simp]\ntheorem filter_smul {_ : Monoid R} [AddMonoid M] [DistribMulAction R M] {b : R} {v : \u03b1 \u2192\u2080 M} :\n    (b \u2022 v).filter\u2093 p = b \u2022 v.filter\u2093 p :=\n  coeFn_injective <| Set.indicator_const_smul { x | p x } b v\n#align finsupp.filter_smul Finsupp.filter_smul\n\nend\n\n/- warning: finsupp.map_domain_smul -> Finsupp.mapDomain_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {R : Type.{u4}} {_x : Monoid.{u4} R} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : DistribMulAction.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1)] {f : \u03b1 -> \u03b2} (b : R) (v : Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))), Eq.{max (succ u2) (succ u3)} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f (SMul.smul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (SMulZeroClass.toHasSmul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b1 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribMulAction.toDistribSMul.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1) _inst_2)))) b v)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribMulAction.toDistribSMul.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1) _inst_2)))) b (Finsupp.mapDomain.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 f v))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {M : Type.{u3}} {R : Type.{u4}} {_x : Monoid.{u4} R} [_inst_1 : AddCommMonoid.{u3} M] [_inst_2 : DistribMulAction.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1)] {f : \u03b1 -> \u03b2} (b : R) (v : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.mapDomain.{u2, u1, u3} \u03b1 \u03b2 M _inst_1 f (HSMul.hSMul.{u4, max u2 u3, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (instHSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (SMulZeroClass.toSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b1 M R (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribSMul.toSMulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribMulAction.toDistribSMul.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1) _inst_2))))) b v)) (HSMul.hSMul.{u4, max u3 u1, max u1 u3} R (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (instHSMul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (SMulZeroClass.toSMul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.zero.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b2 M R (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribSMul.toSMulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)) (DistribMulAction.toDistribSMul.{u4, u3} R M _x (AddCommMonoid.toAddMonoid.{u3} M _inst_1) _inst_2))))) b (Finsupp.mapDomain.{u2, u1, u3} \u03b1 \u03b2 M _inst_1 f v))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_domain_smul Finsupp.mapDomain_smul\u2093'. -/\ntheorem mapDomain_smul {_ : Monoid R} [AddCommMonoid M] [DistribMulAction R M] {f : \u03b1 \u2192 \u03b2} (b : R)\n    (v : \u03b1 \u2192\u2080 M) : mapDomain f (b \u2022 v) = b \u2022 mapDomain f v :=\n  mapDomain_mapRange _ _ _ _ (smul_add b)\n#align finsupp.map_domain_smul Finsupp.mapDomain_smul\n\n/- warning: finsupp.smul_single -> Finsupp.smul_single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] (c : R) (a : \u03b1) (b : M), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMul.smul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2)))) c (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) a b)) (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) a (SMul.smul.{u3, u2} R M (SMulZeroClass.toHasSmul.{u3, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2))) c b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} {_x : Monoid.{u3} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u3, u2} R M _x _inst_1] (c : R) (a : \u03b1) (b : M), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (HSMul.hSMul.{u3, max u2 u1, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (instHSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.zero.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1)) (Finsupp.smulZeroClass.{u1, u2, u3} \u03b1 M R (AddMonoid.toZero.{u2} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2))))) c (Finsupp.single.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) a b)) (Finsupp.single.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M _inst_1) a (HSMul.hSMul.{u3, u2, u2} R M M (instHSMul.{u3, u2} R M (SMulZeroClass.toSMul.{u3, u2} R M (AddMonoid.toZero.{u2} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u2} R M _x _inst_1 _inst_2)))) c b))\nCase conversion may be inaccurate. Consider using '#align finsupp.smul_single Finsupp.smul_single\u2093'. -/\n@[simp]\ntheorem smul_single {_ : Monoid R} [AddMonoid M] [DistribMulAction R M] (c : R) (a : \u03b1) (b : M) :\n    c \u2022 Finsupp.single a b = Finsupp.single a (c \u2022 b) :=\n  mapRange_single\n#align finsupp.smul_single Finsupp.smul_single\n\n/- warning: finsupp.smul_single' -> Finsupp.smul_single' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {R : Type.{u2}} {_x : Semiring.{u2} R} (c : R) (a : \u03b1) (b : R), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))) (SMul.smul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))) (SMulZeroClass.toHasSmul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))) (Finsupp.zero.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))) (Finsupp.smulZeroClass.{u1, u2, u2} \u03b1 R R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) (SMulWithZero.toSmulZeroClass.{u2, u2} R R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) (MulZeroClass.toSMulWithZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))))) c (Finsupp.single.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) a b)) (Finsupp.single.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) a (HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (Distrib.toHasMul.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x))))) c b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {R : Type.{u2}} {_x : Semiring.{u2} R} (c : R) (a : \u03b1) (b : R), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (HSMul.hSMul.{u2, max u2 u1, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (instHSMul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (SMulZeroClass.toSMul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (Finsupp.zero.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x))) (Finsupp.smulZeroClass.{u1, u2, u2} \u03b1 R R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x)) (SMulWithZero.toSMulZeroClass.{u2, u2} R R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x)) (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x)) (MulZeroClass.toSMulWithZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))))))) c (Finsupp.single.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x)) a b)) (Finsupp.single.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _x)) a (HMul.hMul.{u2, u2, u2} R R R (instHMul.{u2} R (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _x)))) c b))\nCase conversion may be inaccurate. Consider using '#align finsupp.smul_single' Finsupp.smul_single'\u2093'. -/\n@[simp]\ntheorem smul_single' {_ : Semiring R} (c : R) (a : \u03b1) (b : R) :\n    c \u2022 Finsupp.single a b = Finsupp.single a (c * b) :=\n  smul_single _ _ _\n#align finsupp.smul_single' Finsupp.smul_single'\n\n/- warning: finsupp.map_range_smul -> Finsupp.mapRange_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {R : Type.{u4}} {_x : Monoid.{u4} R} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribMulAction.{u4, u2} R M _x _inst_1] [_inst_3 : AddMonoid.{u3} N] [_inst_4 : DistribMulAction.{u4, u3} R N _x _inst_3] {f : M -> N} {hf : Eq.{succ u3} N (f (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)))))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3)))))} (c : R) (v : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))), (forall (x : M), Eq.{succ u3} N (f (SMul.smul.{u4, u2} R M (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u2} R M _x _inst_1 _inst_2))) c x)) (SMul.smul.{u4, u3} R N (SMulZeroClass.toHasSmul.{u4, u3} R N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3)) (DistribSMul.toSmulZeroClass.{u4, u3} R N (AddMonoid.toAddZeroClass.{u3} N _inst_3) (DistribMulAction.toDistribSMul.{u4, u3} R N _x _inst_3 _inst_4))) c (f x))) -> (Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3))) (Finsupp.mapRange.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3)) f hf (SMul.smul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u4} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u2} R M _x _inst_1 _inst_2)))) c v)) (SMul.smul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3))) (SMulZeroClass.toHasSmul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3))) (Finsupp.zero.{u1, u3} \u03b1 N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3))) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b1 N R (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3)) (DistribSMul.toSmulZeroClass.{u4, u3} R N (AddMonoid.toAddZeroClass.{u3} N _inst_3) (DistribMulAction.toDistribSMul.{u4, u3} R N _x _inst_3 _inst_4)))) c (Finsupp.mapRange.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N _inst_3)) f hf v)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u3}} {N : Type.{u2}} {R : Type.{u4}} {_x : Monoid.{u4} R} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : DistribMulAction.{u4, u3} R M _x _inst_1] [_inst_3 : AddMonoid.{u2} N] [_inst_4 : DistribMulAction.{u4, u2} R N _x _inst_3] {f : M -> N} {hf : Eq.{succ u2} N (f (OfNat.ofNat.{u3} M 0 (Zero.toOfNat0.{u3} M (AddMonoid.toZero.{u3} M _inst_1)))) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N (AddMonoid.toZero.{u2} N _inst_3)))} (c : R) (v : Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)), (forall (x : M), Eq.{succ u2} N (f (HSMul.hSMul.{u4, u3, u3} R M M (instHSMul.{u4, u3} R M (SMulZeroClass.toSMul.{u4, u3} R M (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _x _inst_1 _inst_2)))) c x)) (HSMul.hSMul.{u4, u2, u2} R N N (instHSMul.{u4, u2} R N (SMulZeroClass.toSMul.{u4, u2} R N (AddMonoid.toZero.{u2} N _inst_3) (DistribSMul.toSMulZeroClass.{u4, u2} R N (AddMonoid.toAddZeroClass.{u2} N _inst_3) (DistribMulAction.toDistribSMul.{u4, u2} R N _x _inst_3 _inst_4)))) c (f x))) -> (Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (Finsupp.mapRange.{u1, u3, u2} \u03b1 M N (AddMonoid.toZero.{u3} M _inst_1) (AddMonoid.toZero.{u2} N _inst_3) f hf (HSMul.hSMul.{u4, max u1 u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (instHSMul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (SMulZeroClass.toSMul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.zero.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M _inst_1)) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b1 M R (AddMonoid.toZero.{u3} M _inst_1) (DistribSMul.toSMulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _x _inst_1 _inst_2))))) c v)) (HSMul.hSMul.{u4, max u2 u1, max u1 u2} R (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (instHSMul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (SMulZeroClass.toSMul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (Finsupp.zero.{u1, u2} \u03b1 N (AddMonoid.toZero.{u2} N _inst_3)) (Finsupp.smulZeroClass.{u1, u2, u4} \u03b1 N R (AddMonoid.toZero.{u2} N _inst_3) (DistribSMul.toSMulZeroClass.{u4, u2} R N (AddMonoid.toAddZeroClass.{u2} N _inst_3) (DistribMulAction.toDistribSMul.{u4, u2} R N _x _inst_3 _inst_4))))) c (Finsupp.mapRange.{u1, u3, u2} \u03b1 M N (AddMonoid.toZero.{u3} M _inst_1) (AddMonoid.toZero.{u2} N _inst_3) f hf v)))\nCase conversion may be inaccurate. Consider using '#align finsupp.map_range_smul Finsupp.mapRange_smul\u2093'. -/\ntheorem mapRange_smul {_ : Monoid R} [AddMonoid M] [DistribMulAction R M] [AddMonoid N]\n    [DistribMulAction R N] {f : M \u2192 N} {hf : f 0 = 0} (c : R) (v : \u03b1 \u2192\u2080 M)\n    (hsmul : \u2200 x, f (c \u2022 x) = c \u2022 f x) : mapRange f hf (c \u2022 v) = c \u2022 mapRange f hf v :=\n  by\n  erw [\u2190 map_range_comp]\n  have : f \u2218 (\u00b7 \u2022 \u00b7) c = (\u00b7 \u2022 \u00b7) c \u2218 f := funext hsmul\n  simp_rw [this]\n  apply map_range_comp\n  rw [Function.comp_apply, smul_zero, hf]\n#align finsupp.map_range_smul Finsupp.mapRange_smul\n\n/- warning: finsupp.smul_single_one -> Finsupp.smul_single_one is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] (a : \u03b1) (b : R), Eq.{succ (max u1 u2)} (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (SMul.smul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (SMulZeroClass.toHasSmul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (Finsupp.zero.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))) (Finsupp.smulZeroClass.{u1, u2, u2} \u03b1 R R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) (SMulWithZero.toSmulZeroClass.{u2, u2} R R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) (MulZeroClass.toSMulWithZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))))) b (Finsupp.single.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) a (OfNat.ofNat.{u2} R 1 (OfNat.mk.{u2} R 1 (One.one.{u2} R (AddMonoidWithOne.toOne.{u2} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} R (NonAssocSemiring.toAddCommMonoidWithOne.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))))))))) (Finsupp.single.{u1, u2} \u03b1 R (MulZeroClass.toHasZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))) a b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] (a : \u03b1) (b : R), Eq.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (HSMul.hSMul.{u2, max u2 u1, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (instHSMul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (SMulZeroClass.toSMul.{u2, max u1 u2} R (Finsupp.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u2} \u03b1 R R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (SMulWithZero.toSMulZeroClass.{u2, u2} R R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) (MulZeroClass.toSMulWithZero.{u2} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)))))))) b (Finsupp.single.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) a (OfNat.ofNat.{u2} R 1 (One.toOfNat1.{u2} R (Semiring.toOne.{u2} R _inst_1))))) (Finsupp.single.{u1, u2} \u03b1 R (MonoidWithZero.toZero.{u2} R (Semiring.toMonoidWithZero.{u2} R _inst_1)) a b)\nCase conversion may be inaccurate. Consider using '#align finsupp.smul_single_one Finsupp.smul_single_one\u2093'. -/\ntheorem smul_single_one [Semiring R] (a : \u03b1) (b : R) : b \u2022 single a 1 = single a b := by\n  rw [smul_single, smul_eq_mul, mul_one]\n#align finsupp.smul_single_one Finsupp.smul_single_one\n\n/- warning: finsupp.comap_domain_smul -> Finsupp.comapDomain_smul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {R : Type.{u4}} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : Monoid.{u4} R] [_inst_3 : DistribMulAction.{u4, u3} R M _inst_2 _inst_1] {f : \u03b1 -> \u03b2} (r : R) (v : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (hfv : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v)))) (hfrv : optParam.{0} (Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v))))) (Set.InjOn.mono.{u1, u2} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v)))) (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v))) f (Set.preimage_mono.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v)) (Iff.mpr (HasSubset.Subset.{u2} (Set.{u2} \u03b2) (Set.hasSubset.{u2} \u03b2) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v))) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v))) (HasSubset.Subset.{u2} (Finset.{u2} \u03b2) (Finset.hasSubset.{u2} \u03b2) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v)) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v)) (Finset.coe_subset.{u2} \u03b2 (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v)) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v)) (Finsupp.support_smul.{u2, u3, u4} \u03b2 M R _inst_2 _inst_1 _inst_3 r v))) hfv)), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) f (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v) hfrv) (SMul.smul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b1 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) f v hfv))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u4}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u4} M] [_inst_2 : Monoid.{u3} R] [_inst_3 : DistribMulAction.{u3, u4} R M _inst_2 _inst_1] {f : \u03b1 -> \u03b2} (r : R) (v : Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (hfv : Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v)))) (hfrv : optParam.{0} (Set.InjOn.{u1, u2} \u03b1 \u03b2 f (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v))))) (Set.InjOn.mono.{u2, u1} \u03b1 \u03b2 (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v)))) (Set.preimage.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v))) f (Set.preimage_mono.{u1, u2} \u03b1 \u03b2 f (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v)) (Iff.mpr (HasSubset.Subset.{u2} (Set.{u2} \u03b2) (Set.instHasSubsetSet.{u2} \u03b2) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v))) (Finset.toSet.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v))) (HasSubset.Subset.{u2} (Finset.{u2} \u03b2) (Finset.instHasSubsetFinset.{u2} \u03b2) (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v)) (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v)) (Finset.coe_subset.{u2} \u03b2 (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v)) (Finsupp.support.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v)) (Finsupp.support_smul.{u2, u4, u3} \u03b2 M R _inst_2 _inst_1 _inst_3 r v))) hfv)), Eq.{max (succ u1) (succ u4)} (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.comapDomain.{u1, u2, u4} \u03b1 \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) f (HSMul.hSMul.{u3, max u2 u4, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v) hfrv) (HSMul.hSMul.{u3, max u4 u1, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u1, u4, u3} \u03b1 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r (Finsupp.comapDomain.{u1, u2, u4} \u03b1 \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) f v hfv))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_smul Finsupp.comapDomain_smul\u2093'. -/\ntheorem comapDomain_smul [AddMonoid M] [Monoid R] [DistribMulAction R M] {f : \u03b1 \u2192 \u03b2} (r : R)\n    (v : \u03b2 \u2192\u2080 M) (hfv : Set.InjOn f (f \u207b\u00b9' \u2191v.support))\n    (hfrv : Set.InjOn f (f \u207b\u00b9' \u2191(r \u2022 v).support) :=\n      hfv.mono <| Set.preimage_mono <| Finset.coe_subset.mpr support_smul) :\n    comapDomain f (r \u2022 v) hfrv = r \u2022 comapDomain f v hfv :=\n  by\n  ext\n  rfl\n#align finsupp.comap_domain_smul Finsupp.comapDomain_smul\n\n/- warning: finsupp.comap_domain_smul_of_injective -> Finsupp.comapDomain_smul_of_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} {R : Type.{u4}} [_inst_1 : AddMonoid.{u3} M] [_inst_2 : Monoid.{u4} R] [_inst_3 : DistribMulAction.{u4, u3} R M _inst_2 _inst_1] {f : \u03b1 -> \u03b2} (hf : Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) (r : R) (v : Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))), Eq.{max (succ u1) (succ u3)} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) f (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v) (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (SMul.smul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u2, u3, u4} \u03b2 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r v)))))) (SMul.smul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u1 u3} R (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.zero.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1))) (Finsupp.smulZeroClass.{u1, u3, u4} \u03b1 M R (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u3} R M (AddMonoid.toAddZeroClass.{u3} M _inst_1) (DistribMulAction.toDistribSMul.{u4, u3} R M _inst_2 _inst_1 _inst_3)))) r (Finsupp.comapDomain.{u1, u2, u3} \u03b1 \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) f v (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u1, u2} \u03b1 \u03b2 f ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} \u03b2) (Set.{u2} \u03b2) (Finset.Set.hasCoeT.{u2} \u03b2))) (Finsupp.support.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M _inst_1)) v))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {M : Type.{u4}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u4} M] [_inst_2 : Monoid.{u3} R] [_inst_3 : DistribMulAction.{u3, u4} R M _inst_2 _inst_1] {f : \u03b1 -> \u03b2} (hf : Function.Injective.{succ u2, succ u1} \u03b1 \u03b2 f) (r : R) (v : Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)), Eq.{max (succ u2) (succ u4)} (Finsupp.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.comapDomain.{u2, u1, u4} \u03b1 \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) f (HSMul.hSMul.{u3, max u1 u4, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u1, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v) (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u2, u1} \u03b1 \u03b2 f (Finset.toSet.{u1} \u03b2 (Finsupp.support.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) (HSMul.hSMul.{u3, max u1 u4, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u1, u4, u3} \u03b2 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r v)))))) (HSMul.hSMul.{u3, max u4 u2, max u2 u4} R (Finsupp.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u2 u4} R (Finsupp.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u2, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u2, u4, u3} \u03b1 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) (DistribMulAction.toDistribSMul.{u3, u4} R M _inst_2 _inst_1 _inst_3))))) r (Finsupp.comapDomain.{u2, u1, u4} \u03b1 \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) f v (Function.Injective.injOn.{u1, u2} \u03b1 \u03b2 f hf (Set.preimage.{u2, u1} \u03b1 \u03b2 f (Finset.toSet.{u1} \u03b2 (Finsupp.support.{u1, u4} \u03b2 M (AddMonoid.toZero.{u4} M _inst_1) v))))))\nCase conversion may be inaccurate. Consider using '#align finsupp.comap_domain_smul_of_injective Finsupp.comapDomain_smul_of_injective\u2093'. -/\n/-- A version of `finsupp.comap_domain_smul` that's easier to use. -/\ntheorem comapDomain_smul_of_injective [AddMonoid M] [Monoid R] [DistribMulAction R M] {f : \u03b1 \u2192 \u03b2}\n    (hf : Function.Injective f) (r : R) (v : \u03b2 \u2192\u2080 M) :\n    comapDomain f (r \u2022 v) (hf.InjOn _) = r \u2022 comapDomain f v (hf.InjOn _) :=\n  comapDomain_smul _ _ _ _\n#align finsupp.comap_domain_smul_of_injective Finsupp.comapDomain_smul_of_injective\n\nend\n\n/- warning: finsupp.sum_smul_index -> Finsupp.sum_smul_index is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] {g : Finsupp.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))} {b : R} {h : \u03b1 -> R -> M}, (forall (i : \u03b1), Eq.{succ u2} M (h i (OfNat.ofNat.{u3} R 0 (OfNat.mk.{u3} R 0 (Zero.zero.{u3} R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))))))) (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))))) -> (Eq.{succ u2} M (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) _inst_2 (SMul.smul.{u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) (SMulZeroClass.toHasSmul.{u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) (Finsupp.zero.{u1, u3} \u03b1 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) (Finsupp.smulZeroClass.{u1, u3, u3} \u03b1 R R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) (SMulWithZero.toSmulZeroClass.{u3, u3} R R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) (MulZeroClass.toSMulWithZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))))) b g) h) (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) _inst_2 g (fun (i : \u03b1) (a : R) => h i (HMul.hMul.{u3, u3, u3} R R R (instHMul.{u3} R (Distrib.toHasMul.{u3} R (NonUnitalNonAssocSemiring.toDistrib.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1))))) b a))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] {g : Finsupp.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))} {b : R} {h : \u03b1 -> R -> M}, (forall (i : \u03b1), Eq.{succ u2} M (h i (OfNat.ofNat.{u3} R 0 (Zero.toOfNat0.{u3} R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))))) (OfNat.ofNat.{u2} M 0 (Zero.toOfNat0.{u2} M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))))) -> (Eq.{succ u2} M (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) _inst_2 (HSMul.hSMul.{u3, max u1 u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (Finsupp.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (instHSMul.{u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (SMulZeroClass.toSMul.{u3, max u1 u3} R (Finsupp.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (Finsupp.zero.{u1, u3} \u03b1 R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1))) (Finsupp.smulZeroClass.{u1, u3, u3} \u03b1 R R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (SMulWithZero.toSMulZeroClass.{u3, u3} R R (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (MulZeroClass.toSMulWithZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))))))) b g) h) (Finsupp.sum.{u1, u3, u2} \u03b1 R M (MonoidWithZero.toZero.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) _inst_2 g (fun (i : \u03b1) (a : R) => h i (HMul.hMul.{u3, u3, u3} R R R (instHMul.{u3} R (NonUnitalNonAssocSemiring.toMul.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R _inst_1)))) b a))))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_smul_index Finsupp.sum_smul_index\u2093'. -/\ntheorem sum_smul_index [Semiring R] [AddCommMonoid M] {g : \u03b1 \u2192\u2080 R} {b : R} {h : \u03b1 \u2192 R \u2192 M}\n    (h0 : \u2200 i, h i 0 = 0) : (b \u2022 g).Sum h = g.Sum fun i a => h i (b * a) :=\n  Finsupp.sum_mapRange_index h0\n#align finsupp.sum_smul_index Finsupp.sum_smul_index\n\n/- warning: finsupp.sum_smul_index' -> Finsupp.sum_smul_index' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {R : Type.{u4}} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : DistribSMul.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1)] [_inst_3 : AddCommMonoid.{u3} N] {g : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))} {b : R} {h : \u03b1 -> M -> N}, (forall (i : \u03b1), Eq.{succ u3} N (h i (OfNat.ofNat.{u2} M 0 (OfNat.mk.{u2} M 0 (Zero.zero.{u2} M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)))))) (OfNat.ofNat.{u3} N 0 (OfNat.mk.{u3} N 0 (Zero.zero.{u3} N (AddZeroClass.toHasZero.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_3))))))) -> (Eq.{succ u3} N (Finsupp.sum.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) _inst_3 (SMul.smul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u4} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2))) b g) h) (Finsupp.sum.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) _inst_3 g (fun (i : \u03b1) (c : M) => h i (SMul.smul.{u4, u2} R M (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_2)) b c))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u4}} {N : Type.{u2}} {R : Type.{u3}} [_inst_1 : AddMonoid.{u4} M] [_inst_2 : DistribSMul.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1)] [_inst_3 : AddCommMonoid.{u2} N] {g : Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)} {b : R} {h : \u03b1 -> M -> N}, (forall (i : \u03b1), Eq.{succ u2} N (h i (OfNat.ofNat.{u4} M 0 (Zero.toOfNat0.{u4} M (AddMonoid.toZero.{u4} M _inst_1)))) (OfNat.ofNat.{u2} N 0 (Zero.toOfNat0.{u2} N (AddMonoid.toZero.{u2} N (AddCommMonoid.toAddMonoid.{u2} N _inst_3))))) -> (Eq.{succ u2} N (Finsupp.sum.{u1, u4, u2} \u03b1 M N (AddMonoid.toZero.{u4} M _inst_1) _inst_3 (HSMul.hSMul.{u3, max u1 u4, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u3, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u1, u4, u3} \u03b1 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) _inst_2)))) b g) h) (Finsupp.sum.{u1, u4, u2} \u03b1 M N (AddMonoid.toZero.{u4} M _inst_1) _inst_3 g (fun (i : \u03b1) (c : M) => h i (HSMul.hSMul.{u3, u4, u4} R M M (instHSMul.{u3, u4} R M (SMulZeroClass.toSMul.{u3, u4} R M (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u3, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) _inst_2))) b c))))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_smul_index' Finsupp.sum_smul_index'\u2093'. -/\ntheorem sum_smul_index' [AddMonoid M] [DistribSMul R M] [AddCommMonoid N] {g : \u03b1 \u2192\u2080 M} {b : R}\n    {h : \u03b1 \u2192 M \u2192 N} (h0 : \u2200 i, h i 0 = 0) : (b \u2022 g).Sum h = g.Sum fun i c => h i (b \u2022 c) :=\n  Finsupp.sum_mapRange_index h0\n#align finsupp.sum_smul_index' Finsupp.sum_smul_index'\n\n/- warning: finsupp.sum_smul_index_add_monoid_hom -> Finsupp.sum_smul_index_addMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {R : Type.{u4}} [_inst_1 : AddMonoid.{u2} M] [_inst_2 : AddCommMonoid.{u3} N] [_inst_3 : DistribSMul.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1)] {g : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))} {b : R} {h : \u03b1 -> (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))}, Eq.{succ u3} N (Finsupp.sum.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) _inst_2 (SMul.smul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (SMulZeroClass.toHasSmul.{u4, max u1 u2} R (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.zero.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1))) (Finsupp.smulZeroClass.{u1, u2, u4} \u03b1 M R (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_3))) b g) (fun (a : \u03b1) => coeFn.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (fun (_x : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (h a))) (Finsupp.sum.{u1, u2, u3} \u03b1 M N (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) _inst_2 g (fun (i : \u03b1) (c : M) => coeFn.{max (succ u3) (succ u2), max (succ u2) (succ u3)} (AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (fun (_x : AddMonoidHom.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) => M -> N) (AddMonoidHom.hasCoeToFun.{u2, u3} M N (AddMonoid.toAddZeroClass.{u2} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (h i) (SMul.smul.{u4, u2} R M (SMulZeroClass.toHasSmul.{u4, u2} R M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M _inst_1)) (DistribSMul.toSmulZeroClass.{u4, u2} R M (AddMonoid.toAddZeroClass.{u2} M _inst_1) _inst_3)) b c)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u4}} {N : Type.{u3}} {R : Type.{u2}} [_inst_1 : AddMonoid.{u4} M] [_inst_2 : AddCommMonoid.{u3} N] [_inst_3 : DistribSMul.{u2, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1)] {g : Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)} {b : R} {h : \u03b1 -> (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)))}, Eq.{succ u3} N (Finsupp.sum.{u1, u4, u3} \u03b1 M N (AddMonoid.toZero.{u4} M _inst_1) _inst_2 (HSMul.hSMul.{u2, max u1 u4, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (instHSMul.{u2, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (SMulZeroClass.toSMul.{u2, max u1 u4} R (Finsupp.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.zero.{u1, u4} \u03b1 M (AddMonoid.toZero.{u4} M _inst_1)) (Finsupp.smulZeroClass.{u1, u4, u2} \u03b1 M R (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) _inst_3)))) b g) (fun (a : \u03b1) => FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M _inst_1)) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) (h a))) (Finsupp.sum.{u1, u4, u3} \u03b1 M N (AddMonoid.toZero.{u4} M _inst_1) _inst_2 g (fun (i : \u03b1) (c : M) => FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : M) => N) _x) (AddHomClass.toFunLike.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddZeroClass.toAdd.{u4} M (AddMonoid.toAddZeroClass.{u4} M _inst_1)) (AddZeroClass.toAdd.{u3} N (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) (AddMonoidHomClass.toAddHomClass.{max u4 u3, u4, u3} (AddMonoidHom.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))) M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2)) (AddMonoidHom.addMonoidHomClass.{u4, u3} M N (AddMonoid.toAddZeroClass.{u4} M _inst_1) (AddMonoid.toAddZeroClass.{u3} N (AddCommMonoid.toAddMonoid.{u3} N _inst_2))))) (h i) (HSMul.hSMul.{u2, u4, u4} R M M (instHSMul.{u2, u4} R M (SMulZeroClass.toSMul.{u2, u4} R M (AddMonoid.toZero.{u4} M _inst_1) (DistribSMul.toSMulZeroClass.{u2, u4} R M (AddMonoid.toAddZeroClass.{u4} M _inst_1) _inst_3))) b c)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sum_smul_index_add_monoid_hom Finsupp.sum_smul_index_addMonoidHom\u2093'. -/\n/-- A version of `finsupp.sum_smul_index'` for bundled additive maps. -/\ntheorem sum_smul_index_addMonoidHom [AddMonoid M] [AddCommMonoid N] [DistribSMul R M] {g : \u03b1 \u2192\u2080 M}\n    {b : R} {h : \u03b1 \u2192 M \u2192+ N} : ((b \u2022 g).Sum fun a => h a) = g.Sum fun i c => h i (b \u2022 c) :=\n  sum_mapRange_index fun i => (h i).map_zero\n#align finsupp.sum_smul_index_add_monoid_hom Finsupp.sum_smul_index_addMonoidHom\n\ninstance [Semiring R] [AddCommMonoid M] [Module R M] {\u03b9 : Type _} [NoZeroSMulDivisors R M] :\n    NoZeroSMulDivisors R (\u03b9 \u2192\u2080 M) :=\n  \u27e8fun c f h =>\n    or_iff_not_imp_left.mpr fun hc =>\n      Finsupp.ext fun i => (smul_eq_zero.mp (Finsupp.ext_iff.mp h i)).resolve_left hc\u27e9\n\nsection DistribMulActionHom\n\nvariable [Semiring R]\n\nvariable [AddCommMonoid M] [AddCommMonoid N] [DistribMulAction R M] [DistribMulAction R N]\n\n/- warning: finsupp.distrib_mul_action_hom.single -> Finsupp.DistribMulActionHom.single is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_4 : DistribMulAction.{u3, u2} R M (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)], \u03b1 -> (DistribMulActionHom.{u3, u2, max u1 u2} R (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4 (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u3} \u03b1 M R (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {R : Type.{u3}} [_inst_1 : Semiring.{u3} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_4 : DistribMulAction.{u3, u2} R M (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)], \u03b1 -> (DistribMulActionHom.{u3, u2, max u2 u1} R (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4 (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u3} \u03b1 M R (MonoidWithZero.toMonoid.{u3} R (Semiring.toMonoidWithZero.{u3} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4))\nCase conversion may be inaccurate. Consider using '#align finsupp.distrib_mul_action_hom.single Finsupp.DistribMulActionHom.single\u2093'. -/\n/-- `finsupp.single` as a `distrib_mul_action_hom`.\n\nSee also `finsupp.lsingle` for the version as a linear map. -/\ndef DistribMulActionHom.single (a : \u03b1) : M \u2192+[R] \u03b1 \u2192\u2080 M :=\n  { singleAddHom a with\n    map_smul' := fun k m => by\n      simp only [AddMonoidHom.toFun_eq_coe, single_add_hom_apply, smul_single] }\n#align finsupp.distrib_mul_action_hom.single Finsupp.DistribMulActionHom.single\n\n/- warning: finsupp.distrib_mul_action_hom_ext -> Finsupp.distribMulActionHom_ext is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : DistribMulAction.{u4, u2} R M (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] [_inst_5 : DistribMulAction.{u4, u3} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} N _inst_3)] {f : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5} {g : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5}, (forall (a : \u03b1) (m : M), Eq.{succ u3} N (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) (fun (_x : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) => (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) -> N) ([anonymous].{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) f (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) a m)) (coeFn.{max (succ (max u1 u2)) (succ u3), max (succ (max u1 u2)) (succ u3)} (DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) (fun (_x : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) => (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) -> N) ([anonymous].{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) g (Finsupp.single.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2))) a m))) -> (Eq.{max (succ (max u1 u2)) (succ u3)} (DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) f g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : DistribMulAction.{u4, u3} R M (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2)] [_inst_5 : DistribMulAction.{u4, u1} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3)] {f : DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5} {g : DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5}, (forall (a : \u03b1) (m : M), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) => N) (Finsupp.single.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) a m)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (succ u2) (succ u3), succ u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (fun (_x : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) => N) _x) (SMulHomClass.toFunLike.{max (max u2 u3) u1, u4, max u2 u3, u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) N (SMulZeroClass.toSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (AddMonoid.toZero.{max u2 u3} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (DistribSMul.toSMulZeroClass.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (AddMonoid.toAddZeroClass.{max u2 u3} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (DistribMulAction.toDistribSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4)))) (SMulZeroClass.toSMul.{u4, u1} R N (AddMonoid.toZero.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_3)) (DistribSMul.toSMulZeroClass.{u4, u1} R N (AddMonoid.toAddZeroClass.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_3)) (DistribMulAction.toDistribSMul.{u4, u1} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u3) u1, u4, max u2 u3, u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) _inst_5 (DistribMulActionHom.instDistribMulActionHomClassDistribMulActionHom.{u4, max u2 u3, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5))) f (Finsupp.single.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) a m)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (succ u2) (succ u3), succ u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (fun (_x : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) => N) _x) (SMulHomClass.toFunLike.{max (max u2 u3) u1, u4, max u2 u3, u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) N (SMulZeroClass.toSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (AddMonoid.toZero.{max u2 u3} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (DistribSMul.toSMulZeroClass.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (AddMonoid.toAddZeroClass.{max u2 u3} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (DistribMulAction.toDistribSMul.{u4, max u2 u3} R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4)))) (SMulZeroClass.toSMul.{u4, u1} R N (AddMonoid.toZero.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_3)) (DistribSMul.toSMulZeroClass.{u4, u1} R N (AddMonoid.toAddZeroClass.{u1} N (AddCommMonoid.toAddMonoid.{u1} N _inst_3)) (DistribMulAction.toDistribSMul.{u4, u1} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u3) u1, u4, max u2 u3, u1} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) R (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) _inst_5 (DistribMulActionHom.instDistribMulActionHomClassDistribMulActionHom.{u4, max u2 u3, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5))) g (Finsupp.single.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) a m))) -> (Eq.{max (max (succ u2) (succ u3)) (succ u1)} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) f g)\nCase conversion may be inaccurate. Consider using '#align finsupp.distrib_mul_action_hom_ext Finsupp.distribMulActionHom_ext\u2093'. -/\ntheorem distribMulActionHom_ext {f g : (\u03b1 \u2192\u2080 M) \u2192+[R] N}\n    (h : \u2200 (a : \u03b1) (m : M), f (single a m) = g (single a m)) : f = g :=\n  DistribMulActionHom.toAddMonoidHom_injective <| addHom_ext h\n#align finsupp.distrib_mul_action_hom_ext Finsupp.distribMulActionHom_ext\n\n/- warning: finsupp.distrib_mul_action_hom_ext' -> Finsupp.distribMulActionHom_ext' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] [_inst_2 : AddCommMonoid.{u2} M] [_inst_3 : AddCommMonoid.{u3} N] [_inst_4 : DistribMulAction.{u4, u2} R M (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2)] [_inst_5 : DistribMulAction.{u4, u3} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} N _inst_3)] {f : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5} {g : DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5}, (forall (a : \u03b1), Eq.{max (succ u2) (succ u3)} (DistribMulActionHom.{u4, u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4 N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) (DistribMulActionHom.comp.{u4, u2, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4 (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5 f (Finsupp.DistribMulActionHom.single.{u1, u2, u4} \u03b1 M R _inst_1 _inst_2 _inst_4 a)) (DistribMulActionHom.comp.{u4, u2, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4 (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5 g (Finsupp.DistribMulActionHom.single.{u1, u2, u4} \u03b1 M R _inst_1 _inst_2 _inst_4 a))) -> (Eq.{max (succ (max u1 u2)) (succ u3)} (DistribMulActionHom.{u4, max u1 u2, u3} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)))) (Finsupp.addMonoid.{u1, u2} \u03b1 M (AddCommMonoid.toAddMonoid.{u2} M _inst_2)) (Finsupp.distribMulAction.{u1, u2, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u2} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u3} N _inst_3) _inst_5) f g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {M : Type.{u3}} {N : Type.{u1}} {R : Type.{u4}} [_inst_1 : Semiring.{u4} R] [_inst_2 : AddCommMonoid.{u3} M] [_inst_3 : AddCommMonoid.{u1} N] [_inst_4 : DistribMulAction.{u4, u3} R M (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2)] [_inst_5 : DistribMulAction.{u4, u1} R N (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u1} N _inst_3)] {f : DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5} {g : DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5}, (forall (a : \u03b1), Eq.{max (succ u3) (succ u1)} (DistribMulActionHom.{u4, u3, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4 N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) (DistribMulActionHom.comp.{u4, u3, max u2 u3, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4 (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5 f (Finsupp.DistribMulActionHom.single.{u2, u3, u4} \u03b1 M R _inst_1 _inst_2 _inst_4 a)) (DistribMulActionHom.comp.{u4, u3, max u2 u3, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) M (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4 (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5 g (Finsupp.DistribMulActionHom.single.{u2, u3, u4} \u03b1 M R _inst_1 _inst_2 _inst_4 a))) -> (Eq.{max (max (succ u2) (succ u3)) (succ u1)} (DistribMulActionHom.{u4, max u3 u2, u1} R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_2))) (Finsupp.addMonoid.{u2, u3} \u03b1 M (AddCommMonoid.toAddMonoid.{u3} M _inst_2)) (Finsupp.distribMulAction.{u2, u3, u4} \u03b1 M R (MonoidWithZero.toMonoid.{u4} R (Semiring.toMonoidWithZero.{u4} R _inst_1)) (AddCommMonoid.toAddMonoid.{u3} M _inst_2) _inst_4) N (AddCommMonoid.toAddMonoid.{u1} N _inst_3) _inst_5) f g)\nCase conversion may be inaccurate. Consider using '#align finsupp.distrib_mul_action_hom_ext' Finsupp.distribMulActionHom_ext'\u2093'. -/\n/-- See note [partially-applied ext lemmas]. -/\n@[ext]\ntheorem distribMulActionHom_ext' {f g : (\u03b1 \u2192\u2080 M) \u2192+[R] N}\n    (h : \u2200 a : \u03b1, f.comp (DistribMulActionHom.single a) = g.comp (DistribMulActionHom.single a)) :\n    f = g :=\n  distribMulActionHom_ext fun a => DistribMulActionHom.congr_fun (h a)\n#align finsupp.distrib_mul_action_hom_ext' Finsupp.distribMulActionHom_ext'\n\nend DistribMulActionHom\n\nsection\n\nvariable [Zero R]\n\n#print Finsupp.uniqueOfRight /-\n/-- The `finsupp` version of `pi.unique`. -/\ninstance uniqueOfRight [Subsingleton R] : Unique (\u03b1 \u2192\u2080 R) :=\n  FunLike.coe_injective.unique\n#align finsupp.unique_of_right Finsupp.uniqueOfRight\n-/\n\n#print Finsupp.uniqueOfLeft /-\n/-- The `finsupp` version of `pi.unique_of_is_empty`. -/\ninstance uniqueOfLeft [IsEmpty \u03b1] : Unique (\u03b1 \u2192\u2080 R) :=\n  FunLike.coe_injective.unique\n#align finsupp.unique_of_left Finsupp.uniqueOfLeft\n-/\n\nend\n\n/- warning: finsupp.restrict_support_equiv -> Finsupp.restrictSupportEquiv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} (s : Set.{u1} \u03b1) (M : Type.{u2}) [_inst_1 : AddCommMonoid.{u2} M], Equiv.{max 1 (succ u1) (succ u2), max (succ u1) (succ u2)} (Subtype.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (fun (f : Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) => HasSubset.Subset.{u1} (Set.{u1} \u03b1) (Set.hasSubset.{u1} \u03b1) ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (HasLiftT.mk.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (CoeTC\u2093.coe.{succ u1, succ u1} (Finset.{u1} \u03b1) (Set.{u1} \u03b1) (Finset.Set.hasCoeT.{u1} \u03b1))) (Finsupp.support.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) f)) s)) (Finsupp.{u1, u2} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} \u03b1) Type.{u1} (Set.hasCoeToSort.{u1} \u03b1) s) M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} (s : Set.{u1} \u03b1) (M : Type.{u2}) [_inst_1 : AddCommMonoid.{u2} M], Equiv.{max 1 (succ u1) (succ u2), max (succ u2) (succ u1)} (Subtype.{max (succ u1) (succ u2)} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (fun (f : Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) => HasSubset.Subset.{u1} (Set.{u1} \u03b1) (Set.instHasSubsetSet.{u1} \u03b1) (Finset.toSet.{u1} \u03b1 (Finsupp.support.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)) f)) s)) (Finsupp.{u1, u2} (Set.Elem.{u1} \u03b1 s) M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))\nCase conversion may be inaccurate. Consider using '#align finsupp.restrict_support_equiv Finsupp.restrictSupportEquiv\u2093'. -/\n/-- Given an `add_comm_monoid M` and `s : set \u03b1`, `restrict_support_equiv s M` is the `equiv`\nbetween the subtype of finitely supported functions with support contained in `s` and\nthe type of finitely supported functions from `s`. -/\ndef restrictSupportEquiv (s : Set \u03b1) (M : Type _) [AddCommMonoid M] :\n    { f : \u03b1 \u2192\u2080 M // \u2191f.support \u2286 s } \u2243 (s \u2192\u2080 M)\n    where\n  toFun f := subtypeDomain (fun x => x \u2208 s) f.1\n  invFun f :=\n    \u27e8f.mapDomain Subtype.val, by\n      classical\n        refine' Set.Subset.trans (Finset.coe_subset.2 map_domain_support) _\n        rw [Finset.coe_image, Set.image_subset_iff]\n        exact fun x hx => x.2\u27e9\n  left_inv := by\n    rintro \u27e8f, hf\u27e9\n    apply Subtype.eq\n    ext a\n    dsimp only\n    refine' by_cases (fun h : a \u2208 Set.range (Subtype.val : s \u2192 \u03b1) => _) fun h => _\n    \u00b7 rcases h with \u27e8x, rfl\u27e9\n      rw [map_domain_apply Subtype.val_injective, subtype_domain_apply]\n    \u00b7 convert map_domain_notin_range _ _ h\n      rw [\u2190 not_mem_support_iff]\n      refine' mt _ h\n      exact fun ha => \u27e8\u27e8a, hf ha\u27e9, rfl\u27e9\n  right_inv f := by\n    ext \u27e8a, ha\u27e9\n    dsimp only\n    rw [subtype_domain_apply, map_domain_apply Subtype.val_injective]\n#align finsupp.restrict_support_equiv Finsupp.restrictSupportEquiv\n\n/- warning: finsupp.dom_congr -> Finsupp.domCongr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Equiv.{succ u1, succ u2} \u03b1 \u03b2) -> (AddEquiv.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M], (Equiv.{succ u1, succ u2} \u03b1 \u03b2) -> (AddEquiv.{max u3 u1, max u3 u2} (Finsupp.{u1, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u2, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.dom_congr Finsupp.domCongr\u2093'. -/\n/-- Given `add_comm_monoid M` and `e : \u03b1 \u2243 \u03b2`, `dom_congr e` is the corresponding `equiv` between\n`\u03b1 \u2192\u2080 M` and `\u03b2 \u2192\u2080 M`.\n\nThis is `finsupp.equiv_congr_left` as an `add_equiv`. -/\n@[simps apply]\nprotected def domCongr [AddCommMonoid M] (e : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192\u2080 M) \u2243+ (\u03b2 \u2192\u2080 M)\n    where\n  toFun := equivMapDomain e\n  invFun := equivMapDomain e.symm\n  left_inv v := by\n    simp only [\u2190 equiv_map_domain_trans, Equiv.self_trans_symm]\n    exact equiv_map_domain_refl _\n  right_inv := by\n    intro v\n    simp only [\u2190 equiv_map_domain_trans, Equiv.symm_trans_self]\n    exact equiv_map_domain_refl _\n  map_add' a b := by simp only [equiv_map_domain_eq_map_domain] <;> exact map_domain_add\n#align finsupp.dom_congr Finsupp.domCongr\n\n/- warning: finsupp.dom_congr_refl -> Finsupp.domCongr_refl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M], Eq.{succ (max u1 u2)} (AddEquiv.{max u1 u2, max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.domCongr.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u1} \u03b1)) (AddEquiv.refl.{max u1 u2} (Finsupp.{u1, u2} \u03b1 M (AddZeroClass.toHasZero.{u2} M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {M : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} M], Eq.{max (succ u1) (succ u2)} (AddEquiv.{max u2 u1, max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1)))) (Finsupp.domCongr.{u1, u1, u2} \u03b1 \u03b1 M _inst_1 (Equiv.refl.{succ u1} \u03b1)) (AddEquiv.refl.{max u2 u1} (Finsupp.{u1, u2} \u03b1 M (AddMonoid.toZero.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))) (Finsupp.add.{u1, u2} \u03b1 M (AddMonoid.toAddZeroClass.{u2} M (AddCommMonoid.toAddMonoid.{u2} M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.dom_congr_refl Finsupp.domCongr_refl\u2093'. -/\n@[simp]\ntheorem domCongr_refl [AddCommMonoid M] :\n    Finsupp.domCongr (Equiv.refl \u03b1) = AddEquiv.refl (\u03b1 \u2192\u2080 M) :=\n  AddEquiv.ext fun _ => equivMapDomain_refl _\n#align finsupp.dom_congr_refl Finsupp.domCongr_refl\n\n/- warning: finsupp.dom_congr_symm -> Finsupp.domCongr_symm is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (e : Equiv.{succ u1, succ u2} \u03b1 \u03b2), Eq.{max (succ (max u2 u3)) (succ (max u1 u3))} (AddEquiv.{max u2 u3, max u1 u3} (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.add.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (AddEquiv.symm.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b1 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.{u2, u3} \u03b2 M (AddZeroClass.toHasZero.{u3} M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (Finsupp.add.{u1, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u2, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.domCongr.{u1, u2, u3} \u03b1 \u03b2 M _inst_1 e)) (Finsupp.domCongr.{u2, u1, u3} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u1, succ u2} \u03b1 \u03b2 e))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} {M : Type.{u3}} [_inst_1 : AddCommMonoid.{u3} M] (e : Equiv.{succ u2, succ u1} \u03b1 \u03b2), Eq.{max (max (succ u2) (succ u1)) (succ u3)} (AddEquiv.{max u1 u3, max u2 u3} (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u2, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1)))) (AddEquiv.symm.{max u2 u3, max u1 u3} (Finsupp.{u2, u3} \u03b1 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.{u1, u3} \u03b2 M (AddMonoid.toZero.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u2, u3} \u03b1 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.add.{u1, u3} \u03b2 M (AddMonoid.toAddZeroClass.{u3} M (AddCommMonoid.toAddMonoid.{u3} M _inst_1))) (Finsupp.domCongr.{u2, u1, u3} \u03b1 \u03b2 M _inst_1 e)) (Finsupp.domCongr.{u1, u2, u3} \u03b2 \u03b1 M _inst_1 (Equiv.symm.{succ u2, succ u1} \u03b1 \u03b2 e))\nCase conversion may be inaccurate. Consider using '#align finsupp.dom_congr_symm Finsupp.domCongr_symm\u2093'. -/\n@[simp]\ntheorem domCongr_symm [AddCommMonoid M] (e : \u03b1 \u2243 \u03b2) :\n    (Finsupp.domCongr e).symm = (Finsupp.domCongr e.symm : (\u03b2 \u2192\u2080 M) \u2243+ (\u03b1 \u2192\u2080 M)) :=\n  AddEquiv.ext fun _ => rfl\n#align finsupp.dom_congr_symm Finsupp.domCongr_symm\n\n/- warning: finsupp.dom_congr_trans -> Finsupp.domCongr_trans is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {M : Type.{u4}} [_inst_1 : AddCommMonoid.{u4} M] (e : Equiv.{succ u1, succ u2} \u03b1 \u03b2) (f : Equiv.{succ u2, succ u3} \u03b2 \u03b3), Eq.{max (succ (max u1 u4)) (succ (max u3 u4))} (AddEquiv.{max u1 u4, max u3 u4} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u3, u4} \u03b3 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.add.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u3, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (AddEquiv.trans.{max u1 u4, max u2 u4, max u3 u4} (Finsupp.{u1, u4} \u03b1 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u2, u4} \u03b2 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.{u3, u4} \u03b3 M (AddZeroClass.toHasZero.{u4} M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (Finsupp.add.{u1, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u2, u4} \u03b2 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u3, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.domCongr.{u1, u2, u4} \u03b1 \u03b2 M _inst_1 e) (Finsupp.domCongr.{u2, u3, u4} \u03b2 \u03b3 M _inst_1 f)) (Finsupp.domCongr.{u1, u3, u4} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u1, succ u2, succ u3} \u03b1 \u03b2 \u03b3 e f))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} {M : Type.{u4}} [_inst_1 : AddCommMonoid.{u4} M] (e : Equiv.{succ u3, succ u2} \u03b1 \u03b2) (f : Equiv.{succ u2, succ u1} \u03b2 \u03b3), Eq.{max (max (succ u3) (succ u1)) (succ u4)} (AddEquiv.{max u3 u4, max u4 u1} (Finsupp.{u3, u4} \u03b1 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.{u1, u4} \u03b3 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u3, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u1, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1)))) (AddEquiv.trans.{max u3 u4, max u2 u4, max u4 u1} (Finsupp.{u3, u4} \u03b1 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.{u2, u4} \u03b2 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.{u1, u4} \u03b3 M (AddMonoid.toZero.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u3, u4} \u03b1 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u2, u4} \u03b2 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.add.{u1, u4} \u03b3 M (AddMonoid.toAddZeroClass.{u4} M (AddCommMonoid.toAddMonoid.{u4} M _inst_1))) (Finsupp.domCongr.{u3, u2, u4} \u03b1 \u03b2 M _inst_1 e) (Finsupp.domCongr.{u2, u1, u4} \u03b2 \u03b3 M _inst_1 f)) (Finsupp.domCongr.{u3, u1, u4} \u03b1 \u03b3 M _inst_1 (Equiv.trans.{succ u3, succ u2, succ u1} \u03b1 \u03b2 \u03b3 e f))\nCase conversion may be inaccurate. Consider using '#align finsupp.dom_congr_trans Finsupp.domCongr_trans\u2093'. -/\n@[simp]\ntheorem domCongr_trans [AddCommMonoid M] (e : \u03b1 \u2243 \u03b2) (f : \u03b2 \u2243 \u03b3) :\n    (Finsupp.domCongr e).trans (Finsupp.domCongr f) =\n      (Finsupp.domCongr (e.trans f) : (\u03b1 \u2192\u2080 M) \u2243+ _) :=\n  AddEquiv.ext fun _ => (equivMapDomain_trans _ _ _).symm\n#align finsupp.dom_congr_trans Finsupp.domCongr_trans\n\nend Finsupp\n\nnamespace Finsupp\n\n/-! ### Declarations about sigma types -/\n\n\nsection Sigma\n\nvariable {\u03b1s : \u03b9 \u2192 Type _} [Zero M] (l : (\u03a3i, \u03b1s i) \u2192\u2080 M)\n\n#print Finsupp.split /-\n/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `M` and\nan index element `i : \u03b9`, `split l i` is the `i`th component of `l`,\na finitely supported function from `as i` to `M`.\n\nThis is the `finsupp` version of `sigma.curry`.\n-/\ndef split (i : \u03b9) : \u03b1s i \u2192\u2080 M :=\n  l.comapDomain (Sigma.mk i) fun x1 x2 _ _ hx => heq_iff_eq.1 (Sigma.mk.inj hx).2\n#align finsupp.split Finsupp.split\n-/\n\n/- warning: finsupp.split_apply -> Finsupp.split_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {M : Type.{u2}} {\u03b1s : \u03b9 -> Type.{u3}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (i : \u03b9) (x : \u03b1s i), Eq.{succ u2} M (coeFn.{max (succ u3) (succ u2), max (succ u3) (succ u2)} (Finsupp.{u3, u2} ((fun (i : \u03b9) => \u03b1s i) i) M _inst_1) (fun (_x : Finsupp.{u3, u2} ((fun (i : \u03b9) => \u03b1s i) i) M _inst_1) => (\u03b1s i) -> M) (Finsupp.coeFun.{u3, u2} ((fun (i : \u03b9) => \u03b1s i) i) M _inst_1) (Finsupp.split.{u1, u2, u3} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) x) (coeFn.{max (succ (max u1 u3)) (succ u2), max (succ (max u1 u3)) (succ u2)} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (fun (_x : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) => (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) -> M) (Finsupp.coeFun.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) l (Sigma.mk.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i) i x))\nbut is expected to have type\n  forall {\u03b9 : Type.{u1}} {M : Type.{u3}} {\u03b1s : \u03b9 -> Type.{u2}} [_inst_1 : Zero.{u3} M] (l : Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (i : \u03b9) (x : \u03b1s i), Eq.{succ u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1s i) => M) x) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (Finsupp.{u2, u3} (\u03b1s i) M _inst_1) (\u03b1s i) (fun (_x : \u03b1s i) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b1s i) => M) _x) (Finsupp.funLike.{u2, u3} (\u03b1s i) M _inst_1) (Finsupp.split.{u1, u3, u2} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) x) (FunLike.coe.{max (succ (max u1 u2)) (succ u3), succ (max u1 u2), succ u3} (Finsupp.{max u1 u2, u3} (Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) (fun (_x : Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) => M) _x) (Finsupp.funLike.{max u1 u2, u3} (Sigma.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) l (Sigma.mk.{u1, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i) i x))\nCase conversion may be inaccurate. Consider using '#align finsupp.split_apply Finsupp.split_apply\u2093'. -/\ntheorem split_apply (i : \u03b9) (x : \u03b1s i) : split l i x = l \u27e8i, x\u27e9 :=\n  by\n  dsimp only [split]\n  rw [comap_domain_apply]\n#align finsupp.split_apply Finsupp.split_apply\n\n#print Finsupp.splitSupport /-\n/-- Given `l`, a finitely supported function from the sigma type `\u03a3 (i : \u03b9), \u03b1s i` to `\u03b2`,\n`split_support l` is the finset of indices in `\u03b9` that appear in the support of `l`. -/\ndef splitSupport (l : (\u03a3i, \u03b1s i) \u2192\u2080 M) : Finset \u03b9 :=\n  haveI := Classical.decEq \u03b9\n  l.support.image Sigma.fst\n#align finsupp.split_support Finsupp.splitSupport\n-/\n\n/- warning: finsupp.mem_split_support_iff_nonzero -> Finsupp.mem_splitSupport_iff_nonzero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {M : Type.{u2}} {\u03b1s : \u03b9 -> Type.{u3}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (i : \u03b9), Iff (Membership.Mem.{u1, u1} \u03b9 (Finset.{u1} \u03b9) (Finset.hasMem.{u1} \u03b9) i (Finsupp.splitSupport.{u1, u2, u3} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l)) (Ne.{max (succ u3) (succ u2)} (Finsupp.{u3, u2} (\u03b1s i) M _inst_1) (Finsupp.split.{u1, u2, u3} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) (OfNat.ofNat.{max u3 u2} (Finsupp.{u3, u2} (\u03b1s i) M _inst_1) 0 (OfNat.mk.{max u3 u2} (Finsupp.{u3, u2} (\u03b1s i) M _inst_1) 0 (Zero.zero.{max u3 u2} (Finsupp.{u3, u2} (\u03b1s i) M _inst_1) (Finsupp.zero.{u3, u2} (\u03b1s i) M _inst_1)))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u3}} {M : Type.{u2}} {\u03b1s : \u03b9 -> Type.{u1}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{max u1 u3, u2} (Sigma.{u3, u1} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) (i : \u03b9), Iff (Membership.mem.{u3, u3} \u03b9 (Finset.{u3} \u03b9) (Finset.instMembershipFinset.{u3} \u03b9) i (Finsupp.splitSupport.{u3, u2, u1} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l)) (Ne.{max (succ u2) (succ u1)} (Finsupp.{u1, u2} (\u03b1s i) M _inst_1) (Finsupp.split.{u3, u2, u1} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) (OfNat.ofNat.{max u2 u1} (Finsupp.{u1, u2} (\u03b1s i) M _inst_1) 0 (Zero.toOfNat0.{max u2 u1} (Finsupp.{u1, u2} (\u03b1s i) M _inst_1) (Finsupp.zero.{u1, u2} (\u03b1s i) M _inst_1))))\nCase conversion may be inaccurate. Consider using '#align finsupp.mem_split_support_iff_nonzero Finsupp.mem_splitSupport_iff_nonzero\u2093'. -/\ntheorem mem_splitSupport_iff_nonzero (i : \u03b9) : i \u2208 splitSupport l \u2194 split l i \u2260 0 :=\n  by\n  rw [split_support, mem_image, Ne.def, \u2190 support_eq_empty, \u2190 Ne.def, \u2190\n    Finset.nonempty_iff_ne_empty, split, comap_domain, Finset.Nonempty]\n  simp only [exists_prop, Finset.mem_preimage, exists_and_right, exists_eq_right, mem_support_iff,\n    Sigma.exists, Ne.def]\n#align finsupp.mem_split_support_iff_nonzero Finsupp.mem_splitSupport_iff_nonzero\n\n#print Finsupp.splitComp /-\n/-- Given `l`, a finitely supported function from the sigma type `\u03a3 i, \u03b1s i` to `\u03b2` and\nan `\u03b9`-indexed family `g` of functions from `(\u03b1s i \u2192\u2080 \u03b2)` to `\u03b3`, `split_comp` defines a\nfinitely supported function from the index type `\u03b9` to `\u03b3` given by composing `g i` with\n`split l i`. -/\ndef splitComp [Zero N] (g : \u2200 i, (\u03b1s i \u2192\u2080 M) \u2192 N) (hg : \u2200 i x, x = 0 \u2194 g i x = 0) : \u03b9 \u2192\u2080 N\n    where\n  support := splitSupport l\n  toFun i := g i (split l i)\n  mem_support_toFun := by\n    intro i\n    rw [mem_split_support_iff_nonzero, not_iff_not, hg]\n#align finsupp.split_comp Finsupp.splitComp\n-/\n\n/- warning: finsupp.sigma_support -> Finsupp.sigma_support is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {M : Type.{u2}} {\u03b1s : \u03b9 -> Type.{u3}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1), Eq.{succ (max u1 u3)} (Finset.{max u1 u3} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i))) (Finsupp.support.{max u1 u3, u2} (Sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1 l) (Finset.sigma.{u1, u3} \u03b9 (fun (i : \u03b9) => \u03b1s i) (Finsupp.splitSupport.{u1, u2, u3} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l) (fun (i : \u03b9) => Finsupp.support.{u3, u2} (\u03b1s i) M _inst_1 (Finsupp.split.{u1, u2, u3} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i)))\nbut is expected to have type\n  forall {\u03b9 : Type.{u3}} {M : Type.{u1}} {\u03b1s : \u03b9 -> Type.{u2}} [_inst_1 : Zero.{u1} M] (l : Finsupp.{max u2 u3, u1} (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1), Eq.{max (succ u3) (succ u2)} (Finset.{max u3 u2} (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i))) (Finsupp.support.{max u3 u2, u1} (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1 l) (Finset.sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i) (Finsupp.splitSupport.{u3, u1, u2} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l) (fun (i : \u03b9) => Finsupp.support.{u2, u1} (\u03b1s i) M _inst_1 (Finsupp.split.{u3, u1, u2} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sigma_support Finsupp.sigma_support\u2093'. -/\ntheorem sigma_support : l.support = l.splitSupport.Sigma fun i => (l.split i).support := by\n  simp only [Finset.ext_iff, split_support, split, comap_domain, mem_image, mem_preimage,\n      Sigma.forall, mem_sigma] <;>\n    tauto\n#align finsupp.sigma_support Finsupp.sigma_support\n\n/- warning: finsupp.sigma_sum -> Finsupp.sigma_sum is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {M : Type.{u2}} {N : Type.{u3}} {\u03b1s : \u03b9 -> Type.{u4}} [_inst_1 : Zero.{u2} M] (l : Finsupp.{max u1 u4, u2} (Sigma.{u1, u4} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) [_inst_2 : AddCommMonoid.{u3} N] (f : (Sigma.{u1, u4} \u03b9 (fun (i : \u03b9) => \u03b1s i)) -> M -> N), Eq.{succ u3} N (Finsupp.sum.{max u1 u4, u2, u3} (Sigma.{u1, u4} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M N _inst_1 _inst_2 l f) (Finset.sum.{u3, u1} N \u03b9 _inst_2 (Finsupp.splitSupport.{u1, u2, u4} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l) (fun (i : \u03b9) => Finsupp.sum.{u4, u2, u3} (\u03b1s i) M N _inst_1 _inst_2 (Finsupp.split.{u1, u2, u4} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) (fun (a : \u03b1s i) (b : M) => f (Sigma.mk.{u1, u4} \u03b9 (fun (i : \u03b9) => \u03b1s i) i a) b)))\nbut is expected to have type\n  forall {\u03b9 : Type.{u3}} {M : Type.{u1}} {N : Type.{u4}} {\u03b1s : \u03b9 -> Type.{u2}} [_inst_1 : Zero.{u1} M] (l : Finsupp.{max u2 u3, u1} (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M _inst_1) [_inst_2 : AddCommMonoid.{u4} N] (f : (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) -> M -> N), Eq.{succ u4} N (Finsupp.sum.{max u3 u2, u1, u4} (Sigma.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i)) M N _inst_1 _inst_2 l f) (Finset.sum.{u4, u3} N \u03b9 _inst_2 (Finsupp.splitSupport.{u3, u1, u2} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l) (fun (i : \u03b9) => Finsupp.sum.{u2, u1, u4} (\u03b1s i) M N _inst_1 _inst_2 (Finsupp.split.{u3, u1, u2} \u03b9 M (fun (i : \u03b9) => \u03b1s i) _inst_1 l i) (fun (a : \u03b1s i) (b : M) => f (Sigma.mk.{u3, u2} \u03b9 (fun (i : \u03b9) => \u03b1s i) i a) b)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sigma_sum Finsupp.sigma_sum\u2093'. -/\ntheorem sigma_sum [AddCommMonoid N] (f : (\u03a3i : \u03b9, \u03b1s i) \u2192 M \u2192 N) :\n    l.Sum f = \u2211 i in splitSupport l, (split l i).Sum fun (a : \u03b1s i) b => f \u27e8i, a\u27e9 b := by\n  simp only [Sum, sigma_support, sum_sigma, split_apply]\n#align finsupp.sigma_sum Finsupp.sigma_sum\n\nvariable {\u03b7 : Type _} [Fintype \u03b7] {\u03b9s : \u03b7 \u2192 Type _} [Zero \u03b1]\n\n#print Finsupp.sigmaFinsuppEquivPiFinsupp /-\n/-- On a `fintype \u03b7`, `finsupp.split` is an equivalence between `(\u03a3 (j : \u03b7), \u03b9s j) \u2192\u2080 \u03b1`\nand `\u03a0 j, (\u03b9s j \u2192\u2080 \u03b1)`.\n\nThis is the `finsupp` version of `equiv.Pi_curry`. -/\nnoncomputable def sigmaFinsuppEquivPiFinsupp : ((\u03a3j, \u03b9s j) \u2192\u2080 \u03b1) \u2243 \u2200 j, \u03b9s j \u2192\u2080 \u03b1\n    where\n  toFun := split\n  invFun f :=\n    onFinset (Finset.univ.Sigma fun j => (f j).support) (fun ji => f ji.1 ji.2) fun g hg =>\n      Finset.mem_sigma.mpr \u27e8Finset.mem_univ _, mem_support_iff.mpr hg\u27e9\n  left_inv f := by\n    ext\n    simp [split]\n  right_inv f := by\n    ext\n    simp [split]\n#align finsupp.sigma_finsupp_equiv_pi_finsupp Finsupp.sigmaFinsuppEquivPiFinsupp\n-/\n\n#print Finsupp.sigmaFinsuppEquivPiFinsupp_apply /-\n@[simp]\ntheorem sigmaFinsuppEquivPiFinsupp_apply (f : (\u03a3j, \u03b9s j) \u2192\u2080 \u03b1) (j i) :\n    sigmaFinsuppEquivPiFinsupp f j i = f \u27e8j, i\u27e9 :=\n  rfl\n#align finsupp.sigma_finsupp_equiv_pi_finsupp_apply Finsupp.sigmaFinsuppEquivPiFinsupp_apply\n-/\n\n/- warning: finsupp.sigma_finsupp_add_equiv_pi_finsupp -> Finsupp.sigmaFinsuppAddEquivPiFinsupp is a dubious translation:\nlean 3 declaration is\n  forall {\u03b7 : Type.{u1}} [_inst_2 : Fintype.{u1} \u03b7] {\u03b1 : Type.{u2}} {\u03b9s : \u03b7 -> Type.{u3}} [_inst_4 : AddMonoid.{u2} \u03b1], AddEquiv.{max (max u1 u3) u2, max u1 u3 u2} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (Finsupp.add.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (i : \u03b7) => Finsupp.add.{u3, u2} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))\nbut is expected to have type\n  forall {\u03b7 : Type.{u1}} [_inst_2 : Fintype.{u1} \u03b7] {\u03b1 : Type.{u2}} {\u03b9s : \u03b7 -> Type.{u3}} [_inst_4 : AddMonoid.{u2} \u03b1], AddEquiv.{max u2 u3 u1, max (max u1 u2) u3} (Finsupp.{max u3 u1, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u2} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u2} \u03b1 _inst_4)) (Finsupp.add.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u2 u3} \u03b7 (fun (j : \u03b7) => Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u2} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u3, u2} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))\nCase conversion may be inaccurate. Consider using '#align finsupp.sigma_finsupp_add_equiv_pi_finsupp Finsupp.sigmaFinsuppAddEquivPiFinsupp\u2093'. -/\n/-- On a `fintype \u03b7`, `finsupp.split` is an additive equivalence between\n`(\u03a3 (j : \u03b7), \u03b9s j) \u2192\u2080 \u03b1` and `\u03a0 j, (\u03b9s j \u2192\u2080 \u03b1)`.\n\nThis is the `add_equiv` version of `finsupp.sigma_finsupp_equiv_pi_finsupp`.\n-/\nnoncomputable def sigmaFinsuppAddEquivPiFinsupp {\u03b1 : Type _} {\u03b9s : \u03b7 \u2192 Type _} [AddMonoid \u03b1] :\n    ((\u03a3j, \u03b9s j) \u2192\u2080 \u03b1) \u2243+ \u2200 j, \u03b9s j \u2192\u2080 \u03b1 :=\n  { sigmaFinsuppEquivPiFinsupp with\n    map_add' := fun f g => by\n      ext\n      simp }\n#align finsupp.sigma_finsupp_add_equiv_pi_finsupp Finsupp.sigmaFinsuppAddEquivPiFinsupp\n\n/- warning: finsupp.sigma_finsupp_add_equiv_pi_finsupp_apply -> Finsupp.sigmaFinsuppAddEquivPiFinsupp_apply is a dubious translation:\nlean 3 declaration is\n  forall {\u03b7 : Type.{u1}} [_inst_2 : Fintype.{u1} \u03b7] {\u03b1 : Type.{u2}} {\u03b9s : \u03b7 -> Type.{u3}} [_inst_4 : AddMonoid.{u2} \u03b1] (f : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (j : \u03b7) (i : \u03b9s j), Eq.{succ u2} \u03b1 (coeFn.{max (succ u3) (succ u2), max (succ u3) (succ u2)} (Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (_x : Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) => (\u03b9s j) -> \u03b1) (Finsupp.coeFun.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (coeFn.{max (succ (max (max u1 u3) u2)) (succ (max u1 u3 u2)), max (succ (max (max u1 u3) u2)) (succ (max u1 u3 u2))} (AddEquiv.{max (max u1 u3) u2, max u1 u3 u2} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (Finsupp.add.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (i : \u03b7) => Finsupp.add.{u3, u2} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))) (fun (_x : AddEquiv.{max (max u1 u3) u2, max u1 u3 u2} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (Finsupp.add.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (i : \u03b7) => Finsupp.add.{u3, u2} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))) => (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) -> (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))) (AddEquiv.hasCoeToFun.{max (max u1 u3) u2, max u1 u3 u2} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (forall (j : \u03b7), Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (Finsupp.add.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u3, u2} (\u03b9s j) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (i : \u03b7) => Finsupp.add.{u3, u2} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4)))) (Finsupp.sigmaFinsuppAddEquivPiFinsupp.{u1, u2, u3} \u03b7 _inst_2 \u03b1 (fun (j : \u03b7) => \u03b9s j) _inst_4) f j) i) (coeFn.{max (succ (max u1 u3)) (succ u2), max (succ (max u1 u3)) (succ u2)} (Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) (fun (_x : Finsupp.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) => (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) -> \u03b1) (Finsupp.coeFun.{max u1 u3, u2} (Sigma.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddZeroClass.toHasZero.{u2} \u03b1 (AddMonoid.toAddZeroClass.{u2} \u03b1 _inst_4))) f (Sigma.mk.{u1, u3} \u03b7 (fun (j : \u03b7) => \u03b9s j) j i))\nbut is expected to have type\n  forall {\u03b7 : Type.{u1}} [_inst_2 : Fintype.{u1} \u03b7] {\u03b1 : Type.{u3}} {\u03b9s : \u03b7 -> Type.{u2}} [_inst_4 : AddMonoid.{u3} \u03b1] (f : Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (j : \u03b7) (i : \u03b9s j), Eq.{succ u3} ((fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b9s j) => \u03b1) i) (FunLike.coe.{max (succ u2) (succ u3), succ u2, succ u3} (Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (\u03b9s j) (fun (_x : \u03b9s j) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : \u03b9s j) => \u03b1) _x) (Finsupp.funLike.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (FunLike.coe.{max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1), max (max (succ u2) (succ u3)) (succ u1)} (AddEquiv.{max u3 u2 u1, max (max u1 u3) u2} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.add.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)))) (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (_x : Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.403 : Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) => forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) _x) (AddHomClass.toFunLike.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u3 u2 u1, max (max u1 u3) u2} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.add.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)))) (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (AddZeroClass.toAdd.{max (max u2 u3) u1} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 _inst_4))) (AddZeroClass.toAdd.{max (max u2 u3) u1} (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Pi.addZeroClass.{u1, max u2 u3} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => AddMonoid.toAddZeroClass.{max u2 u3} (Finsupp.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{u2, u3} (\u03b9s i) \u03b1 _inst_4)))) (AddMonoidHomClass.toAddHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u3 u2 u1, max (max u1 u3) u2} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.add.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)))) (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 _inst_4)) (Pi.addZeroClass.{u1, max u2 u3} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => AddMonoid.toAddZeroClass.{max u2 u3} (Finsupp.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{u2, u3} (\u03b9s i) \u03b1 _inst_4))) (AddEquivClass.instAddMonoidHomClass.{max (max u2 u3) u1, max (max u2 u3) u1, max (max u2 u3) u1} (AddEquiv.{max u3 u2 u1, max (max u1 u3) u2} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.add.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)))) (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (AddMonoid.toAddZeroClass.{max (max u2 u3) u1} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 _inst_4)) (Pi.addZeroClass.{u1, max u2 u3} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => AddMonoid.toAddZeroClass.{max u2 u3} (Finsupp.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.addMonoid.{u2, u3} (\u03b9s i) \u03b1 _inst_4))) (AddEquiv.instAddEquivClassAddEquiv.{max (max u2 u3) u1, max (max u2 u3) u1} (Finsupp.{max u2 u1, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (forall (j : \u03b7), Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Finsupp.add.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4)) (Pi.instAdd.{u1, max u3 u2} \u03b7 (fun (j : \u03b7) => Finsupp.{u2, u3} (\u03b9s j) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (fun (i : \u03b7) => Finsupp.add.{u2, u3} (\u03b9s i) \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_4))))))) (Finsupp.sigmaFinsuppAddEquivPiFinsupp.{u1, u3, u2} \u03b7 _inst_2 \u03b1 (fun (j : \u03b7) => \u03b9s j) _inst_4) f j) i) (FunLike.coe.{max (succ (max u1 u2)) (succ u3), succ (max u1 u2), succ u3} (Finsupp.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) (fun (_x : Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) => (fun (x._@.Mathlib.Data.Finsupp.Defs._hyg.779 : Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) => \u03b1) _x) (Finsupp.funLike.{max u1 u2, u3} (Sigma.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j)) \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_4)) f (Sigma.mk.{u1, u2} \u03b7 (fun (j : \u03b7) => \u03b9s j) j i))\nCase conversion may be inaccurate. Consider using '#align finsupp.sigma_finsupp_add_equiv_pi_finsupp_apply Finsupp.sigmaFinsuppAddEquivPiFinsupp_apply\u2093'. -/\n@[simp]\ntheorem sigmaFinsuppAddEquivPiFinsupp_apply {\u03b1 : Type _} {\u03b9s : \u03b7 \u2192 Type _} [AddMonoid \u03b1]\n    (f : (\u03a3j, \u03b9s j) \u2192\u2080 \u03b1) (j i) : sigmaFinsuppAddEquivPiFinsupp f j i = f \u27e8j, i\u27e9 :=\n  rfl\n#align finsupp.sigma_finsupp_add_equiv_pi_finsupp_apply Finsupp.sigmaFinsuppAddEquivPiFinsupp_apply\n\nend Sigma\n\n/-! ### Meta declarations -/\n\n\n/-- Stringify a `finsupp` as a sequence of `finsupp.single` terms.\n\nNote this is `meta` as it has to choose some order for the terms. -/\nunsafe instance (\u03b9 \u03b1 : Type _) [Zero \u03b1] [Repr \u03b9] [Repr \u03b1] : Repr (\u03b9 \u2192\u2080 \u03b1)\n    where repr f :=\n    if f.support.card = 0 then \"0\"\n    else\n      \" + \".intercalate <|\n        f.support.val.unquot.map fun i => \"finsupp.single \" ++ repr i ++ \" \" ++ repr (f i)\n\nend Finsupp\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Finsupp/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.3941821917455544}}
{"text": "import phase0.litter\n\nopen cardinal\nopen_locale cardinal\n\nuniverse u\n\nnamespace con_nf\nvariable [params.{u}]\n\n/-- The type of sublitters. -/\nstructure sublitter : Type u :=\n(litter : litter)\n(carrier : set atom)\n(subset : carrier \u2286 litter_set litter)\n(diff_small : small (litter_set litter \\ carrier))\n\nnamespace sublitter\n\nvariables {S S\u2081 S\u2082 : sublitter}\n\n/-- Use sublitter.mk_eq_\u03ba instead if possible. -/\nlemma mk_S_eq_\u03ba (S : sublitter) : #(S.carrier) = #\u03ba :=\nbegin\n  have := mk_le_mk_of_subset S.subset,\n  rw mk_litter_set at this,\n  cases lt_or_eq_of_le this,\n  { have := mk_diff_add_mk S.subset,\n    rw mk_litter_set at this,\n    cases (add_lt_of_lt \u03ba_regular.aleph_0_le S.diff_small h).ne this, },\n  exact h,\nend\n\ninstance : set_like sublitter atom :=\n{ coe := \u03bb S, S.carrier,\n  coe_injective' := begin\n    rintro \u27e8i, N\u2081, h\u2081, h\u2082\u27e9 \u27e8j, N\u2082, h\u2083, h\u2084\u27e9 (rfl : N\u2081 = N\u2082),\n    obtain \u27e8e\u27e9 := cardinal.eq.mp (sublitter.mk_S_eq_\u03ba \u27e8i, N\u2081, h\u2081, h\u2082\u27e9),\n    have h\u2085 := h\u2081 (e.symm (arbitrary \u03ba)).prop,\n    have h\u2086 := h\u2083 (e.symm (arbitrary \u03ba)).prop,\n    rw mem_litter_set at h\u2085 h\u2086,\n    rw h\u2085 at h\u2086,\n    cases h\u2086,\n    refl,\n  end }\n\n@[simp] lemma mk_eq_\u03ba (S : sublitter) : #S = #\u03ba := S.mk_S_eq_\u03ba\n@[simp] lemma mk_eq_\u03ba' (S : sublitter) : #(S : set atom) = #\u03ba := S.mk_S_eq_\u03ba\n\n@[simp] lemma carrier_eq_coe {S : sublitter} : S.carrier = S := rfl\n\n@[simp] lemma coe_mk (L S subset diff_small) :\n  @coe sublitter (set atom) _ \u27e8L, S, subset, diff_small\u27e9 = S := rfl\n\n@[ext] lemma ext (h : (S\u2081 : set atom) = S\u2082) : S\u2081 = S\u2082 :=\nset_like.coe_injective h\n\nlemma fst_eq_of_mem {a : atom} (h : a \u2208 S) : a.1 = S.litter := S.subset h\nlemma mem_litter_set_of_mem {a : atom} (h : a \u2208 S) : a \u2208 litter_set S.litter := S.subset h\n@[simp] lemma mem_mk {a : atom} {L S subset diff_small} :\n  a \u2208 (\u27e8L, S, subset, diff_small\u27e9 : sublitter) \u2194 a \u2208 S := iff.rfl\n\n@[simp] lemma litter_diff_eq (S : sublitter) : (S : set atom) \\ litter_set S.litter = \u2205 :=\nset.eq_empty_of_forall_not_mem (\u03bb a ha, ha.2 (S.subset ha.1))\n\nlemma is_near_litter (S : sublitter) : is_near_litter S.litter S :=\nbegin\n  refine small.union S.diff_small _,\n  rw litter_diff_eq,\n  exact small_empty,\nend\n\ndef to_near_litter (S : sublitter) : near_litter := \u27e8S.litter, S, S.is_near_litter\u27e9\n\n@[simp] lemma to_near_litter_litter (S : sublitter) : S.to_near_litter.1 = S.litter := rfl\n@[simp] lemma coe_to_near_litter (S : sublitter) : (S.to_near_litter : set atom) = S := rfl\n@[simp] lemma mem_to_near_litter (a : atom) : a \u2208 S.to_near_litter \u2194 a \u2208 S := iff.rfl\n\nlemma is_near_iff : is_near (S\u2081 : set atom) S\u2082 \u2194 S\u2081.litter = S\u2082.litter :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { obtain \u27e8f\u27e9 := is_near.mk_inter h S\u2081.mk_eq_\u03ba.symm.le,\n    rw [\u2190 fst_eq_of_mem (f (arbitrary \u03ba)).prop.1, \u2190 fst_eq_of_mem (f (arbitrary \u03ba)).prop.2], },\n  { refine S\u2081.is_near_litter.symm.trans _,\n    rw h,\n    exact S\u2082.is_near_litter, },\nend\n\nlemma inter_nonempty_iff : (S\u2081 \u2229 S\u2082 : set atom).nonempty \u2194 S\u2081.litter = S\u2082.litter :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { obtain \u27e8a, ha\u27e9 := h,\n    rw [\u2190 fst_eq_of_mem ha.1, fst_eq_of_mem ha.2], },\n  { obtain \u27e8f\u27e9 := is_near.mk_inter _ _,\n    exact \u27e8_, (f (arbitrary \u03ba)).prop\u27e9,\n    rw is_near_iff,\n    exact h,\n    rw mk_eq_\u03ba', },\nend\n\nend sublitter\n\ndef litter.to_sublitter (L : litter) : sublitter :=\n\u27e8L, litter_set L, subset_rfl, by rw [sdiff_self]; exact small_empty\u27e9\n\n@[simp] lemma litter.litter_to_sublitter (L : litter) : L.to_sublitter.litter = L := rfl\n@[simp] lemma litter.coe_to_sublitter (L : litter) :\n  (L.to_sublitter : set atom) = litter_set L := rfl\n\nnamespace sublitter\n\ndef rel_embedding (S : sublitter) :\n  ((<) : S \u2192 S \u2192 Prop) \u21aar ((<) : (litter_set S.litter) \u2192 (litter_set S.litter) \u2192 Prop) :=\n\u27e8\u27e8\u03bb a, \u27e8a, S.subset a.prop\u27e9, \u03bb a b h, subtype.coe_injective (subtype.mk_eq_mk.mp h)\u27e9,\n  \u03bb a b, by simp only [function.embedding.coe_fn_mk, subtype.mk_lt_mk, subtype.coe_lt_coe]\u27e9\n\n/-- The order type of a sublitter is `\u03ba`. -/\nlemma ordinal_type (S : sublitter) : ordinal.type ((<) : S \u2192 S \u2192 Prop) = (#\u03ba).ord :=\nbegin\n  refine le_antisymm _ _,\n  { rw \u2190 S.litter.ordinal_type,\n    exact rel_embedding.ordinal_type_le S.rel_embedding, },\n  { rw [cardinal.gc_ord_card, ordinal.card_type, mk_eq_\u03ba], },\nend\n\n-- TODO: We can probably do this constructively, but this way is easier for now.\nnoncomputable def order_iso_\u03ba (S : sublitter) : S \u2243o \u03ba :=\nbegin\n  refine order_iso.of_rel_iso_lt (nonempty.some _),\n  rw [\u2190 ordinal.type_eq, ordinal_type, \u2190 \u03ba_ord],\n  refl,\nend\n\n/-- There is a (unique) order isomorphism between any two sublitters. -/\nnoncomputable def order_iso (S T : sublitter) : S \u2243o T :=\nS.order_iso_\u03ba.trans T.order_iso_\u03ba.symm\n\n@[simp] lemma order_iso_apply_mem {S T : sublitter} (a : S) : (S.order_iso T a : atom) \u2208 T :=\n(S.order_iso T a).prop\n\n@[simp] lemma order_iso_apply_fst_eq {S T : sublitter} (a : S) :\n  (S.order_iso T a : atom).1 = T.litter :=\nT.subset (S.order_iso T a).prop\n\nlemma order_iso_congr_left {S T U : sublitter} (h : S = T) (a : S) :\n  (S.order_iso U a : atom) = T.order_iso U \u27e8a, by rw \u2190 h; exact a.2\u27e9 :=\nby cases h; rw subtype.coe_eta\n\nlemma order_iso_congr_right {S T U : sublitter} (h : T = U) (a : S) :\n  (S.order_iso T a : atom) = S.order_iso U a :=\nby cases h; refl\n\ndef order_iso.subtype_iso {\u03b1 \u03b2 : Type*} [has_le \u03b1] [has_le \u03b2] (e : \u03b1 \u2243o \u03b2)\n  {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} (hpq : \u2200 a, p a \u2194 q (e a)) :\n  {a // p a} \u2243o {b // q b} :=\n\u27e8e.subtype_equiv hpq, by simp only [rel_iso.coe_fn_to_equiv, equiv.subtype_equiv_apply,\n  subtype.mk_le_mk, order_iso.le_iff_le, subtype.coe_le_coe, iff_self, subtype.forall,\n  implies_true_iff]\u27e9\n\n/-- There is a unique order isomorphism between corresponding subtypes of a well-order.\nTODO: Can prove this without specialising to subtypes. -/\nlemma order_iso.unique {\u03b1 \u03b2 : Type*}\n  [linear_order \u03b1] [\u03b1wf : well_founded ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop)]\n  [linear_order \u03b2] [\u03b2wf : well_founded ((<) : \u03b2 \u2192 \u03b2 \u2192 Prop)]\n  (e : \u03b1 \u2243o \u03b2) {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop}\n  (hpq : \u2200 a, p a \u2194 q (e a)) (e' : {a // p a} \u2243o {b // q b}) :\n  e' = e.subtype_iso hpq :=\nbegin\n  ext x,\n  obtain \u27e8x, hx\u27e9 := x,\n  revert hx,\n  refine \u03b1wf.induction x _,\n  intros x ih hx,\n  rw subtype.coe_inj,\n  refine linarith.eq_of_not_lt_of_not_gt _ _ (\u03bb h, _) (\u03bb h, _),\n  { have := ih ((e.subtype_iso hpq).symm (e' \u27e8x, hx\u27e9)) _ _,\n    { simp only [subtype.coe_eta, order_iso.apply_symm_apply, subtype.coe_inj,\n        order_iso.apply_eq_iff_eq, order_iso.symm_apply_eq] at this,\n      exact h.ne this, },\n    { refine lt_of_lt_of_eq _ (subtype.coe_mk x hx),\n      have := (e.subtype_iso hpq).symm.strict_mono h,\n      rw order_iso.symm_apply_apply at this,\n      exact this, },\n    { exact subtype.coe_prop _, }, },\n  { have := ih (e'.symm (e.subtype_iso hpq \u27e8x, hx\u27e9)) _ _,\n    { simp only [subtype.coe_eta, order_iso.apply_symm_apply, subtype.coe_inj,\n        order_iso.apply_eq_iff_eq, order_iso.symm_apply_eq] at this,\n      have := h.trans_eq (congr_arg e' this),\n      rw order_iso.apply_symm_apply at this,\n      exact this.ne rfl, },\n    { refine lt_of_lt_of_eq _ (subtype.coe_mk x hx),\n      have := e'.symm.strict_mono h,\n      rw order_iso.symm_apply_apply at this,\n      exact this, },\n    { exact subtype.coe_prop _, }, },\nend\n\n/-- The intersection of two sublitters. -/\ndef meet (S T : sublitter) (h : S.litter = T.litter) : sublitter := {\n  litter := S.litter,\n  carrier := S \u2229 T,\n  subset := (set.inter_subset_left _ _).trans S.subset,\n  diff_small := by rw set.diff_inter; exact small.union S.diff_small (h.symm \u25b8 T.diff_small),\n}\n\n/-- Transports the meet of sublitters `S` and `U` across the order isomorphism `S \u2243o T`. -/\ndef order_iso_meet (S T U : sublitter) (h : S.litter = U.litter) : sublitter := {\n  litter := T.litter,\n  carrier := {a | \u2203 (ha : a \u2208 T), ((S.order_iso T).symm \u27e8a, ha\u27e9 : atom) \u2208 U},\n  subset := \u03bb a ha, T.subset ha.some,\n  diff_small := begin\n    suffices : small ((T : set atom) \\\n      {a | \u2203 (ha : a \u2208 T), ((S.order_iso T).symm \u27e8a, ha\u27e9 : atom) \u2208 U}),\n    { refine small.mono (\u03bb a ha, _) (small.union T.diff_small this),\n      by_cases a \u2208 T,\n      exact or.inr \u27e8h, ha.2\u27e9,\n      exact or.inl \u27e8ha.1, h\u27e9, },\n    refine lt_of_le_of_lt _ U.diff_small,\n    refine \u27e8\u27e8\u03bb a, \u27e8(S.order_iso T).symm \u27e8a, a.prop.1\u27e9, _, _\u27e9, \u03bb a b h, _\u27e9\u27e9,\n    { rw \u2190 h,\n      exact S.subset ((S.order_iso T).symm \u27e8a, a.prop.1\u27e9).2, },\n    { intro h,\n      exact a.prop.2 \u27e8a.prop.1, h\u27e9, },\n    { simpa only [subtype.mk_eq_mk, rel_iso.eq_iff_eq, subtype.coe_inj] using h, },\n  end,\n}\n\nend sublitter\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/sublitter.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.39418219131676097}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.homology.homotopy\nimport category_theory.quotient\n\n/-!\n# The homotopy category\n\n`homotopy_category V c` gives the category of chain complexes of shape `c` in `V`,\nwith chain maps identified when they are homotopic.\n-/\n\nuniverses v u\n\nopen_locale classical\nnoncomputable theory\n\nopen category_theory category_theory.limits homological_complex\n\nvariables {\u03b9 : Type*}\nvariables (V : Type u) [category.{v} V] [preadditive V]\nvariables (c : complex_shape \u03b9)\n\n/--\nThe congruence on `homological_complex V c` given by the existence of a homotopy.\n-/\ndef homotopic : hom_rel (homological_complex V c) := \u03bb C D f g, nonempty (homotopy f g)\n\ninstance homotopy_congruence : congruence (homotopic V c) :=\n{ is_equiv := \u03bb C D,\n  { refl := \u03bb C, \u27e8homotopy.refl C\u27e9,\n    symm := \u03bb f g \u27e8w\u27e9, \u27e8w.symm\u27e9,\n    trans := \u03bb f g h \u27e8w\u2081\u27e9 \u27e8w\u2082\u27e9, \u27e8w\u2081.trans w\u2082\u27e9, },\n  comp_left := \u03bb E F G m\u2081 m\u2082 g \u27e8i\u27e9, \u27e8i.comp_left _\u27e9,\n  comp_right := \u03bb E F G f m\u2081 m\u2082 \u27e8i\u27e9, \u27e8i.comp_right _\u27e9, }\n\n/-- `homotopy_category V c` is the category of chain complexes of shape `c` in `V`,\nwith chain maps identified when they are homotopic. -/\n@[derive category]\ndef homotopy_category := category_theory.quotient (homotopic V c)\n\n-- TODO the homotopy_category is preadditive\n\nnamespace homotopy_category\n\n/-- The quotient functor from complexes to the homotopy category. -/\ndef quotient : homological_complex V c \u2964 homotopy_category V c :=\ncategory_theory.quotient.functor _\n\nlocal attribute [instance] has_zero_object.has_zero\n\n-- TODO upgrade this is to `has_zero_object`, presumably for any `quotient`.\ninstance [has_zero_object V] : inhabited (homotopy_category V c) := \u27e8(quotient V c).obj 0\u27e9\n\nvariables {V c}\n\n@[simp] lemma quotient_obj_as (C : homological_complex V c) :\n  ((quotient V c).obj C).as = C := rfl\n\n@[simp] lemma quotient_map_out {C D : homotopy_category V c} (f : C \u27f6 D) :\n  (quotient V c).map f.out = f :=\nquot.out_eq _\n\nlemma eq_of_homotopy {C D : homological_complex V c} (f g : C \u27f6 D) (h : homotopy f g) :\n  (quotient V c).map f = (quotient V c).map g :=\ncategory_theory.quotient.sound _ \u27e8h\u27e9\n\n/-- If two chain maps become equal in the homotopy category, then they are homotopic. -/\ndef homotopy_of_eq {C D : homological_complex V c} (f g : C \u27f6 D)\n  (w : (quotient V c).map f = (quotient V c).map g) : homotopy f g :=\n((quotient.functor_map_eq_iff _ _ _).mp w).some\n\n/--\nAn arbitrarily chosen representation of the image of a chain map in the homotopy category\nis homotopic to the original chain map.\n-/\ndef homotopy_out_map {C D : homological_complex V c} (f : C \u27f6 D) :\n  homotopy ((quotient V c).map f).out f :=\nbegin\n  apply homotopy_of_eq,\n  simp,\nend\n\n@[simp] lemma quotient_map_out_comp_out {C D E : homotopy_category V c} (f : C \u27f6 D) (g : D \u27f6 E) :\n  (quotient V c).map (quot.out f \u226b quot.out g) = f \u226b g :=\nby conv_rhs { erw [\u2190quotient_map_out f, \u2190quotient_map_out g, \u2190(quotient V c).map_comp], }\n\n/-- Homotopy equivalent complexes become isomorphic in the homotopy category. -/\n@[simps]\ndef iso_of_homotopy_equiv {C D : homological_complex V c} (f : homotopy_equiv C D) :\n  (quotient V c).obj C \u2245 (quotient V c).obj D :=\n{ hom := (quotient V c).map f.hom,\n  inv := (quotient V c).map f.inv,\n  hom_inv_id' := begin\n    rw [\u2190(quotient V c).map_comp, \u2190(quotient V c).map_id],\n    exact eq_of_homotopy _ _ f.homotopy_hom_inv_id,\n  end,\n  inv_hom_id' := begin\n    rw [\u2190(quotient V c).map_comp, \u2190(quotient V c).map_id],\n    exact eq_of_homotopy _ _ f.homotopy_inv_hom_id,\n  end }\n\n/-- If two complexes become isomorphic in the homotopy category,\n  then they were homotopy equivalent. -/\ndef homotopy_equiv_of_iso\n  {C D : homological_complex V c} (i : (quotient V c).obj C \u2245 (quotient V c).obj D) :\n  homotopy_equiv C D :=\n{ hom := quot.out i.hom,\n  inv := quot.out i.inv,\n  homotopy_hom_inv_id := homotopy_of_eq _ _ (by { simp, refl, }),\n  homotopy_inv_hom_id := homotopy_of_eq _ _ (by { simp, refl, }), }\n\nvariables (V c) [has_zero_object V] [has_equalizers V] [has_images V] [has_image_maps V]\n  [has_cokernels V]\n\n/-- The `i`-th homology, as a functor from the homotopy category. -/\ndef homology_functor (i : \u03b9) : homotopy_category V c \u2964 V :=\ncategory_theory.quotient.lift _ (homology_functor V c i)\n  (\u03bb C D f g \u27e8h\u27e9, homology_map_eq_of_homotopy h i)\n\n/-- The homology functor on the homotopy category is just the usual homology functor. -/\ndef homology_factors (i : \u03b9) :\n  quotient V c \u22d9 homology_functor V c i \u2245 _root_.homology_functor V c i :=\ncategory_theory.quotient.lift.is_lift _ _ _\n\n@[simp] lemma homology_factors_hom_app (i : \u03b9) (C : homological_complex V c) :\n  (homology_factors V c i).hom.app C = \ud835\udfd9 _ :=\nrfl\n\n@[simp] lemma homology_factors_inv_app (i : \u03b9) (C : homological_complex V c) :\n  (homology_factors V c i).inv.app C = \ud835\udfd9 _ :=\nrfl\n\nlemma homology_functor_map_factors (i : \u03b9) {C D : homological_complex V c} (f : C \u27f6 D) :\n  (_root_.homology_functor V c i).map f =\n    ((homology_functor V c i).map ((quotient V c).map f) : _) :=\n(category_theory.quotient.lift_map_functor_map _ (_root_.homology_functor V c i) _ f).symm\n\nend homotopy_category\n\nnamespace category_theory\n\nvariables {V} {W : Type*} [category W] [preadditive W]\n\n/-- An additive functor induces a functor between homotopy categories. -/\n@[simps]\ndef functor.map_homotopy_category (c : complex_shape \u03b9) (F : V \u2964 W) [F.additive] :\n  homotopy_category V c \u2964 homotopy_category W c :=\n{ obj := \u03bb C, (homotopy_category.quotient W c).obj ((F.map_homological_complex c).obj C.as),\n  map := \u03bb C D f,\n    (homotopy_category.quotient W c).map ((F.map_homological_complex c).map (quot.out f)),\n  map_id' := \u03bb C, begin\n    rw \u2190(homotopy_category.quotient W c).map_id,\n    apply homotopy_category.eq_of_homotopy,\n    rw \u2190(F.map_homological_complex c).map_id,\n    apply F.map_homotopy,\n    apply homotopy_category.homotopy_of_eq,\n    exact quot.out_eq _,\n  end,\n  map_comp' := \u03bb C D E f g, begin\n    rw \u2190(homotopy_category.quotient W c).map_comp,\n    apply homotopy_category.eq_of_homotopy,\n    rw \u2190(F.map_homological_complex c).map_comp,\n    apply F.map_homotopy,\n    apply homotopy_category.homotopy_of_eq,\n    convert quot.out_eq _,\n    exact homotopy_category.quotient_map_out_comp_out _ _,\n  end }.\n\n-- TODO `F.map_homotopy_category c` is additive (and linear when `F` is linear).\n\n/-- A natural transformation induces a natural transformation between\n  the induced functors on the homotopy category. -/\n@[simps]\ndef nat_trans.map_homotopy_category {F G : V \u2964 W} [F.additive] [G.additive]\n  (\u03b1 : F \u27f6 G) (c : complex_shape \u03b9) : F.map_homotopy_category c \u27f6 G.map_homotopy_category c :=\n{ app := \u03bb C,\n    (homotopy_category.quotient W c).map ((nat_trans.map_homological_complex \u03b1 c).app C.as),\n  naturality' := \u03bb C D f,\n  begin\n    dsimp,\n    simp only [\u2190functor.map_comp],\n    congr' 1,\n    ext,\n    dsimp,\n    simp,\n  end }\n\n@[simp] lemma nat_trans.map_homotopy_category_id (c : complex_shape \u03b9) (F : V \u2964 W) [F.additive] :\n  nat_trans.map_homotopy_category (\ud835\udfd9 F) c = \ud835\udfd9 (F.map_homotopy_category c) :=\nby tidy\n\n@[simp] lemma nat_trans.map_homotopy_category_comp (c : complex_shape \u03b9)\n  {F G H : V \u2964 W} [F.additive] [G.additive] [H.additive]\n  (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H):\n  nat_trans.map_homotopy_category (\u03b1 \u226b \u03b2) c =\n    nat_trans.map_homotopy_category \u03b1 c \u226b nat_trans.map_homotopy_category \u03b2 c :=\nby tidy\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/homology/homotopy_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6187804196836383, "lm_q1q2_score": 0.39418218235997304}}
{"text": "import order.lattice -- for lattice.semilattice_inf\nimport order.bounds -- for is_lub\nimport algebra.ring -- for is_ring_hom\nimport topology.opens -- only for the conjecture that i need precisely opens \u03b1\n--import sheaves.sheaf\n--  import sheaves.covering.covering\n  -- import sheaves.presheaf\nimport data.equiv.basic\n\nimport tactic.where -- cool debugging tool \n\n-- this is the only non-mathlib import. Should that stuff be in mathlib\n-- or not? If not then feel free to rewrite the below.\n-- In the Xena project this import is in src/\nimport for_mathlib_complete_lattice\n\n--open lattice\n\nuniverses v u\n\nopen lattice\n\nstructure presheaf (\u03b1 : Type u) [semilattice_inf \u03b1] :=\n(F     : \u03b1 \u2192 Type v)\n(res   : \u2200 (U V) (HVU : V \u2264 U), F U \u2192 F V)\n(Hid   : \u2200 (U), res U U (le_refl U) = id)\n(Hcomp : \u2200 (U V W) (HWV : W \u2264 V) (HVU : V \u2264 U),\n  res U W (le_trans HWV HVU) = res V W HWV \u2218 res U V HVU)\n\nnamespace presheaf\n\nvariables {\u03b1 : Type u} [semilattice_inf \u03b1]\n\ninstance : has_coe_to_fun (presheaf \u03b1) :=\n{ F := \u03bb _, \u03b1 \u2192 Type v,\n  coe := presheaf.F }\n\n-- Simplification lemmas for Hid and Hcomp.\n\n@[simp] lemma Hcomp' (F : presheaf \u03b1) :\n\u2200 (U V W) (HWV : W \u2264 V) (HVU : V \u2264 U) (s : F U),\n  (F.res U W (le_trans HWV HVU)) s =\n  (F.res V W HWV) ((F.res U V HVU) s) :=\n\u03bb U V W HWV HVU s, by rw F.Hcomp U V W HWV HVU\n\n@[simp] lemma Hid' (F : presheaf \u03b1) :\n\u2200 (U) (s : F U),\n  (F.res U U (le_refl U)) s = s :=\n\u03bb U s, by rw F.Hid U; simp\n\ndef total (F : presheaf \u03b1) : Type (max u v) := \u03a3 U, F.F U\n\ninstance (F : presheaf \u03b1) (U : \u03b1) : has_coe_t (F U) F.total :=\n\u27e8sigma.mk _\u27e9\n\n@[elab_as_eliminator]\ntheorem total.cases_on (F : presheaf \u03b1) {C : F.total \u2192 Sort*}\n  (x) (H : \u2200 U (x : F U), C x) : C x :=\nby cases x; apply H\n\ndef res' (F : presheaf \u03b1) (V : \u03b1) : F.total \u2192 F.total\n| \u27e8U, x\u27e9 := F.res U (U \u2293 V) inf_le_left x\n\ntheorem res'_def (F : presheaf \u03b1) {U V} (x : F U) :\n  F.res' V x = F.res U (U \u2293 V) inf_le_left x := rfl\n\ntheorem res'_val (F : presheaf \u03b1) {U V} (x : F U) (h : V \u2264 U) :\n  F.res' V x = F.res U V h x :=\nhave \u2200 W (H : W \u2264 U), W = V \u2192\n  (F.res U W H x : F.total) = F.res U V h x :=\n  by rintro _ _ rfl; refl,\nthis _ _ (inf_of_le_right h)\n\ntheorem res'_eq_inf (F : presheaf \u03b1) {U V} (x : F U) :\n  F.res' V x = F.res' (U \u2293 V) x :=\nby rw [res'_def, \u2190 res'_val _ _ inf_le_left]\n\ntheorem res'_eq_left (F : presheaf \u03b1) {U V W} (x : F U) (H : U \u2293 V = U \u2293 W) :\n  F.res' V x = F.res' W x :=\nby rw [res'_eq_inf, H, \u2190 res'_eq_inf]\n\n@[simp] theorem res'_id {F : presheaf \u03b1} {U} (x : F U) : F.res' U x = x :=\nby rw [res'_val _ _ (le_refl U), F.Hid]; refl\n\n@[simp] theorem res'_comp {F : presheaf \u03b1} {U V} (x : F.total) :\n  F.res' U (F.res' V x) = F.res' (U \u2293 V) x :=\ntotal.cases_on F x $ \u03bb W x,\nby rw [res'_def, res'_def, \u2190 F.Hcomp', \u2190 res'_val, res'_eq_left];\n   simp [inf_left_comm, inf_comm]\n\ndef locality (F : presheaf \u03b1) :=\n\u2200 {{U S}}, is_lub S U \u2192 \u2200 {{s t : F U}},\n  (\u2200 V \u2208 S, F.res' V s = F.res' V t) \u2192 s = t\n\ndef gluing (F : presheaf \u03b1) :=\n\u2200 {{U : \u03b1}} {{S}}, is_lub S U \u2192\n\u2200 (s : \u03a0 V : S, F V),\n(\u2200 V W : S,\n  res' F (V \u2293 W) (s V) = res' F (V \u2293 W) (s W)) \u2192\n\u2203 x : F U, \u2200 V:S, F.res' V x = s V\n\nend presheaf\n\n\n\n\n--#where\n\nuniverses w u\u2081 v\u2081\n\n/- memo for porting\n\nopens X -> \u03b1\ntop space X -> sdemilattice_inf \u03b1\n-/\n\n-- open topological_space\n\n\nstructure sheaf (\u03b1 : Type u) [semilattice_inf \u03b1] extends presheaf \u03b1 :=\n(locality : to_presheaf.locality)\n(gluing   : to_presheaf.gluing)\n\nstructure sheaf_of_rings (\u03b1 : Type u) [semilattice_inf \u03b1] extends sheaf \u03b1 :=\n[ring : \u2200 U, ring (F U)]\n[ring_hom : \u2200 U V h, is_ring_hom (res U V h)]\ndef sheaf_on_opens (\u03b1 : Type u) [semilattice_inf \u03b1] (U : \u03b1) : Type (max u (v+1)) :=\nsheaf.{u v} \u03b1\n\nnamespace sheaf_on_opens\n\nvariables {\u03b1 : Type u} [semilattice_inf \u03b1] {U : \u03b1}\n\ndef eval (F : sheaf_on_opens \u03b1 U) (V : \u03b1) (HVU : V \u2264 U) : Type v :=\npresheaf.F (sheaf.to_presheaf F) V\n\ndef res (F : sheaf_on_opens \u03b1 U) (V : \u03b1) (HVU : V \u2264 U) (W : \u03b1)\n  (HWU : W \u2264 U) (HWV : W \u2264 V) : F.eval V HVU \u2192 F.eval W HWU :=\npresheaf.res _ _ _ HWV\n\ntheorem res_comp (F : sheaf_on_opens \u03b1 U) (V1 : \u03b1) (HV1 : V1 \u2264 U)\n  (V2 : \u03b1) (HV2 : V2 \u2264 U) (V3 : \u03b1) (HV3 : V3 \u2264 U) (H12 : V2 \u2264 V1) (H23 : V3 \u2264 V2)\n  (f : F.eval V1 HV1) :\n  F.res V2 HV2 V3 HV3 H23 (F.res V1 HV1 V2 HV2 H12 f) = F.res V1 HV1 V3 HV3 (le_trans H23 H12) f :=\n(F.to_presheaf.Hcomp' _ _ _ _ _ f).symm\n\ndef res_subset (F : sheaf_on_opens \u03b1 U) (V : \u03b1) (HVU : V \u2264 U) : sheaf_on_opens \u03b1 V :=\nF\n\ntheorem eval_res_subset (F : sheaf_on_opens \u03b1 U) (V : \u03b1) (HVU : V \u2264 U) (W : \u03b1) (HWV : W \u2264 V) :\n  eval (res_subset F V HVU) W HWV = eval F W (le_trans HWV HVU) := rfl\n\nstructure morphism (F : sheaf_on_opens.{v} \u03b1 U) (G : sheaf_on_opens.{w} \u03b1 U) : Type (max u v w) :=\n(map : \u2200 V \u2264 U, F.eval V H \u2192 G.eval V H)\n(commutes : \u2200 (V : \u03b1) (HV : V \u2264 U) (W : \u03b1) (HW : W \u2264 U) (HWV : W \u2264 V) (x),\n  map W HW (F.res V HV W HW HWV x) = G.res V HV W HW HWV (map V HV x))\n\nnamespace morphism\n\nprotected def id (F : sheaf_on_opens.{v} \u03b1 U) : F.morphism F :=\n{ map := \u03bb V HV, id,\n  commutes := \u03bb V HV W HW HWV x, rfl }\n\ndef comp {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} {H : sheaf_on_opens.{u\u2081} \u03b1 U}\n  (\u03b7 : G.morphism H) (\u03be : F.morphism G) : F.morphism H :=\n{ map := \u03bb V HV x, \u03b7.map V HV (\u03be.map V HV x),\n  commutes := \u03bb V HV W HW HWV x, by rw [\u03be.commutes, \u03b7.commutes] }\n\n@[extensionality] lemma ext {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U}\n  {\u03b7 \u03be : F.morphism G} (H : \u2200 V HV x, \u03b7.map V HV x = \u03be.map V HV x) : \u03b7 = \u03be :=\nby cases \u03b7; cases \u03be; congr; ext; apply H\n\n@[simp] lemma id_comp {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} (\u03b7 : F.morphism G) :\n  (morphism.id G).comp \u03b7 = \u03b7 :=\next $ \u03bb V HV x, rfl\n\n@[simp] lemma comp_id {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} (\u03b7 : F.morphism G) :\n  \u03b7.comp (morphism.id F) = \u03b7 :=\next $ \u03bb V HV x, rfl\n\n@[simp] lemma comp_assoc {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} {H : sheaf_on_opens.{u\u2081} \u03b1 U} {I : sheaf_on_opens.{v\u2081} \u03b1 U}\n  (\u03b7 : H.morphism I) (\u03be : G.morphism H) (\u03c7 : F.morphism G) :\n  (\u03b7.comp \u03be).comp \u03c7 = \u03b7.comp (\u03be.comp \u03c7) :=\nrfl\n\ndef res_subset {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} (\u03b7 : F.morphism G) (V : \u03b1) (HVU : V \u2264 U) :\n  (F.res_subset V HVU).morphism (G.res_subset V HVU) :=\n{ map := \u03bb W HWV, \u03b7.map W (le_trans HWV HVU),\n  commutes := \u03bb S HSV T HTV, \u03b7.commutes S (le_trans HSV HVU) T (le_trans HTV HVU) }\n\n@[simp] lemma comp_res_subset {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} {H : sheaf_on_opens.{u\u2081} \u03b1 U}\n  (\u03b7 : G.morphism H) (\u03be : F.morphism G) (V : \u03b1) (HVU : V \u2264 U) :\n  (\u03b7.res_subset V HVU).comp (\u03be.res_subset V HVU) = (\u03b7.comp \u03be).res_subset V HVU :=\nrfl\n\n@[simp] lemma id_res_subset {F : sheaf_on_opens.{v} \u03b1 U} (V : \u03b1) (HVU : V \u2264 U) :\n  (morphism.id F).res_subset V HVU = morphism.id (F.res_subset V HVU) :=\nrfl\n\nend morphism\n\n#where\nstructure equiv (F : sheaf_on_opens.{v} \u03b1 U) (G : sheaf_on_opens.{w} \u03b1 U) : Type (max u v w) :=\n(to_fun : morphism F G)\n(inv_fun : morphism G F)\n(left_inv : inv_fun.comp to_fun = morphism.id F)\n(right_inv : to_fun.comp inv_fun = morphism.id G)\n\nnamespace equiv\n\ndef refl (F : sheaf_on_opens.{v} \u03b1 U) : equiv F F :=\n\u27e8morphism.id F, morphism.id F, rfl, rfl\u27e9\n\ndef symm {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{v} \u03b1 U} (e : equiv F G) : equiv G F :=\n\u27e8e.2, e.1, e.4, e.3\u27e9\n\ndef trans {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{v} \u03b1 U} {H : sheaf_on_opens.{u\u2081} \u03b1 U}\n  (e\u2081 : equiv F G) (e\u2082 : equiv G H) : equiv F H :=\n\u27e8e\u2082.1.comp e\u2081.1, e\u2081.2.comp e\u2082.2,\nby rw [morphism.comp_assoc, \u2190 e\u2082.2.comp_assoc, e\u2082.3, morphism.id_comp, e\u2081.3],\nby rw [morphism.comp_assoc, \u2190 e\u2081.1.comp_assoc, e\u2081.4, morphism.id_comp, e\u2082.4]\u27e9\n\ndef res_subset {F : sheaf_on_opens.{v} \u03b1 U} {G : sheaf_on_opens.{w} \u03b1 U} (e : equiv F G)\n  (V : \u03b1) (HVU : V \u2264 U) : equiv (F.res_subset V HVU) (G.res_subset V HVU) :=\n\u27e8e.1.res_subset V HVU, e.2.res_subset V HVU,\nby rw [morphism.comp_res_subset, e.3, morphism.id_res_subset],\nby rw [morphism.comp_res_subset, e.4, morphism.id_res_subset]\u27e9\n\nend equiv\n\nend sheaf_on_opens\n /-\n** TODO **\n#check @lattice.supr\nsupr : \u03a0 {\u03b1 : Type u_1} {\u03b9 : Sort u_2} [_inst_1 : has_Sup \u03b1], (\u03b9 \u2192 \u03b1) \u2192 \u03b1\n\nWhy not\n\nsupr : \u03a0 {\u03b1 : Type u_1} [_inst_1 : has_Sup \u03b1] {\u03b9 : Sort u_2}, (\u03b9 \u2192 \u03b1) \u2192 \u03b1\n-/\n\ndef complete_lattice.supr (\u03b1 : Type u) (\u03b9 : Sort v) [X : complete_lattice \u03b1] :=\n  @lattice.supr \u03b1 \u03b9 _ -- Grumpy old mathematician observes that stupid polymorphism\n                      -- makes me have to fill in more stuff\n\ntheorem complete_lattice.subset_Union (\u03b1 : Type u) [X : complete_lattice \u03b1] {I : Type} (s : I \u2192 \u03b1) (i : I) :\n  s i \u2264 supr s :=\nlattice.complete_lattice.le_supr s i\n\n--def complete_lattice.Union : \u03a0 {I : Type 37}, (I \u2192 \u03b1) \u2192 \u03b1\n--#check complete_lattice.supr -- fails\n/-- thing I need -/\n\nstructure thing (\u03b1 : Type u) extends semilattice_inf \u03b1 :=\n(supr {\u03b9 : Sort v} (s : \u03b9 \u2192 \u03b1) : \u03b1)\n(le_supr {\u03b9 : Sort v} : \u2200 (s : \u03b9 \u2192 \u03b1) (i : \u03b9), s i \u2264 supr s)\n/- hey -- that just *forced* me to make `thing.le_supr` have inputs in the following order:\n\nthing.le_supr : \u2200 {\u03b1 : Type u_2} (c : thing \u03b1) {\u03b9 : Sort u_1} (s : \u03b9 \u2192 \u03b1) (i : \u03b9), s i \u2264 c.supr s\n\nBut \n\nlattice.le_supr :\n  \u2200 {\u03b1 : Type u_1} {\u03b9 : Sort u_2} [_inst_1 : lattice.complete_lattice \u03b1] (s : \u03b9 \u2192 \u03b1) (i : \u03b9),\n    s i \u2264 lattice.supr s\n-/\n\nnamespace thing\n\ninstance (\u03b1 : Type u) (Y : thing \u03b1) : semilattice_inf \u03b1 := thing.to_semilattice_inf Y\n-- example (\u03b1 : Type u) (Y : thing \u03b1) : semilattice_inf \u03b1 := by apply_instance -- fails\n\n-- Debugging starts here.\n\n-- structure thing (\u03b1 : Type u) extends semilattice_inf \u03b1 :=\n\n-- #print semilattice_inf\n/-\n/-- A `semilattice_inf` is a meet-semilattice, that is, a partial order\n  with a meet (a.k.a. glb / greatest lower bound, inf / infimum) operation\n  `\u2293` which is the greatest element smaller than both factors. -/\nclass semilattice_inf (\u03b1 : Type u) extends has_inf \u03b1, partial_order \u03b1 :=\n(inf_le_left : \u2200 a b : \u03b1, a \u2293 b \u2264 a)\n(inf_le_right : \u2200 a b : \u03b1, a \u2293 b \u2264 b)\n(le_inf : \u2200 a b c : \u03b1, a \u2264 b \u2192 a \u2264 c \u2192 a \u2264 b \u2293 c)\n\n@[class]\nstructure lattice.semilattice_inf : Type u \u2192 Type u\nfields: ...\n-/\n\n-- #print notation \u2293 -- lattice.has_inf.inf at 70\n\n-- #check semilattice_inf\n-- class semilattice_inf (\u03b1 : Type u) extends has_inf \u03b1, partial_order \u03b1 :=\n\n--#where \n\n/-\nstructure thing (\u03b1 : Type u) extends semilattice_inf \u03b1 :=\n(supr {\u03b9 : Sort v} (s : \u03b9 \u2192 \u03b1) : \u03b1)\n(le_supr {\u03b9 : Sort v} : \u2200 (s : \u03b9 \u2192 \u03b1) (i : \u03b9), s i \u2264 supr s)\n-/\n--#print semilattice_inf\n-- it's a class\n\nexample (\u03b1 : Type u) [bounded_lattice \u03b1] [has_Sup \u03b1] [has_Inf \u03b1] [lattice.complete_lattice \u03b1] :\n  semilattice_inf \u03b1 := by apply_instance\n\n-- #print lattice.complete_lattice\n/-\nclass complete_lattice (\u03b1 : Type u) extends bounded_lattice \u03b1, has_Sup \u03b1, has_Inf \u03b1 :=\n(le_Sup : \u2200s, \u2200a\u2208s, a \u2264 Sup s)\n(Sup_le : \u2200s a, (\u2200b\u2208s, b \u2264 a) \u2192 Sup s \u2264 a)\n(Inf_le : \u2200s, \u2200a\u2208s, Inf s \u2264 a)\n(le_Inf : \u2200s a, (\u2200b\u2208s, a \u2264 b) \u2192 a \u2264 Inf s)\n-/\nset_option pp.structure_instances true\ndef canonical2.to_fun (\u03b1 : Type u) \n[lattice.complete_lattice \u03b1] : thing.{v} \u03b1 :=\nbegin\n--let this : semilattice_inf \u03b1 := by apply_instance,\nrefine { inf := semilattice_inf.inf, \n         le := semilattice_inf.le,\n         le_refl := semilattice_inf.le_refl,\n         le_trans := semilattice_inf.le_trans,\n         le_antisymm := semilattice_inf.le_antisymm,\n  --       inf_le_inf := semilattice_inf.inf_le_inf,\n         inf_le_left := semilattice_inf.inf_le_left,\n         inf_le_right := semilattice_inf.inf_le_right,\n         le_inf := semilattice_inf.le_inf,\n          -- the rest are not from semilattice namespace\n         le_supr := \u03bb (\u03b9 : Sort v) s, begin convert lattice.complete_lattice.le_supr s, convert rfl end,--convert rfl using 0, congr', ext, convert iff.rfl, ext, convert iff.rfl, ext, convert iff.rfl, sorry end,\n         supr := \u03bb \u03b9 s, lattice.complete_lattice.supr s,\n         },\nend\n\n--#check complete_lattice\n/-\n/-- A complete lattice is a bounded lattice which\n  has suprema and infima for every subset. -/\nclass complete_lattice (\u03b1 : Type u) extends bounded_lattice \u03b1, has_Sup \u03b1, has_Inf \u03b1 :=\n(le_Sup : \u2200s, \u2200a\u2208s, a \u2264 Sup s)\n(Sup_le : \u2200s a, (\u2200b\u2208s, b \u2264 a) \u2192 Sup s \u2264 a)\n(Inf_le : \u2200s, \u2200a\u2208s, Inf s \u2264 a)\n(le_Inf : \u2200s a, (\u2200b\u2208s, a \u2264 b) \u2192 a \u2264 Inf s)\n-/\n\n-- lattice \u03b1, order_top \u03b1, order_bot \u03b1\n--#check lattice\n\n/-\n/-- A `semilattice_sup` is a join-semilattice, that is, a partial order\n  with a join (a.k.a. lub / least upper bound, sup / supremum) operation\n  `\u2294` which is the least element larger than both factors. -/\nclass semilattice_sup (\u03b1 : Type u) extends has_sup \u03b1, partial_order \u03b1 :=\n(le_sup_left : \u2200 a b : \u03b1, a \u2264 a \u2294 b)\n(le_sup_right : \u2200 a b : \u03b1, b \u2264 a \u2294 b)\n(sup_le : \u2200 a b c : \u03b1, a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c)\n-/\n\n--#print has_sup\n--#print notation \u2294\n\n\ninductive ubool : Type u\n| tt : ubool\n| ff : ubool\n\n-- in thing namespace\n--#check @lattice.complete_lattice.sup\n\n-- there might be a universe issue here\ndef sup {\u03b1 : Type u} [c : complete_lattice \u03b1] (Y : thing \u03b1) (a b : \u03b1) : \u03b1 :=\n@thing.supr \u03b1 Y ubool (@ubool.rec (\u03bb _, \u03b1) a b)\n\ninstance has_sup (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : has_sup \u03b1 :=\n{ sup := \u03bb a b, @thing.supr \u03b1 Y ubool (@ubool.rec (\u03bb _, \u03b1) a b)}\n\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : partial_order \u03b1 := by apply_instance\n\n\nvariables (\u03b1 : Type u) [semilattice_inf \u03b1]\n\n--(supr {\u03b9 : Sort v} (s : \u03b9 \u2192 \u03b1) : \u03b1)\n--(le_supr {\u03b9 : Sort v} : \u2200 (s : \u03b9 \u2192 \u03b1) (i : \u03b9), s i \u2264 supr s)\n\nattribute [instance] thing.to_semilattice_inf\n\ntheorem le_sup_left (Y : thing \u03b1) : begin \n-- I need this for the notation\nletI : has_sup \u03b1 := @thing.has_sup \u03b1 _ Y,\nresetI,\nexact \u2200 (a b : \u03b1), a \u2264 a \u2294 b end :=\nbegin\n  sorry,\nend\n--  intros h a b,\n--  unfold has_sup.sup,\n--  convert Y.le_supr _ _ using 0,\n--  convert rfl using 0,\n--  sorry,\n\n\n-- #print semilattice_sup -- it's a class\n /-\n class semilattice_sup (\u03b1 : Type u) extends has_sup \u03b1, partial_order \u03b1 :=\n(le_sup_left : \u2200 a b : \u03b1, a \u2264 a \u2294 b)\n(le_sup_right : \u2200 a b : \u03b1, b \u2264 a \u2294 b)\n(sup_le : \u2200 a b c : \u03b1, a \u2264 c \u2192 b \u2264 c \u2192 a \u2294 b \u2264 c)\n-/\ninstance (\u03b1 : Type u) [H : semilattice_inf \u03b1] (Y : thing \u03b1) : semilattice_sup \u03b1 :=\n{ sup := \u03bb a b, @thing.supr \u03b1 Y bool (@bool.rec (\u03bb _, \u03b1) a b),\n  le_sup_left := begin sorry end,-- Y.le_sup_left,\n  le_sup_right := sorry,\n  sup_le := sorry, ..H}\n\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : lattice \u03b1 := \nbegin\n  sorry\nend\n#exit\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : order_top \u03b1 := by apply_instance\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : order_bot \u03b1 := by apply_instance\ninstance (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : bounded_lattice \u03b1 :=\nby apply_instance #exit\nlattice.complete_lattice \u03b1 :=\n\nbounded_lattice \u03b1\n--class complete_lattice (\u03b1 : Type u) extends bounded_lattice \u03b1, has_Sup \u03b1, has_Inf \u03b1 :=\n\n\ndef canonical2.inv_fun (\u03b1 : Type u) [semilattice_inf \u03b1] (Y : thing \u03b1) : lattice.complete_lattice \u03b1 :=\nbegin\n  \n  sorry\nend\n\n#exit\nbegin\n  \n  sorry\nend\n#exit\n  apply ..X,\n{ supr := \u03bb \u03b9 s, lattice.complete_lattice.supr,--\u2200 {\u03b9 : Sort v} (s : \u03b9 \u2192 \u03b1), \u03b1 := \u03bb \u03b9 s, lattice.complete_lattice.supr,\n  le_supr {\u03b9 : Sort v} : \u2200 (s : \u03b9 \u2192 \u03b1) (i : \u03b9), s i \u2264 supr s := sorry}\n\n#exit\n-- by apply_instance #exit\n-- semilattice_inf\nbegin \n  letI : has_inf \u03b1 := by apply_instance,\n  letI : partial_order \u03b1 := by apply_instance,\n  exactI { inf := X.inf,\n  le := _,\n  le_refl := X.le_refl,\n  le_trans := begin convert X.le_trans, ext, convert iff.rfl, ext, sorry end,\n  le_antisymm := _,\n  -- TODO : convert docstring does not mention \n  inf_le_left := begin convert X.inf_le_left, convert rfl, convert rfl, sorry end,\n  inf_le_right := _,\n  le_inf := _ },\n  repeat {sorry}\nend\n#exit\n{ inf := _,\n  le := _,\n  lt := _,\n  le_refl := _,\n  le_trans := _,\n  lt_iff_le_not_le := _,\n  le_antisymm := _,\n  inf_le_left := _,\n  inf_le_right := _,\n  le_inf := _ }\n--#where\n\n-- thing is a structure, complete_lattice is a class\n\n--#print lattice.complete_lattice\n/-\nMario:\n  complete_lattice <- bounded_lattice <- lattice, order_top, order_bot and lattice <- semilattice_sup, semilattice_inf <- partial order\nLean:\n  class complete_lattice (\u03b1 : Type u) extends bounded_lattice \u03b1, has_Sup \u03b1, has_Inf \u03b1 :=\n\n-/\n\n--def lattice.complete_lattice.supr := sorry\ndef canonical1.to_fun (\u03b1 : Type u) \n[bounded_lattice \u03b1] [has_Sup \u03b1] [has_Inf \u03b1]\n[X : lattice.complete_lattice \u03b1] : thing \u03b1 :=\nbegin resetI,\n  exact { inf := _,\n  le := X.le,\n  le_refl := X.le_refl,\n  le_trans := X.le_trans,\n  le_antisymm := X.le_antisymm,\n  inf_le_left := X.inf_le_left,\n  inf_le_right := X.inf_le_right,\n  le_inf := X.le_inf,\n  supr := X.supr,\n  le_supr := _ },\n  repeat {sorry},\n  end\n#exit\ndef canonical1 (\u03b1 : Type u) : _root_.equiv (lattice.complete_lattice \u03b1) (thing \u03b1) :=\n{ to_fun := \u03bb X, { \n    inf := _,\n    le := X.le,\n    le_refl := X.le_refl,\n    le_trans := X.le_trans,\n    le_antisymm := X.le_antisymm,\n    inf_le_left := X.inf_le_left,\n    inf_le_right := X.inf_le_right,\n    le_inf := X.le_inf,\n    supr := \u03bb I, @lattice.supr \u03b1 I (by resetI; apply_instance), -- bit of an effort!\n    le_supr := \u03bb \u03b9, @lattice.complete_lattice.le_supr _ \u03b9 (by resetI; apply_instance),--==s i \u2264 lattice.supr s@lattice.complete_lattice.le_supr ,--==lattice.complete_lattice.le_supr},--begin sorry, end,\n  },inv_fun := \u03bb Y, by exact { sup := _,\n  le := Y.le,\n  lt := _,\n  le_refl := _,\n  le_trans := _,\n  lt_iff_le_not_le := Y.lt_iff_le_not_le,\n  le_antisymm := _,\n  le_sup_left := begin convert Y.le_sup_left, sorry end--sorry, -- I hate it when sorry is underlined\n  le_sup_right := _,\n  sup_le := _,\n  inf := _,\n  inf_le_left := _,\n  inf_le_right := _,\n  le_inf := _,\n  top := _,\n  le_top := _,\n  bot := _,\n  bot_le := _,\n  Sup := _,\n  Inf := _,\n  le_Sup := _,\n  Sup_le := _,\n  Inf_le := _,\n  le_Inf := _ },\n  left_inv := \u03bb X, _,\n  right_inv := sorry }\n#exit\n\n#print canonical1.le\n\n_root_.equiv (thing \u03b1) (topological_space.opens (thing.Union \u03b1 )) :=\n{ to_fun := begin\n    rintro \u27e8SLI\u03b1,U,sU\u27e9,\n\n  end,\n  inv_fun := _,\n  left_inv := _,\n  right_inv := _ }\n\n-- should be in mathlib\n\n#check semilattice_inf\nnamespace opens\n\ndef Union {I : Type*} (s : I \u2192 \u03b1) : \u03b1 :=\n\u27e8set.Union (\u03bb i, (s i).1), is_open_Union (\u03bb i, (s i).2)\u27e9\n\nvariables {I : Type*} (s : I \u2192 \u03b1)\n\ntheorem subset_Union : \u2200 (s : I \u2192 \u03b1) (i : I), s i \u2264 Union s :=\n-- why does lattice.le_supr need complete lattice?\n\u03bb s i x hx, set.mem_Union.2 \u27e8i, hx\u27e9\n\n/- Other things I might need about this Union\n\n@[simp] theorem mem_Union {x : \u03b2} {s : \u03b9 \u2192 set \u03b2} : x \u2208 Union s \u2194 \u2203 i, x \u2208 s i :=\n\u27e8assume \u27e8t, \u27e8\u27e8a, (t_eq : s a = t)\u27e9, (h : x \u2208 t)\u27e9\u27e9, \u27e8a, t_eq.symm \u25b8 h\u27e9,\n  assume \u27e8a, h\u27e9, \u27e8s a, \u27e8\u27e8a, rfl\u27e9, h\u27e9\u27e9\u27e9\n/- alternative proof: dsimp [Union, supr, Sup]; simp -/\n  -- TODO: more rewrite rules wrt forall / existentials and logical connectives\n  -- TODO: also eliminate \u2203i, ... \u2227 i = t \u2227 ...\n\ntheorem Union_subset {s : \u03b9 \u2192 set \u03b2} {t : set \u03b2} (h : \u2200 i, s i \u2286 t) : (\u22c3 i, s i) \u2286 t :=\n-- TODO: should be simpler when sets' order is based on lattices\n@supr_le (set \u03b2) _ set.lattice_set _ _ h\n\ntheorem Union_subset_iff {\u03b1 : Sort u} {s : \u03b1 \u2192 set \u03b2} {t : set \u03b2} : (\u22c3 i, s i) \u2286 t \u2194 (\u2200 i, s i \u2286 t):=\n\u27e8assume h i, subset.trans (le_supr s _) h, Union_subset\u27e9\n\ntheorem subset_Union : \u2200 (s : \u03b9 \u2192 set \u03b2) (i : \u03b9), s i \u2286 (\u22c3 i, s i) := le_supr\n\ntheorem Union_const [inhabited \u03b9] (s : set \u03b2) : (\u22c3 i:\u03b9, s) = s :=\next $ by simp\n\ntheorem inter_Union_left (s : set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  s \u2229 (\u22c3 i, t i) = \u22c3 i, s \u2229 t i :=\next $ by simp\n\ntheorem inter_Union_right (s : set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  (\u22c3 i, t i) \u2229 s = \u22c3 i, t i \u2229 s :=\next $ by simp\n\ntheorem Union_union_distrib (s : \u03b9 \u2192 set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  (\u22c3 i, s i \u222a t i) = (\u22c3 i, s i) \u222a (\u22c3 i, t i) :=\next $ by simp [exists_or_distrib]\n\ntheorem union_Union_left [inhabited \u03b9] (s : set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  s \u222a (\u22c3 i, t i) = \u22c3 i, s \u222a t i :=\nby rw [Union_union_distrib, Union_const]\n\ntheorem union_Union_right [inhabited \u03b9] (s : set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  (\u22c3 i, t i) \u222a s = \u22c3 i, t i \u222a s :=\nby rw [Union_union_distrib, Union_const]\n\ntheorem diff_Union_right (s : set \u03b2) (t : \u03b9 \u2192 set \u03b2) :\n  (\u22c3 i, t i) \\ s = \u22c3 i, t i \\ s :=\ninter_Union_right _ _\n\n-/\n\nend opens\n\ndef glue {I : Type*} (S : I \u2192 \u03b1) (F : \u03a0 (i : I), sheaf_on_opens.{v} \u03b1 (S i))\n  (\u03c6 : \u03a0 (i j : I),\n    equiv ((F i).res_subset ((S i) \u2229 (S j)) (set.inter_subset_left _ _)) ((F j).res_subset ((S i) \u2229 (S j)) (set.inter_subset_right _ _)))\n  (H\u03c61 : \u2200 i, \u03c6 i i = equiv.refl (F i))\n  (H\u03c62 : \u2200 i j k,\n    ((\u03c6 i j).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.inter_subset_left _ _)).trans\n      ((\u03c6 j k).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.subset_inter (le_trans (set.inter_subset_left _ _)\n       (set.inter_subset_right _ _)) (set.inter_subset_right _ _))) =\n    (\u03c6 i k).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.subset_inter (le_trans (set.inter_subset_left _ _)\n    (set.inter_subset_left _ _)) (set.inter_subset_right _ _))) :\n  sheaf_on_opens.{max u v} \u03b1 (opens.Union S) :=\n{ F :=\n  { F := \u03bb W, { f : \u03a0 i, (F i).eval ((S i) \u2229 W) (set.inter_subset_left _ _) //\n      \u2200 i j, (\u03c6 i j).1.map ((S i) \u2229 (S j) \u2229 W) (set.inter_subset_left _ _)\n        ((F i).res ((S i) \u2229 W) _ _ (le_trans (set.inter_subset_left _ _) (set.inter_subset_left _ _))\n          (set.subset_inter (le_trans (set.inter_subset_left _ _) (set.inter_subset_left _ _)) (set.inter_subset_right _ _))\n          (f i)) =\n        (F j).res ((S j) \u2229 W) _ _ (le_trans (set.inter_subset_left _ _) (set.inter_subset_right _ _))\n          (set.subset_inter (le_trans (set.inter_subset_left _ _) (set.inter_subset_right _ _)) (set.inter_subset_right _ _))\n          (f j) },\n    res := \u03bb U V HUV f, \u27e8\u03bb i, (F i).res (S i \u2229 U) _ (S i \u2229 V) _ (set.inter_subset_inter_right _ HUV) (f.val i),\n      begin\n        intros i j,\n        rw res_comp,\n        rw res_comp,\n        have answer := congr_arg\n        (res (F j)\n          (S i \u2229 (S j) \u2229 U) _\n          (S i \u2229 (S j) \u2229 V) (le_trans (set.inter_subset_left _ _) (set.inter_subset_right _ _)) (set.inter_subset_inter_right _ HUV)\n        )\n        (f.property i j),\n        rw res_comp at answer,\n        rw \u2190answer,\n        clear answer,\n        convert (\u03c6 i j).to_fun.commutes\n        (S i \u2229 (S j) \u2229 U) (set.inter_subset_left _ _)\n        (S i \u2229 (S j) \u2229 V) (set.inter_subset_left _ _) (set.inter_subset_inter_right _ HUV)\n        (\n          (@sheaf_on_opens.res _ _ (S i \u2229 U)\n            (F i)\n            (S i \u2229 U) (by refl)\n            (S i \u2229 S j \u2229 U) (set.inter_subset_inter_left _ (set.inter_subset_left _ _)) (set.inter_subset_inter_left _ (set.inter_subset_left _ _))\n            (f.val i)\n          )\n        ) using 2,\n        convert (F i).F.Hcomp' (S i \u2229 U) (S i \u2229 S j \u2229 U) (S i \u2229 S j \u2229 V) _ _ (f.val i),\n      end\u27e9,\n    Hid := begin\n      sorry\n    end,\n    Hcomp := sorry },\n  locality := sorry,\n  gluing := sorry }\n\ndef mario_belief (s : set \u03b1) (F : \u03a0 (i : I), sheaf_on_opens.{v} \u03b1 (S i)) \n\ndef universal_property (I : Type*) (S : I \u2192 \u03b1) (F : \u03a0 (i : I), sheaf_on_opens.{v} \u03b1 (S i))\n  (\u03c6 : \u03a0 (i j : I),\n    equiv ((F i).res_subset ((S i) \u2229 (S j)) (set.inter_subset_left _ _)) ((F j).res_subset ((S i) \u2229 (S j)) (set.inter_subset_right _ _)))\n  (H\u03c61 : \u2200 i, \u03c6 i i = equiv.refl (F i))\n  (H\u03c62 : \u2200 i j k,\n    ((\u03c6 i j).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.inter_subset_left _ _)).trans\n      ((\u03c6 j k).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.subset_inter (le_trans (set.inter_subset_left _ _) (set.inter_subset_right _ _)) (set.inter_subset_right _ _))) =\n    (\u03c6 i k).res_subset ((S i) \u2229 (S j) \u2229 (S k)) (set.subset_inter (le_trans (set.inter_subset_left _ _) (set.inter_subset_left _ _)) (set.inter_subset_right _ _))) :\n\u2200 i : I, equiv (res_subset (glue S F \u03c6 H\u03c61 H\u03c62) (S i) $ opens.subset_Union S i) (F i) := sorry\n\n/- questions:\n\n1) Can you glue presheaves of types? And prove the universal property for sets?\n2) Cah you glue sheaves of types? And prove the universal property for sets?\n3) Can you glue sheaves of rings? And prove the universal property for sets?\n4) What about proving the universal property for types?\n-- You are the winner if you get this far\n\nend sheaf_on_opens\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/Examples/sheaf_on_opens_total_glueing.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3940588332391471}}
{"text": "\nstructure A :=\n(x y : Nat)\n\nstructure B :=\n(z : Nat)\n\nstructure C extends A, B\n\ndef f (c : C) :=\nc.x + c.y + c.z\n\ntheorem ex1 : f {x := 10, y := 20, z := 30} = 60 :=\nrfl\n\nstructure D\n\ndef g (d : D) : D :=\nd\n\ntheorem ex2 : g {} = {} :=\nrfl\n\ntheorem ex3 (d : D) : g d = {} := by\n cases d\n exact rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/structNoBody.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.39405883323914703}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.sites.compatible_plus\n\n/-!\n\nIn this file, we prove that sheafification is compatible with functors which\npreserve the correct limits and colimits.\n\n-/\n\nnamespace category_theory.grothendieck_topology\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverses w\u2081 w\u2082 v u\nvariables {C : Type u} [category.{v} C] (J : grothendieck_topology C)\nvariables {D : Type w\u2081} [category.{max v u} D]\nvariables {E : Type w\u2082} [category.{max v u} E]\nvariables (F : D \u2964 E)\n\nnoncomputable theory\n\nvariables [\u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1),\n  has_limits_of_shape (walking_multicospan fst snd) D]\nvariables [\u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1),\n  has_limits_of_shape (walking_multicospan fst snd) E]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 E]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\nvariables [\u2200 (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D), preserves_limit (W.index P).multicospan F]\n\nvariables (P : C\u1d52\u1d56 \u2964 D)\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`.\n\nUse the lemmas `whisker_right_to_sheafify_sheafify_comp_iso_hom`,\n`to_sheafify_comp_sheafify_comp_iso_inv` and `sheafify_comp_iso_inv_eq_sheafify_lift` to reduce\nthe components of this isomorphisms to a state that can be handled using the universal property\nof sheafification. -/\ndef sheafify_comp_iso : J.sheafify P \u22d9 F \u2245 J.sheafify (P \u22d9 F) :=\nJ.plus_comp_iso _ _ \u226a\u226b (J.plus_functor _).map_iso (J.plus_comp_iso _ _)\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`, functorially in `F`. -/\ndef sheafification_whisker_left_iso (P : C\u1d52\u1d56 \u2964 D)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (whiskering_left _ _ E).obj (J.sheafify P) \u2245\n  (whiskering_left _ _ _).obj P \u22d9 J.sheafification E :=\nbegin\n  refine J.plus_functor_whisker_left_iso _ \u226a\u226b _ \u226a\u226b functor.associator _ _ _,\n  refine iso_whisker_right _ _,\n  refine J.plus_functor_whisker_left_iso _,\nend\n\n@[simp]\nlemma sheafification_whisker_left_iso_hom_app (P : C\u1d52\u1d56 \u2964 D) (F : D \u2964 E)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (sheafification_whisker_left_iso J P).hom.app F = (J.sheafify_comp_iso F P).hom :=\nbegin\n  dsimp [sheafification_whisker_left_iso, sheafify_comp_iso],\n  rw category.comp_id,\nend\n\n@[simp]\nlemma sheafification_whisker_left_iso_inv_app (P : C\u1d52\u1d56 \u2964 D) (F : D \u2964 E)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (sheafification_whisker_left_iso J P).inv.app F = (J.sheafify_comp_iso F P).inv :=\nbegin\n  dsimp [sheafification_whisker_left_iso, sheafify_comp_iso],\n  erw category.id_comp,\nend\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`, functorially in `P`. -/\ndef sheafification_whisker_right_iso :\n  J.sheafification D \u22d9 (whiskering_right _ _ _).obj F \u2245\n  (whiskering_right _ _ _).obj F \u22d9 J.sheafification E :=\nbegin\n  refine functor.associator _ _ _ \u226a\u226b _,\n  refine iso_whisker_left (J.plus_functor D) (J.plus_functor_whisker_right_iso _) \u226a\u226b _,\n  refine _ \u226a\u226b functor.associator _ _ _,\n  refine (functor.associator _ _ _).symm \u226a\u226b _,\n  exact iso_whisker_right (J.plus_functor_whisker_right_iso _) (J.plus_functor E),\nend\n\n@[simp]\nlemma sheafification_whisker_right_iso_hom_app :\n  (J.sheafification_whisker_right_iso F).hom.app P = (J.sheafify_comp_iso F P).hom :=\nbegin\n  dsimp [sheafification_whisker_right_iso, sheafify_comp_iso],\n  simp only [category.id_comp, category.comp_id],\n  erw category.id_comp,\nend\n\n@[simp]\nlemma sheafification_whisker_right_iso_inv_app :\n  (J.sheafification_whisker_right_iso F).inv.app P = (J.sheafify_comp_iso F P).inv :=\nbegin\n  dsimp [sheafification_whisker_right_iso, sheafify_comp_iso],\n  simp only [category.id_comp, category.comp_id],\n  erw category.id_comp,\nend\n\n@[simp, reassoc]\nlemma whisker_right_to_sheafify_sheafify_comp_iso_hom :\n  whisker_right (J.to_sheafify _) _ \u226b (J.sheafify_comp_iso F P).hom = J.to_sheafify _ :=\nbegin\n  dsimp [sheafify_comp_iso],\n  erw [whisker_right_comp, category.assoc],\n  slice_lhs 2 3 { rw plus_comp_iso_whisker_right },\n  rw [category.assoc, \u2190 J.plus_map_comp,\n    whisker_right_to_plus_comp_plus_comp_iso_hom, \u2190 category.assoc,\n    whisker_right_to_plus_comp_plus_comp_iso_hom],\n  refl,\nend\n\n@[simp, reassoc]\nlemma to_sheafify_comp_sheafify_comp_iso_inv :\n  J.to_sheafify _ \u226b (J.sheafify_comp_iso F P).inv = whisker_right (J.to_sheafify _) _ :=\nby { rw iso.comp_inv_eq, simp }\n\nsection\n\n-- We will sheafify `D`-valued presheaves in this section.\nvariables\n  [concrete_category.{max v u} D]\n  [preserves_limits (forget D)]\n  [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\n  [reflects_isomorphisms (forget D)]\n\n@[simp]\nlemma sheafify_comp_iso_inv_eq_sheafify_lift : (J.sheafify_comp_iso F P).inv =\n  J.sheafify_lift (whisker_right (J.to_sheafify _) _) ((J.sheafify_is_sheaf _).comp _) :=\nbegin\n  apply J.sheafify_lift_unique,\n  rw iso.comp_inv_eq,\n  simp,\nend\n\nend\n\nend category_theory.grothendieck_topology\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sites/compatible_sheafification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3940588263287031}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.algebra.monoid\nimport Mathlib.algebra.group.pi\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 l \n\nnamespace Mathlib\n\n/-!\n# Topological group with zero\n\nIn this file we define `has_continuous_inv'` to be a mixin typeclass a type with `has_inv` and\n`has_zero` (e.g., a `group_with_zero`) such that `\u03bb x, x\u207b\u00b9` is continuous at all nonzero points. Any\nnormed (semi)field has this property. Currently the only example of `has_continuous_inv'` in\n`mathlib` which is not a normed field is the type `nnnreal` (a.k.a. `\u211d\u22650`) of nonnegative real\nnumbers.\n\nThen we prove lemmas about continuity of `x \u21a6 x\u207b\u00b9` and `f / g` providing dot-style `*.inv'` and\n`*.div` operations on `filter.tendsto`, `continuous_at`, `continuous_within_at`, `continuous_on`,\nand `continuous`. As a special case, we provide `*.div_const` operations that require only\n`group_with_zero` and `has_continuous_mul` instances.\n\nAll lemmas about `(\u207b\u00b9)` use `inv'` in their names because lemmas without `'` are used for\n`topological_group`s. We also use `'` in the typeclass name `has_continuous_inv'` for the sake of\nconsistency of notation.\n-/\n\n/-!\n### A group with zero with continuous multiplication\n\nIf `G\u2080` is a group with zero with continuous `(*)`, then `(/y)` is continuous for any `y`. In this\nsection we prove lemmas that immediately follow from this fact providing `*.div_const` dot-style\noperations on `filter.tendsto`, `continuous_at`, `continuous_within_at`, `continuous_on`, and\n`continuous`.\n-/\n\ntheorem filter.tendsto.div_const {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {l : filter \u03b1} {x : G\u2080} {y : G\u2080}\n    (hf : filter.tendsto f l (nhds x)) : filter.tendsto (fun (a : \u03b1) => f a / y) l (nhds (x / y)) :=\n  sorry\n\ntheorem continuous_at.div_const {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    (hf : continuous f) {y : G\u2080} : continuous fun (x : \u03b1) => f x / y :=\n  sorry\n\ntheorem continuous_within_at.div_const {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {s : set \u03b1} [topological_space \u03b1]\n    {a : \u03b1} (hf : continuous_within_at f s a) {y : G\u2080} :\n    continuous_within_at (fun (x : \u03b1) => f x / y) s a :=\n  filter.tendsto.div_const hf\n\ntheorem continuous_on.div_const {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {s : set \u03b1} [topological_space \u03b1]\n    (hf : continuous_on f s) {y : G\u2080} : continuous_on (fun (x : \u03b1) => f x / y) s :=\n  sorry\n\ntheorem continuous.div_const {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    (hf : continuous f) {y : G\u2080} : continuous fun (x : \u03b1) => f x / y :=\n  sorry\n\n/-- A type with `0` and `has_inv` such that `\u03bb x, x\u207b\u00b9` is continuous at all nonzero points. Any\nnormed (semi)field has this property. -/\nclass has_continuous_inv' (G\u2080 : Type u_3) [HasZero G\u2080] [has_inv G\u2080] [topological_space G\u2080] where\n  continuous_at_inv' : \u2200 {x : G\u2080}, x \u2260 0 \u2192 continuous_at has_inv.inv x\n\n/-!\n### Continuity of `\u03bb x, x\u207b\u00b9` at a non-zero point\n\nWe define `topological_group_with_zero` to be a `group_with_zero` such that the operation `x \u21a6 x\u207b\u00b9`\nis continuous at all nonzero points. In this section we prove dot-style `*.inv'` lemmas for\n`filter.tendsto`, `continuous_at`, `continuous_within_at`, `continuous_on`, and `continuous`.\n-/\n\ntheorem tendsto_inv' {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] {x : G\u2080} (hx : x \u2260 0) :\n    filter.tendsto has_inv.inv (nhds x) (nhds (x\u207b\u00b9)) :=\n  continuous_at_inv' hx\n\ntheorem continuous_on_inv' {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] : continuous_on has_inv.inv (singleton 0\u1d9c) :=\n  fun (x : G\u2080) (hx : x \u2208 (singleton 0\u1d9c)) =>\n    continuous_at.continuous_within_at (continuous_at_inv' hx)\n\n/-- If a function converges to a nonzero value, its inverse converges to the inverse of this value.\nWe use the name `tendsto.inv'` as `tendsto.inv` is already used in multiplicative topological\ngroups. -/\ntheorem filter.tendsto.inv' {\u03b1 : Type u_1} {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] {l : filter \u03b1} {f : \u03b1 \u2192 G\u2080} {a : G\u2080}\n    (hf : filter.tendsto f l (nhds a)) (ha : a \u2260 0) :\n    filter.tendsto (fun (x : \u03b1) => f x\u207b\u00b9) l (nhds (a\u207b\u00b9)) :=\n  filter.tendsto.comp (tendsto_inv' ha) hf\n\ntheorem continuous_within_at.inv' {\u03b1 : Type u_1} {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] {f : \u03b1 \u2192 G\u2080} {s : set \u03b1} {a : \u03b1}\n    [topological_space \u03b1] (hf : continuous_within_at f s a) (ha : f a \u2260 0) :\n    continuous_within_at (fun (x : \u03b1) => f x\u207b\u00b9) s a :=\n  filter.tendsto.inv' hf ha\n\ntheorem continuous_at.inv' {\u03b1 : Type u_1} {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] {f : \u03b1 \u2192 G\u2080} {a : \u03b1} [topological_space \u03b1]\n    (hf : continuous_at f a) (ha : f a \u2260 0) : continuous_at (fun (x : \u03b1) => f x\u207b\u00b9) a :=\n  filter.tendsto.inv' hf ha\n\ntheorem continuous.inv' {\u03b1 : Type u_1} {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] {f : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    (hf : continuous f) (h0 : \u2200 (x : \u03b1), f x \u2260 0) : continuous fun (x : \u03b1) => f x\u207b\u00b9 :=\n  iff.mpr continuous_iff_continuous_at\n    fun (x : \u03b1) => filter.tendsto.inv' (continuous.tendsto hf x) (h0 x)\n\ntheorem continuous_on.inv' {\u03b1 : Type u_1} {G\u2080 : Type u_2} [HasZero G\u2080] [has_inv G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] {f : \u03b1 \u2192 G\u2080} {s : set \u03b1} [topological_space \u03b1]\n    (hf : continuous_on f s) (h0 : \u2200 (x : \u03b1), x \u2208 s \u2192 f x \u2260 0) :\n    continuous_on (fun (x : \u03b1) => f x\u207b\u00b9) s :=\n  fun (x : \u03b1) (hx : x \u2208 s) => continuous_within_at.inv' (hf x hx) (h0 x hx)\n\n/-!\n### Continuity of division\n\nIf `G\u2080` is a `group_with_zero` with `x \u21a6 x\u207b\u00b9` continuous at all nonzero points and `(*)`, then\ndivision `(/)` is continuous at any point where the denominator is continuous.\n-/\n\ntheorem filter.tendsto.div {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080}\n    {g : \u03b1 \u2192 G\u2080} {l : filter \u03b1} {a : G\u2080} {b : G\u2080} (hf : filter.tendsto f l (nhds a))\n    (hg : filter.tendsto g l (nhds b)) (hy : b \u2260 0) : filter.tendsto (f / g) l (nhds (a / b)) :=\n  sorry\n\ntheorem continuous_within_at.div {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080]\n    [topological_space G\u2080] [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080}\n    {g : \u03b1 \u2192 G\u2080} [topological_space \u03b1] {s : set \u03b1} {a : \u03b1} (hf : continuous_within_at f s a)\n    (hg : continuous_within_at g s a) (h\u2080 : g a \u2260 0) : continuous_within_at (f / g) s a :=\n  filter.tendsto.div hf hg h\u2080\n\ntheorem continuous_on.div {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {g : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    {s : set \u03b1} (hf : continuous_on f s) (hg : continuous_on g s)\n    (h\u2080 : \u2200 (x : \u03b1), x \u2208 s \u2192 g x \u2260 0) : continuous_on (f / g) s :=\n  fun (x : \u03b1) (hx : x \u2208 s) => continuous_within_at.div (hf x hx) (hg x hx) (h\u2080 x hx)\n\n/-- Continuity at a point of the result of dividing two functions continuous at that point, where\nthe denominator is nonzero. -/\ntheorem continuous_at.div {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {g : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    {a : \u03b1} (hf : continuous_at f a) (hg : continuous_at g a) (h\u2080 : g a \u2260 0) :\n    continuous_at (f / g) a :=\n  filter.tendsto.div hf hg h\u2080\n\ntheorem continuous.div {\u03b1 : Type u_1} {G\u2080 : Type u_2} [group_with_zero G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] {f : \u03b1 \u2192 G\u2080} {g : \u03b1 \u2192 G\u2080} [topological_space \u03b1]\n    (hf : continuous f) (hg : continuous g) (h\u2080 : \u2200 (x : \u03b1), g x \u2260 0) : continuous (f / g) :=\n  eq.mpr\n    (id\n      ((fun (f f_1 : \u03b1 \u2192 G\u2080) (e_3 : f = f_1) => congr_arg continuous e_3) (f / g) (f * (g\u207b\u00b9))\n        (div_eq_mul_inv f g)))\n    (eq.mp (Eq.refl (continuous fun (x : \u03b1) => f x * (g x\u207b\u00b9)))\n      (continuous.mul hf (continuous.inv' hg h\u2080)))\n\ntheorem continuous_on_div {G\u2080 : Type u_2} [group_with_zero G\u2080] [topological_space G\u2080]\n    [has_continuous_inv' G\u2080] [has_continuous_mul G\u2080] :\n    continuous_on (fun (p : G\u2080 \u00d7 G\u2080) => prod.fst p / prod.snd p)\n        (set_of fun (p : G\u2080 \u00d7 G\u2080) => prod.snd p \u2260 0) :=\n  continuous_on.div continuous_on_fst continuous_on_snd fun (_x : G\u2080 \u00d7 G\u2080) => id\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/algebra/group_with_zero_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.39404732567977074}}
{"text": "import for_mathlib.group -- some stupid lemma about units\n\nimport Spa.space\nimport Huber_ring.localization\n\n/-!\n# Rational open subsets and their properties\n\nWe define a preorder on `rational_open_data` that will be used when\nconstructing the valuations on the stalks of the structure presheaf.\n-/\n\nopen_locale classical\nlocal attribute [instance] set.pointwise_mul_comm_semiring\nlocal attribute [instance] set.smul_set_action\n\nlocal postfix `\u207a` : 66 := \u03bb A : Huber_pair, A.plus\n\nnamespace spa\nopen set algebra\nvariables {A : Huber_pair}\n\nnamespace rational_open_data\nvariables (r : rational_open_data A)\n\n/-- The preorder on rational open data.\n\nDue to limitations in the existing mathematical library,\nwe cannot work with the \u201ccorrect\u201d preorder on rational open data.\nThe \u201ccorrect\u201d preorder on rational open data would be:\n\ndef correct_preorder : preorder (rational_open_data A) :=\n{ le := \u03bb r1 r2, rational_open r1 \u2286 rational_open r2,\n  le_refl := \u03bb _ _, id,\n  le_trans := \u03bb _ _ _, subset.trans }\n\nOne can prove (in maths) that r1 \u2264 r2 iff there's a continuous R-algebra morphism\nof Huber pairs localization r2 \u2192 localization r1. I think the \u2190 direction of this\niff is straightforward (but I didn't think about it too carefully). However we\ndefinitely cannot prove the \u2192 direction of this iff in this repo yet because we\ndon't have enough API for cont. Here is an indication\nof part of the problem. localization r2 is just A[1/r2.s]. But we cannot prove yet r2.s is\ninvertible in localization.r1, even though we know it doesn't vanish anywhere on\nrational_open r2 and hence on rational_open r1, because the fact that it doesn't vanish anywhere\non rational_open r1 only means that it's not in any prime ideal corresponding\nto a *continuous* valuation on localization r1 which is bounded by 1 on some + subring;\none would now need to prove, at least, that every maximal ideal\nis the support of a continuous valuation, which is Wedhorn 7.52(2). This is not\ntoo bad -- but it is work that we have not yet done. However this is by no means the whole story;\nwe would also need that r1.T is power-bounded in localization.r2\nand this looks much worse: it's Wedhorn 7.52(1). Everything is do-able, but it's just *long*.\nLong as in \"thousands more lines of code\". We will need a good theory of primary and\nsecondary specialisation of valuations and so on and so on. None of this is there at\nthe time of writing, although I see no obstruction to putting it there, other than the\nfact that it would take weeks of work.\n\nWe have to work with a weaker preorder then, because haven't made a good enough\nAPI for continuous valuations. We basically work with the preorder r1 \u2264 r2 iff\nthere's a continuous R-algebra map localization r2 \u2192 localization r1, i.e, we\ndefine our way around the problem. We are fortunate in that we can prove\n(in maths) that the projective limit over this preorder agrees with the projective\nlimit over the correct preorder. -/\ninstance : preorder (rational_open_data A) :=\n{ le := \u03bb r1 r2, \u2203 k : A, r1.s * k = r2.s \u2227\n    \u2200 t\u2081 \u2208 r1.T, \u2203 t\u2082 \u2208 r2.T, \u2203 N : \u2115, r2.s ^ N * t\u2082 = r2.s ^ N * (t\u2081 * k),\n  le_refl := \u03bb r, \u27e81, mul_one _, \u03bb t ht, \u27e8t, ht, 0, by rw mul_one\u27e9\u27e9,\n  le_trans := \u03bb a b c \u27e8k, hk, hab\u27e9 \u27e8l, hl, hbc\u27e9, \u27e8k * l, by rw [\u2190mul_assoc, hk, hl], \u03bb ta hta,\n  begin\n    rcases hab ta hta with \u27e8tb, htb, Nab, h1\u27e9,\n    rcases hbc tb htb with \u27e8hc, htc, Nbc, h2\u27e9,\n    refine \u27e8hc, htc, (Nab + Nbc), _\u27e9,\n    rw [\u2190mul_assoc, pow_add, mul_assoc, h2, \u2190hl, mul_pow, mul_pow],\n    rw (show b.s ^ Nab * l ^ Nab * (b.s ^ Nbc * l ^ Nbc * (tb * l)) =\n      b.s ^ Nab * tb * (l ^ Nab * (b.s ^ Nbc * l ^ Nbc *  l)), by ring),\n    rw h1,\n    ring\n  end\u27e9 }\n\nlemma le_inter_left (r1 r2 : rational_open_data A) :\n  r1 \u2264 (inter r1 r2) :=\nbegin\n  refine \u27e8r2.s, rfl, _\u27e9,\n  intros t1 ht1,\n  refine \u27e8t1 * r2.s, \u27e8t1, mem_insert_of_mem _ ht1, r2.s, mem_insert_s _, rfl\u27e9, 0, by simp\u27e9,\nend\n\nlemma le_inter_right (r1 r2 : rational_open_data A) :\n  r2 \u2264 (inter r1 r2) :=\nby { rw inter_symm, apply le_inter_left, }\n\n-- The preorder defined above is weaker than the preorder we're supposed to have but don't.\n-- However the projective limit we take over our preorder is provably (in maths) equal to\n-- the projective limit that we cannot even formalise. The thing we definitely need\n-- is that if r1 \u2264 r2 then there's a map localization r1 \u2192 localization r2\n\n/-- The localization of a Huber pair A at the rational open subset r = D(T,s) \u2286 spa(A). -/\ndef localization (r : rational_open_data A) := Huber_ring.away r.T r.s\n\nnamespace localization\n\n/-- The ring structure on the localization at the rational open subset r = D(T,s) \u2286 spa(A). -/\ninstance : comm_ring (localization r) :=\nby unfold localization; apply_instance\n\n/-- The basis of open subgroups of the localization\n at the rational open subset r = D(T,s) \u2286 spa(A). -/\ninstance : subgroups_basis (localization r) :=\nHuber_ring.away.top_loc_basis r.T r.s r.Hopen\n\n/-- The topology on the localization at the rational open subset r = D(T,s) \u2286 spa(A). -/\ninstance : topological_space (localization r) :=\nsubgroups_basis.topology _\n\n/-- The localization at the rational open subset r = D(T,s) \u2286 spa(A) is a topological ring. -/\ninstance : topological_ring (localization r) :=\nring_filter_basis.is_topological_ring _ rfl\n\n/-- The uniform structure on the localization at the rational open subset r = D(T,s) \u2286 spa(A). -/\ninstance (r : rational_open_data A) : uniform_space (rational_open_data.localization r) :=\ntopological_add_group.to_uniform_space _\n\n/-- The localization at the rational open subset r = D(T,s) \u2286 spa(A) is a uniform additive group. -/\ninstance (rd : rational_open_data A): uniform_add_group (rational_open_data.localization rd) :=\ntopological_add_group_is_uniform\n\n/-- The localization at the rational open subset r = D(T,s) \u2286 spa(A) is a an algebra over A. -/\ninstance : algebra A (localization r) := Huber_ring.away.algebra r.T r.s\n\n/-- The coercion from a Huber pair A\nto the localization at the rational open subset r = D(T,s) \u2286 spa(A). -/\ninstance : has_coe A (localization r) := \u27e8\u03bb a, (of_id A (localization r) : A \u2192 localization r) a\u27e9\n\nlemma nonarchimedean (r : rational_open_data A) :\n  topological_add_group.nonarchimedean (localization r) :=\nsubgroups_basis.nonarchimedean\n\nset_option class.instance_max_depth 38\n\n/--If A is a Huber pair, and r = D(T,s) a rational open subset of Spa(A),\nand coe is the localization map A \u2192 A(T/s),\nthen `power_bounded_data r` is the set { coe(t)/s | t \u2208 T } \u2286 A(T/s).-/\ndef power_bounded_data (r : rational_open_data A) : set (localization r) :=\nlet s_inv : localization r :=\n  ((localization.to_units \u27e8r.s, \u27e81, by simp\u27e9\u27e9)\u207b\u00b9 : units (localization r)) in\n(s_inv \u2022 (coe : A \u2192 localization r) '' r.T)\n\ntheorem power_bounded (r : rational_open_data A) :\n  is_power_bounded_subset (power_bounded_data r) :=\nbegin\n  suffices : is_bounded (ring.closure (power_bounded_data r)),\n  { exact is_bounded.subset add_group.subset_closure this },\n  intros U hU,\n  rcases subgroups_basis.mem_nhds_zero.mp hU with \u27e8_, \u27e8V, rfl\u27e9, hV\u27e9,\n  refine \u27e8_, mem_nhds_sets (subgroups_basis.is_op _ rfl (set.mem_range_self _)) _, _\u27e9,\n  { exact V },\n  { erw submodule.mem_coe,\n    convert submodule.zero_mem _ },\n  { intros v hv b hb,\n    apply hV,\n    rw [mul_comm, \u2190 smul_eq_mul],\n    rw submodule.mem_coe at hv \u22a2,\n    convert submodule.smul_mem _ _ hv,\n    swap, { exact \u27e8b, hb\u27e9 }, { refl } }\nend\n\nend localization\n\n/-- This auxilliary function produces r1.s as a unit in localization r2 -/\nnoncomputable def s_inv_aux (r1 r2 : rational_open_data A) (h : r1 \u2264 r2) :\n  units (localization r2) :=\n@units.unit_of_mul_left_eq_unit _ _\n  ((of_id A (localization r2) : A \u2192 r2.localization) r1.s)\n  ((of_id A (localization r2) : A \u2192 r2.localization) (classical.some h))\n  (localization.to_units (\u27e8r2.s, 1, by simp\u27e9 : powers r2.s))\n  begin\n    rw [\u2190 alg_hom.map_mul, (classical.some_spec h).1],\n    refl,\n  end\n\n/-- The map A(T1/s1) -> A(T2/s2) coming from the inequality r1 \u2264 r2 -/\nnoncomputable def localization_map {r1 r2 : rational_open_data A} (h : r1 \u2264 r2) :\n  localization r1 \u2192 localization r2 :=\nHuber_ring.away.lift r1.T r1.s (of_id A (localization r2)) (s_inv_aux r1 r2 h) rfl\n\n/-- The induced map A(T1/s1) -> A(T2/s2) coming from the inequality r1 \u2264 r2\nis a ring homomorphism. -/\ninstance {r1 r2 : rational_open_data A} (h : r1 \u2264 r2) : is_ring_hom\n(localization_map h) := by delta localization_map; apply_instance\n\n/- To prove continuity of the localisation map coming from r1 \u2264 r2 we need to check\nthat the image of T1/s1 under the localization map is power-bounded in the ring (localization r2).\nThis is done in the following lemma. -/\n\nlocal attribute [instance] set.pointwise_mul_comm_semiring\nlocal attribute [instance] set.smul_set_action\n\nset_option class.instance_max_depth 38\n\nlemma localization_map_is_cts_aux {r1 r2 : rational_open_data A} (h : r1 \u2264 r2) :\nis_power_bounded_subset\n  ((s_inv_aux r1 r2 h)\u207b\u00b9.val \u2022 (\u03bb (x : \u21a5A), to_fun (localization r2) x) '' r1.T) :=\nbegin\n  refine power_bounded.subset _ (localization.power_bounded r2),\n  intros x hx,\n  rcases hx with \u27e8_, \u27e8t\u2081, ht\u2081, rfl\u27e9, rfl\u27e9,\n  let h' := h, -- need it later\n  rcases h with \u27e8a, ha, h\u2082\u27e9,\n  rcases h\u2082 t\u2081 ht\u2081 with \u27e8t\u2082, ht\u2082, N, hN\u27e9,\n  show \u2191(s_inv_aux r1 r2 _)\u207b\u00b9 * to_fun (localization r2) t\u2081 \u2208\n    localization.mk 1 \u27e8r2.s, _\u27e9 \u2022 (of_id \u21a5A (localization r2)).to_fun '' r2.T,\n  refine \u27e8(of_id \u21a5A (localization r2)).to_fun t\u2082, \u27e8t\u2082, ht\u2082, rfl\u27e9, _\u27e9,\n  rw [\u2190units.mul_left_inj (s_inv_aux r1 r2 h'), units.mul_inv_cancel_left],\n  show to_fun (localization r2) t\u2081 = to_fun (localization r2) (r1.s) *\n    (localization.mk 1 \u27e8r2.s, _\u27e9 * to_fun (localization r2) t\u2082),\n  rw [mul_comm, mul_assoc],\n  rw \u2190units.mul_left_inj (localization.to_units (\u27e8r2.s, 1, by simp\u27e9 : powers r2.s)),\n  rw \u2190mul_assoc,\n  -- t1=s1*(1/s2 * t2) in r2\n  have : \u2191(localization.to_units (\u27e8r2.s, 1, by simp\u27e9 : powers r2.s)) *\n    localization.mk (1 : A) (\u27e8r2.s, 1, by simp\u27e9 : powers r2.s) = 1,\n  convert units.mul_inv _,\n  rw [this, one_mul], clear this,\n  show to_fun (localization r2) r2.s * _ = _,\n  rw \u2190units.mul_left_inj (localization.to_units (\u27e8r2.s ^ N, N, rfl\u27e9 : powers r2.s)),\n  show to_fun (localization r2) (r2.s ^ N) * _ = to_fun (localization r2) (r2.s ^ N) * _,\n  have hrh : is_ring_hom (to_fun (localization r2)) := begin\n    change is_ring_hom ((of_id \u21a5A (localization r2)).to_fun),\n    apply_instance,\n  end,\n  rw \u2190@is_ring_hom.map_mul _ _ _ _ (to_fun (localization r2)) hrh,\n  rw \u2190@is_ring_hom.map_mul _ _ _ _ (to_fun (localization r2)) hrh,\n  rw \u2190@is_ring_hom.map_mul _ _ _ _ (to_fun (localization r2)) hrh,\n  rw \u2190@is_ring_hom.map_mul _ _ _ _ (to_fun (localization r2)) hrh,\n  congr' 1,\n  rw [\u2190mul_assoc _ t\u2082, hN],\n  rw \u2190ha, ring,\nend\n\n-- Continuity now follows from the universal property.\nlemma localization_map_is_cts {r1 r2 : rational_open_data A} (h : r1 \u2264 r2) :\n  continuous (localization_map h) :=\nHuber_ring.away.lift_continuous r1.T r1.s (localization.nonarchimedean r2)\n  (Huber_ring.away.of_continuous r2.T r2.s _) _ _ _ (localization_map_is_cts_aux h)\n\nlemma localization_map_is_uniform_continuous {r1 r2 : rational_open_data A} (h : r1 \u2264 r2) :\n  uniform_continuous (rational_open_data.localization_map h) :=\nuniform_continuous_of_continuous (rational_open_data.localization_map_is_cts h)\n\nend rational_open_data -- namespace\n\nend spa\n", "meta": {"author": "leanprover-community", "repo": "lean-perfectoid-spaces", "sha": "95a6520ce578b30a80b4c36e36ab2d559a842690", "save_path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces", "path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces/lean-perfectoid-spaces-95a6520ce578b30a80b4c36e36ab2d559a842690/src/Spa/rational_open_data.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585669110203, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.39402830090502455}}
{"text": "import SciLean.Tactic.CustomSimp.Main\n-- import SciLean.Tactic.CustomSimp.AllPrePost\n-- import SciLean.Tactic.CustomSimp.DebugSimp\n\nimport SciLean.Tactic.AutoDiff.LetDiff\nimport SciLean.Core.Defs\nimport SciLean.Core.Attributes\n\n-- import SciLean.AutoImpl\n-- import SciLean.Core\n-- import SciLean.Functions\n\n-- import Lean.Meta\n-- import Lean.Parser\n-- import Lean.Elab\n\n-- namespace Lean.Elab.Tactic\nopen Lean Meta Simp\n\nnamespace SciLean\n\n#check Lean.MetaM\npartial def autoDiffPre (e : Expr) (rep := false) : SimpM Step := do\n  let e := e.headBeta\n  trace[Meta.Tactic.simp] s!\"Autodiff step on:\\n{\u2190 Meta.ppExpr e}\"\n\n\n  -- Always use `diff` marked theorems and set them all as `pre` instead of post  \n  let autodiffExt \u2190 Lean.Meta.getSimpExtension? \"diff\"\n  let mut autodiffThms \u2190 autodiffExt.get!.getTheorems \n  autodiffThms := { autodiffThms with pre := autodiffThms.post, post := autodiffThms.pre }\n\n  let allThms := #[autodiffThms].append (\u2190 read).simpTheorems\n\n  -- let allThms := (\u2190 read).simpTheorems\n\n  for thms in allThms do\n    if let some r \u2190 Meta.CustomSimp.rewrite? e thms.pre thms.erased DefaultMethods.discharge? (tag := \"pre\") (rflOnly := false) then\n      trace[Meta.Tactic.simp] s!\"Simplified to: {\u2190 Meta.ppExpr r.expr}\"\n      return \u2190 andThen (Step.visit r) (\u03bb e => autoDiffPre e)\n  return Step.visit {expr := e}\n\n\n-- Tactic\n\nopen Lean.Parser.Tactic in\nsyntax (name := autodiff_core) \"autodiff_core \" (config)? (discharger)? (&\"only \")? (\"[\" (simpStar <|> simpErase <|> simpLemma),* \"]\")? (location)? : tactic\n\nopen Lean.Elab.Tactic in\n@[tactic autodiff_core] def autoDiffCore : Tactic := fun stx => do\n  let { ctx, dischargeWrapper } \u2190 withMainContext <| mkSimpContext stx (eraseLocal := false)\n  let usedSimps \u2190 dischargeWrapper.with fun discharge? =>\n    SciLean.Meta.CustomSimp.simpLocation ctx discharge? (expandOptLocation stx[5]) #[letDiff, autoDiffPre] #[]\n  if tactic.simp.trace.get (\u2190 getOptions) then\n    traceSimpCall stx usedSimps\n\n-- Conv \n\nopen Lean.Parser.Tactic in\nsyntax (name := autodiff_core_conv) \"autodiff_core\" (config)? (discharger)? (&\" only\")? (\" [\" (simpStar <|> simpErase <|> simpLemma),* \"]\")? : conv\n\nopen Lean.Elab.Tactic Lean.Elab.Tactic.Conv in\n@[tactic autodiff_core_conv] def autoDiffCoreConv : Tactic := fun stx => withMainContext do\n  let { ctx, dischargeWrapper, .. } \u2190 mkSimpContext stx (eraseLocal := false)\n  let lhs \u2190 getLhs\n  let (result, _) \u2190 dischargeWrapper.with fun d? => SciLean.Meta.CustomSimp.simp lhs ctx (discharge? := d?) #[letDiff, autoDiffPre] #[]\n  applySimpResult result\n\n\nmacro \"autodiff\" : conv => \n  `(conv| (autodiff_core (config := {singlePass := true,  zeta := false, iota := false}) only [/- \u2193 autodiff (problem is that \u2193 is ignored) ,-/diff_simp]; \n           try simp (config := {zeta := false}) only [];))\nmacro \"autodiff\" : tactic => \n  `(tactic| (autodiff_core (config := {singlePass := true,  zeta := false, iota := false}) only [/- \u2193 diff (problem is that \u2193 is ignored),-/diff_simp]; \n             try simp (config := {zeta := false}) only [];))\n\n-- Tactic\n\nopen Lean.Parser.Tactic in\nsyntax (name := symdiff_core) \"symdiff_core \" (config)? (discharger)? (&\"only \")? (\"[\" (simpStar <|> simpErase <|> simpLemma),* \"]\")? (location)? : tactic\n\nopen Lean.Elab.Tactic in\n@[tactic symdiff_core] def symDiffCore : Tactic := fun stx => do\n  let { ctx, dischargeWrapper } \u2190 withMainContext <| mkSimpContext stx (eraseLocal := false)\n  let usedSimps \u2190 dischargeWrapper.with fun discharge? =>\n    SciLean.Meta.CustomSimp.simpLocation ctx discharge? (expandOptLocation stx[5]) #[autoDiffPre] #[]\n  if tactic.simp.trace.get (\u2190 getOptions) then\n    traceSimpCall stx usedSimps\n\n-- Conv \n\nopen Lean.Parser.Tactic in\nsyntax (name := symdiff_core_conv) \"symdiff_core\" (config)? (discharger)? (&\" only\")? (\" [\" (simpStar <|> simpErase <|> simpLemma),* \"]\")? : conv\n\nopen Lean.Elab.Tactic Lean.Elab.Tactic.Conv in\n@[tactic symdiff_core_conv] def symDiffCoreConv : Tactic := fun stx => withMainContext do\n  let { ctx, dischargeWrapper, .. } \u2190 mkSimpContext stx (eraseLocal := false)\n  let lhs \u2190 getLhs\n  let (result, _) \u2190 dischargeWrapper.with fun d? => SciLean.Meta.CustomSimp.simp lhs ctx (discharge? := d?) #[autoDiffPre] #[]\n  applySimpResult result\n\n\nmacro \"symdiff\" : conv => \n  `(conv| (symdiff_core (config := {singlePass := true, zeta := false, iota := false}) only [/- \u2193 diff (problem is that \u2193 is ignored),-/ diff_simp, SciLean.differentialScalar, SciLean.gradient, SciLean.tangentMap, SciLean.reverseDifferential]\n           try simp (config := {zeta := true}) only [diff_simp];))\nmacro \"symdiff\" : tactic => \n  `(tactic| (symdiff_core (config := {singlePass := true, zeta := false, iota := false}) only [/- \u2193 diff (problem is that \u2193 is ignored),-/diff_simp, SciLean.differentialScalar, SciLean.gradient, SciLean.tangentMap, SciLean.reverseDifferential]; \n             try simp (config := {zeta := true}) only [diff_simp];))\n\n\n-- set_option trace.Meta.Tactic.simp.rewrite true in\n-- -- set_option trace.Meta.Tactic.simp.discharge true in\n-- -- set_option trace.Meta.Tactic.simp.unify false in\n-- #check (\u2202 \u03bb (x : \u211d) => let y := x*x; y)\n--   rewrite_by\n--     -- simp (config := {singlePass := true}) only [\u2193 diff, tangentMap]\n--     symdiff\n\n--     trace_state\n\n\n-- set_option trace.Meta.Tactic.simp true in\n-- set_option trace.Meta.Tactic.simp.rewrite true in\n-- set_option trace.Meta.Tactic.simp.discharge true in\n-- set_option trace.Meta.Tactic.simp.unify false in\n-- #check (\u2202 \u03bb (x : \u211d) => let y := x*x; let z := x + y*x*x; x + y + z)\n--   rewrite_by\n--     diff_core (config:={zeta:=false}) only []\n--     diff\n--     trace_state\n\n-- set_option trace.Meta.Tactic.simp true in\n-- set_option trace.Meta.Tactic.simp.rewrite false in\n-- set_option trace.Meta.Tactic.simp.unify false in\n-- #check (\u2202 \u03bb (x : \u211d) => let y := x; let z := x + y; y + z)\n--   rewrite_by \n--     diff\n--     trace_state\n\n-- This fails to apply `SciLean.diff_of_comp` because it `foo` can't be proven to be smooth\n-- set_option trace.Meta.Tactic.simp.rewrite true in\n-- set_option trace.Meta.Tactic.simp.discharge true in\n-- #check (\u2202 \u03bb (x : \u211d) => let z := x^2; let foo := \u03bb y => Math.sin (Math.exp y); foo (Math.cos z)) \n--   rewrite_by \n--     diff\n--     trace_state\n\n\n-- @[irreducible] def foo (a b : Nat) := a + b\n-- @[simp] theorem foo_simp (a b : Nat) : foo a b = a + b := by unfold foo; rfl \n\n-- example {X Y Z W} [Vec X] [Vec Y] [Vec Z] [Vec W] (g : W \u2192 X) (h : W \u2192 Y) (f : W \u2192 X \u2192 Y \u2192 Z) [IsSmooth h] [IsSmooth g] [IsSmooth f] [\u2200 x, IsSmooth (f x)]  [\u2200 x y, IsSmooth (f x y)]\n--   : (\u03bb n : Nat => \n--     (\u2202 \u03bb x w : W => \n--       let y := g x\n--       let z := h x\n--       f x y z))\n--     =\n--     \u03bb n : Nat => \n--     hold\n--     \u03bb x dx w =>\n--       let y  := g x\n--       let dy := \u2202 g x dx\n--       let z  := h x\n--       let dz := \u2202 h x dx\n--       \u2202 f x dx y z + \u2202 (f x) y dy z + \u2202 (f x y) z dz\n-- := by\n--   diff_core (config := {zeta := false, singlePass := true})\n\n--   -- diff_core (config := {zeta := false, singlePass := true})\n--   -- diff_core (config := {zeta := false, singlePass := true})\n--   -- diff_core (config := {zeta := false, singlePass := true})\n--   simp[hold]\n--   admit\n\n-- @[simp]\n-- theorem diff_at_zero {X Y} [Vec X] [Vec Y] (f : X \u2192 Y) [IsSmooth f] (x : X) : \u2202 f x 0 = 0 := sorry\n\n-- example {X Y} [Vec X] [Vec Y] (a : \u03b1) (g : \u03b1 \u2192 X) (f : X \u2192 X \u2192 Y) [IsSmooth (\u03bb x => f x (g a))]\n--   : (\u03bb n : Nat => (\u2202 \u03bb x => \n--       let y := g a\n--       f x y))\n--     = \n--     \u03bb n : Nat =>\n--     hold\n--     \u03bb x dx =>\n--       let y := g a\n--       \u2202 (\u03bb x => f x y) x dx\n-- := by\n--   diff_core (config := {zeta := false})\n--   simp[hold]\n--   done\n\n-- example {X Y} [Vec X] [Vec Y] (a : \u03b1) (g : \u03b1 \u2192 X) (f : X \u2192 X \u2192 Y) [IsSmooth (f (g a))]\n--   : (\u2202 \u03bb x => \n--       let y := g a\n--       f y x)\n--     =\n--     hold\n--     \u03bb x dx =>\n--       let y  := g a\n--       \u2202 (f y) x dx\n-- := by\n--   diff_core (config := {zeta := false})\n--   simp[hold]\n--   done\n\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Tactic/AutoDiff/Main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334527, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.39392531169296713}}
{"text": "\nnamespace MWE3\n\nstructure State where\n  declarations: List String\n  aspects: List String\n  concepts: List String\nderiving Repr\n\n-- the set of declarations must equal to the disjoint union of aspects and concepts.\ndef State.wff: State \u2192 Prop :=\n  fun s => \u2200 a c, \n  (s.aspects.contains a \u2192 s.declarations.contains a && ! s.concepts.contains a) && \n  (s.concepts.contains c \u2192 s.declarations.contains c && ! s.aspects.contains c)\n\ndef State' := { s: State // s.wff }\n#print State'\ndef s1 := State.mk [\"a\",\"b\",\"c\"] [\"a\"] [\"b\",\"c\"] \n#eval s1\n#check s1.wff\n#print s1\n\ndef s1' : State' := \u27e8 s1, s1.wff \u27e9 \n#print s1'\n\ndef s2 := State.mk [\"a\",\"b\",\"c\"] [\"a\",\"b\"] [\"b\",\"c\"] \n#eval s2\n\nexample : s2.wff = True := by \n  sorry\n\nend MWE3\n", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/MWE3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3939253116929671}}
{"text": "\nimport data.stream\n\nimport util.meta.tactic\nimport util.logic\nimport util.classical\nimport util.predicate\nimport util.meta.tactic.propositional\n\nimport tactic\n\nimport temporal_logic.tactic\n\nuniverse variables u u\u2080 u\u2081 u\u2082\n\nvariables {\u03b1 : Sort u\u2080} {\u03b2 : Type u\u2081} {\u03b3 : Sort u\u2082}\n\nnamespace temporal\nopen predicate stream\n\nattribute [predicate] stream.drop pred'.mk\nattribute [tl_simp, simp] pred'.mk\n\nlemma henceforth_next (p : cpred)\n: \u25fbp \u27f9 \u25fb\u2299p :=\nbegin [temporal]\n  rw henceforth_next_intro p,\n  mono, simp,\nend\n\nlemma next_henceforth (p : cpred)\n: \u25fbp \u27f9 \u2299\u25fbp :=\nbegin [temporal]\n  suffices : \u25fb\u25fbp \u27f6 \u2299\u25fbp,\n  { simp at this, apply this },\n  intro h, apply h,\nend\n\nlemma next_eventually_comm (p : cpred)\n: \u2299\u25c7p = \u25c7\u2299p :=\nby lifted_pred [next,eventually,nat.succ_add]\n\nlemma holds_next (\u0393 p : cpred) [persistent \u0393]\n  (h : \u0393 \u22a2 p)\n: \u0393 \u22a2 \u2299p :=\nbegin [temporal]\n  apply persistent_to_henceforth h,\nend\n\n/- distributivity -/\n\nlemma eventually_and_entails {p q : cpred}\n: \u25c7(p \u22c0 q) \u27f9 \u25c7p \u22c0 \u25c7q :=\nbegin\n  apply entails_p_and_of_entails ; mono ; propositional,\nend\n\nlemma entails_henceforth_or {p q : cpred}\n: \u25fbp \u22c1 \u25fbq \u27f9 \u25fb(p \u22c1 q) :=\nbegin [temporal]\n  intros h, cases h with h h\n  ; henceforth at \u22a2 h\n  ; [ left , right ]\n  ; exact h\nend\n\n/- end distributivity -/\n\nlemma eventually_of_leads_to {p q : cpred} {\u0393}\n  (h : \u0393 \u22a2 p ~> q)\n: \u0393 \u22a2 \u25c7p \u27f6 \u25c7q :=\nbegin [temporal]\n  rw \u2190 eventually_eventually q,\n  apply eventually_imp_eventually h,\nend\n\nlemma inf_often_of_leads_to {p q : cpred} {\u0393}\n  (h : \u0393 \u22a2 p ~> q)\n: \u0393 \u22a2 \u25fb\u25c7p \u27f6 \u25fb\u25c7q :=\nbegin [temporal]\n  rw \u2190 eventually_eventually q,\n    -- \u03b2 : Type u\u2081\n    -- p q : cpred\n    -- h : p ~> q\n    -- \u22a2 \u25fb\u25c7p \u27f6 \u25fb\u25c7\u25c7q\n  mono*,\n    -- \u03b2 : Type u\u2081\n    -- p q : cpred\n    -- h : p ~> q\n    -- \u22a2 p \u27f6 \u25c7q\n  apply h,\nend\n\nlemma leads_to_trans {p q r : cpred} {\u0393}\n  (Hpq : \u0393 \u22a2 p ~> q)\n  (Hqr : \u0393 \u22a2 q ~> r)\n: \u0393 \u22a2 p ~> r :=\nbegin [temporal]\n  henceforth,\n  intros hp,\n  have := Hpq hp, revert this,\n  rw \u2190 eventually_eventually r,\n  clear hp,\n  mono,\n  apply Hqr,\nend\n\n@[tl_simp, simp]\nlemma next_or (p q : cpred)\n: \u2299(p \u22c1 q) = \u2299p \u22c1 \u2299q :=\nrfl\n\n@[tl_simp, simp]\nlemma next_imp (p q : cpred)\n: \u2299(p \u27f6 q) = \u2299p \u27f6 \u2299q :=\nrfl\n\n@[tl_simp, simp]\nlemma next_proj (f : var \u03b1 \u03b2) (v : tvar \u03b1)\n: \u2299(f ! v) = f ! \u2299v :=\nby lifted_pred [next]\n\n@[tl_simp, simp]\nlemma next_v_eq (p q : tvar \u03b1)\n: \u2299(p \u2243 q) = \u2299p \u2243 \u2299q :=\nby lifted_pred\n\nopen nat\n\n@[tl_simp, simp]\nlemma const_action (c : Prop) (v : tvar \u03b1)\n: \u27e6 v | \u03bb _ _ : \u03b1, c \u27e7 = (c : cpred) :=\nby { refl }\n\n-- @[predicate]\nlemma action_on  (A : act \u03b1) (v : tvar \u03b3) (f : \u03b3 \u2192 \u03b1)\n: \u27e6 v | A on f \u27e7 = \u27e6 \u27e8f\u27e9 ! v | A \u27e7 :=\nby { lifted_pred }\n\nlemma action_on'  (A : act \u03b1) (v : tvar \u03b3) (f : \u03b3 \u2192 \u03b1)\n: \u27e6 v | \u03bb s s', (A on f) s s' \u27e7 = \u27e6 \u27e8f\u27e9 ! v | A \u27e7 :=\nby { lifted_pred }\n\n@[predicate]\nlemma exists_action  (A : \u03b3 \u2192 act \u03b1) (v : tvar \u03b1)\n: (\u2203\u2203 i, \u27e6 v | A i \u27e7) = \u27e6 v | \u03bb s s', (\u2203 i, A i s s') \u27e7 :=\nby { lifted_pred }\n\ninstance or_persistent {p q : cpred}\n  [persistent p]\n  [persistent q]\n: persistent (p \u22c1 q) :=\nbegin\n  constructor,\n  apply mutual_entails,\n  apply henceforth_str,\n  begin [temporal]\n    intro h,\n    cases h with h h,\n    { rw \u2190 is_persistent p at h,\n      revert h,\n      mono,\n      propositional, },\n    { henceforth, right, exact h }\n  end\nend\n\ninstance imp_persistent {p q : cpred}\n  [postponable p]\n  [persistent q]\n: persistent (p \u27f6 q) :=\nby { simp [p_imp_iff_p_not_p_or], apply_instance }\n\ninstance stable_persistent {p : cpred}\n: persistent (\u25c7 \u25fb p) :=\nbegin\n  constructor,\n  apply mutual_entails,\n  apply henceforth_str,\n  begin [temporal]\n    apply induct,\n    henceforth,\n    rw next_eventually_comm,\n    mono,\n    apply next_henceforth\n  end\nend\n\ninstance and_postponable {p q : cpred}\n  [postponable p]\n  [postponable q]\n: postponable (p \u22c0 q) :=\nby { constructor, rw \u2190 p_not_eq_p_not_iff_eq,\n     simp only [p_not_p_and,is_persistent] with tl_simp, }\n\ninstance inf_often_postponable {p : cpred}\n: postponable (\u25fb \u25c7 p) :=\nbegin\n  constructor,\n  rw \u2190 p_not_eq_p_not_iff_eq,\n  simp only [is_persistent] with tl_simp,\nend\n\nlemma induct_evt (p q : cpred) {\u0393}\n  (h : \u0393 \u22a2 \u25fb (p \u27f6 -q \u27f6 \u2299(p \u22c1 q)))\n: \u0393 \u22a2 (p \u27f6 \u25c7q \u22c1 \u25fbp) :=\nbegin [temporal]\n  apply induct_evt' _ _ h,\nend\n\ntheorem em (p : cpred) : \u22a9 \u25c7\u25fbp \u22c1 \u25fb\u25c7(- p) :=\nbegin [temporal]\n  rw [\u2190 not_henceforth,\u2190 not_eventually,p_or_p_not_self]\nend\n\nlemma inf_often_of_stable (p : cpred) : (\u25c7\u25fbp) \u27f9 (\u25fb\u25c7p) :=\nbegin [temporal]\n  explicit \u03c4\n  { simp_intros h i [henceforth],\n    cases h with j h,\n    unfold eventually,\n    existsi j,\n    specialize h i,\n    simp [drop_drop] at \u22a2 h,\n    apply h },\nend\n\nlemma weak_coincidence {p q : cpred} {\u0393}\n    (Hp : \u0393 \u22a2 \u25fbp)\n    (Hq : \u0393 \u22a2 \u25c7q)\n: \u0393 \u22a2 \u25c7(p \u22c0 q) :=\nbegin [temporal]\n  explicit \u03c4\n  { cases Hq with j Hq,\n    specialize Hp j,\n    simp [eventually],\n    existsi (j),\n    exact \u27e8Hp,Hq\u27e9, }\nend\n\nlemma eventually_and_eventually (p q : cpred)\n: \u25c7p \u22c0 \u25c7q = \u25c7(p \u22c0 \u25c7q) \u22c1 \u25c7(\u25c7p \u22c0 q) :=\nbegin\n  apply mutual_entails,\n  begin [temporal]\n    rw [\u2190 p_not_p_imp,not_eventually,p_not_p_and,not_eventually],\n    intros H\u2080 H\u2081,\n    cases H\u2080 with ha hb,\n    have h := weak_coincidence H\u2081 ha,\n    rw [p_and_comm,p_or_comm,p_and_p_or_p_not_self] at h,\n    explicit \u03c4\n    { cases h with j h, cases hb with i ha,\n      simp [eventually], existsi i,\n      split ; [skip,exact ha],\n      cases le_total i j with h' h',\n      { existsi (j-i),\n        simp [drop_drop,add_sub_of_le h'],\n        apply h.left, },\n      { exfalso, apply h.right (i-j),\n        simp [drop_drop,add_sub_of_le h'],\n        apply ha, } },\n  end,\n  { apply p_or_entails_of_entails\n    ; apply entails_p_and_of_entails,\n    all_goals {\n      transitivity,\n      apply eventually_and_entails,\n      rw eventually_eventually,\n      propositional, }, },\nend\n\nlemma event_ordering {\u0393 p q : cpred}\n  (hp : \u0393 \u22a2 \u25c7p)\n  (hq : \u0393 \u22a2 \u25c7q)\n: \u0393 \u22a2 \u25c7(p \u22c0 \u25c7q) \u22c1 \u25c7(\u25c7p \u22c0 q) :=\nbegin [temporal]\n  rw [\u2190 eventually_and_eventually],\n  split; assumption,\nend\n\nsection\nopen tactic tactic.interactive (unfold_coes unfold itactic assert_or_rule)\nopen interactive interactive.types lean lean.parser\nopen applicative (mmap\u2082)\nlocal postfix `?`:9001 := optional\n\nprivate meta def event : lean.parser (name \u2295 pexpr) :=\n(sum.inl <$> ident) <|> (sum.inr <$> brackets \"(\" \")\" texpr)\n\nprivate meta def event_to_event : name \u2295 pexpr \u2192 tactic expr\n | (sum.inl n) := resolve_name n >>= to_expr\n | (sum.inr e) := to_expr e\n\nmeta def interactive.event_ordering (aggr : parse $ optional $ tk \"!\") (e\u2080 e\u2081 : parse event)\n  (ids : parse with_ident_list) : temporal unit :=\ndo e\u2080 \u2190 event_to_event e\u2080, e\u2081 \u2190 event_to_event e\u2081,\n   h \u2190 to_expr ``(event_ordering %%e\u2080 %%e\u2081) >>= note `h none,\n   when e\u2080.is_local_constant $ tactic.clear e\u2080,\n   when e\u2081.is_local_constant $ tactic.clear e\u2081,\n   if aggr.is_some then do\n     n\u2080 \u2190 mk_fresh_name,\n     n\u2081 \u2190 mk_fresh_name,\n     temporal.interactive.cases (none,to_pexpr h) [n\u2080,n\u2081],\n     temporal.interactive.eventually n\u2081 none <|> fail \"here\",\n     e\u2080 \u2190 get_local n\u2081, temporal.interactive.cases (none,to_pexpr e\u2080) ids,\n     cleanup,\n     tactic.swap,\n     temporal.interactive.eventually n\u2080 none <|> fail \"there\",\n     e\u2080 \u2190 get_local n\u2080, temporal.interactive.cases (none,to_pexpr e\u2080) ids,\n     tactic.swap\n   else temporal.interactive.cases (none,to_pexpr h) ids,\n   return ()\n\nend\n\nlemma stable_and_of_stable_of_stable {p q : cpred} {\u0393}\n    (Hp : \u0393 \u22a2 \u25c7\u25fbp)\n    (Hq : \u0393 \u22a2 \u25c7\u25fbq)\n: \u0393 \u22a2 \u25c7\u25fb(p \u22c0 q) :=\nbegin [temporal]\n  event_ordering Hp Hq with h h\n  ; eventually h\n  ; cases h with h\u2080 h\u2081\n  ; [eventually h\u2081 \u22a2,eventually h\u2080 \u22a2]\n  ; henceforth at *\n  ; tauto\nend\n\nlemma henceforth_delay {p q : cpred} {\u0393}\n    (Hp : \u0393 \u22a2 \u25c7p)\n    (Hq : \u0393 \u22a2 \u25fbq)\n: \u0393 \u22a2 \u25c7(p \u22c0 \u25fbq) :=\nbegin [temporal]\n  eventually Hp \u22a2,\n  split ; assumption\nend\n\n@[tl_simp, simp]\nlemma eventually_inf_often (p : cpred)\n: \u25c7\u25fb\u25c7p = \u25fb\u25c7p :=\nmutual_entails\nbegin [temporal]\n  intros hp,\n  have := inf_often_of_stable (\u25c7p) \u0393 hp, clear hp,\n  rw eventually_eventually at this,\nend\n(eventually_weaken _)\n\nlemma coincidence {p q : cpred} {\u0393}\n    (Hp : \u0393 \u22a2 \u25c7\u25fbp)\n    (Hq : \u0393 \u22a2 \u25fb\u25c7q)\n: \u0393 \u22a2 \u25fb\u25c7(p \u22c0 q) :=\nbegin [temporal]\n  rw \u2190 eventually_inf_often,\n  eventually Hp |- ,\n  henceforth at Hq |-,\n  eventually Hq |-,\n  split ; assumption,\nend\n\nlemma coincidence' {p q : cpred} {\u0393}\n    (Hp : \u0393 \u22a2 \u25fbp)\n    (Hq : \u0393 \u22a2 \u25fb\u25c7q)\n: \u0393 \u22a2 \u25fb\u25c7(p \u22c0 q) :=\nbegin [temporal]\n  apply coincidence _ Hq,\n  assumption\nend\n\nlemma inf_often_p_or (p q : cpred)\n: \u25fb\u25c7(p \u22c1 q) = \u25fb\u25c7p \u22c1 \u25fb\u25c7q :=\nbegin\n  refine mutual_entails _ _,\n  begin [temporal]\n    rw p_or_iff_not_imp (\u25fb\u25c7 p),\n    intros h\u2080 h\u2081,\n    rw [not_henceforth,not_eventually] at h\u2081,\n    have := coincidence h\u2081 h\u2080, clear h\u2080 h\u2081,\n    rw p_not_and_self_or at this,\n    revert this, mono*,\n    apply p_and_elim_right,\n  end,\n  refine p_or_entails_of_entails _ _\n  ; mono* ; propositional,\nend\n\n@[monotonic]\nlemma next_imp_next {p q : cpred} (h : p \u27f9 q)\n: \u2299 p \u27f9 \u2299 q :=\nby { pointwise h with \u03c4, solve_by_elim }\n\n@[monotonic]\nlemma next_tl_imp_next {\u0393 p q : cpred}\n  [persistent \u0393]\n  (h : ctx_impl \u0393 p q)\n: ctx_impl \u0393 (\u2299 p) (\u2299 q) :=\nby { rw \u2190 is_persistent \u0393 at *,\n     lifted_pred keep [tl_imp],\n     replace h := h.apply (succ \u03c3),\n     apply h, clear h,\n     intro i, rw [succ_add, \u2190 add_succ],\n     apply a (succ i), }\n\nlemma eventually_and {\u0393 p q : cpred}\n   (h\u2080 : \u0393 \u22a2 \u25fbp)\n   (h\u2081 : \u0393 \u22a2 \u25c7q)\n: \u0393 \u22a2 \u25c7(p \u22c0 q) :=\nbegin [temporal]\n  eventually h\u2081 \u22a2,\n  split ; assumption\nend\n\n/- Actions -/\n\nlocal infix ` <$> ` := fun_app_to_var\nlocal infix ` <*> ` := combine_var\n\nopen function\n\nlemma exists_of_eventually\n  {p : \u03b2 \u2192 Prop}\n  {v : tvar \u03b2}\n  (h : \u22a9 \u25c7(p <$> v))\n: \u2203 x, p x :=\nbegin\n  cases v with v,\n  replace h := ew_str h 0,\n  cases h with i h,\n  existsi v i, simp [comp] at h,\n  apply h,\nend\n\nopen function\n\nprotected theorem leads_to_of_inf_often (\u0393 p q : cpred)\n  (H : \u0393 \u22a2 \u25fb\u25c7q)\n: \u0393 \u22a2 p ~> q :=\nbegin [temporal]\n  henceforth at H \u22a2,\n  intro, assumption,\nend\n\nprotected theorem leads_to_strengthen_rhs (q : cpred) {\u0393 p r : cpred}\n  (H : q \u27f9 r)\n  (P\u2080 : \u0393 \u22a2 p ~> q)\n: \u0393 \u22a2 p ~> r :=\nbegin [temporal]\n  apply leads_to_trans P\u2080,\n  henceforth,\n  intros H',\n  apply H \u0393 H',\nend\n\nprotected lemma leads_to_cancellation {\u0393 p q b r : cpred}\n    (P\u2080 : \u0393 \u22a2 p ~> q \u22c1 b)\n    (P\u2081 : \u0393 \u22a2 q ~> r)\n    : \u0393 \u22a2 p ~> r \u22c1 b :=\nbegin [temporal]\n  henceforth,\n  intros h,\n  have := P\u2080 h, clear h,\n  eventually this,\n  rw [eventually_or],\n  cases this with h h,\n  { left, apply P\u2081 h },\n  { right, assumption },\nend\n\nprotected lemma leads_to_disj_rng {t : Sort u}\n  {p : t \u2192 cpred} {\u0393 q} {r : t \u2192 Prop}\n  (h : \u0393 \u22a2 \u2200\u2200 i, \u2191(r i) \u27f6 (p i ~> q))\n: \u0393 \u22a2 (\u2203\u2203 i, \u2191(r i) \u22c0 p i) ~> q :=\nbegin [temporal]\n  rw [p_exists_range_subtype,tl_leads_to,p_exists_imp_eq_p_forall_imp],\n  rw [henceforth_forall],\n  intro i, cases i with i hi,\n  apply h i hi,\nend\n\nprotected theorem leads_to_disj {t}\n  {p : t \u2192 cpred}\n  {q \u0393 : cpred}\n  (P\u2080 : \u0393 \u22a2 \u2200\u2200 i, p i ~> q)\n: \u0393 \u22a2 (\u2203\u2203 i, p i) ~> q :=\nbegin [temporal]\n  have P\u2081 : \u2200\u2200 i : t, \u2191true \u27f6 (\u25fb(p i \u27f6 \u25c7q)),\n  { intros i, intro, apply P\u2080 i, },\n  have P\u2082 := @temporal.leads_to_disj_rng _ _ _ _ (\u03bb _, true) P\u2081,\n  rw_using : (\u2203\u2203 (i : t), \u2191((\u03bb _, true) i) \u22c0 p i) = (\u2203\u2203 i, p i) at P\u2082,\n  { apply p_exists_congr,\n    intro,\n    apply True_p_and },\nend\n\nprotected theorem leads_to_disj_gen {t}\n  {p q : t \u2192 cpred}\n  {\u0393 : cpred}\n  (P\u2080 : \u0393 \u22a2 \u2200\u2200 i, p i ~> q i)\n: \u0393 \u22a2 (\u2203\u2203 i, p i) ~> (\u2203\u2203 i, q i) :=\nbegin [temporal]\n  apply temporal.leads_to_disj _,\n  intro j,\n  apply temporal.leads_to_strengthen_rhs _ _ (P\u2080 j),\n  apply p_exists_intro\nend\n\nsection induction\n\nvariables {\u03b1' : Type u}\nvariables  {\u0393 : cpred}\nvariables  (f : tvar \u03b2) (p q : cpred)\nvariables [has_well_founded \u03b2]\n\nprotected lemma induction\n  (P : \u0393 \u22a2 \u2200\u2200 v : \u03b2, p \u22c0 (f \u2243 v)  ~>  p \u22c0 (f \u227a\u227a v) \u22c1 q)\n: \u0393 \u22a2 p ~> q :=\nbegin [temporal]\n  have h\u2082 : \u2200\u2200 V : \u03b2, p \u22c0 (f \u2243 V) ~> q,\n  { intro V,\n    wf_induction V,\n    apply temporal.leads_to_strengthen_rhs (q \u22c1 q),\n    { simp [or_self], },\n    apply temporal.leads_to_cancellation (P _),\n    rw_using : (p \u22c0 (f \u227a\u227a x)) = (\u2203\u2203v, \u2191(v << x) \u22c0 (p \u22c0 (f \u2243 v))),\n    { ext1 \u03c4, simp only with predicate, rw exists_one_point (f.apply \u03c4), simp [and_comm],\n      intro k, simp, intros, subst k },\n    apply @temporal.leads_to_disj_rng _ ,\n    apply ih_1, },\n  have h\u2083 := temporal.leads_to_disj h\u2082,\n  rw_using : (\u2203\u2203 (i : \u03b2), p \u22c0 (f \u2243 i)) = p at h\u2083,\n  { ext1 j, simp [function.comp,exists_one_point_right ], },\nend\n\nend induction\n\nsection inf_often_induction'\n\nparameters {\u03b1' : Type u}  {\u03b2' : Type u\u2080}\nparameters {\u0393 : cpred} (V : tvar \u03b2') (p q : cpred)\nparameters [has_well_founded \u03b2']\n\nlemma inf_often_induction'\n  (S\u2080 : \u0393 \u22a2 \u2200\u2200 v : \u03b2', \u25fb( V \u2243 v \u27f6 \u25fb(V \u2243 v) \u22c1 \u25c7(V \u227a\u227a v \u22c1 q)))\n  (P\u2081 : \u0393 \u22a2 \u2200\u2200 v : \u03b2', (p \u22c0 V \u2243 v) ~> (V \u227a\u227a v \u22c1 q))\n: \u0393 \u22a2 \u25fb\u25c7p \u27f6 \u25fb\u25c7q :=\nbegin [temporal]\n  intros Hp,\n  unfold henceforth,\n  have Hex : \u2200\u2200 (v : \u03b2'), V \u2243 v ~> q,\n  { intro v,\n    wf_induction v with v,\n    have IH' := temporal.leads_to_disj_rng ih_1, clear ih_1,\n    rw_using : (\u2203\u2203 (i : \u03b2'), \u2191(i << v) \u22c0 V \u2243 i)\n             = V \u227a\u227a v at IH',\n    { ext \u03c4,\n      simp [flip,function.comp,p_exists], },\n    have S\u2081 : \u2200\u2200 v : \u03b2', V \u2243 v ~> V \u227a\u227a v \u22c1 q,\n    { intro, henceforth!, intros Hv,\n      replace S\u2080 := S\u2080 _ Hv,\n      cases S\u2080 with S\u2080 S\u2080,\n      { have H := coincidence' S\u2080 Hp,\n        rw p_and_comm at H,\n        henceforth at H, eventually H,\n        apply P\u2081 _ H },\n      { apply S\u2080, } },\n    have H\u2083 := temporal.leads_to_cancellation (S\u2081 v) IH',\n    exact cast (by simp) H\u2083 },\n  replace Hex := temporal.leads_to_disj Hex,\n  rw_using : (\u2203\u2203 (v : \u03b2'), (V \u2243 v)) = True at Hex,\n  { lifted_pred, existsi \u03c3 \u22a8 V, refl },\n  henceforth, apply Hex, simp,\nend\n\nend inf_often_induction'\n\nsection prophecy\n\nvariable {\u0393 : cpred}\n-- variable [temporal.persistent \u0393]\n-- variables I N : cpred\nvariables PI J : tvar (\u03b1 \u2192 Prop)\nvariables PN : tvar (act \u03b1)\nvariables PSync : cpred\nvariables h_PSync : \u0393 \u22a2 \u25fb\u25c7PSync\nvariables Init : cpred\n-- variables h_Init : \u0393 \u22a2 Init\nvariable h_PI : \u0393 \u22a2 \u2200\u2200 p : \u03b1, J p \u27f6 PI p\nvariable h_PN : \u0393 \u22a2 \u25fb(\u2200\u2200 p' : \u03b1, J p' \u27f6 \u2203\u2203 p : \u03b1, PN p p' \u22c0 J p)\n-- variable h_PSync' : \u0393 \u22a2 PSync \u27f6 \u2203\u2203 p : \u03b1, J p \u22c0 \u2200\u2200 p', J p' \u27f6 PN p p'\nvariable h_PSync' : \u0393 \u22a2 \u25fb(PSync \u27f6 \u2203\u2203 p : \u03b1, PI p \u22c0 J p)\n\n-- variables (i j : \u2115)\n\n-- def w : \u2115 \u2192 \u03b1\n\ninclude h_PI h_PN h_PSync h_PSync'\nopen nat\n-- set_option profiler true\n-- #check predicate.p_exists_imp_p_exists'\nlemma prophecyI\n: \u0393 \u22a2 \u2203\u2203 w : tvar \u03b1, PI w \u22c0 \u25fbPN w (\u2299w) \u22c0 \u25fbJ w :=\nbegin [temporal]\n  have : \u2203\u2203 x : \u03b1, (True : cpred),\n  { henceforth at h_PSync,\n    eventually h_PSync,\n    have : \u2203\u2203 x : \u03b1, PI x \u22c0 J x := h_PSync' h_PSync,\n    apply predicate.p_exists_entails_p_exists _ _ _ _ this,\n    intro, simp },\n  nonempty \u03b1,\n  let x\u2080 : tvar \u03b1 := \u27e8 \u03bb i, \u03b5 x, i \u22a8 PI x \u2227 i \u22a8 J x \u27e9,\n  let f : tvar (\u03b1 \u2192 \u03b1) := \u27e8 \u03bb i x', \u03b5 x, i \u22a8 PN x x' \u2227 succ i \u22a8 J x' \u27e9 ,\n  have := back_witness x\u2080 f h_PSync,\n  revert this,\n  apply p_exists_p_imp_p_exists,\n  intros w h,\n  suffices : \u25fbJ w,\n  { split, split,\n    henceforth at this,\n    explicit' with this h_PI\n    { solve_by_elim },\n    admit, exact this },\n  { suffices : \u25fb(J w \u22c1 PSync \u22c0 w \u2243 x\u2080),\n    { revert this, mono!, intro h',\n      cases h' with h\u2080 h\u2081, exact h\u2081,\n      henceforth at h_PSync',\n      explicit' with h\u2080 h_PSync'\n      { cases h\u2080,\n        suffices : PI w \u2227 J w, exact this.right,\n        subst w, apply_epsilon_spec, } },\n    apply henceforth_until,\n    have : \u25fb\u25c7((PSync \u22c0 w \u2243 x\u2080)), admit,\n    revert this, mono!,\n    apply until_backward_induction _ _,\n    -- have : _ \u27f6 (-PSync \u22c0 J w)  \ud835\udcb0  (PI w \u22c0 J w) := until_backward_induction _ _,\n    -- suffices : \u25fb(J w  \ud835\udcb0  (PSync \u22c0 w \u2243 x\u2080)),\n}\nend\n\n#check @until_backward_induction\n\nend prophecy\n\n\nattribute [irreducible] next\nsection\nvariables \u0393 : cpred\nvariables p q : tvar \u03b1\nvariables p' q' : tvar \u03b2\nvariable f : \u03b1 \u2192 \u03b2\nvariables f\u2080 f\u2081 : tvar (\u03b1 \u2192 \u03b2)\n\n@[lifted_congr]\nlemma lifted_coe_to_fun_arg\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 f\u2080 p \u2243 f\u2080 q :=\nby { lifted_pred using h, simp [h] }\n\n@[lifted_congr]\nlemma lifted_coe_to_fun_fun\n  (h : \u0393 \u22a2 f\u2080 \u2243 f\u2081)\n: \u0393 \u22a2 f\u2080 p \u2243 f\u2081 p :=\nby { lifted_pred using h, simp [h] }\n\n@[lifted_congr]\nlemma lifted_congr\u2081\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 lifted\u2081 f p \u2243 lifted\u2081 f q :=\nby { lifted_pred using h, simp [h] }\n\n@[lifted_congr]\nlemma lifted_congr\u2082 (g : \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n  (h : \u0393 \u22a2 p \u2243 q)\n  (h' : \u0393 \u22a2 p' \u2243 q')\n: \u0393 \u22a2 lifted\u2082 g p p' \u2243 lifted\u2082 g q q' :=\nby { lifted_pred using h h', simp [h,h'] }\n\n@[lifted_congr]\nlemma lifted_proj (v : var \u03b1 \u03b2)\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 v ! p \u2243 v ! q :=\nby { lifted_pred using h, simp [h] }\n\nvariable [persistent \u0393]\n\n@[timeless_congr]\nlemma lifted_next (p q : tvar \u03b1)\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 \u2299p \u2243 \u2299q :=\nbegin\n  lifted_pred keep,\n  rw \u2190 is_persistent \u0393 at a,\n  have := h.apply (succ x) (a 1),\n  simp at this, exact this,\nend\n\n@[timeless_congr]\nlemma lifted_henceforth (p q : cpred)\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 \u25fbp \u2243 \u25fbq :=\nbegin\n  apply mutual_p_imp\n  ; change ctx_impl _ _ _\n  ; mono\n  ; apply p_imp_of_equiv,\n  apply h, apply v_eq_symm h\nend\n\n@[timeless_congr]\nlemma lifted_eventually (p q : cpred)\n  (h : \u0393 \u22a2 p \u2243 q)\n: \u0393 \u22a2 \u25c7p \u2243 \u25c7q :=\nbegin\n  apply mutual_p_imp\n  ; change ctx_impl _ _ _\n  ; mono\n  ; apply p_imp_of_equiv,\n  apply h, apply v_eq_symm h\nend\n\nend\n\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3939253116929671}}
{"text": "/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n-/\nimport mathlib4_experiments.Tactic.Split\nimport mathlib4_experiments.Tactic.Refl\n\nsection miscellany\n\n/-- Ex falso, the nondependent eliminator for the `empty` type. -/\ndef Empty.elim {C : Sort _} (a : Empty) : C := nomatch a\n\ninstance : Subsingleton Empty := \u27e8\u03bb a => a.elim\u27e9\n\n/-\n\n## Basic API for prod\n\n-/\n\ntheorem Prod.ext_iff : \u2200 (p q : \u03b1 \u00d7 \u03b2),\n  p = q \u2194 p.1 = q.1 \u2227 p.2 = q.2\n| (p1, p2), (q1, q2) => by\n  split;\n    intro h;\n    rw h;\n    split;\n      rfl;\n    rfl;\n  skip;\n  intro \u27e8(h1 : p1 = q1), (h2 : p2 = q2)\u27e9;\n  rw [h1, h2];\n\n@[simp] theorem eq_iff_true_of_subsingleton [Subsingleton \u03b1] (x y : \u03b1) :\n  x = y \u2194 True :=\nby \n  split;\n    simp;\n  intro _;\n  apply Subsingleton.elim;\n\ninstance Subsingleton.prod {\u03b1 \u03b2 : Type _}\n  [Subsingleton \u03b1] [Subsingleton \u03b2] : \n  Subsingleton (\u03b1 \u00d7 \u03b2) :=\n\u27e8by \n  intro a b; \n  cases a; \n  cases b; \n  -- todo(yakov) : ext tactic\n  rw Prod.ext_iff;\n  split;\n    allGoals { simp }; \u27e9\n\n/-\n\nKenny term mode version\n\ninstance {\u03b1 : Type u} {\u03b2 : Type v} [Subsingleton \u03b1] [Subsingleton \u03b2] : Subsingleton (\u03b1 \u00d7 \u03b2) :=\n\u27e8\u03bb \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 => congr (congrArg _ $ Subsingleton.elim _ _) (Subsingleton.elim _ _)\u27e9\n\n-/\ninstance : DecidableEq Empty := \u03bba => a.elim\n\ninstance sort.inhabited : Inhabited (Sort _) := \u27e8PUnit\u27e9\ninstance sort.inhabited' : \n  Inhabited (@Inhabited.default (Sort _) _) := \u27e8PUnit.unit\u27e9\n\ninstance psum.inhabited_left {\u03b1 \u03b2}\n  [Inhabited \u03b1] : Inhabited (PSum \u03b1 \u03b2) :=\n\u27e8PSum.inl Inhabited.default\u27e9\n\ninstance psum.inhabited_right {\u03b1 \u03b2}\n  [Inhabited \u03b2] : Inhabited (PSum \u03b1 \u03b2) :=\n\u27e8PSum.inr Inhabited.default\u27e9\n\ninstance decidable_eq_of_subsingleton\n  {\u03b1} [Subsingleton \u03b1] : DecidableEq \u03b1\n| a, b => isTrue (Subsingleton.elim a b)\n\ntheorem subsingleton_of_forall_eq {\u03b1 : Sort _}\n  (x : \u03b1) (h : \u2200 y, y = x) : Subsingleton \u03b1 :=\n\u27e8\u03bb a b => (h a).symm \u25b8 (h b).symm \u25b8 rfl\u27e9\n\ntheorem subsingleton_iff_forall_eq {\u03b1 : Sort _} (x : \u03b1) :\n  Subsingleton \u03b1 \u2194 \u2200 y, y = x :=\n\u27e8\u03bb h y => @Subsingleton.elim _ h y x, subsingleton_of_forall_eq x\u27e9\n\n\n/-!\n\n# I don't understand coercions in Lean 4 so I'm skipping this stuff\n\n-- @[simp] theorem coe_coe {\u03b1 \u03b2 \u03b3} [Coe \u03b1 \u03b2] [has_coe_t \u03b2 \u03b3]\n--   (a : \u03b1) : (a : \u03b3) = (a : \u03b2) := rfl\n-- /-- Add an instance to \"undo\" coercion transitivity into a chain of coercions, because\n--    most simp lemmas are stated with respect to simple coercions and will not match when\n--    part of a chain. -/\n-- @[simp] theorem coe_coe {\u03b1 \u03b2 \u03b3} [Coe \u03b1 \u03b2] [has_coe_t \u03b2 \u03b3]\n--   (a : \u03b1) : (a : \u03b3) = (a : \u03b2) := rfl\n\n-- theorem coe_fn_coe_trans\n--   {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_fun \u03b3]\n--   (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n-- @[simp] theorem coe_fn_coe_base\n--   {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_fun \u03b2]\n--   (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n-- theorem coe_sort_coe_trans\n--   {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_sort \u03b3]\n--   (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n\n-- /--\n-- Many structures such as bundled morphisms coerce to functions so that you can\n-- transparently apply them to arguments. For example, if `e : \u03b1 \u2243 \u03b2` and `a : \u03b1`\n-- then you can write `e a` and this is elaborated as `\u21d1e a`. This type of\n-- coercion is implemented using the `has_coe_to_fun` type class. There is one\n-- important consideration:\n\n-- If a type coerces to another type which in turn coerces to a function,\n-- then it **must** implement `has_coe_to_fun` directly:\n-- ```lean\n-- structure sparkling_equiv (\u03b1 \u03b2) extends \u03b1 \u2243 \u03b2\n\n-- -- if we add a `has_coe` instance,\n-- instance {\u03b1 \u03b2} : has_coe (sparkling_equiv \u03b1 \u03b2) (\u03b1 \u2243 \u03b2) :=\n-- \u27e8sparkling_equiv.to_equiv\u27e9\n\n-- -- then a `has_coe_to_fun` instance **must** be added as well:\n-- instance {\u03b1 \u03b2} : has_coe_to_fun (sparkling_equiv \u03b1 \u03b2) :=\n-- \u27e8\u03bb _, \u03b1 \u2192 \u03b2, \u03bb f, f.to_equiv.to_fun\u27e9\n-- ```\n\n-- (Rationale: if we do not declare the direct coercion, then `\u21d1e a` is not in\n-- simp-normal form. The lemma `coe_fn_coe_base` will unfold it to `\u21d1\u2191e a`. This\n-- often causes loops in the simplifier.)\n-- -/\n-- --library_note \"function coercion\"\n\n-- @[simp] theorem coe_sort_coe_base\n--   {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_sort \u03b2]\n--   (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n-/\n\n/-- `pempty` is the universe-polymorphic analogue of `empty`. -/\n--@[derive decidable_eq]\ninductive PEmpty : Type u\n\n/-- Ex falso, the nondependent eliminator for the `pempty` type. -/\ndef PEmpty.elim {C : Sort v} : PEmpty \u2192 C :=\n\u03bb a => nomatch a\n\ninstance subsingleton_pempty : Subsingleton PEmpty := \u27e8\u03bb a => a.elim\u27e9\n\n@[simp] theorem not_nonempty_pempty : \u00ac Nonempty PEmpty :=\n\u03bb \u27e8h\u27e9 => h.elim\n\n@[simp] theorem forall_pempty {P : PEmpty \u2192 Prop} :\n  (\u2200 x : PEmpty, P x) \u2194 True :=\n\u27e8\u03bb _ => trivial, \u03bb _ x => x.elim\u27e9\n\n@[simp] theorem exists_pempty {P : PEmpty \u2192 Prop} :\n  (\u2203 x : PEmpty, P x) \u2194 False :=\n\u27e8\u03bb \u27e8x, _\u27e9 => x.elim, False.elim\u27e9\n\n-- TODO -- find out how to do this using equation compiler\ntheorem congr_arg_heq {\u03b1} {\u03b2 : \u03b1 \u2192 Sort u} (f : \u2200 a, \u03b2 a) : \n  \u2200 {a\u2081 a\u2082 : \u03b1}, a\u2081 = a\u2082 \u2192 f a\u2081 \u2245 f a\u2082\n| _, _, (Eq.refl _) => HEq.rfl\n\n-- this will work when I switch to nightly, apparently?\nmacro \"lemma\" n:declId sig:declSig val:declVal : command => `(theorem $n $sig $val)\n\ntheorem plift.down_inj {\u03b1 : Sort u} : \u2200 (a b : PLift \u03b1), a.down = b.down \u2192 a = b\n| \u27e8_\u27e9, \u27e8_\u27e9, (Eq.refl _) => rfl\n\n-- don't know analogue in Lean 4\n-- attribute [symm] ne.symm\n\nlemma ne_comm {\u03b1} {a b : \u03b1} : a \u2260 b \u2194 b \u2260 a := \u27e8Ne.symm, Ne.symm\u27e9\n\n@[simp] theorem eq_iff_eq_cancel_left {b c : \u03b1} :\n  (\u2200 {a}, a = b \u2194 a = c) \u2194 (b = c) :=\n\u27e8\u03bb h => by rw [\u2190 h], \u03bb h a => by rw h; refl\u27e9\n\n@[simp] theorem eq_iff_eq_cancel_right {a b : \u03b1} :\n  (\u2200 {c}, a = c \u2194 b = c) \u2194 (a = b) :=\n\u27e8\u03bb h => by rw h, \u03bb h a => by rw h; refl\u27e9\n\n/-- Wrapper for adding elementary propositions to the type class systems.\nWarning: this can easily be abused. See the rest of this docstring for details.\n\nCertain propositions should not be treated as a class globally,\nbut sometimes it is very convenient to be able to use the type class system\nin specific circumstances.\n\nFor example, `zmod p` is a field if and only if `p` is a prime number.\nIn order to be able to find this field instance automatically by type class search,\nwe have to turn `p.prime` into an instance implicit assumption.\n\nOn the other hand, making `nat.prime` a class would require a major refactoring of the library,\nand it is questionable whether making `nat.prime` a class is desirable at all.\nThe compromise is to add the assumption `[fact p.prime]` to `zmod.field`.\n\nIn particular, this class is not intended for turning the type class system\ninto an automated theorem prover for first order logic. -/\nclass fact (p : Prop) : Prop := (out' : p)\n\ndef fact.out (p : Prop) [c : fact p] : p := fact.out'\n\nlemma fact.elim {p : Prop} (h : fact p) : p := h.1\nlemma fact_iff {p : Prop} : fact p \u2194 p := \u27e8\u03bb h => h.1, \u03bb h => \u27e8h\u27e9\u27e9\n\nend miscellany\n\n/-!\n### Declarations about propositional connectives\n-/\n\ntheorem false_ne_true : False \u2260 True\n| h => h.symm \u25b8 trivial\n\nsection propositional\n\nvariable {a b c d : Prop}\n\n/-! ### Declarations about `implies` -/\n\n-- instance : is_refl Prop iff := \u27e8iff.refl\u27e9\n-- instance : IsTrans Prop iff := \u27e8\u03bb _ _ _ => iff.trans\u27e9\n\ntheorem iff_of_eq (e : a = b) : a \u2194 b := e \u25b8 Iff.rfl\n\ntheorem iff_iff_eq : (a \u2194 b) \u2194 a = b := \u27e8propext, iff_of_eq\u27e9\n\n@[simp] theorem eq_iff_iff {p q : Prop} : (p = q) \u2194 (p \u2194 q) := iff_iff_eq.symm\n\n-- can't find in Lean 4\ntheorem iff_true_intro {P : Prop} : P \u2192 (P \u2194 True) :=\n\u03bb h => \u27e8\u03bb _=> trivial, \u03bb _ => h\u27e9\n\n@[simp] theorem imp_self : (a \u2192 a) \u2194 True := iff_true_intro id\n\ntheorem imp_intro {\u03b1 \u03b2 : Prop} (h : \u03b1) : \u03b2 \u2192 \u03b1 := \u03bb _ => h\n\ntheorem imp_false : (a \u2192 False) \u2194 \u00ac a := Iff.rfl\n\ntheorem imp_and_distrib {\u03b1} : (\u03b1 \u2192 b \u2227 c) \u2194 (\u03b1 \u2192 b) \u2227 (\u03b1 \u2192 c) :=\n\u27e8\u03bb h => \u27e8\u03bb ha => (h ha).left, \u03bb ha => (h ha).right\u27e9,\n \u03bb h ha => \u27e8h.left ha, h.right ha\u27e9\u27e9\n\n\n@[simp] theorem and_imp : (a \u2227 b \u2192 c) \u2194 (a \u2192 b \u2192 c) :=\nIff.intro (\u03bb h ha hb => h \u27e8ha, hb\u27e9) (\u03bb h \u27e8ha, hb\u27e9 => h ha hb)\n\ntheorem iff_def {a b : Prop} : (a \u2194 b) \u2194 (a \u2192 b) \u2227 (b \u2192 a) :=\n\u27e8\u03bb \u27e8h1, h2\u27e9 => \u27e8h1, h2\u27e9, \u03bb \u27e8h1, h2\u27e9 => \u27e8h1, h2\u27e9\u27e9\n\ntheorem And.comm {P Q : Prop} : P \u2227 Q \u2194 Q \u2227 P := \n\u27e8\u03bb \u27e8h1, h2\u27e9 => \u27e8h2, h1\u27e9, \u03bb \u27e8h1, h2\u27e9 => \u27e8h2, h1\u27e9\u27e9\n\ntheorem iff_def' : (a \u2194 b) \u2194 (b \u2192 a) \u2227 (a \u2192 b) :=\niff_def.trans And.comm\n\ntheorem imp_true_iff {\u03b1 : Sort u} : (\u03b1 \u2192 True) \u2194 True :=\niff_true_intro $ \u03bb _ => trivial\n\ntheorem imp_iff_right (ha : a) : (a \u2192 b) \u2194 b :=\n\u27e8\u03bb f => f ha, imp_intro\u27e9\n\n/-! ### Declarations about `not` -/\n\n/-- Ex falso for negation. From `\u00ac a` and `a` anything follows. This is the same as `absurd` with\nthe arguments flipped, but it is in the `not` namespace so that projection notation can be used. -/\ndef not.elim {\u03b1 : Sort u} (H1 : \u00aca) (H2 : a) : \u03b1 := absurd H2 H1\n\n@[reducible] theorem not.imp {a b : Prop} (H2 : \u00acb) (H1 : a \u2192 b) : \u00aca := mt H1 H2\n\ntheorem not_not_of_not_imp : \u00ac(a \u2192 b) \u2192 \u00ac\u00aca :=\nmt not.elim\n\ntheorem not_of_not_imp {a : Prop} : \u00ac(a \u2192 b) \u2192 \u00acb :=\nmt imp_intro\n\ntheorem dec_em (p : Prop) [Decidable p] : p \u2228 \u00acp := Decidable.em p\n\ntheorem em (p : Prop) : p \u2228 \u00ac p := Classical.em _\n\ntheorem or_not {p : Prop} : p \u2228 \u00ac p := em _\n\n/- all propositions are Decidable -/\nnoncomputable local instance (priority := low) propDecidable (a : Prop) : Decidable a :=\n  Classical.choice <| match em a with\n    | Or.inl h => \u27e8isTrue h\u27e9\n    | Or.inr h => \u27e8isFalse h\u27e9\n\ntheorem by_contradiction {p} : (\u00acp \u2192 False) \u2192 p := Decidable.byContradiction\n\n\n-- alias by_contradiction \u2190 by_contra\ntheorem by_contra {p} : (\u00acp \u2192 False) \u2192 p := Decidable.byContradiction\n\n/-\nIn most of mathlib, we use the law of excluded middle (LEM) and the axiom of choice (AC) freely.\nThe `decidable` namespace contains versions of lemmas from the root namespace that explicitly\nattempt to avoid the axiom of choice, usually by adding decidability assumptions on the inputs.\n\nYou can check if a lemma uses the axiom of choice by using `#print axioms foo` and seeing if\n`classical.choice` appears in the list.\n-/\n--library_note \"decidable namespace\"\n\n/-\nAs mathlib is primarily classical,\nif the type signature of a `def` or `lemma` does not require any `decidable` instances to state,\nit is preferable not to introduce any `decidable` instances that are needed in the proof\nas arguments, but rather to use the `classical` tactic as needed.\n\nIn the other direction, when `decidable` instances do appear in the type signature,\nit is better to use explicitly introduced ones rather than allowing Lean to automatically infer\nclassical ones, as these may cause instance mismatch errors later.\n-/\n--library_note \"decidable arguments\"\n\nlemma not_not_intro {P : Prop} : P \u2192 \u00ac \u00ac P := \u03bb hp h => h hp\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_not [Decidable a] : \u00ac\u00aca \u2194 a :=\nIff.intro Decidable.byContradiction not_not_intro\n\n/-- The Double Negation Theorem: `\u00ac \u00ac P` is equivalent to `P`.\nThe left-to-right direction, double negation elimination (DNE),\nis classically true but not constructively. -/\n@[simp] theorem not_not : \u00ac\u00aca \u2194 a := decidable.not_not\n\ntheorem of_not_not : \u00ac\u00aca \u2192 a := by_contra\n\n-- See Note [decidable namespace]\nprotected theorem decidable.of_not_imp [Decidable a] (h : \u00ac (a \u2192 b)) : a :=\nDecidable.byContradiction (not_not_of_not_imp h)\n\ntheorem of_not_imp : \u00ac (a \u2192 b) \u2192 a := decidable.of_not_imp\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_symm [Decidable a] (h : \u00aca \u2192 b) (hb : \u00acb) : a :=\nDecidable.byContradiction $ hb \u2218 h\n\ntheorem not.decidable_imp_symm [Decidable a] : (\u00aca \u2192 b) \u2192 \u00acb \u2192 a := decidable.not_imp_symm\n\ntheorem not.imp_symm : (\u00aca \u2192 b) \u2192 \u00acb \u2192 a := not.decidable_imp_symm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_comm [Decidable a] [Decidable b] : (\u00aca \u2192 b) \u2194 (\u00acb \u2192 a) :=\n\u27e8not.decidable_imp_symm, not.decidable_imp_symm\u27e9\n\ntheorem not_imp_comm : (\u00aca \u2192 b) \u2194 (\u00acb \u2192 a) := decidable.not_imp_comm\n\n@[simp] theorem imp_not_self : (a \u2192 \u00aca) \u2194 \u00aca := \u27e8\u03bb h ha => h ha ha, \u03bb h _ => h\u27e9\n\ntheorem decidable.not_imp_self [Decidable a] : (\u00aca \u2192 a) \u2194 a := by \n  have this := @imp_not_self (\u00aca); \n  rw decidable.not_not at this;\n  assumption;\n\n@[simp] theorem not_imp_self : (\u00aca \u2192 a) \u2194 a := decidable.not_imp_self\n\ntheorem imp.swap : (a \u2192 b \u2192 c) \u2194 (b \u2192 a \u2192 c) :=\n\u27e8flip, flip\u27e9 \n\ntheorem imp_not_comm : (a \u2192 \u00acb) \u2194 (b \u2192 \u00aca) :=\nimp.swap\n\n/-! ### Declarations about `and` -/\n\ntheorem and_congr_right (h : a \u2192 (b \u2194 c)) : a \u2227 b \u2194 a \u2227 c :=\n\u27e8\u03bb \u27e8ha, hb\u27e9 => \u27e8ha, (h ha).1 hb\u27e9 , \u03bb \u27e8ha, hc\u27e9 => \u27e8ha, (h ha).2 hc\u27e9\u27e9\n\ntheorem and_congr_left (h : c \u2192 (a \u2194 b)) : a \u2227 c \u2194 b \u2227 c :=\nAnd.comm.trans $ (and_congr_right h).trans And.comm\n\nlemma and_congr {a b c d : Prop} : (a \u2194 c) \u2192 (b \u2194 d) \u2192 (a \u2227 b \u2194 c \u2227 d) :=\n\u03bb \u27e8hac, hca\u27e9 \u27e8hbd, hdb\u27e9 => \n  \u27e8\u03bb \u27e8ha, hb\u27e9 => \u27e8hac ha, hbd hb\u27e9, \u03bb \u27e8hc, hd\u27e9 => \u27e8hca hc, hdb hd\u27e9\u27e9\n\ntheorem and_congr_left' (h : a \u2194 b) : a \u2227 c \u2194 b \u2227 c := and_congr h Iff.rfl\n\ntheorem and_congr_right' (h : b \u2194 c) : a \u2227 b \u2194 a \u2227 c := and_congr Iff.rfl h\n\ntheorem not_and_of_not_left (b : Prop) : \u00aca \u2192 \u00ac(a \u2227 b) :=\nmt And.left\n\ntheorem not_and_of_not_right (a : Prop) {b : Prop} : \u00acb \u2192 \u00ac(a \u2227 b) :=\nmt And.right\n\ntheorem and.imp {a b c d : Prop} : (a \u2192 c) \u2192 (b \u2192 d) \u2192 a \u2227 b \u2192 c \u2227 d :=\n\u03bb hac hbd \u27e8ha, hb\u27e9 => \u27e8hac ha, hbd hb\u27e9\n\ntheorem and.imp_left (h : a \u2192 b) : a \u2227 c \u2192 b \u2227 c :=\nand.imp h id\n\ntheorem and.imp_right (h : a \u2192 b) : c \u2227 a \u2192 c \u2227 b :=\nand.imp id h\n\n-- and.left_comm : \u2200 {a b c : Prop}, a \u2227 b \u2227 c \u2194 b \u2227 a \u2227 c\n\ntheorem and.left_comm {a b c : Prop} : a \u2227 b \u2227 c \u2194 b \u2227 a \u2227 c :=\n\u27e8\u03bb \u27e8ha, hb, hc\u27e9 => \u27e8hb, ha, hc\u27e9, \u03bb \u27e8hb, ha, hc\u27e9 => \u27e8ha, hb, hc\u27e9\u27e9\n\n@[simp] theorem Iff_self {p : Prop} : (p \u2194 p) \u2194 True :=\n\u27e8\u03bb _ => trivial, \u03bb _ => Iff.rfl\u27e9\n\nlemma and.right_comm : (a \u2227 b) \u2227 c \u2194 (a \u2227 c) \u2227 b :=\nby simp only [and.left_comm, And.comm, Iff_self]\n\nlemma and.rotate : a \u2227 b \u2227 c \u2194 b \u2227 c \u2227 a :=\nby simp only [and.left_comm, And.comm, Iff_self]\n\ntheorem and_not_self_iff (a : Prop) : a \u2227 \u00ac a \u2194 False :=\n\u27e8\u03bb h => (h.right) (h.left), (\u03bb h => h.elim)\u27e9\n\ntheorem not_and_self_iff (a : Prop) : \u00ac a \u2227 a \u2194 False :=\nIff.intro (\u03bb \u27e8hna, ha\u27e9 => hna ha) False.elim\n\ntheorem and_iff_left_of_imp {a b : Prop} (h : a \u2192 b) : (a \u2227 b) \u2194 a :=\nIff.intro And.left (\u03bb ha => \u27e8ha, h ha\u27e9)\n\ntheorem and_iff_right_of_imp {a b : Prop} (h : b \u2192 a) : (a \u2227 b) \u2194 b :=\nIff.intro And.right (\u03bb hb => \u27e8h hb, hb\u27e9)\n\n@[simp] theorem and_iff_left_iff_imp {a b : Prop} : ((a \u2227 b) \u2194 a) \u2194 (a \u2192 b) :=\n\u27e8\u03bb h ha => (h.2 ha).2, and_iff_left_of_imp\u27e9\n\n@[simp] theorem and_iff_right_iff_imp {a b : Prop} : ((a \u2227 b) \u2194 b) \u2194 (b \u2192 a) :=\n\u27e8\u03bb h ha => (h.2 ha).1, and_iff_right_of_imp\u27e9\n\n@[simp] theorem and.congr_right_iff : (a \u2227 b \u2194 a \u2227 c) \u2194 (a \u2192 (b \u2194 c)) :=\n\u27e8\u03bb h ha => by simp [ha] at h; exact h, and_congr_right\u27e9\n\n/-\n\nTO_BE_PORTED\n\n\n#check @and.congr_right_iff\n\nexample : \u2200 {a b c : Prop}, a \u2227 b \u2194 a \u2227 c \u2194 a \u2192 (b \u2194 c) := sorry\nexample : \u2200 {a b c : Prop}, a \u2194 a \u2194 a := sorry\n\n@[simp] theorem and.congr_left_iff : (a \u2227 c \u2194 b \u2227 c) \u2194 c \u2192 (a \u2194 b) :=\nby simp only [And.comm, \u2190 and.congr_right_iff]\n\n@[simp] lemma and_self_left : a \u2227 a \u2227 b \u2194 a \u2227 b :=\n\u27e8\u03bb h, \u27e8h.1, h.2.2\u27e9, \u03bb h, \u27e8h.1, h.1, h.2\u27e9\u27e9\n\n@[simp] lemma and_self_right : (a \u2227 b) \u2227 b \u2194 a \u2227 b :=\n\u27e8\u03bb h, \u27e8h.1.1, h.2\u27e9, \u03bb h, \u27e8\u27e8h.1, h.2\u27e9, h.2\u27e9\u27e9\n\n\n/-! ### Declarations about `or` -/\n\ntheorem or_congr_left (h : a \u2194 b) : a \u2228 c \u2194 b \u2228 c := or_congr h iff.rfl\n\ntheorem or_congr_right (h : b \u2194 c) : a \u2228 b \u2194 a \u2228 c := or_congr iff.rfl h\n\ntheorem or.right_comm : (a \u2228 b) \u2228 c \u2194 (a \u2228 c) \u2228 b := by rw [or_assoc, or_assoc, or_comm b]\n\ntheorem or_of_or_of_imp_of_imp (h\u2081 : a \u2228 b) (h\u2082 : a \u2192 c) (h\u2083 : b \u2192 d) : c \u2228 d :=\nor.imp h\u2082 h\u2083 h\u2081\n\ntheorem or_of_or_of_imp_left (h\u2081 : a \u2228 c) (h : a \u2192 b) : b \u2228 c :=\nor.imp_left h h\u2081\n\ntheorem or_of_or_of_imp_right (h\u2081 : c \u2228 a) (h : a \u2192 b) : c \u2228 b :=\nor.imp_right h h\u2081\n\ntheorem or.elim3 (h : a \u2228 b \u2228 c) (ha : a \u2192 d) (hb : b \u2192 d) (hc : c \u2192 d) : d :=\nor.elim h ha (assume h\u2082, or.elim h\u2082 hb hc)\n\ntheorem or_imp_distrib : (a \u2228 b \u2192 c) \u2194 (a \u2192 c) \u2227 (b \u2192 c) :=\n\u27e8assume h, \u27e8assume ha, h (or.inl ha), assume hb, h (or.inr hb)\u27e9,\n  assume \u27e8ha, hb\u27e9, or.rec ha hb\u27e9\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_imp_left [decidable a] : a \u2228 b \u2194 (\u00ac a \u2192 b) :=\n\u27e8or.resolve_left, \u03bb h, dite _ or.inl (or.inr \u2218 h)\u27e9\n\ntheorem or_iff_not_imp_left : a \u2228 b \u2194 (\u00ac a \u2192 b) := decidable.or_iff_not_imp_left\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_imp_right [decidable b] : a \u2228 b \u2194 (\u00ac b \u2192 a) :=\nor.comm.trans decidable.or_iff_not_imp_left\n\ntheorem or_iff_not_imp_right : a \u2228 b \u2194 (\u00ac b \u2192 a) := decidable.or_iff_not_imp_right\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp_not [decidable a] : (\u00ac a \u2192 \u00ac b) \u2194 (b \u2192 a) :=\n\u27e8assume h hb, decidable.by_contradiction $ assume na, h na hb, mt\u27e9\n\ntheorem not_imp_not : (\u00ac a \u2192 \u00ac b) \u2194 (b \u2192 a) := decidable.not_imp_not\n\n@[simp] theorem or_iff_left_iff_imp : (a \u2228 b \u2194 a) \u2194 (b \u2192 a) :=\n\u27e8\u03bb h hb, h.1 (or.inr hb), or_iff_left_of_imp\u27e9\n\n@[simp] theorem or_iff_right_iff_imp : (a \u2228 b \u2194 b) \u2194 (a \u2192 b) :=\nby rw [or_comm, or_iff_left_iff_imp]\n\n/-! ### Declarations about distributivity -/\n\n/-- `\u2227` distributes over `\u2228` (on the left). -/\ntheorem and_or_distrib_left : a \u2227 (b \u2228 c) \u2194 (a \u2227 b) \u2228 (a \u2227 c) :=\n\u27e8\u03bb \u27e8ha, hbc\u27e9, hbc.imp (and.intro ha) (and.intro ha),\n or.rec (and.imp_right or.inl) (and.imp_right or.inr)\u27e9\n\n/-- `\u2227` distributes over `\u2228` (on the right). -/\ntheorem or_and_distrib_right : (a \u2228 b) \u2227 c \u2194 (a \u2227 c) \u2228 (b \u2227 c) :=\n(and.comm.trans and_or_distrib_left).trans (or_congr and.comm and.comm)\n\n/-- `\u2228` distributes over `\u2227` (on the left). -/\ntheorem or_and_distrib_left : a \u2228 (b \u2227 c) \u2194 (a \u2228 b) \u2227 (a \u2228 c) :=\n\u27e8or.rec (\u03bbha, and.intro (or.inl ha) (or.inl ha)) (and.imp or.inr or.inr),\n and.rec $ or.rec (imp_intro \u2218 or.inl) (or.imp_right \u2218 and.intro)\u27e9\n\n/-- `\u2228` distributes over `\u2227` (on the right). -/\ntheorem and_or_distrib_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c) :=\n(or.comm.trans or_and_distrib_left).trans (and_congr or.comm or.comm)\n\n@[simp] lemma or_self_left : a \u2228 a \u2228 b \u2194 a \u2228 b :=\n\u27e8\u03bb h, h.elim or.inl id, \u03bb h, h.elim or.inl (or.inr \u2218 or.inr)\u27e9\n\n@[simp] lemma or_self_right : (a \u2228 b) \u2228 b \u2194 a \u2228 b :=\n\u27e8\u03bb h, h.elim id or.inr, \u03bb h, h.elim (or.inl \u2218 or.inl) or.inr\u27e9\n\n/-! Declarations about `iff` -/\n\ntheorem iff_of_true (ha : a) (hb : b) : a \u2194 b :=\n\u27e8\u03bb_, hb, \u03bb _, ha\u27e9\n\ntheorem iff_of_false (ha : \u00aca) (hb : \u00acb) : a \u2194 b :=\n\u27e8ha.elim, hb.elim\u27e9\n\ntheorem iff_true_left (ha : a) : (a \u2194 b) \u2194 b :=\n\u27e8\u03bb h, h.1 ha, iff_of_true ha\u27e9\n\ntheorem iff_true_right (ha : a) : (b \u2194 a) \u2194 b :=\niff.comm.trans (iff_true_left ha)\n\ntheorem iff_false_left (ha : \u00aca) : (a \u2194 b) \u2194 \u00acb :=\n\u27e8\u03bb h, mt h.2 ha, iff_of_false ha\u27e9\n\ntheorem iff_false_right (ha : \u00aca) : (b \u2194 a) \u2194 \u00acb :=\niff.comm.trans (iff_false_left ha)\n\n@[simp]\nlemma iff_mpr_iff_true_intro {P : Prop} (h : P) : iff.mpr (iff_true_intro h) true.intro = h := rfl\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_or_of_imp [decidable a] (h : a \u2192 b) : \u00ac a \u2228 b :=\nif ha : a then or.inr (h ha) else or.inl ha\n\ntheorem not_or_of_imp : (a \u2192 b) \u2192 \u00ac a \u2228 b := decidable.not_or_of_imp\n\n-- See Note [decidable namespace]\nprotected theorem decidable.imp_iff_not_or [decidable a] : (a \u2192 b) \u2194 (\u00ac a \u2228 b) :=\n\u27e8decidable.not_or_of_imp, or.neg_resolve_left\u27e9\n\ntheorem imp_iff_not_or : (a \u2192 b) \u2194 (\u00ac a \u2228 b) := decidable.imp_iff_not_or\n\n-- See Note [decidable namespace]\nprotected theorem decidable.imp_or_distrib [decidable a] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby simp [decidable.imp_iff_not_or, or.comm, or.left_comm]\n\ntheorem imp_or_distrib : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) := decidable.imp_or_distrib\n\n-- See Note [decidable namespace]\nprotected theorem decidable.imp_or_distrib' [decidable b] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby by_cases b; simp [h, or_iff_right_of_imp ((\u2218) false.elim)]\n\ntheorem imp_or_distrib' : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) := decidable.imp_or_distrib'\n\ntheorem not_imp_of_and_not : a \u2227 \u00ac b \u2192 \u00ac (a \u2192 b)\n| \u27e8ha, hb\u27e9 h := hb $ h ha\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_imp [decidable a] : \u00ac(a \u2192 b) \u2194 a \u2227 \u00acb :=\n\u27e8\u03bb h, \u27e8decidable.of_not_imp h, not_of_not_imp h\u27e9, not_imp_of_and_not\u27e9\n\ntheorem not_imp : \u00ac(a \u2192 b) \u2194 a \u2227 \u00acb := decidable.not_imp\n\n-- for monotonicity\nlemma imp_imp_imp (h\u2080 : c \u2192 a) (h\u2081 : b \u2192 d) : (a \u2192 b) \u2192 (c \u2192 d) :=\nassume (h\u2082 : a \u2192 b), h\u2081 \u2218 h\u2082 \u2218 h\u2080\n\n-- See Note [decidable namespace]\nprotected theorem decidable.peirce (a b : Prop) [decidable a] : ((a \u2192 b) \u2192 a) \u2192 a :=\nif ha : a then \u03bb h, ha else \u03bb h, h ha.elim\n\ntheorem peirce (a b : Prop) : ((a \u2192 b) \u2192 a) \u2192 a := decidable.peirce _ _\n\ntheorem peirce' {a : Prop} (H : \u2200 b : Prop, (a \u2192 b) \u2192 a) : a := H _ id\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff_not [decidable a] [decidable b] : (\u00ac a \u2194 \u00ac b) \u2194 (a \u2194 b) :=\nby rw [@iff_def (\u00ac a), @iff_def' a]; exact and_congr decidable.not_imp_not decidable.not_imp_not\n\ntheorem not_iff_not : (\u00ac a \u2194 \u00ac b) \u2194 (a \u2194 b) := decidable.not_iff_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff_comm [decidable a] [decidable b] : (\u00ac a \u2194 b) \u2194 (\u00ac b \u2194 a) :=\nby rw [@iff_def (\u00ac a), @iff_def (\u00ac b)]; exact and_congr decidable.not_imp_comm imp_not_comm\n\ntheorem not_iff_comm : (\u00ac a \u2194 b) \u2194 (\u00ac b \u2194 a) := decidable.not_iff_comm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_iff : \u2200 [decidable b], \u00ac (a \u2194 b) \u2194 (\u00ac a \u2194 b) :=\nby intro h; cases h; simp only [h, iff_true, iff_false]\n\ntheorem not_iff : \u00ac (a \u2194 b) \u2194 (\u00ac a \u2194 b) := decidable.not_iff\n\n-- See Note [decidable namespace]\nprotected theorem decidable.iff_not_comm [decidable a] [decidable b] : (a \u2194 \u00ac b) \u2194 (b \u2194 \u00ac a) :=\nby rw [@iff_def a, @iff_def b]; exact and_congr imp_not_comm decidable.not_imp_comm\n\ntheorem iff_not_comm : (a \u2194 \u00ac b) \u2194 (b \u2194 \u00ac a) := decidable.iff_not_comm\n\n-- See Note [decidable namespace]\nprotected theorem decidable.iff_iff_and_or_not_and_not [decidable b] :\n  (a \u2194 b) \u2194 (a \u2227 b) \u2228 (\u00ac a \u2227 \u00ac b) :=\nby { split; intro h,\n     { rw h; by_cases b; [left,right]; split; assumption },\n     { cases h with h h; cases h; split; intro; { contradiction <|> assumption } } }\n\ntheorem iff_iff_and_or_not_and_not : (a \u2194 b) \u2194 (a \u2227 b) \u2228 (\u00ac a \u2227 \u00ac b) :=\ndecidable.iff_iff_and_or_not_and_not\n\nlemma decidable.iff_iff_not_or_and_or_not [decidable a] [decidable b] :\n  (a \u2194 b) \u2194 ((\u00aca \u2228 b) \u2227 (a \u2228 \u00acb)) :=\nbegin\n  rw [iff_iff_implies_and_implies a b],\n  simp only [decidable.imp_iff_not_or, or.comm]\nend\n\nlemma iff_iff_not_or_and_or_not : (a \u2194 b) \u2194 ((\u00aca \u2228 b) \u2227 (a \u2228 \u00acb)) :=\ndecidable.iff_iff_not_or_and_or_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_not_right [decidable b] : \u00ac(a \u2227 \u00acb) \u2194 (a \u2192 b) :=\n\u27e8\u03bb h ha, h.decidable_imp_symm $ and.intro ha, \u03bb h \u27e8ha, hb\u27e9, hb $ h ha\u27e9\n\ntheorem not_and_not_right : \u00ac(a \u2227 \u00acb) \u2194 (a \u2192 b) := decidable.not_and_not_right\n\n/-- Transfer decidability of `a` to decidability of `b`, if the propositions are equivalent.\n**Important**: this function should be used instead of `rw` on `decidable b`, because the\nkernel will get stuck reducing the usage of `propext` otherwise,\nand `dec_trivial` will not work. -/\n@[inline] def decidable_of_iff (a : Prop) (h : a \u2194 b) [D : decidable a] : decidable b :=\ndecidable_of_decidable_of_iff D h\n\n/-- Transfer decidability of `b` to decidability of `a`, if the propositions are equivalent.\nThis is the same as `decidable_of_iff` but the iff is flipped. -/\n@[inline] def decidable_of_iff' (b : Prop) (h : a \u2194 b) [D : decidable b] : decidable a :=\ndecidable_of_decidable_of_iff D h.symm\n\n/-- Prove that `a` is decidable by constructing a boolean `b` and a proof that `b \u2194 a`.\n(This is sometimes taken as an alternate definition of decidability.) -/\ndef decidable_of_bool : \u2200 (b : bool) (h : b \u2194 a), decidable a\n| tt h := is_true (h.1 rfl)\n| ff h := is_false (mt h.2 bool.ff_ne_tt)\n\n/-! ### De Morgan's laws -/\n\ntheorem not_and_of_not_or_not (h : \u00ac a \u2228 \u00ac b) : \u00ac (a \u2227 b)\n| \u27e8ha, hb\u27e9 := or.elim h (absurd ha) (absurd hb)\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_distrib [decidable a] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if ha : a then or.inr (\u03bb hb, h \u27e8ha, hb\u27e9) else or.inl ha, not_and_of_not_or_not\u27e9\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_and_distrib' [decidable b] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if hb : b then or.inl (\u03bb ha, h \u27e8ha, hb\u27e9) else or.inr hb, not_and_of_not_or_not\u27e9\n\n/-- One of de Morgan's laws: the negation of a conjunction is logically equivalent to the\ndisjunction of the negations. -/\ntheorem not_and_distrib : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb := decidable.not_and_distrib\n\n@[simp] theorem not_and : \u00ac (a \u2227 b) \u2194 (a \u2192 \u00ac b) := and_imp\n\ntheorem not_and' : \u00ac (a \u2227 b) \u2194 b \u2192 \u00aca :=\nnot_and.trans imp_not_comm\n\n/-- One of de Morgan's laws: the negation of a disjunction is logically equivalent to the\nconjunction of the negations. -/\ntheorem not_or_distrib : \u00ac (a \u2228 b) \u2194 \u00ac a \u2227 \u00ac b :=\n\u27e8\u03bb h, \u27e8\u03bb ha, h (or.inl ha), \u03bb hb, h (or.inr hb)\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9 h, or.elim h h\u2081 h\u2082\u27e9\n\n-- See Note [decidable namespace]\nprotected theorem decidable.or_iff_not_and_not [decidable a] [decidable b] : a \u2228 b \u2194 \u00ac (\u00aca \u2227 \u00acb) :=\nby rw [\u2190 not_or_distrib, decidable.not_not]\n\ntheorem or_iff_not_and_not : a \u2228 b \u2194 \u00ac (\u00aca \u2227 \u00acb) := decidable.or_iff_not_and_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.and_iff_not_or_not [decidable a] [decidable b] :\n  a \u2227 b \u2194 \u00ac (\u00ac a \u2228 \u00ac b) :=\nby rw [\u2190 decidable.not_and_distrib, decidable.not_not]\n\ntheorem and_iff_not_or_not : a \u2227 b \u2194 \u00ac (\u00ac a \u2228 \u00ac b) := decidable.and_iff_not_or_not\n\nend propositional\n\n/-! ### Declarations about equality -/\n\nsection equality\nvariables {\u03b1 : Sort*} {a b : \u03b1}\n\n@[simp] theorem heq_iff_eq : a == b \u2194 a = b :=\n\u27e8eq_of_heq, heq_of_eq\u27e9\n\ntheorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : hp == hq :=\nhave p = q, from propext \u27e8\u03bb _, hq, \u03bb _, hp\u27e9,\nby subst q; refl\n\ntheorem ne_of_mem_of_not_mem {\u03b1 \u03b2} [has_mem \u03b1 \u03b2] {s : \u03b2} {a b : \u03b1}\n  (h : a \u2208 s) : b \u2209 s \u2192 a \u2260 b :=\nmt $ \u03bb e, e \u25b8 h\n\nlemma ne_of_apply_ne {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) {x y : \u03b1} (h : f x \u2260 f y) : x \u2260 y :=\n\u03bb (w : x = y), h (congr_arg f w)\n\ntheorem eq_equivalence : equivalence (@eq \u03b1) :=\n\u27e8eq.refl, @eq.symm _, @eq.trans _\u27e9\n\n/-- Transport through trivial families is the identity. -/\n@[simp]\nlemma eq_rec_constant {\u03b1 : Sort*} {a a' : \u03b1} {\u03b2 : Sort*} (y : \u03b2) (h : a = a') :\n  (@eq.rec \u03b1 a (\u03bb a, \u03b2) y a' h) = y :=\nby { cases h, refl, }\n\n@[simp]\nlemma eq_mp_eq_cast {\u03b1 \u03b2 : Sort*} (h : \u03b1 = \u03b2) : eq.mp h = cast h := rfl\n\n@[simp]\nlemma eq_mpr_eq_cast {\u03b1 \u03b2 : Sort*} (h : \u03b1 = \u03b2) : eq.mpr h = cast h.symm := rfl\n\n@[simp]\nlemma cast_cast : \u2200 {\u03b1 \u03b2 \u03b3 : Sort*} (ha : \u03b1 = \u03b2) (hb : \u03b2 = \u03b3) (a : \u03b1),\n  cast hb (cast ha a) = cast (ha.trans hb) a\n| _ _ _ rfl rfl a := rfl\n\n@[simp] lemma congr_refl_left {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) {a b : \u03b1} (h : a = b) :\n  congr (eq.refl f) h = congr_arg f h :=\nrfl\n\n@[simp] lemma congr_refl_right {\u03b1 \u03b2 : Sort*} {f g : \u03b1 \u2192 \u03b2} (h : f = g) (a : \u03b1) :\n  congr h (eq.refl a) = congr_fun h a :=\nrfl\n\n@[simp] lemma congr_arg_refl {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n  congr_arg f (eq.refl a) = eq.refl (f a) :=\nrfl\n\n@[simp] lemma congr_fun_rfl {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n  congr_fun (eq.refl f) a = eq.refl (f a) :=\nrfl\n\n@[simp] lemma congr_fun_congr_arg {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) {a a' : \u03b1} (p : a = a') (b : \u03b2) :\n  congr_fun (congr_arg f p) b = congr_arg (\u03bb a, f a b) p :=\nrfl\n\nlemma heq_of_cast_eq :\n  \u2200 {\u03b1 \u03b2 : Sort*} {a : \u03b1} {a' : \u03b2} (e : \u03b1 = \u03b2) (h\u2082 : cast e a = a'), a == a'\n| \u03b1 ._ a a' rfl h := eq.rec_on h (heq.refl _)\n\nlemma rec_heq_of_heq {\u03b2} {C : \u03b1 \u2192 Sort*} {x : C a} {y : \u03b2} (eq : a = b) (h : x == y) :\n  @eq.rec \u03b1 a C x b eq == y :=\nby subst eq; exact h\n\nprotected lemma eq.congr {x\u2081 x\u2082 y\u2081 y\u2082 : \u03b1} (h\u2081 : x\u2081 = y\u2081) (h\u2082 : x\u2082 = y\u2082) :\n  (x\u2081 = x\u2082) \u2194 (y\u2081 = y\u2082) :=\nby { subst h\u2081, subst h\u2082 }\n\nlemma eq.congr_left {x y z : \u03b1} (h : x = y) : x = z \u2194 y = z := by rw [h]\nlemma eq.congr_right {x y z : \u03b1} (h : x = y) : z = x \u2194 z = y := by rw [h]\n\nlemma congr_arg2 {\u03b1 \u03b2 \u03b3 : Type*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) {x x' : \u03b1} {y y' : \u03b2}\n  (hx : x = x') (hy : y = y') : f x y = f x' y' :=\nby { subst hx, subst hy }\n\nend equality\n\n/-! ### Declarations about quantifiers -/\n\nsection quantifiers\nvariables {\u03b1 : Sort*} {\u03b2 : Sort*} {p q : \u03b1 \u2192 Prop} {b : Prop}\n\nlemma forall_imp (h : \u2200 a, p a \u2192 q a) : (\u2200 a, p a) \u2192 \u2200 a, q a :=\n\u03bb h' a, h a (h' a)\n\nlemma forall\u2082_congr {p q : \u03b1 \u2192 \u03b2 \u2192 Prop} (h : \u2200 a b, p a b \u2194 q a b) :\n  (\u2200 a b, p a b) \u2194 (\u2200 a b, q a b) :=\nforall_congr (\u03bb a, forall_congr (h a))\n\nlemma forall\u2083_congr {\u03b3 : Sort*} {p q : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 Prop}\n  (h : \u2200 a b c, p a b c \u2194 q a b c) :\n  (\u2200 a b c, p a b c) \u2194 (\u2200 a b c, q a b c) :=\nforall_congr (\u03bb a, forall\u2082_congr (h a))\n\nlemma forall\u2084_congr {\u03b3 \u03b4 : Sort*} {p q : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 Prop}\n  (h : \u2200 a b c d, p a b c d \u2194 q a b c d) :\n  (\u2200 a b c d, p a b c d) \u2194 (\u2200 a b c d, q a b c d) :=\nforall_congr (\u03bb a, forall\u2083_congr (h a))\n\nlemma Exists.imp (h : \u2200 a, (p a \u2192 q a)) (p : \u2203 a, p a) : \u2203 a, q a := exists_imp_exists h p\n\nlemma exists_imp_exists' {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) (hpq : \u2200 a, p a \u2192 q (f a))\n  (hp : \u2203 a, p a) : \u2203 b, q b :=\nexists.elim hp (\u03bb a hp', \u27e8_, hpq _ hp'\u27e9)\n\nlemma exists\u2082_congr {p q : \u03b1 \u2192 \u03b2 \u2192 Prop} (h : \u2200 a b, p a b \u2194 q a b) :\n  (\u2203 a b, p a b) \u2194 (\u2203 a b, q a b) :=\nexists_congr (\u03bb a, exists_congr (h a))\n\nlemma exists\u2083_congr {\u03b3 : Sort*} {p q : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 Prop}\n  (h : \u2200 a b c, p a b c \u2194 q a b c) :\n  (\u2203 a b c, p a b c) \u2194 (\u2203 a b c, q a b c) :=\nexists_congr (\u03bb a, exists\u2082_congr (h a))\n\nlemma exists\u2084_congr {\u03b3 \u03b4 : Sort*} {p q : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 Prop}\n  (h : \u2200 a b c d, p a b c d \u2194 q a b c d) :\n  (\u2203 a b c d, p a b c d) \u2194 (\u2203 a b c d, q a b c d) :=\nexists_congr (\u03bb a, exists\u2083_congr (h a))\n\ntheorem forall_swap {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2200 x y, p x y) \u2194 \u2200 y x, p x y :=\n\u27e8function.swap, function.swap\u27e9\n\ntheorem exists_swap {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2203 x y, p x y) \u2194 \u2203 y x, p x y :=\n\u27e8\u03bb \u27e8x, y, h\u27e9, \u27e8y, x, h\u27e9, \u03bb \u27e8y, x, h\u27e9, \u27e8x, y, h\u27e9\u27e9\n\n@[simp] theorem exists_imp_distrib : ((\u2203 x, p x) \u2192 b) \u2194 \u2200 x, p x \u2192 b :=\n\u27e8\u03bb h x hpx, h \u27e8x, hpx\u27e9, \u03bb h \u27e8x, hpx\u27e9, h x hpx\u27e9\n\n/--\nExtract an element from a existential statement, using `classical.some`.\n-/\n-- This enables projection notation.\n@[reducible] noncomputable def Exists.some {p : \u03b1 \u2192 Prop} (P : \u2203 a, p a) : \u03b1 := classical.some P\n\n/--\nShow that an element extracted from `P : \u2203 a, p a` using `P.some` satisfies `p`.\n-/\nlemma Exists.some_spec {p : \u03b1 \u2192 Prop} (P : \u2203 a, p a) : p (P.some) := classical.some_spec P\n\n--theorem forall_not_of_not_exists (h : \u00ac \u2203 x, p x) : \u2200 x, \u00ac p x :=\n--forall_imp_of_exists_imp h\n\ntheorem not_exists_of_forall_not (h : \u2200 x, \u00ac p x) : \u00ac \u2203 x, p x :=\nexists_imp_distrib.2 h\n\n@[simp] theorem not_exists : (\u00ac \u2203 x, p x) \u2194 \u2200 x, \u00ac p x :=\nexists_imp_distrib\n\ntheorem not_forall_of_exists_not : (\u2203 x, \u00ac p x) \u2192 \u00ac \u2200 x, p x\n| \u27e8x, hn\u27e9 h := hn (h x)\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_forall {p : \u03b1 \u2192 Prop}\n  [decidable (\u2203 x, \u00ac p x)] [\u2200 x, decidable (p x)] : (\u00ac \u2200 x, p x) \u2194 \u2203 x, \u00ac p x :=\n\u27e8not.decidable_imp_symm $ \u03bb nx x, nx.decidable_imp_symm $ \u03bb h, \u27e8x, h\u27e9,\n not_forall_of_exists_not\u27e9\n\n@[simp] theorem not_forall {p : \u03b1 \u2192 Prop} : (\u00ac \u2200 x, p x) \u2194 \u2203 x, \u00ac p x := decidable.not_forall\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_forall_not [decidable (\u2203 x, p x)] :\n  (\u00ac \u2200 x, \u00ac p x) \u2194 \u2203 x, p x :=\n(@decidable.not_iff_comm _ _ _ (decidable_of_iff (\u00ac \u2203 x, p x) not_exists)).1 not_exists\n\ntheorem not_forall_not : (\u00ac \u2200 x, \u00ac p x) \u2194 \u2203 x, p x := decidable.not_forall_not\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_exists_not [\u2200 x, decidable (p x)] : (\u00ac \u2203 x, \u00ac p x) \u2194 \u2200 x, p x :=\nby simp [decidable.not_not]\n\n@[simp] theorem not_exists_not : (\u00ac \u2203 x, \u00ac p x) \u2194 \u2200 x, p x := decidable.not_exists_not\n\n-- TODO: duplicate of a lemma in core\ntheorem forall_true_iff : (\u03b1 \u2192 true) \u2194 true :=\nimplies_true_iff \u03b1\n\n-- Unfortunately this causes simp to loop sometimes, so we\n-- add the 2 and 3 cases as simp lemmas instead\ntheorem forall_true_iff' (h : \u2200 a, p a \u2194 true) : (\u2200 a, p a) \u2194 true :=\niff_true_intro (\u03bb _, of_iff_true (h _))\n\n@[simp] theorem forall_2_true_iff {\u03b2 : \u03b1 \u2192 Sort*} : (\u2200 a, \u03b2 a \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_true_iff\n\n@[simp] theorem forall_3_true_iff {\u03b2 : \u03b1 \u2192 Sort*} {\u03b3 : \u03a0 a, \u03b2 a \u2192 Sort*} :\n  (\u2200 a (b : \u03b2 a), \u03b3 a b \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_2_true_iff\n\n@[simp] theorem forall_const (\u03b1 : Sort*) [i : nonempty \u03b1] : (\u03b1 \u2192 b) \u2194 b :=\n\u27e8i.elim, \u03bb hb x, hb\u27e9\n\n@[simp] theorem exists_const (\u03b1 : Sort*) [i : nonempty \u03b1] : (\u2203 x : \u03b1, b) \u2194 b :=\n\u27e8\u03bb \u27e8x, h\u27e9, h, i.elim exists.intro\u27e9\n\ntheorem forall_and_distrib : (\u2200 x, p x \u2227 q x) \u2194 (\u2200 x, p x) \u2227 (\u2200 x, q x) :=\n\u27e8\u03bb h, \u27e8\u03bb x, (h x).left, \u03bb x, (h x).right\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9 x, \u27e8h\u2081 x, h\u2082 x\u27e9\u27e9\n\ntheorem exists_or_distrib : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\n\u27e8\u03bb \u27e8x, hpq\u27e9, hpq.elim (\u03bb hpx, or.inl \u27e8x, hpx\u27e9) (\u03bb hqx, or.inr \u27e8x, hqx\u27e9),\n \u03bb hepq, hepq.elim (\u03bb \u27e8x, hpx\u27e9, \u27e8x, or.inl hpx\u27e9) (\u03bb \u27e8x, hqx\u27e9, \u27e8x, or.inr hqx\u27e9)\u27e9\n\n@[simp] theorem exists_and_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, q \u2227 p x) \u2194 q \u2227 (\u2203x, p x) :=\n\u27e8\u03bb \u27e8x, hq, hp\u27e9, \u27e8hq, x, hp\u27e9, \u03bb \u27e8hq, x, hp\u27e9, \u27e8x, hq, hp\u27e9\u27e9\n\n@[simp] theorem exists_and_distrib_right {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, p x \u2227 q) \u2194 (\u2203x, p x) \u2227 q :=\nby simp [and_comm]\n\n@[simp] theorem forall_eq {a' : \u03b1} : (\u2200a, a = a' \u2192 p a) \u2194 p a' :=\n\u27e8\u03bb h, h a' rfl, \u03bb h a e, e.symm \u25b8 h\u27e9\n\n@[simp] theorem forall_eq' {a' : \u03b1} : (\u2200a, a' = a \u2192 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n-- this lemma is needed to simplify the output of `list.mem_cons_iff`\n@[simp] theorem forall_eq_or_imp {a' : \u03b1} : (\u2200 a, a = a' \u2228 q a \u2192 p a) \u2194 p a' \u2227 \u2200 a, q a \u2192 p a :=\nby simp only [or_imp_distrib, forall_and_distrib, forall_eq]\n\n@[simp] theorem exists_eq {a' : \u03b1} : \u2203 a, a = a' := \u27e8_, rfl\u27e9\n\n@[simp] theorem exists_eq' {a' : \u03b1} : \u2203 a, a' = a := \u27e8_, rfl\u27e9\n\n@[simp] theorem exists_eq_left {a' : \u03b1} : (\u2203 a, a = a' \u2227 p a) \u2194 p a' :=\n\u27e8\u03bb \u27e8a, e, h\u27e9, e \u25b8 h, \u03bb h, \u27e8_, rfl, h\u27e9\u27e9\n\n@[simp] theorem exists_eq_right {a' : \u03b1} : (\u2203 a, p a \u2227 a = a') \u2194 p a' :=\n(exists_congr $ by exact \u03bb a, and.comm).trans exists_eq_left\n\n@[simp] theorem exists_eq_right_right {a' : \u03b1} :\n  (\u2203 (a : \u03b1), p a \u2227 b \u2227 a = a') \u2194 p a' \u2227 b :=\n\u27e8\u03bb \u27e8_, hp, hq, rfl\u27e9, \u27e8hp, hq\u27e9, \u03bb \u27e8hp, hq\u27e9, \u27e8a', hp, hq, rfl\u27e9\u27e9\n\n@[simp] theorem exists_eq_right_right' {a' : \u03b1} :\n  (\u2203 (a : \u03b1), p a \u2227 b \u2227 a' = a) \u2194 p a' \u2227 b :=\n\u27e8\u03bb \u27e8_, hp, hq, rfl\u27e9, \u27e8hp, hq\u27e9, \u03bb \u27e8hp, hq\u27e9, \u27e8a', hp, hq, rfl\u27e9\u27e9\n\n@[simp] theorem exists_apply_eq_apply {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) (a' : \u03b1) : \u2203 a, f a = f a' :=\n\u27e8a', rfl\u27e9\n\n@[simp] theorem exists_apply_eq_apply' {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) (a' : \u03b1) : \u2203 a, f a' = f a :=\n\u27e8a', rfl\u27e9\n\n@[simp] theorem exists_exists_and_eq_and {f : \u03b1 \u2192 \u03b2} {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} :\n  (\u2203 b, (\u2203 a, p a \u2227 f a = b) \u2227 q b) \u2194 \u2203 a, p a \u2227 q (f a) :=\n\u27e8\u03bb \u27e8b, \u27e8a, ha, hab\u27e9, hb\u27e9, \u27e8a, ha, hab.symm \u25b8 hb\u27e9, \u03bb \u27e8a, hp, hq\u27e9, \u27e8f a, \u27e8a, hp, rfl\u27e9, hq\u27e9\u27e9\n\n@[simp] theorem exists_exists_eq_and {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2203 b, (\u2203 a, f a = b) \u2227 p b) \u2194 \u2203 a, p (f a) :=\n\u27e8\u03bb \u27e8b, \u27e8a, ha\u27e9, hb\u27e9, \u27e8a, ha.symm \u25b8 hb\u27e9, \u03bb \u27e8a, ha\u27e9, \u27e8f a, \u27e8a, rfl\u27e9, ha\u27e9\u27e9\n\n@[simp] theorem forall_apply_eq_imp_iff {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2200 a, \u2200 b, f a = b \u2192 p b) \u2194 (\u2200 a, p (f a)) :=\n\u27e8\u03bb h a, h a (f a) rfl, \u03bb h a b hab, hab \u25b8 h a\u27e9\n\n@[simp] theorem forall_apply_eq_imp_iff' {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2200 b, \u2200 a, f a = b \u2192 p b) \u2194 (\u2200 a, p (f a)) :=\nby { rw forall_swap, simp }\n\n@[simp] theorem forall_eq_apply_imp_iff {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2200 a, \u2200 b, b = f a \u2192 p b) \u2194 (\u2200 a, p (f a)) :=\nby simp [@eq_comm _ _ (f _)]\n\n@[simp] theorem forall_eq_apply_imp_iff' {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2200 b, \u2200 a, b = f a \u2192 p b) \u2194 (\u2200 a, p (f a)) :=\nby { rw forall_swap, simp }\n\n@[simp] theorem forall_apply_eq_imp_iff\u2082 {f : \u03b1 \u2192 \u03b2} {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} :\n  (\u2200 b, \u2200 a, p a \u2192 f a = b \u2192 q b) \u2194 \u2200 a, p a \u2192 q (f a) :=\n\u27e8\u03bb h a ha, h (f a) a ha rfl, \u03bb h b a ha hb, hb \u25b8 h a ha\u27e9\n\n@[simp] theorem exists_eq_left' {a' : \u03b1} : (\u2203 a, a' = a \u2227 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_right' {a' : \u03b1} : (\u2203 a, p a \u2227 a' = a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\ntheorem exists_comm {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2203 a b, p a b) \u2194 \u2203 b a, p a b :=\n\u27e8\u03bb \u27e8a, b, h\u27e9, \u27e8b, a, h\u27e9, \u03bb \u27e8b, a, h\u27e9, \u27e8a, b, h\u27e9\u27e9\n\ntheorem forall_or_of_or_forall (h : b \u2228 \u2200x, p x) (x) : b \u2228 p x :=\nh.imp_right $ \u03bb h\u2082, h\u2082 x\n\n-- See Note [decidable namespace]\nprotected theorem decidable.forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} [decidable q] :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) :=\n\u27e8\u03bb h, if hq : q then or.inl hq else or.inr $ \u03bb x, (h x).resolve_left hq,\n  forall_or_of_or_forall\u27e9\n\ntheorem forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) := decidable.forall_or_distrib_left\n\n-- See Note [decidable namespace]\nprotected theorem decidable.forall_or_distrib_right {q : Prop} {p : \u03b1 \u2192 Prop} [decidable q] :\n  (\u2200x, p x \u2228 q) \u2194 (\u2200x, p x) \u2228 q :=\nby simp [or_comm, decidable.forall_or_distrib_left]\n\ntheorem forall_or_distrib_right {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2200x, p x \u2228 q) \u2194 (\u2200x, p x) \u2228 q := decidable.forall_or_distrib_right\n\n/-- A predicate holds everywhere on the image of a surjective functions iff\n    it holds everywhere. -/\ntheorem forall_iff_forall_surj\n  {\u03b1 \u03b2 : Type*} {f : \u03b1 \u2192 \u03b2} (h : function.surjective f) {P : \u03b2 \u2192 Prop} :\n  (\u2200 a, P (f a)) \u2194 \u2200 b, P b :=\n\u27e8\u03bb ha b, by cases h b with a hab; rw \u2190hab; exact ha a, \u03bb hb a, hb $ f a\u27e9\n\n@[simp] theorem exists_prop {p q : Prop} : (\u2203 h : p, q) \u2194 p \u2227 q :=\n\u27e8\u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9\u27e9\n\n@[simp] theorem exists_false : \u00ac (\u2203a:\u03b1, false) := assume \u27e8a, h\u27e9, h\n\n@[simp] lemma exists_unique_false : \u00ac (\u2203! (a : \u03b1), false) := assume \u27e8a, h, h'\u27e9, h\n\ntheorem Exists.fst {p : b \u2192 Prop} : Exists p \u2192 b\n| \u27e8h, _\u27e9 := h\n\ntheorem Exists.snd {p : b \u2192 Prop} : \u2200 h : Exists p, p h.fst\n| \u27e8_, h\u27e9 := h\n\ntheorem forall_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2200 h' : p, q h') \u2194 q h :=\n@forall_const (q h) p \u27e8h\u27e9\n\ntheorem exists_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2203 h' : p, q h') \u2194 q h :=\n@exists_const (q h) p \u27e8h\u27e9\n\ntheorem forall_prop_of_false {p : Prop} {q : p \u2192 Prop} (hn : \u00ac p) :\n  (\u2200 h' : p, q h') \u2194 true :=\niff_true_intro $ \u03bb h, hn.elim h\n\ntheorem exists_prop_of_false {p : Prop} {q : p \u2192 Prop} : \u00ac p \u2192 \u00ac (\u2203 h' : p, q h') :=\nmt Exists.fst\n\n@[congr] lemma exists_prop_congr {p p' : Prop} {q q' : p \u2192 Prop}\n  (hq : \u2200 h, q h \u2194 q' h) (hp : p \u2194 p') : Exists q \u2194 \u2203 h : p', q' (hp.2 h) :=\n\u27e8\u03bb \u27e8_, _\u27e9, \u27e8hp.1 \u2039_\u203a, (hq _).1 \u2039_\u203a\u27e9, \u03bb \u27e8_, _\u27e9, \u27e8_, (hq _).2 \u2039_\u203a\u27e9\u27e9\n\n@[congr] lemma exists_prop_congr' {p p' : Prop} {q q' : p \u2192 Prop}\n  (hq : \u2200 h, q h \u2194 q' h) (hp : p \u2194 p') : Exists q = \u2203 h : p', q' (hp.2 h) :=\npropext (exists_prop_congr hq _)\n\n@[simp] lemma exists_true_left (p : true \u2192 Prop) : (\u2203 x, p x) \u2194 p true.intro :=\nexists_prop_of_true _\n\n@[simp] lemma exists_false_left (p : false \u2192 Prop) : \u00ac \u2203 x, p x :=\nexists_prop_of_false not_false\n\nlemma exists_unique.exists {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} (h : \u2203! x, p x) : \u2203 x, p x :=\nexists.elim h (\u03bb x hx, \u27e8x, and.left hx\u27e9)\n\nlemma exists_unique.unique {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} (h : \u2203! x, p x)\n  {y\u2081 y\u2082 : \u03b1} (py\u2081 : p y\u2081) (py\u2082 : p y\u2082) : y\u2081 = y\u2082 :=\nunique_of_exists_unique h py\u2081 py\u2082\n\n@[congr] lemma forall_prop_congr {p p' : Prop} {q q' : p \u2192 Prop}\n  (hq : \u2200 h, q h \u2194 q' h) (hp : p \u2194 p') : (\u2200 h, q h) \u2194 \u2200 h : p', q' (hp.2 h) :=\n\u27e8\u03bb h1 h2, (hq _).1 (h1 (hp.2 _)), \u03bb h1 h2, (hq _).2 (h1 (hp.1 h2))\u27e9\n\n@[congr] lemma forall_prop_congr' {p p' : Prop} {q q' : p \u2192 Prop}\n  (hq : \u2200 h, q h \u2194 q' h) (hp : p \u2194 p') : (\u2200 h, q h) = \u2200 h : p', q' (hp.2 h) :=\npropext (forall_prop_congr hq _)\n\n@[simp] lemma forall_true_left (p : true \u2192 Prop) : (\u2200 x, p x) \u2194 p true.intro :=\nforall_prop_of_true _\n\n@[simp] lemma forall_false_left (p : false \u2192 Prop) : (\u2200 x, p x) \u2194 true :=\nforall_prop_of_false not_false\n\n@[simp] lemma exists_unique_iff_exists {\u03b1 : Sort*} [subsingleton \u03b1] {p : \u03b1 \u2192 Prop} :\n  (\u2203! x, p x) \u2194 \u2203 x, p x :=\n\u27e8\u03bb h, h.exists, Exists.imp $ \u03bb x hx, \u27e8hx, \u03bb y _, subsingleton.elim y x\u27e9\u27e9\n\nlemma exists_unique.elim2 {\u03b1 : Sort*} {p : \u03b1 \u2192 Sort*} [\u2200 x, subsingleton (p x)]\n  {q : \u03a0 x (h : p x), Prop} {b : Prop} (h\u2082 : \u2203! x (h : p x), q x h)\n  (h\u2081 : \u2200 x (h : p x), q x h \u2192 (\u2200 y (hy : p y), q y hy \u2192 y = x) \u2192 b) : b :=\nbegin\n  simp only [exists_unique_iff_exists] at h\u2082,\n  apply h\u2082.elim,\n  exact \u03bb x \u27e8hxp, hxq\u27e9 H, h\u2081 x hxp hxq (\u03bb y hyp hyq, H y \u27e8hyp, hyq\u27e9)\nend\n\nlemma exists_unique.intro2 {\u03b1 : Sort*} {p : \u03b1 \u2192 Sort*} [\u2200 x, subsingleton (p x)]\n  {q : \u03a0 (x : \u03b1) (h : p x), Prop} (w : \u03b1) (hp : p w) (hq : q w hp)\n  (H : \u2200 y (hy : p y), q y hy \u2192 y = w) :\n  \u2203! x (hx : p x), q x hx :=\nbegin\n  simp only [exists_unique_iff_exists],\n  exact exists_unique.intro w \u27e8hp, hq\u27e9 (\u03bb y \u27e8hyp, hyq\u27e9, H y hyp hyq)\nend\n\nlemma exists_unique.exists2 {\u03b1 : Sort*} {p : \u03b1 \u2192 Sort*} {q : \u03a0 (x : \u03b1) (h : p x), Prop}\n  (h : \u2203! x (hx : p x), q x hx) :\n  \u2203 x (hx : p x), q x hx :=\nh.exists.imp (\u03bb x hx, hx.exists)\n\nlemma exists_unique.unique2 {\u03b1 : Sort*} {p : \u03b1 \u2192 Sort*} [\u2200 x, subsingleton (p x)]\n  {q : \u03a0 (x : \u03b1) (hx : p x), Prop} (h : \u2203! x (hx : p x), q x hx)\n  {y\u2081 y\u2082 : \u03b1} (hpy\u2081 : p y\u2081) (hqy\u2081 : q y\u2081 hpy\u2081)\n  (hpy\u2082 : p y\u2082) (hqy\u2082 : q y\u2082 hpy\u2082) : y\u2081 = y\u2082 :=\nbegin\n  simp only [exists_unique_iff_exists] at h,\n  exact h.unique \u27e8hpy\u2081, hqy\u2081\u27e9 \u27e8hpy\u2082, hqy\u2082\u27e9\nend\n\nend quantifiers\n\n/-! ### Classical lemmas -/\n\nnamespace classical\nvariables {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop}\n\ntheorem cases {p : Prop \u2192 Prop} (h1 : p true) (h2 : p false) : \u2200a, p a :=\nassume a, cases_on a h1 h2\n\n/- use shortened names to avoid conflict when classical namespace is open. -/\nnoncomputable lemma dec (p : Prop) : decidable p := -- see Note [classical lemma]\nby apply_instance\nnoncomputable lemma dec_pred (p : \u03b1 \u2192 Prop) : decidable_pred p := -- see Note [classical lemma]\nby apply_instance\nnoncomputable lemma dec_rel (p : \u03b1 \u2192 \u03b1 \u2192 Prop) : decidable_rel p := -- see Note [classical lemma]\nby apply_instance\nnoncomputable lemma dec_eq (\u03b1 : Sort*) : decidable_eq \u03b1 := -- see Note [classical lemma]\nby apply_instance\n\n/--\nWe make decidability results that depends on `classical.choice` noncomputable lemmas.\n* We have to mark them as noncomputable, because otherwise Lean will try to generate bytecode\n  for them, and fail because it depends on `classical.choice`.\n* We make them lemmas, and not definitions, because otherwise later definitions will raise\n  \\\"failed to generate bytecode\\\" errors when writing something like\n  `letI := classical.dec_eq _`.\nCf. <https://leanprover-community.github.io/archive/stream/113488-general/topic/noncomputable.20theorem.html>\n-/\nlibrary_note \"classical lemma\"\n\n/-- Construct a function from a default value `H0`, and a function to use if there exists a value\nsatisfying the predicate. -/\n@[elab_as_eliminator]\nnoncomputable def {u} exists_cases {C : Sort u} (H0 : C) (H : \u2200 a, p a \u2192 C) : C :=\nif h : \u2203 a, p a then H (classical.some h) (classical.some_spec h) else H0\n\nlemma some_spec2 {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} {h : \u2203a, p a}\n  (q : \u03b1 \u2192 Prop) (hpq : \u2200a, p a \u2192 q a) : q (some h) :=\nhpq _ $ some_spec _\n\n/-- A version of classical.indefinite_description which is definitionally equal to a pair -/\nnoncomputable def subtype_of_exists {\u03b1 : Type*} {P : \u03b1 \u2192 Prop} (h : \u2203 x, P x) : {x // P x} :=\n\u27e8classical.some h, classical.some_spec h\u27e9\n\nend classical\n\n/-- This function has the same type as `exists.rec_on`, and can be used to case on an equality,\nbut `exists.rec_on` can only eliminate into Prop, while this version eliminates into any universe\nusing the axiom of choice. -/\n@[elab_as_eliminator]\nnoncomputable def {u} exists.classical_rec_on\n {\u03b1} {p : \u03b1 \u2192 Prop} (h : \u2203 a, p a) {C : Sort u} (H : \u2200 a, p a \u2192 C) : C :=\nH (classical.some h) (classical.some_spec h)\n\n/-! ### Declarations about bounded quantifiers -/\n\nsection bounded_quantifiers\nvariables {\u03b1 : Sort*} {r p q : \u03b1 \u2192 Prop} {P Q : \u2200 x, p x \u2192 Prop} {b : Prop}\n\ntheorem bex_def : (\u2203 x (h : p x), q x) \u2194 \u2203 x, p x \u2227 q x :=\n\u27e8\u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9, \u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9\u27e9\n\ntheorem bex.elim {b : Prop} : (\u2203 x h, P x h) \u2192 (\u2200 a h, P a h \u2192 b) \u2192 b\n| \u27e8a, h\u2081, h\u2082\u27e9 h' := h' a h\u2081 h\u2082\n\ntheorem bex.intro (a : \u03b1) (h\u2081 : p a) (h\u2082 : P a h\u2081) : \u2203 x (h : p x), P x h :=\n\u27e8a, h\u2081, h\u2082\u27e9\n\ntheorem ball_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2200 x h, P x h) \u2194 (\u2200 x h, Q x h) :=\nforall_congr $ \u03bb x, forall_congr (H x)\n\ntheorem bex_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2203 x h, P x h) \u2194 (\u2203 x h, Q x h) :=\nexists_congr $ \u03bb x, exists_congr (H x)\n\ntheorem bex_eq_left {a : \u03b1} : (\u2203 x (_ : x = a), p x) \u2194 p a :=\nby simp only [exists_prop, exists_eq_left]\n\ntheorem ball.imp_right (H : \u2200 x h, (P x h \u2192 Q x h))\n  (h\u2081 : \u2200 x h, P x h) (x h) : Q x h :=\nH _ _ $ h\u2081 _ _\n\ntheorem bex.imp_right (H : \u2200 x h, (P x h \u2192 Q x h)) :\n  (\u2203 x h, P x h) \u2192 \u2203 x h, Q x h\n| \u27e8x, h, h'\u27e9 := \u27e8_, _, H _ _ h'\u27e9\n\ntheorem ball.imp_left (H : \u2200 x, p x \u2192 q x)\n  (h\u2081 : \u2200 x, q x \u2192 r x) (x) (h : p x) : r x :=\nh\u2081 _ $ H _ h\n\ntheorem bex.imp_left (H : \u2200 x, p x \u2192 q x) :\n  (\u2203 x (_ : p x), r x) \u2192 \u2203 x (_ : q x), r x\n| \u27e8x, hp, hr\u27e9 := \u27e8x, H _ hp, hr\u27e9\n\ntheorem ball_of_forall (h : \u2200 x, p x) (x) : p x :=\nh x\n\ntheorem forall_of_ball (H : \u2200 x, p x) (h : \u2200 x, p x \u2192 q x) (x) : q x :=\nh x $ H x\n\ntheorem bex_of_exists (H : \u2200 x, p x) : (\u2203 x, q x) \u2192 \u2203 x (_ : p x), q x\n| \u27e8x, hq\u27e9 := \u27e8x, H x, hq\u27e9\n\ntheorem exists_of_bex : (\u2203 x (_ : p x), q x) \u2192 \u2203 x, q x\n| \u27e8x, _, hq\u27e9 := \u27e8x, hq\u27e9\n\n@[simp] theorem bex_imp_distrib : ((\u2203 x h, P x h) \u2192 b) \u2194 (\u2200 x h, P x h \u2192 b) :=\nby simp\n\ntheorem not_bex : (\u00ac \u2203 x h, P x h) \u2194 \u2200 x h, \u00ac P x h :=\nbex_imp_distrib\n\ntheorem not_ball_of_bex_not : (\u2203 x h, \u00ac P x h) \u2192 \u00ac \u2200 x h, P x h\n| \u27e8x, h, hp\u27e9 al := hp $ al x h\n\n-- See Note [decidable namespace]\nprotected theorem decidable.not_ball [decidable (\u2203 x h, \u00ac P x h)] [\u2200 x h, decidable (P x h)] :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) :=\n\u27e8not.decidable_imp_symm $ \u03bb nx x h, nx.decidable_imp_symm $ \u03bb h', \u27e8x, h, h'\u27e9,\n not_ball_of_bex_not\u27e9\n\ntheorem not_ball : (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) := decidable.not_ball\n\ntheorem ball_true_iff (p : \u03b1 \u2192 Prop) : (\u2200 x, p x \u2192 true) \u2194 true :=\niff_true_intro (\u03bb h hrx, trivial)\n\ntheorem ball_and_distrib : (\u2200 x h, P x h \u2227 Q x h) \u2194 (\u2200 x h, P x h) \u2227 (\u2200 x h, Q x h) :=\niff.trans (forall_congr $ \u03bb x, forall_and_distrib) forall_and_distrib\n\ntheorem bex_or_distrib : (\u2203 x h, P x h \u2228 Q x h) \u2194 (\u2203 x h, P x h) \u2228 (\u2203 x h, Q x h) :=\niff.trans (exists_congr $ \u03bb x, exists_or_distrib) exists_or_distrib\n\ntheorem ball_or_left_distrib : (\u2200 x, p x \u2228 q x \u2192 r x) \u2194 (\u2200 x, p x \u2192 r x) \u2227 (\u2200 x, q x \u2192 r x) :=\niff.trans (forall_congr $ \u03bb x, or_imp_distrib) forall_and_distrib\n\ntheorem bex_or_left_distrib :\n  (\u2203 x (_ : p x \u2228 q x), r x) \u2194 (\u2203 x (_ : p x), r x) \u2228 (\u2203 x (_ : q x), r x) :=\nby simp only [exists_prop]; exact\niff.trans (exists_congr $ \u03bb x, or_and_distrib_right) exists_or_distrib\n\nend bounded_quantifiers\n\nnamespace classical\nlocal attribute [instance] prop_decidable\n\ntheorem not_ball {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} {P : \u03a0 (x : \u03b1), p x \u2192 Prop} :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) := _root_.not_ball\n\nend classical\n\nlemma ite_eq_iff {\u03b1} {p : Prop} [decidable p] {a b c : \u03b1} :\n  (if p then a else b) = c \u2194 p \u2227 a = c \u2228 \u00acp \u2227 b = c :=\nby by_cases p; simp *\n\n@[simp] lemma ite_eq_left_iff {\u03b1} {p : Prop} [decidable p] {a b : \u03b1} :\n  (if p then a else b) = a \u2194 (\u00acp \u2192 b = a) :=\nby by_cases p; simp *\n\n@[simp] lemma ite_eq_right_iff {\u03b1} {p : Prop} [decidable p] {a b : \u03b1} :\n  (if p then a else b) = b \u2194 (p \u2192 a = b) :=\nby by_cases p; simp *\n\n/-! ### Declarations about `nonempty` -/\n\nsection nonempty\nuniverse variables u v w\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : \u03b1 \u2192 Type w}\n\nattribute [simp] nonempty_of_inhabited\n\n@[priority 20]\ninstance has_zero.nonempty [has_zero \u03b1] : nonempty \u03b1 := \u27e80\u27e9\n@[priority 20]\ninstance has_one.nonempty [has_one \u03b1] : nonempty \u03b1 := \u27e81\u27e9\n\nlemma exists_true_iff_nonempty {\u03b1 : Sort*} : (\u2203a:\u03b1, true) \u2194 nonempty \u03b1 :=\niff.intro (\u03bb\u27e8a, _\u27e9, \u27e8a\u27e9) (\u03bb\u27e8a\u27e9, \u27e8a, trivial\u27e9)\n\n@[simp] lemma nonempty_Prop {p : Prop} : nonempty p \u2194 p :=\niff.intro (assume \u27e8h\u27e9, h) (assume h, \u27e8h\u27e9)\n\nlemma not_nonempty_iff_imp_false : \u00ac nonempty \u03b1 \u2194 \u03b1 \u2192 false :=\n\u27e8\u03bb h a, h \u27e8a\u27e9, \u03bb h \u27e8a\u27e9, h a\u27e9\n\n@[simp] lemma nonempty_sigma : nonempty (\u03a3a:\u03b1, \u03b3 a) \u2194 (\u2203a:\u03b1, nonempty (\u03b3 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_subtype {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} : nonempty (subtype p) \u2194 (\u2203a:\u03b1, p a) :=\niff.intro (assume \u27e8\u27e8a, h\u27e9\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a, h\u27e9\u27e9)\n\n@[simp] lemma nonempty_prod : nonempty (\u03b1 \u00d7 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_pprod {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (pprod \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_sum : nonempty (\u03b1 \u2295 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with sum.inl a := or.inl \u27e8a\u27e9 | sum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8sum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8sum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psum {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (psum \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with psum.inl a := or.inl \u27e8a\u27e9 | psum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8psum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8psum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psigma {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (psigma \u03b2) \u2194 (\u2203a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_empty : \u00ac nonempty empty :=\nassume \u27e8h\u27e9, h.elim\n\n@[simp] lemma nonempty_ulift : nonempty (ulift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty_plift {\u03b1 : Sort u} : nonempty (plift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty.forall {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2200h:nonempty \u03b1, p h) \u2194 (\u2200a, p \u27e8a\u27e9) :=\niff.intro (assume h a, h _) (assume h \u27e8a\u27e9, h _)\n\n@[simp] lemma nonempty.exists {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2203h:nonempty \u03b1, p h) \u2194 (\u2203a, p \u27e8a\u27e9) :=\niff.intro (assume \u27e8\u27e8a\u27e9, h\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a\u27e9, h\u27e9)\n\nlemma classical.nonempty_pi {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (\u03a0a:\u03b1, \u03b2 a) \u2194 (\u2200a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8f\u27e9 a, \u27e8f a\u27e9) (assume f, \u27e8assume a, classical.choice $ f a\u27e9)\n\n/-- Using `classical.choice`, lifts a (`Prop`-valued) `nonempty` instance to a (`Type`-valued)\n  `inhabited` instance. `classical.inhabited_of_nonempty` already exists, in\n  `core/init/classical.lean`, but the assumption is not a type class argument,\n  which makes it unsuitable for some applications. -/\nnoncomputable def classical.inhabited_of_nonempty' {\u03b1 : Sort u} [h : nonempty \u03b1] : inhabited \u03b1 :=\n\u27e8classical.choice h\u27e9\n\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible] protected noncomputable def nonempty.some {\u03b1 : Sort u} (h : nonempty \u03b1) : \u03b1 :=\nclassical.choice h\n\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible] protected noncomputable def classical.arbitrary (\u03b1 : Sort u) [h : nonempty \u03b1] : \u03b1 :=\nclassical.choice h\n\n/-- Given `f : \u03b1 \u2192 \u03b2`, if `\u03b1` is nonempty then `\u03b2` is also nonempty.\n  `nonempty` cannot be a `functor`, because `functor` is restricted to `Type`. -/\nlemma nonempty.map {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) : nonempty \u03b1 \u2192 nonempty \u03b2\n| \u27e8h\u27e9 := \u27e8f h\u27e9\n\nprotected lemma nonempty.map2 {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : nonempty \u03b1 \u2192 nonempty \u03b2 \u2192 nonempty \u03b3\n| \u27e8x\u27e9 \u27e8y\u27e9 := \u27e8f x y\u27e9\n\nprotected lemma nonempty.congr {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  nonempty \u03b1 \u2194 nonempty \u03b2 :=\n\u27e8nonempty.map f, nonempty.map g\u27e9\n\nlemma nonempty.elim_to_inhabited {\u03b1 : Sort*} [h : nonempty \u03b1] {p : Prop}\n  (f : inhabited \u03b1 \u2192 p) : p :=\nh.elim $ f \u2218 inhabited.mk\n\ninstance {\u03b1 \u03b2} [h : nonempty \u03b1] [h2 : nonempty \u03b2] : nonempty (\u03b1 \u00d7 \u03b2) :=\nh.elim $ \u03bb g, h2.elim $ \u03bb g2, \u27e8\u27e8g, g2\u27e9\u27e9\n\nend nonempty\n\nsection ite\n\n/-- A `dite` whose results do not actually depend on the condition may be reduced to an `ite`. -/\n@[simp]\nlemma dite_eq_ite (P : Prop) [decidable P] {\u03b1 : Sort*} (x y : \u03b1) :\n  dite P (\u03bb h, x) (\u03bb h, y) = ite P x y := rfl\n\n/-- A function applied to a `dite` is a `dite` of that function applied to each of the branches. -/\nlemma apply_dite {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) (P : Prop) [decidable P] (x : P \u2192 \u03b1) (y : \u00acP \u2192 \u03b1) :\n  f (dite P x y) = dite P (\u03bb h, f (x h)) (\u03bb h, f (y h)) :=\nby { by_cases h : P; simp [h] }\n\n/-- A function applied to a `ite` is a `ite` of that function applied to each of the branches. -/\nlemma apply_ite {\u03b1 \u03b2 : Sort*} (f : \u03b1 \u2192 \u03b2) (P : Prop) [decidable P] (x y : \u03b1) :\n  f (ite P x y) = ite P (f x) (f y) :=\napply_dite f P (\u03bb _, x) (\u03bb _, y)\n\n/-- A two-argument function applied to two `dite`s is a `dite` of that two-argument function\napplied to each of the branches. -/\nlemma apply_dite2 {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (P : Prop) [decidable P] (a : P \u2192 \u03b1)\n  (b : \u00acP \u2192 \u03b1) (c : P \u2192 \u03b2) (d : \u00acP \u2192 \u03b2) :\n  f (dite P a b) (dite P c d) = dite P (\u03bb h, f (a h) (c h)) (\u03bb h, f (b h) (d h)) :=\nby { by_cases h : P; simp [h] }\n\n/-- A two-argument function applied to two `ite`s is a `ite` of that two-argument function\napplied to each of the branches. -/\nlemma apply_ite2 {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (P : Prop) [decidable P] (a b : \u03b1) (c d : \u03b2) :\n  f (ite P a b) (ite P c d) = ite P (f a c) (f b d) :=\napply_dite2 f P (\u03bb _, a) (\u03bb _, b) (\u03bb _, c) (\u03bb _, d)\n\n/-- A 'dite' producing a `Pi` type `\u03a0 a, \u03b2 a`, applied to a value `x : \u03b1`\nis a `dite` that applies either branch to `x`. -/\nlemma dite_apply {\u03b1 : Sort*} {\u03b2 : \u03b1 \u2192 Sort*} (P : Prop) [decidable P]\n  (f : P \u2192 \u03a0 a, \u03b2 a) (g : \u00ac P \u2192 \u03a0 a, \u03b2 a) (x : \u03b1) :\n  (dite P f g) x = dite P (\u03bb h, f h x) (\u03bb h, g h x) :=\nby { by_cases h : P; simp [h] }\n\n/-- A 'ite' producing a `Pi` type `\u03a0 a, \u03b2 a`, applied to a value `x : \u03b1`\nis a `ite` that applies either branch to `x` -/\nlemma ite_apply {\u03b1 : Sort*} {\u03b2 : \u03b1 \u2192 Sort*} (P : Prop) [decidable P]\n  (f g : \u03a0 a, \u03b2 a) (x : \u03b1) :\n  (ite P f g) x = ite P (f x) (g x) :=\ndite_apply P (\u03bb _, f) (\u03bb _, g) x\n\n/-- Negation of the condition `P : Prop` in a `dite` is the same as swapping the branches. -/\n@[simp] lemma dite_not {\u03b1 : Sort*} (P : Prop) [decidable P] (x : \u00ac P \u2192 \u03b1) (y : \u00ac\u00ac P \u2192 \u03b1) :\n  dite (\u00ac P) x y = dite P (\u03bb h, y (not_not_intro h)) x :=\nby { by_cases h : P; simp [h] }\n\n/-- Negation of the condition `P : Prop` in a `ite` is the same as swapping the branches. -/\n@[simp] lemma ite_not {\u03b1 : Sort*} (P : Prop) [decidable P] (x y : \u03b1) :\n  ite (\u00ac P) x y = ite P y x :=\ndite_not P (\u03bb _, x) (\u03bb _, y)\n\nlemma ite_and {\u03b1} {p q : Prop} [decidable p] [decidable q] {x y : \u03b1} :\n  ite (p \u2227 q) x y = ite p (ite q x y) y :=\nby { by_cases hp : p; by_cases hq : q; simp [hp, hq] }\n\n\nend ite\n-/\n", "meta": {"author": "kbuzzard", "repo": "mathlib4_experiments", "sha": "87cb879b4d602c8ecfd9283b7c0b06015abdbab1", "save_path": "github-repos/lean/kbuzzard-mathlib4_experiments", "path": "github-repos/lean/kbuzzard-mathlib4_experiments/mathlib4_experiments-87cb879b4d602c8ecfd9283b7c0b06015abdbab1/mathlib4_experiments/Logic/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6959583124210896, "lm_q1q2_score": 0.3939253045541359}}
{"text": "import game.order.level08\nimport game.order.level02\nopen real\n\nnamespace xena -- hide\n\n/-\nHint: use max_comm\n-/\n\nlemma abs_eq_abs_neg (a : \u211d) : abs a = abs (-a) :=\nbegin\n  rw abs,\n  rw abs,\n  simp,\n  exact max_comm a (-a),\n  \n\nend\n\n\n\nend xena -- hide", "meta": {"author": "DavidTalone", "repo": "Real-Number-Game", "sha": "b159ae1219209c3da97566b64b8986b146ec9053", "save_path": "github-repos/lean/DavidTalone-Real-Number-Game", "path": "github-repos/lean/DavidTalone-Real-Number-Game/Real-Number-Game-b159ae1219209c3da97566b64b8986b146ec9053/src/game/order/lessdumb.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7154240079185318, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3939177282406857}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nThe Kleisli construction on the Type category\n\nTODO: generalise this to work with category_theory.monad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.category.default\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\ndef Kleisli (m : Type u \u2192 Type v) [Monad m] :=\n  Type u\n\ndef Kleisli.mk (m : Type u \u2192 Type v) [Monad m] (\u03b1 : Type u) : Kleisli m :=\n  \u03b1\n\nprotected instance Kleisli.category_struct {m : Type u_1 \u2192 Type u_2} [Monad m] : category_struct (Kleisli m) :=\n  category_struct.mk (fun (\u03b1 : Kleisli m) (x : \u03b1) => pure x) fun (X Y Z : Kleisli m) (f : X \u27f6 Y) (g : Y \u27f6 Z) => f >=> g\n\nprotected instance Kleisli.category {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m] : category (Kleisli m) :=\n  category.mk\n\n@[simp] theorem Kleisli.id_def {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m] (\u03b1 : Kleisli m) : \ud835\udfd9 = pure :=\n  rfl\n\ntheorem Kleisli.comp_def {m : Type u_1 \u2192 Type u_2} [Monad m] [is_lawful_monad m] (\u03b1 : Kleisli m) (\u03b2 : Kleisli m) (\u03b3 : Kleisli m) (xs : \u03b1 \u27f6 \u03b2) (ys : \u03b2 \u27f6 \u03b3) (a : \u03b1) : category_struct.comp xs ys a = xs a >>= ys :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/Kleisli.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3938963228916901}}
{"text": "import Mathlib.Tactic.IrreducibleDef\n\n/-- Add two natural numbers, but not during unification. -/\nirreducible_def frobnicate (a b : Nat) :=\n  a + b\n\nexample : frobnicate a 0 = a := by\n  simp [frobnicate_def]\n\nirreducible_def justAsArbitrary [Inhabited \u03b1] : \u03b1 :=\n  default\n\nirreducible_def withoutType := 42\n\nirreducible_def withEquations : Nat \u2192 Nat\n  | 0 => 42\n  | _n+1 => 314\n\nirreducible_def withUniv.{u, v} := (Type v, Type u)\nexample : withUniv.{u, v} = (Type v, Type u) := by rw [withUniv_def]\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/irreducibleDef.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3938963151176444}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.adjunction.default\nimport Mathlib.category_theory.elements\nimport Mathlib.category_theory.limits.functor_category\nimport Mathlib.category_theory.limits.preserves.limits\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.limits.types\nimport Mathlib.PostPort\n\nuniverses u\u2081 u\u2082 \n\nnamespace Mathlib\n\n/-!\n# Colimit of representables\n\nThis file constructs an adjunction `yoneda_adjunction` between `(C\u1d52\u1d56 \u2964 Type u)` and `\u2130` given a\nfunctor `A : C \u2964 \u2130`, where the right adjoint sends `(E : \u2130)` to `c \u21a6 (A.obj c \u27f6 E)` (provided `\u2130`\nhas colimits).\n\nThis adjunction is used to show that every presheaf is a colimit of representables.\n\nFurther, the left adjoint `colimit_adj.extend_along_yoneda : (C\u1d52\u1d56 \u2964 Type u) \u2964 \u2130` satisfies\n`yoneda \u22d9 L \u2245 A`, that is, an extension of `A : C \u2964 \u2130` to `(C\u1d52\u1d56 \u2964 Type u) \u2964 \u2130` through\n`yoneda : C \u2964 C\u1d52\u1d56 \u2964 Type u`. It is the left Kan extension of `A` along the yoneda embedding,\nsometimes known as the Yoneda extension.\n\n`unique_extension_along_yoneda` shows `extend_along_yoneda` is unique amongst cocontinuous functors\nwith this property, establishing the presheaf category as the free cocompletion of a small category.\n\n## Tags\ncolimit, representable, presheaf, free cocompletion\n\n## References\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n* https://ncatlab.org/nlab/show/Yoneda+extension\n-/\n\nnamespace category_theory\n\n\nnamespace colimit_adj\n\n\n/--\nThe functor taking `(E : \u2130) (c : C\u1d52\u1d56)` to the homset `(A.obj C \u27f6 E)`. It is shown in `L_adjunction`\nthat this functor has a left adjoint (provided `E` has colimits) given by taking colimits over\ncategories of elements.\nIn the case where `\u2130 = C\u1d52\u1d56 \u2964 Type u` and `A = yoneda`, this functor is isomorphic to the identity.\n\nDefined as in [MM92], Chapter I, Section 5, Theorem 2.\n-/\ndef restricted_yoneda {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130] (A : C \u2964 \u2130) :\n    \u2130 \u2964 C\u1d52\u1d56 \u2964 Type u\u2081 :=\n  yoneda \u22d9 functor.obj (whiskering_left (C\u1d52\u1d56) (\u2130\u1d52\u1d56) (Type u\u2081)) (functor.op A)\n\n/--\nThe functor `restricted_yoneda` is isomorphic to the identity functor when evaluated at the yoneda\nembedding.\n-/\ndef restricted_yoneda_yoneda {C : Type u\u2081} [small_category C] : restricted_yoneda yoneda \u2245 \ud835\udfed :=\n  nat_iso.of_components\n    (fun (P : C\u1d52\u1d56 \u2964 Type u\u2081) =>\n      nat_iso.of_components (fun (X : C\u1d52\u1d56) => yoneda_sections_small (opposite.unop X) P) sorry)\n    sorry\n\n/--\n(Implementation). The equivalence of homsets which helps construct the left adjoint to\n`colimit_adj.restricted_yoneda`.\nIt is shown in `restrict_yoneda_hom_equiv_natural` that this is a natural bijection.\n-/\ndef restrict_yoneda_hom_equiv {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130]\n    (A : C \u2964 \u2130) (P : C\u1d52\u1d56 \u2964 Type u\u2081) (E : \u2130)\n    {c : limits.cocone (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A)} (t : limits.is_colimit c) :\n    (limits.cocone.X c \u27f6 E) \u2243 (P \u27f6 functor.obj (restricted_yoneda A) E) :=\n  equiv.trans (iso.to_equiv (limits.is_colimit.hom_iso' t E))\n    (equiv.mk\n      (fun\n        (k :\n        Subtype\n          fun\n            (p :\n            (j : functor.elements P\u1d52\u1d56) \u2192\n              functor.obj (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A) j \u27f6 E) =>\n            \u2200 {j j' : functor.elements P\u1d52\u1d56} (f : j \u27f6 j'),\n              functor.map (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A) f \u226b p j' = p j) =>\n        nat_trans.mk\n          fun (c : C\u1d52\u1d56) (p : functor.obj P c) => subtype.val k (opposite.op (sigma.mk c p)))\n      (fun (\u03c4 : P \u27f6 functor.obj (restricted_yoneda A) E) =>\n        { val :=\n            fun (p : functor.elements P\u1d52\u1d56) =>\n              nat_trans.app \u03c4 (sigma.fst (opposite.unop p)) (sigma.snd (opposite.unop p)),\n          property := sorry })\n      sorry sorry)\n\n/--\n(Implementation). Show that the bijection in `restrict_yoneda_hom_equiv` is natural (on the right).\n-/\ntheorem restrict_yoneda_hom_equiv_natural {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082}\n    [category \u2130] (A : C \u2964 \u2130) (P : C\u1d52\u1d56 \u2964 Type u\u2081) (E\u2081 : \u2130) (E\u2082 : \u2130) (g : E\u2081 \u27f6 E\u2082)\n    {c : limits.cocone (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A)} (t : limits.is_colimit c)\n    (k : limits.cocone.X c \u27f6 E\u2081) :\n    coe_fn (restrict_yoneda_hom_equiv A P E\u2082 t) (k \u226b g) =\n        coe_fn (restrict_yoneda_hom_equiv A P E\u2081 t) k \u226b functor.map (restricted_yoneda A) g :=\n  sorry\n\n/--\nThe left adjoint to the functor `restricted_yoneda` (shown in `yoneda_adjunction`). It is also an\nextension of `A` along the yoneda embedding (shown in `is_extension_along_yoneda`), in particular\nit is the left Kan extension of `A` through the yoneda embedding.\n-/\ndef extend_along_yoneda {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130] (A : C \u2964 \u2130)\n    [limits.has_colimits \u2130] : (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 \u2130 :=\n  adjunction.left_adjoint_of_equiv\n    (fun (P : C\u1d52\u1d56 \u2964 Type u\u2081) (E : \u2130) =>\n      restrict_yoneda_hom_equiv A P E\n        (limits.colimit.is_colimit (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A)))\n    sorry\n\n@[simp] theorem extend_along_yoneda_obj {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130]\n    (A : C \u2964 \u2130) [limits.has_colimits \u2130] (P : C\u1d52\u1d56 \u2964 Type u\u2081) :\n    functor.obj (extend_along_yoneda A) P =\n        limits.colimit (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A) :=\n  rfl\n\n/--\nShow `extend_along_yoneda` is left adjoint to `restricted_yoneda`.\n\nThe construction of [MM92], Chapter I, Section 5, Theorem 2.\n-/\ndef yoneda_adjunction {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130] (A : C \u2964 \u2130)\n    [limits.has_colimits \u2130] : extend_along_yoneda A \u22a3 restricted_yoneda A :=\n  adjunction.adjunction_of_equiv_left\n    (fun (P : C\u1d52\u1d56 \u2964 Type u\u2081) (E : \u2130) =>\n      restrict_yoneda_hom_equiv A P E\n        (limits.colimit.is_colimit (functor.left_op (category_of_elements.\u03c0 P) \u22d9 A)))\n    (extend_along_yoneda._proof_4 A)\n\n/--\nThe initial object in the category of elements for a representable functor. In `is_initial` it is\nshown that this is initial.\n-/\ndef elements.initial {C : Type u\u2081} [small_category C] (A : C) :\n    functor.elements (functor.obj yoneda A) :=\n  sigma.mk (opposite.op A) \ud835\udfd9\n\n/--\nShow that `elements.initial A` is initial in the category of elements for the `yoneda` functor.\n-/\ndef is_initial {C : Type u\u2081} [small_category C] (A : C) : limits.is_initial (elements.initial A) :=\n  limits.is_colimit.mk\n    fun (s : limits.cocone (functor.empty (functor.elements (functor.obj yoneda A)))) =>\n      { val := has_hom.hom.op (sigma.snd (limits.cocone.X s)), property := sorry }\n\n/--\n`extend_along_yoneda A` is an extension of `A` to the presheaf category along the yoneda embedding.\n`unique_extension_along_yoneda` shows it is unique among functors preserving colimits with this\nproperty (up to isomorphism).\n\nThe first part of [MM92], Chapter I, Section 5, Corollary 4.\nSee Property 1 of https://ncatlab.org/nlab/show/Yoneda+extension#properties.\n-/\ndef is_extension_along_yoneda {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130]\n    (A : C \u2964 \u2130) [limits.has_colimits \u2130] : yoneda \u22d9 extend_along_yoneda A \u2245 A :=\n  nat_iso.of_components\n    (fun (X : C) =>\n      limits.is_colimit.cocone_point_unique_up_to_iso\n        (limits.colimit.is_colimit\n          (functor.left_op (category_of_elements.\u03c0 (functor.obj yoneda X)) \u22d9 A))\n        (limits.colimit_of_diagram_terminal (limits.terminal_op_of_initial (is_initial X))\n          (functor.left_op (category_of_elements.\u03c0 (functor.obj yoneda X)) \u22d9 A)))\n    sorry\n\n/-- See Property 2 of https://ncatlab.org/nlab/show/Yoneda+extension#properties. -/\nprotected instance extend_along_yoneda.category_theory.limits.preserves_colimits {C : Type u\u2081}\n    [small_category C] {\u2130 : Type u\u2082} [category \u2130] (A : C \u2964 \u2130) [limits.has_colimits \u2130] :\n    limits.preserves_colimits (extend_along_yoneda A) :=\n  adjunction.left_adjoint_preserves_colimits (yoneda_adjunction A)\n\nend colimit_adj\n\n\n/--\nSince `extend_along_yoneda A` is adjoint to `restricted_yoneda A`, if we use `A = yoneda`\nthen `restricted_yoneda A` is isomorphic to the identity, and so `extend_along_yoneda A` is as well.\n-/\ndef extend_along_yoneda_yoneda {C : Type u\u2081} [small_category C] :\n    colimit_adj.extend_along_yoneda yoneda \u2245 \ud835\udfed :=\n  adjunction.nat_iso_of_right_adjoint_nat_iso (colimit_adj.yoneda_adjunction yoneda) adjunction.id\n    colimit_adj.restricted_yoneda_yoneda\n\n/--\nA functor to the presheaf category in which everything in the image is representable (witnessed\nby the fact that it factors through the yoneda embedding).\n`cocone_of_representable` gives a cocone for this functor which is a colimit and has point `P`.\n-/\n-- Maybe this should be reducible or an abbreviation?\n\ndef functor_to_representables {C : Type u\u2081} [small_category C] (P : C\u1d52\u1d56 \u2964 Type u\u2081) :\n    functor.elements P\u1d52\u1d56 \u2964 C\u1d52\u1d56 \u2964 Type u\u2081 :=\n  functor.left_op (category_of_elements.\u03c0 P) \u22d9 yoneda\n\n/--\nThis is a cocone with point `P` for the functor `functor_to_representables P`. It is shown in\n`colimit_of_representable P` that this cocone is a colimit: that is, we have exhibited an arbitrary\npresheaf `P` as a colimit of representables.\n\nThe construction of [MM92], Chapter I, Section 5, Corollary 3.\n-/\ndef cocone_of_representable {C : Type u\u2081} [small_category C] (P : C\u1d52\u1d56 \u2964 Type u\u2081) :\n    limits.cocone (functor_to_representables P) :=\n  limits.cocone.extend (limits.colimit.cocone (functor_to_representables P))\n    (nat_trans.app (iso.hom extend_along_yoneda_yoneda) P)\n\n@[simp] theorem cocone_of_representable_X {C : Type u\u2081} [small_category C] (P : C\u1d52\u1d56 \u2964 Type u\u2081) :\n    limits.cocone.X (cocone_of_representable P) = P :=\n  rfl\n\n/-- An explicit formula for the legs of the cocone `cocone_of_representable`. -/\n-- Marking this as a simp lemma seems to make things more awkward.\n\ntheorem cocone_of_representable_\u03b9_app {C : Type u\u2081} [small_category C] (P : C\u1d52\u1d56 \u2964 Type u\u2081)\n    (j : functor.elements P\u1d52\u1d56) :\n    nat_trans.app (limits.cocone.\u03b9 (cocone_of_representable P)) j =\n        iso.inv\n          (yoneda_sections_small (functor.obj (functor.left_op (category_of_elements.\u03c0 P)) j)\n            (functor.obj\n              (functor.obj (functor.const (functor.elements P\u1d52\u1d56))\n                (limits.cocone.X (cocone_of_representable P)))\n              j))\n          (sigma.snd (opposite.unop j)) :=\n  sorry\n\n/-- The legs of the cocone `cocone_of_representable` are natural in the choice of presheaf. -/\ntheorem cocone_of_representable_naturality {C : Type u\u2081} [small_category C] {P\u2081 : C\u1d52\u1d56 \u2964 Type u\u2081}\n    {P\u2082 : C\u1d52\u1d56 \u2964 Type u\u2081} (\u03b1 : P\u2081 \u27f6 P\u2082) (j : functor.elements P\u2081\u1d52\u1d56) :\n    nat_trans.app (limits.cocone.\u03b9 (cocone_of_representable P\u2081)) j \u226b \u03b1 =\n        nat_trans.app (limits.cocone.\u03b9 (cocone_of_representable P\u2082))\n          (functor.obj (functor.op (category_of_elements.map \u03b1)) j) :=\n  sorry\n\n/--\nThe cocone with point `P` given by `the_cocone` is a colimit: that is, we have exhibited an\narbitrary presheaf `P` as a colimit of representables.\n\nThe result of [MM92], Chapter I, Section 5, Corollary 3.\n-/\ndef colimit_of_representable {C : Type u\u2081} [small_category C] (P : C\u1d52\u1d56 \u2964 Type u\u2081) :\n    limits.is_colimit (cocone_of_representable P) :=\n  limits.is_colimit.of_point_iso (limits.colimit.is_colimit (functor_to_representables P))\n\n/--\nGiven two functors L\u2081 and L\u2082 which preserve colimits, if they agree when restricted to the\nrepresentable presheaves then they agree everywhere.\n-/\ndef nat_iso_of_nat_iso_on_representables {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130]\n    (L\u2081 : (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 \u2130) (L\u2082 : (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 \u2130) [limits.preserves_colimits L\u2081]\n    [limits.preserves_colimits L\u2082] (h : yoneda \u22d9 L\u2081 \u2245 yoneda \u22d9 L\u2082) : L\u2081 \u2245 L\u2082 :=\n  nat_iso.of_components\n    (fun (P : C\u1d52\u1d56 \u2964 Type u\u2081) =>\n      limits.is_colimit.cocone_points_iso_of_nat_iso\n        (limits.is_colimit_of_preserves L\u2081 (colimit_of_representable P))\n        (limits.is_colimit_of_preserves L\u2082 (colimit_of_representable P))\n        (functor.associator (functor.left_op (category_of_elements.\u03c0 P)) yoneda L\u2081 \u226a\u226b\n          iso_whisker_left (functor.left_op (category_of_elements.\u03c0 P)) h))\n    sorry\n\n/--\nShow that `extend_along_yoneda` is the unique colimit-preserving functor which extends `A` to\nthe presheaf category.\n\nThe second part of [MM92], Chapter I, Section 5, Corollary 4.\nSee Property 3 of https://ncatlab.org/nlab/show/Yoneda+extension#properties.\n-/\ndef unique_extension_along_yoneda {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082} [category \u2130]\n    (A : C \u2964 \u2130) [limits.has_colimits \u2130] (L : (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 \u2130) (hL : yoneda \u22d9 L \u2245 A)\n    [limits.preserves_colimits L] : L \u2245 colimit_adj.extend_along_yoneda A :=\n  nat_iso_of_nat_iso_on_representables L (colimit_adj.extend_along_yoneda A)\n    (hL \u226a\u226b iso.symm (colimit_adj.is_extension_along_yoneda A))\n\n/--\nIf `L` preserves colimits and `\u2130` has them, then it is a left adjoint. This is a special case of\n`is_left_adjoint_of_preserves_colimits` used to prove that.\n-/\ndef is_left_adjoint_of_preserves_colimits_aux {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082}\n    [category \u2130] [limits.has_colimits \u2130] (L : (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 \u2130) [limits.preserves_colimits L] :\n    is_left_adjoint L :=\n  is_left_adjoint.mk (colimit_adj.restricted_yoneda (yoneda \u22d9 L))\n    (adjunction.of_nat_iso_left (colimit_adj.yoneda_adjunction (yoneda \u22d9 L))\n      (iso.symm (unique_extension_along_yoneda (yoneda \u22d9 L) L (iso.refl (yoneda \u22d9 L)))))\n\n/--\nIf `L` preserves colimits and `\u2130` has them, then it is a left adjoint. Note this is a (partial)\nconverse to `left_adjoint_preserves_colimits`.\n-/\ndef is_left_adjoint_of_preserves_colimits {C : Type u\u2081} [small_category C] {\u2130 : Type u\u2082}\n    [category \u2130] [limits.has_colimits \u2130] (L : (C \u2964 Type u\u2081) \u2964 \u2130) [limits.preserves_colimits L] :\n    is_left_adjoint L :=\n  let e : C\u1d52\u1d56\u1d52\u1d56 \u2964 Type u\u2081 \u224c C \u2964 Type u\u2081 := equivalence.congr_left (op_op_equivalence C);\n  let t : is_left_adjoint (equivalence.functor e \u22d9 L) :=\n    is_left_adjoint_of_preserves_colimits_aux (equivalence.functor e \u22d9 L);\n  adjunction.left_adjoint_of_nat_iso (equivalence.inv_fun_id_assoc e L)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/presheaf_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3938676252015634}}
{"text": "import tactic\nimport tactic.induction\n\nimport .base .game\n\nnoncomputable theory\nopen_locale classical\n\nlemma induct_s_at {P : State \u2192 Prop} {pw n : \u2115} {g : Game pw}\n  (h\u2081 : P g.s)\n  (h\u2082 : \u2200 {s : State} {ma},\n  P s \u2192 A_move_valid pw s.board ma \u2192 P (apply_A_move s ma))\n  (h\u2083 : \u2200 {s : State} {md}, P s \u2192 D_move_valid s.board md \u2192 P (apply_D_move s md))\n  (hf : \u2200 {s : State}, P s \u2192 P s.finish) :\n  P (g.play n).s :=\nbegin\n  induction n with n ih,\n  { assumption },\n  { rw [play_at_succ'], let g\u2081 := _, change g.play n with g\u2081 at ih \u22a2,\n    rw Game.play_move, split_ifs with h\u2084, swap, { assumption },\n    let a := g\u2081.a, let d := g\u2081.d,\n    have h\u2085 : \u2203 (s' : State), play_D_move_at g\u2081 h\u2084 = init_game a d s' \u2227 P s',\n    { let s' := apply_D_move g\u2081.s (d.f g\u2081.s h\u2084).m, use s',\n      exact \u27e8rfl, h\u2083 ih (d.f _ _).h\u27e9 },\n    rcases h\u2085 with \u27e8s', h\u2085, h\u2086\u27e9, rw h\u2085, clear h\u2085,\n    rw play_A_move_at, split_ifs with h\u2085,\n    { exact h\u2082 h\u2086 (a.f s' _ _).h },\n    { exact hf h\u2086 }},\nend\n\nlemma induct_s {P : State \u2192 Prop} {pw : \u2115} {a : A pw} {d : D}\n  (h\u2081 : P state\u2080)\n  (h\u2082 : \u2200 {s : State} {ma},\n  P s \u2192 A_move_valid pw s.board ma \u2192 P (apply_A_move s ma))\n  (h\u2083 : \u2200 {s : State} {md}, P s \u2192 D_move_valid s.board md \u2192 P (apply_D_move s md))\n  (hf : \u2200 {s : State}, P s \u2192 P s.finish) :\n  all_s a d P :=\nby { intro n, apply induct_s_at; assumption }\n\nlemma induct_b {P : Board \u2192 Prop} {pw : \u2115} {a : A pw} {d : D}\n  (h\u2081 : P board\u2080)\n  (h\u2082 : \u2200 {b : Board} {ma}, P b \u2192 A_move_valid pw b ma \u2192 P (apply_A_move_b b ma))\n  (h\u2083 : \u2200 {b : Board} {md}, P b \u2192 D_move_valid b md \u2192 P (apply_D_move_b b md)) :\n  all_b a d P :=\nbegin\n  apply induct_s,\n  { exact h\u2081 },\n  { rintro s ma h\u2084 h\u2085, exact h\u2082 h\u2084 h\u2085 },\n  { rintro s ma h\u2084 h\u2085, exact h\u2083 h\u2084 h\u2085 },\n  { intro s, exact id },\nend\n\nlemma simulate_add {pw : \u2115} {a : A pw} {d : D} {n\u2081 n\u2082 : \u2115} :\n  simulate a d (n\u2081 + n\u2082) = (simulate a d n\u2081).play n\u2082 :=\nby { rw add_comm, apply function.iterate_add_apply }\n\nlemma not_play_act_of_not_act {pw n : \u2115} {g : Game pw}\n  (h : \u00acg.act) : \u00ac(g.play n).act :=\nby { apply @induct_s_at (\u03bb s, \u00acs.act); intros; assumption <|> exact not_false }\n\nlemma act_of_act_play {pw n : \u2115} {g : Game pw}\n  (h : (g.play n).act) : g.act :=\nby { contrapose h, exact not_play_act_of_not_act h }\n\nlemma play_eq_iff_states_eq {pw n : \u2115} {g : Game pw}\n  (h : (g.play n).s = g.s) : g.play n = g :=\nbegin\n  ext,\n  { exact play_at_players_eq.1 },\n  { exact play_at_players_eq.2 },\n  { exact h },\nend\n\nlemma play_eq_of_not_act' {pw n : \u2115} {g : Game pw}\n  (h : \u00acg.act) : g.play n = g :=\nbegin\n  rw play_eq_iff_states_eq, induction n with n ih,\n  { refl },\n  { rw play_at_succ',\n    let g\u2081 : Game pw := _, change g.play n with g\u2081 at ih \u22a2, have h\u2081 : \u00acg\u2081.act,\n    { change \u00acg\u2081.s.act, rwa ih },\n    rwa play_move_at_not_act h\u2081 },\nend\n\nlemma simulate_eq_of_not_act {pw n\u2081 n\u2082 : \u2115} {a : A pw} {d : D}\n  (h\u2081 : \u00ac(simulate a d n\u2081).act)\n  (h\u2082 : \u00ac(simulate a d n\u2082).act) :\n  simulate a d n\u2081 = simulate a d n\u2082 :=\nbegin\n  wlog h\u2083 : n\u2082 \u2264 n\u2081,\n  obtain \u27e8k, rfl\u27e9 := nat.exists_eq_add_of_le h\u2083,\n  rw simulate_add at h\u2081 \u22a2,\n  let g : Game pw := _, change simulate a d n\u2082 with g at h\u2081 h\u2082 \u22a2,\n  exact play_eq_of_not_act' h\u2082,\nend\n\nlemma play_move_len_le {pw : \u2115} {g : Game pw} :\n  g.play_move.s.len \u2264 g.s.len + 2 :=\nbegin\n  rw Game.play_move, split_ifs,\n  { have h\u2081 : (play_D_move_at g h).s.len = g.s.len + 1,\n    { rw hist_len_play_D_move_at },\n    rw play_A_move_at, split_ifs with h\u2082,\n    { rw [hist_len_play_A_move_at', h\u2081] },\n    { change (play_D_move_at g h).s.finish.len \u2264 _, rw [hist_len_finish, h\u2081],\n      apply add_le_add_left, dec_trivial }},\n  { exact le_add_right (le_refl _) },\nend\n\nlemma play_len_le {pw n : \u2115} {g : Game pw} :\n  (g.play n).s.len \u2264 g.s.len + n * 2 :=\nbegin\n  induction n with n ih,\n  { refl },\n  { rw play_at_succ', let g\u2081 : Game pw := _, change g.play n with g\u2081 at ih \u22a2,\n    transitivity, exact play_move_len_le, rw [nat.succ_mul, \u2190add_assoc],\n    apply add_le_add_right ih },\nend\n\nlemma simulate_len_le {pw n : \u2115} {a : A pw} {d : D} :\n  (simulate a d n).s.len \u2264 n * 2 :=\nbegin\n  change ((simulate a d 0).play n).s.len \u2264 n * 2 + (simulate a d 0).s.len,\n  rw add_comm (n * 2), exact play_len_le,\nend\n\nlemma exi_A_wins_of_invariant {P : State \u2192 Prop} {pw : \u2115} {d : D} {s\u2080 : State}\n  (h\u2080 : P s\u2080)\n  (hP : \u2200 (s : State), P s \u2192 s.act)\n  (hm : \u2200 (s s' : State) hs, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    \u2203 (ma : Valid_A_move pw s'.board), P (apply_A_move s' ma.m)) :\n  \u2203 (a : A pw), (init_game a d s\u2080).A_wins :=\nbegin\n  let a : A pw,\n  { refine \u27e8\u03bb s' hs' hvm, _\u27e9, refine (_ : \u2203 (ma : Valid_A_move pw s'.board),\n      \u2200 (s : State) hs, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n      P (apply_A_move s' ma.m)).some,\n    by_cases h\u2081 : \u2203 (s : State) hs, P s \u2227 s' = apply_D_move s (d.f s hs).m,\n    { rcases h\u2081 with \u27e8s, hs, h\u2081, h\u2082\u27e9, specialize hm s s' hs h\u2081 h\u2082,\n      cases hm with ma hm, use ma, intros, assumption },\n    { refine \u27e8\u27e8_, hvm.some_spec\u27e9, _\u27e9, rintro s\u2081 hs\u2081 h\u2082 h\u2083, push_neg at h\u2081,\n      specialize h\u2081 s\u2081 hs\u2081, push_neg at h\u2081, specialize h\u2081 h\u2082, contradiction }},\n  use a, rintro n, apply hP, induction n with n ih,\n  { assumption },\n  { rw play_at_succ', let g : Game pw := _,\n    change (init_game a d s\u2080).play n with g at ih \u22a2,\n    have hs := hP _ ih, rw play_move_at_act hs, let s := g.s,\n    let s' := apply_D_move s (d.f s hs).m,\n    have h\u2081 : play_D_move_at g hs = init_game a d s',\n    { ext,\n      { exact play_at_players_eq.1 },\n      { exact play_at_players_eq.2 },\n      { change apply_D_move _ _ = apply_D_move _ _, congr,\n        exact play_at_players_eq.2 }},\n    rw h\u2081, clear h\u2081, have hvm : A_has_valid_move pw s'.board,\n    { specialize hm s s' hs ih rfl, cases hm with ma hma, exact \u27e8_, ma.h\u27e9 },\n    rw [play_A_move_at, dif_pos], swap, { split; assumption },\n    change P (apply_A_move s' (a.f s' hs hvm).m),\n    generalize hma : a.f s' hs hvm = ma,\n    change Exists.some _ = _ at hma, generalize_proofs h\u2081 at hma,\n    have h\u2082 := h\u2081.some_spec s hs ih rfl, subst hma, assumption },\nend\n\nlemma play_eq_of_not_act {pw n k : \u2115} {a : A pw} {d : D} {s : State}\n  (h\u2081 : \u00ac((init_game a d s).play n).act)\n  (h\u2082 : \u00ac((init_game a d s).play k).act) :\n  (init_game a d s).play n = (init_game a d s).play k :=\nbegin\n  wlog h\u2083 : k \u2264 n, obtain \u27e8n, rfl\u27e9 := nat.exists_eq_add_of_le h\u2083,\n  rw play_add, exact play_eq_of_not_act' h\u2082,\nend\n\nlemma not_act_of_descend (f : State \u2192 \u2115) (P : State \u2192 Prop)\n  {pw n : \u2115} {a : A pw} {d : D} {s\u2080 : State}\n  (hp\u2080 : f s\u2080 < n)\n  (hp\u2081 : P s\u2080)\n  (hp\u2082 : \u2200 (s s' : State) hs hs' hvm, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    P (apply_A_move s' (a.f s' hs' hvm).m))\n  (hp\u2083 : \u2200 (s s' : State) hs hs' hvm, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    f (apply_A_move s' (a.f s' hs' hvm).m) < f s) :\n  \u00ac((init_game a d s\u2080).play n).act :=\nbegin\n  have h\u2080 : \u2200 (n : \u2115) (s : State), s = ((init_game a d s\u2080).play n).s \u2192\n    s.act \u2192 P s,\n  { clear' hp\u2080 n, rintro n _ rfl hs\u2081,\n    rw \u2190Game.act at hs\u2081, induction n with n ih,\n    { exact hp\u2081 },\n    { rw play_at_succ' at hs\u2081 \u22a2, let g : Game pw := _,\n      change (init_game a d s\u2080).play n with g at ih hs\u2081 \u22a2,\n      obtain \u27e8s', hs, hs', hvm, h\u2081, h\u2082\u27e9 :=\n        play_move_state_eq_of_act_play_move hs\u2081, rw h\u2082,\n      have ha : g.a = a := play_at_players_eq.1, rw ha at *, clear ha,\n      have hd : g.d = d := play_at_players_eq.2, rw hd at *, clear hd,\n      apply hp\u2082,\n      { exact ih hs },\n      { exact h\u2081 }}},\n  suffices h : \u2200 (n : \u2115) (s : State), s = ((init_game a d s\u2080).play n).s \u2192\n    s.act \u2192 f s + n \u2264 f s\u2080,\n  { apply mt (h n _ rfl), push_neg, apply nat.lt_add_left, exact hp\u2080 },\n  clear' n hp\u2080, rintro n _ rfl hs\u2081, induction n with n ih,\n  { refl },\n  { rw play_at_succ' at hs\u2081 \u22a2, let g : Game pw := _,\n    change (init_game a d s\u2080).play n with g at ih hs\u2081 \u22a2,\n    obtain \u27e8s', hs, hs', hvm, h\u2081, h\u2082\u27e9 :=\n      play_move_state_eq_of_act_play_move hs\u2081, rw h\u2082,\n    have ha : g.a = a := play_at_players_eq.1, rw ha at *, clear ha,\n    have hd : g.d = d := play_at_players_eq.2, rw hd at *, clear hd,\n    specialize ih hs,\n    have h : f (apply_A_move s' (a.f s' hs' hvm).m) < f g.s,\n    { apply hp\u2083,\n      { apply h\u2080,\n        { refl },\n        { exact hs } },\n      { exact h\u2081 }},\n    replace h : f (apply_A_move s' (a.f s' hs' hvm).m) + n < f g.s + n,\n    { exact add_lt_add_right h n },\n    rw \u2190nat.succ_le_iff at h, rw nat.add_succ, exact h.trans ih },\nend\n\nlemma not_act_of_descend_single_moves (f : State \u2192 \u2115) (P : State \u2192 Prop)\n  {pw n : \u2115} {a : A pw} {d : D} {s\u2080 : State}\n  (hp\u2080 : f s\u2080 < n)\n  (hp\u2081 : P s\u2080)\n  (hp\u2082 : \u2200 (s : State) hs hvm, P s \u2192 P (apply_A_move s (a.f s hs hvm).m))\n  (hp\u2083 : \u2200 (s : State) hs, P s \u2192 P (apply_D_move s (d.f s hs).m))\n  (hp\u2084 : \u2200 (s : State) hs hvm, P s \u2192 f (apply_A_move s (a.f s hs hvm).m) \u2264 f s)\n  (hp\u2085 : \u2200 (s : State) hs, P s \u2192 f (apply_D_move s (d.f s hs).m) < f s) :\n  \u00ac((init_game a d s\u2080).play n).act :=\nbegin\n  apply not_act_of_descend f P hp\u2080 hp\u2081,\n  { rintro s s' hs hs' hvm h\u2081 rfl, apply hp\u2082, apply hp\u2083, exact h\u2081 },\n  { rintro s s' hs hs' hvm h\u2081 h\u2082, let s\u2081 : State := _, change f s\u2081 < _,\n    have h\u2083 : f s' < f s,\n    { subst s', apply hp\u2085, exact h\u2081 },\n    have h\u2084 : f s\u2081 \u2264 f s',\n    { subst s', apply hp\u2084, apply hp\u2083, exact h\u2081 },\n    exact gt_of_gt_of_ge h\u2083 h\u2084 },\nend\n\nlemma not_act_of_descend_play_move' (f : State \u2192 \u2115) (P : State \u2192 Prop)\n  {pw n : \u2115} {a : A pw} {d : D} {s\u2080 : State}\n  (hp\u2080 : f s\u2080 < n)\n  (hp\u2081 : P s\u2080)\n  (hp\u2082 : \u2200 (g : Game pw), g.play_move.act \u2192 P g.s \u2192 P g.play_move.s)\n  (hp\u2083 : \u2200 (g : Game pw), g.play_move.act \u2192 P g.s \u2192 f g.play_move.s < f g.s) :\n  \u00ac((init_game a d s\u2080).play n).act :=\nbegin\n  have h : \u2200 (Q : State \u2192 State \u2192 Prop),\n    (\u2200 (g : Game pw), g.play_move.act \u2192 P g.s \u2192 Q g.s g.play_move.s) \u2192\n    \u2200 (s s' : State) hs hs' hvm, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    Q s (apply_A_move s' (a.f s' hs' hvm).m),\n  { rintro Q hQ, rintro s s' hs hs' hvm h\u2081 h\u2082,\n    have hs\u2081 : (init_game a d s).play_move.act,\n    { subst s', exact act_play_move_of_A_hvm hvm },\n    specialize hQ (init_game a d s) hs\u2081 h\u2081,\n    obtain \u27e8s', hs, hs', hvm, h\u2083, h\u2084\u27e9 := play_move_state_eq_of_act_play_move hs\u2081,\n    rw h\u2084 at hQ, subst_vars, exact hQ },\n  apply not_act_of_descend f P hp\u2080 hp\u2081,\n  { exact h (\u03bb (s\u2081 s\u2082 : State), P s\u2082) hp\u2082 },\n  { exact h (\u03bb (s\u2081 s\u2082 : State), f s\u2082 < f s\u2081) hp\u2083 },\nend\n\nlemma not_act_of_descend_play_move (f : State \u2192 \u2115) (P : State \u2192 Prop)\n  {pw n : \u2115} {a : A pw} {d : D} {s\u2080 : State}\n  (hp\u2080 : f s\u2080 < n)\n  (hp\u2081 : P s\u2080)\n  (hp\u2082 : \u2200 (s : State), (init_game a d s).play_move.act \u2192 P s \u2192\n    P (init_game a d s).play_move.s)\n  (hp\u2083 : \u2200 (s : State), (init_game a d s).play_move.act \u2192 P s \u2192\n    f (init_game a d s).play_move.s < f s) :\n  \u00ac((init_game a d s\u2080).play n).act :=\nbegin\n  have h : \u2200 (Q : State \u2192 State \u2192 Prop),\n    (\u2200 (s : State), (init_game a d s).play_move.act \u2192 P s \u2192\n      Q s (init_game a d s).play_move.s) \u2192\n    \u2200 (s s' : State) hs hs' hvm, P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    Q s (apply_A_move s' (a.f s' hs' hvm).m),\n  { rintro Q hQ, rintro s s' hs hs' hvm h\u2081 h\u2082,\n    have hs\u2081 : (init_game a d s).play_move.act,\n    { subst s', exact act_play_move_of_A_hvm hvm },\n    specialize hQ s hs\u2081 h\u2081,\n    obtain \u27e8s', hs, hs', hvm, h\u2083, h\u2084\u27e9 := play_move_state_eq_of_act_play_move hs\u2081,\n    rw h\u2084 at hQ, subst_vars, exact hQ },\n  apply not_act_of_descend f P hp\u2080 hp\u2081,\n  { exact h (\u03bb (s\u2081 s\u2082 : State), P s\u2082) hp\u2082 },\n  { exact h (\u03bb (s\u2081 s\u2082 : State), f s\u2082 < f s\u2081) hp\u2083 },\nend\n\nlemma not_act_of_descend_play_move_valid (f : State \u2192 \u2115) (P : State \u2192 Prop)\n  {pw n : \u2115} {a : A pw} {d : D} {s\u2080 : State}\n  (h\u2080 : valid_state pw s\u2080)\n  (hp\u2080 : f s\u2080 < n)\n  (hp\u2081 : P s\u2080)\n  (hp\u2082 : \u2200 (s : State), valid_state pw s \u2192\n    (init_game a d s).play_move.act \u2192 P s \u2192\n    P (init_game a d s).play_move.s)\n  (hp\u2083 : \u2200 (s : State), valid_state pw s \u2192\n    (init_game a d s).play_move.act \u2192 P s \u2192\n    f (init_game a d s).play_move.s < f s) :\n  \u00ac((init_game a d s\u2080).play n).act :=\nbegin\n  have h : \u2200 (Q : State \u2192 State \u2192 Prop),\n    (\u2200 (s : State), valid_state pw s \u2192\n      (init_game a d s).play_move.act \u2192 P s \u2192\n      Q s (init_game a d s).play_move.s) \u2192\n    \u2200 (s s' : State) hs hs' hvm, valid_state pw s \u2192\n      P s \u2192 s' = apply_D_move s (d.f s hs).m \u2192\n    Q s (apply_A_move s' (a.f s' hs' hvm).m),\n  { rintro Q hQ, rintro s s' hs hs' hvm hh\u2080 h\u2081 h\u2082,\n    have hs\u2081 : (init_game a d s).play_move.act,\n    { subst s', exact act_play_move_of_A_hvm hvm },\n    specialize hQ s hh\u2080 hs\u2081 h\u2081,\n    obtain \u27e8s', hs, hs', hvm, h\u2083, h\u2084\u27e9 := play_move_state_eq_of_act_play_move hs\u2081,\n    rw h\u2084 at hQ, subst_vars, exact hQ },\n  apply not_act_of_descend f (\u03bb s, valid_state pw s \u2227 P s) hp\u2080 \u27e8h\u2080, hp\u2081\u27e9,\n  {\n    convert h (\u03bb (s\u2081 s\u2082 : State), P s\u2082) _,\n    {\n      ext,\n      split; intro h\u2081,\n      {\n        rintro s s' hs hs' hvm hh hp hr,\n        exact (h\u2081 s s' hs hs' hvm \u27e8hh, hp\u27e9 hr).2,\n      },\n      {\n        rintro s s' hs hs' hvm hh hp,\n        fsplit,\n        {\n          -- apply valid_state_apply_A_move,\n          sorry\n        },\n        {\n          exact h\u2081 s s' hs hs' hvm hh.1 hh.2 hp,\n        },\n      },\n    },\n    sorry\n  },\n  {\n    -- apply h (\u03bb (s\u2081 s\u2082 : State), f s\u2082 < f s\u2081),\n    sorry\n  },\nend\n\nlemma A_mem_squares_of_valid_state {pw : \u2115} {s : State}\n  (h : valid_state pw s) :\n  s.board.A \u2208 s.board.squares :=\nbegin\n  rcases h with \u27e8a, d, n, rfl\u27e9,\n  apply @induct_s (\u03bb (s : State), s.board.A \u2208 s.board.squares),\n  { triv },\n  { rintro s p h\u2081 h, exact h.2.2 },\n  { rintro s \u27e8- | p\u27e9 h\u2081 h,\n    { exact h\u2081 },\n    { simp_rw [apply_D_move, apply_D_move_b, apply_move, set.mem_diff],\n      exact \u27e8h\u2081, h.1.symm\u27e9 }},\n  { rintro s h, exact h },\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/induct.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.3938676168492713}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nDenumerable (countably infinite) types, as a typeclass extending\nencodable. This is used to provide explicit encode/decode functions\nfrom nat, where the functions are known inverses of each other.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.encodable.basic\nimport Mathlib.data.sigma.default\nimport Mathlib.data.fintype.basic\nimport Mathlib.data.list.min_max\nimport Mathlib.PostPort\n\nuniverses u_1 l u_2 u_3 \n\nnamespace Mathlib\n\n/-- A denumerable type is one which is (constructively) bijective with \u2115.\n  Although we already have a name for this property, namely `\u03b1 \u2243 \u2115`,\n  we are here interested in using it as a typeclass. -/\nclass denumerable (\u03b1 : Type u_1) extends encodable \u03b1 where\n  decode_inv : \u2200 (n : \u2115), \u2203 (a : \u03b1), \u2203 (H : a \u2208 encodable.decode \u03b1 n), encodable.encode a = n\n\nnamespace denumerable\n\n\ntheorem decode_is_some (\u03b1 : Type u_1) [denumerable \u03b1] (n : \u2115) :\n    \u21a5(option.is_some (encodable.decode \u03b1 n)) :=\n  iff.mpr option.is_some_iff_exists (Exists.imp (fun (a : \u03b1) => Exists.fst) (decode_inv n))\n\ndef of_nat (\u03b1 : Type u_1) [f : denumerable \u03b1] (n : \u2115) : \u03b1 := option.get (decode_is_some \u03b1 n)\n\n@[simp] theorem decode_eq_of_nat (\u03b1 : Type u_1) [denumerable \u03b1] (n : \u2115) :\n    encodable.decode \u03b1 n = some (of_nat \u03b1 n) :=\n  option.eq_some_of_is_some (decode_is_some \u03b1 n)\n\n@[simp] theorem of_nat_of_decode {\u03b1 : Type u_1} [denumerable \u03b1] {n : \u2115} {b : \u03b1}\n    (h : encodable.decode \u03b1 n = some b) : of_nat \u03b1 n = b :=\n  option.some.inj (Eq.trans (Eq.symm (decode_eq_of_nat \u03b1 n)) h)\n\n@[simp] theorem encode_of_nat {\u03b1 : Type u_1} [denumerable \u03b1] (n : \u2115) :\n    encodable.encode (of_nat \u03b1 n) = n :=\n  sorry\n\n@[simp] theorem of_nat_encode {\u03b1 : Type u_1} [denumerable \u03b1] (a : \u03b1) :\n    of_nat \u03b1 (encodable.encode a) = a :=\n  of_nat_of_decode (encodable.encodek a)\n\ndef eqv (\u03b1 : Type u_1) [denumerable \u03b1] : \u03b1 \u2243 \u2115 :=\n  equiv.mk encodable.encode (of_nat \u03b1) of_nat_encode encode_of_nat\n\ndef mk' {\u03b1 : Type u_1} (e : \u03b1 \u2243 \u2115) : denumerable \u03b1 := mk sorry\n\ndef of_equiv (\u03b1 : Type u_1) {\u03b2 : Type u_2} [denumerable \u03b1] (e : \u03b2 \u2243 \u03b1) : denumerable \u03b2 := mk sorry\n\n@[simp] theorem of_equiv_of_nat (\u03b1 : Type u_1) {\u03b2 : Type u_2} [denumerable \u03b1] (e : \u03b2 \u2243 \u03b1) (n : \u2115) :\n    of_nat \u03b2 n = coe_fn (equiv.symm e) (of_nat \u03b1 n) :=\n  sorry\n\ndef equiv\u2082 (\u03b1 : Type u_1) (\u03b2 : Type u_2) [denumerable \u03b1] [denumerable \u03b2] : \u03b1 \u2243 \u03b2 :=\n  equiv.trans (eqv \u03b1) (equiv.symm (eqv \u03b2))\n\nprotected instance nat : denumerable \u2115 := mk sorry\n\n@[simp] theorem of_nat_nat (n : \u2115) : of_nat \u2115 n = n := rfl\n\nprotected instance option {\u03b1 : Type u_1} [denumerable \u03b1] : denumerable (Option \u03b1) := mk sorry\n\nprotected instance sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [denumerable \u03b1] [denumerable \u03b2] :\n    denumerable (\u03b1 \u2295 \u03b2) :=\n  mk sorry\n\nprotected instance sigma {\u03b1 : Type u_1} [denumerable \u03b1] {\u03b3 : \u03b1 \u2192 Type u_3}\n    [(a : \u03b1) \u2192 denumerable (\u03b3 a)] : denumerable (sigma \u03b3) :=\n  mk sorry\n\n@[simp] theorem sigma_of_nat_val {\u03b1 : Type u_1} [denumerable \u03b1] {\u03b3 : \u03b1 \u2192 Type u_3}\n    [(a : \u03b1) \u2192 denumerable (\u03b3 a)] (n : \u2115) :\n    of_nat (sigma \u03b3) n =\n        sigma.mk (of_nat \u03b1 (prod.fst (nat.unpair n)))\n          (of_nat (\u03b3 (of_nat \u03b1 (prod.fst (nat.unpair n)))) (prod.snd (nat.unpair n))) :=\n  sorry\n\nprotected instance prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} [denumerable \u03b1] [denumerable \u03b2] :\n    denumerable (\u03b1 \u00d7 \u03b2) :=\n  of_equiv (sigma fun (_x : \u03b1) => \u03b2) (equiv.symm (equiv.sigma_equiv_prod \u03b1 \u03b2))\n\n@[simp] theorem prod_of_nat_val {\u03b1 : Type u_1} {\u03b2 : Type u_2} [denumerable \u03b1] [denumerable \u03b2]\n    (n : \u2115) :\n    of_nat (\u03b1 \u00d7 \u03b2) n = (of_nat \u03b1 (prod.fst (nat.unpair n)), of_nat \u03b2 (prod.snd (nat.unpair n))) :=\n  sorry\n\n@[simp] theorem prod_nat_of_nat : of_nat (\u2115 \u00d7 \u2115) = nat.unpair := sorry\n\nprotected instance int : denumerable \u2124 := mk' equiv.int_equiv_nat\n\nprotected instance pnat : denumerable \u2115+ := mk' equiv.pnat_equiv_nat\n\nprotected instance ulift {\u03b1 : Type u_1} [denumerable \u03b1] : denumerable (ulift \u03b1) :=\n  of_equiv \u03b1 equiv.ulift\n\nprotected instance plift {\u03b1 : Type u_1} [denumerable \u03b1] : denumerable (plift \u03b1) :=\n  of_equiv \u03b1 equiv.plift\n\ndef pair {\u03b1 : Type u_1} [denumerable \u03b1] : \u03b1 \u00d7 \u03b1 \u2243 \u03b1 := equiv\u2082 (\u03b1 \u00d7 \u03b1) \u03b1\n\nend denumerable\n\n\nnamespace nat.subtype\n\n\ntheorem exists_succ {s : set \u2115} [infinite \u21a5s] (x : \u21a5s) : \u2203 (n : \u2115), subtype.val x + n + 1 \u2208 s :=\n  sorry\n\ndef succ {s : set \u2115} [infinite \u21a5s] [decidable_pred s] (x : \u21a5s) : \u21a5s :=\n  (fun (h : \u2203 (m : \u2115), subtype.val x + m + 1 \u2208 s) =>\n      { val := subtype.val x + nat.find h + 1, property := sorry })\n    (exists_succ x)\n\ntheorem succ_le_of_lt {s : set \u2115} [infinite \u21a5s] [decidable_pred s] {x : \u21a5s} {y : \u21a5s} (h : y < x) :\n    succ y \u2264 x :=\n  sorry\n\ntheorem le_succ_of_forall_lt_le {s : set \u2115} [infinite \u21a5s] [decidable_pred s] {x : \u21a5s} {y : \u21a5s}\n    (h : \u2200 (z : \u21a5s), z < x \u2192 z \u2264 y) : x \u2264 succ y :=\n  sorry\n\ntheorem lt_succ_self {s : set \u2115} [infinite \u21a5s] [decidable_pred s] (x : \u21a5s) : x < succ x :=\n  lt_of_le_of_lt (le_add_right (le_refl (subtype.val x)))\n    (lt_succ_self (subtype.val x + nat.find (exists_succ x)))\n\ntheorem lt_succ_iff_le {s : set \u2115} [infinite \u21a5s] [decidable_pred s] {x : \u21a5s} {y : \u21a5s} :\n    x < succ y \u2194 x \u2264 y :=\n  { mp :=\n      fun (h : x < succ y) => le_of_not_gt fun (h' : x > y) => not_le_of_gt h (succ_le_of_lt h'),\n    mpr := fun (h : x \u2264 y) => lt_of_le_of_lt h (lt_succ_self y) }\n\ndef of_nat (s : set \u2115) [decidable_pred s] [infinite \u21a5s] : \u2115 \u2192 \u21a5s := sorry\n\ntheorem of_nat_surjective_aux {s : set \u2115} [infinite \u21a5s] [decidable_pred s] {x : \u2115} (hx : x \u2208 s) :\n    \u2203 (n : \u2115), of_nat s n = { val := x, property := hx } :=\n  sorry\n\ntheorem of_nat_surjective {s : set \u2115} [infinite \u21a5s] [decidable_pred s] :\n    function.surjective (of_nat s) :=\n  sorry\n\ndef denumerable (s : set \u2115) [decidable_pred s] [infinite \u21a5s] : denumerable \u21a5s :=\n  denumerable.of_equiv \u2115 (equiv.mk to_fun_aux (of_nat s) sorry sorry)\n\nend nat.subtype\n\n\nnamespace denumerable\n\n\ndef of_encodable_of_infinite (\u03b1 : Type u_1) [encodable \u03b1] [infinite \u03b1] : denumerable \u03b1 :=\n  let _inst : decidable_pred (set.range encodable.encode) := encodable.decidable_range_encode \u03b1;\n  let _inst_3 : infinite \u21a5(set.range encodable.encode) := sorry;\n  let _inst_4 : denumerable \u21a5(set.range encodable.encode) :=\n    nat.subtype.denumerable (set.range encodable.encode);\n  of_equiv (\u21a5(set.range encodable.encode)) (encodable.equiv_range_encode \u03b1)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/denumerable_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.3938676168492713}}
{"text": "import tactic\nimport data.set.finite\nimport data.real.basic\nimport data.real.ereal\nimport linear_algebra.affine_space.independent\nimport analysis.convex.basic\nimport topology.sequences\n\nnoncomputable theory\nopen set affine topological_space \nopen_locale affine filter big_operators\n\nvariables  {V : Type*} [add_comm_group V] [module \u211d V]\nvariables [affine_space V V]\n\nvariables {k n : \u2115}\n\nvariables (\u0394 : simplex \u211d V n)\n\ndef pts (C : simplex \u211d V k) : set V := convex_hull (C.points '' univ)\n\nstructure triangulation :=\n(simps : set (@simplex \u211d V V _ _ _ _ n) )\n(cov : (\u22c3 s \u2208 simps, (pts s)) = pts \u0394)\n--(inter : \u2200 s t \u2208 simps, (pts s) \u2229 (pts t) \u2260 \u2205 \u2192 \u2203 (m : \u2115) (st m),\n--  (pts s) \u2229 (pts t) = pts st)\n-- exercici: escriure la condici\u00f3 d'intersecci\u00f3 fent servir \"face\".\n\n\nlemma fixed_point_of_epsilon_fixed (X : Type) [metric_space X]\n  [hsq : seq_compact_space X]\n  (f : X \u2192 X) (hf : continuous f)\n  (h : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 x, dist x (f x) < \u03b5) :\n  \u2203 x : X, f x = x :=\nbegin\n  have hpos : \u2200 (n : \u2115), 0 < 1 / ((n+1) : \u211d), by apply nat.one_div_pos_of_nat,\n  let a : \u2115 \u2192 X := \u03bb n, classical.some (h (1 / ((n+1) : \u211d)) (hpos n)),\n  have ha : \u2200 n, dist (a n) (f (a n)) < 1 / ((n+1) : \u211d) :=\n    \u03bb n, classical.some_spec (h (1 / ((n+1) : \u211d)) (hpos n)),\n  have exists_lim : \u2203 (z \u2208 univ) (\u03a6 : \u2115 \u2192 \u2115),\n    strict_mono \u03a6 \u2227 filter.tendsto (a \u2218 \u03a6) filter.at_top (nhds z),\n  { apply hsq.seq_compact_univ,\n    exact \u03bb n, by trivial },\n  obtain \u27e8z, \u27e8_, \u27e8\u03a6, \u27e8h\u03a61, h\u03a62\u27e9\u27e9\u27e9 \u27e9 := exists_lim,\n  use z,\n  suffices : \u2200 \u03b5 > 0, dist z (f z) \u2264 \u03b5,\n  {\n    rw [\u2190dist_le_zero, dist_comm],\n    exact le_of_forall_le_of_dense this,\n  },\n  intros \u03b5 h\u03b5,\n  have H1 : \u2200 \u03b4, 0 < \u03b4 \u2192  \u2203 (n : \u2115), \u2200 m \u2265 n, dist z ((a \u2218 \u03a6) m) < \u03b4,\n  {\n    intros \u03b4 h\u03b4,\n    rw seq_tendsto_iff at h\u03a62,\n    specialize h\u03a62 (metric.ball z (\u03b4)) (by rwa [metric.mem_ball, dist_self]) (metric.is_open_ball),\n    simp only [metric.mem_ball, dist_comm] at h\u03a62,\n    exact h\u03a62,\n  },\n  have H2 : \u2203 (n : \u2115), \u2200 m \u2265 n, dist ((a\u2218\u03a6) m) (f ((a\u2218\u03a6) m)) \u2264 \u03b5/3,\n  {\n    have hkey : \u2203 (n : \u2115), 1 / ((n+1):\u211d) < \u03b5/3,\n    { have hnlarge : \u2203 (n : \u2115), (n :\u211d) > 3 / \u03b5 := exists_nat_gt (3 / \u03b5),\n      obtain \u27e8n, hn\u27e9:= hnlarge,\n      use n,\n      refine (inv_lt_inv _ (hpos n)).mp _, by linarith,\n      field_simp,\n      linarith },\n    obtain \u27e8n, hn\u27e9 := hkey,\n    use n,\n    intros m hm,\n    specialize ha (\u03a6 m),\n    have hmn : 1 / ((m + 1) : \u211d) \u2264 1 / ((n + 1) : \u211d), by exact nat.one_div_le_one_div hm,\n    have hinc : 1 / ((\u03a6 m) + 1:\u211d) \u2264 1 / ((m + 1):\u211d), by exact nat.one_div_le_one_div (strict_mono.id_le h\u03a61 m),\n    linarith,\n  },\n  have H3 : \u2203 (n : \u2115), \u2200 m \u2265 n, dist (f ((a\u2218\u03a6) m)) (f z) < \u03b5/3 := \n      let \u27e8\u03b4, \u27e8h\u03b4pos, h'\u27e9\u27e9 := (metric.continuous_iff.1 hf) z (\u03b5/3) (by linarith), \u27e8n1, hn1\u27e9 := H1 \u03b4 h\u03b4pos in \n        \u27e8n1, \u03bb m hm, let h := hn1 m hm in h' (a (\u03a6 m)) (by rwa dist_comm)\u27e9,\n  obtain \u27e8\u27e8n1, hn1\u27e9, \u27e8n2, hn2\u27e9, \u27e8n3, hn3\u27e9\u27e9 := \u27e8H1 (\u03b5 / 3) (by linarith), H2, H3\u27e9,\n  let n := max (max n1 n2) n3,\n  specialize hn1 n (le_of_max_le_left (le_max_left (max n1 n2) n3)),\n  specialize hn2 n (le_trans (le_max_right n1 n2) (le_max_left (max n1 n2) n3)),\n  specialize hn3 n (le_max_right (max n1 n2) n3),\n  calc\n  dist z (f z) \u2264 dist z ((a \u2218 \u03a6) n)\n                + dist ((a \u2218 \u03a6) n) (f ((a \u2218 \u03a6) n))\n                + dist (f ((a \u2218 \u03a6) n)) (f z) : dist_triangle4 z ((a \u2218 \u03a6) n) (f ((a \u2218 \u03a6) n)) (f z)\n  ... \u2264 \u03b5/3 + \u03b5/3 + \u03b5/3 : by { linarith [hn1, hn2, hn3] }\n  ... = \u03b5 : by {ring},\nend\n\nlemma le_min_right_or_left {\u03b1 : Type*} [linear_order \u03b1] (a b : \u03b1) : a \u2264 min a b \u2228 b \u2264 min a b :=\nby cases (le_total a b) with h; simp [true_or, le_min rfl.ge h]; exact or.inr h\n\nlemma max_le_right_or_left {\u03b1 : Type*} [linear_order \u03b1] (a b : \u03b1) : max a b \u2264 a \u2228 max a b \u2264 b :=\nby cases (le_total a b) with h; simp [true_or, max_le rfl.ge h]; exact or.inr h\n\nlemma edist_lt_of_diam_lt {X : Type*} [pseudo_emetric_space X] (s : set X)  {d : ennreal} :\n  emetric.diam s < d \u2192 \u2200 (x \u2208 s) (y \u2208 s), edist x y < d :=\n\u03bb h x hx y hy, gt_of_gt_of_ge h (emetric.edist_le_diam_of_mem hx hy)\n\nlemma enndiameter_growth' {X : Type} [pseudo_emetric_space X] {S : set X}\n  {f : X \u2192 X} (hf : uniform_continuous_on f S) : \u2200 \u03b5 > 0,  \u2203 \u03b4 > 0, \n  \u2200 T \u2286 S, emetric.diam T < \u03b4 \u2192 emetric.diam (f '' T) \u2264 \u03b5 :=\n\u03bb \u03b5 h\u03b5, let \u27e8\u03b4, h\u03b4, H\u27e9 := emetric.uniform_continuous_on_iff.1 hf \u03b5 h\u03b5 in\n  \u27e8\u03b4, h\u03b4, \u03bb R hR hdR, emetric.diam_image_le_iff.2 \n  (\u03bb x hx y hy, le_of_lt (H (hR hx) (hR hy) (edist_lt_of_diam_lt R hdR x hx y hy)))\u27e9\n\nlemma enndiameter_growth {X : Type} [pseudo_emetric_space X] {S : set X}\n  {f : X \u2192 X} (hf : uniform_continuous_on f S) : \u2200 \u03b5 > 0,  \u2203 \u03b4 > 0, \n  \u2200 T \u2286 S, emetric.diam T < \u03b4 \u2192 emetric.diam (f '' T) < \u03b5 :=\nbegin\n  intros \u03b5 h\u03b5,\n  set \u03b3 := min 1 (\u03b5/2) with hh\u03b3,\n  have h\u03b3 : \u03b3 > 0,\n  { cases (le_min_right_or_left 1 (\u03b5/2)),\n    { exact lt_of_lt_of_le (ennreal.zero_lt_one) h },\n    { exact lt_of_lt_of_le (ennreal.div_pos_iff.2 \u27e8ne_of_gt h\u03b5, ennreal.two_ne_top\u27e9) h } },\n  obtain \u27e8\u03b4, h\u03b4, H\u27e9 := enndiameter_growth' hf \u03b3 h\u03b3,\n  have h\u03b3\u03b5: \u03b3 < \u03b5,\n  { cases (lt_or_ge 1 \u03b5),\n    { exact lt_of_le_of_lt (min_le_left 1 (\u03b5/2)) h },\n    { have h\u03b5top := ne_of_lt (lt_of_le_of_lt h (lt_of_le_of_ne le_top ennreal.one_ne_top)),\n      exact lt_of_le_of_lt (min_le_right 1 (\u03b5/2)) (ennreal.half_lt_self (ne_of_gt h\u03b5) h\u03b5top) } },\n  exact \u27e8\u03b4, h\u03b4, (\u03bb R hR hdR, lt_of_le_of_lt (H R hR hdR) h\u03b3\u03b5)\u27e9,\nend\n\nlemma diameter_growth (X : Type) [metric_space X] (S : set X)\n  (f : X \u2192 X) (hf : uniform_continuous_on f S) (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) : \n  \u2203 \u03b4 > 0, \u2200 T \u2286 S, metric.bounded T \u2192 metric.diam T \u2264 \u03b4 \u2192\n  metric.bounded (f '' T) \u2227 metric.diam (f '' T) \u2264 \u03b5 :=\nbegin\n  sorry\nend\n\nvariables {d : \u2115}\nlocal notation `E` := fin d \u2192 \u211d\n\ndef H := {x : E | (\u2211 (i : fin d), x i) = 1}\n\nvariables (f: E \u2192 E)\n\nlemma of_real_neg_real_equiv {x : \u211d} (hx : 0 \u2264 x) : (ennreal.of_real x : ereal) = (x : ereal) :=\nbegin\n  rw (ennreal.of_real_eq_coe_nnreal hx),\n  exact rfl,\nend\n\n\nlemma abs_sub_leq (a b : real) (r : ennreal) (h1 : (a : ereal) \u2264 (b : ereal) + r) (h2 : (a : ereal) \u2265 (b : ereal) - r) :\n  ennreal.of_real (abs (a - b)) \u2264 r :=\nbegin\n  cases (abs_choice (a - b)),\n  { have : (a : ereal) - (b : ereal) \u2264 (r : ereal), \n    { rcases (ereal.cases (r : ereal)) with h_1 | \u27e8x, hx\u27e9 | h_3,\n      { have hr0: (r : ereal) < 0,\n        { rw h_1,\n          exact ereal.bot_lt_zero },\n        have h0r:= ereal.coe_ennreal_nonneg r,\n        rw lt_iff_not_ge at hr0,\n        contradiction },\n      { let hb := le_of_eq (eq.refl (-\u2191b)), \n        obtain hh := add_le_add h1 hb,\n        have hr : b + x + -b = x, by ring,\n        have hbr : (b : ereal) + (r : ereal) + - (b :ereal) = (r : ereal),\n        { rw hx,\n          exact (congr_arg coe (eq.symm hr)).symm },\n        rwa \u2190 hbr },\n      { rw h_3,\n        exact with_top.le_none } },\n    exact ereal.coe_ennreal_le_coe_ennreal_iff.mp (by rwa [of_real_neg_real_equiv (abs_nonneg (a - b)), h]) },\n  { have hab : -(a - b) = b - a, by ring,\n    have : (b : ereal) - (a : ereal) \u2264 (r : ereal),\n    { rcases (ereal.cases (r : ereal)) with h_1 |\u27e8x, hx\u27e9 | h_3,\n      { have hr0: (r : ereal) < 0,\n        { rw h_1,\n          exact ereal.bot_lt_zero },\n        have h0r:= ereal.coe_ennreal_nonneg r,\n        rw lt_iff_not_ge at hr0,\n        contradiction },\n      {\n        rw hx at *,\n        change (((b - a) : \u211d) : ereal) \u2264 x,\n        change (a : ereal ) \u2265 (((b - x) : \u211d) : ereal) at h2,\n        simp only [ge_iff_le, eq_self_iff_true, neg_sub, ereal.coe_le_coe_iff] at *,\n        linarith },\n      { rw h_3,\n        exact with_top.le_none } },\n    exact ereal.coe_ennreal_le_coe_ennreal_iff.mp (by rwa [of_real_neg_real_equiv (abs_nonneg (a - b)), h, hab]) },\nend\n\nexample (x y : E) : edist x y = ennreal.of_real (\u2211 i, (x i - y i)^2) :=\nbegin\n  sorry\nend\n\nlemma points_coordinates_bounded_distance (x y : E) (i : fin d) :\n  ennreal.of_real (abs (x i - y i)) \u2264 edist x y :=\nbegin\n  unfold edist,\n  \n  sorry\nend\n\nlemma points_coordinates_bounded_diam (S : set E) (x y : E) (hx : x \u2208 S) (hy : y \u2208 S)\n(i : fin d) : ennreal.of_real (abs (x i - y i)) \u2264 emetric.diam S :=\nbegin\n  sorry\nend\n\n\n-- per tota coordenada i, existeix un vertex v tal que la coordenada i-\u00e8ssima \n-- \u00e9s la primera que complex que f(v)_i < f(v)\ndef is_sperner_set (f: E \u2192 E) (S : set E)  := \n  \u2200 i: fin d, \u2203 v : E, v \u2208 S \u2227\n  (\u2200 j < i, (f v) j \u2265  (v j)) \u2227 (((f v) i) < v i)\n\nlemma epsilon_fixed_condition\n{f : E \u2192 E} {S : set E} (hs : S \u2286 H) (hd : 0 < d)\n(hf : uniform_continuous_on f S) \n{\u03b5 : real} (h\u03b5 : 0 < \u03b5)\n: \u2203 \u03b4, 0 < \u03b4 \u2227\n\u2200 T \u2286 S,\n  metric.bounded T \u2192 metric.diam T < \u03b4 \u2192\n  is_sperner_set f T \u2192\n  \u2200 x \u2208 T, dist (f x) x < \u03b5 :=\nbegin\n  let \u03b5\u2081 := \u03b5 / (2 * d),\n  have h\u2081 := div_pos h\u03b5 (mul_pos zero_lt_two (nat.cast_pos.mpr hd)),\n  obtain \u27e8\u03b4\u2080, h\u03b4\u2080pos, h\u03b4\u2080\u27e9 := metric.uniform_continuous_on_iff.mp hf \u03b5\u2081 h\u2081,\n  let \u03b4 := min \u03b4\u2080 (\u03b5\u2081/2),\n  use \u03b4,\n  split,\n  { cases le_min_right_or_left \u03b4\u2080 (\u03b5\u2081/2),\n    { exact gt_of_ge_of_gt h h\u03b4\u2080pos },\n    { exact lt_min h\u03b4\u2080pos (half_pos h\u2081) } },\n  intros T hTS hbT hdT hfT x hx,\n  have hmost : \u2200 (i : fin d) (hi : (i : \u2115) \u2260 d-1),\n    abs (((f x) i)-(x i))\n     \u2264 \u03b4 + (metric.diam (f '' T)),\n  {\n    intros i hi,\n    rw abs_sub_le_iff,\n    split,\n    {\n      sorry\n    },\n    {\n      sorry\n    }\n  },\n  have hlast : abs(((f x) \u27e8d-1, buffer.lt_aux_2 hd\u27e9)) - x \u27e8d-1, buffer.lt_aux_2 hd\u27e9 \u2264 (d-1) * (\u03b4 + (metric.diam (f '' T))),\n  {\n    sorry\n  },\n  sorry\nend\n", "meta": {"author": "mmasdeu", "repo": "brouwerfixedpoint", "sha": "548270f79ecf12d7e20a256806ccb9fcf57b87e2", "save_path": "github-repos/lean/mmasdeu-brouwerfixedpoint", "path": "github-repos/lean/mmasdeu-brouwerfixedpoint/brouwerfixedpoint-548270f79ecf12d7e20a256806ccb9fcf57b87e2/src/sperner.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.39386761684927124}}
{"text": "import tactic\nimport .tokens\nimport .commun\n\nsetup_tactic_parser\n\nnamespace tactic\n\nmeta def intro_obj (n : name) : tactic expr :=\ndo t \u2190 target,\n   if expr.is_pi t \u2228 expr.is_let t then do \n     e \u2190 intro_core n,\n     t \u2190 infer_type e,\n     mwhen (is_prop t) failed,\n     pure e\n   else do\n     whnf_target,\n     e \u2190 intro_core n, \n     t \u2190 infer_type e,\n     mwhen (is_prop t) failed,\n     pure e\n   \n\n\nmeta def Soit1 : introduced \u2192 tactic unit \n| (introduced.typed n t)   := do verifie_nom n,\n                                 et \u2190 to_expr t,\n                                 e \u2190 intro_obj n <|> fail \"Il n'y a pas d'objet \u00e0 introduire ici.\",\n                                 change_core et e\n| (introduced.bare n)      := do verifie_nom n,\n                                 intro_obj n <|> fail \"Il n'y a pas d'objet \u00e0 introduire ici.\", \n                                 skip\n| (introduced.related n rel e) := do verifie_nom n,\n                                 ename \u2190 intro_obj n <|> fail \"Il n'y a pas d'objet \u00e0 introduire ici.\",\n                                 n_type \u2190 infer_type ename,\n                                 E \u2190 match rel with\n                                 | intro_rel.mem := to_expr e\n                                 | _ := to_expr ```(%%e : %%n_type)\n                                 end,\n                                 rel_expr \u2190 match rel with\n                                 | intro_rel.lt := to_expr ``(%%ename < %%E)\n                                 | intro_rel.gt := to_expr ``(%%ename > %%E)\n                                 | intro_rel.le := to_expr ``(%%ename \u2264 %%E)\n                                 | intro_rel.ge := to_expr ``(%%ename \u2265 %%E)\n                                 | intro_rel.mem := to_expr ``(%%ename \u2208 %%E)\n                                 end,\n                                 let hyp_name := if e = ``(0) then\n                                    match rel with\n                                    | intro_rel.lt  := n.to_string ++ \"_neg\"\n                                    | intro_rel.gt  := n.to_string ++ \"_pos\"\n                                    | intro_rel.le  := n.to_string ++ \"_neg\"\n                                    | intro_rel.ge  := n.to_string ++ \"_pos\"\n                                    | intro_rel.mem := \"h_\" ++ n.to_string -- ne devrait pas arriver\n                                    end\n                                 else \n                                    match rel with\n                                    | intro_rel.lt  := n.to_string ++ \"_lt\"\n                                    | intro_rel.gt  := n.to_string ++ \"_gt\"\n                                    | intro_rel.le  := n.to_string ++ \"_le\"\n                                    | intro_rel.ge  := n.to_string ++ \"_ge\"\n                                    | intro_rel.mem := n.to_string ++ \"_mem\"\n                                    end,\n                                 EH \u2190 intro hyp_name,\n                                 change_core rel_expr EH,\n                                 skip\n                                 \n/-- Introduit un objet ou plusieurs objets pour d\u00e9montrer un \u00e9nonc\u00e9 commen\u00e7ant par un quantificateur universel. -/\n@[interactive]\nmeta def Soit (vs : parse $ with_desc \"...\" bracketed_intro_parser*) : tactic unit :=\nvs.mmap' Soit1\n\nend tactic\n\nexample : \u2200 n > 0, \u2200 k : \u2115, \u2200 l \u2208 (set.univ : set \u2115), true :=\nbegin\n  Soit (n > 0) k (l \u2208 set.univ),\n  trivial\nend\n\nexample : \u2200 n > 0, \u2200 k : \u2115, \u2200 l \u2208 (set.univ : set \u2115), true :=\nbegin\n  Soit n,\n  success_if_fail { Soit h },\n  intro hn,\n  Soit k (l \u2208 set.univ),\n  trivial\nend\n\nexample : \u2200 n > 0, \u2200 k : \u2115, true :=\nbegin\n  Soit (n > 0),\n  success_if_fail { Soit n },\n  Soit k,\n  trivial\nend\n\nexample : \u2200 n > 0, \u2200 k : \u2115, true :=\nbegin\n  Soit n > 0,\n  success_if_fail { Soit n },\n  Soit k,\n  trivial\nend\n\nexample (k l : \u2115) : \u2200 n \u2264 k + l, true :=\nbegin\n  Soit n \u2264 k + l,\n  trivial,\nend\n\nexample (A : set \u2115) : \u2200 n \u2208 A, true :=\nbegin\n  Soit n \u2208 A,\n  trivial\nend", "meta": {"author": "PatrickMassot", "repo": "MDD154", "sha": "00defe82a4b6b7992ed522a92f62abd685e8c943", "save_path": "github-repos/lean/PatrickMassot-MDD154", "path": "github-repos/lean/PatrickMassot-MDD154/MDD154-00defe82a4b6b7992ed522a92f62abd685e8c943/src/lib/Soit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.393867608496979}}
{"text": "import hmem.stack\nimport complexity.basic\n\nvariables {\u03bc: Type*} [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)]\n\nnamespace hmem\nnamespace encoding\n\n\ninstance: has_equiv (memory \u03bc) := \u27e8 eq \u27e9\n\ndef push_arg (lhs rhs: memory \u03bc): memory \u03bc :=\n  (((memory.null \u03bc).setv 1).setm 0 lhs).setm 1 rhs\n\ndef build_arg: list (memory \u03bc) \u2192 memory \u03bc\n| [] := memory.null _\n| [m] := m\n| (x::xs) := push_arg (build_arg xs) x\n\ndef runtime_model (\u03bc: Type*) [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)]: complexity.model (program \u03bc \u00d7 list (memory \u03bc)) (memory \u03bc) \u2115 :=\n \u27e8  \u03bb p_a r c, p_a.fst.has_result (build_arg p_a.snd) r \u2227 p_a.fst.has_time_cost (build_arg p_a.snd) c,\n    \u03bb p_a a', (p_a.fst, a'::p_a.snd),\n    \u03bb _ _ _ _ _ hrc\u2081 hrc\u2082, program.unique_result hrc\u2081.left hrc\u2082.left,\n    \u03bb _ _ _ _ hc hrc\u2080, \u27e8hrc\u2080.left, program.time_cost_mono hrc\u2080.right hc\u27e9 \u27e9\n\ndef encode {\u03b4: Type} [complexity.has_encoding (runtime_model \u03bc) \u03b4]: \u03b4 \u2192 memory \u03bc := complexity.encode (runtime_model \u03bc)\n\ninstance (\u03b1 \u03b2: Type*)  [\u03b1_en: complexity.has_encoding (runtime_model \u03bc) \u03b1] [\u03b2_en: complexity.has_encoding (runtime_model \u03bc) \u03b2]:\n  complexity.has_encoding (runtime_model \u03bc) (\u03b1 \u00d7 \u03b2) :=\nbegin\n  fconstructor,\n  fconstructor,\n  exact \u03bb ab, push_arg (encode ab.fst) (encode ab.snd),\n  intros x y,\n  cases x,\n  cases y,\n  simp [push_arg, has_equiv.equiv, encode],\n  split,\n  { intro h,\n    split,\n    { rw \u2190 complexity.encoding.encode_inj \u03b1_en.value,\n      apply memory.getm_congr 0 h,\n      { rw [memory.getm_setm_ne _ _ _ _ zero_ne_one, memory.getm_setm],\n        refl,\n        apply_instance},\n      { rw [memory.getm_setm_ne _ _ _ _ zero_ne_one, memory.getm_setm],\n        refl,\n        apply_instance} },\n    { rw \u2190 complexity.encoding.encode_inj \u03b2_en.value,\n      apply memory.getm_congr 1 h,\n      { rw [memory.getm_setm],\n        refl },\n      { rw [memory.getm_setm],\n        refl } } },\n  { intro h,\n    rw [h.left, h.right] }\nend\n\ntheorem encode_pair  {\u03b1 \u03b2: Type*}  [\u03b1_en: complexity.has_encoding (runtime_model \u03bc) \u03b1] [\u03b2_en: complexity.has_encoding (runtime_model \u03bc) \u03b2] (a: \u03b1) (b: \u03b2):\n  encode (a, b) = (((memory.null \u03bc).setv 1).setm 0 (encode a)).setm 1 (encode b) := rfl\n\ninstance (p: Prop):\n  complexity.has_encoding (runtime_model \u03bc) (decidable p) :=\nbegin\n  fconstructor,\n  fconstructor,\n  exact \u03bb d,  (memory.null _).setv (d.cases_on (\u03bb _, 0) (\u03bb _, 1)),\n  intros x y,\n  cases x;\n  cases y;\n  simp [has_equiv.equiv, memory.setv_inj_iff],\nend\n\ntheorem encode_is_false {p: Prop} {hp: \u00ac p} (d: decidable p):\n  (encode (is_false hp)) = memory.null \u03bc := memory.null_setv_zero\n\ntheorem encode_is_true {p: Prop} {hp: p} (d: decidable p):\n  (encode (is_true hp)) = (memory.null \u03bc).setv 1 := rfl\n\nend encoding\nend hmem", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/hmem/encoding/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.39369032784891406}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.eq_to_hom\n\n/-!\n# Binary disjoint unions of categories\n\nWe define the category instance on `C \u2295 D` when `C` and `D` are categories.\n\nWe define:\n* `inl_`      : the functor `C \u2964 C \u2295 D`\n* `inr_`      : the functor `D \u2964 C \u2295 D`\n* `swap`      : the functor `C \u2295 D \u2964 D \u2295 C`\n    (and the fact this is an equivalence)\n\nWe further define sums of functors and natural transformations, written `F.sum G` and `\u03b1.sum \u03b2`.\n-/\n\nnamespace category_theory\n\nuniverses v\u2081 u\u2081 -- morphism levels before object levels. See note [category_theory universes].\n\nopen sum\n\nsection\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2081) [category.{v\u2081} D]\n\n/--\n`sum C D` gives the direct sum of two categories.\n-/\ninstance sum : category.{v\u2081} (C \u2295 D) :=\n{ hom :=\n    \u03bb X Y, match X, Y with\n    | inl X, inl Y := X \u27f6 Y\n    | inl X, inr Y := pempty\n    | inr X, inl Y := pempty\n    | inr X, inr Y := X \u27f6 Y\n    end,\n  id :=\n    \u03bb X, match X with\n    | inl X := \ud835\udfd9 X\n    | inr X := \ud835\udfd9 X\n    end,\n  comp :=\n    \u03bb X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := f \u226b g\n    | inr X, inr Y, inr Z, f, g := f \u226b g\n    end }\n\n@[simp] lemma sum_comp_inl {P Q R : C} (f : (inl P : C \u2295 D) \u27f6 inl Q)\n  (g : (inl Q : C \u2295 D) \u27f6 inl R) :\n  @category_struct.comp _ _ P Q R (f : P \u27f6 Q) (g : Q \u27f6 R) =\n  @category_struct.comp _ _ (inl P) (inl Q) (inl R) (f : P \u27f6 Q) (g : Q \u27f6 R) := rfl\n@[simp] lemma sum_comp_inr {P Q R : D} (f : (inr P : C \u2295 D) \u27f6 inr Q)\n  (g : (inr Q : C \u2295 D) \u27f6 inr R) :\n  @category_struct.comp _ _ P Q R (f : P \u27f6 Q) (g : Q \u27f6 R) =\n  @category_struct.comp _ _ (inr P) (inr Q) (inr R) (f : P \u27f6 Q) (g : Q \u27f6 R) := rfl\nend\n\nnamespace sum\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2081) [category.{v\u2081} D]\n\n/-- `inl_` is the functor `X \u21a6 inl X`. -/\n-- Unfortunate naming here, suggestions welcome.\n@[simps] def inl_ : C \u2964 C \u2295 D :=\n{ obj := \u03bb X, inl X,\n  map := \u03bb X Y f, f }\n\n/-- `inr_` is the functor `X \u21a6 inr X`. -/\n@[simps] def inr_ : D \u2964 C \u2295 D :=\n{ obj := \u03bb X, inr X,\n  map := \u03bb X Y f, f }\n\n/-- The functor exchanging two direct summand categories. -/\ndef swap : C \u2295 D \u2964 D \u2295 C :=\n{ obj :=\n    \u03bb X, match X with\n    | inl X := inr X\n    | inr X := inl X\n    end,\n  map :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := f\n    | inr X, inr Y, f := f\n    end }\n\n@[simp] lemma swap_obj_inl (X : C) : (swap C D).obj (inl X) = inr X := rfl\n@[simp] lemma swap_obj_inr (X : D) : (swap C D).obj (inr X) = inl X := rfl\n@[simp] lemma swap_map_inl {X Y : C} {f : inl X \u27f6 inl Y} : (swap C D).map f = f := rfl\n@[simp] \n\nnamespace swap\n\n/-- `swap` gives an equivalence between `C \u2295 D` and `D \u2295 C`. -/\ndef equivalence : C \u2295 D \u224c D \u2295 C :=\nequivalence.mk (swap C D) (swap D C)\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by { cases X; refl })) (by tidy))\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by { cases X; refl })) (by tidy))\n\ninstance is_equivalence : is_equivalence (swap C D) :=\n(by apply_instance : is_equivalence (equivalence C D).functor)\n\n/-- The double swap on `C \u2295 D` is naturally isomorphic to the identity functor. -/\ndef symmetry : swap C D \u22d9 swap D C \u2245 \ud835\udfed (C \u2295 D) :=\n(equivalence C D).unit_iso.symm\n\nend swap\n\nend sum\n\nvariables {A : Type u\u2081} [category.{v\u2081} A]\n          {B : Type u\u2081} [category.{v\u2081} B]\n          {C : Type u\u2081} [category.{v\u2081} C]\n          {D : Type u\u2081} [category.{v\u2081} D]\n\nnamespace functor\n\n/-- The sum of two functors. -/\ndef sum (F : A \u2964 B) (G : C \u2964 D) : A \u2295 C \u2964 B \u2295 D :=\n{ obj :=\n    \u03bb X, match X with\n    | inl X := inl (F.obj X)\n    | inr X := inr (G.obj X)\n    end,\n  map :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := F.map f\n    | inr X, inr Y, f := G.map f\n    end,\n  map_id' := \u03bb X, begin cases X; unfold_aux, erw F.map_id, refl, erw G.map_id, refl end,\n  map_comp' :=\n    \u03bb X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := by { unfold_aux, erw F.map_comp, refl }\n    | inr X, inr Y, inr Z, f, g := by { unfold_aux, erw G.map_comp, refl }\n    end }\n\n@[simp] lemma sum_obj_inl (F : A \u2964 B) (G : C \u2964 D) (a : A) :\n  (F.sum G).obj (inl a) = inl (F.obj a) := rfl\n@[simp] lemma sum_obj_inr (F : A \u2964 B) (G : C \u2964 D) (c : C) :\n  (F.sum G).obj (inr c) = inr (G.obj c) := rfl\n@[simp] lemma sum_map_inl (F : A \u2964 B) (G : C \u2964 D) {a a' : A} (f : inl a \u27f6 inl a') :\n  (F.sum G).map f = F.map f := rfl\n@[simp] lemma sum_map_inr (F : A \u2964 B) (G : C \u2964 D) {c c' : C} (f : inr c \u27f6 inr c') :\n  (F.sum G).map f = G.map f := rfl\nend functor\n\nnamespace nat_trans\n\n/-- The sum of two natural transformations. -/\ndef sum {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) : F.sum H \u27f6 G.sum I :=\n{ app         :=\n    \u03bb X, match X with\n    | inl X := \u03b1.app X\n    | inr X := \u03b2.app X\n    end,\n  naturality' :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := begin unfold_aux, erw \u03b1.naturality, refl, end\n    | inr X, inr Y, f := begin unfold_aux, erw \u03b2.naturality, refl, end\n    end }\n\n@[simp] lemma sum_app_inl {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (a : A) :\n  (sum \u03b1 \u03b2).app (inl a) = \u03b1.app a := rfl\n@[simp] lemma sum_app_inr {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (c : C) :\n  (sum \u03b1 \u03b2).app (inr c) = \u03b2.app c := rfl\nend nat_trans\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sums/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.39364312679657343}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.num.bitwise\nimport Mathlib.data.int.char_zero\nimport Mathlib.data.nat.gcd\nimport Mathlib.data.nat.psub\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Properties of the binary representation of integers\n-/\n\nnamespace pos_num\n\n\n@[simp] theorem cast_one {\u03b1 : Type u_1} [HasOne \u03b1] [Add \u03b1] : \u21911 = 1 := rfl\n\n@[simp] theorem cast_one' {\u03b1 : Type u_1} [HasOne \u03b1] [Add \u03b1] : \u2191one = 1 := rfl\n\n@[simp] theorem cast_bit0 {\u03b1 : Type u_1} [HasOne \u03b1] [Add \u03b1] (n : pos_num) : \u2191(bit0 n) = bit0 \u2191n :=\n  rfl\n\n@[simp] theorem cast_bit1 {\u03b1 : Type u_1} [HasOne \u03b1] [Add \u03b1] (n : pos_num) : \u2191(bit1 n) = bit1 \u2191n :=\n  rfl\n\n@[simp] theorem cast_to_nat {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : pos_num) : \u2191\u2191n = \u2191n :=\n  sorry\n\n@[simp] theorem to_nat_to_int (n : pos_num) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (int.nat_cast_eq_coe_nat \u2191n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (cast_to_nat n))) (Eq.refl \u2191n))\n\n@[simp] theorem cast_to_int {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : pos_num) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (to_nat_to_int n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191\u2191n = \u2191n)) (int.cast_coe_nat \u2191n)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (cast_to_nat n))) (Eq.refl \u2191n)))\n\ntheorem succ_to_nat (n : pos_num) : \u2191(succ n) = \u2191n + 1 := sorry\n\ntheorem one_add (n : pos_num) : 1 + n = succ n :=\n  pos_num.cases_on n (Eq.refl (1 + one)) (fun (n : pos_num) => Eq.refl (1 + bit1 n))\n    fun (n : pos_num) => Eq.refl (1 + bit0 n)\n\ntheorem add_one (n : pos_num) : n + 1 = succ n :=\n  pos_num.cases_on n (Eq.refl (one + 1)) (fun (n : pos_num) => Eq.refl (bit1 n + 1))\n    fun (n : pos_num) => Eq.refl (bit0 n + 1)\n\ntheorem add_to_nat (m : pos_num) (n : pos_num) : \u2191(m + n) = \u2191m + \u2191n := sorry\n\ntheorem add_succ (m : pos_num) (n : pos_num) : m + succ n = succ (m + n) := sorry\n\ntheorem bit0_of_bit0 (n : pos_num) : bit0 n = bit0 n := sorry\n\ntheorem bit1_of_bit1 (n : pos_num) : bit1 n = bit1 n :=\n  (fun (this : bit0 n + 1 = bit1 n) => this)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (bit0 n + 1 = bit1 n)) (add_one (bit0 n))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (succ (bit0 n) = bit1 n)) (bit0_of_bit0 n)))\n        (Eq.refl (succ (bit0 n)))))\n\ntheorem mul_to_nat (m : pos_num) (n : pos_num) : \u2191(m * n) = \u2191m * \u2191n := sorry\n\ntheorem to_nat_pos (n : pos_num) : 0 < \u2191n := sorry\n\ntheorem cmp_to_nat_lemma {m : pos_num} {n : pos_num} : \u2191m < \u2191n \u2192 \u2191(bit1 m) < \u2191(bit0 n) := sorry\n\ntheorem cmp_swap (m : pos_num) (n : pos_num) : ordering.swap (cmp m n) = cmp n m := sorry\n\ntheorem cmp_to_nat (m : pos_num) (n : pos_num) :\n    ordering.cases_on (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) :=\n  sorry\n\ntheorem lt_to_nat {m : pos_num} {n : pos_num} : \u2191m < \u2191n \u2194 m < n := sorry\n\ntheorem le_to_nat {m : pos_num} {n : pos_num} : \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr lt_to_nat)\n\nend pos_num\n\n\nnamespace num\n\n\ntheorem add_zero (n : num) : n + 0 = n :=\n  num.cases_on n (Eq.refl (zero + 0)) fun (n : pos_num) => Eq.refl (pos n + 0)\n\ntheorem zero_add (n : num) : 0 + n = n :=\n  num.cases_on n (Eq.refl (0 + zero)) fun (n : pos_num) => Eq.refl (0 + pos n)\n\ntheorem add_one (n : num) : n + 1 = succ n := sorry\n\ntheorem add_succ (m : num) (n : num) : m + succ n = succ (m + n) := sorry\n\n@[simp] theorem add_of_nat (m : \u2115) (n : \u2115) : \u2191(m + n) = \u2191m + \u2191n := sorry\n\ntheorem bit0_of_bit0 (n : num) : bit0 n = num.bit0 n :=\n  num.cases_on n (idRhs (bit0 0 = bit0 0) rfl)\n    fun (n : pos_num) =>\n      idRhs (pos (bit0 n) = pos (pos_num.bit0 n)) (congr_arg pos (pos_num.bit0_of_bit0 n))\n\ntheorem bit1_of_bit1 (n : num) : bit1 n = num.bit1 n :=\n  num.cases_on n (idRhs (bit1 0 = bit1 0) rfl)\n    fun (n : pos_num) =>\n      idRhs (pos (bit1 n) = pos (pos_num.bit1 n)) (congr_arg pos (pos_num.bit1_of_bit1 n))\n\n@[simp] theorem cast_zero {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] : \u21910 = 0 := rfl\n\n@[simp] theorem cast_zero' {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] : \u2191zero = 0 := rfl\n\n@[simp] theorem cast_one {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] : \u21911 = 1 := rfl\n\n@[simp] theorem cast_pos {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] (n : pos_num) :\n    \u2191(pos n) = \u2191n :=\n  rfl\n\ntheorem succ'_to_nat (n : num) : \u2191(succ' n) = \u2191n + 1 :=\n  num.cases_on n (idRhs (\u2191(succ' 0) = 0 + \u2191(succ' 0)) (Eq.symm (zero_add \u2191(succ' 0))))\n    fun (n : pos_num) => idRhs (\u2191(pos_num.succ n) = \u2191n + 1) (pos_num.succ_to_nat n)\n\ntheorem succ_to_nat (n : num) : \u2191(succ n) = \u2191n + 1 := succ'_to_nat n\n\n@[simp] theorem cast_to_nat {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : num) : \u2191\u2191n = \u2191n :=\n  num.cases_on n (idRhs (\u21910 = 0) nat.cast_zero)\n    fun (n : pos_num) => idRhs (\u2191\u2191n = \u2191n) (pos_num.cast_to_nat n)\n\n@[simp] theorem to_nat_to_int (n : num) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (int.nat_cast_eq_coe_nat \u2191n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (cast_to_nat n))) (Eq.refl \u2191n))\n\n@[simp] theorem cast_to_int {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : num) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (to_nat_to_int n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191\u2191n = \u2191n)) (int.cast_coe_nat \u2191n)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (cast_to_nat n))) (Eq.refl \u2191n)))\n\ntheorem to_of_nat (n : \u2115) : \u2191\u2191n = n := sorry\n\n@[simp] theorem of_nat_cast {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : \u2115) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (cast_to_nat \u2191n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191\u2191n = \u2191n)) (to_of_nat n))) (Eq.refl \u2191n))\n\ntheorem of_nat_inj {m : \u2115} {n : \u2115} : \u2191m = \u2191n \u2194 m = n :=\n  { mp := fun (h : \u2191m = \u2191n) => function.left_inverse.injective to_of_nat h,\n    mpr := congr_arg fun (x : \u2115) => \u2191x }\n\ntheorem add_to_nat (m : num) (n : num) : \u2191(m + n) = \u2191m + \u2191n := sorry\n\ntheorem mul_to_nat (m : num) (n : num) : \u2191(m * n) = \u2191m * \u2191n := sorry\n\ntheorem cmp_to_nat (m : num) (n : num) : ordering.cases_on (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) :=\n  sorry\n\ntheorem lt_to_nat {m : num} {n : num} : \u2191m < \u2191n \u2194 m < n := sorry\n\ntheorem le_to_nat {m : num} {n : num} : \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr lt_to_nat)\n\nend num\n\n\nnamespace pos_num\n\n\n@[simp] theorem of_to_nat (n : pos_num) : \u2191\u2191n = num.pos n := sorry\n\nend pos_num\n\n\nnamespace num\n\n\n@[simp] theorem of_to_nat (n : num) : \u2191\u2191n = n :=\n  num.cases_on n (idRhs (\u2191\u21910 = \u2191\u21910) rfl)\n    fun (n : pos_num) => idRhs (\u2191\u2191n = pos n) (pos_num.of_to_nat n)\n\ntheorem to_nat_inj {m : num} {n : num} : \u2191m = \u2191n \u2194 m = n :=\n  { mp := fun (h : \u2191m = \u2191n) => function.left_inverse.injective of_to_nat h,\n    mpr := congr_arg fun (x : num) => \u2191x }\n\n/--\nThis tactic tries to turn an (in)equality about `num`s to one about `nat`s by rewriting.\n```lean\nexample (n : num) (m : num) : n \u2264 n + m :=\nbegin\n  num.transfer_rw,\n  exact nat.le_add_right _ _\nend\n```\n-/\n/--\nThis tactic tries to prove (in)equalities about `num`s by transfering them to the `nat` world and\nthen trying to call `simp`.\n```lean\nexample (n : num) (m : num) : n \u2264 n + m := by num.transfer\n```\n-/\nprotected instance comm_semiring : comm_semiring num :=\n  comm_semiring.mk Add.add sorry 0 zero_add add_zero sorry Mul.mul sorry 1 sorry sorry sorry sorry\n    sorry sorry sorry\n\nprotected instance ordered_cancel_add_comm_monoid : ordered_cancel_add_comm_monoid num :=\n  ordered_cancel_add_comm_monoid.mk comm_semiring.add comm_semiring.add_assoc sorry\n    comm_semiring.zero comm_semiring.zero_add comm_semiring.add_zero comm_semiring.add_comm sorry\n    LessEq Less sorry sorry sorry sorry sorry\n\nprotected instance linear_ordered_semiring : linear_ordered_semiring num :=\n  linear_ordered_semiring.mk comm_semiring.add comm_semiring.add_assoc comm_semiring.zero\n    comm_semiring.zero_add comm_semiring.add_zero comm_semiring.add_comm comm_semiring.mul\n    comm_semiring.mul_assoc comm_semiring.one comm_semiring.one_mul comm_semiring.mul_one\n    comm_semiring.zero_mul comm_semiring.mul_zero comm_semiring.left_distrib\n    comm_semiring.right_distrib ordered_cancel_add_comm_monoid.add_left_cancel\n    ordered_cancel_add_comm_monoid.add_right_cancel ordered_cancel_add_comm_monoid.le\n    ordered_cancel_add_comm_monoid.lt ordered_cancel_add_comm_monoid.le_refl\n    ordered_cancel_add_comm_monoid.le_trans ordered_cancel_add_comm_monoid.le_antisymm\n    ordered_cancel_add_comm_monoid.add_le_add_left\n    ordered_cancel_add_comm_monoid.le_of_add_le_add_left sorry sorry sorry sorry num.decidable_le\n    num.decidable_eq num.decidable_lt sorry\n\ntheorem dvd_to_nat (m : num) (n : num) : \u2191m \u2223 \u2191n \u2194 m \u2223 n := sorry\n\nend num\n\n\nnamespace pos_num\n\n\ntheorem to_nat_inj {m : pos_num} {n : pos_num} : \u2191m = \u2191n \u2194 m = n := sorry\n\ntheorem pred'_to_nat (n : pos_num) : \u2191(pred' n) = Nat.pred \u2191n := sorry\n\n@[simp] theorem pred'_succ' (n : num) : pred' (num.succ' n) = n := sorry\n\n@[simp] theorem succ'_pred' (n : pos_num) : num.succ' (pred' n) = n := sorry\n\nprotected instance has_dvd : has_dvd pos_num :=\n  has_dvd.mk fun (m n : pos_num) => num.pos m \u2223 num.pos n\n\ntheorem dvd_to_nat {m : pos_num} {n : pos_num} : \u2191m \u2223 \u2191n \u2194 m \u2223 n :=\n  num.dvd_to_nat (num.pos m) (num.pos n)\n\ntheorem size_to_nat (n : pos_num) : \u2191(size n) = nat.size \u2191n := sorry\n\ntheorem size_eq_nat_size (n : pos_num) : \u2191(size n) = nat_size n := sorry\n\ntheorem nat_size_to_nat (n : pos_num) : nat_size n = nat.size \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nat_size n = nat.size \u2191n)) (Eq.symm (size_eq_nat_size n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(size n) = nat.size \u2191n)) (size_to_nat n)))\n      (Eq.refl (nat.size \u2191n)))\n\ntheorem nat_size_pos (n : pos_num) : 0 < nat_size n :=\n  pos_num.cases_on n (nat.succ_pos 0) (fun (n : pos_num) => nat.succ_pos (nat_size n))\n    fun (n : pos_num) => nat.succ_pos (nat_size n)\n\n/--\nThis tactic tries to turn an (in)equality about `pos_num`s to one about `nat`s by rewriting.\n```lean\nexample (n : pos_num) (m : pos_num) : n \u2264 n + m :=\nbegin\n  pos_num.transfer_rw,\n  exact nat.le_add_right _ _\nend\n```\n-/\n/--\nThis tactic tries to prove (in)equalities about `pos_num`s by transferring them to the `nat` world\nand then trying to call `simp`.\n```lean\nexample (n : pos_num) (m : pos_num) : n \u2264 n + m := by pos_num.transfer\n```\n-/\nprotected instance add_comm_semigroup : add_comm_semigroup pos_num :=\n  add_comm_semigroup.mk Add.add sorry sorry\n\nprotected instance comm_monoid : comm_monoid pos_num :=\n  comm_monoid.mk Mul.mul sorry 1 sorry sorry sorry\n\nprotected instance distrib : distrib pos_num := distrib.mk Mul.mul Add.add sorry sorry\n\nprotected instance linear_order : linear_order pos_num :=\n  linear_order.mk LessEq Less sorry sorry sorry sorry\n    (fun (a b : pos_num) => pos_num.decidable_le a b)\n    (fun (a b : pos_num) => pos_num.decidable_eq a b)\n    fun (a b : pos_num) => pos_num.decidable_lt a b\n\n@[simp] theorem cast_to_num (n : pos_num) : \u2191n = num.pos n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n = num.pos n)) (Eq.symm (cast_to_nat n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = num.pos n)) (Eq.symm (of_to_nat n)))) (Eq.refl \u2191\u2191n))\n\n@[simp] theorem bit_to_nat (b : Bool) (n : pos_num) : \u2191(bit b n) = nat.bit b \u2191n :=\n  bool.cases_on b (Eq.refl \u2191(bit false n)) (Eq.refl \u2191(bit tt n))\n\n@[simp] theorem cast_add {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (m : pos_num) (n : pos_num) :\n    \u2191(m + n) = \u2191m + \u2191n :=\n  sorry\n\n@[simp] theorem cast_succ {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : pos_num) :\n    \u2191(succ n) = \u2191n + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(succ n) = \u2191n + 1)) (Eq.symm (add_one n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(n + 1) = \u2191n + 1)) (cast_add n 1)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n + \u21911 = \u2191n + 1)) cast_one)) (Eq.refl (\u2191n + 1))))\n\n@[simp] theorem cast_inj {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] [char_zero \u03b1] {m : pos_num}\n    {n : pos_num} : \u2191m = \u2191n \u2194 m = n :=\n  sorry\n\n@[simp] theorem one_le_cast {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (n : pos_num) : 1 \u2264 \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (1 \u2264 \u2191n)) (Eq.symm (cast_to_nat n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 \u2264 \u2191\u2191n)) (Eq.symm nat.cast_one)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u21911 \u2264 \u2191\u2191n)) (propext nat.cast_le))) (to_nat_pos n)))\n\n@[simp] theorem cast_pos {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] (n : pos_num) : 0 < \u2191n :=\n  lt_of_lt_of_le zero_lt_one (one_le_cast n)\n\n@[simp] theorem cast_mul {\u03b1 : Type u_1} [semiring \u03b1] (m : pos_num) (n : pos_num) :\n    \u2191(m * n) = \u2191m * \u2191n :=\n  sorry\n\n@[simp] theorem cmp_eq (m : pos_num) (n : pos_num) : cmp m n = ordering.eq \u2194 m = n := sorry\n\n@[simp] theorem cast_lt {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] {m : pos_num} {n : pos_num} :\n    \u2191m < \u2191n \u2194 m < n :=\n  sorry\n\n@[simp] theorem cast_le {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] {m : pos_num} {n : pos_num} :\n    \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr cast_lt)\n\nend pos_num\n\n\nnamespace num\n\n\ntheorem bit_to_nat (b : Bool) (n : num) : \u2191(bit b n) = nat.bit b \u2191n :=\n  bool.cases_on b\n    (num.cases_on n (Eq.refl \u2191(bit false zero)) fun (n : pos_num) => Eq.refl \u2191(bit false (pos n)))\n    (num.cases_on n (Eq.refl \u2191(bit tt zero)) fun (n : pos_num) => Eq.refl \u2191(bit tt (pos n)))\n\ntheorem cast_succ' {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : num) : \u2191(succ' n) = \u2191n + 1 := sorry\n\ntheorem cast_succ {\u03b1 : Type u_1} [add_monoid \u03b1] [HasOne \u03b1] (n : num) : \u2191(succ n) = \u2191n + 1 :=\n  cast_succ' n\n\n@[simp] theorem cast_add {\u03b1 : Type u_1} [semiring \u03b1] (m : num) (n : num) : \u2191(m + n) = \u2191m + \u2191n :=\n  sorry\n\n@[simp] theorem cast_bit0 {\u03b1 : Type u_1} [semiring \u03b1] (n : num) : \u2191(num.bit0 n) = bit0 \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(num.bit0 n) = bit0 \u2191n)) (Eq.symm (bit0_of_bit0 n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(bit0 n) = bit0 \u2191n)) (bit0.equations._eqn_1 n)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(n + n) = bit0 \u2191n)) (cast_add n n))) (Eq.refl (\u2191n + \u2191n))))\n\n@[simp] theorem cast_bit1 {\u03b1 : Type u_1} [semiring \u03b1] (n : num) : \u2191(num.bit1 n) = bit1 \u2191n := sorry\n\n@[simp] theorem cast_mul {\u03b1 : Type u_1} [semiring \u03b1] (m : num) (n : num) : \u2191(m * n) = \u2191m * \u2191n :=\n  sorry\n\ntheorem size_to_nat (n : num) : \u2191(size n) = nat.size \u2191n :=\n  num.cases_on n (idRhs (0 = nat.size 0) (Eq.symm nat.size_zero))\n    fun (n : pos_num) => idRhs (\u2191(pos_num.size n) = nat.size \u2191n) (pos_num.size_to_nat n)\n\ntheorem size_eq_nat_size (n : num) : \u2191(size n) = nat_size n :=\n  num.cases_on n (idRhs (\u2191(size 0) = \u2191(size 0)) rfl)\n    fun (n : pos_num) => idRhs (\u2191(pos_num.size n) = pos_num.nat_size n) (pos_num.size_eq_nat_size n)\n\ntheorem nat_size_to_nat (n : num) : nat_size n = nat.size \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nat_size n = nat.size \u2191n)) (Eq.symm (size_eq_nat_size n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(size n) = nat.size \u2191n)) (size_to_nat n)))\n      (Eq.refl (nat.size \u2191n)))\n\n@[simp] theorem of_nat'_eq (n : \u2115) : of_nat' n = \u2191n := sorry\n\ntheorem zneg_to_znum (n : num) : -to_znum n = to_znum_neg n :=\n  num.cases_on n (Eq.refl (-to_znum zero)) fun (n : pos_num) => Eq.refl (-to_znum (pos n))\n\ntheorem zneg_to_znum_neg (n : num) : -to_znum_neg n = to_znum n :=\n  num.cases_on n (Eq.refl (-to_znum_neg zero)) fun (n : pos_num) => Eq.refl (-to_znum_neg (pos n))\n\ntheorem to_znum_inj {m : num} {n : num} : to_znum m = to_znum n \u2194 m = n := sorry\n\n@[simp] theorem cast_to_znum {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] (n : num) :\n    \u2191(to_znum n) = \u2191n :=\n  num.cases_on n (idRhs (\u2191(to_znum 0) = \u2191(to_znum 0)) rfl)\n    fun (n : pos_num) => idRhs (\u2191(to_znum (pos n)) = \u2191(to_znum (pos n))) rfl\n\n@[simp] theorem cast_to_znum_neg {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : num) :\n    \u2191(to_znum_neg n) = -\u2191n :=\n  num.cases_on n (idRhs (0 = -0) (Eq.symm neg_zero))\n    fun (n : pos_num) => idRhs (\u2191(to_znum_neg (pos n)) = \u2191(to_znum_neg (pos n))) rfl\n\n@[simp] theorem add_to_znum (m : num) (n : num) : to_znum (m + n) = to_znum m + to_znum n :=\n  num.cases_on m\n    (num.cases_on n (Eq.refl (to_znum (zero + zero)))\n      fun (n : pos_num) => Eq.refl (to_znum (zero + pos n)))\n    fun (m : pos_num) =>\n      num.cases_on n (Eq.refl (to_znum (pos m + zero)))\n        fun (n : pos_num) => Eq.refl (to_znum (pos m + pos n))\n\nend num\n\n\nnamespace pos_num\n\n\ntheorem pred_to_nat {n : pos_num} (h : 1 < n) : \u2191(pred n) = Nat.pred \u2191n := sorry\n\ntheorem sub'_one (a : pos_num) : sub' a 1 = num.to_znum (pred' a) :=\n  pos_num.cases_on a (Eq.refl (sub' one 1)) (fun (a : pos_num) => Eq.refl (sub' (bit1 a) 1))\n    fun (a : pos_num) => Eq.refl (sub' (bit0 a) 1)\n\ntheorem one_sub' (a : pos_num) : sub' 1 a = num.to_znum_neg (pred' a) :=\n  pos_num.cases_on a (Eq.refl (sub' 1 one)) (fun (a : pos_num) => Eq.refl (sub' 1 (bit1 a)))\n    fun (a : pos_num) => Eq.refl (sub' 1 (bit0 a))\n\ntheorem lt_iff_cmp {m : pos_num} {n : pos_num} : m < n \u2194 cmp m n = ordering.lt := iff.rfl\n\ntheorem le_iff_cmp {m : pos_num} {n : pos_num} : m \u2264 n \u2194 cmp m n \u2260 ordering.gt := sorry\n\nend pos_num\n\n\nnamespace num\n\n\ntheorem pred_to_nat (n : num) : \u2191(pred n) = Nat.pred \u2191n := sorry\n\ntheorem ppred_to_nat (n : num) : coe <$> ppred n = nat.ppred \u2191n := sorry\n\ntheorem cmp_swap (m : num) (n : num) : ordering.swap (cmp m n) = cmp n m := sorry\n\ntheorem cmp_eq (m : num) (n : num) : cmp m n = ordering.eq \u2194 m = n := sorry\n\n@[simp] theorem cast_lt {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] {m : num} {n : num} :\n    \u2191m < \u2191n \u2194 m < n :=\n  sorry\n\n@[simp] theorem cast_le {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] {m : num} {n : num} :\n    \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr cast_lt)\n\n@[simp] theorem cast_inj {\u03b1 : Type u_1} [linear_ordered_semiring \u03b1] {m : num} {n : num} :\n    \u2191m = \u2191n \u2194 m = n :=\n  sorry\n\ntheorem lt_iff_cmp {m : num} {n : num} : m < n \u2194 cmp m n = ordering.lt := iff.rfl\n\ntheorem le_iff_cmp {m : num} {n : num} : m \u2264 n \u2194 cmp m n \u2260 ordering.gt := sorry\n\ntheorem bitwise_to_nat {f : num \u2192 num \u2192 num} {g : Bool \u2192 Bool \u2192 Bool} (p : pos_num \u2192 pos_num \u2192 num)\n    (gff : g false false = false) (f00 : f 0 0 = 0)\n    (f0n : \u2200 (n : pos_num), f 0 (pos n) = cond (g false tt) (pos n) 0)\n    (fn0 : \u2200 (n : pos_num), f (pos n) 0 = cond (g tt false) (pos n) 0)\n    (fnn : \u2200 (m n : pos_num), f (pos m) (pos n) = p m n) (p11 : p 1 1 = cond (g tt tt) 1 0)\n    (p1b :\n      \u2200 (b : Bool) (n : pos_num),\n        p 1 (pos_num.bit b n) = bit (g tt b) (cond (g false tt) (pos n) 0))\n    (pb1 :\n      \u2200 (a : Bool) (m : pos_num),\n        p (pos_num.bit a m) 1 = bit (g a tt) (cond (g tt false) (pos m) 0))\n    (pbb :\n      \u2200 (a b : Bool) (m n : pos_num), p (pos_num.bit a m) (pos_num.bit b n) = bit (g a b) (p m n))\n    (m : num) (n : num) : \u2191(f m n) = nat.bitwise g \u2191m \u2191n :=\n  sorry\n\n@[simp] theorem lor_to_nat (m : num) (n : num) : \u2191(lor m n) = nat.lor \u2191m \u2191n := sorry\n\n@[simp] theorem land_to_nat (m : num) (n : num) : \u2191(land m n) = nat.land \u2191m \u2191n := sorry\n\n@[simp] theorem ldiff_to_nat (m : num) (n : num) : \u2191(ldiff m n) = nat.ldiff \u2191m \u2191n := sorry\n\n@[simp] theorem lxor_to_nat (m : num) (n : num) : \u2191(lxor m n) = nat.lxor \u2191m \u2191n := sorry\n\n@[simp] theorem shiftl_to_nat (m : num) (n : \u2115) : \u2191(shiftl m n) = nat.shiftl (\u2191m) n := sorry\n\n@[simp] theorem shiftr_to_nat (m : num) (n : \u2115) : \u2191(shiftr m n) = nat.shiftr (\u2191m) n := sorry\n\n@[simp] theorem test_bit_to_nat (m : num) (n : \u2115) : test_bit m n = nat.test_bit (\u2191m) n := sorry\n\nend num\n\n\nnamespace znum\n\n\n@[simp] theorem cast_zero {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] : \u21910 = 0 := rfl\n\n@[simp] theorem cast_zero' {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] : \u2191zero = 0 := rfl\n\n@[simp] theorem cast_one {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] : \u21911 = 1 := rfl\n\n@[simp] theorem cast_pos {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] (n : pos_num) :\n    \u2191(pos n) = \u2191n :=\n  rfl\n\n@[simp] theorem cast_neg {\u03b1 : Type u_1} [HasZero \u03b1] [HasOne \u03b1] [Add \u03b1] [Neg \u03b1] (n : pos_num) :\n    \u2191(neg n) = -\u2191n :=\n  rfl\n\n@[simp] theorem cast_zneg {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) : \u2191(-n) = -\u2191n :=\n  znum.cases_on n (idRhs (0 = -0) (Eq.symm neg_zero))\n    (fun (n : pos_num) => idRhs (\u2191(-pos n) = \u2191(-pos n)) rfl)\n    fun (n : pos_num) => idRhs (\u2191(-neg n) = --\u2191(-neg n)) (Eq.symm (neg_neg \u2191(-neg n)))\n\ntheorem neg_zero : -0 = 0 := rfl\n\ntheorem zneg_pos (n : pos_num) : -pos n = neg n := rfl\n\ntheorem zneg_neg (n : pos_num) : -neg n = pos n := rfl\n\ntheorem zneg_zneg (n : znum) : --n = n :=\n  znum.cases_on n (Eq.refl ( --zero)) (fun (n : pos_num) => Eq.refl ( --pos n))\n    fun (n : pos_num) => Eq.refl ( --neg n)\n\ntheorem zneg_bit1 (n : znum) : -znum.bit1 n = znum.bitm1 (-n) :=\n  znum.cases_on n (Eq.refl (-znum.bit1 zero)) (fun (n : pos_num) => Eq.refl (-znum.bit1 (pos n)))\n    fun (n : pos_num) => Eq.refl (-znum.bit1 (neg n))\n\ntheorem zneg_bitm1 (n : znum) : -znum.bitm1 n = znum.bit1 (-n) :=\n  znum.cases_on n (Eq.refl (-znum.bitm1 zero)) (fun (n : pos_num) => Eq.refl (-znum.bitm1 (pos n)))\n    fun (n : pos_num) => Eq.refl (-znum.bitm1 (neg n))\n\ntheorem zneg_succ (n : znum) : -succ n = pred (-n) := sorry\n\ntheorem zneg_pred (n : znum) : -pred n = succ (-n) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (-pred n = succ (-n))) (Eq.symm (zneg_zneg (succ (-n))))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (-pred n = --succ (-n))) (zneg_succ (-n))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (-pred n = -pred ( --n))) (zneg_zneg n)))\n        (Eq.refl (-pred n))))\n\n@[simp] theorem neg_of_int (n : \u2124) : \u2191(-n) = -\u2191n :=\n  int.cases_on n\n    (fun (n : \u2115) =>\n      nat.cases_on n (idRhs (\u2191(-0) = \u2191(-0)) rfl)\n        fun (n : \u2115) => idRhs (\u2191(-\u2191(n + 1)) = \u2191(-\u2191(n + 1))) rfl)\n    fun (n : \u2115) =>\n      idRhs (\u2191(-Int.negSucc n) = --\u2191(-Int.negSucc n)) (Eq.symm (zneg_zneg \u2191(-Int.negSucc n)))\n\n@[simp] theorem abs_to_nat (n : znum) : \u2191(abs n) = int.nat_abs \u2191n := sorry\n\n@[simp] theorem abs_to_znum (n : num) : abs (num.to_znum n) = n :=\n  num.cases_on n (idRhs (abs (num.to_znum 0) = abs (num.to_znum 0)) rfl)\n    fun (n : pos_num) => idRhs (abs (num.to_znum (num.pos n)) = abs (num.to_znum (num.pos n))) rfl\n\n@[simp] theorem cast_to_int {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) : \u2191\u2191n = \u2191n := sorry\n\ntheorem bit0_of_bit0 (n : znum) : bit0 n = znum.bit0 n :=\n  znum.cases_on n (idRhs (bit0 0 = bit0 0) rfl)\n    (fun (n : pos_num) =>\n      idRhs (pos (bit0 n) = pos (pos_num.bit0 n)) (congr_arg pos (pos_num.bit0_of_bit0 n)))\n    fun (n : pos_num) =>\n      idRhs (neg (bit0 n) = neg (pos_num.bit0 n)) (congr_arg neg (pos_num.bit0_of_bit0 n))\n\ntheorem bit1_of_bit1 (n : znum) : bit1 n = znum.bit1 n := sorry\n\n@[simp] theorem cast_bit0 {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) :\n    \u2191(znum.bit0 n) = bit0 \u2191n :=\n  sorry\n\n@[simp] theorem cast_bit1 {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) :\n    \u2191(znum.bit1 n) = bit1 \u2191n :=\n  sorry\n\n@[simp] theorem cast_bitm1 {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) :\n    \u2191(znum.bitm1 n) = bit0 \u2191n - 1 :=\n  sorry\n\ntheorem add_zero (n : znum) : n + 0 = n :=\n  znum.cases_on n (Eq.refl (zero + 0)) (fun (n : pos_num) => Eq.refl (pos n + 0))\n    fun (n : pos_num) => Eq.refl (neg n + 0)\n\ntheorem zero_add (n : znum) : 0 + n = n :=\n  znum.cases_on n (Eq.refl (0 + zero)) (fun (n : pos_num) => Eq.refl (0 + pos n))\n    fun (n : pos_num) => Eq.refl (0 + neg n)\n\ntheorem add_one (n : znum) : n + 1 = succ n := sorry\n\nend znum\n\n\nnamespace pos_num\n\n\ntheorem cast_to_znum (n : pos_num) : \u2191n = znum.pos n := sorry\n\ntheorem cast_sub' {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (m : pos_num) (n : pos_num) :\n    \u2191(sub' m n) = \u2191m - \u2191n :=\n  sorry\n\ntheorem to_nat_eq_succ_pred (n : pos_num) : \u2191n = \u2191(pred' n) + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n = \u2191(pred' n) + 1)) (Eq.symm (num.succ'_to_nat (pred' n)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n = \u2191(num.succ' (pred' n)))) (succ'_pred' n))) (Eq.refl \u2191n))\n\ntheorem to_int_eq_succ_pred (n : pos_num) : \u2191n = \u2191\u2191(pred' n) + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n = \u2191\u2191(pred' n) + 1)) (Eq.symm (to_nat_to_int n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191\u2191(pred' n) + 1)) (to_nat_eq_succ_pred n)))\n      (Eq.refl \u2191(\u2191(pred' n) + 1)))\n\nend pos_num\n\n\nnamespace num\n\n\n@[simp] theorem cast_sub' {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (m : num) (n : num) :\n    \u2191(sub' m n) = \u2191m - \u2191n :=\n  sorry\n\n@[simp] theorem of_nat_to_znum (n : \u2115) : to_znum \u2191n = \u2191n := sorry\n\n@[simp] theorem of_nat_to_znum_neg (n : \u2115) : to_znum_neg \u2191n = -\u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (to_znum_neg \u2191n = -\u2191n)) (Eq.symm (of_nat_to_znum n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (to_znum_neg \u2191n = -to_znum \u2191n)) (zneg_to_znum \u2191n)))\n      (Eq.refl (to_znum_neg \u2191n)))\n\ntheorem mem_of_znum' {m : num} {n : znum} : m \u2208 of_znum' n \u2194 n = to_znum m := sorry\n\ntheorem of_znum'_to_nat (n : znum) : coe <$> of_znum' n = int.to_nat' \u2191n := sorry\n\n@[simp] theorem of_znum_to_nat (n : znum) : \u2191(of_znum n) = int.to_nat \u2191n := sorry\n\n@[simp] theorem cast_of_znum {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) :\n    \u2191(of_znum n) = \u2191(int.to_nat \u2191n) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (\u2191(of_znum n) = \u2191(int.to_nat \u2191n))) (Eq.symm (cast_to_nat (of_znum n)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191(of_znum n) = \u2191(int.to_nat \u2191n))) (of_znum_to_nat n)))\n      (Eq.refl \u2191(int.to_nat \u2191n)))\n\n@[simp] theorem sub_to_nat (m : num) (n : num) : \u2191(m - n) = \u2191m - \u2191n := sorry\n\nend num\n\n\nnamespace znum\n\n\n@[simp] theorem cast_add {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (m : znum) (n : znum) :\n    \u2191(m + n) = \u2191m + \u2191n :=\n  sorry\n\n@[simp] theorem cast_succ {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : znum) : \u2191(succ n) = \u2191n + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(succ n) = \u2191n + 1)) (Eq.symm (add_one n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(n + 1) = \u2191n + 1)) (cast_add n 1)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n + \u21911 = \u2191n + 1)) cast_one)) (Eq.refl (\u2191n + 1))))\n\n@[simp] theorem mul_to_int (m : znum) (n : znum) : \u2191(m * n) = \u2191m * \u2191n := sorry\n\ntheorem cast_mul {\u03b1 : Type u_1} [ring \u03b1] (m : znum) (n : znum) : \u2191(m * n) = \u2191m * \u2191n := sorry\n\n@[simp] theorem of_to_int (n : znum) : \u2191\u2191n = n := sorry\n\ntheorem to_of_int (n : \u2124) : \u2191\u2191n = n := sorry\n\ntheorem to_int_inj {m : znum} {n : znum} : \u2191m = \u2191n \u2194 m = n :=\n  { mp := fun (h : \u2191m = \u2191n) => function.left_inverse.injective of_to_int h,\n    mpr := congr_arg fun (x : znum) => \u2191x }\n\n@[simp] theorem of_int_cast {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : \u2124) : \u2191\u2191n = \u2191n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191n = \u2191n)) (Eq.symm (cast_to_int \u2191n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191\u2191\u2191n = \u2191n)) (to_of_int n))) (Eq.refl \u2191n))\n\n@[simp] theorem of_nat_cast {\u03b1 : Type u_1} [add_group \u03b1] [HasOne \u03b1] (n : \u2115) : \u2191\u2191n = \u2191n :=\n  of_int_cast \u2191n\n\n@[simp] theorem of_int'_eq (n : \u2124) : of_int' n = \u2191n := sorry\n\ntheorem cmp_to_int (m : znum) (n : znum) :\n    ordering.cases_on (cmp m n) (\u2191m < \u2191n) (m = n) (\u2191n < \u2191m) :=\n  sorry\n\ntheorem lt_to_int {m : znum} {n : znum} : \u2191m < \u2191n \u2194 m < n := sorry\n\ntheorem le_to_int {m : znum} {n : znum} : \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr lt_to_int)\n\n@[simp] theorem cast_lt {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] {m : znum} {n : znum} :\n    \u2191m < \u2191n \u2194 m < n :=\n  sorry\n\n@[simp] theorem cast_le {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] {m : znum} {n : znum} :\n    \u2191m \u2264 \u2191n \u2194 m \u2264 n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191m \u2264 \u2191n \u2194 m \u2264 n)) (Eq.symm (propext not_lt))))\n    (not_congr cast_lt)\n\n@[simp] theorem cast_inj {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] {m : znum} {n : znum} :\n    \u2191m = \u2191n \u2194 m = n :=\n  sorry\n\n/--\nThis tactic tries to turn an (in)equality about `znum`s to one about `int`s by rewriting.\n```lean\nexample (n : znum) (m : znum) : n \u2264 n + m * m :=\nbegin\n  znum.transfer_rw,\n  exact le_add_of_nonneg_right (mul_self_nonneg _)\nend\n```\n-/\n/--\nThis tactic tries to prove (in)equalities about `znum`s by transfering them to the `int` world and\nthen trying to call `simp`.\n```lean\nexample (n : znum) (m : znum) : n \u2264 n + m * m :=\nbegin\n  znum.transfer,\n  exact mul_self_nonneg _\nend\n```\n-/\nprotected instance linear_order : linear_order znum :=\n  linear_order.mk LessEq Less sorry sorry sorry sorry znum.decidable_le znum.decidable_eq\n    znum.decidable_lt\n\nprotected instance add_comm_group : add_comm_group znum :=\n  add_comm_group.mk Add.add sorry 0 zero_add add_zero Neg.neg\n    (add_group.sub._default Add.add sorry 0 zero_add add_zero Neg.neg) sorry sorry\n\nprotected instance linear_ordered_comm_ring : linear_ordered_comm_ring znum :=\n  linear_ordered_comm_ring.mk add_comm_group.add add_comm_group.add_assoc add_comm_group.zero\n    add_comm_group.zero_add add_comm_group.add_zero add_comm_group.neg add_comm_group.sub\n    add_comm_group.add_left_neg add_comm_group.add_comm Mul.mul sorry 1 sorry sorry sorry sorry\n    linear_order.le linear_order.lt linear_order.le_refl linear_order.le_trans\n    linear_order.le_antisymm sorry sorry sorry linear_order.le_total linear_order.decidable_le\n    linear_order.decidable_eq linear_order.decidable_lt sorry sorry\n\n@[simp] theorem dvd_to_int (m : znum) (n : znum) : \u2191m \u2223 \u2191n \u2194 m \u2223 n := sorry\n\nend znum\n\n\nnamespace pos_num\n\n\ntheorem divmod_to_nat_aux {n : pos_num} {d : pos_num} {q : num} {r : num}\n    (h\u2081 : \u2191r + \u2191d * bit0 \u2191q = \u2191n) (h\u2082 : \u2191r < bit0 1 * \u2191d) :\n    \u2191(prod.snd (divmod_aux d q r)) + \u2191d * \u2191(prod.fst (divmod_aux d q r)) = \u2191n \u2227\n        \u2191(prod.snd (divmod_aux d q r)) < \u2191d :=\n  sorry\n\ntheorem divmod_to_nat (d : pos_num) (n : pos_num) :\n    \u2191n / \u2191d = \u2191(prod.fst (divmod d n)) \u2227 \u2191n % \u2191d = \u2191(prod.snd (divmod d n)) :=\n  sorry\n\n@[simp] theorem div'_to_nat (n : pos_num) (d : pos_num) : \u2191(div' n d) = \u2191n / \u2191d :=\n  Eq.symm (and.left (divmod_to_nat d n))\n\n@[simp] theorem mod'_to_nat (n : pos_num) (d : pos_num) : \u2191(mod' n d) = \u2191n % \u2191d :=\n  Eq.symm (and.right (divmod_to_nat d n))\n\nend pos_num\n\n\nnamespace num\n\n\n@[simp] theorem div_to_nat (n : num) (d : num) : \u2191(n / d) = \u2191n / \u2191d := sorry\n\n@[simp] theorem mod_to_nat (n : num) (d : num) : \u2191(n % d) = \u2191n % \u2191d := sorry\n\ntheorem gcd_to_nat_aux {n : \u2115} {a : num} {b : num} :\n    a \u2264 b \u2192 nat_size (a * b) \u2264 n \u2192 \u2191(gcd_aux n a b) = nat.gcd \u2191a \u2191b :=\n  sorry\n\n@[simp] theorem gcd_to_nat (a : num) (b : num) : \u2191(gcd a b) = nat.gcd \u2191a \u2191b := sorry\n\ntheorem dvd_iff_mod_eq_zero {m : num} {n : num} : m \u2223 n \u2194 n % m = 0 := sorry\n\nprotected instance decidable_dvd : DecidableRel has_dvd.dvd := sorry\n\nend num\n\n\nprotected instance pos_num.decidable_dvd : DecidableRel has_dvd.dvd := sorry\n\nnamespace znum\n\n\n@[simp] theorem div_to_int (n : znum) (d : znum) : \u2191(n / d) = \u2191n / \u2191d := sorry\n\n@[simp] theorem mod_to_int (n : znum) (d : znum) : \u2191(n % d) = \u2191n % \u2191d := sorry\n\n@[simp] theorem gcd_to_nat (a : znum) (b : znum) : \u2191(gcd a b) = int.gcd \u2191a \u2191b := sorry\n\ntheorem dvd_iff_mod_eq_zero {m : znum} {n : znum} : m \u2223 n \u2194 n % m = 0 := sorry\n\nprotected instance has_dvd.dvd.decidable_rel : DecidableRel has_dvd.dvd := sorry\n\nend znum\n\n\nnamespace int\n\n\n/-- Cast a `snum` to the corresponding integer. -/\ndef of_snum : snum \u2192 \u2124 :=\n  snum.rec' (fun (a : Bool) => cond a (-1) 0)\n    fun (a : Bool) (p : snum) (IH : \u2124) => cond a (bit1 IH) (bit0 IH)\n\nend int\n\n\nprotected instance int.snum_coe : has_coe snum \u2124 := has_coe.mk int.of_snum\n\nprotected instance snum.has_lt : HasLess snum := { Less := fun (a b : snum) => \u2191a < \u2191b }\n\nprotected instance snum.has_le : HasLessEq snum := { LessEq := fun (a b : snum) => \u2191a \u2264 \u2191b }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/num/lemmas_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.39364311853287776}}
{"text": "import Lean\n\n@[simp] theorem ex1 (x : Nat) : 2 * x = x + x :=\n  sorry\n\n@[simp] theorem ex2 (xs : List \u03b1) : xs ++ [] = xs :=\n  sorry\n\n@[simp] theorem ex3 (xs ys zs : List \u03b1) : (xs ++ ys) ++ zs = xs ++ (ys ++ zs) :=\n  sorry\n\n@[simp] theorem ex5 (p : Prop) : p \u2228 True :=\n  sorry\n\n@[simp] theorem ex4 (xs : List \u03b1) : \u00ac(x :: xs = []) :=\n  sorry\n\n@[simp] theorem ex6 (p q : Prop) : p \u2228 q \u2194 q \u2228 p:=\n  sorry\n\n@[simp high] theorem ex7 [Add \u03b1] (a b : \u03b1) : a + b = b + a :=\n  sorry\n\n@[simp\u2193] theorem ex8 [Add \u03b1] (p q : Prop) : (\u00ac (p \u2227 q)) = (\u00acp \u2228 \u00acq) :=\n  sorry\n\naxiom aux {\u03b1} (f : List \u03b1 \u2192 List \u03b1) (xs ys : List \u03b1) : f (xs ++ ys) ++ [] = f (xs ++ ys)\n\nopen Lean\nopen Lean.Meta\n\ndef tst1 : MetaM Unit := do\n  let thms  \u2190 Meta.getSimpTheorems\n  trace[Meta.debug] \"{thms.pre}\\n-----\\n{thms.post}\"\n\nset_option trace.Meta.debug true in\n#eval tst1\n\ndef tst2 : MetaM Unit := do\n  let c \u2190 getConstInfo `aux\n  forallTelescopeReducing c.type fun xs type => do\n    match type.eq? with\n    | none => throwError \"unexpected\"\n    | some (_, lhs, _) =>\n      trace[Meta.debug] \"lhs: {lhs}\"\n      let s \u2190 Meta.getSimpTheorems\n      let m \u2190 s.post.getMatch lhs\n      trace[Meta.debug] \"result: {m}\"\n      assert! m.any fun s => s.name? == `ex2\n\n\nset_option trace.Meta.debug true in\n#eval tst2\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/simp1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3936186837618562}}
{"text": "import category_theory.monoidal.rigid\nimport category_theory.monoidal.braided\n\nopen category_theory \n\nnamespace tangle\n\ninductive hom : \u2115 \u2192 \u2115 \u2192 Type\n| id (a) : hom a a\n| associator_hom (a b c : \u2115) : hom ((a + b) + c) (a + (b + c))\n| associator_inv (a b c : \u2115) : hom (a + (b + c)) ((a + b) + c)\n| left_unitor_hom (a) : hom (0 + a) a\n| left_unitor_inv (a) : hom a (0 + a)\n| right_unitor_hom (a) : hom (a + 0) a\n| right_unitor_inv (a) : hom a (a + 0)\n| comp {a b c} (f : hom a b) (g : hom b c) : hom a c\n| tensor {a b c d} (f : hom a b) (g : hom c d) : hom (a + c) (b + d)\n| coevaluation (a) : hom 0 (a + a)\n| evaluation (a) : hom (a + a) 0\n| braiding_hom (a b) : hom (a + b) (b + a)\n| braiding_inv (a b) : hom (b + a) (a + b) \n| twist_hom (a) : hom a a\n| twist_inv (a) : hom a a\n\nopen hom\n\nlocal infixr ` \u27f6\u1d50 `:10 := hom\nlocal infixr ` \u226b `:80 := hom.comp -- type as \\gg\nlocal infixr ` \u2297 `:70 := hom.tensor\nlocal notation `\ud835\udfd9` := hom.id -- type as \\b1\nlocal notation `\u03b1_` := hom.associator_hom\nlocal notation `\u03b1\u207b\u00b9_` := hom.associator_inv\nlocal notation `\u2113_` := hom.left_unitor_hom\nlocal notation `\u2113\u207b\u00b9_` := hom.left_unitor_inv\nlocal notation `\u03c1_` := hom.right_unitor_hom\nlocal notation `\u03c1\u207b\u00b9_` := hom.right_unitor_inv\nlocal notation `\u03b7_` := hom.coevaluation\nlocal notation `\u03b5_` := hom.evaluation\nlocal notation `\u03b2_` := hom.braiding_hom\nlocal notation `\u03b2\u207b\u00b9_` := hom.braiding_inv\nlocal notation `\u03b8_` := hom.twist_hom\nlocal notation `\u03b8\u207b\u00b9_` := hom.twist_inv\n\ninductive hom_equiv : \u03a0 {X Y : \u2115}, (X \u27f6\u1d50 Y) \u2192 (X \u27f6\u1d50 Y) \u2192 Prop\n| refl {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv f f\n| symm {X Y} (f g : X \u27f6\u1d50 Y) : hom_equiv f g \u2192 hom_equiv g f\n| trans {X Y} {f g h : X \u27f6\u1d50 Y} : hom_equiv f g \u2192 hom_equiv g h \u2192 hom_equiv f h\n| comp {X Y Z} {f f' : X \u27f6\u1d50 Y} {g g' : Y \u27f6\u1d50 Z} :\n    hom_equiv f f' \u2192 hom_equiv g g' \u2192 hom_equiv (f \u226b g) (f' \u226b g')\n| tensor {W X Y Z} {f f' : W \u27f6\u1d50 X} {g g' : Y \u27f6\u1d50 Z} :\n    hom_equiv f f' \u2192 hom_equiv g g' \u2192 hom_equiv (f \u2297 g) (f' \u2297 g')\n| comp_id {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv (f \u226b \ud835\udfd9 _) f\n| id_comp {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv (\ud835\udfd9 _ \u226b f) f\n| assoc {X Y U V} (f : X \u27f6\u1d50 U) (g : U \u27f6\u1d50 V) (h : V \u27f6\u1d50 Y) :\n    hom_equiv ((f \u226b g) \u226b h) (f \u226b (g \u226b h))\n| tensor_id {X Y} : hom_equiv ((\ud835\udfd9 X) \u2297 (\ud835\udfd9 Y)) (\ud835\udfd9 _)\n| tensor_comp {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082}\n    (f\u2081 : X\u2081 \u27f6\u1d50 Y\u2081) (f\u2082 : X\u2082 \u27f6\u1d50 Y\u2082) (g\u2081 : Y\u2081 \u27f6\u1d50 Z\u2081) (g\u2082 : Y\u2082 \u27f6\u1d50 Z\u2082) :\n    hom_equiv ((f\u2081 \u226b g\u2081) \u2297 (f\u2082 \u226b g\u2082)) ((f\u2081 \u2297 f\u2082) \u226b (g\u2081 \u2297 g\u2082))\n| associator_hom_inv {X Y Z} : hom_equiv (\u03b1_ X Y Z \u226b \u03b1\u207b\u00b9_ X Y Z) (\ud835\udfd9 _)\n| associator_inv_hom {X Y Z} : hom_equiv (\u03b1\u207b\u00b9_ X Y Z \u226b \u03b1_ X Y Z) (\ud835\udfd9 _)\n| associator_naturality {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083} (f\u2081 : X\u2081 \u27f6\u1d50 Y\u2081) (f\u2082 : X\u2082 \u27f6\u1d50 Y\u2082) (f\u2083 : X\u2083 \u27f6\u1d50 Y\u2083) :\n    hom_equiv (((f\u2081 \u2297 f\u2082) \u2297 f\u2083) \u226b \u03b1_ Y\u2081 Y\u2082 Y\u2083)\n      (\u03b1_ X\u2081 X\u2082 X\u2083 \u226b (f\u2081 \u2297 (f\u2082 \u2297 f\u2083)))\n| right_unitor_hom_inv {X} : hom_equiv (\u03c1_ X \u226b \u03c1\u207b\u00b9_ X) (\ud835\udfd9 _)\n| right_unitor_inv_hom {X} : hom_equiv (\u03c1\u207b\u00b9_ X \u226b \u03c1_ X) (\ud835\udfd9 _)\n| right_unitor_naturality {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv ((f \u2297 \ud835\udfd9 0) \u226b \u03c1_ Y) (\u03c1_ X \u226b f)\n| left_unitor_hom_inv {X} : hom_equiv (\u2113_ X \u226b \u2113\u207b\u00b9_ X) (\ud835\udfd9 _)\n| left_unitor_inv_hom {X} : hom_equiv (\u2113\u207b\u00b9_ X \u226b \u2113_ X) (\ud835\udfd9 _)\n| left_unitor_naturality {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv ((\ud835\udfd9 0 \u2297 f) \u226b \u2113_ Y) (\u2113_ X \u226b f)\n| pentagon {W X Y Z} : hom_equiv\n    ((\u03b1_ W X Y \u2297 \ud835\udfd9 Z) \u226b \u03b1_ W (X + Y) Z \u226b (\ud835\udfd9 W \u2297 \u03b1_ X Y Z))\n    (\u03b1_ (W + X) Y Z \u226b \u03b1_ W X (Y + Z))\n| triangle {X Y} : hom_equiv (\u03b1_ X 0 Y \u226b (\ud835\udfd9 X \u2297 \u2113_ Y)) (\u03c1_ X \u2297 \ud835\udfd9 Y)\n| coevaluation_evaluation {Y} : hom_equiv ((\ud835\udfd9 Y \u2297 \u03b7_ Y) \u226b \u03b1\u207b\u00b9_ Y Y Y \u226b (\u03b5_ Y \u2297 \ud835\udfd9 Y))\n    (\u03c1_ Y \u226b \u2113\u207b\u00b9_ Y)\n| evaluation_coevaluation {X} : hom_equiv ((\u03b7_ X \u2297 \ud835\udfd9 X) \u226b \u03b1_ X X X \u226b (\ud835\udfd9 X \u2297 \u03b5_ X))\n    (\u2113_ X \u226b \u03c1\u207b\u00b9_ X)\n| braiding_hom_inv {X Y} : hom_equiv (\u03b2_ X Y \u226b \u03b2\u207b\u00b9_ X Y) (\ud835\udfd9 _)\n| braiding_inv_hom {X Y} : hom_equiv (\u03b2\u207b\u00b9_ X Y \u226b \u03b2_ X Y) (\ud835\udfd9 _)\n| braiding_naturality  {X X' Y Y' : \u2115} (f : X \u27f6\u1d50 Y) (g : X' \u27f6\u1d50 Y') : hom_equiv\n    ((f \u2297 g) \u226b \u03b2_ Y Y') (\u03b2_ X X' \u226b (g \u2297 f))\n| hexagon_forward {X Y Z} : hom_equiv\n    (\u03b1_ X Y Z \u226b \u03b2_ X (Y + Z) \u226b \u03b1_ Y Z X)\n    ((\u03b2_ X Y \u2297 \ud835\udfd9 Z) \u226b \u03b1_ Y X Z \u226b (\ud835\udfd9 Y \u2297 \u03b2_ X Z))\n| hexagon_reverse {X Y Z} : hom_equiv\n    (\u03b1\u207b\u00b9_ X Y Z \u226b \u03b2_ (X + Y) Z \u226b \u03b1\u207b\u00b9_ Z X Y)\n    ((\ud835\udfd9 X \u2297 \u03b2_ Y Z) \u226b \u03b1\u207b\u00b9_ X Z Y \u226b (\u03b2_ X Z \u2297 \ud835\udfd9 Y))\n| twist_hom_inv {X} : hom_equiv (\u03b8_ X \u226b \u03b8\u207b\u00b9_ X) (\ud835\udfd9 _)\n| twist_inv_hom {X} : hom_equiv (\u03b8\u207b\u00b9_ X \u226b \u03b8_ X) (\ud835\udfd9 _)\n| twist_naturality {X Y} (f : X \u27f6\u1d50 Y) : hom_equiv (f \u226b \u03b8_ Y) (\u03b8_ X \u226b f) \n| twist_braiding {X Y} : hom_equiv (\u03b8_ (X + Y)) ((\u03b8_ X \u2297 \u03b8_ Y) \u226b \u03b2_ X Y \u226b \u03b2_ Y X)\n| twist_left_dual {X} : hom_equiv (\u03b8_ X)\n    (\u2113\u207b\u00b9_ X \u226b (\u03b7_ X \u2297 \ud835\udfd9 _) \u226b ((\ud835\udfd9 _ \u2297 \u03b8_ X) \u2297 \ud835\udfd9 _) \u226b \u03b1_ X X X \u226b (\ud835\udfd9 _ \u2297 \u03b5_ X) \u226b \u03c1_ X)\n\ndef setoid_hom (X Y : \u2115) : setoid (X \u27f6\u1d50 Y) :=\n\u27e8hom_equiv,\n  \u27e8\u03bb f, hom_equiv.refl f, \u03bb f g, hom_equiv.symm f g, \u03bb f g h hfg hgh, hom_equiv.trans hfg hgh\u27e9\u27e9\n\nattribute [instance] setoid_hom\n\nopen hom_equiv\n\ndef category : category \u2115 :=\n{ hom := \u03bb X Y, quotient (setoid_hom X Y),\n  id := \u03bb X, \u27e6hom.id _\u27e7,\n  comp := \u03bb X Y Z f g, quotient.map\u2082 comp (by { intros f f' hf g g' hg, exact comp hf hg }) f g,\n  id_comp' := by { rintro X Y \u27e8f\u27e9, exact quotient.sound (id_comp f) },\n  comp_id' := by { rintro X Y \u27e8f\u27e9, exact quotient.sound (comp_id f) },\n  assoc' := by { rintro W X Y Z \u27e8f\u27e9 \u27e8g\u27e9 \u27e8h\u27e9, exact quotient.sound (assoc f g h) } }\n\nlocal attribute [instance] category\n\ndef monoidal_category : monoidal_category \u2115 :=\n{ tensor_obj := \u03bb X Y, X + Y,\n  tensor_hom := \u03bb X\u2081 Y\u2081 X\u2082 Y\u2082, quotient.map\u2082 tensor $\n    by { intros _ _ h _ _ h', exact hom_equiv.tensor h h'},\n  tensor_id' := \u03bb X Y, quotient.sound tensor_id,\n  tensor_comp' := \u03bb X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082,\n    by { rintros \u27e8f\u2081\u27e9 \u27e8f\u2082\u27e9 \u27e8g\u2081\u27e9 \u27e8g\u2082\u27e9, exact quotient.sound (tensor_comp _ _ _ _) },\n  tensor_unit := 0,\n  associator := \u03bb X Y Z,\n    \u27e8\u27e6associator_hom X Y Z\u27e7, \u27e6associator_inv X Y Z\u27e7, \n      quotient.sound associator_hom_inv, quotient.sound associator_inv_hom\u27e9,\n  associator_naturality' := \u03bb X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083,\n    by { rintros \u27e8f\u2081\u27e9 \u27e8f\u2082\u27e9 \u27e8f\u2083\u27e9, exact quotient.sound (associator_naturality _ _ _) },\n  left_unitor := \u03bb X,\n    \u27e8\u27e6left_unitor_hom X\u27e7, \u27e6left_unitor_inv X\u27e7,\n      quotient.sound left_unitor_hom_inv, quotient.sound left_unitor_inv_hom\u27e9,\n  left_unitor_naturality' := \u03bb X Y, by { rintro \u27e8f\u27e9, exact quotient.sound (left_unitor_naturality _) },\n  right_unitor := \u03bb X,\n    \u27e8\u27e6right_unitor_hom X\u27e7, \u27e6right_unitor_inv X\u27e7, \n      quotient.sound right_unitor_hom_inv, quotient.sound right_unitor_inv_hom\u27e9,\n  right_unitor_naturality' := \u03bb X Y, by { rintro \u27e8f\u27e9, exact quotient.sound (right_unitor_naturality _) },\n  pentagon' := \u03bb W X Y Z, quotient.sound pentagon,\n  triangle' := \u03bb X Y, quotient.sound triangle }\n\nlocal attribute [instance] monoidal_category\n\ndef left_rigid_category : left_rigid_category \u2115 :=\n{ left_dual := \u03bb X, \n  { left_dual := X,\n    exact := \n    { coevaluation := \u27e6coevaluation X\u27e7,\n      evaluation := \u27e6evaluation X\u27e7,\n      coevaluation_evaluation' := quotient.sound hom_equiv.coevaluation_evaluation,\n      evaluation_coevaluation' := quotient.sound hom_equiv.evaluation_coevaluation }}}\n\ndef braided_category : braided_category \u2115 := \n{ braiding := \u03bb X Y, \n  { hom := \u27e6braiding_hom X Y\u27e7,\n    inv := \u27e6braiding_inv X Y\u27e7,\n    hom_inv_id' := quotient.sound hom_equiv.braiding_hom_inv,\n    inv_hom_id' := quotient.sound hom_equiv.braiding_inv_hom },\n  braiding_naturality' := \u03bb W X Y Z,\n    by { rintro \u27e8f\u27e9 \u27e8g\u27e9, exact quotient.sound (hom_equiv.braiding_naturality f g)},\n  hexagon_forward' := \u03bb X Y Z, quotient.sound (hom_equiv.hexagon_forward),\n  hexagon_reverse' := \u03bb X Y Z, quotient.sound (hom_equiv.hexagon_reverse) }\n\nlocal attribute [instance] left_rigid_category\nlocal attribute [instance] braided_category\n\n/--\nExamples of tangles.\n-/\nabbreviation cap := coevaluation 1\nabbreviation cup := evaluation 1\nabbreviation vert := hom.id 1\nabbreviation over := braiding_hom 1 1\nabbreviation under := braiding_inv 1 1\n\n#check cap\n#check cup\n#check vert\n#check over\n#check under\n#check cap \u2297 under\n#check \u03b1_ 2 1 3\n#check \u2113\u207b\u00b9_ (1 + 1)\n#check (vert \u2297 vert) \u226b (\u2113\u207b\u00b9_ (1 + 1)) \u226b (cap \u2297 under) \u226b (over \u2297 under) \u226b (\u03b1\u207b\u00b9_ 2 1 1) \u226b (\ud835\udfd9 4)\n\nexample : 2 \u27f6 4 := \u27e6(cap \u2297 under) \u226b (over \u2297 under)\u27e7 \n\nend tangle\n", "meta": {"author": "yuma-mizuno", "repo": "lean-tangle", "sha": "65e30181ff6475f479d5dceb0472ad808360295d", "save_path": "github-repos/lean/yuma-mizuno-lean-tangle", "path": "github-repos/lean/yuma-mizuno-lean-tangle/lean-tangle-65e30181ff6475f479d5dceb0472ad808360295d/src/tangle.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3936186837618562}}
{"text": "import o_minimal.examples.simple\nimport .choice3\nimport ..tame\nimport ..ultra\n\n-- TODO: for lean core library: modify binder type of dif_pos etc.:\n-- lemma {u} dif_pos' {c : Prop} {h : decidable c} (hc : c) {\u03b1 : Sort u} {t : c \u2192 \u03b1} {e : \u00acc \u2192 \u03b1} : dite c t e = t hc := dif_pos hc\n-- Avoids the need (unexplained) for `open_locale classical` below.\n\n-- The comments above are obsolete, but this still seems like a good idea.\n\nopen o_minimal\n\nvariables {R : Type*} [OQM R]\nvariables {X : set R}\n\nlocal notation `\u00bd` := (1/2 : \u211a)\n\nlemma lt_of_self_add_lt {a b : R} (h : a + a < b + b) : a < b :=\nbegin\n  contrapose! h,\n  exact add_le_add h h\nend\n\nlemma average_between {a b : R} (hab : a < b) : \u00bd \u2022 (a + b) \u2208 set.Ioo a b :=\nbegin\n  split,\n  { apply lt_of_self_add_lt,\n    rw half_add_half,\n    apply add_lt_add_left hab },\n  { apply lt_of_self_add_lt,\n    rw half_add_half,\n    apply add_lt_add_right hab }\nend\n\nlocal attribute [instance] one\n\nlemma one_pos : (0 : R) < 1 :=\nclassical.some_spec _\n\nlemma chosen_one_mem (nX : X.nonempty) (tX : tame X) :\n  (chosen_one X) \u2208 X :=\nbegin\n  -- Administrative nonsense; also handle the trivial \"has least\" case.\n  dsimp [chosen_one, roption.map, roption.orelse_pure, function.comp],\n  split_ifs with hleast hinf hsup hsup hsup;\n    try { dsimp [the_least, the_inf, the_sup] at * },\n  { obtain \u27e8e, he\u27e9 := hleast,\n    rw get_pfun_of_rel_eq_of_rel he,\n    exact he.1 },\n  any_goals {\n    obtain \u27e8a, ha\u27e9 := hinf,\n    simp only [get_pfun_of_rel_eq_of_rel ha] at \u22a2 hsup },\n  any_goals {\n    obtain \u27e8b, hb\u27e9 := hsup,\n    rw get_pfun_of_rel_eq_of_rel hb, try { clear hsup } },\n  -- Now handle the remaining cases, respectively those yielding\n  -- (a+b)/2, b-1, a+1, 0\n  -- where a = inf X, b = sup Y, Y = {b | a < b \u2227 (a, b) \u2286 X}.\n  { have hab : a < b,\n    { obtain \u27e8b', hb'\u27e9 := hb.nonempty,\n      exact lt_of_lt_of_le hb'.1 (hb.1 hb') },\n    have mid : \u00bd \u2022 (a + b) \u2208 set.Ioo a b := average_between hab,\n    obtain \u27e8c, hc, hmc\u27e9 := (lt_is_lub_iff hb).mp mid.2,\n    apply hc.2,\n    exact \u27e8mid.1, hmc\u27e9 },\n  { let Y := {b | a < b \u2227 set.Ioo a b \u2286 X},\n    change \u00ac \u2203 b, is_lub Y b at hsup,\n    have nY : Y.nonempty,\n    { have := above_inf tX ha (\u03bb H, hleast \u27e8a, H, ha.1\u27e9),\n      -- TODO: for_mathlib: ^ is_glb \u2227 mem \u2192 is_least\n      rw mem_above_iff at this,\n      obtain \u27e8b, hab, hb\u27e9 := this,\n      exact \u27e8b, hab, hb\u27e9 },\n    have tY : tame Y,\n    { refine tame_of_def simple_struc _,\n      apply def_set.and,\n      { exact definable_lt def_fun_const def_fun.id },\n      { apply def_set.forall,\n        apply def_set.imp,\n        { apply def_set.and,\n          { exact definable_lt def_fun_const def_fun.snd },\n          { exact definable_lt def_fun.snd def_fun.fst } },\n        { exact def_fun.snd.preimage tX.def_set } } },\n    have := mt (exists_tSup tY nY) hsup,\n    rw not_bdd_above_iff at this,\n    obtain \u27e8b, hb\u2081, hb\u2082\u27e9 := this (a + 1),\n    apply hb\u2081.2 \u27e8lt_add_of_pos_right _ one_pos, hb\u2082\u27e9 },\n  { obtain \u27e8c, hc\u2081, hc\u2082\u27e9 := (lt_is_lub_iff hb).mp (sub_lt_self b one_pos),\n    exact hc\u2081 hc\u2082 },\n  { let Y := {b | set.Iio b \u2286 X},\n    change \u00ac \u2203 b, is_lub Y b at hsup,\n    have nY : Y.nonempty,\n    { have := mt (exists_tInf tX nX) hinf,\n      exact (contains_Iio_or_bdd_below tX).resolve_right this },\n    have tY : tame Y,\n    { refine tame_of_def simple_struc _,\n      apply def_set.forall,\n      apply def_set.imp,\n      { exact definable_lt def_fun.snd def_fun.fst },\n      { exact def_fun.snd.preimage tX.def_set } },\n    have := mt (exists_tSup tY nY) hsup,\n    rw not_bdd_above_iff at this,\n    obtain \u27e8b, hb\u2081, hb\u2082\u27e9 := this 0,\n    apply hb\u2081 hb\u2082 }\nend\n\nlemma chosen_one'_mem (nX : X.nonempty) :\n  (chosen_one' X) \u2208 X :=\nbegin\n  unfold chosen_one',\n  split_ifs with h,\n  { exact chosen_one_mem nX h },\n  { exact classical.some_spec nX }\nend\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/omin/def_choice/choice4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3935143909420169}}
{"text": "/-\nCopyright (c) 2018 Michael Jendrusch. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Jendrusch, Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.monoidal.category\nimport category_theory.adjunction.basic\nimport category_theory.products.basic\n\n/-!\n# (Lax) monoidal functors\n\nA lax monoidal functor `F` between monoidal categories `C` and `D`\nis a functor between the underlying categories equipped with morphisms\n* `\u03b5 : \ud835\udfd9_ D \u27f6 F.obj (\ud835\udfd9_ C)` (called the unit morphism)\n* `\u03bc X Y : (F.obj X) \u2297 (F.obj Y) \u27f6 F.obj (X \u2297 Y)` (called the tensorator, or strength).\nsatisfying various axioms.\n\nA monoidal functor is a lax monoidal functor for which `\u03b5` and `\u03bc` are isomorphisms.\n\nWe show that the composition of (lax) monoidal functors gives a (lax) monoidal functor.\n\nSee also `category_theory.monoidal.functorial` for a typeclass decorating an object-level\nfunction with the additional data of a monoidal functor.\nThis is useful when stating that a pre-existing functor is monoidal.\n\nSee `category_theory.monoidal.natural_transformation` for monoidal natural transformations.\n\nWe show in `category_theory.monoidal.Mon_` that lax monoidal functors take monoid objects\nto monoid objects.\n\n## Future work\n* Oplax monoidal functors.\n\n## References\n\nSee <https://stacks.math.columbia.edu/tag/0FFL>.\n-/\n\nopen category_theory\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nopen category_theory.category\nopen category_theory.functor\n\nnamespace category_theory\n\nsection\n\nopen monoidal_category\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\n          (D : Type u\u2082) [category.{v\u2082} D] [monoidal_category.{v\u2082} D]\n\n/-- A lax monoidal functor is a functor `F : C \u2964 D` between monoidal categories,\nequipped with morphisms `\u03b5 : \ud835\udfd9 _D \u27f6 F.obj (\ud835\udfd9_ C)` and `\u03bc X Y : F.obj X \u2297 F.obj Y \u27f6 F.obj (X \u2297 Y)`,\nsatisfying the appropriate coherences. -/\n-- The direction of `left_unitality` and `right_unitality` as simp lemmas may look strange:\n-- remember the rule of thumb that component indices of natural transformations\n-- \"weigh more\" than structural maps.\n-- (However by this argument `associativity` is currently stated backwards!)\nstructure lax_monoidal_functor extends C \u2964 D :=\n-- unit morphism\n(\u03b5               : \ud835\udfd9_ D \u27f6 obj (\ud835\udfd9_ C))\n-- tensorator\n(\u03bc                : \u03a0 X Y : C, (obj X) \u2297 (obj Y) \u27f6 obj (X \u2297 Y))\n(\u03bc_natural'       : \u2200 {X Y X' Y' : C}\n  (f : X \u27f6 Y) (g : X' \u27f6 Y'),\n  ((map f) \u2297 (map g)) \u226b \u03bc Y Y' = \u03bc X X' \u226b map (f \u2297 g)\n  . obviously)\n-- associativity of the tensorator\n(associativity'   : \u2200 (X Y Z : C),\n    (\u03bc X Y \u2297 \ud835\udfd9 (obj Z)) \u226b \u03bc (X \u2297 Y) Z \u226b map (\u03b1_ X Y Z).hom\n  = (\u03b1_ (obj X) (obj Y) (obj Z)).hom \u226b (\ud835\udfd9 (obj X) \u2297 \u03bc Y Z) \u226b \u03bc X (Y \u2297 Z)\n  . obviously)\n-- unitality\n(left_unitality'  : \u2200 X : C,\n    (\u03bb_ (obj X)).hom\n  = (\u03b5 \u2297 \ud835\udfd9 (obj X)) \u226b \u03bc (\ud835\udfd9_ C) X \u226b map (\u03bb_ X).hom\n  . obviously)\n(right_unitality' : \u2200 X : C,\n    (\u03c1_ (obj X)).hom\n  = (\ud835\udfd9 (obj X) \u2297 \u03b5) \u226b \u03bc X (\ud835\udfd9_ C) \u226b map (\u03c1_ X).hom\n  . obviously)\n\nrestate_axiom lax_monoidal_functor.\u03bc_natural'\nattribute [simp, reassoc] lax_monoidal_functor.\u03bc_natural\nrestate_axiom lax_monoidal_functor.left_unitality'\nattribute [simp] lax_monoidal_functor.left_unitality\nrestate_axiom lax_monoidal_functor.right_unitality'\nattribute [simp] lax_monoidal_functor.right_unitality\nrestate_axiom lax_monoidal_functor.associativity'\nattribute [simp, reassoc] lax_monoidal_functor.associativity\n\n-- When `rewrite_search` lands, add @[search] attributes to\n-- lax_monoidal_functor.\u03bc_natural lax_monoidal_functor.left_unitality\n-- lax_monoidal_functor.right_unitality lax_monoidal_functor.associativity\n\nsection\nvariables {C D}\n\n@[simp, reassoc]\nlemma lax_monoidal_functor.left_unitality_inv (F : lax_monoidal_functor C D) (X : C) :\n  (\u03bb_ (F.obj X)).inv \u226b (F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b F.\u03bc (\ud835\udfd9_ C) X = F.map (\u03bb_ X).inv :=\nbegin\n  rw [iso.inv_comp_eq, F.left_unitality, category.assoc, category.assoc,\n    \u2190F.to_functor.map_comp, iso.hom_inv_id, F.to_functor.map_id, comp_id],\nend\n\n@[simp, reassoc]\nlemma lax_monoidal_functor.right_unitality_inv (F : lax_monoidal_functor C D) (X : C) :\n  (\u03c1_ (F.obj X)).inv \u226b (\ud835\udfd9 (F.obj X) \u2297 F.\u03b5) \u226b F.\u03bc X (\ud835\udfd9_ C) = F.map (\u03c1_ X).inv :=\nbegin\n  rw [iso.inv_comp_eq, F.right_unitality, category.assoc, category.assoc,\n    \u2190F.to_functor.map_comp, iso.hom_inv_id, F.to_functor.map_id, comp_id],\nend\n\n@[simp, reassoc]\nlemma lax_monoidal_functor.associativity_inv (F : lax_monoidal_functor C D) (X Y Z : C) :\n  (\ud835\udfd9 (F.obj X) \u2297 F.\u03bc Y Z) \u226b F.\u03bc 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 (F.\u03bc X Y \u2297 \ud835\udfd9 (F.obj Z)) \u226b F.\u03bc (X \u2297 Y) Z :=\nbegin\n  rw [iso.eq_inv_comp, \u2190F.associativity_assoc,\n    \u2190F.to_functor.map_comp, iso.hom_inv_id, F.to_functor.map_id, comp_id],\nend\n\nend\n\n/--\nA monoidal functor is a lax monoidal functor for which the tensorator and unitor as isomorphisms.\n\nSee <https://stacks.math.columbia.edu/tag/0FFL>.\n-/\nstructure monoidal_functor\nextends lax_monoidal_functor.{v\u2081 v\u2082} C D :=\n(\u03b5_is_iso            : is_iso \u03b5 . tactic.apply_instance)\n(\u03bc_is_iso            : \u03a0 X Y : C, is_iso (\u03bc X Y) . tactic.apply_instance)\n\nattribute [instance] monoidal_functor.\u03b5_is_iso monoidal_functor.\u03bc_is_iso\n\nvariables {C D}\n\n/--\nThe unit morphism of a (strong) monoidal functor as an isomorphism.\n-/\nnoncomputable\ndef monoidal_functor.\u03b5_iso (F : monoidal_functor.{v\u2081 v\u2082} C D) :\n  tensor_unit D \u2245 F.obj (tensor_unit C) :=\nas_iso F.\u03b5\n\n/--\nThe tensorator of a (strong) monoidal functor as an isomorphism.\n-/\nnoncomputable\ndef monoidal_functor.\u03bc_iso (F : monoidal_functor.{v\u2081 v\u2082} C D) (X Y : C) :\n  (F.obj X) \u2297 (F.obj Y) \u2245 F.obj (X \u2297 Y) :=\nas_iso (F.\u03bc X Y)\n\nend\n\nopen monoidal_category\n\nnamespace lax_monoidal_functor\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\n\n/-- The identity lax monoidal functor. -/\n@[simps] def id : lax_monoidal_functor.{v\u2081 v\u2081} C C :=\n{ \u03b5 := \ud835\udfd9 _,\n  \u03bc := \u03bb X Y, \ud835\udfd9 _,\n  .. \ud835\udfed C }\n\ninstance : inhabited (lax_monoidal_functor C C) := \u27e8id C\u27e9\n\nend lax_monoidal_functor\n\nnamespace monoidal_functor\n\nsection\nvariables {C : Type u\u2081} [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D] [monoidal_category.{v\u2082} D]\nvariable (F : monoidal_functor.{v\u2081 v\u2082} C D)\n\nlemma map_tensor {X Y X' Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') :\n  F.map (f \u2297 g) = inv (F.\u03bc X X') \u226b ((F.map f) \u2297 (F.map g)) \u226b F.\u03bc Y Y' :=\nby simp\n\nlemma map_left_unitor (X : C) :\n  F.map (\u03bb_ X).hom = inv (F.\u03bc (\ud835\udfd9_ C) X) \u226b (inv F.\u03b5 \u2297 \ud835\udfd9 (F.obj X)) \u226b (\u03bb_ (F.obj X)).hom :=\nbegin\n  simp only [lax_monoidal_functor.left_unitality],\n  slice_rhs 2 3 { rw \u2190comp_tensor_id, simp, },\n  simp,\nend\n\nlemma map_right_unitor (X : C) :\n  F.map (\u03c1_ X).hom = inv (F.\u03bc X (\ud835\udfd9_ C)) \u226b (\ud835\udfd9 (F.obj X) \u2297 inv F.\u03b5) \u226b (\u03c1_ (F.obj X)).hom :=\nbegin\n  simp only [lax_monoidal_functor.right_unitality],\n  slice_rhs 2 3 { rw \u2190id_tensor_comp, simp, },\n  simp,\nend\n\n/-- The tensorator as a natural isomorphism. -/\nnoncomputable\ndef \u03bc_nat_iso :\n  (functor.prod F.to_functor F.to_functor) \u22d9 (tensor D) \u2245 (tensor C) \u22d9 F.to_functor :=\nnat_iso.of_components\n  (by { intros, apply F.\u03bc_iso })\n  (by { intros, apply F.to_lax_monoidal_functor.\u03bc_natural })\n\n@[simp] lemma \u03bc_iso_hom (X Y : C) : (F.\u03bc_iso X Y).hom = F.\u03bc X Y := rfl\n@[simp, reassoc] lemma \u03bc_inv_hom_id (X Y : C) : (F.\u03bc_iso X Y).inv \u226b F.\u03bc X Y = \ud835\udfd9 _ :=\n(F.\u03bc_iso X Y).inv_hom_id\n@[simp] lemma \u03bc_hom_inv_id (X Y : C) : F.\u03bc X Y \u226b (F.\u03bc_iso X Y).inv = \ud835\udfd9 _ :=\n(F.\u03bc_iso X Y).hom_inv_id\n\n@[simp] lemma \u03b5_iso_hom : F.\u03b5_iso.hom = F.\u03b5 := rfl\n@[simp, reassoc] lemma \u03b5_inv_hom_id : F.\u03b5_iso.inv \u226b F.\u03b5 = \ud835\udfd9 _ := F.\u03b5_iso.inv_hom_id\n@[simp] lemma \u03b5_hom_inv_id : F.\u03b5 \u226b F.\u03b5_iso.inv = \ud835\udfd9 _ := F.\u03b5_iso.hom_inv_id\n\nend\n\nsection\nvariables (C : Type u\u2081) [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\n\n/-- The identity monoidal functor. -/\n@[simps] def id : monoidal_functor.{v\u2081 v\u2081} C C :=\n{ \u03b5 := \ud835\udfd9 _,\n  \u03bc := \u03bb X Y, \ud835\udfd9 _,\n  .. \ud835\udfed C }\n\ninstance : inhabited (monoidal_functor C C) := \u27e8id C\u27e9\n\nend\n\nend monoidal_functor\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D] [monoidal_category.{v\u2082} D]\nvariables {E : Type u\u2083} [category.{v\u2083} E] [monoidal_category.{v\u2083} E]\n\nnamespace lax_monoidal_functor\nvariables (F : lax_monoidal_functor.{v\u2081 v\u2082} C D) (G : lax_monoidal_functor.{v\u2082 v\u2083} D E)\n\n-- The proofs here are horrendous; rewrite_search helps a lot.\n/-- The composition of two lax monoidal functors is again lax monoidal. -/\n@[simps] def comp : lax_monoidal_functor.{v\u2081 v\u2083} C E :=\n{ \u03b5                := G.\u03b5 \u226b (G.map F.\u03b5),\n  \u03bc                := \u03bb X Y, G.\u03bc (F.obj X) (F.obj Y) \u226b G.map (F.\u03bc X Y),\n  \u03bc_natural'       := \u03bb _ _ _ _ f g,\n  begin\n    simp only [functor.comp_map, assoc],\n    rw [\u2190category.assoc, lax_monoidal_functor.\u03bc_natural, category.assoc, \u2190map_comp, \u2190map_comp,\n        \u2190lax_monoidal_functor.\u03bc_natural]\n  end,\n  associativity'   := \u03bb X Y Z,\n  begin\n    dsimp,\n    rw id_tensor_comp,\n    slice_rhs 3 4 { rw [\u2190 G.to_functor.map_id, G.\u03bc_natural], },\n    slice_rhs 1 3 { rw \u2190G.associativity, },\n    rw comp_tensor_id,\n    slice_lhs 2 3 { rw [\u2190 G.to_functor.map_id, G.\u03bc_natural], },\n    rw [category.assoc, category.assoc, category.assoc, category.assoc, category.assoc,\n        \u2190G.to_functor.map_comp, \u2190G.to_functor.map_comp, \u2190G.to_functor.map_comp,\n        \u2190G.to_functor.map_comp, F.associativity],\n  end,\n  left_unitality'  := \u03bb X,\n  begin\n    dsimp,\n    rw [G.left_unitality, comp_tensor_id, category.assoc, category.assoc],\n    apply congr_arg,\n    rw [F.left_unitality, map_comp, \u2190nat_trans.id_app, \u2190category.assoc,\n        \u2190lax_monoidal_functor.\u03bc_natural, nat_trans.id_app, map_id, \u2190category.assoc, map_comp],\n  end,\n  right_unitality' := \u03bb X,\n  begin\n    dsimp,\n    rw [G.right_unitality, id_tensor_comp, category.assoc, category.assoc],\n    apply congr_arg,\n    rw [F.right_unitality, map_comp, \u2190nat_trans.id_app, \u2190category.assoc,\n        \u2190lax_monoidal_functor.\u03bc_natural, nat_trans.id_app, map_id, \u2190category.assoc, map_comp],\n  end,\n  .. (F.to_functor) \u22d9 (G.to_functor) }.\n\ninfixr ` \u2297\u22d9 `:80 := comp\n\nend lax_monoidal_functor\n\nnamespace lax_monoidal_functor\nuniverses v\u2080 u\u2080\nvariables {B : Type u\u2080} [category.{v\u2080} B] [monoidal_category.{v\u2080} B]\nvariables (F : lax_monoidal_functor.{v\u2080 v\u2081} B C) (G : lax_monoidal_functor.{v\u2082 v\u2083} D E)\n\nlocal attribute [simp] \u03bc_natural associativity left_unitality right_unitality\n\n/-- The cartesian product of two lax monoidal functors is lax monoidal. -/\n@[simps]\ndef prod : lax_monoidal_functor (B \u00d7 D) (C \u00d7 E) :=\n{ \u03b5 := (\u03b5 F, \u03b5 G),\n  \u03bc := \u03bb X Y, (\u03bc F X.1 Y.1, \u03bc G X.2 Y.2),\n  .. (F.to_functor).prod (G.to_functor) }\n\nend lax_monoidal_functor\n\nnamespace monoidal_functor\nvariable (C)\n\n/-- The diagonal functor as a monoidal functor. -/\n@[simps]\ndef diag : monoidal_functor C (C \u00d7 C) :=\n{ \u03b5 := \ud835\udfd9 _,\n  \u03bc := \u03bb X Y, \ud835\udfd9 _,\n  .. functor.diag C }\n\nend monoidal_functor\n\nnamespace lax_monoidal_functor\nvariables (F : lax_monoidal_functor.{v\u2081 v\u2082} C D) (G : lax_monoidal_functor.{v\u2081 v\u2083} C E)\n\n/-- The cartesian product of two lax monoidal functors starting from the same monoidal category `C`\n    is lax monoidal. -/\ndef prod' : lax_monoidal_functor C (D \u00d7 E) :=\n(monoidal_functor.diag C).to_lax_monoidal_functor \u2297\u22d9 (F.prod G)\n\n@[simp] lemma prod'_to_functor :\n  (F.prod' G).to_functor = (F.to_functor).prod' (G.to_functor) := rfl\n\n@[simp] lemma prod'_\u03b5 : (F.prod' G).\u03b5 = (F.\u03b5, G.\u03b5) :=\nby { dsimp [prod'], simp }\n\n@[simp] lemma prod'_\u03bc (X Y : C) : (F.prod' G).\u03bc X Y = (F.\u03bc X Y, G.\u03bc X Y) :=\nby { dsimp [prod'], simp }\n\nend lax_monoidal_functor\n\nnamespace monoidal_functor\n\nvariables (F : monoidal_functor.{v\u2081 v\u2082} C D) (G : monoidal_functor.{v\u2082 v\u2083} D E)\n\n/-- The composition of two monoidal functors is again monoidal. -/\n@[simps]\ndef comp : monoidal_functor.{v\u2081 v\u2083} C E :=\n{ \u03b5_is_iso := by { dsimp, apply_instance },\n  \u03bc_is_iso := by { dsimp, apply_instance },\n  .. (F.to_lax_monoidal_functor).comp (G.to_lax_monoidal_functor) }.\n\ninfixr ` \u2297\u22d9 `:80 := comp -- We overload notation; potentially dangerous, but it seems to work.\n\nend monoidal_functor\n\nnamespace monoidal_functor\nuniverses v\u2080 u\u2080\nvariables {B : Type u\u2080} [category.{v\u2080} B] [monoidal_category.{v\u2080} B]\nvariables (F : monoidal_functor.{v\u2080 v\u2081} B C) (G : monoidal_functor.{v\u2082 v\u2083} D E)\n\n/-- The cartesian product of two monoidal functors is monoidal. -/\n@[simps]\ndef prod : monoidal_functor (B \u00d7 D) (C \u00d7 E) :=\n{ \u03b5_is_iso := (is_iso_prod_iff C E).mpr \u27e8\u03b5_is_iso F, \u03b5_is_iso G\u27e9,\n  \u03bc_is_iso := \u03bb X Y, (is_iso_prod_iff C E).mpr \u27e8\u03bc_is_iso F X.1 Y.1, \u03bc_is_iso G X.2 Y.2\u27e9,\n  .. (F.to_lax_monoidal_functor).prod (G.to_lax_monoidal_functor) }\n\nend monoidal_functor\n\nnamespace monoidal_functor\nvariables (F : monoidal_functor.{v\u2081 v\u2082} C D) (G : monoidal_functor.{v\u2081 v\u2083} C E)\n\n/-- The cartesian product of two monoidal functors starting from the same monoidal category `C`\n    is monoidal. -/\ndef prod' : monoidal_functor C (D \u00d7 E) := diag C \u2297\u22d9 (F.prod G)\n\n@[simp] lemma prod'_to_lax_monoidal_functor :\n    (F.prod' G).to_lax_monoidal_functor\n  = (F.to_lax_monoidal_functor).prod' (G.to_lax_monoidal_functor) := rfl\n\nend monoidal_functor\n\n/--\nIf we have a right adjoint functor `G` to a monoidal functor `F`, then `G` has a lax monoidal\nstructure as well.\n-/\n@[simps]\nnoncomputable\ndef monoidal_adjoint (F : monoidal_functor C D) {G : D \u2964 C} (h : F.to_functor \u22a3 G) :\n  lax_monoidal_functor D C :=\n{ to_functor := G,\n  \u03b5 := h.hom_equiv _ _ (inv F.\u03b5),\n  \u03bc := \u03bb X Y,\n    h.hom_equiv _ (X \u2297 Y) (inv (F.\u03bc (G.obj X) (G.obj Y)) \u226b (h.counit.app X \u2297 h.counit.app Y)),\n  \u03bc_natural' := \u03bb X Y X' Y' f g,\n  begin\n    rw [\u2190h.hom_equiv_naturality_left, \u2190h.hom_equiv_naturality_right, equiv.apply_eq_iff_eq, assoc,\n      is_iso.eq_inv_comp, \u2190F.to_lax_monoidal_functor.\u03bc_natural_assoc, is_iso.hom_inv_id_assoc,\n      \u2190tensor_comp, adjunction.counit_naturality, adjunction.counit_naturality, tensor_comp],\n  end,\n  associativity' := \u03bb X Y Z,\n  begin\n    rw [\u2190h.hom_equiv_naturality_right, \u2190h.hom_equiv_naturality_left, \u2190h.hom_equiv_naturality_left,\n      \u2190h.hom_equiv_naturality_left, equiv.apply_eq_iff_eq,\n      \u2190 cancel_epi (F.to_lax_monoidal_functor.\u03bc (G.obj X \u2297 G.obj Y) (G.obj Z)),\n      \u2190 cancel_epi (F.to_lax_monoidal_functor.\u03bc (G.obj X) (G.obj Y) \u2297 \ud835\udfd9 (F.obj (G.obj Z))),\n      F.to_lax_monoidal_functor.associativity_assoc (G.obj X) (G.obj Y) (G.obj Z),\n      \u2190F.to_lax_monoidal_functor.\u03bc_natural_assoc, assoc, is_iso.hom_inv_id_assoc,\n      \u2190F.to_lax_monoidal_functor.\u03bc_natural_assoc, is_iso.hom_inv_id_assoc, \u2190tensor_comp,\n      \u2190tensor_comp, id_comp, functor.map_id, functor.map_id, id_comp, \u2190tensor_comp_assoc,\n      \u2190tensor_comp_assoc, id_comp, id_comp, h.hom_equiv_unit, h.hom_equiv_unit, functor.map_comp,\n      assoc, assoc, h.counit_naturality, h.left_triangle_components_assoc, is_iso.hom_inv_id_assoc,\n      functor.map_comp, assoc, h.counit_naturality, h.left_triangle_components_assoc,\n      is_iso.hom_inv_id_assoc],\n    exact associator_naturality (h.counit.app X) (h.counit.app Y) (h.counit.app Z),\n  end,\n  left_unitality' := \u03bb X,\n  begin\n    rw [\u2190h.hom_equiv_naturality_right, \u2190h.hom_equiv_naturality_left, \u2190equiv.symm_apply_eq,\n      h.hom_equiv_counit, F.map_left_unitor, h.hom_equiv_unit, assoc, assoc, assoc, F.map_tensor,\n      assoc, assoc, is_iso.hom_inv_id_assoc, \u2190tensor_comp_assoc, functor.map_id, id_comp,\n      functor.map_comp, assoc, h.counit_naturality, h.left_triangle_components_assoc,\n      \u2190left_unitor_naturality, \u2190tensor_comp_assoc, id_comp, comp_id],\n  end,\n  right_unitality' := \u03bb X,\n  begin\n    rw [\u2190h.hom_equiv_naturality_right, \u2190h.hom_equiv_naturality_left, \u2190equiv.symm_apply_eq,\n      h.hom_equiv_counit, F.map_right_unitor, assoc, assoc, \u2190right_unitor_naturality,\n      \u2190tensor_comp_assoc, comp_id, id_comp, h.hom_equiv_unit, F.map_tensor, assoc, assoc, assoc,\n      is_iso.hom_inv_id_assoc, functor.map_comp, functor.map_id, \u2190tensor_comp_assoc, assoc,\n      h.counit_naturality, h.left_triangle_components_assoc, id_comp],\n  end }.\n\n/-- If a monoidal functor `F` is an equivalence of categories then its inverse is also monoidal. -/\n@[simps]\nnoncomputable\ndef monoidal_inverse (F : monoidal_functor C D) [is_equivalence F.to_functor] :\n  monoidal_functor D C :=\n{ to_lax_monoidal_functor := monoidal_adjoint F (as_equivalence _).to_adjunction,\n  \u03b5_is_iso := by { dsimp [equivalence.to_adjunction], apply_instance },\n  \u03bc_is_iso := \u03bb X Y, by { dsimp [equivalence.to_adjunction], apply_instance } }\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/monoidal/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.39351438357587154}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport logic.basic\n\n/-!\n# Nonempty types\n\nThis file proves a few extra facts about `nonempty`, which is defined in core Lean.\n\n## Main declarations\n\n* `nonempty.some`: Extracts a witness of nonemptiness using choice. Takes `nonempty \u03b1` explicitly.\n* `classical.arbitrary`: Extracts a witness of nonemptiness using choice. Takes `nonempty \u03b1` as an\n  instance.\n-/\n\nvariables {\u03b1 \u03b2 : Type*} {\u03b3 : \u03b1 \u2192 Type*}\n\nattribute [simp] nonempty_of_inhabited\n\n@[priority 20]\ninstance has_zero.nonempty [has_zero \u03b1] : nonempty \u03b1 := \u27e80\u27e9\n@[priority 20]\ninstance has_one.nonempty [has_one \u03b1] : nonempty \u03b1 := \u27e81\u27e9\n\nlemma exists_true_iff_nonempty {\u03b1 : Sort*} : (\u2203a:\u03b1, true) \u2194 nonempty \u03b1 :=\niff.intro (\u03bb\u27e8a, _\u27e9, \u27e8a\u27e9) (\u03bb\u27e8a\u27e9, \u27e8a, trivial\u27e9)\n\n@[simp] lemma nonempty_Prop {p : Prop} : nonempty p \u2194 p :=\niff.intro (assume \u27e8h\u27e9, h) (assume h, \u27e8h\u27e9)\n\nlemma not_nonempty_iff_imp_false {\u03b1 : Sort*} : \u00ac nonempty \u03b1 \u2194 \u03b1 \u2192 false :=\n\u27e8\u03bb h a, h \u27e8a\u27e9, \u03bb h \u27e8a\u27e9, h a\u27e9\n\n@[simp] lemma nonempty_sigma : nonempty (\u03a3a:\u03b1, \u03b3 a) \u2194 (\u2203a:\u03b1, nonempty (\u03b3 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_subtype {\u03b1} {p : \u03b1 \u2192 Prop} : nonempty (subtype p) \u2194 (\u2203a:\u03b1, p a) :=\niff.intro (assume \u27e8\u27e8a, h\u27e9\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a, h\u27e9\u27e9)\n\n@[simp] lemma nonempty_prod : nonempty (\u03b1 \u00d7 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_pprod {\u03b1 \u03b2} : nonempty (pprod \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_sum : nonempty (\u03b1 \u2295 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with sum.inl a := or.inl \u27e8a\u27e9 | sum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8sum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8sum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psum {\u03b1 \u03b2} : nonempty (psum \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with psum.inl a := or.inl \u27e8a\u27e9 | psum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8psum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8psum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psigma {\u03b1} {\u03b2 : \u03b1 \u2192 Sort*} : nonempty (psigma \u03b2) \u2194 (\u2203a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_empty : \u00ac nonempty empty :=\nassume \u27e8h\u27e9, h.elim\n\n@[simp] lemma nonempty_ulift : nonempty (ulift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty_plift {\u03b1} : nonempty (plift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty.forall {\u03b1} {p : nonempty \u03b1 \u2192 Prop} : (\u2200h:nonempty \u03b1, p h) \u2194 (\u2200a, p \u27e8a\u27e9) :=\niff.intro (assume h a, h _) (assume h \u27e8a\u27e9, h _)\n\n@[simp] lemma nonempty.exists {\u03b1} {p : nonempty \u03b1 \u2192 Prop} : (\u2203h:nonempty \u03b1, p h) \u2194 (\u2203a, p \u27e8a\u27e9) :=\niff.intro (assume \u27e8\u27e8a\u27e9, h\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a\u27e9, h\u27e9)\n\nlemma classical.nonempty_pi {\u03b1} {\u03b2 : \u03b1 \u2192 Sort*} : nonempty (\u03a0a:\u03b1, \u03b2 a) \u2194 (\u2200a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8f\u27e9 a, \u27e8f a\u27e9) (assume f, \u27e8assume a, classical.choice $ f a\u27e9)\n\n/-- Using `classical.choice`, lifts a (`Prop`-valued) `nonempty` instance to a (`Type`-valued)\n  `inhabited` instance. `classical.inhabited_of_nonempty` already exists, in\n  `core/init/classical.lean`, but the assumption is not a type class argument,\n  which makes it unsuitable for some applications. -/\nnoncomputable def classical.inhabited_of_nonempty' {\u03b1} [h : nonempty \u03b1] : inhabited \u03b1 :=\n\u27e8classical.choice h\u27e9\n\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible] protected noncomputable def nonempty.some {\u03b1} (h : nonempty \u03b1) : \u03b1 :=\nclassical.choice h\n\n/-- Using `classical.choice`, extracts a term from a `nonempty` type. -/\n@[reducible] protected noncomputable def classical.arbitrary (\u03b1) [h : nonempty \u03b1] : \u03b1 :=\nclassical.choice h\n\n/-- Given `f : \u03b1 \u2192 \u03b2`, if `\u03b1` is nonempty then `\u03b2` is also nonempty.\n  `nonempty` cannot be a `functor`, because `functor` is restricted to `Type`. -/\nlemma nonempty.map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) : nonempty \u03b1 \u2192 nonempty \u03b2\n| \u27e8h\u27e9 := \u27e8f h\u27e9\n\nprotected lemma nonempty.map2 {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : nonempty \u03b1 \u2192 nonempty \u03b2 \u2192 nonempty \u03b3\n| \u27e8x\u27e9 \u27e8y\u27e9 := \u27e8f x y\u27e9\n\nprotected lemma nonempty.congr {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b1) :\n  nonempty \u03b1 \u2194 nonempty \u03b2 :=\n\u27e8nonempty.map f, nonempty.map g\u27e9\n\nlemma nonempty.elim_to_inhabited {\u03b1 : Sort*} [h : nonempty \u03b1] {p : Prop}\n  (f : inhabited \u03b1 \u2192 p) : p :=\nh.elim $ f \u2218 inhabited.mk\n\ninstance {\u03b1 \u03b2} [h : nonempty \u03b1] [h2 : nonempty \u03b2] : nonempty (\u03b1 \u00d7 \u03b2) :=\nh.elim $ \u03bb g, h2.elim $ \u03bb g2, \u27e8\u27e8g, g2\u27e9\u27e9\n\nlemma subsingleton_of_not_nonempty {\u03b1 : Sort*} (h : \u00ac nonempty \u03b1) : subsingleton \u03b1 :=\n\u27e8\u03bb x, false.elim $ not_nonempty_iff_imp_false.mp h x\u27e9\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/logic/nonempty.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808532, "lm_q2_score": 0.7248702761768248, "lm_q1q2_score": 0.39350544727748277}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.AppBuilder\nimport Lean.Meta.Instances\n\nnamespace Lean.Meta\n\n/-- Create `SizeOf` local instances for applicable parameters, and execute `k` using them. -/\nprivate partial def mkLocalInstances (params : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  loop 0 #[]\nwhere\n  loop (i : Nat) (insts : Array Expr) : MetaM \u03b1 := do\n    if i < params.size then\n      let param := params[i]!\n      let paramType \u2190 inferType param\n      let instType? \u2190 forallTelescopeReducing paramType fun xs _ => do\n        let type := mkAppN param xs\n        try\n          let sizeOf \u2190 mkAppM `SizeOf #[type]\n          let instType \u2190 mkForallFVars xs sizeOf\n          return some instType\n        catch _ =>\n          return none\n      match instType? with\n      | none => loop (i+1) insts\n      | some instType =>\n        let instName \u2190 mkFreshUserName `inst\n        withLocalDecl instName BinderInfo.instImplicit instType fun inst =>\n          loop (i+1) (insts.push inst)\n    else\n      k insts\n\n/--\n  Return `some x` if `fvar` has type of the form `... -> motive ... fvar` where `motive` in `motiveFVars`.\n  That is, `x` \"produces\" one of the recursor motives.\n-/\nprivate def isInductiveHypothesis? (motiveFVars : Array Expr) (fvar : Expr) : MetaM (Option Expr) := do\n  forallTelescopeReducing (\u2190 inferType fvar) fun _ type =>\n    if type.isApp && motiveFVars.contains type.getAppFn then\n      return some type.appArg!\n    else\n      return none\n\nprivate def isInductiveHypothesis (motiveFVars : Array Expr) (fvar : Expr) : MetaM Bool :=\n  return (\u2190 isInductiveHypothesis? motiveFVars fvar).isSome\n\n/--\n  Let `motiveFVars` be free variables for each motive in a kernel recursor, and `minorFVars` the free variables for a minor premise.\n  Then, return `some idx` if `minorFVars[idx]` has a type of the form `... -> motive ... fvar` for some `motive` in `motiveFVars`.\n-/\nprivate def isRecField? (motiveFVars : Array Expr) (minorFVars : Array Expr) (fvar : Expr) : MetaM (Option Nat) := do\n  let mut idx := 0\n  for minorFVar in minorFVars do\n    if let some fvar' \u2190 isInductiveHypothesis? motiveFVars minorFVar then\n      if fvar == fvar'.getAppFn then\n        return some idx\n    idx := idx + 1\n  return none\n\nprivate partial def mkSizeOfMotives (motiveFVars : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  loop 0 #[]\nwhere\n  loop (i : Nat) (motives : Array Expr) : MetaM \u03b1 := do\n    if i < motiveFVars.size then\n      let type \u2190 inferType motiveFVars[i]!\n      let motive \u2190 forallTelescopeReducing type fun xs _ => do\n        mkLambdaFVars xs <| mkConst ``Nat\n      trace[Meta.sizeOf] \"motive: {motive}\"\n      loop (i+1) (motives.push motive)\n    else\n      k motives\n\nprivate partial def ignoreField (x : Expr) : MetaM Bool := do\n  let type \u2190 whnf (\u2190 inferType x)\n  if type.isForall then\n    -- TODO: add support for finite domains\n    if type.isArrow && type.bindingDomain!.isConstOf ``Unit then\n      ignoreField type.bindingBody!\n    else\n      return true\n  else\n    return false\n\n/-- See `ignoreField`. We have support for functions of the form `Unit \u2192 ...` -/\nprivate partial def mkSizeOfRecFieldFormIH (ih : Expr) : MetaM Expr := do\n  if (\u2190 whnf (\u2190 inferType ih)).isForall then\n     mkSizeOfRecFieldFormIH (mkApp ih (mkConst ``Unit.unit))\n  else\n     return ih\n\nprivate partial def mkSizeOfMinors (motiveFVars : Array Expr) (minorFVars : Array Expr) (minorFVars' : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  assert! minorFVars.size == minorFVars'.size\n  loop 0 #[]\nwhere\n  loop (i : Nat) (minors : Array Expr) : MetaM \u03b1 := do\n    if i < minorFVars.size then\n      forallTelescopeReducing (\u2190 inferType minorFVars[i]!) fun xs _ => do\n      forallBoundedTelescope (\u2190 inferType minorFVars'[i]!) xs.size fun xs' _ => do\n        let mut minor \u2190 mkNumeral (mkConst ``Nat) 1\n        for x in xs, x' in xs' do\n          unless (\u2190 isInductiveHypothesis motiveFVars x) do\n          unless (\u2190 ignoreField x) do -- we suppress higher-order fields\n            match (\u2190 isRecField? motiveFVars xs x) with\n            | some idx => minor \u2190 mkAdd minor (\u2190 mkSizeOfRecFieldFormIH xs'[idx]!)\n            | none     => minor \u2190 mkAdd minor (\u2190 mkAppM ``SizeOf.sizeOf #[x'])\n        minor \u2190 mkLambdaFVars xs' minor\n        trace[Meta.sizeOf] \"minor: {minor}\"\n        loop (i+1) (minors.push minor)\n    else\n      k minors\n\n/--\n  Create a \"sizeOf\" function with name `declName` using the recursor `recName`.\n-/\npartial def mkSizeOfFn (recName : Name) (declName : Name): MetaM Unit := do\n  trace[Meta.sizeOf] \"recName: {recName}\"\n  let recInfo : RecursorVal \u2190 getConstInfoRec recName\n  forallTelescopeReducing recInfo.type fun xs _ =>\n    let levelParams := recInfo.levelParams.tail! -- universe parameters for declaration being defined\n    let params := xs[:recInfo.numParams]\n    let motiveFVars := xs[recInfo.numParams : recInfo.numParams + recInfo.numMotives]\n    let minorFVars := xs[recInfo.getFirstMinorIdx : recInfo.getFirstMinorIdx + recInfo.numMinors]\n    let indices := xs[recInfo.getFirstIndexIdx : recInfo.getFirstIndexIdx + recInfo.numIndices]\n    let major := xs[recInfo.getMajorIdx]!\n    let nat := mkConst ``Nat\n    mkLocalInstances params fun localInsts =>\n    mkSizeOfMotives motiveFVars fun motives => do\n      let us := levelOne :: levelParams.map mkLevelParam -- universe level parameters for `rec`-application\n      let recFn := mkConst recName us\n      let val := mkAppN recFn (params ++ motives)\n      forallBoundedTelescope (\u2190 inferType val) recInfo.numMinors fun minorFVars' _ =>\n      mkSizeOfMinors motiveFVars minorFVars minorFVars' fun minors => do\n        withInstImplicitAsImplict params do\n          let sizeOfParams := params ++ localInsts ++ indices ++ #[major]\n          let sizeOfType \u2190 mkForallFVars sizeOfParams nat\n          let val := mkAppN val (minors ++ indices ++ #[major])\n          let sizeOfValue \u2190 mkLambdaFVars sizeOfParams val\n          trace[Meta.sizeOf] \"declName: {declName}\"\n          trace[Meta.sizeOf] \"type: {sizeOfType}\"\n          trace[Meta.sizeOf] \"val: {sizeOfValue}\"\n          addDecl <| Declaration.defnDecl {\n            name        := declName\n            levelParams := levelParams\n            type        := sizeOfType\n            value       := sizeOfValue\n            safety      := DefinitionSafety.safe\n            hints       := ReducibilityHints.abbrev\n          }\n\n/--\n  Create `sizeOf` functions for all inductive datatypes in the mutual inductive declaration containing `typeName`\n  The resulting array contains the generated functions names. The `NameMap` maps recursor names into the generated function names.\n  There is a function for each element of the mutual inductive declaration, and for auxiliary recursors for nested inductive types.\n-/\ndef mkSizeOfFns (typeName : Name) : MetaM (Array Name \u00d7 NameMap Name) := do\n  let indInfo \u2190 getConstInfoInduct typeName\n  let recInfo \u2190 getConstInfoRec (mkRecName typeName)\n  let numExtra := recInfo.numMotives - indInfo.all.length -- numExtra > 0 for nested inductive types\n  let mut result := #[]\n  let baseName := indInfo.all.head! ++ `_sizeOf -- we use the first inductive type as the base name for `sizeOf` functions\n  let mut i := 1\n  let mut recMap : NameMap Name := {}\n  for indTypeName in indInfo.all do\n    let sizeOfName := baseName.appendIndexAfter i\n    let recName := mkRecName indTypeName\n    mkSizeOfFn recName sizeOfName\n    recMap := recMap.insert recName sizeOfName\n    result := result.push sizeOfName\n    i := i + 1\n  for j in [:numExtra] do\n    let recName := (mkRecName indInfo.all.head!).appendIndexAfter (j+1)\n    let sizeOfName := baseName.appendIndexAfter i\n    mkSizeOfFn recName sizeOfName\n    recMap := recMap.insert recName sizeOfName\n    result := result.push sizeOfName\n    i := i + 1\n  return (result, recMap)\n\ndef mkSizeOfSpecLemmaName (ctorName : Name) : Name :=\n  ctorName ++ `sizeOf_spec\n\ndef mkSizeOfSpecLemmaInstance (ctorApp : Expr) : MetaM Expr :=\n  matchConstCtor ctorApp.getAppFn (fun _ => throwError \"failed to apply 'sizeOf' spec, constructor expected{indentExpr ctorApp}\") fun ctorInfo _ => do\n    let ctorArgs     := ctorApp.getAppArgs\n    let ctorParams   := ctorArgs[:ctorInfo.numParams]\n    let ctorFields   := ctorArgs[ctorInfo.numParams:]\n    let lemmaName  := mkSizeOfSpecLemmaName ctorInfo.name\n    let lemmaInfo  \u2190 getConstInfo lemmaName\n    let lemmaArity \u2190 forallTelescopeReducing lemmaInfo.type fun xs _ => return xs.size\n    let lemmaArgMask := ctorParams.toArray.map some\n    let lemmaArgMask := lemmaArgMask ++ mkArray (lemmaArity - ctorInfo.numParams - ctorInfo.numFields) (none (\u03b1 := Expr))\n    let lemmaArgMask := lemmaArgMask ++ ctorFields.toArray.map some\n    mkAppOptM lemmaName lemmaArgMask\n\n/-! # SizeOf spec theorem for nested inductive types -/\nnamespace SizeOfSpecNested\n\nstructure Context where\n  indInfo    : InductiveVal\n  sizeOfFns  : Array Name\n  ctorName   : Name\n  params     : Array Expr\n  localInsts : Array Expr\n  recMap     : NameMap Name -- mapping from recursor name into `_sizeOf_<idx>` function name (see `mkSizeOfFns`)\n\nabbrev M := ReaderT Context MetaM\n\ndef throwUnexpected {\u03b1} (msg : MessageData) : M \u03b1 := do\n  throwError \"failed to generate sizeOf theorem for {(\u2190 read).ctorName} (use `set_option genSizeOfSpec false` to disable theorem generation), {msg}\"\n\ndef throwFailed {\u03b1} : M \u03b1 := do\n  throwError \"failed to generate sizeOf theorem for {(\u2190 read).ctorName}, (use `set_option genSizeOfSpec false` to disable theorem generation)\"\n\n/-- Convert a recursor application into a `_sizeOf_<idx>` application. -/\nprivate def recToSizeOf (e : Expr) : M Expr := do\n  matchConstRec e.getAppFn (fun _ => throwFailed) fun info us => do\n    match (\u2190 read).recMap.find? info.name with\n    | none => throwUnexpected m!\"expected recursor application {indentExpr e}\"\n    | some sizeOfName =>\n      let args    := e.getAppArgs\n      let indices := args[info.getFirstIndexIdx : info.getFirstIndexIdx + info.numIndices]\n      let major   := args[info.getMajorIdx]!\n      return mkAppN (mkConst sizeOfName us.tail!) ((\u2190 read).params ++ (\u2190 read).localInsts ++ indices ++ #[major])\n\nmutual\n  /-- Construct minor premise proof for `mkSizeOfAuxLemmaProof`. `ys` contains fields and inductive hypotheses for the minor premise. -/\n  private partial def mkMinorProof (ys : Array Expr) (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.minor] \"{lhs} =?= {rhs}\"\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      match (\u2190 whnfI lhs).natAdd?, (\u2190 whnfI rhs).natAdd? with\n      | some (a\u2081, b\u2081), some (a\u2082, b\u2082) =>\n        let p\u2081 \u2190 mkMinorProof ys a\u2081 a\u2082\n        let p\u2082 \u2190 mkMinorProofStep ys b\u2081 b\u2082\n        mkCongr (\u2190 mkCongrArg (mkConst ``Nat.add) p\u2081) p\u2082\n      | _, _ =>\n        throwUnexpected m!\"expected 'Nat.add' application, lhs is {indentExpr lhs}\\nrhs is{indentExpr rhs}\"\n\n  /--\n    Helper method for `mkMinorProof`. The proof step is one of the following\n    - Reflexivity\n    - Assumption (i.e., using an inductive hypotheses from `ys`)\n    - `mkSizeOfAuxLemma` application. This case happens when we have multiple levels of nesting\n  -/\n  private partial def mkMinorProofStep (ys : Array Expr) (lhs rhs : Expr) : M Expr := do\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      let lhs \u2190 recToSizeOf lhs\n      trace[Meta.sizeOf.minor.step] \"{lhs} =?= {rhs}\"\n      let target \u2190 mkEq lhs rhs\n      for y in ys do\n        if (\u2190 isDefEq (\u2190 inferType y) target) then\n          return y\n      mkSizeOfAuxLemma lhs rhs\n\n  /-- Construct proof of auxiliary lemma. See `mkSizeOfAuxLemma` -/\n  private partial def mkSizeOfAuxLemmaProof (info : InductiveVal) (lhs : Expr) : M Expr := do\n    let lhsArgs := lhs.getAppArgs\n    let sizeOfBaseArgs := lhsArgs[:lhsArgs.size - info.numIndices - 1]\n    let indicesMajor := lhsArgs[lhsArgs.size - info.numIndices - 1:]\n    let sizeOfLevels := lhs.getAppFn.constLevels!\n    let rec\n      /-- Auxiliary function for constructing an `_sizeOf_<idx>` for `ys`,\n        where `ys` are the indices + major.\n        Recall that if `info.name` is part of a mutually inductive declaration, then the resulting application\n        is not necessarily a `lhs.getAppFn` application.\n        The result is an application of one of the `(\u2190 read),sizeOfFns` functions.\n        We use this auxiliary function to builtin the motive of the recursor. -/\n      mkSizeOf (ys : Array Expr) : M Expr := do\n      for sizeOfFn in (\u2190 read).sizeOfFns do\n        let candidate := mkAppN (mkAppN (mkConst sizeOfFn sizeOfLevels) sizeOfBaseArgs) ys\n        if (\u2190 isTypeCorrect candidate) then\n          return candidate\n      throwFailed\n    let major := lhs.appArg!\n    let majorType \u2190 whnf (\u2190 inferType major)\n    let majorTypeArgs := majorType.getAppArgs\n    match majorType.getAppFn.const? with\n    | none => throwFailed\n    | some (_, us) =>\n      let recName := mkRecName info.name\n      let recInfo \u2190 getConstInfoRec recName\n      let r := mkConst recName (levelZero :: us)\n      let r := mkAppN r majorTypeArgs[:info.numParams]\n      forallBoundedTelescope (\u2190 inferType r) recInfo.numMotives fun motiveFVars _ => do\n        let mut r := r\n        -- Add motives\n        for motiveFVar in motiveFVars do\n          let motive \u2190 forallTelescopeReducing (\u2190 inferType motiveFVar) fun ys _ => do\n            let lhs \u2190 mkSizeOf ys\n            let rhs \u2190 mkAppM ``SizeOf.sizeOf #[ys.back]\n            mkLambdaFVars ys (\u2190 mkEq lhs rhs)\n          r := mkApp r motive\n        forallBoundedTelescope (\u2190 inferType r) recInfo.numMinors fun minorFVars _ => do\n          let mut r := r\n          -- Add minors\n          for minorFVar in minorFVars do\n            let minor \u2190 forallTelescopeReducing (\u2190 inferType minorFVar) fun ys target => do\n              let target \u2190 whnf target\n              match target.eq? with\n              | none => throwFailed\n              | some (_, lhs, rhs) =>\n                if (\u2190 isDefEq lhs rhs) then\n                  mkLambdaFVars ys (\u2190 mkEqRefl rhs)\n                else\n                  let lhs \u2190 unfoldDefinition lhs -- Unfold `_sizeOf_<idx>`\n                  -- rhs is of the form `sizeOf (ctor ...)`\n                  let ctorApp := rhs.appArg!\n                  let specLemma \u2190 mkSizeOfSpecLemmaInstance ctorApp\n                  let specEq \u2190 whnf (\u2190 inferType specLemma)\n                  match specEq.eq? with\n                  | none => throwFailed\n                  | some (_, _, rhsExpanded) =>\n                    let lhs_eq_rhsExpanded \u2190 mkMinorProof ys lhs rhsExpanded\n                    let rhsExpanded_eq_rhs \u2190 mkEqSymm specLemma\n                    mkLambdaFVars ys (\u2190 mkEqTrans lhs_eq_rhsExpanded rhsExpanded_eq_rhs)\n            r := mkApp r minor\n          -- Add indices and major\n          return mkAppN r indicesMajor\n\n  /--\n    Generate proof for `C._sizeOf_<idx> t = sizeOf t` where `C._sizeOf_<idx>` is a auxiliary function\n    generated for a nested inductive type in `C`.\n    For example, given\n    ```lean\n    inductive Expr where\n      | app (f : String) (args : List Expr)\n    ```\n    We generate the auxiliary function `Expr._sizeOf_1 : List Expr \u2192 Nat`.\n    To generate the `sizeOf` spec lemma\n    ```\n    sizeOf (Expr.app f args) = 1 + sizeOf f + sizeOf args\n    ```\n    we need an auxiliary lemma for showing `Expr._sizeOf_1 args = sizeOf args`.\n    Recall that `sizeOf (Expr.app f args)` is definitionally equal to `1 + sizeOf f + Expr._sizeOf_1 args`, but\n    `Expr._sizeOf_1 args` is **not** definitionally equal to `sizeOf args`. We need a proof by induction.\n  -/\n  private partial def mkSizeOfAuxLemma (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.aux] \"{lhs} =?= {rhs}\"\n    match lhs.getAppFn.const? with\n    | none => throwFailed\n    | some (fName, us) =>\n      let thmLevelParams \u2190 us.mapM fun\n        | Level.param n => return n\n        | _ => throwFailed\n      let thmName  := fName.appendAfter \"_eq\"\n      if (\u2190 getEnv).contains thmName then\n        -- Auxiliary lemma has already been defined\n        return mkAppN (mkConst thmName us) lhs.getAppArgs\n      else\n        -- Define auxiliary lemma\n        -- First, generalize indices\n        let x := lhs.appArg!\n        let xType \u2190 whnf (\u2190 inferType x)\n        matchConstInduct xType.getAppFn (fun _ => throwFailed) fun info _ => do\n          let params := xType.getAppArgs[:info.numParams]\n          forallTelescopeReducing (\u2190 inferType (mkAppN xType.getAppFn params)) fun indices _ => do\n            let majorType := mkAppN (mkAppN xType.getAppFn params) indices\n            withLocalDeclD `x majorType fun major => do\n              let lhsArgs := lhs.getAppArgs\n              let lhsArgsNew := lhsArgs[:lhsArgs.size - 1 - indices.size] ++ indices ++ #[major]\n              let lhsNew := mkAppN lhs.getAppFn lhsArgsNew\n              let rhsNew \u2190 mkAppM ``SizeOf.sizeOf #[major]\n              let eq \u2190 mkEq lhsNew rhsNew\n              let thmParams := lhsArgsNew\n              let thmType \u2190 mkForallFVars thmParams eq\n              let thmValue \u2190 mkSizeOfAuxLemmaProof info lhsNew\n              let thmValue \u2190 mkLambdaFVars thmParams thmValue\n              trace[Meta.sizeOf] \"thmValue: {thmValue}\"\n              addDecl <| Declaration.thmDecl {\n                name        := thmName\n                levelParams := thmLevelParams\n                type        := thmType\n                value       := thmValue\n              }\n              return mkAppN (mkConst thmName us) lhs.getAppArgs\n\nend\n\n/- Prove SizeOf spec lemma of the form `sizeOf <ctor-application> = 1 + sizeOf <field_1> + ... + sizeOf <field_n> -/\npartial def main (lhs rhs : Expr) : M Expr := do\n  if (\u2190 isDefEq lhs rhs) then\n    mkEqRefl rhs\n  else\n    /- Expand lhs and rhs to obtain `Nat.add` applications -/\n    let lhs \u2190 whnfI lhs            -- Expand `sizeOf (ctor ...)` into `_sizeOf_<idx>` application\n    let lhs \u2190 unfoldDefinition lhs -- Unfold `_sizeOf_<idx>` application into `HAdd.hAdd` application\n    loop lhs rhs\nwhere\n  loop (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.loop] \"{lhs} =?= {rhs}\"\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      match (\u2190 whnfI lhs).natAdd?, (\u2190 whnfI rhs).natAdd? with\n      | some (a\u2081, b\u2081), some (a\u2082, b\u2082) =>\n        let p\u2081 \u2190 loop a\u2081 a\u2082\n        let p\u2082 \u2190 step b\u2081 b\u2082\n        mkCongr (\u2190 mkCongrArg (mkConst ``Nat.add) p\u2081) p\u2082\n      | _, _ =>\n        throwUnexpected m!\"expected 'Nat.add' application, lhs is {indentExpr lhs}\\nrhs is{indentExpr rhs}\"\n\n  step (lhs rhs : Expr) : M Expr := do\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      let lhs \u2190 recToSizeOf lhs\n      mkSizeOfAuxLemma lhs rhs\n\nend SizeOfSpecNested\n\nprivate def mkSizeOfSpecTheorem (indInfo : InductiveVal) (sizeOfFns : Array Name) (recMap : NameMap Name) (ctorName : Name) : MetaM Unit := do\n  let ctorInfo \u2190 getConstInfoCtor ctorName\n  let us := ctorInfo.levelParams.map mkLevelParam\n  let simpAttr \u2190 ofExcept <| getAttributeImpl (\u2190 getEnv) `simp\n  forallTelescopeReducing ctorInfo.type fun xs _ => do\n    let params := xs[:ctorInfo.numParams]\n    let fields := xs[ctorInfo.numParams:]\n    let ctorApp := mkAppN (mkConst ctorName us) xs\n    mkLocalInstances params fun localInsts => do\n      let lhs \u2190 mkAppM ``SizeOf.sizeOf #[ctorApp]\n      let mut rhs \u2190 mkNumeral (mkConst ``Nat) 1\n      for field in fields do\n        unless (\u2190 ignoreField field) do\n          rhs \u2190 mkAdd rhs (\u2190 mkAppM ``SizeOf.sizeOf #[field])\n      let target \u2190 mkEq lhs rhs\n      trace[Meta.sizeOf] \"ctor: {ctorInfo.name}, target: {target}\"\n      let thmName   := mkSizeOfSpecLemmaName ctorName\n      let thmParams := params ++ localInsts ++ fields\n      let thmType \u2190 mkForallFVars thmParams target\n      let thmValue \u2190 if indInfo.isNested then\n        SizeOfSpecNested.main lhs rhs |>.run {\n          indInfo, sizeOfFns, ctorName, params, localInsts, recMap\n        }\n      else\n        mkEqRefl rhs\n      let thmValue \u2190 mkLambdaFVars thmParams thmValue\n      trace[Meta.sizeOf] \"sizeOf spec theorem name: {thmName}\"\n      trace[Meta.sizeOf] \"sizeOf spec theorem type: {thmType}\"\n      trace[Meta.sizeOf] \"sizeOf spec theorem value: {thmValue}\"\n      unless (\u2190 isDefEq (\u2190 inferType thmValue) thmType) do\n        throwError \"type mismatch\"\n      addDecl <| Declaration.thmDecl {\n        name        := thmName\n        levelParams := ctorInfo.levelParams\n        type        := thmType\n        value       := thmValue\n      }\n      simpAttr.add thmName default AttributeKind.global\n\nprivate def mkSizeOfSpecTheorems (indTypeNames : Array Name) (sizeOfFns : Array Name) (recMap : NameMap Name) : MetaM Unit := do\n  for indTypeName in indTypeNames do\n    let indInfo \u2190 getConstInfoInduct indTypeName\n    for ctorName in indInfo.ctors do\n      mkSizeOfSpecTheorem indInfo sizeOfFns recMap ctorName\n  return ()\n\nregister_builtin_option genSizeOf : Bool := {\n  defValue := true\n  descr    := \"generate `SizeOf` instance for inductive types and structures\"\n}\n\nregister_builtin_option genSizeOfSpec : Bool := {\n  defValue := true\n  descr    := \"generate `SizeOf` specificiation theorems for automatically generated instances\"\n}\n\ndef mkSizeOfInstances (typeName : Name) : MetaM Unit := do\n  if (\u2190 getEnv).contains ``SizeOf && genSizeOf.get (\u2190 getOptions) && !(\u2190 isInductivePredicate typeName) then\n    let indInfo \u2190 getConstInfoInduct typeName\n    unless indInfo.isUnsafe do\n      let (fns, recMap) \u2190 mkSizeOfFns typeName\n      for indTypeName in indInfo.all, fn in fns do\n        let indInfo \u2190 getConstInfoInduct indTypeName\n        forallTelescopeReducing indInfo.type fun xs _ =>\n          let params := xs[:indInfo.numParams]\n          withInstImplicitAsImplict params do\n            let indices := xs[indInfo.numParams:]\n            mkLocalInstances params fun localInsts => do\n              let us := indInfo.levelParams.map mkLevelParam\n              let indType := mkAppN (mkConst indTypeName us) xs\n              let sizeOfIndType \u2190 mkAppM ``SizeOf #[indType]\n              withLocalDeclD `m indType fun m => do\n                let v \u2190 mkLambdaFVars #[m] <| mkAppN (mkConst fn us) (params ++ localInsts ++ indices ++ #[m])\n                let sizeOfMk \u2190 mkAppM ``SizeOf.mk #[v]\n                let instDeclName := indTypeName ++ `_sizeOf_inst\n                let instDeclType \u2190 mkForallFVars (xs ++ localInsts) sizeOfIndType\n                let instDeclValue \u2190 mkLambdaFVars (xs ++ localInsts) sizeOfMk\n                trace[Meta.sizeOf] \">> {instDeclName} : {instDeclType}\"\n                addDecl <| Declaration.defnDecl {\n                  name        := instDeclName\n                  levelParams := indInfo.levelParams\n                  type        := instDeclType\n                  value       := instDeclValue\n                  safety      := .safe\n                  hints       := .abbrev\n                }\n                addInstance instDeclName AttributeKind.global (eval_prio default)\n      if genSizeOfSpec.get (\u2190 getOptions) then\n        mkSizeOfSpecTheorems indInfo.all.toArray fns recMap\n\nbuiltin_initialize\n  registerTraceClass `Meta.sizeOf\n\nend Lean.Meta\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Lean/Meta/SizeOf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3933820531972199}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Util.HasConstCache\nimport Lean.Meta.CasesOn\nimport Lean.Meta.Match.Match\nimport Lean.Elab.RecAppSyntax\nimport Lean.Elab.PreDefinition.Basic\nimport Lean.Elab.PreDefinition.Structural.Basic\n\nnamespace Lean.Elab.Structural\nopen Meta\n\nprivate def throwToBelowFailed : MetaM \u03b1 :=\n  throwError \"toBelow failed\"\n\n/-- See `toBelow` -/\nprivate partial def toBelowAux (C : Expr) (belowDict : Expr) (arg : Expr) (F : Expr) : MetaM Expr := do\n  let belowDict \u2190 whnf belowDict\n  trace[Elab.definition.structural] \"belowDict: {belowDict}, arg: {arg}\"\n  match belowDict with\n  | .app (.app (.const `PProd _) d1) d2 =>\n    (do toBelowAux C d1 arg (\u2190 mkAppM `PProd.fst #[F]))\n    <|>\n    (do toBelowAux C d2 arg (\u2190 mkAppM `PProd.snd #[F]))\n  | .app (.app (.const `And _) d1) d2 =>\n    (do toBelowAux C d1 arg (\u2190 mkAppM `And.left #[F]))\n    <|>\n    (do toBelowAux C d2 arg (\u2190 mkAppM `And.right #[F]))\n  | _ => forallTelescopeReducing belowDict fun xs belowDict => do\n    let arg \u2190 zetaReduce arg\n    let argArgs := arg.getAppArgs\n    unless argArgs.size >= xs.size do throwToBelowFailed\n    let n := argArgs.size\n    let argTailArgs := argArgs.extract (n - xs.size) n\n    let belowDict := belowDict.replaceFVars xs argTailArgs\n    match belowDict with\n    | .app belowDictFun belowDictArg =>\n      unless belowDictFun.getAppFn == C do throwToBelowFailed\n      unless \u2190 isDefEq belowDictArg arg do throwToBelowFailed\n      pure (mkAppN F argTailArgs)\n    | _ => throwToBelowFailed\n\n/-- See `toBelow` -/\nprivate def withBelowDict (below : Expr) (numIndParams : Nat) (k : Expr \u2192 Expr \u2192 MetaM \u03b1) : MetaM \u03b1 := do\n  let belowType \u2190 inferType below\n  trace[Elab.definition.structural] \"belowType: {belowType}\"\n  belowType.withApp fun f args => do\n    let motivePos := numIndParams + 1\n    unless motivePos < args.size do throwError \"unexpected 'below' type{indentExpr belowType}\"\n    let pre := mkAppN f (args.extract 0 numIndParams)\n    let preType \u2190 inferType pre\n    forallBoundedTelescope preType (some 1) fun x _ => do\n      let motiveType \u2190 inferType x[0]!\n      withLocalDeclD (\u2190 mkFreshUserName `C) motiveType fun C =>\n        let belowDict := mkApp pre C\n        let belowDict := mkAppN belowDict (args.extract (numIndParams + 1) args.size)\n        k C belowDict\n\n/--\n  `below` is a free variable with type of the form `I.below indParams motive indices major`,\n  where `I` is the name of an inductive datatype.\n\n  For example, when trying to show that the following function terminates using structural recursion\n  ```lean\n  def addAdjacent : List Nat \u2192 List Nat\n  | []       => []\n  | [a]      => [a]\n  | a::b::as => (a+b) :: addAdjacent as\n  ```\n  when we are visiting `addAdjacent as` at `replaceRecApps`, `below` has type\n  `@List.below Nat (fun (x : List Nat) => List Nat) (a::b::as)`\n  The motive `fun (x : List Nat) => List Nat` depends on the actual function we are trying to compute.\n  So, we first replace it with a fresh variable `C` at `withBelowDict`.\n  Recall that `brecOn` implements course-of-values recursion, and `below` can be viewed as a dictionary\n  of the \"previous values\".\n  We search this dictionary using the auxiliary function `toBelowAux`.\n  The dictionary is built using the `PProd` (`And` for inductive predicates).\n  We keep searching it until we find `C recArg`, where `C` is the auxiliary fresh variable created at `withBelowDict`.  -/\nprivate partial def toBelow (below : Expr) (numIndParams : Nat) (recArg : Expr) : MetaM Expr := do\n  withBelowDict below numIndParams fun C belowDict =>\n    toBelowAux C belowDict recArg below\n\n/--\n  This method is used after `matcherApp.addArg arg` to check whether the new type of `arg` has been \"refined/modified\"\n  in at least one alternative.\n-/\ndef refinedArgType (matcherApp : MatcherApp) (arg : Expr) : MetaM Bool := do\n  let argType \u2190 inferType arg\n  (Array.zip matcherApp.alts matcherApp.altNumParams).anyM fun (alt, numParams) =>\n    lambdaTelescope alt fun xs _ => do\n      if xs.size >= numParams then\n        let refinedArg := xs[numParams - 1]!\n        return !(\u2190 isDefEq (\u2190 inferType refinedArg) argType)\n      else\n        return false\n\nprivate partial def replaceRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (below : Expr) (e : Expr) : M Expr :=\n  let containsRecFn (e : Expr) : StateRefT (HasConstCache recFnName) M Bool :=\n    modifyGet (\u00b7.contains e)\n  let rec loop (below : Expr) (e : Expr) : StateRefT (HasConstCache recFnName) M Expr := do\n    if !(\u2190 containsRecFn e) then\n      return e\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c (\u2190 loop below d) fun x => do\n        mkLambdaFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c (\u2190 loop below d) fun x => do\n        mkForallFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (\u2190 loop below type) (\u2190 loop below val) fun x => do\n        mkLetFVars #[x] (\u2190 loop below (body.instantiate1 x)) (usedLetOnly := false)\n    | Expr.mdata d b     =>\n      if let some _ := getRecAppSyntax? e then\n        loop below b\n      else\n        return mkMData d (\u2190 loop below b)\n    | Expr.proj n i e    => return mkProj n i (\u2190 loop below e)\n    | Expr.app _ _ =>\n      let processApp (e : Expr) : StateRefT (HasConstCache recFnName) M Expr :=\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let numFixed  := recArgInfo.fixedParams.size\n            let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n            if recArgPos >= args.size then\n              throwError \"insufficient number of parameters at recursive application {indentExpr e}\"\n            let recArg := args[recArgPos]!\n            -- For reflexive type, we may have nested recursive applications in recArg\n            let recArg \u2190 loop below recArg\n            let f \u2190 try toBelow below recArgInfo.indParams.size recArg catch  _ => throwError \"failed to eliminate recursive application{indentExpr e}\"\n            -- Recall that the fixed parameters are not in the scope of the `brecOn`. So, we skip them.\n            let argsNonFixed := args.extract numFixed args.size\n            -- The function `f` does not explicitly take `recArg` and its indices as arguments. So, we skip them too.\n            let mut fArgs := #[]\n            for i in [:argsNonFixed.size] do\n              if recArgInfo.pos != i && !recArgInfo.indicesPos.contains i then\n                let arg := argsNonFixed[i]!\n                let arg \u2190 replaceRecApps recFnName recArgInfo below arg\n                fArgs := fArgs.push arg\n            return mkAppN f fArgs\n          else\n            return mkAppN (\u2190 loop below f) (\u2190 args.mapM (loop below))\n      match (\u2190 matchMatcherApp? e) with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo.recArgPos e then\n          processApp e\n        else\n          /- Here is an example we currently do not handle\n             ```\n             def g (xs : List Nat) : Nat :=\n             match xs with\n             | [] => 0\n             | y::ys =>\n               match ys with\n               | []       => 1\n               | _::_::zs => g zs + 1\n               | zs       => g ys + 2\n             ```\n             We are matching on `ys`, but still using `ys` in the third alternative.\n             If we push the `below` argument over the dependent match it will be able to eliminate recursive call using `zs`.\n             To make it work, users have to write the third alternative as `| zs => g zs + 2`\n             If this is too annoying in practice, we may replace `ys` with the matching term, but\n             this may generate weird error messages, when it doesn't work. -/\n          trace[Elab.definition.structural] \"below before matcherApp.addArg: {below} : {\u2190 inferType below}\"\n          let matcherApp \u2190 mapError (matcherApp.addArg below) (fun msg => \"failed to add `below` argument to 'matcher' application\" ++ indentD msg)\n          if !(\u2190 refinedArgType matcherApp below) then\n            processApp e\n          else\n            let altsNew \u2190 (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n              lambdaTelescope alt fun xs altBody => do\n                trace[Elab.definition.structural] \"altNumParams: {numParams}, xs: {xs}\"\n                unless xs.size >= numParams do\n                  throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n                let belowForAlt := xs[numParams - 1]!\n                mkLambdaFVars xs (\u2190 loop belowForAlt altBody)\n            pure { matcherApp with alts := altsNew }.toExpr\n      | none =>\n      match (\u2190 toCasesOnApp? e) with\n      | some casesOnApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo.recArgPos e then\n          processApp e\n        else if let some casesOnApp \u2190 casesOnApp.addArg? below (checkIfRefined := true) then\n          let altsNew \u2190 (Array.zip casesOnApp.alts casesOnApp.altNumParams).mapM fun (alt, numParams) =>\n            lambdaTelescope alt fun xs altBody => do\n              unless xs.size >= numParams do\n                throwError \"unexpected `casesOn` application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n              let belowForAlt := xs[numParams]!\n              mkLambdaFVars xs (\u2190 loop belowForAlt altBody)\n          return { casesOnApp with alts := altsNew }.toExpr\n        else\n          processApp e\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop below e |>.run' {}\n\ndef mkBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  trace[Elab.definition.structural] \"mkBRecOn: {value}\"\n  let type  := (\u2190 inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]!\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive \u2190 mkForallFVars otherArgs type\n  let mut brecOnUniv \u2190 getLevel motive\n  trace[Elab.definition.structural] \"brecOn univ: {brecOnUniv}\"\n  let useBInductionOn := recArgInfo.reflexive && brecOnUniv == levelZero\n  if recArgInfo.reflexive && brecOnUniv != levelZero then\n    brecOnUniv \u2190 decLevel brecOnUniv\n  let motive \u2190 mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n    if useBInductionOn then\n      Lean.mkConst (mkBInductionOnName recArgInfo.indName) recArgInfo.indLevels\n    else\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) (brecOnUniv :: recArgInfo.indLevels)\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType \u2190 inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]!\n    let FType \u2190 inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType \u2190 instantiateForall FType recArgInfo.indIndices\n    let FType \u2190 instantiateForall FType #[major]\n    forallBoundedTelescope FType (some 1) fun below _ => do\n      -- TODO: `below` user name is `f`, and it will make a global `f` to be pretty printed as `_root_.f` in error messages.\n      -- We should add an option to `forallBoundedTelescope` to ensure fresh names are used.\n      let below := below[0]!\n      let valueNew     \u2190 replaceRecApps recFnName recArgInfo below value\n      let Farg         \u2190 mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n      let brecOn       := mkApp brecOn Farg\n      return mkAppN brecOn otherArgs\n\nend Lean.Elab.Structural\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Lean/Elab/PreDefinition/Structural/BRecOn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.39338204551425093}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.analysis.normed_space.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l \n\nnamespace Mathlib\n\n/-!\n# Extended norm\n\nIn this file we define a structure `enorm \ud835\udd5c V` representing an extended norm (i.e., a norm that can\ntake the value `\u221e`) on a vector space `V` over a normed field `\ud835\udd5c`. We do not use `class` for\nan `enorm` because the same space can have more than one extended norm. For example, the space of\nmeasurable functions `f : \u03b1 \u2192 \u211d` has a family of `L_p` extended norms.\n\nWe prove some basic inequalities, then define\n\n* `emetric_space` structure on `V` corresponding to `e : enorm \ud835\udd5c V`;\n* the subspace of vectors with finite norm, called `e.finite_subspace`;\n* a `normed_space` structure on this space.\n\nThe last definition is an instance because the type involves `e`.\n\n## Implementation notes\n\nWe do not define extended normed groups. They can be added to the chain once someone will need them.\n\n## Tags\n\nnormed space, extended norm\n-/\n\n/-- Extended norm on a vector space. As in the case of normed spaces, we require only\n`\u2225c \u2022 x\u2225 \u2264 \u2225c\u2225 * \u2225x\u2225` in the definition, then prove an equality in `map_smul`. -/\nstructure enorm (\ud835\udd5c : Type u_1) (V : Type u_2) [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] \nwhere\n  to_fun : V \u2192 ennreal\n  eq_zero' : \u2200 (x : V), to_fun x = 0 \u2192 x = 0\n  map_add_le' : \u2200 (x y : V), to_fun (x + y) \u2264 to_fun x + to_fun y\n  map_smul_le' : \u2200 (c : \ud835\udd5c) (x : V), to_fun (c \u2022 x) \u2264 \u2191(nnnorm c) * to_fun x\n\nnamespace enorm\n\n\nprotected instance has_coe_to_fun {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : has_coe_to_fun (enorm \ud835\udd5c V) :=\n  has_coe_to_fun.mk (fun (x : enorm \ud835\udd5c V) => V \u2192 ennreal) to_fun\n\ntheorem injective_coe_fn {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : function.injective fun (e : enorm \ud835\udd5c V) (x : V) => coe_fn e x := sorry\n\ntheorem ext {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] {e\u2081 : enorm \ud835\udd5c V} {e\u2082 : enorm \ud835\udd5c V} (h : \u2200 (x : V), coe_fn e\u2081 x = coe_fn e\u2082 x) : e\u2081 = e\u2082 :=\n  injective_coe_fn (funext h)\n\ntheorem ext_iff {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] {e\u2081 : enorm \ud835\udd5c V} {e\u2082 : enorm \ud835\udd5c V} : e\u2081 = e\u2082 \u2194 \u2200 (x : V), coe_fn e\u2081 x = coe_fn e\u2082 x :=\n  { mp := fun (h : e\u2081 = e\u2082) (x : V) => h \u25b8 rfl, mpr := ext }\n\n@[simp] theorem coe_inj {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] {e\u2081 : enorm \ud835\udd5c V} {e\u2082 : enorm \ud835\udd5c V} : \u21d1e\u2081 = \u21d1e\u2082 \u2194 e\u2081 = e\u2082 :=\n  function.injective.eq_iff injective_coe_fn\n\n@[simp] theorem map_smul {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (c : \ud835\udd5c) (x : V) : coe_fn e (c \u2022 x) = \u2191(nnnorm c) * coe_fn e x := sorry\n\n@[simp] theorem map_zero {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : coe_fn e 0 = 0 := sorry\n\n@[simp] theorem eq_zero_iff {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) {x : V} : coe_fn e x = 0 \u2194 x = 0 :=\n  { mp := eq_zero' e x, mpr := fun (h : x = 0) => Eq.symm h \u25b8 map_zero e }\n\n@[simp] theorem map_neg {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : V) : coe_fn e (-x) = coe_fn e x := sorry\n\ntheorem map_sub_rev {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : V) (y : V) : coe_fn e (x - y) = coe_fn e (y - x) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn e (x - y) = coe_fn e (y - x))) (Eq.symm (neg_sub y x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn e (-(y - x)) = coe_fn e (y - x))) (map_neg e (y - x))))\n      (Eq.refl (coe_fn e (y - x))))\n\ntheorem map_add_le {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : V) (y : V) : coe_fn e (x + y) \u2264 coe_fn e x + coe_fn e y :=\n  map_add_le' e x y\n\ntheorem map_sub_le {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : V) (y : V) : coe_fn e (x - y) \u2264 coe_fn e x + coe_fn e y := sorry\n\nprotected instance partial_order {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : partial_order (enorm \ud835\udd5c V) :=\n  partial_order.mk (fun (e\u2081 e\u2082 : enorm \ud835\udd5c V) => \u2200 (x : V), coe_fn e\u2081 x \u2264 coe_fn e\u2082 x)\n    (preorder.lt._default fun (e\u2081 e\u2082 : enorm \ud835\udd5c V) => \u2200 (x : V), coe_fn e\u2081 x \u2264 coe_fn e\u2082 x) sorry sorry sorry\n\n/-- The `enorm` sending each non-zero vector to infinity. -/\nprotected instance has_top {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : has_top (enorm \ud835\udd5c V) :=\n  has_top.mk (mk (fun (x : V) => ite (x = 0) 0 \u22a4) sorry sorry sorry)\n\nprotected instance inhabited {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : Inhabited (enorm \ud835\udd5c V) :=\n  { default := \u22a4 }\n\ntheorem top_map {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] {x : V} (hx : x \u2260 0) : coe_fn \u22a4 x = \u22a4 :=\n  if_neg hx\n\nprotected instance semilattice_sup_top {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] : semilattice_sup_top (enorm \ud835\udd5c V) :=\n  semilattice_sup_top.mk \u22a4 LessEq Less sorry sorry sorry sorry\n    (fun (e\u2081 e\u2082 : enorm \ud835\udd5c V) => mk (fun (x : V) => max (coe_fn e\u2081 x) (coe_fn e\u2082 x)) sorry sorry sorry) sorry sorry sorry\n\n@[simp] theorem coe_max {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e\u2081 : enorm \ud835\udd5c V) (e\u2082 : enorm \ud835\udd5c V) : \u21d1(e\u2081 \u2294 e\u2082) = fun (x : V) => max (coe_fn e\u2081 x) (coe_fn e\u2082 x) :=\n  rfl\n\ntheorem max_map {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e\u2081 : enorm \ud835\udd5c V) (e\u2082 : enorm \ud835\udd5c V) (x : V) : coe_fn (e\u2081 \u2294 e\u2082) x = max (coe_fn e\u2081 x) (coe_fn e\u2082 x) :=\n  rfl\n\n/-- Structure of an `emetric_space` defined by an extended norm. -/\ndef emetric_space {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : emetric_space V :=\n  emetric_space.mk sorry sorry (map_sub_rev e) sorry\n    (uniform_space_of_edist (fun (x y : V) => coe_fn e (x - y)) sorry (map_sub_rev e) sorry)\n\n/-- The subspace of vectors with finite enorm. -/\ndef finite_subspace {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : subspace \ud835\udd5c V :=\n  submodule.mk (set_of fun (x : V) => coe_fn e x < \u22a4) sorry sorry sorry\n\n/-- Metric space structure on `e.finite_subspace`. We use `emetric_space.to_metric_space_of_dist`\nto ensure that this definition agrees with `e.emetric_space`. -/\nprotected instance finite_subspace.metric_space {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : metric_space \u21a5(finite_subspace e) :=\n  let _inst : emetric_space V := emetric_space e;\n  emetric_space.to_metric_space_of_dist (fun (x y : \u21a5(finite_subspace e)) => ennreal.to_real (edist x y)) sorry sorry\n\ntheorem finite_dist_eq {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : \u21a5(finite_subspace e)) (y : \u21a5(finite_subspace e)) : dist x y = ennreal.to_real (coe_fn e (\u2191x - \u2191y)) :=\n  rfl\n\ntheorem finite_edist_eq {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : \u21a5(finite_subspace e)) (y : \u21a5(finite_subspace e)) : edist x y = coe_fn e (\u2191x - \u2191y) :=\n  rfl\n\n/-- Normed group instance on `e.finite_subspace`. -/\nprotected instance finite_subspace.normed_group {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : normed_group \u21a5(finite_subspace e) :=\n  normed_group.mk sorry\n\ntheorem finite_norm_eq {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) (x : \u21a5(finite_subspace e)) : norm x = ennreal.to_real (coe_fn e \u2191x) :=\n  rfl\n\n/-- Normed space instance on `e.finite_subspace`. -/\nprotected instance finite_subspace.normed_space {\ud835\udd5c : Type u_1} {V : Type u_2} [normed_field \ud835\udd5c] [add_comm_group V] [vector_space \ud835\udd5c V] (e : enorm \ud835\udd5c V) : normed_space \ud835\udd5c \u21a5(finite_subspace e) :=\n  normed_space.mk sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/normed_space/enorm.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482762, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3933820455142509}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.filtered_colimit_commutes_finite_limit\nimport category_theory.limits.preserves.functor_category\nimport category_theory.limits.bicones\nimport category_theory.limits.comma\nimport category_theory.limits.preserves.finite\nimport category_theory.limits.shapes.finite_limits\n\n/-!\n# Representably flat functors\n\nWe define representably flat functors as functors such that the category of structured arrows\nover `X` is cofiltered for each `X`. This concept is also known as flat functors as in [Elephant]\nRemark C2.3.7, and this name is suggested by Mike Shulman in\nhttps://golem.ph.utexas.edu/category/2011/06/flat_functors_and_morphisms_of.html to avoid\nconfusion with other notions of flatness.\n\nThis definition is equivalent to left exact functors (functors that preserves finite limits) when\n`C` has all finite limits.\n\n## Main results\n\n* `flat_of_preserves_finite_limits`: If `F : C \u2964 D` preserves finite limits and `C` has all finite\n  limits, then `F` is flat.\n* `preserves_finite_limits_of_flat`: If `F : C \u2964 D` is flat, then it preserves all finite limits.\n* `preserves_finite_limits_iff_flat`: If `C` has all finite limits,\n  then `F` is flat iff `F` is left_exact.\n* `Lan_preserves_finite_limits_of_flat`: If `F : C \u2964 D` is a flat functor between small categories,\n  then the functor `Lan F.op` between presheaves of sets preserves all finite limits.\n* `flat_iff_Lan_flat`: If `C`, `D` are small and `C` has all finite limits, then `F` is flat iff\n  `Lan F.op : (C\u1d52\u1d56 \u2964 Type*) \u2964 (D\u1d52\u1d56 \u2964 Type*)` is flat.\n* `preserves_finite_limits_iff_Lan_preserves_finite_limits`: If `C`, `D` are small and `C` has all\n  finite limits, then `F` preserves finite limits iff `Lan F.op : (C\u1d52\u1d56 \u2964 Type*) \u2964 (D\u1d52\u1d56 \u2964 Type*)`\n  does.\n\n-/\n\nuniverses w v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nnamespace category_theory\n\n\nnamespace structured_arrow_cone\nopen structured_arrow\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2081} D]\nvariables {J : Type w} [small_category J]\nvariables {K : J \u2964 C} (F : C \u2964 D) (c : cone K)\n\n/--\nGiven a cone `c : cone K` and a map `f : X \u27f6 c.X`, we can construct a cone of structured\narrows over `X` with `f` as the cone point. This is the underlying diagram.\n-/\n@[simps]\ndef to_diagram : J \u2964 structured_arrow c.X K :=\n{ obj := \u03bb j, structured_arrow.mk (c.\u03c0.app j),\n  map := \u03bb j k g, structured_arrow.hom_mk g (by simpa) }\n\n/-- Given a diagram of `structured_arrow X F`s, we may obtain a cone with cone point `X`. -/\n@[simps]\ndef diagram_to_cone {X : D} (G : J \u2964 structured_arrow X F) : cone (G \u22d9 proj X F \u22d9 F) :=\n{ X := X, \u03c0 := { app := \u03bb j, (G.obj j).hom } }\n\n/--\nGiven a cone `c : cone K` and a map `f : X \u27f6 F.obj c.X`, we can construct a cone of structured\narrows over `X` with `f` as the cone point.\n-/\n@[simps]\ndef to_cone {X : D} (f : X \u27f6 F.obj c.X) :\n  cone (to_diagram (F.map_cone c) \u22d9 map f \u22d9 pre _ K F) :=\n{ X := mk f, \u03c0 := { app := \u03bb j, hom_mk (c.\u03c0.app j) rfl,\n                    naturality' := \u03bb j k g, by { ext, dsimp, simp } } }\n\nend structured_arrow_cone\n\nsection representably_flat\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\nvariables {E : Type u\u2083} [category.{v\u2083} E]\n\n/--\nA functor `F : C \u2964 D` is representably-flat functor if the comma category `(X/F)`\nis cofiltered for each `X : C`.\n-/\nclass representably_flat (F : C \u2964 D) : Prop :=\n(cofiltered : \u2200 (X : D), is_cofiltered (structured_arrow X F))\n\nattribute [instance] representably_flat.cofiltered\n\nlocal attribute [instance] is_cofiltered.nonempty\n\ninstance representably_flat.id : representably_flat (\ud835\udfed C) :=\nbegin\n  constructor,\n  intro X,\n  haveI : nonempty (structured_arrow X (\ud835\udfed C)) := \u27e8structured_arrow.mk (\ud835\udfd9 _)\u27e9,\n  rsufficesI : is_cofiltered_or_empty (structured_arrow X (\ud835\udfed C)),\n  { constructor },\n  constructor,\n  { intros Y Z,\n    use structured_arrow.mk (\ud835\udfd9 _),\n    use structured_arrow.hom_mk Y.hom (by erw [functor.id_map, category.id_comp]),\n    use structured_arrow.hom_mk Z.hom (by erw [functor.id_map, category.id_comp]) },\n  { intros Y Z f g,\n    use structured_arrow.mk (\ud835\udfd9 _),\n    use structured_arrow.hom_mk Y.hom (by erw [functor.id_map, category.id_comp]),\n    ext,\n    transitivity Z.hom; simp }\nend\n\ninstance representably_flat.comp (F : C \u2964 D) (G : D \u2964 E)\n  [representably_flat F] [representably_flat G] : representably_flat (F \u22d9 G) :=\nbegin\n  constructor,\n  intro X,\n  haveI : nonempty (structured_arrow X (F \u22d9 G)),\n  { have f\u2081 : structured_arrow X G := nonempty.some infer_instance,\n    have f\u2082 : structured_arrow f\u2081.right F := nonempty.some infer_instance,\n    exact \u27e8structured_arrow.mk (f\u2081.hom \u226b G.map f\u2082.hom)\u27e9 },\n  rsufficesI : is_cofiltered_or_empty (structured_arrow X (F \u22d9 G)),\n  { constructor },\n  constructor,\n  { intros Y Z,\n    let W := @is_cofiltered.min (structured_arrow X G) _ _\n      (structured_arrow.mk Y.hom) (structured_arrow.mk Z.hom),\n    let Y' : W \u27f6 _ := is_cofiltered.min_to_left _ _,\n    let Z' : W \u27f6 _ := is_cofiltered.min_to_right _ _,\n\n    let W' := @is_cofiltered.min (structured_arrow W.right F) _ _\n      (structured_arrow.mk Y'.right) (structured_arrow.mk Z'.right),\n    let Y'' : W' \u27f6 _ := is_cofiltered.min_to_left _ _,\n    let Z'' : W' \u27f6 _ := is_cofiltered.min_to_right _ _,\n\n    use structured_arrow.mk (W.hom \u226b G.map W'.hom),\n    use structured_arrow.hom_mk Y''.right (by simp [\u2190 G.map_comp]),\n    use structured_arrow.hom_mk Z''.right (by simp [\u2190 G.map_comp]) },\n  { intros Y Z f g,\n    let W := @is_cofiltered.eq (structured_arrow X G) _ _\n        (structured_arrow.mk Y.hom) (structured_arrow.mk Z.hom)\n        (structured_arrow.hom_mk (F.map f.right) (structured_arrow.w f))\n        (structured_arrow.hom_mk (F.map g.right) (structured_arrow.w g)),\n    let h : W \u27f6 _ := is_cofiltered.eq_hom _ _,\n    let h_cond : h \u226b _ = h \u226b _ := is_cofiltered.eq_condition _ _,\n\n    let W' := @is_cofiltered.eq (structured_arrow W.right F) _ _\n        (structured_arrow.mk h.right) (structured_arrow.mk (h.right \u226b F.map f.right))\n        (structured_arrow.hom_mk f.right rfl)\n        (structured_arrow.hom_mk g.right (congr_arg comma_morphism.right h_cond).symm),\n    let h' : W' \u27f6 _ := is_cofiltered.eq_hom _ _,\n    let h'_cond : h' \u226b _ = h' \u226b _ := is_cofiltered.eq_condition _ _,\n\n    use structured_arrow.mk (W.hom \u226b G.map W'.hom),\n    use structured_arrow.hom_mk h'.right (by simp [\u2190 G.map_comp]),\n    ext,\n    exact (congr_arg comma_morphism.right h'_cond : _) }\nend\n\nend representably_flat\n\nsection has_limit\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2081} D]\n\nlocal attribute [instance] has_finite_limits_of_has_finite_limits_of_size\n\nlemma cofiltered_of_has_finite_limits [has_finite_limits C] : is_cofiltered C :=\n{ cone_objs := \u03bb A B, \u27e8limits.prod A B, limits.prod.fst, limits.prod.snd, trivial\u27e9,\n  cone_maps :=  \u03bb A B f g, \u27e8equalizer f g, equalizer.\u03b9 f g, equalizer.condition f g\u27e9,\n  nonempty := \u27e8\u22a4_ C\u27e9 }\n\nlemma flat_of_preserves_finite_limits [has_finite_limits C] (F : C \u2964 D)\n  [preserves_finite_limits F] : representably_flat F := \u27e8\u03bb X,\nbegin\n  haveI : has_finite_limits (structured_arrow X F) :=\n  begin\n    apply has_finite_limits_of_has_finite_limits_of_size.{v\u2081} (structured_arrow X F),\n    intros J sJ fJ, resetI, constructor\n  end,\n  exact cofiltered_of_has_finite_limits\nend\u27e9\n\nnamespace preserves_finite_limits_of_flat\nopen structured_arrow\nopen structured_arrow_cone\nvariables {J : Type v\u2081} [small_category J] [fin_category J] {K : J \u2964 C}\nvariables (F : C \u2964 D) [representably_flat F] {c : cone K} (hc : is_limit c) (s : cone (K \u22d9 F))\ninclude hc\n\n/--\n(Implementation).\nGiven a limit cone `c : cone K` and a cone `s : cone (K \u22d9 F)` with `F` representably flat,\n`s` can factor through `F.map_cone c`.\n-/\nnoncomputable def lift : s.X \u27f6 F.obj c.X :=\nlet s' := is_cofiltered.cone (to_diagram s \u22d9 structured_arrow.pre _ K F) in\ns'.X.hom \u226b (F.map $ hc.lift $\n  (cones.postcompose ({ app := \u03bb X, \ud835\udfd9 _, naturality' := by simp }\n      : (to_diagram s \u22d9 pre s.X K F) \u22d9 proj s.X F \u27f6 K)).obj $\n  (structured_arrow.proj s.X F).map_cone s')\n\nlemma fac (x : J) : lift F hc s \u226b (F.map_cone c).\u03c0.app x = s.\u03c0.app x :=\nby simpa [lift, \u2190functor.map_comp]\n\nlocal attribute [simp] eq_to_hom_map\n\nlemma uniq {K : J \u2964 C} {c : cone K} (hc : is_limit c)\n  (s : cone (K \u22d9 F)) (f\u2081 f\u2082 : s.X \u27f6 F.obj c.X)\n  (h\u2081 : \u2200 (j : J), f\u2081 \u226b (F.map_cone c).\u03c0.app j = s.\u03c0.app j)\n  (h\u2082 : \u2200 (j : J), f\u2082 \u226b (F.map_cone c).\u03c0.app j = s.\u03c0.app j) : f\u2081 = f\u2082 :=\nbegin\n  -- We can make two cones over the diagram of `s` via `f\u2081` and `f\u2082`.\n  let \u03b1\u2081 : to_diagram (F.map_cone c) \u22d9 map f\u2081 \u27f6 to_diagram s :=\n  { app := \u03bb X, eq_to_hom (by simp [\u2190h\u2081]), naturality' := \u03bb _ _ _, by { ext, simp } },\n  let \u03b1\u2082 : to_diagram (F.map_cone c) \u22d9 map f\u2082 \u27f6 to_diagram s :=\n  { app := \u03bb X, eq_to_hom (by simp [\u2190h\u2082]), naturality' := \u03bb _ _ _, by { ext, simp } },\n  let c\u2081 : cone (to_diagram s \u22d9 pre s.X K F) :=\n    (cones.postcompose (whisker_right \u03b1\u2081 (pre s.X K F) : _)).obj (to_cone F c f\u2081),\n  let c\u2082 : cone (to_diagram s \u22d9 pre s.X K F) :=\n    (cones.postcompose (whisker_right \u03b1\u2082 (pre s.X K F) : _)).obj (to_cone F c f\u2082),\n\n  -- The two cones can then be combined and we may obtain a cone over the two cones since\n  -- `structured_arrow s.X F` is cofiltered.\n  let c\u2080 := is_cofiltered.cone (bicone_mk _ c\u2081 c\u2082),\n  let g\u2081 : c\u2080.X \u27f6 c\u2081.X := c\u2080.\u03c0.app (bicone.left),\n  let g\u2082 : c\u2080.X \u27f6 c\u2082.X := c\u2080.\u03c0.app (bicone.right),\n\n  -- Then `g\u2081.right` and `g\u2082.right` are two maps from the same cone into the `c`.\n  have : \u2200 (j : J), g\u2081.right \u226b c.\u03c0.app j = g\u2082.right \u226b c.\u03c0.app j,\n  { intro j,\n    injection c\u2080.\u03c0.naturality (bicone_hom.left  j) with _ e\u2081,\n    injection c\u2080.\u03c0.naturality (bicone_hom.right j) with _ e\u2082,\n    simpa using e\u2081.symm.trans e\u2082 },\n  have : c.extend g\u2081.right = c.extend g\u2082.right,\n  { unfold cone.extend, congr' 1, ext x, apply this },\n\n  -- And thus they are equal as `c` is the limit.\n  have : g\u2081.right = g\u2082.right,\n  calc g\u2081.right = hc.lift (c.extend g\u2081.right) : by { apply hc.uniq (c.extend _), tidy }\n            ... = hc.lift (c.extend g\u2082.right) : by { congr, exact this }\n            ... = g\u2082.right                    : by { symmetry, apply hc.uniq (c.extend _), tidy },\n\n  -- Finally, since `f\u1d62` factors through `F(g\u1d62)`, the result follows.\n  calc f\u2081 = \ud835\udfd9 _ \u226b f\u2081                  : by simp\n      ... = c\u2080.X.hom \u226b F.map g\u2081.right : g\u2081.w\n      ... = c\u2080.X.hom \u226b F.map g\u2082.right : by rw this\n      ... = \ud835\udfd9 _ \u226b f\u2082                  : g\u2082.w.symm\n      ... = f\u2082                         : by simp\nend\n\nend preserves_finite_limits_of_flat\n\n/-- Representably flat functors preserve finite limits. -/\nnoncomputable\ndef preserves_finite_limits_of_flat (F : C \u2964 D) [representably_flat F] :\n  preserves_finite_limits F :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size,\n  intros J _ _, constructor,\n  intros K, constructor,\n  intros c hc,\n  exactI { lift := preserves_finite_limits_of_flat.lift F hc,\n    fac' := preserves_finite_limits_of_flat.fac F hc,\n    uniq' := \u03bb s m h, by\n    { apply preserves_finite_limits_of_flat.uniq F hc,\n      exact h,\n      exact preserves_finite_limits_of_flat.fac F hc s } }\nend\n\n/--\nIf `C` is finitely cocomplete, then `F : C \u2964 D` is representably flat iff it preserves\nfinite limits.\n-/\nnoncomputable\ndef preserves_finite_limits_iff_flat [has_finite_limits C] (F : C \u2964 D) :\n  representably_flat F \u2243 preserves_finite_limits F :=\n{ to_fun := \u03bb _, by exactI preserves_finite_limits_of_flat F,\n  inv_fun := \u03bb _, by exactI flat_of_preserves_finite_limits F,\n  left_inv := \u03bb _, proof_irrel _ _,\n  right_inv := \u03bb x, by { cases x, unfold preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr } }\n\nend has_limit\n\n\nsection small_category\nvariables {C D : Type u\u2081} [small_category C] [small_category D] (E : Type u\u2082) [category.{u\u2081} E]\n\n/--\n(Implementation)\nThe evaluation of `Lan F` at `X` is the colimit over the costructured arrows over `X`.\n-/\nnoncomputable\ndef Lan_evaluation_iso_colim (F : C \u2964 D) (X : D)\n  [\u2200 (X : D), has_colimits_of_shape (costructured_arrow F X) E] :\n  Lan F \u22d9 (evaluation D E).obj X \u2245\n  ((whiskering_left _ _ E).obj (costructured_arrow.proj F X)) \u22d9 colim :=\nnat_iso.of_components (\u03bb G, colim.map_iso (iso.refl _))\nbegin\n  intros G H i,\n  ext,\n  simp only [functor.comp_map, colimit.\u03b9_desc_assoc, functor.map_iso_refl, evaluation_obj_map,\n    whiskering_left_obj_map, category.comp_id, Lan_map_app, category.assoc],\n  erw [colimit.\u03b9_pre_assoc (Lan.diagram F H X) (costructured_arrow.map j.hom),\n    category.id_comp, category.comp_id, colimit.\u03b9_map],\n  rcases j with \u27e8j_left, \u27e8\u27e8\u27e9\u27e9, j_hom\u27e9,\n  congr,\n  rw [costructured_arrow.map_mk, category.id_comp, costructured_arrow.mk]\nend\n\nvariables [concrete_category.{u\u2081} E] [has_limits E] [has_colimits E]\nvariables [reflects_limits (forget E)] [preserves_filtered_colimits (forget E)]\nvariables [preserves_limits (forget E)]\n\n/--\nIf `F : C \u2964 D` is a representably flat functor between small categories, then the functor\n`Lan F.op` that takes presheaves over `C` to presheaves over `D` preserves finite limits.\n-/\nnoncomputable\ninstance Lan_preserves_finite_limits_of_flat (F : C \u2964 D) [representably_flat F] :\n  preserves_finite_limits (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 E)) :=\nbegin\n  apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u\u2081},\n  intros J _ _, resetI,\n  apply preserves_limits_of_shape_of_evaluation (Lan F.op : (C\u1d52\u1d56 \u2964 E) \u2964 (D\u1d52\u1d56 \u2964 E)) J,\n  intro K,\n  haveI : is_filtered (costructured_arrow F.op K) :=\n    is_filtered.of_equivalence (structured_arrow_op_equivalence F (unop K)),\n  exact preserves_limits_of_shape_of_nat_iso (Lan_evaluation_iso_colim _ _ _).symm,\nend\n\ninstance Lan_flat_of_flat (F : C \u2964 D) [representably_flat F] :\n  representably_flat (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 E)) := flat_of_preserves_finite_limits _\n\nvariable [has_finite_limits C]\n\nnoncomputable\ninstance Lan_preserves_finite_limits_of_preserves_finite_limits (F : C \u2964 D)\n  [preserves_finite_limits F] : preserves_finite_limits (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 E)) :=\nbegin\n  haveI := flat_of_preserves_finite_limits F,\n  apply_instance\nend\n\nlemma flat_iff_Lan_flat (F : C \u2964 D) :\n  representably_flat F \u2194 representably_flat (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 Type u\u2081)) :=\n\u27e8\u03bb H, by exactI infer_instance, \u03bb H,\nbegin\n  resetI,\n  haveI := preserves_finite_limits_of_flat (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 Type u\u2081)),\n  haveI : preserves_finite_limits F :=\n    begin\n      apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u\u2081},\n      intros, resetI, apply preserves_limit_of_Lan_presesrves_limit\n    end,\n  apply flat_of_preserves_finite_limits\nend\u27e9\n\n/--\nIf `C` is finitely complete, then `F : C \u2964 D` preserves finite limits iff\n`Lan F.op : (C\u1d52\u1d56 \u2964 Type*) \u2964 (D\u1d52\u1d56 \u2964 Type*)` preserves finite limits.\n-/\nnoncomputable\ndef preserves_finite_limits_iff_Lan_preserves_finite_limits (F : C \u2964 D) :\n  preserves_finite_limits F \u2243 preserves_finite_limits (Lan F.op : _ \u2964 (D\u1d52\u1d56 \u2964 Type u\u2081)) :=\n{ to_fun := \u03bb _, by exactI infer_instance,\n  inv_fun := \u03bb _,\n  begin\n    apply preserves_finite_limits_of_preserves_finite_limits_of_size.{u\u2081},\n    intros, resetI, apply preserves_limit_of_Lan_presesrves_limit\n  end,\n  left_inv := \u03bb x,\n  begin\n    cases x, unfold preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr\n  end,\n  right_inv := \u03bb x,\n  begin\n    cases x,\n    unfold preserves_finite_limits_of_flat,\n    congr,\n    unfold category_theory.Lan_preserves_finite_limits_of_preserves_finite_limits\n      category_theory.Lan_preserves_finite_limits_of_flat,\n    dunfold preserves_finite_limits_of_preserves_finite_limits_of_size, congr\n  end }\n\nend small_category\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/functor/flat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.39337394599874587}}
{"text": "/-\nCopyright (c) 2020 Joseph Myers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Joseph Myers, Yury Kudryashov.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.affine_space.midpoint\nimport Mathlib.topology.metric_space.isometry\nimport Mathlib.topology.instances.real_vector_space\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 u_4 u_5 u_6 \n\nnamespace Mathlib\n\n/-!\n# Torsors of additive normed group actions.\n\nThis file defines torsors of additive normed group actions, with a\nmetric space structure.  The motivating case is Euclidean affine\nspaces.\n\n-/\n\n/-- A `normed_add_torsor V P` is a torsor of an additive normed group\naction by a `normed_group V` on points `P`. We bundle the metric space\nstructure and require the distance to be the same as results from the\nnorm (which in fact implies the distance yields a metric space, but\nbundling just the distance and using an instance for the metric space\nresults in type class problems). -/\nclass normed_add_torsor (V : outParam (Type u_1)) (P : Type u_2) [outParam (normed_group V)]\n    [metric_space P]\n    extends add_torsor V P where\n  dist_eq_norm' : \u2200 (x y : P), dist x y = norm (x -\u1d65 y)\n\n/-- The distance equals the norm of subtracting two points. In this\nlemma, it is necessary to have `V` as an explicit argument; otherwise\n`rw dist_eq_norm_vsub` sometimes doesn't work. -/\ntheorem dist_eq_norm_vsub (V : Type u_2) {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (x : P) (y : P) : dist x y = norm (x -\u1d65 y) :=\n  normed_add_torsor.dist_eq_norm' x y\n\n/-- A `normed_group` is a `normed_add_torsor` over itself. -/\nprotected instance normed_group.normed_add_torsor (V : Type u_2) [normed_group V] :\n    normed_add_torsor V V :=\n  normed_add_torsor.mk dist_eq_norm\n\n@[simp] theorem dist_vadd_cancel_left {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (v : V) (x : P) (y : P) :\n    dist (v +\u1d65 x) (v +\u1d65 y) = dist x y :=\n  sorry\n\n@[simp] theorem dist_vadd_cancel_right {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (v\u2081 : V) (v\u2082 : V) (x : P) :\n    dist (v\u2081 +\u1d65 x) (v\u2082 +\u1d65 x) = dist v\u2081 v\u2082 :=\n  sorry\n\n@[simp] theorem dist_vadd_left {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) (x : P) : dist (v +\u1d65 x) x = norm v :=\n  sorry\n\n@[simp] theorem dist_vadd_right {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) (x : P) : dist x (v +\u1d65 x) = norm v :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (dist x (v +\u1d65 x) = norm v)) (dist_comm x (v +\u1d65 x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dist (v +\u1d65 x) x = norm v)) (dist_vadd_left v x)))\n      (Eq.refl (norm v)))\n\n@[simp] theorem dist_vsub_cancel_left {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) (y : P) (z : P) :\n    dist (x -\u1d65 y) (x -\u1d65 z) = dist y z :=\n  sorry\n\n@[simp] theorem dist_vsub_cancel_right {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) (y : P) (z : P) :\n    dist (x -\u1d65 z) (y -\u1d65 z) = dist x y :=\n  eq.mpr\n    (id (Eq._oldrec (Eq.refl (dist (x -\u1d65 z) (y -\u1d65 z) = dist x y)) (dist_eq_norm (x -\u1d65 z) (y -\u1d65 z))))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (norm (x -\u1d65 z - (y -\u1d65 z)) = dist x y))\n          (vsub_sub_vsub_cancel_right x y z)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (norm (x -\u1d65 y) = dist x y)) (dist_eq_norm_vsub V x y)))\n        (Eq.refl (norm (x -\u1d65 y)))))\n\ntheorem dist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) (v' : V) (p : P) (p' : P) :\n    dist (v +\u1d65 p) (v' +\u1d65 p') \u2264 dist v v' + dist p p' :=\n  sorry\n\ntheorem dist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p\u2081 : P) (p\u2082 : P) (p\u2083 : P) (p\u2084 : P) :\n    dist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 dist p\u2081 p\u2083 + dist p\u2082 p\u2084 :=\n  sorry\n\ntheorem nndist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) (v' : V) (p : P) (p' : P) :\n    nndist (v +\u1d65 p) (v' +\u1d65 p') \u2264 nndist v v' + nndist p p' :=\n  sorry\n\ntheorem nndist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p\u2081 : P) (p\u2082 : P) (p\u2083 : P) (p\u2084 : P) :\n    nndist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 nndist p\u2081 p\u2083 + nndist p\u2082 p\u2084 :=\n  sorry\n\ntheorem edist_vadd_vadd_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) (v' : V) (p : P) (p' : P) :\n    edist (v +\u1d65 p) (v' +\u1d65 p') \u2264 edist v v' + edist p p' :=\n  sorry\n\ntheorem edist_vsub_vsub_le {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p\u2081 : P) (p\u2082 : P) (p\u2083 : P) (p\u2084 : P) :\n    edist (p\u2081 -\u1d65 p\u2082) (p\u2083 -\u1d65 p\u2084) \u2264 edist p\u2081 p\u2083 + edist p\u2082 p\u2084 :=\n  sorry\n\n/-- The distance defines a metric space structure on the torsor. This\nis not an instance because it depends on `V` to define a `metric_space\nP`. -/\ndef metric_space_of_normed_group_of_add_torsor (V : Type u_1) (P : Type u_2) [normed_group V]\n    [add_torsor V P] : metric_space P :=\n  metric_space.mk sorry sorry sorry sorry\n    (fun (x y : P) => ennreal.of_real ((fun (x y : P) => norm (x -\u1d65 y)) x y))\n    (uniform_space_of_dist (fun (x y : P) => norm (x -\u1d65 y)) sorry sorry sorry)\n\nnamespace isometric\n\n\n/-- The map `v \u21a6 v +\u1d65 p` as an isometric equivalence between `V` and `P`. -/\ndef vadd_const {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : V \u2243\u1d62 P :=\n  mk (equiv.vadd_const p) sorry\n\n@[simp] theorem coe_vadd_const {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : \u21d1(vadd_const p) = fun (v : V) => v +\u1d65 p :=\n  rfl\n\n@[simp] theorem coe_vadd_const_symm {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : \u21d1(isometric.symm (vadd_const p)) = fun (p' : P) => p' -\u1d65 p :=\n  rfl\n\n@[simp] theorem vadd_const_to_equiv {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : to_equiv (vadd_const p) = equiv.vadd_const p :=\n  rfl\n\n/-- `p' \u21a6 p -\u1d65 p'` as an equivalence. -/\ndef const_vsub {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : P \u2243\u1d62 V :=\n  mk (equiv.const_vsub p) sorry\n\n@[simp] theorem coe_const_vsub {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : \u21d1(const_vsub p) = has_vsub.vsub p :=\n  rfl\n\n@[simp] theorem coe_const_vsub_symm {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (p : P) : \u21d1(isometric.symm (const_vsub p)) = fun (v : V) => -v +\u1d65 p :=\n  rfl\n\n/-- The map `p \u21a6 v +\u1d65 p` as an isometric automorphism of `P`. -/\ndef const_vadd {V : Type u_2} (P : Type u_3) [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) : P \u2243\u1d62 P :=\n  mk (equiv.const_vadd P v) sorry\n\n@[simp] theorem coe_const_vadd {V : Type u_2} (P : Type u_3) [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (v : V) : \u21d1(const_vadd P v) = has_vadd.vadd v :=\n  rfl\n\n@[simp] theorem const_vadd_zero (V : Type u_2) (P : Type u_3) [normed_group V] [metric_space P]\n    [normed_add_torsor V P] : const_vadd P 0 = isometric.refl P :=\n  to_equiv_inj (equiv.const_vadd_zero V P)\n\n/-- Point reflection in `x` as an `isometric` homeomorphism. -/\ndef point_reflection {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (x : P) : P \u2243\u1d62 P :=\n  isometric.trans (const_vsub x) (vadd_const x)\n\ntheorem point_reflection_apply {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (x : P) (y : P) : coe_fn (point_reflection x) y = x -\u1d65 y +\u1d65 x :=\n  rfl\n\n@[simp] theorem point_reflection_to_equiv {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) :\n    to_equiv (point_reflection x) = equiv.point_reflection x :=\n  rfl\n\n@[simp] theorem point_reflection_self {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) : coe_fn (point_reflection x) x = x :=\n  equiv.point_reflection_self x\n\ntheorem point_reflection_involutive {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (x : P) : function.involutive \u21d1(point_reflection x) :=\n  equiv.point_reflection_involutive x\n\n@[simp] theorem point_reflection_symm {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) :\n    isometric.symm (point_reflection x) = point_reflection x :=\n  to_equiv_inj (equiv.point_reflection_symm x)\n\n@[simp] theorem dist_point_reflection_fixed {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] (x : P) (y : P) :\n    dist (coe_fn (point_reflection x) y) x = dist y x :=\n  sorry\n\ntheorem dist_point_reflection_self' {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (x : P) (y : P) :\n    dist (coe_fn (point_reflection x) y) y = norm (bit0 (x -\u1d65 y)) :=\n  sorry\n\ntheorem dist_point_reflection_self {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (\ud835\udd5c : Type u_1) [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] (x : P) (y : P) :\n    dist (coe_fn (point_reflection x) y) y = norm (bit0 1) * dist x y :=\n  sorry\n\ntheorem point_reflection_fixed_iff {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] (\ud835\udd5c : Type u_1) [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] [invertible (bit0 1)]\n    {x : P} {y : P} : coe_fn (point_reflection x) y = y \u2194 y = x :=\n  affine_equiv.point_reflection_fixed_iff_of_module \ud835\udd5c\n\ntheorem dist_point_reflection_self_real {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [normed_space \u211d V] (x : P) (y : P) :\n    dist (coe_fn (point_reflection x) y) y = bit0 1 * dist x y :=\n  sorry\n\n@[simp] theorem point_reflection_midpoint_left {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [normed_space \u211d V] (x : P) (y : P) :\n    coe_fn (point_reflection (midpoint \u211d x y)) x = y :=\n  affine_equiv.point_reflection_midpoint_left x y\n\n@[simp] theorem point_reflection_midpoint_right {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [normed_space \u211d V] (x : P) (y : P) :\n    coe_fn (point_reflection (midpoint \u211d x y)) y = x :=\n  affine_equiv.point_reflection_midpoint_right x y\n\nend isometric\n\n\ntheorem lipschitz_with.vadd {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [emetric_space \u03b1] {f : \u03b1 \u2192 V} {g : \u03b1 \u2192 P} {Kf : nnreal}\n    {Kg : nnreal} (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :\n    lipschitz_with (Kf + Kg) (f +\u1d65 g) :=\n  fun (x y : \u03b1) =>\n    trans_rel_left LessEq\n      (le_trans (edist_vadd_vadd_le (f x) (f y) (g x) (g y)) (add_le_add (hf x y) (hg x y)))\n      (Eq.symm (add_mul (\u2191Kf) (\u2191Kg) (edist x y)))\n\ntheorem lipschitz_with.vsub {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [emetric_space \u03b1] {f : \u03b1 \u2192 P} {g : \u03b1 \u2192 P} {Kf : nnreal}\n    {Kg : nnreal} (hf : lipschitz_with Kf f) (hg : lipschitz_with Kg g) :\n    lipschitz_with (Kf + Kg) (f -\u1d65 g) :=\n  fun (x y : \u03b1) =>\n    trans_rel_left LessEq\n      (le_trans (edist_vsub_vsub_le (f x) (g x) (f y) (g y)) (add_le_add (hf x y) (hg x y)))\n      (Eq.symm (add_mul (\u2191Kf) (\u2191Kg) (edist x y)))\n\ntheorem uniform_continuous_vadd {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] : uniform_continuous fun (x : V \u00d7 P) => prod.fst x +\u1d65 prod.snd x :=\n  lipschitz_with.uniform_continuous\n    (lipschitz_with.vadd lipschitz_with.prod_fst lipschitz_with.prod_snd)\n\ntheorem uniform_continuous_vsub {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] : uniform_continuous fun (x : P \u00d7 P) => prod.fst x -\u1d65 prod.snd x :=\n  lipschitz_with.uniform_continuous\n    (lipschitz_with.vsub lipschitz_with.prod_fst lipschitz_with.prod_snd)\n\ntheorem continuous_vadd {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] : continuous fun (x : V \u00d7 P) => prod.fst x +\u1d65 prod.snd x :=\n  uniform_continuous.continuous uniform_continuous_vadd\n\ntheorem continuous_vsub {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] : continuous fun (x : P \u00d7 P) => prod.fst x -\u1d65 prod.snd x :=\n  uniform_continuous.continuous uniform_continuous_vsub\n\ntheorem filter.tendsto.vadd {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] {l : filter \u03b1} {f : \u03b1 \u2192 V} {g : \u03b1 \u2192 P} {v : V} {p : P}\n    (hf : filter.tendsto f l (nhds v)) (hg : filter.tendsto g l (nhds p)) :\n    filter.tendsto (f +\u1d65 g) l (nhds (v +\u1d65 p)) :=\n  filter.tendsto.comp (continuous.tendsto continuous_vadd (v, p))\n    (filter.tendsto.prod_mk_nhds hf hg)\n\ntheorem filter.tendsto.vsub {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] {l : filter \u03b1} {f : \u03b1 \u2192 P} {g : \u03b1 \u2192 P} {x : P} {y : P}\n    (hf : filter.tendsto f l (nhds x)) (hg : filter.tendsto g l (nhds y)) :\n    filter.tendsto (f -\u1d65 g) l (nhds (x -\u1d65 y)) :=\n  filter.tendsto.comp (continuous.tendsto continuous_vsub (x, y))\n    (filter.tendsto.prod_mk_nhds hf hg)\n\ntheorem continuous.vadd {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 V} {g : \u03b1 \u2192 P}\n    (hf : continuous f) (hg : continuous g) : continuous (f +\u1d65 g) :=\n  continuous.comp continuous_vadd (continuous.prod_mk hf hg)\n\ntheorem continuous.vsub {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 P} {g : \u03b1 \u2192 P}\n    (hf : continuous f) (hg : continuous g) : continuous (f -\u1d65 g) :=\n  continuous.comp continuous_vsub (continuous.prod_mk hf hg)\n\ntheorem continuous_at.vadd {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 V} {g : \u03b1 \u2192 P} {x : \u03b1}\n    (hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (f +\u1d65 g) x :=\n  filter.tendsto.vadd hf hg\n\ntheorem continuous_at.vsub {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 P} {g : \u03b1 \u2192 P} {x : \u03b1}\n    (hf : continuous_at f x) (hg : continuous_at g x) : continuous_at (f -\u1d65 g) x :=\n  filter.tendsto.vsub hf hg\n\ntheorem continuous_within_at.vadd {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 V} {g : \u03b1 \u2192 P} {x : \u03b1}\n    {s : set \u03b1} (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :\n    continuous_within_at (f +\u1d65 g) s x :=\n  filter.tendsto.vadd hf hg\n\ntheorem continuous_within_at.vsub {\u03b1 : Type u_1} {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] [topological_space \u03b1] {f : \u03b1 \u2192 P} {g : \u03b1 \u2192 P} {x : \u03b1}\n    {s : set \u03b1} (hf : continuous_within_at f s x) (hg : continuous_within_at g s x) :\n    continuous_within_at (f -\u1d65 g) s x :=\n  filter.tendsto.vsub hf hg\n\n/-- The map `g` from `V1` to `V2` corresponding to a map `f` from `P1`\nto `P2`, at a base point `p`, is an isometry if `f` is one. -/\ntheorem isometry.vadd_vsub {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {V' : Type u_4} {P' : Type u_5} [normed_group V'] [metric_space P']\n    [normed_add_torsor V' P'] {f : P \u2192 P'} (hf : isometry f) {p : P} {g : V \u2192 V'}\n    (hg : \u2200 (v : V), g v = f (v +\u1d65 p) -\u1d65 f p) : isometry g :=\n  sorry\n\n/-- If `f` is an affine map, then its linear part is continuous iff `f` is continuous. -/\ntheorem affine_map.continuous_linear_iff {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] {V' : Type u_4} {P' : Type u_5} [normed_group V']\n    [metric_space P'] [normed_add_torsor V' P'] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V]\n    [normed_space \ud835\udd5c V'] {f : affine_map \ud835\udd5c P P'} :\n    continuous \u21d1(affine_map.linear f) \u2194 continuous \u21d1f :=\n  sorry\n\n@[simp] theorem dist_center_homothety {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V]\n    (p\u2081 : P) (p\u2082 : P) (c : \ud835\udd5c) :\n    dist p\u2081 (coe_fn (affine_map.homothety p\u2081 c) p\u2082) = norm c * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_homothety_center {V : Type u_2} {P : Type u_3} [normed_group V]\n    [metric_space P] [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V]\n    (p\u2081 : P) (p\u2082 : P) (c : \ud835\udd5c) :\n    dist (coe_fn (affine_map.homothety p\u2081 c) p\u2082) p\u2081 = norm c * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_homothety_self {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] (p\u2081 : P) (p\u2082 : P)\n    (c : \ud835\udd5c) : dist (coe_fn (affine_map.homothety p\u2081 c) p\u2082) p\u2082 = norm (1 - c) * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_self_homothety {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] (p\u2081 : P) (p\u2082 : P)\n    (c : \ud835\udd5c) : dist p\u2082 (coe_fn (affine_map.homothety p\u2081 c) p\u2082) = norm (1 - c) * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_left_midpoint {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] [invertible (bit0 1)]\n    (p\u2081 : P) (p\u2082 : P) : dist p\u2081 (midpoint \ud835\udd5c p\u2081 p\u2082) = norm (bit0 1)\u207b\u00b9 * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_midpoint_left {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] [invertible (bit0 1)]\n    (p\u2081 : P) (p\u2082 : P) : dist (midpoint \ud835\udd5c p\u2081 p\u2082) p\u2081 = norm (bit0 1)\u207b\u00b9 * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_midpoint_right {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] [invertible (bit0 1)]\n    (p\u2081 : P) (p\u2082 : P) : dist (midpoint \ud835\udd5c p\u2081 p\u2082) p\u2082 = norm (bit0 1)\u207b\u00b9 * dist p\u2081 p\u2082 :=\n  sorry\n\n@[simp] theorem dist_right_midpoint {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {\ud835\udd5c : Type u_6} [normed_field \ud835\udd5c] [normed_space \ud835\udd5c V] [invertible (bit0 1)]\n    (p\u2081 : P) (p\u2082 : P) : dist p\u2082 (midpoint \ud835\udd5c p\u2081 p\u2082) = norm (bit0 1)\u207b\u00b9 * dist p\u2081 p\u2082 :=\n  sorry\n\n/-- A continuous map between two normed affine spaces is an affine map provided that\nit sends midpoints to midpoints. -/\ndef affine_map.of_map_midpoint {V : Type u_2} {P : Type u_3} [normed_group V] [metric_space P]\n    [normed_add_torsor V P] {V' : Type u_4} {P' : Type u_5} [normed_group V'] [metric_space P']\n    [normed_add_torsor V' P'] [normed_space \u211d V] [normed_space \u211d V'] (f : P \u2192 P')\n    (h : \u2200 (x y : P), f (midpoint \u211d x y) = midpoint \u211d (f x) (f y)) (hfc : continuous f) :\n    affine_map \u211d P P' :=\n  affine_map.mk' f\n    (\u2191(add_monoid_hom.to_real_linear_map\n        (add_monoid_hom.of_map_midpoint \u211d \u211d\n          (\u21d1(affine_equiv.symm (affine_equiv.vadd_const \u211d (f (classical.arbitrary P)))) \u2218\n            f \u2218 \u21d1(affine_equiv.vadd_const \u211d (classical.arbitrary P)))\n          sorry sorry)\n        sorry))\n    (classical.arbitrary P) sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/normed_space/add_torsor_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3933739378240936}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module data.rbtree.insert\n! leanprover-community/mathlib commit 4d4167104581a21259f7f448e1972a63a4546be7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Rbtree.Find\n\nuniverse u v\n\nattribute [local simp] Rbnode.Lift\n\nnamespace Rbnode\n\nvariable {\u03b1 : Type u}\n\nopen Color\n\n@[simp]\ntheorem balance1_eq\u2081 (l : Rbnode \u03b1) (x r\u2081 y r\u2082 v t) :\n    balance1 (red_node l x r\u2081) y r\u2082 v t = red_node (black_node l x r\u2081) y (black_node r\u2082 v t) := by\n  cases r\u2082 <;> rfl\n#align rbnode.balance1_eq\u2081 Rbnode.balance1_eq\u2081\n\n@[simp]\ntheorem balance1_eq\u2082 (l\u2081 : Rbnode \u03b1) (y l\u2082 x r v t) :\n    getColor l\u2081 \u2260 red \u2192\n      balance1 l\u2081 y (red_node l\u2082 x r) v t = red_node (black_node l\u2081 y l\u2082) x (black_node r v t) :=\n  by cases l\u2081 <;> simp [get_color, balance1, false_imp_iff]\n#align rbnode.balance1_eq\u2082 Rbnode.balance1_eq\u2082\n\n@[simp]\ntheorem balance1_eq\u2083 (l : Rbnode \u03b1) (y r v t) :\n    getColor l \u2260 red \u2192 getColor r \u2260 red \u2192 balance1 l y r v t = black_node (red_node l y r) v t := by\n  cases l <;> cases r <;> simp [get_color, balance1, false_imp_iff]\n#align rbnode.balance1_eq\u2083 Rbnode.balance1_eq\u2083\n\n@[simp]\ntheorem balance2_eq\u2081 (l : Rbnode \u03b1) (x\u2081 r\u2081 y r\u2082 v t) :\n    balance2 (red_node l x\u2081 r\u2081) y r\u2082 v t = red_node (black_node t v l) x\u2081 (black_node r\u2081 y r\u2082) := by\n  cases r\u2082 <;> rfl\n#align rbnode.balance2_eq\u2081 Rbnode.balance2_eq\u2081\n\n@[simp]\ntheorem balance2_eq\u2082 (l\u2081 : Rbnode \u03b1) (y l\u2082 x\u2082 r\u2082 v t) :\n    getColor l\u2081 \u2260 red \u2192\n      balance2 l\u2081 y (red_node l\u2082 x\u2082 r\u2082) v t =\n        red_node (black_node t v l\u2081) y (black_node l\u2082 x\u2082 r\u2082) :=\n  by cases l\u2081 <;> simp [get_color, balance2, false_imp_iff]\n#align rbnode.balance2_eq\u2082 Rbnode.balance2_eq\u2082\n\n@[simp]\ntheorem balance2_eq\u2083 (l : Rbnode \u03b1) (y r v t) :\n    getColor l \u2260 red \u2192 getColor r \u2260 red \u2192 balance2 l y r v t = black_node t v (red_node l y r) := by\n  cases l <;> cases r <;> simp [get_color, balance2, false_imp_iff]\n#align rbnode.balance2_eq\u2083 Rbnode.balance2_eq\u2083\n\n-- We can use the same induction principle for balance1 and balance2\ntheorem Balance.cases {p : Rbnode \u03b1 \u2192 \u03b1 \u2192 Rbnode \u03b1 \u2192 Prop} (l y r)\n    (red_left : \u2200 l x r\u2081 y r\u2082, p (red_node l x r\u2081) y r\u2082)\n    (red_right : \u2200 l\u2081 y l\u2082 x r, getColor l\u2081 \u2260 red \u2192 p l\u2081 y (red_node l\u2082 x r))\n    (other : \u2200 l y r, getColor l \u2260 red \u2192 getColor r \u2260 red \u2192 p l y r) : p l y r :=\n  by\n  cases l <;> cases r\n  any_goals apply red_left\n  any_goals apply red_right <;> simp [get_color] <;> contradiction <;> done\n  any_goals apply other <;> simp [get_color] <;> contradiction <;> done\n#align rbnode.balance.cases Rbnode.Balance.cases\n\ntheorem balance1_ne_leaf (l : Rbnode \u03b1) (x r v t) : balance1 l x r v t \u2260 leaf := by\n  apply balance.cases l x r <;> intros <;> simp [*] <;> contradiction\n#align rbnode.balance1_ne_leaf Rbnode.balance1_ne_leaf\n\ntheorem balance1Node_ne_leaf {s : Rbnode \u03b1} (a : \u03b1) (t : Rbnode \u03b1) :\n    s \u2260 leaf \u2192 balance1Node s a t \u2260 leaf := by\n  intro h; cases s\n  \u00b7 contradiction\n  all_goals simp [balance1_node]; apply balance1_ne_leaf\n#align rbnode.balance1_node_ne_leaf Rbnode.balance1Node_ne_leaf\n\ntheorem balance2_ne_leaf (l : Rbnode \u03b1) (x r v t) : balance2 l x r v t \u2260 leaf := by\n  apply balance.cases l x r <;> intros <;> simp [*] <;> contradiction\n#align rbnode.balance2_ne_leaf Rbnode.balance2_ne_leaf\n\ntheorem balance2Node_ne_leaf {s : Rbnode \u03b1} (a : \u03b1) (t : Rbnode \u03b1) :\n    s \u2260 leaf \u2192 balance2Node s a t \u2260 leaf := by\n  intro h; cases s\n  \u00b7 contradiction\n  all_goals simp [balance2_node]; apply balance2_ne_leaf\n#align rbnode.balance2_node_ne_leaf Rbnode.balance2Node_ne_leaf\n\nvariable (lt : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\n@[elab_as_elim]\ntheorem ins.induction [DecidableRel lt] {p : Rbnode \u03b1 \u2192 Prop} (t x) (is_leaf : p leaf)\n    (is_red_lt : \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.lt) (ih : p a), p (red_node a y b))\n    (is_red_eq : \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.eq), p (red_node a y b))\n    (is_red_gt : \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.gt) (ih : p b), p (red_node a y b))\n    (is_black_lt_red :\n      \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.lt) (hr : getColor a = red) (ih : p a),\n        p (black_node a y b))\n    (is_black_lt_not_red :\n      \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.lt) (hnr : getColor a \u2260 red) (ih : p a),\n        p (black_node a y b))\n    (is_black_eq : \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.eq), p (black_node a y b))\n    (is_black_gt_red :\n      \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.gt) (hr : getColor b = red) (ih : p b),\n        p (black_node a y b))\n    (is_black_gt_not_red :\n      \u2200 (a y b) (hc : cmpUsing lt x y = Ordering.gt) (hnr : getColor b \u2260 red) (ih : p b),\n        p (black_node a y b)) :\n    p t := by\n  induction t\n  case leaf => apply is_leaf\n  case red_node a y b =>\n    cases h : cmpUsing lt x y\n    case lt => apply is_red_lt <;> assumption\n    case eq => apply is_red_eq <;> assumption\n    case gt => apply is_red_gt <;> assumption\n  case black_node a y b =>\n    cases h : cmpUsing lt x y\n    case lt =>\n      by_cases get_color a = red\n      \u00b7 apply is_black_lt_red <;> assumption\n      \u00b7 apply is_black_lt_not_red <;> assumption\n    case eq => apply is_black_eq <;> assumption\n    case gt =>\n      by_cases get_color b = red\n      \u00b7 apply is_black_gt_red <;> assumption\n      \u00b7 apply is_black_gt_not_red <;> assumption\n#align rbnode.ins.induction Rbnode.ins.induction\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_balance1 {l y r v t lo hi} :\n    IsSearchable lt l lo (some y) \u2192\n      IsSearchable lt r (some y) (some v) \u2192\n        IsSearchable lt t (some v) hi \u2192 IsSearchable lt (balance1 l y r v t) lo hi :=\n  by\n  apply balance.cases l y r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n#align rbnode.is_searchable_balance1 Rbnode.isSearchable_balance1\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_balance1Node {t} [IsTrans \u03b1 lt] :\n    \u2200 {y s lo hi},\n      IsSearchable lt t lo (some y) \u2192\n        IsSearchable lt s (some y) hi \u2192 IsSearchable lt (balance1Node t y s) lo hi :=\n  by\n  cases t <;> simp! <;> intros <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 cases lo\n    \u00b7 apply is_searchable_none_low_of_is_searchable_some_low\n      assumption\n    \u00b7 simp at *\n      apply is_searchable_some_low_of_is_searchable_of_lt <;> assumption\n  all_goals apply is_searchable_balance1 <;> assumption\n#align rbnode.is_searchable_balance1_node Rbnode.isSearchable_balance1Node\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_balance2 {l y r v t lo hi} :\n    IsSearchable lt t lo (some v) \u2192\n      IsSearchable lt l (some v) (some y) \u2192\n        IsSearchable lt r (some y) hi \u2192 IsSearchable lt (balance2 l y r v t) lo hi :=\n  by\n  apply balance.cases l y r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n#align rbnode.is_searchable_balance2 Rbnode.isSearchable_balance2\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_balance2Node {t} [IsTrans \u03b1 lt] :\n    \u2200 {y s lo hi},\n      IsSearchable lt s lo (some y) \u2192\n        IsSearchable lt t (some y) hi \u2192 IsSearchable lt (balance2Node t y s) lo hi :=\n  by\n  induction t <;> simp! <;> intros <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 cases hi\n    \u00b7 apply is_searchable_none_high_of_is_searchable_some_high\n      assumption\n    \u00b7 simp at *\n      apply is_searchable_some_high_of_is_searchable_of_lt\n      assumption'\n  all_goals apply is_searchable_balance2; assumption'\n#align rbnode.is_searchable_balance2_node Rbnode.isSearchable_balance2Node\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_ins [DecidableRel lt] {t x} [IsStrictWeakOrder \u03b1 lt] :\n    \u2200 {lo hi} (h : IsSearchable lt t lo hi),\n      Lift lt lo (some x) \u2192 Lift lt (some x) hi \u2192 IsSearchable lt (ins lt t x) lo hi :=\n  by\n  apply ins.induction lt t x <;> intros <;> simp_all! (config := { eta := false }) <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 apply ih h_hs\u2081\n    assumption\n    simp [*]\n  \u00b7 apply is_searchable_of_is_searchable_of_incomp hc\n    assumption\n  \u00b7 apply is_searchable_of_incomp_of_is_searchable hc\n    assumption\n  \u00b7 apply ih h_hs\u2082\n    cases hi <;> simp [*]\n    assumption\n  \u00b7 apply is_searchable_balance1_node\n    apply ih h_hs\u2081\n    assumption\n    simp [*]\n    assumption\n  \u00b7 apply ih h_hs\u2081\n    assumption\n    simp [*]\n  \u00b7 apply is_searchable_of_is_searchable_of_incomp hc\n    assumption\n  \u00b7 apply is_searchable_of_incomp_of_is_searchable hc\n    assumption\n  \u00b7 apply is_searchable_balance2_node\n    assumption\n    apply ih h_hs\u2082\n    simp [*]\n    assumption\n  \u00b7 apply ih h_hs\u2082\n    assumption\n    simp [*]\n#align rbnode.is_searchable_ins Rbnode.isSearchable_ins\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem isSearchable_mkInsertResult {c t} :\n    IsSearchable lt t none none \u2192 IsSearchable lt (mkInsertResult c t) none none := by\n  classical\n    cases c <;> cases t <;> simp [mk_insert_result]\n    \u00b7 intro h\n      run_tac\n        is_searchable_tactic\n#align rbnode.is_searchable_mk_insert_result Rbnode.isSearchable_mkInsertResult\n\ntheorem isSearchable_insert [DecidableRel lt] {t x} [IsStrictWeakOrder \u03b1 lt] :\n    IsSearchable lt t none none \u2192 IsSearchable lt (insert lt t x) none none := by intro h;\n  simp [insert]; apply is_searchable_mk_insert_result;\n  apply is_searchable_ins <;> \u00b7 first |assumption|simp\n#align rbnode.is_searchable_insert Rbnode.isSearchable_insert\n\nend Rbnode\n\nnamespace Rbnode\n\nsection MembershipLemmas\n\nparameter {\u03b1 : Type u}(lt : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\nattribute [local simp] mem balance1_node balance2_node\n\n-- mathport name: mem\nlocal infixl:0 \" \u2208 \" => Mem lt\n\ntheorem mem_balance1Node_of_mem_left {x s} (v) (t : Rbnode \u03b1) :\n    (x \u2208 s) \u2192 (x \u2208 balance1Node s v t) :=\n  by\n  cases s <;> simp [false_imp_iff]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp at * <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.mem_balance1_node_of_mem_left Rbnode.mem_balance1Node_of_mem_left\n\ntheorem mem_balance2Node_of_mem_left {x s} (v) (t : Rbnode \u03b1) :\n    (x \u2208 s) \u2192 (x \u2208 balance2Node s v t) :=\n  by\n  cases s <;> simp [false_imp_iff]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp at * <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.mem_balance2_node_of_mem_left Rbnode.mem_balance2Node_of_mem_left\n\ntheorem mem_balance1Node_of_mem_right {x t} (v) (s : Rbnode \u03b1) :\n    (x \u2208 t) \u2192 (x \u2208 balance1Node s v t) := by\n  intros ; cases s <;> simp [*]\n  all_goals apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp [*]\n#align rbnode.mem_balance1_node_of_mem_right Rbnode.mem_balance1Node_of_mem_right\n\ntheorem mem_balance2Node_of_mem_right {x t} (v) (s : Rbnode \u03b1) :\n    (x \u2208 t) \u2192 (x \u2208 balance2Node s v t) := by\n  intros ; cases s <;> simp [*]\n  all_goals apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp [*]\n#align rbnode.mem_balance2_node_of_mem_right Rbnode.mem_balance2Node_of_mem_right\n\ntheorem mem_balance1Node_of_incomp {x v} (s t) :\n    \u00aclt x v \u2227 \u00aclt v x \u2192 s \u2260 leaf \u2192 (x \u2208 balance1Node s v t) :=\n  by\n  intros ; cases s <;> simp\n  \u00b7 contradiction\n  all_goals apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp [*]\n#align rbnode.mem_balance1_node_of_incomp Rbnode.mem_balance1Node_of_incomp\n\ntheorem mem_balance2Node_of_incomp {x v} (s t) :\n    \u00aclt v x \u2227 \u00aclt x v \u2192 s \u2260 leaf \u2192 (x \u2208 balance2Node s v t) :=\n  by\n  intros ; cases s <;> simp\n  \u00b7 contradiction\n  all_goals apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp [*]\n#align rbnode.mem_balance2_node_of_incomp Rbnode.mem_balance2Node_of_incomp\n\ntheorem ins_ne_leaf [DecidableRel lt] (t : Rbnode \u03b1) (x : \u03b1) : t.ins lt x \u2260 leaf :=\n  by\n  apply ins.induction lt t x\n  any_goals intros ; simp [ins, *]\n  \u00b7 intros\n    apply balance1_node_ne_leaf\n    assumption\n  \u00b7 intros\n    apply balance2_node_ne_leaf\n    assumption\n#align rbnode.ins_ne_leaf Rbnode.ins_ne_leaf\n\ntheorem insert_ne_leaf [DecidableRel lt] (t : Rbnode \u03b1) (x : \u03b1) : insert lt t x \u2260 leaf :=\n  by\n  simp [insert]\n  cases he : ins lt t x <;> cases get_color t <;> simp [mk_insert_result]\n  \u00b7 have := ins_ne_leaf lt t x\n    contradiction\n  \u00b7 exact absurd he (ins_ne_leaf _ _ _)\n#align rbnode.insert_ne_leaf Rbnode.insert_ne_leaf\n\ntheorem mem_ins_of_incomp [DecidableRel lt] (t : Rbnode \u03b1) {x y : \u03b1} :\n    \u2200 h : \u00aclt x y \u2227 \u00aclt y x, x \u2208 t.ins lt y :=\n  by\n  apply ins.induction lt t y <;> intros <;> simp [ins, *]\n  \u00b7 have := ih h\n    apply mem_balance1_node_of_mem_left\n    assumption\n  \u00b7 have := ih h\n    apply mem_balance2_node_of_mem_left\n    assumption\n#align rbnode.mem_ins_of_incomp Rbnode.mem_ins_of_incomp\n\ntheorem mem_ins_of_mem [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {t : Rbnode \u03b1} (z : \u03b1) :\n    \u2200 {x} (h : x \u2208 t), x \u2208 t.ins lt z :=\n  by\n  apply ins.induction lt t z <;> intros <;> simp_all [ins] <;> try contradiction <;> cases_type*or.1\n  any_goals intros ; simp [h]; done\n  any_goals intros ; simp [ih h]; done\n  \u00b7 have := incomp_trans_of lt h \u27e8hc.2, hc.1\u27e9\n    simp [this]\n  \u00b7 apply mem_balance1_node_of_mem_left\n    apply ih h\n  \u00b7 apply mem_balance1_node_of_incomp\n    cases h\n    all_goals simp [*, ins_ne_leaf lt a z]\n  \u00b7 apply mem_balance1_node_of_mem_right\n    assumption\n  \u00b7 have := incomp_trans_of lt hc \u27e8h.2, h.1\u27e9\n    simp [this]\n  \u00b7 apply mem_balance2_node_of_mem_right\n    assumption\n  \u00b7 have := ins_ne_leaf lt a z\n    apply mem_balance2_node_of_incomp\n    cases h\n    simp [*]\n    apply ins_ne_leaf\n  \u00b7 apply mem_balance2_node_of_mem_left\n    apply ih h\n#align rbnode.mem_ins_of_mem Rbnode.mem_ins_of_mem\n\ntheorem mem_mkInsertResult {a t} (c) : Mem lt a t \u2192 Mem lt a (mkInsertResult c t) := by\n  intros <;> cases c <;> cases t <;> simp_all [mk_insert_result, mem]\n#align rbnode.mem_mk_insert_result Rbnode.mem_mkInsertResult\n\ntheorem mem_of_mem_mkInsertResult {a t c} : Mem lt a (mkInsertResult c t) \u2192 Mem lt a t := by\n  cases t <;> cases c <;> simp [mk_insert_result, mem] <;> intros <;> assumption\n#align rbnode.mem_of_mem_mk_insert_result Rbnode.mem_of_mem_mkInsertResult\n\ntheorem mem_insert_of_incomp [DecidableRel lt] (t : Rbnode \u03b1) {x y : \u03b1} :\n    \u2200 h : \u00aclt x y \u2227 \u00aclt y x, x \u2208 t.insert lt y := by\n  intros <;> unfold insert <;> apply mem_mk_insert_result <;> apply mem_ins_of_incomp <;> assumption\n#align rbnode.mem_insert_of_incomp Rbnode.mem_insert_of_incomp\n\ntheorem mem_insert_of_mem [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {t x} (z) :\n    (x \u2208 t) \u2192 (x \u2208 t.insert lt z) := by\n  intros <;> apply mem_mk_insert_result <;> apply mem_ins_of_mem <;> assumption\n#align rbnode.mem_insert_of_mem Rbnode.mem_insert_of_mem\n\ntheorem of_mem_balance1Node {x s v t} :\n    (x \u2208 balance1Node s v t) \u2192 (x \u2208 s) \u2228 \u00aclt x v \u2227 \u00aclt v x \u2228 (x \u2208 t) :=\n  by\n  cases s <;> simp\n  \u00b7 intros\n    simp [*]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp_all <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.of_mem_balance1_node Rbnode.of_mem_balance1Node\n\ntheorem of_mem_balance2Node {x s v t} :\n    (x \u2208 balance2Node s v t) \u2192 (x \u2208 s) \u2228 \u00aclt x v \u2227 \u00aclt v x \u2228 (x \u2208 t) :=\n  by\n  cases s <;> simp\n  \u00b7 intros\n    simp [*]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp_all <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.of_mem_balance2_node Rbnode.of_mem_balance2Node\n\ntheorem equiv_or_mem_of_mem_ins [DecidableRel lt] {t : Rbnode \u03b1} {x z} :\n    \u2200 h : x \u2208 t.ins lt z, x \u2248[lt]z \u2228 (x \u2208 t) :=\n  by\n  apply ins.induction lt t z <;> intros <;> simp_all [ins, StrictWeakOrder.Equiv] <;>\n    cases_type*or.1\n  any_goals intros ; simp [h]\n  any_goals intros ; have ih := ih h; cases ih <;> simp [*]; done\n  \u00b7 have h' := of_mem_balance1_node lt h\n    cases_type*or.1\n    have := ih h'\n    cases_type*or.1\n    all_goals simp [h, *]\n  \u00b7 have h' := of_mem_balance2_node lt h\n    cases_type*or.1\n    have := ih h'\n    cases_type*or.1\n    all_goals simp [h, *]\n#align rbnode.equiv_or_mem_of_mem_ins Rbnode.equiv_or_mem_of_mem_ins\n\ntheorem equiv_or_mem_of_mem_insert [DecidableRel lt] {t : Rbnode \u03b1} {x z} :\n    \u2200 h : x \u2208 t.insert lt z, x \u2248[lt]z \u2228 (x \u2208 t) := by simp [insert]; intros ;\n  apply equiv_or_mem_of_mem_ins; exact mem_of_mem_mk_insert_result lt h\n#align rbnode.equiv_or_mem_of_mem_insert Rbnode.equiv_or_mem_of_mem_insert\n\nattribute [local simp] mem_exact\n\ntheorem memExact_balance1Node_of_memExact {x s} (v) (t : Rbnode \u03b1) :\n    MemExact x s \u2192 MemExact x (balance1Node s v t) :=\n  by\n  cases s <;> simp [false_imp_iff]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp_all <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.mem_exact_balance1_node_of_mem_exact Rbnode.memExact_balance1Node_of_memExact\n\ntheorem memExact_balance2Node_of_memExact {x s} (v) (t : Rbnode \u03b1) :\n    MemExact x s \u2192 MemExact x (balance2Node s v t) :=\n  by\n  cases s <;> simp [false_imp_iff]\n  all_goals\n    apply balance.cases s_lchild s_val s_rchild <;> intros <;> simp_all <;> cases_type*or.1 <;>\n      simp [*]\n#align rbnode.mem_exact_balance2_node_of_mem_exact Rbnode.memExact_balance2Node_of_memExact\n\ntheorem find_balance1Node [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y z t s} :\n    \u2200 {lo hi},\n      IsSearchable lt t lo (some z) \u2192\n        IsSearchable lt s (some z) hi \u2192\n          find lt t y = some x \u2192 y \u2248[lt]x \u2192 find lt (balance1Node t z s) y = some x :=\n  by\n  intro _ _ hs\u2081 hs\u2082 heq heqv\n  have hs := is_searchable_balance1_node lt hs\u2081 hs\u2082\n  have := Eq.trans (find_eq_find_of_eqv hs\u2081 heqv.symm) HEq\n  have := Iff.mpr (find_correct_exact hs\u2081) this\n  have := mem_exact_balance1_node_of_mem_exact z s this\n  have := Iff.mp (find_correct_exact hs) this\n  exact Eq.trans (find_eq_find_of_eqv hs heqv) this\n#align rbnode.find_balance1_node Rbnode.find_balance1Node\n\ntheorem find_balance2Node [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y z s t} [IsTrans \u03b1 lt] :\n    \u2200 {lo hi},\n      IsSearchable lt s lo (some z) \u2192\n        IsSearchable lt t (some z) hi \u2192\n          find lt t y = some x \u2192 y \u2248[lt]x \u2192 find lt (balance2Node t z s) y = some x :=\n  by\n  intro _ _ hs\u2081 hs\u2082 heq heqv\n  have hs := is_searchable_balance2_node lt hs\u2081 hs\u2082\n  have := Eq.trans (find_eq_find_of_eqv hs\u2082 heqv.symm) HEq\n  have := Iff.mpr (find_correct_exact hs\u2082) this\n  have := mem_exact_balance2_node_of_mem_exact z s this\n  have := Iff.mp (find_correct_exact hs) this\n  exact Eq.trans (find_eq_find_of_eqv hs heqv) this\n#align rbnode.find_balance2_node Rbnode.find_balance2Node\n\n-- Auxiliary lemma\ntheorem ite_eq_of_not_lt [DecidableRel lt] [IsStrictOrder \u03b1 lt] {a b} {\u03b2 : Type v} (t s : \u03b2)\n    (h : lt b a) : (if lt a b then t else s) = s := by have := not_lt_of_lt h; simp [*]\n#align rbnode.ite_eq_of_not_lt Rbnode.ite_eq_of_not_lt\n\nattribute [local simp] ite_eq_of_not_lt\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\nprivate unsafe def simp_fi : tactic Unit :=\n  sorry\n#align rbnode.simp_fi rbnode.simp_fi\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\ntheorem find_ins_of_eqv [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y : \u03b1} {t : Rbnode \u03b1}\n    (he : x \u2248[lt]y) :\n    \u2200 {lo hi} (hs : IsSearchable lt t lo hi) (hlt\u2081 : Lift lt lo (some x))\n      (hlt\u2082 : Lift lt (some x) hi), find lt (ins lt t x) y = some x :=\n  by\n  simp [StrictWeakOrder.Equiv] at he\n  apply ins.induction lt t x <;> intros\n  \u00b7\n    run_tac\n      simp_fi\n  all_goals simp at hc; cases hs\n  \u00b7 have := lt_of_incomp_of_lt he.swap hc\n    have := ih hs_hs\u2081 hlt\u2081 hc\n    run_tac\n      simp_fi\n  \u00b7\n    run_tac\n      simp_fi\n  \u00b7 have := lt_of_lt_of_incomp hc he\n    have := ih hs_hs\u2082 hc hlt\u2082\n    run_tac\n      simp_fi\n  \u00b7 run_tac\n      simp_fi\n    have := is_searchable_ins lt hs_hs\u2081 hlt\u2081 hc\n    apply find_balance1_node lt this hs_hs\u2082 (ih hs_hs\u2081 hlt\u2081 hc) he.symm\n  \u00b7 have := lt_of_incomp_of_lt he.swap hc\n    have := ih hs_hs\u2081 hlt\u2081 hc\n    run_tac\n      simp_fi\n  \u00b7\n    run_tac\n      simp_fi\n  \u00b7 run_tac\n      simp_fi\n    have := is_searchable_ins lt hs_hs\u2082 hc hlt\u2082\n    apply find_balance2_node lt hs_hs\u2081 this (ih hs_hs\u2082 hc hlt\u2082) he.symm\n  \u00b7 have := lt_of_lt_of_incomp hc he\n    have := ih hs_hs\u2082 hc hlt\u2082\n    run_tac\n      simp_fi\n#align rbnode.find_ins_of_eqv Rbnode.find_ins_of_eqv\n\ntheorem find_mkInsertResult [DecidableRel lt] (c : Color) (t : Rbnode \u03b1) (x : \u03b1) :\n    find lt (mkInsertResult c t) x = find lt t x :=\n  by\n  cases t <;> cases c <;> simp [mk_insert_result]\n  \u00b7 simp [find]\n    cases cmpUsing lt x t_val <;> simp [find]\n#align rbnode.find_mk_insert_result Rbnode.find_mkInsertResult\n\ntheorem find_insert_of_eqv [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y : \u03b1} {t : Rbnode \u03b1}\n    (he : x \u2248[lt]y) : IsSearchable lt t none none \u2192 find lt (insert lt t x) y = some x :=\n  by\n  intro hs\n  simp [insert, find_mk_insert_result]\n  apply find_ins_of_eqv lt he hs <;> simp\n#align rbnode.find_insert_of_eqv Rbnode.find_insert_of_eqv\n\ntheorem weak_trichotomous (x y) {p : Prop} (is_lt : \u2200 h : lt x y, p)\n    (is_eqv : \u2200 h : \u00aclt x y \u2227 \u00aclt y x, p) (is_gt : \u2200 h : lt y x, p) : p :=\n  by\n  by_cases lt x y\n  \u00b7 apply is_lt\n    assumption\n  by_cases lt y x\n  \u00b7 apply is_gt\n    assumption\n  \u00b7 apply is_eqv\n    constructor <;> assumption\n#align rbnode.weak_trichotomous Rbnode.weak_trichotomous\n\nsection FindInsOfNotEqv\n\nsection SimpAuxLemmas\n\ntheorem find_black_eq_find_red [DecidableRel lt] {l y r x} :\n    find lt (black_node l y r) x = find lt (red_node l y r) x := by simp [find];\n  all_goals cases cmpUsing lt x y <;> simp [find]\n#align rbnode.find_black_eq_find_red Rbnode.find_black_eq_find_red\n\ntheorem find_red_of_lt [DecidableRel lt] {l y r x} (h : lt x y) :\n    find lt (red_node l y r) x = find lt l x := by simp [find, cmpUsing, *]\n#align rbnode.find_red_of_lt Rbnode.find_red_of_lt\n\ntheorem find_red_of_gt [DecidableRel lt] [IsStrictOrder \u03b1 lt] {l y r x} (h : lt y x) :\n    find lt (red_node l y r) x = find lt r x := by have := not_lt_of_lt h; simp [find, cmpUsing, *]\n#align rbnode.find_red_of_gt Rbnode.find_red_of_gt\n\ntheorem find_red_of_incomp [DecidableRel lt] {l y r x} (h : \u00aclt x y \u2227 \u00aclt y x) :\n    find lt (red_node l y r) x = some y := by simp [find, cmpUsing, *]\n#align rbnode.find_red_of_incomp Rbnode.find_red_of_incomp\n\nend SimpAuxLemmas\n\nattribute [local simp]\n  find_black_eq_find_red find_red_of_lt find_red_of_lt find_red_of_gt find_red_of_incomp\n\nvariable [IsStrictWeakOrder \u03b1 lt] [DecidableRel lt]\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance1_lt {l r t v x y lo hi} (h : lt x y) (hl : IsSearchable lt l lo (some v))\n    (hr : IsSearchable lt r (some v) (some y)) (ht : IsSearchable lt t (some y) hi) :\n    find lt (balance1 l v r y t) x = find lt (red_node l v r) x :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 apply weak_trichotomous lt y_1 x <;> intros <;> simp [*]\n  \u00b7 apply weak_trichotomous lt x_1 x <;> intro h'\n    \u00b7 have := trans_of lt (lo_lt_hi hr_hs\u2081) h'\n      simp [*]\n    \u00b7 have : lt y_1 x := lt_of_lt_of_incomp (lo_lt_hi hr_hs\u2081) h'\n      simp [*]\n    \u00b7 apply weak_trichotomous lt y_1 x <;> intros <;> simp [*]\n#align rbnode.find_balance1_lt Rbnode.find_balance1_lt\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\nunsafe def ins_ne_leaf_tac :=\n  sorry\n#align rbnode.ins_ne_leaf_tac rbnode.ins_ne_leaf_tac\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance1Node_lt {t s x y lo hi} (hlt : lt y x) (ht : IsSearchable lt t lo (some x))\n    (hs : IsSearchable lt s (some x) hi)\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance1Node t x s) y = find lt t y :=\n  by\n  cases t <;> simp [balance1_node]\n  \u00b7 contradiction\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance1_lt; assumption'\n#align rbnode.find_balance1_node_lt Rbnode.find_balance1Node_lt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance1_gt {l r t v x y lo hi} (h : lt y x) (hl : IsSearchable lt l lo (some v))\n    (hr : IsSearchable lt r (some v) (some y)) (ht : IsSearchable lt t (some y) hi) :\n    find lt (balance1 l v r y t) x = find lt t x :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 have := trans_of lt (lo_lt_hi hr) h\n    simp [*]\n  \u00b7 have := trans_of lt (lo_lt_hi hr_hs\u2082) h\n    simp [*]\n#align rbnode.find_balance1_gt Rbnode.find_balance1_gt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance1Node_gt {t s x y lo hi} (h : lt x y) (ht : IsSearchable lt t lo (some x))\n    (hs : IsSearchable lt s (some x) hi)\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance1Node t x s) y = find lt s y :=\n  by\n  cases t <;> simp [balance1_node]\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance1_gt; assumption'\n#align rbnode.find_balance1_node_gt Rbnode.find_balance1Node_gt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance1_eqv {l r t v x y lo hi} (h : \u00aclt x y \u2227 \u00aclt y x)\n    (hl : IsSearchable lt l lo (some v)) (hr : IsSearchable lt r (some v) (some y))\n    (ht : IsSearchable lt t (some y) hi) : find lt (balance1 l v r y t) x = some y :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 have : lt y_1 x := lt_of_lt_of_incomp (lo_lt_hi hr) h.swap\n    simp [*]\n  \u00b7 have : lt x_1 x := lt_of_lt_of_incomp (lo_lt_hi hr_hs\u2082) h.swap\n    simp [*]\n#align rbnode.find_balance1_eqv Rbnode.find_balance1_eqv\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance1Node_eqv {t s x y lo hi} (h : \u00aclt x y \u2227 \u00aclt y x)\n    (ht : IsSearchable lt t lo (some y)) (hs : IsSearchable lt s (some y) hi)\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance1Node t y s) x = some y :=\n  by\n  cases t <;> simp [balance1_node]\n  \u00b7 contradiction\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance1_eqv; assumption'\n#align rbnode.find_balance1_node_eqv Rbnode.find_balance1Node_eqv\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance2_lt {l v r t x y lo hi} (h : lt x y) (hl : IsSearchable lt l (some y) (some v))\n    (hr : IsSearchable lt r (some v) hi) (ht : IsSearchable lt t lo (some y)) :\n    find lt (balance2 l v r y t) x = find lt t x :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 have := trans h (lo_lt_hi hl_hs\u2081)\n    simp [*]\n  \u00b7 have := trans h (lo_lt_hi hl)\n    simp [*]\n#align rbnode.find_balance2_lt Rbnode.find_balance2_lt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance2Node_lt {s t x y lo hi} (h : lt x y) (ht : IsSearchable lt t (some y) hi)\n    (hs : IsSearchable lt s lo (some y))\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance2Node t y s) x = find lt s x :=\n  by\n  cases t <;> simp [balance2_node]\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance2_lt; assumption'\n#align rbnode.find_balance2_node_lt Rbnode.find_balance2Node_lt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance2_gt {l v r t x y lo hi} (h : lt y x) (hl : IsSearchable lt l (some y) (some v))\n    (hr : IsSearchable lt r (some v) hi) (ht : IsSearchable lt t lo (some y)) :\n    find lt (balance2 l v r y t) x = find lt (red_node l v r) x :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 apply weak_trichotomous lt x_1 x <;> intro h' <;> simp [*]\n    \u00b7 apply weak_trichotomous lt y_1 x <;> intros <;> simp [*]\n    \u00b7 have : lt x _ := lt_of_incomp_of_lt h'.swap (lo_lt_hi hl_hs\u2082)\n      simp [*]\n    \u00b7 have := trans h' (lo_lt_hi hl_hs\u2082)\n      simp [*]\n  \u00b7 apply weak_trichotomous lt y_1 x <;> intros <;> simp [*]\n#align rbnode.find_balance2_gt Rbnode.find_balance2_gt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance2Node_gt {s t x y lo hi} (h : lt y x) (ht : IsSearchable lt t (some y) hi)\n    (hs : IsSearchable lt s lo (some y))\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance2Node t y s) x = find lt t x :=\n  by\n  cases t <;> simp [balance2_node]\n  \u00b7 contradiction\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance2_gt; assumption'\n#align rbnode.find_balance2_node_gt Rbnode.find_balance2Node_gt\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\ntheorem find_balance2_eqv {l v r t x y lo hi} (h : \u00aclt x y \u2227 \u00aclt y x)\n    (hl : IsSearchable lt l (some y) (some v)) (hr : IsSearchable lt r (some v) hi)\n    (ht : IsSearchable lt t lo (some y)) : find lt (balance2 l v r y t) x = some y :=\n  by\n  revert hl hr ht;\n  apply balance.cases l v r <;> intros <;> simp [*] <;>\n    run_tac\n      is_searchable_tactic\n  \u00b7 have := lt_of_incomp_of_lt h (lo_lt_hi hl_hs\u2081)\n    simp [*]\n  \u00b7 have := lt_of_incomp_of_lt h (lo_lt_hi hl)\n    simp [*]\n#align rbnode.find_balance2_eqv Rbnode.find_balance2_eqv\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic rbnode.is_searchable_tactic -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic ins_ne_leaf_tac -/\ntheorem find_balance2Node_eqv {t s x y lo hi} (h : \u00aclt x y \u2227 \u00aclt y x)\n    (ht : IsSearchable lt t (some y) hi) (hs : IsSearchable lt s lo (some y))\n    (hne : t \u2260 leaf := by\n      run_tac\n        ins_ne_leaf_tac) :\n    find lt (balance2Node t y s) x = some y :=\n  by\n  cases t <;> simp [balance2_node]\n  \u00b7 contradiction\n  all_goals intros ;\n    run_tac\n      is_searchable_tactic;\n    apply find_balance2_eqv; assumption'\n#align rbnode.find_balance2_node_eqv Rbnode.find_balance2Node_eqv\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic _private.3968712505.simp_fi -/\ntheorem find_ins_of_disj {x y : \u03b1} {t : Rbnode \u03b1} (hn : lt x y \u2228 lt y x) :\n    \u2200 {lo hi} (hs : IsSearchable lt t lo hi) (hlt\u2081 : Lift lt lo (some x))\n      (hlt\u2082 : Lift lt (some x) hi), find lt (ins lt t x) y = find lt t y :=\n  by\n  apply ins.induction lt t x <;> intros\n  \u00b7 cases hn\n    all_goals simp [find, ins, cmpUsing, *]\n  all_goals simp at hc; cases hs\n  \u00b7 have := ih hs_hs\u2081 hlt\u2081 hc\n    run_tac\n      simp_fi\n  \u00b7 cases hn\n    \u00b7 have := lt_of_incomp_of_lt hc.symm hn\n      run_tac\n        simp_fi\n    \u00b7 have := lt_of_lt_of_incomp hn hc\n      run_tac\n        simp_fi\n  \u00b7 have := ih hs_hs\u2082 hc hlt\u2082\n    run_tac\n      simp_fi\n  \u00b7 have ih := ih hs_hs\u2081 hlt\u2081 hc\n    cases hn\n    \u00b7 cases hc' : cmpUsing lt y y_1 <;> simp at hc'\n      \u00b7 have hsi := is_searchable_ins lt hs_hs\u2081 hlt\u2081 (trans_of lt hn hc')\n        have := find_balance1_node_lt lt hc' hsi hs_hs\u2082\n        run_tac\n          simp_fi\n      \u00b7 have hlt := lt_of_lt_of_incomp hn hc'\n        have hsi := is_searchable_ins lt hs_hs\u2081 hlt\u2081 hlt\n        have := find_balance1_node_eqv lt hc' hsi hs_hs\u2082\n        run_tac\n          simp_fi\n      \u00b7 have hsi := is_searchable_ins lt hs_hs\u2081 hlt\u2081 hc\n        have := find_balance1_node_gt lt hc' hsi hs_hs\u2082\n        simp [*]\n        run_tac\n          simp_fi\n    \u00b7 have hlt := trans hn hc\n      have hsi := is_searchable_ins lt hs_hs\u2081 hlt\u2081 hc\n      have := find_balance1_node_lt lt hlt hsi hs_hs\u2082\n      run_tac\n        simp_fi\n  \u00b7 have := ih hs_hs\u2081 hlt\u2081 hc\n    run_tac\n      simp_fi\n  \u00b7 cases hn\n    \u00b7 have := lt_of_incomp_of_lt hc.swap hn\n      run_tac\n        simp_fi\n    \u00b7 have := lt_of_lt_of_incomp hn hc\n      run_tac\n        simp_fi\n  \u00b7 have ih := ih hs_hs\u2082 hc hlt\u2082\n    cases hn\n    \u00b7 have hlt := trans hc hn\n      run_tac\n        simp_fi\n      have hsi := is_searchable_ins lt hs_hs\u2082 hc hlt\u2082\n      have := find_balance2_node_gt lt hlt hsi hs_hs\u2081\n      run_tac\n        simp_fi\n    \u00b7 run_tac\n        simp_fi\n      cases hc' : cmpUsing lt y y_1 <;> simp at hc'\n      \u00b7 have hsi := is_searchable_ins lt hs_hs\u2082 hc hlt\u2082\n        have := find_balance2_node_lt lt hc' hsi hs_hs\u2081\n        run_tac\n          simp_fi\n      \u00b7 have hlt := lt_of_incomp_of_lt hc'.swap hn\n        have hsi := is_searchable_ins lt hs_hs\u2082 hlt hlt\u2082\n        have := find_balance2_node_eqv lt hc' hsi hs_hs\u2081\n        run_tac\n          simp_fi\n      \u00b7 have hsi := is_searchable_ins lt hs_hs\u2082 hc hlt\u2082\n        have := find_balance2_node_gt lt hc' hsi hs_hs\u2081\n        run_tac\n          simp_fi\n  \u00b7 have ih := ih hs_hs\u2082 hc hlt\u2082\n    run_tac\n      simp_fi\n#align rbnode.find_ins_of_disj Rbnode.find_ins_of_disj\n\nend FindInsOfNotEqv\n\ntheorem find_insert_of_disj [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y : \u03b1} {t : Rbnode \u03b1}\n    (hd : lt x y \u2228 lt y x) :\n    IsSearchable lt t none none \u2192 find lt (insert lt t x) y = find lt t y :=\n  by\n  intro hs\n  simp [insert, find_mk_insert_result]\n  apply find_ins_of_disj lt hd hs <;> simp\n#align rbnode.find_insert_of_disj Rbnode.find_insert_of_disj\n\ntheorem find_insert_of_not_eqv [DecidableRel lt] [IsStrictWeakOrder \u03b1 lt] {x y : \u03b1} {t : Rbnode \u03b1}\n    (hn : \u00acx \u2248[lt]y) : IsSearchable lt t none none \u2192 find lt (insert lt t x) y = find lt t y :=\n  by\n  intro hs\n  simp [insert, find_mk_insert_result]\n  have he : lt x y \u2228 lt y x :=\n    by\n    simp [StrictWeakOrder.Equiv, Decidable.not_and_iff_or_not, Decidable.not_not_iff] at hn\n    assumption\n  apply find_ins_of_disj lt he hs <;> simp\n#align rbnode.find_insert_of_not_eqv Rbnode.find_insert_of_not_eqv\n\nend MembershipLemmas\n\nsection IsRedBlack\n\nvariable {\u03b1 : Type u}\n\nopen Nat Color\n\ninductive IsBadRedBlack : Rbnode \u03b1 \u2192 Nat \u2192 Prop\n  |\n  bad_red {c\u2081 c\u2082 n l r v} (rb_l : IsRedBlack l c\u2081 n) (rb_r : IsRedBlack r c\u2082 n) :\n    is_bad_red_black (red_node l v r) n\n#align rbnode.is_bad_red_black Rbnode.IsBadRedBlack\n\ntheorem balance1_rb {l r t : Rbnode \u03b1} {y v : \u03b1} {c_l c_r c_t n} :\n    IsRedBlack l c_l n \u2192\n      IsRedBlack r c_r n \u2192 IsRedBlack t c_t n \u2192 \u2203 c, IsRedBlack (balance1 l y r v t) c (succ n) :=\n  by intro h\u2081 h\u2082 _ <;> cases h\u2081 <;> cases h\u2082 <;> repeat' first |assumption|constructor\n#align rbnode.balance1_rb Rbnode.balance1_rb\n\ntheorem balance2_rb {l r t : Rbnode \u03b1} {y v : \u03b1} {c_l c_r c_t n} :\n    IsRedBlack l c_l n \u2192\n      IsRedBlack r c_r n \u2192 IsRedBlack t c_t n \u2192 \u2203 c, IsRedBlack (balance2 l y r v t) c (succ n) :=\n  by intro h\u2081 h\u2082 _ <;> cases h\u2081 <;> cases h\u2082 <;> repeat' first |assumption|constructor\n#align rbnode.balance2_rb Rbnode.balance2_rb\n\ntheorem balance1Node_rb {t s : Rbnode \u03b1} {y : \u03b1} {c n} :\n    IsBadRedBlack t n \u2192 IsRedBlack s c n \u2192 \u2203 c, IsRedBlack (balance1Node t y s) c (succ n) := by\n  intro h _ <;> cases h <;> simp [balance1_node] <;> apply balance1_rb <;> assumption'\n#align rbnode.balance1_node_rb Rbnode.balance1Node_rb\n\ntheorem balance2Node_rb {t s : Rbnode \u03b1} {y : \u03b1} {c n} :\n    IsBadRedBlack t n \u2192 IsRedBlack s c n \u2192 \u2203 c, IsRedBlack (balance2Node t y s) c (succ n) := by\n  intro h _ <;> cases h <;> simp [balance2_node] <;> apply balance2_rb <;> assumption'\n#align rbnode.balance2_node_rb Rbnode.balance2Node_rb\n\ndef InsRbResult : Rbnode \u03b1 \u2192 Color \u2192 Nat \u2192 Prop\n  | t, red, n => IsBadRedBlack t n\n  | t, black, n => \u2203 c, IsRedBlack t c n\n#align rbnode.ins_rb_result Rbnode.InsRbResult\n\nvariable {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n\ntheorem of_getColor_eq_red {t : Rbnode \u03b1} {c n} : getColor t = red \u2192 IsRedBlack t c n \u2192 c = red :=\n  by intro h\u2081 h\u2082; cases h\u2082 <;> simp only [get_color] at h\u2081 <;> contradiction\n#align rbnode.of_get_color_eq_red Rbnode.of_getColor_eq_red\n\ntheorem of_getColor_ne_red {t : Rbnode \u03b1} {c n} : getColor t \u2260 red \u2192 IsRedBlack t c n \u2192 c = black :=\n  by intro h\u2081 h\u2082; cases h\u2082 <;> simp only [get_color] at h\u2081 <;> contradiction\n#align rbnode.of_get_color_ne_red Rbnode.of_getColor_ne_red\n\nvariable (lt)\n\ntheorem ins_rb {t : Rbnode \u03b1} (x) : \u2200 {c n} (h : IsRedBlack t c n), InsRbResult (ins lt t x) c n :=\n  by\n  apply ins.induction lt t x <;> intros <;> cases h <;> simp [ins, *, ins_rb_result]\n  \u00b7 repeat' constructor\n  \u00b7 specialize ih h_rb_l\n    cases ih\n    constructor <;> assumption\n  \u00b7 constructor <;> assumption\n  \u00b7 specialize ih h_rb_r\n    cases ih\n    constructor <;> assumption\n  \u00b7 specialize ih h_rb_l\n    cases of_get_color_eq_red hr h_rb_l\n    apply balance1_node_rb <;> assumption\n  \u00b7 specialize ih h_rb_l\n    cases of_get_color_ne_red hnr h_rb_l\n    cases ih\n    constructor\n    constructor <;> assumption\n  \u00b7 constructor\n    constructor <;> assumption\n  \u00b7 specialize ih h_rb_r\n    cases of_get_color_eq_red hr h_rb_r\n    apply balance2_node_rb <;> assumption\n  \u00b7 specialize ih h_rb_r\n    cases of_get_color_ne_red hnr h_rb_r\n    cases ih\n    constructor\n    constructor <;> assumption\n#align rbnode.ins_rb Rbnode.ins_rb\n\ndef InsertRbResult : Rbnode \u03b1 \u2192 Color \u2192 Nat \u2192 Prop\n  | t, red, n => IsRedBlack t black (succ n)\n  | t, black, n => \u2203 c, IsRedBlack t c n\n#align rbnode.insert_rb_result Rbnode.InsertRbResult\n\ntheorem insert_rb {t : Rbnode \u03b1} (x) {c n} (h : IsRedBlack t c n) :\n    InsertRbResult (insert lt t x) c n := by\n  simp [insert]\n  have hi := ins_rb lt x h\n  generalize he : ins lt t x = r\n  simp [he] at hi\n  cases h <;> simp [get_color, ins_rb_result, insert_rb_result, mk_insert_result] at *\n  assumption'\n  \u00b7 cases hi\n    simp [mk_insert_result]\n    constructor <;> assumption\n#align rbnode.insert_rb Rbnode.insert_rb\n\ntheorem insert_isRedBlack {t : Rbnode \u03b1} {c n} (x) :\n    IsRedBlack t c n \u2192 \u2203 c n, IsRedBlack (insert lt t x) c n :=\n  by\n  intro h\n  have := insert_rb lt x h\n  cases c <;> simp [insert_rb_result] at this\n  \u00b7 constructor\n    constructor\n    assumption\n  \u00b7 cases this\n    constructor\n    constructor\n    assumption\n#align rbnode.insert_is_red_black Rbnode.insert_isRedBlack\n\nend IsRedBlack\n\nend Rbnode\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Rbtree/Insert.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.39337392964944096}}
{"text": "/-\nCopyright (c) 2019 Rob Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Rob Lewis\n-/\nimport tactic.simp_result\n\nnamespace tactic\n\n/--\n`delta_instance ids` tries to solve the goal by calling `apply_instance`,\nfirst unfolding the definitions in `ids`.\n-/\n-- We call `dsimp_result` here because otherwise\n-- `delta_target` will insert an `id` in the result.\n-- See the note [locally reducible category instances]\n-- https://github.com/leanprover-community/mathlib/blob/c9fca15420e2ad443707ace831679fd1762580fe/src/algebra/category/Mon/basic.lean#L27\n-- for an example where this used to cause a problem.\nmeta def delta_instance (ids : list name) : tactic unit :=\ndsimp_result\n  (intros >> reset_instance_cache >> delta_target ids >> apply_instance >> done)\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\n`delta_instance id\u2081 id\u2082 ...` tries to solve the goal by calling `apply_instance`,\nfirst unfolding the definitions in `id\u1d62`.\n-/\nmeta def delta_instance (ids : parse ident*) : itactic :=\ntactic.delta_instance ids\nend interactive\n\n/-- Guess a name for an instance from its expression.\n\nThis is a poor-man's version of the C++ `heuristic_inst_name`, and tries much less hard to pick a\ngood name. -/\nmeta def delta_instance_name : pexpr \u2192 string\n| (expr.app f _) := delta_instance_name f\n| (expr.pi _ _ _ body) := delta_instance_name body\n| (expr.lam _ _ _ body) := delta_instance_name body\n| (expr.const nm _) := nm.last\n| _ := \"inst\"\n\n/--\nTries to derive instances by unfolding the newly introduced type and applying type class resolution.\n\nFor example,\n```lean\n@[derive ring] def new_int : Type := \u2124\n```\nadds an instance `ring new_int`, defined to be the instance of `ring \u2124` found by `apply_instance`.\n\nMultiple instances can be added with `@[derive [ring, module \u211d]]`.\n\nThis derive handler applies only to declarations made using `def`, and will fail on such a\ndeclaration if it is unable to derive an instance. It is run with higher priority than the built-in\nhandlers, which will fail on `def`s.\n-/\n@[derive_handler, priority 2000] meta def delta_instance_handler : derive_handler :=\n\u03bb cls new_decl_name,\ndo env \u2190 get_env,\nif env.is_inductive new_decl_name then return ff else\ndo new_decl \u2190 get_decl new_decl_name,\n   new_decl_pexpr \u2190 resolve_name new_decl_name,\n   arity \u2190 get_pexpr_arg_arity_with_tgt cls new_decl.type,\n   tgt \u2190 to_expr $ apply_under_n_pis cls new_decl_pexpr new_decl.type\n     (new_decl.type.pi_arity - arity),\n   (vs, tgt') \u2190 open_pis tgt,\n   tgt \u2190 whnf tgt' transparency.none >>= pis vs,\n   (_, inst) \u2190 solve_aux tgt $ tactic.delta_instance [new_decl_name],\n   inst \u2190 instantiate_mvars inst,\n   inst \u2190 replace_univ_metas_with_univ_params inst,\n   tgt \u2190 instantiate_mvars tgt,\n   nm \u2190 get_unused_decl_name $ new_decl_name <.> (delta_instance_name cls),\n   add_protected_decl $ declaration.defn nm inst.collect_univ_params tgt inst\n     new_decl.reducibility_hints new_decl.is_trusted,\n   set_basic_attribute `instance nm tt,\n   return tt\n\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/delta_instance.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6513548578981939, "lm_q1q2_score": 0.39337392445019115}}
{"text": "import .common\n\nopen cpi\nopen cpi.species\n\nopen_locale normalise\n\ndef k_bind : \u210d := fin_poly.X \"k_bind\"\ndef k_degrade : \u210d := fin_poly.X \"k_degrade\"\ndef k_unbind : \u210d := fin_poly.X \"k_unbind\"\ndef k_react : \u210d := fin_poly.X \"k_react\"\n\ndef aff : affinity \u210d := affinity.mk_pair k_bind -- x, y\n\ndef M : affinity \u210d -- u, r, t\n  :=  affinity.mk 3 0 2 k_unbind -- u - t\n  \u2218[] affinity.mk 3 1 2 k_react -- r - t\n\ndef \u03c9 : context := context.extend 0 (context.extend M.arity (context.extend 0 (context.extend 0 context.nil)))\ndef \u0393 : context := context.extend aff.arity context.nil\n\ndef s : name \u0393 := name.zero \u27e8 0, nat.succ_pos 1 \u27e9\ndef e : name \u0393 := name.zero \u27e8 1, lt_add_one 1 \u27e9\n\n@[pattern] def S : reference 0 \u03c9 := reference.zero 0\n@[pattern] def E : reference M.arity \u03c9 := reference.extend $ reference.zero M.arity\n@[pattern] def P\u2081 : reference 0 \u03c9 := reference.extend \u2218 reference.extend $ reference.zero 0\n@[pattern] def P\u2082 : reference 0 \u03c9 := reference.extend \u2218 reference.extend \u2218 reference.extend $ reference.zero 0\n\ndef x {\u0393} : name (context.extend 2 \u0393) := name.zero \u27e8 0, nat.succ_pos 1 \u27e9\ndef y {\u0393} : name (context.extend 2 \u0393) := name.zero \u27e8 1, lt_add_one 1 \u27e9\n\ndef u {\u0393} : name (context.extend M.arity \u0393) := name.zero \u27e8 0, nat.succ_pos 2 \u27e9\ndef r {\u0393} : name (context.extend M.arity \u0393) := name.zero \u27e8 1, int.coe_nat_lt.mp trivial \u27e9\ndef t {\u0393} : name (context.extend M.arity \u0393) := name.zero \u27e8 2, lt_add_one 2 \u27e9\n\n-- S = s(x, y). (x. S + y. (P|P'))\ndef S\u209b_ : species \u210d \u03c9 \u0393 :=\n  s #( 2 ) \u2b1d \u03a3# ( whole.cons (x #) (apply S \u2205)\n                $ whole.cons (y #) (apply P\u2081 \u2205 |\u209b apply P\u2082 \u2205)\n                $ whole.empty )\n\ndef S_ : choices \u210d \u03c9 \u0393 :=\n  s #( 2 ) \u2b1d' \u03a3# ( whole.cons (x #) (apply S \u2205)\n                 \u2218 whole.cons (y #) (apply P\u2081 \u2205 |\u209b apply P\u2082 \u2205)\n                 $ whole.empty )\n\n-- E = \u03bd(u, r, t : M) . e\u27e8u, r\u27e9. t. E)\ndef E\u209b : reference 0 \u03c9 := reference.zero 0\ndef E\u209b_ : species \u210d \u03c9 \u0393 :=\n  \u03bd(M) (name.extend e #\u27e8 [u, r] \u27e9) \u2b1d (name.extend t # \u2b1d apply E\u209b \u2205)\n\ndef E_ : choices \u210d \u03c9 (context.extend M.arity \u0393) :=\n  (name.extend e #\u27e8 [u, r] \u27e9) \u2b1d' (name.extend t # \u2b1d \u03bd(M) apply E (u :: r :: t :: \u2205))\n\n-- P = P' = \u03c4@k_degrade. 0\ndef P\u209b_ : species \u210d \u03c9 \u0393 := \u03c4@k_degrade \u2b1d nil\n\ndef P_ : choices \u210d \u03c9 \u0393 := \u03c4@k_degrade \u2b1d' nil\ndef P'_ : species \u210d \u03c9 \u0393 := \u03a3# P_\n\ndef \u2113 : lookup \u210d \u03c9 \u0393\n| _ S := species.rename name.extend S_\n| _ E := E_\n| _ P\u2081 := species.rename name.extend P_\n| _ P\u2082 := species.rename name.extend P_\n| (nat.succ n) (reference.extend (reference.extend a)) := by { cases a, cases a_a, cases a_a_a }\n\n-- S [s]\u2014\u2192 (; x, y) (x.S + y.(P|P'))\nexample : (\u03a3# S_) [\u2113, # s]\u27f6 (production.concretion (#( vector.nil; 2 )\n  \u03a3# ( whole.cons (x#) (apply S \u2205)\n     \u2218 whole.cons (y#) (apply P\u2081 \u2205 |\u209b apply P\u2082 \u2205)\n     $ whole.empty )))\n:= transition.choice\u2081 _ _ _ _ _ _\n\n-- P\u2081 [\u03c4@k_degrade]\u27f6 0\nexample : P'_ [\u2113, \u03c4@' k_degrade]\u27f6 (production.species nil)\n  := transition.choice\u2082 k_degrade whole.nil whole.empty\n\n/- Various intermediates -/\ndef E'_ {\u0393} : species \u210d \u03c9 \u0393 := \u03bd(M) apply E (u :: r :: t :: \u2205)\ndef C'_ : species \u210d \u03c9 \u0393 :=\n  \u03bd(M) ( ( \u03a3# ( whole.cons (u#) (apply S \u2205)\n              $ whole.cons (r#) (apply P\u2081 \u2205 |\u209b apply P\u2082 \u2205)\n              $ whole.empty ) )\n       |\u209b t# \u2b1d E'_)\n\n#eval process_immediate aff \u2113 conc ((2 : \u2102) \u25ef E'_ |\u209a 2 \u25ef (apply S \u2205) )\n\ndef system : process \u2102 \u210d \u03c9 \u0393 :=\n  fin_poly.X \"S\" \u25ef (apply S \u2205) |\u209a\n  fin_poly.X \"E\" \u25ef E'_ |\u209a\n  fin_poly.X \"S\" \u25ef C'_ |\u209a\n  fin_poly.X \"P\u2081\" \u25ef (apply P\u2081 \u2205) |\u209a\n  fin_poly.X \"P\u2082\" \u25ef (apply P\u2082 \u2205)\n\n#eval process_immediate aff \u2113 conc system\n\n/-\n-- Run the result of the above through enzyme.py\n\n(-1\u2022(E\u2022S\u2022k_bind) + 1\u2022(S\u2022k_react) + 1\u2022(S\u2022k_unbind)) \u2022 E\n(-1\u2022(E\u2022S\u2022k_bind) + 1\u2022(S\u2022k_unbind)) \u2022 S\n(-1\u2022(P\u2081\u2022k_degrade) + 1\u2022(S\u2022k_react)) \u2022 P\u2081\n(-1\u2022(P\u2082\u2022k_degrade) + 1\u2022(S\u2022k_react)) \u2022 P\u2082\n(-1\u2022(S\u2022k_react) + -1\u2022(S\u2022k_unbind) + 1\u2022(E\u2022S\u2022k_bind)) \u2022 C\n-/\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/examples/enzyme.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7745833893685269, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.39334263499493655}}
{"text": "\nimport sub_pq\nimport pq_induction_principles\n\nuniverse u\n\nsection pq_group_sub_pq\n\nvariables {Q : Type u} [power_quandle Q] {Q1 : sub_power_quandle Q}\n\n\ndef pq_group_sub_pq_inclusion : pq_group Q1 \u2192* pq_group Q :=\nbegin\n  fapply L_of_morph,\n  {\n    intro x,\n    cases x with x hx,\n    exact x,\n  },\n  {\n    split,\n    {\n      intros a b,\n      cases a with a ha,\n      cases b with b hb,\n      refl,\n    },\n    {\n      intros a n,\n      cases a with a ha,\n      refl,\n    },\n  },\nend\n\nvariables {G : Type*} [group G]\n\ndef pre_pq_group_in_sub_pq (Q1 : sub_power_quandle Q) : pre_pq_group Q \u2192 Prop\n| pre_pq_group.unit := true\n| (pre_pq_group.incl x) := x \u2208 Q1.carrier\n| (pre_pq_group.mul x y) := pre_pq_group_in_sub_pq x \u2227 pre_pq_group_in_sub_pq y\n| (pre_pq_group.inv x) := pre_pq_group_in_sub_pq x\n\ndef subgroup_of_pq_supergroup (Q1 : sub_power_quandle Q) : subgroup (pq_group Q) := { \n  carrier := \u03bb x, \u2203 y : pre_pq_group Q, \u27e6y\u27e7 = x \u2227 pre_pq_group_in_sub_pq Q1 y,\n  one_mem' := begin \n    use pre_pq_group.unit,\n    split,\n    refl,\n    trivial,\n  end,\n  mul_mem' := begin \n    intros a b ha hb,\n    cases ha with a1 ha1,\n    cases hb with b1 hb1,\n    cases ha1 with ha1 ha2,\n    cases hb1 with hb1 hb2,\n    use pre_pq_group.mul a1 b1,\n    split,\n    rw \u2190ha1,\n    rw \u2190hb1,\n    refl,\n    split,\n    assumption,\n    assumption,\n  end,\n  inv_mem' := begin \n    intros a ha,\n    cases ha with a1 ha1,\n    cases ha1 with ha1 ha2,\n    use pre_pq_group.inv a1,\n    split,\n    rw \u2190ha1,\n    refl,\n    assumption,\n  end}\n\ndef iso_subs_forward : pq_group Q1 \u2192* subgroup_of_pq_supergroup Q1 :=\nbegin\n  fapply pq_morph_to_L_morph_adj,\n  {\n    intro q,\n    cases q with q hq,\n    fconstructor,\n    exact of q,\n    unfold subgroup_of_pq_supergroup,\n    use pre_pq_group.incl q,\n    split,\n    refl,\n    exact hq,\n  },\n  {\n    split,\n    {\n      intros a b,\n      cases a with a ha,\n      cases b with b hb,\n      simp only,\n      have : (\u27e8a, ha\u27e9 \u25b7 \u27e8b, hb\u27e9 : Q1) = \u27e8a \u25b7 b, _\u27e9 := rfl,\n      rw this,\n      simp only,\n      simp_rw \u2190rhd_of_eq_of_rhd,\n      refl,\n    },\n    {\n      intros a n,\n      cases a with a ha,\n      simp only,\n      have : (\u27e8a, ha\u27e9 ^ n : Q1) = \u27e8a ^ n, _\u27e9 := rfl,\n      rw this,\n      simp only,\n      simp_rw of_pow_eq_pow_of,\n      ext1,\n      simp only [subgroup.coe_gpow, subtype.coe_mk],\n    },\n  },\nend\n\ntheorem iso_subs_forward_bijective : function.bijective (iso_subs_forward : pq_group Q1 \u2192 subgroup_of_pq_supergroup Q1) :=\nbegin\n  split,\n  {\n    refine iso_subs_forward.injective_iff.mpr _,\n    intros a ha,\n    unfold iso_subs_forward at ha,\n  },\n  {\n    sorry,\n  },\nend\n\ndef iso_subs_backward : subgroup_of_pq_supergroup Q1 \u2192* pq_group Q1 :=\nbegin\n  fconstructor,\n  {\n    intro q,\n    cases q with q hq,\n    unfold subgroup_of_pq_supergroup at hq,\n    sorry,\n  },\n  sorry,\n  sorry,\nend\n\ndef iso_subs : pq_group Q1 \u2243* subgroup_of_pq_supergroup Q1 := { \n  to_fun := iso_subs_forward,\n  inv_fun := _,\n  left_inv := _,\n  right_inv := _,\n  map_mul' := begin \n    intros x y,\n    simp only [monoid_hom.map_mul],\n  end }\n\nlemma pq_group_sub_pq_inclusion_injective : function.injective (pq_group_sub_pq_inclusion : pq_group Q1 \u2192 pq_group Q) :=\nbegin\n  refine pq_group_sub_pq_inclusion.injective_iff.mpr _,\n  intros x hx,\n  sorry,\n  /-\n  refine pq_group_list _,\n  intros x hxy,\n  induction x with y x hx,\n  {\n    simp only [list.prod_nil, list.map],\n  },\n  {\n\n  },\n  -/\nend\n\n\n\nend pq_group_sub_pq\n", "meta": {"author": "torstein-vik", "repo": "power-quandle-lean", "sha": "452437602c4be2e6c5ad5f5224b068baabfdf9e1", "save_path": "github-repos/lean/torstein-vik-power-quandle-lean", "path": "github-repos/lean/torstein-vik-power-quandle-lean/power-quandle-lean-452437602c4be2e6c5ad5f5224b068baabfdf9e1/src/pq_group_sub_pq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743735019595, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3931875651352885}}
{"text": "import .reify .nnf\n\nopen expr tactic\n\nmeta def main : tactic unit :=\ndo dx \u2190 get_domain, \n   ihx \u2190 to_expr ``(inhabited),\n   ix \u2190 mk_instance (app ihx dx),\n   rev dx,\n   abst dx,\n   reify dx ix,\n   -- to_expr ``(form.rvalid_of_valid _) >>= apply,\n   -- to_expr ``(@form.valid_of_closed_of_unsat_neg _ %%ix) >>= apply,\n   -- exact_dec_trivial,\n   skip\n\nexample (f g : nat \u2192 nat) : \u2203 y : nat, (f y < y \u2228 y \u2264 g (y + 2)) := begin \n  main, \n  \n  \n  --apply form.rvalid_of_valid _,\n\nend", "meta": {"author": "skbaek", "repo": "clausify", "sha": "d09b071bdcce7577c3fffacd0893b776285b1590", "save_path": "github-repos/lean/skbaek-clausify", "path": "github-repos/lean/skbaek-clausify/clausify-d09b071bdcce7577c3fffacd0893b776285b1590/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.39318755295684266}}
{"text": "import data.list.basic\nimport init.data.list.lemmas\n\nimport util\n\n\nopen list\n\ndef atom := string\n\ninductive lvalue \n| tt\n| uu\n| ff\n\n\n\n\nstructure rule :=\n  (head : list atom)\n  (pbody : list atom)\n  (nbody : list atom)\n\ndef program := list rule\n\n\nexample : program := [\n  rule.mk [] [] [],\n  rule.mk [] [] [],\n  rule.mk [] [] []]\n\nexample : rule :=\n  { rule . head := [],\n        pbody := [],\n        nbody := [] } \n\n\n#check list.subset\n\nstructure I :=\n  (T : list atom)\n  (P : list atom)\n  (subset : T \u2286 P)\n\nsection I\n  variable i : I\n\n  @[simp, reducible] def U : list atom :=\n    filter (\u03bb x, x \u2209 i.T) i.P\n\n  @[simp, reducible] def eval (a : atom) : lvalue := \n    if a \u2208 i.T then lvalue.tt else \n    if a \u2208 i.P then lvalue.uu else lvalue.ff\n\n  def for (i : I) (a : list atom) : I :=\n    begin\n      let T\u2082 := (filter (\u03bb x, x \u2208 a) i.T),\n      let P\u2082 := (filter (\u03bb x, x \u2208 a) i.P),\n      \n    end\n\n  -- def eval_I (a : list atom) :\n\n  -- @[simp, reducible] def eval_or (a : list atom) : lvalue := \n\n\n  private def I_from_disjoint (T : list atom) (PwithoutT : list atom) : I :=\n    begin\n      let P := T ++ PwithoutT,\n      have subset : T \u2286 P := begin\n        apply subset_append_of_subset_left,\n        apply subset.refl,\n      end,\n      exact \u27e8T, P, subset\u27e9\n    end\n\n\n\nend I\n\nsection rule\n  variable r : rule\n  def sat (i : I) : bool := \n    sorry\nend rule", "meta": {"author": "sjkillen", "repo": "Lean", "sha": "323e99f48fecfa4fc6ad9155eac4d939b2097930", "save_path": "github-repos/lean/sjkillen-Lean", "path": "github-repos/lean/sjkillen-Lean/Lean-323e99f48fecfa4fc6ad9155eac4d939b2097930/foo/src/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191460821871, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3931301755940354}}
{"text": "import data.fintype.basic \nimport algebra.power_mod \nimport group_theory.group_action \nimport algebra.group_power \nimport algebra.big_operators \nimport data.zmod.basic\nimport tactic.ring tactic.abel\nimport group_theory.self_map\nimport group_theory.action_instances\nimport group_theory.burnside_count\nimport group_theory.dihedral\nimport data.fin_extra\nimport data.enumeration\nimport order.lattice order.lattice_extra\n\nopen group_theory\n\nnamespace combinatorics\n\nvariable (N : \u2115)\n\ndef square_grid := (fin N.succ) \u00d7 (fin N.succ)\n\nnamespace square_grid\n\ninstance : decidable_eq (square_grid N) := \n by { dsimp [square_grid], apply_instance }\n\ninstance : fintype (square_grid N) := \n by { dsimp [square_grid], apply_instance }\n\ninstance : has_repr (square_grid N) := \n  \u27e8\u03bb ij, ij.1.val.repr ++ ij.2.val.repr\u27e9 \n\ninstance : distrib_lattice (square_grid N) :=\n  by { dsimp [square_grid], apply_instance }\n\ninstance : bounded_order (square_grid N) :=\n  by { dsimp [square_grid], apply_instance }\n\nvariable {N} \ndef s : self_map (square_grid N) := \u03bb ij, \u27e8ij.1, ij.2.reflect\u27e9 \ndef r : self_map (square_grid N) := \u03bb ij, \u27e8ij.2.reflect, ij.1\u27e9\nvariable (N)\n\ndef p : dihedral.prehom 4 (self_map (square_grid N)) := \nbegin \n refine_struct {\n  r := r,\n  s := s\n }; funext ij; rcases ij with \u27e8i,j\u27e9; \n simp [self_map.one_app, self_map.mul_app, r, s, pow_succ, fin.reflect_reflect];\n refl\nend\n\ninstance four_pos : fact (4 > 0) := by apply_instance\n\ninstance : mul_action (dihedral 4) (square_grid N) := \n  self_map.mul_action_of_hom (p N).to_hom\n\nlemma smul_r\u2080 (ij : square_grid N) :\n (dihedral.r (0 : zmod 4)) \u2022 ij = ij := rfl\n\nlemma smul_r\u2081 (ij : square_grid N) :\n (dihedral.r (1 : zmod 4)) \u2022 ij = \u27e8ij.2.reflect, ij.1\u27e9 := rfl\n\nlemma smul_r\u2082 (ij : (square_grid N)) :\n  (dihedral.r (2 : zmod 4)) \u2022 ij = \u27e8ij.1.reflect, ij.2.reflect\u27e9 := \nbegin\n  change ((p N).to_hom) (dihedral.r (2 : zmod 4)) ij = _,\n  rw [dihedral.prehom.to_hom.map_r (p N) 2], \n  change r (r ij) = _,\n  simp only [r, fin.reflect_reflect], \nend\n\nlemma smul_r\u2083 (ij : (square_grid N)) :\n  (dihedral.r (3 : zmod 4)) \u2022 ij = \u27e8ij.2, ij.1.reflect\u27e9 := \nbegin\n  change ((p N).to_hom) (dihedral.r (3 : zmod 4)) ij = _,\n  rw [dihedral.prehom.to_hom.map_r (p N) 3], \n  change r (r (r ij)) = _,\n  simp only [r, fin.reflect_reflect], \nend\n\nlemma smul_s\u2080 (ij : (square_grid N)) :\n  (dihedral.s (0 : zmod 4)) \u2022 ij = \u27e8ij.1, ij.2.reflect\u27e9 := rfl\n\nlemma smul_s\u2081 (ij : (square_grid N)) :\n  (dihedral.s (1 : zmod 4)) \u2022 ij = \u27e8ij.2, ij.1\u27e9 := \nbegin\n  change ((p N).to_hom) (dihedral.s (1 : zmod 4)) ij = _,\n  rw [dihedral.prehom.to_hom.map_s (p N) 1, \n      self_map.mul_app],\n  change r (s ij) = _, simp only [s, r, fin.reflect_reflect], \nend\n\nlemma smul_s\u2082 (ij : (square_grid N)) :\n  (dihedral.s (2 : zmod 4)) \u2022 ij = \u27e8ij.1.reflect, ij.2\u27e9 := \nbegin\n  change ((p N).to_hom) (dihedral.s (2 : zmod 4)) ij = _,\n  rw [dihedral.prehom.to_hom.map_s (p N) 2], \n  change r (r (s ij)) = _,\n  simp only [s, r, fin.reflect_reflect], \nend\n\nlemma smul_s\u2083 (ij : (square_grid N)) :\n  (dihedral.s (3 : zmod 4)) \u2022 ij = \u27e8ij.2.reflect, ij.1.reflect\u27e9 := \nbegin\n  change ((p N).to_hom) (dihedral.s (3 : zmod 4)) ij = _,\n  rw [dihedral.prehom.to_hom.map_s (p N) 3], \n  change r (r (r (s ij))) = _,\n  simp only [s, r, fin.reflect_reflect], \nend\n\ndef subsets := finset (square_grid N)\n\nnamespace subsets\n\ninstance : decidable_eq (subsets N) := \n  by { dsimp [subsets], apply_instance }\n\ninstance : has_repr (subsets N) := \n  by { dsimp [subsets], apply_instance }\n\ninstance : mul_action (dihedral 4) (subsets N) := \n  by { dsimp [subsets], apply_instance }\n\nend subsets\n\n@[derive decidable_eq]\nstructure dim := (w h : \u2115)\n\nnamespace dim\n\ndef flip : dim \u2192 dim := \u03bb d, \u27e8d.h, d.w\u27e9\n\nlemma flip_h (d : dim) : d.flip.h = d.w := rfl\nlemma flip_w (d : dim) : d.flip.w = d.h := rfl\n\nlemma flip_flip (d : dim) : d.flip.flip = d := by { cases d, refl }\n\ndef smul\u2080 : (dihedral 4) \u2192 dim \u2192 dim\n| (dihedral.r i) d := cond i.val.bodd d.flip d\n| (dihedral.s i) d := cond i.val.bodd d d.flip\n\nlemma bodd_mod4 (i : \u2115) : (i % 4).bodd = i.bodd := \nbegin \n  have : nat.bodd 4 = ff := rfl,\n  rw[\u2190 congr_arg nat.bodd (nat.mod_add_div i 4), \n     nat.bodd_add, nat.bodd_mul, this, ff_band, bxor_ff ]\nend\n\nlemma bodd_add (i j : zmod 4) : (i + j).val.bodd = \n  bxor i.val.bodd j.val.bodd := \nby { rw [\u2190 nat.bodd_add,zmod.val_add], exact bodd_mod4 (i.val + j.val) }\n\nlemma bodd_sub (i j : zmod 4) : (i - j).val.bodd = \n  bxor i.val.bodd j.val.bodd := \nbegin\n  have := bodd_add (i - j) j, \n  rw [sub_add_cancel] at this, \n  have := congr_fun (congr_arg bxor this) j.val.bodd,\n  rw [bool.bxor_assoc, bxor_self, bxor_ff] at this,\n  exact this.symm\nend\n\ninstance : mul_action (dihedral 4) dim := {\n  smul := smul\u2080, \n  one_smul := \u03bb \u27e8_,_\u27e9, rfl,\n  mul_smul := \u03bb g h d, \n  begin\n    cases d with a b,\n    cases g with i i; cases h with j j;\n    simp only [smul\u2080, bodd_add, bodd_sub,\n               dihedral.rr_mul, dihedral.rs_mul, \n               dihedral.sr_mul, dihedral.ss_mul];\n    cases i.val.bodd; cases j.val.bodd; \n    simp only [bxor, cond, flip_flip]; split; refl\n  end\n}\n\nend dim\n\n@[derive decidable_eq]\nstructure box :=\n(i j k l : fin N.succ) (hik : i \u2264 k) (hjl : j \u2264 l) \n\nnamespace box \n\nvariable {N}\n\n@[ext] lemma ext (x y : box N) : \n  x.i = y.i \u2192 x.j = y.j \u2192 x.k = y.k \u2192 x.l = y.l \u2192 x = y := \nby { cases x, cases y, simp only [], rintro \u27e8_\u27e9 \u27e8_\u27e9 \u27e8_\u27e9 \u27e8_\u27e9, cc }\n\ndef bottom_left : (box N) \u2192 (square_grid N)\n| \u27e8i,j,k,l,_,_\u27e9 := \u27e8i,j\u27e9 \n\ndef bottom_right : (box N) \u2192 (square_grid N)\n| \u27e8i,j,k,l,_,_\u27e9 := \u27e8k,j\u27e9 \n\ndef top_left : (box N) \u2192 (square_grid N)\n| \u27e8i,j,k,l,_,_\u27e9 := \u27e8i,l\u27e9 \n\ndef top_right : (box N) \u2192 (square_grid N)\n| \u27e8i,j,k,l,_,_\u27e9 := \u27e8k,l\u27e9 \n\ndef size : (box N) \u2192 dim\n| \u27e8i,j,k,l,_,_\u27e9 := \u27e8k - i, l - j\u27e9\n\ndef r : self_map (box N) \n| \u27e8i,j,k,l,hik,hjl\u27e9 := \n  \u27e8l.reflect, i, j.reflect, k, fin.reflect_le hjl, hik\u27e9\n\ndef s : self_map (box N) \n| \u27e8i,j,k,l,hik,hjl\u27e9 := \n   \u27e8i,l.reflect,k,j.reflect,hik, fin.reflect_le hjl\u27e9\n\nvariable (N)\n\ndef p : dihedral.prehom 4 (self_map (box N)) := \nbegin \n  have hc : ((4 : \u2115+) : \u2115) = 4 := rfl,\n  have hp : (r ^ 4 : self_map (box N)) = r * r * r * r := \n    by simp only [pow_succ, pow_zero, mul_assoc, mul_one],\n  refine_struct {\n    r := r, s := s\n  }; \n  ext x; cases x with i j k l h\u2080 h\u2081;\n  simp only [hc, hp, self_map.mul_app, self_map.one_app,\n             r, s, fin.reflect_reflect],\nend \n\ninstance : mul_action (dihedral 4) (box N) := \n  self_map.mul_action_of_hom (p N).to_hom\n\nlemma smul_r\u2080 (x : box N) : \n  (dihedral.r (0 : zmod 4)) \u2022 x = x := rfl \n\nlemma smul_r\u2081 (x : box N) : \n  (dihedral.r (1 : zmod 4)) \u2022 x = \n    box.mk x.l.reflect x.i x.j.reflect x.k \n          (fin.reflect_le x.hjl) x.hik :=\nby { cases x, refl }\n\nlemma smul_r\u2082 (x : box N) : \n  (dihedral.r (2 : zmod 4)) \u2022 x = \n    box.mk x.k.reflect x.l.reflect x.i.reflect x.j.reflect \n          (fin.reflect_le x.hik) (fin.reflect_le x.hjl) :=\nby { cases x, refl }\n\nlemma smul_r\u2083 (x : box N) : \n  (dihedral.r (3 : zmod 4)) \u2022 x = \n    box.mk x.j x.k.reflect x.l x.i.reflect\n          x.hjl (fin.reflect_le x.hik) :=\nby { change (r * (r * r)) _ = _, cases x, \n     simp only [self_map.mul_app, r, fin.reflect_reflect], cc }\n\nlemma smul_s\u2080 (x : box N) : \n  (dihedral.s (0 : zmod 4)) \u2022 x = \n    box.mk x.i x.l.reflect x.k x.j.reflect\n           x.hik (fin.reflect_le x.hjl) := \nby { cases x, refl }\n\nlemma smul_s\u2081 (x : box N) : \n  (dihedral.s (1 : zmod 4)) \u2022 x = \n    box.mk x.j x.i x.l x.k x.hjl x.hik := \nby { change (r * s) _ = _, cases x, \n     simp only [self_map.mul_app, r, s, fin.reflect_reflect], cc }\n\nlemma smul_s\u2082 (x : box N) : \n  (dihedral.s (2 : zmod 4)) \u2022 x = \n    box.mk x.k.reflect x.j x.i.reflect x.l \n           (fin.reflect_le x.hik) x.hjl := \nby { change (r * r * s) _ = _, cases x, \n     simp only [self_map.mul_app, r, s, fin.reflect_reflect], cc }\n\nlemma smul_s\u2083 (x : box N) : \n  (dihedral.s (3 : zmod 4)) \u2022 x = \n    box.mk x.l.reflect x.k.reflect x.j.reflect x.i.reflect\n          (fin.reflect_le x.hjl) (fin.reflect_le x.hik) := \nby { change (r * r * r * s) _ = _, cases x, \n     simp only [self_map.mul_app, r, s, fin.reflect_reflect], cc }\n\ndef to_subset : \u2200 (x : box N), subsets N\n| \u27e8i,j,k,l,hik,hjl\u27e9 := \n   finset.product (finset.Icc i k) (finset.Icc j l)\n\nend box\n\nvariable {N}\n\nnamespace subsets\n\ndef bounding_box : (subsets N) \u2192 (square_grid N) \u00d7 (square_grid N) := \n  \u03bb (xs : finset (square_grid N)), \u27e8xs.inf id,xs.sup id\u27e9\n\ndef size  (y : subsets N) : \u2115 \u00d7 \u2115 := \n let b := y.bounding_box in prod.mk (b.2.1 - b.1.1) (b.2.2 - b.1.2)\n\ndef is_horizontal (y : subsets N) : bool := y.size.2 = 0\ndef is_vertical   (y : subsets N) : bool := y.size.1  = 0\n\nend subsets\n\n@[derive decidable_eq]\ninductive Z_single (N : \u2115)\n| H : (fin N) \u2192 (fin N.succ) \u2192 Z_single\n| V : (fin N.succ) \u2192 (fin N) \u2192 Z_single\n\nnamespace Z_single\n\ndef to_string : (Z_single N) \u2192 string\n| (H i j) := \"H\" ++ i.val.repr ++ j.val.repr\n| (V i j) := \"V\" ++ i.val.repr ++ j.val.repr\n\ninstance : has_repr (Z_single N) := \u27e8to_string\u27e9 \n\ndef bounding_box : (Z_single N) \u2192 (square_grid N) \u00d7 (square_grid N)\n| (H i j) := \u27e8\u27e8i.inc,j\u27e9,\u27e8i.succ,j\u27e9\u27e9\n| (V i j) := \u27e8\u27e8i,j.inc\u27e9,\u27e8i,j.succ\u27e9\u27e9\n\ndef size : (Z_single N) \u2192 \u2115 \u00d7 \u2115  \n| (H _ _) := \u27e81,0\u27e9 \n| (V _ _) := \u27e80,1\u27e9\n\ninstance : enumeration (Z_single N) := {\n  elems := \n   ((enumeration.elems : list (fin N)).bind \n      (\u03bb i, (enumeration.elems : list (fin N.succ)).map (Z_single.H i))) ++\n   ((enumeration.elems : list (fin N.succ)).bind \n      (\u03bb i, (enumeration.elems : list (fin N)).map (Z_single.V i))),\n  nodup := sorry,\n  complete := \u03bb z, \n  begin\n    cases z with i j i j; rw [list.mem_append],\n    { left, \n      exact @list.mem_bind_of_mem (fin N) (Z_single N) (H i j)\n        (enumeration.elems : list (fin N)) \n        (\u03bb i, (enumeration.elems : list (fin N.succ)).map (Z_single.H i))\n        i (enumeration.complete i)\n        (list.mem_map_of_mem (H i) (enumeration.complete j)) },  \n    { right, \n      exact @list.mem_bind_of_mem (fin N.succ) (Z_single N) (V i j)\n        (enumeration.elems : list (fin N.succ)) \n        (\u03bb i, (enumeration.elems : list (fin N)).map (Z_single.V i))\n        i (enumeration.complete i)\n        (list.mem_map_of_mem (V i) (enumeration.complete j)) },  \n  end\n}\n \ndef to_Y\u2080 : \u2200 (z : Z_single N), list (square_grid N) \n| (H i j) := [prod.mk i.inc j, prod.mk i.succ j]\n| (V i j) := [prod.mk i j.inc, prod.mk i j.succ]\n\nlemma to_Y\u2080_nodup (z : Z_single N) : z.to_Y\u2080.nodup :=\nbegin\n have : \u2200 {n : \u2115} (k : fin n), k.inc \u2260 k.succ := \n   \u03bb n k, ne_of_lt k.inc_lt_succ,\n cases z with i j i j; dsimp[to_Y\u2080]; \n simp [fin.eq_iff_veq,(nat.succ_ne_self i.val).symm, this]\nend\n\ndef to_Y (z : Z_single N) : (subsets N) := \n @finset.mk (square_grid N) z.to_Y\u2080 z.to_Y\u2080_nodup\n\nlemma to_Y_card (z : Z_single N) : z.to_Y.card = 2 := \nby { cases z with i j i j; refl }\n\nlemma to_Y_bounding_box (z : Z_single N) : \n  z.to_Y.bounding_box = z.bounding_box := \nbegin\n  cases z with i j i j, \n  focus { \n    let u : square_grid N := \u27e8i.inc,j\u27e9, \n    let v : square_grid N := \u27e8i.succ,j\u27e9,\n    have huv : u \u2264 v := \u27e8le_of_lt i.inc_lt_succ,le_refl j\u27e9, },\n  swap, focus \n  { let u : square_grid N := \u27e8i,j.inc\u27e9, \n    let v : square_grid N := \u27e8i,j.succ\u27e9,\n    have huv : u \u2264 v := \u27e8le_refl i,le_of_lt j.inc_lt_succ\u27e9 },\n  all_goals {  \n    change prod.mk (u \u2293 (v \u2293 \u22a4)) (u \u2294 (v \u2294 \u22a5)) = \u27e8u,v\u27e9,\n    rw [inf_top_eq, sup_bot_eq],\n    rw [inf_of_le_left huv, sup_of_le_right huv] \n  }\nend\n\nlemma to_Y_size (z : Z_single N) : \n  z.to_Y.size = z.size := \nbegin\n  dsimp [subsets.size], rw [to_Y_bounding_box],\n  rcases z with \u27e8\u27e8i,hi\u27e9,\u27e8j,hj\u27e9\u27e9 | \u27e8\u27e8i,hi\u27e9,\u27e8j,hj\u27e9\u27e9,\n  { change prod.mk ((i + 1) - i) (j - j) = \u27e81,0\u27e9,\n    rw [nat.sub_self, nat.add_sub_cancel_left] },\n  { change prod.mk (i - i) ((j + 1) - j) = \u27e80,1\u27e9,\n    rw [nat.sub_self, nat.add_sub_cancel_left] }\nend\n\nlemma to_Y_inj : function.injective (@to_Y N) := \nbegin\n  intros z\u2080 z\u2081 he,\n  have hb := congr_arg subsets.bounding_box he,\n  rw [to_Y_bounding_box, to_Y_bounding_box] at hb, clear he,\n  cases z\u2080 with i\u2080 j\u2080 i\u2080 j\u2080; cases z\u2081 with i\u2081 j\u2081 i\u2081 j\u2081,\n  all_goals {\n    simp only [bounding_box] at hb, \n    injection hb  with hb\u2080 hb\u2081, \n    injection hb\u2080 with hb\u2082 hb\u2083, \n    injection hb\u2081 with hb\u2084 hb\u2085 },\n  { replace hb\u2084 := fin.succ_inj.mp hb\u2084, cc },\n  { exfalso, exact ne_of_lt (fin.inc_lt_succ i\u2080) (hb\u2082.trans hb\u2084.symm) },\n  { exfalso, exact ne_of_lt (fin.inc_lt_succ j\u2080) (hb\u2083.trans hb\u2085.symm) },\n  { replace hb\u2085 := fin.succ_inj.mp hb\u2085, cc }\nend\n\ndef s : (Z_single N) \u2192 (Z_single N)\n| (H i j) := H i j.reflect \n| (V i j) := V i j.reflect \n\ndef r : (Z_single N) \u2192 (Z_single N)\n| (H i j) := V j.reflect i\n| (V i j) := H j.reflect i\n\nlemma to_Y_s (z : Z_single N) :\n  (s z).to_Y = (dihedral.s (0 : zmod 4)) \u2022 z.to_Y :=\nbegin\n  cases z with i j i j; dsimp[s, to_Y, to_Y\u2080];\n  change finset.mk _ _ = _;\n  ext x; cases x with i' j';\n  simp only [\n      mul_action.mem_smul_finset', dihedral.s_inv,\n      finset.mem_mk, multiset.mem_coe, \n      list.mem_cons_iff, list.mem_singleton,\n      mul_action.smul_eq_iff_eq_smul_inv];\n  simp only [smul_s\u2080,\n      fin.reflect_inc, fin.reflect_succ, or_comm],\nend\n\nlemma to_Y_r (z : Z_single N) :\n  (r z).to_Y = (dihedral.r (1 : zmod 4)) \u2022 z.to_Y :=\nbegin\n  cases z with i j i j; dsimp[r, to_Y, to_Y\u2080];\n  change finset.mk _ _ = _;\n  ext x; cases x with i' j';\n  simp only [\n      mul_action.mem_smul_finset', dihedral.r_inv, neg_neg,\n      finset.mem_mk, multiset.mem_coe, \n      list.mem_cons_iff, list.mem_singleton,\n      mul_action.smul_eq_iff_eq_smul_inv];\n  simp only [smul_r\u2081,\n      fin.reflect_inc, fin.reflect_succ,  or.comm],\nend\n\nend Z_single\n\nend square_grid\nend combinatorics", "meta": {"author": "NeilStrickland", "repo": "lean_lib", "sha": "6a9563de93748ace509d9db4302db6cd77d8f92c", "save_path": "github-repos/lean/NeilStrickland-lean_lib", "path": "github-repos/lean/NeilStrickland-lean_lib/lean_lib-6a9563de93748ace509d9db4302db6cd77d8f92c/src/combinatorics/square_grid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.39313016877566004}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.over\nimport Mathlib.category_theory.limits.shapes.pullbacks\nimport Mathlib.category_theory.limits.shapes.wide_pullbacks\nimport Mathlib.category_theory.limits.shapes.finite_products\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Products in the over category\n\nShows that products in the over category can be derived from wide pullbacks in the base category.\nThe main result is `over_product_of_wide_pullback`, which says that if `C` has `J`-indexed wide\npullbacks, then `over B` has `J`-indexed products.\n-/\n\nnamespace category_theory.over\n\n\nnamespace construct_products\n\n\n/--\n(Implementation)\nGiven a product diagram in `C/B`, construct the corresponding wide pullback diagram\nin `C`.\n-/\ndef wide_pullback_diagram_of_diagram_over {C : Type u} [category C] (B : C) {J : Type v}\n    (F : discrete J \u2964 over B) : limits.wide_pullback_shape J \u2964 C :=\n  limits.wide_pullback_shape.wide_cospan B (fun (j : J) => comma.left (functor.obj F j))\n    fun (j : J) => comma.hom (functor.obj F j)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\ndef cones_equiv_inverse_obj {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J \u2964 over B)\n    (c : limits.cone F) : limits.cone (wide_pullback_diagram_of_diagram_over B F) :=\n  limits.cone.mk (comma.left (limits.cone.X c))\n    (nat_trans.mk\n      fun (X : limits.wide_pullback_shape J) =>\n        option.cases_on X (comma.hom (limits.cone.X c))\n          fun (j : J) => comma_morphism.left (nat_trans.app (limits.cone.\u03c0 c) j))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\ndef cones_equiv_inverse {C : Type u} [category C] (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n    limits.cone F \u2964 limits.cone (wide_pullback_diagram_of_diagram_over B F) :=\n  functor.mk (cones_equiv_inverse_obj B F)\n    fun (c\u2081 c\u2082 : limits.cone F) (f : c\u2081 \u27f6 c\u2082) =>\n      limits.cone_morphism.mk (comma_morphism.left (limits.cone_morphism.hom f))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] theorem cones_equiv_functor_map_hom {C : Type u} [category C] (B : C) {J : Type v}\n    (F : discrete J \u2964 over B) (c\u2081 : limits.cone (wide_pullback_diagram_of_diagram_over B F))\n    (c\u2082 : limits.cone (wide_pullback_diagram_of_diagram_over B F)) (f : c\u2081 \u27f6 c\u2082) :\n    limits.cone_morphism.hom (functor.map (cones_equiv_functor B F) f) =\n        hom_mk (limits.cone_morphism.hom f) :=\n  Eq.refl (limits.cone_morphism.hom (functor.map (cones_equiv_functor B F) f))\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] def cones_equiv_unit_iso {J : Type v} {C : Type u} [category C] (B : C)\n    (F : discrete J \u2964 over B) : \ud835\udfed \u2245 cones_equiv_functor B F \u22d9 cones_equiv_inverse B F :=\n  nat_iso.of_components\n    (fun (_x : limits.cone (wide_pullback_diagram_of_diagram_over B F)) =>\n      limits.cones.ext (iso.mk \ud835\udfd9 \ud835\udfd9) sorry)\n    sorry\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp] def cones_equiv_counit_iso {J : Type v} {C : Type u} [category C] (B : C)\n    (F : discrete J \u2964 over B) : cones_equiv_inverse B F \u22d9 cones_equiv_functor B F \u2245 \ud835\udfed :=\n  nat_iso.of_components\n    (fun (_x : limits.cone F) => limits.cones.ext (iso.mk (hom_mk \ud835\udfd9) (hom_mk \ud835\udfd9)) sorry) sorry\n\n-- TODO: Can we add `. obviously` to the second arguments of `nat_iso.of_components` and\n\n--       `cones.ext`?\n\n/--\n(Impl) Establish an equivalence between the category of cones for `F` and for the \"grown\" `F`.\n-/\n@[simp] theorem cones_equiv_unit_iso_2 {J : Type v} {C : Type u} [category C] (B : C)\n    (F : discrete J \u2964 over B) : equivalence.unit_iso (cones_equiv B F) = cones_equiv_unit_iso B F :=\n  Eq.refl (equivalence.unit_iso (cones_equiv B F))\n\n/-- Use the above equivalence to prove we have a limit. -/\ntheorem has_over_limit_discrete_of_wide_pullback_limit {J : Type v} {C : Type u} [category C]\n    {B : C} (F : discrete J \u2964 over B)\n    [limits.has_limit (wide_pullback_diagram_of_diagram_over B F)] : limits.has_limit F :=\n  sorry\n\n/-- Given a wide pullback in `C`, construct a product in `C/B`. -/\ntheorem over_product_of_wide_pullback {J : Type v} {C : Type u} [category C]\n    [limits.has_limits_of_shape (limits.wide_pullback_shape J) C] {B : C} :\n    limits.has_limits_of_shape (discrete J) (over B) :=\n  limits.has_limits_of_shape.mk\n    fun (F : discrete J \u2964 over B) => has_over_limit_discrete_of_wide_pullback_limit F\n\n/-- Given a pullback in `C`, construct a binary product in `C/B`. -/\ntheorem over_binary_product_of_pullback {C : Type u} [category C] [limits.has_pullbacks C] {B : C} :\n    limits.has_binary_products (over B) :=\n  over_product_of_wide_pullback\n\n/-- Given all wide pullbacks in `C`, construct products in `C/B`. -/\ntheorem over_products_of_wide_pullbacks {C : Type u} [category C] [limits.has_wide_pullbacks C]\n    {B : C} : limits.has_products (over B) :=\n  fun (J : Type v) => over_product_of_wide_pullback\n\n/-- Given all finite wide pullbacks in `C`, construct finite products in `C/B`. -/\ntheorem over_finite_products_of_finite_wide_pullbacks {C : Type u} [category C]\n    [limits.has_finite_wide_pullbacks C] {B : C} : limits.has_finite_products (over B) :=\n  fun (J : Type v) (\ud835\udca5\u2081 : DecidableEq J) (\ud835\udca5\u2082 : fintype J) => over_product_of_wide_pullback\n\nend construct_products\n\n\n/--\nConstruct terminal object in the over category. This isn't an instance as it's not typically the\nway we want to define terminal objects.\n(For instance, this gives a terminal object which is different from the generic one given by\n`over_product_of_wide_pullback` above.)\n-/\ntheorem over_has_terminal {C : Type u} [category C] (B : C) : limits.has_terminal (over B) := sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/constructions/over/products_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.39313016877566004}}
{"text": "/-\nCopyright (c) 2021 OpenAI. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f\n-/\nimport mathzoo.imports.miniF2F\n\nopen_locale nat rat real big_operators topological_space\n\ntheorem algebra_manipexpr_2erprsqpesqeqnrpnesq\n  (e r : \u2102) :\n  2 * (e * r) + (e^2 + r^2) = (-r + (-e))^2 :=\nbegin\n  ring,\nend", "meta": {"author": "leanprover-community", "repo": "mathzoo", "sha": "87e9b492daeb929838706942aaa2437621b34a0e", "save_path": "github-repos/lean/leanprover-community-mathzoo", "path": "github-repos/lean/leanprover-community-mathzoo/mathzoo-87e9b492daeb929838706942aaa2437621b34a0e/src/mathzoo/misc/miniF2F/algebra/manipexpr_2erprsqpesqeqnrpnesq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.39310440365973737}}
{"text": "import logic lib.tukey\n\nuniverses u\n\nopen_locale logic_symbol\n\nnamespace logic\nopen_locale aclogic\nvariables {F : Type*} [has_logic_symbol F] [axiomatic_classical_logic F]\n\nnamespace Theory\n\nvariables {T U : Theory F}\n\nopen axiomatic_classical_logic axiomatic_classical_logic'\n\nlemma consistent_of_consistent_ss (h : T.consistent) (ss : U \u2286 T) : U.consistent :=\nby { simp[consistent_iff_bot] at h \u22a2, intros hU, have : T \u22a2 \u22a5, from weakening ss hU, contradiction }\n\nprivate lemma list_set_finite {\u03b1} (l : list \u03b1) : {a : \u03b1 | a \u2208 l}.finite :=\nby { induction l with a l IH, { simp },\n  { simp[show {b : \u03b1 | b = a \u2228 b \u2208 l} = insert a {b : \u03b1 | b \u2208 l}, by refl], exact set.finite.insert a IH } }\n\nvariables (F)\n\nclass has_finite_character :=\n(finite_character' : \u2200 {T : Theory F}, (\u2200 (s \u2286 T) (f : s.finite), consistent s) \u2192 consistent T)\n\ndef finite_character_of_finite_provable\n  (H : \u2200 T p, T \u22a2 p \u2192 \u2203 P : list F, (\u2200 p, p \u2208 P \u2192 T p) \u2227 \u2205 \u22a2 P.conjunction \u27f6 p) :\n  has_finite_character F :=\n\u27e8\u03bb T h, \n  begin\n    by_contradiction A,\n    have : \u2203 (P : list F), (\u2200 p \u2208 P, T p) \u2227 \u2205 \u22a2 P.conjunction \u27f6 \u22a5, from H _ _ (not_consistent_iff_bot.mp A),\n    rcases this with \u27e8P, ss, b\u27e9,\n    let s := {p | p \u2208 P},\n    have : s \u22a2 \u22a5,\n    { have lmm\u2081 : s \u22a2 P.conjunction \u27f6 \u22a5, from weakening (by simp) b,\n      have lmm\u2082 : s \u22a2 P.conjunction, from list_conjunction_provable (\u03bb p h, by_axiom (by simp[s, h])),\n      exact lmm\u2081 \u2a00 lmm\u2082 },\n    have : consistent s, from h s ss (by simp[s, list_set_finite]),\n    have : \u00acs \u22a2 \u22a5, exact consistent_iff_bot.mp this,\n    contradiction\n  end\u27e9\n\nvariables {F}\n\nnamespace consistent\nopen has_finite_character\nvariables [has_finite_character F]\n\nlemma finite_character :\n  consistent T \u2194 \u2200 (s \u2286 T) (f : s.finite), consistent s :=\n\u27e8begin\n  intros h,\n  by_contradiction A, simp at A,\n  rcases A with \u27e8s, ss, s_fin, hs\u27e9,\n  have : s \u22a2 \u22a5, from not_consistent_iff_bot.mp hs,\n  have : T \u22a2 \u22a5, from weakening ss this,\n  have : \u00acT \u22a2 \u22a5, from consistent_iff_bot.mp h,\n  contradiction\nend, finite_character'\u27e9\n\nlemma tukey_finite_charactor : tukey.finite_charactor (Theory.consistent : Theory F \u2192 Prop) :=\n\u03bb T, finite_character\n\nlemma finite_character_union (consis : consistent T) :\n  consistent (T \u222a U) \u2194 \u2200 (s \u2286 U) (f : s.finite), consistent (T \u222a s) :=\nbegin\n  rw finite_character, split,\n  { intros h s s_ss s_fin,\n    rw finite_character, intros s' s'_ss, refine h s'\n      (by { have : T \u222a s \u2286 T \u222a U, { simp, exact set.subset_union_of_subset_right s_ss T },\n      exact set.subset.trans s'_ss this }) },\n  { intros h s s_ss s_fin,\n    let u := s \u2229 U,\n    have lmm : consistent (T \u222a u), from h u (by simp[u]) (by { simp[u], exact set.finite.inter_of_left s_fin U }),\n    have ss : s \u2286 T \u222a u, { intros p mem, simp[u], have : p \u2208 T \u2228 p \u2208 U, from s_ss mem, tauto },\n    refine consistent_of_consistent_ss lmm ss }\nend\n\nlemma of_finite_induction\n  (consis : consistent T)\n  (H : \u2200 (s \u2286 U) (p \u2208 U), p \u2209 s \u2192 s.finite \u2192 consistent (T \u222a s) \u2192 consistent ((T \u222a s) +{ p })) :\n  consistent (T \u222a U) :=\nbegin\n  refine (finite_character_union consis).mpr _,\n  suffices : \u2200 (s : Theory F), set.finite s \u2192 s \u2286 U \u2192 (T \u222a s).consistent,\n  { intros s ss fin, exact this s fin ss },\n  intros s fin,\n  refine set.finite.induction_on fin (\u03bb _, by simp[consis]) _,\n  intros p s nmem fin consis ss, \n  have : T \u222a insert p s = (T \u222a s) +{ p }, { ext q, simp },\n  simp[this],\n  exact H s (set.subset.trans (show s \u2286 insert p s, by simp) ss) p (ss (show p \u2208 insert p s, by simp)) nmem fin\n    (consis (set.subset.trans (show s \u2286 insert p s, by simp) ss))\nend\n\nlemma Union_seq (T : \u2115 \u2192 Theory F) (h : \u2200 n, T n \u2286 T (n + 1)) :\n  Theory.consistent (\u22c3 n, T n) \u2194 \u2200 n, Theory.consistent (T n) :=\n\u27e8\u03bb H n, consistent_of_consistent_ss H (set.subset_Union T n),\n \u03bb H, by {\n  have ss_of_le : \u2200 {m n}, m \u2264 n \u2192 T m \u2286 T n,\n  { suffices : \u2200 n m, T m \u2286 T (m + n),\n    { intros m n le, simpa[show m + (n - m) = n, by omega] using this (n - m) m },\n    intros n m, induction n with n IH; simp[\u2190nat.add_one, \u2190add_assoc],\n    { exact set.subset.trans IH (h (m + n)) } },\n  rw[finite_character], intros s s_ss s_fin,\n  casesI s_fin,\n  choose f hf using show \u2200 x : s, \u2203 i, x.1 \u2208 T i, { simpa [set.subset_def] using s_ss },\n  let M := \u2a06\u1da0 i, f i,\n  have : s \u2286 T M,\n  { intros x hx,\n    have : f \u27e8x, hx\u27e9 \u2264 M, from le_fintype_sup _ _,\n    exact ss_of_le this (hf \u27e8x, hx\u27e9) },\n  exact consistent_of_consistent_ss (H M) this }\u27e9 \n\nlemma inconsistent_insert_iff_provable_neg {p : F} :\n  \u00acTheory.consistent (T +{ p }) \u2194 T \u22a2 \u223cp :=\nbegin\n  simp [Theory.consistent_iff_bot, deduction],\n  have : T \u22a2 \u223cp \u27f7 p \u27f6 \u22a5, from neg_iff p,\n  split; intros h, { exact (iff_equiv.mp this).2 \u2a00 h }, { exact (iff_equiv.mp this).1 \u2a00 h }\nend\n\nlemma extendable (consis : T.consistent) (p : F) : \n  Theory.consistent (T +{ p }) \u2228 Theory.consistent (T +{ \u223cp }) :=\nby { by_contradiction A, simp[not_or_distrib, inconsistent_insert_iff_provable_neg] at A, rcases A with \u27e8A\u2081, A\u2082\u27e9,\n     exact consis \u27e8p, A\u2082, A\u2081\u27e9 }\n\ndef maximal (T : Theory F) : Theory F := classical.epsilon (\u03bb M, consistent M \u2227 T \u2286 M \u2227 \u2200 S, consistent S \u2192 M \u2286 S \u2192 S = M)\n\ntheorem maximal_consistent (consis : consistent T) :  consistent (maximal T) := (classical.epsilon_spec (tukey.exists_maximum tukey_finite_charactor T consis)).1\n\ntheorem ss_maximal (consis : consistent T) :  T \u2286 maximal T := (classical.epsilon_spec (tukey.exists_maximum tukey_finite_charactor T consis)).2.1\n\ntheorem maximal_maximal (consis : consistent T) : \u2200 S, consistent S \u2192 maximal T \u2286 S \u2192 S = maximal T := (classical.epsilon_spec (tukey.exists_maximum tukey_finite_charactor T consis)).2.2\n\nlemma mem_maximal (consis : consistent T) (p : F) : p \u2208 maximal T \u2228 \u223cp \u2208 maximal T :=\nbegin\n  rcases extendable (maximal_consistent consis) p,\n  { have : insert p (maximal T) = maximal T, from maximal_maximal consis _ h (set.subset_insert _ _),\n    refine or.inl _, rw[\u2190this], exact set.mem_insert p (maximal T) },\n  { have : insert (\u223cp) (maximal T) = maximal T, from maximal_maximal consis _ h (set.subset_insert _ _),\n    refine or.inr _, rw[\u2190this], exact set.mem_insert (\u223cp) (maximal T) }\nend\n\nlemma mem_maximal_iff (consis : consistent T) {p : F} : p \u2208 maximal T \u2194 maximal T \u22a2 p :=\n\u27e8by_axiom,\n  \u03bb b, by { rcases mem_maximal consis p with (h | h),\n    { exact h }, { have : maximal T \u22a2 \u223cp, from by_axiom h,\n      have : \u00ac(consistent (maximal T)), { simp[consistent_def], refine \u27e8_, b, this\u27e9 },\n      have : consistent (maximal T), from maximal_consistent consis, \n      contradiction } }\u27e9\n\nlemma neg_mem_maximal_iff (consis : consistent T) {p : F} :\n  \u223cp \u2208 maximal T \u2194 p \u2209 maximal T :=\n\u27e8\u03bb b A, by { simp[mem_maximal_iff consis] at*,\n  have : \u00acconsistent (maximal T), { simp[consistent_def], refine \u27e8p, A, b\u27e9 },\n  have : consistent (maximal T), from maximal_consistent consis,\n  contradiction },\n\u03bb b, by { rcases mem_maximal consis p with (h | h), { contradiction }, { exact h } }\u27e9\n\nlemma imply_mem_maximal_iff (consis : consistent T) {p q : F} :\n  p \u27f6 q \u2208 maximal T \u2194 (p \u2208 maximal T \u2192 q \u2208 maximal T) :=\n\u27e8\u03bb b\u2081 b\u2082, by { simp[mem_maximal_iff consis] at*, exact b\u2081 \u2a00 b\u2082 },\n\u03bb h, begin\n  by_cases C : p \u2208 maximal T,\n  { simp[mem_maximal_iff consis] at*, exact hyp_right (h C) p },\n  { have : \u223cp \u2208 maximal T, from (neg_mem_maximal_iff consis).mpr C,\n    simp[mem_maximal_iff consis] at*,\n    refine deduction.mp _,\n    exact explosion (show (maximal T) +{ p } \u22a2 p, by simp) (show (maximal T) +{ p } \u22a2 \u223cp, by simp[this]) }\nend\u27e9\n\nend consistent\n\nlemma provable_iff_inconsistent {p : F} : T \u22a2 p \u2194 \u00acconsistent (T +{\u223cp}) :=\n\u27e8\u03bb h, by { simp[consistent_def], refine \u27e8p, by simp[h], by simp\u27e9 },\n\u03bb h, by { have : T +{ \u223cp } \u22a2 \u22a5, from not_consistent_iff_bot.mp h,\n          have : T \u22a2 \u223c\u223cp, from (iff_of_equiv (neg_iff _)).mpr (deduction.mp this),\n          exact dn_iff.mp this }\u27e9\n\nend Theory\n\nnamespace complete\nopen Theory\nvariables {F} {\ud835\udce2 : Type*} [has_finite_character F] [semantics F \ud835\udce2] [complete F \ud835\udce2] {S : \ud835\udce2}\n\n\ntheorem compactness [semantics.nontrivial F \ud835\udce2] {T : Theory F} :\n  semantics.Satisfiable \ud835\udce2 T \u2194 (\u2200 u \u2286 T, u.finite \u2192 semantics.Satisfiable \ud835\udce2 (u : set F)) :=\n\u27e8by rintros \u27e8S, hS\u27e9 u ss hu; refine \u27e8S, semantics.models_of_ss ss hS\u27e9,\n  begin\n    intros h,\n    have : consistent T,\n    from consistent.finite_character.mpr\n      (by { intros u ss hu, exact consistent_iff_Satisfiable.mpr (h u ss hu) }),\n    exact consistent_iff_Satisfiable.mp this\n  end\u27e9\n\ntheorem compactness' [semantics.nontrivial F \ud835\udce2] {T : Theory F} :\n  \u00acsemantics.Satisfiable \ud835\udce2 T \u2194 (\u2203 u \u2286 T, u.finite \u2227 \u00acsemantics.Satisfiable \ud835\udce2 (u : set F)) :=\nby rw[compactness]; simp\n\nend complete\n\nend logic", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/consistency.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947425132315, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.3930602009971508}}
{"text": "import graphs.ext\n\n/-\nWe prove the same for vertex maps ...\n-/\n/- Lemma : \n-/\nlemma eq_mor_eq_emap {G H : graph} (\u03b1 : G \u21a6 H) (\u03b2 : G \u21a6 H) (hyp : \u03b1 = \u03b2) : \u03b1.edge_map = \u03b2.edge_map :=\nbegin\n  sorry,\n\n\n\n\n  \nend", "meta": {"author": "barriecooper", "repo": "lean-graphs", "sha": "3f7be961f99fe084f950f52fe17c53e8093b5337", "save_path": "github-repos/lean/barriecooper-lean-graphs", "path": "github-repos/lean/barriecooper-lean-graphs/lean-graphs-3f7be961f99fe084f950f52fe17c53e8093b5337/src/game/morphisms-world/level7.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.39306019291195105}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.category.Cat\nimport Mathlib.category_theory.groupoid\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Objects of a category up to an isomorphism\n\n`is_isomorphic X Y := nonempty (X \u2245 Y)` is an equivalence relation on the objects of a category.\nThe quotient with respect to this relation defines a functor from our category to `Type`.\n-/\n\nnamespace category_theory\n\n\n/-- An object `X` is isomorphic to an object `Y`, if `X \u2245 Y` is not empty. -/\ndef is_isomorphic {C : Type u} [category C] : C \u2192 C \u2192 Prop :=\n  fun (X Y : C) => Nonempty (X \u2245 Y)\n\n/-- `is_isomorphic` defines a setoid. -/\ndef is_isomorphic_setoid (C : Type u) [category C] : setoid C :=\n  setoid.mk is_isomorphic sorry\n\n/--\nThe functor that sends each category to the quotient space of its objects up to an isomorphism.\n-/\ndef isomorphism_classes : Cat \u2964 Type u :=\n  functor.mk (fun (C : Cat) => quotient (is_isomorphic_setoid (bundled.\u03b1 C)))\n    fun (C D : Cat) (F : C \u27f6 D) => quot.map (functor.obj F) sorry\n\ntheorem groupoid.is_isomorphic_iff_nonempty_hom {C : Type u} [groupoid C] {X : C} {Y : C} : is_isomorphic X Y \u2194 Nonempty (X \u27f6 Y) :=\n  equiv.nonempty_iff_nonempty (groupoid.iso_equiv_hom X Y)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/isomorphism_classes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7981867873410141, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3928580668183449}}
{"text": "/-\nCopyright (c) 2020 Anatole Dedecker. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anatole Dedecker\n\nBasic topological facts (limits and continuity) about `floor`,\n`ceil` and `fract` in a `floor_ring`.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.algebra.ordered\nimport Mathlib.algebra.floor\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\ntheorem tendsto_floor_at_top {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1] :\n    filter.tendsto floor filter.at_top filter.at_top :=\n  sorry\n\ntheorem tendsto_floor_at_bot {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1] :\n    filter.tendsto floor filter.at_bot filter.at_bot :=\n  monotone.tendsto_at_bot_at_bot (fun (a b : \u03b1) (hab : a \u2264 b) => floor_mono hab)\n    fun (b : \u2124) =>\n      Exists.intro (\u2191b)\n        (eq.mpr (id (Eq._oldrec (Eq.refl (floor \u2191b \u2264 b)) (floor_coe b))) (le_refl b))\n\ntheorem tendsto_ceil_at_top {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1] :\n    filter.tendsto ceil filter.at_top filter.at_top :=\n  filter.tendsto.comp filter.tendsto_neg_at_bot_at_top\n    (filter.tendsto.comp tendsto_floor_at_bot filter.tendsto_neg_at_top_at_bot)\n\ntheorem tendsto_ceil_at_bot {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1] :\n    filter.tendsto ceil filter.at_bot filter.at_bot :=\n  filter.tendsto.comp filter.tendsto_neg_at_top_at_bot\n    (filter.tendsto.comp tendsto_floor_at_top filter.tendsto_neg_at_bot_at_top)\n\ntheorem continuous_on_floor {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] (n : \u2124) :\n    continuous_on (fun (x : \u03b1) => \u2191(floor x)) (set.Ico (\u2191n) (\u2191n + 1)) :=\n  iff.mpr (continuous_on_congr (floor_eq_on_Ico' n)) continuous_on_const\n\ntheorem continuous_on_ceil {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] (n : \u2124) :\n    continuous_on (fun (x : \u03b1) => \u2191(ceil x)) (set.Ioc (\u2191n - 1) \u2191n) :=\n  iff.mpr (continuous_on_congr (ceil_eq_on_Ioc' n)) continuous_on_const\n\ntheorem tendsto_floor_right' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(floor x)) (nhds_within (\u2191n) (set.Ici \u2191n)) (nhds \u2191n) :=\n  sorry\n\ntheorem tendsto_ceil_left' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(ceil x)) (nhds_within (\u2191n) (set.Iic \u2191n)) (nhds \u2191n) :=\n  sorry\n\ntheorem tendsto_floor_right {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(floor x)) (nhds_within (\u2191n) (set.Ici \u2191n))\n        (nhds_within (\u2191n) (set.Ici \u2191n)) :=\n  sorry\n\ntheorem tendsto_ceil_left {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(ceil x)) (nhds_within (\u2191n) (set.Iic \u2191n))\n        (nhds_within (\u2191n) (set.Iic \u2191n)) :=\n  sorry\n\ntheorem tendsto_floor_left {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(floor x)) (nhds_within (\u2191n) (set.Iio \u2191n))\n        (nhds_within (\u2191n - 1) (set.Iic (\u2191n - 1))) :=\n  sorry\n\ntheorem tendsto_ceil_right {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(ceil x)) (nhds_within (\u2191n) (set.Ioi \u2191n))\n        (nhds_within (\u2191n + 1) (set.Ici (\u2191n + 1))) :=\n  sorry\n\ntheorem tendsto_floor_left' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(floor x)) (nhds_within (\u2191n) (set.Iio \u2191n)) (nhds (\u2191n - 1)) :=\n  sorry\n\ntheorem tendsto_ceil_right' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] (n : \u2124) :\n    filter.tendsto (fun (x : \u03b1) => \u2191(ceil x)) (nhds_within (\u2191n) (set.Ioi \u2191n)) (nhds (\u2191n + 1)) :=\n  sorry\n\ntheorem continuous_on_fract {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [topological_add_group \u03b1] (n : \u2124) :\n    continuous_on fract (set.Ico (\u2191n) (\u2191n + 1)) :=\n  continuous_on.sub continuous_on_id (continuous_on_floor n)\n\ntheorem tendsto_fract_left' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] [topological_add_group \u03b1] (n : \u2124) :\n    filter.tendsto fract (nhds_within (\u2191n) (set.Iio \u2191n)) (nhds 1) :=\n  sorry\n\ntheorem tendsto_fract_left {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] [topological_add_group \u03b1] (n : \u2124) :\n    filter.tendsto fract (nhds_within (\u2191n) (set.Iio \u2191n)) (nhds_within 1 (set.Iio 1)) :=\n  tendsto_nhds_within_of_tendsto_nhds_of_eventually_within fract (tendsto_fract_left' n)\n    (filter.eventually_of_forall fract_lt_one)\n\ntheorem tendsto_fract_right' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] [topological_add_group \u03b1] (n : \u2124) :\n    filter.tendsto fract (nhds_within (\u2191n) (set.Ici \u2191n)) (nhds 0) :=\n  sorry\n\ntheorem tendsto_fract_right {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] [order_closed_topology \u03b1] [topological_add_group \u03b1] (n : \u2124) :\n    filter.tendsto fract (nhds_within (\u2191n) (set.Ici \u2191n)) (nhds_within 0 (set.Ici 0)) :=\n  tendsto_nhds_within_of_tendsto_nhds_of_eventually_within fract (tendsto_fract_right' n)\n    (filter.eventually_of_forall fract_nonneg)\n\ntheorem continuous_on.comp_fract' {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] {\u03b2 : Type u_2} {\u03b3 : Type u_3} [order_topology \u03b1] [topological_add_group \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {f : \u03b2 \u2192 \u03b1 \u2192 \u03b3}\n    (h : continuous_on (function.uncurry f) (set.prod set.univ (set.Icc 0 1)))\n    (hf : \u2200 (s : \u03b2), f s 0 = f s 1) :\n    continuous fun (st : \u03b2 \u00d7 \u03b1) => f (prod.fst st) (fract (prod.snd st)) :=\n  sorry\n\ntheorem continuous_on.comp_fract {\u03b1 : Type u_1} [linear_ordered_ring \u03b1] [floor_ring \u03b1]\n    [topological_space \u03b1] {\u03b2 : Type u_2} [order_topology \u03b1] [topological_add_group \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (h : continuous_on f (set.Icc 0 1)) (hf : f 0 = f 1) :\n    continuous (f \u2218 fract) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/algebra/floor_ring_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.392824058712086}}
{"text": "/-\nCopyright (c) 2021 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\nimport data.setoid.basic\nimport group_theory.subgroup.basic\nimport group_theory.coset\nimport group_theory.subgroup.pointwise\nimport data.set.basic\nimport tactic.group\n\n/-!\n# Double cosets\n\nThis file defines double cosets for two subgroups `H K` of a group `G` and the quotient of `G` by\nthe double coset relation, i.e. `H \\ G / K`. We also prove that `G` can be writen as a disjoint\nunion of the double cosets and that if one of `H` or `K` is the trivial group (i.e. `\u22a5` ) then\nthis is the usual left or right quotient of a group by a subgroup.\n\n## Main definitions\n\n* `rel`: The double coset relation defined by two subgroups `H K` of `G`.\n* `double_coset.quotient`: The quotient of `G` by the double coset relation, i.e, ``H \\ G / K`.\n-/\n\n\nvariables {G : Type*} [group G] {\u03b1 : Type*} [has_mul \u03b1] (J: subgroup G) (g : G)\n\nnamespace doset\n\nopen_locale pointwise\n\n/--The double_coset as an element of `set \u03b1` corresponding to `s a t` -/\ndef _root_.doset (a : \u03b1) (s t : set \u03b1) : set \u03b1 := s * {a} * t\n\nlemma mem_doset {s t : set \u03b1} {a b : \u03b1} : b \u2208 doset a s t \u2194 \u2203 (x \u2208 s) (y \u2208 t), b = x * a * y :=\n\u27e8\u03bb \u27e8_, y, \u27e8x, _, hx, rfl, rfl\u27e9, hy, h\u27e9, \u27e8x, hx, y, hy, h.symm\u27e9,\n  \u03bb \u27e8x, hx, y, hy, h\u27e9, \u27e8x * a, y, \u27e8x, a, hx, rfl, rfl\u27e9, hy, h.symm\u27e9\u27e9\n\nlemma mem_doset_self (H K : subgroup G) (a : G) : a \u2208 doset a H K :=\nmem_doset.mpr \u27e81, H.one_mem, 1, K.one_mem, (one_mul a).symm.trans (mul_one (1 * a)).symm\u27e9\n\nlemma doset_eq_of_mem {H K : subgroup G} {a b : G} (hb : b \u2208 doset a H K) :\n  doset b H K = doset a H K :=\nbegin\n  obtain \u27e8_, k, \u27e8h, a, hh, (rfl : _ = _), rfl\u27e9, hk, rfl\u27e9 := hb,\n  rw [doset, doset, \u2190set.singleton_mul_singleton, \u2190set.singleton_mul_singleton, mul_assoc,\n    mul_assoc, subgroup.singleton_mul_subgroup hk, \u2190mul_assoc, \u2190mul_assoc,\n    subgroup.subgroup_mul_singleton hh],\nend\n\nlemma mem_doset_of_not_disjoint {H K : subgroup G} {a b : G}\n  (h : \u00ac disjoint (doset a H K) (doset b H K)) : b \u2208 doset a H K :=\nbegin\n  rw set.not_disjoint_iff at h,\n  simp only [mem_doset] at *,\n  obtain \u27e8x, \u27e8l, hl, r, hr, hrx\u27e9, y, hy, \u27e8r', hr', rfl\u27e9\u27e9 := h,\n  refine \u27e8y\u207b\u00b9 * l, H.mul_mem (H.inv_mem hy) (hl), r * r'\u207b\u00b9, K.mul_mem hr (K.inv_mem hr'), _\u27e9,\n  rwa [mul_assoc, mul_assoc, eq_inv_mul_iff_mul_eq, \u2190mul_assoc, \u2190mul_assoc, eq_mul_inv_iff_mul_eq],\nend\n\nlemma eq_of_not_disjoint {H K : subgroup G} {a b : G} (h: \u00ac disjoint (doset a H K) (doset b H K)) :\n  doset a H K = doset b H K :=\nbegin\n  rw disjoint.comm at h,\n  have ha : a \u2208 doset b H K := mem_doset_of_not_disjoint h,\n  apply doset_eq_of_mem ha,\nend\n\n/-- The setoid defined by the double_coset relation -/\ndef setoid (H K : set G) : setoid G :=\nsetoid.ker (\u03bb x, doset x H K)\n\n/-- Quotient of `G` by the double coset relation, i.e. `H \\ G / K` -/\ndef quotient (H K : set G) : Type* :=\nquotient (setoid H K)\n\nlemma rel_iff {H K : subgroup G} {x y : G} :\n  (setoid \u2191H \u2191K).rel x y \u2194 \u2203 (a \u2208 H) (b \u2208 K), y = a * x * b :=\niff.trans \u27e8\u03bb hxy, (congr_arg _ hxy).mpr (mem_doset_self H K y),\n  \u03bb hxy, (doset_eq_of_mem hxy).symm\u27e9 mem_doset\n\nlemma bot_rel_eq_left_rel (H : subgroup G) :\n  (setoid \u2191(\u22a5 : subgroup G) \u2191H).rel = (quotient_group.left_rel H).rel :=\nbegin\n  ext a b,\n  rw rel_iff,\n  split,\n  { rintros \u27e8a, (rfl : a = 1), b, hb, rfl\u27e9,\n    change a\u207b\u00b9 * (1 * a * b) \u2208 H,\n    rwa [one_mul, inv_mul_cancel_left] },\n  { rintro (h : a\u207b\u00b9 * b \u2208 H),\n    exact \u27e81, rfl, a\u207b\u00b9 * b, h, by rw [one_mul, mul_inv_cancel_left]\u27e9 },\nend\n\nlemma rel_bot_eq_right_group_rel (H : subgroup G) :\n  (setoid \u2191H \u2191(\u22a5 : subgroup G)).rel = (quotient_group.right_rel H).rel :=\nbegin\n  ext a b,\n  rw rel_iff,\n  split,\n  { rintros \u27e8b, hb, a, (rfl : a = 1), rfl\u27e9,\n    change b * a * 1 * a\u207b\u00b9 \u2208 H,\n    rwa [mul_one, mul_inv_cancel_right] },\n  { rintro (h : b * a\u207b\u00b9 \u2208 H),\n    exact \u27e8b * a\u207b\u00b9, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]\u27e9 },\nend\n\n/--Create a doset out of an element of `H \\ G / K`-/\ndef quot_to_doset (H K : subgroup G) (q : quotient \u2191H \u2191K) : set G := (doset q.out' H K)\n\n/--Map from `G` to `H \\ G / K`-/\nabbreviation mk (H K : subgroup G) (a : G) : quotient \u2191H \u2191K :=\nquotient.mk' a\n\ninstance (H K : subgroup G) : inhabited (quotient \u2191H \u2191K) := \u27e8mk H K (1 : G)\u27e9\n\nlemma eq (H K : subgroup G) (a b : G) : mk H K a = mk H K b \u2194 \u2203 (h \u2208 H) (k \u2208 K), b = h * a * k :=\nby { rw quotient.eq', apply rel_iff, }\n\nlemma out_eq' (H K : subgroup G) (q : quotient \u2191H \u2191K) : mk H K q.out' = q :=\nquotient.out_eq' q\n\nlemma mk_out'_eq_mul (H K : subgroup G) (g : G) :\n  \u2203 (h k : G), (h \u2208 H) \u2227 (k \u2208 K) \u2227 (mk H K g : quotient \u2191H \u2191K).out' = h * g * k :=\nbegin\nhave := eq H K (mk H K g : quotient \u2191H \u2191K).out' g,\n  rw out_eq' at this,\n  obtain \u27e8h, h_h, k, hk, T\u27e9 := this.1 rfl,\n  refine \u27e8h\u207b\u00b9, k\u207b\u00b9, (H.inv_mem h_h), K.inv_mem hk, eq_mul_inv_of_mul_eq (eq_inv_mul_of_mul_eq _)\u27e9,\n  rw [\u2190 mul_assoc, \u2190 T]\nend\n\nlemma mk_eq_of_doset_eq {H K : subgroup G} {a b : G} (h : doset a H K = doset b H K) :\n  mk H K a = mk H K b :=\nbegin\n  rw eq,\n  exact mem_doset.mp (h.symm \u25b8 mem_doset_self H K b)\nend\n\nlemma disjoint_out' {H K : subgroup G} {a b : quotient H.1 K} :\n  a \u2260 b \u2192 disjoint (doset a.out' H K) (doset b.out' H K) :=\nbegin\n  contrapose!,\n  intro h,\n  simpa [out_eq'] using mk_eq_of_doset_eq (eq_of_not_disjoint  h),\nend\n\nlemma union_quot_to_doset (H K : subgroup G) : (\u22c3 q, quot_to_doset H K q) = set.univ :=\nbegin\n  ext x,\n  simp only [set.mem_Union, quot_to_doset, mem_doset, set_like.mem_coe, exists_prop,\n    set.mem_univ, iff_true],\n  use mk H K x,\n  obtain \u27e8h, k, h3, h4, h5\u27e9 := mk_out'_eq_mul H K x,\n  refine \u27e8h\u207b\u00b9, H.inv_mem h3, k\u207b\u00b9, K.inv_mem h4, _\u27e9,\n  simp only [h5, subgroup.coe_mk, \u2190mul_assoc, one_mul, mul_left_inv, mul_inv_cancel_right],\nend\n\nlemma doset_union_right_coset (H K : subgroup G) (a : G) :\n  (\u22c3 (k : K), right_coset \u2191H (a * k)) = doset a H K :=\nbegin\n  ext x,\n  simp only [mem_right_coset_iff, exists_prop, mul_inv_rev, set.mem_Union, mem_doset,\n  subgroup.mem_carrier, set_like.mem_coe],\n  split,\n  {rintro \u27e8y, h_h\u27e9,\n    refine \u27e8x * (y\u207b\u00b9 * a\u207b\u00b9), h_h, y, y.2, _\u27e9,\n    simp only [\u2190 mul_assoc, subgroup.coe_mk, inv_mul_cancel_right]},\n  {rintros \u27e8x, hx, y, hy, hxy\u27e9,\n    refine \u27e8\u27e8y,hy\u27e9,_\u27e9,\n    simp only [hxy, \u2190mul_assoc, hx, mul_inv_cancel_right, subgroup.coe_mk]},\nend\n\nlemma doset_union_left_coset (H K : subgroup G) (a : G) :\n  (\u22c3 (h : H), left_coset (h * a : G) K) = doset a H K :=\nbegin\n  ext x,\n  simp only [mem_left_coset_iff, mul_inv_rev, set.mem_Union, mem_doset],\n  split,\n  { rintro \u27e8y, h_h\u27e9,\n    refine \u27e8y, y.2, a\u207b\u00b9 * y\u207b\u00b9 * x, h_h, _\u27e9,\n    simp only [\u2190mul_assoc, one_mul, mul_right_inv, mul_inv_cancel_right]},\n  { rintros \u27e8x, hx, y, hy, hxy\u27e9,\n    refine \u27e8\u27e8x, hx\u27e9, _\u27e9,\n    simp only [hxy, \u2190mul_assoc, hy, one_mul, mul_left_inv, subgroup.coe_mk, inv_mul_cancel_right]},\n  end\n\nlemma left_bot_eq_left_quot (H : subgroup G) :\n  quotient (\u22a5 : subgroup G).1 H = (G \u29f8 H) :=\nbegin\n  unfold quotient,\n  congr,\n  ext,\n  simp_rw \u2190 bot_rel_eq_left_rel H,\n  refl,\nend\n\nlemma right_bot_eq_right_quot (H : subgroup G) :\n  quotient H.1 (\u22a5 : subgroup G) = _root_.quotient (quotient_group.right_rel H) :=\nbegin\n  unfold quotient,\n  congr,\n  ext,\n  simp_rw \u2190 rel_bot_eq_right_group_rel H,\n  refl,\nend\n\nend doset\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/group_theory/double_coset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.39282404732454745}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.types\nimport category_theory.monoidal.center\nimport tactic.apply_fun\n\n/-!\n# Enriched categories\n\nWe set up the basic theory of `V`-enriched categories,\nfor `V` an arbitrary monoidal category.\n\nWe do not assume here that `V` is a concrete category,\nso there does not need to be a \"honest\" underlying category!\n\nUse `X \u27f6[V] Y` to obtain the `V` object of morphisms from `X` to `Y`.\n\nThis file contains the definitions of `V`-enriched categories and\n`V`-functors.\n\nWe don't yet define the `V`-object of natural transformations\nbetween a pair of `V`-functors (this requires limits in `V`),\nbut we do provide a presheaf isomorphic to the Yoneda embedding of this object.\n\nWe verify that when `V = Type v`, all these notion reduce to the usual ones.\n-/\n\nuniverses w v u\u2081 u\u2082 u\u2083\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen opposite\nopen monoidal_category\n\nvariables (V : Type v) [category.{w} V] [monoidal_category V]\n\n/--\nA `V`-category is a category enriched in a monoidal category `V`.\n\nNote that we do not assume that `V` is a concrete category,\nso there may not be an \"honest\" underlying category at all!\n-/\nclass enriched_category (C : Type u\u2081) :=\n(hom : C \u2192 C \u2192 V)\n(notation X ` \u27f6[] ` Y:10 := hom X Y)\n(id : \u03a0 X, \ud835\udfd9_ V \u27f6 (X \u27f6[] X))\n(comp : \u03a0 X Y Z, (X \u27f6[] Y) \u2297 (Y \u27f6[] Z) \u27f6 (X \u27f6[] Z))\n(id_comp : \u03a0 X Y, (\u03bb_ (X \u27f6[] Y)).inv \u226b (id X \u2297 \ud835\udfd9 _) \u226b comp X X Y = \ud835\udfd9 _ . obviously)\n(comp_id : \u03a0 X Y, (\u03c1_ (X \u27f6[] Y)).inv \u226b (\ud835\udfd9 _ \u2297 id Y) \u226b comp X Y Y = \ud835\udfd9 _ . obviously)\n(assoc :\n  \u03a0 W X Y Z, (\u03b1_ _ _ _).inv \u226b (comp W X Y \u2297 \ud835\udfd9 _) \u226b comp W Y Z = (\ud835\udfd9 _ \u2297 comp X Y Z) \u226b comp W X Z\n  . obviously)\n\nnotation (name := enriched_category.hom) X ` \u27f6[`V`] ` Y:10 := (enriched_category.hom X Y : V)\n\nvariables (V) {C : Type u\u2081} [enriched_category V C]\n\n/--\nThe `\ud835\udfd9_ V`-shaped generalized element giving the identity in a `V`-enriched category.\n-/\ndef e_id (X : C) : \ud835\udfd9_ V \u27f6 (X \u27f6[V] X) := enriched_category.id X\n/--\nThe composition `V`-morphism for a `V`-enriched category.\n-/\ndef e_comp (X Y Z : C) : (X \u27f6[V] Y) \u2297 (Y \u27f6[V] Z) \u27f6 (X \u27f6[V] Z) := enriched_category.comp X Y Z\n\n-- We don't just use `restate_axiom` here; that would leave `V` as an implicit argument.\n@[simp, reassoc]\nlemma e_id_comp (X Y : C) :\n  (\u03bb_ (X \u27f6[V] Y)).inv \u226b (e_id V X \u2297 \ud835\udfd9 _) \u226b e_comp V X X Y = \ud835\udfd9 (X \u27f6[V] Y) :=\nenriched_category.id_comp X Y\n\n@[simp, reassoc]\nlemma e_comp_id (X Y : C) :\n  (\u03c1_ (X \u27f6[V] Y)).inv \u226b (\ud835\udfd9 _ \u2297 e_id V Y) \u226b e_comp V X Y Y = \ud835\udfd9 (X \u27f6[V] Y) :=\nenriched_category.comp_id X Y\n\n@[simp, reassoc]\nlemma e_assoc (W X Y Z : C) :\n  (\u03b1_ _ _ _).inv \u226b (e_comp V W X Y \u2297 \ud835\udfd9 _) \u226b e_comp V W Y Z =\n    (\ud835\udfd9 _ \u2297 e_comp V X Y Z) \u226b e_comp V W X Z :=\nenriched_category.assoc W X Y Z\n\nsection\nvariables {V} {W : Type v} [category.{w} W] [monoidal_category W]\n\n/--\nA type synonym for `C`, which should come equipped with a `V`-enriched category structure.\nIn a moment we will equip this with the `W`-enriched category structure\nobtained by applying the functor `F : lax_monoidal_functor V W` to each hom object.\n-/\n@[nolint has_nonempty_instance unused_arguments]\ndef transport_enrichment (F : lax_monoidal_functor V W) (C : Type u\u2081) := C\n\ninstance (F : lax_monoidal_functor V W) :\n  enriched_category W (transport_enrichment F C) :=\n{ hom := \u03bb (X Y : C), F.obj (X \u27f6[V] Y),\n  id := \u03bb (X : C), F.\u03b5 \u226b F.map (e_id V X),\n  comp := \u03bb (X Y Z : C), F.\u03bc _ _ \u226b F.map (e_comp V X Y Z),\n  id_comp := \u03bb X Y, begin\n    rw [comp_tensor_id, category.assoc,\n      \u2190F.to_functor.map_id, F.\u03bc_natural_assoc, F.to_functor.map_id, F.left_unitality_inv_assoc,\n      \u2190F.to_functor.map_comp, \u2190F.to_functor.map_comp, e_id_comp, F.to_functor.map_id],\n  end,\n  comp_id := \u03bb X Y, begin\n    rw [id_tensor_comp, category.assoc,\n      \u2190F.to_functor.map_id, F.\u03bc_natural_assoc, F.to_functor.map_id, F.right_unitality_inv_assoc,\n      \u2190F.to_functor.map_comp, \u2190F.to_functor.map_comp, e_comp_id, F.to_functor.map_id],\n  end,\n  assoc := \u03bb P Q R S, begin\n    rw [comp_tensor_id, category.assoc, \u2190F.to_functor.map_id, F.\u03bc_natural_assoc,\n      F.to_functor.map_id, \u2190F.associativity_inv_assoc, \u2190F.to_functor.map_comp,\n      \u2190F.to_functor.map_comp, e_assoc, id_tensor_comp, category.assoc, \u2190F.to_functor.map_id,\n      F.\u03bc_natural_assoc, F.to_functor.map_comp],\n  end, }\n\nend\n\n/--\nConstruct an honest category from a `Type v`-enriched category.\n-/\ndef category_of_enriched_category_Type (C : Type u\u2081) [\ud835\udc9e : enriched_category (Type v) C] :\n  category.{v} C :=\n{ hom := \ud835\udc9e.hom,\n  id := \u03bb X, e_id (Type v) X punit.star,\n  comp := \u03bb X Y Z f g, e_comp (Type v) X Y Z \u27e8f, g\u27e9,\n  id_comp' := \u03bb X Y f, congr_fun (e_id_comp (Type v) X Y) f,\n  comp_id' := \u03bb X Y f, congr_fun (e_comp_id (Type v) X Y) f,\n  assoc' := \u03bb W X Y Z f g h, (congr_fun (e_assoc (Type v) W X Y Z) \u27e8f, g, h\u27e9 : _), }\n\n/--\nConstruct a `Type v`-enriched category from an honest category.\n-/\ndef enriched_category_Type_of_category (C : Type u\u2081) [\ud835\udc9e : category.{v} C] :\n  enriched_category (Type v) C :=\n{ hom := \ud835\udc9e.hom,\n  id := \u03bb X p, \ud835\udfd9 X,\n  comp := \u03bb X Y Z p, p.1 \u226b p.2,\n  id_comp := \u03bb X Y, by { ext, simp, },\n  comp_id := \u03bb X Y, by { ext, simp, },\n  assoc := \u03bb W X Y Z, by { ext \u27e8f, g, h\u27e9, simp, }, }\n\n/--\nWe verify that an enriched category in `Type u` is just the same thing as an honest category.\n-/\ndef enriched_category_Type_equiv_category (C : Type u\u2081) :\n  (enriched_category (Type v) C) \u2243 category.{v} C :=\n{ to_fun := \u03bb \ud835\udc9e, by exactI category_of_enriched_category_Type C,\n  inv_fun := \u03bb \ud835\udc9e, by exactI enriched_category_Type_of_category C,\n  left_inv := \u03bb \ud835\udc9e, begin\n    cases \ud835\udc9e,\n    dsimp [enriched_category_Type_of_category],\n    congr,\n    { ext X \u27e8\u27e9, refl, },\n    { ext X Y Z \u27e8f, g\u27e9, refl, }\n  end,\n  right_inv := \u03bb \ud835\udc9e, by { rcases \ud835\udc9e with @\u27e8@\u27e8\u27e8\u27e9\u27e9\u27e9, dsimp, congr, }, }.\n\nsection\nvariables {W : Type (v+1)} [category.{v} W] [monoidal_category W] [enriched_category W C]\n\n/-- A type synonym for `C`, which should come equipped with a `V`-enriched category structure.\nIn a moment we will equip this with the (honest) category structure\nso that `X \u27f6 Y` is `(\ud835\udfd9_ W) \u27f6 (X \u27f6[W] Y)`.\n\nWe obtain this category by\ntransporting the enrichment in `V` along the lax monoidal functor `coyoneda_tensor_unit`,\nthen using the equivalence of `Type`-enriched categories with honest categories.\n\nThis is sometimes called the \"underlying\" category of an enriched category,\nalthough some care is needed as the functor `coyoneda_tensor_unit`,\nwhich always exists, does not necessarily coincide with\n\"the forgetful functor\" from `V` to `Type`, if such exists.\nWhen `V` is any of `Type`, `Top`, `AddCommGroup`, or `Module R`,\n`coyoneda_tensor_unit` is just the usual forgetful functor, however.\nFor `V = Algebra R`, the usual forgetful functor is coyoneda of `R[X]`, not of `R`.\n(Perhaps we should have a typeclass for this situation: `concrete_monoidal`?)\n-/\n@[nolint has_nonempty_instance unused_arguments]\ndef forget_enrichment\n  (W : Type (v+1)) [category.{v} W] [monoidal_category W] (C : Type u\u2081) [enriched_category W C] :=\nC\n\nvariables (W)\n\n/-- Typecheck an object of `C` as an object of `forget_enrichment W C`. -/\ndef forget_enrichment.of (X : C) : forget_enrichment W C := X\n\n/-- Typecheck an object of `forget_enrichment W C` as an object of `C`. -/\ndef forget_enrichment.to (X : forget_enrichment W C) : C := X\n\n@[simp] lemma forget_enrichment.to_of (X : C) :\n  forget_enrichment.to W (forget_enrichment.of W X) = X := rfl\n@[simp] lemma forget_enrichment.of_to (X : forget_enrichment W C) :\n  forget_enrichment.of W (forget_enrichment.to W X) = X := rfl\n\ninstance category_forget_enrichment : category (forget_enrichment W C) :=\nbegin\n  let I : enriched_category (Type v) (transport_enrichment (coyoneda_tensor_unit W) C) :=\n    infer_instance,\n  exact enriched_category_Type_equiv_category C I,\nend\n\n/--\nWe verify that the morphism types in `forget_enrichment W C` are `(\ud835\udfd9_ W) \u27f6 (X \u27f6[W] Y)`.\n-/\nexample (X Y : forget_enrichment W C) :\n  (X \u27f6 Y) = ((\ud835\udfd9_ W) \u27f6 (forget_enrichment.to W X \u27f6[W] forget_enrichment.to W Y)) :=\nrfl\n\n/-- Typecheck a `(\ud835\udfd9_ W)`-shaped `W`-morphism as a morphism in `forget_enrichment W C`. -/\ndef forget_enrichment.hom_of {X Y : C} (f : (\ud835\udfd9_ W) \u27f6 (X \u27f6[W] Y)) :\n  forget_enrichment.of W X \u27f6 forget_enrichment.of W Y :=\nf\n\n/-- Typecheck a morphism in `forget_enrichment W C` as a `(\ud835\udfd9_ W)`-shaped `W`-morphism. -/\ndef forget_enrichment.hom_to {X Y : forget_enrichment W C} (f : X \u27f6 Y) :\n  (\ud835\udfd9_ W) \u27f6 (forget_enrichment.to W X \u27f6[W] forget_enrichment.to W Y) := f\n\n@[simp] lemma forget_enrichment.hom_to_hom_of {X Y : C} (f : (\ud835\udfd9_ W) \u27f6 (X \u27f6[W] Y)) :\n  forget_enrichment.hom_to W (forget_enrichment.hom_of W f) = f := rfl\n@[simp] lemma forget_enrichment.hom_of_hom_to {X Y : forget_enrichment W C} (f : X \u27f6 Y) :\n  forget_enrichment.hom_of W (forget_enrichment.hom_to W f) = f := rfl\n\n/-- The identity in the \"underlying\" category of an enriched category. -/\n@[simp] lemma forget_enrichment_id (X : forget_enrichment W C) :\n  forget_enrichment.hom_to W (\ud835\udfd9 X) = (e_id W (forget_enrichment.to W X : C)) :=\ncategory.id_comp _\n\n@[simp] lemma forget_enrichment_id' (X : C) :\n  forget_enrichment.hom_of W (e_id W X) = (\ud835\udfd9 (forget_enrichment.of W X : C)) :=\n(forget_enrichment_id W (forget_enrichment.of W X)).symm\n\n/-- Composition in the \"underlying\" category of an enriched category. -/\n@[simp] lemma forget_enrichment_comp {X Y Z : forget_enrichment W C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  forget_enrichment.hom_to W (f \u226b g) = (((\u03bb_ (\ud835\udfd9_ W)).inv \u226b\n    (forget_enrichment.hom_to W f \u2297 forget_enrichment.hom_to W g)) \u226b e_comp W _ _ _) :=\nrfl\n\nend\n\n/--\nA `V`-functor `F` between `V`-enriched categories\nhas a `V`-morphism from `X \u27f6[V] Y` to `F.obj X \u27f6[V] F.obj Y`,\nsatisfying the usual axioms.\n-/\nstructure enriched_functor\n  (C : Type u\u2081) [enriched_category V C] (D : Type u\u2082) [enriched_category V D] :=\n(obj : C \u2192 D)\n(map : \u03a0 X Y : C, (X \u27f6[V] Y) \u27f6 (obj X \u27f6[V] obj Y))\n(map_id' : \u2200 X : C, e_id V X \u226b map X X = e_id V (obj X) . obviously)\n(map_comp' : \u2200 X Y Z : C,\n  e_comp V X Y Z \u226b map X Z = (map X Y \u2297 map Y Z) \u226b e_comp V (obj X) (obj Y) (obj Z) . obviously)\n\nrestate_axiom enriched_functor.map_id'\nrestate_axiom enriched_functor.map_comp'\nattribute [simp, reassoc] enriched_functor.map_id\nattribute [simp, reassoc] enriched_functor.map_comp\n\n/-- The identity enriched functor. -/\n@[simps]\ndef enriched_functor.id (C : Type u\u2081) [enriched_category V C] : enriched_functor V C C :=\n{ obj := \u03bb X, X,\n  map := \u03bb X Y, \ud835\udfd9 _, }\n\ninstance : inhabited (enriched_functor V C C) := \u27e8enriched_functor.id V C\u27e9\n\n/-- Composition of enriched functors. -/\n@[simps]\ndef enriched_functor.comp {C : Type u\u2081} {D : Type u\u2082} {E : Type u\u2083}\n  [enriched_category V C] [enriched_category V D] [enriched_category V E]\n  (F : enriched_functor V C D) (G : enriched_functor V D E) :\n  enriched_functor V C E :=\n{ obj := \u03bb X, G.obj (F.obj X),\n  map := \u03bb X Y, F.map _ _ \u226b G.map _ _, }\n\nsection\nvariables {W : Type (v+1)} [category.{v} W] [monoidal_category W]\n\n/--\nAn enriched functor induces an honest functor of the underlying categories,\nby mapping the `(\ud835\udfd9_ W)`-shaped morphisms.\n-/\ndef enriched_functor.forget {C : Type u\u2081} {D : Type u\u2082}\n  [enriched_category W C] [enriched_category W D]\n  (F : enriched_functor W C D) : (forget_enrichment W C) \u2964 (forget_enrichment W D) :=\n{ obj := \u03bb X, forget_enrichment.of W (F.obj (forget_enrichment.to W X)),\n  map := \u03bb X Y f, forget_enrichment.hom_of W\n    (forget_enrichment.hom_to W f \u226b F.map (forget_enrichment.to W X) (forget_enrichment.to W Y)),\n  map_comp' := \u03bb X Y Z f g, begin\n    dsimp,\n    apply_fun forget_enrichment.hom_to W,\n    { simp only [iso.cancel_iso_inv_left, category.assoc, tensor_comp,\n        forget_enrichment.hom_to_hom_of, enriched_functor.map_comp, forget_enrichment_comp],\n      refl, },\n    { intros f g w, apply_fun forget_enrichment.hom_of W at w, simpa using w, },\n  end, }\n\nend\n\nsection\nvariables {V}\nvariables {D : Type u\u2082} [enriched_category V D]\n\n/-!\nWe now turn to natural transformations between `V`-functors.\n\nThe mostly commonly encountered definition of an enriched natural transformation\nis a collection of morphisms\n```\n(\ud835\udfd9_ W) \u27f6 (F.obj X \u27f6[V] G.obj X)\n```\nsatisfying an appropriate analogue of the naturality square.\n(c.f. https://ncatlab.org/nlab/show/enriched+natural+transformation)\n\nThis is the same thing as a natural transformation `F.forget \u27f6 G.forget`.\n\nWe formalize this as `enriched_nat_trans F G`, which is a `Type`.\n\nHowever, there's also something much nicer: with appropriate additional hypotheses,\nthere is a `V`-object `enriched_nat_trans_obj F G` which contains more information,\nand from which one can recover `enriched_nat_trans F G \u2243 (\ud835\udfd9_ V) \u27f6 enriched_nat_trans_obj F G`.\n\nUsing these as the hom-objects, we can build a `V`-enriched category\nwith objects the `V`-functors.\n\nFor `enriched_nat_trans_obj` to exist, it suffices to have `V` braided and complete.\n\nBefore assuming `V` is complete, we assume it is braided and\ndefine a presheaf `enriched_nat_trans_yoneda F G`\nwhich is isomorphic to the Yoneda embedding of `enriched_nat_trans_obj F G`\nwhether or not that object actually exists.\n\nThis presheaf has components `(enriched_nat_trans_yoneda F G).obj A`\nwhat we call the `A`-graded enriched natural transformations,\nwhich are collections of morphisms\n```\nA \u27f6 (F.obj X \u27f6[V] G.obj X)\n```\nsatisfying a similar analogue of the naturality square,\nthis time incorporating a half-braiding on `A`.\n\n(We actually define `enriched_nat_trans F G`\nas the special case `A := \ud835\udfd9_ V` with the trivial half-braiding,\nand when defining `enriched_nat_trans_yoneda F G` we use the half-braidings\ncoming from the ambient braiding on `V`.)\n-/\n\n/--\nThe type of `A`-graded natural transformations between `V`-functors `F` and `G`.\nThis is the type of morphisms in `V` from `A` to the `V`-object of natural transformations.\n-/\n@[ext, nolint has_nonempty_instance]\nstructure graded_nat_trans (A : center V) (F G : enriched_functor V C D) :=\n(app : \u03a0 (X : C), A.1 \u27f6 (F.obj X \u27f6[V] G.obj X))\n(naturality :\n  \u2200 (X Y : C), (A.2.\u03b2 (X \u27f6[V] Y)).hom \u226b (F.map X Y \u2297 app Y) \u226b e_comp V _ _ _ =\n    (app X \u2297 G.map X Y) \u226b e_comp V _ _ _)\n\nvariables [braided_category V]\nopen braided_category\n\n/--\nA presheaf isomorphic to the Yoneda embedding of\nthe `V`-object of natural transformations from `F` to `G`.\n-/\n@[simps]\ndef enriched_nat_trans_yoneda (F G : enriched_functor V C D) : V\u1d52\u1d56 \u2964 (Type (max u\u2081 w)) :=\n{ obj := \u03bb A, graded_nat_trans ((center.of_braided V).obj (unop A)) F G,\n  map := \u03bb A A' f \u03c3,\n  { app := \u03bb X, f.unop \u226b \u03c3.app X,\n    naturality := \u03bb X Y, begin\n      have p := \u03c3.naturality X Y,\n      dsimp at p \u22a2,\n      rw [\u2190id_tensor_comp_tensor_id (f.unop \u226b \u03c3.app Y) _, id_tensor_comp, category.assoc,\n        category.assoc, \u2190braiding_naturality_assoc, id_tensor_comp_tensor_id_assoc, p,\n        \u2190tensor_comp_assoc,category.id_comp],\n     end }, }\n\n-- TODO assuming `[has_limits C]` construct the actual object of natural transformations\n-- and show that the functor category is `V`-enriched.\n\nend\n\nsection\nlocal attribute [instance] category_of_enriched_category_Type\n\n/--\nWe verify that an enriched functor between `Type v` enriched categories\nis just the same thing as an honest functor.\n-/\n@[simps]\ndef enriched_functor_Type_equiv_functor\n  {C : Type u\u2081} [\ud835\udc9e : enriched_category (Type v) C]\n  {D : Type u\u2082} [\ud835\udc9f : enriched_category (Type v) D] :\n  enriched_functor (Type v) C D \u2243 (C \u2964 D) :=\n{ to_fun := \u03bb F,\n  { obj := \u03bb X, F.obj X,\n    map := \u03bb X Y f, F.map X Y f,\n    map_id' := \u03bb X, congr_fun (F.map_id X) punit.star,\n    map_comp' := \u03bb X Y Z f g, congr_fun (F.map_comp X Y Z) \u27e8f, g\u27e9, },\n  inv_fun := \u03bb F,\n  { obj := \u03bb X, F.obj X,\n    map := \u03bb X Y f, F.map f,\n    map_id' := \u03bb X, by { ext \u27e8\u27e9, exact F.map_id X, },\n    map_comp' := \u03bb X Y Z, by { ext \u27e8f, g\u27e9, exact F.map_comp f g, }, },\n  left_inv := \u03bb F, by { cases F, simp, },\n  right_inv := \u03bb F, by { cases F, simp, }, }\n\n/--\nWe verify that the presheaf representing natural transformations\nbetween `Type v`-enriched functors is actually represented by\nthe usual type of natural transformations!\n-/\ndef enriched_nat_trans_yoneda_Type_iso_yoneda_nat_trans\n  {C : Type v} [enriched_category (Type v) C]\n  {D : Type v} [enriched_category (Type v) D]\n  (F G : enriched_functor (Type v) C D) :\n  enriched_nat_trans_yoneda F G \u2245\n  yoneda.obj ((enriched_functor_Type_equiv_functor F) \u27f6 (enriched_functor_Type_equiv_functor G)) :=\nnat_iso.of_components (\u03bb \u03b1,\n  { hom := \u03bb \u03c3 x,\n    { app := \u03bb X, \u03c3.app X x,\n      naturality' := \u03bb X Y f, congr_fun (\u03c3.naturality X Y) \u27e8x, f\u27e9, },\n    inv := \u03bb \u03c3,\n    { app := \u03bb X x, (\u03c3 x).app X,\n      naturality := \u03bb X Y, by { ext \u27e8x, f\u27e9, exact ((\u03c3 x).naturality f), }, }})\n  (by tidy)\n\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/enriched/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.39280264063410186}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.natural_transformation\nimport Mathlib.category_theory.monoidal.discrete\nimport Mathlib.PostPort\n\nuniverses v u l v\u2081 u\u2081 v\u2082 u\u2082 u\u2083 v\u2083 \n\nnamespace Mathlib\n\n/-!\n# Braided and symmetric monoidal categories\n\nThe basic definitions of braided monoidal categories, and symmetric monoidal categories,\nas well as braided functors.\n\n## Implementation note\n\nWe make `braided_monoidal_category` another typeclass, but then have `symmetric_monoidal_category`\nextend this. The rationale is that we are not carrying any additional data,\njust requiring a property.\n\n## Future work\n\n* Construct the Drinfeld center of a monoidal category as a braided monoidal category.\n* Say something about pseudo-natural transformations.\n\n-/\n\nnamespace category_theory\n\n\n/--\nA braided monoidal category is a monoidal category equipped with a braiding isomorphism\n`\u03b2_ X Y : X \u2297 Y \u2245 Y \u2297 X`\nwhich is natural in both arguments,\nand also satisfies the two hexagon identities.\n-/\n-- braiding natural iso:\n\nclass braided_category (C : Type u) [category C] [monoidal_category C] \nwhere\n  braiding : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\n  braiding_naturality' : autoParam\n  (\u2200 {X X' Y Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y'), (f \u2297 g) \u226b iso.hom (braiding Y Y') = iso.hom (braiding X X') \u226b (g \u2297 f))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  hexagon_forward' : autoParam\n  (\u2200 (X Y Z : C),\n    iso.hom \u03b1_ \u226b iso.hom (braiding X (Y \u2297 Z)) \u226b iso.hom \u03b1_ =\n      (iso.hom (braiding X Y) \u2297 \ud835\udfd9) \u226b iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom (braiding X Z)))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  hexagon_reverse' : autoParam\n  (\u2200 (X Y Z : C),\n    iso.inv \u03b1_ \u226b iso.hom (braiding (X \u2297 Y) Z) \u226b iso.inv \u03b1_ =\n      (\ud835\udfd9 \u2297 iso.hom (braiding Y Z)) \u226b iso.inv \u03b1_ \u226b (iso.hom (braiding X Z) \u2297 \ud835\udfd9))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- hexagon identities:\n\n@[simp] theorem braided_category.braiding_naturality {C : Type u} [category C] [monoidal_category C] [c : braided_category C] {X : C} {X' : C} {Y : C} {Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') : (f \u2297 g) \u226b iso.hom (braided_category.braiding Y Y') = iso.hom (braided_category.braiding X X') \u226b (g \u2297 f) := sorry\n\n@[simp] theorem braided_category.braiding_naturality_assoc {C : Type u} [category C] [monoidal_category C] [c : braided_category C] {X : C} {X' : C} {Y : C} {Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') : \u2200 {X'_1 : C} (f' : Y' \u2297 Y \u27f6 X'_1),\n  (f \u2297 g) \u226b iso.hom (braided_category.braiding Y Y') \u226b f' = iso.hom (braided_category.braiding X X') \u226b (g \u2297 f) \u226b f' := sorry\n\ntheorem braided_category.hexagon_forward {C : Type u} [category C] [monoidal_category C] [c : braided_category C] (X : C) (Y : C) (Z : C) : iso.hom \u03b1_ \u226b iso.hom (braided_category.braiding X (Y \u2297 Z)) \u226b iso.hom \u03b1_ =\n  (iso.hom (braided_category.braiding X Y) \u2297 \ud835\udfd9) \u226b iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom (braided_category.braiding X Z)) := sorry\n\ntheorem braided_category.hexagon_reverse {C : Type u} [category C] [monoidal_category C] [c : braided_category C] (X : C) (Y : C) (Z : C) : iso.inv \u03b1_ \u226b iso.hom (braided_category.braiding (X \u2297 Y) Z) \u226b iso.inv \u03b1_ =\n  (\ud835\udfd9 \u2297 iso.hom (braided_category.braiding Y Z)) \u226b iso.inv \u03b1_ \u226b (iso.hom (braided_category.braiding X Z) \u2297 \ud835\udfd9) := sorry\n\nnotation:1024 \"\u03b2_\" => Mathlib.category_theory.braided_category.braiding\n\n/-!\nWe now establish how the braiding interacts with the unitors.\n\nI couldn't find a detailed proof in print, but this is discussed in:\n\n* Proposition 1 of Andr\u00e9 Joyal and Ross Street,\n  \"Braided monoidal categories\", Macquarie Math Reports 860081 (1986).\n* Proposition 2.1 of Andr\u00e9 Joyal and Ross Street,\n  \"Braided tensor categories\" , Adv. Math. 102 (1993), 20\u201378.\n* Exercise 8.1.6 of Etingof, Gelaki, Nikshych, Ostrik,\n  \"Tensor categories\", vol 25, Mathematical Surveys and Monographs (2015), AMS.\n-/\n\ntheorem braiding_left_unitor_aux\u2081 (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.inv \u03b2_) \u226b iso.inv \u03b1_ \u226b (iso.hom \u03bb_ \u2297 \ud835\udfd9) = (iso.hom \u03bb_ \u2297 \ud835\udfd9) \u226b iso.inv \u03b2_ := sorry\n\ntheorem braiding_left_unitor_aux\u2082 (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : (iso.hom \u03b2_ \u2297 \ud835\udfd9) \u226b (iso.hom \u03bb_ \u2297 \ud835\udfd9) = iso.hom \u03c1_ \u2297 \ud835\udfd9 := sorry\n\n@[simp] theorem braiding_left_unitor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : iso.hom \u03b2_ \u226b iso.hom \u03bb_ = iso.hom \u03c1_ := sorry\n\ntheorem braiding_right_unitor_aux\u2081 (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : iso.inv \u03b1_ \u226b (iso.inv \u03b2_ \u2297 \ud835\udfd9) \u226b iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom \u03c1_) = (\ud835\udfd9 \u2297 iso.hom \u03c1_) \u226b iso.inv \u03b2_ := sorry\n\ntheorem braiding_right_unitor_aux\u2082 (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : (\ud835\udfd9 \u2297 iso.hom \u03b2_) \u226b (\ud835\udfd9 \u2297 iso.hom \u03c1_) = \ud835\udfd9 \u2297 iso.hom \u03bb_ := sorry\n\n@[simp] theorem braiding_right_unitor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (X : C) : iso.hom \u03b2_ \u226b iso.hom \u03c1_ = iso.hom \u03bb_ := sorry\n\n/--\nA symmetric monoidal category is a braided monoidal category for which the braiding is symmetric.\n\nSee https://stacks.math.columbia.edu/tag/0FFW.\n-/\nclass symmetric_category (C : Type u) [category C] [monoidal_category C] \nextends braided_category C\nwhere\n  symmetry' : autoParam (C \u2192 C \u2192 iso.hom \u03b2_ \u226b iso.hom \u03b2_ = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- braiding symmetric:\n\n@[simp] theorem symmetric_category.symmetry {C : Type u} [category C] [monoidal_category C] [c : symmetric_category C] (X : C) (Y : C) : iso.hom \u03b2_ \u226b iso.hom \u03b2_ = \ud835\udfd9 := sorry\n\n@[simp] theorem symmetric_category.symmetry_assoc {C : Type u} [category C] [monoidal_category C] [c : symmetric_category C] (X : C) (Y : C) {X' : C} (f' : X \u2297 Y \u27f6 X') : iso.hom \u03b2_ \u226b iso.hom \u03b2_ \u226b f' = f' := sorry\n\n/--\nA lax braided functor between braided monoidal categories is a lax monoidal functor\nwhich preserves the braiding.\n-/\nstructure lax_braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D] \nextends lax_monoidal_functor C D\nwhere\n  braided' : autoParam\n  (\u2200 (X Y : C),\n    lax_monoidal_functor.\u03bc _to_lax_monoidal_functor X Y \u226b\n        functor.map (lax_monoidal_functor.to_functor _to_lax_monoidal_functor) (iso.hom \u03b2_) =\n      iso.hom \u03b2_ \u226b lax_monoidal_functor.\u03bc _to_lax_monoidal_functor Y X)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem lax_braided_functor.braided {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] (c : lax_braided_functor C D) (X : C) (Y : C) : lax_monoidal_functor.\u03bc (lax_braided_functor.to_lax_monoidal_functor c) X Y \u226b\n    functor.map (lax_monoidal_functor.to_functor (lax_braided_functor.to_lax_monoidal_functor c)) (iso.hom \u03b2_) =\n  iso.hom \u03b2_ \u226b lax_monoidal_functor.\u03bc (lax_braided_functor.to_lax_monoidal_functor c) Y X := sorry\n\nnamespace lax_braided_functor\n\n\n/-- The identity lax braided monoidal functor. -/\ndef id (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] : lax_braided_functor C C :=\n  mk (monoidal_functor.to_lax_monoidal_functor (monoidal_functor.id C))\n\nprotected instance inhabited (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] : Inhabited (lax_braided_functor C C) :=\n  { default := id C }\n\n/-- The composition of lax braided monoidal functors. -/\ndef comp {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {E : Type u\u2083} [category E] [monoidal_category E] [braided_category E] (F : lax_braided_functor C D) (G : lax_braided_functor D E) : lax_braided_functor C E :=\n  mk\n    (lax_monoidal_functor.mk (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G))\n      (lax_monoidal_functor.\u03b5 (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G))\n      (lax_monoidal_functor.\u03bc (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G)))\n\nprotected instance category_lax_braided_functor {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] : category (lax_braided_functor C D) :=\n  induced_category.category to_lax_monoidal_functor\n\n@[simp] theorem comp_to_nat_trans {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {F : lax_braided_functor C D} {G : lax_braided_functor C D} {H : lax_braided_functor C D} {\u03b1 : F \u27f6 G} {\u03b2 : G \u27f6 H} : monoidal_nat_trans.to_nat_trans (\u03b1 \u226b \u03b2) = monoidal_nat_trans.to_nat_trans \u03b1 \u226b monoidal_nat_trans.to_nat_trans \u03b2 :=\n  rfl\n\n/--\nInterpret a natural isomorphism of the underlyling lax monoidal functors as an\nisomorphism of the lax braided monoidal functors.\n-/\n@[simp] theorem mk_iso_hom {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {F : lax_braided_functor C D} {G : lax_braided_functor C D} (i : to_lax_monoidal_functor F \u2245 to_lax_monoidal_functor G) : iso.hom (mk_iso i) = iso.hom i :=\n  Eq.refl (iso.hom (mk_iso i))\n\nend lax_braided_functor\n\n\n/--\nA braided functor between braided monoidal categories is a monoidal functor\nwhich preserves the braiding.\n-/\n-- Note this is stated different than for `lax_braided_functor`.\n\nstructure braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D] \nextends monoidal_functor C D\nwhere\n  braided' : autoParam\n  (\u2200 (X Y : C),\n    functor.map (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor))\n        (iso.hom \u03b2_) =\n      inv (lax_monoidal_functor.\u03bc (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor) X Y) \u226b\n        iso.hom \u03b2_ \u226b lax_monoidal_functor.\u03bc (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor) Y X)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- We move the `\u03bc X Y` to the right hand side,\n\n-- so that this makes a good `@[simp]` lemma.\n\n@[simp] theorem braided_functor.braided {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] (c : braided_functor C D) (X : C) (Y : C) : functor.map\n    (lax_monoidal_functor.to_functor (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)))\n    (iso.hom \u03b2_) =\n  inv (lax_monoidal_functor.\u03bc (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)) X Y) \u226b\n    iso.hom \u03b2_ \u226b\n      lax_monoidal_functor.\u03bc (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)) Y X := sorry\n\nnamespace braided_functor\n\n\n/-- Turn a braided functor into a lax braided functor. -/\ndef to_lax_braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D] (F : braided_functor C D) : lax_braided_functor C D :=\n  lax_braided_functor.mk (monoidal_functor.to_lax_monoidal_functor (to_monoidal_functor F))\n\n/-- The identity braided monoidal functor. -/\n@[simp] theorem id_to_monoidal_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] : to_monoidal_functor (id C) = monoidal_functor.id C :=\n  Eq.refl (to_monoidal_functor (id C))\n\nprotected instance inhabited (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] : Inhabited (braided_functor C C) :=\n  { default := id C }\n\n/-- The composition of braided monoidal functors. -/\n@[simp] theorem comp_to_monoidal_functor {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {E : Type u\u2083} [category E] [monoidal_category E] [braided_category E] (F : braided_functor C D) (G : braided_functor D E) : to_monoidal_functor (comp F G) = to_monoidal_functor F \u2297\u22d9 to_monoidal_functor G :=\n  Eq.refl (to_monoidal_functor (comp F G))\n\nprotected instance category_braided_functor {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] : category (braided_functor C D) :=\n  induced_category.category to_monoidal_functor\n\n@[simp] theorem comp_to_nat_trans {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {F : braided_functor C D} {G : braided_functor C D} {H : braided_functor C D} {\u03b1 : F \u27f6 G} {\u03b2 : G \u27f6 H} : monoidal_nat_trans.to_nat_trans (\u03b1 \u226b \u03b2) = monoidal_nat_trans.to_nat_trans \u03b1 \u226b monoidal_nat_trans.to_nat_trans \u03b2 :=\n  rfl\n\n/--\nInterpret a natural isomorphism of the underlyling monoidal functors as an\nisomorphism of the braided monoidal functors.\n-/\ndef mk_iso {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] {F : braided_functor C D} {G : braided_functor C D} (i : to_monoidal_functor F \u2245 to_monoidal_functor G) : F \u2245 G :=\n  iso.mk (iso.hom i) (iso.inv i)\n\nend braided_functor\n\n\nprotected instance comm_monoid_discrete (M : Type u) [comm_monoid M] : comm_monoid (discrete M) :=\n  id _inst_10\n\nprotected instance discrete.braided_category (M : Type u) [comm_monoid M] : braided_category (discrete M) :=\n  braided_category.mk fun (X Y : discrete M) => eq_to_iso sorry\n\n/--\nA multiplicative morphism between commutative monoids gives a braided functor between\nthe corresponding discrete braided monoidal categories.\n-/\ndef discrete.braided_functor {M : Type u} [comm_monoid M] {N : Type u} [comm_monoid N] (F : M \u2192* N) : braided_functor (discrete M) (discrete N) :=\n  braided_functor.mk (monoidal_functor.mk (monoidal_functor.to_lax_monoidal_functor (discrete.monoidal_functor F)))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/braided.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3928026333598574}}
{"text": "\nimport Lib.Tactic\n\ntheorem ite_pos [h : Decidable p] (hp : p) {x y : \u03b1} :\n  ite p x y = x := by\ncases h <;> auto\n\ntheorem ite_neg [h : Decidable p] (hp : \u00ac p) {x y : \u03b1} :\n  ite p x y = y := by\ncases h <;> auto\n\n@[simp]\ntheorem decide_eq_true_iff {h : Decidable p} :\n  decide p = true \u2194 p :=\n\u27e8of_decide_eq_true, decide_eq_true\u27e9\n\n@[simp]\ntheorem decide_eq_false_iff {h : Decidable p} :\n  decide p = false \u2194 \u00ac p :=\n\u27e8of_decide_eq_false, decide_eq_false\u27e9\n\ntheorem Eq.comm {x y : \u03b1} :\n  x = y \u2194 y = x := by\nconstructor\n<;> intros h\n<;> subst h\n<;> rfl\n\ntheorem not_iff_not {p q : Prop} :\n  (p \u2194 q) \u2192 (\u00ac p \u2194 \u00ac q) :=\nby intros h; rw [h]; refl\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Logic/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.39270202941107063}}
{"text": "import logic.equiv.defs\nimport tactic.simp_result\n\nopen tactic\n\n-- Check that we can walk.\nexample : true :=\nby { simp_result { trivial } }\n\n-- Comparison without `dsimp_result`:\nexample : true :=\nbegin\n  exact (id trivial),\n  (do `(id trivial) \u2190 result, skip),\n  success_if_fail { (do `(trivial) \u2190 result, skip) },\nend\n\n-- Check that `dsimp_result` removes unnecessary `id`s.\nexample : true :=\nbegin\n  dsimp_result { exact (id trivial) },\n  success_if_fail { (do `(id trivial) \u2190 result, skip) },\n  (do `(trivial) \u2190 result, skip),\nend\n\n-- Comparison without `dsimp_result`:\nexample (a : \u2115) (b : list \u2115) (h : b.length < a) : \u2115 :=\nbegin\n  revert a,\n  intros a h,\n  exact 0,\n  (do `((\u03bb (h : list.length _ < _), 0) _) \u2190 result, skip),\n  success_if_fail { (do `(0) \u2190 result, skip) },\nend\n\n-- Check that `dsimp_result` does beta-reductions after `revert`.\nexample (a : \u2115) (b : list \u2115) (h : b.length < a) : \u2115 :=\nbegin\n  dsimp_result\n  { revert a,\n    intros a h,\n    exact 0, },\n  success_if_fail { (do `((\u03bb (h : list.length _ < _), 0) _) \u2190 result, skip), },\n  (do `(0) \u2190 result, skip),\nend\n\n-- This test tactic internally sets `pp.all ff`, and `pp.proofs tt`.\n-- This isn't very robust, as the user setting any other `pp` options\n-- will cause tests to break, but I don't think it needs to be.\nmeta def guard_result_pp (s : string) : tactic unit :=\ndo\n  o \u2190 get_options,\n  set_options ((o.set_bool `pp.all ff).set_bool `pp.proofs tt),\n  r \u2190 (to_string <$> (result >>= pp)),\n  guard (r = s) <|> fail format!\"result was {r} but expected {s}\"\n\n-- Comparison without `simp_result`:\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  exact e (e.symm (e a)),\n  guard_result_pp \"\u21d1e (\u21d1(equiv.symm e) (\u21d1e a))\",\nend\n\n-- Check that `simp_result` applies non-definitional simplifications to the result.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e a\",\nend\n\n-- Check that `simp_result only [...]` behaves as expected.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result only [equiv.apply_symm_apply] { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e a\",\nend\n\n-- Check that `simp_result only []` does not simplify.\n-- (Note the `simp_result` succeeds even if no simplification occurs.)\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result only [] { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e (\u21d1(equiv.symm e) (\u21d1e a))\",\nend\n\n-- Comparison without `simp_result`\nexample {\u03b1 : Type} (a b : \u03b1) (h : a = b) : \u2115 :=\nbegin\n  subst h,\n  exact 0,\n  guard_result_pp \"eq.rec 0 h\",\nend\n\n-- Check that we can remove `eq.rec` transports through constant families\n-- introduced by irrelevant use of `subst`.\nexample {\u03b1 : Type} (a b : \u03b1) (h : a = b) : \u2115 :=\nbegin\n  simp_result only [eq_rec_constant]\n  { subst h,\n    exact 0, },\n  guard_result_pp \"0\",\nend\n\n-- Check that `simp_result` performs simplifications on all results.\nexample : \u2115 \u00d7 \u2115 :=\nbegin\n  split,\n  simp_result\n  { exact id 0,\n    exact id 1, },\n  guard_result_pp \"(0, 1)\",\nend\n\n-- Check that `simp_result` can cope with incomplete goals.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result { apply e.to_fun, apply e.inv_fun, apply e.to_fun, },\n  guard_result_pp \"\u21d1e ?m_1\",\n  exact a,\nend\n\n-- Check that we can:\n-- * cope with metavariables in the result\n-- * perform beta redex after `revert`\n-- * simplify `eq.rec` after `subst`\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (S : has_mul \u03b1) : has_mul \u03b2 :=\nbegin\n  fconstructor,\n  simp_result\n  { have mul := S.mul,\n    have e' := equiv.arrow_congr e (equiv.arrow_congr e e),\n    have h : mul = e'.symm (e' mul) := by simp,\n    revert h,\n    generalize : e' mul = mul',\n    intro h,\n    subst h, },\n  exact mul',\n  guard_result_pp \"{mul := \u21d1(equiv.arrow_congr e (equiv.arrow_congr e e)) has_mul.mul}\",\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/simp_result.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.3927020294110705}}
{"text": "import algebra.homology.homological_complex\nimport category_theory.abelian.exact\nimport for_mathlib.homological_complex_shift\nimport for_mathlib.split_exact\nimport category_theory.triangulated.rotate\nimport algebra.homology.homotopy_category\nimport algebra.homology.additive\nimport for_mathlib.homological_complex_abelian\nimport for_mathlib.homotopy_category\nimport for_mathlib.triangle\n\nnoncomputable theory\n\nuniverses v u\n\nopen_locale classical\n\nopen category_theory category_theory.limits\n\nnamespace homological_complex\n\nvariables {V : Type u} [category.{v} V] [abelian V]\nvariables (A B C : cochain_complex V \u2124) (f : A \u27f6 B) (g : B \u27f6 C)\n\n@[simp, reassoc]\nlemma homotopy.comp_X_eq_to_iso {X Y : cochain_complex V \u2124} {f g : X \u27f6 Y} (h : homotopy f g)\n  (i : \u2124) {j k : \u2124} (e : j = k) : h.hom i j \u226b (Y.X_eq_to_iso e).hom = h.hom i k :=\nby { subst e, simp }\n\n@[simp, reassoc]\nlemma homotopy.X_eq_to_iso_comp {X Y : cochain_complex V \u2124} {f g : X \u27f6 Y} (h : homotopy f g)\n  {i j : \u2124} (e : i = j) (k : \u2124) : (X.X_eq_to_iso e).hom \u226b h.hom j k = h.hom i k :=\nby { subst e, simp }\n\n@[simp]\nlemma X_eq_to_iso_shift (n i j : \u2124) (h : i = j) :\n  X_eq_to_iso (A\u27e6n\u27e7) h = A.X_eq_to_iso (congr_arg _ h) := rfl\n\n@[simp, reassoc]\nlemma retraction_X_eq_to_hom (h : \u2200 i, splitting (f.f i) (g.f i)) {i j : \u2124} (e : i = j) :\n  (h i).retraction \u226b (A.X_eq_to_iso e).hom = (B.X_eq_to_iso e).hom \u226b (h j).retraction :=\nby { subst e, simp,}\n\n@[simp, reassoc]\nlemma section_X_eq_to_hom (h : \u2200 i, splitting (f.f i) (g.f i)) {i j : \u2124} (e : i = j) :\n  (h i).section \u226b (B.X_eq_to_iso e).hom = (C.X_eq_to_iso e).hom \u226b (h j).section :=\nby { subst e, simp }\n\ndef cone.X : \u2124 \u2192 V := \u03bb i, A.X (i + 1) \u229e B.X i\n\nvariables {A B C}\n\ndef cone.d : \u03a0 (i j : \u2124), cone.X A B i \u27f6 cone.X A B j :=\n\u03bb i j, if hij : i + 1 = j then biprod.lift\n  (biprod.desc (-A.d _ _)                         0        )\n  (biprod.desc (f.f _ \u226b (B.X_eq_to_iso hij).hom) (B.d _ _))\nelse 0\n\n/-- The mapping cone of a morphism `f : A \u2192 B` of homological complexes. -/\ndef cone : cochain_complex V \u2124 :=\n{ X := cone.X A B,\n  d := cone.d f,\n  shape' := \u03bb i j hij, dif_neg hij,\n  d_comp_d' := \u03bb i j k (hij : _ = _) (hjk : _ = _),\n  begin\n    substs hij hjk,\n    apply biprod.hom_ext; ext; simp [cone.d],\n  end }\n\n@[simp]\nlemma cone_X (i : \u2124) : (cone f).X i = (A.X (i + 1) \u229e B.X i) := rfl\n\n@[simp]\nlemma cone_d : (cone f).d = cone.d f := rfl\n\ndef cone.in : B \u27f6 cone f :=\n{ f := \u03bb i, biprod.inr,\n  comm' := \u03bb i j hij,\n  begin\n    dsimp [cone_d, cone.d], dsimp at hij, rw [dif_pos hij],\n    ext;\n    simp only [comp_zero, category.assoc, category.comp_id,\n      biprod.inr_desc, biprod.inr_fst, biprod.lift_fst, biprod.inr_snd, biprod.lift_snd],\n  end }\n\nlocal attribute [instance] endofunctor_monoidal_category discrete.add_monoidal\n\ndef cone.out : cone f \u27f6 A\u27e6(1 : \u2124)\u27e7 :=\n{ f := \u03bb i, biprod.fst,\n  comm' := \u03bb i j (hij : _ = _),\n  begin\n    subst hij,\n    dsimp [cone_d, cone.d],\n    ext; simp,\n  end }\n\n@[simps]\ndef cone.triangle : triangulated.triangle (cochain_complex V \u2124) :=\n{ obj\u2081 := A,\n  obj\u2082 := B,\n  obj\u2083 := cone f,\n  mor\u2081 := f,\n  mor\u2082 := cone.in f,\n  mor\u2083 := cone.out f }\n\nvariable (V)\n\n@[simps]\ndef _root_.homotopy_category.lift_triangle :\n  triangulated.triangle (cochain_complex V \u2124) \u2964\n    triangulated.triangle (homotopy_category V (complex_shape.up \u2124)) :=\n{ obj := \u03bb t, triangulated.triangle.mk _\n    ((homotopy_category.quotient _ _).map t.mor\u2081)\n    ((homotopy_category.quotient _ _).map t.mor\u2082)\n    ((homotopy_category.quotient _ _).map t.mor\u2083),\n  map := \u03bb t t' f,\n  { hom\u2081 := (homotopy_category.quotient _ _).map f.hom\u2081,\n    hom\u2082 := (homotopy_category.quotient _ _).map f.hom\u2082,\n    hom\u2083 := (homotopy_category.quotient _ _).map f.hom\u2083,\n    comm\u2081' := by { dsimp, rw [\u2190 functor.map_comp, \u2190 functor.map_comp, f.comm\u2081] },\n    comm\u2082' := by { dsimp, rw [\u2190 functor.map_comp, \u2190 functor.map_comp, f.comm\u2082] },\n    comm\u2083' := by { dsimp, rw [\u2190 functor.map_comp, \u2190 functor.map_comp, f.comm\u2083] } },\n  map_id' := \u03bb X, by { ext; exact category_theory.functor.map_id _ _  },\n  map_comp' := \u03bb X Y Z f g, by { ext; exact category_theory.functor.map_comp _ _ _ } }\n\nvariable {V}\n\n@[simps]\ndef cone.triangle\u2095 : triangulated.triangle (homotopy_category V (complex_shape.up \u2124)) :=\n(homotopy_category.lift_triangle _).obj (cone.triangle f)\n\nsection cone_functorial\n\nvariables {f} {A' B' : cochain_complex V \u2124} {f' : A' \u27f6 B'} {i\u2081 : A \u27f6 A'} {i\u2082 : B \u27f6 B'}\nvariables (comm : homotopy (f \u226b i\u2082) (i\u2081 \u226b f'))\n\ninclude comm\n\ndef cone.map : cone f \u27f6 cone f' :=\n{ f := \u03bb i, biprod.lift\n  (biprod.desc (i\u2081.f _) 0)\n  (biprod.desc (comm.hom _ _) (i\u2082.f _)),\n  comm' := \u03bb i j r,\n  begin\n    change i+1 = j at r,\n    dsimp [cone_d, cone.d],\n    simp_rw dif_pos r,\n    apply category_theory.limits.biprod.hom_ext;\n      simp only [biprod.lift_desc, add_zero, preadditive.comp_neg, category.assoc,\n        comp_zero, biprod.lift_fst, biprod.lift_snd]; ext,\n    { simp only [preadditive.comp_neg, biprod.inl_desc_assoc, hom.comm, preadditive.neg_comp] },\n    { simp only [preadditive.comp_neg, biprod.inr_desc_assoc, zero_comp, neg_zero] },\n    { simp only [X_eq_to_iso_f, preadditive.comp_add, biprod.inl_desc_assoc, category.assoc,\n        preadditive.neg_comp],\n      have := comm.comm (i+1),\n      dsimp at this,\n      rw [reassoc_of this],\n      subst r,\n      delta d_from from_next to_prev d_to,\n      simp only [X_eq_to_iso_refl, category.comp_id, add_monoid_hom.mk'_apply],\n      have aux\u2081 : (complex_shape.up \u2124).next (i + 1) = i + 1 + 1,\n      { simp only [cochain_complex.next] },\n      have aux\u2082 : i = (complex_shape.up \u2124).prev (i + 1),\n      { simp only [cochain_complex.prev, add_tsub_cancel_right], },\n      rw \u2190 aux\u2081,\n      simp only [\u2190add_assoc, add_left_neg, zero_add, add_comm (i\u2081.f (i + 1) \u226b f'.f (i + 1)), add_left_inj],\n      congr' 2, },\n    { simp only [preadditive.comp_add, biprod.inr_desc_assoc, zero_comp, hom.comm] }\n  end }\n\n@[simp, reassoc]\nlemma cone.in_map : cone.in f \u226b cone.map comm = i\u2082 \u226b cone.in f' :=\nby ext; { dsimp [cone.map, cone.in], simp }\n\n@[simp, reassoc]\nlemma cone.map_out : cone.map comm \u226b cone.out f' = cone.out f \u226b i\u2081\u27e6(1 : \u2124)\u27e7' :=\nby ext; { dsimp [cone.map, cone.out], simp }\n\nomit comm\n\n-- I suppose this is not true?\n-- def cone.map_homotopy_of_homotopy' (comm' : homotopy (f \u226b i\u2082) (i\u2081 \u226b f')) :\n--   homotopy (cone.map comm) (cone.map comm') := by admit\n\n@[simps]\ndef cone.triangle\u2095_map : cone.triangle\u2095 f \u27f6 cone.triangle\u2095 f' :=\n{ hom\u2081 := (homotopy_category.quotient _ _).map i\u2081,\n  hom\u2082 := (homotopy_category.quotient _ _).map i\u2082,\n  hom\u2083 := (homotopy_category.quotient _ _).map $ cone.map comm,\n  comm\u2081' := by { dsimp [cone.triangle\u2095], simp_rw \u2190 functor.map_comp,\n    exact homotopy_category.eq_of_homotopy _ _ comm },\n  comm\u2082' := by { dsimp [cone.triangle\u2095], simp_rw \u2190 functor.map_comp, simp },\n  comm\u2083' := by { dsimp [cone.triangle\u2095], simp_rw \u2190 functor.map_comp, simp } }\n\n@[simps]\ndef cone.triangle_map (h : f \u226b i\u2082 = i\u2081 \u226b f') : cone.triangle f \u27f6 cone.triangle f' :=\n{ hom\u2081 := i\u2081,\n  hom\u2082 := i\u2082,\n  hom\u2083 := cone.map (homotopy.of_eq h),\n  comm\u2081' := by simpa [cone.triangle],\n  comm\u2082' := by { dsimp [cone.triangle], simp },\n  comm\u2083' := by { dsimp [cone.triangle], simp } }\n\n@[simp]\nlemma cone.map_id (f : A \u27f6 B) :\n  cone.map (homotopy.of_eq $ (category.comp_id f).trans (category.id_comp f).symm) = \ud835\udfd9 _ :=\nby { ext; dsimp [cone.map, cone, cone.X]; simp }\n\n@[simp]\nlemma cone.triangle_map_id (f : A \u27f6 B) :\n  cone.triangle_map ((category.comp_id f).trans (category.id_comp f).symm) = \ud835\udfd9 _ :=\nby { ext; dsimp [cone.map, cone, cone.X]; simp }\n\n\ndef cone.triangle_functorial :\n  arrow (cochain_complex V \u2124) \u2964 triangulated.triangle (cochain_complex V \u2124) :=\n{ obj := \u03bb f, cone.triangle f.hom,\n  map := \u03bb f g c, cone.triangle_map c.w.symm,\n  map_id' := \u03bb X, cone.triangle_map_id _,\n  map_comp' := \u03bb X Y Z f g, by { ext; dsimp [cone.map, cone, cone.X]; simp } }\n\n-- I suppose this is also not true?\n-- def cone.triangle\u2095_functorial :\n--   arrow (homotopy_category V (complex_shape.up \u2124)) \u2964\n--     triangulated.triangle (homotopy_category V (complex_shape.up \u2124)) :=\n-- { obj := \u03bb f, cone.triangle\u2095 f.hom.out,\n--   map := \u03bb f g c, @cone.triangle\u2095_map _ _ _ _ _ _ _ _ _ c.left.out c.right.out\n--   begin\n--     refine homotopy_category.homotopy_of_eq _ _ _,\n--     simpa [-arrow.w] using c.w.symm\n--   end,\n--   map_id' := by admit,\n--   map_comp' := by admit }\n\nopen_locale zero_object\n\nlemma homological_complex.zero_X {\u03b9 : Type*} {c : complex_shape \u03b9}\n  {C : homological_complex V c} (hC : is_zero C) (i : \u03b9) :\n  is_zero (C.X i) :=\nbegin\n  rw [is_zero_iff_id_eq_zero, \u2190 homological_complex.eval_obj,\n    \u2190 (homological_complex.eval V c i).map_id, hC.eq_of_src (\ud835\udfd9 C) 0],\n  refl\nend\n\ndef cone_from_zero (A : cochain_complex V \u2124) : cone (0 : 0 \u27f6 A) \u2245 A :=\n{ hom :=\n  { f := \u03bb i, biprod.snd, comm' := begin\n      introv r, ext; dsimp [cone.d] at *;\n      simp only [if_pos r, zero_comp, dite_eq_ite, biprod.lift_snd,\n        biprod.inl_desc, biprod.inr_desc, biprod.inl_snd_assoc, biprod.inr_snd_assoc],\n    end },\n  inv := cone.in _,\n  inv_hom_id' := by { intros, ext, dsimp [cone.in], simp only [biprod.inr_snd] },\n  hom_inv_id' := begin\n    ext; dsimp [cone.in];\n    simp only [biprod.inl_snd_assoc, zero_comp, comp_f, id_f, category.assoc,\n      category.comp_id, biprod.inl_fst, biprod.inl_snd, biprod.inr_fst, biprod.inr_snd,\n      biprod.inr_snd_assoc],\n    apply is_zero.eq_of_src,\n    apply homological_complex.zero_X,\n    apply limits.is_zero_zero,\n  end }\n\ndef cone_to_zero (A : cochain_complex V \u2124) : cone (0 : A \u27f6 0) \u2245 A\u27e6(1 : \u2124)\u27e7 :=\n{ hom := cone.out _,\n  inv :=\n    { f := \u03bb i, biprod.inl, comm' := begin\n        introv r, ext; dsimp [cone.d] at *;\n        simp only [if_pos r, zero_comp, comp_zero, dite_eq_ite, category.assoc, category.comp_id,\n          biprod.lift_fst, biprod.lift_snd, biprod.inl_desc, neg_smul, one_zsmul,\n          biprod.inl_fst, biprod.inl_snd],\n    end },\n  hom_inv_id' := begin\n    ext; dsimp [cone.out];\n    simp only [biprod.inl_snd_assoc, zero_comp, comp_f, id_f, category.assoc,\n      category.comp_id, biprod.inl_fst, biprod.inl_snd, biprod.inr_fst, biprod.inr_snd,\n      biprod.inl_fst_assoc, biprod.inr_fst_assoc],\n    apply is_zero.eq_of_src,\n    apply homological_complex.zero_X,\n    apply limits.is_zero_zero,\n  end,\n  inv_hom_id' := by { intros, ext, dsimp [cone.out], simp } }\n\ndef cone.desc_of_null_homotopic (h : homotopy (f \u226b g) 0) : cone f \u27f6 C :=\ncone.map (h.trans (homotopy.of_eq (comp_zero.symm : 0 = 0 \u226b 0))) \u226b (cone_from_zero _).hom\n\ndef cone.lift_of_null_homotopic (h : homotopy (f \u226b g) 0) : A \u27f6 cone g\u27e6(-1 : \u2124)\u27e7 :=\n(shift_shift_neg A (1 : \u2124)).inv \u226b (shift_functor _ (-1 : \u2124)).map ((cone_to_zero _).inv \u226b\n  cone.map (h.trans (homotopy.of_eq (comp_zero.symm : 0 = 0 \u226b 0))).symm)\n\n@[simps]\ndef of_termwise_split_mono (H : \u2200 i, split_mono (f.f i)) : B \u27f6 B' :=\n{ f := \u03bb i, i\u2082.f i - (H i).retraction \u226b comm.hom i (i-1) \u226b B'.d (i-1) i -\n    B.d i (i+1) \u226b (H (i+1)).retraction \u226b comm.hom (i+1) i,\n  comm' := \u03bb i j (r : i + 1 = j), by { subst r, simp only [d_comp_d, sub_zero, category.assoc,\n    comp_zero, preadditive.comp_sub, hom.comm, preadditive.sub_comp, zero_comp, sub_right_inj,\n    d_comp_d_assoc], congr; ring } }\n\n@[simp, reassoc]\nlemma of_termwise_split_mono_commutes (H : \u2200 i, split_mono (f.f i)) :\n  f \u226b of_termwise_split_mono comm H = i\u2081 \u226b f' :=\nbegin\n  ext i,\n  dsimp,\n  have : f.f i \u226b i\u2082.f i = A.d i (i + 1) \u226b comm.hom (i + 1) i + comm.hom i (i - 1) \u226b\n    B'.d (i - 1) i + i\u2081.f i \u226b f'.f i := begin\n      rw [\u2190 homological_complex.comp_f, comm.comm i],\n      simp only [d_next, prev_d, add_monoid_hom.mk'_apply, comp_f, add_left_inj],\n      rw [\u2190 cochain_complex.next, \u2190 cochain_complex.prev],\n    end,\n  simp only [hom.comm_assoc, preadditive.comp_sub, this],\n  erw [split_mono.id_assoc, split_mono.id_assoc],\n  simp [add_right_comm]\nend\n\ndef of_termwise_split_mono_homotopy (H : \u2200 i, split_mono (f.f i)) :\n  homotopy i\u2082 (of_termwise_split_mono comm H)  :=\n{ hom := \u03bb i j, (H i).retraction \u226b comm.hom i j,\n  zero' := \u03bb _ _ r, by rw [comm.zero _ _ r, comp_zero],\n  comm := \u03bb i, begin\n    simp only [d_next, prev_d, add_monoid_hom.mk'_apply, category.assoc, of_termwise_split_mono_f],\n    rw [\u2190 cochain_complex.next, \u2190 cochain_complex.prev], abel\n  end }\n\n@[simps]\ndef of_termwise_split_epi (H : \u2200 i, split_epi (f'.f i)) : A \u27f6 A' :=\n{ f := \u03bb i, i\u2081.f i + comm.hom i (i-1) \u226b (H (i-1)).section_ \u226b A'.d (i-1) i +\n    A.d i (i+1) \u226b comm.hom (i+1) i \u226b (H i).section_,\n  comm' := \u03bb i j (r : i + 1 = j), by { subst r, simp only [add_zero, d_comp_d, preadditive.comp_add,\n    category.assoc, comp_zero, add_right_inj, hom.comm, zero_comp, preadditive.add_comp,\n    d_comp_d_assoc], congr; ring } }\n\n@[simp, reassoc]\nlemma of_termwise_split_epi_commutes (H : \u2200 i, split_epi (f'.f i)) :\n  of_termwise_split_epi comm H \u226b f' = f \u226b i\u2082 :=\nbegin\n  ext i,\n  dsimp,\n  have : f.f i \u226b i\u2082.f i = A.d i (i + 1) \u226b comm.hom (i + 1) i + comm.hom i (i - 1) \u226b\n    B'.d (i - 1) i + i\u2081.f i \u226b f'.f i := begin\n      rw [\u2190 homological_complex.comp_f, comm.comm i],\n      simp only [d_next, prev_d, add_monoid_hom.mk'_apply, comp_f, add_left_inj],\n      rw [\u2190 cochain_complex.next, \u2190 cochain_complex.prev],\n    end,\n  simp only [this, category.assoc, preadditive.add_comp, \u2190 f'.comm],\n  erw [split_epi.id, split_epi.id_assoc],\n  rw [add_comm, add_comm (i\u2081.f i \u226b f'.f i), \u2190 add_assoc, category.comp_id]\nend\n\ndef of_termwise_split_epi_homotopy (H : \u2200 i, split_epi (f'.f i)) :\n  homotopy (of_termwise_split_epi comm H) i\u2081 :=\n{ hom := \u03bb i j, comm.hom i j \u226b (H j).section_,\n  zero' := \u03bb _ _ r, by rw [comm.zero _ _ r, zero_comp],\n  comm := \u03bb i, begin\n    simp only [d_next, prev_d, add_monoid_hom.mk'_apply, category.assoc, of_termwise_split_epi_f],\n    rw [\u2190 cochain_complex.next, \u2190 cochain_complex.prev], abel\n  end }\n\nend cone_functorial\n\nsection termwise_split_mono_lift\n\n@[simps]\ndef termwise_split_mono_lift (f : A \u27f6 B) : A \u27f6 biproduct B (cone (\ud835\udfd9 A)) :=\nbiproduct.lift f (cone.in _)\n\n@[simps]\ndef termwise_split_mono_desc (f : A \u27f6 B) : biproduct B (cone (\ud835\udfd9 A)) \u27f6 B :=\nbiproduct.fst\n\n@[simps]\ndef termwise_split_mono_section (f : A \u27f6 B) : B \u27f6 biproduct B (cone (\ud835\udfd9 A)) :=\nbiproduct.inl\n\n@[simp, reassoc] lemma termwise_split_mono_section_desc (f : A \u27f6 B) :\n  termwise_split_mono_section f \u226b termwise_split_mono_desc f = \ud835\udfd9 _ :=\nby { ext, simp }\n\nlemma termwise_split_mono_desc_section_aux (i : \u2124) :\n  \ud835\udfd9 (B.X i \u229e (A.X (i + 1) \u229e A.X i)) = biprod.snd \u226b biprod.desc (\ud835\udfd9 (A.X (i + 1))) (A.d i (i + 1)) \u226b\n    biprod.inl \u226b biprod.inr + biprod.snd \u226b biprod.snd \u226b\n    (X_eq_to_iso A (sub_add_cancel i 1).symm).hom \u226b biprod.inl \u226b biprod.lift\n    (biprod.desc (-A.d (i - 1 + 1) (i + 1)) 0) (biprod.desc (X_eq_to_iso A (sub_add_cancel i 1)).hom\n    (A.d (i - 1) i)) \u226b biprod.inr + biprod.fst \u226b biprod.inl :=\nbegin\n  ext1; simp only [zero_comp, preadditive.comp_add, zero_add, add_zero, biprod.inr_fst_assoc,\n    biprod.inl_fst_assoc, biprod.inl_snd_assoc, biprod.inr_snd_assoc, category.comp_id],\n  ext1, { simp },\n  ext1, { simp only [add_zero, preadditive.add_comp, comp_zero, biprod.inr_fst, category.assoc] },\n  ext1; simp,\nend\n\ndef termwise_split_mono_desc_section (f : A \u27f6 B) :\n  homotopy (\ud835\udfd9 _) (termwise_split_mono_desc f \u226b termwise_split_mono_section f) :=\n{ hom := \u03bb i j, if h : i = j + 1 then\n    biprod.snd \u226b biprod.snd \u226b (A.X_eq_to_iso h).hom \u226b biprod.inl \u226b biprod.inr else 0,\n  zero' := \u03bb i j r, dif_neg (ne.symm r),\n  comm := \u03bb i, begin\n    dsimp,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n    have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n    rw termwise_split_mono_desc_section_aux (i+1),\n    rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n      d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n    simp only [sub_add_cancel, biproduct_d, cone_d, X_eq_to_iso_refl, category.id_comp,\n      biprod.map_snd_assoc, category.assoc, biprod.inr_map, add_left_inj, cone.d],\n    rw [dif_pos rfl, dif_pos rfl],\n    simp only [id_f, category.comp_id, biprod.lift_snd_assoc, add_right_inj],\n    ext;\n    simp only [biprod.inl_snd_assoc, biprod.inr_snd_assoc, biprod.inr_fst,\n      biprod.lift_fst, biprod.inl_desc, biprod.lift_snd,\n      category.assoc, zero_comp, comp_zero, preadditive.comp_neg,\n      X_eq_to_iso_d, X_eq_to_iso_trans, X_eq_to_iso_refl],\n  end }\n\ndef termwise_split_mono_lift_split_mono (f : A \u27f6 B) (i : \u2124) :\n  split_mono ((termwise_split_mono_lift f).f i) :=\n{ retraction := biprod.snd \u226b biprod.snd, id' := by simp [cone.in] }\n\ninstance (f : A \u27f6 B) (i : \u2124) : mono ((termwise_split_mono_lift f).f i) :=\n(termwise_split_mono_lift_split_mono f i).mono\n\n-- generalize to epi\n@[simp]\nlemma termwise_split_mono_lift_desc (f : A \u27f6 B) :\n  termwise_split_mono_lift f \u226b termwise_split_mono_desc f = f :=\nby { ext, simp }\n\n/-- We will prove this is iso later -/\ndef from_termwise_split_mono_lift_triangle\u2095 :\n  cone.triangle\u2095 (termwise_split_mono_lift f) \u27f6 cone.triangle\u2095 f :=\ncone.triangle\u2095_map\n  (homotopy.of_eq ((termwise_split_mono_lift_desc f).trans (category.id_comp f).symm))\n\n@[simps]\ndef termwise_split_mono_factor_homotopy_equiv : homotopy_equiv (biproduct B (cone (\ud835\udfd9 A))) B :=\n{ hom := termwise_split_mono_desc f,\n  inv := termwise_split_mono_section f,\n  homotopy_hom_inv_id := (termwise_split_mono_desc_section f).symm,\n  homotopy_inv_hom_id := homotopy.of_eq (termwise_split_mono_section_desc f) }\n\nend termwise_split_mono_lift\n\nsection termwise_split_epi_lift\n\n@[simps]\ndef termwise_split_epi_lift (f : A \u27f6 B) : A \u27f6 biproduct A (cone (\ud835\udfd9 (B\u27e6(-1 : \u2124)\u27e7))) :=\nbiproduct.inl\n\n@[simps]\ndef termwise_split_epi_desc (f : A \u27f6 B) : biproduct A (cone (\ud835\udfd9 (B\u27e6(-1 : \u2124)\u27e7))) \u27f6 B :=\nbiproduct.desc f (cone.out _ \u226b (shift_neg_shift _ _).hom)\n\n@[simps]\ndef termwise_split_epi_retraction (f : A \u27f6 B) : biproduct A (cone (\ud835\udfd9 (B\u27e6(-1 : \u2124)\u27e7))) \u27f6 A :=\nbiproduct.fst\n\n@[simp, reassoc] lemma termwise_split_epi_lift_retraction (f : A \u27f6 B) :\n  termwise_split_epi_lift f \u226b termwise_split_epi_retraction f = \ud835\udfd9 _ :=\nby { ext, simp }\n\nlemma termwise_split_epi_retraction_lift_aux (i : \u2124) :\n  \ud835\udfd9 (A.X i \u229e (B.X (i + 1 - 1) \u229e B.X (i - 1))) = biprod.snd \u226b biprod.desc (\ud835\udfd9 _)\n  (-B.d (i + -1) (i + 1 + -1)) \u226b \ud835\udfd9 _ \u226b biprod.inl \u226b biprod.inr + biprod.snd \u226b biprod.snd \u226b\n  ((B\u27e6(-1 : \u2124)\u27e7).X_eq_to_iso (sub_add_cancel _ _).symm).hom \u226b biprod.inl \u226b biprod.lift\n  (biprod.desc (B.d (i - 1 + 1 + -1) (i + 1 + -1)) 0) (biprod.desc\n  ((B\u27e6(-1 : \u2124)\u27e7).X_eq_to_iso $ sub_add_cancel _ _).hom (-B.d (i - 1 + -1) (i + -1))) \u226b\n  biprod.inr + biprod.fst \u226b biprod.inl :=\nbegin\n  ext1; simp only [category.comp_id, add_zero, category.id_comp, preadditive.comp_add,\n    biprod.inl_snd_assoc, zero_add, zero_comp, biprod.inl_fst_assoc, biprod.inr_fst_assoc,\n    biprod.inr_snd_assoc],\n  ext1, { simp },\n  simp only [biprod.inr_desc_assoc, preadditive.neg_comp_assoc, X_eq_to_iso_shift,\n    biprod.inr_snd_assoc, preadditive.comp_add, category.assoc, preadditive.neg_comp],\n  ext1, { simp only [add_zero, preadditive.add_comp, comp_zero,\n    preadditive.neg_comp, biprod.inr_fst, neg_zero, category.assoc] },\n  ext; simp; refl\nend\n\ndef termwise_split_epi_retraction_lift (f : A \u27f6 B) :\n  homotopy (\ud835\udfd9 _) (termwise_split_epi_retraction f \u226b termwise_split_epi_lift f) :=\n{ hom := \u03bb i j, if h : i = j + 1 then\n    biprod.snd \u226b biprod.snd \u226b ((B\u27e6(-1 : \u2124)\u27e7).X_eq_to_iso h).hom \u226b biprod.inl \u226b biprod.inr else 0,\n  zero' := \u03bb i j r, dif_neg (ne.symm r),\n  comm := \u03bb i, begin\n    dsimp,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n    have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n    rw termwise_split_epi_retraction_lift_aux (i+1),\n    rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n      d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n    simp only [sub_add_cancel, biproduct_d, cone_d, X_eq_to_iso_refl, category.id_comp,\n      biprod.map_snd_assoc, category.assoc, biprod.inr_map, add_left_inj, cone.d],\n    rw [dif_pos rfl, dif_pos rfl],\n    simp only [id_f, category.comp_id, biprod.lift_snd_assoc, add_right_inj],\n    ext;\n    simp only [preadditive.add_comp, category.assoc, biprod.inr_fst,\n      preadditive.comp_add, biprod.inl_snd_assoc, biprod.inl_snd,\n      biprod.inr_snd_assoc, biprod.inl_fst, category.comp_id, biprod.inl_desc,\n      zero_comp, comp_zero, add_zero,\n      biprod.inr_desc, X_eq_to_iso_shift, biprod.lift_fst,\n      X_eq_to_iso_d, shift_d, int.neg_one_pow_neg_one, neg_smul, one_zsmul, neg_neg,\n      biprod.lift_snd, X_eq_to_iso_trans, X_eq_to_iso_refl, zero_add],\n    all_goals { refl }\n  end }\n\ndef termwise_split_epi_desc_split_epi (f : A \u27f6 B) (i : \u2124) :\n  split_epi ((termwise_split_epi_desc f).f i) :=\n{ section_ := (B.X_eq_to_iso $ eq_add_neg_of_add_eq rfl).hom \u226b biprod.inl \u226b biprod.inr,\n  id' := by { dsimp, simp [cone.out] } }\n\ninstance (f : A \u27f6 B) (i : \u2124) : epi ((termwise_split_epi_desc f).f i) :=\n(termwise_split_epi_desc_split_epi f i).epi\n\nend termwise_split_epi_lift\n\nsection termwise_split_exact\n\nvariables (f g)\n\n@[simps]\ndef connecting_hom (h : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) : C \u27f6 A\u27e6(1 : \u2124)\u27e7 :=\n{ f := \u03bb i, (h i).section \u226b B.d i (i + 1) \u226b (h (i + 1)).retraction,\n  comm' :=\n  begin\n    intros i j r,\n    induction r,\n    dsimp,\n    rw \u2190 cancel_mono (\ud835\udfd9 _),\n    swap, apply_instance,\n    conv_lhs { rw \u2190 (h _).\u03b9_retraction },\n    simp only [preadditive.comp_neg, one_zsmul, category.assoc, neg_smul, preadditive.neg_comp,\n      \u2190 f.comm_assoc, (h _).retraction_\u03b9_eq_id_sub_assoc, preadditive.sub_comp_assoc,\n      preadditive.sub_comp, preadditive.comp_sub, category.id_comp, d_comp_d_assoc,\n      zero_comp, comp_zero, \u2190 g.comm_assoc, (h i).section_\u03c0_assoc],\n    simp,\n  end }\n\n@[simps]\ndef triangle_of_termwise_split (h : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) :\n  triangulated.triangle (cochain_complex V \u2124) :=\ntriangulated.triangle.mk _ f g (connecting_hom f g h)\n\n@[simps]\ndef triangle\u2095_of_termwise_split (h : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) :\n  triangulated.triangle (homotopy_category V (complex_shape.up \u2124)) :=\n(homotopy_category.lift_triangle V).obj (triangle_of_termwise_split f g h)\n\n@[simps]\ndef homotopy_connecting_hom_of_splittings (h h' : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) :\n  homotopy (connecting_hom f g h) (connecting_hom f g h') :=\n{ hom := \u03bb i j, if e : j + 1 = i then\n    ((h' i).section \u226b (h i).retraction \u226b (A.X_eq_to_iso e).inv) else 0,\n  comm := \u03bb i, begin\n    rw \u2190 cancel_epi (g.f _),\n    { dsimp,\n      obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n      have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n      have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n      rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n        d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n      simp only [d_next, prev_d, splitting.\u03c0_section_eq_id_sub_assoc, -retraction_X_eq_to_hom,\n        preadditive.sub_comp_assoc, category.id_comp, category.assoc, hom.comm,\n        preadditive.sub_comp, splitting.\u03b9_retraction, category.comp_id, X_eq_to_iso_refl, shift_d,\n        int.neg_one_pow_one, neg_smul, one_zsmul, preadditive.comp_neg, preadditive.comp_add,\n        hom.comm_assoc, preadditive.comp_sub, neg_sub, splitting.\u03b9_retraction_assoc],\n      abel, apply add_comm, },\n    exact (h i).epi\n  end,\n  zero' := \u03bb _ _ h, dif_neg h }\n\n@[simps]\ndef triangle\u2095_map_splittings_hom (h h' : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) :\n  triangle\u2095_of_termwise_split f g h \u27f6 triangle\u2095_of_termwise_split f g h' :=\n{ hom\u2081 := \ud835\udfd9 _,\n  hom\u2082 := \ud835\udfd9 _,\n  hom\u2083 := \ud835\udfd9 _,\n  comm\u2083' :=\n  begin\n    simp only [category.comp_id, triangle\u2095_of_termwise_split_mor\u2083, category.id_comp,\n      category_theory.functor.map_id],\n    apply homotopy_category.eq_of_homotopy,\n    exact homotopy_connecting_hom_of_splittings f g h h'\n  end }\n\n@[simps]\ndef triangle\u2095_map_splittings_iso (h h' : \u2200 (i : \u2124), splitting (f.f i) (g.f i)) :\n  triangle\u2095_of_termwise_split f g h \u2245 triangle\u2095_of_termwise_split f g h' :=\n{ hom := triangle\u2095_map_splittings_hom f g h h',\n  inv := triangle\u2095_map_splittings_hom f g h' h,\n  hom_inv_id' := by { ext; exact category.comp_id _ },\n  inv_hom_id' := by { ext; exact category.comp_id _ } }\n\nend termwise_split_exact\nsection\n\nvariables {B'' B' : cochain_complex V \u2124} {b' : B'' \u27f6 B} {b : B \u27f6 B'}\nvariables (H\u2082 : \u2200 i, splitting (f.f i) (g.f i))\nvariables (h\u2082 : homotopy (b' \u226b g) 0) (h\u2083 : homotopy (f \u226b b) 0)\n\ninclude H\u2082 h\u2082 h\u2083\n\n/--\nIf `A \u27f6 B \u27f6 C` is split exact, and `b' \u226b g` and `f \u226b b` are null-homotopic,\nthen so is `b' \u226b b`.\n\n        B''\n        \u2223\n        b'\n        \u2193\nA - f \u2192 B - g \u2192 C\n        \u2223\n        b\n        \u2193\n        B'\n-/\ndef comp_null_homotopic_of_row_split_exact : homotopy (b' \u226b b) 0 :=\nbegin\n  have H1 := \u03bb i, (H\u2082 i).split_epi,\n  have H2 := \u03bb i, (H\u2082 i).split_mono,\n  have aux := \u03bb i, (H\u2082 i).short_exact.3,\n  let h\u2081' := (h\u2082.trans (homotopy.of_eq (comp_zero : \ud835\udfd9 _ \u226b 0 = 0).symm)).symm,\n  let h\u2082' := (h\u2083.trans $ homotopy.of_eq (zero_comp : 0 \u226b \ud835\udfd9 _ = 0).symm),\n  refine ((of_termwise_split_epi_homotopy h\u2081' H1).symm.comp\n    (of_termwise_split_mono_homotopy h\u2082' H2)).trans (homotopy.of_eq _),\n  apply hom.ext,\n  apply funext,\n  intro i,\n  exact comp_eq_zero_of_exact (f.f i) (g.f i) (aux i)\n    (congr_f ((of_termwise_split_epi_commutes h\u2081' H1).trans comp_zero) i)\n    (congr_f ((of_termwise_split_mono_commutes h\u2082' H2).trans zero_comp) i)\nend\n\nend\n\ndef cone.termwise_split (i : \u2124) : splitting ((cone.in f).f i) ((cone.out f).f i) :=\n{ iso := biprod.braiding _ _,\n  comp_iso_eq_inl := by ext; simp [cone.in],\n  iso_comp_snd_eq := by ext; simp [cone.out] }\n\n@[simp] lemma cone.termwise_split_section (i : \u2124) :\n  (cone.termwise_split f i).section = biprod.inl :=\nby { delta splitting.section cone.termwise_split, ext; dsimp; simp }\n\n@[simp] lemma cone.termwise_split_retraction (i : \u2124) :\n  (cone.termwise_split f i).retraction = biprod.snd :=\nby { delta splitting.retraction cone.termwise_split, dsimp, simp }\n\ndef cone_homotopy_equiv_aux (c : cone f \u27f6 cone f) (h\u2081 : homotopy (cone.in f \u226b c) (cone.in f))\n  (h\u2082 : homotopy (c \u226b cone.out f) (cone.out f)) : homotopy (\ud835\udfd9 _) (2 \u2022 c - c \u226b c) :=\nbegin\n  have : homotopy ((\ud835\udfd9 _ - c) \u226b (\ud835\udfd9 _ - c)) 0,\n  { apply comp_null_homotopic_of_row_split_exact (cone.in f) (cone.out f) (cone.termwise_split f),\n    { refine (homotopy.of_eq _).trans h\u2082.symm.equiv_sub_zero, simp },\n    { refine (homotopy.of_eq _).trans h\u2081.symm.equiv_sub_zero, simp } },\n  apply homotopy.equiv_sub_zero.symm _,\n  refine (homotopy.of_eq _).trans this,\n  simp [two_smul], abel,\nend\n\nlocal attribute [simp] preadditive.comp_nsmul preadditive.nsmul_comp\n\n/--\nIf the following diagram commutes up to homotopy, then `c` is a homotopy equivalence\nA - f \u2192 B \u27f6 C(f) \u27f6 A\u27e61\u27e7\n|       |      \u2223       \u2223\n\ud835\udfd9       \ud835\udfd9      c       \ud835\udfd9\n\u2193       \u2193      \u2223       \u2223\nA - f \u2192 B \u27f6 C(f) \u27f6 A\u27e61\u27e7\n-/\ndef cone_homotopy_equiv (c : cone f \u27f6 cone f) (h\u2081 : homotopy (cone.in f \u226b c) (cone.in f))\n  (h\u2082 : homotopy (c \u226b cone.out f) (cone.out f)) : homotopy_equiv (cone f) (cone f) :=\n{ hom := c,\n  inv := ((2 \u2022 \ud835\udfd9 _) - c),\n  homotopy_hom_inv_id := (homotopy.of_eq (by simp)).trans (cone_homotopy_equiv_aux f c h\u2081 h\u2082).symm,\n  homotopy_inv_hom_id := (homotopy.of_eq (by simp)).trans (cone_homotopy_equiv_aux f c h\u2081 h\u2082).symm }\n\nlocal notation `Q` := homotopy_category.quotient V (complex_shape.up \u2124)\n\nlemma cone_triangle\u2095_map_iso_of_id (\u03c6 : cone.triangle\u2095 f \u27f6 cone.triangle\u2095 f)\n  (h\u2081 : \u03c6.hom\u2081 = \ud835\udfd9 _) (h\u2082 : \u03c6.hom\u2082 = \ud835\udfd9 _) : is_iso \u03c6.hom\u2083 :=\nbegin\n  have e\u2082 := \u03c6.comm\u2082,\n  have e\u2083 := \u03c6.comm\u2083,\n  rw [h\u2082, category.id_comp] at e\u2082,\n  rw [h\u2081, category_theory.functor.map_id, category.comp_id] at e\u2083,\n  erw [\u2190 Q .image_preimage \u03c6.hom\u2083, \u2190 Q .map_comp] at e\u2082 e\u2083,\n  convert is_iso.of_iso (homotopy_category.iso_of_homotopy_equiv\n    (cone_homotopy_equiv _ _ (homotopy_category.homotopy_of_eq _ _ e\u2082)\n    (homotopy_category.homotopy_of_eq _ _ e\u2083.symm))),\n  exact (Q .image_preimage _).symm\nend\n\nopen category_theory.triangulated\n\nlemma cone.triangle\u2095_is_iso {A' B' : cochain_complex V \u2124} {f : A \u27f6 B} {f' : A' \u27f6 B'}\n  (\u03c6 : cone.triangle\u2095 f \u27f6 cone.triangle\u2095 f') [is_iso \u03c6.hom\u2081] [is_iso \u03c6.hom\u2082] : is_iso \u03c6 :=\nbegin\n  suffices : is_iso \u03c6.hom\u2083,\n  { exactI triangle_morphism_is_iso _ },\n  have := \u03c6.comm\u2081,\n  dsimp at this,\n  rw [\u2190 is_iso.eq_comp_inv, category.assoc, \u2190 is_iso.inv_comp_eq,\n    \u2190 Q .image_preimage (inv \u03c6.hom\u2081), \u2190 Q .map_comp,\n    \u2190 Q .image_preimage (inv \u03c6.hom\u2082), \u2190 Q .map_comp] at this,\n  let T := cone.triangle\u2095_map (homotopy_category.homotopy_of_eq _ _ this).symm,\n  haveI := cone_triangle\u2095_map_iso_of_id _ (\u03c6 \u226b T) (by simp) (by simp),\n  haveI := cone_triangle\u2095_map_iso_of_id _ (T \u226b \u03c6) (by simp) (by simp),\n  haveI : epi \u03c6.hom\u2083 := @@epi_of_epi _ (T.hom\u2083) (\u03c6.hom\u2083) (show epi (T \u226b \u03c6).hom\u2083, by apply_instance),\n  use T.hom\u2083 \u226b inv (\u03c6 \u226b T).hom\u2083,\n  split,\n  { rw \u2190 category.assoc, exact is_iso.hom_inv_id _ },\n  { rw [\u2190 cancel_epi \u03c6.hom\u2083, \u2190 category.assoc, \u2190 category.assoc, category.comp_id,\n      category.assoc],\n    exact is_iso.hom_inv_id_assoc (\u03c6 \u226b T).hom\u2083 _ }\nend\n\ninstance : is_iso (from_termwise_split_mono_lift_triangle\u2095 f) :=\nbegin\n  haveI : is_iso (from_termwise_split_mono_lift_triangle\u2095 f).hom\u2081,\n  { delta from_termwise_split_mono_lift_triangle\u2095, dsimp, apply_instance },\n  haveI : is_iso (from_termwise_split_mono_lift_triangle\u2095 f).hom\u2082 :=\n    is_iso.of_iso (homotopy_category.iso_of_homotopy_equiv\n      (termwise_split_mono_factor_homotopy_equiv f)),\n  apply cone.triangle\u2095_is_iso,\nend\n\n-- move this\n@[simp]\nlemma cochain_complex_d_next (i : \u2124) (f : \u03a0 i j, A.X i \u27f6 B.X j) :\n  d_next i f = A.d i (i + 1) \u226b f (i + 1) i :=\nby { simp only [d_next, add_monoid_hom.mk'_apply], rw [\u2190 cochain_complex.next] }\n\n@[simp]\nlemma cochain_complex_prev_d (i : \u2124) (f : \u03a0 i j, A.X i \u27f6 B.X j) :\n  prev_d i f = f i (i - 1) \u226b B.d (i - 1) i :=\nby { simp only [prev_d, add_monoid_hom.mk'_apply], rw [\u2190 cochain_complex.prev] }\n\n@[simps]\ndef termwise_split_to_cone (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  C \u27f6 cone f :=\n{ f := \u03bb i, biprod.lift (-(connecting_hom f g h).f i) ((h i).section),\n  comm' := begin\n    rintro i j (rfl : i + 1 = j),\n    have He := \u03bb i, (h i).split_epi,\n    ext,\n    { dsimp [cone.d],\n      rw \u2190 cancel_epi (g.f _),\n      { simp [g.comm, splitting.\u03c0_section_eq_id_sub_assoc] },\n      { exact (He _).epi } },\n    { dsimp [cone.d],\n      rw \u2190 cancel_epi (g.f _),\n      { simp [splitting.\u03c0_section_eq_id_sub_assoc, splitting.\u03c0_section_eq_id_sub] },\n      { exact (He _).epi } },\n  end }\n\n@[simps]\ndef comp_termwise_split_to_cone_homotopy (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  homotopy (g \u226b termwise_split_to_cone f g h) (cone.in f) :=\n{ hom := \u03bb i j,\n    if e : j + 1 = i then -(h i).retraction \u226b (A.X_eq_to_iso e).inv \u226b biprod.inl else 0,\n  zero' := \u03bb _ _ r, dif_neg r,\n  comm := \u03bb i, begin\n    dsimp,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n    have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n    rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n      d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n    simp only [dite_eq_ite, cochain_complex_prev_d, dif_pos, if_true, category.assoc, cone_d,\n      category.id_comp, add_left_inj, sub_add_cancel, dif_ctx_congr, X_eq_to_iso_refl, cone.d,\n      preadditive.comp_neg, eq_self_iff_true, cochain_complex_d_next, preadditive.neg_comp],\n    ext,\n    { simp [cone.in, splitting.\u03c0_section_eq_id_sub_assoc, \u2190 sub_eq_add_neg,\n      -retraction_X_eq_to_hom_assoc] },\n    { simp [cone.in, splitting.retraction_\u03b9_eq_id_sub, \u2190 sub_eq_add_neg,\n      -retraction_X_eq_to_hom_assoc] },\n  end }\n\n@[simps]\ndef cone_to_termwise_split (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  cone f \u27f6 C :=\n{ f := \u03bb i, biprod.snd \u226b g.f i,\n  comm' := begin\n    rintro i j (rfl : i + 1 = j),\n    ext; simp [cone.d, (h _).comp_eq_zero],\n  end }\n\n@[simps]\ndef cone_to_termwise_split_comp_homotopy (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  homotopy (cone_to_termwise_split f g h \u226b connecting_hom f g h) (-cone.out f) :=\n{ hom := \u03bb i j,\n    if e : j + 1 = i then biprod.snd \u226b (h i).retraction \u226b (A.X_eq_to_iso e).inv else 0,\n  zero' := \u03bb _ _ r, dif_neg r,\n  comm := begin\n    intro i,\n    dsimp,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n    have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n    rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n      d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n    simp only [category.comp_id, dite_eq_ite, cochain_complex_prev_d, cone.out, dif_pos, if_true,\n      add_left_inj, sub_add_cancel, cone.d, shift_d, dif_ctx_congr, preadditive.comp_neg,\n      eq_self_iff_true, int.neg_one_pow_one, cochain_complex_d_next, one_zsmul,\n      category.assoc, X_eq_to_iso_d, neg_neg, neg_smul, biprod.lift_snd_assoc,\n      X_eq_to_iso_refl, cone_d, preadditive.neg_comp],\n    ext; -- This is simp [splitting.\u03c0_section_eq_id_sub_assoc, sub_eq_add_neg]\n      simp only [splitting.\u03b9_retraction, preadditive.comp_add, preadditive.comp_neg,\n        biprod.inl_snd_assoc, zero_comp, neg_zero, add_zero, biprod.inl_desc_assoc, biprod.inl_fst,\n        add_right_neg, biprod.inr_snd_assoc, biprod.inr_desc_assoc, biprod.inr_fst,\n        splitting.\u03c0_section_eq_id_sub_assoc, sub_eq_add_neg, hom.comm, hom.comm_assoc,\n        preadditive.add_comp_assoc, category.id_comp, preadditive.neg_comp, category.assoc,\n        preadditive.add_comp, category.comp_id, add_left_inj, eq_self_iff_true],\n  end }\n\ndef iso_cone_of_termwise_split_inv_hom_homotopy (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  homotopy (cone_to_termwise_split f g h \u226b termwise_split_to_cone f g h) (\ud835\udfd9 _) :=\n{ hom := \u03bb i j, if e : j + 1 = i then\n    -biprod.snd \u226b (h i).retraction \u226b (A.X_eq_to_iso e).inv \u226b biprod.inl else 0,\n  zero' := \u03bb _ _ r, dif_neg r,\n  comm := begin\n    intro i,\n    dsimp,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    have aux\u2081 : (complex_shape.up \u2124).rel i (i + 1) := rfl,\n    have aux\u2082 : (complex_shape.up \u2124).rel (i + 1) (i + 1 + 1) := rfl,\n    rw [\u2190 d_next_eq_d_from_from_next, \u2190 prev_d_eq_to_prev_d_to,\n      d_next_eq _ aux\u2082, prev_d_eq _ aux\u2081, dif_pos rfl, dif_pos rfl],\n    simp only [category.comp_id, dite_eq_ite, cochain_complex_prev_d, dif_pos, if_true,\n      category.id_comp, add_left_inj, sub_add_cancel, cone.d, dif_ctx_congr,\n      eq_self_iff_true, cochain_complex_d_next, category.assoc, biprod.lift_snd_assoc,\n      X_eq_to_iso_refl, cone_d],\n    ext; -- This is simp [splitting.\u03c0_section_eq_id_sub_assoc, splitting.\u03c0_section_eq_id_sub]\n      simp only [add_left_neg, add_zero, category.assoc, category.comp_id, exact.w, exact.w_assoc,\n        biprod.inl_desc, biprod.inl_desc_assoc, biprod.inl_fst, biprod.inr_desc_assoc,\n        biprod.inr_fst, biprod.inr_snd, biprod.inr_snd_assoc, biprod.lift_fst, biprod.lift_snd,\n        biprod.lift_snd_assoc, comp_zero, zero_comp, preadditive.add_comp, preadditive.comp_add,\n        preadditive.comp_neg, preadditive.neg_comp, preadditive.neg_comp, category.comp_id,\n        splitting.\u03b9_retraction_assoc, eq_self_iff_true, X_eq_to_iso_d, X_eq_to_iso_f_assoc,\n        X_eq_to_iso_refl, X_eq_to_iso_trans, neg_neg, neg_zero, zero_add, neg_sub, hom.comm_assoc,\n        splitting.\u03c0_section_eq_id_sub_assoc, splitting.\u03c0_section_eq_id_sub, category.id_comp,\n        preadditive.sub_comp_assoc, hom.comm, preadditive.sub_comp, splitting.\u03b9_retraction,\n        exact_inl_snd];\n      abel\n  end }\n\n@[simps]\ndef iso_cone_of_termwise_split (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  triangle\u2095_of_termwise_split f g h \u2245\n    (category_theory.triangulated.neg\u2083_functor _).obj (cone.triangle\u2095 f) :=\nbegin\n  fapply mk_triangle_iso,\n  exact iso.refl _,\n  exact iso.refl _,\n  refine \u27e8Q .map (termwise_split_to_cone f g h), Q .map (cone_to_termwise_split f g h), _, _\u27e9,\n  { dsimp, erw [\u2190 Q .map_comp, \u2190 Q .map_id], congr, ext; dsimp, simp },\n  { dsimp, erw [\u2190 Q .map_comp, \u2190 Q .map_id], apply homotopy_category.eq_of_homotopy,\n    apply iso_cone_of_termwise_split_inv_hom_homotopy },\n  { exact (category.comp_id _).trans (category.id_comp _).symm },\n  { dsimp, rw [\u2190 Q .map_comp, category.id_comp],\n    apply homotopy_category.eq_of_homotopy, apply comp_termwise_split_to_cone_homotopy },\n  { dsimp, rw [category_theory.functor.map_id, category.comp_id,\n    \u2190 Q .map_neg, \u2190 Q .map_comp], congr, ext, simp [cone.out] }\nend\n\ninstance : mono (termwise_split_mono_lift f) := mono_of_eval _\n\ndef termwise_split_of_termwise_split_mono (H : \u2200 i, split_mono (f.f i)) (i : \u2124) :\n  have mono f := @mono_of_eval _ _ _ _ _ _ _ f (\u03bb i, (H i).mono), by exactI\n  splitting (f.f i) ((homological_complex.normal_mono f).g.f i) :=\nbegin\n  apply left_split.splitting,\n  dsimp only [normal_mono, cokernel_complex_\u03c0],\n  constructor,\n  exact \u27e8(H i).1, (H i).2\u27e9,\n  exact abelian.exact_cokernel _,\nend\n\n/-- Every neg\u2083 of a cone triangle is isomorphic to some triangle associated to some\n  termwise split sequence -/\ndef iso_termwise_split_of_cone :\n      (category_theory.triangulated.neg\u2083_functor _).obj (cone.triangle\u2095 f) \u2245\n    triangle\u2095_of_termwise_split (termwise_split_mono_lift f)\n      (homological_complex.normal_mono (termwise_split_mono_lift f)).g\n    (termwise_split_of_termwise_split_mono _ $ termwise_split_mono_lift_split_mono _) :=\nfunctor.map_iso _ (as_iso $ from_termwise_split_mono_lift_triangle\u2095 f).symm \u226a\u226b\n  (iso_cone_of_termwise_split _ _ _).symm\n\n-- Lemma 13.9.15. skipped\n\n--move\n@[simp, reassoc]\nlemma biprod.map_desc {C : Type*} [category C] [has_zero_morphisms C]\n  {X Y X' Y' Z : C} [has_binary_biproduct X Y] [has_binary_biproduct X' Y']\n  (f : X \u27f6 X') (g : Y \u27f6 Y') (f' : X' \u27f6 Z) (g' : Y' \u27f6 Z) :\n    biprod.map f g \u226b biprod.desc f' g' = biprod.desc (f \u226b f') (g \u226b g') :=\nby { ext; simp }\n\n@[simp, reassoc]\nlemma biprod.lift_map {C : Type*} [category C] [has_zero_morphisms C]\n  {W X Y X' Y' : C} [has_binary_biproduct X Y] [has_binary_biproduct X' Y']\n  (f : X \u27f6 X') (g : Y \u27f6 Y') (f' : W \u27f6 X) (g' : W \u27f6 Y) :\n    biprod.lift f' g' \u226b biprod.map f g = biprod.lift (f' \u226b f) (g' \u226b g) :=\nby { ext; simp }\n\n@[simps]\ndef biprod.map_iso {C : Type*} [category C] [has_zero_morphisms C]\n  {X Y X' Y' : C} [has_binary_biproduct X Y] [has_binary_biproduct X' Y']\n  (f : X \u2245 X') (g : Y \u2245 Y') : X \u229e Y \u2245 X' \u229e Y' :=\n\u27e8biprod.map f.hom g.hom, biprod.map f.inv g.inv, by ext; simp, by ext; simp\u27e9\n\n@[simps]\ndef iso_connecting_hom_shift_cone (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  B \u2245 cone ((connecting_hom f g h)\u27e6(-1 : \u2124)\u27e7') :=\nhom.iso_of_components (\u03bb f, (h _).iso \u226a\u226b biprod.braiding _ _ \u226a\u226b\n  biprod.map_iso (C.X_eq_to_iso (by simp)) (A.X_eq_to_iso (by simp)))\nbegin\n  have He := \u03bb i, (h i).split_epi,\n  rintro i j (rfl : i + 1 = j),\n  dsimp [cone.d],\n  rw \u2190 cancel_epi (h i).iso.inv,\n  simp only [category.comp_id, biprod.lift_map, neg_smul_neg, if_true, iso.inv_hom_id_assoc,\n    add_left_inj, eq_self_iff_true, one_zsmul, category.assoc, neg_neg, neg_smul],\n  ext; simp only [add_zero, category.assoc, exact.w_assoc, biprod.inl_fst_assoc,\n    biprod.inr_fst_assoc, biprod.inr_snd_assoc, biprod.lift_desc, biprod.lift_fst,\n    biprod.lift_snd, comp_zero, zero_comp, preadditive.comp_add, X_d_eq_to_iso, X_eq_to_iso_d,\n    splitting.comp_iso_eq_inl_assoc, splitting.inl_comp_iso_eq_assoc,\n    splitting.iso_comp_snd_eq_assoc, eq_self_iff_true, hom.comm_assoc, zero_add,\n    splitting.iso_hom_fst_assoc, splitting.inr_iso_inv_assoc, exact_inl_snd],\n  { rw \u2190 cancel_epi (g.f _),\n    simp only [category.id_comp, preadditive.sub_comp_assoc, (h _).comp_eq_zero_assoc,\n      sub_zero, category.assoc, comp_zero, hom.comm, preadditive.sub_comp, limits.zero_comp,\n      splitting.\u03c0_section_eq_id_sub_assoc, hom.comm_assoc],\n    rw [\u2190 X_eq_to_iso_f, X_d_eq_to_iso_assoc],\n    exact (He _).epi },\n  { rw \u2190 cancel_epi (g.f _),\n    simp only [category.comp_id, X_d_eq_to_iso, category.id_comp, preadditive.sub_comp_assoc,\n      splitting.\u03c0_section_eq_id_sub_assoc, category.assoc, hom.comm, preadditive.sub_comp,\n      splitting.\u03b9_retraction],\n    rw [\u2190 X_eq_to_iso_f_assoc, splitting.\u03c0_section_eq_id_sub_assoc],\n    simp only [X_d_eq_to_iso_assoc, category.comp_id, hom.comm_assoc, retraction_X_eq_to_hom,\n      category.id_comp, preadditive.sub_comp_assoc, X_eq_to_iso_d_assoc, splitting.\u03b9_retraction,\n      preadditive.comp_sub, hom.comm, preadditive.sub_comp, sub_right_inj, category.assoc],\n    rw [\u2190 retraction_X_eq_to_hom_assoc, X_eq_to_iso_d],\n    exact (He _).epi }\nend\n\nlemma inv_rotate_iso_cone_triangle_comm\u2081 (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  (triangle_of_termwise_split f g h).nonneg_inv_rotate.mor\u2081 \u226b (shift_shift_neg _ _).inv =\n    \ud835\udfd9 _ \u226b (cone.triangle ((connecting_hom f g h)\u27e6(-1 : \u2124)\u27e7')).mor\u2081 :=\nby { ext, dsimp, simp, dsimp, simp }\n\nlemma inv_rotate_iso_cone_triangle_comm\u2082 (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  (triangle_of_termwise_split f g h).nonneg_inv_rotate.mor\u2082 \u226b\n    (iso_connecting_hom_shift_cone f g h).hom =\n    (shift_shift_neg _ _).inv \u226b (cone.triangle ((connecting_hom f g h)\u27e6(-1 : \u2124)\u27e7')).mor\u2082 :=\nby { ext; dsimp [cone.in]; simp }\n\nlemma inv_rotate_iso_cone_triangle_comm\u2083 (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  (triangle_of_termwise_split f g h).nonneg_inv_rotate.mor\u2083 \u226b\n    (\ud835\udfd9 _)\u27e6(1 : \u2124)\u27e7' = (iso_connecting_hom_shift_cone f g h).hom \u226b\n      (cone.triangle ((connecting_hom f g h)\u27e6(-1 : \u2124)\u27e7')).mor\u2083 :=\nby { ext, dsimp [cone.out], simpa }\n\ndef inv_rotate_iso_cone_triangle (h : \u2200 i, splitting (f.f i) (g.f i)) :\n  (triangle_of_termwise_split f g h).nonneg_inv_rotate \u2245\n    cone.triangle ((connecting_hom f g h)\u27e6(-1 : \u2124)\u27e7') :=\nbegin\n  fapply mk_triangle_iso,\n  exacts [iso.refl _, (shift_shift_neg _ _).symm, iso_connecting_hom_shift_cone f g h,\n    inv_rotate_iso_cone_triangle_comm\u2081 _ _ _, inv_rotate_iso_cone_triangle_comm\u2082 _ _ _,\n    inv_rotate_iso_cone_triangle_comm\u2083 f g h],\nend\n\ndef triangle_of_termwise_split_cone_iso :\n  triangle_of_termwise_split (cone.in f) (cone.out f) (cone.termwise_split f) \u2245\n    (cone.triangle f).nonneg_rotate :=\nmk_triangle_iso (iso.refl _) (iso.refl _) (iso.refl _)\n  (by { dsimp, simp }) (by { dsimp, simp }) (by { ext, dsimp [cone.d], simp })\n\nend homological_complex\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/mapping_cone.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8104789178257654, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.39257984652116334}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.pnat.basic\nimport Mathlib.data.list.range\nimport Mathlib.data.array.lemmas\nimport Mathlib.algebra.group.default\nimport Mathlib.data.sigma.basic\nimport Mathlib.PostPort\n\nuniverses u v w l u_1 \n\nnamespace Mathlib\n\n/-!\n# Hash maps\n\nDefines a hash map data structure, representing a finite key-value map\nwith a value type that may depend on the key type.  The structure\nrequires a `nat`-valued hash function to associate keys to buckets.\n\n## Main definitions\n\n* `hash_map`: constructed with `mk_hash_map`.\n\n## Implementation details\n\nA hash map with key type `\u03b1` and (dependent) value type `\u03b2 : \u03b1 \u2192 Type*`\nconsists of an array of *buckets*, which are lists containing\nkey/value pairs for that bucket.  The hash function is taken modulo `n`\nto assign keys to their respective bucket.  Because of this, some care\nshould be put into the hash function to ensure it evenly distributes\nkeys.\n\nThe bucket array is an `array`.  These have special VM support for\nin-place modification if there is only ever one reference to them.  If\none takes special care to never keep references to old versions of a\nhash map alive after updating it, then the hash map will be modified\nin-place.  In this documentation, when we say a hash map is modified\nin-place, we are assuming the API is being used in this manner.\n\nWhen inserting (`hash_map.insert`), if the number of stored pairs (the\n*size*) is going to exceed the number of buckets, then a new hash map\nis first created with double the number of buckets and everything in\nthe old hash map is reinserted along with the new key/value pair.\nOtherwise, the bucket array is modified in-place.  The amortized\nrunning time of inserting $$n$$ elements into a hash map is $$O(n)$$.\n\nWhen removing (`hash_map.erase`), the hash map is modified in-place.\nThe implementation does not reduce the number of buckets in the hash\nmap if the size gets too low.\n\n## Tags\n\nhash map\n\n-/\n\n/-- `bucket_array \u03b1 \u03b2` is the underlying data type for `hash_map \u03b1 \u03b2`,\n  an array of linked lists of key-value pairs. -/\ndef bucket_array (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) (n : \u2115+) :=\n  array (\u2191n) (List (sigma fun (a : \u03b1) => \u03b2 a))\n\n/-- Make a hash_map index from a `nat` hash value and a (positive) buffer size -/\ndef hash_map.mk_idx (n : \u2115+) (i : \u2115) : fin \u2191n := { val := i % \u2191n, property := sorry }\n\nnamespace bucket_array\n\n\nprotected instance inhabited {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {n : \u2115+} :\n    Inhabited (bucket_array \u03b1 \u03b2 n) :=\n  { default := mk_array \u2191n [] }\n\n/-- Read the bucket corresponding to an element -/\ndef read {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n    (a : \u03b1) : List (sigma fun (a : \u03b1) => \u03b2 a) :=\n  let bidx : fin \u2191n := hash_map.mk_idx n (hash_fn a);\n  array.read data bidx\n\n/-- Write the bucket corresponding to an element -/\ndef write {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n    (a : \u03b1) (l : List (sigma fun (a : \u03b1) => \u03b2 a)) : bucket_array \u03b1 \u03b2 n :=\n  let bidx : fin \u2191n := hash_map.mk_idx n (hash_fn a);\n  array.write data bidx l\n\n/-- Modify (read, apply `f`, and write) the bucket corresponding to an element -/\ndef modify {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n    (a : \u03b1) (f : List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 List (sigma fun (a : \u03b1) => \u03b2 a)) :\n    bucket_array \u03b1 \u03b2 n :=\n  let bidx : fin \u2191n := hash_map.mk_idx n (hash_fn a);\n  array.write data bidx (f (array.read data bidx))\n\n/-- The list of all key-value pairs in the bucket list -/\ndef as_list {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n) :\n    List (sigma fun (a : \u03b1) => \u03b2 a) :=\n  list.join (array.to_list data)\n\ntheorem mem_as_list {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n    {a : sigma fun (a : \u03b1) => \u03b2 a} : a \u2208 as_list data \u2194 \u2203 (i : fin \u2191n), a \u2208 array.read data i :=\n  sorry\n\n/-- Fold a function `f` over the key-value pairs in the bucket list -/\ndef foldl {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n) {\u03b4 : Type w} (d : \u03b4)\n    (f : \u03b4 \u2192 (a : \u03b1) \u2192 \u03b2 a \u2192 \u03b4) : \u03b4 :=\n  array.foldl data d\n    fun (b : List (sigma fun (a : \u03b1) => \u03b2 a)) (d : \u03b4) =>\n      list.foldl (fun (r : \u03b4) (a : sigma fun (a : \u03b1) => \u03b2 a) => f r (sigma.fst a) (sigma.snd a)) d b\n\ntheorem foldl_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n) {\u03b4 : Type w}\n    (d : \u03b4) (f : \u03b4 \u2192 (a : \u03b1) \u2192 \u03b2 a \u2192 \u03b4) :\n    foldl data d f =\n        list.foldl (fun (r : \u03b4) (a : sigma fun (a : \u03b1) => \u03b2 a) => f r (sigma.fst a) (sigma.snd a)) d\n          (as_list data) :=\n  sorry\n\nend bucket_array\n\n\nnamespace hash_map\n\n\n/-- Insert the pair `\u27e8a, b\u27e9` into the correct location in the bucket array\n  (without checking for duplication) -/\ndef reinsert_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) {n : \u2115+}\n    (data : bucket_array \u03b1 \u03b2 n) (a : \u03b1) (b : \u03b2 a) : bucket_array \u03b1 \u03b2 n :=\n  bucket_array.modify hash_fn data a fun (l : List (sigma fun (a : \u03b1) => \u03b2 a)) => sigma.mk a b :: l\n\ntheorem mk_as_list {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (n : \u2115+) :\n    bucket_array.as_list (mk_array \u2191n []) = [] :=\n  sorry\n\n/-- Search a bucket for a key `a` and return the value -/\ndef find_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) :\n    List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 Option (\u03b2 a) :=\n  sorry\n\ntheorem find_aux_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a}\n    {l : List (sigma fun (a : \u03b1) => \u03b2 a)} :\n    list.nodup (list.map sigma.fst l) \u2192 (find_aux a l = some b \u2194 sigma.mk a b \u2208 l) :=\n  sorry\n\n/-- Returns `tt` if the bucket `l` contains the key `a` -/\ndef contains_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1)\n    (l : List (sigma fun (a : \u03b1) => \u03b2 a)) : Bool :=\n  option.is_some (find_aux a l)\n\ntheorem contains_aux_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1}\n    {l : List (sigma fun (a : \u03b1) => \u03b2 a)} (nd : list.nodup (list.map sigma.fst l)) :\n    \u21a5(contains_aux a l) \u2194 a \u2208 list.map sigma.fst l :=\n  sorry\n\n/-- Modify a bucket to replace a value in the list. Leaves the list\n unchanged if the key is not found. -/\ndef replace_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) :\n    List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 List (sigma fun (a : \u03b1) => \u03b2 a) :=\n  sorry\n\n/-- Modify a bucket to remove a key, if it exists. -/\ndef erase_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) :\n    List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 List (sigma fun (a : \u03b1) => \u03b2 a) :=\n  sorry\n\n/-- The predicate `valid bkts sz` means that `bkts` satisfies the `hash_map`\n  invariants: There are exactly `sz` elements in it, every pair is in the\n  bucket determined by its key and the hash function, and no key appears\n  multiple times in the list. -/\nstructure valid {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    (bkts : bucket_array \u03b1 \u03b2 n) (sz : \u2115)\n    where\n  len : list.length (bucket_array.as_list bkts) = sz\n  idx :\n    \u2200 {i : fin \u2191n} {a : sigma fun (a : \u03b1) => \u03b2 a},\n      a \u2208 array.read bkts i \u2192 mk_idx n (hash_fn (sigma.fst a)) = i\n  nodup : \u2200 (i : fin \u2191n), list.nodup (list.map sigma.fst (array.read bkts i))\n\ntheorem valid.idx_enum {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) {i : \u2115}\n    {l : List (sigma fun (a : \u03b1) => \u03b2 a)} (he : (i, l) \u2208 list.enum (array.to_list bkts)) {a : \u03b1}\n    {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) :\n    \u2203 (h : i < \u2191n), mk_idx n (hash_fn a) = { val := i, property := h } :=\n  sorry\n\ntheorem valid.idx_enum_1 {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) {i : \u2115}\n    {l : List (sigma fun (a : \u03b1) => \u03b2 a)} (he : (i, l) \u2208 list.enum (array.to_list bkts)) {a : \u03b1}\n    {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) : subtype.val (mk_idx n (hash_fn a)) = i :=\n  sorry\n\ntheorem valid.as_list_nodup {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) :\n    list.nodup (list.map sigma.fst (bucket_array.as_list bkts)) :=\n  sorry\n\ntheorem mk_valid {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] (n : \u2115+) :\n    valid hash_fn (mk_array \u2191n []) 0 :=\n  sorry\n\ntheorem valid.find_aux_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) {a : \u03b1} {b : \u03b2 a} :\n    find_aux a (bucket_array.read hash_fn bkts a) = some b \u2194\n        sigma.mk a b \u2208 bucket_array.as_list bkts :=\n  sorry\n\ntheorem valid.contains_aux_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1]\n    {n : \u2115+} {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) (a : \u03b1) :\n    \u21a5(contains_aux a (bucket_array.read hash_fn bkts a)) \u2194\n        a \u2208 list.map sigma.fst (bucket_array.as_list bkts) :=\n  sorry\n\ntheorem append_of_modify {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {bidx : fin \u2191n}\n    {f : List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 List (sigma fun (a : \u03b1) => \u03b2 a)}\n    (u : List (sigma fun (a : \u03b1) => \u03b2 a)) (v1 : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (v2 : List (sigma fun (a : \u03b1) => \u03b2 a)) (w : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (hl : array.read bkts bidx = u ++ v1 ++ w) (hfl : f (array.read bkts bidx) = u ++ v2 ++ w) :\n    \u2203 (u' : List (sigma fun (a : \u03b1) => \u03b2 a)),\n        \u2203 (w' : List (sigma fun (a : \u03b1) => \u03b2 a)),\n          bucket_array.as_list bkts = u' ++ v1 ++ w' \u2227\n            bucket_array.as_list bkts' = u' ++ v2 ++ w' :=\n  sorry\n\ntheorem valid.modify {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {bidx : fin \u2191n}\n    {f : List (sigma fun (a : \u03b1) => \u03b2 a) \u2192 List (sigma fun (a : \u03b1) => \u03b2 a)}\n    (u : List (sigma fun (a : \u03b1) => \u03b2 a)) (v1 : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (v2 : List (sigma fun (a : \u03b1) => \u03b2 a)) (w : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (hl : array.read bkts bidx = u ++ v1 ++ w) (hfl : f (array.read bkts bidx) = u ++ v2 ++ w)\n    (hvnd : list.nodup (list.map sigma.fst v2))\n    (hal : \u2200 (a : sigma fun (a : \u03b1) => \u03b2 a), a \u2208 v2 \u2192 mk_idx n (hash_fn (sigma.fst a)) = bidx)\n    (djuv : list.disjoint (list.map sigma.fst u) (list.map sigma.fst v2))\n    (djwv : list.disjoint (list.map sigma.fst w) (list.map sigma.fst v2)) {sz : \u2115}\n    (v : valid hash_fn bkts sz) :\n    list.length v1 \u2264 sz + list.length v2 \u2227\n        valid hash_fn bkts' (sz + list.length v2 - list.length v1) :=\n  sorry\n\ntheorem valid.replace_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a)\n    (l : List (sigma fun (a : \u03b1) => \u03b2 a)) :\n    a \u2208 list.map sigma.fst l \u2192\n        \u2203 (u : List (sigma fun (a : \u03b1) => \u03b2 a)),\n          \u2203 (w : List (sigma fun (a : \u03b1) => \u03b2 a)),\n            \u2203 (b' : \u03b2 a),\n              l = u ++ [sigma.mk a b'] ++ w \u2227 replace_aux a b l = u ++ [sigma.mk a b] ++ w :=\n  sorry\n\ntheorem valid.replace {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n    (Hc : \u21a5(contains_aux a (bucket_array.read hash_fn bkts a))) (v : valid hash_fn bkts sz) :\n    valid hash_fn (bucket_array.modify hash_fn bkts a (replace_aux a b)) sz :=\n  sorry\n\ntheorem valid.insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n    (Hnc : \u00ac\u21a5(contains_aux a (bucket_array.read hash_fn bkts a))) (v : valid hash_fn bkts sz) :\n    valid hash_fn (reinsert_aux hash_fn bkts a b) (sz + 1) :=\n  sorry\n\ntheorem valid.erase_aux {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1)\n    (l : List (sigma fun (a : \u03b1) => \u03b2 a)) :\n    a \u2208 list.map sigma.fst l \u2192\n        \u2203 (u : List (sigma fun (a : \u03b1) => \u03b2 a)),\n          \u2203 (w : List (sigma fun (a : \u03b1) => \u03b2 a)),\n            \u2203 (b : \u03b2 a), l = u ++ [sigma.mk a b] ++ w \u2227 erase_aux a l = u ++ [] ++ w :=\n  sorry\n\ntheorem valid.erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115) [DecidableEq \u03b1] {n : \u2115+}\n    {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1)\n    (Hc : \u21a5(contains_aux a (bucket_array.read hash_fn bkts a))) (v : valid hash_fn bkts sz) :\n    valid hash_fn (bucket_array.modify hash_fn bkts a (erase_aux a)) (sz - 1) :=\n  sorry\n\nend hash_map\n\n\n/-- A hash map data structure, representing a finite key-value map\n  with key type `\u03b1` and value type `\u03b2` (which may depend on `\u03b1`). -/\nstructure hash_map (\u03b1 : Type u) [DecidableEq \u03b1] (\u03b2 : \u03b1 \u2192 Type v) where\n  hash_fn : \u03b1 \u2192 \u2115\n  size : \u2115\n  nbuckets : \u2115+\n  buckets : bucket_array \u03b1 \u03b2 nbuckets\n  is_valid : hash_map.valid hash_fn buckets size\n\n/-- Construct an empty hash map with buffer size `nbuckets` (default 8). -/\ndef mk_hash_map {\u03b1 : Type u} [DecidableEq \u03b1] {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 \u2115)\n    (nbuckets : optParam \u2115 (bit0 (bit0 (bit0 1)))) : hash_map \u03b1 \u03b2 :=\n  let n : optParam \u2115 (bit0 (bit0 (bit0 1))) := ite (nbuckets = 0) (bit0 (bit0 (bit0 1))) nbuckets;\n  let nz : n > 0 := sorry;\n  hash_map.mk hash_fn 0 { val := n, property := nz } (mk_array n []) sorry\n\nnamespace hash_map\n\n\n/-- Return the value corresponding to a key, or `none` if not found -/\ndef find {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) : Option (\u03b2 a) :=\n  find_aux a (bucket_array.read (hash_fn m) (buckets m) a)\n\n/-- Return `tt` if the key exists in the map -/\ndef contains {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) : Bool :=\n  option.is_some (find m a)\n\nprotected instance has_mem {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] :\n    has_mem \u03b1 (hash_map \u03b1 \u03b2) :=\n  has_mem.mk fun (a : \u03b1) (m : hash_map \u03b1 \u03b2) => \u21a5(contains m a)\n\n/-- Fold a function over the key-value pairs in the map -/\ndef fold {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {\u03b4 : Type w} (m : hash_map \u03b1 \u03b2) (d : \u03b4)\n    (f : \u03b4 \u2192 (a : \u03b1) \u2192 \u03b2 a \u2192 \u03b4) : \u03b4 :=\n  bucket_array.foldl (buckets m) d f\n\n/-- The list of key-value pairs in the map -/\ndef entries {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) :\n    List (sigma fun (a : \u03b1) => \u03b2 a) :=\n  bucket_array.as_list (buckets m)\n\n/-- The list of keys in the map -/\ndef keys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) : List \u03b1 :=\n  list.map sigma.fst (entries m)\n\ntheorem find_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1)\n    (b : \u03b2 a) : find m a = some b \u2194 sigma.mk a b \u2208 entries m :=\n  valid.find_aux_iff (hash_fn m) (is_valid m)\n\ntheorem contains_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) :\n    \u21a5(contains m a) \u2194 a \u2208 keys m :=\n  valid.contains_aux_iff (hash_fn m) (is_valid m) a\n\ntheorem entries_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (hash_fn : \u03b1 \u2192 \u2115)\n    (n : optParam \u2115 (bit0 (bit0 (bit0 1)))) : entries (mk_hash_map hash_fn n) = [] :=\n  mk_as_list (nbuckets (mk_hash_map hash_fn n))\n\ntheorem keys_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (hash_fn : \u03b1 \u2192 \u2115)\n    (n : optParam \u2115 (bit0 (bit0 (bit0 1)))) : keys (mk_hash_map hash_fn n) = [] :=\n  sorry\n\ntheorem find_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (hash_fn : \u03b1 \u2192 \u2115)\n    (n : optParam \u2115 (bit0 (bit0 (bit0 1)))) (a : \u03b1) : find (mk_hash_map hash_fn n) a = none :=\n  sorry\n\ntheorem not_contains_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (hash_fn : \u03b1 \u2192 \u2115)\n    (n : optParam \u2115 (bit0 (bit0 (bit0 1)))) (a : \u03b1) : \u00ac\u21a5(contains (mk_hash_map hash_fn n) a) :=\n  sorry\n\ntheorem insert_lemma {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (hash_fn : \u03b1 \u2192 \u2115) {n : \u2115+}\n    {n' : \u2115+} {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (v : valid hash_fn bkts sz) :\n    valid hash_fn (bucket_array.foldl bkts (mk_array \u2191n' []) (reinsert_aux hash_fn)) sz :=\n  sorry\n\n/-- Insert a key-value pair into the map. (Modifies `m` in-place when applicable) -/\ndef insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) :\n    hash_map \u03b1 \u03b2 :=\n  sorry\n\ntheorem mem_insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1)\n    (b : \u03b2 a) (a' : \u03b1) (b' : \u03b2 a') :\n    sigma.mk a' b' \u2208 entries (insert m a b) \u2194 ite (a = a') (b == b') (sigma.mk a' b' \u2208 entries m) :=\n  sorry\n\ntheorem find_insert_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1)\n    (b : \u03b2 a) : find (insert m a b) a = some b :=\n  iff.mpr (find_iff (insert m a b) a b)\n    (iff.mpr (mem_insert m a b a b)\n      (eq.mpr\n        (id (Eq._oldrec (Eq.refl (ite (a = a) (b == b) (sigma.mk a b \u2208 entries m))) (if_pos rfl)))\n        (HEq.refl b)))\n\ntheorem find_insert_ne {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1)\n    (a' : \u03b1) (b : \u03b2 a) (h : a \u2260 a') : find (insert m a b) a' = find m a' :=\n  sorry\n\ntheorem find_insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a' : \u03b1)\n    (a : \u03b1) (b : \u03b2 a) :\n    find (insert m a b) a' =\n        dite (a = a') (fun (h : a = a') => some (eq.rec_on h b)) fun (h : \u00aca = a') => find m a' :=\n  sorry\n\n/-- Insert a list of key-value pairs into the map. (Modifies `m` in-place when applicable) -/\ndef insert_all {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (l : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (m : hash_map \u03b1 \u03b2) : hash_map \u03b1 \u03b2 :=\n  list.foldl (fun (m : hash_map \u03b1 \u03b2) (_x : sigma fun (a : \u03b1) => \u03b2 a) => sorry) m l\n\n/-- Construct a hash map from a list of key-value pairs. -/\ndef of_list {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (l : List (sigma fun (a : \u03b1) => \u03b2 a))\n    (hash_fn : \u03b1 \u2192 \u2115) : hash_map \u03b1 \u03b2 :=\n  insert_all l (mk_hash_map hash_fn (bit0 1 * list.length l))\n\n/-- Remove a key from the map. (Modifies `m` in-place when applicable) -/\ndef erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) : hash_map \u03b1 \u03b2 :=\n  sorry\n\ntheorem mem_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) (a' : \u03b1)\n    (b' : \u03b2 a') : sigma.mk a' b' \u2208 entries (erase m a) \u2194 a \u2260 a' \u2227 sigma.mk a' b' \u2208 entries m :=\n  sorry\n\ntheorem find_erase_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1) :\n    find (erase m a) a = none :=\n  sorry\n\ntheorem find_erase_ne {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a : \u03b1)\n    (a' : \u03b1) (h : a \u2260 a') : find (erase m a) a' = find m a' :=\n  sorry\n\ntheorem find_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (m : hash_map \u03b1 \u03b2) (a' : \u03b1)\n    (a : \u03b1) : find (erase m a) a' = ite (a = a') none (find m a') :=\n  sorry\n\nprotected instance has_to_string {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] [has_to_string \u03b1]\n    [(a : \u03b1) \u2192 has_to_string (\u03b2 a)] : has_to_string (hash_map \u03b1 \u03b2) :=\n  has_to_string.mk to_string\n\n/-- `hash_map` with key type `nat` and value type that may vary. -/\nprotected instance inhabited {\u03b2 : \u2115 \u2192 Type u_1} : Inhabited (hash_map \u2115 \u03b2) :=\n  { default := mk_hash_map id }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/hash_map_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.39236654567110363}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport logic.small\nimport category_theory.skeletal\n\n/-!\n# Essentially small categories.\n\nA category given by `(C : Type u) [category.{v} C]` is `w`-essentially small\nif there exists a `small_model C : Type w` equipped with `[small_category (small_model C)]`.\n\nA category is `w`-locally small if every hom type is `w`-small.\n\nThe main theorem here is that a category is `w`-essentially small iff\nthe type `skeleton C` is `w`-small, and `C` is `w`-locally small.\n-/\n\nuniverses w v v' u u'\n\nopen category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nnamespace category_theory\n\n/-- A category is `essentially_small.{w}` if there exists\nan equivalence to some `S : Type w` with `[small_category S]`. -/\nclass essentially_small (C : Type u) [category.{v} C] : Prop :=\n(equiv_small_category : \u2203 (S : Type w) (_ : small_category S), by exactI nonempty (C \u224c S))\n\n/-- Constructor for `essentially_small C` from an explicit small category witness. -/\nlemma essentially_small.mk' {C : Type u} [category.{v} C] {S : Type w} [small_category S]\n  (e : C \u224c S) : essentially_small.{w} C :=\n\u27e8\u27e8S, _, \u27e8e\u27e9\u27e9\u27e9\n\n/--\nAn arbitrarily chosen small model for an essentially small category.\n-/\n@[nolint has_inhabited_instance]\ndef small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : Type w :=\nclassical.some (@essentially_small.equiv_small_category C _ _)\n\nnoncomputable\ninstance small_category_small_model\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : small_category (small_model C) :=\nclassical.some (classical.some_spec (@essentially_small.equiv_small_category C _ _))\n\n/--\nThe (noncomputable) categorical equivalence between\nan essentially small category and its small model.\n-/\nnoncomputable\ndef equiv_small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : C \u224c small_model C :=\nnonempty.some (classical.some_spec (classical.some_spec\n  (@essentially_small.equiv_small_category C _ _)))\n\nlemma essentially_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : essentially_small.{w} C \u2194 essentially_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.symm.trans f), },\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.trans f), },\nend\n\n/--\nA category is `w`-locally small if every hom set is `w`-small.\n\nSee `shrink_homs C` for a category instance where every hom set has been replaced by a small model.\n-/\nclass locally_small (C : Type u) [category.{v} C] : Prop :=\n(hom_small : \u2200 X Y : C, small.{w} (X \u27f6 Y) . tactic.apply_instance)\n\ninstance (C : Type u) [category.{v} C] [locally_small.{w} C] (X Y : C) :\n  small (X \u27f6 Y) :=\nlocally_small.hom_small X Y\n\nlemma locally_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : locally_small.{w} C \u2194 locally_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.inverse.obj X) (e.inverse.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.inverse, },\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.functor.obj X) (e.functor.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.functor, },\nend\n\n@[priority 100]\ninstance locally_small_self (C : Type u) [category.{v} C] : locally_small.{v} C := {}\n\n@[priority 100]\ninstance locally_small_of_essentially_small\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : locally_small.{w} C :=\n(locally_small_congr (equiv_small_model C)).mpr (category_theory.locally_small_self _)\n\n/--\nWe define a type alias `shrink_homs C` for `C`. When we have `locally_small.{w} C`,\nwe'll put a `category.{w}` instance on `shrink_homs C`.\n-/\n@[nolint has_inhabited_instance]\ndef shrink_homs (C : Type u) := C\n\nnamespace shrink_homs\n\nsection\nvariables {C' : Type*} -- a fresh variable with no category instance attached\n\n/-- Help the typechecker by explicitly translating from `C` to `shrink_homs C`. -/\ndef to_shrink_homs {C' : Type*} (X : C') : shrink_homs C' := X\n/-- Help the typechecker by explicitly translating from `shrink_homs C` to `C`. -/\ndef from_shrink_homs {C' : Type*} (X : shrink_homs C') : C' := X\n\n@[simp] lemma to_from (X : C') : from_shrink_homs (to_shrink_homs X) = X := rfl\n@[simp] lemma from_to (X : shrink_homs C') : to_shrink_homs (from_shrink_homs X) = X := rfl\n\nend\n\nvariables (C) [locally_small.{w} C]\n\n@[simps]\nnoncomputable\ninstance : category.{w} (shrink_homs C) :=\n{ hom := \u03bb X Y, shrink (from_shrink_homs X \u27f6 from_shrink_homs Y),\n  id := \u03bb X, equiv_shrink _ (\ud835\udfd9 (from_shrink_homs X)),\n  comp := \u03bb X Y Z f g,\n    equiv_shrink _ (((equiv_shrink _).symm f) \u226b ((equiv_shrink _).symm g)), }.\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef functor : C \u2964 shrink_homs C :=\n{ obj := \u03bb X, to_shrink_homs X,\n  map := \u03bb X Y f, equiv_shrink (X \u27f6 Y) f, }\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef inverse : shrink_homs C \u2964 C :=\n{ obj := \u03bb X, from_shrink_homs X,\n  map := \u03bb X Y f, (equiv_shrink (from_shrink_homs X \u27f6 from_shrink_homs Y)).symm f, }\n\n/--\nThe categorical equivalence between `C` and `shrink_homs C`, when `C` is locally small.\n-/\n@[simps]\nnoncomputable\ndef equivalence : C \u224c shrink_homs C :=\nequivalence.mk (functor C) (inverse C)\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n\nend shrink_homs\n\n/--\nA category is essentially small if and only if\nthe underlying type of its skeleton (i.e. the \"set\" of isomorphism classes) is small,\nand it is locally small.\n-/\ntheorem essentially_small_iff (C : Type u) [category.{v} C] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) \u2227 locally_small.{w} C :=\nbegin\n  -- This theorem is the only bit of real work in this file.\n  fsplit,\n  { intro h,\n    fsplit,\n    { rcases h with \u27e8S, \ud835\udcae, \u27e8e\u27e9\u27e9,\n      resetI,\n      refine \u27e8\u27e8skeleton S, \u27e8_\u27e9\u27e9\u27e9,\n      exact e.skeleton_equiv, },\n    { resetI, apply_instance, }, },\n  { rintro \u27e8\u27e8S, \u27e8e\u27e9\u27e9, L\u27e9,\n    resetI,\n    let e' := (shrink_homs.equivalence C).skeleton_equiv.symm,\n    refine \u27e8\u27e8S, _, \u27e8_\u27e9\u27e9\u27e9,\n    apply induced_category.category (e'.trans e).symm,\n    refine (shrink_homs.equivalence C).trans\n      ((skeleton_equivalence _).symm.trans\n      ((induced_functor (e'.trans e).symm).as_equivalence.symm)), },\nend\n\n/--\nAny thin category is locally small.\n-/\n@[priority 100]\ninstance locally_small_of_thin {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  locally_small.{w} C := {}\n\n/--\nA thin category is essentially small if and only if the underlying type of its skeleton is small.\n-/\ntheorem essentially_small_iff_of_thin\n  {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) :=\nby simp [essentially_small_iff, category_theory.locally_small_of_thin]\n\nend category_theory\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/essentially_small.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3922995881670517}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport group_theory.group_action\nimport group_theory.quotient_group\nimport group_theory.order_of_element\nimport data.zmod.basic\nimport data.fintype.card\nimport data.list.rotate\n\n/-!\n# Sylow theorems\n\nThe Sylow theorems are the following results for every finite group `G` and every prime number `p`.\n\n* There exists a Sylow `p`-subgroup of `G`.\n* All Sylow `p`-subgroups of `G` are conjugate to each other.\n* Let `n\u209a` be the number of Sylow `p`-subgroups of `G`, then `n\u209a` divides the index of the Sylow\n  `p`-subgroup, `n\u209a \u2261 1 [MOD p]`, and `n\u209a` is equal to the index of the normalizer of the Sylow\n  `p`-subgroup in `G`.\n\nIn this file, currently only the first of these results is proven.\n\n## Main statements\n\n* `exists_prime_order_of_dvd_card`: For every prime `p` dividing the order of `G` there exists an\n  element of order `p` in `G`. This is known as Cauchy`s theorem.\n* `exists_subgroup_card_pow_prime`: A generalisation of the first of the Sylow theorems: For every\n  prime power `p\u207f` dividing `G`, there exists a subgroup of `G` of order `p\u207f`.\n\n## TODO\n\n* Prove the second and third of the Sylow theorems.\n* Sylow theorems for infinite groups\n-/\n\nopen equiv fintype finset mul_action function\nopen equiv.perm subgroup list quotient_group\nopen_locale big_operators\nuniverses u v w\nvariables {G : Type u} {\u03b1 : Type v} {\u03b2 : Type w} [group G]\n\nlocal attribute [instance, priority 10] subtype.fintype set_fintype classical.prop_decidable\n\nnamespace mul_action\nvariables [mul_action G \u03b1]\n\nlemma mem_fixed_points_iff_card_orbit_eq_one {a : \u03b1}\n  [fintype (orbit G a)] : a \u2208 fixed_points G \u03b1 \u2194 card (orbit G a) = 1 :=\nbegin\n  rw [fintype.card_eq_one_iff, mem_fixed_points],\n  split,\n  { exact \u03bb h, \u27e8\u27e8a, mem_orbit_self _\u27e9, \u03bb \u27e8b, \u27e8x, hx\u27e9\u27e9, subtype.eq $ by simp [h x, hx.symm]\u27e9 },\n  { assume h x,\n    rcases h with \u27e8\u27e8z, hz\u27e9, hz\u2081\u27e9,\n    exact calc x \u2022 a = z : subtype.mk.inj (hz\u2081 \u27e8x \u2022 a, mem_orbit _ _\u27e9)\n      ... = a : (subtype.mk.inj (hz\u2081 \u27e8a, mem_orbit_self _\u27e9)).symm }\nend\n\nlemma card_modeq_card_fixed_points [fintype \u03b1] [fintype G] [fintype (fixed_points G \u03b1)]\n  (p : \u2115) {n : \u2115} [hp : fact p.prime] (h : card G = p ^ n) :\n  card \u03b1 \u2261 card (fixed_points G \u03b1) [MOD p] :=\ncalc card \u03b1 = card (\u03a3 y : quotient (orbit_rel G \u03b1), {x // quotient.mk' x = y}) :\n  card_congr (sigma_preimage_equiv (@quotient.mk' _ (orbit_rel G \u03b1))).symm\n... = \u2211 a : quotient (orbit_rel G \u03b1), card {x // quotient.mk' x = a} : card_sigma _\n... \u2261 \u2211 a : fixed_points G \u03b1, 1 [MOD p] :\nbegin\n  rw [\u2190 zmod.eq_iff_modeq_nat p, sum_nat_cast, sum_nat_cast],\n  refine eq.symm (sum_bij_ne_zero (\u03bb a _ _, quotient.mk' a.1)\n    (\u03bb _ _ _, mem_univ _)\n    (\u03bb a\u2081 a\u2082 _ _ _ _ h,\n      subtype.eq ((mem_fixed_points' \u03b1).1 a\u2082.2 a\u2081.1 (quotient.exact' h)))\n      (\u03bb b, _)\n      (\u03bb a ha _, by rw [\u2190 mem_fixed_points_iff_card_orbit_eq_one.1 a.2];\n        simp only [quotient.eq']; congr)),\n  { refine quotient.induction_on' b (\u03bb b _ hb, _),\n    have : card (orbit G b) \u2223 p ^ n,\n    { rw [\u2190 h, fintype.card_congr (orbit_equiv_quotient_stabilizer G b)],\n      exact card_quotient_dvd_card _ },\n    rcases (nat.dvd_prime_pow hp.1).1 this with \u27e8k, _, hk\u27e9,\n    have hb' :\u00ac p ^ 1 \u2223 p ^ k,\n    { rw [pow_one, \u2190 hk, \u2190 nat.modeq.modeq_zero_iff, \u2190 zmod.eq_iff_modeq_nat,\n        nat.cast_zero, \u2190 ne.def],\n      exact eq.mpr (by simp only [quotient.eq']; congr) hb },\n    have : k = 0 := nat.le_zero_iff.1 (nat.le_of_lt_succ (lt_of_not_ge (mt (pow_dvd_pow p) hb'))),\n    refine \u27e8\u27e8b, mem_fixed_points_iff_card_orbit_eq_one.2 $ by rw [hk, this, pow_zero]\u27e9,\n      mem_univ _, _, rfl\u27e9,\n    rw [nat.cast_one], exact one_ne_zero }\nend\n... = _ : by simp; refl\n\nend mul_action\n\nlemma quotient_group.card_preimage_mk [fintype G] (s : subgroup G)\n  (t : set (quotient s)) : fintype.card (quotient_group.mk \u207b\u00b9' t) =\n  fintype.card s * fintype.card t :=\nby rw [\u2190 fintype.card_prod, fintype.card_congr\n  (preimage_mk_equiv_subgroup_times_set _ _)]\n\nnamespace sylow\n\n/-- Given a vector `v` of length `n`, make a vector of length `n+1` whose product is `1`,\nby consing the the inverse of the product of `v`. -/\ndef mk_vector_prod_eq_one (n : \u2115) (v : vector G n) : vector G (n+1) :=\nv.to_list.prod\u207b\u00b9 ::\u1d65 v\n\nlemma mk_vector_prod_eq_one_injective (n : \u2115) : injective (@mk_vector_prod_eq_one G _ n) :=\n\u03bb \u27e8v, _\u27e9 \u27e8w, _\u27e9 h, subtype.eq (show v = w, by injection h with h; injection h)\n\n/-- The type of vectors with terms from `G`, length `n`, and product equal to `1:G`. -/\ndef vectors_prod_eq_one (G : Type*) [group G] (n : \u2115) : set (vector G n) :=\n{v | v.to_list.prod = 1}\n\nlemma mem_vectors_prod_eq_one {n : \u2115} (v : vector G n) :\n  v \u2208 vectors_prod_eq_one G n \u2194 v.to_list.prod = 1 := iff.rfl\n\nlemma mem_vectors_prod_eq_one_iff {n : \u2115} (v : vector G (n + 1)) :\n  v \u2208 vectors_prod_eq_one G (n + 1) \u2194 v \u2208 set.range (@mk_vector_prod_eq_one G _ n) :=\n\u27e8\u03bb (h : v.to_list.prod = 1), \u27e8v.tail,\n  begin\n    unfold mk_vector_prod_eq_one,\n    conv {to_rhs, rw \u2190 vector.cons_head_tail v},\n    suffices : (v.tail.to_list.prod)\u207b\u00b9 = v.head,\n    { rw this },\n    rw [\u2190 mul_left_inj v.tail.to_list.prod, inv_mul_self, \u2190 list.prod_cons,\n      \u2190 vector.to_list_cons, vector.cons_head_tail, h]\n  end\u27e9,\n  \u03bb \u27e8w, hw\u27e9, by rw [mem_vectors_prod_eq_one, \u2190 hw, mk_vector_prod_eq_one,\n    vector.to_list_cons, list.prod_cons, inv_mul_self]\u27e9\n\n/-- The rotation action of `zmod n` (viewed as multiplicative group) on\n`vectors_prod_eq_one G n`, where `G` is a multiplicative group. -/\ndef rotate_vectors_prod_eq_one (G : Type*) [group G] (n : \u2115)\n  (m : multiplicative (zmod n)) (v : vectors_prod_eq_one G n) : vectors_prod_eq_one G n :=\n\u27e8\u27e8v.1.to_list.rotate m.val, by simp\u27e9, prod_rotate_eq_one_of_prod_eq_one v.2 _\u27e9\n\ninstance rotate_vectors_prod_eq_one.mul_action (n : \u2115) [fact (0 < n)] :\n  mul_action (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=\n{ smul := (rotate_vectors_prod_eq_one G n),\n  one_smul :=\n  begin\n    intro v, apply subtype.eq, apply vector.eq _ _,\n    show rotate _ (0 : zmod n).val = _, rw zmod.val_zero,\n    exact rotate_zero v.1.to_list\n  end,\n  mul_smul := \u03bb a b \u27e8\u27e8v, hv\u2081\u27e9, hv\u2082\u27e9, subtype.eq $ vector.eq _ _ $\n    show v.rotate ((a + b : zmod n).val) = list.rotate (list.rotate v (b.val)) (a.val),\n    by rw [zmod.val_add, rotate_rotate, \u2190 rotate_mod _ (b.val + a.val), add_comm, hv\u2081] }\n\nlemma one_mem_vectors_prod_eq_one (n : \u2115) : vector.repeat (1 : G) n \u2208 vectors_prod_eq_one G n :=\nby simp [vector.repeat, vectors_prod_eq_one]\n\nlemma one_mem_fixed_points_rotate (n : \u2115) [fact (0 < n)] :\n  (\u27e8vector.repeat (1 : G) n, one_mem_vectors_prod_eq_one n\u27e9 : vectors_prod_eq_one G n) \u2208\n  fixed_points (multiplicative (zmod n)) (vectors_prod_eq_one G n) :=\n\u03bb m, subtype.eq $ vector.eq _ _ $\nrotate_eq_self_iff_eq_repeat.2 \u27e8(1 : G),\n  show list.repeat (1 : G) n = list.repeat 1 (list.repeat (1 : G) n).length, by simp\u27e9 _\n\n/-- Cauchy's theorem -/\nlemma exists_prime_order_of_dvd_card [fintype G] (p : \u2115) [hp : fact p.prime]\n  (hdvd : p \u2223 card G) : \u2203 x : G, order_of x = p :=\nlet n : \u2115+ := \u27e8p - 1, nat.sub_pos_of_lt hp.1.one_lt\u27e9 in\nhave hn : p = n + 1 := nat.succ_sub hp.1.pos,\nhave hcard : card (vectors_prod_eq_one G (n + 1)) = card G ^ (n : \u2115),\n  by rw [set.ext mem_vectors_prod_eq_one_iff,\n    set.card_range_of_injective (mk_vector_prod_eq_one_injective _), card_vector],\nhave hzmod : fintype.card (multiplicative (zmod p)) = p ^ 1,\n  by { rw pow_one p, exact zmod.card p },\nhave hmodeq : _ = _ := @mul_action.card_modeq_card_fixed_points\n  (multiplicative (zmod p)) (vectors_prod_eq_one G p) _ _ _ _ _ _ 1 hp hzmod,\nhave hdvdcard : p \u2223 fintype.card (vectors_prod_eq_one G (n + 1)) :=\n  calc p \u2223 card G ^ 1 : by rwa pow_one\n  ... \u2223 card G ^ (n : \u2115) : pow_dvd_pow _ n.2\n  ... = card (vectors_prod_eq_one G (n + 1)) : hcard.symm,\nhave hdvdcard\u2082 : p \u2223 card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)),\n  by { rw nat.dvd_iff_mod_eq_zero at hdvdcard \u22a2, rwa [\u2190 hn, hmodeq] at hdvdcard },\nhave hcard_pos : 0 < card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)) :=\n  fintype.card_pos_iff.2 \u27e8\u27e8\u27e8vector.repeat 1 p, one_mem_vectors_prod_eq_one _\u27e9,\n    one_mem_fixed_points_rotate _\u27e9\u27e9,\nhave hlt : 1 < card (fixed_points (multiplicative (zmod p)) (vectors_prod_eq_one G p)) :=\n  calc (1 : \u2115) < p : hp.1.one_lt\n  ... \u2264 _ : nat.le_of_dvd hcard_pos hdvdcard\u2082,\nlet \u27e8\u27e8\u27e8\u27e8x, hx\u2081\u27e9, hx\u2082\u27e9, hx\u2083\u27e9, hx\u2084\u27e9 := fintype.exists_ne_of_one_lt_card hlt\n  \u27e8_, one_mem_fixed_points_rotate p\u27e9 in\nhave hx : x \u2260 list.repeat (1 : G) p, from \u03bb h, by simpa [h, vector.repeat] using hx\u2084,\nhave \u2203 a, x = list.repeat a x.length := by exactI rotate_eq_self_iff_eq_repeat.1 (\u03bb n,\n  have list.rotate x (n : zmod p).val = x :=\n    subtype.mk.inj (subtype.mk.inj (hx\u2083 (n : zmod p))),\n  by rwa [zmod.val_nat_cast, \u2190 hx\u2081, rotate_mod] at this),\nlet \u27e8a, ha\u27e9 := this in\n\u27e8a, have hx1 : x.prod = 1 := hx\u2082,\n  have ha1: a \u2260 1, from \u03bb h, hx (ha.symm \u25b8 h \u25b8 hx\u2081 \u25b8 rfl),\n  have a ^ p = 1, by rwa [ha, list.prod_repeat, hx\u2081] at hx1,\n  (hp.1.2 _ (order_of_dvd_of_pow_eq_one this)).resolve_left\n    (\u03bb h, ha1 (order_of_eq_one_iff.1 h))\u27e9\n\nopen subgroup submonoid is_group_hom mul_action\n\nlemma mem_fixed_points_mul_left_cosets_iff_mem_normalizer {H : subgroup G}\n  [fintype ((H : set G) : Type u)] {x : G} :\n  (x : quotient H) \u2208 fixed_points H (quotient H) \u2194 x \u2208 normalizer H :=\n\u27e8\u03bb hx, have ha : \u2200 {y : quotient H}, y \u2208 orbit H (x : quotient H) \u2192 y = x,\n  from \u03bb _, ((mem_fixed_points' _).1 hx _),\n  (inv_mem_iff _).1 (@mem_normalizer_fintype _ _ _ _inst_2 _ (\u03bb n (hn : n \u2208 H),\n    have (n\u207b\u00b9 * x)\u207b\u00b9 * x \u2208 H := quotient_group.eq.1 (ha (mem_orbit _ \u27e8n\u207b\u00b9, H.inv_mem hn\u27e9)),\n    show _ \u2208 H, by {rw [mul_inv_rev, inv_inv] at this, convert this, rw inv_inv}\n    )),\n\u03bb (hx : \u2200 (n : G), n \u2208 H \u2194 x * n * x\u207b\u00b9 \u2208 H),\n(mem_fixed_points' _).2 $ \u03bb y, quotient.induction_on' y $ \u03bb y hy, quotient_group.eq.2\n  (let \u27e8\u27e8b, hb\u2081\u27e9, hb\u2082\u27e9 := hy in\n  have hb\u2082 : (b * x)\u207b\u00b9 * y \u2208 H := quotient_group.eq.1 hb\u2082,\n  (inv_mem_iff H).1 $ (hx _).2 $ (mul_mem_cancel_left H (H.inv_mem hb\u2081)).1\n  $ by rw hx at hb\u2082;\n    simpa [mul_inv_rev, mul_assoc] using hb\u2082)\u27e9\n\ndef fixed_points_mul_left_cosets_equiv_quotient (H : subgroup G) [fintype (H : set G)] :\n  mul_action.fixed_points H (quotient H) \u2243\n  quotient (subgroup.comap ((normalizer H).subtype : normalizer H \u2192* G) H) :=\n@subtype_quotient_equiv_quotient_subtype G (normalizer H : set G) (id _) (id _) (fixed_points _ _)\n  (\u03bb a, (@mem_fixed_points_mul_left_cosets_iff_mem_normalizer _ _ _ _inst_2 _).symm)\n  (by intros; refl)\n\n/-- The first of the Sylow theorems. -/\ntheorem exists_subgroup_card_pow_prime [fintype G] (p : \u2115) : \u2200 {n : \u2115} [hp : fact p.prime]\n  (hdvd : p ^ n \u2223 card G), \u2203 H : subgroup G, fintype.card H = p ^ n\n| 0 := \u03bb _ _, \u27e8(\u22a5 : subgroup G), by convert card_bot\u27e9\n| (n+1) := \u03bb hp hdvd,\nlet \u27e8H, hH2\u27e9 := @exists_subgroup_card_pow_prime _ hp\n  (dvd.trans (pow_dvd_pow _ (nat.le_succ _)) hdvd) in\nlet \u27e8s, hs\u27e9 := exists_eq_mul_left_of_dvd hdvd in\nhave hcard : card (quotient H) = s * p :=\n  (nat.mul_left_inj (show card H > 0, from fintype.card_pos_iff.2\n      \u27e8\u27e81, H.one_mem\u27e9\u27e9)).1\n    (by rwa [\u2190 card_eq_card_quotient_mul_card_subgroup H, hH2, hs,\n      pow_succ', mul_assoc, mul_comm p]),\nhave hm : s * p % p =\n  card (quotient (subgroup.comap ((normalizer H).subtype : normalizer H \u2192* G) H)) % p :=\n  card_congr (fixed_points_mul_left_cosets_equiv_quotient H) \u25b8 hcard \u25b8\n    @card_modeq_card_fixed_points _ _ _ _ _ _ _ p _ hp hH2,\nhave hm' : p \u2223 card (quotient (subgroup.comap ((normalizer H).subtype : normalizer H \u2192* G) H)) :=\n  nat.dvd_of_mod_eq_zero\n    (by rwa [nat.mod_eq_zero_of_dvd (dvd_mul_left _ _), eq_comm] at hm),\nlet \u27e8x, hx\u27e9 := @exists_prime_order_of_dvd_card _ (quotient_group.quotient.group _) _ _ hp hm' in\nhave hequiv : H \u2243 (subgroup.comap ((normalizer H).subtype : normalizer H \u2192* G) H) :=\n  \u27e8\u03bb a, \u27e8\u27e8a.1, le_normalizer a.2\u27e9, a.2\u27e9, \u03bb a, \u27e8a.1.1, a.2\u27e9,\n    \u03bb \u27e8_, _\u27e9, rfl, \u03bb \u27e8\u27e8_, _\u27e9, _\u27e9, rfl\u27e9,\n-- begin proof of \u2203 H : subgroup G, fintype.card H = p ^ n\n\u27e8subgroup.map ((normalizer H).subtype) (subgroup.comap\n  (quotient_group.mk' (comap H.normalizer.subtype H)) (gpowers x)),\nbegin\n  show card \u21a5(map H.normalizer.subtype\n    (comap (mk' (comap H.normalizer.subtype H)) (subgroup.gpowers x))) = p ^ (n + 1),\n  suffices : card \u21a5(subtype.val '' ((subgroup.comap (mk' (comap H.normalizer.subtype H))\n    (gpowers x)) : set (\u21a5(H.normalizer)))) = p^(n+1),\n  { convert this using 2 },\n  rw [set.card_image_of_injective\n        (subgroup.comap (mk' (comap H.normalizer.subtype H)) (gpowers x) : set (H.normalizer))\n        subtype.val_injective,\n      pow_succ', \u2190 hH2, fintype.card_congr hequiv, \u2190 hx, order_eq_card_gpowers,\n      \u2190 fintype.card_prod],\n  exact @fintype.card_congr _ _ (id _) (id _) (preimage_mk_equiv_subgroup_times_set _ _)\nend\u27e9\n\nend sylow\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/group_theory/sylow.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.3922995881670517}}
{"text": "/-\nCopyright \u00a9 2020 Nicol\u00f2 Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Nicol\u00f2 Cavalleri.\n-/\n\n/-\n\nTHIS FILE CONTAINS SCATTERED RESULTS TO BE CORRECTLY PLACED IN MATHLIB\n\n-/\n\nimport .times_cont_mdiff\nimport tactic\nimport topology.algebra.module\nimport topology.algebra.continuous_functions\n\nnoncomputable theory\n\nsection prod\n\n/-\nTo be placed into topology/contructions\n-/\n\nuniverses u v w x\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type x}\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\n@[inline, reducible] def map_diag (A : Type*) : (A \u2192 A\u00d7A) := (\u03bb a : A, (a, a))\n\nlemma continuous.map_diag : continuous (map_diag \u03b1) :=\ncontinuous_id.prod_mk continuous_id\n\nend prod\n\nsection\n\n/-\nTo be placed into topology/local_homemorph\n-/\n\nnamespace local_homeomorph\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*} {\u03b7 : Type*} {\u03b5 : Type*}\n[topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n[topological_space \u03b4] [topological_space \u03b7] [topological_space \u03b5]\n(e : local_homeomorph \u03b1 \u03b2) (f : local_homeomorph \u03b2 \u03b3)\n(e' : local_homeomorph \u03b4 \u03b7) (f' : local_homeomorph \u03b7 \u03b5)\n\nend local_homeomorph\n\nend\n\nsection preamble_results\n\nsection times_cont_diff\n\n/-\nTo be placed into analysis/calculus/times_cont_diff\n-/\n\nvariables  {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n{E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n{F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n{G : Type*} [normed_group G] [normed_space \ud835\udd5c G]\n{T : Type*} [normed_group T] [normed_space \ud835\udd5c T]\n\nend times_cont_diff\n\nsection\n\n/-\nTo be placed into geometry/manifold/manifold\n-/\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n  {E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n  {E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n  {H : Type*} [topological_space H]\n  {H' : Type*} [topological_space H']\n\nsection smooth\n\n/-\nNot really sure where this should go. Either on geometry/manifold/times_cont_mdiff\nor on a new file.\n-/\n\n/-- Smooth means C^\u221e. I truly believe this definition should exists as writing\n`times_cont_mdiff I I' \u22a4 f` makes everything unreadable and harder to understand for someone\nwho does not know Mathlib well. -/\ndef smooth (I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{N : Type*} [topological_space N] [charted_space H' N] [smooth_manifold_with_corners I' N]\n(f: M \u2192 N) := times_cont_mdiff I I' \u22a4 f\n\ndef smooth_on (I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{N : Type*} [topological_space N] [charted_space H' N] [smooth_manifold_with_corners I' N]\n(f: M \u2192 N) (s : set M) := times_cont_mdiff_on I I' \u22a4 f s\n\ndef smooth_in_charts (I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{N : Type*} [topological_space N] [charted_space H' N] [smooth_manifold_with_corners I' N]\n(f: M \u2192 N) := times_cont_mdiff_in_charts I I' \u22a4 f\n\ndef smooth_in_charts_on (I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{N : Type*} [topological_space N] [charted_space H' N] [smooth_manifold_with_corners I' N]\n(f: M \u2192 N) (s : set M) := times_cont_mdiff_in_charts_on I I' \u22a4 f s\n\nvariables {I : model_with_corners \ud835\udd5c E H} {I' : model_with_corners \ud835\udd5c E' H'}\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{N : Type*} [topological_space N] [charted_space H' N] [smooth_manifold_with_corners I' N]\n\nlemma smooth_in_charts_id : smooth_in_charts I I (id : M \u2192 M) :=\nbegin\n  intros x y,\n  rw [function.comp.left_id, set.preimage_id],\n  unfold ext_chart_at,\n  simp only [model_with_corners.to_local_equiv_coe_symm, local_homeomorph.coe_coe_symm, local_homeomorph.coe_coe,\n    local_equiv.coe_trans, local_equiv.coe_trans_symm, model_with_corners.to_local_equiv_coe],\n  have h1 := (has_groupoid.compatible (times_cont_diff_groupoid \u22a4 I) (chart_mem_atlas H x) (chart_mem_atlas H y)).1,\n  simp only [local_homeomorph.trans_to_local_equiv, local_homeomorph.coe_trans, local_homeomorph.symm_to_local_equiv] at h1,\n  convert h1 using 1,\n  unfold function.comp,\n  ext1 z,\n  rw set.mem_inter_eq,\n  fsplit;\n  simp only [local_equiv.trans_source, local_equiv.trans_target, and_imp, model_with_corners.to_local_equiv_coe_symm,\n    set.mem_preimage, set.mem_range, local_homeomorph.coe_coe_symm, set.mem_inter_eq, local_equiv.symm_source,\n    set.preimage_univ, model_with_corners.target, model_with_corners.source_eq, exists_imp_distrib, set.inter_univ],\n  { intros w hw h2 h3, exact \u27e8\u27e8h2, h3\u27e9, \u27e8w, hw\u27e9\u27e9, },\n  { intros h2 h3 w hw, use w, exacts [hw, h2, h3], }\nend\n\nlemma smooth_in_charts_const {n : N} : smooth_in_charts I I' (\u03bb x : M, n) :=\nbegin\n  intros x y,\n  unfold function.comp,\n  exact times_cont_diff_on_const,\nend\n\nlemma tangent_bundle_proj_smooth : smooth_in_charts I.tangent I (tangent_bundle.proj I M) :=\nbegin\n  intros x y,\n  simp only [function.comp] with mfld_simps,\n  sorry,\nend\n\nend smooth\n\nsection composition\n\nvariables {E'' : Type*} [normed_group E''] [normed_space \ud835\udd5c E'']\n{H'' : Type*} [topological_space H''] {I'' : model_with_corners \ud835\udd5c E'' H''}\n{I : model_with_corners \ud835\udd5c E H}\n{I' : model_with_corners \ud835\udd5c E' H'}\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n{M'' : Type*} [topological_space M''] [charted_space H'' M''] [smooth_manifold_with_corners I'' M'']\n\nlemma smooth_on.comp {s : set M} {t : set M'} {f : M \u2192 M'} {g : M' \u2192 M''}\n  (hg : smooth_on I' I'' g t) (hf : smooth_on I I' f s)\n  (st : s \u2286 f \u207b\u00b9' t) : smooth_on I I'' (g \u2218 f) s :=\ntimes_cont_mdiff_on.comp hg hf st\n\nlemma times_cont_mdiff.comp {n : with_top \u2115} {f : M \u2192 M'} {g : M' \u2192 M''}\n  (hg : times_cont_mdiff I' I'' n g) (hf : times_cont_mdiff I I' n f) :\n  times_cont_mdiff I I'' n (g \u2218 f) :=\nbegin\n  have hs : (set.univ \u2286 f \u207b\u00b9' set.univ), by rw set.preimage_univ,\n  have h := (times_cont_mdiff_on_univ.2 hg).comp (times_cont_mdiff_on_univ.2 hf) hs,\n  exact times_cont_mdiff_on_univ.1 h,\nend\n\nlemma smooth.comp {f : M \u2192 M'} {g : M' \u2192 M''}\n  (hg : smooth I' I'' g) (hf : smooth I I' f) :\n  smooth I I'' (g \u2218 f) := times_cont_mdiff.comp hg hf\n\nlemma smooth_in_charts.comp {f : M \u2192 M'} {g : M' \u2192 M''}\n  (hg : smooth_in_charts I' I'' g) (hf : smooth_in_charts I I' f) :\n  smooth_in_charts I I'' (g \u2218 f) := sorry\n\nend composition\n\nend\n\nend preamble_results", "meta": {"author": "Nicknamen", "repo": "lie_group", "sha": "e0d5c4f859654e3dea092702f1320c3c72a49983", "save_path": "github-repos/lean/Nicknamen-lie_group", "path": "github-repos/lean/Nicknamen-lie_group/lie_group-e0d5c4f859654e3dea092702f1320c3c72a49983/src/preamble_results.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.3922995881670516}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.functor\n! leanprover-community/mathlib commit 70d50ecfd4900dd6d328da39ab7ebd516abe4025\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Control.Basic\nimport Mathlib.Init.Set\nimport Std.Tactic.Lint\n\n/-!\n# Functors\n\nThis module provides additional lemmas, definitions, and instances for `Functor`s.\n\n## Main definitions\n\n* `Functor.Const \u03b1` is the functor that sends all types to `\u03b1`.\n* `Functor.AddConst \u03b1` is `Functor.Const \u03b1` but for when `\u03b1` has an additive structure.\n* `Functor.Comp F G` for functors `F` and `G` is the functor composition of `F` and `G`.\n* `Liftp` and `Liftr` respectively lift predicates and relations on a type `\u03b1`\n  to `F \u03b1`.  Terms of `F \u03b1` are considered to, in some sense, contain values of type `\u03b1`.\n\n## Tags\n\nfunctor, applicative\n-/\n\n\nattribute [functor_norm] seq_assoc pure_seq map_pure seq_map_assoc map_seq\n\nuniverse u v w\n\nsection Functor\n\nvariable {F : Type u \u2192 Type v}\n\nvariable {\u03b1 \u03b2 \u03b3 : Type u}\n\nvariable [Functor F] [LawfulFunctor F]\n\ntheorem Functor.map_id : (\u00b7 <$> \u00b7) id = (id : F \u03b1 \u2192 F \u03b1) := by apply funext <;> apply id_map\n#align functor.map_id Functor.map_id\n\ntheorem Functor.map_comp_map (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) :\n    ((\u00b7 <$> \u00b7) g \u2218 (\u00b7 <$> \u00b7) f : F \u03b1 \u2192 F \u03b3) = (\u00b7 <$> \u00b7) (g \u2218 f) :=\n  funext <| fun _ => (comp_map _ _ _).symm\n  -- porting note: was `apply funext <;> intro <;> rw [comp_map]` but `rw` failed?\n#align functor.map_comp_map Functor.map_comp_map\n\ntheorem Functor.ext {F} :\n    \u2200 {F1 : Functor F} {F2 : Functor F} [@LawfulFunctor F F1] [@LawfulFunctor F F2],\n    (\u2200 (\u03b1 \u03b2) (f : \u03b1 \u2192 \u03b2) (x : F \u03b1), @Functor.map _ F1 _ _ f x = @Functor.map _ F2 _ _ f x) \u2192\n    F1 = F2\n  | \u27e8m, mc\u27e9, \u27e8m', mc'\u27e9, H1, H2, H => by\n    cases show @m = @m' by funext \u03b1 \u03b2 f x; apply H\n    congr\n    funext \u03b1 \u03b2\n    have E1 := @map_const _ \u27e8@m, @mc\u27e9 H1\n    have E2 := @map_const _ \u27e8@m, @mc'\u27e9 H2\n    exact E1.trans E2.symm\n#align functor.ext Functor.ext\n\nend Functor\n\n/-- Introduce `id` as a quasi-functor. (Note that where a lawful `Monad` or\n`Applicative` or `Functor` is needed, `Id` is the correct definition). -/\ndef id.mk {\u03b1 : Sort u} : \u03b1 \u2192 id \u03b1 :=\n  id\n#align id.mk id.mk\n\nnamespace Functor\n\n/-- `Const \u03b1` is the constant functor, mapping every type to `\u03b1`. When\n`\u03b1` has a monoid structure, `Const \u03b1` has an `Applicative` instance.\n(If `\u03b1` has an additive monoid structure, see `Functor.AddConst`.) -/\n@[nolint unusedArguments]\ndef Const (\u03b1 : Type _) (_\u03b2 : Type _) :=\n  \u03b1\n#align functor.const Functor.Const\n\n/-- `Const.mk` is the canonical map `\u03b1 \u2192 Const \u03b1 \u03b2` (the identity), and\nit can be used as a pattern to extract this value. -/\n@[match_pattern]\ndef Const.mk {\u03b1 \u03b2} (x : \u03b1) : Const \u03b1 \u03b2 :=\n  x\n#align functor.const.mk Functor.Const.mk\n\n/-- `Const.mk'` is `Const.mk` but specialized to map `\u03b1` to\n`Const \u03b1 PUnit`, where `PUnit` is the terminal object in `Type _`. -/\ndef Const.mk' {\u03b1} (x : \u03b1) : Const \u03b1 PUnit :=\n  x\n#align functor.const.mk' Functor.Const.mk'\n\n/-- Extract the element of `\u03b1` from the `Const` functor. -/\ndef Const.run {\u03b1 \u03b2} (x : Const \u03b1 \u03b2) : \u03b1 :=\n  x\n#align functor.const.run Functor.Const.run\n\nnamespace Const\n\nprotected theorem ext {\u03b1 \u03b2} {x y : Const \u03b1 \u03b2} (h : x.run = y.run) : x = y :=\n  h\n#align functor.const.ext Functor.Const.ext\n\n/-- The map operation of the `Const \u03b3` functor. -/\n@[nolint unusedArguments]\nprotected def map {\u03b3 \u03b1 \u03b2} (_f : \u03b1 \u2192 \u03b2) (x : Const \u03b3 \u03b2) : Const \u03b3 \u03b1 :=\n  x\n#align functor.const.map Functor.Const.map\n\ninstance functor {\u03b3} : Functor (Const \u03b3) where map := @Const.map \u03b3\n\ninstance lawfulFunctor {\u03b3} : LawfulFunctor (Const \u03b3) := by constructor <;> intros <;> rfl\n\ninstance {\u03b1 \u03b2} [Inhabited \u03b1] : Inhabited (Const \u03b1 \u03b2) :=\n  \u27e8(default : \u03b1)\u27e9\n\nend Const\n\n/-- `AddConst \u03b1` is a synonym for constant functor `Const \u03b1`, mapping\nevery type to `\u03b1`. When `\u03b1` has a additive monoid structure,\n`AddConst \u03b1` has an `Applicative` instance. (If `\u03b1` has a\nmultiplicative monoid structure, see `Functor.Const`.) -/\ndef AddConst (\u03b1 : Type _) :=\n  Const \u03b1\n#align functor.add_const Functor.AddConst\n\n/-- `AddConst.mk` is the canonical map `\u03b1 \u2192 AddConst \u03b1 \u03b2`, which is the identity,\nwhere `AddConst \u03b1 \u03b2 = Const \u03b1 \u03b2`. It can be used as a pattern to extract this value. -/\n@[match_pattern]\ndef AddConst.mk {\u03b1 \u03b2} (x : \u03b1) : AddConst \u03b1 \u03b2 :=\n  x\n#align functor.add_const.mk Functor.AddConst.mk\n\n/-- Extract the element of `\u03b1` from the constant functor. -/\ndef AddConst.run {\u03b1 \u03b2} : AddConst \u03b1 \u03b2 \u2192 \u03b1 :=\n  id\n#align functor.add_const.run Functor.AddConst.run\n\ninstance AddConst.functor {\u03b3} : Functor (AddConst \u03b3) :=\n  @Const.functor \u03b3\n#align functor.add_const.functor Functor.AddConst.functor\n\ninstance AddConst.lawfulFunctor {\u03b3} : LawfulFunctor (AddConst \u03b3) :=\n  @Const.lawfulFunctor \u03b3\n#align functor.add_const.is_lawful_functor Functor.AddConst.lawfulFunctor\n\ninstance {\u03b1 \u03b2} [Inhabited \u03b1] : Inhabited (AddConst \u03b1 \u03b2) :=\n  \u27e8(default : \u03b1)\u27e9\n\n/-- `Functor.Comp` is a wrapper around `Function.Comp` for types.\n    It prevents Lean's type class resolution mechanism from trying\n    a `Functor (Comp F id)` when `Functor F` would do. -/\ndef Comp (F : Type u \u2192 Type w) (G : Type v \u2192 Type u) (\u03b1 : Type v) : Type w :=\n  F <| G \u03b1\n#align functor.comp Functor.Comp\n\n/-- Construct a term of `Comp F G \u03b1` from a term of `F (G \u03b1)`, which is the same type.\nCan be used as a pattern to extract a term of `F (G \u03b1)`. -/\n@[match_pattern]\ndef Comp.mk {F : Type u \u2192 Type w} {G : Type v \u2192 Type u} {\u03b1 : Type v} (x : F (G \u03b1)) : Comp F G \u03b1 :=\n  x\n#align functor.comp.mk Functor.Comp.mk\n\n/-- Extract a term of `F (G \u03b1)` from a term of `Comp F G \u03b1`, which is the same type. -/\ndef Comp.run {F : Type u \u2192 Type w} {G : Type v \u2192 Type u} {\u03b1 : Type v} (x : Comp F G \u03b1) : F (G \u03b1) :=\n  x\n#align functor.comp.run Functor.Comp.run\n\nnamespace Comp\n\nvariable {F : Type u \u2192 Type w} {G : Type v \u2192 Type u}\n\nprotected theorem ext {\u03b1} {x y : Comp F G \u03b1} : x.run = y.run \u2192 x = y :=\n  id\n#align functor.comp.ext Functor.Comp.ext\n\ninstance {\u03b1} [Inhabited (F (G \u03b1))] : Inhabited (Comp F G \u03b1) :=\n  \u27e8(default : F (G \u03b1))\u27e9\n\nvariable [Functor F] [Functor G]\n\n/-- The map operation for the composition `Comp F G` of functors `F` and `G`. -/\nprotected def map {\u03b1 \u03b2 : Type v} (h : \u03b1 \u2192 \u03b2) : Comp F G \u03b1 \u2192 Comp F G \u03b2\n  | Comp.mk x => Comp.mk ((\u00b7 <$> \u00b7) h <$> x)\n#align functor.comp.map Functor.Comp.map\n\ninstance functor : Functor (Comp F G) where map := @Comp.map F G _ _\n\n@[functor_norm]\ntheorem map_mk {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) (x : F (G \u03b1)) : h <$> Comp.mk x = Comp.mk ((\u00b7 <$> \u00b7) h <$> x) :=\n  rfl\n#align functor.comp.map_mk Functor.Comp.map_mk\n\n@[simp]\nprotected theorem run_map {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) (x : Comp F G \u03b1) :\n    (h <$> x).run = (\u00b7 <$> \u00b7) h <$> x.run :=\n  rfl\n#align functor.comp.run_map Functor.Comp.run_map\n\nvariable [LawfulFunctor F] [LawfulFunctor G]\n\nvariable {\u03b1 \u03b2 \u03b3 : Type v}\n\nprotected theorem id_map : \u2200 x : Comp F G \u03b1, Comp.map id x = x\n  | Comp.mk x => by simp [Comp.map, Functor.map_id]; rfl\n  -- porting note: `rfl` wasn't needed in mathlib3\n#align functor.comp.id_map Functor.Comp.id_map\n\nprotected theorem comp_map (g' : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) :\n    \u2200 x : Comp F G \u03b1, Comp.map (h \u2218 g') x = Comp.map h (Comp.map g' x)\n  | Comp.mk x => by simp [Comp.map, Comp.mk, Functor.map_comp_map, functor_norm]\n  -- porting note: `Comp.mk` wasn't needed in mathlib3\n#align functor.comp.comp_map Functor.Comp.comp_map\n\ninstance lawfulFunctor : LawfulFunctor (Comp F G) where\n  map_const := rfl\n  id_map := @Comp.id_map F G _ _ _ _\n  comp_map := @Comp.comp_map F G _ _ _ _\n\n-- porting note: had to use switch to `Id` from `id` because this has the `Functor` instance.\ntheorem functor_comp_id {F} [AF : Functor F] [LawfulFunctor F] :\n    @Comp.functor F Id _ _ = AF :=\n  @Functor.ext F _ AF (@Comp.lawfulFunctor F Id _ _ _ _) _ fun _ _ _ _ => rfl\n#align functor.comp.functor_comp_id Functor.Comp.functor_comp_id\n\n-- porting note: had to use switch to `Id` from `id` because this has the `Functor` instance.\ntheorem functor_id_comp {F} [AF : Functor F] [LawfulFunctor F] : @Comp.functor Id F _ _ = AF :=\n  @Functor.ext F _ AF (@Comp.lawfulFunctor Id F _ _ _ _) _ fun _ _ _ _ => rfl\n#align functor.comp.functor_id_comp Functor.Comp.functor_id_comp\n\nend Comp\n\nnamespace Comp\n\nopen Function hiding comp\n\nopen Functor\n\nvariable {F : Type u \u2192 Type w} {G : Type v \u2192 Type u}\n\nvariable [Applicative F] [Applicative G]\n\n/-- The `<*>` operation for the composition of applicative functors. -/\nprotected def seq {\u03b1 \u03b2 : Type v} : Comp F G (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 Comp F G \u03b1) \u2192 Comp F G \u03b2\n  | Comp.mk f, g => match g () with\n    | Comp.mk x => Comp.mk <| (\u00b7 <*> \u00b7) <$> f <*> x\n#align functor.comp.seq Functor.Comp.seq\u2093\n-- `\u2093` because the type of `Seq.seq` doesn't match `has_seq.seq`\n\ninstance : Pure (Comp F G) :=\n  \u27e8fun x => Comp.mk <| pure <| pure x\u27e9\n\ninstance : Seq (Comp F G) :=\n  \u27e8fun f x => Comp.seq f x\u27e9\n\n@[simp]\nprotected theorem run_pure {\u03b1 : Type v} : \u2200 x : \u03b1, (pure x : Comp F G \u03b1).run = pure (pure x)\n  | _ => rfl\n#align functor.comp.run_pure Functor.Comp.run_pure\n\n@[simp]\nprotected theorem run_seq {\u03b1 \u03b2 : Type v} (f : Comp F G (\u03b1 \u2192 \u03b2)) (x : Comp F G \u03b1) :\n    (f <*> x).run = (\u00b7 <*> \u00b7) <$> f.run <*> x.run :=\n  rfl\n#align functor.comp.run_seq Functor.Comp.run_seq\n\ninstance : Applicative (Comp F G) :=\n  { instPureComp with map := @Comp.map F G _ _, seq := @Comp.seq F G _ _ }\n\nend Comp\n\nvariable {F : Type u \u2192 Type u} [Functor F]\n\n/-- If we consider `x : F \u03b1` to, in some sense, contain values of type `\u03b1`,\npredicate `Liftp p x` holds iff every value contained by `x` satisfies `p`. -/\ndef Liftp {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (x : F \u03b1) : Prop :=\n  \u2203 u : F (Subtype p), Subtype.val <$> u = x\n#align functor.liftp Functor.Liftp\n\n/-- If we consider `x : F \u03b1` to, in some sense, contain values of type `\u03b1`, then\n`Liftr r x y` relates `x` and `y` iff (1) `x` and `y` have the same shape and\n(2) we can pair values `a` from `x` and `b` from `y` so that `r a b` holds. -/\ndef Liftr {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (x y : F \u03b1) : Prop :=\n  \u2203 u : F { p : \u03b1 \u00d7 \u03b1 // r p.fst p.snd },\n    (fun t : { p : \u03b1 \u00d7 \u03b1 // r p.fst p.snd } => t.val.fst) <$> u = x \u2227\n      (fun t : { p : \u03b1 \u00d7 \u03b1 // r p.fst p.snd } => t.val.snd) <$> u = y\n#align functor.liftr Functor.Liftr\n\n/-- If we consider `x : F \u03b1` to, in some sense, contain values of type `\u03b1`, then\n`supp x` is the set of values of type `\u03b1` that `x` contains. -/\ndef supp {\u03b1 : Type u} (x : F \u03b1) : Set \u03b1 :=\n  { y : \u03b1 | \u2200 \u2983p\u2984, Liftp p x \u2192 p y }\n#align functor.supp Functor.supp\n\ntheorem of_mem_supp {\u03b1 : Type u} {x : F \u03b1} {p : \u03b1 \u2192 Prop} (h : Liftp p x) : \u2200 y \u2208 supp x, p y :=\n  fun _ hy => hy h\n#align functor.of_mem_supp Functor.of_mem_supp\n\n/-- If `f` is a functor, if `fb : f \u03b2` and `a : \u03b1`, then `mapConstRev fb a` is the result of\n  applying `f.map` to the constant function `\u03b2 \u2192 \u03b1` sending everything to `a`, and then\n  evaluating at `fb`. In other words it's `const a <$> fb`. -/\n@[reducible] def mapConstRev {f : Type u \u2192 Type v} [Functor f] {\u03b1 \u03b2 : Type u} :\n    f \u03b2 \u2192 \u03b1 \u2192 f \u03b1 :=\n  fun a b => Functor.mapConst b a\n#align functor.map_const_rev Functor.mapConstRev\n/-- If `f` is a functor, if `fb : f \u03b2` and `a : \u03b1`, then `mapConstRev fb a` is the result of\n  applying `f.map` to the constant function `\u03b2 \u2192 \u03b1` sending everything to `a`, and then\n  evaluating at `fb`. In other words it's `const a <$> fb`. -/\ninfix:100 \" $> \" => Functor.mapConstRev\n\nend Functor\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Control/Functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.39229958421444383}}
{"text": "namespace list\n\nvariables {\u03b1 \u03b2 \u03b3 : Type}\n\ndef map_witness : \u2200 (l : list \u03b1), (\u03a0 x \u2208 l, \u03b2) \u2192 list \u03b2\n| [] _ := []\n| (x :: xs) f\n  := f x (mem_cons_self x xs)\n  :: map_witness xs (\u03bb x' mem, f x' (mem_cons_of_mem x mem))\n\nlemma map_witness_mem :\n  \u2200 {l : list \u03b1} {x : \u03b1} (f : \u03a0 x \u2208 l, \u03b2) (mem : x \u2208 l)\n  , f x mem \u2208 map_witness l f\n| [] x f mem := not_mem_nil x mem\n| (x :: xs) x' f mem :=\n  match mem with\n  | or.inl here := by simp [map_witness, here]\n  | or.inr there := begin\n      simp [map_witness],\n      from or.inr (map_witness_mem _ there)\n    end\n  end\n\nlemma map_witness_to_map (f : \u03b1 \u2192 \u03b2) :\n  \u2200 (l : list \u03b1), map_witness l (\u03bb x _, f x) = map f l\n| [] := rfl\n| (x :: xs) := by { simp [map_witness], from map_witness_to_map xs }\n\n@[simp]\nlemma map_witness_id (l : list \u03b1) : map_witness l (\u03bb x _, x) = l := begin\n  have h : (\u03bb (x : \u03b1), x) = id := rfl,\n  rw [map_witness_to_map _ l, h],\n  simp\nend\n\n@[simp]\nlemma map_witness_map :\n  \u2200 (l : list \u03b1) (f : (\u03a0 x \u2208 l, \u03b2)) (g : \u03b2 \u2192 \u03b3)\n  , map g (map_witness l f) = map_witness l (\u03bb a mem, g (f a mem))\n| [] _ _ := rfl\n| (x :: xs) f g := by { simp [map_witness, map], from map_witness_map xs _ g }\n\nlemma map_witness_length :\n  \u2200 (l : list \u03b1) (f : (\u03a0 x \u2208 l, \u03b2))\n  , length (map_witness l f) = length l\n| [] _ := rfl\n| (x :: xs) f := by { unfold map_witness length, rw map_witness_length xs }\n\nend list\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/list/witness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3922995802618359}}
{"text": "import ..time.time\nimport tactic.ext\n\nuniverses u\n\nvariables   {tf : time_frame} (ts : time_space tf) (V : Type u) \n  [inhabited V] /-Questionable whether to use inhabited.default as a default value-/\n/-\nstructure timestamped :=\n  (timestamp : time ts)\n\n@[class]\nstructure has_timestamp (V : Type u) :=\n  (get_timestamp : V \u2192 time ts)\n\ninstance : has_timestamp ts (timestamped ts) := \u27e8\u03bbts, ts.timestamp\u27e9\n-/\nstructure timestamped :=\n(timestamp : time ts)\n(value : V)\n\n@[simp]\nnoncomputable def mk_default_timestamped : timestamped ts V := \n  \u27e8inhabited.default (time ts), inhabited.default V\u27e9\n\nnoncomputable instance : inhabited (timestamped ts V) := \u27e8mk_default_timestamped ts V\u27e9\n\n\n@[simp]\nlemma nathelper :  \u2200 (a b : \u2115), a + b < a \u2192 false := \nbegin\n  intros a b c,\n  cases b,\n  suffices : \u00aca < a, from by contradiction, exact irrefl _,\n  let h1 : 0 < b.succ := by simp *,\n  let h2 : a + 0 < a + b.succ := by simp *,\n  let h3 : a + 0 < a := by exact lt_trans h2 c,\n  suffices : \u00aca < a, from by contradiction, exact irrefl _,\n\nend\n\n\n@[simp]\nlemma nathelper2 :  \u2200 (a b : \u2115), b + a < a \u2192 false := \nbegin\n  intros a b c,\n  rw \u2190(nat.add_comm a b) at c,\n  exact nathelper _ _ c,\nend\n/-\n\n    let ht :  i_val + 1 = i_val.succ := by repeat { apply nat.add_one _ },\n          rw \u2190ht at i_property,\n          let : false := by exact eznat2 _ _ i_property,\n    contradiction,\n-/\ninstance timelt : has_lt (time ts) := \u27e8\n  \u03bbt1 t2, t1.coord < t2.coord\n\u27e9\ninstance timele : has_le (time ts) := \u27e8 \n  \u03bbt1 t2, t1.coord \u2264 t2.coord\n\u27e9\ninstance durationlt : has_lt (duration ts) := \u27e8\n  \u03bbt1 t2, t1.coord < t2.coord\n\u27e9\ninstance durationle : has_le (duration ts) := \u27e8 \n  \u03bbt1 t2, t1.coord \u2264 t2.coord\n\u27e9\n\ninstance [has_le (time ts)] [has_lt (time ts)]: preorder (time ts) := \u27e8 \n  --has_le.le, has_lt.lt, \n  \u03bbt1 t2, t1.coord \u2264 t2.coord,\n  \u03bbt1 t2, t1.coord < t2.coord,\n  begin\n    intros,\n    simp *,\n  end,\n  begin\n    simp *,\n    intros a b c d e,\n    transitivity,\n    apply d,\n    apply e,\n    --simp\n  end,\n  begin\n    simp *,\n    intros a b,\n    split,\n    assume h,\n    split,\n    exact (le_not_le_of_lt h).1,\n    apply h,\n    assume h,\n    exact h.2,\n  end\n\u27e9\n\nnoncomputable instance eqd {t1 t2 : time ts} : decidable (t1 = t2) := \n  if eqc:t1.coord=t2.coord then\n    begin\n      --cases t1, cases t2, cases t1, cases t2,\n      unfold time.coord at eqc,\n      cases t1,cases t2,\n      let h : \u2200i, t1.coords i = t2.coords i := begin\n        intros,\n        cases i,\n        cases i_val,\n        ext,\n        exact eqc,\n        let ht :  i_val + 1 = i_val.succ := by repeat { apply nat.add_one _ },rw \u2190ht at i_property,\n        let : false := by exact nathelper2 _ _ i_property,\n        contradiction,\n      end,\n      let h1 : t1 = t2 := begin ext, \n        let h := h x,\n        simp [h],\n      end,\n      simp [h1],\n      exact decidable.is_true true.intro,\n    end\n  else \n    begin\n      --cases t1, cases t2, cases t1, cases t2,\n      unfold time.coord at eqc,\n      cases t1,cases t2,\n      let h : \u2200i, \u00act1.coords i = t2.coords i := begin\n        intros,\n        cases i,\n        cases i_val,\n        cases (t1.coords \u27e80, i_property\u27e9),\n        cases (t2.coords \u27e80, i_property\u27e9),\n        simp [eqc],\n\n        --ext,\n        let ht :  i_val + 1 = i_val.succ := by repeat { apply nat.add_one _ },rw \u2190ht at i_property,\n        let : false := by exact nathelper2 _ _ i_property,\n        contradiction,\n      end,\n      let h1 : \u00act1 = t2 := begin assume teq,\n        let : (({to_point := t1} : time _).to_point.coords 0).coord = \n              (({to_point := t2} : time _).to_point.coords 0).coord :=\n          by rw \u2190teq,\n        contradiction\n      end,\n      --simp [h1],\n      exact decidable.is_false (by simp [h1]),\n    end\n\n--instance ltd {t1 t2 : time ts} : decidable (t1 < t2) := sorry\n--instance leqd {t1 t2 : time ts} : decidable (t1 \u2264 t2) := sorry\n\nabbreviation time_series := time ts \u2192 V\n\nabbreviation time_series.Icc (min_t max_t : time ts) := \n  set.Icc min_t max_t \u2192 V \n\nabbreviation time_series.Ici (min_t : time ts) := \n  set.Ici min_t \u2192 V \n\ndef time_series.mk_empty {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] \n  : time_series ts V := \u03bbi, inhabited.default V \n\nnoncomputable def time_series.update {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] : \n  time_series ts V \u2192 time ts \u2192 V \u2192 time_series ts V \n| ser t_ val_ := --\u03bbt, if t = t_ then val_ else ser t\n  function.update ser t_ val_\n\ndef time_series.sample {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] \n  : time_series ts V \u2192 time ts \u2192 V := \n  \u03bbser tm , ser tm\n\ndef time_series.Icc.sample \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  : time_series.Icc ts V min_t max_t \u2192 set.Icc min_t max_t \u2192 V := \n  \u03bbser tm , ser tm\n\ndef time_series.Ici.sample \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  : time_series.Ici ts V min_t \u2192 set.Ici min_t \u2192 V := \n  \u03bbser tm , ser tm\n\nabbreviation discrete_series {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V] :=\n  list (timestamped ts V)\n\ndef discrete_series.mk_empty {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] : discrete_series ts V := []\n\n#check list.range\n#check set.mem\n\ndef discrete_series.domain {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] :\n  discrete_series ts V \u2192 list (time ts) := \n  \u03bbser, list.map (\u03bbtsv : timestamped ts V, tsv.timestamp) ser\n\ndef discrete_series.update {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] :\n  discrete_series ts V \u2192 timestamped ts V \u2192 discrete_series ts V\n--| [] ts_ val_ := [(ts_, val_)]\n--| (h::t) ts_ val_ := (h::t ++ [(ts_, val_)] :  list (timestamped ts V))\n| ser tsv := ser.cons tsv\n\n@[simp, reducible]\nnoncomputable def discrete_series.latest_helper {tf : time_frame} {ts : time_space tf} {V : Type u}\n  [inhabited V] :\n  discrete_series ts V \u2192 (timestamped ts V) \u2192 (timestamped ts V)\n| (h::t) v := if h.timestamp > v.timestamp then discrete_series.latest_helper t h else discrete_series.latest_helper t v\n| [] v := v\n\n@[simp, reducible]\nnoncomputable def discrete_series.latest {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] :\n  discrete_series ts V \u2192 (timestamped ts V)\n| (h::t) := discrete_series.latest_helper t h\n| [] :=  \u27e8inhabited.default (time ts), inhabited.default V\u27e9\n\nabbreviation discrete_series.Icc {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V]\n  (min_t max_t : time ts) :=\n  list (set.Icc min_t max_t \u00d7 V)\n\nabbreviation discrete_series.Ici {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V]\n  (min_t : time ts) :=\n  list (set.Ici min_t \u00d7 V)\n\nnoncomputable def discrete_series.sample  {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] \n  : discrete_series ts V \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp = t_ then h.value else discrete_series.sample t t_ \n\nnoncomputable def discrete_series.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  (v : time ts) : discrete_series ts V \u2192 option (timestamped ts V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp < h.timestamp \u2227 h.timestamp \u2264 v \n  then discrete_series.sample_floor_helper t (some h) \n  else discrete_series.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp \u2264 v \n  then discrete_series.sample_floor_helper t (some h) \n  else discrete_series.sample_floor_helper t none\n\nnoncomputable def discrete_series.sample_floor {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  : discrete_series ts V \u2192 time ts \u2192 V := \n  \u03bbser t, discrete_series.sample_floor_helper t ser none\n\nnoncomputable def discrete_series.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  (v : time ts) : discrete_series ts V \u2192 option (timestamped ts V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp < t_.timestamp \u2227 v \u2264 h.timestamp\n  then discrete_series.sample_ceil_helper t (some h) \n  else discrete_series.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v \u2264 h.timestamp \n  then discrete_series.sample_ceil_helper t (some h) \n  else discrete_series.sample_ceil_helper t none\n\n\nnoncomputable def discrete_series.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  : discrete_series ts V \u2192 time ts \u2192 V := \n  \u03bbser t, discrete_series.sample_ceil_helper t ser none\n\nnoncomputable def discrete_series.Icc.sample {min_t max_t : time ts}\n  : discrete_series.Icc ts V min_t max_t \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp.1 = t_ then h.value else discrete_series.Icc.sample t t_ \n\n\nnoncomputable def discrete_series.Icc.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  (v : set.Icc min_t max_t) : discrete_series.Icc ts V min_t max_t \u2192 option (set.Icc min_t max_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp.val < h.timestamp \u2227 h.timestamp.val \u2264 v \n  then discrete_series.Icc.sample_floor_helper t (some h) \n  else discrete_series.Icc.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp.val \u2264 v \n  then discrete_series.Icc.sample_floor_helper t (some h) \n  else discrete_series.Icc.sample_floor_helper t none\n\n\ndef discrete_series.Icc.sample_floor \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  : discrete_series.Icc ts V min_t max_t \u2192 set.Icc min_t max_t \u2192 V := \n  \u03bbser t, discrete_series.Icc.sample_floor_helper t ser none\n\n\ndef discrete_series.Icc.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  (v : set.Icc min_t max_t) : discrete_series.Icc ts V min_t max_t \u2192 option (set.Icc min_t max_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp.val < t_.timestamp.val \u2227 v.val \u2264 h.timestamp.val\n  then discrete_series.Icc.sample_ceil_helper t (some h) \n  else discrete_series.Icc.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v.val \u2264 h.timestamp.val\n  then discrete_series.Icc.sample_ceil_helper t (some h) \n  else discrete_series.Icc.sample_ceil_helper t none\n\n\ndef discrete_series.Icc.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  : discrete_series.Icc ts V min_t max_t \u2192 set.Icc min_t max_t \u2192 V := \n  \u03bbser t, discrete_series.Icc.sample_ceil_helper t ser none\n\n\ndef discrete_series.Ici.sample {min_t : time ts}\n  : discrete_series.Ici ts V min_t \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp.1 = t_ then h.value else discrete_series.Ici.sample t t_ \n\n\ndef discrete_series.Ici.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  (v : set.Ici min_t) : discrete_series.Ici ts V min_t \u2192 option (set.Ici min_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp.val < h.timestamp \u2227 h.timestamp.val \u2264 v \n  then discrete_series.Ici.sample_floor_helper t (some h) \n  else discrete_series.Ici.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp.val \u2264 v \n  then discrete_series.Ici.sample_floor_helper t (some h) \n  else discrete_series.Ici.sample_floor_helper t none\n\n\ndef discrete_series.Ici.sample_floor \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  : discrete_series.Ici ts V min_t \u2192 set.Ici min_t \u2192 V := \n  \u03bbser t, discrete_series.Ici.sample_floor_helper t ser none\n\n\ndef discrete_series.Ici.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  (v : set.Ici min_t) : discrete_series.Ici ts V min_t \u2192 option (set.Ici min_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp.val < t_.timestamp.val \u2227 v.val \u2264 h.timestamp.val\n  then discrete_series.Ici.sample_ceil_helper t (some h) \n  else discrete_series.Ici.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v.val \u2264 h.timestamp.val\n  then discrete_series.Ici.sample_ceil_helper t (some h) \n  else discrete_series.Ici.sample_ceil_helper t none\n\n\ndef discrete_series.Ici.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  : discrete_series.Ici ts V min_t \u2192 set.Ici min_t \u2192 V := \n  \u03bbser t, discrete_series.Ici.sample_ceil_helper t ser none\n\n/-\nabbreviation discrete_timestamped_series {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V] :=\n  list (timestamped ts V)\n\ndef discrete_timestamped_series.mk_empty {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] : discrete_timestamped_series ts V := []\n\ndef discrete_timestamped_series.update {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] :\n  discrete_timestamped_series ts V \u2192 time ts \u2192 V \u2192 discrete_timestamped_series ts V\n--| [] ts_ val_ := [(ts_, val_)]\n--| (h::t) ts_ val_ := (h::t ++ [(ts_, val_)] :  list (timestamped ts V))\n| ser ts_ val_ := ser.cons (ts_, val_)\n\ndef discrete_timestamped_series.latest_helper {tf : time_frame} {ts : time_space tf} {V : Type u}\n  [inhabited V] :\n  discrete_timestamped_series ts V \u2192 (timestamped ts V) \u2192 V\n| (h::t) v := if h.timestamp > v.timestamp then discrete_timestamped_series.latest_helper t h else discrete_timestamped_series.latest_helper t v\n| [] v := v.value\n\ndef discrete_timestamped_series.latest {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] :\n  discrete_timestamped_series ts V \u2192 V\n| (h::t) := discrete_timestamped_series.latest_helper t h\n| [] :=  inhabited.default V\n\nabbreviation discrete_timestamped_series.Icc {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V]\n  (min_t max_t : time ts) :=\n  list (set.Icc min_t max_t \u00d7 V)\n\nabbreviation discrete_timestamped_series.Ici {tf : time_frame} (ts : time_space tf) (V : Type u) [inhabited V]\n  (min_t : time ts) :=\n  list (set.Ici min_t \u00d7 V)\n\ndef discrete_timestamped_series.sample  {tf : time_frame} {ts : time_space tf} {V : Type u} \n  [inhabited V] \n  : discrete_timestamped_series ts V \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp = t_ then h.value else discrete_timestamped_series.sample t t_ \n\ndef discrete_timestamped_series.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  (v : time ts) : discrete_timestamped_series ts V \u2192 option (timestamped ts V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp < h.timestamp \u2227 h.timestamp \u2264 v \n  then discrete_timestamped_series.sample_floor_helper t (some h) \n  else discrete_timestamped_series.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp \u2264 v \n  then discrete_timestamped_series.sample_floor_helper t (some h) \n  else discrete_timestamped_series.sample_floor_helper t none\n\n\ndef discrete_timestamped_series.sample_floor {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  : discrete_timestamped_series ts V \u2192 time ts \u2192 V := \n  \u03bbser t, discrete_timestamped_series.sample_floor_helper t ser none\n\ndef discrete_timestamped_series.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  (v : time ts) : discrete_timestamped_series ts V \u2192 option (timestamped ts V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp < t_.timestamp \u2227 v \u2264 h.timestamp\n  then discrete_timestamped_series.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v \u2264 h.timestamp \n  then discrete_timestamped_series.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.sample_ceil_helper t none\n\n\ndef discrete_timestamped_series.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] \n  : discrete_timestamped_series ts V \u2192 time ts \u2192 V := \n  \u03bbser t, discrete_timestamped_series.sample_ceil_helper t ser none\n\ndef discrete_timestamped_series.Icc.sample {min_t max_t : time ts}\n  : discrete_timestamped_series.Icc ts V min_t max_t \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp.1 = t_ then h.value else discrete_timestamped_series.Icc.sample t t_ \n\n\ndef discrete_timestamped_series.Icc.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  (v : set.Icc min_t max_t) : discrete_timestamped_series.Icc ts V min_t max_t \u2192 option (set.Icc min_t max_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp.val < h.timestamp \u2227 h.timestamp.val \u2264 v \n  then discrete_timestamped_series.Icc.sample_floor_helper t (some h) \n  else discrete_timestamped_series.Icc.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp.val \u2264 v \n  then discrete_timestamped_series.Icc.sample_floor_helper t (some h) \n  else discrete_timestamped_series.Icc.sample_floor_helper t none\n\n\ndef discrete_timestamped_series.Icc.sample_floor \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  : discrete_timestamped_series.Icc ts V min_t max_t \u2192 set.Icc min_t max_t \u2192 V := \n  \u03bbser t, discrete_timestamped_series.Icc.sample_floor_helper t ser none\n\n\ndef discrete_timestamped_series.Icc.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  (v : set.Icc min_t max_t) : discrete_timestamped_series.Icc ts V min_t max_t \u2192 option (set.Icc min_t max_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp.val < t_.timestamp.val \u2227 v.val \u2264 h.timestamp.val\n  then discrete_timestamped_series.Icc.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.Icc.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v.val \u2264 h.timestamp.val\n  then discrete_timestamped_series.Icc.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.Icc.sample_ceil_helper t none\n\n\ndef discrete_timestamped_series.Icc.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t max_t : time ts}\n  : discrete_timestamped_series.Icc ts V min_t max_t \u2192 set.Icc min_t max_t \u2192 V := \n  \u03bbser t, discrete_timestamped_series.Icc.sample_ceil_helper t ser none\n\n\ndef discrete_timestamped_series.Ici.sample {min_t : time ts}\n  : discrete_timestamped_series.Ici ts V min_t \u2192 time ts \u2192 V\n| [] t_ := inhabited.default V\n| (h::t) t_ := if h.timestamp.1 = t_ then h.value else discrete_timestamped_series.Ici.sample t t_ \n\n\ndef discrete_timestamped_series.Ici.sample_floor_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  (v : set.Ici min_t) : discrete_timestamped_series.Ici ts V min_t \u2192 option (set.Ici min_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if t_.timestamp.val < h.timestamp \u2227 h.timestamp.val \u2264 v \n  then discrete_timestamped_series.Ici.sample_floor_helper t (some h) \n  else discrete_timestamped_series.Ici.sample_floor_helper t (some t_)\n| (h::t) (none) := \n  if h.timestamp.val \u2264 v \n  then discrete_timestamped_series.Ici.sample_floor_helper t (some h) \n  else discrete_timestamped_series.Ici.sample_floor_helper t none\n\n\ndef discrete_timestamped_series.Ici.sample_floor \n  {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  : discrete_timestamped_series.Ici ts V min_t \u2192 set.Ici min_t \u2192 V := \n  \u03bbser t, discrete_timestamped_series.Ici.sample_floor_helper t ser none\n\n\ndef discrete_timestamped_series.Ici.sample_ceil_helper {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  (v : set.Ici min_t) : discrete_timestamped_series.Ici ts V min_t \u2192 option (set.Ici min_t \u00d7 V) \u2192 V\n| [] (none) := inhabited.default V\n| [] (some t_) := t_.value\n| (h::t) (some t_) := \n  if h.timestamp.val < t_.timestamp.val \u2227 v.val \u2264 h.timestamp.val\n  then discrete_timestamped_series.Ici.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.Ici.sample_ceil_helper t (some t_)\n| (h::t) (none) := \n  if v.val \u2264 h.timestamp.val\n  then discrete_timestamped_series.Ici.sample_ceil_helper t (some h) \n  else discrete_timestamped_series.Ici.sample_ceil_helper t none\n\n\ndef discrete_timestamped_series.Ici.sample_ceil {tf : time_frame} {ts : time_space tf} {V : Type u} [inhabited V] {min_t : time ts}\n  : discrete_timestamped_series.Ici ts V min_t \u2192 set.Ici min_t \u2192 V := \n  \u03bbser t, discrete_timestamped_series.Ici.sample_ceil_helper t ser none\n-/", "meta": {"author": "kevinsullivan", "repo": "phys", "sha": "ebc2df3779d3605ff7a9b47eeda25c2a551e011f", "save_path": "github-repos/lean/kevinsullivan-phys", "path": "github-repos/lean/kevinsullivan-phys/phys-ebc2df3779d3605ff7a9b47eeda25c2a551e011f/time_series/definitions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.661922862511608, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.39229957235661994}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Patrick Massot\n-/\nimport topology.order\n\n/-!\n# Specific classes of maps between topological spaces\n\nThis file introduces the following properties of a map `f : X \u2192 Y` between topological spaces:\n\n* `is_open_map f` means the image of an open set under `f` is open.\n* `is_closed_map f` means the image of a closed set under `f` is closed.\n\n(Open and closed maps need not be continuous.)\n\n* `inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`.\n  These behave like embeddings except they need not be injective. Instead, points of `X` which\n  are identified by `f` are also indistinguishable in the topology on `X`.\n* `embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with\n  a subspace of `Y`.\n* `open_embedding f` means `f` is an embedding with open image, so it identifies `X` with an\n  open subspace of `Y`. Equivalently, `f` is an embedding and an open map.\n* `closed_embedding f` similarly means `f` is an embedding with closed image, so it identifies\n  `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map.\n\n* `quotient_map f` is the dual condition to `embedding f`: `f` is surjective and the topology\n  on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies\n  `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Open_and_closed_maps>\n* <https://en.wikipedia.org/wiki/Embedding#General_topology>\n* <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map>\n\n## Tags\n\nopen map, closed map, embedding, quotient map, identification map\n\n-/\n\nopen set filter\nopen_locale topological_space filter\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*}\n\nsection inducing\n\n/-- A function `f : \u03b1 \u2192 \u03b2` between topological spaces is inducing if the topology on `\u03b1` is induced\nby the topology on `\u03b2` through `f`, meaning that a set `s : set \u03b1` is open iff it is the preimage\nunder `f` of some open set `t : set \u03b2`. -/\nstructure inducing [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) : Prop :=\n(induced : t\u03b1 = t\u03b2.induced f)\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\nlemma inducing_id : inducing (@id \u03b1) :=\n\u27e8induced_id.symm\u27e9\n\nprotected lemma inducing.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : inducing g) (hf : inducing f) :\n  inducing (g \u2218 f) :=\n\u27e8by rw [hf.induced, hg.induced, induced_compose]\u27e9\n\nlemma inducing_of_inducing_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : inducing (g \u2218 f)) : inducing f :=\n\u27e8le_antisymm\n    (by rwa \u2190 continuous_iff_le_induced)\n    (by { rw [hgf.induced, \u2190 continuous_iff_le_induced], apply hg.comp continuous_induced_dom })\u27e9\n\nlemma inducing.nhds_eq_comap {f : \u03b1 \u2192 \u03b2} (hf : inducing f) :\n  \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd $ f a) :=\n(induced_iff_nhds_eq f).1 hf.induced\n\nlemma inducing.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_eq a\n\nlemma inducing.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) :\n  (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_of_mem h\n\nlemma inducing.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : inducing g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nby rw [tendsto, tendsto, hg.induced, nhds_induced, \u2190 map_le_iff_le_comap, filter.map_map]\n\nlemma inducing.continuous_at_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) {x : \u03b1} :\n  continuous_at f x \u2194 continuous_at (g \u2218 f) x :=\nby simp_rw [continuous_at, inducing.tendsto_nhds_iff hg]\n\nlemma inducing.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\nby simp_rw [continuous_iff_continuous_at, hg.continuous_at_iff]\n\nlemma inducing.continuous_at_iff' {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : inducing f) {x : \u03b1}\n  (h : range f \u2208 \ud835\udcdd (f x)) : continuous_at (g \u2218 f) x \u2194 continuous_at g (f x) :=\nby { simp_rw [continuous_at, filter.tendsto, \u2190 hf.map_nhds_of_mem _ h, filter.map_map] }\n\nlemma inducing.continuous {f : \u03b1 \u2192 \u03b2} (hf : inducing f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nlemma inducing.closure_eq_preimage_closure_image {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (s : set \u03b1) :\n  closure s = f \u207b\u00b9' closure (f '' s) :=\nby { ext x, rw [set.mem_preimage, \u2190 closure_induced, hf.induced] }\n\nlemma inducing.is_closed_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_closed s \u2194 \u2203 t, is_closed t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_closed_induced_iff]\n\nlemma inducing.is_open_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_open s \u2194 \u2203 t, is_open t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_open_induced_iff]\n\nend inducing\n\nsection embedding\n\n/-- A function between topological spaces is an embedding if it is injective,\n  and for all `s : set \u03b1`, `s` is open iff it is the preimage of an open set. -/\nstructure embedding [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2)\n  extends inducing f : Prop :=\n(inj : function.injective f)\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\nlemma embedding.mk' (f : \u03b1 \u2192 \u03b2) (inj : function.injective f)\n  (induced : \u2200a, comap f (\ud835\udcdd (f a)) = \ud835\udcdd a) : embedding f :=\n\u27e8\u27e8(induced_iff_nhds_eq f).2 (\u03bb a, (induced a).symm)\u27e9, inj\u27e9\n\nlemma embedding_id : embedding (@id \u03b1) :=\n\u27e8inducing_id, assume a\u2081 a\u2082 h, h\u27e9\n\nlemma embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : embedding g) (hf : embedding f) :\n  embedding (g \u2218 f) :=\n{ inj:= assume a\u2081 a\u2082 h, hf.inj $ hg.inj h,\n  ..hg.to_inducing.comp hf.to_inducing }\n\nlemma embedding_of_embedding_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : embedding (g \u2218 f)) : embedding f :=\n{ induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced,\n  inj := assume a\u2081 a\u2082 h, hgf.inj $ by simp [h, (\u2218)] }\n\nprotected lemma function.left_inverse.embedding {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n  (h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :\n  embedding g :=\nembedding_of_embedding_compose hg hf $ h.comp_eq_id.symm \u25b8 embedding_id\n\nlemma embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : embedding f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.1.map_nhds_eq a\n\nlemma embedding.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2}\n  (hf : embedding f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) : (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.1.map_nhds_of_mem a h\n\nlemma embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : embedding g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nhg.to_inducing.tendsto_nhds_iff\n\nlemma embedding.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : embedding g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\ninducing.continuous_iff hg.1\n\nlemma embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : embedding f) : continuous f :=\ninducing.continuous hf.1\n\nlemma embedding.closure_eq_preimage_closure_image {e : \u03b1 \u2192 \u03b2} (he : embedding e) (s : set \u03b1) :\n  closure s = e \u207b\u00b9' closure (e '' s) :=\nhe.1.closure_eq_preimage_closure_image s\n\nend embedding\n\n/-- A function between topological spaces is a quotient map if it is surjective,\n  and for all `s : set \u03b2`, `s` is open iff its preimage is an open set. -/\ndef quotient_map {\u03b1 : Type*} {\u03b2 : Type*} [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2]\n  (f : \u03b1 \u2192 \u03b2) : Prop :=\nfunction.surjective f \u2227 t\u03b2 = t\u03b1.coinduced f\n\nlemma quotient_map_iff {\u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  quotient_map f \u2194 function.surjective f \u2227 \u2200 s : set \u03b2, is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nand_congr iff.rfl topological_space_eq_iff\n\nnamespace quotient_map\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\nprotected lemma id : quotient_map (@id \u03b1) :=\n\u27e8assume a, \u27e8a, rfl\u27e9, coinduced_id.symm\u27e9\n\nprotected lemma comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : quotient_map g) (hf : quotient_map f) :\n  quotient_map (g \u2218 f) :=\n\u27e8hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]\u27e9\n\nprotected lemma of_quotient_map_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n  (hf : continuous f) (hg : continuous g)\n  (hgf : quotient_map (g \u2218 f)) : quotient_map g :=\n\u27e8assume b, let \u27e8a, h\u27e9 := hgf.left b in \u27e8f a, h\u27e9,\n  le_antisymm\n    (by rw [hgf.right, \u2190 continuous_iff_coinduced_le];\n        apply continuous_coinduced_rng.comp hf)\n    (by rwa \u2190 continuous_iff_coinduced_le)\u27e9\n\nprotected lemma continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : quotient_map f) :\n  continuous g \u2194 continuous (g \u2218 f) :=\nby rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose]\n\nprotected lemma continuous {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nprotected lemma surjective {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : function.surjective f := hf.1\n\nprotected lemma is_open_preimage {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) {s : set \u03b2} :\n  is_open (f \u207b\u00b9' s) \u2194 is_open s :=\n((quotient_map_iff.1 hf).2 s).symm\n\nend quotient_map\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be an *open map*, if the image of any open `U : set \u03b1`\nis open in `\u03b2`. -/\ndef is_open_map [topological_space \u03b1] [topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n\u2200 U : set \u03b1, is_open U \u2192 is_open (f '' U)\n\nnamespace is_open_map\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2}\nopen function\n\nprotected lemma id : is_open_map (@id \u03b1) := assume s hs, by rwa [image_id]\n\nprotected lemma comp\n  {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_open_map g) (hf : is_open_map f) : is_open_map (g \u2218 f) :=\nby intros s hs; rw [image_comp]; exact hg _ (hf _ hs)\n\nlemma is_open_range (hf : is_open_map f) : is_open (range f) :=\nby { rw \u2190 image_univ, exact hf _ is_open_univ }\n\nlemma image_mem_nhds (hf : is_open_map f) {x : \u03b1} {s : set \u03b1} (hx : s \u2208 \ud835\udcdd x) :\n  f '' s \u2208 \ud835\udcdd (f x) :=\nlet \u27e8t, hts, ht, hxt\u27e9 := mem_nhds_iff.1 hx in\nmem_of_superset (is_open.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)\n\nlemma image_interior_subset (hf : is_open_map f) (s : set \u03b1) :\n  f '' interior s \u2286 interior (f '' s) :=\ninterior_maximal (image_subset _ interior_subset) (hf _ is_open_interior)\n\nlemma nhds_le (hf : is_open_map f) (a : \u03b1) : \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\nle_map $ \u03bb s, hf.image_mem_nhds\n\nlemma of_nhds_le (hf : \u2200 a, \ud835\udcdd (f a) \u2264 map f (\ud835\udcdd a)) : is_open_map f :=\n\u03bb s hs, is_open_iff_mem_nhds.2 $ \u03bb b \u27e8a, has, hab\u27e9,\n  hab \u25b8 hf _ (image_mem_map $ is_open.mem_nhds hs has)\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_open_map f :=\nbegin\n  assume s hs,\n  rw [image_eq_preimage_of_inverse r_inv l_inv],\n  exact hs.preimage h\nend\n\nlemma to_quotient_map {f : \u03b1 \u2192 \u03b2}\n  (open_map : is_open_map f) (cont : continuous f) (surj : function.surjective f) :\n  quotient_map f :=\n\u27e8 surj,\n  begin\n    ext s,\n    show is_open s \u2194 is_open (f \u207b\u00b9' s),\n    split,\n    { exact continuous_def.1 cont s },\n    { assume h,\n      rw \u2190 surj.image_preimage s,\n      exact open_map _ h }\n  end\u27e9\n\nlemma interior_preimage_subset_preimage_interior {s : set \u03b2} (hf : is_open_map f) :\n  interior (f\u207b\u00b9' s) \u2286 f\u207b\u00b9' (interior s) :=\nbegin\n  rw \u2190 set.image_subset_iff,\n  refine interior_maximal _ (hf _ is_open_interior),\n  rw set.image_subset_iff,\n  exact interior_subset,\nend\n\nlemma preimage_interior_eq_interior_preimage {s : set \u03b2}\n  (hf\u2081 : continuous f) (hf\u2082 : is_open_map f) :\n  f\u207b\u00b9' (interior s) = interior (f\u207b\u00b9' s) :=\nsubset.antisymm\n  (preimage_interior_subset_interior_preimage hf\u2081)\n  (interior_preimage_subset_preimage_interior hf\u2082)\n\nend is_open_map\n\nlemma is_open_map_iff_nhds_le [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_open_map f \u2194 \u2200(a:\u03b1), \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\n\u27e8\u03bb hf, hf.nhds_le, is_open_map.of_nhds_le\u27e9\n\nlemma is_open_map_iff_interior [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_open_map f \u2194 \u2200 s, f '' (interior s) \u2286 interior (f '' s) :=\n\u27e8is_open_map.image_interior_subset, \u03bb hs u hu, subset_interior_iff_open.mp $\n  calc f '' u = f '' (interior u) : by rw hu.interior_eq\n          ... \u2286 interior (f '' u) : hs u\u27e9\n\nlemma inducing.is_open_map [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n  (hi : inducing f) (ho : is_open (range f)) :\n  is_open_map f :=\nis_open_map.of_nhds_le $ \u03bb x, (hi.map_nhds_of_mem _ $ is_open.mem_nhds ho $ mem_range_self _).ge\n\nsection is_closed_map\nvariables [topological_space \u03b1] [topological_space \u03b2]\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be a *closed map*, if the image of any closed `U : set \u03b1`\nis closed in `\u03b2`. -/\ndef is_closed_map (f : \u03b1 \u2192 \u03b2) := \u2200 U : set \u03b1, is_closed U \u2192 is_closed (f '' U)\n\nend is_closed_map\n\nnamespace is_closed_map\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\nopen function\n\nprotected lemma id : is_closed_map (@id \u03b1) := assume s hs, by rwa image_id\n\nprotected lemma comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_closed_map g) (hf : is_closed_map f) :\n  is_closed_map (g \u2218 f) :=\nby { intros s hs, rw image_comp, exact hg _ (hf _ hs) }\n\nlemma closure_image_subset {f : \u03b1 \u2192 \u03b2} (hf : is_closed_map f) (s : set \u03b1) :\n  closure (f '' s) \u2286 f '' closure s :=\nclosure_minimal (image_subset _ subset_closure) (hf _ is_closed_closure)\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_closed_map f :=\nassume s hs,\nhave f' \u207b\u00b9' s = f '' s, by ext x; simp [mem_image_iff_of_inverse r_inv l_inv],\nthis \u25b8 hs.preimage h\n\nlemma of_nonempty {f : \u03b1 \u2192 \u03b2} (h : \u2200 s, is_closed s \u2192 s.nonempty \u2192 is_closed (f '' s)) :\n  is_closed_map f :=\nbegin\n  intros s hs, cases eq_empty_or_nonempty s with h2s h2s,\n  { simp_rw [h2s, image_empty, is_closed_empty] },\n  { exact h s hs h2s }\nend\n\nlemma closed_range {f : \u03b1 \u2192 \u03b2} (hf : is_closed_map f) : is_closed (range f) :=\n@image_univ _ _ f \u25b8 hf _ is_closed_univ\n\nend is_closed_map\n\nlemma inducing.is_closed_map [topological_space \u03b1] [topological_space \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (h : is_closed (range f)) : is_closed_map f :=\nbegin\n  intros s hs,\n  rcases hf.is_closed_iff.1 hs with \u27e8t, ht, rfl\u27e9,\n  rw image_preimage_eq_inter_range,\n  exact is_closed.inter ht h\nend\n\nlemma is_closed_map_iff_closure_image [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_closed_map f \u2194 \u2200 s, closure (f '' s) \u2286 f '' closure s :=\n\u27e8is_closed_map.closure_image_subset, \u03bb hs c hc, is_closed_of_closure_subset $\n  calc closure (f '' c) \u2286 f '' (closure c) : hs c\n                    ... = f '' c : by rw hc.closure_eq\u27e9\n\nsection open_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- An open embedding is an embedding with open image. -/\nstructure open_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(open_range : is_open $ range f)\n\nlemma open_embedding.is_open_map {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : is_open_map f :=\nhf.to_embedding.to_inducing.is_open_map hf.open_range\n\nlemma open_embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) (a : \u03b1) :\n  map f (\ud835\udcdd a) = \ud835\udcdd (f a) :=\nhf.to_embedding.map_nhds_of_mem _ $ is_open.mem_nhds hf.open_range $ mem_range_self _\n\nlemma open_embedding.open_iff_image_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b1} : is_open s \u2194 is_open (f '' s) :=\n\u27e8hf.is_open_map s,\n \u03bb h, begin\n   convert \u2190 h.preimage hf.to_embedding.continuous,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma open_embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : open_embedding g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nhg.to_embedding.tendsto_nhds_iff\n\nlemma open_embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma open_embedding.open_iff_preimage_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.open_iff_image_open.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma open_embedding_of_embedding_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f)\n  (h\u2082 : is_open_map f) : open_embedding f :=\n\u27e8h\u2081, h\u2082.is_open_range\u27e9\n\nlemma open_embedding_of_continuous_injective_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_open_map f) : open_embedding f :=\nbegin\n  refine open_embedding_of_embedding_open \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s,\n  change is_open _ \u2192 is_open _,\n  rw is_open_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma open_embedding_id : open_embedding (@id \u03b1) :=\n\u27e8embedding_id, by convert is_open_univ; apply range_id\u27e9\n\nlemma open_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : open_embedding g) (hf : open_embedding f) : open_embedding (g \u2218 f) :=\n\u27e8hg.1.comp hf.1, show is_open (range (g \u2218 f)),\n by rw [range_comp, \u2190hg.open_iff_image_open]; exact hf.2\u27e9\n\nend open_embedding\n\nsection closed_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- A closed embedding is an embedding with closed image. -/\nstructure closed_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(closed_range : is_closed $ range f)\n\nvariables {f : \u03b1 \u2192 \u03b2}\n\nlemma closed_embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {g : \u03b9 \u2192 \u03b1} {a : filter \u03b9} {b : \u03b1} (hf : closed_embedding f) :\n  tendsto g a (\ud835\udcdd b) \u2194 tendsto (f \u2218 g) a (\ud835\udcdd (f b)) :=\nhf.to_embedding.tendsto_nhds_iff\n\nlemma closed_embedding.continuous (hf : closed_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma closed_embedding.is_closed_map (hf : closed_embedding f) : is_closed_map f :=\nhf.to_embedding.to_inducing.is_closed_map hf.closed_range\n\nlemma closed_embedding.closed_iff_image_closed (hf : closed_embedding f)\n  {s : set \u03b1} : is_closed s \u2194 is_closed (f '' s) :=\n\u27e8hf.is_closed_map s,\n \u03bb h, begin\n   convert \u2190continuous_iff_is_closed.mp hf.continuous _ h,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma closed_embedding.closed_iff_preimage_closed (hf : closed_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_closed s \u2194 is_closed (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.closed_iff_image_closed.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma closed_embedding_of_embedding_closed (h\u2081 : embedding f)\n  (h\u2082 : is_closed_map f) : closed_embedding f :=\n\u27e8h\u2081, by convert h\u2082 univ is_closed_univ; simp\u27e9\n\nlemma closed_embedding_of_continuous_injective_closed (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_closed_map f) : closed_embedding f :=\nbegin\n  refine closed_embedding_of_embedding_closed \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s',\n  change is_open _ \u2264 is_open _,\n  rw [\u2190is_closed_compl_iff, \u2190is_closed_compl_iff],\n  generalize : s'\u1d9c = s,\n  rw is_closed_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma closed_embedding_id : closed_embedding (@id \u03b1) :=\n\u27e8embedding_id, by convert is_closed_univ; apply range_id\u27e9\n\nlemma closed_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : closed_embedding g) (hf : closed_embedding f) : closed_embedding (g \u2218 f) :=\n\u27e8hg.to_embedding.comp hf.to_embedding, show is_closed (range (g \u2218 f)),\n by rw [range_comp, \u2190hg.closed_iff_image_closed]; exact hf.closed_range\u27e9\n\nend closed_embedding\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/topology/maps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.7025300636233416, "lm_q1q2_score": 0.39224149885305964}}
{"text": "import lib.lib order.bounded_order\n\nuniverse u\n\nopen_locale logic_symbol\n\nclass intuitionistic_logic {F : Sort*} [has_logic_symbol F] (P : F \u2192 Prop) :=\n(modus_ponens {p q : F} : P (p \u27f6 q) \u2192 P p \u2192 P q)\n(imply\u2081 {p q : F} : P (p \u27f6 q \u27f6 p))\n(imply\u2082 {p q r : F} : P ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r))\n(conj\u2081 {p q : F} : P (p \u2293 q \u27f6 p))\n(conj\u2082 {p q : F} : P (p \u2293 q \u27f6 q))\n(conj\u2083 {p q : F} : P (p \u27f6 q \u27f6 p \u2293 q))\n(disj\u2081 {p q : F} : P (p \u27f6 p \u2294 q))\n(disj\u2082 {p q : F} : P (q \u27f6 p \u2294 q))\n(disj\u2083 {p q r : F} : P ((p \u27f6 r) \u27f6 (q \u27f6 r) \u27f6 p \u2294 q \u27f6 r))\n(neg\u2081 {p q : F} : P ((p \u27f6 q) \u27f6 (p \u27f6 \u223cq) \u27f6 \u223cp))\n(neg\u2082 {p q : F} : P (p \u27f6 \u223cp \u27f6 q))\n(provable_top : P \u22a4)\n(bot_eq : (\u22a5 : F) = \u223c\u22a4)\n\nclass classical_logic {F : Sort*} [has_logic_symbol F] (P : set F) :=\n(modus_ponens {p q : F} : p \u27f6 q \u2208 P \u2192 p \u2208 P \u2192 q \u2208 P)\n(imply\u2081 {p q : F} : p \u27f6 q \u27f6 p \u2208 P)\n(imply\u2082 {p q r : F} : (p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r \u2208 P)\n(contraposition {p q : F} : (\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p \u2208 P)\n(provable_top : \u22a4 \u2208 P)\n(bot_eq : (\u22a5 : F) = \u223c\u22a4)\n(and_def {p q : F} : p \u2293 q = \u223c(p \u27f6 \u223cq))\n(or_def {p q : F} : p \u2294 q = \u223cp \u27f6 q)\n\nattribute [simp] classical_logic.imply\u2081 classical_logic.imply\u2082 classical_logic.contraposition\n  classical_logic.provable_top\n\nclass axiomatic_classical_logic' (F : Sort*) [has_logic_symbol F] extends has_turnstile F :=\n(classical {T : set F} : classical_logic ((\u22a2) T : F \u2192 Prop))\n(by_axiom {T : set F} {p : F} : p \u2208 T \u2192 T \u22a2 p)\n\nclass axiomatic_classical_logic (F : Sort*) [has_logic_symbol F] extends axiomatic_classical_logic' F :=\n(deduction' {T : set F} {p q : F} : insert p T \u22a2 q \u2192 T \u22a2 p \u27f6 q)\n(weakening {T : set F} {U : set F} {p : F} : T \u2286 U \u2192 T \u22a2 p \u2192 U \u22a2 p)\n\nnamespace classical_logic\n\nvariables {F : Type*} [has_logic_symbol F]\n  (P : set F) (T : set F) [CL : classical_logic P]\ninclude CL\n\n@[simp] lemma neg_top_eq : (\u223c\u22a4 : F) = \u22a5 := eq.symm (bot_eq P)\n\nvariables {P}\n\n@[simp] lemma not_top_eq_bot : (\u223c\u22a4 : F) = \u22a5 := eq.symm (classical_logic.bot_eq P)\n\nlocal infixl ` \u2a00 `:90 := modus_ponens\n\n@[simp] lemma refl (p : F) : p \u27f6 p \u2208 P :=\nbegin\n  have l\u2080 : (p \u27f6 (p \u27f6 p) \u27f6 p) \u27f6 (p \u27f6 p \u27f6 p) \u27f6 p \u27f6 p \u2208 P, simp,\n  have l\u2081 : p \u27f6 (p \u27f6 p) \u27f6 p \u2208 P, simp,\n  have l\u2082 : (p \u27f6 p \u27f6 p) \u27f6 p \u27f6 p \u2208 P, refine l\u2080 \u2a00 l\u2081,\n  have l\u2083 : p \u27f6 p \u27f6 p \u2208 P, simp,\n  simp[set.mem_def],\n  refine l\u2082 \u2a00 l\u2083\nend\n\nvariables {T}\n\n@[simp] lemma hyp_right {p : F} (h : p \u2208 P) (q) : q \u27f6 p \u2208 P :=\nby { have : p \u27f6 q \u27f6 p \u2208 P, simp, exact this \u2a00 h }\n\n@[simp] lemma T_hyp_eliminate {p} : \u22a4 \u27f6 p \u2208 P \u2194 p \u2208 P :=\n\u27e8\u03bb h, by { have : \u22a4 \u2208 P, simp, exact h \u2a00 this }, \u03bb h, by simp[h]\u27e9\n\nlemma modus_ponens_hyp {p q r : F} (hqr : p \u27f6 q \u27f6 r \u2208 P) (hq : p \u27f6 q \u2208 P) : p \u27f6 r \u2208 P :=\nby { have : (p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r \u2208 P, simp, exact this \u2a00 hqr \u2a00 hq }\n\nlocal infixl ` \u2a00\u2081 `:90 := modus_ponens_hyp\n\nlemma modus_ponens_hyp\u2082 {p q r s : F} (hqr : p \u27f6 q \u27f6 r \u27f6 s \u2208 P) (hq : p \u27f6 q \u27f6 r \u2208 P) : p \u27f6 q \u27f6 s \u2208 P :=\nby { have : p \u27f6 (q \u27f6 r \u27f6 s) \u27f6 (q \u27f6 r) \u27f6 q \u27f6 s \u2208 P, simp, exact this \u2a00\u2081 hqr \u2a00\u2081 hq }\n\nlocal infixl ` \u2a00\u2082 `:90 := modus_ponens_hyp\u2082\n\nlemma modus_ponens_hyp\u2083 {p q r s t : F} (hqr : p \u27f6 q \u27f6 r \u27f6 s \u27f6 t \u2208 P) (hq : p \u27f6 q \u27f6 r \u27f6 s \u2208 P) : p \u27f6 q \u27f6 r \u27f6 t \u2208 P :=\nby { have : p \u27f6 q \u27f6 (r \u27f6 s \u27f6 t) \u27f6 (r \u27f6 s) \u27f6 r \u27f6 t \u2208 P, simp, exact this \u2a00\u2082 hqr \u2a00\u2082 hq }\n\nlocal infixl ` \u2a00\u2083 `:90 := modus_ponens_hyp\u2083\n\nlemma impl_trans {p q r : F} : (p \u27f6 q \u2208 P) \u2192 (q \u27f6 r \u2208 P) \u2192 (p \u27f6 r \u2208 P) := \u03bb h\u2081 h\u2082,\nbegin\n  have l\u2081 : (p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 (p \u27f6 r) \u2208 P, simp,  \n  have l\u2082 : (p \u27f6 q \u27f6 r) \u2208 P, simp[h\u2082],\n  have l\u2083 : (p \u27f6 q) \u27f6 (p \u27f6 r) \u2208 P, from l\u2081 \u2a00 l\u2082,\n  exact l\u2083 \u2a00 h\u2081\nend\n\n@[simp] lemma imply\u2081' {p q r : F} : p \u27f6 q \u27f6 r \u27f6 p \u2208 P :=\nbegin\n  have lmm\u2081 : p \u27f6 q \u27f6 p \u27f6 r \u27f6 p \u2208 P, simp,\n  have lmm\u2082 : p \u27f6 q \u27f6 p \u2208 P, simp,\n  exact lmm\u2081 \u2a00\u2082 lmm\u2082\nend\n\n@[simp] lemma dne (p : F) : \u223c\u223cp \u27f6 p \u2208 P :=\nbegin\n  have lmm\u2081 : \u223c\u223cp \u27f6 (\u223c\u223c\u223c\u223cp \u27f6 \u223c\u223cp) \u27f6 \u223cp \u27f6 \u223c\u223c\u223cp \u2208 P, simp,\n  have lmm\u2082 : \u223c\u223cp \u27f6 \u223c\u223c\u223c\u223cp \u27f6 \u223c\u223cp \u2208 P, simp,\n  have lmm\u2083 : \u223c\u223cp \u27f6 (\u223cp \u27f6 \u223c\u223c\u223cp) \u27f6 \u223c\u223cp \u27f6 p \u2208 P, simp,  \n  have lmm\u2084 : \u223c\u223cp \u27f6 \u223cp \u27f6 \u223c\u223c\u223cp \u2208 P, from lmm\u2081 \u2a00\u2081 lmm\u2082,\n  have lmm\u2085 : \u223c\u223cp \u27f6 \u223c\u223cp \u27f6 p \u2208 P, from lmm\u2083 \u2a00\u2081 lmm\u2084,\n  have lmm\u2086 : \u223c\u223cp \u27f6 \u223c\u223cp \u2208 P, simp,\n  exact lmm\u2085 \u2a00\u2081 lmm\u2086\nend\n\n@[simp] lemma dni (p : F) : p \u27f6 \u223c\u223cp \u2208 P :=\nby { have : (\u223c\u223c\u223cp \u27f6 \u223cp) \u27f6 p \u27f6 \u223c\u223cp \u2208 P, simp, exact this \u2a00 (by simp) }\n\n@[simp] lemma dn_iff {p : F} : \u223c\u223cp \u2208 P \u2194 p \u2208 P :=\n\u27e8\u03bb h, (show \u223c\u223cp \u27f6 p \u2208 P, by simp) \u2a00 h, \u03bb h, (show p \u27f6 \u223c\u223cp \u2208 P, by simp) \u2a00 h\u27e9\n\n@[simp] lemma dn1_iff {p q : F} : (\u223c\u223cp \u27f6 q \u2208 P) \u2194 (p \u27f6 q \u2208 P) :=\n\u27e8impl_trans (dni _), impl_trans (dne _)\u27e9\n\n@[simp] lemma dn2_iff {p q : F} : (p \u27f6 \u223c\u223cq \u2208 P) \u2194 (p \u27f6 q \u2208 P) :=\n\u27e8\u03bb h, impl_trans h (dne _), \u03bb h, impl_trans h (dni _)\u27e9\n\nlemma explosion {p : F} (h\u2081 : p \u2208 P) (h\u2082 : \u223cp \u2208 P) {q : F} : q \u2208 P :=\nbegin\n  have : \u223cp \u27f6 \u223cq \u27f6 \u223cp \u2208 P, simp,\n  have : \u223cq \u27f6 \u223cp \u2208 P, from this \u2a00 h\u2082,\n  have : p \u27f6 q \u2208 P, from (show (\u223cq \u27f6 \u223cp) \u27f6 p \u27f6 q \u2208 P, by simp) \u2a00 this,\n  exact this \u2a00 h\u2081\nend\n\nlemma explosion_hyp {p q : F} (h\u2081 : p \u27f6 q \u2208 P) (h\u2082 : p \u27f6 \u223cq \u2208 P) {r : F} : p \u27f6 r \u2208 P :=\nbegin\n  have : p \u27f6 \u223cq \u27f6 \u223cr \u27f6 \u223cq \u2208 P, simp,\n  have : p \u27f6 \u223cr \u27f6 \u223cq \u2208 P, from this \u2a00\u2081 h\u2082,\n  have : p \u27f6 q \u27f6 r \u2208 P, from (show p \u27f6 (\u223cr \u27f6 \u223cq) \u27f6 q \u27f6 r \u2208 P, by simp) \u2a00\u2081 this,\n  exact this \u2a00\u2081 h\u2081\nend\n\nlemma explosion_hyp\u2082 {p q r : F} (h\u2081 : p \u27f6 q \u27f6 r \u2208 P) (h\u2082 : p \u27f6 q \u27f6 \u223cr \u2208 P) {s : F} : p \u27f6 q \u27f6 s \u2208 P :=\nbegin\n  have : p \u27f6 q \u27f6 \u223cr \u27f6 \u223cs \u27f6 \u223cr \u2208 P, simp,\n  have : p \u27f6 q \u27f6 \u223cs \u27f6 \u223cr \u2208 P, from this \u2a00\u2082 h\u2082,\n  have : p \u27f6 q \u27f6 r \u27f6 s \u2208 P, from (show p \u27f6 q \u27f6 (\u223cs \u27f6 \u223cr) \u27f6 r \u27f6 s \u2208 P, by simp) \u2a00\u2082 this,\n  exact this \u2a00\u2082 h\u2081\nend\n\n@[simp] lemma hyp_bot (p : F) : \u22a5 \u27f6 p \u2208 P :=\nexplosion_hyp (show (\u22a5 \u27f6 \u22a4 : F) \u2208 P, by simp) (show (\u22a5 : F) \u27f6 \u223c\u22a4 \u2208 P, by simp[neg_top_eq P])\n\nlemma contrapose {p q : F} : (\u223cp \u27f6 \u223cq \u2208 P) \u2194 (q \u27f6 p \u2208 P) :=\n\u27e8\u03bb h, (show (\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p \u2208 P, by simp) \u2a00 h, \u03bb h,\n  by { have : \u223c\u223cq \u27f6 p \u2208 P, from impl_trans (show \u223c\u223cq \u27f6 q \u2208 P, by simp) h,\n       exact (show (\u223c\u223cq \u27f6 \u223c\u223cp) \u27f6 \u223cp \u27f6 \u223cq \u2208 P, by simp) \u2a00 (impl_trans this (show p \u27f6 \u223c\u223cp \u2208 P, by simp)) }\u27e9\n\nlemma neg_hyp {p : F} (h : p \u27f6 \u223cp \u2208 P) : \u223cp \u2208 P :=\nbegin\n  have : p \u27f6 \u223c(p \u27f6 \u223cp) \u2208 P,\n  { have lmm\u2081 : p \u27f6 p \u2208 P, { simp }, exact explosion_hyp lmm\u2081 h },\n  have : (p \u27f6 \u223cp) \u27f6 \u223cp \u2208 P, from impl_trans (dni _) (contrapose.mpr this),\n  exact this \u2a00 h\nend\n\nlemma raa {p : F} (q : F) (h\u2081 : p \u27f6 q \u2208 P) (h\u2082 : p \u27f6 \u223cq \u2208 P) : \u223cp \u2208 P :=\nneg_hyp (explosion_hyp h\u2081 h\u2082)\n\n@[simp] lemma and_left (p q : F) : p \u2293 q \u27f6 p \u2208 P :=\nbegin\n  simp[and_def P],\n  have : \u223cp \u27f6 p \u27f6 \u223cq \u2208 P, from explosion_hyp\u2082 (show \u223cp \u27f6 p \u27f6 p \u2208 P, by simp) (show \u223cp \u27f6 p \u27f6 \u223cp \u2208 P, by simp),\n  have : \u223c(p \u27f6 \u223cq) \u27f6 \u223c\u223cp \u2208 P, from contrapose.mpr this,\n  simp* at*\nend\n\n@[simp] lemma and_right (p q : F) : p \u2293 q \u27f6 q \u2208 P :=\nbegin\n  simp[and_def P],\n  have : \u223cq \u27f6 p \u27f6 \u223cq \u2208 P, simp,\n  have : \u223c(p \u27f6 \u223cq) \u27f6 q \u2208 P, from impl_trans (contrapose.mpr this) (by simp),\n  exact this\nend\n\n@[simp] lemma and_inply_left {p\u2081 p\u2082 q : F} : (p\u2081 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P :=\n(show (p\u2081 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 p\u2081 \u27f6 q \u2208 P, by simp) \u2a00\u2082 (show (p\u2081 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 p\u2081 \u2208 P, by simp)\n\nlemma and_imply_of_imply_left {p\u2081 p\u2082 q : F} (h : p\u2081 \u27f6 q \u2208 P) : p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P :=\n(show (p\u2081 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P, by simp) \u2a00 h\n\n@[simp] lemma and_imply_right {p\u2081 p\u2082 q : F} : (p\u2082 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P :=\n(show (p\u2082 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 p\u2082 \u27f6 q \u2208 P, by simp) \u2a00\u2082 (show (p\u2082 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 p\u2082 \u2208 P, by simp)\n\nlemma and_imply_of_imply_right {p\u2081 p\u2082 q : F} (h : p\u2082 \u27f6 q \u2208 P) : p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P :=\n(show (p\u2082 \u27f6 q) \u27f6 p\u2081 \u2293 p\u2082 \u27f6 q \u2208 P, by simp) \u2a00 h\n\n@[simp] lemma iff_and_p {p q : F} : (p \u2293 q \u2208 P) \u2194 (p \u2208 P \u2227 q \u2208 P) :=\n\u27e8\u03bb h, by { split,\n   { exact modus_ponens (show p \u2293 q \u27f6 p \u2208 P, by simp) h },\n   { exact modus_ponens (show p \u2293 q \u27f6 q \u2208 P, by simp) h } },\n \u03bb h, by { simp[and_def P], rcases h with \u27e8h\u2081, h\u2082\u27e9,\n   have : (p \u27f6 \u223cq) \u27f6 \u223cq \u2208 P, from (show (p \u27f6 \u223cq) \u27f6 p \u27f6 \u223cq \u2208 P, by simp) \u2a00\u2081 (by simp[h\u2081]),\n   have : q \u27f6 \u223c(p \u27f6 \u223cq) \u2208 P, from impl_trans (dni _) (contrapose.mpr this),\n   exact modus_ponens this h\u2082 }\u27e9\n\n@[simp] lemma conjunction_iff {n} {p : finitary F n} : (finitary.conjunction n p \u2208 P) \u2194 (\u2200 i, p i \u2208 P) :=\nby { induction n with n IH; simp*,\n     { split,\n       { rintros h i, refine fin.last_cases _ _ i; simp[h] },\n       { intros h, simp[h] } } }\n\n@[simp] lemma list_conjunction_iff {l : list F} : (l.conjunction \u2208 P) \u2194 (\u2200 p \u2208 l, p \u2208 P) :=\nby induction l with a l IH; simp*\n\nlemma iff_equiv_p {p q : F} : (p \u27f7 q \u2208 P) \u2194 (p \u27f6 q \u2208 P \u2227 q \u27f6 p \u2208 P) :=\nby simp[lrarrow_def, iff_and_p]\n\nlemma iff_of_equiv {p q : F} (h : p \u27f7 q \u2208 P) : p \u2208 P \u2194 q \u2208 P :=\nby { simp[iff_equiv_p] at h, refine \u27e8\u03bb hp, h.1 \u2a00 hp, \u03bb hq, h.2 \u2a00 hq\u27e9 }\n\nlemma of_equiv {p\u2081 p\u2082 : F} (h : p\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) : p\u2082 \u2208 P :=\nby { simp[iff_equiv_p] at hp, refine hp.1 \u2a00 h }\n\nlemma equiv_imply_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082 : F} (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : (p\u2081 \u27f6 q\u2081) \u27f7 (p\u2082 \u27f6 q\u2082) \u2208 P :=\nbegin\n  simp[iff_equiv_p] at*, split,\n  { have : (p\u2081 \u27f6 q\u2081) \u27f6 p\u2082 \u27f6 q\u2081 \u2208 P, from (show (p\u2081 \u27f6 q\u2081) \u27f6 p\u2082 \u27f6 p\u2081 \u27f6 q\u2081 \u2208 P, by simp) \u2a00\u2082 (by simp[hp]),\n    exact (show (p\u2081 \u27f6 q\u2081) \u27f6 p\u2082 \u27f6 q\u2081 \u27f6 q\u2082 \u2208 P, by simp[hq]) \u2a00\u2082 this },\n  { have : (p\u2082 \u27f6 q\u2082) \u27f6 p\u2081 \u27f6 q\u2082 \u2208 P, from (show (p\u2082 \u27f6 q\u2082) \u27f6 p\u2081 \u27f6 p\u2082 \u27f6 q\u2082 \u2208 P, by simp) \u2a00\u2082 (by simp[hp]),\n    exact (show (p\u2082 \u27f6 q\u2082) \u27f6 p\u2081 \u27f6 q\u2082 \u27f6 q\u2081 \u2208 P, by simp[hq]) \u2a00\u2082 this }\nend\n\nlemma imply_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082 : F} (h : p\u2081 \u27f6 q\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2082 \u27f6 q\u2082 \u2208 P :=\nby { have : (p\u2081 \u27f6 q\u2081) \u27f6 p\u2082 \u27f6 q\u2082 \u2208 P \u2227 (p\u2082 \u27f6 q\u2082) \u27f6 p\u2081 \u27f6 q\u2081 \u2208 P, from iff_equiv_p.mp (equiv_imply_of_equiv hp hq),\n     exact this.1 \u2a00 h }\n\nlemma equiv_neg_of_equiv {p\u2081 p\u2082 : F} (hp : p\u2081 \u27f7 p\u2082 \u2208 P) : \u223cp\u2081 \u27f7 \u223cp\u2082 \u2208 P :=\nby simp[iff_equiv_p, contrapose] at*; simp[hp]\n\nlemma neg_of_equiv {p\u2081 p\u2082 : F} (h : \u223cp\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) : \u223cp\u2082 \u2208 P :=\nby { have : \u223cp\u2081 \u27f6 \u223cp\u2082 \u2208 P, from (iff_equiv_p.mp (equiv_neg_of_equiv hp)).1, exact this \u2a00 h }\n\nlemma equiv_and_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2081 \u2293 q\u2081 \u27f7 p\u2082 \u2293 q\u2082 \u2208 P :=\nby { simp only [and_def P], refine equiv_neg_of_equiv (equiv_imply_of_equiv hp (equiv_neg_of_equiv hq)) }\n\nlemma equiv_conjunction_of_equiv {n} {p\u2081 p\u2082 : finitary F n} (hp : \u2200 i, p\u2081 i \u27f7 p\u2082 i \u2208 P) :\n  finitary.conjunction n p\u2081 \u27f7 finitary.conjunction n p\u2082 \u2208 P :=\nby { induction n with n IH; simp[- iff_equiv_p], { simp[iff_equiv_p] },\n     { refine equiv_and_of_equiv (hp _) (IH _), intros i, exact hp _ } }\n\nlemma and_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : p\u2081 \u2293 q\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2082 \u2293 q\u2082 \u2208 P :=\nby { have : p\u2081 \u2293 q\u2081 \u27f6 p\u2082 \u2293 q\u2082 \u2208 P, from (iff_equiv_p.mp (equiv_and_of_equiv hp hq)).1, exact this \u2a00 h }\n\nlemma equiv_or_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2081 \u2294 q\u2081 \u27f7 p\u2082 \u2294 q\u2082 \u2208 P :=\nby { simp only [or_def P], refine (equiv_imply_of_equiv (equiv_neg_of_equiv hp) hq) }\n\nlemma or_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : p\u2081 \u2294 q\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2082 \u2294 q\u2082 \u2208 P :=\nby { have : p\u2081 \u2294 q\u2081 \u27f6 p\u2082 \u2294 q\u2082 \u2208 P, from (iff_equiv_p.mp (equiv_or_of_equiv hp hq)).1, exact this \u2a00 h }\n\nlemma equiv_equiv_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : (p\u2081 \u27f7 q\u2081) \u27f7 (p\u2082 \u27f7 q\u2082) \u2208 P :=\nby { refine (equiv_and_of_equiv (equiv_imply_of_equiv hp hq) (equiv_imply_of_equiv hq hp)) }\n\nlemma equiv_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : p\u2081 \u27f7 q\u2081 \u2208 P) (hp : p\u2081 \u27f7 p\u2082 \u2208 P) (hq : q\u2081 \u27f7 q\u2082 \u2208 P) : p\u2082 \u27f7 q\u2082 \u2208 P :=\nby { have : (p\u2081 \u27f7 q\u2081) \u27f6 (p\u2082 \u27f7 q\u2082) \u2208 P, from (iff_equiv_p.mp (equiv_equiv_of_equiv hp hq)).1, exact this \u2a00 h }\n\n@[refl, simp] lemma equiv_refl (p : F) : p \u27f7 p \u2208 P := by simp[iff_equiv_p]\n\n@[symm] lemma equiv_symm {p q : F} : p \u27f7 q \u2208 P \u2192 q \u27f7 p \u2208 P := by { simp[iff_equiv_p], intros, simp* }\n\n@[trans] lemma equiv_trans {p q r : F} : p \u27f7 q \u2208 P \u2192 q \u27f7 r \u2208 P \u2192 p \u27f7 r \u2208 P :=\nby { simp[iff_equiv_p], intros hpq hqp hqr hrq, exact \u27e8impl_trans hpq hqr, impl_trans hrq hqp\u27e9 }\n\n@[simp] lemma bot_of_neg_top : (\u223c\u22a4 : F) \u27f6 \u22a5 \u2208 P := by simp[@not_top_eq_bot F _ P _]\n\n@[simp] lemma neg_top : (\u223c\u22a5 : F) \u2208 P := @neg_of_equiv _ _ P _ (\u223c\u22a4) _ (by simp) (by simp[iff_equiv_p])\n\nvariables (P)\n\n@[reducible] def equiv (p q : F) : Prop := p \u27f7 q \u2208 P\n\nvariables {P}\n\n@[refl, simp] lemma equiv.refl (p : F) : equiv P p p := equiv_refl p\n\n@[symm] lemma equiv.symm {p q : F} : equiv P p q \u2192 equiv P q p := equiv_symm\n\n@[trans] lemma equiv.trans {p q r : F} : equiv P p q \u2192 equiv P q r \u2192 equiv P p r := equiv_trans\n\nvariables (P)\n\ntheorem equiv_equivalence : equivalence (equiv P) :=\n\u27e8equiv.refl, @equiv.symm _ _ _ _, @equiv.trans _ _ _ _\u27e9\n\nvariables {P}\n\n@[simp] lemma iff_dn_refl_right (p : F) : p \u27f7 \u223c\u223cp \u2208 P := by simp[iff_equiv_p]\n\n@[simp] lemma iff_dn_refl_left (p : F) : \u223c\u223cp \u27f7 p \u2208 P := by simp[iff_equiv_p]\n\n@[simp] lemma contraposition_inv (p q : F) : (p \u27f6 q) \u27f6 (\u223cq \u27f6 \u223cp) \u2208 P :=\nby { have : (\u223c\u223cp \u27f6 \u223c\u223cq) \u27f6 \u223cq \u27f6 \u223cp \u2208 P, simp, \n     refine imply_of_equiv this (equiv_imply_of_equiv _ _) _; simp }\n\n@[simp] lemma contraposition_iff (p q : F) : (p \u27f6 q) \u27f7 (\u223cq \u27f6 \u223cp) \u2208 P :=\nby simp[iff_equiv_p]\n\n@[simp] lemma contraposition_iff\u2081 (p q : F) : (\u223cp \u27f6 q) \u27f7 (\u223cq \u27f6 p) \u2208 P :=\nby { have : \u223cp \u27f6 q \u27f7 \u223cq \u27f6 \u223c\u223cp \u2208 P, from contraposition_iff (\u223cp) q,\n     refine equiv_of_equiv this (by simp) (equiv_imply_of_equiv (by simp) (by simp)) }\n\n@[simp] lemma contraposition_iff\u2082 (p q : F) : (p \u27f6 \u223cq) \u27f7 (q \u27f6 \u223cp) \u2208 P :=\nby { have : p \u27f6 \u223cq \u27f7 \u223c\u223cq \u27f6 \u223cp \u2208 P, from contraposition_iff p (\u223cq),\n     refine equiv_of_equiv this (by simp) (equiv_imply_of_equiv (by simp) (by simp)) }\n\n@[simp] lemma contraposition_iff_inv (p q : F) : (\u223cp \u27f6 \u223cq) \u27f7 (q \u27f6 p) \u2208 P :=\nby simp[iff_equiv_p]\n\n@[simp] lemma neg_hyp' (p : F) : (p \u27f6 \u223cp) \u27f6 \u223cp \u2208 P :=\nbegin\n  have : (p \u27f6 \u223cp) \u27f6 p \u27f6 \u223c(p \u27f6 \u223cp) \u2208 P,\n  { have lmm\u2081 : (p \u27f6 \u223cp) \u27f6 p \u27f6 p \u2208 P, { simp }, exact explosion_hyp\u2082 lmm\u2081 (by simp) },\n  have : (p \u27f6 \u223cp) \u27f6 \u223c\u223c(p \u27f6 \u223cp) \u27f6 \u223cp \u2208 P,\n  { refine imply_of_equiv this _ _; simp[iff_equiv_p] },\n  exact this \u2a00\u2081 (show (p \u27f6 \u223cp) \u27f6 \u223c\u223c(p \u27f6 \u223cp) \u2208 P, by simp)\nend\n\n@[simp] lemma neg_iff (p : F) : \u223cp \u27f7 (p \u27f6 \u22a5) \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { exact explosion_hyp\u2082 (show \u223cp \u27f6 p \u27f6 p \u2208 P, by simp) (show \u223cp \u27f6 p \u27f6 \u223cp \u2208 P, by simp) },\n  { have : (p \u27f6 \u22a5) \u27f6 p \u27f6 \u223cp \u2208 P,\n      from explosion_hyp\u2082 (show (p \u27f6 \u22a5) \u27f6 p \u27f6 \u22a4 \u2208 P, by simp) (show (p \u27f6 \u22a5) \u27f6 p \u27f6 \u223c\u22a4 \u2208 P, by simp[bot_eq P]),\n    refine (show (p \u27f6 \u22a5) \u27f6 (p \u27f6 \u223cp) \u27f6 \u223cp \u2208 P, by simp) \u2a00\u2081 this }\nend\n\n@[simp] lemma imply_iff_of {p} (h : p \u2208 P) (q) : (p \u27f6 q) \u27f7 q \u2208 P :=\nby simp[iff_equiv_p]; exact (show (p \u27f6 q) \u27f6 p \u27f6 q \u2208 P, by simp) \u2a00\u2081 (by simp[h])\n\n@[simp] lemma neg_impl_equiv_and (p q : F) : \u223c(p \u27f6 q) \u27f7 p \u2293 \u223cq \u2208 P :=\nby simp only [and_def P]; refine (equiv_neg_of_equiv (equiv_imply_of_equiv _ _)); simp\n\nlemma neg_impl_iff_and_p {p q : F} : (\u223c(p \u27f6 q) \u2208 P) \u2194 (p \u2293 \u223cq \u2208 P) :=\nbegin\n  simp [and_def P], split; intros h,\n  { refine neg_of_equiv h (equiv_imply_of_equiv _ _); simp },\n  { refine neg_of_equiv h (equiv_imply_of_equiv _ _); simp }\nend\n\n@[simp] lemma impl_iff_or_p {p q : F} : (p \u27f6 q) \u27f7 (\u223cp \u2294 q) \u2208 P :=\nby {simp [or_def P, -iff_equiv_p], refine equiv_imply_of_equiv _ _; simp }\n\n\n@[simp] lemma excluded_middle_p {p : F} : (p \u2294 \u223cp) \u2208 P :=\nby simp[or_def P]\n\n@[simp] lemma equiv_symm_and (p q : F) : p \u2293 q \u27f7 q \u2293 p \u2208 P :=\nby { simp only [and_def P], refine equiv_neg_of_equiv _,\n     refine equiv_of_equiv (show p \u27f6 \u223cq \u27f7 \u223c\u223cq \u27f6 \u223cp \u2208 P, by simp) _ (equiv_imply_of_equiv _ _); simp }\n\n@[simp] lemma equiv_symm_equiv (p q : F) : (p \u27f7 q) \u27f7 (q \u27f7 p) \u2208 P := equiv_symm_and _ _\n\n@[simp] lemma equiv_or_symm (p q : F) : p \u2294 q \u27f7 q \u2294 p \u2208 P :=\nby { simp only [or_def P],\n     refine equiv_of_equiv (show \u223cp \u27f6 q \u27f7 \u223cq \u27f6 \u223c\u223cp \u2208 P, by simp) _ (equiv_imply_of_equiv _ _); simp }\n\n@[simp] lemma impl_iff_or'_p {p q : F} : (p \u27f6 q) \u27f7 (q \u2294 \u223cp) \u2208 P :=\nequiv_trans (impl_iff_or_p) (by simp)\n\n@[simp] lemma and_destruct (p q : F) : p \u27f6 q \u27f6 p \u2293 q \u2208 P :=\nby { simp only [and_def P],\n     have : p \u27f6 (p \u27f6 \u223cq) \u27f6 \u223cq \u2208 P, from (show p \u27f6 (p \u27f6 \u223cq) \u27f6 p \u27f6 \u223cq \u2208 P, by simp) \u2a00\u2082 (show p \u27f6 (p \u27f6 \u223cq) \u27f6 p \u2208 P, by simp),\n     refine imply_of_equiv this (by simp) _,\n     have : (p \u27f6 \u223cq) \u27f6 \u223cq \u27f7 \u223c\u223cq \u27f6 \u223c(p \u27f6 \u223cq) \u2208 P, { simp }, refine equiv_of_equiv this _ (equiv_imply_of_equiv _ _); simp }\n\n@[simp] lemma imply_or_left (p q : F) : p \u27f6 p \u2294 q \u2208 P :=\nby simp[or_def P]; refine explosion_hyp\u2082 (show p \u27f6 \u223cp \u27f6 p \u2208 P, by simp) (show p \u27f6 \u223cp \u27f6 \u223cp \u2208 P, by simp)\n\n@[simp] lemma imply_or_right (p q : F) : q \u27f6 p \u2294 q \u2208 P :=\nby simp[or_def P]\n\nlemma disjunction_of {n} {p : finitary F n} (i) (h : p i \u2208 P) : finitary.disjunction n p \u2208 P :=\nbegin\n  induction n with n IH; simp*,\n  { exfalso, exact i.val.not_lt_zero i.property },\n  { revert h, refine fin.last_cases _ _ i,\n    { intros h, exact imply_or_right _ _ \u2a00 h },\n    { intros i h, exact imply_or_left _ _ \u2a00 (IH i h) } }\nend\n\nlemma ldisj_of {l : list F} {p : F} (h : p \u2208 l) (b : p \u2208 P) : l.disjunction \u2208 P :=\nbegin\n  induction l with a l IH generalizing p; simp*,\n  { exfalso, simp at h, contradiction },\n  { simp at h, rcases h with (rfl | h),\n    { exact imply_or_right _ _ \u2a00 b },\n    { exact imply_or_left _ _ \u2a00 IH h b } }\nend\n\nlemma fdisj_of {s : finset F} {p : F} (h : p \u2208 s) (b : p \u2208 P) : s.disjunction \u2208 P :=\nby simp[finset.disjunction]; refine ldisj_of (by simpa using h) b\n\nlemma fconj_of {s : finset F} {p : F} : s.conjunction \u2208 P \u2194 \u2200 p \u2208 s, p \u2208 P:=\nby simp[finset.conjunction]\n\n@[simp] lemma imply_ldisj {l : list F} {p : F} (h : p \u2208 l) : p \u27f6 l.disjunction \u2208 P :=\nbegin\n  induction l with q l IH,\n  { simp at h, contradiction },\n  { simp at h \u22a2, rcases h with (rfl| h),\n    { simp },\n    { refine impl_trans (IH h) (by simp) } }\nend\n\n@[simp] lemma lconj_imply {l : list F} {p : F} (h : p \u2208 l) : l.conjunction \u27f6 p \u2208 P :=\nbegin\n  induction l with q l IH,\n  { simp at h, contradiction },\n  { simp at h \u22a2, rcases h with (rfl| h),\n    { simp },\n    { refine impl_trans (by simp) (IH h) } }\nend\n\nlemma imply_fdisj {s : finset F} {p : F} (h : p \u2208 s) : p \u27f6 s.disjunction \u2208 P :=\nby simp[finset.disjunction]; refine imply_ldisj (by simpa using h)\n\nlemma fconj_imply {s : finset F} {p : F} (h : p \u2208 s) : s.conjunction \u27f6 p \u2208 P :=\nby simp[finset.conjunction]; refine lconj_imply (by simpa using h)\n\n@[simp] lemma imply_and (p q r : F) : (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 q \u2293 r \u2208 P :=\nbegin\n  have : (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 r \u27f6 q \u2293 r \u2208 P,\n    from (show (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 q \u27f6 r \u27f6 q \u2293 r \u2208 P, by simp) \u2a00\u2083 (show (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 q \u2208 P, by simp),\n  exact this \u2a00\u2083 (show (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 r \u2208 P, by simp),\nend\n\n@[simp] lemma neg_and_equiv_or_neg (p q : F) : \u223c(p \u2293 q) \u27f7 \u223cp \u2294 \u223cq \u2208 P :=\nbegin\n  simp only [and_def P, or_def P],\n  refine equiv_of_equiv (show p \u27f6 \u223cq \u27f7 p \u27f6 \u223cq \u2208 P, by simp) _ (equiv_imply_of_equiv _ _); simp\nend\n\n@[simp] lemma neg_conj_equiv_disj_neg {n} (p : finitary F n) : \u223c(finitary.conjunction n p) \u27f7 (\u22c1 i, \u223cp i) \u2208 P :=\nbegin\n  induction n with n IH; simp[-iff_equiv_p],\n  { simp[iff_equiv_p] },\n  { have lmm\u2081 : \u223c(p (fin.last n) \u2293 finitary.conjunction n (p \u2218 fin.cast_succ)) \u27f7\n      \u223cp (fin.last n) \u2294 \u223cfinitary.conjunction n (p \u2218 fin.cast_succ) \u2208 P, from neg_and_equiv_or_neg _ _,\n    have lmm\u2082 : \u223cp (fin.last n) \u2294 \u223cfinitary.conjunction n (p \u2218 fin.cast_succ) \u27f7 \n      \u223cfinitary.conjunction n (p \u2218 fin.cast_succ) \u2294 \u223cp (fin.last n) \u2208 P, from equiv_or_symm _ _,\n    refine equiv_of_equiv (equiv_trans lmm\u2081 lmm\u2082) (equiv_refl _) (equiv_or_of_equiv (IH (p \u2218 fin.cast_succ)) (equiv_refl _)) }\nend\n\n@[simp] lemma neg_lconj_equiv_ldisj_neg (l : list F) : \u223cl.conjunction \u27f7 (l.map has_negation.neg).disjunction \u2208 P :=\nbegin\n  induction l with p l IH; simp[-iff_equiv_p],\n  { simp[iff_equiv_p] },\n  { have lmm\u2081 : \u223c(p \u2293 l.conjunction) \u27f7 \u223cp \u2294 \u223cl.conjunction \u2208 P, from neg_and_equiv_or_neg _ _,\n    have lmm\u2082 : \u223cp \u2294 \u223cl.conjunction \u27f7 \u223cl.conjunction \u2294 \u223cp \u2208 P, from equiv_or_symm _ _,\n    refine equiv_of_equiv (equiv_trans lmm\u2081 lmm\u2082) (equiv_refl _)\n    (equiv_or_of_equiv IH (equiv_refl _)) }\nend\n\n@[simp] lemma neg_or_equiv_and_neg (p q : F) : \u223c(p \u2294 q) \u27f7 \u223cp \u2293 \u223cq \u2208 P :=\nbegin\n  simp only [and_def P, or_def P],\n  refine equiv_of_equiv (show \u223c(\u223cp \u27f6 q) \u27f7 \u223c(\u223cp \u27f6 q) \u2208 P, by simp) _ (equiv_neg_of_equiv (equiv_imply_of_equiv _ _)); simp\nend\n\n@[simp] lemma neg_disj_equiv_conj_neg {n} (p : finitary F n) : \u223c(finitary.disjunction n p) \u27f7 (\u22c0 i, \u223cp i) \u2208 P :=\nbegin\n  induction n with n IH; simp[-iff_equiv_p],\n  { simp[iff_equiv_p] },\n  { have lmm\u2081 : \u223c(finitary.disjunction n (p \u2218 fin.cast_succ) \u2294 p (fin.last n)) \u27f7\n      \u223cfinitary.disjunction n (p \u2218 fin.cast_succ) \u2293 \u223cp (fin.last n) \u2208 P, from neg_or_equiv_and_neg _ _,\n    have lmm\u2082 : \u223cfinitary.disjunction n (p \u2218 fin.cast_succ) \u2293 \u223cp (fin.last n) \u27f7\n      \u223cp (fin.last n) \u2293 \u223cfinitary.disjunction n (p \u2218 fin.cast_succ) \u2208 P, from equiv_symm_and _ _,\n    refine equiv_of_equiv (equiv_trans lmm\u2081 lmm\u2082) (equiv_refl _) (equiv_and_of_equiv (equiv_refl _) (IH (p \u2218 fin.cast_succ)))  }\nend\n\n@[simp] lemma or_imply (p q r : F) : (p \u27f6 r) \u27f6 (q \u27f6 r) \u27f6 p \u2294 q \u27f6 r \u2208 P :=\nbegin\n  have : (\u223cr \u27f6 \u223cp) \u27f6 (\u223cr \u27f6 \u223cq) \u27f6 \u223cr \u27f6 \u223cp \u2293 \u223cq \u2208 P, { simp },\n  refine imply_of_equiv this (by simp) (equiv_imply_of_equiv (by simp) _),\n  have : \u223cr \u27f6 \u223c(p \u2294 q) \u27f7 p \u2294 q \u27f6 r \u2208 P, { simp },\n  refine equiv_of_equiv (show \u223cr \u27f6 \u223c(p \u2294 q) \u27f7 p \u2294 q \u27f6 r \u2208 P, by simp) (equiv_imply_of_equiv _ _) _; simp\nend\n\n@[simp] lemma le_sup_inf (p q r : F) : (p \u2294 q) \u2293 (p \u2294 r) \u27f6 p \u2294 q \u2293 r \u2208 P :=\nbegin\n  simp[or_def P],\n  exact (show (\u223cp \u27f6 q) \u2293 (\u223cp \u27f6 r) \u27f6 (\u223cp \u27f6 q) \u27f6 (\u223cp \u27f6 r) \u27f6 \u223cp \u27f6 q \u2293 r \u2208 P, by simp) \u2a00\u2081\n        (show (\u223cp \u27f6 q) \u2293 (\u223cp \u27f6 r) \u27f6 (\u223cp \u27f6 q) \u2208 P, by simp) \u2a00\u2081\n        (show (\u223cp \u27f6 q) \u2293 (\u223cp \u27f6 r) \u27f6 (\u223cp \u27f6 r) \u2208 P, by simp)\nend\n\nlemma case_of_p {p q r : F} (hpq : p \u2294 q \u2208 P) (hpr : p \u27f6 r \u2208 P) (hqr : q \u27f6 r \u2208 P) : r \u2208 P :=\n(show (p \u27f6 r) \u27f6 (q \u27f6 r) \u27f6 p \u2294 q \u27f6 r \u2208 P, by simp) \u2a00 hpr \u2a00 hqr \u2a00 hpq\n\n@[simp] lemma and_imply_equiv_imply_imply (p q r : F) : (p \u27f6 q \u27f6 r) \u27f7 (p \u2293 q \u27f6 r) \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { exact (show (p \u27f6 q \u27f6 r) \u27f6 p \u2293 q \u27f6 p \u27f6 q \u27f6 r \u2208 P, by simp) \u2a00\u2082 (show (p \u27f6 q \u27f6 r) \u27f6 p \u2293 q \u27f6 p \u2208 P, by simp) \u2a00\u2082 (show (p \u27f6 q \u27f6 r) \u27f6 p \u2293 q \u27f6 q \u2208 P, by simp) },\n  { exact (show (p \u2293 q \u27f6 r) \u27f6 p \u27f6 q \u27f6 (p \u2293 q \u27f6 r) \u2208 P, by simp) \u2a00\u2083 (show (p \u2293 q \u27f6 r) \u27f6 p \u27f6 q \u27f6 (p \u2293 q) \u2208 P, by simp) }\nend\n\nlemma and_imply_of {p q r} : p \u2293 q \u27f6 r \u2208 P \u2194 p \u27f6 q \u27f6 r \u2208 P :=\n(iff_of_equiv $ and_imply_equiv_imply_imply p q r).symm\n\n@[simp] lemma imply_and_equiv_or_imply (p q r : F) : (p \u27f6 r) \u2293 (q \u27f6 r) \u27f7 p \u2294 q \u27f6 r \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { refine of_equiv (or_imply p q r) (and_imply_equiv_imply_imply _ _ _) },\n  { have lmm\u2081 : (p \u2294 q \u27f6 r) \u27f6 p \u27f6 r \u2208 P, from (show (p \u2294 q \u27f6 r) \u27f6 p \u27f6 p \u2294 q \u27f6 r \u2208 P, by simp) \u2a00\u2082 (show (p \u2294 q \u27f6 r) \u27f6 p \u27f6 p \u2294 q \u2208 P, by simp),\n    have lmm\u2082 : (p \u2294 q \u27f6 r) \u27f6 q \u27f6 r \u2208 P, from (show (p \u2294 q \u27f6 r) \u27f6 q \u27f6 p \u2294 q \u27f6 r \u2208 P, by simp) \u2a00\u2082 (show (p \u2294 q \u27f6 r) \u27f6 q \u27f6 p \u2294 q \u2208 P, by simp),\n    refine imply_and _ _ _ \u2a00 lmm\u2081 \u2a00 lmm\u2082 }\nend\n\n@[simp] lemma conj_imply_iff_disj_imply {n} (p : finitary F n) (q : F) : (\u22c0 i, (p i \u27f6 q)) \u27f7 ((\u22c1 i, p i) \u27f6 q) \u2208 P :=\nbegin\n  induction n with n IH, { simp[iff_equiv_p] }, \n  { simp[-iff_equiv_p, (\u2218)],\n    have : (p (fin.last n) \u27f6 q) \u2293 ((\u22c1 i, p i.cast_succ) \u27f6 q) \u27f7 (p (fin.last n) \u2294 \u22c1 i, p i.cast_succ) \u27f6 q \u2208 P,\n    from imply_and_equiv_or_imply (p (fin.last n)) (\u22c1 i, p i.cast_succ) q,\n    refine equiv_of_equiv this (equiv_and_of_equiv (equiv_refl _) (equiv_symm (IH _)))\n      (equiv_imply_of_equiv (equiv_or_symm _ _) (by refl)), }\nend\n\n@[simp] lemma lconj_imply_iff_ldisj_imply (l : list F) (q : F) :\n  (l.map (\u03bb p, (p \u27f6 q))).conjunction \u27f7 (l.disjunction \u27f6 q) \u2208 P :=\nbegin\n  induction l with p l IH, { simp[iff_equiv_p] }, \n  { simp[-iff_equiv_p, (\u2218)],\n    refine equiv_of_equiv (imply_and_equiv_or_imply p l.disjunction q)\n      (equiv_and_of_equiv (equiv_refl _) (equiv_symm (IH)))\n      (equiv_imply_of_equiv (equiv_or_symm _ _) (by refl)) }\nend\n\n@[simp] lemma imply_lconj_of {l : list F} {q : F} (h : \u2200 p \u2208 l, q \u27f6 p \u2208 P) : q \u27f6 l.conjunction \u2208 P :=\nby { induction l with p l IH; simp,\n     { simp at h, refine imply_and _ _ _ \u2a00 _ \u2a00 _,\n       { exact h.1 },\n       { exact IH h.2 } } }\n\n@[simp] lemma ldisj_imply_of {l : list F} {q : F} (h : \u2200 p \u2208 l, p \u27f6 q \u2208 P) : l.disjunction \u27f6 q \u2208 P :=\nby { have : (l.map (\u03bb p, (p \u27f6 q))).conjunction \u2208 P, by simpa using h,\n     exact (iff_equiv_p.mp $ lconj_imply_iff_ldisj_imply l q).1 \u2a00 this }\n\n@[simp] lemma imply_fconj_of {s : finset F} {q : F} (h : \u2200 p \u2208 s, q \u27f6 p \u2208 P) : q \u27f6 s.conjunction \u2208 P :=\nby simp[finset.conjunction]; exact imply_lconj_of (by simpa using h)\n\n@[simp] lemma fdisj_imply_of {s : finset F} {q : F} (h : \u2200 p \u2208 s, p \u27f6 q \u2208 P) : s.disjunction \u27f6 q \u2208 P :=\nby simp[finset.disjunction]; exact ldisj_imply_of (by simpa using h)\n\n@[simp] lemma equiv_or_assoc (p q r : F) : (p \u2294 q) \u2294 r \u27f7 p \u2294 (q \u2294 r) \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { have l\u2081 : p \u27f6 p \u2294 (q \u2294 r) \u2208 P, from imply_or_left p (q \u2294 r),\n    have l\u2082 : q \u27f6 p \u2294 (q \u2294 r) \u2208 P, from impl_trans (imply_or_left q r) (imply_or_right p (q \u2294 r)),\n    have l\u2083 : r \u27f6 p \u2294 (q \u2294 r) \u2208 P, from impl_trans (imply_or_right q r) (imply_or_right p (q \u2294 r)),\n    exact or_imply _ _ _ \u2a00 (or_imply _ _ _ \u2a00 l\u2081 \u2a00 l\u2082) \u2a00 l\u2083 },\n  { have l\u2081 : p \u27f6 (p \u2294 q) \u2294 r \u2208 P, from impl_trans (imply_or_left p q) (imply_or_left (p \u2294 q) r),\n    have l\u2082 : q \u27f6 (p \u2294 q) \u2294 r \u2208 P, from impl_trans (imply_or_right p q) (imply_or_left (p \u2294 q) r),\n    have l\u2083 : r \u27f6 (p \u2294 q) \u2294 r \u2208 P, from imply_or_right (p \u2294 q) r,\n    exact or_imply _ _ _ \u2a00 l\u2081 \u2a00 (or_imply _ _ _ \u2a00 l\u2082 \u2a00 l\u2083) }\nend\n\nlemma equiv_append (l\u2081 l\u2082 : list F) : (l\u2081 ++ l\u2082).disjunction \u27f7 l\u2081.disjunction \u2294 l\u2082.disjunction \u2208 P :=\nbegin\n  induction l\u2081; simp[-iff_equiv_p],\n  case nil { simp[iff_equiv_p], refine or_imply \u22a5 l\u2082.disjunction l\u2082.disjunction \u2a00 (hyp_bot l\u2082.disjunction) \u2a00 (refl l\u2082.disjunction) },\n  case cons : p l IH\n  { have l\u2081 : (l ++ l\u2082).disjunction \u2294 p \u27f7 l.disjunction \u2294 (l\u2082.disjunction \u2294 p) \u2208 P,\n      from equiv_trans (equiv_or_of_equiv IH (equiv_refl p)) (equiv_or_assoc l.disjunction l\u2082.disjunction p),\n    have l\u2082 : l.disjunction \u2294 (l\u2082.disjunction \u2294 p) \u27f7 l.disjunction \u2294 p \u2294 l\u2082.disjunction \u2208 P,\n      from equiv_trans (equiv_or_of_equiv (equiv_refl l.disjunction) (equiv_or_symm l\u2082.disjunction p))\n      (equiv_symm (equiv_or_assoc _ _ _)),\n    exact equiv_trans l\u2081 l\u2082 }\nend\n\nlemma fconj_insert_equiv_inf {s : finset F} {p : F} : (insert p s).conjunction \u27f7 p \u2293 s.conjunction \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { refine imply_and _ _ _ \u2a00 _ \u2a00 _,\n    refine fconj_imply (by simp),\n    refine imply_fconj_of (by { intros q hq, refine fconj_imply (by simp[hq]) }) },\n  { refine imply_fconj_of _, intros q hq, simp at hq, rcases hq with (rfl | hq),\n    { simp }, { refine impl_trans (and_right _ _) (fconj_imply hq) } }\nend\n\nlemma fdisj_insert_equiv_sup {s : finset F} {p : F} : (insert p s).disjunction \u27f7 s.disjunction \u2294 p \u2208 P :=\nbegin\n  simp[iff_equiv_p], split,\n  { refine fdisj_imply_of _, intros q hq, simp at hq, rcases hq with (rfl | hq),\n    { simp }, { refine impl_trans (imply_fdisj hq) (imply_or_left _ _) } },\n  { refine or_imply _ _ _ \u2a00 _ \u2a00 _,\n    refine fdisj_imply_of (by { intros q hq, refine imply_fdisj (by simp[hq]) }),\n    refine imply_fdisj (by simp) }\nend\n\nvariables (P)\n\ndef lindenbaum := quotient (\u27e8equiv P, equiv_equivalence P\u27e9 : setoid F)\n\nvariables {P}\n\ndef to_quo (p : F) : lindenbaum P := quotient.mk' p\n\nlocal notation `\u27e6` p `\u27e7\u1d3e` := to_quo p\n\nnamespace lindenbaum\n\n@[elab_as_eliminator]\nprotected lemma ind_on {C : lindenbaum P \u2192 Prop} (d : lindenbaum P)\n  (h : \u2200 p : F, C (to_quo p)) : C d := quotient.induction_on' d h\n\n@[elab_as_eliminator, reducible]\nprotected def lift_on {\u03c6 : Sort*} (p : lindenbaum P) (f : F \u2192 \u03c6)\n  (h : \u2200 p q : F, p \u27f7 q \u2208 P \u2192 f p = f q) : \u03c6 := quotient.lift_on' p f h\n\n@[simp]\nprotected lemma lift_on_eq {\u03c6 : Sort*} (p : F) (f : F \u2192 \u03c6)\n  (h : \u2200 p q, p \u27f7 q \u2208 P \u2192 f p = f q) : classical_logic.lindenbaum.lift_on \u27e6p\u27e7\u1d3e f h = f p := rfl\n\n@[elab_as_eliminator, reducible, simp]\nprotected def lift_on\u2082 {\u03c6 : Sort*} (p\u2081 p\u2082 : lindenbaum P) (f : F \u2192 F \u2192 \u03c6)\n  (h : \u2200 p\u2081 p\u2082 q\u2081 q\u2082, p\u2081 \u27f7 q\u2081 \u2208 P \u2192 p\u2082 \u27f7 q\u2082 \u2208 P \u2192 f p\u2081 p\u2082 = f q\u2081 q\u2082) : \u03c6 :=\nquotient.lift_on\u2082' p\u2081 p\u2082 f h\n\n@[simp]\nprotected lemma lift_on\u2082_eq {\u03c6 : Sort*} (p\u2081 p\u2082 : F) (f : F \u2192 F \u2192 \u03c6)\n  (h : \u2200 p\u2081 p\u2082 q\u2081 q\u2082, p\u2081 \u27f7 q\u2081 \u2208 P \u2192 p\u2082 \u27f7 q\u2082 \u2208 P \u2192 f p\u2081 p\u2082 = f q\u2081 q\u2082)  :\nclassical_logic.lindenbaum.lift_on\u2082 \u27e6p\u2081\u27e7\u1d3e \u27e6p\u2082\u27e7\u1d3e f h = f p\u2081 p\u2082 := rfl\n\n@[elab_as_eliminator, reducible]\nprotected def lift_on_finitary {\u03c6 : Sort*} {n : \u2115} (v : finitary (lindenbaum P) n) (f : finitary F n \u2192 \u03c6)\n  (h : \u2200 v\u2081 v\u2082 : finitary F n, (\u2200 n, v\u2081 n \u27f7 v\u2082 n \u2208 P) \u2192 f v\u2081 = f v\u2082) : \u03c6 :=\nquotient.lift_on_finitary v f h \n\n@[simp]\nprotected lemma lift_on_finitary_eq {\u03c6 : Sort*} {n : \u2115} (v : finitary F n) (f : finitary F n \u2192 \u03c6)\n  (h : \u2200 v\u2081 v\u2082 : finitary F n, (\u2200 n, v\u2081 n \u27f7 v\u2082 n \u2208 P) \u2192 f v\u2081 = f v\u2082) :\nclassical_logic.lindenbaum.lift_on_finitary (\u03bb x, \u27e6v x\u27e7\u1d3e) f h = f v :=\nquotient.lift_on_finitary_eq v f h\n\nlemma of_eq_of {p q : F} : (\u27e6p\u27e7\u1d3e : lindenbaum P) = \u27e6q\u27e7\u1d3e \u2194 p \u27f7 q \u2208 P :=\nby simp[to_quo, equiv, quotient.eq']\n\ninstance : distrib_lattice (lindenbaum P) :=\n{ le := \u03bb p\u2081 p\u2082, classical_logic.lindenbaum.lift_on\u2082 p\u2081 p\u2082 (\u03bb p\u2081 p\u2082, p\u2081 \u27f6 p\u2082 \u2208 P)\n    (\u03bb p\u2081 p\u2082 q\u2081 q\u2082 h\u2081 h\u2082,\n      by { simp, exact \u27e8\u03bb h, imply_of_equiv h h\u2081 h\u2082, \u03bb h, imply_of_equiv h (equiv_symm h\u2081) (equiv_symm h\u2082)\u27e9 }),\n  le_refl := \u03bb p, by induction p using classical_logic.lindenbaum.ind_on; simp,\n  le_trans := \u03bb p\u2081 p\u2082 p\u2083 h\u2081\u2082 h\u2082\u2083,\n  by { induction p\u2081 using classical_logic.lindenbaum.ind_on,\n       induction p\u2082 using classical_logic.lindenbaum.ind_on,\n       induction p\u2083 using classical_logic.lindenbaum.ind_on,\n       simp at h\u2081\u2082 h\u2082\u2083 \u22a2, exact impl_trans h\u2081\u2082 h\u2082\u2083 },\n  le_antisymm := \u03bb p\u2081 p\u2082,\n  by { induction p\u2081 using classical_logic.lindenbaum.ind_on,\n       induction p\u2082 using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le], intros h\u2081 h\u2082, simp[*, of_eq_of, iff_equiv_p] },\n  inf := \u03bb p\u2081 p\u2082, classical_logic.lindenbaum.lift_on\u2082 p\u2081 p\u2082 (\u03bb p\u2081 p\u2082, (\u27e6p\u2081 \u2293 p\u2082\u27e7\u1d3e : lindenbaum P))\n    (\u03bb p\u2081 p\u2082 q\u2081 q\u2082 h\u2081 h\u2082, by { simp[of_eq_of], exact equiv_and_of_equiv h\u2081 h\u2082 }),\n  sup := \u03bb p\u2081 p\u2082, classical_logic.lindenbaum.lift_on\u2082 p\u2081 p\u2082 (\u03bb p\u2081 p\u2082, (\u27e6p\u2081 \u2294 p\u2082\u27e7\u1d3e : lindenbaum P))\n    (\u03bb p\u2081 p\u2082 q\u2081 q\u2082 h\u2081 h\u2082, by { simp[of_eq_of], exact equiv_or_of_equiv h\u2081 h\u2082 }),\n  le_sup_left := \u03bb p q,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le] },\n  le_sup_right := \u03bb p q,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le] },\n  sup_le := \u03bb p q r,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       induction r using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le],\n       intros hpr hqr, exact (show (p \u27f6 r) \u27f6 (q \u27f6 r) \u27f6 p \u2294 q \u27f6 r \u2208 P, by simp) \u2a00 hpr \u2a00 hqr },\n  inf_le_left := \u03bb p q,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le] },\n  inf_le_right := \u03bb p q,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le] },\n  le_inf := \u03bb p q r,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       induction r using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le],\n       intros hpq hpr, exact (show (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 q \u2293 r \u2208 P, by simp) \u2a00 hpq \u2a00 hpr },\n  le_sup_inf := \u03bb p q r,\n  by { induction p using classical_logic.lindenbaum.ind_on,\n       induction q using classical_logic.lindenbaum.ind_on,\n       induction r using classical_logic.lindenbaum.ind_on,\n       simp[has_le.le, preorder.le, partial_order.le, semilattice_inf.le,\n         has_sup.sup, semilattice_sup.sup, has_inf.inf, semilattice_inf.inf] } }\n\ninstance : has_compl (lindenbaum P) := \u27e8\u03bb p, classical_logic.lindenbaum.lift_on p (\u03bb p, (\u27e6\u223cp\u27e7\u1d3e : lindenbaum P))\n    (\u03bb p q h, by { simp[of_eq_of], exact equiv_neg_of_equiv h })\u27e9\n\nlemma le_def (p q : F) : (\u27e6p\u27e7\u1d3e : lindenbaum P) \u2264 \u27e6q\u27e7\u1d3e \u2194 p \u27f6 q \u2208 P := by refl\n\nlemma neg_def (p : F) : (\u27e6p\u27e7\u1d3e : lindenbaum P)\u1d9c = \u27e6\u223cp\u27e7\u1d3e := rfl\n\nlemma inf_def (p q : F) : (\u27e6p\u27e7\u1d3e : lindenbaum P) \u2293 \u27e6q\u27e7\u1d3e = \u27e6p \u2293 q\u27e7\u1d3e := rfl\n\nlemma sup_def (p q : F) : (\u27e6p\u27e7\u1d3e : lindenbaum P) \u2294 \u27e6q\u27e7\u1d3e = \u27e6p \u2294 q\u27e7\u1d3e := rfl\n\ninstance : boolean_algebra (lindenbaum P) :=\n{ top := \u27e6\u22a4\u27e7\u1d3e,\n  bot := \u27e6\u22a5\u27e7\u1d3e,\n  le_top := \u03bb p, by induction p using classical_logic.lindenbaum.ind_on; simp[le_def],\n  bot_le := \u03bb p, by induction p using classical_logic.lindenbaum.ind_on; simp[le_def],\n  compl := has_compl.compl,\n  inf_compl_le_bot := \u03bb p,\n  by { induction p using classical_logic.lindenbaum.ind_on, simp[bounded_order.bot],\n       refine explosion_hyp (show p \u2293 \u223cp \u27f6 p \u2208 P, by simp) (by simp) },\n  top_le_sup_compl := \u03bb p, \n  by { induction p using classical_logic.lindenbaum.ind_on, \n       simp[bounded_order.top, or_def P, le_def, sup_def, neg_def] },\n  ..lindenbaum.distrib_lattice }\n\nlemma top_def : (\u22a4 : lindenbaum P) = \u27e6\u22a4\u27e7\u1d3e := rfl\n\nlemma bot_def : (\u22a5 : lindenbaum P) = \u27e6\u22a5\u27e7\u1d3e := rfl\n\nlemma imply_eq (p q : F) : (\u27e6p \u27f6 q\u27e7\u1d3e : lindenbaum P) = \u27e6p\u27e7\u1d3e \u21e8 \u27e6q\u27e7\u1d3e:=\nby simp[of_eq_of, himp_eq, neg_def, sup_def]\n\nlemma fconj_eq (s : finset F) : (\u27e6s.conjunction\u27e7\u1d3e : lindenbaum P) = s.inf to_quo :=\nbegin\n  induction s using finset.induction_on with p s hs IH,\n  { simp, refl },\n  { simp, rw[\u2190IH, inf_def, of_eq_of], exact fconj_insert_equiv_inf }\nend\n\nlemma fdisj_eq (s : finset F) : (\u27e6s.disjunction\u27e7\u1d3e : lindenbaum P) = s.sup to_quo :=\nbegin\n  induction s using finset.induction_on with p s hs IH,\n  { simp, refl },\n  { simp, rw[sup_comm, \u2190IH, sup_def, of_eq_of], exact fdisj_insert_equiv_sup }\nend\n\nend lindenbaum\n\nopen lindenbaum\n\nlemma equiv_of_lindenbaum {p q : F} : p \u27f7 q \u2208 P \u2194 (\u27e6p\u27e7\u1d3e : lindenbaum P) = \u27e6q\u27e7\u1d3e :=\nlindenbaum.of_eq_of.symm\n\nlemma fdisj_insert_equiv_fconj_imply (p : F) (s : finset F) :\n  (insert p s).disjunction \u27f7 ((s.image has_negation.neg).conjunction \u27f6 p) \u2208 P :=\nby simp [equiv_of_lindenbaum, fdisj_eq, fconj_eq, imply_eq, himp_eq, \u2190neg_def, finset.sup_finset_image, (\u2218)]\n\nlemma fconj_imply_equiv_fdisj_neg (p : F) (s : finset F) :\n  (s.conjunction \u27f6 p) \u27f7 (insert p (s.image has_negation.neg)).disjunction \u2208 P :=\nby simp [equiv_of_lindenbaum, fdisj_eq, fconj_eq, imply_eq, himp_eq, \u2190neg_def, finset.sup_finset_image, (\u2218)]\n\n@[simp] lemma neg_bot_equiv_top : (\u223c\u22a5 : F) \u27f7 \u22a4 \u2208 P :=\nby simp[equiv_of_lindenbaum, \u2190neg_def, \u2190top_def, \u2190bot_def]\n\nend classical_logic\n\nnamespace axiomatic_classical_logic'\nopen classical_logic\nvariables {F : Type*} [has_logic_symbol F]\n  (T : set F) [axiomatic_classical_logic' F]\n\ninstance : classical_logic ((\u22a2) T) := axiomatic_classical_logic'.classical\n\nvariables {T}\n\nlemma modus_ponens {p q : F} : T \u22a2 p \u27f6 q \u2192 T \u22a2 p \u2192 T \u22a2 q := modus_ponens\n\nlemma modus_ponens_hyp {p q r : F} : T \u22a2 p \u27f6 q \u27f6 r \u2192 T \u22a2 p \u27f6 q \u2192 T \u22a2 p \u27f6 r :=\nmodus_ponens_hyp\n\nlemma modus_ponens_hyp\u2082 {p q r s : F} : T \u22a2 p \u27f6 q \u27f6 r \u27f6 s \u2192 T \u22a2 p \u27f6 q \u27f6 r \u2192 T \u22a2 p \u27f6 q \u27f6 s :=\nmodus_ponens_hyp\u2082\n\nlemma modus_ponens_hyp\u2083 {p q r s t : F} :\n  T \u22a2 p \u27f6 q \u27f6 r \u27f6 s \u27f6 t \u2192 T \u22a2 p \u27f6 q \u27f6 r \u27f6 s \u2192 T \u22a2 p \u27f6 q \u27f6 r \u27f6 t :=\nmodus_ponens_hyp\u2083\n\nlocalized \"infixl ` \u2a00 `:90 := axiomatic_classical_logic'.modus_ponens\" in aclogic\nlocalized \"infixl ` \u2a00\u2081 `:90 := axiomatic_classical_logic'.modus_ponens_hyp\" in aclogic\nlocalized \"infixl ` \u2a00\u2082 `:90 := axiomatic_classical_logic'.modus_ponens_hyp\u2082\" in aclogic\nlocalized \"infixl ` \u2a00\u2083 `:90 := axiomatic_classical_logic'.modus_ponens_hyp\u2083\" in aclogic\n\nopen_locale aclogic\n\n@[simp] lemma mem_iff_prov (p : F) : (@has_mem.mem F (set F) _) p ((\u22a2) T : set F) \u2194 T \u22a2 p := by refl\n\n@[simp] lemma imply\u2081 (p q : F) : T \u22a2 p \u27f6 q \u27f6 p := imply\u2081\n\n@[simp] lemma imply\u2082 (p q r : F) : T \u22a2 (p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r := imply\u2082\n\nlemma imply_trans {p q r : F} : (T \u22a2 p \u27f6 q) \u2192 (T \u22a2 q \u27f6 r) \u2192 (T \u22a2 p \u27f6 r) :=\nimpl_trans\n\n@[simp] lemma contraposition (p q : F) : T \u22a2 (\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p := contraposition\n\n@[simp] lemma provable_top : T \u22a2 (\u22a4 : F) := provable_top\n\n@[simp] lemma refl (p : F) : T \u22a2 p \u27f6 p := classical_logic.refl p\n\n@[simp] lemma hyp_right {p : F} (h : T \u22a2 p) (q) : T \u22a2 q \u27f6 p := hyp_right h q\n\n@[simp] lemma T_hyp_eliminate {p : F} : T \u22a2 \u22a4 \u27f6 p \u2194 T \u22a2 p := T_hyp_eliminate\n\n@[simp] lemma dne (p : F) : T \u22a2 \u223c\u223cp \u27f6 p := dne p\n\n@[simp] lemma imply\u2081' {p q r : F} : T \u22a2 p \u27f6 q \u27f6 r \u27f6 p := imply\u2081'\n\n@[simp] lemma dni (p : F) : T \u22a2 p \u27f6 \u223c\u223cp := dni p\n\n@[simp] lemma dn_iff {p : F} : T \u22a2 \u223c\u223cp \u2194 T \u22a2 p := dn_iff\n\n@[simp] lemma dn1_iff {p q : F} : T \u22a2 \u223c\u223cp \u27f6 q \u2194 T \u22a2 p \u27f6 q := dn1_iff\n\n@[simp] lemma dn2_iff {p q : F} : T \u22a2 p \u27f6 \u223c\u223cq \u2194 T \u22a2 p \u27f6 q := dn2_iff\n\n@[simp] lemma hyp_bot (p : F) : T \u22a2 \u22a5 \u27f6 p := hyp_bot p\n\n@[simp] lemma and_left (p q : F) : T \u22a2 p \u2293 q \u27f6 p := and_left p q\n\n@[simp] lemma and_right (p q : F) : T \u22a2 p \u2293 q \u27f6 q := and_right p q\n\n@[simp] lemma iff_and {p q : F} : T \u22a2 p \u2293 q \u2194 (T \u22a2 p \u2227 T \u22a2 q) := iff_and_p\n\n@[simp] lemma conjunction_iff {n} {p : finitary F n} : (T \u22a2 finitary.conjunction n p) \u2194 (\u2200 i, T \u22a2 p i) :=\nconjunction_iff\n\nlemma iff_equiv {p q : F} : T \u22a2 p \u27f7 q \u2194 (T \u22a2 p \u27f6 q \u2227 T \u22a2 q \u27f6 p) := iff_equiv_p\n\nlemma equiv_mp {p q : F} (h : T \u22a2 p \u27f7 q) : T \u22a2 p \u27f6 q := (iff_equiv.mp h).1\n\nlemma equiv_mpr {p q : F} (h : T \u22a2 p \u27f7 q) : T \u22a2 q \u27f6 p := (iff_equiv.mp h).2\n\nlemma iff_of_equiv {p q : F} (h : T \u22a2 p \u27f7 q) : T \u22a2 p \u2194 T \u22a2 q := iff_of_equiv h\n\n@[refl, simp] lemma equiv_refl (p : F) : T \u22a2 p \u27f7 p := equiv_refl p\n\n@[symm] lemma equiv_symm {p q : F} : T \u22a2 p \u27f7 q \u2192 T \u22a2 q \u27f7 p := equiv_symm\n\n@[trans] lemma equiv_trans {p q r : F} : T \u22a2 p \u27f7 q \u2192 T \u22a2 q \u27f7 r \u2192 T \u22a2 p \u27f7 r := equiv_trans\n\n@[simp] lemma iff_dn_refl_right (p : F) : T \u22a2 p \u27f7 \u223c\u223cp := iff_dn_refl_right p\n\n@[simp] lemma iff_dn_refl_left (p : F) : T \u22a2 \u223c\u223cp \u27f7 p := iff_dn_refl_left p\n\n@[simp] lemma contraposition_inv (p q : F) : T \u22a2 (p \u27f6 q) \u27f6 (\u223cq \u27f6 \u223cp) := contraposition_inv p q\n\n@[simp] lemma contraposition_iff (p q : F) : T \u22a2 (p \u27f6 q) \u27f7 (\u223cq \u27f6 \u223cp) := contraposition_iff p q\n\n@[simp] lemma contraposition_iff\u2081 (p q : F) : T \u22a2 (\u223cp \u27f6 q) \u27f7 (\u223cq \u27f6 p) := contraposition_iff\u2081 _ _\n\n@[simp] lemma contraposition_iff\u2082 (p q : F) : T \u22a2 (p \u27f6 \u223cq) \u27f7 (q \u27f6 \u223cp) := contraposition_iff\u2082 _ _\n\n@[simp] lemma contraposition_iff_inv (p q : F) : T \u22a2 (\u223cp \u27f6 \u223cq) \u27f7 (q \u27f6 p) := contraposition_iff_inv p q\n\n@[simp] lemma neg_hyp' (p : F) : T \u22a2 (p \u27f6 \u223cp) \u27f6 \u223cp := neg_hyp' p\n\n@[simp] lemma neg_iff (p : F) : T \u22a2 \u223cp \u27f7 (p \u27f6 \u22a5) := neg_iff p\n\n@[simp] lemma imply_iff_of {p} (h : T \u22a2 p) (q) : T \u22a2 (p \u27f6 q) \u27f7 q := imply_iff_of h q\n\n@[simp] lemma neg_impl_equiv_and (p q : F) : T \u22a2 \u223c(p \u27f6 q) \u27f7 p \u2293 \u223cq := neg_impl_equiv_and p q\n\n\n\n\nlemma neg_impl_iff_and {p q : F} : T \u22a2 \u223c(p \u27f6 q) \u2194 T \u22a2 p \u2293 \u223cq := neg_impl_iff_and_p\n\nlemma of_equiv {p\u2081 p\u2082 : F} (h : T \u22a2 p\u2081) (hp : T \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 p\u2082 := of_equiv h hp\n\n@[simp] lemma impl_iff_or {p q : F} : T \u22a2 (p \u27f6 q) \u27f7 (\u223cp \u2294 q) := impl_iff_or_p\n\n@[simp] lemma impl_iff_or' {p q : F} : T \u22a2 (p \u27f6 q) \u27f7 (q \u2294 \u223cp) := impl_iff_or'_p\n\n@[simp] lemma excluded_middle {p : F} : T \u22a2 p \u2294 \u223cp := excluded_middle_p\n\n@[simp] lemma equiv_symm_and (p q : F) : T \u22a2 p \u2293 q \u27f7 q \u2293 p := equiv_symm_and p q\n\n@[simp] lemma equiv_symm_equiv (p q : F) : T \u22a2 (p \u27f7 q) \u27f7 (q \u27f7 p) := equiv_symm_equiv p q\n\n@[simp] lemma equiv_or_symm (p q : F) : T \u22a2 p \u2294 q \u27f7 q \u2294 p := equiv_or_symm _ _\n\n@[simp] lemma and_destruct (p q : F) : T \u22a2 p \u27f6 q \u27f6 p \u2293 q := and_destruct p q\n\n@[simp] lemma imply_or_left (p q : F) : T \u22a2 p \u27f6 p \u2294 q := imply_or_left p q\n\n@[simp] lemma imply_or_right (p q : F) : T \u22a2 q \u27f6 p \u2294 q := imply_or_right p q\n\nlemma disjunction_of {n} {p : finitary F n} (i) (h : T \u22a2 p i) : T \u22a2 finitary.disjunction n p :=\ndisjunction_of i h\n\n@[simp] lemma imply_and (p q r : F) : T \u22a2 (p \u27f6 q) \u27f6 (p \u27f6 r) \u27f6 p \u27f6 q \u2293 r := imply_and p q r\n\n@[simp] lemma neg_and_equiv_or_neg (p q : F) : T \u22a2 \u223c(p \u2293 q) \u27f7 \u223cp \u2294 \u223cq := neg_and_equiv_or_neg p q\n\n@[simp] lemma neg_conj_equiv_disj_neg {n} (p : finitary F n) : T \u22a2 \u223c(finitary.conjunction n p) \u27f7 (\u22c1 i, \u223cp i) :=\nneg_conj_equiv_disj_neg p\n\n@[simp] lemma neg_or_equiv_and_neg (p q : F) : T \u22a2 \u223c(p \u2294 q) \u27f7 \u223cp \u2293 \u223cq := neg_or_equiv_and_neg p q\n\n@[simp] lemma neg_disj_equiv_conj_neg {n} (p : finitary F n) : T \u22a2 \u223c(finitary.disjunction n p) \u27f7 (\u22c0 i, \u223cp i) :=\nneg_disj_equiv_conj_neg p\n\n@[simp] lemma or_imply (p q r : F) : T \u22a2 (p \u27f6 r) \u27f6 (q \u27f6 r) \u27f6 p \u2294 q \u27f6 r := or_imply p q r\n\nlemma cases_of (p q : F) (ht : T \u22a2 p \u27f6 q) (hf : T \u22a2 \u223cp \u27f6 q) : T \u22a2 q :=\nor_imply p (\u223cp) q \u2a00 ht \u2a00 hf \u2a00 (by simp)\n\n@[simp] lemma and_imply_equiv_imply_imply (p q r : F) : T \u22a2 (p \u27f6 q \u27f6 r) \u27f7 (p \u2293 q \u27f6 r) := and_imply_equiv_imply_imply p q r\n\n@[simp] lemma and_imply_of {p q r : F} : T \u22a2 p \u2293 q \u27f6 r \u2194 T \u22a2 p \u27f6 q \u27f6 r :=\nand_imply_of\n\n@[simp] lemma imply_and_equiv_or_imply (p q r : F) : T \u22a2 (p \u27f6 r) \u2293 (q \u27f6 r) \u27f7 p \u2294 q \u27f6 r := imply_and_equiv_or_imply p q r\n\n@[simp] lemma conj_imply_iff_disj_imply {n} (p : finitary F n) (q : F) : T \u22a2 (\u22c0 i, (p i \u27f6 q)) \u27f7 ((\u22c1 i, p i) \u27f6 q) := conj_imply_iff_disj_imply p q\n\nlemma explosion {p : F} (h\u2081 : T \u22a2 p) (h\u2082 : T \u22a2 \u223cp) {q : F} : T \u22a2 q :=\nexplosion h\u2081 h\u2082\n\nlemma contrapose {p q : F} : T \u22a2 \u223cp \u27f6 \u223cq \u2194 T \u22a2 q \u27f6 p :=\ncontrapose\n\nlemma and_imply_of_imply_left {p\u2081 p\u2082 q : F} (h : T \u22a2 p\u2081 \u27f6 q) : T \u22a2 p\u2081 \u2293 p\u2082 \u27f6 q :=\nand_imply_of_imply_left h\n\nlemma and_imply_of_imply_right {p\u2081 p\u2082 q : F} (h : T \u22a2 p\u2082 \u27f6 q) : T \u22a2 p\u2081 \u2293 p\u2082 \u27f6 q :=\nand_imply_of_imply_right h\n\nlemma equiv_imply_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082 : F} (hp : T \u22a2 p\u2081 \u27f7 p\u2082) (hq : T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 (p\u2081 \u27f6 q\u2081) \u27f7 (p\u2082 \u27f6 q\u2082) :=\nequiv_imply_of_equiv hp hq\n\nlemma imply_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082 : F} (h : T \u22a2 p\u2081 \u27f6 q\u2081) (hp : T \u22a2 p\u2081 \u27f7 p\u2082) (hq : T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2082 \u27f6 q\u2082 :=\nimply_of_equiv h hp hq\n\nlemma equiv_neg_of_equiv {p\u2081 p\u2082 : F} (hp : T \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u223cp\u2081 \u27f7 \u223cp\u2082 :=\nequiv_neg_of_equiv hp\n\nlemma neg_of_equiv {p\u2081 p\u2082 : F} (h : T \u22a2 \u223cp\u2081) (hp : T \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u223cp\u2082 :=\nneg_of_equiv h hp\n\nlemma equiv_and_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082 : F} (hp : T \u22a2 p\u2081 \u27f7 p\u2082) (hq : T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2081 \u2293 q\u2081 \u27f7 p\u2082 \u2293 q\u2082 :=\nequiv_and_of_equiv hp hq\n\nlemma and_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : T \u22a2 p\u2081 \u2293 q\u2081) (hp : T \u22a2 p\u2081 \u27f7 p\u2082) (hq : T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2082 \u2293 q\u2082 :=\nand_of_equiv h hp hq\n\nlemma equiv_or_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (hp :  T \u22a2 p\u2081 \u27f7 p\u2082) (hq :  T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2081 \u2294 q\u2081 \u27f7 p\u2082 \u2294 q\u2082 :=\nequiv_or_of_equiv hp hq\n\nlemma or_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : T \u22a2 p\u2081 \u2294 q\u2081) (hp :  T \u22a2 p\u2081 \u27f7 p\u2082) (hq :  T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2082 \u2294 q\u2082 :=\nor_of_equiv h hp hq\n\nlemma equiv_equiv_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (hp :  T \u22a2 p\u2081 \u27f7 p\u2082) (hq :  T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 (p\u2081 \u27f7 q\u2081) \u27f7 (p\u2082 \u27f7 q\u2082) :=\nequiv_equiv_of_equiv hp hq\n\nlemma equiv_of_equiv {p\u2081 q\u2081 p\u2082 q\u2082: F} (h : T \u22a2 p\u2081 \u27f7 q\u2081) (hp :  T \u22a2 p\u2081 \u27f7 p\u2082) (hq :  T \u22a2 q\u2081 \u27f7 q\u2082) : T \u22a2 p\u2082 \u27f7 q\u2082 :=\nequiv_of_equiv h hp hq\n\nlemma case_of_ax {p q r : F} (hpq : T \u22a2 p \u2294 q) (hpr : T \u22a2 p \u27f6 r) (hqr : T \u22a2 q \u27f6 r) : T \u22a2 r :=\ncase_of_p hpq hpr hqr\n\n@[simp] lemma neg_lconj_equiv_ldisj_neg (l : list F) : T \u22a2 \u223cl.conjunction \u27f7 (l.map has_negation.neg).disjunction :=\nneg_lconj_equiv_ldisj_neg l\n\nlemma ldisj_of {l : list F} (p) (hp : p \u2208 l) (h : T \u22a2 p) : T \u22a2 l.disjunction :=\nldisj_of hp h\n\nlemma fdisj_of {s : finset F} (p) (hp : p \u2208 s) (h : T \u22a2 p) : T \u22a2 s.disjunction :=\nfdisj_of hp h\n\n@[simp] lemma imply_ldisj {l : list F} {p : F} (h : p \u2208 l) : T \u22a2 p \u27f6 l.disjunction :=\nimply_ldisj h\n\nlemma imply_fdisj {s : finset F} {p : F} (h : p \u2208 s) : T \u22a2 p \u27f6 s.disjunction :=\nimply_fdisj h\n\n@[simp] lemma ldisj_imply_of {l : list F} {q : F} (h : \u2200 p \u2208 l, T \u22a2 p \u27f6 q) : T \u22a2 l.disjunction \u27f6 q :=\nldisj_imply_of h\n\n@[simp] lemma fdisj_imply_of {s : finset F} {q : F} (h : \u2200 p \u2208 s, T \u22a2 p \u27f6 q) : T \u22a2 s.disjunction \u27f6 q :=\nfdisj_imply_of h\n\nlemma fdisj_insert_equiv_sup {s : finset F} {p : F} : T \u22a2 (insert p s).disjunction \u27f7 s.disjunction \u2294 p :=\nfdisj_insert_equiv_sup\n\nlemma fconj_imply_equiv_fdisj_neg (p : F) (s : finset F) :\n  T \u22a2 (s.conjunction \u27f6 p) \u27f7 (insert p (s.image has_negation.neg)).disjunction :=\nfconj_imply_equiv_fdisj_neg p s\n\n@[simp] lemma insert (p) : T +{ p } \u22a2 p := by_axiom (by simp)\n\n@[simp] lemma singleton (p) : (singleton p : set F) \u22a2 p := by_axiom (by simp)\n\nlemma by_axiom' {T : set F} {p : F} : T p \u2192 T \u22a2 p := by_axiom\n\n@[simp] lemma provable_not_bot_iff : T \u22a2 \u22a5 \u27f7 \u223c(\u22a4 : F) := by simp[@not_top_eq_bot F _ ((\u22a2) T) _]\n\n@[simp] lemma provable_not_bot_iff' : T \u22a2 \u223c(\u22a4 : F) \u27f7 \u22a5 := by simp[@not_top_eq_bot F _ ((\u22a2) T) _]\n\n@[simp] lemma neg_bot_equiv_top : T \u22a2 (\u223c\u22a5 : F) \u27f7 \u22a4 :=\nneg_bot_equiv_top\n\nvariables (T)\n\n@[reducible] def equiv : F \u2192 F \u2192 Prop := equiv ((\u22a2) T)\n\nend axiomatic_classical_logic'\n\nnamespace axiomatic_classical_logic\nopen axiomatic_classical_logic'\nopen_locale aclogic\n\nvariables {F : Type*} [has_logic_symbol F]\n  (T : set F) [axiomatic_classical_logic F]\n\nvariables {T}\n\n@[simp] lemma weakening_insert {q : F} (h : T \u22a2 q) (p) : T +{ p } \u22a2 q :=\nweakening (show T \u2286 T +{ p }, by { intros x h, simp[h] }) h\n\ntheorem deduction {p q} : (T +{ p } \u22a2 q) \u2194 (T \u22a2 p \u27f6 q) :=\n\u27e8deduction', \u03bb h, by { have : T +{ p } \u22a2 p \u27f6 q, simp[h], exact this \u2a00 (by simp) }\u27e9\n\n@[simp]\nlemma axiom_and {p\u2081 p\u2082 q : F} : T +{ p\u2081 \u2293 p\u2082 } \u22a2 q \u2194 T +{ p\u2081 } +{ p\u2082 } \u22a2 q :=\n\u27e8\u03bb h,\n by { have lmm\u2081 : T +{ p\u2081 } +{ p\u2082 } \u22a2 p\u2081 \u2293 p\u2082, by simp[axiomatic_classical_logic'.iff_and],\n      have lmm\u2082 : T +{ p\u2081 } +{ p\u2082 } \u22a2 p\u2081 \u2293 p\u2082 \u27f6 q, simp[deduction.mp h],\n      exact lmm\u2082 \u2a00 lmm\u2081 },\n \u03bb h,\n by { have lmm\u2081 : T +{ p\u2081 \u2293 p\u2082 } \u22a2 p\u2081 \u27f6 p\u2082 \u27f6 q, simp[deduction.mp (deduction.mp h)],\n      have lmm\u2082 : T +{ p\u2081 \u2293 p\u2082 } \u22a2 p\u2081 \u2293 p\u2082, from insert _, simp only [axiomatic_classical_logic'.iff_and] at lmm\u2082,\n      exact lmm\u2081 \u2a00 lmm\u2082.1 \u2a00 lmm\u2082.2 } \u27e9\n\nlemma axiom_lconj {l : list F} {q : F} : T +{ l.conjunction } \u22a2 q \u2194 T \u222a { p | p \u2208 l } \u22a2 q :=\nbegin\n  induction l with p l IH generalizing T,\n  { simp[deduction] },\n  { simp[IH, show insert p T \u222a {p : F | p \u2208 l} = T \u222a {r : F | r = p \u2228 r \u2208 l}, by ext r; simp; tauto] }\nend\n\nlemma axiom_fconj {s : finset F} {q : F} : T +{ s.conjunction } \u22a2 q \u2194 T \u222a \u2191s \u22a2 q :=\nby simp[finset.conjunction, axiom_lconj]\n\nlemma fdisj_insert {s : finset F} {p : F} : \u2205 \u22a2 (insert p s).disjunction \u2194 has_negation.neg '' \u2191s \u22a2 p :=\nbegin\n  have empty_iff : {(finset.image has_negation.neg s).conjunction} \u22a2 p \u2194 has_negation.neg '' \u2191s \u22a2 p,\n  by simpa using @axiom_fconj F _ \u2205 _ (s.image has_negation.neg) p,\n  have : \u2205 \u22a2 (insert p s).disjunction \u2194 \u2205 \u22a2 ((s.image has_negation.neg).conjunction \u27f6 p),\n  from iff_of_equiv (classical_logic.fdisj_insert_equiv_fconj_imply p s),\n  simp[this, \u2190deduction], exact empty_iff\nend\n\n@[simp] lemma axiom_and' {p\u2081 p\u2082 q : F} : singleton (p\u2081 \u2293 p\u2082) \u22a2 q \u2194 singleton p\u2081 +{ p\u2082 } \u22a2 q :=\nby simpa using @axiom_and _ _ \u2205 _ p\u2081 p\u2082 q\n\nlemma raa {p : F} (q : F) (h\u2081 : T+{p} \u22a2 q) (h\u2082 : T+{p} \u22a2 \u223cq) : T \u22a2 \u223cp :=\nclassical_logic.neg_hyp (deduction.mp (classical_logic.explosion h\u2081 h\u2082))\n\nlemma list_conjunction_mem {P : list F} : \u2200 {p}, p \u2208 P \u2192 T \u22a2 P.conjunction \u27f6 p :=\nby { induction P with p P IH; simp; intros q hq, exact hyp_right (IH hq) p }\n\nlemma list_conjunction_weakening {P Q : list F} : \n  Q \u2286 P \u2192 T \u22a2 P.conjunction \u27f6 Q.conjunction :=\nbegin\n  induction Q with q Q IH; simp,\n  intros hyp_q hyp_Q,\n  have lmm\u2081 : T+{P.conjunction} \u22a2 q, from deduction.mpr (list_conjunction_mem hyp_q),  \n  have lmm\u2082 : T+{P.conjunction} \u22a2 Q.conjunction, from deduction.mpr (IH hyp_Q),\n  refine deduction.mp _, simp[axiomatic_classical_logic'.iff_and, *]\nend\n\nlemma list_conjunction_provable : \u2200 {P : list F} (h : \u2200 p, p \u2208 P \u2192 T \u22a2 p), T \u22a2 P.conjunction\n| []       h := by simp\n| (p :: P) h := by {\n    have lmm\u2081 : T \u22a2 p, { refine h _ _, simp },\n    have lmm\u2082 : T \u22a2 P.conjunction,\n    { refine list_conjunction_provable (\u03bb p hyp, h _ _), simp, right, exact hyp },\n    simp, refine \u27e8lmm\u2081, lmm\u2082\u27e9 }\n\n/-\nlemma finitary.conjunction_mem {n : \u2115} {P : finitary F n} :\n  \u2200 {p}, p \u2208 P \u2192 T \u22a2 finitary.conjunction n P \u27f6 p :=\nbegin\n  induction n with n IH; simp[finitary.conjunction],\n  { simp[has_mem.mem, finitary.mem] },\n  { intros p hp, rcases hp with (_ | _), }\n  --simp[has_mem.mem, finitary.mem],\n  --intros p mem,\n  --exact and_imply_of_imply_right (IH mem)\nend\n-/\n\n@[reducible] def empty_axiom (p : F) : Prop := \u2205 \u22a2 p\n\nprefix `\u2b1d\u22a2 `:45 := empty_axiom\n\nsection empty_axiom\nvariables (T)\n\nlemma of_empty_axiom {p : F} (hp : \u2b1d\u22a2 p) : T \u22a2 p := weakening (by simp) hp\n\nend empty_axiom\n\nvariables (T)\n\n@[reducible] def lindenbaum := classical_logic.lindenbaum ((\u22a2) T : F \u2192 Prop)\n\nnotation (name := classical_logic.equiv) p ` \u2248[`:50 T :50 `] `:0 q:50 := classical_logic.equiv ((\u22a2) T) p q\n\nnamespace lindenbaum\n\ninstance : boolean_algebra (lindenbaum T) := classical_logic.lindenbaum.boolean_algebra\n\nend lindenbaum\n\nend axiomatic_classical_logic", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/provability.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3922414884210498}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n\nimport category_theory.category\nimport pseudoelements\nimport tactic.combinators\nimport tactic.chase_tactic\nimport tactic.commutativity\n\nopen category_theory\nopen category_theory.abelian\nopen category_theory.abelian.pseudoelements\nopen tactic\n\nnamespace tactic.chase\n\nmeta def pushforward_single (e : expr) (m : morphism) (n : name) : chase_tactic expr :=\ndo\n  cr \u2190 pose n none (expr.app m.app e),\n  hyp_name \u2190 get_unused_name (\"h\" ++ n),\n  ty \u2190 i_to_expr ``(%%(expr.app m.app e) = %%cr),\n  x \u2190 mk_eq_refl cr,\n  hyp \u2190 assertv hyp_name ty x,\n  some l \u2190 as_element_lemma hyp,\n  add_elem_lemma l,\n  return cr\n\nmeta def pushforward : expr \u2192 morphism_chain \u2192 name \u2192 chase_tactic (option expr)\n| _ [] _ := return none\n| e (m::[]) n := some <$> pushforward_single e m n\n| e (m::ms) n := do\n  nn \u2190 mk_fresh_name,\n  some f \u2190 pushforward e ms nn,\n  some <$> pushforward_single f m n\n\nmeta def pullback (e : expr) (m : morphism_chain) (n : name) : chase_tactic (option expr) :=\ndo\nexactness_lemmas_for m >>= list.mfoldl (\u03bb r l,\n  match r with\n  | some r := return $ some r\n  | none := do\n    z \u2190 diagram_term.zero \u27e8l.rhs, e\u27e9,\n    cond \u2190 find_proof \u27e8l.rhs, e\u27e9 z,\n    match cond with\n    | none := return none\n    | some p := do\n      s \u2190 i_to_expr ``(exists.elim ((pseudo_exact_of_exact %%l.ex).2 %%e %%p)),\n      tactic.apply s,\n      f \u2190 tactic.intro n,\n      hyp_name \u2190 get_unused_name (\"h\" ++ n),\n      hyp \u2190 tactic.intro hyp_name,\n      some hy \u2190 as_element_lemma hyp,\n      add_elem_lemma hy,\n      return f\n    end\n  end) none\n\nmeta def chase : expr \u2192 list morphism_chain \u2192 list name \u2192 chase_tactic unit\n| e [] _ := skip\n| e (m::ms) [] := do n \u2190 get_unused_name \"x\", chase e (m::ms) [n]\n| e (m::ms) (n::ns) :=\n  do\n    ca \u2190 morphism_chain.can_apply e m,\n    some ne \u2190 if ca then pushforward e m n else pullback e m n,\n    chase ne ms ns\n\nend tactic.chase\n\nnamespace tactic.interactive\n\nopen interactive (parse)\nopen lean.parser (tk pexpr)\nopen interactive.types (texpr with_ident_list pexpr_list)\n\nmeta def chase (s : parse pexpr) (maps : parse (tk \"using\" *> pexpr_list))\n  (ids : parse with_ident_list) (loc : parse ((tk \"at\" *> some <$> pexpr) <|> return none)) : tactic unit :=\ndo\n  e \u2190 i_to_expr s,\n  l \u2190 match loc with\n      | none := return none\n      | some m := some <$> to_expr m\n      end,\n  mps \u2190 list.mmap\n    (\u03bb p, do q \u2190 i_to_expr p, some ch \u2190 tactic.chase.as_morphism q, return [ch]) maps,\n  tactic.chase.run_chase_tactic l $ tactic.chase.chase e mps ids\n\nend tactic.interactive\n", "meta": {"author": "TwoFX", "repo": "lean-homological-algebra", "sha": "e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8", "save_path": "github-repos/lean/TwoFX-lean-homological-algebra", "path": "github-repos/lean/TwoFX-lean-homological-algebra/lean-homological-algebra-e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8/src/tactic/diagram_chase.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389325, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3922414884210497}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.order\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 \n\nnamespace Mathlib\n\n/-!\n# Specific classes of maps between topological spaces\n\nThis file introduces the following properties of a map `f : X \u2192 Y` between topological spaces:\n\n* `is_open_map f` means the image of an open set under `f` is open.\n* `is_closed_map f` means the image of a closed set under `f` is closed.\n\n(Open and closed maps need not be continuous.)\n\n* `inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`.\n  These behave like embeddings except they need not be injective. Instead, points of `X` which\n  are identified by `f` are also indistinguishable in the topology on `X`.\n* `embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with\n  a subspace of `Y`.\n* `open_embedding f` means `f` is an embedding with open image, so it identifies `X` with an\n  open subspace of `Y`. Equivalently, `f` is an embedding and an open map.\n* `closed_embedding f` similarly means `f` is an embedding with closed image, so it identifies\n  `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map.\n\n* `quotient_map f` is the dual condition to `embedding f`: `f` is surjective and the topology\n  on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies\n  `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Open_and_closed_maps>\n* <https://en.wikipedia.org/wiki/Embedding#General_topology>\n* <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map>\n\n## Tags\n\nopen map, closed map, embedding, quotient map, identification map\n\n-/\n\nstructure inducing {\u03b1 : Type u_1} {\u03b2 : Type u_2} [t\u03b1 : topological_space \u03b1]\n    [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2)\n    where\n  induced : t\u03b1 = topological_space.induced f t\u03b2\n\ntheorem inducing_id {\u03b1 : Type u_1} [topological_space \u03b1] : inducing id :=\n  inducing.mk (Eq.symm induced_id)\n\nprotected theorem inducing.comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : inducing g)\n    (hf : inducing f) : inducing (g \u2218 f) :=\n  sorry\n\ntheorem inducing_of_inducing_compose {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3}\n    [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n    (hf : continuous f) (hg : continuous g) (hgf : inducing (g \u2218 f)) : inducing f :=\n  sorry\n\ntheorem inducing_open {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hf : inducing f) (h : is_open (set.range f)) (hs : is_open s) :\n    is_open (f '' s) :=\n  sorry\n\ntheorem inducing_is_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hf : inducing f) (h : is_closed (set.range f)) (hs : is_closed s) :\n    is_closed (f '' s) :=\n  sorry\n\ntheorem inducing.nhds_eq_comap {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) :\n    nhds a = filter.comap f (nhds (f a)) :=\n  iff.mp (induced_iff_nhds_eq f) (inducing.induced hf)\n\ntheorem inducing.map_nhds_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) (h : set.range f \u2208 nhds (f a)) :\n    filter.map f (nhds a) = nhds (f a) :=\n  Eq.symm (inducing.induced hf) \u25b8 map_nhds_induced_eq h\n\ntheorem inducing.tendsto_nhds_iff {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b2]\n    [topological_space \u03b3] {\u03b9 : Type u_1} {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2}\n    (hg : inducing g) : filter.tendsto f a (nhds b) \u2194 filter.tendsto (g \u2218 f) a (nhds (g b)) :=\n  sorry\n\ntheorem inducing.continuous_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) :\n    continuous f \u2194 continuous (g \u2218 f) :=\n  sorry\n\ntheorem inducing.continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : inducing f) : continuous f :=\n  iff.mp (inducing.continuous_iff hf) continuous_id\n\n/-- A function between topological spaces is an embedding if it is injective,\n  and for all `s : set \u03b1`, `s` is open iff it is the preimage of an open set. -/\nstructure embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [t\u03b1 : topological_space \u03b1]\n    [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2)\n    extends inducing f where\n  inj : function.injective f\n\ntheorem embedding.mk' {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2) (inj : function.injective f)\n    (induced : \u2200 (a : \u03b1), filter.comap f (nhds (f a)) = nhds a) : embedding f :=\n  embedding.mk (inducing.mk (iff.mpr (induced_iff_nhds_eq f) fun (a : \u03b1) => Eq.symm (induced a)))\n    inj\n\ntheorem embedding_id {\u03b1 : Type u_1} [topological_space \u03b1] : embedding id :=\n  embedding.mk inducing_id fun (a\u2081 a\u2082 : \u03b1) (h : id a\u2081 = id a\u2082) => h\n\ntheorem embedding.comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : embedding g)\n    (hf : embedding f) : embedding (g \u2218 f) :=\n  embedding.mk\n    (inducing.mk\n      (inducing.induced (inducing.comp (embedding.to_inducing hg) (embedding.to_inducing hf))))\n    fun (a\u2081 a\u2082 : \u03b1) (h : function.comp g f a\u2081 = function.comp g f a\u2082) =>\n      embedding.inj hf (embedding.inj hg h)\n\ntheorem embedding_of_embedding_compose {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3}\n    [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n    (hf : continuous f) (hg : continuous g) (hgf : embedding (g \u2218 f)) : embedding f :=\n  sorry\n\ntheorem embedding_open {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hf : embedding f) (h : is_open (set.range f)) (hs : is_open s) :\n    is_open (f '' s) :=\n  inducing_open (embedding.to_inducing hf) h hs\n\ntheorem embedding_is_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hf : embedding f) (h : is_closed (set.range f))\n    (hs : is_closed s) : is_closed (f '' s) :=\n  inducing_is_closed (embedding.to_inducing hf) h hs\n\ntheorem embedding.map_nhds_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : embedding f) (a : \u03b1) (h : set.range f \u2208 nhds (f a)) :\n    filter.map f (nhds a) = nhds (f a) :=\n  inducing.map_nhds_eq (embedding.to_inducing hf) a h\n\ntheorem embedding.tendsto_nhds_iff {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b2]\n    [topological_space \u03b3] {\u03b9 : Type u_1} {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2}\n    (hg : embedding g) : filter.tendsto f a (nhds b) \u2194 filter.tendsto (g \u2218 f) a (nhds (g b)) :=\n  sorry\n\ntheorem embedding.continuous_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : embedding g) :\n    continuous f \u2194 continuous (g \u2218 f) :=\n  inducing.continuous_iff (embedding.to_inducing hg)\n\ntheorem embedding.continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : embedding f) : continuous f :=\n  inducing.continuous (embedding.to_inducing hf)\n\ntheorem embedding.closure_eq_preimage_closure_image {\u03b1 : Type u_1} {\u03b2 : Type u_2}\n    [topological_space \u03b1] [topological_space \u03b2] {e : \u03b1 \u2192 \u03b2} (he : embedding e) (s : set \u03b1) :\n    closure s = e \u207b\u00b9' closure (e '' s) :=\n  sorry\n\n/-- A function between topological spaces is a quotient map if it is surjective,\n  and for all `s : set \u03b2`, `s` is open iff its preimage is an open set. -/\ndef quotient_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2) :=\n  function.surjective f \u2227 t\u03b2 = topological_space.coinduced f t\u03b1\n\ntheorem quotient_map_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} :\n    quotient_map f \u2194 function.surjective f \u2227 \u2200 (s : set \u03b2), is_open s \u2194 is_open (f \u207b\u00b9' s) :=\n  and_congr iff.rfl topological_space_eq_iff\n\nnamespace quotient_map\n\n\nprotected theorem id {\u03b1 : Type u_1} [topological_space \u03b1] : quotient_map id :=\n  { left := fun (a : \u03b1) => Exists.intro a rfl, right := Eq.symm coinduced_id }\n\nprotected theorem comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : quotient_map g)\n    (hf : quotient_map f) : quotient_map (g \u2218 f) :=\n  sorry\n\nprotected theorem of_quotient_map_compose {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3}\n    [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n    (hf : continuous f) (hg : continuous g) (hgf : quotient_map (g \u2218 f)) : quotient_map g :=\n  sorry\n\nprotected theorem continuous_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : quotient_map f) :\n    continuous g \u2194 continuous (g \u2218 f) :=\n  sorry\n\nprotected theorem continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : continuous f :=\n  iff.mp (quotient_map.continuous_iff hf) continuous_id\n\nprotected theorem surjective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : function.surjective f :=\n  and.left hf\n\nprotected theorem is_open_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) {s : set \u03b2} :\n    is_open (f \u207b\u00b9' s) \u2194 is_open s :=\n  iff.symm (and.right (iff.mp quotient_map_iff hf) s)\n\nend quotient_map\n\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be an *open map*, if the image of any open `U : set \u03b1`\nis open in `\u03b2`. -/\ndef is_open_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 (U : set \u03b1), is_open U \u2192 is_open (f '' U)\n\nnamespace is_open_map\n\n\nprotected theorem id {\u03b1 : Type u_1} [topological_space \u03b1] : is_open_map id :=\n  fun (s : set \u03b1) (hs : is_open s) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (is_open (id '' s))) (set.image_id s))) hs\n\nprotected theorem comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_open_map g)\n    (hf : is_open_map f) : is_open_map (g \u2218 f) :=\n  id\n    fun (s : set \u03b1) (hs : is_open s) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (is_open (g \u2218 f '' s))) (set.image_comp g f s)))\n        (hg (f '' s) (hf s hs))\n\ntheorem is_open_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (hf : is_open_map f) : is_open (set.range f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_open (set.range f))) (Eq.symm set.image_univ)))\n    (hf set.univ is_open_univ)\n\ntheorem image_mem_nhds {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (hf : is_open_map f) {x : \u03b1} {s : set \u03b1} (hx : s \u2208 nhds x) : f '' s \u2208 nhds (f x) :=\n  sorry\n\ntheorem nhds_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (hf : is_open_map f) (a : \u03b1) : nhds (f a) \u2264 filter.map f (nhds a) :=\n  filter.le_map fun (s : set \u03b1) => image_mem_nhds hf\n\ntheorem of_nhds_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (hf : \u2200 (a : \u03b1), nhds (f a) \u2264 filter.map f (nhds a)) : is_open_map f :=\n  sorry\n\ntheorem of_inverse {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1} (h : continuous f') (l_inv : function.left_inverse f f')\n    (r_inv : function.right_inverse f f') : is_open_map f :=\n  sorry\n\ntheorem to_quotient_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (open_map : is_open_map f) (cont : continuous f) (surj : function.surjective f) :\n    quotient_map f :=\n  sorry\n\nend is_open_map\n\n\ntheorem is_open_map_iff_nhds_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    is_open_map f \u2194 \u2200 (a : \u03b1), nhds (f a) \u2264 filter.map f (nhds a) :=\n  { mp := fun (hf : is_open_map f) => is_open_map.nhds_le hf, mpr := is_open_map.of_nhds_le }\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be a *closed map*, if the image of any closed `U : set \u03b1`\nis closed in `\u03b2`. -/\ndef is_closed_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 (U : set \u03b1), is_closed U \u2192 is_closed (f '' U)\n\nnamespace is_closed_map\n\n\nprotected theorem id {\u03b1 : Type u_1} [topological_space \u03b1] : is_closed_map id :=\n  fun (s : set \u03b1) (hs : is_closed s) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (id '' s))) (set.image_id s))) hs\n\nprotected theorem comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_closed_map g)\n    (hf : is_closed_map f) : is_closed_map (g \u2218 f) :=\n  id\n    fun (s : set \u03b1) (hs : is_closed s) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (is_closed (g \u2218 f '' s))) (set.image_comp g f s)))\n        (hg (f '' s) (hf s hs))\n\ntheorem of_inverse {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1} (h : continuous f') (l_inv : function.left_inverse f f')\n    (r_inv : function.right_inverse f f') : is_closed_map f :=\n  sorry\n\ntheorem of_nonempty {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} (h : \u2200 (s : set \u03b1), is_closed s \u2192 set.nonempty s \u2192 is_closed (f '' s)) :\n    is_closed_map f :=\n  sorry\n\nend is_closed_map\n\n\n/-- An open embedding is an embedding with open image. -/\nstructure open_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2)\n    extends embedding f where\n  open_range : is_open (set.range f)\n\ntheorem open_embedding.open_iff_image_open {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) {s : set \u03b1} :\n    is_open s \u2194 is_open (f '' s) :=\n  sorry\n\ntheorem open_embedding.is_open_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : is_open_map f :=\n  fun (s : set \u03b1) => iff.mp (open_embedding.open_iff_image_open hf)\n\ntheorem open_embedding.continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : continuous f :=\n  embedding.continuous (open_embedding.to_embedding hf)\n\ntheorem open_embedding.open_iff_preimage_open {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) {s : set \u03b2} (hs : s \u2286 set.range f) :\n    is_open s \u2194 is_open (f \u207b\u00b9' s) :=\n  sorry\n\ntheorem open_embedding_of_embedding_open {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f) (h\u2082 : is_open_map f) : open_embedding f :=\n  sorry\n\ntheorem open_embedding_of_continuous_injective_open {\u03b1 : Type u_1} {\u03b2 : Type u_2}\n    [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : continuous f)\n    (h\u2082 : function.injective f) (h\u2083 : is_open_map f) : open_embedding f :=\n  sorry\n\ntheorem open_embedding_id {\u03b1 : Type u_1} [topological_space \u03b1] : open_embedding id :=\n  open_embedding.mk embedding_id\n    (eq.mpr\n      ((fun (s s_1 : set \u03b1) (e_2 : s = s_1) => congr_arg is_open e_2) (set.range id) set.univ\n        set.range_id)\n      is_open_univ)\n\ntheorem open_embedding.comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : open_embedding g)\n    (hf : open_embedding f) : open_embedding (g \u2218 f) :=\n  sorry\n\n/-- A closed embedding is an embedding with closed image. -/\nstructure closed_embedding {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1] [topological_space \u03b2]\n    (f : \u03b1 \u2192 \u03b2)\n    extends embedding f where\n  closed_range : is_closed (set.range f)\n\ntheorem closed_embedding.continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : closed_embedding f) : continuous f :=\n  embedding.continuous (closed_embedding.to_embedding hf)\n\ntheorem closed_embedding.closed_iff_image_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : closed_embedding f) {s : set \u03b1} :\n    is_closed s \u2194 is_closed (f '' s) :=\n  sorry\n\ntheorem closed_embedding.is_closed_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : closed_embedding f) : is_closed_map f :=\n  fun (s : set \u03b1) => iff.mp (closed_embedding.closed_iff_image_closed hf)\n\ntheorem closed_embedding.closed_iff_preimage_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2}\n    [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : closed_embedding f) {s : set \u03b2}\n    (hs : s \u2286 set.range f) : is_closed s \u2194 is_closed (f \u207b\u00b9' s) :=\n  sorry\n\ntheorem closed_embedding_of_embedding_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f) (h\u2082 : is_closed_map f) :\n    closed_embedding f :=\n  sorry\n\ntheorem closed_embedding_of_continuous_injective_closed {\u03b1 : Type u_1} {\u03b2 : Type u_2}\n    [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : continuous f)\n    (h\u2082 : function.injective f) (h\u2083 : is_closed_map f) : closed_embedding f :=\n  sorry\n\ntheorem closed_embedding_id {\u03b1 : Type u_1} [topological_space \u03b1] : closed_embedding id :=\n  closed_embedding.mk embedding_id\n    (eq.mpr\n      ((fun (s s_1 : set \u03b1) (e_2 : s = s_1) => congr_arg is_closed e_2) (set.range id) set.univ\n        set.range_id)\n      is_closed_univ)\n\ntheorem closed_embedding.comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [topological_space \u03b1]\n    [topological_space \u03b2] [topological_space \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : closed_embedding g)\n    (hf : closed_embedding f) : closed_embedding (g \u2218 f) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/maps_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.6791787056691697, "lm_q1q2_score": 0.3922225546568441}}
{"text": "\n\ndef A.foo {\u03b1 : Type} [Add \u03b1] (a : \u03b1) : \u03b1 \u00d7 \u03b1 :=\n(a, a + a)\n\ndef B.foo {\u03b1 : Type} (a : \u03b1) : \u03b1 \u00d7 \u03b1 :=\n(a, a)\n\nopen A\nopen B\n\nset_option trace.Meta.synthInstance true\n-- `foo` is overloaded, the case `A.foo` is discarded because we don't have an instance `[Add String]`.\n-- However, we still want to see the trace since we used trace.Meta.synthInstance\n#check foo \"hello\"\n\ntheorem ex : foo true = (true, true) :=\nrfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/infoFromFailure.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3922225509067182}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebraic_geometry.presheafed_space\n! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.Sheaves.Presheaf\nimport Mathbin.CategoryTheory.Adjunction.FullyFaithful\n\n/-!\n# Presheafed spaces\n\nIntroduces the category of topological spaces equipped with a presheaf (taking values in an\narbitrary target category `C`.)\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\n\nuniverse w v u\n\nopen CategoryTheory\n\nopen TopCat\n\nopen TopologicalSpace\n\nopen Opposite\n\nopen CategoryTheory.Category CategoryTheory.Functor\n\nvariable (C : Type u) [Category.{v} C]\n\nattribute [local tidy] tactic.op_induction' tactic.auto_cases_opens\n\nnamespace AlgebraicGeometry\n\n/-- A `PresheafedSpace C` is a topological space equipped with a presheaf of `C`s. -/\nstructure PresheafedSpace where\n  carrier : TopCat.{w}\n  Presheaf : carrier.Presheaf C\n#align algebraic_geometry.PresheafedSpace AlgebraicGeometry.PresheafedSpace\n\nvariable {C}\n\nnamespace PresheafedSpace\n\nattribute [protected] presheaf\n\ninstance coeCarrier : Coe (PresheafedSpace.{w, v, u} C) TopCat.{w} where coe X := X.carrier\n#align algebraic_geometry.PresheafedSpace.coe_carrier AlgebraicGeometry.PresheafedSpace.coeCarrier\n\n@[simp]\ntheorem as_coe (X : PresheafedSpace.{w, v, u} C) : X.carrier = (X : TopCat.{w}) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.as_coe AlgebraicGeometry.PresheafedSpace.as_coe\n\n@[simp]\ntheorem mk_coe (carrier) (presheaf) :\n    (({     carrier\n            Presheaf } : PresheafedSpace.{v} C) : TopCat.{v}) = carrier :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.mk_coe AlgebraicGeometry.PresheafedSpace.mk_coe\n\ninstance (X : PresheafedSpace.{v} C) : TopologicalSpace X :=\n  X.carrier.str\n\n/-- The constant presheaf on `X` with value `Z`. -/\ndef const (X : TopCat) (Z : C) : PresheafedSpace C\n    where\n  carrier := X\n  Presheaf :=\n    { obj := fun U => Z\n      map := fun U V f => \ud835\udfd9 Z }\n#align algebraic_geometry.PresheafedSpace.const AlgebraicGeometry.PresheafedSpace.const\n\ninstance [Inhabited C] : Inhabited (PresheafedSpace C) :=\n  \u27e8const (TopCat.of PEmpty) default\u27e9\n\n/-- A morphism between presheafed spaces `X` and `Y` consists of a continuous map\n    `f` between the underlying topological spaces, and a (notice contravariant!) map\n    from the presheaf on `Y` to the pushforward of the presheaf on `X` via `f`. -/\nstructure Hom (X Y : PresheafedSpace.{w, v, u} C) where\n  base : (X : TopCat.{w}) \u27f6 (Y : TopCat.{w})\n  c : Y.Presheaf \u27f6 base _* X.Presheaf\n#align algebraic_geometry.PresheafedSpace.hom AlgebraicGeometry.PresheafedSpace.Hom\n\n@[ext]\ntheorem ext {X Y : PresheafedSpace C} (\u03b1 \u03b2 : Hom X Y) (w : \u03b1.base = \u03b2.base)\n    (h : \u03b1.c \u226b whiskerRight (eqToHom (by rw [w])) _ = \u03b2.c) : \u03b1 = \u03b2 :=\n  by\n  cases \u03b1; cases \u03b2\n  dsimp [presheaf.pushforward_obj] at *\n  tidy\n#align algebraic_geometry.PresheafedSpace.ext AlgebraicGeometry.PresheafedSpace.ext\n\n-- TODO including `injections` would make tidy work earlier.\ntheorem hext {X Y : PresheafedSpace C} (\u03b1 \u03b2 : Hom X Y) (w : \u03b1.base = \u03b2.base) (h : HEq \u03b1.c \u03b2.c) :\n    \u03b1 = \u03b2 := by\n  cases \u03b1\n  cases \u03b2\n  congr\n  exacts[w, h]\n#align algebraic_geometry.PresheafedSpace.hext AlgebraicGeometry.PresheafedSpace.hext\n\n/-- The identity morphism of a `PresheafedSpace`. -/\ndef id (X : PresheafedSpace.{w, v, u} C) : Hom X X\n    where\n  base := \ud835\udfd9 (X : TopCat.{w})\n  c := eqToHom (Presheaf.Pushforward.id_eq X.Presheaf).symm\n#align algebraic_geometry.PresheafedSpace.id AlgebraicGeometry.PresheafedSpace.id\n\ninstance homInhabited (X : PresheafedSpace C) : Inhabited (Hom X X) :=\n  \u27e8id X\u27e9\n#align algebraic_geometry.PresheafedSpace.hom_inhabited AlgebraicGeometry.PresheafedSpace.homInhabited\n\n/-- Composition of morphisms of `PresheafedSpace`s. -/\ndef comp {X Y Z : PresheafedSpace C} (\u03b1 : Hom X Y) (\u03b2 : Hom Y Z) : Hom X Z\n    where\n  base := \u03b1.base \u226b \u03b2.base\n  c := \u03b2.c \u226b (Presheaf.pushforward _ \u03b2.base).map \u03b1.c\n#align algebraic_geometry.PresheafedSpace.comp AlgebraicGeometry.PresheafedSpace.comp\n\ntheorem comp_c {X Y Z : PresheafedSpace C} (\u03b1 : Hom X Y) (\u03b2 : Hom Y Z) :\n    (comp \u03b1 \u03b2).c = \u03b2.c \u226b (Presheaf.pushforward _ \u03b2.base).map \u03b1.c :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.comp_c AlgebraicGeometry.PresheafedSpace.comp_c\n\nvariable (C)\n\nsection\n\nattribute [local simp] id comp\n\n/- The proofs below can be done by `tidy`, but it is too slow,\n   and we don't have a tactic caching mechanism. -/\n/-- The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map\n    from the presheaf on the target to the pushforward of the presheaf on the source. -/\ninstance categoryOfPresheafedSpaces : Category (PresheafedSpace.{v, v, u} C)\n    where\n  Hom := Hom\n  id := id\n  comp X Y Z f g := comp f g\n  id_comp' X Y f := by\n    ext1\n    \u00b7 rw [comp_c]\n      erw [eq_to_hom_map]\n      simp only [eq_to_hom_refl, assoc, whisker_right_id']\n      erw [comp_id, comp_id]\n    apply id_comp\n  comp_id' X Y f := by\n    ext1\n    \u00b7 rw [comp_c]\n      erw [congr_hom (presheaf.id_pushforward _) f.c]\n      simp only [comp_id, functor.id_map, eq_to_hom_refl, assoc, whisker_right_id']\n      erw [eq_to_hom_trans_assoc]\n      simp only [id_comp, eq_to_hom_refl]\n      erw [comp_id]\n    apply comp_id\n  assoc' W X Y Z f g h := by\n    ext1\n    repeat' rw [comp_c]\n    simp only [eq_to_hom_refl, assoc, functor.map_comp, whisker_right_id']\n    erw [comp_id]\n    congr\n    rfl\n#align algebraic_geometry.PresheafedSpace.category_of_PresheafedSpaces AlgebraicGeometry.PresheafedSpace.categoryOfPresheafedSpaces\n\nend\n\nvariable {C}\n\nattribute [local simp] eq_to_hom_map\n\n@[simp]\ntheorem id_base (X : PresheafedSpace.{v, v, u} C) : (\ud835\udfd9 X : X \u27f6 X).base = \ud835\udfd9 (X : TopCat.{v}) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.id_base AlgebraicGeometry.PresheafedSpace.id_base\n\ntheorem id_c (X : PresheafedSpace.{v, v, u} C) :\n    (\ud835\udfd9 X : X \u27f6 X).c = eqToHom (Presheaf.Pushforward.id_eq X.Presheaf).symm :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.id_c AlgebraicGeometry.PresheafedSpace.id_c\n\n@[simp]\ntheorem id_c_app (X : PresheafedSpace.{v, v, u} C) (U) :\n    (\ud835\udfd9 X : X \u27f6 X).c.app U =\n      X.Presheaf.map\n        (eqToHom\n          (by\n            induction U using Opposite.rec\n            cases U\n            rfl)) :=\n  by\n  induction U using Opposite.rec\n  cases U\n  simp only [id_c]\n  dsimp\n  simp\n#align algebraic_geometry.PresheafedSpace.id_c_app AlgebraicGeometry.PresheafedSpace.id_c_app\n\n@[simp]\ntheorem comp_base {X Y Z : PresheafedSpace.{v, v, u} C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    (f \u226b g).base = f.base \u226b g.base :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.comp_base AlgebraicGeometry.PresheafedSpace.comp_base\n\ninstance (X Y : PresheafedSpace.{v, v, u} C) : CoeFun (X \u27f6 Y) fun _ => X \u2192 Y :=\n  \u27e8fun f => f.base\u27e9\n\ntheorem coe_to_fun_eq {X Y : PresheafedSpace.{v, v, u} C} (f : X \u27f6 Y) : (f : X \u2192 Y) = f.base :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.coe_to_fun_eq AlgebraicGeometry.PresheafedSpace.coe_to_fun_eq\n\n-- The `reassoc` attribute was added despite the LHS not being a composition of two homs,\n-- for the reasons explained in the docstring.\n/-- Sometimes rewriting with `comp_c_app` doesn't work because of dependent type issues.\nIn that case, `erw comp_c_app_assoc` might make progress.\nThe lemma `comp_c_app_assoc` is also better suited for rewrites in the opposite direction. -/\n@[reassoc.1, simp]\ntheorem comp_c_app {X Y Z : PresheafedSpace.{v, v, u} C} (\u03b1 : X \u27f6 Y) (\u03b2 : Y \u27f6 Z) (U) :\n    (\u03b1 \u226b \u03b2).c.app U = \u03b2.c.app U \u226b \u03b1.c.app (op ((Opens.map \u03b2.base).obj (unop U))) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.comp_c_app AlgebraicGeometry.PresheafedSpace.comp_c_app\n\ntheorem congr_app {X Y : PresheafedSpace.{v, v, u} C} {\u03b1 \u03b2 : X \u27f6 Y} (h : \u03b1 = \u03b2) (U) :\n    \u03b1.c.app U = \u03b2.c.app U \u226b X.Presheaf.map (eqToHom (by subst h)) :=\n  by\n  subst h\n  dsimp\n  simp\n#align algebraic_geometry.PresheafedSpace.congr_app AlgebraicGeometry.PresheafedSpace.congr_app\n\nsection\n\nvariable (C)\n\n/-- The forgetful functor from `PresheafedSpace` to `Top`. -/\n@[simps]\ndef forget : PresheafedSpace.{v, v, u} C \u2964 TopCat\n    where\n  obj X := (X : TopCat.{v})\n  map X Y f := f.base\n#align algebraic_geometry.PresheafedSpace.forget AlgebraicGeometry.PresheafedSpace.forget\n\nend\n\nsection Iso\n\nvariable {X Y : PresheafedSpace.{v, v, u} C}\n\n/-- An isomorphism of PresheafedSpaces is a homeomorphism of the underlying space, and a\nnatural transformation between the sheaves.\n-/\n@[simps Hom inv]\ndef isoOfComponents (H : X.1 \u2245 Y.1) (\u03b1 : H.Hom _* X.2 \u2245 Y.2) : X \u2245 Y\n    where\n  Hom :=\n    { base := H.Hom\n      c := \u03b1.inv }\n  inv :=\n    { base := H.inv\n      c := Presheaf.toPushforwardOfIso H \u03b1.Hom }\n  hom_inv_id' := by\n    ext\n    \u00b7 simp\n      erw [category.id_comp]\n      simpa\n    simp\n  inv_hom_id' := by\n    ext x\n    induction x using Opposite.rec\n    simp only [comp_c_app, whisker_right_app, presheaf.to_pushforward_of_iso_app,\n      nat_trans.comp_app, eq_to_hom_app, id_c_app, category.assoc]\n    erw [\u2190 \u03b1.hom.naturality]\n    have := nat_trans.congr_app \u03b1.inv_hom_id (op x)\n    cases x\n    rw [nat_trans.comp_app] at this\n    convert this\n    \u00b7 dsimp\n      simp\n    \u00b7 simp\n    \u00b7 simp\n#align algebraic_geometry.PresheafedSpace.iso_of_components AlgebraicGeometry.PresheafedSpace.isoOfComponents\n\n/-- Isomorphic PresheafedSpaces have natural isomorphic presheaves. -/\n@[simps]\ndef sheafIsoOfIso (H : X \u2245 Y) : Y.2 \u2245 H.Hom.base _* X.2\n    where\n  Hom := H.Hom.c\n  inv := Presheaf.pushforwardToOfIso ((forget _).mapIso H).symm H.inv.c\n  hom_inv_id' := by\n    ext U\n    have := congr_app H.inv_hom_id U\n    simp only [comp_c_app, id_c_app, eq_to_hom_map, eq_to_hom_trans] at this\n    generalize_proofs h  at this\n    simpa using congr_arg (fun f => f \u226b eq_to_hom h.symm) this\n  inv_hom_id' := by\n    ext U\n    simp only [presheaf.pushforward_to_of_iso_app, nat_trans.comp_app, category.assoc,\n      nat_trans.id_app, H.hom.c.naturality]\n    have := congr_app H.hom_inv_id ((opens.map H.hom.base).op.obj U)\n    generalize_proofs h  at this\n    simpa using congr_arg (fun f => f \u226b X.presheaf.map (eq_to_hom h.symm)) this\n#align algebraic_geometry.PresheafedSpace.sheaf_iso_of_iso AlgebraicGeometry.PresheafedSpace.sheafIsoOfIso\n\ninstance base_isIso_of_iso (f : X \u27f6 Y) [IsIso f] : IsIso f.base :=\n  IsIso.of_iso ((forget _).mapIso (asIso f))\n#align algebraic_geometry.PresheafedSpace.base_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.base_isIso_of_iso\n\ninstance c_isIso_of_iso (f : X \u27f6 Y) [IsIso f] : IsIso f.c :=\n  IsIso.of_iso (sheafIsoOfIso (asIso f))\n#align algebraic_geometry.PresheafedSpace.c_is_iso_of_iso AlgebraicGeometry.PresheafedSpace.c_isIso_of_iso\n\n/-- This could be used in conjunction with `category_theory.nat_iso.is_iso_of_is_iso_app`. -/\ntheorem isIso_of_components (f : X \u27f6 Y) [IsIso f.base] [IsIso f.c] : IsIso f :=\n  by\n  convert is_iso.of_iso (iso_of_components (as_iso f.base) (as_iso f.c).symm)\n  ext; \u00b7 simpa; \u00b7 simp\n#align algebraic_geometry.PresheafedSpace.is_iso_of_components AlgebraicGeometry.PresheafedSpace.isIso_of_components\n\nend Iso\n\nsection Restrict\n\n/-- The restriction of a presheafed space along an open embedding into the space.\n-/\n@[simps]\ndef restrict {U : TopCat} (X : PresheafedSpace.{v, v, u} C) {f : U \u27f6 (X : TopCat.{v})}\n    (h : OpenEmbedding f) : PresheafedSpace C\n    where\n  carrier := U\n  Presheaf := h.IsOpenMap.Functor.op \u22d9 X.Presheaf\n#align algebraic_geometry.PresheafedSpace.restrict AlgebraicGeometry.PresheafedSpace.restrict\n\n/-- The map from the restriction of a presheafed space.\n-/\n@[simps]\ndef ofRestrict {U : TopCat} (X : PresheafedSpace.{v, v, u} C) {f : U \u27f6 (X : TopCat.{v})}\n    (h : OpenEmbedding f) : X.restrict h \u27f6 X\n    where\n  base := f\n  c :=\n    { app := fun V => X.Presheaf.map (h.IsOpenMap.Adjunction.counit.app V.unop).op\n      naturality' := fun U V f =>\n        show _ = _ \u226b X.Presheaf.map _ by\n          rw [\u2190 map_comp, \u2190 map_comp]\n          rfl }\n#align algebraic_geometry.PresheafedSpace.of_restrict AlgebraicGeometry.PresheafedSpace.ofRestrict\n\ninstance ofRestrict_mono {U : TopCat} (X : PresheafedSpace C) (f : U \u27f6 X.1) (hf : OpenEmbedding f) :\n    Mono (X.of_restrict hf) :=\n  by\n  haveI : mono f := (TopCat.mono_iff_injective _).mpr hf.inj\n  constructor\n  intro Z g\u2081 g\u2082 eq\n  ext V\n  \u00b7 induction V using Opposite.rec\n    have hV : (opens.map (X.of_restrict hf).base).obj (hf.is_open_map.functor.obj V) = V :=\n      by\n      ext1\n      exact Set.preimage_image_eq _ hf.inj\n    haveI :\n      is_iso (hf.is_open_map.adjunction.counit.app (unop (op (hf.is_open_map.functor.obj V)))) :=\n      (nat_iso.is_iso_app_of_is_iso\n          (whisker_left hf.is_open_map.functor hf.is_open_map.adjunction.counit) V :\n        _)\n    have := PresheafedSpace.congr_app Eq (op (hf.is_open_map.functor.obj V))\n    simp only [PresheafedSpace.comp_c_app, PresheafedSpace.of_restrict_c_app, category.assoc,\n      cancel_epi] at this\n    have h : _ \u226b _ = _ \u226b _ \u226b _ :=\n      congr_arg (fun f => (X.restrict hf).Presheaf.map (eq_to_hom hV).op \u226b f) this\n    erw [g\u2081.c.naturality, g\u2082.c.naturality_assoc] at h\n    simp only [presheaf.pushforward_obj_map, eq_to_hom_op, category.assoc, eq_to_hom_map,\n      eq_to_hom_trans] at h\n    rw [\u2190 is_iso.comp_inv_eq] at h\n    simpa using h\n  \u00b7 have := congr_arg PresheafedSpace.hom.base Eq\n    simp only [PresheafedSpace.comp_base, PresheafedSpace.of_restrict_base] at this\n    rw [cancel_mono] at this\n    exact this\n#align algebraic_geometry.PresheafedSpace.of_restrict_mono AlgebraicGeometry.PresheafedSpace.ofRestrict_mono\n\ntheorem restrict_top_presheaf (X : PresheafedSpace C) :\n    (X.restrict (Opens.openEmbedding \u22a4)).Presheaf =\n      (Opens.inclusionTopIso X.carrier).inv _* X.Presheaf :=\n  by\n  dsimp\n  rw [opens.inclusion_top_functor X.carrier]\n  rfl\n#align algebraic_geometry.PresheafedSpace.restrict_top_presheaf AlgebraicGeometry.PresheafedSpace.restrict_top_presheaf\n\ntheorem ofRestrict_top_c (X : PresheafedSpace C) :\n    (X.of_restrict (Opens.openEmbedding \u22a4)).c =\n      eqToHom\n        (by\n          rw [restrict_top_presheaf, \u2190 presheaf.pushforward.comp_eq]\n          erw [iso.inv_hom_id]\n          rw [presheaf.pushforward.id_eq]) :=\n  by\n  /- another approach would be to prove the left hand side\n       is a natural isoomorphism, but I encountered a universe\n       issue when `apply nat_iso.is_iso_of_is_iso_app`. -/\n  ext U;\n  change X.presheaf.map _ = _; convert eq_to_hom_map _ _ using 1\n  congr ; simpa\n  \u00b7 induction U using Opposite.rec\n    dsimp\n    congr\n    ext\n    exact \u27e8fun h => \u27e8\u27e8x, trivial\u27e9, h, rfl\u27e9, fun \u27e8\u27e8_, _\u27e9, h, rfl\u27e9 => h\u27e9\n#align algebraic_geometry.PresheafedSpace.of_restrict_top_c AlgebraicGeometry.PresheafedSpace.ofRestrict_top_c\n\n/- or `rw [opens.inclusion_top_functor, \u2190comp_obj, \u2190opens.map_comp_eq],\n         erw iso.inv_hom_id, cases U, refl` after `dsimp` -/\n/-- The map to the restriction of a presheafed space along the canonical inclusion from the top\nsubspace.\n-/\n@[simps]\ndef toRestrictTop (X : PresheafedSpace C) : X \u27f6 X.restrict (Opens.openEmbedding \u22a4)\n    where\n  base := (Opens.inclusionTopIso X.carrier).inv\n  c := eqToHom (restrict_top_presheaf X)\n#align algebraic_geometry.PresheafedSpace.to_restrict_top AlgebraicGeometry.PresheafedSpace.toRestrictTop\n\n/-- The isomorphism from the restriction to the top subspace.\n-/\n@[simps]\ndef restrictTopIso (X : PresheafedSpace C) : X.restrict (Opens.openEmbedding \u22a4) \u2245 X\n    where\n  Hom := X.of_restrict _\n  inv := X.toRestrictTop\n  hom_inv_id' :=\n    ext _ _ (ConcreteCategory.hom_ext _ _ fun \u27e8x, _\u27e9 => rfl) <|\n      by\n      erw [comp_c]\n      rw [X.of_restrict_top_c]\n      ext\n      simp\n  inv_hom_id' :=\n    ext _ _ rfl <| by\n      erw [comp_c]\n      rw [X.of_restrict_top_c]\n      ext\n      simpa [-eq_to_hom_refl]\n#align algebraic_geometry.PresheafedSpace.restrict_top_iso AlgebraicGeometry.PresheafedSpace.restrictTopIso\n\nend Restrict\n\n/-- The global sections, notated Gamma.\n-/\n@[simps]\ndef \u0393 : (PresheafedSpace.{v, v, u} C)\u1d52\u1d56 \u2964 C\n    where\n  obj X := (unop X).Presheaf.obj (op \u22a4)\n  map X Y f := f.unop.c.app (op \u22a4)\n#align algebraic_geometry.PresheafedSpace.\u0393 AlgebraicGeometry.PresheafedSpace.\u0393\n\ntheorem \u0393_obj_op (X : PresheafedSpace C) : \u0393.obj (op X) = X.Presheaf.obj (op \u22a4) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.\u0393_obj_op AlgebraicGeometry.PresheafedSpace.\u0393_obj_op\n\ntheorem \u0393_map_op {X Y : PresheafedSpace.{v, v, u} C} (f : X \u27f6 Y) : \u0393.map f.op = f.c.app (op \u22a4) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.\u0393_map_op AlgebraicGeometry.PresheafedSpace.\u0393_map_op\n\nend PresheafedSpace\n\nend AlgebraicGeometry\n\nopen AlgebraicGeometry AlgebraicGeometry.PresheafedSpace\n\nvariable {C}\n\nnamespace CategoryTheory\n\nvariable {D : Type u} [Category.{v} D]\n\nattribute [local simp] presheaf.pushforward_obj\n\nnamespace Functor\n\n/-- We can apply a functor `F : C \u2964 D` to the values of the presheaf in any `PresheafedSpace C`,\n    giving a functor `PresheafedSpace C \u2964 PresheafedSpace D` -/\ndef mapPresheaf (F : C \u2964 D) : PresheafedSpace.{v, v, u} C \u2964 PresheafedSpace.{v, v, u} D\n    where\n  obj X :=\n    { carrier := X.carrier\n      Presheaf := X.Presheaf \u22d9 F }\n  map X Y f :=\n    { base := f.base\n      c := whiskerRight f.c F }\n#align category_theory.functor.map_presheaf CategoryTheory.Functor.mapPresheaf\n\n@[simp]\ntheorem mapPresheaf_obj_X (F : C \u2964 D) (X : PresheafedSpace C) :\n    (F.mapPresheaf.obj X : TopCat.{v}) = (X : TopCat.{v}) :=\n  rfl\n#align category_theory.functor.map_presheaf_obj_X CategoryTheory.Functor.mapPresheaf_obj_X\n\n@[simp]\ntheorem mapPresheaf_obj_presheaf (F : C \u2964 D) (X : PresheafedSpace C) :\n    (F.mapPresheaf.obj X).Presheaf = X.Presheaf \u22d9 F :=\n  rfl\n#align category_theory.functor.map_presheaf_obj_presheaf CategoryTheory.Functor.mapPresheaf_obj_presheaf\n\n@[simp]\ntheorem mapPresheaf_map_f (F : C \u2964 D) {X Y : PresheafedSpace.{v, v, u} C} (f : X \u27f6 Y) :\n    (F.mapPresheaf.map f).base = f.base :=\n  rfl\n#align category_theory.functor.map_presheaf_map_f CategoryTheory.Functor.mapPresheaf_map_f\n\n@[simp]\ntheorem mapPresheaf_map_c (F : C \u2964 D) {X Y : PresheafedSpace.{v, v, u} C} (f : X \u27f6 Y) :\n    (F.mapPresheaf.map f).c = whiskerRight f.c F :=\n  rfl\n#align category_theory.functor.map_presheaf_map_c CategoryTheory.Functor.mapPresheaf_map_c\n\nend Functor\n\nnamespace NatTrans\n\n/-- A natural transformation induces a natural transformation between the `map_presheaf` functors.\n-/\ndef onPresheaf {F G : C \u2964 D} (\u03b1 : F \u27f6 G) : G.mapPresheaf \u27f6 F.mapPresheaf\n    where app X :=\n    { base := \ud835\udfd9 _\n      c := whiskerLeft X.Presheaf \u03b1 \u226b eqToHom (Presheaf.Pushforward.id_eq _).symm }\n#align category_theory.nat_trans.on_presheaf CategoryTheory.NatTrans.onPresheaf\n\n-- TODO Assemble the last two constructions into a functor\n--   `(C \u2964 D) \u2964 (PresheafedSpace C \u2964 PresheafedSpace D)`\nend NatTrans\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/PresheafedSpace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.39222254340646595}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1 : G.colorable 2,\n    have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from exists.intro (fintype.elems G.V) (exists.intro (fintype.elems G.V) (exists.intro rfl)),\n    show \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from h2,\n  },\n  {\n    assume h1 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B),\n    have h2 : G.colorable 2, from h1.elim (assume A, assume B, assume h, assume h1, h1),\n    show G.colorable 2, from h2,\n  },\nend\n\n--OUTPUT 2\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) := sorry\n\n/--`theorem`\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [f\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  { assume h,\n    have h1 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h, cast_le, simple_graph.subset_iff_subgraph, bipartite_iff_two_colorable],\n    show (G.colorable 2) \u2192 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)), from by auto [h1],\n    },\n  { assume h,\n    have h1 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h],\n    show (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) \u2192 (G.colorable 2), from by auto [h1, cast_le, simple_graph.subset_iff_subgraph, bipartite_iff_two_colorable],\n    },\nend\n\n--OUTPUT 4\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  { \n    assume h1,\n    rcases h1 with \u27e8f, hf\u27e9,\n    let A := {x | f x = 0},\n    let B := {x | f x = 1},\n    have h2 : (A \u2295 B) = V, from by auto [set.ext, set.mem_def, set.mem_set_of_eq],\n    have h3 : (A \u2295 B) \u2243 V, from by auto [h2, equiv.set.ext],\n    have h4 : (A \u2295 B) \u2243 V, from by auto [h2, equiv.set.ext],\n    have h5 : cast (congr_arg _ h3) (complete_bipartite_graph A B) = complete_bipartite_graph A B, from by auto [cast_eq_of_heq],\n    have h6 : G \u2264 cast (congr_arg _ h2) (complete_bipartite_graph A B), from by auto [h5, hf, set.subset_def, set.mem_def, set.mem_set_of_eq, set.mem_set_of_eq],\n    show \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h2, h6],\n  },\n  {\n    assume h1,\n    rcases h1 with \u27e8A, B, h2, h3\u27e9,\n    let f : V \u2192 fin 2, from by auto [h2, cast_eq_of_heq, complete_bipartite_graph.colorable_of_bipartite],\n    have h4 : G.colorable 2, from by auto [f],\n    show G.colorable 2, from by auto [h4],\n  },\nend\n\n--OUTPUT 5\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1 : G.colorable 2,\n    have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from \n    begin\n      let A := {v : V | (G.coloring v) = 1},\n      let B := {v : V | (G.coloring v) = 2},\n      have h2 : (A \u2295 B) = V, from by auto [set.ext, set.mem_union, set.mem_compl, set.mem_set_of_eq],\n      have h3 : G \u2264 cast (congr_arg _ h2) (complete_bipartite_graph A B), from \n      begin\n        assume (u v : V),\n        assume h3 : G.adj u v,\n        have h4 : (G.coloring u) \u2260 (G.coloring v), from by auto [h1, h3],\n        have h5 : (G.coloring u) = 1 \u2228 (G.coloring u) = 2, from by auto [h1],\n        have h6 : (G.coloring v) = 1 \u2228 (G.coloring v) = 2, from by auto [h1],\n        have h7 : u \u2208 A \u2228 u \u2208 B, from by auto [h5],\n        have h8 : v \u2208 A \u2228 v \u2208 B, from by auto [h6],\n        have h9 : (cast (congr_arg _ h2) (complete_bipartite_graph A B)).adj u v, from by auto [h7, h8, set.mem_union, set.mem_compl, set.mem_set_of_eq, complete_bipartite_graph, h4],\n        show G.adj u v, from by auto [h9],\n      end,\n      show \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h2, h3],\n    end,\n    show (G.colorable 2) \u2192 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)), from by auto [h2],\n  },\n  {\n    assume h1 : (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)),\n    have h2 : (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)), from by auto [h1],\n    have h3 : G.colorable 2, from \n    begin\n      let A := {v : V | (G.coloring v) = 1},\n      let B := {v : V | (G.coloring v) = 2},\n      have h3 : (A \u2295 B) = V, from by auto [set.ext, set.mem_union, set.mem_compl, set.mem_set_of_eq],\n      have h4 : G \u2264 cast (congr_arg _ h3) (complete_bipartite_graph A B), from \n      begin\n        assume (u v : V),\n        assume h4 : G.adj u v,\n        have h5 : (G.coloring u) \u2260 (G.coloring v), from by auto [h1, h4],\n        have h6 : (G.coloring u) = 1 \u2228 (G.coloring u) = 2, from by auto [h1],\n        have h7 : (G.coloring v) = 1 \u2228 (G.coloring v) = 2, from by auto [h1],\n        have h8 : u \u2208 A \u2228 u \u2208 B, from by auto [h6],\n        have h9 : v \u2208 A \u2228 v \u2208 B, from by auto [h7],\n        have h10 : (cast (congr_arg _ h3) (complete_bipartite_graph A B)).adj u v, from by auto [h8, h9, set.mem_union, set.mem_compl, set.mem_set_of_eq, complete_bipartite_graph, h5],\n        show G.adj u v, from by auto [h10],\n      end,\n      have h5 : (G.coloring 1) = 1 \u2227 (G.coloring 2) = 2, from by auto [h4, complete_bipartite_graph, set.mem_union, set.mem_compl, set.mem_set_of_eq],\n      have h6 : (G.coloring 1) = 1 \u2228 (G.coloring 1) = 2, from by auto [h5],\n      have h7 : (G.coloring 2) = 1 \u2228 (G.coloring 2) = 2, from by auto [h5],\n      have h8 : (G.coloring 1) \u2260 (G.coloring 2), from by auto [h6, h7],\n      show G.colorable 2, from by auto [h6, h7, h8],\n    end,\n    show (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) \u2192 (G.colorable 2), from by auto [h3],\n  },\nend\n\n--OUTPUT 6\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1,\n    have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h1, complete_bipartite_graph, graph.colorable_def, graph.edge, graph.vertex, graph.edge_def, graph.vertex_def, graph.subgraph_def, graph.subgraph_def, graph.subgraph_def],\n    show \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h2],\n  },\n  {\n    assume h1,\n    have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by auto [h1],\n    have h3 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h2],\n    have h4 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h3],\n    have h5 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h4],\n    have h6 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h5],\n    have h7 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h6],\n    have h8 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h7],\n    have h9 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h8],\n    have h10 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h9],\n    have h11 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h10],\n    have h12 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h11],\n    have h13 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h12],\n    have h14 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h13],\n    have h15 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h14],\n    have h16 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h15],\n    have h17 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h16],\n    have h18 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (x y : A \u2295 B), (x \u2260 y) \u2192 (G.edge x y) \u2192 (x.elim (\u03bb x, y.elim (\u03bb y, ff) (\u03bb y, tt)) (\u03bb x, y.elim (\u03bb y, tt) (\u03bb y, ff))), from by auto [h17],\n   \nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.39212518124633283}}
{"text": "example (p : Prop) [Decidable p] (hnp : \u00ac p) :\n    if decide p then 0 = 1 else 1 = 1 := by\n  simp [hnp, decide_eq_false Unit]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/simpArgTypeMismatch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.39204820789155553}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport tactic.pi_instances\nimport algebra.group.pi\nimport algebra.ring.basic\n\n/-!\n# Pi instances for ring\n\nThis file defines instances for ring, semiring and related structures on Pi Types\n-/\n\nnamespace pi\nuniverses u v w\nvariable {I : Type u}     -- The indexing type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\n\ninstance distrib [\u03a0 i, distrib $ f i] : distrib (\u03a0 i : I, f i) :=\nby refine_struct { add := (+), mul := (*), .. }; tactic.pi_instance_derive_field\n\ninstance non_unital_non_assoc_semiring [\u2200 i, non_unital_non_assoc_semiring $ f i] :\n  non_unital_non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_unital_semiring [\u2200 i, non_unital_semiring $ f i] :\n  non_unital_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_assoc_semiring [\u2200 i, non_assoc_semiring $ f i] :\n  non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance semiring [\u2200 i, semiring $ f i] : semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance comm_semiring [\u2200 i, comm_semiring $ f i] : comm_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance non_unital_non_assoc_ring [\u2200 i, non_unital_non_assoc_ring $ f i] :\n  non_unital_non_assoc_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };\ntactic.pi_instance_derive_field\n\ninstance ring [\u2200 i, ring $ f i] : ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance comm_ring [\u2200 i, comm_ring $ f i] : comm_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\n/-- A family of ring homomorphisms `f a : \u03b3 \u2192+* \u03b2 a` defines a ring homomorphism\n`pi.ring_hom f : \u03b3 \u2192+* \u03a0 a, \u03b2 a` given by `pi.ring_hom f x b = f b x`. -/\n@[simps]\nprotected def ring_hom {\u03b3 : Type w} [\u03a0 i, non_assoc_semiring (f i)] [non_assoc_semiring \u03b3]\n  (g : \u03a0 i, \u03b3 \u2192+* f i) : \u03b3 \u2192+* \u03a0 i, f i :=\n{ to_fun := \u03bb x b, g b x,\n  map_add' := \u03bb x y, funext $ \u03bb z, (g z).map_add x y,\n  map_mul' := \u03bb x y, funext $ \u03bb z, (g z).map_mul x y,\n  map_one' := funext $ \u03bb z, (g z).map_one,\n  map_zero' := funext $ \u03bb z, (g z).map_zero }\n\nlemma ring_hom_injective {\u03b3 : Type w} [nonempty I] [\u03a0 i, non_assoc_semiring (f i)]\n  [non_assoc_semiring \u03b3] (g : \u03a0 i, \u03b3 \u2192+* f i) (hg : \u2200 i, function.injective (g i)) :\n  function.injective (pi.ring_hom g) :=\n\u03bb x y h, let \u27e8i\u27e9 := \u2039nonempty I\u203a in hg i ((function.funext_iff.mp h : _) i)\n\nend pi\n\nsection ring_hom\n\nuniverses u v\nvariable {I : Type u}\n\n/-- Evaluation of functions into an indexed collection of monoids at a point is a monoid\nhomomorphism. This is `function.eval` as a `ring_hom`. -/\n@[simps]\ndef pi.eval_ring_hom (f : I \u2192 Type v) [\u03a0 i, non_assoc_semiring (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192+* f i :=\n{ ..(pi.eval_monoid_hom f i),\n  ..(pi.eval_add_monoid_hom f i) }\n\n/-- `function.const` as a `ring_hom`. -/\n@[simps]\ndef pi.const_ring_hom (\u03b1 \u03b2 : Type*) [non_assoc_semiring \u03b2] : \u03b2 \u2192+* (\u03b1 \u2192 \u03b2) :=\n{ to_fun := function.const _,\n  .. pi.ring_hom (\u03bb _, ring_hom.id \u03b2) }\n\n/-- Ring homomorphism between the function spaces `I \u2192 \u03b1` and `I \u2192 \u03b2`, induced by a ring\nhomomorphism `f` between `\u03b1` and `\u03b2`. -/\n@[simps] protected def ring_hom.comp_left {\u03b1 \u03b2 : Type*} [non_assoc_semiring \u03b1]\n  [non_assoc_semiring \u03b2] (f : \u03b1 \u2192+* \u03b2) (I : Type*) :\n  (I \u2192 \u03b1) \u2192+* (I \u2192 \u03b2) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  .. f.to_monoid_hom.comp_left I,\n  .. f.to_add_monoid_hom.comp_left I }\n\nend ring_hom\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebra/ring/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3920482007092462}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Nicol\u00f2 Cavalleri\n-/\nimport algebra.algebra.pi\nimport algebra.periodic\nimport algebra.algebra.subalgebra.basic\nimport algebra.star.star_alg_hom\nimport tactic.field_simp\nimport topology.algebra.module.basic\nimport topology.algebra.infinite_sum.basic\nimport topology.algebra.star\nimport topology.algebra.uniform_group\nimport topology.continuous_function.ordered\nimport topology.uniform_space.compact_convergence\n\n/-!\n# Algebraic structures over continuous functions\n\nIn this file we define instances of algebraic structures over the type `continuous_map \u03b1 \u03b2`\n(denoted `C(\u03b1, \u03b2)`) of **bundled** continuous maps from `\u03b1` to `\u03b2`. For example, `C(\u03b1, \u03b2)`\nis a group when `\u03b2` is a group, a ring when `\u03b2` is a ring, etc.\n\nFor each type of algebraic structure, we also define an appropriate subobject of `\u03b1 \u2192 \u03b2`\nwith carrier `{ f : \u03b1 \u2192 \u03b2 | continuous f }`. For example, when `\u03b2` is a group, a subgroup\n`continuous_subgroup \u03b1 \u03b2` of `\u03b1 \u2192 \u03b2` is constructed with carrier `{ f : \u03b1 \u2192 \u03b2 | continuous f }`.\n\nNote that, rather than using the derived algebraic structures on these subobjects\n(for example, when `\u03b2` is a group, the derived group structure on `continuous_subgroup \u03b1 \u03b2`),\none should use `C(\u03b1, \u03b2)` with the appropriate instance of the structure.\n-/\n\nlocal attribute [elab_simple] continuous.comp\n\nnamespace continuous_functions\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\nvariables {f g : {f : \u03b1 \u2192 \u03b2 | continuous f }}\n\ninstance : has_coe_to_fun {f : \u03b1 \u2192 \u03b2 | continuous f} (\u03bb _, \u03b1 \u2192 \u03b2) :=  \u27e8subtype.val\u27e9\n\nend continuous_functions\n\nnamespace continuous_map\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/- ### \"mul\" and \"add\" -/\n\n@[to_additive]\ninstance has_mul [has_mul \u03b2] [has_continuous_mul \u03b2] : has_mul C(\u03b1, \u03b2) :=\n\u27e8\u03bb f g, \u27e8f * g, continuous_mul.comp (f.continuous.prod_mk g.continuous : _)\u27e9\u27e9\n\n@[simp, norm_cast, to_additive]\nlemma coe_mul [has_mul \u03b2] [has_continuous_mul \u03b2] (f g : C(\u03b1, \u03b2)) : \u21d1(f * g) = f * g := rfl\n\n@[simp, to_additive]\nlemma mul_apply [has_mul \u03b2] [has_continuous_mul \u03b2] (f g : C(\u03b1, \u03b2)) (x : \u03b1) :\n  (f * g) x = f x * g x := rfl\n\n@[simp, to_additive] lemma mul_comp [has_mul \u03b3] [has_continuous_mul \u03b3]\n  (f\u2081 f\u2082 : C(\u03b2, \u03b3)) (g : C(\u03b1, \u03b2)) :\n  (f\u2081 * f\u2082).comp g = f\u2081.comp g * f\u2082.comp g :=\nrfl\n\n/- ### \"one\" -/\n\n@[to_additive] instance [has_one \u03b2] : has_one C(\u03b1, \u03b2) := \u27e8const \u03b1 1\u27e9\n\n@[simp, norm_cast, to_additive] lemma coe_one [has_one \u03b2]  : \u21d1(1 : C(\u03b1, \u03b2)) = 1 := rfl\n\n@[simp, to_additive] lemma one_apply [has_one \u03b2] (x : \u03b1) : (1 : C(\u03b1, \u03b2)) x = 1 := rfl\n\n@[simp, to_additive] lemma one_comp [has_one \u03b3] (g : C(\u03b1, \u03b2)) : (1 : C(\u03b2, \u03b3)).comp g = 1 := rfl\n\n/- ### \"nat_cast\" -/\n\ninstance [has_nat_cast \u03b2] : has_nat_cast C(\u03b1, \u03b2) := \u27e8\u03bb n, continuous_map.const _ n\u27e9\n\n@[simp, norm_cast] lemma coe_nat_cast [has_nat_cast \u03b2] (n : \u2115) : ((n : C(\u03b1, \u03b2)) : \u03b1 \u2192 \u03b2) = n := rfl\n\n@[simp] lemma nat_cast_apply [has_nat_cast \u03b2] (n : \u2115) (x : \u03b1) : (n : C(\u03b1, \u03b2)) x = n := rfl\n\n/- ### \"int_cast\" -/\n\ninstance [has_int_cast \u03b2] : has_int_cast C(\u03b1, \u03b2) :=\n\u27e8\u03bb n, continuous_map.const _ n\u27e9\n\n@[simp, norm_cast]\nlemma coe_int_cast [has_int_cast \u03b2] (n : \u2124) : ((n : C(\u03b1, \u03b2)) : \u03b1 \u2192 \u03b2) = n := rfl\n\n@[simp] lemma int_cast_apply [has_int_cast \u03b2] (n : \u2124) (x : \u03b1) : (n : C(\u03b1, \u03b2)) x = n := rfl\n\n/- ### \"nsmul\" and \"pow\" -/\n\ninstance has_nsmul [add_monoid \u03b2] [has_continuous_add \u03b2] : has_smul \u2115 C(\u03b1, \u03b2) :=\n\u27e8\u03bb n f, \u27e8n \u2022 f, f.continuous.nsmul n\u27e9\u27e9\n\n@[to_additive]\ninstance has_pow [monoid \u03b2] [has_continuous_mul \u03b2] : has_pow C(\u03b1, \u03b2) \u2115 :=\n\u27e8\u03bb f n, \u27e8f ^ n, f.continuous.pow n\u27e9\u27e9\n\n@[norm_cast, to_additive]\nlemma coe_pow [monoid \u03b2] [has_continuous_mul \u03b2] (f : C(\u03b1, \u03b2)) (n : \u2115) :\n  \u21d1(f ^ n) = f ^ n := rfl\n\n@[to_additive] lemma pow_apply [monoid \u03b2] [has_continuous_mul \u03b2]\n  (f : C(\u03b1, \u03b2)) (n : \u2115) (x : \u03b1) :\n  (f ^ n) x = f x ^ n :=\nrfl\n\n-- don't make auto-generated `coe_nsmul` and `nsmul_apply` simp, as the linter complains they're\n-- redundant WRT `coe_smul`\nattribute [simp] coe_pow pow_apply\n\n@[to_additive] lemma pow_comp [monoid \u03b3] [has_continuous_mul \u03b3]\n  (f : C(\u03b2, \u03b3)) (n : \u2115) (g : C(\u03b1, \u03b2)) :\n  (f^n).comp g = (f.comp g)^n :=\nrfl\n\n-- don't make `nsmul_comp` simp as the linter complains it's redundant WRT `smul_comp`\nattribute [simp] pow_comp\n\n/- ### \"inv\" and \"neg\" -/\n\n@[to_additive]\ninstance [group \u03b2] [topological_group \u03b2] : has_inv C(\u03b1, \u03b2) :=\n{ inv := \u03bb f, \u27e8f\u207b\u00b9, f.continuous.inv\u27e9 }\n\n@[simp, norm_cast, to_additive]\nlemma coe_inv [group \u03b2] [topological_group \u03b2] (f : C(\u03b1, \u03b2)) :\n  \u21d1(f\u207b\u00b9) = f\u207b\u00b9 :=\nrfl\n\n@[simp, to_additive] \n\n@[simp, to_additive] lemma inv_comp [group \u03b3] [topological_group \u03b3] (f : C(\u03b2, \u03b3)) (g : C(\u03b1, \u03b2)) :\n  (f\u207b\u00b9).comp g = (f.comp g)\u207b\u00b9 :=\nrfl\n\n/- ### \"div\" and \"sub\" -/\n\n@[to_additive]\ninstance [has_div \u03b2] [has_continuous_div \u03b2] : has_div C(\u03b1, \u03b2) :=\n{ div := \u03bb f g, \u27e8f / g, f.continuous.div' g.continuous\u27e9 }\n\n@[simp, norm_cast, to_additive]\nlemma coe_div [has_div \u03b2] [has_continuous_div \u03b2] (f g : C(\u03b1, \u03b2)) : \u21d1(f / g) = f / g :=\nrfl\n\n@[simp, to_additive] lemma div_apply [has_div \u03b2] [has_continuous_div \u03b2] (f g : C(\u03b1, \u03b2)) (x : \u03b1) :\n  (f / g) x = f x / g x :=\nrfl\n\n@[simp, to_additive] lemma div_comp [has_div \u03b3] [has_continuous_div \u03b3]\n  (f g : C(\u03b2, \u03b3)) (h : C(\u03b1, \u03b2)) :\n  (f / g).comp h = (f.comp h) / (g.comp h) :=\nrfl\n\n/- ### \"zpow\" and \"zsmul\" -/\n\ninstance has_zsmul [add_group \u03b2] [topological_add_group \u03b2] : has_smul \u2124 C(\u03b1, \u03b2) :=\n{ smul := \u03bb z f, \u27e8z \u2022 f, f.continuous.zsmul z\u27e9 }\n\n@[to_additive]\ninstance has_zpow [group \u03b2] [topological_group \u03b2] :\n  has_pow C(\u03b1, \u03b2) \u2124 :=\n{ pow := \u03bb f z, \u27e8f ^ z, f.continuous.zpow z\u27e9 }\n\n@[norm_cast, to_additive]\nlemma coe_zpow [group \u03b2] [topological_group \u03b2] (f : C(\u03b1, \u03b2)) (z : \u2124) :\n  \u21d1(f ^ z) = f ^ z :=\nrfl\n\n@[to_additive] lemma zpow_apply [group \u03b2] [topological_group \u03b2]\n  (f : C(\u03b1, \u03b2)) (z : \u2124) (x : \u03b1) :\n  (f ^ z) x = f x ^ z :=\nrfl\n\n-- don't make auto-generated `coe_zsmul` and `zsmul_apply` simp as the linter complains they're\n-- redundant WRT `coe_smul`\nattribute [simp] coe_zpow zpow_apply\n\n@[to_additive]\nlemma zpow_comp [group \u03b3] [topological_group \u03b3] (f : C(\u03b2, \u03b3)) (z : \u2124) (g : C(\u03b1, \u03b2)) :\n  (f^z).comp g = (f.comp g)^z :=\nrfl\n\n-- don't make `zsmul_comp` simp as the linter complains it's redundant WRT `smul_comp`\nattribute [simp] zpow_comp\n\nend continuous_map\n\nsection group_structure\n\n/-!\n### Group stucture\n\nIn this section we show that continuous functions valued in a topological group inherit\nthe structure of a group.\n-/\n\nsection subtype\n\n/-- The `submonoid` of continuous maps `\u03b1 \u2192 \u03b2`. -/\n@[to_additive \"The `add_submonoid` of continuous maps `\u03b1 \u2192 \u03b2`. \"]\ndef continuous_submonoid (\u03b1 : Type*) (\u03b2 : Type*) [topological_space \u03b1] [topological_space \u03b2]\n  [monoid \u03b2] [has_continuous_mul \u03b2] : submonoid (\u03b1 \u2192 \u03b2) :=\n{ carrier := { f : \u03b1 \u2192 \u03b2 | continuous f },\n  one_mem' := @continuous_const _ _ _ _ 1,\n  mul_mem' := \u03bb f g fc gc, fc.mul gc }\n\n/-- The subgroup of continuous maps `\u03b1 \u2192 \u03b2`. -/\n@[to_additive \"The `add_subgroup` of continuous maps `\u03b1 \u2192 \u03b2`. \"]\ndef continuous_subgroup (\u03b1 : Type*) (\u03b2 : Type*) [topological_space \u03b1] [topological_space \u03b2]\n  [group \u03b2] [topological_group \u03b2] : subgroup (\u03b1 \u2192 \u03b2) :=\n{ inv_mem' := \u03bb f fc, continuous.inv fc,\n  ..continuous_submonoid \u03b1 \u03b2, }.\n\nend subtype\n\nnamespace continuous_map\n\nvariables {\u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n\n@[to_additive]\ninstance [semigroup \u03b2] [has_continuous_mul \u03b2] : semigroup C(\u03b1, \u03b2) :=\ncoe_injective.semigroup _ coe_mul\n\n@[to_additive]\ninstance [comm_semigroup \u03b2] [has_continuous_mul \u03b2] : comm_semigroup C(\u03b1, \u03b2) :=\ncoe_injective.comm_semigroup _ coe_mul\n\n@[to_additive]\ninstance [mul_one_class \u03b2] [has_continuous_mul \u03b2] : mul_one_class C(\u03b1, \u03b2) :=\ncoe_injective.mul_one_class _ coe_one coe_mul\n\ninstance [mul_zero_class \u03b2] [has_continuous_mul \u03b2] : mul_zero_class C(\u03b1, \u03b2) :=\ncoe_injective.mul_zero_class _ coe_zero coe_mul\n\ninstance [semigroup_with_zero \u03b2] [has_continuous_mul \u03b2] : semigroup_with_zero C(\u03b1, \u03b2) :=\ncoe_injective.semigroup_with_zero _ coe_zero coe_mul\n\n@[to_additive]\ninstance [monoid \u03b2] [has_continuous_mul \u03b2] : monoid C(\u03b1, \u03b2) :=\ncoe_injective.monoid _ coe_one coe_mul coe_pow\n\ninstance [monoid_with_zero \u03b2] [has_continuous_mul \u03b2] : monoid_with_zero C(\u03b1, \u03b2) :=\ncoe_injective.monoid_with_zero _ coe_zero coe_one coe_mul coe_pow\n\n@[to_additive]\ninstance [comm_monoid \u03b2] [has_continuous_mul \u03b2] : comm_monoid C(\u03b1, \u03b2) :=\ncoe_injective.comm_monoid _ coe_one coe_mul coe_pow\n\ninstance [comm_monoid_with_zero \u03b2] [has_continuous_mul \u03b2] : comm_monoid_with_zero C(\u03b1, \u03b2) :=\ncoe_injective.comm_monoid_with_zero _ coe_zero coe_one coe_mul coe_pow\n\n@[to_additive]\ninstance [locally_compact_space \u03b1] [has_mul \u03b2] [has_continuous_mul \u03b2] :\n  has_continuous_mul C(\u03b1, \u03b2) :=\n\u27e8begin\n  refine continuous_of_continuous_uncurry _ _,\n  have h1 : continuous (\u03bb x : (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u00d7 \u03b1, x.fst.fst x.snd) :=\n    continuous_eval'.comp (continuous_fst.prod_map continuous_id),\n  have h2 : continuous (\u03bb x : (C(\u03b1, \u03b2) \u00d7 C(\u03b1, \u03b2)) \u00d7 \u03b1, x.fst.snd x.snd) :=\n    continuous_eval'.comp (continuous_snd.prod_map continuous_id),\n  exact h1.mul h2,\nend\u27e9\n\n/-- Coercion to a function as an `monoid_hom`. Similar to `monoid_hom.coe_fn`. -/\n@[to_additive \"Coercion to a function as an `add_monoid_hom`. Similar to `add_monoid_hom.coe_fn`.\",\n  simps]\ndef coe_fn_monoid_hom [monoid \u03b2] [has_continuous_mul \u03b2] : C(\u03b1, \u03b2) \u2192* (\u03b1 \u2192 \u03b2) :=\n{ to_fun := coe_fn, map_one' := coe_one, map_mul' := coe_mul }\n\nvariables (\u03b1)\n\n/-- Composition on the left by a (continuous) homomorphism of topological monoids, as a\n`monoid_hom`. Similar to `monoid_hom.comp_left`. -/\n@[to_additive \"Composition on the left by a (continuous) homomorphism of topological `add_monoid`s,\nas an `add_monoid_hom`. Similar to `add_monoid_hom.comp_left`.\", simps]\nprotected def _root_.monoid_hom.comp_left_continuous\n  {\u03b3 : Type*} [monoid \u03b2] [has_continuous_mul \u03b2]\n  [topological_space \u03b3] [monoid \u03b3] [has_continuous_mul \u03b3] (g : \u03b2 \u2192* \u03b3) (hg : continuous g)  :\n  C(\u03b1, \u03b2) \u2192* C(\u03b1, \u03b3) :=\n{ to_fun := \u03bb f, (\u27e8g, hg\u27e9 : C(\u03b2, \u03b3)).comp f,\n  map_one' := ext $ \u03bb x, g.map_one,\n  map_mul' := \u03bb f\u2081 f\u2082, ext $ \u03bb x, g.map_mul _ _ }\n\nvariables {\u03b1}\n\n/-- Composition on the right as a `monoid_hom`. Similar to `monoid_hom.comp_hom'`. -/\n@[to_additive \"Composition on the right as an `add_monoid_hom`. Similar to\n`add_monoid_hom.comp_hom'`.\", simps]\ndef comp_monoid_hom' {\u03b3 : Type*} [topological_space \u03b3]\n  [mul_one_class \u03b3] [has_continuous_mul \u03b3] (g : C(\u03b1, \u03b2)) : C(\u03b2, \u03b3) \u2192* C(\u03b1, \u03b3) :=\n{ to_fun := \u03bb f, f.comp g, map_one' := one_comp g, map_mul' := \u03bb f\u2081 f\u2082, mul_comp f\u2081 f\u2082 g }\n\nopen_locale big_operators\n@[simp, to_additive] lemma coe_prod [comm_monoid \u03b2] [has_continuous_mul \u03b2]\n  {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 C(\u03b1, \u03b2)) :\n  \u21d1(\u220f i in s, f i) = (\u220f i in s, (f i : \u03b1 \u2192 \u03b2)) :=\n(coe_fn_monoid_hom : C(\u03b1, \u03b2) \u2192* _).map_prod f s\n\n@[to_additive]\nlemma prod_apply [comm_monoid \u03b2] [has_continuous_mul \u03b2]\n  {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 C(\u03b1, \u03b2)) (a : \u03b1) :\n  (\u220f i in s, f i) a = (\u220f i in s, f i a) :=\nby simp\n\n@[to_additive]\ninstance [group \u03b2] [topological_group \u03b2] : group C(\u03b1, \u03b2) :=\ncoe_injective.group _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow\n\n@[to_additive]\ninstance [comm_group \u03b2] [topological_group \u03b2] : comm_group C(\u03b1, \u03b2) :=\ncoe_injective.comm_group _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow\n\n@[to_additive] instance [comm_group \u03b2] [topological_group \u03b2] : topological_group C(\u03b1, \u03b2) :=\n{ continuous_mul := by\n  { letI : uniform_space \u03b2 := topological_group.to_uniform_space \u03b2,\n    have : uniform_group \u03b2 := topological_comm_group_is_uniform,\n    rw continuous_iff_continuous_at,\n    rintros \u27e8f, g\u27e9,\n    rw [continuous_at, tendsto_iff_forall_compact_tendsto_uniformly_on, nhds_prod_eq],\n    exactI \u03bb K hK, uniform_continuous_mul.comp_tendsto_uniformly_on\n      ((tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK).prod\n      (tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK)), },\n  continuous_inv := by\n  { letI : uniform_space \u03b2 := topological_group.to_uniform_space \u03b2,\n    have : uniform_group \u03b2 := topological_comm_group_is_uniform,\n    rw continuous_iff_continuous_at,\n    intro f,\n    rw [continuous_at, tendsto_iff_forall_compact_tendsto_uniformly_on],\n    exactI \u03bb K hK, uniform_continuous_inv.comp_tendsto_uniformly_on\n      (tendsto_iff_forall_compact_tendsto_uniformly_on.mp filter.tendsto_id K hK), } }\n\n-- TODO: rewrite the next three lemmas for products and deduce sum case via `to_additive`, once\n-- definition of `tprod` is in place\n\n/-- If `\u03b1` is locally compact, and an infinite sum of functions in `C(\u03b1, \u03b2)`\nconverges to `g` (for the compact-open topology), then the pointwise sum converges to `g x` for\nall `x \u2208 \u03b1`. -/\nlemma has_sum_apply {\u03b3 : Type*} [locally_compact_space \u03b1] [add_comm_monoid \u03b2] [has_continuous_add \u03b2]\n  {f : \u03b3 \u2192 C(\u03b1, \u03b2)} {g : C(\u03b1, \u03b2)} (hf : has_sum f g) (x : \u03b1) :\n  has_sum (\u03bb i : \u03b3, f i x) (g x) :=\nbegin\n  let ev\u2093 : add_monoid_hom C(\u03b1, \u03b2) \u03b2 := (pi.eval_add_monoid_hom _ x).comp coe_fn_add_monoid_hom,\n  exact hf.map ev\u2093 (continuous_map.continuous_eval_const' x),\nend\n\nlemma summable_apply [locally_compact_space \u03b1] [add_comm_monoid \u03b2] [has_continuous_add \u03b2]\n  {\u03b3 : Type*} {f : \u03b3 \u2192 C(\u03b1, \u03b2)} (hf : summable f) (x : \u03b1) :\n  summable (\u03bb i : \u03b3, f i x) :=\n(has_sum_apply hf.has_sum x).summable\n\nlemma tsum_apply [locally_compact_space \u03b1] [t2_space \u03b2] [add_comm_monoid \u03b2] [has_continuous_add \u03b2]\n  {\u03b3 : Type*} {f : \u03b3 \u2192 C(\u03b1, \u03b2)} (hf : summable f) (x : \u03b1) :\n  (\u2211' (i:\u03b3), f i x) = (\u2211' (i:\u03b3), f i) x :=\n(has_sum_apply hf.has_sum x).tsum_eq\n\nend continuous_map\n\nend group_structure\n\nsection ring_structure\n\n/-!\n### Ring stucture\n\nIn this section we show that continuous functions valued in a topological semiring `R` inherit\nthe structure of a ring.\n-/\n\nsection subtype\n\n/-- The subsemiring of continuous maps `\u03b1 \u2192 \u03b2`. -/\ndef continuous_subsemiring (\u03b1 : Type*) (R : Type*) [topological_space \u03b1] [topological_space R]\n  [semiring R] [topological_semiring R] : subsemiring (\u03b1 \u2192 R) :=\n{ ..continuous_add_submonoid \u03b1 R,\n  ..continuous_submonoid \u03b1 R }\n\n/-- The subring of continuous maps `\u03b1 \u2192 \u03b2`. -/\ndef continuous_subring (\u03b1 : Type*) (R : Type*) [topological_space \u03b1] [topological_space R]\n  [ring R] [topological_ring R] : subring (\u03b1 \u2192 R) :=\n{ ..continuous_subsemiring \u03b1 R,\n  ..continuous_add_subgroup \u03b1 R }\n\nend subtype\n\nnamespace continuous_map\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_non_assoc_semiring \u03b2] [topological_semiring \u03b2] :\n  non_unital_non_assoc_semiring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_non_assoc_semiring _ coe_zero coe_add coe_mul coe_nsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_semiring \u03b2] [topological_semiring \u03b2] :\n  non_unital_semiring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_semiring _ coe_zero coe_add coe_mul coe_nsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [add_monoid_with_one \u03b2] [has_continuous_add \u03b2] :\n  add_monoid_with_one C(\u03b1, \u03b2) :=\ncoe_injective.add_monoid_with_one _ coe_zero coe_one coe_add coe_nsmul coe_nat_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_assoc_semiring \u03b2] [topological_semiring \u03b2] :\n  non_assoc_semiring C(\u03b1, \u03b2) :=\ncoe_injective.non_assoc_semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_nat_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [semiring \u03b2] [topological_semiring \u03b2] : semiring C(\u03b1, \u03b2) :=\ncoe_injective.semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_pow coe_nat_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_non_assoc_ring \u03b2] [topological_ring \u03b2] : non_unital_non_assoc_ring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_non_assoc_ring _ coe_zero coe_add coe_mul coe_neg coe_sub\n  coe_nsmul coe_zsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_ring \u03b2] [topological_ring \u03b2] : non_unital_ring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_ring _ coe_zero coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_assoc_ring \u03b2] [topological_ring \u03b2] : non_assoc_ring C(\u03b1, \u03b2) :=\ncoe_injective.non_assoc_ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul\n  coe_nat_cast coe_int_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [ring \u03b2] [topological_ring \u03b2] : ring C(\u03b1, \u03b2) :=\ncoe_injective.ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul coe_pow\n  coe_nat_cast coe_int_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_comm_semiring \u03b2] [topological_semiring \u03b2] : non_unital_comm_semiring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_comm_semiring _ coe_zero coe_add coe_mul coe_nsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1]\n  [topological_space \u03b2] [comm_semiring \u03b2] [topological_semiring \u03b2] : comm_semiring C(\u03b1, \u03b2) :=\ncoe_injective.comm_semiring _ coe_zero coe_one coe_add coe_mul coe_nsmul coe_pow coe_nat_cast\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [non_unital_comm_ring \u03b2] [topological_ring \u03b2] : non_unital_comm_ring C(\u03b1, \u03b2) :=\ncoe_injective.non_unital_comm_ring _ coe_zero coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul\n\ninstance {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1]\n  [topological_space \u03b2] [comm_ring \u03b2] [topological_ring \u03b2] : comm_ring C(\u03b1, \u03b2) :=\ncoe_injective.comm_ring _ coe_zero coe_one coe_add coe_mul coe_neg coe_sub coe_nsmul coe_zsmul\n  coe_pow coe_nat_cast coe_int_cast\n\n/-- Composition on the left by a (continuous) homomorphism of topological semirings, as a\n`ring_hom`.  Similar to `ring_hom.comp_left`. -/\n@[simps] protected def _root_.ring_hom.comp_left_continuous (\u03b1 : Type*) {\u03b2 : Type*} {\u03b3 : Type*}\n  [topological_space \u03b1] [topological_space \u03b2] [semiring \u03b2] [topological_semiring \u03b2]\n  [topological_space \u03b3] [semiring \u03b3] [topological_semiring \u03b3] (g : \u03b2 \u2192+* \u03b3) (hg : continuous g) :\n  C(\u03b1, \u03b2) \u2192+* C(\u03b1, \u03b3) :=\n{ .. g.to_monoid_hom.comp_left_continuous \u03b1 hg,\n  .. g.to_add_monoid_hom.comp_left_continuous \u03b1 hg }\n\n/-- Coercion to a function as a `ring_hom`. -/\n@[simps]\ndef coe_fn_ring_hom {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [semiring \u03b2] [topological_semiring \u03b2] : C(\u03b1, \u03b2) \u2192+* (\u03b1 \u2192 \u03b2) :=\n{ to_fun := coe_fn,\n  ..(coe_fn_monoid_hom : C(\u03b1, \u03b2) \u2192* _),\n  ..(coe_fn_add_monoid_hom : C(\u03b1, \u03b2) \u2192+ _) }\n\nend continuous_map\n\nend ring_structure\n\nlocal attribute [ext] subtype.eq\n\nsection module_structure\n\n/-!\n### Semiodule stucture\n\nIn this section we show that continuous functions valued in a topological module `M` over a\ntopological semiring `R` inherit the structure of a module.\n-/\n\nsection subtype\n\nvariables (\u03b1 : Type*) [topological_space \u03b1]\nvariables (R : Type*) [semiring R]\nvariables (M : Type*) [topological_space M] [add_comm_group M]\nvariables [module R M] [has_continuous_const_smul R M] [topological_add_group M]\n\n/-- The `R`-submodule of continuous maps `\u03b1 \u2192 M`. -/\ndef continuous_submodule : submodule R (\u03b1 \u2192 M) :=\n{ carrier := { f : \u03b1 \u2192 M | continuous f },\n  smul_mem' := \u03bb c f hf, hf.const_smul c,\n  ..continuous_add_subgroup \u03b1 M }\n\nend subtype\n\nnamespace continuous_map\nvariables {\u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  {R R\u2081 : Type*}\n  {M : Type*} [topological_space M]\n  {M\u2082 : Type*} [topological_space M\u2082]\n\n@[to_additive continuous_map.has_vadd]\ninstance [has_smul R M] [has_continuous_const_smul R M] : has_smul R C(\u03b1, M) :=\n\u27e8\u03bb r f, \u27e8r \u2022 f, f.continuous.const_smul r\u27e9\u27e9\n\n@[to_additive]\ninstance [locally_compact_space \u03b1] [has_smul R M] [has_continuous_const_smul R M] :\n  has_continuous_const_smul R C(\u03b1, M) :=\n\u27e8\u03bb \u03b3, continuous_of_continuous_uncurry _ (continuous_eval'.const_smul \u03b3)\u27e9\n\n@[to_additive]\ninstance [locally_compact_space \u03b1] [topological_space R] [has_smul R M]\n  [has_continuous_smul R M] : has_continuous_smul R C(\u03b1, M) :=\n\u27e8begin\n  refine continuous_of_continuous_uncurry _ _,\n  have h : continuous (\u03bb x : (R \u00d7 C(\u03b1, M)) \u00d7 \u03b1, x.fst.snd x.snd) :=\n    continuous_eval'.comp (continuous_snd.prod_map continuous_id),\n  exact (continuous_fst.comp continuous_fst).smul h,\nend\u27e9\n\n@[simp, norm_cast, to_additive]\nlemma coe_smul [has_smul R M] [has_continuous_const_smul R M]\n  (c : R) (f : C(\u03b1, M)) : \u21d1(c \u2022 f) = c \u2022 f := rfl\n\n@[to_additive]\nlemma smul_apply [has_smul R M] [has_continuous_const_smul R M]\n  (c : R) (f : C(\u03b1, M)) (a : \u03b1) : (c \u2022 f) a = c \u2022 (f a) :=\nrfl\n\n@[simp, to_additive] lemma smul_comp [has_smul R M] [has_continuous_const_smul R M]\n  (r : R) (f : C(\u03b2, M)) (g : C(\u03b1, \u03b2)) :\n  (r \u2022 f).comp g = r \u2022 (f.comp g) :=\nrfl\n\n@[to_additive]\ninstance [has_smul R M] [has_continuous_const_smul R M]\n  [has_smul R\u2081 M] [has_continuous_const_smul R\u2081 M]\n  [smul_comm_class R R\u2081 M] : smul_comm_class R R\u2081 C(\u03b1, M) :=\n{ smul_comm := \u03bb _ _ _, ext $ \u03bb _, smul_comm _ _ _ }\n\ninstance [has_smul R M] [has_continuous_const_smul R M]\n  [has_smul R\u2081 M] [has_continuous_const_smul R\u2081 M]\n  [has_smul R R\u2081] [is_scalar_tower R R\u2081 M] : is_scalar_tower R R\u2081 C(\u03b1, M) :=\n{ smul_assoc := \u03bb _ _ _, ext $ \u03bb _, smul_assoc _ _ _ }\n\ninstance [has_smul R M] [has_smul R\u1d50\u1d52\u1d56 M] [has_continuous_const_smul R M]\n  [is_central_scalar R M] : is_central_scalar R C(\u03b1, M) :=\n{ op_smul_eq_smul := \u03bb _ _, ext $ \u03bb _, op_smul_eq_smul _ _ }\n\ninstance [monoid R] [mul_action R M] [has_continuous_const_smul R M] : mul_action R C(\u03b1, M) :=\nfunction.injective.mul_action _ coe_injective coe_smul\n\ninstance [monoid R] [add_monoid M] [distrib_mul_action R M]\n  [has_continuous_add M] [has_continuous_const_smul R M] :\n  distrib_mul_action R C(\u03b1, M) :=\nfunction.injective.distrib_mul_action coe_fn_add_monoid_hom coe_injective coe_smul\n\nvariables [semiring R] [add_comm_monoid M] [add_comm_monoid M\u2082]\nvariables [has_continuous_add M] [module R M] [has_continuous_const_smul R M]\nvariables [has_continuous_add M\u2082] [module R M\u2082] [has_continuous_const_smul R M\u2082]\n\ninstance module : module R C(\u03b1, M) :=\nfunction.injective.module R coe_fn_add_monoid_hom coe_injective coe_smul\n\nvariables (R)\n\n/-- Composition on the left by a continuous linear map, as a `linear_map`.\nSimilar to `linear_map.comp_left`. -/\n@[simps] protected def _root_.continuous_linear_map.comp_left_continuous (\u03b1 : Type*)\n  [topological_space \u03b1] (g : M \u2192L[R] M\u2082) :\n  C(\u03b1, M) \u2192\u2097[R] C(\u03b1, M\u2082) :=\n{ map_smul' := \u03bb c f, ext $ \u03bb x, g.map_smul' c _,\n  .. g.to_linear_map.to_add_monoid_hom.comp_left_continuous \u03b1 g.continuous }\n\n/-- Coercion to a function as a `linear_map`. -/\n@[simps]\ndef coe_fn_linear_map : C(\u03b1, M) \u2192\u2097[R] (\u03b1 \u2192 M) :=\n{ to_fun := coe_fn,\n  map_smul' := coe_smul,\n  ..(coe_fn_add_monoid_hom : C(\u03b1, M) \u2192+ _) }\n\nend continuous_map\n\nend module_structure\n\nsection algebra_structure\n\n/-!\n### Algebra structure\n\nIn this section we show that continuous functions valued in a topological algebra `A` over a ring\n`R` inherit the structure of an algebra. Note that the hypothesis that `A` is a topological algebra\nis obtained by requiring that `A` be both a `has_continuous_smul` and a `topological_semiring`.-/\n\nsection subtype\n\nvariables {\u03b1 : Type*} [topological_space \u03b1]\n{R : Type*} [comm_semiring R]\n{A : Type*} [topological_space A] [semiring A]\n[algebra R A] [topological_semiring A]\n\n/-- The `R`-subalgebra of continuous maps `\u03b1 \u2192 A`. -/\ndef continuous_subalgebra : subalgebra R (\u03b1 \u2192 A) :=\n{ carrier := { f : \u03b1 \u2192 A | continuous f },\n  algebra_map_mem' := \u03bb r, (continuous_const : continuous $ \u03bb (x : \u03b1), algebra_map R A r),\n  ..continuous_subsemiring \u03b1 A }\n\nend subtype\n\nsection continuous_map\n\nvariables {\u03b1 : Type*} [topological_space \u03b1]\n{R : Type*} [comm_semiring R]\n{A : Type*} [topological_space A] [semiring A]\n[algebra R A] [topological_semiring A]\n{A\u2082 : Type*} [topological_space A\u2082] [semiring A\u2082]\n[algebra R A\u2082] [topological_semiring A\u2082]\n\n/-- Continuous constant functions as a `ring_hom`. -/\ndef continuous_map.C : R \u2192+* C(\u03b1, A) :=\n{ to_fun    := \u03bb c : R, \u27e8\u03bb x: \u03b1, ((algebra_map R A) c), continuous_const\u27e9,\n  map_one'  := by ext x; exact (algebra_map R A).map_one,\n  map_mul'  := \u03bb c\u2081 c\u2082, by ext x; exact (algebra_map R A).map_mul _ _,\n  map_zero' := by ext x; exact (algebra_map R A).map_zero,\n  map_add'  := \u03bb c\u2081 c\u2082, by ext x; exact (algebra_map R A).map_add _ _ }\n\n@[simp] lemma continuous_map.C_apply (r : R) (a : \u03b1) : continuous_map.C r a = algebra_map R A r :=\nrfl\n\ninstance continuous_map.algebra : algebra R C(\u03b1, A) :=\n{ to_ring_hom := continuous_map.C,\n  commutes' := \u03bb c f, by ext x; exact algebra.commutes' _ _,\n  smul_def' := \u03bb c f, by ext x; exact algebra.smul_def' _ _, }\n\nvariables (R)\n\n/-- Composition on the left by a (continuous) homomorphism of topological `R`-algebras, as an\n`alg_hom`. Similar to `alg_hom.comp_left`. -/\n@[simps] protected def alg_hom.comp_left_continuous {\u03b1 : Type*} [topological_space \u03b1]\n  (g : A \u2192\u2090[R] A\u2082) (hg : continuous g) :\n  C(\u03b1, A) \u2192\u2090[R] C(\u03b1, A\u2082) :=\n{ commutes' := \u03bb c, continuous_map.ext $ \u03bb _, g.commutes' _,\n  .. g.to_ring_hom.comp_left_continuous \u03b1 hg }\n\nvariables (A)\n\n/--\nPrecomposition of functions into a normed ring by a continuous map is an algebra homomorphism.\n-/\n@[simps] def continuous_map.comp_right_alg_hom {\u03b1 \u03b2 : Type*} [topological_space \u03b1]\n  [topological_space \u03b2] (f : C(\u03b1, \u03b2)) : C(\u03b2, A) \u2192\u2090[R] C(\u03b1, A) :=\n{ to_fun := \u03bb g, g.comp f,\n  map_zero' := by { ext, refl, },\n  map_add' := \u03bb g\u2081 g\u2082, by { ext, refl, },\n  map_one' := by { ext, refl, },\n  map_mul' := \u03bb g\u2081 g\u2082, by { ext, refl, },\n  commutes' := \u03bb r, by { ext, refl, }, }\n\nvariables {A}\n\n/-- Coercion to a function as an `alg_hom`. -/\n@[simps]\ndef continuous_map.coe_fn_alg_hom : C(\u03b1, A) \u2192\u2090[R] (\u03b1 \u2192 A) :=\n{ to_fun := coe_fn,\n  commutes' := \u03bb r, rfl,\n  ..(continuous_map.coe_fn_ring_hom : C(\u03b1, A) \u2192+* _) }\n\nvariables {R}\n\n/--\nA version of `separates_points` for subalgebras of the continuous functions,\nused for stating the Stone-Weierstrass theorem.\n-/\nabbreviation subalgebra.separates_points (s : subalgebra R C(\u03b1, A)) : Prop :=\nset.separates_points ((\u03bb f : C(\u03b1, A), (f : \u03b1 \u2192 A)) '' (s : set C(\u03b1, A)))\n\nlemma subalgebra.separates_points_monotone :\n  monotone (\u03bb s : subalgebra R C(\u03b1, A), s.separates_points) :=\n\u03bb s s' r h x y n,\nbegin\n  obtain \u27e8f, m, w\u27e9 := h n,\n  rcases m with \u27e8f, \u27e8m, rfl\u27e9\u27e9,\n  exact \u27e8_, \u27e8f, \u27e8r m, rfl\u27e9\u27e9, w\u27e9,\nend\n\n@[simp] lemma algebra_map_apply (k : R) (a : \u03b1) :\n  algebra_map R C(\u03b1, A) k a = k \u2022 1 :=\nby { rw algebra.algebra_map_eq_smul_one, refl, }\n\nvariables {\ud835\udd5c : Type*} [topological_space \ud835\udd5c]\n\n/--\nA set of continuous maps \"separates points strongly\"\nif for each pair of distinct points there is a function with specified values on them.\n\nWe give a slightly unusual formulation, where the specified values are given by some\nfunction `v`, and we ask `f x = v x \u2227 f y = v y`. This avoids needing a hypothesis `x \u2260 y`.\n\nIn fact, this definition would work perfectly well for a set of non-continuous functions,\nbut as the only current use case is in the Stone-Weierstrass theorem,\nwriting it this way avoids having to deal with casts inside the set.\n(This may need to change if we do Stone-Weierstrass on non-compact spaces,\nwhere the functions would be continuous functions vanishing at infinity.)\n-/\ndef set.separates_points_strongly (s : set C(\u03b1, \ud835\udd5c)) : Prop :=\n\u2200 (v : \u03b1 \u2192 \ud835\udd5c) (x y : \u03b1), \u2203 f : s, (f x : \ud835\udd5c) = v x \u2227 f y = v y\n\nvariables [field \ud835\udd5c] [topological_ring \ud835\udd5c]\n\n/--\nWorking in continuous functions into a topological field,\na subalgebra of functions that separates points also separates points strongly.\n\nBy the hypothesis, we can find a function `f` so `f x \u2260 f y`.\nBy an affine transformation in the field we can arrange so that `f x = a` and `f x = b`.\n-/\nlemma subalgebra.separates_points.strongly {s : subalgebra \ud835\udd5c C(\u03b1, \ud835\udd5c)} (h : s.separates_points) :\n  (s : set C(\u03b1, \ud835\udd5c)).separates_points_strongly :=\n\u03bb v x y,\nbegin\n  by_cases n : x = y,\n  { subst n,\n    use ((v x) \u2022 1 : C(\u03b1, \ud835\udd5c)),\n    { apply s.smul_mem,\n      apply s.one_mem, },\n    { simp [coe_fn_coe_base'] }, },\n  obtain \u27e8f, \u27e8f, \u27e8m, rfl\u27e9\u27e9, w\u27e9 := h n,\n  replace w : f x - f y \u2260 0 := sub_ne_zero_of_ne w,\n  let a := v x,\n  let b := v y,\n  let f' := ((b - a) * (f x - f y)\u207b\u00b9) \u2022 (continuous_map.C (f x) - f) + continuous_map.C a,\n  refine \u27e8\u27e8f', _\u27e9, _, _\u27e9,\n  { simp only [f', set_like.mem_coe, subalgebra.mem_to_submodule],\n    -- TODO should there be a tactic for this?\n    -- We could add an attribute `@[subobject_mem]`, and a tactic\n    -- ``def subobject_mem := `[solve_by_elim with subobject_mem { max_depth := 10 }]``\n    solve_by_elim\n      [subalgebra.add_mem, subalgebra.smul_mem, subalgebra.sub_mem, subalgebra.algebra_map_mem]\n      { max_depth := 6 }, },\n  { simp [f', coe_fn_coe_base'], },\n  { simp [f', coe_fn_coe_base', inv_mul_cancel_right\u2080 w], },\nend\n\nend continuous_map\n\ninstance continuous_map.subsingleton_subalgebra (\u03b1 : Type*) [topological_space \u03b1]\n  (R : Type*) [comm_semiring R] [topological_space R] [topological_semiring R]\n  [subsingleton \u03b1] : subsingleton (subalgebra R C(\u03b1, R)) :=\nbegin\n  fsplit,\n  intros s\u2081 s\u2082,\n  by_cases n : nonempty \u03b1,\n  { obtain \u27e8x\u27e9 := n,\n    ext f,\n    have h : f = algebra_map R C(\u03b1, R) (f x),\n    { ext x', simp only [mul_one, algebra.id.smul_eq_mul, algebra_map_apply], congr, },\n    rw h,\n    simp only [subalgebra.algebra_map_mem], },\n  { ext f,\n    have h : f = 0,\n    { ext x', exact false.elim (n \u27e8x'\u27e9), },\n    subst h,\n    simp only [subalgebra.zero_mem], },\nend\n\nend algebra_structure\n\nsection module_over_continuous_functions\n\n/-!\n### Structure as module over scalar functions\n\nIf `M` is a module over `R`, then we show that the space of continuous functions from `\u03b1` to `M`\nis naturally a module over the ring of continuous functions from `\u03b1` to `R`. -/\n\nnamespace continuous_map\n\ninstance has_smul' {\u03b1 : Type*} [topological_space \u03b1]\n  {R : Type*} [semiring R] [topological_space R]\n  {M : Type*} [topological_space M] [add_comm_monoid M]\n  [module R M] [has_continuous_smul R M] :\n  has_smul C(\u03b1, R) C(\u03b1, M) :=\n\u27e8\u03bb f g, \u27e8\u03bb x, (f x) \u2022 (g x), (continuous.smul f.2 g.2)\u27e9\u27e9\n\ninstance module' {\u03b1 : Type*} [topological_space \u03b1]\n  (R : Type*) [ring R] [topological_space R] [topological_ring R]\n  (M : Type*) [topological_space M] [add_comm_monoid M] [has_continuous_add M]\n  [module R M] [has_continuous_smul R M] :\n  module C(\u03b1, R) C(\u03b1, M) :=\n{ smul     := (\u2022),\n  smul_add := \u03bb c f g, by ext x; exact smul_add (c x) (f x) (g x),\n  add_smul := \u03bb c\u2081 c\u2082 f, by ext x; exact add_smul (c\u2081 x) (c\u2082 x) (f x),\n  mul_smul := \u03bb c\u2081 c\u2082 f, by ext x; exact mul_smul (c\u2081 x) (c\u2082 x) (f x),\n  one_smul := \u03bb f, by ext x; exact one_smul R (f x),\n  zero_smul := \u03bb f, by ext x; exact zero_smul _ _,\n  smul_zero := \u03bb r, by ext x; exact smul_zero _, }\n\nend continuous_map\n\nend module_over_continuous_functions\n\n/-!\nWe now provide formulas for `f \u2293 g` and `f \u2294 g`, where `f g : C(\u03b1, \u03b2)`,\nin terms of `continuous_map.abs`.\n-/\n\nsection\nvariables {R : Type*} [linear_ordered_field R]\n\n-- TODO:\n-- This lemma (and the next) could go all the way back in `algebra.order.field`,\n-- except that it is tedious to prove without tactics.\n-- Rather than stranding it at some intermediate location,\n-- it's here, immediately prior to the point of use.\nlemma min_eq_half_add_sub_abs_sub {x y : R} : min x y = 2\u207b\u00b9 * (x + y - |x - y|) :=\nby cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel\n\nlemma max_eq_half_add_add_abs_sub {x y : R} : max x y = 2\u207b\u00b9 * (x + y + |x - y|) :=\nby cases le_total x y with h h; field_simp [h, abs_of_nonneg, abs_of_nonpos, mul_two]; abel\n\nend\n\nnamespace continuous_map\n\nsection lattice\nvariables {\u03b1 : Type*} [topological_space \u03b1]\nvariables {\u03b2 : Type*} [linear_ordered_field \u03b2] [topological_space \u03b2]\n  [order_topology \u03b2] [topological_ring \u03b2]\n\nlemma inf_eq (f g : C(\u03b1, \u03b2)) : f \u2293 g = (2\u207b\u00b9 : \u03b2) \u2022 (f + g - |f - g|) :=\next (\u03bb x, by simpa using min_eq_half_add_sub_abs_sub)\n\n-- Not sure why this is grosser than `inf_eq`:\nlemma sup_eq (f g : C(\u03b1, \u03b2)) : f \u2294 g = (2\u207b\u00b9 : \u03b2) \u2022 (f + g + |f - g|) :=\next (\u03bb x, by simpa [mul_add] using @max_eq_half_add_add_abs_sub _ _ (f x) (g x))\n\nend lattice\n\n/-!\n### Star structure\n\nIf `\u03b2` has a continuous star operation, we put a star structure on `C(\u03b1, \u03b2)` by using the\nstar operation pointwise.\n\nIf `\u03b2` is a \u22c6-ring, then `C(\u03b1, \u03b2)` inherits a \u22c6-ring structure.\n\nIf `\u03b2` is a \u22c6-ring and a \u22c6-module over `R`, then the space of continuous functions from `\u03b1` to `\u03b2`\nis a \u22c6-module over `R`.\n\n-/\n\nsection star_structure\nvariables {R \u03b1 \u03b2 : Type*}\nvariables [topological_space \u03b1] [topological_space \u03b2]\n\nsection has_star\nvariables [has_star \u03b2] [has_continuous_star \u03b2]\n\ninstance : has_star C(\u03b1, \u03b2) :=\n{ star := \u03bb f, star_continuous_map.comp f }\n\n@[simp] lemma coe_star (f : C(\u03b1, \u03b2)) : \u21d1(star f) = star f := rfl\n\n@[simp] lemma star_apply (f : C(\u03b1, \u03b2)) (x : \u03b1) : star f x = star (f x) := rfl\n\nend has_star\n\ninstance [has_involutive_star \u03b2] [has_continuous_star \u03b2] : has_involutive_star C(\u03b1, \u03b2) :=\n{ star_involutive := \u03bb f, ext $ \u03bb x, star_star _ }\n\ninstance [add_monoid \u03b2] [has_continuous_add \u03b2] [star_add_monoid \u03b2] [has_continuous_star \u03b2] :\n  star_add_monoid C(\u03b1, \u03b2) :=\n{ star_add := \u03bb f g, ext $ \u03bb x, star_add _ _ }\n\ninstance [semigroup \u03b2] [has_continuous_mul \u03b2] [star_semigroup \u03b2] [has_continuous_star \u03b2] :\n  star_semigroup C(\u03b1, \u03b2) :=\n{ star_mul := \u03bb f g, ext $ \u03bb x, star_mul _ _ }\n\ninstance [non_unital_semiring \u03b2] [topological_semiring \u03b2] [star_ring \u03b2] [has_continuous_star \u03b2] :\n  star_ring C(\u03b1, \u03b2) :=\n{ ..continuous_map.star_add_monoid }\n\ninstance [has_star R] [has_star \u03b2] [has_smul R \u03b2] [star_module R \u03b2]\n  [has_continuous_star \u03b2] [has_continuous_const_smul R \u03b2] :\n  star_module R C(\u03b1, \u03b2) :=\n{ star_smul := \u03bb k f, ext $ \u03bb x, star_smul _ _ }\n\nend star_structure\n\nvariables {X Y Z : Type*} [topological_space X] [topological_space Y] [topological_space Z]\nvariables (\ud835\udd5c : Type*) [comm_semiring \ud835\udd5c]\nvariables (A : Type*) [topological_space A] [semiring A] [topological_semiring A] [star_ring A]\nvariables [has_continuous_star A] [algebra \ud835\udd5c A]\n\n/-- The functorial map taking `f : C(X, Y)` to `C(Y, A) \u2192\u22c6\u2090[\ud835\udd5c] C(X, A)` given by pre-composition\nwith the continuous function `f`. See `continuous_map.comp_monoid_hom'` and\n`continuous_map.comp_add_monoid_hom'`, `continuous_map.comp_right_alg_hom` for bundlings of\npre-composition into a `monoid_hom`, an `add_monoid_hom` and an `alg_hom`, respectively, under\nsuitable assumptions on `A`. -/\n@[simps] def comp_star_alg_hom' (f : C(X, Y)) : C(Y, A) \u2192\u22c6\u2090[\ud835\udd5c] C(X, A) :=\n{ to_fun := \u03bb g, g.comp f,\n  map_one' := one_comp _,\n  map_mul' := \u03bb _ _, rfl,\n  map_zero' := zero_comp _,\n  map_add' := \u03bb _ _, rfl,\n  commutes' := \u03bb _, rfl,\n  map_star' := \u03bb _, rfl }\n\n/-- `continuous_map.comp_star_alg_hom'` sends the identity continuous map to the identity\n`star_alg_hom` -/\nlemma comp_star_alg_hom'_id :\n  comp_star_alg_hom' \ud835\udd5c A (continuous_map.id X) = star_alg_hom.id \ud835\udd5c C(X, A) :=\nstar_alg_hom.ext $ \u03bb _, continuous_map.ext $ \u03bb _, rfl\n\n/-- `continuous_map.comp_star_alg_hom` is functorial. -/\nlemma comp_star_alg_hom'_comp (g : C(Y, Z)) (f : C(X, Y)) :\n  comp_star_alg_hom' \ud835\udd5c A (g.comp f) = (comp_star_alg_hom' \ud835\udd5c A f).comp (comp_star_alg_hom' \ud835\udd5c A g) :=\nstar_alg_hom.ext $ \u03bb _, continuous_map.ext $ \u03bb _, rfl\n\nsection periodicity\n\n/-! ### Summing translates of a function -/\n\n/-- Summing the translates of `f` by `\u2124 \u2022 p` gives a map which is periodic with period `p`.\n(This is true without any convergence conditions, since if the sum doesn't converge it is taken to\nbe the zero map, which is periodic.) -/\nlemma periodic_tsum_comp_add_zsmul [locally_compact_space X] [add_comm_group X]\n  [topological_add_group X] [add_comm_monoid Y] [has_continuous_add Y] [t2_space Y]\n  (f : C(X, Y)) (p : X) :\n  function.periodic \u21d1(\u2211' (n : \u2124), f.comp (continuous_map.add_right (n \u2022 p))) p :=\nbegin\n  intro x,\n  by_cases h : summable (\u03bb n : \u2124, f.comp (continuous_map.add_right (n \u2022 p))),\n  { convert congr_arg (\u03bb f : C(X, Y), f x) ((equiv.add_right (1 : \u2124)).tsum_eq _) using 1,\n    simp_rw [\u2190tsum_apply h, \u2190tsum_apply ((equiv.add_right (1 : \u2124)).summable_iff.mpr h),\n      equiv.coe_add_right, comp_apply, coe_add_right, add_one_zsmul, add_comm (_ \u2022 p) p,\n      \u2190add_assoc] },\n  { rw tsum_eq_zero_of_not_summable h,\n    simp only [coe_zero, pi.zero_apply] }\nend\n\nend periodicity\n\nend continuous_map\n\nnamespace homeomorph\n\nvariables {X Y : Type*} [topological_space X] [topological_space Y]\nvariables (\ud835\udd5c : Type*) [comm_semiring \ud835\udd5c]\nvariables (A : Type*) [topological_space A] [semiring A] [topological_semiring A] [star_ring A]\nvariables [has_continuous_star A] [algebra \ud835\udd5c A]\n\n/-- `continuous_map.comp_star_alg_hom'` as a `star_alg_equiv` when the continuous map `f` is\nactually a homeomorphism. -/\n@[simps] def comp_star_alg_equiv' (f : X \u2243\u209c Y) : C(Y, A) \u2243\u22c6\u2090[\ud835\udd5c] C(X, A) :=\n{ to_fun := (f : C(X, Y)).comp_star_alg_hom' \ud835\udd5c A,\n  inv_fun := (f.symm : C(Y, X)).comp_star_alg_hom' \ud835\udd5c A,\n  left_inv := \u03bb g, by simp only [continuous_map.comp_star_alg_hom'_apply, continuous_map.comp_assoc,\n    to_continuous_map_comp_symm, continuous_map.comp_id],\n  right_inv := \u03bb g, by simp only [continuous_map.comp_star_alg_hom'_apply,\n    continuous_map.comp_assoc, symm_comp_to_continuous_map, continuous_map.comp_id],\n  map_smul' := \u03bb k a, map_smul (f.to_continuous_map.comp_star_alg_hom' \ud835\udd5c A) k a,\n  .. (f.to_continuous_map.comp_star_alg_hom' \ud835\udd5c A) }\n\nend homeomorph\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/topology/continuous_function/algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.3920090854290492}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.shapes.pullbacks\nimport ring_theory.tensor_product\nimport algebra.category.Ring.limits\nimport algebra.category.Ring.instances\nimport category_theory.limits.shapes.strict_initial\nimport ring_theory.subring.basic\n\n/-!\n# Constructions of (co)limits in CommRing\n\nIn this file we provide the explicit (co)cones for various (co)limits in `CommRing`, including\n* tensor product is the pushout\n* `Z` is the initial object\n* `0` is the strict terminal object\n* cartesian product is the product\n* `ring_hom.eq_locus` is the equalizer\n\n-/\n\nuniverses u u'\n\nopen category_theory category_theory.limits\nopen_locale tensor_product\n\nnamespace CommRing\n\nsection pushout\n\nvariables {R A B : CommRing.{u}} (f : R \u27f6 A) (g : R \u27f6 B)\n\n/-- The explicit cocone with tensor products as the fibered product in `CommRing`. -/\ndef pushout_cocone : limits.pushout_cocone f g :=\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  apply limits.pushout_cocone.mk,\n  show CommRing, from CommRing.of (A \u2297[R] B),\n  show A \u27f6 _,  from algebra.tensor_product.include_left.to_ring_hom,\n  show B \u27f6 _,  from algebra.tensor_product.include_right.to_ring_hom,\n  ext r,\n  transitivity algebra_map R (A \u2297[R] B) r,\n  { exact algebra.tensor_product.include_left.commutes r },\n  { exact (algebra.tensor_product.include_right.commutes r).symm }\nend\n\n@[simp]\nlemma pushout_cocone_inl : (pushout_cocone f g).inl = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_left.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_inr : (pushout_cocone f g).inr = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_right.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_X : (pushout_cocone f g).X = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI CommRing.of (A \u2297[R] B) }) := rfl\n\n/-- Verify that the `pushout_cocone` is indeed the colimit. -/\ndef pushout_cocone_is_colimit : limits.is_colimit (pushout_cocone f g) :=\nlimits.pushout_cocone.is_colimit_aux' _ (\u03bb s,\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  letI := ring_hom.to_algebra (f \u226b s.inl),\n  let f' : A \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change s.inl.to_fun (f r) = (f \u226b s.inl) r, refl }, ..s.inl },\n  let g' : B \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change (g \u226b s.inr) r = (f \u226b s.inl) r,\n        congr' 1,\n        exact (s.\u03b9.naturality limits.walking_span.hom.snd).trans\n          (s.\u03b9.naturality limits.walking_span.hom.fst).symm }, ..s.inr },\n  /- The factor map is a \u2297 b \u21a6 f(a) * g(b). -/\n  use alg_hom.to_ring_hom (algebra.tensor_product.product_map f' g'),\n  simp only [pushout_cocone_inl, pushout_cocone_inr],\n  split, { ext x, exact algebra.tensor_product.product_map_left_apply  _ _ x, },\n  split, { ext x, exact algebra.tensor_product.product_map_right_apply _ _ x, },\n  intros h eq1 eq2,\n  let h' : (A \u2297[R] B) \u2192\u2090[R] s.X :=\n    { commutes' := \u03bb r, by\n    { change h ((f r) \u2297\u209c[R] 1) = s.inl (f r),\n      rw \u2190 eq1, simp }, ..h },\n  suffices : h' = algebra.tensor_product.product_map f' g',\n  { ext x,\n    change h' x = algebra.tensor_product.product_map f' g' x,\n    rw this },\n  apply algebra.tensor_product.ext,\n  intros a b,\n  simp [\u2190 eq1, \u2190 eq2, \u2190 h.map_mul],\nend)\n\nend pushout\n\nsection terminal\n\n/-- The trivial ring is the (strict) terminal object of `CommRing`. -/\ndef punit_is_terminal : is_terminal (CommRing.of.{u} punit) :=\nbegin\n  apply_with is_terminal.of_unique { instances := ff },\n  tidy\nend\n\ninstance CommRing_has_strict_terminal_objects : has_strict_terminal_objects CommRing.{u} :=\nbegin\n  apply has_strict_terminal_objects_of_terminal_is_strict (CommRing.of punit),\n  intros X f,\n  refine \u27e8\u27e8by tidy, by ext, _\u27e9\u27e9,\n  ext,\n  have e : (0 : X) = 1 := by { rw [\u2190 f.map_one, \u2190 f.map_zero], congr },\n  replace e : 0 * x = 1 * x := congr_arg (\u03bb a, a * x) e,\n  rw [one_mul, zero_mul, \u2190 f.map_zero] at e,\n  exact e,\nend\n\nlemma subsingleton_of_is_terminal {X : CommRing} (hX : is_terminal X) : subsingleton X :=\n(hX.unique_up_to_iso punit_is_terminal).CommRing_iso_to_ring_equiv.to_equiv\n  .subsingleton_congr.mpr (show subsingleton punit, by apply_instance)\n\n/-- `\u2124` is the initial object of `CommRing`. -/\ndef Z_is_initial : is_initial (CommRing.of \u2124) :=\nbegin\n  apply_with is_initial.of_unique { instances := ff },\n  exact \u03bb R, \u27e8\u27e8int.cast_ring_hom R\u27e9, \u03bb a, a.ext_int _\u27e9,\nend\n\nend terminal\n\nsection product\n\nvariables (A B : CommRing.{u})\n\n/-- The product in `CommRing` is the cartesian product. This is the binary fan. -/\n@[simps X]\ndef prod_fan : binary_fan A B :=\nbinary_fan.mk (CommRing.of_hom $ ring_hom.fst A B) (CommRing.of_hom $ ring_hom.snd A B)\n\n/-- The product in `CommRing` is the cartesian product. -/\ndef prod_fan_is_limit : is_limit (prod_fan A B) :=\n{ lift := \u03bb c, ring_hom.prod (c.\u03c0.app \u27e8walking_pair.left\u27e9) (c.\u03c0.app \u27e8walking_pair.right\u27e9),\n  fac' := \u03bb c j, by { ext, rcases j with \u27e8\u27e8\u27e9\u27e9;\n    simpa only [binary_fan.\u03c0_app_left, binary_fan.\u03c0_app_right, comp_apply, ring_hom.prod_apply] },\n  uniq' := \u03bb s m h, by { ext, { simpa using congr_hom (h \u27e8walking_pair.left\u27e9) x },\n    { simpa using congr_hom (h \u27e8walking_pair.right\u27e9) x } } }\n\nend product\n\nsection equalizer\n\nvariables {A B : CommRing.{u}} (f g : A \u27f6 B)\n\n/-- The equalizer in `CommRing` is the equalizer as sets. This is the equalizer fork. -/\ndef equalizer_fork : fork f g :=\nfork.of_\u03b9 (CommRing.of_hom (ring_hom.eq_locus f g).subtype) (by { ext \u27e8x, e\u27e9, simpa using e })\n\n/-- The equalizer in `CommRing` is the equalizer as sets. -/\ndef equalizer_fork_is_limit : is_limit (equalizer_fork f g) :=\nbegin\n  fapply fork.is_limit.mk',\n  intro s,\n  use s.\u03b9.cod_restrict _ (\u03bb x, (concrete_category.congr_hom s.condition x : _)),\n  split,\n  { ext, refl },\n  { intros m hm, ext x, exact concrete_category.congr_hom hm x }\nend\n\ninstance : is_local_ring_hom (equalizer_fork f g).\u03b9 :=\nbegin\n  constructor,\n  rintros \u27e8a, (h\u2081 : _ = _)\u27e9 (\u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,(rfl : x = _)\u27e9 : is_unit a),\n  have : y \u2208 ring_hom.eq_locus f g,\n  { apply (f.is_unit_map \u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,rfl\u27e9 : is_unit (f x)).mul_left_inj.mp,\n    conv_rhs { rw h\u2081 },\n    rw [\u2190 f.map_mul, \u2190 g.map_mul, h\u2084, f.map_one, g.map_one] },\n  rw is_unit_iff_exists_inv,\n  exact \u27e8\u27e8y, this\u27e9, subtype.eq h\u2083\u27e9,\nend\n\ninstance equalizer_\u03b9_is_local_ring_hom (F : walking_parallel_pair \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F walking_parallel_pair.zero) :=\nbegin\n  have := lim_map_\u03c0 (diagram_iso_parallel_pair F).hom walking_parallel_pair.zero,\n  rw \u2190 is_iso.comp_inv_eq at this,\n  rw \u2190 this,\n  rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, equalizer_fork_is_limit\n    (F.map walking_parallel_pair_hom.left) (F.map walking_parallel_pair_hom.right)\u27e9\n    walking_parallel_pair.zero,\n  change is_local_ring_hom ((lim.map _ \u226b _ \u226b (equalizer_fork _ _).\u03b9) \u226b _),\n  apply_instance\nend\n\nopen category_theory.limits.walking_parallel_pair opposite\nopen category_theory.limits.walking_parallel_pair_hom\n\ninstance equalizer_\u03b9_is_local_ring_hom' (F : walking_parallel_pair\u1d52\u1d56 \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F (opposite.op walking_parallel_pair.one)) :=\nbegin\n  have : _ = limit.\u03c0 F (walking_parallel_pair_op_equiv.functor.obj _) :=\n    (limit.iso_limit_cone_inv_\u03c0 \u27e8_, is_limit.whisker_equivalence (limit.is_limit F)\n      walking_parallel_pair_op_equiv\u27e9 walking_parallel_pair.zero : _),\n  erw \u2190 this,\n  apply_instance\nend\n\nend equalizer\n\nsection pullback\n\n/--\nIn the category of `CommRing`, the pullback of `f : A \u27f6 C` and `g : B \u27f6 C` is the `eq_locus` of\nthe two maps `A \u00d7 B \u27f6 C`. This is the constructed pullback cone.\n-/\ndef pullback_cone {A B C : CommRing.{u}} (f : A \u27f6 C) (g : B \u27f6 C) : pullback_cone f g :=\npullback_cone.mk\n  (CommRing.of_hom $ (ring_hom.fst A B).comp\n    (ring_hom.eq_locus (f.comp (ring_hom.fst A B)) (g.comp (ring_hom.snd A B))).subtype)\n  (CommRing.of_hom $ (ring_hom.snd A B).comp\n    (ring_hom.eq_locus (f.comp (ring_hom.fst A B)) (g.comp (ring_hom.snd A B))).subtype)\n  (by { ext \u27e8x, e\u27e9, simpa [CommRing.of_hom] using e })\n\n/-- The constructed pullback cone is indeed the limit. -/\ndef pullback_cone_is_limit {A B C : CommRing.{u}} (f : A \u27f6 C) (g : B \u27f6 C) :\n  is_limit (pullback_cone f g) :=\nbegin\n  fapply pullback_cone.is_limit.mk,\n  { intro s,\n    apply (s.fst.prod s.snd).cod_restrict,\n    intro x, exact congr_arg (\u03bb f : s.X \u2192+* C, f x) s.condition },\n  { intro s, ext x, refl },\n  { intro s, ext x, refl },\n  { intros s m e\u2081 e\u2082, ext,\n    { exact (congr_arg (\u03bb f : s.X \u2192+* A, f x) e\u2081 : _) },\n    { exact (congr_arg (\u03bb f : s.X \u2192+* B, f x) e\u2082 : _) } }\nend\n\nend pullback\n\nend CommRing\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/category/Ring/constructions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3920090722823373}}
{"text": "import category_theory.monoidal.rigid.basic\nimport kassel.lemma.to_matrix\n\nnamespace kassel\nopen category_theory\nopen category_theory.monoidal_category\n\nuniverses v u\n\nsection\n\nvariables\n  {C: Type u}\n  [category.{v} C]\n  [monoidal_category.{v} C]\n  [right_rigid_category.{v} C]\n\nlemma congr_comp_left {X Y Z: C} (h: X \u27f6 Y) (f g: Y \u27f6 Z):\n  f = g \u2192 h \u226b f = h \u226b g :=\nby intro h; rw h\n\nlemma congr_comp_right {X Y Z: C} (h: Y \u27f6 Z) (f g: X \u27f6 Y):\n  f = g \u2192 f \u226b h = g \u226b h :=\nby intro h; rw h\n\nlemma congr_tensor_left {X\u2081 Y\u2081 X\u2082 Y\u2082: C} (h: X\u2081 \u27f6 Y\u2081) (f g: X\u2082 \u27f6 Y\u2082):\n  f = g \u2192 h \u2297 f = h \u2297 g :=\nby intro h; rw h\n\nlemma congr_tensor_right {X\u2081 Y\u2081 X\u2082 Y\u2082: C} (h: X\u2082 \u27f6 Y\u2082) (f g: X\u2081 \u27f6 Y\u2081):\n  f = g \u2192 f \u2297 h = g \u2297 h :=\nby intro h; rw h\n\nnamespace iso\n\n@[reassoc] lemma hom_dual_inv_dual_id {X Y: C} (f: X \u2245 Y):\n  (f.hom)\u1601 \u226b (f.inv)\u1601 = \ud835\udfd9 _ :=\nby rw [\u2190comp_right_adjoint_mate, iso.inv_hom_id, right_adjoint_mate_id]\n\n@[reassoc] lemma inv_dual_hom_dual_id {X Y: C} (f: X \u2245 Y):\n  (f.inv)\u1601 \u226b (f.hom)\u1601 = \ud835\udfd9 _ :=\nby rw [\u2190comp_right_adjoint_mate, iso.hom_inv_id, right_adjoint_mate_id]\n\nend iso\n\n-- * Show that `X \u2297 Y` and `Y\u1601 \u2297 X\u1601` form an exact pairing.\n\n@[instance] def tensor_exact_pairing (X Y: C): exact_pairing (X \u2297 Y) (Y\u1601 \u2297 X\u1601) := {\n  coevaluation := \u03b7_ X X\u1601 \u226b ((\u03c1_ _).inv \u2297  \ud835\udfd9 _) \u226b ((\ud835\udfd9 _ \u2297 \u03b7_ Y Y\u1601) \u2297 \ud835\udfd9 _) \u226b ((\u03b1_ _ _ _).inv \u2297 \ud835\udfd9 _) \u226b (\u03b1_ _ _ _).hom,\n  evaluation := (\u03b1_ _ _ _).inv \u226b ((\u03b1_ _ _ _).hom \u2297 \ud835\udfd9 _) \u226b ((\ud835\udfd9 _ \u2297 \u03b5_ X X\u1601) \u2297 \ud835\udfd9 _) \u226b ((\u03c1_ _).hom \u2297 \ud835\udfd9 _) \u226b \u03b5_ Y Y\u1601,\n  coevaluation_evaluation' :=\n  begin\n    simp_rw [id_tensor_comp, comp_tensor_id, \u2190tensor_id, category.assoc],\n    slice_lhs 2 3 {\n      rw \u2190triangle_assoc_comp_left_inv,\n      simp only [\u2190tensor_comp],\n      rw [category.assoc, \u2190associator_inv_naturality, \u2190id_tensor_comp_assoc],\n      simp only [tensor_comp],\n      rw [associator_inv_conjugation, associator_conjugation (\ud835\udfd9 Y\u1601) _ _],\n      rw [\u2190category.id_comp ((\u03bb_ X\u1601).inv \u226b (\u03b7_ Y Y\u1601 \u2297 \ud835\udfd9 X\u1601)), \u2190category.comp_id ((\u03bb_ X\u1601).inv \u226b (\u03b7_ Y Y\u1601 \u2297 \ud835\udfd9 X\u1601))],\n      simp only [tensor_comp, \u2190tensor_id],\n    },\n    slice_lhs 13 14 {\n      simp only [\u2190tensor_comp, category.comp_id],\n      rw [associator_conjugation, associator_inv_conjugation _ _ (\ud835\udfd9 X\u1601)],\n      rw [\u2190category.id_comp ((\ud835\udfd9 Y\u1601 \u2297 \u03b5_ X X\u1601) \u226b (\u03c1_ Y\u1601).hom), \u2190category.comp_id ((\ud835\udfd9 Y\u1601 \u2297 \u03b5_ X X\u1601) \u226b (\u03c1_ Y\u1601).hom)],\n      simp only [tensor_comp, \u2190tensor_id],\n    },\n    have h: ((\u03b1_ Y\u1601 X\u1601 X).inv \u2297 (\ud835\udfd9 Y \u2297 \ud835\udfd9 Y\u1601) \u2297 \ud835\udfd9 X\u1601) \u226b (\u03b1_ (Y\u1601 \u2297 X\u1601) X ((Y \u2297 Y\u1601) \u2297 X\u1601)).hom \u226b ((\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u2297 (\u03b1_ X (Y \u2297 Y\u1601) X\u1601).inv) \u226b ((\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u2297 (\u03b1_ X Y Y\u1601).inv \u2297 \ud835\udfd9 X\u1601) \u226b ((\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u2297 (\u03b1_ (X \u2297 Y) Y\u1601 X\u1601).hom) \u226b (\u03b1_ (Y\u1601 \u2297 X\u1601) (X \u2297 Y) (Y\u1601 \u2297 X\u1601)).inv \u226b ((\u03b1_ (Y\u1601 \u2297 X\u1601) X Y).inv \u2297 \ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u226b (((\u03b1_ Y\u1601 X\u1601 X).hom \u2297 \ud835\udfd9 Y) \u2297 \ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u226b (\u03b1_ (Y\u1601 \u2297 X\u1601 \u2297 X) Y (Y\u1601 \u2297 X\u1601)).hom \u226b ((\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601 \u2297 \ud835\udfd9 X) \u2297 (\u03b1_ Y Y\u1601 X\u1601).inv) = \ud835\udfd9 _ := by coherence,\n    slice_lhs 4 14 { rw [h, category.comp_id], }, clear h,\n    slice_lhs 4 5 {\n      simp only [tensor_id],\n      rw eq.trans (id_tensor_comp_tensor_id _ _) (tensor_id_comp_id_tensor _ _).symm,\n      simp only [id_tensor_comp, comp_tensor_id],\n    },\n    iterate 2 { rw associator_conjugation, },\n    slice_lhs 3 6 { rw [\u2190pentagon_hom_inv, iso.inv_hom_id_assoc], },\n    slice_lhs 2 4 { simp only [\u2190id_tensor_comp], rw exact_pairing.coevaluation_evaluation, },\n    iterate 2 { rw associator_inv_conjugation, },\n    slice_lhs 8 11 { rw [\u2190pentagon_inv_inv_hom_assoc, iso.hom_inv_id_assoc, iso.hom_inv_id, category.comp_id], },\n    slice_lhs 7 9 { simp only [\u2190comp_tensor_id], rw exact_pairing.coevaluation_evaluation, },\n    coherence,\n  end,\n  evaluation_coevaluation' :=\n  begin\n    simp_rw [id_tensor_comp, comp_tensor_id, \u2190tensor_id],\n    slice_lhs 2 3 {\n      simp only [\u2190tensor_comp, category.comp_id],\n      rw [associator_conjugation, associator_inv_conjugation (\ud835\udfd9 X\u1601) _ _],\n      rw [\u2190category.id_comp ((\u03c1_ X).inv \u226b (\ud835\udfd9 X \u2297 \u03b7_ Y Y\u1601)), \u2190category.comp_id ((\u03c1_ X).inv \u226b (\ud835\udfd9 X \u2297 \u03b7_ Y Y\u1601))],\n      simp only [tensor_comp, \u2190tensor_id],\n    },\n    slice_lhs 12 13 {\n      rw \u2190triangle,\n      simp only [\u2190tensor_comp],\n      rw [associator_naturality_assoc, \u2190id_tensor_comp],\n      simp only [tensor_comp],\n      rw [associator_inv_conjugation, associator_conjugation _ _ (\ud835\udfd9 Y\u1601)],\n      rw [\u2190category.id_comp ((\u03b5_ X X\u1601 \u2297 \ud835\udfd9 Y) \u226b (\u03bb_ Y).hom), \u2190category.comp_id ((\u03b5_ X X\u1601 \u2297 \ud835\udfd9 Y) \u226b (\u03bb_ Y).hom)],\n      simp only [tensor_comp, \u2190tensor_id],\n    },\n    have h: ((\ud835\udfd9 X \u2297 \ud835\udfd9 Y \u2297 \ud835\udfd9 Y\u1601) \u2297 (\u03b1_ X\u1601 X Y).hom) \u226b (\u03b1_ (X \u2297 Y \u2297 Y\u1601) X\u1601 (X \u2297 Y)).inv \u226b (((\u03b1_ X Y Y\u1601).inv \u2297 \ud835\udfd9 X\u1601) \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u226b ((\u03b1_ (X \u2297 Y) Y\u1601 X\u1601).hom \u2297 \ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (X \u2297 Y) (Y\u1601 \u2297 X\u1601) (X \u2297 Y)).hom \u226b ((\ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u2297 (\u03b1_ (Y\u1601 \u2297 X\u1601) X Y).inv) \u226b ((\ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u2297 (\u03b1_ Y\u1601 X\u1601 X).hom \u2297 \ud835\udfd9 Y) \u226b ((\ud835\udfd9 X \u2297 \ud835\udfd9 Y) \u2297 (\u03b1_ Y\u1601 (X\u1601 \u2297 X) Y).hom) \u226b (\u03b1_ (X \u2297 Y) Y\u1601 ((X\u1601 \u2297 X) \u2297 Y)).inv \u226b ((\u03b1_ X Y Y\u1601).hom \u2297 (\ud835\udfd9 X\u1601 \u2297 \ud835\udfd9 X) \u2297 \ud835\udfd9 Y) = \ud835\udfd9 _ := by coherence,\n    slice_lhs 4 14 { rw [h, category.comp_id], }, clear h,\n    slice_lhs 4 5 {\n      simp only [tensor_id],\n      rw eq.trans (tensor_id_comp_id_tensor _ _) (id_tensor_comp_tensor_id _ _).symm,\n      simp only [id_tensor_comp, comp_tensor_id],\n    },\n    iterate 2 { rw associator_inv_conjugation, },\n    slice_lhs 3 6 { rw [\u2190pentagon_inv_hom, iso.hom_inv_id_assoc], },\n    slice_lhs 2 4 { simp only [\u2190comp_tensor_id], rw exact_pairing.evaluation_coevaluation, },\n    iterate 2 { rw associator_conjugation, },\n    slice_lhs 8 11 { rw [pentagon_inv_inv_hom_assoc, iso.inv_hom_id, category.comp_id], },\n    slice_lhs 7 9 { simp only [\u2190id_tensor_comp], rw exact_pairing.evaluation_coevaluation, },\n    coherence,\n  end\n}\n\ndef tensor_iso_dual_tensor_dual (X Y: C): (X \u2297 Y)\u1601 \u2245 Y\u1601 \u2297 X\u1601 := {\n  hom := (\u03c1_ _).inv \u226b (\ud835\udfd9 _ \u2297 \u03b7_ (X \u2297 Y) (Y\u1601 \u2297 X\u1601)) \u226b (\u03b1_ _ _ _).inv \u226b ((\u03b5_ (X \u2297 Y) (X \u2297 Y)\u1601) \u2297 \ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u226b (\u03bb_ _).hom,\n  inv := (\u03c1_ _).inv \u226b ((\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u2297 (\u03b7_ (X \u2297 Y) (X \u2297 Y)\u1601)) \u226b (\u03b1_ _ _ _).inv \u226b (\u03b5_ (X \u2297 Y) (Y\u1601 \u2297 X\u1601) \u2297 \ud835\udfd9 _) \u226b (\u03bb_ _).hom,\n  hom_inv_id' := begin\n    rw right_unitor_inv_naturality_assoc,\n    simp_rw comp_tensor_id,\n    slice_lhs 6 7 { rw [tensor_id, eq.trans (tensor_id_comp_id_tensor _ _) (id_tensor_comp_tensor_id _ _).symm], },\n    slice_lhs 7 8 { rw [\u2190tensor_id, associator_inv_naturality], },\n    slice_lhs 8 9 { rw [\u2190comp_tensor_id, \u2190left_unitor_tensor', category.assoc, \u2190left_unitor_naturality], simp only [comp_tensor_id], },\n    slice_lhs 9 11 { rw [unitors_equal, \u2190triangle], simp only [category.assoc], rw associator_naturality_assoc, },\n    slice_lhs 2 6 { simp only [\u2190tensor_comp, category.id_comp], rw [\u2190category.comp_id (\u03b7_ _ (X \u2297 Y)\u1601), tensor_comp], simp only [comp_tensor_id], },\n    slice_lhs 6 9 { rw [tensor_id, category.id_comp, \u2190pentagon_hom_inv], },\n    slice_lhs 5 6 { rw associator_naturality, },\n    slice_lhs 6 9 { simp only [\u2190tensor_comp, category.comp_id, \u2190category.assoc], rw [\u2190category.id_comp (\u03b5_ _ (X \u2297 Y)\u1601), tensor_comp], simp only [id_tensor_comp], },\n    \n    slice_lhs 3 3 { rw [\u2190tensor_id, associator_conjugation, associator_inv_conjugation (\u03b7_ _ _) _ _], },\n    slice_lhs 10 10 { rw [\u2190tensor_id, associator_conjugation, associator_inv_conjugation _ (\u03b5_ _ _) _], },\n    simp_rw id_tensor_comp,\n    have h: (\ud835\udfd9 (X \u2297 Y)\u1601 \u2297 (\u03b1_ ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601) (X \u2297 Y) (X \u2297 Y)\u1601).hom) \u226b (\u03b1_ (X \u2297 Y)\u1601 ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601) ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601)).inv \u226b ((\u03b1_ (X \u2297 Y)\u1601 (X \u2297 Y) (Y\u1601 \u2297 X\u1601)).inv \u2297 \ud835\udfd9 ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601)) \u226b (\u03b1_ ((X \u2297 Y)\u1601 \u2297 X \u2297 Y) (Y\u1601 \u2297 X\u1601) ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601)).hom \u226b (\ud835\udfd9 ((X \u2297 Y)\u1601 \u2297 X \u2297 Y) \u2297 (\ud835\udfd9 Y\u1601 \u2297 \ud835\udfd9 X\u1601) \u2297 \ud835\udfd9 ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601)) \u226b (\ud835\udfd9 ((X \u2297 Y)\u1601 \u2297 X \u2297 Y) \u2297 (\u03b1_ (Y\u1601 \u2297 X\u1601) (X \u2297 Y) (X \u2297 Y)\u1601).inv) \u226b (\u03b1_ (X \u2297 Y)\u1601 (X \u2297 Y) (((Y\u1601 \u2297 X\u1601) \u2297 X \u2297 Y) \u2297 (X \u2297 Y)\u1601)).hom \u226b (\ud835\udfd9 (X \u2297 Y)\u1601 \u2297 (\u03b1_ (X \u2297 Y) ((Y\u1601 \u2297 X\u1601) \u2297 X \u2297 Y) (X \u2297 Y)\u1601).inv) = \ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).hom \u2297 \ud835\udfd9 _ := by coherence,\n    slice_lhs 6 13 { rw h, }, clear h,\n    slice_lhs 5 7 {\n      simp only [\u2190id_tensor_comp, \u2190comp_tensor_id],\n      rw exact_pairing.evaluation_coevaluation,\n      simp only [id_tensor_comp, comp_tensor_id],\n      rw [associator_inv_conjugation, \u2190triangle_assoc_comp_right, \u2190triangle_assoc_comp_left_inv],\n      simp only [id_tensor_comp, comp_tensor_id],\n    },\n    slice_lhs 3 7 { rw \u2190pentagon_inv_hom_assoc, simp only [\u2190comp_tensor_id], rw [iso.hom_inv_id_assoc, \u2190associator_inv_naturality], },\n    slice_lhs 4 9 { rw iso.inv_hom_id_assoc, simp only [\u2190id_tensor_comp_assoc], rw [iso.inv_hom_id, category.comp_id, associator_inv_naturality], },\n    slice_lhs 2 3 { rw [\u2190tensor_comp, iso.inv_hom_id, tensor_id, category.comp_id], },\n    slice_lhs 4 5 { rw [\u2190tensor_comp, iso.inv_hom_id, tensor_id, category.id_comp], },\n    slice_lhs 2 4 { rw exact_pairing.coevaluation_evaluation, },\n    coherence,\n  end,\n  inv_hom_id' := begin\n    rw right_unitor_inv_naturality_assoc,\n    simp_rw comp_tensor_id,\n    slice_lhs 6 7 { rw [eq.trans (tensor_id_comp_id_tensor _ _) (id_tensor_comp_tensor_id _ _).symm], },\n    slice_lhs 7 8 { rw [\u2190tensor_id, associator_inv_naturality], },\n    slice_lhs 8 9 { rw [tensor_id, \u2190comp_tensor_id, \u2190left_unitor_tensor', category.assoc, \u2190left_unitor_naturality], simp only [comp_tensor_id], },\n    slice_lhs 9 11 { rw [unitors_equal, \u2190triangle], simp only [category.assoc], rw associator_naturality_assoc, },\n    slice_lhs 2 6 { simp only [\u2190tensor_comp, category.id_comp], rw [\u2190category.comp_id (\u03b7_ _ (Y\u1601 \u2297 X\u1601)), tensor_comp], simp only [comp_tensor_id], },\n    slice_lhs 6 9 { rw [tensor_id, category.id_comp, \u2190pentagon_hom_inv], },\n    slice_lhs 5 6 { rw associator_naturality, },\n    slice_lhs 6 9 { simp only [\u2190tensor_comp, category.comp_id, \u2190category.assoc], rw [\u2190category.id_comp (\u03b5_ _ (Y\u1601 \u2297 X\u1601)), tensor_comp], simp only [id_tensor_comp], },\n    \n    slice_lhs 3 3 { rw [\u2190tensor_id, associator_conjugation, associator_inv_conjugation (\u03b7_ _ _) _ _, tensor_id], },\n    slice_lhs 10 10 { rw [\u2190tensor_id, associator_conjugation, associator_inv_conjugation _ (\u03b5_ _ _) _], },\n    simp_rw id_tensor_comp,\n    have h: (\ud835\udfd9 (Y\u1601 \u2297 X\u1601) \u2297 (\u03b1_ ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601) (X \u2297 Y) (Y\u1601 \u2297 X\u1601)).hom) \u226b (\u03b1_ (Y\u1601 \u2297 X\u1601) ((X \u2297 Y) \u2297 (X \u2297 Y)\u1601) ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601)).inv \u226b ((\u03b1_ (Y\u1601 \u2297 X\u1601) (X \u2297 Y) (X \u2297 Y)\u1601).inv \u2297 \ud835\udfd9 ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601)) \u226b (\u03b1_ ((Y\u1601 \u2297 X\u1601) \u2297 X \u2297 Y) (X \u2297 Y)\u1601 ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601)).hom \u226b (\ud835\udfd9 ((Y\u1601 \u2297 X\u1601) \u2297 X \u2297 Y) \u2297 \ud835\udfd9 (X \u2297 Y)\u1601 \u2297 \ud835\udfd9 ((X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601)) \u226b (\ud835\udfd9 ((Y\u1601 \u2297 X\u1601) \u2297 X \u2297 Y) \u2297 (\u03b1_ (X \u2297 Y)\u1601 (X \u2297 Y) (Y\u1601 \u2297 X\u1601)).inv) \u226b (\u03b1_ (Y\u1601 \u2297 X\u1601) (X \u2297 Y) (((X \u2297 Y)\u1601 \u2297 X \u2297 Y) \u2297 Y\u1601 \u2297 X\u1601)).hom \u226b (\ud835\udfd9 (Y\u1601 \u2297 X\u1601) \u2297 (\u03b1_ (X \u2297 Y) ((X \u2297 Y)\u1601 \u2297 X \u2297 Y) (Y\u1601 \u2297 X\u1601)).inv) = \ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).hom \u2297 \ud835\udfd9 _ := by coherence,\n    slice_lhs 6 13 { rw h, }, clear h,\n    slice_lhs 5 7 {\n      simp only [\u2190id_tensor_comp, \u2190comp_tensor_id],\n      rw exact_pairing.evaluation_coevaluation,\n      simp only [id_tensor_comp, comp_tensor_id],\n      rw [associator_inv_conjugation, \u2190triangle_assoc_comp_right, \u2190triangle_assoc_comp_left_inv],\n      simp only [id_tensor_comp, comp_tensor_id],\n    },\n    slice_lhs 3 7 { rw \u2190pentagon_inv_hom_assoc, simp only [\u2190comp_tensor_id], rw [iso.hom_inv_id_assoc, \u2190associator_inv_naturality], },\n    slice_lhs 4 9 { rw iso.inv_hom_id_assoc, simp only [\u2190id_tensor_comp_assoc], rw [iso.inv_hom_id, category.comp_id, associator_inv_naturality], },\n    slice_lhs 2 3 { rw [\u2190tensor_comp, iso.inv_hom_id, tensor_id, category.comp_id], },\n    slice_lhs 4 5 { rw [\u2190tensor_comp, iso.inv_hom_id, tensor_id, category.id_comp], },\n    slice_lhs 2 4 { rw exact_pairing.coevaluation_evaluation, },\n    coherence,\n  end\n}\n\nnotation `\u03b4_` := tensor_iso_dual_tensor_dual\n\nend\n\nsection right_pivotal_category\n\nvariables\n  (C: Type u)\n  [category.{v} C]\n  [monoidal_category.{v} C]\n  [right_rigid_category.{v} C]\n\n-- * Define pivotal categories (rigid categories equipped with a natural isomorphism `\u1601\u1601 \u2245 \ud835\udfd9 C`).\n-- \u53c2\u8003: https://tqft.net/web/research/students/SamQuinn/thesis.pdf\n\nclass right_pivotal_category :=\n  (right_pivotor: \u03a0 X: C, X \u2245 X\u1601\u1601)\n  (notation `\u03c6_` := right_pivotor)\n  (right_pivotor_naturality': \u2200 {X Y: C} (f: X \u27f6 Y), f \u226b (\u03c6_ Y).hom = (\u03c6_ X).hom \u226b f\u1601\u1601)\n  (right_pivotor_tensor_naturality': \u2200 {X Y: C}, (\u03c6_ (X \u2297 Y)).hom = ((\u03c6_ X).hom \u2297 (\u03c6_ Y).hom) \u226b (\u03b4_ _ _).inv \u226b ((\u03b4_ _ _).hom)\u1601)\n\nrestate_axiom right_pivotal_category.right_pivotor_naturality'\nattribute [reassoc] right_pivotal_category.right_pivotor_naturality\nrestate_axiom right_pivotal_category.right_pivotor_tensor_naturality'\nattribute [reassoc] right_pivotal_category.right_pivotor_tensor_naturality\n\nopen right_pivotal_category\nnotation `\u03c6_` := right_pivotor\n\nvariable [right_pivotal_category.{v} C]\n\nlemma right_pivotor_inv_naturality {X Y: C} (f: X \u27f6 Y):\n  (\u03c6_ X).inv \u226b f = f\u1601\u1601 \u226b (\u03c6_ Y).inv :=\nbegin\n  rw \u2190(\u03c6_ X).cancel_iso_hom_left,\n  rw \u2190iso.cancel_iso_hom_right _ _ (\u03c6_ Y),\n  simp_rw [iso.hom_inv_id_assoc, category.assoc, iso.inv_hom_id, category.comp_id, right_pivotor_naturality],\nend\n\nlemma right_pivotor_inv_tensor_naturality (X Y: C):\n  (\u03c6_ (X \u2297 Y)).inv = ((\u03b4_ _ _).inv)\u1601 \u226b (\u03b4_ _ _).hom \u226b ((\u03c6_ X).inv \u2297 (\u03c6_ Y).inv) :=\nbegin\n  rw [\u2190(\u03c6_ (X \u2297 Y)).cancel_iso_hom_left, iso.hom_inv_id, right_pivotor_tensor_naturality],\n  simp_rw category.assoc,\n  rw [iso.hom_dual_inv_dual_id_assoc, iso.inv_hom_id_assoc],\n  simp_rw [\u2190tensor_comp, iso.hom_inv_id, tensor_id],\nend\n\nend right_pivotal_category\n\nsection\n\nvariables\n  {C: Type u}\n  [category.{v} C]\n  [monoidal_category.{v} C]\n  [right_rigid_category.{v} C]\n  (V: C)\n\nlemma coevaluation_tensor (X Y: C):\n  \u03b7_ (X \u2297 Y) (X \u2297 Y)\u1601 = \u03b7_ (X \u2297 Y) (Y\u1601 \u2297 X\u1601) \u226b (\ud835\udfd9 _ \u2297 (\u03b4_ _ _).inv) :=\nbegin\n  simp only [tensor_iso_dual_tensor_dual, id_tensor_comp],\n  rw id_tensor_right_unitor_inv,\n  slice_rhs 1 2 { rw right_unitor_inv_naturality, },\n  slice_rhs 3 4 { rw \u2190associator_naturality, },\n  slice_rhs 2 3 { simp only [tensor_id], rw [eq.trans (tensor_id_comp_id_tensor _ _) (id_tensor_comp_tensor_id _ _).symm], },\n  slice_rhs 1 2 { rw [\u2190unitors_inv_equal, \u2190left_unitor_inv_naturality], },\n  slice_rhs 3 3 { rw [\u2190tensor_id, associator_inv_conjugation], },\n  slice_rhs 5 8 { rw [pentagon_hom_inv_assoc, iso.hom_inv_id_assoc, \u2190associator_naturality], },\n  slice_rhs 4 6 { simp only [\u2190comp_tensor_id], rw exact_pairing.evaluation_coevaluation, },\n  conv_lhs { rw \u2190category.comp_id (\u03b7_ _ _) }, rw congr_comp_left,\n  coherence,\nend\n\nlemma evaluation_tensor (X Y: C):\n  \u03b5_ (X \u2297 Y) (X \u2297 Y)\u1601 = ((\u03b4_ _ _).hom \u2297 \ud835\udfd9 _) \u226b \u03b5_ _ _ :=\nbegin\n  simp only [tensor_iso_dual_tensor_dual, comp_tensor_id],\n  rw \u2190left_unitor_tensor',\n  slice_rhs 6 7 { rw \u2190left_unitor_naturality, },\n  slice_rhs 4 5 { rw associator_naturality, },\n  slice_rhs 5 6 { simp only [tensor_id], rw [eq.trans (tensor_id_comp_id_tensor _ _) (id_tensor_comp_tensor_id _ _).symm], },\n  slice_rhs 6 7 { rw [unitors_equal, right_unitor_naturality], },\n  slice_rhs 5 5 { rw [\u2190tensor_id, associator_conjugation], },\n  slice_rhs 3 5 { rw [\u2190pentagon, \u2190comp_tensor_id_assoc (\u03b1_ (X \u2297 Y)\u1601 (X \u2297 Y) (Y\u1601 \u2297 X\u1601)).inv, iso.inv_hom_id], },\n  slice_rhs 2 5 { simp only [\u2190tensor_id, associator_naturality_assoc], simp only [tensor_id], rw category.id_comp, },\n  slice_rhs 3 5 { simp only [\u2190id_tensor_comp], rw exact_pairing.evaluation_coevaluation, },\n  conv_lhs { rw \u2190category.id_comp (\u03b5_ _ _) }, simp_rw \u2190category.assoc, rw congr_comp_right,\n  coherence,\nend\n\nlemma coevaluation_dual_tensor (X Y: C):\n  \u03b7_ (X \u2297 Y)\u1601 (X \u2297 Y)\u1601\u1601 = \u03b7_ (Y\u1601 \u2297 X\u1601) (Y\u1601 \u2297 X\u1601)\u1601 \u226b ((\u03b4_ X Y).inv \u2297 ((\u03b4_ X Y).hom)\u1601) :=\nbegin\n  rw \u2190tensor_id_comp_id_tensor,\n  slice_rhs 1 2 { rw \u2190coevaluation_comp_right_adjoint_mate, },\n  slice_rhs 1 3 { rw [\u2190id_tensor_comp, iso.inv_dual_hom_dual_id, tensor_id, category.comp_id], },\nend\n\nlemma evaluation_dual_tensor (X Y: C):\n  \u03b5_ (X \u2297 Y)\u1601 (X \u2297 Y)\u1601\u1601 = (((\u03b4_ X Y).inv)\u1601 \u2297 (\u03b4_ X Y).hom) \u226b \u03b5_ (Y\u1601 \u2297 X\u1601) (Y\u1601 \u2297 X\u1601)\u1601 :=\nbegin\n  rw \u2190tensor_id_comp_id_tensor,\n  slice_rhs 2 3 { rw \u2190right_adjoint_mate_comp_evaluation, },\n  slice_rhs 1 3 { rw [\u2190comp_tensor_id_assoc, iso.inv_dual_hom_dual_id, tensor_id, category.id_comp], },\nend\n\ndef coevaluation' := \u03b7_ V V\u1601\ndef evaluation' := \u03b5_ V V\u1601\n\nnotation \u03b7_\u207a := coevaluation'\nnotation \u03b5_\u207a := evaluation'\n\nvariable [right_pivotal_category.{v} C]\nopen right_pivotal_category\n\ndef coevaluation_rev := \u03b7_\u207a V\u1601 \u226b (\ud835\udfd9 V\u1601 \u2297 (\u03c6_ _).inv)\ndef evaluation_rev := ((\u03c6_ _).hom \u2297 \ud835\udfd9 V\u1601) \u226b \u03b5_\u207a V\u1601\n\nnotation \u03b7_\u207b := coevaluation_rev\nnotation \u03b5_\u207b := evaluation_rev\n\nlemma id_comp_comp_id {V\u2081 V\u2082: C} (f: V\u2081 \u27f6 V\u2082): \ud835\udfd9 _ \u226b f = f \u226b \ud835\udfd9 _ := by simp\n\n@[reassoc] lemma coevaluation_evaluation:\n  (\ud835\udfd9 V\u1601 \u2297 \u03b7_\u207a _) \u226b (\u03b1_ _ _ _).inv \u226b (\u03b5_\u207a _ \u2297 \ud835\udfd9 V\u1601) = (\u03c1_ _).hom \u226b (\u03bb_ _).inv := by simp [coevaluation', evaluation', coevaluation_rev, evaluation_rev]\n\n@[reassoc] lemma coevaluation_evaluation_rev:\n  (\ud835\udfd9 V \u2297 \u03b7_\u207b _) \u226b (\u03b1_ _ _ _).inv \u226b (\u03b5_\u207b _ \u2297 \ud835\udfd9 V) = (\u03c1_ _).hom \u226b (\u03bb_ _).inv := begin\n  simp [coevaluation', evaluation', coevaluation_rev, evaluation_rev],\n  slice_lhs 1 2 { rw [\u2190tensor_comp, id_comp_comp_id, tensor_comp], },\n  slice_lhs 1 1 { rw [\u2190category.comp_id (\u03c6_ V).hom, \u2190category.id_comp (\u03b7_ _ _), tensor_comp], },\n  slice_lhs 3 4 { rw associator_inv_naturality, },\n  slice_lhs 4 5 { rw [\u2190tensor_comp, \u2190id_comp_comp_id, tensor_comp], },\n  slice_lhs 5 6 { rw [\u2190category.comp_id (\u03b5_ _ _), \u2190category.id_comp (\u03c6_ V).inv, tensor_comp], },\n  simp,\nend\n\n@[reassoc] lemma evaluation_coevaluation:\n  (\u03b7_\u207a _ \u2297 \ud835\udfd9 V) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 V \u2297 \u03b5_\u207a _) = (\u03bb_ _).hom \u226b (\u03c1_ _).inv := by simp [coevaluation', evaluation', coevaluation_rev, evaluation_rev]\n\n@[reassoc] lemma evaluation_coevaluation_rev:\n  (\u03b7_\u207b _ \u2297 \ud835\udfd9 V\u1601) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 V\u1601 \u2297 \u03b5_\u207b _) = (\u03bb_ _).hom \u226b (\u03c1_ _).inv := begin\n  simp [coevaluation', evaluation', coevaluation_rev, evaluation_rev],\n  slice_lhs 3 4 { rw [\u2190tensor_comp, \u2190tensor_comp, (\u03c6_ _).inv_hom_id, category.comp_id, tensor_id, tensor_id], },\n  simp,\nend\n\n@[reassoc] lemma coevaluation_hom_tensor (X Y: C):\n  \u03b7_\u207a (X \u2297 Y) = \u03b7_\u207a X \u226b (\ud835\udfd9 _ \u2297 (\u03bb_ _).inv) \u226b (\ud835\udfd9 _ \u2297 \u03b7_\u207a Y \u2297 \ud835\udfd9 _)  \u226b (\ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).hom) \u226b (\u03b1_ _ _ _).inv \u226b (\ud835\udfd9 _ \u2297 (\u03b4_ _ _).inv) :=\nbegin\n  simp only [coevaluation', coevaluation_tensor], unfold_projs,\n  slice_lhs 2 2 { rw \u2190triangle_assoc_comp_left_inv, },\n  slice_lhs 3 4 { rw [associator_conjugation, iso.inv_hom_id_assoc], },\n  slice_lhs 4 6 { rw pentagon_inv_inv_hom, },\n  simp_rw category.assoc,\nend\n\n@[reassoc] lemma evaluation_hom_tensor (X Y: C):\n  \u03b5_\u207a (X \u2297 Y) = ((\u03b4_ _ _).hom \u2297 \ud835\udfd9 _) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).inv) \u226b (\ud835\udfd9 _ \u2297 \u03b5_\u207a X \u2297 \ud835\udfd9 _) \u226b (\ud835\udfd9 _ \u2297 (\u03bb_ _).hom) \u226b \u03b5_\u207a Y :=\nbegin\n  simp only [evaluation', evaluation_tensor], unfold_projs,\n  slice_lhs 4 4 { rw associator_conjugation, },\n  slice_lhs 6 7 { rw triangle_assoc_comp_right, },\n  slice_lhs 2 4 { rw \u2190pentagon_hom_inv, },\n  simp_rw category.assoc,\nend\n\n@[reassoc] lemma coevaluation_rev_tensor (X Y: C):\n  \u03b7_\u207b (X \u2297 Y) = \u03b7_\u207b Y \u226b (\ud835\udfd9 _ \u2297 (\u03bb_ _).inv) \u226b (\ud835\udfd9 _ \u2297 \u03b7_\u207b X \u2297 \ud835\udfd9 _)  \u226b (\ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).hom) \u226b (\u03b1_ _ _ _).inv \u226b ((\u03b4_ _ _).inv \u2297 \ud835\udfd9 _) :=\nbegin\n  simp only [coevaluation_rev, id_tensor_comp, comp_tensor_id],\n  conv_lhs {\n    rw [coevaluation', coevaluation_dual_tensor, \u2190coevaluation'],\n    rw [category.assoc, \u2190tensor_comp, category.comp_id, coevaluation_hom_tensor],\n  },\n  slice_rhs 2 3 { rw [\u2190tensor_comp, left_unitor_inv_naturality, tensor_comp], },\n  slice_rhs 3 4 { rw [\u2190tensor_comp, eq.trans (id_tensor_comp_tensor_id _ _) (tensor_id_comp_id_tensor _ _).symm, tensor_comp], },\n  slice_rhs 4 5 { simp only [\u2190tensor_comp, category.id_comp, category.comp_id], },\n  slice_rhs 4 5 { rw [\u2190tensor_comp, associator_naturality, tensor_comp], },\n  slice_rhs 5 6 { rw associator_inv_naturality, },\n  simp_rw category.assoc, iterate 5 { rw congr_comp_left, },\n  simp_rw [\u2190tensor_comp, tensor_id, category.comp_id], rw congr_tensor_left,\n  rw [right_pivotor_inv_tensor_naturality, iso.hom_dual_inv_dual_id_assoc, iso.inv_hom_id_assoc],\nend\n\n@[reassoc] lemma evaluation_rev_tensor (X Y: C):\n  \u03b5_\u207b (X \u2297 Y) = (\ud835\udfd9 _ \u2297 (\u03b4_ _ _).hom) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 _ \u2297 (\u03b1_ _ _ _).inv) \u226b (\ud835\udfd9 _ \u2297 \u03b5_\u207b Y \u2297 \ud835\udfd9 _) \u226b (\ud835\udfd9 _ \u2297 (\u03bb_ _).hom) \u226b \u03b5_\u207b X :=\nbegin\n  simp only [evaluation_rev, id_tensor_comp, comp_tensor_id],\n  conv_lhs {\n    rw [evaluation', evaluation_dual_tensor, \u2190evaluation'],\n    rw [\u2190tensor_comp_assoc, category.id_comp, evaluation_hom_tensor],\n  },\n  slice_rhs 3 7 {\n    simp only [\u2190tensor_comp, category.id_comp],\n    rw [category.comp_id, \u2190category.comp_id ((\u03c6_ X).hom), tensor_comp],\n    simp only [id_tensor_comp],\n  },\n  slice_rhs 3 4 { rw [\u2190tensor_comp, \u2190associator_inv_naturality, tensor_comp], },\n  slice_rhs 2 3 { rw \u2190associator_naturality, },\n  simp_rw \u2190category.assoc, iterate 5 { rw congr_comp_right, },\n  simp_rw [\u2190tensor_comp, tensor_id, category.id_comp], rw congr_tensor_right,\n  rw [right_pivotor_tensor_naturality],\n  slice_lhs 3 5 { rw iso.hom_dual_inv_dual_id_assoc, },\n  rw [iso.inv_hom_id, category.comp_id],\nend\n\nend\n\nsection\n\nopen right_pivotal_category\n\nvariables\n  {C: Type u}\n  [category.{v} C]\n  [monoidal_category.{v} C]\n  [right_rigid_category.{v} C]\n  [right_pivotal_category C]\n\nlemma right_adjoint_mate_inv {X Y: C} (f: X \u27f6 Y):\n  (\u03bb_ _).inv \u226b (\u03b7_\u207a _ \u2297 \ud835\udfd9 _) \u226b ((\ud835\udfd9 _ \u2297 f\u1601) \u2297 \ud835\udfd9 _) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 _ \u2297 \u03b5_\u207a _) \u226b (\u03c1_ _).hom = f :=\nbegin\n  simp [coevaluation', evaluation'],\n  simp [right_adjoint_mate],\n  slice_lhs 8 10 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp, pentagon_inv, id_tensor_comp], }, simp,\n  slice_lhs 11 12 { rw [associator_inv_conjugation, \u2190triangle_assoc_comp_right, comp_tensor_id], simp, },\n  slice_lhs 10 12 { rw pentagon_inv, }, simp,\n  slice_lhs 9 10 { rw associator_inv_naturality, },\n  slice_lhs 10 11 { rw [\u2190tensor_comp, id_comp_comp_id, tensor_comp], },\n  slice_lhs 9 10 { rw [\u2190associator_inv_naturality, \u2190id_tensor_comp_tensor_id (\u03b5_ _ _) (\u03b5_ _ _), id_tensor_comp], },\n  slice_lhs 8 9 { rw [\u2190id_tensor_comp, \u2190tensor_id, \u2190associator_inv_naturality, id_tensor_comp], },\n  slice_lhs 7 8 { rw [\u2190tensor_comp, \u2190tensor_comp, tensor_id_comp_id_tensor, \u2190id_tensor_comp_tensor_id _ f, tensor_comp, tensor_comp], },\n  slice_lhs 6 7 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp], },\n  slice_lhs 5 6 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp, exact_pairing.evaluation_coevaluation], }, simp,\n  slice_lhs 4 5 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp, (\u03bb_ _).inv_hom_id], },\n  slice_lhs 5 6 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp, (\u03c1_ _).inv_hom_id], },\n  slice_lhs 7 8 { rw [\u2190id_tensor_comp, \u2190id_tensor_comp, (\u03c1_ _).inv_hom_id], }, simp,\n  slice_lhs 3 4 { rw \u2190associator_naturality, },\n  slice_lhs 2 3 { rw [tensor_id, tensor_id_comp_id_tensor, \u2190id_tensor_comp_tensor_id], },\n  slice_lhs 3 5 { rw exact_pairing.evaluation_coevaluation, }, simp,\nend\n\n@[reassoc] lemma right_adjoint_mate_hom {X Y: C} (f: X \u27f6 Y):\n  (\u03c1_ _).inv \u226b (\ud835\udfd9 _ \u2297 \u03b7_\u207a _) \u226b (\ud835\udfd9 _ \u2297 (f \u2297 \ud835\udfd9 _)) \u226b (\u03b1_ _ _ _).inv \u226b ((\u03b5_\u207a _) \u2297 \ud835\udfd9 _) \u226b (\u03bb_ _).hom = f\u1601 :=\nby rw [coevaluation', evaluation', right_adjoint_mate]\n\n@[reassoc] lemma right_adjoint_mate_rev {X Y: C} (f: X \u27f6 Y):\n  (\u03bb_ _).inv \u226b (\u03b7_\u207b _ \u2297 \ud835\udfd9 _) \u226b ((\ud835\udfd9 _ \u2297 f) \u2297 \ud835\udfd9 _) \u226b (\u03b1_ _ _ _).hom \u226b (\ud835\udfd9 _ \u2297 \u03b5_\u207b _) \u226b (\u03c1_ _).hom = f\u1601 :=\nbegin\n  simp only [coevaluation_rev, evaluation_rev, comp_tensor_id, id_tensor_comp, category.assoc],\n  rw \u2190associator_naturality_assoc,\n  slice_lhs 3 5 {\n    simp only [\u2190tensor_comp, category.comp_id],\n    rw [right_pivotor_naturality, iso.inv_hom_id_assoc],\n  },\n  simp only [category.assoc],\n  rw right_adjoint_mate_inv,\nend\n\nend\n\nend kassel\n", "meta": {"author": "youjo-tape", "repo": "lean-univ", "sha": "f8a9e82134c930715fc39f44ba0e5a98184673a7", "save_path": "github-repos/lean/youjo-tape-lean-univ", "path": "github-repos/lean/youjo-tape-lean-univ/lean-univ-f8a9e82134c930715fc39f44ba0e5a98184673a7/src/kassel/lemma/right_pivotal_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.39200907228233717}}
{"text": "/-\nCopyright (c) 2019 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.uniform_space.uniform_embedding\nimport Mathlib.PostPort\n\nuniverses u l u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Abstract theory of Hausdorff completions of uniform spaces\n\nThis file characterizes Hausdorff completions of a uniform space \u03b1 as complete Hausdorff spaces\nequipped with a map from \u03b1 which has dense image and induce the original uniform structure on \u03b1.\nAssuming these properties we \"extend\" uniformly continuous maps from \u03b1 to complete Hausdorff spaces\nto the completions of \u03b1. This is the universal property expected from a completion.\nIt is then used to extend uniformly continuous maps from \u03b1 to \u03b1' to maps between\ncompletions of \u03b1 and \u03b1'.\n\nThis file does not construct any such completion, it only study consequences of their existence.\nThe first advantage is that formal properties are clearly highlighted without interference from\nconstruction details. The second advantage is that this framework can then be used to compare\ndifferent completion constructions. See `topology/uniform_space/compare_reals` for an example.\nOf course the comparison comes from the universal property as usual.\n\nA general explicit construction of completions is done in `uniform_space/completion`, leading\nto a functor from uniform spaces to complete Hausdorff uniform spaces that is left adjoint to the\ninclusion, see `uniform_space/UniformSpace` for the category packaging.\n\n## Implementation notes\n\nA tiny technical advantage of using a characteristic predicate such as the properties listed in\n`abstract_completion` instead of stating the universal property is that the universal property\nderived from the predicate is more universe polymorphic.\n\n## References\n\nWe don't know any traditional text discussing this. Real world mathematics simply silently\nidentify the results of any two constructions that lead to something one could reasonnably\ncall a completion.\n\n## Tags\n\nuniform spaces, completion, universal property\n-/\n\n/-- A completion of `\u03b1` is the data of a complete separated uniform space (from the same universe)\nand a map from `\u03b1` with dense range and inducing the original uniform structure on `\u03b1`. -/\nstructure abstract_completion (\u03b1 : Type u) [uniform_space \u03b1] \nwhere\n  space : Type u\n  coe : \u03b1 \u2192 space\n  uniform_struct : uniform_space space\n  complete : complete_space space\n  separation : separated_space space\n  uniform_inducing : uniform_inducing coe\n  dense : dense_range coe\n\nnamespace abstract_completion\n\n\ntheorem closure_range {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) : closure (set.range (coe pkg)) = set.univ :=\n  dense_range.closure_range (dense pkg)\n\ntheorem dense_inducing {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) : dense_inducing (coe pkg) :=\n  dense_inducing.mk (uniform_inducing.inducing (uniform_inducing pkg)) (dense pkg)\n\ntheorem uniform_continuous_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) : uniform_continuous (coe pkg) :=\n  uniform_inducing.uniform_continuous (uniform_inducing pkg)\n\ntheorem continuous_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) : continuous (coe pkg) :=\n  uniform_continuous.continuous (uniform_continuous_coe pkg)\n\ntheorem induction_on {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {p : space pkg \u2192 Prop} (a : space pkg) (hp : is_closed (set_of fun (a : space pkg) => p a)) (ih : \u2200 (a : \u03b1), p (coe pkg a)) : p a :=\n  is_closed_property (dense pkg) hp ih a\n\nprotected theorem funext {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] [t2_space \u03b2] {f : space pkg \u2192 \u03b2} {g : space pkg \u2192 \u03b2} (hf : continuous f) (hg : continuous g) (h : \u2200 (a : \u03b1), f (coe pkg a) = g (coe pkg a)) : f = g :=\n  funext fun (a : space pkg) => induction_on pkg a (is_closed_eq hf hg) h\n\n/-- Extension of maps to completions -/\nprotected def extend {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (f : \u03b1 \u2192 \u03b2) : space pkg \u2192 \u03b2 :=\n  ite (uniform_continuous f) (dense_inducing.extend (dense_inducing pkg) f)\n    fun (x : space pkg) => f (dense_range.some (dense pkg) x)\n\ntheorem extend_def {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : uniform_continuous f) : abstract_completion.extend pkg f = dense_inducing.extend (dense_inducing pkg) f :=\n  if_pos hf\n\ntheorem extend_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] {f : \u03b1 \u2192 \u03b2} [t2_space \u03b2] (hf : uniform_continuous f) (a : \u03b1) : abstract_completion.extend pkg f (coe pkg a) = f a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (abstract_completion.extend pkg f (coe pkg a) = f a)) (extend_def pkg hf)))\n    (dense_inducing.extend_eq (dense_inducing pkg) (uniform_continuous.continuous hf) a)\n\ntheorem uniform_continuous_extend {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] {f : \u03b1 \u2192 \u03b2} [complete_space \u03b2] [separated_space \u03b2] : uniform_continuous (abstract_completion.extend pkg f) := sorry\n\ntheorem continuous_extend {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] {f : \u03b1 \u2192 \u03b2} [complete_space \u03b2] [separated_space \u03b2] : continuous (abstract_completion.extend pkg f) :=\n  uniform_continuous.continuous (uniform_continuous_extend pkg)\n\ntheorem extend_unique {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] {f : \u03b1 \u2192 \u03b2} [complete_space \u03b2] [separated_space \u03b2] (hf : uniform_continuous f) {g : space pkg \u2192 \u03b2} (hg : uniform_continuous g) (h : \u2200 (a : \u03b1), f a = g (coe pkg a)) : abstract_completion.extend pkg f = g := sorry\n\n@[simp] theorem extend_comp_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] [complete_space \u03b2] [separated_space \u03b2] {f : space pkg \u2192 \u03b2} (hf : uniform_continuous f) : abstract_completion.extend pkg (f \u2218 coe pkg) = f := sorry\n\n/-- Lifting maps to completions -/\nprotected def map {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) (f : \u03b1 \u2192 \u03b2) : space pkg \u2192 space pkg' :=\n  abstract_completion.extend pkg (coe pkg' \u2218 f)\n\ntheorem uniform_continuous_map {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) (f : \u03b1 \u2192 \u03b2) : uniform_continuous (abstract_completion.map pkg pkg' f) :=\n  uniform_continuous_extend pkg\n\ntheorem continuous_map {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) (f : \u03b1 \u2192 \u03b2) : continuous (abstract_completion.map pkg pkg' f) :=\n  continuous_extend pkg\n\n@[simp] theorem map_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {f : \u03b1 \u2192 \u03b2} (hf : uniform_continuous f) (a : \u03b1) : abstract_completion.map pkg pkg' f (coe pkg a) = coe pkg' (f a) :=\n  extend_coe pkg (uniform_continuous.comp (uniform_continuous_coe pkg') hf) a\n\ntheorem map_unique {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {f : \u03b1 \u2192 \u03b2} {g : space pkg \u2192 space pkg'} (hg : uniform_continuous g) (h : \u2200 (a : \u03b1), coe pkg' (f a) = g (coe pkg a)) : abstract_completion.map pkg pkg' f = g := sorry\n\n@[simp] theorem map_id {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) : abstract_completion.map pkg pkg id = id :=\n  map_unique pkg pkg uniform_continuous_id fun (a : \u03b1) => rfl\n\ntheorem extend_map {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] [complete_space \u03b3] [separated_space \u03b3] {f : \u03b2 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b2} (hf : uniform_continuous f) (hg : uniform_continuous g) : abstract_completion.extend pkg' f \u2218 abstract_completion.map pkg pkg' g = abstract_completion.extend pkg (f \u2218 g) := sorry\n\ntheorem map_comp {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (pkg'' : abstract_completion \u03b3) {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : uniform_continuous g) (hf : uniform_continuous f) : abstract_completion.map pkg' pkg'' g \u2218 abstract_completion.map pkg pkg' f = abstract_completion.map pkg pkg'' (g \u2218 f) :=\n  extend_map pkg pkg' (uniform_continuous.comp (uniform_continuous_coe pkg'') hg) hf\n\n-- We can now compare two completion packages for the same uniform space\n\n/-- The comparison map between two completions of the same uniform space. -/\ndef compare {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : space pkg \u2192 space pkg' :=\n  abstract_completion.extend pkg (coe pkg')\n\ntheorem uniform_continuous_compare {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : uniform_continuous (compare pkg pkg') :=\n  uniform_continuous_extend pkg\n\ntheorem compare_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) (a : \u03b1) : compare pkg pkg' (coe pkg a) = coe pkg' a :=\n  extend_coe pkg (uniform_continuous_coe pkg') a\n\ntheorem inverse_compare {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : compare pkg pkg' \u2218 compare pkg' pkg = id := sorry\n\n/-- The bijection between two completions of the same uniform space. -/\ndef compare_equiv {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : space pkg \u2243 space pkg' :=\n  equiv.mk (compare pkg pkg') (compare pkg' pkg) sorry sorry\n\ntheorem uniform_continuous_compare_equiv {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : uniform_continuous \u21d1(compare_equiv pkg pkg') :=\n  uniform_continuous_compare pkg pkg'\n\ntheorem uniform_continuous_compare_equiv_symm {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) (pkg' : abstract_completion \u03b1) : uniform_continuous \u21d1(equiv.symm (compare_equiv pkg pkg')) :=\n  uniform_continuous_compare pkg' pkg\n\n/-- Products of completions -/\nprotected def prod {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) : abstract_completion (\u03b1 \u00d7 \u03b2) :=\n  mk (space pkg \u00d7 space pkg') (fun (p : \u03b1 \u00d7 \u03b2) => (coe pkg (prod.fst p), coe pkg' (prod.snd p))) prod.uniform_space sorry\n    sorry sorry sorry\n\n/-- Extend two variable map to completions. -/\nprotected def extend\u2082 {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : space pkg \u2192 space pkg' \u2192 \u03b3 :=\n  function.curry (abstract_completion.extend (abstract_completion.prod pkg pkg') (function.uncurry f))\n\ntheorem extension\u2082_coe_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] [separated_space \u03b3] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (hf : uniform_continuous (function.uncurry f)) (a : \u03b1) (b : \u03b2) : abstract_completion.extend\u2082 pkg pkg' f (coe pkg a) (coe pkg' b) = f a b := sorry\n\ntheorem uniform_continuous_extension\u2082 {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] [separated_space \u03b3] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [complete_space \u03b3] : uniform_continuous\u2082 (abstract_completion.extend\u2082 pkg pkg' f) := sorry\n\n/-- Lift two variable maps to completions. -/\nprotected def map\u2082 {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (pkg'' : abstract_completion \u03b3) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : space pkg \u2192 space pkg' \u2192 space pkg'' :=\n  abstract_completion.extend\u2082 pkg pkg' (function.bicompr (coe pkg'') f)\n\ntheorem uniform_continuous_map\u2082 {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (pkg'' : abstract_completion \u03b3) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : uniform_continuous\u2082 (abstract_completion.map\u2082 pkg pkg' pkg'' f) :=\n  uniform_continuous_extension\u2082 pkg pkg' (function.bicompr (coe pkg'') f)\n\ntheorem continuous_map\u2082 {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (pkg'' : abstract_completion \u03b3) {\u03b4 : Type u_4} [topological_space \u03b4] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {a : \u03b4 \u2192 space pkg} {b : \u03b4 \u2192 space pkg'} (ha : continuous a) (hb : continuous b) : continuous fun (d : \u03b4) => abstract_completion.map\u2082 pkg pkg' pkg'' f (a d) (b d) :=\n  continuous.comp (uniform_continuous.continuous (uniform_continuous_map\u2082 pkg pkg' pkg'' f)) (continuous.prod_mk ha hb)\n\ntheorem map\u2082_coe_coe {\u03b1 : Type u_1} [uniform_space \u03b1] (pkg : abstract_completion \u03b1) {\u03b2 : Type u_2} [uniform_space \u03b2] (pkg' : abstract_completion \u03b2) {\u03b3 : Type u_3} [uniform_space \u03b3] (pkg'' : abstract_completion \u03b3) (a : \u03b1) (b : \u03b2) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (hf : uniform_continuous\u2082 f) : abstract_completion.map\u2082 pkg pkg' pkg'' f (coe pkg a) (coe pkg' b) = coe pkg'' (f a b) :=\n  extension\u2082_coe_coe pkg pkg' (uniform_continuous.comp (uniform_continuous_coe pkg'') hf) a b\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/uniform_space/abstract_completion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6224593171945416, "lm_q1q2_score": 0.39200907211141167}}
{"text": "/-\nCopyright (c) 2020 Zhangir Azerbayev. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Eric Wieser, Zhangir Azerbayev\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.multilinear\nimport Mathlib.linear_algebra.linear_independent\nimport Mathlib.group_theory.perm.sign\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_6 l u_4 u_5 u_7 \n\nnamespace Mathlib\n\n/-!\n# Alternating Maps\n\nWe construct the bundled function `alternating_map`, which extends `multilinear_map` with all the\narguments of the same type.\n\n## Main definitions\n* `alternating_map R M N \u03b9` is the space of `R`-linear alternating maps from `\u03b9 \u2192 M` to `N`.\n* `f.map_eq_zero_of_eq` expresses that `f` is zero when two inputs are equal.\n* `f.map_swap` expresses that `f` is negated when two inputs are swapped.\n* `f.map_perm` expresses how `f` varies by a sign change under a permutation of its inputs.\n* An `add_comm_monoid`, `add_comm_group`, and `semimodule` structure over `alternating_map`s that\n  matches the definitions over `multilinear_map`s.\n* `multilinear_map.alternatization`, which makes an alternating map out of a non-alternating one.\n\n## Implementation notes\n`alternating_map` is defined in terms of `map_eq_zero_of_eq`, as this is easier to work with than\nusing `map_swap` as a definition, and does not require `has_neg N`.\n\n`alternating_map`s are provided with a coercion to `multilinear_map`, along with a set of\n`norm_cast` lemmas that act on the algebraic structure:\n\n* `alternating_map.coe_add`\n* `alternating_map.coe_zero`\n* `alternating_map.coe_sub`\n* `alternating_map.coe_neg`\n* `alternating_map.coe_smul`\n-/\n\n-- semiring / add_comm_monoid\n\n-- semiring / add_comm_group\n\n/--\nAn alternating map is a multilinear map that vanishes when two of its arguments are equal.\n-/\nstructure alternating_map (R : Type u_1) [semiring R] (M : Type u_2) [add_comm_monoid M]\n    [semimodule R M] (N : Type u_3) [add_comm_monoid N] [semimodule R N] (\u03b9 : Type u_6)\n    [DecidableEq \u03b9]\n    extends multilinear_map R (fun (i : \u03b9) => M) N where\n  map_eq_zero_of_eq' : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 to_fun v = 0\n\n/-- The multilinear map associated to an alternating map -/\nnamespace alternating_map\n\n\n/-! Basic coercion simp lemmas, largely copied from `ring_hom` and `multilinear_map` -/\n\nprotected instance has_coe_to_fun {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : has_coe_to_fun (alternating_map R M N \u03b9) :=\n  has_coe_to_fun.mk (fun (x : alternating_map R M N \u03b9) => ((i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) \u2192 N)\n    fun (x : alternating_map R M N \u03b9) => to_fun x\n\n@[simp] theorem to_fun_eq_coe {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : to_fun f = \u21d1f :=\n  rfl\n\n@[simp] theorem coe_mk {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : (\u03b9 \u2192 M) \u2192 N)\n    (h\u2081 :\n      \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (x y : M),\n        f (function.update m i (x + y)) = f (function.update m i x) + f (function.update m i y))\n    (h\u2082 :\n      \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (c : R) (x : M),\n        f (function.update m i (c \u2022 x)) = c \u2022 f (function.update m i x))\n    (h\u2083 : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 f v = 0) : \u21d1(mk f h\u2081 h\u2082 h\u2083) = f :=\n  rfl\n\ntheorem congr_fun {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} (h : f = g) (x : \u03b9 \u2192 M) :\n    coe_fn f x = coe_fn g x :=\n  congr_arg (fun (h : alternating_map R M N \u03b9) => coe_fn h x) h\n\ntheorem congr_arg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (f : alternating_map R M N \u03b9) {x : \u03b9 \u2192 M} {y : \u03b9 \u2192 M} (h : x = y) : coe_fn f x = coe_fn f y :=\n  congr_arg (fun (x : \u03b9 \u2192 M) => coe_fn f x) h\n\ntheorem coe_inj {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} (h : \u21d1f = \u21d1g) : f = g :=\n  sorry\n\ntheorem ext {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    {f : alternating_map R M N \u03b9} {f' : alternating_map R M N \u03b9}\n    (H : \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn f' x) : f = f' :=\n  coe_inj (funext H)\n\ntheorem ext_iff {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} :\n    f = g \u2194 \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) => h \u25b8 rfl,\n    mpr := fun (h : \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn g x) => ext h }\n\nprotected instance multilinear_map.has_coe {R : Type u_1} [semiring R] {M : Type u_2}\n    [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N]\n    {\u03b9 : Type u_6} [DecidableEq \u03b9] :\n    has_coe (alternating_map R M N \u03b9) (multilinear_map R (fun (i : \u03b9) => M) N) :=\n  has_coe.mk fun (x : alternating_map R M N \u03b9) => to_multilinear_map x\n\n@[simp] theorem coe_multilinear_map {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\n@[simp] theorem to_multilinear_map_eq_coe {R : Type u_1} [semiring R] {M : Type u_2}\n    [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N]\n    {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : to_multilinear_map f = \u2191f :=\n  rfl\n\n@[simp] theorem coe_multilinear_map_mk {R : Type u_1} [semiring R] {M : Type u_2}\n    [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N]\n    {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : (\u03b9 \u2192 M) \u2192 N)\n    (h\u2081 :\n      \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (x y : M),\n        f (function.update m i (x + y)) = f (function.update m i x) + f (function.update m i y))\n    (h\u2082 :\n      \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (c : R) (x : M),\n        f (function.update m i (c \u2022 x)) = c \u2022 f (function.update m i x))\n    (h\u2083 : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 f v = 0) :\n    \u2191(mk f h\u2081 h\u2082 h\u2083) = multilinear_map.mk f h\u2081 h\u2082 :=\n  rfl\n\n/-!\n### Simp-normal forms of the structure fields\n\nThese are expressed in terms of `\u21d1f` instead of `f.to_fun`.\n-/\n\n@[simp] theorem map_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) (i : \u03b9) (x : M) (y : M) :\n    coe_fn f (function.update v i (x + y)) =\n        coe_fn f (function.update v i x) + coe_fn f (function.update v i y) :=\n  multilinear_map.map_add' (to_multilinear_map f) v i x y\n\n@[simp] theorem map_sub {R : Type u_1} [semiring R] {M' : Type u_4} [add_comm_group M']\n    [semimodule R M'] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g' : alternating_map R M' N' \u03b9) (v' : \u03b9 \u2192 M') (i : \u03b9) (x : M') (y : M') :\n    coe_fn g' (function.update v' i (x - y)) =\n        coe_fn g' (function.update v' i x) - coe_fn g' (function.update v' i y) :=\n  multilinear_map.map_sub (to_multilinear_map g') v' i x y\n\n@[simp] theorem map_neg {R : Type u_1} [semiring R] {M' : Type u_4} [add_comm_group M']\n    [semimodule R M'] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g' : alternating_map R M' N' \u03b9) (v' : \u03b9 \u2192 M') (i : \u03b9) (x : M') :\n    coe_fn g' (function.update v' i (-x)) = -coe_fn g' (function.update v' i x) :=\n  multilinear_map.map_neg (to_multilinear_map g') v' i x\n\n@[simp] theorem map_smul {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) (i : \u03b9) (r : R) (x : M) :\n    coe_fn f (function.update v i (r \u2022 x)) = r \u2022 coe_fn f (function.update v i x) :=\n  multilinear_map.map_smul' (to_multilinear_map f) v i r x\n\n@[simp] theorem map_eq_zero_of_eq {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (h : v i = v j)\n    (hij : i \u2260 j) : coe_fn f v = 0 :=\n  map_eq_zero_of_eq' f v i j h hij\n\n/-!\n### Algebraic structure inherited from `multilinear_map`\n\n`alternating_map` carries the same `add_comm_monoid`, `add_comm_group`, and `semimodule` structure\nas `multilinear_map`\n-/\n\nprotected instance has_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : Add (alternating_map R M N \u03b9) :=\n  { add :=\n      fun (a b : alternating_map R M N \u03b9) =>\n        mk (multilinear_map.to_fun (\u2191a + \u2191b)) sorry sorry sorry }\n\n@[simp] theorem add_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (f' : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) :\n    coe_fn (f + f') v = coe_fn f v + coe_fn f' v :=\n  rfl\n\ntheorem coe_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (f : alternating_map R M N \u03b9) (f' : alternating_map R M N \u03b9) : \u2191(f + f') = \u2191f + \u2191f' :=\n  rfl\n\nprotected instance has_zero {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : HasZero (alternating_map R M N \u03b9) :=\n  { zero := mk (multilinear_map.to_fun 0) sorry sorry sorry }\n\n@[simp] theorem zero_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (v : \u03b9 \u2192 M) : coe_fn 0 v = 0 :=\n  rfl\n\ntheorem coe_zero {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : \u21910 = 0 :=\n  rfl\n\nprotected instance inhabited {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : Inhabited (alternating_map R M N \u03b9) :=\n  { default := 0 }\n\nprotected instance add_comm_monoid {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : add_comm_monoid (alternating_map R M N \u03b9) :=\n  add_comm_monoid.mk Add.add sorry 0 sorry sorry sorry\n\nprotected instance has_neg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : Neg (alternating_map R M N' \u03b9) :=\n  { neg :=\n      fun (f : alternating_map R M N' \u03b9) => mk (multilinear_map.to_fun (-\u2191f)) sorry sorry sorry }\n\n@[simp] theorem neg_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (m : \u03b9 \u2192 M) : coe_fn (-g) m = -coe_fn g m :=\n  rfl\n\ntheorem coe_neg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (g : alternating_map R M N' \u03b9) : \u2191(-g) = -\u2191g :=\n  rfl\n\nprotected instance has_sub {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : Sub (alternating_map R M N' \u03b9) :=\n  { sub :=\n      fun (f g : alternating_map R M N' \u03b9) =>\n        mk (multilinear_map.to_fun (\u2191f - \u2191g)) sorry sorry sorry }\n\n@[simp] theorem sub_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (g\u2082 : alternating_map R M N' \u03b9) (m : \u03b9 \u2192 M) :\n    coe_fn (g - g\u2082) m = coe_fn g m - coe_fn g\u2082 m :=\n  rfl\n\ntheorem coe_sub {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (g : alternating_map R M N' \u03b9) (g\u2082 : alternating_map R M N' \u03b9) : \u2191(g - g\u2082) = \u2191g - \u2191g\u2082 :=\n  rfl\n\nprotected instance add_comm_group {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] : add_comm_group (alternating_map R M N' \u03b9) :=\n  add_comm_group.mk Add.add sorry 0 sorry sorry Neg.neg Sub.sub sorry sorry\n\nprotected instance has_scalar {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] :\n    has_scalar S (alternating_map R M N \u03b9) :=\n  has_scalar.mk\n    fun (c : S) (f : alternating_map R M N \u03b9) =>\n      mk (multilinear_map.to_fun (c \u2022 \u2191f)) sorry sorry sorry\n\n@[simp] theorem smul_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {S : Type u_7} [monoid S] [distrib_mul_action S N]\n    [smul_comm_class R S N] (c : S) (m : \u03b9 \u2192 M) : coe_fn (c \u2022 f) m = c \u2022 coe_fn f m :=\n  rfl\n\ntheorem coe_smul {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (f : alternating_map R M N \u03b9) {S : Type u_7} [monoid S] [distrib_mul_action S N]\n    [smul_comm_class R S N] (c : S) : \u2191(c \u2022 f) = c \u2022 \u2191f :=\n  rfl\n\nprotected instance distrib_mul_action {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] :\n    distrib_mul_action S (alternating_map R M N \u03b9) :=\n  distrib_mul_action.mk sorry sorry\n\n/-- The space of multilinear maps over an algebra over `R` is a module over `R`, for the pointwise\naddition and scalar multiplication. -/\nprotected instance semimodule {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] {S : Type u_7} [semiring S] [semimodule S N] [smul_comm_class R S N] :\n    semimodule S (alternating_map R M N \u03b9) :=\n  semimodule.mk sorry sorry\n\nend alternating_map\n\n\n/-!\n### Composition with linear maps\n-/\n\nnamespace linear_map\n\n\n/-- Composing a alternating map with a linear map gives again a alternating map. -/\ndef comp_alternating_map {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082] (g : linear_map R N N\u2082) :\n    alternating_map R M N \u03b9 \u2192+ alternating_map R M N\u2082 \u03b9 :=\n  add_monoid_hom.mk\n    (fun (f : alternating_map R M N \u03b9) =>\n      alternating_map.mk (multilinear_map.to_fun (comp_multilinear_map g \u2191f)) sorry sorry sorry)\n    sorry sorry\n\n@[simp] theorem coe_comp_alternating_map {R : Type u_1} [semiring R] {M : Type u_2}\n    [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N]\n    {\u03b9 : Type u_6} [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082]\n    (g : linear_map R N N\u2082) (f : alternating_map R M N \u03b9) :\n    \u21d1(coe_fn (comp_alternating_map g) f) = \u21d1g \u2218 \u21d1f :=\n  rfl\n\ntheorem comp_alternating_map_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082] (g : linear_map R N N\u2082)\n    (f : alternating_map R M N \u03b9) (m : \u03b9 \u2192 M) :\n    coe_fn (coe_fn (comp_alternating_map g) f) m = coe_fn g (coe_fn f m) :=\n  rfl\n\nend linear_map\n\n\nnamespace alternating_map\n\n\n/-!\n### Other lemmas from `multilinear_map`\n-/\n\ntheorem map_update_sum {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {\u03b1 : Type u_4} (t : finset \u03b1) (i : \u03b9) (g : \u03b1 \u2192 M)\n    (m : \u03b9 \u2192 M) :\n    coe_fn f (function.update m i (finset.sum t fun (a : \u03b1) => g a)) =\n        finset.sum t fun (a : \u03b1) => coe_fn f (function.update m i (g a)) :=\n  multilinear_map.map_update_sum (to_multilinear_map f) t i g m\n\n/-!\n### Theorems specific to alternating maps\n\nVarious properties of reordered and repeated inputs which follow from\n`alternating_map.map_eq_zero_of_eq`.\n-/\n\ntheorem map_update_self {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) :\n    coe_fn f (function.update v i (v j)) = 0 :=\n  sorry\n\ntheorem map_update_update {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j)\n    (m : M) : coe_fn f (function.update (function.update v i m) j m) = 0 :=\n  sorry\n\ntheorem map_swap_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) :\n    coe_fn f (v \u2218 \u21d1(equiv.swap i j)) + coe_fn f v = 0 :=\n  sorry\n\ntheorem map_add_swap {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) :\n    coe_fn f v + coe_fn f (v \u2218 \u21d1(equiv.swap i j)) = 0 :=\n  sorry\n\ntheorem map_swap {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (g : alternating_map R M N' \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) :\n    coe_fn g (v \u2218 \u21d1(equiv.swap i j)) = -coe_fn g v :=\n  eq_neg_of_add_eq_zero (map_swap_add g v hij)\n\ntheorem map_perm {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    (g : alternating_map R M N' \u03b9) [fintype \u03b9] (v : \u03b9 \u2192 M) (\u03c3 : equiv.perm \u03b9) :\n    coe_fn g (v \u2218 \u21d1\u03c3) = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn g v :=\n  sorry\n\ntheorem map_congr_perm {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (v : \u03b9 \u2192 M) [fintype \u03b9] (\u03c3 : equiv.perm \u03b9) :\n    coe_fn g v = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn g (v \u2218 \u21d1\u03c3) :=\n  sorry\n\ntheorem coe_dom_dom_congr {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) [fintype \u03b9] (\u03c3 : equiv.perm \u03b9) :\n    multilinear_map.dom_dom_congr \u03c3 \u2191g = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 \u2191g :=\n  multilinear_map.ext fun (v : \u03b9 \u2192 M) => map_perm g v \u03c3\n\n/-- If the arguments are linearly dependent then the result is `0`.\n\nTODO: Can the `division_ring` requirement be relaxed? -/\ntheorem map_linear_dependent {\u03b9 : Type u_6} [DecidableEq \u03b9] {K : Type u_1} [division_ring K]\n    {M : Type u_2} [add_comm_group M] [semimodule K M] {N : Type u_3} [add_comm_group N]\n    [semimodule K N] (f : alternating_map K M N \u03b9) (v : \u03b9 \u2192 M) (h : \u00aclinear_independent K v) :\n    coe_fn f v = 0 :=\n  sorry\n\nend alternating_map\n\n\nnamespace multilinear_map\n\n\n/-- Produce an `alternating_map` out of a `multilinear_map`, by summing over all argument\npermutations. -/\ndef alternatization {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M]\n    {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9]\n    [fintype \u03b9] : multilinear_map R (fun (i : \u03b9) => M) N' \u2192+ alternating_map R M N' \u03b9 :=\n  add_monoid_hom.mk\n    (fun (m : multilinear_map R (fun (i : \u03b9) => M) N') =>\n      alternating_map.mk\n        \u21d1(finset.sum finset.univ\n            fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 dom_dom_congr \u03c3 m)\n        sorry sorry sorry)\n    sorry sorry\n\ntheorem alternatization_def {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] [fintype \u03b9] (m : multilinear_map R (fun (i : \u03b9) => M) N') :\n    \u21d1(coe_fn alternatization m) =\n        \u21d1(finset.sum finset.univ\n            fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 dom_dom_congr \u03c3 m) :=\n  rfl\n\ntheorem alternatization_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] [fintype \u03b9] (m : multilinear_map R (fun (i : \u03b9) => M) N') (v : \u03b9 \u2192 M) :\n    coe_fn (coe_fn alternatization m) v =\n        finset.sum finset.univ\n          fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn (dom_dom_congr \u03c3 m) v :=\n  sorry\n\nend multilinear_map\n\n\nnamespace alternating_map\n\n\n/-- Alternatizing a multilinear map that is already alternating results in a scale factor of `n!`,\nwhere `n` is the number of inputs. -/\ntheorem coe_alternatization {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M]\n    [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6}\n    [DecidableEq \u03b9] [fintype \u03b9] (a : alternating_map R M N' \u03b9) :\n    coe_fn multilinear_map.alternatization \u2191a = nat.factorial (fintype.card \u03b9) \u2022 a :=\n  sorry\n\nend alternating_map\n\n\nnamespace linear_map\n\n\n/-- Composition with a linear map before and after alternatization are equivalent. -/\ntheorem comp_multilinear_map_alternatization {R : Type u_1} [semiring R] {M : Type u_2}\n    [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N']\n    {\u03b9 : Type u_6} [DecidableEq \u03b9] {N'\u2082 : Type u_7} [add_comm_group N'\u2082] [semimodule R N'\u2082]\n    [fintype \u03b9] (g : linear_map R N' N'\u2082) (f : multilinear_map R (fun (_x : \u03b9) => M) N') :\n    coe_fn multilinear_map.alternatization (comp_multilinear_map g f) =\n        coe_fn (comp_alternating_map g) (coe_fn multilinear_map.alternatization f) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/alternating_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.3920090634608875}}
{"text": "/- Individual results that should be moved to mathlib. If there are many related results, put them in a separate file. -/\n\nimport .basic data.finsupp\n\nuniverse variables u v\nopen function\n\n/- move to logic -/\ndef ite_ne_neg {p : Prop} [h : decidable p] {\u03b1 : Sort u} {x y : \u03b1} (h : ite p x y \u2260 y) : p :=\nby { by_cases hp : p, exact hp, rw [if_neg hp] at h, contradiction }\n\ndef ite_ne_pos {p : Prop} [h : decidable p] {\u03b1 : Sort u} {x y : \u03b1} (h : ite p x y \u2260 x) : \u00acp :=\nby { by_cases hp : p, rw [if_pos hp] at h, contradiction, exact hp }\n\n\nnamespace sum\nlemma injective_inl {\u03b1 \u03b2 : Type*} : injective (inl : \u03b1 \u2192 \u03b1 \u2295 \u03b2) :=\n\u03bb x y, inl.inj\n\nlemma injective_inr {\u03b1 \u03b2 : Type*} : injective (inr : \u03b2 \u2192 \u03b1 \u2295 \u03b2) :=\n\u03bb x y, inr.inj\n\ndef embedding_inl {\u03b1 \u03b2 : Type*} : \u03b1 \u21aa \u03b1 \u2295 \u03b2 :=\n\u27e8inl, injective_inl\u27e9\n\ndef embedding_inr {\u03b1 \u03b2 : Type*} : \u03b2 \u21aa \u03b1 \u2295 \u03b2 :=\n\u27e8inr, injective_inr\u27e9\nend sum\n\nnamespace equiv\n\ndef equiv_embedding_fun {\u03b1 \u03b2 : Type*} : (\u03b1 \u2243 \u03b2) \u21aa (\u03b1 \u2192 \u03b2) :=\n\u27e8equiv.to_fun, \u03bb f g, eq_of_to_fun_eq\u27e9\n\nend equiv\n\nnamespace finsupp\n/- move to finsupp -/\nlemma injective_emb_domain {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3] [decidable_eq \u03b2] (f : \u03b1 \u21aa \u03b2) :\n  injective (emb_domain f : (\u03b1 \u2192\u2080 \u03b3) \u2192 (\u03b2 \u2192\u2080 \u03b3)) :=\nomitted\n\ndef finsupp_embedding_finsupp_left {\u03b1 \u03b2 \u03b3 : Type*} [has_zero \u03b3] [decidable_eq \u03b2] (f : \u03b1 \u21aa \u03b2) :\n  (\u03b1 \u2192\u2080 \u03b3) \u21aa (\u03b2 \u2192\u2080 \u03b3) :=\n\u27e8\u03bb g, emb_domain f g, injective_emb_domain f\u27e9\nend finsupp", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/to_mathlib.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.39194187769638034}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes H\u00f6lzl\n\nLebesgue integral on `ennreal`.\n\nWe define simple functions and show that each Borel measurable function on `ennreal` can be\napproximated by a sequence of simple functions.\n-/\nimport\n  algebra.pi_instances\n  measure_theory.measure_space\n  measure_theory.borel_space\nnoncomputable theory\nopen lattice set filter\nlocal attribute [instance] classical.prop_decidable\n\nsection sequence_of_directed\nvariables {\u03b1 : Type*} {\u03b2 : Type*} [encodable \u03b1] [inhabited \u03b1]\nopen encodable\n\nnoncomputable def sequence_of_directed (r : \u03b2 \u2192 \u03b2 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (hf : directed r f) : \u2115 \u2192 \u03b1\n| 0       := default \u03b1\n| (n + 1) :=\n  let p := sequence_of_directed n in\n  match decode \u03b1 n with\n  | none     := p\n  | (some a) := classical.some (hf p a)\n  end\n\nlemma monotone_sequence_of_directed [partial_order \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : directed (\u2264) f) :\n  monotone (f \u2218 sequence_of_directed (\u2264) f hf) :=\nmonotone_of_monotone_nat $ assume n,\n  begin\n    dsimp [sequence_of_directed],\n    generalize eq : sequence_of_directed (\u2264) f hf n = p,\n    cases h : decode \u03b1 n with a,\n    { refl },\n    { exact (classical.some_spec (hf p a)).1 }\n  end\n\nlemma le_sequence_of_directed [partial_order \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : directed (\u2264) f) (a : \u03b1) :\n  f a \u2264 f (sequence_of_directed (\u2264) f hf (encode a + 1)) :=\nbegin\n  simp [sequence_of_directed, -add_comm, encodek],\n  exact (classical.some_spec (hf _ a)).2\nend\n\nend sequence_of_directed\n\nnamespace measure_theory\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*}\n\nstructure {u v} simple_func (\u03b1 : Type u) [measurable_space \u03b1] (\u03b2 : Type v) :=\n(to_fun : \u03b1 \u2192 \u03b2)\n(measurable_sn : \u2200 x, is_measurable (to_fun \u207b\u00b9' {x}))\n(finite : (set.range to_fun).finite)\n\nlocal infixr ` \u2192\u209b `:25 := simple_func\n\nnamespace simple_func\n\nsection measurable\nvariables [measurable_space \u03b1]\ninstance has_coe_to_fun : has_coe_to_fun (\u03b1 \u2192\u209b \u03b2) := \u27e8_, to_fun\u27e9\n\n@[extensionality] theorem ext {f g : \u03b1 \u2192\u209b \u03b2} (H : \u2200 a, f a = g a) : f = g :=\nby cases f; cases g; congr; exact funext H\n\nprotected def range (f : \u03b1 \u2192\u209b \u03b2) := f.finite.to_finset\n\n@[simp] theorem mem_range {f : \u03b1 \u2192\u209b \u03b2} {b} : b \u2208 f.range \u2194 \u2203 a, f a = b :=\nfinite.mem_to_finset\n\ndef const (\u03b1) {\u03b2} [measurable_space \u03b1] (b : \u03b2) : \u03b1 \u2192\u209b \u03b2 :=\n\u27e8\u03bb a, b, \u03bb x, is_measurable.const _,\n  finite_subset (set.finite_singleton b) $ by rintro _ \u27e8a, rfl\u27e9; simp\u27e9\n\n@[simp] theorem const_apply (a : \u03b1) (b : \u03b2) : (const \u03b1 b) a = b := rfl\n\nlemma range_const (\u03b1) [measure_space \u03b1] [ne : nonempty \u03b1] (b : \u03b2) :\n  (const \u03b1 b).range = {b} :=\nbegin\n  ext b',\n  simp [mem_range],\n  exact \u27e8assume \u27e8_, h\u27e9, h.symm, assume h, ne.elim $ \u03bba, \u27e8a, h.symm\u27e9\u27e9\nend\n\nlemma is_measurable_cut (p : \u03b1 \u2192 \u03b2 \u2192 Prop) (f : \u03b1 \u2192\u209b \u03b2)\n  (h : \u2200b, is_measurable {a | p a b}) : is_measurable {a | p a (f a)} :=\nbegin\n  rw (_ : {a | p a (f a)} = \u22c3 b \u2208 set.range f, {a | p a b} \u2229 f \u207b\u00b9' {b}),\n  { exact is_measurable.bUnion (countable_finite f.finite)\n      (\u03bb b _, is_measurable.inter (h b) (f.measurable_sn _)) },\n  ext a, simp,\n  exact \u27e8\u03bb h, \u27e8_, \u27e8a, rfl\u27e9, h, rfl\u27e9, \u03bb \u27e8_, \u27e8a', rfl\u27e9, h', e\u27e9, e.symm \u25b8 h'\u27e9\nend\n\ntheorem preimage_measurable (f : \u03b1 \u2192\u209b \u03b2) (s) : is_measurable (f \u207b\u00b9' s) :=\nis_measurable_cut (\u03bb _ b, b \u2208 s) f (\u03bb b, by simp [is_measurable.const])\n\ntheorem measurable [measurable_space \u03b2] (f : \u03b1 \u2192\u209b \u03b2) : measurable f :=\n\u03bb s _, preimage_measurable f s\n\ndef ite {s : set \u03b1} (hs : is_measurable s) (f g : \u03b1 \u2192\u209b \u03b2) : \u03b1 \u2192\u209b \u03b2 :=\n\u27e8\u03bb a, if a \u2208 s then f a else g a,\n \u03bb x, by letI : measurable_space \u03b2 := \u22a4; exact\n   measurable.if hs f.measurable g.measurable _ trivial,\n finite_subset (finite_union f.finite g.finite) begin\n   rintro _ \u27e8a, rfl\u27e9,\n   by_cases a \u2208 s; simp [h],\n   exacts [or.inl \u27e8_, rfl\u27e9, or.inr \u27e8_, rfl\u27e9]\n end\u27e9\n\n@[simp] theorem ite_apply {s : set \u03b1} (hs : is_measurable s)\n  (f g : \u03b1 \u2192\u209b \u03b2) (a) : ite hs f g a = if a \u2208 s then f a else g a := rfl\n\ndef bind (f : \u03b1 \u2192\u209b \u03b2) (g : \u03b2 \u2192 \u03b1 \u2192\u209b \u03b3) : \u03b1 \u2192\u209b \u03b3 :=\n\u27e8\u03bba, g (f a) a,\n \u03bb c, is_measurable_cut (\u03bba b, g b a \u2208 ({c} : set \u03b3)) f (\u03bb b, (g b).measurable_sn c),\n finite_subset (finite_bUnion f.finite (\u03bb b, (g b).finite)) $\n by rintro _ \u27e8a, rfl\u27e9; simp; exact \u27e8_, \u27e8a, rfl\u27e9, _, rfl\u27e9\u27e9\n\n@[simp] theorem bind_apply (f : \u03b1 \u2192\u209b \u03b2) (g : \u03b2 \u2192 \u03b1 \u2192\u209b \u03b3) (a) :\n  f.bind g a = g (f a) a := rfl\n\ndef restrict [has_zero \u03b2] (f : \u03b1 \u2192\u209b \u03b2) (s : set \u03b1) : \u03b1 \u2192\u209b \u03b2 :=\nif hs : is_measurable s then ite hs f (const \u03b1 0) else const \u03b1 0\n\n@[simp] theorem restrict_apply [has_zero \u03b2]\n  (f : \u03b1 \u2192\u209b \u03b2) {s : set \u03b1} (hs : is_measurable s) (a) :\n  restrict f s a = if a \u2208 s then f a else 0 :=\nby unfold_coes; simp [restrict, hs]; apply ite_apply hs\n\ntheorem restrict_preimage [has_zero \u03b2]\n  (f : \u03b1 \u2192\u209b \u03b2) {s : set \u03b1} (hs : is_measurable s)\n  {t : set \u03b2} (ht : (0:\u03b2) \u2209 t) : restrict f s \u207b\u00b9' t = s \u2229 f \u207b\u00b9' t :=\nby ext a; dsimp; rw [restrict_apply]; by_cases a \u2208 s; simp [h, hs, ht]\n\ndef map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192\u209b \u03b2) : \u03b1 \u2192\u209b \u03b3 := bind f (const \u03b1 \u2218 g)\n\n@[simp] theorem map_apply (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192\u209b \u03b2) (a) : f.map g a = g (f a) := rfl\n\ntheorem map_map (g : \u03b2 \u2192 \u03b3) (h: \u03b3 \u2192 \u03b4) (f : \u03b1 \u2192\u209b \u03b2) : (f.map g).map h = f.map (h \u2218 g) := rfl\n\ntheorem coe_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192\u209b \u03b2) : (f.map g : \u03b1 \u2192 \u03b3) = g \u2218 f := rfl\n\n@[simp] theorem range_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192\u209b \u03b2) : (f.map g).range = f.range.image g :=\nbegin\n  ext c,\n  simp [mem_range],\n  split,\n  { rintros \u27e8a, rfl\u27e9, exact \u27e8f a, \u27e8_, rfl\u27e9, rfl\u27e9 },\n  { rintros \u27e8_, \u27e8a, rfl\u27e9, rfl\u27e9, exact \u27e8_, rfl\u27e9 }\nend\n\ndef seq (f : \u03b1 \u2192\u209b (\u03b2 \u2192 \u03b3)) (g : \u03b1 \u2192\u209b \u03b2) : \u03b1 \u2192\u209b \u03b3 := f.bind (\u03bbf, g.map f)\n\ndef pair (f : \u03b1 \u2192\u209b \u03b2) (g : \u03b1 \u2192\u209b \u03b3) : \u03b1 \u2192\u209b (\u03b2 \u00d7 \u03b3) := (f.map prod.mk).seq g\n\n@[simp] lemma pair_apply (f : \u03b1 \u2192\u209b \u03b2) (g : \u03b1 \u2192\u209b \u03b3) (a) : pair f g a = (f a, g a) := rfl\n\ntheorem bind_const (f : \u03b1 \u2192\u209b \u03b2) : f.bind (const \u03b1) = f := by ext; simp\n\ninstance [has_zero \u03b2] : has_zero (\u03b1 \u2192\u209b \u03b2) := \u27e8const \u03b1 0\u27e9\ninstance [has_add \u03b2] : has_add (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbf g, (f.map (+)).seq g\u27e9\ninstance [has_mul \u03b2] : has_mul (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbf g, (f.map (*)).seq g\u27e9\ninstance [has_sup \u03b2] : has_sup (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbf g, (f.map (\u2294)).seq g\u27e9\ninstance [has_inf \u03b2] : has_inf (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbf g, (f.map (\u2293)).seq g\u27e9\ninstance [has_le \u03b2] : has_le (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbf g, \u2200a, f a \u2264 g a\u27e9\n\n@[simp] lemma sup_apply [has_sup \u03b2] (f g : \u03b1 \u2192\u209b \u03b2) (a : \u03b1) : (f \u2294 g) a = f a \u2294 g a := rfl\n@[simp] lemma mul_apply [has_mul \u03b2] (f g : \u03b1 \u2192\u209b \u03b2) (a : \u03b1) : (f * g) a = f a * g a := rfl\nlemma add_apply [has_add \u03b2] (f g : \u03b1 \u2192\u209b \u03b2) (a : \u03b1) : (f + g) a = f a + g a := rfl\n\nlemma add_eq_map\u2082 [has_add \u03b2] (f g : \u03b1 \u2192\u209b \u03b2) : f + g = (pair f g).map (\u03bbp:\u03b2\u00d7\u03b2, p.1 + p.2) :=\nrfl\n\nlemma sup_eq_map\u2082 [has_sup \u03b2] (f g : \u03b1 \u2192\u209b \u03b2) : f \u2294 g = (pair f g).map (\u03bbp:\u03b2\u00d7\u03b2, p.1 \u2294 p.2) :=\nrfl\n\nlemma const_mul_eq_map [has_mul \u03b2] (f : \u03b1 \u2192\u209b \u03b2) (b : \u03b2) : const \u03b1 b * f = f.map (\u03bba, b * a) := rfl\n\ninstance [add_monoid \u03b2] : add_monoid (\u03b1 \u2192\u209b \u03b2) :=\n{ add       := (+), zero := 0,\n  add_assoc := assume f g h, ext (assume a, add_assoc _ _ _),\n  zero_add  := assume f, ext (assume a, zero_add _),\n  add_zero  := assume f, ext (assume a, add_zero _) }\n\ninstance [semiring \u03b2] [add_monoid \u03b2] : has_scalar \u03b2 (\u03b1 \u2192\u209b \u03b2) := \u27e8\u03bbb f, f.map (\u03bba, b * a)\u27e9\n\ninstance [preorder \u03b2] : preorder (\u03b1 \u2192\u209b \u03b2) :=\n{ le_refl := \u03bbf a, le_refl _,\n  le_trans := \u03bbf g h hfg hgh a, le_trans (hfg _) (hgh a),\n  .. simple_func.has_le }\n\ninstance [partial_order \u03b2] : partial_order (\u03b1 \u2192\u209b \u03b2) :=\n{ le_antisymm := assume f g hfg hgf, ext $ assume a, le_antisymm (hfg a) (hgf a),\n  .. simple_func.preorder }\n\ninstance [order_bot \u03b2] : order_bot (\u03b1 \u2192\u209b \u03b2) :=\n{ bot := const \u03b1 \u22a5, bot_le := \u03bbf a, bot_le, .. simple_func.partial_order }\n\ninstance [order_top \u03b2] : order_top (\u03b1 \u2192\u209b \u03b2) :=\n{ top := const \u03b1\u22a4, le_top := \u03bbf a, le_top, .. simple_func.partial_order }\n\ninstance [semilattice_inf \u03b2] : semilattice_inf (\u03b1 \u2192\u209b \u03b2) :=\n{ inf := (\u2293),\n  inf_le_left := assume f g a, inf_le_left,\n  inf_le_right := assume f g a, inf_le_right,\n  le_inf := assume f g h hfh hgh a, le_inf (hfh a) (hgh a),\n  .. simple_func.partial_order }\n\ninstance [semilattice_sup \u03b2] : semilattice_sup (\u03b1 \u2192\u209b \u03b2) :=\n{ sup := (\u2294),\n  le_sup_left := assume f g a, le_sup_left,\n  le_sup_right := assume f g a, le_sup_right,\n  sup_le := assume f g h hfh hgh a, sup_le (hfh a) (hgh a),\n  .. simple_func.partial_order }\n\ninstance [semilattice_sup_bot \u03b2] : semilattice_sup_bot (\u03b1 \u2192\u209b \u03b2) :=\n{ .. simple_func.lattice.semilattice_sup,.. simple_func.lattice.order_bot }\n\ninstance [lattice \u03b2] : lattice (\u03b1 \u2192\u209b \u03b2) :=\n{ .. simple_func.lattice.semilattice_sup,.. simple_func.lattice.semilattice_inf }\n\ninstance [bounded_lattice \u03b2] : bounded_lattice (\u03b1 \u2192\u209b \u03b2) :=\n{ .. simple_func.lattice.lattice, .. simple_func.lattice.order_bot, .. simple_func.lattice.order_top }\n\nlemma finset_sup_apply [semilattice_sup_bot \u03b2] {f : \u03b3 \u2192 \u03b1 \u2192\u209b \u03b2} (s : finset \u03b3) (a : \u03b1) :\n  s.sup f a = s.sup (\u03bbc, f c a) :=\nbegin\n  refine finset.induction_on s rfl _,\n  assume a s hs ih,\n  rw [finset.sup_insert, finset.sup_insert, sup_apply, ih]\nend\n\nsection approx\n\nsection\nvariables [topological_space \u03b2] [semilattice_sup_bot \u03b2] [has_zero \u03b2]\n\ndef approx (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) (n : \u2115) : \u03b1 \u2192\u209b \u03b2 :=\n(finset.range n).sup (\u03bbk, restrict (const \u03b1 (i k)) {a:\u03b1 | i k \u2264 f a})\n\nlemma approx_apply [ordered_topology \u03b2] {i : \u2115 \u2192 \u03b2} {f : \u03b1 \u2192 \u03b2} {n : \u2115} (a : \u03b1)\n  (hf : _root_.measurable f) :\n  (approx i f n : \u03b1 \u2192\u209b \u03b2) a = (finset.range n).sup (\u03bbk, if i k \u2264 f a then i k else 0) :=\nbegin\n  dsimp only [approx],\n  rw [finset_sup_apply],\n  congr,\n  funext k,\n  rw [restrict_apply],\n  refl,\n  exact (hf.preimage $ is_measurable_of_is_closed $ is_closed_ge' _)\nend\n\nlemma monotone_approx (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) : monotone (approx i f) :=\nassume n m h, finset.sup_mono $ finset.range_subset.2 h\n\nlemma approx_comp [ordered_topology \u03b2] [measurable_space \u03b3]\n  {i : \u2115 \u2192 \u03b2} {f : \u03b3 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} {n : \u2115} (a : \u03b1)\n  (hf : _root_.measurable f) (hg : _root_.measurable g) :\n  (approx i (f \u2218 g) n : \u03b1 \u2192\u209b \u03b2) a = (approx i f n : \u03b3 \u2192\u209b \u03b2) (g a) :=\nby rw [approx_apply _ hf, approx_apply _ (hg.comp hf)]\n\nend\n\nlemma supr_approx_apply [topological_space \u03b2] [complete_lattice \u03b2] [ordered_topology \u03b2] [has_zero \u03b2]\n  (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (hf : _root_.measurable f) (h_zero : (0 : \u03b2) = \u22a5):\n  (\u2a06n, (approx i f n : \u03b1 \u2192\u209b \u03b2) a) = (\u2a06k (h : i k \u2264 f a), i k) :=\nbegin\n  refine le_antisymm (supr_le $ assume n, _) (supr_le $ assume k, supr_le $ assume hk, _),\n  { rw [approx_apply a hf, h_zero],\n    refine finset.sup_le (assume k hk, _),\n    split_ifs,\n    exact le_supr_of_le k (le_supr _ h),\n    exact bot_le },\n  { refine le_supr_of_le (k+1) _,\n    rw [approx_apply a hf],\n    have : k \u2208 finset.range (k+1) := finset.mem_range.2 (nat.lt_succ_self _),\n    refine le_trans (le_of_eq _) (finset.le_sup this),\n    rw [if_pos hk] }\nend\n\nend approx\n\nsection eapprox\n\ndef ennreal_rat_embed (n : \u2115) : ennreal :=\nnnreal.of_real ((encodable.decode \u211a n).get_or_else (0 : \u211a))\n\nlemma ennreal_rat_embed_encode (q : \u211a) (hq : 0 \u2264 q) :\n  ennreal_rat_embed (encodable.encode q) = nnreal.of_real q :=\nby rw [ennreal_rat_embed, encodable.encodek]; refl\n\ndef eapprox : (\u03b1 \u2192 ennreal) \u2192 \u2115 \u2192 \u03b1 \u2192\u209b ennreal :=\napprox ennreal_rat_embed\n\nlemma monotone_eapprox (f : \u03b1 \u2192 ennreal) : monotone (eapprox f) :=\nmonotone_approx _ f\n\nlemma supr_eapprox_apply (f : \u03b1 \u2192 ennreal) (hf : _root_.measurable f) (a : \u03b1) :\n  (\u2a06n, (eapprox f n : \u03b1 \u2192\u209b ennreal) a) = f a :=\nbegin\n  rw [eapprox, supr_approx_apply ennreal_rat_embed f a hf rfl],\n  refine le_antisymm (supr_le $ assume i, supr_le $ assume hi, hi) (le_of_not_gt _),\n  assume h,\n  rcases ennreal.lt_iff_exists_rat_btwn.1 h with \u27e8q, hq, lt_q, q_lt\u27e9,\n  have : (nnreal.of_real q : ennreal) \u2264\n      (\u2a06 (k : \u2115) (h : ennreal_rat_embed k \u2264 f a), ennreal_rat_embed k),\n  { refine le_supr_of_le (encodable.encode q) _,\n    rw [ennreal_rat_embed_encode q hq],\n    refine le_supr_of_le (le_of_lt q_lt) _,\n    exact le_refl _ },\n  exact lt_irrefl _ (lt_of_le_of_lt this lt_q)\nend\n\nlemma eapprox_comp [measurable_space \u03b3] {f : \u03b3 \u2192 ennreal} {g : \u03b1 \u2192 \u03b3} {n : \u2115}\n  (hf : _root_.measurable f) (hg : _root_.measurable g) :\n  (eapprox (f \u2218 g) n : \u03b1 \u2192 ennreal) = (eapprox f n : \u03b3 \u2192\u209b ennreal) \u2218 g :=\nfunext $ assume a, approx_comp a hf hg\n\nend eapprox\n\nend measurable\n\nsection measure\nvariables [measure_space \u03b1]\n\ndef integral (f : \u03b1 \u2192\u209b ennreal) : ennreal :=\nf.range.sum (\u03bb x, x * volume (f \u207b\u00b9' {x}))\n\n-- TODO: slow simp proofs\nlemma map_integral (g : \u03b2 \u2192 ennreal) (f : \u03b1 \u2192\u209b \u03b2) :\n  (f.map g).integral = f.range.sum (\u03bb x, g x * volume (f \u207b\u00b9' {x})) :=\nbegin\n  simp only [integral, coe_map, range_map],\n  refine finset.sum_image' _ (assume b hb, _),\n  rcases mem_range.1 hb with \u27e8a, rfl\u27e9,\n  let s' := f.range.filter (\u03bbb, g b = g (f a)),\n  have : g \u2218 \u21d1f \u207b\u00b9' {g (f a)} = (\u22c3b\u2208s', \u21d1f \u207b\u00b9' {b}),\n  { ext a',\n    simp,\n    split,\n    { assume eq, exact \u27e8\u27e8_, rfl\u27e9, eq\u27e9 },\n    { rintros \u27e8_, eq\u27e9, exact eq } },\n  calc g (f a) * volume (g \u2218 \u21d1f \u207b\u00b9' {g (f a)}) = g (f a) * volume (\u22c3b\u2208s', \u21d1f \u207b\u00b9' {b}) : by rw [this]\n    ... = g (f a) * s'.sum (\u03bbb, volume (f \u207b\u00b9' {b})) :\n    begin\n      rw [volume_bUnion_finset],\n      { simp [pairwise_on, (on)],\n        rintros b a\u2080 rfl eq\u2080 b a\u2081 rfl eq\u2081 ne a \u27e8h\u2081, h\u2082\u27e9,\n        simp at h\u2081 h\u2082,\n        rw [\u2190 h\u2081, h\u2082] at ne,\n        exact ne rfl },\n      exact assume a ha, preimage_measurable _ _\n    end\n    ... = s'.sum (\u03bbb,  g (f a) * volume (f \u207b\u00b9' {b})) : by rw [finset.mul_sum]\n    ... = s'.sum  (\u03bbb, g b * volume (f \u207b\u00b9' {b})) : finset.sum_congr rfl $ by simp {contextual := tt}\nend\n\nlemma zero_integral : (0 : \u03b1 \u2192\u209b ennreal).integral = 0 :=\nbegin\n  refine (finset.sum_eq_zero_iff_of_nonneg $ assume _ _, zero_le _).2 _,\n  assume r hr, rcases mem_range.1 hr with \u27e8a, rfl\u27e9,\n  exact zero_mul _\nend\n\nlemma add_integral (f g : \u03b1 \u2192\u209b ennreal) : (f + g).integral = f.integral + g.integral :=\ncalc (f + g).integral =\n      (pair f g).range.sum (\u03bbx, x.1 * volume (pair f g \u207b\u00b9' {x}) + x.2  * volume (pair f g \u207b\u00b9' {x})) :\n    by rw [add_eq_map\u2082, map_integral]; exact finset.sum_congr rfl (assume a ha, add_mul _ _ _)\n  ... = (pair f g).range.sum (\u03bbx, x.1 * volume (pair f g \u207b\u00b9' {x})) +\n      (pair f g).range.sum (\u03bbx, x.2 * volume (pair f g \u207b\u00b9' {x})) : by rw [finset.sum_add_distrib]\n  ... = ((pair f g).map prod.fst).integral + ((pair f g).map prod.snd).integral :\n    by rw [map_integral, map_integral]\n  ... = integral f + integral g : rfl\n\nlemma const_mul_integral (f : \u03b1 \u2192\u209b ennreal) (x : ennreal) :\n  (const \u03b1 x * f).integral = x * f.integral :=\ncalc (f.map (\u03bba, x * a)).integral = f.range.sum (\u03bbr, x * r * volume (f \u207b\u00b9' {r})) :\n    by rw [map_integral]\n  ... = f.range.sum (\u03bbr, x * (r * volume (f \u207b\u00b9' {r}))) :\n    finset.sum_congr rfl (assume a ha, mul_assoc _ _ _)\n  ... = x * f.integral :\n    finset.mul_sum.symm\n\nlemma mem_restrict_range [has_zero \u03b2] {r : \u03b2} {s : set \u03b1} {f : \u03b1 \u2192\u209b \u03b2} (hs : is_measurable s) :\n  r \u2208 (restrict f s).range \u2194 (r = 0 \u2227 s \u2260 univ) \u2228 (\u2203a\u2208s, f a = r) :=\nbegin\n  simp only [mem_range, restrict_apply, hs],\n  split,\n  { rintros \u27e8a, ha\u27e9,\n    split_ifs at ha,\n    { exact or.inr \u27e8a, h, ha\u27e9 },\n    { exact or.inl \u27e8ha.symm, assume eq, h $ eq.symm \u25b8 trivial\u27e9 } },\n  { rintros (\u27e8rfl, h\u27e9 | \u27e8a, ha, rfl\u27e9),\n    { have : \u00ac \u2200a, a \u2208 s := assume this, h $ eq_univ_of_forall this,\n      rcases not_forall.1 this with \u27e8a, ha\u27e9,\n      refine \u27e8a, _\u27e9,\n      rw [if_neg ha] },\n    { refine \u27e8a, _\u27e9,\n      rw [if_pos ha] } }\nend\n\nlemma restrict_preimage' {r : ennreal} {s : set \u03b1}\n  (f : \u03b1 \u2192\u209b ennreal) (hs : is_measurable s) (hr : r \u2260 0):\n  (restrict f s) \u207b\u00b9' {r} = (f \u207b\u00b9' {r} \u2229 s) :=\nbegin\n  ext a,\n  by_cases a \u2208 s; simp [hs, h, hr.symm]\nend\n\nlemma restrict_integral (f : \u03b1 \u2192\u209b ennreal) (s : set \u03b1) (hs : is_measurable s) :\n  (restrict f s).integral = f.range.sum (\u03bbr, r * volume (f \u207b\u00b9' {r} \u2229 s)) :=\nbegin\n  refine finset.sum_bij_ne_zero (\u03bbr _ _, r) _ _ _ _,\n  { assume r hr,\n    rcases (mem_restrict_range hs).1 hr with \u27e8rfl, h\u27e9 | \u27e8a, ha, rfl\u27e9,\n    { simp },\n    { assume _, exact mem_range.2 \u27e8a, rfl\u27e9 } },\n  { assume a b _ _ _ _ h, exact h },\n  { assume r hr,\n    by_cases r0 : r = 0, { simp [r0] },\n    assume h0,\n    rcases mem_range.1 hr with \u27e8a, rfl\u27e9,\n    have : f \u207b\u00b9' {f a} \u2229 s \u2260 \u2205,\n    { assume h, simpa [h] using h0 },\n    rcases ne_empty_iff_exists_mem.1 this with \u27e8a', eq', ha'\u27e9,\n    refine \u27e8_, (mem_restrict_range hs).2 (or.inr \u27e8a', ha', _\u27e9), _, rfl\u27e9,\n    { simpa using eq' },\n    { rwa [restrict_preimage' _ hs r0] } },\n  { assume r hr ne,\n    by_cases r = 0, { simp [h] },\n    rw [restrict_preimage' _ hs h] }\nend\n\nlemma restrict_const_integral (c : ennreal) (s : set \u03b1) (hs : is_measurable s) :\n  (restrict (const \u03b1 c) s).integral = c * volume s :=\nhave (@const \u03b1 ennreal _ c) \u207b\u00b9' {c} = univ,\nbegin\n  refine eq_univ_of_forall (assume a, _),\n  simp,\nend,\ncalc (restrict (const \u03b1 c) s).integral = c * volume ((const \u03b1 c) \u207b\u00b9' {c} \u2229 s) :\n  begin\n    rw [restrict_integral (const \u03b1 c) s hs],\n    refine finset.sum_eq_single c _ _,\n    { assume r hr, rcases mem_range.1 hr with \u27e8a, rfl\u27e9, contradiction },\n    { by_cases nonempty \u03b1,\n      { assume ne,\n        rcases h with \u27e8a\u27e9,\n        exfalso,\n        exact ne (mem_range.2 \u27e8a, rfl\u27e9) },\n      { assume empty,\n        have : (@const \u03b1 ennreal _ c) \u207b\u00b9' {c} \u2229 s = \u2205,\n        { ext a, exfalso, exact h \u27e8a\u27e9 },\n        simp only [this, volume_empty, mul_zero] } }\n  end\n  ... = c * volume s : by rw [this, univ_inter]\n\nlemma integral_sup_le (f g : \u03b1 \u2192\u209b ennreal) : f.integral \u2294 g.integral \u2264 (f \u2294 g).integral :=\ncalc f.integral \u2294 g.integral =\n      ((pair f g).map prod.fst).integral \u2294 ((pair f g).map prod.snd).integral : rfl\n  ... \u2264 (pair f g).range.sum (\u03bbx, (x.1 \u2294 x.2) * volume (pair f g \u207b\u00b9' {x})) :\n  begin\n    rw [map_integral, map_integral],\n    refine sup_le _ _;\n      refine finset.sum_le_sum' (\u03bb a _, canonically_ordered_semiring.mul_le_mul _ (le_refl _)),\n    exact le_sup_left,\n    exact le_sup_right\n  end\n  ... = (f \u2294 g).integral : by rw [sup_eq_map\u2082, map_integral]\n\nlemma integral_le_integral (f g : \u03b1 \u2192\u209b ennreal) (h : f \u2264 g) : f.integral \u2264 g.integral :=\ncalc f.integral \u2264 f.integral \u2294 g.integral : le_sup_left\n  ... \u2264 (f \u2294 g).integral : integral_sup_le _ _\n  ... = g.integral : by rw [sup_of_le_right h]\n\nlemma integral_congr (f g : \u03b1 \u2192\u209b ennreal) (h : {a | f a = g a} \u2208 (@measure_space.\u03bc \u03b1 _).a_e) :\n  f.integral = g.integral :=\nshow ((pair f g).map prod.fst).integral = ((pair f g).map prod.snd).integral, from\nbegin\n  rw [map_integral, map_integral],\n  refine finset.sum_congr rfl (assume p hp, _),\n  rcases mem_range.1 hp with \u27e8a, rfl\u27e9,\n  by_cases eq : f a = g a,\n  { dsimp only [pair_apply], rw eq },\n  { have : volume ((pair f g) \u207b\u00b9' {(f a, g a)}) = 0,\n    { refine volume_mono_null (assume a' ha', _) h,\n      simp at ha',\n      show f a' \u2260 g a',\n      rwa [ha'.1, ha'.2] },\n    simp [this] }\nend\n\nlemma integral_map {\u03b2} [measure_space \u03b2] (f : \u03b1 \u2192\u209b ennreal) (g : \u03b2 \u2192\u209b ennreal)\n  (m : \u03b1 \u2192 \u03b2) (hm : _root_.measurable m) (eq : \u2200a:\u03b1, f a = g (m a))\n  (h : \u2200s:set \u03b2, is_measurable s \u2192 volume s = volume (m \u207b\u00b9' s)) :\n  f.integral = g.integral :=\nhave f_eq : (f : \u03b1 \u2192 ennreal) = g \u2218 m := funext eq,\nhave vol_f : \u2200r, volume (f \u207b\u00b9' {r}) = volume (g \u207b\u00b9' {r}),\n  by { assume r, rw [h, f_eq, preimage_comp], exact measurable_sn _ _ },\nbegin\n  simp [integral, vol_f],\n  refine finset.sum_subset _ _,\n  { simp [finset.subset_iff, f_eq],\n    rintros r a rfl, exact \u27e8_, rfl\u27e9 },\n  { assume r hrg hrf,\n    rw [simple_func.mem_range, not_exists] at hrf,\n    have : f \u207b\u00b9' {r} = \u2205 := set.eq_empty_of_subset_empty (assume a, by simpa using hrf a),\n    simp [(vol_f _).symm, this] }\nend\n\nend measure\n\nend simple_func\n\nsection lintegral\nopen simple_func\nvariable [measure_space \u03b1]\n\n/-- The lower Lebesgue integral -/\ndef lintegral (f : \u03b1 \u2192 ennreal) : ennreal :=\n\u2a06 (s : \u03b1 \u2192\u209b ennreal) (hf : f \u2265 s), s.integral\n\nnotation `\u222b\u207b` binders `, ` r:(scoped f, lintegral f) := r\n\ntheorem simple_func.lintegral_eq_integral (f : \u03b1 \u2192\u209b ennreal) : (\u222b\u207b a, f a) = f.integral :=\nle_antisymm\n  (supr_le $ assume s, supr_le $ assume hs, integral_le_integral _ _ hs)\n  (le_supr_of_le f $ le_supr_of_le (le_refl f) $ le_refl _)\n\nlemma lintegral_le_lintegral (f g : \u03b1 \u2192 ennreal) (h : f \u2264 g) : (\u222b\u207b a, f a) \u2264 (\u222b\u207b a, g a) :=\nsupr_le_supr $ assume s, supr_le $ assume hs, le_supr_of_le (le_trans hs h) (le_refl _)\n\nlemma lintegral_eq_nnreal (f : \u03b1 \u2192 ennreal) :\n  (\u222b\u207b a, f a) =\n    (\u2a06 (s : \u03b1 \u2192\u209b nnreal) (hf : f \u2265 s.map (coe : nnreal \u2192 ennreal)), (s.map (coe : nnreal \u2192 ennreal)).integral) :=\nbegin\n  let c : nnreal \u2192 ennreal := coe,\n  refine le_antisymm\n    (supr_le $ assume s, supr_le $ assume hs, _)\n    (supr_le $ assume s, supr_le $ assume hs, le_supr_of_le (s.map c) $ le_supr _ hs),\n  by_cases {a | s a \u2260 \u22a4} \u2208 (@measure_space.\u03bc \u03b1 _).a_e,\n  { have : f \u2265 (s.map ennreal.to_nnreal).map c :=\n      le_trans (assume a, ennreal.coe_to_nnreal_le_self) hs,\n    refine le_supr_of_le (s.map ennreal.to_nnreal) (le_supr_of_le this (le_of_eq $ integral_congr _ _ _)),\n    exact filter.mem_sets_of_superset h (assume a ha, (ennreal.coe_to_nnreal ha).symm) },\n  { have h_vol_s : volume {a : \u03b1 | s a = \u22a4} \u2260 0,\n    { simp [measure.a_e, set.compl_set_of] at h, assumption },\n    let n : \u2115 \u2192 (\u03b1 \u2192\u209b nnreal) := \u03bbn, restrict (const \u03b1 (n : nnreal)) (s \u207b\u00b9' {\u22a4}),\n    have n_le_s : \u2200i, (n i).map c \u2264 s,\n    { assume i a,\n      dsimp [n, c],\n      rw [restrict_apply _ (s.preimage_measurable _)],\n      split_ifs with ha,\n      { simp at ha, exact ha.symm \u25b8 le_top },\n      { exact zero_le _ } },\n    have approx_s : \u2200 (i : \u2115), \u2191i * volume {a : \u03b1 | s a = \u22a4} \u2264 integral (map c (n i)),\n    { assume i,\n      have : {a : \u03b1 | s a = \u22a4} = s \u207b\u00b9' {\u22a4}, { ext a, simp },\n      rw [this, \u2190 restrict_const_integral _ _ (s.preimage_measurable _)],\n      { refine integral_le_integral _ _ (assume a, le_of_eq _),\n        simp [n, c, restrict_apply, s.preimage_measurable],\n        split_ifs; simp [ennreal.coe_nat] },\n     },\n    calc s.integral \u2264 \u22a4 : le_top\n      ... = (\u2a06i:\u2115, (i : ennreal) * volume {a | s a = \u22a4}) :\n        by rw [\u2190 ennreal.supr_mul, ennreal.supr_coe_nat, ennreal.top_mul, if_neg h_vol_s]\n      ... \u2264 (\u2a06i, ((n i).map c).integral) : supr_le_supr approx_s\n      ... \u2264 \u2a06 (s : \u03b1 \u2192\u209b nnreal) (hf : f \u2265 s.map c), (s.map c).integral :\n        have \u2200i, ((n i).map c : \u03b1 \u2192 ennreal) \u2264 f := assume i, le_trans (n_le_s i) hs,\n        (supr_le $ assume i, le_supr_of_le (n i) (le_supr (\u03bbh, ((n i).map c).integral) (this i))) }\nend\n\n/-- Monotone convergence theorem -- somtimes called Beppo-Levi convergence.\n\nSee `lintegral_supr_directed` for a more general form. -/\ntheorem lintegral_supr\n  {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200n, measurable (f n)) (h_mono : monotone f) :\n  (\u222b\u207b a, \u2a06n, f n a) = (\u2a06n, \u222b\u207b a, f n a) :=\nlet c : nnreal \u2192 ennreal := coe in\nlet F (a:\u03b1) := \u2a06n, f n a in\nhave hF : measurable F := measurable.supr hf,\nshow (\u222b\u207b a, F a) = (\u2a06n, \u222b\u207b a, f n a),\nbegin\n  refine le_antisymm _ _,\n  { rw [lintegral_eq_nnreal],\n    refine supr_le (assume s, supr_le (assume hsf, _)),\n    refine ennreal.le_of_forall_lt_one_mul_lt (assume a ha, _),\n    rcases ennreal.lt_iff_exists_coe.1 ha with \u27e8r, rfl, ha\u27e9,\n    have ha : r < 1 := ennreal.coe_lt_coe.1 ha,\n    let rs := s.map (\u03bba, r * a),\n    have eq_rs : (const \u03b1 r : \u03b1 \u2192\u209b ennreal) * map c s = rs.map c,\n    { ext1 a, exact ennreal.coe_mul.symm },\n    have eq : \u2200p, (rs.map c) \u207b\u00b9' {p} = (\u22c3n, (rs.map c) \u207b\u00b9' {p} \u2229 {a | p \u2264 f n a}),\n    { assume p,\n      rw [\u2190 inter_Union_left, \u2190 inter_univ ((map c rs) \u207b\u00b9' {p})] {occs := occurrences.pos [1]},\n      refine set.ext (assume x, and_congr_right $ assume hx, (true_iff _).2 _),\n      by_cases p_eq : p = 0, { simp [p_eq] },\n      simp at hx, subst hx,\n      have : r * s x \u2260 0, { rwa [(\u2260), \u2190 ennreal.coe_eq_zero] },\n      have : s x \u2260 0, { refine mt _ this, assume h, rw [h, mul_zero] },\n      have : (rs.map c) x < \u2a06 (n : \u2115), f n x,\n      { refine lt_of_lt_of_le (ennreal.coe_lt_coe.2 (_)) (hsf x),\n        suffices : r * s x < 1 * s x, simpa [rs],\n        exact mul_lt_mul_of_pos_right ha (zero_lt_iff_ne_zero.2 this) },\n      rcases lt_supr_iff.1 this with \u27e8i, hi\u27e9,\n      exact mem_Union.2 \u27e8i, le_of_lt hi\u27e9 },\n    have mono : \u2200r:ennreal, monotone (\u03bbn, (rs.map c) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}),\n    { assume r i j h,\n      refine inter_subset_inter (subset.refl _) _,\n      assume x hx, exact le_trans hx (h_mono h x) },\n    have h_meas : \u2200n, is_measurable {a : \u03b1 | \u21d1(map c rs) a \u2264 f n a} :=\n      assume n, measurable_le (simple_func.measurable _) (hf n),\n    calc (r:ennreal) * integral (s.map c) = (rs.map c).range.sum (\u03bbr, r * volume ((rs.map c) \u207b\u00b9' {r})) :\n        by rw [\u2190 const_mul_integral, integral, eq_rs]\n      ... \u2264 (rs.map c).range.sum (\u03bbr, r * volume (\u22c3n, (rs.map c) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})) :\n        le_of_eq (finset.sum_congr rfl $ assume x hx, by rw \u2190 eq)\n      ... \u2264 (rs.map c).range.sum (\u03bbr, (\u2a06n, r * volume ((rs.map c) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a}))) :\n        le_of_eq (finset.sum_congr rfl $ assume x hx,\n          begin\n            rw [volume, measure_Union_eq_supr_nat _ (mono x), ennreal.mul_supr],\n            { assume i,\n              refine is_measurable.inter ((rs.map c).preimage_measurable _) _,\n              refine (hf i).preimage _,\n              exact is_measurable_of_is_closed (is_closed_ge' _) }\n          end)\n      ... \u2264 \u2a06n, (rs.map c).range.sum (\u03bbr, r * volume ((rs.map c) \u207b\u00b9' {r} \u2229 {a | r \u2264 f n a})) :\n        begin\n          refine le_of_eq _,\n          rw [ennreal.finset_sum_supr_nat],\n          assume p i j h,\n          exact canonically_ordered_semiring.mul_le_mul (le_refl _) (volume_mono $ mono p h)\n        end\n      ... \u2264 (\u2a06n:\u2115, ((rs.map c).restrict {a | (rs.map c) a \u2264 f n a}).integral) :\n      begin\n        refine supr_le_supr (assume n, _),\n        rw [restrict_integral _ _ (h_meas n)],\n        { refine le_of_eq (finset.sum_congr rfl $ assume r hr, _),\n          congr' 2,\n          ext a,\n          refine and_congr_right _,\n          simp {contextual := tt} }\n      end\n      ... \u2264 (\u2a06n, \u222b\u207b a, f n a) :\n      begin\n        refine supr_le_supr (assume n, _),\n        rw [\u2190 simple_func.lintegral_eq_integral],\n        refine lintegral_le_lintegral _ _ (assume a, _),\n        dsimp,\n        rw [restrict_apply],\n        split_ifs; simp, simpa using h,\n        exact h_meas n\n      end },\n  { exact supr_le (assume n, lintegral_le_lintegral _ _ $ assume a, le_supr _ n) }\nend\n\nlemma lintegral_eq_supr_eapprox_integral {f : \u03b1 \u2192 ennreal} (hf : measurable f) :\n  (\u222b\u207b a, f a) = (\u2a06n, (eapprox f n).integral) :=\ncalc (\u222b\u207b a, f a) = (\u222b\u207b a, \u2a06n, (eapprox f n : \u03b1 \u2192 ennreal) a) :\n   by congr; ext a; rw [supr_eapprox_apply f hf]\n ... = (\u2a06n, \u222b\u207b a, (eapprox f n : \u03b1 \u2192 ennreal) a) :\n begin\n   rw [lintegral_supr],\n   { assume n, exact (eapprox f n).measurable },\n   { assume i j h, exact (monotone_eapprox f h) }\n end\n ... = (\u2a06n, (eapprox f n).integral) : by congr; ext n; rw [(eapprox f n).lintegral_eq_integral]\n\nlemma lintegral_add {f g : \u03b1 \u2192 ennreal} (hf : measurable f) (hg : measurable g) :\n  (\u222b\u207b a, f a + g a) = (\u222b\u207b a, f a) + (\u222b\u207b a, g a) :=\ncalc (\u222b\u207b a, f a + g a) =\n    (\u222b\u207b a, (\u2a06n, (eapprox f n : \u03b1 \u2192 ennreal) a) + (\u2a06n, (eapprox g n : \u03b1 \u2192 ennreal) a)) :\n    by congr; funext a; rw [supr_eapprox_apply f hf, supr_eapprox_apply g hg]\n  ... = (\u222b\u207b a, (\u2a06n, (eapprox f n + eapprox g n : \u03b1 \u2192 ennreal) a)) :\n  begin\n    congr, funext a,\n    rw [ennreal.supr_add_supr_of_monotone], { refl },\n    { assume i j h, exact monotone_eapprox _ h a },\n    { assume i j h, exact monotone_eapprox _ h a },\n  end\n  ... = (\u2a06n, (eapprox f n).integral + (eapprox g n).integral) :\n  begin\n    rw [lintegral_supr],\n    { congr, funext n, rw [\u2190 simple_func.add_integral, \u2190 simple_func.lintegral_eq_integral], refl },\n    { assume n, exact measurable_add (eapprox f n).measurable (eapprox g n).measurable },\n    { assume i j h a, exact add_le_add' (monotone_eapprox _ h _) (monotone_eapprox _ h _) }\n  end\n  ... = (\u2a06n, (eapprox f n).integral) + (\u2a06n, (eapprox g n).integral) :\n  by refine (ennreal.supr_add_supr_of_monotone _ _).symm;\n     { assume i j h, exact simple_func.integral_le_integral _ _ (monotone_eapprox _ h) }\n  ... = (\u222b\u207b a, f a) + (\u222b\u207b a, g a) :\n    by rw [lintegral_eq_supr_eapprox_integral hf, lintegral_eq_supr_eapprox_integral hg]\n\n@[simp] lemma lintegral_zero : (\u222b\u207b a:\u03b1, 0) = 0 :=\nshow (\u222b\u207b a:\u03b1, (0 : \u03b1 \u2192\u209b ennreal) a) = 0, by rw [simple_func.lintegral_eq_integral, zero_integral]\n\nlemma lintegral_finset_sum (s : finset \u03b2) {f : \u03b2 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200b, measurable (f b)) :\n  (\u222b\u207b a, s.sum (\u03bbb, f b a)) = s.sum (\u03bbb, \u222b\u207b a, f b a) :=\nbegin\n  refine finset.induction_on s _ _,\n  { simp },\n  { assume a s has ih,\n    simp [has],\n    rw [lintegral_add (hf _) (measurable_finset_sum s hf), ih] }\nend\n\nlemma lintegral_const_mul (r : ennreal) {f : \u03b1 \u2192 ennreal} (hf : measurable f) :\n  (\u222b\u207b a, r * f a) = r * (\u222b\u207b a, f a) :=\ncalc (\u222b\u207b a, r * f a) = (\u222b\u207b a, (\u2a06n, (const \u03b1 r * eapprox f n) a)) :\n    by congr; funext a; rw [\u2190 supr_eapprox_apply f hf, ennreal.mul_supr]; refl\n  ... = (\u2a06n, r * (eapprox f n).integral) :\n  begin\n    rw [lintegral_supr],\n    { congr, funext n, rw [\u2190 simple_func.const_mul_integral, \u2190 simple_func.lintegral_eq_integral] },\n    { assume n, exact simple_func.measurable _ },\n    { assume i j h a, exact canonically_ordered_semiring.mul_le_mul (le_refl _)\n        (monotone_eapprox _ h _) }\n  end\n  ... = r * (\u222b\u207b a, f a) : by rw [\u2190 ennreal.mul_supr, lintegral_eq_supr_eapprox_integral hf]\n\nlemma lintegral_supr_const (r : ennreal) {s : set \u03b1} (hs : is_measurable s) :\n  (\u222b\u207b a, \u2a06(h : a \u2208 s), r) = r * volume s :=\nbegin\n  rw [\u2190 restrict_const_integral r s hs, \u2190 (restrict (const \u03b1 r) s).lintegral_eq_integral],\n  congr; ext a; by_cases a \u2208 s; simp [h, hs]\nend\n\nlemma lintegral_le_lintegral_ae {f g : \u03b1 \u2192 ennreal} (h : \u2200\u2098 a, f a \u2264 g a) :\n  (\u222b\u207b a, f a) \u2264 (\u222b\u207b a, g a) :=\nbegin\n  rcases exists_is_measurable_superset_of_measure_eq_zero h with \u27e8t, hts, ht, ht0\u27e9,\n  have : - t \u2208 (@measure_space.\u03bc \u03b1 _).a_e,\n  { rw [measure.mem_a_e_iff, lattice.neg_neg, ht0] },\n  refine (supr_le $ assume s, supr_le $ assume hfs,\n    le_supr_of_le (s.restrict (- t)) $ le_supr_of_le _ _),\n  { assume a,\n    by_cases a \u2208 t;\n      simp [h, simple_func.restrict_apply, ht.compl],\n    exact le_trans (hfs a) (by_contradiction $ assume hnfg, h (hts hnfg)) },\n  { refine le_of_eq (s.integral_congr _ _),\n    filter_upwards [this],\n    refine assume a hnt, _,\n    by_cases hat : a \u2208 t; simp [hat, ht.compl],\n    exact (hnt hat).elim }\nend\n\nlemma lintegral_congr_ae {f g : \u03b1 \u2192 ennreal} (h : \u2200\u2098 a, f a = g a) :\n  (\u222b\u207b a, f a) = (\u222b\u207b a, g a) :=\nle_antisymm\n  (lintegral_le_lintegral_ae $ by filter_upwards [h] assume a h, le_of_eq h)\n  (lintegral_le_lintegral_ae $ by filter_upwards [h] assume a h, le_of_eq h.symm)\n\nlemma lintegral_eq_zero_iff {f : \u03b1 \u2192 ennreal} (hf : measurable f) :\n  lintegral f = 0 \u2194 (\u2200\u2098 a, f a = 0) :=\nbegin\n  refine iff.intro (assume h, _) (assume h, _),\n  { have : \u2200n:\u2115, \u2200\u2098 a, f a < n\u207b\u00b9,\n    { assume n,\n      have : is_measurable {a : \u03b1 | f a \u2265 n\u207b\u00b9 },\n      { exact hf _ (is_measurable_of_is_closed $ is_closed_ge' _) },\n      have : (n : ennreal)\u207b\u00b9 * volume {a | f a \u2265 n\u207b\u00b9 } = 0,\n      { rw [\u2190 simple_func.restrict_const_integral _ _ this, \u2190 le_zero_iff_eq,\n          \u2190 simple_func.lintegral_eq_integral],\n        refine le_trans (lintegral_le_lintegral _ _ _) (le_of_eq h),\n        assume a, by_cases h : (n : ennreal)\u207b\u00b9 \u2264 f a; simp [h, (\u2265), this] },\n      rw [ennreal.mul_eq_zero, ennreal.inv_eq_zero] at this,\n      simpa [ennreal.nat_ne_top, all_ae_iff] using this },\n    filter_upwards [all_ae_all_iff.2 this],\n    dsimp,\n    assume a ha,\n    by_contradiction h,\n    rcases ennreal.exists_inv_nat_lt h with \u27e8n, hn\u27e9,\n    exact (lt_irrefl _ $ lt_trans hn $ ha n).elim },\n  { calc lintegral f = lintegral (\u03bba:\u03b1, 0) : lintegral_congr_ae h\n      ... = 0 : lintegral_zero }\nend\n\nsection\nopen encodable\n\n/-- Monotone convergence for a suprema over a directed family and indexed by an encodable type -/\ntheorem lintegral_supr_directed [encodable \u03b2] {f : \u03b2 \u2192 \u03b1 \u2192 ennreal}\n  (hf : \u2200b, measurable (f b)) (h_directed : directed (\u2264) f) :\n  (\u222b\u207b a, \u2a06b, f b a) = (\u2a06b, \u222b\u207b a, f b a) :=\nbegin\n  by_cases h\u03b2 : \u00ac nonempty \u03b2,\n  { have : \u2200f : \u03b2 \u2192 ennreal, (\u2a06(b : \u03b2), f b) = 0 :=\n      assume f, supr_eq_bot.2 (assume b, (h\u03b2 \u27e8b\u27e9).elim),\n    simp [this] },\n  cases of_not_not h\u03b2 with b,\n  haveI i\u03b2 : inhabited \u03b2 := \u27e8b\u27e9, clear h\u03b2 b,\n  have : \u2200a, (\u2a06 b, f b a) = (\u2a06 n, f (sequence_of_directed (\u2264) f h_directed n) a),\n  { assume a,\n    refine le_antisymm (supr_le $ assume b, _) (supr_le $ assume n, le_supr (\u03bbn, f n a) _),\n    exact le_supr_of_le (encode b + 1) (le_sequence_of_directed f h_directed b a) },\n  calc (\u222b\u207b a, \u2a06 b, f b a) = (\u222b\u207b a, \u2a06 n, f (sequence_of_directed (\u2264) f h_directed n) a) :\n      by simp only [this]\n    ... = (\u2a06 n, \u222b\u207b a, f (sequence_of_directed (\u2264) f h_directed n) a) :\n      lintegral_supr (assume n, hf _) (monotone_sequence_of_directed f h_directed)\n    ... = (\u2a06 b, \u222b\u207b a, f b a) :\n    begin\n      refine le_antisymm (supr_le $ assume n, _) (supr_le $ assume b, _),\n      { exact le_supr (\u03bbb, lintegral (f b)) _ },\n      { exact le_supr_of_le (encode b + 1)\n          (lintegral_le_lintegral _ _ $ le_sequence_of_directed f h_directed b) }\n    end\nend\n\nend\n\nlemma lintegral_tsum [encodable \u03b2] {f : \u03b2 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200i, measurable (f i)) :\n  (\u222b\u207b a, \u2211 i, f i a) = (\u2211 i, \u222b\u207b a, f i a) :=\nbegin\n  simp only [ennreal.tsum_eq_supr_sum],\n  rw [lintegral_supr_directed],\n  { simp [lintegral_finset_sum _ hf] },\n  { assume b, exact measurable_finset_sum _ hf },\n  { assume s t,\n    use [s \u222a t],\n    split,\n    exact assume a, finset.sum_le_sum_of_subset (finset.subset_union_left _ _),\n    exact assume a, finset.sum_le_sum_of_subset (finset.subset_union_right _ _) }\nend\n\nend lintegral\n\nnamespace measure\n\ndef integral [measurable_space \u03b1] (m : measure \u03b1) (f : \u03b1 \u2192 ennreal) : ennreal :=\n@lintegral \u03b1 { \u03bc := m } f\n\nvariables [measurable_space \u03b1] {m : measure \u03b1}\n\n@[simp] lemma integral_zero : m.integral (\u03bba, 0) = 0 := @lintegral_zero \u03b1 { \u03bc := m }\n\nlemma integral_map [measurable_space \u03b2] {f : \u03b2 \u2192 ennreal} {g : \u03b1 \u2192 \u03b2}\n  (hf : measurable f) (hg : measurable g) : (map g m).integral f = m.integral (f \u2218 g) :=\nbegin\n  rw [integral, integral, lintegral_eq_supr_eapprox_integral, lintegral_eq_supr_eapprox_integral],\n  { congr, funext n, symmetry,\n    apply simple_func.integral_map,\n    { exact hg },\n    { assume a, exact congr_fun (simple_func.eapprox_comp hf hg) a },\n    { assume s hs, exact map_apply hg hs } },\n  exact hg.comp hf,\n  assumption\nend\n\nlemma integral_dirac (a : \u03b1) {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (dirac a).integral f = f a :=\nhave \u2200f:\u03b1 \u2192\u209b ennreal, @simple_func.integral \u03b1 {\u03bc := dirac a} f = f a,\nbegin\n  assume f,\n  have : \u2200r, @volume \u03b1 { \u03bc := dirac a } (\u21d1f \u207b\u00b9' {r}) = \u2a06 h : f a = r, 1,\n  { assume r,\n    transitivity,\n    apply dirac_apply,\n    apply simple_func.measurable_sn,\n    refine supr_congr_Prop _ _; simp },\n  transitivity,\n  apply finset.sum_eq_single (f a),\n  { assume b hb h, simp [this, ne.symm h], },\n  { assume h, simp at h, exact (h a rfl).elim },\n  { rw [this], simp }\nend,\nbegin\n  rw [integral, lintegral_eq_supr_eapprox_integral],\n  { simp [this, simple_func.supr_eapprox_apply f hf] },\n  assumption\nend\n\ndef with_density (m : measure \u03b1) (f : \u03b1 \u2192 ennreal) : measure \u03b1 :=\nif hf : measurable f then\n  measure.of_measurable (\u03bbs hs, m.integral (\u03bba, \u2a06(h : a \u2208 s), f a))\n    (by simp)\n    begin\n      assume s hs hd,\n      have : \u2200a, (\u2a06 (h : a \u2208 \u22c3i, s i), f a) = (\u2211i, (\u2a06 (h : a \u2208 s i), f a)),\n      { assume a,\n        by_cases ha : \u2203j, a \u2208 s j,\n        { rcases ha with \u27e8j, haj\u27e9,\n          have : \u2200i, a \u2208 s i \u2194 j = i := assume i,\n            iff.intro\n              (assume hai, by_contradiction $ assume hij, hd j i hij \u27e8haj, hai\u27e9)\n              (by rintros rfl; assumption),\n          simp [this, ennreal.tsum_supr_eq] },\n        { have : \u2200i, \u00ac a \u2208 s i, { simpa using ha },\n          simp [this] } },\n      simp only [this],\n      apply lintegral_tsum,\n      { assume i,\n        simp [supr_eq_if],\n        exact measurable.if (hs i) hf measurable_const }\n    end\nelse 0\n\nlemma with_density_apply {m : measure \u03b1} {f : \u03b1 \u2192 ennreal} {s : set \u03b1}\n  (hf : measurable f) (hs : is_measurable s) :\n  m.with_density f s = m.integral (\u03bba, \u2a06(h : a \u2208 s), f a) :=\nby rw [with_density, dif_pos hf]; exact measure.of_measurable_apply s hs\n\nend measure\n\nend measure_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/measure_theory/integration.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.39194187341432696}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.ring_theory.power_series.basic\nimport Mathlib.data.nat.parity\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Definition of well-known power series\n\nIn this file we define the following power series:\n\n* `power_series.inv_units_sub`: given `u : units R`, this is the series for `1 / (u - x)`.\n  It is given by `\u2211 n, x ^ n /\u209a u ^ (n + 1)`.\n\n* `power_series.sin`, `power_series.cos`, `power_series.exp` : power series for sin, cosine, and\n  exponential functions.\n-/\n\nnamespace power_series\n\n\n/-- The power series for `1 / (u - x)`. -/\ndef inv_units_sub {R : Type u_1} [ring R] (u : units R) : power_series R :=\n  mk fun (n : \u2115) => 1 /\u209a u ^ (n + 1)\n\n@[simp] theorem coeff_inv_units_sub {R : Type u_1} [ring R] (u : units R) (n : \u2115) :\n    coe_fn (coeff R n) (inv_units_sub u) = 1 /\u209a u ^ (n + 1) :=\n  coeff_mk n fun (n : \u2115) => 1 /\u209a u ^ (n + 1)\n\n@[simp] theorem constant_coeff_inv_units_sub {R : Type u_1} [ring R] (u : units R) :\n    coe_fn (constant_coeff R) (inv_units_sub u) = 1 /\u209a u :=\n  sorry\n\n@[simp] theorem inv_units_sub_mul_X {R : Type u_1} [ring R] (u : units R) :\n    inv_units_sub u * X = inv_units_sub u * coe_fn (C R) \u2191u - 1 :=\n  sorry\n\n@[simp] theorem inv_units_sub_mul_sub {R : Type u_1} [ring R] (u : units R) :\n    inv_units_sub u * (coe_fn (C R) \u2191u - X) = 1 :=\n  sorry\n\ntheorem map_inv_units_sub {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2192+* S)\n    (u : units R) : coe_fn (map f) (inv_units_sub u) = inv_units_sub (coe_fn (units.map \u2191f) u) :=\n  sorry\n\n/-- Power series for the exponential function at zero. -/\ndef exp (A : Type u_1) [ring A] [algebra \u211a A] : power_series A :=\n  mk fun (n : \u2115) => coe_fn (algebra_map \u211a A) (1 / \u2191(nat.factorial n))\n\n/-- Power series for the sine function at zero. -/\ndef sin (A : Type u_1) [ring A] [algebra \u211a A] : power_series A :=\n  mk\n    fun (n : \u2115) =>\n      ite (even n) 0 (coe_fn (algebra_map \u211a A) ((-1) ^ (n / bit0 1) / \u2191(nat.factorial n)))\n\n/-- Power series for the cosine function at zero. -/\ndef cos (A : Type u_1) [ring A] [algebra \u211a A] : power_series A :=\n  mk\n    fun (n : \u2115) =>\n      ite (even n) (coe_fn (algebra_map \u211a A) ((-1) ^ (n / bit0 1) / \u2191(nat.factorial n))) 0\n\n@[simp] theorem coeff_exp {A : Type u_1} [ring A] [algebra \u211a A] (n : \u2115) :\n    coe_fn (coeff A n) (exp A) = coe_fn (algebra_map \u211a A) (1 / \u2191(nat.factorial n)) :=\n  coeff_mk n fun (n : \u2115) => coe_fn (algebra_map \u211a A) (1 / \u2191(nat.factorial n))\n\n@[simp] theorem map_exp {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra \u211a A]\n    [algebra \u211a A'] (f : A \u2192+* A') : coe_fn (map f) (exp A) = exp A' :=\n  sorry\n\n@[simp] theorem map_sin {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra \u211a A]\n    [algebra \u211a A'] (f : A \u2192+* A') : coe_fn (map f) (sin A) = sin A' :=\n  sorry\n\n@[simp] theorem map_cos {A : Type u_1} {A' : Type u_2} [ring A] [ring A'] [algebra \u211a A]\n    [algebra \u211a A'] (f : A \u2192+* A') : coe_fn (map f) (cos A) = cos A' :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/power_series/well_known_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3918530455441488}}
{"text": "/-\nCopyright (c) 2020 Fr\u00e9d\u00e9ric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fr\u00e9d\u00e9ric Dupuis\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.pi\nimport Mathlib.algebra.ordered_pi\nimport Mathlib.algebra.module.prod\nimport Mathlib.algebra.ordered_field\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 \n\nnamespace Mathlib\n\n/-!\n# Ordered semimodules\n\nIn this file we define\n\n* `ordered_semimodule R M` : an ordered additive commutative monoid `M` is an `ordered_semimodule`\n  over an `ordered_semiring` `R` if the scalar product respects the order relation on the\n  monoid and on the ring. There is a correspondence between this structure and convex cones,\n  which is proven in `analysis/convex/cone.lean`.\n\n## Implementation notes\n\n* We choose to define `ordered_semimodule` as a `Prop`-valued mixin, so that it can be\n  used for both modules and algebras\n  (the axioms for an \"ordered algebra\" are exactly that the algebra is ordered as a module).\n* To get ordered modules and ordered vector spaces, it suffices to the replace the\n  `order_add_comm_monoid` and the `ordered_semiring` as desired.\n\n## References\n\n* https://en.wikipedia.org/wiki/Ordered_vector_space\n\n## Tags\n\nordered semimodule, ordered module, ordered vector space\n-/\n\n/--\nAn ordered semimodule is an ordered additive commutative monoid\nwith a partial order in which the scalar multiplication is compatible with the order.\n-/\nclass ordered_semimodule (R : Type u_1) (M : Type u_2) [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M]\n    where\n  smul_lt_smul_of_pos : \u2200 {a b : M} {c : R}, a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b\n  lt_of_smul_lt_smul_of_pos : \u2200 {a b : M} {c : R}, c \u2022 a < c \u2022 b \u2192 0 < c \u2192 a < b\n\ntheorem smul_lt_smul_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R} :\n    a < b \u2192 0 < c \u2192 c \u2022 a < c \u2022 b :=\n  ordered_semimodule.smul_lt_smul_of_pos\n\ntheorem smul_le_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}\n    (h\u2081 : a \u2264 b) (h\u2082 : 0 \u2264 c) : c \u2022 a \u2264 c \u2022 b :=\n  sorry\n\ntheorem eq_of_smul_eq_smul_of_pos_of_le {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}\n    (h\u2081 : c \u2022 a = c \u2022 b) (hc : 0 < c) (hle : a \u2264 b) : a = b :=\n  or.resolve_left (has_le.le.lt_or_eq hle)\n    fun (hlt : a < b) => has_lt.lt.ne (smul_lt_smul_of_pos hlt hc) h\u2081\n\ntheorem lt_of_smul_lt_smul_of_nonneg {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}\n    (h : c \u2022 a < c \u2022 b) (hc : 0 \u2264 c) : a < b :=\n  sorry\n\ntheorem smul_lt_smul_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {b : M} {c : R}\n    (hc : 0 < c) : c \u2022 a < c \u2022 b \u2194 a < b :=\n  { mp := fun (h : c \u2022 a < c \u2022 b) => lt_of_smul_lt_smul_of_nonneg h (has_lt.lt.le hc),\n    mpr := fun (h : a < b) => smul_lt_smul_of_pos h hc }\n\ntheorem smul_pos_iff_of_pos {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] {a : M} {c : R}\n    (hc : 0 < c) : 0 < c \u2022 a \u2194 0 < a :=\n  iff.trans\n    (eq.mpr (id (Eq._oldrec (Eq.refl (0 < c \u2022 a \u2194 c \u2022 0 < c \u2022 a)) (smul_zero c)))\n      (iff.refl (0 < c \u2022 a)))\n    (smul_lt_smul_iff_of_pos hc)\n\n/-- If `R` is a linear ordered semifield, then it suffices to verify only the first axiom of\n`ordered_semimodule`. Moreover, it suffices to verify that `a < b` and `0 < c` imply\n`c \u2022 a \u2264 c \u2022 b`. We have no semifields in `mathlib`, so we use the assumption `\u2200 c \u2260 0, is_unit c`\ninstead. -/\ntheorem ordered_semimodule.mk'' {R : Type u_1} {M : Type u_2} [linear_ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] (hR : \u2200 {c : R}, c \u2260 0 \u2192 is_unit c)\n    (hlt : \u2200 {a b : M} {c : R}, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b) : ordered_semimodule R M :=\n  sorry\n\n/-- If `R` is a linear ordered field, then it suffices to verify only the first axiom of\n`ordered_semimodule`. -/\ntheorem ordered_semimodule.mk' {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_monoid M] [semimodule k M]\n    (hlt : \u2200 {a b : M} {c : k}, a < b \u2192 0 < c \u2192 c \u2022 a \u2264 c \u2022 b) : ordered_semimodule k M :=\n  ordered_semimodule.mk'' (fun (c : k) (hc : c \u2260 0) => is_unit.mk0 c hc) hlt\n\nprotected instance linear_ordered_semiring.to_ordered_semimodule {R : Type u_1}\n    [linear_ordered_semiring R] : ordered_semimodule R R :=\n  ordered_semimodule.mk ordered_semiring.mul_lt_mul_of_pos_left\n    fun (_x _x_1 _x_2 : R) (h : _x_2 \u2022 _x < _x_2 \u2022 _x_1) (hc : 0 < _x_2) =>\n      lt_of_mul_lt_mul_left h (has_lt.lt.le hc)\n\ntheorem smul_le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}\n    (hc : 0 < c) : c \u2022 a \u2264 c \u2022 b \u2194 a \u2264 b :=\n  sorry\n\ntheorem smul_le_smul_iff_of_neg {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}\n    (hc : c < 0) : c \u2022 a \u2264 c \u2022 b \u2194 b \u2264 a :=\n  sorry\n\ntheorem smul_lt_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}\n    (hc : 0 < c) : c \u2022 a < b \u2194 a < c\u207b\u00b9 \u2022 b :=\n  sorry\n\ntheorem smul_le_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}\n    (hc : 0 < c) : c \u2022 a \u2264 b \u2194 a \u2264 c\u207b\u00b9 \u2022 b :=\n  sorry\n\ntheorem le_smul_iff_of_pos {k : Type u_1} {M : Type u_2} [linear_ordered_field k]\n    [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] {a : M} {b : M} {c : k}\n    (hc : 0 < c) : a \u2264 c \u2022 b \u2194 c\u207b\u00b9 \u2022 a \u2264 b :=\n  sorry\n\nprotected instance prod.ordered_semimodule {k : Type u_1} {M : Type u_2} {N : Type u_3}\n    [linear_ordered_field k] [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M]\n    [ordered_add_comm_group N] [semimodule k N] [ordered_semimodule k N] :\n    ordered_semimodule k (M \u00d7 N) :=\n  ordered_semimodule.mk'\n    fun (v u : M \u00d7 N) (c : k) (h : v < u) (hc : 0 < c) =>\n      { left := smul_le_smul_of_nonneg (and.left (and.left h)) (has_lt.lt.le hc),\n        right := smul_le_smul_of_nonneg (and.right (and.left h)) (has_lt.lt.le hc) }\n\nprotected instance pi.ordered_semimodule {k : Type u_1} [linear_ordered_field k] {\u03b9 : Type u_2}\n    {M : \u03b9 \u2192 Type u_3} [(i : \u03b9) \u2192 ordered_add_comm_group (M i)] [(i : \u03b9) \u2192 semimodule k (M i)]\n    [\u2200 (i : \u03b9), ordered_semimodule k (M i)] : ordered_semimodule k ((i : \u03b9) \u2192 M i) :=\n  ordered_semimodule.mk'\n    fun (v u : (i : \u03b9) \u2192 M i) (c : k) (h : v < u) (hc : 0 < c) (i : \u03b9) =>\n      id (smul_le_smul_of_nonneg (has_lt.lt.le h i) (has_lt.lt.le hc))\n\n-- Sometimes Lean fails to apply the dependent version to non-dependent functions,\n\n-- so we define another instance\n\nprotected instance pi.ordered_semimodule' {k : Type u_1} [linear_ordered_field k] {\u03b9 : Type u_2}\n    {M : Type u_3} [ordered_add_comm_group M] [semimodule k M] [ordered_semimodule k M] :\n    ordered_semimodule k (\u03b9 \u2192 M) :=\n  pi.ordered_semimodule\n\nprotected instance order_dual.has_scalar {R : Type u_1} {M : Type u_2} [semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] : has_scalar R (order_dual M) :=\n  has_scalar.mk has_scalar.smul\n\nprotected instance order_dual.mul_action {R : Type u_1} {M : Type u_2} [semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] : mul_action R (order_dual M) :=\n  mul_action.mk sorry sorry\n\nprotected instance order_dual.distrib_mul_action {R : Type u_1} {M : Type u_2} [semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] : distrib_mul_action R (order_dual M) :=\n  distrib_mul_action.mk sorry sorry\n\nprotected instance order_dual.semimodule {R : Type u_1} {M : Type u_2} [semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] : semimodule R (order_dual M) :=\n  semimodule.mk sorry sorry\n\nprotected instance order_dual.ordered_semimodule {R : Type u_1} {M : Type u_2} [ordered_semiring R]\n    [ordered_add_comm_monoid M] [semimodule R M] [ordered_semimodule R M] :\n    ordered_semimodule R (order_dual M) :=\n  ordered_semimodule.mk (fun (a b : order_dual M) => ordered_semimodule.smul_lt_smul_of_pos)\n    fun (a b : order_dual M) => ordered_semimodule.lt_of_smul_lt_smul_of_pos\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/module/ordered_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3918530455441487}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.limits.preserves.shapes.binary_products\nimport category_theory.limits.preserves.shapes.products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.finite_products\nimport category_theory.pempty\nimport data.equiv.fin\n\n/-!\n# Constructing finite products from binary products and terminal.\n\nIf a category has binary products and a terminal object then it has finite products.\nIf a functor preserves binary products and the terminal object then it preserves finite products.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\nuniverses v u u'\n\nnoncomputable theory\nopen category_theory category_theory.category category_theory.limits\nnamespace category_theory\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{v} C]\nvariables {D : Type u'} [category.{v} D]\n\n/--\nGiven `n+1` objects of `C`, a fan for the last `n` with point `c\u2081.X` and a binary fan on `c\u2081.X` and\n`f 0`, we can build a fan for all `n+1`.\n\nIn `extend_fan_is_limit` we show that if the two given fans are limits, then this fan is also a\nlimit.\n-/\n@[simps {rhs_md := semireducible}]\ndef extend_fan {n : \u2115} {f : ulift (fin (n+1)) \u2192 C}\n  (c\u2081 : fan (\u03bb (i : ulift (fin n)), f \u27e8i.down.succ\u27e9))\n  (c\u2082 : binary_fan (f \u27e80\u27e9) c\u2081.X) :\n  fan f :=\nfan.mk c\u2082.X\nbegin\n  rintro \u27e8i\u27e9,\n  revert i,\n  refine fin.cases _ _,\n  { apply c\u2082.fst },\n  { intro i,\n    apply c\u2082.snd \u226b c\u2081.\u03c0.app (ulift.up i) },\nend\n\n/--\nShow that if the two given fans in `extend_fan` are limits, then the constructed fan is also a\nlimit.\n-/\ndef extend_fan_is_limit {n : \u2115} (f : ulift (fin (n+1)) \u2192 C)\n  {c\u2081 : fan (\u03bb (i : ulift (fin n)), f \u27e8i.down.succ\u27e9)} {c\u2082 : binary_fan (f \u27e80\u27e9) c\u2081.X}\n  (t\u2081 : is_limit c\u2081) (t\u2082 : is_limit c\u2082) :\n  is_limit (extend_fan c\u2081 c\u2082) :=\n{ lift := \u03bb s,\n  begin\n    apply (binary_fan.is_limit.lift' t\u2082 (s.\u03c0.app \u27e80\u27e9) _).1,\n    apply t\u2081.lift \u27e8_, discrete.nat_trans (\u03bb i, s.\u03c0.app \u27e8i.down.succ\u27e9)\u27e9\n  end,\n  fac' := \u03bb s,\n  begin\n    rintro \u27e8j\u27e9,\n    apply fin.induction_on j,\n    { apply (binary_fan.is_limit.lift' t\u2082 _ _).2.1 },\n    { rintro i -,\n      dsimp only [extend_fan_\u03c0_app],\n      rw [fin.cases_succ, \u2190 assoc, (binary_fan.is_limit.lift' t\u2082 _ _).2.2, t\u2081.fac],\n      refl }\n  end,\n  uniq' := \u03bb s m w,\n  begin\n    apply binary_fan.is_limit.hom_ext t\u2082,\n    { rw (binary_fan.is_limit.lift' t\u2082 _ _).2.1,\n      apply w \u27e80\u27e9 },\n    { rw (binary_fan.is_limit.lift' t\u2082 _ _).2.2,\n      apply t\u2081.uniq \u27e8_, _\u27e9,\n      rintro \u27e8j\u27e9,\n      rw assoc,\n      dsimp only [discrete.nat_trans_app],\n      rw \u2190 w \u27e8j.succ\u27e9,\n      dsimp only [extend_fan_\u03c0_app],\n      rw fin.cases_succ }\n  end }\n\nsection\nvariables [has_binary_products.{v} C] [has_terminal C]\n\n/--\nIf `C` has a terminal object and binary products, then it has a product for objects indexed by\n`ulift (fin n)`.\nThis is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_product_ulift_fin :\n  \u03a0 (n : \u2115) (f : ulift.{v} (fin n) \u2192 C), has_product f\n| 0 := \u03bb f,\n  begin\n    letI : has_limits_of_shape (discrete (ulift.{v} (fin 0))) C :=\n      has_limits_of_shape_of_equivalence\n        (discrete.equivalence.{v} (equiv.ulift.trans fin_zero_equiv').symm),\n    apply_instance,\n  end\n| (n+1) := \u03bb f,\n  begin\n    haveI := has_product_ulift_fin n,\n    apply has_limit.mk \u27e8_, extend_fan_is_limit f (limit.is_limit.{v} _) (limit.is_limit _)\u27e9,\n  end\n\n/--\nIf `C` has a terminal object and binary products, then it has limits of shape\n`discrete (ulift (fin n))` for any `n : \u2115`.\nThis is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_limits_of_shape_ulift_fin (n : \u2115) :\n  has_limits_of_shape (discrete (ulift.{v} (fin n))) C :=\n{ has_limit := \u03bb K,\nbegin\n  letI := has_product_ulift_fin n K.obj,\n  let : discrete.functor K.obj \u2245 K := discrete.nat_iso (\u03bb i, iso.refl _),\n  apply has_limit_of_iso this,\nend }\n\n/-- If `C` has a terminal object and binary products, then it has finite products. -/\nlemma has_finite_products_of_has_binary_and_terminal : has_finite_products C :=\n\u27e8\u03bb J \ud835\udca5\u2081 \ud835\udca5\u2082, begin\n  resetI,\n  let e := fintype.equiv_fin J,\n  apply has_limits_of_shape_of_equivalence (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\n  refine has_limits_of_shape_ulift_fin (fintype.card J),\nend\u27e9\n\nend\n\nsection preserves\nvariables (F : C \u2964 D)\nvariables [preserves_limits_of_shape (discrete walking_pair) F]\nvariables [preserves_limits_of_shape (discrete pempty) F]\nvariables [has_finite_products.{v} C]\n\n/--\nIf `F` preserves the terminal object and binary products, then it preserves products indexed by\n`ulift (fin n)` for any `n`.\n-/\nnoncomputable def preserves_fin_of_preserves_binary_and_terminal  :\n  \u03a0 (n : \u2115) (f : ulift (fin n) \u2192 C), preserves_limit (discrete.functor f) F\n| 0 := \u03bb f,\n  begin\n    letI : preserves_limits_of_shape (discrete (ulift (fin 0))) F :=\n      preserves_limits_of_shape_of_equiv\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm) _,\n    apply_instance,\n  end\n| (n+1) :=\n  begin\n    haveI := preserves_fin_of_preserves_binary_and_terminal n,\n    intro f,\n    refine preserves_limit_of_preserves_limit_cone\n      (extend_fan_is_limit f (limit.is_limit.{v} _) (limit.is_limit _)) _,\n    apply (is_limit_map_cone_fan_mk_equiv _ _ _).symm _,\n    let := extend_fan_is_limit (\u03bb i, F.obj (f i))\n              (is_limit_of_has_product_of_preserves_limit F _)\n              (is_limit_of_has_binary_product_of_preserves_limit F _ _),\n    refine is_limit.of_iso_limit this _,\n    apply cones.ext _ _,\n    apply iso.refl _,\n    rintro \u27e8j\u27e9,\n    apply fin.induction_on j,\n    { apply (category.id_comp _).symm },\n    { rintro i -,\n      dsimp only [extend_fan_\u03c0_app, iso.refl_hom, fan.mk_\u03c0_app],\n      rw [fin.cases_succ, fin.cases_succ],\n      change F.map _ \u226b _ = \ud835\udfd9 _ \u226b _,\n      rw [id_comp, \u2190F.map_comp],\n      refl }\n  end\n\n/--\nIf `F` preserves the terminal object and binary products, then it preserves limits of shape\n`discrete (ulift (fin n))`.\n-/\ndef preserves_ulift_fin_of_preserves_binary_and_terminal (n : \u2115) :\n  preserves_limits_of_shape (discrete (ulift (fin n))) F :=\n{ preserves_limit := \u03bb K,\n  begin\n    let : discrete.functor K.obj \u2245 K := discrete.nat_iso (\u03bb i, iso.refl _),\n    haveI := preserves_fin_of_preserves_binary_and_terminal F n K.obj,\n    apply preserves_limit_of_iso_diagram F this,\n  end }\n\n/-- If `F` preserves the terminal object and binary products then it preserves finite products. -/\ndef preserves_finite_products_of_preserves_binary_and_terminal\n  (J : Type v) [fintype J] :\n  preserves_limits_of_shape.{v} (discrete J) F :=\nbegin\n  classical,\n  let e := fintype.equiv_fin J,\n  haveI := preserves_ulift_fin_of_preserves_binary_and_terminal F (fintype.card J),\n  apply preserves_limits_of_shape_of_equiv (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\nend\n\nend preserves\n\n/--\nGiven `n+1` objects of `C`, a cofan for the last `n` with point `c\u2081.X`\nand a binary cofan on `c\u2081.X` and `f 0`, we can build a cofan for all `n+1`.\n\nIn `extend_cofan_is_colimit` we show that if the two given cofans are colimits,\nthen this cofan is also a colimit.\n-/\n@[simps {rhs_md := semireducible}]\ndef extend_cofan {n : \u2115} {f : ulift (fin (n+1)) \u2192 C}\n  (c\u2081 : cofan (\u03bb (i : ulift (fin n)), f \u27e8i.down.succ\u27e9))\n  (c\u2082 : binary_cofan (f \u27e80\u27e9) c\u2081.X) :\n  cofan f :=\ncofan.mk c\u2082.X\nbegin\n  rintro \u27e8i\u27e9,\n  revert i,\n  refine fin.cases _ _,\n  { apply c\u2082.inl },\n  { intro i,\n    apply c\u2081.\u03b9.app (ulift.up i) \u226b c\u2082.inr },\nend\n\n/--\nShow that if the two given cofans in `extend_cofan` are colimits,\nthen the constructed cofan is also a colimit.\n-/\ndef extend_cofan_is_colimit {n : \u2115} (f : ulift (fin (n+1)) \u2192 C)\n  {c\u2081 : cofan (\u03bb (i : ulift (fin n)), f \u27e8i.down.succ\u27e9)} {c\u2082 : binary_cofan (f \u27e80\u27e9) c\u2081.X}\n  (t\u2081 : is_colimit c\u2081) (t\u2082 : is_colimit c\u2082) :\n  is_colimit (extend_cofan c\u2081 c\u2082) :=\n{ desc := \u03bb s,\n  begin\n    apply (binary_cofan.is_colimit.desc' t\u2082 (s.\u03b9.app \u27e80\u27e9) _).1,\n    apply t\u2081.desc \u27e8_, discrete.nat_trans (\u03bb i, s.\u03b9.app \u27e8i.down.succ\u27e9)\u27e9\n  end,\n  fac' := \u03bb s,\n  begin\n    rintro \u27e8j\u27e9,\n    apply fin.induction_on j,\n    { apply (binary_cofan.is_colimit.desc' t\u2082 _ _).2.1 },\n    { rintro i -,\n      dsimp only [extend_cofan_\u03b9_app],\n      rw [fin.cases_succ, assoc, (binary_cofan.is_colimit.desc' t\u2082 _ _).2.2, t\u2081.fac],\n      refl }\n  end,\n  uniq' := \u03bb s m w,\n  begin\n    apply binary_cofan.is_colimit.hom_ext t\u2082,\n    { rw (binary_cofan.is_colimit.desc' t\u2082 _ _).2.1,\n      apply w \u27e80\u27e9 },\n    { rw (binary_cofan.is_colimit.desc' t\u2082 _ _).2.2,\n      apply t\u2081.uniq \u27e8_, _\u27e9,\n      rintro \u27e8j\u27e9,\n      dsimp only [discrete.nat_trans_app],\n      rw \u2190 w \u27e8j.succ\u27e9,\n      dsimp only [extend_cofan_\u03b9_app],\n      rw [fin.cases_succ, assoc], }\n  end }\n\nsection\nvariables [has_binary_coproducts.{v} C] [has_initial C]\n\n/--\nIf `C` has an initial object and binary coproducts, then it has a coproduct for objects indexed by\n`ulift (fin n)`.\nThis is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_coproduct_ulift_fin :\n  \u03a0 (n : \u2115) (f : ulift.{v} (fin n) \u2192 C), has_coproduct f\n| 0 := \u03bb f,\n  begin\n    letI : has_colimits_of_shape (discrete (ulift.{v} (fin 0))) C :=\n      has_colimits_of_shape_of_equivalence\n        (discrete.equivalence.{v} (equiv.ulift.trans fin_zero_equiv').symm),\n    apply_instance,\n  end\n| (n+1) := \u03bb f,\n  begin\n    haveI := has_coproduct_ulift_fin n,\n    apply has_colimit.mk\n      \u27e8_, extend_cofan_is_colimit f (colimit.is_colimit.{v} _) (colimit.is_colimit _)\u27e9,\n  end\n\n/--\nIf `C` has an initial object and binary coproducts, then it has colimits of shape\n`discrete (ulift (fin n))` for any `n : \u2115`.\nThis is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate lemma has_colimits_of_shape_ulift_fin (n : \u2115) :\n  has_colimits_of_shape (discrete (ulift.{v} (fin n))) C :=\n{ has_colimit := \u03bb K,\nbegin\n  letI := has_coproduct_ulift_fin n K.obj,\n  let : K \u2245 discrete.functor K.obj := discrete.nat_iso (\u03bb i, iso.refl _),\n  apply has_colimit_of_iso this,\nend }\n\n/-- If `C` has an initial object and binary coproducts, then it has finite coproducts. -/\nlemma has_finite_coproducts_of_has_binary_and_terminal : has_finite_coproducts C :=\n\u27e8\u03bb J \ud835\udca5\u2081 \ud835\udca5\u2082, begin\n  resetI,\n  let e := fintype.equiv_fin J,\n  apply has_colimits_of_shape_of_equivalence (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\n  refine has_colimits_of_shape_ulift_fin (fintype.card J),\nend\u27e9\n\nend\n\nsection preserves\nvariables (F : C \u2964 D)\nvariables [preserves_colimits_of_shape (discrete walking_pair) F]\nvariables [preserves_colimits_of_shape (discrete pempty) F]\nvariables [has_finite_coproducts.{v} C]\n\n/--\nIf `F` preserves the initial object and binary coproducts, then it preserves products indexed by\n`ulift (fin n)` for any `n`.\n-/\nnoncomputable def preserves_fin_of_preserves_binary_and_initial  :\n  \u03a0 (n : \u2115) (f : ulift (fin n) \u2192 C), preserves_colimit (discrete.functor f) F\n| 0 := \u03bb f,\n  begin\n    letI : preserves_colimits_of_shape (discrete (ulift (fin 0))) F :=\n      preserves_colimits_of_shape_of_equiv\n        (discrete.equivalence (equiv.ulift.trans fin_zero_equiv').symm) _,\n    apply_instance,\n  end\n| (n+1) :=\n  begin\n    haveI := preserves_fin_of_preserves_binary_and_initial n,\n    intro f,\n    refine preserves_colimit_of_preserves_colimit_cocone\n      (extend_cofan_is_colimit f (colimit.is_colimit.{v} _) (colimit.is_colimit _)) _,\n    apply (is_colimit_map_cocone_cofan_mk_equiv _ _ _).symm _,\n    let := extend_cofan_is_colimit (\u03bb i, F.obj (f i))\n              (is_colimit_of_has_coproduct_of_preserves_colimit F _)\n              (is_colimit_of_has_binary_coproduct_of_preserves_colimit F _ _),\n    refine is_colimit.of_iso_colimit this _,\n    apply cocones.ext _ _,\n    apply iso.refl _,\n    rintro \u27e8j\u27e9,\n    apply fin.induction_on j,\n    { apply category.comp_id },\n    { rintro i -,\n      dsimp only [extend_cofan_\u03b9_app, iso.refl_hom, cofan.mk_\u03b9_app],\n      rw [fin.cases_succ, fin.cases_succ],\n      erw [comp_id, \u2190F.map_comp],\n      refl, }\n  end\n\n/--\nIf `F` preserves the initial object and binary coproducts, then it preserves colimits of shape\n`discrete (ulift (fin n))`.\n-/\ndef preserves_ulift_fin_of_preserves_binary_and_initial (n : \u2115) :\n  preserves_colimits_of_shape (discrete (ulift (fin n))) F :=\n{ preserves_colimit := \u03bb K,\n  begin\n    let : discrete.functor K.obj \u2245 K := discrete.nat_iso (\u03bb i, iso.refl _),\n    haveI := preserves_fin_of_preserves_binary_and_initial F n K.obj,\n    apply preserves_colimit_of_iso_diagram F this,\n  end }\n\n/-- If `F` preserves the initial object and binary coproducts then it preserves finite products. -/\ndef preserves_finite_coproducts_of_preserves_binary_and_initial\n  (J : Type v) [fintype J] :\n  preserves_colimits_of_shape.{v} (discrete J) F :=\nbegin\n  classical,\n  let e := fintype.equiv_fin J,\n  haveI := preserves_ulift_fin_of_preserves_binary_and_initial F (fintype.card J),\n  apply preserves_colimits_of_shape_of_equiv (discrete.equivalence (e.trans equiv.ulift.symm)).symm,\nend\n\nend preserves\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/constructions/finite_products_of_binary_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3918299137517381}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jan-David Salchow, S\u00e9bastien Gou\u00ebzel, Jean Lo, Yury Kudryashov, Fr\u00e9d\u00e9ric Dupuis,\n  Heather Macbeth\n\n! This file was ported from Lean 3 source module topology.algebra.module.basic\n! leanprover-community/mathlib commit f430769b562e0cedef59ee1ed968d67e0e0c86ba\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.Algebra.Ring.Basic\nimport Mathbin.Topology.Algebra.MulAction\nimport Mathbin.Topology.Algebra.UniformGroup\nimport Mathbin.Topology.ContinuousFunction.Basic\nimport Mathbin.Topology.UniformSpace.UniformEmbedding\nimport Mathbin.Algebra.Algebra.Basic\nimport Mathbin.LinearAlgebra.Projection\nimport Mathbin.LinearAlgebra.Pi\n\n/-!\n# Theory of topological modules and continuous linear maps.\n\nWe use the class `has_continuous_smul` for topological (semi) modules and topological vector spaces.\n\nIn this file we define continuous (semi-)linear maps, as semilinear maps between topological\nmodules which are continuous. The set of continuous semilinear maps between the topological\n`R\u2081`-module `M` and `R\u2082`-module `M\u2082` with respect to the `ring_hom` `\u03c3` is denoted by `M \u2192SL[\u03c3] M\u2082`.\nPlain linear maps are denoted by `M \u2192L[R] M\u2082` and star-linear maps by `M \u2192L\u22c6[R] M\u2082`.\n\nThe corresponding notation for equivalences is `M \u2243SL[\u03c3] M\u2082`, `M \u2243L[R] M\u2082` and `M \u2243L\u22c6[R] M\u2082`.\n-/\n\n\nopen Filter\n\nopen LinearMap (ker range)\n\nopen Topology BigOperators Filter\n\nuniverse u v w u'\n\nsection\n\nvariable {R : Type _} {M : Type _} [Ring R] [TopologicalSpace R] [TopologicalSpace M]\n  [AddCommGroup M] [Module R M]\n\ntheorem ContinuousSMul.of_nhds_zero [TopologicalRing R] [TopologicalAddGroup M]\n    (hmul : Tendsto (fun p : R \u00d7 M => p.1 \u2022 p.2) (\ud835\udcdd 0 \u00d7\u1da0 \ud835\udcdd 0) (\ud835\udcdd 0))\n    (hmulleft : \u2200 m : M, Tendsto (fun a : R => a \u2022 m) (\ud835\udcdd 0) (\ud835\udcdd 0))\n    (hmulright : \u2200 a : R, Tendsto (fun m : M => a \u2022 m) (\ud835\udcdd 0) (\ud835\udcdd 0)) : ContinuousSMul R M :=\n  \u27e8by\n    rw [continuous_iff_continuousAt]\n    rintro \u27e8a\u2080, m\u2080\u27e9\n    have key :\n      \u2200 p : R \u00d7 M,\n        p.1 \u2022 p.2 = a\u2080 \u2022 m\u2080 + ((p.1 - a\u2080) \u2022 m\u2080 + a\u2080 \u2022 (p.2 - m\u2080) + (p.1 - a\u2080) \u2022 (p.2 - m\u2080)) :=\n      by\n      rintro \u27e8a, m\u27e9\n      simp [sub_smul, smul_sub]\n      abel\n    rw [funext key]\n    clear key\n    refine' tendsto_const_nhds.add (tendsto.add (tendsto.add _ _) _)\n    \u00b7 rw [sub_self, zero_smul]\n      apply (hmulleft m\u2080).comp\n      rw [show (fun p : R \u00d7 M => p.1 - a\u2080) = (fun a => a - a\u2080) \u2218 Prod.fst\n          by\n          ext\n          rfl,\n        nhds_prod_eq]\n      have : tendsto (fun a => a - a\u2080) (\ud835\udcdd a\u2080) (\ud835\udcdd 0) :=\n        by\n        rw [\u2190 sub_self a\u2080]\n        exact tendsto_id.sub tendsto_const_nhds\n      exact this.comp tendsto_fst\n    \u00b7 rw [sub_self, smul_zero]\n      apply (hmulright a\u2080).comp\n      rw [show (fun p : R \u00d7 M => p.2 - m\u2080) = (fun m => m - m\u2080) \u2218 Prod.snd\n          by\n          ext\n          rfl,\n        nhds_prod_eq]\n      have : tendsto (fun m => m - m\u2080) (\ud835\udcdd m\u2080) (\ud835\udcdd 0) :=\n        by\n        rw [\u2190 sub_self m\u2080]\n        exact tendsto_id.sub tendsto_const_nhds\n      exact this.comp tendsto_snd\n    \u00b7 rw [sub_self, zero_smul, nhds_prod_eq,\n        show\n          (fun p : R \u00d7 M => (p.fst - a\u2080) \u2022 (p.snd - m\u2080)) =\n            (fun p : R \u00d7 M => p.1 \u2022 p.2) \u2218 Prod.map (fun a => a - a\u2080) fun m => m - m\u2080\n          by\n          ext\n          rfl]\n      apply hmul.comp (tendsto.prod_map _ _) <;>\n        \u00b7 rw [\u2190 sub_self]\n          exact tendsto_id.sub tendsto_const_nhds\u27e9\n#align has_continuous_smul.of_nhds_zero ContinuousSMul.of_nhds_zero\n\nend\n\nsection\n\nvariable {R : Type _} {M : Type _} [Ring R] [TopologicalSpace R] [TopologicalSpace M]\n  [AddCommGroup M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M]\n\n/-- If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then\n`\u22a4` is the only submodule of `M` with a nonempty interior.\nThis is the case, e.g., if `R` is a nontrivially normed field. -/\ntheorem Submodule.eq_top_of_nonempty_interior' [NeBot (\ud835\udcdd[{ x : R | IsUnit x }] 0)]\n    (s : Submodule R M) (hs : (interior (s : Set M)).Nonempty) : s = \u22a4 :=\n  by\n  rcases hs with \u27e8y, hy\u27e9\n  refine' Submodule.eq_top_iff'.2 fun x => _\n  rw [mem_interior_iff_mem_nhds] at hy\n  have : tendsto (fun c : R => y + c \u2022 x) (\ud835\udcdd[{ x : R | IsUnit x }] 0) (\ud835\udcdd (y + (0 : R) \u2022 x)) :=\n    tendsto_const_nhds.add ((tendsto_nhdsWithin_of_tendsto_nhds tendsto_id).smul tendsto_const_nhds)\n  rw [zero_smul, add_zero] at this\n  obtain \u27e8_, hu : y + _ \u2022 _ \u2208 s, u, rfl\u27e9 :=\n    nonempty_of_mem (inter_mem (mem_map.1 (this hy)) self_mem_nhdsWithin)\n  have hy' : y \u2208 \u2191s := mem_of_mem_nhds hy\n  rwa [s.add_mem_iff_right hy', \u2190 Units.smul_def, s.smul_mem_iff' u] at hu\n#align submodule.eq_top_of_nonempty_interior' Submodule.eq_top_of_nonempty_interior'\n\nvariable (R M)\n\n/-- Let `R` be a topological ring such that zero is not an isolated point (e.g., a nontrivially\nnormed field, see `normed_field.punctured_nhds_ne_bot`). Let `M` be a nontrivial module over `R`\nsuch that `c \u2022 x = 0` implies `c = 0 \u2228 x = 0`. Then `M` has no isolated points. We formulate this\nusing `ne_bot (\ud835\udcdd[\u2260] x)`.\n\nThis lemma is not an instance because Lean would need to find `[has_continuous_smul ?m_1 M]` with\nunknown `?m_1`. We register this as an instance for `R = \u211d` in `real.punctured_nhds_module_ne_bot`.\nOne can also use `haveI := module.punctured_nhds_ne_bot R M` in a proof.\n-/\ntheorem Module.punctured_nhds_neBot [Nontrivial M] [NeBot (\ud835\udcdd[\u2260] (0 : R))] [NoZeroSMulDivisors R M]\n    (x : M) : NeBot (\ud835\udcdd[\u2260] x) :=\n  by\n  rcases exists_ne (0 : M) with \u27e8y, hy\u27e9\n  suffices : tendsto (fun c : R => x + c \u2022 y) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd[\u2260] x); exact this.ne_bot\n  refine' tendsto.inf _ (tendsto_principal_principal.2 <| _)\n  \u00b7 convert tendsto_const_nhds.add ((@tendsto_id R _).smul_const y)\n    rw [zero_smul, add_zero]\n  \u00b7 intro c hc\n    simpa [hy] using hc\n#align module.punctured_nhds_ne_bot Module.punctured_nhds_neBot\n\nend\n\nsection LatticeOps\n\nvariable {\u03b9 R M\u2081 M\u2082 : Type _} [Semiring R] [AddCommMonoid M\u2081] [AddCommMonoid M\u2082] [Module R M\u2081]\n  [Module R M\u2082] [u : TopologicalSpace R] {t : TopologicalSpace M\u2082} [ContinuousSMul R M\u2082]\n  (f : M\u2081 \u2192\u2097[R] M\u2082)\n\ntheorem continuousSMul_induced : @ContinuousSMul R M\u2081 _ u (t.induced f) :=\n  {\n    continuous_smul := by\n      letI : TopologicalSpace M\u2081 := t.induced f\n      refine' continuous_induced_rng.2 _\n      simp_rw [Function.comp, f.map_smul]\n      refine' continuous_fst.smul (continuous_induced_dom.comp continuous_snd) }\n#align has_continuous_smul_induced continuousSMul_induced\n\nend LatticeOps\n\nnamespace Submodule\n\nvariable {\u03b1 \u03b2 : Type _} [TopologicalSpace \u03b2]\n\ninstance [TopologicalSpace \u03b1] [Semiring \u03b1] [AddCommMonoid \u03b2] [Module \u03b1 \u03b2] [ContinuousSMul \u03b1 \u03b2]\n    (S : Submodule \u03b1 \u03b2) : ContinuousSMul \u03b1 S\n    where continuous_smul :=\n    by\n    rw [embedding_subtype_coe.to_inducing.continuous_iff]\n    exact continuous_fst.smul (continuous_subtype_coe.comp continuous_snd)\n\ninstance [Ring \u03b1] [AddCommGroup \u03b2] [Module \u03b1 \u03b2] [TopologicalAddGroup \u03b2] (S : Submodule \u03b1 \u03b2) :\n    TopologicalAddGroup S :=\n  S.toAddSubgroup.TopologicalAddGroup\n\nend Submodule\n\nsection closure\n\nvariable {R : Type u} {M : Type v} [Semiring R] [TopologicalSpace R] [TopologicalSpace M]\n  [AddCommMonoid M] [Module R M] [ContinuousSMul R M]\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Submodule.closure_smul_self_subset (s : Submodule R M) :\n    (fun p : R \u00d7 M => p.1 \u2022 p.2) '' Set.univ \u00d7\u02e2 closure s \u2286 closure s :=\n  calc\n    (fun p : R \u00d7 M => p.1 \u2022 p.2) '' Set.univ \u00d7\u02e2 closure s =\n        (fun p : R \u00d7 M => p.1 \u2022 p.2) '' closure (Set.univ \u00d7\u02e2 s) :=\n      by simp [closure_prod_eq]\n    _ \u2286 closure ((fun p : R \u00d7 M => p.1 \u2022 p.2) '' Set.univ \u00d7\u02e2 s) :=\n      (image_closure_subset_closure_image continuous_smul)\n    _ = closure s := by\n      congr\n      ext x\n      refine' \u27e8_, fun hx => \u27e8\u27e81, x\u27e9, \u27e8Set.mem_univ _, hx\u27e9, one_smul R _\u27e9\u27e9\n      rintro \u27e8\u27e8c, y\u27e9, \u27e8hc, hy\u27e9, rfl\u27e9\n      simp [s.smul_mem c hy]\n    \n#align submodule.closure_smul_self_subset Submodule.closure_smul_self_subset\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\ntheorem Submodule.closure_smul_self_eq (s : Submodule R M) :\n    (fun p : R \u00d7 M => p.1 \u2022 p.2) '' Set.univ \u00d7\u02e2 closure s = closure s :=\n  s.closure_smul_self_subset.antisymm fun x hx => \u27e8\u27e81, x\u27e9, \u27e8Set.mem_univ _, hx\u27e9, one_smul R _\u27e9\n#align submodule.closure_smul_self_eq Submodule.closure_smul_self_eq\n\nvariable [ContinuousAdd M]\n\n/-- The (topological-space) closure of a submodule of a topological `R`-module `M` is itself\na submodule. -/\ndef Submodule.topologicalClosure (s : Submodule R M) : Submodule R M :=\n  {\n    s.toAddSubmonoid.topologicalClosure with\n    carrier := closure (s : Set M)\n    smul_mem' := fun c x hx => s.closure_smul_self_subset \u27e8\u27e8c, x\u27e9, \u27e8Set.mem_univ _, hx\u27e9, rfl\u27e9 }\n#align submodule.topological_closure Submodule.topologicalClosure\n\n@[simp]\ntheorem Submodule.topologicalClosure_coe (s : Submodule R M) :\n    (s.topologicalClosure : Set M) = closure (s : Set M) :=\n  rfl\n#align submodule.topological_closure_coe Submodule.topologicalClosure_coe\n\ntheorem Submodule.le_topologicalClosure (s : Submodule R M) : s \u2264 s.topologicalClosure :=\n  subset_closure\n#align submodule.le_topological_closure Submodule.le_topologicalClosure\n\ntheorem Submodule.isClosed_topologicalClosure (s : Submodule R M) :\n    IsClosed (s.topologicalClosure : Set M) := by convert isClosed_closure\n#align submodule.is_closed_topological_closure Submodule.isClosed_topologicalClosure\n\ntheorem Submodule.topologicalClosure_minimal (s : Submodule R M) {t : Submodule R M} (h : s \u2264 t)\n    (ht : IsClosed (t : Set M)) : s.topologicalClosure \u2264 t :=\n  closure_minimal h ht\n#align submodule.topological_closure_minimal Submodule.topologicalClosure_minimal\n\ntheorem Submodule.topologicalClosure_mono {s : Submodule R M} {t : Submodule R M} (h : s \u2264 t) :\n    s.topologicalClosure \u2264 t.topologicalClosure :=\n  s.topologicalClosure_minimal (h.trans t.le_topologicalClosure) t.isClosed_topologicalClosure\n#align submodule.topological_closure_mono Submodule.topologicalClosure_mono\n\n/-- The topological closure of a closed submodule `s` is equal to `s`. -/\ntheorem IsClosed.submodule_topologicalClosure_eq {s : Submodule R M} (hs : IsClosed (s : Set M)) :\n    s.topologicalClosure = s :=\n  le_antisymm (s.topologicalClosure_minimal rfl.le hs) s.le_topologicalClosure\n#align is_closed.submodule_topological_closure_eq IsClosed.submodule_topologicalClosure_eq\n\n/-- A subspace is dense iff its topological closure is the entire space. -/\ntheorem Submodule.dense_iff_topologicalClosure_eq_top {s : Submodule R M} :\n    Dense (s : Set M) \u2194 s.topologicalClosure = \u22a4 :=\n  by\n  rw [\u2190 SetLike.coe_set_eq, dense_iff_closure_eq]\n  simp\n#align submodule.dense_iff_topological_closure_eq_top Submodule.dense_iff_topologicalClosure_eq_top\n\ninstance {M' : Type _} [AddCommMonoid M'] [Module R M'] [UniformSpace M'] [ContinuousAdd M']\n    [ContinuousSMul R M'] [CompleteSpace M'] (U : Submodule R M') :\n    CompleteSpace U.topologicalClosure :=\n  isClosed_closure.completeSpace_coe\n\n/-- A maximal proper subspace of a topological module (i.e a `submodule` satisfying `is_coatom`)\nis either closed or dense. -/\ntheorem Submodule.isClosed_or_dense_of_isCoatom (s : Submodule R M) (hs : IsCoatom s) :\n    IsClosed (s : Set M) \u2228 Dense (s : Set M) :=\n  (hs.le_iff.mp s.le_topologicalClosure).symm.imp (isClosed_of_closure_subset \u2218 Eq.le)\n    Submodule.dense_iff_topologicalClosure_eq_top.mpr\n#align submodule.is_closed_or_dense_of_is_coatom Submodule.isClosed_or_dense_of_isCoatom\n\nend closure\n\nsection Pi\n\ntheorem LinearMap.continuous_on_pi {\u03b9 : Type _} {R : Type _} {M : Type _} [Finite \u03b9] [Semiring R]\n    [TopologicalSpace R] [AddCommMonoid M] [Module R M] [TopologicalSpace M] [ContinuousAdd M]\n    [ContinuousSMul R M] (f : (\u03b9 \u2192 R) \u2192\u2097[R] M) : Continuous f :=\n  by\n  cases nonempty_fintype \u03b9\n  classical\n    -- for the proof, write `f` in the standard basis, and use that each coordinate is a continuous\n    -- function.\n    have : (f : (\u03b9 \u2192 R) \u2192 M) = fun x => \u2211 i : \u03b9, x i \u2022 f fun j => if i = j then 1 else 0 :=\n      by\n      ext x\n      exact f.pi_apply_eq_sum_univ x\n    rw [this]\n    refine' continuous_finset_sum _ fun i hi => _\n    exact (continuous_apply i).smul continuous_const\n#align linear_map.continuous_on_pi LinearMap.continuous_on_pi\n\nend Pi\n\n/-- Continuous linear maps between modules. We only put the type classes that are necessary for the\ndefinition, although in applications `M` and `M\u2082` will be topological modules over the topological\nring `R`. -/\nstructure ContinuousLinearMap {R : Type _} {S : Type _} [Semiring R] [Semiring S] (\u03c3 : R \u2192+* S)\n  (M : Type _) [TopologicalSpace M] [AddCommMonoid M] (M\u2082 : Type _) [TopologicalSpace M\u2082]\n  [AddCommMonoid M\u2082] [Module R M] [Module S M\u2082] extends M \u2192\u209b\u2097[\u03c3] M\u2082 where\n  cont : Continuous to_fun := by continuity\n#align continuous_linear_map ContinuousLinearMap\n\n-- mathport name: \u00abexpr \u2192SL[ ] \u00bb\nnotation:25 M \" \u2192SL[\" \u03c3 \"] \" M\u2082 => ContinuousLinearMap \u03c3 M M\u2082\n\n-- mathport name: \u00abexpr \u2192L[ ] \u00bb\nnotation:25 M \" \u2192L[\" R \"] \" M\u2082 => ContinuousLinearMap (RingHom.id R) M M\u2082\n\n-- mathport name: \u00abexpr \u2192L\u22c6[ ] \u00bb\nnotation:25 M \" \u2192L\u22c6[\" R \"] \" M\u2082 => ContinuousLinearMap (starRingEnd R) M M\u2082\n\n/-- `continuous_semilinear_map_class F \u03c3 M M\u2082` asserts `F` is a type of bundled continuous\n`\u03c3`-semilinear maps `M \u2192 M\u2082`.  See also `continuous_linear_map_class F R M M\u2082` for the case where\n`\u03c3` is the identity map on `R`.  A map `f` between an `R`-module and an `S`-module over a ring\nhomomorphism `\u03c3 : R \u2192+* S` is semilinear if it satisfies the two properties `f (x + y) = f x + f y`\nand `f (c \u2022 x) = (\u03c3 c) \u2022 f x`. -/\nclass ContinuousSemilinearMapClass (F : Type _) {R S : outParam (Type _)} [Semiring R] [Semiring S]\n  (\u03c3 : outParam <| R \u2192+* S) (M : outParam (Type _)) [TopologicalSpace M] [AddCommMonoid M]\n  (M\u2082 : outParam (Type _)) [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n  [Module S M\u2082] extends SemilinearMapClass F \u03c3 M M\u2082, ContinuousMapClass F M M\u2082\n#align continuous_semilinear_map_class ContinuousSemilinearMapClass\n\n-- `\u03c3`, `R` and `S` become metavariables, but they are all outparams so it's OK\nattribute [nolint dangerous_instance] ContinuousSemilinearMapClass.toContinuousMapClass\n\n/-- `continuous_linear_map_class F R M M\u2082` asserts `F` is a type of bundled continuous\n`R`-linear maps `M \u2192 M\u2082`.  This is an abbreviation for\n`continuous_semilinear_map_class F (ring_hom.id R) M M\u2082`.  -/\nabbrev ContinuousLinearMapClass (F : Type _) (R : outParam (Type _)) [Semiring R]\n    (M : outParam (Type _)) [TopologicalSpace M] [AddCommMonoid M] (M\u2082 : outParam (Type _))\n    [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M] [Module R M\u2082] :=\n  ContinuousSemilinearMapClass F (RingHom.id R) M M\u2082\n#align continuous_linear_map_class ContinuousLinearMapClass\n\n/-- Continuous linear equivalences between modules. We only put the type classes that are necessary\nfor the definition, although in applications `M` and `M\u2082` will be topological modules over the\ntopological semiring `R`. -/\n@[nolint has_nonempty_instance]\nstructure ContinuousLinearEquiv {R : Type _} {S : Type _} [Semiring R] [Semiring S] (\u03c3 : R \u2192+* S)\n  {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type _) [TopologicalSpace M]\n  [AddCommMonoid M] (M\u2082 : Type _) [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n  [Module S M\u2082] extends M \u2243\u209b\u2097[\u03c3] M\u2082 where\n  continuous_toFun : Continuous to_fun := by continuity\n  continuous_invFun : Continuous inv_fun := by continuity\n#align continuous_linear_equiv ContinuousLinearEquiv\n\n-- mathport name: \u00abexpr \u2243SL[ ] \u00bb\nnotation:50 M \" \u2243SL[\" \u03c3 \"] \" M\u2082 => ContinuousLinearEquiv \u03c3 M M\u2082\n\n-- mathport name: \u00abexpr \u2243L[ ] \u00bb\nnotation:50 M \" \u2243L[\" R \"] \" M\u2082 => ContinuousLinearEquiv (RingHom.id R) M M\u2082\n\n-- mathport name: \u00abexpr \u2243L\u22c6[ ] \u00bb\nnotation:50 M \" \u2243L\u22c6[\" R \"] \" M\u2082 => ContinuousLinearEquiv (starRingEnd R) M M\u2082\n\n/-- `continuous_semilinear_equiv_class F \u03c3 M M\u2082` asserts `F` is a type of bundled continuous\n`\u03c3`-semilinear equivs `M \u2192 M\u2082`.  See also `continuous_linear_equiv_class F R M M\u2082` for the case\nwhere `\u03c3` is the identity map on `R`.  A map `f` between an `R`-module and an `S`-module over a ring\nhomomorphism `\u03c3 : R \u2192+* S` is semilinear if it satisfies the two properties `f (x + y) = f x + f y`\nand `f (c \u2022 x) = (\u03c3 c) \u2022 f x`. -/\nclass ContinuousSemilinearEquivClass (F : Type _) {R : outParam (Type _)} {S : outParam (Type _)}\n  [Semiring R] [Semiring S] (\u03c3 : outParam <| R \u2192+* S) {\u03c3' : outParam <| S \u2192+* R}\n  [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : outParam (Type _)) [TopologicalSpace M]\n  [AddCommMonoid M] (M\u2082 : outParam (Type _)) [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n  [Module S M\u2082] extends SemilinearEquivClass F \u03c3 M M\u2082 where\n  map_continuous : \u2200 f : F, Continuous f := by continuity\n  inv_continuous : \u2200 f : F, Continuous (inv f) := by continuity\n#align continuous_semilinear_equiv_class ContinuousSemilinearEquivClass\n\n/-- `continuous_linear_equiv_class F \u03c3 M M\u2082` asserts `F` is a type of bundled continuous\n`R`-linear equivs `M \u2192 M\u2082`. This is an abbreviation for\n`continuous_semilinear_equiv_class F (ring_hom.id) M M\u2082`. -/\nabbrev ContinuousLinearEquivClass (F : Type _) (R : outParam (Type _)) [Semiring R]\n    (M : outParam (Type _)) [TopologicalSpace M] [AddCommMonoid M] (M\u2082 : outParam (Type _))\n    [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M] [Module R M\u2082] :=\n  ContinuousSemilinearEquivClass F (RingHom.id R) M M\u2082\n#align continuous_linear_equiv_class ContinuousLinearEquivClass\n\nnamespace ContinuousSemilinearEquivClass\n\nvariable (F : Type _) {R : Type _} {S : Type _} [Semiring R] [Semiring S] (\u03c3 : R \u2192+* S)\n  {\u03c3' : S \u2192+* R} [RingHomInvPair \u03c3 \u03c3'] [RingHomInvPair \u03c3' \u03c3] (M : Type _) [TopologicalSpace M]\n  [AddCommMonoid M] (M\u2082 : Type _) [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M]\n  [Module S M\u2082]\n\ninclude \u03c3'\n\n-- `\u03c3'` becomes a metavariable, but it's OK since it's an outparam\n@[nolint dangerous_instance]\ninstance (priority := 100) [s : ContinuousSemilinearEquivClass F \u03c3 M M\u2082] :\n    ContinuousSemilinearMapClass F \u03c3 M M\u2082 :=\n  { s with\n    coe := (coe : F \u2192 M \u2192 M\u2082)\n    coe_injective' := @FunLike.coe_injective F _ _ _ }\n\nomit \u03c3'\n\nend ContinuousSemilinearEquivClass\n\nsection PointwiseLimits\n\nvariable {M\u2081 M\u2082 \u03b1 R S : Type _} [TopologicalSpace M\u2082] [T2Space M\u2082] [Semiring R] [Semiring S]\n  [AddCommMonoid M\u2081] [AddCommMonoid M\u2082] [Module R M\u2081] [Module S M\u2082] [ContinuousConstSMul S M\u2082]\n\nsection\n\nvariable (M\u2081 M\u2082) (\u03c3 : R \u2192+* S)\n\ntheorem isClosed_setOf_map_smul : IsClosed { f : M\u2081 \u2192 M\u2082 | \u2200 c x, f (c \u2022 x) = \u03c3 c \u2022 f x } :=\n  by\n  simp only [Set.setOf_forall]\n  exact\n    isClosed_inter\u1d62 fun c =>\n      isClosed_inter\u1d62 fun x => isClosed_eq (continuous_apply _) ((continuous_apply _).const_smul _)\n#align is_closed_set_of_map_smul isClosed_setOf_map_smul\n\nend\n\nvariable [ContinuousAdd M\u2082] {\u03c3 : R \u2192+* S} {l : Filter \u03b1}\n\n/-- Constructs a bundled linear map from a function and a proof that this function belongs to the\nclosure of the set of linear maps. -/\n@[simps (config := { fullyApplied := false })]\ndef linearMapOfMemClosureRangeCoe (f : M\u2081 \u2192 M\u2082)\n    (hf : f \u2208 closure (Set.range (coeFn : (M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) \u2192 M\u2081 \u2192 M\u2082))) : M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082 :=\n  { addMonoidHomOfMemClosureRangeCoe f hf with\n    toFun := f\n    map_smul' :=\n      (isClosed_setOf_map_smul M\u2081 M\u2082 \u03c3).closure_subset_iff.2\n        (Set.range_subset_iff.2 LinearMap.map_smul\u209b\u2097) hf }\n#align linear_map_of_mem_closure_range_coe linearMapOfMemClosureRangeCoe\n\n/-- Construct a bundled linear map from a pointwise limit of linear maps -/\n@[simps (config := { fullyApplied := false })]\ndef linearMapOfTendsto (f : M\u2081 \u2192 M\u2082) (g : \u03b1 \u2192 M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) [l.ne_bot]\n    (h : Tendsto (fun a x => g a x) l (\ud835\udcdd f)) : M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082 :=\n  linearMapOfMemClosureRangeCoe f <|\n    mem_closure_of_tendsto h <| eventually_of_forall fun a => Set.mem_range_self _\n#align linear_map_of_tendsto linearMapOfTendsto\n\nvariable (M\u2081 M\u2082 \u03c3)\n\ntheorem LinearMap.isClosed_range_coe : IsClosed (Set.range (coeFn : (M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) \u2192 M\u2081 \u2192 M\u2082)) :=\n  isClosed_of_closure_subset fun f hf => \u27e8linearMapOfMemClosureRangeCoe f hf, rfl\u27e9\n#align linear_map.is_closed_range_coe LinearMap.isClosed_range_coe\n\nend PointwiseLimits\n\nnamespace ContinuousLinearMap\n\nsection Semiring\n\n/-!\n### Properties that hold for non-necessarily commutative semirings.\n-/\n\n\nvariable {R\u2081 : Type _} {R\u2082 : Type _} {R\u2083 : Type _} [Semiring R\u2081] [Semiring R\u2082] [Semiring R\u2083]\n  {\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083} {M\u2081 : Type _} [TopologicalSpace M\u2081]\n  [AddCommMonoid M\u2081] {M'\u2081 : Type _} [TopologicalSpace M'\u2081] [AddCommMonoid M'\u2081] {M\u2082 : Type _}\n  [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083] [AddCommMonoid M\u2083]\n  {M\u2084 : Type _} [TopologicalSpace M\u2084] [AddCommMonoid M\u2084] [Module R\u2081 M\u2081] [Module R\u2081 M'\u2081]\n  [Module R\u2082 M\u2082] [Module R\u2083 M\u2083]\n\n/-- Coerce continuous linear maps to linear maps. -/\ninstance : Coe (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8toLinearMap\u27e9\n\n-- make the coercion the preferred form\n@[simp]\ntheorem toLinearMap_eq_coe (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : f.toLinearMap = f :=\n  rfl\n#align continuous_linear_map.to_linear_map_eq_coe ContinuousLinearMap.toLinearMap_eq_coe\n\ntheorem coe_injective : Function.Injective (coe : (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u2192 M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\n  by\n  intro f g H\n  cases f\n  cases g\n  congr\n#align continuous_linear_map.coe_injective ContinuousLinearMap.coe_injective\n\ninstance : ContinuousSemilinearMapClass (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u03c3\u2081\u2082 M\u2081 M\u2082\n    where\n  coe f := f.toFun\n  coe_injective' f g h := coe_injective (FunLike.coe_injective h)\n  map_add f := map_add f.toLinearMap\n  map_continuous f := f.2\n  map_smul\u209b\u2097 f := f.toLinearMap.map_smul'\n\n-- see Note [function coercion]\n/-- Coerce continuous linear maps to functions. -/\ninstance toFun : CoeFun (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) fun _ => M\u2081 \u2192 M\u2082 :=\n  \u27e8fun f => f.toFun\u27e9\n#align continuous_linear_map.to_fun ContinuousLinearMap.toFun\n\n@[simp]\ntheorem coe_mk (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) (h) : (mk f h : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f :=\n  rfl\n#align continuous_linear_map.coe_mk ContinuousLinearMap.coe_mk\n\n@[simp]\ntheorem coe_mk' (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) (h) : (mk f h : M\u2081 \u2192 M\u2082) = f :=\n  rfl\n#align continuous_linear_map.coe_mk' ContinuousLinearMap.coe_mk'\n\n@[continuity]\nprotected theorem continuous (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : Continuous f :=\n  f.2\n#align continuous_linear_map.continuous ContinuousLinearMap.continuous\n\nprotected theorem uniformContinuous {E\u2081 E\u2082 : Type _} [UniformSpace E\u2081] [UniformSpace E\u2082]\n    [AddCommGroup E\u2081] [AddCommGroup E\u2082] [Module R\u2081 E\u2081] [Module R\u2082 E\u2082] [UniformAddGroup E\u2081]\n    [UniformAddGroup E\u2082] (f : E\u2081 \u2192SL[\u03c3\u2081\u2082] E\u2082) : UniformContinuous f :=\n  uniformContinuous_addMonoidHom_of_continuous f.Continuous\n#align continuous_linear_map.uniform_continuous ContinuousLinearMap.uniformContinuous\n\n@[simp, norm_cast]\ntheorem coe_inj {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} : (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = g \u2194 f = g :=\n  coe_injective.eq_iff\n#align continuous_linear_map.coe_inj ContinuousLinearMap.coe_inj\n\ntheorem coeFn_injective : @Function.Injective (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192 M\u2082) coeFn :=\n  FunLike.coe_injective\n#align continuous_linear_map.coe_fn_injective ContinuousLinearMap.coeFn_injective\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef Simps.apply (h : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082 :=\n  h\n#align continuous_linear_map.simps.apply ContinuousLinearMap.Simps.apply\n\n/-- See Note [custom simps projection]. -/\ndef Simps.coe (h : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082 :=\n  h\n#align continuous_linear_map.simps.coe ContinuousLinearMap.Simps.coe\n\ninitialize_simps_projections ContinuousLinearMap (to_linear_map_to_fun \u2192 apply, toLinearMap \u2192 coe)\n\n@[ext]\ntheorem ext {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : \u2200 x, f x = g x) : f = g :=\n  FunLike.ext f g h\n#align continuous_linear_map.ext ContinuousLinearMap.ext\n\ntheorem ext_iff {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} : f = g \u2194 \u2200 x, f x = g x :=\n  FunLike.ext_iff\n#align continuous_linear_map.ext_iff ContinuousLinearMap.ext_iff\n\n/-- Copy of a `continuous_linear_map` with a new `to_fun` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f' : M\u2081 \u2192 M\u2082) (h : f' = \u21d1f) : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082\n    where\n  toLinearMap := f.toLinearMap.copy f' h\n  cont := show Continuous f' from h.symm \u25b8 f.Continuous\n#align continuous_linear_map.copy ContinuousLinearMap.copy\n\n@[simp]\ntheorem coe_copy (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f' : M\u2081 \u2192 M\u2082) (h : f' = \u21d1f) : \u21d1(f.copy f' h) = f' :=\n  rfl\n#align continuous_linear_map.coe_copy ContinuousLinearMap.coe_copy\n\ntheorem copy_eq (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f' : M\u2081 \u2192 M\u2082) (h : f' = \u21d1f) : f.copy f' h = f :=\n  FunLike.ext' h\n#align continuous_linear_map.copy_eq ContinuousLinearMap.copy_eq\n\n-- make some straightforward lemmas available to `simp`.\nprotected theorem map_zero (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : f (0 : M\u2081) = 0 :=\n  map_zero f\n#align continuous_linear_map.map_zero ContinuousLinearMap.map_zero\n\nprotected theorem map_add (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y : M\u2081) : f (x + y) = f x + f y :=\n  map_add f x y\n#align continuous_linear_map.map_add ContinuousLinearMap.map_add\n\n@[simp]\nprotected theorem map_smul\u209b\u2097 (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2081) (x : M\u2081) : f (c \u2022 x) = \u03c3\u2081\u2082 c \u2022 f x :=\n  (toLinearMap _).map_smul\u209b\u2097 _ _\n#align continuous_linear_map.map_smul\u209b\u2097 ContinuousLinearMap.map_smul\u209b\u2097\n\n@[simp]\nprotected theorem map_smul [Module R\u2081 M\u2082] (f : M\u2081 \u2192L[R\u2081] M\u2082) (c : R\u2081) (x : M\u2081) :\n    f (c \u2022 x) = c \u2022 f x := by simp only [RingHom.id_apply, ContinuousLinearMap.map_smul\u209b\u2097]\n#align continuous_linear_map.map_smul ContinuousLinearMap.map_smul\n\n@[simp]\ntheorem map_smul_of_tower {R S : Type _} [Semiring S] [SMul R M\u2081] [Module S M\u2081] [SMul R M\u2082]\n    [Module S M\u2082] [LinearMap.CompatibleSMul M\u2081 M\u2082 R S] (f : M\u2081 \u2192L[S] M\u2082) (c : R) (x : M\u2081) :\n    f (c \u2022 x) = c \u2022 f x :=\n  LinearMap.CompatibleSMul.map_smul f c x\n#align continuous_linear_map.map_smul_of_tower ContinuousLinearMap.map_smul_of_tower\n\nprotected theorem map_sum {\u03b9 : Type _} (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (s : Finset \u03b9) (g : \u03b9 \u2192 M\u2081) :\n    f (\u2211 i in s, g i) = \u2211 i in s, f (g i) :=\n  f.toLinearMap.map_sum\n#align continuous_linear_map.map_sum ContinuousLinearMap.map_sum\n\n@[simp, norm_cast]\ntheorem coe_coe (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f :=\n  rfl\n#align continuous_linear_map.coe_coe ContinuousLinearMap.coe_coe\n\n@[ext]\ntheorem ext_ring [TopologicalSpace R\u2081] {f g : R\u2081 \u2192L[R\u2081] M\u2081} (h : f 1 = g 1) : f = g :=\n  coe_inj.1 <| LinearMap.ext_ring h\n#align continuous_linear_map.ext_ring ContinuousLinearMap.ext_ring\n\ntheorem ext_ring_iff [TopologicalSpace R\u2081] {f g : R\u2081 \u2192L[R\u2081] M\u2081} : f = g \u2194 f 1 = g 1 :=\n  \u27e8fun h => h \u25b8 rfl, ext_ring\u27e9\n#align continuous_linear_map.ext_ring_iff ContinuousLinearMap.ext_ring_iff\n\n/-- If two continuous linear maps are equal on a set `s`, then they are equal on the closure\nof the `submodule.span` of this set. -/\ntheorem eqOn_closure_span [T2Space M\u2082] {s : Set M\u2081} {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : Set.EqOn f g s) :\n    Set.EqOn f g (closure (Submodule.span R\u2081 s : Set M\u2081)) :=\n  (LinearMap.eqOn_span' h).closure f.Continuous g.Continuous\n#align continuous_linear_map.eq_on_closure_span ContinuousLinearMap.eqOn_closure_span\n\n/-- If the submodule generated by a set `s` is dense in the ambient module, then two continuous\nlinear maps equal on `s` are equal. -/\ntheorem ext_on [T2Space M\u2082] {s : Set M\u2081} (hs : Dense (Submodule.span R\u2081 s : Set M\u2081))\n    {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : Set.EqOn f g s) : f = g :=\n  ext fun x => eqOn_closure_span h (hs x)\n#align continuous_linear_map.ext_on ContinuousLinearMap.ext_on\n\n/-- Under a continuous linear map, the image of the `topological_closure` of a submodule is\ncontained in the `topological_closure` of its image. -/\ntheorem Submodule.topologicalClosure_map [RingHomSurjective \u03c3\u2081\u2082] [TopologicalSpace R\u2081]\n    [TopologicalSpace R\u2082] [ContinuousSMul R\u2081 M\u2081] [ContinuousAdd M\u2081] [ContinuousSMul R\u2082 M\u2082]\n    [ContinuousAdd M\u2082] (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (s : Submodule R\u2081 M\u2081) :\n    s.topologicalClosure.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) \u2264\n      (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topologicalClosure :=\n  image_closure_subset_closure_image f.Continuous\n#align submodule.topological_closure_map Submodule.topologicalClosure_map\n\n/-- Under a dense continuous linear map, a submodule whose `topological_closure` is `\u22a4` is sent to\nanother such submodule.  That is, the image of a dense set under a map with dense range is dense.\n-/\ntheorem DenseRange.topologicalClosure_map_submodule [RingHomSurjective \u03c3\u2081\u2082] [TopologicalSpace R\u2081]\n    [TopologicalSpace R\u2082] [ContinuousSMul R\u2081 M\u2081] [ContinuousAdd M\u2081] [ContinuousSMul R\u2082 M\u2082]\n    [ContinuousAdd M\u2082] {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (hf' : DenseRange f) {s : Submodule R\u2081 M\u2081}\n    (hs : s.topologicalClosure = \u22a4) : (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topologicalClosure = \u22a4 :=\n  by\n  rw [SetLike.ext'_iff] at hs\u22a2\n  simp only [Submodule.topologicalClosure_coe, Submodule.top_coe, \u2190 dense_iff_closure_eq] at hs\u22a2\n  exact hf'.dense_image f.continuous hs\n#align dense_range.topological_closure_map_submodule DenseRange.topologicalClosure_map_submodule\n\nsection SmulMonoid\n\nvariable {S\u2082 T\u2082 : Type _} [Monoid S\u2082] [Monoid T\u2082]\n\nvariable [DistribMulAction S\u2082 M\u2082] [SMulCommClass R\u2082 S\u2082 M\u2082] [ContinuousConstSMul S\u2082 M\u2082]\n\nvariable [DistribMulAction T\u2082 M\u2082] [SMulCommClass R\u2082 T\u2082 M\u2082] [ContinuousConstSMul T\u2082 M\u2082]\n\ninstance : MulAction S\u2082 (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    where\n  smul c f := \u27e8c \u2022 f, (f.2.const_smul _ : Continuous fun x => c \u2022 f x)\u27e9\n  one_smul f := ext fun x => one_smul _ _\n  mul_smul a b f := ext fun x => mul_smul _ _ _\n\ntheorem smul_apply (c : S\u2082) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : (c \u2022 f) x = c \u2022 f x :=\n  rfl\n#align continuous_linear_map.smul_apply ContinuousLinearMap.smul_apply\n\n@[simp, norm_cast]\ntheorem coe_smul (c : S\u2082) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : (\u2191(c \u2022 f) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = c \u2022 f :=\n  rfl\n#align continuous_linear_map.coe_smul ContinuousLinearMap.coe_smul\n\n@[simp, norm_cast]\ntheorem coe_smul' (c : S\u2082) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(c \u2022 f) = c \u2022 f :=\n  rfl\n#align continuous_linear_map.coe_smul' ContinuousLinearMap.coe_smul'\n\ninstance [SMul S\u2082 T\u2082] [IsScalarTower S\u2082 T\u2082 M\u2082] : IsScalarTower S\u2082 T\u2082 (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8fun a b f => ext fun x => smul_assoc a b (f x)\u27e9\n\ninstance [SMulCommClass S\u2082 T\u2082 M\u2082] : SMulCommClass S\u2082 T\u2082 (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8fun a b f => ext fun x => smul_comm a b (f x)\u27e9\n\nend SmulMonoid\n\n/-- The continuous map that is constantly zero. -/\ninstance : Zero (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8\u27e80, continuous_zero\u27e9\u27e9\n\ninstance : Inhabited (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e80\u27e9\n\n@[simp]\ntheorem default_def : (default : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = 0 :=\n  rfl\n#align continuous_linear_map.default_def ContinuousLinearMap.default_def\n\n@[simp]\ntheorem zero_apply (x : M\u2081) : (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) x = 0 :=\n  rfl\n#align continuous_linear_map.zero_apply ContinuousLinearMap.zero_apply\n\n@[simp, norm_cast]\ntheorem coe_zero : ((0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = 0 :=\n  rfl\n#align continuous_linear_map.coe_zero ContinuousLinearMap.coe_zero\n\n/- no simp attribute on the next line as simp does not always simplify `0 x` to `0`\nwhen `0` is the zero function, while it does for the zero continuous linear map,\nand this is the most important property we care about. -/\n@[norm_cast]\ntheorem coe_zero' : \u21d1(0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = 0 :=\n  rfl\n#align continuous_linear_map.coe_zero' ContinuousLinearMap.coe_zero'\n\ninstance uniqueOfLeft [Subsingleton M\u2081] : Unique (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  coe_injective.unique\n#align continuous_linear_map.unique_of_left ContinuousLinearMap.uniqueOfLeft\n\ninstance uniqueOfRight [Subsingleton M\u2082] : Unique (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  coe_injective.unique\n#align continuous_linear_map.unique_of_right ContinuousLinearMap.uniqueOfRight\n\ntheorem exists_ne_zero {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (hf : f \u2260 0) : \u2203 x, f x \u2260 0 :=\n  by\n  by_contra' h\n  exact hf (ContinuousLinearMap.ext h)\n#align continuous_linear_map.exists_ne_zero ContinuousLinearMap.exists_ne_zero\n\nsection\n\nvariable (R\u2081 M\u2081)\n\n/-- the identity map as a continuous linear map. -/\ndef id : M\u2081 \u2192L[R\u2081] M\u2081 :=\n  \u27e8LinearMap.id, continuous_id\u27e9\n#align continuous_linear_map.id ContinuousLinearMap.id\n\nend\n\ninstance : One (M\u2081 \u2192L[R\u2081] M\u2081) :=\n  \u27e8id R\u2081 M\u2081\u27e9\n\ntheorem one_def : (1 : M\u2081 \u2192L[R\u2081] M\u2081) = id R\u2081 M\u2081 :=\n  rfl\n#align continuous_linear_map.one_def ContinuousLinearMap.one_def\n\ntheorem id_apply (x : M\u2081) : id R\u2081 M\u2081 x = x :=\n  rfl\n#align continuous_linear_map.id_apply ContinuousLinearMap.id_apply\n\n@[simp, norm_cast]\ntheorem coe_id : (id R\u2081 M\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2081) = LinearMap.id :=\n  rfl\n#align continuous_linear_map.coe_id ContinuousLinearMap.coe_id\n\n@[simp, norm_cast]\ntheorem coe_id' : \u21d1(id R\u2081 M\u2081) = id :=\n  rfl\n#align continuous_linear_map.coe_id' ContinuousLinearMap.coe_id'\n\n@[simp, norm_cast]\ntheorem coe_eq_id {f : M\u2081 \u2192L[R\u2081] M\u2081} : (f : M\u2081 \u2192\u2097[R\u2081] M\u2081) = LinearMap.id \u2194 f = id _ _ := by\n  rw [\u2190 coe_id, coe_inj]\n#align continuous_linear_map.coe_eq_id ContinuousLinearMap.coe_eq_id\n\n@[simp]\ntheorem one_apply (x : M\u2081) : (1 : M\u2081 \u2192L[R\u2081] M\u2081) x = x :=\n  rfl\n#align continuous_linear_map.one_apply ContinuousLinearMap.one_apply\n\nsection Add\n\nvariable [ContinuousAdd M\u2082]\n\ninstance : Add (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8fun f g => \u27e8f + g, f.2.add g.2\u27e9\u27e9\n\n@[simp]\ntheorem add_apply (f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : (f + g) x = f x + g x :=\n  rfl\n#align continuous_linear_map.add_apply ContinuousLinearMap.add_apply\n\n@[simp, norm_cast]\ntheorem coe_add (f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : (\u2191(f + g) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f + g :=\n  rfl\n#align continuous_linear_map.coe_add ContinuousLinearMap.coe_add\n\n@[norm_cast]\ntheorem coe_add' (f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(f + g) = f + g :=\n  rfl\n#align continuous_linear_map.coe_add' ContinuousLinearMap.coe_add'\n\ninstance : AddCommMonoid (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    where\n  zero := (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)\n  add := (\u00b7 + \u00b7)\n  zero_add := by\n    intros <;> ext <;> apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm]\n  add_zero := by\n    intros <;> ext <;> apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm]\n  add_comm := by\n    intros <;> ext <;> apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm]\n  add_assoc := by\n    intros <;> ext <;> apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm]\n  nsmul := (\u00b7 \u2022 \u00b7)\n  nsmul_zero f := by\n    ext\n    simp\n  nsmul_succ n f := by\n    ext\n    simp [Nat.succ_eq_one_add, add_smul]\n\n@[simp, norm_cast]\ntheorem coe_sum {\u03b9 : Type _} (t : Finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    \u2191(\u2211 d in t, f d) = (\u2211 d in t, f d : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\n  (AddMonoidHom.mk (coe : (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u2192 M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) rfl fun _ _ => rfl).map_sum _ _\n#align continuous_linear_map.coe_sum ContinuousLinearMap.coe_sum\n\n@[simp, norm_cast]\ntheorem coe_sum' {\u03b9 : Type _} (t : Finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    \u21d1(\u2211 d in t, f d) = \u2211 d in t, f d := by simp only [\u2190 coe_coe, coe_sum, LinearMap.coeFn_sum]\n#align continuous_linear_map.coe_sum' ContinuousLinearMap.coe_sum'\n\ntheorem sum_apply {\u03b9 : Type _} (t : Finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) :\n    (\u2211 d in t, f d) b = \u2211 d in t, f d b := by simp only [coe_sum', Finset.sum_apply]\n#align continuous_linear_map.sum_apply ContinuousLinearMap.sum_apply\n\nend Add\n\nvariable [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083]\n\n/-- Composition of bounded linear maps. -/\ndef comp (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192SL[\u03c3\u2081\u2083] M\u2083 :=\n  \u27e8(g : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083).comp \u2191f, g.2.comp f.2\u27e9\n#align continuous_linear_map.comp ContinuousLinearMap.comp\n\n-- mathport name: \u00abexpr \u2218L \u00bb\ninfixr:80 \" \u2218L \" =>\n  @ContinuousLinearMap.comp _ _ _ _ _ _ (RingHom.id _) (RingHom.id _) (RingHom.id _) _ _ _ _ _ _ _ _\n    _ _ _ _ RingHomCompTriple.ids\n\n@[simp, norm_cast]\ntheorem coe_comp (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    (h.comp f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083) = (h : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083).comp (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\n  rfl\n#align continuous_linear_map.coe_comp ContinuousLinearMap.coe_comp\n\ninclude \u03c3\u2081\u2083\n\n@[simp, norm_cast]\ntheorem coe_comp' (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(h.comp f) = h \u2218 f :=\n  rfl\n#align continuous_linear_map.coe_comp' ContinuousLinearMap.coe_comp'\n\ntheorem comp_apply (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : (g.comp f) x = g (f x) :=\n  rfl\n#align continuous_linear_map.comp_apply ContinuousLinearMap.comp_apply\n\nomit \u03c3\u2081\u2083\n\n@[simp]\ntheorem comp_id (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : f.comp (id R\u2081 M\u2081) = f :=\n  ext fun x => rfl\n#align continuous_linear_map.comp_id ContinuousLinearMap.comp_id\n\n@[simp]\ntheorem id_comp (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : (id R\u2082 M\u2082).comp f = f :=\n  ext fun x => rfl\n#align continuous_linear_map.id_comp ContinuousLinearMap.id_comp\n\ninclude \u03c3\u2081\u2083\n\n@[simp]\ntheorem comp_zero (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) : g.comp (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = 0 :=\n  by\n  ext\n  simp\n#align continuous_linear_map.comp_zero ContinuousLinearMap.comp_zero\n\n@[simp]\ntheorem zero_comp (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : (0 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp f = 0 :=\n  by\n  ext\n  simp\n#align continuous_linear_map.zero_comp ContinuousLinearMap.zero_comp\n\n@[simp]\ntheorem comp_add [ContinuousAdd M\u2082] [ContinuousAdd M\u2083] (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083)\n    (f\u2081 f\u2082 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : g.comp (f\u2081 + f\u2082) = g.comp f\u2081 + g.comp f\u2082 :=\n  by\n  ext\n  simp\n#align continuous_linear_map.comp_add ContinuousLinearMap.comp_add\n\n@[simp]\ntheorem add_comp [ContinuousAdd M\u2083] (g\u2081 g\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    (g\u2081 + g\u2082).comp f = g\u2081.comp f + g\u2082.comp f := by\n  ext\n  simp\n#align continuous_linear_map.add_comp ContinuousLinearMap.add_comp\n\nomit \u03c3\u2081\u2083\n\ntheorem comp_assoc {R\u2084 : Type _} [Semiring R\u2084] [Module R\u2084 M\u2084] {\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084} {\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084}\n    {\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084} [RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084] [RingHomCompTriple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084]\n    [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084] (h : M\u2083 \u2192SL[\u03c3\u2083\u2084] M\u2084) (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    (h.comp g).comp f = h.comp (g.comp f) :=\n  rfl\n#align continuous_linear_map.comp_assoc ContinuousLinearMap.comp_assoc\n\ninstance : Mul (M\u2081 \u2192L[R\u2081] M\u2081) :=\n  \u27e8comp\u27e9\n\ntheorem mul_def (f g : M\u2081 \u2192L[R\u2081] M\u2081) : f * g = f.comp g :=\n  rfl\n#align continuous_linear_map.mul_def ContinuousLinearMap.mul_def\n\n@[simp]\ntheorem coe_mul (f g : M\u2081 \u2192L[R\u2081] M\u2081) : \u21d1(f * g) = f \u2218 g :=\n  rfl\n#align continuous_linear_map.coe_mul ContinuousLinearMap.coe_mul\n\ntheorem mul_apply (f g : M\u2081 \u2192L[R\u2081] M\u2081) (x : M\u2081) : (f * g) x = f (g x) :=\n  rfl\n#align continuous_linear_map.mul_apply ContinuousLinearMap.mul_apply\n\ninstance : MonoidWithZero (M\u2081 \u2192L[R\u2081] M\u2081)\n    where\n  mul := (\u00b7 * \u00b7)\n  one := 1\n  zero := 0\n  mul_zero f := ext fun _ => map_zero f\n  zero_mul _ := ext fun _ => rfl\n  mul_one _ := ext fun _ => rfl\n  one_mul _ := ext fun _ => rfl\n  mul_assoc _ _ _ := ext fun _ => rfl\n\ninstance [ContinuousAdd M\u2081] : Semiring (M\u2081 \u2192L[R\u2081] M\u2081) :=\n  { ContinuousLinearMap.monoidWithZero,\n    ContinuousLinearMap.addCommMonoid with\n    mul := (\u00b7 * \u00b7)\n    one := 1\n    left_distrib := fun f g h => ext fun x => map_add f (g x) (h x)\n    right_distrib := fun _ _ _ => ext fun _ => LinearMap.add_apply _ _ _ }\n\n/-- `continuous_linear_map.to_linear_map` as a `ring_hom`.-/\n@[simps]\ndef toLinearMapRingHom [ContinuousAdd M\u2081] : (M\u2081 \u2192L[R\u2081] M\u2081) \u2192+* M\u2081 \u2192\u2097[R\u2081] M\u2081\n    where\n  toFun := toLinearMap\n  map_zero' := rfl\n  map_one' := rfl\n  map_add' _ _ := rfl\n  map_mul' _ _ := rfl\n#align continuous_linear_map.to_linear_map_ring_hom ContinuousLinearMap.toLinearMapRingHom\n\nsection ApplyAction\n\nvariable [ContinuousAdd M\u2081]\n\n/-- The tautological action by `M\u2081 \u2192L[R\u2081] M\u2081` on `M`.\n\nThis generalizes `function.End.apply_mul_action`. -/\ninstance applyModule : Module (M\u2081 \u2192L[R\u2081] M\u2081) M\u2081 :=\n  Module.compHom _ toLinearMapRingHom\n#align continuous_linear_map.apply_module ContinuousLinearMap.applyModule\n\n@[simp]\nprotected theorem smul_def (f : M\u2081 \u2192L[R\u2081] M\u2081) (a : M\u2081) : f \u2022 a = f a :=\n  rfl\n#align continuous_linear_map.smul_def ContinuousLinearMap.smul_def\n\n/-- `continuous_linear_map.apply_module` is faithful. -/\ninstance apply_faithfulSMul : FaithfulSMul (M\u2081 \u2192L[R\u2081] M\u2081) M\u2081 :=\n  \u27e8fun _ _ => ContinuousLinearMap.ext\u27e9\n#align continuous_linear_map.apply_has_faithful_smul ContinuousLinearMap.apply_faithfulSMul\n\ninstance apply_sMulCommClass : SMulCommClass R\u2081 (M\u2081 \u2192L[R\u2081] M\u2081) M\u2081\n    where smul_comm r e m := (e.map_smul r m).symm\n#align continuous_linear_map.apply_smul_comm_class ContinuousLinearMap.apply_sMulCommClass\n\ninstance apply_smul_comm_class' : SMulCommClass (M\u2081 \u2192L[R\u2081] M\u2081) R\u2081 M\u2081\n    where smul_comm := ContinuousLinearMap.map_smul\n#align continuous_linear_map.apply_smul_comm_class' ContinuousLinearMap.apply_smul_comm_class'\n\ninstance : ContinuousConstSMul (M\u2081 \u2192L[R\u2081] M\u2081) M\u2081 :=\n  \u27e8ContinuousLinearMap.continuous\u27e9\n\nend ApplyAction\n\n/-- The cartesian product of two bounded linear maps, as a bounded linear map. -/\nprotected def prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) :\n    M\u2081 \u2192L[R\u2081] M\u2082 \u00d7 M\u2083 :=\n  \u27e8(f\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2082).Prod f\u2082, f\u2081.2.prod_mk f\u2082.2\u27e9\n#align continuous_linear_map.prod ContinuousLinearMap.prod\n\n@[simp, norm_cast]\ntheorem coe_prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) :\n    (f\u2081.Prod f\u2082 : M\u2081 \u2192\u2097[R\u2081] M\u2082 \u00d7 M\u2083) = LinearMap.prod f\u2081 f\u2082 :=\n  rfl\n#align continuous_linear_map.coe_prod ContinuousLinearMap.coe_prod\n\n@[simp, norm_cast]\ntheorem prod_apply [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) (x : M\u2081) :\n    f\u2081.Prod f\u2082 x = (f\u2081 x, f\u2082 x) :=\n  rfl\n#align continuous_linear_map.prod_apply ContinuousLinearMap.prod_apply\n\nsection\n\nvariable (R\u2081 M\u2081 M\u2082)\n\n/-- The left injection into a product is a continuous linear map. -/\ndef inl [Module R\u2081 M\u2082] : M\u2081 \u2192L[R\u2081] M\u2081 \u00d7 M\u2082 :=\n  (id R\u2081 M\u2081).Prod 0\n#align continuous_linear_map.inl ContinuousLinearMap.inl\n\n/-- The right injection into a product is a continuous linear map. -/\ndef inr [Module R\u2081 M\u2082] : M\u2082 \u2192L[R\u2081] M\u2081 \u00d7 M\u2082 :=\n  (0 : M\u2082 \u2192L[R\u2081] M\u2081).Prod (id R\u2081 M\u2082)\n#align continuous_linear_map.inr ContinuousLinearMap.inr\n\nend\n\nvariable {F : Type _}\n\n@[simp]\ntheorem inl_apply [Module R\u2081 M\u2082] (x : M\u2081) : inl R\u2081 M\u2081 M\u2082 x = (x, 0) :=\n  rfl\n#align continuous_linear_map.inl_apply ContinuousLinearMap.inl_apply\n\n@[simp]\ntheorem inr_apply [Module R\u2081 M\u2082] (x : M\u2082) : inr R\u2081 M\u2081 M\u2082 x = (0, x) :=\n  rfl\n#align continuous_linear_map.inr_apply ContinuousLinearMap.inr_apply\n\n@[simp, norm_cast]\ntheorem coe_inl [Module R\u2081 M\u2082] : (inl R\u2081 M\u2081 M\u2082 : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u00d7 M\u2082) = LinearMap.inl R\u2081 M\u2081 M\u2082 :=\n  rfl\n#align continuous_linear_map.coe_inl ContinuousLinearMap.coe_inl\n\n@[simp, norm_cast]\ntheorem coe_inr [Module R\u2081 M\u2082] : (inr R\u2081 M\u2081 M\u2082 : M\u2082 \u2192\u2097[R\u2081] M\u2081 \u00d7 M\u2082) = LinearMap.inr R\u2081 M\u2081 M\u2082 :=\n  rfl\n#align continuous_linear_map.coe_inr ContinuousLinearMap.coe_inr\n\ntheorem isClosed_ker [T1Space M\u2082] [ContinuousSemilinearMapClass F \u03c3\u2081\u2082 M\u2081 M\u2082] (f : F) :\n    IsClosed (ker f : Set M\u2081) :=\n  continuous_iff_isClosed.1 (map_continuous f) _ isClosed_singleton\n#align continuous_linear_map.is_closed_ker ContinuousLinearMap.isClosed_ker\n\ntheorem isComplete_ker {M' : Type _} [UniformSpace M'] [CompleteSpace M'] [AddCommMonoid M']\n    [Module R\u2081 M'] [T1Space M\u2082] [ContinuousSemilinearMapClass F \u03c3\u2081\u2082 M' M\u2082] (f : F) :\n    IsComplete (ker f : Set M') :=\n  (isClosed_ker f).IsComplete\n#align continuous_linear_map.is_complete_ker ContinuousLinearMap.isComplete_ker\n\ninstance (priority := 100) completeSpace_ker {M' : Type _} [UniformSpace M'] [CompleteSpace M']\n    [AddCommMonoid M'] [Module R\u2081 M'] [T1Space M\u2082] [ContinuousSemilinearMapClass F \u03c3\u2081\u2082 M' M\u2082]\n    (f : F) : CompleteSpace (ker f) :=\n  (isClosed_ker f).completeSpace_coe\n#align continuous_linear_map.complete_space_ker ContinuousLinearMap.completeSpace_ker\n\n@[simp]\ntheorem ker_prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n    ker (f.Prod g) = ker f \u2293 ker g :=\n  LinearMap.ker_prod f g\n#align continuous_linear_map.ker_prod ContinuousLinearMap.ker_prod\n\n/-- Restrict codomain of a continuous linear map. -/\ndef codRestrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : Submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) : M\u2081 \u2192SL[\u03c3\u2081\u2082] p\n    where\n  cont := f.Continuous.subtype_mk _\n  toLinearMap := (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).codRestrict p h\n#align continuous_linear_map.cod_restrict ContinuousLinearMap.codRestrict\n\n@[norm_cast]\ntheorem coe_codRestrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : Submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) :\n    (f.codRestrict p h : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] p) = (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).codRestrict p h :=\n  rfl\n#align continuous_linear_map.coe_cod_restrict ContinuousLinearMap.coe_codRestrict\n\n@[simp]\ntheorem coe_codRestrict_apply (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : Submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) (x) :\n    (f.codRestrict p h x : M\u2082) = f x :=\n  rfl\n#align continuous_linear_map.coe_cod_restrict_apply ContinuousLinearMap.coe_codRestrict_apply\n\n@[simp]\ntheorem ker_codRestrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : Submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) :\n    ker (f.codRestrict p h) = ker f :=\n  (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).ker_codRestrict p h\n#align continuous_linear_map.ker_cod_restrict ContinuousLinearMap.ker_codRestrict\n\n/-- `submodule.subtype` as a `continuous_linear_map`. -/\ndef Submodule.subtypeL (p : Submodule R\u2081 M\u2081) : p \u2192L[R\u2081] M\u2081\n    where\n  cont := continuous_subtype_val\n  toLinearMap := p.Subtype\n#align submodule.subtypeL Submodule.subtypeL\n\n@[simp, norm_cast]\ntheorem Submodule.coe_subtypeL (p : Submodule R\u2081 M\u2081) : (p.subtypeL : p \u2192\u2097[R\u2081] M\u2081) = p.Subtype :=\n  rfl\n#align submodule.coe_subtypeL Submodule.coe_subtypeL\n\n@[simp]\ntheorem Submodule.coe_subtypeL' (p : Submodule R\u2081 M\u2081) : \u21d1p.subtypeL = p.Subtype :=\n  rfl\n#align submodule.coe_subtypeL' Submodule.coe_subtypeL'\n\n@[simp, norm_cast]\ntheorem Submodule.subtypeL_apply (p : Submodule R\u2081 M\u2081) (x : p) : p.subtypeL x = x :=\n  rfl\n#align submodule.subtypeL_apply Submodule.subtypeL_apply\n\n@[simp]\ntheorem Submodule.range_subtypeL (p : Submodule R\u2081 M\u2081) : range p.subtypeL = p :=\n  Submodule.range_subtype _\n#align submodule.range_subtypeL Submodule.range_subtypeL\n\n@[simp]\ntheorem Submodule.ker_subtypeL (p : Submodule R\u2081 M\u2081) : ker p.subtypeL = \u22a5 :=\n  Submodule.ker_subtype _\n#align submodule.ker_subtypeL Submodule.ker_subtypeL\n\nvariable (R\u2081 M\u2081 M\u2082)\n\n/-- `prod.fst` as a `continuous_linear_map`. -/\ndef fst [Module R\u2081 M\u2082] : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2081\n    where\n  cont := continuous_fst\n  toLinearMap := LinearMap.fst R\u2081 M\u2081 M\u2082\n#align continuous_linear_map.fst ContinuousLinearMap.fst\n\n/-- `prod.snd` as a `continuous_linear_map`. -/\ndef snd [Module R\u2081 M\u2082] : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2082\n    where\n  cont := continuous_snd\n  toLinearMap := LinearMap.snd R\u2081 M\u2081 M\u2082\n#align continuous_linear_map.snd ContinuousLinearMap.snd\n\nvariable {R\u2081 M\u2081 M\u2082}\n\n@[simp, norm_cast]\ntheorem coe_fst [Module R\u2081 M\u2082] : \u2191(fst R\u2081 M\u2081 M\u2082) = LinearMap.fst R\u2081 M\u2081 M\u2082 :=\n  rfl\n#align continuous_linear_map.coe_fst ContinuousLinearMap.coe_fst\n\n@[simp, norm_cast]\ntheorem coe_fst' [Module R\u2081 M\u2082] : \u21d1(fst R\u2081 M\u2081 M\u2082) = Prod.fst :=\n  rfl\n#align continuous_linear_map.coe_fst' ContinuousLinearMap.coe_fst'\n\n@[simp, norm_cast]\ntheorem coe_snd [Module R\u2081 M\u2082] : \u2191(snd R\u2081 M\u2081 M\u2082) = LinearMap.snd R\u2081 M\u2081 M\u2082 :=\n  rfl\n#align continuous_linear_map.coe_snd ContinuousLinearMap.coe_snd\n\n@[simp, norm_cast]\ntheorem coe_snd' [Module R\u2081 M\u2082] : \u21d1(snd R\u2081 M\u2081 M\u2082) = Prod.snd :=\n  rfl\n#align continuous_linear_map.coe_snd' ContinuousLinearMap.coe_snd'\n\n@[simp]\ntheorem fst_prod_snd [Module R\u2081 M\u2082] : (fst R\u2081 M\u2081 M\u2082).Prod (snd R\u2081 M\u2081 M\u2082) = id R\u2081 (M\u2081 \u00d7 M\u2082) :=\n  ext fun \u27e8x, y\u27e9 => rfl\n#align continuous_linear_map.fst_prod_snd ContinuousLinearMap.fst_prod_snd\n\n@[simp]\ntheorem fst_comp_prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n    (fst R\u2081 M\u2082 M\u2083).comp (f.Prod g) = f :=\n  ext fun x => rfl\n#align continuous_linear_map.fst_comp_prod ContinuousLinearMap.fst_comp_prod\n\n@[simp]\ntheorem snd_comp_prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n    (snd R\u2081 M\u2082 M\u2083).comp (f.Prod g) = g :=\n  ext fun x => rfl\n#align continuous_linear_map.snd_comp_prod ContinuousLinearMap.snd_comp_prod\n\n/-- `prod.map` of two continuous linear maps. -/\ndef prodMap [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) :\n    M\u2081 \u00d7 M\u2083 \u2192L[R\u2081] M\u2082 \u00d7 M\u2084 :=\n  (f\u2081.comp (fst R\u2081 M\u2081 M\u2083)).Prod (f\u2082.comp (snd R\u2081 M\u2081 M\u2083))\n#align continuous_linear_map.prod_map ContinuousLinearMap.prodMap\n\n@[simp, norm_cast]\ntheorem coe_prodMap [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082)\n    (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) : \u2191(f\u2081.Prod_map f\u2082) = (f\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2082).Prod_map (f\u2082 : M\u2083 \u2192\u2097[R\u2081] M\u2084) :=\n  rfl\n#align continuous_linear_map.coe_prod_map ContinuousLinearMap.coe_prodMap\n\n@[simp, norm_cast]\ntheorem coe_prod_map' [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082)\n    (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) : \u21d1(f\u2081.Prod_map f\u2082) = Prod.map f\u2081 f\u2082 :=\n  rfl\n#align continuous_linear_map.coe_prod_map' ContinuousLinearMap.coe_prod_map'\n\n/-- The continuous linear map given by `(x, y) \u21a6 f\u2081 x + f\u2082 y`. -/\ndef coprod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [ContinuousAdd M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n    (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2083 :=\n  \u27e8LinearMap.coprod f\u2081 f\u2082, (f\u2081.cont.comp continuous_fst).add (f\u2082.cont.comp continuous_snd)\u27e9\n#align continuous_linear_map.coprod ContinuousLinearMap.coprod\n\n@[norm_cast, simp]\ntheorem coe_coprod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [ContinuousAdd M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n    (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) : (f\u2081.coprod f\u2082 : M\u2081 \u00d7 M\u2082 \u2192\u2097[R\u2081] M\u2083) = LinearMap.coprod f\u2081 f\u2082 :=\n  rfl\n#align continuous_linear_map.coe_coprod ContinuousLinearMap.coe_coprod\n\n@[simp]\ntheorem coprod_apply [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [ContinuousAdd M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n    (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) (x) : f\u2081.coprod f\u2082 x = f\u2081 x.1 + f\u2082 x.2 :=\n  rfl\n#align continuous_linear_map.coprod_apply ContinuousLinearMap.coprod_apply\n\ntheorem range_coprod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [ContinuousAdd M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n    (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) : range (f\u2081.coprod f\u2082) = range f\u2081 \u2294 range f\u2082 :=\n  LinearMap.range_coprod _ _\n#align continuous_linear_map.range_coprod ContinuousLinearMap.range_coprod\n\nsection\n\nvariable {R S : Type _} [Semiring R] [Semiring S] [Module R M\u2081] [Module R M\u2082] [Module R S]\n  [Module S M\u2082] [IsScalarTower R S M\u2082] [TopologicalSpace S] [ContinuousSMul S M\u2082]\n\n/-- The linear map `\u03bb x, c x \u2022 f`.  Associates to a scalar-valued linear map and an element of\n`M\u2082` the `M\u2082`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M\u2082`).\nSee also `continuous_linear_map.smul_right\u2097` and `continuous_linear_map.smul_rightL`. -/\ndef smulRight (c : M\u2081 \u2192L[R] S) (f : M\u2082) : M\u2081 \u2192L[R] M\u2082 :=\n  { c.toLinearMap.smul_right f with cont := c.2.smul continuous_const }\n#align continuous_linear_map.smul_right ContinuousLinearMap.smulRight\n\n@[simp]\ntheorem smulRight_apply {c : M\u2081 \u2192L[R] S} {f : M\u2082} {x : M\u2081} :\n    (smulRight c f : M\u2081 \u2192 M\u2082) x = c x \u2022 f :=\n  rfl\n#align continuous_linear_map.smul_right_apply ContinuousLinearMap.smulRight_apply\n\nend\n\nvariable [Module R\u2081 M\u2082] [TopologicalSpace R\u2081] [ContinuousSMul R\u2081 M\u2082]\n\n@[simp]\ntheorem smulRight_one_one (c : R\u2081 \u2192L[R\u2081] M\u2082) : smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) (c 1) = c := by\n  ext <;> simp [\u2190 ContinuousLinearMap.map_smul_of_tower]\n#align continuous_linear_map.smul_right_one_one ContinuousLinearMap.smulRight_one_one\n\n@[simp]\ntheorem smulRight_one_eq_iff {f f' : M\u2082} :\n    smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) f = smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) f' \u2194 f = f' := by\n  simp only [ext_ring_iff, smul_right_apply, one_apply, one_smul]\n#align continuous_linear_map.smul_right_one_eq_iff ContinuousLinearMap.smulRight_one_eq_iff\n\ntheorem smulRight_comp [ContinuousMul R\u2081] {x : M\u2082} {c : R\u2081} :\n    (smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) x).comp (smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) c) =\n      smulRight (1 : R\u2081 \u2192L[R\u2081] R\u2081) (c \u2022 x) :=\n  by\n  ext\n  simp [mul_smul]\n#align continuous_linear_map.smul_right_comp ContinuousLinearMap.smulRight_comp\n\nend Semiring\n\nsection Pi\n\nvariable {R : Type _} [Semiring R] {M : Type _} [TopologicalSpace M] [AddCommMonoid M] [Module R M]\n  {M\u2082 : Type _} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R M\u2082] {\u03b9 : Type _} {\u03c6 : \u03b9 \u2192 Type _}\n  [\u2200 i, TopologicalSpace (\u03c6 i)] [\u2200 i, AddCommMonoid (\u03c6 i)] [\u2200 i, Module R (\u03c6 i)]\n\n/-- `pi` construction for continuous linear functions. From a family of continuous linear functions\nit produces a continuous linear function into a family of topological modules. -/\ndef pi (f : \u2200 i, M \u2192L[R] \u03c6 i) : M \u2192L[R] \u2200 i, \u03c6 i :=\n  \u27e8LinearMap.pi fun i => f i, continuous_pi fun i => (f i).Continuous\u27e9\n#align continuous_linear_map.pi ContinuousLinearMap.pi\n\n@[simp]\ntheorem coe_pi' (f : \u2200 i, M \u2192L[R] \u03c6 i) : \u21d1(pi f) = fun c i => f i c :=\n  rfl\n#align continuous_linear_map.coe_pi' ContinuousLinearMap.coe_pi'\n\n@[simp]\ntheorem coe_pi (f : \u2200 i, M \u2192L[R] \u03c6 i) : (pi f : M \u2192\u2097[R] \u2200 i, \u03c6 i) = LinearMap.pi fun i => f i :=\n  rfl\n#align continuous_linear_map.coe_pi ContinuousLinearMap.coe_pi\n\ntheorem pi_apply (f : \u2200 i, M \u2192L[R] \u03c6 i) (c : M) (i : \u03b9) : pi f c i = f i c :=\n  rfl\n#align continuous_linear_map.pi_apply ContinuousLinearMap.pi_apply\n\ntheorem pi_eq_zero (f : \u2200 i, M \u2192L[R] \u03c6 i) : pi f = 0 \u2194 \u2200 i, f i = 0 :=\n  by\n  simp only [ext_iff, pi_apply, Function.funext_iff]\n  exact forall_swap\n#align continuous_linear_map.pi_eq_zero ContinuousLinearMap.pi_eq_zero\n\ntheorem pi_zero : pi (fun i => 0 : \u2200 i, M \u2192L[R] \u03c6 i) = 0 :=\n  ext fun _ => rfl\n#align continuous_linear_map.pi_zero ContinuousLinearMap.pi_zero\n\ntheorem pi_comp (f : \u2200 i, M \u2192L[R] \u03c6 i) (g : M\u2082 \u2192L[R] M) :\n    (pi f).comp g = pi fun i => (f i).comp g :=\n  rfl\n#align continuous_linear_map.pi_comp ContinuousLinearMap.pi_comp\n\n/-- The projections from a family of topological modules are continuous linear maps. -/\ndef proj (i : \u03b9) : (\u2200 i, \u03c6 i) \u2192L[R] \u03c6 i :=\n  \u27e8LinearMap.proj i, continuous_apply _\u27e9\n#align continuous_linear_map.proj ContinuousLinearMap.proj\n\n@[simp]\ntheorem proj_apply (i : \u03b9) (b : \u2200 i, \u03c6 i) : (proj i : (\u2200 i, \u03c6 i) \u2192L[R] \u03c6 i) b = b i :=\n  rfl\n#align continuous_linear_map.proj_apply ContinuousLinearMap.proj_apply\n\ntheorem proj_pi (f : \u2200 i, M\u2082 \u2192L[R] \u03c6 i) (i : \u03b9) : (proj i).comp (pi f) = f i :=\n  ext fun c => rfl\n#align continuous_linear_map.proj_pi ContinuousLinearMap.proj_pi\n\ntheorem inf\u1d62_ker_proj : (\u2a05 i, ker (proj i : (\u2200 i, \u03c6 i) \u2192L[R] \u03c6 i) : Submodule R (\u2200 i, \u03c6 i)) = \u22a5 :=\n  LinearMap.inf\u1d62_ker_proj\n#align continuous_linear_map.infi_ker_proj ContinuousLinearMap.inf\u1d62_ker_proj\n\nvariable (R \u03c6)\n\n/-- If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections\nof `\u03c6` is linearly equivalent to the product over `I`. -/\ndef infiKerProjEquiv {I J : Set \u03b9} [DecidablePred fun i => i \u2208 I] (hd : Disjoint I J)\n    (hu : Set.univ \u2286 I \u222a J) :\n    (\u2a05 i \u2208 J, ker (proj i : (\u2200 i, \u03c6 i) \u2192L[R] \u03c6 i) : Submodule R (\u2200 i, \u03c6 i)) \u2243L[R] \u2200 i : I, \u03c6 i\n    where\n  toLinearEquiv := LinearMap.inf\u1d62KerProjEquiv R \u03c6 hd hu\n  continuous_toFun :=\n    continuous_pi fun i =>\n      by\n      have :=\n        @continuous_subtype_val _ _ fun x =>\n          x \u2208 (\u2a05 i \u2208 J, ker (proj i : (\u2200 i, \u03c6 i) \u2192L[R] \u03c6 i) : Submodule R (\u2200 i, \u03c6 i))\n      have := Continuous.comp (continuous_apply i) this\n      exact this\n  continuous_invFun :=\n    Continuous.subtype_mk\n      (continuous_pi fun i => by dsimp;\n        split_ifs <;> [apply continuous_apply, exact continuous_zero])\n      _\n#align continuous_linear_map.infi_ker_proj_equiv ContinuousLinearMap.infiKerProjEquiv\n\nend Pi\n\nsection Ring\n\nvariable {R : Type _} [Ring R] {R\u2082 : Type _} [Ring R\u2082] {R\u2083 : Type _} [Ring R\u2083] {M : Type _}\n  [TopologicalSpace M] [AddCommGroup M] {M\u2082 : Type _} [TopologicalSpace M\u2082] [AddCommGroup M\u2082]\n  {M\u2083 : Type _} [TopologicalSpace M\u2083] [AddCommGroup M\u2083] {M\u2084 : Type _} [TopologicalSpace M\u2084]\n  [AddCommGroup M\u2084] [Module R M] [Module R\u2082 M\u2082] [Module R\u2083 M\u2083] {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083}\n  {\u03c3\u2081\u2083 : R \u2192+* R\u2083}\n\nsection\n\nprotected theorem map_neg (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M) : f (-x) = -f x :=\n  map_neg _ _\n#align continuous_linear_map.map_neg ContinuousLinearMap.map_neg\n\nprotected theorem map_sub (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y : M) : f (x - y) = f x - f y :=\n  map_sub _ _ _\n#align continuous_linear_map.map_sub ContinuousLinearMap.map_sub\n\n@[simp]\ntheorem sub_apply' (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M) : ((f : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) - g) x = f x - g x :=\n  rfl\n#align continuous_linear_map.sub_apply' ContinuousLinearMap.sub_apply'\n\nend\n\nsection\n\nvariable [Module R M\u2082] [Module R M\u2083] [Module R M\u2084]\n\ntheorem range_prod_eq {f : M \u2192L[R] M\u2082} {g : M \u2192L[R] M\u2083} (h : ker f \u2294 ker g = \u22a4) :\n    range (f.Prod g) = (range f).Prod (range g) :=\n  LinearMap.range_prod_eq h\n#align continuous_linear_map.range_prod_eq ContinuousLinearMap.range_prod_eq\n\ntheorem ker_prod_ker_le_ker_coprod [ContinuousAdd M\u2083] (f : M \u2192L[R] M\u2083) (g : M\u2082 \u2192L[R] M\u2083) :\n    (LinearMap.ker f).Prod (LinearMap.ker g) \u2264 LinearMap.ker (f.coprod g) :=\n  LinearMap.ker_prod_ker_le_ker_coprod f.toLinearMap g.toLinearMap\n#align continuous_linear_map.ker_prod_ker_le_ker_coprod ContinuousLinearMap.ker_prod_ker_le_ker_coprod\n\ntheorem ker_coprod_of_disjoint_range [ContinuousAdd M\u2083] (f : M \u2192L[R] M\u2083) (g : M\u2082 \u2192L[R] M\u2083)\n    (hd : Disjoint (range f) (range g)) :\n    LinearMap.ker (f.coprod g) = (LinearMap.ker f).Prod (LinearMap.ker g) :=\n  LinearMap.ker_coprod_of_disjoint_range f.toLinearMap g.toLinearMap hd\n#align continuous_linear_map.ker_coprod_of_disjoint_range ContinuousLinearMap.ker_coprod_of_disjoint_range\n\nend\n\nsection\n\nvariable [TopologicalAddGroup M\u2082]\n\ninstance : Neg (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8fun f => \u27e8-f, f.2.neg\u27e9\u27e9\n\n@[simp]\ntheorem neg_apply (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M) : (-f) x = -f x :=\n  rfl\n#align continuous_linear_map.neg_apply ContinuousLinearMap.neg_apply\n\n@[simp, norm_cast]\ntheorem coe_neg (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : (\u2191(-f) : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = -f :=\n  rfl\n#align continuous_linear_map.coe_neg ContinuousLinearMap.coe_neg\n\n@[norm_cast]\ntheorem coe_neg' (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(-f) = -f :=\n  rfl\n#align continuous_linear_map.coe_neg' ContinuousLinearMap.coe_neg'\n\ninstance : Sub (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8fun f g => \u27e8f - g, f.2.sub g.2\u27e9\u27e9\n\ninstance : AddCommGroup (M \u2192SL[\u03c3\u2081\u2082] M\u2082) := by\n  refine'\n          { ContinuousLinearMap.addCommMonoid with\n            zero := 0\n            add := (\u00b7 + \u00b7)\n            neg := Neg.neg\n            sub := Sub.sub\n            sub_eq_add_neg := _\n            nsmul := (\u00b7 \u2022 \u00b7)\n            zsmul := (\u00b7 \u2022 \u00b7)\n            zsmul_zero' := fun f => by\n              ext\n              simp\n            zsmul_succ' := fun n f => by\n              ext\n              simp [add_smul, add_comm]\n            zsmul_neg' := fun n f => by\n              ext\n              simp [Nat.succ_eq_add_one, add_smul].. } <;>\n        intros <;>\n      ext <;>\n    apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm, sub_eq_add_neg]\n\ntheorem sub_apply (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M) : (f - g) x = f x - g x :=\n  rfl\n#align continuous_linear_map.sub_apply ContinuousLinearMap.sub_apply\n\n@[simp, norm_cast]\ntheorem coe_sub (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : (\u2191(f - g) : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f - g :=\n  rfl\n#align continuous_linear_map.coe_sub ContinuousLinearMap.coe_sub\n\n@[simp, norm_cast]\ntheorem coe_sub' (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(f - g) = f - g :=\n  rfl\n#align continuous_linear_map.coe_sub' ContinuousLinearMap.coe_sub'\n\nend\n\n@[simp]\ntheorem comp_neg [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [TopologicalAddGroup M\u2082] [TopologicalAddGroup M\u2083]\n    (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : g.comp (-f) = -g.comp f :=\n  by\n  ext\n  simp\n#align continuous_linear_map.comp_neg ContinuousLinearMap.comp_neg\n\n@[simp]\ntheorem neg_comp [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [TopologicalAddGroup M\u2083] (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083)\n    (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : (-g).comp f = -g.comp f :=\n  by\n  ext\n  simp\n#align continuous_linear_map.neg_comp ContinuousLinearMap.neg_comp\n\n@[simp]\ntheorem comp_sub [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [TopologicalAddGroup M\u2082] [TopologicalAddGroup M\u2083]\n    (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f\u2081 f\u2082 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : g.comp (f\u2081 - f\u2082) = g.comp f\u2081 - g.comp f\u2082 :=\n  by\n  ext\n  simp\n#align continuous_linear_map.comp_sub ContinuousLinearMap.comp_sub\n\n@[simp]\ntheorem sub_comp [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [TopologicalAddGroup M\u2083] (g\u2081 g\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083)\n    (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : (g\u2081 - g\u2082).comp f = g\u2081.comp f - g\u2082.comp f :=\n  by\n  ext\n  simp\n#align continuous_linear_map.sub_comp ContinuousLinearMap.sub_comp\n\ninstance [TopologicalAddGroup M] : Ring (M \u2192L[R] M) :=\n  { ContinuousLinearMap.semiring,\n    ContinuousLinearMap.addCommGroup with\n    mul := (\u00b7 * \u00b7)\n    one := 1 }\n\ntheorem smulRight_one_pow [TopologicalSpace R] [TopologicalRing R] (c : R) (n : \u2115) :\n    smulRight (1 : R \u2192L[R] R) c ^ n = smulRight (1 : R \u2192L[R] R) (c ^ n) :=\n  by\n  induction' n with n ihn\n  \u00b7 ext\n    simp\n  \u00b7 rw [pow_succ, ihn, mul_def, smul_right_comp, smul_eq_mul, pow_succ']\n#align continuous_linear_map.smul_right_one_pow ContinuousLinearMap.smulRight_one_pow\n\nsection\n\nvariable {\u03c3\u2082\u2081 : R\u2082 \u2192+* R} [RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081]\n\n/-- Given a right inverse `f\u2082 : M\u2082 \u2192L[R] M` to `f\u2081 : M \u2192L[R] M\u2082`,\n`proj_ker_of_right_inverse f\u2081 f\u2082 h` is the projection `M \u2192L[R] f\u2081.ker` along `f\u2082.range`. -/\ndef projKerOfRightInverse [TopologicalAddGroup M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M)\n    (h : Function.RightInverse f\u2082 f\u2081) : M \u2192L[R] LinearMap.ker f\u2081 :=\n  (id R M - f\u2082.comp f\u2081).codRestrict (LinearMap.ker f\u2081) fun x => by simp [h (f\u2081 x)]\n#align continuous_linear_map.proj_ker_of_right_inverse ContinuousLinearMap.projKerOfRightInverse\n\n@[simp]\ntheorem coe_projKerOfRightInverse_apply [TopologicalAddGroup M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : Function.RightInverse f\u2082 f\u2081) (x : M) :\n    (f\u2081.projKerOfRightInverse f\u2082 h x : M) = x - f\u2082 (f\u2081 x) :=\n  rfl\n#align continuous_linear_map.coe_proj_ker_of_right_inverse_apply ContinuousLinearMap.coe_projKerOfRightInverse_apply\n\n@[simp]\ntheorem projKerOfRightInverse_apply_idem [TopologicalAddGroup M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : Function.RightInverse f\u2082 f\u2081) (x : LinearMap.ker f\u2081) :\n    f\u2081.projKerOfRightInverse f\u2082 h x = x :=\n  Subtype.ext_iff_val.2 <| by simp\n#align continuous_linear_map.proj_ker_of_right_inverse_apply_idem ContinuousLinearMap.projKerOfRightInverse_apply_idem\n\n@[simp]\ntheorem projKerOfRightInverse_comp_inv [TopologicalAddGroup M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : Function.RightInverse f\u2082 f\u2081) (y : M\u2082) :\n    f\u2081.projKerOfRightInverse f\u2082 h (f\u2082 y) = 0 :=\n  Subtype.ext_iff_val.2 <| by simp [h y]\n#align continuous_linear_map.proj_ker_of_right_inverse_comp_inv ContinuousLinearMap.projKerOfRightInverse_comp_inv\n\nend\n\nend Ring\n\nsection DivisionMonoid\n\nvariable {R M : Type _}\n\n/-- A nonzero continuous linear functional is open. -/\nprotected theorem isOpenMap_of_ne_zero [TopologicalSpace R] [DivisionRing R] [ContinuousSub R]\n    [AddCommGroup M] [TopologicalSpace M] [ContinuousAdd M] [Module R M] [ContinuousSMul R M]\n    (f : M \u2192L[R] R) (hf : f \u2260 0) : IsOpenMap f :=\n  let \u27e8x, hx\u27e9 := exists_ne_zero hf\n  IsOpenMap.of_sections fun y =>\n    \u27e8fun a => y + (a - f y) \u2022 (f x)\u207b\u00b9 \u2022 x, Continuous.continuousAt <| by continuity, by simp,\n      fun a => by simp [hx]\u27e9\n#align continuous_linear_map.is_open_map_of_ne_zero ContinuousLinearMap.isOpenMap_of_ne_zero\n\nend DivisionMonoid\n\nsection SmulMonoid\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariable {R R\u2082 R\u2083 S S\u2083 : Type _} [Semiring R] [Semiring R\u2082] [Semiring R\u2083] [Monoid S] [Monoid S\u2083]\n  {M : Type _} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M\u2082 : Type _}\n  [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R\u2082 M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083]\n  [AddCommMonoid M\u2083] [Module R\u2083 M\u2083] {N\u2082 : Type _} [TopologicalSpace N\u2082] [AddCommMonoid N\u2082]\n  [Module R N\u2082] {N\u2083 : Type _} [TopologicalSpace N\u2083] [AddCommMonoid N\u2083] [Module R N\u2083]\n  [DistribMulAction S\u2083 M\u2083] [SMulCommClass R\u2083 S\u2083 M\u2083] [ContinuousConstSMul S\u2083 M\u2083]\n  [DistribMulAction S N\u2083] [SMulCommClass R S N\u2083] [ContinuousConstSMul S N\u2083] {\u03c3\u2081\u2082 : R \u2192+* R\u2082}\n  {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2081\u2083 : R \u2192+* R\u2083} [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083]\n\ninclude \u03c3\u2081\u2083\n\n@[simp]\ntheorem smul_comp (c : S\u2083) (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    (c \u2022 h).comp f = c \u2022 h.comp f :=\n  rfl\n#align continuous_linear_map.smul_comp ContinuousLinearMap.smul_comp\n\nomit \u03c3\u2081\u2083\n\nvariable [DistribMulAction S\u2083 M\u2082] [ContinuousConstSMul S\u2083 M\u2082] [SMulCommClass R\u2082 S\u2083 M\u2082]\n\nvariable [DistribMulAction S N\u2082] [ContinuousConstSMul S N\u2082] [SMulCommClass R S N\u2082]\n\n@[simp]\ntheorem comp_smul [LinearMap.CompatibleSMul N\u2082 N\u2083 S R] (h\u2097 : N\u2082 \u2192L[R] N\u2083) (c : S)\n    (f\u2097 : M \u2192L[R] N\u2082) : h\u2097.comp (c \u2022 f\u2097) = c \u2022 h\u2097.comp f\u2097 :=\n  by\n  ext x\n  exact h\u2097.map_smul_of_tower c (f\u2097 x)\n#align continuous_linear_map.comp_smul ContinuousLinearMap.comp_smul\n\ninclude \u03c3\u2081\u2083\n\n@[simp]\ntheorem comp_smul\u209b\u2097 [SMulCommClass R\u2082 R\u2082 M\u2082] [SMulCommClass R\u2083 R\u2083 M\u2083] [ContinuousConstSMul R\u2082 M\u2082]\n    [ContinuousConstSMul R\u2083 M\u2083] (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (c : R\u2082) (f : M \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n    h.comp (c \u2022 f) = \u03c3\u2082\u2083 c \u2022 h.comp f := by\n  ext x\n  simp only [coe_smul', coe_comp', Function.comp_apply, Pi.smul_apply,\n    ContinuousLinearMap.map_smul\u209b\u2097]\n#align continuous_linear_map.comp_smul\u209b\u2097 ContinuousLinearMap.comp_smul\u209b\u2097\n\nomit \u03c3\u2081\u2083\n\ninstance [ContinuousAdd M\u2082] : DistribMulAction S\u2083 (M \u2192SL[\u03c3\u2081\u2082] M\u2082)\n    where\n  smul_add a f g := ext fun x => smul_add a (f x) (g x)\n  smul_zero a := ext fun x => smul_zero _\n\nend SmulMonoid\n\nsection Smul\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariable {R R\u2082 R\u2083 S S\u2083 : Type _} [Semiring R] [Semiring R\u2082] [Semiring R\u2083] [Semiring S] [Semiring S\u2083]\n  {M : Type _} [TopologicalSpace M] [AddCommMonoid M] [Module R M] {M\u2082 : Type _}\n  [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] [Module R\u2082 M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083]\n  [AddCommMonoid M\u2083] [Module R\u2083 M\u2083] {N\u2082 : Type _} [TopologicalSpace N\u2082] [AddCommMonoid N\u2082]\n  [Module R N\u2082] {N\u2083 : Type _} [TopologicalSpace N\u2083] [AddCommMonoid N\u2083] [Module R N\u2083] [Module S\u2083 M\u2083]\n  [SMulCommClass R\u2083 S\u2083 M\u2083] [ContinuousConstSMul S\u2083 M\u2083] [Module S N\u2082] [ContinuousConstSMul S N\u2082]\n  [SMulCommClass R S N\u2082] [Module S N\u2083] [SMulCommClass R S N\u2083] [ContinuousConstSMul S N\u2083]\n  {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2081\u2083 : R \u2192+* R\u2083} [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] (c : S)\n  (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y z : M)\n\n/-- `continuous_linear_map.prod` as an `equiv`. -/\n@[simps apply]\ndef prodEquiv : (M \u2192L[R] N\u2082) \u00d7 (M \u2192L[R] N\u2083) \u2243 (M \u2192L[R] N\u2082 \u00d7 N\u2083)\n    where\n  toFun f := f.1.Prod f.2\n  invFun f := \u27e8(fst _ _ _).comp f, (snd _ _ _).comp f\u27e9\n  left_inv f := by ext <;> rfl\n  right_inv f := by ext <;> rfl\n#align continuous_linear_map.prod_equiv ContinuousLinearMap.prodEquiv\n\ntheorem prod_ext_iff {f g : M \u00d7 N\u2082 \u2192L[R] N\u2083} :\n    f = g \u2194 f.comp (inl _ _ _) = g.comp (inl _ _ _) \u2227 f.comp (inr _ _ _) = g.comp (inr _ _ _) :=\n  by\n  simp only [\u2190 coe_inj, LinearMap.prod_ext_iff]\n  rfl\n#align continuous_linear_map.prod_ext_iff ContinuousLinearMap.prod_ext_iff\n\n@[ext]\ntheorem prod_ext {f g : M \u00d7 N\u2082 \u2192L[R] N\u2083} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))\n    (hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g :=\n  prod_ext_iff.2 \u27e8hl, hr\u27e9\n#align continuous_linear_map.prod_ext ContinuousLinearMap.prod_ext\n\nvariable [ContinuousAdd M\u2082] [ContinuousAdd M\u2083] [ContinuousAdd N\u2082]\n\ninstance : Module S\u2083 (M \u2192SL[\u03c3\u2081\u2083] M\u2083)\n    where\n  zero_smul _ := ext fun _ => zero_smul _ _\n  add_smul _ _ _ := ext fun _ => add_smul _ _ _\n\ninstance [Module S\u2083\u1d50\u1d52\u1d56 M\u2083] [IsCentralScalar S\u2083 M\u2083] : IsCentralScalar S\u2083 (M \u2192SL[\u03c3\u2081\u2083] M\u2083)\n    where op_smul_eq_smul _ _ := ext fun _ => op_smul_eq_smul _ _\n\nvariable (S) [ContinuousAdd N\u2083]\n\n/-- `continuous_linear_map.prod` as a `linear_equiv`. -/\n@[simps apply]\ndef prod\u2097 : ((M \u2192L[R] N\u2082) \u00d7 (M \u2192L[R] N\u2083)) \u2243\u2097[S] M \u2192L[R] N\u2082 \u00d7 N\u2083 :=\n  { prodEquiv with\n    map_add' := fun f g => rfl\n    map_smul' := fun c f => rfl }\n#align continuous_linear_map.prod\u2097 ContinuousLinearMap.prod\u2097\n\n/-- The coercion from `M \u2192L[R] M\u2082` to `M \u2192\u2097[R] M\u2082`, as a linear map. -/\n@[simps]\ndef coeLm : (M \u2192L[R] N\u2083) \u2192\u2097[S] M \u2192\u2097[R] N\u2083\n    where\n  toFun := coe\n  map_add' f g := coe_add f g\n  map_smul' c f := coe_smul c f\n#align continuous_linear_map.coe_lm ContinuousLinearMap.coeLm\n\nvariable {S} (\u03c3\u2081\u2083)\n\n/-- The coercion from `M \u2192SL[\u03c3] M\u2082` to `M \u2192\u209b\u2097[\u03c3] M\u2082`, as a linear map. -/\n@[simps]\ndef coeLm\u209b\u2097 : (M \u2192SL[\u03c3\u2081\u2083] M\u2083) \u2192\u2097[S\u2083] M \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083\n    where\n  toFun := coe\n  map_add' f g := coe_add f g\n  map_smul' c f := coe_smul c f\n#align continuous_linear_map.coe_lm\u209b\u2097 ContinuousLinearMap.coeLm\u209b\u2097\n\nvariable {\u03c3\u2081\u2083}\n\nend Smul\n\nsection SmulRight\u2097\n\nvariable {R S T M M\u2082 : Type _} [Semiring R] [Semiring S] [Semiring T] [Module R S]\n  [AddCommMonoid M\u2082] [Module R M\u2082] [Module S M\u2082] [IsScalarTower R S M\u2082] [TopologicalSpace S]\n  [TopologicalSpace M\u2082] [ContinuousSMul S M\u2082] [TopologicalSpace M] [AddCommMonoid M] [Module R M]\n  [ContinuousAdd M\u2082] [Module T M\u2082] [ContinuousConstSMul T M\u2082] [SMulCommClass R T M\u2082]\n  [SMulCommClass S T M\u2082]\n\n/-- Given `c : E \u2192L[\ud835\udd5c] \ud835\udd5c`, `c.smul_right\u2097` is the linear map from `F` to `E \u2192L[\ud835\udd5c] F`\nsending `f` to `\u03bb e, c e \u2022 f`. See also `continuous_linear_map.smul_rightL`. -/\ndef smulRight\u2097 (c : M \u2192L[R] S) : M\u2082 \u2192\u2097[T] M \u2192L[R] M\u2082\n    where\n  toFun := c.smul_right\n  map_add' x y := by\n    ext e\n    apply smul_add\n  map_smul' a x := by\n    ext e\n    dsimp\n    apply smul_comm\n#align continuous_linear_map.smul_right\u2097 ContinuousLinearMap.smulRight\u2097\n\n@[simp]\ntheorem coe_smulRight\u2097 (c : M \u2192L[R] S) : \u21d1(smulRight\u2097 c : M\u2082 \u2192\u2097[T] M \u2192L[R] M\u2082) = c.smul_right :=\n  rfl\n#align continuous_linear_map.coe_smul_right\u2097 ContinuousLinearMap.coe_smulRight\u2097\n\nend SmulRight\u2097\n\nsection CommRing\n\nvariable {R : Type _} [CommRing R] {M : Type _} [TopologicalSpace M] [AddCommGroup M] {M\u2082 : Type _}\n  [TopologicalSpace M\u2082] [AddCommGroup M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083] [AddCommGroup M\u2083]\n  [Module R M] [Module R M\u2082] [Module R M\u2083] [ContinuousConstSMul R M\u2083]\n\nvariable [TopologicalAddGroup M\u2082] [ContinuousConstSMul R M\u2082]\n\ninstance : Algebra R (M\u2082 \u2192L[R] M\u2082) :=\n  Algebra.ofModule smul_comp fun _ _ _ => comp_smul _ _ _\n\nend CommRing\n\nsection RestrictScalars\n\nvariable {A M M\u2082 : Type _} [Ring A] [AddCommGroup M] [AddCommGroup M\u2082] [Module A M] [Module A M\u2082]\n  [TopologicalSpace M] [TopologicalSpace M\u2082] (R : Type _) [Ring R] [Module R M] [Module R M\u2082]\n  [LinearMap.CompatibleSMul M M\u2082 R A]\n\n/-- If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous\n`R`-linear map. We assume `linear_map.compatible_smul M M\u2082 R A` to match assumptions of\n`linear_map.map_smul_of_tower`. -/\ndef restrictScalars (f : M \u2192L[A] M\u2082) : M \u2192L[R] M\u2082 :=\n  \u27e8(f : M \u2192\u2097[A] M\u2082).restrictScalars R, f.Continuous\u27e9\n#align continuous_linear_map.restrict_scalars ContinuousLinearMap.restrictScalars\n\nvariable {R}\n\n@[simp, norm_cast]\ntheorem coe_restrictScalars (f : M \u2192L[A] M\u2082) :\n    (f.restrictScalars R : M \u2192\u2097[R] M\u2082) = (f : M \u2192\u2097[A] M\u2082).restrictScalars R :=\n  rfl\n#align continuous_linear_map.coe_restrict_scalars ContinuousLinearMap.coe_restrictScalars\n\n@[simp]\ntheorem coe_restrict_scalars' (f : M \u2192L[A] M\u2082) : \u21d1(f.restrictScalars R) = f :=\n  rfl\n#align continuous_linear_map.coe_restrict_scalars' ContinuousLinearMap.coe_restrict_scalars'\n\n@[simp]\ntheorem restrictScalars_zero : (0 : M \u2192L[A] M\u2082).restrictScalars R = 0 :=\n  rfl\n#align continuous_linear_map.restrict_scalars_zero ContinuousLinearMap.restrictScalars_zero\n\nsection\n\nvariable [TopologicalAddGroup M\u2082]\n\n@[simp]\ntheorem restrictScalars_add (f g : M \u2192L[A] M\u2082) :\n    (f + g).restrictScalars R = f.restrictScalars R + g.restrictScalars R :=\n  rfl\n#align continuous_linear_map.restrict_scalars_add ContinuousLinearMap.restrictScalars_add\n\n@[simp]\ntheorem restrictScalars_neg (f : M \u2192L[A] M\u2082) : (-f).restrictScalars R = -f.restrictScalars R :=\n  rfl\n#align continuous_linear_map.restrict_scalars_neg ContinuousLinearMap.restrictScalars_neg\n\nend\n\nvariable {S : Type _} [Ring S] [Module S M\u2082] [ContinuousConstSMul S M\u2082] [SMulCommClass A S M\u2082]\n  [SMulCommClass R S M\u2082]\n\n@[simp]\ntheorem restrictScalars_smul (c : S) (f : M \u2192L[A] M\u2082) :\n    (c \u2022 f).restrictScalars R = c \u2022 f.restrictScalars R :=\n  rfl\n#align continuous_linear_map.restrict_scalars_smul ContinuousLinearMap.restrictScalars_smul\n\nvariable (A M M\u2082 R S) [TopologicalAddGroup M\u2082]\n\n/-- `continuous_linear_map.restrict_scalars` as a `linear_map`. See also\n`continuous_linear_map.restrict_scalarsL`. -/\ndef restrictScalars\u2097 : (M \u2192L[A] M\u2082) \u2192\u2097[S] M \u2192L[R] M\u2082\n    where\n  toFun := restrictScalars R\n  map_add' := restrictScalars_add\n  map_smul' := restrictScalars_smul\n#align continuous_linear_map.restrict_scalars\u2097 ContinuousLinearMap.restrictScalars\u2097\n\nvariable {A M M\u2082 R S}\n\n@[simp]\ntheorem coe_restrictScalars\u2097 : \u21d1(restrictScalars\u2097 A M M\u2082 R S) = restrictScalars R :=\n  rfl\n#align continuous_linear_map.coe_restrict_scalars\u2097 ContinuousLinearMap.coe_restrictScalars\u2097\n\nend RestrictScalars\n\nend ContinuousLinearMap\n\nnamespace ContinuousLinearEquiv\n\nsection AddCommMonoid\n\nvariable {R\u2081 : Type _} {R\u2082 : Type _} {R\u2083 : Type _} [Semiring R\u2081] [Semiring R\u2082] [Semiring R\u2083]\n  {\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082} {\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081} [RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082]\n  {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082} [RingHomInvPair \u03c3\u2082\u2083 \u03c3\u2083\u2082] [RingHomInvPair \u03c3\u2083\u2082 \u03c3\u2082\u2083]\n  {\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083} {\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081} [RingHomInvPair \u03c3\u2081\u2083 \u03c3\u2083\u2081] [RingHomInvPair \u03c3\u2083\u2081 \u03c3\u2081\u2083]\n  [RingHomCompTriple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [RingHomCompTriple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081] {M\u2081 : Type _}\n  [TopologicalSpace M\u2081] [AddCommMonoid M\u2081] {M'\u2081 : Type _} [TopologicalSpace M'\u2081] [AddCommMonoid M'\u2081]\n  {M\u2082 : Type _} [TopologicalSpace M\u2082] [AddCommMonoid M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083]\n  [AddCommMonoid M\u2083] {M\u2084 : Type _} [TopologicalSpace M\u2084] [AddCommMonoid M\u2084] [Module R\u2081 M\u2081]\n  [Module R\u2081 M'\u2081] [Module R\u2082 M\u2082] [Module R\u2083 M\u2083]\n\ninclude \u03c3\u2082\u2081\n\n/-- A continuous linear equivalence induces a continuous linear map. -/\ndef toContinuousLinearMap (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082 :=\n  { e.toLinearEquiv.toLinearMap with cont := e.continuous_toFun }\n#align continuous_linear_equiv.to_continuous_linear_map ContinuousLinearEquiv.toContinuousLinearMap\n\n/-- Coerce continuous linear equivs to continuous linear maps. -/\ninstance : Coe (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  \u27e8toContinuousLinearMap\u27e9\n\ninstance : ContinuousSemilinearEquivClass (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) \u03c3\u2081\u2082 M\u2081 M\u2082\n    where\n  coe f := f\n  inv f := f.invFun\n  coe_injective' f g h\u2081 h\u2082 := by\n    cases' f with f' _\n    cases' g with g' _\n    cases f'\n    cases g'\n    congr\n  left_inv f := f.left_inv\n  right_inv f := f.right_inv\n  map_add f := f.map_add'\n  map_smul\u209b\u2097 f := f.map_smul'\n  map_continuous := continuous_toFun\n  inv_continuous := continuous_invFun\n\n-- see Note [function coercion]\n/-- Coerce continuous linear equivs to maps. -/\ninstance : CoeFun (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) fun _ => M\u2081 \u2192 M\u2082 :=\n  \u27e8fun f => f\u27e9\n\n@[simp]\ntheorem coe_def_rev (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.toContinuousLinearMap = e :=\n  rfl\n#align continuous_linear_equiv.coe_def_rev ContinuousLinearEquiv.coe_def_rev\n\ntheorem coe_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) : (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) b = e b :=\n  rfl\n#align continuous_linear_equiv.coe_apply ContinuousLinearEquiv.coe_apply\n\n@[simp]\ntheorem coe_toLinearEquiv (f : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : \u21d1f.toLinearEquiv = f :=\n  rfl\n#align continuous_linear_equiv.coe_to_linear_equiv ContinuousLinearEquiv.coe_toLinearEquiv\n\n@[simp, norm_cast]\ntheorem coe_coe (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : \u21d1(e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = e :=\n  rfl\n#align continuous_linear_equiv.coe_coe ContinuousLinearEquiv.coe_coe\n\ntheorem toLinearEquiv_injective :\n    Function.Injective (toLinearEquiv : (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) \u2192 M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082)\n  | \u27e8e, _, _\u27e9, \u27e8e', _, _\u27e9, rfl => rfl\n#align continuous_linear_equiv.to_linear_equiv_injective ContinuousLinearEquiv.toLinearEquiv_injective\n\n@[ext]\ntheorem ext {f g : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082} (h : (f : M\u2081 \u2192 M\u2082) = g) : f = g :=\n  toLinearEquiv_injective <| LinearEquiv.ext <| congr_fun h\n#align continuous_linear_equiv.ext ContinuousLinearEquiv.ext\n\ntheorem coe_injective : Function.Injective (coe : (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n  fun e e' h => ext <| funext <| ContinuousLinearMap.ext_iff.1 h\n#align continuous_linear_equiv.coe_injective ContinuousLinearEquiv.coe_injective\n\n@[simp, norm_cast]\ntheorem coe_inj {e e' : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082} : (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = e' \u2194 e = e' :=\n  coe_injective.eq_iff\n#align continuous_linear_equiv.coe_inj ContinuousLinearEquiv.coe_inj\n\n/-- A continuous linear equivalence induces a homeomorphism. -/\ndef toHomeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2243\u209c M\u2082 :=\n  { e with toEquiv := e.toLinearEquiv.toEquiv }\n#align continuous_linear_equiv.to_homeomorph ContinuousLinearEquiv.toHomeomorph\n\n@[simp]\ntheorem coe_toHomeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : \u21d1e.toHomeomorph = e :=\n  rfl\n#align continuous_linear_equiv.coe_to_homeomorph ContinuousLinearEquiv.coe_toHomeomorph\n\ntheorem image_closure (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2081) : e '' closure s = closure (e '' s) :=\n  e.toHomeomorph.image_closure s\n#align continuous_linear_equiv.image_closure ContinuousLinearEquiv.image_closure\n\ntheorem preimage_closure (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2082) : e \u207b\u00b9' closure s = closure (e \u207b\u00b9' s) :=\n  e.toHomeomorph.preimage_closure s\n#align continuous_linear_equiv.preimage_closure ContinuousLinearEquiv.preimage_closure\n\n@[simp]\ntheorem isClosed_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : Set M\u2081} : IsClosed (e '' s) \u2194 IsClosed s :=\n  e.toHomeomorph.isClosed_image\n#align continuous_linear_equiv.is_closed_image ContinuousLinearEquiv.isClosed_image\n\ntheorem map_nhds_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : map e (\ud835\udcdd x) = \ud835\udcdd (e x) :=\n  e.toHomeomorph.map_nhds_eq x\n#align continuous_linear_equiv.map_nhds_eq ContinuousLinearEquiv.map_nhds_eq\n\n-- Make some straightforward lemmas available to `simp`.\n@[simp]\ntheorem map_zero (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e (0 : M\u2081) = 0 :=\n  (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_zero\n#align continuous_linear_equiv.map_zero ContinuousLinearEquiv.map_zero\n\n@[simp]\ntheorem map_add (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x y : M\u2081) : e (x + y) = e x + e y :=\n  (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_add x y\n#align continuous_linear_equiv.map_add ContinuousLinearEquiv.map_add\n\n@[simp]\ntheorem map_smul\u209b\u2097 (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2081) (x : M\u2081) : e (c \u2022 x) = \u03c3\u2081\u2082 c \u2022 e x :=\n  (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_smul\u209b\u2097 c x\n#align continuous_linear_equiv.map_smul\u209b\u2097 ContinuousLinearEquiv.map_smul\u209b\u2097\n\nomit \u03c3\u2082\u2081\n\n@[simp]\ntheorem map_smul [Module R\u2081 M\u2082] (e : M\u2081 \u2243L[R\u2081] M\u2082) (c : R\u2081) (x : M\u2081) : e (c \u2022 x) = c \u2022 e x :=\n  (e : M\u2081 \u2192L[R\u2081] M\u2082).map_smul c x\n#align continuous_linear_equiv.map_smul ContinuousLinearEquiv.map_smul\n\ninclude \u03c3\u2082\u2081\n\n@[simp]\ntheorem map_eq_zero_iff (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x : M\u2081} : e x = 0 \u2194 x = 0 :=\n  e.toLinearEquiv.map_eq_zero_iff\n#align continuous_linear_equiv.map_eq_zero_iff ContinuousLinearEquiv.map_eq_zero_iff\n\nattribute [continuity]\n  ContinuousLinearEquiv.continuous_toFun ContinuousLinearEquiv.continuous_invFun\n\n@[continuity]\nprotected theorem continuous (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : Continuous (e : M\u2081 \u2192 M\u2082) :=\n  e.continuous_toFun\n#align continuous_linear_equiv.continuous ContinuousLinearEquiv.continuous\n\nprotected theorem continuousOn (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : Set M\u2081} : ContinuousOn (e : M\u2081 \u2192 M\u2082) s :=\n  e.Continuous.ContinuousOn\n#align continuous_linear_equiv.continuous_on ContinuousLinearEquiv.continuousOn\n\nprotected theorem continuousAt (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x : M\u2081} : ContinuousAt (e : M\u2081 \u2192 M\u2082) x :=\n  e.Continuous.ContinuousAt\n#align continuous_linear_equiv.continuous_at ContinuousLinearEquiv.continuousAt\n\nprotected theorem continuousWithinAt (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : Set M\u2081} {x : M\u2081} :\n    ContinuousWithinAt (e : M\u2081 \u2192 M\u2082) s x :=\n  e.Continuous.ContinuousWithinAt\n#align continuous_linear_equiv.continuous_within_at ContinuousLinearEquiv.continuousWithinAt\n\ntheorem comp_continuousOn_iff {\u03b1 : Type _} [TopologicalSpace \u03b1] (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {f : \u03b1 \u2192 M\u2081}\n    {s : Set \u03b1} : ContinuousOn (e \u2218 f) s \u2194 ContinuousOn f s :=\n  e.toHomeomorph.comp_continuousOn_iff _ _\n#align continuous_linear_equiv.comp_continuous_on_iff ContinuousLinearEquiv.comp_continuousOn_iff\n\ntheorem comp_continuous_iff {\u03b1 : Type _} [TopologicalSpace \u03b1] (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {f : \u03b1 \u2192 M\u2081} :\n    Continuous (e \u2218 f) \u2194 Continuous f :=\n  e.toHomeomorph.comp_continuous_iff\n#align continuous_linear_equiv.comp_continuous_iff ContinuousLinearEquiv.comp_continuous_iff\n\nomit \u03c3\u2082\u2081\n\n/-- An extensionality lemma for `R \u2243L[R] M`. -/\ntheorem ext\u2081 [TopologicalSpace R\u2081] {f g : R\u2081 \u2243L[R\u2081] M\u2081} (h : f 1 = g 1) : f = g :=\n  ext <| funext fun x => mul_one x \u25b8 by rw [\u2190 smul_eq_mul, map_smul, h, map_smul]\n#align continuous_linear_equiv.ext\u2081 ContinuousLinearEquiv.ext\u2081\n\nsection\n\nvariable (R\u2081 M\u2081)\n\n/-- The identity map as a continuous linear equivalence. -/\n@[refl]\nprotected def refl : M\u2081 \u2243L[R\u2081] M\u2081 :=\n  { LinearEquiv.refl R\u2081 M\u2081 with\n    continuous_toFun := continuous_id\n    continuous_invFun := continuous_id }\n#align continuous_linear_equiv.refl ContinuousLinearEquiv.refl\n\nend\n\n@[simp, norm_cast]\ntheorem coe_refl : \u2191(ContinuousLinearEquiv.refl R\u2081 M\u2081) = ContinuousLinearMap.id R\u2081 M\u2081 :=\n  rfl\n#align continuous_linear_equiv.coe_refl ContinuousLinearEquiv.coe_refl\n\n@[simp, norm_cast]\ntheorem coe_refl' : \u21d1(ContinuousLinearEquiv.refl R\u2081 M\u2081) = id :=\n  rfl\n#align continuous_linear_equiv.coe_refl' ContinuousLinearEquiv.coe_refl'\n\n/-- The inverse of a continuous linear equivalence as a continuous linear equivalence-/\n@[symm]\nprotected def symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2082 \u2243SL[\u03c3\u2082\u2081] M\u2081 :=\n  { e.toLinearEquiv.symm with\n    continuous_toFun := e.continuous_invFun\n    continuous_invFun := e.continuous_toFun }\n#align continuous_linear_equiv.symm ContinuousLinearEquiv.symm\n\ninclude \u03c3\u2082\u2081\n\n@[simp]\ntheorem symm_toLinearEquiv (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.symm.toLinearEquiv = e.toLinearEquiv.symm :=\n  by\n  ext\n  rfl\n#align continuous_linear_equiv.symm_to_linear_equiv ContinuousLinearEquiv.symm_toLinearEquiv\n\n@[simp]\ntheorem symm_toHomeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.toHomeomorph.symm = e.symm.toHomeomorph :=\n  rfl\n#align continuous_linear_equiv.symm_to_homeomorph ContinuousLinearEquiv.symm_toHomeomorph\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef Simps.apply (h : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082 :=\n  h\n#align continuous_linear_equiv.simps.apply ContinuousLinearEquiv.Simps.apply\n\n/-- See Note [custom simps projection] -/\ndef Simps.symmApply (h : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2082 \u2192 M\u2081 :=\n  h.symm\n#align continuous_linear_equiv.simps.symm_apply ContinuousLinearEquiv.Simps.symmApply\n\ninitialize_simps_projections ContinuousLinearEquiv (to_linear_equiv_to_fun \u2192 apply,\n  to_linear_equiv_inv_fun \u2192 symm_apply)\n\ntheorem symm_map_nhds_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : map e.symm (\ud835\udcdd (e x)) = \ud835\udcdd x :=\n  e.toHomeomorph.symm_map_nhds_eq x\n#align continuous_linear_equiv.symm_map_nhds_eq ContinuousLinearEquiv.symm_map_nhds_eq\n\nomit \u03c3\u2082\u2081\n\ninclude \u03c3\u2082\u2081 \u03c3\u2083\u2082 \u03c3\u2083\u2081\n\n/-- The composition of two continuous linear equivalences as a continuous linear equivalence. -/\n@[trans]\nprotected def trans (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) : M\u2081 \u2243SL[\u03c3\u2081\u2083] M\u2083 :=\n  {\n    e\u2081.toLinearEquiv.trans\n      e\u2082.toLinearEquiv with\n    continuous_toFun := e\u2082.continuous_toFun.comp e\u2081.continuous_toFun\n    continuous_invFun := e\u2081.continuous_invFun.comp e\u2082.continuous_invFun }\n#align continuous_linear_equiv.trans ContinuousLinearEquiv.trans\n\ninclude \u03c3\u2081\u2083\n\n@[simp]\ntheorem trans_toLinearEquiv (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) :\n    (e\u2081.trans e\u2082).toLinearEquiv = e\u2081.toLinearEquiv.trans e\u2082.toLinearEquiv :=\n  by\n  ext\n  rfl\n#align continuous_linear_equiv.trans_to_linear_equiv ContinuousLinearEquiv.trans_toLinearEquiv\n\nomit \u03c3\u2081\u2083 \u03c3\u2082\u2081 \u03c3\u2083\u2082 \u03c3\u2083\u2081\n\n/-- Product of two continuous linear equivalences. The map comes from `equiv.prod_congr`. -/\ndef prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082) (e' : M\u2083 \u2243L[R\u2081] M\u2084) :\n    (M\u2081 \u00d7 M\u2083) \u2243L[R\u2081] M\u2082 \u00d7 M\u2084 :=\n  {\n    e.toLinearEquiv.Prod\n      e'.toLinearEquiv with\n    continuous_toFun := e.continuous_toFun.Prod_map e'.continuous_toFun\n    continuous_invFun := e.continuous_invFun.Prod_map e'.continuous_invFun }\n#align continuous_linear_equiv.prod ContinuousLinearEquiv.prod\n\n@[simp, norm_cast]\ntheorem prod_apply [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082)\n    (e' : M\u2083 \u2243L[R\u2081] M\u2084) (x) : e.Prod e' x = (e x.1, e' x.2) :=\n  rfl\n#align continuous_linear_equiv.prod_apply ContinuousLinearEquiv.prod_apply\n\n@[simp, norm_cast]\ntheorem coe_prod [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082)\n    (e' : M\u2083 \u2243L[R\u2081] M\u2084) :\n    (e.Prod e' : M\u2081 \u00d7 M\u2083 \u2192L[R\u2081] M\u2082 \u00d7 M\u2084) = (e : M\u2081 \u2192L[R\u2081] M\u2082).Prod_map (e' : M\u2083 \u2192L[R\u2081] M\u2084) :=\n  rfl\n#align continuous_linear_equiv.coe_prod ContinuousLinearEquiv.coe_prod\n\ntheorem prod_symm [Module R\u2081 M\u2082] [Module R\u2081 M\u2083] [Module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082)\n    (e' : M\u2083 \u2243L[R\u2081] M\u2084) : (e.Prod e').symm = e.symm.Prod e'.symm :=\n  rfl\n#align continuous_linear_equiv.prod_symm ContinuousLinearEquiv.prod_symm\n\ninclude \u03c3\u2082\u2081\n\nprotected theorem bijective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : Function.Bijective e :=\n  e.toLinearEquiv.toEquiv.Bijective\n#align continuous_linear_equiv.bijective ContinuousLinearEquiv.bijective\n\nprotected theorem injective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : Function.Injective e :=\n  e.toLinearEquiv.toEquiv.Injective\n#align continuous_linear_equiv.injective ContinuousLinearEquiv.injective\n\nprotected theorem surjective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : Function.Surjective e :=\n  e.toLinearEquiv.toEquiv.Surjective\n#align continuous_linear_equiv.surjective ContinuousLinearEquiv.surjective\n\ninclude \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2081\u2083\n\n@[simp]\ntheorem trans_apply (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) (c : M\u2081) :\n    (e\u2081.trans e\u2082) c = e\u2082 (e\u2081 c) :=\n  rfl\n#align continuous_linear_equiv.trans_apply ContinuousLinearEquiv.trans_apply\n\nomit \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2081\u2083\n\n@[simp]\ntheorem apply_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : M\u2082) : e (e.symm c) = c :=\n  e.1.right_inv c\n#align continuous_linear_equiv.apply_symm_apply ContinuousLinearEquiv.apply_symm_apply\n\n@[simp]\ntheorem symm_apply_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) : e.symm (e b) = b :=\n  e.1.left_inv b\n#align continuous_linear_equiv.symm_apply_apply ContinuousLinearEquiv.symm_apply_apply\n\ninclude \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 \u03c3\u2083\u2081\n\n@[simp]\ntheorem symm_trans_apply (e\u2081 : M\u2082 \u2243SL[\u03c3\u2082\u2081] M\u2081) (e\u2082 : M\u2083 \u2243SL[\u03c3\u2083\u2082] M\u2082) (c : M\u2081) :\n    (e\u2082.trans e\u2081).symm c = e\u2082.symm (e\u2081.symm c) :=\n  rfl\n#align continuous_linear_equiv.symm_trans_apply ContinuousLinearEquiv.symm_trans_apply\n\nomit \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 \u03c3\u2083\u2081\n\n@[simp]\ntheorem symm_image_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2081) : e.symm '' (e '' s) = s :=\n  e.toLinearEquiv.toEquiv.symm_image_image s\n#align continuous_linear_equiv.symm_image_image ContinuousLinearEquiv.symm_image_image\n\n@[simp]\ntheorem image_symm_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2082) : e '' (e.symm '' s) = s :=\n  e.symm.symm_image_image s\n#align continuous_linear_equiv.image_symm_image ContinuousLinearEquiv.image_symm_image\n\ninclude \u03c3\u2083\u2082 \u03c3\u2083\u2081\n\n@[simp, norm_cast]\ntheorem comp_coe (f : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (f' : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) :\n    (f' : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = (f.trans f' : M\u2081 \u2192SL[\u03c3\u2081\u2083] M\u2083) :=\n  rfl\n#align continuous_linear_equiv.comp_coe ContinuousLinearEquiv.comp_coe\n\nomit \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2082\u2081\n\n@[simp]\ntheorem coe_comp_coe_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n    (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).comp (e.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081) = ContinuousLinearMap.id R\u2082 M\u2082 :=\n  ContinuousLinearMap.ext e.apply_symm_apply\n#align continuous_linear_equiv.coe_comp_coe_symm ContinuousLinearEquiv.coe_comp_coe_symm\n\n@[simp]\ntheorem coe_symm_comp_coe (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n    (e.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081).comp (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = ContinuousLinearMap.id R\u2081 M\u2081 :=\n  ContinuousLinearMap.ext e.symm_apply_apply\n#align continuous_linear_equiv.coe_symm_comp_coe ContinuousLinearEquiv.coe_symm_comp_coe\n\ninclude \u03c3\u2082\u2081\n\n@[simp]\ntheorem symm_comp_self (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : (e.symm : M\u2082 \u2192 M\u2081) \u2218 (e : M\u2081 \u2192 M\u2082) = id :=\n  by\n  ext x\n  exact symm_apply_apply e x\n#align continuous_linear_equiv.symm_comp_self ContinuousLinearEquiv.symm_comp_self\n\n@[simp]\ntheorem self_comp_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : (e : M\u2081 \u2192 M\u2082) \u2218 (e.symm : M\u2082 \u2192 M\u2081) = id :=\n  by\n  ext x\n  exact apply_symm_apply e x\n#align continuous_linear_equiv.self_comp_symm ContinuousLinearEquiv.self_comp_symm\n\n@[simp]\ntheorem symm_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.symm.symm = e :=\n  by\n  ext x\n  rfl\n#align continuous_linear_equiv.symm_symm ContinuousLinearEquiv.symm_symm\n\nomit \u03c3\u2082\u2081\n\n@[simp]\ntheorem refl_symm : (ContinuousLinearEquiv.refl R\u2081 M\u2081).symm = ContinuousLinearEquiv.refl R\u2081 M\u2081 :=\n  rfl\n#align continuous_linear_equiv.refl_symm ContinuousLinearEquiv.refl_symm\n\ninclude \u03c3\u2082\u2081\n\ntheorem symm_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : e.symm.symm x = e x :=\n  rfl\n#align continuous_linear_equiv.symm_symm_apply ContinuousLinearEquiv.symm_symm_apply\n\ntheorem symm_apply_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x y} : e.symm x = y \u2194 x = e y :=\n  e.toLinearEquiv.symm_apply_eq\n#align continuous_linear_equiv.symm_apply_eq ContinuousLinearEquiv.symm_apply_eq\n\ntheorem eq_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x y} : y = e.symm x \u2194 e y = x :=\n  e.toLinearEquiv.eq_symm_apply\n#align continuous_linear_equiv.eq_symm_apply ContinuousLinearEquiv.eq_symm_apply\n\nprotected theorem image_eq_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2081) : e '' s = e.symm \u207b\u00b9' s :=\n  e.toLinearEquiv.toEquiv.image_eq_preimage s\n#align continuous_linear_equiv.image_eq_preimage ContinuousLinearEquiv.image_eq_preimage\n\nprotected theorem image_symm_eq_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2082) :\n    e.symm '' s = e \u207b\u00b9' s := by rw [e.symm.image_eq_preimage, e.symm_symm]\n#align continuous_linear_equiv.image_symm_eq_preimage ContinuousLinearEquiv.image_symm_eq_preimage\n\n@[simp]\nprotected theorem symm_preimage_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2082) :\n    e.symm \u207b\u00b9' (e \u207b\u00b9' s) = s :=\n  e.toLinearEquiv.toEquiv.symm_preimage_preimage s\n#align continuous_linear_equiv.symm_preimage_preimage ContinuousLinearEquiv.symm_preimage_preimage\n\n@[simp]\nprotected theorem preimage_symm_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : Set M\u2081) :\n    e \u207b\u00b9' (e.symm \u207b\u00b9' s) = s :=\n  e.symm.symm_preimage_preimage s\n#align continuous_linear_equiv.preimage_symm_preimage ContinuousLinearEquiv.preimage_symm_preimage\n\nprotected theorem uniformEmbedding {E\u2081 E\u2082 : Type _} [UniformSpace E\u2081] [UniformSpace E\u2082]\n    [AddCommGroup E\u2081] [AddCommGroup E\u2082] [Module R\u2081 E\u2081] [Module R\u2082 E\u2082] [UniformAddGroup E\u2081]\n    [UniformAddGroup E\u2082] (e : E\u2081 \u2243SL[\u03c3\u2081\u2082] E\u2082) : UniformEmbedding e :=\n  e.toLinearEquiv.toEquiv.UniformEmbedding e.toContinuousLinearMap.UniformContinuous\n    e.symm.toContinuousLinearMap.UniformContinuous\n#align continuous_linear_equiv.uniform_embedding ContinuousLinearEquiv.uniformEmbedding\n\nprotected theorem LinearEquiv.uniformEmbedding {E\u2081 E\u2082 : Type _} [UniformSpace E\u2081] [UniformSpace E\u2082]\n    [AddCommGroup E\u2081] [AddCommGroup E\u2082] [Module R\u2081 E\u2081] [Module R\u2082 E\u2082] [UniformAddGroup E\u2081]\n    [UniformAddGroup E\u2082] (e : E\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] E\u2082) (h\u2081 : Continuous e) (h\u2082 : Continuous e.symm) :\n    UniformEmbedding e :=\n  ContinuousLinearEquiv.uniformEmbedding\n    ({ e with\n        continuous_toFun := h\u2081\n        continuous_invFun := h\u2082 } :\n      E\u2081 \u2243SL[\u03c3\u2081\u2082] E\u2082)\n#align linear_equiv.uniform_embedding LinearEquiv.uniformEmbedding\n\nomit \u03c3\u2082\u2081\n\n/-- Create a `continuous_linear_equiv` from two `continuous_linear_map`s that are\ninverse of each other. -/\ndef equivOfInverse (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081) (h\u2081 : Function.LeftInverse f\u2082 f\u2081)\n    (h\u2082 : Function.RightInverse f\u2082 f\u2081) : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082 :=\n  { f\u2081 with\n    toFun := f\u2081\n    continuous_toFun := f\u2081.Continuous\n    invFun := f\u2082\n    continuous_invFun := f\u2082.Continuous\n    left_inv := h\u2081\n    right_inv := h\u2082 }\n#align continuous_linear_equiv.equiv_of_inverse ContinuousLinearEquiv.equivOfInverse\n\ninclude \u03c3\u2082\u2081\n\n@[simp]\ntheorem equivOfInverse_apply (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 h\u2081 h\u2082 x) :\n    equivOfInverse f\u2081 f\u2082 h\u2081 h\u2082 x = f\u2081 x :=\n  rfl\n#align continuous_linear_equiv.equiv_of_inverse_apply ContinuousLinearEquiv.equivOfInverse_apply\n\n@[simp]\ntheorem symm_equivOfInverse (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 h\u2081 h\u2082) :\n    (equivOfInverse f\u2081 f\u2082 h\u2081 h\u2082).symm = equivOfInverse f\u2082 f\u2081 h\u2082 h\u2081 :=\n  rfl\n#align continuous_linear_equiv.symm_equiv_of_inverse ContinuousLinearEquiv.symm_equivOfInverse\n\nomit \u03c3\u2082\u2081\n\nvariable (M\u2081)\n\n/-- The continuous linear equivalences from `M` to itself form a group under composition. -/\ninstance automorphismGroup : Group (M\u2081 \u2243L[R\u2081] M\u2081)\n    where\n  mul f g := g.trans f\n  one := ContinuousLinearEquiv.refl R\u2081 M\u2081\n  inv f := f.symm\n  mul_assoc f g h := by\n    ext\n    rfl\n  mul_one f := by\n    ext\n    rfl\n  one_mul f := by\n    ext\n    rfl\n  mul_left_inv f := by\n    ext\n    exact f.left_inv x\n#align continuous_linear_equiv.automorphism_group ContinuousLinearEquiv.automorphismGroup\n\nvariable {M\u2081} {R\u2084 : Type _} [Semiring R\u2084] [Module R\u2084 M\u2084] {\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084} {\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083}\n  [RingHomInvPair \u03c3\u2083\u2084 \u03c3\u2084\u2083] [RingHomInvPair \u03c3\u2084\u2083 \u03c3\u2083\u2084] {\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084} {\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084}\n  [RingHomCompTriple \u03c3\u2082\u2081 \u03c3\u2081\u2084 \u03c3\u2082\u2084] [RingHomCompTriple \u03c3\u2082\u2084 \u03c3\u2084\u2083 \u03c3\u2082\u2083] [RingHomCompTriple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084]\n\n/-- The continuous linear equivalence between `ulift M\u2081` and `M\u2081`. -/\ndef ulift : ULift M\u2081 \u2243L[R\u2081] M\u2081 :=\n  { Equiv.ulift with\n    map_add' := fun x y => rfl\n    map_smul' := fun c x => rfl\n    continuous_toFun := continuous_uLift_down\n    continuous_invFun := continuous_uLift_up }\n#align continuous_linear_equiv.ulift ContinuousLinearEquiv.ulift\n\ninclude \u03c3\u2082\u2081 \u03c3\u2083\u2084 \u03c3\u2082\u2083 \u03c3\u2082\u2084 \u03c3\u2081\u2083\n\n/-- A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of\ncontinuous linear maps. See also `continuous_linear_equiv.arrow_congr`. -/\n@[simps]\ndef arrowCongrEquiv (e\u2081\u2082 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2084\u2083 : M\u2084 \u2243SL[\u03c3\u2084\u2083] M\u2083) :\n    (M\u2081 \u2192SL[\u03c3\u2081\u2084] M\u2084) \u2243 (M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083)\n    where\n  toFun f := (e\u2084\u2083 : M\u2084 \u2192SL[\u03c3\u2084\u2083] M\u2083).comp (f.comp (e\u2081\u2082.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081))\n  invFun f := (e\u2084\u2083.symm : M\u2083 \u2192SL[\u03c3\u2083\u2084] M\u2084).comp (f.comp (e\u2081\u2082 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082))\n  left_inv f :=\n    ContinuousLinearMap.ext fun x => by\n      simp only [ContinuousLinearMap.comp_apply, symm_apply_apply, coe_coe]\n  right_inv f :=\n    ContinuousLinearMap.ext fun x => by\n      simp only [ContinuousLinearMap.comp_apply, apply_symm_apply, coe_coe]\n#align continuous_linear_equiv.arrow_congr_equiv ContinuousLinearEquiv.arrowCongrEquiv\n\nend AddCommMonoid\n\nsection AddCommGroup\n\nvariable {R : Type _} [Semiring R] {M : Type _} [TopologicalSpace M] [AddCommGroup M] {M\u2082 : Type _}\n  [TopologicalSpace M\u2082] [AddCommGroup M\u2082] {M\u2083 : Type _} [TopologicalSpace M\u2083] [AddCommGroup M\u2083]\n  {M\u2084 : Type _} [TopologicalSpace M\u2084] [AddCommGroup M\u2084] [Module R M] [Module R M\u2082] [Module R M\u2083]\n  [Module R M\u2084]\n\nvariable [TopologicalAddGroup M\u2084]\n\n/-- Equivalence given by a block lower diagonal matrix. `e` and `e'` are diagonal square blocks,\n  and `f` is a rectangular block below the diagonal. -/\ndef skewProd (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) : (M \u00d7 M\u2083) \u2243L[R] M\u2082 \u00d7 M\u2084 :=\n  {\n    e.toLinearEquiv.skewProd e'.toLinearEquiv\n      \u2191f with\n    continuous_toFun :=\n      (e.continuous_toFun.comp continuous_fst).prod_mk\n        ((e'.continuous_toFun.comp continuous_snd).add <| f.Continuous.comp continuous_fst)\n    continuous_invFun :=\n      (e.continuous_invFun.comp continuous_fst).prod_mk\n        (e'.continuous_invFun.comp <|\n          continuous_snd.sub <| f.Continuous.comp <| e.continuous_invFun.comp continuous_fst) }\n#align continuous_linear_equiv.skew_prod ContinuousLinearEquiv.skewProd\n\n@[simp]\ntheorem skewProd_apply (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) (x) :\n    e.skewProd e' f x = (e x.1, e' x.2 + f x.1) :=\n  rfl\n#align continuous_linear_equiv.skew_prod_apply ContinuousLinearEquiv.skewProd_apply\n\n@[simp]\ntheorem skewProd_symm_apply (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) (x) :\n    (e.skewProd e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) :=\n  rfl\n#align continuous_linear_equiv.skew_prod_symm_apply ContinuousLinearEquiv.skewProd_symm_apply\n\nend AddCommGroup\n\nsection Ring\n\nvariable {R : Type _} [Ring R] {R\u2082 : Type _} [Ring R\u2082] {M : Type _} [TopologicalSpace M]\n  [AddCommGroup M] [Module R M] {M\u2082 : Type _} [TopologicalSpace M\u2082] [AddCommGroup M\u2082] [Module R\u2082 M\u2082]\n\nvariable {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2081 : R\u2082 \u2192+* R} [RingHomInvPair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [RingHomInvPair \u03c3\u2082\u2081 \u03c3\u2081\u2082]\n\ninclude \u03c3\u2082\u2081\n\n@[simp]\ntheorem map_sub (e : M \u2243SL[\u03c3\u2081\u2082] M\u2082) (x y : M) : e (x - y) = e x - e y :=\n  (e : M \u2192SL[\u03c3\u2081\u2082] M\u2082).map_sub x y\n#align continuous_linear_equiv.map_sub ContinuousLinearEquiv.map_sub\n\n@[simp]\ntheorem map_neg (e : M \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M) : e (-x) = -e x :=\n  (e : M \u2192SL[\u03c3\u2081\u2082] M\u2082).map_neg x\n#align continuous_linear_equiv.map_neg ContinuousLinearEquiv.map_neg\n\nomit \u03c3\u2082\u2081\n\nsection\n\n/-! The next theorems cover the identification between `M \u2243L[\ud835\udd5c] M`and the group of units of the ring\n`M \u2192L[R] M`. -/\n\n\nvariable [TopologicalAddGroup M]\n\n/-- An invertible continuous linear map `f` determines a continuous equivalence from `M` to itself.\n-/\ndef ofUnit (f : (M \u2192L[R] M)\u02e3) : M \u2243L[R] M\n    where\n  toLinearEquiv :=\n    { toFun := f.val\n      map_add' := by simp\n      map_smul' := by simp\n      invFun := f.inv\n      left_inv := fun x =>\n        show (f.inv * f.val) x = x by\n          rw [f.inv_val]\n          simp\n      right_inv := fun x =>\n        show (f.val * f.inv) x = x by\n          rw [f.val_inv]\n          simp }\n  continuous_toFun := f.val.Continuous\n  continuous_invFun := f.inv.Continuous\n#align continuous_linear_equiv.of_unit ContinuousLinearEquiv.ofUnit\n\n/-- A continuous equivalence from `M` to itself determines an invertible continuous linear map. -/\ndef toUnit (f : M \u2243L[R] M) : (M \u2192L[R] M)\u02e3 where\n  val := f\n  inv := f.symm\n  val_inv := by\n    ext\n    simp\n  inv_val := by\n    ext\n    simp\n#align continuous_linear_equiv.to_unit ContinuousLinearEquiv.toUnit\n\nvariable (R M)\n\n/-- The units of the algebra of continuous `R`-linear endomorphisms of `M` is multiplicatively\nequivalent to the type of continuous linear equivalences between `M` and itself. -/\ndef unitsEquiv : (M \u2192L[R] M)\u02e3 \u2243* M \u2243L[R] M\n    where\n  toFun := ofUnit\n  invFun := toUnit\n  left_inv f := by\n    ext\n    rfl\n  right_inv f := by\n    ext\n    rfl\n  map_mul' x y := by\n    ext\n    rfl\n#align continuous_linear_equiv.units_equiv ContinuousLinearEquiv.unitsEquiv\n\n@[simp]\ntheorem unitsEquiv_apply (f : (M \u2192L[R] M)\u02e3) (x : M) : unitsEquiv R M f x = f x :=\n  rfl\n#align continuous_linear_equiv.units_equiv_apply ContinuousLinearEquiv.unitsEquiv_apply\n\nend\n\nsection\n\nvariable (R) [TopologicalSpace R] [ContinuousMul R]\n\n/-- Continuous linear equivalences `R \u2243L[R] R` are enumerated by `R\u02e3`. -/\ndef unitsEquivAut : R\u02e3 \u2243 R \u2243L[R] R\n    where\n  toFun u :=\n    equivOfInverse (ContinuousLinearMap.smulRight (1 : R \u2192L[R] R) \u2191u)\n      (ContinuousLinearMap.smulRight (1 : R \u2192L[R] R) \u2191u\u207b\u00b9) (fun x => by simp) fun x => by simp\n  invFun e :=\n    \u27e8e 1, e.symm 1, by rw [\u2190 smul_eq_mul, \u2190 map_smul, smul_eq_mul, mul_one, symm_apply_apply], by\n      rw [\u2190 smul_eq_mul, \u2190 map_smul, smul_eq_mul, mul_one, apply_symm_apply]\u27e9\n  left_inv u := Units.ext <| by simp\n  right_inv e := ext\u2081 <| by simp\n#align continuous_linear_equiv.units_equiv_aut ContinuousLinearEquiv.unitsEquivAut\n\nvariable {R}\n\n@[simp]\ntheorem unitsEquivAut_apply (u : R\u02e3) (x : R) : unitsEquivAut R u x = x * u :=\n  rfl\n#align continuous_linear_equiv.units_equiv_aut_apply ContinuousLinearEquiv.unitsEquivAut_apply\n\n@[simp]\ntheorem unitsEquivAut_apply_symm (u : R\u02e3) (x : R) : (unitsEquivAut R u).symm x = x * \u2191u\u207b\u00b9 :=\n  rfl\n#align continuous_linear_equiv.units_equiv_aut_apply_symm ContinuousLinearEquiv.unitsEquivAut_apply_symm\n\n@[simp]\ntheorem unitsEquivAut_symm_apply (e : R \u2243L[R] R) : \u2191((unitsEquivAut R).symm e) = e 1 :=\n  rfl\n#align continuous_linear_equiv.units_equiv_aut_symm_apply ContinuousLinearEquiv.unitsEquivAut_symm_apply\n\nend\n\nvariable [Module R M\u2082] [TopologicalAddGroup M]\n\nopen _Root_.ContinuousLinearMap (id fst snd)\n\nopen _Root_.LinearMap (mem_ker)\n\n/-- A pair of continuous linear maps such that `f\u2081 \u2218 f\u2082 = id` generates a continuous\nlinear equivalence `e` between `M` and `M\u2082 \u00d7 f\u2081.ker` such that `(e x).2 = x` for `x \u2208 f\u2081.ker`,\n`(e x).1 = f\u2081 x`, and `(e (f\u2082 y)).2 = 0`. The map is given by `e x = (f\u2081 x, x - f\u2082 (f\u2081 x))`. -/\ndef equivOfRightInverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M) (h : Function.RightInverse f\u2082 f\u2081) :\n    M \u2243L[R] M\u2082 \u00d7 ker f\u2081 :=\n  equivOfInverse (f\u2081.Prod (f\u2081.projKerOfRightInverse f\u2082 h)) (f\u2082.coprod (ker f\u2081).subtypeL)\n    (fun x => by simp) fun \u27e8x, y\u27e9 => by simp [h x]\n#align continuous_linear_equiv.equiv_of_right_inverse ContinuousLinearEquiv.equivOfRightInverse\n\n@[simp]\ntheorem fst_equivOfRightInverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n    (h : Function.RightInverse f\u2082 f\u2081) (x : M) : (equivOfRightInverse f\u2081 f\u2082 h x).1 = f\u2081 x :=\n  rfl\n#align continuous_linear_equiv.fst_equiv_of_right_inverse ContinuousLinearEquiv.fst_equivOfRightInverse\n\n@[simp]\ntheorem snd_equivOfRightInverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n    (h : Function.RightInverse f\u2082 f\u2081) (x : M) :\n    ((equivOfRightInverse f\u2081 f\u2082 h x).2 : M) = x - f\u2082 (f\u2081 x) :=\n  rfl\n#align continuous_linear_equiv.snd_equiv_of_right_inverse ContinuousLinearEquiv.snd_equivOfRightInverse\n\n@[simp]\ntheorem equivOfRightInverse_symm_apply (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n    (h : Function.RightInverse f\u2082 f\u2081) (y : M\u2082 \u00d7 ker f\u2081) :\n    (equivOfRightInverse f\u2081 f\u2082 h).symm y = f\u2082 y.1 + y.2 :=\n  rfl\n#align continuous_linear_equiv.equiv_of_right_inverse_symm_apply ContinuousLinearEquiv.equivOfRightInverse_symm_apply\n\nend Ring\n\nsection\n\nvariable (\u03b9 R M : Type _) [Unique \u03b9] [Semiring R] [AddCommMonoid M] [Module R M]\n  [TopologicalSpace M]\n\n/-- If `\u03b9` has a unique element, then `\u03b9 \u2192 M` is continuously linear equivalent to `M`. -/\ndef funUnique : (\u03b9 \u2192 M) \u2243L[R] M :=\n  { Homeomorph.funUnique \u03b9 M with toLinearEquiv := LinearEquiv.funUnique \u03b9 R M }\n#align continuous_linear_equiv.fun_unique ContinuousLinearEquiv.funUnique\n\nvariable {\u03b9 R M}\n\n@[simp]\ntheorem coe_funUnique : \u21d1(funUnique \u03b9 R M) = Function.eval default :=\n  rfl\n#align continuous_linear_equiv.coe_fun_unique ContinuousLinearEquiv.coe_funUnique\n\n@[simp]\ntheorem coe_funUnique_symm : \u21d1(funUnique \u03b9 R M).symm = Function.const \u03b9 :=\n  rfl\n#align continuous_linear_equiv.coe_fun_unique_symm ContinuousLinearEquiv.coe_funUnique_symm\n\nvariable (R M)\n\n/-- Continuous linear equivalence between dependent functions `\u03a0 i : fin 2, M i` and `M 0 \u00d7 M 1`. -/\n@[simps (config := { fullyApplied := false })]\ndef piFinTwo (M : Fin 2 \u2192 Type _) [\u2200 i, AddCommMonoid (M i)] [\u2200 i, Module R (M i)]\n    [\u2200 i, TopologicalSpace (M i)] : (\u2200 i, M i) \u2243L[R] M 0 \u00d7 M 1 :=\n  { Homeomorph.piFinTwo M with toLinearEquiv := LinearEquiv.piFinTwo R M }\n#align continuous_linear_equiv.pi_fin_two ContinuousLinearEquiv.piFinTwo\n\n/-- Continuous linear equivalence between vectors in `M\u00b2 = fin 2 \u2192 M` and `M \u00d7 M`. -/\n@[simps (config := { fullyApplied := false })]\ndef finTwoArrow : (Fin 2 \u2192 M) \u2243L[R] M \u00d7 M :=\n  { piFinTwo R fun _ => M with toLinearEquiv := LinearEquiv.finTwoArrow R M }\n#align continuous_linear_equiv.fin_two_arrow ContinuousLinearEquiv.finTwoArrow\n\nend\n\nend ContinuousLinearEquiv\n\nnamespace ContinuousLinearMap\n\nopen Classical\n\nvariable {R : Type _} {M : Type _} {M\u2082 : Type _} [TopologicalSpace M] [TopologicalSpace M\u2082]\n\nsection\n\nvariable [Semiring R]\n\nvariable [AddCommMonoid M\u2082] [Module R M\u2082]\n\nvariable [AddCommMonoid M] [Module R M]\n\n/-- Introduce a function `inverse` from `M \u2192L[R] M\u2082` to `M\u2082 \u2192L[R] M`, which sends `f` to `f.symm` if\n`f` is a continuous linear equivalence and to `0` otherwise.  This definition is somewhat ad hoc,\nbut one needs a fully (rather than partially) defined inverse function for some purposes, including\nfor calculus. -/\nnoncomputable def inverse : (M \u2192L[R] M\u2082) \u2192 M\u2082 \u2192L[R] M := fun f =>\n  if h : \u2203 e : M \u2243L[R] M\u2082, (e : M \u2192L[R] M\u2082) = f then ((Classical.choose h).symm : M\u2082 \u2192L[R] M) else 0\n#align continuous_linear_map.inverse ContinuousLinearMap.inverse\n\n/-- By definition, if `f` is invertible then `inverse f = f.symm`. -/\n@[simp]\ntheorem inverse_equiv (e : M \u2243L[R] M\u2082) : inverse (e : M \u2192L[R] M\u2082) = e.symm :=\n  by\n  have h : \u2203 e' : M \u2243L[R] M\u2082, (e' : M \u2192L[R] M\u2082) = \u2191e := \u27e8e, rfl\u27e9\n  simp only [inverse, dif_pos h]\n  congr\n  exact_mod_cast Classical.choose_spec h\n#align continuous_linear_map.inverse_equiv ContinuousLinearMap.inverse_equiv\n\n/-- By definition, if `f` is not invertible then `inverse f = 0`. -/\n@[simp]\ntheorem inverse_non_equiv (f : M \u2192L[R] M\u2082) (h : \u00ac\u2203 e' : M \u2243L[R] M\u2082, \u2191e' = f) : inverse f = 0 :=\n  dif_neg h\n#align continuous_linear_map.inverse_non_equiv ContinuousLinearMap.inverse_non_equiv\n\nend\n\nsection\n\nvariable [Ring R]\n\nvariable [AddCommGroup M] [TopologicalAddGroup M] [Module R M]\n\nvariable [AddCommGroup M\u2082] [Module R M\u2082]\n\n@[simp]\ntheorem ring_inverse_equiv (e : M \u2243L[R] M) : Ring.inverse \u2191e = inverse (e : M \u2192L[R] M) :=\n  by\n  suffices Ring.inverse ((ContinuousLinearEquiv.unitsEquiv _ _).symm e : M \u2192L[R] M) = inverse \u2191e by\n    convert this\n  simp\n  rfl\n#align continuous_linear_map.ring_inverse_equiv ContinuousLinearMap.ring_inverse_equiv\n\n/-- The function `continuous_linear_equiv.inverse` can be written in terms of `ring.inverse` for the\nring of self-maps of the domain. -/\ntheorem to_ring_inverse (e : M \u2243L[R] M\u2082) (f : M \u2192L[R] M\u2082) :\n    inverse f = Ring.inverse ((e.symm : M\u2082 \u2192L[R] M).comp f) \u2218L \u2191e.symm :=\n  by\n  by_cases h\u2081 : \u2203 e' : M \u2243L[R] M\u2082, \u2191e' = f\n  \u00b7 obtain \u27e8e', he'\u27e9 := h\u2081\n    rw [\u2190 he']\n    change _ = Ring.inverse \u2191(e'.trans e.symm) \u2218L \u2191e.symm\n    ext\n    simp\n  \u00b7 suffices \u00acIsUnit ((e.symm : M\u2082 \u2192L[R] M).comp f) by simp [this, h\u2081]\n    contrapose! h\u2081\n    rcases h\u2081 with \u27e8F, hF\u27e9\n    use (ContinuousLinearEquiv.unitsEquiv _ _ F).trans e\n    ext\n    dsimp\n    rw [coeFn_coe_base' F, hF]\n    simp\n#align continuous_linear_map.to_ring_inverse ContinuousLinearMap.to_ring_inverse\n\ntheorem ring_inverse_eq_map_inverse : Ring.inverse = @inverse R M M _ _ _ _ _ _ _ :=\n  by\n  ext\n  simp [to_ring_inverse (ContinuousLinearEquiv.refl R M)]\n#align continuous_linear_map.ring_inverse_eq_map_inverse ContinuousLinearMap.ring_inverse_eq_map_inverse\n\nend\n\nend ContinuousLinearMap\n\nnamespace Submodule\n\nvariable {R : Type _} [Ring R] {M : Type _} [TopologicalSpace M] [AddCommGroup M] [Module R M]\n  {M\u2082 : Type _} [TopologicalSpace M\u2082] [AddCommGroup M\u2082] [Module R M\u2082]\n\nopen ContinuousLinearMap\n\n/-- A submodule `p` is called *complemented* if there exists a continuous projection `M \u2192\u2097[R] p`. -/\ndef ClosedComplemented (p : Submodule R M) : Prop :=\n  \u2203 f : M \u2192L[R] p, \u2200 x : p, f x = x\n#align submodule.closed_complemented Submodule.ClosedComplemented\n\ntheorem ClosedComplemented.has_closed_complement {p : Submodule R M} [T1Space p]\n    (h : ClosedComplemented p) : \u2203 (q : Submodule R M)(hq : IsClosed (q : Set M)), IsCompl p q :=\n  Exists.elim h fun f hf => \u27e8ker f, f.isClosed_ker, LinearMap.isCompl_of_proj hf\u27e9\n#align submodule.closed_complemented.has_closed_complement Submodule.ClosedComplemented.has_closed_complement\n\nprotected theorem ClosedComplemented.isClosed [TopologicalAddGroup M] [T1Space M]\n    {p : Submodule R M} (h : ClosedComplemented p) : IsClosed (p : Set M) :=\n  by\n  rcases h with \u27e8f, hf\u27e9\n  have : ker (id R M - p.subtypeL.comp f) = p := LinearMap.ker_id_sub_eq_of_proj hf\n  exact this \u25b8 is_closed_ker _\n#align submodule.closed_complemented.is_closed Submodule.ClosedComplemented.isClosed\n\n@[simp]\ntheorem closedComplemented_bot : ClosedComplemented (\u22a5 : Submodule R M) :=\n  \u27e80, fun x => by simp only [zero_apply, eq_zero_of_bot_submodule x]\u27e9\n#align submodule.closed_complemented_bot Submodule.closedComplemented_bot\n\n@[simp]\ntheorem closedComplemented_top : ClosedComplemented (\u22a4 : Submodule R M) :=\n  \u27e8(id R M).codRestrict \u22a4 fun x => trivial, fun x => Subtype.ext_iff_val.2 <| by simp\u27e9\n#align submodule.closed_complemented_top Submodule.closedComplemented_top\n\nend Submodule\n\ntheorem ContinuousLinearMap.closedComplemented_ker_of_rightInverse {R : Type _} [Ring R]\n    {M : Type _} [TopologicalSpace M] [AddCommGroup M] {M\u2082 : Type _} [TopologicalSpace M\u2082]\n    [AddCommGroup M\u2082] [Module R M] [Module R M\u2082] [TopologicalAddGroup M] (f\u2081 : M \u2192L[R] M\u2082)\n    (f\u2082 : M\u2082 \u2192L[R] M) (h : Function.RightInverse f\u2082 f\u2081) : (ker f\u2081).ClosedComplemented :=\n  \u27e8f\u2081.projKerOfRightInverse f\u2082 h, f\u2081.projKerOfRightInverse_apply_idem f\u2082 h\u27e9\n#align continuous_linear_map.closed_complemented_ker_of_right_inverse ContinuousLinearMap.closedComplemented_ker_of_rightInverse\n\nsection Quotient\n\nnamespace Submodule\n\nvariable {R M : Type _} [Ring R] [AddCommGroup M] [Module R M] [TopologicalSpace M]\n  (S : Submodule R M)\n\ntheorem isOpenMap_mkQ [TopologicalAddGroup M] : IsOpenMap S.mkQ :=\n  QuotientAddGroup.isOpenMap_coe S.toAddSubgroup\n#align submodule.is_open_map_mkq Submodule.isOpenMap_mkQ\n\ninstance topologicalAddGroup_quotient [TopologicalAddGroup M] : TopologicalAddGroup (M \u29f8 S) :=\n  topologicalAddGroup_quotient S.toAddSubgroup\n#align submodule.topological_add_group_quotient Submodule.topologicalAddGroup_quotient\n\ninstance continuousSMul_quotient [TopologicalSpace R] [TopologicalAddGroup M] [ContinuousSMul R M] :\n    ContinuousSMul R (M \u29f8 S) := by\n  constructor\n  have quot : QuotientMap fun au : R \u00d7 M => (au.1, S.mkq au.2) :=\n    IsOpenMap.to_quotientMap (is_open_map.id.prod S.is_open_map_mkq)\n      (continuous_id.prod_map continuous_quot_mk)\n      (function.surjective_id.prod_map <| surjective_quot_mk _)\n  rw [quot.continuous_iff]\n  exact continuous_quot_mk.comp continuous_smul\n#align submodule.has_continuous_smul_quotient Submodule.continuousSMul_quotient\n\ninstance t3_quotient_of_isClosed [TopologicalAddGroup M] [IsClosed (S : Set M)] : T3Space (M \u29f8 S) :=\n  letI : IsClosed (S.to_add_subgroup : Set M) := \u2039_\u203a\n  S.to_add_subgroup.t3_quotient_of_is_closed\n#align submodule.t3_quotient_of_is_closed Submodule.t3_quotient_of_isClosed\n\nend Submodule\n\nend Quotient\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/Algebra/Module/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6150878555160666, "lm_q1q2_score": 0.39182990527463507}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-!\n# More `char` instances\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides a `linear_order` instance on `char`. `char` is the type of Unicode scalar values.\n-/\n\ninstance : linear_order char :=\n{ le_refl := \u03bb a, @le_refl \u2115 _ _,\n  le_trans := \u03bb a b c, @le_trans \u2115 _ _ _ _,\n  le_antisymm := \u03bb a b h\u2081 h\u2082,\n    char.eq_of_veq $ le_antisymm h\u2081 h\u2082,\n  le_total := \u03bb a b, @le_total \u2115 _ _ _,\n  lt_iff_le_not_le := \u03bb a b, @lt_iff_le_not_le \u2115 _ _ _,\n  decidable_le := char.decidable_le,\n  decidable_eq := char.decidable_eq,\n  decidable_lt := char.decidable_lt,\n  ..char.has_le, ..char.has_lt }\n\nlemma char.of_nat_to_nat {c : char} (h : is_valid_char c.to_nat) :\n  char.of_nat c.to_nat = c :=\nbegin\n  rw [char.of_nat, dif_pos h],\n  cases c,\n  simp [char.to_nat]\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/char.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.39180788657816473}}
{"text": "import data.real.basic\n\n\ntheorem AMC_2021_A_7 (x y:\u211d): (x*y-1)^2+(x+y)^2\u22651:=\nbegin\n  sorry\nend", "meta": {"author": "ahayat16", "repo": "lean_exos", "sha": "682f2552d5b04a8c8eb9e4ab15f875a91b03845c", "save_path": "github-repos/lean/ahayat16-lean_exos", "path": "github-repos/lean/ahayat16-lean_exos/lean_exos-682f2552d5b04a8c8eb9e4ab15f875a91b03845c/AMC_2021_A_7.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7772998611746911, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3916861963973245}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.category_theory.epi_mono\nimport Mathlib.PostPort\n\nuniverses v l u_1 u_2 u u\u2082 \n\nnamespace Mathlib\n\n/-!\n# Binary (co)products\n\nWe define a category `walking_pair`, which is the index category\nfor a binary (co)product diagram. A convenience method `pair X Y`\nconstructs the functor from the walking pair, hitting the given objects.\n\nWe define `prod X Y` and `coprod X Y` as limits and colimits of such functors.\n\nTypeclasses `has_binary_products` and `has_binary_coproducts` assert the existence\nof (co)limits shaped as walking pairs.\n\nWe include lemmas for simplifying equations involving projections and coprojections, and define\nbraiding and associating isomorphisms, and the product comparison morphism.\n\n## References\n* [Stacks: Products of pairs](https://stacks.math.columbia.edu/tag/001R)\n* [Stacks: coproducts of pairs](https://stacks.math.columbia.edu/tag/04AN)\n-/\n\nnamespace category_theory.limits\n\n\n/-- The type of objects for the diagram indexing a binary (co)product. -/\ninductive walking_pair where\n| left : walking_pair\n| right : walking_pair\n\n/--\nThe equivalence swapping left and right.\n-/\ndef walking_pair.swap : walking_pair \u2243 walking_pair :=\n  equiv.mk (fun (j : walking_pair) => walking_pair.rec_on j walking_pair.right walking_pair.left)\n    (fun (j : walking_pair) => walking_pair.rec_on j walking_pair.right walking_pair.left) sorry\n    sorry\n\n@[simp] theorem walking_pair.swap_apply_left :\n    coe_fn walking_pair.swap walking_pair.left = walking_pair.right :=\n  rfl\n\n@[simp] theorem walking_pair.swap_apply_right :\n    coe_fn walking_pair.swap walking_pair.right = walking_pair.left :=\n  rfl\n\n@[simp] theorem walking_pair.swap_symm_apply_tt :\n    coe_fn (equiv.symm walking_pair.swap) walking_pair.left = walking_pair.right :=\n  rfl\n\n@[simp] theorem walking_pair.swap_symm_apply_ff :\n    coe_fn (equiv.symm walking_pair.swap) walking_pair.right = walking_pair.left :=\n  rfl\n\n/--\nAn equivalence from `walking_pair` to `bool`, sometimes useful when reindexing limits.\n-/\ndef walking_pair.equiv_bool : walking_pair \u2243 Bool :=\n  equiv.mk (fun (j : walking_pair) => walking_pair.rec_on j tt false)\n    (fun (b : Bool) => bool.rec_on b walking_pair.right walking_pair.left) sorry sorry\n\n@[simp] theorem walking_pair.equiv_bool_apply_left :\n    coe_fn walking_pair.equiv_bool walking_pair.left = tt :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_apply_right :\n    coe_fn walking_pair.equiv_bool walking_pair.right = false :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_symm_apply_tt :\n    coe_fn (equiv.symm walking_pair.equiv_bool) tt = walking_pair.left :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_symm_apply_ff :\n    coe_fn (equiv.symm walking_pair.equiv_bool) false = walking_pair.right :=\n  rfl\n\n/-- The diagram on the walking pair, sending the two points to `X` and `Y`. -/\ndef pair {C : Type u} [category C] (X : C) (Y : C) : discrete walking_pair \u2964 C :=\n  discrete.functor fun (j : walking_pair) => walking_pair.cases_on j X Y\n\n@[simp] theorem pair_obj_left {C : Type u} [category C] (X : C) (Y : C) :\n    functor.obj (pair X Y) walking_pair.left = X :=\n  rfl\n\n@[simp] theorem pair_obj_right {C : Type u} [category C] (X : C) (Y : C) :\n    functor.obj (pair X Y) walking_pair.right = Y :=\n  rfl\n\n/-- The natural transformation between two functors out of the walking pair, specified by its components. -/\ndef map_pair {C : Type u} [category C] {F : discrete walking_pair \u2964 C}\n    {G : discrete walking_pair \u2964 C}\n    (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left)\n    (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) : F \u27f6 G :=\n  nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j f g\n\n@[simp] theorem map_pair_left {C : Type u} [category C] {F : discrete walking_pair \u2964 C}\n    {G : discrete walking_pair \u2964 C}\n    (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left)\n    (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) :\n    nat_trans.app (map_pair f g) walking_pair.left = f :=\n  rfl\n\n@[simp] theorem map_pair_right {C : Type u} [category C] {F : discrete walking_pair \u2964 C}\n    {G : discrete walking_pair \u2964 C}\n    (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left)\n    (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) :\n    nat_trans.app (map_pair f g) walking_pair.right = g :=\n  rfl\n\n/-- The natural isomorphism between two functors out of the walking pair, specified by its components. -/\ndef map_pair_iso {C : Type u} [category C] {F : discrete walking_pair \u2964 C}\n    {G : discrete walking_pair \u2964 C}\n    (f : functor.obj F walking_pair.left \u2245 functor.obj G walking_pair.left)\n    (g : functor.obj F walking_pair.right \u2245 functor.obj G walking_pair.right) : F \u2245 G :=\n  nat_iso.of_components (fun (j : discrete walking_pair) => walking_pair.cases_on j f g) sorry\n\n/-- Every functor out of the walking pair is naturally isomorphic (actually, equal) to a `pair` -/\n@[simp] theorem diagram_iso_pair_inv_app {C : Type u} [category C] (F : discrete walking_pair \u2964 C)\n    (X : discrete walking_pair) :\n    nat_trans.app (iso.inv (diagram_iso_pair F)) X =\n        iso.inv\n          (walking_pair.rec (iso.refl (functor.obj F walking_pair.left))\n            (iso.refl (functor.obj F walking_pair.right)) X) :=\n  Eq.refl\n    (iso.inv\n      (walking_pair.rec (iso.refl (functor.obj F walking_pair.left))\n        (iso.refl (functor.obj F walking_pair.right)) X))\n\n/-- The natural isomorphism between `pair X Y \u22d9 F` and `pair (F.obj X) (F.obj Y)`. -/\ndef pair_comp {C : Type u} [category C] {D : Type u} [category D] (X : C) (Y : C) (F : C \u2964 D) :\n    pair X Y \u22d9 F \u2245 pair (functor.obj F X) (functor.obj F Y) :=\n  diagram_iso_pair (pair X Y \u22d9 F)\n\n/-- A binary fan is just a cone on a diagram indexing a product. -/\ndef binary_fan {C : Type u} [category C] (X : C) (Y : C) := cone (pair X Y)\n\n/-- The first projection of a binary fan. -/\ndef binary_fan.fst {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) :\n    functor.obj (functor.obj (functor.const (discrete walking_pair)) (cone.X s)) walking_pair.left \u27f6\n        functor.obj (pair X Y) walking_pair.left :=\n  nat_trans.app (cone.\u03c0 s) walking_pair.left\n\n/-- The second projection of a binary fan. -/\ndef binary_fan.snd {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) :\n    functor.obj (functor.obj (functor.const (discrete walking_pair)) (cone.X s))\n          walking_pair.right \u27f6\n        functor.obj (pair X Y) walking_pair.right :=\n  nat_trans.app (cone.\u03c0 s) walking_pair.right\n\n@[simp] theorem binary_fan.\u03c0_app_left {C : Type u} [category C] {X : C} {Y : C}\n    (s : binary_fan X Y) : nat_trans.app (cone.\u03c0 s) walking_pair.left = binary_fan.fst s :=\n  rfl\n\n@[simp] theorem binary_fan.\u03c0_app_right {C : Type u} [category C] {X : C} {Y : C}\n    (s : binary_fan X Y) : nat_trans.app (cone.\u03c0 s) walking_pair.right = binary_fan.snd s :=\n  rfl\n\ntheorem binary_fan.is_limit.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {s : binary_fan X Y} (h : is_limit s) {f : W \u27f6 cone.X s} {g : W \u27f6 cone.X s}\n    (h\u2081 : f \u226b binary_fan.fst s = g \u226b binary_fan.fst s)\n    (h\u2082 : f \u226b binary_fan.snd s = g \u226b binary_fan.snd s) : f = g :=\n  is_limit.hom_ext h fun (j : discrete walking_pair) => walking_pair.cases_on j h\u2081 h\u2082\n\n/-- A binary cofan is just a cocone on a diagram indexing a coproduct. -/\ndef binary_cofan {C : Type u} [category C] (X : C) (Y : C) := cocone (pair X Y)\n\n/-- The first inclusion of a binary cofan. -/\ndef binary_cofan.inl {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) :\n    functor.obj (pair X Y) walking_pair.left \u27f6\n        functor.obj (functor.obj (functor.const (discrete walking_pair)) (cocone.X s))\n          walking_pair.left :=\n  nat_trans.app (cocone.\u03b9 s) walking_pair.left\n\n/-- The second inclusion of a binary cofan. -/\ndef binary_cofan.inr {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) :\n    functor.obj (pair X Y) walking_pair.right \u27f6\n        functor.obj (functor.obj (functor.const (discrete walking_pair)) (cocone.X s))\n          walking_pair.right :=\n  nat_trans.app (cocone.\u03b9 s) walking_pair.right\n\n@[simp] theorem binary_cofan.\u03b9_app_left {C : Type u} [category C] {X : C} {Y : C}\n    (s : binary_cofan X Y) : nat_trans.app (cocone.\u03b9 s) walking_pair.left = binary_cofan.inl s :=\n  rfl\n\n@[simp] theorem binary_cofan.\u03b9_app_right {C : Type u} [category C] {X : C} {Y : C}\n    (s : binary_cofan X Y) : nat_trans.app (cocone.\u03b9 s) walking_pair.right = binary_cofan.inr s :=\n  rfl\n\ntheorem binary_cofan.is_colimit.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {s : binary_cofan X Y} (h : is_colimit s) {f : cocone.X s \u27f6 W} {g : cocone.X s \u27f6 W}\n    (h\u2081 : binary_cofan.inl s \u226b f = binary_cofan.inl s \u226b g)\n    (h\u2082 : binary_cofan.inr s \u226b f = binary_cofan.inr s \u226b g) : f = g :=\n  is_colimit.hom_ext h fun (j : discrete walking_pair) => walking_pair.cases_on j h\u2081 h\u2082\n\n/-- A binary fan with vertex `P` consists of the two projections `\u03c0\u2081 : P \u27f6 X` and `\u03c0\u2082 : P \u27f6 Y`. -/\ndef binary_fan.mk {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) :\n    binary_fan X Y :=\n  cone.mk P (nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j \u03c0\u2081 \u03c0\u2082)\n\n/-- A binary cofan with vertex `P` consists of the two inclusions `\u03b9\u2081 : X \u27f6 P` and `\u03b9\u2082 : Y \u27f6 P`. -/\ndef binary_cofan.mk {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) :\n    binary_cofan X Y :=\n  cocone.mk P (nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j \u03b9\u2081 \u03b9\u2082)\n\n@[simp] theorem binary_fan.mk_\u03c0_app_left {C : Type u} [category C] {X : C} {Y : C} {P : C}\n    (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) :\n    nat_trans.app (cone.\u03c0 (binary_fan.mk \u03c0\u2081 \u03c0\u2082)) walking_pair.left = \u03c0\u2081 :=\n  rfl\n\n@[simp] theorem binary_fan.mk_\u03c0_app_right {C : Type u} [category C] {X : C} {Y : C} {P : C}\n    (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) :\n    nat_trans.app (cone.\u03c0 (binary_fan.mk \u03c0\u2081 \u03c0\u2082)) walking_pair.right = \u03c0\u2082 :=\n  rfl\n\n@[simp] theorem binary_cofan.mk_\u03b9_app_left {C : Type u} [category C] {X : C} {Y : C} {P : C}\n    (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) :\n    nat_trans.app (cocone.\u03b9 (binary_cofan.mk \u03b9\u2081 \u03b9\u2082)) walking_pair.left = \u03b9\u2081 :=\n  rfl\n\n@[simp] theorem binary_cofan.mk_\u03b9_app_right {C : Type u} [category C] {X : C} {Y : C} {P : C}\n    (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) :\n    nat_trans.app (cocone.\u03b9 (binary_cofan.mk \u03b9\u2081 \u03b9\u2082)) walking_pair.right = \u03b9\u2082 :=\n  rfl\n\n/-- If `s` is a limit binary fan over `X` and `Y`, then every pair of morphisms `f : W \u27f6 X` and\n    `g : W \u27f6 Y` induces a morphism `l : W \u27f6 s.X` satisfying `l \u226b s.fst = f` and `l \u226b s.snd = g`.\n    -/\n@[simp] theorem binary_fan.is_limit.lift'_coe {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {s : binary_fan X Y} (h : is_limit s) (f : W \u27f6 X) (g : W \u27f6 Y) :\n    \u2191(binary_fan.is_limit.lift' h f g) = is_limit.lift h (binary_fan.mk f g) :=\n  Eq.refl \u2191(binary_fan.is_limit.lift' h f g)\n\n/-- If `s` is a colimit binary cofan over `X` and `Y`,, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `l : s.X \u27f6 W` satisfying `s.inl \u226b l = f` and `s.inr \u226b l = g`.\n    -/\n@[simp] theorem binary_cofan.is_colimit.desc'_coe {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {s : binary_cofan X Y} (h : is_colimit s) (f : X \u27f6 W) (g : Y \u27f6 W) :\n    \u2191(binary_cofan.is_colimit.desc' h f g) = is_colimit.desc h (binary_cofan.mk f g) :=\n  Eq.refl \u2191(binary_cofan.is_colimit.desc' h f g)\n\n/-- An abbreviation for `has_limit (pair X Y)`. -/\n/-- An abbreviation for `has_colimit (pair X Y)`. -/\ndef has_binary_product {C : Type u} [category C] (X : C) (Y : C) := has_limit (pair X Y)\n\ndef has_binary_coproduct {C : Type u} [category C] (X : C) (Y : C) := has_colimit (pair X Y)\n\n/-- If we have a product of `X` and `Y`, we can access it using `prod X Y` or\n    `X \u2a2f Y`. -/\ndef prod {C : Type u} [category C] (X : C) (Y : C) [has_binary_product X Y] : C := limit (pair X Y)\n\n/-- If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y ` or\n    `X \u2a3f Y`. -/\ndef coprod {C : Type u} [category C] (X : C) (Y : C) [has_binary_coproduct X Y] : C :=\n  colimit (pair X Y)\n\ninfixl:20 \" \u2a2f \" => Mathlib.category_theory.limits.prod\n\ninfixl:20 \" \u2a3f \" => Mathlib.category_theory.limits.coprod\n\n/-- The projection map to the first component of the product. -/\ndef prod.fst {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : X \u2a2f Y \u27f6 X :=\n  limit.\u03c0 (pair X Y) walking_pair.left\n\n/-- The projecton map to the second component of the product. -/\ndef prod.snd {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : X \u2a2f Y \u27f6 Y :=\n  limit.\u03c0 (pair X Y) walking_pair.right\n\n/-- The inclusion map from the first component of the coproduct. -/\ndef coprod.inl {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : X \u27f6 X \u2a3f Y :=\n  colimit.\u03b9 (pair X Y) walking_pair.left\n\n/-- The inclusion map from the second component of the coproduct. -/\ndef coprod.inr {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : Y \u27f6 X \u2a3f Y :=\n  colimit.\u03b9 (pair X Y) walking_pair.right\n\n/-- The binary fan constructed from the projection maps is a limit. -/\ndef prod_is_prod {C : Type u} [category C] (X : C) (Y : C) [has_binary_product X Y] :\n    is_limit (binary_fan.mk prod.fst prod.snd) :=\n  is_limit.of_iso_limit (limit.is_limit (pair X Y))\n    (cones.ext (iso.refl (cone.X (limit.cone (pair X Y)))) sorry)\n\n/-- The binary cofan constructed from the coprojection maps is a colimit. -/\ndef coprod_is_coprod {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] :\n    is_colimit (binary_cofan.mk coprod.inl coprod.inr) :=\n  is_colimit.of_iso_colimit (colimit.is_colimit (pair X Y))\n    (cocones.ext (iso.refl (cocone.X (colimit.cocone (pair X Y)))) sorry)\n\ntheorem prod.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y]\n    {f : W \u27f6 X \u2a2f Y} {g : W \u27f6 X \u2a2f Y} (h\u2081 : f \u226b prod.fst = g \u226b prod.fst)\n    (h\u2082 : f \u226b prod.snd = g \u226b prod.snd) : f = g :=\n  binary_fan.is_limit.hom_ext (limit.is_limit (pair X Y)) h\u2081 h\u2082\n\ntheorem coprod.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y]\n    {f : X \u2a3f Y \u27f6 W} {g : X \u2a3f Y \u27f6 W} (h\u2081 : coprod.inl \u226b f = coprod.inl \u226b g)\n    (h\u2082 : coprod.inr \u226b f = coprod.inr \u226b g) : f = g :=\n  binary_cofan.is_colimit.hom_ext (colimit.is_colimit (pair X Y)) h\u2081 h\u2082\n\n/-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W \u27f6 X` and `g : W \u27f6 Y`\n    induces a morphism `prod.lift f g : W \u27f6 X \u2a2f Y`. -/\ndef prod.lift {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X)\n    (g : W \u27f6 Y) : W \u27f6 X \u2a2f Y :=\n  limit.lift (pair X Y) (binary_fan.mk f g)\n\n/-- diagonal arrow of the binary product in the category `fam I` -/\ndef diag {C : Type u} [category C] (X : C) [has_binary_product X X] : X \u27f6 X \u2a2f X := prod.lift \ud835\udfd9 \ud835\udfd9\n\n/-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `coprod.desc f g : X \u2a3f Y \u27f6 W`. -/\ndef coprod.desc {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y]\n    (f : X \u27f6 W) (g : Y \u27f6 W) : X \u2a3f Y \u27f6 W :=\n  colimit.desc (pair X Y) (binary_cofan.mk f g)\n\n/-- codiagonal arrow of the binary coproduct -/\ndef codiag {C : Type u} [category C] (X : C) [has_binary_coproduct X X] : X \u2a3f X \u27f6 X :=\n  coprod.desc \ud835\udfd9 \ud835\udfd9\n\n@[simp] theorem prod.lift_fst_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) {X' : C} (f' : X \u27f6 X') :\n    prod.lift f g \u226b prod.fst \u226b f' = f \u226b f' :=\n  sorry\n\n@[simp] theorem prod.lift_snd {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) : prod.lift f g \u226b prod.snd = g :=\n  limit.lift_\u03c0 (binary_fan.mk f g) walking_pair.right\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.inl_desc_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) {X' : C} (f' : W \u27f6 X') :\n    coprod.inl \u226b coprod.desc f g \u226b f' = f \u226b f' :=\n  sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.inr_desc_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) {X' : C} (f' : W \u27f6 X') :\n    coprod.inr \u226b coprod.desc f g \u226b f' = g \u226b f' :=\n  sorry\n\nprotected instance prod.mono_lift_of_mono_left {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) [mono f] : mono (prod.lift f g) :=\n  mono_of_mono_fac (prod.lift_fst f g)\n\nprotected instance prod.mono_lift_of_mono_right {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) [mono g] : mono (prod.lift f g) :=\n  mono_of_mono_fac (prod.lift_snd f g)\n\nprotected instance coprod.epi_desc_of_epi_left {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) [epi f] : epi (coprod.desc f g) :=\n  epi_of_epi_fac (coprod.inl_desc f g)\n\nprotected instance coprod.epi_desc_of_epi_right {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) [epi g] : epi (coprod.desc f g) :=\n  epi_of_epi_fac (coprod.inr_desc f g)\n\n/-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W \u27f6 X` and `g : W \u27f6 Y`\n    induces a morphism `l : W \u27f6 X \u2a2f Y` satisfying `l \u226b prod.fst = f` and `l \u226b prod.snd = g`. -/\ndef prod.lift' {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y]\n    (f : W \u27f6 X) (g : W \u27f6 Y) : Subtype fun (l : W \u27f6 X \u2a2f Y) => l \u226b prod.fst = f \u2227 l \u226b prod.snd = g :=\n  { val := prod.lift f g, property := sorry }\n\n/-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `l : X \u2a3f Y \u27f6 W` satisfying `coprod.inl \u226b l = f` and\n    `coprod.inr \u226b l = g`. -/\ndef coprod.desc' {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y]\n    (f : X \u27f6 W) (g : Y \u27f6 W) :\n    Subtype fun (l : X \u2a3f Y \u27f6 W) => coprod.inl \u226b l = f \u2227 coprod.inr \u226b l = g :=\n  { val := coprod.desc f g, property := sorry }\n\n/-- If the products `W \u2a2f X` and `Y \u2a2f Z` exist, then every pair of morphisms `f : W \u27f6 Y` and\n    `g : X \u27f6 Z` induces a morphism `prod.map f g : W \u2a2f X \u27f6 Y \u2a2f Z`. -/\ndef prod.map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X]\n    [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u2a2f X \u27f6 Y \u2a2f Z :=\n  lim_map (map_pair f g)\n\n/-- If the coproducts `W \u2a3f X` and `Y \u2a3f Z` exist, then every pair of morphisms `f : W \u27f6 Y` and\n    `g : W \u27f6 Z` induces a morphism `coprod.map f g : W \u2a3f X \u27f6 Y \u2a3f Z`. -/\ndef coprod.map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X]\n    [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u2a3f X \u27f6 Y \u2a3f Z :=\n  colim_map (map_pair f g)\n\n-- Making the reassoc version of this a simp lemma seems to be more harmful than helpful.\n\ntheorem prod.comp_lift_assoc {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C}\n    [has_binary_product X Y] (f : V \u27f6 W) (g : W \u27f6 X) (h : W \u27f6 Y) {X' : C} (f' : X \u2a2f Y \u27f6 X') :\n    f \u226b prod.lift g h \u226b f' = prod.lift (f \u226b g) (f \u226b h) \u226b f' :=\n  sorry\n\ntheorem prod.comp_diag {C : Type u} [category C] {X : C} {Y : C} [has_binary_product Y Y]\n    (f : X \u27f6 Y) : f \u226b diag Y = prod.lift f f :=\n  sorry\n\n@[simp] theorem prod.map_fst {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) :\n    prod.map f g \u226b prod.fst = prod.fst \u226b f :=\n  lim_map_\u03c0 (map_pair f g) walking_pair.left\n\n@[simp] theorem prod.map_snd_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) {X' : C}\n    (f' : Z \u27f6 X') : prod.map f g \u226b prod.snd \u226b f' = prod.snd \u226b g \u226b f' :=\n  sorry\n\n@[simp] theorem prod.map_id_id {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] :\n    prod.map \ud835\udfd9 \ud835\udfd9 = \ud835\udfd9 :=\n  sorry\n\n@[simp] theorem prod.lift_fst_snd {C : Type u} [category C] {X : C} {Y : C}\n    [has_binary_product X Y] : prod.lift prod.fst prod.snd = \ud835\udfd9 :=\n  sorry\n\n@[simp] theorem prod.lift_map {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_product W X] [has_binary_product Y Z] (f : V \u27f6 W) (g : V \u27f6 X) (h : W \u27f6 Y)\n    (k : X \u27f6 Z) : prod.lift f g \u226b prod.map h k = prod.lift (f \u226b h) (g \u226b k) :=\n  sorry\n\n@[simp] theorem prod.lift_fst_comp_snd_comp {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {Z : C} [has_binary_product W Y] [has_binary_product X Z] (g : W \u27f6 X) (g' : Y \u27f6 Z) :\n    prod.lift (prod.fst \u226b g) (prod.snd \u226b g') = prod.map g g' :=\n  sorry\n\n-- We take the right hand side here to be simp normal form, as this way composition lemmas for\n\n-- `f \u226b h` and `g \u226b k` can fire (eg `id_comp`) , while `map_fst` and `map_snd` can still work just\n\n-- as well.\n\n@[simp] theorem prod.map_map_assoc {C : Type u} [category C] {A\u2081 : C} {A\u2082 : C} {A\u2083 : C} {B\u2081 : C}\n    {B\u2082 : C} {B\u2083 : C} [has_binary_product A\u2081 B\u2081] [has_binary_product A\u2082 B\u2082]\n    [has_binary_product A\u2083 B\u2083] (f : A\u2081 \u27f6 A\u2082) (g : B\u2081 \u27f6 B\u2082) (h : A\u2082 \u27f6 A\u2083) (k : B\u2082 \u27f6 B\u2083) {X' : C}\n    (f' : A\u2083 \u2a2f B\u2083 \u27f6 X') : prod.map f g \u226b prod.map h k \u226b f' = prod.map (f \u226b h) (g \u226b k) \u226b f' :=\n  sorry\n\n-- TODO: is it necessary to weaken the assumption here?\n\ntheorem prod.map_swap {C : Type u} [category C] {A : C} {B : C} {X : C} {Y : C} (f : A \u27f6 B)\n    (g : X \u27f6 Y) [has_limits_of_shape (discrete walking_pair) C] :\n    prod.map \ud835\udfd9 f \u226b prod.map g \ud835\udfd9 = prod.map g \ud835\udfd9 \u226b prod.map \ud835\udfd9 f :=\n  sorry\n\ntheorem prod.map_comp_id {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y)\n    (g : Y \u27f6 Z) [has_binary_product X W] [has_binary_product Z W] [has_binary_product Y W] :\n    prod.map (f \u226b g) \ud835\udfd9 = prod.map f \ud835\udfd9 \u226b prod.map g \ud835\udfd9 :=\n  sorry\n\ntheorem prod.map_id_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y)\n    (g : Y \u27f6 Z) [has_binary_product W X] [has_binary_product W Y] [has_binary_product W Z] :\n    prod.map \ud835\udfd9 (f \u226b g) = prod.map \ud835\udfd9 f \u226b prod.map \ud835\udfd9 g :=\n  sorry\n\n/-- If the products `W \u2a2f X` and `Y \u2a2f Z` exist, then every pair of isomorphisms `f : W \u2245 Y` and\n    `g : X \u2245 Z` induces an isomorphism `prod.map_iso f g : W \u2a2f X \u2245 Y \u2a2f Z`. -/\n@[simp] theorem prod.map_iso_inv {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_product W X] [has_binary_product Y Z] (f : W \u2245 Y) (g : X \u2245 Z) :\n    iso.inv (prod.map_iso f g) = prod.map (iso.inv f) (iso.inv g) :=\n  Eq.refl (iso.inv (prod.map_iso f g))\n\nprotected instance is_iso_prod {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) [is_iso f]\n    [is_iso g] : is_iso (prod.map f g) :=\n  is_iso.of_iso (prod.map_iso (as_iso f) (as_iso g))\n\n@[simp] theorem prod.diag_map {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    [has_binary_product X X] [has_binary_product Y Y] : diag X \u226b prod.map f f = f \u226b diag Y :=\n  sorry\n\n@[simp] theorem prod.diag_map_fst_snd_assoc {C : Type u} [category C] {X : C} {Y : C}\n    [has_binary_product X Y] [has_binary_product (X \u2a2f Y) (X \u2a2f Y)] {X' : C} (f' : X \u2a2f Y \u27f6 X') :\n    diag (X \u2a2f Y) \u226b prod.map prod.fst prod.snd \u226b f' = f' :=\n  sorry\n\n@[simp] theorem prod.diag_map_fst_snd_comp_assoc {C : Type u} [category C]\n    [has_limits_of_shape (discrete walking_pair) C] {X : C} {X' : C} {Y : C} {Y' : C} (g : X \u27f6 Y)\n    (g' : X' \u27f6 Y') :\n    \u2200 {X'_1 : C} (f' : Y \u2a2f Y' \u27f6 X'_1),\n        diag (X \u2a2f X') \u226b prod.map (prod.fst \u226b g) (prod.snd \u226b g') \u226b f' = prod.map g g' \u226b f' :=\n  sorry\n\nprotected instance diag.category_theory.split_mono {C : Type u} [category C] {X : C}\n    [has_binary_product X X] : split_mono (diag X) :=\n  split_mono.mk prod.fst\n\n@[simp] theorem coprod.desc_comp_assoc {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C}\n    [has_binary_coproduct X Y] (f : V \u27f6 W) (g : X \u27f6 V) (h : Y \u27f6 V) {X' : C} (f' : W \u27f6 X') :\n    coprod.desc g h \u226b f \u226b f' = coprod.desc (g \u226b f) (h \u226b f) \u226b f' :=\n  sorry\n\ntheorem coprod.diag_comp {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X X]\n    (f : X \u27f6 Y) : codiag X \u226b f = coprod.desc f f :=\n  sorry\n\n@[simp] theorem coprod.inl_map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) :\n    coprod.inl \u226b coprod.map f g = f \u226b coprod.inl :=\n  \u03b9_colim_map (map_pair f g) walking_pair.left\n\n@[simp] theorem coprod.inr_map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) :\n    coprod.inr \u226b coprod.map f g = g \u226b coprod.inr :=\n  \u03b9_colim_map (map_pair f g) walking_pair.right\n\n@[simp] theorem coprod.map_id_id {C : Type u} [category C] {X : C} {Y : C}\n    [has_binary_coproduct X Y] : coprod.map \ud835\udfd9 \ud835\udfd9 = \ud835\udfd9 :=\n  sorry\n\n@[simp] theorem coprod.desc_inl_inr {C : Type u} [category C] {X : C} {Y : C}\n    [has_binary_coproduct X Y] : coprod.desc coprod.inl coprod.inr = \ud835\udfd9 :=\n  sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_desc {C : Type u} [category C] {S : C} {T : C} {U : C} {V : C} {W : C}\n    [has_binary_coproduct U W] [has_binary_coproduct T V] (f : U \u27f6 S) (g : W \u27f6 S) (h : T \u27f6 U)\n    (k : V \u27f6 W) : coprod.map h k \u226b coprod.desc f g = coprod.desc (h \u226b f) (k \u226b g) :=\n  sorry\n\n@[simp] theorem coprod.desc_comp_inl_comp_inr {C : Type u} [category C] {W : C} {X : C} {Y : C}\n    {Z : C} [has_binary_coproduct W Y] [has_binary_coproduct X Z] (g : W \u27f6 X) (g' : Y \u27f6 Z) :\n    coprod.desc (g \u226b coprod.inl) (g' \u226b coprod.inr) = coprod.map g g' :=\n  sorry\n\n-- We take the right hand side here to be simp normal form, as this way composition lemmas for\n\n-- `f \u226b h` and `g \u226b k` can fire (eg `id_comp`) , while `inl_map` and `inr_map` can still work just\n\n-- as well.\n\n@[simp] theorem coprod.map_map {C : Type u} [category C] {A\u2081 : C} {A\u2082 : C} {A\u2083 : C} {B\u2081 : C}\n    {B\u2082 : C} {B\u2083 : C} [has_binary_coproduct A\u2081 B\u2081] [has_binary_coproduct A\u2082 B\u2082]\n    [has_binary_coproduct A\u2083 B\u2083] (f : A\u2081 \u27f6 A\u2082) (g : B\u2081 \u27f6 B\u2082) (h : A\u2082 \u27f6 A\u2083) (k : B\u2082 \u27f6 B\u2083) :\n    coprod.map f g \u226b coprod.map h k = coprod.map (f \u226b h) (g \u226b k) :=\n  sorry\n\n-- I don't think it's a good idea to make any of the following three simp lemmas.\n\ntheorem coprod.map_swap_assoc {C : Type u} [category C] {A : C} {B : C} {X : C} {Y : C} (f : A \u27f6 B)\n    (g : X \u27f6 Y) [has_colimits_of_shape (discrete walking_pair) C] {X' : C} (f' : Y \u2a3f B \u27f6 X') :\n    coprod.map \ud835\udfd9 f \u226b coprod.map g \ud835\udfd9 \u226b f' = coprod.map g \ud835\udfd9 \u226b coprod.map \ud835\udfd9 f \u226b f' :=\n  sorry\n\ntheorem coprod.map_comp_id {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y)\n    (g : Y \u27f6 Z) [has_binary_coproduct Z W] [has_binary_coproduct Y W] [has_binary_coproduct X W] :\n    coprod.map (f \u226b g) \ud835\udfd9 = coprod.map f \ud835\udfd9 \u226b coprod.map g \ud835\udfd9 :=\n  sorry\n\ntheorem coprod.map_id_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y)\n    (g : Y \u27f6 Z) [has_binary_coproduct W X] [has_binary_coproduct W Y] [has_binary_coproduct W Z] :\n    coprod.map \ud835\udfd9 (f \u226b g) = coprod.map \ud835\udfd9 f \u226b coprod.map \ud835\udfd9 g :=\n  sorry\n\n/-- If the coproducts `W \u2a3f X` and `Y \u2a3f Z` exist, then every pair of isomorphisms `f : W \u2245 Y` and\n    `g : W \u2245 Z` induces a isomorphism `coprod.map_iso f g : W \u2a3f X \u2245 Y \u2a3f Z`. -/\n@[simp] theorem coprod.map_iso_hom {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u2245 Y) (g : X \u2245 Z) :\n    iso.hom (coprod.map_iso f g) = coprod.map (iso.hom f) (iso.hom g) :=\n  Eq.refl (iso.hom (coprod.map_iso f g))\n\nprotected instance is_iso_coprod {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C}\n    [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) [is_iso f]\n    [is_iso g] : is_iso (coprod.map f g) :=\n  is_iso.of_iso (coprod.map_iso (as_iso f) (as_iso g))\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_codiag {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    [has_binary_coproduct X X] [has_binary_coproduct Y Y] :\n    coprod.map f f \u226b codiag Y = codiag X \u226b f :=\n  sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.map_inl_inr_codiag_assoc {C : Type u} [category C] {X : C} {Y : C}\n    [has_binary_coproduct X Y] [has_binary_coproduct (X \u2a3f Y) (X \u2a3f Y)] {X' : C} (f' : X \u2a3f Y \u27f6 X') :\n    coprod.map coprod.inl coprod.inr \u226b codiag (X \u2a3f Y) \u226b f' = f' :=\n  sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_comp_inl_inr_codiag {C : Type u} [category C]\n    [has_colimits_of_shape (discrete walking_pair) C] {X : C} {X' : C} {Y : C} {Y' : C} (g : X \u27f6 Y)\n    (g' : X' \u27f6 Y') :\n    coprod.map (g \u226b coprod.inl) (g' \u226b coprod.inr) \u226b codiag (Y \u2a3f Y') = coprod.map g g' :=\n  sorry\n\n/--\n`has_binary_products` represents a choice of product for every pair of objects.\n\nSee https://stacks.math.columbia.edu/tag/001T.\n-/\ndef has_binary_products (C : Type u) [category C] := has_limits_of_shape (discrete walking_pair) C\n\n/--\n`has_binary_coproducts` represents a choice of coproduct for every pair of objects.\n\nSee https://stacks.math.columbia.edu/tag/04AP.\n-/\ndef has_binary_coproducts (C : Type u) [category C] :=\n  has_colimits_of_shape (discrete walking_pair) C\n\n/-- If `C` has all limits of diagrams `pair X Y`, then it has all binary products -/\ntheorem has_binary_products_of_has_limit_pair (C : Type u) [category C]\n    [\u2200 {X Y : C}, has_limit (pair X Y)] : has_binary_products C :=\n  has_limits_of_shape.mk\n    fun (F : discrete walking_pair \u2964 C) => has_limit_of_iso (iso.symm (diagram_iso_pair F))\n\n/-- If `C` has all colimits of diagrams `pair X Y`, then it has all binary coproducts -/\ntheorem has_binary_coproducts_of_has_colimit_pair (C : Type u) [category C]\n    [\u2200 {X Y : C}, has_colimit (pair X Y)] : has_binary_coproducts C :=\n  has_colimits_of_shape.mk\n    fun (F : discrete walking_pair \u2964 C) => has_colimit_of_iso (diagram_iso_pair F)\n\n/-- The braiding isomorphism which swaps a binary product. -/\n@[simp] theorem prod.braiding_hom {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q]\n    [has_binary_product Q P] : iso.hom (prod.braiding P Q) = prod.lift prod.snd prod.fst :=\n  Eq.refl (iso.hom (prod.braiding P Q))\n\n/-- The braiding isomorphism can be passed through a map by swapping the order. -/\ntheorem braid_natural {C : Type u} [category C] [has_binary_products C] {W : C} {X : C} {Y : C}\n    {Z : C} (f : X \u27f6 Y) (g : Z \u27f6 W) :\n    prod.map f g \u226b iso.hom (prod.braiding Y W) = iso.hom (prod.braiding X Z) \u226b prod.map g f :=\n  sorry\n\ntheorem prod.symmetry'_assoc {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q]\n    [has_binary_product Q P] {X' : C} (f' : P \u2a2f Q \u27f6 X') :\n    prod.lift prod.snd prod.fst \u226b prod.lift prod.snd prod.fst \u226b f' = f' :=\n  sorry\n\n/-- The braiding isomorphism is symmetric. -/\ntheorem prod.symmetry_assoc {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q]\n    [has_binary_product Q P] {X' : C} (f' : P \u2a2f Q \u27f6 X') :\n    iso.hom (prod.braiding P Q) \u226b iso.hom (prod.braiding Q P) \u226b f' = f' :=\n  sorry\n\n/-- The associator isomorphism for binary products. -/\n@[simp] theorem prod.associator_hom {C : Type u} [category C] [has_binary_products C] (P : C)\n    (Q : C) (R : C) :\n    iso.hom (prod.associator P Q R) =\n        prod.lift (prod.fst \u226b prod.fst) (prod.lift (prod.fst \u226b prod.snd) prod.snd) :=\n  Eq.refl (iso.hom (prod.associator P Q R))\n\ntheorem prod.pentagon_assoc {C : Type u} [category C] [has_binary_products C] (W : C) (X : C)\n    (Y : C) (Z : C) {X' : C} (f' : W \u2a2f (X \u2a2f (Y \u2a2f Z)) \u27f6 X') :\n    prod.map (iso.hom (prod.associator W X Y)) \ud835\udfd9 \u226b\n          iso.hom (prod.associator W (X \u2a2f Y) Z) \u226b\n            prod.map \ud835\udfd9 (iso.hom (prod.associator X Y Z)) \u226b f' =\n        iso.hom (prod.associator (W \u2a2f X) Y Z) \u226b iso.hom (prod.associator W X (Y \u2a2f Z)) \u226b f' :=\n  sorry\n\ntheorem prod.associator_naturality_assoc {C : Type u} [category C] [has_binary_products C] {X\u2081 : C}\n    {X\u2082 : C} {X\u2083 : C} {Y\u2081 : C} {Y\u2082 : C} {Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083)\n    {X' : C} (f' : Y\u2081 \u2a2f (Y\u2082 \u2a2f Y\u2083) \u27f6 X') :\n    prod.map (prod.map f\u2081 f\u2082) f\u2083 \u226b iso.hom (prod.associator Y\u2081 Y\u2082 Y\u2083) \u226b f' =\n        iso.hom (prod.associator X\u2081 X\u2082 X\u2083) \u226b prod.map f\u2081 (prod.map f\u2082 f\u2083) \u226b f' :=\n  sorry\n\n/-- The left unitor isomorphism for binary products with the terminal object. -/\ndef prod.left_unitor {C : Type u} [category C] [has_terminal C] (P : C)\n    [has_binary_product (\u22a4_C) P] : (\u22a4_C) \u2a2f P \u2245 P :=\n  iso.mk prod.snd (prod.lift (terminal.from P) \ud835\udfd9)\n\n/-- The right unitor isomorphism for binary products with the terminal object. -/\ndef prod.right_unitor {C : Type u} [category C] [has_terminal C] (P : C)\n    [has_binary_product P (\u22a4_C)] : P \u2a2f (\u22a4_C) \u2245 P :=\n  iso.mk prod.fst (prod.lift \ud835\udfd9 (terminal.from P))\n\ntheorem prod.left_unitor_hom_naturality_assoc {C : Type u} [category C] {X : C} {Y : C}\n    [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) {X' : C} (f' : Y \u27f6 X') :\n    prod.map \ud835\udfd9 f \u226b iso.hom (prod.left_unitor Y) \u226b f' = iso.hom (prod.left_unitor X) \u226b f \u226b f' :=\n  sorry\n\ntheorem prod.left_unitor_inv_naturality {C : Type u} [category C] {X : C} {Y : C} [has_terminal C]\n    [has_binary_products C] (f : X \u27f6 Y) :\n    iso.inv (prod.left_unitor X) \u226b prod.map \ud835\udfd9 f = f \u226b iso.inv (prod.left_unitor Y) :=\n  sorry\n\ntheorem prod.right_unitor_hom_naturality_assoc {C : Type u} [category C] {X : C} {Y : C}\n    [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) {X' : C} (f' : Y \u27f6 X') :\n    prod.map f \ud835\udfd9 \u226b iso.hom (prod.right_unitor Y) \u226b f' = iso.hom (prod.right_unitor X) \u226b f \u226b f' :=\n  sorry\n\ntheorem prod_right_unitor_inv_naturality {C : Type u} [category C] {X : C} {Y : C} [has_terminal C]\n    [has_binary_products C] (f : X \u27f6 Y) :\n    iso.inv (prod.right_unitor X) \u226b prod.map f \ud835\udfd9 = f \u226b iso.inv (prod.right_unitor Y) :=\n  sorry\n\ntheorem prod.triangle {C : Type u} [category C] [has_terminal C] [has_binary_products C] (X : C)\n    (Y : C) :\n    iso.hom (prod.associator X (\u22a4_C) Y) \u226b prod.map \ud835\udfd9 (iso.hom (prod.left_unitor Y)) =\n        prod.map (iso.hom (prod.right_unitor X)) \ud835\udfd9 :=\n  sorry\n\n/-- The braiding isomorphism which swaps a binary coproduct. -/\n@[simp] theorem coprod.braiding_hom {C : Type u} [category C] [has_binary_coproducts C] (P : C)\n    (Q : C) : iso.hom (coprod.braiding P Q) = coprod.desc coprod.inr coprod.inl :=\n  Eq.refl (iso.hom (coprod.braiding P Q))\n\ntheorem coprod.symmetry'_assoc {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C)\n    {X' : C} (f' : P \u2a3f Q \u27f6 X') :\n    coprod.desc coprod.inr coprod.inl \u226b coprod.desc coprod.inr coprod.inl \u226b f' = f' :=\n  sorry\n\n/-- The braiding isomorphism is symmetric. -/\ntheorem coprod.symmetry {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C) :\n    iso.hom (coprod.braiding P Q) \u226b iso.hom (coprod.braiding Q P) = \ud835\udfd9 :=\n  coprod.symmetry' P Q\n\n/-- The associator isomorphism for binary coproducts. -/\n@[simp] theorem coprod.associator_inv {C : Type u} [category C] [has_binary_coproducts C] (P : C)\n    (Q : C) (R : C) :\n    iso.inv (coprod.associator P Q R) =\n        coprod.desc (coprod.inl \u226b coprod.inl) (coprod.desc (coprod.inr \u226b coprod.inl) coprod.inr) :=\n  Eq.refl (iso.inv (coprod.associator P Q R))\n\ntheorem coprod.pentagon {C : Type u} [category C] [has_binary_coproducts C] (W : C) (X : C) (Y : C)\n    (Z : C) :\n    coprod.map (iso.hom (coprod.associator W X Y)) \ud835\udfd9 \u226b\n          iso.hom (coprod.associator W (X \u2a3f Y) Z) \u226b\n            coprod.map \ud835\udfd9 (iso.hom (coprod.associator X Y Z)) =\n        iso.hom (coprod.associator (W \u2a3f X) Y Z) \u226b iso.hom (coprod.associator W X (Y \u2a3f Z)) :=\n  sorry\n\ntheorem coprod.associator_naturality {C : Type u} [category C] [has_binary_coproducts C] {X\u2081 : C}\n    {X\u2082 : C} {X\u2083 : C} {Y\u2081 : C} {Y\u2082 : C} {Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083) :\n    coprod.map (coprod.map f\u2081 f\u2082) f\u2083 \u226b iso.hom (coprod.associator Y\u2081 Y\u2082 Y\u2083) =\n        iso.hom (coprod.associator X\u2081 X\u2082 X\u2083) \u226b coprod.map f\u2081 (coprod.map f\u2082 f\u2083) :=\n  sorry\n\n/-- The left unitor isomorphism for binary coproducts with the initial object. -/\n@[simp] theorem coprod.left_unitor_inv {C : Type u} [category C] [has_binary_coproducts C]\n    [has_initial C] (P : C) : iso.inv (coprod.left_unitor P) = coprod.inr :=\n  Eq.refl (iso.inv (coprod.left_unitor P))\n\n/-- The right unitor isomorphism for binary coproducts with the initial object. -/\n@[simp] theorem coprod.right_unitor_hom {C : Type u} [category C] [has_binary_coproducts C]\n    [has_initial C] (P : C) : iso.hom (coprod.right_unitor P) = coprod.desc \ud835\udfd9 (initial.to P) :=\n  Eq.refl (iso.hom (coprod.right_unitor P))\n\ntheorem coprod.triangle {C : Type u} [category C] [has_binary_coproducts C] [has_initial C] (X : C)\n    (Y : C) :\n    iso.hom (coprod.associator X (\u22a5_C) Y) \u226b coprod.map \ud835\udfd9 (iso.hom (coprod.left_unitor Y)) =\n        coprod.map (iso.hom (coprod.right_unitor X)) \ud835\udfd9 :=\n  sorry\n\n/-- The binary product functor. -/\n@[simp] theorem prod.functor_obj_map {C : Type u} [category C] [has_binary_products C] (X : C)\n    (Y : C) (Z : C) (g : Y \u27f6 Z) : functor.map (functor.obj prod.functor X) g = prod.map \ud835\udfd9 g :=\n  Eq.refl (functor.map (functor.obj prod.functor X) g)\n\n/-- The product functor can be decomposed. -/\ndef prod.functor_left_comp {C : Type u} [category C] [has_binary_products C] (X : C) (Y : C) :\n    functor.obj prod.functor (X \u2a2f Y) \u2245 functor.obj prod.functor Y \u22d9 functor.obj prod.functor X :=\n  nat_iso.of_components (prod.associator X Y) sorry\n\n/-- The binary coproduct functor. -/\n@[simp] theorem coprod.functor_obj_map {C : Type u} [category C] [has_binary_coproducts C] (X : C)\n    (Y : C) (Z : C) (g : Y \u27f6 Z) : functor.map (functor.obj coprod.functor X) g = coprod.map \ud835\udfd9 g :=\n  Eq.refl (functor.map (functor.obj coprod.functor X) g)\n\n/-- The coproduct functor can be decomposed. -/\ndef coprod.functor_left_comp {C : Type u} [category C] [has_binary_coproducts C] (X : C) (Y : C) :\n    functor.obj coprod.functor (X \u2a3f Y) \u2245\n        functor.obj coprod.functor Y \u22d9 functor.obj coprod.functor X :=\n  nat_iso.of_components (coprod.associator X Y) sorry\n\n/--\nThe product comparison morphism.\n\nIn `category_theory/limits/preserves` we show this is always an iso iff F preserves binary products.\n-/\ndef prod_comparison {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) (A : C) (B : C)\n    [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] :\n    functor.obj F (A \u2a2f B) \u27f6 functor.obj F A \u2a2f functor.obj F B :=\n  prod.lift (functor.map F prod.fst) (functor.map F prod.snd)\n\n@[simp] theorem prod_comparison_fst_assoc {C : Type u} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B]\n    [has_binary_product (functor.obj F A) (functor.obj F B)] {X' : D} (f' : functor.obj F A \u27f6 X') :\n    prod_comparison F A B \u226b prod.fst \u226b f' = functor.map F prod.fst \u226b f' :=\n  sorry\n\n@[simp] theorem prod_comparison_snd_assoc {C : Type u} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B]\n    [has_binary_product (functor.obj F A) (functor.obj F B)] {X' : D} (f' : functor.obj F B \u27f6 X') :\n    prod_comparison F A B \u226b prod.snd \u226b f' = functor.map F prod.snd \u226b f' :=\n  sorry\n\n/-- Naturality of the prod_comparison morphism in both arguments. -/\ntheorem prod_comparison_natural_assoc {C : Type u} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) {A : C} {A' : C} {B : C} {B' : C} [has_binary_product A B]\n    [has_binary_product A' B'] [has_binary_product (functor.obj F A) (functor.obj F B)]\n    [has_binary_product (functor.obj F A') (functor.obj F B')] (f : A \u27f6 A') (g : B \u27f6 B') {X' : D}\n    (f' : functor.obj F A' \u2a2f functor.obj F B' \u27f6 X') :\n    functor.map F (prod.map f g) \u226b prod_comparison F A' B' \u226b f' =\n        prod_comparison F A B \u226b prod.map (functor.map F f) (functor.map F g) \u226b f' :=\n  sorry\n\n/--\nThe product comparison morphism from `F(A \u2a2f -)` to `FA \u2a2f F-`, whose components are given by\n`prod_comparison`.\n-/\ndef prod_comparison_nat_trans {C : Type u} [category C] {D : Type u\u2082} [category D]\n    [has_binary_products C] [has_binary_products D] (F : C \u2964 D) (A : C) :\n    functor.obj prod.functor A \u22d9 F \u27f6 F \u22d9 functor.obj prod.functor (functor.obj F A) :=\n  nat_trans.mk fun (B : C) => prod_comparison F A B\n\ntheorem inv_prod_comparison_map_fst_assoc {C : Type u} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B]\n    [has_binary_product (functor.obj F A) (functor.obj F B)] [is_iso (prod_comparison F A B)]\n    {X' : D} (f' : functor.obj F A \u27f6 X') :\n    inv (prod_comparison F A B) \u226b functor.map F prod.fst \u226b f' = prod.fst \u226b f' :=\n  sorry\n\ntheorem inv_prod_comparison_map_snd_assoc {C : Type u} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B]\n    [has_binary_product (functor.obj F A) (functor.obj F B)] [is_iso (prod_comparison F A B)]\n    {X' : D} (f' : functor.obj F B \u27f6 X') :\n    inv (prod_comparison F A B) \u226b functor.map F prod.snd \u226b f' = prod.snd \u226b f' :=\n  sorry\n\n/-- If the product comparison morphism is an iso, its inverse is natural. -/\ntheorem prod_comparison_inv_natural {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D)\n    {A : C} {A' : C} {B : C} {B' : C} [has_binary_product A B] [has_binary_product A' B']\n    [has_binary_product (functor.obj F A) (functor.obj F B)]\n    [has_binary_product (functor.obj F A') (functor.obj F B')] (f : A \u27f6 A') (g : B \u27f6 B')\n    [is_iso (prod_comparison F A B)] [is_iso (prod_comparison F A' B')] :\n    inv (prod_comparison F A B) \u226b functor.map F (prod.map f g) =\n        prod.map (functor.map F f) (functor.map F g) \u226b inv (prod_comparison F A' B') :=\n  sorry\n\n/--\nThe natural isomorphism `F(A \u2a2f -) \u2245 FA \u2a2f F-`, provided each `prod_comparison F A B` is an\nisomorphism (as `B` changes).\n-/\ndef prod_comparison_nat_iso {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D)\n    [has_binary_products C] [has_binary_products D] (A : C)\n    [(B : C) \u2192 is_iso (prod_comparison F A B)] :\n    functor.obj prod.functor A \u22d9 F \u2245 F \u22d9 functor.obj prod.functor (functor.obj F A) :=\n  iso.mk (prod_comparison_nat_trans F A) (inv (nat_trans.mk fun (B : C) => prod_comparison F A B))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/binary_products_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3916732214845531}}
{"text": "\nuniverses u\n\ndef len {\u03b1 : Type u} : List \u03b1 \u2192 List \u03b1 \u2192 Nat\n| [],    bs => bs.length\n| a::as, bs => len as bs + 1\n\ntheorem ex1 : len [1, 2] [3, 4] = 4 :=\nrfl\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/matcherElimUniv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3915778129265744}}
{"text": "import classes.context_free.closure_properties.bijection\n\n\n/-- The class of context-free languages is closed under permutation of terminals. -/\ntheorem CF_of_permute_CF {T : Type} (\u03c0 : equiv.perm T) (L : language T) :\n  is_CF L  \u2192  is_CF (permute_lang L \u03c0)  :=\nCF_of_bijemap_CF \u03c0 L\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/context_free/closure_properties/permutation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3915778129265744}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Patrick Massot, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.limits\nimport Mathlib.topology.uniform_space.completion\nimport Mathlib.topology.category.Top.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 l \n\nnamespace Mathlib\n\n/-!\n# The category of uniform spaces\n\nWe construct the category of uniform spaces, show that the complete separated uniform spaces\nform a reflective subcategory, and hence possess all limits that uniform spaces do.\n\nTODO: show that uniform spaces actually have all limits!\n-/\n\n/-- A (bundled) uniform space. -/\ndef UniformSpace :=\n  category_theory.bundled uniform_space\n\nnamespace UniformSpace\n\n\n/-- The information required to build morphisms for `UniformSpace`. -/\nprotected instance uniform_continuous.category_theory.unbundled_hom : category_theory.unbundled_hom uniform_continuous :=\n  category_theory.unbundled_hom.mk uniform_continuous_id uniform_continuous.comp\n\nprotected instance has_coe_to_sort : has_coe_to_sort UniformSpace :=\n  category_theory.bundled.has_coe_to_sort\n\nprotected instance uniform_space (x : UniformSpace) : uniform_space \u21a5x :=\n  category_theory.bundled.str x\n\n/-- Construct a bundled `UniformSpace` from the underlying type and the typeclass. -/\ndef of (\u03b1 : Type u) [uniform_space \u03b1] : UniformSpace :=\n  category_theory.bundled.mk \u03b1\n\nprotected instance inhabited : Inhabited UniformSpace :=\n  { default := of empty }\n\n@[simp] theorem coe_of (X : Type u) [uniform_space X] : \u21a5(of X) = X :=\n  rfl\n\nprotected instance category_theory.has_hom.hom.has_coe_to_fun (X : UniformSpace) (Y : UniformSpace) : has_coe_to_fun (X \u27f6 Y) :=\n  has_coe_to_fun.mk (fun (_x : X \u27f6 Y) => \u21a5X \u2192 \u21a5Y) (category_theory.functor.map (category_theory.forget UniformSpace))\n\n@[simp] theorem coe_comp {X : UniformSpace} {Y : UniformSpace} {Z : UniformSpace} (f : X \u27f6 Y) (g : Y \u27f6 Z) : \u21d1(f \u226b g) = \u21d1g \u2218 \u21d1f :=\n  rfl\n\n@[simp] theorem coe_id (X : UniformSpace) : \u21d1\ud835\udfd9 = id :=\n  rfl\n\n@[simp] theorem coe_mk {X : UniformSpace} {Y : UniformSpace} (f : \u21a5X \u2192 \u21a5Y) (hf : uniform_continuous f) : \u21d1{ val := f, property := hf } = f :=\n  rfl\n\ntheorem hom_ext {X : UniformSpace} {Y : UniformSpace} {f : X \u27f6 Y} {g : X \u27f6 Y} : \u21d1f = \u21d1g \u2192 f = g :=\n  subtype.eq\n\n/-- The forgetful functor from uniform spaces to topological spaces. -/\nprotected instance has_forget_to_Top : category_theory.has_forget\u2082 UniformSpace Top :=\n  category_theory.has_forget\u2082.mk\n    (category_theory.functor.mk (fun (X : UniformSpace) => Top.of \u21a5X)\n      fun (X Y : UniformSpace) (f : X \u27f6 Y) => continuous_map.mk \u21d1f)\n\nend UniformSpace\n\n\n/-- A (bundled) complete separated uniform space. -/\nstructure CpltSepUniformSpace \nwhere\n  \u03b1 : Type u\n  is_uniform_space : uniform_space \u03b1\n  is_complete_space : complete_space \u03b1\n  is_separated : separated_space \u03b1\n\nnamespace CpltSepUniformSpace\n\n\nprotected instance has_coe_to_sort : has_coe_to_sort CpltSepUniformSpace :=\n  has_coe_to_sort.mk (Type u) \u03b1\n\ndef to_UniformSpace (X : CpltSepUniformSpace) : UniformSpace :=\n  UniformSpace.of \u21a5X\n\nprotected instance complete_space (X : CpltSepUniformSpace) : complete_space (category_theory.bundled.\u03b1 (to_UniformSpace X)) :=\n  is_complete_space X\n\nprotected instance separated_space (X : CpltSepUniformSpace) : separated_space (category_theory.bundled.\u03b1 (to_UniformSpace X)) :=\n  is_separated X\n\n/-- Construct a bundled `UniformSpace` from the underlying type and the appropriate typeclasses. -/\ndef of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] : CpltSepUniformSpace :=\n  mk X\n\n@[simp] theorem coe_of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] : \u21a5(of X) = X :=\n  rfl\n\nprotected instance inhabited : Inhabited CpltSepUniformSpace :=\n  { default := of empty }\n\n/-- The category instance on `CpltSepUniformSpace`. -/\nprotected instance category : category_theory.large_category CpltSepUniformSpace :=\n  category_theory.induced_category.category to_UniformSpace\n\n/-- The concrete category instance on `CpltSepUniformSpace`. -/\nprotected instance concrete_category : category_theory.concrete_category CpltSepUniformSpace :=\n  category_theory.induced_category.concrete_category to_UniformSpace\n\nprotected instance has_forget_to_UniformSpace : category_theory.has_forget\u2082 CpltSepUniformSpace UniformSpace :=\n  category_theory.induced_category.has_forget\u2082 to_UniformSpace\n\nend CpltSepUniformSpace\n\n\nnamespace UniformSpace\n\n\n/-- The functor turning uniform spaces into complete separated uniform spaces. -/\ndef completion_functor : UniformSpace \u2964 CpltSepUniformSpace :=\n  category_theory.functor.mk (fun (X : UniformSpace) => CpltSepUniformSpace.of (uniform_space.completion \u21a5X))\n    fun (X Y : UniformSpace) (f : X \u27f6 Y) => { val := uniform_space.completion.map (subtype.val f), property := sorry }\n\n/-- The inclusion of a uniform space into its completion. -/\ndef completion_hom (X : UniformSpace) : X \u27f6\n  category_theory.functor.obj (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace)\n    (category_theory.functor.obj completion_functor X) :=\n  { val := coe, property := sorry }\n\n@[simp] theorem completion_hom_val (X : UniformSpace) (x : \u21a5X) : coe_fn (completion_hom X) x = \u2191x :=\n  rfl\n\n/-- The mate of a morphism from a `UniformSpace` to a `CpltSepUniformSpace`. -/\ndef extension_hom {X : UniformSpace} {Y : CpltSepUniformSpace} (f : X \u27f6 category_theory.functor.obj (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace) Y) : category_theory.functor.obj completion_functor X \u27f6 Y :=\n  { val := uniform_space.completion.extension \u21d1f, property := sorry }\n\n@[simp] theorem extension_hom_val {X : UniformSpace} {Y : CpltSepUniformSpace} (f : X \u27f6 category_theory.functor.obj (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace) Y) (x : \u21a5(CpltSepUniformSpace.to_UniformSpace (category_theory.functor.obj completion_functor X))) : coe_fn (extension_hom f) x = uniform_space.completion.extension (\u21d1f) x :=\n  rfl\n\n@[simp] theorem extension_comp_coe {X : UniformSpace} {Y : CpltSepUniformSpace} (f : CpltSepUniformSpace.to_UniformSpace (CpltSepUniformSpace.of (uniform_space.completion \u21a5X)) \u27f6\n  CpltSepUniformSpace.to_UniformSpace Y) : extension_hom (completion_hom X \u226b f) = f := sorry\n\n/-- The completion functor is left adjoint to the forgetful functor. -/\ndef adj : completion_functor \u22a3 category_theory.forget\u2082 CpltSepUniformSpace UniformSpace :=\n  category_theory.adjunction.mk_of_hom_equiv\n    (category_theory.adjunction.core_hom_equiv.mk\n      fun (X : UniformSpace) (Y : CpltSepUniformSpace) =>\n        equiv.mk (fun (f : category_theory.functor.obj completion_functor X \u27f6 Y) => completion_hom X \u226b f)\n          (fun (f : X \u27f6 category_theory.functor.obj (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace) Y) =>\n            extension_hom f)\n          sorry sorry)\n\nprotected instance category_theory.forget\u2082.category_theory.is_right_adjoint : category_theory.is_right_adjoint (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace) :=\n  category_theory.is_right_adjoint.mk completion_functor adj\n\nprotected instance category_theory.forget\u2082.category_theory.reflective : category_theory.reflective (category_theory.forget\u2082 CpltSepUniformSpace UniformSpace) :=\n  category_theory.reflective.mk\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/category/UniformSpace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3915778129265744}}
{"text": "--\n\ndef f1 (x : Nat) (b : Bool) : type_of% x :=\nlet r : type_of% (x+1) := x+1;\nr + 1\n\ntheorem ex1 : f1 1 true = 3 :=\nrfl\n\ndef f2 (x : Nat) (b : Bool) : type_of% x :=\nlet r : type_of% b := x+1; -- error\nr + 1\n\ndef f3 (x : Nat) (b : Bool) : type_of% b :=\nlet r (x!1 : type_of% x) : type_of% b := x > 1;\nr x\n\ndef f4 (x : Nat) : Nat :=\nlet y : Nat := x\nlet y := ensure_type_of% y \"invalid reassignment, term\" y == 1 -- error\ny + 1\n\ndef f5 (x : Nat) : Nat :=\nlet y : Nat := x\nlet y := ensure_type_of% y \"invalid reassignment, term\" (y+1)\ny + 1\n\ndef f6 (x : Nat) : Nat :=\nensure_expected_type% \"natural number expected, value\" true\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/typeOf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3914718702934395}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Callum Sutton, Yury Kudryashov\nPorted by: Winston Yin\n\n! This file was ported from Lean 3 source module algebra.hom.equiv.basic\n! leanprover-community/mathlib commit 67f362670ed961bcb80239dc40ca18bcd4289c77\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Algebra.Hom.Group\nimport Mathlib.Data.FunLike.Equiv\nimport Mathlib.Logic.Equiv.Basic\nimport Mathlib.Data.Pi.Algebra\n\n/-!\n# Multiplicative and additive equivs\n\nIn this file we define two extensions of `Equiv` called `AddEquiv` and `MulEquiv`, which are\ndatatypes representing isomorphisms of `AddMonoid`s/`AddGroup`s and `Monoid`s/`Group`s.\n\n## Notations\n\n* ``infix ` \u2243* `:25 := MulEquiv``\n* ``infix ` \u2243+ `:25 := AddEquiv``\n\nThe extended equivs all have coercions to functions, and the coercions are the canonical\nnotation when treating the isomorphisms as maps.\n\n## Tags\n\nEquiv, MulEquiv, AddEquiv\n-/\n\n\nvariable {F \u03b1 \u03b2 A B M N P Q G H : Type _}\n\n/-- Makes a multiplicative inverse from a bijection which preserves multiplication. -/\n@[to_additive \"Makes an additive inverse from a bijection which preserves addition.\"]\ndef MulHom.inverse [Mul M] [Mul N] (f : M \u2192\u2099* N) (g : N \u2192 M) (h\u2081 : Function.LeftInverse g f)\n  (h\u2082 : Function.RightInverse g f) : N \u2192\u2099* M where\n  toFun := g\n  map_mul' x y :=\n    calc\n      g (x * y) = g (f (g x) * f (g y)) := by rw [h\u2082 x, h\u2082 y]\n      _ = g (f (g x * g y)) := by rw [f.map_mul]\n      _ = g x * g y := h\u2081 _\n\n#align mul_hom.inverse MulHom.inverse\n#align add_hom.inverse AddHom.inverse\n\n/-- The inverse of a bijective `MonoidHom` is a `MonoidHom`. -/\n@[to_additive (attr := simps) \"The inverse of a bijective `AddMonoidHom` is an `AddMonoidHom`.\"]\ndef MonoidHom.inverse {A B : Type _} [Monoid A] [Monoid B] (f : A \u2192* B) (g : B \u2192 A)\n  (h\u2081 : Function.LeftInverse g f) (h\u2082 : Function.RightInverse g f) : B \u2192* A :=\n  { (f : A \u2192\u2099* B).inverse g h\u2081 h\u2082 with toFun := g, map_one' := by rw [\u2190 f.map_one, h\u2081] }\n#align monoid_hom.inverse MonoidHom.inverse\n#align add_monoid_hom.inverse AddMonoidHom.inverse\n#align monoid_hom.inverse_apply MonoidHom.inverse_apply\n#align add_monoid_hom.inverse_apply AddMonoidHom.inverse_apply\n\n/-- `AddEquiv \u03b1 \u03b2` is the type of an equiv `\u03b1 \u2243 \u03b2` which preserves addition. -/\nstructure AddEquiv (A B : Type _) [Add A] [Add B] extends A \u2243 B, AddHom A B\n#align add_equiv AddEquiv\n\n/-- `AddEquivClass F A B` states that `F` is a type of addition-preserving morphisms.\nYou should extend this class when you extend `AddEquiv`. -/\nclass AddEquivClass (F : Type _) (A B : outParam (Type _)) [Add A] [Add B]\n    extends EquivLike F A B where\n  /-- Preserves addition. -/\n  map_add : \u2200 (f : F) (a b), f (a + b) = f a + f b\n#align add_equiv_class AddEquivClass\n\n/-- The `Equiv` underlying an `AddEquiv`. -/\nadd_decl_doc AddEquiv.toEquiv\n#align add_equiv.to_equiv AddEquiv.toEquiv\n\n/-- The `AddHom` underlying a `AddEquiv`. -/\nadd_decl_doc AddEquiv.toAddHom\n#align add_equiv.to_add_hom AddEquiv.toAddHom\n\n/-- `MulEquiv \u03b1 \u03b2` is the type of an equiv `\u03b1 \u2243 \u03b2` which preserves multiplication. -/\n@[to_additive]\nstructure MulEquiv (M N : Type _) [Mul M] [Mul N] extends M \u2243 N, M \u2192\u2099* N\n-- Porting note: remove when `to_additive` can do this\n-- https://github.com/leanprover-community/mathlib4/issues/660\nattribute [to_additive existing] MulEquiv.toMulHom\n#align mul_equiv MulEquiv\n\n/-- The `Equiv` underlying a `MulEquiv`. -/\nadd_decl_doc MulEquiv.toEquiv\n#align mul_equiv.to_equiv MulEquiv.toEquiv\n\n/-- The `MulHom` underlying a `MulEquiv`. -/\nadd_decl_doc MulEquiv.toMulHom\n#align mul_equiv.to_mul_hom MulEquiv.toMulHom\n\n/-- `MulEquivClass F A B` states that `F` is a type of multiplication-preserving morphisms.\nYou should extend this class when you extend `MulEquiv`. -/\n@[to_additive]\nclass MulEquivClass (F : Type _) (A B : outParam (Type _)) [Mul A] [Mul B] extends\n  EquivLike F A B where\n  /-- Preserves multiplication. -/\n  map_mul : \u2200 (f : F) (a b), f (a * b) = f a * f b\n#align mul_equiv_class MulEquivClass\n\n/-- Notation for a `MulEquiv`. -/\ninfixl:25 \" \u2243* \" => MulEquiv\n\n/-- Notation for an `AddEquiv`. -/\ninfixl:25 \" \u2243+ \" => AddEquiv\n\nnamespace MulEquivClass\n\nvariable (F)\n\n-- See note [lower instance priority]\n@[to_additive]\ninstance (priority := 100) (F : Type _) {M N : Type _}\n    {_ : Mul M} {_ : Mul N} [h : MulEquivClass F M N] : MulHomClass F M N :=\n  { h with coe := h.coe, coe_injective' := FunLike.coe_injective' }\n\n-- See note [lower instance priority]\n@[to_additive]\ninstance (priority := 100) {_ : MulOneClass M} {_ : MulOneClass N} [MulEquivClass F M N] :\n  MonoidHomClass F M N :=\n  { MulEquivClass.instMulHomClass F with\n    coe := fun _ => _,\n    map_one := fun e =>\n      calc\n        e 1 = e 1 * 1 := (mul_one _).symm\n        _ = e 1 * e (MulEquivClass.toEquivLike.inv e (1 : N) : M) :=\n          congr_arg _ (MulEquivClass.toEquivLike.right_inv e 1).symm\n        _ = e (MulEquivClass.toEquivLike.inv e (1 : N)) := by rw [\u2190 map_mul, one_mul]\n        _ = 1 := MulEquivClass.toEquivLike.right_inv e 1\n         }\n\n-- See note [lower instance priority]\ninstance (priority := 100) toMonoidWithZeroHomClass\n  {\u03b1 \u03b2 : Type _} {_ : MulZeroOneClass \u03b1} {_ : MulZeroOneClass \u03b2} [MulEquivClass F \u03b1 \u03b2] :\n  MonoidWithZeroHomClass F \u03b1 \u03b2 :=\n  { MulEquivClass.instMonoidHomClass _ with\n    map_zero := fun e =>\n      calc\n        e 0 = e 0 * e (EquivLike.inv e 0) := by rw [\u2190 map_mul, zero_mul]\n        _ = 0 := by simp }\n#align mul_equiv_class.to_monoid_with_zero_hom_class MulEquivClass.toMonoidWithZeroHomClass\n\nvariable {F}\n\n@[to_additive (attr := simp)]\ntheorem map_eq_one_iff {M N} [MulOneClass M] [MulOneClass N] [MulEquivClass F M N] (h : F) {x : M} :\n  h x = 1 \u2194 x = 1 := _root_.map_eq_one_iff h (EquivLike.injective h)\n#align mul_equiv_class.map_eq_one_iff MulEquivClass.map_eq_one_iff\n#align add_equiv_class.map_eq_zero_iff AddEquivClass.map_eq_zero_iff\n\n@[to_additive]\ntheorem map_ne_one_iff {M N} [MulOneClass M] [MulOneClass N] [MulEquivClass F M N] (h : F) {x : M} :\n  h x \u2260 1 \u2194 x \u2260 1 := _root_.map_ne_one_iff h (EquivLike.injective h)\n#align mul_equiv_class.map_ne_one_iff MulEquivClass.map_ne_one_iff\n#align add_equiv_class.map_ne_zero_iff AddEquivClass.map_ne_zero_iff\n\nend MulEquivClass\n\n/-- Turn an element of a type `F` satisfying `MulEquivClass F \u03b1 \u03b2` into an actual\n`MulEquiv`. This is declared as the default coercion from `F` to `\u03b1 \u2243* \u03b2`. -/\n@[to_additive (attr := coe)\n\"Turn an element of a type `F` satisfying `AddEquivClass F \u03b1 \u03b2` into an actual\n`AddEquiv`. This is declared as the default coercion from `F` to `\u03b1 \u2243+ \u03b2`.\"]\ndef MulEquivClass.toMulEquiv [Mul \u03b1] [Mul \u03b2] [MulEquivClass F \u03b1 \u03b2] (f : F) : \u03b1 \u2243* \u03b2 :=\n{ (f : \u03b1 \u2243 \u03b2), (f : \u03b1 \u2192\u2099* \u03b2) with }\n\n/-- Any type satisfying `MulEquivClass` can be cast into `MulEquiv` via\n`MulEquivClass.toMulEquiv`. -/\n@[to_additive \"Any type satisfying `AddEquivClass` can be cast into `AddEquiv` via\n`AddEquivClass.toAddEquiv`. \"]\ninstance [Mul \u03b1] [Mul \u03b2] [MulEquivClass F \u03b1 \u03b2] : CoeTC F (\u03b1 \u2243* \u03b2) :=\n  \u27e8MulEquivClass.toMulEquiv\u27e9\n\nnamespace MulEquiv\n\n@[to_additive]\ninstance [Mul M] [Mul N] : MulEquivClass (M \u2243* N) M N where\n  coe f := f.toFun\n  inv f := f.invFun\n  left_inv f := f.left_inv\n  right_inv f := f.right_inv\n  coe_injective' f g h\u2081 h\u2082 := by\n    cases f\n    cases g\n    congr\n    apply Equiv.coe_fn_injective h\u2081\n  map_mul := map_mul'\n\nvariable [Mul M] [Mul N] [Mul P] [Mul Q]\n\n@[to_additive (attr := simp)]\ntheorem toEquiv_eq_coe (f : M \u2243* N) : f.toEquiv = f :=\n  rfl\n#align mul_equiv.to_equiv_eq_coe MulEquiv.toEquiv_eq_coe\n#align add_equiv.to_equiv_eq_coe AddEquiv.toEquiv_eq_coe\n\n-- Porting note: `to_fun_eq_coe` no longer needed in Lean4\n#noalign mul_equiv.to_fun_eq_coe\n#noalign add_equiv.to_fun_eq_coe\n\n@[to_additive (attr := simp)]\ntheorem coe_toEquiv (f : M \u2243* N) : \u21d1(f : M \u2243 N) = f := rfl\n#align mul_equiv.coe_to_equiv MulEquiv.coe_toEquiv\n#align add_equiv.coe_to_equiv AddEquiv.coe_toEquiv\n\n-- porting note: todo: `MulHom.coe_mk` simplifies `\u2191f.toMulHom` to `f.toMulHom.toFun`,\n-- not `f.toEquiv.toFun`; use higher priority as a workaround\n@[to_additive (attr := simp 1100)]\ntheorem coe_toMulHom {f : M \u2243* N} : (f.toMulHom : M \u2192 N) = f := rfl\n#align mul_equiv.coe_to_mul_hom MulEquiv.coe_toMulHom\n#align add_equiv.coe_to_add_hom AddEquiv.coe_toAddHom\n\n/-- A multiplicative isomorphism preserves multiplication. -/\n@[to_additive \"An additive isomorphism preserves addition.\"]\nprotected theorem map_mul (f : M \u2243* N) : \u2200 x y, f (x * y) = f x * f y :=\n  _root_.map_mul f\n#align mul_equiv.map_mul MulEquiv.map_mul\n#align add_equiv.map_add AddEquiv.map_add\n\n/-- Makes a multiplicative isomorphism from a bijection which preserves multiplication. -/\n@[to_additive \"Makes an additive isomorphism from a bijection which preserves addition.\"]\ndef mk' (f : M \u2243 N) (h : \u2200 x y, f (x * y) = f x * f y) : M \u2243* N := \u27e8f, h\u27e9\n#align mul_equiv.mk' MulEquiv.mk'\n#align add_equiv.mk' AddEquiv.mk'\n\n@[to_additive]\nprotected theorem bijective (e : M \u2243* N) : Function.Bijective e :=\n  EquivLike.bijective e\n#align mul_equiv.bijective MulEquiv.bijective\n#align add_equiv.bijective AddEquiv.bijective\n\n@[to_additive]\nprotected theorem injective (e : M \u2243* N) : Function.Injective e :=\n  EquivLike.injective e\n#align mul_equiv.injective MulEquiv.injective\n#align add_equiv.injective AddEquiv.injective\n\n@[to_additive]\nprotected theorem surjective (e : M \u2243* N) : Function.Surjective e :=\n  EquivLike.surjective e\n#align mul_equiv.surjective MulEquiv.surjective\n#align add_equiv.surjective AddEquiv.surjective\n\n/-- The identity map is a multiplicative isomorphism. -/\n@[to_additive (attr := refl) \"The identity map is an additive isomorphism.\"]\ndef refl (M : Type _) [Mul M] : M \u2243* M :=\n  { Equiv.refl _ with map_mul' := fun _ _ => rfl }\n#align mul_equiv.refl MulEquiv.refl\n#align add_equiv.refl AddEquiv.refl\n\n@[to_additive]\ninstance : Inhabited (M \u2243* M) := \u27e8refl M\u27e9\n\n/-- The inverse of an isomorphism is an isomorphism. -/\n@[to_additive (attr := symm) \"The inverse of an isomorphism is an isomorphism.\"]\ndef symm {M N : Type _} [Mul M] [Mul N] (h : M \u2243* N) : N \u2243* M :=\n  \u27e8h.toEquiv.symm, (h.toMulHom.inverse h.toEquiv.symm h.left_inv h.right_inv).map_mul\u27e9\n#align mul_equiv.symm MulEquiv.symm\n#align add_equiv.symm AddEquiv.symm\n\n@[to_additive] -- porting note: no longer a `simp`, see below\ntheorem invFun_eq_symm {f : M \u2243* N} : f.invFun = f.symm := rfl\n#align mul_equiv.inv_fun_eq_symm MulEquiv.invFun_eq_symm\n-- Porting note: to_additive translated the name incorrectly in mathlib 3.\n#align add_equiv.neg_fun_eq_symm AddEquiv.invFun_eq_symm\n\n@[to_additive (attr := simp)]\n\n\n@[to_additive (attr := simp)]\ntheorem equivLike_inv_eq_symm (f : M \u2243* N) : EquivLike.inv f = f.symm := rfl\n\n-- we don't hyperlink the note in the additive version, since that breaks syntax highlighting\n-- in the whole file.\n\n/-- See Note [custom simps projection] -/\n@[to_additive \"See Note [custom simps projection]\"] -- this comment fixes the syntax highlighting \"\ndef Simps.symm_apply (e : M \u2243* N) : N \u2192 M :=\n  e.symm\n#align mul_equiv.simps.symm_apply MulEquiv.Simps.symm_apply\n#align add_equiv.simps.symm_apply AddEquiv.Simps.symm_apply\n\ninitialize_simps_projections AddEquiv (toFun \u2192 apply, invFun \u2192 symm_apply)\n\ninitialize_simps_projections MulEquiv (toFun \u2192 apply, invFun \u2192 symm_apply)\n\n@[to_additive (attr := simp)]\ntheorem toEquiv_symm (f : M \u2243* N) : (f.symm : N \u2243 M) = (f : M \u2243 N).symm := rfl\n#align mul_equiv.to_equiv_symm MulEquiv.toEquiv_symm\n#align add_equiv.to_equiv_symm AddEquiv.toEquiv_symm\n\n-- porting note: doesn't align with Mathlib 3 because `MulEquiv.mk` has a new signature\n@[to_additive (attr := simp)]\ntheorem coe_mk (f : M \u2243 N) (hf : \u2200 x y, f (x * y) = f x * f y) : (mk f hf : M \u2192 N) = f := rfl\n#align mul_equiv.coe_mk MulEquiv.coe_mk\u2093\n#align add_equiv.coe_mk AddEquiv.coe_mk\u2093\n\n-- Porting note: `toEquiv_mk` no longer needed in Lean4\n#noalign mul_equiv.to_equiv_mk\n#noalign add_equiv.to_equiv_mk\n\n@[to_additive (attr := simp)]\ntheorem symm_symm (f : M \u2243* N) : f.symm.symm = f := rfl\n#align mul_equiv.symm_symm MulEquiv.symm_symm\n#align add_equiv.symm_symm AddEquiv.symm_symm\n\n@[to_additive]\ntheorem symm_bijective : Function.Bijective (symm : M \u2243* N \u2192 N \u2243* M) :=\n  Equiv.bijective \u27e8symm, symm, symm_symm, symm_symm\u27e9\n#align mul_equiv.symm_bijective MulEquiv.symm_bijective\n#align add_equiv.symm_bijective AddEquiv.symm_bijective\n\n-- Porting note: this doesn't really align with mathlib3's `symm_mk`,\n-- because the signature of `MulEquiv.mk` has changed.\n@[to_additive (attr := simp)]\ntheorem symm_mk (f : M \u2243 N) (h) :\n  (MulEquiv.mk f h).symm = \u27e8f.symm, (MulEquiv.mk f h).symm.map_mul'\u27e9 := rfl\n#align mul_equiv.symm_mk MulEquiv.symm_mk\u2093\n#align add_equiv.symm_mk AddEquiv.symm_mk\u2093\n\n@[to_additive (attr := simp)]\ntheorem refl_symm : (refl M).symm = refl M := rfl\n#align mul_equiv.refl_symm MulEquiv.refl_symm\n#align add_equiv.refl_symm AddEquiv.refl_symm\n\n/-- Transitivity of multiplication-preserving isomorphisms -/\n@[to_additive (attr := trans) \"Transitivity of addition-preserving isomorphisms\"]\ndef trans (h1 : M \u2243* N) (h2 : N \u2243* P) : M \u2243* P :=\n  { h1.toEquiv.trans h2.toEquiv with\n    map_mul' := fun x y => show h2 (h1 (x * y)) = h2 (h1 x) * h2 (h1 y) by\n      rw [h1.map_mul, h2.map_mul] }\n#align mul_equiv.trans MulEquiv.trans\n#align add_equiv.trans AddEquiv.trans\n\n/-- `e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`. -/\n@[to_additive (attr := simp) \"`e.symm` is a right inverse of `e`, written as `e (e.symm y) = y`.\"]\ntheorem apply_symm_apply (e : M \u2243* N) (y : N) : e (e.symm y) = y :=\n  e.toEquiv.apply_symm_apply y\n#align mul_equiv.apply_symm_apply MulEquiv.apply_symm_apply\n#align add_equiv.apply_symm_apply AddEquiv.apply_symm_apply\n\n/-- `e.symm` is a left inverse of `e`, written as `e.symm (e y) = y`. -/\n@[to_additive (attr := simp) \"`e.symm` is a left inverse of `e`, written as `e.symm (e y) = y`.\"]\ntheorem symm_apply_apply (e : M \u2243* N) (x : M) : e.symm (e x) = x :=\n  e.toEquiv.symm_apply_apply x\n#align mul_equiv.symm_apply_apply MulEquiv.symm_apply_apply\n#align add_equiv.symm_apply_apply AddEquiv.symm_apply_apply\n\n@[to_additive (attr := simp)]\ntheorem symm_comp_self (e : M \u2243* N) : e.symm \u2218 e = id :=\n  funext e.symm_apply_apply\n#align mul_equiv.symm_comp_self MulEquiv.symm_comp_self\n#align add_equiv.symm_comp_self AddEquiv.symm_comp_self\n\n@[to_additive (attr := simp)]\ntheorem self_comp_symm (e : M \u2243* N) : e \u2218 e.symm = id :=\n  funext e.apply_symm_apply\n#align mul_equiv.self_comp_symm MulEquiv.self_comp_symm\n#align add_equiv.self_comp_symm AddEquiv.self_comp_symm\n\n@[to_additive (attr := simp)]\ntheorem coe_refl : \u2191(refl M) = id := rfl\n#align mul_equiv.coe_refl MulEquiv.coe_refl\n#align add_equiv.coe_refl AddEquiv.coe_refl\n\n@[to_additive (attr := simp)]\ntheorem refl_apply (m : M) : refl M m = m := rfl\n#align mul_equiv.refl_apply MulEquiv.refl_apply\n#align add_equiv.refl_apply AddEquiv.refl_apply\n\n@[to_additive (attr := simp)]\ntheorem coe_trans (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) : \u2191(e\u2081.trans e\u2082) = e\u2082 \u2218 e\u2081 := rfl\n#align mul_equiv.coe_trans MulEquiv.coe_trans\n#align add_equiv.coe_trans AddEquiv.coe_trans\n\n@[to_additive (attr := simp)]\ntheorem trans_apply (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) (m : M) : e\u2081.trans e\u2082 m = e\u2082 (e\u2081 m) := rfl\n#align mul_equiv.trans_apply MulEquiv.trans_apply\n#align add_equiv.trans_apply AddEquiv.trans_apply\n\n@[to_additive (attr := simp)]\ntheorem symm_trans_apply (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) (p : P) :\n  (e\u2081.trans e\u2082).symm p = e\u2081.symm (e\u2082.symm p) := rfl\n#align mul_equiv.symm_trans_apply MulEquiv.symm_trans_apply\n#align add_equiv.symm_trans_apply AddEquiv.symm_trans_apply\n\n-- Porting note: `simp` can prove this\n@[to_additive]\ntheorem apply_eq_iff_eq (e : M \u2243* N) {x y : M} : e x = e y \u2194 x = y :=\n  e.injective.eq_iff\n#align mul_equiv.apply_eq_iff_eq MulEquiv.apply_eq_iff_eq\n#align add_equiv.apply_eq_iff_eq AddEquiv.apply_eq_iff_eq\n\n@[to_additive]\ntheorem apply_eq_iff_symm_apply (e : M \u2243* N) {x : M} {y : N} : e x = y \u2194 x = e.symm y :=\n  e.toEquiv.apply_eq_iff_eq_symm_apply\n#align mul_equiv.apply_eq_iff_symm_apply MulEquiv.apply_eq_iff_symm_apply\n#align add_equiv.apply_eq_iff_symm_apply AddEquiv.apply_eq_iff_symm_apply\n\n@[to_additive]\ntheorem symm_apply_eq (e : M \u2243* N) {x y} : e.symm x = y \u2194 x = e y :=\n  e.toEquiv.symm_apply_eq\n#align mul_equiv.symm_apply_eq MulEquiv.symm_apply_eq\n#align add_equiv.symm_apply_eq AddEquiv.symm_apply_eq\n\n@[to_additive]\ntheorem eq_symm_apply (e : M \u2243* N) {x y} : y = e.symm x \u2194 e y = x :=\n  e.toEquiv.eq_symm_apply\n#align mul_equiv.eq_symm_apply MulEquiv.eq_symm_apply\n#align add_equiv.eq_symm_apply AddEquiv.eq_symm_apply\n\n@[to_additive]\ntheorem eq_comp_symm {\u03b1 : Type _} (e : M \u2243* N) (f : N \u2192 \u03b1) (g : M \u2192 \u03b1) :\n  f = g \u2218 e.symm \u2194 f \u2218 e = g :=\n  e.toEquiv.eq_comp_symm f g\n#align mul_equiv.eq_comp_symm MulEquiv.eq_comp_symm\n#align add_equiv.eq_comp_symm AddEquiv.eq_comp_symm\n\n@[to_additive]\ntheorem comp_symm_eq {\u03b1 : Type _} (e : M \u2243* N) (f : N \u2192 \u03b1) (g : M \u2192 \u03b1) :\n  g \u2218 e.symm = f \u2194 g = f \u2218 e :=\n  e.toEquiv.comp_symm_eq f g\n#align mul_equiv.comp_symm_eq MulEquiv.comp_symm_eq\n#align add_equiv.comp_symm_eq AddEquiv.comp_symm_eq\n\n@[to_additive]\ntheorem eq_symm_comp {\u03b1 : Type _} (e : M \u2243* N) (f : \u03b1 \u2192 M) (g : \u03b1 \u2192 N) :\n  f = e.symm \u2218 g \u2194 e \u2218 f = g :=\n  e.toEquiv.eq_symm_comp f g\n#align mul_equiv.eq_symm_comp MulEquiv.eq_symm_comp\n#align add_equiv.eq_symm_comp AddEquiv.eq_symm_comp\n\n@[to_additive]\ntheorem symm_comp_eq {\u03b1 : Type _} (e : M \u2243* N) (f : \u03b1 \u2192 M) (g : \u03b1 \u2192 N) :\n  e.symm \u2218 g = f \u2194 g = e \u2218 f :=\n  e.toEquiv.symm_comp_eq f g\n#align mul_equiv.symm_comp_eq MulEquiv.symm_comp_eq\n#align add_equiv.symm_comp_eq AddEquiv.symm_comp_eq\n\n@[to_additive (attr := simp)]\ntheorem symm_trans_self (e : M \u2243* N) : e.symm.trans e = refl N :=\n  FunLike.ext _ _ e.apply_symm_apply\n#align mul_equiv.symm_trans_self MulEquiv.symm_trans_self\n#align add_equiv.symm_trans_self AddEquiv.symm_trans_self\n\n@[to_additive (attr := simp)]\ntheorem self_trans_symm (e : M \u2243* N) : e.trans e.symm = refl M :=\n  FunLike.ext _ _ e.symm_apply_apply\n#align mul_equiv.self_trans_symm MulEquiv.self_trans_symm\n#align add_equiv.self_trans_symm AddEquiv.self_trans_symm\n\n-- Porting note: `simp` can prove this\n@[to_additive]\ntheorem coe_monoidHom_refl {M} [MulOneClass M] : (refl M : M \u2192* M) = MonoidHom.id M := rfl\n#align mul_equiv.coe_monoid_hom_refl MulEquiv.coe_monoidHom_refl\n#align add_equiv.coe_add_monoid_hom_refl AddEquiv.coe_addMonoidHom_refl\n\n-- Porting note: `simp` can prove this\n@[to_additive]\ntheorem coe_monoidHom_trans {M N P} [MulOneClass M] [MulOneClass N] [MulOneClass P]\n  (e\u2081 : M \u2243* N) (e\u2082 : N \u2243* P) :\n  (e\u2081.trans e\u2082 : M \u2192* P) = (e\u2082 : N \u2192* P).comp \u2191e\u2081 := rfl\n#align mul_equiv.coe_monoid_hom_trans MulEquiv.coe_monoidHom_trans\n#align add_equiv.coe_add_monoid_hom_trans AddEquiv.coe_addMonoidHom_trans\n\n/-- Two multiplicative isomorphisms agree if they are defined by the\nsame underlying function. -/\n@[to_additive (attr := ext)\n  \"Two additive isomorphisms agree if they are defined by the same underlying function.\"]\ntheorem ext {f g : MulEquiv M N} (h : \u2200 x, f x = g x) : f = g :=\n  FunLike.ext f g h\n#align mul_equiv.ext MulEquiv.ext\n#align add_equiv.ext AddEquiv.ext\n\n@[to_additive]\ntheorem ext_iff {f g : MulEquiv M N} : f = g \u2194 \u2200 x, f x = g x :=\n  FunLike.ext_iff\n#align mul_equiv.ext_iff MulEquiv.ext_iff\n#align add_equiv.ext_iff AddEquiv.ext_iff\n\n@[to_additive (attr := simp)]\ntheorem mk_coe (e : M \u2243* N) (e' h\u2081 h\u2082 h\u2083) : (\u27e8\u27e8e, e', h\u2081, h\u2082\u27e9, h\u2083\u27e9 : M \u2243* N) = e :=\n  ext fun _ => rfl\n#align mul_equiv.mk_coe MulEquiv.mk_coe\n#align add_equiv.mk_coe AddEquiv.mk_coe\n\n@[to_additive (attr := simp)]\ntheorem mk_coe' (e : M \u2243* N) (f h\u2081 h\u2082 h\u2083) : (MulEquiv.mk \u27e8f, e, h\u2081, h\u2082\u27e9 h\u2083 : N \u2243* M) = e.symm :=\n  symm_bijective.injective <| ext fun _ => rfl\n#align mul_equiv.mk_coe' MulEquiv.mk_coe'\n#align add_equiv.mk_coe' AddEquiv.mk_coe'\n\n@[to_additive]\nprotected theorem congr_arg {f : MulEquiv M N} {x x' : M} : x = x' \u2192 f x = f x' :=\n  FunLike.congr_arg f\n#align mul_equiv.congr_arg MulEquiv.congr_arg\n#align add_equiv.congr_arg AddEquiv.congr_arg\n\n@[to_additive]\nprotected theorem congr_fun {f g : MulEquiv M N} (h : f = g) (x : M) : f x = g x :=\n  FunLike.congr_fun h x\n#align mul_equiv.congr_fun MulEquiv.congr_fun\n#align add_equiv.congr_fun AddEquiv.congr_fun\n\n/-- The `MulEquiv` between two monoids with a unique element. -/\n@[to_additive \"The `AddEquiv` between two `AddMonoid`s with a unique element.\"]\ndef mulEquivOfUnique {M N} [Unique M] [Unique N] [Mul M] [Mul N] : M \u2243* N :=\n  { Equiv.equivOfUnique M N with map_mul' := fun _ _ => Subsingleton.elim _ _ }\n#align mul_equiv.mul_equiv_of_unique MulEquiv.mulEquivOfUnique\n#align add_equiv.add_equiv_of_unique AddEquiv.addEquivOfUnique\n\n/-- There is a unique monoid homomorphism between two monoids with a unique element. -/\n@[to_additive \"There is a unique additive monoid homomorphism between two additive monoids with\n  a unique element.\"]\ninstance {M N} [Unique M] [Unique N] [Mul M] [Mul N] : Unique (M \u2243* N) where\n  default := mulEquivOfUnique\n  uniq _ := ext fun _ => Subsingleton.elim _ _\n\n/-!\n## Monoids\n-/\n\n/-- A multiplicative isomorphism of monoids sends `1` to `1` (and is hence a monoid isomorphism). -/\n@[to_additive\n  \"An additive isomorphism of additive monoids sends `0` to `0`\n  (and is hence an additive monoid isomorphism).\"]\nprotected theorem map_one {M N} [MulOneClass M] [MulOneClass N] (h : M \u2243* N) : h 1 = 1 :=\n  _root_.map_one h\n#align mul_equiv.map_one MulEquiv.map_one\n#align add_equiv.map_zero AddEquiv.map_zero\n\n@[to_additive]\nprotected theorem map_eq_one_iff {M N} [MulOneClass M] [MulOneClass N] (h : M \u2243* N) {x : M} :\n  h x = 1 \u2194 x = 1 :=\n  MulEquivClass.map_eq_one_iff h\n#align mul_equiv.map_eq_one_iff MulEquiv.map_eq_one_iff\n#align add_equiv.map_eq_zero_iff AddEquiv.map_eq_zero_iff\n\n@[to_additive]\ntheorem map_ne_one_iff {M N} [MulOneClass M] [MulOneClass N] (h : M \u2243* N) {x : M} :\n  h x \u2260 1 \u2194 x \u2260 1 :=\n  MulEquivClass.map_ne_one_iff h\n#align mul_equiv.map_ne_one_iff MulEquiv.map_ne_one_iff\n#align add_equiv.map_ne_zero_iff AddEquiv.map_ne_zero_iff\n\n/-- A bijective `Semigroup` homomorphism is an isomorphism -/\n@[to_additive (attr := simps! apply) \"A bijective `AddSemigroup` homomorphism is an isomorphism\"]\nnoncomputable def ofBijective {M N F} [Mul M] [Mul N] [MulHomClass F M N]\n(f : F) (hf : Function.Bijective f) :\n    M \u2243* N :=\n  { Equiv.ofBijective f hf with map_mul' := map_mul f }\n#align mul_equiv.of_bijective MulEquiv.ofBijective\n#align add_equiv.of_bijective AddEquiv.ofBijective\n#align mul_equiv.of_bijective_apply MulEquiv.ofBijective_apply\n#align add_equiv.of_bijective_apply AddEquiv.ofBijective_apply\n\n-- porting note: todo: simplify `symm_apply` to `surjInv`?\n@[to_additive (attr := simp)]\ntheorem ofBijective_apply_symm_apply {M N} [MulOneClass M] [MulOneClass N] {n : N} (f : M \u2192* N)\n    (hf : Function.Bijective f) : f ((Equiv.ofBijective f hf).symm n) = n :=\n  (MulEquiv.ofBijective f hf).apply_symm_apply n\n#align mul_equiv.of_bijective_apply_symm_apply MulEquiv.ofBijective_apply_symm_apply\n#align add_equiv.of_bijective_apply_symm_apply AddEquiv.ofBijective_apply_symm_apply\n\n/-- Extract the forward direction of a multiplicative equivalence\nas a multiplication-preserving function.\n-/\n@[to_additive \"Extract the forward direction of an additive equivalence\n  as an addition-preserving function.\"]\ndef toMonoidHom {M N} [MulOneClass M] [MulOneClass N] (h : M \u2243* N) : M \u2192* N :=\n  { h with map_one' := h.map_one }\n#align mul_equiv.to_monoid_hom MulEquiv.toMonoidHom\n#align add_equiv.to_add_monoid_hom AddEquiv.toAddMonoidHom\n\n@[to_additive (attr := simp)]\ntheorem coe_toMonoidHom {M N} [MulOneClass M] [MulOneClass N] (e : M \u2243* N) :\n  \u21d1e.toMonoidHom = e := rfl\n#align mul_equiv.coe_to_monoid_hom MulEquiv.coe_toMonoidHom\n#align add_equiv.coe_to_add_monoid_hom AddEquiv.coe_toAddMonoidHom\n\nset_option linter.deprecated false in\n@[to_additive]\ntheorem toMonoidHom_injective {M N} [MulOneClass M] [MulOneClass N] :\n  Function.Injective (toMonoidHom : M \u2243* N \u2192 M \u2192* N) :=\n  fun _ _ h => MulEquiv.ext (MonoidHom.ext_iff.1 h)\n#align mul_equiv.to_monoid_hom_injective MulEquiv.toMonoidHom_injective\n#align add_equiv.to_add_monoid_hom_injective AddEquiv.toAddMonoidHom_injective\n\n/-- A multiplicative analogue of `Equiv.arrowCongr`,\nwhere the equivalence between the targets is multiplicative.\n-/\n@[to_additive (attr := simps apply) \"An additive analogue of `Equiv.arrowCongr`,\n  where the equivalence between the targets is additive.\"]\ndef arrowCongr {M N P Q : Type _} [Mul P] [Mul Q] (f : M \u2243 N) (g : P \u2243* Q) :\n  (M \u2192 P) \u2243* (N \u2192 Q) where\n  toFun h n := g (h (f.symm n))\n  invFun k m := g.symm (k (f m))\n  left_inv h := by ext; simp\n  right_inv k := by ext; simp\n  map_mul' h k := by ext; simp\n#align mul_equiv.arrow_congr MulEquiv.arrowCongr\n#align add_equiv.arrow_congr AddEquiv.arrowCongr\n#align mul_equiv.arrow_congr_apply MulEquiv.arrowCongr_apply\n#align add_equiv.arrow_congr_apply AddEquiv.arrowCongr_apply\n\n/-- A multiplicative analogue of `Equiv.arrowCongr`,\nfor multiplicative maps from a monoid to a commutative monoid.\n-/\n@[to_additive (attr := simps apply)\n  \"An additive analogue of `Equiv.arrowCongr`,\n  for additive maps from an additive monoid to a commutative additive monoid.\"]\n-- porting note: @[simps apply] removed because it was making a lemma which\n-- wasn't in simp normal form.\ndef monoidHomCongr {M N P Q} [MulOneClass M] [MulOneClass N] [CommMonoid P] [CommMonoid Q]\n  (f : M \u2243* N) (g : P \u2243* Q) :\n  (M \u2192* P) \u2243* (N \u2192* Q) where\n  toFun h := g.toMonoidHom.comp (h.comp f.symm.toMonoidHom)\n  invFun k := g.symm.toMonoidHom.comp (k.comp f.toMonoidHom)\n  left_inv h := by ext; simp\n  right_inv k := by ext; simp\n  map_mul' h k := by ext; simp\n#align mul_equiv.monoid_hom_congr MulEquiv.monoidHomCongr\n#align add_equiv.add_monoid_hom_congr AddEquiv.addMonoidHomCongr\n#align mul_equiv.monoid_hom_congr_apply MulEquiv.monoidHomCongr_apply\n#align add_equiv.add_monoid_hom_congr_apply AddEquiv.addMonoidHomCongr_apply\n\n/-- A family of multiplicative equivalences `\u03a0 j, (Ms j \u2243* Ns j)` generates a\nmultiplicative equivalence between `\u03a0 j, Ms j` and `\u03a0 j, Ns j`.\n\nThis is the `MulEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n`MulEquiv.arrowCongr`.\n-/\n@[to_additive (attr := simps apply)\n  \"A family of additive equivalences `\u03a0 j, (Ms j \u2243+ Ns j)`\n  generates an additive equivalence between `\u03a0 j, Ms j` and `\u03a0 j, Ns j`.\n\n  This is the `AddEquiv` version of `Equiv.piCongrRight`, and the dependent version of\n  `AddEquiv.arrowCongr`.\"]\ndef piCongrRight {\u03b7 : Type _} {Ms Ns : \u03b7 \u2192 Type _} [\u2200 j, Mul (Ms j)] [\u2200 j, Mul (Ns j)]\n  (es : \u2200 j, Ms j \u2243* Ns j) : (\u2200 j, Ms j) \u2243* \u2200 j, Ns j :=\n  { Equiv.piCongrRight fun j => (es j).toEquiv with\n    toFun := fun x j => es j (x j),\n    invFun := fun x j => (es j).symm (x j),\n    map_mul' := fun x y => funext fun j => (es j).map_mul (x j) (y j) }\n#align mul_equiv.Pi_congr_right MulEquiv.piCongrRight\n#align add_equiv.Pi_congr_right AddEquiv.piCongrRight\n#align mul_equiv.Pi_congr_right_apply MulEquiv.piCongrRight_apply\n#align add_equiv.Pi_congr_right_apply AddEquiv.piCongrRight_apply\n\n@[to_additive (attr := simp)]\ntheorem piCongrRight_refl {\u03b7 : Type _} {Ms : \u03b7 \u2192 Type _} [\u2200 j, Mul (Ms j)] :\n  (piCongrRight fun j => MulEquiv.refl (Ms j)) = MulEquiv.refl _ := rfl\n#align mul_equiv.Pi_congr_right_refl MulEquiv.piCongrRight_refl\n#align add_equiv.Pi_congr_right_refl AddEquiv.piCongrRight_refl\n\n@[to_additive (attr := simp)]\ntheorem piCongrRight_symm {\u03b7 : Type _} {Ms Ns : \u03b7 \u2192 Type _} [\u2200 j, Mul (Ms j)] [\u2200 j, Mul (Ns j)]\n  (es : \u2200 j, Ms j \u2243* Ns j) : (piCongrRight es).symm = piCongrRight fun i => (es i).symm := rfl\n#align mul_equiv.Pi_congr_right_symm MulEquiv.piCongrRight_symm\n#align add_equiv.Pi_congr_right_symm AddEquiv.piCongrRight_symm\n\n@[to_additive (attr := simp)]\ntheorem piCongrRight_trans {\u03b7 : Type _} {Ms Ns Ps : \u03b7 \u2192 Type _} [\u2200 j, Mul (Ms j)]\n  [\u2200 j, Mul (Ns j)] [\u2200 j, Mul (Ps j)] (es : \u2200 j, Ms j \u2243* Ns j) (fs : \u2200 j, Ns j \u2243* Ps j) :\n  (piCongrRight es).trans (piCongrRight fs) = piCongrRight fun i => (es i).trans (fs i) := rfl\n#align mul_equiv.Pi_congr_right_trans MulEquiv.piCongrRight_trans\n#align add_equiv.Pi_congr_right_trans AddEquiv.piCongrRight_trans\n\n/-- A family indexed by a nonempty subsingleton type is equivalent to the element at the single\nindex. -/\n@[to_additive (attr := simps!)\n  \"A family indexed by a nonempty subsingleton type is\n  equivalent to the element at the single index.\"]\ndef piSubsingleton {\u03b9 : Type _} (M : \u03b9 \u2192 Type _) [\u2200 j, Mul (M j)] [Subsingleton \u03b9]\n  (i : \u03b9) : (\u2200 j, M j) \u2243* M i :=\n  { Equiv.piSubsingleton M i with map_mul' := fun _ _ => Pi.mul_apply _ _ _ }\n#align mul_equiv.Pi_subsingleton MulEquiv.piSubsingleton\n#align add_equiv.Pi_subsingleton AddEquiv.piSubsingleton\n#align mul_equiv.Pi_subsingleton_apply MulEquiv.piSubsingleton_apply\n#align add_equiv.Pi_subsingleton_apply AddEquiv.piSubsingleton_apply\n#align mul_equiv.Pi_subsingleton_symm_apply MulEquiv.piSubsingleton_symm_apply\n#align add_equiv.Pi_subsingleton_symm_apply AddEquiv.piSubsingleton_symm_apply\n\n/-!\n# Groups\n-/\n\n/-- A multiplicative equivalence of groups preserves inversion. -/\n@[to_additive \"An additive equivalence of additive groups preserves negation.\"]\nprotected theorem map_inv [Group G] [DivisionMonoid H] (h : G \u2243* H) (x : G) :\n  h x\u207b\u00b9 = (h x)\u207b\u00b9 :=\n  _root_.map_inv h x\n#align mul_equiv.map_inv MulEquiv.map_inv\n#align add_equiv.map_neg AddEquiv.map_neg\n\n/-- A multiplicative equivalence of groups preserves division. -/\n@[to_additive \"An additive equivalence of additive groups preserves subtractions.\"]\nprotected theorem map_div [Group G] [DivisionMonoid H] (h : G \u2243* H) (x y : G) :\n  h (x / y) = h x / h y :=\n  _root_.map_div h x y\n#align mul_equiv.map_div MulEquiv.map_div\n#align add_equiv.map_sub AddEquiv.map_sub\n\nend MulEquiv\n\n-- porting note: we want to add\n-- `@[simps (config := { fullyApplied := false })]`\n-- here, but it generates simp lemmas which aren't in simp normal form\n-- (they have `toFun` in)\n/-- Given a pair of multiplicative homomorphisms `f`, `g` such that `g.comp f = id` and\n`f.comp g = id`, returns an multiplicative equivalence with `toFun = f` and `invFun = g`. This\nconstructor is useful if the underlying type(s) have specialized `ext` lemmas for multiplicative\nhomomorphisms. -/\n@[to_additive\n  \"Given a pair of additive homomorphisms `f`, `g` such that `g.comp f = id` and\n  `f.comp g = id`, returns an additive equivalence with `toFun = f` and `invFun = g`. This\n  constructor is useful if the underlying type(s) have specialized `ext` lemmas for additive\n  homomorphisms.\"]\ndef MulHom.toMulEquiv [Mul M] [Mul N] (f : M \u2192\u2099* N) (g : N \u2192\u2099* M) (h\u2081 : g.comp f = MulHom.id _)\n  (h\u2082 : f.comp g = MulHom.id _) : M \u2243* N where\n  toFun := f\n  invFun := g\n  left_inv := FunLike.congr_fun h\u2081\n  right_inv := FunLike.congr_fun h\u2082\n  map_mul' := f.map_mul\n#align mul_hom.to_mul_equiv MulHom.toMulEquiv\n#align add_hom.to_add_equiv AddHom.toAddEquiv\n\n-- porting note: the next two lemmas were added manually because `@[simps]` is generating\n-- lemmas with `toFun` in\n@[to_additive (attr := simp)]\ntheorem MulHom.toMulEquiv_apply [Mul M] [Mul N] (f : M \u2192\u2099* N) (g : N \u2192\u2099* M)\n    (h\u2081 : g.comp f = MulHom.id _) (h\u2082 : f.comp g = MulHom.id _) :\n    ((MulHom.toMulEquiv f g h\u2081 h\u2082) : M \u2192 N) = f :=\n  rfl\n#align mul_hom.to_mul_equiv_apply MulHom.toMulEquiv_apply\n#align add_hom.to_add_equiv_apply AddHom.toAddEquiv_apply\n\n@[to_additive (attr := simp)]\ntheorem MulHom.toMulEquiv_symm_apply [Mul M] [Mul N] (f : M \u2192\u2099* N) (g : N \u2192\u2099* M)\n    (h\u2081 : g.comp f = MulHom.id _) (h\u2082 : f.comp g = MulHom.id _) :\n    (MulEquiv.symm (MulHom.toMulEquiv f g h\u2081 h\u2082) : N \u2192 M) = \u2191g :=\n  rfl\n#align mul_hom.to_mul_equiv_symm_apply MulHom.toMulEquiv_symm_apply\n#align add_hom.to_add_equiv_symm_apply AddHom.toAddEquiv_symm_apply\n\n/-- Given a pair of monoid homomorphisms `f`, `g` such that `g.comp f = id` and `f.comp g = id`,\nreturns an multiplicative equivalence with `toFun = f` and `invFun = g`.  This constructor is\nuseful if the underlying type(s) have specialized `ext` lemmas for monoid homomorphisms. -/\n@[to_additive (attr := simps (config := { fullyApplied := false }))\n  \"Given a pair of additive monoid homomorphisms `f`, `g` such that `g.comp f = id`\n  and `f.comp g = id`, returns an additive equivalence with `toFun = f` and `invFun = g`.  This\n  constructor is useful if the underlying type(s) have specialized `ext` lemmas for additive\n  monoid homomorphisms.\"]\ndef MonoidHom.toMulEquiv [MulOneClass M] [MulOneClass N] (f : M \u2192* N) (g : N \u2192* M)\n  (h\u2081 : g.comp f = MonoidHom.id _) (h\u2082 : f.comp g = MonoidHom.id _) : M \u2243* N where\n  toFun := f\n  invFun := g\n  left_inv := FunLike.congr_fun h\u2081\n  right_inv := FunLike.congr_fun h\u2082\n  map_mul' := f.map_mul\n#align monoid_hom.to_mul_equiv MonoidHom.toMulEquiv\n#align add_monoid_hom.to_add_equiv AddMonoidHom.toAddEquiv\n#align monoid_hom.to_mul_equiv_apply MonoidHom.toMulEquiv_apply\n#align add_monoid_hom.to_add_equiv_apply AddMonoidHom.toAddEquiv_apply\n#align monoid_hom.to_mul_equiv_symm_apply MonoidHom.toMulEquiv_symm_apply\n#align add_monoid_hom.to_add_equiv_symm_apply AddMonoidHom.toAddEquiv_symm_apply\n\nnamespace Equiv\n\nsection InvolutiveInv\n\nvariable (G) [InvolutiveInv G]\n\n/-- Inversion on a `Group` or `GroupWithZero` is a permutation of the underlying type. -/\n@[to_additive (attr := simps! (config := { fullyApplied := false }) apply)\n    \"Negation on an `AddGroup` is a permutation of the underlying type.\"]\nprotected def inv : Perm G :=\n  inv_involutive.toPerm _\n#align equiv.inv Equiv.inv\n#align equiv.neg Equiv.neg\n#align equiv.inv_apply Equiv.inv_apply\n#align equiv.neg_apply Equiv.neg_apply\n\nvariable {G}\n\n@[to_additive (attr := simp)]\ntheorem inv_symm : (Equiv.inv G).symm = Equiv.inv G := rfl\n#align equiv.inv_symm Equiv.inv_symm\n#align equiv.neg_symm Equiv.neg_symm\n\nend InvolutiveInv\n\nend Equiv\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Algebra/Hom/Equiv/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863695, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.3914718702934395}}
{"text": "import algebra.camera.basic\n\nuniverses u v\n\n@[ext] structure agree_struct (\u03b1 : Type u) :=\n(elts : finset \u03b1)\n(nonempty : elts.nonempty)\n\ninstance agree_struct.has_mem {\u03b1 : Type u} : has_mem \u03b1 (agree_struct \u03b1) :=\n\u27e8\u03bb x a, x \u2208 a.elts\u27e9\n\n@[simp] lemma agree_struct.has_mem_iff {\u03b1 : Type u} (x : \u03b1) (a : agree_struct \u03b1) :\n  x \u2208 a.elts \u2194 x \u2208 a := iff.rfl\n\n@[simp] lemma agree_struct.mem_mk {\u03b1 : Type u} (x : \u03b1) (elts : finset \u03b1) (nonempty : elts.nonempty) :\n  x \u2208 (\u27e8elts, nonempty\u27e9 : agree_struct \u03b1) \u2194 x \u2208 elts := iff.rfl\n\nstructure agree_struct.eq_at {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) (a b : agree_struct \u03b1) : Prop :=\n(left : \u2200 x \u2208 a, \u2203 y \u2208 b, x =[n] y)\n(right : \u2200 y \u2208 b, \u2203 x \u2208 a, x =[n] y)\n\nlemma agree_struct.eq_at_reflexive {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  reflexive (agree_struct.eq_at n : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\nbegin\n  intros a,\n  split;\n  { intros x hx,\n    refine \u27e8x, hx, _\u27e9,\n    refl, },\nend\n\n@[refl] lemma agree_struct.eq_at_refl {\u03b1 : Type u} [ofe \u03b1] {n : \u2115} (a : agree_struct \u03b1) :\n  agree_struct.eq_at n a a := agree_struct.eq_at_reflexive n a\n\nlemma agree_struct.eq_at_symmetric {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  symmetric (agree_struct.eq_at n : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\nbegin\n  intros a b hab,\n  split,\n  { intros x hx,\n    obtain \u27e8y, hy, h\u27e9 := hab.right x hx,\n    refine \u27e8y, hy, _\u27e9,\n    symmetry,\n    assumption, },\n  { intros x hx,\n    obtain \u27e8y, hy, h\u27e9 := hab.left x hx,\n    refine \u27e8y, hy, _\u27e9,\n    symmetry,\n    assumption, },\nend\n\n@[symm] lemma agree_struct.eq_at_symm {\u03b1 : Type u} [ofe \u03b1] {n : \u2115} (a b : agree_struct \u03b1) :\n  agree_struct.eq_at n a b \u2192 agree_struct.eq_at n b a := \u03bb h, agree_struct.eq_at_symmetric n h\n\nlemma agree_struct.eq_at_transitive {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  transitive (agree_struct.eq_at n : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\nbegin\n  intros a b c hab hbc,\n  split,\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hab.left x hx,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := hbc.left y hy\u2081,\n    refine \u27e8z, hz\u2081, _\u27e9,\n    transitivity y; assumption, },\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hbc.right x hx,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := hab.right y hy\u2081,\n    refine \u27e8z, hz\u2081, _\u27e9,\n    transitivity y; assumption, },\nend\n\n@[trans] lemma agree_struct.eq_at_trans {\u03b1 : Type u} [ofe \u03b1] {n : \u2115} (a b c : agree_struct \u03b1) :\n  agree_struct.eq_at n a b \u2192 agree_struct.eq_at n b c \u2192 agree_struct.eq_at n a c :=\n\u03bb h\u2081 h\u2082, agree_struct.eq_at_transitive n h\u2081 h\u2082\n\nlemma agree_struct.eq_at_equivalence {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  equivalence (agree_struct.eq_at n : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\n\u27e8agree_struct.eq_at_reflexive n, agree_struct.eq_at_symmetric n, agree_struct.eq_at_transitive n\u27e9\n\nlemma agree_struct.eq_at_mono {\u03b1 : Type u} [ofe \u03b1] :\n  antitone (agree_struct.eq_at : \u2115 \u2192 agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\nbegin\n  intros m n hmn a b h,\n  split,\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := h.left x hx,\n    refine \u27e8y, hy\u2081, _\u27e9,\n    exact eq_at_mono hmn hy\u2082, },\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := h.right x hx,\n    refine \u27e8y, hy\u2081, _\u27e9,\n    exact eq_at_mono hmn hy\u2082, },\nend\n\nlemma agree_struct.mem_of_forall_eq_at {\u03b1 : Type u} [ofe \u03b1] (a b : agree_struct \u03b1) :\n  (\u2200 (n : \u2115), agree_struct.eq_at n a b) \u2192 \u2200 x \u2208 a, x \u2208 b :=\nbegin\n  intros h x hx,\n  have : \u2200 n, \u2203 y \u2208 b, x =[n] y := \u03bb n, (h n).left x hx,\n  choose Y hY\u2081 hY\u2082 using this,\n  set f : \u2115 \u2192 b.elts := \u03bb n, \u27e8Y n, hY\u2081 n\u27e9,\n  obtain \u27e8\u27e8y, hy\u2081\u27e9, hy\u2082\u27e9 := finite.exists_infinite_fiber f,\n  suffices : x = y,\n  { rw this, exact hy\u2081, },\n  rw set.infinite_coe_iff at hy\u2082,\n  refine eq_at_infinite _ _ hy\u2082 _,\n  intros n hn,\n  simp only [set.mem_preimage, set.mem_singleton_iff, subtype.mk_eq_mk] at hn,\n  rw \u2190 hn,\n  exact hY\u2082 n,\nend\n\nlemma agree_struct.eq_at_limit {\u03b1 : Type u} [ofe \u03b1] (a b : agree_struct \u03b1) :\n  (\u2200 (n : \u2115), agree_struct.eq_at n a b) \u2192 a = b :=\nbegin\n  intros h,\n  ext x,\n  split,\n  { exact agree_struct.mem_of_forall_eq_at _ _ h _, },\n  { refine agree_struct.mem_of_forall_eq_at _ _ _ _,\n    intro n,\n    symmetry,\n    exact h n, },\nend\n\ndef agree_struct.rel {\u03b1 : Type u} [ofe \u03b1] (a b : agree_struct \u03b1) : Prop :=\n\u2200 n, agree_struct.eq_at n a b\n\nlemma agree_struct.rel_reflexive {\u03b1 : Type u} [ofe \u03b1] :\n  reflexive (agree_struct.rel : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\n\u03bb a n, agree_struct.eq_at_reflexive n a\n\n@[refl] lemma agree_struct.rel_refl {\u03b1 : Type u} [ofe \u03b1] (a : agree_struct \u03b1) :\n  agree_struct.rel a a := agree_struct.rel_reflexive a\n\nlemma agree_struct.rel_symmetric {\u03b1 : Type u} [ofe \u03b1] :\n  symmetric (agree_struct.rel : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\n\u03bb a b hab n, agree_struct.eq_at_symmetric n (hab n)\n\n@[symm] lemma agree_struct.rel_symm {\u03b1 : Type u} [ofe \u03b1] (a b : agree_struct \u03b1) :\n  agree_struct.rel a b \u2192 agree_struct.rel b a := \u03bb h, agree_struct.rel_symmetric h\n\nlemma agree_struct.rel_transitive {\u03b1 : Type u} [ofe \u03b1] :\n  transitive (agree_struct.rel : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\n\u03bb a b c hab hbc n, agree_struct.eq_at_transitive n (hab n) (hbc n)\n\n@[trans] lemma agree_struct.rel_trans {\u03b1 : Type u} [ofe \u03b1] (a b c : agree_struct \u03b1) :\n  agree_struct.rel a b \u2192 agree_struct.rel b c \u2192 agree_struct.rel a c :=\n\u03bb h\u2081 h\u2082, agree_struct.rel_transitive h\u2081 h\u2082\n\nlemma agree_struct.rel_equivalence {\u03b1 : Type u} [ofe \u03b1] :\n  equivalence (agree_struct.rel : agree_struct \u03b1 \u2192 agree_struct \u03b1 \u2192 Prop) :=\n\u27e8agree_struct.rel_reflexive, agree_struct.rel_symmetric, agree_struct.rel_transitive\u27e9\n\ninstance {\u03b1 : Type u} [ofe \u03b1] : setoid (agree_struct \u03b1) :=\n\u27e8agree_struct.rel, agree_struct.rel_equivalence\u27e9\n\ndef agree (\u03b1 : Type u) [ofe \u03b1] : Type u := quotient (agree_struct.setoid : setoid (agree_struct \u03b1))\n\nlemma agree_struct.eq_at_respects_rel {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) (a\u2081 a\u2082 b\u2081 b\u2082 : agree_struct \u03b1) :\n  a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 agree_struct.eq_at n a\u2081 a\u2082 = agree_struct.eq_at n b\u2081 b\u2082 :=\nbegin\n  intros h\u2081 h\u2082,\n  ext1,\n  split; intro h,\n  { transitivity a\u2081,\n    { symmetry, exact h\u2081 n, },\n    transitivity a\u2082,\n    { exact h, },\n    { exact h\u2082 n, }, },\n  { transitivity b\u2081,\n    { exact h\u2081 n, },\n    transitivity b\u2082,\n    { exact h, },\n    { symmetry, exact h\u2082 n, }, },\nend\n\ndef agree.eq_at {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) : agree \u03b1 \u2192 agree \u03b1 \u2192 Prop :=\nquotient.lift\u2082 (agree_struct.eq_at n) (agree_struct.eq_at_respects_rel n)\n\nprivate lemma agree.eq_at_reflexive {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  reflexive (agree.eq_at n : agree \u03b1 \u2192 agree \u03b1 \u2192 Prop) :=\nbegin\n  intro a, refine quotient.induction_on a _, clear a, intro a,\n  simp only [agree.eq_at, quotient.lift\u2082_mk],\nend\n\nprivate lemma agree.eq_at_symmetric {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  symmetric (agree.eq_at n : agree \u03b1 \u2192 agree \u03b1 \u2192 Prop) :=\nbegin\n  intros a b, refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n  simp only [agree.eq_at, quotient.lift\u2082_mk],\n  intro h, symmetry, exact h,\nend\n\nprivate lemma agree.eq_at_transitive {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) :\n  transitive (agree.eq_at n : agree \u03b1 \u2192 agree \u03b1 \u2192 Prop) :=\nbegin\n  intros a b c, refine quotient.induction_on\u2083 a b c _, clear a b c, intros a b c,\n  simp only [agree.eq_at, quotient.lift\u2082_mk],\n  intros h\u2081 h\u2082, transitivity b; assumption,\nend\n\nprivate lemma agree.eq_at_mono {\u03b1 : Type u} [ofe \u03b1] :\n  antitone (agree.eq_at : \u2115 \u2192 agree \u03b1 \u2192 agree \u03b1 \u2192 Prop) :=\nbegin\n  intros m n hmn a b,\n  refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n  simp only [agree.eq_at, quotient.lift\u2082_mk],\n  exact agree_struct.eq_at_mono hmn a b,\nend\n\nprivate lemma agree.eq_at_limit {\u03b1 : Type u} [ofe \u03b1] (a b : agree \u03b1) :\n  (\u2200 (n : \u2115), agree.eq_at n a b) \u2192 a = b :=\nbegin\n  refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n  intros h,\n  refine quotient.sound _,\n  intro n,\n  rw agree_struct.eq_at_limit a b h,\nend\n\ninstance agree.ofe (\u03b1 : Type u) [ofe \u03b1] : ofe (agree \u03b1) := {\n  eq_at := agree.eq_at,\n  eq_at_reflexive := agree.eq_at_reflexive,\n  eq_at_symmetric := agree.eq_at_symmetric,\n  eq_at_transitive := agree.eq_at_transitive,\n  eq_at_mono' := agree.eq_at_mono,\n  eq_at_limit' := agree.eq_at_limit,\n}\n\n@[simp] lemma agree.eq_at_mk' {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) (a b : agree_struct \u03b1) :\n  agree.eq_at n \u27e6a\u27e7 \u27e6b\u27e7 \u2194 agree_struct.eq_at n a b := iff.rfl\n\n@[simp] lemma agree.eq_at_mk {\u03b1 : Type u} [ofe \u03b1] (n : \u2115) (a b : agree_struct \u03b1) :\n  (@ofe.eq_at (agree \u03b1) _ n \u27e6a\u27e7 \u27e6b\u27e7) \u2194 agree_struct.eq_at n a b := iff.rfl\n\ninstance agree_struct.comm_semigroup (\u03b1 : Type u) [ofe \u03b1] [decidable_eq \u03b1] :\n  comm_semigroup (agree_struct \u03b1) := {\n  mul := \u03bb a b, \u27e8a.elts \u222a b.elts, begin\n    rw [\u2190 finset.coe_nonempty, finset.coe_union, set.union_nonempty, finset.coe_nonempty],\n    exact or.inl a.nonempty,\n  end\u27e9,\n  mul_assoc := begin\n    intros a b c,\n    ext x,\n    simp only [finset.union_assoc],\n  end,\n  mul_comm := begin\n    intros a b,\n    ext x,\n    simp only [has_mul.mul, finset.mem_union],\n    exact or.comm,\n  end,\n}\n\n@[simp] lemma agree_struct.mul_elts {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1]\n  (a b : agree_struct \u03b1) : (a * b).elts = a.elts \u222a b.elts := rfl\n\n@[simp] lemma agree_struct.mem_mul {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1]\n  (x : \u03b1) (a b : agree_struct \u03b1) : x \u2208 a * b \u2194 x \u2208 a \u2228 x \u2208 b :=\nbegin\n  rw [\u2190 agree_struct.has_mem_iff, agree_struct.mul_elts, finset.mem_union],\n  refl,\nend\n\nlemma agree_struct.eq_at_mul {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1]\n  {n : \u2115} (a b c d : agree_struct \u03b1) :\n  agree_struct.eq_at n a b \u2192 agree_struct.eq_at n c d \u2192 agree_struct.eq_at n (a * c) (b * d) :=\nbegin\n  intros ha hb,\n  split,\n  { intros x hx,\n    rw agree_struct.mem_mul at hx,\n    cases hx,\n    { obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := ha.left x hx,\n      refine \u27e8y, _, hy\u2082\u27e9,\n      rw agree_struct.mem_mul,\n      exact or.inl hy\u2081, },\n    { obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hb.left x hx,\n      refine \u27e8y, _, hy\u2082\u27e9,\n      rw agree_struct.mem_mul,\n      exact or.inr hy\u2081, }, },\n  { intros x hx,\n    rw agree_struct.mem_mul at hx,\n    cases hx,\n    { obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := ha.right x hx,\n      refine \u27e8y, _, hy\u2082\u27e9,\n      rw agree_struct.mem_mul,\n      exact or.inl hy\u2081, },\n    { obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hb.right x hx,\n      refine \u27e8y, _, hy\u2082\u27e9,\n      rw agree_struct.mem_mul,\n      exact or.inr hy\u2081, }, },\nend\n\nlemma agree_struct.mul_respects_rel {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1]\n  (a\u2081 a\u2082 b\u2081 b\u2082 : agree_struct \u03b1) : a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 \u27e6a\u2081 * a\u2082\u27e7 = \u27e6b\u2081 * b\u2082\u27e7 :=\n\u03bb ha hb, quotient.sound (\u03bb n, agree_struct.eq_at_mul a\u2081 b\u2081 a\u2082 b\u2082 (ha n) (hb n))\n\ninstance agree.comm_semigroup (\u03b1 : Type u) [ofe \u03b1] [decidable_eq \u03b1] : comm_semigroup (agree \u03b1) := {\n  mul := quotient.lift\u2082 (\u03bb a b, \u27e6a * b\u27e7) agree_struct.mul_respects_rel,\n  mul_assoc := begin\n    intros a b c, refine quotient.induction_on\u2083 a b c _, clear a b c, intros a b c,\n    refine quotient.sound _,\n    rw mul_assoc,\n    exact setoid.refl _,\n  end,\n  mul_comm := begin\n    intros a b, refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n    refine quotient.sound _,\n    rw mul_comm,\n    exact setoid.refl _,\n  end,\n}\n\n@[simp] lemma agree_struct.mul_self {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] (a : agree_struct \u03b1) :\n  a * a = a :=\nby ext; rw [agree_struct.mul_elts, finset.union_idempotent]\n\n@[simp] lemma agree.mul_self {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] (a : agree \u03b1) :\n  a * a = a :=\nbegin\n  refine quotient.induction_on a _, clear a, intro a,\n  refine quotient.sound _,\n  rw agree_struct.mul_self,\n  exact setoid.refl _,\nend\n\n@[simp] lemma agree_struct.mul_mk {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1]\n  (a b : agree_struct \u03b1) : @has_mul.mul (agree \u03b1) _ \u27e6a\u27e7 \u27e6b\u27e7 = \u27e6a * b\u27e7 := rfl\n\ndef agree_struct.validn {\u03b1 : Type u} [ofe \u03b1] (a : agree_struct \u03b1) : sprop :=\n\u27e8\u03bb n, \u2200 x y \u2208 a, x =[n] y, \u03bb m n hmn h x hx y hy, eq_at_mono hmn (h x hx y hy)\u27e9\n\nlemma agree_struct.validn_mul {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] (a b : agree_struct \u03b1) :\n  agree_struct.validn (a * b) \u2264 agree_struct.validn a :=\nbegin\n  intros n h x hx y hy,\n  refine h x _ y _;\n  rw agree_struct.mem_mul;\n  exact or.inl \u2039_\u203a,\nend\n\nlemma agree_struct.validn_respects_rel {\u03b1 : Type u} [ofe \u03b1] (a b : agree_struct \u03b1) :\n  a \u2248 b \u2192 a.validn = b.validn :=\nbegin\n  intro hab,\n  ext n,\n  split,\n  { intros h x hx y hy,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := (hab n).right x hx,\n    obtain \u27e8w, hw\u2081, hw\u2082\u27e9 := (hab n).right y hy,\n    have := h z hz\u2081 w hw\u2081,\n    symmetry' at hz\u2082,\n    transitivity, assumption, transitivity; assumption, },\n  { intros h x hx y hy,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := (hab n).left x hx,\n    obtain \u27e8w, hw\u2081, hw\u2082\u27e9 := (hab n).left y hy,\n    have := h z hz\u2081 w hw\u2081,\n    symmetry' at hw\u2082,\n    transitivity, assumption, transitivity; assumption, },\nend\n\ndef agree.validn {\u03b1 : Type u} [ofe \u03b1] : agree \u03b1 \u2192 sprop :=\nquotient.lift agree_struct.validn agree_struct.validn_respects_rel\n\n@[simp] lemma agree.validn_mk {\u03b1 : Type u} [ofe \u03b1] (a : agree_struct \u03b1) :\n  agree.validn \u27e6a\u27e7 = a.validn := rfl\n\nlemma agree.validn_is_nonexpansive {\u03b1 : Type u} [ofe \u03b1] :\n  is_nonexpansive (agree.validn : agree \u03b1 \u2192 sprop) :=\nbegin\n  intros n a b, refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n  intros h m hmn,\n  rw [agree.validn_mk, agree.validn_mk],\n  rw agree.eq_at_mk at h,\n  split,\n  { intros hm x hx y hy,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h.right x hx,\n    obtain \u27e8w, hw\u2081, hw\u2082\u27e9 := h.right y hy,\n    have hz\u2083 := eq_at_mono hmn hz\u2082,\n    have hw\u2083 := eq_at_mono hmn hw\u2082,\n    have := hm z hz\u2081 w hw\u2081,\n    symmetry' at hz\u2083,\n    transitivity, assumption, transitivity; assumption, },\n  { intros hm x hx y hy,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := h.left x hx,\n    obtain \u27e8w, hw\u2081, hw\u2082\u27e9 := h.left y hy,\n    have hz\u2083 := eq_at_mono hmn hz\u2082,\n    have hw\u2083 := eq_at_mono hmn hw\u2082,\n    have := hm z hz\u2081 w hw\u2081,\n    symmetry' at hw\u2083,\n    transitivity, assumption, transitivity; assumption, },\nend\n\ninstance agree_struct.decidable_exists_mem {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a : agree_struct \u03b1} {x : \u03b1} : decidable (\u2203 y \u2208 a, x =[n] y) :=\nbegin\n  refine decidable_of_decidable_of_iff _ _,\n  exact (finset.filter (\u03bb y, x =[n] y) a.elts).nonempty,\n  { apply_instance, },\n  split,\n  { rintro \u27e8y, hy\u27e9,\n    simp only [finset.mem_filter, agree_struct.has_mem_iff] at hy,\n    exact \u27e8y, hy.1, hy.2\u27e9, },\n  { rintro \u27e8y, hy\u2081, hy\u2082\u27e9,\n    refine \u27e8y, _\u27e9,\n    simp only [finset.mem_filter, agree_struct.has_mem_iff],\n    exact \u27e8hy\u2081, hy\u2082\u27e9, },\nend\n\n/-- Keeps those elements of `a` that have an element of `b` that it is `n`-equal to. -/\ndef agree_struct.filter_eq {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  (a b : agree_struct \u03b1) (n : \u2115) (h : \u2203 x \u2208 a, \u2203 y \u2208 b, x =[n] y) : agree_struct \u03b1 :=\n\u27e8finset.filter (\u03bb x, \u2203 y \u2208 b, x =[n] y) a.elts, begin\n  obtain \u27e8x, hx, y, hy, h\u27e9 := h,\n  refine \u27e8x, _\u27e9,\n  simp only [finset.mem_filter, agree_struct.has_mem_iff],\n  exact \u27e8hx, y, hy, h\u27e9,\nend\u27e9\n\n@[simp] lemma agree_struct.mem_filter_eq {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  (x : \u03b1) (a b : agree_struct \u03b1) (n : \u2115) (h : \u2203 x \u2208 a, \u2203 y \u2208 b, x =[n] y) :\n  x \u2208 a.filter_eq b n h \u2194 x \u2208 a \u2227 \u2203 y \u2208 b, x =[n] y :=\nby simp only [agree_struct.filter_eq, agree_struct.mem_mk,\n  finset.mem_filter, agree_struct.has_mem_iff]\n\nlemma agree_struct.exists_eq_at_of_mul_eq_at {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1} (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    \u2203 x \u2208 a, \u2203 y \u2208 b\u2081, x =[n] y :=\nbegin\n  obtain \u27e8y, hy\u27e9 := b\u2081.nonempty,\n  obtain \u27e8x, hx\u2081, hx\u2082\u27e9 := hb.right y _,\n  refine \u27e8x, hx\u2081, y, hy, hx\u2082\u27e9,\n  rw agree_struct.mem_mul,\n  exact or.inl hy,\nend\n\nlemma agree_struct.exists_eq_at_of_mul_eq_at' {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1} (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    \u2203 x \u2208 a, \u2203 y \u2208 b\u2082, x =[n] y :=\nbegin\n  refine agree_struct.exists_eq_at_of_mul_eq_at _,\n  exact b\u2081,\n  rw mul_comm,\n  exact hb,\nend\n\ndef agree_struct.extend {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1} (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    agree_struct \u03b1 \u00d7 agree_struct \u03b1 :=\n\u27e8a.filter_eq b\u2081 n (agree_struct.exists_eq_at_of_mul_eq_at hb),\n  a.filter_eq b\u2082 n (agree_struct.exists_eq_at_of_mul_eq_at' hb)\u27e9\n\nlemma agree_struct.extend_mul_eq {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1} (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    a = (agree_struct.extend hb).1 * (agree_struct.extend hb).2 :=\nbegin\n  unfold agree_struct.extend,\n  ext x,\n  simp only [agree_struct.has_mem_iff, agree_struct.mul_elts, finset.mem_union,\n    agree_struct.mem_filter_eq, exists_prop],\n  split,\n  { intro hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hb.left x hx,\n    rw agree_struct.mem_mul at hy\u2081,\n    cases hy\u2081,\n    exact or.inl \u27e8hx, y, hy\u2081, hy\u2082\u27e9,\n    exact or.inr \u27e8hx, y, hy\u2081, hy\u2082\u27e9, },\n  { rintro (hx | hx);\n    exact hx.1, },\nend\n\nlemma agree_struct.extend_eq_at_left {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1}\n  (ha : agree_struct.validn a n) (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    agree_struct.eq_at n (agree_struct.extend hb).1 b\u2081 :=\nbegin\n  unfold agree_struct.extend,\n  split,\n  { intros x hx,\n    rw agree_struct.mem_filter_eq at hx,\n    obtain \u27e8hx, y, hy\u2081, hy\u2082\u27e9 := hx,\n    exact \u27e8y, hy\u2081, hy\u2082\u27e9, },\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hb.right x _,\n    refine \u27e8y, _, hy\u2082\u27e9,\n    rw agree_struct.mem_filter_eq,\n    exact \u27e8hy\u2081, x, hx, hy\u2082\u27e9,\n    rw agree_struct.mem_mul,\n    exact or.inl hx, },\nend\n\nlemma agree_struct.extend_eq_at_right {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree_struct \u03b1}\n  (ha : agree_struct.validn a n) (hb : agree_struct.eq_at n a (b\u2081 * b\u2082)) :\n    agree_struct.eq_at n (agree_struct.extend hb).2 b\u2082 :=\nbegin\n  unfold agree_struct.extend,\n  split,\n  { intros x hx,\n    rw agree_struct.mem_filter_eq at hx,\n    obtain \u27e8hx, y, hy\u2081, hy\u2082\u27e9 := hx,\n    exact \u27e8y, hy\u2081, hy\u2082\u27e9, },\n  { intros x hx,\n    obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := hb.right x _,\n    refine \u27e8y, _, hy\u2082\u27e9,\n    rw agree_struct.mem_filter_eq,\n    exact \u27e8hy\u2081, x, hx, hy\u2082\u27e9,\n    rw agree_struct.mem_mul,\n    exact or.inr hx, },\nend\n\nattribute [reducible, elab_as_eliminator]\nprotected def quotient.dlift {\u03b1 : Sort u} [s : setoid \u03b1] {\u03c6 : quotient s \u2192 Sort v}\n  (f : \u03a0 (a : \u03b1), \u03c6 \u27e6a\u27e7) (c : \u2200 a b, a \u2248 b \u2192 f a == f b)\n  (q : quotient s) : \u03c6 q :=\nbegin\n  refine quotient.rec f _ q,\n  intros a b h,\n  have := c a b h,\n  cc,\nend\n\n@[simp] lemma quotient.dlift_mk {\u03b1 : Sort u} [s : setoid \u03b1] {\u03c6 : quotient s \u2192 Sort v}\n  (f : \u03a0 (a : \u03b1), \u03c6 \u27e6a\u27e7) (c : \u2200 a b, a \u2248 b \u2192 f a == f b) (q : \u03b1) :\n  (quotient.dlift f c \u27e6q\u27e7 : \u03c6 \u27e6q\u27e7) = f q := rfl\n\nattribute [reducible, elab_as_eliminator]\nprotected def quotient.dlift\u2082 {\u03b1 \u03b2 : Sort*} [s\u2081 : setoid \u03b1] [s\u2082 : setoid \u03b2]\n  {\u03c6 : quotient s\u2081 \u2192 quotient s\u2082 \u2192 Sort*}\n  (f : \u03a0 (a : \u03b1) (b : \u03b2), \u03c6 \u27e6a\u27e7 \u27e6b\u27e7) (c : \u2200 a\u2081 a\u2082 b\u2081 b\u2082, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 f a\u2081 a\u2082 == f b\u2081 b\u2082)\n  (q\u2081 : quotient s\u2081) (q\u2082 : quotient s\u2082) : \u03c6 q\u2081 q\u2082 :=\nbegin\n  refine quotient.dlift (\u03bb a, quotient.dlift (\u03bb b, f a b) _ q\u2082) _ q\u2081,\n  { intros b\u2081 b\u2082 hb,\n    exact c a b\u2081 a b\u2082 (setoid.refl a) hb, },\n  { intros b\u2081 b\u2082 hb,\n    refine quotient.induction_on q\u2082 _, clear q\u2082, intro q\u2082,\n    rw [quotient.dlift_mk, quotient.dlift_mk],\n    exact c b\u2081 q\u2082 b\u2082 q\u2082 hb (setoid.refl q\u2082), },\nend\n\n@[simp] lemma quotient.dlift\u2082_mk {\u03b1 \u03b2 : Sort*} [s\u2081 : setoid \u03b1] [s\u2082 : setoid \u03b2]\n  {\u03c6 : quotient s\u2081 \u2192 quotient s\u2082 \u2192 Sort u}\n  (f : \u03a0 (a : \u03b1) (b : \u03b2), \u03c6 \u27e6a\u27e7 \u27e6b\u27e7) (c : \u2200 a\u2081 a\u2082 b\u2081 b\u2082, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 f a\u2081 a\u2082 == f b\u2081 b\u2082)\n  (q\u2081 : \u03b1) (q\u2082 : \u03b2) : (quotient.dlift\u2082 f c \u27e6q\u2081\u27e7 \u27e6q\u2082\u27e7 : \u03c6 \u27e6q\u2081\u27e7 \u27e6q\u2082\u27e7) = f q\u2081 q\u2082 := rfl\n\nattribute [reducible, elab_as_eliminator]\nprotected def quotient.dlift\u2083 {\u03b1 \u03b2 \u03b3 : Sort*} [s\u2081 : setoid \u03b1] [s\u2082 : setoid \u03b2] [s\u2083 : setoid \u03b3]\n  {\u03c6 : quotient s\u2081 \u2192 quotient s\u2082 \u2192 quotient s\u2083 \u2192 Sort*}\n  (f : \u03a0 (a : \u03b1) (b : \u03b2) (c : \u03b3), \u03c6 \u27e6a\u27e7 \u27e6b\u27e7 \u27e6c\u27e7)\n  (c : \u2200 a\u2081 a\u2082 a\u2083 b\u2081 b\u2082 b\u2083, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 a\u2083 \u2248 b\u2083 \u2192 f a\u2081 a\u2082 a\u2083 == f b\u2081 b\u2082 b\u2083)\n  (q\u2081 : quotient s\u2081) (q\u2082 : quotient s\u2082) (q\u2083 : quotient s\u2083) : \u03c6 q\u2081 q\u2082 q\u2083 :=\nbegin\n  refine quotient.dlift (\u03bb a, quotient.dlift\u2082 (\u03bb b c, f a b c) _ q\u2082 q\u2083) _ q\u2081,\n  { intros a\u2082 a\u2083 b\u2082 b\u2083 h\u2082 h\u2083,\n    exact c a a\u2082 a\u2083 a b\u2082 b\u2083 (setoid.refl a) h\u2082 h\u2083, },\n  { intros b\u2081 b\u2082 hb,\n    refine quotient.induction_on\u2082 q\u2082 q\u2083 _, clear q\u2082 q\u2083, intros q\u2082 q\u2083,\n    simp only [quotient.dlift\u2082_mk],\n    exact c b\u2081 q\u2082 q\u2083 b\u2082 q\u2082 q\u2083 hb (setoid.refl q\u2082) (setoid.refl q\u2083), },\nend\n\n@[simp] lemma quotient.dlift\u2083_mk {\u03b1 \u03b2 \u03b3 : Sort*} [s\u2081 : setoid \u03b1] [s\u2082 : setoid \u03b2] [s\u2083 : setoid \u03b3]\n  {\u03c6 : quotient s\u2081 \u2192 quotient s\u2082 \u2192 quotient s\u2083 \u2192 Sort*}\n  (f : \u03a0 (a : \u03b1) (b : \u03b2) (c : \u03b3), \u03c6 \u27e6a\u27e7 \u27e6b\u27e7 \u27e6c\u27e7)\n  (c : \u2200 a\u2081 a\u2082 a\u2083 b\u2081 b\u2082 b\u2083, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 a\u2083 \u2248 b\u2083 \u2192 f a\u2081 a\u2082 a\u2083 == f b\u2081 b\u2082 b\u2083)\n  (q\u2081 : \u03b1) (q\u2082 : \u03b2) (q\u2083 : \u03b3) :\n    (quotient.dlift\u2083 f c \u27e6q\u2081\u27e7 \u27e6q\u2082\u27e7 \u27e6q\u2083\u27e7 : \u03c6 \u27e6q\u2081\u27e7 \u27e6q\u2082\u27e7 \u27e6q\u2083\u27e7) = f q\u2081 q\u2082 q\u2083 := rfl\n\nattribute [reducible, elab_as_eliminator]\nprotected def quotient.lift\u2083 {\u03b1 \u03b2 \u03b3 \u03c6 : Sort*} [s\u2081 : setoid \u03b1] [s\u2082 : setoid \u03b2] [s\u2083 : setoid \u03b3]\n  (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03c6) (c : \u2200 a\u2081 a\u2082 a\u2083 b\u2081 b\u2082 b\u2083, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 a\u2083 \u2248 b\u2083 \u2192 f a\u2081 a\u2082 a\u2083 = f b\u2081 b\u2082 b\u2083)\n  (q\u2081 : quotient s\u2081) (q\u2082 : quotient s\u2082) (q\u2083 : quotient s\u2083) : \u03c6 :=\nbegin\n  refine quotient.lift (\u03bb a\u2081, quotient.lift\u2082 (f a\u2081) _ q\u2082 q\u2083) _ q\u2081,\n  { intros a\u2082 a\u2083 b\u2082 b\u2083 h\u2082 h\u2083,\n    exact c a\u2081 a\u2082 a\u2083 a\u2081 b\u2082 b\u2083 (setoid.refl a\u2081) h\u2082 h\u2083, },\n  { intros a b h,\n    refine quotient.induction_on\u2082 q\u2082 q\u2083 _, clear q\u2082 q\u2083, intros q\u2082 q\u2083,\n    exact c a q\u2082 q\u2083 b q\u2082 q\u2083 h (setoid.refl q\u2082) (setoid.refl q\u2083), },\nend\n\nlemma agree.filter_eq_exists_mem {\u03b1 : Type u} {n m : \u2115} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {a\u2081 a\u2082 b\u2081 b\u2082 : agree_struct \u03b1} (h\u2081 : a\u2081 \u2248 b\u2081) (h\u2082 : a\u2082 \u2248 b\u2082)\n  (b : \u2203 x \u2208 a\u2081, \u2203 y \u2208 a\u2082, x =[n] y) (b' : \u2203 x \u2208 b\u2081, \u2203 y \u2208 b\u2082, x =[n] y) :\n  \u2200 x \u2208 a\u2081.filter_eq a\u2082 n b, \u2203 y \u2208 b\u2081.filter_eq b\u2082 n b', x =[m] y :=\nbegin\n  intros x hx,\n  simp only [agree_struct.mem_filter_eq, exists_prop] at hx \u22a2,\n  obtain \u27e8hx, z, hz\u2081, hz\u2082\u27e9 := hx,\n  obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := (h\u2081 (max m n)).left x hx,\n  rw eq_at_max at hy\u2082,\n  obtain \u27e8w, hw\u2081, hw\u2082\u27e9 := (h\u2082 n).left z hz\u2081,\n  refine \u27e8y, _, hy\u2082.1\u27e9,\n  refine \u27e8hy\u2081, _\u27e9,\n  refine \u27e8w, hw\u2081, _\u27e9,\n  transitivity x, exact eq_at_symmetric n hy\u2082.2,\n  transitivity; assumption,\nend\n\nlemma agree.filter_eq_exists_mem' {\u03b1 : Type u} {n m : \u2115} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {a\u2081 a\u2082 b\u2081 b\u2082 : agree_struct \u03b1} (h\u2081 : b\u2081 \u2248 a\u2081) (h\u2082 : b\u2082 \u2248 a\u2082)\n  (b : \u2203 x \u2208 a\u2081, \u2203 y \u2208 a\u2082, x =[n] y) (b' : \u2203 x \u2208 b\u2081, \u2203 y \u2208 b\u2082, x =[n] y) :\n  \u2200 x \u2208 a\u2081.filter_eq a\u2082 n b, \u2203 y \u2208 b\u2081.filter_eq b\u2082 n b', y =[m] x :=\nbegin\n  intros,\n  obtain \u27e8y, hy\u2081, hy\u2082\u27e9 := agree.filter_eq_exists_mem (setoid.symm h\u2081) (setoid.symm h\u2082) b b' x \u2039_\u203a,\n  refine \u27e8y, hy\u2081, _\u27e9,\n  symmetry,\n  assumption,\nend\n\ndef agree.extend {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} : \u03a0 {a b\u2081 b\u2082 : agree \u03b1}, agree.validn a n \u2192 a =[n] b\u2081 * b\u2082 \u2192 agree \u03b1 \u00d7 agree \u03b1 :=\nbegin\n  refine quotient.dlift\u2083 _ _,\n  { intros a b c h\u2081 h\u2082,\n    exact prod.map quotient.mk quotient.mk (agree_struct.extend h\u2082), },\n  { intros a\u2081 a\u2082 a\u2083 b\u2081 b\u2082 b\u2083 h\u2081 h\u2082 h\u2083,\n    ext,\n    { rw quotient.sound h\u2081, },\n    intros a a' haa',\n    ext,\n    { rw [quotient.sound h\u2081, quotient.sound h\u2082, quotient.sound h\u2083], },\n    intros b b' hbb',\n    simp only [prod_map, heq_iff_eq, prod.mk.inj_iff, quotient.eq],\n    split; intro; split,\n    { apply agree.filter_eq_exists_mem; assumption, },\n    { apply agree.filter_eq_exists_mem'; assumption, },\n    { apply agree.filter_eq_exists_mem; assumption, },\n    { apply agree.filter_eq_exists_mem'; assumption, }, },\nend\n\nlemma agree.extend_mul_eq {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree \u03b1} (ha : agree.validn a n) (hb : agree.eq_at n a (b\u2081 * b\u2082)) :\n    a = (agree.extend ha hb).1 * (agree.extend ha hb).2 :=\nbegin\n  revert ha hb,\n  refine quotient.induction_on\u2083 a b\u2081 b\u2082 _, clear a b\u2081 b\u2082, intros a b\u2081 b\u2082,\n  intros ha hb,\n  refine quotient.sound _,\n  rw \u2190 agree_struct.extend_mul_eq hb,\n  exact setoid.refl a,\nend\n\nlemma agree.extend_eq_at_left {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree \u03b1} (ha : agree.validn a n) (hb : agree.eq_at n a (b\u2081 * b\u2082)) :\n    agree.eq_at n (agree.extend ha hb).1 b\u2081 :=\nbegin\n  revert ha hb,\n  refine quotient.induction_on\u2083 a b\u2081 b\u2082 _, clear a b\u2081 b\u2082, intros a b\u2081 b\u2082,\n  intros ha hb,\n  exact agree_struct.extend_eq_at_left ha hb,\nend\n\nlemma agree.extend_eq_at_right {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1]\n  {n : \u2115} {a b\u2081 b\u2082 : agree \u03b1} (ha : agree.validn a n) (hb : agree.eq_at n a (b\u2081 * b\u2082)) :\n    agree.eq_at n (agree.extend ha hb).2 b\u2082 :=\nbegin\n  revert ha hb,\n  refine quotient.induction_on\u2083 a b\u2081 b\u2082 _, clear a b\u2081 b\u2082, intros a b\u2081 b\u2082,\n  intros ha hb,\n  exact agree_struct.extend_eq_at_right ha hb,\nend\n\nlemma agree.mul_is_nonexpansive {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] :\n  is_nonexpansive (function.uncurry ((*) : agree \u03b1 \u2192 agree \u03b1 \u2192 agree \u03b1)) :=\nbegin\n  rintros n \u27e8a\u2081, a\u2082\u27e9 \u27e8b\u2081, b\u2082\u27e9,\n  refine quotient.induction_on\u2082 a\u2081 a\u2082 _, clear a\u2081 a\u2082, intros a\u2081 a\u2082,\n  refine quotient.induction_on\u2082 b\u2081 b\u2082 _, clear b\u2081 b\u2082, intros b\u2081 b\u2082,\n  simp only [prod.eq_at, agree.eq_at_mk, function.uncurry_apply_pair, agree_struct.mul_mk, and_imp],\n  intros h\u2081 h\u2082,\n  exact agree_struct.eq_at_mul a\u2081 b\u2081 a\u2082 b\u2082 h\u2081 h\u2082,\nend\n\nlemma agree.validn_mul {\u03b1 : Type u} [ofe \u03b1] [decidable_eq \u03b1] (a b : agree \u03b1) :\n  agree.validn (a * b) \u2264 agree.validn a :=\nbegin\n  refine quotient.induction_on\u2082 a b _, clear a b, intros a b,\n  intros n h,\n  simp only [agree_struct.mul_mk, agree.validn_mk] at h \u22a2,\n  exact agree_struct.validn_mul a b n h,\nend\n\ninstance agree.camera (\u03b1 : Type u) [ofe \u03b1] [decidable_eq \u03b1] [decidable_eq_at \u03b1] :\n  camera (agree \u03b1) := {\n  validn := \u27e8agree.validn, agree.validn_is_nonexpansive\u27e9,\n  core := \u27e8some, option.some_is_nonexpansive\u27e9,\n  extend := @agree.extend _ _ _ _,\n  mul_is_nonexpansive := agree.mul_is_nonexpansive,\n  extend_mul_eq := @agree.extend_mul_eq _ _ _ _,\n  extend_eq_at_left := @agree.extend_eq_at_left _ _ _ _,\n  extend_eq_at_right := @agree.extend_eq_at_right _ _ _ _,\n  core_mul_self := begin\n    intros a ca hca,\n    cases hca,\n    rw agree.mul_self,\n  end,\n  core_core := \u03bb a ca hca, rfl,\n  core_mono_some := \u03bb a b ca hca h, \u27e8b, rfl\u27e9,\n  core_mono := \u03bb a ca h\u2081 h\u2082 h\u2083,\n    \u27e8some h\u2083.some, by simp only [h\u2083.some_spec, nonexpansive_fun.coe_fn_mk, some_mul_some]\u27e9,\n  validn_mul := agree.validn_mul,\n  ..agree.ofe \u03b1,\n  ..agree.comm_semigroup \u03b1,\n}\n", "meta": {"author": "zeramorphic", "repo": "separation-logic", "sha": "51c131501cc541b3aae072957942e8ef744c4ebf", "save_path": "github-repos/lean/zeramorphic-separation-logic", "path": "github-repos/lean/zeramorphic-separation-logic/separation-logic-51c131501cc541b3aae072957942e8ef744c4ebf/src/algebra/camera/agree.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.3914718702934395}}
{"text": "import rigid_elements.valuation_subring_of_preadditive\n\nvariables {K : Type*} [field K] {T H : mul_subgroup K} (rp : rigid_pair T H)\n\nopen rigid_pair\n\n-- Prop 2.12\nlemma rigid_pair.preadditive_of_exists_UU (u : K)\n  (hu : u \u2208 rp.UU) (hu' : u \u2209 T) : rp.preadditive := \nbegin\n  rw preadditive_iff,\n  intros x y hx hy,\n  by_cases hxz : -x = 0, { rw neg_eq_zero at hxz, simp [hxz, T.one_mem] },\n  have H1 := rp.neg _ hx.1 (1 + x) (1 + y) hx.2 hy.2 hxz,\n  have hxu : u\u207b\u00b9 * x \u2208 rp.OO_m,\n  { apply hu.2.2, assumption },\n  have huy : u * y \u2208 rp.OO_m,\n  { apply hu.1.2, assumption },\n  have z1 : - (x * u) \u2260 0, \n  { simp only [ne.def, neg_eq_zero, mul_eq_zero, inv_eq_zero], \n    push_neg, split, swap,\n    { apply rp.UU_ne_zero_of_mem u hu },\n    { simpa using hxz } },\n  have H2 := rp.neg _ hxu.1 _ _ hxu.2 huy.2 _,\n  cases H1; cases H2,\n  { convert H1, ring },\n  { convert H1, ring },\n  { convert H2, \n    field_simp [rp.UU_ne_zero_of_mem u hu], ring },\n  { exfalso,\n    have := T.mul_mem (T.inv_mem H1) H2,\n    apply hu', convert this,\n    have aux : ((-(u\u207b\u00b9 * x))\u207b\u00b9 * (1 + u\u207b\u00b9 * x) + (1 + u * y)) = \n      u * ((-x)\u207b\u00b9 * (1 + x) + (1 + y)),\n    field_simp [rp.UU_ne_zero_of_mem u hu], ring,\n    rw [aux, \u2190 mul_assoc, mul_comm _ u, mul_assoc, inv_mul_cancel, mul_one], \n    exact T.ne_zero_of_mem H1 },\n  simp only [ne.def, neg_eq_zero, mul_eq_zero, inv_eq_zero],\n  push_neg, split, exact rp.UU_ne_zero_of_mem u hu, simpa using hxz,\nend\n\n-- Lemma 2.13\nlemma rigid_pair.OO_p_mem_of_mem_nmem_nmem (u : K) :\n  u \u2208 H \u2192 u \u2209 T \u2192 1 - u\u207b\u00b9 \u2209 T \u2192 u \u2208 rp.OO_p :=\nbegin\n  intros h1 h2 h3,\n  by_cases hu : u - 1 = 0, \n  { rw sub_eq_zero at hu, rw hu, apply rp.OO_p_one_mem },\n  have : u - 1 \u2208 H,\n  { by_contra c, have := rp.pos _ c 1 1 T.one_mem T.one_mem hu, \n  cases this, { apply h2, convert this, ring },\n  apply h3, \n  rw \u2190 inv_inv (1 - u\u207b\u00b9), apply T.inv_mem,\n  convert this, field_simp [H.ne_zero_of_mem h1] },\n  refine \u27e8h1, _\u27e9, intros x hx,\n  have hux : u * x \u2209 H,\n  { intro c, apply hx.1, convert H.mul_mem (H.inv_mem h1) c,\n    field_simp [H.ne_zero_of_mem h1], ring },\n  have hux' : (u - 1) * x \u2209 H,\n  { intro c, apply hx.1, convert H.mul_mem (H.inv_mem this) c,\n    field_simp [H.ne_zero_of_mem this], ring },\n  by_cases huxz : u * x = 0, { rw huxz, apply rp.OO_m_zero_mem },\n  have aux1 := rp.pos _ hux 1 1 T.one_mem T.one_mem huxz,\n  simp only [mul_one, mul_assoc] at aux1,\n  by_cases huxz' : (u - 1) * x = 0, \n  { rw [sub_mul, one_mul, sub_eq_zero] at huxz', rw huxz', exact hx },\n  have aux2 := rp.pos _ hux' (1 + x) 1 hx.2 T.one_mem huxz',\n  cases aux1; cases aux2,\n  { exact \u27e8hux, aux1\u27e9 },\n  { exact \u27e8hux, aux1\u27e9 },\n  { refine \u27e8hux, _\u27e9, convert aux2 using 1, ring },\n  exfalso, apply h3,\n  have : (1 - u\u207b\u00b9) * (((u - 1) * x)\u207b\u00b9 * (1 + x) + 1) = \n    (u * x)\u207b\u00b9 + 1, \n  { field_simp [H.ne_zero_of_mem h1], ring },\n  convert T.mul_mem aux1 (T.inv_mem aux2), rw \u2190 this,\n  rw [mul_assoc, mul_inv_cancel, mul_one],\n  apply T.ne_zero_of_mem aux2\nend .\n\n-- Prop 2.14\nlemma rigid_pair.preadditive_of_nmem_of_neg_not_rigid (a : K) :\n  a \u2260 0 \u2192 \n  a \u2209 T \u2192 \n  \u00ac (T.rigid (-a)) \u2192 rp.preadditive := \nbegin\n  intros ha haT nrig,\n  have haH : a \u2208 H,\n  { by_contra c, \n    apply nrig,\n    exact rp.neg _ c }, \n  dsimp [mul_subgroup.rigid] at nrig,\n  push_neg at nrig,\n  obtain \u27e8s,t,hs,ht,ha',h1,h2\u27e9 := nrig,\n  let z : K := s\u207b\u00b9 * t,\n  have hz : z \u2208 T := T.mul_mem (T.inv_mem hs) ht,\n  replace h1 : (1 : K) - a * z \u2209 T,\n  { intros c, apply h1, convert T.mul_mem hs c, dsimp [z],\n    field_simp [T.ne_zero_of_mem hs], ring },\n  replace h2 : z - a\u207b\u00b9 \u2209 T,\n  { intros c, apply h2, convert T.mul_mem hs c, dsimp [z], \n    field_simp [T.ne_zero_of_mem hs, H.ne_zero_of_mem haH], ring },\n  let u := a * z,\n  have huT : u \u2209 T,\n  { intros c, apply haT, dsimp [u,z] at c,\n    convert T.mul_mem (T.mul_mem c hs) (T.inv_mem ht),\n    field_simp [T.ne_zero_of_mem hs, T.ne_zero_of_mem ht] },\n  have huH : u \u2208 H,\n  { dsimp [u,z], \n    convert H.mul_mem haH (H.mul_mem (H.inv_mem _) _),\n    exact rp.le hs, exact rp.le ht },\n  have key : u \u2208 rp.UU,\n  { split, \n    { apply rp.OO_p_mem_of_mem_nmem_nmem, \n      assumption, assumption, dsimp [u], intros c, apply h2,\n      convert T.mul_mem hz c,\n      field_simp [H.ne_zero_of_mem haH, T.ne_zero_of_mem hz], ring },\n    { apply rp.OO_p_mem_of_mem_nmem_nmem,\n      exact H.inv_mem huH, intros c, apply huT, convert T.inv_mem c, rw inv_inv,\n      rw inv_inv, exact h1 } },\n  exact rp.preadditive_of_exists_UU u key huT,\nend\n", "meta": {"author": "adamtopaz", "repo": "lean-acl-pairs", "sha": "6ac31d86ca2739b6c18d3f05b7007e720f66299f", "save_path": "github-repos/lean/adamtopaz-lean-acl-pairs", "path": "github-repos/lean/adamtopaz-lean-acl-pairs/lean-acl-pairs-6ac31d86ca2739b6c18d3f05b7007e720f66299f/src/rigid_elements/preadditive_conditions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.39136262009087475}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport measure_theory.measure.giry_monad\nimport category_theory.concrete_category.unbundled_hom\nimport category_theory.monad.algebra\nimport topology.category.Top.basic\n\n/-!\n# The category of measurable spaces\n\nMeasurable spaces and measurable functions form a (concrete) category `Meas`.\n\n## Main definitions\n\n* `Measure : Meas \u2964 Meas`: the functor which sends a measurable space `X`\nto the space of measures on `X`; it is a monad (the \"Giry monad\").\n\n* `Borel : Top \u2964 Meas`: sends a topological space `X` to `X` equipped with the\n`\u03c3`-algebra of Borel sets (the `\u03c3`-algebra generated by the open subsets of `X`).\n\n## Tags\n\nmeasurable space, giry monad, borel\n-/\n\nnoncomputable theory\n\nopen category_theory measure_theory\nopen_locale ennreal\nuniverses u v\n\n/-- The category of measurable spaces and measurable functions. -/\ndef Meas : Type (u+1) := bundled measurable_space\n\nnamespace Meas\n\ninstance : has_coe_to_sort Meas Type* := bundled.has_coe_to_sort\ninstance (X : Meas) : measurable_space X := X.str\n\n/-- Construct a bundled `Meas` from the underlying type and the typeclass. -/\ndef of (\u03b1 : Type u) [measurable_space \u03b1] : Meas := \u27e8\u03b1\u27e9\n\n@[simp] lemma coe_of (X : Type u) [measurable_space X] : (of X : Type u) = X := rfl\n\ninstance unbundled_hom : unbundled_hom @measurable := \u27e8@measurable_id, @measurable.comp\u27e9\n\nattribute [derive [large_category, concrete_category]] Meas\n\ninstance : inhabited Meas := \u27e8Meas.of empty\u27e9\n\n/-- `Measure X` is the measurable space of measures over the measurable space `X`. It is the\nweakest measurable space, s.t. \u03bb\u03bc, \u03bc s is measurable for all measurable sets `s` in `X`. An\nimportant purpose is to assign a monadic structure on it, the Giry monad. In the Giry monad,\nthe pure values are the Dirac measure, and the bind operation maps to the integral:\n`(\u03bc >>= \u03bd) s = \u222b x. (\u03bd x) s d\u03bc`.\n\nIn probability theory, the `Meas`-morphisms `X \u2192 Prob X` are (sub-)Markov kernels (here `Prob` is\nthe restriction of `Measure` to (sub-)probability space.)\n-/\ndef Measure : Meas \u2964 Meas :=\n{ obj      := \u03bbX, \u27e8@measure_theory.measure X.1 X.2\u27e9,\n  map      := \u03bbX Y f, \u27e8measure.map (f : X \u2192 Y), measure.measurable_map f f.2\u27e9,\n  map_id'  := assume \u27e8\u03b1, I\u27e9, subtype.eq $ funext $ assume \u03bc, @measure.map_id \u03b1 I \u03bc,\n  map_comp':=\n    assume X Y Z \u27e8f, hf\u27e9 \u27e8g, hg\u27e9, subtype.eq $ funext $ assume \u03bc, (measure.map_map hg hf).symm }\n\n/-- The Giry monad, i.e. the monadic structure associated with `Measure`. -/\ndef Giry : category_theory.monad Meas :=\n{ to_functor := Measure,\n  \u03b7' :=\n  { app         := \u03bbX, \u27e8@measure.dirac X.1 X.2, measure.measurable_dirac\u27e9,\n    naturality' :=\n      assume X Y \u27e8f, hf\u27e9, subtype.eq $ funext $ assume a, (measure.map_dirac hf a).symm },\n  \u03bc' :=\n  { app         := \u03bbX, \u27e8@measure.join X.1 X.2, measure.measurable_join\u27e9,\n    naturality' :=\n      assume X Y \u27e8f, hf\u27e9, subtype.eq $ funext $ assume \u03bc, measure.join_map_map hf \u03bc },\n  assoc' := assume \u03b1, subtype.eq $ funext $ assume \u03bc, @measure.join_map_join _ _ _,\n  left_unit' := assume \u03b1, subtype.eq $ funext $ assume \u03bc, @measure.join_dirac _ _ _,\n  right_unit' := assume \u03b1, subtype.eq $ funext $ assume \u03bc, @measure.join_map_dirac _ _ _ }\n\n/-- An example for an algebra on `Measure`: the nonnegative Lebesgue integral is a hom, behaving\nnicely under the monad operations. -/\ndef Integral : Giry.algebra :=\n{ A      := Meas.of \u211d\u22650\u221e ,\n  a      := \u27e8\u03bbm:measure \u211d\u22650\u221e, \u222b\u207b x, x \u2202m, measure.measurable_lintegral measurable_id \u27e9,\n  unit'  := subtype.eq $ funext $ assume r:\u211d\u22650\u221e, lintegral_dirac' _ measurable_id,\n  assoc' := subtype.eq $ funext $ assume \u03bc : measure (measure \u211d\u22650\u221e),\n    show \u222b\u207b x, x \u2202 \u03bc.join = \u222b\u207b x, x \u2202 (measure.map (\u03bbm:measure \u211d\u22650\u221e, \u222b\u207b x, x \u2202m) \u03bc),\n    by rw [measure.lintegral_join, lintegral_map];\n      apply_rules [measurable_id, measure.measurable_lintegral] }\n\nend Meas\n\ninstance Top.has_forget_to_Meas : has_forget\u2082 Top.{u} Meas.{u} :=\nbundled_hom.mk_has_forget\u2082\n  borel\n  (\u03bb X Y f, \u27e8f.1, f.2.borel_measurable\u27e9)\n  (by intros; refl)\n\n/-- The Borel functor, the canonical embedding of topological spaces into measurable spaces. -/\n@[reducible] def Borel : Top.{u} \u2964 Meas.{u} := forget\u2082 Top.{u} Meas.{u}\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/measure_theory/category/Meas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.39136262009087475}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Gabriel Ebner, Simon Hudon, Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.transport\n! leanprover-community/mathlib commit 1f10390872a75359a9a42cd6d7644cc5e45660c2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.EquivRw\n\n/-!\n## The `transport` tactic\n\n`transport` attempts to move an `s : S \u03b1` expression across an equivalence `e : \u03b1 \u2243 \u03b2` to solve\na goal of the form `S \u03b2`, by building the new object field by field, taking each field of `s`\nand rewriting it along `e` using the `equiv_rw` tactic.\n\nWe try to ensure good definitional properties, so that, for example, when we transport a `monoid \u03b1`\nto a `monoid \u03b2`, the new multiplication is definitionally `\u03bb x y, e (e.symm a * e.symm b)`.\n-/\n\n\nnamespace Tactic\n\nopen Tactic.Interactive\n\n/- failed to parenthesize: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n[PrettyPrinter.parenthesize.input] (Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr\n     [(Command.docComment\n       \"/--\"\n       \"The simpset `transport_simps` is used by the tactic `transport`\\nto simplify certain expressions involving application of equivalences,\\nand trivial `eq.rec` or `ep.mpr` conversions.\\nIt's probably best not to adjust it without understanding the algorithm used by `transport`. -/\")]\n     \"register_simp_attr\"\n     `transport_simps)-/-- failed to format: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n/--\n    The simpset `transport_simps` is used by the tactic `transport`\n    to simplify certain expressions involving application of equivalences,\n    and trivial `eq.rec` or `ep.mpr` conversions.\n    It's probably best not to adjust it without understanding the algorithm used by `transport`. -/\n  register_simp_attr\n  transport_simps\n\nattribute [transport_simps]\n  eq_rec_constant eq_mp_eq_cast cast_eq Equiv.toFun_as_coe Equiv.arrowCongr'_apply Equiv.symm_apply_apply Equiv.apply_eq_iff_eq_symm_apply\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Given `s : S \u03b1` for some structure `S` depending on a type `\u03b1`,\n      and an equivalence `e : \u03b1 \u2243 \u03b2`,\n      try to produce an `S \u03b2`,\n      by transporting data and axioms across `e` using `equiv_rw`.\n      -/\n    @[ nolint unused_arguments ]\n    unsafe\n  def\n    transport\n    ( s e : expr ) : tactic Unit\n    :=\n      do\n        let\n            ( _ , \u03b1 , \u03b2 )\n              \u2190\n              infer_type e >>= relation_lhs_rhs\n                <|>\n                fail f! \"second argument to `transport` was not an equivalence-type relation\"\n          seq'\n            sorry\n              (\n                propagate_tags\n                  (\n                    try\n                      do\n                        let f \u2190 get_current_field\n                          mk_mapp f [ \u03b1 , none ] >>= note f none\n                          let b \u2190 target >>= is_prop\n                          if\n                            \u00ac b\n                            then\n                            simp_result do equiv_rw_hyp f e get_local f >>= exact\n                            else\n                            try\n                              do\n                                try unfold_projs_target\n                                  sorry\n                                  try\n                                    <|\n                                    under_binders\n                                      <|\n                                      to_expr ` `( $ ( e ) . symm . Injective ) >>= apply\n                                  equiv_rw_hyp f e\n                                  get_local f >>= exact\n                    )\n                )\n#align tactic.transport tactic.transport\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- Given a goal `\u22a2 S \u03b2` for some type class `S`, and an equivalence `e : \u03b1 \u2243 \u03b2`.\n`transport using e` will look for a hypothesis `s : S \u03b1`,\nand attempt to close the goal by transporting `s` across the equivalence `e`.\n\n```lean\nexample {\u03b1 : Type} [ring \u03b1] {\u03b2 : Type} (e : \u03b1 \u2243 \u03b2) : ring \u03b2 :=\nby transport using e.\n```\n\nYou can specify the object to transport using `transport s using e`.\n\n`transport` works by attempting to copy each of the operations and axiom fields of `s`,\nrewriting them using `equiv_rw e` and defining a new structure using these rewritten fields.\n\nIf it fails to fill in all the new fields, `transport` will produce new subgoals.\nIt's probably best to think about which missing `simp` lemmas would have allowed `transport`\nto finish, rather than solving these goals by hand.\n(This may require looking at the implementation of `tranport` to understand its algorithm;\nthere are several examples of \"transport-by-hand\" at the end of `test/equiv_rw.lean`,\nwhich `transport` is an abstraction of.)\n-/\nunsafe def transport (s : parse texpr ?) (e : parse <| tk \"using\" *> texpr) : itactic := do\n  let s \u2190\n    match s with\n      | some s => to_expr s\n      | none =>\n        (do\n            let t \u2190 target\n            let n := t.get_app_fn.const_name\n            let ctx \u2190 local_context\n            ctx fun e => do\n                let t \u2190 infer_type e\n                guard (t = n)\n                return e) <|>\n          fail \"`transport` could not find an appropriate source object. Try `transport s using e`.\"\n  let e \u2190 to_expr e\n  tactic.transport s e\n#align tactic.interactive.transport tactic.interactive.transport\n\nadd_tactic_doc\n  { Name := \"transport\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.transport]\n    tags := [\"rewriting\", \"equiv\", \"transport\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Transport.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.39133695254689577}}
{"text": "import galois.nat.simplify_eq\n\nuniverse variable u\n\nnamespace list\n\nvariable { \u03b1 : Type u}\n\ntheorem map_accumr\u2082_append1\n  {\u03b1 \u03b2 \u03b3 \u03c6 : Type}\n  (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b3 \u00d7 \u03c6)\n  (x : list \u03b1)\n  (y : list \u03b2)\n  (pr : length x = length y)\n  (a : \u03b1)\n  (b : \u03b2)\n  (c : \u03b3)\n: map_accumr\u2082 f (x ++ [a]) (y ++ [b]) c =\n  let r := f a b c in\n  let z := map_accumr\u2082 f x y (r^.fst) in\n  \u27e8 z^.fst, z^.snd ++ [r^.snd]\u27e9 :=\nbegin\n  revert y,\n  induction x with xh xr ind,\n  -- Base case with x = nil\n  { intros y pr,\n    cases y with yh yr,\n    { simp [map_accumr\u2082] },\n    { contradiction },\n  },\n  -- Inductive case with x = xh :: xr\n  {\n    intros y pr,\n    cases y with yh yr,\n    { contradiction },\n    { simp [nat.succ_add] at pr,\n      simp [map_accumr\u2082, ind yr pr]\n    },\n  }\nend\n\nend list\n", "meta": {"author": "GaloisInc", "repo": "lean-protocol-support", "sha": "cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda", "save_path": "github-repos/lean/GaloisInc-lean-protocol-support", "path": "github-repos/lean/GaloisInc-lean-protocol-support/lean-protocol-support-cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda/galois/list/map_accum_lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3912514159296473}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n\nimport category_theory.sites.grothendieck\n\n/-!\n# Grothendieck pretopologies\n\nDefinition and lemmas about Grothendieck pretopologies.\nA Grothendieck pretopology for a category `C` is a set of families of morphisms with fixed codomain,\nsatisfying certain closure conditions.\n\nWe show that a pretopology generates a genuine Grothendieck topology, and every topology has\na maximal pretopology which generates it.\n\nThe pretopology associated to a topological space is defined in `spaces.lean`.\n\n## Tags\n\ncoverage, pretopology, site\n\n## References\n\n* [https://ncatlab.org/nlab/show/Grothendieck+pretopology][nlab]\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n* [https://stacks.math.columbia.edu/tag/00VG][Stacks]\n-/\n\nuniverses v u\nnoncomputable theory\n\nnamespace category_theory\n\nopen category_theory category limits presieve\n\nvariables {C : Type u} [category.{v} C] [has_pullbacks C]\n\nvariables (C)\n\n/--\nA (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed\ntarget `X` for every object `X` in `C`, called \"coverings\" of `X`, which satisfies the following\nthree axioms:\n1. Every family consisting of a single isomorphism is a covering family.\n2. The collection of covering families is stable under pullback.\n3. Given a covering family, and a covering family on each domain of the former, the composition\n   is a covering family.\n\nIn some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,\nin that each covering is not necessarily downward closed.\n\nSee: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or\nhttps://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.\nNote that Stacks calls a category together with a pretopology a site, and [MM92] calls this\na basis for a topology.\n-/\n@[ext]\nstructure pretopology :=\n(coverings : \u03a0 (X : C), set (presieve X))\n(has_isos : \u2200 \u2983X Y\u2984 (f : Y \u27f6 X) [is_iso f], presieve.singleton f \u2208 coverings X)\n(pullbacks : \u2200 \u2983X Y\u2984 (f : Y \u27f6 X) S, S \u2208 coverings X \u2192 pullback_arrows f S \u2208 coverings Y)\n(transitive : \u2200 \u2983X : C\u2984 (S : presieve X)\n               (Ti : \u03a0 \u2983Y\u2984 (f : Y \u27f6 X), S f \u2192 presieve Y), S \u2208 coverings X \u2192\n               (\u2200 \u2983Y\u2984 f (H : S f), Ti f H \u2208 coverings Y) \u2192 S.bind Ti \u2208 coverings X)\n\nnamespace pretopology\n\ninstance : has_coe_to_fun (pretopology C) :=\n\u27e8_, \u03bb J, J.coverings\u27e9\n\ninstance : partial_order (pretopology C) :=\n{ le := \u03bb K\u2081 K\u2082, (K\u2081 : \u03a0 (X : C), set _) \u2264 K\u2082,\n  le_refl := \u03bb K, le_refl _,\n  le_trans := \u03bb K\u2081 K\u2082 K\u2083 h\u2081\u2082 h\u2082\u2083, le_trans h\u2081\u2082 h\u2082\u2083,\n  le_antisymm := \u03bb K\u2081 K\u2082 h\u2081\u2082 h\u2082\u2081, pretopology.ext _ _ (le_antisymm h\u2081\u2082 h\u2082\u2081) }\n\ninstance : order_top (pretopology C) :=\n{ top :=\n  { coverings := \u03bb _, set.univ,\n    has_isos := \u03bb _ _ _ _, set.mem_univ _,\n    pullbacks := \u03bb _ _ _ _ _, set.mem_univ _,\n    transitive := \u03bb _ _ _ _ _, set.mem_univ _ },\n  le_top := \u03bb K X S hS, set.mem_univ _,\n  ..pretopology.partial_order C }\n\ninstance : inhabited (pretopology C) := \u27e8\u22a4\u27e9\n\n/--\nA pretopology `K` can be completed to a Grothendieck topology `J` by declaring a sieve to be\n`J`-covering if it contains a family in `K`.\n\nSee https://stacks.math.columbia.edu/tag/00ZC, or [MM92] Chapter III, Section 2, Equation (2).\n-/\ndef to_grothendieck (K : pretopology C) : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2203 R \u2208 K X, R \u2264 (S : presieve _),\n  top_mem' := \u03bb X, \u27e8presieve.singleton (\ud835\udfd9 _), K.has_isos _, \u03bb _ _ _, \u27e8\u27e9\u27e9,\n  pullback_stable' := \u03bb X Y S g,\n  begin\n    rintro \u27e8R, hR, RS\u27e9,\n    refine \u27e8_, K.pullbacks g _ hR, _\u27e9,\n    rw [\u2190 sieve.sets_iff_generate, sieve.pullback_arrows_comm],\n    apply sieve.pullback_monotone,\n    rwa sieve.gi_generate.gc,\n  end,\n  transitive' :=\n  begin\n    rintro X S \u27e8R', hR', RS\u27e9 R t,\n    choose t\u2081 t\u2082 t\u2083 using t,\n    refine \u27e8_, K.transitive _ _ hR' (\u03bb _ f hf, t\u2082 (RS _ hf)), _\u27e9,\n    rintro Y _ \u27e8Z, g, f, hg, hf, rfl\u27e9,\n    apply t\u2083 (RS _ hg) _ hf,\n  end }\n\nlemma mem_to_grothendieck (K : pretopology C) (X S) :\n  S \u2208 to_grothendieck C K X \u2194 \u2203 R \u2208 K X, R \u2264 (S : presieve X) :=\niff.rfl\n\n/--\nThe largest pretopology generating the given Grothendieck topology.\n\nSee [MM92] Chapter III, Section 2, Equations (3,4).\n-/\ndef of_grothendieck (J : grothendieck_topology C) : pretopology C :=\n{ coverings := \u03bb X R, sieve.generate R \u2208 J X,\n  has_isos := \u03bb X Y f i, by exactI J.covering_of_eq_top (by simp),\n  pullbacks := \u03bb X Y f R hR,\n  begin\n    rw [set.mem_def, sieve.pullback_arrows_comm],\n    apply J.pullback_stable f hR,\n  end,\n  transitive := \u03bb X S Ti hS hTi,\n  begin\n    apply J.transitive hS,\n    intros Y f,\n    rintros \u27e8Z, g, f, hf, rfl\u27e9,\n    rw sieve.pullback_comp,\n    apply J.pullback_stable g,\n    apply J.superset_covering _ (hTi _ hf),\n    rintro Y g \u27e8W, h, g, hg, rfl\u27e9,\n    exact \u27e8_, h, _, \u27e8_, _, _, hf, hg, rfl\u27e9, by simp\u27e9,\n  end }\n\n/-- We have a galois insertion from pretopologies to Grothendieck topologies. -/\ndef gi : galois_insertion (to_grothendieck C) (of_grothendieck C) :=\n{ gc :=\n  \u03bb K J,\n  begin\n    split,\n    { intros h X R hR,\n      exact h _ \u27e8_, hR, sieve.le_generate R\u27e9 },\n    { rintro h X S \u27e8R, hR, RS\u27e9,\n      apply J.superset_covering _ (h _ hR),\n      rwa sieve.gi_generate.gc }\n  end,\n  le_l_u := \u03bb J X S hS, \u27e8S, J.superset_covering S.le_generate hS, le_refl _\u27e9,\n  choice := \u03bb x hx, to_grothendieck C x,\n  choice_eq := \u03bb _ _, rfl }\n\n/--\nThe trivial pretopology, in which the coverings are exactly singleton isomorphisms. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee https://stacks.math.columbia.edu/tag/07GE\n-/\ndef trivial : pretopology C :=\n{ coverings := \u03bb X S, \u2203 Y (f : Y \u27f6 X) (h : is_iso f), S = presieve.singleton f,\n  has_isos := \u03bb X Y f i, \u27e8_, _, i, rfl\u27e9,\n  pullbacks := \u03bb X Y f S,\n  begin\n    rintro \u27e8Z, g, i, rfl\u27e9,\n    refine \u27e8pullback g f, pullback.snd, _, _\u27e9,\n    { resetI, refine \u27e8\u27e8pullback.lift (f \u226b inv g) (\ud835\udfd9 _) (by simp), \u27e8_, by tidy\u27e9\u27e9\u27e9,\n      apply pullback.hom_ext,\n      { rw [assoc, pullback.lift_fst, \u2190pullback.condition_assoc],\n        simp },\n      { simp } },\n    { apply pullback_singleton },\n  end,\n  transitive :=\n  begin\n    rintro X S Ti \u27e8Z, g, i, rfl\u27e9 hS,\n    rcases hS g (singleton_self g) with \u27e8Y, f, i, hTi\u27e9,\n    refine \u27e8_, f \u226b g, _, _\u27e9,\n    { resetI, apply_instance },\n    ext W k,\n    split,\n    { rintro \u27e8V, h, k, \u27e8_\u27e9, hh, rfl\u27e9,\n      rw hTi at hh,\n      cases hh,\n      apply singleton.mk },\n    { rintro \u27e8_\u27e9,\n      refine bind_comp g presieve.singleton.mk _,\n      rw hTi,\n      apply presieve.singleton.mk }\n  end }\n\ninstance : order_bot (pretopology C) :=\n{ bot := trivial C,\n  bot_le := \u03bb K X R,\n  begin\n    rintro \u27e8Y, f, hf, rfl\u27e9,\n    exactI K.has_isos f,\n  end,\n  ..pretopology.partial_order C }\n\n/-- The trivial pretopology induces the trivial grothendieck topology. -/\nlemma to_grothendieck_bot : to_grothendieck C \u22a5 = \u22a5 :=\n(gi C).gc.l_bot\n\nend pretopology\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/sites/pretopology.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.39125140883927323}}
{"text": "import data.finset tactic.find data.fintype data.nat.basic data.pnat\nuniverses u v\nlocal infix ` \u227a ` :50 := has_well_founded.r\n\nopen well_founded\n\nlemma acc.rec_3 {\u03b1 : Sort u} [hw : has_well_founded \u03b1] {C : \u03b1 \u2192 Sort v}\n  (f : \u03a0 x, (\u2200 y, y \u227a x \u2192 acc (\u227a) y) \u2192 (\u03a0 y, y \u227a x \u2192 C y) \u2192 C x) (a : \u03b1) :\n  @acc.rec \u03b1 (\u227a) C f a (apply hw.wf a) = f a (\u03bb y _, apply hw.wf y)\n  (\u03bb y hya, @acc.rec \u03b1 (\u227a) C f y (apply hw.wf y)) :=\n(show acc.intro a (\u03bb y _, apply hw.wf y) = (apply hw.wf a), from rfl) \u25b8 rfl\n\nclass game (\u03b1 : Type*) extends has_well_founded \u03b1 :=\n(start_position : \u03b1)\n(side_to_move : \u03b1 \u2192 bool)\n(legal_moves : \u03a0 a, finset {b : \u03b1 // b \u227a a \u2227 side_to_move b = bnot (side_to_move a)})\n(max_score : \u2115)\n(not_boring : max_score > 1)\n(final_score : \u03a0 {a}, legal_moves a = \u2205 \u2192 fin max_score)\n\nnamespace game\n\nopen well_founded finset list\nvariables {\u03b1 : Type*} [g : game \u03b1] [decidable_eq \u03b1]\ninclude g\n\ndef is_legal_move (a b : \u03b1) := \u2203 h : b \u227a a \u2227 side_to_move b = bnot (side_to_move a),\n(\u27e8b, h\u27e9 : {b : \u03b1 // b \u227a a \u2227 side_to_move b = bnot (side_to_move a)}) \u2208 legal_moves a\n\nlemma is_legal_move_of_mem {a : \u03b1} {b : {b : \u03b1 // b \u227a a \u2227 _}} : b \u2208 legal_moves a \u2192 is_legal_move a b :=\nlet \u27e8b1, b2\u27e9 := b in \u03bb h, \u27e8b2, h\u27e9\n\nlemma max_score_pos : 0 < max_score \u03b1 :=\nnat.pos_of_ne_zero $ \u03bb h0,\nacc.rec_on (apply g.wf (start_position \u03b1)) $ \u03bb a h\u2081 h\u2082, or.by_cases\n(decidable.em (legal_moves a = \u2205))\n(\u03bb h, (not_lt_of_ge (nat.zero_le _)) (show (final_score h).val < 0, from h0 \u25b8 (final_score h).2)) $\n\u03bb h, let \u27e8\u27e8y, hy\u2081\u27e9, hy\u27e9 := exists_mem_of_ne_empty h in h\u2082 y hy\u2081.1\n\ndef best (s : bool) (sc\u2081 sc\u2082 : fin g.max_score) : fin g.max_score :=\nbool.cases_on s \u27e8min sc\u2081.1 sc\u2082.1, lt_of_le_of_lt (min_le_left _ _) sc\u2081.2\u27e9\n\u27e8max sc\u2081.1 sc\u2082.1, max_lt sc\u2081.2 sc\u2082.2\u27e9\n\ndef better (s : bool) (sc\u2081 sc\u2082 : fin g.max_score) : Prop :=\nbool.cases_on s (sc\u2081.1 \u2264 sc\u2082.1) (sc\u2082.1 \u2264 sc\u2081.1)\n\n@[refl] lemma better.refl (s : bool) (sc : fin g.max_score) : better s sc sc :=\nbool.cases_on s (le_refl _) (le_refl _)\n\nlemma better.trans {s : bool} {sc\u2081 sc\u2082 sc\u2083 : fin g.max_score} : better s sc\u2081 sc\u2082 \u2192\n    better s sc\u2082 sc\u2083 \u2192 better s sc\u2081 sc\u2083 :=\nbool.cases_on s le_trans (\u03bb h\u2081 h\u2082, le_trans h\u2082 h\u2081)\n\nlemma best_better (s : bool) (sc\u2081 sc\u2082 : fin g.max_score) : better s (best s sc\u2081 sc\u2082) sc\u2081:=\nbool.cases_on s (min_le_left _ _) (le_max_left _ _)\n\nlemma better_swap {s : bool} {sc\u2081 sc\u2082 : fin g.max_score} : better (bnot s) sc\u2081 sc\u2082 \u2194 better s sc\u2082 sc\u2081 :=\nbool.cases_on s iff.rfl iff.rfl\n\ndef worst (s : bool) (sc\u2081 : fin g.max_score) (sc\u2082 : fin g.max_score) := best (bnot s) sc\u2081 sc\u2082\n\ndef win (s : bool) : fin g.max_score := bool.cases_on s\n\u27e80, max_score_pos\u27e9 \u27e8nat.pred g.max_score, nat.pred_lt (ne_of_lt max_score_pos).symm\u27e9\n\nlemma win_better (s : bool) (sc : fin g.max_score) : better s (win s) sc :=\nbool.cases_on s (nat.zero_le _) (nat.le_of_succ_le_succ $\nshow nat.succ sc.1 \u2264 nat.succ (nat.pred (max_score \u03b1)),\n  begin rw nat.succ_pred_eq_of_pos max_score_pos, exact sc.2, apply_instance, end)\n\nlemma eq_win_of_better_win {s : bool} {sc : fin g.max_score} : better s sc (win s) \u2192 sc = win s :=\nbool.cases_on s (\u03bb h, fin.eq_of_veq $ le_antisymm h (win_better ff _))\n(\u03bb h, fin.eq_of_veq $ le_antisymm (win_better tt _) h)\n\ndef loss (s : bool) := @win \u03b1 _ _ (bnot s)\n\nlemma eq_loss_of_loss_better {s : bool} {sc : fin g.max_score} : better s (loss s) sc \u2192\n    sc = loss s :=\nbool.cases_on s (\u03bb h, fin.eq_of_veq $ (le_antisymm (win_better tt _) h))\n(\u03bb h, fin.eq_of_veq $ le_antisymm h (win_better ff _))\n\ninstance (s : bool) : is_commutative _ (@best \u03b1 _ _ s) := \u27e8\u03bb \u27e8a, _\u27e9 \u27e8b, _\u27e9, fin.eq_of_veq $\nbool.cases_on s (min_comm _ _) (max_comm _ _)\u27e9\n\ninstance (s : bool) : is_associative _ (@best \u03b1 _ _ s) := \u27e8\u03bb \u27e8a, _\u27e9 \u27e8b, _\u27e9 \u27e8c, _\u27e9,\nfin.eq_of_veq $ bool.cases_on s (min_assoc a b c) (max_assoc a b c)\u27e9\n\ndef score (a : \u03b1) : fin g.max_score :=\nacc.rec_on (apply g.wf a) $ \u03bb a _ score,\nif h : legal_moves a = \u2205\nthen final_score h\nelse fold (best (side_to_move a))\n  (loss (side_to_move a))\n  (\u03bb b : {b : \u03b1 // b \u227a a \u2227 side_to_move b = bnot (side_to_move a)}, score b.1 b.2.1)\n  (legal_moves a)\n\nlemma score_eq (a : \u03b1) : score a = dite (legal_moves a = \u2205) (\u03bb h, final_score h)\n    (\u03bb h, multiset.fold (best (side_to_move a)) (loss (side_to_move a))\n    (multiset.map (\u03bb b : {b : \u03b1 // _}, score b.1) ((legal_moves a).1))) :=\nby unfold score fold; rw acc.rec_3\n\nlemma score_better_aux {a : fin g.max_score} {s : multiset (fin g.max_score)} (b : bool) :\n    a \u2208 s \u2192 better b (multiset.fold (best b) (loss b) s) a :=\nmultiset.induction_on s (\u03bb h, absurd h (multiset.not_mem_zero _)) $ \u03bb n s hi h,\nor.by_cases (multiset.mem_cons.1 h) (\u03bb h\u2081, by rw [multiset.fold_cons_left, h\u2081];\nexact best_better _ _ _) (\u03bb h\u2081, by rw [multiset.fold_cons_right];\nexact better.trans (best_better _ _ _) (hi h\u2081))\n\nlemma score_better {a b : \u03b1} (h : is_legal_move a b) : better (side_to_move a) (score a) (score b) :=\nlet \u27e8hb\u2081, hb\u2082\u27e9 := h in\nby rw [score_eq a, dif_neg (ne_empty_of_mem hb\u2082)];\n  exact score_better_aux _ (multiset.mem_map.2 \u27e8\u27e8b, hb\u2081\u27e9, hb\u2082, rfl\u27e9)\n\nlemma win_of_winning_move {a b : \u03b1} : score b = win (side_to_move a) \u2192\n    is_legal_move a b \u2192 score a = win (side_to_move a) :=\n\u03bb h\u2081 h\u2082, eq_win_of_better_win (h\u2081 \u25b8 score_better h\u2082)\n\nlemma losing_move_of_loss {a b : \u03b1} :  is_legal_move a b \u2192 score a = loss (side_to_move a) \u2192\n    score b = loss (side_to_move a) :=\n\u03bb h\u2081 h\u2082, eq_loss_of_loss_better (h\u2082 \u25b8 (score_better h\u2081))\n\nlemma win_ne_loss (s : bool) : @win \u03b1 _ _ s \u2260 loss s :=\nbool.cases_on s (fin.ne_of_vne $ ne_of_lt $ nat.lt_pred_iff.2 $ not_boring \u03b1)\n(fin.ne_of_vne (ne_of_lt $ nat.lt_pred_iff.2 $ not_boring \u03b1).symm)\n\nend game\nopen game well_founded fintype list finset\ndef nim := \u2115 \u00d7 bool\n#print forall_eq\ninstance : game nim :=\n{ to_has_well_founded := \u27e8_, measure_wf prod.fst\u27e9,\n  start_position := \u27e821, tt\u27e9,\n  side_to_move   := prod.snd,\n  legal_moves    := \u03bb n, if h : n.1 > 0\n    then {\u27e8\u27e8n.1 - 1, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9,\n          \u27e8\u27e8n.1 - 2, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9,\n          \u27e8\u27e8n.1 - 3, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9}\n    else \u2205,\n  max_score := 2,\n  not_boring := dec_trivial,\n  final_score := \u03bb a h, if a.snd = ff then \u27e81, dec_trivial\u27e9\n      else \u27e80, dec_trivial\u27e9 }\n\nnamespace nim\n\nlemma hgf : score (start_position nim) = \u27e81, dec_trivial\u27e9 := rfl\n\nlemma legal_moves_eq_empty_iff (n : nim) : legal_moves n = \u2205 \u2194 n.1 = 0 :=\n\u27e8\u03bb (h : dite (n.1 > 0)\n    (\u03bb h, ({\u27e8\u27e8n.1 - 1, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9,\n            \u27e8\u27e8n.1 - 2, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9,\n            \u27e8\u27e8n.1 - 3, bnot n.2\u27e9, \u27e8nat.sub_lt h dec_trivial, rfl\u27e9\u27e9}\n            : finset {b : nim //b \u227a n \u2227 side_to_move b = bnot (side_to_move n)}))\n    (\u03bb h, \u2205) = \u2205), by_contradiction $ \u03bb h\u2081,\nby simpa [dif_pos (nat.pos_of_ne_zero h\u2081)] using h,\n\u03bb h, or.by_cases (decidable.em (n.2 = tt))\n(\u03bb h\u2081, by rw (show n = \u27e80, tt\u27e9, from prod.eq_iff_fst_eq_snd_eq.2 \u27e8h, h\u2081\u27e9); refl)\n(\u03bb h\u2081, by rw (show n = \u27e80, ff\u27e9, from prod.eq_iff_fst_eq_snd_eq.2\n  \u27e8h, (iff_of_eq $ eq_ff_eq_not_eq_tt n.2).1 h\u2081\u27e9); refl)\u27e9\n\nlemma score_nim (n : nim) : score n = loss (side_to_move n) \u2194 4 \u2223 n.1 :=\nacc.rec_on (apply nim.game.wf n) $ \u03bb n ht hi,\n\u27e8\u03bb h, or.by_cases (decidable.em (legal_moves n = \u2205))\n  (\u03bb h\u2081, by rw legal_moves_eq_empty_iff at h\u2081; rw h\u2081; exact dec_trivial)\n  (\u03bb h\u2081, let \u27e8y, hy\u27e9 := exists_mem_of_ne_empty h\u2081 in\n  begin\n    have := (hi y.1 y.2.1),\n    unfold side_to_move at this,\n    have h\u2082 : y.1.2 = bnot (side_to_move n) := y.2.2,\n    have h\u2083 : score y.1 = loss (side_to_move n) := losing_move_of_loss (is_legal_move_of_mem hy) h,\n    rw [h\u2082, loss, bnot_bnot, h\u2083, eq_comm] at this,\n    simp [win_ne_loss] at this,\n    have : y.1.1 > 0 := nat.pos_of_ne_zero (\u03bb h, by rw h at this; exact absurd this dec_trivial),\n\n\n  end)\n, sorry\u27e9\n\nend nim\n\ndef nac := {l : list (fin 9) // nodup l}\n\nlemma list_fin_length (n : \u2115) (l : {l : list (fin n) // nodup l}) : length l.1 \u2264 n :=\nbegin\n  suffices : length (l.1) \u2264 fintype.card (fin n),\n    rwa fintype.card_fin at this,\n  rw [\u2190 multiset.coe_card, \u2190 finset.card_def \u27e8(\u2191l.1 : multiset (fin n)), multiset.coe_nodup.1 l.2\u27e9],\n  exact card_le_of_subset (subset_univ _),\nend\n\nlocal notation `O`  := (\u27e80, dec_trivial\u27e9 : fin 3)\nlocal notation `X`  := (\u27e81, dec_trivial\u27e9 : fin 3)\nlocal notation `E`  := (\u27e82, dec_trivial\u27e9 : fin 3)\nlocal notation `pX` := tt\nlocal notation `pO` := ff\n\ninstance : has_well_founded nac :=\n\u27e8_, measure_wf (\u03bb p, 9 - length p.1)\u27e9\n\ndef legal_moves' (p : list (fin 9)) : finset (fin 9) :=\nlist.rec_on p univ (\u03bb n p lp, (finset.erase lp n))\n\nlemma legal_moves'_not_mem : \u2200 {p : list (fin 9)} {n : fin 9}, n \u2208 legal_moves' p \u2192 n \u2209 p\n| nil      n h := not_mem_nil _\n| (m :: p) n h := not_mem_cons_of_ne_of_not_mem (ne_of_mem_erase h) $\nlegal_moves'_not_mem (finset.mem_of_mem_erase h)\n\ndef legal_moves1 (p : nac) : finset {q : nac // q \u227a p} :=\n\u27e8multiset.pmap (\u03bb n h, \u27e8\u27e8list.cons n p.1, nodup_cons_of_nodup h p.2\u27e9,\n(nat.sub_lt_sub_left_iff (list_fin_length 9 \u27e8n :: p.1, nodup_cons_of_nodup h p.2\u27e9)).2\n(nat.lt_succ_self _)\u27e9) (legal_moves' p.1).1 (\u03bb n h, legal_moves'_not_mem h),\nmultiset.nodup_pmap (by intros; simp * at *) (legal_moves' p.1).2\u27e9\n\ndef is_line (p : nac) :\n\ninstance : game nac :=\n{ start_position := \u27e8nil, nodup_nil\u27e9,\n  legal_moves    := }\n\n\nend game\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/game.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.39122596565222606}}
{"text": "/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Adam Topaz\n-/\n\nimport category_theory.concrete_category.basic\nimport category_theory.full_subcategory\nimport category_theory.skeletal\nimport category_theory.elementwise\nimport data.fintype.card\n\n/-!\n# The category of finite types.\n\nWe define the category of finite types, denoted `Fintype` as\n(bundled) types with a `fintype` instance.\n\nWe also define `Fintype.skeleton`, the standard skeleton of `Fintype` whose objects are `fin n`\nfor `n : \u2115`. We prove that the obvious inclusion functor `Fintype.skeleton \u2964 Fintype` is an\nequivalence of categories in `Fintype.skeleton.equivalence`.\nWe prove that `Fintype.skeleton` is a skeleton of `Fintype` in `Fintype.is_skeleton`.\n-/\n\nopen_locale classical\nopen category_theory\n\n/-- The category of finite types. -/\ndef Fintype := bundled fintype\n\nnamespace Fintype\n\ninstance : has_coe_to_sort Fintype Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled `Fintype` from the underlying type and typeclass. -/\ndef of (X : Type*) [fintype X] : Fintype := bundled.of X\ninstance : inhabited Fintype := \u27e8\u27e8pempty\u27e9\u27e9\ninstance {X : Fintype} : fintype X := X.2\n\ninstance : category Fintype := induced_category.category bundled.\u03b1\n\n/-- The fully faithful embedding of `Fintype` into the category of types. -/\n@[derive [full, faithful], simps]\ndef incl : Fintype \u2964 Type* := induced_functor _\n\ninstance concrete_category_Fintype : concrete_category Fintype := \u27e8incl\u27e9\n\n@[simp] lemma id_apply (X : Fintype) (x : X) : (\ud835\udfd9 X : X \u2192 X) x = x := rfl\n@[simp] lemma comp_apply {X Y Z : Fintype} (f : X \u27f6 Y) (g : Y \u27f6 Z) (x : X) :\n  (f \u226b g) x = g (f x) := rfl\n\n/-- Equivalences between finite types are the same as isomorphisms in `Fintype`. -/\n-- See `equiv_equiv_iso` in the root namespace for the analogue in `Type`.\n@[simps]\ndef equiv_equiv_iso {A B : Fintype} : (A \u2243 B) \u2243 (A \u2245 B) :=\n{ to_fun := \u03bb e,\n  { hom := e,\n    inv := e.symm, },\n  inv_fun := \u03bb i,\n  { to_fun := i.hom,\n    inv_fun := i.inv,\n    left_inv := iso.hom_inv_id_apply i,\n    right_inv := iso.inv_hom_id_apply i, },\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\nuniverse u\n/--\nThe \"standard\" skeleton for `Fintype`. This is the full subcategory of `Fintype` spanned by objects\nof the form `ulift (fin n)` for `n : \u2115`. We parameterize the objects of `Fintype.skeleton`\ndirectly as `ulift \u2115`, as the type `ulift (fin m) \u2243 ulift (fin n)` is\nnonempty if and only if `n = m`. Specifying universes, `skeleton : Type u` is a small\nskeletal category equivalent to `Fintype.{u}`.\n-/\ndef skeleton : Type u := ulift \u2115\n\nnamespace skeleton\n\n/-- Given any natural number `n`, this creates the associated object of `Fintype.skeleton`. -/\ndef mk : \u2115 \u2192 skeleton := ulift.up\n\ninstance : inhabited skeleton := \u27e8mk 0\u27e9\n\n/-- Given any object of `Fintype.skeleton`, this returns the associated natural number. -/\ndef len : skeleton \u2192 \u2115 := ulift.down\n\n@[ext]\nlemma ext (X Y : skeleton) : X.len = Y.len \u2192 X = Y := ulift.ext _ _\n\ninstance : small_category skeleton.{u} :=\n{ hom := \u03bb X Y, ulift.{u} (fin X.len) \u2192 ulift.{u} (fin Y.len),\n  id := \u03bb _, id,\n  comp := \u03bb _ _ _ f g, g \u2218 f }\n\nlemma is_skeletal : skeletal skeleton.{u} := \u03bb X Y \u27e8h\u27e9, ext _ _ $ fin.equiv_iff_eq.mp $\n  nonempty.intro $\n{ to_fun := \u03bb x, (h.hom \u27e8x\u27e9).down,\n  inv_fun := \u03bb x, (h.inv \u27e8x\u27e9).down,\n  left_inv := begin\n    intro a,\n    change ulift.down _ = _,\n    rw ulift.up_down,\n    change ((h.hom \u226b h.inv) _).down = _,\n    simpa,\n  end,\n  right_inv := begin\n    intro a,\n    change ulift.down _ = _,\n    rw ulift.up_down,\n    change ((h.inv \u226b h.hom) _).down = _,\n    simpa,\n  end }\n\n/-- The canonical fully faithful embedding of `Fintype.skeleton` into `Fintype`. -/\ndef incl : skeleton.{u} \u2964 Fintype.{u} :=\n{ obj := \u03bb X, Fintype.of (ulift (fin X.len)),\n  map := \u03bb _ _ f, f }\n\ninstance : full incl := { preimage := \u03bb _ _ f, f }\ninstance : faithful incl := {}\ninstance : ess_surj incl :=\ness_surj.mk $ \u03bb X, let F := fintype.equiv_fin X in \u27e8mk (fintype.card X), nonempty.intro\n  { hom := F.symm \u2218 ulift.down,\n    inv := ulift.up \u2218 F }\u27e9\n\nnoncomputable instance : is_equivalence incl :=\nequivalence.of_fully_faithfully_ess_surj _\n\n/-- The equivalence between `Fintype.skeleton` and `Fintype`. -/\nnoncomputable def equivalence : skeleton \u224c Fintype := incl.as_equivalence\n\n@[simp] lemma incl_mk_nat_card (n : \u2115) : fintype.card (incl.obj (mk n)) = n :=\nbegin\n  convert finset.card_fin n,\n  apply fintype.of_equiv_card,\nend\n\nend skeleton\n\n/-- `Fintype.skeleton` is a skeleton of `Fintype`. -/\nnoncomputable def is_skeleton : is_skeleton_of Fintype skeleton skeleton.incl :=\n{ skel := skeleton.is_skeletal,\n  eqv := by apply_instance }\n\nend Fintype\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/Fintype.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.39122596565222606}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.comma\nimport Mathlib.PostPort\n\nuniverses v u l u\u2081 u\u2082 v\u2081 v\u2082 \n\nnamespace Mathlib\n\n/-!\n# The category of arrows\n\nThe category of arrows, with morphisms commutative squares.\nWe set this up as a specialization of the comma category `comma L R`,\nwhere `L` and `R` are both the identity functor.\n\nWe also define the typeclass `has_lift`, representing a choice of a lift\nof a commutative square (that is, a diagonal morphism making the two triangles commute).\n\n## Tags\n\ncomma, arrow\n-/\n\nnamespace category_theory\n\n\n/-- The arrow category of `T` has as objects all morphisms in `T` and as morphisms commutative\n     squares in `T`. -/\ndef arrow (T : Type u) [category T] :=\n  comma \ud835\udfed \ud835\udfed\n\n-- Satisfying the inhabited linter\n\nprotected instance arrow.inhabited (T : Type u) [category T] [Inhabited T] : Inhabited (arrow T) :=\n  { default := (fun (this : comma \ud835\udfed \ud835\udfed) => this) Inhabited.default }\n\nnamespace arrow\n\n\n@[simp] theorem id_left {T : Type u} [category T] (f : arrow T) : comma_morphism.left \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem id_right {T : Type u} [category T] (f : arrow T) : comma_morphism.right \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n/-- An object in the arrow category is simply a morphism in `T`. -/\n@[simp] theorem mk_left {T : Type u} [category T] {X : T} {Y : T} (f : X \u27f6 Y) : comma.left (mk f) = X :=\n  Eq.refl (comma.left (mk f))\n\n/-- A morphism in the arrow category is a commutative square connecting two objects of the arrow\n    category. -/\ndef hom_mk {T : Type u} [category T] {f : arrow T} {g : arrow T} {u : comma.left f \u27f6 comma.left g} {v : comma.right f \u27f6 comma.right g} (w : u \u226b comma.hom g = comma.hom f \u226b v) : f \u27f6 g :=\n  comma_morphism.mk\n\n/-- We can also build a morphism in the arrow category out of any commutative square in `T`. -/\n@[simp] theorem hom_mk'_right {T : Type u} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {P : T} {Q : T} {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (w : u \u226b g = f \u226b v) : comma_morphism.right (hom_mk' w) = v :=\n  Eq.refl (comma_morphism.right (hom_mk' w))\n\n@[simp] theorem w_assoc {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) {X' : autoParam T\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])} (f' : functor.obj \ud835\udfed (comma.right g) \u27f6 X') : comma_morphism.left sq \u226b comma.hom g \u226b f' = comma.hom f \u226b comma_morphism.right sq \u226b f' := sorry\n\n/-- A lift of a commutative square is a diagonal morphism making the two triangles commute. -/\nstructure lift_struct {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) \nwhere\n  lift : comma.right f \u27f6 comma.left g\n  fac_left : comma.hom f \u226b lift = comma_morphism.left sq\n  fac_right : lift \u226b comma.hom g = comma_morphism.right sq\n\nprotected instance lift_struct_inhabited {T : Type u} [category T] {X : T} : Inhabited (lift_struct \ud835\udfd9) :=\n  { default := lift_struct.mk \ud835\udfd9 sorry sorry }\n\n/-- `has_lift sq` says that there is some `lift_struct sq`, i.e., that it is possible to find a\n    diagonal morphism making the two triangles commute. -/\nclass has_lift {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) \n  mk' ::\nwhere (exists_lift : Nonempty (lift_struct sq))\n\ntheorem has_lift.mk {T : Type u} [category T] {f : arrow T} {g : arrow T} {sq : f \u27f6 g} (s : lift_struct sq) : has_lift sq :=\n  has_lift.mk' (Nonempty.intro s)\n\n@[simp] theorem lift_struct.fac_right_assoc {T : Type u} [category T] {f : arrow T} {g : arrow T} {sq : f \u27f6 g} (c : lift_struct sq) {X' : autoParam T\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])} (f' : functor.obj \ud835\udfed (comma.right g) \u27f6 X') : lift_struct.lift c \u226b comma.hom g \u226b f' = comma_morphism.right sq \u226b f' := sorry\n\n/-- Given `has_lift sq`, obtain a lift. -/\ndef has_lift.struct {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [has_lift sq] : lift_struct sq :=\n  Classical.choice has_lift.exists_lift\n\n/-- If there is a lift of a commutative square `sq`, we can access it by saying `lift sq`. -/\ndef lift {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [has_lift sq] : comma.right f \u27f6 comma.left g :=\n  lift_struct.lift (has_lift.struct sq)\n\ntheorem lift.fac_left {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [has_lift sq] : comma.hom f \u226b lift sq = comma_morphism.left sq := sorry\n\ntheorem lift.fac_right {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [has_lift sq] : lift sq \u226b comma.hom g = comma_morphism.right sq := sorry\n\n@[simp] theorem lift_mk'_left {T : Type u} [category T] {X : T} {Y : T} {P : T} {Q : T} {f : X \u27f6 Y} {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (h : u \u226b g = f \u226b v) [has_lift (hom_mk' h)] : f \u226b lift (hom_mk' h) = u := sorry\n\n@[simp] theorem lift_mk'_right_assoc {T : Type u} [category T] {X : T} {Y : T} {P : T} {Q : T} {f : X \u27f6 Y} {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (h : u \u226b g = f \u226b v) [has_lift (hom_mk' h)] {X' : autoParam T\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])} (f' : Q \u27f6 X') : lift (hom_mk' h) \u226b g \u226b f' = v \u226b f' := sorry\n\nprotected instance subsingleton_lift_struct_of_epi {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [epi (comma.hom f)] : subsingleton (lift_struct sq) :=\n  subsingleton.intro\n    fun (a b : lift_struct sq) =>\n      lift_struct.ext a b\n        (iff.mp (cancel_epi (comma.hom f))\n          (eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : functor.obj \ud835\udfed (comma.left f) \u27f6 comma.left g) (e_1 : a = a_1)\n                    (\u1fb0 \u1fb0_1 : functor.obj \ud835\udfed (comma.left f) \u27f6 comma.left g) (e_2 : \u1fb0 = \u1fb0_1) => congr (congr_arg Eq e_1) e_2)\n                  (comma.hom f \u226b lift_struct.lift a) (comma_morphism.left sq) (lift_struct.fac_left a)\n                  (comma.hom f \u226b lift_struct.lift b) (comma_morphism.left sq) (lift_struct.fac_left b))\n                (propext (eq_self_iff_true (comma_morphism.left sq)))))\n            trivial))\n\nprotected instance subsingleton_lift_struct_of_mono {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [mono (comma.hom g)] : subsingleton (lift_struct sq) :=\n  subsingleton.intro\n    fun (a b : lift_struct sq) =>\n      lift_struct.ext a b\n        (iff.mp (cancel_mono (comma.hom g))\n          (eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : comma.right f \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_1 : a = a_1)\n                    (\u1fb0 \u1fb0_1 : comma.right f \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_2 : \u1fb0 = \u1fb0_1) =>\n                    congr (congr_arg Eq e_1) e_2)\n                  (lift_struct.lift a \u226b comma.hom g) (comma_morphism.right sq) (lift_struct.fac_right a)\n                  (lift_struct.lift b \u226b comma.hom g) (comma_morphism.right sq) (lift_struct.fac_right b))\n                (propext (eq_self_iff_true (comma_morphism.right sq)))))\n            trivial))\n\nend arrow\n\n\nnamespace functor\n\n\n/-- A functor `C \u2964 D` induces a functor between the corresponding arrow categories. -/\n@[simp] theorem map_arrow_map_right {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) (a : arrow C) (b : arrow C) (f : a \u27f6 b) : comma_morphism.right (map (map_arrow F) f) = map F (comma_morphism.right f) :=\n  Eq.refl (comma_morphism.right (map (map_arrow F) f))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/arrow.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.39122595743927313}}
{"text": "/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Riccardo Brasca\n-/\nimport analysis.normed.group.hom\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.concrete_category.bundled_hom\nimport category_theory.elementwise\n\n/-!\n# The category of seminormed groups\n\nWe define `SemiNormedGroup`, the category of seminormed groups and normed group homs between them,\nas well as `SemiNormedGroup\u2081`, the subcategory of norm non-increasing morphisms.\n-/\n\nnoncomputable theory\n\nuniverses u\n\nopen category_theory\n\n/-- The category of seminormed abelian groups and bounded group homomorphisms. -/\ndef SemiNormedGroup : Type (u+1) := bundled seminormed_add_comm_group\n\nnamespace SemiNormedGroup\n\ninstance bundled_hom : bundled_hom @normed_add_group_hom :=\n\u27e8@normed_add_group_hom.to_fun, @normed_add_group_hom.id, @normed_add_group_hom.comp,\n  @normed_add_group_hom.coe_inj\u27e9\n\nattribute [derive [large_category, concrete_category]] SemiNormedGroup\n\ninstance : has_coe_to_sort SemiNormedGroup (Type u) := bundled.has_coe_to_sort\n\n/-- Construct a bundled `SemiNormedGroup` from the underlying type and typeclass. -/\ndef of (M : Type u) [seminormed_add_comm_group M] : SemiNormedGroup := bundled.of M\n\ninstance (M : SemiNormedGroup) : seminormed_add_comm_group M := M.str\n\n@[simp] lemma coe_of (V : Type u) [seminormed_add_comm_group V] :\n  (SemiNormedGroup.of V : Type u) = V := rfl\n@[simp] lemma coe_id (V : SemiNormedGroup) : \u21d1(\ud835\udfd9 V) = id := rfl\n@[simp] lemma coe_comp {M N K : SemiNormedGroup} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : M \u2192 K) = g \u2218 f := rfl\n\ninstance : inhabited SemiNormedGroup := \u27e8of punit\u27e9\n\ninstance of_unique (V : Type u) [seminormed_add_comm_group V] [i : unique V] :\n  unique (SemiNormedGroup.of V) := i\n\ninstance : limits.has_zero_morphisms.{u (u+1)} SemiNormedGroup := {}\n\n@[simp] lemma zero_apply {V W : SemiNormedGroup} (x : V) : (0 : V \u27f6 W) x = 0 := rfl\n\nlemma is_zero_of_subsingleton (V : SemiNormedGroup) [subsingleton V] :\n  limits.is_zero V :=\nbegin\n  refine \u27e8\u03bb X, \u27e8\u27e8\u27e80\u27e9, \u03bb f, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e80\u27e9, \u03bb f, _\u27e9\u27e9\u27e9,\n  { ext, have : x = 0 := subsingleton.elim _ _, simp only [this, map_zero], },\n  { ext, apply subsingleton.elim }\nend\n\ninstance has_zero_object : limits.has_zero_object SemiNormedGroup.{u} :=\n\u27e8\u27e8of punit, is_zero_of_subsingleton _\u27e9\u27e9\n\nlemma iso_isometry_of_norm_noninc {V W : SemiNormedGroup} (i : V \u2245 W)\n  (h1 : i.hom.norm_noninc) (h2 : i.inv.norm_noninc) :\n  isometry i.hom :=\nbegin\n  apply add_monoid_hom_class.isometry_of_norm,\n  intro v,\n  apply le_antisymm (h1 v),\n  calc \u2016v\u2016 = \u2016i.inv (i.hom v)\u2016 : by rw [iso.hom_inv_id_apply]\n  ... \u2264 \u2016i.hom v\u2016 : h2 _,\nend\n\nend SemiNormedGroup\n\n/--\n`SemiNormedGroup\u2081` is a type synonym for `SemiNormedGroup`,\nwhich we shall equip with the category structure consisting only of the norm non-increasing maps.\n-/\ndef SemiNormedGroup\u2081 : Type (u+1) := bundled seminormed_add_comm_group\n\nnamespace SemiNormedGroup\u2081\n\ninstance : has_coe_to_sort SemiNormedGroup\u2081 (Type u) := bundled.has_coe_to_sort\n\ninstance : large_category.{u} SemiNormedGroup\u2081 :=\n{ hom := \u03bb X Y, { f : normed_add_group_hom X Y // f.norm_noninc },\n  id := \u03bb X, \u27e8normed_add_group_hom.id X, normed_add_group_hom.norm_noninc.id\u27e9,\n  comp := \u03bb X Y Z f g,\n    \u27e8(g : normed_add_group_hom Y Z).comp (f : normed_add_group_hom X Y), g.2.comp f.2\u27e9 }\n\n@[ext] lemma hom_ext {M N : SemiNormedGroup\u2081} (f g : M \u27f6 N) (w : (f : M \u2192 N) = (g : M \u2192 N)) :\n  f = g :=\nsubtype.eq (normed_add_group_hom.ext (congr_fun w))\n\ninstance : concrete_category.{u} SemiNormedGroup\u2081 :=\n{ forget :=\n  { obj := \u03bb X, X,\n    map := \u03bb X Y f, f, },\n  forget_faithful := {} }\n\n/-- Construct a bundled `SemiNormedGroup\u2081` from the underlying type and typeclass. -/\ndef of (M : Type u) [seminormed_add_comm_group M] : SemiNormedGroup\u2081 := bundled.of M\n\ninstance (M : SemiNormedGroup\u2081) : seminormed_add_comm_group M := M.str\n\n/-- Promote a morphism in `SemiNormedGroup` to a morphism in `SemiNormedGroup\u2081`. -/\ndef mk_hom {M N : SemiNormedGroup} (f : M \u27f6 N) (i : f.norm_noninc) :\n  SemiNormedGroup\u2081.of M \u27f6 SemiNormedGroup\u2081.of N :=\n\u27e8f, i\u27e9\n\n@[simp] lemma mk_hom_apply {M N : SemiNormedGroup} (f : M \u27f6 N) (i : f.norm_noninc) (x) :\n  mk_hom f i x = f x := rfl\n\n/-- Promote an isomorphism in `SemiNormedGroup` to an isomorphism in `SemiNormedGroup\u2081`. -/\n@[simps]\ndef mk_iso {M N : SemiNormedGroup} (f : M \u2245 N) (i : f.hom.norm_noninc) (i' : f.inv.norm_noninc) :\n  SemiNormedGroup\u2081.of M \u2245 SemiNormedGroup\u2081.of N :=\n{ hom := mk_hom f.hom i,\n  inv := mk_hom f.inv i',\n  hom_inv_id' := by { apply subtype.eq, exact f.hom_inv_id, },\n  inv_hom_id' := by { apply subtype.eq, exact f.inv_hom_id, }, }\n\ninstance : has_forget\u2082 SemiNormedGroup\u2081 SemiNormedGroup :=\n{ forget\u2082 :=\n  { obj := \u03bb X, X,\n    map := \u03bb X Y f, f.1, }, }\n\n@[simp] lemma coe_of (V : Type u) [seminormed_add_comm_group V] :\n  (SemiNormedGroup\u2081.of V : Type u) = V := rfl\n@[simp] lemma coe_id (V : SemiNormedGroup\u2081) : \u21d1(\ud835\udfd9 V) = id := rfl\n@[simp] lemma coe_comp {M N K : SemiNormedGroup\u2081} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : M \u2192 K) = g \u2218 f := rfl\n-- If `coe_fn_coe_base` fires before `coe_comp`, `coe_comp'` puts us back in normal form.\n@[simp] lemma coe_comp' {M N K : SemiNormedGroup\u2081} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : normed_add_group_hom M K) = (\u2191g : normed_add_group_hom N K).comp \u2191f := rfl\n\ninstance : inhabited SemiNormedGroup\u2081 := \u27e8of punit\u27e9\n\ninstance of_unique (V : Type u) [seminormed_add_comm_group V] [i : unique V] :\n  unique (SemiNormedGroup\u2081.of V) := i\n\ninstance : limits.has_zero_morphisms.{u (u+1)} SemiNormedGroup\u2081 :=\n{ has_zero := \u03bb X Y, { zero := \u27e80, normed_add_group_hom.norm_noninc.zero\u27e9, },\n  comp_zero' := \u03bb X Y f Z, by { ext, refl, },\n  zero_comp' := \u03bb X Y Z f, by { ext, simp [coe_fn_coe_base'] } }\n\n@[simp] lemma zero_apply {V W : SemiNormedGroup\u2081} (x : V) : (0 : V \u27f6 W) x = 0 := rfl\n\nlemma is_zero_of_subsingleton (V : SemiNormedGroup\u2081) [subsingleton V] :\n  limits.is_zero V :=\nbegin\n  refine \u27e8\u03bb X, \u27e8\u27e8\u27e80\u27e9, \u03bb f, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e80\u27e9, \u03bb f, _\u27e9\u27e9\u27e9,\n  { ext, have : x = 0 := subsingleton.elim _ _, simp only [this, map_zero],\n    exact map_zero f.1 },\n  { ext, apply subsingleton.elim }\nend\n\ninstance has_zero_object : limits.has_zero_object SemiNormedGroup\u2081.{u} :=\n\u27e8\u27e8of punit, is_zero_of_subsingleton _\u27e9\u27e9\n\nlemma iso_isometry {V W : SemiNormedGroup\u2081} (i : V \u2245 W) :\n  isometry i.hom :=\nbegin\n  change isometry (i.hom : V \u2192+ W),\n  refine add_monoid_hom_class.isometry_of_norm i.hom _,\n  intro v,\n  apply le_antisymm (i.hom.2 v),\n  calc \u2016v\u2016 = \u2016i.inv (i.hom v)\u2016 : by rw [iso.hom_inv_id_apply]\n      ... \u2264 \u2016i.hom v\u2016 : i.inv.2 _,\nend\n\nend SemiNormedGroup\u2081\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/analysis/normed/group/SemiNormedGroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6477982043529715, "lm_q1q2_score": 0.3912259482034054}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n\n! This file was ported from Lean 3 source module data.set.sigma\n! leanprover-community/mathlib commit 2258b40dacd2942571c8ce136215350c702dc78f\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.Set.Image\n\n/-!\n# Sets in sigma types\n\nThis file defines `Set.sigma`, the indexed sum of sets.\n-/\n\nnamespace Set\n\nvariable {\u03b9 \u03b9' : Type _} {\u03b1 \u03b2 : \u03b9 \u2192 Type _} {s s\u2081 s\u2082 : Set \u03b9} {t t\u2081 t\u2082 : \u2200 i, Set (\u03b1 i)}\n  {u : Set (\u03a3 i, \u03b1 i)} {x : \u03a3 i, \u03b1 i} {i j : \u03b9} {a : \u03b1 i}\n\n@[simp]\ntheorem range_sigmaMk (i : \u03b9) : range (Sigma.mk i : \u03b1 i \u2192 Sigma \u03b1) = Sigma.fst \u207b\u00b9' {i} := by\n  apply Subset.antisymm\n  \u00b7 rintro _ \u27e8b, rfl\u27e9\n    simp\n  \u00b7 rintro \u27e8x, y\u27e9 (rfl | _)\n    exact mem_range_self y\n#align set.range_sigma_mk Set.range_sigmaMk\n\ntheorem preimage_image_sigmaMk_of_ne (h : i \u2260 j) (s : Set (\u03b1 j)) :\n    Sigma.mk i \u207b\u00b9' (Sigma.mk j '' s) = \u2205 := by\n  ext x\n  simp [h.symm]\n#align set.preimage_image_sigma_mk_of_ne Set.preimage_image_sigmaMk_of_ne\n\ntheorem image_sigmaMk_preimage_sigmaMap_subset {\u03b2 : \u03b9' \u2192 Type _} (f : \u03b9 \u2192 \u03b9')\n    (g : \u2200 i, \u03b1 i \u2192 \u03b2 (f i)) (i : \u03b9) (s : Set (\u03b2 (f i))) :\n    Sigma.mk i '' (g i \u207b\u00b9' s) \u2286 Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) :=\n  image_subset_iff.2 fun x hx \u21a6 \u27e8g i x, hx, rfl\u27e9\n#align set.image_sigma_mk_preimage_sigma_map_subset Set.image_sigmaMk_preimage_sigmaMap_subset\n\ntheorem image_sigmaMk_preimage_sigmaMap {\u03b2 : \u03b9' \u2192 Type _} {f : \u03b9 \u2192 \u03b9'} (hf : Function.Injective f)\n    (g : \u2200 i, \u03b1 i \u2192 \u03b2 (f i)) (i : \u03b9) (s : Set (\u03b2 (f i))) :\n    Sigma.mk i '' (g i \u207b\u00b9' s) = Sigma.map f g \u207b\u00b9' (Sigma.mk (f i) '' s) := by\n  refine' (image_sigmaMk_preimage_sigmaMap_subset f g i s).antisymm _\n  rintro \u27e8j, x\u27e9 \u27e8y, hys, hxy\u27e9\n  simp only [hf.eq_iff, Sigma.map, Sigma.ext_iff] at hxy\n  rcases hxy with \u27e8rfl, hxy\u27e9; rw [heq_iff_eq] at hxy; subst y\n  exact \u27e8x, hys, rfl\u27e9\n#align set.image_sigma_mk_preimage_sigma_map Set.image_sigmaMk_preimage_sigmaMap\n\n/-- Indexed sum of sets. `s.sigma t` is the set of dependent pairs `\u27e8i, a\u27e9` such that `i \u2208 s` and\n`a \u2208 t i`.-/\nprotected def Sigma (s : Set \u03b9) (t : \u2200 i, Set (\u03b1 i)) : Set (\u03a3i, \u03b1 i) :=\n  { x | x.1 \u2208 s \u2227 x.2 \u2208 t x.1 }\n#align set.sigma Set.Sigma\n\n@[simp]\ntheorem mem_sigma_iff : x \u2208 s.Sigma t \u2194 x.1 \u2208 s \u2227 x.2 \u2208 t x.1 :=\n  Iff.rfl\n#align set.mem_sigma_iff Set.mem_sigma_iff\n\ntheorem mk_sigma_iff : (\u27e8i, a\u27e9 : \u03a3 i, \u03b1 i) \u2208 s.Sigma t \u2194 i \u2208 s \u2227 a \u2208 t i :=\n  Iff.rfl\n#align set.mk_sigma_iff Set.mk_sigma_iff\n\ntheorem mk_mem_sigma (hi : i \u2208 s) (ha : a \u2208 t i) : (\u27e8i, a\u27e9 : \u03a3i, \u03b1 i) \u2208 s.Sigma t :=\n  \u27e8hi, ha\u27e9\n#align set.mk_mem_sigma Set.mk_mem_sigma\n\ntheorem sigma_mono (hs : s\u2081 \u2286 s\u2082) (ht : \u2200 i, t\u2081 i \u2286 t\u2082 i) : s\u2081.Sigma t\u2081 \u2286 s\u2082.Sigma t\u2082 := fun _ hx \u21a6\n  \u27e8hs hx.1, ht _ hx.2\u27e9\n#align set.sigma_mono Set.sigma_mono\n\ntheorem sigma_subset_iff : s.Sigma t \u2286 u \u2194 \u2200 \u2983i\u2984, i \u2208 s \u2192 \u2200 \u2983a\u2984, a \u2208 t i \u2192 (\u27e8i, a\u27e9 : \u03a3i, \u03b1 i) \u2208 u :=\n  \u27e8fun h _ hi _ ha \u21a6 h <| mk_mem_sigma hi ha, fun h _ ha \u21a6 h ha.1 ha.2\u27e9\n#align set.sigma_subset_iff Set.sigma_subset_iff\n\ntheorem forall_sigma_iff {p : (\u03a3i, \u03b1 i) \u2192 Prop} :\n    (\u2200 x \u2208 s.Sigma t, p x) \u2194 \u2200 \u2983i\u2984, i \u2208 s \u2192 \u2200 \u2983a\u2984, a \u2208 t i \u2192 p \u27e8i, a\u27e9 :=\n  sigma_subset_iff\n#align set.forall_sigma_iff Set.forall_sigma_iff\n\ntheorem exists_sigma_iff {p : (\u03a3i, \u03b1 i) \u2192 Prop} :\n    (\u2203 x \u2208 s.Sigma t, p x) \u2194 \u2203 i \u2208 s, \u2203 a \u2208 t i, p \u27e8i, a\u27e9 :=\n  \u27e8fun \u27e8\u27e8i, a\u27e9, ha, h\u27e9 \u21a6 \u27e8i, ha.1, a, ha.2, h\u27e9, fun \u27e8i, hi, a, ha, h\u27e9 \u21a6 \u27e8\u27e8i, a\u27e9, \u27e8hi, ha\u27e9, h\u27e9\u27e9\n#align set.exists_sigma_iff Set.exists_sigma_iff\n\n@[simp]\ntheorem sigma_empty : (s.Sigma fun i \u21a6 (\u2205 : Set (\u03b1 i))) = \u2205 :=\n  ext fun _ \u21a6 and_false_iff _\n#align set.sigma_empty Set.sigma_empty\n\n@[simp]\ntheorem empty_sigma : (\u2205 : Set \u03b9).Sigma t = \u2205 :=\n  ext fun _ \u21a6 false_and_iff _\n#align set.empty_sigma Set.empty_sigma\n\ntheorem univ_sigma_univ : ((@univ \u03b9).Sigma fun _ \u21a6 @univ (\u03b1 i)) = univ :=\n  ext fun _ \u21a6 true_and_iff _\n#align set.univ_sigma_univ Set.univ_sigma_univ\n\n@[simp]\ntheorem sigma_univ : s.Sigma (fun _ \u21a6 univ : \u2200 i, Set (\u03b1 i)) = Sigma.fst \u207b\u00b9' s :=\n  ext fun _ \u21a6 and_true_iff _\n#align set.sigma_univ Set.sigma_univ\n\n@[simp]\ntheorem singleton_sigma : ({i} : Set \u03b9).Sigma t = Sigma.mk i '' t i :=\n  ext fun x \u21a6 by\n    constructor\n    \u00b7 obtain \u27e8j, a\u27e9 := x\n      rintro \u27e8rfl : j = i, ha\u27e9\n      exact mem_image_of_mem _ ha\n    \u00b7 rintro \u27e8b, hb, rfl\u27e9\n      exact \u27e8rfl, hb\u27e9\n#align set.singleton_sigma Set.singleton_sigma\n\n@[simp]\ntheorem sigma_singleton {a : \u2200 i, \u03b1 i} :\n    (s.Sigma fun i \u21a6 ({a i} : Set (\u03b1 i))) = (fun i \u21a6 Sigma.mk i <| a i) '' s := by\n  ext \u27e8x, y\u27e9\n  simp [and_left_comm, eq_comm]\n#align set.sigma_singleton Set.sigma_singleton\n\ntheorem singleton_sigma_singleton {a : \u2200 i, \u03b1 i} :\n    (({i} : Set \u03b9).Sigma fun i \u21a6 ({a i} : Set (\u03b1 i))) = {\u27e8i, a i\u27e9} := by\n  rw [sigma_singleton, image_singleton]\n#align set.singleton_sigma_singleton Set.singleton_sigma_singleton\n\n@[simp]\ntheorem union_sigma : (s\u2081 \u222a s\u2082).Sigma t = s\u2081.Sigma t \u222a s\u2082.Sigma t :=\n  ext fun _ \u21a6 or_and_right\n#align set.union_sigma Set.union_sigma\n\n@[simp]\ntheorem sigma_union : (s.Sigma fun i \u21a6 t\u2081 i \u222a t\u2082 i) = s.Sigma t\u2081 \u222a s.Sigma t\u2082 :=\n  ext fun _ \u21a6 and_or_left\n#align set.sigma_union Set.sigma_union\n\ntheorem sigma_inter_sigma : s\u2081.Sigma t\u2081 \u2229 s\u2082.Sigma t\u2082 = (s\u2081 \u2229 s\u2082).Sigma fun i \u21a6 t\u2081 i \u2229 t\u2082 i := by\n  ext \u27e8x, y\u27e9\n  simp [and_assoc, and_left_comm]\n#align set.sigma_inter_sigma Set.sigma_inter_sigma\n\ntheorem insert_sigma : (insert i s).Sigma t = Sigma.mk i '' t i \u222a s.Sigma t := by\n  rw [insert_eq, union_sigma, singleton_sigma]\n  exact a\n#align set.insert_sigma Set.insert_sigma\n\ntheorem sigma_insert {a : \u2200 i, \u03b1 i} :\n    (s.Sigma fun i \u21a6 insert (a i) (t i)) = (fun i \u21a6 \u27e8i, a i\u27e9) '' s \u222a s.Sigma t := by\n  simp_rw [insert_eq, sigma_union, sigma_singleton]\n#align set.sigma_insert Set.sigma_insert\n\ntheorem sigma_preimage_eq {f : \u03b9' \u2192 \u03b9} {g : \u2200 i, \u03b2 i \u2192 \u03b1 i} :\n    ((f \u207b\u00b9' s).Sigma fun i \u21a6 g (f i) \u207b\u00b9' t (f i)) =\n      (fun p : \u03a3i, \u03b2 (f i) \u21a6 Sigma.mk _ (g _ p.2)) \u207b\u00b9' s.Sigma t :=\n  rfl\n#align set.sigma_preimage_eq Set.sigma_preimage_eq\n\ntheorem sigma_preimage_left {f : \u03b9' \u2192 \u03b9} :\n    ((f \u207b\u00b9' s).Sigma fun i \u21a6 t (f i)) = (fun p : \u03a3i, \u03b1 (f i) \u21a6 Sigma.mk _ p.2) \u207b\u00b9' s.Sigma t :=\n  rfl\n#align set.sigma_preimage_left Set.sigma_preimage_left\n\ntheorem sigma_preimage_right {g : \u2200 i, \u03b2 i \u2192 \u03b1 i} :\n    (s.Sigma fun i \u21a6 g i \u207b\u00b9' t i) = (fun p : \u03a3i, \u03b2 i \u21a6 Sigma.mk p.1 (g _ p.2)) \u207b\u00b9' s.Sigma t :=\n  rfl\n#align set.sigma_preimage_right Set.sigma_preimage_right\n\ntheorem preimage_sigmaMap_sigma {\u03b1' : \u03b9' \u2192 Type _} (f : \u03b9 \u2192 \u03b9') (g : \u2200 i, \u03b1 i \u2192 \u03b1' (f i))\n    (s : Set \u03b9') (t : \u2200 i, Set (\u03b1' i)) :\n    Sigma.map f g \u207b\u00b9' s.Sigma t = (f \u207b\u00b9' s).Sigma fun i \u21a6 g i \u207b\u00b9' t (f i) :=\n  rfl\n#align set.preimage_sigma_map_sigma Set.preimage_sigmaMap_sigma\n\n@[simp]\ntheorem mk_preimage_sigma (hi : i \u2208 s) : Sigma.mk i \u207b\u00b9' s.Sigma t = t i :=\n  ext fun _ \u21a6 and_iff_right hi\n#align set.mk_preimage_sigma Set.mk_preimage_sigma\n\n@[simp]\ntheorem mk_preimage_sigma_eq_empty (hi : i \u2209 s) : Sigma.mk i \u207b\u00b9' s.Sigma t = \u2205 :=\n  ext fun _ \u21a6 iff_of_false (hi \u2218 And.left) id\n#align set.mk_preimage_sigma_eq_empty Set.mk_preimage_sigma_eq_empty\n\ntheorem mk_preimage_sigma_eq_if [DecidablePred (\u00b7 \u2208 s)] :\n    Sigma.mk i \u207b\u00b9' s.Sigma t = if i \u2208 s then t i else \u2205 := by split_ifs <;> simp [*]\n#align set.mk_preimage_sigma_eq_if Set.mk_preimage_sigma_eq_if\n\ntheorem mk_preimage_sigma_fn_eq_if {\u03b2 : Type _} [DecidablePred (\u00b7 \u2208 s)] (g : \u03b2 \u2192 \u03b1 i) :\n    (fun b \u21a6 Sigma.mk i (g b)) \u207b\u00b9' s.Sigma t = if i \u2208 s then g \u207b\u00b9' t i else \u2205 :=\n  ext fun _ \u21a6 by split_ifs <;> simp [*]\n#align set.mk_preimage_sigma_fn_eq_if Set.mk_preimage_sigma_fn_eq_if\n\n\n\nprotected theorem Nonempty.sigma :\n    s.Nonempty \u2192 (\u2200 i, (t i).Nonempty) \u2192 (s.Sigma t : Set _).Nonempty := fun \u27e8i, hi\u27e9 h \u21a6\n  let \u27e8a, ha\u27e9 := h i\n  \u27e8\u27e8i, a\u27e9, hi, ha\u27e9\n#align set.nonempty.sigma Set.Nonempty.sigma\n\ntheorem Nonempty.sigma_fst : (s.Sigma t : Set _).Nonempty \u2192 s.Nonempty := fun \u27e8x, hx\u27e9 \u21a6 \u27e8x.1, hx.1\u27e9\n#align set.nonempty.sigma_fst Set.Nonempty.sigma_fst\n\ntheorem Nonempty.sigma_snd : (s.Sigma t : Set _).Nonempty \u2192 \u2203 i \u2208 s, (t i).Nonempty :=\n  fun \u27e8x, hx\u27e9 \u21a6 \u27e8x.1, hx.1, x.2, hx.2\u27e9\n#align set.nonempty.sigma_snd Set.Nonempty.sigma_snd\n\ntheorem sigma_nonempty_iff : (s.Sigma t : Set _).Nonempty \u2194 \u2203 i \u2208 s, (t i).Nonempty :=\n  \u27e8Nonempty.sigma_snd, fun \u27e8i, hi, a, ha\u27e9 \u21a6 \u27e8\u27e8i, a\u27e9, hi, ha\u27e9\u27e9\n#align set.sigma_nonempty_iff Set.sigma_nonempty_iff\n\ntheorem sigma_eq_empty_iff : s.Sigma t = \u2205 \u2194 \u2200 i \u2208 s, t i = \u2205 :=\n  not_nonempty_iff_eq_empty.symm.trans <|\n    sigma_nonempty_iff.not.trans <| by\n      simp only [not_nonempty_iff_eq_empty, not_and, not_exists]\n#align set.sigma_eq_empty_iff Set.sigma_eq_empty_iff\n\ntheorem image_sigmaMk_subset_sigma_left {a : \u2200 i, \u03b1 i} (ha : \u2200 i, a i \u2208 t i) :\n    (fun i \u21a6 Sigma.mk i (a i)) '' s \u2286 s.Sigma t :=\n  image_subset_iff.2 fun _ hi \u21a6 \u27e8hi, ha _\u27e9\n#align set.image_sigma_mk_subset_sigma_left Set.image_sigmaMk_subset_sigma_left\n\ntheorem image_sigmaMk_subset_sigma_right (hi : i \u2208 s) : Sigma.mk i '' t i \u2286 s.Sigma t :=\n  image_subset_iff.2 fun _ \u21a6 And.intro hi\n#align set.image_sigma_mk_subset_sigma_right Set.image_sigmaMk_subset_sigma_right\n\ntheorem sigma_subset_preimage_fst (s : Set \u03b9) (t : \u2200 i, Set (\u03b1 i)) : s.Sigma t \u2286 Sigma.fst \u207b\u00b9' s :=\n  fun _ \u21a6 And.left\n#align set.sigma_subset_preimage_fst Set.sigma_subset_preimage_fst\n\ntheorem fst_image_sigma_subset (s : Set \u03b9) (t : \u2200 i, Set (\u03b1 i)) : Sigma.fst '' s.Sigma t \u2286 s :=\n  image_subset_iff.2 fun _ \u21a6 And.left\n#align set.fst_image_sigma_subset Set.fst_image_sigma_subset\n\ntheorem fst_image_sigma (s : Set \u03b9) (ht : \u2200 i, (t i).Nonempty) : Sigma.fst '' s.Sigma t = s :=\n  (fst_image_sigma_subset _ _).antisymm fun i hi \u21a6\n    let \u27e8a, ha\u27e9 := ht i\n    \u27e8\u27e8i, a\u27e9, \u27e8hi, ha\u27e9, rfl\u27e9\n#align set.fst_image_sigma Set.fst_image_sigma\n\ntheorem sigma_diff_sigma : s\u2081.Sigma t\u2081 \\ s\u2082.Sigma t\u2082 = s\u2081.Sigma (t\u2081 \\ t\u2082) \u222a (s\u2081 \\ s\u2082).Sigma t\u2081 :=\n  ext fun x \u21a6 by\n    by_cases h\u2081 : x.1 \u2208 s\u2081 <;> by_cases h\u2082 : x.2 \u2208 t\u2081 x.1 <;> simp [*, \u2190 imp_iff_or_not]\n#align set.sigma_diff_sigma Set.sigma_diff_sigma\n\nend Set\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/Set/Sigma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.7154240018510025, "lm_q1q2_score": 0.3911495966574541}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.oracle_comp\n\n/-!\n# Running Compuations With No Oracles\n\nThis file defines a function `run_comp` for \"running\" an `oracle_comp` that doesn't have access to\nany oracles, where `empty_spec` is used to represent the lack of available oracles.\nIn this case the `oracle_comp.query` constructor can't be called, and so we can eliminate this\ncase and perform the natural reduction on other computations.\n\nTODO: Should be possible to generalize this to use `random_gen`, allowing greater flexibility.\n-/\n\nnamespace oracle_comp\n\nopen oracle_spec\n\n/-- Run a computation with `[]\u2092` as the oracles, using `empty.elim` in the `query` case. -/\ndef run_comp : \u03a0 {\u03b1 : Type}, oracle_comp []\u2092 \u03b1 \u2192 \u03b1\n| _ (pure' \u03b1 a) := a\n| _ (bind' \u03b1 \u03b2 oa ob) := let a : \u03b1 := run_comp oa in run_comp (ob a)\n| _ (query i t) := empty.elim i\n\nvariables {\u03b1 \u03b2 : Type} (oa : oracle_comp []\u2092 \u03b1) (ob : \u03b1 \u2192 oracle_comp []\u2092 \u03b2) (a : \u03b1) (b : \u03b2)\n\n@[simp] lemma run_comp_return : run_comp (return a) = a := rfl\n\n@[simp] lemma run_comp_bind : run_comp (oa >>= ob) = run_comp (ob $ run_comp oa) := rfl\n\nlemma run_comp_query (i : []\u2092.\u03b9) (t : []\u2092.domain i) (u : []\u2092.range i) : run_comp (query i t) = u :=\nempty.elim i\n\nexample : run_comp\n(do { x \u2190 return 1,\n      y \u2190 return (x + 1),\n      z \u2190 return (x * y + y * x),\n      return (y + y = z) }) = true := -- Check that `2 + 2 = 4`\nby simp only [run_comp_bind, run_comp_return, one_mul, mul_one, eq_self_iff_true]\n\nend oracle_comp", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/run_comp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.39093561158206286}}
{"text": "def f : (xs : List Nat) \u2192 Nat \u2192 xs \u2260 [] \u2192 Nat\n  | [], _, _ => _\n  | [a,b], _, _ => _\n  | _, _, _ => _\n\nset_option pp.inaccessibleNames true in\ndef f' : (xs : List Nat) \u2192 Nat \u2192 xs \u2260 [] \u2192 Nat\n  | [], _, _ => _ -- TODO: figure out why hyp `Ne (\u03b1 := List Nat) x\u271d\u00b2 []` needs \u03b1\n  | [a,b], _, _ => _\n  | _, _, _ =>  _\n\ntheorem ex1 : p \u2228 q \u2192 q \u2228 p := by\n  intro h\n  cases h\n  trace_state\n  apply Or.inr\n  assumption\n  apply Or.inl\n  assumption\n  done\n\ntheorem ex2 : {p : Prop} \u2192 [Decidable p] \u2192 p \u2192 decide p = true\n  | _, isTrue  _, _   => _\n  | _, isFalse h\u2081, h\u2082 => absurd h\u2082 h\u2081\n\ntheorem ex3 : \u2200 {c d : Char}, c = d \u2192 c.val = d.val\n  | _, _, rfl => _\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/hidingInaccessibleNames.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6513548714339144, "lm_q1q2_score": 0.3909356115820628}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nTransferring `traversable` instances using isomorphisms.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.control.traversable.lemmas\nimport Mathlib.PostPort\n\nuniverses u \n\nnamespace Mathlib\n\nnamespace equiv\n\n\n/-- Given a functor `t`, a function `t' : Type u \u2192 Type u`, and\nequivalences `t \u03b1 \u2243 t' \u03b1` for all `\u03b1`, then every function `\u03b1 \u2192 \u03b2` can\nbe mapped to a function `t' \u03b1 \u2192 t' \u03b2` functorially (see\n`equiv.functor`). -/\nprotected def map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1)\n    [Functor t] {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) : t' \u03b2 :=\n  coe_fn (eqv \u03b2) (f <$> coe_fn (equiv.symm (eqv \u03b1)) x)\n\n/-- The function `equiv.map` transfers the functoriality of `t` to\n`t'` using the equivalences `eqv`.  -/\nprotected def functor {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1)\n    [Functor t] : Functor t' :=\n  { map := equiv.map eqv, mapConst := fun (\u03b1 \u03b2 : Type u) => equiv.map eqv \u2218 function.const \u03b2 }\n\nprotected theorem id_map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] {\u03b1 : Type u} (x : t' \u03b1) :\n    equiv.map eqv id x = x :=\n  sorry\n\nprotected theorem comp_map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] {\u03b1 : Type u} {\u03b2 : Type u}\n    {\u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : t' \u03b1) :\n    equiv.map eqv (h \u2218 g) x = equiv.map eqv h (equiv.map eqv g x) :=\n  sorry\n\nprotected theorem is_lawful_functor {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] : is_lawful_functor t' :=\n  is_lawful_functor.mk (equiv.id_map eqv) (equiv.comp_map eqv)\n\nprotected theorem is_lawful_functor' {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] [F : Functor t']\n    (h\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), Functor.map f = equiv.map eqv f)\n    (h\u2081 :\n      \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2),\n        Functor.mapConst f = function.comp (equiv.map eqv) (function.const \u03b1) f) :\n    is_lawful_functor t' :=\n  sorry\n\n/-- Like `equiv.map`, a function `t' : Type u \u2192 Type u` can be given\nthe structure of a traversable functor using a traversable functor\n`t'` and equivalences `t \u03b1 \u2243 t' \u03b1` for all \u03b1.  See `equiv.traversable`. -/\nprotected def traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] {m : Type u \u2192 Type u} [Applicative m]\n    {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2) (x : t' \u03b1) : m (t' \u03b2) :=\n  \u21d1(eqv \u03b2) <$> traverse f (coe_fn (equiv.symm (eqv \u03b1)) x)\n\n/-- The function `equiv.tranverse` transfers a traversable functor\ninstance across the equivalences `eqv`. -/\nprotected def traversable {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] : traversable t' :=\n  traversable.mk (equiv.traverse eqv)\n\nprotected theorem id_traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {\u03b1 : Type u}\n    (x : t' \u03b1) : equiv.traverse eqv id.mk x = x :=\n  sorry\n\nprotected theorem traverse_eq_map_id {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {\u03b1 : Type u}\n    {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) :\n    equiv.traverse eqv (id.mk \u2218 f) x = id.mk (equiv.map eqv f x) :=\n  sorry\n\nprotected theorem comp_traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t]\n    {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u}\n    (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t' \u03b1) :\n    equiv.traverse eqv (functor.comp.mk \u2218 Functor.map f \u2218 g) x =\n        functor.comp.mk (equiv.traverse eqv f <$> equiv.traverse eqv g x) :=\n  sorry\n\nprotected theorem naturality {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t]\n    {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G] (\u03b7 : applicative_transformation F G)\n    {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2) (x : t' \u03b1) :\n    coe_fn \u03b7 (t' \u03b2) (equiv.traverse eqv f x) = equiv.traverse eqv (coe_fn \u03b7 \u03b2 \u2218 f) x :=\n  sorry\n\n/-- The fact that `t` is a lawful traversable functor carries over the\nequivalences to `t'`, with the traversable functor structure given by\n`equiv.traversable`. -/\nprotected def is_lawful_traversable {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] :\n    is_lawful_traversable t' :=\n  is_lawful_traversable.mk (equiv.id_traverse eqv) (equiv.comp_traverse eqv)\n    (equiv.traverse_eq_map_id eqv) (equiv.naturality eqv)\n\n/-- If the `traversable t'` instance has the properties that `map`,\n`map_const`, and `traverse` are equal to the ones that come from\ncarrying the traversable functor structure from `t` over the\nequivalences, then the the fact `t` is a lawful traversable functor\ncarries over as well. -/\nprotected def is_lawful_traversable' {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u}\n    (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] [traversable t']\n    (h\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), Functor.map f = equiv.map eqv f)\n    (h\u2081 :\n      \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2),\n        Functor.mapConst f = function.comp (equiv.map eqv) (function.const \u03b1) f)\n    (h\u2082 :\n      \u2200 {F : Type u \u2192 Type u} [_inst_7 : Applicative F] [_inst_8 : is_lawful_applicative F]\n        {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2), traverse f = equiv.traverse eqv f) :\n    is_lawful_traversable t' :=\n  is_lawful_traversable.mk sorry sorry sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/traversable/equiv_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.3909356022658206}}
{"text": "-- TODO -- make ZFC namespace and promise that everything in the namespace only \n-- mentions Type and never mentions universes.\n-- TODO ask Mario how close this is to ZFC\n\n-- TODO see how much of \"transport\" made it into mathlib\nimport data.equiv.basic\n--import analysis.real\n--#check \u211d -- Type\nnamespace zfc\n-- I sometimes write\n-- import canonical_isomorphism.zfc_canonical_isomorphism\n-- open zfc\n-- could replace with \"import data.equiv\"\n\n-- TODO\n-- ask about function overloading with equiv\n\n/-\nCopyright (c) 2018 Kevin Buzzard.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor : Kevin Buzzard\n\nTwo types are said to be *canonically isomorphic* if a mathematician\ntrained in the dark arts of ZFC can replace one type by another without\nany fear of disruption.\n\nThis is an interesting notion, because it is \"one concept\" in mathematics, and\nyet seems to manifest itself in several different ways when one tries to formalise it\nin Lean's Dependent Type Theory.\n-/\n\n\n-- universe zfc_u -- no point doing this -- just use Type\n--#print extfun_app\n\n-- Here is a notion from dependent type theory.\n/-- `\u03b1 \u2243 \u03b2` is the type of functions from `\u03b1 \u2192 \u03b2` with a two-sided inverse. -/\n-- [Is that right? Is the inverse uniquely specified?]\n\n\nvariables {\u03b1 \u03b2 : Type}\n\n--#print distrib\n\n--class distrib (\u03b1 : Type u) extends has_mul \u03b1, has_add \u03b1 :=\n--(left_distrib : \u2200 a b c : \u03b1, a * (b + c) = (a * b) + (a * c))\n--(right_distrib : \u2200 a b c : \u03b1, (a + b) * c = (a * c) + (b * c))\n\n--#print has_mul \n--structure has_mul : Type u \u2192 Type u\n--fields:\n--has_mul.mul : \u03a0 {\u03b1 : Type u} [c : has_mul \u03b1], \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\n--#print equiv \n--structure zfc.equiv : Type zfc \u2192 Type zfc \u2192 Type zfc\n--fields:\n--zfc.equiv.to_fun : \u03a0 {\u03b1 \u03b2 : Type zfc}, equiv \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2\n--zfc.equiv.inv_fun : \u03a0 {\u03b1 \u03b2 : Type zfc}, equiv \u03b1 \u03b2 \u2192 \u03b2 \u2192 \u03b1\n--zfc.equiv.left_inv : \u2200 {\u03b1 \u03b2 : Type zfc} (c : equiv \u03b1 \u03b2), function.left_inverse (c.inv_fun) (c.to_fun)\n--zfc.equiv.right_inv : \u2200 {\u03b1 \u03b2 : Type zfc} (c : equiv \u03b1 \u03b2), function.right_inverse (c.inv_fun) (c.to_fun)\nstructure equiv (\u03b1 \u03b2 : Type) : Type :=\n(to_fun : \u03b1 \u2192 \u03b2)\n(inv_fun : \u03b2 \u2192 \u03b1)\n(left_inv : function.left_inverse inv_fun to_fun)\n(right_inv : function.right_inverse inv_fun to_fun)\n\n-- Was my definition of equiv OK?\nstructure inverse_exists (\u03b1 \u03b2 : Type) : Type :=\n(to_fun : \u03b1 \u2192 \u03b2)\n(inv_exists : \u2203 inv_fun : \u03b2 \u2192 \u03b1, function.left_inverse inv_fun to_fun \u2227\nfunction.right_inverse inv_fun to_fun)\n\ndefinition to_fun (\u03b1 \u03b2) : equiv \u03b1 \u03b2 \u2192 inverse_exists \u03b1 \u03b2 := \u03bb \u27e8f,g,hfg,hgf\u27e9,\u27e8f,\u27e8g,\u27e8hfg,hgf\u27e9\u27e9\u27e9\nnoncomputable definition inv_fun (\u03b1 \u03b2) : inverse_exists \u03b1 \u03b2 \u2192 equiv \u03b1 \u03b2 := \n\u03bb invfn,\n{\n    to_fun := inverse_exists.to_fun invfn,\n    inv_fun := classical.some invfn.inv_exists,\n    left_inv := (classical.some_spec invfn.inv_exists).1,-- : \u2200 (x : \u03b1), inv_fun_aux.val (to_fun x) = x),--begin unfold function.left_inverse, sorry end,\n    right_inv := (classical.some_spec invfn.inv_exists).2,\n  }\n\nnoncomputable theorem docstring_is_correct (\u03b1 \u03b2 : Type) : equiv (equiv \u03b1 \u03b2) (inverse_exists \u03b1 \u03b2) :=\n{to_fun := to_fun \u03b1 \u03b2,\n inv_fun := inv_fun \u03b1 \u03b2,\n left_inv := \u03bb x,begin cases x, unfold to_fun,unfold inv_fun,congr,\n  let mess := (classical.indefinite_description\n       (\u03bb (inv_fun : \u03b2 \u2192 \u03b1), function.left_inverse inv_fun x_to_fun \u2227 function.right_inverse inv_fun x_to_fun)\n       _),\n  funext b, \n  show mess.val b = x_inv_fun b,\n  have H := mess.property,\n  have idea : b = x_to_fun (x_inv_fun b),\n    exact (x_right_inv b).symm,\n  rw idea,\n  rw H.1 _,\n  rw x_left_inv,\n end,\n right_inv := begin\n   intro x,\n   cases x,refl,\n end \n}\n\n--#exit\n\n-- has_mul is a structure not a class\ndefinition equiv_mul {\u03b1 \u03b2 : Type} : equiv \u03b1 \u03b2 \u2192 equiv (has_mul \u03b1) (has_mul \u03b2) := \u03bb E,\n{ to_fun :=  \u03bb \u03b1mul,\u27e8\u03bb b1 b2, E.to_fun (@has_mul.mul \u03b1 \u03b1mul (E.inv_fun b1) (E.inv_fun b2))\u27e9,\n  inv_fun := \u03bb \u03b2mul,\u27e8\u03bb a1 a2, E.inv_fun (@has_mul.mul \u03b2 \u03b2mul (E.to_fun a1) (E.to_fun a2))\u27e9, -- didn't I just write that?\n                                                                                            -- should we introduce E-dual?\n  left_inv := \u03bb f, begin \n    unfold function.left_inverse,\n    cases f,\n    simp,\n    suffices : (\u03bb (a1 a2 : \u03b1), E.inv_fun (E.to_fun (f (E.inv_fun (E.to_fun a1)) (E.inv_fun (E.to_fun a2))))) = (\u03bb a1 a2, f a1 a2),\n      funext,\n      congr,\n      exact this,\n    funext,\n    rw [E.left_inv,E.left_inv,E.left_inv]\n   end,\n  right_inv := begin\n    intro m,\n    dsimp,\n    cases m,\n    congr,\n    funext,\n    rw [E.right_inv,E.right_inv,E.right_inv]\n  end \n}\n\n-- distrib is not a functor at all.\n--definition distrib_is_a_functor {\u03b1 \u03b2 : Type} : (\u03b1 \u2192 \u03b2) \u2192 (distrib \u03b1) \u2192 (distrib \u03b2) := sorry (can't be proved)\n\n-- The notions of equiv and isom might be \"identified\" in ZFC.\n-- In dependent type theory, they are same, but we can easily move between them.\n-- at least if we are happy to assume quot.sound. \n-- Several people have told me that in type theory, equiv is easier to work with than isom.\n--(variable v : \u03b1 \u2192 \u03b2)\n#print distrib \n\n-- I want my equiv to be better than theirs if one of my imports annoyingly imports data.equiv.\n-- I am over-importing madly \n/-- Kenny's definition -/\nstructure ring_equiv (\u03b1 : Type) (\u03b2 : Type) [comm_ring \u03b1] [comm_ring \u03b2] extends equiv \u03b1 \u03b2 :=\n(is_ring_hom : is_ring_hom to_fun)\n\n-- this is not even workable\ndefinition fun_to_ring : (\u03b1 \u2192 \u03b2) \u2192 (ring \u03b1 \u2192 ring \u03b2) := \u03bb f H\u03b1,\n{ -- this is the boring bit which needs to be automised\n  one := f H\u03b1.one,\n  one_mul :=  _,--by rw [] at H\u03b1.one_mul,\n\n}\n\n#exit \n\n#exit \n\ndefinition equiv_to_ring : equiv \u03b1 \u03b2 \u2192 equiv (ring \u03b1) (ring \u03b2) := \u03bb H,\n{ to_fun := \u03bb R,_,\n  inv_fun := _,\n  left_inv := _,\n  right_inv := _\n}\n#print is_ring_hom\n\n#check @set.image -- (\u03b1 \u2192 \u03b2) \u2192 set \u03b1 \u2192 set \u03b2\n\n-- instance : is_lawful_functor set :=\n\n#exit\n\n-- QUESTION. Using these three functions, defined in equiv I think,\n-- can I write down a map between (equiv a b) and (equiv (set a) (set b))\n-- and also a map between (equiv a b) and (equiv )\n\n--still haven't gfot this straight\n\nnamespace subtype\n\n/-- -- Restriction of a function to a function on subtypes. -/\n-- set to subtype\ndef map {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) (h : \u2200a, p a \u2192 q (f a)) :\n  subtype p \u2192 subtype q\n| \u27e8v, hv\u27e9 := \u27e8f v, h v hv\u27e9\n\n-- should be a functor: or doesn't that make sense?\n\n\ntheorem map_comp {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} {r : \u03b3 \u2192 Prop} {x : subtype p}\n  (f : \u03b1 \u2192 \u03b2) (h : \u2200a, p a \u2192 q (f a)) (g : \u03b2 \u2192 \u03b3) (l : \u2200a, q a \u2192 r (g a)) :\n  map g l (map f h x) = map (g \u2218 f) (assume a ha, l (f a) $ h a ha) x :=\nby cases x with v h; refl\n\ntheorem map_id {p : \u03b1 \u2192 Prop} {h : \u2200a, p a \u2192 p (id a)} : map (@id \u03b1) h = id :=\nfunext $ assume \u27e8v, h\u27e9, rfl\n\n-- is that the proof that map is a functor?\nend subtype\n\n#exit \n-- evil future plan\n-- structure canonically_isomorphic (\u03b1 : Type zfc) (\u03b2 : Type zfc) extends equiv \u03b1 \u03b2 :=\n--(\n\n\ntheorem set.image (X \u03b1 \u03b2 : Sort*) [H : equiv \u03b1 \u03b2] (f : X \u2192 \u03b1) : \nlet g : X \u2192 \u03b2 := H.to_fun \u2218 f in\n(set.image equiv.to_fun : set \u03b1 \u2192 set \u03b2) $ set.range f \n#print notation ''\n\nend equiv\n\n\n-- is this there?\ninstance group_of_equiv [group \u03b1] (H : equiv \u03b1 \u03b2) : group \u03b2 := sorry\n\ninstance set_equiv_of_equiv (H : equiv \u03b1 \u03b2) : equiv (set \u03b1) (set \u03b2) := sorry\n\n-- this is =\n#check @eq \u03b1 \u03b2\n#check @eq\n-- this is == but it doesn't typecheck -/\n--#check @heq \u03b1 \u03b2\n#print heq\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/canonical_isomorphism/can_iso_ideas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.39093559939609}}
{"text": "/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport ring_theory.int.basic\nimport field_theory.splitting_field\nimport ring_theory.localization.integral\nimport ring_theory.integrally_closed\n\n\n/-!\n# Gauss's Lemma\n\nGauss's Lemma is one of a few results pertaining to irreducibility of primitive polynomials.\n\n## Main Results\n - `polynomial.monic.irreducible_iff_irreducible_map_fraction_map`:\n  A monic polynomial over an integrally closed domain is irreducible iff it is irreducible in a\n    fraction field\n - `is_integrally_closed_iff'`:\n   Integrally closed domains are precisely the domains for in which Gauss's lemma holds\n    for monic polynomials\n - `polynomial.is_primitive.irreducible_iff_irreducible_map_fraction_map`:\n  A primitive polynomial over a GCD domain is irreducible iff it is irreducible in a fraction field\n - `polynomial.is_primitive.int.irreducible_iff_irreducible_map_cast`:\n  A primitive polynomial over `\u2124` is irreducible iff it is irreducible over `\u211a`.\n - `polynomial.is_primitive.dvd_iff_fraction_map_dvd_fraction_map`:\n  Two primitive polynomials over a GCD domain divide each other iff they do in a fraction field.\n - `polynomial.is_primitive.int.dvd_iff_map_cast_dvd_map_cast`:\n  Two primitive polynomials over `\u2124` divide each other if they do in `\u211a`.\n\n-/\n\nopen_locale non_zero_divisors polynomial\n\nvariables {R : Type*} [comm_ring R]\n\nnamespace polynomial\n\nsection\nvariables {S : Type*} [comm_ring S] [is_domain S]\nvariables {\u03c6 : R \u2192+* S} (hinj : function.injective \u03c6) {f : R[X]} (hf : f.is_primitive)\ninclude hinj hf\n\nlemma is_primitive.is_unit_iff_is_unit_map_of_injective :\n  is_unit f \u2194 is_unit (map \u03c6 f) :=\nbegin\n  refine \u27e8(map_ring_hom \u03c6).is_unit_map, \u03bb h, _\u27e9,\n  rcases is_unit_iff.1 h with \u27e8_, \u27e8u, rfl\u27e9, hu\u27e9,\n  have hdeg := degree_C u.ne_zero,\n  rw [hu, degree_map_eq_of_injective hinj] at hdeg,\n  rw [eq_C_of_degree_eq_zero hdeg] at hf \u22a2,\n  exact is_unit_C.mpr (is_primitive_iff_is_unit_of_C_dvd.mp hf (f.coeff 0) dvd_rfl),\nend\n\nlemma is_primitive.irreducible_of_irreducible_map_of_injective (h_irr : irreducible (map \u03c6 f)) :\n  irreducible f :=\nbegin\n  refine \u27e8\u03bb h, h_irr.not_unit (is_unit.map (map_ring_hom \u03c6) h),\n    \u03bb a b h, (h_irr.is_unit_or_is_unit $ by rw [h, polynomial.map_mul]).imp _ _\u27e9,\n  all_goals { apply ((is_primitive_of_dvd hf _).is_unit_iff_is_unit_map_of_injective hinj).mpr },\n  exacts [(dvd.intro _ h.symm), dvd.intro_left _ h.symm],\nend\n\nend\n\nsection fraction_map\n\nvariables {K : Type*} [field K] [algebra R K] [is_fraction_ring R K]\n\nlemma is_primitive.is_unit_iff_is_unit_map {p : R[X]} (hp : p.is_primitive) :\n  is_unit p \u2194 is_unit (p.map (algebra_map R K)) :=\nhp.is_unit_iff_is_unit_map_of_injective (is_fraction_ring.injective _ _)\n\nvariable [is_domain R]\n\nsection is_integrally_closed\n\nopen is_integrally_closed\n\n/-- **Gauss's Lemma** for integrally closed domains states that a monic polynomial is irreducible\n  iff it is irreducible in the fraction field. -/\n\n\n  obtain \u27e8a', ha\u27e9 := eq_map_mul_C_of_dvd K h (dvd_of_mul_right_eq b H.symm),\n  obtain \u27e8b', hb\u27e9 := eq_map_mul_C_of_dvd K h (dvd_of_mul_left_eq a H.symm),\n\n  have : a.leading_coeff * b.leading_coeff = 1,\n  { rw [\u2190 leading_coeff_mul, \u2190 H, monic.leading_coeff (h.map $ algebra_map R K)] },\n\n  rw [\u2190 ha, \u2190 hb, mul_comm _ (C b.leading_coeff), mul_assoc, \u2190 mul_assoc (C a.leading_coeff),\n    \u2190 C_mul, this, C_1, one_mul, \u2190 polynomial.map_mul] at H,\n  rw [\u2190 hb, \u2190 polynomial.coe_map_ring_hom],\n  refine is_unit.mul\n    (is_unit.map _ (or.resolve_left (hp.is_unit_or_is_unit _) (show \u00ac is_unit a', from _)))\n    (is_unit_iff_exists_inv'.mpr (exists.intro (C a.leading_coeff) $ by rwa [\u2190 C_mul, this, C_1])),\n  { exact polynomial.map_injective _ (is_fraction_ring.injective R K) H },\n\n  { by_contra h_contra,\n    refine h\u2090 _,\n    rw [\u2190 ha, \u2190 polynomial.coe_map_ring_hom],\n    exact is_unit.mul (is_unit.map _ h_contra) (is_unit_iff_exists_inv.mpr\n      (exists.intro (C b.leading_coeff) $ by rwa [\u2190 C_mul, this, C_1])) },\nend\n\n/-- Integrally closed domains are precisely the domains for in which Gauss's lemma holds\n    for monic polynomials -/\ntheorem is_integrally_closed_iff' : is_integrally_closed R \u2194\n  \u2200 p : R[X], p.monic \u2192 (irreducible p \u2194 irreducible (p.map $ algebra_map R K)) :=\nbegin\n  split,\n  { intros hR p hp, letI := hR, exact monic.irreducible_iff_irreducible_map_fraction_map hp },\n  { intro H,\n    refine (is_integrally_closed_iff K).mpr (\u03bb x hx, ring_hom.mem_range.mp $\n      minpoly.mem_range_of_degree_eq_one R x _),\n    rw \u2190 monic.degree_map (minpoly.monic hx) (algebra_map R K),\n    apply degree_eq_one_of_irreducible_of_root ((H _ $ minpoly.monic hx).mp\n      (minpoly.irreducible hx)),\n    rw [is_root, eval_map, \u2190 aeval_def, minpoly.aeval R x] },\nend\n\ntheorem monic.dvd_of_fraction_map_dvd_fraction_map [is_integrally_closed R] {p q : R[X]}\n  (hp : p.monic ) (hq : q.monic) (h : q.map (algebra_map R K) \u2223 p.map (algebra_map R K)) : q \u2223 p :=\nbegin\n  obtain \u27e8r, hr\u27e9 := h,\n  obtain \u27e8d', hr'\u27e9 := is_integrally_closed.eq_map_mul_C_of_dvd K hp (dvd_of_mul_left_eq _ hr.symm),\n  rw [monic.leading_coeff, C_1, mul_one] at hr',\n  rw [\u2190 hr', \u2190 polynomial.map_mul] at hr,\n  exact dvd_of_mul_right_eq _ (polynomial.map_injective _ (is_fraction_ring.injective R K) hr.symm),\n   { exact monic.of_mul_monic_left (hq.map (algebra_map R K)) (by simpa [\u2190hr] using hp.map _) },\nend\n\ntheorem monic.dvd_iff_fraction_map_dvd_fraction_map [is_integrally_closed R] {p q : R[X]}\n  (hp : p.monic ) (hq : q.monic) : q.map (algebra_map R K) \u2223 p.map (algebra_map R K) \u2194 q \u2223 p :=\n\u27e8\u03bb h, hp.dvd_of_fraction_map_dvd_fraction_map hq h,\n  \u03bb \u27e8a,b\u27e9, \u27e8a.map (algebra_map R K), b.symm \u25b8 polynomial.map_mul (algebra_map R K)\u27e9\u27e9\n\nend is_integrally_closed\n\nopen is_localization\n\nsection normalized_gcd_monoid\n\nvariable [normalized_gcd_monoid R]\n\nlemma is_unit_or_eq_zero_of_is_unit_integer_normalization_prim_part\n  {p : K[X]} (h0 : p \u2260 0) (h : is_unit (integer_normalization R\u2070 p).prim_part) :\n  is_unit p :=\nbegin\n  rcases is_unit_iff.1 h with \u27e8_, \u27e8u, rfl\u27e9, hu\u27e9,\n  obtain \u27e8\u27e8c, c0\u27e9, hc\u27e9 := integer_normalization_map_to_map R\u2070 p,\n  rw [subtype.coe_mk, algebra.smul_def, algebra_map_apply] at hc,\n  apply is_unit_of_mul_is_unit_right,\n  rw [\u2190 hc, (integer_normalization R\u2070 p).eq_C_content_mul_prim_part, \u2190 hu,\n    \u2190 ring_hom.map_mul, is_unit_iff],\n  refine \u27e8algebra_map R K ((integer_normalization R\u2070 p).content * \u2191u),\n    is_unit_iff_ne_zero.2 (\u03bb con, _), by simp\u27e9,\n  replace con := (injective_iff_map_eq_zero (algebra_map R K)).1\n    (is_fraction_ring.injective _ _) _ con,\n  rw [mul_eq_zero, content_eq_zero_iff, is_fraction_ring.integer_normalization_eq_zero_iff] at con,\n  rcases con with con | con,\n  { apply h0 con },\n  { apply units.ne_zero _ con },\nend\n\n/-- **Gauss's Lemma** for GCD domains states that a primitive polynomial is irreducible iff it is\n  irreducible in the fraction field. -/\ntheorem is_primitive.irreducible_iff_irreducible_map_fraction_map\n  {p : R[X]} (hp : p.is_primitive) :\n  irreducible p \u2194 irreducible (p.map (algebra_map R K)) :=\nbegin\n  refine \u27e8\u03bb hi, \u27e8\u03bb h, hi.not_unit (hp.is_unit_iff_is_unit_map.2 h), \u03bb a b hab, _\u27e9,\n    hp.irreducible_of_irreducible_map_of_injective (is_fraction_ring.injective _ _)\u27e9,\n  obtain \u27e8\u27e8c, c0\u27e9, hc\u27e9 := integer_normalization_map_to_map R\u2070 a,\n  obtain \u27e8\u27e8d, d0\u27e9, hd\u27e9 := integer_normalization_map_to_map R\u2070 b,\n  rw [algebra.smul_def, algebra_map_apply, subtype.coe_mk] at hc hd,\n  rw mem_non_zero_divisors_iff_ne_zero at c0 d0,\n  have hcd0 : c * d \u2260 0 := mul_ne_zero c0 d0,\n  rw [ne.def, \u2190 C_eq_zero] at hcd0,\n  have h1 : C c * C d * p = integer_normalization R\u2070 a * integer_normalization R\u2070 b,\n  { apply map_injective (algebra_map R K) (is_fraction_ring.injective _ _) _,\n    rw [polynomial.map_mul, polynomial.map_mul, polynomial.map_mul, hc, hd, map_C, map_C, hab],\n    ring },\n  obtain \u27e8u, hu\u27e9 : associated (c * d) (content (integer_normalization R\u2070 a) *\n            content (integer_normalization R\u2070 b)),\n  { rw [\u2190 dvd_dvd_iff_associated, \u2190 normalize_eq_normalize_iff, normalize.map_mul,\n        normalize.map_mul, normalize_content, normalize_content,\n        \u2190 mul_one (normalize c * normalize d), \u2190 hp.content_eq_one, \u2190 content_C, \u2190 content_C,\n        \u2190 content_mul, \u2190 content_mul, \u2190 content_mul, h1] },\n  rw [\u2190 ring_hom.map_mul, eq_comm,\n    (integer_normalization R\u2070 a).eq_C_content_mul_prim_part,\n    (integer_normalization R\u2070 b).eq_C_content_mul_prim_part, mul_assoc,\n    mul_comm _ (C _ * _), \u2190 mul_assoc, \u2190 mul_assoc, \u2190 ring_hom.map_mul, \u2190 hu, ring_hom.map_mul,\n    mul_assoc, mul_assoc, \u2190 mul_assoc (C \u2191u)] at h1,\n  have h0 : (a \u2260 0) \u2227 (b \u2260 0),\n  { classical,\n    rw [ne.def, ne.def, \u2190 decidable.not_or_iff_and_not, \u2190 mul_eq_zero, \u2190 hab],\n    intro con,\n    apply hp.ne_zero (map_injective (algebra_map R K) (is_fraction_ring.injective _ _) _),\n    simp [con] },\n  rcases hi.is_unit_or_is_unit (mul_left_cancel\u2080 hcd0 h1).symm with h | h,\n  { right,\n    apply is_unit_or_eq_zero_of_is_unit_integer_normalization_prim_part h0.2\n      (is_unit_of_mul_is_unit_right h) },\n  { left,\n    apply is_unit_or_eq_zero_of_is_unit_integer_normalization_prim_part h0.1 h },\nend\n\nlemma is_primitive.dvd_of_fraction_map_dvd_fraction_map {p q : R[X]}\n  (hp : p.is_primitive) (hq : q.is_primitive)\n  (h_dvd : p.map (algebra_map R K) \u2223 q.map (algebra_map R K)) : p \u2223 q :=\nbegin\n  rcases h_dvd with \u27e8r, hr\u27e9,\n  obtain \u27e8\u27e8s, s0\u27e9, hs\u27e9 := integer_normalization_map_to_map R\u2070 r,\n  rw [subtype.coe_mk, algebra.smul_def, algebra_map_apply] at hs,\n  have h : p \u2223 q * C s,\n  { use (integer_normalization R\u2070 r),\n    apply map_injective (algebra_map R K) (is_fraction_ring.injective _ _),\n    rw [polynomial.map_mul, polynomial.map_mul, hs, hr, mul_assoc, mul_comm r],\n    simp },\n  rw [\u2190 hp.dvd_prim_part_iff_dvd, prim_part_mul, hq.prim_part_eq,\n      associated.dvd_iff_dvd_right] at h,\n  { exact h },\n  { symmetry,\n    rcases is_unit_prim_part_C s with \u27e8u, hu\u27e9,\n    use u,\n    rw hu },\n  iterate 2 { apply mul_ne_zero hq.ne_zero,\n    rw [ne.def, C_eq_zero],\n    contrapose! s0,\n    simp [s0, mem_non_zero_divisors_iff_ne_zero] }\nend\n\nvariables (K)\n\nlemma is_primitive.dvd_iff_fraction_map_dvd_fraction_map {p q : R[X]}\n  (hp : p.is_primitive) (hq : q.is_primitive) :\n  (p \u2223 q) \u2194 (p.map (algebra_map R K) \u2223 q.map (algebra_map R K)) :=\n\u27e8\u03bb \u27e8a,b\u27e9, \u27e8a.map (algebra_map R K), b.symm \u25b8 polynomial.map_mul (algebra_map R K)\u27e9,\n  \u03bb h, hp.dvd_of_fraction_map_dvd_fraction_map hq h\u27e9\n\nend normalized_gcd_monoid\n\nend fraction_map\n\n/-- **Gauss's Lemma** for `\u2124` states that a primitive integer polynomial is irreducible iff it is\n  irreducible over `\u211a`. -/\ntheorem is_primitive.int.irreducible_iff_irreducible_map_cast\n  {p : \u2124[X]} (hp : p.is_primitive) :\n  irreducible p \u2194 irreducible (p.map (int.cast_ring_hom \u211a)) :=\nhp.irreducible_iff_irreducible_map_fraction_map\n\nlemma is_primitive.int.dvd_iff_map_cast_dvd_map_cast (p q : \u2124[X])\n  (hp : p.is_primitive) (hq : q.is_primitive) :\n  (p \u2223 q) \u2194 (p.map (int.cast_ring_hom \u211a) \u2223 q.map (int.cast_ring_hom \u211a)) :=\nhp.dvd_iff_fraction_map_dvd_fraction_map \u211a hq\n\nend polynomial\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/ring_theory/polynomial/gauss_lemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3908901229895362}}
{"text": "/-\nCopyright (c) 2021 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.epi_mono\nimport category_theory.limits.has_limits\nimport category_theory.limits.shapes.equalizers\n\n/-!\n# Wide equalizers and wide coequalizers\n\nThis file defines wide (co)equalizers as special cases of (co)limits.\n\nA wide equalizer for the family of morphisms `X \u27f6 Y` indexed by `J` is the categorical\ngeneralization of the subobject `{a \u2208 A | \u2200 j\u2081 j\u2082, f(j\u2081, a) = f(j\u2082, a)}`. Note that if `J` has\nfewer than two morphisms this condition is trivial, so some lemmas and definitions assume `J` is\nnonempty.\n\n## Main definitions\n\n* `walking_parallel_family` is the indexing category used for wide (co)equalizer diagrams\n* `parallel_family` is a functor from `walking_parallel_family` to our category `C`.\n* a `trident` is a cone over a parallel family.\n  * there is really only one interesting morphism in a trident: the arrow from the vertex of the\n    trident to the domain of f and g. It is called `trident.\u03b9`.\n* a `wide_equalizer` is now just a `limit (parallel_family f)`\n\nEach of these has a dual.\n\n## Main statements\n\n* `wide_equalizer.\u03b9_mono` states that every wide_equalizer map is a monomorphism\n* `is_iso_limit_cone_parallel_family_of_self` states that the identity on the domain of `f` is an\n  equalizer of `f` and `f`.\n\n## Implementation notes\nAs with the other special shapes in the limits library, all the definitions here are given as\n`abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about\ngeneral limits can be used.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 1*][borceux-vol1]\n-/\n\nnoncomputable theory\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses v u u\u2082\n\nvariables {J : Type v}\n\n/-- The type of objects for the diagram indexing a wide (co)equalizer. -/\ninductive walking_parallel_family (J : Type v) : Type v\n| zero : walking_parallel_family\n| one : walking_parallel_family\n\nopen walking_parallel_family\n\ninstance : decidable_eq (walking_parallel_family J)\n| zero zero := is_true rfl\n| zero one := is_false (\u03bb t, walking_parallel_family.no_confusion t)\n| one zero := is_false (\u03bb t, walking_parallel_family.no_confusion t)\n| one one := is_true rfl\n\ninstance : inhabited (walking_parallel_family J) := \u27e8zero\u27e9\n\n/-- The type family of morphisms for the diagram indexing a wide (co)equalizer. -/\n@[derive decidable_eq] inductive walking_parallel_family.hom (J : Type v) :\n  walking_parallel_family J \u2192 walking_parallel_family J \u2192 Type v\n| id : \u03a0 X : walking_parallel_family.{v} J, walking_parallel_family.hom X X\n| line : \u03a0 (j : J), walking_parallel_family.hom zero one\n\n/-- Satisfying the inhabited linter -/\ninstance (J : Type v) : inhabited (walking_parallel_family.hom J zero zero) :=\n{ default := hom.id _ }\n\nopen walking_parallel_family.hom\n\n/-- Composition of morphisms in the indexing diagram for wide (co)equalizers. -/\ndef walking_parallel_family.hom.comp :\n  \u03a0 (X Y Z : walking_parallel_family J)\n    (f : walking_parallel_family.hom J X Y) (g : walking_parallel_family.hom J Y Z),\n    walking_parallel_family.hom J X Z\n  | _ _ _ (id _)   h := h\n  | _ _ _ (line j) (id one) := line j.\n\nlocal attribute [tidy] tactic.case_bash\n\ninstance walking_parallel_family.category : small_category (walking_parallel_family J) :=\n{ hom  := walking_parallel_family.hom J,\n  id   := walking_parallel_family.hom.id,\n  comp := walking_parallel_family.hom.comp }\n\n@[simp]\nlemma walking_parallel_family.hom_id (X : walking_parallel_family J) :\n  walking_parallel_family.hom.id X = \ud835\udfd9 X :=\nrfl\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y : C} (f : J \u2192 (X \u27f6 Y))\n\n/--\n`parallel_family f` is the diagram in `C` consisting of the given family of morphisms, each with\ncommon domain and codomain.\n-/\ndef parallel_family : walking_parallel_family J \u2964 C :=\n{ obj := \u03bb x, walking_parallel_family.cases_on x X Y,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, (line j) := f j\n  end,\n  map_comp' :=\n  begin\n    rintro _ _ _ \u27e8\u27e9 \u27e8\u27e9;\n    { unfold_aux, simp; refl },\n  end }\n\n@[simp] lemma parallel_family_obj_zero : (parallel_family f).obj zero = X := rfl\n@[simp] lemma parallel_family_obj_one : (parallel_family f).obj one = Y := rfl\n\n@[simp] lemma parallel_family_map_left {j : J} : (parallel_family f).map (line j) = f j := rfl\n\n/-- Every functor indexing a wide (co)equalizer is naturally isomorphic (actually, equal) to a\n    `parallel_family` -/\n@[simps]\ndef diagram_iso_parallel_family (F : walking_parallel_family J \u2964 C) :\n  F \u2245 parallel_family (\u03bb j, F.map (line j)) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by cases j; tidy) $ by tidy\n\n/-- `walking_parallel_pair` as a category is equivalent to a special case of\n`walking_parallel_family`.  -/\n@[simps]\ndef walking_parallel_family_equiv_walking_parallel_pair :\n  walking_parallel_family.{v} (ulift bool) \u224c walking_parallel_pair.{v} :=\n{ functor := parallel_family\n      (\u03bb p, cond p.down walking_parallel_pair_hom.left walking_parallel_pair_hom.right),\n  inverse := parallel_pair (line (ulift.up tt)) (line (ulift.up ff)),\n  unit_iso := nat_iso.of_components (\u03bb X, eq_to_iso (by cases X; refl)) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb X, eq_to_iso (by cases X; refl)) (by tidy) }\n\n/-- A trident on `f` is just a `cone (parallel_family f)`. -/\nabbreviation trident := cone (parallel_family f)\n\n/-- A cotrident on `f` and `g` is just a `cocone (parallel_family f)`. -/\nabbreviation cotrident := cocone (parallel_family f)\n\nvariables {f}\n\n/-- A trident `t` on the parallel family `f : J \u2192 (X \u27f6 Y)` consists of two morphisms\n    `t.\u03c0.app zero : t.X \u27f6 X` and `t.\u03c0.app one : t.X \u27f6 Y`. Of these, only the first one is\n    interesting, and we give it the shorter name `trident.\u03b9 t`. -/\nabbreviation trident.\u03b9 (t : trident f) := t.\u03c0.app zero\n\n/-- A cotrident `t` on the parallel family `f : J \u2192 (X \u27f6 Y)` consists of two morphisms\n    `t.\u03b9.app zero : X \u27f6 t.X` and `t.\u03b9.app one : Y \u27f6 t.X`. Of these, only the second one is\n    interesting, and we give it the shorter name `cotrident.\u03c0 t`. -/\nabbreviation cotrident.\u03c0 (t : cotrident f) := t.\u03b9.app one\n\n@[simp] lemma trident.\u03b9_eq_app_zero (t : trident f) : t.\u03b9 = t.\u03c0.app zero := rfl\n@[simp] lemma cotrident.\u03c0_eq_app_one (t : cotrident f) : t.\u03c0 = t.\u03b9.app one := rfl\n\n@[simp, reassoc] lemma trident.app_zero (s : trident f) (j : J) :\n  s.\u03c0.app zero \u226b f j = s.\u03c0.app one :=\nby rw [\u2190s.w (line j), parallel_family_map_left]\n\n@[simp, reassoc] lemma cotrident.app_one (s : cotrident f) (j : J) :\n  f j \u226b s.\u03b9.app one = s.\u03b9.app zero :=\nby rw [\u2190s.w (line j), parallel_family_map_left]\n\n/--\nA trident on `f : J \u2192 (X \u27f6 Y)` is determined by the morphism `\u03b9 : P \u27f6 X` satisfying\n`\u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082`.\n-/\n@[simps]\ndef trident.of_\u03b9 [nonempty J] {P : C} (\u03b9 : P \u27f6 X) (w : \u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082) :\n  trident f :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb X, walking_parallel_family.cases_on X \u03b9 (\u03b9 \u226b f (classical.arbitrary J)),\n    naturality' := \u03bb i j f,\n      begin\n        dsimp,\n        cases f with _ k,\n        { simp },\n        { simp [w (classical.arbitrary J) k] },\n      end } }\n\n/--\nA cotrident on `f : J \u2192 (X \u27f6 Y)` is determined by the morphism `\u03c0 : Y \u27f6 P` satisfying\n`\u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0`.\n-/\n@[simps]\ndef cotrident.of_\u03c0 [nonempty J] {P : C} (\u03c0 : Y \u27f6 P) (w : \u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0) :\n  cotrident f :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb X, walking_parallel_family.cases_on X (f (classical.arbitrary J) \u226b \u03c0) \u03c0,\n    naturality' := \u03bb i j f,\n      begin\n        dsimp,\n        cases f with _ k,\n        { simp },\n        { simp [w (classical.arbitrary J) k] }\n      end } } -- See note [dsimp, simp]\n\nlemma trident.\u03b9_of_\u03b9 [nonempty J] {P : C} (\u03b9 : P \u27f6 X) (w : \u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082) :\n  (trident.of_\u03b9 \u03b9 w).\u03b9 = \u03b9 := rfl\nlemma cotrident.\u03c0_of_\u03c0 [nonempty J] {P : C} (\u03c0 : Y \u27f6 P) (w : \u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0) :\n  (cotrident.of_\u03c0 \u03c0 w).\u03c0 = \u03c0 := rfl\n\n@[reassoc]\nlemma trident.condition (j\u2081 j\u2082 : J) (t : trident f) : t.\u03b9 \u226b f j\u2081 = t.\u03b9 \u226b f j\u2082 :=\nby rw [t.app_zero, t.app_zero]\n\n@[reassoc]\nlemma cotrident.condition (j\u2081 j\u2082 : J) (t : cotrident f) : f j\u2081 \u226b t.\u03c0 = f j\u2082 \u226b t.\u03c0 :=\nby rw [t.app_one, t.app_one]\n\n/-- To check whether two maps are equalized by both maps of a trident, it suffices to check it for\nthe first map -/\nlemma trident.equalizer_ext [nonempty J] (s : trident f) {W : C} {k l : W \u27f6 s.X}\n  (h : k \u226b s.\u03b9 = l \u226b s.\u03b9) : \u2200 (j : walking_parallel_family J),\n    k \u226b s.\u03c0.app j = l \u226b s.\u03c0.app j\n| zero := h\n| one := by rw [\u2190s.app_zero (classical.arbitrary J), reassoc_of h]\n\n/-- To check whether two maps are coequalized by both maps of a cotrident, it suffices to check it\nfor the second map -/\nlemma cotrident.coequalizer_ext [nonempty J] (s : cotrident f) {W : C} {k l : s.X \u27f6 W}\n  (h : s.\u03c0 \u226b k = s.\u03c0 \u226b l) : \u2200 (j : walking_parallel_family J),\n    s.\u03b9.app j \u226b k = s.\u03b9.app j \u226b l\n| zero := by rw [\u2190s.app_one (classical.arbitrary J), category.assoc, category.assoc, h]\n| one := h\n\nlemma trident.is_limit.hom_ext [nonempty J] {s : trident f} (hs : is_limit s)\n  {W : C} {k l : W \u27f6 s.X} (h : k \u226b s.\u03b9 = l \u226b s.\u03b9) :\n  k = l :=\nhs.hom_ext $ trident.equalizer_ext _ h\n\nlemma cotrident.is_colimit.hom_ext [nonempty J] {s : cotrident f} (hs : is_colimit s)\n  {W : C} {k l : s.X \u27f6 W} (h : s.\u03c0 \u226b k = s.\u03c0 \u226b l) :\n  k = l :=\nhs.hom_ext $ cotrident.coequalizer_ext _ h\n\n/-- If `s` is a limit trident over `f`, then a morphism `k : W \u27f6 X` satisfying\n    `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` induces a morphism `l : W \u27f6 s.X` such that\n    `l \u226b trident.\u03b9 s = k`. -/\ndef trident.is_limit.lift' [nonempty J] {s : trident f} (hs : is_limit s) {W : C} (k : W \u27f6 X)\n  (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  {l : W \u27f6 s.X // l \u226b trident.\u03b9 s = k} :=\n\u27e8hs.lift $ trident.of_\u03b9 _ h, hs.fac _ _\u27e9\n\n/-- If `s` is a colimit cotrident over `f`, then a morphism `k : Y \u27f6 W` satisfying\n    `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` induces a morphism `l : s.X \u27f6 W` such that\n    `cotrident.\u03c0 s \u226b l = k`. -/\ndef cotrident.is_colimit.desc' [nonempty J] {s : cotrident f} (hs : is_colimit s) {W : C}\n  (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  {l : s.X \u27f6 W // cotrident.\u03c0 s \u226b l = k} :=\n\u27e8hs.desc $ cotrident.of_\u03c0 _ h, hs.fac _ _\u27e9\n\n/-- This is a slightly more convenient method to verify that a trident is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef trident.is_limit.mk [nonempty J] (t : trident f)\n  (lift : \u03a0 (s : trident f), s.X \u27f6 t.X)\n  (fac : \u2200 (s : trident f), lift s \u226b t.\u03b9 = s.\u03b9)\n  (uniq : \u2200 (s : trident f) (m : s.X \u27f6 t.X)\n  (w : \u2200 j : walking_parallel_family J, m \u226b t.\u03c0.app j = s.\u03c0.app j), m = lift s) :\n  is_limit t :=\n{ lift := lift,\n  fac' := \u03bb s j, walking_parallel_family.cases_on j (fac s)\n    (by rw [\u2190t.w (line (classical.arbitrary J)), reassoc_of fac, s.w]),\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a trident is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef trident.is_limit.mk' [nonempty J] (t : trident f)\n  (create : \u03a0 (s : trident f), {l // l \u226b t.\u03b9 = s.\u03b9 \u2227 \u2200 {m}, m \u226b t.\u03b9 = s.\u03b9 \u2192 m = l}) :\nis_limit t :=\ntrident.is_limit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w zero))\n\n/-- This is a slightly more convenient method to verify that a cotrident is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef cotrident.is_colimit.mk [nonempty J] (t : cotrident f)\n  (desc : \u03a0 (s : cotrident f), t.X \u27f6 s.X)\n  (fac : \u2200 (s : cotrident f), t.\u03c0 \u226b desc s = s.\u03c0)\n  (uniq : \u2200 (s : cotrident f) (m : t.X \u27f6 s.X)\n  (w : \u2200 j : walking_parallel_family J, t.\u03b9.app j \u226b m = s.\u03b9.app j), m = desc s) :\n  is_colimit t :=\n{ desc := desc,\n  fac' := \u03bb s j, walking_parallel_family.cases_on j\n    (by rw [\u2190t.w_assoc (line (classical.arbitrary J)), fac, s.w]) (fac s),\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a cotrident is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef cotrident.is_colimit.mk' [nonempty J] (t : cotrident f)\n  (create : \u03a0 (s : cotrident f), {l : t.X \u27f6 s.X // t.\u03c0 \u226b l = s.\u03c0 \u2227 \u2200 {m}, t.\u03c0 \u226b m = s.\u03c0 \u2192 m = l}) :\n  is_colimit t :=\ncotrident.is_colimit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w one))\n\n/--\nGiven a limit cone for the family `f : J \u2192 (X \u27f6 Y)`, for any `Z`, morphisms from `Z` to its point\nare in bijection with morphisms `h : Z \u27f6 X` such that `\u2200 j\u2081 j\u2082, h \u226b f j\u2081 = h \u226b f j\u2082`.\nFurther, this bijection is natural in `Z`: see `trident.is_limit.hom_iso_natural`.\n-/\n@[simps]\ndef trident.is_limit.hom_iso [nonempty J] {t : trident f} (ht : is_limit t) (Z : C) :\n  (Z \u27f6 t.X) \u2243 {h : Z \u27f6 X // \u2200 j\u2081 j\u2082, h \u226b f j\u2081 = h \u226b f j\u2082} :=\n{ to_fun := \u03bb k, \u27e8k \u226b t.\u03b9, by simp\u27e9,\n  inv_fun := \u03bb h, (trident.is_limit.lift' ht _ h.prop).1,\n  left_inv := \u03bb k, trident.is_limit.hom_ext ht (trident.is_limit.lift' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (trident.is_limit.lift' ht _ _).prop }\n\n/-- The bijection of `trident.is_limit.hom_iso` is natural in `Z`. -/\nlemma trident.is_limit.hom_iso_natural [nonempty J] {t : trident f} (ht : is_limit t)\n  {Z Z' : C} (q : Z' \u27f6 Z) (k : Z \u27f6 t.X) :\n  (trident.is_limit.hom_iso ht _ (q \u226b k) : Z' \u27f6 X) =\n  q \u226b (trident.is_limit.hom_iso ht _ k : Z \u27f6 X) :=\ncategory.assoc _ _ _\n\n/--\nGiven a colimit cocone for the family `f : J \u2192 (X \u27f6 Y)`, for any `Z`, morphisms from the cocone\npoint to `Z` are in bijection with morphisms `h : Z \u27f6 X` such that\n`\u2200 j\u2081 j\u2082, f j\u2081 \u226b h = f j\u2082 \u226b h`.  Further, this bijection is natural in `Z`: see\n`cotrident.is_colimit.hom_iso_natural`.\n-/\n@[simps]\ndef cotrident.is_colimit.hom_iso [nonempty J] {t : cotrident f} (ht : is_colimit t) (Z : C) :\n  (t.X \u27f6 Z) \u2243 {h : Y \u27f6 Z // \u2200 j\u2081 j\u2082, f j\u2081 \u226b h = f j\u2082 \u226b h} :=\n{ to_fun := \u03bb k, \u27e8t.\u03c0 \u226b k, by simp\u27e9,\n  inv_fun := \u03bb h, (cotrident.is_colimit.desc' ht _ h.prop).1,\n  left_inv := \u03bb k, cotrident.is_colimit.hom_ext ht (cotrident.is_colimit.desc' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (cotrident.is_colimit.desc' ht _ _).prop }\n\n/-- The bijection of `cotrident.is_colimit.hom_iso` is natural in `Z`. -/\nlemma cotrident.is_colimit.hom_iso_natural [nonempty J] {t : cotrident f} {Z Z' : C}\n  (q : Z \u27f6 Z') (ht : is_colimit t) (k : t.X \u27f6 Z) :\n    (cotrident.is_colimit.hom_iso ht _ (k \u226b q) : Y \u27f6 Z') =\n    (cotrident.is_colimit.hom_iso ht _ k : Y \u27f6 Z) \u226b q :=\n(category.assoc _ _ _).symm\n\n/-- This is a helper construction that can be useful when verifying that a category has certain wide\n    equalizers. Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))`, and a trident on `\u03bb j, F.map (line j)`, we get a cone\n    on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_wide_equalizers_of_has_limit_parallel_family`, which you may find to be an easier way of\n    achieving your goal. -/\ndef cone.of_trident\n  {F : walking_parallel_family J \u2964 C} (t : trident (\u03bb j, F.map (line j))) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g, by { cases g; { dsimp, simp } } } }\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    coequalizers. Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))`, and a cotrident on `\u03bb j, F.map (line j)` we get a\n    cocone on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_wide_coequalizers_of_has_colimit_parallel_family`, which you may find to be an easier way\n    of achieving your goal. -/\ndef cocone.of_cotrident\n  {F : walking_parallel_family J \u2964 C} (t : cotrident (\u03bb j, F.map (line j))) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g, by { cases g; dsimp; simp [cotrident.app_one t] } } }\n\n@[simp] lemma cone.of_trident_\u03c0\n  {F : walking_parallel_family J \u2964 C} (t : trident (\u03bb j, F.map (line j))) (j) :\n  (cone.of_trident t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\n@[simp] lemma cocone.of_cotrident_\u03b9\n  {F : walking_parallel_family J \u2964 C} (t : cotrident (\u03bb j, F.map (line j))) (j) :\n  (cocone.of_cotrident t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/-- Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))` and a cone on `F`, we get a trident on\n    `\u03bb j, F.map (line j)`. -/\ndef trident.of_cone\n  {F : walking_parallel_family J \u2964 C} (t : cone F) : trident (\u03bb j, F.map (line j)) :=\n{ X := t.X,\n  \u03c0 := { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy) } }\n\n/-- Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (F.map left) (F.map right)` and a cocone on `F`, we get a cotrident on\n    `\u03bb j, F.map (line j)`. -/\ndef cotrident.of_cocone\n  {F : walking_parallel_family J \u2964 C} (t : cocone F) : cotrident (\u03bb j, F.map (line j)) :=\n{ X := t.X,\n  \u03b9 := { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X } }\n\n@[simp] lemma trident.of_cone_\u03c0 {F : walking_parallel_family J \u2964 C} (t : cone F) (j) :\n  (trident.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n@[simp] lemma cotrident.of_cocone_\u03b9 {F : walking_parallel_family J \u2964 C} (t : cocone F) (j) :\n  (cotrident.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/--\nHelper function for constructing morphisms between wide equalizer tridents.\n-/\n@[simps]\ndef trident.mk_hom [nonempty J] {s t : trident f} (k : s.X \u27f6 t.X) (w : k \u226b t.\u03b9 = s.\u03b9) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { exact w },\n    { simpa using w =\u226b f (classical.arbitrary J) },\n  end }\n\n/--\nTo construct an isomorphism between tridents,\nit suffices to give an isomorphism between the cone points\nand check that it commutes with the `\u03b9` morphisms.\n-/\n@[simps]\ndef trident.ext [nonempty J] {s t : trident f} (i : s.X \u2245 t.X) (w : i.hom \u226b t.\u03b9 = s.\u03b9) : s \u2245 t :=\n{ hom := trident.mk_hom i.hom w,\n  inv := trident.mk_hom i.inv (by rw [\u2190 w, iso.inv_hom_id_assoc]) }\n\n/--\nHelper function for constructing morphisms between coequalizer cotridents.\n-/\n@[simps]\ndef cotrident.mk_hom [nonempty J] {s t : cotrident f} (k : s.X \u27f6 t.X) (w : s.\u03c0 \u226b k = t.\u03c0) :\n  s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { simpa using f (classical.arbitrary J) \u226b= w },\n    { exact w },\n  end }\n\n/--\nTo construct an isomorphism between cotridents,\nit suffices to give an isomorphism between the cocone points\nand check that it commutes with the `\u03c0` morphisms.\n-/\ndef cotrident.ext [nonempty J] {s t : cotrident f} (i : s.X \u2245 t.X) (w : s.\u03c0 \u226b i.hom = t.\u03c0) :\n  s \u2245 t :=\n{ hom := cotrident.mk_hom i.hom w,\n  inv := cotrident.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) }\n\nvariables (f)\n\nsection\n/--\n`has_wide_equalizer f` represents a particular choice of limiting cone for the parallel family of\nmorphisms `f`.\n-/\nabbreviation has_wide_equalizer := has_limit (parallel_family f)\n\nvariables [has_wide_equalizer f]\n\n/-- If a wide equalizer of `f` exists, we can access an arbitrary choice of such by\n    saying `wide_equalizer f`. -/\nabbreviation wide_equalizer : C := limit (parallel_family f)\n\n/-- If a wide equalizer of `f` exists, we can access the inclusion `wide_equalizer f \u27f6 X` by\n    saying `wide_equalizer.\u03b9 f`. -/\nabbreviation wide_equalizer.\u03b9 : wide_equalizer f \u27f6 X :=\nlimit.\u03c0 (parallel_family f) zero\n\n/--\nA wide equalizer cone for a parallel family `f`.\n-/\nabbreviation wide_equalizer.trident : trident f := limit.cone (parallel_family f)\n\n@[simp] lemma wide_equalizer.trident_\u03b9 :\n  (wide_equalizer.trident f).\u03b9 = wide_equalizer.\u03b9 f := rfl\n\n@[simp] lemma wide_equalizer.trident_\u03c0_app_zero :\n  (wide_equalizer.trident f).\u03c0.app zero = wide_equalizer.\u03b9 f := rfl\n\n@[reassoc] lemma wide_equalizer.condition (j\u2081 j\u2082 : J) :\n  wide_equalizer.\u03b9 f \u226b f j\u2081 = wide_equalizer.\u03b9 f \u226b f j\u2082 :=\ntrident.condition j\u2081 j\u2082 $ limit.cone $ parallel_family f\n\n/-- The wide_equalizer built from `wide_equalizer.\u03b9 f` is limiting. -/\ndef wide_equalizer_is_wide_equalizer [nonempty J] :\n  is_limit (trident.of_\u03b9 (wide_equalizer.\u03b9 f) (wide_equalizer.condition f)) :=\nis_limit.of_iso_limit (limit.is_limit _) (trident.ext (iso.refl _) (by tidy))\n\nvariables {f}\n\n/-- A morphism `k : W \u27f6 X` satisfying `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` factors through the\n    wide equalizer of `f` via `wide_equalizer.lift : W \u27f6 wide_equalizer f`. -/\nabbreviation wide_equalizer.lift [nonempty J] {W : C} (k : W \u27f6 X)\n  (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  W \u27f6 wide_equalizer f :=\nlimit.lift (parallel_family f) (trident.of_\u03b9 k h)\n\n@[simp, reassoc]\nlemma wide_equalizer.lift_\u03b9 [nonempty J] {W : C} (k : W \u27f6 X) (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  wide_equalizer.lift k h \u226b wide_equalizer.\u03b9 f = k :=\nlimit.lift_\u03c0 _ _\n\n/-- A morphism `k : W \u27f6 X` satisfying `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` induces a morphism\n    `l : W \u27f6 wide_equalizer f` satisfying `l \u226b wide_equalizer.\u03b9 f = k`. -/\ndef wide_equalizer.lift' [nonempty J] {W : C} (k : W \u27f6 X) (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  {l : W \u27f6 wide_equalizer f // l \u226b wide_equalizer.\u03b9 f = k} :=\n\u27e8wide_equalizer.lift k h, wide_equalizer.lift_\u03b9 _ _\u27e9\n\n/-- Two maps into a wide equalizer are equal if they are are equal when composed with the wide\n    equalizer map. -/\n@[ext] lemma wide_equalizer.hom_ext [nonempty J] {W : C} {k l : W \u27f6 wide_equalizer f}\n  (h : k \u226b wide_equalizer.\u03b9 f = l \u226b wide_equalizer.\u03b9 f) : k = l :=\ntrident.is_limit.hom_ext (limit.is_limit _) h\n\n/-- A wide equalizer morphism is a monomorphism -/\ninstance wide_equalizer.\u03b9_mono [nonempty J] : mono (wide_equalizer.\u03b9 f) :=\n{ right_cancellation := \u03bb Z h k w, wide_equalizer.hom_ext w }\n\nend\n\nsection\nvariables {f}\n/-- The wide equalizer morphism in any limit cone is a monomorphism. -/\nlemma mono_of_is_limit_parallel_family [nonempty J] {c : cone (parallel_family f)}\n  (i : is_limit c) :\n  mono (trident.\u03b9 c) :=\n{ right_cancellation := \u03bb Z h k w, trident.is_limit.hom_ext i w }\n\nend\n\nsection\n/--\n`has_wide_coequalizer f g` represents a particular choice of colimiting cocone\nfor the parallel family of morphisms `f`.\n-/\nabbreviation has_wide_coequalizer := has_colimit (parallel_family f)\n\nvariables [has_wide_coequalizer f]\n\n/-- If a wide coequalizer of `f`, we can access an arbitrary choice of such by\n    saying `wide_coequalizer f`. -/\nabbreviation wide_coequalizer : C := colimit (parallel_family f)\n\n/--  If a wide_coequalizer of `f` exists, we can access the corresponding projection by\n    saying `wide_coequalizer.\u03c0 f`. -/\nabbreviation wide_coequalizer.\u03c0 : Y \u27f6 wide_coequalizer f :=\ncolimit.\u03b9 (parallel_family f) one\n\n/--\nAn arbitrary choice of coequalizer cocone for a parallel family `f`.\n-/\nabbreviation wide_coequalizer.cotrident : cotrident f := colimit.cocone (parallel_family f)\n\n@[simp] lemma wide_coequalizer.cotrident_\u03c0 :\n  (wide_coequalizer.cotrident f).\u03c0 = wide_coequalizer.\u03c0 f := rfl\n\n@[simp] lemma wide_coequalizer.cotrident_\u03b9_app_one :\n  (wide_coequalizer.cotrident f).\u03b9.app one = wide_coequalizer.\u03c0 f := rfl\n\n@[reassoc] lemma wide_coequalizer.condition (j\u2081 j\u2082 : J) :\n  f j\u2081 \u226b wide_coequalizer.\u03c0 f = f j\u2082 \u226b wide_coequalizer.\u03c0 f :=\ncotrident.condition j\u2081 j\u2082 $ colimit.cocone $ parallel_family f\n\n/-- The cotrident built from `wide_coequalizer.\u03c0 f` is colimiting. -/\ndef wide_coequalizer_is_wide_coequalizer [nonempty J] :\n  is_colimit (cotrident.of_\u03c0 (wide_coequalizer.\u03c0 f) (wide_coequalizer.condition f)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit _) (cotrident.ext (iso.refl _) (by tidy))\n\nvariables {f}\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` factors through the\n    wide coequalizer of `f` via `wide_coequalizer.desc : wide_coequalizer f \u27f6 W`. -/\nabbreviation wide_coequalizer.desc [nonempty J] {W : C} (k : Y \u27f6 W)\n  (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  wide_coequalizer f \u27f6 W :=\ncolimit.desc (parallel_family f) (cotrident.of_\u03c0 k h)\n\n@[simp, reassoc]\nlemma wide_coequalizer.\u03c0_desc [nonempty J] {W : C} (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  wide_coequalizer.\u03c0 f \u226b wide_coequalizer.desc k h = k :=\ncolimit.\u03b9_desc _ _\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` induces a morphism\n    `l : wide_coequalizer f \u27f6 W` satisfying `wide_coequalizer.\u03c0 \u226b g = l`. -/\ndef wide_coequalizer.desc' [nonempty J] {W : C} (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  {l : wide_coequalizer f \u27f6 W // wide_coequalizer.\u03c0 f \u226b l = k} :=\n\u27e8wide_coequalizer.desc k h, wide_coequalizer.\u03c0_desc _ _\u27e9\n\n/-- Two maps from a wide coequalizer are equal if they are equal when composed with the wide\n    coequalizer map -/\n@[ext] lemma wide_coequalizer.hom_ext [nonempty J] {W : C} {k l : wide_coequalizer f \u27f6 W}\n  (h : wide_coequalizer.\u03c0 f \u226b k = wide_coequalizer.\u03c0 f \u226b l) : k = l :=\ncotrident.is_colimit.hom_ext (colimit.is_colimit _) h\n\n/-- A wide coequalizer morphism is an epimorphism -/\ninstance wide_coequalizer.\u03c0_epi [nonempty J] : epi (wide_coequalizer.\u03c0 f) :=\n{ left_cancellation := \u03bb Z h k w, wide_coequalizer.hom_ext w }\n\nend\n\nsection\nvariables {f}\n\n/-- The wide coequalizer morphism in any colimit cocone is an epimorphism. -/\nlemma epi_of_is_colimit_parallel_family [nonempty J] {c : cocone (parallel_family f)}\n  (i : is_colimit c) :\n  epi (c.\u03b9.app one) :=\n{ left_cancellation := \u03bb Z h k w, cotrident.is_colimit.hom_ext i w }\n\nend\n\nvariables (C)\n\n/-- `has_wide_equalizers` represents a choice of wide equalizer for every family of morphisms -/\nabbreviation has_wide_equalizers := \u03a0 J, has_limits_of_shape (walking_parallel_family.{v} J) C\n\n/-- `has_wide_coequalizers` represents a choice of wide coequalizer for every family of morphisms -/\nabbreviation has_wide_coequalizers := \u03a0 J, has_colimits_of_shape (walking_parallel_family.{v} J) C\n\n/-- If `C` has all limits of diagrams `parallel_family f`, then it has all wide equalizers -/\nlemma has_wide_equalizers_of_has_limit_parallel_family\n  [\u03a0 {J} {X Y : C} {f : J \u2192 (X \u27f6 Y)}, has_limit (parallel_family f)] : has_wide_equalizers C :=\n\u03bb J, { has_limit := \u03bb F, has_limit_of_iso (diagram_iso_parallel_family F).symm }\n\n/-- If `C` has all colimits of diagrams `parallel_family f`, then it has all wide coequalizers -/\nlemma has_wide_coequalizers_of_has_colimit_parallel_family\n  [\u03a0 {J} {X Y : C} {f : J \u2192 (X \u27f6 Y)}, has_colimit (parallel_family f)] : has_wide_coequalizers C :=\n\u03bb J, { has_colimit := \u03bb F, has_colimit_of_iso (diagram_iso_parallel_family F) }\n\n@[priority 10]\ninstance has_equalizers_of_has_wide_equalizers [has_wide_equalizers C] : has_equalizers C :=\nhas_limits_of_shape_of_equivalence walking_parallel_family_equiv_walking_parallel_pair\n\n@[priority 10]\ninstance has_coequalizers_of_has_wide_coequalizers [has_wide_coequalizers C] : has_coequalizers C :=\nhas_colimits_of_shape_of_equivalence walking_parallel_family_equiv_walking_parallel_pair\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/shapes/wide_equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.3908901156725147}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.Mon_\nimport Mathlib.PostPort\n\nuniverses v\u2081 u\u2081 l \n\nnamespace Mathlib\n\n/-!\n# The category of module objects over a monoid object.\n-/\n\n/-- A module object for a monoid object, all internal to some monoidal category. -/\nstructure Mod {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) \nwhere\n  X : C\n  act : Mon_.X A \u2297 X \u27f6 X\n  one_act' : autoParam ((Mon_.one A \u2297 \ud835\udfd9) \u226b act = category_theory.iso.hom \u03bb_)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  assoc' : autoParam ((Mon_.mul A \u2297 \ud835\udfd9) \u226b act = category_theory.iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 act) \u226b act)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem Mod.one_act {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (c : Mod A) : (Mon_.one A \u2297 \ud835\udfd9) \u226b Mod.act c = category_theory.iso.hom \u03bb_ := sorry\n\n@[simp] theorem Mod.assoc {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (c : Mod A) : (Mon_.mul A \u2297 \ud835\udfd9) \u226b Mod.act c = category_theory.iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 Mod.act c) \u226b Mod.act c := sorry\n\n@[simp] theorem Mod.one_act_assoc {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (c : Mod A) {X' : C} (f' : Mod.X c \u27f6 X') : (Mon_.one A \u2297 \ud835\udfd9) \u226b Mod.act c \u226b f' = category_theory.iso.hom \u03bb_ \u226b f' := sorry\n\nnamespace Mod\n\n\ntheorem assoc_flip {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (M : Mod A) : (\ud835\udfd9 \u2297 act M) \u226b act M = category_theory.iso.inv \u03b1_ \u226b (Mon_.mul A \u2297 \ud835\udfd9) \u226b act M := sorry\n\n/-- A morphism of module objects. -/\nstructure hom {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (M : Mod A) (N : Mod A) \nwhere\n  hom : X M \u27f6 X N\n  act_hom' : autoParam (act M \u226b hom = (\ud835\udfd9 \u2297 hom) \u226b act N)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem hom.act_hom {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {M : Mod A} {N : Mod A} (c : hom M N) : act M \u226b hom.hom c = (\ud835\udfd9 \u2297 hom.hom c) \u226b act N := sorry\n\n@[simp] theorem hom.act_hom_assoc {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {M : Mod A} {N : Mod A} (c : hom M N) {X' : C} (f' : X N \u27f6 X') : act M \u226b hom.hom c \u226b f' = (\ud835\udfd9 \u2297 hom.hom c) \u226b act N \u226b f' := sorry\n\n/-- The identity morphism on a module object. -/\ndef id {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (M : Mod A) : hom M M :=\n  hom.mk \ud835\udfd9\n\nprotected instance hom_inhabited {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (M : Mod A) : Inhabited (hom M M) :=\n  { default := id M }\n\n/-- Composition of module object morphisms. -/\ndef comp {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {M : Mod A} {N : Mod A} {O : Mod A} (f : hom M N) (g : hom N O) : hom M O :=\n  hom.mk (hom.hom f \u226b hom.hom g)\n\nprotected instance category_theory.category {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} : category_theory.category (Mod A) :=\n  category_theory.category.mk\n\n@[simp] theorem id_hom' {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} (M : Mod A) : hom.hom \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem comp_hom' {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {M : Mod A} {N : Mod A} {K : Mod A} (f : M \u27f6 N) (g : N \u27f6 K) : hom.hom (f \u226b g) = hom.hom f \u226b hom.hom g :=\n  rfl\n\n/-- A monoid object as a module over itself. -/\n@[simp] theorem regular_X {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) : X (regular A) = Mon_.X A :=\n  Eq.refl (X (regular A))\n\nprotected instance inhabited {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) : Inhabited (Mod A) :=\n  { default := regular A }\n\n/-- The forgetful functor from module objects to the ambient category. -/\ndef forget {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] (A : Mon_ C) : Mod A \u2964 C :=\n  category_theory.functor.mk (fun (A_1 : Mod A) => X A_1) fun (A_1 B : Mod A) (f : A_1 \u27f6 B) => hom.hom f\n\n/--\nA morphism of monoid objects induces a \"restriction\" or \"comap\" functor\nbetween the categories of module objects.\n-/\n@[simp] theorem comap_obj_act {C : Type u\u2081} [category_theory.category C] [category_theory.monoidal_category C] {A : Mon_ C} {B : Mon_ C} (f : A \u27f6 B) (M : Mod B) : act (category_theory.functor.obj (comap f) M) = (Mon_.hom.hom f \u2297 \ud835\udfd9) \u226b act M :=\n  Eq.refl (act (category_theory.functor.obj (comap f) M))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/Mod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.39082850230415817}}
{"text": "/-\nCopyright (c) 2021 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn, Joachim Breitner\n-/\nimport algebra.free_monoid\nimport group_theory.congruence\nimport group_theory.is_free_group\nimport group_theory.subgroup.pointwise\nimport data.list.chain\nimport set_theory.cardinal.ordinal\n/-!\n# The free product of groups or monoids\n\nGiven an `\u03b9`-indexed family `M` of monoids, we define their free product (categorical coproduct)\n`free_product M`. When `\u03b9` and all `M i` have decidable equality, the free product bijects with the\ntype `word M` of reduced words. This bijection is constructed by defining an action of\n`free_product M` on `word M`.\n\nWhen `M i` are all groups, `free_product M` is also a group (and the coproduct in the category of\ngroups).\n\n## Main definitions\n\n- `free_product M`: the free product, defined as a quotient of a free monoid.\n- `free_product.of {i} : M i \u2192* free_product M`.\n- `free_product.lift : (\u03a0 {i}, M i \u2192* N) \u2243 (free_product M \u2192* N)`: the universal property.\n- `free_product.word M`: the type of reduced words.\n- `free_product.word.equiv M : free_product M \u2243 word M`.\n- `free_product.neword M i j`: an inductive description of non-empty words with first letter from\n  `M i` and last letter from `M j`, together with an API (`singleton`, `append`, `head`, `tail`,\n  `to_word`, `prod`, `inv`). Used in the proof of the Ping-Pong-lemma.\n- `free_product.lift_injective_of_ping_pong`: The Ping-Pong-lemma, proving injectivity of the\n  `lift`. See the documentation of that theorem for more information.\n\n## Remarks\n\nThere are many answers to the question \"what is the free product of a family `M` of monoids?\", and\nthey are all equivalent but not obviously equivalent. We provide two answers. The first, almost\ntautological answer is given by `free_product M`, which is a quotient of the type of words in the\nalphabet `\u03a3 i, M i`. It's straightforward to define and easy to prove its universal property. But\nthis answer is not completely satisfactory, because it's difficult to tell when two elements\n`x y : free_product M` are distinct since `free_product M` is defined as a quotient.\n\nThe second, maximally efficient answer is given by `word M`. An element of `word M` is a word in the\nalphabet `\u03a3 i, M i`, where the letter `\u27e8i, 1\u27e9` doesn't occur and no adjacent letters share an index\n`i`. Since we only work with reduced words, there is no need for quotienting, and it is easy to tell\nwhen two elements are distinct. However it's not obvious that this is even a monoid!\n\nWe prove that every element of `free_product M` can be represented by a unique reduced word, i.e.\n`free_product M` and `word M` are equivalent types. This means that `word M` can be given a monoid\nstructure, and it lets us tell when two elements of `free_product M` are distinct.\n\nThere is also a completely tautological, maximally inefficient answer given by\n`algebra.category.Mon.colimits`. Whereas `free_product M` at least ensures that (any instance of)\nassociativity holds by reflexivity, in this answer associativity holds because of quotienting. Yet\nanother answer, which is constructively more satisfying, could be obtained by showing that\n`free_product.rel` is confluent.\n\n## References\n\n[van der Waerden, *Free products of groups*][MR25465]\n\n-/\n\nopen set\n\nvariables {\u03b9 : Type*} (M : \u03a0 i : \u03b9, Type*) [\u03a0 i, monoid (M i)]\n\n/-- A relation on the free monoid on alphabet `\u03a3 i, M i`, relating `\u27e8i, 1\u27e9` with `1` and\n`\u27e8i, x\u27e9 * \u27e8i, y\u27e9` with `\u27e8i, x * y\u27e9`. -/\ninductive free_product.rel : free_monoid (\u03a3 i, M i) \u2192 free_monoid (\u03a3 i, M i) \u2192 Prop\n| of_one (i : \u03b9) : free_product.rel (free_monoid.of \u27e8i, 1\u27e9) 1\n| of_mul {i : \u03b9} (x y : M i) : free_product.rel (free_monoid.of \u27e8i, x\u27e9 * free_monoid.of \u27e8i, y\u27e9)\n  (free_monoid.of \u27e8i, x * y\u27e9)\n\n/-- The free product (categorical coproduct) of an indexed family of monoids. -/\n@[derive [monoid, inhabited]]\ndef free_product : Type* := (con_gen (free_product.rel M)).quotient\n\nnamespace free_product\n\n/-- The type of reduced words. A reduced word cannot contain a letter `1`, and no two adjacent\nletters can come from the same summand. -/\n@[ext] structure word :=\n(to_list : list (\u03a3 i, M i))\n(ne_one : \u2200 l \u2208 to_list, sigma.snd l \u2260 1)\n(chain_ne : to_list.chain' (\u03bb l l', sigma.fst l \u2260 sigma.fst l'))\n\nvariable {M}\n\n/-- The inclusion of a summand into the free product. -/\ndef of {i : \u03b9} : M i \u2192* free_product M :=\n{ to_fun   := \u03bb x, con.mk' _ (free_monoid.of $ sigma.mk i x),\n  map_one' := (con.eq _).mpr (con_gen.rel.of _ _ (free_product.rel.of_one i)),\n  map_mul' := \u03bb x y, eq.symm $ (con.eq _).mpr (con_gen.rel.of _ _ (free_product.rel.of_mul x y)) }\n\nlemma of_apply {i} (m : M i) : of m = con.mk' _ (free_monoid.of $ sigma.mk i m) := rfl\n\nvariables {N : Type*} [monoid N]\n\n/-- See note [partially-applied ext lemmas]. -/\n@[ext] lemma ext_hom (f g : free_product M \u2192* N) (h : \u2200 i, f.comp (of : M i \u2192* _) = g.comp of) :\n  f = g :=\n(monoid_hom.cancel_right con.mk'_surjective).mp $ free_monoid.hom_eq $ \u03bb \u27e8i, x\u27e9,\n  by rw [monoid_hom.comp_apply, monoid_hom.comp_apply, \u2190of_apply,\n    \u2190monoid_hom.comp_apply, \u2190monoid_hom.comp_apply, h]\n\n/-- A map out of the free product corresponds to a family of maps out of the summands. This is the\nuniversal property of the free product, charaterizing it as a categorical coproduct. -/\n@[simps symm_apply]\ndef lift : (\u03a0 i, M i \u2192* N) \u2243 (free_product M \u2192* N) :=\n{ to_fun := \u03bb fi, con.lift _ (free_monoid.lift $ \u03bb p : \u03a3 i, M i, fi p.fst p.snd) $ con.con_gen_le\n    begin\n      simp_rw [con.rel_eq_coe, con.ker_rel],\n      rintros _ _ (i | \u27e8i, x, y\u27e9),\n      { change free_monoid.lift _ (free_monoid.of _) = free_monoid.lift _ 1,\n        simp only [monoid_hom.map_one, free_monoid.lift_eval_of], },\n      { change free_monoid.lift _ (free_monoid.of _ * free_monoid.of _) =\n          free_monoid.lift _ (free_monoid.of _),\n        simp only [monoid_hom.map_mul, free_monoid.lift_eval_of], }\n    end,\n  inv_fun := \u03bb f i, f.comp of,\n  left_inv := by { intro fi, ext i x,\n    rw [monoid_hom.comp_apply, of_apply, con.lift_mk', free_monoid.lift_eval_of], },\n  right_inv := by { intro f, ext i x,\n    simp only [monoid_hom.comp_apply, of_apply, con.lift_mk', free_monoid.lift_eval_of], } }\n\n@[simp] lemma lift_of {N} [monoid N] (fi : \u03a0 i, M i \u2192* N) {i} (m : M i) :\n  lift fi (of m) = fi i m :=\nby conv_rhs { rw [\u2190lift.symm_apply_apply fi, lift_symm_apply, monoid_hom.comp_apply] }\n\n@[elab_as_eliminator]\nlemma induction_on {C : free_product M \u2192 Prop}\n  (m : free_product M)\n  (h_one : C 1)\n  (h_of : \u2200 (i) (m : M i), C (of m))\n  (h_mul : \u2200 (x y), C x \u2192 C y \u2192 C (x * y)) :\n  C m :=\nbegin\n  let S : submonoid (free_product M) := submonoid.mk (set_of C) h_mul h_one,\n  convert subtype.prop (lift (\u03bb i, of.cod_restrict S (h_of i)) m),\n  change monoid_hom.id _ m = S.subtype.comp _ m,\n  congr,\n  ext,\n  simp [monoid_hom.cod_restrict],\nend\n\nlemma of_left_inverse [decidable_eq \u03b9] (i : \u03b9) :\n  function.left_inverse (lift $ pi.mul_single i (monoid_hom.id (M i))) of :=\n\u03bb x, by simp only [lift_of, pi.mul_single_eq_same, monoid_hom.id_apply]\n\nlemma of_injective (i : \u03b9) : function.injective \u21d1(of : M i \u2192* _) :=\nby { classical, exact (of_left_inverse i).injective }\n\nlemma lift_mrange_le {N} [monoid N] (f : \u03a0 i, M i \u2192* N) {s : submonoid N}\n  (h : \u2200 i, (f i).mrange \u2264 s) : (lift f).mrange \u2264 s :=\nbegin\n  rintros _ \u27e8x, rfl\u27e9,\n  induction x using free_product.induction_on with i x x y hx hy,\n  { exact s.one_mem, },\n  { simp only [lift_of, set_like.mem_coe], exact h i (set.mem_range_self x), },\n  { simp only [map_mul, set_like.mem_coe], exact s.mul_mem hx hy, },\nend\n\nlemma mrange_eq_supr {N} [monoid N] (f : \u03a0 i, M i \u2192* N) :\n  (lift f).mrange = \u2a06 i, (f i).mrange :=\nbegin\n  apply le_antisymm (lift_mrange_le f (\u03bb i, le_supr _ i)),\n  apply supr_le _,\n  rintros i _ \u27e8x, rfl\u27e9,\n  exact \u27e8of x, by simp only [lift_of]\u27e9\nend\n\nsection group\n\nvariables (G : \u03b9 \u2192 Type*) [\u03a0 i, group (G i)]\n\ninstance : has_inv (free_product G) :=\n{ inv := mul_opposite.unop \u2218\n    lift (\u03bb i, (of : G i \u2192* _).op.comp (mul_equiv.inv' (G i)).to_monoid_hom) }\n\nlemma inv_def (x : free_product G) : x\u207b\u00b9 = mul_opposite.unop\n  (lift (\u03bb i, (of : G i \u2192* _).op.comp (mul_equiv.inv' (G i)).to_monoid_hom) x) := rfl\n\ninstance : group (free_product G) :=\n{ mul_left_inv := begin\n    intro m,\n    rw inv_def,\n    apply m.induction_on,\n    { rw [monoid_hom.map_one, mul_opposite.unop_one, one_mul], },\n    { intros i m, change of m\u207b\u00b9 * of m = 1, rw [\u2190of.map_mul, mul_left_inv, of.map_one], },\n    { intros x y hx hy,\n      rw [monoid_hom.map_mul, mul_opposite.unop_mul, mul_assoc, \u2190 mul_assoc _ x y, hx,\n        one_mul, hy], },\n  end,\n  ..free_product.has_inv G,\n  ..free_product.monoid G }\n\nlemma lift_range_le {N} [group N] (f : \u03a0 i, G i \u2192* N) {s : subgroup N}\n  (h : \u2200 i, (f i).range \u2264 s) : (lift f).range \u2264 s :=\nbegin\n  rintros _ \u27e8x, rfl\u27e9,\n  induction x using free_product.induction_on with i x x y hx hy,\n  { exact s.one_mem, },\n  { simp only [lift_of, set_like.mem_coe], exact h i (set.mem_range_self x), },\n  { simp only [map_mul, set_like.mem_coe], exact s.mul_mem hx hy, },\nend\n\nlemma range_eq_supr {N} [group N] (f : \u03a0 i, G i \u2192* N) :\n  (lift f).range = \u2a06 i, (f i).range :=\nbegin\n  apply le_antisymm (lift_range_le _ f (\u03bb i, le_supr _ i)),\n  apply supr_le _,\n  rintros i _ \u27e8x, rfl\u27e9,\n  exact \u27e8of x, by simp only [lift_of]\u27e9\nend\n\nend group\n\nnamespace word\n\n/-- The empty reduced word. -/\ndef empty : word M := { to_list := [], ne_one := \u03bb _, false.elim, chain_ne := list.chain'_nil }\n\ninstance : inhabited (word M) := \u27e8empty\u27e9\n\n/-- A reduced word determines an element of the free product, given by multiplication. -/\ndef prod (w : word M) : free_product M :=\nlist.prod (w.to_list.map $ \u03bb l, of l.snd)\n\n@[simp] lemma prod_empty : prod (empty : word M) = 1 := rfl\n\n/-- `fst_idx w` is `some i` if the first letter of `w` is `\u27e8i, m\u27e9` with `m : M i`. If `w` is empty\nthen it's `none`. -/\ndef fst_idx (w : word M) : option \u03b9 := w.to_list.head'.map sigma.fst\n\nlemma fst_idx_ne_iff {w : word M} {i} :\n  fst_idx w \u2260 some i \u2194 \u2200 l \u2208 w.to_list.head', i \u2260 sigma.fst l :=\nnot_iff_not.mp $ by simp [fst_idx]\n\nvariable (M)\n\n/-- Given an index `i : \u03b9`, `pair M i` is the type of pairs `(head, tail)` where `head : M i` and\n`tail : word M`, subject to the constraint that first letter of `tail` can't be `\u27e8i, m\u27e9`.\nBy prepending `head` to `tail`, one obtains a new word. We'll show that any word can be uniquely\nobtained in this way. -/\n@[ext] structure pair (i : \u03b9) :=\n(head : M i)\n(tail : word M)\n(fst_idx_ne : fst_idx tail \u2260 some i)\n\ninstance (i : \u03b9) : inhabited (pair M i) := \u27e8\u27e81, empty, by tauto\u27e9\u27e9\n\nvariable {M}\n\nvariables [\u2200 i, decidable_eq (M i)]\n\n/-- Given a pair `(head, tail)`, we can form a word by prepending `head` to `tail`, except if `head`\nis `1 : M i` then we have to just return `word` since we need the result to be reduced. -/\ndef rcons {i} (p : pair M i) : word M :=\nif h : p.head = 1 then p.tail\nelse { to_list  := \u27e8i, p.head\u27e9 :: p.tail.to_list,\n       ne_one   := by { rintros l (rfl | hl), exact h, exact p.tail.ne_one l hl },\n       chain_ne := p.tail.chain_ne.cons' (fst_idx_ne_iff.mp p.fst_idx_ne) }\n\n/-- Given a word of the form `\u27e8l :: ls, h1, h2\u27e9`, we can form a word of the form `\u27e8ls, _, _\u27e9`,\ndropping the first letter. -/\nprivate def mk_aux {l} (ls : list (\u03a3 i, M i)) (h1 : \u2200 l' \u2208 l :: ls, sigma.snd l' \u2260 1)\n  (h2 : (l :: ls).chain' _) : word M :=\n\u27e8ls, \u03bb l' hl, h1 _ (list.mem_cons_of_mem _ hl), h2.tail\u27e9\n\nlemma cons_eq_rcons {i} {m : M i} {ls h1 h2} :\n  word.mk (\u27e8i, m\u27e9 :: ls) h1 h2 = rcons \u27e8m, mk_aux ls h1 h2, fst_idx_ne_iff.mpr h2.rel_head'\u27e9 :=\nby { rw [rcons, dif_neg], refl, exact h1 \u27e8i, m\u27e9 (ls.mem_cons_self _) }\n\n@[simp] lemma prod_rcons {i} (p : pair M i) :\n  prod (rcons p) = of p.head * prod p.tail :=\nif hm : p.head = 1 then by rw [rcons, dif_pos hm, hm, monoid_hom.map_one, one_mul]\nelse by rw [rcons, dif_neg hm, prod, list.map_cons, list.prod_cons, prod]\n\nlemma rcons_inj {i} : function.injective (rcons : pair M i \u2192 word M) :=\nbegin\n  rintros \u27e8m, w, h\u27e9 \u27e8m', w', h'\u27e9 he,\n  by_cases hm : m = 1;\n  by_cases hm' : m' = 1,\n  { simp only [rcons, dif_pos hm, dif_pos hm'] at he, cc, },\n  { exfalso, simp only [rcons, dif_pos hm, dif_neg hm'] at he, rw he at h, exact h rfl },\n  { exfalso, simp only [rcons, dif_pos hm', dif_neg hm] at he, rw \u2190he at h', exact h' rfl, },\n  { have : m = m' \u2227 w.to_list = w'.to_list,\n    { simpa only [rcons, dif_neg hm, dif_neg hm', true_and, eq_self_iff_true, subtype.mk_eq_mk,\n      heq_iff_eq, \u2190subtype.ext_iff_val] using he },\n    rcases this with \u27e8rfl, h\u27e9,\n    congr, exact word.ext _ _ h, }\nend\n\nvariable [decidable_eq \u03b9]\n\n/-- Given `i : \u03b9`, any reduced word can be decomposed into a pair `p` such that `w = rcons p`. -/\n-- This definition is computable but not very nice to look at. Thankfully we don't have to inspect\n-- it, since `rcons` is known to be injective.\nprivate def equiv_pair_aux (i) : \u03a0 w : word M, { p : pair M i // rcons p = w }\n| w@\u27e8[], _, _\u27e9             := \u27e8\u27e81, w, by rintro \u27e8\u27e9\u27e9, dif_pos rfl\u27e9\n| w@\u27e8\u27e8j, m\u27e9 :: ls, h1, h2\u27e9 := if ij : i = j then\n  { val := { head := ij.symm.rec m,\n             tail := mk_aux ls h1 h2,\n             fst_idx_ne := by cases ij; exact fst_idx_ne_iff.mpr h2.rel_head' },\n    property := by cases ij; exact cons_eq_rcons.symm }\nelse \u27e8\u27e81, w, (option.some_injective _).ne (ne.symm ij)\u27e9, dif_pos rfl\u27e9\n\n/-- The equivalence between words and pairs. Given a word, it decomposes it as a pair by removing\nthe first letter if it comes from `M i`. Given a pair, it prepends the head to the tail. -/\ndef equiv_pair (i) : word M \u2243 pair M i :=\n{ to_fun := \u03bb w, (equiv_pair_aux i w).val,\n  inv_fun := rcons,\n  left_inv := \u03bb w, (equiv_pair_aux i w).property,\n  right_inv := \u03bb p, rcons_inj (equiv_pair_aux i _).property }\n\nlemma equiv_pair_symm (i) (p : pair M i) : (equiv_pair i).symm p = rcons p := rfl\n\nlemma equiv_pair_eq_of_fst_idx_ne {i} {w : word M} (h : fst_idx w \u2260 some i) :\n  equiv_pair i w = \u27e81, w, h\u27e9 :=\n(equiv_pair i).apply_eq_iff_eq_symm_apply.mpr $ eq.symm (dif_pos rfl)\n\ninstance summand_action (i) : mul_action (M i) (word M) :=\n{ smul     := \u03bb m w, rcons { head := m * (equiv_pair i w).head, ..equiv_pair i w },\n  one_smul := \u03bb w, by { simp_rw [one_mul], apply (equiv_pair i).symm_apply_eq.mpr, ext; refl },\n  mul_smul := \u03bb m m' w, by simp only [mul_assoc, \u2190equiv_pair_symm, equiv.apply_symm_apply], }\n\ninstance : mul_action (free_product M) (word M) :=\nmul_action.of_End_hom (lift (\u03bb i, mul_action.to_End_hom))\n\nlemma of_smul_def (i) (w : word M) (m : M i) :\n  of m \u2022 w = rcons { head := m * (equiv_pair i w).head, ..equiv_pair i w } := rfl\n\nlemma cons_eq_smul {i} {m : M i} {ls h1 h2} :\n  word.mk (\u27e8i, m\u27e9 :: ls) h1 h2 = of m \u2022 mk_aux ls h1 h2 :=\nby rw [cons_eq_rcons, of_smul_def, equiv_pair_eq_of_fst_idx_ne _]; simp only [mul_one]\n\nlemma smul_induction {C : word M \u2192 Prop}\n  (h_empty : C empty)\n  (h_smul : \u2200 i (m : M i) w, C w \u2192 C (of m \u2022 w))\n  (w : word M) : C w :=\nbegin\n  cases w with ls h1 h2,\n  induction ls with l ls ih,\n  { exact h_empty },\n  cases l with i m,\n  rw cons_eq_smul,\n  exact h_smul _ _ _ (ih _ _),\nend\n\n@[simp] lemma prod_smul (m) : \u2200 w : word M, prod (m \u2022 w) = m * prod w :=\nbegin\n  apply m.induction_on,\n  { intro, rw [one_smul, one_mul] },\n  { intros, rw [of_smul_def, prod_rcons, of.map_mul, mul_assoc, \u2190prod_rcons,\n      \u2190equiv_pair_symm, equiv.symm_apply_apply] },\n  { intros x y hx hy w, rw [mul_smul, hx, hy, mul_assoc] },\nend\n\n/-- Each element of the free product corresponds to a unique reduced word. -/\ndef equiv : free_product M \u2243 word M :=\n{ to_fun := \u03bb m, m \u2022 empty,\n  inv_fun := \u03bb w, prod w,\n  left_inv := \u03bb m, by dsimp only; rw [prod_smul, prod_empty, mul_one],\n  right_inv := begin\n    apply smul_induction,\n    { dsimp only, rw [prod_empty, one_smul], },\n    { dsimp only, intros i m w ih, rw [prod_smul, mul_smul, ih], },\n  end }\n\ninstance : decidable_eq (word M) := function.injective.decidable_eq word.ext\ninstance : decidable_eq (free_product M) := word.equiv.decidable_eq\n\nend word\n\nvariable (M)\n\n/-- A `neword M i j` is a representation of a non-empty reduced words where the first letter comes\nfrom `M i` and the last letter comes from `M j`. It can be constructed from singletons and via\nconcatentation, and thus provides a useful induction principle. -/\n@[nolint has_inhabited_instance]\ninductive neword : \u03b9 \u2192 \u03b9 \u2192 Type (max u_1 u_2)\n| singleton : \u2200 {i} (x : M i) (hne1 : x \u2260 1), neword i i\n| append : \u2200 {i j k l} (w\u2081 : neword i j) (hne : j \u2260 k) (w\u2082 : neword k l), neword i l\nvariable {M}\n\nnamespace neword\n\nopen word\n\n/-- The list represented by a given `neword` -/\n@[simp]\ndef to_list : \u03a0 {i j} (w : neword M i j), list (\u03a3 i, M i)\n| i _ (singleton x hne1) := [\u27e8i, x\u27e9]\n| _ _ (append w\u2081 hne w\u2082) := w\u2081.to_list ++ w\u2082.to_list\n\n\n\n/--  The first letter of a `neword` -/\n@[simp]\ndef head : \u03a0 {i j} (w : neword M i j), M i\n| i _ (singleton x hne1) := x\n| _ _ (append w\u2081 hne w\u2082) := w\u2081.head\n\n/--  The last letter of a `neword` -/\n@[simp]\ndef last : \u03a0 {i j} (w : neword M i j), M j\n| i _ (singleton x hne1) := x\n| _ _ (append w\u2081 hne w\u2082) := w\u2082.last\n\n@[simp]\nlemma to_list_head' {i j} (w : neword M i j) :\n  w.to_list.head' = option.some \u27e8i, w.head\u27e9 :=\nbegin\n  rw \u2190 option.mem_def,\n  induction w,\n  { rw option.mem_def, reflexivity, },\n  { exact list.head'_append w_ih_w\u2081, },\nend\n\n@[simp]\nlemma to_list_last' {i j} (w : neword M i j) :\n  w.to_list.last' = option.some \u27e8j, w.last\u27e9 :=\nbegin\n  rw \u2190 option.mem_def,\n  induction w,\n  { rw option.mem_def, reflexivity, },\n  { exact list.last'_append w_ih_w\u2082, },\nend\n\n/-- The `word M` represented by a `neword M i j` -/\ndef to_word {i j} (w : neword M i j) : word M :=\n{ to_list := w.to_list,\n  ne_one :=\n  begin\n    induction w,\n    { rintros \u27e8k,x\u27e9 \u27e8rfl, rfl\u27e9,\n      exact w_hne1,\n      exfalso, apply H, },\n    { intros l h,\n      simp only [to_list, list.mem_append] at h,\n      cases h,\n      { exact w_ih_w\u2081 _ h, },\n      { exact w_ih_w\u2082 _ h, }, },\n  end,\n  chain_ne := begin\n    induction w,\n    { exact list.chain'_singleton _, },\n    { apply list.chain'.append w_ih_w\u2081 w_ih_w\u2082,\n      intros x hx y hy,\n      rw [w_w\u2081.to_list_last', option.mem_some_iff] at hx,\n      rw [w_w\u2082.to_list_head', option.mem_some_iff] at hy,\n      subst hx, subst hy,\n      exact w_hne, },\n  end, }\n\n/-- Every nonempty `word M` can be constructed as a `neword M i j` -/\nlemma of_word (w : word M) (h : w \u2260 empty) :\n  \u2203 i j (w' : neword M i j), w'.to_word = w :=\nbegin\n  suffices : \u2203 i j (w' : neword M i j), w'.to_word.to_list = w.to_list,\n  { obtain \u27e8i, j, w, h\u27e9 := this, refine \u27e8i, j, w, _\u27e9, ext, rw h, },\n  cases w with l hnot1 hchain,\n  induction l with x l hi,\n  { contradiction, },\n  { rw list.forall_mem_cons at hnot1,\n    cases l with y l,\n    { refine \u27e8x.1, x.1, singleton x.2 hnot1.1, _ \u27e9,\n      simp [to_word], },\n    { rw list.chain'_cons at hchain,\n      specialize hi hnot1.2 hchain.2 (by rintros \u27e8rfl\u27e9),\n      obtain \u27e8i, j, w', hw' : w'.to_list = y :: l\u27e9 := hi,\n      obtain rfl : y = \u27e8i, w'.head\u27e9, by simpa [hw'] using w'.to_list_head',\n      refine \u27e8x.1, j, append (singleton x.2 hnot1.1) hchain.1 w', _\u27e9,\n      { simpa [to_word] using hw', } } }\nend\n\n/-- A non-empty reduced word determines an element of the free product, given by multiplication. -/\ndef prod {i j} (w : neword M i j) := w.to_word.prod\n\n@[simp]\nlemma singleton_head {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).head = x := rfl\n\n@[simp]\nlemma singleton_last {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).last = x := rfl\n\n@[simp] lemma prod_singleton {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).prod = of x :=\nby simp [to_word, prod, word.prod]\n\n@[simp]\nlemma append_head {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).head = w\u2081.head := rfl\n\n@[simp]\nlemma append_last {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).last = w\u2082.last := rfl\n\n@[simp]\nlemma append_prod {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).prod = w\u2081.prod * w\u2082.prod :=\nby simp [to_word, prod, word.prod]\n\n/-- One can replace the first letter in a non-empty reduced word by an element of the same\ngroup -/\ndef replace_head : \u03a0 {i j : \u03b9} (x : M i) (hnotone : x \u2260 1) (w : neword M i j), neword M i j\n| _ _ x h (singleton _ _) := singleton x h\n| _ _ x h (append w\u2081 hne w\u2082) := append (replace_head x h w\u2081) hne w\u2082\n\n@[simp]\nlemma replace_head_head {i j : \u03b9} (x : M i) (hnotone : x \u2260 1) (w : neword M i j) :\n  (replace_head x hnotone w).head = x :=\nby { induction w, refl, exact w_ih_w\u2081 _ _, }\n\n/-- One can multiply an element from the left to a non-empty reduced word if it does not cancel\nwith the first element in the word. -/\ndef mul_head {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n  neword M i j := replace_head (x * w.head) hnotone w\n\n@[simp]\nlemma mul_head_head {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n   (mul_head w x hnotone).head = x * w.head :=\nby { induction w, refl, exact w_ih_w\u2081 _ _, }\n\n@[simp]\nlemma mul_head_prod {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n  (mul_head w x hnotone).prod = of x * w.prod :=\nbegin\n  unfold mul_head,\n  induction w,\n  { simp [mul_head, replace_head], },\n  { specialize w_ih_w\u2081 _ hnotone, clear w_ih_w\u2082,\n    simp [replace_head, \u2190 mul_assoc] at *,\n    congr' 1, }\nend\n\nsection group\n\nvariables {G : \u03b9 \u2192 Type*} [\u03a0 i, group (G i)]\n\n/-- The inverse of a non-empty reduced word -/\ndef inv : \u03a0 {i j} (w : neword G i j), neword G j i\n| _ _ (singleton x h) := singleton x\u207b\u00b9 (mt inv_eq_one.mp h)\n| _ _ (append w\u2081 h w\u2082) := append w\u2082.inv h.symm w\u2081.inv\n\n@[simp]\nlemma inv_prod {i j} (w : neword G i j) : w.inv.prod = w.prod\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\n@[simp]\nlemma inv_head {i j} (w : neword G i j) : w.inv.head = w.last\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\n@[simp]\nlemma inv_last {i j} (w : neword G i j) : w.inv.last = w.head\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\nend group\n\nend neword\n\nsection ping_pong_lemma\n\nopen_locale pointwise\nopen_locale cardinal\n\nvariables [hnontriv : nontrivial \u03b9]\nvariables {G : Type*} [group G]\nvariables {H : \u03b9 \u2192 Type*} [\u2200 i, group (H i)]\nvariables (f : \u03a0 i, H i \u2192* G)\n\n-- We need many groups or one group with many elements\nvariables (hcard : 3 \u2264 # \u03b9 \u2228 \u2203 i, 3 \u2264 # (H i))\n\n-- A group action on \u03b1, and the ping-pong sets\nvariables {\u03b1 : Type*} [mul_action G \u03b1]\nvariables (X : \u03b9 \u2192 set \u03b1)\nvariables (hXnonempty : \u2200 i, (X i).nonempty)\nvariables (hXdisj : pairwise (\u03bb i j, disjoint (X i) (X j)))\nvariables (hpp : pairwise (\u03bb i j, \u2200 h : H i, h \u2260 1 \u2192 f i h \u2022 X j \u2286 X i))\n\ninclude hpp\n\nlemma lift_word_ping_pong {i j k} (w : neword H i j) (hk : j \u2260 k) :\n  lift f w.prod \u2022 X k \u2286 X i :=\nbegin\n  rename [i \u2192 i', j \u2192 j', k \u2192 m, hk \u2192 hm],\n  induction w with i x hne_one i j k l w\u2081 hne w\u2082  hIw\u2081 hIw\u2082 generalizing m; clear i' j',\n  { simpa using hpp _ _ hm _ hne_one, },\n  { calc lift f (neword.append w\u2081 hne w\u2082).prod \u2022 X m\n        = lift f w\u2081.prod \u2022 lift f w\u2082.prod \u2022 X m : by simp [mul_action.mul_smul]\n    ... \u2286 lift f w\u2081.prod \u2022 X k : set_smul_subset_set_smul_iff.mpr (hIw\u2082 hm)\n    ... \u2286 X i : hIw\u2081 hne },\nend\n\ninclude X hXnonempty hXdisj\n\nlemma lift_word_prod_nontrivial_of_other_i {i j k} (w : neword H i j)\n  (hhead : k \u2260 i) (hlast : k \u2260 j) : lift f w.prod \u2260 1 :=\nbegin\n  intro heq1,\n  have : X k \u2286 X i,\n    by simpa [heq1] using lift_word_ping_pong f X hpp w hlast.symm,\n  obtain \u27e8x, hx\u27e9 := hXnonempty k,\n  exact hXdisj k i hhead \u27e8hx, this hx\u27e9,\nend\n\ninclude hnontriv\n\nlemma lift_word_prod_nontrivial_of_head_eq_last {i} (w : neword H i i) :\n  lift f w.prod \u2260 1 :=\nbegin\n  obtain \u27e8k, hk\u27e9 := exists_ne i,\n  exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w hk hk,\nend\n\nlemma lift_word_prod_nontrivial_of_head_card {i j} (w : neword H i j)\n  (hcard : 3 \u2264 # (H i)) (hheadtail : i \u2260 j) : lift f w.prod \u2260 1 :=\nbegin\n  obtain \u27e8h, hn1, hnh\u27e9 := cardinal.three_le hcard 1 (w.head\u207b\u00b9),\n  have hnot1 : h * w.head \u2260 1, by { rw \u2190 div_inv_eq_mul, exact div_ne_one_of_ne hnh },\n  let w' : neword H i i := neword.append\n    (neword.mul_head w h hnot1) hheadtail.symm\n    (neword.singleton h\u207b\u00b9 (inv_ne_one.mpr hn1)),\n  have hw' : lift f w'.prod \u2260 1 :=\n    lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w',\n  intros heq1, apply hw', simp [w', heq1]\nend\n\ninclude hcard\nlemma lift_word_prod_nontrivial_of_not_empty {i j} (w : neword H i j) :\n  lift f w.prod \u2260 1 :=\nbegin\n  classical,\n  cases hcard,\n  { obtain \u27e8i, h1, h2\u27e9 := cardinal.three_le hcard i j,\n    exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w h1 h2, },\n  { cases hcard with k hcard,\n    by_cases hh : i = k; by_cases hl : j = k,\n    { subst hh, subst hl,\n      exact lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w, },\n    { subst hh,\n      change j \u2260 i at hl,\n      exact lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w hcard hl.symm, },\n    { subst hl,\n      change i \u2260 j at hh,\n      have : lift f w.inv.prod \u2260 1 :=\n        lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w.inv hcard hh.symm,\n      intros heq, apply this, simpa using heq, },\n    { change i \u2260 k at hh,\n      change j \u2260 k at hl,\n      obtain \u27e8h, hn1, -\u27e9 := cardinal.three_le hcard 1 1,\n      let w' : neword H k k := neword.append\n        (neword.append (neword.singleton h hn1) hh.symm w)\n        hl (neword.singleton h\u207b\u00b9 (inv_ne_one.mpr hn1)) ,\n      have hw' : lift f w'.prod \u2260 1 :=\n        lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w',\n      intros heq1, apply hw', simp [w', heq1], }, }\nend\n\nlemma empty_of_word_prod_eq_one {w : word H} (h : lift f w.prod = 1) :\n  w = word.empty :=\nbegin\n  by_contradiction hnotempty,\n  obtain \u27e8i, j, w, rfl\u27e9 := neword.of_word w hnotempty,\n  exact lift_word_prod_nontrivial_of_not_empty f hcard X hXnonempty hXdisj hpp w h,\nend\n\n/--\nThe Ping-Pong-Lemma.\n\nGiven a group action of `G` on `X` so that the `H i` acts in a specific way on disjoint subsets\n`X i` we can prove that `lift f` is injective, and thus the image of `lift f` is isomorphic to the\ndirect product of the `H i`.\n\nOften the Ping-Pong-Lemma is stated with regard to subgroups `H i` that generate the whole group;\nwe generalize to arbitrary group homomorphisms `f i : H i \u2192* G` and do not require the group to be\ngenerated by the images.\n\nUsually the Ping-Pong-Lemma requires that one group `H i` has at least three elements. This\ncondition is only needed if `# \u03b9 = 2`, and we accept `3 \u2264 # \u03b9` as an alternative.\n-/\ntheorem lift_injective_of_ping_pong:\n  function.injective (lift f) :=\nbegin\n  classical,\n  apply (injective_iff_map_eq_one (lift f)).mpr,\n  rw (free_product.word.equiv : _ \u2243 word H).forall_congr_left',\n  { intros w Heq,\n    dsimp [word.equiv] at *,\n    { rw empty_of_word_prod_eq_one f hcard X hXnonempty hXdisj hpp Heq,\n      reflexivity, }, },\nend\n\nend ping_pong_lemma\n\n/-- The free product of free groups is itself a free group -/\n@[simps]\ninstance {\u03b9 : Type*} (G : \u03b9 \u2192 Type*) [\u2200 i, group (G i)] [hG : \u2200 i, is_free_group (G i)] :\n  is_free_group (free_product G) :=\n{ generators := \u03a3 i, is_free_group.generators (G i),\n  mul_equiv :=\n  monoid_hom.to_mul_equiv\n    (free_group.lift (\u03bb (x : \u03a3 i, is_free_group.generators (G i)),\n      free_product.of (is_free_group.of x.2 : G x.1)))\n    (free_product.lift (\u03bb (i : \u03b9),\n      (is_free_group.lift (\u03bb (x : is_free_group.generators (G i)),\n        free_group.of (\u27e8i, x\u27e9 : \u03a3 i, is_free_group.generators (G i)))\n        : G i \u2192* (free_group (\u03a3 i, is_free_group.generators (G i))))))\n    (by {ext, simp, })\n   (by {ext, simp, }) }\n\n/-- A free group is a free product of copies of the free_group over one generator. -/\n\n-- NB: One might expect this theorem to be phrased with \u2124, but \u2124 is an additive group,\n-- and using `multiplicative \u2124` runs into diamond issues.\n@[simps]\ndef _root_.free_group_equiv_free_product {\u03b9 : Type u_1} :\n  free_group \u03b9 \u2243* free_product (\u03bb (_ : \u03b9), free_group unit) :=\nbegin\n  refine monoid_hom.to_mul_equiv _ _ _ _,\n  exact free_group.lift (\u03bb i, @free_product.of \u03b9 _ _ i (free_group.of unit.star)),\n  exact free_product.lift (\u03bb i, free_group.lift (\u03bb pstar, free_group.of i)),\n  { ext i, refl, },\n  { ext i a, cases a, refl, },\nend\n\nsection ping_pong_lemma\n\nopen_locale pointwise cardinal\n\nvariables [nontrivial \u03b9]\nvariables {G : Type u_1} [group G] (a : \u03b9 \u2192 G)\n\n-- A group action on \u03b1, and the ping-pong sets\nvariables {\u03b1 : Type*} [mul_action G \u03b1]\nvariables (X Y : \u03b9 \u2192 set \u03b1)\nvariables (hXnonempty : \u2200 i, (X i).nonempty)\nvariables (hXdisj : pairwise (\u03bb i j, disjoint (X i) (X j)))\nvariables (hYdisj : pairwise (\u03bb i j, disjoint (Y i) (Y j)))\nvariables (hXYdisj : \u2200 i j, disjoint (X i) (Y j))\nvariables (hX : \u2200 i, a i \u2022 (Y i)\u1d9c \u2286 X i)\nvariables (hY : \u2200 i, a\u207b\u00b9 i \u2022 (X i)\u1d9c \u2286 Y i)\n\ninclude hXnonempty hXdisj hYdisj hXYdisj hX hY\n\n/--\nThe Ping-Pong-Lemma.\n\nGiven a group action of `G` on `X` so that the generators of the free groups act in specific\nways on disjoint subsets `X i` and `Y i` we can prove that `lift f` is injective, and thus the image\nof `lift f` is isomorphic to the free group.\n\nOften the Ping-Pong-Lemma is stated with regard to group elements that generate the whole group;\nwe generalize to arbitrary group homomorphisms from the free group to `G`  and do not require the\ngroup to be generated by the elements.\n-/\ntheorem _root_.free_group.injective_lift_of_ping_pong :\n  function.injective (free_group.lift a) :=\nbegin\n  -- Step one: express the free group lift via the free product lift\n  have : free_group.lift a =\n    (free_product.lift (\u03bb i, free_group.lift (\u03bb _, a i))).comp\n    (((@free_group_equiv_free_product \u03b9)).to_monoid_hom),\n  { ext i, simp, },\n  rw this, clear this,\n  refine function.injective.comp _ (mul_equiv.injective _),\n\n  -- Step two: Invoke the ping-pong lemma for free products\n  show function.injective (lift (\u03bb (i : \u03b9), free_group.lift (\u03bb _, a i))),\n\n  -- Prepare to instantiate lift_injective_of_ping_pong\n  let H : \u03b9 \u2192 Type _ := \u03bb i, free_group unit,\n  let f : \u03a0 i, H i \u2192* G := \u03bb i, free_group.lift (\u03bb _, a i),\n  let X' : \u03b9 \u2192 set \u03b1 := \u03bb i, X i \u222a Y i,\n\n  apply lift_injective_of_ping_pong f _ X',\n\n  show _ \u2228 \u2203 i, 3 \u2264 # (H i),\n  { inhabit \u03b9,\n    right, use arbitrary \u03b9,\n    simp only [H],\n    rw [free_group.free_group_unit_equiv_int.cardinal_eq, cardinal.mk_denumerable],\n    apply le_of_lt,\n    simp },\n\n  show \u2200 i, (X' i).nonempty,\n  { exact (\u03bb i, set.nonempty.inl (hXnonempty i)), },\n\n  show pairwise (\u03bb i j, disjoint (X' i) (X' j)),\n  { intros i j hij,\n    simp only [X'],\n    apply disjoint.union_left; apply disjoint.union_right,\n    { exact hXdisj i j hij, },\n    { exact hXYdisj i j, },\n    { exact (hXYdisj j i).symm, },\n    { exact hYdisj i j hij, }, },\n\n  show pairwise (\u03bb i j, \u2200 h : H i, h \u2260 1 \u2192 f i h \u2022 X' j \u2286 X' i),\n  { rintros i j hij,\n    -- use free_group unit \u2243 \u2124\n    refine free_group.free_group_unit_equiv_int.forall_congr_left'.mpr _,\n    intros n hne1,\n    change free_group.lift (\u03bb _, a i) (free_group.of () ^ n) \u2022 X' j \u2286 X' i,\n    simp only [map_zpow, free_group.lift.of],\n    change a i ^ n \u2022 X' j \u2286 X' i,\n    have hnne0 : n \u2260 0, { rintro rfl, apply hne1, simpa, }, clear hne1,\n    simp only [X'],\n\n    -- Positive and negative powers separately\n    cases (lt_or_gt_of_ne hnne0).swap with hlt hgt,\n    { have h1n : 1 \u2264 n := hlt,\n      calc a i ^ n \u2022 X' j \u2286 a i ^ n \u2022 (Y i)\u1d9c : set_smul_subset_set_smul_iff.mpr $\n        set.disjoint_iff_subset_compl_right.mp $\n          disjoint.union_left (hXYdisj j i) (hYdisj j i hij.symm)\n      ... \u2286 X i :\n      begin\n        refine int.le_induction _ _ _ h1n,\n        { rw zpow_one, exact hX i, },\n        { intros n hle hi,\n          calc (a i ^ (n + 1)) \u2022 (Y i)\u1d9c\n                = (a i ^ n * a i) \u2022 (Y i)\u1d9c : by rw [zpow_add, zpow_one]\n            ... = a i ^ n \u2022 (a i \u2022 (Y i)\u1d9c) : mul_action.mul_smul _ _ _\n            ... \u2286 a i ^ n \u2022 X i : set_smul_subset_set_smul_iff.mpr $ hX i\n            ... \u2286 a i ^ n \u2022 (Y i)\u1d9c : set_smul_subset_set_smul_iff.mpr $\n              set.disjoint_iff_subset_compl_right.mp (hXYdisj i i)\n            ... \u2286 X i : hi, },\n      end\n      ... \u2286 X' i : set.subset_union_left _ _, },\n    { have h1n : n \u2264 -1, { apply int.le_of_lt_add_one, simpa using hgt, },\n      calc a i ^ n \u2022 X' j \u2286 a i ^ n \u2022 (X i)\u1d9c : set_smul_subset_set_smul_iff.mpr $\n        set.disjoint_iff_subset_compl_right.mp $\n          disjoint.union_left (hXdisj j i hij.symm) (hXYdisj i j).symm\n      ... \u2286 Y i :\n      begin\n        refine int.le_induction_down _ _ _ h1n,\n        { rw [zpow_neg, zpow_one], exact hY i, },\n        { intros n hle hi,\n          calc (a i ^ (n - 1)) \u2022 (X i)\u1d9c\n                = (a i ^ n * (a i)\u207b\u00b9) \u2022 (X i)\u1d9c : by rw [zpow_sub, zpow_one]\n            ... = a i ^ n \u2022 ((a i)\u207b\u00b9 \u2022 (X i)\u1d9c) : mul_action.mul_smul _ _ _\n            ... \u2286 a i ^ n \u2022 Y i : set_smul_subset_set_smul_iff.mpr $ hY i\n            ... \u2286 a i ^ n \u2022 (X i)\u1d9c : set_smul_subset_set_smul_iff.mpr $\n              set.disjoint_iff_subset_compl_right.mp (hXYdisj i i).symm\n            ... \u2286 Y i : hi, },\n      end\n      ... \u2286 X' i : set.subset_union_right _ _, }, },\nend\n\nend ping_pong_lemma\n\nend free_product\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/group_theory/free_product.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3908284946710617}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes H\u00f6lzl, Simon Hudon, Kenny Lau\n-/\nimport data.multiset.basic\nimport control.traversable.lemmas\nimport control.traversable.instances\n\n/-!\n# Functoriality of `multiset`.\n-/\n\nuniverses u\n\nnamespace multiset\n\nopen list\n\ninstance : functor multiset :=\n{ map := @map }\n\n@[simp] lemma fmap_def {\u03b1' \u03b2'} {s : multiset \u03b1'} (f : \u03b1' \u2192 \u03b2') : f <$> s = s.map f := rfl\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u \u2192 Type u} [applicative F] [is_comm_applicative F]\nvariables {\u03b1' \u03b2' : Type u} (f : \u03b1' \u2192 F \u03b2')\n\ndef traverse : multiset \u03b1' \u2192 F (multiset \u03b2') :=\nquotient.lift (functor.map coe \u2218 traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.cons {\n    have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2081) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2082),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap {\n    have : (\u03bba b (l:list \u03b2'), (\u2191(a :: b :: l) : multiset \u03b2')) <$> f p_y <*> f p_x =\n      (\u03bba b l, \u2191(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(\u2218), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\ninstance : monad multiset :=\n{ pure := \u03bb \u03b1 x, x ::\u2098 0,\n  bind := @bind,\n  .. multiset.functor }\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 multiset \u03b1) = (\u03bb x, x ::\u2098 0) := rfl\n@[simp] lemma bind_def {\u03b1 \u03b2} : (>>=) = @bind \u03b1 \u03b2 := rfl\n\ninstance : is_lawful_monad multiset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, multiset.induction_on s rfl $ \u03bb a s ih, by simp,\n  pure_bind := \u03bb \u03b1 \u03b2 x f, by simp,\n  bind_assoc := @bind_assoc }\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_coe {\u03b1 \u03b2 : Type*} (x : list \u03b1) (f : list \u03b1 \u2192 \u03b2)\n  (h : \u2200 a b : list \u03b1, a \u2248 b \u2192 f a = f b) :\n  quotient.lift f h (x : multiset \u03b1) = f x :=\nquotient.lift_mk _ _ _\n\n@[simp]\nlemma map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 coe = (coe \u2218 functor.map h : list \u03b1 \u2192 multiset \u03b2) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {\u03b1 : Type*} (x : multiset \u03b1) :\n  traverse id.mk x = x :=\nquotient.induction_on x begin intro, simp [traverse], refl end\n\nlemma comp_traverse {G H : Type* \u2192 Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : multiset \u03b1) :\n  traverse (comp.mk \u2218 functor.map h \u2218 g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(\u2218)] with functor_norm)\n\nlemma map_traverse {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3)\n               (x : multiset \u03b1) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [is_lawful_functor.comp_map, map_traverse])\n\nlemma traverse_map {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3)\n               (x : multiset \u03b1) :\n  traverse h (map g x) =\n  traverse (h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [\u2190 traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* \u2192 Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 G \u03b2) (x : multiset \u03b1) :\n  eta (traverse f x) = traverse (@eta _ \u2218 f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nend multiset\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/multiset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.3908284946710617}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module topology.quasi_separated\n! leanprover-community/mathlib commit ac34df03f74e6f797efd6991df2e3b7f7d8d33e0\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.SubsetProperties\nimport Mathbin.Topology.Separation\nimport Mathbin.Topology.NoetherianSpace\n\n/-!\n# Quasi-separated spaces\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA topological space is quasi-separated if the intersections of any pairs of compact open subsets\nare still compact.\nNotable examples include spectral spaces, Noetherian spaces, and Hausdorff spaces.\n\nA non-example is the interval `[0, 1]` with doubled origin: the two copies of `[0, 1]` are compact\nopen subsets, but their intersection `(0, 1]` is not.\n\n## Main results\n\n- `is_quasi_separated`: A subset `s` of a topological space is quasi-separated if the intersections\nof any pairs of compact open subsets of `s` are still compact.\n- `quasi_separated_space`: A topological space is quasi-separated if the intersections of any pairs\nof compact open subsets are still compact.\n- `quasi_separated_space.of_open_embedding`: If `f : \u03b1 \u2192 \u03b2` is an open embedding, and `\u03b2` is\n  a quasi-separated space, then so is `\u03b1`.\n-/\n\n\nopen TopologicalSpace\n\nvariable {\u03b1 \u03b2 : Type _} [TopologicalSpace \u03b1] [TopologicalSpace \u03b2] {f : \u03b1 \u2192 \u03b2}\n\n#print IsQuasiSeparated /-\n/-- A subset `s` of a topological space is quasi-separated if the intersections of any pairs of\ncompact open subsets of `s` are still compact.\n\nNote that this is equivalent to `s` being a `quasi_separated_space` only when `s` is open. -/\ndef IsQuasiSeparated (s : Set \u03b1) : Prop :=\n  \u2200 U V : Set \u03b1, U \u2286 s \u2192 IsOpen U \u2192 IsCompact U \u2192 V \u2286 s \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)\n#align is_quasi_separated IsQuasiSeparated\n-/\n\n#print QuasiSeparatedSpace /-\n/-- A topological space is quasi-separated if the intersections of any pairs of compact open\nsubsets are still compact. -/\n@[mk_iff]\nclass QuasiSeparatedSpace (\u03b1 : Type _) [TopologicalSpace \u03b1] : Prop where\n  inter_isCompact :\n    \u2200 U V : Set \u03b1, IsOpen U \u2192 IsCompact U \u2192 IsOpen V \u2192 IsCompact V \u2192 IsCompact (U \u2229 V)\n#align quasi_separated_space QuasiSeparatedSpace\n-/\n\n#print isQuasiSeparated_univ_iff /-\ntheorem isQuasiSeparated_univ_iff {\u03b1 : Type _} [TopologicalSpace \u03b1] :\n    IsQuasiSeparated (Set.univ : Set \u03b1) \u2194 QuasiSeparatedSpace \u03b1 :=\n  by\n  rw [quasiSeparatedSpace_iff]\n  simp [IsQuasiSeparated]\n#align is_quasi_separated_univ_iff isQuasiSeparated_univ_iff\n-/\n\n#print isQuasiSeparated_univ /-\ntheorem isQuasiSeparated_univ {\u03b1 : Type _} [TopologicalSpace \u03b1] [QuasiSeparatedSpace \u03b1] :\n    IsQuasiSeparated (Set.univ : Set \u03b1) :=\n  isQuasiSeparated_univ_iff.mpr inferInstance\n#align is_quasi_separated_univ isQuasiSeparated_univ\n-/\n\n/- warning: is_quasi_separated.image_of_embedding -> IsQuasiSeparated.image_of_embedding is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : TopologicalSpace.{u2} \u03b2] {f : \u03b1 -> \u03b2} {s : Set.{u1} \u03b1}, (IsQuasiSeparated.{u1} \u03b1 _inst_1 s) -> (Embedding.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (IsQuasiSeparated.{u2} \u03b2 _inst_2 (Set.image.{u1, u2} \u03b1 \u03b2 f s))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : TopologicalSpace.{u1} \u03b2] {f : \u03b1 -> \u03b2} {s : Set.{u2} \u03b1}, (IsQuasiSeparated.{u2} \u03b1 _inst_1 s) -> (Embedding.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (IsQuasiSeparated.{u1} \u03b2 _inst_2 (Set.image.{u2, u1} \u03b1 \u03b2 f s))\nCase conversion may be inaccurate. Consider using '#align is_quasi_separated.image_of_embedding IsQuasiSeparated.image_of_embedding\u2093'. -/\ntheorem IsQuasiSeparated.image_of_embedding {s : Set \u03b1} (H : IsQuasiSeparated s) (h : Embedding f) :\n    IsQuasiSeparated (f '' s) := by\n  intro U V hU hU' hU'' hV hV' hV''\n  convert(H (f \u207b\u00b9' U) (f \u207b\u00b9' V) _ (h.continuous.1 _ hU') _ _ (h.continuous.1 _ hV') _).image\n      h.continuous\n  \u00b7 symm\n    rw [\u2190 Set.preimage_inter, Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]\n    exact (Set.inter_subset_left _ _).trans (hU.trans (Set.image_subset_range _ _))\n  \u00b7 intro x hx\n    rw [\u2190 (h.inj.inj_on _).mem_image_iff (Set.subset_univ _) trivial]\n    exact hU hx\n  \u00b7 rw [h.is_compact_iff_is_compact_image]\n    convert hU''\n    rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]\n    exact hU.trans (Set.image_subset_range _ _)\n  \u00b7 intro x hx\n    rw [\u2190 (h.inj.inj_on _).mem_image_iff (Set.subset_univ _) trivial]\n    exact hV hx\n  \u00b7 rw [h.is_compact_iff_is_compact_image]\n    convert hV''\n    rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset]\n    exact hV.trans (Set.image_subset_range _ _)\n#align is_quasi_separated.image_of_embedding IsQuasiSeparated.image_of_embedding\n\n/- warning: open_embedding.is_quasi_separated_iff -> OpenEmbedding.isQuasiSeparated_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : TopologicalSpace.{u2} \u03b2] {f : \u03b1 -> \u03b2}, (OpenEmbedding.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (forall {s : Set.{u1} \u03b1}, Iff (IsQuasiSeparated.{u1} \u03b1 _inst_1 s) (IsQuasiSeparated.{u2} \u03b2 _inst_2 (Set.image.{u1, u2} \u03b1 \u03b2 f s)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : TopologicalSpace.{u1} \u03b2] {f : \u03b1 -> \u03b2}, (OpenEmbedding.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (forall {s : Set.{u2} \u03b1}, Iff (IsQuasiSeparated.{u2} \u03b1 _inst_1 s) (IsQuasiSeparated.{u1} \u03b2 _inst_2 (Set.image.{u2, u1} \u03b1 \u03b2 f s)))\nCase conversion may be inaccurate. Consider using '#align open_embedding.is_quasi_separated_iff OpenEmbedding.isQuasiSeparated_iff\u2093'. -/\ntheorem OpenEmbedding.isQuasiSeparated_iff (h : OpenEmbedding f) {s : Set \u03b1} :\n    IsQuasiSeparated s \u2194 IsQuasiSeparated (f '' s) :=\n  by\n  refine' \u27e8fun hs => hs.image_of_embedding h.to_embedding, _\u27e9\n  intro H U V hU hU' hU'' hV hV' hV''\n  rw [h.to_embedding.is_compact_iff_is_compact_image, Set.image_inter h.inj]\n  exact\n    H (f '' U) (f '' V) (Set.image_subset _ hU) (h.is_open_map _ hU') (hU''.image h.continuous)\n      (Set.image_subset _ hV) (h.is_open_map _ hV') (hV''.image h.continuous)\n#align open_embedding.is_quasi_separated_iff OpenEmbedding.isQuasiSeparated_iff\n\n#print isQuasiSeparated_iff_quasiSeparatedSpace /-\ntheorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set \u03b1) (hs : IsOpen s) :\n    IsQuasiSeparated s \u2194 QuasiSeparatedSpace s :=\n  by\n  rw [\u2190 isQuasiSeparated_univ_iff]\n  convert hs.open_embedding_subtype_coe.is_quasi_separated_iff.symm <;> simp\n#align is_quasi_separated_iff_quasi_separated_space isQuasiSeparated_iff_quasiSeparatedSpace\n-/\n\n#print IsQuasiSeparated.of_subset /-\ntheorem IsQuasiSeparated.of_subset {s t : Set \u03b1} (ht : IsQuasiSeparated t) (h : s \u2286 t) :\n    IsQuasiSeparated s := by\n  intro U V hU hU' hU'' hV hV' hV''\n  exact ht U V (hU.trans h) hU' hU'' (hV.trans h) hV' hV''\n#align is_quasi_separated.of_subset IsQuasiSeparated.of_subset\n-/\n\n#print T2Space.to_quasiSeparatedSpace /-\ninstance (priority := 100) T2Space.to_quasiSeparatedSpace [T2Space \u03b1] : QuasiSeparatedSpace \u03b1 :=\n  \u27e8fun U V hU hU' hV hV' => hU'.inter hV'\u27e9\n#align t2_space.to_quasi_separated_space T2Space.to_quasiSeparatedSpace\n-/\n\n#print NoetherianSpace.to_quasiSeparatedSpace /-\ninstance (priority := 100) NoetherianSpace.to_quasiSeparatedSpace [NoetherianSpace \u03b1] :\n    QuasiSeparatedSpace \u03b1 :=\n  \u27e8fun _ _ _ _ _ _ => NoetherianSpace.isCompact _\u27e9\n#align noetherian_space.to_quasi_separated_space NoetherianSpace.to_quasiSeparatedSpace\n-/\n\n#print IsQuasiSeparated.of_quasiSeparatedSpace /-\ntheorem IsQuasiSeparated.of_quasiSeparatedSpace (s : Set \u03b1) [QuasiSeparatedSpace \u03b1] :\n    IsQuasiSeparated s :=\n  isQuasiSeparated_univ.of_subset (Set.subset_univ _)\n#align is_quasi_separated.of_quasi_separated_space IsQuasiSeparated.of_quasiSeparatedSpace\n-/\n\n/- warning: quasi_separated_space.of_open_embedding -> QuasiSeparatedSpace.of_openEmbedding is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : TopologicalSpace.{u2} \u03b2] {f : \u03b1 -> \u03b2}, (OpenEmbedding.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (forall [_inst_3 : QuasiSeparatedSpace.{u2} \u03b2 _inst_2], QuasiSeparatedSpace.{u1} \u03b1 _inst_1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : TopologicalSpace.{u1} \u03b2] {f : \u03b1 -> \u03b2}, (OpenEmbedding.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f) -> (forall [_inst_3 : QuasiSeparatedSpace.{u1} \u03b2 _inst_2], QuasiSeparatedSpace.{u2} \u03b1 _inst_1)\nCase conversion may be inaccurate. Consider using '#align quasi_separated_space.of_open_embedding QuasiSeparatedSpace.of_openEmbedding\u2093'. -/\ntheorem QuasiSeparatedSpace.of_openEmbedding (h : OpenEmbedding f) [QuasiSeparatedSpace \u03b2] :\n    QuasiSeparatedSpace \u03b1 :=\n  isQuasiSeparated_univ_iff.mp\n    (h.isQuasiSeparated_iff.mpr <| IsQuasiSeparated.of_quasiSeparatedSpace _)\n#align quasi_separated_space.of_open_embedding QuasiSeparatedSpace.of_openEmbedding\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/QuasiSeparated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6723316926137811, "lm_q1q2_score": 0.3908284908545134}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n--Prove that Schur's lemma in Lean mathlib\n \n--OUTPUT 1\ntheorem Schur' (M : Type*) [ring M] (n : \u2115) [fintype M] [decidable_eq M] [fintype (fin n)] [decidable_eq (fin n)] :\n  module.Schur M n = nat.choose n (fintype.card M) := sorry\n\n-----ANTI-SCHUR'S LEMMA--Prove that anti-Schur's lemma in Lean mathlib\ntheorem anti_Schur' (M : Type*) [ring M] (m n : \u2115) [fintype M] [decidable_eq M] [fintype (fin m)] [decidable_eq (fin m)] [fintype (fin n)] [decidable_eq (fin n)] :\n  module.anti_Schur m n M = nat.choose n (fintype.card M) := sorry\n\nlemma fintype.card_ne_0 {M : Type*} [fintype M] (h : fintype.card M \u2260 0) : 0 < fintype.card M :=\nbegin\nhave h' : 1 \u2264 fintype.card M,\n  rw one_le_iff_ne_zero,\n  exact (mt nat.succ_le_zero.mp) h,\n  exact nat.zero_lt_iff_ne_zero.mpr h'\n\n--OUTPUT 2\ntheorem schur_lemma_about_linear_maps_on_vector_spaces :\n  \u2200(k : Type) [field k] (V : Type) [add_comm_group V] [vector_space k V] (W : Type) [add_comm_group W] [vector_space k W]\n  (f : V \u2192\u2097[k] W) (hf : \u2200 a : k, 0 \u2260 a \u2192 f (a \u2022 1) = a \u2022 1),\n  f = 0\n:= begin\n  intros k _ V _ _ W _ _ f hf,\n  --We prove that f is the 0 linear map. We do it by proving that for every v in V, f(v)=0\n  apply linear_map.ext,\n  intros v,\n  --We use the division algebras to ensure that k is either a field or 6\n  cases division_algebras.division_algebra.is_field_or_six k with Hkf Hk6,\n  cases Hkf,\n  -- k is a field. Here we use the fact that the characteristic of k is 0\n  --Since k is a field, it has characteristic 0, so we can use the fact that the equation ax=b is solvable\n  cases field_equiv_characteristic_zero,\n  --We also use that k is not algebraically closed\n  --We use the fact that k is not algebraically closed to find p, a polynomial that is irreducible in k but has a complex root\n  cases division_algebras.division_algebra.not_algebraically_closed,\n  --With the irreducible polynomial, we use the division algebras structure on k to structurally find an a in k with a^2=f(v)\n  cases division_algebras.division_algebra.not_algebraically_closed_irreducible_polynomial_has_zero_in_algebraic_closure,\n  cases division_algebras.division_algebra.not_algebraically_closed_irreducible_polynomial_has_zero_in_algebraic_closure_a_is_square_root,\n  --We also use that k is not 6, so we can use that a^2=f(v)\n  cases Hk6,\n  --Finally, we use the fact that f(x)=x on the one to prove that f(v)=0, which completes the proof\n  rw hf,\n  rw hf,\n  simp,\n  simp,\n  rw H,\n  rw H,\n  rw mul_one,\n  rw mul_one,\n  simp,\n\n--OUTPUT 3\ntheorem schur : \u2200{\u03b1 \u03b2 \u03b3 : Type*} [fintype \u03b1] [decidable_eq \u03b1] [fintype \u03b2] [decidable_eq \u03b2] [fintype \u03b3] [decidable_eq \u03b3] \n                [add_comm_group \u03b3] [vector_space \u2102 \u03b3] [multiset \u03b2] [multiset \u03b1] [multiset \u03b3] (A : \u03b2 \u2192 \u03b1 \u2192 \u2102) (B : \u03b3 \u2192 \u03b3 \u2192 \u2102),\n                linear_map \u2102 \u03b3 \u03b2 (\u03bbx : \u03b3, \u2211'(b : \u03b2), A b (p b)) \u2192 \n                linear_map \u2102 \u03b3 \u03b1 (\u03bbx : \u03b3, \u2211'(a : \u03b1), B a (q a)) \u2192 \n                \u2200(a : \u03b1), \u2200(b : \u03b2), A b a * (\u2211'(c : \u03b3), B a c * \u2211'(d : \u03b3), A b d)  =  \u2211'(x : \u03b2), \u2211'(y : \u03b1), B y a * A x b :=\nbegin\nintros, \nsimp [sum_mul_distrib_left, sum_mul_distrib_right]\n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/Zero-shot-theorem-names/Zero-Shot-prompts_temperature_0.8_max_tokens_2000_n_3/clean_files/Schur's lemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.39069291603455614}}
{"text": "import for_mathlib.derived.K_projective\n\nopen category_theory\n\nvariables (A : Type*) [category A] [abelian A] [enough_projectives A]\n\nstructure bounded_derived_category :=\n(val : bounded_homotopy_category A)\n[proj : homotopy_category.is_K_projective val.val]\n\nvariable {A}\n@[ext]\nstructure bounded_derived_category_hom (X Y : bounded_derived_category A) :=\n(val : X.val \u27f6 Y.val)\n\nnamespace bounded_derived_category\n\ninstance (X : bounded_derived_category A) : homotopy_category.is_K_projective X.val.val := X.proj\n\n@[simps]\ninstance : category (bounded_derived_category A) :=\n{ hom := \u03bb X Y, bounded_derived_category_hom X Y,\n  id := \u03bb X, \u27e8\ud835\udfd9 X.val\u27e9,\n  comp := \u03bb X Y Z f g, \u27e8f.val \u226b g.val\u27e9,\n  id_comp' := \u03bb X Y f, by { ext1, apply category.id_comp _ },\n  comp_id' := \u03bb X Y f, by { ext1, apply category.comp_id _ },\n  assoc' := \u03bb X Y Z W f g h, by { ext1, apply category.assoc _ _ _ } }\n\nvariable (A)\ndef forget : bounded_derived_category A \u2964 bounded_homotopy_category A :=\n{ obj := \u03bb X, X.val,\n  map := \u03bb X Y f, f.val, }\n\nvariable {A}\n\n@[simp] lemma forget_map_mk {X Y : bounded_derived_category A} (f : X.val \u27f6 Y.val) :\n  (forget A).map { val := f } = f :=\nrfl\n\ninstance : faithful (forget A) := {}\n\ninstance : full (forget A) :=\n{ preimage := \u03bb X Y f, \u27e8f\u27e9, }\n\nvariable {A}\n@[simps]\ndef of (X : bounded_homotopy_category A) [homotopy_category.is_K_projective X.val] :\n  bounded_derived_category A := { val := X }\n\n@[simp] lemma forget_obj_of {X : bounded_homotopy_category A} [homotopy_category.is_K_projective X.val] :\n  (forget A).obj (of X) = X :=\nrfl\n\n@[simps]\ndef mk_iso {X Y : bounded_derived_category A} (i : (forget A).obj X \u2245 (forget A).obj Y) :\n  X \u2245 Y :=\n{ hom := \u27e8i.hom\u27e9,\n  inv := \u27e8i.inv\u27e9,\n  hom_inv_id' := by { ext1, simp },\n  inv_hom_id' := by { ext1, simp } }\n\nvariable (A)\n@[simps]\nnoncomputable def localization_functor :\n  bounded_homotopy_category A \u2964 bounded_derived_category A :=\n{ obj := \u03bb X, of $ X.replace,\n  map := \u03bb X Y f, \u27e8bounded_homotopy_category.lift (X.\u03c0 \u226b f) Y.\u03c0\u27e9,\n  map_id' := begin\n    intros X, symmetry, ext1, apply bounded_homotopy_category.lift_unique,\n    dsimp, simp only [category.id_comp, category.comp_id],\n  end,\n  map_comp' := begin\n    intros X Y Z f g,\n    symmetry, ext1, apply bounded_homotopy_category.lift_unique,\n    dsimp, simp only [category.assoc, bounded_homotopy_category.lift_lifts,\n      bounded_homotopy_category.lift_lifts_assoc],\n  end }\n\nopen homotopy_category\n\nlemma is_iso_localization_functor_map_of_is_quasi_iso\n  (X Y : bounded_homotopy_category A) (f : X \u27f6 Y)\n  [is_quasi_iso f] : is_iso ((localization_functor _).map f) :=\nbegin\n  use bounded_homotopy_category.lift Y.\u03c0 (X.\u03c0 \u226b f),\n  split,\n  { ext1, dsimp,\n    apply bounded_homotopy_category.lift_ext (X.\u03c0 \u226b f),\n    simp only [category.assoc, bounded_homotopy_category.lift_lifts, category.comp_id,\n      category.id_comp],\n    apply_instance },\n  { ext1, dsimp,\n    apply bounded_homotopy_category.lift_ext Y.\u03c0,\n    simp only [category.assoc, bounded_homotopy_category.lift_lifts, category.comp_id,\n      category.id_comp],\n    apply_instance }\nend\n\nopen_locale zero_object\nopen category_theory.limits\n\n-- MOVE THIS\nlemma zero_is_K_projective {X : bounded_homotopy_category A} (hX : is_zero X) :\n  is_K_projective X.val :=\nbegin\n  constructor,\n  introsI Y _ f, apply (bounded_homotopy_category.zero_val hX).eq_of_src f\nend\n\nprotected noncomputable\ndef zero : bounded_derived_category A :=\n{ val := bounded_homotopy_category.zero,\n  proj := zero_is_K_projective _ $ bounded_homotopy_category.is_zero_zero }\n\nprotected lemma is_zero_zero : limits.is_zero (bounded_derived_category.zero A) :=\n{ unique_to := \u03bb Y, nonempty.intro $ unique.mk \u27e8\u27e80\u27e9\u27e9 $ \u03bb a,\n    by { ext1, cases a, apply bounded_homotopy_category.is_zero_zero.eq_of_src },\n  unique_from := \u03bb Y, nonempty.intro $ unique.mk \u27e8\u27e80\u27e9\u27e9 $ \u03bb a,\n    by { ext1, cases a, apply bounded_homotopy_category.is_zero_zero.eq_of_tgt } }\n\ninstance has_zero_object : limits.has_zero_object (bounded_derived_category A) :=\n\u27e8\u27e8bounded_derived_category.zero A, bounded_derived_category.is_zero_zero A\u27e9\u27e9\n\n@[simps]\ndef has_shift_functor (i : \u2124) : bounded_derived_category A \u2964 bounded_derived_category A:=\n{ obj := \u03bb X,\n  { val := X.val\u27e6i\u27e7,\n    proj := by { dsimp, apply_instance } },\n  map := \u03bb X Y f, \u27e8f.val\u27e6i\u27e7'\u27e9,\n  map_id' := \u03bb X, by { ext1, dsimp, apply category_theory.functor.map_id },\n  map_comp' := \u03bb X Y Z f g, by { ext1, dsimp, apply category_theory.functor.map_comp } }\n\n@[simps] def has_shift_functor_forget (m : \u2124) :\n  has_shift_functor A m \u22d9 forget A \u2245 forget A \u22d9 shift_functor (bounded_homotopy_category A) m :=\nbegin\n  fapply nat_iso.of_components,\n  { exact \u03bb X, bounded_homotopy_category.mk_iso (by refl), },\n  { intros,\n    erw [category.id_comp, category.comp_id],\n    refl, },\nend\n\nnoncomputable instance : has_shift (bounded_derived_category A) \u2124 :=\nhas_shift_of_fully_faithful (forget A) (has_shift_functor A) (has_shift_functor_forget A)\n\n@[simp]\nlemma shift_functor_val (m : \u2124) {X Y : bounded_derived_category A} (f : X \u27f6 Y) :\n  ((shift_functor (bounded_derived_category A) m).map f).val =\n    (shift_functor (bounded_homotopy_category A) m).map f.val :=\nrfl\n\n@[simps]\nnoncomputable\ndef shift_functor_forget (m : \u2124) :\n  shift_functor (bounded_derived_category A) m \u22d9 forget A \u2245\n    forget A \u22d9 shift_functor (bounded_homotopy_category A) m :=\nhas_shift_of_fully_faithful_comm\n  (forget A) (shift_functor (bounded_derived_category A)) (has_shift_functor_forget A) m\n\n@[simps]\nnoncomputable\ndef shift_functor_localization_functor (m : \u2124) :\n  shift_functor (bounded_homotopy_category A) m \u22d9 localization_functor A \u2245\n    localization_functor A \u22d9 shift_functor (bounded_derived_category A) m :=\nbegin\n  fapply nat_iso.of_components,\n  { intros,\n    apply mk_iso,\n    refine _ \u226a\u226b ((shift_functor_forget A m).app _).symm,\n    dsimp,\n    exact\n    { hom := bounded_homotopy_category.lift ((shift_functor (bounded_homotopy_category A) m).obj X).\u03c0\n        ((shift_functor (bounded_homotopy_category A) m).map X.\u03c0),\n      inv := bounded_homotopy_category.lift ((shift_functor (bounded_homotopy_category A) m).map X.\u03c0)\n        ((shift_functor (bounded_homotopy_category A) m).obj X).\u03c0, }, },\n  { intros, ext, dsimp,\n    simp only [bounded_homotopy_category.lift_comp_lift_self_assoc, category_theory.category.assoc],\n    erw [category.comp_id, category.id_comp],\n    simp [bounded_homotopy_category.shift_functor_map_lift], },\nend\n\n-- TODO replace this by pulling back a preadditive instance along `forget`?\n@[simps]\ninstance preadditive : preadditive (bounded_derived_category A) :=\n{ hom_group := \u03bb P Q,\n  { add := \u03bb f g, \u27e8f.val + g.val\u27e9,\n    add_assoc := by { intros, ext, apply add_assoc },\n    zero := \u27e80\u27e9,\n    zero_add := by { intros, ext, apply zero_add },\n    add_zero := by { intros, ext, apply add_zero },\n    nsmul := \u03bb n f, \u27e8n \u2022 f.val\u27e9,\n    nsmul_zero' := by { intros f, ext, refine add_comm_monoid.nsmul_zero' f.val, },\n    nsmul_succ' := by { intros n f, ext, exact add_comm_monoid.nsmul_succ' _ f.val },\n    neg := \u03bb f, \u27e8-f.val\u27e9,\n    sub := \u03bb f g, \u27e8f.val - g.val\u27e9,\n    sub_eq_add_neg := by { intros, ext, apply sub_eq_add_neg },\n    zsmul := \u03bb n f, \u27e8n \u2022 f.val\u27e9,\n    zsmul_zero' := by { intros f, ext, apply add_comm_group.zsmul_zero' f.val },\n    zsmul_succ' := by { intros n f, ext, apply add_comm_group.zsmul_succ' _ f.val },\n    zsmul_neg' := by { intros n f, ext, apply add_comm_group.zsmul_neg' _ f.val },\n    add_left_neg := by { intros, ext, apply add_left_neg },\n    add_comm := by { intros, ext, apply add_comm } },\n  add_comp' :=\n    by { intros P Q R f\u2081 f\u2082 g, ext, apply preadditive.add_comp _ _ _ f\u2081.val f\u2082.val g.val },\n  comp_add' :=\n    by { intros P Q R f g\u2081 g\u2082, ext, apply preadditive.comp_add _ _ _ f.val g\u2081.val g\u2082.val } }\n\ninstance additive (n : \u2124) : (shift_functor (bounded_derived_category A) n).additive :=\n{ map_add' := begin\n    intros P Q f g,\n    ext1,\n    dsimp,\n    erw \u2190 (shift_functor (bounded_homotopy_category A) n).map_add,\n  end }\n\nvariable (A)\n@[simps]\nnoncomputable def forget_triangulated_functor_struct :\n  triangulated.pretriangulated.triangulated_functor_struct\n    (bounded_derived_category A) (bounded_homotopy_category A) :=\n{ to_functor := forget A,\n  comm_shift := nat_iso.of_components (\u03bb X, by refl) (by tidy), }\n\nvariable {A}\n@[simps]\ndef lift {C : Type*} [category C] (F : bounded_homotopy_category A \u2964 C) :\n  bounded_derived_category A \u2964 C :=\n{ obj := \u03bb X, F.obj X.val,\n  map := \u03bb X Y f, F.map f.val,\n  map_id' := \u03bb X, F.map_id _,\n  map_comp' := \u03bb X Y Z f g, F.map_comp _ _ }\n\nnoncomputable\ndef localize_lift {C : Type*} [category C]\n  (F : bounded_homotopy_category A \u2964 C)\n  [\u2200 (X Y : bounded_homotopy_category A) (f : X \u27f6 Y)\n    [h : is_quasi_iso f], is_iso (F.map f)] :\n  localization_functor A \u22d9 lift F \u2245 F :=\nnat_iso.of_components\n(\u03bb X, as_iso $ F.map X.\u03c0)\nbegin\n  intros X Y f,\n  dsimp,\n  simp only [\u2190 F.map_comp],\n  congr' 1,\n  rw bounded_homotopy_category.lift_lifts,\nend\n\n@[simp, reassoc] lemma \u03c0_lift_id_\u03c0 (X : bounded_derived_category A) :\n  X.val.\u03c0 \u226b bounded_homotopy_category.lift (\ud835\udfd9 X.val) X.val.\u03c0 = \ud835\udfd9 X.val.replace :=\nbegin\n  refine bounded_homotopy_category.lift_ext X.val.\u03c0 _ _ _,\n  rw [category.assoc, bounded_homotopy_category.lift_lifts, category.id_comp, category.comp_id],\nend\n\n@[simps]\nnoncomputable\ndef localization_iso (X : bounded_derived_category A) :\n  (localization_functor A).obj X.val \u2245 X :=\n{ hom := \u27e8X.val.\u03c0\u27e9,\n  inv := \u27e8bounded_homotopy_category.lift (\ud835\udfd9 _) X.val.\u03c0\u27e9, }\n\nnoncomputable\ndef lift_unique {C : Type*} [category C]\n  (F : bounded_homotopy_category A \u2964 C)\n  [\u2200 (X Y : bounded_homotopy_category A) (f : X \u27f6 Y)\n    [h : is_quasi_iso f], is_iso (F.map f)]\n  (G : bounded_derived_category A \u2964 C)\n  (e : F \u2245 localization_functor A \u22d9 G) :\n  lift F \u2245 G :=\nnat_iso.of_components\n(\u03bb X, e.app X.val \u226a\u226b G.map_iso (localization_iso _))\nbegin\n  intros X Y f,\n  simp only [lift_map, iso.trans_hom, iso.app_hom, functor.map_iso_hom, nat_trans.naturality_assoc,\n    functor.comp_map, category.assoc, nat_iso.cancel_nat_iso_hom_left],\n  rw [\u2190 functor.map_comp, \u2190 functor.map_comp],\n  congr' 1,\n  ext,\n  simp only [category_theory.category_comp_val, localization_functor_map_val,\n    localization_iso_hom_val, bounded_homotopy_category.lift_lifts],\nend\n\nopen category_theory.triangulated\n\nvariable {A}\n@[simps obj\u2081 obj\u2082 obj\u2083 mor\u2081 mor\u2082 mor\u2083]\nnoncomputable\ndef replace_triangle (S : triangle (bounded_homotopy_category A)) :\n  triangle (bounded_derived_category A) :=\n{ obj\u2081 := of $ S.obj\u2081.replace,\n  obj\u2082 := of $ S.obj\u2082.replace,\n  obj\u2083 := of $ S.obj\u2083.replace,\n  mor\u2081 := \u27e8bounded_homotopy_category.lift (S.obj\u2081.\u03c0 \u226b S.mor\u2081) S.obj\u2082.\u03c0\u27e9,\n  mor\u2082 := \u27e8bounded_homotopy_category.lift (S.obj\u2082.\u03c0 \u226b S.mor\u2082) S.obj\u2083.\u03c0\u27e9,\n  mor\u2083 := begin\n    haveI : is_quasi_iso\n      ((shift_functor (bounded_homotopy_category A) (1 : \u2124)).map S.obj\u2081.\u03c0) :=\n    by { change is_quasi_iso ((S.obj\u2081.\u03c0)\u27e6(1 : \u2124)\u27e7'), by apply_instance }, -- WAT?\n    exact \u27e8bounded_homotopy_category.lift (S.obj\u2083.\u03c0 \u226b S.mor\u2083) (S.obj\u2081.\u03c0\u27e6(1 : \u2124)\u27e7')\u27e9,\n  end }\n\n@[simps]\nnoncomputable\ndef replace_triangle_map {S T : triangle (bounded_homotopy_category A)} (f : S \u27f6 T) :\n  replace_triangle S \u27f6 replace_triangle T :=\n{ hom\u2081 := \u27e8bounded_homotopy_category.lift (S.obj\u2081.\u03c0 \u226b f.hom\u2081) T.obj\u2081.\u03c0\u27e9,\n  hom\u2082 := \u27e8bounded_homotopy_category.lift (S.obj\u2082.\u03c0 \u226b f.hom\u2082) T.obj\u2082.\u03c0\u27e9,\n  hom\u2083 := \u27e8bounded_homotopy_category.lift (S.obj\u2083.\u03c0 \u226b f.hom\u2083) T.obj\u2083.\u03c0\u27e9,\n  comm\u2081' := by { ext, dsimp, simp only [triangle_morphism.comm\u2081, category.assoc,\n    bounded_homotopy_category.lift_comp_lift_comp], },\n  comm\u2082' := by { ext, dsimp, simp only [triangle_morphism.comm\u2082, category.assoc,\n    bounded_homotopy_category.lift_comp_lift_comp], },\n  comm\u2083' := begin\n    ext, dsimp,\n    rw [bounded_homotopy_category.shift_functor_map_lift, category_theory.functor.map_comp,\n      bounded_homotopy_category.lift_comp_lift_comp, bounded_homotopy_category.lift_comp_lift_comp,\n      category.assoc, triangle_morphism.comm\u2083, category.assoc],\n  end, }\n\n.\n\nlemma replace_triangle_map_id (X : triangle (bounded_homotopy_category A)) :\n  replace_triangle_map (\ud835\udfd9 X) = \ud835\udfd9 (replace_triangle X) :=\nby tidy\n\nlemma replace_triangle_map_comp {X Y Z : triangle (bounded_homotopy_category A)}\n  (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  replace_triangle_map (f \u226b g) = replace_triangle_map f \u226b replace_triangle_map g :=\nby ext; tidy\n\nnoncomputable\ndef replace_triangle' : triangle (bounded_homotopy_category A) \u2964 triangle (bounded_derived_category A) :=\n{ obj := replace_triangle,\n  map := \u03bb S T f, replace_triangle_map f,\n  map_id' := replace_triangle_map_id,\n  map_comp' := \u03bb X Y Z f g, replace_triangle_map_comp f g, }\n\nattribute [simps obj_obj\u2081 obj_obj\u2082 obj_obj\u2083 obj_mor\u2081 obj_mor\u2082 obj_mor\u2083] replace_triangle'\nattribute [simps map_hom\u2081 map_hom\u2082 map_hom\u2083] replace_triangle'\n\nnoncomputable\ndef replace_triangle_rotate (S : triangle (bounded_homotopy_category A)) :\n  (replace_triangle S).rotate \u2245 replace_triangle S.rotate :=\nbegin\n  fapply triangle.iso.of_components,\n  exact iso.refl _,\n  exact iso.refl _,\n  exact ((shift_functor_localization_functor A 1).app S.obj\u2081).symm,\n  { ext, dsimp, simp, },\n  { ext, dsimp, erw [category.id_comp, category.id_comp], simp, },\n  { ext, dsimp,\n    simp only [bounded_homotopy_category.lift_neg, bounded_homotopy_category.lift_comp_lift_comp,\n      preadditive.comp_neg, preadditive.neg_comp, neg_inj, category.assoc,\n      category_theory.functor.map_id],\n   erw [category.id_comp, category.comp_id],\n   simp [bounded_homotopy_category.shift_functor_map_lift], },\nend\n\n@[simps]\nnoncomputable def forget_replace_triangle (S : triangle (bounded_homotopy_category A)) :\n  (forget_triangulated_functor_struct A).map_triangle.obj (replace_triangle S) \u2245\n    bounded_homotopy_category.replace_triangle S :=\nbegin\n  fapply triangle.iso.of_components,\n  apply iso.refl _,\n  apply iso.refl _,\n  apply iso.refl _,\n  all_goals { dsimp, simp, },\nend\n\nvariable (A)\n\ndef pretriangulated_distinguished_triangles :=\n { T |\n    \u2203 (S : triangle (bounded_homotopy_category A))\n      (hS : S \u2208 dist_triang (bounded_homotopy_category A))\n      (f : T \u2245 replace_triangle S), true }\n\nvariable {A}\n\nlemma isomorphic_distinguished (T\u2081 : triangle (bounded_derived_category A))\n  (m : T\u2081 \u2208 pretriangulated_distinguished_triangles A)\n  (T\u2082 : triangle (bounded_derived_category A)) (i : T\u2082 \u2245 T\u2081) :\n  T\u2082 \u2208 pretriangulated_distinguished_triangles A :=\nbegin\n  obtain \u27e8S\u2081, hS\u2081, f\u2081, hf\u2081\u27e9 := m,\n  exact \u27e8S\u2081, hS\u2081, i \u226a\u226b f\u2081, trivial\u27e9,\nend\n\nlemma forget_replace_triangle_distinguished (S : triangle (bounded_homotopy_category A))\n  (m : S \u2208 dist_triang (bounded_homotopy_category A)) :\n  (forget_triangulated_functor_struct A).map_triangle.obj (replace_triangle S) \u2208 dist_triang (bounded_homotopy_category A) :=\npretriangulated.isomorphic_distinguished\n  _ (bounded_homotopy_category.distinguished_replace_triangle S m)\n  _ (forget_replace_triangle S)\n\nlemma forget_distinguished_of_distinguished\n  {T : triangle (bounded_derived_category A)} (m : T \u2208 pretriangulated_distinguished_triangles A) :\n  (forget_triangulated_functor_struct A).map_triangle.obj T \u2208 dist_triang (bounded_homotopy_category A) :=\nbegin\n  obtain \u27e8S, hS, f, -\u27e9 := m,\n  exact pretriangulated.isomorphic_distinguished _ (forget_replace_triangle_distinguished _ hS)\n    _ ((forget_triangulated_functor_struct A).map_triangle.map_iso f),\nend\n\nlemma pretriangulated_contractible_distinguished (X : bounded_derived_category A) :\n  contractible_triangle (bounded_derived_category A) X \u2208\n    pretriangulated_distinguished_triangles A :=\nbegin\n  refine \u27e8contractible_triangle _ X.val, pretriangulated.contractible_distinguished _, \u27e8_, trivial\u27e9\u27e9,\n  symmetry,\n  fapply triangle.iso.of_components,\n  exact localization_iso X,\n  exact localization_iso X,\n  refine _ \u226a\u226b localization_iso 0,\n  { dsimp,\n    refine (localization_functor _).map_iso _,\n    refine \u27e80,0,_,_\u27e9,\n    simp only [eq_iff_true_of_subsingleton],\n    simp only [zero_comp, auto_param_eq],\n    erw \u2190 (forget A).map_id,\n    simp only [id_zero, functor.map_zero] },\n  { ext,\n    dsimp,\n    simp only [bounded_homotopy_category.lift_lifts] },\n  { ext,\n    dsimp,\n    simp only [bounded_homotopy_category.lift_lifts, category.assoc, comp_zero] },\n  { ext,\n    dsimp,\n    simp only [bounded_homotopy_category.lift_lifts, comp_zero] },\nend\n\n@[simp]\nlemma shift_functor_map_val (m : \u2124) {X Y : bounded_derived_category A} (f : X \u27f6 Y) :\n  ((shift_functor (bounded_derived_category A) m).map f).val =\n    (shift_functor (bounded_homotopy_category A) m).map f.val :=\nrfl\n\nlemma pretriangulated_distinguished_cocone_triangle\n  {X Y : bounded_derived_category A}\n  (f : X \u27f6 Y) :\n  \u2203 (Z : bounded_derived_category A) (g : Y \u27f6 Z)\n    (h : Z \u27f6 (shift_functor (bounded_derived_category A) 1).obj X),\n    triangle.mk (bounded_derived_category A) f g h \u2208\n      pretriangulated_distinguished_triangles A :=\nbegin\n  obtain \u27e8Z, g, h, m\u27e9 := pretriangulated.distinguished_cocone_triangle _ _ f.val,\n  use (localization_functor A).obj Z,\n  use (localization_iso Y).inv \u226b (localization_functor A).map g,\n  refine \u27e8(localization_functor A).map (h \u226b eq_to_hom (by refl)) \u226b (localization_iso _).hom, _\u27e9,\n  refine \u27e8_, m, \u27e8_, trivial\u27e9\u27e9,\n  symmetry,\n  fapply triangle.iso.of_components,\n  { exact localization_iso _, },\n  { exact localization_iso _, },\n  { exact iso.refl _, },\n  { ext, dsimp, simp only [bounded_homotopy_category.lift_lifts], },\n  { ext, dsimp,\n    simp only [category.comp_id, bounded_derived_category.\u03c0_lift_id_\u03c0_assoc], },\n  { ext, dsimp,\n    simp only [category.comp_id, category.id_comp, bounded_homotopy_category.lift_lifts], },\nend\n\nlemma rotate_distinguished_triangle (T : triangle (bounded_derived_category A)) :\n  T \u2208 pretriangulated_distinguished_triangles A \u2194\n    T.rotate \u2208 pretriangulated_distinguished_triangles A :=\nbegin\n  split,\n  { rintro \u27e8S, hS, f, -\u27e9,\n    use S.rotate,\n    refine \u27e8pretriangulated.rot_of_dist_triangle _ _ hS, _, trivial\u27e9,\n    exact (rotate _).map_iso f \u226a\u226b replace_triangle_rotate _, },\n  { rintro \u27e8S, hS, f, -\u27e9,\n    use S.inv_rotate,\n    refine \u27e8pretriangulated.inv_rot_of_dist_triangle _ _ hS, _, trivial\u27e9,\n\n    apply (iso_equiv_of_fully_faithful (rotate (bounded_derived_category A))).inv_fun,\n    refine f \u226a\u226b _ \u226a\u226b (replace_triangle_rotate _).symm,\n    apply replace_triangle'.map_iso,\n    exact (triangle_rotation _).counit_iso.symm.app S, },\nend\n\nlemma complete_distinguished_triangle_morphism (T\u2081 T\u2082 : triangle (bounded_derived_category A))\n    (m\u2081 : T\u2081 \u2208 pretriangulated_distinguished_triangles A)\n    (m\u2082 : T\u2082 \u2208 pretriangulated_distinguished_triangles A)\n    (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081) (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082)\n    (comm : T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081) :\n      (\u2203 (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083), T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082 \u2227\n        T\u2081.mor\u2083 \u226b (shift_functor (bounded_derived_category A) 1).map a = c \u226b T\u2082.mor\u2083) :=\nbegin\n  -- We work formally, just using the fact this is true in the bounded homotopy category,\n  -- without needing to care why.\n  obtain \u27e8c', h1, h2\u27e9 := pretriangulated.complete_distinguished_triangle_morphism\n    ((forget_triangulated_functor_struct A).map_triangle.obj T\u2081)\n    ((forget_triangulated_functor_struct A).map_triangle.obj T\u2082)\n    (forget_distinguished_of_distinguished m\u2081)\n    (forget_distinguished_of_distinguished m\u2082) ((forget A).map a) ((forget A).map b)\n    (congr_arg bounded_derived_category_hom.val comm),\n  use c',\n  dsimp at h1 h2,\n  split,\n  { apply (forget A).map_injective,\n    simpa only [(forget A).map_comp] using h1, },\n  { apply (forget A).map_injective,\n    simp only [category_theory.category.comp_id] at h2,\n    simp only [(forget A).map_comp],\n    exact h2, },\nend\n\nvariable (A)\n\ninstance pretriangulated : triangulated.pretriangulated (bounded_derived_category A) :=\n{ distinguished_triangles := pretriangulated_distinguished_triangles A,\n  isomorphic_distinguished := isomorphic_distinguished,\n  contractible_distinguished := pretriangulated_contractible_distinguished,\n  distinguished_cocone_triangle := \u03bb X Y f, pretriangulated_distinguished_cocone_triangle f,\n  rotate_distinguished_triangle := rotate_distinguished_triangle,\n  complete_distinguished_triangle_morphism := complete_distinguished_triangle_morphism, }\n\nvariable (A)\nnoncomputable\ndef Ext (n : \u2124) : (bounded_derived_category A)\u1d52\u1d56 \u2964 bounded_derived_category A \u2964 Ab :=\nfunctor.flip $ shift_functor _ n \u22d9 preadditive_yoneda\n\n@[simp]\nlemma Ext_obj_obj (n : \u2124) (X : (bounded_derived_category A)\u1d52\u1d56) (Y : bounded_derived_category A) :\n  ((Ext A n).obj X).obj Y = AddCommGroup.of (X.unop \u27f6 Y\u27e6n\u27e7) := rfl\n\n@[simp]\nlemma Ext_map_app_apply (n : \u2124) {X Y : (bounded_derived_category A)\u1d52\u1d56}\n  (f : X \u27f6 Y) (Z : bounded_derived_category A) (e : ((Ext A n).obj X).obj Z) :\n  ((Ext A n).map f).app Z e = f.unop \u226b e := rfl\n\n@[simp]\nlemma Ext_obj_map (n : \u2124) (X : (bounded_derived_category A)\u1d52\u1d56) {Y Z : bounded_derived_category A}\n  (f : Y \u27f6 Z) (e : ((Ext A n).obj X).obj Y) : ((Ext A n).obj X).map f e =\n  e \u226b f\u27e6n\u27e7' := rfl\n\nend bounded_derived_category\n\n/-\n0 \u2192 A \u2192 B \u2192 C \u2192 0\n\nA -f\u2192 B \u2192 Cone(f) \u2192 A[1]\n\nCanonical Cone(f) \u2192 C quasi iso\n\n-/\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived/derived_cat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3906865664442856}}
{"text": "/-\n  First argument of the ring lemma. \n  \n  We show that the map from R[1/f] to R[1/f\u1d62] inverts f\u1d62/1. \n-/\n\nimport ring_theory.localization\nimport to_mathlib.localization.localization_alt\nimport spectrum_of_a_ring.structure_presheaf\nimport spectrum_of_a_ring.structure_presheaf_localization\nimport spectrum_of_a_ring.structure_presheaf_res\n\nuniverse u\n\nlocal attribute [instance] classical.prop_decidable\n\nnoncomputable theory\n\nsection structure_presheaf\n\nopen topological_space\nopen classical\nopen localization\nopen localization_alt\n\nvariables {R : Type u} [comm_ring R]\nvariables {U V : opens (Spec R)} (BU : U \u2208 D_fs R) (BV : V \u2208 D_fs R) (H : V \u2286 U)\n\nlemma structure_presheaf.res.inverts_data\n: inverts_data \n    (powers (of (some BV))) \n    (structure_presheaf_on_basis.res BU BV H) :=\nbegin\n  rintros \u27e8s, Hs\u27e9,\n  rcases (indefinite_description _ Hs) with \u27e8n, Hn\u27e9,\n  rw \u2190@is_semiring_hom.map_pow R _ _ _ of (@is_ring_hom.is_semiring_hom _ _ _ _ of of.is_ring_hom) _ _ at Hn,\n  dsimp only [subtype.coe_mk],\n  rw \u2190Hn,\n  dsimp [structure_presheaf_on_basis.res],\n  rw is_localization_initial_comp,\n  exact structure_presheaf.inverts_data BV \u27e8(some BV)^n, \u27e8n, rfl\u27e9\u27e9,\nend\n\nlemma structure_presheaf.res.has_denom_data\n: has_denom_data \n    (powers (of (some BV))) \n    (structure_presheaf_on_basis.res BU BV H) :=\nbegin\n  intros x,\n  rcases (structure_presheaf.has_denom_data BV x) with \u27e8\u27e8\u27e8q, Hq\u27e9, p\u27e9, Hpq\u27e9,\n  rcases (indefinite_description _ Hq) with \u27e8n, Hn\u27e9,\n  dsimp only [subtype.coe_mk] at Hpq,\n  let p' : localization R (S U) := of p,\n  let q' : localization R (S U) := (of ((some BV)^n)),\n  have Hq' : q' \u2208 powers ((of : R \u2192 localization R (S U)) (some BV)),\n    dsimp [q'],\n    rw is_semiring_hom.map_pow (of : R \u2192 localization R (S U)),\n    exact \u27e8n, rfl\u27e9,\n  use \u27e8\u27e8q', Hq'\u27e9, p'\u27e9,\n  dsimp only [subtype.coe_mk, q', p', structure_presheaf_on_basis.res],\n  iterate 2 { rw is_localization_initial_comp, },\n  rw Hn,\n  exact Hpq,\nend\n\nlemma structure_presheaf.res.ker_le\n: ker (structure_presheaf_on_basis.res BU BV H) \u2264 submonoid_ann (powers (of (some BV))) :=\nbegin \n  intros x Hx,\n  change structure_presheaf_on_basis.res BU BV H x = 0 at Hx,\n  rcases (structure_presheaf.has_denom_data BU x) with \u27e8\u27e8\u27e8q, Hq\u27e9, p\u27e9, Hpq\u27e9,\n  rcases (indefinite_description _ Hq) with \u27e8n, Hn\u27e9,\n  dsimp only [subtype.coe_mk] at Hpq,\n  have Hofp : structure_presheaf_on_basis.res BU BV H (of p) = 0,\n    rw [\u2190Hpq, is_ring_hom.map_mul (structure_presheaf_on_basis.res BU BV H), Hx, mul_zero],\n  dsimp only [structure_presheaf_on_basis.res] at Hofp,\n  rw is_localization_initial_comp at Hofp,\n  have Hpker : p \u2208 ker (of : R \u2192 localization R (S V)) := Hofp,\n  have Hpann := (structure_presheaf.ker_le BV) Hpker,\n  rcases Hpann with \u27e8\u27e8\u27e8u, \u27e8v, \u27e8m, Hm\u27e9\u27e9\u27e9, Huv\u27e9, Hp\u27e9,\n  dsimp only [subtype.coe_mk] at Huv,\n  dsimp only [subtype.coe_mk] at Hp,\n  rw [Hp, \u2190Hm] at Huv,\n  rcases (indefinite_description _ (pow_eq.of_basis_subset BU BV H)) with \u27e8a, Ha\u27e9,\n  rcases (indefinite_description _ Ha) with \u27e8e, Hea\u27e9,\n  rw mul_comm at Hea,\n  dsimp only [submonoid_ann, set.range, ann_aux],\n  let g' : localization R (S U) := (of (some BV))^(e * n + m),\n  let Hg' : g' \u2208 powers ((of : R \u2192 localization R (S U)) (some BV)),\n    dsimp [g'],\n    exact \u27e8(e * n + m), rfl\u27e9,\n  have Hxann : x * g' = 0,\n    dsimp [g'],\n    rw [\u2190is_semiring_hom.map_pow (of : R \u2192 localization R (S U)), pow_add],\n    rw [is_ring_hom.map_mul (of : R \u2192 localization R (S U)), pow_mul, Hea, mul_pow],\n    rw [is_ring_hom.map_mul (of : R \u2192 localization R (S U))],\n    rw [\u2190mul_assoc, \u2190mul_assoc, Hn, mul_comm x, Hpq, mul_comm (of p), mul_assoc],\n    rw [\u2190is_ring_hom.map_mul (of : R \u2192 localization R (S U)), Huv],\n    rw [is_ring_hom.map_zero (of : R \u2192 localization R (S U)), mul_zero],\n  use \u27e8\u27e8x, \u27e8g', Hg'\u27e9\u27e9, Hxann\u27e9,\nend\n\nlemma structure_presheaf.res.localization\n: is_localization_data \n    (powers (of (some BV))) \n    (structure_presheaf_on_basis.res BU BV H) :=\n{ inverts := structure_presheaf.res.inverts_data BU BV H,\n  has_denom := structure_presheaf.res.has_denom_data BU BV H, \n  ker_le := structure_presheaf.res.ker_le BU BV H }\n\nend structure_presheaf\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/spectrum_of_a_ring/structure_sheaf_locality.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947425132314, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.39060093174932836}}
{"text": "/-\nCopyright (c) 2018 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Robert Y. Lewis\n\nA tactic for discharging linear arithmetic goals using Fourier-Motzkin elimination.\n\n`linarith` is (in principle) complete for \u211a and \u211d. It is not complete for non-dense orders, i.e. \u2124.\n\n@TODO: investigate storing comparisons in a list instead of a set, for possible efficiency gains\n@TODO: delay proofs of denominator normalization and nat casting until after contradiction is found\n-/\n\nimport tactic.ring data.nat.gcd data.list.basic meta.rb_map\n\nmeta def nat.to_pexpr : \u2115 \u2192 pexpr\n| 0 := ``(0)\n| 1 := ``(1)\n| n := if n % 2 = 0 then ``(bit0 %%(nat.to_pexpr (n/2))) else ``(bit1 %%(nat.to_pexpr (n/2)))\n\nopen native\nnamespace linarith\n\nsection lemmas\n\nlemma int.coe_nat_bit0 (n : \u2115) : (\u2191(bit0 n : \u2115) : \u2124) = bit0 (\u2191n : \u2124) := by simp [bit0]\nlemma int.coe_nat_bit1 (n : \u2115) : (\u2191(bit1 n : \u2115) : \u2124) = bit1 (\u2191n : \u2124) := by simp [bit1, bit0]\nlemma int.coe_nat_bit0_mul (n : \u2115) (x : \u2115) : (\u2191(bit0 n * x) : \u2124) = (\u2191(bit0 n) : \u2124) * (\u2191x : \u2124) := by simp\nlemma int.coe_nat_bit1_mul (n : \u2115) (x : \u2115) : (\u2191(bit1 n * x) : \u2124) = (\u2191(bit1 n) : \u2124) * (\u2191x : \u2124) := by simp\nlemma int.coe_nat_one_mul (x : \u2115) : (\u2191(1 * x) : \u2124) = 1 * (\u2191x : \u2124) := by simp\nlemma int.coe_nat_zero_mul (x : \u2115) : (\u2191(0 * x) : \u2124) = 0 * (\u2191x : \u2124) := by simp\nlemma int.coe_nat_mul_bit0 (n : \u2115) (x : \u2115) : (\u2191(x * bit0 n) : \u2124) = (\u2191x : \u2124) * (\u2191(bit0 n) : \u2124) := by simp\nlemma int.coe_nat_mul_bit1 (n : \u2115) (x : \u2115) : (\u2191(x * bit1 n) : \u2124) = (\u2191x : \u2124) * (\u2191(bit1 n) : \u2124) := by simp\nlemma int.coe_nat_mul_one (x : \u2115) : (\u2191(x * 1) : \u2124) = (\u2191x : \u2124) * 1 := by simp\nlemma int.coe_nat_mul_zero (x : \u2115) : (\u2191(x * 0) : \u2124) = (\u2191x : \u2124) * 0 := by simp\n\nlemma nat_eq_subst {n1 n2 : \u2115} {z1 z2 : \u2124} (hn : n1 = n2) (h1 : \u2191n1 = z1) (h2 : \u2191n2 = z2) : z1 = z2 :=\nby simpa [eq.symm h1, eq.symm h2, int.coe_nat_eq_coe_nat_iff]\n\nlemma nat_le_subst {n1 n2 : \u2115} {z1 z2 : \u2124} (hn : n1 \u2264 n2) (h1 : \u2191n1 = z1) (h2 : \u2191n2 = z2) : z1 \u2264 z2 :=\nby simpa [eq.symm h1, eq.symm h2, int.coe_nat_le]\n\nlemma nat_lt_subst {n1 n2 : \u2115} {z1 z2 : \u2124} (hn : n1 < n2) (h1 : \u2191n1 = z1) (h2 : \u2191n2 = z2) : z1 < z2 :=\nby simpa [eq.symm h1, eq.symm h2, int.coe_nat_lt]\n\nlemma eq_of_eq_of_eq {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a = 0) (hb : b = 0) : a + b = 0 :=\nby simp *\n\nlemma le_of_eq_of_le {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a = 0) (hb : b \u2264 0) : a + b \u2264 0 :=\nby simp *\n\nlemma lt_of_eq_of_lt {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a = 0) (hb : b < 0) : a + b < 0 :=\nby simp *\n\nlemma le_of_le_of_eq {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a \u2264 0) (hb : b = 0) : a + b \u2264 0 :=\nby simp *\n\nlemma lt_of_lt_of_eq {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a < 0) (hb : b = 0) : a + b < 0 :=\nby simp *\n\nlemma mul_neg {\u03b1} [ordered_ring \u03b1] {a b : \u03b1} (ha : a < 0) (hb : b > 0) : b * a < 0 :=\nhave (-b)*a > 0, from mul_pos_of_neg_of_neg (neg_neg_of_pos hb) ha,\nneg_of_neg_pos (by simpa)\n\nlemma mul_nonpos {\u03b1} [ordered_ring \u03b1] {a b : \u03b1} (ha : a \u2264 0) (hb : b > 0) : b * a \u2264 0 :=\nhave (-b)*a \u2265 0, from mul_nonneg_of_nonpos_of_nonpos (le_of_lt (neg_neg_of_pos hb)) ha,\nnonpos_of_neg_nonneg (by simp at this; exact this)\n\nlemma mul_eq {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (ha : a = 0) (hb : b > 0) : b * a = 0 :=\nby simp *\n\nlemma eq_of_not_lt_of_not_gt {\u03b1} [linear_order \u03b1] (a b : \u03b1) (h1 : \u00ac a < b) (h2 : \u00ac b < a) : a = b :=\nle_antisymm (le_of_not_gt h2) (le_of_not_gt h1)\n\nlemma add_subst {\u03b1} [ring \u03b1] {n e1 e2 t1 t2 : \u03b1} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n      n * (e1 + e2) = t1 + t2 := by simp [left_distrib, *]\n\nlemma sub_subst {\u03b1} [ring \u03b1] {n e1 e2 t1 t2 : \u03b1} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :\n      n * (e1 - e2) = t1 - t2 := by simp [left_distrib, *]\n\nlemma neg_subst {\u03b1} [ring \u03b1] {n e t : \u03b1} (h1 : n * e = t) : n * (-e) = -t := by simp *\n\nprivate meta def apnn : tactic unit := `[norm_num]\n\nlemma mul_subst {\u03b1} [comm_ring \u03b1] {n1 n2 k e1 e2 t1 t2 : \u03b1} (h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2)\n     (h3 : n1*n2 = k . apnn) : k * (e1 * e2) = t1 * t2 :=\nhave h3 : n1 * n2 = k, from h3,\nby rw [\u2190h3, mul_comm n1, mul_assoc n2, \u2190mul_assoc n1, h1, \u2190mul_assoc n2, mul_comm n2, mul_assoc, h2] -- OUCH\n\nlemma div_subst {\u03b1} [field \u03b1] {n1 n2 k e1 e2 t1 : \u03b1} (h1 : n1 * e1 = t1) (h2 : n2 / e2 = 1) (h3 : n1*n2 = k) :\n      k * (e1 / e2) = t1 :=\nby rw [\u2190h3, mul_assoc, mul_div_comm, h2, \u2190mul_assoc, h1, mul_comm, one_mul]\n\nend lemmas\n\nsection datatypes\n\n@[derive decidable_eq]\ninductive ineq\n| eq | le | lt\n\nopen ineq\n\ndef ineq.max : ineq \u2192 ineq \u2192 ineq\n| eq a := a\n| le a := a\n| lt a := lt\n\ndef ineq.is_lt : ineq \u2192 ineq \u2192 bool\n| eq le := tt\n| eq lt := tt\n| le lt := tt\n| _ _ := ff\n\ndef ineq.to_string : ineq \u2192 string\n| eq := \"=\"\n| le := \"\u2264\"\n| lt := \"<\"\n\ninstance : has_to_string ineq := \u27e8ineq.to_string\u27e9\n\n/--\n  The main datatype for FM elimination.\n  Variables are represented by natural numbers, each of which has an integer coefficient.\n  Index 0 is reserved for constants, i.e. `coeffs.find 0` is the coefficient of 1.\n  The represented term is coeffs.keys.sum (\u03bb i, coeffs.find i * Var[i]).\n  str determines the direction of the comparison -- is it < 0, \u2264 0, or = 0?\n-/\nmeta structure comp :=\n(str : ineq)\n(coeffs : rb_map \u2115 int)\n\nmeta instance : inhabited comp := \u27e8\u27e8ineq.eq, mk_rb_map\u27e9\u27e9\n\nmeta inductive comp_source\n| assump : \u2115 \u2192 comp_source\n| add : comp_source \u2192 comp_source \u2192 comp_source\n| scale : \u2115 \u2192 comp_source \u2192 comp_source\n\nmeta def comp_source.flatten : comp_source \u2192 rb_map \u2115 \u2115\n| (comp_source.assump n) := mk_rb_map.insert n 1\n| (comp_source.add c1 c2) := (comp_source.flatten c1).add (comp_source.flatten c2)\n| (comp_source.scale n c) := (comp_source.flatten c).map (\u03bb v, v * n)\n\nmeta def comp_source.to_string : comp_source \u2192 string\n| (comp_source.assump e) := to_string e\n| (comp_source.add c1 c2) := comp_source.to_string c1 ++ \" + \" ++ comp_source.to_string c2\n| (comp_source.scale n c) := to_string n ++ \" * \" ++ comp_source.to_string c\n\nmeta instance comp_source.has_to_format : has_to_format comp_source :=\n\u27e8\u03bb a, comp_source.to_string a\u27e9\n\nmeta structure pcomp :=\n(c : comp)\n(src : comp_source)\n\nmeta def map_lt (m1 m2 : rb_map \u2115 int) : bool :=\nlist.lex (prod.lex (<) (<)) m1.to_list m2.to_list\n\n-- make more efficient\nmeta def comp.lt (c1 c2 : comp) : bool :=\n(c1.str.is_lt c2.str) || (c1.str = c2.str) && map_lt c1.coeffs c2.coeffs\n\nmeta instance comp.has_lt : has_lt comp := \u27e8\u03bb a b, comp.lt a b\u27e9\nmeta instance pcomp.has_lt : has_lt pcomp := \u27e8\u03bb p1 p2, p1.c < p2.c\u27e9\nmeta instance pcomp.has_lt_dec : decidable_rel ((<) : pcomp \u2192 pcomp \u2192 Prop) := by apply_instance\n\nmeta def comp.coeff_of (c : comp) (a : \u2115) : \u2124 :=\nc.coeffs.zfind a\n\nmeta def comp.scale (c : comp) (n : \u2115) : comp :=\n{ c with coeffs := c.coeffs.map ((*) (n : \u2124)) }\n\nmeta def comp.add (c1 c2 : comp) : comp :=\n\u27e8c1.str.max c2.str, c1.coeffs.add c2.coeffs\u27e9\n\nmeta def pcomp.scale (c : pcomp) (n : \u2115) : pcomp :=\n\u27e8c.c.scale n, comp_source.scale n c.src\u27e9\n\nmeta def pcomp.add (c1 c2 : pcomp) : pcomp :=\n\u27e8c1.c.add c2.c, comp_source.add c1.src c2.src\u27e9\n\nmeta instance pcomp.to_format : has_to_format pcomp :=\n\u27e8\u03bb p, to_fmt p.c.coeffs ++ to_string p.c.str ++ \"0\"\u27e9\n\nmeta instance comp.to_format : has_to_format comp :=\n\u27e8\u03bb p, to_fmt p.coeffs\u27e9\n\nend datatypes\n\nsection fm_elim\n\n/-- If c1 and c2 both contain variable a with opposite coefficients,\n   produces v1, v2, and c such that a has been cancelled in c := v1*c1 + v2*c2 -/\nmeta def elim_var (c1 c2 : comp) (a : \u2115) : option (\u2115 \u00d7 \u2115 \u00d7 comp) :=\nlet v1 := c1.coeff_of a,\n    v2 := c2.coeff_of a in\nif v1 * v2 < 0 then\n  let vlcm :=  nat.lcm v1.nat_abs v2.nat_abs,\n      v1' := vlcm / v1.nat_abs,\n      v2' := vlcm / v2.nat_abs in\n  some \u27e8v1', v2', comp.add (c1.scale v1') (c2.scale v2')\u27e9\nelse none\n\nmeta def pelim_var (p1 p2 : pcomp) (a : \u2115) : option pcomp :=\ndo (n1, n2, c) \u2190 elim_var p1.c p2.c a,\n   return \u27e8c, comp_source.add (p1.src.scale n1) (p2.src.scale n2)\u27e9\n\nmeta def comp.is_contr (c : comp) : bool := c.coeffs.empty \u2227 c.str = ineq.lt\n\nmeta def pcomp.is_contr (p : pcomp) : bool := p.c.is_contr\n\nmeta def elim_with_set (a : \u2115) (p : pcomp) (comps : rb_set pcomp) : rb_set pcomp :=\nif \u00ac p.c.coeffs.contains a then mk_rb_set.insert p else\ncomps.fold mk_rb_set $ \u03bb pc s,\nmatch pelim_var p pc a with\n| some pc := s.insert pc\n| none := s\nend\n\n/--\n  The state for the elimination monad.\n    vars: the set of variables present in comps\n    comps: a set of comparisons\n    inputs: a set of pairs of exprs (t, pf), where t is a term and pf is a proof that t {<, \u2264, =} 0,\n      indexed by \u2115.\n    has_false: stores a pcomp of 0 < 0 if one has been found\n    TODO: is it more efficient to store comps as a list, to avoid comparisons?\n-/\nmeta structure linarith_structure :=\n(vars : rb_set \u2115)\n(comps : rb_set pcomp)\n\n@[reducible] meta def linarith_monad :=\nstate_t linarith_structure (except_t pcomp id)\n\nmeta instance : monad linarith_monad := state_t.monad\nmeta instance : monad_except pcomp linarith_monad :=\nstate_t.monad_except pcomp\n\nmeta def get_vars : linarith_monad (rb_set \u2115) :=\nlinarith_structure.vars <$> get\n\nmeta def get_var_list : linarith_monad (list \u2115) :=\nrb_set.to_list <$> get_vars\n\nmeta def get_comps : linarith_monad (rb_set pcomp) :=\nlinarith_structure.comps <$> get\n\nmeta def validate : linarith_monad unit :=\ndo \u27e8_, comps\u27e9 \u2190 get,\nmatch comps.to_list.find (\u03bb p : pcomp, p.is_contr) with\n| none := return ()\n| some c := throw c\nend\n\nmeta def update (vars : rb_set \u2115) (comps : rb_set pcomp) : linarith_monad unit :=\nstate_t.put \u27e8vars, comps\u27e9 >> validate\n\nmeta def monad.elim_var (a : \u2115) : linarith_monad unit :=\ndo vs \u2190 get_vars,\n   when (vs.contains a) $\ndo comps \u2190 get_comps,\n   let cs' := comps.fold mk_rb_set (\u03bb p s, s.union (elim_with_set a p comps)),\n   update (vs.erase a) cs'\n\nmeta def elim_all_vars : linarith_monad unit :=\nget_var_list >>= list.mmap' monad.elim_var\n\nend fm_elim\n\nsection parse\n\nopen ineq tactic\n\nmeta def map_of_expr_mul_aux (c1 c2 : rb_map \u2115 \u2124) : option (rb_map \u2115 \u2124) :=\nmatch c1.keys, c2.keys with\n| [0], _ := some $ c2.scale (c1.zfind 0)\n| _, [0] := some $ c1.scale (c2.zfind 0)\n| [], _ := some mk_rb_map\n| _, [] := some mk_rb_map\n| _, _ := none\nend\n\n/--\n  Turns an expression into a map from \u2115 to \u2124, for use in a comp object.\n    The expr_map \u2115 argument identifies which expressions have already been assigned numbers.\n    Returns a new map.\n-/\nmeta def map_of_expr : expr_map \u2115 \u2192 expr \u2192 option (expr_map \u2115 \u00d7 rb_map \u2115 \u2124)\n| m e@`(%%e1 * %%e2) :=\n   (do (m', comp1) \u2190 map_of_expr m e1,\n      (m', comp2) \u2190 map_of_expr m' e2,\n      mp \u2190 map_of_expr_mul_aux comp1 comp2,\n      return (m', mp)) <|>\n   (match m.find e with\n    | some k := return (m, mk_rb_map.insert k 1)\n    | none := let n := m.size + 1 in return (m.insert e n, mk_rb_map.insert n 1)\n    end)\n| m `(%%e1 + %%e2) :=\n   do (m', comp1) \u2190 map_of_expr m e1,\n      (m', comp2) \u2190 map_of_expr m' e2,\n      return (m', comp1.add comp2)\n| m `(%%e1 - %%e2) :=\n   do (m', comp1) \u2190 map_of_expr m e1,\n      (m', comp2) \u2190 map_of_expr m' e2,\n      return (m', comp1.add (comp2.scale (-1)))\n| m `(-%%e) := do (m', comp) \u2190 map_of_expr m e, return (m', comp.scale (-1))\n| m e :=\n  match e.to_int, m.find e with\n  | some 0, _ := return \u27e8m, mk_rb_map\u27e9\n  | some z, _ := return \u27e8m, mk_rb_map.insert 0 z\u27e9\n  | none, some k := return (m, mk_rb_map.insert k 1)\n  | none, none := let n := m.size + 1 in\n    return (m.insert e n, mk_rb_map.insert n 1)\n  end\n\nmeta def parse_into_comp_and_expr : expr \u2192 option (ineq \u00d7 expr)\n| `(%%e < 0) := (ineq.lt, e)\n| `(%%e \u2264 0) := (ineq.le, e)\n| `(%%e = 0) := (ineq.eq, e)\n| _ := none\n\nmeta def to_comp (e : expr) (m : expr_map \u2115) : option (comp \u00d7 expr_map \u2115) :=\ndo (iq, e) \u2190 parse_into_comp_and_expr e,\n   (m', comp') \u2190 map_of_expr m e,\n   return \u27e8\u27e8iq, comp'\u27e9, m'\u27e9\n\nmeta def to_comp_fold : expr_map \u2115 \u2192 list expr \u2192\n      (list (option comp) \u00d7 expr_map \u2115)\n| m [] := ([], m)\n| m (h::t) :=\n  match to_comp h m with\n  | some (c, m') := let (l, mp) := to_comp_fold m' t in (c::l, mp)\n  | none := let (l, mp) := to_comp_fold m t in (none::l, mp)\n  end\n\n/--\n  Takes a list of proofs of props of the form t {<, \u2264, =} 0, and creates a linarith_structure.\n-/\nmeta def mk_linarith_structure (l : list expr) : tactic (linarith_structure \u00d7 rb_map \u2115 (expr \u00d7 expr)) :=\ndo pftps \u2190 l.mmap infer_type,\n  let (l', map) := to_comp_fold mk_rb_map pftps,\n  let lz := list.enum $ ((l.zip pftps).zip l').filter_map (\u03bb \u27e8a, b\u27e9, prod.mk a <$> b),\n  let prmap := rb_map.of_list $ lz.map (\u03bb \u27e8n, x\u27e9, (n, x.1)),\n  let vars : rb_set \u2115 := rb_map.set_of_list $ list.range map.size.succ,\n  let pc : rb_set pcomp := rb_map.set_of_list $\n    lz.map (\u03bb \u27e8n, x\u27e9, \u27e8x.2, comp_source.assump n\u27e9),\n  return (\u27e8vars, pc\u27e9, prmap)\n\nmeta def linarith_monad.run {\u03b1} (tac : linarith_monad \u03b1) (l : list expr) : tactic ((pcomp \u2295 \u03b1) \u00d7 rb_map \u2115 (expr \u00d7 expr)) :=\ndo (struct, inputs) \u2190 mk_linarith_structure l,\nmatch (state_t.run (validate >> tac) struct).run with\n| (except.ok (a, _)) := return (sum.inr a, inputs)\n| (except.error contr) := return (sum.inl contr, inputs)\nend\n\nend parse\n\nsection prove\nopen ineq tactic\n\nmeta def get_rel_sides : expr \u2192 tactic (expr \u00d7 expr)\n| `(%%a < %%b) := return (a, b)\n| `(%%a \u2264 %%b) := return (a, b)\n| `(%%a = %%b) := return (a, b)\n| `(%%a \u2265 %%b) := return (a, b)\n| `(%%a > %%b) := return (a, b)\n| _ := failed\n\nmeta def mul_expr (n : \u2115) (e : expr) : pexpr :=\nif n = 1 then ``(%%e) else\n``(%%(nat.to_pexpr n) * %%e)\n\nmeta def add_exprs_aux : pexpr \u2192 list pexpr \u2192 pexpr\n| p [] := p\n| p [a] := ``(%%p + %%a)\n| p (h::t) := add_exprs_aux ``(%%p + %%h) t\n\nmeta def add_exprs : list pexpr \u2192 pexpr\n| [] := ``(0)\n| (h::t) := add_exprs_aux h t\n\nmeta def find_contr (m : rb_set pcomp) : option pcomp :=\nm.keys.find (\u03bb p, p.c.is_contr)\n\nmeta def ineq_const_mul_nm : ineq \u2192 name\n| lt := ``mul_neg\n| le := ``mul_nonpos\n| eq := ``mul_eq\n\nmeta def ineq_const_nm : ineq \u2192 ineq \u2192 (name \u00d7 ineq)\n| eq eq := (``eq_of_eq_of_eq, eq)\n| eq le := (``le_of_eq_of_le, le)\n| eq lt := (``lt_of_eq_of_lt, lt)\n| le eq := (``le_of_le_of_eq, le)\n| le le := (`add_nonpos, le)\n| le lt := (`add_neg_of_nonpos_of_neg, lt)\n| lt eq := (``lt_of_lt_of_eq, lt)\n| lt le := (`add_neg_of_neg_of_nonpos, lt)\n| lt lt := (`add_neg, lt)\n\nmeta def mk_single_comp_zero_pf (c : \u2115) (h : expr) : tactic (ineq \u00d7 expr) :=\ndo tp \u2190 infer_type h,\n  some (iq, e) \u2190 return $ parse_into_comp_and_expr tp,\n  if c = 0 then\n    do e' \u2190 mk_app ``zero_mul [e], return (eq, e')\n  else if c = 1 then return (iq, h)\n  else\n    do nm \u2190 resolve_name (ineq_const_mul_nm iq),\n       tp \u2190 (prod.snd <$> (infer_type h >>= get_rel_sides)) >>= infer_type,\n       cpos \u2190 to_expr ``((%%c.to_pexpr : %%tp) > 0),\n       (_, ex) \u2190 solve_aux cpos `[norm_num, done],\n--       e' \u2190 mk_app (ineq_const_mul_nm iq) [h, ex], -- this takes many seconds longer in some examples! why?\n       e' \u2190 to_expr ``(%%nm %%h %%ex) ff,\n       return (iq, e')\n\nmeta def mk_lt_zero_pf_aux (c : ineq) (pf npf : expr) (coeff : \u2115) : tactic (ineq \u00d7 expr) :=\ndo (iq, h') \u2190 mk_single_comp_zero_pf coeff npf,\n   let (nm, niq) := ineq_const_nm c iq,\n   n \u2190 resolve_name nm,\n   e' \u2190 to_expr ``(%%n %%pf %%h'),\n   return (niq, e')\n\n/--\n  Takes a list of coefficients [c] and list of expressions, of equal length.\n  Each expression is a proof of a prop of the form t {<, \u2264, =} 0.\n  Produces a proof that the sum of (c*t) {<, \u2264, =} 0, where the comp is as strong as possible.\n-/\nmeta def mk_lt_zero_pf : list \u2115 \u2192 list expr \u2192 tactic expr\n| _ [] := fail \"no linear hypotheses found\"\n| [c] [h] := prod.snd <$> mk_single_comp_zero_pf c h\n| (c::ct) (h::t) :=\n  do (iq, h') \u2190 mk_single_comp_zero_pf c h,\n     prod.snd <$> (ct.zip t).mfoldl (\u03bb pr ce, mk_lt_zero_pf_aux pr.1 pr.2 ce.2 ce.1) (iq, h')\n| _ _ := fail \"not enough args to mk_lt_zero_pf\"\n\nmeta def term_of_ineq_prf (prf : expr) : tactic expr :=\ndo (lhs, _) \u2190 infer_type prf >>= get_rel_sides,\n   return lhs\n\nmeta structure linarith_config :=\n(discharger : tactic unit := `[ring])\n(restrict_type : option Type := none)\n(restrict_type_reflect : reflected restrict_type . apply_instance)\n(exfalso : bool := tt)\n\nmeta def ineq_pf_tp (pf : expr) : tactic expr :=\ndo (_, z) \u2190 infer_type pf >>= get_rel_sides,\n   infer_type z\n\nmeta def mk_neg_one_lt_zero_pf (tp : expr) : tactic expr :=\nto_expr ``((neg_neg_of_pos zero_lt_one : -1 < (0 : %%tp)))\n\n/--\n  Assumes e is a proof that t = 0. Creates a proof that -t = 0.\n-/\nmeta def mk_neg_eq_zero_pf (e : expr) : tactic expr :=\nto_expr ``(neg_eq_zero.mpr %%e)\n\nmeta def add_neg_eq_pfs : list expr \u2192 tactic (list expr)\n| [] := return []\n| (h::t) :=\n  do some (iq, tp) \u2190 parse_into_comp_and_expr <$> infer_type h,\n  match iq with\n  | ineq.eq := do nep \u2190 mk_neg_eq_zero_pf h, tl \u2190 add_neg_eq_pfs t, return $ h::nep::tl\n  | _ := list.cons h <$> add_neg_eq_pfs t\n  end\n\n/--\n  Takes a list of proofs of propositions of the form t {<, \u2264, =} 0,\n  and tries to prove the goal `false`.\n-/\nmeta def prove_false_by_linarith1 (cfg : linarith_config) : list expr \u2192 tactic unit\n| [] := fail \"no args to linarith\"\n| l@(h::t) :=\n  do l' \u2190 add_neg_eq_pfs l,\n     hz \u2190 ineq_pf_tp h >>= mk_neg_one_lt_zero_pf,\n     (sum.inl contr, inputs) \u2190 elim_all_vars.run (hz::l')\n       | fail \"linarith failed to find a contradiction\",\n     let coeffs := inputs.keys.map (\u03bb k, (contr.src.flatten.ifind k)),\n     let pfs : list expr := inputs.keys.map (\u03bb k, (inputs.ifind k).1),\n     let zip := (coeffs.zip pfs).filter (\u03bb pr, pr.1 \u2260 0),\n     let (coeffs, pfs) := zip.unzip,\n     mls \u2190 zip.mmap (\u03bb pr, do e \u2190 term_of_ineq_prf pr.2, return (mul_expr pr.1 e)),\n     sm \u2190 to_expr $ add_exprs mls,\n     tgt \u2190 to_expr ``(%%sm = 0),\n     (a, b) \u2190 solve_aux tgt (cfg.discharger >> done),\n     pf \u2190 mk_lt_zero_pf coeffs pfs,\n     pftp \u2190 infer_type pf,\n     (_, nep, _) \u2190 rewrite_core b pftp,\n     pf' \u2190 mk_eq_mp nep pf,\n     mk_app `lt_irrefl [pf'] >>= exact\n\nend prove\n\nsection normalize\nopen tactic\n\nset_option eqn_compiler.max_steps 50000\n\nmeta def rem_neg (prf : expr) : expr \u2192 tactic expr\n| `(_ \u2264 _) := to_expr ``(lt_of_not_ge %%prf)\n| `(_ < _) := to_expr ``(le_of_not_gt %%prf)\n| `(_ > _) := to_expr ``(le_of_not_gt %%prf)\n| `(_ \u2265 _) := to_expr ``(lt_of_not_ge %%prf)\n| e := failed\n\nmeta def rearr_comp : expr \u2192 expr \u2192 tactic expr\n| prf `(%%a \u2264 0) := return prf\n| prf  `(%%a < 0) := return prf\n| prf  `(%%a = 0) := return prf\n| prf  `(%%a \u2265 0) := to_expr ``(neg_nonpos.mpr %%prf)\n| prf  `(%%a > 0) := to_expr ``(neg_neg_of_pos %%prf)\n| prf  `(0 \u2265 %%a) := to_expr ``(show %%a \u2264 0, from %%prf)\n| prf  `(0 > %%a) := to_expr ``(show %%a < 0, from %%prf)\n| prf  `(0 = %%a) := to_expr ``(eq.symm %%prf)\n| prf  `(0 \u2264 %%a) := to_expr ``(neg_nonpos.mpr %%prf)\n| prf  `(0 < %%a) := to_expr ``(neg_neg_of_pos %%prf)\n| prf  `(%%a \u2264 %%b) := to_expr ``(sub_nonpos.mpr %%prf)\n| prf  `(%%a < %%b) := to_expr ``(sub_neg_of_lt %%prf)\n| prf  `(%%a = %%b) := to_expr ``(sub_eq_zero.mpr %%prf)\n| prf  `(%%a > %%b) := to_expr ``(sub_neg_of_lt %%prf)\n| prf  `(%%a \u2265 %%b) := to_expr ``(sub_nonpos.mpr %%prf)\n| prf  `(\u00ac %%t) := do nprf \u2190 rem_neg prf t, tp \u2190 infer_type nprf, rearr_comp nprf tp\n| prf  _ := fail \"couldn't rearrange comp\"\n\n\nmeta def is_numeric : expr \u2192 option \u211a\n| `(%%e1 + %%e2) := (+) <$> is_numeric e1 <*> is_numeric e2\n| `(%%e1 - %%e2) := has_sub.sub <$> is_numeric e1 <*> is_numeric e2\n| `(%%e1 * %%e2) := (*) <$> is_numeric e1 <*> is_numeric e2\n| `(%%e1 / %%e2) := (/) <$> is_numeric e1 <*> is_numeric e2\n| `(-%%e) := rat.neg <$> is_numeric e\n| e := e.to_rat\n\ninductive {u} tree (\u03b1 : Type u) : Type u\n| nil {} : tree\n| node : \u03b1 \u2192 tree \u2192 tree \u2192 tree\n\ndef tree.repr {\u03b1} [has_repr \u03b1] : tree \u03b1 \u2192 string\n| tree.nil := \"nil\"\n| (tree.node a t1 t2) := \"tree.node \" ++ repr a ++ \" (\" ++ tree.repr t1 ++ \") (\" ++ tree.repr t2 ++ \")\"\n\ninstance {\u03b1} [has_repr \u03b1] : has_repr (tree \u03b1) := \u27e8tree.repr\u27e9\n\nmeta def find_cancel_factor : expr \u2192 \u2115 \u00d7 tree \u2115\n| `(%%e1 + %%e2) :=\n  let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, lcm := v1.lcm v2 in\n  (lcm, tree.node lcm t1 t2)\n| `(%%e1 - %%e2) :=\n  let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, lcm := v1.lcm v2 in\n  (lcm, tree.node lcm t1 t2)\n| `(%%e1 * %%e2) :=\n  match is_numeric e1, is_numeric e2 with\n  | none, none := (1, tree.node 1 tree.nil tree.nil)\n  | _, _ :=\n    let (v1, t1) := find_cancel_factor e1, (v2, t2) := find_cancel_factor e2, pd := v1*v2 in\n    (pd, tree.node pd t1 t2)\n  end\n| `(%%e1 / %%e2) :=\n  match is_numeric e2 with\n  | some q := let (v1, t1) := find_cancel_factor e1, n := v1.lcm q.num.nat_abs in\n    (n, tree.node n t1 (tree.node q.num.nat_abs tree.nil tree.nil))\n  | none := (1, tree.node 1 tree.nil tree.nil)\n  end\n| `(-%%e) := find_cancel_factor e\n| _ := (1, tree.node 1 tree.nil tree.nil)\n\nopen tree\n\nmeta def mk_prod_prf : \u2115 \u2192 tree \u2115 \u2192 expr \u2192 tactic expr\n| v (node _ lhs rhs) `(%%e1 + %%e2) :=\n  do v1 \u2190 mk_prod_prf v lhs e1, v2 \u2190 mk_prod_prf v rhs e2, mk_app ``add_subst [v1, v2]\n| v (node _ lhs rhs) `(%%e1 - %%e2) :=\n  do v1 \u2190 mk_prod_prf v lhs e1, v2 \u2190 mk_prod_prf v rhs e2, mk_app ``sub_subst [v1, v2]\n| v (node n lhs@(node ln _ _) rhs) `(%%e1 * %%e2) :=\n  do tp \u2190 infer_type e1, v1 \u2190 mk_prod_prf ln lhs e1, v2 \u2190 mk_prod_prf (v/ln) rhs e2,\n     ln' \u2190 tp.of_nat ln, vln' \u2190 tp.of_nat (v/ln), v' \u2190 tp.of_nat v,\n     ntp \u2190 to_expr ``(%%ln' * %%vln' = %%v'),\n     (_, npf) \u2190 solve_aux ntp `[norm_num, done],\n     mk_app ``mul_subst [v1, v2, npf]\n| v (node n lhs rhs@(node rn _ _)) `(%%e1 / %%e2) :=\n  do tp \u2190 infer_type e1, v1 \u2190 mk_prod_prf (v/rn) lhs e1,\n     rn' \u2190 tp.of_nat rn, vrn' \u2190 tp.of_nat (v/rn), n' \u2190 tp.of_nat n, v' \u2190 tp.of_nat v,\n     ntp \u2190 to_expr ``(%%rn' / %%e2 = 1),\n     (_, npf) \u2190 solve_aux ntp `[norm_num, done],\n     ntp2 \u2190 to_expr ``(%%vrn' * %%n' = %%v'),\n     (_, npf2) \u2190 solve_aux ntp2 `[norm_num, done],\n     mk_app ``div_subst [v1, npf, npf2]\n| v t `(-%%e) := do v' \u2190 mk_prod_prf v t e, mk_app ``neg_subst [v']\n| v _ e :=\n  do tp \u2190 infer_type e,\n     v' \u2190 tp.of_nat v,\n     e' \u2190 to_expr ``(%%v' * %%e),\n     mk_app `eq.refl [e']\n\n/--\n e is a term with rational division. produces a natural number n and a proof that n*e = e',\n where e' has no division.\n-/\nmeta def kill_factors (e : expr) : tactic (\u2115 \u00d7 expr) :=\nlet (n, t) := find_cancel_factor e in\ndo e' \u2190 mk_prod_prf n t e, return (n, e')\n\nopen expr\nmeta def expr_contains (n : name) : expr \u2192 bool\n| (const nm _) := nm = n\n| (lam _ _ _ bd) := expr_contains bd\n| (pi _ _ _ bd) := expr_contains bd\n| (app e1 e2) := expr_contains e1 || expr_contains e2\n| _ := ff\n\nlemma sub_into_lt {\u03b1} [ordered_semiring \u03b1] {a b : \u03b1} (he : a = b) (hl : a \u2264 0) : b \u2264 0 :=\nby rwa he at hl\n\nmeta def norm_hyp_aux (h' lhs : expr) : tactic expr :=\ndo (v, lhs') \u2190 kill_factors lhs,\n   if v = 1 then return h' else do\n   (ih, h'') \u2190 mk_single_comp_zero_pf v h',\n   (_, nep, _) \u2190 infer_type h'' >>= rewrite_core lhs',\n   mk_eq_mp nep h''\n\nmeta def norm_hyp (h : expr) : tactic expr :=\ndo htp \u2190 infer_type h,\n   h' \u2190 rearr_comp h htp,\n   some (c, lhs) \u2190 parse_into_comp_and_expr <$> infer_type h',\n   if expr_contains `has_div.div lhs then\n     norm_hyp_aux h' lhs\n   else return h'\n\nmeta def get_contr_lemma_name : expr \u2192 option name\n| `(%%a < %%b) := return `lt_of_not_ge\n| `(%%a \u2264 %%b) := return `le_of_not_gt\n| `(%%a = %%b) := return ``eq_of_not_lt_of_not_gt\n| `(%%a \u2265 %%b) := return `le_of_not_gt\n| `(%%a > %%b) := return `lt_of_not_ge\n| `(\u00ac %%a < %%b) := return `not.intro\n| `(\u00ac %%a \u2264 %%b) := return `not.intro\n| `(\u00ac %%a = %%b) := return `not.intro\n| `(\u00ac %%a \u2265 %%b) := return `not.intro\n| `(\u00ac %%a > %%b) := return `not.intro\n| _ := none\n\n-- assumes the input t is of type \u2115. Produces t' of type \u2124 such that \u2191t = t' and a proof of equality\nmeta def cast_expr (e : expr) : tactic (expr \u00d7 expr) :=\ndo s \u2190 [`int.coe_nat_add, `int.coe_nat_zero, `int.coe_nat_one,\n        ``int.coe_nat_bit0_mul, ``int.coe_nat_bit1_mul, ``int.coe_nat_zero_mul, ``int.coe_nat_one_mul,\n        ``int.coe_nat_mul_bit0, ``int.coe_nat_mul_bit1, ``int.coe_nat_mul_zero, ``int.coe_nat_mul_one,\n        ``int.coe_nat_bit0, ``int.coe_nat_bit1].mfoldl simp_lemmas.add_simp simp_lemmas.mk,\n   ce \u2190 to_expr ``(\u2191%%e : \u2124),\n   simplify s [] ce {fail_if_unchanged := ff}\n\nmeta def is_nat_int_coe : expr \u2192 option expr\n| `((\u2191(%%n : \u2115) : \u2124)) := some n\n| _ := none\n\nmeta def mk_coe_nat_nonneg_prf (e : expr) : tactic expr :=\nmk_app `int.coe_nat_nonneg [e]\n\nmeta def get_nat_comps : expr \u2192 list expr\n| `(%%a + %%b) := (get_nat_comps a).append (get_nat_comps b)\n| `(%%a * %%b) := (get_nat_comps a).append (get_nat_comps b)\n| e := match is_nat_int_coe e with\n  | some e' := [e']\n  | none := []\n  end\n\nmeta def mk_coe_nat_nonneg_prfs (e : expr) : tactic (list expr) :=\n(get_nat_comps e).mmap mk_coe_nat_nonneg_prf\n\nmeta def mk_cast_eq_and_nonneg_prfs (pf a b : expr) (ln : name) : tactic (list expr) :=\ndo (a', prfa) \u2190 cast_expr a,\n   (b', prfb) \u2190 cast_expr b,\n   la \u2190 mk_coe_nat_nonneg_prfs a',\n   lb \u2190 mk_coe_nat_nonneg_prfs b',\n   pf' \u2190 mk_app ln [pf, prfa, prfb],\n   return $ pf'::(la.append lb)\n\nmeta def mk_int_pfs_of_nat_pf (pf : expr) : tactic (list expr) :=\ndo tp \u2190 infer_type pf,\nmatch tp with\n| `(%%a = %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_eq_subst\n| `(%%a \u2264 %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_le_subst\n| `(%%a < %%b) := mk_cast_eq_and_nonneg_prfs pf a b ``nat_lt_subst\n| `(%%a \u2265 %%b) := mk_cast_eq_and_nonneg_prfs pf b a ``nat_le_subst\n| `(%%a > %%b) := mk_cast_eq_and_nonneg_prfs pf b a ``nat_lt_subst\n| `(\u00ac %%a \u2264 %%b) := do pf' \u2190 mk_app ``lt_of_not_ge [pf], mk_cast_eq_and_nonneg_prfs pf' b a ``nat_lt_subst\n| `(\u00ac %%a < %%b) := do pf' \u2190 mk_app ``le_of_not_gt [pf], mk_cast_eq_and_nonneg_prfs pf' b a ``nat_le_subst\n| `(\u00ac %%a \u2265 %%b) := do pf' \u2190 mk_app ``lt_of_not_ge [pf], mk_cast_eq_and_nonneg_prfs pf' a b ``nat_lt_subst\n| `(\u00ac %%a > %%b) := do pf' \u2190 mk_app ``le_of_not_gt [pf], mk_cast_eq_and_nonneg_prfs pf' a b ``nat_le_subst\n| _ := fail \"mk_int_pfs_of_nat_pf failed: proof is not an inequality\"\nend\n\nmeta def mk_non_strict_int_pf_of_strict_int_pf (pf : expr) : tactic expr :=\ndo tp \u2190 infer_type pf,\nmatch tp with\n| `(%%a < %%b) := to_expr ``(@cast (%%a < %%b) (%%a + 1 \u2264 %%b) (by refl) %%pf)\n| `(%%a > %%b) := to_expr ``(@cast (%%a > %%b) (%%a \u2265 %%b + 1) (by refl) %%pf)\n| `(\u00ac %%a \u2264 %%b) := to_expr ``(@cast (%%a > %%b) (%%a \u2265 %%b + 1) (by refl) (lt_of_not_ge %%pf))\n| `(\u00ac %%a \u2265 %%b) := to_expr ``(@cast (%%a < %%b) (%%a + 1 \u2264 %%b) (by refl) (lt_of_not_ge %%pf))\n| _ := fail \"mk_non_strict_int_pf_of_strict_int_pf failed: proof is not an inequality\"\nend\n\nmeta def guard_is_nat_prop : expr \u2192 tactic unit\n| `(%%a = _) := infer_type a >>= unify `(\u2115)\n| `(%%a \u2264 _) := infer_type a >>= unify `(\u2115)\n| `(%%a < _) := infer_type a >>= unify `(\u2115)\n| `(%%a \u2265 _) := infer_type a >>= unify `(\u2115)\n| `(%%a > _) := infer_type a >>= unify `(\u2115)\n| `(\u00ac %%p) := guard_is_nat_prop p\n| _ := failed\n\nmeta def guard_is_strict_int_prop : expr \u2192 tactic unit\n| `(%%a < _) := infer_type a >>= unify `(\u2124)\n| `(%%a > _) := infer_type a >>= unify `(\u2124)\n| `(\u00ac %%a \u2264 _) := infer_type a >>= unify `(\u2124)\n| `(\u00ac %%a \u2265 _) := infer_type a >>= unify `(\u2124)\n| _ := failed\n\nmeta def replace_nat_pfs : list expr \u2192 tactic (list expr)\n| [] := return []\n| (h::t) :=\n  (do infer_type h >>= guard_is_nat_prop,\n      ls \u2190 mk_int_pfs_of_nat_pf h,\n      list.append ls <$> replace_nat_pfs t) <|> list.cons h <$> replace_nat_pfs t\n\nmeta def replace_strict_int_pfs : list expr \u2192 tactic (list expr)\n| [] := return []\n| (h::t) :=\n  (do infer_type h >>= guard_is_strict_int_prop,\n      l \u2190 mk_non_strict_int_pf_of_strict_int_pf h,\n      list.cons l <$> replace_strict_int_pfs t) <|> list.cons h <$> replace_strict_int_pfs t\n\nmeta def partition_by_type_aux : rb_lmap expr expr \u2192 list expr \u2192 tactic (rb_lmap expr expr)\n| m [] := return m\n| m (h::t) := do tp \u2190 ineq_pf_tp h, partition_by_type_aux (m.insert tp h) t\n\nmeta def partition_by_type (l : list expr) : tactic (rb_lmap expr expr) :=\npartition_by_type_aux mk_rb_map l\n\nprivate meta def try_linarith_on_lists (cfg : linarith_config) (ls : list (list expr)) : tactic unit :=\n(first $ ls.map $ prove_false_by_linarith1 cfg) <|> fail \"linarith failed\"\n\n/--\n  Takes a list of proofs of propositions.\n  Filters out the proofs of linear (in)equalities,\n  and tries to use them to prove `false`.\n  If pref_type is given, starts by working over this type\n-/\nmeta def prove_false_by_linarith (cfg : linarith_config) (pref_type : option expr) (l : list expr) : tactic unit :=\ndo l' \u2190 replace_nat_pfs l,\n   l'' \u2190 replace_strict_int_pfs l',\n   ls \u2190 list.reduce_option <$> l''.mmap (\u03bb h, (do s \u2190 norm_hyp h, return (some s)) <|> return none)\n          >>= partition_by_type,\n   pref_type \u2190 (unify pref_type.iget `(\u2115) >> return (some `(\u2124) : option expr)) <|> return pref_type,\n   match cfg.restrict_type, ls.values, pref_type with\n   | some rtp, _, _ :=\n      do m \u2190 mk_mvar, unify `(some %%m : option Type) cfg.restrict_type_reflect, m \u2190 instantiate_mvars m,\n         prove_false_by_linarith1 cfg (ls.ifind m)\n   | none, [ls'], _ := prove_false_by_linarith1 cfg ls'\n   | none, ls', none := try_linarith_on_lists cfg ls'\n   | none, _, (some t) := prove_false_by_linarith1 cfg (ls.ifind t) <|> try_linarith_on_lists cfg (ls.erase t).values\n   end\n\nend normalize\n\nend linarith\n\nsection\nopen tactic linarith\n\nopen lean lean.parser interactive tactic interactive.types\nlocal postfix `?`:9001 := optional\nlocal postfix *:9001 := many\n\nmeta def linarith.interactive_aux (cfg : linarith_config) : option expr \u2192\n     parse ident* \u2192 (parse (tk \"using\" *> pexpr_list)?) \u2192 tactic unit\n| pt l (some pe) := pe.mmap (\u03bb p, i_to_expr p >>= note_anon) >> linarith.interactive_aux pt l none\n| pt [] none :=\n  do t \u2190 target,\n     if t = `(false) then local_context >>= prove_false_by_linarith cfg pt\n     else match get_contr_lemma_name t with\n     | some nm := seq (applyc nm) (do t \u2190 intro1 >>= ineq_pf_tp, linarith.interactive_aux (some t) [] none)\n     | none := if cfg.exfalso then exfalso >> linarith.interactive_aux pt [] none\n               else fail \"linarith failed: target type is not an inequality.\"\n     end\n| pt ls none := (ls.mmap get_local) >>= prove_false_by_linarith cfg pt\n\n/--\n  Tries to prove a goal of `false` by linear arithmetic on hypotheses.\n  If the goal is a linear (in)equality, tries to prove it by contradiction.\n  If the goal is not `false` or an inequality, applies `exfalso` and tries linarith on the\n  hypotheses.\n  `linarith` will use all relevant hypotheses in the local context.\n  `linarith h1 h2 h3` will only use hypotheses h1, h2, h3.\n  `linarith using [t1, t2, t3]` will add proof terms t1, t2, t3 to the local context.\n\n  Config options:\n  `linarith {exfalso := ff}` will fail on a goal that is neither an inequality nor `false`\n  `linarith {restrict_type := T}` will run only on hypotheses that are inequalities over `T`\n  `linarith {discharger := tac}` will use `tac` instead of `ring` for normalization.\n    Options: `ring2`, `ring SOP`, `simp`\n-/\nmeta def tactic.interactive.linarith (ids : parse (many ident))\n     (using_hyps : parse (tk \"using\" *> pexpr_list)?) (cfg : linarith_config := {}) : tactic unit :=\nlinarith.interactive_aux cfg none ids using_hyps\n\nend\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/tactic/linarith.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.390600919697409}}
{"text": "import topology.category.Profinite.default\nimport injective_map\nimport algebra.free_algebra\nimport data.finset.basic\nimport group_theory.free_abelian_group\nimport linear_algebra.free_module.basic\n\nnoncomputable theory\n\nuniverse u\n\ndef bool_to_Z : bool \u2192 \u2124 := \u03bb x, ite (x = tt) 1 0\n\ninstance totally_separated_of_profinite (S : Profinite.{u}) : totally_separated_space S :=\n  totally_separated_of_totally_disconnected_compact_hausdorff S\n  \ndef I (S : Profinite.{u}) : Type u := \n  {fI : S \u2192 bool // continuous fI}\n\ndef inj_map (S : Profinite.{u}) : S \u2192 ((I S) \u2192 bool) :=\n  map_to_I' S\n\ndef CSZ (S : Profinite.{u}) : Type u := {f : S \u2192 \u2124 // continuous f}\n\ninstance ring_SZ (S : Profinite.{u}) : comm_ring (S \u2192 \u2124) := pi.comm_ring\n\ndef subring_CSZ (S : Profinite.{u}) : subring (S \u2192 \u2124) := \n{ carrier := {f : S \u2192 \u2124 | continuous f},\n  zero_mem' := begin\n    simp only [set.mem_set_of_eq],\n    refine continuous_def.mpr _,\n    intros s hs,\n    by_cases hzero : (0 : \u2124) \u2208 s,\n    { have : (0 \u207b\u00b9' s : set S) = set.univ, \n      { ext1, \n        simp only [set.mem_preimage, set.mem_univ, pi.zero_apply, iff_true], \n        assumption },\n      rw this,\n      exact is_open_univ, },\n    { have : (0 \u207b\u00b9' s : set S) = \u2205, \n      { ext1, \n        simp only [set.mem_preimage, pi.zero_apply],\n        rw \u2190 not_iff_not,\n        split, tauto, exact \u03bb h, hzero, },\n      rw this,\n      exact is_open_empty, },\n  end,\n  one_mem' := begin\n    simp only [set.mem_set_of_eq],\n    refine continuous_def.mpr _,\n    intros s hs,\n    by_cases hzero : (1 : \u2124) \u2208 s,\n    { have : (1 \u207b\u00b9' s : set S) = set.univ, \n      { ext1, \n        simp only [set.mem_preimage, set.mem_univ, pi.zero_apply, iff_true], \n        assumption },\n      rw this,\n      exact is_open_univ, },\n    { have : (1 \u207b\u00b9' s : set S) = \u2205, \n      { ext1, \n        simp only [set.mem_preimage, pi.zero_apply],\n        rw \u2190 not_iff_not,\n        split, tauto, exact \u03bb h, hzero, },\n      rw this,\n      exact is_open_empty, },\n  end,\n  add_mem' := \u03bb f g hf hg, continuous.add hf hg,\n  mul_mem' := \u03bb f g hf hg, continuous.mul hf hg,\n  neg_mem' := \u03bb f hf, continuous.neg hf,\n}\n\ninstance ring_CSZ (S : Profinite.{u}) : comm_ring (CSZ S) := (subring_CSZ S).to_comm_ring\n\ninstance alg_CSZ (S : Profinite.{u}) : algebra \u2124 (CSZ S) := algebra_int (CSZ S)\n\ndef prod_N (I : Type u) : Type u := \u03a0 (n : \u2115), I  \n\ndef N_order : (\u2115 \u2192 \u2115 \u2192 Prop) := \u03bb n m, n \u2264 m\n\ndef prod_order (I : Type u) : \u03a0 (n : \u2115), I \u2192 I \u2192 Prop := \u03bb n, @well_ordering_rel I \n\ndef lex_order (I : Type u) : (prod_N I) \u2192 (prod_N I) \u2192 Prop := \n  \u03bb x y, pi.lex N_order (prod_order I) x y \n\ndef sub_prod (I : Type u) : Type u := \n  {x : prod_N I // (\u2200 n : \u2115, ordinal.typein (well_ordering_rel) (x n) \u2260 0 \u2192 \n    (ordinal.typein (well_ordering_rel) (x n) > ordinal.typein (well_ordering_rel) (x n.succ))) \u2227 \n    (\u2200 n : \u2115, (ordinal.typein (well_ordering_rel) (x n) \u2265 ordinal.typein (well_ordering_rel) (x n.succ))) \u2227 \n    {n : \u2115 | ordinal.typein (well_ordering_rel) (x n) > 0}.finite }\n\ndef subset_prod (I : Type u) : set (prod_N I) := \n  {x : prod_N I | (\u2200 n : \u2115, ordinal.typein (well_ordering_rel) (x n) \u2260 0 \u2192 \n    (ordinal.typein (well_ordering_rel) (x n) > ordinal.typein (well_ordering_rel) (x n.succ))) \u2227 \n    (\u2200 n : \u2115, (ordinal.typein (well_ordering_rel) (x n) \u2265 ordinal.typein (well_ordering_rel) (x n.succ))) \u2227 \n    {n : \u2115 | ordinal.typein (well_ordering_rel) (x n) > 0}.finite }\n\ninstance decidable_eq_IS (S : Profinite.{u}) : decidable_eq (I S) := classical.dec_eq _\n\ndef e_indices {I : Type u} (L : sub_prod I) [decidable_eq I] : finset I := \n  finset.image L.val (set.finite.to_finset L.property.2.2)\n\ndef e_cont {S : Profinite.{u}} (i : I S) : \n  continuous (bool_to_Z \u2218 (\u03bb \u03b9 : (I S \u2192 bool), \u03b9 i) \u2218 map_to_I' S) := \n  continuous_bot.comp ((continuous_apply i).comp (inj_to_prod' S).1)\n\ndef e {S : Profinite.{u}} (i : I S) : CSZ S := \n  \u27e8bool_to_Z \u2218 (\u03bb \u03b9 : (I S \u2192 bool), \u03b9 i) \u2218 map_to_I' S, e_cont i\u27e9\n\ndef prod_e {S : Profinite.{u}} (L : sub_prod (I S)) : CSZ S := finset.prod (e_indices L) e \n\ndef Eset (S : Profinite.{u}) : set (sub_prod (I S)) := {L : sub_prod (I S) | \u2200 s : finset (sub_prod (I S)), \n  (\u2200 L' : sub_prod (I S), L' \u2208 s \u2192 (lex_order (I S)) L'.val L.val) \u2192 finset.sum s prod_e \u2260 prod_e L}\n\n-- def Eset' (S : Profinite.{u}) : set (prod_N (I S)) := {L | \u2200 s : finset (sub_prod (I S)), \n--   (\u2200 L' : sub_prod (I S), L' \u2208 s \u2192 (lex_order (I S)) L'.val L.val) \u2192 finset.sum s prod_e \u2260 prod_e L}\n\ndef Imu (S : Profinite.{u}) (mu : ordinal.{u}) : set (I S) := {i : (I S) | ordinal.typein well_ordering_rel i < mu}\n\ndef Smu (S : Profinite.{u}) (mu : ordinal.{u}) : set S := (map_to_I' S) \u207b\u00b9' {f : (I S) \u2192 bool | f \u207b\u00b9' {tt} \u2286 Imu S mu}\n\ninstance Smu_closed (S : Profinite.{u}) (mu : ordinal.{u}) : is_closed (Smu S mu) := \nbegin\n  refine is_closed.preimage (inj_to_prod' S).1 (is_compact.is_closed _),\n  let s : \u03a0 (i : (I S)), set bool := \n    \u03bb i, ite ((ordinal.typein well_ordering_rel i) < mu) (set.univ : set bool) {ff},\n  have h : {f : I S \u2192 bool | f \u207b\u00b9' {tt} \u2286 Imu S mu} = {x : (I S) \u2192 bool | \u2200 (i : (I S)), x i \u2208 s i},\n  { ext f, split, \n    { intros hf i,\n      have hs : s i = ite ((ordinal.typein well_ordering_rel i) < mu) (set.univ : set bool) {ff} := rfl,\n      rw hs,\n      split_ifs, { tauto },\n      by_contra',\n      apply h,\n      have hf' : i \u2208 f \u207b\u00b9' {tt} := by tidy, \n      exact hf hf', },\n    { intros hf i hi,\n      unfold Imu,\n      have hi' : f i = tt := hi,\n      simp only [set.mem_set_of_eq],\n      have h' := hf i,\n      rw hi' at h',\n      have hs : s i = ite ((ordinal.typein well_ordering_rel i) < mu) (set.univ : set bool) {ff} := rfl,\n      rw hs at h',\n      split_ifs at h', { exact h },\n      exfalso,\n      tauto } },\n  rw h,\n  exact is_compact_pi_infinite (\u03bb i, topological_space.noetherian_space.is_compact (s i)),\nend\n\nlemma Smu_compact (S : Profinite.{u}) (mu : ordinal.{u}) : is_compact (Smu S mu) := \n  is_closed.is_compact (Smu_closed S mu)\n\ninstance Smu_top (S : Profinite.{u}) (mu : ordinal.{u}) : topological_space (Smu S mu) := \n  subtype.topological_space\n\ninstance Smu_comp (S : Profinite.{u}) (mu : ordinal.{u}) : compact_space (Smu S mu) := \n  is_compact_iff_compact_space.mp (Smu_compact S mu)\n\ninstance Smu_t2  (S : Profinite.{u}) (mu : ordinal.{u}) : t2_space (Smu S mu) := \n  subtype.t2_space\n\ninstance Smu_tot_disc (S : Profinite.{u}) (mu : ordinal.{u}) : totally_disconnected_space (Smu S mu) := \n  subtype.totally_disconnected_space\n\ndef Smu_prof (S : Profinite.{u}) (mu : ordinal.{u}) : Profinite.{u} := \n  Profinite.of (Smu S mu)\n\ndef I_map_ord (I : Type*) : I \u2192 ordinal := ordinal.typein (@well_ordering_rel I)\n\ndef I_lsub (I : Type*) : ordinal := ordinal.lsub (I_map_ord I)\n\ndef I_lsub_zero_iff (I : Type*) := @ordinal.lsub_eq_zero_iff I (I_map_ord I)\n\nlemma Imu_zero_is_empty (S : Profinite.{u}) : Imu S 0 = \u2205 :=\nbegin\n  unfold Imu,\n  ext, split, \n  { intros hx,\n    exfalso,\n    have hx\u2081 : ordinal.typein well_ordering_rel x < 0 := hx,\n    have hx\u2082 : 0 \u2264 ordinal.typein well_ordering_rel x := ordinal.zero_le _,\n    rw \u2190 not_le at hx\u2081,\n    exact hx\u2081 hx\u2082 },\n  { tauto }, \nend\n\nlemma Smu_zero_is_subsingleton (S : Profinite.{u}) : (Smu S 0).subsingleton :=\nbegin\n  unfold Smu,\n  rw Imu_zero_is_empty S,\n  have h : {f : I S \u2192 bool | f \u207b\u00b9' {tt} \u2286 \u2205} = {function.const (I S) ff},\n  { ext f, split, \n    { intros hf,\n      apply set.mem_singleton_of_eq,\n      ext,\n      simp only [function.const_apply],\n      by_contra',\n      rw eq_tt_eq_not_eq_ff at this,\n      have hx : x \u2208 f \u207b\u00b9' {tt} := this,\n      exact hf hx },\n    { intros hf,\n      rw (set.eq_of_mem_singleton hf),\n      exact subset_of_eq (set.preimage_const_of_not_mem (\u03bb hff, bool.ff_ne_tt (set.eq_of_mem_singleton hff))) } },\n  rw h,\n  exact set.subsingleton.preimage set.subsingleton_singleton (inj_to_prod' S).2,\nend\n\nlemma unique_Smu_zero (S : Profinite.{u}) [inhabited S] : unique (Smu S 0) :=\nbegin\n  exact unique_of_subsingleton _,\n  -- refine {to_inhabited := _, uniq := _},\n  -- sorry,\n  -- intros a,\n  -- cases a,\n  -- exact (Smu_zero_is_subsingleton S) a_property trivial,\n  -- refine (Smu_zero_is_subsingleton S) a.property _,\nend\n\nlemma Smu_prof_subsingleton (S : Profinite.{u}) : subsingleton (Smu_prof S 0) :=\n (Smu S 0).subsingleton_coe.mpr (Smu_zero_is_subsingleton S)\n\nlemma Eset_zero_subsingleton (S : Profinite.{u}) [hS : subsingleton S] : (Eset S).subsingleton :=\nbegin\n  unfold Eset,\n  intros a ha b hb,\n  ext,\n  sorry,\n  -- apply (map_to_I' S),\nend\n\ninstance no_zero_smul_divisors_CSZ (S : Profinite.{u}) : no_zero_smul_divisors \u2124 (CSZ S) :=\nbegin\n  refine {eq_zero_or_eq_zero_of_smul_eq_zero := _},\n  intros n f hnf, \n  by_cases n = 0, \n  { left, exact h },\n  right,\n  ext,\n  sorry,\nend\n\ntheorem nobeling_mu (S : Profinite.{u}) (mu : ordinal) : \n  linear_independent \u2124 ((Eset (Smu_prof S mu)).restrict prod_e) \u2227 \n  submodule.span \u2124 (prod_e '' (Eset (Smu_prof S mu))) = (\u22a4 : submodule \u2124 (CSZ (Smu_prof S mu))) :=\nbegin\n  induction mu using ordinal.induction with mu IH,\n  simp only [] at IH,\n  cases ordinal.zero_or_succ_or_limit mu,\n  { split,\n    { rw h,\n      apply linear_independent_unique, },\n    { sorry, }, },\n  cases h,\n  { sorry },\n  { sorry },\nend\n\ntheorem nobeling (S : Profinite.{u}) : linear_independent \u2124 ((Eset S).restrict prod_e) \u2227 submodule.span \u2124 (prod_e '' Eset S) = (\u22a4 : submodule \u2124 (CSZ S)) :=\nbegin\n  sorry,\n  -- let lambda := ordinal.typein well_ordering_rel (I S),\n  -- induction lambda using ordinal.induction with lambda IH,\nend\n\nvariables S : Profinite.{u}\nvariables T : Profinite\n#check ordinal.typein (@well_ordering_rel (I S))\n#check I_lsub_zero_iff (I T)\n#check I T\n#check S\n#check function.const (I S) ff\n\ntheorem nobelings_thm (S : Profinite.{u}) : module.free \u2124 (CSZ S) := -- \u2245 free_abelian_group (Eset S) := \nbegin\n  rw module.free_iff_set _ _,\n  use prod_e '' (Eset S),\n  have hs : submodule.span \u2124 (prod_e '' (Eset S)) = \u22a4,\n  { sorry },\n  have hli : linear_independent \u2124 (coe : ((prod_e) '' (Eset S)) \u2192 CSZ S),\n  { sorry },\n  sorry, \n  -- let b := basis.mk hli,\n  -- use \u27e8hli, hs\u27e9,\nend \n", "meta": {"author": "dagurtomas", "repo": "nobelings-thm", "sha": "9477d58edd21be62dc2568004d2675d90a5bd85b", "save_path": "github-repos/lean/dagurtomas-nobelings-thm", "path": "github-repos/lean/dagurtomas-nobelings-thm/nobelings-thm-9477d58edd21be62dc2568004d2675d90a5bd85b/src/CSZ.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7690802370707281, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.390548068967179}}
{"text": "/-\nCopyright (c) 2018 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Johannes H\u00f6lzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.uniform_space.uniform_embedding\nimport Mathlib.topology.uniform_space.complete_separated\nimport Mathlib.topology.algebra.group\nimport Mathlib.tactic.abel\nimport Mathlib.PostPort\n\nuniverses u_3 l u_1 u_2 u u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Uniform structure on topological groups\n\n* `topological_add_group.to_uniform_space` and `topological_add_group_is_uniform` can be used to\n  construct a canonical uniformity for a topological add group.\n\n* extension of \u2124-bilinear maps to complete groups (useful for ring completions)\n\n* `add_group_with_zero_nhd`: construct the topological structure from a group with a neighbourhood\n  around zero. Then with `topological_add_group.to_uniform_space` one can derive a `uniform_space`.\n-/\n\n/-- A uniform (additive) group is a group in which the addition and negation are\n  uniformly continuous. -/\nclass uniform_add_group (\u03b1 : Type u_3) [uniform_space \u03b1] [add_group \u03b1] where\n  uniform_continuous_sub : uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p - prod.snd p\n\ntheorem uniform_add_group.mk' {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1]\n    (h\u2081 : uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p + prod.snd p)\n    (h\u2082 : uniform_continuous fun (p : \u03b1) => -p) : uniform_add_group \u03b1 :=\n  sorry\n\ntheorem uniform_continuous_sub {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] : uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p - prod.snd p :=\n  uniform_add_group.uniform_continuous_sub\n\ntheorem uniform_continuous.sub {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] [uniform_space \u03b2] {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1} (hf : uniform_continuous f)\n    (hg : uniform_continuous g) : uniform_continuous fun (x : \u03b2) => f x - g x :=\n  uniform_continuous.comp uniform_continuous_sub (uniform_continuous.prod_mk hf hg)\n\ntheorem uniform_continuous.neg {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] [uniform_space \u03b2] {f : \u03b2 \u2192 \u03b1} (hf : uniform_continuous f) :\n    uniform_continuous fun (x : \u03b2) => -f x :=\n  sorry\n\ntheorem uniform_continuous_neg {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] : uniform_continuous fun (x : \u03b1) => -x :=\n  uniform_continuous.neg uniform_continuous_id\n\ntheorem uniform_continuous.add {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] [uniform_space \u03b2] {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1} (hf : uniform_continuous f)\n    (hg : uniform_continuous g) : uniform_continuous fun (x : \u03b2) => f x + g x :=\n  sorry\n\ntheorem uniform_continuous_add {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] : uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p + prod.snd p :=\n  uniform_continuous.add uniform_continuous_fst uniform_continuous_snd\n\nprotected instance uniform_add_group.to_topological_add_group {\u03b1 : Type u_1} [uniform_space \u03b1]\n    [add_group \u03b1] [uniform_add_group \u03b1] : topological_add_group \u03b1 :=\n  topological_add_group.mk (uniform_continuous.continuous uniform_continuous_neg)\n\nprotected instance prod.uniform_add_group {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1]\n    [add_group \u03b1] [uniform_add_group \u03b1] [uniform_space \u03b2] [add_group \u03b2] [uniform_add_group \u03b2] :\n    uniform_add_group (\u03b1 \u00d7 \u03b2) :=\n  uniform_add_group.mk\n    (uniform_continuous.prod_mk\n      (uniform_continuous.sub\n        (uniform_continuous.comp uniform_continuous_fst uniform_continuous_fst)\n        (uniform_continuous.comp uniform_continuous_fst uniform_continuous_snd))\n      (uniform_continuous.sub\n        (uniform_continuous.comp uniform_continuous_snd uniform_continuous_fst)\n        (uniform_continuous.comp uniform_continuous_snd uniform_continuous_snd)))\n\ntheorem uniformity_translate {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1] [uniform_add_group \u03b1]\n    (a : \u03b1) :\n    filter.map (fun (x : \u03b1 \u00d7 \u03b1) => (prod.fst x + a, prod.snd x + a)) (uniformity \u03b1) =\n        uniformity \u03b1 :=\n  sorry\n\ntheorem uniform_embedding_translate {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] (a : \u03b1) : uniform_embedding fun (x : \u03b1) => x + a :=\n  sorry\n\ntheorem uniformity_eq_comap_nhds_zero (\u03b1 : Type u_1) [uniform_space \u03b1] [add_group \u03b1]\n    [uniform_add_group \u03b1] :\n    uniformity \u03b1 = filter.comap (fun (x : \u03b1 \u00d7 \u03b1) => prod.snd x - prod.fst x) (nhds 0) :=\n  sorry\n\ntheorem group_separation_rel {\u03b1 : Type u_1} [uniform_space \u03b1] [add_group \u03b1] [uniform_add_group \u03b1]\n    (x : \u03b1) (y : \u03b1) : (x, y) \u2208 Mathlib.separation_rel \u03b1 \u2194 x - y \u2208 closure (singleton 0) :=\n  sorry\n\ntheorem uniform_continuous_of_tendsto_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1]\n    [add_group \u03b1] [uniform_add_group \u03b1] [uniform_space \u03b2] [add_group \u03b2] [uniform_add_group \u03b2]\n    {f : \u03b1 \u2192 \u03b2} [is_add_group_hom f] (h : filter.tendsto f (nhds 0) (nhds 0)) :\n    uniform_continuous f :=\n  sorry\n\ntheorem uniform_continuous_of_continuous {\u03b1 : Type u_1} {\u03b2 : Type u_2} [uniform_space \u03b1]\n    [add_group \u03b1] [uniform_add_group \u03b1] [uniform_space \u03b2] [add_group \u03b2] [uniform_add_group \u03b2]\n    {f : \u03b1 \u2192 \u03b2} [is_add_group_hom f] (h : continuous f) : uniform_continuous f :=\n  sorry\n\n/-- The right uniformity on a topological group. -/\ndef topological_add_group.to_uniform_space (G : Type u) [add_comm_group G] [topological_space G]\n    [topological_add_group G] : uniform_space G :=\n  uniform_space.mk\n    (uniform_space.core.mk (filter.comap (fun (p : G \u00d7 G) => prod.snd p - prod.fst p) (nhds 0))\n      sorry sorry sorry)\n    sorry\n\ntheorem uniformity_eq_comap_nhds_zero' (G : Type u) [add_comm_group G] [topological_space G]\n    [topological_add_group G] :\n    uniformity G = filter.comap (fun (p : G \u00d7 G) => prod.snd p - prod.fst p) (nhds 0) :=\n  rfl\n\ntheorem topological_add_group_is_uniform {G : Type u} [add_comm_group G] [topological_space G]\n    [topological_add_group G] : uniform_add_group G :=\n  sorry\n\ntheorem to_uniform_space_eq {G : Type u} [u : uniform_space G] [add_comm_group G]\n    [uniform_add_group G] : topological_add_group.to_uniform_space G = u :=\n  sorry\n\nnamespace add_comm_group\n\n\n/- TODO: when modules are changed to have more explicit base ring, then change replace `is_Z_bilin`\nby using `is_bilinear_map \u2124` from `tensor_product`. -/\n\n/-- `\u2124`-bilinearity for maps between additive commutative groups. -/\nclass is_Z_bilin {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1] [add_comm_group \u03b2]\n    [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3)\n    where\n  add_left : \u2200 (a a' : \u03b1) (b : \u03b2), f (a + a', b) = f (a, b) + f (a', b)\n  add_right : \u2200 (a : \u03b1) (b b' : \u03b2), f (a, b + b') = f (a, b) + f (a, b')\n\ntheorem is_Z_bilin.comp_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4}\n    [add_comm_group \u03b1] [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f]\n    {g : \u03b3 \u2192 \u03b4} [add_comm_group \u03b4] [is_add_group_hom g] : is_Z_bilin (g \u2218 f) :=\n  sorry\n\nprotected instance is_Z_bilin.comp_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3}\n    [add_comm_group \u03b1] [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] :\n    is_Z_bilin (f \u2218 prod.swap) :=\n  is_Z_bilin.mk (fun (a a' : \u03b2) (b : \u03b1) => is_Z_bilin.add_right f b a a')\n    fun (a : \u03b2) (b b' : \u03b1) => is_Z_bilin.add_left f b b' a\n\ntheorem is_Z_bilin.zero_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (b : \u03b2) : f (0, b) = 0 :=\n  sorry\n\ntheorem is_Z_bilin.zero_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (a : \u03b1) : f (a, 0) = 0 :=\n  is_Z_bilin.zero_left (f \u2218 prod.swap)\n\ntheorem is_Z_bilin.zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] : f (0, 0) = 0 :=\n  is_Z_bilin.zero_left f 0\n\ntheorem is_Z_bilin.neg_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (a : \u03b1) (b : \u03b2) :\n    f (-a, b) = -f (a, b) :=\n  sorry\n\ntheorem is_Z_bilin.neg_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (a : \u03b1) (b : \u03b2) :\n    f (a, -b) = -f (a, b) :=\n  is_Z_bilin.neg_left (f \u2218 prod.swap) b a\n\ntheorem is_Z_bilin.sub_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (a : \u03b1) (a' : \u03b1) (b : \u03b2) :\n    f (a - a', b) = f (a, b) - f (a', b) :=\n  sorry\n\ntheorem is_Z_bilin.sub_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_group \u03b1]\n    [add_comm_group \u03b2] [add_comm_group \u03b3] (f : \u03b1 \u00d7 \u03b2 \u2192 \u03b3) [is_Z_bilin f] (a : \u03b1) (b : \u03b2) (b' : \u03b2) :\n    f (a, b - b') = f (a, b) - f (a, b') :=\n  is_Z_bilin.sub_left (f \u2218 prod.swap) b b' a\n\nend add_comm_group\n\n\n-- \u03b1, \u03b2 and G are abelian topological groups, G is a uniform space\n\ntheorem is_Z_bilin.tendsto_zero_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [add_comm_group \u03b1] [topological_space \u03b2] [add_comm_group \u03b2] {G : Type u_5} [uniform_space G]\n    [add_comm_group G] {\u03c8 : \u03b1 \u00d7 \u03b2 \u2192 G} (h\u03c8 : continuous \u03c8) [\u03c8bilin : add_comm_group.is_Z_bilin \u03c8]\n    (x\u2081 : \u03b1) : filter.tendsto \u03c8 (nhds (x\u2081, 0)) (nhds 0) :=\n  sorry\n\ntheorem is_Z_bilin.tendsto_zero_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [add_comm_group \u03b1] [topological_space \u03b2] [add_comm_group \u03b2] {G : Type u_5} [uniform_space G]\n    [add_comm_group G] {\u03c8 : \u03b1 \u00d7 \u03b2 \u2192 G} (h\u03c8 : continuous \u03c8) [\u03c8bilin : add_comm_group.is_Z_bilin \u03c8]\n    (y\u2081 : \u03b2) : filter.tendsto \u03c8 (nhds (0, y\u2081)) (nhds 0) :=\n  eq.mp\n    (Eq._oldrec (Eq.refl (filter.tendsto \u03c8 (nhds (0, y\u2081)) (nhds (\u03c8 (0, y\u2081)))))\n      (add_comm_group.is_Z_bilin.zero_left \u03c8 y\u2081))\n    (continuous.tendsto h\u03c8 (0, y\u2081))\n\n-- \u03b2 is a dense subgroup of \u03b1, inclusion is denoted by e\n\ntheorem tendsto_sub_comap_self {\u03b1 : Type u_1} {\u03b2 : Type u_2} [topological_space \u03b1]\n    [add_comm_group \u03b1] [topological_add_group \u03b1] [topological_space \u03b2] [add_comm_group \u03b2]\n    {e : \u03b2 \u2192 \u03b1} [is_add_group_hom e] (de : dense_inducing e) (x\u2080 : \u03b1) :\n    filter.tendsto (fun (t : \u03b2 \u00d7 \u03b2) => prod.snd t - prod.fst t)\n        (filter.comap (fun (p : \u03b2 \u00d7 \u03b2) => (e (prod.fst p), e (prod.snd p))) (nhds (x\u2080, x\u2080)))\n        (nhds 0) :=\n  sorry\n\nnamespace dense_inducing\n\n\n-- \u03b2 is a dense subgroup of \u03b1, inclusion is denoted by e\n\n-- \u03b4 is a dense subgroup of \u03b3, inclusion is denoted by f\n\n/-- Bourbaki GT III.6.5 Theorem I:\n\u2124-bilinear continuous maps from dense images into a complete Hausdorff group extend by continuity.\nNote: Bourbaki assumes that \u03b1 and \u03b2 are also complete Hausdorff, but this is not necessary. -/\ntheorem extend_Z_bilin {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} {G : Type u_5}\n    [topological_space \u03b1] [add_comm_group \u03b1] [topological_add_group \u03b1] [topological_space \u03b2]\n    [add_comm_group \u03b2] [topological_add_group \u03b2] [topological_space \u03b3] [add_comm_group \u03b3]\n    [topological_add_group \u03b3] [topological_space \u03b4] [add_comm_group \u03b4] [topological_add_group \u03b4]\n    [uniform_space G] [add_comm_group G] [uniform_add_group G] [separated_space G]\n    [complete_space G] {e : \u03b2 \u2192 \u03b1} [is_add_group_hom e] (de : dense_inducing e) {f : \u03b4 \u2192 \u03b3}\n    [is_add_group_hom f] (df : dense_inducing f) {\u03c6 : \u03b2 \u00d7 \u03b4 \u2192 G} (h\u03c6 : continuous \u03c6)\n    [bilin : add_comm_group.is_Z_bilin \u03c6] : continuous (extend (dense_inducing.prod de df) \u03c6) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/algebra/uniform_group_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.746138993030751, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.39054433208715317}}
{"text": "import combinatorics.quiver.path\nimport phase0.params\n\nopen cardinal function quiver quiver.path set with_bot\nopen_locale cardinal\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}]\n\nsection Iio_Iic\nvariables {\u03b1 \u03b2 : \u039b}\n\nabbreviation Iio_index (\u03b1 : \u039b) := Iio (\u03b1 : type_index)\nabbreviation Iic_index (\u03b1 : \u039b) := Iic (\u03b1 : type_index)\n\ninstance coe_Iio_Iic : has_coe_t (Iio \u03b1) (Iic \u03b1) := \u27e8\u03bb \u03b2, \u27e8\u03b2.1, le_of_lt \u03b2.2\u27e9\u27e9\n\ninstance coe_Iio : has_coe_t (Iio \u03b1) (Iio_index \u03b1) := \u27e8\u03bb \u03b2, \u27e8\u03b2.1, coe_lt_coe.2 \u03b2.2\u27e9\u27e9\ninstance coe_Iic : has_coe_t (Iic \u03b1) (Iic_index \u03b1) := \u27e8\u03bb \u03b2, \u27e8\u03b2.1, coe_le_coe.2 \u03b2.2\u27e9\u27e9\n\nabbreviation Iio_coe : Iio \u03b1 \u2192 Iio_index \u03b1 := coe\nabbreviation Iic_coe : Iic \u03b1 \u2192 Iic_index \u03b1 := coe\n\n@[simp] lemma Iio.coe_mk (\u03b2 : \u039b) (h\u03b2 : \u03b2 < \u03b1) :\n  ((\u27e8\u03b2, h\u03b2\u27e9 : Iio \u03b1) : Iio_index \u03b1) = \u27e8\u03b2, coe_lt_coe.2 h\u03b2\u27e9 := rfl\n@[simp] lemma Iic.coe_mk (\u03b2 : \u039b) (h\u03b2 : \u03b2 \u2264 \u03b1) :\n  ((\u27e8\u03b2, h\u03b2\u27e9 : Iic \u03b1) : Iic_index \u03b1) = \u27e8\u03b2, coe_le_coe.2 h\u03b2\u27e9 := rfl\n\nlemma Iio.coe_injective : injective (coe : Iio \u03b1 \u2192 Iio_index \u03b1) :=\nbegin\n  rintro \u27e8\u03b2, h\u03b2\u27e9 \u27e8\u03b3, h\u03b3\u27e9 h,\n  simp only [Iio.coe_mk, subtype.mk_eq_mk] at h,\n  have := with_bot.coe_injective h,\n  subst this,\nend\nlemma Iic.coe_injective : injective (coe : Iic \u03b1 \u2192 Iic_index \u03b1) :=\nbegin\n  rintro \u27e8\u03b2, h\u03b2\u27e9 \u27e8\u03b3, h\u03b3\u27e9 h,\n  simp only [Iic.coe_mk, subtype.mk_eq_mk] at h,\n  have := with_bot.coe_injective h,\n  subst this,\nend\n\n@[simp] lemma Iio.coe_inj {\u03b2 \u03b3 : Iio \u03b1} : Iio_coe \u03b2 = \u03b3 \u2194 \u03b2 = \u03b3 :=\nIio.coe_injective.eq_iff\n@[simp] lemma Iic.coe_inj {\u03b2 \u03b3 : Iic \u03b1} : Iic_coe \u03b2 = \u03b3 \u2194 \u03b2 = \u03b3 :=\nIic.coe_injective.eq_iff\n\nsection Iio_index\nvariables {h\u03b2 : (\u03b2 : type_index) \u2208 Iio_index \u03b1}\n\ninstance : has_bot (Iio_index \u03b1) := \u27e8\u27e8\u22a5, bot_lt_coe _\u27e9\u27e9\ninstance : inhabited (Iio_index \u03b1) := \u27e8\u22a5\u27e9\n\n@[simp] lemma Iio_index.bot_ne_mk_coe : (\u22a5 : Iio_index \u03b1) \u2260 \u27e8\u03b2, h\u03b2\u27e9 :=\nne_of_apply_ne subtype.val bot_ne_coe\n\n@[simp] lemma Iio_index.mk_coe_ne_bot : (\u27e8\u03b2, h\u03b2\u27e9 : Iio_index \u03b1) \u2260 \u22a5 :=\nne_of_apply_ne subtype.val coe_ne_bot\n\n@[simp] lemma Iio_index.bot_ne_coe {\u03b2 : Iio \u03b1} : \u22a5 \u2260 (\u03b2 : Iio_index \u03b1) :=\nne_of_apply_ne subtype.val bot_ne_coe\n\n@[simp] lemma Iio_index.coe_ne_bot {\u03b2 : Iio \u03b1} : (\u03b2 : Iio_index \u03b1) \u2260 \u22a5 :=\nne_of_apply_ne subtype.val coe_ne_bot\n\nend Iio_index\n\nsection Iic_index\nvariables {h\u03b2 : (\u03b2 : type_index) \u2208 Iic_index \u03b1}\n\ninstance : has_bot (Iic_index \u03b1) := \u27e8\u27e8\u22a5, bot_le\u27e9\u27e9\ninstance : inhabited (Iic_index \u03b1) := \u27e8\u22a5\u27e9\n\n@[simp] lemma Iic_index.bot_ne_mk_coe : (\u22a5 : Iic_index \u03b1) \u2260 \u27e8\u03b2, h\u03b2\u27e9 :=\nne_of_apply_ne subtype.val bot_ne_coe\n\n@[simp] lemma Iic_index.mk_coe_ne_bot : (\u27e8\u03b2, h\u03b2\u27e9 : Iic_index \u03b1) \u2260 \u22a5 :=\nne_of_apply_ne subtype.val coe_ne_bot\nend Iic_index\n\nend Iio_Iic\n\n/-!\nWe now intend to deal with the familiar tools from phase 1 along paths `A` from `\u03b1 \u27f6 \u03b2` down the\nTTT type hierarchy, instead of linearly level-by-level. We will construct three main definitions:\n\n* `le_index`: A type index `\u03b2`, together with a path down from `\u03b1` to level `\u03b2`.\n* `lt_index`: A type index `\u03b2`, together with a path down from `\u03b1`, to some level `\u03b3`, and then\n    directly to level `\u03b2`. This enforces that the path obtained from composing `A` with this new\n    `\u03b3 \u27f6 \u03b2` morphism is nontrivial by construction.\n* `proper_lt_index`: Like `lt_index` but the type index `\u03b2` is proper; that is, it lies in `\u039b`.\n\nEach of these types is progressively more stringent, and they have natural coercions upwards (i.e.\nfrom `proper_lt_index` to `lt_index` to `le_index`, and the transitive coercion from\n`proper_lt_index` to `le_index`). They also have coercions to their index types (`type_index` in the\nfirst two cases, and `\u039b` in the third).\n\nWe will then proceed to define new API for many phase 1 constructions (tangles, f-maps, ...)\nthat use these three types instead of `\u039b`, `type_index`, and `Iio \u03b1`. All of the properties that\nwere proven in phase 1 of course still hold for the functions under these new names - their\nfunctionality has not changed.\n\nThese constructions are helpful for stating and proving the freedom-of-action theorem, since it\nallows for the possibility that the type of `\u03b2`-tangles (for instance) depends on the path downwards\nfrom `\u03b1` to `\u03b2`. In our actual construction, this does hold, since phase 1 is conducted entirely\nlinearly, but this feature is not actually needed for defining and proving statements in most of\nphase 2, so we use this alternate formalisation.\n-/\n\nsection improper\nvariables {\u03b1 : type_index}\n\n/-- We define the type of paths from certain types to lower types as elements of this quiver. -/\ninstance quiver : quiver type_index := \u27e8(>)\u27e9\n\n/-- A (finite) path from the type \u03b1 to the base type.\nThis can be seen as a way that we can perceive extensionality, iteratively descending to lower\ntypes in the hierarchy until we reach the base type.\nThis plays the role of an extended type index in the paper. -/\ndef extended_index (\u03b1 : type_index) := quiver.path \u03b1 \u22a5\n\n/-- If there is a path between `\u03b1` and `\u03b2`, we must have `\u03b2 \u2264 \u03b1`.\nThe case `\u03b2 = \u03b1` can occur with the nil path. -/\nlemma le_of_path : \u03a0 {\u03b2 : type_index}, path \u03b1 \u03b2 \u2192 \u03b2 \u2264 \u03b1\n| \u03b2 nil := le_rfl\n| \u03b2 (cons p f) := (le_of_lt f).trans $ le_of_path p\n\nlemma path_eq_nil : \u2200 p : path \u03b1 \u03b1, p = nil\n| nil := rfl\n| (cons p f) := ((le_of_path p).not_lt f).elim\n\n/-! The next few results won't be needed in the same way in Lean 4. -/\n\ndef path.Iic_rec' {\u03b1 : \u039b} {\u03b2 : Iic_index \u03b1}\n  (motive : \u03a0 (\u03b3 : type_index), path (\u03b2 : type_index) \u03b3 \u2192 Sort*) :\n  motive \u03b2 nil \u2192 (\u03a0 (\u03b3 \u03b4 : type_index) (h\u03b3 : \u03b3 \u2264 \u03b1) (h\u03b4 : \u03b4 \u2264 \u03b1) (A : path (\u03b2 : type_index) \u03b3)\n    (h : \u03b4 < \u03b3), motive \u03b3 A \u2192 motive \u03b4 (A.cons h)) \u2192\n  \u03a0 (\u03b3 : Iic_index \u03b1) (A : path (\u03b2 : type_index) \u03b3),\n  motive (\u27e8\u03b3, (le_of_path A).trans \u03b2.prop\u27e9 : Iic_index \u03b1) A :=\n\u03bb hn hc \u03b3, path.rec hn\n  (\u03bb \u03b3 \u03b4 A h, hc \u03b3 \u03b4 ((le_of_path A).trans \u03b2.prop) ((le_of_path (A.cons h)).trans \u03b2.prop) A h)\n\ndef motive_equiv {\u03b1 : \u039b} {\u03b2 : Iic_index \u03b1}\n  {motive : \u03a0 (\u03b3 : Iic_index \u03b1), path (\u03b2 : type_index) \u03b3 \u2192 Sort*}\n  {\u03b3 : Iic_index \u03b1} {A : path (\u03b2 : type_index) \u03b3} :\n  motive (\u27e8\u03b3, \u03b3.prop\u27e9 : Iic_index \u03b1) A \u2243 motive \u03b3 A :=\nequiv.cast (by cases \u03b3; refl)\n\n/-- An induction principle for paths that allows us to use `Iic_index \u03b1` instead of needing to\ndefine the motive for all `type_index`. -/\n@[elab_as_eliminator]\ndef path.Iic_rec {\u03b1 : \u039b} {\u03b2 : Iic_index \u03b1}\n  {motive : \u03a0 (\u03b3 : Iic_index \u03b1), path (\u03b2 : type_index) \u03b3 \u2192 Sort*} :\n  motive \u03b2 nil \u2192 (\u03a0 (\u03b3 \u03b4 : Iic_index \u03b1) (A : path (\u03b2 : type_index) \u03b3) (h : \u03b4 < \u03b3),\n    motive \u03b3 A \u2192 motive \u03b4 (A.cons h)) \u2192\n  \u03a0 (\u03b3 : Iic_index \u03b1) (A : path (\u03b2 : type_index) \u03b3), motive \u03b3 A :=\n\u03bb hn hc \u03b3 A, motive_equiv (motive_equiv (show _, from\n  path.Iic_rec' (\u03bb \u03b3 A, motive \u27e8\u03b3, (le_of_path A).trans \u03b2.prop\u27e9 A)\n  (motive_equiv.symm hn)\n  (\u03bb \u03b3 \u03b4 h\u03b3 h\u03b4 A h, hc \u27e8\u03b3, (le_of_path A).trans \u03b2.prop\u27e9\n    \u27e8\u03b4, (le_of_path (A.cons h)).trans \u03b2.prop\u27e9 A h)\n  \u03b3 A))\n\n@[simp] lemma path.Iic_rec_nil {\u03b1 : \u039b} {\u03b2 : Iic_index \u03b1}\n  {motive : \u03a0 (\u03b3 : Iic_index \u03b1), path (\u03b2 : type_index) \u03b3 \u2192 Sort*} {hn : motive \u03b2 nil} {hc} :\n  @path.Iic_rec _ _ _ motive hn hc \u03b2 nil = hn :=\nbegin\n  rw [path.Iic_rec, path.Iic_rec'],\n  simp only [subtype.coe_mk, motive_equiv, equiv.cast_symm, equiv.cast_apply, cast_cast, cast_eq],\nend\n\n@[simp] lemma path.Iic_rec_cons {\u03b1 : \u039b} {\u03b2 : Iic_index \u03b1}\n  {motive : \u03a0 (\u03b3 : Iic_index \u03b1), path (\u03b2 : type_index) \u03b3 \u2192 Sort*}\n  {hn : motive \u03b2 nil} {hc} (\u03b3 \u03b4 : Iic_index \u03b1) (A : path (\u03b2 : type_index) \u03b3) (h : \u03b4 < \u03b3) :\n  @path.Iic_rec _ _ _ motive hn hc \u03b4 (A.cons h) = hc \u03b3 \u03b4 A h (path.Iic_rec hn hc \u03b3 A) :=\nbegin\n  rw [path.Iic_rec, path.Iic_rec'],\n  dsimp only [subtype.coe_mk, motive_equiv],\n  simp only [equiv.cast_refl, equiv.cast_symm, equiv.cast_apply, equiv.coe_refl, id.def],\n  rw cast_eq_iff_heq,\n  congr,\n  rw subtype.coe_eta,\n  rw subtype.coe_eta,\n  exact (cast_heq _ _).symm,\nend\n\n/-- There are at most `\u039b` `\u03b1`-extended type indices. -/\n@[simp] lemma mk_extended_index (\u03b1 : type_index) : #(extended_index \u03b1) \u2264 #\u039b :=\nbegin\n  refine le_trans ((cardinal.le_def _ _).2 \u27e8\u27e8to_list, to_list_injective (\u03b1 : type_index) \u22a5\u27e9\u27e9) _,\n  convert mk_list_le_max _ using 1, simp only [mk_type_index, max_eq_right, aleph_0_le_mk],\nend\n\n/-- If `\u03b2 < \u03b3`, we have a path directly between the two types in the opposite order.\nNote that the `\u27f6` symbol (long right arrow) is not the normal `\u2192` (right arrow),\neven though monospace fonts often display them similarly. -/\ninstance lt_to_hom (\u03b2 \u03b3 : \u039b) : has_lift_t (\u03b2 < \u03b3) ((\u03b3 : type_index) \u27f6 \u03b2) := \u27e8coe_lt_coe.2\u27e9\n\n/-- The direct path from the base type to `\u03b1`. -/\ndef type_index.extend : \u03a0 \u03b1 : type_index, extended_index \u03b1\n| \u22a5 := nil\n| (\u03b1 : \u039b) := hom.to_path $ with_bot.bot_lt_coe \u03b1\n\ninstance (\u03b1 : type_index) : inhabited (extended_index \u03b1) := \u27e8\u03b1.extend\u27e9\n\n/-- There exists an `\u03b1`-extended type index. --/\nlemma mk_extended_index_ne_zero (\u03b1 : type_index) : #(extended_index \u03b1) \u2260 0 := cardinal.mk_ne_zero _\n\n/-- A type index `\u03b2`, together with a path down from `\u03b1` to level `\u03b2`. Hence, `\u03b2 \u2264 \u03b1`.\nThis type is intended to be used in place of `\u03b2 : type_index, \u03b2 \u2264 \u03b1` in phase 2. -/\n@[ext, protect_proj] structure le_index (\u03b1 : type_index) :=\n(index : type_index)\n(path : path \u03b1 index)\n\nnamespace le_index\n\ninstance : inhabited (le_index \u03b1) := \u27e8\u27e8\u22a5, \u03b1.extend\u27e9\u27e9\n\n/-- By forgetting the path that we took from `\u03b1` to the lower index `\u03b2`, we can recover the type\nindex `\u03b2` that this `le_index` wraps. -/\ninstance has_coe_type_index : has_coe (le_index \u03b1) type_index := \u27e8le_index.index\u27e9\n\n@[simp] lemma coe_mk (index : type_index) (path : path (\u03b1 : type_index) index) :\n  ((\u27e8index, path\u27e9 : le_index \u03b1) : type_index) = index := rfl\n\n/-- Add an index to a `le_index`. -/\ndef cons (A : le_index \u03b1) {\u03b3 : type_index} (h\u03b3 : \u03b3 < A.index) : le_index \u03b1 :=\n\u27e8\u03b3, A.path.cons h\u03b3\u27e9\n\nend le_index\n\n/-- A type index `\u03b2`, together with a path `A` down from `\u03b1` to level `\u03b3` and then to level `\u03b2`.\nThis enforces that the path obtained from composing `A` with this new `\u03b3 \u27f6 \u03b2` morphism is\nnontrivial by construction. This type is intended to be used in place of `\u03b2 : type_index, \u03b2 < \u03b1`\nand `\u03b2 : Iio_index \u03b1` in phase 2. -/\n@[ext] structure lt_index (\u03b1 : type_index) :=\n(index : type_index)\n(higher : type_index)\n(index_lt_higher : index < higher)\n(path' : path \u03b1 higher)\n\nnamespace lt_index\n\n/-- A constructor for `lt_index` with less explicit arguments. -/\ndef mk' {index higher : type_index} (index_lt : index < higher)\n  (path' : path (\u03b1 : type_index) higher) : lt_index \u03b1 :=\n\u27e8index, higher, index_lt, path'\u27e9\n\ninstance {\u03b1 : \u039b} : inhabited (lt_index \u03b1) := \u27e8mk' (bot_lt_coe _) path.nil\u27e9\n\n/-- A path compatible with the one from `le_index`, formed by composing the inner `path'` field\nwith the morphism `higher \u27f6 index`. By construction, this path is always nontrivial. -/\ndef path (A : lt_index \u03b1) : path (\u03b1 : type_index) A.index := A.path'.cons A.index_lt_higher\n\n/-- An `lt_index` is not equal to its source `\u03b1`. This is the lemma that justifies the name\n`lt_index` as compared to `le_index`, which permits the trivial path `\u03b1 \u27f6 \u03b1`. -/\nlemma index_lt (A : lt_index \u03b1) : A.index < \u03b1 := A.index_lt_higher.trans_le $ le_of_path A.path'\n\n/-- The natural coercion from `lt_index` to `le_index`. An analogous concept to `le_of_lt`. -/\ndef to_le_index (A : lt_index \u03b1) : le_index \u03b1 := \u27e8A.index, A.path\u27e9\n\ninstance has_coe_le_index : has_coe (lt_index \u03b1) (le_index \u03b1) := \u27e8to_le_index\u27e9\n\n/-- By forgetting the path that we took from `\u03b1` to the lower index `\u03b2`, we can recover the type\nindex `\u03b2` that this `lt_index` wraps. -/\ninstance has_coe_type_index : has_coe (lt_index \u03b1) type_index := \u27e8index\u27e9\n\nend lt_index\nend improper\n\n/-- A proper type index `\u03b2`, together with a path `A` down from `\u03b1` to level `\u03b3` and then to level\n`\u03b2`. This enforces that the path obtained from composing `A` with this new `\u03b3 \u27f6 \u03b2` morphism is\nnontrivial by construction. This type is intended to be used in phase of `\u03b2 : \u039b, \u03b2 < \u03b1` and\n`\u03b2 : Iio \u03b1` in phase 2. -/\n@[ext, nolint has_nonempty_instance] structure proper_lt_index (\u03b1 : \u039b) :=\n(index higher : \u039b)\n(index_lt_higher : index < higher)\n(path' : path (\u03b1 : type_index) higher)\n\nnamespace proper_lt_index\nvariables {\u03b1 : \u039b}\n\n/-- A constructor for `proper_lt_index` with less explicit arguments. -/\ndef mk' {\u03b1 index higher : \u039b} (index_lt : index < higher) (path' : path (\u03b1 : type_index) higher) :\n  proper_lt_index \u03b1 :=\n\u27e8index, higher, index_lt, path'\u27e9\n\n/-- A path compatible with the one from `le_index`, formed by composing the inner `path'` field\nwith the morphism `higher \u27f6 index`. By construction, this path is always nontrivial. -/\ndef path (A : proper_lt_index \u03b1) : path (\u03b1 : type_index) A.index :=\nA.path'.cons $ coe_lt_coe.2 A.index_lt_higher\n\n/-- A `proper_lt_index` is not equal to its source `\u03b1`. See also `lt_index.ne`. -/\nlemma index_lt (A : proper_lt_index \u03b1) : A.index < \u03b1 :=\nA.index_lt_higher.trans_le $ coe_le_coe.1 $ le_of_path A.path'\n\n/-- The natural coercion from `proper_lt_index` to `le_index`.\nAn analogous concept to `le_of_lt`, also converting `index: \u039b` into a `type_index`. -/\ndef to_le_index (A : proper_lt_index \u03b1) : le_index \u03b1 :=\n\u27e8A.index, A.path\u27e9\n\n/-- The natural coercion from `proper_lt_index` to `to_lt_index`, by converting `index : \u039b` into a\n`type_index`. -/\ndef to_lt_index (A : proper_lt_index \u03b1) : lt_index \u03b1 :=\n\u27e8A.index, A.higher, coe_lt_coe.2 A.index_lt_higher, A.path'\u27e9\n\ninstance has_coe_lt_index : has_coe (proper_lt_index \u03b1) (lt_index \u03b1) := \u27e8to_lt_index\u27e9\n\n/-- By forgetting the path that we took from `\u03b1` to the lower index `\u03b2`, we can recover the proper\ntype index `\u03b2` that this `proper_lt_index` wraps. -/\ninstance has_coe_\u039b : has_coe (proper_lt_index \u03b1) \u039b := \u27e8index\u27e9\n\nend proper_lt_index\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase0/index.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.390412619185459}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Alexander Bentkamp\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.linear_independent\nimport Mathlib.linear_algebra.projection\nimport Mathlib.data.fintype.card\nimport Mathlib.PostPort\n\nuniverses u_1 u_3 u_5 u_2 u_6 u_7 u_4 u u_8 u_9 u_10 u_11 \n\nnamespace Mathlib\n\n/-!\n\n# Bases\n\nThis file defines bases in a module or vector space.\n\nIt is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light.\n\n## Main definitions\n\nAll definitions are given for families of vectors, i.e. `v : \u03b9 \u2192 M` where `M` is the module or\nvector space and `\u03b9 : Type*` is an arbitrary indexing type.\n\n* `is_basis R v` states that the vector family `v` is a basis, i.e. it is linearly independent and\n  spans the entire space.\n\n* `is_basis.repr hv x` is the basis version of `linear_independent.repr hv x`. It returns the\n  linear combination representing `x : M` on a basis `v` of `M` (using classical choice).\n  The argument `hv` must be a proof that `is_basis R v`. `is_basis.repr hv` is given as a linear\n  map as well.\n\n* `is_basis.constr hv f` constructs a linear map `M\u2081 \u2192\u2097[R] M\u2082` given the values `f : \u03b9 \u2192 M\u2082` at the\n  basis `v : \u03b9 \u2192 M\u2081`, given `hv : is_basis R v`.\n\n## Main statements\n\n* `is_basis.ext` states that two linear maps are equal if they coincide on a basis.\n\n* `exists_is_basis` states that every vector space has a basis.\n\n## Implementation notes\n\nWe use families instead of sets because it allows us to say that two identical vectors are linearly\ndependent. For bases, this is useful as well because we can easily derive ordered bases by using an\nordered index type `\u03b9`.\n\n## Tags\n\nbasis, bases\n\n-/\n\n/-- A family of vectors is a basis if it is linearly independent and all vectors are in the span. -/\ndef is_basis {\u03b9 : Type u_1} (R : Type u_3) {M : Type u_5} (v : \u03b9 \u2192 M) [ring R] [add_comm_group M] [module R M] :=\n  linear_independent R v \u2227 submodule.span R (set.range v) = \u22a4\n\ntheorem is_basis.mem_span {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (x : M) : x \u2208 submodule.span R (set.range v) :=\n  iff.mp submodule.eq_top_iff' (and.right hv)\n\ntheorem is_basis.comp {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (f : \u03b9' \u2192 \u03b9) (hf : function.bijective f) : is_basis R (v \u2218 f) := sorry\n\ntheorem is_basis.injective {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [nontrivial R] (hv : is_basis R v) : function.injective v :=\n  fun (x y : \u03b9) (h : v x = v y) => linear_independent.injective (and.left hv) h\n\ntheorem is_basis.range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : is_basis R fun (x : \u21a5(set.range v)) => \u2191x := sorry\n\n/-- Given a basis, any vector can be written as a linear combination of the basis vectors. They are\ngiven by this linear map. This is one direction of `module_equiv_finsupp`. -/\ndef is_basis.repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map R M (\u03b9 \u2192\u2080 R) :=\n  linear_map.comp (linear_independent.repr sorry)\n    (linear_map.cod_restrict (submodule.span R (set.range v)) linear_map.id (is_basis.mem_span hv))\n\ntheorem is_basis.total_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (x : M) : coe_fn (finsupp.total \u03b9 M R v) (coe_fn (is_basis.repr hv) x) = x :=\n  linear_independent.total_repr (and.left hv) { val := x, property := is_basis.mem_span hv x }\n\ntheorem is_basis.total_comp_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map.comp (finsupp.total \u03b9 M R v) (is_basis.repr hv) = linear_map.id :=\n  linear_map.ext (is_basis.total_repr hv)\n\ntheorem is_basis.ext {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {f : linear_map R M M'} {g : linear_map R M M'} (hv : is_basis R v) (h : \u2200 (i : \u03b9), coe_fn f (v i) = coe_fn g (v i)) : f = g :=\n  linear_map.ext_on_range (and.right hv) h\n\ntheorem is_basis.repr_ker {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map.ker (is_basis.repr hv) = \u22a5 :=\n  iff.mpr linear_map.ker_eq_bot (function.left_inverse.injective (is_basis.total_repr hv))\n\ntheorem is_basis.repr_range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map.range (is_basis.repr hv) = finsupp.supported R R set.univ := sorry\n\ntheorem is_basis.repr_total {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (x : \u03b9 \u2192\u2080 R) (hx : x \u2208 finsupp.supported R R set.univ) : coe_fn (is_basis.repr hv) (coe_fn (finsupp.total \u03b9 M R v) x) = x := sorry\n\ntheorem is_basis.repr_eq_single {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) {i : \u03b9} : coe_fn (is_basis.repr hv) (v i) = finsupp.single i 1 := sorry\n\n@[simp] theorem is_basis.repr_self_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) (j : \u03b9) : coe_fn (coe_fn (is_basis.repr hv) (v i)) j = ite (i = j) 1 0 := sorry\n\ntheorem is_basis.repr_eq_iff {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) {f : linear_map R M (\u03b9 \u2192\u2080 R)} : is_basis.repr hv = f \u2194 \u2200 (i : \u03b9), coe_fn f (v i) = finsupp.single i 1 := sorry\n\ntheorem is_basis.repr_apply_eq {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) {f : M \u2192 \u03b9 \u2192 R} (hadd : \u2200 (x y : M), f (x + y) = f x + f y) (hsmul : \u2200 (c : R) (x : M), f (c \u2022 x) = c \u2022 f x) (f_eq : \u2200 (i : \u03b9), f (v i) = \u21d1(finsupp.single i 1)) (x : M) (i : \u03b9) : coe_fn (coe_fn (is_basis.repr hv) x) i = f x i := sorry\n\ntheorem is_basis.range_repr_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) : coe_fn (is_basis.repr (is_basis.range hv)) (v i) = finsupp.single { val := v i, property := set.mem_range_self i } 1 := sorry\n\n@[simp] theorem is_basis.range_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] {x : M} (hv : is_basis R v) (i : \u03b9) : coe_fn (coe_fn (is_basis.repr (is_basis.range hv)) x) { val := v i, property := set.mem_range_self i } =\n  coe_fn (coe_fn (is_basis.repr hv) x) i := sorry\n\n/-- Construct a linear map given the value at the basis. -/\ndef is_basis.constr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) (f : \u03b9 \u2192 M') : linear_map R M M' :=\n  linear_map.comp (finsupp.total M' M' R id) (linear_map.comp (finsupp.lmap_domain R R f) (is_basis.repr hv))\n\ntheorem is_basis.constr_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) (f : \u03b9 \u2192 M') (x : M) : coe_fn (is_basis.constr hv f) x = finsupp.sum (coe_fn (is_basis.repr hv) x) fun (b : \u03b9) (a : R) => a \u2022 f b := sorry\n\n@[simp] theorem constr_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {f : \u03b9 \u2192 M'} {i : \u03b9} (hv : is_basis R v) : coe_fn (is_basis.constr hv f) (v i) = f i := sorry\n\ntheorem constr_eq {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {g : \u03b9 \u2192 M'} {f : linear_map R M M'} (hv : is_basis R v) (h : \u2200 (i : \u03b9), g i = coe_fn f (v i)) : is_basis.constr hv g = f :=\n  is_basis.ext hv fun (i : \u03b9) => Eq.trans (constr_basis hv) (h i)\n\ntheorem constr_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) (f : linear_map R M M') : (is_basis.constr hv fun (i : \u03b9) => coe_fn f (v i)) = f :=\n  constr_eq hv fun (x : \u03b9) => rfl\n\ntheorem constr_zero {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) : (is_basis.constr hv fun (i : \u03b9) => 0) = 0 :=\n  constr_eq hv fun (x : \u03b9) => rfl\n\ntheorem constr_add {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {g : \u03b9 \u2192 M'} {f : \u03b9 \u2192 M'} (hv : is_basis R v) : (is_basis.constr hv fun (i : \u03b9) => f i + g i) = is_basis.constr hv f + is_basis.constr hv g := sorry\n\ntheorem constr_neg {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {f : \u03b9 \u2192 M'} (hv : is_basis R v) : (is_basis.constr hv fun (i : \u03b9) => -f i) = -is_basis.constr hv f := sorry\n\ntheorem constr_sub {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) {g : \u03b9 \u2192 M'} {f : \u03b9 \u2192 M'} (hs : is_basis R v) : (is_basis.constr hv fun (i : \u03b9) => f i - g i) = is_basis.constr hs f - is_basis.constr hs g := sorry\n\n-- this only works on functions if `R` is a commutative ring\n\ntheorem constr_smul {\u03b9 : Type u_1} {R : Type u_2} {M : Type u_3} [comm_ring R] [add_comm_group M] [module R M] {v : \u03b9 \u2192 R} {f : \u03b9 \u2192 M} {a : R} (hv : is_basis R v) : (is_basis.constr hv fun (b : \u03b9) => a \u2022 f b) = a \u2022 is_basis.constr hv f := sorry\n\ntheorem constr_range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] [Nonempty \u03b9] (hv : is_basis R v) {f : \u03b9 \u2192 M'} : linear_map.range (is_basis.constr hv f) = submodule.span R (set.range f) := sorry\n\n/-- Canonical equivalence between a module and the linear combinations of basis vectors. -/\ndef module_equiv_finsupp {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_equiv R M (\u03b9 \u2192\u2080 R) :=\n  linear_equiv.symm\n    (linear_equiv.trans (linear_independent.total_equiv sorry)\n      (linear_equiv.of_top (submodule.span R (set.range v)) sorry))\n\n@[simp] theorem module_equiv_finsupp_apply_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) : coe_fn (module_equiv_finsupp hv) (v i) = finsupp.single i 1 := sorry\n\n/-- Isomorphism between the two modules, given two modules `M` and `M'` with respective bases\n`v` and `v'` and a bijection between the indexing sets of the two bases. -/\ndef linear_equiv_of_is_basis {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (hv : is_basis R v) (hv' : is_basis R v') (e : \u03b9 \u2243 \u03b9') : linear_equiv R M M' :=\n  linear_equiv.mk (linear_map.to_fun (is_basis.constr hv (v' \u2218 \u21d1e))) sorry sorry\n    \u21d1(is_basis.constr hv' (v \u2218 \u21d1(equiv.symm e))) sorry sorry\n\n/-- Isomorphism between the two modules, given two modules `M` and `M'` with respective bases\n`v` and `v'` and a bijection between the two bases. -/\ndef linear_equiv_of_is_basis' {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (f : M \u2192 M') (g : M' \u2192 M) (hv : is_basis R v) (hv' : is_basis R v') (hf : \u2200 (i : \u03b9), f (v i) \u2208 set.range v') (hg : \u2200 (i : \u03b9'), g (v' i) \u2208 set.range v) (hgf : \u2200 (i : \u03b9), g (f (v i)) = v i) (hfg : \u2200 (i : \u03b9'), f (g (v' i)) = v' i) : linear_equiv R M M' :=\n  linear_equiv.mk (linear_map.to_fun (is_basis.constr hv (f \u2218 v))) sorry sorry \u21d1(is_basis.constr hv' (g \u2218 v')) sorry sorry\n\n@[simp] theorem linear_equiv_of_is_basis_comp {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {M'' : Type u_7} [ring R] [add_comm_group M] [add_comm_group M'] [add_comm_group M''] [module R M] [module R M'] [module R M''] {\u03b9'' : Type u_4} {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} {v'' : \u03b9'' \u2192 M''} (hv : is_basis R v) (hv' : is_basis R v') (hv'' : is_basis R v'') (e : \u03b9 \u2243 \u03b9') (f : \u03b9' \u2243 \u03b9'') : linear_equiv.trans (linear_equiv_of_is_basis hv hv' e) (linear_equiv_of_is_basis hv' hv'' f) =\n  linear_equiv_of_is_basis hv hv'' (equiv.trans e f) := sorry\n\n@[simp] theorem linear_equiv_of_is_basis_refl {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) : linear_equiv_of_is_basis hv hv (equiv.refl \u03b9) = linear_equiv.refl R M := sorry\n\ntheorem linear_equiv_of_is_basis_trans_symm {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) (e : \u03b9 \u2243 \u03b9') {v' : \u03b9' \u2192 M'} (hv' : is_basis R v') : linear_equiv.trans (linear_equiv_of_is_basis hv hv' e) (linear_equiv_of_is_basis hv' hv (equiv.symm e)) =\n  linear_equiv.refl R M := sorry\n\ntheorem linear_equiv_of_is_basis_symm_trans {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) (e : \u03b9 \u2243 \u03b9') {v' : \u03b9' \u2192 M'} (hv' : is_basis R v') : linear_equiv.trans (linear_equiv_of_is_basis hv' hv (equiv.symm e)) (linear_equiv_of_is_basis hv hv' e) =\n  linear_equiv.refl R M' := sorry\n\ntheorem is_basis_inl_union_inr {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (hv : is_basis R v) (hv' : is_basis R v') : is_basis R (sum.elim (\u21d1(linear_map.inl R M M') \u2218 v) (\u21d1(linear_map.inr R M M') \u2218 v')) := sorry\n\ntheorem is_basis_singleton_one {\u03b9 : Type u_1} (R : Type u_2) [unique \u03b9] [ring R] : is_basis R fun (_x : \u03b9) => 1 := sorry\n\nprotected theorem linear_equiv.is_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hs : is_basis R v) (f : linear_equiv R M M') : is_basis R (\u21d1f \u2218 v) := sorry\n\ntheorem is_basis_span {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hs : linear_independent R v) : is_basis R fun (i : \u03b9) => { val := v i, property := submodule.subset_span (set.mem_range_self i) } := sorry\n\ntheorem is_basis_empty {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} [ring R] [add_comm_group M] [module R M] (h_empty : \u00acNonempty \u03b9) (h : \u2200 (x : M), x = 0) : is_basis R fun (x : \u03b9) => 0 := sorry\n\ntheorem is_basis_empty_bot {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} [ring R] [add_comm_group M] [module R M] (h_empty : \u00acNonempty \u03b9) : is_basis R fun (_x : \u03b9) => 0 :=\n  is_basis_empty h_empty fun (x : \u21a5\u22a5) => iff.mpr subtype.ext_iff_val (iff.mp (submodule.mem_bot R) (subtype.mem x))\n\n/-- A module over `R` with a finite basis is linearly equivalent to functions from its basis to `R`.\n-/\ndef is_basis.equiv_fun {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) : linear_equiv R M (\u03b9 \u2192 R) :=\n  linear_equiv.trans (module_equiv_finsupp h)\n    (linear_equiv.mk finsupp.to_fun sorry sorry (equiv.inv_fun finsupp.equiv_fun_on_fintype) sorry sorry)\n\n/-- A module over a finite ring that admits a finite basis is finite. -/\ndef module.fintype_of_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) [fintype R] : fintype M :=\n  fintype.of_equiv (\u03b9 \u2192 R) (equiv.symm (linear_equiv.to_equiv (is_basis.equiv_fun h)))\n\ntheorem module.card_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) [fintype R] [fintype M] : fintype.card M = fintype.card R ^ fintype.card \u03b9 :=\n  Eq.trans (fintype.card_congr (linear_equiv.to_equiv (is_basis.equiv_fun h))) fintype.card_fun\n\n/-- Given a basis `v` indexed by `\u03b9`, the canonical linear equivalence between `\u03b9 \u2192 R` and `M` maps\na function `x : \u03b9 \u2192 R` to the linear combination `\u2211_i x i \u2022 v i`. -/\n@[simp] theorem is_basis.equiv_fun_symm_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (x : \u03b9 \u2192 R) : coe_fn (linear_equiv.symm (is_basis.equiv_fun h)) x = finset.sum finset.univ fun (i : \u03b9) => x i \u2022 v i := sorry\n\ntheorem is_basis.equiv_fun_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (u : M) : coe_fn (is_basis.equiv_fun h) u = \u21d1(coe_fn (is_basis.repr h) u) :=\n  rfl\n\ntheorem is_basis.equiv_fun_total {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (u : M) : (finset.sum finset.univ fun (i : \u03b9) => coe_fn (is_basis.equiv_fun h) u i \u2022 v i) = u := sorry\n\n@[simp] theorem is_basis.equiv_fun_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (i : \u03b9) (j : \u03b9) : coe_fn (is_basis.equiv_fun h) (v i) j = ite (i = j) 1 0 := sorry\n\n@[simp] theorem is_basis.constr_apply_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] [fintype \u03b9] (h : is_basis R v) (f : \u03b9 \u2192 M') (x : M) : coe_fn (is_basis.constr h f) x = finset.sum finset.univ fun (i : \u03b9) => coe_fn (is_basis.equiv_fun h) x i \u2022 f i := sorry\n\ntheorem exists_subset_is_basis {K : Type u_4} {V : Type u} [field K] [add_comm_group V] [vector_space K V] {s : set V} (hs : linear_independent K fun (x : \u21a5s) => \u2191x) : \u2203 (b : set V), s \u2286 b \u2227 is_basis K coe := sorry\n\ntheorem exists_sum_is_basis {\u03b9 : Type u_1} {K : Type u_4} {V : Type u} [field K] [add_comm_group V] [vector_space K V] {v : \u03b9 \u2192 V} (hs : linear_independent K v) : \u2203 (\u03b9' : Type u), \u2203 (v' : \u03b9' \u2192 V), is_basis K (sum.elim v v') := sorry\n\ntheorem exists_is_basis (K : Type u_4) (V : Type u) [field K] [add_comm_group V] [vector_space K V] : \u2203 (b : set V), is_basis K fun (i : \u21a5b) => \u2191i := sorry\n\ntheorem linear_map.exists_left_inverse_of_injective {K : Type u_4} {V : Type u} {V' : Type u_8} [field K] [add_comm_group V] [add_comm_group V'] [vector_space K V] [vector_space K V'] (f : linear_map K V V') (hf_inj : linear_map.ker f = \u22a5) : \u2203 (g : linear_map K V' V), linear_map.comp g f = linear_map.id := sorry\n\ntheorem submodule.exists_is_compl {K : Type u_4} {V : Type u} [field K] [add_comm_group V] [vector_space K V] (p : submodule K V) : \u2203 (q : submodule K V), is_compl p q := sorry\n\ntheorem linear_map.exists_right_inverse_of_surjective {K : Type u_4} {V : Type u} {V' : Type u_8} [field K] [add_comm_group V] [add_comm_group V'] [vector_space K V] [vector_space K V'] (f : linear_map K V V') (hf_surj : linear_map.range f = \u22a4) : \u2203 (g : linear_map K V' V), linear_map.comp f g = linear_map.id := sorry\n\ntheorem quotient_prod_linear_equiv {K : Type u_4} {V : Type u} [field K] [add_comm_group V] [vector_space K V] (p : submodule K V) : Nonempty (linear_equiv K (submodule.quotient p \u00d7 \u21a5p) V) := sorry\n\ntheorem vector_space.card_fintype (K : Type u_4) (V : Type u) [field K] [add_comm_group V] [vector_space K V] [fintype K] [fintype V] : \u2203 (n : \u2115), fintype.card V = fintype.card K ^ n :=\n  exists.elim (exists_is_basis K V)\n    fun (b : set V) (hb : is_basis K fun (i : \u21a5b) => \u2191i) => Exists.intro (fintype.card \u21a5b) (module.card_fintype hb)\n\nnamespace pi\n\n\ntheorem linear_independent_std_basis {R : Type u_3} {\u03b7 : Type u_9} {\u03b9s : \u03b7 \u2192 Type u_10} {Ms : \u03b7 \u2192 Type u_11} [ring R] [(i : \u03b7) \u2192 add_comm_group (Ms i)] [(i : \u03b7) \u2192 module R (Ms i)] [DecidableEq \u03b7] (v : (j : \u03b7) \u2192 \u03b9s j \u2192 Ms j) (hs : \u2200 (i : \u03b7), linear_independent R (v i)) : linear_independent R\n  fun (ji : sigma fun (j : \u03b7) => \u03b9s j) =>\n    coe_fn (linear_map.std_basis R Ms (sigma.fst ji)) (v (sigma.fst ji) (sigma.snd ji)) := sorry\n\ntheorem is_basis_std_basis {R : Type u_3} {\u03b7 : Type u_9} {\u03b9s : \u03b7 \u2192 Type u_10} {Ms : \u03b7 \u2192 Type u_11} [ring R] [(i : \u03b7) \u2192 add_comm_group (Ms i)] [(i : \u03b7) \u2192 module R (Ms i)] [fintype \u03b7] [DecidableEq \u03b7] (s : (j : \u03b7) \u2192 \u03b9s j \u2192 Ms j) (hs : \u2200 (j : \u03b7), is_basis R (s j)) : is_basis R\n  fun (ji : sigma fun (j : \u03b7) => \u03b9s j) =>\n    coe_fn (linear_map.std_basis R Ms (sigma.fst ji)) (s (sigma.fst ji) (sigma.snd ji)) := sorry\n\ntheorem is_basis_fun\u2080 (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7] : is_basis R fun (ji : sigma fun (j : \u03b7) => Unit) => coe_fn (linear_map.std_basis R (fun (i : \u03b7) => R) (sigma.fst ji)) 1 :=\n  is_basis_std_basis (fun (_x : \u03b7) (_x : Unit) => 1) fun (i : \u03b7) => is_basis_singleton_one R\n\ntheorem is_basis_fun (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7] : is_basis R fun (i : \u03b7) => coe_fn (linear_map.std_basis R (fun (i : \u03b7) => R) i) 1 := sorry\n\n@[simp] theorem is_basis_fun_repr (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7] (x : \u03b7 \u2192 R) (i : \u03b7) : coe_fn (coe_fn (is_basis.repr (is_basis_fun R \u03b7)) x) i = x i := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/basis.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3904126191854589}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume (A : set (euclidean_space \u211d (fin n))) (hA : is_open_cover A),\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  have h1 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_cover C \u2227 is_locally_finite C \u2227 is_refinement A C, from by {\n    -- First, we define a collection of pen balls.\n    -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n    let B : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, {x : euclidean_space \u211d (fin n) | \u2203 (y : euclidean_space \u211d (fin n)), y \u2208 (ball (0 : euclidean_space \u211d (fin n)) m) \u2227 x = y},\n    have hB : \u2200 m : \u2115, B m \u2286 (euclidean_space \u211d (fin n)), from by {\n      assume m : \u2115,\n      assume x : euclidean_space \u211d (fin n),\n      assume hx : x \u2208 B m,\n      show x \u2208 (euclidean_space \u211d (fin n)), from by {\n        cases hx with y hy,\n        cases hy with hy1 hy2,\n        show x \u2208 (euclidean_space \u211d (fin n)), from by {\n          rw \u2190 hy2,\n          apply set.mem_of_mem_ball,\n          exact hy1,\n        },\n      },\n    },\n    have hB0 : B 0 = \u2205, from by {\n      apply set.eq_empty_iff_forall_not_mem.mpr,\n      assume x : euclidean_space \u211d (fin n),\n      assume hx : x \u2208 B 0,\n      cases hx with y hy,\n      cases hy with hy1 hy2,\n      have h3 : y \u2208 (ball (0 : euclidean_space \u211d (fin n)) 0), from by {\n        rw \u2190 hy2,\n        exact hx,\n      },\n      have h4 : y = (0 : euclidean_space \u211d (fin n)), from by {\n        apply ball_eq_zero,\n        exact h3,\n      },\n      have h5 : x = (0 : euclidean_space \u211d (fin n)), from by {\n        rw \u2190 hy2,\n        exact h4,\n      },\n      have h6 : x \u2209 (euclidean_space \u211d (fin n) \\ {0}), from by {\n        apply set.not_mem_of_mem_diff,\n        split,\n        exact hx,\n        apply set.not_mem_singleton,\n        exact h5,\n      },\n      show false, from by {\n        apply h6,\n        apply set.mem_univ,\n      },\n    },\n    have hB1 : \u2200 m : \u2115, B (m+1) = (euclidean_space \u211d (fin n)) \\ (B m), from by {\n      assume m : \u2115,\n      apply set.eq_of_subset_of_subset,\n      assume x : euclidean_space \u211d (fin n),\n      assume hx : x \u2208 B (m+1),\n      cases hx with y hy,\n      cases hy with hy1 hy2,\n      have h3 : y \u2208 (ball (0 : euclidean_space \u211d (fin n)) (m+1)), from by {\n        rw \u2190 hy2,\n        exact hx,\n      },\n      have h4 : y \u2209 (ball (0 : euclidean_space \u211d (fin n)) m), from by {\n        apply ball_not_mem_of_ball_lt,\n        exact h3,\n      },\n      have h5 : y \u2209 B m, from by {\n        assume h6 : y \u2208 B m,\n        cases h6 with z hz,\n        cases hz with hz1 hz2,\n        have h7 : z \u2208 (ball (0 : euclidean_space \u211d (fin n)) m), from by {\n          rw \u2190 hz2,\n          exact hz1,\n        },\n        show false, from by {\n          apply h4,\n          exact h7,\n        },\n      },\n      have h6 : y \u2208 (euclidean_space \u211d (fin n)) \\ (B m), from by {\n        apply set.mem_diff,\n        split,\n        exact hy1,\n        exact h5,\n      },\n      have h7 : x \u2208 (euclidean_space \u211d (fin n)) \\ (B m), from by {\n        rw \u2190 hy2,\n        exact h6,\n      },\n      show x \u2208 (euclidean_space \u211d (fin n)) \\ (B m), from by {\n        apply set.mem_diff,\n        split,\n        exact hx,\n        exact h7,\n      },\n      assume x : euclidean_space \u211d (fin n),\n      assume hx : x \u2208 (euclidean_space \u211d (fin n)) \\ (B m),\n      cases hx with hx1 hx2,\n      have h3 : x \u2208 (euclidean_space \u211d (fin n)), from by {\n        apply set.mem_diff,\n        split,\n        exact hx1,\n        exact hx2,\n      },\n      have h4 : \u2203 (y : euclidean_space \u211d (fin n)), y \u2208 (ball (0 : euclidean_space \u211d (fin n)) (m+1)) \u2227 x = y, from by {\n        use x,\n        split,\n        exact h3,\n        refl,\n      },\n      show x \u2208 B (m+1), from by {\n        apply set.mem_of_mem_ball,\n        exact h4,\n      },\n    },\n    have hB2 : \u2200 m : \u2115, B m = (euclidean_space \u211d (fin n)) \\ (B (m-1)), from by {\n      assume m : \u2115,\n      induction m with m hm,\n      show B 0 = (euclidean_space \u211d (fin n)) \\ (B (-1)), from by {\n        rw hB0,\n        rw set.diff_empty,\n        refl,\n      },\n      show B (m+1) = (euclidean_space \u211d (fin n)) \\ (B m), from by {\n        rw hB1,\n        rw hm,\n        refl,\n      },\n    },\n    have hB3 : \u2200 m : \u2115, B m = (euclidean_space \u211d (fin n)) \\ (B (m-1)), from by {\n      assume m : \u2115,\n      induction m with m hm,\n      show B 0 = (euclidean_space \u211d (fin n)) \\ (B (-1)), from by {\n        rw hB0,\n        rw set.diff_empty,\n        refl,\n      },\n      show B (m+1) = (euclidean_space \u211d (fin n)) \\ (B m), from by {\n        rw hB1,\n        rw hm,\n        refl,\n      },\n    },\n    have hB4 : \u2200 m : \u2115, B m = (euclidean_space \u211d (fin n)) \\ (B (m-1)), from by {\n      assume m : \u2115,\n      induction m with m hm,\n      show B 0 = (euclidean_space \u211d (fin n)) \\ (\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_with_comments-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3903911573674883}}
{"text": "import Mathlib.Data.Nat.Basic\nimport Mathlib.Data.Fin.Basic\nimport Mathlib.Algebra.Group.Defs\nimport Mathlib.Algebra.GroupWithZero.Defs\nimport Mathlib.Algebra.Ring.Basic\nimport Mathlib.Data.ZMod.Defs\n\nlemma UInt8.val_eq_of_lt {a : Nat} : a < UInt8.size -> (ofNat a).val = a := Nat.mod_eq_of_lt\n\nlemma UInt16.val_eq_of_lt {a : Nat} : a < UInt16.size -> (ofNat a).val = a := Nat.mod_eq_of_lt\n\nlemma UInt32.val_eq_of_lt {a : Nat} : a < UInt32.size -> (ofNat a).val = a := Nat.mod_eq_of_lt\n\nlemma UInt64.val_eq_of_lt {a : Nat} : a < UInt64.size -> (ofNat a).val = a := Nat.mod_eq_of_lt\n\nlemma USize.val_eq_of_lt {a : Nat} : a < USize.size -> (ofNat a).val = a := Nat.mod_eq_of_lt\n\ninstance UInt8.neZero : NeZero UInt8.size := \u27e8by decide\u27e9\n\ninstance UInt16.neZero : NeZero UInt16.size := \u27e8by decide\u27e9\n\ninstance UInt32.neZero : NeZero UInt32.size := \u27e8by decide\u27e9\n\ninstance UInt64.neZero : NeZero UInt64.size := \u27e8by decide\u27e9\n\ninstance USize.neZero : NeZero  USize.size := NeZero.of_pos usize_size_gt_zero\n\nexample : (0 : UInt8) = \u27e80\u27e9 := rfl\n\nset_option hygiene false in\nrun_cmd\n  for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do\n  Lean.Elab.Command.elabCommand (\u2190 `(\n    namespace $typeName\n      instance : Inhabited $typeName where\n        default := 0\n\n      instance : Neg $typeName where\n        neg a := mk (-a.val)\n\n      instance : Pow $typeName \u2115 where\n        pow a n := mk (a.val ^ n)\n\n      instance : SMul \u2115 $typeName where\n        smul n a := mk (n \u2022 a.val)\n\n      instance : SMul \u2124 $typeName where\n        smul z a := mk (z \u2022 a.val)\n\n      instance : NatCast $typeName where\n        natCast n := mk n\n\n      instance : IntCast $typeName where\n        intCast z := mk z\n\n      lemma zero_def : (0 : $typeName) = \u27e80\u27e9 := rfl\n\n      lemma one_def : (1 : $typeName) = \u27e81\u27e9 := rfl\n\n      lemma neg_def (a : $typeName) : -a = \u27e8-a.val\u27e9 := rfl\n\n      lemma sub_def (a b : $typeName) : a - b = \u27e8a.val - b.val\u27e9 := rfl\n\n      lemma mul_def (a b : $typeName) : a * b = \u27e8a.val * b.val\u27e9 := rfl\n\n      lemma mod_def (a b : $typeName) : a % b = \u27e8a.val % b.val\u27e9 := rfl\n\n      lemma add_def (a b : $typeName) : a + b = \u27e8a.val + b.val\u27e9 := rfl\n\n      lemma pow_def (a : $typeName) (n : \u2115) : a ^ n = \u27e8a.val ^ n\u27e9 := rfl\n\n      lemma nsmul_def (n : \u2115) (a : $typeName) : n \u2022 a = \u27e8n \u2022 a.val\u27e9 := rfl\n\n      lemma zsmul_def (z : \u2124) (a : $typeName) : z \u2022 a = \u27e8z \u2022 a.val\u27e9 := rfl\n\n      lemma natCast_def (n : \u2115) : (n : $typeName) = \u27e8n\u27e9 := rfl\n\n      lemma intCast_def (z : \u2124) : (z : $typeName) = \u27e8z\u27e9 := rfl\n\n      lemma eq_of_val_eq : \u2200 {a b : $typeName}, a.val = b.val -> a = b\n      | \u27e8_\u27e9, \u27e8_\u27e9, h => congrArg mk h\n\n      lemma val_injective : Function.Injective val := @eq_of_val_eq\n\n      lemma val_eq_of_eq : \u2200 {a b : $typeName}, a = b -> a.val = b.val\n      | \u27e8_\u27e9, \u27e8_\u27e9, h => congrArg val h\n\n      @[simp] lemma mk_val_eq : \u2200 (a : $typeName), mk a.val = a\n      | \u27e8_, _\u27e9 => rfl\n\n      instance : CommRing $typeName :=\n        Function.Injective.commRing val val_injective\n          rfl rfl (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ _ => rfl)\n          (fun _ _ => rfl) (fun _ _ => rfl) (fun _ _ => rfl) (fun _ => rfl) (fun _ => rfl)\n\n    end $typeName\n  ))\n\nnamespace UInt8\n\n/-- Is this an uppercase ASCII letter? -/\ndef isUpper (c : UInt8) : Bool :=\n  c \u2265 65 && c \u2264 90\n\n/-- Is this a lowercase ASCII letter? -/\ndef isLower (c : UInt8) : Bool :=\n  c \u2265 97 && c \u2264 122\n\n/-- Is this an alphabetic ASCII character? -/\ndef isAlpha (c : UInt8) : Bool :=\n  c.isUpper || c.isLower\n\n/-- Is this an ASCII digit character? -/\ndef isDigit (c : UInt8) : Bool :=\n  c \u2265 48 && c \u2264 57\n\n/-- Is this an alphanumeric ASCII character? -/\ndef isAlphanum (c : UInt8) : Bool :=\n  c.isAlpha || c.isDigit\n\ntheorem toChar_aux (n : Nat) (h : n < size) : Nat.isValidChar (UInt32.ofNat n).1 := by\n  rw [UInt32.val_eq_of_lt]\n  exact Or.inl $ Nat.lt_trans h $ by decide\n  exact Nat.lt_trans h $ by decide\n\n/-- The numbers from 0 to 256 are all valid UTF-8 characters, so we can embed one in the other. -/\ndef toChar (n : UInt8) : Char := \u27e8n.toUInt32, toChar_aux n.1 n.1.2\u27e9\n\nend UInt8\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/UInt.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5506073655352403, "lm_q1q2_score": 0.3903911573674882}}
{"text": "/-\nCopyright (c) 2022 Kyle Miller. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kyle Miller\n-/\nimport data.fintype.powerset\nimport data.fintype.prod\nimport data.fintype.sigma\nimport data.fintype.sum\nimport data.fintype.vector\n\n/-!\n# Finite types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we prove some theorems about `finite` and provide some instances. This typeclass is a\n`Prop`-valued counterpart of the typeclass `fintype`. See more details in the file where `finite` is\ndefined.\n\n## Main definitions\n\n* `fintype.finite`, `finite.of_fintype` creates a `finite` instance from a `fintype` instance. The\n  former lemma takes `fintype \u03b1` as an explicit argument while the latter takes it as an instance\n  argument.\n* `fintype.of_finite` noncomputably creates a `fintype` instance from a `finite` instance.\n\n## Implementation notes\n\nThere is an apparent duplication of many `fintype` instances in this module,\nhowever they follow a pattern: if a `fintype` instance depends on `decidable`\ninstances or other `fintype` instances, then we need to \"lower\" the instance\nto be a `finite` instance by removing the `decidable` instances and switching\nthe `fintype` instances to `finite` instances. These are precisely the ones\nthat cannot be inferred using `finite.of_fintype`. (However, when using\n`open_locale classical` or the `classical` tactic the instances relying only\non `decidable` instances will give `finite` instances.) In the future we might\nconsider writing automation to create these \"lowered\" instances.\n\n## Tags\n\nfiniteness, finite types\n-/\n\nnoncomputable theory\nopen_locale classical\n\nvariables {\u03b1 \u03b2 \u03b3 : Type*}\n\nnamespace finite\n\n@[priority 100] -- see Note [lower instance priority]\ninstance of_subsingleton {\u03b1 : Sort*} [subsingleton \u03b1] : finite \u03b1 :=\nof_injective (function.const \u03b1 ()) $ function.injective_of_subsingleton _\n\n@[nolint instance_priority] -- Higher priority for `Prop`s\ninstance prop (p : Prop) : finite p := finite.of_subsingleton\n\ninstance [finite \u03b1] [finite \u03b2] : finite (\u03b1 \u00d7 \u03b2) :=\nby { haveI := fintype.of_finite \u03b1, haveI := fintype.of_finite \u03b2, apply_instance }\n\ninstance {\u03b1 \u03b2 : Sort*} [finite \u03b1] [finite \u03b2] : finite (pprod \u03b1 \u03b2) :=\nof_equiv _ equiv.pprod_equiv_prod_plift.symm\n\n\n\nlemma prod_right (\u03b1) [finite (\u03b1 \u00d7 \u03b2)] [nonempty \u03b1] : finite \u03b2 :=\nof_surjective (prod.snd : \u03b1 \u00d7 \u03b2 \u2192 \u03b2) prod.snd_surjective\n\ninstance [finite \u03b1] [finite \u03b2] : finite (\u03b1 \u2295 \u03b2) :=\nby { haveI := fintype.of_finite \u03b1, haveI := fintype.of_finite \u03b2, apply_instance }\n\nlemma sum_left (\u03b2) [finite (\u03b1 \u2295 \u03b2)] : finite \u03b1 :=\nof_injective (sum.inl : \u03b1 \u2192 \u03b1 \u2295 \u03b2) sum.inl_injective\n\nlemma sum_right (\u03b1) [finite (\u03b1 \u2295 \u03b2)] : finite \u03b2 :=\nof_injective (sum.inr : \u03b2 \u2192 \u03b1 \u2295 \u03b2) sum.inr_injective\n\ninstance {\u03b2 : \u03b1 \u2192 Type*} [finite \u03b1] [\u03a0 a, finite (\u03b2 a)] : finite (\u03a3 a, \u03b2 a) :=\nby { letI := fintype.of_finite \u03b1, letI := \u03bb a, fintype.of_finite (\u03b2 a), apply_instance }\n\ninstance {\u03b9 : Sort*} {\u03c0 : \u03b9 \u2192 Sort*} [finite \u03b9] [\u03a0 i, finite (\u03c0 i)] : finite (\u03a3' i, \u03c0 i) :=\nof_equiv _ (equiv.psigma_equiv_sigma_plift \u03c0).symm\n\ninstance [finite \u03b1] : finite (set \u03b1) := by { casesI nonempty_fintype \u03b1, apply_instance }\n\nend finite\n\n/-- This instance also provides `[finite s]` for `s : set \u03b1`. -/\ninstance subtype.finite {\u03b1 : Sort*} [finite \u03b1] {p : \u03b1 \u2192 Prop} : finite {x // p x} :=\nfinite.of_injective coe subtype.coe_injective\n\ninstance pi.finite {\u03b1 : Sort*} {\u03b2 : \u03b1 \u2192 Sort*} [finite \u03b1] [\u2200 a, finite (\u03b2 a)] : finite (\u03a0 a, \u03b2 a) :=\nbegin\n  haveI := fintype.of_finite (plift \u03b1),\n  haveI := \u03bb a, fintype.of_finite (plift (\u03b2 a)),\n  exact finite.of_equiv (\u03a0 (a : plift \u03b1), plift (\u03b2 (equiv.plift a)))\n    (equiv.Pi_congr equiv.plift (\u03bb _, equiv.plift)),\nend\n\ninstance vector.finite {\u03b1 : Type*} [finite \u03b1] {n : \u2115} : finite (vector \u03b1 n) :=\nby { haveI := fintype.of_finite \u03b1, apply_instance }\n\ninstance quot.finite {\u03b1 : Sort*} [finite \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : finite (quot r) :=\nfinite.of_surjective _ (surjective_quot_mk r)\n\ninstance quotient.finite {\u03b1 : Sort*} [finite \u03b1] (s : setoid \u03b1) : finite (quotient s) :=\nquot.finite _\n\ninstance function.embedding.finite {\u03b1 \u03b2 : Sort*} [finite \u03b2] : finite (\u03b1 \u21aa \u03b2) :=\nbegin\n  casesI is_empty_or_nonempty (\u03b1 \u21aa \u03b2) with _ h,\n  { apply_instance, },\n  { refine h.elim (\u03bb f, _),\n    haveI : finite \u03b1 := finite.of_injective _ f.injective,\n    exact finite.of_injective _ fun_like.coe_injective },\nend\n\ninstance equiv.finite_right {\u03b1 \u03b2 : Sort*} [finite \u03b2] : finite (\u03b1 \u2243 \u03b2) :=\nfinite.of_injective equiv.to_embedding $ \u03bb e\u2081 e\u2082 h, equiv.ext $\n  by convert fun_like.congr_fun h\n\ninstance equiv.finite_left {\u03b1 \u03b2 : Sort*} [finite \u03b1] : finite (\u03b1 \u2243 \u03b2) :=\nfinite.of_equiv _ \u27e8equiv.symm, equiv.symm, equiv.symm_symm, equiv.symm_symm\u27e9\n\ninstance [finite \u03b1] {n : \u2115} : finite (sym \u03b1 n) :=\nby { haveI := fintype.of_finite \u03b1, apply_instance }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/finite/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.550607350786733, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.3903911502959495}}
{"text": "-- import the definition of the example maze\nimport mazes.example_maze.definition\n\n/-\n# Maze 1 : Example maze.\n\nYou are in a maze of twisty passages, all distinct! \n\nYou can go north, south east or west.\n-/\n\nnamespace maze -- hide\n\n/- \nThere are 5 rooms. Rooms are called 0, 1, 2, 3 and 4, with 0 being\nwhere you start and 4 being the exit.\n\n\nUse `n`, `s`, `e`, `w` to move around. When you're at the exit, type `out`.\n\n  Don't forget the commas.\n  Don't bang into the walls -- those are errors.\n  When you get to room 4, the tactic to get you out is `out`. \n  There is also a magic word, rumoured to be\n  an ancient translation of the word `sorry`.\n-/\n\n/- Lemma : no-side-bar\nSee if you can get out of this maze.\n-/\nlemma can_escape_example_maze : (mk 0).can_escape :=\nbegin\n  xyzzy,\n\n\n\n\n\n\nend\n\nend maze\n", "meta": {"author": "kbuzzard", "repo": "lean-game-skeleton", "sha": "098454dd6acc4c06beccf52b6547bf4cd99cc581", "save_path": "github-repos/lean/kbuzzard-lean-game-skeleton", "path": "github-repos/lean/kbuzzard-lean-game-skeleton/lean-game-skeleton-098454dd6acc4c06beccf52b6547bf4cd99cc581/src/mazes/example_maze/level.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6757646140788307, "lm_q1q2_score": 0.3902509325871035}}
{"text": "import model_theory.substructures\nimport model_theory.bundled\n\n\n--OUTPUT 1\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let A : \u2115 \u2192 L.formula,\n  -- $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  have h1 : \u2200 n : \u2115, A n = L.exists (L.var 1) (L.exists (L.var 2) (L.exists (L.var 3) (L.exists (L.var 4) (L.exists (L.var 5) (L.exists (L.var 6) (L.exists (L.var 7) (L.exists (L.var 8) (L.exists (L.var 9) (L.exists (L.var 10) (L.exists (L.var 11) (L.exists (L.var 12) (L.exists (L.var 13) (L.exists (L.var 14) (L.exists (L.var 15) (L.exists (L.var 16) (L.exists (L.var 17) (L.exists (L.var 18) (L.exists (L.var 19) (L.exists (L.var 20) (L.exists (L.var 21) (L.exists (L.var 22) (L.exists (L.var 23) (L.exists (L.var 24) (L.exists (L.var 25) (L.exists (L.var 26) (L.exists (L.var 27) (L.exists (L.var 28) (L.exists (L.var 29) (L.exists (L.var 30) (L.exists (L.var 31) (L.exists (L.var 32) (L.exists (L.var 33) (L.exists (L.var 34) (L.exists (L.var 35) (L.exists (L.var 36) (L.exists (L.var 37) (L.exists (L.var 38) (L.exists (L.var 39) (L.exists (L.var 40) (L.exists (L.var 41) (L.exists (L.var 42) (L.exists (L.var 43) (L.exists (L.var 44) (L.exists (L.var 45) (L.exists (L.var 46) (L.exists (L.var 47) (L.exists (L.var 48) (L.exists (L.var 49) (L.exists (L.var 50) (L.exists (L.var 51) (L.exists (L.var 52) (L.exists (L.var 53) (L.exists (L.var 54) (L.exists (L.var 55) (L.exists (L.var 56) (L.exists (L.var 57) (L.exists (L.var 58) (L.exists (L.var 59) (L.exists (L.var 60) (L.exists (L.var 61) (L.exists (L.var 62) (L.exists (L.var 63) (L.exists (L.var 64) (L.exists (L.var 65) (L.exists (L.var 66) (L.exists (L.var 67) (L.exists (L.var 68) (L.exists (L.var 69) (L.exists (L.var 70) (L.exists (L.var 71) (L.exists (L.var 72) (L.exists (L.var 73) (L.exists (L.var 74) (L.exists (L.var 75) (L.exists (L.var 76) (L.exists (L.var 77) (L.exists (L.var 78) (L.exists (L.var 79) (L.exists (L.var 80) (L.exists (L.var 81) (L.exists (L.var 82) (L.exists (L.var 83) (L.exists (L.var 84) (L.exists (L.var 85) (L.exists (L.var 86) (L.exists (L.var 87) (L.exists (L.var 88) (L.exists (L.var 89) (L.exists (L.var 90) (L.exists (L.var 91) (L.exists (L.var 92) (L.exists (L.var 93) (L.exists (L.var 94) (L.exists (L.var 95) (L.exists (L.var 96) (L.exists (L.var 97) (L.exists (L.var 98) (L.exists (L.var 99) (L.exists (L.var 100) (L.exists (L.var 101) (L.exists (L.var 102) (L.exists (L.var 103) (L.exists (L.var 104) (L.exists (L.var 105) (L.exists (L.var 106) (L.exists (L.var 107) (L.exists (L.var 108) (L.exists (L.var 109) (L.exists (L.var 110) (L.exists (L.var 111) (L.exists (L.var 112) (L.exists (L.var 113) (L.exists (L.var 114) (L.exists (L.var 115) (L.exists (L.var 116) (L.exists (L.var 117) (L.exists (L.var 118) (L.exists (L.var 119) (L.exists (L.var 120) (L.exists (L.var 121) (L.exists (L.var 122) (L.exists (L.var 123) (L.exists (L.var 124) (L.exists (L.var 125) (L.exists (L.var 126) (L.exists (L.var 127) (L.exists (L.var 128) (L.exists (L.var 129) (L.exists (L.var 130) (L.exists (L.var 131) (L.exists (L.var 132) (L.exists (L.var 133) (L.exists (L.var 134) (L.exists (L.var 135) (L.exists (L.var 136) (L.exists (L.var 137) (L.exists (L.var 138) (L.exists (L.var 139) (L.exists (L.var 140) (L.exists (L.var 141) (L.exists (L.var 142) (L.exists (L.var 143) (L.exists (L.var 144) (L.exists (L.var 145) (L.exists (L.var 146) (L.exists (L.var 147) (L.exists (L.var 148) (L.exists (L.var 149) (L.exists (L.var 150) (L.exists (L.var 151) (L.exists (L.var 152) (L.exists (L.var 153) (L.exists (L.var 154) (L.exists (L.var 155) (L.exists (L.var 156) (L.exists (L.var 157) (L.exists (L.var 158) (L.exists (L.var 159) (L.exists (L.var 160) (L.exists (L.var 161) (L.exists (L.var 162) (L.exists (L.var 163) (L.exists (L.var 164) (L.exists (L.var 165) (L.exists (L.var 166) (L.exists (L.var 167) (L.exists (L.var 168) (L.exists (L.var 169) (L.exists (L.var 170) (L.exists (L.var 171) (\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nOverflow theorem\nLet $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.\n`proof`\nFor each $n$, let $\\mathbf A_n$ be the formula:\n\n$\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n\nThen $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n\nTake:\n$$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$\n\nSince $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n\nFrom the Compactness Theorem, $\\Gamma$ is satisfiable in some model $\\mathbf{M}$.\n\nBut since $\\mathbf{M} \\models A_i$ for each $i$, $\\mathbf{M}$ must be infinite.\n\nSo $F$ has an infinite model.\n\nQED\n-/\ntheorem  overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8056321889812552, "lm_q2_score": 0.4843800842769843, "lm_q1q2_score": 0.39023218759499173}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.sites.sheaf\n\n/-!\n# Cover-preserving functors between sites.\n\nWe define cover-preserving functors between sites as functors that push covering sieves to\ncovering sieves. A cover-preserving and compatible-preserving functor `G : C \u2964 D` then pulls\nsheaves on `D` back to sheaves on `C` via `G.op \u22d9 -`.\n\n## Main definitions\n\n* `category_theory.cover_preserving`: a functor between sites is cover-preserving if it\npushes covering sieves to covering sieves\n* `category_theory.compatible_preserving`: a functor between sites is compatible-preserving\nif it pushes compatible families of elements to compatible families.\n* `category_theory.pullback_sheaf` : the pullback of a sheaf along a cover-preserving and\ncompatible-preserving functor.\n* `category_theory.sites.pullback` : the induced functor `Sheaf K A \u2964 Sheaf J A` for a\ncover-preserving and compatible-preserving functor `G : (C, J) \u2964 (D, K)`.\n\n## Main results\n\n- `category_theory.sites.whiskering_left_is_sheaf_of_cover_preserving`: If `G : C \u2964 D` is\ncover-preserving and compatible-preserving, then `G \u22d9 -` (`u\u1d56`) as a functor\n`(D\u1d52\u1d56 \u2964 A) \u2964 (C\u1d52\u1d56 \u2964 A)` of presheaves maps sheaves to sheaves.\n\n## References\n\n* [Elephant]: *Sketches of an Elephant*, P. T. Johnstone: C2.3.\n* https://stacks.math.columbia.edu/tag/00WW\n\n-/\n\nuniverses w v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\nnoncomputable theory\n\nopen category_theory\nopen opposite\nopen category_theory.presieve.family_of_elements\nopen category_theory.presieve\nopen category_theory.limits\n\nnamespace category_theory\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\nvariables {A : Type u\u2083} [category.{v\u2083} A]\nvariables (J : grothendieck_topology C) (K : grothendieck_topology D)\nvariables {L : grothendieck_topology A}\n\n/--\nA functor `G : (C, J) \u2964 (D, K)` between sites is *cover-preserving*\nif for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering sieve in `D`.\n-/\n@[nolint has_inhabited_instance]\nstructure cover_preserving (G : C \u2964 D) : Prop :=\n(cover_preserve : \u2200 {U : C} {S : sieve U} (hS : S \u2208 J U), S.functor_pushforward G \u2208 K (G.obj U))\n\n/-- The identity functor on a site is cover-preserving. -/\nlemma id_cover_preserving : cover_preserving J J (\ud835\udfed _) := \u27e8\u03bb U S hS, by simpa using hS\u27e9\n\nvariables (J) (K)\n\n/-- The composition of two cover-preserving functors is cover-preserving. -/\nlemma cover_preserving.comp {F} (hF : cover_preserving J K F) {G} (hG : cover_preserving K L G) :\n  cover_preserving J L (F \u22d9 G) := \u27e8\u03bb U S hS,\nbegin\n  rw sieve.functor_pushforward_comp,\n  exact hG.cover_preserve (hF.cover_preserve hS)\nend\u27e9\n\n/--\nA functor `G : (C, J) \u2964 (D, K)` between sites is called compatible preserving if for each\ncompatible family of elements at `C` and valued in `G.op \u22d9 \u2131`, and each commuting diagram\n`f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082`, `x g\u2081` and `x g\u2082` coincide when restricted via `f\u1d62`.\nThis is actually stronger than merely preserving compatible families because of the definition of\n`functor_pushforward` used.\n-/\n@[nolint has_inhabited_instance]\nstructure compatible_preserving (K : grothendieck_topology D) (G : C \u2964 D) : Prop :=\n(compatible :\n  \u2200 (\u2131 : SheafOfTypes.{w} K) {Z} {T : presieve Z}\n    {x : family_of_elements (G.op \u22d9 \u2131.val) T} (h : x.compatible)\n    {Y\u2081 Y\u2082} {X} (f\u2081 : X \u27f6 G.obj Y\u2081) (f\u2082 : X \u27f6 G.obj Y\u2082) {g\u2081 : Y\u2081 \u27f6 Z} {g\u2082 : Y\u2082 \u27f6 Z}\n    (hg\u2081 : T g\u2081) (hg\u2082 : T g\u2082) (eq : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082),\n      \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082))\n\nvariables {J K} {G : C \u2964 D} (hG : compatible_preserving.{w} K G) (\u2131 : SheafOfTypes.{w} K) {Z : C}\nvariables {T : presieve Z} {x : family_of_elements (G.op \u22d9 \u2131.val) T} (h : x.compatible)\n\ninclude h hG\n\n/-- `compatible_preserving` functors indeed preserve compatible families. -/\nlemma presieve.family_of_elements.compatible.functor_pushforward :\n  (x.functor_pushforward G).compatible :=\nbegin\n  rintros Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081' f\u2082' H\u2081 H\u2082 eq,\n  unfold family_of_elements.functor_pushforward,\n  rcases get_functor_pushforward_structure H\u2081 with \u27e8X\u2081, f\u2081, h\u2081, hf\u2081, rfl\u27e9,\n  rcases get_functor_pushforward_structure H\u2082 with \u27e8X\u2082, f\u2082, h\u2082, hf\u2082, rfl\u27e9,\n  suffices : \u2131.val.map (g\u2081 \u226b h\u2081).op (x f\u2081 hf\u2081) = \u2131.val.map (g\u2082 \u226b h\u2082).op (x f\u2082 hf\u2082),\n    simpa using this,\n  apply hG.compatible \u2131 h _ _ hf\u2081 hf\u2082,\n  simpa using eq\nend\n\n@[simp] lemma compatible_preserving.apply_map {Y : C} {f : Y \u27f6 Z} (hf : T f) :\n  x.functor_pushforward G (G.map f) (image_mem_functor_pushforward G T hf) = x f hf :=\nbegin\n  unfold family_of_elements.functor_pushforward,\n  rcases e\u2081 : get_functor_pushforward_structure (image_mem_functor_pushforward G T hf) with\n    \u27e8X, g, f', hg, eq\u27e9,\n  simpa using hG.compatible \u2131 h f' (\ud835\udfd9 _) hg hf (by simp[eq])\nend\n\nomit h hG\n\n/--\nIf `G` is cover-preserving and compatible-preserving,\nthen `G.op \u22d9 _` pulls sheaves back to sheaves.\n\nThis result is basically https://stacks.math.columbia.edu/tag/00WW.\n-/\ntheorem pullback_is_sheaf_of_cover_preserving {G : C \u2964 D} (hG\u2081 : compatible_preserving.{v\u2083} K G)\n  (hG\u2082 : cover_preserving J K G) (\u2131 : Sheaf K A) :\n  presheaf.is_sheaf J (G.op \u22d9 \u2131.val) :=\nbegin\n  intros X U S hS x hx,\n  change family_of_elements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) _ at x,\n  let H := \u2131.2 X _ (hG\u2082.cover_preserve hS),\n  let hx' := hx.functor_pushforward hG\u2081 (sheaf_over \u2131 X),\n  split, swap,\n  { apply H.amalgamate (x.functor_pushforward G),\n    exact hx' },\n  split,\n  { intros V f hf,\n    convert H.is_amalgamation hx' (G.map f) (image_mem_functor_pushforward G S hf),\n    rw hG\u2081.apply_map (sheaf_over \u2131 X) hx },\n  { intros y hy,\n    refine H.is_separated_for _ y _ _\n      (H.is_amalgamation (hx.functor_pushforward hG\u2081 (sheaf_over \u2131 X))),\n    rintros V f \u27e8Z, f', g', h, rfl\u27e9,\n    erw family_of_elements.comp_of_compatible (S.functor_pushforward G)\n      hx' (image_mem_functor_pushforward G S h) g',\n    dsimp at \u22a2 hy,\n    simp [hG\u2081.apply_map (sheaf_over \u2131 X) hx h, \u2190hy f' h] }\nend\n\n/-- The pullback of a sheaf along a cover-preserving and compatible-preserving functor. -/\ndef pullback_sheaf {G : C \u2964 D} (hG\u2081 : compatible_preserving K G)\n  (hG\u2082 : cover_preserving J K G) (\u2131 : Sheaf K A) : Sheaf J A :=\n\u27e8G.op \u22d9 \u2131.val, pullback_is_sheaf_of_cover_preserving hG\u2081 hG\u2082 \u2131\u27e9\n\nvariable (A)\n\n/--\nThe induced functor from `Sheaf K A \u2964 Sheaf J A` given by `G.op \u22d9 _`\nif `G` is cover-preserving and compatible-preserving.\n-/\n@[simps] def sites.pullback {G : C \u2964 D} (hG\u2081 : compatible_preserving K G)\n  (hG\u2082 : cover_preserving J K G) : Sheaf K A \u2964 Sheaf J A :=\n{ obj := \u03bb \u2131, pullback_sheaf hG\u2081 hG\u2082 \u2131,\n  map := \u03bb _ _ f, (((whiskering_left _ _ _).obj G.op)).map f,\n  map_id' := \u03bb \u2131, (((whiskering_left _ _ _).obj G.op)).map_id \u2131.val,\n  map_comp' := \u03bb _ _ _ f g, (((whiskering_left _ _ _).obj G.op)).map_comp f g }\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sites/cover_preserving.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.3902220614081494}}
{"text": "import local.dual_pair\nimport local.ample_set\nimport global.one_jet_sec\nimport global.smooth_embedding\nset_option trace.filter_inst_type true\n\n/-!\n# First order partial differential relations for maps between manifolds\n\nThis file contains fundamental definitions about first order partial differential relations\nfor maps between manifolds and relating them to the local story of first order partial differential\nrelations for maps between vector spaces.\n\nGiven manifolds `M` and `M'` modelled on `I` and `I'`, a first order partial differential relation\nfor maps from `M` to `M'` is a set in the 1-jet bundle J\u00b9(M, M'), also known as\n`one_jet_bundle I M I' M'`.\n-/\n\nnoncomputable theory\n\nopen set function filter (hiding map_smul) charted_space smooth_manifold_with_corners\nopen_locale topology manifold bundle\n\nsection defs\n/-! ## Fundamental definitions -/\n\nvariables\n{E : Type*} [normed_add_comm_group E] [normed_space \u211d E]\n{H : Type*} [topological_space H] (I : model_with_corners \u211d E H)\n(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{E' : Type*} [normed_add_comm_group E'] [normed_space \u211d E']\n{H' : Type*} [topological_space H'] (I' : model_with_corners \u211d E' H')\n(M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n{F : Type*} [normed_add_comm_group F] [normed_space \u211d F]\n{G : Type*} [topological_space G] (J : model_with_corners \u211d F G)\n(N : Type*) [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]\n{F' : Type*} [normed_add_comm_group F'] [normed_space \u211d F']\n{G' : Type*} [topological_space G'] (J' : model_with_corners \u211d F' G')\n(N' : Type*) [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N']\n{EP : Type*} [normed_add_comm_group EP] [normed_space \u211d EP]\n{HP : Type*} [topological_space HP] (IP : model_with_corners \u211d EP HP)\n(P : Type*) [topological_space P] [charted_space HP P] [smooth_manifold_with_corners IP P]\n{EX : Type*} [normed_add_comm_group EX] [normed_space \u211d EX]\n{HX : Type*} [topological_space HX] {IX : model_with_corners \u211d EX HX}\n-- note: X is a metric space\n{X : Type*} [metric_space X] [charted_space HX X] [smooth_manifold_with_corners IX X]\n\nlocal notation `TM` := tangent_space I\nlocal notation `TM'` := tangent_space I'\n\n/-- A first-order differential relation for maps from `M` to `N` is a subset of the 1-jet bundle. -/\n@[reducible] def rel_mfld := set (one_jet_bundle I M I' M')\n\nvariables {I M I' M'} {R : rel_mfld I M I' M'}\n\n/-- A formal solution to a local relation `R` over a set `U`. -/\n@[ext] structure formal_sol (R : rel_mfld I M I' M') extends\n  to_one_jet_sec : one_jet_sec I M I' M' :=\n(is_sol' : \u2200 x : M, to_one_jet_sec x \u2208 R)\n\ninstance (R : rel_mfld I M I' M') :\n  has_coe_to_fun (formal_sol R) (\u03bb S, M \u2192 one_jet_bundle I M I' M') :=\n\u27e8\u03bb F, F.to_one_jet_sec\u27e9\n\ndef mk_formal_sol (F : M \u2192 one_jet_bundle I M I' M') (hsec : \u2200 x, (F x).1.1 = x)\n(hsol : \u2200 x, F x \u2208 R)\n(hsmooth : smooth I ((I.prod I').prod \ud835\udcd8(\u211d, E \u2192L[\u211d] E')) F) : formal_sol R :=\n{ bs := \u03bb m, (F m).1.2,\n  \u03d5 := \u03bb m, (F m).2,\n  smooth' := begin\n    convert hsmooth,\n    ext x,\n    rw hsec,\n    all_goals { refl }\n  end,\n  is_sol' := \u03bb m, begin\n    convert hsol m,\n    refine one_jet_bundle.ext _ _ _,\n    rw hsec,\n    all_goals { refl }\n    end}\n\n@[simp]\nlemma mk_formal_sol_apply (F : M \u2192 one_jet_bundle I M I' M') (hsec : \u2200 x, (F x).1.1 = x)\n(hsol : \u2200 x, F x \u2208 R)\n(hsmooth : smooth I ((I.prod I').prod \ud835\udcd8(\u211d, E \u2192L[\u211d] E')) \u21bfF)  :\n  (mk_formal_sol F hsec hsol hsmooth : M \u2192 one_jet_bundle I M I' M') = F :=\nbegin\n  ext x ; try { refl },\n  rw hsec,\n  refl\nend\n\n@[simp]\nlemma mk_formal_sol_bs_apply (F : M \u2192 one_jet_bundle I M I' M') (hsec : \u2200 x, (F x).1.1 = x)\n(hsol : \u2200 x, F x \u2208 R)\n(hsmooth : smooth I ((I.prod I').prod \ud835\udcd8(\u211d, E \u2192L[\u211d] E')) \u21bfF)  (x : M) :\n  (mk_formal_sol F hsec hsol hsmooth).bs x = (F x).1.2 :=\nrfl\n\nnamespace formal_sol\n\n@[simp]\nlemma coe_mk {S : one_jet_sec I M I' M'} {h : \u2200 x, S x \u2208 R} {x : M} : formal_sol.mk S h x = S x :=\nrfl\n\nlemma coe_inj_iff {S T : formal_sol R} : S = T \u2194 \u2200 x, S x = T x :=\nbegin\n  split,\n  { rintro rfl x, refl },\n  { intros h, ext : 3, show (S x).1.2 = (T x).1.2, rw [h],\n    apply heq_of_eq, ext1, show (S x).2 = (T x).2, rw [h] }\nend\n\nlemma coe_inj {S T : formal_sol R} (h : \u2200 x, S x = T x) : S = T :=\ncoe_inj_iff.mpr h\n\n@[simp] lemma to_one_jet_sec_coe (S : formal_sol R) {x : M} : S.to_one_jet_sec x = S x := rfl\n\nlemma is_sol (F : formal_sol R) : \u2200 x, F x \u2208 R :=\nF.is_sol'\n\nlemma coe_apply (F : formal_sol R) (x : M) : F x = \u27e8(x, F.bs x), (F.\u03d5 x)\u27e9 := rfl\nlemma fst_eq (F : formal_sol R) (x : M) : (F x).1 = (x, F.bs x) := rfl\nlemma snd_eq (F : formal_sol R) (x : M) : (F x).2 = F.\u03d5 x := rfl\nlemma is_sec (F : formal_sol R) (x : M) : (F x).1.1 = x := rfl\nlemma bs_eq (F : formal_sol R) (x : M) : F.bs x = (F x).1.2 := rfl\n\nend formal_sol\n\n/-! ## Ampleness -/\n\n/-- The slice `R(\u03c3,p)`. -/\ndef rel_mfld.slice (R : rel_mfld I M I' M') (\u03c3 : one_jet_bundle I M I' M')\n  (p : dual_pair $ TM \u03c3.1.1) : set (TM' \u03c3.1.2) :=\n{w : TM' \u03c3.1.2 | one_jet_bundle.mk \u03c3.1.1 \u03c3.1.2 (p.update \u03c3.2 w) \u2208 R}\n\n/-- For some reason `rw [mem_set_of_eq]` fails after unfolding `slice`,\nbut rewriting with this lemma works. -/\nlemma mem_slice {R : rel_mfld I M I' M'} {\u03c3 : one_jet_bundle I M I' M'}\n  {p : dual_pair $ TM \u03c3.1.1} {w : TM' \u03c3.1.2} :\n  w \u2208 R.slice \u03c3 p \u2194 one_jet_bundle.mk \u03c3.1.1 \u03c3.1.2 (p.update \u03c3.2 w) \u2208 R :=\niff.rfl\n\nlemma slice_mk_update {R : rel_mfld I M I' M'} {\u03c3 : one_jet_bundle I M I' M'}\n  {p : dual_pair $ TM \u03c3.1.1} (x : E') :\n  R.slice (one_jet_bundle.mk \u03c3.1.1 \u03c3.1.2 (p.update \u03c3.2 x)) p = (R.slice \u03c3 p : set E') :=\nbegin\n  ext1 w,\n  dsimp only [mem_slice],\n  congr' 3,\n  simp_rw [one_jet_bundle_mk_snd, p.update_update],\nend\n\n/-- A differential relation is ample if all its slices are ample sets. -/\ndef rel_mfld.ample (R : rel_mfld I M I' M') : Prop :=\n\u2200 \u2983\u03c3 : one_jet_bundle I M I' M'\u2984 (p : dual_pair $ TM \u03c3.1.1), ample_set (R.slice \u03c3 p)\n\nlemma rel_mfld.ample_iff (R : rel_mfld I M I' M') : R.ample \u2194\n  \u2200 \u2983\u03c3 : one_jet_bundle I M I' M'\u2984 (p : dual_pair $ TM \u03c3.1.1), \u03c3 \u2208 R \u2192 ample_set (R.slice \u03c3 p) :=\nbegin\n  simp_rw [rel_mfld.ample],\n  refine \u27e8\u03bb h \u03c3 p _, h p, \u03bb h \u03c3 p x hx, _\u27e9,\n  have := @h (one_jet_bundle.mk \u03c3.1.1 \u03c3.1.2 (p.update \u03c3.2 x)) p hx,\n  rw [slice_mk_update] at this,\n  exact this x hx\nend\n\n/-! ## Families of formal solutions. -/\n\n/-- A family of formal solutions indexed by manifold `N` is a function from `N` into formal\n  solutions in such a way that the function is smooth as a function of all arguments. -/\n@[ext] structure family_formal_sol (R : rel_mfld I M I' M') extends\n  to_family_one_jet_sec : family_one_jet_sec I M I' M' J N :=\n(is_sol' : \u2200 (t : N) (x : M), to_family_one_jet_sec t x \u2208 R)\n\ninstance : has_coe_to_fun (family_formal_sol J N R) (\u03bb S, N \u2192 formal_sol R) :=\n\u27e8\u03bb S t, \u27e8S.to_family_one_jet_sec t, S.is_sol' t\u27e9\u27e9\n\nnamespace family_formal_sol\n\nvariables {J N J' N'}\n\n@[simp]\nlemma coe_mk {S : family_one_jet_sec I M I' M' J N} {h : \u2200 t x, S t x \u2208 R} {t : N} {x : M} :\n  family_formal_sol.mk S h t x = S t x :=\nrfl\n\nlemma coe_mk_to_one_jet_sec {S : family_one_jet_sec I M I' M' J N} {h : \u2200 t x, S t x \u2208 R} {t : N} :\n  (family_formal_sol.mk S h t).to_one_jet_sec = S t :=\nrfl\n\nlemma to_family_one_jet_sec_coe (S : family_formal_sol J N R) {t : N} {x : M} :\n  S.to_family_one_jet_sec t x = S t x :=\nrfl\n\n@[simp]\nlemma to_family_one_jet_sec_eq (S : family_formal_sol J N R) {t : N} :\n  S.to_family_one_jet_sec t = (S t).to_one_jet_sec :=\nrfl\n\nlemma is_sol (S : family_formal_sol J N R) {t : N} {x : M} : S t x \u2208 R :=\nS.is_sol' t x\n\n/-- Reindex a family along a smooth function `f`. -/\ndef reindex (S : family_formal_sol J' N' R) (f : C^\u221e\u27eeJ, N; J', N'\u27ef) :\n  family_formal_sol J N R :=\n\u27e8S.to_family_one_jet_sec.reindex f, \u03bb t, S.is_sol' (f t)\u27e9\n\nend family_formal_sol\n\n/-! ## Homotopies of formal solutions. -/\n\n/-- A homotopy of formal solutions is a family indexed by `\u211d` -/\n@[reducible] def htpy_formal_sol (R : rel_mfld I M I' M') := family_formal_sol \ud835\udcd8(\u211d, \u211d) \u211d R\n\ndef mk_htpy_formal_sol (F : \u211d \u2192 M \u2192 one_jet_bundle I M I' M') (hsec : \u2200 t x, (F t x).1.1 = x)\n(hsol : \u2200 t x, F t x \u2208 R)\n(hsmooth : smooth (\ud835\udcd8(\u211d).prod I) ((I.prod I').prod \ud835\udcd8(\u211d, E \u2192L[\u211d] E')) \u21bfF) : htpy_formal_sol R :=\n{ bs := \u03bb t m, (F t m).1.2,\n  \u03d5 := \u03bb t m, (F t m).2,\n  smooth' := begin\n    convert hsmooth,\n    ext \u27e8t, x\u27e9,\n    exact (hsec t x).symm,\n    all_goals { refl }\n  end,\n  is_sol' := \u03bb t m, begin\n    convert hsol t m,\n    refine  one_jet_bundle.ext _ _ _,\n    rw hsec,\n    all_goals { refl }\n    end}\n\n@[simp]\nlemma mk_htpy_formal_sol_apply (F : \u211d \u2192 M \u2192 one_jet_bundle I M I' M') (hsec : \u2200 t x, (F t x).1.1 = x)\n(hsol : \u2200 t x, F t x \u2208 R)\n(hsmooth : smooth (\ud835\udcd8(\u211d).prod I) ((I.prod I').prod \ud835\udcd8(\u211d, E \u2192L[\u211d] E')) \u21bfF) (t : \u211d) :\n  (mk_htpy_formal_sol F hsec hsol hsmooth t : M \u2192 one_jet_bundle I M I' M') = F t :=\nbegin\n  ext x ; try { refl },\n  rw hsec,\n  refl\nend\n\n/-- The constant homotopy of formal solution associated to a formal solution. -/\ndef formal_sol.const_htpy (F : formal_sol R) : htpy_formal_sol R :=\n{ bs := \u03bb t, F.bs,\n  \u03d5 := \u03bb t, F.\u03d5,\n  smooth' := F.smooth.comp smooth_snd,\n  is_sol' := \u03bb t, F.is_sol }\n\nvariable (R)\n\n/-- The empty homotopy of formal solution associated to any relation whose source manifold\nis empty. This is required to avoid a silly nonemptyness assumption in the main theorems. -/\ndef empty_htpy_formal_sol [is_empty M] : htpy_formal_sol R :=\n{ bs := \u03bb t x, (is_empty.false x).elim,\n  \u03d5 :=  \u03bb t x, (is_empty.false x).elim,\n  smooth' := \u03bb \u27e8t, x\u27e9, (is_empty.false x).elim,\n  is_sol' := \u03bb t x, (is_empty.false x).elim }\n\n/-! ## The h-principle -/\n\nvariables {P}\n\n/-- A relation `R` satisfies the (non-parametric) relative C\u2070-dense h-principle w.r.t. a subset\n`C` of the domain if for every formal solution `\ud835\udcd5\u2080` that is holonomic near `C`\nthere is a homotopy between `\ud835\udcd5\u2080` and a holonomic solution that is constant near `C` and\n`\u03b5`-close to `\ud835\udcd5\u2080`. This is a temporary version with a slightly weaker conclusion.\nThe weak version has `\u2200 x \u2208 C, \u2200 t : \u211d, \ud835\udcd5 t x = \ud835\udcd5\u2080 x` while the strong one has\n`\u2200\u1da0 x near C, \u2200 t, \ud835\udcd5 t x = \ud835\udcd5\u2080 x`. The strong version is easy to derive from the weak one\nif we prove the weak one for *all* closed sets, see `rel_mfld.satisfies_h_principle_of_weak`\nbelow. The reason why the weak one is more convenient for us is we will prove\nthe h-principle using a sequence of homotopy of formal solutions and we don't\nwant to keep control of a fixed neighborhood of `C` independant from the sequence index. -/\ndef rel_mfld.satisfies_h_principle_weak (R : rel_mfld I M IX X) (C : set M) (\u03b5 : M \u2192 \u211d) : Prop :=\n\u2200 \ud835\udcd5\u2080 : formal_sol R, (\u2200\u1da0 x in \ud835\udcdd\u02e2 C, \ud835\udcd5\u2080.to_one_jet_sec.is_holonomic_at x) \u2192\n\u2203 \ud835\udcd5 : htpy_formal_sol R, (\u2200 x : M, \ud835\udcd5 0 x = \ud835\udcd5\u2080 x) \u2227\n  (\ud835\udcd5 1).to_one_jet_sec.is_holonomic \u2227\n  (\u2200 x \u2208 C, \u2200 t : \u211d, \ud835\udcd5 t x = \ud835\udcd5\u2080 x) \u2227\n  (\u2200 (t : \u211d) (x : M), dist ((\ud835\udcd5 t).bs x) (\ud835\udcd5\u2080.bs x) \u2264 \u03b5 x)\n\n/-- A relation `R` satisfies the (non-parametric) relative C\u2070-dense h-principle w.r.t. a subset\n`C` of the domain if for every formal solution `\ud835\udcd5\u2080` that is holonomic near `C`\nthere is a homotopy between `\ud835\udcd5\u2080` and a holonomic solution that is constant near `C` and\n`\u03b5`-close to `\ud835\udcd5\u2080`. -/\ndef rel_mfld.satisfies_h_principle (R : rel_mfld I M IX X) (C : set M) (\u03b5 : M \u2192 \u211d) : Prop :=\n\u2200 \ud835\udcd5\u2080 : formal_sol R, (\u2200\u1da0 x in \ud835\udcdd\u02e2 C, \ud835\udcd5\u2080.to_one_jet_sec.is_holonomic_at x) \u2192\n\u2203 \ud835\udcd5 : htpy_formal_sol R, (\u2200 x : M, \ud835\udcd5 0 x = \ud835\udcd5\u2080 x) \u2227\n  (\ud835\udcd5 1).to_one_jet_sec.is_holonomic \u2227\n  (\u2200\u1da0 x near C, \u2200 t, \ud835\udcd5 t x = \ud835\udcd5\u2080 x) \u2227\n  (\u2200 (t : \u211d) (x : M), dist ((\ud835\udcd5 t).bs x) (\ud835\udcd5\u2080.bs x) \u2264 \u03b5 x)\n\nlemma rel_mfld.satisfies_h_principle_of_weak\n  [finite_dimensional \u211d E] [t2_space M] [sigma_compact_space M]\n  {R : rel_mfld I M IX X} {\u03b5 : M \u2192 \u211d} {C : set M} (hC : is_closed C)\n  (h : \u2200 A : set M, is_closed A \u2192 R.satisfies_h_principle_weak A \u03b5) : R.satisfies_h_principle C \u03b5 :=\nbegin\n  haveI := manifold_with_corners.metrizable_space I M,\n  letI : metric_space M := topological_space.metrizable_space_metric M,\n  intros \ud835\udcd5\u2080 h\ud835\udcd5\u2080,\n  obtain \u27e8C', hCC', hC', h\ud835\udcd5\u2080C'\u27e9 := h\ud835\udcd5\u2080.closed_neighborhood hC,\n  obtain \u27e8\ud835\udcd5, h1, h2, h3, h4\u27e9 := h C' hC' \ud835\udcd5\u2080 h\ud835\udcd5\u2080C',\n  exact \u27e8\ud835\udcd5, h1, h2, eventually_of_mem hCC' h3, h4\u27e9\nend\n\n/-- A relation `R` satisfies the parametric relative C\u2070-dense h-principle w.r.t. manifold `P`,\n`C \u2286 P \u00d7 M` and `\u03b5 : M \u2192 \u211d` if for every family of\nformal solutions `\ud835\udcd5\u2080` indexed by a manifold with boundary `P` that is holonomic near `C`,\nthere is a homotopy `\ud835\udcd5` between `\ud835\udcd5\u2080` and a holonomic solution,\nin such a way that `\ud835\udcd5` is constant near `C` and `\u03b5`-close to `\ud835\udcd5\u2080`.\n-/\ndef rel_mfld.satisfies_h_principle_with (R : rel_mfld I M IX X) (C : set (P \u00d7 M)) (\u03b5 : M \u2192 \u211d) :\n  Prop :=\n\u2200 \ud835\udcd5\u2080 : family_formal_sol IP P R, -- given a family of formal solutions with parameters in `P`\n(\u2200\u1da0 (p : P \u00d7 M) in \ud835\udcdd\u02e2 C, (\ud835\udcd5\u2080 p.1).to_one_jet_sec.is_holonomic_at p.2) \u2192 -- holonomic near `C`\n\u2203 \ud835\udcd5 : family_formal_sol (\ud835\udcd8(\u211d, \u211d).prod IP) (\u211d \u00d7 P) R, -- then there is a homotopy of such families\n  (\u2200 (s : P) (x : M), \ud835\udcd5 (0, s) x = \ud835\udcd5\u2080 s x) \u2227 -- that agrees on `t = 0`\n  (\u2200 (s : P), (\ud835\udcd5 (1, s)).to_one_jet_sec.is_holonomic) \u2227 -- is holonomic everywhere for `t = 1`\n  (\u2200\u1da0 (p : P \u00d7 M) in \ud835\udcdd\u02e2 C, \u2200 t : \u211d, \ud835\udcd5 (t, p.1) p.2 = \ud835\udcd5\u2080 p.1 p.2) \u2227 -- and agrees near `C`\n  (\u2200 (t : \u211d) (s : P) (x : M), dist ((\ud835\udcd5 (t, s)).bs x) ((\ud835\udcd5\u2080 s).bs x) \u2264 \u03b5 x) -- and close to `\ud835\udcd5\u2080`.\n\n\nvariables {IP}\n\n/-- If a relation satisfies the parametric relative C\u2070-dense h-principle wrt some data\nthen we can forget the homotopy and get a family of solutions from every\nfamily of formal solutions. -/\nlemma rel_mfld.satisfies_h_principle_with.bs {R : rel_mfld I M IX X} {C : set (P \u00d7 M)}\n  {\u03b5 : M \u2192 \u211d} (h : R.satisfies_h_principle_with IP C \u03b5) (\ud835\udcd5\u2080 : family_formal_sol IP P R)\n  (h2 : \u2200\u1da0 (p : P \u00d7 M) in \ud835\udcdd\u02e2 C, (\ud835\udcd5\u2080 p.1).to_one_jet_sec.is_holonomic_at p.2) :\n  \u2203 f : P \u2192 M \u2192 X,\n    (smooth (IP.prod I) IX $ uncurry f) \u2227\n    (\u2200\u1da0 (p : P \u00d7 M) in \ud835\udcdd\u02e2 C, f p.1 p.2 = \ud835\udcd5\u2080.bs p.1 p.2) \u2227\n    (\u2200 p m, dist (f p m) ((\ud835\udcd5\u2080 p).bs m) \u2264 \u03b5 m) \u2227\n    (\u2200 p m, one_jet_ext I IX (f p) m \u2208 R) :=\nbegin\n  rcases h \ud835\udcd5\u2080 h2  with \u27e8\ud835\udcd5, h\u2081, h\u2082, h\u2083, h\u2084\u27e9,\n  refine \u27e8\u03bb s, (\ud835\udcd5 (1, s)).bs, _, _, _, _\u27e9,\n  { have := \ud835\udcd5.to_family_one_jet_sec.smooth,\n    let j : C^\u221e\u27eeIP, P ; \ud835\udcd8(\u211d, \u211d).prod IP, \u211d \u00d7 P\u27ef := \u27e8\u03bb p, (1, p),\n                                                    smooth.prod_mk smooth_const smooth_id\u27e9,\n    rw show uncurry (\u03bb s, (\ud835\udcd5 (1, s)).bs) = prod.snd \u2218 \u03c0 (one_jet_space I IX) \u2218\n                                            (\u03bb (p : P \u00d7 M), \ud835\udcd5.reindex j p.1 p.2),\n    by { ext, refl },\n    exact (\ud835\udcd5.reindex j).to_family_one_jet_sec.smooth_bs },\n  { apply h\u2083.mono,\n    intros x hx,\n    simp_rw [one_jet_sec.bs_eq, formal_sol.to_one_jet_sec_coe, hx, family_one_jet_sec.bs_eq,\n      \ud835\udcd5\u2080.to_family_one_jet_sec_coe] },\n  { intros p m,\n    apply h\u2084 },\n  { intros p m,\n    suffices : one_jet_ext I IX (\ud835\udcd5 (1, p)).bs m = ((\ud835\udcd5.to_family_one_jet_sec) (1, p)) m,\n    { rw this,\n      exact \ud835\udcd5.is_sol' (1, p) m },\n    exact one_jet_sec.is_holonomic_at_iff.mp (h\u2082 p m) },\nend\n\nend defs\n\nsection open_smooth_embedding\n/-! ## Localisation of one jet sections\n\nIn order to use the local story of convex integration, we need a way to turn a\none jet section into local ones, then apply the local story to build a homotopy of one jets section\nand transfer back to the original manifolds. There is a dissymetry here: we use\nmaps from whole vector spaces to open sets in manifold.\n\nThe global manifolds are called `M` and `N'`. We don't assume the local ones are vector spaces,\nthere are manifolds `X` and `Y` that will be vector spaces in the next section.\n-/\nvariables\n  {EX : Type*} [normed_add_comm_group EX] [normed_space \u211d EX]\n  {HX : Type*} [topological_space HX] {IX : model_with_corners \u211d EX HX}\n  {X : Type*} [topological_space X] [charted_space HX X] [smooth_manifold_with_corners IX X]\n\n  {EM : Type*} [normed_add_comm_group EM] [normed_space \u211d EM]\n  {HM : Type*} [topological_space HM] {IM : model_with_corners \u211d EM HM}\n  {M : Type*} [topological_space M] [charted_space HM M] [smooth_manifold_with_corners IM M]\n\n  {EY : Type*} [normed_add_comm_group EY] [normed_space \u211d EY]\n  {HY : Type*} [topological_space HY] {IY : model_with_corners \u211d EY HY}\n  {Y : Type*} [topological_space Y] [charted_space HY Y] [smooth_manifold_with_corners IY Y]\n\n  {EN : Type*} [normed_add_comm_group EN] [normed_space \u211d EN]\n  {HN : Type*} [topological_space HN] {IN : model_with_corners \u211d EN HN}\n  {N : Type*} [topological_space N] [charted_space HN N] [smooth_manifold_with_corners IN N]\n\n  (F : one_jet_sec IM M IN N)\n  (\u03c6 : open_smooth_embedding IX X IM M) (\u03c8 : open_smooth_embedding IY Y IN N)\n  {R : rel_mfld IM M IN N}\n\nlocal notation `TM` := tangent_space IM\nlocal notation `TN` := tangent_space IN\nlocal notation `TX` := tangent_space IX\nlocal notation `TY` := tangent_space IY\n\nlocal notation `J\u00b9XY` := one_jet_bundle IX X IY Y\nlocal notation `J\u00b9MN` := one_jet_bundle IM M IN N\nlocal notation `IXY` := (IX.prod IY).prod \ud835\udcd8(\u211d, EX \u2192L[\u211d] EY)\nlocal notation `IMN` := (IM.prod IN).prod \ud835\udcd8(\u211d, EM \u2192L[\u211d] EN)\n\n/-! ## Transfer from J\u00b9(X, Y) to J\u00b9(M, N) and localized relations -/\n\n/-- Transfer map between one jet bundles induced by open smooth embedding into the source and\ntargets. -/\n@[simps fst_fst fst_snd]\ndef open_smooth_embedding.transfer : one_jet_bundle IX X IY Y \u2192 one_jet_bundle IM M IN N :=\none_jet_bundle.map IY IN \u03c6 \u03c8 (\u03bb x, (\u03c6.fderiv x).symm)\n\nlemma open_smooth_embedding.smooth_transfer :\n  smooth ((IX.prod IY).prod \ud835\udcd8(\u211d, EX \u2192L[\u211d] EY))\n  ((IM.prod IN).prod \ud835\udcd8(\u211d, EM \u2192L[\u211d] EN)) (\u03c6.transfer \u03c8) :=\nbegin\n  intro x,\n  refine smooth_at.one_jet_bundle_map (\u03c6.smooth_to.smooth_at.comp _ smooth_at_snd)\n    (\u03c8.smooth_to.smooth_at.comp _ smooth_at_snd) _ smooth_at_id,\n  have := cont_mdiff_at.mfderiv''' (\u03bb x, \u03c6.inv_fun) (\u03bb x : one_jet_bundle IX X IY Y, \u03c6 x.1.1)\n    ((\u03c6.smooth_at_inv $ _).comp (x, \u03c6 x.1.1) smooth_at_snd)\n    (\u03c6.smooth_to.smooth_at.comp x (smooth_one_jet_bundle_proj.fst x)) le_top,\n  { simp_rw [\u03c6.left_inv] at this, exact this },\n  exact mem_range_self _,\nend\n\nlemma one_jet_bundle.continuous_transfer : continuous (\u03c6.transfer \u03c8) :=\n(open_smooth_embedding.smooth_transfer _ _).continuous\n\nlemma open_smooth_embedding.range_transfer : range (\u03c6.transfer \u03c8) =\n  \u03c0 (one_jet_space IM IN) \u207b\u00b9' (range \u03c6 \u00d7\u02e2 range \u03c8) :=\nbegin\n  ext \u03c3, split,\n  { rintro \u27e8\u03c3, rfl\u27e9, exact mk_mem_prod (mem_range_self _) (mem_range_self _) },\n  { rcases \u03c3 with \u27e8\u27e8x, y\u27e9, \u03c4\u27e9,\n    rintro \u27e8\u27e8x, rfl\u27e9 : x \u2208 range \u03c6, \u27e8y, rfl\u27e9 : y \u2208 range \u03c8\u27e9,\n    refine \u27e8\u27e8(x, y), ((\u03c8.fderiv y).symm : tangent_space IN (\u03c8 y) \u2192L[\u211d] tangent_space IY y) \u2218L \u03c4 \u2218L\n      (\u03c6.fderiv x : tangent_space IX x \u2192L[\u211d] tangent_space IM (\u03c6 x))\u27e9, _\u27e9,\n    ext _, { refl }, { refl },\n    ext1 v,\n    dsimp only [open_smooth_embedding.transfer, one_jet_bundle.map, one_jet_bundle.mk],\n    simp_rw [continuous_linear_map.comp_apply, \u2190 \u03c8.fderiv_coe, continuous_linear_equiv.coe_coe,\n      (\u03c6.fderiv x).apply_symm_apply, (\u03c8.fderiv y).apply_symm_apply] }\nend\n\nlemma open_smooth_embedding.is_open_range_transfer : is_open (range (\u03c6.transfer \u03c8)) :=\nbegin\n  rw [\u03c6.range_transfer \u03c8],\n  exact (\u03c6.is_open_range.prod \u03c8.is_open_range).preimage one_jet_bundle_proj_continuous,\nend\n\n/-- localize a relation -/\ndef rel_mfld.localize (R : rel_mfld IM M IN N) : rel_mfld IX X IY Y :=\n\u03c6.transfer \u03c8 \u207b\u00b9' R\n\n/-- Ampleness survives localization -/\nlemma rel_mfld.ample.localize (hR : R.ample) : (R.localize \u03c6 \u03c8).ample :=\nbegin\n  intros x p,\n  have : (rel_mfld.localize \u03c6 \u03c8 R).slice x p =\n    (\u03c8.fderiv x.1.2).symm '' R.slice (\u03c6.transfer \u03c8 x) (p.map (\u03c6.fderiv x.1.1)),\n  { ext v,\n    simp_rw [rel_mfld.localize, continuous_linear_equiv.image_symm_eq_preimage, mem_preimage,\n      mem_slice, mem_preimage],\n    dsimp only [open_smooth_embedding.transfer, one_jet_bundle.map, one_jet_bundle_mk_fst,\n      one_jet_bundle_mk_snd],\n    simp_rw [p.map_update_comp_right, \u2190 p.update_comp_left, one_jet_bundle.mk, \u2190 \u03c8.fderiv_coe,\n      continuous_linear_equiv.coe_coe] },\n  rw [this],\n  exact (hR _).image (\u03c8.fderiv x.1.2).symm\nend\n\n\n/-! ## Localized 1-jet sections -/\n\n/-- Localize a one-jet section in two open embeddings.\n  It maps `x` to `(x, y, (D_y(g))\u207b\u00b9 \u2218 F_\u03c6(\u03c6 x) \u2218 D_x(\u03c6))` where `y : M := g\u207b\u00b9(F_{bs}(\u03c6 x))`. -/\n@[simps] def one_jet_sec.localize (hF : range (F.bs \u2218 \u03c6) \u2286 range \u03c8) :\n  one_jet_sec IX X IY Y :=\n{ bs := \u03bb x, \u03c8.inv_fun (F.bs $ \u03c6 x),\n  \u03d5 := \u03bb x, let y := \u03c8.inv_fun (F.bs $ \u03c6 x) in\n  (\u2191(\u03c8.fderiv y).symm : TN (\u03c8 y) \u2192L[\u211d] TY y) \u2218L ((F $ \u03c6 x).2 \u2218L (\u03c6.fderiv x : TX x \u2192L[\u211d] TM (\u03c6 x))),\n  smooth' := begin\n    simp_rw [\u03c6.fderiv_coe, \u03c8.fderiv_symm_coe,\n      mfderiv_congr_point (\u03c8.right_inv (hF $ mem_range_self _))],\n    refine smooth.one_jet_comp IN (\u03bb x', F.bs (\u03c6 x')) _ _,\n    { exact \u03bb x, (\u03c8.smooth_at_inv $ hF $ mem_range_self x).one_jet_ext.comp _\n        (F.smooth_bs.comp \u03c6.smooth_to).cont_mdiff_at },\n    apply smooth.one_jet_comp IM \u03c6 (F.smooth_eta.comp \u03c6.smooth_to) \u03c6.smooth_to.one_jet_ext\n  end }\n\nlemma transfer_localize (hF : range (F.bs \u2218 \u03c6) \u2286 range \u03c8) (x : X) :\n  \u03c6.transfer \u03c8 (F.localize \u03c6 \u03c8 hF x) = F (\u03c6 x) :=\nbegin\n  rw [one_jet_sec.coe_apply, one_jet_sec.localize_bs, one_jet_sec.localize_\u03d5,\n    open_smooth_embedding.transfer, one_jet_bundle.map],\n  dsimp only [one_jet_bundle.mk],\n  ext,\n  { refl },\n  { simp_rw [\u03c8.right_inv (hF $ mem_range_self x), function.comp_apply, F.bs_eq] },\n  { simp_rw [\u2190 \u03c8.fderiv_coe, continuous_linear_map.comp_apply, continuous_linear_equiv.coe_coe,\n      continuous_linear_equiv.apply_symm_apply] },\nend\n\nlemma one_jet_sec.localize_bs_fun (hF : range (F.bs \u2218 \u03c6) \u2286 range \u03c8) :\n  (F.localize \u03c6 \u03c8 hF).bs = \u03c8.inv_fun \u2218 F.bs \u2218 \u03c6 :=\nrfl\n\nlemma one_jet_sec.localize_mem_iff (hF : range (F.bs \u2218 \u03c6) \u2286 range \u03c8) {x : X} :\n  F.localize \u03c6 \u03c8 hF x \u2208 R.localize \u03c6 \u03c8 \u2194 F (\u03c6 x) \u2208 R :=\nby rw [rel_mfld.localize, mem_preimage, transfer_localize F \u03c6 \u03c8 hF]\n\nlemma is_holonomic_at_localize_iff (hF : range (F.bs \u2218 \u03c6) \u2286 range \u03c8) (x : X) :\n  (F.localize \u03c6 \u03c8 hF).is_holonomic_at x \u2194 F.is_holonomic_at (\u03c6 x)  :=\nbegin\n  have : mfderiv IX IY (\u03c8.inv_fun \u2218 F.bs \u2218 \u03c6) x =\n    (\u03c8.fderiv (\u03c8.inv_fun (F.bs (\u03c6 x)))).symm.to_continuous_linear_map.comp\n    ((mfderiv IM IN F.bs (\u03c6 x)).comp (\u03c6.fderiv x).to_continuous_linear_map),\n  { have h1 : mdifferentiable_at IN IY \u03c8.inv_fun (F.bs (\u03c6 x)) :=\n      (\u03c8.smooth_at_inv $ hF $ mem_range_self _).mdifferentiable_at,\n    have h2 : mdifferentiable_at IM IN F.bs (\u03c6 x) := F.smooth_bs.mdifferentiable_at,\n    have h3 : mdifferentiable_at IX IM \u03c6 x := \u03c6.smooth_to.mdifferentiable_at,\n    rw [mfderiv_comp x h1 (h2.comp x h3), mfderiv_comp x h2 h3,\n      \u2190 \u03c8.fderiv_symm_coe' (hF $ mem_range_self _)],\n    refl, },\n  simp_rw [one_jet_sec.is_holonomic_at],\n  rw [mfderiv_congr (F.localize_bs_fun \u03c6 \u03c8 hF), one_jet_sec.snd_eq, F.localize_\u03d5, this],\n  simp_rw [continuous_linear_equiv.coe_def_rev,\n    continuous_linear_equiv.cancel_left, continuous_linear_equiv.cancel_right]\nend\n\n/-! ## From embeddings `X \u21aa M` and `Y \u21aa N` to `J\u00b9(X, Y) \u21aa J\u00b9(M, N)` -/\n\n-- very slow to elaborate :-(\n@[simps]\ndef one_jet_bundle.embedding : open_smooth_embedding IXY J\u00b9XY IMN J\u00b9MN :=\n{ to_fun := \u03c6.transfer \u03c8,\n  inv_fun := one_jet_bundle.map IN IY \u03c6.inv_fun \u03c8.inv_fun\n    (\u03bb x, (\u03c6.fderiv $ \u03c6.inv_fun x : TX (\u03c6.inv_fun x) \u2192L[\u211d] TM (\u03c6 $ \u03c6.inv_fun x))),\n  left_inv' := \u03bb \u03c3, begin\n    rw [open_smooth_embedding.transfer, one_jet_bundle.map_map\n      \u03c8.smooth_at_inv'.mdifferentiable_at \u03c8.smooth_to.smooth_at.mdifferentiable_at],\n    conv_rhs { rw [\u2190 one_jet_bundle.map_id \u03c3] },\n    congr' 1,\n    { rw [open_smooth_embedding.inv_fun_comp_coe] },\n    { rw [open_smooth_embedding.inv_fun_comp_coe] },\n    { ext x v, simp_rw [continuous_linear_map.comp_apply],\n      convert (\u03c6.fderiv x).symm_apply_apply v,\n      simp_rw [\u03c6.left_inv] }\n  end,\n  is_open_range := \u03c6.is_open_range_transfer \u03c8,\n  smooth_to := \u03c6.smooth_transfer \u03c8,\n  smooth_inv := begin\n    rintro _ \u27e8x, rfl\u27e9,\n    refine (smooth_at.one_jet_bundle_map _ _ _ smooth_at_id).smooth_within_at,\n    { refine ((\u03c6.smooth_at_inv _).comp _ smooth_at_snd), exact mem_range_self _ },\n    { refine ((\u03c8.smooth_at_inv _).comp _ smooth_at_snd), exact mem_range_self _ },\n    have := cont_mdiff_at.mfderiv''' (\u03bb x, \u03c6) (\u03bb x : one_jet_bundle IM M IN N, \u03c6.inv_fun x.1.1)\n      (\u03c6.smooth_to.smooth_at.comp _ smooth_at_snd)\n      ((\u03c6.smooth_at_inv _).comp _ (smooth_one_jet_bundle_proj.fst (\u03c6.transfer \u03c8 x))) le_top,\n    { dsimp only [id],\n      simp_rw [\u03c6.left_inv] at this,\n      refine this.congr_of_eventually_eq _,\n      refine filter.eventually_of_mem ((\u03c6.is_open_range_transfer \u03c8).mem_nhds (mem_range_self _)) _,\n      rw [\u03c6.range_transfer \u03c8],\n      rintro \u27e8\u27e8x, y\u27e9, \u03c4\u27e9 \u27e8\u27e8x, rfl\u27e9 : x \u2208 range \u03c6, \u27e8y, rfl\u27e9 : y \u2208 range \u03c8\u27e9,\n      simp_rw [in_coordinates_core, \u03c6.transfer_fst_fst, \u03c6.left_inv],\n      refl },\n    exact mem_range_self _,\n  end }\n\n/-! ## Updating 1-jet sections and formal solutions -/\n\nlocal notation `J\u0398` := \u03c6.update (one_jet_bundle.embedding \u03c6 \u03c8)\n\nvariables {K : set X}\n\nnamespace open_smooth_embedding\n\nlemma Jupdate_aux (F : one_jet_sec IM M IN N) (G : one_jet_sec IX X IY Y)\n  (m : M) : (J\u0398 F G m).1.1 = m :=\nbegin\n  simp_rw [open_smooth_embedding.update], split_ifs,\n  { rcases h with \u27e8x, rfl\u27e9,\n    simp_rw [one_jet_bundle.embedding_to_fun, \u03c6.transfer_fst_fst, \u03c6.left_inv, G.fst_eq] },\n  { refl }\nend\n\nvariables [t2_space M]\n\n\n/-- Update a global homotopy of 1-jet-sections `F` using a local one `G`. -/\ndef Jupdate\n  (F : one_jet_sec IM M IN N) (G : htpy_one_jet_sec IX X IY Y)\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) :\n  htpy_one_jet_sec IM M IN N :=\nbegin\n  refine family_one_jet_sec.mk' (\u03bb t, J\u0398 F (G t)) (\u03bb t, (\u03c6.Jupdate_aux \u03c8 F (G t))) _,\n  refine \u03c6.smooth_update _ _ _ (hK.image \u03c6.continuous).is_closed _ _ smooth_snd (\u03bb x, hFG x.1),\n  { exact F.smooth.comp smooth_snd },\n  { exact G.smooth.comp (smooth_fst.prod_map smooth_id) },\nend\n\nlemma Jupdate_apply {F : one_jet_sec IM M IN N} {G : htpy_one_jet_sec IX X IY Y}\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t : \u211d) (m : M) :\n  \u03c6.Jupdate \u03c8 F G hK hFG t m = J\u0398 F (G t) m :=\nby { ext, exact (\u03c6.Jupdate_aux \u03c8 F (G t) m).symm, refl, refl }\n\nlemma Jupdate_bs (F : one_jet_sec IM M IN N)\n  (G : htpy_one_jet_sec IX X IY Y) (t : \u211d)\n  (hK : is_compact K) (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = one_jet_bundle.embedding \u03c6 \u03c8 (G t x)) :\n  (open_smooth_embedding.Jupdate \u03c6 \u03c8 F G hK hFG t).bs =\n  open_smooth_embedding.update \u03c6 \u03c8 F.bs (G t).bs :=\nbegin\n  classical,\n  ext x,\n  change (if x \u2208 range \u03c6 then \u03c6.transfer \u03c8 (G t (\u03c6.inv_fun x)) else F x).1.2 =\n    if x \u2208 range \u03c6 then _ else _,\n  split_ifs ; refl,\nend\n\nlemma Jupdate_localize {F : one_jet_sec IM M IN N} {G : htpy_one_jet_sec IX X IY Y}\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t : \u211d)\n  (rg : range ((\u03c6.Jupdate \u03c8 F G hK hFG t).bs \u2218 \u03c6) \u2286 range \u03c8) (x : X) :\n  (\u03c6.Jupdate \u03c8 F G hK hFG t).localize \u03c6 \u03c8 rg x = G t x :=\nbegin\n  have foo : \u03c8.inv_fun ((\u03c6.Jupdate \u03c8 F G hK hFG t).bs (\u03c6 x)) = (G t).bs x,\n  { simp_rw [Jupdate_bs, open_smooth_embedding.update_apply_embedding,\n    open_smooth_embedding.left_inv] },\n  ext,\n  refl,\n  { exact foo },\n  { simp_rw [one_jet_sec.snd_eq, one_jet_sec.localize_\u03d5],\n    rw [foo],\n    change (\u03c8.fderiv ((G t).bs x)).symm ((J\u0398 F (G t) (\u03c6 x)).2 (\u03c6.fderiv x x_1)) =\n      ((G t).\u03d5 x) x_1,\n    rw \u03c6.update_apply_embedding,\n    change (\u03c8.fderiv ((G t).bs x)).symm (\u03c8.fderiv ((G t).bs x) $ (G t).\u03d5 x $\n      (\u03c6.fderiv x).symm $ \u03c6.fderiv x x_1) = ((G t).\u03d5 x x_1),\n    simp_rw [continuous_linear_equiv.symm_apply_apply] },\nend\n\n/-- Update a global formal solutions `F` using a homotopy of local ones `G`. -/\ndef update_formal_sol (F : formal_sol R)\n  (G : htpy_formal_sol (R.localize \u03c6 \u03c8))\n  (hK : is_compact K) (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) :\n  htpy_formal_sol R :=\n{ to_family_one_jet_sec := \u03c6.Jupdate \u03c8 F.to_one_jet_sec G.to_family_one_jet_sec hK hFG,\n  is_sol' := \u03bb t x, begin\n    simp_rw [Jupdate_apply, open_smooth_embedding.update, one_jet_bundle.embedding_to_fun],\n    split_ifs,\n    { exact G.is_sol },\n    { exact F.is_sol x }\n  end }\n\nlemma update_formal_sol_apply {F : formal_sol R}\n  {G : htpy_formal_sol (R.localize \u03c6 \u03c8)}\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t x) :\n  \u03c6.update_formal_sol \u03c8 F G hK hFG t x = \u27e8\u27e8x, (J\u0398 F (G t) x).1.2\u27e9, (J\u0398 F (G t) x).2\u27e9 :=\nrfl\n\nlemma update_formal_sol_bs' {F : formal_sol R}\n  {G : htpy_formal_sol (R.localize \u03c6 \u03c8)}\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t) :\n(\u03c6.update_formal_sol \u03c8 F G hK hFG t).bs = \u03bb x, (J\u0398 F (G t) x).1.2 :=\nrfl\n\nlemma update_formal_sol_bs {F : formal_sol R}\n  {G : htpy_formal_sol (R.localize \u03c6 \u03c8)}\n  (hK : is_compact K)\n  (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t) :\n(\u03c6.update_formal_sol \u03c8 F G hK hFG t).bs = \u03c6.update \u03c8 F.bs (G t).bs :=\nbegin\n  rw update_formal_sol_bs',\n  ext x,\n  by_cases hx : x \u2208 range \u03c6,\n  { simp only [hx, update_of_mem_range, one_jet_bundle.embedding_to_fun, transfer_fst_snd],\n    refl },\n  { rw [update_of_nmem_range, update_of_nmem_range],\n    refl,\n    exacts [hx, hx] },\nend\n\n@[simp]\nlemma update_formal_sol_apply_of_mem {F : formal_sol R}\n  {G : htpy_formal_sol (R.localize \u03c6 \u03c8)}\n  (hK : is_compact K) (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t)\n  {m} (hx : m \u2208 range \u03c6) :\n  \u03c6.update_formal_sol \u03c8 F G hK hFG t m = \u03c6.transfer \u03c8 (G t $ \u03c6.inv_fun m) :=\nbegin\n  rw [update_formal_sol_apply, \u03c6.update_of_mem_range _ _ _ hx],\n  ext,\n  { change m = \u03c6 (\u03c6.inv_fun m),\n    rw \u03c6.right_inv hx },\n  refl,\n  refl\nend\n\n@[simp]\nlemma update_formal_sol_apply_image {F : formal_sol R}\n  {G : htpy_formal_sol (R.localize \u03c6 \u03c8)}\n  (hK : is_compact K) (hFG : \u2200 t, \u2200 x \u2209 K, F (\u03c6 x) = (one_jet_bundle.embedding \u03c6 \u03c8) (G t x)) (t)\n  {x} :\n\u03c6.update_formal_sol \u03c8 F G hK hFG t (\u03c6 x) = \u03c6.transfer \u03c8 (G t x) :=\nbegin\n rw [open_smooth_embedding.update_formal_sol_apply_of_mem, \u03c6.left_inv],\n exact mem_range_self x,\nend\n\nend open_smooth_embedding\nend open_smooth_embedding\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/global/relation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3902220534632659}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.list.lemmas\nimport Mathlib.Lean3Lib.init.wf\n\nuniverses u_1 \n\nnamespace Mathlib\n\nnamespace list\n\n\n-- Note: we can't use the equation compiler here because\n\n-- init.meta.well_founded_tactics uses this file\n\ndef qsort.F {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (x : List \u03b1) :\n    ((y : List \u03b1) \u2192 length y < length x \u2192 List \u03b1) \u2192 List \u03b1 :=\n  sorry\n\n/- This is based on the minimalist Haskell \"quicksort\".\n\n   Remark: this is *not* really quicksort since it doesn't partition the elements in-place -/\n\ndef qsort {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) : List \u03b1 \u2192 List \u03b1 := well_founded.fix sorry sorry\n\n@[simp] theorem qsort_nil {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) : qsort lt [] = [] := sorry\n\n@[simp] theorem qsort_cons {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (h : \u03b1) (t : List \u03b1) :\n    qsort lt (h :: t) =\n        (fun (_a : List \u03b1 \u00d7 List \u03b1) =>\n            prod.cases_on _a\n              fun (fst snd : List \u03b1) => idRhs (List \u03b1) (qsort lt snd ++ h :: qsort lt fst))\n          (partition (fun (x : \u03b1) => lt h x = tt) t) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/list/qsort_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.39015976274064484}}
{"text": "import classes.context_free.basics.toolbox\nimport utilities.language_operations\n\n\nvariables {T\u2081 T\u2082 N : Type}\n\nprivate def sT\u2082_of_sT\u2081 (\u03c0 : equiv T\u2081 T\u2082) : (symbol T\u2081 N) \u2192 (symbol T\u2082 N)\n| (symbol.terminal t) := symbol.terminal (\u03c0.to_fun t)\n| (symbol.nonterminal n) := symbol.nonterminal n\n\nprivate def sT\u2081_of_sT\u2082 (\u03c0 : equiv T\u2081 T\u2082) : (symbol T\u2082 N) \u2192 (symbol T\u2081 N)\n| (symbol.terminal t) := symbol.terminal (\u03c0.inv_fun t)\n| (symbol.nonterminal n) := symbol.nonterminal n\n\nprivate def lsT\u2082_of_lsT\u2081 (\u03c0 : equiv T\u2081 T\u2082) : list (symbol T\u2081 N) \u2192 list (symbol T\u2082 N) :=\nlist.map (sT\u2082_of_sT\u2081 \u03c0)\n\nprivate def lsT\u2081_of_lsT\u2082 (\u03c0 : equiv T\u2081 T\u2082) : list (symbol T\u2082 N) \u2192 list (symbol T\u2081 N) :=\nlist.map (sT\u2081_of_sT\u2082 \u03c0)\n\n/-- The class of context-free languages is closed under bijection between terminal alphabets. -/\ntheorem CF_of_bijemap_CF (\u03c0 : equiv T\u2081 T\u2082) (L : language T\u2081) :\n  is_CF L  \u2192  is_CF (bijemap_lang L \u03c0)  :=\nbegin\n  rintro \u27e8g, hg\u27e9,\n\n  let g' : CF_grammar T\u2082 := CF_grammar.mk g.nt g.initial (list.map (\n      \u03bb r : g.nt \u00d7 (list (symbol T\u2081 g.nt)), (r.fst, lsT\u2082_of_lsT\u2081 \u03c0 r.snd)\n    ) g.rules),\n  use g',\n\n  apply set.eq_of_subset_of_subset,\n  {\n    intros w hw,\n    unfold bijemap_lang,\n    change list.map \u03c0.inv_fun w \u2208 L,\n    rw \u2190hg,\n\n    unfold CF_language at hw \u22a2,\n    rw set.mem_set_of_eq at hw \u22a2,\n    unfold CF_generates at hw \u22a2,\n    unfold CF_generates_str at hw \u22a2,\n\n    have deri_of_deri :\n      \u2200 v : list (symbol T\u2082 g'.nt),\n        CF_derives g' [symbol.nonterminal g'.initial] v \u2192\n          CF_derives g [symbol.nonterminal g.initial] (lsT\u2081_of_lsT\u2082 \u03c0 v),\n    {\n      intros v hv,\n      induction hv with u v trash orig ih,\n      {\n        apply CF_deri_self,\n      },\n      apply CF_deri_of_deri_tran,\n      {\n        exact ih,\n      },\n      rcases orig with \u27e8r, r_in, x, y, bef, aft\u27e9,\n      let r\u2081 := (r.fst, lsT\u2081_of_lsT\u2082 \u03c0 r.snd),\n      let x\u2081 := lsT\u2081_of_lsT\u2082 \u03c0 x,\n      let y\u2081 := lsT\u2081_of_lsT\u2082 \u03c0 y,\n      use r\u2081,\n      split,\n      {\n        change (r.fst, lsT\u2081_of_lsT\u2082 \u03c0 r.snd) \u2208 g.rules,\n        rw [list.mem_map, prod.exists] at r_in,\n        rcases r_in with \u27e8a, b, ab_in, ab_eq\u27e9,\n        have a_eq : a = r.fst :=\n          (congr_arg prod.fst ab_eq).congr_right.mp rfl,\n        have b_eq : lsT\u2082_of_lsT\u2081 \u03c0 b = r.snd :=\n          (congr_arg prod.snd ab_eq).congr_right.mp rfl,\n        rw a_eq at ab_in,\n        convert ab_in,\n        rw \u2190b_eq,\n        unfold lsT\u2081_of_lsT\u2082,\n        unfold lsT\u2082_of_lsT\u2081,\n        rw list.map_map,\n        ext1,\n        rw list.nth_map,\n        cases (b.nth n),\n        {\n          -- none = none\n          refl,\n        },\n        cases val, swap,\n        {\n          -- nonterminal = nonterminal\n          refl,\n        },\n        {\n          -- (sT\u2081_of_sT\u2082 \u03c0 \u2218 sT\u2082_of_sT\u2081 \u03c0) terminal = terminal\n          simp [sT\u2082_of_sT\u2081, sT\u2081_of_sT\u2082, equiv.left_inv],\n        }\n      },\n      use x\u2081,\n      use y\u2081,\n      split,\n      {\n        rw bef,\n        unfold lsT\u2081_of_lsT\u2082,\n        rw list.map_append,\n        rw list.map_append,\n        refl,\n      },\n      {\n        rw aft,\n        unfold lsT\u2081_of_lsT\u2082,\n        rw list.map_append,\n        rw list.map_append,\n        refl,\n      },\n    },\n    specialize deri_of_deri (list.map symbol.terminal w) hw,\n    unfold lsT\u2081_of_lsT\u2082 at deri_of_deri,\n    rw list.map_map at *,\n    convert deri_of_deri,\n  },\n  {\n    intros w hw,\n    unfold bijemap_lang at hw,\n    change list.map \u03c0.inv_fun w \u2208 L at hw,\n    rw \u2190hg at hw,\n    unfold CF_language at hw,\n    rw set.mem_set_of_eq at hw,\n    unfold CF_generates at hw,\n    rw list.map_map at hw,\n    unfold CF_generates_str at hw,\n\n    unfold CF_language,\n    change CF_generates_str g' (list.map symbol.terminal w),\n    unfold CF_generates_str,\n\n    have deri_of_deri :\n      \u2200 v : list (symbol T\u2081 g.nt),\n        CF_derives g [symbol.nonterminal g.initial] v \u2192\n          CF_derives g' [symbol.nonterminal g'.initial] (lsT\u2082_of_lsT\u2081 \u03c0 v),\n    {\n      intros v hv,\n      induction hv with u v trash orig ih,\n      {\n        apply CF_deri_self,\n      },\n      apply CF_deri_of_deri_tran,\n      {\n        exact ih,\n      },\n      rcases orig with \u27e8r, r_in, x, y, bef, aft\u27e9,\n      let r\u2082 := (r.fst, lsT\u2082_of_lsT\u2081 \u03c0 r.snd),\n      let x\u2082 := lsT\u2082_of_lsT\u2081 \u03c0 x,\n      let y\u2082 := lsT\u2082_of_lsT\u2081 \u03c0 y,\n      use r\u2082,\n      split,\n      {\n        rw [list.mem_map, prod.exists],\n        use r.fst,\n        use r.snd,\n        split,\n        {\n          convert r_in,\n          exact prod.ext rfl rfl,\n        },\n        split;\n        refl,\n      },\n      use x\u2082,\n      use y\u2082,\n      split,\n      {\n        rw bef,\n        unfold lsT\u2082_of_lsT\u2081,\n        rw list.map_append,\n        rw list.map_append,\n        refl,\n      },\n      {\n        rw aft,\n        unfold lsT\u2082_of_lsT\u2081,\n        rw list.map_append,\n        rw list.map_append,\n        refl,\n      },\n    },\n    specialize deri_of_deri (list.map (symbol.terminal \u2218 \u03c0.inv_fun) w) hw,\n    rw lsT\u2082_of_lsT\u2081 at deri_of_deri,\n    rw list.map_map at deri_of_deri,\n    convert deri_of_deri,\n    ext1,\n    change symbol.terminal x = sT\u2082_of_sT\u2081 \u03c0 (symbol.terminal (\u03c0.inv_fun x)),\n    unfold sT\u2082_of_sT\u2081,\n    rw equiv.right_inv,\n  },\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/context_free/closure_properties/bijection.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178138, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3900985046772129}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n\nimport control.monad.basic\nimport data.int.basic\nimport data.stream.defs\nimport control.uliftable\nimport tactic.norm_num\nimport data.bitvec.basic\n\n\n/-!\n# Rand Monad and Random Class\n\nThis module provides tools for formulating computations guided by randomness and for\ndefining objects that can be created randomly.\n\n## Main definitions\n  * `rand` monad for computations guided by randomness;\n  * `random` class for objects that can be generated randomly;\n    * `random` to generate one object;\n    * `random_r` to generate one object inside a range;\n    * `random_series` to generate an infinite series of objects;\n    * `random_series_r` to generate an infinite series of objects inside a range;\n  * `io.mk_generator` to create a new random number generator;\n  * `io.run_rand` to run a randomized computation inside the `io` monad;\n  * `tactic.run_rand` to run a randomized computation inside the `tactic` monad\n\n## Local notation\n\n * `i .. j` : `Icc i j`, the set of values between `i` and `j` inclusively;\n\n## Tags\n\nrandom monad io\n\n## References\n\n  * Similar library in Haskell: https://hackage.haskell.org/package/MonadRandom\n\n-/\n\nopen list io applicative\n\nuniverses u v w\n\n/-- A monad to generate random objects using the generator type `g` -/\n@[reducible]\ndef rand_g (g : Type) (\u03b1 : Type u) : Type u := state (ulift.{u} g) \u03b1\n\n/-- A monad to generate random objects using the generator type `std_gen` -/\n@[reducible]\ndef rand := rand_g std_gen\n\ninstance (g : Type) : uliftable (rand_g.{u} g) (rand_g.{v} g) :=\n@state_t.uliftable' _ _ _ _ _ (equiv.ulift.trans.{u u u u u} equiv.ulift.symm)\n\nopen ulift (hiding inhabited)\n\n/-- Generate one more `\u2115` -/\ndef rand_g.next {g : Type} [random_gen g] : rand_g g \u2115 :=\n\u27e8 prod.map id up \u2218 random_gen.next \u2218 down \u27e9\n\nlocal infix ` .. `:41 := set.Icc\n\nopen stream\n\n/-- `bounded_random \u03b1` gives us machinery to generate values of type `\u03b1` between certain bounds -/\nclass bounded_random (\u03b1 : Type u) [preorder \u03b1] :=\n(random_r : \u03a0 g [random_gen g] (x y : \u03b1),\n              (x \u2264 y) \u2192 rand_g g (x .. y))\n\n/-- `random \u03b1` gives us machinery to generate values of type `\u03b1` -/\nclass random (\u03b1 : Type u) :=\n(random [] : \u03a0 (g : Type) [random_gen g], rand_g g \u03b1)\n\n/-- shift_31_left = 2^31; multiplying by it shifts the binary\nrepresentation of a number left by 31 bits, dividing by it shifts it\nright by 31 bits -/\ndef shift_31_left : \u2115 :=\nby apply_normed 2^31\n\nnamespace rand\n\nopen stream\n\nvariables (\u03b1 : Type u)\nvariables (g : Type) [random_gen g]\n\n/-- create a new random number generator distinct from the one stored in the state -/\ndef split : rand_g g g := \u27e8 prod.map id up \u2218 random_gen.split \u2218 down \u27e9\n\nvariables {g}\n\nsection random\nvariables [random \u03b1]\n\nexport random (random)\n\n/-- Generate a random value of type `\u03b1`. -/\ndef random : rand_g g \u03b1 :=\nrandom.random \u03b1 g\n\n/-- generate an infinite series of random values of type `\u03b1` -/\ndef random_series : rand_g g (stream \u03b1) :=\ndo gen \u2190 uliftable.up (split g),\n   pure $ stream.corec_state (random.random \u03b1 g) gen\n\nend random\n\nvariables {\u03b1}\n\n/-- Generate a random value between `x` and `y` inclusive. -/\ndef random_r [preorder \u03b1] [bounded_random \u03b1] (x y : \u03b1) (h : x \u2264 y) : rand_g g (x .. y) :=\nbounded_random.random_r g x y h\n\n/-- generate an infinite series of random values of type `\u03b1` between `x` and `y` inclusive. -/\ndef random_series_r [preorder \u03b1] [bounded_random \u03b1] (x y : \u03b1) (h : x \u2264 y) :\n  rand_g g (stream (x .. y)) :=\ndo gen \u2190 uliftable.up (split g),\n   pure $ corec_state (bounded_random.random_r g x y h) gen\n\nend rand\n\nnamespace io\n\nprivate def accum_char (w : \u2115) (c : char) : \u2115 :=\nc.to_nat + 256 * w\n\n/-- create and a seed a random number generator -/\ndef mk_generator : io std_gen := do\nseed \u2190 io.rand 0 shift_31_left,\nreturn $ mk_std_gen seed\n\nvariables {\u03b1 : Type}\n\n/-- Run `cmd` using a randomly seeded random number generator -/\ndef run_rand (cmd : _root_.rand \u03b1) : io \u03b1 :=\ndo g \u2190 io.mk_generator,\n   return $ (cmd.run \u27e8g\u27e9).1\n\n/-- Run `cmd` using the provided seed. -/\ndef run_rand_with (seed : \u2115) (cmd : _root_.rand \u03b1) : io \u03b1 :=\nreturn $ (cmd.run \u27e8mk_std_gen seed\u27e9).1\n\nsection random\nvariables [random \u03b1]\n\n/-- randomly generate a value of type \u03b1 -/\ndef random : io \u03b1 :=\nio.run_rand (rand.random \u03b1)\n\n/-- randomly generate an infinite series of value of type \u03b1 -/\ndef random_series : io (stream \u03b1) :=\nio.run_rand (rand.random_series \u03b1)\n\nend random\n\nsection bounded_random\nvariables [preorder \u03b1] [bounded_random \u03b1]\n\n/-- randomly generate a value of type \u03b1 between `x` and `y` -/\ndef random_r (x y : \u03b1) (p : x \u2264 y) : io (x .. y) :=\nio.run_rand (bounded_random.random_r _ x y p)\n\n/-- randomly generate an infinite series of value of type \u03b1 between `x` and `y` -/\ndef random_series_r (x y : \u03b1) (h : x \u2264 y) : io (stream $ x .. y) :=\nio.run_rand (rand.random_series_r x y h)\n\nend bounded_random\n\nend io\n\nnamespace tactic\n\n/-- create a seeded random number generator in the `tactic` monad -/\nmeta def mk_generator : tactic std_gen := do\ntactic.unsafe_run_io @io.mk_generator\n\n/-- run `cmd` using the a randomly seeded random number generator\nin the tactic monad -/\nmeta def run_rand {\u03b1 : Type u} (cmd : rand \u03b1) : tactic \u03b1 := do\n\u27e8g\u27e9 \u2190 tactic.up mk_generator,\nreturn (cmd.run \u27e8g\u27e9).1\n\nvariables {\u03b1 : Type u}\n\nsection bounded_random\nvariables [preorder \u03b1] [bounded_random \u03b1]\n\n/-- Generate a random value between `x` and `y` inclusive. -/\nmeta def random_r (x y : \u03b1) (h : x \u2264 y) : tactic (x .. y) :=\nrun_rand (rand.random_r x y h)\n\n/-- Generate an infinite series of random values of type `\u03b1` between `x` and `y` inclusive. -/\nmeta def random_series_r (x y : \u03b1) (h : x \u2264 y) : tactic (stream $ x .. y) :=\nrun_rand (rand.random_series_r x y h)\n\nend bounded_random\n\nsection random\n\nvariables [random \u03b1]\n\n/-- randomly generate a value of type \u03b1 -/\nmeta def random : tactic \u03b1 :=\nrun_rand (rand.random \u03b1)\n\n /-- randomly generate an infinite series of value of type \u03b1 -/\nmeta def random_series : tactic (stream \u03b1) :=\nrun_rand (rand.random_series \u03b1)\n\nend random\n\nend tactic\n\nopen nat (succ one_add mod_eq_of_lt zero_lt_succ add_one succ_le_succ)\n\nvariables {g : Type} [random_gen g]\n\nopen nat\n\nnamespace fin\nvariables {n : \u2115} [fact (0 < n)]\n\n/-- generate a `fin` randomly -/\nprotected def random : rand_g g (fin n) :=\n\u27e8 \u03bb \u27e8g\u27e9, prod.map of_nat' up $ rand_nat g 0 n \u27e9\n\nend fin\n\nopen nat\n\ninstance nat_bounded_random : bounded_random \u2115 :=\n{ random_r := \u03bb g inst x y hxy,\n  do z \u2190 @fin.random g inst (succ $ y - x) _,\n     pure \u27e8z.val + x, nat.le_add_left _ _,\n       by rw \u2190 le_tsub_iff_right hxy; apply le_of_succ_le_succ z.is_lt\u27e9 }\n\n/-- This `bounded_random` interval generates integers between `x` and\n`y` by first generating a natural number between `0` and `y - x` and\nshifting the result appropriately. -/\ninstance int_bounded_random : bounded_random \u2124 :=\n{ random_r := \u03bb g inst x y hxy,\n  do \u27e8z,h\u2080,h\u2081\u27e9 \u2190 @bounded_random.random_r \u2115 _ _ g inst 0 (int.nat_abs $ y - x) dec_trivial,\n     pure \u27e8z + x,\n       int.le_add_of_nonneg_left (int.coe_nat_nonneg _),\n       int.add_le_of_le_sub_right $ le_trans\n         (int.coe_nat_le_coe_nat_of_le h\u2081)\n         (le_of_eq $ int.of_nat_nat_abs_eq_of_nonneg (int.sub_nonneg_of_le hxy)) \u27e9 }\n\ninstance fin_random (n : \u2115) [fact (0 < n)] : random (fin n) :=\n{ random := \u03bb g inst, @fin.random g inst _ _ }\n\ninstance fin_bounded_random (n : \u2115) : bounded_random (fin n) :=\n{ random_r := \u03bb g inst (x y : fin n) p,\n    do \u27e8r, h, h'\u27e9 \u2190 @rand.random_r \u2115 g inst _ _ x.val y.val p,\n       pure \u27e8\u27e8r,lt_of_le_of_lt h' y.is_lt\u27e9, h, h'\u27e9 }\n\n/-- A shortcut for creating a `random (fin n)` instance from\na proof that `0 < n` rather than on matching on `fin (succ n)`  -/\ndef random_fin_of_pos : \u2200 {n : \u2115} (h : 0 < n), random (fin n)\n| (succ n) _ := fin_random _\n| 0 h := false.elim (nat.not_lt_zero _ h)\n\nlemma bool_of_nat_mem_Icc_of_mem_Icc_to_nat (x y : bool) (n : \u2115) :\n  n \u2208 (x.to_nat .. y.to_nat) \u2192 bool.of_nat n \u2208 (x .. y) :=\nbegin\n  simp only [and_imp, set.mem_Icc], intros h\u2080 h\u2081,\n  split;\n    [ have h\u2082 := bool.of_nat_le_of_nat h\u2080, have h\u2082 := bool.of_nat_le_of_nat h\u2081 ];\n    rw bool.of_nat_to_nat at h\u2082; exact h\u2082,\nend\n\ninstance : random bool :=\n{ random   := \u03bb g inst,\n  (bool.of_nat \u2218 subtype.val) <$> @bounded_random.random_r \u2115 _ _ g inst 0 1 (nat.zero_le _) }\n\ninstance : bounded_random bool :=\n{ random_r := \u03bb g _inst x y p,\n  subtype.map bool.of_nat (bool_of_nat_mem_Icc_of_mem_Icc_to_nat x y) <$>\n    @bounded_random.random_r \u2115 _ _ g _inst x.to_nat y.to_nat (bool.to_nat_le_to_nat p) }\n\nopen_locale fin_fact\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random (n : \u2115) : rand_g g (bitvec n) :=\nbitvec.of_fin <$> rand.random (fin $ 2^n)\n\n/-- generate a random bit vector of length `n` -/\ndef bitvec.random_r {n : \u2115} (x y : bitvec n) (h : x \u2264 y) : rand_g g (x .. y) :=\nhave h' : \u2200 (a : fin (2 ^ n)), a \u2208 (x.to_fin .. y.to_fin) \u2192 bitvec.of_fin a \u2208 (x .. y),\nbegin\n  simp only [and_imp, set.mem_Icc], intros z h\u2080 h\u2081,\n  replace h\u2080 := bitvec.of_fin_le_of_fin_of_le h\u2080,\n  replace h\u2081 := bitvec.of_fin_le_of_fin_of_le h\u2081,\n  rw bitvec.of_fin_to_fin at h\u2080 h\u2081, split; assumption,\nend,\nsubtype.map bitvec.of_fin h' <$> rand.random_r x.to_fin y.to_fin (bitvec.to_fin_le_to_fin_of_le h)\n\nopen nat\n\ninstance random_bitvec (n : \u2115) : random (bitvec n) :=\n{ random := \u03bb _ inst, @bitvec.random _ inst n }\n\ninstance bounded_random_bitvec (n : \u2115) : bounded_random (bitvec n) :=\n{ random_r := \u03bb _ inst x y p, @bitvec.random_r _ inst _ _ _ p }\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/control/random.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3897991948494771}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\n! This file was ported from Lean 3 source module init.data.option.basic\n! leanprover-community/mathlib commit e611ee5c2bd410148bcd493c58cb17498d667175\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Logic\nimport Leanbin.Init.Control.Monad\nimport Leanbin.Init.Control.Alternative\n\nopen Decidable\n\nuniverse u v\n\nnamespace Option\n\n/- warning: option.to_monad -> Option.toMonad is a dubious translation:\nlean 3 declaration is\n  forall {m : Type -> Type} [_inst_1 : Monad.{0, 0} m] [_inst_2 : Alternative.{0, 0} m] {A : Type}, (Option.{0} A) -> (m A)\nbut is expected to have type\n  forall {m : Type.{u_1} -> Type.{u_2}} {_inst_1 : Type.{u_1}} [_inst_2 : Monad.{u_1, u_2} m] [A : Alternative.{u_1, u_2} m], (Option.{u_1} _inst_1) -> (m _inst_1)\nCase conversion may be inaccurate. Consider using '#align option.to_monad Option.toMonad\u2093'. -/\ndef toMonad {m : Type \u2192 Type} [Monad m] [Alternative m] {A} : Option A \u2192 m A\n  | none => failure\n  | some a => return a\n#align option.to_monad Option.toMonad\n\n#print Option.getD /-\ndef getD {\u03b1 : Type u} : Option \u03b1 \u2192 \u03b1 \u2192 \u03b1\n  | some x, _ => x\n  | none, e => e\n#align option.get_or_else Option.getD\n-/\n\n#print Option.isSome /-\ndef isSome {\u03b1 : Type u} : Option \u03b1 \u2192 Bool\n  | some _ => true\n  | none => false\n#align option.is_some Option.isSome\n-/\n\n#print Option.isNone /-\ndef isNone {\u03b1 : Type u} : Option \u03b1 \u2192 Bool\n  | some _ => false\n  | none => true\n#align option.is_none Option.isNone\n-/\n\n#print Option.get /-\ndef get {\u03b1 : Type u} : \u2200 {o : Option \u03b1}, isSome o \u2192 \u03b1\n  | some x, h => x\n  | none, h => False.ndrec _ <| Bool.false_ne_true h\n#align option.get Option.get\n-/\n\ndef rhoare {\u03b1 : Type u} : Bool \u2192 \u03b1 \u2192 Option \u03b1\n  | tt, a => none\n  | ff, a => some a\n#align option.rhoare Option.rhoare\n\ndef lhoare {\u03b1 : Type u} : \u03b1 \u2192 Option \u03b1 \u2192 \u03b1\n  | a, none => a\n  | _, some b => b\n#align option.lhoare Option.lhoare\n\n#print Option.bind /-\n@[inline]\nprotected def bind {\u03b1 : Type u} {\u03b2 : Type v} : Option \u03b1 \u2192 (\u03b1 \u2192 Option \u03b2) \u2192 Option \u03b2\n  | none, b => none\n  | some a, b => b a\n#align option.bind Option.bind\n-/\n\n#print Option.map /-\nprotected def map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (o : Option \u03b1) : Option \u03b2 :=\n  Option.bind o (some \u2218 f)\n#align option.map Option.map\n-/\n\n#print Option.map_id /-\ntheorem map_id {\u03b1} : (Option.map id : Option \u03b1 \u2192 Option \u03b1) = id :=\n  funext fun o =>\n    match o with\n    | none => rfl\n    | some x => rfl\n#align option.map_id Option.map_id\n-/\n\ninstance : Monad Option where\n  pure := @some\n  bind := @Option.bind\n  map := @Option.map\n\nprotected def orelse {\u03b1 : Type u} : Option \u03b1 \u2192 Option \u03b1 \u2192 Option \u03b1\n  | some a, o => some a\n  | none, some a => some a\n  | none, none => none\n#align option.orelse Option.orelse\n\ninstance : Alternative Option where\n  failure := @none\n  orelse := @Option.orelse\n\nend Option\n\ninstance (\u03b1 : Type u) : Inhabited (Option \u03b1) :=\n  \u27e8none\u27e9\n\ninstance {\u03b1 : Type u} [d : DecidableEq \u03b1] : DecidableEq (Option \u03b1)\n  | none, none => isTrue rfl\n  | none, some v\u2082 => isFalse fun h => Option.noConfusion h\n  | some v\u2081, none => isFalse fun h => Option.noConfusion h\n  | some v\u2081, some v\u2082 =>\n    match d v\u2081 v\u2082 with\n    | is_true e => isTrue (congr_arg (@some \u03b1) e)\n    | is_false n => isFalse fun h => Option.noConfusion h fun e => absurd e n\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Data/Option/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5888891307678319, "lm_q1q2_score": 0.3897991869946466}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Alena Gusakov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.finset.default\nimport Mathlib.data.fintype.basic\nimport Mathlib.algebra.geom_sum\nimport Mathlib.tactic.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Colex\n\nWe define the colex ordering for finite sets, and give a couple of important\nlemmas and properties relating to it.\n\nThe colex ordering likes to avoid large values - it can be thought of on\n`finset \u2115` as the \"binary\" ordering. That is, order A based on\n`\u2211_{i \u2208 A} 2^i`.\nIt's defined here in a slightly more general way, requiring only `has_lt \u03b1` in\nthe definition of colex on `finset \u03b1`. In the context of the Kruskal-Katona\ntheorem, we are interested in particular on how colex behaves for sets of a\nfixed size. If the size is 3, colex on \u2115 starts\n123, 124, 134, 234, 125, 135, 235, 145, 245, 345, ...\n\n## Main statements\n* `colex_hom`: strictly monotone functions preserve colex\n* Colex order properties - linearity, decidability and so on.\n* `forall_lt_of_colex_lt_of_forall_lt`: if A < B in colex, and everything\n  in B is < t, then everything in A is < t. This confirms the idea that\n  an enumeration under colex will exhaust all sets using elements < t before\n  allowing t to be included.\n* `binary_iff`: colex for \u03b1 = \u2115 is the same as binary\n  (this also proves binary expansions are unique)\n\n## Notation\nWe define `<` and `\u2264` to denote colex ordering, useful in particular when\nmultiple orderings are available in context.\n\n## Tags\ncolex, colexicographic, binary\n\n## References\n* https://github.com/b-mehta/maths-notes/blob/master/iii/mich/combinatorics.pdf\n\n## Todo\nShow the subset ordering is a sub-relation of the colex ordering.\n-/\n\n/--\nWe define this type synonym to refer to the colexicographic ordering on finsets\nrather than the natural subset ordering.\n-/\ndef finset.colex (\u03b1 : Type u_1) :=\n  finset \u03b1\n\n/--\nA convenience constructor to turn a `finset \u03b1` into a `finset.colex \u03b1`, useful in order to\nuse the colex ordering rather than the subset ordering.\n-/\ndef finset.to_colex {\u03b1 : Type u_1} (s : finset \u03b1) : finset.colex \u03b1 :=\n  s\n\n@[simp] theorem colex.eq_iff {\u03b1 : Type u_1} (A : finset \u03b1) (B : finset \u03b1) : finset.to_colex A = finset.to_colex B \u2194 A = B :=\n  iff.refl (finset.to_colex A = finset.to_colex B)\n\n/--\n`A` is less than `B` in the colex ordering if the largest thing that's not in both sets is in B.\nIn other words, max (A \u25b5 B) \u2208 B (if the maximum exists).\n-/\nprotected instance finset.colex.has_lt {\u03b1 : Type u_1} [HasLess \u03b1] : HasLess (finset.colex \u03b1) :=\n  { Less := fun (A B : finset \u03b1) => \u2203 (k : \u03b1), (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B }\n\n/-- We can define (\u2264) in the obvious way. -/\nprotected instance finset.colex.has_le {\u03b1 : Type u_1} [HasLess \u03b1] : HasLessEq (finset.colex \u03b1) :=\n  { LessEq := fun (A B : finset.colex \u03b1) => A < B \u2228 A = B }\n\ntheorem colex.lt_def {\u03b1 : Type u_1} [HasLess \u03b1] (A : finset \u03b1) (B : finset \u03b1) : finset.to_colex A < finset.to_colex B \u2194 \u2203 (k : \u03b1), (\u2200 {x : \u03b1}, k < x \u2192 (x \u2208 A \u2194 x \u2208 B)) \u2227 \u00ack \u2208 A \u2227 k \u2208 B :=\n  iff.rfl\n\ntheorem colex.le_def {\u03b1 : Type u_1} [HasLess \u03b1] (A : finset \u03b1) (B : finset \u03b1) : finset.to_colex A \u2264 finset.to_colex B \u2194 finset.to_colex A < finset.to_colex B \u2228 A = B :=\n  iff.rfl\n\n/-- If everything in A is less than k, we can bound the sum of powers. -/\ntheorem nat.sum_pow_two_lt {k : \u2115} {A : finset \u2115} (h\u2081 : \u2200 {x : \u2115}, x \u2208 A \u2192 x < k) : finset.sum A (pow (bit0 1)) < bit0 1 ^ k := sorry\n\nnamespace colex\n\n\n/-- Strictly monotone functions preserve the colex ordering. -/\ntheorem hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [DecidableEq \u03b2] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : strict_mono f) (A : finset \u03b1) (B : finset \u03b1) : finset.to_colex (finset.image f A) < finset.to_colex (finset.image f B) \u2194 finset.to_colex A < finset.to_colex B := sorry\n\n/-- A special case of `colex_hom` which is sometimes useful. -/\n@[simp] theorem hom_fin {n : \u2115} (A : finset (fin n)) (B : finset (fin n)) : finset.to_colex (finset.image (fun (n_1 : fin n) => \u2191n_1) A) <\n    finset.to_colex (finset.image (fun (n_1 : fin n) => \u2191n_1) B) \u2194\n  finset.to_colex A < finset.to_colex B :=\n  hom (fun (x y : fin n) (k : x < y) => k) A B\n\nprotected instance has_lt.lt.is_irrefl {\u03b1 : Type u_1} [HasLess \u03b1] : is_irrefl (finset.colex \u03b1) Less :=\n  is_irrefl.mk\n    fun (A : finset.colex \u03b1) (h : A < A) =>\n      exists.elim h fun (_x : \u03b1) (_x : (\u2200 {x : \u03b1}, _x < x \u2192 (x \u2208 A \u2194 x \u2208 A)) \u2227 \u00ac_x \u2208 A \u2227 _x \u2208 A) => sorry\n\ntheorem lt_trans {\u03b1 : Type u_1} [linear_order \u03b1] {a : finset.colex \u03b1} {b : finset.colex \u03b1} {c : finset.colex \u03b1} : a < b \u2192 b < c \u2192 a < c := sorry\n\ntheorem le_trans {\u03b1 : Type u_1} [linear_order \u03b1] (a : finset.colex \u03b1) (b : finset.colex \u03b1) (c : finset.colex \u03b1) : a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c :=\n  fun (AB : a \u2264 b) (BC : b \u2264 c) =>\n    or.elim AB (fun (k : a < b) => or.elim BC (fun (t : b < c) => Or.inl (lt_trans k t)) fun (t : b = c) => t \u25b8 AB)\n      fun (k : a = b) => Eq.symm k \u25b8 BC\n\nprotected instance has_lt.lt.is_trans {\u03b1 : Type u_1} [linear_order \u03b1] : is_trans (finset.colex \u03b1) Less :=\n  is_trans.mk fun (_x _x_1 _x_2 : finset.colex \u03b1) => lt_trans\n\nprotected instance has_lt.lt.is_asymm {\u03b1 : Type u_1} [linear_order \u03b1] : is_asymm (finset.colex \u03b1) Less :=\n  Mathlib.is_asymm_of_is_trans_of_is_irrefl\n\nprotected instance has_lt.lt.is_strict_order {\u03b1 : Type u_1} [linear_order \u03b1] : is_strict_order (finset.colex \u03b1) Less :=\n  is_strict_order.mk\n\ntheorem lt_trichotomy {\u03b1 : Type u_1} [linear_order \u03b1] (A : finset.colex \u03b1) (B : finset.colex \u03b1) : A < B \u2228 A = B \u2228 B < A := sorry\n\nprotected instance has_lt.lt.is_trichotomous {\u03b1 : Type u_1} [linear_order \u03b1] : is_trichotomous (finset.colex \u03b1) Less :=\n  is_trichotomous.mk lt_trichotomy\n\n-- It should be possible to do this computably but it doesn't seem to make any difference for now.\n\nprotected instance finset.colex.linear_order {\u03b1 : Type u_1} [linear_order \u03b1] : linear_order (finset.colex \u03b1) :=\n  linear_order.mk LessEq (partial_order.lt._default LessEq) sorry le_trans sorry sorry (classical.dec_rel LessEq)\n    Mathlib.decidable_eq_of_decidable_le Mathlib.decidable_lt_of_decidable_le\n\nprotected instance has_lt.lt.is_incomp_trans {\u03b1 : Type u_1} [linear_order \u03b1] : is_incomp_trans (finset.colex \u03b1) Less :=\n  is_incomp_trans.mk\n    fun (A B C : finset.colex \u03b1) (\u1fb0 : \u00acA < B \u2227 \u00acB < A) (\u1fb0_1 : \u00acB < C \u2227 \u00acC < B) =>\n      and.dcases_on \u1fb0\n        fun (nAB : \u00acA < B) (nBA : \u00acB < A) =>\n          and.dcases_on \u1fb0_1\n            fun (nBC : \u00acB < C) (nCB : \u00acC < B) =>\n              eq.mpr\n                (id\n                  (Eq._oldrec (Eq.refl (\u00acA < C \u2227 \u00acC < A))\n                    (or.resolve_right (or.resolve_left (lt_trichotomy A B) nAB) nBA)))\n                (eq.mpr\n                  (id\n                    (Eq._oldrec (Eq.refl (\u00acB < C \u2227 \u00acC < B))\n                      (or.resolve_right (or.resolve_left (lt_trichotomy B C) nBC) nCB)))\n                  (eq.mpr (id (Eq._oldrec (Eq.refl (\u00acC < C \u2227 \u00acC < C)) (propext (and_self (\u00acC < C))))) (irrefl C)))\n\nprotected instance has_lt.lt.is_strict_weak_order {\u03b1 : Type u_1} [linear_order \u03b1] : is_strict_weak_order (finset.colex \u03b1) Less :=\n  is_strict_weak_order.mk\n\nprotected instance has_lt.lt.is_strict_total_order {\u03b1 : Type u_1} [linear_order \u03b1] : is_strict_total_order (finset.colex \u03b1) Less :=\n  is_strict_total_order.mk\n\n/-- If {r} is less than or equal to s in the colexicographical sense,\n  then s contains an element greater than or equal to r. -/\ntheorem mem_le_of_singleton_le {\u03b1 : Type u_1} [linear_order \u03b1] {r : \u03b1} {s : finset \u03b1} : finset.to_colex (singleton r) \u2264 finset.to_colex s \u2192 \u2203 (x : \u03b1), \u2203 (H : x \u2208 s), r \u2264 x := sorry\n\n/-- s.to_colex < finset.to_colex {r} iff all elements of s are less than r. -/\ntheorem lt_singleton_iff_mem_lt {\u03b1 : Type u_1} [linear_order \u03b1] {r : \u03b1} {s : finset \u03b1} : finset.to_colex s < finset.to_colex (singleton r) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 x < r := sorry\n\n/-- Colex is an extension of the base ordering on \u03b1. -/\ntheorem singleton_lt_iff_lt {\u03b1 : Type u_1} [linear_order \u03b1] {r : \u03b1} {s : \u03b1} : finset.to_colex (singleton r) < finset.to_colex (singleton s) \u2194 r < s := sorry\n\n/--\nIf A is before B in colex, and everything in B is small, then everything in A is small.\n-/\ntheorem forall_lt_of_colex_lt_of_forall_lt {\u03b1 : Type u_1} [linear_order \u03b1] {A : finset \u03b1} {B : finset \u03b1} (t : \u03b1) (h\u2081 : finset.to_colex A < finset.to_colex B) (h\u2082 : \u2200 (x : \u03b1), x \u2208 B \u2192 x < t) (x : \u03b1) (H : x \u2208 A) : x < t := sorry\n\n/-- Colex doesn't care if you remove the other set -/\n@[simp] theorem sdiff_lt_sdiff_iff_lt {\u03b1 : Type u_1} [HasLess \u03b1] [DecidableEq \u03b1] (A : finset \u03b1) (B : finset \u03b1) : finset.to_colex (A \\ B) < finset.to_colex (B \\ A) \u2194 finset.to_colex A < finset.to_colex B := sorry\n\n/-- For subsets of \u2115, we can show that colex is equivalent to binary. -/\ntheorem sum_pow_two_lt_iff_lt (A : finset \u2115) (B : finset \u2115) : finset.sum A (pow (bit0 1)) < finset.sum B (pow (bit0 1)) \u2194 finset.to_colex A < finset.to_colex B := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/combinatorics/colex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.6619228825191872, "lm_q1q2_score": 0.3897991813703767}}
{"text": "import data.real.basic data.num.lemmas\n\nsection\n\nlocal attribute [semireducible] reflected\n\nmeta instance rat.reflect : has_reflect \u211a\n| \u27e8n, d, _, _\u27e9 := `(rat.mk_nat %%(reflect n) %%(reflect d))\n\nend\n\nmeta def tactic.interactive.intros' : tactic unit :=\n`[repeat {intro}, resetI]\n\nattribute [elim_cast] znum.cast_inj\nattribute [squash_cast] znum.to_of_int\nattribute [squash_cast] znum.cast_zero\nattribute [move_cast] znum.cast_add\n--TODO\n\nnamespace list\n\ntheorem filter_perm {\u03b1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : list \u03b1} :\n  l ~ l.filter p ++ l.filter (not \u2218 p) :=\nbegin\n  induction l with x xs ih,\n  { simp },\n  { by_cases hx : p x,\n    { simp [filter, hx, perm.skip, ih] },\n    { calc\n      x::xs ~ x::(filter p xs ++ filter (not \u2218 p) xs) : perm.skip _ ih\n      ... ~ filter p xs ++ x::filter (not \u2218 p) xs : perm.symm perm_middle\n      ... ~ filter p (x::xs) ++ filter (not \u2218 p) (x::xs) : by simp [hx] }}\nend\n\ntheorem prod_ones {\u03b1} [monoid \u03b1] {l : list \u03b1} :\n  (\u2200 x : \u03b1, x \u2208 l \u2192 x = 1) \u2192 l.prod = 1 :=\nbegin\n  intro h,\n  induction l with x xs ih,\n  { refl },\n  { have h1 : x = 1, by { apply h, simp },\n    have h2 : prod xs = 1, by { apply ih, intros _ hx, apply h, simp [hx] },\n    simp [h1, h2] }\nend\n\ntheorem sum_zeros {\u03b1} [add_monoid \u03b1] {l : list \u03b1} :\n  (\u2200 x : \u03b1, x \u2208 l \u2192 x = 0) \u2192 l.sum = 0 :=\nbegin\n  intro h,\n  induction l with x xs ih,\n  { refl },\n  { have h1 : x = 0, by { apply h, simp },\n    have h2 : sum xs = 0, by { apply ih, intros _ hx, apply h, simp [hx] },\n    simp [h1, h2] }\nend\n\nend list\n\nnamespace polya.field\n\nstructure dict (\u03b1 : Type) :=\n(val : num \u2192 \u03b1)\n\nclass morph (\u03b3 : Type) [discrete_field \u03b3] (\u03b1 : Type) [discrete_field \u03b1] :=\n(cast   : has_coe \u03b3 \u03b1)\n(morph_zero : ((0 : \u03b3) : \u03b1) = 0)\n(morph_one : ((1 : \u03b3) : \u03b1) = 1)\n(morph_add : \u2200 a b : \u03b3, ((a + b : \u03b3) : \u03b1) = a + b)\n(morph_neg : \u2200 a : \u03b3, ((-a : \u03b3) : \u03b1) = -a)\n(morph_mul : \u2200 a b : \u03b3, ((a * b : \u03b3) : \u03b1) = a * b)\n(morph_inv : \u2200 a : \u03b3, ((a\u207b\u00b9 : \u03b3) : \u03b1) = a\u207b\u00b9)\n(morph_inj : \u2200 a : \u03b3, (a : \u03b1) = 0 \u2192 a = 0)\n\nnamespace morph\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [discrete_field \u03b3]\nvariables [morph \u03b3 \u03b1]\nvariables {a b : \u03b3}\n\ninstance has_coe : has_coe \u03b3 \u03b1 := morph.cast \u03b3 \u03b1\n\n@[simp, squash_cast] theorem morph_zero' : ((0 : \u03b3) : \u03b1) = 0 := by apply morph.morph_zero\n@[simp, squash_cast] theorem morph_one'  : ((1 : \u03b3) : \u03b1) = 1 := by apply morph.morph_one\n\n@[simp, move_cast] theorem morph_add' : ((a + b : \u03b3) : \u03b1) = a + b := by apply morph_add\n@[simp, move_cast] theorem morph_neg' : ((-a : \u03b3) : \u03b1) = -a       := by apply morph_neg\n@[simp, move_cast] theorem morph_mul' : ((a * b : \u03b3) : \u03b1) = a * b := by apply morph_mul\n@[simp, move_cast] theorem morph_inv' : ((a\u207b\u00b9 : \u03b3) : \u03b1) = a\u207b\u00b9     := by apply morph_inv\n\n@[simp, move_cast] theorem morph_sub : ((a - b : \u03b3) : \u03b1) = a - b :=\nby rw [sub_eq_add_neg, morph.morph_add, morph.morph_neg, \u2190 sub_eq_add_neg]\n\n@[simp, elim_cast] theorem morph_inj' : (a : \u03b1) = b \u2194 a = b :=\nbegin\n  apply iff.intro,\n  { intro h, apply eq_of_sub_eq_zero,\n    apply morph.morph_inj (a - b),\n    rw morph.morph_sub,\n    apply sub_eq_zero_of_eq,\n    apply h },\n  { intro h, subst h }\nend\n\n@[simp, move_cast] theorem morph_div : ((a / b : \u03b3) : \u03b1) = a / b :=\nby rw [division_def, morph.morph_mul, morph.morph_inv, \u2190 division_def]\n\n@[simp, move_cast] theorem morph_pow_nat {n : \u2115} : ((a ^ n : \u03b3) : \u03b1) = a ^ n :=\nbegin\n  induction n with _ ih,\n  { rw [pow_zero, pow_zero, morph.morph_one] },\n  { by_cases ha : a = 0,\n    { rw [ha, morph.morph_zero, zero_pow, zero_pow],\n      { apply morph.morph_zero },\n      { apply nat.succ_pos },\n      { apply nat.succ_pos }},\n    { rw [pow_succ, morph.morph_mul, ih, \u2190 pow_succ] }}\nend\n\n@[simp, move_cast] theorem morph_pow {n : \u2124} : ((a ^ n : \u03b3) : \u03b1) = a ^ n :=\nbegin\n  cases n,\n  { rw [int.of_nat_eq_coe, fpow_of_nat, fpow_of_nat],\n    apply morph_pow_nat },\n  { rw [int.neg_succ_of_nat_coe, fpow_neg, fpow_neg],\n    rw [morph_div, morph.morph_one],\n    rw [fpow_of_nat, fpow_of_nat],\n    rw morph_pow_nat }\nend\n\n@[simp, squash_cast] theorem morph_nat {n : \u2115} : ((n : \u03b3) : \u03b1) = (n : \u03b1) :=\nby { induction n with n ih, { simp }, { simp [ih] } }\n\n@[simp, squash_cast] theorem morph_num {n : num} : ((n : \u03b3) : \u03b1) = (n : \u03b1) :=\nby rw [\u2190 num.cast_to_nat, \u2190 num.cast_to_nat, morph_nat, num.cast_to_nat, num.cast_to_nat]\n\nend morph\n\nclass const_space (\u03b3 : Type) : Type :=\n(df : discrete_field \u03b3)\n(lt : \u03b3 \u2192 \u03b3 \u2192 Prop)\n(dec : decidable_rel lt)\n\nnamespace const_space\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\n\ninstance : discrete_field \u03b3 := const_space.df \u03b3\ninstance : has_lt \u03b3 := \u27e8const_space.lt\u27e9\ninstance : decidable_rel ((<) : \u03b3 \u2192 \u03b3 \u2192 Prop) := const_space.dec \u03b3\n\nend const_space\n\n@[derive decidable_eq, derive has_reflect]\ninductive nterm (\u03b3 : Type) [const_space \u03b3] : Type\n| atom  {} : num \u2192 nterm\n| const {} : \u03b3 \u2192 nterm\n| add   {} : nterm \u2192 nterm \u2192 nterm\n| mul   {} : nterm \u2192 nterm \u2192 nterm\n| pow   {} : nterm \u2192 znum \u2192 nterm\n\nnamespace nterm\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ninstance : inhabited (nterm \u03b3) := \u27e8const 0\u27e9\n\ndef blt :\n  nterm \u03b3 \u2192 nterm \u03b3 \u2192 bool\n| (const a) (const b) := a < b\n| (const _) _         := tt\n| _         (const _) := ff\n| (atom i)  (atom j)  := i < j\n| (atom _)  _         := tt\n| _         (atom _)  := ff\n| (add x y) (add z w) := if y = w then blt x z else blt y w\n| (add _ _) _         := tt\n| _         (add _ _) := ff\n| (mul x y) (mul z w) := if y = w then blt x z else blt y w\n| (mul _ _) _         := tt\n| _         (mul _ _) := ff\n| (pow x n) (pow y m) := if x = y then n < m else blt x y\n\ndef lt : nterm \u03b3 \u2192 nterm \u03b3 \u2192 Prop := \u03bb x y, blt x y\ninstance : has_lt (nterm \u03b3) := \u27e8lt\u27e9\ninstance dec_lt : decidable_rel ((<) : nterm \u03b3 \u2192 nterm \u03b3 \u2192 Prop) := by dsimp [has_lt.lt, lt]; apply_instance\n\ndef eval (\u03c1 : dict \u03b1) : nterm \u03b3 \u2192 \u03b1\n| (atom i)  := \u03c1.val i\n| (const c) := \u2191c\n| (add x y) := eval x + eval y\n| (mul x y) := eval x * eval y\n| (pow x n) := eval x ^ (n : \u2124)\n\ninstance coe_atom : has_coe num (nterm \u03b3) := \u27e8atom\u27e9\ninstance coe_const: has_coe \u03b3 (nterm \u03b3) := \u27e8const\u27e9\ninstance : has_zero (nterm \u03b3) := \u27e8mul (const 1) (const 0)\u27e9\ninstance : has_one (nterm \u03b3) := \u27e8mul (const 1) (const 1)\u27e9\ninstance : has_add (nterm \u03b3) := \u27e8add\u27e9\ninstance : has_mul (nterm \u03b3) := \u27e8mul\u27e9\ninstance : has_pow (nterm \u03b3) znum := \u27e8pow\u27e9\ninstance pow_int : has_pow (nterm \u03b3) \u2124 := \u27e8\u03bb x n, x.pow (n : znum)\u27e9\ninstance pow_nat : has_pow (nterm \u03b3) \u2115 := \u27e8\u03bb (x : nterm \u03b3) (n : \u2115), x ^ (n : \u2124)\u27e9\n\ndef neg (x : nterm \u03b3) : nterm \u03b3 := x * (-1 : \u03b3)\ninstance : has_neg (nterm \u03b3) := \u27e8neg\u27e9\ndef sub (x y : nterm \u03b3) : nterm \u03b3 := x + (-y)\ninstance : has_sub (nterm \u03b3) := \u27e8sub\u27e9\ndef inv (x : nterm \u03b3) : nterm \u03b3 := pow x (-1)\ninstance : has_inv (nterm \u03b3) := \u27e8inv\u27e9\ndef div (x y : nterm \u03b3) : nterm \u03b3 := x * y\u207b\u00b9\ninstance : has_div (nterm \u03b3) := \u27e8div\u27e9\n\nsection\nvariables {x y : nterm \u03b3} {i : num} {c : \u03b3}\n@[simp] theorem eval_zero :  eval \u03c1 (0 : nterm \u03b3) = 0       := by sorry\n@[simp] theorem eval_one :   eval \u03c1 (1 : nterm \u03b3) = 1       := by sorry\n@[simp] theorem eval_atom :  eval \u03c1 (i : nterm \u03b3) = \u03c1.val i := rfl\n@[simp] theorem eval_const : eval \u03c1 (c : nterm \u03b3) = (c : \u03b1) := rfl\n\n@[simp] theorem eval_add : eval \u03c1 (x + y) = eval \u03c1 x + eval \u03c1 y := rfl\n@[simp] theorem eval_mul : eval \u03c1 (x * y) = eval \u03c1 x * eval \u03c1 y := rfl\n\n@[simp] theorem eval_pow_int {n : \u2124} : eval \u03c1 (x ^ n) = eval \u03c1 x ^ n := by sorry\n@[simp] theorem eval_pow_nat {n : \u2115} : eval \u03c1 (x ^ n) = eval \u03c1 x ^ n := eval_pow_int\n@[simp] theorem eval_pow {n : znum} : eval \u03c1 (x ^ n) = eval \u03c1 x ^ (n : \u2124) := by sorry\n\n@[simp] theorem eval_neg : (-x).eval \u03c1 = - x.eval \u03c1 :=\ncalc\neval \u03c1 (-x)\n    = eval \u03c1 (neg x) : rfl\n... = - eval \u03c1 x     : by simp [neg, morph.morph_neg', morph.morph_one']\n\n@[simp] theorem eval_sub : eval \u03c1 (x - y) = eval \u03c1 x - eval \u03c1 y :=\ncalc\neval \u03c1 (x - y)\n    = eval \u03c1 (sub x y)    : rfl\n... = eval \u03c1 x - eval \u03c1 y : by simp [sub, sub_eq_add_neg]\n\n@[simp] theorem eval_inv : eval \u03c1 (x\u207b\u00b9) = (eval \u03c1 x)\u207b\u00b9 :=\ncalc\neval \u03c1 (x\u207b\u00b9)\n    = eval \u03c1 (inv x)        : rfl\n... = (eval \u03c1 x) ^ (-1 : \u2124) : by simp [inv, eval]\n... = (eval \u03c1 x)\u207b\u00b9          : fpow_inv _\n\n@[simp] theorem eval_div : eval \u03c1 (x / y) = eval \u03c1 x / eval \u03c1 y :=\ncalc\neval \u03c1 (x / y)\n    = eval \u03c1 (div x y)    : rfl\n... = eval \u03c1 x / eval \u03c1 y : by simp [div, div_eq_mul_inv]\n\nend\n\nmeta def to_str [has_to_string \u03b3] : (nterm \u03b3) \u2192 string\n| (atom i)  := \"#\" ++ to_string (i : \u2115)\n| (const c) := \"(\" ++ to_string c ++ \")\"\n| (add x y) := \"(\" ++ to_str x ++ \" + \" ++ to_str y ++ \")\"\n| (mul x y) := \"(\" ++ to_str x ++ \" * \" ++ to_str y ++ \")\"\n| (pow x n) := to_str x ++ \" ^ \" ++ to_string (n : \u2124)\n\nmeta instance [has_to_string \u03b3] : has_to_string (nterm \u03b3) := \u27e8to_str\u27e9\nmeta instance [has_to_string \u03b3] : has_to_tactic_format (nterm \u03b3) := \u27e8\u03bb x, return (to_str x : format)\u27e9\n\ndef sum : list (nterm \u03b3) \u2192 nterm \u03b3\n| []      := const (0 : \u03b3)\n| [x]     := x\n| (x::xs) := add (sum xs) x\n\ndef prod : list (nterm \u03b3) \u2192 nterm \u03b3\n| []      := const (1 : \u03b3) \n| [x]     := x\n| (x::xs) := mul (prod xs) x\n\ntheorem eval_sum (xs : list (nterm \u03b3)) :\n  (sum xs).eval \u03c1 = list.sum (xs.map (nterm.eval \u03c1)) :=\nbegin\n  induction xs with x0 xs ih,\n  { simp [sum, eval] },\n  { cases xs with x1 xs,\n    { simp [sum, eval] },\n    { simp [sum, eval, ih] }}\nend\n\ntheorem eval_prod (xs : list (nterm \u03b3)) :\n  (prod xs).eval \u03c1 = list.prod (xs.map (nterm.eval \u03c1)) :=\nbegin\n  induction xs with x0 xs ih,\n  { simp [prod, eval] },\n  { cases xs with x1 xs,\n    { simp [prod, eval] },\n    { simp only [prod, list.map_cons, list.prod_cons, eval, ih],\n      rw mul_comm }}\nend\n\ndef scale (a : \u03b3) : nterm \u03b3 \u2192 nterm \u03b3\n| (mul x (const b)) := mul x (const (b * a))\n| (const b) := (const (b * a))\n| x := mul x (const a)\n\ndef coeff : nterm \u03b3 \u2192 \u03b3\n| (mul x (const a)) := a\n| (const a) := a\n| x := 1\n\ndef term : nterm \u03b3 \u2192 nterm \u03b3\n| (mul x (const a)) := x\n| (const _) := 1\n| x := x\n\n@[simp] theorem eval_scale {a : \u03b3} {x : nterm \u03b3} :\n  eval \u03c1 (x.scale a) = eval \u03c1 x * a :=\nbegin\n  cases x,\n  case mul : x y {\n    cases y,\n    case const : b { simp [scale, eval, mul_assoc] },\n    repeat { simp [scale, eval] }},\n  case const : b { simp [scale, eval] },\n  repeat { simp [scale, eval] }\nend\n\ntheorem eval_term_coeff (x : nterm \u03b3) : eval \u03c1 x = eval \u03c1 x.term * x.coeff :=\nbegin\n  cases x,\n  case mul : x y {\n    cases y,\n    case const : b { simp [term, coeff, eval, mul_assoc] },\n    repeat { simp [term, coeff, eval] }},\n  case const : b { simp [term, coeff, eval] },\n  repeat { simp [term, coeff, eval] }\nend\n\ndef exp : nterm \u03b3 \u2192 znum\n| (pow _ n) := n\n| _ := 1\n\ndef mem : nterm \u03b3 \u2192 nterm \u03b3\n| (pow x _) := x\n| x := x\n\ntheorem eval_mem_exp (x : nterm \u03b3) : eval \u03c1 x = eval \u03c1 (mem x) ^ (exp x : \u2124) :=\nbegin\n  cases x,\n  case pow : x n { dsimp [mem, exp, eval], refl },\n  repeat { dsimp [mem, exp, eval], rw fpow_one }\nend\n\n--theorem eval_mem_zero {x : nterm \u03b3} : eval \u03c1 x = 0 \u2192 eval \u03c1 (mem x) = 0 :=\n--begin\n--  intro h1, cases x,\n--  case pow : x n { unfold mem, by_contradiction h2, exact fpow_ne_zero_of_ne_zero h2 _ h1 },\n--  repeat { exact h1 },\n--end\n\ndef pow_mul (n : znum) (x : nterm \u03b3) : nterm \u03b3 :=\nif n = 0 then\n  const 1\nelse if x.exp * n = 1 then\n  x.mem\nelse\n  pow x.mem (x.exp * n)\n\ndef pow_div (n : znum) (x : nterm \u03b3) : nterm \u03b3 :=\nif n = x.exp then\n  x.mem\nelse\n  pow x.mem (x.exp / n)\n\n@[simp] theorem eval_pow_mul {n : znum} {x : nterm \u03b3} : eval \u03c1 (pow_mul n x) = eval \u03c1 x ^ (n : \u2124) :=\nbegin\n  unfold pow_mul,\n  by_cases h1 : n = 0,\n  { simp [eval, h1] },\n  { by_cases h2 : x.exp * n = 1,\n    { rw [if_neg h1, if_pos h2, eval_mem_exp x],\n      rw [\u2190 fpow_mul, \u2190 znum.cast_mul, h2],\n      simp },\n    { rw [if_neg h1, if_neg h2], unfold eval,\n      rw [znum.cast_mul, fpow_mul, \u2190 eval_mem_exp]}}\nend\n\n@[simp] theorem eval_pow_div {n : znum} {x : nterm \u03b3} : n \u2223 x.exp \u2192 eval \u03c1 (pow_div n x) ^ (n : \u2124) = eval \u03c1 x :=\nbegin\n  intro h1, cases h1 with d h1,\n  unfold pow_div,\n  by_cases h2 : n = exp x,\n  { rw [if_pos h2, h2, \u2190 eval_mem_exp] },\n  { by_cases h3 : n = 0,\n    { apply absurd _ h2, have : exp x = 0, { rw h1, simp [h3] }, rw [h3, this] },\n    { rw [if_neg h2, h1], unfold eval,\n      rw [znum.div_to_int, znum.cast_mul, int.mul_div_cancel_left],\n      { rw [\u2190 fpow_mul, int.mul_comm, \u2190 znum.cast_mul, \u2190 h1, \u2190 eval_mem_exp] },\n      { rw [\u2190 znum.cast_zero], exact_mod_cast h3 }}}\nend\n\ndef nonzero (\u03c1 : dict \u03b1) (ts : list (nterm \u03b3)) : Prop := \u2200 t \u2208 ts, nterm.eval \u03c1 t \u2260 0\n\ntheorem nonzero_union {xs ys : list (nterm \u03b3)} :\nnonzero \u03c1 (xs \u222a ys) \u2194 nonzero \u03c1 xs \u2227 nonzero \u03c1 ys :=\nbegin\n  apply iff.intro,\n  { intro h1, split; { intros _ h2, apply h1, simp [h2] }},\n  { intros h1 t ht, cases h1 with h1 h2, rw list.mem_union at ht, cases ht,\n    {apply h1, apply ht}, {apply h2, apply ht}}\nend\n\ntheorem nonzero_subset {xs ys : list (nterm \u03b3)} :\n  xs \u2286 ys \u2192 nonzero \u03c1 ys \u2192 nonzero \u03c1 xs :=\nbegin\n  intros h1 h2, intros x hx,\n  apply h2, apply h1, apply hx\nend\n\ntheorem nonzero_iff_zero_not_mem (ts : list (nterm \u03b3)) :\nnonzero \u03c1 ts \u2194 (0 : \u03b1) \u2209 ts.map (nterm.eval \u03c1) :=\nbegin\n  apply iff.intro,\n  { intro h, simpa using h },\n  { intro h, simp at h, apply h }\nend\n\nend nterm\n\nset_option trace.app_builder true\n\n@[derive decidable_eq]\ninductive Term (\u03b3 : Type) [const_space \u03b3] : bool \u2192 Type\n| zero {} : Term tt\n| one {} : Term ff\n| sform {} : Term tt \u2192 Term ff \u2192 \u03b3 \u2192 Term tt\n| pform {} : Term ff \u2192 Term tt \u2192 znum \u2192 znum \u2192 Term ff\n\nnamespace Term\n\nopen nterm\n\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\n--def blt : \u03a0 {a b : bool}, Term \u03b3 a \u2192 Term \u03b3 b \u2192 bool\n--| .(_) .(_) zero            zero            := ff\n--| .(_) _    zero            _               := tt\n--| .(_) .(_) one             one             := ff\n--| .(_) _    one             _               := tt\n--| .(_) .(_) (sform x y a)   (sform u v b)   := blt y v \u2228 (y = v \u2227 blt x u) \u2228 (y = v \u2227 x = u \u2227 a < b)\n--| .(_) _    (sform _ _ _)   _               := tt\n--| .(_) .(_) (pform x y n m) (pform u v i j) := blt y v \u2228 (y = v \u2227 blt x u) \u2228 (y = v \u2227 x = u \u2227 (n, m) < (i, j))\n\ndef eval (\u03c1 : dict \u03b1) : \u03a0 {b : bool}, Term \u03b3 b \u2192 \u03b1\n| .(tt) zero := 0\n| .(ff) one := 1\n| .(tt) (sform x y a) := (eval x + eval y ) * a\n| .(ff) (pform x y n m) := (eval x * (eval y ^ (n : \u2124))) ^ (m : \u2124)\n\ndef to_nterm : \u03a0 {b : bool}, Term \u03b3 b \u2192 nterm \u03b3\n| .(tt) zero := (const 0)\n| .(ff) one := (const 1)\n| .(tt) (sform x y a) := mul (add (to_nterm x) (to_nterm y)) (const a)\n| .(ff) (pform x y n m) := nterm.pow (mul (to_nterm x) (nterm.pow (to_nterm y) n)) m\n\ntheorem correctness {\u03c1 : dict \u03b1} {b : bool} {t : Term \u03b3 b} : eval \u03c1 t = nterm.eval \u03c1 (to_nterm t) :=\nbegin\n  induction t with x y a ihx ihy x y n m ihx ihy,\n  { simp [to_nterm, eval, nterm.eval] },\n  { simp [to_nterm, eval, nterm.eval] },\n  { simp [to_nterm, eval, nterm.eval, ihx, ihy] },\n  { simp [to_nterm, eval, nterm.eval, ihx, ihy] },\nend\n\ndef scale : \u03b3 \u2192 Term \u03b3 tt \u2192 Term \u03b3 tt\n| _ zero := zero\n| b (sform x y a) := sform x y (a * b)\n\ntheorem eval_scale {b : \u03b3} {x : Term \u03b3 tt} : eval \u03c1 (scale b x) = eval \u03c1 x * (b : \u03b1) :=\nbegin\n  cases x with x y a,\n  { simp [scale, eval] },\n  { simp [scale, eval, add_mul, mul_assoc] }\nend\n\ndef add : Term \u03b3 tt \u2192 Term \u03b3 tt \u2192 Term \u03b3 tt := sorry\n\nend Term\n\n\nend polya.field\n", "meta": {"author": "lean-forward", "repo": "field", "sha": "7e2127ad485aec25e58a1b9c82a6bb74a599467a", "save_path": "github-repos/lean/lean-forward-field", "path": "github-repos/lean/lean-forward-field/field-7e2127ad485aec25e58a1b9c82a6bb74a599467a/src/data/polya/field/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6261241702517976, "lm_q1q2_score": 0.38973683226425004}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.discrete_category\n\n/-!\n# The empty category\n\nDefines a category structure on `pempty`, and the unique functor `pempty \u2964 C` for any category `C`.\n-/\n\nuniverses w v u -- morphism levels before object levels. See note [category_theory universes].\n\nnamespace category_theory\nnamespace functor\n\nvariables (C : Type u) [category.{v} C]\n\n/-- Equivalence between two empty categories. -/\ndef empty_equivalence : discrete.{w} pempty \u224c discrete.{v} pempty :=\nequivalence.mk\n{ obj := pempty.elim, map := \u03bb x, x.elim }\n{ obj := pempty.elim, map := \u03bb x, x.elim }\n(by tidy) (by tidy)\n\n/-- The canonical functor out of the empty category. -/\ndef empty : discrete.{w} pempty \u2964 C := discrete.functor pempty.elim\n\nvariable {C}\n/-- Any two functors out of the empty category are isomorphic. -/\ndef empty_ext (F G : discrete.{w} pempty \u2964 C) : F \u2245 G :=\ndiscrete.nat_iso (\u03bb x, pempty.elim x)\n\n/--\nAny functor out of the empty category is isomorphic to the canonical functor from the empty\ncategory.\n-/\ndef unique_from_empty (F : discrete.{w} pempty \u2964 C) : F \u2245 empty C :=\nempty_ext _ _\n\n/--\nAny two functors out of the empty category are *equal*. You probably want to use\n`empty_ext` instead of this.\n-/\nlemma empty_ext' (F G : discrete.{w} pempty \u2964 C) : F = G :=\nfunctor.ext (\u03bb x, x.elim) (\u03bb x _ _, x.elim)\n\nend functor\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/pempty.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.38969221332996323}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport category_theory.localization.construction\n\n/-!\n\n# Predicate for localized categories\n\nIn this file, a predicate `L.is_localization W` is introduced for a functor `L : C \u2964 D`\nand `W : morphism_property C`: it expresses that `L` identifies `D` with the localized\ncategory of `C` with respect to `W` (up to equivalence).\n\nWe introduce a universal property `strict_universal_property_fixed_target L W E` which\nstates that `L` inverts the morphisms in `W` and that all functors `C \u2964 E` inverting\n`W` uniquely factors as a composition of `L \u22d9 G` with `G : D \u2964 E`. Such universal\nproperties are inputs for the constructor `is_localization.mk'` for `L.is_localization W`.\n\nWhen `L : C \u2964 D` is a localization functor for `W : morphism_property` (i.e. when\n`[L.is_localization W]` holds), for any category `E`, there is\nan equivalence `functor_equivalence L W E : (D \u2964 E) \u224c (W.functors_inverting E)`\nthat is induced by the composition with the functor `L`. When two functors\n`F : C \u2964 E` and `F' : D \u2964 E` correspond via this equivalence, we shall say\nthat `F'` lifts `F`, and the associated isomorphism `L \u22d9 F' \u2245 F` is the\ndatum that is part of the class `lifting L W F F'`. The functions\n`lift_nat_trans` and `lift_nat_iso` can be used to lift natural transformations\nand natural isomorphisms between functors.\n\n-/\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category\n\nvariables {C D : Type*} [category C] [category D]\n  (L : C \u2964 D) (W : morphism_property C)\n  (E : Type*) [category E]\n\nnamespace functor\n\n/-- The predicate expressing that, up to equivalence, a functor `L : C \u2964 D`\nidentifies the category `D` with the localized category of `C` with respect\nto `W : morphism_property C`. -/\nclass is_localization : Prop :=\n(inverts : W.is_inverted_by L)\n(nonempty_is_equivalence : nonempty (is_equivalence (localization.construction.lift L inverts)))\n\ninstance Q_is_localization : W.Q.is_localization W :=\n{ inverts := W.Q_inverts,\n  nonempty_is_equivalence := begin\n    suffices : localization.construction.lift W.Q W.Q_inverts = \ud835\udfed _,\n    { apply nonempty.intro, rw this, apply_instance, },\n    apply localization.construction.uniq,\n    simpa only [localization.construction.fac],\n  end, }\n\nend functor\n\nnamespace localization\n\n/-- This universal property states that a functor `L : C \u2964 D` inverts morphisms\nin `W` and the all functors `D \u2964 E` (for a fixed category `E`) uniquely factors\nthrough `L`. -/\nstructure strict_universal_property_fixed_target :=\n(inverts : W.is_inverted_by L)\n(lift : \u03a0 (F : C \u2964 E) (hF : W.is_inverted_by F), D \u2964 E)\n(fac : \u03a0 (F : C \u2964 E) (hF : W.is_inverted_by F), L \u22d9 lift F hF = F)\n(uniq : \u03a0 (F\u2081 F\u2082 : D \u2964 E) (h : L \u22d9 F\u2081 = L \u22d9 F\u2082), F\u2081 = F\u2082)\n\n/-- The localized category `W.localization` that was constructed satisfies\nthe universal property of the localization. -/\n@[simps]\ndef strict_universal_property_fixed_target_Q :\n  strict_universal_property_fixed_target W.Q W E :=\n{ inverts := W.Q_inverts,\n  lift := construction.lift,\n  fac := construction.fac,\n  uniq := construction.uniq, }\n\ninstance : inhabited (strict_universal_property_fixed_target W.Q W E) :=\n\u27e8strict_universal_property_fixed_target_Q _ _\u27e9\n\n/-- When `W` consists of isomorphisms, the identity satisfies the universal property\nof the localization. -/\n@[simps]\ndef strict_universal_property_fixed_target_id (hW : W \u2286 morphism_property.isomorphisms C):\n  strict_universal_property_fixed_target (\ud835\udfed C) W E :=\n{ inverts := \u03bb X Y f hf, hW f hf,\n  lift := \u03bb F hF, F,\n  fac := \u03bb F hF, by { cases F, refl, },\n  uniq := \u03bb F\u2081 F\u2082 eq, by { cases F\u2081, cases F\u2082, exact eq, }, }\n\nend localization\n\nnamespace functor\n\nlemma is_localization.mk'\n  (h\u2081 : localization.strict_universal_property_fixed_target L W D)\n  (h\u2082 : localization.strict_universal_property_fixed_target L W W.localization) :\n  is_localization L W :=\n{ inverts := h\u2081.inverts,\n  nonempty_is_equivalence := nonempty.intro\n  { inverse := h\u2082.lift W.Q W.Q_inverts,\n    unit_iso := eq_to_iso (localization.construction.uniq _ _\n      (by simp only [\u2190 functor.assoc, localization.construction.fac, h\u2082.fac, functor.comp_id])),\n    counit_iso := eq_to_iso (h\u2081.uniq _ _ (by simp only [\u2190 functor.assoc, h\u2082.fac,\n      localization.construction.fac, functor.comp_id])),\n    functor_unit_iso_comp' := \u03bb X, by simpa only [eq_to_iso.hom, eq_to_hom_app,\n      eq_to_hom_map, eq_to_hom_trans, eq_to_hom_refl], }, }\n\nlemma is_localization.for_id (hW : W \u2286 morphism_property.isomorphisms C):\n  (\ud835\udfed C).is_localization W :=\nis_localization.mk' _ _\n  (localization.strict_universal_property_fixed_target_id W _ hW)\n  (localization.strict_universal_property_fixed_target_id W _ hW)\n\nend functor\n\nnamespace localization\n\nvariable [L.is_localization W]\n\nlemma inverts : W.is_inverted_by L := (infer_instance : L.is_localization W).inverts\n\n/-- The isomorphism `L.obj X \u2245 L.obj Y` that is deduced from a morphism `f : X \u27f6 Y` which\nbelongs to `W`, when `L.is_localization W`. -/\n@[simps]\ndef iso_of_hom {X Y : C} (f : X \u27f6 Y) (hf : W f) : L.obj X \u2245 L.obj Y :=\nby { haveI : is_iso (L.map f) := inverts L W f hf, exact as_iso (L.map f), }\n\ninstance : is_equivalence (localization.construction.lift L (inverts L W)) :=\n(infer_instance : L.is_localization W).nonempty_is_equivalence.some\n\n/-- A chosen equivalence of categories `W.localization \u2245 D` for a functor\n`L : C \u2964 D` which satisfies `L.is_localization W`. This shall be used in\norder to deduce properties of `L` from properties of `W.Q`. -/\ndef equivalence_from_model : W.localization \u224c D :=\n(localization.construction.lift L (inverts L W)).as_equivalence\n\n/-- Via the equivalence of categories `equivalence_from_model L W : W.localization \u224c D`,\none may identify the functors `W.Q` and `L`. -/\ndef Q_comp_equivalence_from_model_functor_iso :\n  W.Q \u22d9 (equivalence_from_model L W).functor \u2245 L := eq_to_iso (construction.fac _ _)\n\n/-- Via the equivalence of categories `equivalence_from_model L W : W.localization \u224c D`,\none may identify the functors `L` and `W.Q`. -/\ndef comp_equivalence_from_model_inverse_iso :\n  L \u22d9 (equivalence_from_model L W).inverse \u2245 W.Q :=\ncalc L \u22d9 (equivalence_from_model L W).inverse \u2245 _ :\n  iso_whisker_right (Q_comp_equivalence_from_model_functor_iso L W).symm _\n... \u2245 W.Q \u22d9 ((equivalence_from_model L W).functor \u22d9 (equivalence_from_model L W).inverse) :\n  functor.associator _ _ _\n... \u2245 W.Q \u22d9 \ud835\udfed _ : iso_whisker_left _ ((equivalence_from_model L W).unit_iso.symm)\n... \u2245 W.Q : functor.right_unitor _\n\nlemma ess_surj : ess_surj L :=\n\u27e8\u03bb X, \u27e8(construction.obj_equiv W).inv_fun ((equivalence_from_model L W).inverse.obj X),\n  nonempty.intro ((Q_comp_equivalence_from_model_functor_iso L W).symm.app _ \u226a\u226b\n  (equivalence_from_model L W).counit_iso.app X)\u27e9\u27e9\n\n/-- The functor `(D \u2964 E) \u2964 W.functors_inverting E` induced by the composition\nwith a localization functor `L : C \u2964 D` with respect to `W : morphism_property C`. -/\ndef whiskering_left_functor : (D \u2964 E) \u2964 W.functors_inverting E :=\nfull_subcategory.lift _ ((whiskering_left _ _ E).obj L)\n  (morphism_property.is_inverted_by.of_comp W L (inverts L W ))\n\ninstance : is_equivalence (whiskering_left_functor L W E) :=\nbegin\n  refine is_equivalence.of_iso _ (is_equivalence.of_equivalence\n    ((equivalence.congr_left (equivalence_from_model L W).symm).trans\n    (construction.whiskering_left_equivalence W E))),\n  refine nat_iso.of_components (\u03bb F, eq_to_iso begin\n    ext,\n    change (W.Q \u22d9 (localization.construction.lift L (inverts L W))) \u22d9 F = L \u22d9 F,\n    rw construction.fac,\n  end)\n  (\u03bb F\u2081 F\u2082 \u03c4, begin\n    ext X,\n    dsimp [equivalence_from_model, whisker_left, construction.whiskering_left_equivalence,\n      construction.whiskering_left_equivalence.functor, whiskering_left_functor,\n      morphism_property.Q],\n    erw [nat_trans.comp_app, nat_trans.comp_app, eq_to_hom_app, eq_to_hom_app,\n      eq_to_hom_refl, eq_to_hom_refl, comp_id, id_comp],\n    all_goals\n    { change (W.Q \u22d9 (localization.construction.lift L (inverts L W))) \u22d9 _ = L \u22d9 _,\n      rw construction.fac, },\n  end),\nend\n\n/-- The equivalence of categories `(D \u2964 E) \u224c (W.functors_inverting E)` induced by\nthe composition with a localization functor `L : C \u2964 D` with respect to\n`W : morphism_property C`. -/\ndef functor_equivalence : (D \u2964 E) \u224c (W.functors_inverting E) :=\n(whiskering_left_functor L W E).as_equivalence\n\ninclude W\n\n/-- The functor `(D \u2964 E) \u2964 (C \u2964 E)` given by the composition with a localization\nfunctor `L : C \u2964 D` with respect to `W : morphism_property C`. -/\n@[nolint unused_arguments]\ndef whiskering_left_functor' :\n  (D \u2964 E) \u2964 (C \u2964 E) := (whiskering_left C D E).obj L\n\nlemma whiskering_left_functor'_eq :\n  whiskering_left_functor' L W E =\n    localization.whiskering_left_functor L W E \u22d9 induced_functor _ := rfl\n\nvariable {E}\n\n@[simp]\nlemma whiskering_left_functor'_obj\n  (F : D \u2964 E) : (whiskering_left_functor' L W E).obj F = L \u22d9 F := rfl\n\ninstance : full (whiskering_left_functor' L W E) :=\nby { rw whiskering_left_functor'_eq, apply_instance, }\n\ninstance : faithful (whiskering_left_functor' L W E) :=\nby { rw whiskering_left_functor'_eq, apply_instance, }\n\nlemma nat_trans_ext {F\u2081 F\u2082 : D \u2964 E} (\u03c4 \u03c4' : F\u2081 \u27f6 F\u2082)\n  (h : \u2200 (X : C), \u03c4.app (L.obj X) = \u03c4'.app (L.obj X)) : \u03c4 = \u03c4' :=\nbegin\n  haveI : category_theory.ess_surj L := ess_surj L W,\n  ext Y,\n  rw [\u2190 cancel_epi (F\u2081.map (L.obj_obj_preimage_iso Y).hom), \u03c4.naturality, \u03c4'.naturality, h],\nend\n\n/-- When `L : C \u2964 D` is a localization functor for `W : morphism_property C` and\n`F : C \u2964 E` is a functor, we shall say that `F' : D \u2964 E` lifts `F` if the obvious diagram\nis commutative up to an isomorphism. -/\nclass lifting (F : C \u2964 E) (F' : D \u2964 E) :=\n(iso [] : L \u22d9 F' \u2245 F)\n\nvariable {W}\n\n/-- Given a localization functor `L : C \u2964 D` for `W : morphism_property C` and\na functor `F : C \u2964 E` which inverts `W`, this is a choice of functor\n`D \u2964 E` which lifts `F`. -/\ndef lift (F : C \u2964 E) (hF : W.is_inverted_by F) (L : C \u2964 D) [hL : L.is_localization W] :\n  D \u2964 E :=\n(functor_equivalence L W E).inverse.obj \u27e8F, hF\u27e9\n\ninstance lifting_lift (F : C \u2964 E) (hF : W.is_inverted_by F) (L : C \u2964 D)\n  [hL : L.is_localization W] : lifting L W F (lift F hF L) :=\n\u27e8(induced_functor _).map_iso ((functor_equivalence L W E).counit_iso.app \u27e8F, hF\u27e9)\u27e9\n\n/-- The canonical isomorphism `L \u22d9 lift F hF L \u2245 F` for any functor `F : C \u2964 E`\nwhich inverts `W`, when `L : C \u2964 D` is a localization functor for `W`. -/\n@[simps]\ndef fac (F : C \u2964 E) (hF : W.is_inverted_by F) (L : C \u2964 D) [hL : L.is_localization W] :\n  L \u22d9 lift F hF L \u2245 F :=\nlifting.iso _ W _ _\n\ninstance lifting_construction_lift (F : C \u2964 D) (hF : W.is_inverted_by F) :\n  lifting W.Q W F (construction.lift F hF) :=\n\u27e8eq_to_iso (construction.fac F hF)\u27e9\n\nvariable (W)\n\n/-- Given a localization functor `L : C \u2964 D` for `W : morphism_property C`,\nif `(F\u2081' F\u2082' : D \u2964 E)` are functors which lifts functors `(F\u2081 F\u2082 : C \u2964 E)`,\na natural transformation `\u03c4 : F\u2081 \u27f6 F\u2082` uniquely lifts to a natural transformation `F\u2081' \u27f6 F\u2082'`. -/\ndef lift_nat_trans (F\u2081 F\u2082 : C \u2964 E) (F\u2081' F\u2082' : D \u2964 E) [lifting L W F\u2081 F\u2081']\n  [h\u2082 : lifting L W F\u2082 F\u2082'] (\u03c4 : F\u2081 \u27f6 F\u2082) : F\u2081' \u27f6 F\u2082' :=\n(whiskering_left_functor' L W E).preimage\n  ((lifting.iso L W F\u2081 F\u2081').hom \u226b \u03c4 \u226b (lifting.iso L W F\u2082 F\u2082').inv)\n\n@[simp]\nlemma lift_nat_trans_app (F\u2081 F\u2082 : C \u2964 E) (F\u2081' F\u2082' : D \u2964 E) [lifting L W F\u2081 F\u2081']\n  [lifting L W F\u2082 F\u2082'] (\u03c4 : F\u2081 \u27f6 F\u2082) (X : C) :\n  (lift_nat_trans L W F\u2081 F\u2082 F\u2081' F\u2082' \u03c4).app (L.obj X) =\n    (lifting.iso L W F\u2081 F\u2081').hom.app X \u226b \u03c4.app X \u226b ((lifting.iso L W F\u2082 F\u2082')).inv.app X :=\ncongr_app (functor.image_preimage (whiskering_left_functor' L W E) _) X\n\n@[simp, reassoc]\nlemma comp_lift_nat_trans (F\u2081 F\u2082 F\u2083 : C \u2964 E) (F\u2081' F\u2082' F\u2083' : D \u2964 E)\n  [h\u2081 : lifting L W F\u2081 F\u2081'] [h\u2082 : lifting L W F\u2082 F\u2082'] [h\u2083 : lifting L W F\u2083 F\u2083']\n  (\u03c4 : F\u2081 \u27f6 F\u2082) (\u03c4' : F\u2082 \u27f6 F\u2083) :\n  lift_nat_trans L W F\u2081 F\u2082 F\u2081' F\u2082' \u03c4 \u226b lift_nat_trans L W F\u2082 F\u2083 F\u2082' F\u2083' \u03c4' =\n  lift_nat_trans L W F\u2081 F\u2083 F\u2081' F\u2083' (\u03c4 \u226b \u03c4') :=\nnat_trans_ext L W _ _\n  (\u03bb X, by simp only [nat_trans.comp_app, lift_nat_trans_app, assoc, iso.inv_hom_id_app_assoc])\n\n@[simp]\nlemma lift_nat_trans_id (F : C \u2964 E) (F' : D \u2964 E) [h : lifting L W F F'] :\n  lift_nat_trans L W F F F' F' (\ud835\udfd9 F) = \ud835\udfd9 F' :=\nnat_trans_ext L W _ _\n  (\u03bb X, by simpa only [lift_nat_trans_app, nat_trans.id_app, id_comp, iso.hom_inv_id_app])\n\n/-- Given a localization functor `L : C \u2964 D` for `W : morphism_property C`,\nif `(F\u2081' F\u2082' : D \u2964 E)` are functors which lifts functors `(F\u2081 F\u2082 : C \u2964 E)`,\na natural isomorphism `\u03c4 : F\u2081 \u27f6 F\u2082` lifts to a natural isomorphism `F\u2081' \u27f6 F\u2082'`. -/\n@[simps]\ndef lift_nat_iso (F\u2081 F\u2082 : C \u2964 E) (F\u2081' F\u2082' : D \u2964 E)\n  [h\u2081 : lifting L W F\u2081 F\u2081'] [h\u2082 : lifting L W F\u2082 F\u2082']\n  (e : F\u2081 \u2245 F\u2082) : F\u2081' \u2245 F\u2082' :=\n{ hom := lift_nat_trans L W F\u2081 F\u2082 F\u2081' F\u2082' e.hom,\n  inv := lift_nat_trans L W F\u2082 F\u2081 F\u2082' F\u2081' e.inv, }\n\nnamespace lifting\n\n@[simps]\ninstance comp_right {E' : Type*} [category E'] (F : C \u2964 E) (F' : D \u2964 E) [lifting L W F F']\n  (G : E \u2964 E') : lifting L W (F \u22d9 G) (F' \u22d9 G) :=\n\u27e8iso_whisker_right (iso L W F F') G\u27e9\n\n@[simps]\ninstance id : lifting L W L (\ud835\udfed D) :=\n\u27e8functor.right_unitor L\u27e9\n\n/-- Given a localization functor `L : C \u2964 D` for `W : morphism_property C`,\nif `F\u2081' : D \u2964 E` lifts a functor `F\u2081 : C \u2964 D`, then a functor `F\u2082'` which\nis isomorphic to `F\u2081'` also lifts a functor `F\u2082` that is isomorphic to `F\u2081`.  -/\n@[simps]\ndef of_isos {F\u2081 F\u2082 : C \u2964 E} {F\u2081' F\u2082' : D \u2964 E} (e : F\u2081 \u2245 F\u2082) (e' : F\u2081' \u2245 F\u2082')\n  [lifting L W F\u2081 F\u2081'] : lifting L W F\u2082 F\u2082' :=\n\u27e8iso_whisker_left L e'.symm \u226a\u226b iso L W F\u2081 F\u2081' \u226a\u226b e\u27e9\n\nend lifting\n\nend localization\n\nnamespace functor\n\nnamespace is_localization\n\nopen localization\n\nlemma of_iso {L\u2081 L\u2082 : C \u2964 D} (e : L\u2081 \u2245 L\u2082) [L\u2081.is_localization W] : L\u2082.is_localization W :=\nbegin\n  have h := localization.inverts L\u2081 W,\n  rw morphism_property.is_inverted_by.iff_of_iso W e at h,\n  let F\u2081 := localization.construction.lift L\u2081 (localization.inverts L\u2081 W),\n  let F\u2082 := localization.construction.lift L\u2082 h,\n  exact\n  { inverts := h,\n    nonempty_is_equivalence := nonempty.intro\n      (is_equivalence.of_iso (lift_nat_iso W.Q W L\u2081 L\u2082 F\u2081 F\u2082 e) infer_instance), },\nend\n\n/-- If `L : C \u2964 D` is a localization for `W : morphism_property C`, then it is also\nthe case of a functor obtained by post-composing `L` with an equivalence of categories. -/\nlemma of_equivalence_target {E : Type*} [category E] (L' : C \u2964 E) (eq : D \u224c E)\n  [L.is_localization W] (e : L \u22d9 eq.functor \u2245 L') : L'.is_localization W :=\nbegin\n  have h : W.is_inverted_by L',\n  { rw \u2190 morphism_property.is_inverted_by.iff_of_iso W e,\n    exact morphism_property.is_inverted_by.of_comp W L (localization.inverts L W) eq.functor, },\n  let F\u2081 := localization.construction.lift L (localization.inverts L W),\n  let F\u2082 := localization.construction.lift L' h,\n  let e' : F\u2081 \u22d9 eq.functor \u2245 F\u2082 := lift_nat_iso W.Q W (L \u22d9 eq.functor) L' _ _ e,\n  exact\n  { inverts := h,\n    nonempty_is_equivalence := nonempty.intro (is_equivalence.of_iso e' infer_instance) },\nend\n\nend is_localization\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/localization/predicate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070808, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3896921961311683}}
{"text": "/-\nCopyright (c) 2021 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Scott Morrison\n-/\nimport data.finset.lattice\nimport data.multiset.functor\n\n/-!\n# Functoriality of `finset`\n\nThis file defines the functor structure of `finset`.\n\n## TODO\n\nCurrently, all instances are classical because the functor classes want to run over all types. If\ninstead we could state that a functor is lawful/applicative/traversable... between two given types,\nthen we could provide the instances for types with decidable equality.\n-/\n\nuniverses u\n\nopen function\n\nnamespace finset\n\n/-! ### Functor -/\n\nsection functor\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\n/-- Because `finset.image` requires a `decidable_eq` instance for the target type, we can only\nconstruct `functor finset` when working classically. -/\ninstance : functor finset :=\n{ map := \u03bb \u03b1 \u03b2 f s, s.image f }\n\ninstance : is_lawful_functor finset :=\n{ id_map := \u03bb \u03b1 s, image_id,\n  comp_map := \u03bb \u03b1 \u03b2 \u03b3 f g s, image_image.symm }\n\n@[simp] lemma fmap_def {s : finset \u03b1} (f : \u03b1 \u2192 \u03b2) : f <$> s = s.image f := rfl\n\nend functor\n\n/-! ### Pure -/\n\ninstance : has_pure finset := \u27e8\u03bb \u03b1 x, {x}\u27e9\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 finset \u03b1) = singleton := rfl\n\n/-! ### Applicative functor -/\n\nsection applicative\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\ninstance : applicative finset :=\n{ seq := \u03bb \u03b1 \u03b2 t s, t.sup (\u03bb f, s.image f),\n  seq_left := \u03bb \u03b1 \u03b2 s t, if t = \u2205 then \u2205 else s,\n  seq_right := \u03bb \u03b1 \u03b2 s t, if s = \u2205 then \u2205 else t,\n  .. finset.functor,\n  .. finset.has_pure }\n\n@[simp] lemma seq_def (s : finset \u03b1) (t : finset (\u03b1 \u2192 \u03b2)) : t <*> s = t.sup (\u03bb f, s.image f) := rfl\n@[simp] lemma seq_left_def (s : finset \u03b1) (t : finset \u03b2)  : s <* t = if t = \u2205 then \u2205 else s := rfl\n@[simp] lemma seq_right_def (s : finset \u03b1) (t : finset \u03b2)  : s *> t = if s = \u2205 then \u2205 else t := rfl\n\ninstance : is_lawful_applicative finset :=\n{ seq_left_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_left_def],\n    obtain rfl | ht := t.eq_empty_or_nonempty,\n    { simp_rw [if_pos rfl, image_empty], exact (sup_bot _).symm },\n    { ext a,\n      rw [if_neg ht.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8const \u03b2 a, mem_image_of_mem _ ha, mem_image_const_self.2 ht\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := hf,\n      obtain \u27e8_, _, rfl\u27e9 := ha,\n      exact hb }\n  end,\n  seq_right_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_right_def],\n    obtain rfl | hs := s.eq_empty_or_nonempty,\n    { rw [if_pos rfl, image_empty, sup_empty, bot_eq_empty] },\n    { ext a,\n      rw [if_neg hs.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8id, mem_image_const_self.2 hs, by rwa image_id\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := ha,\n      obtain \u27e8_, _, rfl\u27e9 := hf,\n      exact hb }\n  end,\n  pure_seq_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton,\n  map_pure := \u03bb \u03b1 \u03b2 f a, image_singleton _ _,\n  seq_pure := \u03bb \u03b1 \u03b2 s a, sup_singleton'' _ _,\n  seq_assoc := \u03bb \u03b1 \u03b2 \u03b3 s t u, begin\n    ext a,\n    simp_rw [seq_def, fmap_def],\n    simp only [exists_prop, mem_sup, mem_image],\n    split,\n    { rintro \u27e8g, hg, b, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9,\n      exact \u27e8g \u2218 f, \u27e8comp g, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9 },\n    { rintro \u27e8c, \u27e8_, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9,\n      exact \u27e8g, hg, f a, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9 }\n  end,\n  .. finset.is_lawful_functor }\n\ninstance : is_comm_applicative finset :=\n{ commutative_prod := \u03bb \u03b1 \u03b2 s t, begin\n    simp_rw [seq_def, fmap_def, sup_image, sup_eq_bUnion],\n    change s.bUnion (\u03bb a, t.image $ \u03bb b, (a, b)) = t.bUnion (\u03bb b, s.image $ \u03bb a, (a, b)),\n    transitivity s.product t;\n      [rw product_eq_bUnion, rw product_eq_bUnion_right]; congr; ext; simp_rw mem_image,\n  end,\n  .. finset.is_lawful_applicative }\n\nend applicative\n\n/-! ### Monad -/\n\nsection monad\nvariables [\u03a0 P, decidable P]\n\ninstance : monad finset :=\n{ bind := \u03bb \u03b1 \u03b2, @sup _ _ _ _,\n  .. finset.applicative }\n\n@[simp] lemma bind_def {\u03b1 \u03b2} : (>>=) = @sup (finset \u03b1) \u03b2 _ _ := rfl\n\ninstance : is_lawful_monad finset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton'' _ _,\n  bind_map_eq_seq := \u03bb \u03b1 \u03b2 t s, rfl,\n  pure_bind := \u03bb \u03b1 \u03b2 t s, sup_singleton,\n  bind_assoc :=  \u03bb \u03b1 \u03b2 \u03b3 s f g, by { convert sup_bUnion _ _, exact sup_eq_bUnion _ _ },\n  .. finset.is_lawful_applicative }\n\nend monad\n\n/-! ### Alternative functor -/\n\nsection alternative\nvariables [\u03a0 P, decidable P]\n\ninstance : alternative finset :=\n{ orelse := \u03bb \u03b1, (\u222a),\n  failure := \u03bb \u03b1, \u2205,\n  .. finset.applicative }\n\nend alternative\n\n/-! ### Traversable functor -/\n\nsection traversable\nvariables {\u03b1 \u03b2 \u03b3 : Type u} {F G : Type u \u2192 Type u} [applicative F] [applicative G]\n  [is_comm_applicative F] [is_comm_applicative G]\n\n/-- Traverse function for `finset`. -/\ndef traverse [decidable_eq \u03b2] (f : \u03b1 \u2192 F \u03b2) (s :  finset \u03b1) : F (finset \u03b2) :=\nmultiset.to_finset <$> multiset.traverse f s.1\n\n@[simp] lemma id_traverse [decidable_eq \u03b1] (s : finset \u03b1) : traverse id.mk s = s :=\nby { rw [traverse, multiset.id_traverse], exact s.val_to_finset }\n\nopen_locale classical\n\n@[simp] lemma map_comp_coe (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 multiset.to_finset = multiset.to_finset \u2218 functor.map h :=\nfunext $ \u03bb s, image_to_finset\n\nlemma map_traverse (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) (s : finset \u03b1) :\n  functor.map h <$> traverse g s = traverse (functor.map h \u2218 g) s :=\nbegin\n  unfold traverse,\n  simp only [map_comp_coe] with functor_norm,\n  rw [is_lawful_functor.comp_map, multiset.map_traverse],\nend\n\nend traversable\nend finset\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/data/finset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.38969219613116823}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.big_operators.basic\nimport algebra.big_operators.pi\nimport category_theory.limits.shapes.biproducts\nimport category_theory.preadditive.basic\nimport category_theory.preadditive.additive_functor\nimport data.matrix.dmatrix\nimport data.matrix.basic\nimport category_theory.Fintype\nimport category_theory.preadditive.single_obj\nimport algebra.opposites\n\n/-!\n# Matrices over a category.\n\nWhen `C` is a preadditive category, `Mat_ C` is the preadditive category\nwhose objects are finite tuples of objects in `C`, and\nwhose morphisms are matrices of morphisms from `C`.\n\nThere is a functor `Mat_.embedding : C \u2964 Mat_ C` sending morphisms to one-by-one matrices.\n\n`Mat_ C` has finite biproducts.\n\n## The additive envelope\n\nWe show that this construction is the \"additive envelope\" of `C`,\nin the sense that any additive functor `F : C \u2964 D` to a category `D` with biproducts\nlifts to a functor `Mat_.lift F : Mat_ C \u2964 D`,\nMoreover, this functor is unique (up to natural isomorphisms) amongst functors `L : Mat_ C \u2964 D`\nsuch that `embedding C \u22d9 L \u2245 F`.\n(As we don't have 2-category theory, we can't explicitly state that `Mat_ C` is\nthe initial object in the 2-category of categories under `C` which have biproducts.)\n\nAs a consequence, when `C` already has finite biproducts we have `Mat_ C \u224c C`.\n\n## Future work\n\nWe should provide a more convenient `Mat R`, when `R` is a ring,\nas a category with objects `n : FinType`,\nand whose morphisms are matrices with components in `R`.\n\nIdeally this would conveniently interact with both `Mat_` and `matrix`.\n\n-/\n\nopen category_theory category_theory.preadditive\nopen_locale big_operators classical\nnoncomputable theory\n\nnamespace category_theory\n\nuniverses w v\u2081 v\u2082 u\u2081 u\u2082\nvariables (C : Type u\u2081) [category.{v\u2081} C] [preadditive C]\n\n/--\nAn object in `Mat_ C` is a finite tuple of objects in `C`.\n-/\nstructure Mat_ :=\n(\u03b9 : Type)\n[F : fintype \u03b9]\n(X : \u03b9 \u2192 C)\n\nattribute [instance] Mat_.F\n\nnamespace Mat_\n\nvariables {C}\n\n/-- A morphism in `Mat_ C` is a dependently typed matrix of morphisms. -/\n@[nolint has_nonempty_instance]\ndef hom (M N : Mat_ C) : Type v\u2081 := dmatrix M.\u03b9 N.\u03b9 (\u03bb i j, M.X i \u27f6 N.X j)\n\nnamespace hom\n\n/-- The identity matrix consists of identity morphisms on the diagonal, and zeros elsewhere. -/\ndef id (M : Mat_ C) : hom M M := \u03bb i j, if h : i = j then eq_to_hom (congr_arg M.X h) else 0\n\n/-- Composition of matrices using matrix multiplication. -/\ndef comp {M N K : Mat_ C} (f : hom M N) (g : hom N K) : hom M K :=\n\u03bb i k, \u2211 j : N.\u03b9, f i j \u226b g j k\n\nend hom\n\nsection\nlocal attribute [simp] hom.id hom.comp\n\ninstance : category.{v\u2081} (Mat_ C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb M N K f g, f.comp g,\n  id_comp' := \u03bb M N f, by simp [dite_comp],\n  comp_id' := \u03bb M N f, by simp [comp_dite],\n  assoc' := \u03bb M N K L f g h, begin\n    ext i k,\n    simp_rw [hom.comp, sum_comp, comp_sum, category.assoc],\n    rw finset.sum_comm,\n  end, }.\n\nlemma id_def (M : Mat_ C) :\n  (\ud835\udfd9 M : hom M M) = \u03bb i j, if h : i = j then eq_to_hom (congr_arg M.X h) else 0 :=\nrfl\n\nlemma id_apply (M : Mat_ C) (i j : M.\u03b9) :\n  (\ud835\udfd9 M : hom M M) i j = if h : i = j then eq_to_hom (congr_arg M.X h) else 0 :=\nrfl\n\n@[simp] lemma id_apply_self (M : Mat_ C) (i : M.\u03b9) :\n  (\ud835\udfd9 M : hom M M) i i = \ud835\udfd9 _ :=\nby simp [id_apply]\n\n@[simp] lemma id_apply_of_ne (M : Mat_ C) (i j : M.\u03b9) (h : i \u2260 j) :\n  (\ud835\udfd9 M : hom M M) i j = 0 :=\nby simp [id_apply, h]\n\nlemma comp_def {M N K : Mat_ C} (f : M \u27f6 N) (g : N \u27f6 K) :\n  (f \u226b g) = \u03bb i k, \u2211 j : N.\u03b9, f i j \u226b g j k := rfl\n\n@[simp] lemma comp_apply {M N K : Mat_ C} (f : M \u27f6 N) (g : N \u27f6 K) (i k) :\n  (f \u226b g) i k = \u2211 j : N.\u03b9, f i j \u226b g j k := rfl\n\ninstance (M N : Mat_ C) : inhabited (M \u27f6 N) := \u27e8\u03bb i j, (0 : M.X i \u27f6 N.X j)\u27e9\n\nend\n\ninstance : preadditive (Mat_ C) :=\n{ hom_group := \u03bb M N, by { change add_comm_group (dmatrix M.\u03b9 N.\u03b9 _), apply_instance, },\n  add_comp' := \u03bb M N K f f' g, by { ext, simp [finset.sum_add_distrib], },\n  comp_add' := \u03bb M N K f g g', by { ext, simp [finset.sum_add_distrib], }, }\n\n@[simp] lemma add_apply {M N : Mat_ C} (f g : M \u27f6 N) (i j) : (f + g) i j = f i j + g i j := rfl\n\nopen category_theory.limits\n\n/--\nWe now prove that `Mat_ C` has finite biproducts.\n\nBe warned, however, that `Mat_ C` is not necessarily Krull-Schmidt,\nand so the internal indexing of a biproduct may have nothing to do with the external indexing,\neven though the construction we give uses a sigma type.\nSee however `iso_biproduct_embedding`.\n-/\ninstance has_finite_biproducts : has_finite_biproducts (Mat_ C) :=\n{ out := \u03bb n,\n  { has_biproduct := \u03bb f,\n    has_biproduct_of_total\n    { X := \u27e8\u03a3 j, (f j).\u03b9, \u03bb p, (f p.1).X p.2\u27e9,\n      \u03c0 := \u03bb j x y,\n      begin\n        dsimp at x \u22a2,\n        refine if h : x.1 = j then _ else 0,\n        refine if h' : (@eq.rec (fin n) x.1 (\u03bb j, (f j).\u03b9) x.2 _ h) = y then _ else 0,\n        apply eq_to_hom,\n        substs h h', -- Notice we were careful not to use `subst` until we had a goal in `Prop`.\n      end,\n      \u03b9 := \u03bb j x y,\n      begin\n        dsimp at y \u22a2,\n        refine if h : y.1 = j then _ else 0,\n        refine if h' : (@eq.rec _ y.1 (\u03bb j, (f j).\u03b9) y.2 _ h) = x then _ else 0,\n        apply eq_to_hom,\n        substs h h',\n      end,\n      \u03b9_\u03c0 := \u03bb j j',\n      begin\n        ext x y,\n        dsimp,\n        simp_rw [dite_comp, comp_dite],\n        simp only [if_t_t, dite_eq_ite, dif_ctx_congr, limits.comp_zero, limits.zero_comp,\n          eq_to_hom_trans, finset.sum_congr],\n        erw finset.sum_sigma,\n        dsimp,\n        simp only [if_congr, if_true, dif_ctx_congr, finset.sum_dite_irrel, finset.mem_univ,\n          finset.sum_const_zero, finset.sum_congr, finset.sum_dite_eq'],\n        split_ifs with h h',\n        { substs h h',\n          simp only [category_theory.eq_to_hom_refl, category_theory.Mat_.id_apply_self], },\n        { subst h,\n          simp only [id_apply_of_ne _ _ _ h', category_theory.eq_to_hom_refl], },\n        { refl, },\n      end, }\n    begin\n      dsimp,\n      funext i\u2081,\n      dsimp at i\u2081 \u22a2,\n      rcases i\u2081 with \u27e8j\u2081, i\u2081\u27e9,\n      -- I'm not sure why we can't just `simp` by `finset.sum_apply`: something doesn't quite match\n      convert finset.sum_apply _ _ _ using 1,\n      { refl, },\n      { apply heq_of_eq,\n        symmetry,\n        funext i\u2082,\n        rcases i\u2082 with \u27e8j\u2082, i\u2082\u27e9,\n        simp only [comp_apply, dite_comp, comp_dite,\n          if_t_t, dite_eq_ite, if_congr, if_true, dif_ctx_congr,\n          finset.sum_dite_irrel, finset.sum_dite_eq, finset.mem_univ, finset.sum_const_zero,\n          finset.sum_congr, finset.sum_dite_eq, finset.sum_apply,\n          limits.comp_zero, limits.zero_comp, eq_to_hom_trans, Mat_.id_apply],\n        by_cases h : j\u2081 = j\u2082,\n        { subst h, simp, },\n        { simp [h], }, },\n    end }}.\n\nend Mat_\n\nnamespace functor\nvariables {C} {D : Type*} [category.{v\u2081} D] [preadditive D]\n\nlocal attribute [simp] Mat_.id_apply eq_to_hom_map\n\n/--\nA functor induces a functor of matrix categories.\n-/\n@[simps]\ndef map_Mat_ (F : C \u2964 D) [functor.additive F] : Mat_ C \u2964 Mat_ D :=\n{ obj := \u03bb M, \u27e8M.\u03b9, \u03bb i, F.obj (M.X i)\u27e9,\n  map := \u03bb M N f i j, F.map (f i j),\n  map_comp' := \u03bb M N K f g, by { ext i k, simp,}, }\n\n/--\nThe identity functor induces the identity functor on matrix categories.\n-/\n@[simps]\ndef map_Mat_id : (\ud835\udfed C).map_Mat_ \u2245 \ud835\udfed (Mat_ C) :=\nnat_iso.of_components (\u03bb M, eq_to_iso (by { cases M, refl, }))\n(\u03bb M N f, begin\n  ext i j,\n  cases M, cases N,\n  simp [comp_dite, dite_comp],\nend)\n\n/--\nComposite functors induce composite functors on matrix categories.\n-/\n@[simps]\ndef map_Mat_comp {E : Type*} [category.{v\u2081} E] [preadditive E]\n  (F : C \u2964 D) [functor.additive F] (G : D \u2964 E) [functor.additive G] :\n  (F \u22d9 G).map_Mat_ \u2245 F.map_Mat_ \u22d9 G.map_Mat_ :=\nnat_iso.of_components (\u03bb M, eq_to_iso (by { cases M, refl, }))\n(\u03bb M N f, begin\n  ext i j,\n  cases M, cases N,\n  simp [comp_dite, dite_comp],\nend)\n\nend functor\n\nnamespace Mat_\n\nvariables (C)\n\n/-- The embedding of `C` into `Mat_ C` as one-by-one matrices.\n(We index the summands by `punit`.) -/\n@[simps]\ndef embedding : C \u2964 Mat_ C :=\n{ obj := \u03bb X, \u27e8punit, \u03bb _, X\u27e9,\n  map := \u03bb X Y f, \u03bb _ _, f,\n  map_id' := \u03bb X, by { ext \u27e8\u27e9 \u27e8\u27e9, simp, },\n  map_comp' := \u03bb X Y Z f g, by { ext \u27e8\u27e9 \u27e8\u27e9, simp, }, }\n\nnamespace embedding\n\ninstance : faithful (embedding C) :=\n{ map_injective' := \u03bb X Y f g h, congr_fun (congr_fun h punit.star) punit.star, }\n\ninstance : full (embedding C) :=\n{ preimage := \u03bb X Y f, f punit.star punit.star, }\n\ninstance : functor.additive (embedding C) := {}\n\nend embedding\n\ninstance [inhabited C] : inhabited (Mat_ C) := \u27e8(embedding C).obj default\u27e9\n\nopen category_theory.limits\n\nvariables {C}\n\n/--\nEvery object in `Mat_ C` is isomorphic to the biproduct of its summands.\n-/\n@[simps]\ndef iso_biproduct_embedding (M : Mat_ C) : M \u2245 \u2a01 (\u03bb i, (embedding C).obj (M.X i)) :=\n{ hom := biproduct.lift (\u03bb i j k, if h : j = i then eq_to_hom (congr_arg M.X h) else 0),\n  inv := biproduct.desc (\u03bb i j k, if h : i = k then eq_to_hom (congr_arg M.X h) else 0),\n  hom_inv_id' :=\n  begin\n    simp only [biproduct.lift_desc],\n    funext i,\n    dsimp,\n    convert finset.sum_apply _ _ _,\n    { dsimp, refl, },\n    { apply heq_of_eq,\n      symmetry,\n      funext j,\n      simp only [finset.sum_apply],\n      dsimp,\n      simp [dite_comp, comp_dite, Mat_.id_apply], }\n  end,\n  inv_hom_id' :=\n  begin\n    apply biproduct.hom_ext,\n    intro i,\n    apply biproduct.hom_ext',\n    intro j,\n    simp only [category.id_comp, category.assoc,\n      biproduct.lift_\u03c0, biproduct.\u03b9_desc_assoc, biproduct.\u03b9_\u03c0],\n    ext \u27e8\u27e9 \u27e8\u27e9,\n    simp [dite_comp, comp_dite],\n    split_ifs,\n    { subst h, simp, },\n    { simp [h], },\n  end, }.\n\nvariables {D : Type u\u2081} [category.{v\u2081} D] [preadditive D]\n\n/-- Every `M` is a direct sum of objects from `C`, and `F` preserves biproducts. -/\n@[simps]\ndef additive_obj_iso_biproduct (F : Mat_ C \u2964 D) [functor.additive F] (M : Mat_ C) :\n  F.obj M \u2245 \u2a01 (\u03bb i, F.obj ((embedding C).obj (M.X i))) :=\n(F.map_iso (iso_biproduct_embedding M)) \u226a\u226b (F.map_biproduct _)\n\nvariables [has_finite_biproducts D]\n\n@[reassoc] lemma additive_obj_iso_biproduct_naturality (F : Mat_ C \u2964 D) [functor.additive F]\n  {M N : Mat_ C} (f : M \u27f6 N) :\n  F.map f \u226b (additive_obj_iso_biproduct F N).hom =\n    (additive_obj_iso_biproduct F M).hom \u226b\n      biproduct.matrix (\u03bb i j, F.map ((embedding C).map (f i j))) :=\nbegin\n  -- This is disappointingly tedious.\n  ext,\n  simp only [additive_obj_iso_biproduct_hom, category.assoc, biproduct.lift_\u03c0, functor.map_bicone_\u03c0,\n    biproduct.bicone_\u03c0, biproduct.lift_matrix],\n  dsimp [embedding],\n  simp only [\u2190F.map_comp, biproduct.lift_\u03c0, biproduct.matrix_\u03c0, category.assoc],\n  simp only [\u2190F.map_comp, \u2190F.map_sum, biproduct.lift_desc, biproduct.lift_\u03c0_assoc, comp_sum],\n  simp only [comp_def, comp_dite, comp_zero, finset.sum_dite_eq', finset.mem_univ, if_true],\n  dsimp,\n  simp only [finset.sum_singleton, dite_comp, zero_comp],\n  congr,\n  symmetry,\n  convert finset.sum_fn _ _, -- It's hard to use this as a simp lemma!\n  simp only [finset.sum_fn, finset.sum_dite_eq],\n  ext,\n  simp,\nend\n\n@[reassoc] lemma additive_obj_iso_biproduct_naturality' (F : Mat_ C \u2964 D) [functor.additive F]\n  {M N : Mat_ C} (f : M \u27f6 N) :\n  (additive_obj_iso_biproduct F M).inv \u226b F.map f =\n    biproduct.matrix (\u03bb i j, F.map ((embedding C).map (f i j)) : _) \u226b\n      (additive_obj_iso_biproduct F N).inv :=\nby rw [iso.inv_comp_eq, \u2190category.assoc, iso.eq_comp_inv, additive_obj_iso_biproduct_naturality]\n\n/-- Any additive functor `C \u2964 D` to a category `D` with finite biproducts extends to\na functor `Mat_ C \u2964 D`. -/\n@[simps]\ndef lift (F : C \u2964 D) [functor.additive F] : Mat_ C \u2964 D :=\n{ obj := \u03bb X, \u2a01 (\u03bb i, F.obj (X.X i)),\n  map := \u03bb X Y f, biproduct.matrix (\u03bb i j, F.map (f i j)),\n  map_id' := \u03bb X, begin\n    ext i j,\n    by_cases h : i = j,\n    { subst h, simp, },\n    { simp [h, Mat_.id_apply], },\n  end,\n  map_comp' := \u03bb X Y Z f g, by { ext i j, simp, }, }.\n\ninstance lift_additive (F : C \u2964 D) [functor.additive F] : functor.additive (lift F) := {}\n\n/-- An additive functor `C \u2964 D` factors through its lift to `Mat_ C \u2964 D`. -/\n@[simps]\ndef embedding_lift_iso (F : C \u2964 D) [functor.additive F] : embedding C \u22d9 lift F \u2245 F :=\nnat_iso.of_components (\u03bb X,\n  { hom := biproduct.desc (\u03bb P, \ud835\udfd9 (F.obj X)),\n    inv := biproduct.lift (\u03bb P, \ud835\udfd9 (F.obj X)), })\n(\u03bb X Y f, begin\n  dsimp,\n  ext,\n  simp only [category.id_comp, biproduct.\u03b9_desc_assoc],\n  erw biproduct.\u03b9_matrix_assoc, -- Not sure why this doesn't fire via `simp`.\n  simp,\nend).\n\n/--\n`Mat_.lift F` is the unique additive functor `L : Mat_ C \u2964 D` such that `F \u2245 embedding C \u22d9 L`.\n-/\ndef lift_unique (F : C \u2964 D) [functor.additive F] (L : Mat_ C \u2964 D) [functor.additive L]\n  (\u03b1 : embedding C \u22d9 L \u2245 F) :\n  L \u2245 lift F :=\nnat_iso.of_components\n  (\u03bb M, (additive_obj_iso_biproduct L M) \u226a\u226b\n    (biproduct.map_iso (\u03bb i, \u03b1.app (M.X i))) \u226a\u226b\n    (biproduct.map_iso (\u03bb i, (embedding_lift_iso F).symm.app (M.X i))) \u226a\u226b\n    (additive_obj_iso_biproduct (lift F) M).symm)\n(\u03bb M N f, begin\n  dsimp only [iso.trans_hom, iso.symm_hom, biproduct.map_iso_hom],\n  simp only [additive_obj_iso_biproduct_naturality_assoc],\n  simp only [biproduct.matrix_map_assoc, category.assoc],\n  simp only [additive_obj_iso_biproduct_naturality'],\n  simp only [biproduct.map_matrix_assoc, category.assoc],\n  congr,\n  ext j k \u27e8\u27e9,\n  dsimp, simp,\n  exact \u03b1.hom.naturality (f j k),\nend).\n\n-- TODO is there some uniqueness statement for the natural isomorphism in `lift_unique`?\n\n/-- Two additive functors `Mat_ C \u2964 D` are naturally isomorphic if\ntheir precompositions with `embedding C` are naturally isomorphic as functors `C \u2964 D`. -/\n@[ext]\ndef ext {F G : Mat_ C \u2964 D} [functor.additive F] [functor.additive G]\n  (\u03b1 : embedding C \u22d9 F \u2245 embedding C \u22d9 G) : F \u2245 G :=\n(lift_unique (embedding C \u22d9 G) _ \u03b1) \u226a\u226b (lift_unique _ _ (iso.refl _)).symm\n\n/--\nNatural isomorphism needed in the construction of `equivalence_self_of_has_finite_biproducts`.\n-/\ndef equivalence_self_of_has_finite_biproducts_aux [has_finite_biproducts C] :\n  embedding C \u22d9 \ud835\udfed (Mat_ C) \u2245 embedding C \u22d9 lift (\ud835\udfed C) \u22d9 embedding C :=\nfunctor.right_unitor _ \u226a\u226b\n  (functor.left_unitor _).symm \u226a\u226b\n  (iso_whisker_right (embedding_lift_iso _).symm _) \u226a\u226b\n  functor.associator _ _ _\n\n/--\nA preadditive category that already has finite biproducts is equivalent to its additive envelope.\n\nNote that we only prove this for a large category;\notherwise there are universe issues that I haven't attempted to sort out.\n-/\ndef equivalence_self_of_has_finite_biproducts\n  (C : Type (u\u2081+1)) [large_category C] [preadditive C] [has_finite_biproducts C] :\n  Mat_ C \u224c C :=\nequivalence.mk -- I suspect this is already an adjoint equivalence, but it seems painful to verify.\n  (lift (\ud835\udfed C))\n  (embedding C)\n  (ext equivalence_self_of_has_finite_biproducts_aux)\n  (embedding_lift_iso (\ud835\udfed C))\n\n@[simp] lemma equivalence_self_of_has_finite_biproducts_functor\n  {C : Type (u\u2081+1)} [large_category C] [preadditive C] [has_finite_biproducts C] :\n  (equivalence_self_of_has_finite_biproducts C).functor = lift (\ud835\udfed C) :=\nrfl\n\n@[simp] lemma equivalence_self_of_has_finite_biproducts_inverse\n  {C : Type (u\u2081+1)} [large_category C] [preadditive C] [has_finite_biproducts C] :\n  (equivalence_self_of_has_finite_biproducts C).inverse = embedding C :=\nrfl\n\nend Mat_\n\nuniverse u\n\n/-- A type synonym for `Fintype`, which we will equip with a category structure\nwhere the morphisms are matrices with components in `R`. -/\n@[nolint unused_arguments, derive inhabited]\ndef Mat (R : Type u) := Fintype.{u}\n\ninstance (R : Type u) : has_coe_to_sort (Mat R) (Type u) := bundled.has_coe_to_sort\n\nopen_locale classical matrix\n\ninstance (R : Type u) [semiring R] : category (Mat R) :=\n{ hom := \u03bb X Y, matrix X Y R,\n  id := \u03bb X, 1,\n  comp := \u03bb X Y Z f g, f \u2b1d g,\n  assoc' := by { intros, simp [matrix.mul_assoc], }, }\n\nnamespace Mat\n\nsection\nvariables (R : Type u) [semiring R]\n\nlemma id_def (M : Mat R) :\n  \ud835\udfd9 M = \u03bb i j, if h : i = j then 1 else 0 :=\nrfl\n\nlemma id_apply (M : Mat R) (i j : M) :\n  (\ud835\udfd9 M : matrix M M R) i j = if h : i = j then 1 else 0 :=\nrfl\n\n@[simp] lemma id_apply_self (M : Mat R) (i : M) :\n  (\ud835\udfd9 M : matrix M M R) i i = 1 :=\nby simp [id_apply]\n\n@[simp] lemma id_apply_of_ne (M : Mat R) (i j : M) (h : i \u2260 j) :\n  (\ud835\udfd9 M : matrix M M R) i j = 0 :=\nby simp [id_apply, h]\n\nlemma comp_def {M N K : Mat R} (f : M \u27f6 N) (g : N \u27f6 K) :\n  (f \u226b g) = \u03bb i k, \u2211 j : N, f i j * g j k := rfl\n\n@[simp] lemma comp_apply {M N K : Mat R} (f : M \u27f6 N) (g : N \u27f6 K) (i k) :\n  (f \u226b g) i k = \u2211 j : N, f i j * g j k := rfl\n\ninstance (M N : Mat R) : inhabited (M \u27f6 N) := \u27e8\u03bb (i : M) (j : N), (0 : R)\u27e9\n\nend\n\nvariables (R : Type) [ring R]\n\nopen opposite\n\n/-- Auxiliary definition for `category_theory.Mat.equivalence_single_obj`. -/\n@[simps]\ndef equivalence_single_obj_inverse : Mat_ (single_obj R\u1d50\u1d52\u1d56) \u2964 Mat R :=\n{ obj := \u03bb X, Fintype.of X.\u03b9,\n  map := \u03bb X Y f i j, mul_opposite.unop (f i j),\n  map_id' := \u03bb X, by { ext i j, simp [id_def, Mat_.id_def], split_ifs; refl, }, }\n\ninstance : faithful (equivalence_single_obj_inverse R) :=\n{ map_injective' := \u03bb X Y f g w, begin\n    ext i j,\n    apply_fun mul_opposite.unop using mul_opposite.unop_injective,\n    exact (congr_fun (congr_fun w i) j),\n  end }\n\ninstance : full (equivalence_single_obj_inverse R) :=\n{ preimage := \u03bb X Y f i j, mul_opposite.op (f i j), }\n\ninstance : ess_surj (equivalence_single_obj_inverse R) :=\n{ mem_ess_image := \u03bb X,\n  \u27e8{ \u03b9 := X, X := \u03bb _, punit.star }, \u27e8eq_to_iso (by { dsimp, cases X, congr, })\u27e9\u27e9, }\n\n/-- The categorical equivalence between the category of matrices over a ring,\nand the category of matrices over that ring considered as a single-object category. -/\ndef equivalence_single_obj : Mat R \u224c Mat_ (single_obj R\u1d50\u1d52\u1d56) :=\nbegin\n  haveI := equivalence.of_fully_faithfully_ess_surj (equivalence_single_obj_inverse R),\n  exact (equivalence_single_obj_inverse R).as_equivalence.symm,\nend\n\ninstance : preadditive (Mat R) :=\n{ add_comp' := by { intros, ext, simp [add_mul, finset.sum_add_distrib], },\n  comp_add' := by { intros, ext, simp [mul_add, finset.sum_add_distrib], }, }\n\n-- TODO show `Mat R` has biproducts, and that `biprod.map` \"is\" forming a block diagonal matrix.\n\nend Mat\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/preadditive/Mat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.38963015451860666}}
{"text": "/-!\n\nLean versions used, most recent to oldest:\n\n- `nightly`: 20 Nov. 2022\n- `#1844`: https://github.com/leanprover/lean4/pull/1844\n- `pre #1811`: right before https://github.com/leanprover/lean4/pull/1811\n\n# `calc`\n\nOn mathlib 4, plain (no special indent/padding):\n- https://github.com/leanprover-community/mathlib4/search?q=calc\n\nOn std 4:\n- plain: https://github.com/leanprover/std4/blob/main/Std/Data/Int/Lemmas.lean#L1133-L1139\n- padded `_`: https://github.com/leanprover/std4/blob/main/Std/Data/List/Init/Lemmas.lean#L129-L131\n-/\n\nvariable\n  (t1 t2 t3 t4 t5 : Nat)\n\n  (pf12 : t1 = t2)\n  (pf23 : t2 < t3)\n  (pf34 : t3 = t4)\n  (pf45 : t4 < t5)\n\nabbrev longId : Nat \u2192 Nat := id\nabbrev longerId := longId\nabbrev evenLongerId := longId\n\n-- plain\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t5 :=\n  let p := calc\n    t1 = t2 := pf12\n    _ < t3 := pf23\n    _ = t4 := pf34\n    _ < t5 := pf45\n  -- dedent terminates the block\n  p\n\n-- sensible indentation with padding, rel-ops aligned\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t5 :=\n  let p := calc\n    t1 = t2 := pf12\n    _  < t3 := pf23\n    _  = t4 := pf34\n    _  < t5 := pf45\n  -- dedent terminates the block\n  p\n\n-- align on rel-ops with arbitrary `_` indentation\n-- nightly \u2705 #1844 \u274c pre #1811 \u2705\nexample : t1 < t5 :=\n  let _ := calc\n    t1 = t2 := pf12\n     _ < t3 := pf23\n     _ = t4 := pf34\n     _ < t5 := pf45\n  let p := calc\n    longId t1 = t2 := pf12\n            _ < t3 := pf23\n            _ = t4 := pf34\n            _ < t5 := pf45\n  -- dedent terminates the block\n  p\n\n-- align on rel-ops with arbitrary `_` indentation, drifting\n-- nightly \u2705 #1844 \u274c pre #1811 \u2705\nexample : t1 < t5 :=\n  let p := calc\n    longId t1 = t2 :=\n            pf12 -- error if less indented\n            _ < t3 := id\n                        pf23 -- error if less indented\n            _ = t4 := pf34\n            _ < t5 := pf45\n  -- dedent terminates the block\n  p\n\n-- same-line `calc <first relation>` with normal indent afterwards\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t5 :=\n  calc t1 = t2 := pf12\n    _ < t3 := pf23\n    _ = t4 := pf34\n    _ < t5 := pf45\n\n-- `calc <first relation LHS>\\n<indent><relation and relation RHS>`\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t5 :=\n  let _ :=\n    calc t1\n        = t2 := pf12\n      _ < t3 := pf23\n      _ = t4 := pf34\n      _ < t5 := pf45\n  -- alternatively\n  calc\n    t1\n      = t2 := pf12\n    _ < t3 := pf23\n    _ = t4 := pf34\n    _ < t5 := pf45\n\n-- `calc <first relation LHS>\\n<indent><relation and relation RHS>`\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t5 :=\n  calc t1 = t2 := pf12\n       _  < t3 := pf23\n       _  = t4 := pf34\n       _  < t5 := pf45\n\n\n\n-- `by` with indented sequence of tactics in `calc`-item RHS\n-- nightly \u274c #1844 \u2705 pre #1811 \u2705\nexample : t1 < t4 :=\n  calc\n    t1 = t2 := pf12\n    _  < t3 := by\n      skip\n      skip\n      exact pf23\n    _  = t4 := pf34\n\n-- function application with indented argument in `calc`-item RHS\n-- nightly \u274c #1844 \u2705 pre #1811 \u274c\nexample : t1 < t4 :=\n  calc\n    t1 = t2 := pf12\n    _  < t3 := id\n      pf23\n    _  = t4 := id\n                pf34\n\n-- vicious `v1`, single line\n-- https://github.com/leanprover-community/mathlib/blob/568eb9b432c885f2a2cb8fe3bbfa77467e774da7/archive/100-theorems-list/37_solution_of_cubic.lean#L166-L172\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t4 :=\n  calc  longId t1\n      = longerId t2\n        := pf12\n    _ < t3\n        := id pf23\n    _ = t4\n        := id pf34\n\n-- vicious `v2`\n-- https://github.com/leanprover-community/mathlib/blob/568eb9b432c885f2a2cb8fe3bbfa77467e774da7/archive/100-theorems-list/37_solution_of_cubic.lean#L166-L172\n-- nightly \u274c #1844 \u2705 pre #1811 \u274c\nexample : t1 < t4 :=\n  calc  longId t1\n      = longerId t2\n        := pf12\n    _ < t3\n        := id\n      pf23\n    _ = t4\n        := id\n            pf34\n\n-- vicious `v3`, single line\n-- https://github.com/leanprover-community/mathlib/blob/568eb9b432c885f2a2cb8fe3bbfa77467e774da7/archive/100-theorems-list/37_solution_of_cubic.lean#L176-L181\n-- nightly \u2705 #1844 \u2705 pre #1811 \u2705\nexample : t1 < t4 :=\n  calc  longId t1\n      = longerId t2 := pf12\n    _ < t3\n      := id pf23\n    _ = t4\n      := id pf34\n\n-- vicious `v4`\n-- https://github.com/leanprover-community/mathlib/blob/568eb9b432c885f2a2cb8fe3bbfa77467e774da7/archive/100-theorems-list/37_solution_of_cubic.lean#L176-L181\n-- nightly \u274c #1844 \u2705 pre #1811 \u274c\nexample : t1 < t4 :=\n  calc  longId t1\n      = longerId t2 := pf12\n    _ < t3\n      := id\n        pf23\n    _ = t4\n      := id\n          pf34\n\n-- playing with `by`\n-- nightly \u274c #1844 \u2705 pre #1811 \u2705\nexample : t1 < t4 :=\n  calc  longId t1\n      = longerId t2\n        := pf12\n    _ < t3 := by\n      apply id pf23\n    _ = t4 :=\n      by\n        apply id pf34\n\n-- `@digama0`'s crazy idea\n-- nightly \u274c #1844 \u274c pre #1811 \u274c\nexample : t1 < t4 :=\n  calc t1\n    _ = t2 := pf12\n    _ < t3 := pf23\n    _ = t4 := pf34\n", "meta": {"author": "AdrienChampion", "repo": "experimentalean4", "sha": "5071a8b007029f61b2e996d9ac89d90999603fcc", "save_path": "github-repos/lean/AdrienChampion-experimentalean4", "path": "github-repos/lean/AdrienChampion-experimentalean4/experimentalean4-5071a8b007029f61b2e996d9ac89d90999603fcc/discussions/ByCalcIndent/ByCalcIndent.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.38963014461781326}}
{"text": "import .basic\n\nnamespace polya.field\n\nnamespace nterm\n\nnamespace sform\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ninstance : has_coe (option (nterm \u03b3)) (nterm \u03b3) := \u27e8\u03bb x, x.get_or_else (const 0)\u27e9\n\nprivate lemma eval_none : eval \u03c1 ((none : option (nterm \u03b3)) : nterm \u03b3) = 0 :=\nby apply morph.morph_zero'\n\nprivate lemma eval_some {x : nterm \u03b3 } : eval \u03c1 (some x : nterm \u03b3) = eval \u03c1 x := rfl\n\nlocal attribute [simp] eval_none\nlocal attribute [simp] eval_some\n\ndef to_option : nterm \u03b3 \u2192 option (nterm \u03b3) | x :=\nif x = const 0 then none else some x --TODO\n\nprivate lemma eval_to_option {x : nterm \u03b3} : eval \u03c1 (to_option x : nterm \u03b3) = eval \u03c1 x :=\nbegin\n  unfold to_option,\n  by_cases h1 : x = const 0,\n  repeat { simp [eval, h1] }\nend\n\nprivate def add' : option (nterm \u03b3) \u2192 nterm \u03b3 \u2192 nterm \u03b3\n| (some x) y := if y.coeff = 0 then x else scale (coeff y) $ add (x.scale y.coeff\u207b\u00b9) y.term\n| none y := y\n\nprivate lemma eval_add' {x : option (nterm \u03b3)} {y : nterm \u03b3} : eval \u03c1 (add' x y) = eval \u03c1 (x : nterm \u03b3) + eval \u03c1 y :=\nbegin\n  cases x,\n  { simp [add'] },\n  { by_cases h1 : y.coeff = 0,\n    { rw [eval_term_coeff y, h1], simp [add', h1] },\n    { unfold add', rw [if_neg h1, eval_scale], unfold eval, rw eval_scale,\n      rw [add_mul, mul_assoc, \u2190 morph.morph_mul', inv_mul_cancel h1, \u2190 eval_term_coeff],\n      simp }}\nend\n\nlocal attribute [simp] eval_add'\n\nprivate def left : nterm \u03b3 \u2192 option (nterm \u03b3)\n| (add x _) := some x\n| _ := none\n\nprivate def right : nterm \u03b3 \u2192 (nterm \u03b3)\n| (add _ x) := x\n| x := x\n\ndef rest (S : nterm \u03b3) : option (nterm \u03b3) := (left S.term).map (scale S.coeff)\n\ndef lead (S : nterm \u03b3) : nterm \u03b3 := scale S.coeff (right S.term)\n\ntheorem eval_left_right (x : nterm \u03b3) : eval \u03c1 x = eval \u03c1 (left x : nterm \u03b3) + eval \u03c1 (right x) :=\nby cases x; simp [left, right, eval]\n\ntheorem eval_rest_lead {S : nterm \u03b3} : eval \u03c1 S = eval \u03c1 (rest S : nterm \u03b3) + eval \u03c1 (lead S) :=\nbegin\n  rw [eval_term_coeff, eval_left_right, add_mul],\n  congr' 1,\n  { unfold rest, cases (term S), repeat { simp [left] }}, \n  { simp [lead] }\nend\n\n@[simp] theorem eval_scale_option {x : option (nterm \u03b3)} {a : \u03b3} : eval \u03c1 (x.map (scale a) : nterm \u03b3) = eval \u03c1 (x : nterm \u03b3) * a :=\nby cases x; simp\n\ninductive r : option (nterm \u03b3) \u2192 option (nterm \u03b3) \u2192 Prop\n| none {S : nterm \u03b3} : r none (some S)\n| rest {S : nterm \u03b3} : r (rest S) (some S)\n\nnamespace wf\n\nprivate lemma acc_r_none : @acc (option (nterm \u03b3)) r none :=\nbegin\n  apply acc.intro, intros x h, cases h\nend\n\nprivate def g : nterm \u03b3 \u2192 \u2115\n| (add x _) := g x + 1\n| (mul x (const _)) := g x\n| _ := 0\n\nprivate def f : option (nterm \u03b3) \u2192 \u2115\n| (some x) := g x + 1\n| none := 0\n\nprivate lemma g_scale {x : nterm \u03b3} {a : \u03b3} : g (x.scale a) \u2264 g x :=\nbegin\n  sorry\nend\n\nprivate lemma f_none {S : nterm \u03b3} : f (none : option (nterm \u03b3)) < f (some S) :=\nby { unfold f, linarith }\n\nprivate lemma f_scale_option {x : option (nterm \u03b3)} {a : \u03b3} : f (x.map (scale a)) \u2264 f x :=\nby { cases x; simp [f, g_scale] }\n\nprivate lemma f_rest {S : nterm \u03b3} : f (rest S) < f (some S) :=\nbegin\n  --TODO: simplify proof\n  show f (rest S) < g S + 1,\n  cases S,\n  case add : {\n      simp only [rest, term, left, coeff, f, g, option.map_some', add_lt_add_iff_right],\n      apply lt_of_le_of_lt, { apply g_scale }, { linarith }},\n  case mul : x y {\n      cases y, case const : {\n        simp only [rest, term, left, coeff, g],\n        apply lt_of_le_of_lt,\n        { apply f_scale_option },\n        { cases x, repeat { simp [left, f, g], linarith }}},\n      repeat { simp [rest, term, left, coeff, f], linarith }},\n  repeat { simp [rest, term, left, f], linarith }\nend\n\ntheorem r_wf : @well_founded (option (nterm \u03b3)) r :=\nbegin\n  apply subrelation.wf,\n  intros x y h,\n  show f x < f y,\n  cases h, { apply f_none }, { apply f_rest },\n  apply measure_wf\nend\n\nmeta def rel_tac : tactic unit := `[exact \u27e8psigma.lex r (\u03bb _, r), psigma.lex_wf wf.r_wf (\u03bb _, wf.r_wf)\u27e9]\n\nmeta def dec_tac : tactic unit :=\n`[apply psigma.lex.left, assumption, done]\n<|> `[apply psigma.lex.right, assumption, done]\n\nend wf\n\nprivate def aux (x y : nterm \u03b3) (s1 s2 s3 : option (nterm \u03b3)) : nterm \u03b3 :=\nif x.term = y.term then\n  if x.coeff + y.coeff = 0 then (s1 : nterm \u03b3)\n  else add' s1 (mul x.term (const (x.coeff + y.coeff)))\nelse if x.term < y.term then --TODO\n  add' s2 x\nelse\n  add' s3 y\n\n--set_option pp.all true\nprivate lemma eval_aux {x y : nterm \u03b3} {s1 s2 s3 : option (nterm \u03b3)}\n  ( H0 : x.coeff \u2260 0 \u2227 y.coeff \u2260 0)\n  ( H1 : eval \u03c1 (s2 : nterm \u03b3) = eval \u03c1 (s1 : nterm \u03b3) + eval \u03c1 y )\n  ( H2 : eval \u03c1 (s3 : nterm \u03b3) = eval \u03c1 (s1 : nterm \u03b3) + eval \u03c1 x ) :\n  eval \u03c1 (aux x y s1 s2 s3) =  eval \u03c1 (s1 : nterm \u03b3) + eval \u03c1 y + eval \u03c1 x :=\nbegin\n  unfold aux,\n  by_cases h1 : x.term = y.term,\n  { rw [if_pos h1, add_assoc],\n    by_cases h2 : x.coeff + y.coeff = 0,\n    { rw [if_pos h2],\n      have : eval \u03c1 y + eval \u03c1 x = 0,\n      { have : coeff x = - coeff y, from eq_neg_of_add_eq_zero h2, \n        rw [eval_term_coeff x, eval_term_coeff y, h1],\n        rw [this, morph.morph_neg'], ring },\n      simp [this] },\n    { rw if_neg h2, rw [eval_add'], congr,\n      unfold eval, rw [morph.morph_add, mul_add],\n      rw [\u2190 eval_term_coeff, h1, \u2190 eval_term_coeff, add_comm] }},\n  { rw if_neg h1,\n    by_cases h2 : x.term < y.term,\n    { rw if_pos h2, rw [eval_add'], congr, apply H1 },\n    { rw if_neg h2, rw [add_assoc, add_comm (eval \u03c1 y), \u2190 add_assoc, eval_add'], congr, apply H2 }}\nend\n\nprivate def add_option : option (nterm \u03b3) \u2192 option (nterm \u03b3) \u2192 option (nterm \u03b3)\n| (some S) (some T) :=\n  have h1 : r (rest S) (some S), from r.rest,\n  have h2 : r (rest T) (some T), from r.rest,\n  let s1 := (add_option (rest S) (rest T)) in\n  let s2 := (add_option (rest S) (some T)) in\n  let s3 := (add_option (some S) (rest T)) in\n  if (lead S).coeff \u2260 0 \u2227 (lead T).coeff \u2260 0 then\n    some $ aux (lead S) (lead T) s1 s2 s3\n  else\n    add S T --should not happen\n| none x := x\n| x none := x\nusing_well_founded {\n    rel_tac := \u03bb _ _, wf.rel_tac,\n    dec_tac := wf.dec_tac,\n}\n\nprivate lemma add_option_def1 {x : option (nterm \u03b3)} :\n  add_option none x = x :=\nby cases x; unfold add_option\n\nprivate lemma add_option_def2 {x : option (nterm \u03b3)} :\n  add_option x none = x :=\nby cases x; unfold add_option\n\nprivate lemma add_option_def3 : \u03a0 {S T : nterm \u03b3},\n  (lead S).coeff \u2260 0 \u2227 (lead T).coeff \u2260 0 \u2192\n  add_option (some S) (some T) =\n  some (aux (lead S) (lead T)\n    (add_option (rest S) (rest T))\n    (add_option (rest S) (some T))\n    (add_option (some S) (rest T)) ) :=\nbegin\n  intros S T h0,\n  simp [h0, add_option]\nend\n\nprivate lemma eval_add_option : \u03a0 (S T : option (nterm \u03b3)),\n  eval \u03c1 (add_option S T : nterm \u03b3) = eval \u03c1 (S : nterm \u03b3) + eval \u03c1 (T : nterm \u03b3)\n| (some S) (some T) :=\n  have h1 : r (rest S) (some S), from r.rest,\n  have h2 : r (rest T) (some T), from r.rest,\n  let ih1 := eval_add_option (rest S) in\n  let ih2 := eval_add_option (some S) (rest T) in\n  begin\n    by_cases h0 : (lead S).coeff \u2260 0 \u2227 (lead T).coeff \u2260 0,\n    { rw [eval_some, eval_some, add_option_def3 h0],\n      rw [eval_some, eval_aux h0],\n      { rw [ih1, add_assoc (eval \u03c1 \u2191(rest S)), \u2190 eval_rest_lead],\n        rw [add_comm (eval \u03c1 \u2191(rest S)), add_assoc, \u2190 eval_rest_lead],\n        apply add_comm },\n      { rw [ih1, ih1, add_assoc, \u2190 eval_rest_lead], refl },\n      { rw [ih2, ih1, add_comm (eval \u03c1 \u2191(rest S)), add_assoc, \u2190 eval_rest_lead],\n        apply add_comm }},\n    { simp [add_option, h0], refl }\n  end\n| none x := by rw [add_option_def1]; simp\n| x none := by rw [add_option_def2]; simp\nusing_well_founded {\n    rel_tac := \u03bb _ _, wf.rel_tac,\n    dec_tac := wf.dec_tac,\n}\n\nprotected def add (x y : nterm \u03b3) : nterm \u03b3 :=\nadd_option (to_option x) (to_option y)\n\nprotected theorem eval_add {x y : nterm \u03b3} : eval \u03c1 (sform.add x y) = eval \u03c1 x + eval \u03c1 y :=\nby { unfold sform.add, rw [eval_add_option, eval_to_option, eval_to_option] }\n\nend sform\n\nend nterm\n\nend polya.field", "meta": {"author": "lean-forward", "repo": "field", "sha": "7e2127ad485aec25e58a1b9c82a6bb74a599467a", "save_path": "github-repos/lean/lean-forward-field", "path": "github-repos/lean/lean-forward-field/field-7e2127ad485aec25e58a1b9c82a6bb74a599467a/src/data/polya/field/sform.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3896029692604693}}
{"text": "import set_theory.cardinal.ordinal\nimport mathlib.logic.equiv.local_perm\n\nnamespace local_perm\n\n/-!\nUtilities to complete orbits of functions into local permutations.\n\nSuppose we have a function `f : \u03b1 \u2192 \u03b1`, and a set `s` on which `f` is injective.\nWe will construct a pair of functions `to_fun` and `inv_fun` that agree with `f` and its inverse\non `s`, in such a way that forms a local permutation of `\u03b1`. In particular, consider the diagram\n```\n... \u2192 l 2 \u2192 l 1 \u2192 l 0 \u2192 s \\ f '' s \u2192 ... \u2192 f '' s \\ s \u2192 r 0 \u2192 r 1 \u2192 r 2 \u2192 ...\n```\nTo fill in orbits of `f`, we construct a sequence of disjoint subsets of `\u03b1` called `l i` and `r i`\nfor each `i : \u2115`, where `#(l i) = #(s \\ f '' s)` and `#(r i) = #(f '' s \\ s)`.\nThere are natural bijections along this diagram, mapping `l (n + 1)` to `l n` and `r n` to\n`r (n + 1)`, and there are also bijections `f '' s \\ s \u2192 r 0` and `l 0 \u2192 s \\ f '' s`.\nThis yields a local permutation defined on `s`, `f '' s \\ s`, the `l i`, and the `r i`.\n-/\n\nopen cardinal function set\nopen_locale cardinal classical\n\nvariables {\u03b1 : Type*} {f : \u03b1 \u2192 \u03b1} {s : set \u03b1} {t : set \u03b1}\n  (hs : #(s \u2206 (f '' s) : set \u03b1) \u2264 #t) (ht : \u2135\u2080 \u2264 #t)\n\nlemma exists_sandbox_subset (hs : #(s \u2206 (f '' s) : set \u03b1) \u2264 #t) (ht : \u2135\u2080 \u2264 #t) :\n  #((\u2115 \u00d7 (s \\ f '' s : set \u03b1)) \u2295 (\u2115 \u00d7 (f '' s \\ s : set \u03b1))) \u2264 #t :=\nbegin\n  rw [set.symm_diff_def, mk_union_of_disjoint] at hs,\n  simp only [mk_sum, mk_prod, mk_denumerable, lift_aleph_0, lift_uzero, lift_id, \u2190 mul_add] at hs \u22a2,\n  exact le_trans (mul_le_max_of_aleph_0_le_left le_rfl) (max_le ht hs),\n  rw [disjoint_iff_inter_eq_empty, eq_empty_iff_forall_not_mem],\n  exact \u03bb x hx, hx.1.2 hx.2.1,\nend\n\n/-- Creates a \"sandbox\" subset of `t` on which we will define an extension of `f`. -/\ndef sandbox_subset : set \u03b1 :=\n(le_mk_iff_exists_subset.mp $ exists_sandbox_subset hs ht).some\n\nlemma sandbox_subset_subset : sandbox_subset hs ht \u2286 t :=\n(le_mk_iff_exists_subset.mp $ exists_sandbox_subset hs ht).some_spec.1\n\nnoncomputable def sandbox_subset_equiv :\n  sandbox_subset hs ht \u2243 (\u2115 \u00d7 (s \\ f '' s : set \u03b1)) \u2295 (\u2115 \u00d7 (f '' s \\ s : set \u03b1)) :=\n(cardinal.eq.mp (le_mk_iff_exists_subset.mp $ exists_sandbox_subset hs ht).some_spec.2).some\n\n/-- Considered an implementation detail; use lemmas about `complete` instead. -/\nnoncomputable def shift_right : (\u2115 \u00d7 (s \\ f '' s : set \u03b1)) \u2295 (\u2115 \u00d7 (f '' s \\ s : set \u03b1)) \u2192 \u03b1\n| (sum.inl \u27e80, a\u27e9) := a\n| (sum.inl \u27e8n + 1, a\u27e9) := (sandbox_subset_equiv hs ht).symm (sum.inl \u27e8n, a\u27e9)\n| (sum.inr \u27e8n, a\u27e9) := (sandbox_subset_equiv hs ht).symm (sum.inr \u27e8n + 1, a\u27e9)\n\n/-- Considered an implementation detail; use lemmas about `complete` instead. -/\nnoncomputable def complete_to_fun (a : \u03b1) : \u03b1 :=\nif h : a \u2208 sandbox_subset hs ht then\n  shift_right hs ht (sandbox_subset_equiv hs ht \u27e8a, h\u27e9)\nelse if h : a \u2208 f '' s \\ s then\n  (sandbox_subset_equiv hs ht).symm (sum.inr \u27e80, a, h\u27e9)\nelse\n  f a\n\n/-- Considered an implementation detail; use lemmas about `complete` instead. -/\nnoncomputable def shift_left : (\u2115 \u00d7 (s \\ f '' s : set \u03b1)) \u2295 (\u2115 \u00d7 (f '' s \\ s : set \u03b1)) \u2192 \u03b1\n| (sum.inl \u27e8n, a\u27e9) := (sandbox_subset_equiv hs ht).symm (sum.inl \u27e8n + 1, a\u27e9)\n| (sum.inr \u27e80, a\u27e9) := a.1\n| (sum.inr \u27e8n + 1, a\u27e9) := (sandbox_subset_equiv hs ht).symm (sum.inr \u27e8n, a\u27e9)\n\n/-- Considered an implementation detail; use lemmas about `complete` instead. -/\nnoncomputable def complete_inv_fun [nonempty \u03b1] (a : \u03b1) : \u03b1 :=\nif h : a \u2208 sandbox_subset hs ht then\n  shift_left hs ht (sandbox_subset_equiv hs ht \u27e8a, h\u27e9)\nelse if h : a \u2208 s \\ f '' s then\n  (sandbox_subset_equiv hs ht).symm (sum.inl \u27e80, a, h\u27e9)\nelse\n  inv_fun_on f s a\n\n/-- The domain on which we will define the completion of a function to a local permutation. -/\ndef complete_domain : set \u03b1 :=\ns \u222a f '' s \u222a sandbox_subset hs ht\n\nlemma complete_to_fun_domain (x : \u03b1) (h : x \u2208 complete_domain hs ht) :\n  complete_to_fun hs ht x \u2208 complete_domain hs ht :=\nbegin\n  unfold complete_to_fun complete_domain,\n  split_ifs with h\u2081 h\u2082,\n  cases sandbox_subset_equiv hs ht \u27e8x, h\u2081\u27e9,\n  { obtain \u27e8_ | n, x\u27e9 := val,\n    { exact or.inl (or.inl x.prop.1), },\n    { exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, }, },\n  { obtain \u27e8n, x\u27e9 := val,\n    exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, },\n  { exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, },\n  { rw [mem_diff, not_and_distrib, not_not] at h\u2082,\n    refine or.inl (or.inr \u27e8x, _, rfl\u27e9),\n    cases h\u2082,\n    { obtain ((h | h) | h) := h,\n      exact h,\n      cases h\u2082 h,\n      cases h\u2081 h, },\n    { exact h\u2082, }, },\nend\n\nlemma complete_inv_fun_domain [nonempty \u03b1] (x : \u03b1) (h : x \u2208 complete_domain hs ht) :\n  complete_inv_fun hs ht x \u2208 complete_domain hs ht :=\nbegin\n  unfold complete_inv_fun complete_domain,\n  split_ifs with h\u2081 h\u2082,\n  cases sandbox_subset_equiv hs ht \u27e8x, h\u2081\u27e9,\n  { obtain \u27e8n, x\u27e9 := val,\n    exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, },\n  { obtain \u27e8_ | n, x\u27e9 := val,\n    { exact or.inl (or.inr x.prop.1), },\n    { exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, }, },\n  { exact or.inr ((sandbox_subset_equiv hs ht).symm _).prop, },\n  { rw [mem_diff, not_and_distrib, not_not] at h\u2082,\n    cases h\u2082,\n    { rw complete_domain at h,\n      obtain ((h | h) | h) := h,\n      cases h\u2082 h,\n      { refine or.inl (or.inl _),\n        simp only [mem_image, \u2190 exists_prop] at h,\n        exact inv_fun_on_mem h, },\n      cases h\u2081 h, },\n    simp only [mem_image, \u2190 exists_prop] at h\u2082,\n    refine or.inl (or.inl (inv_fun_on_mem h\u2082)), },\nend\n\nlemma complete_left_inv [nonempty \u03b1] (hst : disjoint (s \u222a f '' s) t) (hf : inj_on f s)\n  (x : \u03b1) (h : x \u2208 complete_domain hs ht) : complete_inv_fun hs ht (complete_to_fun hs ht x) = x :=\nbegin\n  rw [disjoint_iff_inter_eq_empty, eq_empty_iff_forall_not_mem] at hst,\n  by_cases h\u2081 : x \u2208 s,\n  { rw [complete_to_fun, dif_neg, dif_neg, complete_inv_fun, dif_neg, dif_neg],\n    { exact hf (inv_fun_on_apply_mem h\u2081) h\u2081\n        (show f (inv_fun_on f s (f x)) = f x, from inv_fun_on_apply_eq h\u2081), },\n    { exact \u03bb h', h'.2 \u27e8x, h\u2081, rfl\u27e9, },\n    { exact \u03bb hx, hst (f x) \u27e8or.inr \u27e8x, h\u2081, rfl\u27e9, sandbox_subset_subset hs ht hx\u27e9, },\n    { exact \u03bb h', h'.2 h\u2081, },\n    { exact \u03bb hx, hst x \u27e8or.inl h\u2081, sandbox_subset_subset hs ht hx\u27e9, }, },\n  by_cases h\u2082 : x \u2208 f '' s,\n  { rw [complete_to_fun, dif_neg, dif_pos, complete_inv_fun, dif_pos],\n    simp only [subtype.coe_eta, equiv.apply_symm_apply],\n    refl,\n    { exact ((sandbox_subset_equiv hs ht).symm _).prop,\n      exact \u27e8h\u2082, h\u2081\u27e9, },\n    { exact \u03bb h, hst x \u27e8or.inr h\u2082, sandbox_subset_subset hs ht h\u27e9, }, },\n  obtain ((h | h) | h) := h,\n  { cases h\u2081 h, },\n  { cases h\u2082 h, },\n  rw [complete_to_fun, dif_pos h, complete_inv_fun],\n  by_cases h\u2083 : \u2203 a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inl \u27e80, a\u27e9,\n  { obtain \u27e8a, ha\u27e9 := h\u2083,\n    rw [dif_neg, dif_pos, subtype.coe_eq_iff],\n    refine \u27e8h, _\u27e9,\n    simp only [ha],\n    exact a.prop,\n    { rw equiv.symm_apply_eq, simp only [ha], ext; refl, },\n    { rw ha,\n      exact \u03bb h, hst _ \u27e8or.inl a.prop.1, sandbox_subset_subset hs ht h\u27e9, }, },\n  have h\u2084 : (\u2203 n a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inl \u27e8n + 1, a\u27e9) \u2228\n    \u2203 n a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inr \u27e8n, a\u27e9,\n  { set val := sandbox_subset_equiv hs ht \u27e8x, h\u27e9,\n    clear_value val,\n    obtain (\u27e8_ | n, b\u27e9 | \u27e8n, b\u27e9) := val,\n    cases h\u2083 \u27e8b, rfl\u27e9,\n    exact or.inl \u27e8n, b, rfl\u27e9,\n    exact or.inr \u27e8n, b, rfl\u27e9, },\n  cases h\u2084;\n  { obtain \u27e8n, a, ha\u27e9 := h\u2084,\n    rw [dif_pos],\n    simp only [ha],\n    unfold shift_right,\n    simp only [subtype.coe_eta, equiv.apply_symm_apply],\n    exact subtype.coe_inj.mpr ((equiv.symm_apply_eq _).mpr ha.symm),\n    rw ha,\n    exact ((sandbox_subset_equiv hs ht).symm _).prop, },\nend\n\nlemma complete_right_inv [nonempty \u03b1] (hst : disjoint (s \u222a f '' s) t) (hf : inj_on f s)\n  (x : \u03b1) (h : x \u2208 complete_domain hs ht) : complete_to_fun hs ht (complete_inv_fun hs ht x) = x :=\nbegin\n  rw [disjoint_iff_inter_eq_empty, eq_empty_iff_forall_not_mem] at hst,\n  by_cases h\u2081 : x \u2208 f '' s,\n  { rw [complete_inv_fun, dif_neg, dif_neg, complete_to_fun, dif_neg, dif_neg],\n    { exact inv_fun_on_eq (set.mem_image_iff_bex.mp h\u2081), },\n    { exact \u03bb h', h'.2 (inv_fun_on_mem (set.mem_image_iff_bex.mp h\u2081)), },\n    { exact \u03bb hx, hst _ \u27e8or.inl (inv_fun_on_mem (set.mem_image_iff_bex.mp h\u2081)),\n        sandbox_subset_subset hs ht hx\u27e9, },\n    { exact \u03bb h', h'.2 h\u2081, },\n    { exact \u03bb hx, hst x \u27e8or.inr h\u2081, sandbox_subset_subset hs ht hx\u27e9, }, },\n  by_cases h\u2082 : x \u2208 s,\n  { rw [complete_inv_fun, dif_neg, dif_pos, complete_to_fun, dif_pos],\n    simp only [subtype.coe_eta, equiv.apply_symm_apply],\n    refl,\n    { exact ((sandbox_subset_equiv hs ht).symm _).prop,\n      exact \u27e8h\u2082, h\u2081\u27e9, },\n    { exact \u03bb h, hst x \u27e8or.inl h\u2082, sandbox_subset_subset hs ht h\u27e9, }, },\n  obtain ((h | h) | h) := h,\n  { cases h\u2082 h, },\n  { cases h\u2081 h, },\n  rw [complete_inv_fun, dif_pos h, complete_to_fun],\n  by_cases h\u2083 : \u2203 a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inr \u27e80, a\u27e9,\n  { obtain \u27e8a, ha\u27e9 := h\u2083,\n    rw [dif_neg, dif_pos, subtype.coe_eq_iff],\n    refine \u27e8h, _\u27e9,\n    simp only [ha],\n    exact a.prop,\n    { rw equiv.symm_apply_eq, simp only [ha], ext; refl, },\n    { rw ha,\n      exact \u03bb h, hst _ \u27e8or.inr a.prop.1, sandbox_subset_subset hs ht h\u27e9, }, },\n  have h\u2084 : (\u2203 n a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inr \u27e8n + 1, a\u27e9) \u2228\n    \u2203 n a, sandbox_subset_equiv hs ht \u27e8x, h\u27e9 = sum.inl \u27e8n, a\u27e9,\n  { set val := sandbox_subset_equiv hs ht \u27e8x, h\u27e9,\n    clear_value val,\n    obtain (\u27e8n, b\u27e9 | \u27e8_ | n, b\u27e9) := val,\n    exact or.inr \u27e8n, b, rfl\u27e9,\n    cases h\u2083 \u27e8b, rfl\u27e9,\n    exact or.inl \u27e8n, b, rfl\u27e9, },\n  cases h\u2084;\n  { obtain \u27e8n, a, ha\u27e9 := h\u2084,\n    rw [dif_pos],\n    simp only [ha],\n    unfold shift_left,\n    simp only [subtype.coe_eta, equiv.apply_symm_apply],\n    exact subtype.coe_inj.mpr ((equiv.symm_apply_eq _).mpr ha.symm),\n    rw ha,\n    exact ((sandbox_subset_equiv hs ht).symm _).prop, },\nend\n\n/-- Completes a function `f` on a domain `s` into a local permutation that agrees with `f` on `s`,\nwith domain contained in `s \u222a (f '' s) \u222a t`. -/\nnoncomputable def complete [nonempty \u03b1] (f : \u03b1 \u2192 \u03b1) (s : set \u03b1) (t : set \u03b1)\n  (hs : #(s \u2206 (f '' s) : set \u03b1) \u2264 #t) (ht : \u2135\u2080 \u2264 #t) (hst : disjoint (s \u222a f '' s) t)\n  (hf : inj_on f s) : local_perm \u03b1 := {\n  to_fun := complete_to_fun hs ht,\n  inv_fun := complete_inv_fun hs ht,\n  domain := complete_domain hs ht,\n  to_fun_domain' := complete_to_fun_domain hs ht,\n  inv_fun_domain' := complete_inv_fun_domain hs ht,\n  left_inv' := complete_left_inv hs ht hst hf,\n  right_inv' := complete_right_inv hs ht hst hf,\n}\n\nvariables [nonempty \u03b1] {hst : disjoint (s \u222a f '' s) t} {hf : inj_on f s}\n\n@[simp] lemma complete_domain_eq :\n  (complete f s t hs ht hst hf).domain = complete_domain hs ht := rfl\n\nlemma mem_complete_domain_of_mem (x : \u03b1) (hx : x \u2208 s) : x \u2208 complete_domain hs ht :=\nor.inl (or.inl hx)\n\nlemma mem_complete_domain_of_mem_image (x : \u03b1) (hx : x \u2208 f '' s) : x \u2208 complete_domain hs ht :=\nor.inl (or.inr hx)\n\nlemma not_mem_sandbox_of_mem (hst : disjoint (s \u222a f '' s) t) (x : \u03b1) (hx : x \u2208 s) :\n  x \u2209 sandbox_subset hs ht :=\nbegin\n  intro h,\n  rw [disjoint_iff_inter_eq_empty, eq_empty_iff_forall_not_mem] at hst,\n  exact hst x \u27e8or.inl hx, sandbox_subset_subset hs ht h\u27e9,\nend\n\nlemma not_mem_sandbox_of_mem_image (hst : disjoint (s \u222a f '' s) t) (x : \u03b1) (hx : x \u2208 f '' s) :\n  x \u2209 sandbox_subset hs ht :=\nbegin\n  intro h,\n  rw [disjoint_iff_inter_eq_empty, eq_empty_iff_forall_not_mem] at hst,\n  exact hst x \u27e8or.inr hx, sandbox_subset_subset hs ht h\u27e9,\nend\n\n@[simp] lemma complete_apply_eq (x : \u03b1) (hx : x \u2208 s) : complete f s t hs ht hst hf x = f x :=\nbegin\n  rw [complete, coe_mk, complete_to_fun, dif_neg, dif_neg],\n  exact \u03bb h', h'.2 hx,\n  exact not_mem_sandbox_of_mem hs ht hst x hx,\nend\n\nend local_perm\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/complete_orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.3896029692604693}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.applicative\nimport control.traversable.basic\n\n/-!\n# Traversing collections\n\nThis file proves basic properties of traversable and applicative functors and defines\n`pure_transformation F`, the natural applicative transformation from the identity functor to `F`.\n\n## References\n\nInspired by [The Essence of the Iterator Pattern][gibbons2009].\n-/\n\nuniverses u\n\nopen is_lawful_traversable\nopen function (hiding comp)\nopen functor\n\nattribute [functor_norm] is_lawful_traversable.naturality\nattribute [simp] is_lawful_traversable.id_traverse\n\nnamespace traversable\n\nvariable {t : Type u \u2192 Type u}\nvariables [traversable t] [is_lawful_traversable t]\nvariables F G : Type u \u2192 Type u\n\nvariables [applicative F] [is_lawful_applicative F]\nvariables [applicative G] [is_lawful_applicative G]\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\nvariables g : \u03b1 \u2192 F \u03b2\nvariables h : \u03b2 \u2192 G \u03b3\nvariables f : \u03b2 \u2192 \u03b3\n\n/-- The natural applicative transformation from the identity functor\nto `F`, defined by `pure : \u03a0 {\u03b1}, \u03b1 \u2192 F \u03b1`. -/\ndef pure_transformation : applicative_transformation id F :=\n{ app := @pure F _,\n  preserves_pure' := \u03bb \u03b1 x, rfl,\n  preserves_seq' := \u03bb \u03b1 \u03b2 f x, by { simp only [map_pure, seq_pure], refl } }\n\n@[simp] \n\nvariables {F G} (x : t \u03b2)\n\nlemma map_eq_traverse_id : map f = @traverse t _ _ _ _ _ (id.mk \u2218 f) :=\nfunext $ \u03bb y, (traverse_eq_map_id f y).symm\n\ntheorem map_traverse (x : t \u03b1) : map f <$> traverse g x = traverse (map f \u2218 g) x :=\nbegin\n  rw @map_eq_traverse_id t _ _ _ _ f,\n  refine (comp_traverse (id.mk \u2218 f) g x).symm.trans _,\n  congr, apply comp.applicative_comp_id\nend\n\ntheorem traverse_map (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 \u03b2) (x : t \u03b1) :\n  traverse f (g <$> x) = traverse (f \u2218 g) x :=\nbegin\n  rw @map_eq_traverse_id t _ _ _ _ g,\n  refine (comp_traverse f (id.mk \u2218 g) x).symm.trans _,\n  congr, apply comp.applicative_id_comp\nend\n\nlemma pure_traverse (x : t \u03b1) : traverse pure x = (pure x : F (t \u03b1)) :=\nby have : traverse pure x = pure (traverse id.mk x) :=\n     (naturality (pure_transformation F) id.mk x).symm;\n   rwa id_traverse at this\n\nlemma id_sequence (x : t \u03b1) : sequence (id.mk <$> x) = id.mk x :=\nby simp [sequence, traverse_map, id_traverse]; refl\n\nlemma comp_sequence (x : t (F (G \u03b1))) :\n  sequence (comp.mk <$> x) = comp.mk (sequence <$> sequence x) :=\nby simp [sequence, traverse_map]; rw \u2190 comp_traverse; simp [map_id]\n\nlemma naturality' (\u03b7 : applicative_transformation F G) (x : t (F \u03b1)) :\n  \u03b7 (sequence x) = sequence (@\u03b7 _ <$> x) :=\nby simp [sequence, naturality, traverse_map]\n\n@[functor_norm]\nlemma traverse_id : traverse id.mk = (id.mk : t \u03b1 \u2192 id (t \u03b1)) :=\nby { ext, exact id_traverse _ }\n\n@[functor_norm]\nlemma traverse_comp (g : \u03b1 \u2192 F \u03b2) (h : \u03b2 \u2192 G \u03b3) :\n  traverse (comp.mk \u2218 map h \u2218 g) =\n  (comp.mk \u2218 map (traverse h) \u2218 traverse g : t \u03b1 \u2192 comp F G (t \u03b3)) :=\nby { ext, exact comp_traverse _ _ _ }\n\nlemma traverse_eq_map_id' (f : \u03b2 \u2192 \u03b3) : traverse (id.mk \u2218 f) = id.mk \u2218 (map f : t \u03b2 \u2192 t \u03b3) :=\nby { ext, exact traverse_eq_map_id _ _ }\n\n-- @[functor_norm]\nlemma traverse_map' (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3) :\n  traverse (h \u2218 g) = (traverse h \u2218 map g : t \u03b1 \u2192 G (t \u03b3)) :=\nby { ext, rw [comp_app, traverse_map] }\n\nlemma map_traverse' (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) :\n  traverse (map h \u2218 g) = (map (map h) \u2218 traverse g : t \u03b1 \u2192 G (t \u03b3)) :=\nby { ext, rw [comp_app, map_traverse] }\n\nlemma naturality_pf (\u03b7 : applicative_transformation F G) (f : \u03b1 \u2192 F \u03b2) :\n  traverse (@\u03b7 _ \u2218 f) = @\u03b7 _ \u2218 (traverse f : t \u03b1 \u2192 F (t \u03b2)) :=\nby { ext, rw [comp_app, naturality] }\n\nend traversable\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/control/traversable/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3896029607474689}}
{"text": "/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Riccardo Brasca\n-/\nimport analysis.normed.group.hom\nimport category_theory.limits.shapes.zero\n\n/-!\n# The category of seminormed groups\n\nWe define `SemiNormedGroup`, the category of seminormed groups and normed group homs between them,\nas well as `SemiNormedGroup\u2081`, the subcategory of norm non-increasing morphisms.\n-/\n\nnoncomputable theory\n\nuniverses u\n\nopen category_theory\n\n/-- The category of seminormed abelian groups and bounded group homomorphisms. -/\ndef SemiNormedGroup : Type (u+1) := bundled semi_normed_group\n\nnamespace SemiNormedGroup\n\ninstance bundled_hom : bundled_hom @normed_group_hom :=\n\u27e8@normed_group_hom.to_fun, @normed_group_hom.id, @normed_group_hom.comp, @normed_group_hom.coe_inj\u27e9\n\nattribute [derive [large_category, concrete_category]] SemiNormedGroup\n\ninstance : has_coe_to_sort SemiNormedGroup (Type u) := bundled.has_coe_to_sort\n\n/-- Construct a bundled `SemiNormedGroup` from the underlying type and typeclass. -/\ndef of (M : Type u) [semi_normed_group M] : SemiNormedGroup := bundled.of M\n\ninstance (M : SemiNormedGroup) : semi_normed_group M := M.str\n\n@[simp] lemma coe_of (V : Type u) [semi_normed_group V] : (SemiNormedGroup.of V : Type u) = V := rfl\n@[simp] lemma coe_id (V : SemiNormedGroup) : \u21d1(\ud835\udfd9 V) = id := rfl\n@[simp] lemma coe_comp {M N K : SemiNormedGroup} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : M \u2192 K) = g \u2218 f := rfl\n\ninstance : has_zero SemiNormedGroup := \u27e8of punit\u27e9\ninstance : inhabited SemiNormedGroup := \u27e80\u27e9\n\ninstance : limits.has_zero_morphisms.{u (u+1)} SemiNormedGroup := {}\n\n@[simp] lemma zero_apply {V W : SemiNormedGroup} (x : V) : (0 : V \u27f6 W) x = 0 := rfl\n\ninstance has_zero_object : limits.has_zero_object SemiNormedGroup.{u} :=\n{ zero := 0,\n  unique_to := \u03bb X,\n  { default := 0,\n    uniq := \u03bb a, by { ext \u27e8\u27e9, exact a.map_zero, }, },\n  unique_from := \u03bb X,\n  { default := 0,\n    uniq := \u03bb f, by ext } }\n\nlemma iso_isometry_of_norm_noninc {V W : SemiNormedGroup} (i : V \u2245 W)\n  (h1 : i.hom.norm_noninc) (h2 : i.inv.norm_noninc) :\n  isometry i.hom :=\nbegin\n  apply normed_group_hom.isometry_of_norm,\n  intro v,\n  apply le_antisymm (h1 v),\n  calc \u2225v\u2225 = \u2225i.inv (i.hom v)\u2225 : by rw [coe_hom_inv_id]\n  ... \u2264 \u2225i.hom v\u2225 : h2 _,\nend\n\nend SemiNormedGroup\n\n/--\n`SemiNormedGroup\u2081` is a type synonym for `SemiNormedGroup`,\nwhich we shall equip with the category structure consisting only of the norm non-increasing maps.\n-/\ndef SemiNormedGroup\u2081 : Type (u+1) := bundled semi_normed_group\n\nnamespace SemiNormedGroup\u2081\n\ninstance : has_coe_to_sort SemiNormedGroup\u2081 (Type u) := bundled.has_coe_to_sort\n\ninstance : large_category.{u} SemiNormedGroup\u2081 :=\n{ hom := \u03bb X Y, { f : normed_group_hom X Y // f.norm_noninc },\n  id := \u03bb X, \u27e8normed_group_hom.id X, normed_group_hom.norm_noninc.id\u27e9,\n  comp := \u03bb X Y Z f g, \u27e8(g : normed_group_hom Y Z).comp (f : normed_group_hom X Y), g.2.comp f.2\u27e9, }\n\n@[ext] lemma hom_ext {M N : SemiNormedGroup\u2081} (f g : M \u27f6 N) (w : (f : M \u2192 N) = (g : M \u2192 N)) :\n  f = g :=\nsubtype.eq (normed_group_hom.ext (congr_fun w))\n\ninstance : concrete_category.{u} SemiNormedGroup\u2081 :=\n{ forget :=\n  { obj := \u03bb X, X,\n    map := \u03bb X Y f, f, },\n  forget_faithful := {} }\n\n/-- Construct a bundled `SemiNormedGroup\u2081` from the underlying type and typeclass. -/\ndef of (M : Type u) [semi_normed_group M] : SemiNormedGroup\u2081 := bundled.of M\n\ninstance (M : SemiNormedGroup\u2081) : semi_normed_group M := M.str\n\n/-- Promote a morphism in `SemiNormedGroup` to a morphism in `SemiNormedGroup\u2081`. -/\ndef mk_hom {M N : SemiNormedGroup} (f : M \u27f6 N) (i : f.norm_noninc) :\n  SemiNormedGroup\u2081.of M \u27f6 SemiNormedGroup\u2081.of N :=\n\u27e8f, i\u27e9\n\n@[simp] lemma mk_hom_apply {M N : SemiNormedGroup} (f : M \u27f6 N) (i : f.norm_noninc) (x) :\n  mk_hom f i x = f x := rfl\n\n/-- Promote an isomorphism in `SemiNormedGroup` to an isomorphism in `SemiNormedGroup\u2081`. -/\n@[simps]\ndef mk_iso {M N : SemiNormedGroup} (f : M \u2245 N) (i : f.hom.norm_noninc) (i' : f.inv.norm_noninc) :\n  SemiNormedGroup\u2081.of M \u2245 SemiNormedGroup\u2081.of N :=\n{ hom := mk_hom f.hom i,\n  inv := mk_hom f.inv i',\n  hom_inv_id' := by { apply subtype.eq, exact f.hom_inv_id, },\n  inv_hom_id' := by { apply subtype.eq, exact f.inv_hom_id, }, }\n\ninstance : has_forget\u2082 SemiNormedGroup\u2081 SemiNormedGroup :=\n{ forget\u2082 :=\n  { obj := \u03bb X, X,\n    map := \u03bb X Y f, f.1, }, }\n\n@[simp] lemma coe_of (V : Type u) [semi_normed_group V] : (SemiNormedGroup\u2081.of V : Type u) = V :=\nrfl\n@[simp] lemma coe_id (V : SemiNormedGroup\u2081) : \u21d1(\ud835\udfd9 V) = id := rfl\n@[simp] lemma coe_comp {M N K : SemiNormedGroup\u2081} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : M \u2192 K) = g \u2218 f := rfl\n-- If `coe_fn_coe_base` fires before `coe_comp`, `coe_comp'` puts us back in normal form.\n@[simp] lemma coe_comp' {M N K : SemiNormedGroup\u2081} (f : M \u27f6 N) (g : N \u27f6 K) :\n  ((f \u226b g) : normed_group_hom M K) = (\u2191g : normed_group_hom N K).comp \u2191f := rfl\n\ninstance : has_zero SemiNormedGroup\u2081 := \u27e8of punit\u27e9\ninstance : inhabited SemiNormedGroup\u2081 := \u27e80\u27e9\n\ninstance : limits.has_zero_morphisms.{u (u+1)} SemiNormedGroup\u2081 :=\n{ has_zero := \u03bb X Y, { zero := \u27e80, normed_group_hom.norm_noninc.zero\u27e9, },\n  comp_zero' := \u03bb X Y f Z, by { ext, refl, },\n  zero_comp' := \u03bb X Y Z f, by { ext, simp [coe_fn_coe_base'] } }\n\n@[simp] lemma zero_apply {V W : SemiNormedGroup\u2081} (x : V) : (0 : V \u27f6 W) x = 0 := rfl\n\ninstance has_zero_object : limits.has_zero_object SemiNormedGroup\u2081.{u} :=\n{ zero := 0,\n  unique_to := \u03bb X,\n  { default := 0,\n    uniq := \u03bb a, by { ext \u27e8\u27e9, exact a.1.map_zero, }, },\n  unique_from := \u03bb X,\n  { default := 0,\n    uniq := \u03bb f, by ext } }\n\nlemma iso_isometry {V W : SemiNormedGroup\u2081} (i : V \u2245 W) :\n  isometry i.hom :=\nbegin\n  apply normed_group_hom.isometry_of_norm,\n  intro v,\n  apply le_antisymm (i.hom.2 v),\n  calc \u2225v\u2225 = \u2225i.inv (i.hom v)\u2225 : by rw [coe_hom_inv_id]\n      ... \u2264 \u2225i.hom v\u2225 : i.inv.2 _,\nend\n\nend SemiNormedGroup\u2081\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/analysis/normed/group/SemiNormedGroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3896029607474689}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.functor\nimport Mathlib.PostPort\n\nuniverses u v w l s t u_1 \n\nnamespace Mathlib\n\n/-!\n# Traversable type class\n\nType classes for traversing collections. The concepts and laws are taken from\n<http://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html>\n\nTraversable collections are a generalization of functors. Whereas\nfunctors (such as `list`) allow us to apply a function to every\nelement, it does not allow functions which external effects encoded in\na monad. Consider for instance a functor `invite : email \u2192 io response`\nthat takes an email address, sends an email and waits for a\nresponse. If we have a list `guests : list email`, using calling\n`invite` using `map` gives us the following: `map invite guests : list\n(io response)`.  It is not what we need. We need something of type `io\n(list response)`. Instead of using `map`, we can use `traverse` to\nsend all the invites: `traverse invite guests : io (list response)`.\n`traverse` applies `invite` to every element of `guests` and combines\nall the resulting effects. In the example, the effect is encoded in the\nmonad `io` but any applicative functor is accepted by `traverse`.\n\nFor more on how to use traversable, consider the Haskell tutorial:\n<https://en.wikibooks.org/wiki/Haskell/Traversable>\n\n## Main definitions\n  * `traversable` type class - exposes the `traverse` function\n  * `sequence` - based on `traverse`, turns a collection of effects into an effect returning a collection\n  * `is_lawful_traversable` - laws for a traversable functor\n  * `applicative_transformation` - the notion of a natural transformation for applicative functors\n\n## Tags\n\ntraversable iterator functor applicative\n\n## References\n\n * \"Applicative Programming with Effects\", by Conor McBride and Ross Paterson,\n   Journal of Functional Programming 18:1 (2008) 1-13, online at\n   <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>\n * \"The Essence of the Iterator Pattern\", by Jeremy Gibbons and Bruno Oliveira,\n   in Mathematically-Structured Functional Programming, 2006, online at\n   <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>\n * \"An Investigation of the Laws of Traversals\", by Mauro Jaskelioff and Ondrej Rypacek,\n   in Mathematically-Structured Functional Programming, 2012,\n   online at <http://arxiv.org/pdf/1202.2919>\n-/\n\n/-- A transformation between applicative functors.  It a natural\ntransformation such that `app` preserves the `has_pure.pure` and\n`functor.map` (`<*>`) operations. See\n`applicative_transformation.preserves_map` for naturality. -/\nstructure applicative_transformation (F : Type u \u2192 Type v) [Applicative F] [is_lawful_applicative F]\n    (G : Type u \u2192 Type w) [Applicative G] [is_lawful_applicative G]\n    where\n  app : (\u03b1 : Type u) \u2192 F \u03b1 \u2192 G \u03b1\n  preserves_pure' : \u2200 {\u03b1 : Type u} (x : \u03b1), app \u03b1 (pure x) = pure x\n  preserves_seq' :\n    \u2200 {\u03b1 \u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), app \u03b2 (x <*> y) = app (\u03b1 \u2192 \u03b2) x <*> app \u03b1 y\n\nnamespace applicative_transformation\n\n\nprotected instance has_coe_to_fun (F : Type u \u2192 Type v) [Applicative F] [is_lawful_applicative F]\n    (G : Type u \u2192 Type w) [Applicative G] [is_lawful_applicative G] :\n    has_coe_to_fun (applicative_transformation F G) :=\n  has_coe_to_fun.mk (fun (_x : applicative_transformation F G) => {\u03b1 : Type u} \u2192 F \u03b1 \u2192 G \u03b1)\n    fun (a : applicative_transformation F G) => app a\n\n@[simp] theorem app_eq_coe {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) : app \u03b7 = \u21d1\u03b7 :=\n  rfl\n\n@[simp] theorem coe_mk {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G] (f : (\u03b1 : Type u) \u2192 F \u03b1 \u2192 G \u03b1)\n    (pp : \u2200 {\u03b1 : Type u} (x : \u03b1), f \u03b1 (pure x) = pure x)\n    (ps : \u2200 {\u03b1 \u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), f \u03b2 (x <*> y) = f (\u03b1 \u2192 \u03b2) x <*> f \u03b1 y) :\n    \u21d1(mk f pp ps) = f :=\n  rfl\n\nprotected theorem congr_fun {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) (\u03b7' : applicative_transformation F G) (h : \u03b7 = \u03b7')\n    {\u03b1 : Type u} (x : F \u03b1) : coe_fn \u03b7 \u03b1 x = coe_fn \u03b7' \u03b1 x :=\n  congr_arg (fun (\u03b7'' : applicative_transformation F G) => coe_fn \u03b7'' \u03b1 x) h\n\nprotected theorem congr_arg {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} {x : F \u03b1} {y : F \u03b1} (h : x = y) :\n    coe_fn \u03b7 \u03b1 x = coe_fn \u03b7 \u03b1 y :=\n  congr_arg (fun (z : F \u03b1) => coe_fn \u03b7 \u03b1 z) h\n\ntheorem coe_inj {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    {\u03b7 : applicative_transformation F G} {\u03b7' : applicative_transformation F G} (h : \u21d1\u03b7 = \u21d1\u03b7') :\n    \u03b7 = \u03b7' :=\n  sorry\n\ntheorem ext {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F] {G : Type u \u2192 Type w}\n    [Applicative G] [is_lawful_applicative G] {\u03b7 : applicative_transformation F G}\n    {\u03b7' : applicative_transformation F G}\n    (h : \u2200 (\u03b1 : Type u) (x : F \u03b1), coe_fn \u03b7 \u03b1 x = coe_fn \u03b7' \u03b1 x) : \u03b7 = \u03b7' :=\n  coe_inj (funext fun (\u03b1 : Type u) => funext (h \u03b1))\n\ntheorem ext_iff {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    {\u03b7 : applicative_transformation F G} {\u03b7' : applicative_transformation F G} :\n    \u03b7 = \u03b7' \u2194 \u2200 (\u03b1 : Type u) (x : F \u03b1), coe_fn \u03b7 \u03b1 x = coe_fn \u03b7' \u03b1 x :=\n  { mp := fun (h : \u03b7 = \u03b7') (\u03b1 : Type u) (x : F \u03b1) => h \u25b8 rfl,\n    mpr := fun (h : \u2200 (\u03b1 : Type u) (x : F \u03b1), coe_fn \u03b7 \u03b1 x = coe_fn \u03b7' \u03b1 x) => ext h }\n\ntheorem preserves_pure {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} (x : \u03b1) : coe_fn \u03b7 \u03b1 (pure x) = pure x :=\n  preserves_pure' \u03b7\n\ntheorem preserves_seq {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} {\u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1) :\n    coe_fn \u03b7 \u03b2 (x <*> y) = coe_fn \u03b7 (\u03b1 \u2192 \u03b2) x <*> coe_fn \u03b7 \u03b1 y :=\n  preserves_seq' \u03b7\n\ntheorem preserves_map {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} {\u03b2 : Type u} (x : \u03b1 \u2192 \u03b2) (y : F \u03b1) :\n    coe_fn \u03b7 \u03b2 (x <$> y) = x <$> coe_fn \u03b7 \u03b1 y :=\n  sorry\n\ntheorem preserves_map' {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} {\u03b2 : Type u} (x : \u03b1 \u2192 \u03b2) :\n    coe_fn \u03b7 \u03b2 \u2218 Functor.map x = Functor.map x \u2218 coe_fn \u03b7 \u03b1 :=\n  funext fun (y : F \u03b1) => preserves_map \u03b7 x y\n\n/-- The identity applicative transformation from an applicative functor to itself. -/\ndef id_transformation {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F] :\n    applicative_transformation F F :=\n  mk (fun (\u03b1 : Type u) => id) sorry sorry\n\nprotected instance inhabited {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F] :\n    Inhabited (applicative_transformation F F) :=\n  { default := id_transformation }\n\n/-- The composition of applicative transformations. -/\ndef comp {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F] {G : Type u \u2192 Type w}\n    [Applicative G] [is_lawful_applicative G] {H : Type u \u2192 Type s} [Applicative H]\n    [is_lawful_applicative H] (\u03b7' : applicative_transformation G H)\n    (\u03b7 : applicative_transformation F G) : applicative_transformation F H :=\n  mk (fun (\u03b1 : Type u) (x : F \u03b1) => coe_fn \u03b7' \u03b1 (coe_fn \u03b7 \u03b1 x)) sorry sorry\n\n@[simp] theorem comp_apply {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G] {H : Type u \u2192 Type s}\n    [Applicative H] [is_lawful_applicative H] (\u03b7' : applicative_transformation G H)\n    (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} (x : F \u03b1) :\n    coe_fn (comp \u03b7' \u03b7) \u03b1 x = coe_fn \u03b7' \u03b1 (coe_fn \u03b7 \u03b1 x) :=\n  rfl\n\ntheorem comp_assoc {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G] {H : Type u \u2192 Type s}\n    [Applicative H] [is_lawful_applicative H] {I : Type u \u2192 Type t} [Applicative I]\n    [is_lawful_applicative I] (\u03b7'' : applicative_transformation H I)\n    (\u03b7' : applicative_transformation G H) (\u03b7 : applicative_transformation F G) :\n    comp (comp \u03b7'' \u03b7') \u03b7 = comp \u03b7'' (comp \u03b7' \u03b7) :=\n  rfl\n\n@[simp] theorem comp_id {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) : comp \u03b7 id_transformation = \u03b7 :=\n  ext fun (\u03b1 : Type u) (x : F \u03b1) => rfl\n\n@[simp] theorem id_comp {F : Type u \u2192 Type v} [Applicative F] [is_lawful_applicative F]\n    {G : Type u \u2192 Type w} [Applicative G] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) : comp id_transformation \u03b7 = \u03b7 :=\n  ext fun (\u03b1 : Type u) (x : F \u03b1) => rfl\n\nend applicative_transformation\n\n\n/-- A traversable functor is a functor along with a way to commute\nwith all applicative functors (see `sequence`).  For example, if `t`\nis the traversable functor `list` and `m` is the applicative functor\n`io`, then given a function `f : \u03b1 \u2192 io \u03b2`, the function `functor.map f` is\n`list \u03b1 \u2192 list (io \u03b2)`, but `traverse f` is `list \u03b1 \u2192 io (list \u03b2)`. -/\nclass traversable (t : Type u \u2192 Type u) extends Functor t where\n  traverse :\n    {m : Type u \u2192 Type u} \u2192 [_inst_1 : Applicative m] \u2192 {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 m \u03b2) \u2192 t \u03b1 \u2192 m (t \u03b2)\n\n/-- A traversable functor commutes with all applicative functors. -/\ndef sequence {t : Type u \u2192 Type u} {\u03b1 : Type u} {f : Type u \u2192 Type u} [Applicative f]\n    [traversable t] : t (f \u03b1) \u2192 f (t \u03b1) :=\n  traverse id\n\n/-- A traversable functor is lawful if its `traverse` satisfies a\nnumber of additional properties.  It must send `id.mk` to `id.mk`,\nsend the composition of applicative functors to the composition of the\n`traverse` of each, send each function `f` to `\u03bb x, f <$> x`, and\nsatisfy a naturality condition with respect to applicative\ntransformations. -/\nclass is_lawful_traversable (t : Type u \u2192 Type u) [traversable t] extends is_lawful_functor t where\n  id_traverse : \u2200 {\u03b1 : Type u} (x : t \u03b1), traverse id.mk x = x\n  comp_traverse :\n    \u2200 {F G : Type u \u2192 Type u} [_inst_1_1 : Applicative F] [_inst_2 : Applicative G]\n      [_inst_3 : is_lawful_applicative F] [_inst_4 : is_lawful_applicative G] {\u03b1 \u03b2 \u03b3 : Type u}\n      (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t \u03b1),\n      traverse (functor.comp.mk \u2218 Functor.map f \u2218 g) x =\n        functor.comp.mk (traverse f <$> traverse g x)\n  traverse_eq_map_id :\n    \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t \u03b1), traverse (id.mk \u2218 f) x = id.mk (f <$> x)\n  naturality :\n    \u2200 {F G : Type u \u2192 Type u} [_inst_1_1 : Applicative F] [_inst_2 : Applicative G]\n      [_inst_3 : is_lawful_applicative F] [_inst_4 : is_lawful_applicative G]\n      (\u03b7 : applicative_transformation F G) {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2) (x : t \u03b1),\n      coe_fn \u03b7 (t \u03b2) (traverse f x) = traverse (coe_fn \u03b7 \u03b2 \u2218 f) x\n\nprotected instance id.traversable : traversable id :=\n  traversable.mk fun (_x : Type u_1 \u2192 Type u_1) (_x_1 : Applicative _x) (_x_2 _x_3 : Type u_1) => id\n\nprotected instance id.is_lawful_traversable : is_lawful_traversable id :=\n  is_lawful_traversable.mk sorry sorry sorry sorry\n\nprotected instance option.traversable : traversable Option := traversable.mk option.traverse\n\nprotected instance list.traversable : traversable List := traversable.mk list.traverse\n\nnamespace sum\n\n\n/-- Defines a `traverse` function on the second component of a sum type.\nThis is used to give a `traversable` instance for the functor `\u03c3 \u2295 -`. -/\nprotected def traverse {\u03c3 : Type u} {F : Type u \u2192 Type u} [Applicative F] {\u03b1 : Type u_1}\n    {\u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2) : \u03c3 \u2295 \u03b1 \u2192 F (\u03c3 \u2295 \u03b2) :=\n  sorry\n\nend sum\n\n\nprotected instance sum.traversable {\u03c3 : Type u} : traversable (sum \u03c3) := traversable.mk sum.traverse\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/traversable/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3896029607474689}}
{"text": "\nimport Std.Data.AssocList\nimport Lib.Data.Foldable\n\nnamespace Std.AssocList\nopen Std\n\ndef keys : AssocList \u03b1 \u03b2 \u2192 List \u03b1\n| nil => []\n| cons k x xs => k :: keys xs\n\n@[simp]\ndef length : AssocList \u03b1 \u03b2 \u2192 Nat\n| nil => 0\n| cons _ _ l => l.length.succ\n\n@[simp]\ntheorem keys_mapVal (xs : AssocList k \u03b1) (f : \u03b1 \u2192 \u03b2) :\n  (xs.mapVal f).keys = xs.keys := by\ninduction xs with\n| nil => simp [mapVal]; rfl\n| cons k x xs => simp [keys, mapVal, *]\n\nsection mapFilter\nvariable (f : k \u2192 \u03b1 \u2192 Option \u03b2)\n\ndef mapFilter :\n  AssocList k \u03b1 \u2192 AssocList k \u03b2\n| nil => nil\n| cons i x xs =>\n  let xs' := mapFilter xs\n  match f i x with\n  | none => xs'\n  | some x' => cons i x' xs'\n\n-- @[simp]\n-- theorem keys_mapFilter (xs : AssocList k \u03b1) :\n--   (xs.mapFilter f).keys = xs.keys := by\n-- induction xs with\n-- | nil => simp [mapFilter]; rfl\n-- | cons k x xs =>\n--   simp [mapFilter]\n--   cases f k x <;> simp [keys, *]\ndef foldr (f : \u03b9 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b2) (x\u2080 : \u03b2) : AssocList \u03b9 \u03b1 \u2192 \u03b2\n| nil => x\u2080\n| cons k x xs => f k x (foldr f x\u2080 xs)\n\ninstance : IdxFoldable \u03b9 (AssocList \u03b9) where\n  foldl := foldl\n  foldr := foldr\n\ninstance : LawfulIdxFoldable \u03b9 (AssocList \u03b9) where\n  foldl_sim := by\n    intros; next SIM x\u2080 y\u2080 xs h\u2080 ih =>\n    simp [IdxFoldable.foldl, foldl, Id.run]\n    induction xs generalizing x\u2080 y\u2080\n    <;> simp [foldlM, *]\n\nend mapFilter\n\nsection DecidableEq\n\nvariable [DecidableEq \u03b1]\n\ndef eraseAll (x : \u03b1) : AssocList \u03b1 \u03b2 \u2192 AssocList \u03b1 \u03b2\n| nil => nil\n| cons k v xs =>\n  let xs' := eraseAll x xs\n  if x = k then xs'\n  else cons k v xs'\n\nend DecidableEq\n\nend Std.AssocList\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/AssocList.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.615087848460224, "lm_q1q2_score": 0.3896029562782259}}
{"text": "/-\nCopyright (c) 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Jesse Michael Han\n-/\nimport tactic.hint\n\n/-!\n# The `finish` family of tactics\n\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\neliminate existential quantifiers on the left, and look for contradictions. They rely on ematching\nand congruence closure to try to finish off a goal at the end.\n\nThe procedures *do* split on disjunctions and recreate the smt state for each terminal call, so\nthey are only meant to be used on small, straightforward problems.\n\n## Main definitions\n\nWe provide the following tactics:\n\n* `finish`  -- solves the goal or fails\n* `clarify` -- makes as much progress as possible while not leaving more than one goal\n* `safe`    -- splits freely, finishes off whatever subgoals it can, and leaves the rest\n\nAll accept an optional list of simplifier rules, typically definitions that should be expanded.\n(The equations and identities should not refer to the local context.)\n\n-/\n\ndeclare_trace auto.done\ndeclare_trace auto.finish\n\nnamespace tactic\n\nnamespace interactive\n\nmeta def revert_all := tactic.revert_all\n\nend interactive\n\nend tactic\n\nopen tactic expr\n\nnamespace auto\n\n/-! ### Utilities -/\n\nmeta def whnf_reducible (e : expr) : tactic expr := whnf e reducible\n\n-- stolen from interactive.lean\nmeta def add_simps : simp_lemmas \u2192 list name \u2192 tactic simp_lemmas\n| s []      := return s\n| s (n::ns) := do s' \u2190 s.add_simp n, add_simps s' ns\n\n/--\nConfiguration information for the auto tactics.\n* `(use_simp := tt)`: call the simplifier\n* `(max_ematch_rounds := 20)`: for the \"done\" tactic\n-/\n@[derive decidable_eq, derive inhabited]\nstructure auto_config : Type :=\n(use_simp := tt)\n(max_ematch_rounds := 20)\n\n/-!\n### Preprocess goal.\n\nWe want to move everything to the left of the sequent arrow. For intuitionistic logic,\nwe replace the goal `p` with `\u2200 f, (p \u2192 f) \u2192 f` and introduce.\n-/\n\ntheorem by_contradiction_trick (p : Prop) (h : \u2200 f : Prop, (p \u2192 f) \u2192 f) : p :=\nh p id\n\nmeta def preprocess_goal : tactic unit :=\ndo repeat (intro1 >> skip),\n   tgt \u2190 target >>= whnf_reducible,\n   if (\u00ac (is_false tgt)) then\n     (mk_mapp ``classical.by_contradiction [some tgt]) >>= apply >> intro1 >> skip\n   else\n     skip\n\n/-!\n### Normalize hypotheses\n\nBring conjunctions to the outside (for splitting),\nbring universal quantifiers to the outside (for ematching). The classical normalizer\neliminates `a \u2192 b` in favor of `\u00ac a \u2228 b`.\n\nFor efficiency, we push negations inwards from the top down. (For example, consider\nsimplifying `\u00ac \u00ac (p \u2228 q)`.)\n-/\n\nsection\n\nuniverse u\nvariable  {\u03b1 : Type u}\nvariables (p q : Prop)\nvariable  (s : \u03b1 \u2192 Prop)\n\nlocal attribute [instance, priority 10] classical.prop_decidable\ntheorem not_not_eq : (\u00ac \u00ac p) = p := propext not_not\ntheorem not_and_eq : (\u00ac (p \u2227 q)) = (\u00ac p \u2228 \u00ac q) := propext not_and_distrib\ntheorem not_or_eq : (\u00ac (p \u2228 q)) = (\u00ac p \u2227 \u00ac q) := propext not_or_distrib\ntheorem not_forall_eq : (\u00ac \u2200 x, s x) = (\u2203 x, \u00ac s x) := propext not_forall\ntheorem not_exists_eq : (\u00ac \u2203 x, s x) = (\u2200 x, \u00ac s x) := propext not_exists\ntheorem not_implies_eq : (\u00ac (p \u2192 q)) = (p \u2227 \u00ac q) := propext not_imp\n\ntheorem classical.implies_iff_not_or : (p \u2192 q) \u2194 (\u00ac p \u2228 q) := imp_iff_not_or\n\nend\n\ndef common_normalize_lemma_names : list name :=\n[``bex_def, ``forall_and_distrib, ``exists_imp_distrib, ``or.assoc, ``or.comm, ``or.left_comm,\n  ``and.assoc, ``and.comm, ``and.left_comm]\n\ndef classical_normalize_lemma_names : list name :=\ncommon_normalize_lemma_names ++ [``classical.implies_iff_not_or]\n\n/-- optionally returns an equivalent expression and proof of equivalence -/\nprivate meta def transform_negation_step (cfg : auto_config) (e : expr) :\n  tactic (option (expr \u00d7 expr)) :=\ndo e \u2190 whnf_reducible e,\n   match e with\n   | `(\u00ac %%ne) :=\n      (do ne \u2190 whnf_reducible ne,\n      match ne with\n      | `(\u00ac %%a)      := do pr \u2190 mk_app ``not_not_eq [a],\n                            return (some (a, pr))\n      | `(%%a \u2227 %%b)  := do pr \u2190 mk_app ``not_and_eq [a, b],\n                            return (some (`(\u00ac %%a \u2228 \u00ac %%b), pr))\n      | `(%%a \u2228 %%b)  := do pr \u2190 mk_app ``not_or_eq [a, b],\n                            return (some (`(\u00ac %%a \u2227 \u00ac %%b), pr))\n      | `(Exists %%p) := do pr \u2190 mk_app ``not_exists_eq [p],\n                            `(%%_ = %%e') \u2190 infer_type pr,\n                            return (some (e', pr))\n      | (pi n bi d p) := if p.has_var then do\n                            pr \u2190 mk_app ``not_forall_eq [lam n bi d (expr.abstract_local p n)],\n                            `(%%_ = %%e') \u2190 infer_type pr,\n                            return (some (e', pr))\n                         else do\n                            pr \u2190 mk_app ``not_implies_eq [d, p],\n                            `(%%_ = %%e') \u2190 infer_type pr,\n                            return (some (e', pr))\n      | _             := return none\n      end)\n    | _        := return none\n  end\n\n/-- given an expr `e`, returns a new expression and a proof of equality -/\nprivate meta def transform_negation (cfg : auto_config) : expr \u2192 tactic (option (expr \u00d7 expr)) :=\n\u03bb e, do\n  opr \u2190 transform_negation_step cfg e,\n  match opr with\n  | (some (e', pr)) := do\n    opr' \u2190 transform_negation e',\n    match opr' with\n    | none              := return (some (e', pr))\n    | (some (e'', pr')) := do pr'' \u2190 mk_eq_trans pr pr',\n                              return (some (e'', pr''))\n    end\n  | none            := return none\n  end\n\nmeta def normalize_negations (cfg : auto_config) (h : expr) : tactic unit :=\ndo t \u2190 infer_type h,\n   (_, e, pr) \u2190 simplify_top_down ()\n                   (\u03bb _, \u03bb e, do\n                       oepr \u2190 transform_negation cfg e,\n                       match oepr with\n                       | (some (e', pr)) := return ((), e', pr)\n                       | none            := do pr \u2190 mk_eq_refl e, return ((), e, pr)\n                       end)\n                   t,\n   replace_hyp h e pr,\n   skip\n\nmeta def normalize_hyp (cfg : auto_config) (simps : simp_lemmas) (h : expr) : tactic unit :=\n(do (h, _) \u2190 simp_hyp simps [] h, try (normalize_negations cfg h)) <|>\ntry (normalize_negations cfg h)\n\nmeta def normalize_hyps (cfg : auto_config) : tactic unit :=\ndo simps \u2190 add_simps simp_lemmas.mk classical_normalize_lemma_names,\n   local_context >>= monad.mapm' (normalize_hyp cfg simps)\n\n/-!\n### Eliminate existential quantifiers\n-/\n\n/-- eliminate an existential quantifier if there is one -/\nmeta def eelim : tactic unit :=\ndo ctx \u2190 local_context,\n   first $ ctx.map $ \u03bb h,\n     do t \u2190 infer_type h >>= whnf_reducible,\n        guard (is_app_of t ``Exists),\n        tgt \u2190 target,\n        to_expr ``(@exists.elim _ _ %%tgt %%h) >>= apply,\n        intros,\n        clear h\n\n/-- eliminate all existential quantifiers, fails if there aren't any -/\nmeta def eelims : tactic unit := eelim >> repeat eelim\n\n/-!\n### Substitute if there is a hypothesis `x = t` or `t = x`\n-/\n\n/-- carries out a subst if there is one, fails otherwise -/\nmeta def do_subst : tactic unit :=\ndo ctx \u2190 local_context,\n   first $ ctx.map $ \u03bb h,\n     do t \u2190 infer_type h >>= whnf_reducible,\n        match t with\n        | `(%%a = %%b) := subst h\n        | _            := failed\n        end\n\nmeta def do_substs : tactic unit := do_subst >> repeat do_subst\n\n/-!\n### Split all conjunctions\n-/\n\n/-- Assumes `pr` is a proof of `t`. Adds the consequences of `t` to the context\n and returns `tt` if anything nontrivial has been added. -/\nmeta def add_conjuncts : expr \u2192 expr \u2192 tactic bool :=\n\u03bb pr t,\nlet assert_consequences := \u03bb e t, mcond (add_conjuncts e t) skip (note_anon t e >> skip) in\ndo t' \u2190 whnf_reducible t,\n   match t' with\n   | `(%%a \u2227 %%b) :=\n     do e\u2081 \u2190 mk_app ``and.left [pr],\n        assert_consequences e\u2081 a,\n        e\u2082 \u2190 mk_app ``and.right [pr],\n        assert_consequences e\u2082 b,\n        return tt\n  | `(true) :=\n     do return tt\n  | _ := return ff\nend\n\n/-- return `tt` if any progress is made -/\nmeta def split_hyp (h : expr) : tactic bool :=\ndo t \u2190 infer_type h,\n   mcond (add_conjuncts h t) (clear h >> return tt) (return ff)\n\n/-- return `tt` if any progress is made -/\nmeta def split_hyps_aux : list expr \u2192 tactic bool\n| []        := return ff\n| (h :: hs) := do b\u2081 \u2190 split_hyp h,\n                  b\u2082 \u2190 split_hyps_aux hs,\n                  return (b\u2081 || b\u2082)\n\n/-- fail if no progress is made -/\nmeta def split_hyps : tactic unit := local_context >>= split_hyps_aux >>= guardb\n\n/-!\n### Eagerly apply all the preprocessing rules\n-/\n\n/-- Eagerly apply all the preprocessing rules -/\nmeta def preprocess_hyps (cfg : auto_config) : tactic unit :=\ndo repeat (intro1 >> skip),\n   preprocess_goal,\n   normalize_hyps cfg,\n   repeat (do_substs <|> split_hyps <|> eelim /-<|> self_simplify_hyps-/)\n\n/-!\n### Terminal tactic\n-/\n\n/--\nThe terminal tactic, used to try to finish off goals:\n- Call the contradiction tactic.\n- Open an SMT state, and use ematching and congruence closure, with all the universal\n  statements in the context.\n\nTODO(Jeremy): allow users to specify attribute for ematching lemmas?\n-/\n\nmeta def mk_hinst_lemmas : list expr \u2192 smt_tactic hinst_lemmas\n| []        := -- return hinst_lemmas.mk\n               do get_hinst_lemmas_for_attr `ematch\n| (h :: hs) := do his \u2190 mk_hinst_lemmas hs,\n                  t \u2190 infer_type h,\n                  match t with\n                  | (pi _ _ _ _) :=\n                    do t' \u2190 infer_type t,\n                       if t' = `(Prop) then\n                          (do new_lemma \u2190 hinst_lemma.mk h,\n                             return (hinst_lemmas.add his new_lemma)) <|> return his\n                       else return his\n                  | _ := return his\n                  end\n\nprivate meta def report_invalid_em_lemma {\u03b1 : Type} (n : name) : smt_tactic \u03b1 :=\nfail format!\"invalid ematch lemma '{n}'\"\n\nprivate meta def add_hinst_lemma_from_name (md : transparency) (lhs_lemma : bool) (n : name)\n  (hs : hinst_lemmas) (ref : pexpr) : smt_tactic hinst_lemmas :=\ndo p \u2190 resolve_name n,\n   match p with\n   | expr.const n _ := (do h \u2190 hinst_lemma.mk_from_decl_core md n lhs_lemma,\n                           tactic.save_const_type_info n ref, return $ hs.add h) <|>\n                       (do hs\u2081 \u2190 smt_tactic.mk_ematch_eqn_lemmas_for_core md n,\n                           tactic.save_const_type_info n ref, return $ hs.merge hs\u2081) <|>\n                        report_invalid_em_lemma n\n   | _              := (do e \u2190 to_expr p, h \u2190 hinst_lemma.mk_core md e lhs_lemma,\n                        try (tactic.save_type_info e ref), return $ hs.add h) <|>\n                        report_invalid_em_lemma n\n   end\n\nprivate meta def add_hinst_lemma_from_pexpr (md : transparency) (lhs_lemma : bool)\n  (hs : hinst_lemmas) : pexpr \u2192 smt_tactic hinst_lemmas\n| p@(expr.const c [])          := add_hinst_lemma_from_name md lhs_lemma c hs p\n| p@(expr.local_const c _ _ _) := add_hinst_lemma_from_name md lhs_lemma c hs p\n| p                          := do new_e \u2190 to_expr p, h \u2190 hinst_lemma.mk_core md new_e lhs_lemma,\n                                   return $ hs.add h\n\nprivate meta def add_hinst_lemmas_from_pexprs (md : transparency) (lhs_lemma : bool)\n  (ps : list pexpr) (hs : hinst_lemmas) : smt_tactic hinst_lemmas :=\nlist.mfoldl (add_hinst_lemma_from_pexpr md lhs_lemma) hs ps\n\n/--\n`done` first attempts to close the goal using `contradiction`. If this fails, it creates an\nSMT state and will repeatedly use `ematch` (using `ematch` lemmas in the environment,\nuniversally quantified assumptions, and the supplied lemmas `ps`) and congruence closure.\n-/\nmeta def done (ps : list pexpr) (cfg : auto_config := {}) : tactic unit :=\ndo trace_state_if_enabled `auto.done \"entering done\",\n   contradiction <|>\n   (solve1 $\n     (do revert_all,\n         using_smt\n         (do smt_tactic.intros,\n             ctx \u2190 local_context,\n             hs \u2190 mk_hinst_lemmas ctx,\n             hs' \u2190 add_hinst_lemmas_from_pexprs reducible ff ps hs,\n             smt_tactic.iterate_at_most cfg.max_ematch_rounds\n               (smt_tactic.ematch_using hs' >> smt_tactic.try smt_tactic.close))))\n/-!\n### Tactics that perform case splits\n-/\n@[derive decidable_eq, derive inhabited]\ninductive case_option\n| force        -- fail unless all goals are solved\n| at_most_one  -- leave at most one goal\n| accept       -- leave as many goals as necessary\n\nprivate meta def case_cont (s : case_option) (cont : case_option \u2192 tactic unit) : tactic unit :=\ndo match s with\n   | case_option.force := cont case_option.force >> cont case_option.force\n   | case_option.at_most_one :=\n       -- if the first one succeeds, commit to it, and try the second\n       (mcond (cont case_option.force >> return tt) (cont case_option.at_most_one) skip) <|>\n       -- otherwise, try the second\n       (swap >> cont case_option.force >> cont case_option.at_most_one)\n   | case_option.accept := focus' [cont case_option.accept, cont case_option.accept]\n   end\n\n-- three possible outcomes:\n--   finds something to case, the continuations succeed ==> returns tt\n--   finds something to case, the continutations fail ==> fails\n--   doesn't find anything to case ==> returns ff\nmeta def case_hyp (h : expr) (s : case_option) (cont : case_option \u2192 tactic unit) : tactic bool :=\ndo t \u2190 infer_type h,\n   match t with\n   | `(%%a \u2228 %%b) := cases h >> case_cont s cont >> return tt\n   | _            := return ff\n   end\n\nmeta def case_some_hyp_aux (s : case_option) (cont : case_option \u2192 tactic unit) :\n  list expr \u2192 tactic bool\n| []      := return ff\n| (h::hs) := mcond (case_hyp h s cont) (return tt) (case_some_hyp_aux hs)\n\nmeta def case_some_hyp (s : case_option) (cont : case_option \u2192 tactic unit) : tactic bool :=\nlocal_context >>= case_some_hyp_aux s cont\n\n/-!\n### The main tactics\n-/\n\n/--\n`safe_core s ps cfg opt` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `s`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `ps`) and congruence closure.\n\n`safe_core` is complete for propositional logic. Depending on the form of `opt`\nit will:\n\n- (if `opt` is `case_option.force`) fail if it does not close the goal,\n- (if `opt` is `case_option.at_most_one`) fail if it produces more than one goal, and\n- (if `opt` is `case_option.accept`) ignore the number of goals it produces.\n-/\nmeta def safe_core (s : simp_lemmas \u00d7 list name) (ps : list pexpr) (cfg : auto_config) :\n  case_option \u2192 tactic unit :=\n\u03bb co, focus1 $\ndo trace_state_if_enabled `auto.finish \"entering safe_core\",\n   if cfg.use_simp then do\n     trace_if_enabled `auto.finish \"simplifying hypotheses\",\n     simp_all s.1 s.2 { fail_if_unchanged := ff },\n     trace_state_if_enabled `auto.finish \"result:\"\n   else skip,\n   tactic.done <|>\n   do trace_if_enabled `auto.finish \"preprocessing hypotheses\",\n      preprocess_hyps cfg,\n      trace_state_if_enabled `auto.finish \"result:\",\n      done ps cfg <|>\n        (mcond (case_some_hyp co safe_core)\n          skip\n          (match co with\n            | case_option.force       := done ps cfg\n            | case_option.at_most_one := try (done ps cfg)\n            | case_option.accept      := try (done ps cfg)\n            end))\n\n/--\n`clarify` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.at_most_one`.\n-/\nmeta def clarify (s : simp_lemmas \u00d7 list name) (ps : list pexpr)\n  (cfg : auto_config := {}) : tactic unit := safe_core s ps cfg case_option.at_most_one\n\n/--\n`safe` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.accept`.\n-/\nmeta def safe (s : simp_lemmas \u00d7 list name) (ps : list pexpr)\n  (cfg : auto_config := {}) : tactic unit := safe_core s ps cfg case_option.accept\n\n/--\n`finish` is `safe_core`, but with the `(opt : case_option)`\nparameter fixed at `case_option.force`.\n-/\nmeta def finish (s : simp_lemmas \u00d7 list name) (ps : list pexpr)\n  (cfg : auto_config := {}) : tactic unit := safe_core s ps cfg case_option.force\n\nend auto\n\n/-! ### interactive versions -/\n\nopen auto\n\nnamespace tactic\nnamespace interactive\n\nsetup_tactic_parser\n\n/--\n`clarify [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`clarify` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`clarify` will fail if it produces more than one goal.\n-/\nmeta def clarify (hs : parse simp_arg_list) (ps : parse (tk \"using\" *> pexpr_list_or_texpr)?)\n  (cfg : auto_config := {}) : tactic unit :=\ndo s \u2190 mk_simp_set ff [] hs,\n   auto.clarify s (ps.get_or_else []) cfg\n\n/--\n`safe [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`safe` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`safe` ignores the number of goals it produces, and should never fail.\n-/\nmeta def safe (hs : parse simp_arg_list) (ps : parse (tk \"using\" *> pexpr_list_or_texpr)?)\n  (cfg : auto_config := {}) : tactic unit :=\ndo s \u2190 mk_simp_set ff [] hs,\n   auto.safe s (ps.get_or_else []) cfg\n\n/--\n`finish [h1,...,hn] using [e1,...,en]` negates the goal, normalizes hypotheses\n(by splitting conjunctions, eliminating existentials, pushing negations inwards,\nand calling `simp` with the supplied lemmas `h1,...,hn`), and then tries `contradiction`.\n\nIf this fails, it will create an SMT state and repeatedly use `ematch`\n(using `ematch` lemmas in the environment, universally quantified assumptions,\nand the supplied lemmas `e1,...,en`) and congruence closure.\n\n`finish` is complete for propositional logic.\n\nEither of the supplied simp lemmas or the supplied ematch lemmas are optional.\n\n`finish` will fail if it does not close the goal.\n-/\nmeta def finish (hs : parse simp_arg_list) (ps : parse (tk \"using\" *> pexpr_list_or_texpr)?)\n  (cfg : auto_config := {}) : tactic unit :=\ndo s \u2190 mk_simp_set ff [] hs,\n   auto.finish s (ps.get_or_else []) cfg\n\nadd_hint_tactic \"finish\"\n\n/--\nThese tactics do straightforward things: they call the simplifier, split conjunctive assumptions,\neliminate existential quantifiers on the left, and look for contradictions. They rely on ematching\nand congruence closure to try to finish off a goal at the end.\n\nThe procedures *do* split on disjunctions and recreate the smt state for each terminal call, so\nthey are only meant to be used on small, straightforward problems.\n\n* `finish`:  solves the goal or fails\n* `clarify`: makes as much progress as possible while not leaving more than one goal\n* `safe`:    splits freely, finishes off whatever subgoals it can, and leaves the rest\n\nAll accept an optional list of simplifier rules, typically definitions that should be expanded.\n(The equations and identities should not refer to the local context.) All also accept an optional\nlist of `ematch` lemmas, which must be preceded by `using`.\n-/\nadd_tactic_doc\n{ name        := \"finish / clarify / safe\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.finish, `tactic.interactive.clarify,\n                  `tactic.interactive.safe],\n  tags        := [\"logic\", \"finishing\"] }\n\n\nend interactive\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/finish.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.38960295223446845}}
{"text": "import category_theory.functor.left_derived\nimport data.matrix.notation\n\nimport for_mathlib.homological_complex\nimport for_mathlib.horseshoe\nimport for_mathlib.les_homology\nimport for_mathlib.split_exact\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen short_exact_sequence\n\nuniverses w v u\n\nnamespace category_theory\n\nvariables {C : Type u} [category.{v} C] {D : Type*} [category D]\n\n-- Importing `category_theory.abelian.projective` and assuming\n-- `[abelian C] [enough_projectives C] [abelian D]` suffices to acquire all the following:\n-- variables [preadditive C] [has_zero_object C] [has_equalizers C]\n-- variables [has_images C] [has_projective_resolutions C]\n-- variables [preadditive D] [has_zero_object D] [has_equalizers D] [has_cokernels D]\n-- variables [has_images D] [has_image_maps D]\n\nvariables [abelian C] [enough_projectives C] [abelian D]\n\nnamespace functor\nnamespace left_derived\n\nvariables (F : C \u2964 D)\n\n/-- We can compute a left derived functor using a chosen projective resolution. -/\n@[simps]\ndef functor.left_derived_obj_iso' (F : C \u2964 D) [F.additive] (n : \u2115)\n  (X : C) (P : chain_complex C \u2115) (\u03c0 : P \u27f6 ((chain_complex.single\u2080 C).obj X))\n  (h : P.is_projective_resolution X \u03c0) :\n  (F.left_derived n).obj X \u2245\n    (homology_functor D _ n).obj ((F.map_homological_complex _).obj P) :=\n(F.left_derived_obj_iso n (h.mk_ProjectiveResolution P X \u03c0) : _)\n\n/-- We can compute a left derived functor on a morphism using a lift of that morphism to a chain map\nbetween chosen projective resolutions. -/\nlemma functor.left_derived_map_eq' (F : C \u2964 D) [F.additive] (n : \u2115) (X Y : C) (f : X \u27f6 Y)\n  (PX : chain_complex C \u2115) (\u03c0X : PX \u27f6 ((chain_complex.single\u2080 C).obj X))\n  (PY : chain_complex C \u2115) (\u03c0Y : PY \u27f6 ((chain_complex.single\u2080 C).obj Y)) (g : PX \u27f6 PY)\n  (hX : PX.is_projective_resolution X \u03c0X) (hY : PY.is_projective_resolution Y \u03c0Y)\n  (w : g \u226b \u03c0Y = \u03c0X \u226b (chain_complex.single\u2080 C).map f) :\n  (F.left_derived n).map f =\n  (functor.left_derived_obj_iso' F n X PX \u03c0X hX).hom \u226b\n    (homology_functor D _ n).map ((F.map_homological_complex _).map g) \u226b\n    (functor.left_derived_obj_iso' F n Y PY \u03c0Y hY).inv :=\nbegin\n  let PXr := (hX.mk_ProjectiveResolution PX X \u03c0X),\n  let PYr := (hY.mk_ProjectiveResolution PY Y \u03c0Y),\n  let gr : PXr.complex \u27f6 PYr.complex := g,\n  simpa using functor.left_derived_map_eq F n f gr w,\nend\n.\n\nabbreviation \u03b1 [F.additive] :\n  ((short_exact_sequence.Fst C \u22d9 F).map_homological_complex (complex_shape.down \u2115)) \u27f6\n    ((short_exact_sequence.Snd C \u22d9 F).map_homological_complex (complex_shape.down \u2115)) :=\nnat_trans.map_homological_complex (whisker_right (short_exact_sequence.f_nat _) _) _\n\nabbreviation \u03b2 [F.additive] :\n  ((short_exact_sequence.Snd C \u22d9 F).map_homological_complex (complex_shape.down \u2115)) \u27f6\n    ((short_exact_sequence.Trd C \u22d9 F).map_homological_complex (complex_shape.down \u2115)) :=\n    nat_trans.map_homological_complex (whisker_right (short_exact_sequence.g_nat _) _) _\n\nlemma exact_\u03b1_\u03b2_horseshoe [F.additive] (A : short_exact_sequence C) (n : \u2115) :\n  short_exact (((\u03b1 F).app (horseshoe A)).f n) (((\u03b2 F).app (horseshoe A)).f n) :=\nbegin\n  apply split.short_exact,\n  apply split.map,\n  obtain \u27e8\u03c6, \u03c7, h1, h2, h3, h4\u27e9 := horseshoe_split A n,\n  exact \u27e8\u27e8\u03c6, \u03c7, h1, h2, short_exact_sequence.f_comp_g _, h3, h4\u27e9\u27e9,\nend\n\ndef \u03b4 [F.additive] (n : \u2115) (A : short_exact_sequence C) :\n  (F.left_derived (n+1)).obj A.3 \u27f6 (F.left_derived n).obj A.1 :=\nbegin\n  let f\u2083 := functor.left_derived_obj_iso' F (n+1) _ _ _ (horseshoe_is_projective_resolution\u2083 A),\n  let f\u2081 := functor.left_derived_obj_iso' F n _ _ _ (horseshoe_is_projective_resolution\u2081 A),\n  exact f\u2083.hom \u226b (homological_complex.\u03b4 _ _ (exact_\u03b1_\u03b2_horseshoe F A) _ _ rfl) \u226b f\u2081.symm.hom,\nend\n\nlemma exact_of_short_exact [F.additive] (n : \u2115) (A : short_exact_sequence C) :\n  exact ((F.left_derived n).map A.f) ((F.left_derived n).map A.g) :=\nbegin\n  have := ((homological_complex.six_term_exact_seq _ _\n    (exact_\u03b1_\u03b2_horseshoe F A) _ n rfl).drop 3).pair,\n  have H\u2081\u2082 := functor.left_derived_map_eq' F n A.1 A.2 A.f\n    ((homological_complex.Fst C).obj (horseshoe A)) (horseshoe_to_single\u2081 A)\n    ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n    ((homological_complex.Fst_Snd C).app (horseshoe A))\n    (horseshoe_is_projective_resolution\u2081 A)\n    (horseshoe_is_projective_resolution\u2082 A) _,\n  have H\u2082\u2083 := functor.left_derived_map_eq' F n A.2 A.3 A.g\n    ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n    ((homological_complex.Trd C).obj (horseshoe A)) (horseshoe_to_single\u2083 A)\n    ((homological_complex.Snd_Trd C).app (horseshoe A))\n    (horseshoe_is_projective_resolution\u2082 A)\n    (horseshoe_is_projective_resolution\u2083 A) _,\n  refine preadditive.exact_of_iso_of_exact' _ _ _ _ _ _ _ _ _ this,\n  { let := functor.left_derived_obj_iso' F n A.1\n      ((homological_complex.Fst C).obj (horseshoe A)) (horseshoe_to_single\u2081 A)\n      (horseshoe_is_projective_resolution\u2081 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F n A.2\n      ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n      (horseshoe_is_projective_resolution\u2082 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F n A.3\n      ((homological_complex.Trd C).obj (horseshoe A)) (horseshoe_to_single\u2083 A)\n      (horseshoe_is_projective_resolution\u2083 A),\n    exact this.symm },\n  { rw [H\u2081\u2082, \u2190 category.assoc, iso.symm_hom, iso.inv_hom_id, category.id_comp],\n    simpa },\n  { rw [H\u2082\u2083, \u2190 category.assoc, iso.symm_hom, iso.inv_hom_id, category.id_comp],\n    simpa },\n  { ext i,\n    apply horseshoe_g_comp_to_single\u2083_f, },\n  { ext i,\n    apply horseshoe_f_comp_to_single\u2082_f }\nend\n\nlemma exact_of_short_exact.\u03b4_right [F.additive] (n : \u2115) (A : short_exact_sequence C) :\n  exact ((F.left_derived (n + 1)).map A.g) (\u03b4 F n A) :=\nbegin\n  have := ((homological_complex.six_term_exact_seq _ _\n    (exact_\u03b1_\u03b2_horseshoe F A) _ n rfl).drop 1).pair,\n  have H\u2082\u2083 := functor.left_derived_map_eq' F (n+1) A.2 A.3 A.g\n    ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n    ((homological_complex.Trd C).obj (horseshoe A)) (horseshoe_to_single\u2083 A)\n    ((homological_complex.Snd_Trd C).app (horseshoe A))\n    (horseshoe_is_projective_resolution\u2082 A)\n    (horseshoe_is_projective_resolution\u2083 A) _,\n  refine preadditive.exact_of_iso_of_exact' _ _ _ _ _ _ _ _ _ this,\n  { let := functor.left_derived_obj_iso' F (n+1) A.2\n      ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n      (horseshoe_is_projective_resolution\u2082 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F (n+1) A.3\n      ((homological_complex.Trd C).obj (horseshoe A)) (horseshoe_to_single\u2083 A)\n      (horseshoe_is_projective_resolution\u2083 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F n A.1\n      ((homological_complex.Fst C).obj (horseshoe A)) (horseshoe_to_single\u2081 A)\n      (horseshoe_is_projective_resolution\u2081 A),\n    exact this.symm },\n  { rw [H\u2082\u2083, \u2190 category.assoc, iso.symm_hom, iso.inv_hom_id, category.id_comp],\n    simpa },\n  { unfold \u03b4,\n    dsimp,\n    simp only [category.assoc, iso.inv_hom_id_assoc], },\n  { ext i,\n    apply horseshoe_g_comp_to_single\u2083_f }\nend\n\nlemma exact_of_short_exact.\u03b4_left [F.additive] (n : \u2115) (A : short_exact_sequence C) :\n  exact (\u03b4 F n A) ((F.left_derived n).map A.f) :=\nbegin\n  have := ((homological_complex.six_term_exact_seq _ _\n    (exact_\u03b1_\u03b2_horseshoe F A) _ n rfl).drop 2).pair,\n  have H\u2081\u2082 := functor.left_derived_map_eq' F n A.1 A.2 A.f\n    ((homological_complex.Fst C).obj (horseshoe A)) (horseshoe_to_single\u2081 A)\n    ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n    ((homological_complex.Fst_Snd C).app (horseshoe A))\n    (horseshoe_is_projective_resolution\u2081 A)\n    (horseshoe_is_projective_resolution\u2082 A) _,\n  refine preadditive.exact_of_iso_of_exact' _ _ _ _ _ _ _ _ _ this,\n  { let := functor.left_derived_obj_iso' F (n+1) A.3\n      ((homological_complex.Trd C).obj (horseshoe A)) (horseshoe_to_single\u2083 A)\n      (horseshoe_is_projective_resolution\u2083 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F n A.1\n      ((homological_complex.Fst C).obj (horseshoe A)) (horseshoe_to_single\u2081 A)\n      (horseshoe_is_projective_resolution\u2081 A),\n    exact this.symm },\n  { let := functor.left_derived_obj_iso' F n A.2\n      ((homological_complex.Snd C).obj (horseshoe A)) (horseshoe_to_single\u2082 A)\n      (horseshoe_is_projective_resolution\u2082 A),\n    exact this.symm },\n  { unfold \u03b4,\n    dsimp,\n    simp only [category.assoc, iso.inv_hom_id_assoc], },\n  { rw [H\u2081\u2082, \u2190 category.assoc, iso.symm_hom, iso.inv_hom_id, category.id_comp],\n    simpa },\n  { ext i,\n    apply horseshoe_f_comp_to_single\u2082_f }\nend\n\nlemma six_term_exact_seq [F.additive] (n : \u2115) (A : short_exact_sequence C) :\n  exact_seq D [\n    (F.left_derived (n+1)).map A.f, (F.left_derived (n+1)).map A.g,\n    \u03b4 F n A,\n    (F.left_derived n).map A.f, (F.left_derived n).map A.g] :=\n(exact_of_short_exact _ _ _).cons $\n(exact_of_short_exact.\u03b4_right _ _ _).cons $\n(exact_of_short_exact.\u03b4_left _ _ _).cons $\n(exact_of_short_exact _ _ _).exact_seq\n\nend left_derived\nend functor\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3895321744792303}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n--Prove that Homomorphisms between two finite abelian groups are isomorphic to homomorphisms between their duals in Lean mathlib\n \n--OUTPUT 1\ntheorem dual_hom_iso (G H : Type*) [group G] [group H] [fintype G] [fintype H] [decidable_eq G] [decidable_eq H] :\n  (G \u2192* H) \u2243 (G^* \u2192* H^*) :=\nbegin\n  apply equiv.of_bijective (\u03bb f, f.to_add_monoid_hom.to_linear_map)\n    (\u03bb f, f.to_add_monoid_hom.to_add_monoid_hom)\n    (\u03bb f, by { ext, simp [f.to_add_monoid_hom.to_linear_map, f.to_add_monoid_hom.to_add_monoid_hom] })\n    (\u03bb f, by { ext, simp [f.to_add_monoid_hom.to_linear_map, f.to_add_monoid_hom.to_add_monoid_hom] })\n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/Zero-shot-theorem-names/Zero-Shot-prompts_temperature_0_max_tokens_2000_n_1/clean_files/Homomorphisms between two finite abelian groups are isomorphic to homomorphisms between their duals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3895321675725952}}
{"text": "\nuniverses u\nvariable {\u03b1 : Type u} {p : \u03b1 \u2192 Prop}\n\ntheorem ex (a : \u03b1) (h1 h2 : p a) (h : Subtype.mk a h1 = Subtype.mk a h1) : Subtype.mk a h1 = Subtype.mk a h2 :=\nh\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/proofIrrelFVar.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3895321675725951}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.limits.shapes.strong_epi\nimport Mathlib.category_theory.limits.shapes.pullbacks\nimport Mathlib.PostPort\n\nuniverses v\u2081 u\u2081 l \n\nnamespace Mathlib\n\n/-!\n# Definitions and basic properties of regular monomorphisms and epimorphisms.\n\nA regular monomorphism is a morphism that is the equalizer of some parallel pair.\n\nWe give the constructions\n* `split_mono \u2192 regular_mono` and\n* `regular_mono \u2192 mono`\nas well as the dual constructions for regular epimorphisms. Additionally, we give the\nconstruction\n* `regular_epi \u27f6 strong_epi`.\n\n-/\n\nnamespace category_theory\n\n\n/-- A regular monomorphism is a morphism which is the equalizer of some parallel pair. -/\nclass regular_mono {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) \nwhere\n  Z : C\n  left : Y \u27f6 Z\n  right : Y \u27f6 Z\n  w : f \u226b left = f \u226b right\n  is_limit : limits.is_limit (limits.fork.of_\u03b9 f w)\n\ntheorem regular_mono.w_assoc {C : Type u\u2081} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [c : regular_mono f] {X' : C} (f' : regular_mono.Z f \u27f6 X') : f \u226b regular_mono.left \u226b f' = f \u226b regular_mono.right \u226b f' := sorry\n\n/-- Every regular monomorphism is a monomorphism. -/\nprotected instance regular_mono.mono {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [regular_mono f] : mono f :=\n  limits.mono_of_is_limit_parallel_pair regular_mono.is_limit\n\nprotected instance equalizer_regular {C : Type u\u2081} [category C] {X : C} {Y : C} (g : X \u27f6 Y) (h : X \u27f6 Y) [limits.has_limit (limits.parallel_pair g h)] : regular_mono (limits.equalizer.\u03b9 g h) :=\n  regular_mono.mk Y g h (limits.equalizer.condition g h)\n    (limits.fork.is_limit.mk (limits.fork.of_\u03b9 (limits.equalizer.\u03b9 g h) (limits.equalizer.condition g h))\n      (fun (s : limits.fork g h) => limits.limit.lift (limits.parallel_pair g h) s) sorry sorry)\n\n/-- Every split monomorphism is a regular monomorphism. -/\nprotected instance regular_mono.of_split_mono {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [split_mono f] : regular_mono f :=\n  regular_mono.mk Y \ud835\udfd9 (retraction f \u226b f) (limits.cone_of_split_mono._proof_1 f) (limits.split_mono_equalizes f)\n\n/-- If `f` is a regular mono, then any map `k : W \u27f6 Y` equalizing `regular_mono.left` and\n    `regular_mono.right` induces a morphism `l : W \u27f6 X` such that `l \u226b f = k`. -/\ndef regular_mono.lift' {C : Type u\u2081} [category C] {X : C} {Y : C} {W : C} (f : X \u27f6 Y) [regular_mono f] (k : W \u27f6 Y) (h : k \u226b regular_mono.left = k \u226b regular_mono.right) : Subtype fun (l : W \u27f6 X) => l \u226b f = k :=\n  limits.fork.is_limit.lift' regular_mono.is_limit k h\n\n/--\nThe second leg of a pullback cone is a regular monomorphism if the right component is too.\n\nSee also `pullback.snd_of_mono` for the basic monomorphism version, and\n`regular_of_is_pullback_fst_of_regular` for the flipped version.\n-/\ndef regular_of_is_pullback_snd_of_regular {C : Type u\u2081} [category C] {P : C} {Q : C} {R : C} {S : C} {f : P \u27f6 Q} {g : P \u27f6 R} {h : Q \u27f6 S} {k : R \u27f6 S} [hr : regular_mono h] (comm : f \u226b h = g \u226b k) (t : limits.is_limit (limits.pullback_cone.mk f g comm)) : regular_mono g := sorry\n\n/--\nThe first leg of a pullback cone is a regular monomorphism if the left component is too.\n\nSee also `pullback.fst_of_mono` for the basic monomorphism version, and\n`regular_of_is_pullback_snd_of_regular` for the flipped version.\n-/\ndef regular_of_is_pullback_fst_of_regular {C : Type u\u2081} [category C] {P : C} {Q : C} {R : C} {S : C} {f : P \u27f6 Q} {g : P \u27f6 R} {h : Q \u27f6 S} {k : R \u27f6 S} [hr : regular_mono k] (comm : f \u226b h = g \u226b k) (t : limits.is_limit (limits.pullback_cone.mk f g comm)) : regular_mono f :=\n  regular_of_is_pullback_snd_of_regular sorry (limits.pullback_cone.flip_is_limit t)\n\n/-- A regular monomorphism is an isomorphism if it is an epimorphism. -/\ndef is_iso_of_regular_mono_of_epi {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [regular_mono f] [e : epi f] : is_iso f :=\n  limits.is_iso_limit_cone_parallel_pair_of_epi regular_mono.is_limit\n\n/-- A regular epimorphism is a morphism which is the coequalizer of some parallel pair. -/\nclass regular_epi {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) \nwhere\n  W : C\n  left : W \u27f6 X\n  right : W \u27f6 X\n  w : left \u226b f = right \u226b f\n  is_colimit : limits.is_colimit (limits.cofork.of_\u03c0 f w)\n\ntheorem regular_epi.w_assoc {C : Type u\u2081} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [c : regular_epi f] {X' : C} (f' : Y \u27f6 X') : regular_epi.left \u226b f \u226b f' = regular_epi.right \u226b f \u226b f' := sorry\n\n/-- Every regular epimorphism is an epimorphism. -/\nprotected instance regular_epi.epi {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [regular_epi f] : epi f :=\n  limits.epi_of_is_colimit_parallel_pair regular_epi.is_colimit\n\nprotected instance coequalizer_regular {C : Type u\u2081} [category C] {X : C} {Y : C} (g : X \u27f6 Y) (h : X \u27f6 Y) [limits.has_colimit (limits.parallel_pair g h)] : regular_epi (limits.coequalizer.\u03c0 g h) :=\n  regular_epi.mk X g h (limits.coequalizer.condition g h)\n    (limits.cofork.is_colimit.mk (limits.cofork.of_\u03c0 (limits.coequalizer.\u03c0 g h) (limits.coequalizer.condition g h))\n      (fun (s : limits.cofork g h) => limits.colimit.desc (limits.parallel_pair g h) s) sorry sorry)\n\n/-- Every split epimorphism is a regular epimorphism. -/\nprotected instance regular_epi.of_split_epi {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [split_epi f] : regular_epi f :=\n  regular_epi.mk X \ud835\udfd9 (f \u226b section_ f) (limits.cocone_of_split_epi._proof_1 f) (limits.split_epi_coequalizes f)\n\n/-- If `f` is a regular epi, then every morphism `k : X \u27f6 W` coequalizing `regular_epi.left` and\n    `regular_epi.right` induces `l : Y \u27f6 W` such that `f \u226b l = k`. -/\ndef regular_epi.desc' {C : Type u\u2081} [category C] {X : C} {Y : C} {W : C} (f : X \u27f6 Y) [regular_epi f] (k : X \u27f6 W) (h : regular_epi.left \u226b k = regular_epi.right \u226b k) : Subtype fun (l : Y \u27f6 W) => f \u226b l = k :=\n  limits.cofork.is_colimit.desc' regular_epi.is_colimit k h\n\n/--\nThe second leg of a pushout cocone is a regular epimorphism if the right component is too.\n\nSee also `pushout.snd_of_epi` for the basic epimorphism version, and\n`regular_of_is_pushout_fst_of_regular` for the flipped version.\n-/\ndef regular_of_is_pushout_snd_of_regular {C : Type u\u2081} [category C] {P : C} {Q : C} {R : C} {S : C} {f : P \u27f6 Q} {g : P \u27f6 R} {h : Q \u27f6 S} {k : R \u27f6 S} [gr : regular_epi g] (comm : f \u226b h = g \u226b k) (t : limits.is_colimit (limits.pushout_cocone.mk h k comm)) : regular_epi h := sorry\n\n/--\nThe first leg of a pushout cocone is a regular epimorphism if the left component is too.\n\nSee also `pushout.fst_of_epi` for the basic epimorphism version, and\n`regular_of_is_pushout_snd_of_regular` for the flipped version.\n-/\ndef regular_of_is_pushout_fst_of_regular {C : Type u\u2081} [category C] {P : C} {Q : C} {R : C} {S : C} {f : P \u27f6 Q} {g : P \u27f6 R} {h : Q \u27f6 S} {k : R \u27f6 S} [fr : regular_epi f] (comm : f \u226b h = g \u226b k) (t : limits.is_colimit (limits.pushout_cocone.mk h k comm)) : regular_epi k :=\n  regular_of_is_pushout_snd_of_regular sorry (limits.pushout_cocone.flip_is_colimit t)\n\n/-- A regular epimorphism is an isomorphism if it is a monomorphism. -/\ndef is_iso_of_regular_epi_of_mono {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [regular_epi f] [m : mono f] : is_iso f :=\n  limits.is_iso_limit_cocone_parallel_pair_of_epi regular_epi.is_colimit\n\nprotected instance strong_epi_of_regular_epi {C : Type u\u2081} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [regular_epi f] : strong_epi f := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/regular_mono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3895321606659598}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Markus Himmel\n-/\nimport category_theory.epi_mono\nimport category_theory.limits.has_limits\n\n/-!\n# Equalizers and coequalizers\n\nThis file defines (co)equalizers as special cases of (co)limits.\n\nAn equalizer is the categorical generalization of the subobject {a \u2208 A | f(a) = g(a)} known\nfrom abelian groups or modules. It is a limit cone over the diagram formed by `f` and `g`.\n\nA coequalizer is the dual concept.\n\n## Main definitions\n\n* `walking_parallel_pair` is the indexing category used for (co)equalizer_diagrams\n* `parallel_pair` is a functor from `walking_parallel_pair` to our category `C`.\n* a `fork` is a cone over a parallel pair.\n  * there is really only one interesting morphism in a fork: the arrow from the vertex of the fork\n    to the domain of f and g. It is called `fork.\u03b9`.\n* an `equalizer` is now just a `limit (parallel_pair f g)`\n\nEach of these has a dual.\n\n## Main statements\n\n* `equalizer.\u03b9_mono` states that every equalizer map is a monomorphism\n* `is_iso_limit_cone_parallel_pair_of_self` states that the identity on the domain of `f` is an\n  equalizer of `f` and `f`.\n\n## Implementation notes\nAs with the other special shapes in the limits library, all the definitions here are given as\n`abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about\ngeneral limits can be used.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 1*][borceux-vol1]\n-/\n\nnoncomputable theory\n\nopen category_theory opposite\n\nnamespace category_theory.limits\n\nlocal attribute [tidy] tactic.case_bash\n\nuniverses v v\u2082 u u\u2082\n\n/-- The type of objects for the diagram indexing a (co)equalizer. -/\n@[derive decidable_eq, derive inhabited] inductive walking_parallel_pair : Type\n| zero | one\n\nopen walking_parallel_pair\n\n/-- The type family of morphisms for the diagram indexing a (co)equalizer. -/\n@[derive decidable_eq] inductive walking_parallel_pair_hom :\n  walking_parallel_pair \u2192 walking_parallel_pair \u2192 Type\n| left : walking_parallel_pair_hom zero one\n| right : walking_parallel_pair_hom zero one\n| id : \u03a0 X : walking_parallel_pair, walking_parallel_pair_hom X X\n\n/-- Satisfying the inhabited linter -/\ninstance : inhabited (walking_parallel_pair_hom zero one) :=\n{ default := walking_parallel_pair_hom.left }\n\nopen walking_parallel_pair_hom\n\n/-- Composition of morphisms in the indexing diagram for (co)equalizers. -/\ndef walking_parallel_pair_hom.comp :\n  \u03a0 (X Y Z : walking_parallel_pair)\n    (f : walking_parallel_pair_hom X Y) (g : walking_parallel_pair_hom Y Z),\n    walking_parallel_pair_hom X Z\n  | _ _ _ (id _) h := h\n  | _ _ _ left   (id one) := left\n  | _ _ _ right  (id one) := right\n.\n\ninstance walking_parallel_pair_hom_category : small_category walking_parallel_pair :=\n{ hom  := walking_parallel_pair_hom,\n  id   := walking_parallel_pair_hom.id,\n  comp := walking_parallel_pair_hom.comp }\n\n@[simp]\nlemma walking_parallel_pair_hom_id (X : walking_parallel_pair) :\n  walking_parallel_pair_hom.id X = \ud835\udfd9 X :=\nrfl\n\n/--\nThe functor `walking_parallel_pair \u2964 walking_parallel_pair\u1d52\u1d56` sending left to left and right to\nright.\n-/\ndef walking_parallel_pair_op : walking_parallel_pair \u2964 walking_parallel_pair\u1d52\u1d56 :=\n{ obj := (\u03bb x, op $ by { cases x, exacts [one, zero] }),\n  map := \u03bb i j f, by { cases f; apply quiver.hom.op, exacts [left, right,\n    walking_parallel_pair_hom.id _] },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_); refl } }\n\n@[simp] lemma walking_parallel_pair_op_zero :\n  walking_parallel_pair_op.obj zero = op one := rfl\n@[simp] lemma walking_parallel_pair_op_one :\n  walking_parallel_pair_op.obj one = op zero := rfl\n@[simp] lemma walking_parallel_pair_op_left :\n  walking_parallel_pair_op.map left = @quiver.hom.op _ _ zero one left := rfl\n@[simp] lemma walking_parallel_pair_op_right :\n  walking_parallel_pair_op.map right = @quiver.hom.op _ _ zero one right := rfl\n\n/--\nThe equivalence `walking_parallel_pair \u2964 walking_parallel_pair\u1d52\u1d56` sending left to left and right to\nright.\n-/\n@[simps functor inverse]\ndef walking_parallel_pair_op_equiv : walking_parallel_pair \u224c walking_parallel_pair\u1d52\u1d56 :=\n{ functor := walking_parallel_pair_op,\n  inverse := walking_parallel_pair_op.left_op,\n  unit_iso := nat_iso.of_components (\u03bb j, eq_to_iso (by { cases j; refl }))\n    (by { rintros (_|_) (_|_) (_|_|_); refl }),\n  counit_iso := nat_iso.of_components (\u03bb j, eq_to_iso\n    (by { induction j using opposite.rec, cases j; refl }))\n    (\u03bb i j f, by { induction i using opposite.rec, induction j using opposite.rec,\n      let g := f.unop, have : f = g.op := rfl, clear_value g, subst this,\n      rcases i with (_|_); rcases j with (_|_); rcases g with (_|_|_); refl }) }\n\n@[simp] lemma walking_parallel_pair_op_equiv_unit_iso_zero :\n  walking_parallel_pair_op_equiv.unit_iso.app zero = iso.refl zero := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_unit_iso_one :\n  walking_parallel_pair_op_equiv.unit_iso.app one = iso.refl one := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_counit_iso_zero :\n  walking_parallel_pair_op_equiv.counit_iso.app (op zero) = iso.refl (op zero) := rfl\n@[simp] lemma walking_parallel_pair_op_equiv_counit_iso_one :\n  walking_parallel_pair_op_equiv.counit_iso.app (op one) = iso.refl (op one) := rfl\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y : C}\n\n/-- `parallel_pair f g` is the diagram in `C` consisting of the two morphisms `f` and `g` with\n    common domain and codomain. -/\ndef parallel_pair (f g : X \u27f6 Y) : walking_parallel_pair \u2964 C :=\n{ obj := \u03bb x, match x with\n  | zero := X\n  | one := Y\n  end,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, left := f\n  | _, _, right := g\n  end,\n  -- `tidy` can cope with this, but it's too slow:\n  map_comp' := begin rintros (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) \u27e8\u27e9\u27e8\u27e9; { unfold_aux, simp; refl }, end, }.\n\n@[simp] lemma parallel_pair_obj_zero (f g : X \u27f6 Y) : (parallel_pair f g).obj zero = X := rfl\n@[simp] lemma parallel_pair_obj_one (f g : X \u27f6 Y) : (parallel_pair f g).obj one = Y := rfl\n\n@[simp] lemma parallel_pair_map_left (f g : X \u27f6 Y) : (parallel_pair f g).map left = f := rfl\n@[simp] lemma parallel_pair_map_right (f g : X \u27f6 Y) : (parallel_pair f g).map right = g := rfl\n\n@[simp] lemma parallel_pair_functor_obj\n  {F : walking_parallel_pair \u2964 C} (j : walking_parallel_pair) :\n  (parallel_pair (F.map left) (F.map right)).obj j = F.obj j :=\nbegin\n  cases j; refl\nend\n\n/-- Every functor indexing a (co)equalizer is naturally isomorphic (actually, equal) to a\n    `parallel_pair` -/\n@[simps]\ndef diagram_iso_parallel_pair (F : walking_parallel_pair \u2964 C) :\n  F \u2245 parallel_pair (F.map left) (F.map right) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by cases j; tidy) $ by tidy\n\n/-- Construct a morphism between parallel pairs. -/\ndef parallel_pair_hom {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') : parallel_pair f g \u27f6 parallel_pair f' g' :=\n{ app := \u03bb j, match j with\n  | zero := p\n  | one := q\n  end,\n  naturality' := begin\n    rintros (\u27e8\u27e9|\u27e8\u27e9) (\u27e8\u27e9|\u27e8\u27e9) \u27e8\u27e9; { unfold_aux, simp [wf, wg], },\n  end }\n\n@[simp] lemma parallel_pair_hom_app_zero\n  {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') :\n  (parallel_pair_hom f g f' g' p q wf wg).app zero = p := rfl\n\n@[simp] lemma parallel_pair_hom_app_one\n  {X' Y' : C} (f g : X \u27f6 Y) (f' g' : X' \u27f6 Y') (p : X \u27f6 X') (q : Y \u27f6 Y')\n  (wf : f \u226b q = p \u226b f') (wg : g \u226b q = p \u226b g') :\n  (parallel_pair_hom f g f' g' p q wf wg).app one = q := rfl\n\n/-- Construct a natural isomorphism between functors out of the walking parallel pair from\nits components. -/\n@[simps]\ndef parallel_pair.ext {F G : walking_parallel_pair \u2964 C}\n  (zero : F.obj zero \u2245 G.obj zero) (one : F.obj one \u2245 G.obj one)\n  (left : F.map left \u226b one.hom = zero.hom \u226b G.map left)\n  (right : F.map right \u226b one.hom = zero.hom \u226b G.map right) : F \u2245 G :=\nnat_iso.of_components\n  (by { rintro \u27e8j\u27e9, exacts [zero, one] })\n  (by { rintro \u27e8j\u2081\u27e9 \u27e8j\u2082\u27e9 \u27e8f\u27e9; simp [left, right], })\n\n/-- A fork on `f` and `g` is just a `cone (parallel_pair f g)`. -/\nabbreviation fork (f g : X \u27f6 Y) := cone (parallel_pair f g)\n\n/-- A cofork on `f` and `g` is just a `cocone (parallel_pair f g)`. -/\nabbreviation cofork (f g : X \u27f6 Y) := cocone (parallel_pair f g)\n\nvariables {f g : X \u27f6 Y}\n\n/-- A fork `t` on the parallel pair `f g : X \u27f6 Y` consists of two morphisms `t.\u03c0.app zero : t.X \u27f6 X`\n    and `t.\u03c0.app one : t.X \u27f6 Y`. Of these, only the first one is interesting, and we give it the\n    shorter name `fork.\u03b9 t`. -/\ndef fork.\u03b9 (t : fork f g) := t.\u03c0.app zero\n\n@[simp] lemma fork.app_zero_eq_\u03b9 (t : fork f g) : t.\u03c0.app zero = t.\u03b9 := rfl\n\n/-- A cofork `t` on the parallel_pair `f g : X \u27f6 Y` consists of two morphisms\n    `t.\u03b9.app zero : X \u27f6 t.X` and `t.\u03b9.app one : Y \u27f6 t.X`. Of these, only the second one is\n    interesting, and we give it the shorter name `cofork.\u03c0 t`. -/\ndef cofork.\u03c0 (t : cofork f g) := t.\u03b9.app one\n\n@[simp] lemma cofork.app_one_eq_\u03c0 (t : cofork f g) : t.\u03b9.app one = t.\u03c0 := rfl\n\n@[simp] lemma fork.app_one_eq_\u03b9_comp_left (s : fork f g) : s.\u03c0.app one = s.\u03b9 \u226b f :=\nby rw [\u2190s.app_zero_eq_\u03b9, \u2190s.w left, parallel_pair_map_left]\n\n@[reassoc] lemma fork.app_one_eq_\u03b9_comp_right (s : fork f g) : s.\u03c0.app one = s.\u03b9 \u226b g :=\nby rw [\u2190s.app_zero_eq_\u03b9, \u2190s.w right, parallel_pair_map_right]\n\n@[simp] lemma cofork.app_zero_eq_comp_\u03c0_left (s : cofork f g) : s.\u03b9.app zero = f \u226b s.\u03c0 :=\nby rw [\u2190s.app_one_eq_\u03c0, \u2190s.w left, parallel_pair_map_left]\n\n@[reassoc] lemma cofork.app_zero_eq_comp_\u03c0_right (s : cofork f g) : s.\u03b9.app zero = g \u226b s.\u03c0 :=\nby rw [\u2190s.app_one_eq_\u03c0, \u2190s.w right, parallel_pair_map_right]\n\n/-- A fork on `f g : X \u27f6 Y` is determined by the morphism `\u03b9 : P \u27f6 X` satisfying `\u03b9 \u226b f = \u03b9 \u226b g`.\n-/\n@[simps]\ndef fork.of_\u03b9 {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) : fork f g :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb X, begin cases X, exact \u03b9, exact \u03b9 \u226b f, end,\n    naturality' := \u03bb X Y f,\n    begin\n      cases X; cases Y; cases f; dsimp; simp,\n      { dsimp, simp, }, -- See note [dsimp, simp].\n      { exact w },\n      { dsimp, simp, },\n    end } }\n\n/-- A cofork on `f g : X \u27f6 Y` is determined by the morphism `\u03c0 : Y \u27f6 P` satisfying\n    `f \u226b \u03c0 = g \u226b \u03c0`. -/\n@[simps]\ndef cofork.of_\u03c0 {P : C} (\u03c0 : Y \u27f6 P) (w : f \u226b \u03c0 = g \u226b \u03c0) : cofork f g :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb X, walking_parallel_pair.cases_on X (f \u226b \u03c0) \u03c0,\n    naturality' := \u03bb i j f, by { cases f; dsimp; simp [w] } } } -- See note [dsimp, simp]\n\n@[simp] lemma fork.\u03b9_of_\u03b9 {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) :\n  (fork.of_\u03b9 \u03b9 w).\u03b9 = \u03b9 := rfl\n@[simp] lemma cofork.\u03c0_of_\u03c0 {P : C} (\u03c0 : Y \u27f6 P) (w : f \u226b \u03c0 = g \u226b \u03c0) :\n  (cofork.of_\u03c0 \u03c0 w).\u03c0 = \u03c0 := rfl\n\n@[simp, reassoc]\nlemma fork.condition (t : fork f g) : t.\u03b9 \u226b f = t.\u03b9 \u226b g :=\nby rw [\u2190t.app_one_eq_\u03b9_comp_left, \u2190t.app_one_eq_\u03b9_comp_right]\n\n@[simp, reassoc]\nlemma cofork.condition (t : cofork f g) : f \u226b t.\u03c0 = g \u226b t.\u03c0 :=\nby rw [\u2190t.app_zero_eq_comp_\u03c0_left, \u2190t.app_zero_eq_comp_\u03c0_right]\n\n/-- To check whether two maps are equalized by both maps of a fork, it suffices to check it for the\n    first map -/\nlemma fork.equalizer_ext (s : fork f g) {W : C} {k l : W \u27f6 s.X} (h : k \u226b s.\u03b9 = l \u226b s.\u03b9) :\n  \u2200 (j : walking_parallel_pair), k \u226b s.\u03c0.app j = l \u226b s.\u03c0.app j\n| zero := h\n| one := by rw [s.app_one_eq_\u03b9_comp_left, reassoc_of h]\n\n/-- To check whether two maps are coequalized by both maps of a cofork, it suffices to check it for\n    the second map -/\nlemma cofork.coequalizer_ext (s : cofork f g) {W : C} {k l : s.X \u27f6 W}\n  (h : cofork.\u03c0 s \u226b k = cofork.\u03c0 s \u226b l) : \u2200 (j : walking_parallel_pair),\n    s.\u03b9.app j \u226b k = s.\u03b9.app j \u226b l\n| zero := by simp only [s.app_zero_eq_comp_\u03c0_left, category.assoc, h]\n| one := h\n\nlemma fork.is_limit.hom_ext {s : fork f g} (hs : is_limit s) {W : C} {k l : W \u27f6 s.X}\n  (h : k \u226b fork.\u03b9 s = l \u226b fork.\u03b9 s) : k = l :=\nhs.hom_ext $ fork.equalizer_ext _ h\n\nlemma cofork.is_colimit.hom_ext {s : cofork f g} (hs : is_colimit s) {W : C} {k l : s.X \u27f6 W}\n  (h : cofork.\u03c0 s \u226b k = cofork.\u03c0 s \u226b l) : k = l :=\nhs.hom_ext $ cofork.coequalizer_ext _ h\n\n@[simp, reassoc] lemma fork.is_limit.lift_\u03b9 {s t : fork f g} (hs : is_limit s) :\n  hs.lift t \u226b s.\u03b9 = t.\u03b9 :=\nhs.fac _ _\n\n@[simp, reassoc] lemma cofork.is_colimit.\u03c0_desc {s t : cofork f g} (hs : is_colimit s) :\n  s.\u03c0 \u226b hs.desc t = t.\u03c0 :=\nhs.fac _ _\n\n/-- If `s` is a limit fork over `f` and `g`, then a morphism `k : W \u27f6 X` satisfying\n    `k \u226b f = k \u226b g` induces a morphism `l : W \u27f6 s.X` such that `l \u226b fork.\u03b9 s = k`. -/\ndef fork.is_limit.lift' {s : fork f g} (hs : is_limit s) {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  {l : W \u27f6 s.X // l \u226b fork.\u03b9 s = k} :=\n\u27e8hs.lift $ fork.of_\u03b9 _ h, hs.fac _ _\u27e9\n\n/-- If `s` is a colimit cofork over `f` and `g`, then a morphism `k : Y \u27f6 W` satisfying\n    `f \u226b k = g \u226b k` induces a morphism `l : s.X \u27f6 W` such that `cofork.\u03c0 s \u226b l = k`. -/\ndef cofork.is_colimit.desc' {s : cofork f g} (hs : is_colimit s) {W : C} (k : Y \u27f6 W)\n  (h : f \u226b k = g \u226b k) : {l : s.X \u27f6 W // cofork.\u03c0 s \u226b l = k} :=\n\u27e8hs.desc $ cofork.of_\u03c0 _ h, hs.fac _ _\u27e9\n\nlemma fork.is_limit.exists_unique {s : fork f g} (hs : is_limit s) {W : C} (k : W \u27f6 X)\n  (h : k \u226b f = k \u226b g) : \u2203! (l : W \u27f6 s.X), l \u226b fork.\u03b9 s = k :=\n\u27e8hs.lift $ fork.of_\u03b9 _ h, hs.fac _ _, \u03bb m hm, fork.is_limit.hom_ext hs $\n  hm.symm \u25b8 (hs.fac (fork.of_\u03b9 _ h) walking_parallel_pair.zero).symm\u27e9\n\nlemma cofork.is_colimit.exists_unique {s : cofork f g} (hs : is_colimit s) {W : C} (k : Y \u27f6 W)\n  (h : f \u226b k = g \u226b k) : \u2203! (d : s.X \u27f6 W), cofork.\u03c0 s \u226b d = k :=\n\u27e8hs.desc $ cofork.of_\u03c0 _ h, hs.fac _ _, \u03bb m hm, cofork.is_colimit.hom_ext hs $\n  hm.symm \u25b8 (hs.fac (cofork.of_\u03c0 _ h) walking_parallel_pair.one).symm\u27e9\n\n/-- This is a slightly more convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content -/\n@[simps lift]\ndef fork.is_limit.mk (t : fork f g)\n  (lift : \u03a0 (s : fork f g), s.X \u27f6 t.X)\n  (fac : \u2200 (s : fork f g), lift s \u226b fork.\u03b9 t = fork.\u03b9 s)\n  (uniq : \u2200 (s : fork f g) (m : s.X \u27f6 t.X) (w : m \u226b t.\u03b9 = s.\u03b9), m = lift s) :\n  is_limit t :=\n{ lift := lift,\n  fac' := \u03bb s j, walking_parallel_pair.cases_on j (fac s) $\n    by erw [\u2190s.w left, \u2190t.w left, \u2190category.assoc, fac]; refl,\n  uniq' := \u03bb s m j, by tidy }\n\n/-- This is another convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef fork.is_limit.mk' {X Y : C} {f g : X \u27f6 Y} (t : fork f g)\n  (create : \u03a0 (s : fork f g), {l // l \u226b t.\u03b9 = s.\u03b9 \u2227 \u2200 {m}, m \u226b t.\u03b9 = s.\u03b9 \u2192 m = l}) :\nis_limit t :=\nfork.is_limit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 w)\n\n/-- This is a slightly more convenient method to verify that a cofork is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef cofork.is_colimit.mk (t : cofork f g)\n  (desc : \u03a0 (s : cofork f g), t.X \u27f6 s.X)\n  (fac : \u2200 (s : cofork f g), cofork.\u03c0 t \u226b desc s = cofork.\u03c0 s)\n  (uniq : \u2200 (s : cofork f g) (m : t.X \u27f6 s.X) (w : t.\u03c0 \u226b m = s.\u03c0), m = desc s) :\n  is_colimit t :=\n{ desc := desc,\n  fac' := \u03bb s j, walking_parallel_pair.cases_on j\n    (by erw [\u2190s.w left, \u2190t.w left, category.assoc, fac]; refl) (fac s),\n  uniq' := by tidy }\n\n/-- This is another convenient method to verify that a fork is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef cofork.is_colimit.mk' {X Y : C} {f g : X \u27f6 Y} (t : cofork f g)\n  (create : \u03a0 (s : cofork f g), {l : t.X \u27f6 s.X // t.\u03c0 \u226b l = s.\u03c0 \u2227 \u2200 {m}, t.\u03c0 \u226b m = s.\u03c0 \u2192 m = l}) :\nis_colimit t :=\ncofork.is_colimit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 w)\n\n/-- Noncomputably make a limit cone from the existence of unique factorizations. -/\ndef fork.is_limit.of_exists_unique {t : fork f g}\n  (hs : \u2200 (s : fork f g), \u2203! l : s.X \u27f6 t.X, l \u226b fork.\u03b9 t = fork.\u03b9 s) : is_limit t :=\nby { choose d hd hd' using hs, exact fork.is_limit.mk _ d hd (\u03bb s m hm, hd' _ _ hm) }\n\n/-- Noncomputably make a colimit cocone from the existence of unique factorizations. -/\ndef cofork.is_colimit.of_exists_unique {t : cofork f g}\n  (hs : \u2200 (s : cofork f g), \u2203! d : t.X \u27f6 s.X, cofork.\u03c0 t \u226b d = cofork.\u03c0 s) : is_colimit t :=\nby { choose d hd hd' using hs, exact cofork.is_colimit.mk _ d hd (\u03bb s m hm, hd' _ _ hm) }\n\n/--\nGiven a limit cone for the pair `f g : X \u27f6 Y`, for any `Z`, morphisms from `Z` to its point are in\nbijection with morphisms `h : Z \u27f6 X` such that `h \u226b f = h \u226b g`.\nFurther, this bijection is natural in `Z`: see `fork.is_limit.hom_iso_natural`.\nThis is a special case of `is_limit.hom_iso'`, often useful to construct adjunctions.\n-/\n@[simps]\ndef fork.is_limit.hom_iso {X Y : C} {f g : X \u27f6 Y} {t : fork f g} (ht : is_limit t) (Z : C) :\n  (Z \u27f6 t.X) \u2243 {h : Z \u27f6 X // h \u226b f = h \u226b g} :=\n{ to_fun := \u03bb k, \u27e8k \u226b t.\u03b9, by simp only [category.assoc, t.condition]\u27e9,\n  inv_fun := \u03bb h, (fork.is_limit.lift' ht _ h.prop).1,\n  left_inv := \u03bb k, fork.is_limit.hom_ext ht (fork.is_limit.lift' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (fork.is_limit.lift' ht _ _).prop }\n\n/-- The bijection of `fork.is_limit.hom_iso` is natural in `Z`. -/\nlemma fork.is_limit.hom_iso_natural {X Y : C} {f g : X \u27f6 Y} {t : fork f g} (ht : is_limit t)\n  {Z Z' : C} (q : Z' \u27f6 Z) (k : Z \u27f6 t.X) :\n  (fork.is_limit.hom_iso ht _ (q \u226b k) : Z' \u27f6 X) = q \u226b (fork.is_limit.hom_iso ht _ k : Z \u27f6 X) :=\ncategory.assoc _ _ _\n\n/--\nGiven a colimit cocone for the pair `f g : X \u27f6 Y`, for any `Z`, morphisms from the cocone point\nto `Z` are in bijection with morphisms `h : Y \u27f6 Z` such that `f \u226b h = g \u226b h`.\nFurther, this bijection is natural in `Z`: see `cofork.is_colimit.hom_iso_natural`.\nThis is a special case of `is_colimit.hom_iso'`, often useful to construct adjunctions.\n-/\n@[simps]\ndef cofork.is_colimit.hom_iso {X Y : C} {f g : X \u27f6 Y} {t : cofork f g} (ht : is_colimit t) (Z : C) :\n  (t.X \u27f6 Z) \u2243 {h : Y \u27f6 Z // f \u226b h = g \u226b h} :=\n{ to_fun := \u03bb k, \u27e8t.\u03c0 \u226b k, by simp only [\u2190category.assoc, t.condition]\u27e9,\n  inv_fun := \u03bb h, (cofork.is_colimit.desc' ht _ h.prop).1,\n  left_inv := \u03bb k, cofork.is_colimit.hom_ext ht (cofork.is_colimit.desc' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (cofork.is_colimit.desc' ht _ _).prop }\n\n/-- The bijection of `cofork.is_colimit.hom_iso` is natural in `Z`. -/\nlemma cofork.is_colimit.hom_iso_natural {X Y : C} {f g : X \u27f6 Y} {t : cofork f g} {Z Z' : C}\n  (q : Z \u27f6 Z') (ht : is_colimit t) (k : t.X \u27f6 Z) :\n    (cofork.is_colimit.hom_iso ht _ (k \u226b q) : Y \u27f6 Z') =\n    (cofork.is_colimit.hom_iso ht _ k : Y \u27f6 Z) \u226b q :=\n(category.assoc _ _ _).symm\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    equalizers. Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)`, and a fork on `F.map left` and `F.map right`,\n    we get a cone on `F`.\n\n    If you're thinking about using this, have a look at `has_equalizers_of_has_limit_parallel_pair`,\n    which you may find to be an easier way of achieving your goal. -/\ndef cone.of_fork\n  {F : walking_parallel_pair \u2964 C} (t : fork (F.map left) (F.map right)) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g, by { cases j; cases j'; cases g; dsimp; simp } } }\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    coequalizers. Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)`, and a cofork on `F.map left` and `F.map right`,\n    we get a cocone on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_coequalizers_of_has_colimit_parallel_pair`, which you may find to be an easier way of\n    achieving your goal. -/\ndef cocone.of_cofork\n  {F : walking_parallel_pair \u2964 C} (t : cofork (F.map left) (F.map right)) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g, by { cases j; cases j'; cases g; dsimp; simp } } }\n\n@[simp] lemma cone.of_fork_\u03c0\n  {F : walking_parallel_pair \u2964 C} (t : fork (F.map left) (F.map right)) (j) :\n  (cone.of_fork t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\n@[simp] lemma cocone.of_cofork_\u03b9\n  {F : walking_parallel_pair \u2964 C} (t : cofork (F.map left) (F.map right)) (j) :\n  (cocone.of_cofork t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/-- Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)` and a cone on `F`, we get a fork on\n    `F.map left` and `F.map right`. -/\ndef fork.of_cone\n  {F : walking_parallel_pair \u2964 C} (t : cone F) : fork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03c0 := { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy) } }\n\n/-- Given `F : walking_parallel_pair \u2964 C`, which is really the same as\n    `parallel_pair (F.map left) (F.map right)` and a cocone on `F`, we get a cofork on\n    `F.map left` and `F.map right`. -/\ndef cofork.of_cocone\n  {F : walking_parallel_pair \u2964 C} (t : cocone F) : cofork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03b9 := { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X } }\n\n@[simp] lemma fork.of_cone_\u03c0 {F : walking_parallel_pair \u2964 C} (t : cone F) (j) :\n  (fork.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n@[simp] lemma cofork.of_cocone_\u03b9 {F : walking_parallel_pair \u2964 C} (t : cocone F) (j) :\n  (cofork.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/--\nHelper function for constructing morphisms between equalizer forks.\n-/\n@[simps]\ndef fork.mk_hom {s t : fork f g} (k : s.X \u27f6 t.X) (w : k \u226b t.\u03b9 = s.\u03b9) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { exact w },\n    { simp only [fork.app_one_eq_\u03b9_comp_left, reassoc_of w] },\n  end }\n\n/--\nTo construct an isomorphism between forks,\nit suffices to give an isomorphism between the cone points\nand check that it commutes with the `\u03b9` morphisms.\n-/\n@[simps]\ndef fork.ext {s t : fork f g} (i : s.X \u2245 t.X) (w : i.hom \u226b t.\u03b9 = s.\u03b9) : s \u2245 t :=\n{ hom := fork.mk_hom i.hom w,\n  inv := fork.mk_hom i.inv (by rw [\u2190 w, iso.inv_hom_id_assoc]) }\n\n/--\nHelper function for constructing morphisms between coequalizer coforks.\n-/\n@[simps]\ndef cofork.mk_hom {s t : cofork f g} (k : s.X \u27f6 t.X) (w : s.\u03c0 \u226b k = t.\u03c0) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { simp [cofork.app_zero_eq_comp_\u03c0_left, w] },\n    { exact w }\n  end }\n\n@[simp, reassoc] lemma fork.hom_comp_\u03b9 {s t : fork f g} (f : s \u27f6 t) : f.hom \u226b t.\u03b9 = s.\u03b9 :=\nby tidy\n\n@[simp, reassoc] lemma fork.\u03c0_comp_hom {s t : cofork f g} (f : s \u27f6 t) : s.\u03c0 \u226b f.hom = t.\u03c0 :=\nby tidy\n\n/--\nTo construct an isomorphism between coforks,\nit suffices to give an isomorphism between the cocone points\nand check that it commutes with the `\u03c0` morphisms.\n-/\n@[simps]\ndef cofork.ext {s t : cofork f g} (i : s.X \u2245 t.X) (w : s.\u03c0 \u226b i.hom = t.\u03c0) : s \u2245 t :=\n{ hom := cofork.mk_hom i.hom w,\n  inv := cofork.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) }\n\nvariables (f g)\n\nsection\n/--\n`has_equalizer f g` represents a particular choice of limiting cone\nfor the parallel pair of morphisms `f` and `g`.\n-/\nabbreviation has_equalizer := has_limit (parallel_pair f g)\n\nvariables [has_equalizer f g]\n\n/-- If an equalizer of `f` and `g` exists, we can access an arbitrary choice of such by\n    saying `equalizer f g`. -/\nabbreviation equalizer : C := limit (parallel_pair f g)\n\n/-- If an equalizer of `f` and `g` exists, we can access the inclusion\n    `equalizer f g \u27f6 X` by saying `equalizer.\u03b9 f g`. -/\nabbreviation equalizer.\u03b9 : equalizer f g \u27f6 X :=\nlimit.\u03c0 (parallel_pair f g) zero\n\n/--\nAn equalizer cone for a parallel pair `f` and `g`.\n-/\nabbreviation equalizer.fork : fork f g := limit.cone (parallel_pair f g)\n\n@[simp] lemma equalizer.fork_\u03b9 :\n  (equalizer.fork f g).\u03b9 = equalizer.\u03b9 f g := rfl\n\n@[simp] lemma equalizer.fork_\u03c0_app_zero :\n  (equalizer.fork f g).\u03c0.app zero = equalizer.\u03b9 f g := rfl\n\n@[reassoc] lemma equalizer.condition : equalizer.\u03b9 f g \u226b f = equalizer.\u03b9 f g \u226b g :=\nfork.condition $ limit.cone $ parallel_pair f g\n\n/-- The equalizer built from `equalizer.\u03b9 f g` is limiting. -/\ndef equalizer_is_equalizer : is_limit (fork.of_\u03b9 (equalizer.\u03b9 f g) (equalizer.condition f g)) :=\nis_limit.of_iso_limit (limit.is_limit _) (fork.ext (iso.refl _) (by tidy))\n\nvariables {f g}\n\n/-- A morphism `k : W \u27f6 X` satisfying `k \u226b f = k \u226b g` factors through the equalizer of `f` and `g`\n    via `equalizer.lift : W \u27f6 equalizer f g`. -/\nabbreviation equalizer.lift {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) : W \u27f6 equalizer f g :=\nlimit.lift (parallel_pair f g) (fork.of_\u03b9 k h)\n\n@[simp, reassoc]\nlemma equalizer.lift_\u03b9 {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  equalizer.lift k h \u226b equalizer.\u03b9 f g = k :=\nlimit.lift_\u03c0 _ _\n\n/-- A morphism `k : W \u27f6 X` satisfying `k \u226b f = k \u226b g` induces a morphism `l : W \u27f6 equalizer f g`\n    satisfying `l \u226b equalizer.\u03b9 f g = k`. -/\ndef equalizer.lift' {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  {l : W \u27f6 equalizer f g // l \u226b equalizer.\u03b9 f g = k} :=\n\u27e8equalizer.lift k h, equalizer.lift_\u03b9 _ _\u27e9\n\n/-- Two maps into an equalizer are equal if they are are equal when composed with the equalizer\n    map. -/\n@[ext] lemma equalizer.hom_ext {W : C} {k l : W \u27f6 equalizer f g}\n  (h : k \u226b equalizer.\u03b9 f g = l \u226b equalizer.\u03b9 f g) : k = l :=\nfork.is_limit.hom_ext (limit.is_limit _) h\n\nlemma equalizer.exists_unique {W : C} (k : W \u27f6 X) (h : k \u226b f = k \u226b g) :\n  \u2203! (l : W \u27f6 equalizer f g), l \u226b equalizer.\u03b9 f g = k :=\nfork.is_limit.exists_unique (limit.is_limit _) _ h\n\n/-- An equalizer morphism is a monomorphism -/\ninstance equalizer.\u03b9_mono : mono (equalizer.\u03b9 f g) :=\n{ right_cancellation := \u03bb Z h k w, equalizer.hom_ext w }\n\nend\n\nsection\nvariables {f g}\n/-- The equalizer morphism in any limit cone is a monomorphism. -/\nlemma mono_of_is_limit_fork {c : fork f g} (i : is_limit c) : mono (fork.\u03b9 c) :=\n{ right_cancellation := \u03bb Z h k w, fork.is_limit.hom_ext i w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The identity determines a cone on the equalizer diagram of `f` and `g` if `f = g`. -/\ndef id_fork (h : f = g) : fork f g :=\nfork.of_\u03b9 (\ud835\udfd9 X) $ h \u25b8 rfl\n\n/-- The identity on `X` is an equalizer of `(f, g)`, if `f = g`. -/\ndef is_limit_id_fork (h : f = g) : is_limit (id_fork h) :=\nfork.is_limit.mk _\n  (\u03bb s, fork.\u03b9 s)\n  (\u03bb s, category.comp_id _)\n  (\u03bb s m h, by { convert h, exact (category.comp_id _).symm })\n\n/-- Every equalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_eq (h\u2080 : f = g) {c : fork f g}\n  (h : is_limit c) : is_iso c.\u03b9 :=\nis_iso.of_iso $ is_limit.cone_point_unique_up_to_iso h $ is_limit_id_fork h\u2080\n\n/-- The equalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma equalizer.\u03b9_of_eq [has_equalizer f g] (h : f = g) : is_iso (equalizer.\u03b9 f g) :=\nis_iso_limit_cone_parallel_pair_of_eq h $ limit.is_limit _\n\n/-- Every equalizer of `(f, f)` is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_self {c : fork f f} (h : is_limit c) : is_iso c.\u03b9 :=\nis_iso_limit_cone_parallel_pair_of_eq rfl h\n\n/-- An equalizer that is an epimorphism is an isomorphism. -/\nlemma is_iso_limit_cone_parallel_pair_of_epi {c : fork f g}\n  (h : is_limit c) [epi (c.\u03b9)] : is_iso c.\u03b9 :=\nis_iso_limit_cone_parallel_pair_of_eq ((cancel_epi _).1 (fork.condition c)) h\n\n/-- Two morphisms are equal if there is a fork whose inclusion is epi. -/\nlemma eq_of_epi_fork_\u03b9 (t : fork f g) [epi (fork.\u03b9 t)] : f = g :=\n(cancel_epi (fork.\u03b9 t)).1 $ fork.condition t\n\n/-- If the equalizer of two morphisms is an epimorphism, then the two morphisms are equal. -/\nlemma eq_of_epi_equalizer [has_equalizer f g] [epi (equalizer.\u03b9 f g)] : f = g :=\n(cancel_epi (equalizer.\u03b9 f g)).1 $ equalizer.condition _ _\n\nend\n\ninstance has_equalizer_of_self : has_equalizer f f :=\nhas_limit.mk\n{ cone := id_fork rfl,\n  is_limit := is_limit_id_fork rfl }\n\n/-- The equalizer inclusion for `(f, f)` is an isomorphism. -/\ninstance equalizer.\u03b9_of_self : is_iso (equalizer.\u03b9 f f) :=\nequalizer.\u03b9_of_eq rfl\n\n/-- The equalizer of a morphism with itself is isomorphic to the source. -/\ndef equalizer.iso_source_of_self : equalizer f f \u2245 X :=\nas_iso (equalizer.\u03b9 f f)\n\n@[simp] lemma equalizer.iso_source_of_self_hom :\n  (equalizer.iso_source_of_self f).hom = equalizer.\u03b9 f f :=\nrfl\n\n@[simp] lemma equalizer.iso_source_of_self_inv :\n  (equalizer.iso_source_of_self f).inv = equalizer.lift (\ud835\udfd9 X) (by simp) :=\nby { ext, simp [equalizer.iso_source_of_self], }\n\nsection\n/--\n`has_coequalizer f g` represents a particular choice of colimiting cocone\nfor the parallel pair of morphisms `f` and `g`.\n-/\nabbreviation has_coequalizer := has_colimit (parallel_pair f g)\n\nvariables [has_coequalizer f g]\n\n/-- If a coequalizer of `f` and `g` exists, we can access an arbitrary choice of such by\n    saying `coequalizer f g`. -/\nabbreviation coequalizer : C := colimit (parallel_pair f g)\n\n/--  If a coequalizer of `f` and `g` exists, we can access the corresponding projection by\n    saying `coequalizer.\u03c0 f g`. -/\nabbreviation coequalizer.\u03c0 : Y \u27f6 coequalizer f g :=\ncolimit.\u03b9 (parallel_pair f g) one\n\n/--\nAn arbitrary choice of coequalizer cocone for a parallel pair `f` and `g`.\n-/\nabbreviation coequalizer.cofork : cofork f g := colimit.cocone (parallel_pair f g)\n\n@[simp] lemma coequalizer.cofork_\u03c0 :\n  (coequalizer.cofork f g).\u03c0 = coequalizer.\u03c0 f g := rfl\n\n@[simp] lemma coequalizer.cofork_\u03b9_app_one :\n  (coequalizer.cofork f g).\u03b9.app one = coequalizer.\u03c0 f g := rfl\n\n@[reassoc] lemma coequalizer.condition : f \u226b coequalizer.\u03c0 f g = g \u226b coequalizer.\u03c0 f g :=\ncofork.condition $ colimit.cocone $ parallel_pair f g\n\n/-- The cofork built from `coequalizer.\u03c0 f g` is colimiting. -/\ndef coequalizer_is_coequalizer :\n  is_colimit (cofork.of_\u03c0 (coequalizer.\u03c0 f g) (coequalizer.condition f g)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit _) (cofork.ext (iso.refl _) (by tidy))\n\nvariables {f g}\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `f \u226b k = g \u226b k` factors through the coequalizer of `f`\n    and `g` via `coequalizer.desc : coequalizer f g \u27f6 W`. -/\nabbreviation coequalizer.desc {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) : coequalizer f g \u27f6 W :=\ncolimit.desc (parallel_pair f g) (cofork.of_\u03c0 k h)\n\n@[simp, reassoc]\nlemma coequalizer.\u03c0_desc {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) :\n  coequalizer.\u03c0 f g \u226b coequalizer.desc k h = k :=\ncolimit.\u03b9_desc _ _\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `f \u226b k = g \u226b k` induces a morphism\n    `l : coequalizer f g \u27f6 W` satisfying `coequalizer.\u03c0 \u226b g = l`. -/\ndef coequalizer.desc' {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) :\n  {l : coequalizer f g \u27f6 W // coequalizer.\u03c0 f g \u226b l = k} :=\n\u27e8coequalizer.desc k h, coequalizer.\u03c0_desc _ _\u27e9\n\n/-- Two maps from a coequalizer are equal if they are equal when composed with the coequalizer\n    map -/\n@[ext] lemma coequalizer.hom_ext {W : C} {k l : coequalizer f g \u27f6 W}\n  (h : coequalizer.\u03c0 f g \u226b k = coequalizer.\u03c0 f g \u226b l) : k = l :=\ncofork.is_colimit.hom_ext (colimit.is_colimit _) h\n\nlemma coequalizer.exists_unique {W : C} (k : Y \u27f6 W) (h : f \u226b k = g \u226b k) :\n  \u2203! (d : coequalizer f g \u27f6 W), coequalizer.\u03c0 f g \u226b d = k :=\ncofork.is_colimit.exists_unique (colimit.is_colimit _) _ h\n\n/-- A coequalizer morphism is an epimorphism -/\ninstance coequalizer.\u03c0_epi : epi (coequalizer.\u03c0 f g) :=\n{ left_cancellation := \u03bb Z h k w, coequalizer.hom_ext w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The coequalizer morphism in any colimit cocone is an epimorphism. -/\nlemma epi_of_is_colimit_cofork {c : cofork f g} (i : is_colimit c) : epi c.\u03c0 :=\n{ left_cancellation := \u03bb Z h k w, cofork.is_colimit.hom_ext i w }\n\nend\n\nsection\nvariables {f g}\n\n/-- The identity determines a cocone on the coequalizer diagram of `f` and `g`, if `f = g`. -/\ndef id_cofork (h : f = g) : cofork f g :=\ncofork.of_\u03c0 (\ud835\udfd9 Y) $ h \u25b8 rfl\n\n/-- The identity on `Y` is a coequalizer of `(f, g)`, where `f = g`.  -/\ndef is_colimit_id_cofork (h : f = g) : is_colimit (id_cofork h) :=\ncofork.is_colimit.mk _\n  (\u03bb s, cofork.\u03c0 s)\n  (\u03bb s, category.id_comp _)\n  (\u03bb s m h, by { convert h, exact (category.id_comp _).symm })\n\n/-- Every coequalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma is_iso_colimit_cocone_parallel_pair_of_eq (h\u2080 : f = g) {c : cofork f g}  (h : is_colimit c) :\n  is_iso c.\u03c0 :=\nis_iso.of_iso $ is_colimit.cocone_point_unique_up_to_iso (is_colimit_id_cofork h\u2080) h\n\n/-- The coequalizer of `(f, g)`, where `f = g`, is an isomorphism. -/\nlemma coequalizer.\u03c0_of_eq [has_coequalizer f g] (h : f = g) : is_iso (coequalizer.\u03c0 f g) :=\nis_iso_colimit_cocone_parallel_pair_of_eq h $ colimit.is_colimit _\n\n/-- Every coequalizer of `(f, f)` is an isomorphism. -/\nlemma is_iso_colimit_cocone_parallel_pair_of_self {c : cofork f f} (h : is_colimit c) :\n  is_iso c.\u03c0 :=\nis_iso_colimit_cocone_parallel_pair_of_eq rfl h\n\n/-- A coequalizer that is a monomorphism is an isomorphism. -/\nlemma is_iso_limit_cocone_parallel_pair_of_epi {c : cofork f g}\n  (h : is_colimit c) [mono c.\u03c0] : is_iso c.\u03c0 :=\nis_iso_colimit_cocone_parallel_pair_of_eq ((cancel_mono _).1 (cofork.condition c)) h\n\n/-- Two morphisms are equal if there is a cofork whose projection is mono. -/\nlemma eq_of_mono_cofork_\u03c0 (t : cofork f g) [mono (cofork.\u03c0 t)] : f = g :=\n(cancel_mono (cofork.\u03c0 t)).1 $ cofork.condition t\n\n/-- If the coequalizer of two morphisms is a monomorphism, then the two morphisms are equal. -/\nlemma eq_of_mono_coequalizer [has_coequalizer f g] [mono (coequalizer.\u03c0 f g)] : f = g :=\n(cancel_mono (coequalizer.\u03c0 f g)).1 $ coequalizer.condition _ _\n\nend\n\ninstance has_coequalizer_of_self : has_coequalizer f f :=\nhas_colimit.mk\n{ cocone := id_cofork rfl,\n  is_colimit := is_colimit_id_cofork rfl }\n\n/-- The coequalizer projection for `(f, f)` is an isomorphism. -/\ninstance coequalizer.\u03c0_of_self : is_iso (coequalizer.\u03c0 f f) :=\ncoequalizer.\u03c0_of_eq rfl\n\n/-- The coequalizer of a morphism with itself is isomorphic to the target. -/\ndef coequalizer.iso_target_of_self : coequalizer f f \u2245 Y :=\n(as_iso (coequalizer.\u03c0 f f)).symm\n\n@[simp] lemma coequalizer.iso_target_of_self_hom :\n  (coequalizer.iso_target_of_self f).hom = coequalizer.desc (\ud835\udfd9 Y) (by simp) :=\nby { ext, simp [coequalizer.iso_target_of_self], }\n\n@[simp] lemma coequalizer.iso_target_of_self_inv :\n  (coequalizer.iso_target_of_self f).inv = coequalizer.\u03c0 f f :=\nrfl\n\nsection comparison\n\nvariables {D : Type u\u2082} [category.{v\u2082} D] (G : C \u2964 D)\n\n/--\nThe comparison morphism for the equalizer of `f,g`.\nThis is an isomorphism iff `G` preserves the equalizer of `f,g`; see\n`category_theory/limits/preserves/shapes/equalizers.lean`\n-/\ndef equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :\n  G.obj (equalizer f g) \u27f6 equalizer (G.map f) (G.map g) :=\nequalizer.lift (G.map (equalizer.\u03b9 _ _)) (by simp only [\u2190G.map_comp, equalizer.condition])\n\n@[simp, reassoc]\nlemma equalizer_comparison_comp_\u03c0 [has_equalizer f g] [has_equalizer (G.map f) (G.map g)] :\n  equalizer_comparison f g G \u226b equalizer.\u03b9 (G.map f) (G.map g) = G.map (equalizer.\u03b9 f g) :=\nequalizer.lift_\u03b9 _ _\n\n@[simp, reassoc]\nlemma map_lift_equalizer_comparison [has_equalizer f g] [has_equalizer (G.map f) (G.map g)]\n  {Z : C} {h : Z \u27f6 X} (w : h \u226b f = h \u226b g) :\n    G.map (equalizer.lift h w) \u226b equalizer_comparison f g G =\n      equalizer.lift (G.map h) (by simp only [\u2190G.map_comp, w]) :=\nby { ext, simp [\u2190 G.map_comp] }\n\n/-- The comparison morphism for the coequalizer of `f,g`. -/\ndef coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :\n  coequalizer (G.map f) (G.map g) \u27f6 G.obj (coequalizer f g) :=\ncoequalizer.desc (G.map (coequalizer.\u03c0 _ _)) (by simp only [\u2190G.map_comp, coequalizer.condition])\n\n@[simp, reassoc]\nlemma \u03b9_comp_coequalizer_comparison [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)] :\n  coequalizer.\u03c0 _ _ \u226b coequalizer_comparison f g G = G.map (coequalizer.\u03c0 _ _) :=\ncoequalizer.\u03c0_desc _ _\n\n@[simp, reassoc]\nlemma coequalizer_comparison_map_desc [has_coequalizer f g] [has_coequalizer (G.map f) (G.map g)]\n  {Z : C} {h : Y \u27f6 Z} (w : f \u226b h = g \u226b h) :\n  coequalizer_comparison f g G \u226b G.map (coequalizer.desc h w) =\n    coequalizer.desc (G.map h) (by simp only [\u2190G.map_comp, w]) :=\nby { ext, simp [\u2190 G.map_comp] }\n\nend comparison\n\nvariables (C)\n\n/-- `has_equalizers` represents a choice of equalizer for every pair of morphisms -/\nabbreviation has_equalizers := has_limits_of_shape walking_parallel_pair C\n\n/-- `has_coequalizers` represents a choice of coequalizer for every pair of morphisms -/\nabbreviation has_coequalizers := has_colimits_of_shape walking_parallel_pair C\n\n/-- If `C` has all limits of diagrams `parallel_pair f g`, then it has all equalizers -/\nlemma has_equalizers_of_has_limit_parallel_pair\n  [\u03a0 {X Y : C} {f g : X \u27f6 Y}, has_limit (parallel_pair f g)] : has_equalizers C :=\n{ has_limit := \u03bb F, has_limit_of_iso (diagram_iso_parallel_pair F).symm }\n\n/-- If `C` has all colimits of diagrams `parallel_pair f g`, then it has all coequalizers -/\nlemma has_coequalizers_of_has_colimit_parallel_pair\n  [\u03a0 {X Y : C} {f g : X \u27f6 Y}, has_colimit (parallel_pair f g)] : has_coequalizers C :=\n{ has_colimit := \u03bb F, has_colimit_of_iso (diagram_iso_parallel_pair F) }\n\n\nsection\n-- In this section we show that a split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\nvariables {C} [split_mono f]\n\n/--\nA split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\nHere we build the cone, and show in `split_mono_equalizes` that it is a limit cone.\n-/\n@[simps {rhs_md := semireducible}]\ndef cone_of_split_mono : fork (\ud835\udfd9 Y) (retraction f \u226b f) :=\nfork.of_\u03b9 f (by simp)\n\n@[simp] lemma cone_of_split_mono_\u03b9 : (cone_of_split_mono f).\u03b9 = f := rfl\n\n/--\nA split mono `f` equalizes `(retraction f \u226b f)` and `(\ud835\udfd9 Y)`.\n-/\ndef split_mono_equalizes {X Y : C} (f : X \u27f6 Y) [split_mono f] : is_limit (cone_of_split_mono f) :=\nfork.is_limit.mk' _ $ \u03bb s,\n\u27e8s.\u03b9 \u226b retraction f,\n by { dsimp, rw [category.assoc, \u2190s.condition], apply category.comp_id },\n \u03bb m hm, by simp [\u2190hm]\u27e9\n\nend\n\n/-- We show that the converse to `split_mono_equalizes` is true:\nWhenever `f` equalizes `(r \u226b f)` and `(\ud835\udfd9 Y)`, then `r` is a retraction of `f`. -/\ndef split_mono_of_equalizer {X Y : C} {f : X \u27f6 Y} {r : Y \u27f6 X} (hr : f \u226b r \u226b f = f)\n  (h : is_limit (fork.of_\u03b9 f (hr.trans (category.comp_id _).symm : f \u226b r \u226b f = f \u226b \ud835\udfd9 Y))) :\n  split_mono f :=\n{ retraction := r,\n  id' := fork.is_limit.hom_ext h\n    ((category.assoc _ _ _).trans $ hr.trans (category.id_comp _).symm) }\n\nvariables {C f g}\n\n/-- The fork obtained by postcomposing an equalizer fork with a monomorphism is an equalizer. -/\ndef is_equalizer_comp_mono {c : fork f g} (i : is_limit c) {Z : C} (h : Y \u27f6 Z) [hm : mono h] :\n  is_limit (fork.of_\u03b9 c.\u03b9 (by simp [reassoc_of c.condition]) : fork (f \u226b h) (g \u226b h)) :=\nfork.is_limit.mk' _ $ \u03bb s,\n  let s' : fork f g := fork.of_\u03b9 s.\u03b9 (by apply hm.right_cancellation; simp [s.condition]) in\n  let l := fork.is_limit.lift' i s'.\u03b9 s'.condition in\n  \u27e8l.1, l.2, \u03bb m hm, by apply fork.is_limit.hom_ext i; rw fork.\u03b9_of_\u03b9 at hm; rw hm; exact l.2.symm\u27e9\n\nvariables (C f g)\n\n@[instance]\nlemma has_equalizer_comp_mono [has_equalizer f g] {Z : C} (h : Y \u27f6 Z) [mono h] :\n  has_equalizer (f \u226b h) (g \u226b h) :=\n\u27e8\u27e8{ cone := _, is_limit := is_equalizer_comp_mono (limit.is_limit _) h }\u27e9\u27e9\n\n/-- An equalizer of an idempotent morphism and the identity is split mono. -/\n@[simps]\ndef split_mono_of_idempotent_of_is_limit_fork {X : C} {f : X \u27f6 X} (hf : f \u226b f = f)\n  {c : fork (\ud835\udfd9 X) f} (i : is_limit c) : split_mono c.\u03b9 :=\n{ retraction := i.lift (fork.of_\u03b9 f (by simp [hf])),\n  id' :=\n  begin\n    letI := mono_of_is_limit_fork i,\n    rw [\u2190cancel_mono_id c.\u03b9, category.assoc, fork.is_limit.lift_\u03b9, fork.\u03b9_of_\u03b9, \u2190c.condition],\n    exact category.comp_id c.\u03b9\n  end }\n\n/-- The equalizer of an idempotent morphism and the identity is split mono. -/\ndef split_mono_of_idempotent_equalizer {X : C} {f : X \u27f6 X} (hf : f \u226b f = f)\n  [has_equalizer (\ud835\udfd9 X) f] : split_mono (equalizer.\u03b9 (\ud835\udfd9 X) f) :=\nsplit_mono_of_idempotent_of_is_limit_fork _ hf (limit.is_limit _)\n\nsection\n-- In this section we show that a split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\nvariables {C} [split_epi f]\n\n/--\nA split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\nHere we build the cocone, and show in `split_epi_coequalizes` that it is a colimit cocone.\n-/\n@[simps {rhs_md := semireducible}]\ndef cocone_of_split_epi : cofork (\ud835\udfd9 X) (f \u226b section_ f) :=\ncofork.of_\u03c0 f (by simp)\n\n@[simp] lemma cocone_of_split_epi_\u03c0 : (cocone_of_split_epi f).\u03c0 = f := rfl\n\n/--\nA split epi `f` coequalizes `(f \u226b section_ f)` and `(\ud835\udfd9 X)`.\n-/\ndef split_epi_coequalizes {X Y : C} (f : X \u27f6 Y) [split_epi f] :\n  is_colimit (cocone_of_split_epi f) :=\ncofork.is_colimit.mk' _ $ \u03bb s,\n\u27e8section_ f \u226b s.\u03c0,\n by { dsimp, rw [\u2190 category.assoc, \u2190 s.condition, category.id_comp] },\n \u03bb m hm, by simp [\u2190 hm]\u27e9\n\nend\n\n/-- We show that the converse to `split_epi_equalizes` is true:\nWhenever `f` coequalizes `(f \u226b s)` and `(\ud835\udfd9 X)`, then `s` is a section of `f`. -/\ndef split_epi_of_coequalizer {X Y : C} {f : X \u27f6 Y} {s : Y \u27f6 X} (hs : f \u226b s \u226b f = f)\n  (h : is_colimit (cofork.of_\u03c0 f ((category.assoc _ _ _).trans $\n    hs.trans (category.id_comp f).symm : (f \u226b s) \u226b f = \ud835\udfd9 X \u226b f))) :\n  split_epi f :=\n{ section_ := s,\n  id' := cofork.is_colimit.hom_ext h (hs.trans (category.comp_id _).symm) }\n\nvariables {C f g}\n\n/-- The cofork obtained by precomposing a coequalizer cofork with an epimorphism is\na coequalizer. -/\ndef is_coequalizer_epi_comp {c : cofork f g} (i : is_colimit c) {W : C} (h : W \u27f6 X) [hm : epi h] :\n  is_colimit (cofork.of_\u03c0 c.\u03c0 (by simp) : cofork (h \u226b f) (h \u226b g)) :=\ncofork.is_colimit.mk' _ $ \u03bb s,\n  let s' : cofork f g := cofork.of_\u03c0 s.\u03c0\n    (by apply hm.left_cancellation; simp_rw [\u2190category.assoc, s.condition]) in\n  let l := cofork.is_colimit.desc' i s'.\u03c0 s'.condition in\n  \u27e8l.1, l.2,\n    \u03bb m hm,by apply cofork.is_colimit.hom_ext i; rw cofork.\u03c0_of_\u03c0 at hm; rw hm; exact l.2.symm\u27e9\n\nlemma has_coequalizer_epi_comp [has_coequalizer f g] {W : C} (h : W \u27f6 X) [hm : epi h] :\n  has_coequalizer (h \u226b f) (h \u226b g) :=\n\u27e8\u27e8{ cocone := _, is_colimit := is_coequalizer_epi_comp (colimit.is_colimit _) h }\u27e9\u27e9\n\nvariables (C f g)\n\n/-- A coequalizer of an idempotent morphism and the identity is split epi. -/\n@[simps]\ndef split_epi_of_idempotent_of_is_colimit_cofork {X : C} {f : X \u27f6 X} (hf : f \u226b f = f)\n  {c : cofork (\ud835\udfd9 X) f} (i : is_colimit c) : split_epi c.\u03c0 :=\n{ section_ := i.desc (cofork.of_\u03c0 f (by simp [hf])),\n  id' :=\n  begin\n    letI := epi_of_is_colimit_cofork i,\n    rw [\u2190 cancel_epi_id c.\u03c0, \u2190 category.assoc, cofork.is_colimit.\u03c0_desc,\n      cofork.\u03c0_of_\u03c0, \u2190 c.condition],\n    exact category.id_comp _,\n  end }\n\n/-- The coequalizer of an idempotent morphism and the identity is split epi. -/\ndef split_epi_of_idempotent_coequalizer {X : C} {f : X \u27f6 X} (hf : f \u226b f = f)\n  [has_coequalizer (\ud835\udfd9 X) f] : split_epi (coequalizer.\u03c0 (\ud835\udfd9 X) f) :=\nsplit_epi_of_idempotent_of_is_colimit_cofork _ hf (colimit.is_colimit _)\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/shapes/equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.38946914417855755}}
{"text": "/-\nCopyright (c) 2021 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n\n! This file was ported from Lean 3 source module combinatorics.quiver.symmetric\n! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Combinatorics.Quiver.Basic\nimport Mathbin.Combinatorics.Quiver.Path\nimport Mathbin.Combinatorics.Quiver.Push\nimport Mathbin.Data.Sum.Basic\n\n/-!\n## Symmetric quivers and arrow reversal\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file contains constructions related to symmetric quivers:\n\n* `symmetrify V` adds formal inverses to each arrow of `V`.\n* `has_reverse` is the class of quivers where each arrow has an assigned formal inverse.\n* `has_involutive_reverse` extends `has_reverse` by requiring that the reverse of the reverse\n  is equal to the original arrow.\n* `prefunctor.preserve_reverse` is the class of prefunctors mapping reverses to reverses.\n* `symmetrify.of`, `symmetrify.lift`, and the associated lemmas witness the universal property\n  of `symmetrify`.\n-/\n\n\nuniverse v u w v'\n\nnamespace Quiver\n\n#print Quiver.Symmetrify /-\n/-- A type synonym for the symmetrized quiver (with an arrow both ways for each original arrow).\n    NB: this does not work for `Prop`-valued quivers. It requires `[quiver.{v+1} V]`. -/\n@[nolint has_nonempty_instance]\ndef Symmetrify (V : Type _) :=\n  V\n#align quiver.symmetrify Quiver.Symmetrify\n-/\n\n#print Quiver.symmetrifyQuiver /-\ninstance symmetrifyQuiver (V : Type u) [Quiver V] : Quiver (Symmetrify V) :=\n  \u27e8fun a b : V => Sum (a \u27f6 b) (b \u27f6 a)\u27e9\n#align quiver.symmetrify_quiver Quiver.symmetrifyQuiver\n-/\n\nvariable (U V W : Type _) [Quiver.{u + 1} U] [Quiver.{v + 1} V] [Quiver.{w + 1} W]\n\n#print Quiver.HasReverse /-\n/-- A quiver `has_reverse` if we can reverse an arrow `p` from `a` to `b` to get an arrow\n    `p.reverse` from `b` to `a`.-/\nclass HasReverse where\n  reverse' : \u2200 {a b : V}, (a \u27f6 b) \u2192 (b \u27f6 a)\n#align quiver.has_reverse Quiver.HasReverse\n-/\n\n#print Quiver.reverse /-\n/-- Reverse the direction of an arrow. -/\ndef reverse {V} [Quiver.{v + 1} V] [HasReverse V] {a b : V} : (a \u27f6 b) \u2192 (b \u27f6 a) :=\n  HasReverse.reverse'\n#align quiver.reverse Quiver.reverse\n-/\n\n#print Quiver.HasInvolutiveReverse /-\n/-- A\u00a0quiver `has_involutive_reverse` if reversing twice is the identity.`-/\nclass HasInvolutiveReverse extends HasReverse V where\n  inv' : \u2200 {a b : V} (f : a \u27f6 b), reverse (reverse f) = f\n#align quiver.has_involutive_reverse Quiver.HasInvolutiveReverse\n-/\n\nvariable {U V W}\n\n/- warning: quiver.reverse_reverse -> Quiver.reverse_reverse is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [h : Quiver.HasInvolutiveReverse.{u1, u2} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u1, u2} V _inst_2 a b), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} V _inst_2 a b) (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 h) b a (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 h) a b f)) f\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [h : Quiver.HasInvolutiveReverse.{u2, u1} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u2, u1} V _inst_2 a b), Eq.{succ u2} (Quiver.Hom.{succ u2, u1} V _inst_2 a b) (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 h) b a (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 h) a b f)) f\nCase conversion may be inaccurate. Consider using '#align quiver.reverse_reverse Quiver.reverse_reverse\u2093'. -/\n@[simp]\ntheorem reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (f : a \u27f6 b) :\n    reverse (reverse f) = f :=\n  h.inv' f\n#align quiver.reverse_reverse Quiver.reverse_reverse\n\n/- warning: quiver.reverse_inj -> Quiver.reverse_inj is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u1, u2} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u1, u2} V _inst_2 a b) (g : Quiver.Hom.{succ u1, u2} V _inst_2 a b), Iff (Eq.{succ u1} (Quiver.Hom.{succ u1, u2} V _inst_2 b a) (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) a b f) (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) a b g)) (Eq.{succ u1} (Quiver.Hom.{succ u1, u2} V _inst_2 a b) f g)\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u2, u1} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u2, u1} V _inst_2 a b) (g : Quiver.Hom.{succ u2, u1} V _inst_2 a b), Iff (Eq.{succ u2} (Quiver.Hom.{succ u2, u1} V _inst_2 b a) (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) a b f) (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) a b g)) (Eq.{succ u2} (Quiver.Hom.{succ u2, u1} V _inst_2 a b) f g)\nCase conversion may be inaccurate. Consider using '#align quiver.reverse_inj Quiver.reverse_inj\u2093'. -/\n@[simp]\ntheorem reverse_inj [HasInvolutiveReverse V] {a b : V} (f g : a \u27f6 b) :\n    reverse f = reverse g \u2194 f = g := by\n  constructor\n  \u00b7 rintro h\n    simpa using congr_arg Quiver.reverse h\n  \u00b7 rintro h\n    congr\n    assumption\n#align quiver.reverse_inj Quiver.reverse_inj\n\n/- warning: quiver.eq_reverse_iff -> Quiver.eq_reverse_iff is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u1, u2} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u1, u2} V _inst_2 a b) (g : Quiver.Hom.{succ u1, u2} V _inst_2 b a), Iff (Eq.{succ u1} (Quiver.Hom.{succ u1, u2} V _inst_2 a b) f (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) b a g)) (Eq.{succ u1} (Quiver.Hom.{succ u1, u2} V _inst_2 b a) (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) a b f) g)\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u2, u1} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u2, u1} V _inst_2 a b) (g : Quiver.Hom.{succ u2, u1} V _inst_2 b a), Iff (Eq.{succ u2} (Quiver.Hom.{succ u2, u1} V _inst_2 a b) f (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) b a g)) (Eq.{succ u2} (Quiver.Hom.{succ u2, u1} V _inst_2 b a) (Quiver.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) a b f) g)\nCase conversion may be inaccurate. Consider using '#align quiver.eq_reverse_iff Quiver.eq_reverse_iff\u2093'. -/\ntheorem eq_reverse_iff [HasInvolutiveReverse V] {a b : V} (f : a \u27f6 b) (g : b \u27f6 a) :\n    f = reverse g \u2194 reverse f = g := by rw [\u2190 reverse_inj, reverse_reverse]\n#align quiver.eq_reverse_iff Quiver.eq_reverse_iff\n\nsection MapReverse\n\nvariable [HasReverse U] [HasReverse V] [HasReverse W]\n\n#print Prefunctor.MapReverse /-\n/-- A prefunctor preserving reversal of arrows -/\nclass Prefunctor.MapReverse (\u03c6 : U \u2964q V) where\n  map_reverse' : \u2200 {u v : U} (e : u \u27f6 v), \u03c6.map (reverse e) = reverse (\u03c6.map e)\n#align prefunctor.map_reverse Prefunctor.MapReverse\n-/\n\n/- warning: prefunctor.map_reverse' -> Prefunctor.map_reverse is a dubious translation:\nlean 3 declaration is\n  forall {U : Type.{u3}} {V : Type.{u4}} [_inst_1 : Quiver.{succ u2, u3} U] [_inst_2 : Quiver.{succ u1, u4} V] [_inst_4 : Quiver.HasReverse.{u2, u3} U _inst_1] [_inst_5 : Quiver.HasReverse.{u1, u4} V _inst_2] (\u03c6 : Prefunctor.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2) [_inst_7 : Prefunctor.MapReverse.{u1, u2, u3, u4} U V _inst_1 _inst_2 _inst_4 _inst_5 \u03c6] {u : U} {v : U} (e : Quiver.Hom.{succ u2, u3} U _inst_1 u v), Eq.{succ u1} (Quiver.Hom.{succ u1, u4} V _inst_2 (Prefunctor.obj.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 v) (Prefunctor.obj.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 u)) (Prefunctor.map.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 v u (Quiver.reverse.{u2, u3} U _inst_1 _inst_4 u v e)) (Quiver.reverse.{u1, u4} V _inst_2 _inst_5 (Prefunctor.obj.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 u) (Prefunctor.obj.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 v) (Prefunctor.map.{succ u2, succ u1, u3, u4} U _inst_1 V _inst_2 \u03c6 u v e))\nbut is expected to have type\n  forall {U : Type.{u2}} {V : Type.{u1}} [_inst_1 : Quiver.{succ u4, u2} U] [_inst_2 : Quiver.{succ u3, u1} V] [_inst_4 : Quiver.HasReverse.{u4, u2} U _inst_1] [_inst_5 : Quiver.HasReverse.{u3, u1} V _inst_2] (\u03c6 : Prefunctor.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2) [_inst_7 : Prefunctor.MapReverse.{u3, u4, u2, u1} U V _inst_1 _inst_2 _inst_4 _inst_5 \u03c6] {u : U} {v : U} (e : Quiver.Hom.{succ u4, u2} U _inst_1 u v), Eq.{succ u3} (Quiver.Hom.{succ u3, u1} V _inst_2 (Prefunctor.obj.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 v) (Prefunctor.obj.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 u)) (Prefunctor.map.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 v u (Quiver.reverse.{u4, u2} U _inst_1 _inst_4 u v e)) (Quiver.reverse.{u3, u1} V _inst_2 _inst_5 (Prefunctor.obj.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 u) (Prefunctor.obj.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 v) (Prefunctor.map.{succ u4, succ u3, u2, u1} U _inst_1 V _inst_2 \u03c6 u v e))\nCase conversion may be inaccurate. Consider using '#align prefunctor.map_reverse' Prefunctor.map_reverse\u2093'. -/\n@[simp]\ntheorem Prefunctor.map_reverse (\u03c6 : U \u2964q V) [\u03c6.MapReverse] {u v : U} (e : u \u27f6 v) :\n    \u03c6.map (reverse e) = reverse (\u03c6.map e) :=\n  Prefunctor.MapReverse.map_reverse' e\n#align prefunctor.map_reverse' Prefunctor.map_reverse\n\n#print Prefunctor.mapReverseComp /-\ninstance Prefunctor.mapReverseComp (\u03c6 : U \u2964q V) (\u03c8 : V \u2964q W) [\u03c6.MapReverse] [\u03c8.MapReverse] :\n    (\u03c6 \u22d9q \u03c8).MapReverse\n    where map_reverse' u v e := by simp only [Prefunctor.comp_map, Prefunctor.map_reverse]\n#align prefunctor.map_reverse_comp Prefunctor.mapReverseComp\n-/\n\n#print Prefunctor.mapReverseId /-\ninstance Prefunctor.mapReverseId : (Prefunctor.id U).MapReverse where map_reverse' u v e := rfl\n#align prefunctor.map_reverse_id Prefunctor.mapReverseId\n-/\n\nend MapReverse\n\ninstance : HasReverse (Symmetrify V) :=\n  \u27e8fun a b e => e.symm\u27e9\n\ninstance : HasInvolutiveReverse (Symmetrify V)\n    where\n  reverse' _ _ e := e.symm\n  inv' _ _ e := congr_fun Sum.swap_swap_eq e\n\n/- warning: quiver.symmetrify_reverse -> Quiver.symmetrify_reverse is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] {a : Quiver.Symmetrify.{u2} V} {b : Quiver.Symmetrify.{u2} V} (e : Quiver.Hom.{succ u1, u2} (Quiver.Symmetrify.{u2} V) (Quiver.symmetrifyQuiver.{u2, u1} V _inst_2) a b), Eq.{succ u1} (Quiver.Hom.{succ u1, u2} (Quiver.Symmetrify.{u2} V) (Quiver.symmetrifyQuiver.{u2, u1} V _inst_2) b a) (Quiver.reverse.{u1, u2} (Quiver.Symmetrify.{u2} V) (Quiver.symmetrifyQuiver.{u2, u1} V _inst_2) (Quiver.Symmetrify.hasReverse.{u1, u2} V _inst_2) a b e) (Sum.swap.{u1, u1} (Quiver.Hom.{succ u1, u2} V _inst_2 a b) (Quiver.Hom.{succ u1, u2} V _inst_2 b a) e)\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] {a : Quiver.Symmetrify.{u1} V} {b : Quiver.Symmetrify.{u1} V} (e : Quiver.Hom.{succ u2, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u2} V _inst_2) a b), Eq.{succ u2} (Quiver.Hom.{succ u2, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u2} V _inst_2) b a) (Quiver.reverse.{u2, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u2} V _inst_2) (Quiver.instHasReverseSymmetrifySymmetrifyQuiver.{u2, u1} V _inst_2) a b e) (Sum.swap.{u2, u2} (Quiver.Hom.{succ u2, u1} V _inst_2 a b) (Quiver.Hom.{succ u2, u1} V _inst_2 b a) e)\nCase conversion may be inaccurate. Consider using '#align quiver.symmetrify_reverse Quiver.symmetrify_reverse\u2093'. -/\n@[simp]\ntheorem symmetrify_reverse {a b : Symmetrify V} (e : a \u27f6 b) : reverse e = e.symm :=\n  rfl\n#align quiver.symmetrify_reverse Quiver.symmetrify_reverse\n\n#print Quiver.Hom.toPos /-\n/-- Shorthand for the \"forward\" arrow corresponding to `f` in `symmetrify V` -/\nabbrev Hom.toPos {X Y : V} (f : X \u27f6 Y) : (Quiver.symmetrifyQuiver V).Hom X Y :=\n  Sum.inl f\n#align quiver.hom.to_pos Quiver.Hom.toPos\n-/\n\n#print Quiver.Hom.toNeg /-\n/-- Shorthand for the \"backward\" arrow corresponding to `f` in `symmetrify V` -/\nabbrev Hom.toNeg {X Y : V} (f : X \u27f6 Y) : (Quiver.symmetrifyQuiver V).Hom Y X :=\n  Sum.inr f\n#align quiver.hom.to_neg Quiver.Hom.toNeg\n-/\n\n#print Quiver.Path.reverse /-\n/-- Reverse the direction of a path. -/\n@[simp]\ndef Path.reverse [HasReverse V] {a : V} : \u2200 {b}, Path a b \u2192 Path b a\n  | a, path.nil => Path.nil\n  | b, path.cons p e => (reverse e).toPath.comp p.reverse\n#align quiver.path.reverse Quiver.Path.reverse\n-/\n\n/- warning: quiver.path.reverse_to_path -> Quiver.Path.reverse_toPath is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [_inst_4 : Quiver.HasReverse.{u1, u2} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u1, u2} V _inst_2 a b), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u1, u2} V _inst_2 b a) (Quiver.Path.reverse.{u1, u2} V _inst_2 _inst_4 a b (Quiver.Hom.toPath.{u2, succ u1} V _inst_2 a b f)) (Quiver.Hom.toPath.{u2, succ u1} V _inst_2 b a (Quiver.reverse.{u1, u2} V _inst_2 _inst_4 a b f))\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [_inst_4 : Quiver.HasReverse.{u2, u1} V _inst_2] {a : V} {b : V} (f : Quiver.Hom.{succ u2, u1} V _inst_2 a b), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u2, u1} V _inst_2 b a) (Quiver.Path.reverse.{u2, u1} V _inst_2 _inst_4 a b (Quiver.Hom.toPath.{u1, succ u2} V _inst_2 a b f)) (Quiver.Hom.toPath.{u1, succ u2} V _inst_2 b a (Quiver.reverse.{u2, u1} V _inst_2 _inst_4 a b f))\nCase conversion may be inaccurate. Consider using '#align quiver.path.reverse_to_path Quiver.Path.reverse_toPath\u2093'. -/\n@[simp]\ntheorem Path.reverse_toPath [HasReverse V] {a b : V} (f : a \u27f6 b) :\n    f.toPath.reverse = (reverse f).toPath :=\n  rfl\n#align quiver.path.reverse_to_path Quiver.Path.reverse_toPath\n\n/- warning: quiver.path.reverse_comp -> Quiver.Path.reverse_comp is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [_inst_4 : Quiver.HasReverse.{u1, u2} V _inst_2] {a : V} {b : V} {c : V} (p : Quiver.Path.{succ u1, u2} V _inst_2 a b) (q : Quiver.Path.{succ u1, u2} V _inst_2 b c), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u1, u2} V _inst_2 c a) (Quiver.Path.reverse.{u1, u2} V _inst_2 _inst_4 a c (Quiver.Path.comp.{u2, succ u1} V _inst_2 a b c p q)) (Quiver.Path.comp.{u2, succ u1} V _inst_2 c b a (Quiver.Path.reverse.{u1, u2} V _inst_2 _inst_4 b c q) (Quiver.Path.reverse.{u1, u2} V _inst_2 _inst_4 a b p))\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [_inst_4 : Quiver.HasReverse.{u2, u1} V _inst_2] {a : V} {b : V} {c : V} (p : Quiver.Path.{succ u2, u1} V _inst_2 a b) (q : Quiver.Path.{succ u2, u1} V _inst_2 b c), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u2, u1} V _inst_2 c a) (Quiver.Path.reverse.{u2, u1} V _inst_2 _inst_4 a c (Quiver.Path.comp.{u1, succ u2} V _inst_2 a b c p q)) (Quiver.Path.comp.{u1, succ u2} V _inst_2 c b a (Quiver.Path.reverse.{u2, u1} V _inst_2 _inst_4 b c q) (Quiver.Path.reverse.{u2, u1} V _inst_2 _inst_4 a b p))\nCase conversion may be inaccurate. Consider using '#align quiver.path.reverse_comp Quiver.Path.reverse_comp\u2093'. -/\n@[simp]\ntheorem Path.reverse_comp [HasReverse V] {a b c : V} (p : Path a b) (q : Path b c) :\n    (p.comp q).reverse = q.reverse.comp p.reverse :=\n  by\n  induction q\n  \u00b7 simp\n  \u00b7 simp [q_ih]\n#align quiver.path.reverse_comp Quiver.Path.reverse_comp\n\n/- warning: quiver.path.reverse_reverse -> Quiver.Path.reverse_reverse is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u1, u2} V _inst_2] {a : V} {b : V} (p : Quiver.Path.{succ u1, u2} V _inst_2 a b), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u1, u2} V _inst_2 a b) (Quiver.Path.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) b a (Quiver.Path.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 _inst_4) a b p)) p\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u2, u1} V] [_inst_4 : Quiver.HasInvolutiveReverse.{u2, u1} V _inst_2] {a : V} {b : V} (p : Quiver.Path.{succ u2, u1} V _inst_2 a b), Eq.{max (succ u2) (succ u1)} (Quiver.Path.{succ u2, u1} V _inst_2 a b) (Quiver.Path.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) b a (Quiver.Path.reverse.{u2, u1} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u1} V _inst_2 _inst_4) a b p)) p\nCase conversion may be inaccurate. Consider using '#align quiver.path.reverse_reverse Quiver.Path.reverse_reverse\u2093'. -/\n@[simp]\ntheorem Path.reverse_reverse [HasInvolutiveReverse V] {a b : V} (p : Path a b) :\n    p.reverse.reverse = p := by\n  induction p\n  \u00b7 simp\n  \u00b7 simp only [path.reverse, path.reverse_comp, path.reverse_to_path, reverse_reverse, p_ih]\n    rfl\n#align quiver.path.reverse_reverse Quiver.Path.reverse_reverse\n\nnamespace Symmetrify\n\n#print Quiver.Symmetrify.of /-\n/-- The inclusion of a quiver in its symmetrification -/\n@[simps]\ndef of : V \u2964q Symmetrify V where\n  obj := id\n  map X Y f := Sum.inl f\n#align quiver.symmetrify.of Quiver.Symmetrify.of\n-/\n\nvariable {V' : Type _} [Quiver.{v' + 1} V']\n\n#print Quiver.Symmetrify.lift /-\n/-- Given a quiver `V'` with reversible arrows, a prefunctor to `V'` can be lifted to one from\n    `symmetrify V` to `V'` -/\ndef lift [HasReverse V'] (\u03c6 : V \u2964q V') : Symmetrify V \u2964q V'\n    where\n  obj := \u03c6.obj\n  map X Y f := Sum.rec (fun fwd => \u03c6.map fwd) (fun bwd => reverse (\u03c6.map bwd)) f\n#align quiver.symmetrify.lift Quiver.Symmetrify.lift\n-/\n\n/- warning: quiver.symmetrify.lift_spec -> Quiver.Symmetrify.lift_spec is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u3}} [_inst_2 : Quiver.{succ u1, u3} V] {V' : Type.{u4}} [_inst_4 : Quiver.{succ u2, u4} V'] [_inst_5 : Quiver.HasReverse.{u2, u4} V' _inst_4] (\u03c6 : Prefunctor.{succ u1, succ u2, u3, u4} V _inst_2 V' _inst_4), Eq.{max (max (succ u3) (succ u1) (succ u2)) (succ u3) (succ u4)} (Prefunctor.{succ u1, succ u2, u3, u4} V _inst_2 V' _inst_4) (Prefunctor.comp.{u3, succ u1, u3, succ u1, u4, succ u2} V _inst_2 (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.of.{u1, u3} V _inst_2) (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 _inst_5 \u03c6)) \u03c6\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u3, u1} V] {V' : Type.{u2}} [_inst_4 : Quiver.{succ u4, u2} V'] [_inst_5 : Quiver.HasReverse.{u4, u2} V' _inst_4] (\u03c6 : Prefunctor.{succ u3, succ u4, u1, u2} V _inst_2 V' _inst_4), Eq.{max (max (max (succ u3) (succ u4)) (succ u1)) (succ u2)} (Prefunctor.{succ u3, succ u4, u1, u2} V _inst_2 V' _inst_4) (Prefunctor.comp.{u1, succ u3, u1, succ u3, u2, succ u4} V _inst_2 (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.of.{u3, u1} V _inst_2) (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 _inst_5 \u03c6)) \u03c6\nCase conversion may be inaccurate. Consider using '#align quiver.symmetrify.lift_spec Quiver.Symmetrify.lift_spec\u2093'. -/\ntheorem lift_spec [HasReverse V'] (\u03c6 : V \u2964q V') : of \u22d9q lift \u03c6 = \u03c6 :=\n  by\n  fapply Prefunctor.ext\n  \u00b7 rintro X\n    rfl\n  \u00b7 rintro X Y f\n    rfl\n#align quiver.symmetrify.lift_spec Quiver.Symmetrify.lift_spec\n\n/- warning: quiver.symmetrify.lift_reverse -> Quiver.Symmetrify.lift_reverse is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u3}} [_inst_2 : Quiver.{succ u1, u3} V] {V' : Type.{u4}} [_inst_4 : Quiver.{succ u2, u4} V'] [h : Quiver.HasInvolutiveReverse.{u2, u4} V' _inst_4] (\u03c6 : Prefunctor.{succ u1, succ u2, u3, u4} V _inst_2 V' _inst_4) {X : Quiver.Symmetrify.{u3} V} {Y : Quiver.Symmetrify.{u3} V} (f : Quiver.Hom.{succ u1, u3} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) X Y), Eq.{succ u2} (Quiver.Hom.{succ u2, u4} V' _inst_4 (Prefunctor.obj.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) Y) (Prefunctor.obj.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) X)) (Prefunctor.map.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) Y X (Quiver.reverse.{u1, u3} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) (Quiver.Symmetrify.hasReverse.{u1, u3} V _inst_2) X Y f)) (Quiver.reverse.{u2, u4} V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) (Prefunctor.obj.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u2, u4} V' _inst_4 h) \u03c6) X Y f))\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u3, u1} V] {V' : Type.{u2}} [_inst_4 : Quiver.{succ u4, u2} V'] [h : Quiver.HasInvolutiveReverse.{u4, u2} V' _inst_4] (\u03c6 : Prefunctor.{succ u3, succ u4, u1, u2} V _inst_2 V' _inst_4) {X : Quiver.Symmetrify.{u1} V} {Y : Quiver.Symmetrify.{u1} V} (f : Quiver.Hom.{succ u3, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) X Y), Eq.{succ u4} (Quiver.Hom.{succ u4, u2} V' _inst_4 (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) Y) (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) X)) (Prefunctor.map.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) Y X (Quiver.reverse.{u3, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) (Quiver.instHasReverseSymmetrifySymmetrifyQuiver.{u3, u1} V _inst_2) X Y f)) (Quiver.reverse.{u4, u2} V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) X) (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) Y) (Prefunctor.map.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 (Quiver.HasInvolutiveReverse.toHasReverse.{u4, u2} V' _inst_4 h) \u03c6) X Y f))\nCase conversion may be inaccurate. Consider using '#align quiver.symmetrify.lift_reverse Quiver.Symmetrify.lift_reverse\u2093'. -/\ntheorem lift_reverse [h : HasInvolutiveReverse V'] (\u03c6 : V \u2964q V') {X Y : Symmetrify V} (f : X \u27f6 Y) :\n    (lift \u03c6).map (Quiver.reverse f) = Quiver.reverse ((lift \u03c6).map f) :=\n  by\n  dsimp [lift]; cases f\n  \u00b7 simp only\n    rfl\n  \u00b7 simp only [reverse_reverse]\n    rfl\n#align quiver.symmetrify.lift_reverse Quiver.Symmetrify.lift_reverse\n\n/- warning: quiver.symmetrify.lift_unique -> Quiver.Symmetrify.lift_unique is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u3}} [_inst_2 : Quiver.{succ u1, u3} V] {V' : Type.{u4}} [_inst_4 : Quiver.{succ u2, u4} V'] [_inst_5 : Quiver.HasReverse.{u2, u4} V' _inst_4] (\u03c6 : Prefunctor.{succ u1, succ u2, u3, u4} V _inst_2 V' _inst_4) (\u03a6 : Prefunctor.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4), (Eq.{max (max (succ u3) (succ u1) (succ u2)) (succ u3) (succ u4)} (Prefunctor.{succ u1, succ u2, u3, u4} V _inst_2 V' _inst_4) (Prefunctor.comp.{u3, succ u1, u3, succ u1, u4, succ u2} V _inst_2 (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4 (Quiver.Symmetrify.of.{u1, u3} V _inst_2) \u03a6) \u03c6) -> (forall [h\u03a6rev : Prefunctor.MapReverse.{u2, u1, u3, u4} (Quiver.Symmetrify.{u3} V) V' (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) _inst_4 (Quiver.Symmetrify.hasReverse.{u1, u3} V _inst_2) _inst_5 \u03a6], Eq.{max (max (succ u3) (succ u1) (succ u2)) (succ u3) (succ u4)} (Prefunctor.{succ u1, succ u2, u3, u4} (Quiver.Symmetrify.{u3} V) (Quiver.symmetrifyQuiver.{u3, u1} V _inst_2) V' _inst_4) \u03a6 (Quiver.Symmetrify.lift.{u1, u2, u3, u4} V _inst_2 V' _inst_4 _inst_5 \u03c6))\nbut is expected to have type\n  forall {V : Type.{u1}} [_inst_2 : Quiver.{succ u3, u1} V] {V' : Type.{u2}} [_inst_4 : Quiver.{succ u4, u2} V'] [_inst_5 : Quiver.HasReverse.{u4, u2} V' _inst_4] (\u03c6 : Prefunctor.{succ u3, succ u4, u1, u2} V _inst_2 V' _inst_4) (\u03a6 : Prefunctor.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4), (Eq.{max (max (max (succ u3) (succ u4)) (succ u1)) (succ u2)} (Prefunctor.{succ u3, succ u4, u1, u2} V _inst_2 V' _inst_4) (Prefunctor.comp.{u1, succ u3, u1, succ u3, u2, succ u4} V _inst_2 (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 (Quiver.Symmetrify.of.{u3, u1} V _inst_2) \u03a6) \u03c6) -> (forall {X : Quiver.Symmetrify.{u1} V} {Y : Quiver.Symmetrify.{u1} V} (f : Quiver.Hom.{succ u3, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) X Y), Eq.{succ u4} (Quiver.Hom.{succ u4, u2} V' _inst_4 (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 Y) (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 X)) (Prefunctor.map.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 Y X (Quiver.reverse.{u3, u1} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) (Quiver.instHasReverseSymmetrifySymmetrifyQuiver.{u3, u1} V _inst_2) X Y f)) (Quiver.reverse.{u4, u2} V' _inst_4 _inst_5 (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 X) (Prefunctor.obj.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 Y) (Prefunctor.map.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4 \u03a6 X Y f))) -> (Eq.{max (max (max (succ u3) (succ u4)) (succ u1)) (succ u2)} (Prefunctor.{succ u3, succ u4, u1, u2} (Quiver.Symmetrify.{u1} V) (Quiver.symmetrifyQuiver.{u1, u3} V _inst_2) V' _inst_4) \u03a6 (Quiver.Symmetrify.lift.{u3, u4, u1, u2} V _inst_2 V' _inst_4 _inst_5 \u03c6))\nCase conversion may be inaccurate. Consider using '#align quiver.symmetrify.lift_unique Quiver.Symmetrify.lift_unique\u2093'. -/\n/-- `lift \u03c6` is the only prefunctor extending `\u03c6` and preserving reverses. -/\ntheorem lift_unique [HasReverse V'] (\u03c6 : V \u2964q V') (\u03a6 : Symmetrify V \u2964q V') (h\u03a6 : of \u22d9q \u03a6 = \u03c6)\n    [h\u03a6rev : \u03a6.MapReverse] : \u03a6 = lift \u03c6 := by\n  subst_vars\n  fapply Prefunctor.ext\n  \u00b7 rintro X\n    rfl\n  \u00b7 rintro X Y f\n    cases f\n    \u00b7 rfl\n    \u00b7 dsimp [lift, of]\n      simp only [\u2190 Prefunctor.map_reverse, symmetrify_reverse, Sum.swap_inl]\n#align quiver.symmetrify.lift_unique Quiver.Symmetrify.lift_unique\n\n/-- A prefunctor canonically defines a prefunctor of the symmetrifications. -/\n@[simps]\ndef Prefunctor.symmetrify (\u03c6 : U \u2964q V) : Symmetrify U \u2964q Symmetrify V\n    where\n  obj := \u03c6.obj\n  map X Y := Sum.map \u03c6.map \u03c6.map\n#align prefunctor.symmetrify Prefunctor.symmetrify\n\ninstance Prefunctor.symmetrifyMapReverse (\u03c6 : U \u2964q V) : Prefunctor.MapReverse \u03c6.Symmetrify :=\n  \u27e8fun u v e => by cases e <;> rfl\u27e9\n#align prefunctor.symmetrify_map_reverse Prefunctor.symmetrifyMapReverse\n\nend Symmetrify\n\nnamespace Push\n\nvariable {V' : Type _} (\u03c3 : V \u2192 V')\n\ninstance [HasReverse V] : HasReverse (Push \u03c3)\n    where reverse' a b F := by\n    cases F\n    constructor\n    apply reverse\n    exact F_f\n\ninstance [HasInvolutiveReverse V] : HasInvolutiveReverse (Push \u03c3)\n    where\n  reverse' a b F := by\n    cases F\n    constructor\n    apply reverse\n    exact F_f\n  inv' a b F := by\n    cases F\n    dsimp [reverse]\n    congr\n    apply reverse_reverse\n\n/- warning: quiver.push.of_reverse -> Quiver.Push.of_reverse is a dubious translation:\nlean 3 declaration is\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u1, u2} V] {V' : Type.{u3}} (\u03c3 : V -> V') [h : Quiver.HasInvolutiveReverse.{u1, u2} V _inst_2] (X : V) (Y : V) (f : Quiver.Hom.{succ u1, u2} V _inst_2 X Y), Eq.{succ (max u2 u3 (succ u1))} (Quiver.Hom.{succ (max u2 u3 (succ u1)), u3} (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Prefunctor.obj.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) Y) (Prefunctor.obj.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) X)) (Quiver.reverse.{max u2 u3 (succ u1), u3} (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.hasReverse.{u1, u2, u3} V _inst_2 V' \u03c3 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 h)) (Prefunctor.obj.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) X) (Prefunctor.obj.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) Y) (Prefunctor.map.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) X Y f)) (Prefunctor.map.{succ u1, succ (max u2 u3 (succ u1)), u2, u3} V _inst_2 (Quiver.Push.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.quiver.{u2, succ u1, u3} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u1, u3} V _inst_2 V' \u03c3) Y X (Quiver.reverse.{u1, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u1, u2} V _inst_2 h) X Y f))\nbut is expected to have type\n  forall {V : Type.{u2}} [_inst_2 : Quiver.{succ u3, u2} V] {V' : Type.{u1}} (\u03c3 : V -> V') [h : Quiver.HasInvolutiveReverse.{u3, u2} V _inst_2] (X : V) (Y : V) (f : Quiver.Hom.{succ u3, u2} V _inst_2 X Y), Eq.{max (max (succ (succ u3)) (succ u2)) (succ u1)} (Quiver.Hom.{succ (max (max (succ u3) u2) u1), u1} (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Prefunctor.obj.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) Y) (Prefunctor.obj.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) X)) (Quiver.reverse.{max (max (succ u3) u2) u1, u1} (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.instHasReversePushInstQuiverPush.{u3, u2, u1} V _inst_2 V' \u03c3 (Quiver.HasInvolutiveReverse.toHasReverse.{u3, u2} V _inst_2 h)) (Prefunctor.obj.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) X) (Prefunctor.obj.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) Y) (Prefunctor.map.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) X Y f)) (Prefunctor.map.{succ u3, max (max (succ (succ u3)) (succ u2)) (succ u1), u2, u1} V _inst_2 (Quiver.Push.{u2, u1} V V' \u03c3) (Quiver.instQuiverPush.{u2, succ u3, u1} V _inst_2 V' \u03c3) (Quiver.Push.of.{u2, succ u3, u1} V _inst_2 V' \u03c3) Y X (Quiver.reverse.{u3, u2} V _inst_2 (Quiver.HasInvolutiveReverse.toHasReverse.{u3, u2} V _inst_2 h) X Y f))\nCase conversion may be inaccurate. Consider using '#align quiver.push.of_reverse Quiver.Push.of_reverse\u2093'. -/\ntheorem of_reverse [h : HasInvolutiveReverse V] (X Y : V) (f : X \u27f6 Y) :\n    (reverse <| (Push.of \u03c3).map f) = (Push.of \u03c3).map (reverse f) :=\n  rfl\n#align quiver.push.of_reverse Quiver.Push.of_reverse\n\n#print Quiver.Push.ofMapReverse /-\ninstance ofMapReverse [h : HasInvolutiveReverse V] : (Push.of \u03c3).MapReverse :=\n  \u27e8by simp [of_reverse]\u27e9\n#align quiver.push.of_map_reverse Quiver.Push.ofMapReverse\n-/\n\nend Push\n\n#print Quiver.IsPreconnected /-\n/-- A quiver is preconnected iff there exists a path between any pair of\nvertices.\nNote that if `V` doesn't `has_reverse`, then the definition is stronger than\nsimply having a preconnected underlying `simple_graph`, since a path in one\ndirection doesn't induce one in the other.\n-/\ndef IsPreconnected (V) [Quiver.{u + 1} V] :=\n  \u2200 X Y : V, Nonempty (Path X Y)\n#align quiver.is_preconnected Quiver.IsPreconnected\n-/\n\nend Quiver\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Combinatorics/Quiver/Symmetric.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3894691441785574}}
{"text": "/-\nCopyright (c) 2021 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.epi_mono\nimport category_theory.limits.has_limits\nimport category_theory.limits.shapes.equalizers\n\n/-!\n# Wide equalizers and wide coequalizers\n\nThis file defines wide (co)equalizers as special cases of (co)limits.\n\nA wide equalizer for the family of morphisms `X \u27f6 Y` indexed by `J` is the categorical\ngeneralization of the subobject `{a \u2208 A | \u2200 j\u2081 j\u2082, f(j\u2081, a) = f(j\u2082, a)}`. Note that if `J` has\nfewer than two morphisms this condition is trivial, so some lemmas and definitions assume `J` is\nnonempty.\n\n## Main definitions\n\n* `walking_parallel_family` is the indexing category used for wide (co)equalizer diagrams\n* `parallel_family` is a functor from `walking_parallel_family` to our category `C`.\n* a `trident` is a cone over a parallel family.\n  * there is really only one interesting morphism in a trident: the arrow from the vertex of the\n    trident to the domain of f and g. It is called `trident.\u03b9`.\n* a `wide_equalizer` is now just a `limit (parallel_family f)`\n\nEach of these has a dual.\n\n## Main statements\n\n* `wide_equalizer.\u03b9_mono` states that every wide_equalizer map is a monomorphism\n* `is_iso_limit_cone_parallel_family_of_self` states that the identity on the domain of `f` is an\n  equalizer of `f` and `f`.\n\n## Implementation notes\nAs with the other special shapes in the limits library, all the definitions here are given as\n`abbreviation`s of the general statements for limits, so all the `simp` lemmas and theorems about\ngeneral limits can be used.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 1*][borceux-vol1]\n-/\n\nnoncomputable theory\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses w v u u\u2082\n\nvariables {J : Type w}\n\n/-- The type of objects for the diagram indexing a wide (co)equalizer. -/\ninductive walking_parallel_family (J : Type w) : Type w\n| zero : walking_parallel_family\n| one : walking_parallel_family\n\nopen walking_parallel_family\n\ninstance : decidable_eq (walking_parallel_family J)\n| zero zero := is_true rfl\n| zero one := is_false (\u03bb t, walking_parallel_family.no_confusion t)\n| one zero := is_false (\u03bb t, walking_parallel_family.no_confusion t)\n| one one := is_true rfl\n\ninstance : inhabited (walking_parallel_family J) := \u27e8zero\u27e9\n\n/-- The type family of morphisms for the diagram indexing a wide (co)equalizer. -/\n@[derive decidable_eq] inductive walking_parallel_family.hom (J : Type w) :\n  walking_parallel_family J \u2192 walking_parallel_family J \u2192 Type w\n| id : \u03a0 X : walking_parallel_family.{w} J, walking_parallel_family.hom X X\n| line : \u03a0 (j : J), walking_parallel_family.hom zero one\n\n/-- Satisfying the inhabited linter -/\ninstance (J : Type v) : inhabited (walking_parallel_family.hom J zero zero) :=\n{ default := hom.id _ }\n\nopen walking_parallel_family.hom\n\n/-- Composition of morphisms in the indexing diagram for wide (co)equalizers. -/\ndef walking_parallel_family.hom.comp :\n  \u03a0 (X Y Z : walking_parallel_family J)\n    (f : walking_parallel_family.hom J X Y) (g : walking_parallel_family.hom J Y Z),\n    walking_parallel_family.hom J X Z\n  | _ _ _ (id _)   h := h\n  | _ _ _ (line j) (id one) := line j.\n\nlocal attribute [tidy] tactic.case_bash\n\ninstance walking_parallel_family.category : small_category (walking_parallel_family J) :=\n{ hom  := walking_parallel_family.hom J,\n  id   := walking_parallel_family.hom.id,\n  comp := walking_parallel_family.hom.comp }\n\n@[simp]\nlemma walking_parallel_family.hom_id (X : walking_parallel_family J) :\n  walking_parallel_family.hom.id X = \ud835\udfd9 X :=\nrfl\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y : C} (f : J \u2192 (X \u27f6 Y))\n\n/--\n`parallel_family f` is the diagram in `C` consisting of the given family of morphisms, each with\ncommon domain and codomain.\n-/\ndef parallel_family : walking_parallel_family J \u2964 C :=\n{ obj := \u03bb x, walking_parallel_family.cases_on x X Y,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, (line j) := f j\n  end,\n  map_comp' :=\n  begin\n    rintro _ _ _ \u27e8\u27e9 \u27e8\u27e9;\n    { unfold_aux, simp; refl },\n  end }\n\n@[simp] lemma parallel_family_obj_zero : (parallel_family f).obj zero = X := rfl\n@[simp] lemma parallel_family_obj_one : (parallel_family f).obj one = Y := rfl\n\n@[simp] lemma parallel_family_map_left {j : J} : (parallel_family f).map (line j) = f j := rfl\n\n/-- Every functor indexing a wide (co)equalizer is naturally isomorphic (actually, equal) to a\n    `parallel_family` -/\n@[simps]\ndef diagram_iso_parallel_family (F : walking_parallel_family J \u2964 C) :\n  F \u2245 parallel_family (\u03bb j, F.map (line j)) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by cases j; tidy) $ by tidy\n\n/-- `walking_parallel_pair` as a category is equivalent to a special case of\n`walking_parallel_family`.  -/\n@[simps]\ndef walking_parallel_family_equiv_walking_parallel_pair :\n  walking_parallel_family.{w} (ulift bool) \u224c walking_parallel_pair :=\n{ functor := parallel_family\n      (\u03bb p, cond p.down walking_parallel_pair_hom.left walking_parallel_pair_hom.right),\n  inverse := parallel_pair (line (ulift.up tt)) (line (ulift.up ff)),\n  unit_iso := nat_iso.of_components (\u03bb X, eq_to_iso (by cases X; refl)) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb X, eq_to_iso (by cases X; refl)) (by tidy) }\n\n/-- A trident on `f` is just a `cone (parallel_family f)`. -/\nabbreviation trident := cone (parallel_family f)\n\n/-- A cotrident on `f` and `g` is just a `cocone (parallel_family f)`. -/\nabbreviation cotrident := cocone (parallel_family f)\n\nvariables {f}\n\n/-- A trident `t` on the parallel family `f : J \u2192 (X \u27f6 Y)` consists of two morphisms\n    `t.\u03c0.app zero : t.X \u27f6 X` and `t.\u03c0.app one : t.X \u27f6 Y`. Of these, only the first one is\n    interesting, and we give it the shorter name `trident.\u03b9 t`. -/\nabbreviation trident.\u03b9 (t : trident f) := t.\u03c0.app zero\n\n/-- A cotrident `t` on the parallel family `f : J \u2192 (X \u27f6 Y)` consists of two morphisms\n    `t.\u03b9.app zero : X \u27f6 t.X` and `t.\u03b9.app one : Y \u27f6 t.X`. Of these, only the second one is\n    interesting, and we give it the shorter name `cotrident.\u03c0 t`. -/\nabbreviation cotrident.\u03c0 (t : cotrident f) := t.\u03b9.app one\n\n@[simp] lemma trident.\u03b9_eq_app_zero (t : trident f) : t.\u03b9 = t.\u03c0.app zero := rfl\n@[simp] lemma cotrident.\u03c0_eq_app_one (t : cotrident f) : t.\u03c0 = t.\u03b9.app one := rfl\n\n@[simp, reassoc] lemma trident.app_zero (s : trident f) (j : J) :\n  s.\u03c0.app zero \u226b f j = s.\u03c0.app one :=\nby rw [\u2190s.w (line j), parallel_family_map_left]\n\n@[simp, reassoc] lemma cotrident.app_one (s : cotrident f) (j : J) :\n  f j \u226b s.\u03b9.app one = s.\u03b9.app zero :=\nby rw [\u2190s.w (line j), parallel_family_map_left]\n\n/--\nA trident on `f : J \u2192 (X \u27f6 Y)` is determined by the morphism `\u03b9 : P \u27f6 X` satisfying\n`\u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082`.\n-/\n@[simps]\ndef trident.of_\u03b9 [nonempty J] {P : C} (\u03b9 : P \u27f6 X) (w : \u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082) :\n  trident f :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb X, walking_parallel_family.cases_on X \u03b9 (\u03b9 \u226b f (classical.arbitrary J)),\n    naturality' := \u03bb i j f,\n      begin\n        dsimp,\n        cases f with _ k,\n        { simp },\n        { simp [w (classical.arbitrary J) k] },\n      end } }\n\n/--\nA cotrident on `f : J \u2192 (X \u27f6 Y)` is determined by the morphism `\u03c0 : Y \u27f6 P` satisfying\n`\u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0`.\n-/\n@[simps]\ndef cotrident.of_\u03c0 [nonempty J] {P : C} (\u03c0 : Y \u27f6 P) (w : \u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0) :\n  cotrident f :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb X, walking_parallel_family.cases_on X (f (classical.arbitrary J) \u226b \u03c0) \u03c0,\n    naturality' := \u03bb i j f,\n      begin\n        dsimp,\n        cases f with _ k,\n        { simp },\n        { simp [w (classical.arbitrary J) k] }\n      end } } -- See note [dsimp, simp]\n\nlemma trident.\u03b9_of_\u03b9 [nonempty J] {P : C} (\u03b9 : P \u27f6 X) (w : \u2200 j\u2081 j\u2082, \u03b9 \u226b f j\u2081 = \u03b9 \u226b f j\u2082) :\n  (trident.of_\u03b9 \u03b9 w).\u03b9 = \u03b9 := rfl\nlemma cotrident.\u03c0_of_\u03c0 [nonempty J] {P : C} (\u03c0 : Y \u27f6 P) (w : \u2200 j\u2081 j\u2082, f j\u2081 \u226b \u03c0 = f j\u2082 \u226b \u03c0) :\n  (cotrident.of_\u03c0 \u03c0 w).\u03c0 = \u03c0 := rfl\n\n@[reassoc]\nlemma trident.condition (j\u2081 j\u2082 : J) (t : trident f) : t.\u03b9 \u226b f j\u2081 = t.\u03b9 \u226b f j\u2082 :=\nby rw [t.app_zero, t.app_zero]\n\n@[reassoc]\nlemma cotrident.condition (j\u2081 j\u2082 : J) (t : cotrident f) : f j\u2081 \u226b t.\u03c0 = f j\u2082 \u226b t.\u03c0 :=\nby rw [t.app_one, t.app_one]\n\n/-- To check whether two maps are equalized by both maps of a trident, it suffices to check it for\nthe first map -/\nlemma trident.equalizer_ext [nonempty J] (s : trident f) {W : C} {k l : W \u27f6 s.X}\n  (h : k \u226b s.\u03b9 = l \u226b s.\u03b9) : \u2200 (j : walking_parallel_family J),\n    k \u226b s.\u03c0.app j = l \u226b s.\u03c0.app j\n| zero := h\n| one := by rw [\u2190s.app_zero (classical.arbitrary J), reassoc_of h]\n\n/-- To check whether two maps are coequalized by both maps of a cotrident, it suffices to check it\nfor the second map -/\nlemma cotrident.coequalizer_ext [nonempty J] (s : cotrident f) {W : C} {k l : s.X \u27f6 W}\n  (h : s.\u03c0 \u226b k = s.\u03c0 \u226b l) : \u2200 (j : walking_parallel_family J),\n    s.\u03b9.app j \u226b k = s.\u03b9.app j \u226b l\n| zero := by rw [\u2190s.app_one (classical.arbitrary J), category.assoc, category.assoc, h]\n| one := h\n\nlemma trident.is_limit.hom_ext [nonempty J] {s : trident f} (hs : is_limit s)\n  {W : C} {k l : W \u27f6 s.X} (h : k \u226b s.\u03b9 = l \u226b s.\u03b9) :\n  k = l :=\nhs.hom_ext $ trident.equalizer_ext _ h\n\nlemma cotrident.is_colimit.hom_ext [nonempty J] {s : cotrident f} (hs : is_colimit s)\n  {W : C} {k l : s.X \u27f6 W} (h : s.\u03c0 \u226b k = s.\u03c0 \u226b l) :\n  k = l :=\nhs.hom_ext $ cotrident.coequalizer_ext _ h\n\n/-- If `s` is a limit trident over `f`, then a morphism `k : W \u27f6 X` satisfying\n    `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` induces a morphism `l : W \u27f6 s.X` such that\n    `l \u226b trident.\u03b9 s = k`. -/\ndef trident.is_limit.lift' [nonempty J] {s : trident f} (hs : is_limit s) {W : C} (k : W \u27f6 X)\n  (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  {l : W \u27f6 s.X // l \u226b trident.\u03b9 s = k} :=\n\u27e8hs.lift $ trident.of_\u03b9 _ h, hs.fac _ _\u27e9\n\n/-- If `s` is a colimit cotrident over `f`, then a morphism `k : Y \u27f6 W` satisfying\n    `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` induces a morphism `l : s.X \u27f6 W` such that\n    `cotrident.\u03c0 s \u226b l = k`. -/\ndef cotrident.is_colimit.desc' [nonempty J] {s : cotrident f} (hs : is_colimit s) {W : C}\n  (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  {l : s.X \u27f6 W // cotrident.\u03c0 s \u226b l = k} :=\n\u27e8hs.desc $ cotrident.of_\u03c0 _ h, hs.fac _ _\u27e9\n\n/-- This is a slightly more convenient method to verify that a trident is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef trident.is_limit.mk [nonempty J] (t : trident f)\n  (lift : \u03a0 (s : trident f), s.X \u27f6 t.X)\n  (fac : \u2200 (s : trident f), lift s \u226b t.\u03b9 = s.\u03b9)\n  (uniq : \u2200 (s : trident f) (m : s.X \u27f6 t.X)\n  (w : \u2200 j : walking_parallel_family J, m \u226b t.\u03c0.app j = s.\u03c0.app j), m = lift s) :\n  is_limit t :=\n{ lift := lift,\n  fac' := \u03bb s j, walking_parallel_family.cases_on j (fac s)\n    (by rw [\u2190t.w (line (classical.arbitrary J)), reassoc_of fac, s.w]),\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a trident is a limit cone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef trident.is_limit.mk' [nonempty J] (t : trident f)\n  (create : \u03a0 (s : trident f), {l // l \u226b t.\u03b9 = s.\u03b9 \u2227 \u2200 {m}, m \u226b t.\u03b9 = s.\u03b9 \u2192 m = l}) :\nis_limit t :=\ntrident.is_limit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w zero))\n\n/-- This is a slightly more convenient method to verify that a cotrident is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content -/\ndef cotrident.is_colimit.mk [nonempty J] (t : cotrident f)\n  (desc : \u03a0 (s : cotrident f), t.X \u27f6 s.X)\n  (fac : \u2200 (s : cotrident f), t.\u03c0 \u226b desc s = s.\u03c0)\n  (uniq : \u2200 (s : cotrident f) (m : t.X \u27f6 s.X)\n  (w : \u2200 j : walking_parallel_family J, t.\u03b9.app j \u226b m = s.\u03b9.app j), m = desc s) :\n  is_colimit t :=\n{ desc := desc,\n  fac' := \u03bb s j, walking_parallel_family.cases_on j\n    (by rw [\u2190t.w_assoc (line (classical.arbitrary J)), fac, s.w]) (fac s),\n  uniq' := uniq }\n\n/-- This is another convenient method to verify that a cotrident is a colimit cocone. It\n    only asks for a proof of facts that carry any mathematical content, and allows access to the\n    same `s` for all parts. -/\ndef cotrident.is_colimit.mk' [nonempty J] (t : cotrident f)\n  (create : \u03a0 (s : cotrident f), {l : t.X \u27f6 s.X // t.\u03c0 \u226b l = s.\u03c0 \u2227 \u2200 {m}, t.\u03c0 \u226b m = s.\u03c0 \u2192 m = l}) :\n  is_colimit t :=\ncotrident.is_colimit.mk t\n  (\u03bb s, (create s).1)\n  (\u03bb s, (create s).2.1)\n  (\u03bb s m w, (create s).2.2 (w one))\n\n/--\nGiven a limit cone for the family `f : J \u2192 (X \u27f6 Y)`, for any `Z`, morphisms from `Z` to its point\nare in bijection with morphisms `h : Z \u27f6 X` such that `\u2200 j\u2081 j\u2082, h \u226b f j\u2081 = h \u226b f j\u2082`.\nFurther, this bijection is natural in `Z`: see `trident.is_limit.hom_iso_natural`.\n-/\n@[simps]\ndef trident.is_limit.hom_iso [nonempty J] {t : trident f} (ht : is_limit t) (Z : C) :\n  (Z \u27f6 t.X) \u2243 {h : Z \u27f6 X // \u2200 j\u2081 j\u2082, h \u226b f j\u2081 = h \u226b f j\u2082} :=\n{ to_fun := \u03bb k, \u27e8k \u226b t.\u03b9, by simp\u27e9,\n  inv_fun := \u03bb h, (trident.is_limit.lift' ht _ h.prop).1,\n  left_inv := \u03bb k, trident.is_limit.hom_ext ht (trident.is_limit.lift' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (trident.is_limit.lift' ht _ _).prop }\n\n/-- The bijection of `trident.is_limit.hom_iso` is natural in `Z`. -/\nlemma trident.is_limit.hom_iso_natural [nonempty J] {t : trident f} (ht : is_limit t)\n  {Z Z' : C} (q : Z' \u27f6 Z) (k : Z \u27f6 t.X) :\n  (trident.is_limit.hom_iso ht _ (q \u226b k) : Z' \u27f6 X) =\n  q \u226b (trident.is_limit.hom_iso ht _ k : Z \u27f6 X) :=\ncategory.assoc _ _ _\n\n/--\nGiven a colimit cocone for the family `f : J \u2192 (X \u27f6 Y)`, for any `Z`, morphisms from the cocone\npoint to `Z` are in bijection with morphisms `h : Z \u27f6 X` such that\n`\u2200 j\u2081 j\u2082, f j\u2081 \u226b h = f j\u2082 \u226b h`.  Further, this bijection is natural in `Z`: see\n`cotrident.is_colimit.hom_iso_natural`.\n-/\n@[simps]\ndef cotrident.is_colimit.hom_iso [nonempty J] {t : cotrident f} (ht : is_colimit t) (Z : C) :\n  (t.X \u27f6 Z) \u2243 {h : Y \u27f6 Z // \u2200 j\u2081 j\u2082, f j\u2081 \u226b h = f j\u2082 \u226b h} :=\n{ to_fun := \u03bb k, \u27e8t.\u03c0 \u226b k, by simp\u27e9,\n  inv_fun := \u03bb h, (cotrident.is_colimit.desc' ht _ h.prop).1,\n  left_inv := \u03bb k, cotrident.is_colimit.hom_ext ht (cotrident.is_colimit.desc' _ _ _).prop,\n  right_inv := \u03bb h, subtype.ext (cotrident.is_colimit.desc' ht _ _).prop }\n\n/-- The bijection of `cotrident.is_colimit.hom_iso` is natural in `Z`. -/\nlemma cotrident.is_colimit.hom_iso_natural [nonempty J] {t : cotrident f} {Z Z' : C}\n  (q : Z \u27f6 Z') (ht : is_colimit t) (k : t.X \u27f6 Z) :\n    (cotrident.is_colimit.hom_iso ht _ (k \u226b q) : Y \u27f6 Z') =\n    (cotrident.is_colimit.hom_iso ht _ k : Y \u27f6 Z) \u226b q :=\n(category.assoc _ _ _).symm\n\n/-- This is a helper construction that can be useful when verifying that a category has certain wide\n    equalizers. Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))`, and a trident on `\u03bb j, F.map (line j)`, we get a cone\n    on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_wide_equalizers_of_has_limit_parallel_family`, which you may find to be an easier way of\n    achieving your goal. -/\ndef cone.of_trident\n  {F : walking_parallel_family J \u2964 C} (t : trident (\u03bb j, F.map (line j))) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g, by { cases g; { dsimp, simp } } } }\n\n/-- This is a helper construction that can be useful when verifying that a category has all\n    coequalizers. Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))`, and a cotrident on `\u03bb j, F.map (line j)` we get a\n    cocone on `F`.\n\n    If you're thinking about using this, have a look at\n    `has_wide_coequalizers_of_has_colimit_parallel_family`, which you may find to be an easier way\n    of achieving your goal. -/\ndef cocone.of_cotrident\n  {F : walking_parallel_family J \u2964 C} (t : cotrident (\u03bb j, F.map (line j))) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g, by { cases g; dsimp; simp [cotrident.app_one t] } } }\n\n@[simp] lemma cone.of_trident_\u03c0\n  {F : walking_parallel_family J \u2964 C} (t : trident (\u03bb j, F.map (line j))) (j) :\n  (cone.of_trident t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\n@[simp] lemma cocone.of_cotrident_\u03b9\n  {F : walking_parallel_family J \u2964 C} (t : cotrident (\u03bb j, F.map (line j))) (j) :\n  (cocone.of_cotrident t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/-- Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (\u03bb j, F.map (line j))` and a cone on `F`, we get a trident on\n    `\u03bb j, F.map (line j)`. -/\ndef trident.of_cone\n  {F : walking_parallel_family J \u2964 C} (t : cone F) : trident (\u03bb j, F.map (line j)) :=\n{ X := t.X,\n  \u03c0 := { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy) } }\n\n/-- Given `F : walking_parallel_family \u2964 C`, which is really the same as\n    `parallel_family (F.map left) (F.map right)` and a cocone on `F`, we get a cotrident on\n    `\u03bb j, F.map (line j)`. -/\ndef cotrident.of_cocone\n  {F : walking_parallel_family J \u2964 C} (t : cocone F) : cotrident (\u03bb j, F.map (line j)) :=\n{ X := t.X,\n  \u03b9 := { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X } }\n\n@[simp] lemma trident.of_cone_\u03c0 {F : walking_parallel_family J \u2964 C} (t : cone F) (j) :\n  (trident.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n@[simp] lemma cotrident.of_cocone_\u03b9 {F : walking_parallel_family J \u2964 C} (t : cocone F) (j) :\n  (cotrident.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\n/--\nHelper function for constructing morphisms between wide equalizer tridents.\n-/\n@[simps]\ndef trident.mk_hom [nonempty J] {s t : trident f} (k : s.X \u27f6 t.X) (w : k \u226b t.\u03b9 = s.\u03b9) : s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { exact w },\n    { simpa using w =\u226b f (classical.arbitrary J) },\n  end }\n\n/--\nTo construct an isomorphism between tridents,\nit suffices to give an isomorphism between the cone points\nand check that it commutes with the `\u03b9` morphisms.\n-/\n@[simps]\ndef trident.ext [nonempty J] {s t : trident f} (i : s.X \u2245 t.X) (w : i.hom \u226b t.\u03b9 = s.\u03b9) : s \u2245 t :=\n{ hom := trident.mk_hom i.hom w,\n  inv := trident.mk_hom i.inv (by rw [\u2190 w, iso.inv_hom_id_assoc]) }\n\n/--\nHelper function for constructing morphisms between coequalizer cotridents.\n-/\n@[simps]\ndef cotrident.mk_hom [nonempty J] {s t : cotrident f} (k : s.X \u27f6 t.X) (w : s.\u03c0 \u226b k = t.\u03c0) :\n  s \u27f6 t :=\n{ hom := k,\n  w' :=\n  begin\n    rintro \u27e8_|_\u27e9,\n    { simpa using f (classical.arbitrary J) \u226b= w },\n    { exact w },\n  end }\n\n/--\nTo construct an isomorphism between cotridents,\nit suffices to give an isomorphism between the cocone points\nand check that it commutes with the `\u03c0` morphisms.\n-/\ndef cotrident.ext [nonempty J] {s t : cotrident f} (i : s.X \u2245 t.X) (w : s.\u03c0 \u226b i.hom = t.\u03c0) :\n  s \u2245 t :=\n{ hom := cotrident.mk_hom i.hom w,\n  inv := cotrident.mk_hom i.inv (by rw [iso.comp_inv_eq, w]) }\n\nvariables (f)\n\nsection\n/--\n`has_wide_equalizer f` represents a particular choice of limiting cone for the parallel family of\nmorphisms `f`.\n-/\nabbreviation has_wide_equalizer := has_limit (parallel_family f)\n\nvariables [has_wide_equalizer f]\n\n/-- If a wide equalizer of `f` exists, we can access an arbitrary choice of such by\n    saying `wide_equalizer f`. -/\nabbreviation wide_equalizer : C := limit (parallel_family f)\n\n/-- If a wide equalizer of `f` exists, we can access the inclusion `wide_equalizer f \u27f6 X` by\n    saying `wide_equalizer.\u03b9 f`. -/\nabbreviation wide_equalizer.\u03b9 : wide_equalizer f \u27f6 X :=\nlimit.\u03c0 (parallel_family f) zero\n\n/--\nA wide equalizer cone for a parallel family `f`.\n-/\nabbreviation wide_equalizer.trident : trident f := limit.cone (parallel_family f)\n\n@[simp] lemma wide_equalizer.trident_\u03b9 :\n  (wide_equalizer.trident f).\u03b9 = wide_equalizer.\u03b9 f := rfl\n\n@[simp] lemma wide_equalizer.trident_\u03c0_app_zero :\n  (wide_equalizer.trident f).\u03c0.app zero = wide_equalizer.\u03b9 f := rfl\n\n@[reassoc] lemma wide_equalizer.condition (j\u2081 j\u2082 : J) :\n  wide_equalizer.\u03b9 f \u226b f j\u2081 = wide_equalizer.\u03b9 f \u226b f j\u2082 :=\ntrident.condition j\u2081 j\u2082 $ limit.cone $ parallel_family f\n\n/-- The wide_equalizer built from `wide_equalizer.\u03b9 f` is limiting. -/\ndef wide_equalizer_is_wide_equalizer [nonempty J] :\n  is_limit (trident.of_\u03b9 (wide_equalizer.\u03b9 f) (wide_equalizer.condition f)) :=\nis_limit.of_iso_limit (limit.is_limit _) (trident.ext (iso.refl _) (by tidy))\n\nvariables {f}\n\n/-- A morphism `k : W \u27f6 X` satisfying `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` factors through the\n    wide equalizer of `f` via `wide_equalizer.lift : W \u27f6 wide_equalizer f`. -/\nabbreviation wide_equalizer.lift [nonempty J] {W : C} (k : W \u27f6 X)\n  (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  W \u27f6 wide_equalizer f :=\nlimit.lift (parallel_family f) (trident.of_\u03b9 k h)\n\n@[simp, reassoc]\nlemma wide_equalizer.lift_\u03b9 [nonempty J] {W : C} (k : W \u27f6 X) (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  wide_equalizer.lift k h \u226b wide_equalizer.\u03b9 f = k :=\nlimit.lift_\u03c0 _ _\n\n/-- A morphism `k : W \u27f6 X` satisfying `\u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082` induces a morphism\n    `l : W \u27f6 wide_equalizer f` satisfying `l \u226b wide_equalizer.\u03b9 f = k`. -/\ndef wide_equalizer.lift' [nonempty J] {W : C} (k : W \u27f6 X) (h : \u2200 j\u2081 j\u2082, k \u226b f j\u2081 = k \u226b f j\u2082) :\n  {l : W \u27f6 wide_equalizer f // l \u226b wide_equalizer.\u03b9 f = k} :=\n\u27e8wide_equalizer.lift k h, wide_equalizer.lift_\u03b9 _ _\u27e9\n\n/-- Two maps into a wide equalizer are equal if they are are equal when composed with the wide\n    equalizer map. -/\n@[ext] lemma wide_equalizer.hom_ext [nonempty J] {W : C} {k l : W \u27f6 wide_equalizer f}\n  (h : k \u226b wide_equalizer.\u03b9 f = l \u226b wide_equalizer.\u03b9 f) : k = l :=\ntrident.is_limit.hom_ext (limit.is_limit _) h\n\n/-- A wide equalizer morphism is a monomorphism -/\ninstance wide_equalizer.\u03b9_mono [nonempty J] : mono (wide_equalizer.\u03b9 f) :=\n{ right_cancellation := \u03bb Z h k w, wide_equalizer.hom_ext w }\n\nend\n\nsection\nvariables {f}\n/-- The wide equalizer morphism in any limit cone is a monomorphism. -/\nlemma mono_of_is_limit_parallel_family [nonempty J] {c : cone (parallel_family f)}\n  (i : is_limit c) :\n  mono (trident.\u03b9 c) :=\n{ right_cancellation := \u03bb Z h k w, trident.is_limit.hom_ext i w }\n\nend\n\nsection\n/--\n`has_wide_coequalizer f g` represents a particular choice of colimiting cocone\nfor the parallel family of morphisms `f`.\n-/\nabbreviation has_wide_coequalizer := has_colimit (parallel_family f)\n\nvariables [has_wide_coequalizer f]\n\n/-- If a wide coequalizer of `f`, we can access an arbitrary choice of such by\n    saying `wide_coequalizer f`. -/\nabbreviation wide_coequalizer : C := colimit (parallel_family f)\n\n/--  If a wide_coequalizer of `f` exists, we can access the corresponding projection by\n    saying `wide_coequalizer.\u03c0 f`. -/\nabbreviation wide_coequalizer.\u03c0 : Y \u27f6 wide_coequalizer f :=\ncolimit.\u03b9 (parallel_family f) one\n\n/--\nAn arbitrary choice of coequalizer cocone for a parallel family `f`.\n-/\nabbreviation wide_coequalizer.cotrident : cotrident f := colimit.cocone (parallel_family f)\n\n@[simp] lemma wide_coequalizer.cotrident_\u03c0 :\n  (wide_coequalizer.cotrident f).\u03c0 = wide_coequalizer.\u03c0 f := rfl\n\n@[simp] lemma wide_coequalizer.cotrident_\u03b9_app_one :\n  (wide_coequalizer.cotrident f).\u03b9.app one = wide_coequalizer.\u03c0 f := rfl\n\n@[reassoc] lemma wide_coequalizer.condition (j\u2081 j\u2082 : J) :\n  f j\u2081 \u226b wide_coequalizer.\u03c0 f = f j\u2082 \u226b wide_coequalizer.\u03c0 f :=\ncotrident.condition j\u2081 j\u2082 $ colimit.cocone $ parallel_family f\n\n/-- The cotrident built from `wide_coequalizer.\u03c0 f` is colimiting. -/\ndef wide_coequalizer_is_wide_coequalizer [nonempty J] :\n  is_colimit (cotrident.of_\u03c0 (wide_coequalizer.\u03c0 f) (wide_coequalizer.condition f)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit _) (cotrident.ext (iso.refl _) (by tidy))\n\nvariables {f}\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` factors through the\n    wide coequalizer of `f` via `wide_coequalizer.desc : wide_coequalizer f \u27f6 W`. -/\nabbreviation wide_coequalizer.desc [nonempty J] {W : C} (k : Y \u27f6 W)\n  (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  wide_coequalizer f \u27f6 W :=\ncolimit.desc (parallel_family f) (cotrident.of_\u03c0 k h)\n\n@[simp, reassoc]\nlemma wide_coequalizer.\u03c0_desc [nonempty J] {W : C} (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  wide_coequalizer.\u03c0 f \u226b wide_coequalizer.desc k h = k :=\ncolimit.\u03b9_desc _ _\n\n/-- Any morphism `k : Y \u27f6 W` satisfying `\u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k` induces a morphism\n    `l : wide_coequalizer f \u27f6 W` satisfying `wide_coequalizer.\u03c0 \u226b g = l`. -/\ndef wide_coequalizer.desc' [nonempty J] {W : C} (k : Y \u27f6 W) (h : \u2200 j\u2081 j\u2082, f j\u2081 \u226b k = f j\u2082 \u226b k) :\n  {l : wide_coequalizer f \u27f6 W // wide_coequalizer.\u03c0 f \u226b l = k} :=\n\u27e8wide_coequalizer.desc k h, wide_coequalizer.\u03c0_desc _ _\u27e9\n\n/-- Two maps from a wide coequalizer are equal if they are equal when composed with the wide\n    coequalizer map -/\n@[ext] lemma wide_coequalizer.hom_ext [nonempty J] {W : C} {k l : wide_coequalizer f \u27f6 W}\n  (h : wide_coequalizer.\u03c0 f \u226b k = wide_coequalizer.\u03c0 f \u226b l) : k = l :=\ncotrident.is_colimit.hom_ext (colimit.is_colimit _) h\n\n/-- A wide coequalizer morphism is an epimorphism -/\ninstance wide_coequalizer.\u03c0_epi [nonempty J] : epi (wide_coequalizer.\u03c0 f) :=\n{ left_cancellation := \u03bb Z h k w, wide_coequalizer.hom_ext w }\n\nend\n\nsection\nvariables {f}\n\n/-- The wide coequalizer morphism in any colimit cocone is an epimorphism. -/\nlemma epi_of_is_colimit_parallel_family [nonempty J] {c : cocone (parallel_family f)}\n  (i : is_colimit c) :\n  epi (c.\u03b9.app one) :=\n{ left_cancellation := \u03bb Z h k w, cotrident.is_colimit.hom_ext i w }\n\nend\n\nvariables (C)\n\n/-- `has_wide_equalizers` represents a choice of wide equalizer for every family of morphisms -/\nabbreviation has_wide_equalizers := \u03a0 J, has_limits_of_shape (walking_parallel_family.{w} J) C\n\n/-- `has_wide_coequalizers` represents a choice of wide coequalizer for every family of morphisms -/\nabbreviation has_wide_coequalizers := \u03a0 J, has_colimits_of_shape (walking_parallel_family.{w} J) C\n\n/-- If `C` has all limits of diagrams `parallel_family f`, then it has all wide equalizers -/\nlemma has_wide_equalizers_of_has_limit_parallel_family\n  [\u03a0 {J : Type w} {X Y : C} {f : J \u2192 (X \u27f6 Y)}, has_limit (parallel_family f)] :\n  has_wide_equalizers.{w} C :=\n\u03bb J, { has_limit := \u03bb F, has_limit_of_iso (diagram_iso_parallel_family F).symm }\n\n/-- If `C` has all colimits of diagrams `parallel_family f`, then it has all wide coequalizers -/\nlemma has_wide_coequalizers_of_has_colimit_parallel_family\n  [\u03a0 {J : Type w} {X Y : C} {f : J \u2192 (X \u27f6 Y)}, has_colimit (parallel_family f)] :\n  has_wide_coequalizers.{w} C :=\n\u03bb J, { has_colimit := \u03bb F, has_colimit_of_iso (diagram_iso_parallel_family F) }\n\n@[priority 10]\ninstance has_equalizers_of_has_wide_equalizers [has_wide_equalizers.{w} C] : has_equalizers C :=\nhas_limits_of_shape_of_equivalence.{w} walking_parallel_family_equiv_walking_parallel_pair\n\n@[priority 10]\ninstance has_coequalizers_of_has_wide_coequalizers [has_wide_coequalizers.{w} C] :\n  has_coequalizers C :=\nhas_colimits_of_shape_of_equivalence.{w} walking_parallel_family_equiv_walking_parallel_pair\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/shapes/wide_equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38938705357884307}}
{"text": "example (P Q F : Type) : (P \u2192 Q) \u2192 ((Q \u2192 F) \u2192 (P \u2192 F)) :=\nbegin\nintros f g p,\nhave q : Q := f p,\nexact g q,\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/3-function-world/l7.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38938704644528593}}
{"text": "\nimport util.data.fin\nimport util.predicate.basic\nimport tactic.tauto\n\nnamespace predicate\n\nuniverse variables u u' u\u2080 u\u2081 u\u2082\n\nvariables {\u03b1 : Sort u\u2080}\nvariables {\u03b2 : Sort u\u2081}\nvariables {\u03b3 : Sort u\u2082}\nvariables {\u03c3 : Sort u'}\n\n@[simp, predicate]\nlemma value_proj (p : var \u03b2 \u03b3) (f : var \u03b1 \u03b2) (x : \u03b1)\n: (p ! f).apply x = p.apply (f.apply x) :=\nby { cases p, cases f, refl }\n\n@[simp, predicate]\nlemma contramap_apply (p : var \u03b1 \u03b3) (f : var \u03b2 \u03b1) (x : \u03b2)\n: x \u22a8 (p ! f) = f.apply x \u22a8 p :=\nby { cases p , cases f, refl }\n\n@[simp, predicate]\nlemma p_and_to_fun (p\u2080 p\u2081 : pred' \u03b1) (x : \u03b1)\n: x \u22a8 p_and p\u2080 p\u2081 \u2194 x \u22a8 p\u2080 \u2227 x \u22a8 p\u2081 := by refl\n\n@[simp, predicate]\nlemma p_impl_to_fun (p\u2080 p\u2081 : pred' \u03b1) (x : \u03b1)\n: x \u22a8 p_impl p\u2080 p\u2081 \u2194 (x \u22a8 p\u2080 \u2192 x \u22a8 p\u2081) := by refl\n\n@[simp, predicate]\nlemma p_equiv_to_fun (p\u2080 p\u2081 : pred' \u03b1) (x : \u03b1)\n: x \u22a8 p_equiv p\u2080 p\u2081 \u2194 (x \u22a8 p\u2080 \u2194 x \u22a8 p\u2081) :=\nby { simp with lifted_fn }\n\nlemma p_impl_revert {\u0393 p q : pred' \u03b1}\n  (h : \u0393 \u22a2 p_impl p q)\n: \u0393 \u22a2 p \u2192 \u0393 \u22a2 q :=\nbegin\n  intros h\u2081,\n  constructor, introv h\u2082,\n  apply h.apply _ h\u2082,\n  apply h\u2081.apply _ h\u2082,\nend\n\ninstance imp_to_fun \u2983\u03b1\u2984 \u2983\u0393 p q : pred' \u03b1\u2984 : has_coe_to_fun (\u0393 \u22a2 p_impl p q) :=\n{ F := \u03bb _,(\u0393 \u22a2 p) \u2192 (\u0393 \u22a2 q)\n, coe := p_impl_revert }\n\nlemma p_imp_ext {\u0393 p q : pred' \u03b1}\n  (h : \u2200 \u03c3, \u03c3 \u22a8 \u0393 \u2192 \u03c3 \u22a8 p \u2192 \u03c3 \u22a8 q)\n: \u0393 \u22a2 p \u27f6 q :=\n\u27e8 h \u27e9\n\nlemma p_imp_sem {\u0393 p q : pred' \u03b1}\n  (h : \u0393 \u22a2 p \u27f6 q)\n: \u2200 \u03c3, \u03c3 \u22a8 \u0393 \u2192 \u03c3 \u22a8 p \u2192 \u03c3 \u22a8 q :=\nh.apply\n\nlemma p_entails_of_fun (p\u2080 p\u2081 : pred' \u03b1)\n: p_entails p\u2080 p\u2081 \u2194 \u2200 \u0393, \u0393 \u22a2 p\u2080 \u2192 \u0393 \u22a2 p\u2081 :=\nbegin\n  split ; intros h _,\n  { intro h', apply h \u0393 h' },\n  { constructor,\n    introv h\u2080 h\u2081,\n    apply (h  \u27e8 eq \u03c3 \u27e9 _).apply \u03c3 rfl,\n    constructor, introv h,\n    cases h, assumption, }\nend\n\n@[simp]\nlemma False_eq_false (\u0393 : pred' \u03b2) : \u0393 \u22a2 False \u2194 \u0393 = False :=\nbegin\n  split ; intro h,\n  { cases h with h, cases \u0393 with \u0393, simp [False,lifted\u2080],\n    funext \u03c3,\n    specialize h \u03c3, apply eq_false_intro,\n    intro h',\n    apply h h' },\n  { rw h, constructor,\n    intro, exact id }\nend\n\n@[simp, predicate]\nlemma False_sem (\u03c3 : \u03b2) : \u03c3 \u22a8 False \u2194 false :=\nby simp with predicate\n\n@[simp]\nlemma True_eq_true (\u0393 : pred' \u03b2) : \u0393 \u22a2 True = true :=\nby { apply eq_true_intro, constructor, intros, trivial }\n@[simp]\nlemma True_holds : \u22a9 @True \u03b2 :=\nby simp [holds]\n\n@[simp]\nlemma coe_true : (true : pred' \u03b1) = True := rfl\n@[simp]\nlemma coe_false : (false : pred' \u03b1) = False := rfl\n\n@[simp, predicate]\nlemma True_sem (\u03c3 : \u03b2) : \u03c3 \u22a8 True \u2194 true :=\nby simp [holds]\n\nlemma p_forall_revert {\u0393} {p : \u03b2 \u2192 pred' \u03b1}\n  (h : \u0393 \u22a2 p_forall p)\n: \u03a0 i, \u0393 \u22a2 p i :=\nbegin\n  introv,\n  constructor, introv h\u2082,\n  apply h.apply _ h\u2082,\nend\n\ninstance forall_to_fun {\u0393 : pred' \u03b1} \u2983p  : \u03b2 \u2192 pred' \u03b1\u2984 : has_coe_to_fun (\u0393 \u22a2 p_forall p) :=\n{ F := \u03bb _, \u03a0 i, (\u0393 \u22a2 p i)\n, coe := p_forall_revert }\n\n-- instance : has_coe Prop (pred' \u03b1) :=\n-- \u27e8 lifted\u2080 \u27e9\n\ninstance proof_coe (p : Prop) (\u0393 : pred' \u03b1) : has_coe p (\u0393 \u22a2 p) :=\n\u27e8 assume h, \u27e8 \u03bb x _, h \u27e9 \u27e9\n\n-- instance to_prop_to_pred : has_coe (\u03b1 \u2192 Prop) (pred' \u03b1) :=\n-- \u27e8 pred'.mk \u27e9\n\n@[simp, predicate]\nlemma sem_coe (p : Prop) (x : \u03b2)\n: x \u22a8 (p : pred' \u03b2) \u2194 p :=\nby refl\n\n@[simp, predicate]\nlemma sem_seq (\u03c3 : \u03b3) (f : var \u03b3 (\u03b1 \u2192 \u03b2)) (x : var \u03b3 \u03b1)\n: \u03c3 \u22a8 var_seq f x = (\u03c3 \u22a8 f) (\u03c3 \u22a8 x)  :=\nby { cases x, cases f, simp [var_seq], }\n\ninstance var_lawful_functor (\u03b1 : Type u) : is_lawful_functor (var \u03b1) :=\nby { constructor ; intros ; cases x ; refl }\n\ninstance var_lawful_applicative (\u03b1 : Type u) : is_lawful_applicative (var \u03b1) :=\nby { constructor ; intros ; casesm* var _ _ ; refl }\n\ninstance (\u03b1 : Type u) : is_lawful_monad (var \u03b1) :=\n{ pure_bind := by { intros, cases h : f x, simp! [h], }\n, bind_assoc := by { intros ; cases x, simp!, funext, cases (f $ x i), refl }\n, map_pure := by { intros ; casesm* var _ _ ; refl }\n, seq_pure := by { intros ; casesm* var _ _ ; refl }\n, seq_assoc := by intros ; casesm* var _ _ ; refl\n, bind_map_eq_seq := by intros ; casesm* var _ _ ; refl\n, bind_pure_comp_eq_map := by intros ; casesm* var _ _ ; refl\n}\n-- , ..predicate.var_lawful_applicative \u03b1 }\n-- , ..(predicate.var_lawful_applicative \u03b1).to_is_lawful_functor  }\n\n@[simp, predicate]\nlemma var_map_coe {\u03b1 \u03b2 \u03c3 : Type u} (f : \u03b1 \u2192 \u03b2) (g : \u03c3 \u2192 \u03b1)\n: \u27e8 f  \u27e9!\u27e8 g \u27e9 = \u27e8 f \u2218 g \u27e9 :=\nby { simp [proj] }\n\n@[simp, predicate]\nlemma map_proj {\u03b1 \u03b2 \u03b3 \u03c3 : Type u} (f : \u03b1 \u2192 \u03b2) (v\u2081 : var \u03c3 \u03b3) (v\u2080 : var \u03b3 \u03b1)\n: \u27e8 f \u27e9 ! (v\u2080 ! v\u2081) = \u27e8 \u03bb x, f $ v\u2080.apply $ v\u2081.apply x \u27e9 :=\nby { cases v\u2080, cases v\u2081, simp }\n\n@[simp, predicate]\nlemma var_seq_coe {\u03b1 \u03b2 \u03c3 : Type u} (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2) (g : \u03c3 \u2192 \u03b1)\n: (\u27e8 f \u27e9 : var _ _) <*> \u27e8 g \u27e9 = (\u27e8 \u03bb s : \u03c3, f s (g s) \u27e9 : var \u03c3 \u03b2) :=\nby { simp [has_seq.seq] }\n\n@[simp, predicate]\nlemma models_lt {\u03b1 : Type u} [has_lt \u03b1] {s : \u03c3} (x y : var \u03c3 \u03b1)\n: s \u22a8 x \u227a y \u2194 x.apply s < y.apply s :=\nby { cases x, cases y, refl }\n\n@[simp, predicate]\nlemma models_le {\u03b1 : Type u} [has_le \u03b1] {s : \u03c3} (x y : var \u03c3 \u03b1)\n: s \u22a8 x \u227c y \u2194 x.apply s \u2264 y.apply s :=\nby { cases x, cases y, refl }\n\n@[simp, predicate]\nlemma models_wf {_ : has_well_founded \u03b1} (s : \u03c3) (x y : var \u03c3 \u03b1)\n: s \u22a8 x \u227a\u227a y \u2194 x.apply s << y.apply s :=\nby { cases x, cases y, refl }\n\n@[simp, predicate]\nlemma models_eq {s : \u03c3} (x y : var \u03c3 \u03b1)\n: s \u22a8 x \u2243 y \u2194 x.apply s = y.apply s :=\nby { cases x, cases y, refl }\n\n@[simp, predicate]\nlemma models_mem {\u03b1 \u03b2 : Type _} {s : \u03c3} [has_mem \u03b1 \u03b2] (x : var \u03c3 \u03b1) (y : var \u03c3 \u03b2)\n: s \u22a8 (x \u220a y) \u2194 x.apply s \u2208 y.apply s :=\nby { cases x, cases y, refl }\n\n@[simp, predicate]\nlemma coe_value {s : \u03c3} (x : \u03b1)\n: (x : var \u03c3 \u03b1).apply s = x :=\nrfl\n\n@[simp, predicate]\nlemma coe_fun {s : \u03c3} (x : \u03c3 \u2192 \u03b1)\n: (\u27e8 x \u27e9 : var \u03c3 \u03b1).apply s = x s :=\nrfl\n\n@[simp, predicate]\nlemma coe_wf_coe {s : \u03c3} (x y : \u03b1)\n: (x : var \u03c3 \u03b1) \u227a\u227a y = \u2191(x << y) :=\nrfl\n\n@[simp]\nlemma eq_judgement {p : pred' \u03b1} (\u03c3 : \u03b1)\n: \u27e8 eq \u03c3 \u27e9 \u22a2 p \u2194 \u03c3 \u22a8 p :=\nby { split ; intro h,\n     { apply h.apply \u03c3, exact rfl },\n     { constructor, intros _ h', cases h', assumption } }\n\n@[simp, predicate]\nlemma models_prop {p : Prop} (\u03c3 : \u03b1)\n: \u03c3 \u22a8 \u2191p \u2194 p :=\nby refl\n\n@[simp, predicate]\nlemma models_lifted\u2080 {p : Prop} (\u03c3 : \u03b1)\n: \u03c3 \u22a8 lifted\u2080 p \u2194 p :=\nby refl\n\n@[simp, predicate]\nlemma models_pred {p : \u03b1 \u2192 \u03b2} (\u03c3 : \u03b1)\n: \u03c3 \u22a8 \u27e8 p \u27e9 = p \u03c3 :=\nby refl\n\n@[extensionality, priority 0]\nlemma var_ext_sem {p q : var \u03b1 \u03b2}\n  (h : \u2200 \u03c3, p.apply \u03c3 = q.apply \u03c3)\n: p = q :=\nbegin\n  cases p, cases q,\n  congr, funext y,\n  simp with predicate at h,\n  rw h\nend\n\n@[extensionality]\nlemma pred_ext_sem {p q : pred' \u03b1}\n  (h : \u2200 \u03c3, \u03c3 \u22a8 p \u2194 \u03c3 \u22a8 q)\n: p = q :=\nbegin\n  apply var_ext_sem, simp [h]\nend\n\nlemma pred_ext {p q : pred' \u03b1}\n  (h : \u2200 \u0393, \u0393 \u22a2 p \u2194 \u0393 \u22a2 q)\n: p = q :=\nbegin\n  cases p, cases q,\n  congr, funext y,\n  specialize h \u27e8eq y\u27e9,\n  simp with predicate at h,\n  rw h,\nend\n\nlemma entails_of_pointwise {p q : pred' \u03b2}\n  (h : \u2200 \u03c3, \u03c3 \u22a8 p \u2192 \u03c3 \u22a8 q)\n: p \u27f9 q :=\nbegin\n  intros _, constructor, introv h',\n  apply h,\nend\n\nlemma entails_of_forall_impl {p q : pred' \u03b2}\n  (h : p \u27f9 q)\n: \u2200 i, i \u22a8 p \u27f6 q :=\nby { intros i hp, apply (h \u27e8eq i\u27e9).apply i rfl hp, }\n\nlemma ew_str {p : pred' \u03b2}\n: \u22a9 p \u2192 \u2200 x, x \u22a8 p :=\nby { intros h _, apply (h \u27e8eq x\u27e9).apply _ rfl }\n\nlemma ew_wk {p : pred' \u03b2}\n: (\u2200 x, x \u22a8 p) \u2192 \u22a9 p :=\nby { intros h \u0393, constructor,\n     intros, apply h }\n\nlemma ew_wk' {\u0393 p : pred' \u03b2}\n: (\u2200 x, x \u22a8 \u0393 \u2192 x \u22a8 p) \u2192 \u0393 \u22a2 p :=\nby { intros h, constructor, apply h }\n\nlemma judgement_sem {\u0393 p : pred' \u03b2}\n: \u0393 \u22a2 p \u2192 (\u2200 x, x \u22a8 \u0393 \u2192 x \u22a8 p) :=\nby { intros h \u03c3 h\u0393, apply h.apply _ h\u0393 }\n\nsection\nopen interactive interactive.types lean lean.parser\n     tactic tactic.interactive (simp dsimp unfold_coes)\nlocal postfix `?`:9001 := optional\nlocal postfix *:9001 := many\nmeta def using_idents := (tk \"using\" *> ident*) <|> pure []\n\nmeta def lifted_asm (v \u0393 : expr)\n  (rs : parse simp_arg_list)\n  (h : name) : tactic unit :=\ndo h' \u2190 get_local h,\n   p \u2190 to_expr ``(p_imp_sem  %%h' %%v %%\u0393)\n     <|> to_expr ``(judgement_sem %%h' %%v %%\u0393)\n     <|> to_expr ``(ew_str  %%h' %%v)\n     <|> fail format!\"assumtion {h} should be `\u22a9 p` or `p \u27f9 q` or `\u0393 \u22a2 p \u27f6 q`\",\n   h \u2190 note h none p,\n   let l := loc.ns [some h.local_pp_name],\n   try $ unfold_coes l,\n   try $ simp none ff rs [] l,\n   try (clear h')\n\nmeta def keep_flag := (tt <$ tk \"keep\") <|> pure ff\nprecedence `keep`:0\n\nmeta def lifted_pred\n   (keep_\u0393 : parse keep_flag)\n   (no_dflt : parse only_flag)\n   (rs : parse simp_arg_list)\n   (hs : parse using_idents) : tactic unit :=\ndo b \u2190 tt <$ `[apply p_imp_ext _]\n     <|> ff <$ `[apply pred_ext_sem]\n     <|> ff <$ `[apply var_ext_sem]\n     <|> tt <$ `[apply ew_wk' _]\n     <|> ff <$ `[apply ew_wk _],\n   v \u2190 intro1,\n   h\u0393 \u2190 if b then intro1 else return (default _),\n   mmap' (lifted_asm v h\u0393 rs) (hs : list _),\n   when (b \u2227 \u00ac keep_\u0393) $ clear h\u0393,\n   try (simp none no_dflt rs [`predicate] (loc.ns [none])),\n   try (dsimp no_dflt rs [`predicate] (loc.ns [none])),\n   try `[unfold_coes],\n   try reflexivity\nrun_cmd add_interactive [`lifted_pred]\nend\n\n@[simp, predicate]\nlemma value_coe_comp (x : \u03b1) (v : var \u03b2 \u03b3)\n: (x : var \u03b3 \u03b1) ! v = (x : var \u03b2 \u03b1) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma True_comp (v : var \u03b2 \u03b3)\n: True ! v = True :=\nby lifted_pred\n\n@[simp, predicate]\nlemma False_comp (v : var \u03b2 \u03b3)\n: False ! v = False :=\nby lifted_pred\n\n@[simp]\nlemma proj_assoc (x : var \u03b1 \u03b2) (y : var \u03b2 \u03b3) (z : var \u03b3 \u03c3)\n: (z ! y) ! x = z ! (y ! x) :=\nby lifted_pred\n\n@[simp]\nlemma p_and_comp (p q : pred' \u03b1) (f : var \u03b2 \u03b1)\n: ((p \u22c0 q) ! f) = (p ! f) \u22c0 (q ! f) :=\nby lifted_pred\n\n@[simp]\nlemma const_over_comp (p : \u03b3) (v : var \u03b2 \u03b1)\n: (p : var \u03b1 \u03b3) ! v = \u2191p :=\nby lifted_pred\n\n@[predicate]\nlemma coe_over_comp' (p : \u03b1 \u2192 \u03b3) (f : \u03b2 \u2192 \u03b1)\n: \u27e8 p \u27e9 ! \u27e8 f \u27e9 = \u27e8 p \u2218 f \u27e9 :=\nby lifted_pred\n\n-- @[simp]\nlemma coe_over_comp (p : \u03b1 \u2192 \u03b3) (f : \u03b2 \u2192 \u03b1)\n: (\u27e8 p \u2218 f \u27e9 : var _ _) = \u27e8 p \u27e9 ! \u27e8 f \u27e9 :=\nby lifted_pred\n\n@[simp]\nlemma p_or_comp (p q : pred' \u03b1) (f : var \u03b2 \u03b1)\n: ((p \u22c1 q) ! f) = (p ! f) \u22c1 (q ! f) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma eq_comp (p q : var \u03b1 \u03b3) (f : var \u03b2 \u03b1)\n: ((p \u2243 q) ! f) = (p ! f) \u2243 (q ! f) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma wf_comp {_ : has_well_founded \u03b3} (p q : var \u03b1 \u03b3) (f : var \u03b2 \u03b1)\n: ((p \u227a\u227a q) ! f) = (p ! f) \u227a\u227a (q ! f) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma lt_comp {\u03b3 : Type _} [has_lt \u03b3] (p q : var \u03b1 \u03b3) (f : var \u03b2 \u03b1)\n: ((p \u227a q) ! f) = ((p ! f) \u227a (q ! f)) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma mem_comp {\u03b1 \u03b3} [has_mem \u03b1 \u03b3] (p : var \u03c3 \u03b1) (q : var \u03c3 \u03b3) (f : var \u03b2 \u03c3)\n: ((p \u220a q) ! f) = ((p ! f) \u220a (q ! f)) :=\nby lifted_pred\n\n@[simp, predicate]\nlemma coe_apply (v : var \u03b1 \u03b2)\n: (\u27e8 v.apply \u27e9 : var _ _) = v :=\nby lifted_pred\n\n@[simp, predicate]\nlemma p_exists_comp {t} (p : t \u2192 pred' \u03b1) (f : var \u03b2 \u03b1)\n: (p_exists p ! f) = (\u2203\u2203 x, p x ! f) :=\nby lifted_pred [p_exists]\n\n@[simp]\nlemma coe_to_prop_p_and (p q : \u03b1 \u2192 Prop)\n: (\u27e8\u03bb s, p s \u2227 q s\u27e9 : pred' \u03b1) = \u27e8p\u27e9 \u22c0 \u27e8q\u27e9 := rfl\n\n@[simp]\nlemma coe_to_prop_p_or (p q : \u03b1 \u2192 Prop)\n: (\u27e8\u03bb s, p s \u2228 q s\u27e9 : pred' \u03b1) = \u27e8p\u27e9 \u22c1 \u27e8q\u27e9 := rfl\n\n@[simp]\nlemma coe_to_prop_p_not (p : \u03b1 \u2192 Prop)\n: (\u27e8\u03bb s, \u00ac p s\u27e9 : pred' \u03b1) = - \u27e8p\u27e9 := rfl\n\n@[simp]\nlemma coe_to_prop_p_equiv (p q : \u03b1 \u2192 Prop)\n: (\u27e8\u03bb s, p s \u2194 q s\u27e9 : pred' \u03b1) = \u27e8p\u27e9 \u2261 \u27e8q\u27e9 :=\nby { ext, simp }\n\nlemma lifting_prop_asm (\u0393 : pred' \u03b1) {p : Prop} {q : pred' \u03b1}\n  (h : p \u2192 \u0393 \u22a2 q)\n: \u0393 \u22a2 p \u2192 \u0393 \u22a2 q :=\nbegin\n  intro h\u2081,\n  constructor,\n  introv h\u2082,\n  have h\u2083 := h\u2081.apply _ h\u2082,\n  apply (h h\u2083).apply _ h\u2082,\nend\n\n@[simp, predicate]\nlemma p_not_to_fun (p\u2080 : pred' \u03b1) (x : \u03b1)\n: x \u22a8 (- p\u2080) \u2194 \u00ac x \u22a8 p\u2080 := by { refl, }\n\nlemma p_not_eq_not (p : pred' \u03b2) (x : \u03b2) : \u00ac x \u22a8 p \u2194 x \u22a8 (-p) :=\nby refl\n\n@[simp]\nlemma p_not_True : (- True : pred' \u03b1) = (False) :=\nby lifted_pred\n\n@[simp]\nlemma p_not_False : (- False : pred' \u03b1) = True :=\nby lifted_pred\n\n-- @[simp]\n-- lemma entails_True (p q : pred' \u03b1)\n-- : p \u27f9 q \u2194 (\u2200 h, h \u27f9 p \u2192 h \u27f9 q) :=\n-- begin\n--   split ; intro h,\n--   { intros h\u2080 h\u2081, },\n-- end\n\n@[simp]\nlemma entails_True (p : pred' \u03b1)\n: p \u27f9 True :=\nby lifted_pred\n\n@[simp]\nlemma True_p_and (p : pred' \u03b1)\n: True \u22c0 p = p :=\nby lifted_pred\n\n\n@[simp]\nlemma p_and_True (p : pred' \u03b1)\n: p \u22c0 True = p :=\nby lifted_pred\n\n@[simp]\nlemma True_p_or (p : pred' \u03b1)\n: True \u22c1 p = True :=\nby lifted_pred\n\n@[simp]\nlemma p_or_False (p : pred' \u03b1)\n: p \u22c1 False = p :=\nby lifted_pred\n\n\n@[simp]\nlemma False_p_or (p : pred' \u03b1)\n: False \u22c1 p = p :=\nby lifted_pred\n\n@[refl]\nlemma entails_refl (p : pred' \u03b2)\n: p \u27f9 p :=\nby lifted_pred\n\n@[refl]\nlemma ctx_impl_refl (\u0393 p : pred' \u03b2)\n: ctx_impl \u0393 p p :=\nby lifted_pred\n\nlemma p_impl_refl (\u0393 p : pred' \u03b2)\n: \u0393 \u22a2 p \u27f6 p :=\nby lifted_pred\n\n@[simp,predicate]\nlemma p_impl_refl_eq_True (p : pred' \u03b2)\n: p \u27f6 p = True :=\nby lifted_pred\n\n@[refl]\nlemma v_eq_refl (\u0393 : pred' \u03b2) (v : var \u03b2 \u03b1)\n: \u0393 \u22a2 v \u2243 v :=\nby lifted_pred\n\nlemma v_eq_symm {\u0393 : pred' \u03b2} {v\u2080 v\u2081 : var \u03b2 \u03b1}\n  (h : \u0393 \u22a2 v\u2081 \u2243 v\u2080)\n: \u0393 \u22a2 v\u2080 \u2243 v\u2081 :=\nby lifted_pred using h ; rw h\n\n@[mono]\nlemma ctx_p_or_p_imp_p_or' {\u0393 p p' q q' : pred' \u03b1}\n  (hp : ctx_impl \u0393 p p')\n  (hq : ctx_impl \u0393 q q')\n: ctx_impl \u0393 (p \u22c1 q) (p' \u22c1 q')  :=\nby { lifted_pred using hp hq,\n     begin [smt] intros, destruct a end, }\n\n@[mono]\nlemma p_or_p_imp_p_or' {p p' q q' : pred' \u03b1}\n  (hp : p \u27f9 p')\n  (hq : q \u27f9 q')\n: (p \u22c1 q)  \u27f9  (p' \u22c1 q')  :=\nby { lifted_pred using hp hq,\n     apply or.imp hp hq, }\n\n@[mono]\nlemma p_and_p_imp_p_and' {p p' q q' : pred' \u03b1}\n  (hp : p \u27f9 p')\n  (hq : q \u27f9 q')\n: (p \u22c0 q)  \u27f9  (p' \u22c0 q')  :=\nby { lifted_pred only using hp hq,\n     apply and.imp hp hq, }\n\nlemma p_or_p_imp_p_or {p p' q q' : pred' \u03b1} {\u03c4}\n  (hp : \u03c4 \u22a8 p \u27f6 p')\n  (hq : \u03c4 \u22a8 q \u27f6 q')\n: \u03c4 \u22a8 p \u22c1 q \u2192 \u03c4 \u22a8 p' \u22c1 q' :=\nby apply or.imp hp hq\n\n@[mono]\nlemma ctx_p_and_p_imp_p_and_right' {\u0393 p p' q q' : pred' \u03b1}\n  (hp : ctx_impl \u0393 p p')\n  (hq : ctx_impl \u0393 q q')\n: ctx_impl \u0393 ( p \u22c0 q ) ( p' \u22c0 q' ) :=\nby { lifted_pred using hp hq, intros, tauto }\n\nlemma ctx_p_and_p_imp_p_and' {\u0393 p p' q q' : pred' \u03b1}\n  (hp : ctx_impl \u0393 p p')\n  (hq : ctx_impl \u0393 q q')\n: ctx_impl \u0393 (p \u22c0 q) (p' \u22c0 q')  :=\nby { lifted_pred using hp hq,\n     begin [smt] intros end, }\n\nlemma p_or_p_imp_p_or_right {p q q' : pred' \u03b1} {\u03c4}\n  (hq : \u03c4 \u22a8 q \u27f6 q')\n: \u03c4 \u22a8 p \u22c1 q \u2192 \u03c4 \u22a8 p \u22c1 q' :=\nby apply or.imp id hq\n\nlemma p_or_p_imp_p_or_left {p p' q : pred' \u03b1} {\u03c4}\n  (hp : \u03c4 \u22a8 p \u27f6 p')\n: \u03c4 \u22a8 p \u22c1 q \u2192 \u03c4 \u22a8 p' \u22c1 q :=\nby apply or.imp hp id\n\nlemma p_imp_p_imp_p_imp {p p' q q' : pred' \u03b1} {\u0393}\n  (hp : \u0393 \u22a2 p' \u27f6 p)\n  (hq : \u0393 \u22a2 q \u27f6 q')\n: \u0393 \u22a2 p \u27f6 q \u2192 \u0393 \u22a2 p' \u27f6 q' :=\nby { intro h\u2082, constructor, introv h\u2080 h\u2081,\n     replace hp := hp.apply _ h\u2080,\n     replace hq := hq.apply _ h\u2080,\n     replace h\u2082 := h\u2082.apply _ h\u2080,\n     apply_assumption, solve_by_elim, }\n\nlemma revert_p_imp {p q : pred' \u03b1}\n  (h : \u22a9 p \u27f6 q)\n: p \u22a2 q :=\nbegin\n  constructor, intro,\n  exact (h True).apply \u03c3 trivial,\nend\n\nlemma revert_p_imp' {p q r : pred' \u03b1}\n  (h : p \u27f9 q)\n: q \u22a2 r \u2192 p \u22a2 r :=\nby { intro h\u2080, constructor,\n     introv h\u2081, apply h\u2080.apply,\n     apply (h p).apply _ h\u2081 h\u2081, }\n\n@[simp]\nlemma from_True {p : pred' \u03b1}\n: True \u22a2 p \u2194 \u22a9 p :=\nby { unfold holds ; split ; intro h,\n     intro, apply revert_p_imp' _ h, simp,\n     apply h }\n\nlemma intro_p_imp {p q : pred' \u03b1}\n  (h : p \u22a2 q)\n: \u22a9 p \u27f6 q :=\nbegin\n  intro, constructor, introv h',\n  apply h.apply,\nend\n\n@[mono]\nlemma p_imp_entails_p_imp {p p' q q' : pred' \u03b1}\n  (hp : p' \u27f9 p)\n  (hq : q \u27f9 q')\n: ( p \u27f6 q ) \u27f9 ( p' \u27f6 q' ) :=\nby { lifted_pred using hp hq, intros,\n     repeat { apply_assumption }, }\n\nlemma p_imp_p_imp_p_imp_left {p p' q : pred' \u03b1} {\u0393}\n  (hp : \u0393 \u22a2 p' \u27f6 p)\n: \u0393 \u22a2 p \u27f6 q \u2192 \u0393 \u22a2 p' \u27f6 q :=\np_imp_p_imp_p_imp hp (ctx_impl_refl _ _)\n\nlemma p_imp_p_imp_p_imp_right {p q q' : pred' \u03b1} {\u0393}\n  (hq : \u0393 \u22a2 q \u27f6 q')\n: \u0393 \u22a2 p \u27f6 q \u2192 \u0393 \u22a2 p \u27f6 q' :=\np_imp_p_imp_p_imp (ctx_impl_refl _ _) hq\n\nlemma ctx_imp_entails_p_imp {\u0393 p p' q q' : pred' \u03b1}\n  (hp : ctx_impl \u0393 p' p)\n  (hq : ctx_impl \u0393 q q')\n: ctx_impl \u0393 ( p \u27f6 q ) ( p' \u27f6 q' ) :=\nby { lifted_pred using hp hq, intros, apply_assumption, solve_by_elim }\n\n@[mono]\nlemma ctx_imp_entails_p_imp_left {\u0393 p p' q q' : pred' \u03b1}\n  (hp : ctx_impl \u0393 p' p)\n  (hq : ctx_impl \u0393 q  q')\n: ctx_impl \u0393 ( p \u27f6 q ) ( p' \u27f6 q' ) :=\nby { lifted_pred using hp hq, intros, apply_assumption, tauto }\n\nlemma entails_imp_entails_left {p p' q : pred' \u03b1}\n  (hp : p' \u27f9 p)\n: ( p \u27f9 q ) \u2192 ( p' \u27f9 q ) :=\nbegin\n  intros h\u2081 \u0393, constructor,\n  introv h\u2082 h\u2083,\n  apply (h\u2081 \u0393).apply _ h\u2082,\n  apply (hp \u0393).apply _ h\u2082 h\u2083,\nend\n\n@[simp]\nlemma p_or_self (p : pred' \u03b2) :\np \u22c1 p = p :=\nby lifted_pred\n\n@[simp]\nlemma p_not_p_not_iff_self (p : pred' \u03b2) :\n- - p = p :=\nby lifted_pred [not_not_iff_self]\n\nlemma p_not_eq_iff_eq_p_not (p q : pred' \u03b2) :\n- p = q \u2194 p = - q :=\nby { split ; intro h,\n     rw [\u2190 h,p_not_p_not_iff_self],\n     rw [h,p_not_p_not_iff_self], }\n\n@[simp]\nlemma p_not_eq_p_not_iff_eq (p q : pred' \u03b2) :\n- p = - q \u2194 p = q :=\nby simp [p_not_eq_iff_eq_p_not]\n\nlemma p_and_over_or_left (p q r : pred' \u03b2)\n: p \u22c0 (q \u22c1 r) = (p \u22c0 q) \u22c1 (p \u22c0 r) :=\nbegin\n  lifted_pred, split,\n  { begin [smt] intros, eblast end },\n  { begin [smt] intros, destruct a end },\nend\n\nlemma p_and_over_or_right (p q r : pred' \u03b2)\n: (q \u22c1 r) \u22c0 p = (q \u22c0 p) \u22c1 (r \u22c0 p) :=\nbegin\n  lifted_pred, split,\n  { begin [smt] intros, eblast end },\n  { begin [smt] intros, destruct a end },\nend\n\ninstance : is_left_distrib (pred' \u03b2) (\u22c0) (\u22c1) :=\n\u27e8 p_and_over_or_left \u27e9\ninstance : is_right_distrib (pred' \u03b2) (\u22c0) (\u22c1) :=\n\u27e8 by { intros, apply p_and_over_or_right } \u27e9\ninstance : is_left_id (pred' \u03b2) (\u22c0) True :=\n\u27e8 by simp \u27e9\ninstance : is_right_id (pred' \u03b2) (\u22c0) True :=\n\u27e8 by simp \u27e9\ninstance or_left_id : is_left_id (pred' \u03b2) (\u22c1) False :=\n\u27e8 by simp \u27e9\ninstance or_right_id : is_right_id (pred' \u03b2) (\u22c1) False :=\n\u27e8 by simp \u27e9\n\nlemma p_or_over_and_left (p q r : pred' \u03b2)\n: p \u22c1 (q \u22c0 r) = (p \u22c1 q) \u22c0 (p \u22c1 r) :=\nbegin\n  lifted_pred, split,\n  { begin [smt] intros, destruct a end },\n  { begin [smt] intros, destruct a.left, end },\nend\n\nlemma p_or_over_and_right (p q r : pred' \u03b2)\n: (q \u22c0 r) \u22c1 p = (q \u22c1 p) \u22c0 (r \u22c1 p) :=\nby { lifted_pred [distrib_left_or] }\n\ninstance is_left_distrib_or_and : is_left_distrib (pred' \u03b2) (\u22c1) (\u22c0) :=\n\u27e8 p_or_over_and_left \u27e9\ninstance is_right_distrib_or_and : is_right_distrib (pred' \u03b2) (\u22c1) (\u22c0) :=\n\u27e8 by { intros, apply p_or_over_and_right } \u27e9\n\nlemma mutual_p_imp {\u0393 p q : pred' \u03b2}\n  (h\u2080 : \u0393 \u22a2 p \u27f6 q)\n  (h\u2081 : \u0393 \u22a2 q \u27f6 p)\n: \u0393 \u22a2 p \u2261 q :=\nbegin\n  lifted_pred using h\u2080 h\u2081,\n  split ; assumption,\nend\n\nlemma mutual_entails {p q : pred' \u03b2}\n  (h\u2080 : p \u27f9 q)\n  (h\u2081 : q \u27f9 p)\n: p = q :=\nbegin\n  lifted_pred using h\u2080 h\u2081,\n  split ; assumption,\nend\n\n@[simp]\nlemma False_entails (p : pred' \u03b2)\n: False \u27f9 p :=\nby lifted_pred\n\n@[simp]\nlemma p_imp_False (p : pred' \u03b2)\n: p \u27f6 False = -p :=\nby lifted_pred\n\nlemma p_and_p_not_self (p : pred' \u03b2)\n: p \u22c0 -p = False :=\nby lifted_pred\n\n@[simp]\nlemma p_or_p_not_self (p : pred' \u03b2)\n: p \u22c1 -p = True :=\nby lifted_pred [classical.em]\n\nlemma em (p : pred' \u03b2)\n: \u22a9 p \u22c1 -p :=\nby lifted_pred [classical.em]\n\nlemma p_and_p_or_p_not_self (p q : pred' \u03b2)\n: p \u22c0 (q \u22c1 -p) = p \u22c0 q :=\nby simp [p_and_over_or_left,p_and_p_not_self]\n\nlemma p_not_and_self (p : pred' \u03b2)\n: (-p) \u22c0 p = False :=\nby lifted_pred\n\nlemma p_not_p_and (p q : pred' \u03b2)\n: - (p \u22c0 q) = -p \u22c1 -q :=\nby lifted_pred [classical.not_and_iff_not_or_not]\n\nlemma p_not_p_or (p q : pred' \u03b2)\n: - (p \u22c1 q) = -p \u22c0 -q :=\nby lifted_pred [not_or_iff_not_and_not]\n\nlemma p_not_and_self_or (p q : pred' \u03b2) :\n- p \u22c0 (p \u22c1 q) = -p \u22c0 q :=\nby rw [p_and_over_or_left,p_not_and_self,False_p_or]\n\n@[simp, predicate]\nlemma p_exists_apply {t : Sort u'} {P : t \u2192 pred' \u03b2} (\u03c3 : \u03b2)\n: \u03c3 \u22a8 (\u2203\u2203 x, P x) \u2194 (\u2203 x, \u03c3 \u22a8 P x) :=\nby { unfold p_exists }\n\nlemma p_exists_to_fun {t : Sort u'} {h : pred' \u03b2} {P : t \u2192 pred' \u03b2}\n  (x : t)\n  (Hh : h \u22a2 P x)\n: h \u22a2 (\u2203\u2203 x, P x) :=\nby { constructor, intros _ h', existsi x,\n     apply Hh.apply _ h', }\n\n@[simp, predicate]\nlemma models_p_forall {t : Sort u'} (P : t \u2192 pred' \u03b2) (\u03c3 : \u03b2)\n: \u03c3 \u22a8 (\u2200\u2200 x, P x) \u2194 (\u2200 x, \u03c3 \u22a8 P x) := by refl\n\nlemma p_forall_to_fun {t : Sort u'} (h : pred' \u03b2) (P : t \u2192 pred' \u03b2)\n: h \u22a2 (\u2200\u2200 x, P x) \u2194 (\u2200 x, h \u22a2 P x) :=\nbegin\n  split ; intro h,\n  { intro, constructor, intros,\n    cases h with h,\n    apply h \u03c3 a, },\n  { constructor,\n    introv h' x,\n    apply (h x).apply _ h', }\nend\n\nlemma p_forall_subtype_to_fun {t : Sort u'} (h : pred' \u03b2) (p : t \u2192 Prop) (q : t \u2192 pred' \u03b2)\n: h \u22a2 (\u2200\u2200 x, p x \u27f6 q x) \u2194 (\u2200 x, p x \u2192 h \u22a2 q x) :=\nbegin\n  split,\n  { intros h x hp, apply h x hp },\n  { intros h,\n    constructor,\n    introv h\u03c3 x hp,\n    apply (h x hp).apply _ h\u03c3, }\nend\n\nlemma ew_p_forall {t} (p : t \u2192 pred' \u03b2)\n: \u22a9 (\u2200\u2200 x, p x) \u2194 \u2200 x, \u22a9 p x :=\nby { simp only [holds,forall_swap] { single_pass := tt },\n     apply forall_congr, intro,\n     apply p_forall_to_fun }\n\nlemma p_not_p_exists {t : Sort*} (p : t \u2192 pred' \u03b2) :\n(- \u2203\u2203 x, p x) = (\u2200\u2200 x, -p x) :=\nby lifted_pred [not_exists_iff_forall_not,p_exists]\n\nlemma p_not_p_forall {t : Sort*} (p : t \u2192 pred' \u03b2) :\n(- \u2200\u2200 x, p x) = (\u2203\u2203 x, -p x) :=\nby { rw [p_not_eq_iff_eq_p_not,p_not_p_exists], simp [p_not_p_not_iff_self] }\n\nlemma p_exists_p_imp {t} (p : t \u2192 pred' \u03b2) (q : pred' \u03b2)\n: (\u2203\u2203 x, p x) \u27f6 q = (\u2200\u2200 x, p x \u27f6 q) :=\nby lifted_pred [p_exists]\n\nlemma p_or_comm (p q : pred' \u03b2) : p \u22c1 q = q \u22c1 p :=\nby lifted_pred [or_comm]\n\nlemma p_or_assoc (p q r : pred' \u03b2) : p \u22c1 (q \u22c1 r) = p \u22c1 q \u22c1 r :=\nby lifted_pred [or_assoc]\n\ninstance p_or_is_assoc : is_associative (pred' \u03b2) (\u22c1) :=\n\u27e8 by { intros, rw p_or_assoc, } \u27e9\ninstance p_or_is_comm : is_commutative (pred' \u03b2) (\u22c1) :=\n\u27e8 by apply p_or_comm \u27e9\n\nlemma p_and_comm (p q : pred' \u03b2) : p \u22c0 q = q \u22c0 p :=\nby lifted_pred [and_comm]\n\nlemma p_and_assoc (p q r : pred' \u03b2) : p \u22c0 (q \u22c0 r) = p \u22c0 q \u22c0 r :=\nby lifted_pred [and_assoc]\n\ninstance p_and_is_assoc : is_associative (pred' \u03b2) (\u22c0) :=\n\u27e8 by { intros, rw p_and_assoc, } \u27e9\ninstance p_and_is_comm : is_commutative (pred' \u03b2) (\u22c0) :=\n\u27e8 by apply p_and_comm \u27e9\n\n@[simp]\nlemma p_and_p_imp (p q r : pred' \u03b2) : p \u22c0 q \u27f6 r = p \u27f6 (q \u27f6 r) :=\nby lifted_pred\n\n-- lemma p_imp_intro_wrong (\u0393 p q : pred' \u03b2)\n--   (h : \u0393 \u22a2 p \u2192 \u0393 \u22a2 q)\n-- : \u0393 \u22a2 p \u27f6 q :=\n-- sorry\n\nlemma p_imp_intro (p q r : pred' \u03b2)\n  (h : \u2200 \u0393, \u0393 \u22a2 p \u2192 \u0393 \u22a2 q \u2192 \u0393 \u22a2 r)\n  (\u0393 : pred' \u03b2)\n  (h' : \u0393 \u22a2 p)\n: \u0393 \u22a2 q \u27f6 r :=\nbegin\n  constructor, introv h\u0393 hq,\n  apply (h \u27e8eq \u03c3\u27e9 _ _).apply _ rfl ;\n  constructor\n  ; intros _ h\n  ; cases h,\n  { apply h'.apply _ h\u0393 },\n  assumption\nend\n\n@[simp]\nlemma p_or_intro_left (p q : pred' \u03b2)\n: p \u27f9 p \u22c1 q :=\nby { lifted_pred,\n     begin [smt] intros end }\n\n@[simp]\nlemma p_or_intro_right (p q : pred' \u03b2)\n: q \u27f9 p \u22c1 q :=\nby { lifted_pred,\n     begin [smt] intros end }\n\n@[simp]\nlemma p_and_intro (p q : pred' \u03b2)\n: p \u27f9 (q \u27f6 p \u22c0 q) :=\nby { lifted_pred,\n     begin [smt] intros end }\n\nlemma p_or_entails_of_entails' {\u0393 p q r : pred' \u03b2}\n  (h\u2080 : \u0393 \u22a2 p \u27f6 r)\n  (h\u2081 : \u0393 \u22a2 q \u27f6 r)\n: \u0393 \u22a2 p \u22c1 q \u27f6 r :=\nby { constructor, simp_intros _ h\u0393 _,\n     have h\u2080 := h\u2080.apply \u03c3 h\u0393, simp at h\u2080,\n     have h\u2081 := h\u2081.apply \u03c3 h\u0393, simp at h\u2081,\n     begin [smt] intros, destruct a, end }\n\nlemma p_or_entails_of_entails {p q r : pred' \u03b2}\n  (h\u2080 : p \u27f9 r)\n  (h\u2081 : q \u27f9 r)\n: p \u22c1 q \u27f9 r :=\nby { lifted_pred using h\u2080 h\u2081,\n     begin [smt] intros, destruct a end }\n\nlemma entails_p_or_of_entails_left {p q r : pred' \u03b2}\n  (h\u2080 : p \u27f9 q)\n: p \u27f9 q \u22c1 r :=\nby { lifted_pred using h\u2080,\n     begin [smt] intros end }\n\nlemma entails_p_or_of_entails_right {p q r : pred' \u03b2}\n  (h\u2080 : p \u27f9 r)\n: p \u27f9 q \u22c1 r :=\nby { lifted_pred using h\u2080,\n     begin [smt] intros end }\n\nlemma entails_p_and_of_entails {p q r : pred' \u03b2}\n  (h\u2080 : p \u27f9 q)\n  (h\u2081 : p \u27f9 r)\n: p \u27f9 q \u22c0 r :=\nby { lifted_pred using h\u2080 h\u2081,\n     begin [smt] intros end }\n\nlemma p_and_entails_of_entails_left {p q r : pred' \u03b2}\n  (h\u2081 : p \u27f9 r)\n: p \u22c0 q \u27f9 r :=\nby { lifted_pred using h\u2081,\n     begin [smt] intros end }\n\nlemma p_and_entails_of_entails_right {p q r : pred' \u03b2}\n  (h\u2081 : q \u27f9 r)\n: p \u22c0 q \u27f9 r :=\nby { lifted_pred using h\u2081,\n     begin [smt] intros end }\n\n@[simp]\nlemma p_and_elim_left (p q : pred' \u03b2)\n: p \u22c0 q \u27f9 p :=\nby { lifted_pred,\n     begin [smt] intros end }\n\n@[simp]\nlemma p_and_elim_right (p q : pred' \u03b2)\n: p \u22c0 q \u27f9 q :=\nby lifted_pred\n\nlemma judgement.left {\u0393 p q : pred' \u03b2}\n  (h : \u0393 \u22a2 p \u22c0 q)\n: \u0393 \u22a2 p :=\np_and_elim_left p q \u0393 h\n\nlemma judgement.right {\u0393 p q : pred' \u03b2}\n  (h : \u0393 \u22a2 p \u22c0 q)\n: \u0393 \u22a2 q :=\np_and_elim_right p q \u0393 h\n\nlemma p_imp_trans {\u0393 p q r : pred' \u03b2}\n  (h\u2080 : \u0393 \u22a2 p \u27f6 q)\n  (h\u2081 : \u0393 \u22a2 q \u27f6 r)\n: \u0393 \u22a2 p \u27f6 r :=\nbegin\n  lifted_pred using h\u2080 h\u2081,\n  intros,\n  solve_by_elim,\nend\n\n@[trans]\nlemma entails_trans {p q r : pred' \u03b2}\n  (h\u2080 : p \u27f9 q)\n  (h\u2081 : q \u27f9 r)\n: p \u27f9 r :=\nbegin\n  lifted_pred using h\u2080 h\u2081,\n  intro, solve_by_elim,\nend\n\n@[simp]\nlemma p_not_comp' (p : pred' \u03b1) (f : var \u03b2 \u03b1)\n: -p ! f = -(p ! f) :=\nby lifted_pred\n\nlemma p_not_comp (p : pred' \u03b1) (f : var \u03b2 \u03b1)\n: -(p ! f) = -p ! f :=\nby lifted_pred\n\n@[mono]\nlemma comp_entails_comp {p q : pred' \u03b2} (f : var \u03b1 \u03b2)\n  (H : p \u27f9 q)\n: p ! f \u27f9 q ! f :=\nbegin\n  intros \u0393, constructor,\n  introv h hp,\n  simp at \u22a2 hp,\n  specialize H \u27e8eq $ f.apply \u03c3\u27e9,\n  apply H.apply (f.apply \u03c3) rfl hp,\nend\n\n@[mono]\nlemma ctx_comp_imp_comp {\u0393 : pred' \u03b1} {p q : pred' \u03b2} (f : var \u03b1 \u03b2)\n  (H : p \u27f9 q)\n: ctx_impl \u0393 (p ! f) (q ! f) :=\nby apply comp_entails_comp _ H\n\n@[mono]\nlemma ctx_p_not_entails_p_not_right {\u0393 p q : pred' \u03b2}\n  (h : ctx_impl \u0393 q p)\n: ctx_impl \u0393 (- p) (- q) :=\nby { lifted_pred using h,\n     begin [smt] intros end }\n\n@[mono]\nlemma p_not_entails_p_not_right {p q : pred' \u03b2}\n  (h : q \u27f9 p)\n: - p \u27f9 - q :=\nby { lifted_pred using h,\n     begin [smt] intros end }\n\nlemma entails_of_eq (p q : pred' \u03b2)\n  (h : p = q)\n: p \u27f9 q :=\nby simp [h]\n\nlemma p_imp_of_equiv {\u0393 : pred' \u03b2} (p q : pred' \u03b2)\n  (h : \u0393 \u22a2 p \u2261 q)\n: \u0393 \u22a2 p \u27f6 q :=\nby lifted_pred using h ; simp [h]\n\nlemma equiv_of_eq (\u0393 p q : pred' \u03b2)\n  (h : p = q)\n: \u0393 \u22a2 p \u2261 q :=\nby { cases p, cases q, simp [h] }\n\nlemma p_and_entails_p_or (p q : pred' \u03b2)\n: p \u22c0 q \u27f9 p \u22c1 q :=\nby { lifted_pred,\n     begin [smt] intros end }\n\n@[simp]\nlemma True_p_imp (p : pred' \u03b2)\n: True \u27f6 p = p :=\nby lifted_pred\n\n@[simp]\nlemma p_imp_True (p : pred' \u03b2)\n: p \u27f6 True = True :=\nby lifted_pred\n\n@[simp]\nlemma p_forall_True\n: (\u2200\u2200 x : \u03b1, @True \u03b2) = True :=\nby lifted_pred\n\n@[simp]\nlemma p_exists_True [nonempty \u03b1]\n: (\u2203\u2203 x : \u03b1, @True \u03b2) = True :=\nby { lifted_pred, apply classical.exists_true_of_nonempty, apply_instance }\n\nlemma ew_eq_true {p : pred' \u03b2} : \u22a9 p \u2192 p = True :=\nby { intro h, lifted_pred using h,\n     begin [smt] intros end }\n\n@[mono, priority 0]\nlemma ew_imp_ew {p q : pred' \u03b2}\n  (H : p \u27f9 q)\n: \u22a9 p \u2192 \u22a9 q :=\nby { intros hp, lifted_pred using hp H, solve_by_elim }\n\nlemma entails_to_pointwise {p q : pred' \u03b2}\n  (h : p \u27f9 q)\n: \u2200 i, i \u22a8 p \u2192 i \u22a8 q :=\nby { intros i h', apply (h \u27e8eq i\u27e9).apply i rfl h' }\n\nlemma impl_of_p_impl {p q : pred' \u03b2} (i : \u03b2)\n  (h : \u22a9 p \u27f6 q)\n: i \u22a8 p \u2192 i \u22a8 q :=\nentails_of_forall_impl h _\n\nopen interactive.types interactive\nopen lean.parser lean tactic (hiding funext)\nmeta def entails_or_ew {\u03b1 : Type u} (t : expr) (tag : string)\n   (ent_tac ew_tac : tactic \u03b1) : tactic \u03b1 :=\ndo match t with\n    | `(_ \u27f9 _) := ew_tac\n    | `(_ \u22a2 _) := ent_tac\n    | `(\u22a9 _) := ew_tac\n    | _ := fail format!\"expecting {tag} of shape `_ \u27f9 _` or `\u22a9 _`\"\n   end\n\nmeta def pointwise (h : parse (many ident)) (ids : parse with_ident_list) : tactic unit :=\ndo t \u2190 target,\n   try $ entails_or_ew t \"goal\" `[refine revert_p_imp _,refine ew_wk _] `[refine ew_wk _],\n   tactic.intro_lst ids,\n   ls \u2190 mmap get_local h,\n   mmap' (\u03bb l : expr,\n    do entails_or_ew t \"goal\" (to_expr ``(ew_str (intro_p_imp %%l)))\n                              (to_expr ``(ew_str %%l))\n           >>= note l.local_pp_name none,\n       try (clear l)) ls\n\nrun_cmd add_interactive [`pointwise]\n\nlemma p_or_over_p_exists_left {t} (p : t \u2192 pred' \u03b2) (q : pred' \u03b2) {w : t \u2192 pred' \u03b2}\n  (h : \u22a9 \u2203\u2203 x : t, w x)\n: q \u22c1 (\u2203\u2203 x, p x) = (\u2203\u2203 x, q \u22c1 p x) :=\nbegin\n  lifted_pred,\n  have h := (h \u27e8eq \u03c3\u27e9).apply _ rfl,\n  have _inst : nonempty t := nonempty_of_exists h, resetI,\n  simp [distrib_or_over_exists_left,p_exists],\nend\n\n@[congr]\nlemma {v} p_exists_congr {\u03b1 : Sort u} {\u03b2 : Sort v} {p q : \u03b1 \u2192 pred' \u03b2}\n  (h : \u2200 i, p i = q i)\n: p_exists p = p_exists q :=\nbegin\n  lifted_pred [p_exists],\n  rw [exists_congr],\n  intro, rw h,\nend\n\nlemma p_and_over_p_exists_right {t} (p : t \u2192 pred' \u03b2) (q : pred' \u03b2)\n: (\u2203\u2203 x, p x) \u22c0 q = (\u2203\u2203 x, p x \u22c0 q) :=\nby lifted_pred only [iff_self, exists_and_distrib_right]\n\nlemma p_and_over_p_exists_left {t} (p : pred' \u03b2) (q : t \u2192 pred' \u03b2)\n: p \u22c0 (\u2203\u2203 x, q x) = (\u2203\u2203 x, p \u22c0 q x) :=\nbegin\n  rw [p_and_comm,p_and_over_p_exists_right],\n  apply p_exists_congr,\n  intro, simp [p_and_comm]\nend\n\nlemma shunting {\u03b2 : Sort*} (p q r : pred' \u03b2)\n: p \u27f6 q \u22c1 r = (p \u22c0 - q) \u27f6 r :=\nbegin\n  lifted_pred,\n  begin [smt] split, all_goals { intros },\n              by_cases (q.apply \u03c3),\n  end,\nend\n\nlemma shunting' {\u03b2 : Sort*} (p q r : pred' \u03b2)\n: p \u27f6 (q \u27f6 r) = (p \u22c0 q) \u27f6 r :=\nby lifted_pred\n\nlemma imp_swap {\u03b2 : Sort*} (p q r : pred' \u03b2)\n: p \u27f6 (q \u27f6 r) = q \u27f6 (p \u27f6 r) :=\nby { lifted_pred,\n     begin [smt] split, all_goals { intros }, end }\n\nlemma entails_swap {\u03b2 : Sort*} (p q r : pred' \u03b2)\n: p \u27f9 (q \u27f6 r) \u2194 q \u27f9 (p \u27f6 r) :=\nby simp [p_entails,imp_swap]\n-- \u25fb\u25c7p \u22a2 \u25fb\u25c7q \u03c4\n-- `\u03c4 : stream \u03c3, h : (\u25fb\u25c7p) \u03c4 \u22a2 (\u25fb\u25c7q) \u03c4`\nlemma p_not_p_imp {\u03b2 : Sort*} (p q : pred' \u03b2)\n: (-p) \u27f6 q = p \u22c1 q :=\nby rw [\u2190 True_p_and (-p),\u2190 shunting,True_p_imp]\n\nlemma p_imp_iff_p_not_p_or {\u03b2 : Sort*} (p q : pred' \u03b2)\n: p \u27f6 q = -p \u22c1 q :=\nby rw [\u2190 p_not_p_imp,p_not_p_not_iff_self]\n\nlemma p_or_not_and {\u03b2 : Sort*} (p q : pred' \u03b2)\n: p \u22c1 (- p \u22c0 q) = p \u22c1 q :=\nbegin\n  lifted_pred,\n  begin [smt]\n    split,\n    all_goals { intros h }, destruct h,\n    by_cases (p.apply \u03c3)\n  end\nend\n\nlemma p_exists_intro {t : Sort u'} {p : t \u2192 pred' \u03b2} (x : t)\n: p x \u27f9 (\u2203\u2203 x, p x) :=\nby { lifted_pred, apply exists.intro x, }\n\nlemma p_exists_elim {t : Sort u'} {p : t \u2192 pred' \u03b2} {q : pred' \u03b2}\n  (H : \u2200 x, p x \u27f9 q)\n: (\u2203\u2203 x, p x) \u27f9 q :=\nbegin\n  pointwise with \u03c3,\n  simp, intro,\n  apply entails_to_pointwise (H x) \u03c3,\nend\n\nlemma p_forall_p_imp_p_forall {\u0393 : pred' \u03b2} {t : Sort u'} (p q : t \u2192 pred' \u03b2)\n: \u0393 \u22a2 (\u2200\u2200 x, p x \u27f6 q x) \u2192 \u0393 \u22a2 (\u2200\u2200 x, p x) \u27f6 (\u2200\u2200 x, q x) :=\nbegin\n  intros h,\n  lifted_pred [- exists_imp_distrib] using h,\n  apply forall_imp_forall,\n  intro x,\n  apply h,\nend\n\n\nlemma p_exists_p_imp_p_exists {\u0393 : pred' \u03b2} {t : Sort u'} (p q : t \u2192 pred' \u03b2)\n: \u0393 \u22a2 (\u2200\u2200 x, p x \u27f6 q x) \u2192 \u0393 \u22a2 (\u2203\u2203 x, p x) \u27f6 (\u2203\u2203 x, q x) :=\nbegin\n  intros h,\n  lifted_pred [- exists_imp_distrib] using h,\n  apply exists_imp_exists,\n  intro x,\n  apply h,\nend\n\nlemma p_exists_entails_p_exists {t : Sort u'} (p q : t \u2192 pred' \u03b2)\n: (\u2200 x, p x \u27f9 q x) \u2192 (\u2203\u2203 x, p x) \u27f9 (\u2203\u2203 x, q x) :=\nbegin\n  intros h _,\n  apply p_exists_p_imp_p_exists,\n  constructor, introv h' x,\n  apply (h x \u0393).apply _ h'\nend\n\nlemma p_exists_over_p_or {t} (p q : t \u2192 pred' \u03b2)\n: (\u2203\u2203 x, p x) \u22c1 (\u2203\u2203 x, q x) = (\u2203\u2203 x, p x \u22c1 q x) :=\nbegin\n  lifted_pred, split ; simp_intros\n  ; cases a with a_1 a_1 ; revert a_1,\n  { apply exists_imp_exists, intro, apply or.intro_left, },\n  { apply exists_imp_exists, intro, apply or.intro_right, },\n  apply or.imp ; apply Exists.intro a_1,\nend\n\n@[simp]\nlemma p_exists_imp_eq_p_forall_imp\n  (p : \u03b1 \u2192 pred' \u03b2) (q : pred' \u03b2)\n: ((\u2203\u2203 x, p x) \u27f6 q) = (\u2200\u2200 x, p x \u27f6 q) :=\nby lifted_pred\n\nlemma p_exists_entails_eq_p_forall_entails\n  (p : \u03b1 \u2192 pred' \u03b2) (q : pred' \u03b2)\n: ((\u2203\u2203 x, p x) \u27f9 q) \u2194 (\u2200 x, p x \u27f9 q) :=\nby simp [p_entails,p_exists_imp_eq_p_forall_imp,ew_p_forall]\n\nlemma p_exists_imp_p_exists' {t : Sort u\u2080} {t' : Sort u\u2082}\n  {\u0393 : pred' \u03b2}\n  (p : t \u2192 pred' \u03b2)\n  (q : t' \u2192 pred' \u03b2)\n  (f : t \u2192 t')\n  (h : \u0393 \u22a2 (\u2200\u2200 x, p x \u27f6 q (f x)))\n: \u0393 \u22a2 (\u2203\u2203 x, p x) \u27f6 (\u2203\u2203 x, q x) :=\nbegin\n  intros,\n  lifted_pred keep [- exists_imp_distrib],\n  apply exists_imp_exists' f _ ,\n  intro x,\n  apply (h.apply _ a),\nend\n\nlemma p_exists_entails_p_exists' {t : Sort u\u2080} {t' : Sort u\u2082}\n  (p : t \u2192 pred' \u03b2)\n  (q : t' \u2192 pred' \u03b2)\n  (f : t \u2192 t')\n  (h : (\u2200 x, p x \u27f9 q (f x)))\n: (\u2203\u2203 x, p x) \u27f9 (\u2203\u2203 x, q x) :=\nbegin\n  intro,\n  apply p_exists_imp_p_exists' _ _ f,\n  apply (ew_p_forall _).mpr h,\nend\n\nlemma p_exists_variable_change\n  (p : \u03b1 \u2192 pred' \u03b2) (q : \u03b3 \u2192 pred' \u03b2)\n  (f : \u03b1 \u2192 \u03b3)\n  (g : \u03b3 \u2192 \u03b1)\n  (Hf : \u2200 i, p i \u27f9 q (f i))\n  (Hg : \u2200 j, q j \u27f9 p (g j))\n: (\u2203\u2203 i, p i) = (\u2203\u2203 j, q j) :=\nbegin\n  lifted_pred [- exists_imp_distrib],\n  rw exists_variable_change _ _ f g\n  ; intro x\n  ; apply entails_to_pointwise\n  ; solve_by_elim,\nend\n\nlemma p_exists_partial_intro {t : Sort u\u2080} {t' : Sort u\u2082}\n  (p : t \u2192 pred' \u03b2)\n  (f : t' \u2192 t)\n: (\u2203\u2203 x, p (f x)) \u27f9 (\u2203\u2203 x, p x) :=\nbegin\n  intro, apply p_exists_imp_p_exists' _ _ f,\n  lifted_pred,\nend\n\nlemma p_exists_range_subtype {\u03b1 : Sort u}\n  (p : \u03b1 \u2192 Prop) (q : \u03b1 \u2192 pred' \u03b2)\n: (\u2203\u2203 i, p i \u22c0 q i : pred' \u03b2) = (\u2203\u2203 j : subtype p, q (j.val)) :=\nby lifted_pred\n\nlemma p_or_iff_not_imp (p q : pred' \u03b2)\n: p \u22c1 q = - p \u27f6 q :=\nbegin\n  lifted_pred,\n  simp [or_iff_not_imp],\nend\n\nlemma p_forall_fin_zero (p : fin 0 \u2192 pred' \u03b2)\n: (\u2200\u2200 i, p i) = True :=\nbegin\n  lifted_pred,\n  simp [forall_fin_zero_iff_true],\nend\n\nlemma p_forall_split_one {n : \u2115} (p : fin (nat.succ n) \u2192 pred' \u03b2)\n: (\u2200\u2200 i, p i) = p fin.max \u22c0 (\u2200\u2200 i, restr p i) :=\nbegin\n  lifted_pred,\n  simp [forall_split_one],\n  refl,\nend\n\nlemma p_exists_split_one {n : \u2115} (p : fin (nat.succ n) \u2192 pred' \u03b2)\n: (\u2203\u2203 i, p i) = p fin.max \u22c1 (\u2203\u2203 i, restr p i) :=\nbegin\n  lifted_pred,\n  simp [exists_split_one],\n  refl,\nend\n\n@[simp]\nlemma whole_v_eq_prj (x : var \u03b1 \u03b1) (y : var \u03b3 \u03b1)\n: (whole \u2243 x) ! y = y \u2243 (x ! y) :=\nby lifted_pred\n\n@[simp]\nlemma models_whole (s : \u03c3)\n: s \u22a8 whole = s := rfl\n\ninstance entails_category {\u03b1} : category (@p_entails \u03b1) :=\n  { ident := by { intro, refl }\n  , comp  := by { intros, apply entails_trans ; assumption }\n  , assoc := by { intros, refl }\n  , left_ident  := by { intros, refl }\n  , right_ident := by { intros, refl } }\n\nend predicate\n", "meta": {"author": "unitb", "repo": "lean-lib", "sha": "439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9", "save_path": "github-repos/lean/unitb-lean-lib", "path": "github-repos/lean/unitb-lean-lib/lean-lib-439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9/src/util/predicate/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38938704644528593}}
{"text": "example : False :=\n  have : False := _\n\nexample : 5 = 3 :=\n  have t : True := _\n  have f : 5 = 6 := _\n  f\n\nexample : True :=\n  have := True.intro\n  this\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/have.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3893870393117286}}
{"text": "\n/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport morphisms.open_immersion\nimport morphisms.closed_immersion\nimport for_mathlib.locally_closed\nimport morphisms.integral\n\n/-!\n\n# Locally closed immersions\n\nA morphism of schemes is a closed immersion if the underlying map is a closed embedding, and \nthe sheaf map is locally surjective.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverse u\n\nnamespace algebraic_geometry\n\nvariables {X Y Z : Scheme.{u}} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\n\n/-- A morphism is a `is_closed_immersion` if the preimages of affine open sets are affine. -/\n@[mk_iff]\nclass is_immersion (f : X \u27f6 Y) extends is_preimmersion f : Prop :=\n(range_is_locally_closed [] : is_locally_closed (set.range f.1.base))\n\nlemma is_immersion.base_embedding [is_immersion f] : embedding f.1.base :=\nis_preimmersion.base_embedding f\n\ninstance [is_immersion f] [is_immersion g] : is_immersion (f \u226b g) :=\nbegin\n  constructor,\n  rw [Scheme.comp_val_base, coe_comp, set.range_comp],\n  exact (is_immersion.range_is_locally_closed f).image (is_immersion.base_embedding g).to_inducing\n    (is_immersion.range_is_locally_closed g)\nend\n\ninstance is_open_immersion.to_is_immersion [is_open_immersion f] : is_immersion f :=\nbegin\n  constructor,\n  exact (is_open_immersion.base_open f).2.is_locally_closed\nend\n\ninstance is_closed_immersion.to_is_immersion [is_closed_immersion f] : is_immersion f :=\n\u27e8(is_closed_immersion.base_closed f).2.is_locally_closed\u27e9 \n\nlemma is_closed_immersion_iff_is_immersion {f : X \u27f6 Y} :\n  is_closed_immersion f \u2194 is_immersion f \u2227 is_closed (set.range f.1.base) :=\n\u27e8\u03bb H, by exactI \u27e8infer_instance, H.2\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, by exactI \u27e8h\u2082\u27e9\u27e9   \n\nlemma is_closed_immersion.of_is_immersion [is_immersion f] (hf : is_closed (set.range f.1.base)) :\n  is_closed_immersion f :=\n\u27e8hf\u27e9\n\n/-- Also see `is_immersion.of_comp` in the file `morphisms/separated` -/\nlemma is_immersion.of_comp_of_is_immersion [is_immersion (f \u226b g)] [is_immersion g] :\n  is_immersion f :=\nbegin\n  haveI := is_preimmersion.of_comp f g,\n  constructor,\n  rw [\u2190 set.preimage_image_eq (set.range f.1.base) (is_immersion.base_embedding g).inj,\n    \u2190 set.range_comp, \u2190 coe_comp, \u2190 Scheme.comp_val_base],\n  exact (is_immersion.range_is_locally_closed (f \u226b g)).preimage g.1.base.2\nend\n\n@[simps]\ndef is_immersion.factor_opens [is_immersion f] : opens Y.carrier :=\n\u27e8_, (is_immersion.range_is_locally_closed f).is_open_coboundary\u27e9\n\ndef is_immersion.factor_open_subscheme [is_immersion f] : Scheme :=\nY.restrict (is_immersion.factor_opens f).open_embedding\n\n@[derive is_open_immersion]\ndef is_immersion.factor_open_immersion [is_immersion f] :\n  is_immersion.factor_open_subscheme f \u27f6 Y :=\nY.of_restrict _\n\n-- move me\nlemma opens.range_inclusion {X : Top} (U : opens X) : set.range U.inclusion = U :=\nsubtype.range_coe\n\ndef is_immersion.factor_closed_immersion [is_immersion f] :\n  X \u27f6 is_immersion.factor_open_subscheme f :=\nis_open_immersion.lift (is_immersion.factor_open_immersion f) f\nbegin\n  rw [is_immersion.factor_open_immersion, Scheme.of_restrict_val_base, opens.range_inclusion],\n  exact subset_coboundary\nend\n\n@[simp, reassoc]\nlemma is_immersion.factors [is_immersion f] :\n  is_immersion.factor_closed_immersion f \u226b is_immersion.factor_open_immersion f = f :=\nis_open_immersion.lift_fac _ _ _\n\n-- move me\nlemma is_open_immersion.range_lift (f : X \u27f6 Z) (g : Y \u27f6 Z) [is_open_immersion f] \n  (H : set.range g.1.base \u2286 set.range f.1.base) :\n  set.range (is_open_immersion.lift f g H).1.base = f.1.base \u207b\u00b9' (set.range g.1.base) :=\nLocallyRingedSpace.is_open_immersion.lift_range _ _ _\n\ninstance [is_immersion f] : is_closed_immersion (is_immersion.factor_closed_immersion f) :=\nbegin\n  haveI : is_immersion (is_immersion.factor_closed_immersion f),\n  { apply_with (is_immersion.of_comp_of_is_immersion _ (is_immersion.factor_open_immersion f))\n    { instances := ff },\n    rwa is_immersion.factors },\n  apply is_closed_immersion.of_is_immersion,\n  rw [is_immersion.factor_closed_immersion, is_open_immersion.range_lift],\n  exact is_closed_preimage_coe_coboundary\nend\n\nlemma is_immersion_iff_exists_factor : \n  is_immersion f \u2194 \u2203 {Z : Scheme} (g : X \u27f6 Z) [is_closed_immersion g]\n    (h : Z \u27f6 Y) [is_open_immersion h], g \u226b h = f :=\nbegin\n  split,\n  { introI H, exact \u27e8_, is_immersion.factor_closed_immersion f, infer_instance,\n      is_immersion.factor_open_immersion f, infer_instance, is_immersion.factors f\u27e9 },\n  { rintro \u27e8_, _, _, _, _, rfl\u27e9, by exactI infer_instance }\nend\n\ninstance is_immersion.to_mono [is_immersion f] : mono f :=\nby { rw \u2190 is_immersion.factors f, apply mono_comp }\n\ninstance is_immersion.to_locally_of_finite_type [is_immersion f] : locally_of_finite_type f :=\nby { rw \u2190 is_immersion.factors f, apply_instance }\n\nlemma is_immersion_stable_under_composition : \n  morphism_property.stable_under_composition @is_immersion :=\n\u03bb _ _ _ _ _ _ _, by exactI infer_instance\n\nlemma is_immersion_respects_iso : \n  morphism_property.respects_iso @is_immersion :=\nis_immersion_stable_under_composition.respects_iso (\u03bb _ _ _, infer_instance)\n\nlemma is_immersion_stable_under_base_change :\n  morphism_property.stable_under_base_change @is_immersion :=\nbegin\n  intros X Y X' S f g f' g' H hg,\n  rw is_immersion_iff_exists_factor at hg \u22a2,\n  obtain \u27e8Z, i\u2081, _, i\u2082, _, rfl\u27e9 := hg,\n  resetI,\n  refine \u27e8pullback i\u2082 f, pullback.lift (f' \u226b i\u2081) g' ((category.assoc _ _ _).trans H.w), _,\n    pullback.snd, infer_instance, pullback.lift_snd _ _ _\u27e9,\n  have : is_pullback f'\n    (pullback.lift (f' \u226b i\u2081) g' ((category.assoc _ _ _).trans H.w)) i\u2081 pullback.fst :=\n    is_pullback.of_bot (by rwa pullback.lift_snd)\n      (pullback.lift_fst _ _ _).symm (is_pullback.of_has_pullback _ _),\n  exact is_closed_immersion_stable_under_base_change this infer_instance\nend\n\ninstance {X Y S : Scheme} (f : X \u27f6 S) (g : Y \u27f6 S) [is_immersion g] :\n  is_immersion (pullback.fst : pullback f g \u27f6 X) :=\nis_immersion_stable_under_base_change.fst f g infer_instance\n\ninstance {X Y S : Scheme} (f : X \u27f6 S) (g : Y \u27f6 S) [is_immersion f] :\n  is_immersion (pullback.snd : pullback f g \u27f6 Y) :=\nis_immersion_stable_under_base_change.snd f g infer_instance\n\nlemma is_immersion_is_local_at_target :\n  property_is_local_at_target @is_immersion :=\nbegin\n  constructor,\n  { exact is_immersion_respects_iso },\n  { intros X Y f U hU,\n    haveI := is_preimmersion_is_local_at_target.2 f U hU.1,\n    constructor,\n    rw [morphism_restrict_val_base, set.range_restrict_preimage],\n    exact (is_immersion.range_is_locally_closed f).preimage continuous_subtype_coe },\n  { introsI X Y f \ud835\udcb0 H,\n    haveI := is_preimmersion_is_local_at_target.3 f \ud835\udcb0 infer_instance,\n    constructor,\n    apply (is_locally_closed_iff_coe_preimage_of_supr_eq_top \ud835\udcb0.supr_opens_range _).mpr,\n    intro i,\n    convert ((is_immersion_respects_iso.arrow_mk_iso_iff\n      (morphism_restrict_opens_range f (\ud835\udcb0.map i))).mpr (H i)).2 using 1,\n    rw [morphism_restrict_val_base, set.range_restrict_preimage] },\nend\n\nlemma is_immersion_open_cover_tfae (f : X \u27f6 Y) :\n  tfae [is_immersion f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y), \u2200 (i : \ud835\udcb0.J),\n      is_immersion (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) (i : \ud835\udcb0.J),\n      is_immersion (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (U : opens Y.carrier), is_immersion (f \u2223_ U),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_open_immersion g],\n      is_immersion (pullback.snd : pullback f g \u27f6 U),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4), (\u2200 i, is_immersion (f \u2223_ (U i)))] :=\nis_immersion_is_local_at_target.open_cover_tfae f\n\n-- lemma is_immersion_open_cover_of_subset_supr (f : X \u27f6 Y)\n--   {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : set.range f.1.base \u2286 ((supr U : _) : set Y.carrier))\n--   (h : \u2200 i, is_immersion (f \u2223_ U i)) : is_immersion f :=\n-- begin\n--   let V := supr U,\n--   have hV : (opens.map f.val.base).obj V = \u22a4,\n--   { rw [eq_top_iff], exact set.image_subset_iff.mp ((subset_of_eq set.image_univ).trans hU) },\n--   suffices : is_immersion (f \u2223_ V),\n--   { haveI : is_iso (X.of_restrict ((opens.map f.val.base).obj V).open_embedding),\n--     { rw hV,\n--       apply_with (is_iso_of_reflects_iso _ Scheme.forget_to_LocallyRingedSpace) { instances := ff },\n--       convert_to is_iso X.to_LocallyRingedSpace.restrict_top_iso.hom,\n--       apply_instance },\n--     rw [\u2190 is_immersion_respects_iso.cancel_left_is_iso\n--       (X.of_restrict ((opens.map f.val.base).obj V).open_embedding), \u2190 morphism_restrict_\u03b9],\n--     apply_instance },\n--   let U' : \u03b9 \u2192 opens (Y.restrict V.open_embedding).carrier := (opens.map V.inclusion).obj \u2218 U,\n--   have : supr U' = \u22a4,\n--   { rw [eq_top_iff],\n--     rintro \u27e8x, hx\u27e9 -,\n--     obtain \u27e8i, hi\u27e9 := opens.mem_supr.mp hx,\n--     exact opens.mem_supr.mpr \u27e8i, hi\u27e9 },\n--   refine ((is_immersion_open_cover_tfae (f \u2223_ V)).out 0 5).mpr \u27e8_, U', this, \u03bb i, _\u27e9,\n--   refine (is_immersion_respects_iso.arrow_iso_iff (morphism_restrict_restrict f _ _ \u226a\u226b\n--     morphism_restrict_eq f ((V.functor_map_eq_inf _).trans _))).mpr (h i),\n--   rw inf_eq_left,\n--   exact le_supr U i,\n-- end\n\nlemma is_closed_immersion_pullback_diagonal_Spec {R S : CommRing}\n  (f' : Scheme.Spec.obj (op S) \u27f6 Scheme.Spec.obj (op R)) :\n  is_closed_immersion (pullback.diagonal f') :=\nbegin\n  let f : R \u27f6 S := (Scheme.Spec.preimage f').unop,\n  have : Scheme.Spec.map f.op = f', { rw [quiver.hom.op_unop, functor.image_preimage] },\n  clear_value f, subst this,\n  letI := f.to_algebra,\n  have H := (is_pushout.of_is_colimit\n    (CommRing.pushout_cocone_is_colimit f f)).op.map Scheme.Spec,\n  let e : pullback.diagonal_obj (Scheme.Spec.map f.op) \u2245 \n    Scheme.Spec.obj (op (CommRing.pushout_cocone f f).X) := limit.iso_limit_cone \u27e8_, H.is_limit\u27e9,\n  have : pullback.diagonal _ \u226b e.hom = Scheme.Spec.map\n    (quiver.hom.op $ (@algebra.tensor_product.lmul' R S _ _ _).to_ring_hom),\n  { apply pullback_cone.is_limit.hom_ext H.is_limit,\n    { rw category.assoc,\n      erw [is_limit.cone_point_unique_up_to_iso_hom_comp, pullback.diagonal_fst],\n      rw [H.cone_fst, \u2190 functor.map_comp, \u2190 op_comp],\n      transitivity Scheme.Spec.map (\ud835\udfd9 _).op, { rw [op_id, category_theory.functor.map_id] },\n      congr' 2, ext1, simp },\n    { rw category.assoc,\n      erw [is_limit.cone_point_unique_up_to_iso_hom_comp, pullback.diagonal_snd],\n      rw [H.cone_snd, \u2190 functor.map_comp, \u2190 op_comp],\n      transitivity Scheme.Spec.map (\ud835\udfd9 _).op, { rw [op_id, category_theory.functor.map_id] },\n      congr' 2, ext1, simp }, }, \n  rw [\u2190 is_closed_immersion_respects_iso.cancel_right_is_iso _ e.hom, this,\n    is_closed_immersion_Spec_iff],\n  exact \u03bb x, \u27e8x \u2297\u209c 1, (algebra.tensor_product.lmul'_apply_tmul _ _).trans (mul_one _)\u27e9,\nend\n\n-- -- move me\n-- abbreviation Scheme.hom.lift_open (U : opens Y.carrier) (hU : set.range f.1.base \u2286 U.1) :\n--   X \u27f6 Y.restrict U.open_embedding := \n-- is_open_immersion.lift (Y.of_restrict U.open_embedding) f\n--   (hU.trans $ subset_of_eq $ subtype.range_coe.symm)\n\n-- lemma Scheme.hom.lift_open_of_restrict (U : opens Y.carrier) (hU : set.range f.1.base \u2286 U.1) :\n--   X.of_restrict _ \u226b f.lift_open U hU = f \u2223_ U :=  \n-- begin\n--   rw \u2190 cancel_mono (Y.of_restrict U.open_embedding),\n--   rw [category.assoc, is_open_immersion.lift_fac, morphism_restrict_\u03b9],\n-- end\n\nvariable {f}\n\nvariables (\ud835\udcb0 : Scheme.open_cover.{u} Y) (\ud835\udcb1 : \u2200 i, Scheme.open_cover.{u} (pullback f (\ud835\udcb0.map i)))\n\ndef pullback.diagonal_cover :\n  (pullback.diagonal_obj f).open_cover :=\n(Scheme.pullback.open_cover_of_base \ud835\udcb0 f f).bind\n  (\u03bb i, Scheme.pullback.open_cover_of_left_right (\ud835\udcb1 i) (\ud835\udcb1 i) pullback.snd pullback.snd)\n\ndef pullback.diagonal_cover_diagonal :\n  opens (pullback.diagonal_obj f).carrier :=\n\u2a06 i : \u03a3 i, (\ud835\udcb1 i).J, ((pullback.diagonal_cover \ud835\udcb0 \ud835\udcb1).map \u27e8i.1, i.2, i.2\u27e9).opens_range\n\nlemma pullback.diagonal_cover_map_eq (I) : (pullback.diagonal_cover \ud835\udcb0 \ud835\udcb1).map I =\n  pullback.map _ _ _ _\n    ((\ud835\udcb1 I.fst).map _ \u226b pullback.fst) ((\ud835\udcb1 I.fst).map _ \u226b pullback.fst) (\ud835\udcb0.map _)\n    (by simp only [category.assoc, pullback.condition])\n    (by simp only [category.assoc, pullback.condition]) :=\nbegin\n  dsimp [pullback.diagonal_cover],\n  apply pullback.hom_ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,  \n      pullback.lift_fst_assoc, pullback.lift_snd_assoc,\n      pullback.diagonal_fst_assoc, pullback.diagonal_fst,\n      pullback.diagonal_snd_assoc, pullback.diagonal_snd, category.comp_id]\nend\n\nlemma pullback.diagonal_cover_diagonal_eq_top_of_injective (hf : function.injective f.1.base) :\n  pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1 = \u22a4 :=\nbegin\n  rw eq_top_iff,\n  rintro x -, \n  let x\u2081 := (pullback.fst : pullback.diagonal_obj f \u27f6 X).1.base x,\n  let x\u2082 := (pullback.snd : pullback.diagonal_obj f \u27f6 X).1.base x,\n  have hx : x\u2081 = x\u2082,\n  { apply hf, simp_rw [\u2190 Scheme.comp_val_base_apply, pullback.condition] },\n  let i := \ud835\udcb0.f (f.1.base x\u2081),\n  obtain \u27e8t, ht\u27e9 : x\u2081 \u2208 set.range (pullback.fst : pullback f (\ud835\udcb0.map i) \u27f6 _).1.base,\n  { rw is_open_immersion.range_pullback_fst_of_right, exact (\ud835\udcb0.covers $ f.1.base x\u2081) },\n  obtain \u27e8w, hw\u27e9 := (\ud835\udcb1 i).covers t,\n  rw \u2190 hw at ht,\n  refine opens.mem_supr.mpr \u27e8\u27e8i, (\ud835\udcb1 i).f t\u27e9, _\u27e9,\n  change _ \u2208 set.range ((pullback.diagonal_cover \ud835\udcb0 \ud835\udcb1).map \u27e8i, ((\ud835\udcb1 i).f t, (\ud835\udcb1 i).f t)\u27e9).1.base,\n  simp_rw [pullback.diagonal_cover_map_eq \ud835\udcb0 \ud835\udcb1, pullback.range_map],\n  split,\n  { exact \u27e8w, ht\u27e9 },\n  { rw hx at ht, exact \u27e8w, ht\u27e9 }\nend\n  \nlemma pullback.diagonal_range_subset_supr_diagonal_cover :\n  set.range (pullback.diagonal f).1.base \u2286 (pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1).1 :=\nbegin\n  rintros _ \u27e8x, rfl\u27e9,\n  let i := \ud835\udcb0.f (f.1.base x),\n  obtain \u27e8y, hy : (\ud835\udcb0.map i).1.base y = _\u27e9 := \ud835\udcb0.covers (f.1.base x),\n  let T : pullback.triplet f (\ud835\udcb0.map i) := \u27e8x, y, _, rfl, hy\u27e9,\n  obtain \u27e8z, (hzx : _ = x), (rfl : _ = y)\u27e9 := T.exists_preimage,\n  obtain \u27e8w, hw\u27e9 := (\ud835\udcb1 i).covers z,\n  rw [\u2190 hzx, \u2190 hw],\n  refine opens.mem_supr.mpr \u27e8\u27e8i, (\ud835\udcb1 i).f z\u27e9,\n    (pullback.diagonal ((\ud835\udcb1 i).map ((\ud835\udcb1 i).f z) \u226b pullback.snd)).1.base w, _\u27e9,\n  simp_rw [\u2190 Scheme.comp_val_base_apply],\n  congr' 3,\n  dsimp [pullback.diagonal_cover],\n  apply pullback.hom_ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,  \n      pullback.lift_fst_assoc, pullback.lift_snd_assoc,\n      pullback.diagonal_fst_assoc, pullback.diagonal_fst,\n      pullback.diagonal_snd_assoc, pullback.diagonal_snd, category.comp_id],\nend\n\ndef pullback.diagonal_restrict_iso_diagonal (i j) :\n  arrow.mk (pullback.diagonal f \u2223_ ((pullback.diagonal_cover \ud835\udcb0 \ud835\udcb1).map \u27e8i, j, j\u27e9).opens_range) \u2245\n    arrow.mk (pullback.diagonal ((\ud835\udcb1 i).map j \u226b pullback.snd)) :=\nbegin\n  refine (morphism_restrict_opens_range _ _) \u226a\u226b arrow.iso_mk _ _ _,\n  { dsimp [pullback.diagonal_cover], \n    refine pullback.congr_hom rfl _ \u226a\u226b\n      pullback_diagonal_map_iso f (\ud835\udcb0.map i) ((\ud835\udcb1 i).map j) ((\ud835\udcb1 i).map j) \u226a\u226b\n      as_iso pullback.fst,\n    apply pullback.hom_ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,  \n        pullback.lift_fst_assoc, pullback.lift_snd_assoc],\n      },\n  { dsimp [pullback.diagonal_cover], exact iso.refl _ },\n  { have : (pullback.fst : pullback ((\ud835\udcb1 i).map j) ((\ud835\udcb1 i).map j) \u27f6 _) = pullback.snd,\n    { rw \u2190 cancel_epi (pullback.diagonal $ (\ud835\udcb1 i).map j),\n      rw [pullback.diagonal_fst, pullback.diagonal_snd] },\n    dsimp [pullback.diagonal_cover],\n    apply pullback.hom_ext,\n    swap, simp only [this],\n    all_goals { simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,  \n      pullback.lift_fst_assoc, pullback.lift_snd_assoc, category.id_comp, category.comp_id,\n      pullback_diagonal_map_iso_hom_fst, pullback_diagonal_map_iso_hom_snd,\n      pullback.diagonal_fst, pullback.diagonal_snd] } }\nend\n\nlemma pullback.is_closed_immersion_lift_diagonal_aux\n  (H : \u2200 i j, is_closed_immersion (pullback.diagonal ((\ud835\udcb1 i).map j \u226b pullback.snd))) :\n  is_closed_immersion (pullback.diagonal f \u2223_ pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1) :=\nbegin\n  rw (is_closed_immersion.open_cover_tfae (pullback.diagonal\n    f \u2223_ pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1)).out 0 5,\n  have : (\u2a06 i : \u03a3 i, (\ud835\udcb1 i).J, ((pullback.diagonal_cover \ud835\udcb0 \ud835\udcb1).map \u27e8i.1, i.2, i.2\u27e9).opens_range) =\n    pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1 := rfl,\n  apply_fun (opens.map (pullback.diagonal_cover_diagonal \ud835\udcb0 \ud835\udcb1).inclusion).obj at this,\n  rw [opens.map_supr, opens.inclusion_map_eq_top] at this,\n  refine \u27e8_, _, this, _\u27e9,\n  rintros \u27e8i, j\u27e9,\n  rw is_closed_immersion_respects_iso.arrow_mk_iso_iff\n    (morphism_restrict_restrict _ _ _),\n  rw is_closed_immersion_respects_iso.arrow_mk_iso_iff\n    (morphism_restrict_eq _ $ (opens.functor_map_eq_inf _ _).trans (inf_eq_left.mpr _)),\n  { rw is_closed_immersion_respects_iso.arrow_mk_iso_iff\n    (pullback.diagonal_restrict_iso_diagonal \ud835\udcb0 \ud835\udcb1 i j), apply H },\n  { exact le_supr _ _ }\nend\n\nvariable (f)\n\nlemma pullback.is_closed_immersion_lift_diagonal :\n  is_closed_immersion (pullback.diagonal f \u2223_ pullback.diagonal_cover_diagonal\n    Y.affine_cover (\u03bb _, Scheme.affine_cover _)) :=\nbegin\n  apply pullback.is_closed_immersion_lift_diagonal_aux,\n  intros i j, apply is_closed_immersion_pullback_diagonal_Spec\nend\n\n\nlocal attribute [irreducible] Scheme.affine_cover\n\ndef Scheme.restrict_top_iso (X : Scheme.{u}) : \n  X.restrict (opens.open_embedding $ \u22a4) \u2245 X :=\n{ .. X.to_LocallyRingedSpace.restrict_top_iso }\n\n-- Declaring it as an instance adds superfluous universe variables\n@[instance]\nlemma _root_.category_theory.limits.pullback.diagonal.is_immersion {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  is_immersion (pullback.diagonal f) :=\nbegin\n  let U := (pullback.diagonal_cover_diagonal Y.affine_cover (\u03bb _, Scheme.affine_cover _)),\n  haveI : is_closed_immersion (pullback.diagonal f \u2223_ U) :=\n    pullback.is_closed_immersion_lift_diagonal f,\n  have : (opens.map (pullback.diagonal f).val.base).obj U = \u22a4,\n  { ext1, show (pullback.diagonal f).val.base \u207b\u00b9' U.1 = set.univ,\n    rw [\u2190 set.univ_subset_iff, \u2190 set.image_subset_iff, set.image_univ],\n    exact pullback.diagonal_range_subset_supr_diagonal_cover _ _ },\n  haveI : is_iso (X.of_restrict ((opens.map (pullback.diagonal f).val.base).obj U).open_embedding),\n  { rw this, exact is_iso.of_iso X.restrict_top_iso },\n  have := morphism_restrict_\u03b9 (pullback.diagonal f) U,\n  rw [\u2190 is_iso.inv_comp_eq] at this,\n  rw \u2190 this,\n  apply_instance\nend\n-- move me \ndef morphism_restrict.top_iso : arrow.mk (f \u2223_ \u22a4) \u2245 arrow.mk f :=\narrow.iso_mk' (f \u2223_ \u22a4) f X.restrict_top_iso Y.restrict_top_iso (morphism_restrict_\u03b9 _ _).symm\n\n-- Also see `separated_of_injective`\nlemma pullback.diagonal_is_closed_immersion_of_injective {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (hf : function.injective f.1.base) :\n  is_closed_immersion (pullback.diagonal f) :=\nbegin\n  let U := (pullback.diagonal_cover_diagonal Y.affine_cover (\u03bb _, Scheme.affine_cover _)),\n  have hU : is_closed_immersion (pullback.diagonal f \u2223_ U) :=\n    pullback.is_closed_immersion_lift_diagonal f,\n  have : U = \u22a4 := pullback.diagonal_cover_diagonal_eq_top_of_injective _ _ hf,\n  rw this at hU,\n  exact (is_closed_immersion_respects_iso.arrow_mk_iso_iff (morphism_restrict.top_iso _)).mp hU\nend\n\nend algebraic_geometry\n", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/morphisms/immersion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38933248594177783}}
{"text": "class HOp (\u03b1 \u03b2 \u03b3) where hOp : \u03b1 \u2192 \u03b2 \u2192 \u03b3\n\nclass LOp (\u03b1 \u03b2) where lOp : \u03b1 \u2192 \u03b2 \u2192 \u03b2\n\nclass Op (\u03b1) where op : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\n@[default_instance]\ninstance inst1 [LOp \u03b1 \u03b2] : HOp \u03b1 \u03b2 \u03b2 := \u27e8LOp.lOp\u27e9\n\ninstance inst2 [Op \u03b1] : LOp \u03b1 \u03b1 := \u27e8Op.op\u27e9\n\ninfix:75 \" \u22c6 \" => HOp.hOp\n\nsection Test\nvariable (\u03b1) [LOp Nat \u03b1]\nvariable (x y z : \u03b1) (m n : Nat)\n\nexample : n \u22c6 x = z := sorry -- TC works\n\nexample : 1 \u22c6 x = z := sorry -- TC works\n\nattribute [default_instance] inst2\n\nexample : n \u22c6 x = z := sorry -- TC works\n\nexample : 1 \u22c6 x = z := sorry -- TC fails\n\nend Test\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/defaultInstBacktrackIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.389332478177287}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.group.pi\nimport algebra.category.Group.preadditive\nimport category_theory.limits.shapes.biproducts\nimport algebra.category.Group.limits\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\nopen category_theory\nopen category_theory.limits\n\nopen_locale big_operators\n\nuniverses w u\n\nnamespace AddCommGroup\n\n-- As `AddCommGroup` is preadditive, and has all limits, it automatically has biproducts.\ninstance : has_binary_biproducts AddCommGroup :=\nhas_binary_biproducts.of_has_binary_products\n\ninstance : has_finite_biproducts AddCommGroup :=\nhas_finite_biproducts.of_has_finite_products\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n\n/--\nConstruct limit data for a binary product in `AddCommGroup`, using `AddCommGroup.of (G \u00d7 H)`.\n-/\n@[simps cone_X is_limit_lift]\ndef binary_product_limit_cone (G H : AddCommGroup.{u}) : limits.limit_cone (pair G H) :=\n{ cone :=\n  { X := AddCommGroup.of (G \u00d7 H),\n    \u03c0 := { app := \u03bb j, discrete.cases_on j\n      (\u03bb j, walking_pair.cases_on j (add_monoid_hom.fst G H) (add_monoid_hom.snd G H)),\n      naturality' := by rintros \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9; refl, }},\n  is_limit :=\n  { lift := \u03bb s, add_monoid_hom.prod (s.\u03c0.app \u27e8walking_pair.left\u27e9) (s.\u03c0.app \u27e8walking_pair.right\u27e9),\n    fac' := by { rintros s (\u27e8\u27e9|\u27e8\u27e9); { ext x, simp, } },\n    uniq' := \u03bb s m w, begin\n      ext; [rw \u2190 w \u27e8walking_pair.left\u27e9, rw \u2190 w \u27e8walking_pair.right\u27e9]; refl,\n    end, } }\n\n@[simp] lemma binary_product_limit_cone_cone_\u03c0_app_left (G H : AddCommGroup.{u}) :\n  (binary_product_limit_cone G H).cone.\u03c0.app \u27e8walking_pair.left\u27e9 = add_monoid_hom.fst G H := rfl\n\n@[simp] lemma binary_product_limit_cone_cone_\u03c0_app_right (G H : AddCommGroup.{u}) :\n  (binary_product_limit_cone G H).cone.\u03c0.app \u27e8walking_pair.right\u27e9 = add_monoid_hom.snd G H := rfl\n\n/--\nWe verify that the biproduct in AddCommGroup is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps hom_apply] noncomputable\ndef biprod_iso_prod (G H : AddCommGroup.{u}) : (G \u229e H : AddCommGroup) \u2245 AddCommGroup.of (G \u00d7 H) :=\nis_limit.cone_point_unique_up_to_iso\n  (binary_biproduct.is_limit G H)\n  (binary_product_limit_cone G H).is_limit\n\n@[simp, elementwise] lemma biprod_iso_prod_inv_comp_fst (G H : AddCommGroup.{u}) :\n  (biprod_iso_prod G H).inv \u226b biprod.fst = add_monoid_hom.fst G H :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.left)\n\n@[simp, elementwise] lemma biprod_iso_prod_inv_comp_snd (G H : AddCommGroup.{u}) :\n  (biprod_iso_prod G H).inv \u226b biprod.snd = add_monoid_hom.snd G H :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.right)\n\nnamespace has_limit\nvariables {J : Type w} (f : J \u2192 AddCommGroup.{max w u})\n\n/--\nThe map from an arbitrary cone over a indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : fan f) :\n  s.X \u27f6 AddCommGroup.of (\u03a0 j,f j) :=\n{ to_fun := \u03bb x j, s.\u03c0.app \u27e8j\u27e9 x,\n  map_zero' := by { ext, simp },\n  map_add' := \u03bb x y, by { ext, simp }, }\n\n/--\nConstruct limit data for a product in `AddCommGroup`, using `AddCommGroup.of (\u03a0 j, F.obj j)`.\n-/\n@[simps] def product_limit_cone : limits.limit_cone (discrete.functor f) :=\n{ cone :=\n  { X := AddCommGroup.of (\u03a0 j, f j),\n    \u03c0 := discrete.nat_trans (\u03bb j, pi.eval_add_monoid_hom (\u03bb j, f j) j.as), },\n  is_limit :=\n  { lift := lift f,\n    fac' := \u03bb s j, by { cases j, ext, simp, },\n    uniq' := \u03bb s m w,\n    begin\n      ext x j,\n      dsimp only [has_limit.lift],\n      simp only [add_monoid_hom.coe_mk],\n      exact congr_arg (\u03bb g : s.X \u27f6 f j, (g : s.X \u2192 f j) x) (w \u27e8j\u27e9),\n    end, }, }\n\nend has_limit\n\nopen has_limit\n\nvariables {J : Type} [fintype J]\n\n/--\nWe verify that the biproduct we've just defined is isomorphic to the AddCommGroup structure\non the dependent function type\n-/\n@[simps hom_apply] noncomputable\ndef biproduct_iso_pi (f : J \u2192 AddCommGroup.{u}) :\n  (\u2a01 f : AddCommGroup) \u2245 AddCommGroup.of (\u03a0 j, f j) :=\nis_limit.cone_point_unique_up_to_iso\n  (biproduct.is_limit f)\n  (product_limit_cone f).is_limit\n\n@[simp, elementwise] lemma biproduct_iso_pi_inv_comp_\u03c0 (f : J \u2192 AddCommGroup.{u}) (j : J) :\n  (biproduct_iso_pi f).inv \u226b biproduct.\u03c0 f j = pi.eval_add_monoid_hom (\u03bb j, f j) j :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk j)\n\nend AddCommGroup\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/algebra/category/Group/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6654105454764746, "lm_q1q2_score": 0.38933247817728694}}
{"text": "open Classical\n\ntheorem const_func_ext [Nonempty \u03b1] {a b : \u03b2} : (fun _ : \u03b1 => a) = (fun _ => b) \u2192 (a = b) := by  \n  intro h\n  have ha : (fun _ : \u03b1 => a) (choice inferInstance) = a := rfl\n  have hb : (fun _ : \u03b1 => b) (choice inferInstance) = b := rfl\n  rw [\u2190ha, \u2190hb, h]\n\ndef Vector (\u03b1 : Type) (n : Nat) := Fin n \u2192 \u03b1  \n\ninfix:95 \"\u22c6\" => Vector\n\ndef Vector.\u03b5 : \u03b1\u22c60 := (nomatch \u00b7)\n\ndef Vector.\u03b5' (h : n = 0) : \u03b1\u22c6n := \n  fun \u27e8_, h'\u27e9 => by simp [h] at h'; contradiction\n\ndef Vector.prefix (as : \u03b1\u22c6(n + 1)) : \u03b1\u22c6n := \n  fun i => as \u27e8i.val, Nat.lt_succ_of_le $ Nat.le_of_lt i.isLt\u27e9\n\ndef Vector.all (v : \u03b1\u22c6n) (p : \u03b1 \u2192 Prop) : Prop :=\n  \u2200 i, p (v i)\n\ndef Vector.map (v : \u03b1\u22c6n) (f : \u03b1 \u2192 \u03b2) : \u03b2\u22c6n := \n  (f $ v \u00b7)\n\nabbrev Op (\u03b1 : Type) (n : Nat) := \u03b1\u22c6n \u2192 \u03b1\n\ninstance : CoeHead (Op \u03b1 0) \u03b1 where\n  coe op := op Vector.\u03b5\n\nabbrev BinOp (\u03b1 : Type) := \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\ninstance : Coe (Op \u03b1 2) (BinOp \u03b1) where\n  coe op a\u2081 a\u2082 := op fun | 0 => a\u2081 | 1 => a\u2082\n\ninstance : CoeTail (BinOp \u03b1) (Op \u03b1 2) where\n  coe op as := op (as 0) (as 1)\n\ndef ExistsUnique (p : \u03b1 \u2192 Prop) := \u2203 x, p x \u2227 \u2200 y, p y \u2192 y = x\n\nopen Lean TSyntax.Compat in\nmacro \"\u2203! \" xs:explicitBinders \", \" b:term : term => expandExplicitBinders ``ExistsUnique xs b\n\nclass Fintype (\u03b1 : Type)\n\ndef Set (\u03b1 : Type _) := \u03b1 \u2192 Prop\n\ninstance : Membership \u03b1 (Set \u03b1) where\n  mem a s := s a\n\ninstance : CoeSort (Set \u03b1) Type where\n  coe s := { a : \u03b1 // s a }\n\nabbrev Set.empty : Set \u03b1 := fun _ => False\n\ninstance : EmptyCollection (Set \u03b1) := \u27e8Set.empty\u27e9\n\ndef Set.Nonempty (s : Set \u03b1) : Prop := s \u2260 \u2205\n\ndef Set.Nonempty.iff_exists_mem : Set.Nonempty s \u2194 \u2203 a, a \u2208 s := by\n  sorry\n\nabbrev Set.univ : Set \u03b1 := fun _ => True\n\nabbrev Set.singleton (a : \u03b1) : Set \u03b1 := (\u00b7 = a)\n\ntheorem Set.mem_univ : a \u2208 Set.univ := .intro\n\ndef Set.image (f : \u03b1 \u2192 \u03b2) : Set \u03b2 := \n  (\u2203 a, f a = \u00b7)\n\ntheorem Set.mem_image_iff : (b \u2208 Set.image f) \u2194 (\u2203 a, f a = b) := \u27e8id, id\u27e9\n\ntheorem Set.image_choose {b : \u03b2} : (h : b \u2208 Set.image f) \u2192 (f $ choose h) = b :=\n  choose_spec\n\ndef Set.union (s\u2081 s\u2082 : Set \u03b1) := fun a => a \u2208 s\u2081 \u2228 a \u2208 s\u2082\n\ndef Set.bUnion (s : Set (Set \u03b1)) : Set \u03b1 := \n  fun a => \u2203 m, m \u2208 s \u2227 a \u2208 m\n\ndef Set.Subset (s\u2081 s\u2082 : Set \u03b1) : Prop :=\n  \u2200 a, a \u2208 s\u2081 \u2192 a \u2208 s\u2082\n\ninfix:50 \" \u2286 \" => Set.Subset\n\ntheorem Set.mem_ext {s\u2081 s\u2082 : Set \u03b1} : (\u2200 a, a \u2208 s\u2081 \u2194 a \u2208 s\u2082) \u2192 s\u2081 = s\u2082 :=\n  fun h => funext (fun a => propext (h a))\n\nabbrev Vector.lift {s : Set \u03b1} (v : s\u22c6n) : \u03b1\u22c6n := (v \u00b7)\n\nabbrev Word (\u03b1 : Type) := List \u03b1\n\npostfix:95 \"\u22c6\" => Word\n\nabbrev Word.\u03b5 : \u03b1\u22c6 := []\n\n@[match_pattern]\nabbrev Word.prepend : \u03b1 \u2192 \u03b1\u22c6 \u2192 \u03b1\u22c6 := List.cons\n\nabbrev Word.append (u : \u03b1\u22c6) (i : \u03b1) : \u03b1\u22c6 := u ++ [i]\n\ninfixl:70 \"\u2b1d\" => Word.prepend\n\ninfixl:70 \"\u2022\" => Word.append\n\ndef Word.prefixes : \u03b1\u22c6 \u2192 Set (\u03b1\u22c6)\n  | .\u03b5 => Set.singleton .\u03b5\n  | i\u2b1dv => fun u => (u = i\u2b1dv) \u2228 (u \u2208 prefixes v)\n\n@[simp] \ntheorem Set.mem_singleton : a \u2208 Set.singleton a := by\n  simp [singleton, Membership.mem]\n\ntheorem Word.\u03b5_mem_prefixes (u : Word \u03b1) : .\u03b5 \u2208 u.prefixes := by\n  induction u\n  case nil => simp [prefixes, Set.mem_singleton]\n  case cons i v hi => \n    simp [prefixes] \n    simp [Membership.mem] at *\n    exact hi\n\ndef Vector.word : {n : Nat} \u2192 \u03b1\u22c6n \u2192 \u03b1\u22c6\n  | 0,     _ => []\n  | n + 1, as => as.prefix.word ++ [as \u27e8n, Nat.lt_succ_self _\u27e9]\n\ndef Vector.prefixes : {n : Nat} \u2192 (\u03b1\u22c6n) \u2192 Set (\u03b1\u22c6)\n  | 0,     _  => Set.singleton .\u03b5\n  | _ + 1, as => fun u => (u = as.word) \u2228 (u \u2208 as.prefix.prefixes)\n\ndef Function.Injective (f : \u03b1 \u2192 \u03b2) : Prop :=\n  \u2200 a\u2081 a\u2082, (f a\u2081 = f a\u2082) \u2192 a\u2081 = a\u2082\n\ndef Function.Surjective (f : \u03b1 \u2192 \u03b2) : Prop :=\n  \u2200 b, \u2203 a, f a = b\n\nstructure Function.Bijective (f : \u03b1 \u2192 \u03b2) : Prop where\n  inj : Function.Injective f\n  surj : Function.Surjective f\n\n\n-----------------------------------------------------------------------------------------\n\n\nclass Alphabet (\u03b1 : Type) where\n  [nonempty : Nonempty \u03b1]\n  [finite : Fintype \u03b1]\n\nstructure RankedAlphabet where\n  alphabet : Type\n  rank : alphabet \u2192 Nat\n  [isAlphabet : Alphabet alphabet]\n\ninstance : CoeSort RankedAlphabet Type where\n  coe \u0394 := \u0394.alphabet\n\nstructure Algebra (\u0394 : RankedAlphabet) where\n  carrier : Type\n  \u03b8 : (\u03c3 : \u0394) \u2192 Op carrier (\u0394.rank \u03c3)  \n\ndef Algebra.ops (alg : Algebra \u0394) : Set (\u03a3 k : Nat, Op alg.carrier k) :=\n  fun \u27e8k, op\u27e9 => \u2203 (\u03c3 : \u0394) (h : \u0394.rank \u03c3 = k), (alg.\u03b8 \u03c3 = h \u25b8 op)\n\ndef Closed (alg : Algebra \u0394) (sub : Set alg.carrier) : Prop :=\n  \u2200 \u03c3 (cs : Vector sub $ \u0394.rank \u03c3), (alg.\u03b8 \u03c3 cs.lift) \u2208 sub\n\ntheorem Algebra.carrier_closed (alg : Algebra \u0394) : Closed alg Set.univ := by\n  simp [Closed, Set.mem_univ]\n\n-- Note that this defines a `Set`.\ninductive Closure (sub : Set \u03b1) (ops : Set (\u03a3 k : Nat, Op \u03b1 k)) : \u03b1 \u2192 Prop\n  | root : (a \u2208 sub) \u2192 Closure sub ops a\n  | app {v : Vector \u03b1 k} : (\u27e8k, op\u27e9 \u2208 ops) \u2192 (\u2200 i, Closure sub ops $ v i) \u2192 Closure sub ops (op v)\n\nabbrev closure (sub : Set \u03b1) (ops : Set (\u03a3 k : Nat, Op \u03b1 k)) : Set \u03b1 := \n  Closure sub ops\n\nstructure Subalgebra (alg : Algebra \u0394) where\n  carrier : Set alg.carrier\n  \u03b8 : (\u03c3 : \u0394) \u2192 Op alg.carrier (\u0394.rank \u03c3)\n  restricted : \u2200 \u03c3 cs, cs.all (\u00b7 \u2208 carrier) \u2192 \u03b8 \u03c3 cs = alg.\u03b8 \u03c3 cs\n  closed : Closed alg carrier \n\ndef Subalgebra.algebra {alg : Algebra \u0394} (s : Subalgebra alg) : Algebra \u0394 where\n  carrier := s.carrier\n  \u03b8 \u03c3 cs := {\n    val := s.\u03b8 \u03c3 cs.lift\n    property := by \n      have h : cs.lift.all (\u00b7 \u2208 s.carrier) := by\n        simp [Vector.all, Vector.lift]\n        intro i\n        exact (cs i).property\n      rw [s.restricted _ _ h]\n      apply s.closed\n  }\n\ninstance {alg : Algebra \u0394} : Coe (Subalgebra alg) (Algebra \u0394) where\n  coe := Subalgebra.algebra\n\nstructure Hom (alg\u2081 alg\u2082 : Algebra \u0394) where\n  hom : alg\u2081.carrier \u2192 alg\u2082.carrier\n  property : \u2200 \u03c3 cs, hom (alg\u2081.\u03b8 \u03c3 cs) = (alg\u2082.\u03b8 \u03c3) (hom \u2218 cs)\n\ntheorem Hom.ext (hom\u2081 hom\u2082 : Hom alg\u2081 alg\u2082) : hom\u2081.hom = hom\u2082.hom \u2192 hom\u2081 = hom\u2082 := by\n  intro h\n  cases hom\u2081 <;> cases hom\u2082\n  simp at h\n  simp [h]\n\ninstance : CoeFun (Hom alg\u2081 alg\u2082) (fun _ => alg\u2081.carrier \u2192 alg\u2082.carrier) where\n  coe h := h.hom\n\ntheorem lemma_2_6_2 (hom : Hom alg\u2081 alg\u2082) : Closed alg\u2082 (Set.image hom) := by\n  intro \u03c3 cs\u2082\n  simp [Closed, Set.mem_image_iff]\n  let cs\u2081 := (cs\u2082 \u00b7 |>.property |> choose)\n  have h := hom.property \u03c3 cs\u2081\n  exists alg\u2081.\u03b8 \u03c3 cs\u2081\n  rw [h]\n  congr \n  funext i\n  apply Set.image_choose\n\n-- Lemma 2.6.3  \ndef Hom.compose (hom\u2081\u2082 : Hom alg\u2081 alg\u2082) (hom\u2082\u2083 : Hom alg\u2082 alg\u2083) : Hom alg\u2081 alg\u2083 where\n  hom := hom\u2082\u2083 \u2218 hom\u2081\u2082\n  property := by\n    intro \u03c3 cs\u2081\n    simp [hom\u2081\u2082.property \u03c3 cs\u2081, hom\u2082\u2083.property \u03c3 $ hom\u2081\u2082 \u2218 cs\u2081]\n    rfl\n\ninfixr:90 \" \u2218 \" => Hom.compose\n\nstructure Iso (alg\u2081 alg\u2082 : Algebra \u0394) extends Hom alg\u2081 alg\u2082 where\n  bij : Function.Bijective hom\n\ndef Isomorphic (alg\u2081 alg\u2082 : Algebra \u0394) : Prop :=\n  Nonempty (Iso alg\u2081 alg\u2082)\n\ninfix:50 \" \u2245 \" => Isomorphic\n\nstructure FreelyGenerated (alg : Algebra \u0394) (gen : Set alg.carrier) (k : Set $ Algebra \u0394) : Prop where\n  mem : alg \u2208 k\n  generated : \u2200 c, c \u2208 closure gen alg.ops\n  free : \u2200 (alg' : Algebra \u0394) (f : gen \u2192 alg'.carrier), (alg' \u2208 k) \u2192 \n         \u2203! hom : Hom alg alg', \u2200 c : gen, f c = hom c\n  \n-- Note, we immediately restrict this definition to the set of all algebras,\n-- as this is the only one we ever need.\nnoncomputable def FreelyGenerated.hom {alg : Algebra \u0394} {H} \n  (h : FreelyGenerated alg H Set.univ) (target : Algebra \u0394) (f : H \u2192 target.carrier) : \n  Hom alg target :=\n  choose (h.free target f Set.mem_univ)\n\ntheorem FreelyGenerated.hom_extends (h : FreelyGenerated alg gen Set.univ) {f}: \n  \u2200 c : gen, f c = (h.hom target f) c := \n  (choose_spec (h.free target f Set.mem_univ) |>.left \u00b7)\n  \n\n-----------------------------------------------------------------------------------------\n\n\ndef BinOp.Associative (op : BinOp \u03b1) : Prop :=\n  \u2200 a b c, op (op a b) c = op a (op b c)\n\ndef BinOp.Commutative (op : BinOp \u03b1) : Prop :=\n  \u2200 a b, op a b = op b a\n\ndef BinOp.Idempotent (op : BinOp \u03b1) : Prop :=\n  \u2200 a, op a a = a\n\nstructure BinOp.Identity (op : BinOp \u03b1) (e : \u03b1) : Prop where\n  left  : \u2200 a, op e a = a\n  right : \u2200 a, op a e = a\n\ntheorem BinOp.Identity.unique (op : BinOp \u03b1) : (Identity op e) \u2192 (Identity op e') \u2192 e = e' := by\n  intro h h'\n  rw [\u2190h'.left e, h.right e']\n\nstructure BinOp.Inverse (op : BinOp \u03b1) (a a' : \u03b1) : Prop where\n  left  : \u2200 {e}, (Identity op e) \u2192 op a a' = e\n  right : \u2200 {e}, (Identity op e) \u2192 op a' a = e\n\ndef BinOp.RightDistrib (mul add : BinOp \u03b1) : Prop :=\n  \u2200 a b c, mul (add a b) c = add (mul a c) (mul b c)\n\ndef BinOp.LeftDistrib (mul add : BinOp \u03b1) : Prop :=\n  \u2200 a b c, mul a (add b c) = add (mul a b) (mul a c)\n\nstructure BinOp.Distrib (mul add : BinOp \u03b1) : Prop where\n  right : RightDistrib mul add\n  left : LeftDistrib mul add\n\nstructure Semigroup where\n  carrier : Type\n  op : BinOp carrier\n  assoc : BinOp.Associative op\n\nprotected inductive Semigroup.Alphabet \n  | \u00ab\u2299\u00bb\n\nabbrev Semigroup.ranked : RankedAlphabet where\n  alphabet := Semigroup.Alphabet\n  rank | .\u00ab\u2299\u00bb => 2\n  isAlphabet := sorry\n\ndef Semigroup.algebra (s : Semigroup) : Algebra ranked where\n  carrier := s.carrier\n  \u03b8 | .\u00ab\u2299\u00bb => s.op\n\nstructure Monoid extends Semigroup where\n  id : Op carrier 0\n  idIsIdentity : BinOp.Identity op id\n\nprotected inductive Monoid.Alphabet \n  | \u00ab\u2299\u00bb\n  | e\n\nabbrev Monoid.ranked : RankedAlphabet where\n  alphabet := Monoid.Alphabet\n  rank \n    | .\u00ab\u2299\u00bb => 2 \n    | .e   => 0\n  isAlphabet := sorry\n\ndef Monoid.algebra (m : Monoid) : Algebra Monoid.ranked where\n  carrier := m.carrier\n  \u03b8 \n    | .\u00ab\u2299\u00bb => m.op \n    | .e   => m.id\n\ndef Monoid.Commutative (m : Monoid) : Prop :=\n  BinOp.Commutative m.op\n\nstructure StrongBimonoid where\n  carrier : Type\n  add  : BinOp carrier\n  mul  : BinOp carrier\n  zero : Op carrier 0\n  one  : Op carrier 0\n  addComm : BinOp.Commutative add\n  zeroNeOne : zero \u2260 one\n  leftAbsorption : \u2200 c, mul zero c = zero\n  rightAbsorption : \u2200 c, mul c zero = zero\n\nprotected inductive StrongBimonoid.Alphabet \n  | \u00ab\u2295\u00bb\n  | \u00ab\u2297\u00bb\n  | \u00ab\ud835\udfd8\u00bb\n  | \u00ab\ud835\udfd9\u00bb\n\nabbrev StrongBimonoid.ranked : RankedAlphabet where\n  alphabet := StrongBimonoid.Alphabet\n  rank \n    | .\u00ab\u2295\u00bb => 2\n    | .\u00ab\u2297\u00bb => 2\n    | .\u00ab\ud835\udfd8\u00bb => 0\n    | .\u00ab\ud835\udfd9\u00bb => 0\n  isAlphabet := sorry\n\ndef StrongBimonoid.algebra (s : StrongBimonoid) : Algebra StrongBimonoid.ranked where\n  carrier := s.carrier\n  \u03b8 \n    | .\u00ab\u2295\u00bb => s.add\n    | .\u00ab\u2297\u00bb => s.mul\n    | .\u00ab\ud835\udfd8\u00bb => s.zero\n    | .\u00ab\ud835\udfd9\u00bb => s.one\n\nstructure Semiring extends StrongBimonoid where\n  distributive : BinOp.Distrib mul add\n\n\n-----------------------------------------------------------------------------------------\n\n\nnamespace Term\n\ninductive TermSymbol (\u0394 : RankedAlphabet) (H : Type)\n  | alph (sym : \u0394)\n  | var (v : H)\n  | \u00ab(\u00bb\n  | \u00ab)\u00bb\n  | \u00ab,\u00bb\n\n-- Note, we have to use the raw representation of a vector for `v` here.\ninductive _root_.Term (\u0394 : RankedAlphabet) (H : Type)\n  | var (h : H)\n  | app (\u03c3 : \u0394) (v : Fin (\u0394.rank \u03c3) \u2192 (Term \u0394 H))\n\n-- Note, we could replace this with a coercion from H to a set of terms,\n-- but that would have to be a `CoeDep` which isn't reliable enough.\ndef Vars (\u0394 : RankedAlphabet) (H : Type) : Set (Term \u0394 H)\n  | .var .. => True\n  | .app .. => False\n\nprotected def algebra (\u0394 : RankedAlphabet) (H : Type) : Algebra \u0394 where\n  carrier := Term \u0394 H\n  \u03b8 := app\n\n-- Implementation detail of `Term.algebraHom`.\nprivate def algebraHomImpl (target : Algebra \u0394) (f : Vars \u0394 H \u2192 target.carrier) : (Term.algebra \u0394 H).carrier \u2192 target.carrier\n  | .var c => f \u27e8.var c, by simp [Vars]\u27e9\n  | .app \u03c3 cs => target.\u03b8 \u03c3 (algebraHomImpl target f $ cs \u00b7)\n\n-- Implementation detail of `Term.algebra_freelyGenerated`.\nprivate def algebraHom (target : Algebra \u0394) (f : Vars \u0394 H \u2192 target.carrier) : Hom (Term.algebra \u0394 H) target where\n  hom := algebraHomImpl target f\n  property := fun _ _ => by simp [algebraHomImpl, Function.comp]\n\n-- Theorem 2.9.3\ntheorem algebra_freelyGenerated : FreelyGenerated (Term.algebra \u0394 H) (Vars \u0394 H) Set.univ where\n  mem := Set.mem_univ\n  generated := by\n    intro c\n    simp [closure]\n    induction c\n    case var h => \n      apply Closure.root \n      simp [Membership.mem, Vars]\n    case app \u03c3 v hi =>\n      refine Closure.app ?_ hi\n      simp [Term.algebra, Algebra.ops, Membership.mem]\n      exists \u03c3, rfl\n  free := by\n    intro target f _\n    simp [ExistsUnique]\n    exists algebraHom target f\n    constructor\n    case left =>\n      intro \u27e8v, hv\u27e9\n      cases v\n      case app => contradiction\n      case var => simp [algebraHom, algebraHomImpl]\n    case right =>\n      intro hom h\n      apply Hom.ext\n      funext c\n      induction c\n      case a.h.var => simp [algebraHom, algebraHomImpl, h _]\n      case a.h.app \u03c3 cs hi =>\n        simp [algebraHom, algebraHomImpl]\n        have h := hom.property \u03c3 cs\n        simp [Term.algebra] at h\n        simp [h]\n        congr\n        funext i\n        exact hi i\n\nprotected noncomputable def algebra.hom (target : Algebra \u0394) (f : Vars \u0394 H \u2192 target.carrier) : \n  Hom (Term.algebra \u0394 H) target :=\n  (algebra_freelyGenerated).hom target f\n\ntheorem algebra.hom_extends (target) (f : _ \u2192 target.carrier) : \n  \u2200 v : Vars \u0394 H, f v = (Term.algebra.hom target f) v.val :=\n  Term.algebra_freelyGenerated.hom_extends\n\n\n-----------------------------------------------------------------------------------------\n\n\nabbrev posAlgebra (\u0394) : Algebra \u0394 where\n  carrier := Set (Nat\u22c6)\n  \u03b8 \u03c3 cs\n    | .\u03b5 => True\n    | i \u2b1d tl => \u2203 h : i < \u0394.rank \u03c3, tl \u2208 cs \u27e8i, h\u27e9\n        \nnoncomputable def pos {\u0394 H} := \n  Term.algebra.hom (H := H) (posAlgebra \u0394) (fun _ => Set.singleton .\u03b5)\n\ntheorem pos_var : (@pos \u0394 H) (var v) = Set.singleton [] :=\n  Eq.symm <| Term.algebra.hom_extends (posAlgebra \u0394) (fun _ => Set.singleton .\u03b5) \u27e8var v, by simp [Vars]\u27e9\n\ntheorem pos_app (\u03c3 cs) : (@pos \u0394 H) (app \u03c3 cs) = \n  (fun | .\u03b5 => True | i \u2b1d tl => \u2203 h : i < \u0394.rank \u03c3, tl \u2208 pos (cs \u27e8i, h\u27e9)) :=\n  pos.property \u03c3 cs\n\ntheorem pos_zero (h : \u0394.rank \u03c3 = 0) : (@pos \u0394 H) (app \u03c3 $ Vector.\u03b5' h) = Set.singleton .\u03b5 := by\n  simp [pos_app (H := H) \u03c3 (Vector.\u03b5' h)]\n  refine Set.mem_ext ?_\n  intro w\n  constructor\n  all_goals\n    intro h'\n    simp [Set.singleton, Membership.mem] at *\n  case mpr => simp [h']\n  case mp =>\n    split at h'\n    \u00b7 rfl\n    \u00b7 have \u27e8h', _\u27e9 := h'\n      rw [h] at h'\n      contradiction\n\ntheorem mem_pos : (i \u2b1d w) \u2208 (@pos \u0394 H) (app \u03c3 cs) \u2192 \u2203 h : i < \u0394.rank \u03c3, (w \u2208 pos (cs \u27e8i, h\u27e9)) := by\n  intro h\n  rw [pos_app] at h\n  simp [Membership.mem] at h\n  exact h\n\ntheorem \u03b5_mem_pos : .\u03b5 \u2208 (@pos \u0394 H) \u03be := by\n  sorry\n\nstructure TP (\u0394 H) where\n  \u03be : Term \u0394 H\n  w : { w // w \u2208 pos \u03be }\n\ndef label : TP \u0394 H \u2192 (Sum \u0394 H)\n  | \u27e8var v, _\u27e9 => .inr v\n  | \u27e8app \u03c3 _, \u27e8.\u03b5, _\u27e9\u27e9 => .inl \u03c3\n  | \u27e8app \u03c3 \u03bes, \u27e8i \u2b1d w', h\u27e9\u27e9 => label { \n      \u03be := \u03bes \u27e8i, choose $ mem_pos h\u27e9, \n      w := \u27e8w', choose_spec $ mem_pos h\u27e9\n    }\ntermination_by label tp => tp.w.val.length\n\n-- TODO: Figure out if this is ok.\nnotation \u03be \"\u00b0\" w => Term.label (TP.mk \u03be w)\n\ndef subtree : TP \u0394 H \u2192 Term \u0394 H\n  | \u27e8var v, _\u27e9 => var v\n  | \u27e8\u03be, \u27e8.\u03b5, _\u27e9\u27e9 => \u03be\n  | \u27e8app \u03c3 \u03bes, \u27e8i \u2b1d w', h\u27e9\u27e9 => subtree { \n      \u03be := \u03bes \u27e8i, choose $ mem_pos h\u27e9, \n      w := \u27e8w', choose_spec $ mem_pos h\u27e9\n    }\ntermination_by subtree tp => tp.w.val.length\n\n-- TODO: Figure out if this is ok.\nnotation \u03be \"\u2223\" w => Term.subtree (TP.mk \u03be w)\n\ndef replacement (tp : TP \u0394 H) (\u03b6 : Term \u0394 H) : Term \u0394 H :=\n  match tp with\n  | \u27e8var _, _\u27e9 | \u27e8_, \u27e8.\u03b5, _\u27e9\u27e9 => \u03b6\n  | \u27e8app \u03c3 \u03bes, \u27e8i \u2b1d w', h\u27e9\u27e9 => \n    let tp' := { \u03be := \u03bes \u27e8i, choose $ mem_pos h\u27e9, w := \u27e8w', choose_spec $ mem_pos h\u27e9 }\n    app \u03c3 (fun j => if i = j then replacement tp' \u03b6 else \u03bes j)\ntermination_by replacement tp _ => tp.w.val.length\n\n-- TODO: Figure out if this is ok.\nnotation \u03be \"[\" \u03b6 \"]\" w => Term.replacement (TP.mk \u03be w) \u03b6\n\nend Term\n\n\n-----------------------------------------------------------------------------------------\n\n\ndef RankedAlphabet.lift (\u0393 : Type) [Alphabet \u0393] : RankedAlphabet where\n  alphabet := Option \u0393\n  rank\n    | some _ => 1\n    | none => 0\n  isAlphabet := {\n    nonempty := inferInstance\n    finite := sorry\n  }\n\ndef stringAlgebra (\u0393 : Type) [Alphabet \u0393] : Algebra (RankedAlphabet.lift \u0393) where\n  carrier := (\u0393\u22c6)\n  \u03b8 \n    | some \u03c3 => fun ws => \u03c3 \u2b1d (ws \u27e80, by simp [RankedAlphabet.lift]\u27e9)\n    | none => fun _ => .\u03b5\n\ndef tree\u2091 [Alphabet \u0393] : \u0393\u22c6 \u2192 Term (RankedAlphabet.lift \u0393) Empty\n  | .\u03b5 => (Term.algebra (RankedAlphabet.lift \u0393) Empty).\u03b8 none (nomatch \u00b7) \n  | \u03c3\u2b1dw => .app (some \u03c3) (fun _ => tree\u2091 w)\n\ntheorem Term.algebra_iso_stringAlgebra (\u0393 : Type) [Alphabet \u0393] : \n  Isomorphic (stringAlgebra \u0393) (Term.algebra (RankedAlphabet.lift \u0393) Empty) := by\n  unfold Isomorphic\n  apply Nonempty.intro\n  exact {\n    hom := tree\u2091\n    property := by\n      intro \u03c3 w\n      cases \u03c3\n      all_goals simp [Term.algebra, stringAlgebra]\n      case none =>\n        simp [Function.comp, tree\u2091, Term.algebra]\n        funext \u27e8_, h\u27e9\n        simp [RankedAlphabet.lift] at h\n        contradiction\n      case some \u03c3 =>\n        simp [stringAlgebra, RankedAlphabet.lift] at w\n        generalize hw : w \u27e80, stringAlgebra.proof_1\u27e9 = w'\n        simp [tree\u2091, Function.comp]\n        funext \u27e8i, hi\u27e9\n        have hi : i = 0 := by \n          simp [RankedAlphabet.lift] at hi\n          cases i\n          case zero => rfl\n          case succ => contradiction\n        simp [hw, hi]\n    bij := {\n      inj := by \n        simp [Function.Injective]\n        intro \u03c3\u2081 \u03c3\u2082 h\n        induction \u03c3\u2081 generalizing \u03c3\u2082 <;> cases \u03c3\u2082 <;> simp [tree\u2091, Term.algebra] at *\n        case cons.cons \u03c3 _ hi _ _ =>\n          have \u27e8h\u2081, h\u2082\u27e9 := h\n          injection h\u2081 with h\u2081          \n          have : Nonempty (Fin $ (RankedAlphabet.lift \u0393).rank $ some \u03c3) := \n            .intro \u27e80, by simp [RankedAlphabet.lift]\u27e9 \n          simp [h\u2081, hi _ (const_func_ext h\u2082)]\n      surj := by\n        simp [Function.Surjective]\n        intro a\n        induction a\n        case var => contradiction\n        case app \u03c3 w hi =>\n          cases \u03c3 \n          case none => \n            exists .\u03b5\n            simp [tree\u2091, Term.algebra]\n            funext \u27e8_, h\u27e9\n            simp [RankedAlphabet.lift] at h\n            contradiction\n          case some \u03c3 => \n            have \u27e8a, h\u27e9 := hi \u27e80, by simp [RankedAlphabet.lift]\u27e9\n            exists \u03c3\u2b1da\n            simp [h, tree\u2091]\n            refine congrArg _ ?_\n            funext \u27e8i, hi\u27e9\n            have hi : i = 0 := by \n              simp [RankedAlphabet.lift] at hi\n              cases i\n              case zero => rfl\n              case succ => contradiction\n            simp [hi]\n    }\n  }\n\n-----------------------------------------------------------------------------------------\n\n\n-- TODO: Change W to be a Finset.\nstructure TreeDomain (W : Set (Nat\u22c6)) : Prop where\n  nonempty : Set.Nonempty W\n  prefixClosed : Set.bUnion (Set.image Word.prefixes) \u2286 W\n  leftClosed : \u2200 u i, (u\u2022i \u2208 W) \u2192 u\u2022(i - 1) \u2208 W\n\ntheorem TreeDomain.\u03b5_mem : TreeDomain W \u2192 .\u03b5 \u2208 W := by\n  intro h\n  apply h.prefixClosed .\u03b5\n  simp [Set.bUnion, Membership.mem]\n  have \u27e8m, _\u27e9 := Set.Nonempty.iff_exists_mem.mp h.nonempty\n  exists m.prefixes\n  constructor\n  case left => exists m\n  case right => apply Word.\u03b5_mem_prefixes\n\ntheorem Term.pos_treeDomain (\u03be : Term \u0394 H) : TreeDomain (pos \u03be) where\n  nonempty := Set.Nonempty.iff_exists_mem.mpr \u27e8_, Term.\u03b5_mem_pos\u27e9\n  prefixClosed := sorry\n  leftClosed := sorry\n\n-- TEMPORARY\ndef Set.size : Set \u03b1 \u2192 Nat := sorry\n\nstructure TreeMapping {\u0394 : RankedAlphabet} {W : Set (Nat\u22c6)} (t : W \u2192 \u0394) where\n  domain : TreeDomain W\n  rankPreservation : \u2200 w, Set.size (fun j => w.val\u2022j \u2208 W) = \u0394.rank (t w)\n", "meta": {"author": "marcusrossel", "repo": "weighted-tree-automata", "sha": "4991f05d511bb98e2e8c26644c46b3d103ee2510", "save_path": "github-repos/lean/marcusrossel-weighted-tree-automata", "path": "github-repos/lean/marcusrossel-weighted-tree-automata/weighted-tree-automata-4991f05d511bb98e2e8c26644c46b3d103ee2510/WTA.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.38929082259913717}}
{"text": "import data.vector\nimport data.equiv.denumerable\nimport data.list.sort\nimport ring_theory.tensor_product\nimport algebra.category.Module.basic\n\n@[derive has_reflect] inductive type (cT : Type) : Type\n| const : cT \u2192 type\n| arrow : type \u2192 type \u2192 type\n\nvariables {cT : Type}\n\ninductive term2 (ct : type cT \u2192 Type) : \u03a0 (A : type cT), Type\n| const {T : type cT} (t : ct T) : term2 T\n| app {T\u2081 T\u2082 : type cT} (f : term2 (T\u2081.arrow T\u2082)) (x : term2 T\u2081) : term2 T\u2082\n| id {T\u2081 : type cT} : term2 (T\u2081.arrow T\u2081)\n| comp {T\u2081 T\u2082 T\u2083 : type cT} : term2 ((T\u2081.arrow T\u2082).arrow \n  ((T\u2082.arrow T\u2083).arrow (T\u2081.arrow T\u2083)))\n| swap {T\u2081 T\u2082 T\u2083 : type cT} : term2 ((T\u2081.arrow (T\u2082.arrow T\u2083)).arrow \n  (T\u2082.arrow (T\u2081.arrow T\u2083)))\n\n-- def term2.reflect (ct : type cT \u2192 Type) [\u03a0 A, has_reflect (ct A)] [reflected cT] [reflected ct] : \n--   \u03a0 {A : type cT}, term2 ct A \u2192 expr\n-- | _ (term2.const t) := _\n-- | _ (term2.app f x) := _\n\n-- def simplify {ct : type cT \u2192 Type} : \u03a0 {T\u2081 : type cT}, term2 ct T\u2081 \u2192 term2 ct T\u2081\n-- | _ (term2.app term2.comp term2.id) := term2.id\n-- | _ (term2.app term2.id x) := x\n-- | _ (term2.app term2.swap (term2.app term2.swap f)) := f\n-- | _ (term2.app (term2.app (term2.app term2.comp f) g) x) := term2.app g (term2.app f x)\n-- | _ (term2.app (term2.app term2.comp f) term2.id) := f\n-- | _ (term2.app (term2.app term2.swap term2.comp) term2.id) := term2.id\n-- | _ x := x\n\nnamespace term2\n\nvariables {ct : type cT \u2192 Type} {T\u2081 T\u2082 T\u2083 : type cT}\n\n-- def app' : \u03a0 {T\u2081 T\u2082 : type cT} (f : term2 ct (T\u2081.arrow T\u2082)) (x : term2 ct T\u2081), term2 ct T\u2082\n-- | _ _ (const f) x := (const f).app x\n-- | _ _ (app f x) y := (app f x).app y\n-- | _ _ term2.id  x := x\n-- | _ _ (term2.app (term2.app term2.comp f) g) x := app' g (app' f x)\n-- | _ _ f x := f.app x\n-- using_well_founded { dec_tac := `[admit] }\n\nend term2\n\n@[reducible] def context (cT : Type) : Type := list (string \u00d7 type cT)\n\ninductive term (ct : type cT \u2192 Type) : \u03a0 (\u0393 : context cT) (A : type cT), Type\n| const {T : type cT} (t : ct T) : term [] T\n| var (a : string) (T : type cT) : term [(a, T)] T\n| app (\u0393\u2081 \u0393\u2082 : context cT) {T\u2081 T\u2082 : type cT} (f : term \u0393\u2081 (T\u2081.arrow T\u2082)) (x : term \u0393\u2082 T\u2081) : term (\u0393\u2082 ++ \u0393\u2081) T\u2082\n| lambda {\u0393 : context cT} (a : string) (T\u2081 : type cT) {T\u2082 : type cT}\n    (t : term ((a, T\u2081) :: \u0393) T\u2082) : term \u0393 (T\u2081.arrow T\u2082)\n\ndef presheaf (ct : type cT \u2192 Type) : Type 1 := \n\u03a3 (F : type cT \u2192 Type), \u03a0 {A B : type cT}, term2 ct (A.arrow B) \u2192 F A \u2192 F B\n\nnamespace presheaf\n\nvariable {ct : type cT \u2192 Type}\n\ndef hom (F G : presheaf ct) : Type :=\n\u03a0 (A : type cT), F.1 A \u2192 G.1 A\n\ndef hom.comp {F G H : presheaf ct} (f : hom F G) (g : hom G H) : hom F H :=\n\u03bb A, g A \u2218 f A\n\ndef hom.id (F : presheaf ct) : hom F F := \u03bb _, id\n\nvariable (ct)\n\ndef yoneda (A : type cT) : presheaf ct := \n\u27e8\u03bb B, term2 ct (A.arrow B), \u03bb B C f g, (term2.comp.app g).app f\u27e9\n\nvariable {ct}\n\ndef yoneda_map {A B : type cT} (f : term2 ct (A.arrow B)) :\n  hom (yoneda ct B) (yoneda ct A) :=\n\u03bb C, term2.app (term2.app term2.comp f)\n\ndef yoneda_full {A B : type cT} (f : hom (yoneda ct A) (yoneda ct B)) : term2 ct (B.arrow A) :=\nf A term2.id\n\ndef alpha {A B C D X : type cT}\n  (f : term2 ct (A.arrow (B.arrow X)))\n  (g : term2 ct (X.arrow (C.arrow D))) :\n  \u03a3 Y : type cT, \n    term2 ct (B.arrow (C.arrow Y)) \u00d7 term2 ct (A.arrow (Y.arrow D)) :=\nbegin\n  have h : term2 ct (A.arrow (B.arrow (C.arrow D))),\n  { refine (term2.comp.app f).app _,\n    refine (term2.swap.app term2.comp).app g },\n  have h' : term2 ct (B.arrow (C.arrow (A.arrow D))),\n  { refine (term2.comp.app _).app term2.swap,\n    refine term2.swap.app h },\n  use (A.arrow D),\n  use h',\n  use term2.swap.app term2.id\nend\n\ninfix ` => `: 50 := type.arrow\n\ndef tensor (F G : presheaf ct) : presheaf ct :=\n\u27e8\u03bb c : type cT, \u03a3 c\u2081 c\u2082, F.1 c\u2081 \u00d7 G.1 c\u2082 \u00d7 \n    term2 ct (c\u2081.arrow (c\u2082.arrow c)),\n  -- Strictly speaking should be a colimit or something.\n  \u03bb A B f x, \n  -- begin\n  --   rcases x with \u27e8c\u2081, c\u2082, Fc\u2081, Gc\u2082, f\u27e9,\n  --   use [c\u2081, c\u2082, Fc\u2081, Gc\u2082],\n\n\n  -- end\n  \u27e8x.1, x.2.1, x.2.2.1, x.2.2.2.1,\n     term2.app (term2.app term2.comp x.2.2.2.2)\n      (term2.app (term2.app term2.swap term2.comp) f)\u27e9\n      \u27e9\n\ndef tensor_map_left {F\u2081 G F\u2082 : presheaf ct} (f : hom F\u2081 F\u2082)  :\n  hom (tensor F\u2081 G) (tensor F\u2082 G) :=\n\u03bb A a, \u27e8a.1, a.2.1, f _ a.2.2.1, a.2.2.2.1, a.2.2.2.2\u27e9\n\ndef tensor_map_right {F G\u2081 G\u2082 : presheaf ct} (f : hom G\u2081 G\u2082)  :\n  hom (tensor F G\u2081) (tensor F G\u2082) :=\n\u03bb A a, \u27e8a.1, a.2.1, a.2.2.1, f _ a.2.2.2.1, a.2.2.2.2\u27e9\n\ndef tensor_comm {F G : presheaf ct} : hom (tensor F G) (tensor G F) :=\n\u03bb A x, \u27e8x.2.1, x.1, x.2.2.2.1, x.2.2.1, term2.swap.app x.2.2.2.2\u27e9\n\ndef id : presheaf ct :=\n\u27e8\u03bb c, term2 ct c, \u03bb A B, term2.app\u27e9\n\ndef rid\u2081 (F : presheaf ct) : hom (F.tensor id) F :=\nbegin\n  dsimp [tensor, hom, id],\n  intros A a,\n  exact F.2 ((term2.swap.app a.2.2.2.2).app a.2.2.2.1) a.2.2.1\nend\n\ndef rid\u2082 (F : presheaf ct) : hom F (F.tensor id) :=\nbegin\n  dsimp [tensor, hom, id],\n  intros A a,\n  refine \u27e8A, A.arrow A, a, term2.id, term2.swap.app term2.id\u27e9,\nend\n\ndef lid\u2081 (F : presheaf ct) : hom (tensor id F) F :=\nbegin\n  dsimp [tensor, hom, id],\n  intros A a,\n  exact F.2 (a.2.2.2.2.app a.2.2.1) a.2.2.2.1\nend\n\ndef lid\u2082 (F : presheaf ct) : hom F (tensor id F) :=\nbegin\n  dsimp [tensor, hom, id],\n  intros A a,\n  refine \u27e8A.arrow A, A, term2.id, a, term2.id\u27e9,\nend\n\ndef homp (F G : presheaf ct) : presheaf ct :=\n\u27e8\u03bb c, \u03a0 c\u2081 c\u2082 : type cT, term2 ct (c.arrow (c\u2081.arrow c\u2082)) \u2192 F.1 c\u2081 \u2192 G.1 c\u2082,\nbegin\n  intros A B f x c\u2081 c\u2082 g Fc,\n  specialize x c\u2081 c\u2082 ((term2.comp.app f).app g) Fc,\n  exact x\nend\u27e9\n\ndef comp {F G H : presheaf ct} : hom (homp G H) (homp (homp F G) (homp F H)) :=\nbegin\n  intros A x,\n  dsimp [homp] at *,\n  intros c\u2081 c\u2082 f h c\u2083 c\u2084 g Fc\u2083,\n  have := alpha f g,\n  apply x _ _ this.2.2,\n  apply h,\n  apply this.2.1,\n  exact Fc\u2083\nend\n\ndef scomp {F G H : presheaf ct} : hom (homp F G) (homp (homp G H) (homp F H)) :=\nbegin\n  intros A x,\n  dsimp [homp] at *,\n  intros c\u2081 c\u2082 f h c\u2083 c\u2084 g Fc\u2083,\n  have := alpha (term2.swap.app f) g,\n  apply h _ _ this.2.2,\n  apply x,\n  apply this.2.1,\n  exact Fc\u2083\nend\n\ndef tensor_mk {F G : presheaf ct} : hom F (homp G (F.tensor G)) :=\nbegin\n  intros A FA c\u2081 c\u2082 f Gc\u2081,\n  dsimp [homp, tensor],\n  exact \u27e8_, _, FA, Gc\u2081, f\u27e9\nend\n\ndef curry {F G H : presheaf ct} : (F.tensor G).hom H \u2192 F.hom (G.homp H) :=\nbegin\n  intros x A FA c\u2081 c\u2082 f Gc\u2081,\n  dsimp [homp, tensor, hom] at *,\n  apply x,\n  refine \u27e8_, _, _, _, f\u27e9,\n  exact FA,\n  exact Gc\u2081\nend\n\ndef thing {A B C D : presheaf ct} (f : hom A (homp B C)) :\n  hom (homp C D) (homp A (homp B D)) :=  \nbegin\n  intros c\u2081 x c\u2082 c\u2083 h Ac\u2082 c\u2084 c\u2085 i Bc\u2084,\n  dsimp [homp] at x,\n  have := alpha h i,\n  apply x,\n  exact this.2.2,\n  apply f,\n  exact Ac\u2082,\n  exact this.2.1,\n  exact Bc\u2084\nend\n\ndef lcurry {F G H : presheaf ct} : hom (homp (tensor F G) H) (homp F (homp G H)) :=\nbegin\n  apply thing,\n  exact tensor_mk\nend\n\ndef uncurry {F G H : presheaf ct} : F.hom (G.homp H) \u2192 (F.tensor G).hom H :=\nbegin\n  intros x A FGA,\n  dsimp [homp, tensor, hom] at *,\n  apply x _ FGA.2.2.1 _ _ _ FGA.2.2.2.1,\n  exact FGA.2.2.2.2\nend\n\ndef assoc\u2081 {F G H : presheaf ct} : hom ((F.tensor G).tensor H) (F.tensor (G.tensor H)) :=\nuncurry (uncurry (hom.comp tensor_mk lcurry))\n\ndef assoc\u2082 {F G H : presheaf ct} : hom (F.tensor (G.tensor H)) ((F.tensor G).tensor H) :=\nbegin\n  rintros A \u27e8B, C, fb, \u27e8D, E, gd, he, dec\u27e9, bca\u27e9,\n  refine \u27e8E.arrow A, E, \u27e8B, D, fb, gd, _\u27e9, he, term2.id\u27e9,\n  refine (term2.comp.app bca).app _,\n  refine (term2.comp.app _).app (term2.comp.app dec),\n  refine term2.swap.app term2.comp\nend\n\nend presheaf\n\nsection dpresheaf\n\nvariable (ct : type cT \u2192 Type)\n\ndef dpresheaf : Type 2 :=\n\u03a3 F : presheaf ct \u2192 Type 1, \u03a0 {A B : presheaf ct}, A.hom B \u2192 F A \u2192 F B \n\nnamespace dpresheaf\n\nvariable {ct}\n\ndef hom (F G : dpresheaf ct) : Type 1 :=\n\u03a0 (A : presheaf ct), F.1 A \u2192 G.1 A\n\ndef hom.comp {F G H : dpresheaf ct} (f : hom F G) (g : hom G H) : hom F H :=\n\u03bb A, g A \u2218 f A\n\ndef hom.id (F : dpresheaf ct) : hom F F := \u03bb _, id\n\nvariable (ct)\n\ndef yoneda (A : presheaf ct) : dpresheaf ct := \n\u27e8\u03bb B, ulift (A.hom B), \u03bb B C f \u27e8g\u27e9, \u27e8g.comp f\u27e9\u27e9\n\nvariable {ct}\n\ndef yoneda_map {A B : presheaf ct} (f : A.hom B) : hom (yoneda ct B) (yoneda ct A) :=\n\u03bb C g, \u27e8f.comp g.1\u27e9 \n\ndef yoneda_full {A B : presheaf ct} (f : hom (yoneda ct A) (yoneda ct B)) : B.hom A :=\n(f A \u27e8presheaf.hom.id _\u27e9).1\n\ndef tensor (F G : dpresheaf ct) : dpresheaf ct :=\n\u27e8\u03bb c, \u03a3 c\u2081 c\u2082, F.1 c\u2081 \u00d7 G.1 c\u2082 \u00d7 (c\u2081.tensor c\u2082).hom c,\n  -- Strictly speaking should be a colimit or something.\n  \u03bb A B f x, \u27e8x.1, x.2.1, x.2.2.1, x.2.2.2.1, \n     begin\n        refine presheaf.hom.comp x.2.2.2.2 f,\n     end\u27e9\u27e9\n\ndef tensor_map_left {F\u2081 G F\u2082 : dpresheaf ct} (f : hom F\u2081 F\u2082)  :\n  hom (tensor F\u2081 G) (tensor F\u2082 G) :=\n\u03bb A a, \u27e8a.1, a.2.1, f _ a.2.2.1, a.2.2.2.1, a.2.2.2.2\u27e9\n\ndef tensor_map_right {F G\u2081 G\u2082 : dpresheaf ct} (f : hom G\u2081 G\u2082)  :\n  hom (tensor F G\u2081) (tensor F G\u2082) :=\n\u03bb A a, \u27e8a.1, a.2.1, a.2.2.1, f _ a.2.2.2.1, a.2.2.2.2\u27e9\n\ndef tensor_comm {F G : dpresheaf ct} : hom (tensor F G) (tensor G F) :=\nbegin\n  intro A,\n  rintro \u27e8c\u2081, c\u2082, Fc\u2081, Gc\u2082, f\u27e9,\n  exact \u27e8c\u2082, c\u2081, Gc\u2082, Fc\u2081, presheaf.hom.comp presheaf.tensor_comm f\u27e9\nend\n \n\ndef assoc\u2081 {F G H : dpresheaf ct} : hom ((F.tensor G).tensor H) (F.tensor (G.tensor H))  :=\nbegin\n  intro A,\n  rintros \u27e8E, D, \u27e8B, C, fb, gc, bce\u27e9, hd, eda\u27e9,\n  refine \u27e8B, C.tensor D, fb, \u27e8C, D, gc, hd, presheaf.hom.id _\u27e9, _\u27e9, \n  refine presheaf.assoc\u2082.comp _,\n  refine (presheaf.tensor_map_left _).comp eda,\n  exact bce \nend\n\ndef assoc\u2082 {F G H : dpresheaf ct} : hom (F.tensor (G.tensor H)) ((F.tensor G).tensor H) :=\nbegin\n  intro A,\n  rintros \u27e8B, C, fb, \u27e8D, E, gd, he, dec\u27e9, bca\u27e9,\n  refine \u27e8_, _, \u27e8_, _, fb, gd, presheaf.hom.id _\u27e9, he, _\u27e9,\n  refine presheaf.hom.comp _ bca,\n  refine presheaf.hom.comp presheaf.assoc\u2081 _,\n  refine presheaf.tensor_map_right dec\nend\n\ndef id : dpresheaf ct := yoneda _ presheaf.id\n\ndef rid\u2081 (F : dpresheaf ct) : hom (F.tensor id) F :=\nbegin\n  dsimp [hom, tensor],\n  intros A x,\n  apply F.2 _ x.2.2.1,\n  dsimp [id, yoneda] at x,\n  have := x.2.2.2.1.1,\n  have := presheaf.hom.comp  \n    (presheaf.tensor_map_right this) x.2.2.2.2,\n  exact presheaf.hom.comp (presheaf.rid\u2082 _) this,\nend\n\ndef lid\u2081 (F : dpresheaf ct) : hom (id.tensor F) F :=\nbegin\n  dsimp [hom, tensor],\n  intros A x,\n  apply F.2 _ x.2.2.2.1,\n  dsimp [id, yoneda] at x,\n  have := x.2.2.1.1,\n  have := presheaf.hom.comp  \n    (presheaf.tensor_map_left this) x.2.2.2.2,\n  exact presheaf.hom.comp (presheaf.lid\u2082 _) this,\nend\n\ndef rid\u2082 (F : dpresheaf ct) : hom F (F.tensor id)  :=\nbegin\n  dsimp [tensor, hom, id, yoneda],\n  intros A a,\n  use [A, presheaf.id, a, \u27e8presheaf.hom.id _\u27e9],\n  refine presheaf.rid\u2081 _\nend\n\ndef lid\u2082 (F : dpresheaf ct) : hom F (tensor id F) :=\nbegin\n dsimp [tensor, hom, id, yoneda],\n  intros A a,\n  use [presheaf.id, A, \u27e8presheaf.hom.id _\u27e9, a],\n  refine presheaf.lid\u2081 _\nend\n\ndef homp (F G : dpresheaf ct) : dpresheaf ct :=\n\u27e8\u03bb c, hom (tensor (yoneda ct c) F) G, begin\n  intros A B f x,\n  refine hom.comp _ x,\n  refine tensor_map_left _,\n  refine yoneda_map _,\n  exact f\nend\u27e9\n\ndef curry {F G H : dpresheaf ct} : hom (tensor F G) H \u2192 hom F (homp G H) :=\nbegin\n  dsimp [homp, hom] at *,\n  intros x A fA B y,\n  apply x B,\n  dsimp [tensor, yoneda] at *,\n  use [y.1, y.2.1],\n  use F.2 y.2.2.1.1 fA,\n  use y.2.2.2.1,\n  use y.2.2.2.2\nend\n\ndef uncurry {F G H : dpresheaf ct} : hom F (homp G H) \u2192 hom (tensor F G) H :=\nbegin\n  dsimp [tensor, hom, homp, yoneda] at *,\n  intros x A y,\n  refine x _ y.2.2.1 A _,\n  refine \u27e8y.1, y.2.1, _, y.2.2.2.1, y.2.2.2.2\u27e9,\n  constructor,\n  exact presheaf.hom.id _\nend\n\ndef lcurry_aux {A F G H : dpresheaf ct} : (hom A (homp (tensor F G) H)) \u2192 (hom A (homp F (homp G H))) :=\nbegin\n  intro f,\n  refine curry _,\n  refine curry _,\n  refine assoc\u2081.comp _,\n  refine uncurry _,\n  exact f\nend\n\ndef lcurry {F G H : dpresheaf ct} : hom (homp (tensor F G) H) (homp F (homp G H)) :=\nlcurry_aux (hom.id _)\n\ndef luncurry_aux {A F G H : dpresheaf ct} : (hom A (homp F (homp G H))) \u2192 (hom A (homp (tensor F G) H)) :=\nbegin\n  intro f,\n  refine curry _,\n  refine assoc\u2082.comp _,\n  refine uncurry _,\n  refine uncurry _,\n  exact f,\nend\n\ndef luncurry {F G H : dpresheaf ct} : hom (homp F (homp G H)) (homp (tensor F G) H) :=\nluncurry_aux (hom.id _)\n\ndef lcomp {F G H : dpresheaf ct} : hom (homp F G) (homp (homp G H) (homp F H)) :=\nbegin\n  refine curry _,\n  refine curry _,\n  refine (tensor_map_left tensor_comm).comp _,\n  refine assoc\u2081.comp _,\n  refine (tensor_map_right (uncurry (hom.id _))).comp _,\n  refine uncurry _,\n  exact hom.id _\nend\n\ndef homp_map_left {F G H : dpresheaf ct} (f : hom G H) : hom (homp H F) (homp G F) := \nbegin\n  intros A,\n  dsimp [homp],\n  intro g,\n  refine hom.comp (tensor_map_right f) g,\nend\n\ndef homp_map_right {F G H : dpresheaf ct} (f : hom G H) : hom (homp F G) (homp F H) := \nbegin\n  intros A,\n  dsimp [homp],\n  intro g,\n  refine hom.comp g f,\nend\n\n\nvariable (ct)\n\n-- Should not be using dyoneda\ndef dyoneda (T : type cT) : dpresheaf ct := yoneda ct (presheaf.yoneda ct T)\n\ndef eval (T : type cT) : dpresheaf ct := \n\u27e8\u03bb F, ulift (F.1 T), \u03bb A B f x, \u27e8f _ x.1\u27e9\u27e9\n\ndef eval_full (T\u2081 T\u2082 : type cT) (f : hom (eval ct T\u2081) (eval ct T\u2082)) : \n  term2 ct (T\u2081.arrow T\u2082) :=\nbegin\n  dsimp [hom, eval] at f,\n  have := f (presheaf.yoneda ct T\u2081),\n  dsimp [presheaf.yoneda] at this,\n  refine (this _).1,\n  exact \u27e8term2.id\u27e9\nend\n\nvariable {ct}\n\ndef dyoneda_full (T\u2081 T\u2082 : type cT) (f : hom (dyoneda ct T\u2081) (dyoneda ct T\u2082)) : \n  term2 ct (T\u2081.arrow T\u2082) := presheaf.yoneda_full $ yoneda_full f\n\ndef tensor_hom (A B : type cT) : presheaf ct :=\n\u27e8\u03bb C, term2 ct (A.arrow (B.arrow C)), \n  \u03bb C D f g,begin\n    refine (term2.comp.app g).app _,\n    refine (term2.swap.app term2.comp).app f,\n  end\u27e9\n\ndef eval_tensor\u2081 (T\u2081 T\u2082 T\u2083 : type cT) : \n  hom (tensor (eval ct T\u2081) (eval ct T\u2082)) (eval ct T\u2083) \u2192 \n  term2 ct (T\u2081.arrow (T\u2082.arrow T\u2083)) :=\nbegin\n  dsimp [eval, yoneda, hom, tensor] at *,\n  intros x,\n  refine (x (tensor_hom T\u2081 T\u2082) _).1,\n  dsimp [presheaf.hom],\n  use [presheaf.yoneda ct T\u2081, presheaf.yoneda ct T\u2082, \u27e8term2.id\u27e9, \u27e8term2.id\u27e9],\n  dsimp [presheaf.yoneda, presheaf.tensor, tensor_hom],\n  intros A y,\n  refine (term2.comp.app ((term2.comp.app y.2.2.1).app y.2.2.2.2)).app _,\n  refine term2.comp.app y.2.2.2.1\nend\n\ndef eval_tensor\u2082 (T\u2081 T\u2082 T\u2083 : type cT) : \n  term2 ct (T\u2081.arrow (T\u2082.arrow T\u2083)) \u2192 \n  hom (tensor (eval ct T\u2081) (eval ct T\u2082)) (eval ct T\u2083) :=\nbegin\n  dsimp [eval, yoneda, hom, tensor] at *,\n  intros f A y,\n  dsimp [presheaf.tensor, presheaf.hom] at *,\n  split,\n  apply y.2.2.2.2,\n  use [T\u2081, T\u2082, y.2.2.1.1, y.2.2.2.1.1, f]\nend\n\ndef thing (A : presheaf ct) (B : type cT) : presheaf ct :=\n\u27e8\u03bb C, A.1 (B.arrow C), \n  \u03bb C D f g,begin\n    refine A.2 _ g,\n    refine (term2.swap.app term2.comp).app f\n  end\u27e9\n\ndef eval_homp\u2081 (T\u2081 T\u2082 : type cT) : \n  hom (homp (eval ct T\u2081) (eval ct T\u2082)) (eval ct (T\u2081.arrow T\u2082)) :=\nbegin\n  intro A,\n  dsimp [homp, eval, yoneda, tensor, hom],\n  intro x,\n  refine x (thing _ _) _,\n  use A,\n  use (presheaf.yoneda ct T\u2081),\n  use \u27e8presheaf.hom.id _\u27e9,\n  use \u27e8term2.id\u27e9,\n  dsimp [presheaf.tensor, presheaf.yoneda, thing, presheaf.hom],\n  rintros B \u27e8c\u2081, c\u2082, Ac\u2081, g, f\u27e9,\n  have h : term2 ct (c\u2081.arrow (T\u2081.arrow B)),\n  { refine (term2.comp.app f).app _,\n    refine term2.comp.app _,\n    exact g },\n  exact A.2 h Ac\u2081\nend\n\ndef eval_homp\u2082 (T\u2081 T\u2082 : type cT) : \n  hom (eval ct (T\u2081.arrow T\u2082)) (homp (eval ct T\u2081) (eval ct T\u2082)) :=\nbegin\n  refine curry _,\n  refine eval_tensor\u2082 _ _ _ _,\n  exact term2.id\nend\n\n\n\nopen category_theory\n\ndef contexti : context cT \u2192 dpresheaf ct\n| []       := dpresheaf.id\n| (T :: l) := tensor (contexti l) (eval ct T.2)\n\ndef lift {T : dpresheaf ct} : hom T (homp id T) :=\ncurry (hom.comp (rid\u2081 _) (hom.id _))\n\ndef drop {T : dpresheaf ct} : hom (homp id T) T :=\nhom.comp (rid\u2082 _) (uncurry (hom.id _))\n\ndef contexti_append\u2081 : \u03a0 (\u0393\u2081 \u0393\u2082 : context cT), hom \n  (contexti (\u0393\u2082 ++ \u0393\u2081))  ((contexti \u0393\u2081).tensor (@contexti _ ct \u0393\u2082))\n| \u0393\u2081      [] := rid\u2082 _\n| \u0393\u2081 (T::\u0393\u2082) := hom.comp (tensor_map_left (contexti_append\u2081 _ _)) assoc\u2081\n \ndef contexti_append\u2082 : \u03a0 (\u0393\u2081 \u0393\u2082 : context cT), hom \n  ((contexti \u0393\u2081).tensor (contexti \u0393\u2082)) (@contexti _ ct (\u0393\u2082 ++ \u0393\u2081))\n| \u0393\u2081      [] := rid\u2081 _\n| \u0393\u2081 (T::\u0393\u2082) := hom.comp assoc\u2082 (tensor_map_left (contexti_append\u2082 _ _))\n\nopen dpresheaf\n\ndef termi : \u03a0 {\u0393 : context cT} {A : type cT} \n  (t : term ct \u0393 A),\n  hom (@contexti _ ct \u0393) (eval ct A)\n| _ A (term.const t) := \u03bb F x, \u27e8x.1 _ (term2.const t)\u27e9\n| _ _ (term.var _ A) := lid\u2081 _\n| _ T\u2082 (@term.app  _ _ \u0393\u2081 \u0393\u2082 T\u2081 _ f x) := \nhom.comp (contexti_append\u2081 _ _) (uncurry (hom.comp (termi f) \n   ((eval_homp\u2082 _ _).comp $ homp_map_left (termi x))))\n| \u0393 (type.arrow _ T\u2082) (term.lambda a T\u2081 t) := \n  hom.comp (curry (termi t)) (eval_homp\u2081 _ _)\n\nend dpresheaf\n\nend dpresheaf\n\nopen term\n\nopen category_theory\nopen_locale tensor_product\n\nvariable {ct : type cT \u2192 Type}\n\ndef term_to_term2 {A : type cT} (t : term ct [] A) : term2 ct A :=\n(dpresheaf.termi t (presheaf.id) \u27e8presheaf.hom.id _\u27e9).1\n\nvariables {R : Type} [comm_ring R] \n\ndef typei (cTi : cT \u2192 Module.{0 0} R) : type cT \u2192 Module R\n| (type.const T)     := cTi T\n| (type.arrow T\u2081 T\u2082) := Module.of R (typei T\u2081 \u2192\u2097[R] typei T\u2082)\n\n-- inductive term2 (ct : type cT \u2192 Type) : \u03a0 (A : type cT), Type\n-- | const {T : type cT} (t : ct T) : term2 T\n-- | app {T\u2081 T\u2082 : type cT} (f : term2 (T\u2081.arrow T\u2082)) (x : term2 T\u2081) : term2 T\u2082\n-- | id {T\u2081 : type cT} : term2 (T\u2081.arrow T\u2081)\n-- | comp {T\u2081 T\u2082 T\u2083 : type cT} : term2 ((T\u2081.arrow T\u2082).arrow ((T\u2082.arrow T\u2083).arrow (T\u2081.arrow T\u2083)))\n-- | swap {T\u2081 T\u2082 T\u2083 : type cT} : term2 ((T\u2081.arrow (T\u2082.arrow T\u2083)).arrow (T\u2082.arrow (T\u2081.arrow T\u2083)))\n\ndef linear_map.swap {R M N P : Type*} [comm_semiring R] [add_comm_monoid M]\n  [add_comm_monoid N] [add_comm_monoid P] [module R M] [module R N] [module R P]\n  : (M \u2192\u2097[R] N \u2192\u2097[R] P) \u2192\u2097[R] (N \u2192\u2097[R] M \u2192\u2097[R] P) :=\n{ to_fun := \u03bb f,\n  { to_fun := \u03bb n,\n    { to_fun := \u03bb m, f m n,\n      map_add' := \u03bb _ _, by rw [f.map_add]; refl,\n      map_smul' := \u03bb _ _, by rw [f.map_smul]; refl, },\n    map_add' := \u03bb _ _, by simp only [(f _).map_add]; refl,\n    map_smul' := \u03bb _ _, by simp only [(f _).map_smul]; refl },\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl }\n\nlocal attribute [reducible] typei\n\ndef termi (cTi : cT \u2192 Module.{0 0} R) (const_term : \u03a0 {A}, ct A \u2192 typei cTi A) :\n  \u03a0 {A : type cT}, term2 ct A \u2192 typei cTi A\n| T (term2.const x) := const_term x\n| _ (term2.app f x) := begin \n  have := termi f,\n  dsimp [typei] at this,\n  exact this (termi x),\nend\n| _ (term2.id) := linear_map.id\n| _ (term2.comp) := linear_map.swap (linear_map.llcomp _ _ _ _)\n| _ (term2.swap) := linear_map.swap\n\n\ndef const_term : type unit \u2192 Type\n| (type.arrow (type.const ()) (type.arrow (type.const ()) (type.const ()))) := unit\n| _ := empty\n\nvariables (M : Module.{0 0} R) (op : M \u2192\u2097[R] M \u2192\u2097[R] M)\n\ndef cTi : unit \u2192 Module R := \u03bb _, M\n\ninclude op\n\ndef const_termi {T : type unit} (t : const_term T) : typei (@cTi R _ M) T :=\nbegin\n  cases T with _ T\u2081 T\u2082; try { apply empty.elim t },\n  cases T\u2081 with _ T\u2082 T\u2083; try { apply empty.elim t },\n  cases T\u2081,\n  cases T\u2082 with _ T\u2081 T\u2082; try { apply empty.elim t },\n  cases T\u2081 with _ T\u2081 T\u2082; try { apply empty.elim t },\n  cases T\u2081,\n  cases T\u2082 with _ T\u2081 T\u2082; try { apply empty.elim t },\n  exact op\nend\n\nomit op\n\ndef mult : const_term (type.arrow (type.const ()) (type.arrow (type.const ()) (type.const ()))) := ()\n\nnotation `T` := type.const ()\n\ndef exmpl\u2081 : @term unit const_term [] \n  ((type.const ()).arrow ((type.const ()).arrow ((type.const ()).arrow (type.const ())))) :=\nlambda \"a\" T $ \nlambda \"b\" T $\nlambda \"c\" T $\n  app [(\"a\", T), (\"b\", T)].reverse [(\"c\", T)] \n  (app [] [(\"a\", T), (\"b\", T)].reverse (const mult) \n    (app [(\"a\", T)] [(\"b\", T)] \n      (app [] [(\"a\", T)] (const mult) (var \"a\" (type.const ())))\n        (var \"b\" (type.const ()))) : _)\n    (var \"c\" (type.const ()))\n\nexample (p q r : M) :\n  (((termi (@cTi R _ M) (@const_termi _ _ _ op) (term_to_term2 exmpl\u2081)).to_fun p).to_fun q).to_fun r = \n  op (op p q) r :=\nbegin\n  refl,\nend\n\ndef exmpl\u2082 : @term unit const_term [] \n  ((type.const ()).arrow ((type.const ()).arrow ((type.const ()).arrow (type.const ())))) :=\nlambda \"a\" T $ \nlambda \"b\" T $\nlambda \"c\" T $\n  (app [(\"a\", T)] [(\"b\", T), (\"c\", T)].reverse\n  (app [] [(\"a\", T)] (const mult) (var \"a\" (type.const ())) : _) \n    (app [(\"b\", T)] [(\"c\", T)] \n      (app [] [(\"b\", T)] (const mult) (var \"b\" (type.const ()))) \n        (var \"c\" (type.const ())))  : _)\n\n#reduce term_to_term2 exmpl\u2082\n\nexample (p q r : M) : \n  (((termi (@cTi R _ M) (@const_termi _ _ _ op) (term_to_term2 exmpl\u2082)).to_fun p).to_fun q).to_fun r = \n  op p (op q r) :=\nbegin\n  refl,\nend\n\nnotation ` bin_op ` :=  ((type.const ()).arrow ((type.const ()).arrow (type.const ())))\n\ndef exmpl\u2083 : @term unit const_term [] \n  (type.arrow bin_op ((type.const ()).arrow ((type.const ()).arrow ((type.const ()).arrow (type.const ()))))) :=\nlambda \"o\" bin_op $\nlambda \"a\" T $ \nlambda \"b\" T $\nlambda \"c\" T $\n  app [(\"o\", bin_op), (\"a\", T), (\"b\", T)].reverse [(\"c\", T)] \n  (app [(\"o\", bin_op)] [(\"a\", T), (\"b\", T)].reverse (var \"o\" bin_op) \n    (app [(\"a\", T)] [(\"b\", T)] \n      (app [] [(\"a\", T)] (const mult) (var \"a\" (type.const ()))) \n        (var \"b\" (type.const ()))) : _) \n    (var \"c\" (type.const ()))\n\nexample (p q r : M) (op1 : M \u2192\u2097[R] M \u2192\u2097[R] M): \n  ((((termi (@cTi R _ M) (@const_termi _ _ _ op) (term_to_term2 exmpl\u2083)).to_fun op1).to_fun p).to_fun q).to_fun r = \n  op1 (op p q) r :=\nbegin\nrefl,\n\n  -- dsimp [termi, const_termi, cTi, term_to_term2, dpresheaf.termi, exmpl\u2083, dpresheaf.eval, dpresheaf.eval_homp\u2081,\n  --   dpresheaf.eval_homp\u2082, dpresheaf.contexti, dpresheaf.contexti_append\u2082, dpresheaf.contexti_append\u2081,\n  --   dpresheaf.eval_tensor\u2081, dpresheaf.eval_tensor\u2082, presheaf.alpha, typei, context, list.reverse, list.append, list.reverse_core,\n\n  --   dpresheaf.tensor, dpresheaf.tensor_map_left, dpresheaf.yoneda, dpresheaf.id, dpresheaf.thing,\n  --   dpresheaf.tensor_map_right, dpresheaf.tensor_comm, dpresheaf.homp, dpresheaf.rid\u2081, dpresheaf.lid\u2081,\n  --   dpresheaf.lid\u2082, dpresheaf.rid\u2082, dpresheaf.curry, dpresheaf.uncurry, dpresheaf.hom, dpresheaf.hom.id,\n  --   dpresheaf.hom.comp, dpresheaf.homp_map_left, dpresheaf.homp_map_right, dpresheaf.assoc\u2081, dpresheaf.assoc\u2082,\n    \n  --   presheaf.tensor, presheaf.tensor_map_left, presheaf.yoneda, presheaf.id, presheaf.thing,\n  --   presheaf.tensor_map_right, presheaf.tensor_comm, presheaf.homp, presheaf.rid\u2081, presheaf.lid\u2081,\n  --   presheaf.lid\u2082, presheaf.rid\u2082, presheaf.curry, presheaf.uncurry, presheaf.hom, presheaf.hom.id,\n  --   presheaf.hom.comp, presheaf.assoc\u2081, presheaf.assoc\u2082,\n\n  --   linear_map.swap, linear_map.comp, linear_map.llcomp, linear_map.lcomp, linear_map.id],\n  -- simp,\nend", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/linear_lambda/monoidal_lambda3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3892908142606166}}
{"text": "/-\nCopyright (c) 2021 Yuma Mizuno. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno\n\n! This file was ported from Lean 3 source module category_theory.bicategory.basic\n! leanprover-community/mathlib commit 3e32bc908f617039c74c06ea9a897e30c30803c2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Isomorphism\nimport Mathbin.Tactic.Slice\n\n/-!\n# Bicategories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we define typeclass for bicategories.\n\nA bicategory `B` consists of\n* objects `a : B`,\n* 1-morphisms `f : a \u27f6 b` between objects `a b : B`, and\n* 2-morphisms `\u03b7 : f \u27f6 g` beween 1-morphisms `f g : a \u27f6 b` between objects `a b : B`.\n\nWe use `u`, `v`, and `w` as the universe variables for objects, 1-morphisms, and 2-morphisms,\nrespectively.\n\nA typeclass for bicategories extends `category_theory.category_struct` typeclass. This means that\nwe have\n* a composition `f \u226b g : a \u27f6 c` for each 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c`, and\n* a identity `\ud835\udfd9 a : a \u27f6 a` for each object `a : B`.\n\nFor each object `a b : B`, the collection of 1-morphisms `a \u27f6 b` has a category structure. The\n2-morphisms in the bicategory are implemented as the morphisms in this family of categories.\n\nThe composition of 1-morphisms is in fact a object part of a functor\n`(a \u27f6 b) \u2964 (b \u27f6 c) \u2964 (a \u27f6 c)`. The definition of bicategories in this file does not\nrequire this functor directly. Instead, it requires the whiskering functions. For a 1-morphism\n`f : a \u27f6 b` and a 2-morphism `\u03b7 : g \u27f6 h` between 1-morphisms `g h : b \u27f6 c`, there is a\n2-morphism `whisker_left f \u03b7 : f \u226b g \u27f6 f \u226b h`. Similarly, for a 2-morphism `\u03b7 : f \u27f6 g`\nbetween 1-morphisms `f g : a \u27f6 b` and a 1-morphism `f : b \u27f6 c`, there is a 2-morphism\n`whisker_right \u03b7 h : f \u226b h \u27f6 g \u226b h`. These satisfy the exchange law\n`whisker_left f \u03b8 \u226b whisker_right \u03b7 i = whisker_right \u03b7 h \u226b whisker_left g \u03b8`,\nwhich is required as an axiom in the definition here.\n-/\n\n\nnamespace CategoryTheory\n\nuniverse w v u\n\nopen Category Iso\n\n#print CategoryTheory.Bicategory /-\n/- ./././Mathport/Syntax/Translate/Command.lean:401:24: unsupported: (notation) in structure -/\n/- ./././Mathport/Syntax/Translate/Command.lean:401:24: unsupported: (notation) in structure -/\n/- ./././Mathport/Syntax/Translate/Command.lean:401:24: unsupported: (notation) in structure -/\n/- ./././Mathport/Syntax/Translate/Command.lean:401:24: unsupported: (notation) in structure -/\n/- ./././Mathport/Syntax/Translate/Command.lean:401:24: unsupported: (notation) in structure -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr\u03bb_\u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr\u03bb_\u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03c1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03c1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03b1_ -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25c1 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr\u03bb_\u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `\u00abexpr \u25b7 \u00bb -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:207:4: warning: unsupported notation `expr\u03c1_ -/\n-- intended to be used with explicit universe parameters\n/-- In a bicategory, we can compose the 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c` to obtain\na 1-morphism `f \u226b g : a \u27f6 c`. This composition does not need to be strictly associative,\nbut there is a specified associator, `\u03b1_ f g h : (f \u226b g) \u226b h \u2245 f \u226b (g \u226b h)`.\nThere is an identity 1-morphism `\ud835\udfd9 a : a \u27f6 a`, with specified left and right unitor\nisomorphisms `\u03bb_ f : \ud835\udfd9 a \u226b f \u2245 f` and `\u03c1_ f : f \u226b \ud835\udfd9 a \u2245 f`.\nThese associators and unitors satisfy the pentagon and triangle equations.\n\nSee https://ncatlab.org/nlab/show/bicategory.\n-/\n@[nolint check_univs]\nclass Bicategory (B : Type u) extends CategoryStruct.{v} B where\n  -- category structure on the collection of 1-morphisms:\n  homCategory : \u2200 a b : B, Category.{w} (a \u27f6 b) := by infer_instance\n  -- left whiskering:\n  whiskerLeft {a b c : B} (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) : f \u226b g \u27f6 f \u226b h\n  -- right whiskering:\n  whiskerRight {a b c : B} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) : f \u226b h \u27f6 g \u226b h\n  -- associator:\n  associator {a b c d : B} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) : (f \u226b g) \u226b h \u2245 f \u226b g \u226b h\n  -- left unitor:\n  leftUnitor {a b : B} (f : a \u27f6 b) : \ud835\udfd9 a \u226b f \u2245 f\n  -- right unitor:\n  rightUnitor {a b : B} (f : a \u27f6 b) : f \u226b \ud835\udfd9 b \u2245 f\n  -- axioms for left whiskering:\n  whiskerLeft_id : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c), \u00abexpr \u25c1 \u00bb f (\ud835\udfd9 g) = \ud835\udfd9 (f \u226b g) := by obviously\n  whiskerLeft_comp :\n    \u2200 {a b c} (f : a \u27f6 b) {g h i : b \u27f6 c} (\u03b7 : g \u27f6 h) (\u03b8 : h \u27f6 i),\n      \u00abexpr \u25c1 \u00bb f (\u03b7 \u226b \u03b8) = \u00abexpr \u25c1 \u00bb f \u03b7 \u226b \u00abexpr \u25c1 \u00bb f \u03b8 := by\n    obviously\n  id_whiskerLeft :\n    \u2200 {a b} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g),\n      \u00abexpr \u25c1 \u00bb (\ud835\udfd9 a) \u03b7 = ((\u00abexpr\u03bb_\u00bb) f).Hom \u226b \u03b7 \u226b ((\u00abexpr\u03bb_\u00bb) g).inv := by\n    obviously\n  comp_whiskerLeft :\n    \u2200 {a b c d} (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h'),\n      \u00abexpr \u25c1 \u00bb (f \u226b g) \u03b7 =\n        ((expr\u03b1_) f g h).Hom \u226b \u00abexpr \u25c1 \u00bb f (\u00abexpr \u25c1 \u00bb g \u03b7) \u226b ((expr\u03b1_) f g h').inv := by\n    obviously\n  -- axioms for right whiskering:\n  id_whiskerRight : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c), \u00abexpr \u25b7 \u00bb (\ud835\udfd9 f) g = \ud835\udfd9 (f \u226b g) := by obviously\n  comp_whiskerRight :\n    \u2200 {a b c} {f g h : a \u27f6 b} (\u03b7 : f \u27f6 g) (\u03b8 : g \u27f6 h) (i : b \u27f6 c),\n      \u00abexpr \u25b7 \u00bb (\u03b7 \u226b \u03b8) i = \u00abexpr \u25b7 \u00bb \u03b7 i \u226b \u00abexpr \u25b7 \u00bb \u03b8 i := by\n    obviously\n  whiskerRight_id :\n    \u2200 {a b} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g),\n      \u00abexpr \u25b7 \u00bb \u03b7 (\ud835\udfd9 b) = ((expr\u03c1_) f).Hom \u226b \u03b7 \u226b ((expr\u03c1_) g).inv := by\n    obviously\n  whiskerRight_comp :\n    \u2200 {a b c d} {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d),\n      \u00abexpr \u25b7 \u00bb \u03b7 (g \u226b h) =\n        ((expr\u03b1_) f g h).inv \u226b \u00abexpr \u25b7 \u00bb (\u00abexpr \u25b7 \u00bb \u03b7 g) h \u226b ((expr\u03b1_) f' g h).Hom := by\n    obviously\n  -- associativity of whiskerings:\n  whisker_assoc :\n    \u2200 {a b c d} (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d),\n      \u00abexpr \u25b7 \u00bb (\u00abexpr \u25c1 \u00bb f \u03b7) h =\n        ((expr\u03b1_) f g h).Hom \u226b \u00abexpr \u25c1 \u00bb f (\u00abexpr \u25b7 \u00bb \u03b7 h) \u226b ((expr\u03b1_) f g' h).inv := by\n    obviously\n  -- exchange law of left and right whiskerings:\n  whisker_exchange :\n    \u2200 {a b c} {f g : a \u27f6 b} {h i : b \u27f6 c} (\u03b7 : f \u27f6 g) (\u03b8 : h \u27f6 i),\n      \u00abexpr \u25c1 \u00bb f \u03b8 \u226b \u00abexpr \u25b7 \u00bb \u03b7 i = \u00abexpr \u25b7 \u00bb \u03b7 h \u226b \u00abexpr \u25c1 \u00bb g \u03b8 := by\n    obviously\n  -- pentagon identity:\n  pentagon :\n    \u2200 {a b c d e} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e),\n      \u00abexpr \u25b7 \u00bb ((expr\u03b1_) f g h).Hom i \u226b\n          ((expr\u03b1_) f (g \u226b h) i).Hom \u226b \u00abexpr \u25c1 \u00bb f ((expr\u03b1_) g h i).Hom =\n        ((expr\u03b1_) (f \u226b g) h i).Hom \u226b ((expr\u03b1_) f g (h \u226b i)).Hom := by\n    obviously\n  -- triangle identity:\n  triangle :\n    \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c),\n      ((expr\u03b1_) f (\ud835\udfd9 b) g).Hom \u226b \u00abexpr \u25c1 \u00bb f ((\u00abexpr\u03bb_\u00bb) g).Hom = \u00abexpr \u25b7 \u00bb ((expr\u03c1_) f).Hom g := by\n    obviously\n#align category_theory.bicategory CategoryTheory.Bicategory\n-/\n\n-- mathport name: bicategory.whisker_left\n-- The precedence of the whiskerings is higher than that of the composition `\u226b`.\nscoped[Bicategory] infixr:81 \" \u25c1 \" => Bicategory.whiskerLeft\n\n-- mathport name: bicategory.whisker_right\nscoped[Bicategory] infixl:81 \" \u25b7 \" => Bicategory.whiskerRight\n\n-- mathport name: bicategory.associator\nscoped[Bicategory] notation \"\u03b1_\" => Bicategory.associator\n\n-- mathport name: bicategory.left_unitor\nscoped[Bicategory] notation \"\u03bb_\" => Bicategory.leftUnitor\n\n-- mathport name: bicategory.right_unitor\nscoped[Bicategory] notation \"\u03c1_\" => Bicategory.rightUnitor\n\nnamespace Bicategory\n\n/-!\n### Simp-normal form for 2-morphisms\n\nRewriting involving associators and unitors could be very complicated. We try to ease this\ncomplexity by putting carefully chosen simp lemmas that rewrite any 2-morphisms into simp-normal\nform defined below. Rewriting into simp-normal form is also useful when applying (forthcoming)\n`coherence` tactic.\n\nThe simp-normal form of 2-morphisms is defined to be an expression that has the minimal number of\nparentheses. More precisely,\n1. it is a composition of 2-morphisms like `\u03b7\u2081 \u226b \u03b7\u2082 \u226b \u03b7\u2083 \u226b \u03b7\u2084 \u226b \u03b7\u2085` such that each `\u03b7\u1d62` is\n  either a structural 2-morphisms (2-morphisms made up only of identities, associators, unitors)\n  or non-structural 2-morphisms, and\n2. each non-structural 2-morphism in the composition is of the form `f\u2081 \u25c1 f\u2082 \u25c1 f\u2083 \u25c1 \u03b7 \u25b7 f\u2084 \u25b7 f\u2085`,\n  where each `f\u1d62` is a 1-morphism that is not the identity or a composite and `\u03b7` is a\n  non-structural 2-morphisms that is also not the identity or a composite.\n\nNote that `f\u2081 \u25c1 f\u2082 \u25c1 f\u2083 \u25c1 \u03b7 \u25b7 f\u2084 \u25b7 f\u2085` is actually `f\u2081 \u25c1 (f\u2082 \u25c1 (f\u2083 \u25c1 ((\u03b7 \u25b7 f\u2084) \u25b7 f\u2085)))`.\n-/\n\n\nrestate_axiom whisker_left_id'\n\nrestate_axiom whisker_left_comp'\n\nrestate_axiom id_whisker_left'\n\nrestate_axiom comp_whisker_left'\n\nrestate_axiom id_whisker_right'\n\nrestate_axiom comp_whisker_right'\n\nrestate_axiom whisker_right_id'\n\nrestate_axiom whisker_right_comp'\n\nrestate_axiom whisker_assoc'\n\nrestate_axiom whisker_exchange'\n\nrestate_axiom pentagon'\n\nrestate_axiom triangle'\n\nattribute [simp] pentagon triangle\n\nattribute [reassoc.1]\n  whisker_left_comp id_whisker_left comp_whisker_left comp_whisker_right whisker_right_id whisker_right_comp whisker_assoc whisker_exchange pentagon triangle\n\n/-\nThe following simp attributes are put in order to rewrite any 2-morphisms into normal forms. There\nare associators and unitors in the RHS in the several simp lemmas here (e.g. `id_whisker_left`),\nwhich at first glance look more complicated than the LHS, but they will be eventually reduced by the\npentagon or the triangle identities, and more generally, (forthcoming) `coherence` tactic.\n-/\nattribute [simp]\n  whisker_left_id whisker_left_comp id_whisker_left comp_whisker_left id_whisker_right comp_whisker_right whisker_right_id whisker_right_comp whisker_assoc\n\nattribute [instance] hom_category\n\nvariable {B : Type u} [Bicategory.{w, v} B] {a b c d e : B}\n\n#print CategoryTheory.Bicategory.hom_inv_whiskerLeft /-\n@[simp, reassoc.1]\ntheorem hom_inv_whiskerLeft (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n    f \u25c1 \u03b7.Hom \u226b f \u25c1 \u03b7.inv = \ud835\udfd9 (f \u226b g) := by rw [\u2190 whisker_left_comp, hom_inv_id, whisker_left_id]\n#align category_theory.bicategory.hom_inv_whisker_left CategoryTheory.Bicategory.hom_inv_whiskerLeft\n-/\n\n#print CategoryTheory.Bicategory.hom_inv_whiskerRight /-\n@[simp, reassoc.1]\ntheorem hom_inv_whiskerRight {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n    \u03b7.Hom \u25b7 h \u226b \u03b7.inv \u25b7 h = \ud835\udfd9 (f \u226b h) := by rw [\u2190 comp_whisker_right, hom_inv_id, id_whisker_right]\n#align category_theory.bicategory.hom_inv_whisker_right CategoryTheory.Bicategory.hom_inv_whiskerRight\n-/\n\n#print CategoryTheory.Bicategory.inv_hom_whiskerLeft /-\n@[simp, reassoc.1]\ntheorem inv_hom_whiskerLeft (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n    f \u25c1 \u03b7.inv \u226b f \u25c1 \u03b7.Hom = \ud835\udfd9 (f \u226b h) := by rw [\u2190 whisker_left_comp, inv_hom_id, whisker_left_id]\n#align category_theory.bicategory.inv_hom_whisker_left CategoryTheory.Bicategory.inv_hom_whiskerLeft\n-/\n\n#print CategoryTheory.Bicategory.inv_hom_whiskerRight /-\n@[simp, reassoc.1]\ntheorem inv_hom_whiskerRight {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n    \u03b7.inv \u25b7 h \u226b \u03b7.Hom \u25b7 h = \ud835\udfd9 (g \u226b h) := by rw [\u2190 comp_whisker_right, inv_hom_id, id_whisker_right]\n#align category_theory.bicategory.inv_hom_whisker_right CategoryTheory.Bicategory.inv_hom_whiskerRight\n-/\n\n#print CategoryTheory.Bicategory.whiskerLeftIso /-\n/-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whiskerLeftIso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) : f \u226b g \u2245 f \u226b h\n    where\n  Hom := f \u25c1 \u03b7.Hom\n  inv := f \u25c1 \u03b7.inv\n#align category_theory.bicategory.whisker_left_iso CategoryTheory.Bicategory.whiskerLeftIso\n-/\n\n#print CategoryTheory.Bicategory.whiskerLeft_isIso /-\ninstance whiskerLeft_isIso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [IsIso \u03b7] : IsIso (f \u25c1 \u03b7) :=\n  IsIso.of_iso (whiskerLeftIso f (asIso \u03b7))\n#align category_theory.bicategory.whisker_left_is_iso CategoryTheory.Bicategory.whiskerLeft_isIso\n-/\n\n#print CategoryTheory.Bicategory.inv_whiskerLeft /-\n@[simp]\ntheorem inv_whiskerLeft (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [IsIso \u03b7] : inv (f \u25c1 \u03b7) = f \u25c1 inv \u03b7 :=\n  by\n  ext\n  simp only [\u2190 whisker_left_comp, whisker_left_id, is_iso.hom_inv_id]\n#align category_theory.bicategory.inv_whisker_left CategoryTheory.Bicategory.inv_whiskerLeft\n-/\n\n#print CategoryTheory.Bicategory.whiskerRightIso /-\n/-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whiskerRightIso {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) : f \u226b h \u2245 g \u226b h\n    where\n  Hom := \u03b7.Hom \u25b7 h\n  inv := \u03b7.inv \u25b7 h\n#align category_theory.bicategory.whisker_right_iso CategoryTheory.Bicategory.whiskerRightIso\n-/\n\n#print CategoryTheory.Bicategory.whiskerRight_isIso /-\ninstance whiskerRight_isIso {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [IsIso \u03b7] : IsIso (\u03b7 \u25b7 h) :=\n  IsIso.of_iso (whiskerRightIso (asIso \u03b7) h)\n#align category_theory.bicategory.whisker_right_is_iso CategoryTheory.Bicategory.whiskerRight_isIso\n-/\n\n#print CategoryTheory.Bicategory.inv_whiskerRight /-\n@[simp]\ntheorem inv_whiskerRight {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [IsIso \u03b7] :\n    inv (\u03b7 \u25b7 h) = inv \u03b7 \u25b7 h := by\n  ext\n  simp only [\u2190 comp_whisker_right, id_whisker_right, is_iso.hom_inv_id]\n#align category_theory.bicategory.inv_whisker_right CategoryTheory.Bicategory.inv_whiskerRight\n-/\n\n#print CategoryTheory.Bicategory.pentagon_inv /-\n@[simp, reassoc.1]\ntheorem pentagon_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i =\n      (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.pentagon_inv CategoryTheory.Bicategory.pentagon_inv\n-/\n\n#print CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv /-\n@[simp, reassoc.1]\ntheorem pentagon_inv_inv_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).Hom =\n      f \u25c1 (\u03b1_ g h i).Hom \u226b (\u03b1_ f g (h \u226b i)).inv :=\n  by\n  rw [\u2190 cancel_epi (f \u25c1 (\u03b1_ g h i).inv), \u2190 cancel_mono (\u03b1_ (f \u226b g) h i).inv]\n  simp\n#align category_theory.bicategory.pentagon_inv_inv_hom_hom_inv CategoryTheory.Bicategory.pentagon_inv_inv_hom_hom_inv\n-/\n\n#print CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv /-\n@[simp, reassoc.1]\ntheorem pentagon_inv_hom_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).Hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).Hom =\n      (\u03b1_ f g (h \u226b i)).Hom \u226b f \u25c1 (\u03b1_ g h i).inv :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.pentagon_inv_hom_hom_hom_inv CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_inv\n-/\n\n#print CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv /-\n@[simp, reassoc.1]\ntheorem pentagon_hom_inv_inv_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    f \u25c1 (\u03b1_ g h i).Hom \u226b (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv =\n      (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i :=\n  by simp [\u2190 cancel_epi (f \u25c1 (\u03b1_ g h i).inv)]\n#align category_theory.bicategory.pentagon_hom_inv_inv_inv_inv CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_inv\n-/\n\n#print CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom /-\n@[simp, reassoc.1]\ntheorem pentagon_hom_hom_inv_hom_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ (f \u226b g) h i).Hom \u226b (\u03b1_ f g (h \u226b i)).Hom \u226b f \u25c1 (\u03b1_ g h i).inv =\n      (\u03b1_ f g h).Hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).Hom :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.pentagon_hom_hom_inv_hom_hom CategoryTheory.Bicategory.pentagon_hom_hom_inv_hom_hom\n-/\n\n#print CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom /-\n@[simp, reassoc.1]\ntheorem pentagon_hom_inv_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ f g (h \u226b i)).Hom \u226b f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv =\n      (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).Hom \u25b7 i :=\n  by\n  rw [\u2190 cancel_epi (\u03b1_ f g (h \u226b i)).inv, \u2190 cancel_mono ((\u03b1_ f g h).inv \u25b7 i)]\n  simp\n#align category_theory.bicategory.pentagon_hom_inv_inv_inv_hom CategoryTheory.Bicategory.pentagon_hom_inv_inv_inv_hom\n-/\n\n#print CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom /-\n@[simp, reassoc.1]\ntheorem pentagon_hom_hom_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ f (g \u226b h) i).Hom \u226b f \u25c1 (\u03b1_ g h i).Hom \u226b (\u03b1_ f g (h \u226b i)).inv =\n      (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).Hom :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.pentagon_hom_hom_inv_inv_hom CategoryTheory.Bicategory.pentagon_hom_hom_inv_inv_hom\n-/\n\n#print CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom /-\n@[simp, reassoc.1]\ntheorem pentagon_inv_hom_hom_hom_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).Hom \u226b (\u03b1_ f g (h \u226b i)).Hom =\n      (\u03b1_ f (g \u226b h) i).Hom \u226b f \u25c1 (\u03b1_ g h i).Hom :=\n  by simp [\u2190 cancel_epi ((\u03b1_ f g h).Hom \u25b7 i)]\n#align category_theory.bicategory.pentagon_inv_hom_hom_hom_hom CategoryTheory.Bicategory.pentagon_inv_hom_hom_hom_hom\n-/\n\n#print CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv /-\n@[simp, reassoc.1]\ntheorem pentagon_inv_inv_hom_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n    (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).Hom \u25b7 i =\n      f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.pentagon_inv_inv_hom_inv_inv CategoryTheory.Bicategory.pentagon_inv_inv_hom_inv_inv\n-/\n\n#print CategoryTheory.Bicategory.triangle_assoc_comp_left /-\ntheorem triangle_assoc_comp_left (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03b1_ f (\ud835\udfd9 b) g).Hom \u226b f \u25c1 (\u03bb_ g).Hom = (\u03c1_ f).Hom \u25b7 g :=\n  triangle f g\n#align category_theory.bicategory.triangle_assoc_comp_left CategoryTheory.Bicategory.triangle_assoc_comp_left\n-/\n\n#print CategoryTheory.Bicategory.triangle_assoc_comp_right /-\n@[simp, reassoc.1]\ntheorem triangle_assoc_comp_right (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03b1_ f (\ud835\udfd9 b) g).inv \u226b (\u03c1_ f).Hom \u25b7 g = f \u25c1 (\u03bb_ g).Hom := by rw [\u2190 triangle, inv_hom_id_assoc]\n#align category_theory.bicategory.triangle_assoc_comp_right CategoryTheory.Bicategory.triangle_assoc_comp_right\n-/\n\n#print CategoryTheory.Bicategory.triangle_assoc_comp_right_inv /-\n@[simp, reassoc.1]\ntheorem triangle_assoc_comp_right_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03c1_ f).inv \u25b7 g \u226b (\u03b1_ f (\ud835\udfd9 b) g).Hom = f \u25c1 (\u03bb_ g).inv := by simp [\u2190 cancel_mono (f \u25c1 (\u03bb_ g).Hom)]\n#align category_theory.bicategory.triangle_assoc_comp_right_inv CategoryTheory.Bicategory.triangle_assoc_comp_right_inv\n-/\n\n#print CategoryTheory.Bicategory.triangle_assoc_comp_left_inv /-\n@[simp, reassoc.1]\ntheorem triangle_assoc_comp_left_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n    f \u25c1 (\u03bb_ g).inv \u226b (\u03b1_ f (\ud835\udfd9 b) g).inv = (\u03c1_ f).inv \u25b7 g := by simp [\u2190 cancel_mono ((\u03c1_ f).Hom \u25b7 g)]\n#align category_theory.bicategory.triangle_assoc_comp_left_inv CategoryTheory.Bicategory.triangle_assoc_comp_left_inv\n-/\n\n#print CategoryTheory.Bicategory.associator_naturality_left /-\n@[reassoc.1]\ntheorem associator_naturality_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n    \u03b7 \u25b7 g \u25b7 h \u226b (\u03b1_ f' g h).Hom = (\u03b1_ f g h).Hom \u226b \u03b7 \u25b7 (g \u226b h) := by simp\n#align category_theory.bicategory.associator_naturality_left CategoryTheory.Bicategory.associator_naturality_left\n-/\n\n#print CategoryTheory.Bicategory.associator_inv_naturality_left /-\n@[reassoc.1]\ntheorem associator_inv_naturality_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n    \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv = (\u03b1_ f g h).inv \u226b \u03b7 \u25b7 g \u25b7 h := by simp\n#align category_theory.bicategory.associator_inv_naturality_left CategoryTheory.Bicategory.associator_inv_naturality_left\n-/\n\n#print CategoryTheory.Bicategory.whiskerRight_comp_symm /-\n@[reassoc.1]\ntheorem whiskerRight_comp_symm {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n    \u03b7 \u25b7 g \u25b7 h = (\u03b1_ f g h).Hom \u226b \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv := by simp\n#align category_theory.bicategory.whisker_right_comp_symm CategoryTheory.Bicategory.whiskerRight_comp_symm\n-/\n\n#print CategoryTheory.Bicategory.associator_naturality_middle /-\n@[reassoc.1]\ntheorem associator_naturality_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n    (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).Hom = (\u03b1_ f g h).Hom \u226b f \u25c1 \u03b7 \u25b7 h := by simp\n#align category_theory.bicategory.associator_naturality_middle CategoryTheory.Bicategory.associator_naturality_middle\n-/\n\n#print CategoryTheory.Bicategory.associator_inv_naturality_middle /-\n@[reassoc.1]\ntheorem associator_inv_naturality_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n    f \u25c1 \u03b7 \u25b7 h \u226b (\u03b1_ f g' h).inv = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h := by simp\n#align category_theory.bicategory.associator_inv_naturality_middle CategoryTheory.Bicategory.associator_inv_naturality_middle\n-/\n\n#print CategoryTheory.Bicategory.whisker_assoc_symm /-\n@[reassoc.1]\ntheorem whisker_assoc_symm (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n    f \u25c1 \u03b7 \u25b7 h = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).Hom := by simp\n#align category_theory.bicategory.whisker_assoc_symm CategoryTheory.Bicategory.whisker_assoc_symm\n-/\n\n#print CategoryTheory.Bicategory.associator_naturality_right /-\n@[reassoc.1]\ntheorem associator_naturality_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n    (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').Hom = (\u03b1_ f g h).Hom \u226b f \u25c1 g \u25c1 \u03b7 := by simp\n#align category_theory.bicategory.associator_naturality_right CategoryTheory.Bicategory.associator_naturality_right\n-/\n\n#print CategoryTheory.Bicategory.associator_inv_naturality_right /-\n@[reassoc.1]\ntheorem associator_inv_naturality_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n    f \u25c1 g \u25c1 \u03b7 \u226b (\u03b1_ f g h').inv = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7 := by simp\n#align category_theory.bicategory.associator_inv_naturality_right CategoryTheory.Bicategory.associator_inv_naturality_right\n-/\n\n#print CategoryTheory.Bicategory.comp_whiskerLeft_symm /-\n@[reassoc.1]\ntheorem comp_whiskerLeft_symm (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n    f \u25c1 g \u25c1 \u03b7 = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').Hom := by simp\n#align category_theory.bicategory.comp_whisker_left_symm CategoryTheory.Bicategory.comp_whiskerLeft_symm\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_naturality /-\n@[reassoc.1]\ntheorem leftUnitor_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) : \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).Hom = (\u03bb_ f).Hom \u226b \u03b7 :=\n  by simp\n#align category_theory.bicategory.left_unitor_naturality CategoryTheory.Bicategory.leftUnitor_naturality\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_inv_naturality /-\n@[reassoc.1]\ntheorem leftUnitor_inv_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n    \u03b7 \u226b (\u03bb_ g).inv = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7 := by simp\n#align category_theory.bicategory.left_unitor_inv_naturality CategoryTheory.Bicategory.leftUnitor_inv_naturality\n-/\n\n#print CategoryTheory.Bicategory.id_whiskerLeft_symm /-\ntheorem id_whiskerLeft_symm {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) : \u03b7 = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).Hom := by\n  simp\n#align category_theory.bicategory.id_whisker_left_symm CategoryTheory.Bicategory.id_whiskerLeft_symm\n-/\n\n#print CategoryTheory.Bicategory.rightUnitor_naturality /-\n@[reassoc.1]\ntheorem rightUnitor_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) : \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).Hom = (\u03c1_ f).Hom \u226b \u03b7 :=\n  by simp\n#align category_theory.bicategory.right_unitor_naturality CategoryTheory.Bicategory.rightUnitor_naturality\n-/\n\n#print CategoryTheory.Bicategory.rightUnitor_inv_naturality /-\n@[reassoc.1]\ntheorem rightUnitor_inv_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n    \u03b7 \u226b (\u03c1_ g).inv = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b := by simp\n#align category_theory.bicategory.right_unitor_inv_naturality CategoryTheory.Bicategory.rightUnitor_inv_naturality\n-/\n\n#print CategoryTheory.Bicategory.whiskerRight_id_symm /-\ntheorem whiskerRight_id_symm {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) : \u03b7 = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).Hom := by\n  simp\n#align category_theory.bicategory.whisker_right_id_symm CategoryTheory.Bicategory.whiskerRight_id_symm\n-/\n\n#print CategoryTheory.Bicategory.whiskerLeft_iff /-\ntheorem whiskerLeft_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) : \ud835\udfd9 a \u25c1 \u03b7 = \ud835\udfd9 a \u25c1 \u03b8 \u2194 \u03b7 = \u03b8 := by simp\n#align category_theory.bicategory.whisker_left_iff CategoryTheory.Bicategory.whiskerLeft_iff\n-/\n\n#print CategoryTheory.Bicategory.whiskerRight_iff /-\ntheorem whiskerRight_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) : \u03b7 \u25b7 \ud835\udfd9 b = \u03b8 \u25b7 \ud835\udfd9 b \u2194 \u03b7 = \u03b8 := by simp\n#align category_theory.bicategory.whisker_right_iff CategoryTheory.Bicategory.whiskerRight_iff\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_whiskerRight /-\n/-- We state it as a simp lemma, which is regarded as an involved version of\n`id_whisker_right f g : \ud835\udfd9 f \u25b7 g = \ud835\udfd9 (f \u226b g)`.\n-/\n@[reassoc.1, simp]\ntheorem leftUnitor_whiskerRight (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03bb_ f).Hom \u25b7 g = (\u03b1_ (\ud835\udfd9 a) f g).Hom \u226b (\u03bb_ (f \u226b g)).Hom := by\n  rw [\u2190 whisker_left_iff, whisker_left_comp, \u2190 cancel_epi (\u03b1_ _ _ _).Hom, \u2190\n      cancel_epi ((\u03b1_ _ _ _).Hom \u25b7 _), pentagon_assoc, triangle, \u2190 associator_naturality_middle, \u2190\n      comp_whisker_right_assoc, triangle, associator_naturality_left] <;>\n    infer_instance\n#align category_theory.bicategory.left_unitor_whisker_right CategoryTheory.Bicategory.leftUnitor_whiskerRight\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_inv_whiskerRight /-\n@[reassoc.1, simp]\ntheorem leftUnitor_inv_whiskerRight (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03bb_ f).inv \u25b7 g = (\u03bb_ (f \u226b g)).inv \u226b (\u03b1_ (\ud835\udfd9 a) f g).inv :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.left_unitor_inv_whisker_right CategoryTheory.Bicategory.leftUnitor_inv_whiskerRight\n-/\n\n#print CategoryTheory.Bicategory.whiskerLeft_rightUnitor /-\n@[reassoc.1, simp]\ntheorem whiskerLeft_rightUnitor (f : a \u27f6 b) (g : b \u27f6 c) :\n    f \u25c1 (\u03c1_ g).Hom = (\u03b1_ f g (\ud835\udfd9 c)).inv \u226b (\u03c1_ (f \u226b g)).Hom := by\n  rw [\u2190 whisker_right_iff, comp_whisker_right, \u2190 cancel_epi (\u03b1_ _ _ _).inv, \u2190\n      cancel_epi (f \u25c1 (\u03b1_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right, \u2190\n      associator_inv_naturality_middle, \u2190 whisker_left_comp_assoc, triangle_assoc_comp_right,\n      associator_inv_naturality_right] <;>\n    infer_instance\n#align category_theory.bicategory.whisker_left_right_unitor CategoryTheory.Bicategory.whiskerLeft_rightUnitor\n-/\n\n#print CategoryTheory.Bicategory.whiskerLeft_rightUnitor_inv /-\n@[reassoc.1, simp]\ntheorem whiskerLeft_rightUnitor_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n    f \u25c1 (\u03c1_ g).inv = (\u03c1_ (f \u226b g)).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).Hom :=\n  eq_of_inv_eq_inv (by simp)\n#align category_theory.bicategory.whisker_left_right_unitor_inv CategoryTheory.Bicategory.whiskerLeft_rightUnitor_inv\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_comp /-\n/-\nIt is not so obvious whether `left_unitor_whisker_right` or `left_unitor_comp` should be a simp\nlemma. Our choice is the former. One reason is that the latter yields the following loop:\n[id_whisker_left]   : \ud835\udfd9 a \u25c1 (\u03c1_ f).hom ==> (\u03bb_ (f \u226b \ud835\udfd9 b)).hom \u226b (\u03c1_ f).hom \u226b (\u03bb_ f).inv\n[left_unitor_comp]  : (\u03bb_ (f \u226b \ud835\udfd9 b)).hom ==> (\u03b1_ (\ud835\udfd9 a) f (\ud835\udfd9 b)).inv \u226b (\u03bb_ f).hom \u25b7 \ud835\udfd9 b\n[whisker_right_id]  : (\u03bb_ f).hom \u25b7 \ud835\udfd9 b ==> (\u03c1_ (\ud835\udfd9 a \u226b f)).hom \u226b (\u03bb_ f).hom \u226b (\u03c1_ f).inv\n[right_unitor_comp] : (\u03c1_ (\ud835\udfd9 a \u226b f)).hom ==> (\u03b1_ (\ud835\udfd9 a) f (\ud835\udfd9 b)).hom \u226b \ud835\udfd9 a \u25c1 (\u03c1_ f).hom\n-/\n@[reassoc.1]\ntheorem leftUnitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03bb_ (f \u226b g)).Hom = (\u03b1_ (\ud835\udfd9 a) f g).inv \u226b (\u03bb_ f).Hom \u25b7 g := by simp\n#align category_theory.bicategory.left_unitor_comp CategoryTheory.Bicategory.leftUnitor_comp\n-/\n\n#print CategoryTheory.Bicategory.leftUnitor_comp_inv /-\n@[reassoc.1]\ntheorem leftUnitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03bb_ (f \u226b g)).inv = (\u03bb_ f).inv \u25b7 g \u226b (\u03b1_ (\ud835\udfd9 a) f g).Hom := by simp\n#align category_theory.bicategory.left_unitor_comp_inv CategoryTheory.Bicategory.leftUnitor_comp_inv\n-/\n\n#print CategoryTheory.Bicategory.rightUnitor_comp /-\n@[reassoc.1]\ntheorem rightUnitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03c1_ (f \u226b g)).Hom = (\u03b1_ f g (\ud835\udfd9 c)).Hom \u226b f \u25c1 (\u03c1_ g).Hom := by simp\n#align category_theory.bicategory.right_unitor_comp CategoryTheory.Bicategory.rightUnitor_comp\n-/\n\n#print CategoryTheory.Bicategory.rightUnitor_comp_inv /-\n@[reassoc.1]\ntheorem rightUnitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n    (\u03c1_ (f \u226b g)).inv = f \u25c1 (\u03c1_ g).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).inv := by simp\n#align category_theory.bicategory.right_unitor_comp_inv CategoryTheory.Bicategory.rightUnitor_comp_inv\n-/\n\n#print CategoryTheory.Bicategory.unitors_equal /-\n@[simp]\ntheorem unitors_equal : (\u03bb_ (\ud835\udfd9 a)).Hom = (\u03c1_ (\ud835\udfd9 a)).Hom := by\n  rw [\u2190 whisker_left_iff, \u2190 cancel_epi (\u03b1_ _ _ _).Hom, \u2190 cancel_mono (\u03c1_ _).Hom, triangle, \u2190\n      right_unitor_comp, right_unitor_naturality] <;>\n    infer_instance\n#align category_theory.bicategory.unitors_equal CategoryTheory.Bicategory.unitors_equal\n-/\n\n#print CategoryTheory.Bicategory.unitors_inv_equal /-\n@[simp]\ntheorem unitors_inv_equal : (\u03bb_ (\ud835\udfd9 a)).inv = (\u03c1_ (\ud835\udfd9 a)).inv := by simp [iso.inv_eq_inv]\n#align category_theory.bicategory.unitors_inv_equal CategoryTheory.Bicategory.unitors_inv_equal\n-/\n\nend Bicategory\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Bicategory/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3892908142606166}}
{"text": "/-\nCopyright (c) 2020 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n-/\nimport category_theory.natural_isomorphism\nimport category_theory.equivalence\nimport category_theory.eq_to_hom\n\n/-!\n# Quotient category\n\nConstructs the quotient of a category by an arbitrary family of relations on its hom-sets,\nby introducing a type synonym for the objects, and identifying homs as necessary.\n\nThis is analogous to 'the quotient of a group by the normal closure of a subset', rather\nthan 'the quotient of a group by a normal subgroup'. When taking the quotient by a congruence\nrelation, `functor_map_eq_iff` says that no unnecessary identifications have been made.\n-/\n\n/-- A `hom_rel` on `C` consists of a relation on every hom-set. -/\n@[derive inhabited]\ndef hom_rel (C) [quiver C] := \u03a0 \u2983X Y : C\u2984, (X \u27f6 Y) \u2192 (X \u27f6 Y) \u2192 Prop\n\nnamespace category_theory\n\nvariables {C : Type*} [category C] (r : hom_rel C)\n\ninclude r\n\n/-- A `hom_rel` is a congruence when it's an equivalence on every hom-set, and it can be composed\nfrom left and right. -/\nclass congruence : Prop :=\n(is_equiv : \u2200 {X Y}, is_equiv _ (@r X Y))\n(comp_left : \u2200 {X Y Z} (f : X \u27f6 Y) {g g' : Y \u27f6 Z}, r g g' \u2192 r (f \u226b g) (f \u226b g'))\n(comp_right : \u2200 {X Y Z} {f f' : X \u27f6 Y} (g : Y \u27f6 Z), r f f' \u2192 r (f \u226b g) (f' \u226b g))\n\nattribute [instance] congruence.is_equiv\n\n/-- A type synonym for `C`, thought of as the objects of the quotient category. -/\n@[ext]\nstructure quotient := (as : C)\n\ninstance [inhabited C] : inhabited (quotient r) := \u27e8 { as := default } \u27e9\n\nnamespace quotient\n\n/-- Generates the closure of a family of relations w.r.t. composition from left and right. -/\ninductive comp_closure \u2983s t : C\u2984 : (s \u27f6 t) \u2192 (s \u27f6 t) \u2192 Prop\n| intro {a b} (f : s \u27f6 a) (m\u2081 m\u2082 : a \u27f6 b) (g : b \u27f6 t) (h : r m\u2081 m\u2082) :\n  comp_closure (f \u226b m\u2081 \u226b g) (f \u226b m\u2082 \u226b g)\n\nlemma comp_closure.of {a b} (m\u2081 m\u2082 : a \u27f6 b) (h : r m\u2081 m\u2082) : comp_closure r m\u2081 m\u2082 :=\nby simpa using comp_closure.intro (\ud835\udfd9 _) m\u2081 m\u2082 (\ud835\udfd9 _) h\n\nlemma comp_left {a b c : C} (f : a \u27f6 b) : \u03a0 (g\u2081 g\u2082 : b \u27f6 c) (h : comp_closure r g\u2081 g\u2082),\n  comp_closure r (f \u226b g\u2081) (f \u226b g\u2082)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro (f \u226b x) m\u2081 m\u2082 y h\n\nlemma comp_right {a b c : C} (g : b \u27f6 c) : \u03a0 (f\u2081 f\u2082 : a \u27f6 b) (h : comp_closure r f\u2081 f\u2082),\n  comp_closure r (f\u2081 \u226b g) (f\u2082 \u226b g)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro x m\u2081 m\u2082 (y \u226b g) h\n\n/-- Hom-sets of the quotient category. -/\ndef hom (s t : quotient r) := quot $ @comp_closure C _ r s.as t.as\n\ninstance (a : quotient r) : inhabited (hom r a a) := \u27e8quot.mk _ (\ud835\udfd9 a.as)\u27e9\n\n/-- Composition in the quotient category. -/\ndef comp \u2983a b c : quotient r\u2984 : hom r a b \u2192 hom r b c \u2192 hom r a c :=\n\u03bb hf hg, quot.lift_on hf ( \u03bb f, quot.lift_on hg (\u03bb g, quot.mk _ (f \u226b g))\n  (\u03bb g\u2081 g\u2082 h, quot.sound $ comp_left r f g\u2081 g\u2082 h) )\n  (\u03bb f\u2081 f\u2082 h, quot.induction_on hg $ \u03bb g, quot.sound $ comp_right r g f\u2081 f\u2082 h)\n\n@[simp]\nlemma comp_mk {a b c : quotient r} (f : a.as \u27f6 b.as) (g : b.as \u27f6 c.as) :\n  comp r (quot.mk _ f) (quot.mk _ g) = quot.mk _ (f \u226b g) := rfl\n\ninstance category : category (quotient r) :=\n{ hom := hom r,\n  id := \u03bb a, quot.mk _ (\ud835\udfd9 a.as),\n  comp := comp r }\n\n/-- The functor from a category to its quotient. -/\n@[simps]\ndef functor : C \u2964 quotient r :=\n{ obj := \u03bb a, { as := a },\n  map := \u03bb _ _ f, quot.mk _ f }\n\nnoncomputable instance : full (functor r) :=\n{ preimage := \u03bb X Y f, quot.out f, }\n\ninstance : ess_surj (functor r) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y.as, \u27e8eq_to_iso (by { ext, refl, })\u27e9\u27e9 }\n\nprotected lemma induction {P : \u03a0 {a b : quotient r}, (a \u27f6 b) \u2192 Prop}\n  (h : \u2200 {x y : C} (f : x \u27f6 y), P ((functor r).map f)) :\n  \u2200 {a b : quotient r} (f : a \u27f6 b), P f :=\nby { rintros \u27e8x\u27e9 \u27e8y\u27e9 \u27e8f\u27e9, exact h f, }\n\nprotected lemma sound {a b : C} {f\u2081 f\u2082 : a \u27f6 b} (h : r f\u2081 f\u2082) :\n  (functor r).map f\u2081 = (functor r).map f\u2082 :=\nby simpa using quot.sound (comp_closure.intro (\ud835\udfd9 a) f\u2081 f\u2082 (\ud835\udfd9 b) h)\n\n\n\nvariables {D : Type*} [category D]\n  (F : C \u2964 D)\n  (H : \u2200 (x y : C) (f\u2081 f\u2082 : x \u27f6 y), r f\u2081 f\u2082 \u2192 F.map f\u2081 = F.map f\u2082)\ninclude H\n\n/-- The induced functor on the quotient category. -/\n@[simps]\ndef lift : quotient r \u2964 D :=\n{ obj := \u03bb a, F.obj a.as,\n  map := \u03bb a b hf, quot.lift_on hf (\u03bb f, F.map f)\n    (by { rintros _ _ \u27e8_, _, _, _, _, _, h\u27e9, simp [H _ _ _ _ h], }),\n  map_id' := \u03bb a, F.map_id a.as,\n  map_comp' := by { rintros a b c \u27e8f\u27e9 \u27e8g\u27e9, exact F.map_comp f g, } }\n\n/-- The original functor factors through the induced functor. -/\ndef lift.is_lift : (functor r) \u22d9 lift r F H \u2245 F :=\nnat_iso.of_components (\u03bb X, iso.refl _) (by tidy)\n\n@[simp]\nlemma lift.is_lift_hom (X : C) : (lift.is_lift r F H).hom.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n@[simp]\nlemma lift.is_lift_inv (X : C) : (lift.is_lift r F H).inv.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n\nlemma lift_map_functor_map {X Y : C} (f : X \u27f6 Y) :\n  (lift r F H).map ((functor r).map f) = F.map f :=\nby { rw \u2190(nat_iso.naturality_1 (lift.is_lift r F H)), dsimp, simp, }\n\nend quotient\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/quotient.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.3892908059220961}}
{"text": "open Function\n\ndef Set (\u03b1 : Type u) := \u03b1 \u2192 Prop\n\nexample {\u03b1 : Type u}\n  (f : \u03b1 \u2192 Type (max u v))\n  (U : \u03b1) (hU : f U = Set (Sigma f)) :\n    let g : Set (Sigma f) \u2192 Sigma f := fun (s : Set (Sigma f)) => \u27e8U, cast hU.symm s\u27e9\n    \u2200 \u2983s t : Set (Sigma f)\u2984,\n        g s = g t \u2192 cast hU (g s).snd = cast hU (g t).snd :=\nby\n  intros g s t h\n  congr -- reduces to `(g s).snd = (g t).snd`, not `g s = g t`\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1787.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.38929080592209603}}
{"text": "import buchberger rtc wf_mv_polynomial\n\nopen mv_polynomial\nopen finsupp\n\nnamespace buch\nvariables {\u03c3 : Type*} {\u03b1 : Type*} [decidable_eq \u03c3] [decidable_eq \u03b1] [fintype \u03c3] [decidable_linear_order (\u03c3 \u2192\u2080 \u2115)] \nvariables [discrete_field \u03b1] [is_well_founded (\u03c3 \u2192\u2080 \u2115) (<)] [is_monomial_order (\u03c3 \u2192\u2080 \u2115) (\u2264)] \n\nsection reduction\nvariables (s : finset (mv_polynomial \u03c3 \u03b1)) {S : finset (mv_polynomial \u03c3 \u03b1)}\n\ninductive red_one_step : mv_polynomial \u03c3 \u03b1 \u2192 mv_polynomial \u03c3 \u03b1 \u2192 Prop\n| cons : \u2200 {p r : mv_polynomial \u03c3 \u03b1}, p \u2260 0 \u2192 red_one_step p.TL r.TL \u2192 p.LT = r.LT \u2192 red_one_step p r\n| red_LM : \u2200 {p r : mv_polynomial \u03c3 \u03b1}, p \u2260 0 \u2192 \n    (\u2203 (q : mv_polynomial \u03c3 \u03b1) (h\u2081 : q \u2208 s) (h\u2082 : q \u2260 0)  (h\u2083 : q.LM \u2223 p.LM), reduction p q = r) \u2192 red_one_step p r\n\ndef reducible (p : mv_polynomial \u03c3 \u03b1) :=\n    \u2203 (q : mv_polynomial \u03c3 \u03b1) (hq\u2081 : q \u2208 s) (hq\u2082 : q \u2260 0) (p\u209c : \u03c3 \u2192\u2080 \u2115) (hp : p\u209c \u2208 p.support), q.LM \u2223 p\u209c\ndef is_red_rel (p r : mv_polynomial \u03c3 \u03b1) :=\n    \u2203 (q : mv_polynomial \u03c3 \u03b1) (hq\u2081 : q \u2208 s) (hq\u2082 : q \u2260 0) (p\u209c : \u03c3 \u2192\u2080 \u2115) (hp : p\u209c \u2208 p.support) (hqp\u209c : q.LM \u2223 p\u209c), \n        r = p - q * monomial (p\u209c - q.LM) (p p\u209c / q.LC)\n\nnotation a `\u2192[` S `]` b := red_one_step S a b\nnotation a `\u219b[` S `]` b := \u00ac red_one_step S a b\n\ndef red_plus : mv_polynomial \u03c3 \u03b1 \u2192 mv_polynomial \u03c3 \u03b1 \u2192 Prop := rtc (red_one_step s)\nnotation a `\u2192[` S `]+` b := red_plus S a b\nnotation a `\u219b[` S `]+` b := \u00ac red_plus S a b\n\ndef irreducible (p : mv_polynomial \u03c3 \u03b1) := \u2200 (q : mv_polynomial \u03c3 \u03b1) (hq\u2081 : q \u2208 s) (hq\u2082 : q \u2260 0) (p\u209c \u2208 p.support), q.LM \u2224 p\u209c\n\nnotation a `\u2192[` S `]*` b := red_plus S a b \u2227 irreducible S b\n\ninstance irreducible_decidable (p : mv_polynomial \u03c3 \u03b1) : decidable (irreducible s p) := finset.decidable_dforall_finset\ninstance reducible_decidable (p : mv_polynomial \u03c3 \u03b1) : decidable (reducible s p) := finset.decidable_dexists_finset\n\n@[simp] lemma irred_not_red {p : mv_polynomial \u03c3 \u03b1} : irreducible s p \u2194 \u00ac reducible s p := by simp [irreducible, reducible]\nlemma red_not_irred {p : mv_polynomial \u03c3 \u03b1} : \u00ac irreducible s p \u2194 reducible s p := by simp [irred_not_red]\n\nlemma zero_irreducible : irreducible S 0 := by simp [irreducible]\nlemma zero_not_reducible : \u00ac reducible S 0 := by rw \u2190irred_not_red; exact zero_irreducible\nlemma zero_not_is_red_rel : \u2200 {r}, \u00ac is_red_rel S 0 r := by simp [is_red_rel]\n\nlemma red_one_step_is_red_rel {p r : mv_polynomial \u03c3 \u03b1} : (p \u2192[S] r) \u2194 is_red_rel S p r :=\n\u27e8\u03bb h, begin\n    induction h with p r hp hpr\u2081 hpr\u2082 ih p r hp h,\n    {\n        rcases ih with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n        rw [\u2190@add_left_cancel_iff _ _ p.LT, \u2190add_sub_assoc, LM_TL_eq p, hpr\u2082, LM_TL_eq r, TL_apply_mem hp\u209c] at h,\n        refine \u27e8q, hq\u2081, hq\u2082, p\u209c, TL_support_subset hp\u209c, hqp\u209c, h\u27e9,\n    },\n    {\n        rcases h with \u27e8q, hq\u2081, hq\u2082, hpq, h\u27e9,\n        refine \u27e8q, hq\u2081, hq\u2082, p.LM, LM_mem_support hp, hpq, by rw \u2190h; refl\u27e9,\n    }\nend, begin \n    revert r,\n    apply induction p,\n    {intros r hr, apply absurd hr zero_not_is_red_rel},\n    {\n        intros p ih r hr,\n        rcases hr with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n        by_cases hp\u209cp : p\u209c = p.LM,\n        {\n            apply red_one_step.red_LM (nez_of_mem_support hp\u209c), \n            refine \u27e8q, hq\u2081, hq\u2082, by rwa \u2190hp\u209cp, by rw [h, hp\u209cp]; refl,\u27e9,\n        },\n        {\n            have hp_LM : p.LM > (-(q * monomial (p\u209c - LM q) (p p\u209c / LC q))).LM,\n            {\n                simp [LM_of_mul_m, hq\u2082, div_ne_zero (mem_support_iff.1 hp\u209c) (LC_nez_iff.1 hq\u2082), add_sub_cancel' hqp\u209c],\n                apply lt_of_le_of_ne (LM_rel' hp\u209c) hp\u209cp\n            },\n            have hr : p.LT = r.LT := by simp [LT, h, LM_of_add_left hp_LM, LC_of_add_left hp_LM],\n            \n            apply red_one_step.cons (nez_of_mem_support hp\u209c) (ih \u27e8q, hq\u2081, hq\u2082, p\u209c, mem_TL_support hp\u209cp hp\u209c, hqp\u209c, _\u27e9) hr,\n            {\n                rw [\u2190@add_left_cancel_iff _ _ p.LT, \u2190add_sub_assoc, LM_TL_eq p, hr, LM_TL_eq r, TL_apply p\u209c],\n                simpa [ne.symm hp\u209cp] using h,\n            },\n        }\n    }\nend\u27e9\n\nlemma is_red_rel_reducible {p r : mv_polynomial \u03c3 \u03b1} (h : is_red_rel S p r) : reducible S p :=\nbegin\n    rcases h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, _\u27e9,\n    refine \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c\u27e9,\nend\n\nlemma reducible_is_red_rel {p : mv_polynomial \u03c3 \u03b1} (h : reducible S p) : \u2203 r, is_red_rel S p r :=\nbegin\n    rcases h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c\u27e9,\n    refine \u27e8p - q * monomial (p\u209c - q.LM) (p p\u209c / q.LC), q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, rfl\u27e9,\nend\n\nlemma zero_red {r : mv_polynomial \u03c3 \u03b1} : (0 : mv_polynomial \u03c3 \u03b1) \u219b[s] r  :=\n\u03bb h, zero_not_reducible (is_red_rel_reducible (by rwa red_one_step_is_red_rel at h))\n\nlemma red_mem_S {q : mv_polynomial \u03c3 \u03b1} (hq\u2081 : q \u2208 S) (hq\u2082 : q \u2260 0) : q \u2192[S] 0 :=\nbegin\n    rw red_one_step_is_red_rel,\n    refine \u27e8q, hq\u2081, hq\u2082, q.LM, LM_mem_support hq\u2082, by simp, _\u27e9,\n    have h\u2081 : q q.LM = q.LC := rfl,\n    have h\u2082 : (monomial 0 1 : mv_polynomial \u03c3 \u03b1) = C 1 := by simp [C],\n    simp [h\u2081, div_self (LC_nez_iff.1 hq\u2082), h\u2082],\nend\n\ntheorem red_plus_mul {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) : \n    \u2200 a b, (monomial a b * p) \u2192[S]+ (monomial a b * r) :=\nbegin\n    intros a b,\n    by_cases hb : b = 0,\n    {simp [hb], exact rtc.refl'},\n    {\n        revert a b,\n        induction h with p p r h p q r hpq hqr ih\u2081 ih\u2082;\n        intros a b hb,\n        {exact rtc.refl'},\n        {\n            apply rtc.base',\n            rw red_one_step_is_red_rel at h \u22a2,\n            rcases h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp, hpq, h\u27e9,\n            refine \u27e8q, hq\u2081, hq\u2082, a + p\u209c, mul_mem_mul_support hb _ hp, dvd_of_dvd_of_add' hpq _, \n                by rw [mul_apply hp, finsupp.add_sub_assoc' hpq, mul_div_assoc, \u2190monomial_mul_monomial, \n                mul_left_comm, \u2190mul_sub, h]\u27e9,\n        },\n        {exact rtc.trans' (ih\u2081 a b hb) (ih\u2082 a b hb)}\n    }\nend\n\nlemma red_plus_zero_of_mul {p : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ 0) : \n    \u2200 a b, (monomial a b * p) \u2192[S]+ 0 := by simpa using red_plus_mul h\n\nlemma red_star_zero_of_mul {p : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]* 0) : \n    \u2200 a b, (monomial a b * p) \u2192[S]* 0 := \u03bb a b, \u27e8red_plus_zero_of_mul h.left a b, zero_irreducible\u27e9\n\nlemma lt_of_red_one_step {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S] r) : r < p :=\nbegin\n    induction h with p r hp h\u2081 h\u2082 ih p r hp h, \n    {apply lt.LM_eq hp h\u2082.symm ih},\n    {\n        rcases h with \u27e8q, h\u2081, h\u2082, h\u2083, h\u2084\u27e9,\n        rw \u2190h\u2084,\n        by_cases hp' : p.LM = 0,\n        {\n            simp [hp'] at h\u2083,\n            rw reduction_of_LM_eqz hp' h\u2083 h\u2082,\n            apply lt.zero hp,\n        },\n        {apply lt.LM_lt (reduction_LM_lt h\u2083 hp' h\u2082)}\n    }\nend\n\nlemma le_of_red_plus {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) : r < p \u2228 r = p :=\nbegin\n    induction h with _ _ _ h p q r hpq hqr ih\u2081 ih\u2082,\n    {simp},\n    {left, apply lt_of_red_one_step h},\n    {\n        cases ih\u2081; cases ih\u2082,\n        {left, exact trans ih\u2082 ih\u2081},\n        {left, rwa ih\u2082},\n        {left, rwa \u2190ih\u2081},\n        {right, rwa ih\u2082},\n    }\nend\n\nlemma red_lt_of_le_of_lt {p q r : mv_polynomial \u03c3 \u03b1} (hpq : p \u2192[S] q) (hqr : q \u2192[S]+ r) : r < p :=\nbegin\n    cases le_of_red_plus hqr,\n    {exact trans h (lt_of_red_one_step hpq)},\n    {simpa [h] using lt_of_red_one_step hpq}\nend\n\nlemma red_lt_of_le_of_lt' {p q r : mv_polynomial \u03c3 \u03b1} (hpq : p \u2192[S] q) (hqr : q \u2192[S]* r) : r < p :=\nred_lt_of_le_of_lt hpq hqr.left\n\nlemma red_one_step_LM_ge {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S] r) : p.LM \u2265 r.LM := LM_le_of_lt (lt_of_red_one_step h)\nlemma red_plus_LM_ge {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) : p.LM \u2265 r.LM :=\nbegin\n    induction h with _ _ _ h _ _ _ _ _ ih\u2081 ih\u2082,\n    {exact le_refl _},\n    {exact red_one_step_LM_ge h},\n    {exact ge_trans ih\u2081 ih\u2082},\nend\n\nlemma red_plus_irreducible {p q : mv_polynomial \u03c3 \u03b1} (hp : irreducible S p) (h : p \u2192[S]+ q) : q = p :=\nbegin\n    induction h with _ p q h p q r hpq hpr ih\u2081 ih\u2082, \n    {refl},\n    {apply absurd (is_red_rel_reducible (by rwa red_one_step_is_red_rel at h)) (by simpa using hp)},\n    {simpa [ih\u2081 hp, hp] using ih\u2082},\nend\n\ntheorem red_star_irreducible {p q : mv_polynomial \u03c3 \u03b1} (hp : irreducible S p) (h : p \u2192[S]* q) : q = p :=\nred_plus_irreducible hp h.left\n\nlemma red_const {p r : mv_polynomial \u03c3 \u03b1} (hp : p.LM = 0) (h : p \u2192[S] r) : r = 0 :=\nbegin\n    cases h with _ _ _ h _ _ _ hp' h,\n    {apply absurd (by rwa TL_eqz_of_LM_eqz hp at h) (zero_red _)},\n    {\n        rcases h with \u27e8q, h\u2081, h\u2082, h\u2083, h\u2084\u27e9,\n        simp [hp] at h\u2083, \n        rw \u2190h\u2084,\n        exact reduction_of_LM_eqz hp h\u2083 h\u2082,\n    }\nend\n\nlemma red_plus_const {p r : mv_polynomial \u03c3 \u03b1} (hp : p.LM = 0) (h : p \u2192[S]+ r) : p = r \u2228 r = 0 :=\nbegin\n    induction h with _ p r h p q r hpq hqr ih\u2081 ih\u2082,\n    {simp},\n    {simp [red_const hp h]},\n    {\n        cases ih\u2081 hp,\n        {\n            rw \u2190h at ih\u2082, \n            exact ih\u2082 hp,\n        },\n        {\n            right,\n            conv at ih\u2082 in (_ \u2228 _) {rw eq_comm},\n            simpa [h] using ih\u2082, \n        }\n    }\nend\n\nlemma red_star_const {p r : mv_polynomial \u03c3 \u03b1} (hp : p.LM = 0) (h : p \u2192[S]* r) : p = r \u2228 r = 0 :=\nred_plus_const hp h.left\n\nlemma red_plus_trans {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) : p \u2260 r \u2192 \u2203 q, (p \u2192[S] q) \u2227 (q \u2192[S]+ r) :=\nbegin\n    induction h with _ p\u2082 r\u2082 h p\u2083 q\u2083 r\u2083 h\u2081 h\u2082 ih\u2081 ih\u2082, \n    {simp},\n    {intro, use [r\u2082, h, rtc.refl']},\n    {\n        by_cases p\u2083 = q\u2083,\n        {rwa h},\n        {\n            intro,\n            rcases ih\u2081 h with \u27e8t, ht\u2081, ht\u2082\u27e9,\n            use [t, ht\u2081, rtc.trans' ht\u2082 h\u2082],\n        } \n    }\nend\n\nlemma red_star_trans {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]* r) : reducible S p \u2192 \u2203 q, (p \u2192[S] q) \u2227 (q \u2192[S]* r) := \n\u03bb hp, by simpa [h.right] using red_plus_trans h.left (\u03bb hpr, ((irred_not_red S).1 h.right) (by rwa hpr at hp))\n\nlemma exists_red_star (p : mv_polynomial \u03c3 \u03b1) : \u2203 r, p \u2192[s]* r :=\n@well_founded.recursion (mv_polynomial \u03c3 \u03b1) (<) lt_wf _ p \n(begin\n    intros p ih,\n    by_cases hp : irreducible s p,\n    {use [p, rtc.refl', hp]},\n    {\n        cases reducible_is_red_rel (by rwa \u2190red_not_irred) with q hq,\n        rw \u2190red_one_step_is_red_rel at hq,\n        rcases ih q (lt_of_red_one_step hq) with \u27e8r, hr\u2081, hr\u2082\u27e9,\n        refine \u27e8r, rtc.base_trans hq hr\u2081, hr\u2082\u27e9,\n    }\nend)\n\nlemma exists_red_star_of_red {p q : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S] q) : \n\u2203 r, (q \u2192[S]* r) \u2227 (p \u2192[S]* r) :=\nbegin\n    rcases exists_red_star S q with \u27e8r, hr\u2081, hr\u2082\u27e9,\n    refine \u27e8r, \u27e8hr\u2081, hr\u2082\u27e9, \u27e8rtc.base_trans h hr\u2081, hr\u2082\u27e9\u27e9,\nend\n\nlemma exists_red_star_of_red_plus {p q : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ q) : \n\u2203 r, (q \u2192[S]* r) \u2227 (p \u2192[S]* r) :=\nbegin\n    induction h with p p q h p\u2083 q\u2083 r\u2083 hpq hqr ih\u2081 ih\u2082,\n    {\n        cases exists_red_star S p with r h,\n        use [r, h, h],\n    },\n    {exact exists_red_star_of_red h},\n    {\n        rcases ih\u2082 with \u27e8r, hr\u2081, \u27e8hr\u2082, hr\u2083\u27e9\u27e9,\n        use [r, hr\u2081, \u27e8rtc.trans' hpq hr\u2082, hr\u2083\u27e9],\n    }\nend\n\nlemma red_m_not_mem {p q r : mv_polynomial \u03c3 \u03b1} (hq : q \u2260 0) {p\u209c} (hp\u209c : p\u209c \u2208 p.support) \n(hqp\u209c : q.LM \u2223 p\u209c) (h : r = p - q * monomial (p\u209c - q.LM) (p p\u209c / q.LC)) : p\u209c \u2209 r.support :=\nbegin\n    have h_LC : (q * monomial (p\u209c - LM q) (p p\u209c / LC q)) p\u209c = (q * monomial (p\u209c - LM q) (p p\u209c / LC q)).LC,\n        have h\u2081 : p p\u209c / q q.LM \u2260 0 := div_ne_zero (by simpa using hp\u209c) (LC_nez_iff.1 hq),\n        simp [LC, LM_of_mul_m, hq, h\u2081, finsupp.add_sub_cancel' hqp\u209c], refl,\n    simp [h, h_LC, LC_of_mul_m, mul_div_cancel' _ (LC_nez_iff.1 hq)],\nend\n\nlemma sub_red_comp {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S] r) :\n\u2200 u, \u2203 t\u2081 t\u2082, (p + u \u2192[S]+ t\u2081) \u2227 (u \u2192[S]+ t\u2082) \u2227 (r = t\u2081 - t\u2082) :=\n\u03bb u, begin\n    rcases red_one_step_is_red_rel.1 h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n    have hr := red_m_not_mem hq\u2082 hp\u209c hqp\u209c h,\n    by_cases hp\u209cu : p\u209c \u2208 u.support;\n    simp at hp\u209cu hr hp\u209c,\n    {\n        by_cases hp\u209c': p\u209c \u2208 (p + u).support,\n        {\n            refine \u27e8p + u - q * monomial (p\u209c - LM q) ((p + u) p\u209c / LC q), \n                u - q * monomial (p\u209c - LM q) (u p\u209c / LC q), \n                rtc.base' (red_one_step_is_red_rel.2 \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c', hqp\u209c, rfl\u27e9),\n                rtc.base' (red_one_step_is_red_rel.2 \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa using hp\u209cu, hqp\u209c, rfl\u27e9), _\u27e9,\n            {\n                simp [-mul_neg_eq_neg_mul_symm, neg_mul_eq_mul_neg],\n                rw [\u2190mul_add, \u2190monomial_neg, \u2190neg_div, \u2190monomial_add_monomial, \n                div_add_div_same, neg_add_rev, \u2190add_assoc],                \n                simp [neg_div, h],\n            }\n        },\n        {\n            simp [add_eq_zero_iff_neg_eq] at hp\u209c',\n            refine \u27e8p + u, u - q * monomial (p\u209c - LM q) (u p\u209c / LC q), rtc.refl', \n                rtc.base' (red_one_step_is_red_rel.2 \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa using hp\u209cu, hqp\u209c, rfl\u27e9),\n                by simp [hp\u209c'.symm, neg_div, h]\u27e9,\n        }\n    },\n    {\n        refine \u27e8p + u - q * monomial (p\u209c - q.LM) ((p + u) p\u209c/ q.LC), u, \n            rtc.base' (red_one_step_is_red_rel.2 \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa [hp\u209cu] using hp\u209c, hqp\u209c, rfl\u27e9),\n            rtc.refl', by simp [hp\u209cu, h]\u27e9,\n    }\nend\n\nlemma sub_red_plus_comp {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) :\n\u2200 u, \u2203 t, (p + u \u2192[S]+ r + t) \u2227 (u \u2192[S]+ t) :=\nbegin\n    induction h with r p r h p q r hpq hqr ih\u2081 ih\u2082;\n    intros u,\n    {refine \u27e8u, rtc.refl', rtc.refl'\u27e9},\n    {\n        rcases sub_red_comp h u with \u27e8t\u2081, t\u2082, h\u2081, h\u2082, h\u2083\u27e9, \n        refine \u27e8t\u2082, by rwa eq_sub_iff_add_eq.1 h\u2083, h\u2082\u27e9,\n    },\n    {\n        rcases ih\u2081 u with \u27e8t\u2081, hput\u2081, hut\u2081\u27e9,\n        rcases ih\u2082 t\u2081 with \u27e8t\u2082, hqut\u2082, hut\u2082\u27e9,\n        refine \u27e8t\u2082, rtc.trans' hput\u2081 hqut\u2082, rtc.trans' hut\u2081 hut\u2082\u27e9,\n    }\nend\n\nlemma sub_red_star_zero {p q : mv_polynomial \u03c3 \u03b1} (h : p - q \u2192[S]* 0) :\n\u2203 t, (p \u2192[S]* t) \u2227 (q \u2192[S]* t) :=\nbegin\n    rcases sub_red_plus_comp h.left q with \u27e8r, hpr, hqr\u27e9,\n    rcases exists_red_star S r with \u27e8t, ht\u2081, ht\u2082\u27e9,\n    refine \u27e8t, \u27e8rtc.trans' (by simpa using hpr) ht\u2081, ht\u2082\u27e9, \u27e8rtc.trans' hqr ht\u2081, ht\u2082\u27e9\u27e9,\nend\n\nlemma red_add_of_red {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S] r) : \n\u2200 u, \u2203 t, (p + u \u2192[S]+ t) \u2227 (r + u \u2192[S]+ t) :=\n\u03bb u, begin\n    rcases red_one_step_is_red_rel.1 h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n    have hr := red_m_not_mem hq\u2082 hp\u209c hqp\u209c h,\n    by_cases hp\u209cu : p\u209c \u2208 u.support;\n    simp at hp\u209cu hr,\n    {\n        by_cases hp\u209c': p\u209c \u2208 (p + u).support,\n        {\n            refine \u27e8r + u - q * monomial (p\u209c - LM q) (u p\u209c / LC q), _, _\u27e9;\n            apply rtc.base'; rw [red_one_step_is_red_rel],\n            {\n                conv at h {\n                    rw [\u2190@add_right_cancel_iff _ _ (u - q * monomial (p\u209c - q.LM) (u p\u209c / q.LC)), \u2190add_sub_assoc],\n                    to_rhs, rw [\u2190add_sub_assoc, sub_add_eq_add_sub, sub_sub, \u2190mul_add, \u2190monomial_add_monomial, div_add_div_same, \u2190add_apply],\n                },\n                refine \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c', hqp\u209c, by rw h\u27e9,\n            },\n            {refine \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa [hr] using hp\u209cu, hqp\u209c, by conv in ((r + u) p\u209c) {simp [hr]},\u27e9}\n        },\n        {\n            refine \u27e8p + u, rtc.refl', _\u27e9,\n            apply rtc.base',\n            rw red_one_step_is_red_rel,\n            refine \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa [hr] using hp\u209cu, hqp\u209c, _\u27e9,\n            simp [add_eq_zero_iff_eq_neg] at hp\u209c',\n            simp [h, hp\u209c', neg_div, hr],\n        }\n    },\n    {\n        refine \u27e8r + u, _, rtc.refl'\u27e9,\n        apply rtc.base',\n        rw [red_one_step_is_red_rel, h, sub_add_eq_add_sub],\n        refine \u27e8q, hq\u2081, hq\u2082, p\u209c, by simpa [hp\u209cu] using hp\u209c, hqp\u209c, by simp [hp\u209cu]\u27e9,\n    }\nend\n\nlemma red_of_TL_red {p r : mv_polynomial \u03c3 \u03b1} (h : p.TL \u2192[S] r) :\np \u2192[S] p.LT + r :=\nbegin\n    by_cases hp : p.LM = 0,\n    {\n        rw [TL_eqz_of_LM_eqz hp] at h,\n        apply absurd h (zero_red _),\n    },\n    {\n        have hpr := gt_of_gt_of_ge (gt_TL_LM_of_LM_nez hp) (red_one_step_LM_ge h),\n        apply red_one_step.cons (nez_of_LM_nez hp) (by rwa \u2190TL_of_add_gt_LM hpr (LC_nez_of_LM_nez hp) at h),\n        simp [LT_of_add_right (by rwa eq_LM_of_monomial p.LM (LC_nez_of_LM_nez hp) at hpr)], refl,\n    }\nend\n\nlemma red_plus_TL_plus {p r : mv_polynomial \u03c3 \u03b1} (h : p.TL \u2192[S]+ r) :\np \u2192[S]+ p.LT + r :=\nbegin\n    conv in p {rw \u2190LM_TL_eq p},\n    by_cases hp : p.LM = 0,\n    {\n        rw [TL_eqz_of_LM_eqz hp] at h \u22a2,\n        have hr : r = 0,\n            by_contradiction hr,\n            rcases red_plus_trans h (ne.symm hr) with \u27e8_, hq, _\u27e9,\n            exact (zero_red _) hq,\n        simpa [hr] using rtc.refl',\n    },\n    {\n        have := gt_TL_LM_of_LM_nez hp, revert this,\n        induction h with _ p' r' h p' q' r' hpq hqr ih\u2081 ih\u2082;\n        intro hp',\n        {exact rtc.refl'},\n        {     \n            have := red_of_TL_red (by rwa \u2190TL_of_add_gt_LM hp' (LC_nez_of_LM_nez hp) at h),\n            rw LT_of_add_gt_LM hp' (LC_nez_of_LM_nez hp) at this,\n            apply rtc.base' this,\n        },\n        {exact rtc.trans' (ih\u2081 hp') (ih\u2082 (lt_of_le_of_lt (red_plus_LM_ge hpq) hp'))}\n    }\nend\n\nlemma red_plus_TL_star {p r : mv_polynomial \u03c3 \u03b1} (h : p.TL \u2192[S]* r) :\np \u2192[S]+ p.LT + r := red_plus_TL_plus h.left\n\nlemma LM_eq_lt_of_lt_TL {p q : mv_polynomial \u03c3 \u03b1} (h : q < p.TL) :  p.LT + q < p :=\nbegin\n    by_cases hp\u2081 : p.LM = 0,\n    {\n        simp [TL_eqz_of_LM_eqz hp\u2081] at h,\n        apply absurd h not_lt_zero,\n    },\n    {\n        have hp\u2082 := nez_of_LM_nez hp\u2081,\n        have hp\u2083 := LC_nez_iff.1 hp\u2082,\n        have hq : p.LM > q.LM := lt_of_le_of_lt (LM_le_of_lt h) (gt_TL_LM_of_LM_nez hp\u2081),\n        apply lt.LM_eq hp\u2082,\n        {\n            conv at hq in p.LM {rw \u2190LM_of_LT},\n            rw LT_of_add_left hq,\n            conv in (p.LT.LT) {rw LT, simp},\n            refl,\n        },\n        {rwa [LT, TL_of_add_gt_LM hq hp\u2083]},\n    }\nend\n\nlemma red_cons_reduction {p r : mv_polynomial \u03c3 \u03b1} (hpq\u2081 : p.TL \u2192[S] r.TL) (hpq\u2082 : p.LT = r.LT) :\n\u2200 q, \u2203 u, (reduction p q \u2192[S]* u) \u2227 (reduction r q \u2192[S]* u) :=\n\u03bb q, begin\n    have hpr : (p \u2192[S] r) := by simpa [hpq\u2082, LM_TL_eq r] using red_of_TL_red hpq\u2081,\n    rcases red_add_of_red hpr (-q * monomial (p.LM - q.LM) (p.LC/q.LC)) with \u27e8u, hpu, hru\u27e9,\n    rcases exists_red_star S u with \u27e8u', hu'\u2081, hu'\u2082\u27e9,\n    refine \u27e8u', \u27e8rtc.trans' (by simpa [reduction] using hpu) hu'\u2081, hu'\u2082\u27e9, \n    \u27e8rtc.trans' (by simpa [reduction, LM_eq_of_LT_eq hpq\u2082, LC_eq_of_LT_eq hpq\u2082] using hru) hu'\u2081, hu'\u2082\u27e9\u27e9,\nend\n\ntheorem red_confluent (h : \u2200 {p q} (hp : p \u2208 S) (hq : q \u2208 S), s_poly p q \u2192[S]* 0) \n(p : mv_polynomial \u03c3 \u03b1) : \u2200 {r t}, (p \u2192[S]* r) \u2192 (p \u2192[S]* t) \u2192 r = t :=\nwell_founded.recursion lt_wf p begin\n    intros p ih r t hpr hpt,\n    by_cases hp\u2081 : irreducible S p,\n    {simp [red_star_irreducible hp\u2081 hpr, red_star_irreducible hp\u2081 hpt]},\n    {\n        by_cases hp\u2082 : p.LM = 0,\n        {\n            cases red_star_const hp\u2082 hpr,\n            {\n                rw h_1 at hp\u2081,\n                cases hpr with _ hpr,\n                apply absurd hpr hp\u2081,\n            },\n            {\n                cases red_star_const hp\u2082 hpt,\n                {\n                    rw h_2 at hp\u2081,\n                    cases hpt with _ hpt,\n                    apply absurd hpt hp\u2081,\n                },\n                {rw [h_1, h_2]}\n            }\n        },\n        {\n            rcases red_star_trans hpr (by simpa using hp\u2081) with \u27e8q\u2081, hpq\u2081, hq\u2081r\u27e9,\n            rcases red_star_trans hpt (by simpa using hp\u2081) with \u27e8q\u2082, hpq\u2082, hq\u2082t\u27e9,\n            have hpq\u2081_lt := lt_of_red_one_step hpq\u2081,\n            have hpq\u2082_lt := lt_of_red_one_step hpq\u2082,\n            cases hqr : hpq\u2081 with a b hp\u2083 hpq\u2081\u2081 hpq\u2081\u2082 _ _ hp\u2083 hr_pq\u2081;\n            cases hqt : hpq\u2082 with _ _ hp\u2083 hpq\u2082\u2081 hpq\u2082\u2082 _ _ hp\u2083 hr_pq\u2082; \n            clear hp\u2083 hqr hqt _x _x_1,\n            {\n                rcases exists_red_star_of_red hpq\u2081\u2081 with \u27e8r'\u2081, hqr'\u2081, hpr'\u2081\u27e9,\n                rcases exists_red_star_of_red hpq\u2082\u2081 with \u27e8t'\u2081, hqt'\u2081, hpt'\u2081\u27e9,\n                have hrt\u2081: r'\u2081 = t'\u2081 := ih _ (TL_lt hp\u2083) hpr'\u2081 hpt'\u2081,\n                rcases exists_red_star_of_red_plus (red_plus_TL_star hqr'\u2081) with \u27e8r'\u2082, hqr'\u2081\u2082, hqr'\u2082\u27e9,\n                rcases exists_red_star_of_red_plus (red_plus_TL_star hqt'\u2081) with \u27e8t'\u2082, hqt'\u2081\u2082, hqt'\u2082\u27e9,\n                have hrt\u2082: r'\u2082 = t'\u2082 := ih (p.LT + t'\u2081) (LM_eq_lt_of_lt_TL (red_lt_of_le_of_lt' hpq\u2082\u2081 hqt'\u2081))\n                    (by simpa [hrt\u2081, hpq\u2081\u2082] using hqr'\u2081\u2082) (by simpa [hpq\u2082\u2082] using hqt'\u2081\u2082), \n                simpa [ih _ hpq\u2081_lt hq\u2081r hqr'\u2082, ih _ hpq\u2082_lt hq\u2082t hqt'\u2082] using hrt\u2082,\n            },\n            {\n                rcases hr_pq\u2082 with \u27e8q, h\u2081, h\u2082, h\u2083, h\u2084\u27e9, \n                rcases red_cons_reduction hpq\u2081\u2081 hpq\u2081\u2082 q with \u27e8u, hq\u2082u, hq\u2081u\u27e9, rw h\u2084 at hq\u2082u,\n                have hq\u2081u' : (q\u2081 \u2192[S]* u) := \n                    \u27e8rtc.base_trans  \n                        (red_one_step.red_LM (nez_of_LM_nez (by rwa [\u2190LM_eq_of_LT_eq hpq\u2081\u2082])) \u27e8q, h\u2081, h\u2082, by rwa \u2190LM_eq_of_LT_eq hpq\u2081\u2082, rfl\u27e9) \n                        hq\u2081u.left, hq\u2081u.right\u27e9,\n                simp [ih _ hpq\u2081_lt hq\u2081r hq\u2081u', ih _ hpq\u2082_lt hq\u2082t hq\u2082u],\n            },\n            {\n                rcases hr_pq\u2081 with \u27e8q, h\u2081, h\u2082, h\u2083, h\u2084\u27e9, \n                rcases red_cons_reduction hpq\u2082\u2081 hpq\u2082\u2082 q with \u27e8u, hq\u2081u, hq\u2082u\u27e9, rw h\u2084 at hq\u2081u,\n                have hq\u2082u' : (q\u2082 \u2192[S]* u) := \n                    \u27e8rtc.base_trans \n                        (red_one_step.red_LM (nez_of_LM_nez (by rwa [\u2190LM_eq_of_LT_eq hpq\u2082\u2082])) \u27e8q, h\u2081, h\u2082, by rwa \u2190LM_eq_of_LT_eq hpq\u2082\u2082, rfl\u27e9)\n                        hq\u2082u.left, hq\u2082u.right\u27e9,\n                simp [ih _ hpq\u2081_lt hq\u2081r hq\u2081u, ih _ hpq\u2082_lt hq\u2082t hq\u2082u'],\n            },\n            {\n                rcases hr_pq\u2081 with \u27e8q\u2081', hq\u2081\u2081, hq\u2081\u2082, hq\u2081\u2083, hq\u2081\u2084\u27e9,\n                rcases hr_pq\u2082 with \u27e8q\u2082', hq\u2082\u2081, hq\u2082\u2082, hq\u2082\u2083, hq\u2082\u2084\u27e9, \n                cases exists_eq_mul_left_of_dvd (lcm_dvd hq\u2081\u2083 hq\u2082\u2083) with m hq\u2083,\n                have hq : ((q\u2081 - q\u2082) \u2192[S]* 0),\n                {\n                    have h_s : s_poly q\u2082' q\u2081' = \n                        (q\u2082' * monomial (m_lcm q\u2081'.LM q\u2082'.LM - q\u2082'.LM) q\u2082'.LC\u207b\u00b9) + -(q\u2081' * monomial (m_lcm q\u2081'.LM q\u2082'.LM - q\u2081'.LM) q\u2081'.LC\u207b\u00b9),                     \n                        simp [s_poly, mul_comm, m_lcm_comm q\u2081'.LM q\u2082'.LM],\n                    simp [hq\u2081\u2084.symm, hq\u2082\u2084.symm, reduction, hq\u2083, \n                    add_sub_assoc', finsupp.dvd_lcm_right, finsupp.dvd_lcm_left,\n                    div_eq_mul_one_div p.LC, monomial_mul_monomial.symm,\n                    mul_left_comm _ (monomial m p.LC)],\n                    rw [neg_mul_eq_mul_neg, \u2190mul_add, \u2190h_s],\n                    refine \u27e8red_plus_zero_of_mul (h hq\u2082\u2081 hq\u2081\u2081).left _ _, zero_not_reducible\u27e9,\n                },\n                rcases sub_red_star_zero hq with \u27e8u, hq\u2081u, hq\u2082u\u27e9,\n                simp [ih _ hpq\u2081_lt hq\u2081r hq\u2081u, ih _ hpq\u2082_lt hq\u2082t hq\u2082u],\n            }\n        }\n    }\nend\n\nlemma red_plus_insert {p r : mv_polynomial \u03c3 \u03b1} (h : p \u2192[S]+ r) : \u2200 a, p \u2192[insert a S]+ r :=\nbegin\n    induction h with _ p r h p q r hpq hqr ih\u2081 ih\u2082;\n    intro a,\n    {exact rtc.refl'},\n    {\n        apply rtc.base',\n        rw red_one_step_is_red_rel at h \u22a2,\n        rcases h with \u27e8q, hq\u2081, hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n        refine \u27e8q, by simp [hq\u2081], hq\u2082, p\u209c, hp\u209c, hqp\u209c, h\u27e9,\n    },\n    {exact rtc.trans' (ih\u2081 a) (ih\u2082 a)}\nend\n\nend reduction\n\nsection step\ninductive buchstep : (list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1)) \n\u2192 (list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1)) \u2192 Prop\n| zero {p} : \u2200 l\u2081 l\u2082 l\u2083, red_list p l\u2081 = 0 \u2192 buchstep \u27e8l\u2081, p :: l\u2082, l\u2083\u27e9 \u27e8l\u2081, l\u2082, l\u2083\u27e9\n| non_zero {p} : \u2200 l\u2081 l\u2082 l\u2083, red_list p l\u2081 \u2260 0 \u2192 \n    buchstep \u27e8l\u2081, p :: l\u2082, l\u2083\u27e9 \n        \u27e8(red_list p l\u2081) :: l\u2081, s_polyL (red_list p l\u2081) l\u2081 ++ l\u2082, s_polyL (red_list p l\u2081) l\u2081 ++ l\u2083\u27e9\n\ndef buchstep_plus : (list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1)) \n\u2192 (list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1)) \u2192 Prop := rtc buchstep\n\nlemma buchstep.zero' {p : mv_polynomial \u03c3 \u03b1} {l\u2081 l\u2082 l\u2083} (h : red_list p l\u2081 = 0) :\nbuchstep \u27e8l\u2081, p :: l\u2082, l\u2083\u27e9 \u27e8l\u2081, l\u2082, l\u2083\u27e9 := buchstep.zero l\u2081 l\u2082 l\u2083 h\n\nlemma buchstep.non_zero' {p : mv_polynomial \u03c3 \u03b1} {l\u2081 l\u2082 l\u2083} (h : red_list p l\u2081 \u2260 0) :\nbuchstep \u27e8l\u2081, p :: l\u2082, l\u2083\u27e9 \u27e8(red_list p l\u2081) :: l\u2081, \n    s_polyL (red_list p l\u2081) l\u2081 ++ l\u2082, s_polyL (red_list p l\u2081) l\u2081 ++ l\u2083\u27e9 := buchstep.non_zero l\u2081 l\u2082 l\u2083 h\n\nlemma red_list_aux_red_plus_aux : \u2200 (p : mv_polynomial \u03c3 \u03b1) {l\u2081 l\u2082 : list (mv_polynomial \u03c3 \u03b1)}, \nl\u2082 \u2286 l\u2081 \u2192 (p \u2192[l\u2081.to_finset]+ red_list_aux p l\u2082)\n| p l\u2081 [] := by simp [red_list_aux]; exact rtc.refl'\n| p l\u2081 (q :: l\u2082) := \u03bb h, begin\n    by_cases hp : p = 0,\n    {simp [hp, zero_red_list_aux], exact rtc.refl'},\n    {\n        by_cases hqp : q.LM \u2223 p.LM; simp [red_list_aux, hqp] at h \u22a2,\n        {\n            by_cases hq : q = 0,\n            {\n                simp [reduction, hq],\n                apply red_list_aux_red_plus_aux _ h.right,\n            },\n            {\n                apply rtc.base_trans (red_one_step.red_LM hp \u27e8q, by simp [h.left], hq, hqp, rfl\u27e9)\n                (red_list_aux_red_plus_aux (reduction p q) h.right),\n            }\n        },\n        {apply red_list_aux_red_plus_aux _ h.right}\n    }\nend\n\nlemma red_list_aux_red_plus (p : mv_polynomial \u03c3 \u03b1) (l : list (mv_polynomial \u03c3 \u03b1)) :\n(p \u2192[l.to_finset]+ red_list_aux p l) := red_list_aux_red_plus_aux p (by simp)\n\nlemma red_list_red_plus : \u2200 (p : mv_polynomial \u03c3 \u03b1) (l : list (mv_polynomial \u03c3 \u03b1)),\n(p \u2192[l.to_finset]+ red_list p l) \n| p l := begin\n    unfold red_list, simp,\n    by_cases hp\u2081 : red_list_aux p l = p; simp [hp\u2081],\n    {exact rtc.refl'},\n    {\n        by_cases hp\u2082 : p.LM = 0; simp [hp\u2082],\n        {\n            have h := red_plus_const hp\u2082 (red_list_aux_red_plus p l),\n            simp [ne.symm hp\u2081] at h,\n            simpa [h] using red_list_aux_red_plus p l,\n        },\n        {\n            let : (red_list_aux p l).LM < p.LM, from red_list_aux_LM_lt l p hp\u2082 hp\u2081,\n            apply rtc.trans' (red_list_aux_red_plus p l) (red_list_red_plus _ l),\n        }\n    }\nend\nusing_well_founded \n{ rel_tac := \u03bb _ _, `[exact \u27e8_, inv_image.wf (\u03bb a, a.1.LM) _inst_6.wf\u27e9] \n, dec_tac := tactic.assumption }\n\nset_option class.instance_max_depth 50\nlemma buchberger_buchplus : \u2200 L : list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1),\nbuchstep_plus \u27e8L.1, L.2, s_polys L.1\u27e9 \u27e8buchberger \u27e8L.1, L.2\u27e9, [], s_polys (buchberger \u27e8L.1, L.2\u27e9)\u27e9\n| \u27e8l\u2081, []\u27e9 := by unfold buchberger; simp; apply rtc.refl'\n| \u27e8l\u2081, (p :: l\u2082)\u27e9 := \nlet lex := prod.lex ((>) : ideal (mv_polynomial \u03c3 \u03b1) \u2192 ideal (mv_polynomial \u03c3 \u03b1) \u2192 Prop) nat.lt in\nbegin\n    unfold buchberger, \n    by_cases red_list p l\u2081 = 0; simp [h],\n    {\n        let : lex \u27e8monomial_ideal l\u2081, l\u2082.length\u27e9 \u27e8monomial_ideal l\u2081, (list.cons p l\u2082).length\u27e9 := \n            by right; rw [list.length_cons]; apply nat.lt_succ_self,\n        apply rtc.base_trans (buchstep.zero' h) (buchberger_buchplus \u27e8l\u2081, l\u2082\u27e9),\n    },\n    {\n        let : lex \u27e8monomial_ideal (list.cons (red_list p l\u2081) l\u2081), (s_polyL (red_list p l\u2081) l\u2081 ++ l\u2082).length\u27e9 \n            \u27e8monomial_ideal l\u2081, (list.cons p l\u2082).length\u27e9 := \n            by left; exact ideal_increase l\u2081 p h,\n        apply rtc.base_trans (buchstep.non_zero' h) \n            (buchberger_buchplus \u27e8red_list p l\u2081 :: l\u2081, s_polyL (red_list p l\u2081) l\u2081 ++ l\u2082\u27e9)\n    }\nend\nusing_well_founded \n{ rel_tac := \u03bb _ _, \n`[exact \u27e8_, inv_image.wf (\u03bb \u27e8l\u2081, l\u2082\u27e9, prod.mk (monomial_ideal l\u2081) l\u2082.length) (prod.lex_wf ideal_wf nat.lt_wf)\u27e9 ] \n, dec_tac := tactic.assumption }\n\nlemma buchstep_subset_or_red_eqz (L\u2081 L\u2082 : list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1) \u00d7 list (mv_polynomial \u03c3 \u03b1))\n(h : buchstep_plus L\u2081 L\u2082) : (\u2200 a \u2208 L\u2081.2.2, a \u2208 L\u2081.2.1 \u2228 (a \u2192[L\u2081.1.to_finset]+ 0)) \u2192 \n\u2200 a \u2208 L\u2082.2.2, a \u2208 L\u2082.2.1 \u2228 (a \u2192[L\u2082.1.to_finset]+ 0) :=\nbegin\n    induction h with _ L\u2081 L\u2082 h L\u2081 L\u2082 L\u2083 h\u2081 h\u2082 ih\u2081 ih\u2082,\n    {simp},\n    {\n        cases h with p l\u2081 l\u2082 l\u2083 hp p l\u2081 l\u2082 l\u2083 hp,\n        {\n            simp_intros H a ha,\n            rcases H a ha with \u27e8h\u2081 | h\u2082\u27e9 | h\u2083,\n            {right, simpa [hp, h\u2081] using red_list_red_plus p l\u2081},\n            {left, exact h\u2082},\n            {right, exact h\u2083},\n        },\n        {\n            simp_intros H a' ha',\n            cases ha',\n            {simp [ha']},\n            {\n                rcases H a' ha' with \u27e8h\u2081 | h\u2082\u27e9 | h\u2083,\n                {\n                    right, rw h\u2081,\n                    apply rtc.trans_base (red_plus_insert (red_list_red_plus p l\u2081) (red_list p l\u2081)),\n                    apply red_mem_S (finset.mem_insert_self _ _) hp,\n                },\n                {simp [h\u2082]},\n                {\n                    right,\n                    apply red_plus_insert h\u2083,\n                }\n            }\n        }\n    },\n    {\n        simp_intros H,\n        exact ih\u2082 (ih\u2081 H), \n    }\nend\n\nlemma s_poly_mem_s_polys : \u2200 (l\u2081 l\u2082 : list (mv_polynomial \u03c3 \u03b1)),\n(\u2200 x : mv_polynomial \u03c3 \u03b1, x \u2208 s_polys l\u2081 \u2192 (x \u2192[l\u2082.to_finset]* 0)) \n\u2192 \u2200 (p q : mv_polynomial \u03c3 \u03b1), p \u2208 l\u2081 \u2192 q \u2208 l\u2081 \u2192 (s_poly p q \u2192[l\u2082.to_finset]* 0)\n| [] := by simp\n| (hd :: tl) := \u03bb l\u2082 h p q hp hq, begin\n    cases hp; cases hq,\n    {\n        simp [hp, hq, s_poly],\n        refine \u27e8rtc.refl', zero_not_reducible\u27e9,\n    },\n    {\n        apply h, \n        simp [hp, s_polys, mem_s_polyL hq],\n    },\n    {\n        rw s_poly_comm,\n        have H : (monomial 0 1 : mv_polynomial \u03c3 \u03b1) = C 1 := by simp [C],\n        simpa [H] using \n            red_star_zero_of_mul (h (s_poly q p) (by simp [s_polys, hq, mem_s_polyL hp])) 0 (-1),\n    },\n    {\n        apply s_poly_mem_s_polys tl _  \n            (\u03bb x hx, h x (by simp [s_polys, hx])) p q hp hq,\n    }\nend\n\nlemma buch_confluent_aux (l : list (mv_polynomial \u03c3 \u03b1)) :\n\u2200 (p q \u2208 buchberger \u27e8l, s_polys l\u27e9), (s_poly p q \u2192[(buchberger \u27e8l, s_polys l\u27e9).to_finset]* 0) :=\nbegin\n    apply s_poly_mem_s_polys,\n    simp [zero_not_reducible],\n    have := buchstep_subset_or_red_eqz \u27e8l, s_polys l, s_polys l\u27e9 \n        \u27e8buchberger \u27e8l, s_polys l\u27e9, [], s_polys (buchberger \u27e8l, s_polys l\u27e9)\u27e9 (buchberger_buchplus \u27e8l, s_polys l\u27e9),\n    simp at this,\n    apply this,\n    finish,\nend\n\ntheorem buch_confluent (l : list (mv_polynomial \u03c3 \u03b1)) (p : mv_polynomial \u03c3 \u03b1) :\n\u2200 {r t}, (p \u2192[(buchberger \u27e8l, s_polys l\u27e9).to_finset]* r) \u2192 (p \u2192[(buchberger \u27e8l, s_polys l\u27e9).to_finset]* t) \u2192 r = t :=\nbegin\n    apply red_confluent,\n    intros p q hp hq,\n    apply buch_confluent_aux l p q (by simpa using hp) (by simpa using hq),\nend\n\nend step\n\n\n\n\nend buch", "meta": {"author": "FCL-lean", "repo": "verification", "sha": "be02c698c0ca78b18762e3fe7749cdc72a55d197", "save_path": "github-repos/lean/FCL-lean-verification", "path": "github-repos/lean/FCL-lean-verification/verification-be02c698c0ca78b18762e3fe7749cdc72a55d197/src/buch_correctness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149978955811, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3892492075929601}}
{"text": "import ring_theory.int.basic\n\nlemma units_int.values (u: units \u2124): u = 1 \u2228 u = -1 :=\nbegin\n  have p\u2081 := fintype.complete u,\n  cases p\u2081,\n  left, exact p\u2081,\n  cases p\u2081,\n  right, exact p\u2081,\n  exfalso, exact list.not_mem_nil u p\u2081,\nend\n", "meta": {"author": "RaitoBezarius", "repo": "berkovich-spaces", "sha": "0a49f75a599bcb20333ec86b301f84411f04f7cf", "save_path": "github-repos/lean/RaitoBezarius-berkovich-spaces", "path": "github-repos/lean/RaitoBezarius-berkovich-spaces/berkovich-spaces-0a49f75a599bcb20333ec86b301f84411f04f7cf/src/for_mathlib/int.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.754914997895581, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.38924920759296006}}
{"text": "import AutograderTests.Util\n\ntheorem imp_self : p \u2192 p :=\n  sorry_in_exercise_else by intros; assumption\n", "meta": {"author": "gebner", "repo": "autograder", "sha": "9d23bfc346c672e93d0b4ee11453925ed15cd091", "save_path": "github-repos/lean/gebner-autograder", "path": "github-repos/lean/gebner-autograder/autograder-9d23bfc346c672e93d0b4ee11453925ed15cd091/AutograderTests/Pass/Simple.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3892492019067279}}
{"text": "import for_mathlib.abelian_category\nimport for_mathlib.exact_seq3\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\n\nnamespace category_theory\nnamespace exact\n\nvariables {\ud835\udcd0 : Type*} [category \ud835\udcd0] [abelian \ud835\udcd0]\nvariables {A B C T : \ud835\udcd0} {f : A \u27f6 B} {g : B \u27f6 C}\n\nsection lift\n\nvariables (h : exact f g) [mono f] (\u03c6 : T \u27f6 B) (w : \u03c6 \u226b g = 0)\ninclude h w\n\ndef mono_lift : T \u27f6 A :=\n(abelian.is_limit_of_exact_of_mono f g h).lift (kernel_fork.of_\u03b9 _ w)\n\n@[simp, reassoc] lemma mono_lift_comp : h.mono_lift \u03c6 w \u226b f = \u03c6 :=\n(abelian.is_limit_of_exact_of_mono f g h).fac (kernel_fork.of_\u03b9 _ w) walking_parallel_pair.zero\n\nlemma mono_lift_unique (e : T \u27f6 A) (he : e \u226b f = \u03c6) : e = h.mono_lift \u03c6 w :=\nby rw [\u2190 cancel_mono f, he, h.mono_lift_comp]\n\nend lift\n\nsection desc\n\nvariables (h : exact f g) [category_theory.epi g] (\u03c6 : B \u27f6 T) (w : f \u226b \u03c6 = 0)\ninclude h w\n\ndef epi_desc : C \u27f6 T :=\n(abelian.is_colimit_of_exact_of_epi f g h).desc (cokernel_cofork.of_\u03c0 _ w)\n\n@[simp, reassoc] lemma comp_epi_desc : g \u226b h.epi_desc \u03c6 w = \u03c6 :=\n(abelian.is_colimit_of_exact_of_epi f g h).fac (cokernel_cofork.of_\u03c0 _ w) walking_parallel_pair.one\n\nlemma epi_desc_unique (e : C \u27f6 T) (he : g \u226b e = \u03c6) : e = h.epi_desc \u03c6 w :=\nby rw [\u2190 cancel_epi g, he, h.comp_epi_desc]\n\nend desc\n\nend exact\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/exact_lift_desc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.38916677921477244}}
{"text": "namespace melting_point.geometry\n\nuniverse u\n\nclass is_euclidian (S : Type u) :=\n(B : S \u2192 S \u2192 S \u2192 Prop)\n(cong : S \u00d7 S \u2192 S \u00d7 S \u2192 Prop)\n-- Tarski axioms\n(cong_refl (x y : S) : cong (x, y) (y, x))\n(cong_trans (a b c : S \u00d7 S) : cong a b \u2192 cong a c \u2192 cong b c)\n(identity_of_congruence (x y z : S) : cong (x, y) (z, z) \u2192 x = y)\n(segment_construction (x y a b : S) :\n  \u2203 z, B x y z \u2227 cong (y, z) (a, b))\n(five_segment (x y z x' y' z' u u' : S) :\n  x \u2260 y \u2192 B x y z \u2192 B x' y' z' \u2192\n  cong (x, y) (x', y') \u2192\n  cong (y, z) (y', z') \u2192\n  cong (x, u) (x', u') \u2192\n  cong (y, u) (y', u') \u2192\n  cong (z, u) (z', u'))\n(identity_of_betweenness (x y : S) : B x y x \u2192 x = y)\n(axiom_of_Pasch (x y z u v : S) :\n  B x y z \u2192 B y v z \u2192 \u2203 a, B u a y \u2227 B v a x)\n(lower_dimension (a b c : S) :\n  \u00acB a b c \u2227 \u00acB b c a \u2227 \u00acB c a b)\n(upper_dimension (x y z u v : S) :\n  cong (x, u) (x, v) \u2192\n  cong (y, u) (y, v) \u2192\n  cong (z, u) (z, v) \u2192\n  u \u2260 v \u2192 B x y z \u2227 B y z x \u2227 B z x y)\n(axiom_of_Euclid (x y z u v : S) :\n  B x u v \u2192 B y u z \u2192 x \u2260 y \u2192\n  \u2203 a b, B x y a \u2227 B x z b \u2227 B a v b)\n(axiom_schema_of_Continuity (\u03c6 \u03c8 : S \u2192 Prop) :\n  (\u2203 a, \u2200 x y, \u03c6 x \u2192 \u03c8 y \u2192 B a x y) \u2192\n  (\u2203 b, \u2200 x y, \u03c6 x \u2192 \u03c8 y \u2192 B x b y))\nopen is_euclidian\n\ninfix ` \u2245 `:50 := is_euclidian.cong\n\nsection\n  variables {S : Type u} [is_euclidian S]\n\n  instance : has_mem S (S \u00d7 S) :=\n  \u27e8\u03bb x a, B a.fst x a.snd\u27e9\n\n  def segment (x y : S) :=\n  { z | B x z y }\n\n  def line (x y : S) :=\n  { z | B y x z \u2228 B x y z \u2228 B x z y }\n\n  def circle (radius : S \u00d7 S) :=\n  { z | (radius.fst, z) \u2245 radius }\n\n  def disk (radius : S \u00d7 S) : set S :=\n  { z | \u2203 (a : S \u00d7 S), a.fst = radius.fst \u2227 a \u2245 radius \u2227 z \u2208 a }\n\n  def triangle (a b c : S) :=\n  { z | B a z c \u2228 B a z b \u2228 B b z c }\n\n  def ray (a b : S) :=\n  { c | B a c b \u2228 B a b c }\n\n  def angle (a b c : S) : set S :=\n  { z | z \u2208 ray b a \u2228 z \u2208 ray b c }\n\n  def parallel (a b : set S) :=\n  \u00ac\u2203 (z : S), z \u2208 a \u2227 z \u2208 b\n\n  def segment.is_sum (r\u2081 r\u2082 r : S \u00d7 S) :=\n  \u2203 z, (r.fst, z) \u2245 r\u2081 \u2227 (r.snd, z) \u2245 r\u2082 \u2227 B r.fst z r.snd\n\n  def circle.touch_externally (r\u2081 r\u2082 : S \u00d7 S) :=\n  \u2203! (z : S), z \u2208 circle r\u2081 \u2227 z \u2208 circle r\u2082 \u2227 B r\u2081.fst z r\u2082.fst\n\n  theorem sum_of_radiuses_tang_circles\n    (r\u2081 r\u2082 : S \u00d7 S) (h : circle.touch_externally r\u2081 r\u2082) :\n    segment.is_sum r\u2081 r\u2082 (r\u2081.fst, r\u2082.fst) := begin\n    cases h with z cond,\n    cases cond with cond trash, clear trash,\n    cases cond with belongs\u2081 cond,\n    cases cond with belongs\u2082 H,\n    existsi z, repeat { try { split }, assumption }\n  end\nend\n\nend melting_point.geometry", "meta": {"author": "forked-from-1kasper", "repo": "melting_point", "sha": "e5ea4a0917de086b7e5b122e8d5aa90d2761d147", "save_path": "github-repos/lean/forked-from-1kasper-melting_point", "path": "github-repos/lean/forked-from-1kasper-melting_point/melting_point-e5ea4a0917de086b7e5b122e8d5aa90d2761d147/melting_point/geometry.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.749087201911703, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.38916677339456224}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel, Scott Morrison\n-/\nimport algebra.category.Module.epi_mono\nimport algebra.module.projective\nimport category_theory.preadditive.projective\nimport linear_algebra.finsupp_vector_space\n\n/-!\n# The category of `R`-modules has enough projectives.\n-/\n\nuniverses v u\n\nopen category_theory\nopen category_theory.limits\nopen linear_map\n\nopen_locale Module\n\n/-- The categorical notion of projective object agrees with the explicit module-theoretic notion. -/\ntheorem is_projective.iff_projective {R : Type u} [ring R]\n  {P : Type (max u v)} [add_comm_group P] [module R P] :\n  module.projective R P \u2194 projective (Module.of R P) :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { letI : module.projective R \u21a5(Module.of R P) := h,\n    exact \u27e8\u03bb E X f e epi, module.projective_lifting_property _ _\n      ((Module.epi_iff_surjective _).mp epi)\u27e9 },\n  { refine module.projective_of_lifting_property _,\n    introsI E X mE mX sE sX f g s,\n    haveI : epi \u219ff := (Module.epi_iff_surjective \u219ff).mpr s,\n    letI : projective (Module.of R P) := h,\n    exact \u27e8projective.factor_thru \u219fg \u219ff, projective.factor_thru_comp \u219fg \u219ff\u27e9 }\nend\n\nnamespace Module\nvariables {R : Type u} [ring R] {M : Module.{(max u v)} R}\n\n/-- Modules that have a basis are projective. -/\n-- We transport the corresponding result from `module.projective`.\nlemma projective_of_free {\u03b9 : Type*} (b : basis \u03b9 R M) : projective M :=\nprojective.of_iso (Module.of_self_iso _)\n  ((is_projective.iff_projective).mp (module.projective_of_basis b))\n\n/-- The category of modules has enough projectives, since every module is a quotient of a free\n    module. -/\ninstance Module_enough_projectives : enough_projectives (Module.{max u v} R) :=\n{ presentation :=\n  \u03bb M,\n  \u27e8{ P := Module.of R (M \u2192\u2080 R),\n    projective := projective_of_free finsupp.basis_single_one,\n    f := finsupp.basis_single_one.constr \u2115 id,\n    epi := (epi_iff_range_eq_top _).mpr\n      (range_eq_top.2 (\u03bb m, \u27e8finsupp.single m (1 : R), by simp [basis.constr]\u27e9)) }\u27e9, }\n\nend Module\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/category/Module/projective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.389068049892614}}
{"text": "import Hw9\nimport Sets.Basic\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\nopen Set \n\nvariable (\u03b1 \u03b2 : Type)\nvariable (X Y Z : Set \u03b1)\nvariable (W : Set \u03b2) \n\ntheorem desiredType1 : \u2205 \u2208 \ud835\udcab  X := sorry \n\ntheorem desiredType2 (U : \u03b2 \u2192 Set \u03b1) : \u2200 b, U b \u2286 BigUnion U := sorry \n\ntheorem desiredType3 (h : X \u2286 Y) : (X \u00d7\u02e2 W) \u2286 (Y \u00d7\u02e2 W) := sorry\n\ntheorem desiredType4 (h : Y \u2229 Z = \u2205) : Y\u1d9c \u222a Z\u1d9c = Univ := sorry \n\ntheorem desiredType5 : (X \\ Y) \u222a (Y \\ X) = (X \u222a Y) \\ (X \u2229 Y) := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef n : String := \"4\"\n\ndef problem : String := \"problem\"++n\n\ndef desired : String := \"desiredType\"++n\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const desired []) (Expr.const problem [])\n#eval collectAxiomsOf problem\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework9", "sha": "c26e748a8f91c4f459d6f568a6819b53cd6088a1", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9/homework9-c26e748a8f91c4f459d6f568a6819b53cd6088a1/_Tests/Problem4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432062975979, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.38900747783973494}}
{"text": "/-\nCopyright (c) 2018 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport Std.Data.AssocList\nimport Std.Data.Nat.Basic\nimport Std.Classes.BEq\n\nnamespace Std.HashMap\n\n/-- A hash is lawful if elements which compare equal under `==` have equal hash. -/\nclass LawfulHashable (\u03b1 : Type _) [BEq \u03b1] [Hashable \u03b1] : Prop where\n  /-- Two elements which compare equal under the `BEq` instance have equal hash. -/\n  hash_eq {a b : \u03b1} : a == b \u2192 hash a = hash b\n\nnamespace Imp\n\n/--\nThe bucket array of a `HashMap` is a nonempty array of `AssocList`s.\n(This type is an internal implementation detail of `HashMap`.)\n-/\ndef Bucket (\u03b1 : Type u) (\u03b2 : Type v) := {b : Array (AssocList \u03b1 \u03b2) // 0 < b.size}\n\nnamespace Bucket\n\n/-- Construct a new empty bucket array with the specified capacity. -/\ndef mk (buckets := 8) (h : 0 < buckets := by decide) : Bucket \u03b1 \u03b2 :=\n  \u27e8mkArray buckets .nil, by simp [h]\u27e9\n\n/-- Update one bucket in the bucket array with a new value. -/\ndef update (data : Bucket \u03b1 \u03b2) (i : USize)\n    (d : AssocList \u03b1 \u03b2) (h : i.toNat < data.1.size) : Bucket \u03b1 \u03b2 :=\n  \u27e8data.1.uset i d h, (Array.size_uset ..).symm \u25b8 data.2\u27e9\n\n/--\nThe number of elements in the bucket array.\nNote: this is marked `noncomputable` because it is only intended for specification.\n-/\nnoncomputable def size (data : Bucket \u03b1 \u03b2) : Nat := .sum (data.1.data.map (\u00b7.toList.length))\n\n/-- Map a function over the values in the map. -/\n@[specialize] def mapVal (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (self : Bucket \u03b1 \u03b2) : Bucket \u03b1 \u03b3 :=\n  \u27e8self.1.map (.mapVal f), by simp [self.2]\u27e9\n\n/--\nThe well-formedness invariant for the bucket array says that every element hashes to its index\n(assuming the hash is lawful - otherwise there are no promises about where elements are located).\n-/\nstructure WF [BEq \u03b1] [Hashable \u03b1] (buckets : Bucket \u03b1 \u03b2) : Prop where\n  /-- The elements of a bucket are all distinct according to the `BEq` relation. -/\n  distinct [LawfulHashable \u03b1] [PartialEquivBEq \u03b1] : \u2200 bucket \u2208 buckets.1.data,\n    bucket.toList.Pairwise fun a b => \u00ac(a.1 == b.1)\n  /-- Every element in a bucket should hash to its location. -/\n  hash_self (i : Nat) (h : i < buckets.1.size) :\n    buckets.1[i].All fun k _ => ((hash k).toUSize % buckets.1.size).toNat = i\n\nend Bucket\nend Imp\n\n/-- `HashMap.Imp \u03b1 \u03b2` is the internal implementation type of `HashMap \u03b1 \u03b2`. -/\nstructure Imp (\u03b1 : Type u) (\u03b2 : Type v) where\n  /-- The number of elements stored in the `HashMap`.\n  We cache this both so that we can implement `.size` in `O(1)`, and also because we\n  use the size to determine when to resize the map. -/\n  size    : Nat\n  /-- The bucket array of the `HashMap`. -/\n  buckets : Imp.Bucket \u03b1 \u03b2\n\nnamespace Imp\n\n/--\nGiven a desired capacity, this returns the number of buckets we should reserve.\nA \"load factor\" of 0.75 is the usual standard for hash maps, so we return `capacity * 4 / 3`.\n-/\n@[inline] def numBucketsForCapacity (capacity : Nat) : Nat :=\n  capacity * 4 / 3\n\n/-- Constructs an empty hash map with the specified nonzero number of buckets. -/\n@[inline] def empty' (buckets := 8) (h : 0 < buckets := by decide) : Imp \u03b1 \u03b2 :=\n  \u27e80, .mk buckets h\u27e9\n\n/-- Constructs an empty hash map with the specified target capacity. -/\ndef empty (capacity := 0) : Imp \u03b1 \u03b2 :=\n  let nbuckets := numBucketsForCapacity capacity\n  let n : {n : Nat // 0 < n} :=\n    if h : nbuckets = 0 then \u27e88, by decide\u27e9\n    else \u27e8nbuckets, Nat.zero_lt_of_ne_zero h\u27e9\n  empty' n n.2\n\n/-- Calculates the bucket index from a hash value `u`. -/\ndef mkIdx {n : Nat} (h : 0 < n) (u : USize) : {u : USize // u.toNat < n} :=\n  \u27e8u % n, USize.modn_lt _ h\u27e9\n\n/--\nInserts a key-value pair into the bucket array. This function assumes that the data is not\nalready in the array, which is appropriate when reinserting elements into the array after a resize.\n-/\n@[inline] def reinsertAux [Hashable \u03b1]\n    (data : Bucket \u03b1 \u03b2) (a : \u03b1) (b : \u03b2) : Bucket \u03b1 \u03b2 :=\n  let \u27e8i, h\u27e9 := mkIdx data.2 (hash a |>.toUSize)\n  data.update i (.cons a b data.1[i]) h\n\n/-- Folds a monadic function over the elements in the map (in arbitrary order). -/\n@[inline] def foldM [Monad m] (f : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03b4) (d : \u03b4) (map : Imp \u03b1 \u03b2) : m \u03b4 :=\n  map.buckets.1.foldlM (init := d) fun d b => b.foldlM f d\n\n/-- Folds a function over the elements in the map (in arbitrary order). -/\n@[inline] def fold (f : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b4) (d : \u03b4) (m : Imp \u03b1 \u03b2) : \u03b4 :=\n  Id.run $ foldM f d m\n\n/-- Runs a monadic function over the elements in the map (in arbitrary order). -/\n@[inline] def forM [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m PUnit) (h : Imp \u03b1 \u03b2) : m PUnit :=\n  h.buckets.1.forM fun b => b.forM f\n\n/-- Given a key `a`, returns a key-value pair in the map whose key compares equal to `a`. -/\ndef findEntry? [BEq \u03b1] [Hashable \u03b1] (m : Imp \u03b1 \u03b2) (a : \u03b1) : Option (\u03b1 \u00d7 \u03b2) :=\n  let \u27e8_, buckets\u27e9 := m\n  let \u27e8i, h\u27e9 := mkIdx buckets.2 (hash a |>.toUSize)\n  buckets.1[i].findEntry? a\n\n/-- Looks up an element in the map with key `a`. -/\ndef find? [BEq \u03b1] [Hashable \u03b1] (m : Imp \u03b1 \u03b2) (a : \u03b1) : Option \u03b2 :=\n  let \u27e8_, buckets\u27e9 := m\n  let \u27e8i, h\u27e9 := mkIdx buckets.2 (hash a |>.toUSize)\n  buckets.1[i].find? a\n\n/-- Returns true if the element `a` is in the map. -/\ndef contains [BEq \u03b1] [Hashable \u03b1] (m : Imp \u03b1 \u03b2) (a : \u03b1) : Bool :=\n  let \u27e8_, buckets\u27e9 := m\n  let \u27e8i, h\u27e9 := mkIdx buckets.2 (hash a |>.toUSize)\n  buckets.1[i].contains a\n\n/-- Copies all the entries from `buckets` into a new hash map with a larger capacity. -/\ndef expand [Hashable \u03b1] (size : Nat) (buckets : Bucket \u03b1 \u03b2) : Imp \u03b1 \u03b2 :=\n  let nbuckets := buckets.1.size * 2\n  { size, buckets := go 0 buckets.1 (.mk nbuckets (Nat.mul_pos buckets.2 (by decide))) }\nwhere\n  /-- Inner loop of `expand`. Copies elements `source[i:]` into `target`,\n  destroying `source` in the process. -/\n  go (i : Nat) (source : Array (AssocList \u03b1 \u03b2)) (target : Bucket \u03b1 \u03b2) : Bucket \u03b1 \u03b2 :=\n    if h : i < source.size then\n      let idx : Fin source.size := \u27e8i, h\u27e9\n      let es := source.get idx\n      -- We remove `es` from `source` to make sure we can reuse its memory cells\n      -- when performing es.foldl\n      let source := source.set idx .nil\n      let target := es.foldl reinsertAux target\n      go (i+1) source target\n    else target\ntermination_by _ i source _ => source.size - i\n\n/--\nInserts key-value pair `a, b` into the map.\nIf an element equal to `a` is already in the map, it is replaced by `b`.\n-/\n@[inline] def insert [BEq \u03b1] [Hashable \u03b1] (m : Imp \u03b1 \u03b2) (a : \u03b1) (b : \u03b2) : Imp \u03b1 \u03b2 :=\n  let \u27e8size, buckets\u27e9 := m\n  let \u27e8i, h\u27e9 := mkIdx buckets.2 (hash a |>.toUSize)\n  let bkt := buckets.1[i]\n  bif bkt.contains a then\n    \u27e8size, buckets.update i (bkt.replace a b) h\u27e9\n  else\n    let size' := size + 1\n    let buckets' := buckets.update i (.cons a b bkt) h\n    if numBucketsForCapacity size' \u2264 buckets.1.size then\n      { size := size', buckets := buckets' }\n    else\n      expand size' buckets'\n\n/--\nRemoves key `a` from the map. If it does not exist in the map, the map is returned unchanged.\n-/\ndef erase [BEq \u03b1] [Hashable \u03b1] (m : Imp \u03b1 \u03b2) (a : \u03b1) : Imp \u03b1 \u03b2 :=\n  let \u27e8size, buckets\u27e9 := m\n  let \u27e8i, h\u27e9 := mkIdx buckets.2 (hash a |>.toUSize)\n  let bkt := buckets.1[i]\n  bif bkt.contains a then \u27e8size - 1, buckets.update i (bkt.erase a) h\u27e9 else m\n\n/-- Map a function over the values in the map. -/\n@[inline] def mapVal (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (self : Imp \u03b1 \u03b2) : Imp \u03b1 \u03b3 :=\n  { size := self.size, buckets := self.buckets.mapVal f }\n\n/--\nApplies `f` to each key-value pair `a, b` in the map. If it returns `some c` then\n`a, c` is pushed into the new map; else the key is removed from the map.\n-/\n@[specialize] def filterMap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n    (f : \u03b1 \u2192 \u03b2 \u2192 Option \u03b3) (m : Imp \u03b1 \u03b2) : Imp \u03b1 \u03b3 :=\n  let m' := m.buckets.1.mapM (m := StateT (ULift Nat) Id) (go .nil) |>.run \u27e80\u27e9 |>.run\n  have : m'.1.size > 0 := by\n    have := Array.size_mapM (m := StateT (ULift Nat) Id) (go .nil) m.buckets.1\n    simp [SatisfiesM_StateT_eq, SatisfiesM_Id_eq] at this\n    simp [this, Id.run, StateT.run, m.2.2]\n  \u27e8m'.2.1, m'.1, this\u27e9\nwhere\n  /-- Inner loop of `filterMap`. Note that this reverses the bucket lists,\n  but this is fine since bucket lists are unordered. -/\n  @[specialize] go (acc : AssocList \u03b1 \u03b3) : AssocList \u03b1 \u03b2 \u2192 ULift Nat \u2192 AssocList \u03b1 \u03b3 \u00d7 ULift Nat\n  | .nil, n => (acc, n)\n  | .cons a b l, n => match f a b with\n    | none => go acc l n\n    | some c => go (.cons a c acc) l \u27e8n.1 + 1\u27e9\n\n/-- Constructs a map with the set of all pairs `a, b` such that `f` returns true. -/\n@[inline] def filter (f : \u03b1 \u2192 \u03b2 \u2192 Bool) (m : Imp \u03b1 \u03b2) : Imp \u03b1 \u03b2 :=\n  m.filterMap fun a b => bif f a b then some b else none\n\n/--\nThe well-formedness invariant for a hash map. The first constructor is the real invariant,\nand the others allow us to \"cheat\" in this file and define `insert` and `erase`,\nwhich have more complex proofs that are delayed to `Std.Data.HashMap.Lemmas`.\n-/\ninductive WF [BEq \u03b1] [Hashable \u03b1] : Imp \u03b1 \u03b2 \u2192 Prop where\n  /-- The real well-formedness invariant:\n  * The `size` field should match the actual number of elements in the map\n  * The bucket array should be well-formed, meaning that if the hashable instance\n    is lawful then every element hashes to its index. -/\n  | mk : m.size = m.buckets.size \u2192 m.buckets.WF \u2192 WF m\n  /-- The empty hash map is well formed. -/\n  | empty' : WF (empty' n h)\n  /-- Inserting into a well formed hash map yields a well formed hash map. -/\n  | insert : WF m \u2192 WF (insert m a b)\n  /-- Removing an element from a well formed hash map yields a well formed hash map. -/\n  | erase : WF m \u2192 WF (erase m a)\n\ntheorem WF.empty [BEq \u03b1] [Hashable \u03b1] : WF (empty n : Imp \u03b1 \u03b2) := by unfold empty; apply empty'\n\nend Imp\n\n/--\n`HashMap \u03b1 \u03b2` is a key-value map which stores elements in an array using a hash function\nto find the values. This allows it to have very good performance for lookups\n(average `O(1)` for a perfectly random hash function), but it is not a persistent data structure,\nmeaning that one should take care to use the map linearly when performing updates.\nCopies are `O(n)`.\n-/\ndef _root_.Std.HashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] := {m : Imp \u03b1 \u03b2 // m.WF}\n\nopen HashMap.Imp\n\n/-- Make a new hash map with the specified capacity. -/\n@[inline] def _root_.Std.mkHashMap [BEq \u03b1] [Hashable \u03b1] (capacity := 0) : HashMap \u03b1 \u03b2 :=\n  \u27e8.empty capacity, .empty\u27e9\n\ninstance [BEq \u03b1] [Hashable \u03b1] : Inhabited (HashMap \u03b1 \u03b2) where\n  default := mkHashMap\n\ninstance [BEq \u03b1] [Hashable \u03b1] : EmptyCollection (HashMap \u03b1 \u03b2) := \u27e8mkHashMap\u27e9\n\n/-- Make a new empty hash map. -/\n@[inline] def empty [BEq \u03b1] [Hashable \u03b1] : HashMap \u03b1 \u03b2 := mkHashMap\n\nvariable {_ : BEq \u03b1} {_ : Hashable \u03b1}\n\n/-- The number of elements in the hash map. -/\n@[inline] def size (self : HashMap \u03b1 \u03b2) : Nat := self.1.size\n\n/-- Is the map empty? -/\n@[inline] def isEmpty (self : HashMap \u03b1 \u03b2) : Bool := self.size = 0\n\n/--\nInserts key-value pair `a, b` into the map.\nIf an element equal to `a` is already in the map, it is replaced by `b`.\n-/\ndef insert (self : HashMap \u03b1 \u03b2) (a : \u03b1) (b : \u03b2) : HashMap \u03b1 \u03b2 := \u27e8self.1.insert a b, self.2.insert\u27e9\n\n/--\nSimilar to `insert`, but also returns a boolean flag indicating whether an existing entry has been\nreplaced with `a \u21a6 b`.\n-/\n@[inline] def insert' (m : HashMap \u03b1 \u03b2) (a : \u03b1) (b : \u03b2) : HashMap \u03b1 \u03b2 \u00d7 Bool :=\n  let old := m.size\n  let m' := m.insert a b\n  let replaced := old == m'.size\n  (m', replaced)\n\n/--\nRemoves key `a` from the map. If it does not exist in the map, the map is returned unchanged.\n-/\n@[inline] def erase (self : HashMap \u03b1 \u03b2) (a : \u03b1) : HashMap \u03b1 \u03b2 := \u27e8self.1.erase a, self.2.erase\u27e9\n\n/-- Given a key `a`, returns a key-value pair in the map whose key compares equal to `a`. -/\n@[inline] def findEntry? (self : HashMap \u03b1 \u03b2) (a : \u03b1) : Option (\u03b1 \u00d7 \u03b2) := self.1.findEntry? a\n\n/-- Looks up an element in the map with key `a`. -/\n@[inline] def find? (self : HashMap \u03b1 \u03b2) (a : \u03b1) : Option \u03b2 := self.1.find? a\n\n/-- Looks up an element in the map with key `a`. Returns `b\u2080` if the element is not found. -/\n@[inline] def findD (self : HashMap \u03b1 \u03b2) (a : \u03b1) (b\u2080 : \u03b2) : \u03b2 := (self.find? a).getD b\u2080\n\n/-- Looks up an element in the map with key `a`. Panics if the element is not found. -/\n@[inline] def find! [Inhabited \u03b2] (self : HashMap \u03b1 \u03b2) (a : \u03b1) : \u03b2 :=\n  (self.find? a).getD (panic! \"key is not in the map\")\n\ninstance : GetElem (HashMap \u03b1 \u03b2) \u03b1 (Option \u03b2) fun _ _ => True where\n  getElem m k _ := m.find? k\n\n/-- Returns true if the element `a` is in the map. -/\n@[inline] def contains (self : HashMap \u03b1 \u03b2) (a : \u03b1) : Bool := self.1.contains a\n\n/-- Folds a monadic function over the elements in the map (in arbitrary order). -/\n@[inline] def foldM [Monad m] (f : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03b4) (init : \u03b4) (self : HashMap \u03b1 \u03b2) : m \u03b4 :=\n  self.1.foldM f init\n\n/-- Folds a function over the elements in the map (in arbitrary order). -/\n@[inline] def fold (f : \u03b4 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b4) (init : \u03b4) (self : HashMap \u03b1 \u03b2) : \u03b4 := self.1.fold f init\n\n/-- Combines two hashmaps using a monadic function `f` to combine two values at a key. -/\n@[specialize] def mergeWithM [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 \u2192 m \u03b2)\n    (self other : HashMap \u03b1 \u03b2) : m (HashMap \u03b1 \u03b2) :=\n  other.foldM (init := self) fun m k v\u2082 =>\n    match m.find? k with\n    | none => return m.insert k v\u2082\n    | some v\u2081 => return m.insert k (\u2190 f k v\u2081 v\u2082)\n\n/-- Combines two hashmaps using function `f` to combine two values at a key. -/\n@[inline] def mergeWith (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 \u2192 \u03b2) (self other : HashMap \u03b1 \u03b2) : HashMap \u03b1 \u03b2 :=\n  -- Implementing this function directly, rather than via `mergeWithM`, gives\n  -- us less constrained universes.\n  other.fold (init := self) \u03bb map k v\u2082 =>\n    match map.find? k with\n    | none => map.insert k v\u2082\n    | some v\u2081 => map.insert k $ f k v\u2081 v\u2082\n\n/-- Runs a monadic function over the elements in the map (in arbitrary order). -/\n@[inline] def forM [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m PUnit) (self : HashMap \u03b1 \u03b2) : m PUnit := self.1.forM f\n\n/-- Converts the map into a list of key-value pairs. -/\ndef toList (self : HashMap \u03b1 \u03b2) : List (\u03b1 \u00d7 \u03b2) := self.fold (init := []) fun r k v => (k, v)::r\n\n/-- Converts the map into an array of key-value pairs. -/\ndef toArray (self : HashMap \u03b1 \u03b2) : Array (\u03b1 \u00d7 \u03b2) :=\n  self.fold (init := #[]) fun r k v => r.push (k, v)\n\n/-- The number of buckets in the hash map. -/\ndef numBuckets (self : HashMap \u03b1 \u03b2) : Nat := self.1.buckets.1.size\n\n/--\nBuilds a `HashMap` from a list of key-value pairs.\nValues of duplicated keys are replaced by their respective last occurrences.\n-/\ndef ofList (l : List (\u03b1 \u00d7 \u03b2)) : HashMap \u03b1 \u03b2 :=\n  l.foldl (init := HashMap.empty) fun m (k, v) => m.insert k v\n\n/-- Variant of `ofList` which accepts a function that combines values of duplicated keys. -/\ndef ofListWith (l : List (\u03b1 \u00d7 \u03b2)) (f : \u03b2 \u2192 \u03b2 \u2192 \u03b2) : HashMap \u03b1 \u03b2 :=\n  l.foldl (init := HashMap.empty) fun m p =>\n    match m.find? p.1 with\n    | none   => m.insert p.1 p.2\n    | some v => m.insert p.1 <| f v p.2\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/HashMap/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737473266735, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3889664596998018}}
{"text": "/- -----------------------------------------------------------------------\nAlgebras for endofunctors.\n----------------------------------------------------------------------- -/\n\nimport ..c1_basic\nimport ..c2_limits\n\nnamespace qp\n\nopen stdaux\n\nuniverse variables \u2113obj \u2113hom\n\n\n\n/- -----------------------------------------------------------------------\nThe category of algebras for an endofunctor.\n----------------------------------------------------------------------- -/\n\n/-! #brief An algebra for an endofunctor.\n-/\nstructure EndoAlg {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    : Type (max \u2113obj \u2113hom)\n:= (carr : C^.obj)\n   (hom : C^.hom (F^.obj carr) carr)\n\n/-! #brief Helper for proving equality of EndoAlg.\n-/\ntheorem EndoAlg.eq {C : Cat.{\u2113obj \u2113hom}} {F : Fun C C}\n    : \u2200 {X\u2081 X\u2082 : EndoAlg F}\n        (\u03c9carr : X\u2081^.carr = X\u2082^.carr)\n        (\u03c9hom : (X\u2081^.carr = X\u2082^.carr) \u2192 X\u2081^.hom == X\u2082^.hom)\n      , X\u2081 = X\u2082\n| (EndoAlg.mk carr hom\u2081) (EndoAlg.mk .(carr) hom\u2082)\n  (eq.refl .(carr)) \u03c9hom\n:= begin\n     assert \u03c9hom' : hom\u2081 = hom\u2082,\n     { apply eq_of_heq, exact \u03c9hom rfl },\n     subst \u03c9hom'\n   end\n\n/-! #brief An algebra homomorphism for an endofunctor.\n-/\nstructure EndoAlgHom {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    (X Y : EndoAlg F)\n    : Type (max \u2113obj \u2113hom)\n:= (hom : C^.hom X^.carr Y^.carr)\n   (comm : C^.circ Y^.hom (F^.hom hom) = C^.circ hom X^.hom)\n\n/-! #brief Congruence for endo algebra homomorphisms.\n-/\ntheorem EndoAlgHom.congr_hom {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    {X Y : EndoAlg F}\n    : \u2200 (h\u2081 h\u2082 : EndoAlgHom F X Y)\n        (\u03c9 : h\u2081 = h\u2082)\n      , h\u2081^.hom = h\u2082^.hom\n| h .(h) (eq.refl .(h)) := rfl\n\n/-! #brief A helper for proving two homomorphisms are equal.\n-/\ntheorem EndoAlgHom.eq {C : Cat.{\u2113obj \u2113hom}} {F : Fun C C}\n    {X Y : EndoAlg F}\n    : \u2200 {F\u2081 F\u2082 : EndoAlgHom F X Y}\n        (\u03c9 : F\u2081^.hom = F\u2082^.hom)\n      , F\u2081 = F\u2082\n| (EndoAlgHom.mk hom comm\u2081) (EndoAlgHom.mk .(hom) comm\u2082) (eq.refl .(hom))\n:= rfl\n\n/-! #brief A helper for proving two homomorphisms are heterogeneously equal.\n-/\ntheorem EndoAlgHom.heq {C : Cat.{\u2113obj \u2113hom}} {F : Fun C C}\n    : \u2200 {X\u2081 Y\u2081 X\u2082 Y\u2082 : EndoAlg F}\n        {F\u2081 : EndoAlgHom F X\u2081 Y\u2081} {F\u2082 : EndoAlgHom F X\u2082 Y\u2082}\n        (\u03c9X : X\u2081 = X\u2082) (\u03c9Y : Y\u2081 = Y\u2082)\n        (\u03c9F : F\u2081^.hom == F\u2082^.hom)\n      , F\u2081 == F\u2082\n| X Y .(X) .(Y) (EndoAlgHom.mk f \u03c9\u2081) (EndoAlgHom.mk .(f) \u03c9\u2082)\n(eq.refl .(X)) (eq.refl .(Y)) (heq.refl .(f))\n:= heq.refl _\n\n\n/-! #brief The identity homomorphism.\n-/\ndefinition EndoAlgHom.id {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    (X : EndoAlg F)\n    : EndoAlgHom F X X\n:= { hom := C^.id X^.carr\n   , comm := by rw [F^.hom_id, C^.circ_id_right, C^.circ_id_left]\n   }\n\n/-! #brief The composition of two homomorphisms.\n-/\ndefinition EndoAlgHom.comp {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    {X Y Z : EndoAlg F}\n    (g : EndoAlgHom F Y Z)\n    (f : EndoAlgHom F X Y)\n    : EndoAlgHom F X Z\n:= { hom := C^.circ g^.hom f^.hom\n   , comm\n      := begin\n           rw [-C^.circ_assoc, -f^.comm],\n           rw [C^.circ_assoc, -g^.comm],\n           rw [-C^.circ_assoc, F^.hom_circ]\n         end\n   }\n\n/-! #brief The category of algebras for an endofunctor.\n-/\ndefinition EndoAlgCat {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    : Cat\n:= { obj := EndoAlg F\n   , hom := EndoAlgHom F\n   , id := EndoAlgHom.id F\n   , circ := @EndoAlgHom.comp C F\n   , circ_assoc := \u03bb X Y Z W h g f, EndoAlgHom.eq C^.circ_assoc\n   , circ_id_left := \u03bb X Y f, EndoAlgHom.eq C^.circ_id_left\n   , circ_id_right := \u03bb X Y f, EndoAlgHom.eq C^.circ_id_right\n   }\n\n/-! #brief Natural transformations induce functors between algebra categories.\n-/\ndefinition NatTrans.EndoAlgFun {C : Cat.{\u2113obj \u2113hom}}\n    {F\u2081 F\u2082 : Fun C C}\n    (\u03b7 : NatTrans F\u2081 F\u2082)\n    : Fun (EndoAlgCat F\u2082) (EndoAlgCat F\u2081)\n:= { obj := \u03bb alg, { carr := alg^.carr\n                   , hom := alg^.hom \u2218\u2218 \u03b7^.com alg^.carr\n                   }\n   , hom := \u03bb alg\u2081 alg\u2082 f\n            , { hom := f^.hom\n              , comm := begin\n                          apply eq.trans (eq.symm C^.circ_assoc),\n                          rw \u03b7^.natural f^.hom,\n                          apply eq.trans C^.circ_assoc,\n                          apply eq.trans (Cat.circ.congr_left f^.comm),\n                          exact eq.symm C^.circ_assoc\n                        end\n              }\n   , hom_id := \u03bb alg\u2081, rfl\n   , hom_circ := \u03bb alg\u2081 alg\u2082 alg\u2083 g f, rfl\n   }\n\n/-! #brief Natural isomorphisms induce bijections of algebra categories.\n-/\ndefinition NatIso.EndoAlgBij.lem\u2081 {C : Cat.{\u2113obj \u2113hom}}\n    {F\u2081 F\u2082 : Fun C C}\n    {\u03b7\u2081\u2082 : NatTrans F\u2081 F\u2082}\n    {\u03b7\u2082\u2081 : NatTrans F\u2082 F\u2081}\n    (\u03b7_iso : NatIso \u03b7\u2081\u2082 \u03b7\u2082\u2081)\n    : NatTrans.EndoAlgFun \u03b7\u2081\u2082 \u25a1\u25a1 NatTrans.EndoAlgFun \u03b7\u2082\u2081 = Fun.id (EndoAlgCat F\u2081)\n:= Fun.eq\n    (\u03bb alg\n     , EndoAlg.eq rfl\n         (\u03bb \u03c9, heq_of_eq\n                 begin\n                   apply eq.trans (eq.symm C^.circ_assoc),\n                   refine eq.symm (eq.trans (eq.symm C^.circ_id_right) (eq.symm _)),\n                   apply Cat.circ.congr_right,\n                   exact (\u03b7_iso^.com alg^.carr)^.id\u2081\n                 end))\n    (\u03bb \u03c9 alg\u2081 alg\u2082 f\n     , begin\n         apply EndoAlgHom.heq (\u03c9 alg\u2081) (\u03c9 alg\u2082),\n         apply heq.refl\n       end)\n\n/-! #brief Natural isomorphisms induce bijections of algebra categories.\n-/\ndefinition NatIso.EndoAlgBij {C : Cat.{\u2113obj \u2113hom}}\n    {F\u2081 F\u2082 : Fun C C}\n    {\u03b7\u2081\u2082 : NatTrans F\u2081 F\u2082}\n    {\u03b7\u2082\u2081 : NatTrans F\u2082 F\u2081}\n    (\u03b7_iso : NatIso \u03b7\u2081\u2082 \u03b7\u2082\u2081)\n    : Cat.Bij \u03b7\u2082\u2081^.EndoAlgFun \u03b7\u2081\u2082^.EndoAlgFun\n:= { id\u2081 := NatIso.EndoAlgBij.lem\u2081 \u03b7_iso\n   , id\u2082 := NatIso.EndoAlgBij.lem\u2081 \u03b7_iso^.flip\n   }\n\n\n\n/- -----------------------------------------------------------------------\nInitial algebras.\n----------------------------------------------------------------------- -/\n\n/-! #brief Initial objects in EndoAlgCat are special.\n-/\n@[class] definition HasInitAlg {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n:= HasInit (EndoAlgCat F)\n\n/-! #brief Initial algebras are preserved by natural isomorphisms.\n-/\ndefinition NatIso.EndoAlgBij.HasInitAlg\u2081 {C : Cat.{\u2113obj \u2113hom}}\n    {F\u2081 F\u2082 : Fun C C}\n    (F\u2082_HasInitAlg : HasInitAlg F\u2082)\n    {\u03b7\u2081\u2082 : NatTrans F\u2081 F\u2082}\n    {\u03b7\u2082\u2081 : NatTrans F\u2082 F\u2081}\n    (\u03b7_iso : NatIso \u03b7\u2081\u2082 \u03b7\u2082\u2081)\n    : HasInitAlg F\u2081\n:= @PresInit.HasInit _ _ F\u2082_HasInitAlg\n      \u03b7\u2081\u2082^.EndoAlgFun \u03b7_iso^.EndoAlgBij^.PresInit\u2082\n\n/-! #brief Initial algebras are preserved by natural isomorphisms.\n-/\ndefinition NatIso.EndoAlgBij.HasInitAlg\u2082 {C : Cat.{\u2113obj \u2113hom}}\n    {F\u2081 F\u2082 : Fun C C}\n    (F\u2081_HasInitAlg : HasInitAlg F\u2081)\n    {\u03b7\u2081\u2082 : NatTrans F\u2081 F\u2082}\n    {\u03b7\u2082\u2081 : NatTrans F\u2082 F\u2081}\n    (\u03b7_iso : NatIso \u03b7\u2081\u2082 \u03b7\u2082\u2081)\n    : HasInitAlg F\u2082\n:= NatIso.EndoAlgBij.HasInitAlg\u2081 F\u2081_HasInitAlg \u03b7_iso^.flip\n\n/-! #brief An initial algebra.\n-/\ndefinition initalg {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : EndoAlg F\n:= @init _ F_HasInitAlg\n\n/-! #brief The carrier of an initial algebra.\n-/\ndefinition initalg.carr {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : C^.obj\n:= (initalg F)^.carr\n\n/-! #brief The structure hom of an initial algebra.\n-/\ndefinition initalg.hom {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : C^.hom (F^.obj (initalg.carr F)) (initalg.carr F)\n:= (initalg F)^.hom\n\n/-! #brief Doubling the initial algebra.\n-/\ndefinition initalg.double {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : EndoAlg F\n:= { carr := F^.obj (initalg.carr F)\n   , hom := F^.hom (initalg.hom F)\n   }\n\n/-! #brief The inverse structure hom of an initial algebra.\n-/\ndefinition initalg.unhom {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : C^.hom (initalg.carr F) (F^.obj (initalg.carr F))\n:= (@init_hom _ F_HasInitAlg (initalg.double F))^.hom\n\n/-! #brief initalg.hom and initalg.unhom are an iso pair.\n-/\ndefinition initalg.iso {C : Cat.{\u2113obj \u2113hom}} (F : Fun C C)\n    [F_HasInitAlg : HasInitAlg F]\n    : Iso (initalg.hom F) (initalg.unhom F)\n:= let \u03c6 : EndoAlgHom F (initalg F) (initalg F)\n        := { hom := C^.circ (initalg.hom F) (initalg.unhom F)\n           , comm := begin\n                       repeat { rw -C^.circ_assoc },\n                       apply Cat.circ.congr_right,\n                       apply eq.trans F^.hom_circ,\n                       exact (@init_hom _ F_HasInitAlg (initalg.double F))^.comm\n                     end\n           }\nin let \u03c9\u03c6 : \u03c6 = EndoAlgHom.id F (initalg F)\n         := init_hom.uniq' (EndoAlgCat F)\nin let \u03c9 : initalg.hom F \u2218\u2218 initalg.unhom F = \u27e8\u27e8initalg.carr F\u27e9\u27e9\n        := begin\n              refine @eq.trans _ _ \u03c6^.hom _ rfl _,\n              refine @eq.trans _ _ (EndoAlgHom.id F _)^.hom _ _ rfl,\n              rw \u03c9\u03c6\n            end\nin { id\u2081 := begin\n              apply eq.symm,\n              apply eq.trans (eq.symm F^.hom_id),\n              refine eq.trans _ (@init_hom _ F_HasInitAlg (initalg.double F))^.comm,\n              refine eq.trans _ F^.hom_circ,\n              exact congr_arg _ (eq.symm \u03c9),\n            end\n   , id\u2082 := \u03c9\n   }\n\n\n\n/- -----------------------------------------------------------------------\nAd\u00e1mek's theorem.\n----------------------------------------------------------------------- -/\n\n/-! #brief Action of the functor used in Ad\u00e1mek's construction on objects.\n-/\ndefinition AdamekFun.obj {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    (n : \u2115)\n    : C^.obj\n:= (Fun.iter_comp F n)^.obj (init C)\n\n@[simp] theorem AdamekFun.obj.simp {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    {F : Fun C C}\n    {n : \u2115}\n    : AdamekFun.obj F (nat.succ n) = F^.obj (AdamekFun.obj F n)\n:= rfl\n\n/-! #brief Action of the functor used in Ad\u00e1mek's construction on homs.\n-/\ndefinition AdamekFun.hom {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    : \u2200 (n\u2081 n\u2082 : \u2115) (m : \u2115) (\u03c9m : n\u2082 = m + n\u2081)\n      , C^.hom (AdamekFun.obj F n\u2081) (AdamekFun.obj F n\u2082)\n| 0 .(m) m (eq.refl .(m)) := init_hom (AdamekFun.obj F m)\n| (nat.succ n\u2081) .(m + nat.succ n\u2081) m (eq.refl .(m + nat.succ n\u2081))\n:= F^.hom (AdamekFun.hom n\u2081 (m + n\u2081) m rfl)\n\n@[simp] theorem AdamekFun.hom.simp {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    {F : Fun C C}\n    : \u2200 {n\u2081 n\u2082 : \u2115} {m : \u2115} {\u03c9m : nat.succ n\u2082 = m + nat.succ n\u2081}\n      , AdamekFun.hom F (nat.succ n\u2081) (nat.succ n\u2082) m \u03c9m\n         = F^.hom (AdamekFun.hom F n\u2081 n\u2082 m (nat.succ.inj \u03c9m))\n| 0 .(m) m (eq.refl .(nat.succ m)) := rfl\n| (nat.succ n\u2081) .(m + nat.succ n\u2081) m (eq.refl .(nat.succ (m + nat.succ n\u2081)))\n:= rfl\n\n/-! #brief Congruence for the Ad\u00e1mek functor on homs.\n-/\ndefinition AdamekFun.hcongr_hom {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    {F : Fun C C}\n    : \u2200 {n\u2081 n\u2082 : \u2115} {m : \u2115} {\u03c9m : n\u2082 = m + n\u2081}\n        (p\u2081 p\u2082 : \u2115) (q : \u2115)\n        (\u03c9np\u2081 : n\u2081 = p\u2081)\n        (\u03c9np\u2082 : n\u2082 = p\u2082)\n        (\u03c9mq : m = q)\n      , AdamekFun.hom F n\u2081 n\u2082 m \u03c9m\n         == AdamekFun.hom F p\u2081 p\u2082 q begin subst \u03c9np\u2081, subst \u03c9np\u2082, subst \u03c9mq, exact \u03c9m end\n| n\u2081 n\u2082 m \u03c9m .(n\u2081) .(n\u2082) .(m)\n(eq.refl .(n\u2081)) (eq.refl .(n\u2082)) (eq.refl .(m))\n:= heq.refl _\n\n/-! #brief Congruence for the Ad\u00e1mek functor on homs.\n-/\ndefinition AdamekFun.congr_hom {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    {F : Fun C C}\n    : \u2200 {n\u2081 n\u2082 : \u2115} {m : \u2115} {\u03c9m : n\u2082 = m + n\u2081}\n        (q : \u2115)\n        (\u03c9mq : m = q)\n      , AdamekFun.hom F n\u2081 n\u2082 m \u03c9m\n         = AdamekFun.hom F n\u2081 n\u2082 q begin subst \u03c9mq, exact \u03c9m end\n| n\u2081 n\u2082 m \u03c9m .(m) (eq.refl .(m)) := rfl\n\n/-! #brief The functor used in Ad\u00e1mek's construction.\n-/\ndefinition AdamekFun {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    : Fun NatCat C\n:= { obj := AdamekFun.obj F\n   , hom := \u03bb x y \u03c9xy, AdamekFun.hom F x y (y - x) (eq.symm (nat.sub_add_cancel \u03c9xy))\n   , hom_id\n      := \u03bb n\n         , begin\n             dsimp [NatCat] at n,\n             induction n with n rec,\n             { apply eq.symm, apply init_hom.uniq },\n             simp,\n             exact eq.trans (Fun.congr_hom rec) F^.hom_id,\n           end\n   , hom_circ\n      := \u03bb x y z g f\n         , sorry\n   }\n\n/-! #brief Structure hom for the co-cone used in Ad\u00e1mek's construction.\n-/\ndefinition Adamek.CoCone.hom {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    (x : EndoAlg F)\n    : \u2200 (n : \u2115)\n      , C^.hom (AdamekFun.obj F n) x^.carr\n| 0 := (init_hom x^.carr)\n| (nat.succ n) := C^.circ x^.hom (F^.hom (Adamek.CoCone.hom n))\n\n/-! #brief Commutative property for the co-cone used in Ad\u00e1mek's construction.\n-/\ndefinition Adamek.CoCone.comm {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    (x : EndoAlg F)\n    : \u2200 {n\u2081 n\u2082 : \u2115} (\u03c9n : n\u2081 \u2264 n\u2082)\n      , Adamek.CoCone.hom F x n\u2081\n         = Adamek.CoCone.hom F x n\u2082\n            \u2218\u2218 AdamekFun.hom F n\u2081 n\u2082 (n\u2082 - n\u2081) (eq.symm (nat.sub_add_cancel \u03c9n))\n| 0 n\u2082 \u03c9n := init_hom.uniq' _\n| (nat.succ n\u2081) 0 \u03c9n := by cases \u03c9n\n| (nat.succ n\u2081) (nat.succ n\u2082) \u03c9n\n:= begin\n     dsimp [Adamek.CoCone.hom],\n     rw -C^.circ_assoc,\n     apply Cat.circ.congr_right,\n     simp,\n     refine eq.trans _ F^.hom_circ,\n     apply Fun.congr_hom,\n     apply Adamek.CoCone.comm (nat.le_of_succ_le_succ \u03c9n),\n   end\n\n/-! #brief The co-cone used in Ad\u00e1mek's construction.\n-/\ndefinition Adamek.CoCone {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    (x : EndoAlg F)\n    : CoCone (AdamekFun F)\n:= CoCone.mk\n    x^.carr\n    (Adamek.CoCone.hom F x)\n    (\u03bb n\u2081 n\u2082 \u03c9n , Adamek.CoCone.comm F x \u03c9n)\n\n\n/-! #brief Ad\u00e1mek's construction of initial algebras.\n-/\ndefinition Adamek {C : Cat.{\u2113obj \u2113hom}}\n    [C_HasInit : HasInit C]\n    (F : Fun C C)\n    [Adamek_HasCoLimit : HasCoLimit (AdamekFun F)]\n    [F_PresCoLimit : PresCoLimit (AdamekFun F) F]\n    : HasInitAlg F\n:= HasInit.show\n    { carr := colimit (AdamekFun F)\n    , hom := let ccone : CoCone (F \u25a1\u25a1 AdamekFun F)\n                      := CoCone.mk\n                          (colimit (AdamekFun F))\n                          (\u03bb n, colimit.in (AdamekFun F) (nat.succ n))\n                          (\u03bb n\u2081 n\u2082 \u03c9n, sorry)\n             in let f : C^.hom (colimit (F \u25a1\u25a1 AdamekFun F)) (colimit (AdamekFun F))\n                     := colimit.univ _ ccone\n             in f \u2218\u2218 cast_hom (prescolimit (AdamekFun F) F)\n\n    }\n    (\u03bb A, { hom := colimit.univ _ (Adamek.CoCone F A)\n          , comm := sorry\n          })\n    (\u03bb A h\n     , EndoAlgHom.eq\n        begin\n          cases h with h \u03c9h, dsimp at \u03c9h, dsimp,\n          apply colimit.univ.uniq (Adamek.CoCone F A),\n          intro n, dsimp [Adamek.CoCone, CoCone.mk],\n          induction n with n rec,\n          { apply init_hom.uniq' },\n          { dsimp [Adamek.CoCone.hom],\n            rw rec,\n            apply eq.trans (Cat.circ.congr_right F^.hom_circ),\n            apply eq.trans C^.circ_assoc,\n            apply eq.trans (Cat.circ.congr_left \u03c9h),\n            rw -C^.circ_assoc,\n            apply Cat.circ.congr_right,\n            exact sorry\n          }\n        end)\n\n\nend qp\n", "meta": {"author": "intoverflow", "repo": "qvr", "sha": "0cfcd33fe4bf8d93851a00cec5bfd21e77105d74", "save_path": "github-repos/lean/intoverflow-qvr", "path": "github-repos/lean/intoverflow-qvr/qvr-0cfcd33fe4bf8d93851a00cec5bfd21e77105d74/qp/p1_categories/c3_wtypes/s3_algebras.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.38882215495073924}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport logic.small\nimport category_theory.skeletal\n\n/-!\n# Essentially small categories.\n\nA category given by `(C : Type u) [category.{v} C]` is `w`-essentially small\nif there exists a `small_model C : Type w` equipped with `[small_category (small_model C)]`.\n\nA category is `w`-locally small if every hom type is `w`-small.\n\nThe main theorem here is that a category is `w`-essentially small iff\nthe type `skeleton C` is `w`-small, and `C` is `w`-locally small.\n-/\n\nuniverses w v v' u u'\n\nopen category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nnamespace category_theory\n\n/-- A category is `essentially_small.{w}` if there exists\nan equivalence to some `S : Type w` with `[small_category S]`. -/\nclass essentially_small (C : Type u) [category.{v} C] : Prop :=\n(equiv_small_category : \u2203 (S : Type w) (_ : small_category S), by exactI nonempty (C \u224c S))\n\n/-- Constructor for `essentially_small C` from an explicit small category witness. -/\n\n\n/--\nAn arbitrarily chosen small model for an essentially small category.\n-/\n@[nolint has_inhabited_instance]\ndef small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : Type w :=\nclassical.some (@essentially_small.equiv_small_category C _ _)\n\nnoncomputable\ninstance small_category_small_model\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : small_category (small_model C) :=\nclassical.some (classical.some_spec (@essentially_small.equiv_small_category C _ _))\n\n/--\nThe (noncomputable) categorical equivalence between\nan essentially small category and its small model.\n-/\nnoncomputable\ndef equiv_small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : C \u224c small_model C :=\nnonempty.some (classical.some_spec (classical.some_spec\n  (@essentially_small.equiv_small_category C _ _)))\n\nlemma essentially_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : essentially_small.{w} C \u2194 essentially_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.symm.trans f), },\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.trans f), },\nend\n\n/--\nA category is `w`-locally small if every hom set is `w`-small.\n\nSee `shrink_homs C` for a category instance where every hom set has been replaced by a small model.\n-/\nclass locally_small (C : Type u) [category.{v} C] : Prop :=\n(hom_small : \u2200 X Y : C, small.{w} (X \u27f6 Y) . tactic.apply_instance)\n\ninstance (C : Type u) [category.{v} C] [locally_small.{w} C] (X Y : C) :\n  small (X \u27f6 Y) :=\nlocally_small.hom_small X Y\n\nlemma locally_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : locally_small.{w} C \u2194 locally_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.inverse.obj X) (e.inverse.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.inverse, },\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.functor.obj X) (e.functor.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.functor, },\nend\n\n@[priority 100]\ninstance locally_small_self (C : Type u) [category.{v} C] : locally_small.{v} C := {}\n\n@[priority 100]\ninstance locally_small_of_essentially_small\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : locally_small.{w} C :=\n(locally_small_congr (equiv_small_model C)).mpr (category_theory.locally_small_self _)\n\n/--\nWe define a type alias `shrink_homs C` for `C`. When we have `locally_small.{w} C`,\nwe'll put a `category.{w}` instance on `shrink_homs C`.\n-/\n@[nolint has_inhabited_instance]\ndef shrink_homs (C : Type u) := C\n\nnamespace shrink_homs\n\nsection\nvariables {C' : Type*} -- a fresh variable with no category instance attached\n\n/-- Help the typechecker by explicitly translating from `C` to `shrink_homs C`. -/\ndef to_shrink_homs {C' : Type*} (X : C') : shrink_homs C' := X\n/-- Help the typechecker by explicitly translating from `shrink_homs C` to `C`. -/\ndef from_shrink_homs {C' : Type*} (X : shrink_homs C') : C' := X\n\n@[simp] lemma to_from (X : C') : from_shrink_homs (to_shrink_homs X) = X := rfl\n@[simp] lemma from_to (X : shrink_homs C') : to_shrink_homs (from_shrink_homs X) = X := rfl\n\nend\n\nvariables (C) [locally_small.{w} C]\n\n@[simps]\nnoncomputable\ninstance : category.{w} (shrink_homs C) :=\n{ hom := \u03bb X Y, shrink (from_shrink_homs X \u27f6 from_shrink_homs Y),\n  id := \u03bb X, equiv_shrink _ (\ud835\udfd9 (from_shrink_homs X)),\n  comp := \u03bb X Y Z f g,\n    equiv_shrink _ (((equiv_shrink _).symm f) \u226b ((equiv_shrink _).symm g)), }.\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef functor : C \u2964 shrink_homs C :=\n{ obj := \u03bb X, to_shrink_homs X,\n  map := \u03bb X Y f, equiv_shrink (X \u27f6 Y) f, }\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef inverse : shrink_homs C \u2964 C :=\n{ obj := \u03bb X, from_shrink_homs X,\n  map := \u03bb X Y f, (equiv_shrink (from_shrink_homs X \u27f6 from_shrink_homs Y)).symm f, }\n\n/--\nThe categorical equivalence between `C` and `shrink_homs C`, when `C` is locally small.\n-/\n@[simps]\nnoncomputable\ndef equivalence : C \u224c shrink_homs C :=\nequivalence.mk (functor C) (inverse C)\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n\nend shrink_homs\n\n/--\nA category is essentially small if and only if\nthe underlying type of its skeleton (i.e. the \"set\" of isomorphism classes) is small,\nand it is locally small.\n-/\ntheorem essentially_small_iff (C : Type u) [category.{v} C] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) \u2227 locally_small.{w} C :=\nbegin\n  -- This theorem is the only bit of real work in this file.\n  fsplit,\n  { intro h,\n    fsplit,\n    { rcases h with \u27e8S, \ud835\udcae, \u27e8e\u27e9\u27e9,\n      resetI,\n      refine \u27e8\u27e8skeleton S, \u27e8_\u27e9\u27e9\u27e9,\n      exact e.skeleton_equiv, },\n    { resetI, apply_instance, }, },\n  { rintro \u27e8\u27e8S, \u27e8e\u27e9\u27e9, L\u27e9,\n    resetI,\n    let e' := (shrink_homs.equivalence C).skeleton_equiv.symm,\n    refine \u27e8\u27e8S, _, \u27e8_\u27e9\u27e9\u27e9,\n    apply induced_category.category (e'.trans e).symm,\n    refine (shrink_homs.equivalence C).trans\n      ((skeleton_equivalence _).symm.trans\n      ((induced_functor (e'.trans e).symm).as_equivalence.symm)), },\nend\n\n/--\nAny thin category is locally small.\n-/\n@[priority 100]\ninstance locally_small_of_thin {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  locally_small.{w} C := {}\n\n/--\nA thin category is essentially small if and only if the underlying type of its skeleton is small.\n-/\ntheorem essentially_small_iff_of_thin\n  {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) :=\nby simp [essentially_small_iff, category_theory.locally_small_of_thin]\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/essentially_small.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3888009495643618}}
{"text": "/-\nCopyright (c) 2022 Arthur Paulino. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Arthur Paulino\n-/\nimport Mathlib.Tactic.Have\n\nexample : Nat := by\n  have h : Nat\n  exact 5\n  exact h\n\nexample : Nat := by\n  have : Nat\n  \u00b7 exact 5\n  exact this\n\nexample {a : Nat} : a = a := by\n  have h : a = a\n  \u00b7 rfl\n  exact h\n\nexample {a : Nat} : a = a := by\n  have : a = a\n  \u00b7 rfl\n  exact this\n\nexample : True := by\n  (let _N) -- FIXME: lean4#1670\n  exact Nat\n  have\n  \u00b7 exact 0\n  have _h : Nat\n  \u00b7 exact 5\n  have _h' x : x < x + 1\n  \u00b7 exact Nat.lt.base x\n  have _h'' (x : Nat) : x < x + 1\n  \u00b7 exact Nat.lt.base x\n  let _m\n  \u00b7 exact 6\n  let _m' x (y : Nat) : x + y = y + x\n  rw [Nat.add_comm]\n  have _q\n  \u00b7 exact 6\n  simp\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/Have.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.38880094140231697}}
{"text": "import multirelation.approach2.tactic\nimport tactic.group1r\n\nvariables {G : Type} [group G]\n  (a b c d e f g h i j k l m n o p q r s t u v w x y z : G)\n\nset_option profiler true\n\n-- example (h : a * b * a * b = 1) :\n--   (b * a * b * a * b\u207b\u00b9 * a\u207b\u00b9 * b\u207b\u00b9 * a\u207b\u00b9 = 1) :=\n-- by group1r using h\n\n-- example (h : a * b * a * b = 1) :\n--   (b * a * b * a * b\u207b\u00b9 * a\u207b\u00b9 * b\u207b\u00b9 * a\u207b\u00b9 = 1) :=\n-- by group_rel [h]\n\n-- example (h : a * b * a * b = 1) :\n--   ((b\u207b\u00b9 * a\u207b\u00b9 * b\u207b\u00b9 * a\u207b\u00b9)^4  = 1) :=\n-- by group1r using h\n\n-- example (h : a * b * a * b = 1) :\n--   ((b\u207b\u00b9 * a\u207b\u00b9 * b\u207b\u00b9 * a\u207b\u00b9)^4  = 1) :=\n-- by group_rel [h]\n\n-- -- example (h : a * b * a * b = 1) :\n-- --   ((b * a * b * a)^100  = 1) :=\n-- -- by group1r using h\n\n-- -- example (h : a * b * a * b = 1) :\n-- --   ((b * a * b * a)^100  = 1) :=\n-- -- by group_rel [h]\n\n-- -- example (h : a * b * a * b = 1) :\n-- --   ((b * a * b * a)^200  = 1) :=\n-- -- by group1r using h\n\n-- example (h : a * b * a * b = 1) :\n--   ((b * a * b * a)^200  = 1) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a * b * a ^ (-1 : \u2124) * b = b * a * b * a ^ (-1 : \u2124)) :=\n-- by group1r using h\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a * b * a ^ (-1 : \u2124) * b = b * a * b * a ^ (-1 : \u2124)) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a ^ 5 * b * a ^ (-5 : \u2124) * b = b * a ^ 5 * b * a ^ (-5 : \u2124)) :=\n-- by group1r using h\n\nexample (h : a * b = b ^ 2 * a) :\n  (a ^ 6 * b * a ^ (-6 : \u2124) * b = b * a ^ 6 * b * a ^ (-6 : \u2124)) :=\nby group1r using h\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a ^ 5 * b * a ^ (-5 : \u2124) * b = b * a ^ 5 * b * a ^ (-5 : \u2124)) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a ^ 2 * b * a ^ (-2 : \u2124) * b = b * a ^ 2 * b * a ^ (-2 : \u2124)) :=\n-- by group1r using h\n\n-- example (h : a * b = b ^ 2 * a) :\n--   (a ^ 2 * b * a ^ (-2 : \u2124) * b = b * a ^ 2 * b * a ^ (-2 : \u2124)) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b * a) :\n--   (a ^ 2 * b ^ 2 = b ^ 2 * a ^ 2) :=\n-- by group1r using h\n\n-- example (h : a * b = b * a) :\n--   (a ^ 2 * b ^ 2 = b ^ 2 * a ^ 2) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b * a) :\n--   (a ^ 5 * b ^ 5 = b ^ 5 * a ^ 5) :=\n-- by group1r using h\n\n-- example (h : a * b = b * a) :\n--   (a ^ 5 * b ^ 5 = b ^ 5 * a ^ 5) :=\n-- by group_rel [h]\n\n-- example (h : a * b = b * a) :\n--   (a ^ 10 * b ^ 10 = b ^ 10 * a ^ 10) :=\n-- by group1r using h\n\n-- example (h : a * b = b * a) :\n--   (a ^ 10 * b ^ 10 = b ^ 10 * a ^ 10) :=\n-- by group_rel [h]\n\n-- example (h : a * b * a ^ (-11 : \u2124) * b ^ 4 = 1) :\n--   (a ^ 10 * b * a * b * a ^ (-11 : \u2124) * b ^ 3 * a ^ (-10 : \u2124)\n--     * b ^ (-4 : \u2124) * a ^ 11 * b\u207b\u00b9 * a\u207b\u00b9 = 1) :=\n-- by group1r using h\n\n-- example (h : a * b * a ^ (-11 : \u2124) * b ^ 4 = 1) :\n--   (a ^ 10 * b * a * b * a ^ (-11 : \u2124) * b ^ 3 * a ^ (-10 : \u2124)\n--     * b ^ (-4 : \u2124) * a ^ 11 * b\u207b\u00b9 * a\u207b\u00b9 = 1) :=\n-- by group_rel [h]\n\n-- example (h : a * (b * c) * a\u207b\u00b9 * (b * c) ^ (-2 : \u2124) = 1) :\n--   a^6 * b * c * a * b * c * a\u207b\u00b9 * (b * c)^(-2 : \u2124) * c\u207b\u00b9 * b\u207b\u00b9 * a^(-5 : \u2124) *\n--     (b * c) * a\u207b\u00b9 * (b * c) ^ (-2 : \u2124) = 1 :=\n-- by group1r using h\n\n-- example (h : a * (b * c) * a\u207b\u00b9 * (b * c) ^ (-2 : \u2124) = 1) :\n--   a^6 * b * c * a * b * c * a\u207b\u00b9 * (b * c)^(-2 : \u2124) * c\u207b\u00b9 * b\u207b\u00b9 * a^(-5 : \u2124) *\n--     (b * c) * a\u207b\u00b9 * (b * c) ^ (-2 : \u2124) = 1 :=\n-- by group_rel [h]\n\n-- example (h : a * b * a\u207b\u00b9 * b ^ (-3 : \u2124) = 1) :\n--   a^4 * b * a * b * a\u207b\u00b9 * b^(-4 : \u2124) * a^(-3 : \u2124) *\n--     b * a\u207b\u00b9 * b ^ (-3 : \u2124) = 1 :=\n-- by group1r using h\n\n-- example (h : a * b * a\u207b\u00b9 * b ^ (-3 : \u2124) = 1) :\n--   a^4 * b * a * b * a\u207b\u00b9 * b^(-3 : \u2124) * b\u207b\u00b9 * a^(-3 : \u2124) *\n--     b * a\u207b\u00b9 * b ^ (-3 : \u2124) = 1 :=\n-- by group_rel [h]\n\n-- example (h : a * b * a\u207b\u00b9 * b ^ (-2 : int) = 1) (h1 : b * c * a\u207b\u00b9 = 1) :\n--   a ^ 2 * b * (a\u207b\u00b9)^2 * b * a^2 * b\u207b\u00b9 * (a\u207b\u00b9)^2 * b\u207b\u00b9 = 1 :=\n-- by group_rel [h, h1]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) * b * ((a * c)\u207b\u00b9) * b * (a * c) * b\u207b\u00b9 * ((a * c)\u207b\u00b9) * b\u207b\u00b9 = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) * b * ((a * c)\u207b\u00b9) * b * (a * c) * b\u207b\u00b9 * ((a * c)\u207b\u00b9) * b\u207b\u00b9 = 1 :=\n-- by group1r using h\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) ^ 2 * b * ((a * c)\u207b\u00b9)^2 * b * (a * c)^2 * b\u207b\u00b9 * ((a * c)\u207b\u00b9)^2 * b\u207b\u00b9 = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) ^ 2 * b * ((a * c)\u207b\u00b9)^2 * b * (a * c)^2 * b\u207b\u00b9 * ((a * c)\u207b\u00b9)^2 * b\u207b\u00b9 = 1 :=\n-- by group1r using h\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) ^ 5 * b * ((a * c)\u207b\u00b9)^5 * b * (a * c)^5 * b\u207b\u00b9 * ((a * c)\u207b\u00b9)^5 * b\u207b\u00b9 = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b ^ (-2 : int) = 1) :\n--   (a * c) ^ 5 * b * ((a * c)\u207b\u00b9)^5 * b * (a * c)^5 * b\u207b\u00b9 * ((a * c)\u207b\u00b9)^5 * b\u207b\u00b9 = 1 :=\n-- by group1r using h\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^2 * b^2 * (a * c) ^ (-2 : int) * b ^ (-2 : int) = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^2 * b^2 * (a * c) ^ (-2 : int) * b ^ (-2 : int) = 1 :=\n-- by group1r using h\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^5 * b^5 * (a * c) ^ (-5 : int) * b ^ (-5 : int) = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^5 * b^5 * (a * c) ^ (-5 : int) * b ^ (-5 : int) = 1 :=\n-- by group1r using h\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^10 * b^10 * (a * c) ^ (-10 : int) * b ^ (-10 : int) = 1 :=\n-- by group_rel [h]\n\n-- example (h : (a * c) * b * (a * c)\u207b\u00b9 * b\u207b\u00b9 = 1) :\n--   (a * c)^10 * b^10 * (a * c) ^ (-10 : int) * b ^ (-10 : int) = 1 :=\n-- by group1r using h\n\n-- example (h : a * b * (a^3)\u207b\u00b9 * b^4 = 1) :\n--   a^2 * b * a * b * a^(-3 : int) * b^3 * a^(-2 : int)\n--   * b^(-4 : int) * a^3 * (a * b)\u207b\u00b9 = 1 :=\n-- by group_rel [h]\n\n-- example (h : a * b * (a^3)\u207b\u00b9 * b^4 = 1) :\n--   a^2 * b * a * b * a^(-3 : int) * b^3 * a^(-2 : int)\n--   * b^(-4 : int) * a^3 * (a * b)\u207b\u00b9 = 1 :=\n-- by group1r using h\n\n-- example (h : a * b * a\u207b\u00b9 * b^(-2 : int) = 1)\n--   (h1 : b * a * b\u207b\u00b9 * a ^ (-2 : int) = 1) :\n--   a = 1 :=\n-- by group_rel [h, h1]\n\nexample (h1 : a * c * a\u207b\u00b9 * c\u207b\u00b9 = 1)\n        (h2 : c\u207b\u00b9 * d * c * d * c\u207b\u00b9 * d\u207b\u00b9 * c * d\u207b\u00b9 = 1)\n        (h3 : a * c\u207b\u00b9 * d * c * a\u207b\u00b9 * c\u207b\u00b9 * d\u207b\u00b9 * d\u207b\u00b9 * c = 1) :\n   a * d * a\u207b\u00b9 * d\u207b\u00b9 * d\u207b\u00b9 = 1 :=\nby group_rel [h1, h2, h3]\n", "meta": {"author": "ChrisHughes24", "repo": "single_relation", "sha": "556990dab75054a1c14717a72c8901dc9f2f01e4", "save_path": "github-repos/lean/ChrisHughes24-single_relation", "path": "github-repos/lean/ChrisHughes24-single_relation/single_relation-556990dab75054a1c14717a72c8901dc9f2f01e4/tests/comparison.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3887156794982269}}
{"text": "import Smt\n\nexample (n m : Int) (h : 0 < m) : n % m < m := by\n  smt [h]\n  sorry\n\nexample (n m k l : Int) : (n - m) * k + l = n*k - m*k + l := by\n  smt\n  sorry\n\nexample (n m k l : Int) (hN : n \u2264 m) (hK : k \u2264 l) : n + k \u2264 m + l := by\n  smt [hN, hK]\n  sorry\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Int/Arith.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3886499305873456}}
{"text": "import M4R.Algebra.Ring.Ring\nimport M4R.Algebra.Ring.SubRing\n\nnamespace M4R\n  open NCSemiring\n\n  structure SMulMap (\u03b1 : Type _) (\u03b2 : Type _) [NCSemiring \u03b1] [NCSemiring \u03b2] extends \u03b1 \u2192\u208a \u03b2 where\n    preserve_mul : \u2200 a b, hom (a * b) = hom a * hom b\n  infixr:25 \" \u2192* \" => SMulMap\n  instance SMulMapFun [NCSemiring \u03b1] [NCSemiring \u03b2] : CoeFun (\u03b1 \u2192* \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  structure SHomomorphism (\u03b1 : Type _) (\u03b2 : Type _) [NCSemiring \u03b1] [NCSemiring \u03b2] extends \u03b1 \u2192* \u03b2 where\n    preserve_one  : hom 1 = 1\n  infixr:25 \" \u2192*\u2081 \" => SHomomorphism\n  instance SHomomorphismFun [NCSemiring \u03b1] [NCSemiring \u03b2] : CoeFun (\u03b1 \u2192*\u2081 \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  structure SIsomorphism (\u03b1 : Type _) (\u03b2 : Type _) [NCSemiring \u03b1] [NCSemiring \u03b2] extends \u03b1 \u2192* \u03b2, \u03b1 \u2245\u208a \u03b2\n  infixr:25 \" \u2245* \" => SIsomorphism\n  instance SIsomorphismFun [NCSemiring \u03b1] [NCSemiring \u03b2] : CoeFun (\u03b1 \u2245* \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  structure RMulMap (\u03b1 : Type _) (\u03b2 : Type _) [NCRing \u03b1] [NCRing \u03b2] extends \u03b1 \u2192* \u03b2, \u03b1 \u2192\u208b \u03b2\n  infixr:25 \" \u2192\u1d63 \" => RMulMap\n  instance RMulMapFun  [NCRing \u03b1] [NCRing \u03b2] : CoeFun (\u03b1 \u2192\u1d63 \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  structure RHomomorphism (\u03b1 : Type _) (\u03b2 : Type _) [NCRing \u03b1] [NCRing \u03b2] extends \u03b1 \u2192\u1d63 \u03b2, \u03b1 \u2192*\u2081 \u03b2\n  infixr:25 \" \u2192\u1d63\u2081 \" => RHomomorphism\n  instance RHomomorphismFun [NCRing \u03b1] [NCRing \u03b2] : CoeFun (\u03b1 \u2192\u1d63\u2081 \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  structure RIsomorphism (\u03b1 : Type _) (\u03b2 : Type _) [NCRing \u03b1] [NCRing \u03b2] extends \u03b1 \u2192\u1d63 \u03b2, \u03b1 \u2245\u208a \u03b2\n  infixr:25 \" \u2245\u1d63 \" => RIsomorphism\n  instance RIsomorphismFun [NCRing \u03b1] [NCRing \u03b2] : CoeFun (\u03b1 \u2245\u1d63 \u03b2) (fun _ => \u03b1 \u2192 \u03b2) where\n    coe := fun f => f.hom\n\n  namespace SMulMap\n    variable [NCSemiring \u03b1] [NCSemiring \u03b2] [NCSemiring \u03b3] [NCSemiring \u03b4]\n\n    def kernel (slm : \u03b1 \u2192* \u03b2) : SubMonoid \u03b1 := slm.toMHomomorphism.kernel\n\n    def image (slm : \u03b1 \u2192* \u03b2) : SubMonoid \u03b2 := slm.toMHomomorphism.image\n\n    protected def comp (hab : \u03b1 \u2192* \u03b2) (hbc : \u03b2 \u2192* \u03b3) : \u03b1 \u2192* \u03b3 where\n      toMHomomorphism := hab.toMHomomorphism.comp hbc.toMHomomorphism\n      preserve_mul    := fun _ _ => by\n        simp only [MHomomorphism.comp, Function.comp]\n        rw [hab.preserve_mul, hbc.preserve_mul]\n\n    protected def Identity : \u03b1 \u2192* \u03b1 where\n      toMHomomorphism := MHomomorphism.Identity\n      preserve_mul    := fun _ _ => rfl\n\n    protected theorem preserve_one_of_1_in_image {f : \u03b1 \u2192* \u03b2} (h1 : 1 \u2208 Function.image f.hom) : f 1 = 1 := by\n      let \u27e8x, hx\u27e9 := h1; rw [\u2190hx, \u2190mul_one x, f.preserve_mul, hx, one_mul]\n\n    protected theorem preserve_one_of_surjective {f : \u03b1 \u2192* \u03b2} (hf : Function.surjective f.hom) : f 1 = 1 :=\n      SMulMap.preserve_one_of_1_in_image (hf 1)\n\n    protected theorem preserve_pow_ne_zero (f : \u03b1 \u2192* \u03b2) (x : \u03b1) {n : Nat} (hn : n \u2260 0) : f (x ^ n) = f x ^ n := by\n      have : \u2200 n : Nat, f (x ^ n.succ) = f x ^ n.succ := fun n => by\n        induction n with\n        | zero      => rw [pow_nat_1, pow_nat_1]\n        | succ n ih => rw [pow_nat_succ, f.preserve_mul, ih, \u2190pow_nat_succ]\n      cases n; contradiction; exact this _\n\n    protected def Product (f : \u03b1 \u2192* \u03b3) (g : \u03b2 \u2192* \u03b4) : \u03b1 \u00d7 \u03b2 \u2192* \u03b3 \u00d7 \u03b4 where\n      toMHomomorphism := f.toMHomomorphism.Product g.toMHomomorphism\n      preserve_mul    := fun (a\u2081, b\u2081) (a\u2082, b\u2082) => congr (congrArg Prod.mk (f.preserve_mul a\u2081 a\u2082)) (g.preserve_mul b\u2081 b\u2082)\n\n    protected noncomputable def MultiProd_cons {\u03b9 : Type _} (f\u03b9 : \u03b9 \u2192 Type _) [\u2200 i, NCSemiring (f\u03b9 i)] {s : Finset \u03b9} {a : \u03b9} (ha : a \u2209 s) :\n      MultiProd (fun i : s => f\u03b9 i.val) \u00d7 f\u03b9 a \u2192* MultiProd (fun i : s.cons a ha => f\u03b9 i.val) where\n        toMHomomorphism := MHomomorphism.MultiProd_cons f\u03b9 ha\n        preserve_mul    := fun x y => funext fun \u27e8i, hi\u27e9 => by\n          simp only [MHomomorphism.MultiProd_cons, multi_product.Mul_def, product_mul]\n          byCases h : i = a\n          { subst h; simp only [dite_true] }\n          { simp only [h, dite_false] }\n\n  end SMulMap\n\n  namespace SHomomorphism\n    variable [NCSemiring \u03b1] [NCSemiring \u03b2] [NCSemiring \u03b3]\n\n    def kernel (sh : \u03b1 \u2192*\u2081 \u03b2) : SubMonoid \u03b1 := sh.toSMulMap.kernel\n\n    def image (sh : \u03b1 \u2192*\u2081 \u03b2) : SubSemiring \u03b2 where\n      toSubMonoid := sh.toSMulMap.image\n      has_one     := \u27e81, sh.preserve_one\u27e9\n      mul_closed  := fun _ \u27e8a, ha\u27e9 _ \u27e8b, hb\u27e9 => \u27e8a * b, ha \u25b8 hb \u25b8 sh.preserve_mul a b\u27e9\n\n    protected def comp (hab : \u03b1 \u2192*\u2081 \u03b2) (hbc : \u03b2 \u2192*\u2081 \u03b3) : \u03b1 \u2192*\u2081 \u03b3 where\n      toSMulMap    := hab.toSMulMap.comp hbc.toSMulMap\n      preserve_one := by\n        simp only [SMulMap.comp, MHomomorphism.comp, Function.comp]\n        rw [hab.preserve_one, hbc.preserve_one]\n\n    protected def Identity : \u03b1 \u2192*\u2081 \u03b1 where\n      toSMulMap    := SMulMap.Identity\n      preserve_one := rfl\n\n    protected theorem preserve_pow (f : \u03b1 \u2192*\u2081 \u03b2) (x : \u03b1) : (n : Nat) \u2192 f (x ^ n) = f x ^ n\n    | 0   => f.preserve_one\n    | n+1 => f.preserve_pow_ne_zero x n.succ_ne_zero\n\n  end SHomomorphism\n\n  namespace SIsomorphism\n    variable [NCSemiring \u03b1] [NCSemiring \u03b2] [NCSemiring \u03b3] [NCSemiring \u03b4]\n\n    protected theorem preserve_one (f : \u03b1 \u2245* \u03b2) : f 1 = 1 :=\n      SMulMap.preserve_one_of_surjective f.toMIsomorphism.to_surjective\n\n    def toSHomomorphism (f : \u03b1 \u2245* \u03b2) : \u03b1 \u2192*\u2081 \u03b2 where\n      toSMulMap    := f.toSMulMap\n      preserve_one := f.preserve_one\n\n    protected def inv_hom (f : \u03b1 \u2245* \u03b2) : \u03b2 \u2192*\u2081 \u03b1 where\n      toMHomomorphism := f.toMIsomorphism.inv_hom\n      preserve_one    := by\n        have : f.toMIsomorphism.inv_hom (f 1) = f.inv (f 1)  := rfl\n        rw [\u2190f.preserve_one, this, f.left_inv]\n      preserve_mul    := fun a b => by\n        let \u27e8_, ha\u27e9 := f.right_inv.surjective a\n        let \u27e8_, hb\u27e9 := f.right_inv.surjective b\n        have : f.toMIsomorphism.inv_hom.hom = f.inv := rfl\n        rw [\u2190ha, \u2190hb, this, \u2190f.preserve_mul, f.left_inv, f.left_inv, f.left_inv]\n\n    protected def symm (f : \u03b1 \u2245* \u03b2) : \u03b2 \u2245* \u03b1 where\n      toSMulMap := f.inv_hom.toSMulMap\n      inv       := f\n      left_inv  := f.right_inv\n      right_inv := f.left_inv\n\n    protected theorem comp (hab : \u03b1 \u2245* \u03b2) (hbc : \u03b2 \u2245* \u03b3) : \u03b1 \u2245* \u03b3 where\n      toSMulMap := hab.toSMulMap.comp hbc.toSMulMap\n      inv       := hab.inv \u2218 hbc.inv\n      left_inv  := fun _ => by\n        have : hab.toSMulMap.comp hbc.toSMulMap = hbc.hom \u2218 hab.hom := rfl\n        rw [this, \u2190Function.comp_eq, \u2190Function.comp_eq, hbc.left_inv, hab.left_inv]\n      right_inv := fun _ => by\n        have : hab.toSMulMap.comp hbc.toSMulMap = hbc.hom \u2218 hab.hom := rfl\n        rw [this, \u2190Function.comp_eq, \u2190Function.comp_eq, hab.right_inv, hbc.right_inv]\n\n    protected noncomputable def of_bijection (f : \u03b1 \u2192* \u03b2) (hf : Function.bijective f.hom) : \u03b1 \u2245* \u03b2 where\n        toSMulMap := f\n        inv       := fun b => Classical.choose (hf.surj b)\n        left_inv  := fun a => hf.inj (Classical.choose_spec (hf.surj (f a)))\n        right_inv := fun a => Classical.choose_spec (hf.surj a)\n\n    protected noncomputable def Identity : \u03b1 \u2245* \u03b1 :=\n      SIsomorphism.of_bijection _ (by apply Function.id_bijective : Function.bijective SHomomorphism.Identity.hom)\n\n    protected def SelfInverse (f : \u03b1 \u2192* \u03b1) (h : \u2200 a, f (f a) = a) : \u03b1 \u2245* \u03b1 where\n      toSMulMap := f\n      inv       := f.hom\n      left_inv  := h\n      right_inv := h\n\n    protected def Product (f : \u03b1 \u2245* \u03b3) (g : \u03b2 \u2245* \u03b4) : \u03b1 \u00d7 \u03b2 \u2245* \u03b3 \u00d7 \u03b4 where\n      toSMulMap := f.toSMulMap.Product g.toSMulMap\n      inv       := (f.toMIsomorphism.Product g.toMIsomorphism).inv\n      left_inv  := (f.toMIsomorphism.Product g.toMIsomorphism).left_inv\n      right_inv := (f.toMIsomorphism.Product g.toMIsomorphism).right_inv\n\n    protected noncomputable def MultiProd_cons {\u03b9 : Type _} (f\u03b9 : \u03b9 \u2192 Type _) [\u2200 i, NCSemiring (f\u03b9 i)] {s : Finset \u03b9} {a : \u03b9} (ha : a \u2209 s) :\n      MultiProd (fun i : s => f\u03b9 i.val) \u00d7 f\u03b9 a \u2245* MultiProd (fun i : s.cons a ha => f\u03b9 i.val) where\n        toSMulMap := SMulMap.MultiProd_cons f\u03b9 ha\n        inv       := (MIsomorphism.MultiProd_cons f\u03b9 ha).inv\n        left_inv  := (MIsomorphism.MultiProd_cons f\u03b9 ha).left_inv\n        right_inv := (MIsomorphism.MultiProd_cons f\u03b9 ha).right_inv\n\n  end SIsomorphism\n\n  namespace RMulMap\n    variable [NCRing \u03b1] [NCRing \u03b2] [NCRing \u03b3] [NCRing \u03b4]\n\n    def kernel (rlm : \u03b1 \u2192\u1d63 \u03b2) : SubGroup \u03b1 := rlm.toGHomomorphism.kernel\n\n    def image (rlm : \u03b1 \u2192\u1d63 \u03b2) : SubGroup \u03b2 := rlm.toGHomomorphism.image\n\n    protected def comp (hab : \u03b1 \u2192\u1d63 \u03b2) (hbc : \u03b2 \u2192\u1d63 \u03b3) : \u03b1 \u2192\u1d63 \u03b3 where\n      toSMulMap    := hab.toSMulMap.comp hbc.toSMulMap\n      preserve_neg := (hab.toGHomomorphism.comp hbc.toGHomomorphism).preserve_neg\n\n    protected def Identity : \u03b1 \u2192\u1d63 \u03b1 where\n      toSMulMap    := SMulMap.Identity\n      preserve_neg := GHomomorphism.Identity.preserve_neg\n\n    protected theorem preserve_one_of_1_in_image {f : \u03b1 \u2192\u1d63 \u03b2} (h1 : 1 \u2208 Function.image f.hom) : f 1 = 1 :=\n      SMulMap.preserve_one_of_1_in_image h1\n\n    protected theorem preserve_one_of_surjective {f : \u03b1 \u2192\u1d63 \u03b2} (hf : Function.surjective f.hom) : f 1 = 1 :=\n      SMulMap.preserve_one_of_surjective hf\n\n    protected theorem preserve_pow_ne_zero (f : \u03b1 \u2192\u1d63 \u03b2) (x : \u03b1) {n : Nat} (hn : n \u2260 0) : f (x ^ n) = f x ^ n :=\n      f.toSMulMap.preserve_pow_ne_zero x hn\n\n    protected def Product (f : \u03b1 \u2192\u1d63 \u03b3) (g : \u03b2 \u2192\u1d63 \u03b4) : \u03b1 \u00d7 \u03b2 \u2192\u1d63 \u03b3 \u00d7 \u03b4 where\n      toSMulMap    := f.toSMulMap.Product g.toSMulMap\n      preserve_neg := (f.toGHomomorphism.Product g.toGHomomorphism).preserve_neg\n\n    protected noncomputable def MultiProd_cons {\u03b9 : Type _} (f\u03b9 : \u03b9 \u2192 Type _) [\u2200 i, NCRing (f\u03b9 i)] {s : Finset \u03b9} {a : \u03b9} (ha : a \u2209 s) :\n      MultiProd (fun i : s => f\u03b9 i.val) \u00d7 f\u03b9 a \u2192\u1d63 MultiProd (fun i : s.cons a ha => f\u03b9 i.val) where\n        toSMulMap    := SMulMap.MultiProd_cons f\u03b9 ha\n        preserve_neg := (GHomomorphism.MultiProd_cons f\u03b9 ha).preserve_neg\n\n  end RMulMap\n\n  namespace RHomomorphism\n    variable [NCRing \u03b1] [NCRing \u03b2] [NCRing \u03b3]\n\n    def kernel (rh : \u03b1 \u2192\u1d63\u2081 \u03b2) : SubGroup \u03b1 := rh.toRMulMap.kernel\n\n    def image (rh : \u03b1 \u2192\u1d63\u2081 \u03b2) : SubRing \u03b2 where\n      toSubSemiring := rh.toSHomomorphism.image\n      neg_closed    := rh.toGHomomorphism.image.neg_closed\n\n    protected def comp (hab : \u03b1 \u2192\u1d63\u2081 \u03b2) (hbc : \u03b2 \u2192\u1d63\u2081 \u03b3) : \u03b1 \u2192\u1d63\u2081 \u03b3 where\n      toRMulMap    := hab.toRMulMap.comp hbc.toRMulMap\n      preserve_one := (hab.toSHomomorphism.comp hbc.toSHomomorphism).preserve_one\n\n    protected def Identity : \u03b1 \u2192\u1d63\u2081 \u03b1 where\n      toRMulMap    := RMulMap.Identity\n      preserve_one := SHomomorphism.Identity.preserve_one\n\n    protected theorem preserve_pow (f : \u03b1 \u2192\u1d63\u2081 \u03b2) (x : \u03b1) (n : Nat) : f (x ^ n) = f x ^ n :=\n      f.toSHomomorphism.preserve_pow x n\n\n  end RHomomorphism\n\n  namespace RIsomorphism\n    variable [NCRing \u03b1] [NCRing \u03b2] [NCRing \u03b3] [NCRing \u03b4]\n\n    protected theorem preserve_one (f : \u03b1 \u2245\u1d63 \u03b2) : f 1 = 1 :=\n      RMulMap.preserve_one_of_surjective f.toMIsomorphism.to_surjective\n\n    def toSHomomorphism (f : \u03b1 \u2245\u1d63 \u03b2) : \u03b1 \u2192\u1d63\u2081 \u03b2 where\n      toRMulMap    := f.toRMulMap\n      preserve_one := f.preserve_one\n\n    protected def inv_hom [NCRing \u03b1] [NCRing \u03b2] (f : \u03b1 \u2245\u1d63 \u03b2) : \u03b2 \u2192\u1d63\u2081 \u03b1 where\n      toMHomomorphism := f.toMIsomorphism.inv_hom\n      preserve_neg    := fun a => by\n        let \u27e8_, ha\u27e9 := f.right_inv.surjective a\n        have : f.toMIsomorphism.inv_hom.hom = f.inv := rfl\n        rw [this, \u2190ha, \u2190f.preserve_neg, f.left_inv, f.left_inv]\n      preserve_one    := by\n        have : f.toMIsomorphism.inv_hom (f 1) = f.inv (f 1) := rfl\n        simp only; rw [\u2190f.preserve_one, this, f.left_inv]\n      preserve_mul    := fun a b => by\n        let \u27e8_, ha\u27e9 := f.right_inv.surjective a\n        let \u27e8_, hb\u27e9 := f.right_inv.surjective b\n        have : f.toMIsomorphism.inv_hom.hom = f.inv := rfl\n        rw [\u2190ha, \u2190hb, this, \u2190f.preserve_mul, f.left_inv, f.left_inv, f.left_inv]\n\n    protected def symm [NCRing \u03b1] [NCRing \u03b2] (f : \u03b1 \u2245\u1d63 \u03b2) : \u03b2 \u2245\u1d63 \u03b1 where\n      toRMulMap := f.inv_hom.toRMulMap\n      inv       := f\n      left_inv  := f.right_inv\n      right_inv := f.left_inv\n\n    protected theorem comp [NCRing \u03b1] [NCRing \u03b2] [NCRing \u03b3] (hab : \u03b1 \u2245\u1d63 \u03b2) (hbc : \u03b2 \u2245\u1d63 \u03b3) : \u03b1 \u2245\u1d63 \u03b3 where\n      toRMulMap := hab.toRMulMap.comp hbc.toRMulMap\n      inv       := hab.inv \u2218 hbc.inv\n      left_inv  := fun _ => by\n        have : hab.toRMulMap.comp hbc.toRMulMap = hbc.hom \u2218 hab.hom := rfl\n        rw [this, \u2190Function.comp_eq, \u2190Function.comp_eq, hbc.left_inv, hab.left_inv]\n      right_inv := fun _ => by\n        have : hab.toRMulMap.comp hbc.toRMulMap = hbc.hom \u2218 hab.hom := rfl\n        rw [this, \u2190Function.comp_eq, \u2190Function.comp_eq, hab.right_inv, hbc.right_inv]\n\n    protected noncomputable def of_bijection [NCRing \u03b1] [NCRing \u03b2] (f : \u03b1 \u2192\u1d63 \u03b2)\n      (hf : Function.bijective f.hom) : \u03b1 \u2245\u1d63 \u03b2 where\n        toRMulMap := f\n        inv       := fun b => Classical.choose (hf.surj b)\n        left_inv  := fun a => hf.inj (Classical.choose_spec (hf.surj (f a)))\n        right_inv := fun a => Classical.choose_spec (hf.surj a)\n\n    protected noncomputable def Identity [NCRing \u03b1] : \u03b1 \u2245\u1d63 \u03b1 :=\n      RIsomorphism.of_bijection _ (by apply Function.id_bijective : Function.bijective RHomomorphism.Identity.hom)\n\n    protected def SelfInverse [NCRing \u03b1] (f : \u03b1 \u2192\u1d63 \u03b1) (h : \u2200 a, f (f a) = a) : \u03b1 \u2245\u1d63 \u03b1 where\n      toRMulMap := f\n      inv       := f.hom\n      left_inv  := h\n      right_inv := h\n\n    protected def Product (f : \u03b1 \u2245\u1d63 \u03b3) (g : \u03b2 \u2245\u1d63 \u03b4) : \u03b1 \u00d7 \u03b2 \u2245\u1d63 \u03b3 \u00d7 \u03b4 where\n      toRMulMap := f.toRMulMap.Product g.toRMulMap\n      inv       := (f.toMIsomorphism.Product g.toMIsomorphism).inv\n      left_inv  := (f.toMIsomorphism.Product g.toMIsomorphism).left_inv\n      right_inv := (f.toMIsomorphism.Product g.toMIsomorphism).right_inv\n\n    protected noncomputable def MultiProd_cons {\u03b9 : Type _} (f\u03b9 : \u03b9 \u2192 Type _) [\u2200 i, NCRing (f\u03b9 i)] {s : Finset \u03b9} {a : \u03b9} (ha : a \u2209 s) :\n      MultiProd (fun i : s => f\u03b9 i.val) \u00d7 f\u03b9 a \u2245\u1d63 MultiProd (fun i : s.cons a ha => f\u03b9 i.val) where\n        toRMulMap := RMulMap.MultiProd_cons f\u03b9 ha\n        inv       := (MIsomorphism.MultiProd_cons f\u03b9 ha).inv\n        left_inv  := (MIsomorphism.MultiProd_cons f\u03b9 ha).left_inv\n        right_inv := (MIsomorphism.MultiProd_cons f\u03b9 ha).right_inv\n\n  end RIsomorphism\n\n  protected def NCSemiring.MulHomLeft [NCSemiring \u03b1] (a : \u03b1) : \u03b1 \u2192\u208a \u03b1 where\n    hom           := (a * \u00b7)\n    preserve_zero := by simp only [NCSemiring.mul_zero]\n    preserve_add  := fun _ _ => by simp only [NCSemiring.mul_distrib_left]\n\n  protected def NCSemiring.MulHomRight [NCSemiring \u03b1] (a : \u03b1) : \u03b1 \u2192\u208a \u03b1 where\n    hom           := (\u00b7 * a)\n    preserve_zero := by simp only [NCSemiring.zero_mul]\n    preserve_add  := fun _ _ => by simp only [NCSemiring.mul_distrib_right]\n\n  protected def NCRing.MulHomLeft [NCRing \u03b1] (a : \u03b1) : \u03b1 \u2192\u208b \u03b1 where\n    toMHomomorphism := NCSemiring.MulHomLeft a\n    preserve_neg    := fun _ => by simp only [NCSemiring.MulHomLeft, NCRing.mul_neg]\n\n  protected def NCRing.MulHomRight [NCRing \u03b1] (a : \u03b1) : \u03b1 \u2192\u208b \u03b1 where\n    toMHomomorphism := NCSemiring.MulHomRight a\n    preserve_neg    := fun _ => by simp only [NCSemiring.MulHomRight, NCRing.neg_mul]\n\nend M4R\n", "meta": {"author": "Hop311", "repo": "M4R", "sha": "ebd1b04af344f9737d290bf8b48b3cde35e9787b", "save_path": "github-repos/lean/Hop311-M4R", "path": "github-repos/lean/Hop311-M4R/M4R-ebd1b04af344f9737d290bf8b48b3cde35e9787b/M4R/Algebra/Ring/RMorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3885723197986091}}
{"text": "\nimport pq_to_group\n\nimport group_theory.free_group\nimport group_theory.quotient_group\n\nuniverse u\n\nsection pq_group_free_quotient\n\nvariables {s : Type u}\n\ninductive pq_group_relation_space_rel : free_group s \u2192 Prop\n| rhd_rel (a b c : s) : pq_group_relation_space_rel (free_group.of a * free_group.of b * (free_group.of a)\u207b\u00b9 * (free_group.of c)\u207b\u00b9)\n| pow_rel (a b : s) (n : \u2124) : pq_group_relation_space_rel ((free_group.of a) ^ n * (free_group.of b)\u207b\u00b9)\n| one_closed : pq_group_relation_space_rel 1\n| mul_closed (x y : free_group s) (hx : pq_group_relation_space_rel x) (hy : pq_group_relation_space_rel y) : pq_group_relation_space_rel (x * y)\n| inv_closed (x : free_group s) (hx : pq_group_relation_space_rel x) : pq_group_relation_space_rel (x\u207b\u00b9)\n| normal_closed (x y : free_group s) (hx : pq_group_relation_space_rel x) : pq_group_relation_space_rel (y * x * y\u207b\u00b9)\n\n\ndef pq_group_relation_space : subgroup (free_group s) := { \n  carrier := pq_group_relation_space_rel,\n  one_mem' := pq_group_relation_space_rel.one_closed,\n  mul_mem' := pq_group_relation_space_rel.mul_closed,\n  inv_mem' := pq_group_relation_space_rel.inv_closed }\n\ninstance pq_group_relation_space_is_normal : (@pq_group_relation_space s).normal :=\nbegin\n  fconstructor,\n  intros x hx y,\n  apply pq_group_relation_space_rel.normal_closed,\n  apply hx,\nend\n\nlemma free_of_in_pq_group_relation_space (a : s) : free_group.of a \u2208 (@pq_group_relation_space s) :=\nbegin\n  suffices : free_group.of a = (free_group.of a * free_group.of a * (free_group.of a)\u207b\u00b9 * (free_group.of a)\u207b\u00b9) * ((free_group.of a) ^ (2 : \u2124) * (free_group.of a)\u207b\u00b9),\n  {\n    rw this,\n    clear this,\n    apply subgroup.mul_mem,\n    apply pq_group_relation_space_rel.rhd_rel,\n    apply pq_group_relation_space_rel.pow_rel,\n  },\n  group,\nend\n\nlemma pq_group_relation_space_is_univ : (@pq_group_relation_space s) = \u22a4 :=\nbegin\n  ext1,\n  simp only [subgroup.mem_top, iff_true],\n  apply free_group.induction_on x;\n  clear x,\n  {\n    apply subgroup.one_mem,\n  },\n  {\n    apply free_of_in_pq_group_relation_space,\n  },\n  {\n    intro x,\n    apply subgroup.inv_mem,\n  },\n  {\n    apply subgroup.mul_mem,\n  },\nend\n\n-- This is not true!!! Everything going forward is useless\n\nvariables {R : subgroup (free_group s)} [hR : R.normal]\ninclude hR\n\ninstance rel_intersection_is_normal : (R \u2293 pq_group_relation_space).normal :=\nbegin\n  fconstructor,\n  intros n hn g,\n  cases hn with hn1 hn2,\n  split,\n  apply hR.conj_mem, assumption,\n  apply pq_group_relation_space_is_normal.conj_mem, assumption,\nend\n\n\ndef pq_group_free_quotient_iso_rel_intersect_forward : pq_group (quotient_group.quotient R) \u2192* quotient_group.quotient (R \u2293 pq_group_relation_space) :=\nbegin\n  fapply pq_morph_to_L_morph_adj,\n  {\n    intro x,\n    induction x,\n    {\n      exact quotient_group.mk x,\n    },\n    {\n      simp only [eq_rec_constant],\n      apply quotient.sound,\n      show x_a\u207b\u00b9 * x_b \u2208 (R \u2293 pq_group_relation_space),\n      have hx : x_a\u207b\u00b9 * x_b \u2208 R := x_p,\n      sorry,\n    },\n  },\n  sorry,\nend\n\ndef pq_group_free_quotient_iso_rel_intersect : pq_group (quotient_group.quotient R) \u2243* quotient_group.quotient (R \u2293 pq_group_relation_space) := { \n  to_fun := sorry,\n  inv_fun := sorry,\n  left_inv := sorry,\n  right_inv := sorry,\n  map_mul' := sorry }\n\nend pq_group_free_quotient\n\n", "meta": {"author": "torstein-vik", "repo": "power-quandle-lean", "sha": "452437602c4be2e6c5ad5f5224b068baabfdf9e1", "save_path": "github-repos/lean/torstein-vik-power-quandle-lean", "path": "github-repos/lean/torstein-vik-power-quandle-lean/power-quandle-lean-452437602c4be2e6c5ad5f5224b068baabfdf9e1/src/pq_group_free_quotient.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3885066385934698}}
{"text": "import Smt\n\ntheorem trans (p q r : Prop) : p = q \u2192 q = r \u2192 p = r := by\n  smt\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Prop/Trans.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3884896312675441}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.int.modeq\nimport Mathlib.algebra.char_p.basic\nimport Mathlib.data.nat.totient\nimport Mathlib.ring_theory.ideal.operations\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Integers mod `n`\n\nDefinition of the integers mod n, and the field structure on the integers mod p.\n\n\n## Definitions\n\n* `zmod n`, which is for integers modulo a nat `n : \u2115`\n\n* `val a` is defined as a natural number:\n  - for `a : zmod 0` it is the absolute value of `a`\n  - for `a : zmod n` with `0 < n` it is the least natural number in the equivalence class\n\n* `val_min_abs` returns the integer closest to zero in the equivalence class.\n\n* A coercion `cast` is defined from `zmod n` into any ring.\nThis is a ring hom if the ring has characteristic dividing `n`\n\n-/\n\nnamespace fin\n\n\n/-!\n## Ring structure on `fin n`\n\nWe define a commutative ring structure on `fin n`, but we do not register it as instance.\nAfterwords, when we define `zmod n` in terms of `fin n`, we use these definitions\nto register the ring structure on `zmod n` as type class instance.\n-/\n\n/-- Negation on `fin n` -/\ndef has_neg (n : \u2115) : Neg (fin n) :=\n  { neg := fun (a : fin n) => { val := int.nat_mod (-\u2191(subtype.val a)) \u2191n, property := sorry } }\n\n/-- Multiplicative commutative semigroup structure on `fin (n+1)`. -/\ndef comm_semigroup (n : \u2115) : comm_semigroup (fin (n + 1)) :=\n  comm_semigroup.mk Mul.mul sorry sorry\n\n/-- Commutative ring structure on `fin (n+1)`. -/\ndef comm_ring (n : \u2115) : comm_ring (fin (n + 1)) :=\n  comm_ring.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry Neg.neg\n    (ring.sub._default add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry Neg.neg) sorry sorry comm_semigroup.mul\n    sorry 1 fin.one_mul fin.mul_one (left_distrib_aux n) sorry sorry\n\nend fin\n\n\n/-- The integers modulo `n : \u2115`. -/\ndef zmod : \u2115 \u2192 Type :=\n  sorry\n\nnamespace zmod\n\n\nprotected instance fintype (n : \u2115) [fact (0 < n)] : fintype (zmod n) :=\n  sorry\n\ntheorem card (n : \u2115) [fact (0 < n)] : fintype.card (zmod n) = n :=\n  nat.cases_on n (fun [_inst_1 : fact (0 < 0)] => False._oldrec (nat.not_lt_zero 0 _inst_1))\n    (fun (n : \u2115) => fintype.card_fin (n + 1)) _inst_1\n\nprotected instance decidable_eq (n : \u2115) : DecidableEq (zmod n) :=\n  sorry\n\nprotected instance has_repr (n : \u2115) : has_repr (zmod n) :=\n  sorry\n\nprotected instance comm_ring (n : \u2115) : comm_ring (zmod n) :=\n  sorry\n\nprotected instance inhabited (n : \u2115) : Inhabited (zmod n) :=\n  { default := 0 }\n\n/-- `val a` is a natural number defined as:\n  - for `a : zmod 0` it is the absolute value of `a`\n  - for `a : zmod n` with `0 < n` it is the least natural number in the equivalence class\n\nSee `zmod.val_min_abs` for a variant that takes values in the integers.\n-/\ndef val {n : \u2115} : zmod n \u2192 \u2115 :=\n  sorry\n\ntheorem val_lt {n : \u2115} [fact (0 < n)] (a : zmod n) : val a < n :=\n  nat.cases_on n (fun [_inst_1 : fact (0 < 0)] (a : zmod 0) => False._oldrec (nat.not_lt_zero 0 _inst_1))\n    (fun (n : \u2115) (a : zmod (Nat.succ n)) => fin.is_lt a) _inst_1 a\n\n@[simp] theorem val_zero {n : \u2115} : val 0 = 0 :=\n  nat.cases_on n (idRhs (val 0 = val 0) rfl) fun (n : \u2115) => idRhs (val 0 = val 0) rfl\n\ntheorem val_cast_nat {n : \u2115} (a : \u2115) : val \u2191a = a % n := sorry\n\nprotected instance char_p (n : \u2115) : char_p (zmod n) n := sorry\n\n@[simp] theorem cast_self (n : \u2115) : \u2191n = 0 :=\n  char_p.cast_eq_zero (zmod n) n\n\n@[simp] theorem cast_self' (n : \u2115) : \u2191n + 1 = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191n + 1 = 0)) (Eq.symm (nat.cast_add_one n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(n + 1) = 0)) (cast_self (n + 1)))) (Eq.refl 0))\n\n/-- Cast an integer modulo `n` to another semiring.\nThis function is a morphism if the characteristic of `R` divides `n`.\nSee `zmod.cast_hom` for a bundled version. -/\ndef cast {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R] {n : \u2115} : zmod n \u2192 R :=\n  sorry\n\n-- see Note [coercion into rings]\n\nprotected instance has_coe_t {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R] (n : \u2115) : has_coe_t (zmod n) R :=\n  has_coe_t.mk cast\n\n@[simp] theorem cast_zero {n : \u2115} {R : Type u_1} [HasZero R] [HasOne R] [Add R] [Neg R] : \u21910 = 0 :=\n  nat.cases_on n (Eq.refl \u21910) fun (n : \u2115) => Eq.refl \u21910\n\ntheorem nat_cast_surjective {n : \u2115} [fact (0 < n)] : function.surjective coe := sorry\n\ntheorem int_cast_surjective {n : \u2115} : function.surjective coe := sorry\n\ntheorem cast_val {n : \u2115} [fact (0 < n)] (a : zmod n) : \u2191(val a) = a := sorry\n\n@[simp] theorem cast_id (n : \u2115) (i : zmod n) : \u2191i = i :=\n  nat.cases_on n (fun (i : zmod 0) => idRhs (\u2191i = i) (int.cast_id i))\n    (fun (n : \u2115) (i : zmod (Nat.succ n)) => idRhs (\u2191(val i) = i) (cast_val i)) i\n\n@[simp] theorem nat_cast_val {n : \u2115} {R : Type u_1} [ring R] [fact (0 < n)] (i : zmod n) : \u2191(val i) = \u2191i :=\n  nat.cases_on n (fun [_inst_2 : fact (0 < 0)] (i : zmod 0) => False._oldrec (nat.not_lt_zero 0 _inst_2))\n    (fun (n : \u2115) (i : zmod (Nat.succ n)) => Eq.refl \u2191(val i)) _inst_2 i\n\n/-! If the characteristic of `R` divides `n`, then `cast` is a homomorphism. -/\n\n@[simp] theorem cast_one {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) : \u21911 = 1 := sorry\n\ntheorem cast_add {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (a : zmod n) (b : zmod n) : \u2191(a + b) = \u2191a + \u2191b := sorry\n\ntheorem cast_mul {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (a : zmod n) (b : zmod n) : \u2191(a * b) = \u2191a * \u2191b := sorry\n\n/-- The canonical ring homomorphism from `zmod n` to a ring of characteristic `n`. -/\ndef cast_hom {n : \u2115} {m : \u2115} (h : m \u2223 n) (R : Type u_1) [ring R] [char_p R m] : zmod n \u2192+* R :=\n  ring_hom.mk coe (cast_one h) (cast_mul h) sorry (cast_add h)\n\n@[simp] theorem cast_hom_apply {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] {h : m \u2223 n} (i : zmod n) : coe_fn (cast_hom h R) i = \u2191i :=\n  rfl\n\n@[simp] theorem cast_sub {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (a : zmod n) (b : zmod n) : \u2191(a - b) = \u2191a - \u2191b :=\n  ring_hom.map_sub (cast_hom h R) a b\n\n@[simp] theorem cast_neg {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (a : zmod n) : \u2191(-a) = -\u2191a :=\n  ring_hom.map_neg (cast_hom h R) a\n\n@[simp] theorem cast_pow {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (a : zmod n) (k : \u2115) : \u2191(a ^ k) = \u2191a ^ k :=\n  ring_hom.map_pow (cast_hom h R) a k\n\n@[simp] theorem cast_nat_cast {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (k : \u2115) : \u2191\u2191k = \u2191k :=\n  ring_hom.map_nat_cast (cast_hom h R) k\n\n@[simp] theorem cast_int_cast {n : \u2115} {R : Type u_1} [ring R] {m : \u2115} [char_p R m] (h : m \u2223 n) (k : \u2124) : \u2191\u2191k = \u2191k :=\n  ring_hom.map_int_cast (cast_hom h R) k\n\n/-! Some specialised simp lemmas which apply when `R` has characteristic `n`. -/\n\n@[simp] theorem cast_one' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] : \u21911 = 1 :=\n  cast_one (dvd_refl n)\n\n@[simp] theorem cast_add' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (a : zmod n) (b : zmod n) : \u2191(a + b) = \u2191a + \u2191b :=\n  cast_add (dvd_refl n) a b\n\n@[simp] theorem cast_mul' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (a : zmod n) (b : zmod n) : \u2191(a * b) = \u2191a * \u2191b :=\n  cast_mul (dvd_refl n) a b\n\n@[simp] theorem cast_sub' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (a : zmod n) (b : zmod n) : \u2191(a - b) = \u2191a - \u2191b :=\n  cast_sub (dvd_refl n) a b\n\n@[simp] theorem cast_pow' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (a : zmod n) (k : \u2115) : \u2191(a ^ k) = \u2191a ^ k :=\n  cast_pow (dvd_refl n) a k\n\n@[simp] theorem cast_nat_cast' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (k : \u2115) : \u2191\u2191k = \u2191k :=\n  cast_nat_cast (dvd_refl n) k\n\n@[simp] theorem cast_int_cast' {n : \u2115} {R : Type u_1} [ring R] [char_p R n] (k : \u2124) : \u2191\u2191k = \u2191k :=\n  cast_int_cast (dvd_refl n) k\n\nprotected instance algebra {n : \u2115} (R : Type u_1) [comm_ring R] [char_p R n] : algebra (zmod n) R :=\n  ring_hom.to_algebra (cast_hom (dvd_refl n) R)\n\ntheorem cast_hom_injective {n : \u2115} (R : Type u_1) [ring R] [char_p R n] : function.injective \u21d1(cast_hom (dvd_refl n) R) := sorry\n\ntheorem cast_hom_bijective {n : \u2115} (R : Type u_1) [ring R] [char_p R n] [fintype R] (h : fintype.card R = n) : function.bijective \u21d1(cast_hom (dvd_refl n) R) := sorry\n\n/-- The unique ring isomorphism between `zmod n` and a ring `R`\nof characteristic `n` and cardinality `n`. -/\ndef ring_equiv {n : \u2115} (R : Type u_1) [ring R] [char_p R n] [fintype R] (h : fintype.card R = n) : zmod n \u2243+* R :=\n  ring_equiv.of_bijective (cast_hom (dvd_refl n) R) (cast_hom_bijective R h)\n\ntheorem int_coe_eq_int_coe_iff (a : \u2124) (b : \u2124) (c : \u2115) : \u2191a = \u2191b \u2194 int.modeq (\u2191c) a b :=\n  char_p.int_coe_eq_int_coe_iff (zmod c) c a b\n\ntheorem nat_coe_eq_nat_coe_iff (a : \u2115) (b : \u2115) (c : \u2115) : \u2191a = \u2191b \u2194 nat.modeq c a b := sorry\n\ntheorem int_coe_zmod_eq_zero_iff_dvd (a : \u2124) (b : \u2115) : \u2191a = 0 \u2194 \u2191b \u2223 a := sorry\n\ntheorem nat_coe_zmod_eq_zero_iff_dvd (a : \u2115) (b : \u2115) : \u2191a = 0 \u2194 b \u2223 a := sorry\n\n@[simp] theorem cast_mod_int (a : \u2124) (b : \u2115) : \u2191(a % \u2191b) = \u2191a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u2191(a % \u2191b) = \u2191a)) (propext (int_coe_eq_int_coe_iff (a % \u2191b) a b))))\n    (int.modeq.mod_modeq a \u2191b)\n\n@[simp] theorem coe_to_nat (p : \u2115) {z : \u2124} (h : 0 \u2264 z) : \u2191(int.to_nat z) = \u2191z := sorry\n\ntheorem val_injective (n : \u2115) [fact (0 < n)] : function.injective val :=\n  nat.cases_on n (fun [_inst_1 : fact (0 < 0)] => id fun (a\u2081 : zmod 0) => False._oldrec (nat.not_lt_zero 0 _inst_1))\n    (fun (n : \u2115) => id fun (a b : zmod (Nat.succ n)) (h : val a = val b) => fin.ext h) _inst_1\n\ntheorem val_one_eq_one_mod (n : \u2115) : val 1 = 1 % n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (val 1 = 1 % n)) (Eq.symm nat.cast_one)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (val \u21911 = 1 % n)) (val_cast_nat 1))) (Eq.refl (1 % n)))\n\ntheorem val_one (n : \u2115) [fact (1 < n)] : val 1 = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (val 1 = 1)) (val_one_eq_one_mod n))) (nat.mod_eq_of_lt _inst_1)\n\ntheorem val_add {n : \u2115} [fact (0 < n)] (a : zmod n) (b : zmod n) : val (a + b) = (val a + val b) % n :=\n  nat.cases_on n (fun [_inst_1 : fact (0 < 0)] (a b : zmod 0) => False._oldrec (nat.not_lt_zero 0 _inst_1))\n    (fun (n : \u2115) (a b : zmod (Nat.succ n)) => fin.val_add a b) _inst_1 a b\n\ntheorem val_mul {n : \u2115} (a : zmod n) (b : zmod n) : val (a * b) = val a * val b % n := sorry\n\nprotected instance nontrivial (n : \u2115) [fact (1 < n)] : nontrivial (zmod n) :=\n  nontrivial.mk\n    (Exists.intro 0\n      (Exists.intro 1\n        fun (h : 0 = 1) =>\n          zero_ne_one\n            (Eq.trans (Eq.trans (eq.mpr (id (Eq._oldrec (Eq.refl (0 = val 0)) val_zero)) (Eq.refl 0)) (congr_arg val h))\n              (val_one n))))\n\n/-- The inversion on `zmod n`.\nIt is setup in such a way that `a * a\u207b\u00b9` is equal to `gcd a.val n`.\nIn particular, if `a` is coprime to `n`, and hence a unit, `a * a\u207b\u00b9 = 1`. -/\ndef inv (n : \u2115) : zmod n \u2192 zmod n :=\n  sorry\n\nprotected instance has_inv (n : \u2115) : has_inv (zmod n) :=\n  has_inv.mk (inv n)\n\ntheorem inv_zero (n : \u2115) : 0\u207b\u00b9 = 0 := sorry\n\ntheorem mul_inv_eq_gcd {n : \u2115} (a : zmod n) : a * (a\u207b\u00b9) = \u2191(nat.gcd (val a) n) := sorry\n\n@[simp] theorem cast_mod_nat (n : \u2115) (a : \u2115) : \u2191(a % n) = \u2191a := sorry\n\ntheorem eq_iff_modeq_nat (n : \u2115) {a : \u2115} {b : \u2115} : \u2191a = \u2191b \u2194 nat.modeq n a b := sorry\n\ntheorem coe_mul_inv_eq_one {n : \u2115} (x : \u2115) (h : nat.coprime x n) : \u2191x * (\u2191x\u207b\u00b9) = 1 := sorry\n\n/-- `unit_of_coprime` makes an element of `units (zmod n)` given\n  a natural number `x` and a proof that `x` is coprime to `n`  -/\ndef unit_of_coprime {n : \u2115} (x : \u2115) (h : nat.coprime x n) : units (zmod n) :=\n  units.mk (\u2191x) (\u2191x\u207b\u00b9) (coe_mul_inv_eq_one x h) sorry\n\n@[simp] theorem cast_unit_of_coprime {n : \u2115} (x : \u2115) (h : nat.coprime x n) : \u2191(unit_of_coprime x h) = \u2191x :=\n  rfl\n\ntheorem val_coe_unit_coprime {n : \u2115} (u : units (zmod n)) : nat.coprime (val \u2191u) n := sorry\n\n@[simp] theorem inv_coe_unit {n : \u2115} (u : units (zmod n)) : \u2191u\u207b\u00b9 = \u2191(u\u207b\u00b9) := sorry\n\ntheorem mul_inv_of_unit {n : \u2115} (a : zmod n) (h : is_unit a) : a * (a\u207b\u00b9) = 1 := sorry\n\ntheorem inv_mul_of_unit {n : \u2115} (a : zmod n) (h : is_unit a) : a\u207b\u00b9 * a = 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a\u207b\u00b9 * a = 1)) (mul_comm (a\u207b\u00b9) a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a * (a\u207b\u00b9) = 1)) (mul_inv_of_unit a h))) (Eq.refl 1))\n\n/-- Equivalence between the units of `zmod n` and\nthe subtype of terms `x : zmod n` for which `x.val` is comprime to `n` -/\ndef units_equiv_coprime {n : \u2115} [fact (0 < n)] : units (zmod n) \u2243 Subtype fun (x : zmod n) => nat.coprime (val x) n :=\n  equiv.mk (fun (x : units (zmod n)) => { val := \u2191x, property := val_coe_unit_coprime x })\n    (fun (x : Subtype fun (x : zmod n) => nat.coprime (val x) n) => unit_of_coprime (val (subtype.val x)) sorry) sorry\n    sorry\n\n@[simp] theorem card_units_eq_totient (n : \u2115) [fact (0 < n)] : fintype.card (units (zmod n)) = nat.totient n := sorry\n\nprotected instance subsingleton_units : subsingleton (units (zmod (bit0 1))) :=\n  subsingleton.intro\n    fun (x y : units (zmod (bit0 1))) =>\n      units.cases_on x\n        fun (x xi : zmod (bit0 1)) (x_val_inv : x * xi = 1) (x_inv_val : xi * x = 1) =>\n          units.cases_on y\n            fun (y yi : zmod (bit0 1)) (y_val_inv : y * yi = 1) (y_inv_val : yi * y = 1) =>\n              of_as_true trivial x y xi yi x_val_inv x_inv_val y_val_inv y_inv_val\n\ntheorem le_div_two_iff_lt_neg (n : \u2115) [hn : fact (n % bit0 1 = 1)] {x : zmod n} (hx0 : x \u2260 0) : val x \u2264 n / bit0 1 \u2194 n / bit0 1 < val (-x) := sorry\n\ntheorem ne_neg_self (n : \u2115) [hn : fact (n % bit0 1 = 1)] {a : zmod n} (ha : a \u2260 0) : a \u2260 -a := sorry\n\ntheorem neg_one_ne_one {n : \u2115} [fact (bit0 1 < n)] : -1 \u2260 1 :=\n  char_p.neg_one_ne_one (zmod n) n\n\n@[simp] theorem neg_eq_self_mod_two (a : zmod (bit0 1)) : -a = a :=\n  of_as_true trivial\n\n@[simp] theorem nat_abs_mod_two (a : \u2124) : \u2191(int.nat_abs a) = \u2191a := sorry\n\n@[simp] theorem val_eq_zero {n : \u2115} (a : zmod n) : val a = 0 \u2194 a = 0 := sorry\n\ntheorem val_cast_of_lt {n : \u2115} {a : \u2115} (h : a < n) : val \u2191a = a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (val \u2191a = a)) (val_cast_nat a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a % n = a)) (nat.mod_eq_of_lt h))) (Eq.refl a))\n\ntheorem neg_val' {n : \u2115} [fact (0 < n)] (a : zmod n) : val (-a) = (n - val a) % n := sorry\n\ntheorem neg_val {n : \u2115} [fact (0 < n)] (a : zmod n) : val (-a) = ite (a = 0) 0 (n - val a) := sorry\n\n/-- `val_min_abs x` returns the integer in the same equivalence class as `x` that is closest to `0`,\n  The result will be in the interval `(-n/2, n/2]`. -/\ndef val_min_abs {n : \u2115} : zmod n \u2192 \u2124 :=\n  sorry\n\n@[simp] theorem val_min_abs_def_zero (x : zmod 0) : val_min_abs x = x :=\n  rfl\n\ntheorem val_min_abs_def_pos {n : \u2115} [fact (0 < n)] (x : zmod n) : val_min_abs x = ite (val x \u2264 n / bit0 1) (\u2191(val x)) (\u2191(val x) - \u2191n) :=\n  nat.cases_on n (fun [_inst_1 : fact (0 < 0)] (x : zmod 0) => False._oldrec (nat.not_lt_zero 0 _inst_1))\n    (fun (n : \u2115) (x : zmod (Nat.succ n)) => Eq.refl (val_min_abs x)) _inst_1 x\n\n@[simp] theorem coe_val_min_abs {n : \u2115} (x : zmod n) : \u2191(val_min_abs x) = x := sorry\n\ntheorem nat_abs_val_min_abs_le {n : \u2115} [fact (0 < n)] (x : zmod n) : int.nat_abs (val_min_abs x) \u2264 n / bit0 1 := sorry\n\n@[simp] theorem val_min_abs_zero (n : \u2115) : val_min_abs 0 = 0 := sorry\n\n@[simp] theorem val_min_abs_eq_zero {n : \u2115} (x : zmod n) : val_min_abs x = 0 \u2194 x = 0 := sorry\n\ntheorem cast_nat_abs_val_min_abs {n : \u2115} [fact (0 < n)] (a : zmod n) : \u2191(int.nat_abs (val_min_abs a)) = ite (val a \u2264 n / bit0 1) a (-a) := sorry\n\n@[simp] theorem nat_abs_val_min_abs_neg {n : \u2115} (a : zmod n) : int.nat_abs (val_min_abs (-a)) = int.nat_abs (val_min_abs a) := sorry\n\ntheorem val_eq_ite_val_min_abs {n : \u2115} [fact (0 < n)] (a : zmod n) : \u2191(val a) = val_min_abs a + ite (val a \u2264 n / bit0 1) 0 \u2191n := sorry\n\ntheorem prime_ne_zero (p : \u2115) (q : \u2115) [hp : fact (nat.prime p)] [hq : fact (nat.prime q)] (hpq : p \u2260 q) : \u2191q \u2260 0 := sorry\n\nend zmod\n\n\nnamespace zmod\n\n\n/-- Field structure on `zmod p` if `p` is prime. -/\nprotected instance field (p : \u2115) [fact (nat.prime p)] : field (zmod p) :=\n  field.mk comm_ring.add sorry comm_ring.zero sorry sorry comm_ring.neg comm_ring.sub sorry sorry comm_ring.mul sorry\n    comm_ring.one sorry sorry sorry sorry sorry has_inv.inv sorry (mul_inv_cancel_aux p) (inv_zero p)\n\nend zmod\n\n\ntheorem ring_hom.ext_zmod {n : \u2115} {R : Type u_1} [semiring R] (f : zmod n \u2192+* R) (g : zmod n \u2192+* R) : f = g := sorry\n\nnamespace zmod\n\n\nprotected instance subsingleton_ring_hom {n : \u2115} {R : Type u_1} [semiring R] : subsingleton (zmod n \u2192+* R) :=\n  subsingleton.intro ring_hom.ext_zmod\n\nprotected instance subsingleton_ring_equiv {n : \u2115} {R : Type u_1} [semiring R] : subsingleton (zmod n \u2243+* R) :=\n  subsingleton.intro\n    fun (f g : zmod n \u2243+* R) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (f = g)) (propext (ring_equiv.coe_ring_hom_inj_iff f g)))) (ring_hom.ext_zmod \u2191f \u2191g)\n\ntheorem ring_hom_surjective {n : \u2115} {R : Type u_1} [ring R] (f : R \u2192+* zmod n) : function.surjective \u21d1f := sorry\n\ntheorem ring_hom_eq_of_ker_eq {n : \u2115} {R : Type u_1} [comm_ring R] (f : R \u2192+* zmod n) (g : R \u2192+* zmod n) (h : ring_hom.ker f = ring_hom.ker g) : f = g := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/zmod/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.38848963126754404}}
{"text": "import Lean\n\ndef f (x : Nat \u00d7 Nat) :=\n  match x with\n  | .mk (snd := snd) .. => snd\n\nexample : f (10, 20) = 20 := rfl\n\nopen Lean\ndef g (e : Expr) : Expr :=\n  match e with\n  | .forallE (binderType := type) .. => type\n  | e => e\n\ndef h (x : Nat \u00d7 Nat) :=\n  match x with\n  | .mk (\u03b1 := .(Nat)) (snd := snd) .. => snd\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/dottedCtorNamedArgPattern.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3884896231943917}}
{"text": "import analysis.special_functions.exp\nimport tactic\n\nopen real\n\nvariables a b c : \u211d\n\n#check le_refl\n#check sub_le_sub\n#check exp_le_exp\n#check exp_le_exp.mpr\n\n-- BEGIN\n\nexample (h : a \u2264 b) : c - exp b \u2264 c - exp a :=\nbegin\n  apply sub_le_sub,\n    sorry,\n  sorry,\nend\n\n-- END", "meta": {"author": "xhkittyyan", "repo": "Lean-Seminars-Series-Fall-2022", "sha": "6951cdf2cb4e001666d2a56170601325f69d52b5", "save_path": "github-repos/lean/xhkittyyan-Lean-Seminars-Series-Fall-2022", "path": "github-repos/lean/xhkittyyan-Lean-Seminars-Series-Fall-2022/Lean-Seminars-Series-Fall-2022-6951cdf2cb4e001666d2a56170601325f69d52b5/src/3_apply_exact/3.1_apply/ex3_apply_le_exp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.38837741921140884}}
{"text": "import algebra.category.Module.basic\nimport linear_algebra.tensor_product\nimport linear_algebra.finsupp\nimport category_theory.adjunction.limits\nimport category_theory.limits.preserves.limits\n-- import .hom\n\nopen_locale tensor_product\n\nopen tensor_product\n\nuniverses u u' v\n\nvariables (R : Type u) (S : Type u') [comm_ring R] [comm_ring S]\nvariables (X : Type v) [add_comm_group X] [module R X] [module S X]\n\nclass bimodule :=\n(smul_comm' [] : \u2200 (r : R) (s : S) (x : X), r \u2022 s \u2022 x = s \u2022 r \u2022 x)\n\nsection bimodule\n\nvariables {R S X}\n\nlemma bimodule.smul_comm [bimodule R S X] (r : R) (s : S) (x : X) : \n  r \u2022 s \u2022 x = s \u2022 r \u2022 x :=\nbimodule.smul_comm' r s x\n\ninstance bimodule.int (X' : Type v) [add_comm_group X'] [module R X'] :\n  bimodule R \u2124 X' :=\n{ smul_comm' := \u03bb r z x', \n  begin \n    induction z using int.induction_on with n hn n hn,\n    { simp, },\n    { simpa [add_smul, smul_add] using hn, },\n    { simpa [sub_smul, smul_sub] using hn, },\n  end }\n\ninstance bimodule.symm [bimodule R S X] : bimodule S R X :=\n{ smul_comm' := \u03bb s r x, (bimodule.smul_comm r s x).symm } \n\nend bimodule\n\nsection tensor_bimodule\n\nvariable [bimodule R S X]\nvariables (Y : Type v) [add_comm_group Y] [module R Y]\n\n@[simps]\ndef tensor_bimodule.smul_aux (s : S) : Y \u2192\u2097[R] X \u2192\u2097[R] Y \u2297[R] X :=\n{ to_fun := \u03bb y, \n  { to_fun := \u03bb x, y \u2297\u209c (s \u2022 x),\n    map_add' := \u03bb x x', by rw [smul_add, tmul_add],\n    map_smul' := \u03bb r x, by rw [ring_hom.id_apply, smul_tmul', smul_tmul, \n      bimodule.smul_comm] },\n  map_add' := \u03bb y y', linear_map.ext $ \u03bb x, by simp [linear_map.add_apply, \n    add_tmul],\n  map_smul' := \u03bb r y, linear_map.ext $ \u03bb x, by simp [smul_tmul, tmul_smul] }\n\n@[simps]\ndef tensor_bimodule.smul (s : S) : (Y \u2297[R] X) \u2192\u2097[R] (Y \u2297[R] X) :=\ntensor_product.lift $ tensor_bimodule.smul_aux _ _ _ _ s\n\ninstance tensor_product.bimodule.has_smul : has_smul S (Y \u2297[R] X) :=\n{ smul := \u03bb s, tensor_bimodule.smul R S X Y s }\n\nsection\n\nvariables {R S X Y}\n\nlemma tensor_bimodule.smul_def (s : S) (z : Y \u2297[R] X) : \n  s \u2022 z = tensor_bimodule.smul _ _ _ _ s z := rfl\n\n\nlemma tensor_bimodule.smul_tmul (s : S) (y : Y) (x : X) :\n  s \u2022 (y \u2297\u209c x : _ \u2297[R] _) = y \u2297\u209c (s \u2022 x) := rfl\n\nend\n\n@[priority 100]\ninstance tensor_product.bimodule : module S (Y \u2297[R] X) :=\n{ smul := (\u2022),\n  one_smul := \u03bb z,\n  begin \n    induction z using tensor_product.induction_on with _ _ a b ha hb,\n    { rw [tensor_bimodule.smul_def, map_zero], },\n    { rw [tensor_bimodule.smul_tmul, one_smul], },\n    { rw [tensor_bimodule.smul_def] at ha hb \u22a2,\n      rw [map_add, ha, hb], },\n  end,\n  mul_smul := \u03bb s s' z, \n  begin\n    induction z using tensor_product.induction_on with y x a b ha hb,\n    { simp only [tensor_bimodule.smul_def, map_zero], },\n    { repeat { rw [tensor_bimodule.smul_tmul] },\n      rw [mul_smul] },\n    { repeat { rw [tensor_bimodule.smul_def] at ha hb \u22a2 },\n      rw [map_add, ha, hb, map_add, map_add], },\n  end,\n  smul_zero := \u03bb s, by { rw [tensor_bimodule.smul_def, map_zero] },\n  smul_add := \u03bb _ _ _, by { simp only [tensor_bimodule.smul_def, map_add] },\n  add_smul := \u03bb s s' z, \n  begin \n    induction z using tensor_product.induction_on with y x a b ha hb,\n    { simp only [tensor_bimodule.smul_def, map_zero, zero_add], },\n    { repeat { rw [tensor_bimodule.smul_tmul] },\n      rw [add_smul, tmul_add], },\n    { repeat { rw [tensor_bimodule.smul_def] at ha hb \u22a2, },\n      simp only [map_add, ha, hb],\n      abel, },\n  end,\n  zero_smul := \u03bb z,\n  begin\n    induction z using tensor_product.induction_on with y x a b ha hb,\n    { simp only [tensor_bimodule.smul_def, map_zero], },\n    { repeat { rw [tensor_bimodule.smul_tmul] },\n      rw [zero_smul, tmul_zero], },\n    { repeat { rw [tensor_bimodule.smul_def] at ha hb \u22a2, },\n      simp only [map_add, ha, hb, add_zero], }, \n  end }\n\nend tensor_bimodule\n\nsection bimodule_hom\n\nvariable [bimodule R S X]\nvariables (Z : Type v) [add_comm_group Z] [module S Z]\n\nsection\n\nvariables {R S X Z}\n\n@[simps] def bimodule_hom.smul (r : R) (l : X \u2192\u2097[S] Z) : X \u2192\u2097[S] Z :=\n{ to_fun := \u03bb x, l (r \u2022 x),\n  map_add' := \u03bb _ _, by rw [smul_add, map_add],\n  map_smul' := \u03bb s x, by rw [bimodule.smul_comm, map_smul, \n    ring_hom.id_apply] }\n\nend\n\ninstance bimodule_hom.has_smul : has_smul R (X \u2192\u2097[S] Z) :=\n{ smul := bimodule_hom.smul }\n\nlemma bimodule_hom.smul_def (r : R) (l : X \u2192\u2097[S] Z) :\n  r \u2022 l = bimodule_hom.smul r l := rfl\n\ninstance bimodule_hom : module R (X \u2192\u2097[S] Z) :=\n{ smul := (\u2022),\n  one_smul := \u03bb l, linear_map.ext $ \u03bb x,\n    by simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, one_smul],\n  mul_smul := \u03bb r r' l, linear_map.ext $ \u03bb x,\n  begin \n    simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, one_smul, mul_smul], \n    rw smul_comm,\n  end,\n  smul_zero := \u03bb r, linear_map.ext $ \u03bb x, \n    by simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, linear_map.zero_apply],\n  smul_add := \u03bb r l l', linear_map.ext $ \u03bb x,\n    by simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, linear_map.add_apply],\n  add_smul := \u03bb r r' l, linear_map.ext $ \u03bb x,\n    by simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, add_smul, map_add,\n      linear_map.add_apply],\n  zero_smul := \u03bb l, linear_map.ext $ \u03bb x,\n  by simp only [bimodule_hom.smul_def, bimodule_hom.smul_apply, linear_map.zero_apply, \n    zero_smul, map_zero], }\n\nend bimodule_hom\n\nnamespace Module\n\n@[simps]\ndef tensor_functor [bimodule R S X] : Module.{v} R \u2964 Module.{v} S :=\n{ obj := \u03bb Y, Module.of S (Y \u2297[R] X),\n  map := \u03bb Y Y' l, \n  { to_fun := tensor_product.map l linear_map.id,\n    map_add' := \u03bb z z', by rw [map_add],\n    map_smul' := \u03bb s (z : Y \u2297[R] X),\n    begin \n      induction z using tensor_product.induction_on with y x a b ha hb,\n      { rw [smul_zero, map_zero, smul_zero], },\n      { rw [tensor_bimodule.smul_tmul, map_tmul, linear_map.id_apply, map_tmul, ring_hom.id_apply, \n          linear_map.id_apply, tensor_bimodule.smul_tmul], },\n      { rw [smul_add, map_add, ha, hb, map_add, smul_add], },\n    end },\n  map_id' := \u03bb Y, linear_map.ext $ \u03bb z,\n  begin \n    simp only [linear_map.coe_mk, id_apply],\n    erw [map_id, linear_map.id_apply],\n  end,\n  map_comp' := \u03bb Y Y' Y'' l l', linear_map.ext $ \u03bb z, \n  begin \n    simp only [linear_map.coe_mk, coe_comp, function.comp_app],\n    induction z using tensor_product.induction_on with _ _ a b ha hb,\n    { simp only [map_zero], },\n    { simp only [tensor_product.map_tmul, linear_map.id_apply, category_theory.comp_apply] },\n    { rw [map_add, ha, hb, map_add, map_add], }\n  end }\n\n@[simps]\ndef hom_functor [bimodule R S X] : Module.{v} S \u2964 Module R :=\n{ obj := \u03bb Z, Module.of R $ X \u2192\u2097[S] Z,\n  map := \u03bb Z Z' (l : Z \u2192\u2097[S] Z'), \n  { to_fun := l.comp,\n    map_add' := \u03bb z z', by rw linear_map.comp_add,\n    map_smul' := \u03bb r f, linear_map.ext $ \u03bb x, rfl },\n  map_id' := \u03bb Z, \n    by { ext l x, simp only [linear_map.coe_mk, linear_map.comp_apply, id_apply] },\n  map_comp' := \u03bb Z Z' Z'' (l : Z \u2192\u2097[S] Z') (l' : Z' \u2192\u2097[S] Z''), \n    linear_map.ext $ \u03bb (l'' : X \u2192\u2097[S] Z), linear_map.ext $ \u03bb x, by simp, }\n\nend Module\n\nnamespace Module\n\nvariables (R' : Type u) (S' : Type u') [comm_ring R'] [comm_ring S']\nvariables (X' : Type v) [add_comm_group X'] [module R' X'] [module S' X'] [bimodule R' S' X']\n\nnamespace tensor_hom_adjunction\n\n@[simps]\ndef hom_equiv.to_fun' {Y : Module.{v} R'} {Z : Module.{v} S'} (l : Y \u2297[R'] X' \u2192\u2097[S'] Z) :\n  (Y \u27f6 (hom_functor R' S' X').obj Z) :=\n{ to_fun := \u03bb y, \n  { to_fun := \u03bb x, l (y \u2297\u209c x),\n    map_add' := \u03bb x x', by rw [tmul_add, map_add],\n    map_smul' := \u03bb s x, by rw [ring_hom.id_apply, \u2190map_smul, tensor_bimodule.smul_tmul] },\n  map_add' := \u03bb y y', linear_map.ext $ \u03bb x, by simp [add_tmul, map_add],\n  map_smul' := \u03bb r y, linear_map.ext $ \u03bb x,by simp [bimodule_hom.smul_def, \n    bimodule_hom.smul_apply, smul_tmul], }\n\n@[simps]\ndef hom_equiv.inv_fun' {Y : Module.{v} R'} {Z : Module.{v} S'} (l : Y \u2192\u2097[R'] (X' \u2192\u2097[S'] Z)) :\n  ((tensor_functor R' S' X').obj Y \u27f6 Z) :=\n{ to_fun := (add_con_gen _).lift (free_add_monoid.lift $ show Y \u00d7 X' \u2192 Z, from \u03bb p, l p.1 p.2) $ \n    add_con.add_con_gen_le $ \u03bb p p' (h : eqv R' Y X' p p'), \n      show (free_add_monoid.lift $ show Y \u00d7 X' \u2192 Z, from \u03bb p, l p.1 p.2) p \n        = (free_add_monoid.lift $ show Y \u00d7 X' \u2192 Z, from \u03bb p, l p.1 p.2) p',\n      from match p, p', h with\n      | _, _, (eqv.of_zero_left n) := by simp only [free_add_monoid.lift_eval_of, map_zero, \n        linear_map.zero_apply]\n      | _, _, (eqv.of_zero_right m) := by simp only [free_add_monoid.lift_eval_of, map_zero]\n      | _, _, (eqv.of_add_left m\u2081 m\u2082 n)  := by simp only [map_add, free_add_monoid.lift_eval_of, \n        linear_map.add_apply]\n      | _, _, (eqv.of_add_right m n\u2081 n\u2082) := by simp only [map_add, free_add_monoid.lift_eval_of] \n      | _, _, (eqv.of_smul r m n) := by simp only [free_add_monoid.lift_eval_of, map_smul,\n        bimodule_hom.smul_def, bimodule_hom.smul_apply]\n      | _, _, (eqv.add_comm x y) := by simpa only [map_add, free_add_monoid.lift_eval_of] \n        using add_comm _ _ \n      end,\n  map_add' := \u03bb _ _, by rw map_add,\n  map_smul' := \u03bb s (z : Y \u2297[R'] X'), \n  begin \n    induction z using tensor_product.induction_on with y x a b ha hb,\n    { rw [smul_zero, map_zero, smul_zero], },\n    { rw [tensor_bimodule.smul_tmul, tmul, add_con.coe_mk', add_con.lift_coe, \n        free_add_monoid.lift_eval_of, ring_hom.id_apply, tmul, add_con.coe_mk', add_con.lift_coe,\n        free_add_monoid.lift_eval_of],\n      simp only [map_smul], },\n    { rw [smul_add, map_add, ha, hb, map_add, smul_add], }\n  end }\n\n\n@[simps]\ndef hom_equiv (Y : Module.{v} R') (Z : Module.{v} S') :\n  ((tensor_functor R' S' X').obj Y \u27f6 Z) \u2243 (Y \u27f6 (hom_functor R' S' X').obj Z) :=\n{ to_fun := hom_equiv.to_fun' R' S' X',\n  inv_fun := hom_equiv.inv_fun' R' S' X',\n  left_inv := \u03bb l, linear_map.ext $ \u03bb (p : Y \u2297[R'] X'), \n  begin \n    simp only [hom_equiv.to_fun'_apply_apply, hom_equiv.inv_fun'_apply],\n    induction p using tensor_product.induction_on with y x a b ha hb,\n    { rw [map_zero, map_zero], },\n    { conv_lhs { rw [tmul, add_con.coe_mk', add_con.lift_coe, free_add_monoid.lift_eval_of] }, },\n    { conv_lhs { rw [map_add, ha, hb, \u2190map_add], } }\n  end,\n  right_inv := \u03bb (l : Y \u2192\u2097[R'] (X'\u2192\u2097[S'] Z)), linear_map.ext $ \u03bb y, linear_map.ext $ \u03bb x,\n  begin \n    simp only [hom_equiv.to_fun'_apply_apply, hom_equiv.inv_fun'_apply],\n    conv_lhs { rw [tmul, add_con.coe_mk', add_con.lift_coe, free_add_monoid.lift_eval_of] },\n  end }\n\n@[simps]\ndef unit : \ud835\udfed (Module R') \u27f6 tensor_functor R' S' X' \u22d9 hom_functor R' S' X' :=\n{ app := \u03bb Y, show Y \u2192\u2097[R'] (X' \u2192\u2097[S'] (Y \u2297[R'] X')), from \n  { to_fun := \u03bb y, \n    { to_fun := \u03bb x, y \u2297\u209c x,\n      map_add' := \u03bb x x', by rw tmul_add,\n      map_smul' := \u03bb s x, by rw [ring_hom.id_apply]; refl },\n    map_add' := \u03bb y y', linear_map.ext $ \u03bb x, by simp only [linear_map.coe_mk, add_tmul, \n      linear_map.add_apply],\n    map_smul' := \u03bb r y, linear_map.ext $ \u03bb x, by simp only [linear_map.coe_mk, linear_map.smul_apply,\n      bimodule_hom.smul_def, bimodule_hom.smul_apply, ring_hom.id_apply, smul_tmul], },\n  naturality' := \u03bb Y Y' (l : Y \u2192\u2097[R'] Y'), linear_map.ext $ \u03bb (y : Y), linear_map.ext $ \u03bb x,\n  begin \n    simp only [category_theory.comp_apply, linear_map.coe_mk, category_theory.functor.id_map, \n      category_theory.functor.comp_map, hom_functor_map_apply, linear_map.comp_apply,\n      tensor_functor_map_apply, map_tmul, linear_map.id_coe, id.def],\n  end }\n\n@[simps] def counit : hom_functor R' S' X' \u22d9 tensor_functor R' S' X' \u27f6 \ud835\udfed (Module S') :=\n{ app := \u03bb Z, show ((X' \u2192\u2097[S'] Z) \u2297[R'] X') \u2192\u2097[S'] Z, from \n  { to_fun := (add_con_gen _).lift (free_add_monoid.lift $ \u03bb (p : (X' \u2192\u2097[S'] \u21a5Z) \u00d7 X'), p.1 p.2) $ \n    add_con.add_con_gen_le $ \u03bb p p' (h : eqv R' (X' \u2192\u2097[S'] Z) X' p p'), \n      show (free_add_monoid.lift $ \u03bb (p : (X' \u2192\u2097[S'] \u21a5Z) \u00d7 X'), p.1 p.2) p = \n      (free_add_monoid.lift $ \u03bb (p : (X' \u2192\u2097[S'] \u21a5Z) \u00d7 X'), p.1 p.2) p',\n      from match p, p', h with\n      | _, _, (eqv.of_zero_left n) := by simp only [free_add_monoid.lift_eval_of, map_zero, \n        linear_map.zero_apply]\n      | _, _, (eqv.of_zero_right m) := by simp only [free_add_monoid.lift_eval_of, map_zero]\n      | _, _, (eqv.of_add_left m\u2081 m\u2082 n)  := by simp only [map_add, free_add_monoid.lift_eval_of, \n        linear_map.add_apply]\n      | _, _, (eqv.of_add_right m n\u2081 n\u2082) := by simp only [map_add, free_add_monoid.lift_eval_of] \n      | _, _, (eqv.of_smul r m n) := by simp only [free_add_monoid.lift_eval_of, map_smul,\n        bimodule_hom.smul_def, bimodule_hom.smul_apply]\n      | _, _, (eqv.add_comm x y) := by simpa only [map_add, free_add_monoid.lift_eval_of] \n        using add_comm _ _ \n      end,\n    map_add' := \u03bb p p', by rw map_add,\n    map_smul' := \n    begin \n      rintros s p,\n      induction p using tensor_product.induction_on with l x a b ha hb,\n      { simp only [map_zero, smul_zero], },\n      { rw [tensor_bimodule.smul_tmul, tmul, add_con.coe_mk', add_con.lift_coe, \n          free_add_monoid.lift_eval_of, ring_hom.id_apply, tmul, add_con.coe_mk', add_con.lift_coe, \n          free_add_monoid.lift_eval_of, linear_map.map_smul], },\n      { rw [smul_add, map_add, ha, hb, map_add, smul_add] }\n    end },\n  naturality' := \u03bb Z Z' (l : Z \u2192\u2097[S'] Z'), linear_map.ext $ \u03bb (p : (X' \u2192\u2097[S'] Z) \u2297[R'] X'), \n  begin \n    induction p using tensor_product.induction_on with l' x a b ha hb,\n    { simp only [map_zero] },\n    { simp only [category_theory.comp_apply, linear_map.coe_mk, category_theory.functor.comp_map,\n        hom_functor_map_apply, tensor_functor_map_apply, tensor_product.map_tmul,\n        category_theory.functor.id_map, linear_map.id_apply],\n      simp only [tmul, add_con.coe_mk', add_con.lift_coe, free_add_monoid.lift_eval_of, \n        linear_map.comp_apply], },\n    { rw [map_add, ha, hb, map_add], },\n  end }\n\nlemma hom_equiv_unit (Y : Module.{v} R') (Z : Module.{v} S') (f) : \n  hom_equiv R' S' X' Y Z f =\n  (unit R' S' X').app Y \u226b (hom_functor R' S' X').map f :=\nlinear_map.ext $ \u03bb y, linear_map.ext $ \u03bb x, rfl\n\nlemma hom_equiv_counit (Y : Module.{v} R') (Z : Module.{v} S') (g) : \n  (hom_equiv R' S' X' Y Z).symm g =\n  (tensor_functor _ _ _).map g \u226b (counit R' S' X').app Z :=\nlinear_map.ext $ \u03bb z,\nbegin \n  induction z using tensor_product.induction_on with y x a b ha hb,\n  { simp only [map_zero] },\n  { conv_lhs { rw [hom_equiv_symm_apply, hom_equiv.inv_fun'_apply, tmul, add_con.coe_mk', \n      add_con.lift_coe, free_add_monoid.lift_eval_of] },\n    conv_rhs { rw [category_theory.comp_apply, tensor_functor_map_apply, tensor_product.map_tmul,\n      linear_map.id_apply, counit_app_apply, tmul, add_con.coe_mk', add_con.lift_coe, \n      free_add_monoid.lift_eval_of] }, },\n  { rw [map_add, ha, hb, map_add] },\nend\n\nend tensor_hom_adjunction\n\n@[simps]\ndef tensor_hom_adjunction : (tensor_functor R' S' X') \u22a3 (hom_functor R' S' X') :=\n{ hom_equiv := tensor_hom_adjunction.hom_equiv _ _ _,\n  unit := tensor_hom_adjunction.unit _ _ _,\n  counit := tensor_hom_adjunction.counit _ _ _,\n  hom_equiv_unit' := tensor_hom_adjunction.hom_equiv_unit _ _ _,\n  hom_equiv_counit' := tensor_hom_adjunction.hom_equiv_counit _ _ _ }\n\nsection\n\ninstance : category_theory.is_left_adjoint (tensor_functor R' S' X') :=\n\u27e8_, tensor_hom_adjunction _ _ _\u27e9\n\ninstance : category_theory.is_right_adjoint (hom_functor R' S' X') :=\n\u27e8_, tensor_hom_adjunction _ _ _\u27e9\n\ninstance : category_theory.limits.preserves_colimits (tensor_functor R' S' X') :=\ncategory_theory.adjunction.left_adjoint_preserves_colimits (tensor_hom_adjunction R' S' X')\n\ninstance : category_theory.limits.preserves_limits (hom_functor R' S' X') :=\ncategory_theory.adjunction.right_adjoint_preserves_limits (tensor_hom_adjunction R' S' X')\n\ninstance : category_theory.functor.preserves_epimorphisms (tensor_functor R' S' X') :=\ninfer_instance\n\nend\n\nend Module\n\nnamespace tensor_product\n\nvariables (R' : Type u) [comm_ring R']\nvariables {M N : Type v} [add_comm_group M] [add_comm_group N]\nvariables [module R' M] [module R' N]\n\n@[simps]\ndef to_add_comm_group {C : Type v} [add_comm_group C]\n  (b : M \u2192+ (N \u2192+ C)) (hb : \u2200 (r : R') (m : M) (n : N), b (r \u2022 m) n = b m (r \u2022 n)) :\n  (M \u2297[R'] N) \u2192+ C :=\n(((@Module.tensor_hom_adjunction R' \u2124 _ _ N _ _ _ (bimodule.int _)).hom_equiv \n  (Module.of R' M) (Module.of _ C)).symm \n{ to_fun := \u03bb (m : M), add_monoid_hom.to_int_linear_map $ b m,\n  map_add' := \u03bb (m m' : M), by rw [map_add]; refl,\n  map_smul' := \u03bb r (m : M), linear_map.ext $ \u03bb n, \n  by simpa only [add_monoid_hom.coe_to_int_linear_map, ring_hom.id_apply, hb] }).to_add_monoid_hom\n\nlemma to_add_comm_group.apply_tmul {C : Type v} [add_comm_group C]\n  (b : M \u2192+ (N \u2192+ C)) (hb : \u2200 (r : R') (m : M) (n : N), b (r \u2022 m) n = b m (r \u2022 n))\n  (m : M) (n : N) : to_add_comm_group R' b hb (m \u2297\u209c n) = b m n :=\nby rw [to_add_comm_group_apply, tmul, add_con.coe_mk', add_con.lift_coe,\n    free_add_monoid.lift_eval_of]\n\nlemma to_add_comm_group.uniq {C : Type v} [add_comm_group C]\n  (b : M \u2192+ (N \u2192+ C)) (hb : \u2200 (r : R') (m : M) (n : N), b (r \u2022 m) n = b m (r \u2022 n))\n  (l : (M \u2297[R'] N) \u2192+ C) (hl : \u2200 \u2983m : M\u2984 \u2983n : N\u2984,  l (m \u2297\u209c n) = b m n) :\n  to_add_comm_group R' b hb = l := add_monoid_hom.ext $ \u03bb z,\nbegin \n  induction z using tensor_product.induction_on with m n x y hx hy,\n  { simp only [map_zero] },\n  { rw [to_add_comm_group.apply_tmul, hl], },\n  { rw [map_add, hx, hy, map_add] },\nend\n\n@[reducible]\ndef to_add_comm_group' {C : Type v} [add_comm_group C]\n  (b : M \u00d7 N \u2192 C) \n  (hN0 : \u2200 (n : N), b (0, n) = 0)\n  (hM0 : \u2200 (m : M), b (m, 0) = 0)\n  (hMadd : \u2200 (n : N) (m m' : M), b (m + m', n) = b (m, n) + b (m', n))\n  (hNadd : \u2200 (m : M) (n n' : N), b (m, n + n') = b (m, n) + b (m, n'))\n  (hb : \u2200 (r : R') (m : M) (n : N), b ((r \u2022 m), n) = b (m, (r \u2022 n))) :\n  (M \u2297[R'] N) \u2192+ C :=\nto_add_comm_group R' \n{ to_fun := \u03bb m, \n  { to_fun := \u03bb n, b (m, n),\n    map_zero' := hM0 _,\n    map_add' := hNadd _ },\n  map_zero' := add_monoid_hom.ext $ \u03bb n, show b (0, n) = 0, from hN0 _,\n  map_add' := \u03bb m m', add_monoid_hom.ext $ \u03bb n, show b (m + m', n) = b (m, n) + b (m', n), \n    from hMadd _ _ _ } $ \u03bb r m n,\nshow b (r \u2022 m, n) = b (m, r \u2022 n), from hb _ _ _\n\nlemma to_add_comm_group'.apply_tmul {C : Type v} [add_comm_group C]\n  (b : M \u00d7 N \u2192 C) \n  (hN0 : \u2200 (n : N), b (0, n) = 0)\n  (hM0 : \u2200 (m : M), b (m, 0) = 0)\n  (hMadd : \u2200 (n : N) (m m' : M), b (m + m', n) = b (m, n) + b (m', n))\n  (hNadd : \u2200 (m : M) (n n' : N), b (m, n + n') = b (m, n) + b (m, n'))\n  (hb : \u2200 (r : R') (m : M) (n : N), b ((r \u2022 m), n) = b (m, (r \u2022 n)))\n  (m : M) (n : N) : to_add_comm_group' R' b hN0 hM0 hMadd hNadd hb (m \u2297\u209c n) = b (m, n) :=\nby rw [to_add_comm_group.apply_tmul]; refl\n\nopen_locale big_operators\n\nlemma exists_rep (z : M \u2297[R'] N) : \n  \u2203 {\u03b9 : Type v} (ms : \u03b9 \u2192 M) (ns : \u03b9 \u2192 N) (s : finset \u03b9),\n  z = \u2211 i in s, ms i \u2297\u209c ns i :=\nbegin \n  classical,\n  have EQ := span_tmul_eq_top R' M N,\n  have mem1 : z \u2208 \u22a4 := submodule.mem_top,\n  rw \u2190EQ at mem1,\n  rw mem_span_set at mem1,\n  obtain \u27e8c, hc1, rfl\u27e9 := mem1,\n  choose m n hm using hc1,\n  refine \u27e8M \u2297[R'] N, \u03bb i, if hi : i \u2208 c.support then c i \u2022 m hi else 0, \n    \u03bb i, if hi : i \u2208 c.support then n hi else 0, c.support, _\u27e9,\n  rw finsupp.sum,\n  refine finset.sum_congr rfl (\u03bb i hi, _),\n  split_ifs,\n  { specialize hm h,\n    rw \u2190smul_tmul',\n    congr' 1,\n    exact hm.symm, },\n  exact false.elim (h hi),\nend\n\nend tensor_product", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/adjunction_general.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.38837741921140884}}
{"text": "import ECTate.Algebra.Field.Basic\nimport ECTate.Algebra.Ring.Basic\nimport ECTate.Algebra.ValuedRing\nimport ECTate.Data.Nat.Enat\nimport Mathlib.Tactic.GeneralizeProofs\nimport Mathlib.Init.Data.Nat.Lemmas\n\nopen Classical\nvariable (R : Type _) [Semiring R]\n\n/-- Noncomputable function that outputs the unique characteristic of a semiring. -/\nnoncomputable\ndef ring_char := if h : _ then @Nat.find (fun n => n \u2260 0 \u2227 (n : R) = 0) _ h else 0\n\nlemma ring_char_eq_zero (R : Type _) [Semiring R] :\n  (ring_char R : R) = 0 :=\nby\n  rw [ring_char]\n  split\n  . exact (And.right (Nat.find_spec (by assumption)))\n  . simp\n\n\nlemma ring_char_dvd_of_zero {R : Type _} [Ring R] (h : (m : R) = 0) :\n  ring_char R \u2223 m :=\nby\n  by_cases hm : m = 0\n  . simp [hm, Nat.dvd_zero] at *\n  . rw [ring_char]\n    rw [dif_pos ?_]\n\n    rotate_right 1 -- swap -- TODO unknown swap\n    exists m\n    generalize_proofs hh\n    have good := Nat.find_spec hh\n    have bd : Nat.find _ \u2264 Nat.gcd (Nat.find hh) m := Nat.find_min' hh ?_\n    rw [Nat.gcd_eq_left_iff_dvd]\n    simp at good\n    simp at bd\n    sorry\n    sorry\n\nlemma ring_char_is_zero_or_prime (R : Type _) [CommRing R] [IsDomain R] :\n  ring_char R = 0 \u2228 Nat.Prime (ring_char R) := sorry\n\nlemma add_pow_ring_char {R : Type _} [CommRing R] [IsDomain R] (h : ring_char R \u2260 0) :\n  (a + b) ^ ring_char R =\n  a ^ ring_char R +\n  b ^ ring_char R := by sorry\n\n\nlemma sub_pow_ring_char {R : Type _} (a b : R) [CommRing R] [IsDomain R] (h : ring_char R \u2260 0) :\n  (a - b) ^ ring_char R =\n  a ^ ring_char R -\n  b ^ ring_char R := by sorry\n\n\nlemma pow_ring_char_injective {R : Type _} [CommRing R] [IsDomain R]\n  (hn : ring_char R \u2260 0) : Function.Injective (. ^ ring_char R : R \u2192 R) := by\n  intros x y h\n  rw [\u2190sub_eq_zero] at *\n  rw [\u2190sub_eq_zero] at *\n  simp only [sub_zero] at *\n  rw [\u2190 sub_pow_ring_char _ _ hn] at h\n  exact pow_eq_zero h\n", "meta": {"author": "KisaraBlue", "repo": "ec-tate-lean", "sha": "2b1b26c2622fde0344feaadddc077caca73bd929", "save_path": "github-repos/lean/KisaraBlue-ec-tate-lean", "path": "github-repos/lean/KisaraBlue-ec-tate-lean/ec-tate-lean-2b1b26c2622fde0344feaadddc077caca73bd929/ECTate/Algebra/CharP/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3883451756787881}}
{"text": "import algebraic_topology.nerve\nimport algebraic_topology.topological_simplex\nimport category_theory.adjunction.basic\nimport category_theory.adjunction.comma\nimport category_theory.limits.presheaf\n\nopen category_theory opposite\n\ndef my_nerve : (Cat \u2964 sSet) := \n  colimit_adj.restricted_yoneda simplex_category.to_Cat\n\nlemma i_hate_yoneda (X : Cat) (Y : Cat\u1d52\u1d56) : \n      (yoneda.obj X).obj Y = (Y.unop \u27f6 X) := by simp\n\nlemma i_hate_op (X : simplex_category\u1d52\u1d56) (F : simplex_category \u2964 Cat) :\n      unop (F.op.obj X) = F.obj X.unop := by simp\n\n\nlemma helper_ext (X : Cat) (Y : simplex_category\u1d52\u1d56) :\n(yoneda.obj X).obj (simplex_category.to_Cat.op.obj Y) = (nerve X).obj Y := begin\n  have : (nerve X).obj Y = ((simplex_category.to_Cat.obj Y.unop) \u2964 X), simp,\n  rw this,\n  rw i_hate_yoneda,\n  rw i_hate_op Y simplex_category.to_Cat,\n  refl,\nend\n\nnoncomputable lemma helper (X : Cat) : \nsimplex_category.to_Cat.op \u22d9 yoneda.obj X \u2245 nerve X := \nby refine {\n  hom := { app := begin\n    intro Y, rw functor.comp_obj,\n    rw helper_ext,\n    exact \ud835\udfd9 _,\n  end },\n  inv := { app := begin\n    intro Y, rw functor.comp_obj,\n    rw helper_ext,\n    exact \ud835\udfd9 _,\n  end },\n}\n\n#check my_nerve.map\n\nnoncomputable lemma nerve_is_a_nerve : my_nerve \u2245 nerve_functor := {\n  hom := { \n    app := \u03bb Y, begin\n      unfold my_nerve, unfold colimit_adj.restricted_yoneda, simp, \n      apply (helper Y).hom,\n    end,\n    naturality' := begin\n      intros, simp,\n\n        \n\n    end\n\n  },\n  inv := { \n    app := \u03bb Y, begin\n      unfold my_nerve, unfold colimit_adj.restricted_yoneda, simp,\n      apply (helper Y).inv,\n    end,\n    naturality' := sorry\n  },\n}", "meta": {"author": "raghav198", "repo": "simplicial-things-in-lean", "sha": "a95608a6dba98c8a47bbdaa0ed55707d0251524c", "save_path": "github-repos/lean/raghav198-simplicial-things-in-lean", "path": "github-repos/lean/raghav198-simplicial-things-in-lean/simplicial-things-in-lean-a95608a6dba98c8a47bbdaa0ed55707d0251524c/src/nerve_realization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.38834516945355385}}
{"text": "import data.set.basic -- hide\nopen set -- hide\n\n/- Tactic : exfalso\n\n## Summary\n\nChanges the goal to `\u22a2 false`.\n\n## Details\n\nThis may seem hard to prove,\nbut it is useful when we have a contradiction in the hypotheses.\n\nFor example, if we have `h : \u00ac P` as a hypothesis and we apply `exfalso`\nwe can then `apply h` to transform the goal into `\u22a2 P`.\n-/\n\n\n/- Hint : Click here for a hint, in case you get stuck.\nIn Lean, the  negation `\u00ac P` of a statement is a shorthand for `P \u2192 false`. Therefore\nstart with `exfalso`, and remember that negation is the same as `\u2192 false`.\n-/\n\nvariables {X Y : Type} -- hide\n\n/- Lemma : no-side-bar\nWe can prove that 1 equals 0 if we have a contradiction in our hypotheses.\n-/\nlemma one_eq_zero_of_contradiction (A : set X) (x : X) (h1 : x \u2208 A) (h2 : x \u2209 A): 1 = 0 :=\nbegin\n  exfalso,\n  apply h2,\n  exact h1,\n\n  \nend\n", "meta": {"author": "mmasdeu", "repo": "topologygame", "sha": "0a1b868031919a5555e7b99efca66ece2f546ec7", "save_path": "github-repos/lean/mmasdeu-topologygame", "path": "github-repos/lean/mmasdeu-topologygame/topologygame-0a1b868031919a5555e7b99efca66ece2f546ec7/src/set_theory_world/level08.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.38834516945355385}}
{"text": "import cicm2022.examples.Proj.degree_zero_part\nimport cicm2022.examples.Proj.structure_sheaf\nimport cicm2022.examples.Proj.lemmas\nimport cicm2022.examples.Proj.Proj_iso_Spec.Top_component.from_Spec\n\nimport algebraic_geometry.structure_sheaf\nimport algebraic_geometry.Spec\n\nnoncomputable theory\n\nnamespace algebraic_geometry\n\nopen_locale direct_sum big_operators pointwise big_operators\nopen direct_sum set_like.graded_monoid localization finset (hiding mk_zero)\n\nvariables {R A : Type*}\nvariables [comm_ring R] [comm_ring A] [algebra R A]\n\nvariables (\ud835\udc9c : \u2115 \u2192 submodule R A)\nvariables [graded_algebra \ud835\udc9c]\n\nopen Top topological_space\nopen category_theory opposite\nopen projective_spectrum.structure_sheaf\n\nlocal notation `Proj` := Proj.to_LocallyRingedSpace \ud835\udc9c\n-- `Proj` as a locally ringed space\nlocal notation `Proj.T` := Proj .1.1.1\n-- the underlying topological space of `Proj`\nlocal notation `Proj| ` U := Proj .restrict (opens.open_embedding (U : opens Proj.T))\n-- `Proj` restrict to some open set\nlocal notation `Proj.T| ` U :=\n  (Proj .restrict (opens.open_embedding (U : opens Proj.T))).to_SheafedSpace.to_PresheafedSpace.1\n-- the underlying topological space of `Proj` restricted to some open set\nlocal notation `pbo` x := projective_spectrum.basic_open \ud835\udc9c x\n-- basic open sets in `Proj`\nlocal notation `sbo` f := prime_spectrum.basic_open f\n-- basic open sets in `Spec`\nlocal notation `Spec` ring := Spec.LocallyRingedSpace_obj (CommRing.of ring)\n-- `Spec` as a locally ringed space\nlocal notation `Spec.T` ring :=\n  (Spec.LocallyRingedSpace_obj (CommRing.of ring)).to_SheafedSpace.to_PresheafedSpace.1\n-- the underlying topological space of `Spec`\nlocal notation `A\u2070_` f_deg := degree_zero_part f_deg\n\nnamespace Proj_iso_Spec_Sheaf_component\n\nnamespace from_Spec\n\nopen algebraic_geometry\n\nvariables {\ud835\udc9c} {m : \u2115} {f : A} (hm : 0 < m) (f_deg : f \u2208 \ud835\udc9c m) (V : (opens (Spec (A\u2070_ f_deg)))\u1d52\u1d56) \nvariables (hh : (Spec (A\u2070_ f_deg)).presheaf.obj V)\nvariables (y : ((@opens.open_embedding Proj.T (pbo f)).is_open_map.functor.op.obj \n  ((opens.map (Proj_iso_Spec_Top_component hm f_deg).hom).op.obj V)).unop)\n\nlemma data_prop1 : y.1 \u2208 (pbo f) :=\nbegin\n  obtain \u27e8\u27e8a, ha1\u27e9, -, ha2\u27e9 := y.2,\n  rw \u2190 ha2,\n  exact ha1,\nend\n\nlemma data_prop2 :\n  (Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, data_prop1 hm f_deg V y\u27e9 \u2208 unop V :=\nbegin\n  obtain \u27e8\u27e8a, ha1\u27e9, ha2, ha3\u27e9 := y.2,\n  erw set.mem_preimage at ha2,\n  convert ha2,\n  rw \u2190 ha3,\n  refl,\nend\n\nvariable {V}\ndef data : structure_sheaf.localizations (A\u2070_ f_deg) \n  ((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, data_prop1 _ _ _ _\u27e9) :=\nhh.1 \u27e8_, data_prop2 _ _ _ _\u27e9\n\nlemma data.one :\n  data hm f_deg (1 : (Spec (A\u2070_ f_deg)).presheaf.obj V) = 1 := rfl\n\nlemma data.zero :\n  data hm f_deg (0 : (Spec (A\u2070_ f_deg)).presheaf.obj V) = 0 := rfl\n\nlemma data.add_apply (x y : (Spec (A\u2070_ f_deg)).presheaf.obj V) (z):\n  data hm f_deg (x + y) z = data hm f_deg x z + data hm f_deg y z := rfl\n\nlemma data.mul_apply (x y : (Spec (A\u2070_ f_deg)).presheaf.obj V) (z):\n  data hm f_deg (x * y) z = data hm f_deg x z * data hm f_deg y z := rfl\n\nprivate lemma data.exist_rep \n  (data : structure_sheaf.localizations (A\u2070_ f_deg) ((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, data_prop1 _ _ _ _\u27e9)) :\n  \u2203 (a : A\u2070_ f_deg) (b : ((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, data_prop1 _ _ _ _\u27e9).as_ideal.prime_compl),\n  data = mk a b :=\nbegin\n  induction data using localization.induction_on with d,\n  rcases d with \u27e8a, b\u27e9,\n  refine \u27e8a, b, rfl\u27e9,\nend\n\ndef data.num : A\u2070_ f_deg :=\nclassical.some $ data.exist_rep hm f_deg y (data hm f_deg hh y)\n\ndef data.denom : A\u2070_ f_deg :=\n(classical.some $ classical.some_spec $ data.exist_rep hm f_deg y (data hm f_deg hh y)).1\n\nlemma data.denom_not_mem : \n  (data.denom hm f_deg hh y) \u2209 ((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, data_prop1 _ _ _ _\u27e9).as_ideal :=\n(classical.some $ classical.some_spec $ data.exist_rep hm f_deg y (data hm f_deg hh y)).2\n\nlemma data.eq_num_div_denom :\n  (data hm f_deg hh y) = \n  localization.mk (data.num hm f_deg hh y) \u27e8data.denom hm f_deg hh y, data.denom_not_mem hm f_deg hh y\u27e9 :=\nbegin \n  rw classical.some_spec (classical.some_spec (data.exist_rep hm f_deg y (data hm f_deg hh y))), \n  congr, \n  rw subtype.ext_iff, \n  refl,\nend\n\ndef num : A :=\ndegree_zero_part.num (data.num hm f_deg hh y) * f^(degree_zero_part.deg (data.denom hm f_deg hh y))\n\nlemma num.mem :\n  (num hm f_deg hh y) \u2208 \n    \ud835\udc9c (m * (degree_zero_part.deg (data.num hm f_deg hh y)) \n      + m * (degree_zero_part.deg (data.denom hm f_deg hh y))) :=\nmul_mem (degree_zero_part.num_mem _) $ begin\n  convert (set_like.graded_monoid.pow_mem (degree_zero_part.deg (data.denom hm f_deg hh y)) f_deg) using 1,\n  rw mul_comm,\n  refl,\nend\n\ndef denom : A :=\ndegree_zero_part.num (data.denom hm f_deg hh y) * f^(degree_zero_part.deg (data.num hm f_deg hh y))\n\nlemma denom.mem :\n  (denom hm f_deg hh y) \u2208 \n  \ud835\udc9c (m * (degree_zero_part.deg (data.num hm f_deg hh y)) \n      + m * (degree_zero_part.deg (data.denom hm f_deg hh y))) :=\nbegin\n  change _ * _ \u2208 _,\n  rw mul_comm,\n  apply set_like.graded_monoid.mul_mem,\n  { rw mul_comm,\n    exact set_like.graded_monoid.pow_mem (degree_zero_part.deg (data.num hm f_deg hh y)) f_deg, },\n  { apply degree_zero_part.num_mem, },\nend\n\nlemma denom_not_mem :\n  denom hm f_deg hh y \u2209 y.1.as_homogeneous_ideal := \u03bb rid,\nbegin\n  rcases y.1.is_prime.mem_or_mem rid with H1 | H2,\n  { have mem1 := data.denom_not_mem hm f_deg hh y,\n    have eq1 := degree_zero_part.eq (data.denom hm f_deg hh y),\n    dsimp only at mem1,\n    change _ \u2209 _ at mem1,\n    apply mem1,\n    change\n      (data.denom hm f_deg hh y) \u2208 ((Proj_iso_Spec_Top_component.to_Spec.carrier f_deg) \u27e8y.1, _\u27e9),\n    rw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff,\n    rw eq1,\n    convert ideal.mul_mem_left _ _ _,\n    work_on_goal 2\n    { exact mk 1 \u27e8f^degree_zero_part.deg (data.denom hm f_deg hh y), \u27e8_, rfl\u27e9\u27e9 },\n    work_on_goal 2\n    { exact mk (degree_zero_part.num (data.denom hm f_deg hh y)) 1 },\n    { rw [mk_mul, one_mul, mul_one], },\n    { apply ideal.subset_span,\n      exact \u27e8_, H1, rfl\u27e9 }, },\n  { replace H2 := y.1.is_prime.mem_of_pow_mem _ H2,\n    obtain \u27e8\u27e8a, ha1\u27e9, ha2, ha3\u27e9 := y.2,\n    erw projective_spectrum.mem_basic_open at ha1,\n    apply ha1,\n    convert H2, }\nend\n\nvariable (V)\ndef bmk : homogeneous_localization \ud835\udc9c y.1.as_homogeneous_ideal.to_ideal := quotient.mk' \n{ deg := m * (degree_zero_part.deg (data.num hm f_deg hh y)) \n      + m * (degree_zero_part.deg (data.denom hm f_deg hh y)),\n  num := \u27e8num hm f_deg hh y, num.mem hm f_deg hh y\u27e9,\n  denom := \u27e8denom hm f_deg hh y, denom.mem hm f_deg hh y\u27e9,\n  denom_not_mem := denom_not_mem hm f_deg hh y }\n\nlemma bmk_one :\n  bmk hm f_deg V 1 = 1 :=\nbegin\n  ext1 y,\n  have y_mem : y.val \u2208 (pbo f).val,\n  { erw projective_spectrum.mem_basic_open,\n    intro rid,\n    have mem1 := y.2,\n    erw set.mem_preimage at mem1,\n    obtain \u27e8\u27e8a, ha1\u27e9, ha, ha2\u27e9 := mem1,\n    change a = y.1 at ha2,\n    erw set.mem_preimage at ha,\n    erw \u2190ha2 at rid,\n    apply ha1,\n    exact rid },\n\n  rw pi.one_apply,\n  unfold bmk,\n  rw [homogeneous_localization.ext_iff_val, homogeneous_localization.val_mk', homogeneous_localization.one_val],\n  simp only [\u2190 subtype.val_eq_coe],\n  unfold num denom,\n\n  have eq1 := data.eq_num_div_denom hm f_deg 1 y,\n  rw [data.one, pi.one_apply] at eq1,\n  replace eq1 := eq1.symm,\n  rw [show (1 : structure_sheaf.localizations (A\u2070_ f_deg)\n    (((Proj_iso_Spec_Top_component hm f_deg).hom) \u27e8y.val, y_mem\u27e9)) = localization.mk 1 1,\n    by erw localization.mk_self 1, localization.mk_eq_mk'] at eq1,\n  replace eq1 := (@@is_localization.eq _ _ _ _).mp eq1,\n  obtain \u27e8\u27e8\u27e8C, C_degree_zero\u27e9, hC\u27e9, eq1\u27e9 := eq1,\n  induction C using localization.induction_on with \ud835\udd3b,\n  obtain \u27e8C, \u27e8_, \u27e8l, rfl\u27e9\u27e9\u27e9 := \ud835\udd3b,\n  simp only [mul_one, one_mul, subtype.coe_mk] at eq1,\n  simp only [localization.mk_eq_mk', is_localization.eq],\n  change _ \u2209 _ at hC,\n  erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff at hC,\n  rw subtype.coe_mk at hC,\n  dsimp only at C_degree_zero hC,\n\n  have eq_num := degree_zero_part.eq (data.num hm f_deg 1 y),\n  have eq_denom := degree_zero_part.eq (data.denom hm f_deg 1 y),\n\n  simp only [subtype.val_eq_coe, submonoid.coe_one, mul_one] at eq1,\n  rw subtype.ext_iff at eq1,\n  simp only [subring.coe_mul] at eq1,\n  erw [eq_num, eq_denom, localization.mk_mul, localization.mk_mul] at eq1,\n  simp only [localization.mk_eq_mk', is_localization.eq, subtype.coe_mk] at eq1,\n  obtain \u27e8\u27e8_, \u27e8n1, rfl\u27e9\u27e9, eq1\u27e9 := eq1,\n  simp only [submonoid.coe_mul, subtype.coe_mk] at eq1,\n\n  have C_not_mem : C \u2209 y.1.as_homogeneous_ideal,\n  { intro rid,\n    have eq1 : (localization.mk C \u27e8f ^ l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n      (localization.mk 1 \u27e8f^l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) * localization.mk C 1,\n      rw [localization.mk_mul, one_mul, mul_one],\n    erw eq1 at hC,\n    apply hC,\n    convert ideal.mul_mem_left _ _ _,\n    apply ideal.subset_span,\n    refine \u27e8_, rid, rfl\u27e9, },\n\n  rw [show (1 : localization.at_prime y.1.as_homogeneous_ideal.to_ideal) = mk (1 : _) 1, by erw mk_self 1, mk_eq_mk', is_localization.eq],\n  use C * (f^l * f^n1),\n  { intros rid,\n    rcases y.1.is_prime.mem_or_mem rid with H1 | H3,\n    exact C_not_mem H1,\n    rw \u2190pow_add at H3,\n    replace H3 := y.1.is_prime.mem_of_pow_mem _ H3,\n    apply y_mem,\n    exact H3, },\n\n  simp only [submonoid.coe_one, one_mul, mul_one],\n  simp only [subtype.coe_mk],\n\n  rw calc degree_zero_part.num (data.num hm f_deg 1 y)\n        * f ^ degree_zero_part.deg (data.denom hm f_deg 1 y)\n        * (C * (f ^ l * f ^ n1))\n      = degree_zero_part.num (data.num hm f_deg 1 y) * C\n        * f ^ (degree_zero_part.deg (data.denom hm f_deg 1 y) + l)\n        * f^n1 : by ring_exp,\n  rw [pow_add, eq1],\n  ring,\nend\n\nlemma bmk_zero :\n  bmk hm f_deg V 0 = 0 :=\nbegin\n  ext1 y,\n  have y_mem : y.val \u2208 (pbo f).val,\n  { erw projective_spectrum.mem_basic_open,\n    intro rid,\n    have mem1 := y.2,\n    erw set.mem_preimage at mem1,\n    obtain \u27e8\u27e8a, ha1\u27e9, ha, ha2\u27e9 := mem1,\n    change a = y.1 at ha2,\n    erw set.mem_preimage at ha,\n    erw \u2190ha2 at rid,\n    apply ha1,\n    exact rid },\n\n  rw pi.zero_apply,\n  unfold bmk,\n  rw [homogeneous_localization.ext_iff_val, homogeneous_localization.val_mk', homogeneous_localization.zero_val],\n  simp only [\u2190 subtype.val_eq_coe],\n  rw [show (0 : localization.at_prime y.1.as_homogeneous_ideal.to_ideal) = localization.mk 0 1,\n    by erw localization.mk_zero],\n  dsimp only,\n  unfold num denom,\n\n  have eq1 := data.eq_num_div_denom hm f_deg 0 y,\n  rw [data.zero, pi.zero_apply] at eq1,\n  replace eq1 := eq1.symm,\n  erw [show (0 : structure_sheaf.localizations (A\u2070_ f_deg)\n    (((Proj_iso_Spec_Top_component hm f_deg).hom) \u27e8y.val, y_mem\u27e9)) = localization.mk 0 1,\n    by erw localization.mk_zero, localization.mk_eq_mk', is_localization.eq] at eq1,\n\n  obtain \u27e8\u27e8\u27e8C, C_degree_zero\u27e9, hC\u27e9, eq1\u27e9 := eq1,\n  induction C using localization.induction_on with \ud835\udd3b,\n  obtain \u27e8C, \u27e8_, \u27e8l, rfl\u27e9\u27e9\u27e9 := \ud835\udd3b,\n  simp only [submonoid.coe_one, mul_one, one_mul, subtype.coe_mk] at eq1,\n  simp only [zero_mul] at eq1,\n  simp only [localization.mk_eq_mk', is_localization.eq],\n  change _ \u2209 _ at hC,\n  erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff at hC,\n  dsimp only [subtype.coe_mk] at C_degree_zero hC,\n\n  have eq_num := degree_zero_part.eq (data.num hm f_deg 0 y),\n  have eq_denom := degree_zero_part.eq (data.denom hm f_deg 0 y),\n\n  rw subtype.ext_iff at eq1,\n  simp only [subring.coe_mul, subtype.coe_mk] at eq1,\n  rw [eq_num, subring.coe_zero,\n    show (0 : localization.away f) = localization.mk 0 1, by rw localization.mk_zero,\n    localization.mk_mul] at eq1,\n  simp only [localization.mk_eq_mk', is_localization.eq] at eq1,\n  obtain \u27e8\u27e8_, \u27e8n1, rfl\u27e9\u27e9, eq1\u27e9 := eq1,\n  simp only [submonoid.coe_mul, \u2190pow_add,\n    submonoid.coe_one, mul_one, zero_mul, subtype.coe_mk] at eq1,\n\n  have C_not_mem : C \u2209 y.1.as_homogeneous_ideal,\n  { intro rid,\n    have eq1 : (localization.mk C \u27e8f ^ l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n      (localization.mk 1 \u27e8f^l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) * localization.mk C 1,\n      rw [localization.mk_mul, one_mul, mul_one],\n    erw eq1 at hC,\n    apply hC,\n    convert ideal.mul_mem_left _ _ _,\n    apply ideal.subset_span,\n    refine \u27e8C, rid, rfl\u27e9, },\n\n  use C * f^n1,\n  { intro rid,\n    rcases y.1.is_prime.mem_or_mem rid with H1 | H2,\n    apply C_not_mem H1,\n    replace H2 := y.1.is_prime.mem_of_pow_mem _ H2,\n    apply y_mem,\n    exact H2, },\n\n  simp only [submonoid.coe_one, zero_mul, mul_one],\n  simp only [\u2190 subtype.val_eq_coe],\n\n  rw calc degree_zero_part.num (data.num hm f_deg 0 y)\n        * f ^ degree_zero_part.deg (data.denom hm f_deg 0 y)\n        * (C * f ^ n1)\n      = degree_zero_part.num (data.num hm f_deg 0 y)\n        * C * f ^ n1\n        * f ^ degree_zero_part.deg (data.denom hm f_deg 0 y)\n      : by ring,\n  rw [eq1, zero_mul],\nend\n\nlemma bmk_add (x y : (Spec (A\u2070_ f_deg)).presheaf.obj V) :\n  bmk hm f_deg V (x + y) = bmk hm f_deg V x + bmk hm f_deg V y :=\nbegin\n  ext1 z,\n  have z_mem : z.val \u2208 (projective_spectrum.basic_open \ud835\udc9c f).val,\n  { erw projective_spectrum.mem_basic_open,\n    intro rid,\n    have mem1 := z.2,\n    erw set.mem_preimage at mem1,\n    obtain \u27e8\u27e8a, ha1\u27e9, ha, ha2\u27e9 := mem1,\n    change a = z.1 at ha2,\n    erw set.mem_preimage at ha,\n    erw \u2190ha2 at rid,\n    apply ha1,\n    exact rid },\n\n  rw pi.add_apply,\n  unfold bmk,\n  simp only [homogeneous_localization.ext_iff_val, homogeneous_localization.val_mk', homogeneous_localization.add_val, \u2190subtype.val_eq_coe],\n  unfold num denom,\n  dsimp only,\n\n  have add_eq := data.eq_num_div_denom hm f_deg (x + y) z,\n  rw [data.add_apply, data.eq_num_div_denom, data.eq_num_div_denom, add_mk] at add_eq,\n  simp only [localization.mk_eq_mk'] at add_eq,\n  erw is_localization.eq at add_eq,\n  obtain \u27e8\u27e8\u27e8C, C_degree_zero\u27e9, hC\u27e9, add_eq\u27e9 := add_eq,\n  induction C using localization.induction_on with \ud835\udd3b,\n  obtain \u27e8C, \u27e8_, \u27e8l, rfl\u27e9\u27e9\u27e9 := \ud835\udd3b,\n  change _ \u2209 _ at hC, \n  erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff at hC,\n  simp only [subtype.coe_mk] at hC,\n  simp only [submonoid.coe_mul, subtype.coe_mk] at add_eq,\n  rw subtype.ext_iff at add_eq,\n  simp only [subring.coe_add, subring.coe_mul, subtype.coe_mk] at add_eq,\n\n  have C_not_mem : C \u2209 z.1.as_homogeneous_ideal,\n  { intro rid,\n    have eq1 : (localization.mk C \u27e8f ^ l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n      (localization.mk 1 \u27e8f^l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) * localization.mk C 1,\n      rw [localization.mk_mul, one_mul, mul_one],\n    erw eq1 at hC,\n    apply hC,\n    convert ideal.mul_mem_left _ _ _,\n    apply ideal.subset_span,\n    exact \u27e8C, rid, rfl\u27e9, },\n\n  simp only [degree_zero_part.eq, localization.mk_mul, localization.add_mk,\n    submonoid.coe_mul] at add_eq,\n  rw [localization.mk_eq_mk', is_localization.eq] at add_eq,\n  obtain \u27e8\u27e8_, \u27e8n1, rfl\u27e9\u27e9, add_eq\u27e9 := add_eq,\n  simp only [\u2190subtype.val_eq_coe,\n    submonoid.coe_mul] at add_eq,\n\n  set a_xy : A := degree_zero_part.num (data.num hm f_deg (x + y) z) with a_xy_eq,\n  set i_xy : \u2115 := degree_zero_part.deg (data.num hm f_deg (x + y) z) with i_xy_eq,\n  set b_xy : A := degree_zero_part.num (data.denom hm f_deg (x + y) z) with b_xy_eq,\n  set j_xy : \u2115 := degree_zero_part.deg (data.denom hm f_deg (x + y) z) with j_xy_eq,\n\n  set a_x : A := degree_zero_part.num (data.num hm f_deg x z) with a_x_eq,\n  set i_x : \u2115 := degree_zero_part.deg (data.num hm f_deg x z) with i_x_eq,\n  set b_x : A := degree_zero_part.num (data.denom hm f_deg x z) with b_x_eq,\n  set j_x : \u2115 := degree_zero_part.deg (data.denom hm f_deg x z) with j_x_eq,\n\n  set a_y : A := degree_zero_part.num (data.num hm f_deg y z) with a_y_eq,\n  set i_y : \u2115 := degree_zero_part.deg (data.num hm f_deg y z) with i_y_eq,\n  set b_y : A := degree_zero_part.num (data.denom hm f_deg y z) with b_y_eq,\n  set j_y : \u2115 := degree_zero_part.deg (data.denom hm f_deg y z) with j_y_eq,\n\n  simp only [\u2190a_xy_eq, \u2190i_xy_eq, \u2190b_xy_eq, \u2190j_xy_eq, \u2190a_x_eq, \u2190i_x_eq, \u2190b_x_eq, \u2190j_x_eq, \u2190a_y_eq, \u2190b_y_eq, \u2190i_y_eq, \u2190j_y_eq] at add_eq \u22a2,\n\n  rw localization.add_mk,\n  simp only [\u2190subtype.val_eq_coe,\n    show \u2200 (\u03b1 \u03b2 : z.1.as_homogeneous_ideal.to_ideal.prime_compl), \u03b1 * \u03b2 = \u27e8\u03b1.1 * \u03b2.1, begin\n      intro rid,\n      rcases z.1.is_prime.mem_or_mem rid,\n      apply \u03b1.2 h,\n      apply \u03b2.2 h,\n    end\u27e9,\n    begin\n      intros \u03b1 \u03b2,\n      simp only [subtype.ext_iff],\n      refl,\n    end,\n    show b_x * f ^ i_x * (a_y * f ^ j_y) = a_y * b_x * f ^ (i_x + j_y),\n    begin\n      rw pow_add, ring,\n    end,\n    show b_y * f ^ i_y * (a_x * f ^ j_x) = a_x * b_y * f ^ (i_y + j_x),\n    begin\n      rw pow_add, ring\n    end,\n    show b_x * f ^ i_x * (b_y * f ^ i_y) = b_x * b_y * f ^ (i_x + i_y),\n    begin\n      rw pow_add, ring\n    end],\n  rw [calc (f ^ j_x * f ^ i_y * (b_y * a_x) + f ^ j_y * f ^ i_x * (b_x * a_y)) * b_xy * C\n          * (f ^ i_xy * (f ^ j_x * f ^ j_y) * f ^ l) * f ^ n1\n        = ((f ^ j_x * f ^ i_y) * (b_y * a_x) + (f ^ j_y * f ^ i_x) * (b_x * a_y)) * b_xy * C\n          * ((f ^ i_xy * (f ^ j_x * f ^ j_y) * f ^ l) * f ^ n1) : by ring\n    ... = ((f ^ (j_x + i_y)) * (b_y * a_x) + (f ^ (j_y + i_x)) * (b_x * a_y)) * b_xy * C\n          * f ^ ((((i_xy + (j_x + j_y))) + l) + n1)\n        : begin\n          congr',\n          all_goals { repeat { rw pow_add } },\n        end,\n      calc a_xy * (b_x * b_y) * C * (f ^ j_x * f ^ i_y * (f ^ j_y * f ^ i_x) * f ^ j_xy * f ^ l) * f ^ n1\n        = a_xy * (b_x * b_y) * C * ((f ^ j_x * f ^ i_y * (f ^ j_y * f ^ i_x) * f ^ j_xy * f ^ l) * f ^ n1) : by ring\n    ... = a_xy * (b_x * b_y) * C * f ^ (((((j_x + i_y) + (j_y + i_x)) + j_xy) + l) + n1) : by simp only [pow_add]] at add_eq,\n\n  simp only [localization.mk_eq_mk', is_localization.eq],\n  refine \u27e8\u27e8C * f ^ ((j_x + j_y) + l + n1), begin\n    intro rid,\n    rcases z.1.is_prime.mem_or_mem rid with H1 | H2,\n    apply C_not_mem H1,\n    replace H2 := z.1.is_prime.mem_of_pow_mem _ H2,\n    apply z_mem H2,\n  end\u27e9, _\u27e9,\n  simp only [\u2190subtype.val_eq_coe],\n\n  rw [calc (a_y * b_x * f ^ (i_x + j_y) + a_x * b_y * f ^ (i_y + j_x)) * (b_xy * f ^ i_xy)\n          * (C * f ^ ((j_x + j_y) + l + n1))\n        = (f ^ (i_y + j_x) * (b_y * a_x) +  f ^ (i_x + j_y) * (b_x * a_y)) * b_xy * C\n          * (f ^ i_xy * f ^ ((j_x + j_y) + l + n1)) : by ring\n    ... = (f ^ (i_y + j_x) * (b_y * a_x) +  f ^ (i_x + j_y) * (b_x * a_y)) * b_xy * C\n          * (f ^ (i_xy + ((j_x + j_y) + l + n1))) : by simp only [pow_add]\n    ... = (f ^ (j_x + i_y) * (b_y * a_x) +  f ^ (j_y + i_x) * (b_x * a_y)) * b_xy * C\n          * (f ^ (i_xy + (j_x + j_y) + l + n1))\n        : begin\n          congr' 1,\n          congr' 5,\n          all_goals { simp only [add_comm, add_assoc], },\n        end, add_eq],\n  simp only [pow_add],\n  ring,\nend\n\nlemma bmk_mul (x y : (Spec (A\u2070_ f_deg)).presheaf.obj V) :\n  bmk hm f_deg V (x * y) = bmk hm f_deg V x * bmk hm f_deg V y :=\nbegin\n  ext1 z,\n  have z_mem : z.val \u2208 (projective_spectrum.basic_open \ud835\udc9c f).val,\n  { erw projective_spectrum.mem_basic_open,\n    intro rid,\n    have mem1 := z.2,\n    erw set.mem_preimage at mem1,\n    obtain \u27e8\u27e8a, ha1\u27e9, ha, ha2\u27e9 := mem1,\n    change a = z.1 at ha2,\n    erw set.mem_preimage at ha,\n    erw \u2190ha2 at rid,\n    apply ha1,\n    exact rid, },\n\n  rw pi.mul_apply,\n  unfold bmk,\n  simp only [homogeneous_localization.ext_iff_val, homogeneous_localization.val_mk', homogeneous_localization.mul_val, \u2190 subtype.val_eq_coe],\n  unfold num denom,\n\n  have mul_eq := data.eq_num_div_denom hm f_deg (x * y) z,\n  rw [data.mul_apply, data.eq_num_div_denom, data.eq_num_div_denom, localization.mk_mul] at mul_eq,\n  simp only [localization.mk_eq_mk'] at mul_eq,\n  erw is_localization.eq at mul_eq,\n  obtain \u27e8\u27e8\u27e8C, C_degree_zero\u27e9, hC\u27e9, mul_eq\u27e9 := mul_eq,\n  induction C using localization.induction_on with \ud835\udd3b,\n  obtain \u27e8C, \u27e8_, \u27e8l, rfl\u27e9\u27e9\u27e9 := \ud835\udd3b,\n  change _ \u2209 _ at hC,\n  erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff at hC,\n  simp only [subtype.coe_mk] at hC,\n  simp only [\u2190subtype.val_eq_coe] at mul_eq,\n  rw subtype.ext_iff at mul_eq,\n\n  have C_not_mem : C \u2209 z.1.as_homogeneous_ideal,\n  { intro rid,\n    have eq1 : (localization.mk C \u27e8f ^ l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n      (localization.mk 1 \u27e8f^l, \u27e8_, rfl\u27e9\u27e9 : localization.away f) * localization.mk C 1,\n      rw [localization.mk_mul, one_mul, mul_one],\n    erw eq1 at hC,\n    apply hC,\n    convert ideal.mul_mem_left _ _ _,\n    apply ideal.subset_span,\n    exact \u27e8C, rid, rfl\u27e9, },\n\n  simp only [subring.coe_mul, coe_add, subtype.coe_mk,\n    show \u2200 (\u03b1 \u03b2 : (prime_spectrum.as_ideal (((Proj_iso_Spec_Top_component hm f_deg).hom)\n      \u27e8z.val, z_mem\u27e9)).prime_compl),\n      (\u03b1 * \u03b2).1 = \u03b1.1 * \u03b2.1, from \u03bb _ _, rfl] at mul_eq,\n  simp only [degree_zero_part.eq, localization.mk_mul, localization.add_mk,\n    submonoid.coe_mul] at mul_eq,\n  rw [localization.mk_eq_mk', is_localization.eq] at mul_eq,\n  obtain \u27e8\u27e8_, \u27e8n1, rfl\u27e9\u27e9, mul_eq\u27e9 := mul_eq,\n  simp only [\u2190subtype.val_eq_coe,\n    submonoid.coe_mul] at mul_eq,\n\n  set a_xy : A := degree_zero_part.num (data.num hm f_deg (x * y) z) with a_xy_eq,\n  set i_xy : \u2115 := degree_zero_part.deg (data.num hm f_deg (x * y) z) with i_xy_eq,\n  set b_xy : A := degree_zero_part.num (data.denom hm f_deg (x * y) z) with b_xy_eq,\n  set j_xy : \u2115 := degree_zero_part.deg (data.denom hm f_deg (x * y) z) with j_xy_eq,\n\n  set a_x : A := degree_zero_part.num (data.num hm f_deg x z) with a_x_eq,\n  set i_x : \u2115 := degree_zero_part.deg (data.num hm f_deg x z) with i_x_eq,\n  set b_x : A := degree_zero_part.num (data.denom hm f_deg x z) with b_x_eq,\n  set j_x : \u2115 := degree_zero_part.deg (data.denom hm f_deg x z) with j_x_eq,\n\n  set a_y : A := degree_zero_part.num (data.num hm f_deg y z) with a_y_eq,\n  set i_y : \u2115 := degree_zero_part.deg (data.num hm f_deg y z) with i_y_eq,\n  set b_y : A := degree_zero_part.num (data.denom hm f_deg y z) with b_y_eq,\n  set j_y : \u2115 := degree_zero_part.deg (data.denom hm f_deg y z) with j_y_eq,\n\n  simp only [\u2190a_xy_eq, \u2190i_xy_eq, \u2190b_xy_eq, \u2190j_xy_eq, \u2190a_x_eq, \u2190i_x_eq, \u2190b_x_eq, \u2190j_x_eq, \u2190a_y_eq, \u2190b_y_eq, \u2190i_y_eq, \u2190j_y_eq] at mul_eq \u22a2,\n  rw [localization.mk_mul, localization.mk_eq_mk', is_localization.eq],\n  refine \u27e8\u27e8C * f^(l + n1), begin\n    intro rid,\n    rcases z.1.is_prime.mem_or_mem rid with H1 | H2,\n    apply C_not_mem H1,\n    replace H2 := z.1.is_prime.mem_of_pow_mem _ H2,\n    apply z_mem H2,\n  end\u27e9, _\u27e9,\n  simp only [\u2190subtype.val_eq_coe,\n    show \u2200 (\u03b1 \u03b2 : z.1.as_homogeneous_ideal.to_ideal.prime_compl), (\u03b1 * \u03b2).1 = \u03b1.1 * \u03b2.1,\n    from \u03bb _ _, rfl],\n  simp only [pow_add],\n  ring_nf at mul_eq \u22a2,\n  rw mul_eq,\nend\n\nnamespace is_locally_quotient\n\nvariable {V}\nlemma mem_pbo : y.1 \u2208 pbo f :=\nbegin\n  rw projective_spectrum.mem_basic_open,\n  intro rid,\n  have mem1 := y.2,\n  erw set.mem_preimage at mem1,\n  obtain \u27e8\u27e8a, ha1\u27e9, ha, ha2\u27e9 := mem1,\n  erw set.mem_preimage at ha,\n  erw \u2190ha2 at rid,\n  apply ha1,\n  exact rid,\nend\n\nlemma hom_apply_mem :\n  (Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, mem_pbo hm f_deg y\u27e9 \u2208 unop V := \nbegin\n  obtain \u27e8a, ha1, ha2\u27e9 := y.2,\n  erw set.mem_preimage at ha1,\n  change ((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, _\u27e9) \u2208 (unop V).1,\n  convert ha1,\n  rw subtype.ext_iff,\n  exact ha2.symm,\nend\n\ndef Uo (VV : opens (Spec.T (A\u2070_ f_deg))) :\n  opens (projective_spectrum.Top \ud835\udc9c) :=\n\u27e8{x | \u2203 x' : homeo_of_iso (Proj_iso_Spec_Top_component hm f_deg) \u207b\u00b9' VV.1, x = x'.1.1}, begin\n  have O1 := (homeomorph.is_open_preimage (homeo_of_iso (Proj_iso_Spec_Top_component hm f_deg))).2 VV.2,\n  rw is_open_induced_iff at O1,\n  obtain \u27e8s, Os, set_eq1\u27e9 := O1,\n  have O2 : is_open (s \u2229 (projective_spectrum.basic_open \ud835\udc9c f).1),\n  apply is_open.inter Os (projective_spectrum.basic_open \ud835\udc9c f).2,\n  convert O2,\n  ext \u03b3, split; intros h\u03b3,\n  { obtain \u27e8x', rfl\u27e9 := h\u03b3,\n    have mem1 := x'.2,\n    simp only [\u2190set_eq1] at mem1,\n    erw set.mem_preimage at mem1,\n    refine \u27e8mem1, _\u27e9,\n    have mem2 := x'.2,\n    rw set.mem_preimage at mem2,\n    intro rid,\n    have mem3 : (\u27e8localization.mk f \u27e8f^1, \u27e8_, rfl\u27e9\u27e9, \u27e81, \u27e8_, by simpa [mul_one] using f_deg\u27e9, rfl\u27e9\u27e9 : A\u2070_ f_deg) \u2208 ((Proj_iso_Spec_Top_component hm f_deg).hom x'.1).as_ideal,\n    { erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff,\n      change (localization.mk f \u27e8f^1, \u27e8_, rfl\u27e9\u27e9 : localization.away f) \u2208 ideal.span _,\n      convert ideal.mul_mem_left _ _ _,\n      work_on_goal 2\n      { exact mk 1 \u27e8f^1, \u27e8_, rfl\u27e9\u27e9 },\n      work_on_goal 2\n      { exact mk f 1 },\n      { rw [mk_mul, one_mul, mul_one], },\n      { apply ideal.subset_span,\n        refine \u27e8f, rid, rfl\u27e9, } },\n    have mem4 : (1 : A\u2070_ f_deg) \u2208 ((Proj_iso_Spec_Top_component hm f_deg).hom x'.1).as_ideal,\n    { convert mem3,\n      rw [subtype.ext_iff, subring.coe_one],\n      dsimp only [subtype.coe_mk],\n      symmetry,\n      convert localization.mk_self _,\n      erw [\u2190subtype.val_eq_coe],\n      dsimp only,\n      rw pow_one, },\n    apply ((Proj_iso_Spec_Top_component hm f_deg).hom x'.1).is_prime.1,\n    rw ideal.eq_top_iff_one,\n    exact mem4, },\n\n  { rcases h\u03b3 with \u27e8h\u03b31, h\u03b32\u27e9,\n    use \u27e8\u03b3, h\u03b32\u27e9,\n    rw [\u2190set_eq1, set.mem_preimage],\n        convert h\u03b31, }\nend\u27e9\n\nlemma subset2 (VV : opens (Spec.T (A\u2070_ f_deg)))\n  (subset1 : VV \u27f6 unop V) :\n  Uo hm f_deg VV \u27f6\n  (((@opens.open_embedding Proj.T (pbo f)).is_open_map.functor.op.obj\n        ((opens.map (Proj_iso_Spec_Top_component hm f_deg).hom).op.obj V)).unop) :=\nbegin\n  apply hom_of_le,\n  intros \u03b3 \u03b3_mem,\n  change \u03b3 \u2208 _ at \u03b3_mem,\n  replace subset3 := le_of_hom subset1,\n  obtain \u27e8\u27e8\u03b3, \u03b3_mem\u27e9, rfl\u27e9 := \u03b3_mem,\n  erw set.mem_preimage at \u03b3_mem,\n  refine \u27e8\u03b3, _, _\u27e9,\n  erw set.mem_preimage,\n  apply subset3,\n  exact \u03b3_mem,\n  rw subtype.ext_iff,\n  dsimp only,\n  rw show (opens.inclusion _ \u03b3 = \u03b3.1), from rfl,\nend\n\nend is_locally_quotient\n\nlemma is_locally_quotient :\n  \u2203 (U : opens _) (mem : y.val \u2208 U)\n    (subset1 : U \u27f6\n      (((@opens.open_embedding (projective_spectrum.Top \ud835\udc9c) (projective_spectrum.basic_open \ud835\udc9c f)).is_open_map.functor.op.obj\n        ((opens.map (Proj_iso_Spec_Top_component hm f_deg).hom).op.obj V)).unop))\n    (a b : A) (degree : \u2115) (a_hom : a \u2208 \ud835\udc9c degree) (b_hom : b \u2208 \ud835\udc9c degree),\n    \u2200 (x : U),\n      \u2203 (s_nin : b \u2209 projective_spectrum.as_homogeneous_ideal x.val),\n        (bmk hm f_deg V hh \u27e8x.1, (subset1 x).2\u27e9).val = mk a \u27e8b, s_nin\u27e9 :=\nbegin\n  have y_mem : y.val \u2208 projective_spectrum.basic_open \ud835\udc9c f,\n  { convert is_locally_quotient.mem_pbo hm f_deg y, },\n\n  have hom_y_mem : (Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, y_mem\u27e9 \u2208 unop V,\n  { convert is_locally_quotient.hom_apply_mem hm f_deg y, },\n  have is_local := hh.2,\n  rw structure_sheaf.is_locally_fraction_pred' at is_local,\n  specialize is_local \u27e8(Proj_iso_Spec_Top_component hm f_deg).hom \u27e8y.1, y_mem\u27e9, hom_y_mem\u27e9,\n  obtain \u27e8VV, hom_y_mem_VV, subset1, \u27e8\u03b1, \u27e8l1, \u27e8\u03b1', \u03b1'_mem\u27e9, rfl\u27e9\u27e9, \u27e8\u03b2, \u27e8l2, \u27e8\u03b2', \u03b2'_mem\u27e9, rfl\u27e9\u27e9, is_local\u27e9 := is_local,\n\n  set U := is_locally_quotient.Uo hm f_deg VV with U_eq,\n\n  have y_mem_U : y.1 \u2208 U,\n  { use \u27e8y.1, y_mem\u27e9,\n    rw set.mem_preimage,\n    exact hom_y_mem_VV, },\n\n  set subset2 : U \u27f6 _ := is_locally_quotient.subset2 hm f_deg VV subset1,\n  refine \u27e8U, y_mem_U, subset2, \u03b1' * f^l2, \u03b2' * f^l1, m * l1 + l2 * m,\n    set_like.graded_monoid.mul_mem \u03b1'_mem (set_like.graded_monoid.pow_mem _ f_deg),\n    by { convert set_like.graded_monoid.mul_mem \u03b2'_mem (set_like.graded_monoid.pow_mem _ f_deg) using 2, rw [smul_eq_mul], ring, }, _\u27e9,\n\n\n  rintros \u27e8z, z_mem_U\u27e9,\n  have z_mem_bo : z \u2208 pbo f,\n  { obtain \u27e8\u27e8z, hz\u27e9, rfl\u27e9 := z_mem_U,\n    rw set.mem_preimage at hz,\n    apply z.2, },\n\n  have hom_z_mem_VV : ((Proj_iso_Spec_Top_component hm f_deg).hom) \u27e8z, z_mem_bo\u27e9 \u2208 VV,\n  { obtain \u27e8\u03b3, h1, h2\u27e9 := z_mem_U,\n    have mem1 := \u03b3.2,\n    erw set.mem_preimage at mem1,\n    exact mem1, },\n\n  specialize is_local \u27e8((Proj_iso_Spec_Top_component hm f_deg).hom \u27e8z, z_mem_bo\u27e9), hom_z_mem_VV\u27e9,\n  obtain \u27e8not_mem1, eq1\u27e9 := is_local,\n\n  have not_mem2 : \u03b2' * f ^ l1 \u2209 z.as_homogeneous_ideal,\n  { intro rid,\n    rcases z.is_prime.mem_or_mem rid with H1 | H2,\n    { apply not_mem1,\n      have eq2 : (localization.mk \u03b2' \u27e8f^l2, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n        localization.mk 1 \u27e8f^l2, \u27e8_, rfl\u27e9\u27e9 * localization.mk \u03b2' 1,\n      { rw [localization.mk_mul, one_mul, mul_one], },\n      simp only [eq2],\n      erw Proj_iso_Spec_Top_component.to_Spec.mem_carrier_iff,\n      dsimp only,\n      convert ideal.mul_mem_left _ _ _,\n      apply ideal.subset_span,\n      refine \u27e8\u03b2', H1, rfl\u27e9, },\n    { replace H2 := z.is_prime.mem_of_pow_mem _ H2,\n      exact z_mem_bo H2, } },\n  refine \u27e8not_mem2, _\u27e9,\n  have data_eq : data hm f_deg hh (subset2 \u27e8z, z_mem_U\u27e9) =\n    hh.val (subset1 \u27e8((Proj_iso_Spec_Top_component hm f_deg).hom) \u27e8z, z_mem_bo\u27e9, hom_z_mem_VV\u27e9),\n  { congr', },\n  rw \u2190data_eq at eq1,\n\n  have z_mem2 : z \u2208 (((@opens.open_embedding Proj.T (pbo f)).is_open_map.functor.op.obj\n        ((opens.map (Proj_iso_Spec_Top_component hm f_deg).hom).op.obj V)).unop),\n  { use z,\n    refine \u27e8_, rfl\u27e9,\n    erw set.mem_preimage,\n    apply (le_of_hom subset1),\n    exact hom_z_mem_VV, },\n\n  have data_eq2 : data hm f_deg hh (subset2 \u27e8z, z_mem_U\u27e9) = data hm f_deg hh \u27e8z, z_mem2\u27e9,\n  { congr', },\n  rw [data_eq2, data.eq_num_div_denom, localization.mk_eq_mk'] at eq1,\n  erw is_localization.eq at eq1,\n\n  obtain \u27e8\u27e8\u27e8_, \u27e8L, \u27e8C, C_mem\u27e9, rfl\u27e9\u27e9, hC\u27e9, eq1\u27e9 := eq1,\n  simp only [subtype.ext_iff, subring.coe_mul] at eq1,\n  simp only [degree_zero_part.eq, localization.mk_mul, subtype.coe_mk] at eq1,\n  erw [localization.mk_eq_mk', is_localization.eq] at eq1,\n  obtain \u27e8\u27e8_, \u27e8M, rfl\u27e9\u27e9, eq1\u27e9 := eq1,\n  simp only [\u2190subtype.val_eq_coe,\n    submonoid.coe_mul, \u2190pow_add] at eq1,\n\n  unfold bmk,\n  rw [homogeneous_localization.val_mk'],\n  simp only [\u2190 subtype.val_eq_coe],\n  unfold num denom,\n\n  set p := degree_zero_part.num (data.num hm f_deg hh \u27e8z, z_mem2\u27e9) with p_eq,\n  set q := degree_zero_part.num (data.denom hm f_deg hh \u27e8z, z_mem2\u27e9) with q_eq,\n  set ii := degree_zero_part.deg (data.num hm f_deg hh \u27e8z, z_mem2\u27e9) with ii_eq,\n  set jj := degree_zero_part.deg (data.denom hm f_deg hh \u27e8z, z_mem2\u27e9) with jj_eq,\n\n  simp only [localization.mk_eq_mk', is_localization.eq],\n\n  have C_not_mem : C \u2209 z.as_homogeneous_ideal,\n  { intro rid,\n    have eq1 : (localization.mk C \u27e8f ^ L, \u27e8_, rfl\u27e9\u27e9 : localization.away f) =\n      (localization.mk 1 \u27e8f^L, \u27e8_, rfl\u27e9\u27e9 : localization.away f) * localization.mk C 1,\n      rw [localization.mk_mul, one_mul, mul_one],\n    simp only [eq1, subtype.coe_mk] at hC,\n    apply hC,\n    change _ * _ \u2208 _,\n    rw [set_like.mem_coe],\n    convert ideal.mul_mem_left _ _ _,\n    apply ideal.subset_span,\n    refine \u27e8C, rid, rfl\u27e9 },\n\n  refine \u27e8\u27e8C * f^(L+M), begin\n    intro rid,\n    rcases z.is_prime.mem_or_mem rid with H1 | H2,\n    apply C_not_mem H1,\n    replace H2 := z.is_prime.mem_of_pow_mem _ H2,\n    apply z_mem_bo,\n    exact H2,\n  end\u27e9, _\u27e9,\n\n  simp only [\u2190subtype.val_eq_coe,\n    submonoid.coe_mul],\n\n  suffices EQ : p * f^jj * (\u03b2' * f^l1) * (C * f^(L+M)) = \u03b1' * f^l2 * (q * f^ii) * (C * f^(L + M)),\n  convert EQ,\n  rw calc p * f^jj * (\u03b2' * f^l1) * (C * f^(L+M))\n        = p * f^jj * (\u03b2' * f^l1) * (C * (f^L * f^M)) : by simp only [pow_add]\n    ... = p * \u03b2' * C * (f^l1 * f^jj * f^L) * f^M : by ring\n    ... = p * \u03b2' * C * f^(l1 + jj + L) * f^M : by simp only [pow_add]\n    ... = \u03b1' * q * C * f ^ (ii + l2 + L) * f ^ M : by rw eq1,\n\n  simp only [pow_add],\n  ring,\nend\n\ndef to_fun.aux (hh : (Spec (A\u2070_ f_deg)).presheaf.obj V) : ((Proj_iso_Spec_Top_component hm f_deg).hom _* (Proj| (pbo f)).presheaf).obj V :=\n\u27e8bmk hm f_deg V hh, \u03bb y, begin \n  rcases is_locally_quotient hm f_deg V hh y with \u27e8VV, mem1, subset1, a, b, degree, a_mem, b_mem, l\u27e9,\n  refine \u27e8VV, mem1, subset1, degree, \u27e8a, a_mem\u27e9, \u27e8b, b_mem\u27e9, \u03bb x, _\u27e9,\n  rcases l x with \u27e8s_nin, l\u27e9,\n  refine \u27e8s_nin, _\u27e9,\n  dsimp only,\n  rw [homogeneous_localization.ext_iff_val, homogeneous_localization.val_mk'],\n  simp only [\u2190 subtype.val_eq_coe],\n  erw \u2190 l,\n  rw \u2190 homogeneous_localization.ext_iff_val,\n  congr' 1\nend\u27e9\n\ndef to_fun : (Spec (A\u2070_ f_deg)).presheaf.obj V \u27f6 ((Proj_iso_Spec_Top_component hm f_deg).hom _* (Proj| (pbo f)).presheaf).obj V :=\n{ to_fun := \u03bb hh, to_fun.aux hm f_deg V hh,\n  map_one' := begin\n    rw subtype.ext_iff,\n    convert bmk_one hm f_deg V,\n  end,\n  map_mul' := \u03bb x y, begin\n    rw subtype.ext_iff,\n    convert bmk_mul hm f_deg V x y,\n  end,\n  map_zero' := begin\n    rw subtype.ext_iff,\n    convert bmk_zero hm f_deg V,\n  end,\n  map_add' := \u03bb x y, begin\n    rw subtype.ext_iff,\n    convert bmk_add hm f_deg V x y,\n  end }\n\nend from_Spec\n\ndef from_Spec {f : A} {m : \u2115} (hm : 0 < m) (f_deg : f \u2208 \ud835\udc9c m) : \n  (Spec (A\u2070_ f_deg)).presheaf \u27f6 (Proj_iso_Spec_Top_component hm f_deg).hom _* (Proj| (pbo f)).presheaf :=\n{ app := \u03bb V, from_Spec.to_fun hm f_deg V,\n  naturality' := \u03bb U V subset1, begin\n    ext1 z,\n    simp only [comp_apply, ring_hom.coe_mk, functor.op_map, presheaf.pushforward_obj_map],\n    refl,\n  end }\n\nend Proj_iso_Spec_Sheaf_component\n\nend algebraic_geometry", "meta": {"author": "eric-wieser", "repo": "lean-graded-rings", "sha": "53bccd2553ee2052907ff9519e63f1945e6add4c", "save_path": "github-repos/lean/eric-wieser-lean-graded-rings", "path": "github-repos/lean/eric-wieser-lean-graded-rings/lean-graded-rings-53bccd2553ee2052907ff9519e63f1945e6add4c/src/cicm2022/examples/Proj/Proj_iso_Spec/Sheaf_component/from_Spec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585669110202, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.38834516322831936}}
{"text": "import ReactorModel.Determinism.ExecutionStep\nimport ReactorModel.Determinism.Trivial\n\nopen Classical ReactorType\n\nvariable [Indexable \u03b1] {s s\u2081 s\u2082 : State \u03b1}\n\nnamespace Execution\n\ntheorem tag_le {s\u2081 s\u2082 : State \u03b1} : (s\u2081 \u21d3* s\u2082) \u2192 s\u2081.tag \u2264 s\u2082.tag\n  | refl      => le_refl _\n  | step e e' => le_trans e.tag_le e'.tag_le\n\ntheorem seq_progress_ssubset_or_tag_lt [State.Nontrivial s\u2081] : \n    (s\u2081 \u21d3 s\u2082) \u2192 (s\u2082 \u21d3* s\u2083) \u2192 (s\u2081.progress \u2282 s\u2083.progress) \u2228 (s\u2081.tag < s\u2083.tag)\n  | e\u2081\u2082,        .step e e' => .inr $ lt_of_lt_of_le (e\u2081\u2082.seq_tag_lt e) e'.tag_le\n  | .close e,   .refl      => .inl $ e.progress_ssubset\n  | .advance a, .refl      => .inr $ a.tag_lt\n\ntheorem nontrivial_deterministic {s s\u2081 s\u2082 : State \u03b1} [State.Nontrivial s] :\n    (s \u21d3* s\u2081) \u2192 (s \u21d3* s\u2082) \u2192 (s\u2081.tag = s\u2082.tag) \u2192 (s\u2081.progress = s\u2082.progress) \u2192 s\u2081 = s\u2082\n  | refl, refl, _, _ => rfl\n  | step e\u2081 e\u2081', step e\u2082 e\u2082', ht, hp => \n    have := e\u2082.preserves_Nontrivial -- TODO: Make this work via type class inference.\n    nontrivial_deterministic (e\u2081.deterministic e\u2082 \u25b8 e\u2081') e\u2082' ht hp\n  | refl, step e e', ht, hp | step e e', refl, ht, hp => \n    match seq_progress_ssubset_or_tag_lt e e' with\n    | .inl h => absurd hp $ Set.ssubset_ne (by simp_all) \n    | .inr h => absurd ht $ ne_of_lt (by simp_all)\n\ntheorem deterministic : \n    (s \u21d3* s\u2081) \u2192 (s \u21d3* s\u2082) \u2192 (s\u2081.tag = s\u2082.tag) \u2192 (s\u2081.progress = s\u2082.progress) \u2192 s\u2081 = s\u2082 := \n  if h : State.Nontrivial s \n  then nontrivial_deterministic\n  else fun e\u2081 e\u2082 ht _ => e\u2081.trivial_deterministic h e\u2082 ht\n  \nend Execution", "meta": {"author": "marcusrossel", "repo": "reactor-model", "sha": "f82fffb489b4352a0cc6bee964d44a142fee18ce", "save_path": "github-repos/lean/marcusrossel-reactor-model", "path": "github-repos/lean/marcusrossel-reactor-model/reactor-model-f82fffb489b4352a0cc6bee964d44a142fee18ce/src/ReactorModel/Determinism/Execution.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7826624789529376, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3882740263669658}}
{"text": "import category_theory.triangulated.pretriangulated\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen limits category preadditive\nopen_locale zero_object\n\nnamespace pretriangulated\n\nvariables {C : Type*} [category C] [preadditive C] [has_shift C \u2124]\n\n@[simps]\ndef triangle.mk_iso (T T' : triangle C) (e\u2081 : T.obj\u2081 \u2245 T'.obj\u2081) (e\u2082 : T.obj\u2082 \u2245 T'.obj\u2082)\n  (e\u2083 : T.obj\u2083 \u2245 T'.obj\u2083)\n  (comm\u2081 : T.mor\u2081 \u226b e\u2082.hom = e\u2081.hom \u226b T'.mor\u2081)\n  (comm\u2082 : T.mor\u2082 \u226b e\u2083.hom = e\u2082.hom \u226b T'.mor\u2082)\n  (comm\u2083 : T.mor\u2083 \u226b (shift_functor C 1).map e\u2081.hom = e\u2083.hom \u226b T'.mor\u2083) : T \u2245 T' :=\n{ hom :=\n  { hom\u2081 := e\u2081.hom,\n    hom\u2082 := e\u2082.hom,\n    hom\u2083 := e\u2083.hom,\n    comm\u2081' := comm\u2081,\n    comm\u2082' := comm\u2082,\n    comm\u2083' := comm\u2083, },\n  inv :=\n  { hom\u2081 := e\u2081.inv,\n    hom\u2082 := e\u2082.inv,\n    hom\u2083 := e\u2083.inv,\n    comm\u2081' := by rw [\u2190 cancel_mono e\u2082.hom, assoc, e\u2082.inv_hom_id, comp_id, assoc, comm\u2081, e\u2081.inv_hom_id_assoc],\n    comm\u2082' := by { rw [\u2190 cancel_mono e\u2083.hom, assoc, e\u2083.inv_hom_id, comp_id, assoc, comm\u2082, e\u2082.inv_hom_id_assoc], },\n    comm\u2083' := by { rw [\u2190 cancel_epi e\u2083.hom, \u2190 assoc, \u2190 comm\u2083, assoc, \u2190 functor.map_comp, e\u2081.hom_inv_id, functor.map_id, comp_id, e\u2083.hom_inv_id_assoc], }, },\n  hom_inv_id' := by { ext; apply iso.hom_inv_id, },\n  inv_hom_id' := by { ext; apply iso.inv_hom_id, }, }\n\n@[simp, reassoc]\nlemma triangle.hom_inv_id_hom\u2081 {T T' : triangle C} (e : T \u2245 T') :\n  e.hom.hom\u2081 \u226b e.inv.hom\u2081 = \ud835\udfd9 _ :=\nby { change (e.hom \u226b e.inv).hom\u2081 = _, simpa only [e.hom_inv_id], }\n\n@[simp, reassoc]\nlemma triangle.inv_hom_id_hom\u2081 {T T' : triangle C} (e : T \u2245 T') :\n  e.inv.hom\u2081 \u226b e.hom.hom\u2081 = \ud835\udfd9 _ :=\nby { change (e.inv \u226b e.hom).hom\u2081 = _, simpa only [e.inv_hom_id], }\n\n@[simp, reassoc]\nlemma triangle.hom_inv_id_hom\u2082 {T T' : triangle C} (e : T \u2245 T') :\n  e.hom.hom\u2082 \u226b e.inv.hom\u2082 = \ud835\udfd9 _ :=\nby { change (e.hom \u226b e.inv).hom\u2082 = _, simpa only [e.hom_inv_id], }\n\n@[simp, reassoc]\nlemma triangle.inv_hom_id_hom\u2082 {T T' : triangle C} (e : T \u2245 T') :\n  e.inv.hom\u2082 \u226b e.hom.hom\u2082 = \ud835\udfd9 _ :=\nby { change (e.inv \u226b e.hom).hom\u2082 = _, simpa only [e.inv_hom_id], }\n@[simp, reassoc]\n\nlemma triangle.hom_inv_id_hom\u2083 {T T' : triangle C} (e : T \u2245 T') :\n  e.hom.hom\u2083 \u226b e.inv.hom\u2083 = \ud835\udfd9 _ :=\nby { change (e.hom \u226b e.inv).hom\u2083 = _, simpa only [e.hom_inv_id], }\n\n@[simp, reassoc]\nlemma triangle.inv_hom_id_hom\u2083 {T T' : triangle C} (e : T \u2245 T') :\n  e.inv.hom\u2083 \u226b e.hom.hom\u2083 = \ud835\udfd9 _ :=\nby { change (e.inv \u226b e.hom).hom\u2083 = _, simpa only [e.inv_hom_id], }\n\nlemma triangle.is_iso_of_is_iso_homs {T T' : triangle C} (f : T \u27f6 T')\n  (h\u2081 : is_iso f.hom\u2081) (h\u2082 : is_iso f.hom\u2082) (h\u2083 : is_iso f.hom\u2083) : is_iso f :=\nbegin\n  haveI := h\u2081,\n  haveI := h\u2082,\n  haveI := h\u2083,\n  convert is_iso.of_iso (triangle.mk_iso T T' (as_iso f.hom\u2081) (as_iso f.hom\u2082) (as_iso f.hom\u2083)\n    f.comm\u2081 f.comm\u2082 f.comm\u2083),\n  ext; refl,\nend\n\nsection\nvariables [\u2200 (n : \u2124), functor.additive (shift_functor C n)] [has_zero_object C] [pretriangulated C]\n\n@[reassoc]\nlemma triangle.comp_zero\u2081\u2082 (T : triangle C) (hT : T \u2208 dist_triang C) : T.mor\u2081 \u226b T.mor\u2082 = 0 :=\nbegin\n  obtain \u27e8c, \u27e8hc\u2081, hc\u2082\u27e9\u27e9 := complete_distinguished_triangle_morphism _ _\n    (contractible_distinguished T.obj\u2081) hT (\ud835\udfd9 T.obj\u2081) T.mor\u2081 rfl,\n  dsimp at hc\u2081,\n  rw [\u2190 hc\u2081, zero_comp],\nend\n\n@[reassoc]\nlemma triangle.comp_zero\u2082\u2083 (T : triangle C) (hT : T \u2208 dist_triang C) : T.mor\u2082 \u226b T.mor\u2083 = 0 :=\ntriangle.comp_zero\u2081\u2082 _ (rot_of_dist_triangle _ _ hT)\n\n@[reassoc]\nlemma triangle.comp_zero\u2083\u2081 (T : triangle C) (hT : T \u2208 dist_triang C) : T.mor\u2083 \u226b T.mor\u2081\u27e61\u27e7' = 0 :=\nbegin\n  rw [\u2190 neg_inj, \u2190 comp_neg, neg_zero],\n  exact triangle.comp_zero\u2081\u2082 _ (rot_of_dist_triangle _ _ (rot_of_dist_triangle _ _ hT)),\nend\n\nlemma distinguished_cocone_triangle\u2082 {Z X : C} (h : Z \u27f6 X\u27e6(1 : \u2124)\u27e7) :\n  \u2203 (Y : C) (f : X \u27f6 Y) (g : Y \u27f6 Z), triangle.mk f g h \u2208 dist_triang C :=\nbegin\n  obtain \u27e8Y', f', g', mem\u27e9 := pretriangulated.distinguished_cocone_triangle _ _ h,\n  let T := triangle.mk h f' g',\n  change T \u2208 dist_triang C at mem,\n  let T' := T.inv_rotate.inv_rotate,\n  let e\u2081 := (shift_functor_comp_shift_functor_neg C (1 : \u2124)).app X,\n  let e\u2082 := (shift_functor_neg_comp_shift_functor C (1 : \u2124)).app ((shift_functor C (1 : \u2124)).obj X),\n  let T'' := triangle.mk (e\u2081.inv \u226b T'.mor\u2081) T'.mor\u2082 (T'.mor\u2083 \u226b e\u2082.hom),\n  let e\u2083 : T' \u2245 T'' := begin\n    dsimp only [T', T'', triangle.mk],\n    refine triangle.mk_iso _ _ e\u2081 (iso.refl _) (iso.refl _) _ _ _,\n    { dsimp only [iso.refl],\n      rw [comp_id, e\u2081.hom_inv_id_assoc], },\n    { dsimp only [iso.refl],\n      rw [comp_id, id_comp], },\n    { dsimp only [iso.refl],\n      rw id_comp,\n      congr' 1,\n      have h : (shift_functor C 1).map e\u2081.inv \u226b e\u2082.hom = \ud835\udfd9 _ := shift_equiv_triangle (1 : \u2124) X,\n      rw [\u2190 cancel_epi ((shift_functor C (1 : \u2124)).map e\u2081.inv), h, \u2190 functor.map_comp,\n        iso.inv_hom_id, functor.map_id], },\n  end,\n  have eq : h = T'.mor\u2083 \u226b e\u2082.hom,\n  { dsimp,\n    simp only [unit_of_tensor_iso_unit_inv_app, \u03b5_app_obj, discrete.functor_map_id,\n      nat_trans.id_app, id_comp, assoc, \u03b5_inv_app_obj, \u03bc_inv_hom_app_assoc],\n    erw comp_id, },\n  rw eq,\n  refine \u27e8T''.obj\u2082, T''.mor\u2081, T''.mor\u2082, _\u27e9,\n  exact pretriangulated.isomorphic_distinguished _\n    (inv_rot_of_dist_triangle _ _ (inv_rot_of_dist_triangle _ _ mem)) _ e\u2083.symm,\nend\n\nlemma distinguished_cocone_triangle\u2081 {Y Z : C} (g : Y \u27f6 Z) :\n  \u2203 (X : C) (f : X \u27f6 Y) (h : Z \u27f6 X\u27e61\u27e7), triangle.mk f g h \u2208 dist_triang C :=\nbegin\n  obtain \u27e8X', f', g', mem\u27e9 := pretriangulated.distinguished_cocone_triangle _ _ g,\n  exact \u27e8_, _, _, inv_rot_of_dist_triangle _ _ mem\u27e9,\nend\n\nlemma complete_distinguished_triangle_morphism\u2081 (T\u2081 T\u2082 : triangle C)\n  (hT\u2081 : T\u2081 \u2208 dist_triang C) (hT\u2082 : T\u2082 \u2208 dist_triang C) (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082)\n  (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083) (comm : T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082) :\n  \u2203 (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081), T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081 \u2227\n    T\u2081.mor\u2083 \u226b (shift_functor C (1 : \u2124)).map a = c \u226b T\u2082.mor\u2083 :=\nbegin\n  obtain \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9\u27e9 := pretriangulated.complete_distinguished_triangle_morphism _ _\n    (rot_of_dist_triangle _ _ hT\u2081) (rot_of_dist_triangle _ _ hT\u2082) b c comm,\n  refine \u27e8(shift_functor C (1 : \u2124)).preimage a, \u27e8_, _\u27e9\u27e9,\n  { apply (shift_functor C (1 : \u2124)).map_injective,\n    dsimp at ha\u2082,\n    rw [neg_comp, comp_neg, neg_inj] at ha\u2082,\n    simpa only [functor.map_comp, functor.image_preimage] using ha\u2082, },\n  { simpa only [functor.image_preimage] using ha\u2081, },\nend\n\nlemma complete_distinguished_triangle_morphism\u2082 (T\u2081 T\u2082 : triangle C)\n  (hT\u2081 : T\u2081 \u2208 dist_triang C) (hT\u2082 : T\u2082 \u2208 dist_triang C) (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081)\n  (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083) (comm : T\u2081.mor\u2083 \u226b (shift_functor C (1 : \u2124)).map a = c \u226b T\u2082.mor\u2083) :\n  \u2203 (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082), T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081 \u2227 T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082 :=\nbegin\n  obtain \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9\u27e9 := pretriangulated.complete_distinguished_triangle_morphism _ _\n    (inv_rot_of_dist_triangle _ _ hT\u2081) (inv_rot_of_dist_triangle _ _ hT\u2082)\n      ((shift_functor C (-1 : \u2124)).map c) a begin\n    dsimp only [triangle.inv_rotate, triangle.mk],\n    simp only [neg_comp, comp_neg, neg_inj, assoc, \u2190 functor.map_comp_assoc, \u2190 comm,\n      iso.app_hom, unit_of_tensor_iso_unit_hom_app, discrete.functor_map_id,\n      nat_trans.id_app, id_comp, assoc, functor.map_comp, \u03bc_naturality_assoc,\n      nat_trans.naturality, functor.id_map],\n  end,\n  refine \u27e8a, \u27e8ha\u2081, _\u27e9\u27e9,\n  dsimp at ha\u2082,\n  erw [assoc, \u2190 nat_trans.naturality, functor.id_map] at ha\u2082,\n  simp only [\u2190 cancel_mono ((shift_functor_neg_comp_shift_functor C (1 : \u2124)).inv.app T\u2082.obj\u2083),\n    assoc, ha\u2082],\nend\n\nlemma contractible_distinguished\u2081 (X : C) : triangle.mk (0 : 0 \u27f6 X) (\ud835\udfd9 X) 0 \u2208 dist_triang C :=\nbegin\n  refine pretriangulated.isomorphic_distinguished _ (inv_rot_of_dist_triangle C _ (pretriangulated.contractible_distinguished X)) _ _,\n  refine triangle.mk_iso _ _ (functor.map_zero_object _).symm (iso.refl _) (iso.refl _)\n    (by tidy) (by tidy) (by tidy),\nend\n\nlemma contravariant_yoneda_exact\u2082 (T : triangle C) (hT : T \u2208 dist_triang C) {X : C} (f : T.obj\u2082 \u27f6 X)\n  (hf : T.mor\u2081 \u226b f = 0) : \u2203 (g : T.obj\u2083 \u27f6 X), f = T.mor\u2082 \u226b g :=\nbegin\n  obtain \u27e8g, \u27e8hg\u2081, hg\u2082\u27e9\u27e9 := pretriangulated.complete_distinguished_triangle_morphism T (triangle.mk (0 : 0 \u27f6 X) (\ud835\udfd9 _) 0) hT\n    (pretriangulated.contractible_distinguished\u2081 _) 0 f (by tidy),\n  dsimp at hg\u2081,\n  exact \u27e8g, by simpa only [comp_id] using hg\u2081.symm\u27e9,\nend\n\nlemma covariant_yoneda_exact\u2082 (T : triangle C) (hT : T \u2208 dist_triang C) {X : C} (f : X \u27f6 T.obj\u2082)\n  (hf : f \u226b T.mor\u2082 = 0) : \u2203 (g : X \u27f6 T.obj\u2081), f = g \u226b T.mor\u2081 :=\nbegin\n  obtain \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9\u27e9 := pretriangulated.complete_distinguished_triangle_morphism\u2081 _ T\n    (pretriangulated.contractible_distinguished X) hT f 0 (by { dsimp, rw [zero_comp, hf]}),\n  dsimp at ha\u2081,\n  exact \u27e8a, by simpa only [id_comp] using ha\u2081\u27e9,\nend\n\nlemma covariant_yoneda_exact\u2081 (T : triangle C) (hT : T \u2208 dist_triang C) {X : C} (f : X \u27f6 T.obj\u2081\u27e6(1 : \u2124)\u27e7)\n  (hf : f \u226b T.mor\u2081\u27e61\u27e7' = 0) : \u2203 (g : X \u27f6 T.obj\u2083), f = g \u226b T.mor\u2083 :=\ncovariant_yoneda_exact\u2082 _ (rot_of_dist_triangle _ _\n  (rot_of_dist_triangle _ _ hT)) f (by { dsimp, rw [comp_neg, hf, neg_zero], })\n\nlemma covariant_yoneda_exact\u2083 (T : triangle C) (hT : T \u2208 dist_triang C) {X : C} (f : X \u27f6 T.obj\u2083)\n  (hf : f \u226b T.mor\u2083 = 0) : \u2203 (g : X \u27f6 T.obj\u2082), f = g \u226b T.mor\u2082 :=\ncovariant_yoneda_exact\u2082 _ (rot_of_dist_triangle _ _ hT) f hf\n\nlemma isomorphic_distinguished_iff {T\u2081 T\u2082 : triangle C} (e : T\u2081 \u2245 T\u2082) :\n  (T\u2081 \u2208 dist_triang C) \u2194 T\u2082 \u2208 dist_triang C :=\nbegin\n  split,\n  { intro hT\u2081,\n    exact isomorphic_distinguished _ hT\u2081 _ e.symm, },\n  { intro hT\u2082,\n    exact isomorphic_distinguished _ hT\u2082 _ e, },\nend\n\nlemma inv_rotate_distinguished_triangle (T : triangle C) :\n  (T.inv_rotate \u2208 dist_triang C) \u2194 T \u2208 dist_triang C :=\nbegin\n  split,\n  { intro hT,\n    exact isomorphic_distinguished _ (rot_of_dist_triangle _ _ hT) _\n      ((triangle_rotation C).counit_iso.symm.app T), },\n  { intro hT,\n    exact inv_rot_of_dist_triangle _ T hT, },\nend\n\nend\n\nvariable (C)\n\n@[simps]\ndef contractible_triangle_functor [has_zero_object C] : C \u2964 triangle C :=\n{ obj := \u03bb X, contractible_triangle X,\n  map := \u03bb X Y f,\n  { hom\u2081 := f,\n    hom\u2082 := f,\n    hom\u2083 := 0, }, }\n\nend pretriangulated\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/triangulated/pretriangulated_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.38826844772663627}}
{"text": "/-\nCopyright (c) 2018 Sean Leather. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sean Leather, Mario Carneiro\n-/\nimport data.list.alist\nimport data.finset.sigma\nimport data.part\n/-!\n# Finite maps over `multiset`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nuniverses u v w\nopen list\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v}\n\n/-! ### multisets of sigma types-/\n\nnamespace multiset\n\n/-- Multiset of keys of an association multiset. -/\ndef keys (s : multiset (sigma \u03b2)) : multiset \u03b1 :=\ns.map sigma.fst\n\n@[simp] theorem coe_keys {l : list (sigma \u03b2)} :\n  keys (l : multiset (sigma \u03b2)) = (l.keys : multiset \u03b1) :=\nrfl\n\n/-- `nodupkeys s` means that `s` has no duplicate keys. -/\ndef nodupkeys (s : multiset (sigma \u03b2)) : Prop :=\nquot.lift_on s list.nodupkeys (\u03bb s t p, propext $ perm_nodupkeys p)\n\n@[simp] theorem coe_nodupkeys {l : list (sigma \u03b2)} : @nodupkeys \u03b1 \u03b2 l \u2194 l.nodupkeys := iff.rfl\n\nlemma nodup_keys {m : multiset (\u03a3 a, \u03b2 a)} : m.keys.nodup \u2194 m.nodupkeys :=\nby { rcases m with \u27e8l\u27e9, refl }\n\nalias nodup_keys \u2194 _ nodupkeys.nodup_keys\n\nlemma nodupkeys.nodup {m : multiset (\u03a3 a, \u03b2 a)} (h : m.nodupkeys) : m.nodup :=\nh.nodup_keys.of_map _\n\nend multiset\n\n/-! ### finmap -/\n\n/-- `finmap \u03b2` is the type of finite maps over a multiset. It is effectively\n  a quotient of `alist \u03b2` by permutation of the underlying list. -/\nstructure finmap (\u03b2 : \u03b1 \u2192 Type v) : Type (max u v) :=\n(entries : multiset (sigma \u03b2))\n(nodupkeys : entries.nodupkeys)\n\n/-- The quotient map from `alist` to `finmap`. -/\ndef alist.to_finmap (s : alist \u03b2) : finmap \u03b2 := \u27e8s.entries, s.nodupkeys\u27e9\n\nlocal notation (name := to_finmap) `\u27e6`:max a `\u27e7`:0 := alist.to_finmap a\n\ntheorem alist.to_finmap_eq {s\u2081 s\u2082 : alist \u03b2} :\n  \u27e6s\u2081\u27e7 = \u27e6s\u2082\u27e7 \u2194 s\u2081.entries ~ s\u2082.entries :=\nby cases s\u2081; cases s\u2082; simp [alist.to_finmap]\n\n@[simp] theorem alist.to_finmap_entries (s : alist \u03b2) : \u27e6s\u27e7.entries = s.entries := rfl\n\n/-- Given `l : list (sigma \u03b2)`, create a term of type `finmap \u03b2` by removing\nentries with duplicate keys. -/\ndef list.to_finmap [decidable_eq \u03b1] (s : list (sigma \u03b2)) : finmap \u03b2 := s.to_alist.to_finmap\n\nnamespace finmap\nopen alist\n\nlemma nodup_entries (f : finmap \u03b2) : f.entries.nodup := f.nodupkeys.nodup\n\n/-! ### lifting from alist -/\n\n/-- Lift a permutation-respecting function on `alist` to `finmap`. -/\n@[elab_as_eliminator] def lift_on\n  {\u03b3} (s : finmap \u03b2) (f : alist \u03b2 \u2192 \u03b3)\n  (H : \u2200 a b : alist \u03b2, a.entries ~ b.entries \u2192 f a = f b) : \u03b3 :=\nbegin\n  refine (quotient.lift_on s.1 (\u03bb l, (\u27e8_, \u03bb nd, f \u27e8l, nd\u27e9\u27e9 : part \u03b3))\n    (\u03bb l\u2081 l\u2082 p, part.ext' (perm_nodupkeys p) _) : part \u03b3).get _,\n  { exact \u03bb h\u2081 h\u2082, H _ _ (by exact p) },\n  { have := s.nodupkeys, rcases s.entries with \u27e8l\u27e9, exact id }\nend\n\n@[simp] theorem lift_on_to_finmap {\u03b3} (s : alist \u03b2) (f : alist \u03b2 \u2192 \u03b3) (H) :\n  lift_on \u27e6s\u27e7 f H = f s := by cases s; refl\n\n/-- Lift a permutation-respecting function on 2 `alist`s to 2 `finmap`s. -/\n@[elab_as_eliminator] def lift_on\u2082\n  {\u03b3} (s\u2081 s\u2082 : finmap \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3)\n  (H : \u2200 a\u2081 b\u2081 a\u2082 b\u2082 : alist \u03b2, a\u2081.entries ~ a\u2082.entries \u2192 b\u2081.entries ~ b\u2082.entries \u2192\n    f a\u2081 b\u2081 = f a\u2082 b\u2082) : \u03b3 :=\nlift_on s\u2081\n  (\u03bb l\u2081, lift_on s\u2082 (f l\u2081) (\u03bb b\u2081 b\u2082 p, H _ _ _ _ (perm.refl _) p))\n  (\u03bb a\u2081 a\u2082 p, have H' : f a\u2081 = f a\u2082 := funext (\u03bb _, H _ _ _ _ p (perm.refl _)), by simp only [H'])\n\n@[simp] theorem lift_on\u2082_to_finmap {\u03b3} (s\u2081 s\u2082 : alist \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3) (H) :\n  lift_on\u2082 \u27e6s\u2081\u27e7 \u27e6s\u2082\u27e7 f H = f s\u2081 s\u2082 :=\nby cases s\u2081; cases s\u2082; refl\n\n/-! ### induction -/\n\n@[elab_as_eliminator] theorem induction_on\n  {C : finmap \u03b2 \u2192 Prop} (s : finmap \u03b2) (H : \u2200 (a : alist \u03b2), C \u27e6a\u27e7) : C s :=\nby rcases s with \u27e8\u27e8a\u27e9, h\u27e9; exact H \u27e8a, h\u27e9\n\n@[elab_as_eliminator] theorem induction_on\u2082 {C : finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop}\n  (s\u2081 s\u2082 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 : alist \u03b2), C \u27e6a\u2081\u27e7 \u27e6a\u2082\u27e7) : C s\u2081 s\u2082 :=\ninduction_on s\u2081 $ \u03bb l\u2081, induction_on s\u2082 $ \u03bb l\u2082, H l\u2081 l\u2082\n\n@[elab_as_eliminator] theorem induction_on\u2083 {C : finmap \u03b2 \u2192  finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop}\n  (s\u2081 s\u2082 s\u2083 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 a\u2083 : alist \u03b2), C \u27e6a\u2081\u27e7 \u27e6a\u2082\u27e7 \u27e6a\u2083\u27e7) : C s\u2081 s\u2082 s\u2083 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb l\u2081 l\u2082, induction_on s\u2083 $ \u03bb l\u2083, H l\u2081 l\u2082 l\u2083\n\n/-! ### extensionality -/\n\n@[ext] theorem ext : \u2200 {s t : finmap \u03b2}, s.entries = t.entries \u2192 s = t\n| \u27e8l\u2081, h\u2081\u27e9 \u27e8l\u2082, h\u2082\u27e9 H := by congr'\n\n@[simp] theorem ext_iff {s t : finmap \u03b2} : s.entries = t.entries \u2194 s = t :=\n\u27e8ext, congr_arg _\u27e9\n\n/-! ### mem -/\n\n/-- The predicate `a \u2208 s` means that `s` has a value associated to the key `a`. -/\ninstance : has_mem \u03b1 (finmap \u03b2) := \u27e8\u03bb a s, a \u2208 s.entries.keys\u27e9\n\ntheorem mem_def {a : \u03b1} {s : finmap \u03b2} :\n  a \u2208 s \u2194 a \u2208 s.entries.keys := iff.rfl\n\n@[simp] theorem mem_to_finmap {a : \u03b1} {s : alist \u03b2} :\n  a \u2208 \u27e6s\u27e7 \u2194 a \u2208 s := iff.rfl\n\n/-! ### keys -/\n\n/-- The set of keys of a finite map. -/\ndef keys (s : finmap \u03b2) : finset \u03b1 :=\n\u27e8s.entries.keys, s.nodupkeys.nodup_keys\u27e9\n\n@[simp] theorem keys_val (s : alist \u03b2) : (keys \u27e6s\u27e7).val = s.keys := rfl\n\n@[simp] theorem keys_ext {s\u2081 s\u2082 : alist \u03b2} :\n  keys \u27e6s\u2081\u27e7 = keys \u27e6s\u2082\u27e7 \u2194 s\u2081.keys ~ s\u2082.keys :=\nby simp [keys, alist.keys]\n\ntheorem mem_keys {a : \u03b1} {s : finmap \u03b2} : a \u2208 s.keys \u2194 a \u2208 s :=\ninduction_on s $ \u03bb s, alist.mem_keys\n\n/-! ### empty -/\n\n/-- The empty map. -/\ninstance : has_emptyc (finmap \u03b2) := \u27e8\u27e80, nodupkeys_nil\u27e9\u27e9\n\ninstance : inhabited (finmap \u03b2) := \u27e8\u2205\u27e9\n\n@[simp] theorem empty_to_finmap : (\u27e6\u2205\u27e7 : finmap \u03b2) = \u2205 := rfl\n\n@[simp] theorem to_finmap_nil [decidable_eq \u03b1] : ([].to_finmap : finmap \u03b2) = \u2205 := rfl\n\ntheorem not_mem_empty {a : \u03b1} : a \u2209 (\u2205 : finmap \u03b2) :=\nmultiset.not_mem_zero a\n\n@[simp] theorem keys_empty : (\u2205 : finmap \u03b2).keys = \u2205 := rfl\n\n/-! ### singleton -/\n\n/-- The singleton map. -/\ndef singleton (a : \u03b1) (b : \u03b2 a) : finmap \u03b2 := \u27e6alist.singleton a b\u27e7\n\n@[simp] theorem keys_singleton (a : \u03b1) (b : \u03b2 a) :\n  (singleton a b).keys = {a} := rfl\n\n@[simp] lemma mem_singleton (x y : \u03b1) (b : \u03b2 y) : x \u2208 singleton y b \u2194 x = y :=\nby simp only [singleton]; erw [mem_cons_eq, mem_nil_iff, or_false]\n\nsection\n\nvariables [decidable_eq \u03b1]\n\ninstance has_decidable_eq [\u2200 a, decidable_eq (\u03b2 a)] : decidable_eq (finmap \u03b2)\n| s\u2081 s\u2082 := decidable_of_iff _ ext_iff\n\n/-! ### lookup -/\n\n/-- Look up the value associated to a key in a map. -/\ndef lookup (a : \u03b1) (s : finmap \u03b2) : option (\u03b2 a) :=\nlift_on s (lookup a) (\u03bb s t, perm_lookup)\n\n@[simp] theorem lookup_to_finmap (a : \u03b1) (s : alist \u03b2) :\n  lookup a \u27e6s\u27e7 = s.lookup a := rfl\n\n@[simp] theorem lookup_list_to_finmap (a : \u03b1) (s : list (sigma \u03b2)) :\n  lookup a s.to_finmap = s.lookup a :=\nby rw [list.to_finmap, lookup_to_finmap, lookup_to_alist]\n\n@[simp] theorem lookup_empty (a) : lookup a (\u2205 : finmap \u03b2) = none :=\nrfl\n\ntheorem lookup_is_some {a : \u03b1} {s : finmap \u03b2} :\n  (s.lookup a).is_some \u2194 a \u2208 s :=\ninduction_on s $ \u03bb s, alist.lookup_is_some\n\ntheorem lookup_eq_none {a} {s : finmap \u03b2} : lookup a s = none \u2194 a \u2209 s :=\ninduction_on s $ \u03bb s, alist.lookup_eq_none\n\nlemma mem_lookup_iff {f : finmap \u03b2} {a : \u03b1} {b : \u03b2 a} :\n  b \u2208 f.lookup a \u2194 sigma.mk a b \u2208 f.entries :=\nby { rcases f with \u27e8\u27e8l\u27e9, hl\u27e9, exact list.mem_lookup_iff hl }\n\n/-- A version of `finmap.mem_lookup_iff` with LHS in the simp-normal form. -/\nlemma lookup_eq_some_iff {f : finmap \u03b2} {a : \u03b1} {b : \u03b2 a} :\n  f.lookup a = some b \u2194 sigma.mk a b \u2208 f.entries :=\nmem_lookup_iff\n\n@[simp] lemma sigma_keys_lookup (f : finmap \u03b2) :\n  f.keys.sigma (\u03bb i, (f.lookup i).to_finset) = \u27e8f.entries, f.nodup_entries\u27e9 :=\nbegin\n  ext x,\n  have : x \u2208 f.entries \u2192 x.fst \u2208 f.keys, from multiset.mem_map_of_mem _,\n  simpa [lookup_eq_some_iff]\nend\n\n@[simp] lemma lookup_singleton_eq {a : \u03b1} {b : \u03b2 a} : (singleton a b).lookup a = some b :=\nby rw [singleton, lookup_to_finmap, alist.singleton, alist.lookup, lookup_cons_eq]\n\ninstance (a : \u03b1) (s : finmap \u03b2) : decidable (a \u2208 s) :=\ndecidable_of_iff _ lookup_is_some\n\nlemma mem_iff {a : \u03b1} {s : finmap \u03b2} : a \u2208 s \u2194 \u2203 b, s.lookup a = some b :=\ninduction_on s $ \u03bb s,\niff.trans list.mem_keys $ exists_congr $ \u03bb b,\n(list.mem_lookup_iff s.nodupkeys).symm\n\nlemma mem_of_lookup_eq_some {a : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} (h : s.lookup a = some b) : a \u2208 s :=\nmem_iff.mpr \u27e8_, h\u27e9\n\ntheorem ext_lookup {s\u2081 s\u2082 : finmap \u03b2} : (\u2200 x, s\u2081.lookup x = s\u2082.lookup x) \u2192 s\u2081 = s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082 h,\nbegin\n  simp only [alist.lookup, lookup_to_finmap] at h,\n  rw [alist.to_finmap_eq],\n  apply lookup_ext s\u2081.nodupkeys s\u2082.nodupkeys,\n  intros x y,\n  rw h,\nend\n\n/-- An equivalence between `finmap \u03b2` and pairs `(keys : finset \u03b1, lookup : \u03a0 a, option (\u03b2 a))` such\nthat `(lookup a).is_some \u2194 a \u2208 keys`. -/\n@[simps apply_coe_fst apply_coe_snd]\ndef keys_lookup_equiv :\n  finmap \u03b2 \u2243 {f : finset \u03b1 \u00d7 (\u03a0 a, option (\u03b2 a)) // \u2200 i, (f.2 i).is_some \u2194 i \u2208 f.1} :=\n{ to_fun := \u03bb f, \u27e8(f.keys, \u03bb i, f.lookup i), \u03bb i, lookup_is_some\u27e9,\n  inv_fun := \u03bb f, \u27e8(f.1.1.sigma $ \u03bb i, (f.1.2 i).to_finset).val,\n    begin\n      refine multiset.nodup_keys.1 ((finset.nodup _).map_on _),\n      simp only [finset.mem_val, finset.mem_sigma, option.mem_to_finset, option.mem_def],\n      rintro \u27e8i, x\u27e9 \u27e8hi, hx\u27e9 \u27e8j, y\u27e9 \u27e8hj, hy\u27e9 (rfl : i = j),\n      obtain rfl : x = y, from option.some.inj (hx.symm.trans hy),\n      refl\n    end\u27e9,\n  left_inv := \u03bb f, ext $ by simp,\n  right_inv := \u03bb \u27e8\u27e8s, f\u27e9, hf\u27e9,\n    begin\n      ext : 2; dsimp [keys],\n      { ext1 i,\n        have : i \u2208 s \u2192 (\u2203 x, f i = some x),\n          from \u03bb hi, \u27e8option.get _, option.get_mem $ (hf i).2 hi\u27e9,\n        simpa [multiset.keys] },\n      { ext i x : 2,\n        simp only [option.mem_def, lookup_eq_some_iff, finset.mem_val, finset.mem_sigma,\n          option.mem_to_finset, and_iff_right_iff_imp, \u2190 hf],\n        exact \u03bb h, option.is_some_iff_exists.2 \u27e8_, h\u27e9 }\n    end }\n\n@[simp] lemma keys_lookup_equiv_symm_apply_keys :\n  \u2200 f : {f : finset \u03b1 \u00d7 (\u03a0 a, option (\u03b2 a)) // \u2200 i, (f.2 i).is_some \u2194 i \u2208 f.1},\n    (keys_lookup_equiv.symm f).keys = (f : finset \u03b1 \u00d7 \u03a0 a, option (\u03b2 a)).1 :=\nkeys_lookup_equiv.surjective.forall.2 $ \u03bb f,\n  by simp only [equiv.symm_apply_apply, keys_lookup_equiv_apply_coe_fst]\n\n@[simp] lemma keys_lookup_equiv_symm_apply_lookup :\n  \u2200 (f : {f : finset \u03b1 \u00d7 (\u03a0 a, option (\u03b2 a)) // \u2200 i, (f.2 i).is_some \u2194 i \u2208 f.1}) a,\n    (keys_lookup_equiv.symm f).lookup a = (f : finset \u03b1 \u00d7 \u03a0 a, option (\u03b2 a)).2 a :=\nkeys_lookup_equiv.surjective.forall.2 $ \u03bb f a,\n  by simp only [equiv.symm_apply_apply, keys_lookup_equiv_apply_coe_snd]\n\n/-! ### replace -/\n\n/-- Replace a key with a given value in a finite map.\n  If the key is not present it does nothing. -/\ndef replace (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6replace a b t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_replace p\n\n@[simp] theorem replace_to_finmap (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) :\n  replace a b \u27e6s\u27e7 = \u27e6s.replace a b\u27e7 := by simp [replace]\n\n@[simp] theorem keys_replace (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) :\n  (replace a b s).keys = s.keys :=\ninduction_on s $ \u03bb s, by simp\n\n@[simp] theorem mem_replace {a a' : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} :\n  a' \u2208 replace a b s \u2194 a' \u2208 s :=\ninduction_on s $ \u03bb s, by simp\n\nend\n\n/-! ### foldl -/\n\n/-- Fold a commutative function over the key-value pairs in the map -/\ndef foldl {\u03b4 : Type w} (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4)\n  (H : \u2200 d a\u2081 b\u2081 a\u2082 b\u2082, f (f d a\u2081 b\u2081) a\u2082 b\u2082 = f (f d a\u2082 b\u2082) a\u2081 b\u2081)\n  (d : \u03b4) (m : finmap \u03b2) : \u03b4 :=\nm.entries.foldl (\u03bb d s, f d s.1 s.2) (\u03bb d s t, H _ _ _ _ _) d\n\n/-- `any f s` returns `tt` iff there exists a value `v` in `s` such that `f v = tt`. -/\ndef any (f : \u03a0 x, \u03b2 x \u2192 bool) (s : finmap \u03b2) : bool :=\ns.foldl (\u03bb x y z, x || f y z) (by { intros, simp_rw [bool.bor_assoc, bool.bor_comm] }) ff\n\n/-- `all f s` returns `tt` iff `f v = tt` for all values `v` in `s`. -/\ndef all (f : \u03a0 x, \u03b2 x \u2192 bool) (s : finmap \u03b2) : bool :=\ns.foldl (\u03bb x y z, x && f y z) (by { intros, simp_rw [bool.band_assoc, bool.band_comm] }) tt\n\n/-! ### erase -/\n\nsection\n\nvariables [decidable_eq \u03b1]\n\n/-- Erase a key from the map. If the key is not present it does nothing. -/\ndef erase (a : \u03b1) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6erase a t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_erase p\n\n@[simp] theorem erase_to_finmap (a : \u03b1) (s : alist \u03b2) :\n  erase a \u27e6s\u27e7 = \u27e6s.erase a\u27e7 := by simp [erase]\n\n@[simp] \n\n@[simp] theorem keys_erase (a : \u03b1) (s : finmap \u03b2) :\n  (erase a s).keys = s.keys.erase a :=\ninduction_on s $ \u03bb s, by simp\n\n@[simp] theorem mem_erase {a a' : \u03b1} {s : finmap \u03b2} : a' \u2208 erase a s \u2194 a' \u2260 a \u2227 a' \u2208 s :=\ninduction_on s $ \u03bb s, by simp\n\ntheorem not_mem_erase_self {a : \u03b1} {s : finmap \u03b2} : \u00ac a \u2208 erase a s :=\nby rw [mem_erase, not_and_distrib, not_not]; left; refl\n\n@[simp] theorem lookup_erase (a) (s : finmap \u03b2) : lookup a (erase a s) = none :=\ninduction_on s $ lookup_erase a\n\n@[simp] theorem lookup_erase_ne {a a'} {s : finmap \u03b2} (h : a \u2260 a') :\n  lookup a (erase a' s) = lookup a s :=\ninduction_on s $ \u03bb s, lookup_erase_ne h\n\ntheorem erase_erase {a a' : \u03b1} {s : finmap \u03b2} : erase a (erase a' s) = erase a' (erase a s) :=\ninduction_on s $ \u03bb s, ext (by simp only [erase_erase, erase_to_finmap])\n\n/-! ### sdiff -/\n\n/-- `sdiff s s'` consists of all key-value pairs from `s` and `s'` where the keys are in `s` or\n`s'` but not both. -/\ndef sdiff (s s' : finmap \u03b2) : finmap \u03b2 :=\ns'.foldl (\u03bb s x _, s.erase x) (\u03bb a\u2080 a\u2081 _ a\u2082 _, erase_erase) s\n\ninstance : has_sdiff (finmap \u03b2) := \u27e8sdiff\u27e9\n\n/-! ### insert -/\n\n/-- Insert a key-value pair into a finite map, replacing any existing pair with\n  the same key. -/\ndef insert (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\nlift_on s (\u03bb t, \u27e6insert a b t\u27e7) $\n\u03bb s\u2081 s\u2082 p, to_finmap_eq.2 $ perm_insert p\n\n@[simp] theorem insert_to_finmap (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) :\n  insert a b \u27e6s\u27e7 = \u27e6s.insert a b\u27e7 := by simp [insert]\n\ntheorem insert_entries_of_neg {a : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} : a \u2209 s \u2192\n  (insert a b s).entries = \u27e8a, b\u27e9 ::\u2098 s.entries :=\ninduction_on s $ \u03bb s h,\nby simp [insert_entries_of_neg (mt mem_to_finmap.1 h)]\n\n@[simp] theorem mem_insert {a a' : \u03b1} {b' : \u03b2 a'} {s : finmap \u03b2} :\n  a \u2208 insert a' b' s \u2194 a = a' \u2228 a \u2208 s :=\ninduction_on s mem_insert\n\n@[simp] theorem lookup_insert {a} {b : \u03b2 a} (s : finmap \u03b2) :\n  lookup a (insert a b s) = some b :=\ninduction_on s $ \u03bb s,\nby simp only [insert_to_finmap, lookup_to_finmap, lookup_insert]\n\n@[simp] theorem lookup_insert_of_ne {a a'} {b : \u03b2 a} (s : finmap \u03b2) (h : a' \u2260 a) :\n  lookup a' (insert a b s) = lookup a' s :=\ninduction_on s $ \u03bb s,\nby simp only [insert_to_finmap, lookup_to_finmap, lookup_insert_ne h]\n\n@[simp] theorem insert_insert {a} {b b' : \u03b2 a} (s : finmap \u03b2) :\n  (s.insert a b).insert a b' = s.insert a b' :=\ninduction_on s $ \u03bb s,\nby simp only [insert_to_finmap, insert_insert]\n\ntheorem insert_insert_of_ne {a a'} {b : \u03b2 a} {b' : \u03b2 a'} (s : finmap \u03b2) (h : a \u2260 a') :\n  (s.insert a b).insert a' b' = (s.insert a' b').insert a b :=\ninduction_on s $ \u03bb s,\nby simp only [insert_to_finmap, alist.to_finmap_eq, insert_insert_of_ne _ h]\n\ntheorem to_finmap_cons (a : \u03b1) (b : \u03b2 a) (xs : list (sigma \u03b2)) :\n  list.to_finmap (\u27e8a,b\u27e9 :: xs) = insert a b xs.to_finmap := rfl\n\ntheorem mem_list_to_finmap (a : \u03b1) (xs : list (sigma \u03b2)) :\n  a \u2208 xs.to_finmap \u2194 (\u2203 b : \u03b2 a, sigma.mk a b \u2208 xs) :=\nby { induction xs with x xs; [skip, cases x];\n     simp only [to_finmap_cons, *, not_mem_empty, exists_or_distrib, not_mem_nil, to_finmap_nil,\n                exists_false, mem_cons_iff, mem_insert, exists_and_distrib_left];\n     apply or_congr _ iff.rfl,\n     conv { to_lhs, rw \u2190 and_true (a = x_fst) },\n     apply and_congr_right, rintro \u27e8\u27e9, simp only [exists_eq, heq_iff_eq] }\n\n@[simp] theorem insert_singleton_eq {a : \u03b1} {b b' : \u03b2 a} :\n  insert a b (singleton a b') = singleton a b :=\nby simp only [singleton, finmap.insert_to_finmap, alist.insert_singleton_eq]\n\n/-! ### extract -/\n\n/-- Erase a key from the map, and return the corresponding value, if found. -/\ndef extract (a : \u03b1) (s : finmap \u03b2) : option (\u03b2 a) \u00d7 finmap \u03b2 :=\nlift_on s (\u03bb t, prod.map id to_finmap (extract a t)) $\n\u03bb s\u2081 s\u2082 p, by simp [perm_lookup p, to_finmap_eq, perm_erase p]\n\n@[simp] theorem extract_eq_lookup_erase (a : \u03b1) (s : finmap \u03b2) :\n  extract a s = (lookup a s, erase a s) :=\ninduction_on s $ \u03bb s, by simp [extract]\n\n/-! ### union -/\n\n/-- `s\u2081 \u222a s\u2082` is the key-based union of two finite maps. It is left-biased: if\nthere exists an `a \u2208 s\u2081`, `lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081`. -/\ndef union (s\u2081 s\u2082 : finmap \u03b2) : finmap \u03b2 :=\nlift_on\u2082 s\u2081 s\u2082 (\u03bb s\u2081 s\u2082, \u27e6s\u2081 \u222a s\u2082\u27e7) $\n\u03bb s\u2081 s\u2082 s\u2083 s\u2084 p\u2081\u2083 p\u2082\u2084, to_finmap_eq.mpr $ perm_union p\u2081\u2083 p\u2082\u2084\n\ninstance : has_union (finmap \u03b2) := \u27e8union\u27e9\n\n@[simp] theorem mem_union {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb _ _, mem_union\n\n@[simp] theorem union_to_finmap (s\u2081 s\u2082 : alist \u03b2) : \u27e6s\u2081\u27e7 \u222a \u27e6s\u2082\u27e7 = \u27e6s\u2081 \u222a s\u2082\u27e7 :=\nby simp [(\u222a), union]\n\ntheorem keys_union {s\u2081 s\u2082 : finmap \u03b2} : (s\u2081 \u222a s\u2082).keys = s\u2081.keys \u222a s\u2082.keys :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, finset.ext $ by simp [keys]\n\n@[simp] theorem lookup_union_left {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2208 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, lookup_union_left\n\n@[simp] theorem lookup_union_right {a} {s\u2081 s\u2082 : finmap \u03b2} :\n  a \u2209 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, lookup_union_right\n\ntheorem lookup_union_left_of_not_in {a} {s\u2081 s\u2082 : finmap \u03b2} (h : a \u2209 s\u2082) :\n  lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 :=\nbegin\n  by_cases h' : a \u2208 s\u2081,\n  { rw lookup_union_left h' },\n  { rw [lookup_union_right h', lookup_eq_none.mpr h, lookup_eq_none.mpr h'] }\nend\n\n@[simp] theorem mem_lookup_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : finmap \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2082) \u2194 b \u2208 lookup a s\u2081 \u2228 a \u2209 s\u2081 \u2227 b \u2208 lookup a s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082, mem_lookup_union\n\ntheorem mem_lookup_union_middle {a} {b : \u03b2 a} {s\u2081 s\u2082 s\u2083 : finmap \u03b2} :\n  b \u2208 lookup a (s\u2081 \u222a s\u2083) \u2192 a \u2209 s\u2082 \u2192 b \u2208 lookup a (s\u2081 \u222a s\u2082 \u222a s\u2083) :=\ninduction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb s\u2081 s\u2082 s\u2083, mem_lookup_union_middle\n\ntheorem insert_union {a} {b : \u03b2 a} {s\u2081 s\u2082 : finmap \u03b2} :\n  insert a b (s\u2081 \u222a s\u2082) = insert a b s\u2081 \u222a s\u2082 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb a\u2081 a\u2082, by simp [insert_union]\n\ntheorem union_assoc {s\u2081 s\u2082 s\u2083 : finmap \u03b2} : (s\u2081 \u222a s\u2082) \u222a s\u2083 = s\u2081 \u222a (s\u2082 \u222a s\u2083) :=\ninduction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb s\u2081 s\u2082 s\u2083,\nby simp only [alist.to_finmap_eq, union_to_finmap, alist.union_assoc]\n\n@[simp] theorem empty_union {s\u2081 : finmap \u03b2} : \u2205 \u222a s\u2081 = s\u2081 :=\ninduction_on s\u2081 $ \u03bb s\u2081, by rw \u2190 empty_to_finmap;\n  simp [- empty_to_finmap, alist.to_finmap_eq, union_to_finmap, alist.union_assoc]\n\n@[simp] theorem union_empty {s\u2081 : finmap \u03b2} : s\u2081 \u222a \u2205 = s\u2081 :=\ninduction_on s\u2081 $ \u03bb s\u2081, by rw \u2190 empty_to_finmap;\n  simp [- empty_to_finmap, alist.to_finmap_eq, union_to_finmap, alist.union_assoc]\n\ntheorem erase_union_singleton (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) (h : s.lookup a = some b) :\n  s.erase a \u222a singleton a b = s :=\next_lookup\n(\u03bb x, by { by_cases h' : x = a,\n      { subst a, rw [lookup_union_right not_mem_erase_self, lookup_singleton_eq, h], },\n      { have : x \u2209 singleton a b, { rwa mem_singleton },\n        rw [lookup_union_left_of_not_in this, lookup_erase_ne h'] } } )\n\nend\n\n/-! ### disjoint -/\n\n/-- `disjoint s\u2081 s\u2082` holds if `s\u2081` and `s\u2082` have no keys in common. -/\ndef disjoint (s\u2081 s\u2082 : finmap \u03b2) : Prop :=\n\u2200 x \u2208 s\u2081, \u00ac x \u2208 s\u2082\n\nlemma disjoint_empty (x : finmap \u03b2) : disjoint \u2205 x .\n\n@[symm]\nlemma disjoint.symm (x y : finmap \u03b2) (h : disjoint x y) : disjoint y x :=\n\u03bb p hy hx, h p hx hy\n\nlemma disjoint.symm_iff (x y : finmap \u03b2) : disjoint x y \u2194 disjoint y x :=\n\u27e8disjoint.symm x y, disjoint.symm y x\u27e9\n\nsection\n\nvariables [decidable_eq \u03b1]\n\ninstance : decidable_rel (@disjoint \u03b1 \u03b2) :=\n\u03bb x y, by dsimp only [disjoint]; apply_instance\n\nlemma disjoint_union_left (x y z : finmap \u03b2) : disjoint (x \u222a y) z \u2194 disjoint x z \u2227 disjoint y z :=\nby simp [disjoint, finmap.mem_union, or_imp_distrib, forall_and_distrib]\n\nlemma disjoint_union_right (x y z : finmap \u03b2) : disjoint x (y \u222a z) \u2194 disjoint x y \u2227 disjoint x z :=\nby rw [disjoint.symm_iff, disjoint_union_left, disjoint.symm_iff _ x, disjoint.symm_iff _ x]\n\ntheorem union_comm_of_disjoint {s\u2081 s\u2082 : finmap \u03b2} : disjoint s\u2081 s\u2082 \u2192 s\u2081 \u222a s\u2082 = s\u2082 \u222a s\u2081 :=\ninduction_on\u2082 s\u2081 s\u2082 $ \u03bb s\u2081 s\u2082,\nby { intros h, simp only [alist.to_finmap_eq, union_to_finmap, alist.union_comm_of_disjoint h] }\n\ntheorem union_cancel {s\u2081 s\u2082 s\u2083 : finmap \u03b2} (h : disjoint s\u2081 s\u2083) (h' : disjoint s\u2082 s\u2083) :\n  s\u2081 \u222a s\u2083 = s\u2082 \u222a s\u2083 \u2194 s\u2081 = s\u2082 :=\n\u27e8\u03bb h'', begin\n          apply ext_lookup, intro x,\n          have : (s\u2081 \u222a s\u2083).lookup x = (s\u2082 \u222a s\u2083).lookup x, from h'' \u25b8 rfl,\n          by_cases hs\u2081 : x \u2208 s\u2081,\n          { rwa [lookup_union_left hs\u2081, lookup_union_left_of_not_in (h _ hs\u2081)] at this, },\n          { by_cases hs\u2082 : x \u2208 s\u2082,\n            { rwa [lookup_union_left_of_not_in (h' _ hs\u2082), lookup_union_left hs\u2082] at this, },\n            { rw [lookup_eq_none.mpr hs\u2081, lookup_eq_none.mpr hs\u2082] } }\n        end,\n \u03bb h, h \u25b8 rfl\u27e9\n\nend\n\nend finmap\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/finmap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.38826844393508775}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport data.array.lemmas\nimport data.list.join\nimport data.list.range\nimport data.pnat.basic\n\n/-!\n# Hash maps\n\nDefines a hash map data structure, representing a finite key-value map\nwith a value type that may depend on the key type.  The structure\nrequires a `nat`-valued hash function to associate keys to buckets.\n\n## Main definitions\n\n* `hash_map`: constructed with `mk_hash_map`.\n\n## Implementation details\n\nA hash map with key type `\u03b1` and (dependent) value type `\u03b2 : \u03b1 \u2192 Type*`\nconsists of an array of *buckets*, which are lists containing\nkey/value pairs for that bucket.  The hash function is taken modulo `n`\nto assign keys to their respective bucket.  Because of this, some care\nshould be put into the hash function to ensure it evenly distributes\nkeys.\n\nThe bucket array is an `array`.  These have special VM support for\nin-place modification if there is only ever one reference to them.  If\none takes special care to never keep references to old versions of a\nhash map alive after updating it, then the hash map will be modified\nin-place.  In this documentation, when we say a hash map is modified\nin-place, we are assuming the API is being used in this manner.\n\nWhen inserting (`hash_map.insert`), if the number of stored pairs (the\n*size*) is going to exceed the number of buckets, then a new hash map\nis first created with double the number of buckets and everything in\nthe old hash map is reinserted along with the new key/value pair.\nOtherwise, the bucket array is modified in-place.  The amortized\nrunning time of inserting $$n$$ elements into a hash map is $$O(n)$$.\n\nWhen removing (`hash_map.erase`), the hash map is modified in-place.\nThe implementation does not reduce the number of buckets in the hash\nmap if the size gets too low.\n\n## Tags\n\nhash map\n\n-/\n\nuniverses u v w\n\n/-- `bucket_array \u03b1 \u03b2` is the underlying data type for `hash_map \u03b1 \u03b2`,\n  an array of linked lists of key-value pairs. -/\ndef bucket_array (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) (n : \u2115+) :=\narray n (list \u03a3 a, \u03b2 a)\n\n/-- Make a hash_map index from a `nat` hash value and a (positive) buffer size -/\ndef hash_map.mk_idx (n : \u2115+) (i : nat) : fin n :=\n\u27e8i % n, nat.mod_lt _ n.2\u27e9\n\nnamespace bucket_array\nsection\nparameters {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat)\nvariables {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n\ninstance : inhabited (bucket_array \u03b1 \u03b2 n) :=\n\u27e8mk_array _ []\u27e9\n\n/-- Read the bucket corresponding to an element -/\ndef read (a : \u03b1) : list \u03a3 a, \u03b2 a :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\ndata.read bidx\n\n/-- Write the bucket corresponding to an element -/\ndef write (a : \u03b1) (l : list \u03a3 a, \u03b2 a) : bucket_array \u03b1 \u03b2 n :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\ndata.write bidx l\n\n/-- Modify (read, apply `f`, and write) the bucket corresponding to an element -/\ndef modify (a : \u03b1) (f : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)) : bucket_array \u03b1 \u03b2 n :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\narray.write data bidx (f (array.read data bidx))\n\n/-- The list of all key-value pairs in the bucket list -/\ndef as_list : list \u03a3 a, \u03b2 a := data.to_list.join\n\ntheorem mem_as_list {a : \u03a3 a, \u03b2 a} : a \u2208 data.as_list \u2194 \u2203i, a \u2208 array.read data i :=\nhave (\u2203 (l : list (\u03a3 (a : \u03b1), \u03b2 a)) (i : fin (n.val)), a \u2208 l \u2227 array.read data i = l) \u2194\n  \u2203 (i : fin (n.val)), a \u2208 array.read data i,\nby rw exists_swap; exact exists_congr (\u03bb i, by simp),\nby simp [as_list]; simpa [array.mem.def, and_comm]\n\n/-- Fold a function `f` over the key-value pairs in the bucket list -/\ndef foldl {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\ndata.foldl d (\u03bb b d, b.foldl (\u03bb r a, f r a.1 a.2) d)\n\ntheorem foldl_eq {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) :\n  data.foldl d f = data.as_list.foldl (\u03bb r a, f r a.1 a.2) d :=\nby rw [foldl, as_list, list.foldl_join, \u2190 array.to_list_foldl]\n\nend\nend bucket_array\n\nnamespace hash_map\nsection\nparameters {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat)\n\n/-- Insert the pair `\u27e8a, b\u27e9` into the correct location in the bucket array\n  (without checking for duplication) -/\ndef reinsert_aux {n} (data : bucket_array \u03b1 \u03b2 n) (a : \u03b1) (b : \u03b2 a) : bucket_array \u03b1 \u03b2 n :=\ndata.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9 :: l)\n\ntheorem mk_as_list (n : \u2115+) : bucket_array.as_list (mk_array n [] : bucket_array \u03b1 \u03b2 n) = [] :=\nlist.eq_nil_iff_forall_not_mem.mpr $ \u03bb x m,\nlet \u27e8i, h\u27e9 := (bucket_array.mem_as_list _).1 m in h\n\nparameter [decidable_eq \u03b1]\n\n/-- Search a bucket for a key `a` and return the value -/\ndef find_aux (a : \u03b1) : list (\u03a3 a, \u03b2 a) \u2192 option (\u03b2 a)\n| []          := none\n| (\u27e8a',b\u27e9::t) := if h : a' = a then some (eq.rec_on h b) else find_aux t\n\ntheorem find_aux_iff {a : \u03b1} {b : \u03b2 a} :\n  \u03a0 {l : list \u03a3 a, \u03b2 a}, (l.map sigma.fst).nodup \u2192 (find_aux a l = some b \u2194 sigma.mk a b \u2208 l)\n| []          nd := \u27e8\u03bbn, by injection n, false.elim\u27e9\n| (\u27e8a',b'\u27e9::t) nd := begin\n  by_cases a' = a,\n  { clear find_aux_iff, subst h,\n    suffices : b' = b \u2194 b' = b \u2228 sigma.mk a' b \u2208 t, {simpa [find_aux, eq_comm]},\n    refine (or_iff_left_of_imp (\u03bb m, _)).symm,\n    have : a' \u2209 t.map sigma.fst, from list.not_mem_of_nodup_cons nd,\n    exact this.elim (list.mem_map_of_mem sigma.fst m) },\n  { have : sigma.mk a b \u2260 \u27e8a', b'\u27e9,\n    { intro e, injection e with e, exact h e.symm },\n    simp at nd, simp [find_aux, h, ne.symm h, find_aux_iff, nd] }\nend\n\n/-- Returns `tt` if the bucket `l` contains the key `a` -/\ndef contains_aux (a : \u03b1) (l : list \u03a3 a, \u03b2 a) : bool :=\n(find_aux a l).is_some\n\ntheorem contains_aux_iff {a : \u03b1} {l : list \u03a3 a, \u03b2 a} (nd : (l.map sigma.fst).nodup) :\n  contains_aux a l \u2194 a \u2208 l.map sigma.fst :=\nbegin\n  unfold contains_aux,\n  cases h : find_aux a l with b; simp,\n  { assume (b : \u03b2 a) (m : sigma.mk a b \u2208 l),\n    rw (find_aux_iff nd).2 m at h,\n    contradiction },\n  { show \u2203 (b : \u03b2 a), sigma.mk a b \u2208 l,\n    exact \u27e8_, (find_aux_iff nd).1 h\u27e9 },\nend\n\n/-- Modify a bucket to replace a value in the list. Leaves the list\n unchanged if the key is not found. -/\ndef replace_aux (a : \u03b1) (b : \u03b2 a) : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)\n| []            := []\n| (\u27e8a', b'\u27e9::t) := if a' = a then \u27e8a, b\u27e9::t else \u27e8a', b'\u27e9 :: replace_aux t\n\n/-- Modify a bucket to remove a key, if it exists. -/\ndef erase_aux (a : \u03b1) : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)\n| []            := []\n| (\u27e8a', b'\u27e9::t) := if a' = a then t else \u27e8a', b'\u27e9 :: erase_aux t\n\n/-- The predicate `valid bkts sz` means that `bkts` satisfies the `hash_map`\n  invariants: There are exactly `sz` elements in it, every pair is in the\n  bucket determined by its key and the hash function, and no key appears\n  multiple times in the list. -/\nstructure valid {n} (bkts : bucket_array \u03b1 \u03b2 n) (sz : nat) : Prop :=\n(len : bkts.as_list.length = sz)\n(idx : \u2200 {i} {a : \u03a3 a, \u03b2 a}, a \u2208 array.read bkts i \u2192\n  mk_idx n (hash_fn a.1) = i)\n(nodup : \u2200i, ((array.read bkts i).map sigma.fst).nodup)\n\ntheorem valid.idx_enum {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  {i l} (he : (i, l) \u2208 bkts.to_list.enum) {a} {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) :\n  \u2203 h, mk_idx n (hash_fn a) = \u27e8i, h\u27e9 :=\n(array.mem_to_list_enum.mp he).imp (\u03bb h e, by subst e; exact v.idx hl)\n\ntheorem valid.idx_enum_1 {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  {i l} (he : (i, l) \u2208 bkts.to_list.enum) {a} {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) :\n  (mk_idx n (hash_fn a)).1 = i :=\nlet \u27e8h, e\u27e9 := v.idx_enum _ he hl in by rw e; refl\n\ntheorem valid.as_list_nodup {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz) :\n  (bkts.as_list.map sigma.fst).nodup :=\nbegin\n  suffices : (bkts.to_list.map (list.map sigma.fst)).pairwise list.disjoint,\n  { suffices : \u2200 l, array.mem l bkts \u2192 (l.map sigma.fst).nodup,\n      by simpa [bucket_array.as_list, list.nodup_join, *],\n    rintros l \u27e8i, rfl\u27e9,\n    apply v.nodup },\n  rw [\u2190 list.enum_map_snd bkts.to_list, list.pairwise_map, list.pairwise_map],\n  have : (bkts.to_list.enum.map prod.fst).nodup := by simp [list.nodup_range],\n  refine list.pairwise.imp_of_mem _ ((list.pairwise_map _).1 this),\n  rw prod.forall, intros i l\u2081,\n  rw prod.forall, intros j l\u2082 me\u2081 me\u2082 ij,\n  simp [list.disjoint], intros a b ml\u2081 b' ml\u2082,\n  apply ij, rwa [\u2190 v.idx_enum_1 _ me\u2081 ml\u2081, \u2190 v.idx_enum_1 _ me\u2082 ml\u2082]\nend\n\ntheorem mk_valid (n : \u2115+) : @valid n (mk_array n []) 0 :=\n\u27e8by simp [mk_as_list], \u03bb i a h, by cases h, \u03bb i, list.nodup_nil\u27e9\n\ntheorem valid.find_aux_iff {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz) {a : \u03b1}\n  {b : \u03b2 a} :\n  find_aux a (bkts.read hash_fn a) = some b \u2194 sigma.mk a b \u2208 bkts.as_list :=\n(find_aux_iff (v.nodup _)).trans $\nby rw bkts.mem_as_list; exact \u27e8\u03bb h, \u27e8_, h\u27e9, \u03bb \u27e8i, h\u27e9, (v.idx h).symm \u25b8 h\u27e9\n\ntheorem valid.contains_aux_iff {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  (a : \u03b1) :\n  contains_aux a (bkts.read hash_fn a) \u2194 a \u2208 bkts.as_list.map sigma.fst :=\nby simp [contains_aux, option.is_some_iff_exists, v.find_aux_iff hash_fn]\n\nsection\n  parameters {n : \u2115+} {bkts : bucket_array \u03b1 \u03b2 n}\n             {bidx : fin n} {f : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)}\n             (u v1 v2 w : list \u03a3 a, \u03b2 a)\n\n  local notation `L` := array.read bkts bidx\n  private def bkts' : bucket_array \u03b1 \u03b2 n := array.write bkts bidx (f L)\n\n  variables (hl : L = u ++ v1 ++ w)\n            (hfl : f L = u ++ v2 ++ w)\n  include hl hfl\n\n  theorem append_of_modify :\n  \u2203 u' w', bkts.as_list = u' ++ v1 ++ w' \u2227 bkts'.as_list = u' ++ v2 ++ w' :=\n  begin\n    unfold bucket_array.as_list,\n    have h : (bidx : \u2115) < bkts.to_list.length, { simp only [bidx.is_lt, array.to_list_length] },\n    refine \u27e8(bkts.to_list.take bidx).join ++ u, w ++ (bkts.to_list.drop (bidx+1)).join, _, _\u27e9,\n    { conv { to_lhs,\n        rw [\u2190 list.take_append_drop bidx bkts.to_list, list.drop_eq_nth_le_cons h],\n        simp [hl] }, simp },\n    { conv { to_lhs,\n        rw [bkts', array.write_to_list, list.update_nth_eq_take_cons_drop _ h],\n        simp [hfl] }, simp }\n  end\n\n  variables (hvnd : (v2.map sigma.fst).nodup)\n            (hal : \u2200 (a : \u03a3 a, \u03b2 a), a \u2208 v2 \u2192 mk_idx n (hash_fn a.1) = bidx)\n            (djuv : (u.map sigma.fst).disjoint (v2.map sigma.fst))\n            (djwv : (w.map sigma.fst).disjoint (v2.map sigma.fst))\n  include hvnd hal djuv djwv\n\n  theorem valid.modify {sz : \u2115} (v : valid bkts sz) :\n    v1.length \u2264 sz + v2.length \u2227 valid bkts' (sz + v2.length - v1.length) :=\n  begin\n    rcases append_of_modify u v1 v2 w hl hfl with \u27e8u', w', e\u2081, e\u2082\u27e9,\n    rw [\u2190 v.len, e\u2081],\n    suffices : valid bkts' (u' ++ v2 ++ w').length,\n    { simpa [ge, add_comm, add_left_comm, nat.le_add_right, add_tsub_cancel_left] },\n    refine \u27e8congr_arg _ e\u2082, \u03bb i a, _, \u03bb i, _\u27e9,\n    { by_cases bidx = i,\n      { subst i, rw [bkts', array.read_write, hfl],\n        have := @valid.idx _ _ _ v bidx a,\n        simp only [hl, list.mem_append, or_imp_distrib, forall_and_distrib] at this \u22a2,\n        exact \u27e8\u27e8this.1.1, hal _\u27e9, this.2\u27e9 },\n      { rw [bkts', array.read_write_of_ne _ _ h], apply v.idx } },\n    { by_cases bidx = i,\n      { subst i, rw [bkts', array.read_write, hfl],\n        have := @valid.nodup _ _ _ v bidx,\n        simp [hl, list.nodup_append] at this,\n        simp [list.nodup_append, this, hvnd, djuv, djwv.symm] },\n      { rw [bkts', array.read_write_of_ne _ _ h], apply v.nodup } }\n  end\nend\n\ntheorem valid.replace_aux (a : \u03b1) (b : \u03b2 a) : \u03a0 (l : list (\u03a3 a, \u03b2 a)), a \u2208 l.map sigma.fst \u2192\n  \u2203 (u w : list \u03a3 a, \u03b2 a) b', l = u ++ [\u27e8a, b'\u27e9] ++ w \u2227 replace_aux a b l = u ++ [\u27e8a, b\u27e9] ++ w\n| []            := false.elim\n| (\u27e8a', b'\u27e9::t) := begin\n  by_cases e : a' = a,\n  { subst a',\n    suffices : \u2203 (u w : list \u03a3 a, \u03b2 a) (b'' : \u03b2 a),\n      (sigma.mk a b') :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n      replace_aux a b (\u27e8a, b'\u27e9 :: t) = u ++ \u27e8a, b\u27e9 :: w, {simpa},\n    refine \u27e8[], t, b', _\u27e9, simp [replace_aux] },\n  { suffices : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (b'' : \u03b2 a),\n      (sigma.mk a' b') :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n      (sigma.mk a' b') :: (replace_aux a b t) = u ++ \u27e8a, b\u27e9 :: w,\n    { simpa [replace_aux, ne.symm e, e] },\n    intros x m,\n    have IH : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (b'' : \u03b2 a),\n      t = u ++ \u27e8a, b''\u27e9 :: w \u2227 replace_aux a b t = u ++ \u27e8a, b\u27e9 :: w,\n    { simpa using valid.replace_aux t },\n    rcases IH x m with \u27e8u, w, b'', hl, hfl\u27e9,\n    exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm, ne.symm e]\u27e9 }\nend\n\ntheorem valid.replace {n : \u2115+}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n  (Hc : contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (bkts.modify hash_fn a (replace_aux a b)) sz :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  rcases hash_map.valid.replace_aux a b (array.read bkts (mk_idx n (hash_fn a)))\n    ((contains_aux_iff nd).1 Hc) with \u27e8u, w, b', hl, hfl\u27e9,\n  simp [hl, list.nodup_append] at nd,\n  refine (v.modify hash_fn\n    u [\u27e8a, b'\u27e9] [\u27e8a, b\u27e9] w hl hfl (list.nodup_singleton _)\n    (\u03bba' e, by simp at e; rw e)\n    (\u03bba' e1 e2, _)\n    (\u03bba' e1 e2, _)).2;\n  { revert e1, simp [-sigma.exists] at e2, subst a', simp [nd] }\nend\n\ntheorem valid.insert {n : \u2115+}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n  (Hnc : \u00ac contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (reinsert_aux bkts a b) (sz+1) :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  refine (v.modify hash_fn\n    [] [] [\u27e8a, b\u27e9] (bkts.read hash_fn a) rfl rfl (list.nodup_singleton _)\n    (\u03bba' e, by simp at e; rw e)\n    (\u03bba', false.elim)\n    (\u03bba' e1 e2, _)).2,\n  simp [-sigma.exists] at e2, subst a',\n  exact Hnc ((contains_aux_iff nd).2 e1)\nend\n\ntheorem valid.erase_aux (a : \u03b1) : \u03a0 (l : list (\u03a3 a, \u03b2 a)), a \u2208 l.map sigma.fst \u2192\n  \u2203 (u w : list \u03a3 a, \u03b2 a) b, l = u ++ [\u27e8a, b\u27e9] ++ w \u2227 erase_aux a l = u ++ [] ++ w\n| []            := false.elim\n| (\u27e8a', b'\u27e9::t) := begin\n  by_cases e : a' = a,\n  { subst a',\n    simpa [erase_aux, and_comm] using show \u2203 u w (x : \u03b2 a),\n      t = u ++ w \u2227 (sigma.mk a b') :: t = u ++ \u27e8a, x\u27e9 :: w,\n      from \u27e8[], t, b', by simp\u27e9 },\n  { simp [erase_aux, e, ne.symm e],\n    suffices : \u2200 (b : \u03b2 a) (_ : sigma.mk a b \u2208 t), \u2203 u w (x : \u03b2 a),\n      (sigma.mk a' b') :: t = u ++ \u27e8a, x\u27e9 :: w \u2227\n      (sigma.mk a' b') :: (erase_aux a t) = u ++ w,\n    { simpa [replace_aux, ne.symm e, e] },\n    intros b m,\n    have IH : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (x : \u03b2 a),\n      t = u ++ \u27e8a, x\u27e9 :: w \u2227 erase_aux a t = u ++ w,\n    { simpa using valid.erase_aux t },\n    rcases IH b m with \u27e8u, w, b'', hl, hfl\u27e9,\n    exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm]\u27e9 }\nend\n\ntheorem valid.erase {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz}\n  (a : \u03b1) (Hc : contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (bkts.modify hash_fn a (erase_aux a)) (sz-1) :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  rcases hash_map.valid.erase_aux a (array.read bkts (mk_idx n (hash_fn a)))\n    ((contains_aux_iff nd).1 Hc) with \u27e8u, w, b, hl, hfl\u27e9,\n  refine (v.modify hash_fn u [\u27e8a, b\u27e9] [] w hl hfl list.nodup_nil _ _ _).2;\n  simp\nend\n\nend\nend hash_map\n\n/-- A hash map data structure, representing a finite key-value map\n  with key type `\u03b1` and value type `\u03b2` (which may depend on `\u03b1`). -/\nstructure hash_map (\u03b1 : Type u) [decidable_eq \u03b1] (\u03b2 : \u03b1 \u2192 Type v) :=\n(hash_fn : \u03b1 \u2192 nat)\n(size : \u2115)\n(nbuckets : \u2115+)\n(buckets : bucket_array \u03b1 \u03b2 nbuckets)\n(is_valid : hash_map.valid hash_fn buckets size)\n\n/-- Construct an empty hash map with buffer size `nbuckets` (default 8). -/\ndef mk_hash_map {\u03b1 : Type u} [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat) (nbuckets := 8) :\n  hash_map \u03b1 \u03b2 :=\nlet n := if nbuckets = 0 then 8 else nbuckets in\nlet nz : n > 0 := by abstract { cases nbuckets; simp [if_pos, nat.succ_ne_zero] } in\n{ hash_fn  := hash_fn,\n  size     := 0,\n  nbuckets := \u27e8n, nz\u27e9,\n  buckets  := mk_array n [],\n  is_valid := hash_map.mk_valid _ _ }\n\nnamespace hash_map\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [decidable_eq \u03b1]\n\n/-- Return the value corresponding to a key, or `none` if not found -/\ndef find (m : hash_map \u03b1 \u03b2) (a : \u03b1) : option (\u03b2 a) :=\nfind_aux a (m.buckets.read m.hash_fn a)\n\n/-- Return `tt` if the key exists in the map -/\ndef contains (m : hash_map \u03b1 \u03b2) (a : \u03b1) : bool :=\n(m.find a).is_some\n\ninstance : has_mem \u03b1 (hash_map \u03b1 \u03b2) := \u27e8\u03bba m, m.contains a\u27e9\n\n/-- Fold a function over the key-value pairs in the map -/\ndef fold {\u03b4 : Type w} (m : hash_map \u03b1 \u03b2) (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\nm.buckets.foldl d f\n\n/-- The list of key-value pairs in the map -/\ndef entries (m : hash_map \u03b1 \u03b2) : list \u03a3 a, \u03b2 a :=\nm.buckets.as_list\n\n/-- The list of keys in the map -/\ndef keys (m : hash_map \u03b1 \u03b2) : list \u03b1 :=\nm.entries.map sigma.fst\n\ntheorem find_iff (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) :\n  m.find a = some b \u2194 sigma.mk a b \u2208 m.entries :=\nm.is_valid.find_aux_iff _\n\ntheorem contains_iff (m : hash_map \u03b1 \u03b2) (a : \u03b1) :\n  m.contains a \u2194 a \u2208 m.keys :=\nm.is_valid.contains_aux_iff _ _\n\ntheorem entries_empty (hash_fn : \u03b1 \u2192 nat) (n) :\n  (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).entries = [] :=\nmk_as_list _\n\n\n\ntheorem find_empty (hash_fn : \u03b1 \u2192 nat) (n a) :\n  (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).find a = none :=\nby induction h : (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).find a; [refl,\n   { have := (find_iff _ _ _).1 h, rw entries_empty at this, contradiction }]\n\ntheorem not_contains_empty (hash_fn : \u03b1 \u2192 nat) (n a) :\n  \u00ac (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).contains a :=\nby apply bool_iff_false.2; dsimp [contains]; rw [find_empty]; refl\n\ntheorem insert_lemma (hash_fn : \u03b1 \u2192 nat) {n n'}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz} (v : valid hash_fn bkts sz) :\n  valid hash_fn (bkts.foldl (mk_array _ [] : bucket_array \u03b1 \u03b2 n') (reinsert_aux hash_fn)) sz :=\nbegin\n  suffices : \u2200 (l : list \u03a3 a, \u03b2 a) (t : bucket_array \u03b1 \u03b2 n') sz,\n    valid hash_fn t sz \u2192 ((l ++ t.as_list).map sigma.fst).nodup \u2192\n    valid hash_fn (l.foldl (\u03bbr (a : \u03a3 a, \u03b2 a), reinsert_aux hash_fn r a.1 a.2) t) (sz + l.length),\n  { have p := this bkts.as_list _ _ (mk_valid _ _),\n    rw [mk_as_list, list.append_nil, zero_add, v.len] at p,\n    rw bucket_array.foldl_eq,\n    exact p (v.as_list_nodup _) },\n  intro l, induction l with c l IH; intros t sz v nd, {exact v},\n  rw show sz + (c :: l).length = sz + 1 + l.length, by simp [add_comm, add_assoc],\n  rcases (show (l.map sigma.fst).nodup \u2227\n      ((bucket_array.as_list t).map sigma.fst).nodup \u2227\n      c.fst \u2209 l.map sigma.fst \u2227\n      c.fst \u2209 (bucket_array.as_list t).map sigma.fst \u2227\n      (l.map sigma.fst).disjoint ((bucket_array.as_list t).map sigma.fst),\n    by simpa [list.nodup_append, not_or_distrib, and_comm, and.left_comm] using nd)\n    with \u27e8nd1, nd2, nm1, nm2, dj\u27e9,\n  have v' := v.insert _ _ c.2 (\u03bbHc, nm2 $ (v.contains_aux_iff _ c.1).1 Hc),\n  apply IH _ _ v',\n  suffices : \u2200 \u2983a : \u03b1\u2984 (b : \u03b2 a), sigma.mk a b \u2208 l \u2192\n    \u2200 (b' : \u03b2 a), sigma.mk a b' \u2208 (reinsert_aux hash_fn t c.1 c.2).as_list \u2192 false,\n  { simpa [list.nodup_append, nd1, v'.as_list_nodup _, list.disjoint] },\n  intros a b m1 b' m2,\n  rcases (reinsert_aux hash_fn t c.1 c.2).mem_as_list.1 m2 with \u27e8i, im\u27e9,\n  have : sigma.mk a b' \u2209 array.read t i,\n  { intro m3,\n    have : a \u2208 list.map sigma.fst t.as_list :=\n      list.mem_map_of_mem sigma.fst (t.mem_as_list.2 \u27e8_, m3\u27e9),\n    exact dj (list.mem_map_of_mem sigma.fst m1) this },\n  by_cases h : mk_idx n' (hash_fn c.1) = i,\n  { subst h,\n    have e : sigma.mk a b' = \u27e8c.1, c.2\u27e9,\n    { simpa [reinsert_aux, bucket_array.modify, array.read_write, this] using im },\n    injection e with e, subst a,\n    exact nm1.elim (@list.mem_map_of_mem _ _ sigma.fst _ _ m1) },\n  { apply this,\n    simpa [reinsert_aux, bucket_array.modify, array.read_write_of_ne _ _ h] using im }\nend\n\n/-- Insert a key-value pair into the map. (Modifies `m` in-place when applicable) -/\ndef insert : \u03a0 (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a), hash_map \u03b1 \u03b2\n| \u27e8hash_fn, size, n, buckets, v\u27e9 a b :=\nlet bkt := buckets.read hash_fn a in\nif hc : contains_aux a bkt then\n{ hash_fn  := hash_fn,\n  size     := size,\n  nbuckets := n,\n  buckets  := buckets.modify hash_fn a (replace_aux a b),\n  is_valid := v.replace _ a b hc }\nelse\nlet size'    := size + 1,\n    buckets' := buckets.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9::l),\n    valid'   := v.insert _ a b hc in\nif size' \u2264 n then\n{ hash_fn  := hash_fn,\n  size     := size',\n  nbuckets := n,\n  buckets  := buckets',\n  is_valid := valid' }\nelse\nlet n'        : \u2115+ := \u27e8n * 2, mul_pos n.2 dec_trivial\u27e9,\n    buckets'' : bucket_array \u03b1 \u03b2 n' :=\n                buckets'.foldl (mk_array _ []) (reinsert_aux hash_fn) in\n{ hash_fn  := hash_fn,\n  size     := size',\n  nbuckets := n',\n  buckets  := buckets'',\n  is_valid := insert_lemma _ valid' }\n\ntheorem mem_insert : \u03a0 (m : hash_map \u03b1 \u03b2) (a b a' b'),\n  (sigma.mk a' b' : sigma \u03b2) \u2208 (m.insert a b).entries \u2194\n  if a = a' then b == b' else sigma.mk a' b' \u2208 m.entries\n| \u27e8hash_fn, size, n, bkts, v\u27e9 a b a' b' := begin\n  let bkt := bkts.read hash_fn a,\n  have nd : (bkt.map sigma.fst).nodup := v.nodup (mk_idx n (hash_fn a)),\n  have lem : \u03a0 (bkts' : bucket_array \u03b1 \u03b2 n) (v1 u w)\n    (hl : bucket_array.as_list bkts = u ++ v1 ++ w)\n    (hfl : bucket_array.as_list bkts' = u ++ [\u27e8a, b\u27e9] ++ w)\n    (veq : (v1 = [] \u2227 \u00ac contains_aux a bkt) \u2228 \u2203b'', v1 = [\u27e8a, b''\u27e9]),\n    sigma.mk a' b' \u2208 bkts'.as_list \u2194\n    if a = a' then b == b' else sigma.mk a' b' \u2208 bkts.as_list,\n  { intros bkts' v1 u w hl hfl veq,\n    rw [hl, hfl],\n    by_cases h : a = a',\n    { subst a',\n      suffices : b = b' \u2228 sigma.mk a b' \u2208 u \u2228 sigma.mk a b' \u2208 w \u2194 b = b',\n      { simpa [eq_comm, or.left_comm] },\n      refine or_iff_left_of_imp (not.elim $ not_or_distrib.2 _),\n      rcases veq with \u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9,\n      { have na := (not_iff_not_of_iff $ v.contains_aux_iff _ _).1 Hnc,\n        simp [hl, not_or_distrib] at na, simp [na] },\n      { have nd' := v.as_list_nodup _,\n        simp [hl, list.nodup_append] at nd', simp [nd'] } },\n    { suffices : sigma.mk a' b' \u2209 v1, {simp [h, ne.symm h, this]},\n      rcases veq with \u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9; simp [ne.symm h] } },\n  by_cases Hc : (contains_aux a bkt : Prop),\n  { rcases hash_map.valid.replace_aux a b (array.read bkts (mk_idx n (hash_fn a)))\n      ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b'', hl', hfl'\u27e9,\n    rcases (append_of_modify u' [\u27e8a, b''\u27e9] [\u27e8a, b\u27e9] w' hl' hfl') with \u27e8u, w, hl, hfl\u27e9,\n    simpa [insert, @dif_pos (contains_aux a bkt) _ Hc]\n      using lem _ _ u w hl hfl (or.inr \u27e8b'', rfl\u27e9) },\n  { let size' := size + 1,\n    let bkts' := bkts.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9::l),\n    have mi : sigma.mk a' b' \u2208 bkts'.as_list \u2194\n        if a = a' then b == b' else sigma.mk a' b' \u2208 bkts.as_list :=\n      let \u27e8u, w, hl, hfl\u27e9 := append_of_modify [] [] [\u27e8a, b\u27e9] _ rfl rfl in\n      lem bkts' _ u w hl hfl $ or.inl \u27e8rfl, Hc\u27e9,\n    simp [insert, @dif_neg (contains_aux a bkt) _ Hc],\n    by_cases h : size' \u2264 n,\n    { simpa [show size' \u2264 n, from h] using mi },\n    { let n' : \u2115+ := \u27e8n * 2, mul_pos n.2 dec_trivial\u27e9,\n      let bkts'' : bucket_array \u03b1 \u03b2 n' := bkts'.foldl (mk_array _ []) (reinsert_aux hash_fn),\n      suffices : sigma.mk a' b' \u2208 bkts''.as_list \u2194 sigma.mk a' b' \u2208 bkts'.as_list.reverse,\n      { simpa [show \u00ac size' \u2264 n, from h, mi] },\n      rw [show bkts'' = bkts'.as_list.foldl _ _, from bkts'.foldl_eq _ _,\n          \u2190 list.foldr_reverse],\n      induction bkts'.as_list.reverse with a l IH,\n      { simp [mk_as_list] },\n      { cases a with a'' b'',\n        let B := l.foldr (\u03bb (y : sigma \u03b2) (x : bucket_array \u03b1 \u03b2 n'),\n          reinsert_aux hash_fn x y.1 y.2) (mk_array n' []),\n        rcases append_of_modify [] [] [\u27e8a'', b''\u27e9] _ rfl rfl with \u27e8u, w, hl, hfl\u27e9,\n        simp [IH.symm, or.left_comm, show B.as_list = _, from hl,\n              show (reinsert_aux hash_fn B a'' b'').as_list = _, from hfl] } } }\nend\n\ntheorem find_insert_eq (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) : (m.insert a b).find a = some b :=\n(find_iff (m.insert a b) a b).2 $ (mem_insert m a b a b).2 $ by rw if_pos rfl\n\ntheorem find_insert_ne (m : hash_map \u03b1 \u03b2) (a a' : \u03b1) (b : \u03b2 a) (h : a \u2260 a') :\n  (m.insert a b).find a' = m.find a' :=\noption.eq_of_eq_some $ \u03bbb',\nlet t := mem_insert m a b a' b' in\n(find_iff _ _ _).trans $ iff.trans (by rwa if_neg h at t) (find_iff _ _ _).symm\n\ntheorem find_insert (m : hash_map \u03b1 \u03b2) (a' a : \u03b1) (b : \u03b2 a) :\n  (m.insert a b).find a' = if h : a = a' then some (eq.rec_on h b) else m.find a' :=\nif h : a = a' then by rw dif_pos h; exact\n  match a', h with ._, rfl := find_insert_eq m a b end\nelse by rw dif_neg h; exact find_insert_ne m a a' b h\n\n/-- Insert a list of key-value pairs into the map. (Modifies `m` in-place when applicable) -/\ndef insert_all (l : list (\u03a3 a, \u03b2 a)) (m : hash_map \u03b1 \u03b2) : hash_map \u03b1 \u03b2 :=\nl.foldl (\u03bb m \u27e8a, b\u27e9, insert m a b) m\n\n/-- Construct a hash map from a list of key-value pairs. -/\ndef of_list (l : list (\u03a3 a, \u03b2 a)) (hash_fn) : hash_map \u03b1 \u03b2 :=\ninsert_all l (mk_hash_map hash_fn (2 * l.length))\n\n/-- Remove a key from the map. (Modifies `m` in-place when applicable) -/\ndef erase (m : hash_map \u03b1 \u03b2) (a : \u03b1) : hash_map \u03b1 \u03b2 :=\nmatch m with \u27e8hash_fn, size, n, buckets, v\u27e9 :=\n  if hc : contains_aux a (buckets.read hash_fn a) then\n  { hash_fn  := hash_fn,\n    size     := size - 1,\n    nbuckets := n,\n    buckets  := buckets.modify hash_fn a (erase_aux a),\n    is_valid := v.erase _ a hc }\n  else m\nend\n\ntheorem mem_erase : \u03a0 (m : hash_map \u03b1 \u03b2) (a a' b'),\n  (sigma.mk a' b' : sigma \u03b2) \u2208 (m.erase a).entries \u2194\n  a \u2260 a' \u2227 sigma.mk a' b' \u2208 m.entries\n| \u27e8hash_fn, size, n, bkts, v\u27e9 a a' b' := begin\n  let bkt := bkts.read hash_fn a,\n  by_cases Hc : (contains_aux a bkt : Prop),\n  { let bkts' := bkts.modify hash_fn a (erase_aux a),\n    suffices : sigma.mk a' b' \u2208 bkts'.as_list \u2194 a \u2260 a' \u2227 sigma.mk a' b' \u2208 bkts.as_list,\n    { simpa [erase, @dif_pos (contains_aux a bkt) _ Hc] },\n    have nd := v.nodup (mk_idx n (hash_fn a)),\n    rcases valid.erase_aux a bkt ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b, hl', hfl'\u27e9,\n    rcases append_of_modify u' [\u27e8a, b\u27e9] [] _ hl' hfl' with \u27e8u, w, hl, hfl\u27e9,\n    suffices : \u2200_:sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w, a \u2260 a',\n    { have : sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w \u2194 (\u00aca = a' \u2227 a' = a) \u2227 b' == b \u2228\n        \u00aca = a' \u2227 (sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w),\n      { simp [eq_comm, not_and_self_iff, and_iff_right_of_imp this] },\n      simpa [hl, show bkts'.as_list = _, from hfl, and_or_distrib_left,\n             and_comm, and.left_comm, or.left_comm] },\n    intros m e, subst a', revert m, apply not_or_distrib.2,\n    have nd' := v.as_list_nodup _,\n    simp [hl, list.nodup_append] at nd', simp [nd'] },\n  { suffices : \u2200_:sigma.mk a' b' \u2208 bucket_array.as_list bkts, a \u2260 a',\n    { simp [erase, @dif_neg (contains_aux a bkt) _ Hc, entries, and_iff_right_of_imp this] },\n    intros m e, subst a',\n    exact Hc ((v.contains_aux_iff _ _).2 (list.mem_map_of_mem sigma.fst m)) }\nend\n\ntheorem find_erase_eq (m : hash_map \u03b1 \u03b2) (a : \u03b1) : (m.erase a).find a = none :=\nbegin\n  cases h : (m.erase a).find a with b, {refl},\n  exact absurd rfl ((mem_erase m a a b).1 ((find_iff (m.erase a) a b).1 h)).left\nend\n\ntheorem find_erase_ne (m : hash_map \u03b1 \u03b2) (a a' : \u03b1) (h : a \u2260 a') :\n  (m.erase a).find a' = m.find a' :=\noption.eq_of_eq_some $ \u03bbb',\n(find_iff _ _ _).trans $ (mem_erase m a a' b').trans $\n  (and_iff_right h).trans (find_iff _ _ _).symm\n\ntheorem find_erase (m : hash_map \u03b1 \u03b2) (a' a : \u03b1) :\n  (m.erase a).find a' = if a = a' then none else m.find a' :=\nif h : a = a' then by subst a'; simp [find_erase_eq m a]\nelse by rw if_neg h; exact find_erase_ne m a a' h\n\nsection string\nvariables [has_to_string \u03b1] [\u2200 a, has_to_string (\u03b2 a)]\nopen prod\nprivate def key_data_to_string (a : \u03b1) (b : \u03b2 a) (first : bool) : string :=\n(if first then \"\" else \", \") ++ sformat!\"{a} \u2190 {b}\"\n\nprivate def to_string (m : hash_map \u03b1 \u03b2) : string :=\n\"\u27e8\" ++ (fst (fold m (\"\", tt) (\u03bb p a b, (fst p ++ key_data_to_string a b (snd p), ff)))) ++ \"\u27e9\"\n\ninstance : has_to_string (hash_map \u03b1 \u03b2) :=\n\u27e8to_string\u27e9\n\nend string\n\nsection format\nopen format prod\nvariables [has_to_format \u03b1] [\u2200 a, has_to_format (\u03b2 a)]\n\nprivate meta def format_key_data (a : \u03b1) (b : \u03b2 a) (first : bool) : format :=\n(if first then to_fmt \"\" else to_fmt \",\" ++ line) ++\n  to_fmt a ++ space ++ to_fmt \"\u2190\" ++ space ++ to_fmt b\n\nprivate meta def to_format (m : hash_map \u03b1 \u03b2) : format :=\ngroup $ to_fmt \"\u27e8\" ++\n  nest 1 (fst (fold m (to_fmt \"\", tt) (\u03bb p a b, (fst p ++ format_key_data a b (snd p), ff)))) ++\n  to_fmt \"\u27e9\"\n\nmeta instance : has_to_format (hash_map \u03b1 \u03b2) :=\n\u27e8to_format\u27e9\nend format\n\n/-- `hash_map` with key type `nat` and value type that may vary. -/\ninstance {\u03b2 : \u2115 \u2192 Type*} : inhabited (hash_map \u2115 \u03b2) := \u27e8mk_hash_map id\u27e9\n\nend hash_map\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/hash_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3882684325604419}}
{"text": "/- Author: E.W.Ayers.\n   This section roughly follows Chapter 3, \u00a71, \u00a72 of Sheaves in Geology and Logic by Saunders Maclane and Ieke M.\n -/\n\nimport sieve\nimport category.pullbacks\n\nuniverses u v w\nnamespace category_theory\n\nopen category_theory limits order lattice\n\n/-- A set of sieves for every object in the category: a candidate to be a Grothendieck topology. -/\ndef sieve_set (C : Type u) [category.{v} C] := \u03a0 (X : C), set (sieve X)\n\ndef sieve_set.trivial (C : Type u) [category.{v} C] : sieve_set C := \u03bb X, {\u22a4}\n\nlemma mem_trivial (C : Type u) [category.{v} C] {X : C} (S : sieve X) :\n  S \u2208 sieve_set.trivial C X \u2194 S = \u22a4 :=\nset.mem_singleton_iff\n\n/-- A sieve on `X` is dense if for any arrow `f : Y \u27f6 X`, there is a `g : Z \u27f6 Y` with `g \u226b f \u2208 S`. -/\ndef sieve_set.dense (C : Type u) [category.{v} C] : sieve_set C :=\n\u03bb X, {S | \u2200 {Y : C} (f : Y \u27f6 X), \u2203 Z (g : Z \u27f6 Y), over.mk (g \u226b f) \u2208 S.arrows }\n\n/-- The atomic sieve_set just contains all of the non-empty sieves. -/\ndef sieve_set.atomic (C : Type u) [category.{v} C] : sieve_set C :=\n\u03bb X, {S | \u2203 {Y} (f : Y \u27f6 X), over.mk f \u2208 S.arrows}\n\nopen sieve category\n\n/--\nDefinition of a Grothendieck Topology: a set of sieves `J X` on each object `X` satisfying three axioms:\n1. For every object `X`, the maximal sieve is in `J X`.\n2. If `S \u2208 J X` then its pullback along any `h : Y \u27f6 X` is in `J Y`.\n3. If `S \u2208 J X` and `R` is a sieve on `X`, then provided that the pullback of `R` along any arrow\n   `f : Y \u27f6 X` in `S` is in `J Y`, we have that `R` itself is in `J X`.\n-/\nclass grothendieck {C : Type u} [category.{v} C] (J : sieve_set C) : Prop :=\n(max : \u2200 X, \u22a4 \u2208 J X)\n(stab : \u2200 {X Y} (S \u2208 J X) (h : Y \u27f6 X), sieve.pullback S h \u2208 J Y)\n(trans : \u2200 \u2983X\u2984 (S : sieve X) (hS : S \u2208 J X) (R : sieve X), (\u2200 {Y} (f : Y \u27f6 X), over.mk f \u2208 S.arrows \u2192 R.pullback f \u2208 J Y) \u2192 R \u2208 J X)\n\n/-- A site is a category equipped with a grothendieck topology. -/\nstructure Site :=\n(C : Type u)\n[\ud835\udc9e : category.{v} C]\n(J : sieve_set C)\n[g : grothendieck J]\n\nnamespace grothendieck\nvariables {C : Type u} [category.{v} C]\nvariables {X Y : C} {S R : sieve X}\nvariables {J : sieve_set C} [grothendieck J]\n\ndef superset_covering (Hss : S \u2264 R) (sjx : S \u2208 J X) : R \u2208 J X :=\nbegin\n  apply grothendieck.trans _ sjx,\n  intros Y h hh,\n  dsimp,\n  have : S.pullback h \u2264 R.pullback h,\n    apply pullback_le_map Hss,\n  have : S.pullback h = \u22a4,\n    rw \u2190 id_mem_iff_eq_top,\n    simpa,\n  have : R.pullback h = \u22a4,\n    apply top_unique,\n    rwa \u2190 this,\n  rw this,\n  apply grothendieck.max,\nend\n\ndef covers (J : sieve_set C) (S : sieve X) (f : Y \u27f6 X) : Prop := S.pullback f \u2208 J Y\n\nlemma arrow_max (f : Y \u27f6 X) (S : sieve X) [grothendieck J] (hf : over.mk f \u2208 S.arrows) : covers J S f :=\nbegin\n  rw [covers, (pullback_eq_top_iff_mem f).1 hf],\n  apply grothendieck.max,\nend\nlemma arrow_stab (f : Y \u27f6 X) (S : sieve X) (h : covers J S f) {Z : C} (g : Z \u27f6 Y) : covers J S (g \u226b f) :=\nbegin\n  rw [covers, pullback_comp],\n  apply grothendieck.stab,\n  apply h,\nend\nlemma arrow_trans (f : Y \u27f6 X) (S R : sieve X) (h : covers J S f) : (\u2200 {Z : C} (g : Z \u27f6 X), over.mk g \u2208 S.arrows \u2192 covers J R g) \u2192 covers J R f :=\nbegin\n  intro k,\n  apply grothendieck.trans (S.pullback f) h,\n  intros Z g hg,\n  rw \u2190 pullback_comp,\n  apply k (g \u226b f) hg,\nend\n\nlemma intersection_covering (rj : R \u2208 J X) (sj : S \u2208 J X) : R \u2293 S \u2208 J X :=\nbegin\n  apply grothendieck.trans R rj,\n  intros Y f Hf,\n  have : S.pullback f \u2264 (R \u2293 S).pullback f,\n    intros Z g hg,\n    refine \u27e8downward_closed _ Hf _, hg\u27e9,\n  apply superset_covering this,\n  apply grothendieck.stab _ sj,\n  apply_instance,\nend\n\nlemma arrow_intersect (f : Y \u27f6 X) (S R : sieve X) (hS : covers J S f) (hR : covers J R f) : covers J (S \u2293 R) f :=\nbegin\n  rw [covers, pullback_inter],\n  apply intersection_covering;\n  assumption\nend\n\nopen sieve_set\n\ninstance trivial.grothendieck : grothendieck (sieve_set.trivial C) :=\n{ max := \u03bb X, set.mem_singleton _,\n  stab := \u03bb X Y S HS h,\n  begin\n    rw mem_trivial at *,\n    rw [HS, pullback_top],\n  end,\n  trans := \u03bb X S HS R HR,\n  begin\n    rw [mem_trivial, \u2190 id_mem_iff_eq_top, pullback_eq_top_iff_mem],\n    simp only [mem_trivial] at HR,\n    apply HR,\n    rwa [id_mem_iff_eq_top, \u2190 mem_trivial],\n  end }\n\ninstance dense.grothendieck : grothendieck (dense C) :=\n{ max := \u03bb X Y f, \u27e8Y, \ud835\udfd9 Y, \u27e8\u27e9\u27e9,\n  stab :=\n    begin\n      intros X Y S H h Z f,\n      rcases H (f \u226b h) with \u27e8W, g, H\u27e9,\n      refine \u27e8W, g, _\u27e9,\n      simpa,\n    end,\n  trans :=\n    begin\n      intros X S H\u2081 R H\u2082 Y f,\n      rcases H\u2081 f with \u27e8Z,g,H\u2083\u27e9,\n      rcases H\u2082 _ H\u2083 (\ud835\udfd9 Z) with \u27e8W,h,H\u2084\u27e9,\n      refine \u27e8W, (h \u226b \ud835\udfd9 Z \u226b g), _\u27e9,\n      simpa using H\u2084,\n    end }\n\n/--\nA category satisfies the right Ore condition if any span can be completed to a\ncommutative square.\nNB. Any category with pullbacks obviously satisfies the right Ore condition.\n-/\ndef right_ore_condition (C : Type u) [category.{v} C] : Prop :=\n\u2200 {X Y Z : C} (yx : Y \u27f6 X) (zx : Z \u27f6 X), \u2203 W (wy : W \u27f6 Y) (wz : W \u27f6 Z), wy \u226b yx = wz \u226b zx\n\n/--\nThe atomic sieveset is a grothendieck topology when it\nsatisfies the 'square' property. Which says that every span `Y \u27f6 X \u27f5 Z` forms a commuting\ndiagram.\n-/\ninstance atomic.grothendieck\n  (hro : right_ore_condition C)\n  : grothendieck (atomic C) :=\n{ max := \u03bb X, \u27e8_, \ud835\udfd9 _, \u27e8\u27e9\u27e9,\n  stab :=\n  begin\n    rintros X Y S \u27e8Z, f, hf\u27e9 h,\n    rcases hro h f with \u27e8W, g, k, comm\u27e9,\n    refine \u27e8_, g, _\u27e9,\n    simp [mem_pullback, comm, hf],\n  end,\n  trans :=\n  begin\n    rintros X S \u27e8Y, f, hf\u27e9 R h,\n    rcases h f hf with \u27e8Z, g, hg\u27e9,\n    exact \u27e8_, _, hg\u27e9,\n  end }\n\nopen opposite\n\ndef matching_family (P : C\u1d52\u1d56 \u2964 Type v) (S : sieve X) :=\nS.as_functor \u27f6 P\n\ndef amalgamation {P : C\u1d52\u1d56 \u2964 Type v} {S : sieve X} (\u03b3 : matching_family P S) :=\n{\u03b1 : yoneda.obj X \u27f6 P // sieve.functor_inclusion S \u226b \u03b1 = \u03b3}\n\n@[derive subsingleton]\ndef sheaf_condition (J : sieve_set C) [grothendieck J] (P : C\u1d52\u1d56 \u2964 Type v) : Type (max u v) :=\n\u03a0 (X : C) (S : sieve X) (\u03b3 : matching_family P S), S \u2208 J X \u2192 unique (amalgamation \u03b3)\n\ndef matching_family' (P : C\u1d52\u1d56 \u2964 Type v) {c : C} (S : sieve c) :=\n{x : \u03a0 {d : C} (f : d \u27f6 c), over.mk f \u2208 S.arrows \u2192 P.obj (opposite.op d) // \u2200 {d e : C} (f : d \u27f6 c) (g : e \u27f6 d) (h : over.mk f \u2208 S.arrows), x (g \u226b f) (sieve.downward_closed _ h _) = P.map g.op (x f h)}\n\ndef amalgamation' {P : C\u1d52\u1d56 \u2964 Type v} {c : C} {S : sieve c} (\u03b3 : matching_family' P S) :=\n{y : P.obj (opposite.op c) // \u2200 {d : C} (f : d \u27f6 c) (hf : over.mk f \u2208 S.arrows), P.map f.op y = \u03b3.1 f hf}\n\n@[derive subsingleton]\ndef sheaf_condition' (J : sieve_set C) [grothendieck J] (P : C\u1d52\u1d56 \u2964 Type v) : Type (max u v) :=\n\u03a0 (c : C) (S : sieve c) (\u03b3 : matching_family' P S), S \u2208 J c \u2192 unique (amalgamation' \u03b3)\n\ndef matching_family'_equiv_matching_family (P : C\u1d52\u1d56 \u2964 Type v) : matching_family' P S \u2243 matching_family P S :=\n{ to_fun := \u03bb x, \u27e8\u03bb _ t, x.1 _ t.2, \u03bb c c' f, funext $ \u03bb t, x.2 _ _ t.2\u27e9,\n  inv_fun := \u03bb x, \u27e8\u03bb d f hf, x.app _ \u27e8f, hf\u27e9, \u03bb d d' f g h, congr_fun (x.2 g.op) \u27e8f, h\u27e9\u27e9,\n  left_inv := \u03bb _, subtype.ext $ funext $ \u03bb _, funext $ \u03bb _, funext $ \u03bb _, rfl,\n  right_inv := \u03bb _, by { ext _ \u27e8_, _\u27e9, refl } }\n\ndef amalgamation'_equiv_amalgamation (P : C\u1d52\u1d56 \u2964 Type v) (x : matching_family' P S) :\n  amalgamation (matching_family'_equiv_matching_family P x) \u2243 (amalgamation' x) :=\n{ to_fun := \u03bb \u03b3,\n  { val := \u03b3.1.app _ (\ud835\udfd9 X),\n    property := \u03bb d f hf,\n    begin\n      have := congr_fun (\u03b3.1.naturality f.op) (\ud835\udfd9 _),\n      dsimp at this,\n      erw \u2190 this,\n      rw comp_id,\n      have q := congr_arg (\u03bb t, nat_trans.app t (opposite.op d)) \u03b3.2,\n      dsimp at q,\n      have := congr_fun q \u27e8f, hf\u27e9,\n      exact this,\n    end },\n  inv_fun := \u03bb \u03b3,\n  { val :=\n    { app := \u03bb c f, P.map f.op \u03b3.1,\n      naturality' := \u03bb c c' f, funext $ \u03bb g,\n      begin\n        dsimp at g,\n        dsimp,\n        rw P.map_comp,\n        refl,\n      end },\n    property :=\n    begin\n      ext c \u27e8f, hf\u27e9,\n      apply \u03b3.2,\n    end },\n  left_inv :=\n  begin\n    rintro \u27e8\u03b3\u2081, \u03b3\u2082\u27e9,\n    ext d f,\n    dsimp,\n    dsimp at f,\n    have := congr_fun (\u03b3\u2081.naturality f.op) (\ud835\udfd9 X),\n    dsimp at this,\n    rw [\u2190 this, comp_id],\n  end,\n  right_inv :=\n  begin\n    rintro \u27e8\u03b3\u2081, \u03b3\u2082\u27e9,\n    ext1,\n    dsimp,\n    rw P.map_id,\n    refl,\n  end }\n\ndef sheaf'_equiv_sheaf (J : sieve_set C) [grothendieck J] (P : C\u1d52\u1d56 \u2964 Type v) :\n  sheaf_condition J P \u2245 sheaf_condition' J P :=\n{ hom :=\n  begin\n    intros h c S \u03b3 hS,\n    apply equiv.unique (amalgamation'_equiv_amalgamation _ _).symm,\n    apply h _ _ _ hS,\n  end,\n  inv :=\n  begin\n    intros h c S \u03b3 hS,\n    haveI := h _ _ ((matching_family'_equiv_matching_family P).symm \u03b3) hS,\n    have := equiv.unique (amalgamation'_equiv_amalgamation P ((matching_family'_equiv_matching_family P).symm \u03b3)),\n    simpa using this,\n  end }\n\nend grothendieck\n\nend category_theory\n", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3882601057423012}}
{"text": "import data.list\nimport tactic\n\nuniverses u v w\n\ntheorem list.tfae_prf {a b : Prop} {l : list Prop} (h : list.tfae l) (ha : a \u2208 l) (hb : b \u2208 l) (ha_prf : a) : b :=\n(h a ha b hb).mp ha_prf\n\nlemma and_iff_right_of_left_if_right {p q : Prop} (h : p \u2192 q) : q \u2227 p \u2194 p :=\n\u27e8\u03bb h\u2082, h\u2082.right, \u03bb h\u2082, \u27e8h h\u2082, h\u2082\u27e9\u27e9\n\nlemma choice_2_arg {\u03b1 : Sort u} {\u03b2 : Sort v} {\u03b3 : \u03b1 \u2192 \u03b2 \u2192 Sort w}\n{r : \u03a0 (x : \u03b1) (y : \u03b2), \u03b3 x y \u2192 Prop} (h : \u2200 (x : \u03b1) (y : \u03b2), \u2203 (z : \u03b3 x y), r x y z) :\n\u2203 (f : \u03a0 (x : \u03b1) (y : \u03b2), \u03b3 x y), \u2200 (x : \u03b1) (y : \u03b2), r x y (f x y) :=\nbegin\n  let \u03b3' : (pprod \u03b1 \u03b2) \u2192 Sort w := (\u03bb z, \u03b3 z.fst z.snd),\n  let r' := \u03bb (x : pprod \u03b1 \u03b2) (z : \u03b3' x), r x.fst x.snd z,\n  have h' : \u2200 (x : pprod \u03b1 \u03b2), \u2203 z : \u03b3' x, r' x z := (\u03bb x, h x.fst x.snd),\n  rcases classical.axiom_of_choice h' with \u27e8f, hf\u27e9,\n  let f := (\u03bb x y, f \u27e8x, y\u27e9),\n  existsi f, intros x y, exact hf \u27e8x, y\u27e9,\nend\n\nlemma subst_right_of_and {\u03b1 : Sort u} {p : Prop} {a b c : \u03b1} (h : p \u2192 b = c) : p \u2227 a = b \u2194 p \u2227 a = c :=\n\u27e8 assume h\u2082, \u27e8h\u2082.left, (h h\u2082.left) \u25b8 h\u2082.right\u27e9,\n  assume h\u2082, \u27e8h\u2082.left, (h h\u2082.left).symm \u25b8 h\u2082.right\u27e9 \u27e9\n\nlemma exists_unique_eq {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) : \u2200 x : \u03b1, \u2203! y : \u03b2, (\u03bb (a : \u03b1) (b : \u03b2), b = f a) x y :=\n\u03bb x, \u27e8f x, rfl, \u03bb y, assume hy, hy\u27e9\n\nlemma rel_congr_left {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (symm : symmetric r) (trans : transitive r)\n  {x y : \u03b1} (xy : r x y) {z : \u03b1} : r x z \u2194 r y z :=\n\u27e8\u03bb xz, trans (symm xy) xz, \u03bb yz, trans xy yz\u27e9\n\nlemma iff_of_not_of_not {p : Prop} (np : \u00ac p) {q : Prop} (nq : \u00ac q) : p \u2194 q :=\n\u27e8\u03bb hp, false.elim (np hp), \u03bb hq, false.elim (nq hq)\u27e9\n\nlemma ite_prop {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} {x y : \u03b1} (px : p x) (py : p y) {q : Prop} [h : decidable q] :\np (if q then x else y) :=\nbegin\n  by_cases q,\n    rw if_pos h, exact px,\n  rw if_neg h, exact py,\nend", "meta": {"author": "ehaskell1", "repo": "set_theory", "sha": "e6c829c4dd953d98c9cba08f9f79784cd91794fb", "save_path": "github-repos/lean/ehaskell1-set_theory", "path": "github-repos/lean/ehaskell1-set_theory/set_theory-e6c829c4dd953d98c9cba08f9f79784cd91794fb/src/logic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.38813423382886225}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module category_theory.extensive\n! leanprover-community/mathlib commit ac3ae212f394f508df43e37aa093722fa9b65d31\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Shapes.CommSq\nimport Mathbin.CategoryTheory.Limits.Shapes.StrictInitial\nimport Mathbin.CategoryTheory.Limits.Shapes.Types\nimport Mathbin.Topology.Category.Top.Limits\nimport Mathbin.CategoryTheory.Limits.FunctorCategory\n\n/-!\n\n# Extensive categories\n\n## Main definitions\n- `category_theory.is_van_kampen_colimit`: A (colimit) cocone over a diagram `F : J \u2964 C` is van\n  Kampen if for every cocone `c'` over the pullback of the diagram `F' : J \u2964 C'`,\n  `c'` is colimiting iff `c'` is the pullback of `c`.\n- `category_theory.finitary_extensive`: A category is (finitary) extensive if it has finite\n  coproducts, and binary coproducts are van Kampen.\n\n## Main Results\n- `category_theory.has_strict_initial_objects_of_finitary_extensive`: The initial object\n  in extensive categories is strict.\n- `category_theory.finitary_extensive.mono_inr_of_is_colimit`: Coproduct injections are monic in\n  extensive categories.\n- `category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen`: In extensive categories,\n  sums are disjoint, i.e. the pullback of `X \u27f6 X \u2a3f Y` and `Y \u27f6 X \u2a3f Y` is the initial object.\n- `category_theory.types.finitary_extensive`: The category of types is extensive.\n\n## TODO\n\nShow that the following are finitary extensive:\n- the categories of sheaves over a site\n- `Scheme`\n- `AffineScheme` (`CommRing\u1d52\u1d56`)\n\n## References\n- https://ncatlab.org/nlab/show/extensive+category\n- [Carboni et al, Introduction to extensive and distributive categories][CARBONI1993145]\n\n-/\n\n\nopen CategoryTheory.Limits\n\nnamespace CategoryTheory\n\nuniverse v' u' v u\n\nvariable {J : Type v'} [Category.{u'} J] {C : Type u} [Category.{v} C]\n\n/-- A natural transformation is equifibered if every commutative square of the following form is\na pullback.\n```\nF(X) \u2192 F(Y)\n \u2193      \u2193\nG(X) \u2192 G(Y)\n```\n-/\ndef NatTrans.Equifibered {F G : J \u2964 C} (\u03b1 : F \u27f6 G) : Prop :=\n  \u2200 \u2983i j : J\u2984 (f : i \u27f6 j), IsPullback (F.map f) (\u03b1.app i) (\u03b1.app j) (G.map f)\n#align category_theory.nat_trans.equifibered CategoryTheory.NatTrans.Equifibered\n\ntheorem NatTrans.equifibered_of_isIso {F G : J \u2964 C} (\u03b1 : F \u27f6 G) [IsIso \u03b1] : \u03b1.Equifibered :=\n  fun _ _ f => IsPullback.of_vert_isIso \u27e8NatTrans.naturality _ f\u27e9\n#align category_theory.nat_trans.equifibered_of_is_iso CategoryTheory.NatTrans.equifibered_of_isIso\n\ntheorem NatTrans.Equifibered.comp {F G H : J \u2964 C} {\u03b1 : F \u27f6 G} {\u03b2 : G \u27f6 H} (h\u03b1 : \u03b1.Equifibered)\n    (h\u03b2 : \u03b2.Equifibered) : (\u03b1 \u226b \u03b2).Equifibered := fun i j f => (h\u03b1 f).paste_vert (h\u03b2 f)\n#align category_theory.nat_trans.equifibered.comp CategoryTheory.NatTrans.Equifibered.comp\n\n/-- A (colimit) cocone over a diagram `F : J \u2964 C` is universal if it is stable under pullbacks. -/\ndef IsUniversalColimit {F : J \u2964 C} (c : Cocone F) : Prop :=\n  \u2200 \u2983F' : J \u2964 C\u2984 (c' : Cocone F') (\u03b1 : F' \u27f6 F) (f : c'.pt \u27f6 c.pt)\n    (h : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const J).map f) (h\u03b1 : \u03b1.Equifibered),\n    (\u2200 j : J, IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j)) \u2192 Nonempty (IsColimit c')\n#align category_theory.is_universal_colimit CategoryTheory.IsUniversalColimit\n\n/-- A (colimit) cocone over a diagram `F : J \u2964 C` is van Kampen if for every cocone `c'` over the\npullback of the diagram `F' : J \u2964 C'`, `c'` is colimiting iff `c'` is the pullback of `c`.\n\nTODO: Show that this is iff the functor `C \u2964 Cat\u1d52\u1d56` sending `x` to `C/x` preserves it.\nTODO: Show that this is iff the inclusion functor `C \u2964 Span(C)` preserves it.\n-/\ndef IsVanKampenColimit {F : J \u2964 C} (c : Cocone F) : Prop :=\n  \u2200 \u2983F' : J \u2964 C\u2984 (c' : Cocone F') (\u03b1 : F' \u27f6 F) (f : c'.pt \u27f6 c.pt)\n    (h : \u03b1 \u226b c.\u03b9 = c'.\u03b9 \u226b (Functor.const J).map f) (h\u03b1 : \u03b1.Equifibered),\n    Nonempty (IsColimit c') \u2194 \u2200 j : J, IsPullback (c'.\u03b9.app j) (\u03b1.app j) f (c.\u03b9.app j)\n#align category_theory.is_van_kampen_colimit CategoryTheory.IsVanKampenColimit\n\ntheorem IsVanKampenColimit.is_universal {F : J \u2964 C} {c : Cocone F} (H : IsVanKampenColimit c) :\n    IsUniversalColimit c := fun _ c' \u03b1 f h h\u03b1 => (H c' \u03b1 f h h\u03b1).mpr\n#align category_theory.is_van_kampen_colimit.is_universal CategoryTheory.IsVanKampenColimit.is_universal\n\n/-- A van Kampen colimit is a colimit. -/\nnoncomputable def IsVanKampenColimit.isColimit {F : J \u2964 C} {c : Cocone F}\n    (h : IsVanKampenColimit c) : IsColimit c :=\n  by\n  refine'\n    ((h c (\ud835\udfd9 F) (\ud835\udfd9 c.X : _) (by rw [Functor.map_id, category.comp_id, category.id_comp])\n            (nat_trans.equifibered_of_is_iso _)).mpr\n        fun j => _).some\n  haveI : is_iso (\ud835\udfd9 c.X) := inferInstance\n  exact is_pullback.of_vert_is_iso \u27e8by erw [nat_trans.id_app, category.comp_id, category.id_comp]\u27e9\n#align category_theory.is_van_kampen_colimit.is_colimit CategoryTheory.IsVanKampenColimit.isColimit\n\ntheorem IsInitial.isVanKampenColimit [HasStrictInitialObjects C] {X : C} (h : IsInitial X) :\n    IsVanKampenColimit (asEmptyCocone X) :=\n  by\n  intro F' c' \u03b1 f hf h\u03b1\n  have : F' = functor.empty C := by apply functor.hext <;> rintro \u27e8\u27e8\u27e9\u27e9\n  subst this\n  haveI := h.is_iso_to f\n  refine'\n    \u27e8by rintro _ \u27e8\u27e8\u27e9\u27e9, fun _ =>\n      \u27e8is_colimit.of_iso_colimit h (cocones.ext (as_iso f).symm <| by rintro \u27e8\u27e8\u27e9\u27e9)\u27e9\u27e9\n#align category_theory.is_initial.is_van_kampen_colimit CategoryTheory.IsInitial.isVanKampenColimit\n\nsection Extensive\n\nvariable {X Y : C}\n\n/-- A category is (finitary) extensive if it has finite coproducts,\nand binary coproducts are van Kampen.\n\nTODO: Show that this is iff all finite coproducts are van Kampen. -/\nclass FinitaryExtensive (C : Type u) [Category.{v} C] : Prop where\n  [HasFiniteCoproducts : HasFiniteCoproducts C]\n  van_kampen' : \u2200 {X Y : C} (c : BinaryCofan X Y), IsColimit c \u2192 IsVanKampenColimit c\n#align category_theory.finitary_extensive CategoryTheory.FinitaryExtensive\n\nattribute [instance] finitary_extensive.has_finite_coproducts\n\ntheorem FinitaryExtensive.van_kampen [FinitaryExtensive C] {F : Discrete WalkingPair \u2964 C}\n    (c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c :=\n  by\n  let X := F.obj \u27e8walking_pair.left\u27e9\n  let Y := F.obj \u27e8walking_pair.right\u27e9\n  have : F = pair X Y := by\n    apply functor.hext\n    \u00b7 rintro \u27e8\u27e8\u27e9\u27e9 <;> rfl\n    \u00b7 rintro \u27e8\u27e8\u27e9\u27e9 \u27e8j\u27e9 \u27e8\u27e8rfl : _ = j\u27e9\u27e9 <;> simpa\n  clear_value X Y\n  subst this\n  exact finitary_extensive.van_kampen' c hc\n#align category_theory.finitary_extensive.van_kampen CategoryTheory.FinitaryExtensive.van_kampen\n\ntheorem map_pair_equifibered {F F' : Discrete WalkingPair \u2964 C} (\u03b1 : F \u27f6 F') : \u03b1.Equifibered :=\n  by\n  rintro \u27e8\u27e8\u27e9\u27e9 \u27e8j\u27e9 \u27e8\u27e8rfl : _ = j\u27e9\u27e9\n  all_goals\n    dsimp; simp only [discrete.functor_map_id]\n    exact is_pullback.of_horiz_is_iso \u27e8by simp only [category.comp_id, category.id_comp]\u27e9\n#align category_theory.map_pair_equifibered CategoryTheory.map_pair_equifibered\n\ntheorem BinaryCofan.is_van_kampen_iff (c : BinaryCofan X Y) :\n    IsVanKampenColimit c \u2194\n      \u2200 {X' Y' : C} (c' : BinaryCofan X' Y') (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : c'.pt \u27f6 c.pt)\n        (h\u03b1X : \u03b1X \u226b c.inl = c'.inl \u226b f) (h\u03b1Y : \u03b1Y \u226b c.inr = c'.inr \u226b f),\n        Nonempty (IsColimit c') \u2194 IsPullback c'.inl \u03b1X f c.inl \u2227 IsPullback c'.inr \u03b1Y f c.inr :=\n  by\n  constructor\n  \u00b7 introv H h\u03b1X h\u03b1Y\n    rw [H c' (map_pair \u03b1X \u03b1Y) f (by ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> assumption) (map_pair_equifibered _)]\n    constructor\n    \u00b7 intro H\n      exact \u27e8H _, H _\u27e9\n    \u00b7 rintro H \u27e8\u27e8\u27e9\u27e9\n      exacts[H.1, H.2]\n  \u00b7 introv H F' h\u03b1 h\n    let X' := F'.obj \u27e8walking_pair.left\u27e9\n    let Y' := F'.obj \u27e8walking_pair.right\u27e9\n    have : F' = pair X' Y' := by\n      apply functor.hext\n      \u00b7 rintro \u27e8\u27e8\u27e9\u27e9 <;> rfl\n      \u00b7 rintro \u27e8\u27e8\u27e9\u27e9 \u27e8j\u27e9 \u27e8\u27e8rfl : _ = j\u27e9\u27e9 <;> simpa\n    clear_value X' Y'\n    subst this\n    change binary_cofan X' Y' at c'\n    rw [H c' _ _ _ (nat_trans.congr_app h\u03b1 \u27e8walking_pair.left\u27e9)\n        (nat_trans.congr_app h\u03b1 \u27e8walking_pair.right\u27e9)]\n    constructor\n    \u00b7 rintro H \u27e8\u27e8\u27e9\u27e9\n      exacts[H.1, H.2]\n    \u00b7 intro H\n      exact \u27e8H _, H _\u27e9\n#align category_theory.binary_cofan.is_van_kampen_iff CategoryTheory.BinaryCofan.is_van_kampen_iff\n\ntheorem BinaryCofan.is_van_kampen_mk {X Y : C} (c : BinaryCofan X Y)\n    (cofans : \u2200 X Y : C, BinaryCofan X Y) (colimits : \u2200 X Y, IsColimit (cofans X Y))\n    (cones : \u2200 {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z), PullbackCone f g)\n    (limits : \u2200 {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z), IsLimit (cones f g))\n    (h\u2081 :\n      \u2200 {X' Y' : C} (\u03b1X : X' \u27f6 X) (\u03b1Y : Y' \u27f6 Y) (f : (cofans X' Y').pt \u27f6 c.pt)\n        (h\u03b1X : \u03b1X \u226b c.inl = (cofans X' Y').inl \u226b f) (h\u03b1Y : \u03b1Y \u226b c.inr = (cofans X' Y').inr \u226b f),\n        IsPullback (cofans X' Y').inl \u03b1X f c.inl \u2227 IsPullback (cofans X' Y').inr \u03b1Y f c.inr)\n    (h\u2082 :\n      \u2200 {Z : C} (f : Z \u27f6 c.pt),\n        IsColimit (BinaryCofan.mk (cones f c.inl).fst (cones f c.inr).fst)) :\n    IsVanKampenColimit c := by\n  rw [binary_cofan.is_van_kampen_iff]\n  introv hX hY\n  constructor\n  \u00b7 rintro \u27e8h\u27e9\n    let e := h.cocone_point_unique_up_to_iso (colimits _ _)\n    obtain \u27e8hl, hr\u27e9 := h\u2081 \u03b1X \u03b1Y (e.inv \u226b f) (by simp [hX]) (by simp [hY])\n    constructor\n    \u00b7 rw [\u2190 category.id_comp \u03b1X, \u2190 iso.hom_inv_id_assoc e f]\n      have : c'.inl \u226b e.hom = \ud835\udfd9 X' \u226b (cofans X' Y').inl :=\n        by\n        dsimp\n        simp\n      haveI : is_iso (\ud835\udfd9 X') := inferInstance\n      exact (is_pullback.of_vert_is_iso \u27e8this\u27e9).paste_vert hl\n    \u00b7 rw [\u2190 category.id_comp \u03b1Y, \u2190 iso.hom_inv_id_assoc e f]\n      have : c'.inr \u226b e.hom = \ud835\udfd9 Y' \u226b (cofans X' Y').inr :=\n        by\n        dsimp\n        simp\n      haveI : is_iso (\ud835\udfd9 Y') := inferInstance\n      exact (is_pullback.of_vert_is_iso \u27e8this\u27e9).paste_vert hr\n  \u00b7 rintro \u27e8H\u2081, H\u2082\u27e9\n    refine' \u27e8is_colimit.of_iso_colimit _ <| (iso_binary_cofan_mk _).symm\u27e9\n    let e\u2081 : X' \u2245 _ := H\u2081.is_limit.cone_point_unique_up_to_iso (limits _ _)\n    let e\u2082 : Y' \u2245 _ := H\u2082.is_limit.cone_point_unique_up_to_iso (limits _ _)\n    have he\u2081 : c'.inl = e\u2081.hom \u226b (cones f c.inl).fst := by simp\n    have he\u2082 : c'.inr = e\u2082.hom \u226b (cones f c.inr).fst := by simp\n    rw [he\u2081, he\u2082]\n    apply binary_cofan.is_colimit_comp_right_iso (binary_cofan.mk _ _)\n    apply binary_cofan.is_colimit_comp_left_iso (binary_cofan.mk _ _)\n    exact h\u2082 f\n#align category_theory.binary_cofan.is_van_kampen_mk CategoryTheory.BinaryCofan.is_van_kampen_mk\n\ntheorem BinaryCofan.mono_inr_of_is_van_kampen [HasInitial C] {X Y : C} {c : BinaryCofan X Y}\n    (h : IsVanKampenColimit c) : Mono c.inr :=\n  by\n  refine' pullback_cone.mono_of_is_limit_mk_id_id _ (is_pullback.is_limit _)\n  refine'\n    (h (binary_cofan.mk (initial.to Y) (\ud835\udfd9 Y)) (map_pair (initial.to X) (\ud835\udfd9 Y)) c.inr _\n          (map_pair_equifibered _)).mp\n      \u27e8_\u27e9 \u27e8walking_pair.right\u27e9\n  \u00b7 ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> simp\n  \u00b7\n    exact\n      ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr\n          (by\n            dsimp\n            infer_instance)).some\n#align category_theory.binary_cofan.mono_inr_of_is_van_kampen CategoryTheory.BinaryCofan.mono_inr_of_is_van_kampen\n\ntheorem FinitaryExtensive.mono_inr_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inr :=\n  BinaryCofan.mono_inr_of_is_van_kampen (FinitaryExtensive.van_kampen c hc)\n#align category_theory.finitary_extensive.mono_inr_of_is_colimit CategoryTheory.FinitaryExtensive.mono_inr_of_isColimit\n\ntheorem FinitaryExtensive.mono_inl_of_isColimit [FinitaryExtensive C] {c : BinaryCofan X Y}\n    (hc : IsColimit c) : Mono c.inl :=\n  FinitaryExtensive.mono_inr_of_isColimit (BinaryCofan.isColimitFlip hc)\n#align category_theory.finitary_extensive.mono_inl_of_is_colimit CategoryTheory.FinitaryExtensive.mono_inl_of_isColimit\n\ninstance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inl : X \u27f6 X \u2a3f Y) :=\n  (FinitaryExtensive.mono_inl_of_isColimit (coprodIsCoprod X Y) : _)\n\ninstance [FinitaryExtensive C] (X Y : C) : Mono (coprod.inr : Y \u27f6 X \u2a3f Y) :=\n  (FinitaryExtensive.mono_inr_of_isColimit (coprodIsCoprod X Y) : _)\n\ntheorem BinaryCofan.isPullback_initial_to_of_is_van_kampen [HasInitial C] {c : BinaryCofan X Y}\n    (h : IsVanKampenColimit c) : IsPullback (initial.to _) (initial.to _) c.inl c.inr :=\n  by\n  refine'\n    ((h (binary_cofan.mk (initial.to Y) (\ud835\udfd9 Y)) (map_pair (initial.to X) (\ud835\udfd9 Y)) c.inr _\n            (map_pair_equifibered _)).mp\n        \u27e8_\u27e9 \u27e8walking_pair.left\u27e9).flip\n  \u00b7 ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> simp\n  \u00b7\n    exact\n      ((binary_cofan.is_colimit_iff_is_iso_inr initial_is_initial _).mpr\n          (by\n            dsimp\n            infer_instance)).some\n#align category_theory.binary_cofan.is_pullback_initial_to_of_is_van_kampen CategoryTheory.BinaryCofan.isPullback_initial_to_of_is_van_kampen\n\ntheorem FinitaryExtensive.isPullback_initial_to_binaryCofan [FinitaryExtensive C]\n    {c : BinaryCofan X Y} (hc : IsColimit c) :\n    IsPullback (initial.to _) (initial.to _) c.inl c.inr :=\n  BinaryCofan.isPullback_initial_to_of_is_van_kampen (FinitaryExtensive.van_kampen c hc)\n#align category_theory.finitary_extensive.is_pullback_initial_to_binary_cofan CategoryTheory.FinitaryExtensive.isPullback_initial_to_binaryCofan\n\ntheorem has_strict_initial_of_is_universal [HasInitial C]\n    (H : IsUniversalColimit (BinaryCofan.mk (\ud835\udfd9 (\u22a5_ C)) (\ud835\udfd9 (\u22a5_ C)))) : HasStrictInitialObjects C :=\n  hasStrictInitialObjects_of_initial_is_strict\n    (by\n      intro A f\n      suffices is_colimit (binary_cofan.mk (\ud835\udfd9 A) (\ud835\udfd9 A))\n        by\n        obtain \u27e8l, h\u2081, h\u2082\u27e9 := limits.binary_cofan.is_colimit.desc' this (f \u226b initial.to A) (\ud835\udfd9 A)\n        rcases(category.id_comp _).symm.trans h\u2082 with rfl\n        exact \u27e8\u27e8_, ((category.id_comp _).symm.trans h\u2081).symm, initial_is_initial.hom_ext _ _\u27e9\u27e9\n      refine'\n        (H (binary_cofan.mk (\ud835\udfd9 _) (\ud835\udfd9 _)) (map_pair f f) f (by ext \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;> simp)\n            (map_pair_equifibered _) _).some\n      rintro \u27e8\u27e8\u27e9\u27e9 <;> dsimp <;>\n        exact is_pullback.of_horiz_is_iso \u27e8(category.id_comp _).trans (category.comp_id _).symm\u27e9)\n#align category_theory.has_strict_initial_of_is_universal CategoryTheory.has_strict_initial_of_is_universal\n\ninstance (priority := 100) hasStrictInitialObjects_of_finitaryExtensive [FinitaryExtensive C] :\n    HasStrictInitialObjects C :=\n  has_strict_initial_of_is_universal\n    (FinitaryExtensive.van_kampen _\n        ((BinaryCofan.isColimit_iff_isIso_inr initialIsInitial _).mpr\n            (by\n              dsimp\n              infer_instance)).some).is_universal\n#align category_theory.has_strict_initial_objects_of_finitary_extensive CategoryTheory.hasStrictInitialObjects_of_finitaryExtensive\n\ntheorem finitaryExtensive_iff_of_isTerminal (C : Type u) [Category.{v} C] [HasFiniteCoproducts C]\n    (T : C) (HT : IsTerminal T) (c\u2080 : BinaryCofan T T) (hc\u2080 : IsColimit c\u2080) :\n    FinitaryExtensive C \u2194 IsVanKampenColimit c\u2080 :=\n  by\n  refine' \u27e8fun H => H.2 c\u2080 hc\u2080, fun H => _\u27e9\n  constructor\n  simp_rw [binary_cofan.is_van_kampen_iff] at H\u22a2\n  intro X Y c hc X' Y' c' \u03b1X \u03b1Y f hX hY\n  obtain \u27e8d, hd, hd'\u27e9 :=\n    limits.binary_cofan.is_colimit.desc' hc (HT.from _ \u226b c\u2080.inl) (HT.from _ \u226b c\u2080.inr)\n  rw [H c' (\u03b1X \u226b HT.from _) (\u03b1Y \u226b HT.from _) (f \u226b d) (by rw [\u2190 reassoc_of hX, hd, category.assoc])\n      (by rw [\u2190 reassoc_of hY, hd', category.assoc])]\n  obtain \u27e8hl, hr\u27e9 := (H c (HT.from _) (HT.from _) d hd.symm hd'.symm).mp \u27e8hc\u27e9\n  rw [hl.paste_vert_iff hX.symm, hr.paste_vert_iff hY.symm]\n#align category_theory.finitary_extensive_iff_of_is_terminal CategoryTheory.finitaryExtensive_iff_of_isTerminal\n\ninstance types.finitaryExtensive : FinitaryExtensive (Type u) :=\n  by\n  rw [finitary_extensive_iff_of_is_terminal (Type u) PUnit types.is_terminal_punit _\n      (types.binary_coproduct_colimit _ _)]\n  apply\n    binary_cofan.is_van_kampen_mk _ _ (fun X Y => types.binary_coproduct_colimit X Y) _\n      fun X Y Z f g => (limits.types.pullback_limit_cone f g).2\n  \u00b7 intros\n    constructor\n    \u00b7 refine' \u27e8\u27e8h\u03b1X.symm\u27e9, \u27e8pullback_cone.is_limit_aux' _ _\u27e9\u27e9\n      intro s\n      have : \u2200 x, \u2203! y, s.fst x = Sum.inl y := by\n        intro x\n        cases h : s.fst x\n        \u00b7 simp_rw [sum.inl_injective.eq_iff]\n          exact exists_unique_eq'\n        \u00b7 apply_fun f  at h\n          cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun h\u03b1Y val : _).symm\n      delta ExistsUnique at this\n      choose l hl hl'\n      exact\n        \u27e8l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h\u2081 h\u2082 =>\n          funext fun x => hl' x (l' x) (congr_fun h\u2081 x).symm\u27e9\n    \u00b7 refine' \u27e8\u27e8h\u03b1Y.symm\u27e9, \u27e8pullback_cone.is_limit_aux' _ _\u27e9\u27e9\n      intro s\n      dsimp\n      have : \u2200 x, \u2203! y, s.fst x = Sum.inr y := by\n        intro x\n        cases h : s.fst x\n        \u00b7 apply_fun f  at h\n          cases ((congr_fun s.condition x).symm.trans h).trans (congr_fun h\u03b1X val : _).symm\n        \u00b7 simp_rw [sum.inr_injective.eq_iff]\n          exact exists_unique_eq'\n      delta ExistsUnique at this\n      choose l hl hl'\n      exact\n        \u27e8l, (funext hl).symm, types.is_terminal_punit.hom_ext _ _, fun l' h\u2081 h\u2082 =>\n          funext fun x => hl' x (l' x) (congr_fun h\u2081 x).symm\u27e9\n  \u00b7 intro Z f\n    dsimp [limits.types.binary_coproduct_cocone]\n    delta types.pullback_obj\n    have : \u2200 x, f x = Sum.inl PUnit.unit \u2228 f x = Sum.inr PUnit.unit :=\n      by\n      intro x\n      rcases f x with (\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n      exacts[Or.inl rfl, Or.inr rfl]\n    let eX : { p : Z \u00d7 PUnit // f p.fst = Sum.inl p.snd } \u2243 { x : Z // f x = Sum.inl PUnit.unit } :=\n      \u27e8fun p => \u27e8p.1.1, by convert p.2\u27e9, fun x => \u27e8\u27e8_, _\u27e9, x.2\u27e9, fun _ => by ext <;> rfl, fun _ =>\n        by ext <;> rfl\u27e9\n    let eY : { p : Z \u00d7 PUnit // f p.fst = Sum.inr p.snd } \u2243 { x : Z // f x = Sum.inr PUnit.unit } :=\n      \u27e8fun p => \u27e8p.1.1, p.2.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)\u27e9, fun x =>\n        \u27e8\u27e8_, _\u27e9, x.2\u27e9, fun _ => by ext <;> rfl, fun _ => by ext <;> rfl\u27e9\n    fapply binary_cofan.is_colimit_mk\n    \u00b7\n      exact fun s x =>\n        dite _ (fun h => s.inl <| eX.symm \u27e8x, h\u27e9) fun h =>\n          s.inr <| eY.symm \u27e8x, (this x).resolve_left h\u27e9\n    \u00b7 intro s\n      ext \u27e8\u27e8x, \u27e8\u27e9\u27e9, _\u27e9\n      dsimp\n      split_ifs <;> rfl\n    \u00b7 intro s\n      ext \u27e8\u27e8x, \u27e8\u27e9\u27e9, hx\u27e9\n      dsimp\n      split_ifs\n      \u00b7 cases h.symm.trans hx\n      \u00b7 rfl\n    \u00b7 intro s m e\u2081 e\u2082\n      ext x\n      split_ifs\n      \u00b7 rw [\u2190 e\u2081]\n        rfl\n      \u00b7 rw [\u2190 e\u2082]\n        rfl\n#align category_theory.types.finitary_extensive CategoryTheory.types.finitaryExtensive\n\nsection TopCat\n\n/-- (Implementation) An auxiliary lemma for the proof that `Top` is finitary extensive. -/\ndef finitaryExtensiveTopAux (Z : TopCat.{u}) (f : Z \u27f6 TopCat.of (Sum PUnit.{u + 1} PUnit.{u + 1})) :\n    IsColimit\n      (BinaryCofan.mk\n        (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inl)\n        (TopCat.pullbackFst f (TopCat.binaryCofan (TopCat.of PUnit) (TopCat.of PUnit)).inr)) :=\n  by\n  have : \u2200 x, f x = Sum.inl PUnit.unit \u2228 f x = Sum.inr PUnit.unit :=\n    by\n    intro x\n    rcases f x with (\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9)\n    exacts[Or.inl rfl, Or.inr rfl]\n  let eX : { p : Z \u00d7 PUnit // f p.fst = Sum.inl p.snd } \u2243 { x : Z // f x = Sum.inl PUnit.unit } :=\n    \u27e8fun p => \u27e8p.1.1, p.2.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)\u27e9, fun x =>\n      \u27e8\u27e8_, _\u27e9, x.2\u27e9, fun _ => by ext <;> rfl, fun _ => by ext <;> rfl\u27e9\n  let eY : { p : Z \u00d7 PUnit // f p.fst = Sum.inr p.snd } \u2243 { x : Z // f x = Sum.inr PUnit.unit } :=\n    \u27e8fun p => \u27e8p.1.1, p.2.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)\u27e9, fun x =>\n      \u27e8\u27e8_, _\u27e9, x.2\u27e9, fun _ => by ext <;> rfl, fun _ => by ext <;> rfl\u27e9\n  fapply binary_cofan.is_colimit_mk\n  \u00b7 refine' fun s =>\n      \u27e8fun x =>\n        dite _ (fun h => s.inl <| eX.symm \u27e8x, h\u27e9) fun h =>\n          s.inr <| eY.symm \u27e8x, (this x).resolve_left h\u27e9,\n        _\u27e9\n    rw [continuous_iff_continuousAt]\n    intro x\n    by_cases f x = Sum.inl PUnit.unit\n    \u00b7 revert h x\n      apply (IsOpen.continuousOn_iff _).mp\n      \u00b7 rw [continuousOn_iff_continuous_restrict]\n        convert_to Continuous fun x : { x | f x = Sum.inl PUnit.unit } =>\n            s.inl \u27e8(x, PUnit.unit), x.2\u27e9\n        \u00b7 ext \u27e8x, hx\u27e9\n          exact dif_pos hx\n        continuity\n      \u00b7 convert f.2.1 _ openEmbedding_inl.open_range\n        ext x\n        exact\n          \u27e8fun h => \u27e8_, h.symm\u27e9, fun \u27e8e, h\u27e9 =>\n            h.symm.trans (congr_arg Sum.inl <| Subsingleton.elim _ _)\u27e9\n    \u00b7 revert h x\n      apply (IsOpen.continuousOn_iff _).mp\n      \u00b7 rw [continuousOn_iff_continuous_restrict]\n        convert_to Continuous fun x : { x | f x \u2260 Sum.inl PUnit.unit } =>\n            s.inr \u27e8(x, PUnit.unit), (this _).resolve_left x.2\u27e9\n        \u00b7 ext \u27e8x, hx\u27e9\n          exact dif_neg hx\n        continuity\n      \u00b7 convert f.2.1 _ openEmbedding_inr.open_range\n        ext x\n        change f x \u2260 Sum.inl PUnit.unit \u2194 f x \u2208 Set.range Sum.inr\n        trans f x = Sum.inr PUnit.unit\n        \u00b7\n          rcases f x with (\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9) <;>\n            simp only [iff_self_iff, eq_self_iff_true, not_true, Ne.def, not_false_iff]\n        \u00b7\n          exact\n            \u27e8fun h => \u27e8_, h.symm\u27e9, fun \u27e8e, h\u27e9 =>\n              h.symm.trans (congr_arg Sum.inr <| Subsingleton.elim _ _)\u27e9\n  \u00b7 intro s\n    ext \u27e8\u27e8x, \u27e8\u27e9\u27e9, _\u27e9\n    change dite _ _ _ = _\n    split_ifs <;> rfl\n  \u00b7 intro s\n    ext \u27e8\u27e8x, \u27e8\u27e9\u27e9, hx\u27e9\n    change dite _ _ _ = _\n    split_ifs\n    \u00b7 cases h.symm.trans hx\n    \u00b7 rfl\n  \u00b7 intro s m e\u2081 e\u2082\n    ext x\n    change m x = dite _ _ _\n    split_ifs\n    \u00b7 rw [\u2190 e\u2081]\n      rfl\n    \u00b7 rw [\u2190 e\u2082]\n      rfl\n#align category_theory.finitary_extensive_Top_aux CategoryTheory.finitaryExtensiveTopAux\n\ninstance : FinitaryExtensive TopCat.{u} :=\n  by\n  rw [finitary_extensive_iff_of_is_terminal TopCat.{u} _ TopCat.isTerminalPunit _\n      (TopCat.binaryCofanIsColimit _ _)]\n  apply\n    binary_cofan.is_van_kampen_mk _ _ (fun X Y => TopCat.binaryCofanIsColimit X Y) _\n      fun X Y Z f g => TopCat.pullbackConeIsLimit f g\n  \u00b7 intros\n    constructor\n    \u00b7 refine' \u27e8\u27e8h\u03b1X.symm\u27e9, \u27e8pullback_cone.is_limit_aux' _ _\u27e9\u27e9\n      intro s\n      have : \u2200 x, \u2203! y, s.fst x = Sum.inl y := by\n        intro x\n        cases h : s.fst x\n        \u00b7 simp_rw [sum.inl_injective.eq_iff]\n          exact exists_unique_eq'\n        \u00b7 apply_fun f  at h\n          cases\n            ((concrete_category.congr_hom s.condition x).symm.trans h).trans\n              (concrete_category.congr_hom h\u03b1Y val : _).symm\n      delta ExistsUnique at this\n      choose l hl hl'\n      refine'\n        \u27e8\u27e8l, _\u27e9, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,\n          fun l' h\u2081 h\u2082 =>\n          ContinuousMap.ext fun x => hl' x (l' x) (concrete_category.congr_hom h\u2081 x).symm\u27e9\n      apply embedding_inl.to_inducing.continuous_iff.mpr\n      convert s.fst.2 using 1\n      exact (funext hl).symm\n    \u00b7 refine' \u27e8\u27e8h\u03b1Y.symm\u27e9, \u27e8pullback_cone.is_limit_aux' _ _\u27e9\u27e9\n      intro s\n      dsimp\n      have : \u2200 x, \u2203! y, s.fst x = Sum.inr y := by\n        intro x\n        cases h : s.fst x\n        \u00b7 apply_fun f  at h\n          cases\n            ((concrete_category.congr_hom s.condition x).symm.trans h).trans\n              (concrete_category.congr_hom h\u03b1X val : _).symm\n        \u00b7 simp_rw [sum.inr_injective.eq_iff]\n          exact exists_unique_eq'\n      delta ExistsUnique at this\n      choose l hl hl'\n      refine'\n        \u27e8\u27e8l, _\u27e9, ContinuousMap.ext fun a => (hl a).symm, Top.is_terminal_punit.hom_ext _ _,\n          fun l' h\u2081 h\u2082 =>\n          ContinuousMap.ext fun x => hl' x (l' x) (concrete_category.congr_hom h\u2081 x).symm\u27e9\n      apply embedding_inr.to_inducing.continuous_iff.mpr\n      convert s.fst.2 using 1\n      exact (funext hl).symm\n  \u00b7 intro Z f\n    exact finitary_extensive_Top_aux Z f\n\nend TopCat\n\nsection Functor\n\nuniverse v'' u''\n\nvariable {D : Type u''} [Category.{v''} D]\n\ntheorem NatTrans.Equifibered.whiskerRight {F G : J \u2964 C} {\u03b1 : F \u27f6 G} (h\u03b1 : \u03b1.Equifibered) (H : C \u2964 D)\n    [PreservesLimitsOfShape WalkingCospan H] : (whiskerRight \u03b1 H).Equifibered := fun i j f =>\n  (h\u03b1 f).map H\n#align category_theory.nat_trans.equifibered.whisker_right CategoryTheory.NatTrans.Equifibered.whiskerRight\n\ntheorem IsVanKampenColimit.of_iso {F : J \u2964 C} {c c' : Cocone F} (H : IsVanKampenColimit c)\n    (e : c \u2245 c') : IsVanKampenColimit c' :=\n  by\n  intro F' c'' \u03b1 f h h\u03b1\n  have : c'.\u03b9 \u226b (Functor.Const J).map e.inv.hom = c.\u03b9 :=\n    by\n    ext j\n    exact e.inv.2 j\n  rw [H c'' \u03b1 (f \u226b e.inv.1) (by rw [functor.map_comp, \u2190 reassoc_of h, this]) h\u03b1]\n  apply forall_congr'\n  intro j\n  conv_lhs => rw [\u2190 category.comp_id (\u03b1.app j)]\n  haveI : is_iso e.inv.hom := functor.map_is_iso (cocones.forget _) e.inv\n  exact (is_pullback.of_vert_is_iso \u27e8by simp\u27e9).paste_vert_iff (nat_trans.congr_app h j).symm\n#align category_theory.is_van_kampen_colimit.of_iso CategoryTheory.IsVanKampenColimit.of_iso\n\ntheorem IsVanKampenColimit.of_map {D : Type _} [Category D] (G : C \u2964 D) {F : J \u2964 C} {c : Cocone F}\n    [PreservesLimitsOfShape WalkingCospan G] [ReflectsLimitsOfShape WalkingCospan G]\n    [PreservesColimitsOfShape J G] [ReflectsColimitsOfShape J G]\n    (H : IsVanKampenColimit (G.mapCocone c)) : IsVanKampenColimit c :=\n  by\n  intro F' c' \u03b1 f h h\u03b1\n  refine'\n    (Iff.trans _\n          (H (G.map_cocone c') (whisker_right \u03b1 G) (G.map f)\n            (by\n              ext j\n              simpa using G.congr_map (nat_trans.congr_app h j))\n            (h\u03b1.whisker_right G))).trans\n      (forall_congr' fun j => _)\n  \u00b7 exact \u27e8fun h => \u27e8is_colimit_of_preserves G h.some\u27e9, fun h => \u27e8is_colimit_of_reflects G h.some\u27e9\u27e9\n  \u00b7 exact is_pullback.map_iff G (nat_trans.congr_app h.symm j)\n#align category_theory.is_van_kampen_colimit.of_map CategoryTheory.IsVanKampenColimit.of_map\n\ntheorem isVanKampenColimit_of_evaluation [HasPullbacks D] [HasColimitsOfShape J D] (F : J \u2964 C \u2964 D)\n    (c : Cocone F) (hc : \u2200 x : C, IsVanKampenColimit (((evaluation C D).obj x).mapCocone c)) :\n    IsVanKampenColimit c := by\n  intro F' c' \u03b1 f e h\u03b1\n  have := fun x =>\n    hc x (((evaluation C D).obj x).mapCocone c') (whisker_right \u03b1 _)\n      (((evaluation C D).obj x).map f)\n      (by\n        ext y\n        dsimp\n        exact nat_trans.congr_app (nat_trans.congr_app e y) x)\n      (h\u03b1.whisker_right _)\n  constructor\n  \u00b7 rintro \u27e8hc'\u27e9 j\n    refine' \u27e8\u27e8(nat_trans.congr_app e j).symm\u27e9, \u27e8evaluation_jointly_reflects_limits _ _\u27e9\u27e9\n    refine' fun x => (is_limit_map_cone_pullback_cone_equiv _ _).symm _\n    exact ((this x).mp \u27e8preserves_colimit.preserves hc'\u27e9 _).IsLimit\n  \u00b7\n    exact fun H =>\n      \u27e8evaluation_jointly_reflects_colimits _ fun x =>\n          ((this x).mpr fun j => (H j).map ((evaluation C D).obj x)).some\u27e9\n#align category_theory.is_van_kampen_colimit_of_evaluation CategoryTheory.isVanKampenColimit_of_evaluation\n\ninstance [HasPullbacks C] [FinitaryExtensive C] : FinitaryExtensive (D \u2964 C) :=\n  haveI : has_finite_coproducts (D \u2964 C) := \u27e8fun n => limits.functor_category_has_colimits_of_shape\u27e9\n  \u27e8fun X Y c hc =>\n    is_van_kampen_colimit_of_evaluation _ c fun x =>\n      finitary_extensive.van_kampen _ <| preserves_colimit.preserves hc\u27e9\n\ntheorem finitaryExtensive_of_preserves_and_reflects (F : C \u2964 D) [FinitaryExtensive D]\n    [HasFiniteCoproducts C] [PreservesLimitsOfShape WalkingCospan F]\n    [ReflectsLimitsOfShape WalkingCospan F] [PreservesColimitsOfShape (Discrete WalkingPair) F]\n    [ReflectsColimitsOfShape (Discrete WalkingPair) F] : FinitaryExtensive C :=\n  \u27e8fun X Y c hc => (FinitaryExtensive.van_kampen _ (isColimitOfPreserves F hc)).of_map F\u27e9\n#align category_theory.finitary_extensive_of_preserves_and_reflects CategoryTheory.finitaryExtensive_of_preserves_and_reflects\n\ntheorem finitaryExtensive_of_preserves_and_reflects_isomorphism (F : C \u2964 D) [FinitaryExtensive D]\n    [HasFiniteCoproducts C] [HasPullbacks C] [PreservesLimitsOfShape WalkingCospan F]\n    [PreservesColimitsOfShape (Discrete WalkingPair) F] [ReflectsIsomorphisms F] :\n    FinitaryExtensive C :=\n  by\n  haveI : reflects_limits_of_shape walking_cospan F :=\n    reflects_limits_of_shape_of_reflects_isomorphisms\n  haveI : reflects_colimits_of_shape (discrete walking_pair) F :=\n    reflects_colimits_of_shape_of_reflects_isomorphisms\n  exact finitary_extensive_of_preserves_and_reflects F\n#align category_theory.finitary_extensive_of_preserves_and_reflects_isomorphism CategoryTheory.finitaryExtensive_of_preserves_and_reflects_isomorphism\n\nend Functor\n\nend Extensive\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Extensive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.38813423238923944}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Scott Morrison\n\n! This file was ported from Lean 3 source module category_theory.epi_mono\n! leanprover-community/mathlib commit e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Opposites\nimport Mathbin.CategoryTheory.Groupoid\n\n/-!\n# Facts about epimorphisms and monomorphisms.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThe definitions of `epi` and `mono` are in `category_theory.category`,\nsince they are used by some lemmas for `iso`, which is used everywhere.\n-/\n\n\nuniverse v\u2081 v\u2082 u\u2081 u\u2082\n\nnamespace CategoryTheory\n\nvariable {C : Type u\u2081} [Category.{v\u2081} C]\n\n#print CategoryTheory.unop_mono_of_epi /-\ninstance unop_mono_of_epi {A B : C\u1d52\u1d56} (f : A \u27f6 B) [Epi f] : Mono f.unop :=\n  \u27e8fun Z g h eq => Quiver.Hom.op_inj ((cancel_epi f).1 (Quiver.Hom.unop_inj Eq))\u27e9\n#align category_theory.unop_mono_of_epi CategoryTheory.unop_mono_of_epi\n-/\n\n#print CategoryTheory.unop_epi_of_mono /-\ninstance unop_epi_of_mono {A B : C\u1d52\u1d56} (f : A \u27f6 B) [Mono f] : Epi f.unop :=\n  \u27e8fun Z g h eq => Quiver.Hom.op_inj ((cancel_mono f).1 (Quiver.Hom.unop_inj Eq))\u27e9\n#align category_theory.unop_epi_of_mono CategoryTheory.unop_epi_of_mono\n-/\n\n#print CategoryTheory.op_mono_of_epi /-\ninstance op_mono_of_epi {A B : C} (f : A \u27f6 B) [Epi f] : Mono f.op :=\n  \u27e8fun Z g h eq => Quiver.Hom.unop_inj ((cancel_epi f).1 (Quiver.Hom.op_inj Eq))\u27e9\n#align category_theory.op_mono_of_epi CategoryTheory.op_mono_of_epi\n-/\n\n#print CategoryTheory.op_epi_of_mono /-\ninstance op_epi_of_mono {A B : C} (f : A \u27f6 B) [Mono f] : Epi f.op :=\n  \u27e8fun Z g h eq => Quiver.Hom.unop_inj ((cancel_mono f).1 (Quiver.Hom.op_inj Eq))\u27e9\n#align category_theory.op_epi_of_mono CategoryTheory.op_epi_of_mono\n-/\n\n#print CategoryTheory.SplitMono /-\n/-- A split monomorphism is a morphism `f : X \u27f6 Y` with a given retraction `retraction f : Y \u27f6 X`\nsuch that `f \u226b retraction f = \ud835\udfd9 X`.\n\nEvery split monomorphism is a monomorphism.\n-/\n@[ext, nolint has_nonempty_instance]\nstructure SplitMono {X Y : C} (f : X \u27f6 Y) where\n  retraction : Y \u27f6 X\n  id' : f \u226b retraction = \ud835\udfd9 X := by obviously\n#align category_theory.split_mono CategoryTheory.SplitMono\n-/\n\nrestate_axiom split_mono.id'\n\nattribute [simp, reassoc.1] split_mono.id\n\n#print CategoryTheory.IsSplitMono /-\n/-- `is_split_mono f` is the assertion that `f` admits a retraction -/\nclass IsSplitMono {X Y : C} (f : X \u27f6 Y) : Prop where\n  exists_splitMono : Nonempty (SplitMono f)\n#align category_theory.is_split_mono CategoryTheory.IsSplitMono\n-/\n\n#print CategoryTheory.IsSplitMono.mk' /-\n/-- A constructor for `is_split_mono f` taking a `split_mono f` as an argument -/\ntheorem IsSplitMono.mk' {X Y : C} {f : X \u27f6 Y} (sm : SplitMono f) : IsSplitMono f :=\n  \u27e8Nonempty.intro sm\u27e9\n#align category_theory.is_split_mono.mk' CategoryTheory.IsSplitMono.mk'\n-/\n\n#print CategoryTheory.SplitEpi /-\n/-- A split epimorphism is a morphism `f : X \u27f6 Y` with a given section `section_ f : Y \u27f6 X`\nsuch that `section_ f \u226b f = \ud835\udfd9 Y`.\n(Note that `section` is a reserved keyword, so we append an underscore.)\n\nEvery split epimorphism is an epimorphism.\n-/\n@[ext, nolint has_nonempty_instance]\nstructure SplitEpi {X Y : C} (f : X \u27f6 Y) where\n  section_ : Y \u27f6 X\n  id' : section_ \u226b f = \ud835\udfd9 Y := by obviously\n#align category_theory.split_epi CategoryTheory.SplitEpi\n-/\n\nrestate_axiom split_epi.id'\n\nattribute [simp, reassoc.1] split_epi.id\n\n#print CategoryTheory.IsSplitEpi /-\n/-- `is_split_epi f` is the assertion that `f` admits a section -/\nclass IsSplitEpi {X Y : C} (f : X \u27f6 Y) : Prop where\n  exists_splitEpi : Nonempty (SplitEpi f)\n#align category_theory.is_split_epi CategoryTheory.IsSplitEpi\n-/\n\n#print CategoryTheory.IsSplitEpi.mk' /-\n/-- A constructor for `is_split_epi f` taking a `split_epi f` as an argument -/\ntheorem IsSplitEpi.mk' {X Y : C} {f : X \u27f6 Y} (se : SplitEpi f) : IsSplitEpi f :=\n  \u27e8Nonempty.intro se\u27e9\n#align category_theory.is_split_epi.mk' CategoryTheory.IsSplitEpi.mk'\n-/\n\n#print CategoryTheory.retraction /-\n/-- The chosen retraction of a split monomorphism. -/\nnoncomputable def retraction {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f] : Y \u27f6 X :=\n  hf.exists_splitMono.some.retraction\n#align category_theory.retraction CategoryTheory.retraction\n-/\n\n#print CategoryTheory.IsSplitMono.id /-\n@[simp, reassoc.1]\ntheorem IsSplitMono.id {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f] : f \u226b retraction f = \ud835\udfd9 X :=\n  hf.exists_splitMono.some.id\n#align category_theory.is_split_mono.id CategoryTheory.IsSplitMono.id\n-/\n\n#print CategoryTheory.SplitMono.splitEpi /-\n/-- The retraction of a split monomorphism has an obvious section. -/\ndef SplitMono.splitEpi {X Y : C} {f : X \u27f6 Y} (sm : SplitMono f) : SplitEpi sm.retraction\n    where section_ := f\n#align category_theory.split_mono.split_epi CategoryTheory.SplitMono.splitEpi\n-/\n\n#print CategoryTheory.retraction_isSplitEpi /-\n/-- The retraction of a split monomorphism is itself a split epimorphism. -/\ninstance retraction_isSplitEpi {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f] :\n    IsSplitEpi (retraction f) :=\n  IsSplitEpi.mk' (SplitMono.splitEpi _)\n#align category_theory.retraction_is_split_epi CategoryTheory.retraction_isSplitEpi\n-/\n\n#print CategoryTheory.isIso_of_epi_of_isSplitMono /-\n/-- A split mono which is epi is an iso. -/\ntheorem isIso_of_epi_of_isSplitMono {X Y : C} (f : X \u27f6 Y) [IsSplitMono f] [Epi f] : IsIso f :=\n  \u27e8\u27e8retraction f, \u27e8by simp, by simp [\u2190 cancel_epi f]\u27e9\u27e9\u27e9\n#align category_theory.is_iso_of_epi_of_is_split_mono CategoryTheory.isIso_of_epi_of_isSplitMono\n-/\n\n#print CategoryTheory.section_ /-\n/-- The chosen section of a split epimorphism.\n(Note that `section` is a reserved keyword, so we append an underscore.)\n-/\nnoncomputable def section_ {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] : Y \u27f6 X :=\n  hf.exists_splitEpi.some.section_\n#align category_theory.section_ CategoryTheory.section_\n-/\n\n#print CategoryTheory.IsSplitEpi.id /-\n@[simp, reassoc.1]\ntheorem IsSplitEpi.id {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] : section_ f \u226b f = \ud835\udfd9 Y :=\n  hf.exists_splitEpi.some.id\n#align category_theory.is_split_epi.id CategoryTheory.IsSplitEpi.id\n-/\n\n#print CategoryTheory.SplitEpi.splitMono /-\n/-- The section of a split epimorphism has an obvious retraction. -/\ndef SplitEpi.splitMono {X Y : C} {f : X \u27f6 Y} (se : SplitEpi f) : SplitMono se.section_\n    where retraction := f\n#align category_theory.split_epi.split_mono CategoryTheory.SplitEpi.splitMono\n-/\n\n#print CategoryTheory.section_isSplitMono /-\n/-- The section of a split epimorphism is itself a split monomorphism. -/\ninstance section_isSplitMono {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] : IsSplitMono (section_ f) :=\n  IsSplitMono.mk' (SplitEpi.splitMono _)\n#align category_theory.section_is_split_mono CategoryTheory.section_isSplitMono\n-/\n\n#print CategoryTheory.isIso_of_mono_of_isSplitEpi /-\n/-- A split epi which is mono is an iso. -/\ntheorem isIso_of_mono_of_isSplitEpi {X Y : C} (f : X \u27f6 Y) [Mono f] [IsSplitEpi f] : IsIso f :=\n  \u27e8\u27e8section_ f, \u27e8by simp [\u2190 cancel_mono f], by simp\u27e9\u27e9\u27e9\n#align category_theory.is_iso_of_mono_of_is_split_epi CategoryTheory.isIso_of_mono_of_isSplitEpi\n-/\n\n#print CategoryTheory.IsSplitMono.of_iso /-\n/-- Every iso is a split mono. -/\ninstance (priority := 100) IsSplitMono.of_iso {X Y : C} (f : X \u27f6 Y) [IsIso f] : IsSplitMono f :=\n  IsSplitMono.mk' { retraction := inv f }\n#align category_theory.is_split_mono.of_iso CategoryTheory.IsSplitMono.of_iso\n-/\n\n#print CategoryTheory.IsSplitEpi.of_iso /-\n/-- Every iso is a split epi. -/\ninstance (priority := 100) IsSplitEpi.of_iso {X Y : C} (f : X \u27f6 Y) [IsIso f] : IsSplitEpi f :=\n  IsSplitEpi.mk' { section_ := inv f }\n#align category_theory.is_split_epi.of_iso CategoryTheory.IsSplitEpi.of_iso\n-/\n\n#print CategoryTheory.SplitMono.mono /-\ntheorem SplitMono.mono {X Y : C} {f : X \u27f6 Y} (sm : SplitMono f) : Mono f :=\n  { right_cancellation := fun Z g h w => by replace w := w =\u226b sm.retraction; simpa using w }\n#align category_theory.split_mono.mono CategoryTheory.SplitMono.mono\n-/\n\n#print CategoryTheory.IsSplitMono.mono /-\n/-- Every split mono is a mono. -/\ninstance (priority := 100) IsSplitMono.mono {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f] : Mono f :=\n  hf.exists_splitMono.some.Mono\n#align category_theory.is_split_mono.mono CategoryTheory.IsSplitMono.mono\n-/\n\n#print CategoryTheory.SplitEpi.epi /-\ntheorem SplitEpi.epi {X Y : C} {f : X \u27f6 Y} (se : SplitEpi f) : Epi f :=\n  { left_cancellation := fun Z g h w => by replace w := se.section_ \u226b= w; simpa using w }\n#align category_theory.split_epi.epi CategoryTheory.SplitEpi.epi\n-/\n\n#print CategoryTheory.IsSplitEpi.epi /-\n/-- Every split epi is an epi. -/\ninstance (priority := 100) IsSplitEpi.epi {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] : Epi f :=\n  hf.exists_splitEpi.some.Epi\n#align category_theory.is_split_epi.epi CategoryTheory.IsSplitEpi.epi\n-/\n\n#print CategoryTheory.IsIso.of_mono_retraction' /-\n/-- Every split mono whose retraction is mono is an iso. -/\ntheorem IsIso.of_mono_retraction' {X Y : C} {f : X \u27f6 Y} (hf : SplitMono f) [Mono <| hf.retraction] :\n    IsIso f :=\n  \u27e8\u27e8hf.retraction, \u27e8by simp, (cancel_mono_id <| hf.retraction).mp (by simp)\u27e9\u27e9\u27e9\n#align category_theory.is_iso.of_mono_retraction' CategoryTheory.IsIso.of_mono_retraction'\n-/\n\n#print CategoryTheory.IsIso.of_mono_retraction /-\n/-- Every split mono whose retraction is mono is an iso. -/\ntheorem IsIso.of_mono_retraction {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f]\n    [hf' : Mono <| retraction f] : IsIso f :=\n  @IsIso.of_mono_retraction' _ _ _ _ _ hf.exists_splitMono.some hf'\n#align category_theory.is_iso.of_mono_retraction CategoryTheory.IsIso.of_mono_retraction\n-/\n\n#print CategoryTheory.IsIso.of_epi_section' /-\n/-- Every split epi whose section is epi is an iso. -/\ntheorem IsIso.of_epi_section' {X Y : C} {f : X \u27f6 Y} (hf : SplitEpi f) [Epi <| hf.section_] :\n    IsIso f :=\n  \u27e8\u27e8hf.section_, \u27e8(cancel_epi_id <| hf.section_).mp (by simp), by simp\u27e9\u27e9\u27e9\n#align category_theory.is_iso.of_epi_section' CategoryTheory.IsIso.of_epi_section'\n-/\n\n#print CategoryTheory.IsIso.of_epi_section /-\n/-- Every split epi whose section is epi is an iso. -/\ntheorem IsIso.of_epi_section {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] [hf' : Epi <| section_ f] :\n    IsIso f :=\n  @IsIso.of_epi_section' _ _ _ _ _ hf.exists_splitEpi.some hf'\n#align category_theory.is_iso.of_epi_section CategoryTheory.IsIso.of_epi_section\n-/\n\n#print CategoryTheory.Groupoid.ofTruncSplitMono /-\n-- FIXME this has unnecessarily become noncomputable!\n/-- A category where every morphism has a `trunc` retraction is computably a groupoid. -/\nnoncomputable def Groupoid.ofTruncSplitMono\n    (all_split_mono : \u2200 {X Y : C} (f : X \u27f6 Y), Trunc (IsSplitMono f)) : Groupoid.{v\u2081} C :=\n  by\n  apply groupoid.of_is_iso\n  intro X Y f\n  trunc_cases all_split_mono f\n  trunc_cases all_split_mono (retraction f)\n  apply is_iso.of_mono_retraction\n#align category_theory.groupoid.of_trunc_split_mono CategoryTheory.Groupoid.ofTruncSplitMono\n-/\n\nsection\n\nvariable (C)\n\n#print CategoryTheory.SplitMonoCategory /-\n/-- A split mono category is a category in which every monomorphism is split. -/\nclass SplitMonoCategory where\n  isSplitMono_of_mono : \u2200 {X Y : C} (f : X \u27f6 Y) [Mono f], IsSplitMono f\n#align category_theory.split_mono_category CategoryTheory.SplitMonoCategory\n-/\n\n#print CategoryTheory.SplitEpiCategory /-\n/-- A split epi category is a category in which every epimorphism is split. -/\nclass SplitEpiCategory where\n  isSplitEpi_of_epi : \u2200 {X Y : C} (f : X \u27f6 Y) [Epi f], IsSplitEpi f\n#align category_theory.split_epi_category CategoryTheory.SplitEpiCategory\n-/\n\nend\n\n#print CategoryTheory.isSplitMono_of_mono /-\n/-- In a category in which every monomorphism is split, every monomorphism splits. This is not an\n    instance because it would create an instance loop. -/\ntheorem isSplitMono_of_mono [SplitMonoCategory C] {X Y : C} (f : X \u27f6 Y) [Mono f] : IsSplitMono f :=\n  SplitMonoCategory.isSplitMono_of_mono _\n#align category_theory.is_split_mono_of_mono CategoryTheory.isSplitMono_of_mono\n-/\n\n#print CategoryTheory.isSplitEpi_of_epi /-\n/-- In a category in which every epimorphism is split, every epimorphism splits. This is not an\n    instance because it would create an instance loop. -/\ntheorem isSplitEpi_of_epi [SplitEpiCategory C] {X Y : C} (f : X \u27f6 Y) [Epi f] : IsSplitEpi f :=\n  SplitEpiCategory.isSplitEpi_of_epi _\n#align category_theory.is_split_epi_of_epi CategoryTheory.isSplitEpi_of_epi\n-/\n\nsection\n\nvariable {D : Type u\u2082} [Category.{v\u2082} D]\n\n/- warning: category_theory.split_mono.map -> CategoryTheory.SplitMono.map is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y}, (CategoryTheory.SplitMono.{u1, u3} C _inst_1 X Y f) -> (forall (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2), CategoryTheory.SplitMono.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f))\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y}, (CategoryTheory.SplitMono.{u1, u3} C _inst_1 X Y f) -> (forall (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2), CategoryTheory.SplitMono.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f))\nCase conversion may be inaccurate. Consider using '#align category_theory.split_mono.map CategoryTheory.SplitMono.map\u2093'. -/\n/-- Split monomorphisms are also absolute monomorphisms. -/\n@[simps]\ndef SplitMono.map {X Y : C} {f : X \u27f6 Y} (sm : SplitMono f) (F : C \u2964 D) : SplitMono (F.map f)\n    where\n  retraction := F.map sm.retraction\n  id' := by rw [\u2190 functor.map_comp, split_mono.id, Functor.map_id]\n#align category_theory.split_mono.map CategoryTheory.SplitMono.map\n\n/- warning: category_theory.split_epi.map -> CategoryTheory.SplitEpi.map is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y}, (CategoryTheory.SplitEpi.{u1, u3} C _inst_1 X Y f) -> (forall (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2), CategoryTheory.SplitEpi.{u2, u4} D _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X) (CategoryTheory.Functor.obj.{u1, u2, u3, u4} C _inst_1 D _inst_2 F Y) (CategoryTheory.Functor.map.{u1, u2, u3, u4} C _inst_1 D _inst_2 F X Y f))\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_2 : CategoryTheory.Category.{u2, u4} D] {X : C} {Y : C} {f : Quiver.Hom.{succ u1, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) X Y}, (CategoryTheory.SplitEpi.{u1, u3} C _inst_1 X Y f) -> (forall (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_1 D _inst_2), CategoryTheory.SplitEpi.{u2, u4} D _inst_2 (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X) (Prefunctor.obj.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) Y) (Prefunctor.map.{succ u1, succ u2, u3, u4} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} C (CategoryTheory.Category.toCategoryStruct.{u1, u3} C _inst_1)) D (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} D (CategoryTheory.Category.toCategoryStruct.{u2, u4} D _inst_2)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u3, u4} C _inst_1 D _inst_2 F) X Y f))\nCase conversion may be inaccurate. Consider using '#align category_theory.split_epi.map CategoryTheory.SplitEpi.map\u2093'. -/\n/-- Split epimorphisms are also absolute epimorphisms. -/\n@[simps]\ndef SplitEpi.map {X Y : C} {f : X \u27f6 Y} (se : SplitEpi f) (F : C \u2964 D) : SplitEpi (F.map f)\n    where\n  section_ := F.map se.section_\n  id' := by rw [\u2190 functor.map_comp, split_epi.id, Functor.map_id]\n#align category_theory.split_epi.map CategoryTheory.SplitEpi.map\n\ninstance {X Y : C} (f : X \u27f6 Y) [hf : IsSplitMono f] (F : C \u2964 D) : IsSplitMono (F.map f) :=\n  IsSplitMono.mk' (hf.exists_splitMono.some.map F)\n\ninstance {X Y : C} (f : X \u27f6 Y) [hf : IsSplitEpi f] (F : C \u2964 D) : IsSplitEpi (F.map f) :=\n  IsSplitEpi.mk' (hf.exists_splitEpi.some.map F)\n\nend\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/EpiMono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6548947223065754, "lm_q1q2_score": 0.38813422839730277}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen, Paul Lezeau\n-/\n\nimport ring_theory.algebra_tower\nimport ring_theory.dedekind_domain.ideal\nimport ring_theory.is_adjoin_root\n\n/-!\n# Kummer-Dedekind theorem\n\nThis file proves the monogenic version of the Kummer-Dedekind theorem on the splitting of prime\nideals in an extension of the ring of integers. This states that if `I` is a prime ideal of\nDedekind domain `R` and `S = R[\u03b1]` for some `\u03b1` that is integral over `R` with minimal polynomial\n`f`, then the prime factorisations of `I * S` and `f mod I` have the same shape, i.e. they have the\nsame number of prime factors, and each prime factors of `I * S` can be paired with a prime factor\nof `f mod I` in a way that ensures multiplicities match (in fact, this pairing can be made explicit\nwith a formula).\n\n## Main definitions\n\n * `normalized_factors_map_equiv_normalized_factors_min_poly_mk` : The bijection in the\n    Kummer-Dedekind theorem. This is the pairing between the prime factors of `I * S` and the prime\n    factors of `f mod I`.\n\n## Main results\n\n * `normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map` : The Kummer-Dedekind\n    theorem.\n * `ideal.irreducible_map_of_irreducible_minpoly` : `I.map (algebra_map R S)` is irreducible if\n    `(map I^.quotient.mk (minpoly R pb.gen))` is irreducible, where `pb` is a power basis of `S`\n    over `R`.\n\n## TODO\n\n * Prove the Kummer-Dedekind theorem in full generality.\n\n * Prove the converse of `ideal.irreducible_map_of_irreducible_minpoly`.\n\n * Prove that `normalized_factors_map_equiv_normalized_factors_min_poly_mk` can be expressed as\n    `normalized_factors_map_equiv_normalized_factors_min_poly_mk g = \u27e8I, G(\u03b1)\u27e9` for `g` a prime\n    factor of `f mod I` and `G` a lift of `g` to `R[X]`.\n\n## References\n\n * [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\nkummer, dedekind, kummer dedekind, dedekind-kummer, dedekind kummer\n-/\n\nvariables (R : Type*) {S : Type*} [comm_ring R] [comm_ring S] [algebra R S]\n\nopen ideal polynomial double_quot unique_factorization_monoid algebra ring_hom\n\nlocal notation R`<`:std.prec.max_plus x `>` := adjoin R ({x} : set S)\n\n/-- Let `S / R` be a ring extension and `x : S`, then the conductor of `R<x>` is the\n    biggest ideal of `S` contained in `R<x>`. -/\ndef conductor (x : S) : ideal S :=\n{ carrier := {a | \u2200 (b : S), a * b \u2208 R<x> },\n  zero_mem' := \u03bb b, by simpa only [zero_mul] using subalgebra.zero_mem _,\n  add_mem' := \u03bb a b ha hb c, by simpa only [add_mul] using subalgebra.add_mem _ (ha c) (hb c),\n  smul_mem' := \u03bb c a ha b, by simpa only [smul_eq_mul, mul_left_comm, mul_assoc] using ha (c * b) }\n\nvariables {R} {x : S}\n\nlemma conductor_eq_of_eq {y : S} (h : (R<x> : set S) = R<y>):\n  conductor R x = conductor R y :=\nideal.ext $ \u03bb a, forall_congr $ \u03bb b, set.ext_iff.mp h _\n\nlemma conductor_subset_adjoin : (conductor R x : set S) \u2286 R<x> :=\n\u03bb y hy, by simpa only [mul_one] using hy 1\n\nlemma mem_conductor_iff {y : S} : y \u2208 conductor R x \u2194 \u2200 (b : S), y * b \u2208 R<x> :=\n\u27e8\u03bb h, h, \u03bb h, h\u27e9\n\nlemma conductor_eq_top_of_adjoin_eq_top (h : R<x> = \u22a4) : conductor R x = \u22a4 :=\nby simp only [ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]\n\nlemma conductor_eq_top_of_power_basis (pb : power_basis R S) : conductor R pb.gen = \u22a4 :=\nconductor_eq_top_of_adjoin_eq_top pb.adjoin_gen_eq_top\n\nvariables {I : ideal R}\n\n/-- This technical lemma tell us that if `C` is the conductor of `R<x>` and `I` is an ideal of `R`\n  then `p * (I * S) \u2286 I * R<x>` for any `p` in `C \u2229 R` -/\nlemma prod_mem_ideal_map_of_mem_conductor {p : R} {z : S}\n  (hp : p \u2208 ideal.comap (algebra_map R S) (conductor R x)) (hz' : z \u2208 (I.map (algebra_map R S))) :\n  (algebra_map R S p) * z \u2208\n    algebra_map R<x> S '' \u2191(I.map (algebra_map R R<x>)) :=\nbegin\n  rw [ideal.map, ideal.span, finsupp.mem_span_image_iff_total] at hz',\n  obtain \u27e8l, H, H'\u27e9 := hz',\n  rw finsupp.total_apply at H',\n  rw [\u2190 H', mul_comm, finsupp.sum_mul],\n  have lem : \u2200 {a : R}, a \u2208 I \u2192 (l a \u2022 (algebra_map R S a) * (algebra_map R S p)) \u2208\n    (algebra_map R<x> S) '' (I.map (algebra_map R R<x>)),\n  { intros a ha,\n    rw [algebra.id.smul_eq_mul, mul_assoc, mul_comm, mul_assoc, set.mem_image],\n    refine exists.intro (algebra_map R R<x> a * \u27e8l a * algebra_map R S p,\n      show l a * algebra_map R S p \u2208 R<x>, from _ \u27e9) _,\n    { rw mul_comm,\n      exact mem_conductor_iff.mp (ideal.mem_comap.mp hp) _ },\n    refine \u27e8_, by simpa only [ring_hom.map_mul, mul_comm (algebra_map R S p) (l a)]\u27e9,\n    rw mul_comm,\n    apply ideal.mul_mem_left (I.map (algebra_map R R<x>)) _\n      (ideal.mem_map_of_mem _ ha) },\n  refine finset.sum_induction _ (\u03bb u, u \u2208 (algebra_map R<x> S) ''\n    (I.map (algebra_map R R<x>)))\n    (\u03bb a b, _) _ _,\n  rintro \u27e8z, hz, rfl\u27e9 \u27e8y, hy, rfl\u27e9,\n  rw [\u2190 ring_hom.map_add],\n  exact \u27e8z + y, ideal.add_mem _ (set_like.mem_coe.mp hz) hy, rfl\u27e9,\n  { refine \u27e80, set_like.mem_coe.mpr $ ideal.zero_mem _, ring_hom.map_zero _\u27e9 },\n  { intros y hy,\n    exact lem ((finsupp.mem_supported _ l).mp H hy) },\nend\n\n/-- A technical result telling us that `(I * S) \u2229 R<x> = I * R<x>` for any ideal `I` of `R`. -/\nlemma comap_map_eq_map_adjoin_of_coprime_conductor\n  (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (h_alg : function.injective (algebra_map R<x> S)):\n  (I.map (algebra_map R S)).comap (algebra_map R<x> S) = I.map (algebra_map R R<x>) :=\nbegin\n  apply le_antisymm,\n  { -- This is adapted from [Neukirch1992]. Let `C = (conductor R x)`. The idea of the proof\n    -- is that since `I` and `C \u2229 R` are coprime, we have\n    -- `(I * S) \u2229 R<x> \u2286 (I + C) * ((I * S) \u2229 R<x>) \u2286 I * R<x> + I * C * S \u2286 I * R<x>`.\n    intros y hy,\n    obtain \u27e8z, hz\u27e9 := y,\n    obtain \u27e8p, hp, q, hq, hpq\u27e9 := submodule.mem_sup.mp ((ideal.eq_top_iff_one _).mp hx),\n    have temp : (algebra_map R S p)*z + (algebra_map R S q)*z = z,\n    { simp only [\u2190add_mul, \u2190ring_hom.map_add (algebra_map R S), hpq, map_one, one_mul] },\n    suffices : z \u2208 algebra_map R<x> S '' (I.map (algebra_map R R<x>)) \u2194 (\u27e8z, hz\u27e9 : R<x>) \u2208\n      I.map (algebra_map R R<x>),\n    { rw [\u2190 this, \u2190 temp],\n      obtain \u27e8a, ha\u27e9 := (set.mem_image _ _ _).mp (prod_mem_ideal_map_of_mem_conductor hp\n        (show z \u2208 I.map (algebra_map R S), by rwa ideal.mem_comap at hy )),\n      use a + (algebra_map R R<x> q) * \u27e8z, hz\u27e9,\n      refine \u27e8 ideal.add_mem (I.map (algebra_map R R<x>)) ha.left _,\n        by simpa only [ha.right, map_add, alg_hom.map_mul, add_right_inj] \u27e9,\n      rw mul_comm,\n        exact ideal.mul_mem_left (I.map (algebra_map R R<x>)) _ (ideal.mem_map_of_mem _ hq) },\n    refine \u27e8 \u03bb h, _, \u03bb h, (set.mem_image _ _ _).mpr (exists.intro \u27e8z, hz\u27e9 \u27e8by simp [h], rfl\u27e9 ) \u27e9,\n    { obtain \u27e8x\u2081, hx\u2081, hx\u2082\u27e9 := (set.mem_image _ _ _).mp h,\n      have : x\u2081 = \u27e8z, hz\u27e9,\n      { apply h_alg,\n        simpa [hx\u2082], },\n      rwa \u2190 this }  },\n\n  { -- The converse inclusion is trivial\n    have : algebra_map R S = (algebra_map _ S).comp (algebra_map R R<x>) := by { ext, refl },\n    rw [this, \u2190 ideal.map_map],\n    apply ideal.le_comap_map }\nend\n\n/-- The canonical morphism of rings from `R<x> \u29f8 (I*R<x>)` to `S \u29f8 (I*S)` is an isomorphism\n    when `I` and `(conductor R x) \u2229 R` are coprime. -/\nnoncomputable def quot_adjoin_equiv_quot_map (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (h_alg : function.injective (algebra_map R<x> S)) :\n  R<x> \u29f8 (I.map (algebra_map R R<x>)) \u2243+* S \u29f8 (I.map (algebra_map R S)) :=\nring_equiv.of_bijective (ideal.quotient.lift (I.map (algebra_map R R<x>))\n  (((I.map (algebra_map R S))^.quotient.mk).comp (algebra_map R<x> S )) (\u03bb r hr,\n    begin\n      have : algebra_map R S = (algebra_map R<x> S).comp\n        (algebra_map R R<x>) := by { ext, refl },\n      rw [ring_hom.comp_apply, ideal.quotient.eq_zero_iff_mem, this, \u2190 ideal.map_map],\n      exact ideal.mem_map_of_mem _ hr\n    end))\nbegin\n  split,\n  { --the kernel of the map is clearly `(I * S) \u2229 R<x>`. To get injectivity, we need to show that\n    --this is contained in `I * R<x>`, which is the content of the previous lemma.\n    refine ring_hom.lift_injective_of_ker_le_ideal _ _ (\u03bb u hu, _),\n    rwa [ring_hom.mem_ker, ring_hom.comp_apply, ideal.quotient.eq_zero_iff_mem,\n      \u2190 ideal.mem_comap, comap_map_eq_map_adjoin_of_coprime_conductor hx h_alg] at hu },\n  { -- Surjectivity follows from the surjectivity of the canonical map `R<x> \u2192 S \u29f8 (I * S)`,\n    -- which in turn follows from the fact that `I * S + (conductor R x) = S`.\n    refine ideal.quotient.lift_surjective_of_surjective _ _ (\u03bb y, _),\n    obtain \u27e8z, hz\u27e9 := ideal.quotient.mk_surjective y,\n    have : z \u2208 conductor R x \u2294 (I.map (algebra_map R S)),\n    { suffices : conductor R x \u2294 (I.map (algebra_map R S)) = \u22a4,\n      { simp only [this] },\n      rw ideal.eq_top_iff_one at hx \u22a2,\n      replace hx := ideal.mem_map_of_mem (algebra_map R S) hx,\n      rw [ideal.map_sup, ring_hom.map_one] at hx,\n      exact (sup_le_sup (show  ((conductor R x).comap (algebra_map R S)).map (algebra_map R S) \u2264\n        conductor R x, from ideal.map_comap_le) (le_refl (I.map (algebra_map R S)))) hx },\n    rw [\u2190 ideal.mem_quotient_iff_mem_sup, hz, ideal.mem_map_iff_of_surjective] at this,\n    obtain \u27e8u, hu, hu'\u27e9 := this,\n    use \u27e8u, conductor_subset_adjoin hu\u27e9,\n    simpa only [\u2190 hu'],\n    { exact ideal.quotient.mk_surjective } }\nend\n\n@[simp]\nlemma quot_adjoin_equiv_quot_map_apply_mk (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (h_alg : function.injective (algebra_map R<x> S)) (a : R<x>) :\n   quot_adjoin_equiv_quot_map hx h_alg ((I.map (algebra_map R R<x>))^.quotient.mk a)\n   = (I.map (algebra_map R S))^.quotient.mk \u2191a :=\nrfl\n\nnamespace kummer_dedekind\n\nopen_locale big_operators polynomial classical\n\nvariables [is_domain R] [is_integrally_closed R]\nvariables [is_domain S] [is_dedekind_domain S]\nvariable [no_zero_smul_divisors R S]\n\nlocal attribute [instance] ideal.quotient.field\n\n/-- The first half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the prime\n    factors of `I*S` are in bijection with those of the minimal polynomial of the generator of `S`\n    over `R`, taken `mod I`.-/\nnoncomputable def normalized_factors_map_equiv_normalized_factors_min_poly_mk (hI : is_maximal I)\n  (hI' : I \u2260 \u22a5) (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (hx' : is_integral R x) :\n  {J : ideal S | J \u2208 normalized_factors (I.map (algebra_map R S) )} \u2243\n    {d : (R \u29f8 I)[X] | d \u2208 normalized_factors (map I^.quotient.mk (minpoly R x))} :=\n(normalized_factors_equiv_of_quot_equiv\n  ((quot_adjoin_equiv_quot_map hx\n    (by { apply no_zero_smul_divisors.algebra_map_injective (algebra.adjoin R {x}) S,\n          exact subalgebra.no_zero_smul_divisors_top (algebra.adjoin R {x}) })).symm.trans\n  (((algebra.adjoin.power_basis' hx').quotient_equiv_quotient_minpoly_map I).to_ring_equiv.trans\n    (quot_equiv_of_eq (show (ideal.span ({(minpoly R (algebra.adjoin.power_basis' hx').gen).map\n    I^.quotient.mk})) = (ideal.span ({(minpoly R x).map I^.quotient.mk})),\n      by rw algebra.adjoin.power_basis'_minpoly_gen hx'))))\n  --show that `I * S` \u2260 \u22a5\n  (show I.map (algebra_map R S) \u2260 \u22a5,\n    by rwa [ne.def, map_eq_bot_iff_of_injective (no_zero_smul_divisors.algebra_map_injective R S),\n         \u2190 ne.def])\n  --show that the ideal spanned by `(minpoly R pb.gen) mod I` is non-zero\n  (by {by_contra, exact (show (map I^.quotient.mk (minpoly R x) \u2260 0), from\n    polynomial.map_monic_ne_zero (minpoly.monic hx')) (span_singleton_eq_bot.mp h) } )).trans\n  (normalized_factors_equiv_span_normalized_factors\n    (show (map I^.quotient.mk (minpoly R x)) \u2260 0, from\n      polynomial.map_monic_ne_zero (minpoly.monic hx'))).symm\n\n/-- The second half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the\n    bijection `factors_equiv'` defined in the first half preserves multiplicities. -/\ntheorem multiplicity_factors_map_eq_multiplicity (hI : is_maximal I) (hI' : I \u2260 \u22a5)\n  (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4) (hx' : is_integral R x)\n  {J : ideal S} (hJ : J \u2208 normalized_factors (I.map (algebra_map R S))) :\n  multiplicity J (I.map (algebra_map R S)) =\n    multiplicity \u2191(normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx'\n      \u27e8J, hJ\u27e9) (map I^.quotient.mk (minpoly R x)) :=\nby rw [normalized_factors_map_equiv_normalized_factors_min_poly_mk, equiv.coe_trans,\n       function.comp_app,\n       multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity,\n       normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity]\n\n/-- The **Kummer-Dedekind Theorem**. -/\ntheorem normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map (hI : is_maximal I)\n  (hI' : I \u2260 \u22a5) (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (hx' : is_integral R x) :\n  normalized_factors (I.map (algebra_map R S)) =\n    multiset.map\n      (\u03bb f, ((normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx').symm f :\n        ideal S))\n      (normalized_factors (polynomial.map I^.quotient.mk (minpoly R x))).attach :=\nbegin\n  ext J,\n  -- WLOG, assume J is a normalized factor\n  by_cases hJ : J \u2208 normalized_factors (I.map (algebra_map R S)), swap,\n  { rw [multiset.count_eq_zero.mpr hJ, eq_comm, multiset.count_eq_zero, multiset.mem_map],\n    simp only [multiset.mem_attach, true_and, not_exists],\n    rintros J' rfl,\n    exact hJ\n      ((normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx').symm J').prop },\n\n  -- Then we just have to compare the multiplicities, which we already proved are equal.\n  have := multiplicity_factors_map_eq_multiplicity hI hI' hx hx' hJ,\n  rw [multiplicity_eq_count_normalized_factors, multiplicity_eq_count_normalized_factors,\n      unique_factorization_monoid.normalize_normalized_factor _ hJ,\n      unique_factorization_monoid.normalize_normalized_factor,\n      part_enat.coe_inj]\n    at this,\n  refine this.trans _,\n  -- Get rid of the `map` by applying the equiv to both sides.\n  generalize hJ' : (normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx')\n    \u27e8J, hJ\u27e9 = J',\n  have : ((normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx').symm J' :\n    ideal S) = J,\n  { rw [\u2190 hJ', equiv.symm_apply_apply _ _, subtype.coe_mk] },\n  subst this,\n  -- Get rid of the `attach` by applying the subtype `coe` to both sides.\n  rw [multiset.count_map_eq_count' (\u03bb f,\n      ((normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx').symm f\n        : ideal S)),\n      multiset.attach_count_eq_count_coe],\n  { exact subtype.coe_injective.comp (equiv.injective _) },\n  { exact (normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx' _).prop},\n  { exact irreducible_of_normalized_factor _\n    (normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx' _).prop },\n  { exact polynomial.map_monic_ne_zero (minpoly.monic hx') },\n  { exact irreducible_of_normalized_factor _ hJ },\n  { rwa [\u2190 bot_eq_zero, ne.def, map_eq_bot_iff_of_injective\n    (no_zero_smul_divisors.algebra_map_injective R S)] },\nend\n\ntheorem ideal.irreducible_map_of_irreducible_minpoly (hI : is_maximal I) (hI' : I \u2260 \u22a5)\n  (hx : (conductor R x).comap (algebra_map R S) \u2294 I = \u22a4)\n  (hx' : is_integral R x) (hf : irreducible (map I^.quotient.mk (minpoly R x))) :\n  irreducible (I.map (algebra_map R S)) :=\nbegin\n  have mem_norm_factors : normalize (map I^.quotient.mk (minpoly R x)) \u2208 normalized_factors\n    (map I^.quotient.mk (minpoly R x)) := by simp [normalized_factors_irreducible hf],\n  suffices : \u2203 y, normalized_factors (I.map (algebra_map R S)) = {y},\n  { obtain \u27e8y, hy\u27e9 := this,\n    have h := normalized_factors_prod (show I.map (algebra_map R S) \u2260 0, by rwa [\u2190 bot_eq_zero,\n      ne.def, map_eq_bot_iff_of_injective (no_zero_smul_divisors.algebra_map_injective R S)]),\n    rw [associated_iff_eq, hy, multiset.prod_singleton] at h,\n    rw \u2190 h,\n    exact irreducible_of_normalized_factor y\n      (show y \u2208 normalized_factors (I.map (algebra_map R S)), by simp [hy]) },\n  rw normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map hI hI' hx hx',\n  use ((normalized_factors_map_equiv_normalized_factors_min_poly_mk hI hI' hx hx').symm\n    \u27e8normalize (map I^.quotient.mk (minpoly R x)), mem_norm_factors\u27e9 : ideal S),\n  rw multiset.map_eq_singleton,\n  use \u27e8normalize (map I^.quotient.mk (minpoly R x)), mem_norm_factors\u27e9,\n  refine \u27e8_, rfl\u27e9,\n  apply multiset.map_injective subtype.coe_injective,\n  rw [multiset.attach_map_coe, multiset.map_singleton, subtype.coe_mk],\n  exact normalized_factors_irreducible hf\nend\n\nend kummer_dedekind\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/number_theory/kummer_dedekind.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.388134224405366}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n\n! This file was ported from Lean 3 source module algebraic_topology.dold_kan.n_comp_gamma\n! leanprover-community/mathlib commit 19d6240dcc5e5c8bd6e1e3c588b92e837af76f9e\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicTopology.DoldKan.GammaCompN\nimport Mathbin.AlgebraicTopology.DoldKan.NReflectsIso\n\n/-! The unit isomorphism of the Dold-Kan equivalence\n\nIn order to construct the unit isomorphism of the Dold-Kan equivalence,\nwe first construct natural transformations\n`\u0393\u2082N\u2081.nat_trans : N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)` and\n`\u0393\u2082N\u2082.nat_trans : N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`.\nIt is then shown that `\u0393\u2082N\u2082.nat_trans` is an isomorphism by using\nthat it becomes an isomorphism after the application of the functor\n`N\u2082 : karoubi (simplicial_object C) \u2964 karoubi (chain_complex C \u2115)`\nwhich reflects isomorphisms.\n\n-/\n\n\nnoncomputable section\n\nopen\n  CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Idempotents SimplexCategory Opposite SimplicialObject\n\nopen Simplicial DoldKan\n\nnamespace AlgebraicTopology\n\nnamespace DoldKan\n\nvariable {C : Type _} [Category C] [Preadditive C]\n\ntheorem pInfty_comp_map_mono_eq_zero (X : SimplicialObject C) {n : \u2115} {\u0394' : SimplexCategory}\n    (i : \u0394' \u27f6 [n]) [hi : Mono i] (h\u2081 : \u0394'.len \u2260 n) (h\u2082 : \u00acIs\u03b4\u2080 i) : pInfty.f n \u226b X.map i.op = 0 :=\n  by\n  induction' \u0394' using SimplexCategory.rec with m\n  obtain \u27e8k, hk\u27e9 :=\n    Nat.exists_eq_add_of_lt\n      (len_lt_of_mono i fun h => by\n        rw [\u2190 h] at h\u2081\n        exact h\u2081 rfl)\n  simp only [len_mk] at hk\n  cases k\n  \u00b7 change n = m + 1 at hk\n    subst hk\n    obtain \u27e8j, rfl\u27e9 := eq_\u03b4_of_mono i\n    rw [is_\u03b4\u2080.iff] at h\u2082\n    have h\u2083 : 1 \u2264 (j : \u2115) := by\n      by_contra\n      exact h\u2082 (by simpa only [Fin.ext_iff, not_le, Nat.lt_one_iff] using h)\n    exact (higher_faces_vanish.of_P (m + 1) m).comp_\u03b4_eq_zero j h\u2082 (by linarith)\n  \u00b7 simp only [Nat.succ_eq_add_one, \u2190 add_assoc] at hk\n    clear h\u2082 hi\n    subst hk\n    obtain \u27e8j\u2081, i, rfl\u27e9 :=\n      eq_comp_\u03b4_of_not_surjective i fun h =>\n        by\n        have h' := len_le_of_epi (SimplexCategory.epi_iff_surjective.2 h)\n        dsimp at h'\n        linarith\n    obtain \u27e8j\u2082, i, rfl\u27e9 :=\n      eq_comp_\u03b4_of_not_surjective i fun h =>\n        by\n        have h' := len_le_of_epi (SimplexCategory.epi_iff_surjective.2 h)\n        dsimp at h'\n        linarith\n    by_cases hj\u2081 : j\u2081 = 0\n    \u00b7 subst hj\u2081\n      rw [assoc, \u2190 SimplexCategory.\u03b4_comp_\u03b4'' (Fin.zero_le _)]\n      simp only [op_comp, X.map_comp, assoc, P_infty_f]\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ j\u2082.succ_ne_zero, zero_comp]\n      rw [Fin.val_succ]\n      linarith\n    \u00b7 simp only [op_comp, X.map_comp, assoc, P_infty_f]\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ hj\u2081, zero_comp]\n      by_contra\n      exact\n        hj\u2081\n          (by\n            simp only [Fin.ext_iff, Fin.val_zero]\n            linarith)\n#align algebraic_topology.dold_kan.P_infty_comp_map_mono_eq_zero AlgebraicTopology.DoldKan.pInfty_comp_map_mono_eq_zero\n\n@[reassoc.1]\ntheorem \u0393\u2080_obj_termwise_mapMono_comp_pInfty (X : SimplicialObject C) {\u0394 \u0394' : SimplexCategory}\n    (i : \u0394 \u27f6 \u0394') [Mono i] :\n    \u0393\u2080.Obj.Termwise.mapMono (AlternatingFaceMapComplex.obj X) i \u226b pInfty.f \u0394.len =\n      pInfty.f \u0394'.len \u226b X.map i.op :=\n  by\n  induction' \u0394 using SimplexCategory.rec with n\n  induction' \u0394' using SimplexCategory.rec with n'\n  dsimp\n  -- We start with the case `i` is an identity\n  by_cases n = n'\n  \u00b7 subst h\n    simp only [SimplexCategory.eq_id_of_mono i, \u0393\u2080.obj.termwise.map_mono_id, op_id, X.map_id]\n    dsimp\n    simp only [id_comp, comp_id]\n  by_cases hi : is_\u03b4\u2080 i\n  -- The case `i = \u03b4 0`\n  \u00b7 have h' : n' = n + 1 := hi.left\n    subst h'\n    simp only [\u0393\u2080.obj.termwise.map_mono_\u03b4\u2080' _ i hi]\n    dsimp\n    rw [\u2190 P_infty.comm' _ n rfl, alternating_face_map_complex.obj_d_eq]\n    simp only [eq_self_iff_true, id_comp, if_true, preadditive.comp_sum]\n    rw [Finset.sum_eq_single (0 : Fin (n + 2))]\n    rotate_left\n    \u00b7 intro b hb hb'\n      rw [preadditive.comp_zsmul]\n      erw [P_infty_comp_map_mono_eq_zero X (SimplexCategory.\u03b4 b) h\n          (by\n            rw [is_\u03b4\u2080.iff]\n            exact hb'),\n        zsmul_zero]\n    \u00b7 simp only [Finset.mem_univ, not_true, IsEmpty.forall_iff]\n    \u00b7 simpa only [hi.eq_\u03b4\u2080, Fin.val_zero, pow_zero, one_zsmul]\n  -- The case `i \u2260 \u03b4 0`\n  \u00b7 rw [\u0393\u2080.obj.termwise.map_mono_eq_zero _ i _ hi, zero_comp]\n    swap\n    \u00b7 by_contra h'\n      exact h (congr_arg SimplexCategory.len h'.symm)\n    rw [P_infty_comp_map_mono_eq_zero]\n    \u00b7 exact h\n    \u00b7 by_contra h'\n      exact hi h'\n#align algebraic_topology.dold_kan.\u0393\u2080_obj_termwise_map_mono_comp_P_infty AlgebraicTopology.DoldKan.\u0393\u2080_obj_termwise_mapMono_comp_pInfty\n\nvariable [HasFiniteCoproducts C]\n\nnamespace \u0393\u2082N\u2081\n\n/-- The natural transformation `N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)`. -/\n@[simps]\ndef natTrans : (n\u2081 : SimplicialObject C \u2964 _) \u22d9 \u0393\u2082 \u27f6 toKaroubi _\n    where\n  app X :=\n    { f :=\n        { app := fun \u0394 => (\u0393\u2080.splitting K[X]).desc \u0394 fun A => pInfty.f A.1.unop.len \u226b X.map A.e.op\n          naturality' := fun \u0394 \u0394' \u03b8 =>\n            by\n            apply (\u0393\u2080.splitting K[X]).hom_ext'\n            intro A\n            change _ \u226b (\u0393\u2080.obj K[X]).map \u03b8 \u226b _ = _\n            simp only [splitting.\u03b9_desc_assoc, assoc, \u0393\u2080.obj.map_on_summand'_assoc,\n              splitting.\u03b9_desc]\n            erw [\u0393\u2080_obj_termwise_map_mono_comp_P_infty_assoc X (image.\u03b9 (\u03b8.unop \u226b A.e))]\n            dsimp only [to_karoubi]\n            simp only [\u2190 X.map_comp]\n            congr 2\n            simp only [eq_to_hom_refl, id_comp, comp_id, \u2190 op_comp]\n            exact Quiver.Hom.unop_inj (A.fac_pull \u03b8) }\n      comm := by\n        apply (\u0393\u2080.splitting K[X]).hom_ext\n        intro n\n        dsimp [N\u2081]\n        simp only [\u2190 splitting.\u03b9_summand_id, splitting.\u03b9_desc, comp_id, splitting.\u03b9_desc_assoc,\n          assoc, P_infty_f_idem_assoc] }\n  naturality' X Y f := by\n    ext1\n    apply (\u0393\u2080.splitting K[X]).hom_ext\n    intro n\n    dsimp [N\u2081, to_karoubi]\n    simpa only [\u2190 splitting.\u03b9_summand_id, splitting.\u03b9_desc, splitting.\u03b9_desc_assoc, assoc,\n      P_infty_f_idem_assoc, karoubi.comp_f, nat_trans.comp_app, \u0393\u2082_map_f_app,\n      HomologicalComplex.comp_f, alternating_face_map_complex.map_f, P_infty_f_naturality_assoc,\n      nat_trans.naturality]\n#align algebraic_topology.dold_kan.\u0393\u2082N\u2081.nat_trans AlgebraicTopology.DoldKan.\u0393\u2082N\u2081.natTrans\n\nend \u0393\u2082N\u2081\n\n/-- The compatibility isomorphism relating `N\u2082 \u22d9 \u0393\u2082` and `N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082 : toKaroubi (SimplicialObject C) \u22d9 n\u2082 \u22d9 \u0393\u2082 \u2245 n\u2081 \u22d9 \u0393\u2082 :=\n  eqToIso (Functor.congr_obj (functorExtension\u2081_comp_whiskeringLeft_toKaroubi _ _) (n\u2081 \u22d9 \u0393\u2082))\n#align algebraic_topology.dold_kan.compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082 AlgebraicTopology.DoldKan.compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082\n\nnamespace \u0393\u2082N\u2082\n\n/-- The natural transformation `N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`. -/\ndef natTrans : (n\u2082 : Karoubi (SimplicialObject C) \u2964 _) \u22d9 \u0393\u2082 \u27f6 \ud835\udfed _ :=\n  ((whiskeringLeft _ _ _).obj _).Preimage (compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082.Hom \u226b \u0393\u2082N\u2081.natTrans)\n#align algebraic_topology.dold_kan.\u0393\u2082N\u2082.nat_trans AlgebraicTopology.DoldKan.\u0393\u2082N\u2082.natTrans\n\ntheorem natTrans_app_f_app (P : Karoubi (SimplicialObject C)) :\n    \u0393\u2082N\u2082.natTrans.app P =\n      (n\u2082 \u22d9 \u0393\u2082).map P.decompIdI \u226b\n        (compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082.Hom \u226b \u0393\u2082N\u2081.natTrans).app P.pt \u226b P.decompIdP :=\n  whiskeringLeft_obj_preimage_app (compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082.Hom \u226b \u0393\u2082N\u2081.natTrans) P\n#align algebraic_topology.dold_kan.\u0393\u2082N\u2082.nat_trans_app_f_app AlgebraicTopology.DoldKan.\u0393\u2082N\u2082.natTrans_app_f_app\n\nend \u0393\u2082N\u2082\n\ntheorem compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082_natTrans (X : SimplicialObject C) :\n    \u0393\u2082N\u2081.natTrans.app X =\n      (compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082.app X).inv \u226b \u0393\u2082N\u2082.natTrans.app ((toKaroubi _).obj X) :=\n  by\n  rw [\u2190 cancel_epi (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).Hom, iso.hom_inv_id_assoc]\n  exact\n    congr_app\n      (((whiskering_left _ _ _).obj _).image_preimage\n          (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans : _ \u27f6 to_karoubi _ \u22d9 \ud835\udfed _)).symm\n      X\n#align algebraic_topology.dold_kan.compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans AlgebraicTopology.DoldKan.compatibility\u0393\u2082N\u2081\u0393\u2082N\u2082_natTrans\n\ntheorem identity_n\u2082_objectwise (P : Karoubi (SimplicialObject C)) :\n    n\u2082\u0393\u2082.inv.app (n\u2082.obj P) \u226b n\u2082.map (\u0393\u2082N\u2082.natTrans.app P) = \ud835\udfd9 (n\u2082.obj P) :=\n  by\n  ext n\n  have eq\u2081 :\n    (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f.f n =\n      P_infty.f n \u226b\n        P.p.app (op [n]) \u226b\n          (\u0393\u2080.splitting (N\u2082.obj P).pt).\u03b9Summand (splitting.index_set.id (op [n])) :=\n    by simp only [N\u2082\u0393\u2082_inv_app_f_f, N\u2082_obj_p_f, assoc]\n  have eq\u2082 :\n    (\u0393\u2080.splitting (N\u2082.obj P).pt).\u03b9Summand (splitting.index_set.id (op [n])) \u226b\n        (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)).f.f n =\n      P_infty.f n \u226b P.p.app (op [n]) :=\n    by\n    dsimp [N\u2082]\n    simp only [\u0393\u2082N\u2082.nat_trans_app_f_app, P_infty_on_\u0393\u2080_splitting_summand_eq_self_assoc,\n      functor.comp_map, compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_hom, nat_trans.comp_app, eq_to_hom_app, assoc,\n      karoubi.comp_f, karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id, karoubi.decomp_id_p_f,\n      karoubi.comp_p_assoc, \u0393\u2082_map_f_app, N\u2082_map_f_f, karoubi.decomp_id_i_f,\n      \u0393\u2082N\u2081.nat_trans_app_f_app]\n    erw [splitting.\u03b9_desc_assoc, assoc, assoc, splitting.\u03b9_desc_assoc]\n    dsimp [splitting.index_set.id, splitting.index_set.e]\n    simp only [assoc, nat_trans.naturality, P_infty_f_naturality_assoc, app_idem_assoc,\n      P_infty_f_idem_assoc]\n    erw [P.X.map_id, comp_id]\n  simp only [karoubi.comp_f, HomologicalComplex.comp_f, karoubi.id_eq, N\u2082_obj_p_f, assoc, eq\u2081, eq\u2082,\n    P_infty_f_naturality_assoc, app_idem, P_infty_f_idem_assoc]\n#align algebraic_topology.dold_kan.identity_N\u2082_objectwise AlgebraicTopology.DoldKan.identity_n\u2082_objectwise\n\ntheorem identity_n\u2082 :\n    ((\ud835\udfd9 (n\u2082 : Karoubi (SimplicialObject C) \u2964 _) \u25eb n\u2082\u0393\u2082.inv) \u226b \u0393\u2082N\u2082.natTrans \u25eb \ud835\udfd9 n\u2082 : n\u2082 \u27f6 n\u2082) =\n      \ud835\udfd9 n\u2082 :=\n  by\n  ext P : 2\n  dsimp\n  rw [\u0393\u2082.map_id, N\u2082.map_id, comp_id, id_comp, identity_N\u2082_objectwise P]\n#align algebraic_topology.dold_kan.identity_N\u2082 AlgebraicTopology.DoldKan.identity_n\u2082\n\ninstance : IsIso (\u0393\u2082N\u2082.natTrans : (n\u2082 : Karoubi (SimplicialObject C) \u2964 _) \u22d9 _ \u27f6 _) :=\n  by\n  have : \u2200 P : karoubi (simplicial_object C), is_iso (\u0393\u2082N\u2082.nat_trans.app P) :=\n    by\n    intro P\n    have : is_iso (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)) :=\n      by\n      have h := identity_N\u2082_objectwise P\n      erw [hom_comp_eq_id] at h\n      rw [h]\n      infer_instance\n    exact is_iso_of_reflects_iso _ N\u2082\n  apply nat_iso.is_iso_of_is_iso_app\n\ninstance : IsIso (\u0393\u2082N\u2081.natTrans : (n\u2081 : SimplicialObject C \u2964 _) \u22d9 _ \u27f6 _) :=\n  by\n  have : \u2200 X : simplicial_object C, is_iso (\u0393\u2082N\u2081.nat_trans.app X) :=\n    by\n    intro X\n    rw [compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans]\n    infer_instance\n  apply nat_iso.is_iso_of_is_iso_app\n\n/-- The unit isomorphism of the Dold-Kan equivalence. -/\n@[simp]\ndef \u0393\u2082N\u2082 : \ud835\udfed _ \u2245 (n\u2082 : Karoubi (SimplicialObject C) \u2964 _) \u22d9 \u0393\u2082 :=\n  (asIso \u0393\u2082N\u2082.natTrans).symm\n#align algebraic_topology.dold_kan.\u0393\u2082N\u2082 AlgebraicTopology.DoldKan.\u0393\u2082N\u2082\n\n/-- The natural isomorphism `to_karoubi (simplicial_object C) \u2245 N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef \u0393\u2082N\u2081 : toKaroubi _ \u2245 (n\u2081 : SimplicialObject C \u2964 _) \u22d9 \u0393\u2082 :=\n  (asIso \u0393\u2082N\u2081.natTrans).symm\n#align algebraic_topology.dold_kan.\u0393\u2082N\u2081 AlgebraicTopology.DoldKan.\u0393\u2082N\u2081\n\nend DoldKan\n\nend AlgebraicTopology\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicTopology/DoldKan/NCompGamma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3878769457403568}}
{"text": "import rescale.basic\nimport locally_constant.Vhat\n\nimport category_theory.preadditive.additive_functor\n\nimport facts.nnreal\n\nnoncomputable theory\nopen_locale big_operators classical nnreal\n\nnamespace rescale\n\nvariables {N : \u211d\u22650} {V : Type*}\n\ninstance [has_norm V] : has_norm (rescale N V) :=\n{ norm := \u03bb v, \u2225of.symm v\u2225/N }\n\nlemma norm_def [has_norm V] (v : rescale N V) : \u2225v\u2225 = \u2225of.symm v\u2225/N := rfl\n\ninstance [hN : fact (0 < N)] [seminormed_add_comm_group V] : seminormed_add_comm_group (rescale N V) :=\nseminormed_add_comm_group.of_core (rescale N V)\n{ norm_zero := show \u2225(0 : V)\u2225/N = 0, by rw [norm_zero, zero_div],\n  triangle := \u03bb v w,\n  begin\n    simp only [norm_def, \u2190 add_div],\n    exact div_le_div_of_le hN.out.le (norm_add_le _ _), -- defeq abuse\n  end,\n  norm_neg := \u03bb v, by { simp only [norm_def], congr' 1, exact norm_neg _ /- defeq abuse -/ } }\n\ninstance [hN : fact (0 < N)] [normed_add_comm_group V] : normed_add_comm_group (rescale N V) :=\nnormed_add_comm_group.of_core (rescale N V)\n{ norm_eq_zero_iff := \u03bb v,\n  begin\n    have aux : (N:\u211d) \u2260 0 := ne_of_gt hN.out,\n    simp only [norm_def, div_eq_zero_iff, aux, or_false],\n    exact norm_eq_zero -- defeq abuse\n  end,\n  triangle := \u03bb v w,\n  begin\n    simp only [norm_def, \u2190 add_div],\n    exact div_le_div_of_le hN.out.le (norm_add_le _ _), -- defeq abuse\n  end,\n  norm_neg := \u03bb v, by { simp only [norm_def], congr' 1, exact norm_neg _ /- defeq abuse -/ } }\n\nlemma nnnorm_def [hN : fact (0 < N)] [seminormed_add_comm_group V] (v : rescale N V) :\n  \u2225v\u2225\u208a = \u2225of.symm v\u2225\u208a / N := rfl\n\nend rescale\n\nnamespace SemiNormedGroup\n\nvariables (r r\u2081 r\u2082 : \u211d\u22650) [fact (0 < r\u2081)] [fact (0 < r\u2082)]\n\n@[simps]\ndef rescale (r : \u211d\u22650) [hr : fact (0 < r)] : SemiNormedGroup \u2964 SemiNormedGroup :=\n{ obj := \u03bb V, of $ rescale r V,\n  map := \u03bb V\u2081 V\u2082 f,\n  { to_fun := \u03bb v, @rescale.of r V\u2082 $ f ((@rescale.of r V\u2081).symm v),\n    map_add' := f.map_add', -- defeq abuse\n    bound' :=\n    begin\n      obtain \u27e8C, C_pos, hC\u27e9 := f.bound,\n      use C,\n      intro v,\n      have := hC ((@rescale.of r V\u2081).symm v),\n      rw [\u2190 div_le_div_right (show 0 < (r:\u211d), from hr.1), mul_div_assoc] at this,\n      exact this,\n    end },\n  map_id' := \u03bb V, rfl, -- defeq abuse\n  map_comp' := \u03bb V\u2081 V\u2082 V\u2083 f g, rfl /- defeq abuse -/ }\n\ninstance rescale.additive [fact (0 < r)] : (rescale r).additive :=\n{ map_add' := \u03bb V W f g, rfl /- defeq abuse -/ }\n\nlemma norm_rescale_map_le [fact (0 < r)] {V\u2081 V\u2082 : SemiNormedGroup}\n  {f : V\u2081 \u27f6 V\u2082} {C : \u211d} (hf : \u2225f\u2225 \u2264 C) :\n  \u2225(rescale r).map f\u2225 \u2264 C :=\nbegin\n  refine normed_add_group_hom.op_norm_le_bound _ (le_trans (norm_nonneg _) hf) (\u03bb v, _),\n  dsimp,\n  erw [rescale.norm_def, rescale.norm_def, equiv.symm_apply_apply, \u2190 mul_div_assoc],\n  refine div_le_div (mul_nonneg (le_trans (norm_nonneg _) hf) (norm_nonneg _))\n    (normed_add_group_hom.le_of_op_norm_le _ hf _) _ le_rfl,\n  rw nnreal.coe_pos, exact \u2039fact (0 < r)\u203a.out\nend\n\nlemma rescale_map_isometry [fact (0 < r)]\n  {V\u2081 V\u2082 : SemiNormedGroup} {f : V\u2081 \u27f6 V\u2082} (hf : isometry f) :\n  isometry ((rescale r).map f) :=\nbegin\n  rw add_monoid_hom_class.isometry_iff_norm at hf \u22a2,\n  intro v,\n  erw [rescale.norm_def, rescale.norm_def, hf ((@rescale.of r _).symm v)],\nend\n\nlemma rescale_exact [fact (0 < r)] {V\u2081 V\u2082 V\u2083 : SemiNormedGroup} (f : V\u2081 \u27f6 V\u2082) (g : V\u2082 \u27f6 V\u2083)\n  (hfg : f.range = g.ker) :\n  ((rescale r).map f).range = ((rescale r).map g).ker :=\nbegin\n  ext x,\n  calc x \u2208 ((rescale r).map f).range \u2194 x \u2208 f.range : iff.rfl\n  ... \u2194 x \u2208 g.ker : by rw hfg\n  ... \u2194 x \u2208 ((rescale r).map g).ker : iff.rfl,\nend\n\nlemma rescale_exists_norm_le [fact (0 < r)] {V\u2081 V\u2082 : SemiNormedGroup} (f : V\u2081 \u27f6 V\u2082) (C : \u211d\u22650)\n  (hf : \u2200 y, \u2203 x, f x = y \u2227 \u2225x\u2225 \u2264 C * \u2225y\u2225) :\n  \u2200 y, \u2203 x, (rescale r).map f x = y \u2227 \u2225x\u2225 \u2264 C * \u2225y\u2225 :=\nbegin\n  intro y,\n  obtain \u27e8x, h1, h2\u27e9 := hf ((@rescale.of r _).symm y),\n  refine \u27e8@rescale.of r _ x, h1, _\u27e9,\n  erw [rescale.norm_def, rescale.norm_def],\n  simp only [div_eq_mul_inv, \u2190 mul_assoc, equiv.symm_apply_apply, \u2190 coe_nnnorm],\n  norm_cast, exact mul_le_mul' h2 le_rfl,\nend\n\nlemma nnnorm_to_rescale {V : SemiNormedGroup} (v : V) : \u2225(@rescale.of r V) v\u2225 \u2264 r\u207b\u00b9 * \u2225v\u2225 :=\nby { rw \u2190 div_eq_inv_mul, refl }\n\ndef to_rescale [fact (0 < r)] : \ud835\udfed _ \u27f6 rescale r :=\n{ app := \u03bb V,\n  add_monoid_hom.mk_normed_add_group_hom'\n    (add_monoid_hom.mk' (@rescale.of r V) $ \u03bb _ _, rfl) r\u207b\u00b9 (\u03bb v, nnnorm_to_rescale _ v),\n  naturality' := \u03bb V W f, rfl /- defeq abuse -/ }\n\ndef of_rescale [hr : fact (0 < r)] : rescale r \u27f6 \ud835\udfed _ :=\n{ app := \u03bb V,\n  add_monoid_hom.mk_normed_add_group_hom' (add_monoid_hom.mk' (@rescale.of r V) .symm $ \u03bb _ _, rfl) r\n  begin\n    intro v,\n    erw [rescale.nnnorm_def, mul_div_cancel' _ hr.1.ne'],\n    exact le_rfl\n  end,\n  naturality' := \u03bb V W f, rfl /- defeq abuse -/ }\n\n@[simps]\ndef iso_rescale [fact (0 < r)] : \ud835\udfed _ \u2245 (rescale r) :=\n{ hom := to_rescale r,\n  inv := of_rescale r, }\n\nopen _root_.category_theory\n\nlemma iso_rescale_isometry [fact (0 < r)] (h : r = 1) (V : SemiNormedGroup) :\n  isometry ((iso_rescale r).app V).hom :=\nbegin\n  unfreezingI { cases h },\n  dsimp only [nat_iso.app_hom, iso_rescale_hom],\n  apply add_monoid_hom_class.isometry_of_norm,\n  intro v,\n  erw [rescale.norm_def],\n  simp only [div_one, subtype.coe_mk],\n  refl\nend\n\nlemma norm_to_rescale_le [fact (0 < r)] (V : SemiNormedGroup) : \u2225(to_rescale r).app V\u2225 \u2264 r\u207b\u00b9 :=\nnormed_add_group_hom.mk_normed_add_group_hom_norm_le _\n  (inv_nonneg.2 (nnreal.zero_le_coe)) (\u03bb v, nnnorm_to_rescale _ v)\n\nlemma nnnorm_rescale_rescale_symm {V : SemiNormedGroup} (v : (rescale r\u2081).obj V) :\n  \u2225(@rescale.of r\u2082 V) ((@rescale.of r\u2081 V).symm v)\u2225\u208a \u2264 r\u2081 / r\u2082 * \u2225v\u2225\u208a :=\nbegin\n  apply le_of_eq,\n  show _ = r\u2081 / r\u2082 * (\u2225(@rescale.of r\u2081 V).symm v\u2225\u208a / r\u2081),\n  simp only [add_monoid_hom.mk'_apply, div_eq_inv_mul, rescale.nnnorm_def],\n  rw [mul_assoc, mul_inv_cancel_left\u2080 (show r\u2081 \u2260 0, from ne_of_gt $ fact.out _)],\n  refl\nend\n\ndef scale : rescale r\u2081 \u27f6 rescale r\u2082 :=\n{ app := \u03bb V,\n  add_monoid_hom.mk_normed_add_group_hom'\n    (add_monoid_hom.mk' (\u03bb v, (@rescale.of r\u2082 V) $ (@rescale.of r\u2081 V).symm v) $\n      \u03bb _ _, rfl) (r\u2081 / r\u2082) (\u03bb v, nnnorm_rescale_rescale_symm r\u2081 r\u2082 v),\n  naturality' := \u03bb V W f, rfl /- defeq abuse -/ }\n\nlemma norm_scale_le (V : SemiNormedGroup) : \u2225(scale r\u2081 r\u2082).app V\u2225 \u2264 (r\u2081 / r\u2082) :=\nnormed_add_group_hom.mk_normed_add_group_hom_norm_le _ (div_nonneg (nnreal.coe_nonneg _)\n    (nnreal.coe_nonneg _)) (\u03bb v, nnnorm_rescale_rescale_symm r\u2081 r\u2082 v)\n\nlemma scale_comm {V\u2081 V\u2082 W\u2081 W\u2082 : SemiNormedGroup}\n  (f\u2081 : V\u2081 \u27f6 W\u2081) (f\u2082 : V\u2082 \u27f6 W\u2082) (\u03c6 : V\u2081 \u27f6 V\u2082) (\u03c8 : W\u2081 \u27f6 W\u2082) (h : f\u2081 \u226b \u03c8 = \u03c6 \u226b f\u2082) :\n  (rescale r\u2081).map f\u2081 \u226b ((rescale r\u2081).map \u03c8 \u226b (scale r\u2081 r\u2082).app W\u2082) =\n  ((rescale r\u2081).map \u03c6 \u226b (scale r\u2081 r\u2082).app V\u2082) \u226b (rescale r\u2082).map f\u2082 :=\nby rw [\u2190 category.assoc, \u2190 category_theory.functor.map_comp, nat_trans.naturality,\n    nat_trans.naturality, category.assoc, \u2190 category_theory.functor.map_comp, h]\n\nend SemiNormedGroup\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/rescale/normed_group.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702642896702, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3878769393795591}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from by {\n    assume (i j : \u2124) (hij : i \u2260 j),\n    have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 \u00ac (int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j)), from by {\n      assume (i j : \u2124) (hij : i \u2260 j),\n      assume h3 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j),\n      have h4 : \u03b1 = (\u2191i)\u207b\u00b9 * \u2191(int.nat_abs (\u03b1 * \u2191i)) / (\u2191j)\u207b\u00b9 * \u2191(int.nat_abs (\u03b1 * \u2191j)), from by {\n        rw h3,\n        rw int.fract_eq_div_nat_abs,\n        rw int.fract_eq_div_nat_abs,\n        ring,\n      },\n      have h5 : \u03b1 * \u2191j = \u2191i * \u2191(int.nat_abs (\u03b1 * \u2191i)), from by {\n        rw h4,\n        rw mul_assoc,\n        rw mul_assoc,\n        rw mul_comm (\u2191i) (\u2191j)\u207b\u00b9,\n        rw mul_assoc,\n        rw mul_inv_cancel,\n        rw mul_one,\n        ring,\n      },\n      have h6 : \u03b1 * \u2191j = \u2191(int.nat_abs (\u03b1 * \u2191i)) * \u2191i, from by {\n        rw h5,\n        rw mul_comm (\u2191i) (int.nat_abs (\u03b1 * \u2191i)),\n      },\n      have h7 : \u03b1 * \u2191j = \u2191(int.nat_abs (\u03b1 * \u2191i) * i), from by {\n        rw h6,\n        rw int.coe_nat_mul,\n      },\n      have h8 : \u03b1 * \u2191j = \u2191(int.nat_abs (\u03b1 * \u2191j * j)), from by {\n        rw mul_comm (\u03b1 * \u2191j) j,\n        rw int.nat_abs_mul,\n        rw int.coe_nat_mul,\n      },\n      have h9 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i) * i, from by {\n        rw h7,\n        rw h8,\n      },\n      have h10 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h11 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h10,\n      },\n      have h12 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h13 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h12,\n      },\n      have h14 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h15 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h14,\n      },\n      have h16 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h17 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h16,\n      },\n      have h18 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h19 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h18,\n      },\n      have h20 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h21 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h20,\n      },\n      have h22 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h23 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h22,\n      },\n      have h24 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h25 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h24,\n      },\n      have h26 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h27 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h26,\n      },\n      have h28 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h29 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h28,\n      },\n      have h30 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_abs_mul,\n      },\n      have h31 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw h9,\n        rw h30,\n      },\n      have h32 : int.nat_abs (\u03b1 * \u2191j * j) = int.nat_abs (\u03b1 * \u2191i * i), from by {\n        rw mul_comm (\u03b1 * \u2191i) i,\n        rw int.nat_\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from by {\n    assume i j : \u2124, assume h1 : i \u2260 j,\n    have h2 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from by {\n      assume h2 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j),\n      have h3 : (i - j) * \u03b1 = int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j), from by {rw h2, ring},\n      have h4 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j), from by {rw h3, ring},\n      have h5 : \u03b1 * \u2191i = \u03b1 * \u2191j, from by {rw h4, ring},\n      have h6 : \u2191i = \u2191j, from by {apply int.irrational_mul_ne_zero h\u03b1_irrat, rw h5, ring},\n      exact h1 h6,\n    },\n    have h3 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j) \u2208 \u211a, from by {rw \u2190 int.fract_eq_iff_eq_int, ring},\n    show int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from by {apply h2, exact h3},\n  },\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {\n    assume i j : \u2124, assume h2 : i \u2260 j,\n    have h3 : int.fract (\u03b1 * \u2191i) \u2208 set.Icc 0 1, from by {rw \u2190 int.fract_eq_iff_eq_int, ring},\n    have h4 : int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1, from by {rw \u2190 int.fract_eq_iff_eq_int, ring},\n    have h5 : abs (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {rw abs_of_nonneg, rw abs_of_nonneg, apply set.subset.trans h3.1 h4.2, apply set.subset.trans h4.1 h3.2},\n    show (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {rw \u2190 abs_neg, rw \u2190 abs_neg, apply set.subset.trans h5.1 h5.2},\n  },\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {\n    assume i j : \u2124, assume h3 : i \u2260 j,\n    have h4 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {apply h2, exact h3},\n    show (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {apply set.subset.trans h4.1 h4.2},\n  },\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191j) - int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by {\n    assume i j : \u2124, assume h4 : i \u2260 j,\n    have h5 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {apply h2, exact h4},\n    have h6 : (int.fract (\u03b1 * \u2191j) - int.fract (\u03b1 * \u2191i)) \u2208 set.Icc (-1) 1, from by {apply h2, exact h4},\n    have h7 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {apply set.subset.trans h5.1 h5.2},\n    have h8 : (int.fract (\u03b1 * \u2191j) - int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by {apply set.subset.trans h6.1 h6.2},\n    show (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191j) - int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by {split,exact h7,exact h8},\n  },\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {\n    assume i j : \u2124, assume h5 : i \u2260 j,\n    have h6 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {apply h2, exact h5},\n    have h7 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {apply set.subset.trans h6.1 h6.2},\n    show (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {split,exact h7,exact h7},\n  },\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {\n    assume i j : \u2124, assume h6 : i \u2260 j,\n    have h7 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc (-1) 1, from by {apply h2, exact h6},\n    have h8 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {apply set.subset.trans h7.1 h7.2},\n    show (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1 \u2227 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {split,exact h8,exact h8,exact h8},\n  },\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, (i \u2260 j) \u2192 (int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j)), from assume i j : \u2124,\n    assume hneq : i \u2260 j,\n    have h1 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j) \u2192 \u03b1 \u2208 \u211a, from assume h : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j),\n      have h1 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from\n        by {rw [h,int.fract_sub,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract_mul,int.fract\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 m n : \u2124, (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from\n    assume (m n : \u2124), have h1 : (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191n)) \u2192 (\u03b1 \u2208 \u211a), from\n      assume h2 : (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191n)),\n      have h3 : (int.fract (\u03b1 * \u2191m)) = (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)), from by simp,\n      have h4 : (int.fract (\u03b1 * \u2191n)) = (\u03b1 * \u2191n - int.nat_abs (\u03b1 * \u2191n)), from by simp,\n      have h5 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191n - int.nat_abs (\u03b1 * \u2191n)), from by {rw [h3,h4],exact h2},\n      have h6 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191n)), from by {rw [h5],ring},\n      have h7 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191m - (int.nat_abs (\u03b1 * \u2191m) - int.nat_abs (\u03b1 * \u2191n))), from by {rw [h6],ring},\n      have h8 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191m - (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191n))), from by {rw [h7],ring},\n      have h9 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (int.nat_abs (\u03b1 * \u2191n)), from by {rw [h8],ring},\n      have h10 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (int.nat_abs (\u03b1 * \u2191m)), from by {rw [h9],ring},\n      have h11 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191m), from by {rw [h10],ring},\n      have h12 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191m - (\u03b1 * \u2191m - \u03b1 * \u2191n)), from by {rw [h11],ring},\n      have h13 : (\u03b1 * \u2191m - int.nat_abs (\u03b1 * \u2191m)) = (\u03b1 * \u2191n), from by {rw [h12],ring},\n      have h14 : (\u03b1 * \u2191m) = (\u03b1 * \u2191n), from by {rw [h13],ring},\n      have h15 : (\u03b1 * \u2191m) = (m * \u03b1), from by {rw [mul_comm],exact h14},\n      have h16 : (\u03b1 * \u2191n) = (n * \u03b1), from by {rw [mul_comm],exact h14},\n      have h17 : (m * \u03b1) = (n * \u03b1), from by {rw [h15,h16],ring},\n      have h18 : (m * \u03b1) = (\u03b1 * n), from by {rw [mul_comm],exact h17},\n      have h19 : (\u03b1 * \u2191m) = (\u03b1 * \u2191n), from by {rw [h15,h16],ring},\n      have h20 : (\u03b1 * \u2191m) = (\u03b1 * \u2191n), from by {rw [h19],exact h18},\n      have h21 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h20},\n      have h22 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h21],exact h18},\n      have h23 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h22},\n      have h24 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h23],exact h18},\n      have h25 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h24},\n      have h26 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h25],exact h18},\n      have h27 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h26},\n      have h28 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h27],exact h18},\n      have h29 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h28},\n      have h30 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h29],exact h18},\n      have h31 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h30},\n      have h32 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h31],exact h18},\n      have h33 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h32},\n      have h34 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h33],exact h18},\n      have h35 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h34},\n      have h36 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h35],exact h18},\n      have h37 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h36},\n      have h38 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h37],exact h18},\n      have h39 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h38},\n      have h40 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h39],exact h18},\n      have h41 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h40},\n      have h42 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h41],exact h18},\n      have h43 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h42},\n      have h44 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h43],exact h18},\n      have h45 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h44},\n      have h46 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h45],exact h18},\n      have h47 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h46},\n      have h48 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h47],exact h18},\n      have h49 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h48},\n      have h50 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h49],exact h18},\n      have h51 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h50},\n      have h52 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h51],exact h18},\n      have h53 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [mul_comm],exact h52},\n      have h54 : (\u03b1 * \u2191m) = (\u03b1 * \u2191m), from by {rw [h53],exact h18},\n      have h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from \n    assume (i j : \u2124) (hij : i \u2260 j),\n    have h2 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from by {\n      assume h3 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j),\n      have h4 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j) \u2208 \u211a, from by {\n        apply int.fract_in_rat,\n        have h5 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 \u2124, from by {\n          have h6 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n            rw [int.fract_sub],\n            have h7 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n              rw [int.fract_sub],\n              have h8 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                rw [int.fract_sub],\n                have h9 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                  rw [int.fract_sub],\n                  have h10 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                    rw [int.fract_sub],\n                    have h11 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                      rw [int.fract_sub],\n                      have h12 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                        rw [int.fract_sub],\n                        have h13 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                          rw [int.fract_sub],\n                          have h14 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                            rw [int.fract_sub],\n                            have h15 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                              rw [int.fract_sub],\n                              have h16 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                rw [int.fract_sub],\n                                have h17 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                  rw [int.fract_sub],\n                                  have h18 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                    rw [int.fract_sub],\n                                    have h19 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                      rw [int.fract_sub],\n                                      have h20 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                        rw [int.fract_sub],\n                                        have h21 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                          rw [int.fract_sub],\n                                          have h22 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                            rw [int.fract_sub],\n                                            have h23 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                              rw [int.fract_sub],\n                                              have h24 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                                rw [int.fract_sub],\n                                                have h25 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                                  rw [int.fract_sub],\n                                                  have h26 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                                    rw [int.fract_sub],\n                                                    have h27 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)), from by {\n                                                      rw [int.fract_sub],\n                                                      have h28 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) = \u03b1 * \u2191i - \u03b1 * \u2191j - (int.fract (\u03b1 * \u2191i\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from assume i j hneq,\n    by {\n      assume h,\n      have h2 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j))/(i-j), from by {rw [h,int.fract_sub_fract,int.fract_mul],ring},\n      have h3 : \u03b1 \u2208 \u211a, from by {apply quotient.exact h2},\n      contradiction,\n    },\n\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2260 0, from assume i j hneq,\n    by {\n      assume h,\n      have h2 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j), from by {rw [h,sub_self]},\n      contradiction,\n    },\n\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from assume i j hneq,\n    by {\n      have h4 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by {\n        have h5 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = int.fract (\u03b1 * \u2191(i-j)), from by {rw [int.fract_sub_fract,int.fract_mul], ring},\n        have h6 : int.fract (\u03b1 * \u2191(i-j)) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by {apply set.mem_image_of_mem, apply set.mem_univ,},\n        exact h6,\n      },\n      exact h4,\n    },\n\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from assume i j hneq,\n    by {\n      have h5 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by {\n        have h6 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) = int.fract (\u03b1 * \u2191(i-j)), from by {rw [int.fract_sub_fract,int.fract_mul], ring},\n        have h7 : int.fract (\u03b1 * \u2191(i-j)) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by {apply set.mem_image_of_mem, apply set.mem_univ,},\n        exact h7,\n      },\n      exact h5,\n    },\n\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)), from assume i j hneq,\n    by {\n      have h6 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)), from by {\n        have h7 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = abs ((int.fract (\u03b1 * \u2191j)) - (int.fract (\u03b1 * \u2191i))), from by {rw [abs_sub,abs_sub]},\n        have h8 : abs ((int.fract (\u03b1 * \u2191j)) - (int.fract (\u03b1 * \u2191i))) = (int.fract (\u03b1 * \u2191j)) - (int.fract (\u03b1 * \u2191i)), from by {rw [abs_of_nonneg (sub_nonneg.2 (h2 i j hneq))]},\n        have h9 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = (int.fract (\u03b1 * \u2191j)) - (int.fract (\u03b1 * \u2191i)), from by {rw [h7,h8]},\n        have h10 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = -((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))), from by {rw [h9,sub_eq_neg_add]},\n        exact h10,\n      },\n      exact h6,\n    },\n\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from assume i j hneq,\n    by {\n      have h7 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {\n        have h8 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) = abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))), from by {rw [h5 i j hneq]},\n        have h9 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) \u2208 set.Icc 0 1, from by {rw [h8], apply abs_nonneg (int.fract_nonneg (\u03b1 * \u2191i)),},\n        exact h9,\n      },\n      exact h7,\n    },\n\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) \u2208 set.Icc 0 1, from assume i j hneq,\n    by {\n      have h8 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) \u2208 set.Icc 0 1, from by {\n        have h9 : abs ((int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j))) = (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)), from by {rw [h5 i j hneq]},\n        have h10 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from by {rw [h9], apply h6 i j hneq},\n        exact h10,\n      },\n      exact h8,\n    },\n\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from assume i j hneq,\n    by {\n      have h9 : (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by {\n        have h10 : (int.fract (\u03b1 * \u2191i)) - (\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7879311856832191, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3878103813576996}}
{"text": "import ..mcrl2_basic.mcrl2_basic\nimport ..transition.par_comm\nimport ..add\n\nopen mcrl2\n\nvariable {\u03b1 : Type}\nvariable [comm_semigroup_with_zero \u03b1]\n\n/- This is where the axioms are proved for the quotient.-/\n\nlemma mcrl2.dead_alt {x : mcrl2 \u03b1} : x + \u03b4 \u2248 x :=\nby exact R_add_congr (transition.alt_deadlock x)\n\nlemma mcrl2.dead_seq {x : mcrl2 \u03b1} : \u03b4 \u2b1d x \u2248 \u03b4 :=\nby exact R_add_congr (transition.seq_deadlock x)\n\nlemma mcrl2.dead_parl {x : mcrl2 \u03b1} : \u03b4 |_ x \u2248 \u03b4 :=\nby exact R_add_congr (transition.parl_deadlock x)\n\nlemma mcrl2.dead_comml {x : mcrl2 \u03b1} : \u03b4 \u2223 x \u2248 \u03b4 :=\nby exact R_add_congr  (transition.comm_deadlockl x) \n\nlemma mcrl2.dead_commr {x : mcrl2 \u03b1} : x \u2223 \u03b4 \u2248 \u03b4 :=\nby exact R_add_congr (transition.comm_deadlockr x)\n\ninductive R_add_par {x y : mcrl2 \u03b1} :\nmcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop\n| basel : R_add_par x y\n| baser : R_add_par y x\n| par {a b} : R_add_par (a || b) (b || a) \n| refl {a} : R_add_par a a \n\nlemma R_add_par.symm {x y : mcrl2 \u03b1} : \nsymmetric (@R_add_par \u03b1 _ x y) :=\nbegin\n  intros x\u2081 y\u2081 h,\n  cases h,\n  { exact R_add_par.baser},\n  { exact R_add_par.basel},\n  { exact R_add_par.par},\n  { assumption}\nend\n\nlemma R_add_par_refl {x y} : \u2200x\u2081, (@R_add_par \u03b1 _ x y) x\u2081 x\u2081 := by intro x\u2081; exact R_add_par.refl\n\nlemma R_add_par.lift_rel  {x y} {x\u2081 y\u2081 : option (mcrl2 \u03b1)} :\noption.rel (@R_add_par \u03b1 _ x y) (par' x\u2081 y\u2081) (par'  y\u2081 x\u2081) :=\nbegin\n  cases x\u2081,\n  { cases y\u2081,\n    { apply option.rel.none},\n    { apply option.rel.some, exact R_add_par.refl}},\n  { cases y\u2081,\n    { apply option.rel.some, exact R_add_par.refl},\n    { apply option.rel.some, exact R_add_par.par}}\nend\n\nlemma mcrl2.par_def {x y : mcrl2 \u03b1} : x || y \u2248 x |_ y + y |_ x + x \u2223 y :=\nbegin\n  apply exists.intro R_add_par,\n  apply and.intro,\n  exact R_add_par.basel,\n  apply and.intro,\n  { intros x\u2081 y\u2081 x\u2081' a\u2081 h\u2081 h\u2082,\n    cases h\u2081,\n    { cases h\u2082,\n      { apply exists.intro (par' h\u2082_x' (some y)),\n        apply and.intro,\n        { apply transition.altl,\n          apply transition.altl,\n          apply transition.parl,\n          assumption},\n        { cases h\u2082_x',\n          { apply option.rel.some, exact R_add_par.refl},\n          { apply option.rel.some, exact R_add_par.refl}}},\n      { apply exists.intro (par' h\u2082_y' (some x)),\n        apply and.intro,\n        { apply transition.altl,\n          apply transition.altr,\n          apply transition.parl,\n          assumption},\n        { cases h\u2082_y',\n          { apply option.rel.some, exact R_add_par.refl},\n          { apply option.rel.some, exact R_add_par.par}}},\n      { apply exists.intro (par' h\u2082_x' h\u2082_y'),\n        apply and.intro,\n        { apply transition.altr,\n          apply transition.comm; assumption},\n        { cases h\u2082_x',\n          { cases h\u2082_y',\n            { apply option.rel.none},\n            { apply option.rel.some, exact R_add_par.refl}},\n          { cases h\u2082_y',\n            { apply option.rel.some, exact R_add_par.refl},\n            { apply option.rel.some, exact R_add_par.refl}}}}},\n    { cases h\u2082,\n      { cases h\u2082_h,\n        { cases h\u2082_h_h,\n          apply exists.intro (par' h\u2082_h_h_x' (some y)),\n          apply and.intro,\n          { apply transition.par_l; assumption},\n          { cases h\u2082_h_h_x',\n            { apply option.rel.some, exact R_add_par.refl},\n            { apply option.rel.some, exact R_add_par.refl}}},\n        { cases h\u2082_h_h,\n          apply exists.intro (par' (some x) h\u2082_h_h_x'),\n          apply and.intro,\n          { apply transition.par_r; assumption},\n          { cases h\u2082_h_h_x',\n            { apply option.rel.some, exact R_add_par.refl},\n            { apply option.rel.some, exact R_add_par.par}}}},\n      { cases h\u2082_h,\n        apply exists.intro (par' h\u2082_h_x' h\u2082_h_y'),\n        apply and.intro,\n        { apply transition.par_comm; assumption},\n        { cases h\u2082_h_x',\n          { cases h\u2082_h_y',\n            { apply option.rel.none},\n            { apply option.rel.some, exact R_add_par.refl}},\n          { cases h\u2082_h_y',\n            { apply option.rel.some, exact R_add_par.refl},\n            { apply option.rel.some, exact R_add_par.refl}}}}},\n    { cases h\u2082,\n      { apply exists.intro (par' (some h\u2081_b) h\u2082_x'),\n        apply and.intro,\n        { apply transition.par_r; assumption},\n        { exact R_add_par.lift_rel}},\n      { apply exists.intro (par' h\u2082_y' (some h\u2081_a)),\n        apply and.intro,\n        { apply transition.par_l; assumption},\n        { exact R_add_par.lift_rel}},\n      { apply exists.intro (par' h\u2082_y' h\u2082_x'),\n        apply and.intro,\n        { change comm_semigroup_with_zero.mul h\u2082_a h\u2082_b with h\u2082_a * h\u2082_b,\n          rw mul_comm at *, apply transition.par_comm; assumption},\n        { exact R_add_par.lift_rel}}},\n    { apply exists.intro x\u2081',\n      apply and.intro h\u2082,\n      cases x\u2081',\n      exact option.rel.none,\n      apply option.rel.some,\n      exact R_add_par.refl}},\n  { exact R_add_par.symm}\nend\n\nlemma mcrl2.parl_seq_atom {a : \u03b1} {x : mcrl2 \u03b1} : (atom a) |_ x \u2248 (atom a) \u2b1d x :=\nby exact R_add_congr (transition.parl_seq_atom a x)\n\nlemma mcrl2.parl_seq {a : \u03b1} {x y} : (atom a) \u2b1d x |_ y \u2248 (atom a) \u2b1d (x || y) :=\nbegin\n  apply exists.intro R_add,\n  apply and.intro R_add.basel,\n  apply and.intro,\n  { intros x\u2081 y\u2081 x\u2081' a\u2081 h\u2081 h\u2082,\n    cases h\u2081,\n    { cases h\u2082,\n      cases h\u2082_h,\n      cases h\u2082_h_h,\n      apply exists.intro (par' \u2191x \u2191y),\n      apply and.intro,\n      { simp [transition.seq_iff], \n        apply exists.intro none,\n        apply and.intro rfl h\u2082_h_h},\n      { apply option.rel.some, exact R_add.refl}},\n    { cases h\u2082,\n      cases h\u2082_h,\n      apply exists.intro (some (x || y)),\n      apply and.intro,\n      { simp [transition.parl_iff],\n        apply exists.intro (some x),\n        apply and.intro rfl,\n        simp [transition.seq_iff],\n        apply exists.intro none,\n        apply and.intro rfl h\u2082_h},\n      { apply option.rel.some, exact R_add.refl}},\n    { apply exists.intro x\u2081',\n      apply and.intro h\u2082,\n      cases x\u2081',\n      exact option.rel.none,\n      apply option.rel.some,\n      exact R_add.refl}},\n  { exact R_add.symm}\nend\n\nlemma mcrl2.parl_alt {x y z : mcrl2 \u03b1} : (x + y) |_ z \u2248 x |_ z + y |_ z :=\nby exact R_add_congr (transition.parl_alt x y z)\n\nlemma mcrl2.comm_success {a b c : \u03b1} (h : a * b = c) : ((atom a) \u2223 (atom b)) \u2248 atom c :=\nby exact R_add_congr (transition.comm_success a b c h)\n\nlemma mcrl2.comm_fail {a b : \u03b1} (h : a * b = 0) : ((atom a) \u2223 (atom b)) \u2248 \u03b4 :=\nby exact R_add_congr (transition.comm_fail a b h)\n\nlemma mcrl2.comm_seq_distl {a b : \u03b1} {x : mcrl2 \u03b1} : (atom a \u2b1d x) \u2223 (atom b) \u2248 ((atom a) \u2223 (atom b)) \u2b1d x :=\nby exact R_add_congr (transition.comm_seq_distl a b x)\n\nlemma mcrl2.comm_seq_distr {a b : \u03b1} {x : mcrl2 \u03b1} : (atom a) \u2223 (atom b \u2b1d x) \u2248 ((atom a) \u2223 (atom b)) \u2b1d x :=\nby exact R_add_congr (transition.comm_seq_distr a b x)\n\nlemma mcrl2.comm_seq_dist {a b : \u03b1} {x y : mcrl2 \u03b1} : (atom a \u2b1d x) \u2223 (atom b \u2b1d y) \u2248 (atom a \u2223 atom b) \u2b1d (x || y) := \nby exact R_add_congr (transition.comm_seq_dist a b x y)\n\nlemma mcrl2.comm_alt_distl {x y z : mcrl2 \u03b1} : ((x + y) \u2223 z) \u2248 x \u2223 z + y \u2223 z :=\nby exact R_add_congr (transition.comm_alt_distl x y z)\n\nlemma mcrl2.comm_alt_distr {x y z : mcrl2 \u03b1} : (x \u2223 (y + z)) \u2248 x \u2223 y + x \u2223 z :=\nby exact R_add_congr (transition.comm_alt_distr x y z)", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/mcrl2_mrg/par_axioms.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.38777596273258674}}
{"text": "example : Nat \u2192 Nat\n  | 0 => 0\n  | n + 1 => by exact n", "meta": {"author": "leanprover", "repo": "LeanInk", "sha": "499cf46f571562bebee0c8c193a7f9dcf5a30187", "save_path": "github-repos/lean/leanprover-LeanInk", "path": "github-repos/lean/leanprover-LeanInk/LeanInk-499cf46f571562bebee0c8c193a7f9dcf5a30187/test/bugs/GH_15.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.752012562644147, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.38775265417322413}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.groupoid\nimport control.equiv_functor\nimport category_theory.types\n\n/-!\n# The core of a category\n\nThe core of a category `C` is the (non-full) subcategory of `C` consisting of all objects,\nand all isomorphisms. We construct it as a `groupoid`.\n\n`core.inclusion : core C \u2964 C` gives the faithful inclusion into the original category.\n\nAny functor `F` from a groupoid `G` into `C` factors through `core C`,\nbut this is not functorial with respect to `F`.\n-/\n\nnamespace category_theory\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 -- morphism levels before object levels. See note [category_theory universes].\n\n/-- The core of a category C is the groupoid whose morphisms are all the\nisomorphisms of C. -/\n@[nolint has_inhabited_instance]\ndef core (C : Type u\u2081) := C\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n\ninstance core_category : groupoid.{v\u2081} (core C) :=\n{ hom  := \u03bb X Y : C, X \u2245 Y,\n  inv  := \u03bb X Y f, iso.symm f,\n  id   := \u03bb X, iso.refl X,\n  comp := \u03bb X Y Z f g, iso.trans f g }\n\nnamespace core\n@[simp] lemma id_hom (X : core C) : iso.hom (\ud835\udfd9 X) = \ud835\udfd9 X := rfl\n@[simp] lemma comp_hom {X Y Z : core C} (f : X \u27f6 Y) (g : Y \u27f6 Z) : (f \u226b g).hom = f.hom \u226b g.hom :=\nrfl\n\nvariables (C)\n\n/-- The core of a category is naturally included in the category. -/\ndef inclusion : core C \u2964 C :=\n{ obj := id,\n  map := \u03bb X Y f, f.hom }\n\ninstance : faithful (inclusion C) := {}\n\nvariables {C} {G : Type u\u2082} [groupoid.{v\u2082} G]\n\n/-- A functor from a groupoid to a category C factors through the core of C. -/\n-- Note that this function is not functorial\n-- (consider the two functors from [0] to [1], and the natural transformation between them).\nnoncomputable\ndef functor_to_core (F : G \u2964 C) : G \u2964 core C :=\n{ obj := \u03bb X, F.obj X,\n  map := \u03bb X Y f, \u27e8F.map f, F.map (inv f)\u27e9 }\n\n/--\nWe can functorially associate to any functor from a groupoid to the core of a category `C`,\na functor from the groupoid to `C`, simply by composing with the embedding `core C \u2964 C`.\n-/\ndef forget_functor_to_core : (G \u2964 core C) \u2964 (G \u2964 C) := (whiskering_right _ _ _).obj (inclusion C)\nend core\n\n/--\n`of_equiv_functor m` lifts a type-level `equiv_functor`\nto a categorical functor `core (Type u\u2081) \u2964 core (Type u\u2082)`.\n-/\ndef of_equiv_functor (m : Type u\u2081 \u2192 Type u\u2082) [equiv_functor m] :\n  core (Type u\u2081) \u2964 core (Type u\u2082) :=\n{ obj       := m,\n  map       := \u03bb \u03b1 \u03b2 f, (equiv_functor.map_equiv m f.to_equiv).to_iso,\n  -- These are not very pretty.\n  map_id' := \u03bb \u03b1, begin ext, exact (congr_fun (equiv_functor.map_refl _) x), end,\n  map_comp' := \u03bb \u03b1 \u03b2 \u03b3 f g,\n  begin\n    ext,\n    simp only [equiv_functor.map_equiv_apply, equiv.to_iso_hom,\n      function.comp_app, core.comp_hom, types_comp],\n    erw [iso.to_equiv_comp, equiv_functor.map_trans],\n  end, }\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3877349174823433}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro, Simon Hudon\n-/\nimport data.fin.fin2\nimport data.typevec\n\n/-!\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n\nFunctors between the category of tuples of types, and the category Type\n\nFeatures:\n\n`mvfunctor n` : the type class of multivariate functors\n`f <$$> x`    : notation for map\n\n-/\n\nuniverses u v w\n\nopen_locale mvfunctor\n\n/-- multivariate functors, i.e. functor between the category of type vectors\nand the category of Type -/\nclass mvfunctor {n : \u2115} (F : typevec n \u2192 Type*) :=\n(map : \u03a0 {\u03b1 \u03b2 : typevec n}, (\u03b1 \u27f9 \u03b2) \u2192 (F \u03b1 \u2192 F \u03b2))\n\nlocalized \"infixr (name := mvfunctor.map) ` <$$> `:100 := mvfunctor.map\" in mvfunctor\n\nvariables {n : \u2115}\n\nnamespace mvfunctor\n\nvariables {\u03b1 \u03b2 \u03b3 : typevec.{u} n} {F : typevec.{u} n \u2192 Type v} [mvfunctor F]\n\n/-- predicate lifting over multivariate functors -/\ndef liftp {\u03b1 : typevec n} (p : \u03a0 i, \u03b1 i \u2192 Prop) (x : F \u03b1) : Prop :=\n\u2203 u : F (\u03bb i, subtype (p i)), (\u03bb i, @subtype.val _ (p i)) <$$> u = x\n\n/-- relational lifting over multivariate functors -/\ndef liftr {\u03b1 : typevec n} (r : \u03a0 {i}, \u03b1 i \u2192 \u03b1 i \u2192 Prop) (x y : F \u03b1) : Prop :=\n\u2203 u : F (\u03bb i, {p : \u03b1 i \u00d7 \u03b1 i // r p.fst p.snd}),\n  (\u03bb i (t : {p : \u03b1 i \u00d7 \u03b1 i // r p.fst p.snd}), t.val.fst) <$$> u = x \u2227\n  (\u03bb i (t : {p : \u03b1 i \u00d7 \u03b1 i // r p.fst p.snd}), t.val.snd) <$$> u = y\n\n/-- given `x : F \u03b1` and a projection `i` of type vector `\u03b1`, `supp x i` is the set\nof `\u03b1.i` contained in `x` -/\ndef supp {\u03b1 : typevec n} (x : F \u03b1) (i : fin2 n) : set (\u03b1 i) :=\n{ y : \u03b1 i | \u2200 \u2983p\u2984, liftp p x \u2192 p i y }\n\ntheorem of_mem_supp {\u03b1 : typevec n} {x : F \u03b1} {p : \u03a0 \u2983i\u2984, \u03b1 i \u2192 Prop} (h : liftp p x) (i : fin2 n):\n  \u2200 y \u2208 supp x i, p y :=\n\u03bb y hy, hy h\n\nend mvfunctor\n\n/-- laws for `mvfunctor` -/\nclass is_lawful_mvfunctor {n : \u2115} (F : typevec n \u2192 Type*) [mvfunctor F] : Prop :=\n(id_map       : \u2200 {\u03b1 : typevec n} (x : F \u03b1), typevec.id <$$> x = x)\n(comp_map     : \u2200 {\u03b1 \u03b2 \u03b3 : typevec n} (g : \u03b1 \u27f9 \u03b2) (h : \u03b2 \u27f9 \u03b3) (x : F \u03b1),\n                    (h \u229a g) <$$> x = h <$$> g <$$> x)\n\nopen nat typevec\n\nnamespace mvfunctor\n\nexport is_lawful_mvfunctor (comp_map)\nopen is_lawful_mvfunctor\n\nvariables {\u03b1 \u03b2 \u03b3 : typevec.{u} n}\nvariables {F : typevec.{u} n \u2192 Type v} [mvfunctor F]\n\nvariables (p : \u03b1 \u27f9 repeat n Prop) (r : \u03b1 \u2297 \u03b1 \u27f9 repeat n Prop)\n\n/-- adapt `mvfunctor.liftp` to accept predicates as arrows -/\ndef liftp' : F \u03b1 \u2192 Prop :=\nmvfunctor.liftp $ \u03bb i x, of_repeat $ p i x\n\n/-- adapt `mvfunctor.liftp` to accept relations as arrows -/\ndef liftr' : F \u03b1 \u2192 F \u03b1 \u2192 Prop :=\nmvfunctor.liftr $ \u03bb i x y, of_repeat $ r i $ typevec.prod.mk _ x y\n\nvariables [is_lawful_mvfunctor F]\n\n@[simp]\nlemma id_map (x : F \u03b1) :\n  typevec.id <$$> x = x :=\nid_map x\n\n@[simp]\nlemma id_map' (x : F \u03b1) :\n  (\u03bb i a, a) <$$> x = x :=\nid_map x\n\nlemma map_map (g : \u03b1 \u27f9 \u03b2) (h : \u03b2 \u27f9 \u03b3) (x : F \u03b1) :\n  h <$$> g <$$> x = (h \u229a g) <$$> x :=\neq.symm $ comp_map _ _ _\n\nsection liftp'\n\nvariables (F)\n\nlemma exists_iff_exists_of_mono {p : F \u03b1 \u2192 Prop} {q : F \u03b2 \u2192 Prop} (f : \u03b1 \u27f9 \u03b2) (g : \u03b2 \u27f9 \u03b1)\n  (h\u2080 : f \u229a g = id)\n  (h\u2081 : \u2200 u : F \u03b1, p u \u2194 q (f <$$> u)) :\n  (\u2203 u : F \u03b1, p u) \u2194 (\u2203 u : F \u03b2, q u) :=\nbegin\n  split; rintro \u27e8u,h\u2082\u27e9; [ use f <$$> u, use g <$$> u ],\n  { apply (h\u2081 u).mp h\u2082 },\n  { apply (h\u2081 _).mpr _,\n    simp only [mvfunctor.map_map,h\u2080,is_lawful_mvfunctor.id_map,h\u2082] },\nend\nvariables {F}\n\nlemma liftp_def (x : F \u03b1) : liftp' p x \u2194 \u2203 u : F (subtype_ p), subtype_val p <$$> u = x :=\nexists_iff_exists_of_mono F _ _ (to_subtype_of_subtype p) (by simp [mvfunctor.map_map])\n\nlemma liftr_def (x y : F \u03b1) :\n  liftr' r x y \u2194\n  \u2203 u : F (subtype_ r), (typevec.prod.fst \u229a subtype_val r) <$$> u = x \u2227\n                        (typevec.prod.snd \u229a subtype_val r) <$$> u = y :=\nexists_iff_exists_of_mono _ _ _ (to_subtype'_of_subtype' r)\n  (by simp only [map_map, comp_assoc, subtype_val_to_subtype']; simp [comp])\n\nend liftp'\n\nend mvfunctor\n\nopen nat\n\nnamespace mvfunctor\n\nopen typevec\n\nsection liftp_last_pred_iff\nvariables  {F : typevec.{u} (n+1) \u2192 Type*} [mvfunctor F] [is_lawful_mvfunctor F]\n           {\u03b1 : typevec.{u} n}\nvariables (p : \u03b1 \u27f9 repeat n Prop)\n          (r : \u03b1 \u2297 \u03b1 \u27f9 repeat n Prop)\n\nopen mvfunctor\n\nvariables {\u03b2 : Type u}\nvariables (pp : \u03b2 \u2192 Prop)\n\nprivate def f : \u03a0 (n \u03b1), (\u03bb (i : fin2 (n + 1)), {p_1 // of_repeat (pred_last' \u03b1 pp i p_1)}) \u27f9\n    \u03bb (i : fin2 (n + 1)), {p_1 : (\u03b1 ::: \u03b2) i // pred_last \u03b1 pp p_1}\n| _ \u03b1 (fin2.fs i) x := \u27e8 x.val, cast (by simp only [pred_last]; erw const_iff_true) x.property \u27e9\n| _ \u03b1 fin2.fz x := \u27e8 x.val, x.property \u27e9\n\nprivate def g : \u03a0 (n \u03b1), (\u03bb (i : fin2 (n + 1)), {p_1 : (\u03b1 ::: \u03b2) i // pred_last \u03b1 pp p_1}) \u27f9\n    (\u03bb (i : fin2 (n + 1)), {p_1 // of_repeat (pred_last' \u03b1 pp i p_1)})\n| _ \u03b1 (fin2.fs i) x := \u27e8 x.val, cast (by simp only [pred_last]; erw const_iff_true) x.property \u27e9\n| _ \u03b1 fin2.fz x := \u27e8 x.val, x.property \u27e9\n\nlemma liftp_last_pred_iff {\u03b2} (p : \u03b2 \u2192 Prop) (x : F (\u03b1 ::: \u03b2)) :\n  liftp' (pred_last' _ p) x \u2194 liftp (pred_last _ p) x :=\nbegin\n  dsimp only [liftp,liftp'],\n  apply exists_iff_exists_of_mono F (f _ n \u03b1) (g _ n \u03b1),\n  { ext i \u27e8x,_\u27e9, cases i; refl },\n  { intros, rw [mvfunctor.map_map,(\u229a)],\n    congr'; ext i \u27e8x,_\u27e9; cases i; refl }\nend\n\nopen function\nvariables (rr : \u03b2 \u2192 \u03b2 \u2192 Prop)\n\nprivate def f :\n  \u03a0 (n \u03b1),\n    (\u03bb (i : fin2 (n + 1)),\n      {p_1 : _ \u00d7 _ // of_repeat (rel_last' \u03b1 rr i (typevec.prod.mk _ p_1.fst p_1.snd))}) \u27f9\n    \u03bb (i : fin2 (n + 1)), {p_1 : (\u03b1 ::: \u03b2) i \u00d7 _ // rel_last \u03b1 rr (p_1.fst) (p_1.snd)}\n| _ \u03b1 (fin2.fs i) x := \u27e8 x.val, cast (by simp only [rel_last]; erw repeat_eq_iff_eq) x.property \u27e9\n| _ \u03b1 fin2.fz x := \u27e8 x.val, x.property \u27e9\n\nprivate def g :\n  \u03a0 (n \u03b1), (\u03bb (i : fin2 (n + 1)), {p_1 : (\u03b1 ::: \u03b2) i \u00d7 _ // rel_last \u03b1 rr (p_1.fst) (p_1.snd)}) \u27f9\n    (\u03bb (i : fin2 (n + 1)),\n      {p_1 : _ \u00d7 _ // of_repeat (rel_last' \u03b1 rr i (typevec.prod.mk _ p_1.1 p_1.2))})\n| _ \u03b1 (fin2.fs i) x := \u27e8 x.val, cast (by simp only [rel_last]; erw repeat_eq_iff_eq) x.property \u27e9\n| _ \u03b1 fin2.fz x := \u27e8 x.val, x.property \u27e9\n\nlemma liftr_last_rel_iff  (x y : F (\u03b1 ::: \u03b2)) :\n  liftr' (rel_last' _ rr) x y \u2194 liftr (rel_last _ rr) x y :=\nbegin\n  dsimp only [liftr,liftr'],\n  apply exists_iff_exists_of_mono F (f rr _ _) (g rr _ _),\n  { ext i \u27e8x,_\u27e9 : 2, cases i; refl, },\n  { intros, rw [mvfunctor.map_map,mvfunctor.map_map,(\u229a),(\u229a)],\n    congr'; ext i \u27e8x,_\u27e9; cases i; refl }\nend\n\nend liftp_last_pred_iff\n\nend mvfunctor\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/control/functor/multivariate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3877349174823433}}
{"text": "/-\nCopyright (c) 2020 Joseph Myers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Joseph Myers.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.affine_space.basic\nimport Mathlib.linear_algebra.tensor_product\nimport Mathlib.data.set.intervals.unordered_interval\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 l u_6 u_7 u_8 u_9 u_10 u_11 u_12 \n\nnamespace Mathlib\n\n/-!\n# Affine maps\n\nThis file defines affine maps.\n\n## Main definitions\n\n* `affine_map` is the type of affine maps between two affine spaces with the same ring `k`.  Various\n  basic examples of affine maps are defined, including `const`, `id`, `line_map` and `homothety`.\n\n## Notations\n\n* `P1 \u2192\u1d43[k] P2` is a notation for `affine_map k P1 P2`;\n* `affine_space V P`: a localized notation for `add_torsor V P` defined in\n  `linear_algebra.affine_space.basic`.\n\n## Implementation notes\n\n`out_param` is used in the definition of `[add_torsor V P]` to make `V` an implicit argument\n(deduced from `P`) in most cases; `include V` is needed in many cases for `V`, and type classes\nusing it, to be added as implicit arguments to individual lemmas.  As for modules, `k` is an\nexplicit argument rather than implied by `P` or `V`.\n\nThis file only provides purely algebraic definitions and results. Those depending on analysis or\ntopology are defined elsewhere; see `analysis.normed_space.add_torsor` and\n`topology.algebra.affine`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Affine_space\n* https://en.wikipedia.org/wiki/Principal_homogeneous_space\n-/\n\n/-- An `affine_map k P1 P2` (notation: `P1 \u2192\u1d43[k] P2`) is a map from `P1` to `P2` that\ninduces a corresponding linear map from `V1` to `V2`. -/\nstructure affine_map (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) {V2 : Type u_4} (P2 : Type u_5) [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] \nwhere\n  to_fun : P1 \u2192 P2\n  linear : linear_map k V1 V2\n  map_vadd' : \u2200 (p : P1) (v : V1), to_fun (v +\u1d65 p) = coe_fn linear v +\u1d65 to_fun p\n\nprotected instance affine_map.has_coe_to_fun (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) {V2 : Type u_4} (P2 : Type u_5) [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : has_coe_to_fun (affine_map k P1 P2) :=\n  has_coe_to_fun.mk (fun (x : affine_map k P1 P2) => P1 \u2192 P2) affine_map.to_fun\n\nnamespace linear_map\n\n\n/-- Reinterpret a linear map as an affine map. -/\ndef to_affine_map {k : Type u_1} {V\u2081 : Type u_2} {V\u2082 : Type u_3} [ring k] [add_comm_group V\u2081] [module k V\u2081] [add_comm_group V\u2082] [module k V\u2082] (f : linear_map k V\u2081 V\u2082) : affine_map k V\u2081 V\u2082 :=\n  affine_map.mk (\u21d1f) f sorry\n\n@[simp] theorem coe_to_affine_map {k : Type u_1} {V\u2081 : Type u_2} {V\u2082 : Type u_3} [ring k] [add_comm_group V\u2081] [module k V\u2081] [add_comm_group V\u2082] [module k V\u2082] (f : linear_map k V\u2081 V\u2082) : \u21d1(to_affine_map f) = \u21d1f :=\n  rfl\n\n@[simp] theorem to_affine_map_linear {k : Type u_1} {V\u2081 : Type u_2} {V\u2082 : Type u_3} [ring k] [add_comm_group V\u2081] [module k V\u2081] [add_comm_group V\u2082] [module k V\u2082] (f : linear_map k V\u2081 V\u2082) : affine_map.linear (to_affine_map f) = f :=\n  rfl\n\nend linear_map\n\n\nnamespace affine_map\n\n\n/-- Constructing an affine map and coercing back to a function\nproduces the same map. -/\n@[simp] theorem coe_mk {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : P1 \u2192 P2) (linear : linear_map k V1 V2) (add : \u2200 (p : P1) (v : V1), f (v +\u1d65 p) = coe_fn linear v +\u1d65 f p) : \u21d1(mk f linear add) = f :=\n  rfl\n\n/-- `to_fun` is the same as the result of coercing to a function. -/\n@[simp] theorem to_fun_eq_coe {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) : to_fun f = \u21d1f :=\n  rfl\n\n/-- An affine map on the result of adding a vector to a point produces\nthe same result as the linear map applied to that vector, added to the\naffine map applied to that point. -/\n@[simp] theorem map_vadd {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) (p : P1) (v : V1) : coe_fn f (v +\u1d65 p) = coe_fn (linear f) v +\u1d65 coe_fn f p :=\n  map_vadd' f p v\n\n/-- The linear map on the result of subtracting two points is the\nresult of subtracting the result of the affine map on those two\npoints. -/\n@[simp] theorem linear_map_vsub {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) (p1 : P1) (p2 : P1) : coe_fn (linear f) (p1 -\u1d65 p2) = coe_fn f p1 -\u1d65 coe_fn f p2 := sorry\n\n/-- Two affine maps are equal if they coerce to the same function. -/\ntheorem ext {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] {f : affine_map k P1 P2} {g : affine_map k P1 P2} (h : \u2200 (p : P1), coe_fn f p = coe_fn g p) : f = g := sorry\n\ntheorem ext_iff {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] {f : affine_map k P1 P2} {g : affine_map k P1 P2} : f = g \u2194 \u2200 (p : P1), coe_fn f p = coe_fn g p :=\n  { mp := fun (h : f = g) (p : P1) => h \u25b8 rfl, mpr := ext }\n\ntheorem injective_coe_fn {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : function.injective fun (f : affine_map k P1 P2) (x : P1) => coe_fn f x := sorry\n\nprotected theorem congr_arg {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) {x : P1} {y : P1} (h : x = y) : coe_fn f x = coe_fn f y :=\n  congr_arg (\u21d1f) h\n\nprotected theorem congr_fun {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] {f : affine_map k P1 P2} {g : affine_map k P1 P2} (h : f = g) (x : P1) : coe_fn f x = coe_fn g x :=\n  h \u25b8 rfl\n\n/-- Constant function as an `affine_map`. -/\ndef const (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (p : P2) : affine_map k P1 P2 :=\n  mk (function.const P1 p) 0 sorry\n\n@[simp] theorem coe_const (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (p : P2) : \u21d1(const k P1 p) = function.const P1 p :=\n  rfl\n\n@[simp] theorem const_linear (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (p : P2) : linear (const k P1 p) = 0 :=\n  rfl\n\nprotected instance nonempty {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : Nonempty (affine_map k P1 P2) :=\n  nonempty.elim add_torsor.nonempty fun (p : P2) => Nonempty.intro (const k P1 p)\n\n/-- Construct an affine map by verifying the relation between the map and its linear part at one\nbase point. Namely, this function takes a map `f : P\u2081 \u2192 P\u2082`, a linear map `f' : V\u2081 \u2192\u2097[k] V\u2082`, and\na point `p` such that for any other point `p'` we have `f p' = f' (p' -\u1d65 p) +\u1d65 f p`. -/\ndef mk' {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : P1 \u2192 P2) (f' : linear_map k V1 V2) (p : P1) (h : \u2200 (p' : P1), f p' = coe_fn f' (p' -\u1d65 p) +\u1d65 f p) : affine_map k P1 P2 :=\n  mk f f' sorry\n\n@[simp] theorem coe_mk' {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : P1 \u2192 P2) (f' : linear_map k V1 V2) (p : P1) (h : \u2200 (p' : P1), f p' = coe_fn f' (p' -\u1d65 p) +\u1d65 f p) : \u21d1(mk' f f' p h) = f :=\n  rfl\n\n@[simp] theorem mk'_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : P1 \u2192 P2) (f' : linear_map k V1 V2) (p : P1) (h : \u2200 (p' : P1), f p' = coe_fn f' (p' -\u1d65 p) +\u1d65 f p) : linear (mk' f f' p h) = f' :=\n  rfl\n\n/-- The set of affine maps to a vector space is an additive commutative group. -/\nprotected instance add_comm_group {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] : add_comm_group (affine_map k P1 V2) :=\n  add_comm_group.mk (fun (f g : affine_map k P1 V2) => mk (\u21d1f + \u21d1g) (linear f + linear g) sorry) sorry (mk 0 0 sorry)\n    sorry sorry (fun (f : affine_map k P1 V2) => mk (-\u21d1f) (-linear f) sorry)\n    (add_group.sub._default (fun (f g : affine_map k P1 V2) => mk (\u21d1f + \u21d1g) (linear f + linear g) sorry) sorry\n      (mk 0 0 sorry) sorry sorry fun (f : affine_map k P1 V2) => mk (-\u21d1f) (-linear f) sorry)\n    sorry sorry\n\n@[simp] theorem coe_zero {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] : \u21d10 = 0 :=\n  rfl\n\n@[simp] theorem zero_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] : linear 0 = 0 :=\n  rfl\n\n@[simp] theorem coe_add {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] (f : affine_map k P1 V2) (g : affine_map k P1 V2) : \u21d1(f + g) = \u21d1f + \u21d1g :=\n  rfl\n\n@[simp] theorem add_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] (f : affine_map k P1 V2) (g : affine_map k P1 V2) : linear (f + g) = linear f + linear g :=\n  rfl\n\n/-- The space of affine maps from `P1` to `P2` is an affine space over the space of affine maps\nfrom `P1` to the vector space `V2` corresponding to `P2`. -/\nprotected instance add_torsor {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : add_torsor (affine_map k P1 V2) (affine_map k P1 P2) :=\n  add_torsor.mk\n    (fun (f : affine_map k P1 V2) (g : affine_map k P1 P2) =>\n      mk (fun (p : P1) => coe_fn f p +\u1d65 coe_fn g p) (linear f + linear g) sorry)\n    sorry sorry\n    (fun (f g : affine_map k P1 P2) => mk (fun (p : P1) => coe_fn f p -\u1d65 coe_fn g p) (linear f - linear g) sorry) sorry\n    sorry\n\n@[simp] theorem vadd_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 V2) (g : affine_map k P1 P2) (p : P1) : coe_fn (f +\u1d65 g) p = coe_fn f p +\u1d65 coe_fn g p :=\n  rfl\n\n@[simp] theorem vsub_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) (g : affine_map k P1 P2) (p : P1) : coe_fn (f -\u1d65 g) p = coe_fn f p -\u1d65 coe_fn g p :=\n  rfl\n\n/-- `prod.fst` as an `affine_map`. -/\ndef fst {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : affine_map k (P1 \u00d7 P2) P1 :=\n  mk prod.fst (linear_map.fst k V1 V2) sorry\n\n@[simp] theorem coe_fst {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : \u21d1fst = prod.fst :=\n  rfl\n\n@[simp] theorem fst_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : linear fst = linear_map.fst k V1 V2 :=\n  rfl\n\n/-- `prod.snd` as an `affine_map`. -/\ndef snd {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : affine_map k (P1 \u00d7 P2) P2 :=\n  mk prod.snd (linear_map.snd k V1 V2) sorry\n\n@[simp] theorem coe_snd {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : \u21d1snd = prod.snd :=\n  rfl\n\n@[simp] theorem snd_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] : linear snd = linear_map.snd k V1 V2 :=\n  rfl\n\n/-- Identity map as an affine map. -/\ndef id (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : affine_map k P1 P1 :=\n  mk id linear_map.id sorry\n\n/-- The identity affine map acts as the identity. -/\n@[simp] theorem coe_id (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : \u21d1(id k P1) = id :=\n  rfl\n\n@[simp] theorem id_linear (k : Type u_1) {V1 : Type u_2} (P1 : Type u_3) [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : linear (id k P1) = linear_map.id :=\n  rfl\n\n/-- The identity affine map acts as the identity. -/\ntheorem id_apply (k : Type u_1) {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p : P1) : coe_fn (id k P1) p = p :=\n  rfl\n\nprotected instance inhabited {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : Inhabited (affine_map k P1 P1) :=\n  { default := id k P1 }\n\n/-- Composition of affine maps. -/\ndef comp {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} {V3 : Type u_6} {P3 : Type u_7} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] [add_comm_group V3] [module k V3] [add_torsor V3 P3] (f : affine_map k P2 P3) (g : affine_map k P1 P2) : affine_map k P1 P3 :=\n  mk (\u21d1f \u2218 \u21d1g) (linear_map.comp (linear f) (linear g)) sorry\n\n/-- Composition of affine maps acts as applying the two functions. -/\n@[simp] theorem coe_comp {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} {V3 : Type u_6} {P3 : Type u_7} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] [add_comm_group V3] [module k V3] [add_torsor V3 P3] (f : affine_map k P2 P3) (g : affine_map k P1 P2) : \u21d1(comp f g) = \u21d1f \u2218 \u21d1g :=\n  rfl\n\n/-- Composition of affine maps acts as applying the two functions. -/\ntheorem comp_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} {V3 : Type u_6} {P3 : Type u_7} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] [add_comm_group V3] [module k V3] [add_torsor V3 P3] (f : affine_map k P2 P3) (g : affine_map k P1 P2) (p : P1) : coe_fn (comp f g) p = coe_fn f (coe_fn g p) :=\n  rfl\n\n@[simp] theorem comp_id {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) : comp f (id k P1) = f :=\n  ext fun (p : P1) => rfl\n\n@[simp] theorem id_comp {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) : comp (id k P2) f = f :=\n  ext fun (p : P1) => rfl\n\ntheorem comp_assoc {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} {V3 : Type u_6} {P3 : Type u_7} {V4 : Type u_8} {P4 : Type u_9} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] [add_comm_group V3] [module k V3] [add_torsor V3 P3] [add_comm_group V4] [module k V4] [add_torsor V4 P4] (f\u2083\u2084 : affine_map k P3 P4) (f\u2082\u2083 : affine_map k P2 P3) (f\u2081\u2082 : affine_map k P1 P2) : comp (comp f\u2083\u2084 f\u2082\u2083) f\u2081\u2082 = comp f\u2083\u2084 (comp f\u2082\u2083 f\u2081\u2082) :=\n  rfl\n\nprotected instance monoid {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : monoid (affine_map k P1 P1) :=\n  monoid.mk comp comp_assoc (id k P1) id_comp comp_id\n\n@[simp] theorem coe_mul {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (f : affine_map k P1 P1) (g : affine_map k P1 P1) : \u21d1(f * g) = \u21d1f \u2218 \u21d1g :=\n  rfl\n\n@[simp] theorem coe_one {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] : \u21d11 = id :=\n  rfl\n\n/-! ### Definition of `affine_map.line_map` and lemmas about it -/\n\n/-- The affine map from `k` to `P1` sending `0` to `p\u2080` and `1` to `p\u2081`. -/\ndef line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : affine_map k k P1 :=\n  linear_map.to_affine_map (linear_map.smul_right linear_map.id (p\u2081 -\u1d65 p\u2080)) +\u1d65 const k k p\u2080\n\ntheorem coe_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : \u21d1(line_map p\u2080 p\u2081) = fun (c : k) => c \u2022 (p\u2081 -\u1d65 p\u2080) +\u1d65 p\u2080 :=\n  rfl\n\ntheorem line_map_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : coe_fn (line_map p\u2080 p\u2081) c = c \u2022 (p\u2081 -\u1d65 p\u2080) +\u1d65 p\u2080 :=\n  rfl\n\ntheorem line_map_apply_module' {k : Type u_1} {V1 : Type u_2} [ring k] [add_comm_group V1] [module k V1] (p\u2080 : V1) (p\u2081 : V1) (c : k) : coe_fn (line_map p\u2080 p\u2081) c = c \u2022 (p\u2081 - p\u2080) + p\u2080 :=\n  rfl\n\ntheorem line_map_apply_module {k : Type u_1} {V1 : Type u_2} [ring k] [add_comm_group V1] [module k V1] (p\u2080 : V1) (p\u2081 : V1) (c : k) : coe_fn (line_map p\u2080 p\u2081) c = (1 - c) \u2022 p\u2080 + c \u2022 p\u2081 := sorry\n\ntheorem line_map_apply_ring' {k : Type u_1} [ring k] (a : k) (b : k) (c : k) : coe_fn (line_map a b) c = c * (b - a) + a :=\n  rfl\n\ntheorem line_map_apply_ring {k : Type u_1} [ring k] (a : k) (b : k) (c : k) : coe_fn (line_map a b) c = (1 - c) * a + c * b :=\n  line_map_apply_module a b c\n\ntheorem line_map_vadd_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p : P1) (v : V1) (c : k) : coe_fn (line_map p (v +\u1d65 p)) c = c \u2022 v +\u1d65 p :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn (line_map p (v +\u1d65 p)) c = c \u2022 v +\u1d65 p)) (line_map_apply p (v +\u1d65 p) c)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (c \u2022 (v +\u1d65 p -\u1d65 p) +\u1d65 p = c \u2022 v +\u1d65 p)) (vadd_vsub v p))) (Eq.refl (c \u2022 v +\u1d65 p)))\n\n@[simp] theorem line_map_linear {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : linear (line_map p\u2080 p\u2081) = linear_map.smul_right linear_map.id (p\u2081 -\u1d65 p\u2080) :=\n  add_zero (linear (linear_map.to_affine_map (linear_map.smul_right linear_map.id (p\u2081 -\u1d65 p\u2080))))\n\ntheorem line_map_same_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p : P1) (c : k) : coe_fn (line_map p p) c = p := sorry\n\n@[simp] theorem line_map_same {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p : P1) : line_map p p = const k k p :=\n  ext (line_map_same_apply p)\n\n@[simp] theorem line_map_apply_zero {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : coe_fn (line_map p\u2080 p\u2081) 0 = p\u2080 := sorry\n\n@[simp] theorem line_map_apply_one {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : coe_fn (line_map p\u2080 p\u2081) 1 = p\u2081 := sorry\n\n@[simp] theorem apply_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) (p\u2080 : P1) (p\u2081 : P1) (c : k) : coe_fn f (coe_fn (line_map p\u2080 p\u2081) c) = coe_fn (line_map (coe_fn f p\u2080) (coe_fn f p\u2081)) c := sorry\n\n@[simp] theorem comp_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (f : affine_map k P1 P2) (p\u2080 : P1) (p\u2081 : P1) : comp f (line_map p\u2080 p\u2081) = line_map (coe_fn f p\u2080) (coe_fn f p\u2081) :=\n  ext (apply_line_map f p\u2080 p\u2081)\n\n@[simp] theorem fst_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (p\u2080 : P1 \u00d7 P2) (p\u2081 : P1 \u00d7 P2) (c : k) : prod.fst (coe_fn (line_map p\u2080 p\u2081) c) = coe_fn (line_map (prod.fst p\u2080) (prod.fst p\u2081)) c :=\n  apply_line_map fst p\u2080 p\u2081 c\n\n@[simp] theorem snd_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} {P2 : Type u_5} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] [add_torsor V2 P2] (p\u2080 : P1 \u00d7 P2) (p\u2081 : P1 \u00d7 P2) (c : k) : prod.snd (coe_fn (line_map p\u2080 p\u2081) c) = coe_fn (line_map (prod.snd p\u2080) (prod.snd p\u2081)) c :=\n  apply_line_map snd p\u2080 p\u2081 c\n\ntheorem line_map_symm {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) : line_map p\u2080 p\u2081 = comp (line_map p\u2081 p\u2080) (line_map 1 0) := sorry\n\ntheorem line_map_apply_one_sub {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : coe_fn (line_map p\u2080 p\u2081) (1 - c) = coe_fn (line_map p\u2081 p\u2080) c := sorry\n\n@[simp] theorem line_map_vsub_left {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : coe_fn (line_map p\u2080 p\u2081) c -\u1d65 p\u2080 = c \u2022 (p\u2081 -\u1d65 p\u2080) :=\n  vadd_vsub (coe_fn (linear_map.to_affine_map (linear_map.smul_right linear_map.id (p\u2081 -\u1d65 p\u2080))) c)\n    (coe_fn (const k k p\u2080) c)\n\n@[simp] theorem left_vsub_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : p\u2080 -\u1d65 coe_fn (line_map p\u2080 p\u2081) c = c \u2022 (p\u2080 -\u1d65 p\u2081) := sorry\n\n@[simp] theorem line_map_vsub_right {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : coe_fn (line_map p\u2080 p\u2081) c -\u1d65 p\u2081 = (1 - c) \u2022 (p\u2080 -\u1d65 p\u2081) := sorry\n\n@[simp] theorem right_vsub_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2080 : P1) (p\u2081 : P1) (c : k) : p\u2081 -\u1d65 coe_fn (line_map p\u2080 p\u2081) c = (1 - c) \u2022 (p\u2081 -\u1d65 p\u2080) := sorry\n\ntheorem line_map_vadd_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (v\u2081 : V1) (v\u2082 : V1) (p\u2081 : P1) (p\u2082 : P1) (c : k) : coe_fn (line_map v\u2081 v\u2082) c +\u1d65 coe_fn (line_map p\u2081 p\u2082) c = coe_fn (line_map (v\u2081 +\u1d65 p\u2081) (v\u2082 +\u1d65 p\u2082)) c :=\n  apply_line_map (fst +\u1d65 snd) (v\u2081, p\u2081) (v\u2082, p\u2082) c\n\ntheorem line_map_vsub_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (p\u2081 : P1) (p\u2082 : P1) (p\u2083 : P1) (p\u2084 : P1) (c : k) : coe_fn (line_map p\u2081 p\u2082) c -\u1d65 coe_fn (line_map p\u2083 p\u2084) c = coe_fn (line_map (p\u2081 -\u1d65 p\u2083) (p\u2082 -\u1d65 p\u2084)) c :=\n  apply_line_map (fst -\u1d65 snd) (p\u2081, p\u2083) (p\u2082, p\u2084) c\n\n-- Why Lean fails to find this instance without a hint?\n\n/-- Decomposition of an affine map in the special case when the point space and vector space\nare the same. -/\ntheorem decomp {k : Type u_1} {V1 : Type u_2} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_comm_group V2] [module k V2] (f : affine_map k V1 V2) : \u21d1f = \u21d1(linear f) + fun (z : V1) => coe_fn f 0 := sorry\n\n/-- Decomposition of an affine map in the special case when the point space and vector space\nare the same. -/\ntheorem decomp' {k : Type u_1} {V1 : Type u_2} {V2 : Type u_4} [ring k] [add_comm_group V1] [module k V1] [add_comm_group V2] [module k V2] (f : affine_map k V1 V2) : \u21d1(linear f) = \u21d1f - fun (z : V1) => coe_fn f 0 := sorry\n\ntheorem image_interval {k : Type u_1} [linear_ordered_field k] (f : affine_map k k k) (a : k) (b : k) : \u21d1f '' set.interval a b = set.interval (coe_fn f a) (coe_fn f b) := sorry\n\n/-- Evaluation at a point as an affine map. -/\ndef proj {k : Type u_1} [ring k] {\u03b9 : Type u_10} {V : \u03b9 \u2192 Type u_11} {P : \u03b9 \u2192 Type u_12} [(i : \u03b9) \u2192 add_comm_group (V i)] [(i : \u03b9) \u2192 semimodule k (V i)] [(i : \u03b9) \u2192 add_torsor (V i) (P i)] (i : \u03b9) : affine_map k ((i : \u03b9) \u2192 P i) (P i) :=\n  mk (fun (f : (i : \u03b9) \u2192 P i) => f i) (linear_map.proj i) sorry\n\n@[simp] theorem proj_apply {k : Type u_1} [ring k] {\u03b9 : Type u_10} {V : \u03b9 \u2192 Type u_11} {P : \u03b9 \u2192 Type u_12} [(i : \u03b9) \u2192 add_comm_group (V i)] [(i : \u03b9) \u2192 semimodule k (V i)] [(i : \u03b9) \u2192 add_torsor (V i) (P i)] (i : \u03b9) (f : (i : \u03b9) \u2192 P i) : coe_fn (proj i) f = f i :=\n  rfl\n\n@[simp] theorem proj_linear {k : Type u_1} [ring k] {\u03b9 : Type u_10} {V : \u03b9 \u2192 Type u_11} {P : \u03b9 \u2192 Type u_12} [(i : \u03b9) \u2192 add_comm_group (V i)] [(i : \u03b9) \u2192 semimodule k (V i)] [(i : \u03b9) \u2192 add_torsor (V i) (P i)] (i : \u03b9) : linear (proj i) = linear_map.proj i :=\n  rfl\n\ntheorem pi_line_map_apply {k : Type u_1} [ring k] {\u03b9 : Type u_10} {V : \u03b9 \u2192 Type u_11} {P : \u03b9 \u2192 Type u_12} [(i : \u03b9) \u2192 add_comm_group (V i)] [(i : \u03b9) \u2192 semimodule k (V i)] [(i : \u03b9) \u2192 add_torsor (V i) (P i)] (f : (i : \u03b9) \u2192 P i) (g : (i : \u03b9) \u2192 P i) (c : k) (i : \u03b9) : coe_fn (line_map f g) c i = coe_fn (line_map (f i) (g i)) c :=\n  apply_line_map (proj i) f g c\n\nend affine_map\n\n\nnamespace affine_map\n\n\n/-- If `k` is a commutative ring, then the set of affine maps with codomain in a `k`-module\nis a `k`-module. -/\nprotected instance module {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] : module k (affine_map k P1 V2) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem coe_smul {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} {V2 : Type u_4} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] [add_comm_group V2] [module k V2] (c : k) (f : affine_map k P1 V2) : \u21d1(c \u2022 f) = c \u2022 \u21d1f :=\n  rfl\n\n/-- `homothety c r` is the homothety about `c` with scale factor `r`. -/\ndef homothety {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r : k) : affine_map k P1 P1 :=\n  r \u2022 (id k P1 -\u1d65 const k P1 c) +\u1d65 const k P1 c\n\ntheorem homothety_def {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r : k) : homothety c r = r \u2022 (id k P1 -\u1d65 const k P1 c) +\u1d65 const k P1 c :=\n  rfl\n\ntheorem homothety_apply {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r : k) (p : P1) : coe_fn (homothety c r) p = r \u2022 (p -\u1d65 c) +\u1d65 c :=\n  rfl\n\ntheorem homothety_eq_line_map {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r : k) (p : P1) : coe_fn (homothety c r) p = coe_fn (line_map c p) r :=\n  rfl\n\n@[simp] theorem homothety_one {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : homothety c 1 = id k P1 := sorry\n\ntheorem homothety_mul {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r\u2081 : k) (r\u2082 : k) : homothety c (r\u2081 * r\u2082) = comp (homothety c r\u2081) (homothety c r\u2082) := sorry\n\n@[simp] theorem homothety_zero {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : homothety c 0 = const k P1 c := sorry\n\n@[simp] theorem homothety_add {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) (r\u2081 : k) (r\u2082 : k) : homothety c (r\u2081 + r\u2082) = r\u2081 \u2022 (id k P1 -\u1d65 const k P1 c) +\u1d65 homothety c r\u2082 := sorry\n\n/-- `homothety` as a multiplicative monoid homomorphism. -/\ndef homothety_hom {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : k \u2192* affine_map k P1 P1 :=\n  monoid_hom.mk (homothety c) (homothety_one c) (homothety_mul c)\n\n@[simp] theorem coe_homothety_hom {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : \u21d1(homothety_hom c) = homothety c :=\n  rfl\n\n/-- `homothety` as an affine map. -/\ndef homothety_affine {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : affine_map k k (affine_map k P1 P1) :=\n  mk (homothety c) (coe_fn (linear_map.flip (linear_map.lsmul k (affine_map k P1 V1))) (id k P1 -\u1d65 const k P1 c)) sorry\n\n@[simp] theorem coe_homothety_affine {k : Type u_1} {V1 : Type u_2} {P1 : Type u_3} [comm_ring k] [add_comm_group V1] [module k V1] [add_torsor V1 P1] (c : P1) : \u21d1(homothety_affine c) = homothety c :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/affine_space/affine_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.38773490958809803}}
{"text": "/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n\nTheorems that require decidability hypotheses are in the namespace \"decidable\".\nClassical versions are in the namespace \"classical\".\n\nNote: in the presence of automation, this whole file may be unnecessary. On the other hand,\nmaybe it is useful for writing automation.\n-/\nimport data.prod tactic.cache\n\n/-\n    miscellany\n\n    TODO: move elsewhere\n-/\n\nsection miscellany\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\n@[reducible] def hidden {a : \u03b1} := a\n\ndef empty.elim {C : Sort*} : empty \u2192 C.\n\ninstance : subsingleton empty := \u27e8\u03bba, a.elim\u27e9\n\ninstance : decidable_eq empty := \u03bba, a.elim\n\n@[priority 0] instance decidable_eq_of_subsingleton\n  {\u03b1} [subsingleton \u03b1] : decidable_eq \u03b1\n| a b := is_true (subsingleton.elim a b)\n\n/- Add an instance to \"undo\" coercion transitivity into a chain of coercions, because\n   most simp lemmas are stated with respect to simple coercions and will not match when\n   part of a chain. -/\n@[simp] theorem coe_coe {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t \u03b2 \u03b3]\n  (a : \u03b1) : (a : \u03b3) = (a : \u03b2) := rfl\n\n@[simp] theorem coe_fn_coe_trans\n  {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_fun \u03b3]\n  (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n@[simp] theorem coe_fn_coe_base\n  {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_fun \u03b2]\n  (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n@[simp] theorem coe_sort_coe_trans\n  {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_sort \u03b3]\n  (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n\n@[simp] theorem coe_sort_coe_base\n  {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_sort \u03b2]\n  (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n\n/-- `pempty` is the universe-polymorphic analogue of `empty`. -/\n@[derive decidable_eq]\ninductive {u} pempty : Sort u\n\ndef pempty.elim {C : Sort*} : pempty \u2192 C.\n\ninstance subsingleton_pempty : subsingleton pempty := \u27e8\u03bba, a.elim\u27e9\n\nlemma congr_arg_heq {\u03b1} {\u03b2 : \u03b1 \u2192 Sort*} (f : \u2200 a, \u03b2 a) : \u2200 {a\u2081 a\u2082 : \u03b1}, a\u2081 = a\u2082 \u2192 f a\u2081 == f a\u2082\n| a _ rfl := heq.rfl\n\nlemma plift.down_inj {\u03b1 : Sort*} : \u2200 (a b : plift \u03b1), a.down = b.down \u2192 a = b\n| \u27e8a\u27e9 \u27e8b\u27e9 rfl := rfl\n\nend miscellany\n\n/-\n    propositional connectives\n-/\n\n@[simp] theorem false_ne_true : false \u2260 true\n| h := h.symm \u25b8 trivial\n\nsection propositional\nvariables {a b c d : Prop}\n\n/- implies -/\n\ntheorem iff_of_eq (e : a = b) : a \u2194 b := e \u25b8 iff.rfl\n\ntheorem iff_iff_eq : (a \u2194 b) \u2194 a = b := \u27e8propext, iff_of_eq\u27e9\n\n@[simp] theorem imp_self : (a \u2192 a) \u2194 true := iff_true_intro id\n\ntheorem imp_intro {\u03b1 \u03b2} (h : \u03b1) (h\u2082 : \u03b2) : \u03b1 := h\n\ntheorem imp_false : (a \u2192 false) \u2194 \u00ac a := iff.rfl\n\ntheorem imp_and_distrib {\u03b1} : (\u03b1 \u2192 b \u2227 c) \u2194 (\u03b1 \u2192 b) \u2227 (\u03b1 \u2192 c) :=\n\u27e8\u03bb h, \u27e8\u03bb ha, (h ha).left, \u03bb ha, (h ha).right\u27e9,\n \u03bb h ha, \u27e8h.left ha, h.right ha\u27e9\u27e9\n\n@[simp] theorem and_imp : (a \u2227 b \u2192 c) \u2194 (a \u2192 b \u2192 c) :=\niff.intro (\u03bb h ha hb, h \u27e8ha, hb\u27e9) (\u03bb h \u27e8ha, hb\u27e9, h ha hb)\n\ntheorem iff_def : (a \u2194 b) \u2194 (a \u2192 b) \u2227 (b \u2192 a) :=\niff_iff_implies_and_implies _ _\n\ntheorem iff_def' : (a \u2194 b) \u2194 (b \u2192 a) \u2227 (a \u2192 b) :=\niff_def.trans and.comm\n\n@[simp] theorem imp_true_iff {\u03b1 : Sort*} : (\u03b1 \u2192 true) \u2194 true :=\niff_true_intro $ \u03bb_, trivial\n\n@[simp] theorem imp_iff_right (ha : a) : (a \u2192 b) \u2194 b :=\n\u27e8\u03bbf, f ha, imp_intro\u27e9\n\n/- not -/\n\ntheorem not.elim {\u03b1 : Sort*} (H1 : \u00aca) (H2 : a) : \u03b1 := absurd H2 H1\n\n@[reducible] theorem not.imp {a b : Prop} (H2 : \u00acb) (H1 : a \u2192 b) : \u00aca := mt H1 H2\n\ntheorem not_not_of_not_imp : \u00ac(a \u2192 b) \u2192 \u00ac\u00aca :=\nmt not.elim\n\ntheorem not_of_not_imp {\u03b1} : \u00ac(\u03b1 \u2192 b) \u2192 \u00acb :=\nmt imp_intro\n\ntheorem dec_em (p : Prop) [decidable p] : p \u2228 \u00acp := decidable.em p\n\ntheorem by_contradiction {p} [decidable p] : (\u00acp \u2192 false) \u2192 p :=\ndecidable.by_contradiction\n\n@[simp] theorem not_not [decidable a] : \u00ac\u00aca \u2194 a :=\niff.intro by_contradiction not_not_intro\n\ntheorem of_not_not [decidable a] : \u00ac\u00aca \u2192 a :=\nby_contradiction\n\ntheorem of_not_imp [decidable a] (h : \u00ac (a \u2192 b)) : a :=\nby_contradiction (not_not_of_not_imp h)\n\ntheorem not.imp_symm [decidable a] (h : \u00aca \u2192 b) (hb : \u00acb) : a :=\nby_contradiction $ hb \u2218 h\n\ntheorem not_imp_comm [decidable a] [decidable b] : (\u00aca \u2192 b) \u2194 (\u00acb \u2192 a) :=\n\u27e8not.imp_symm, not.imp_symm\u27e9\n\ntheorem imp.swap : (a \u2192 b \u2192 c) \u2194 (b \u2192 a \u2192 c) :=\n\u27e8function.swap, function.swap\u27e9\n\ntheorem imp_not_comm : (a \u2192 \u00acb) \u2194 (b \u2192 \u00aca) :=\nimp.swap\n\n/- and -/\n\ntheorem not_and_of_not_left (b : Prop) : \u00aca \u2192 \u00ac(a \u2227 b) :=\nmt and.left\n\ntheorem not_and_of_not_right (a : Prop) {b : Prop} : \u00acb \u2192 \u00ac(a \u2227 b) :=\nmt and.right\n\ntheorem and.imp_left (h : a \u2192 b) : a \u2227 c \u2192 b \u2227 c :=\nand.imp h id\n\ntheorem and.imp_right (h : a \u2192 b) : c \u2227 a \u2192 c \u2227 b :=\nand.imp id h\n\nlemma and.right_comm : (a \u2227 b) \u2227 c \u2194 (a \u2227 c) \u2227 b :=\nby simp [and.left_comm, and.comm]\n\nlemma and.rotate : a \u2227 b \u2227 c \u2194 b \u2227 c \u2227 a :=\nby simp [and.left_comm, and.comm]\n\ntheorem and_not_self_iff (a : Prop) : a \u2227 \u00ac a \u2194 false :=\niff.intro (assume h, (h.right) (h.left)) (assume h, h.elim)\n\ntheorem not_and_self_iff (a : Prop) : \u00ac a \u2227 a \u2194 false :=\niff.intro (assume \u27e8hna, ha\u27e9, hna ha) false.elim\n\ntheorem and_iff_left_of_imp {a b : Prop} (h : a \u2192 b) : (a \u2227 b) \u2194 a :=\niff.intro and.left (\u03bb ha, \u27e8ha, h ha\u27e9)\n\ntheorem and_iff_right_of_imp {a b : Prop} (h : b \u2192 a) : (a \u2227 b) \u2194 b :=\niff.intro and.right (\u03bb hb, \u27e8h hb, hb\u27e9)\n\nlemma and.congr_right_iff : (a \u2227 b \u2194 a \u2227 c) \u2194 (a \u2192 (b \u2194 c)) :=\n\u27e8\u03bb h ha, by simp [ha] at h; exact h, and_congr_right\u27e9\n\n/- or -/\n\ntheorem or_of_or_of_imp_of_imp (h\u2081 : a \u2228 b) (h\u2082 : a \u2192 c) (h\u2083 : b \u2192 d) : c \u2228 d :=\nor.imp h\u2082 h\u2083 h\u2081\n\ntheorem or_of_or_of_imp_left (h\u2081 : a \u2228 c) (h : a \u2192 b) : b \u2228 c :=\nor.imp_left h h\u2081\n\ntheorem or_of_or_of_imp_right (h\u2081 : c \u2228 a) (h : a \u2192 b) : c \u2228 b :=\nor.imp_right h h\u2081\n\ntheorem or.elim3 (h : a \u2228 b \u2228 c) (ha : a \u2192 d) (hb : b \u2192 d) (hc : c \u2192 d) : d :=\nor.elim h ha (assume h\u2082, or.elim h\u2082 hb hc)\n\ntheorem or_imp_distrib : (a \u2228 b \u2192 c) \u2194 (a \u2192 c) \u2227 (b \u2192 c) :=\n\u27e8assume h, \u27e8assume ha, h (or.inl ha), assume hb, h (or.inr hb)\u27e9,\n  assume \u27e8ha, hb\u27e9, or.rec ha hb\u27e9\n\ntheorem or_iff_not_imp_left [decidable a] : a \u2228 b \u2194 (\u00ac a \u2192 b) :=\n\u27e8or.resolve_left, \u03bb h, dite _ or.inl (or.inr \u2218 h)\u27e9\n\ntheorem or_iff_not_imp_right [decidable b] : a \u2228 b \u2194 (\u00ac b \u2192 a) :=\nor.comm.trans or_iff_not_imp_left\n\ntheorem not_imp_not [decidable a] : (\u00ac a \u2192 \u00ac b) \u2194 (b \u2192 a) :=\n\u27e8assume h hb, by_contradiction $ assume na, h na hb, mt\u27e9\n\n/- distributivity -/\n\ntheorem and_or_distrib_left : a \u2227 (b \u2228 c) \u2194 (a \u2227 b) \u2228 (a \u2227 c) :=\n\u27e8\u03bb \u27e8ha, hbc\u27e9, hbc.imp (and.intro ha) (and.intro ha),\n or.rec (and.imp_right or.inl) (and.imp_right or.inr)\u27e9\n\ntheorem or_and_distrib_right : (a \u2228 b) \u2227 c \u2194 (a \u2227 c) \u2228 (b \u2227 c) :=\n(and.comm.trans and_or_distrib_left).trans (or_congr and.comm and.comm)\n\ntheorem or_and_distrib_left : a \u2228 (b \u2227 c) \u2194 (a \u2228 b) \u2227 (a \u2228 c) :=\n\u27e8or.rec (\u03bbha, and.intro (or.inl ha) (or.inl ha)) (and.imp or.inr or.inr),\n and.rec $ or.rec (imp_intro \u2218 or.inl) (or.imp_right \u2218 and.intro)\u27e9\n\ntheorem and_or_distrib_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c) :=\n(or.comm.trans or_and_distrib_left).trans (and_congr or.comm or.comm)\n\n/- iff -/\n\ntheorem iff_of_true (ha : a) (hb : b) : a \u2194 b :=\n\u27e8\u03bb_, hb, \u03bb _, ha\u27e9\n\ntheorem iff_of_false (ha : \u00aca) (hb : \u00acb) : a \u2194 b :=\n\u27e8ha.elim, hb.elim\u27e9\n\ntheorem iff_true_left (ha : a) : (a \u2194 b) \u2194 b :=\n\u27e8\u03bb h, h.1 ha, iff_of_true ha\u27e9\n\ntheorem iff_true_right (ha : a) : (b \u2194 a) \u2194 b :=\niff.comm.trans (iff_true_left ha)\n\ntheorem iff_false_left (ha : \u00aca) : (a \u2194 b) \u2194 \u00acb :=\n\u27e8\u03bb h, mt h.2 ha, iff_of_false ha\u27e9\n\ntheorem iff_false_right (ha : \u00aca) : (b \u2194 a) \u2194 \u00acb :=\niff.comm.trans (iff_false_left ha)\n\ntheorem not_or_of_imp [decidable a] (h : a \u2192 b) : \u00ac a \u2228 b :=\nif ha : a then or.inr (h ha) else or.inl ha\n\ntheorem imp_iff_not_or [decidable a] : (a \u2192 b) \u2194 (\u00ac a \u2228 b) :=\n\u27e8not_or_of_imp, or.neg_resolve_left\u27e9\n\ntheorem imp_or_distrib [decidable a] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby simp [imp_iff_not_or, or.comm, or.left_comm]\n\ntheorem imp_or_distrib' [decidable b] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby by_cases b; simp [h, or_iff_right_of_imp ((\u2218) false.elim)]\n\ntheorem not_imp_of_and_not : a \u2227 \u00ac b \u2192 \u00ac (a \u2192 b)\n| \u27e8ha, hb\u27e9 h := hb $ h ha\n\n@[simp] theorem not_imp [decidable a] : \u00ac(a \u2192 b) \u2194 a \u2227 \u00acb :=\n\u27e8\u03bb h, \u27e8of_not_imp h, not_of_not_imp h\u27e9, not_imp_of_and_not\u27e9\n\n-- for monotonicity\nlemma imp_imp_imp\n  (h\u2080 : c \u2192 a) (h\u2081 : b \u2192 d) :\n  (a \u2192 b) \u2192 (c \u2192 d) :=\nassume (h\u2082 : a \u2192 b),\nh\u2081 \u2218 h\u2082 \u2218 h\u2080\n\ntheorem peirce (a b : Prop) [decidable a] : ((a \u2192 b) \u2192 a) \u2192 a :=\nif ha : a then \u03bb h, ha else \u03bb h, h ha.elim\n\ntheorem peirce' {a : Prop} (H : \u2200 b : Prop, (a \u2192 b) \u2192 a) : a := H _ id\n\ntheorem not_iff_not [decidable a] [decidable b] : (\u00ac a \u2194 \u00ac b) \u2194 (a \u2194 b) :=\nby rw [@iff_def (\u00ac a), @iff_def' a]; exact and_congr not_imp_not not_imp_not\n\ntheorem not_iff_comm [decidable a] [decidable b] : (\u00ac a \u2194 b) \u2194 (\u00ac b \u2194 a) :=\nby rw [@iff_def (\u00ac a), @iff_def (\u00ac b)]; exact and_congr not_imp_comm imp_not_comm\n\ntheorem not_iff [decidable a] [decidable b] : \u00ac (a \u2194 b) \u2194 (\u00ac a \u2194 b) :=\nby split; intro h; [split, skip]; intro h'; [by_contradiction,intro,skip];\n   try { refine h _; simp [*] }; rw [h',not_iff_self] at h; exact h\n\ntheorem iff_not_comm [decidable a] [decidable b] : (a \u2194 \u00ac b) \u2194 (b \u2194 \u00ac a) :=\nby rw [@iff_def a, @iff_def b]; exact and_congr imp_not_comm not_imp_comm\n\ntheorem iff_iff_and_or_not_and_not [decidable b] : (a \u2194 b) \u2194 (a \u2227 b) \u2228 (\u00ac a \u2227 \u00ac b) :=\nby { split; intro h,\n     { rw h; by_cases b; [left,right]; split; assumption },\n     { cases h with h h; cases h; split; intro; { contradiction <|> assumption } } }\n\n@[simp] theorem not_and_not_right [decidable b] : \u00ac(a \u2227 \u00acb) \u2194 (a \u2192 b) :=\n\u27e8\u03bb h ha, h.imp_symm $ and.intro ha, \u03bb h \u27e8ha, hb\u27e9, hb $ h ha\u27e9\n\n@[inline] def decidable_of_iff (a : Prop) (h : a \u2194 b) [D : decidable a] : decidable b :=\ndecidable_of_decidable_of_iff D h\n\n@[inline] def decidable_of_iff' (b : Prop) (h : a \u2194 b) [D : decidable b] : decidable a :=\ndecidable_of_decidable_of_iff D h.symm\n\ndef decidable_of_bool : \u2200 (b : bool) (h : b \u2194 a), decidable a\n| tt h := is_true (h.1 rfl)\n| ff h := is_false (mt h.2 bool.ff_ne_tt)\n\n/- de morgan's laws -/\n\ntheorem not_and_of_not_or_not (h : \u00ac a \u2228 \u00ac b) : \u00ac (a \u2227 b)\n| \u27e8ha, hb\u27e9 := or.elim h (absurd ha) (absurd hb)\n\ntheorem not_and_distrib [decidable a] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if ha : a then or.inr (\u03bb hb, h \u27e8ha, hb\u27e9) else or.inl ha, not_and_of_not_or_not\u27e9\n\ntheorem not_and_distrib' [decidable b] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if hb : b then or.inl (\u03bb ha, h \u27e8ha, hb\u27e9) else or.inr hb, not_and_of_not_or_not\u27e9\n\n@[simp] theorem not_and : \u00ac (a \u2227 b) \u2194 (a \u2192 \u00ac b) := and_imp\n\ntheorem not_and' : \u00ac (a \u2227 b) \u2194 b \u2192 \u00aca :=\nnot_and.trans imp_not_comm\n\ntheorem not_or_distrib : \u00ac (a \u2228 b) \u2194 \u00ac a \u2227 \u00ac b :=\n\u27e8\u03bb h, \u27e8\u03bb ha, h (or.inl ha), \u03bb hb, h (or.inr hb)\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9 h, or.elim h h\u2081 h\u2082\u27e9\n\ntheorem or_iff_not_and_not [decidable a] [decidable b] : a \u2228 b \u2194 \u00ac (\u00aca \u2227 \u00acb) :=\nby rw [\u2190 not_or_distrib, not_not]\n\ntheorem and_iff_not_or_not [decidable a] [decidable b] : a \u2227 b \u2194 \u00ac (\u00ac a \u2228 \u00ac b) :=\nby rw [\u2190 not_and_distrib, not_not]\n\nend propositional\n\n/- equality -/\n\nsection equality\nvariables {\u03b1 : Sort*} {a b : \u03b1}\n\n@[simp] theorem heq_iff_eq : a == b \u2194 a = b :=\n\u27e8eq_of_heq, heq_of_eq\u27e9\n\ntheorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : hp == hq :=\nhave p = q, from propext \u27e8\u03bb _, hq, \u03bb _, hp\u27e9,\nby subst q; refl\n\ntheorem ne_of_mem_of_not_mem {\u03b1 \u03b2} [has_mem \u03b1 \u03b2] {s : \u03b2} {a b : \u03b1}\n  (h : a \u2208 s) : b \u2209 s \u2192 a \u2260 b :=\nmt $ \u03bb e, e \u25b8 h\n\ntheorem eq_equivalence : equivalence (@eq \u03b1) :=\n\u27e8eq.refl, @eq.symm _, @eq.trans _\u27e9\n\nlemma heq_of_eq_mp :\n  \u2200 {\u03b1 \u03b2 : Sort*} {a : \u03b1} {a' : \u03b2} (e : \u03b1 = \u03b2) (h\u2082 : (eq.mp e a) = a'), a == a'\n| \u03b1 ._ a a' rfl h := eq.rec_on h (heq.refl _)\n\nlemma rec_heq_of_heq {\u03b2} {C : \u03b1 \u2192 Sort*} {x : C a} {y : \u03b2} (eq : a = b) (h : x == y) :\n  @eq.rec \u03b1 a C x b eq == y :=\nby subst eq; exact h\n\n@[simp] lemma {u} eq_mpr_heq {\u03b1 \u03b2 : Sort u} (h : \u03b2 = \u03b1) (x : \u03b1) : eq.mpr h x == x :=\nby subst h; refl\nend equality\n\n/-\n  quantifiers\n-/\n\nsection quantifiers\nvariables {\u03b1 : Sort*} {p q : \u03b1 \u2192 Prop} {b : Prop}\n\ndef Exists.imp := @exists_imp_exists\n\ntheorem forall_swap {\u03b1 \u03b2} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2200 x y, p x y) \u2194 \u2200 y x, p x y :=\n\u27e8function.swap, function.swap\u27e9\n\ntheorem exists_swap {\u03b1 \u03b2} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2203 x y, p x y) \u2194 \u2203 y x, p x y :=\n\u27e8\u03bb \u27e8x, y, h\u27e9, \u27e8y, x, h\u27e9, \u03bb \u27e8y, x, h\u27e9, \u27e8x, y, h\u27e9\u27e9\n\n@[simp] theorem exists_imp_distrib : ((\u2203 x, p x) \u2192 b) \u2194 \u2200 x, p x \u2192 b :=\n\u27e8\u03bb h x hpx, h \u27e8x, hpx\u27e9, \u03bb h \u27e8x, hpx\u27e9, h x hpx\u27e9\n\n--theorem forall_not_of_not_exists (h : \u00ac \u2203 x, p x) : \u2200 x, \u00ac p x :=\n--forall_imp_of_exists_imp h\n\ntheorem not_exists_of_forall_not (h : \u2200 x, \u00ac p x) : \u00ac \u2203 x, p x :=\nexists_imp_distrib.2 h\n\n@[simp] theorem not_exists : (\u00ac \u2203 x, p x) \u2194 \u2200 x, \u00ac p x :=\nexists_imp_distrib\n\ntheorem not_forall_of_exists_not : (\u2203 x, \u00ac p x) \u2192 \u00ac \u2200 x, p x\n| \u27e8x, hn\u27e9 h := hn (h x)\n\ntheorem not_forall {p : \u03b1 \u2192 Prop}\n    [decidable (\u2203 x, \u00ac p x)] [\u2200 x, decidable (p x)] :\n  (\u00ac \u2200 x, p x) \u2194 \u2203 x, \u00ac p x :=\n\u27e8not.imp_symm $ \u03bb nx x, nx.imp_symm $ \u03bb h, \u27e8x, h\u27e9,\n not_forall_of_exists_not\u27e9\n\n@[simp] theorem not_forall_not [decidable (\u2203 x, p x)] :\n  (\u00ac \u2200 x, \u00ac p x) \u2194 \u2203 x, p x :=\nby haveI := decidable_of_iff (\u00ac \u2203 x, p x) not_exists;\n   exact not_iff_comm.1 not_exists\n\n@[simp] theorem not_exists_not [\u2200 x, decidable (p x)] :\n  (\u00ac \u2203 x, \u00ac p x) \u2194 \u2200 x, p x :=\nby simp\n\n@[simp] theorem forall_true_iff : (\u03b1 \u2192 true) \u2194 true :=\niff_true_intro (\u03bb _, trivial)\n\n-- Unfortunately this causes simp to loop sometimes, so we\n-- add the 2 and 3 cases as simp lemmas instead\ntheorem forall_true_iff' (h : \u2200 a, p a \u2194 true) : (\u2200 a, p a) \u2194 true :=\niff_true_intro (\u03bb _, of_iff_true (h _))\n\n@[simp] theorem forall_2_true_iff {\u03b2 : \u03b1 \u2192 Sort*} : (\u2200 a, \u03b2 a \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_true_iff\n\n@[simp] theorem forall_3_true_iff {\u03b2 : \u03b1 \u2192 Sort*} {\u03b3 : \u03a0 a, \u03b2 a \u2192 Sort*} :\n  (\u2200 a (b : \u03b2 a), \u03b3 a b \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_2_true_iff\n\n@[simp] theorem forall_const (\u03b1 : Sort*) [inhabited \u03b1] : (\u03b1 \u2192 b) \u2194 b :=\n\u27e8\u03bb h, h (arbitrary \u03b1), \u03bb hb x, hb\u27e9\n\n@[simp] theorem exists_const (\u03b1 : Sort*) [inhabited \u03b1] : (\u2203 x : \u03b1, b) \u2194 b :=\n\u27e8\u03bb \u27e8x, h\u27e9, h, \u03bb h, \u27e8arbitrary \u03b1, h\u27e9\u27e9\n\ntheorem forall_and_distrib : (\u2200 x, p x \u2227 q x) \u2194 (\u2200 x, p x) \u2227 (\u2200 x, q x) :=\n\u27e8\u03bb h, \u27e8\u03bb x, (h x).left, \u03bb x, (h x).right\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9 x, \u27e8h\u2081 x, h\u2082 x\u27e9\u27e9\n\ntheorem exists_or_distrib : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\n\u27e8\u03bb \u27e8x, hpq\u27e9, hpq.elim (\u03bb hpx, or.inl \u27e8x, hpx\u27e9) (\u03bb hqx, or.inr \u27e8x, hqx\u27e9),\n \u03bb hepq, hepq.elim (\u03bb \u27e8x, hpx\u27e9, \u27e8x, or.inl hpx\u27e9) (\u03bb \u27e8x, hqx\u27e9, \u27e8x, or.inr hqx\u27e9)\u27e9\n\n@[simp] theorem exists_and_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, q \u2227 p x) \u2194 q \u2227 (\u2203x, p x) :=\n\u27e8\u03bb \u27e8x, hq, hp\u27e9, \u27e8hq, x, hp\u27e9, \u03bb \u27e8hq, x, hp\u27e9, \u27e8x, hq, hp\u27e9\u27e9\n\n@[simp] theorem exists_and_distrib_right {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, p x \u2227 q) \u2194 (\u2203x, p x) \u2227 q :=\nby simp [and_comm]\n\n@[simp] theorem forall_eq {a' : \u03b1} : (\u2200a, a = a' \u2192 p a) \u2194 p a' :=\n\u27e8\u03bb h, h a' rfl, \u03bb h a e, e.symm \u25b8 h\u27e9\n\n@[simp] theorem exists_eq {a' : \u03b1} : \u2203 a, a = a' := \u27e8_, rfl\u27e9\n\n@[simp] theorem exists_eq_left {a' : \u03b1} : (\u2203 a, a = a' \u2227 p a) \u2194 p a' :=\n\u27e8\u03bb \u27e8a, e, h\u27e9, e \u25b8 h, \u03bb h, \u27e8_, rfl, h\u27e9\u27e9\n\n@[simp] theorem exists_eq_right {a' : \u03b1} : (\u2203 a, p a \u2227 a = a') \u2194 p a' :=\n(exists_congr $ by exact \u03bb a, and.comm).trans exists_eq_left\n\n@[simp] theorem forall_eq' {a' : \u03b1} : (\u2200a, a' = a \u2192 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_left' {a' : \u03b1} : (\u2203 a, a' = a \u2227 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_right' {a' : \u03b1} : (\u2203 a, p a \u2227 a' = a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\ntheorem forall_or_of_or_forall (h : b \u2228 \u2200x, p x) (x) : b \u2228 p x :=\nh.imp_right $ \u03bb h\u2082, h\u2082 x\n\ntheorem forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} [decidable q] :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) :=\n\u27e8\u03bb h, if hq : q then or.inl hq else or.inr $ \u03bb x, (h x).resolve_left hq,\n  forall_or_of_or_forall\u27e9\n\n@[simp] theorem exists_prop {p q : Prop} : (\u2203 h : p, q) \u2194 p \u2227 q :=\n\u27e8\u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9\u27e9\n\n@[simp] theorem exists_false : \u00ac (\u2203a:\u03b1, false) := assume \u27e8a, h\u27e9, h\n\ntheorem Exists.fst {p : b \u2192 Prop} : Exists p \u2192 b\n| \u27e8h, _\u27e9 := h\n\ntheorem Exists.snd {p : b \u2192 Prop} : \u2200 h : Exists p, p h.fst\n| \u27e8_, h\u27e9 := h\n\n@[simp] theorem forall_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2200 h' : p, q h') \u2194 q h :=\n@forall_const (q h) p \u27e8h\u27e9\n\n@[simp] theorem exists_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2203 h' : p, q h') \u2194 q h :=\n@exists_const (q h) p \u27e8h\u27e9\n\n@[simp] theorem forall_prop_of_false {p : Prop} {q : p \u2192 Prop} (hn : \u00ac p) : (\u2200 h' : p, q h') \u2194 true :=\niff_true_intro $ \u03bb h, hn.elim h\n\n@[simp] theorem exists_prop_of_false {p : Prop} {q : p \u2192 Prop} : \u00ac p \u2192 \u00ac (\u2203 h' : p, q h') :=\nmt Exists.fst\n\nend quantifiers\n\n/- classical versions -/\n\nnamespace classical\nvariables {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop}\n\nlocal attribute [instance] prop_decidable\n\nprotected theorem not_forall : (\u00ac \u2200 x, p x) \u2194 (\u2203 x, \u00ac p x) := not_forall\n\nprotected theorem forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) :=\nforall_or_distrib_left\n\ntheorem cases {p : Prop \u2192 Prop} (h1 : p true) (h2 : p false) : \u2200a, p a :=\nassume a, cases_on a h1 h2\n\ntheorem or_not {p : Prop} : p \u2228 \u00ac p :=\nby_cases or.inl or.inr\n\nprotected theorem or_iff_not_imp_left {p q : Prop} : p \u2228 q \u2194 (\u00ac p \u2192 q) :=\nor_iff_not_imp_left\n\nprotected theorem or_iff_not_imp_right {p q : Prop} : q \u2228 p \u2194 (\u00ac p \u2192 q) :=\nor_iff_not_imp_right\n\nprotected lemma not_not {p : Prop} : \u00ac\u00acp \u2194 p := not_not\n\n/- use shortened names to avoid conflict when classical namespace is open -/\nnoncomputable theorem dec (p : Prop) : decidable p := by apply_instance\nnoncomputable theorem dec_pred (p : \u03b1 \u2192 Prop) : decidable_pred p := by apply_instance\nnoncomputable theorem dec_rel (p : \u03b1 \u2192 \u03b1 \u2192 Prop) : decidable_rel p := by apply_instance\nnoncomputable theorem dec_eq (\u03b1 : Sort*) : decidable_eq \u03b1 := by apply_instance\n\n@[elab_as_eliminator]\nnoncomputable def {u} exists_cases {C : Sort u} (H0 : C) (H : \u2200 a, p a \u2192 C) : C :=\nif h : \u2203 a, p a then H (classical.some h) (classical.some_spec h) else H0\n\nlemma some_spec2 {\u03b1 : Type*} {p : \u03b1 \u2192 Prop} {h : \u2203a, p a}\n  (q : \u03b1 \u2192 Prop) (hpq : \u2200a, p a \u2192 q a) : q (some h) :=\nhpq _ $ some_spec _\n\nend classical\n\n@[elab_as_eliminator]\nnoncomputable def {u} exists.classical_rec_on\n {\u03b1} {p : \u03b1 \u2192 Prop} (h : \u2203 a, p a) {C : Sort u} (H : \u2200 a, p a \u2192 C) : C :=\nH (classical.some h) (classical.some_spec h)\n\n/-\n   bounded quantifiers\n-/\n\nsection bounded_quantifiers\nvariables {\u03b1 : Sort*} {r p q : \u03b1 \u2192 Prop} {P Q : \u2200 x, p x \u2192 Prop} {b : Prop}\n\ntheorem bex_def : (\u2203 x (h : p x), q x) \u2194 \u2203 x, p x \u2227 q x :=\n\u27e8\u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9, \u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9\u27e9\n\ntheorem bex.elim {b : Prop} : (\u2203 x h, P x h) \u2192 (\u2200 a h, P a h \u2192 b) \u2192 b\n| \u27e8a, h\u2081, h\u2082\u27e9 h' := h' a h\u2081 h\u2082\n\ntheorem bex.intro (a : \u03b1) (h\u2081 : p a) (h\u2082 : P a h\u2081) : \u2203 x (h : p x), P x h :=\n\u27e8a, h\u2081, h\u2082\u27e9\n\ntheorem ball_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2200 x h, P x h) \u2194 (\u2200 x h, Q x h) :=\nforall_congr $ \u03bb x, forall_congr (H x)\n\ntheorem bex_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2203 x h, P x h) \u2194 (\u2203 x h, Q x h) :=\nexists_congr $ \u03bb x, exists_congr (H x)\n\ntheorem ball.imp_right (H : \u2200 x h, (P x h \u2192 Q x h))\n  (h\u2081 : \u2200 x h, P x h) (x h) : Q x h :=\nH _ _ $ h\u2081 _ _\n\ntheorem bex.imp_right (H : \u2200 x h, (P x h \u2192 Q x h)) :\n  (\u2203 x h, P x h) \u2192 \u2203 x h, Q x h\n| \u27e8x, h, h'\u27e9 := \u27e8_, _, H _ _ h'\u27e9\n\ntheorem ball.imp_left (H : \u2200 x, p x \u2192 q x)\n  (h\u2081 : \u2200 x, q x \u2192 r x) (x) (h : p x) : r x :=\nh\u2081 _ $ H _ h\n\ntheorem bex.imp_left (H : \u2200 x, p x \u2192 q x) :\n  (\u2203 x (_ : p x), r x) \u2192 \u2203 x (_ : q x), r x\n| \u27e8x, hp, hr\u27e9 := \u27e8x, H _ hp, hr\u27e9\n\ntheorem ball_of_forall (h : \u2200 x, p x) (x) (_ : q x) : p x :=\nh x\n\ntheorem forall_of_ball (H : \u2200 x, p x) (h : \u2200 x, p x \u2192 q x) (x) : q x :=\nh x $ H x\n\ntheorem bex_of_exists (H : \u2200 x, p x) : (\u2203 x, q x) \u2192 \u2203 x (_ : p x), q x\n| \u27e8x, hq\u27e9 := \u27e8x, H x, hq\u27e9\n\ntheorem exists_of_bex : (\u2203 x (_ : p x), q x) \u2192 \u2203 x, q x\n| \u27e8x, _, hq\u27e9 := \u27e8x, hq\u27e9\n\n@[simp] theorem bex_imp_distrib : ((\u2203 x h, P x h) \u2192 b) \u2194 (\u2200 x h, P x h \u2192 b) :=\nby simp\n\ntheorem not_bex : (\u00ac \u2203 x h, P x h) \u2194 \u2200 x h, \u00ac P x h :=\nbex_imp_distrib\n\ntheorem not_ball_of_bex_not : (\u2203 x h, \u00ac P x h) \u2192 \u00ac \u2200 x h, P x h\n| \u27e8x, h, hp\u27e9 al := hp $ al x h\n\ntheorem not_ball [decidable (\u2203 x h, \u00ac P x h)] [\u2200 x h, decidable (P x h)] :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) :=\n\u27e8not.imp_symm $ \u03bb nx x h, nx.imp_symm $ \u03bb h', \u27e8x, h, h'\u27e9,\n not_ball_of_bex_not\u27e9\n\ntheorem ball_true_iff (p : \u03b1 \u2192 Prop) : (\u2200 x, p x \u2192 true) \u2194 true :=\niff_true_intro (\u03bb h hrx, trivial)\n\ntheorem ball_and_distrib : (\u2200 x h, P x h \u2227 Q x h) \u2194 (\u2200 x h, P x h) \u2227 (\u2200 x h, Q x h) :=\niff.trans (forall_congr $ \u03bb x, forall_and_distrib) forall_and_distrib\n\ntheorem bex_or_distrib : (\u2203 x h, P x h \u2228 Q x h) \u2194 (\u2203 x h, P x h) \u2228 (\u2203 x h, Q x h) :=\niff.trans (exists_congr $ \u03bb x, exists_or_distrib) exists_or_distrib\n\nend bounded_quantifiers\n\nnamespace classical\nlocal attribute [instance] prop_decidable\n\ntheorem not_ball {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} {P : \u03a0 (x : \u03b1), p x \u2192 Prop} :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) := _root_.not_ball\n\nend classical\n\nsection nonempty\nuniverses u v w\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : \u03b1 \u2192 Type w}\n\nattribute [simp] nonempty_of_inhabited\n\nlemma exists_true_iff_nonempty {\u03b1 : Sort*} : (\u2203a:\u03b1, true) \u2194 nonempty \u03b1 :=\niff.intro (\u03bb\u27e8a, _\u27e9, \u27e8a\u27e9) (\u03bb\u27e8a\u27e9, \u27e8a, trivial\u27e9)\n\n@[simp] lemma nonempty_Prop {p : Prop} : nonempty p \u2194 p :=\niff.intro (assume \u27e8h\u27e9, h) (assume h, \u27e8h\u27e9)\n\nlemma not_nonempty_iff_imp_false {p : Prop} : \u00ac nonempty \u03b1 \u2194 \u03b1 \u2192 false :=\n\u27e8\u03bb h a, h \u27e8a\u27e9, \u03bb h \u27e8a\u27e9, h a\u27e9\n\n@[simp] lemma nonempty_sigma : nonempty (\u03a3a:\u03b1, \u03b3 a) \u2194 (\u2203a:\u03b1, nonempty (\u03b3 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_subtype {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} : nonempty (subtype p) \u2194 (\u2203a:\u03b1, p a) :=\niff.intro (assume \u27e8\u27e8a, h\u27e9\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a, h\u27e9\u27e9)\n\n@[simp] lemma nonempty_prod : nonempty (\u03b1 \u00d7 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_pprod {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (pprod \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_sum : nonempty (\u03b1 \u2295 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with sum.inl a := or.inl \u27e8a\u27e9 | sum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8sum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8sum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psum {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (psum \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with psum.inl a := or.inl \u27e8a\u27e9 | psum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8psum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8psum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psigma {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (psigma \u03b2) \u2194 (\u2203a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_empty : \u00ac nonempty empty :=\nassume \u27e8h\u27e9, h.elim\n\n@[simp] lemma nonempty_ulift : nonempty (ulift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty_plift {\u03b1 : Sort u} : nonempty (plift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty.forall {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2200h:nonempty \u03b1, p h) \u2194 (\u2200a, p \u27e8a\u27e9) :=\niff.intro (assume h a, h _) (assume h \u27e8a\u27e9, h _)\n\n@[simp] lemma nonempty.exists {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2203h:nonempty \u03b1, p h) \u2194 (\u2203a, p \u27e8a\u27e9) :=\niff.intro (assume \u27e8\u27e8a\u27e9, h\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a\u27e9, h\u27e9)\n\nlemma classical.nonempty_pi {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (\u03a0a:\u03b1, \u03b2 a) \u2194 (\u2200a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8f\u27e9 a, \u27e8f a\u27e9) (assume f, \u27e8assume a, classical.choice $ f a\u27e9)\n\n-- inhabited_of_nonempty already exists, in core/init/classical.lean, but the\n-- assumption is not [...], which makes it unsuitable for some applications\nnoncomputable def classical.inhabited_of_nonempty' {\u03b1 : Sort u} [h : nonempty \u03b1] : inhabited \u03b1 :=\n\u27e8classical.choice h\u27e9\n\n-- `nonempty` cannot be a `functor`, because `functor` is restricted to Types.\nlemma nonempty.map {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) : nonempty \u03b1 \u2192 nonempty \u03b2\n| \u27e8h\u27e9 := \u27e8f h\u27e9\n\nend nonempty\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/logic/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.6992544210587585, "lm_q1q2_score": 0.3877159342661247}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Lucas Allen, Scott Morrison\n-/\nimport tactic.interactive\nimport tactic.converter.interactive\n\n/-!\n## Introduce the `apply_congr` conv mode tactic.\n\n`apply_congr` will apply congruence lemmas inside `conv` mode.\nIt is particularly useful when the automatically generated congruence lemmas\nare not of the optimal shape. An example, described in the doc-string is\nrewriting inside the operand of a `finset.sum`.\n-/\n\nopen tactic\n\nnamespace conv.interactive\nopen interactive interactive.types lean.parser\n\nlocal postfix (name := parser.optional) `?`:9001 := optional\n\n\n/--\nApply a congruence lemma inside `conv` mode.\n\nWhen called without an argument `apply_congr` will try applying all lemmas marked with `@[congr]`.\nOtherwise `apply_congr e` will apply the lemma `e`.\n\nRecall that a goal that appears as `\u2223 X` in `conv` mode\nrepresents a goal of `\u22a2 X = ?m`,\ni.e. an equation with a metavariable for the right hand side.\n\nTo successfully use `apply_congr e`, `e` will need to be an equation\n(possibly after function arguments),\nwhich can be unified with a goal of the form `X = ?m`.\nThe right hand side of `e` will then determine the metavariable,\nand `conv` will subsequently replace `X` with that right hand side.\n\nAs usual, `apply_congr` can create new goals;\nany of these which are _not_ equations with a metavariable on the right hand side\nwill be hard to deal with in `conv` mode.\nThus `apply_congr` automatically calls `intros` on any new goals,\nand fails if they are not then equations.\n\nIn particular it is useful for rewriting inside the operand of a `finset.sum`,\nas it provides an extra hypothesis asserting we are inside the domain.\n\nFor example:\n\n```lean\nexample (f g : \u2124 \u2192 \u2124) (S : finset \u2124) (h : \u2200 m \u2208 S, f m = g m) :\n  finset.sum S f = finset.sum S g :=\nbegin\n  conv_lhs\n  { -- If we just call `congr` here, in the second goal we're helpless,\n    -- because we are only given the opportunity to rewrite `f`.\n    -- However `apply_congr` uses the appropriate `@[congr]` lemma,\n    -- so we get to rewrite `f x`, in the presence of the crucial `H : x \u2208 S` hypothesis.\n    apply_congr,\n    skip,\n    simp [h, H], }\nend\n```\n\nIn the above example, when the `apply_congr` tactic is called it gives the hypothesis `H : x \u2208 S`\nwhich is then used to rewrite the `f x` to `g x`.\n-/\nmeta def apply_congr (q : parse texpr?) : conv unit :=\ndo\n  congr_lemmas \u2190 match q with\n  -- If the user specified a lemma, use that one,\n  | some e := do\n    gs \u2190 get_goals,\n    e \u2190 to_expr e, -- to_expr messes with the goals? (see tests)\n    set_goals gs,\n    return [e]\n  -- otherwise, look up everything tagged `@[congr]`\n  | none := do\n    congr_lemma_names \u2190 attribute.get_instances `congr,\n    congr_lemma_names.mmap mk_const\n  end,\n  -- For every lemma:\n  congr_lemmas.any_of (\u03bb n,\n    -- Call tactic.eapply\n    seq' (tactic.eapply n >> tactic.skip)\n    -- and then call `intros` on each resulting goal, and require that afterwards it's an equation.\n        (tactic.intros >> (do `(_ = _) \u2190 target, tactic.skip)))\n\nadd_tactic_doc\n{ name := \"apply_congr\",\n  category := doc_category.tactic,\n  decl_names := [`conv.interactive.apply_congr],\n  tags := [\"conv\", \"congruence\", \"rewriting\"] }\n\nend conv.interactive\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/converter/apply_congr.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5544704796847396, "lm_q2_score": 0.6992544085240401, "lm_q1q2_score": 0.38771592731599336}}
{"text": "def add (n m : Nat) : Nat := n + m\n@[simp] theorem addZero x : add 0 x = x := by\n  simp [add]\n\nsyntax \"bigAdd0Seq! \" num : term\n\nmacro_rules\n  | `(bigAdd0Seq! $n) =>\n    let n := n.toNat\n    if n == 0 then\n      `(0)\n    else\n      `(add 0 (bigAdd0Seq! $(Lean.quote (n - 1))))\n\nset_option maxRecDepth 10000\n\ntheorem ex : bigAdd0Seq! 20 = 0 := by\n  simp\n\n#print ex\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/playground/add_zeros.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7634837743174789, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.38770611878308237}}
{"text": "import data.real.basic\nimport data.fin.vec_notation\nimport data.fin_enum\n\n\nuniverse u\n\n/-! \n### Dimension type classes\n-/\n\nclass has_time (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(time [] : \u03b1)\n(h : [time].nodup)\n\nclass has_length (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(length [] : \u03b1)\n(h : [length].nodup)\n\nclass has_mass (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(mass [] : \u03b1)\n(h : [mass].nodup)\n\nclass has_amount_of_substance (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(amount_of_substance [] : \u03b1)\n(h : [amount_of_substance].nodup)\n\nclass has_electric_current (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(electric_current [] : \u03b1)\n(h : [electric_current].nodup)\n\nclass has_temperature (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(temperature [] : \u03b1)\n(h : [temperature].nodup)\n\nclass has_luminous_intensity (\u03b1 : Type u) :=\n[dec : decidable_eq \u03b1]\n(luminous_intensity [] : \u03b1)\n(h : [luminous_intensity].nodup)\n\nattribute [instance] has_time.dec\nattribute [instance] has_length.dec\nattribute [instance] has_mass.dec\nattribute [instance] has_amount_of_substance.dec\nattribute [instance] has_electric_current.dec\nattribute [instance] has_temperature.dec\nattribute [instance] has_luminous_intensity.dec\n\n\n/-! \n### Def of dimensions and its properties\n-/\n\n\ndef dimension (\u03b1 : Type u) := \u03b1 \u2192 \u211a\n\nnamespace dimension\ndef dimensionless (\u03b1) : dimension \u03b1 := \u03bb i, 0\ninstance {\u03b1} : has_one (dimension \u03b1) := \u27e8dimension.dimensionless \u03b1\u27e9\ninstance {\u03b1} : nonempty (dimension \u03b1) := has_one.nonempty\n\nprotected noncomputable def add {\u03b1} [decidable_eq (dimension \u03b1)]: dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1 := \nclassical.epsilon $ \u03bb f, \u2200 a b, a = b \u2192 f a b = a\nprotected noncomputable def sub {\u03b1} [decidable_eq (dimension \u03b1)]: dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1 := \nclassical.epsilon $ \u03bb f, \u2200 a b, a = b \u2192 f a b = a\nprotected def mul {\u03b1} : dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1 \n| a b := \u03bb (i : \u03b1), a i + b i\nprotected def div {\u03b1} : dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1 \n| a b := \u03bb (i : \u03b1), a i - b i \n\nprotected def qpow {\u03b1} : dimension \u03b1 \u2192 \u211a \u2192 dimension \u03b1 \n| a n := \u03bb (i : \u03b1), n \u2022 (a i)\nprotected def npow {\u03b1} : dimension \u03b1 \u2192 \u2115 \u2192 dimension \u03b1 \n| a n := a.qpow \u2191n\nprotected def zpow {\u03b1} : dimension \u03b1 \u2192 \u2124 \u2192 dimension \u03b1 \n| a n := a.qpow \u2191n\nprotected def inv {\u03b1} : dimension \u03b1 \u2192 dimension \u03b1 \n| a := a.zpow (-1)\n\nnoncomputable instance {\u03b1} [decidable_eq (dimension \u03b1)] : has_add (dimension \u03b1) := \u27e8dimension.add\u27e9 \nnoncomputable instance {\u03b1} [decidable_eq (dimension \u03b1)] : has_sub (dimension \u03b1) := \u27e8dimension.sub\u27e9 \ninstance {\u03b1} : has_mul (dimension \u03b1) := \u27e8dimension.mul\u27e9 \ninstance {\u03b1} : has_div (dimension \u03b1) := \u27e8dimension.div\u27e9\ninstance {\u03b1} : has_pow (dimension \u03b1) \u2115 := \u27e8dimension.npow\u27e9\ninstance {\u03b1} : has_pow (dimension \u03b1) \u2124 := \u27e8dimension.zpow\u27e9\ninstance {\u03b1} : has_pow (dimension \u03b1) \u211a := \u27e8dimension.qpow\u27e9 \ninstance {\u03b1} : has_inv (dimension \u03b1) := \u27e8dimension.inv\u27e9\n\n\n--I would love to add unicode to make specific globabl notation for dimension derivatives and integrals, \n--but thats more fluff than important\n\nprotected def derivative {\u03b1} : \u2115 \u2192 dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1\n| n a b := a / (b ^ n)\nprotected def integral {\u03b1} : dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1\n| a b := a * b\n\n@[simp] lemma add_def {\u03b1} (a b : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a.add b = a + b := by refl\n@[simp] lemma add_def' {\u03b1} (a : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a.add a = a :=\nbegin\n  generalize hb : a = b, symmetry' at hb,\n  nth_rewrite 1 hb, revert b a hb, unfold dimension.add,\n  apply classical.epsilon_spec (\u27e8\u03bb a _, a, \u03bb _ _ _, rfl\u27e9 :\n    \u2203 (f : dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1), \u2200 a b, a = b \u2192 f a b = a),\nend\n@[simp] lemma add_def'' {\u03b1} (a : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a + a = a := by {rw [\u2190 add_def, add_def'],}\n@[simp] lemma sub_def {\u03b1} (a b : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a.sub b = a - b := by refl\n@[simp] lemma sub_def' {\u03b1} (a : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a.sub a = a :=\nbegin\n  generalize hb : a = b, symmetry' at hb,\n  nth_rewrite 1 hb, revert b a hb, unfold dimension.sub,\n  apply classical.epsilon_spec (\u27e8\u03bb a _, a, \u03bb _ _ _, rfl\u27e9 :\n    \u2203 (f : dimension \u03b1 \u2192 dimension \u03b1 \u2192 dimension \u03b1), \u2200 a b, a = b \u2192 f a b = a),\nend\n@[simp] lemma sub_def'' {\u03b1} (a : dimension \u03b1) [decidable_eq (dimension \u03b1)] : a - a = a := by {rw [\u2190 sub_def, sub_def'],}\n@[simp] lemma mul_def {\u03b1} (a b : dimension \u03b1) : a.mul b = a * b := by refl\n@[simp] lemma mul_def' {\u03b1} (a b : dimension \u03b1) : a * b = \u03bb (i : \u03b1), a i + b i := by refl\n@[simp] lemma div_def {\u03b1} (a b : dimension \u03b1) : a.div b = a / b := by refl\n@[simp] lemma div_def' {\u03b1} (a b : dimension \u03b1) : a / b = \u03bb (i : \u03b1), a i - b i := by refl\n@[simp] lemma qpow_def {\u03b1} (a : dimension \u03b1) (b : \u211a) : a.qpow b = a^b := by refl\n@[simp] lemma qpow_def' {\u03b1} (a : dimension \u03b1) (b : \u211a) : a ^ b = \u03bb (i : \u03b1), b \u2022 (a i):= by refl\n@[simp] lemma pow_def {\u03b1} (a : dimension \u03b1) (b : \u2115) : a.npow b = a^b := by refl\n@[simp] lemma pow_def' {\u03b1} (a : dimension \u03b1) (b : \u2115) : a ^ b = \u03bb (i : \u03b1), b \u2022 (a i) := by {simp, refl}\n@[simp] lemma zpow_def {\u03b1} (a : dimension \u03b1) (b : \u2124) : a.zpow b = a^b := by refl\n@[simp] lemma zpow_def' {\u03b1} (a : dimension \u03b1) (b : \u2124) : a ^ b = \u03bb (i : \u03b1), b \u2022 (a i) := by {simp, refl}\n@[simp] lemma inv_def {\u03b1} (a : dimension \u03b1) : a.inv = a\u207b\u00b9 := by refl\n@[simp] lemma inv_def' {\u03b1} (a : dimension \u03b1) : a\u207b\u00b9 = \u03bb (i : \u03b1), (-1 : \u2124) \u2022 (a i) := by {rw [\u2190 inv_def, dimension.inv, dimension.zpow, dimension.qpow], simp}\n\n/-!\n### Definition of the base dimensions\n-/\ndef length (\u03b1) [has_length \u03b1] : dimension \u03b1 :=\npi.single (has_length.length \u03b1) 1\n\ndef time (\u03b1) [has_time \u03b1] : dimension \u03b1 :=\npi.single (has_time.time \u03b1) 1\n\ndef mass (\u03b1) [has_mass \u03b1] : dimension \u03b1 :=\npi.single (has_mass.mass \u03b1) 1\n\ndef amount_of_substance (\u03b1) [has_amount_of_substance \u03b1] : dimension \u03b1 :=\npi.single (has_amount_of_substance.amount_of_substance \u03b1) 1\n\ndef electric_current (\u03b1) [has_electric_current \u03b1] : dimension \u03b1 :=\npi.single (has_electric_current.electric_current \u03b1) 1\n\ndef temperature (\u03b1) [has_temperature \u03b1] : dimension \u03b1 :=\npi.single (has_temperature.temperature \u03b1) 1\n\ndef luminous_intensity (\u03b1) [has_luminous_intensity \u03b1] : dimension \u03b1 :=\npi.single (has_luminous_intensity.luminous_intensity \u03b1) 1\n\n\n\nprotected def numbers_are_dimensionless (\u03b1 : Type*) [ordered_semiring \u03b1] [nontrivial \u03b1] {\u03b2}: \u03b1 \u2192 dimension \u03b2\n|a := dimension.dimensionless \u03b2 \ninstance {\u03b1} [ordered_semiring \u03b1] [nontrivial \u03b1] {\u03b2}: has_coe \u03b1 (dimension \u03b2):= \u27e8dimension.numbers_are_dimensionless \u03b1\u27e9\n\n@[simp] lemma one_eq_dimensionless {\u03b1} : 1 = dimensionless \u03b1 := rfl\n@[simp] lemma dimensionless_def' {\u03b1} : dimensionless \u03b1 = \u03bb i, 0 := rfl\n\nprotected theorem mul_comm {\u03b1} (a b : dimension \u03b1) : a * b = b * a := by {simp, funext, rw add_comm}\nprotected theorem div_mul_comm {\u03b1} (a b c : dimension \u03b1) : a / c * b  = b / c * a := by {simp, funext, rw sub_add_comm}\nprotected theorem mul_assoc {\u03b1} (a b c : dimension \u03b1) : a * b * c = a * (b * c) := by {simp, funext, rw add_assoc}\nprotected theorem mul_one {\u03b1} (a : dimension \u03b1) : a*1 = a := by simp\nprotected theorem one_mul {\u03b1} (a : dimension \u03b1) : 1*a = a := by simp\nprotected theorem div_eq_mul_inv {\u03b1} (a b : dimension \u03b1) : a / b = a * b\u207b\u00b9 := by {simp, funext, rw sub_eq_add_neg}\nprotected theorem mul_left_inv {\u03b1} (a : dimension \u03b1) : a\u207b\u00b9*a = 1 := by {simp}\nprotected theorem mul_right_inv {\u03b1} (a : dimension \u03b1) : a*a\u207b\u00b9 = 1 := by {simp}\n@[simp] protected lemma nat_numbers_are_dimensionless {\u03b1} {n : \u2115}: \u2191n = (1 : dimension \u03b1) := rfl\n@[simp] protected lemma int_numbers_are_dimensionless {\u03b1} {z : \u2124}: \u2191z = (1 : dimension \u03b1) := rfl\n@[simp] protected lemma rat_numbers_are_dimensionless {\u03b1} {q : \u211a}: \u2191q = (1 : dimension \u03b1) := rfl\n@[simp] protected lemma real_numbers_are_dimensionless {\u03b1} {r : \u211d}: \u2191r = (1 : dimension \u03b1) := rfl\n\n\ninstance {\u03b1} : comm_group (dimension \u03b1) :=\nbegin\n  refine_struct { mul := dimension.mul,\n                  div := dimension.div,\n                  inv := dimension.inv,\n                  mul_assoc := dimension.mul_assoc,\n                  one := dimensionless \u03b1,\n                  npow := @npow_rec (dimension \u03b1) dimension.has_one dimension.has_mul,\n                  zpow := @zpow_rec (dimension \u03b1) dimension.has_one dimension.has_mul dimension.has_inv,\n                  one_mul := dimension.one_mul,\n                  mul_one := dimension.mul_one,\n                  mul_comm := dimension.mul_comm,\n                  div_eq_mul_inv := dimension.div_eq_mul_inv,\n                  mul_left_inv := dimension.mul_left_inv,}, \n  repeat {rintro \u27e8_\u27e9, },\n  iterate 8 {intro, refl,},\nend\nnoncomputable theorem fun_equiv {\u03b1 \u03b2 c} (H : \u03b1 \u2243 \u03b2) : (\u03b1 \u2192 c) \u2192 (\u03b2 \u2192 c) :=\nbegin\n  intros h h1,\n  apply h (H.inv_fun h1),\nend\n\nnoncomputable def dimension.to_tuple {\u03b1} [fintype \u03b1] [decidable_eq \u03b1] (a : dimension \u03b1) : fin (fintype.card \u03b1) \u2192 \u211a :=\nbegin\n  have h := fintype.trunc_equiv_fin \u03b1,\n  have h1 : \u03b1 \u2243 fin (fintype.card \u03b1) := trunc.out h,\n  intro h2,\n  apply fun_equiv h1 a h2,\nend\n\n/-! \n### Other dimensions\n-/\n--physics\ndef velocity (\u03b1) [has_length \u03b1] [has_time \u03b1] : dimension \u03b1 := length \u03b1 / time \u03b1\n\ndef acceleration (\u03b1) [has_length \u03b1] [has_time \u03b1] : dimension \u03b1 := length \u03b1 / ((time \u03b1) ^ 2)\n\ndef force (\u03b1) [has_length \u03b1] [has_time \u03b1] [has_mass \u03b1] : dimension \u03b1 := length \u03b1 / ((time \u03b1) ^ 2) * mass \u03b1\n\ntheorem accel_eq_vel_div_time {\u03b1} [has_length \u03b1] [has_time \u03b1] : acceleration \u03b1 = velocity \u03b1 / time \u03b1 :=\nbegin\n  field_simp [velocity, acceleration],\n  funext,\n  ring_nf,\nend\n\ntheorem force_eq_mass_mul_accel {\u03b1} [has_length \u03b1] [has_time \u03b1] [has_mass \u03b1] : force \u03b1 = mass \u03b1 * acceleration \u03b1 :=\nbegin\n  simp [force, acceleration],\n  funext,\n  ring_nf,\nend\n\nend dimension\n/-! \n### examples for personal understanding\n-/\ninductive system1\n| time | length \n\ninstance : decidable_eq system1 \n| system1.time system1.time := is_true rfl\n| system1.time system1.length := is_false (\u03bb h, system1.no_confusion h)\n| system1.length system1.time := is_false (\u03bb h, system1.no_confusion h)\n| system1.length system1.length := is_true rfl\n\nlemma system1.time_nodup : [system1.time].nodup := by finish\nlemma system1.length_nodup : [system1.length].nodup := by finish\n \ninstance : has_time system1 := {dec := system1.decidable_eq, time := system1.time, h := system1.time_nodup}\ninstance : has_length system1 := {dec := system1.decidable_eq, length := system1.length, h := system1.length_nodup}\n\nlemma system1_length_to_has_length : system1.length = has_length.length system1:= by refl\nlemma system1_time_to_has_time : system1.time = has_time.time system1:= by refl\n\n--Working on cardinality to convert dimension as system1 \u2192 \u211a to fin n \u2192 \u211a for matrix\ninstance : fintype system1 := \u27e8\u27e8multiset.cons system1.time (multiset.cons system1.length \u2205), by simp\u27e9, \u03bb x, by cases x; simp\u27e9 \n\nnoncomputable def system1.dimension_equiv_rat_tuple : dimension system1 \u2243 (fin 2 \u2192 \u211a) := equiv.arrow_congr (fintype.equiv_fin system1) (equiv.refl \u211a)\n\nnoncomputable instance : fin_enum system1 := \u27e8fintype.card system1, (fintype.equiv_fin system1)\u27e9\n\ntheorem system1.dimension_eq_tuple (a : dimension system1) : system1.dimension_equiv_rat_tuple a = ![a system1.length, a system1.time] :=\nbegin\n\n  \nend\n\nexample (h) :   vector.nth h = ![1,2,3,4] :=\nbegin\n  funext,\n  \nend\nprotected def system1.repr : system1 \u2192 string\n| system1.length := \"length\"\n| system1.time := \"time\"\n\ninstance : has_repr system1 := \u27e8system1.repr\u27e9 \n\nopen dimension\ntheorem system1.accel_eq_vel_div_time : acceleration system1 = velocity system1 / time system1 := accel_eq_vel_div_time\n\n\n--This show that we index our tuple through the specific base dimension rather than the previous way of vector number\n\nexample : (dimension.time system1) system1.length = 0 :=\nbegin\n  simp [dimension.time],\n  apply pi.single_eq_of_ne,\n  finish,\nend\n\nexample : (dimension.length system1) * (dimension.length system1) = pi.single (has_length.length system1) 2 :=\nbegin\n  simp [dimension.length],\n  ext1,\n  cases x,\n  rw [pi.single_eq_of_ne, pi.single_eq_of_ne],\n  iterate 4 {finish},\nend\n\nexample : ((dimension.length system1) * (dimension.length system1)) system1.length = 2 :=\nbegin\n  simp [dimension.length],\n  finish,\nend\n\n\n", "meta": {"author": "ATOMSLab", "repo": "LeanChemicalTheories", "sha": "c2b15363c1e0ea0e52c1ae86abd1650670ff9044", "save_path": "github-repos/lean/ATOMSLab-LeanChemicalTheories", "path": "github-repos/lean/ATOMSLab-LeanChemicalTheories/LeanChemicalTheories-c2b15363c1e0ea0e52c1ae86abd1650670ff9044/src/dimensional_analysis/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757646140788308, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.38767156761750343}}
{"text": "import applications.topologies\nimport opens\n\n/-!\nThe category `Sheaf` of (set-valued) sheaves on a site is defined, using the sheaf condition as\ndefined in `grothendieck.lean`. It's defined somewhat abstractly, but an equivalent condition\n(`grothendieck.sheaf_condition'`) is given there too, which is more concrete.\n-/\nuniverses v u\n\nnoncomputable theory\n\nnamespace category_theory\nnamespace site_sheaf\n\nopen category limits\n\nvariables (C : Type u) [small_category C] (J : sieve_set C) [grothendieck J]\n\nstructure Sheaf :=\n(P : C\u1d52\u1d56 \u2964 Type u)\n(sheaf_cond : grothendieck.sheaf_condition J P)\n\ninstance : category (Sheaf C J) := induced_category.category Sheaf.P\n\n/--\nThe category of sheaves (in the Grothendieck sense) is equivalent to a special case of sheaves on\na local operator. We use this equivalence to transfer properties from the abstract topos theory\nto the geometric case.\n-/\ndef equiv_lt_sheaf : Sheaf C J \u224c sheaf (j J) :=\n{ functor :=\n  { obj := \u03bb P, sheaf.mk P.P (equivalent_sheaf_conditions _ _ P.sheaf_cond),\n    map := \u03bb P Q f, f },\n  inverse :=\n  { obj := \u03bb P, \u27e8P.A, (equivalent_sheaf_conditions _ _).symm (get_condition P)\u27e9,\n    map := \u03bb P Q f, f },\n  unit_iso := nat_iso.of_components (\u03bb P, {hom := \ud835\udfd9 _, inv := \ud835\udfd9 _}) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb P, {hom := \ud835\udfd9 _, inv := \ud835\udfd9 _}) (by tidy) }\n\n/-- The forgetful functor from sheaves to presheaves. -/\ndef forget_Sheaf : Sheaf C J \u2964 (C\u1d52\u1d56 \u2964 Type u) := induced_functor _\n\ninstance : full (forget_Sheaf C J) := induced_category.full _\ninstance : faithful (forget_Sheaf C J) := induced_category.faithful _\n\n/-- The sheafification functor for sheaves on a site. -/\ndef sheafify : (C\u1d52\u1d56 \u2964 Type u) \u2964 Sheaf C J := sheafification (j J) \u22d9 (equiv_lt_sheaf _ _).inverse\n\n/-- The equivalence between Sheaf and sheaf commutes with their respective forgetful functors. -/\nlemma forget_comm : forget_Sheaf C J = (equiv_lt_sheaf _ _).functor \u22d9 sheaf.forget (j J) := rfl\n\n/-- Sheafification is left adjoint to the inclusion into presheaves. -/\ndef Sheafy_adjoint : sheafify C J \u22a3 forget_Sheaf C J :=\nadjunction.comp _ _ (sheafification_is_adjoint (j J)) (equiv_lt_sheaf _ _).symm.to_adjunction\n\ninstance : is_right_adjoint (forget_Sheaf C J) :=\n{ left := _, adj := Sheafy_adjoint C J }\n\n/-- The category of sheaves is a reflective subcategory of presheaves -/\ninstance : reflective (forget_Sheaf C J) := {}.\n\n/-- The forgetful functor creates limits. -/\ninstance : creates_limits (forget_Sheaf C J) :=\n{ creates_limits_of_shape := \u03bb D \ud835\udc9f,\n  { creates_limit := \u03bb K,\n    begin\n      change creates_limit _ ((equiv_lt_sheaf _ _).functor \u22d9 sheaf.forget (j J)),\n      apply_instance,\n    end } }\n\n/-- Sheafification preserves finite products -/\ninstance preserve_fin_prod (D : Type u) [decidable_eq D] [fintype D] : preserves_limits_of_shape (discrete D) (sheafify C J) :=\n{ preserves_limit := \u03bb K,\nbegin\n  unfold sheafify,\n  haveI := sheafification_preserves_finite_products (j J) D,\n  apply_instance,\nend }\n\n/-- Sheafification preserves equalizers -/\ninstance preserve_equalizer : preserves_limits_of_shape walking_parallel_pair (sheafify C J) :=\n{ preserves_limit := \u03bb K,\nbegin\n  unfold sheafify,\n  haveI := sheafification_preserves_equalizers (j J),\n  apply_instance,\nend }\n\nend site_sheaf\n\nopen topological_space topological_space.opens\n\n/-- Sheaves on a space. -/\nabbreviation Sh (X : Type u) [topological_space X] := site_sheaf.Sheaf (opens X) (covering X)\n\nend category_theory", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/applications/sheaves_on_a_site.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190477, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.38767156012538784}}
{"text": "import order.conditionally_complete_lattice\n\nopen set\n\n-- TODO: Move: conditionally_complete_lattice\nprivate lemma le_csupr_iff.mpr\n{\u03b1 : Type*} {\u03b9 : Sort*} [conditionally_complete_lattice \u03b1] \n{s : \u03b9 \u2192 \u03b1} {a : \u03b1} (hs : bdd_above (range s)) (h : \u2200 (b : \u03b1), (\u2200 (i : \u03b9), s i \u2264 b) \u2192 a \u2264 b)\n: a \u2264 supr s := \nh (supr s) (\u03bb i, le_csupr hs i)\n", "meta": {"author": "ramonfmir", "repo": "lean-experiments", "sha": "041c8727bb540fb8d1519c1ad84924d473885c27", "save_path": "github-repos/lean/ramonfmir-lean-experiments", "path": "github-repos/lean/ramonfmir-lean-experiments/lean-experiments-041c8727bb540fb8d1519c1ad84924d473885c27/src/picard_lindelof/other/conditionally_complete_lattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3876715601253878}}
{"text": "\nimport  data.set.function\nopen  set\n\nvariables \u03b1 \u03b2 : Type\nvariables  A :set \u03b1 \nvariables  B :set \u03b2  \nvariable f:\u03b1 \u2192 \u03b2 \nvariable g:\u03b2 \u2192 \u03b1 \ntheorem Schroder_Bernstein: \ninj_on f A \u2227 inj_on g B \u2227 maps_to f A B \u2227 maps_to g B A \u2192 \u2203 h:\u03b1 \u2192 \u03b2, bij_on h A B \n:=\nsorry\n", "meta": {"author": "truonghoangle", "repo": "formalabstracts", "sha": "b889ec60143315053a51b1829a5dc4d82ba503b3", "save_path": "github-repos/lean/truonghoangle-formalabstracts", "path": "github-repos/lean/truonghoangle-formalabstracts/formalabstracts-b889ec60143315053a51b1829a5dc4d82ba503b3/Schroder_Bernstein.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.826711776992821, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.38755473197575996}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 \u2203 m : \u2124, m \u2260 0 \u2227 (int.fract (\u03b1 * \u2191i)) = int.fract (\u03b1 * \u2191j), from \n  begin\n    assume (i j : \u2124) (hij : i \u2260 j),\n    cases int.eq_or_lt_of_ne hij with h1 h2,\n    {\n      cases (h\u03b1_irrat i j h1) with h3 h4,\n      have h5 : \u00ac(i * \u03b1 = j * \u03b1), from by auto [h3],\n      have h6 : (int.fract (i * \u03b1)) \u2260 (int.fract (j * \u03b1)), from \n      begin\n        assume h7 : (int.fract (i * \u03b1)) = (int.fract (j * \u03b1)),\n        have h8 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h7],\n        have h9 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h8, sub_eq_zero],\n        have h10 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h9],\n        have h11 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h10],\n        have h12 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h11],\n        have h13 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h12],\n        have h14 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h13],\n        have h15 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h14],\n        have h16 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h15],\n        have h17 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h16],\n        have h18 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h17],\n        have h19 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h18],\n        have h20 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h19],\n        have h21 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h20],\n        have h22 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h21],\n        have h23 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h22],\n        have h24 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h23],\n        have h25 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h24],\n        have h26 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h25],\n        have h27 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h26],\n        have h28 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h27],\n        have h29 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h28],\n        have h30 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h29],\n        have h31 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h30],\n        have h32 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h31],\n        have h33 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h32],\n        have h34 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h33],\n        have h35 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h34],\n        have h36 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h35],\n        have h37 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h36],\n        have h38 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h37],\n        have h39 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h38],\n        have h40 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h39],\n        have h41 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h40],\n        have h42 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h41],\n        have h43 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h42],\n        have h44 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h43],\n        have h45 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h44],\n        have h46 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h45],\n        have h47 : (i * \u03b1) - (int.floor (i * \u03b1)) = (j * \u03b1) - (int.floor (j * \u03b1)), from by auto [h46],\n        have h48 : (i * \u03b1) - (int.floor (i * \u03b1)) = (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 n : \u2124, n \u2260 0 \u2192 int.fract (\u03b1 * n) \u2260 0, from\n  begin\n    assume (n : \u2124) (hn_ne_0 : n \u2260 0),\n    have h1 : (\u03b1 * n) \u2260 (0 : \u211d), from mul_ne_zero hn_ne_0 h\u03b1_irrat,\n    show int.fract (\u03b1 * n) \u2260 0, from by auto [int.fract, ne_iff_not_congr],\n  end,\n\n  have h2 : \u2200 n : \u2124, int.fract (\u03b1 * n) \u2260 int.fract (\u03b1 * \u2191(n + 1)), from\n  begin\n    assume (n : \u2124),\n    have h1 : (\u03b1 * \u2191n) \u2260 (\u03b1 * \u2191(n + 1)), from by auto [int.coe_nat_add, mul_ne_zero, h\u03b1_irrat],\n    show int.fract (\u03b1 * n) \u2260 int.fract (\u03b1 * \u2191(n + 1)), from by auto [int.fract, ne_iff_not_congr, h1],\n  end,\n\n  have h3 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) \u2208 set.Icc 0 1, from\n  begin\n    assume (n : \u2124),\n    have h1 : (0 : \u211d) \u2264 int.fract (\u03b1 * n), from by auto [int.fract, le_add_right],\n    have h2 : int.fract (\u03b1 * n) \u2264 (1 : \u211d), from by auto [int.fract, add_le_to_le_sub],\n    show int.fract (\u03b1 * \u2191n) \u2208 set.Icc 0 1, from by auto [set.mem_Icc],\n  end,\n\n  have h4 : (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from\n  begin\n    assume (m : \u2124) (h_m_mem : m \u2208 (@set.univ \u2124)),\n    show int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from by auto [h3],\n  end,\n\n  have h5 : set.Icc 0 1 \u2286 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from\n  begin\n    assume (x : \u211d) (h_x_mem : x \u2208 set.Icc 0 1),\n    have h1 : 0 \u2264 x, from by auto [set.mem_Icc],\n    have h2 : x \u2264 1, from by auto [set.mem_Icc],\n    have h3 : int.fract x \u2208 set.Icc 0 1, from by auto [int.fract, set.mem_Icc],\n    show x \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [set.mem_closure_iff] using [exists_nat_gt, h1, h2, h3],\n  end,\n\n  show closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from by auto [h4, h5, set.subset.antisymm],\nend\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [h\u03b1_irrat, irrational.irrational_iff_int_mul_ne],\n  have h2 : \u2200 (i j : \u2124), int.fract ((i : \u211d) * \u03b1) \u2260 int.fract ((j : \u211d) * \u03b1), from by auto [h1, int.fract_eq, eq_iff_iff_eq_int_mul_irrat],\n  have h3 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [h\u03b1_irrat, irrational.irrational_iff_int_mul_ne],\n  have h4 : \u2200 (i j : \u2124), int.fract ((i : \u211d) * \u03b1) \u2260 int.fract ((j : \u211d) * \u03b1), from by auto [h1, int.fract_eq, eq_iff_iff_eq_int_mul_irrat],\n\n  have h5 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (i * \u03b1) \u2260 int.fract (j * \u03b1), from by auto [h4],\n  have h6 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (i * \u03b1) \u2260 int.fract (j * \u03b1), from by auto [h4],\n\n  have h7 : \u2203! e : \u2124, \u2200 a : \u2124, int.fract (a * \u03b1) = int.fract (e * \u03b1), from by auto [exists_unique_int.intro, h5, h6, int.fract_eq],\n  have h8 : \u2203! e : \u2124, \u2200 a : \u2124, int.fract (a * \u03b1) = int.fract (e * \u03b1), from by auto [exists_unique_int.intro, h5, h6, int.fract_eq],\n\n  have h9 : (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) = {int.fract (\u03b1 * \u2191(classical.some h8.exists))}, from by auto [set.image_univ, h7, classical.some_spec, exists_unique.exists],\n  have h10 : (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) = {int.fract (\u03b1 * \u2191(classical.some h8.exists))}, from by auto [set.image_univ, h7, classical.some_spec, exists_unique.exists],\n\n  have h11 : int.fract (\u03b1 * \u2191(classical.some h8.exists)) \u2208 set.Icc 0 1, from by auto [int.fract_mem_Icc, int.fract_eq, eq_iff_iff_eq_int_mul_irrat],\n  have h12 : int.fract (\u03b1 * \u2191(classical.some h8.exists)) \u2208 set.Icc 0 1, from by auto [int.fract_mem_Icc, int.fract_eq, eq_iff_iff_eq_int_mul_irrat],\n\n  have h13 : (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from by auto [h11],\n  have h14 : (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from by auto [h12],\n\n  have h15 : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = {int.fract (\u03b1 * \u2191(classical.some h8.exists))}, from by auto [h9, closure_singleton],\n  have h16 : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = {int.fract (\u03b1 * \u2191(classical.some h8.exists))}, from by auto [h10, closure_singleton],\n\n  show closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from by auto [h15, h13, set.subset_def, h16, h14, set.subset_def] using [set.ext, set.mem_Icc],\nend\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (i * \u03b1) \u2260 int.fract (j * \u03b1), from by auto [int.fract_eq_iff_of_irrational h\u03b1_irrat.ne_zero],\n  have h2 : \u2203 l : \u211d, l \u2208 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [exists_limit_point],\n  have h3 : \u2200 m : \u2124, int.fract (\u03b1 * \u2191m) \u2208 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [closure_subset_iff],\n  have h4 : \u2200 m : \u2124, int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from by auto [int.fract_lt_one],\n  have h5 : \u2200 y \u2208 set.Icc 0 1, \u2203 m : \u2124, int.fract (\u03b1 * \u2191m) = y, from by auto [int.fract_eq_iff_of_irrational h\u03b1_irrat.ne_zero],\n  have h6 : closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 set.Icc 0 1, from by auto [closure_subset_iff],\n  have h7 : set.Icc 0 1 \u2286 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [closure_subset_iff],\n  show closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from by auto [closure_eq_iff_eq_of_is_closed, set.is_closed_Icc] using [int.fract_lt_one],\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 n : \u2124, \u2200 i : \u2124, n \u2260 i \u2192 (int.fract (\u03b1 * \u2191n)) \u2260 (int.fract (\u03b1 * \u2191i)), from by auto using [irrational.int_fract_ne, h\u03b1_irrat],\n  have h2 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by auto,\n  have h3 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [set.closure_eq, set.mem_closure_iff_nhds, set.mem_image_of_mem, set.mem_univ, set.mem_nhds_right, set.mem_Icc_self, set.mem_Icc_self, set.mem_Icc_self],\n  have h4 : \u2200 i : \u2124, \u2200 j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from by auto [h1],\n  have h5 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by auto [h2],\n  have h6 : set.Icc 0 1 \u2286 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [set.subset.trans, h5, h3],\n\n  have h7 : \u2200 i : \u2124, \u2200 j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from by auto [h4],\n  have h8 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by auto [h2],\n  have h9 : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 set.Icc 0 1, from by auto [set.subset.trans, h8, h3],\n\n  show closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from by auto [set.eq_of_subset_of_subset, h6, h9],\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from by auto [h\u03b1_irrat, int.fract_eq_iff_eq_of_rat],\n  have h2 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2260 \u2205, from by auto [set.univ_ne_empty, exists.intro 0],\n  have h3 : \u2200 i : \u2124, \u2203 j : \u2124, j \u2260 i \u2227 int.fract (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191i), from by auto [exists.intro (i+1)],\n  have h4 : \u2200 m : \u2124, int.fract (\u03b1 * \u2191m) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by auto [exists.intro m],\n\n  have h5 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from by auto [int.fract_range, int.fract_pos],\n  have h6 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 closure (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m) '' (@set.univ \u2124)), from by auto [set.subset_closure],\n  have h7 : closure (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m) '' (@set.univ \u2124)) \u2286 closure (set.Icc 0 1), from by auto [set.closure_mono],\n  have h8 : closure (set.Icc 0 1) \u2286 set.Icc 0 1, from by auto,\n\n  have h9 : closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 set.Icc 0 1, from by auto [set.subset.trans, set.subset.trans],\n\n  have h10 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 y \u2208 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [set.mem_closure_iff, set.mem_Icc.mp, exists.intro 1],\n  have h11 : set.Icc 0 1 \u2286 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from by auto [set.subset_closure],\n\n  have h12 : closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 set.Icc 0 1, from by auto [set.subset_closure, set.subset.trans, set.subset.trans],\n\n  show closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from by auto [set.subset.antisymm],\nend\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 a b : \u2124, \u00ac(int.fract (\u03b1 * \u2191a) = int.fract (\u03b1 * \u2191b)), from by auto [int.fract_eq_iff, h\u03b1_irrat],\n  have h2 : \u2200 a b : \u2124, \u00ac(a = b), from by auto [h1],\n  have h3 : \u2203! e : \u2124, int.fract (\u03b1 * \u2191e) = 0, from by auto using [use 0],\n  have h4 : \u2203! e : \u2124, int.fract (\u03b1 * \u2191e) = 1, from by auto using [use 1],\n  have h5 : \u2200 n : \u2124, 0 \u2264 int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) \u2264 1, from by auto using [int.fract_nonneg_iff, int.fract_lt_one],\n  have h6 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) = 1, from by auto [h5],\n  have h7 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 0 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5],\n  have h8 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 1 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h9 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 0 \u2264 int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h10 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 0 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) \u2264 1, from by auto [h5],\n  have h11 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 1 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) \u2264 1, from by auto [h5, int.fract_lt_one],\n  have h12 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1 * \u2191n) > 1, from by auto [h5, int.fract_lt_one],\n  have h13 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) > 0 \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h14 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) > 1 \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h15 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1 * \u2191n) > 0, from by auto [h5, int.fract_lt_one],\n  have h16 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 0 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h17 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 1 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h18 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 0 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1 * \u2191n) > 1, from by auto [h5, int.fract_lt_one],\n  have h19 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 0 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h20 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 1 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h21 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1 * \u2191n) > 1, from by auto [h5, int.fract_lt_one],\n  have h22 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 0 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h23 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 1 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h24 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 0 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h25 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 1 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h26 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1 * \u2191n) > 1, from by auto [h5, int.fract_lt_one],\n  have h27 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 0 \u2264 int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) < 1, from by auto [h5, int.fract_lt_one],\n  have h28 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 1 < int.fract (\u03b1 * \u2191n) \u2227 int.fract (\u03b1 * \u2191n) \u2264 1, from by auto [h5, int.fract_lt_one],\n  have h29 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 0 \u2264 int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h30 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 1 < int.fract (\u03b1 * \u2191n), from by auto [h5, int.fract_lt_one],\n  have h31 : \u2200 n : \u2124, int.fract (\u03b1 * \u2191n) = 1 \u2228 int.fract (\u03b1 * \u2191n) < 0 \u2228 int.fract (\u03b1\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8198933315126792, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3875500303973223}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro\n-/\nimport Mathlib.Tactic.Alias\nimport Mathlib.Tactic.IrreducibleDef\nimport Mathlib.Mathport.Rename\nimport Mathlib.Init.Logic\n\n/-! ### alignments from lean 3 `init.classical` -/\n\nnamespace Classical\n\n#align classical.inhabited_of_nonempty Classical.inhabited_of_nonempty\n#align classical.inhabited_of_exists Classical.inhabited_of_exists\n\nattribute [local instance] propDecidable\nattribute [local instance] decidableInhabited\n\nalias axiomOfChoice \u2190 axiom_of_choice -- TODO: fix in core\nalias propComplete \u2190 prop_complete -- TODO: fix in core\n\n@[elab_as_elim] theorem cases_true_false (p : Prop \u2192 Prop)\n    (h1 : p True) (h2 : p False) (a : Prop) : p a :=\n  Or.elim (prop_complete a) (fun ht : a = True \u21a6 ht.symm \u25b8 h1) fun hf : a = False \u21a6 hf.symm \u25b8 h2\n\ntheorem cases_on (a : Prop) {p : Prop \u2192 Prop} (h1 : p True) (h2 : p False) : p a :=\n  @cases_true_false p h1 h2 a\n\ntheorem cases {p : Prop \u2192 Prop} (h1 : p True) (h2 : p False) (a) : p a := cases_on a h1 h2\n#align classical.cases Classical.cases\n\nalias byCases \u2190 by_cases\nalias byContradiction \u2190 by_contradiction\n\ntheorem eq_false_or_eq_true (a : Prop) : a = False \u2228 a = True := (prop_complete a).symm\n\nend Classical\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Init/Classical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.38745561900026}}
{"text": "\nimport field_theory.polynomial_galois_group\n\nopen polynomial\nopen_locale polynomial big_operators\n\nnamespace mul_action\n\n@[to_additive]\ninstance (\u03b1 : Type*) {\u03b2 : Type*} [monoid \u03b1] [fintype \u03b1] [mul_action \u03b1 \u03b2] [decidable_eq \u03b2] (b : \u03b2) :\n  fintype (orbit \u03b1 b) := set.fintype_range _\n\n@[to_additive]\ninstance (\u03b1 : Type*) {\u03b2 : Type*} [group \u03b1] [fintype \u03b1] [mul_action \u03b1 \u03b2] [decidable_eq \u03b2]\n  (x : mul_action.orbit_rel.quotient \u03b1 \u03b2) :\n  fintype x.orbit :=\nquotient.rec_on_subsingleton' x (\u03bb a, set.fintype_range _)\n\nend mul_action\n\nnamespace minpoly\n\nlemma eq_of_alg_hom_eq {K S T : Type*} [field K] [ring S] [ring T]\n  [algebra K S] [algebra K T]\n  (f : S \u2192\u2090[K] T) (hf : function.injective f)\n  {x : S} {y : T} (hx : is_integral K x) (h : y = f x) :\n  minpoly K x = minpoly K y :=\nminpoly.unique _ _ (minpoly.monic hx)\n  (by rw [h, aeval_alg_hom_apply, minpoly.aeval, alg_hom.map_zero])\n  (\u03bb q q_monic root_q, minpoly.min _ _ q_monic\n    (by rwa [h, aeval_alg_hom_apply, map_eq_zero_iff _ hf] at root_q))\n\nend minpoly\n\nsection heq\nuniverses u\u2081 u\u2082 u\u2083\n\nnamespace fun_like\n\nvariables {F F' : Sort u\u2081} {\u03b1 \u03b1' : Sort u\u2082} {\u03b2 : \u03b1 \u2192 Sort u\u2083} {\u03b2' : \u03b1' \u2192 Sort u\u2083}\n  [i : fun_like F \u03b1 \u03b2] [i' : fun_like F' \u03b1' \u03b2']\n\nlemma ext_heq {f : F} {f' : F'}\n  (h\u2081 : F = F') (h\u2082 : \u03b1 = \u03b1') (h\u2083 : \u03b2 == \u03b2') (h\u2084 : i == i')\n  (h : \u2200 x x', x == x' \u2192 f x == f' x') :\n  f == f' := \nby { unfreezingI { cases h\u2081, cases h\u2082, cases h\u2083, cases h\u2084, },\n  exact heq_of_eq (fun_like.ext f f' (\u03bb x, eq_of_heq (h x x heq.rfl))), }\n\nlemma congr_heq {f : F} {f' : F'} {x : \u03b1} {x' : \u03b1'}\n  (h\u2081 : f == f') (h\u2082 : x == x') (h\u2083 : \u03b2 == \u03b2') (h\u2084 : i == i') :\n  f x == f' x' :=\nby { unfreezingI { cases h\u2081, cases h\u2082, cases h\u2083, cases h\u2084, }, refl, }\n\nend fun_like\n\nuniverse u\n\nlemma cast_heq' {\u03b1 \u03b2 \u03b1' : Sort u} (h : \u03b1 = \u03b2) {a : \u03b1} {a' : \u03b1'} (h' : a == a') : cast h a == a' :=\nby { cases h, cases h', refl, }\n\nend heq\n\nnamespace alg_equiv\nvariables {R : Type*} [comm_semiring R] {A\u2081 A\u2082 : Type*}\nvariables [semiring A\u2081] [semiring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082]\nvariables (e : A\u2081 \u2243\u2090[R] A\u2082)\n\nlemma symm_apply_eq {x y} : e.symm x = y \u2194 x = e y :=\ne.to_equiv.symm_apply_eq\n\nend alg_equiv\n\nnamespace intermediate_field\n\nvariables (F : Type*) [field F] {E : Type*} [field E] [algebra F E] {\u03b1 : E}\n\nlemma adjoin_root_equiv_adjoin_symm_apply_gen (h : is_integral F \u03b1) :\n  (adjoin_root_equiv_adjoin F h).symm (adjoin_simple.gen F \u03b1) =\n    adjoin_root.root (minpoly F \u03b1) :=\nby rw [alg_equiv.symm_apply_eq, adjoin_root_equiv_adjoin_apply_root]\n\nend intermediate_field\n\nnamespace polynomial\n\nvariables {T : Type*} [comm_ring T]\n\nnoncomputable abbreviation aroots (p : T[X]) (S) [comm_ring S] [is_domain S] [algebra T S] : multiset S :=\n(p.map (algebra_map T S)).roots\n\nlemma aroots_def (p : T[X]) (S) [comm_ring S] [is_domain S] [algebra T S] :\n  p.aroots S = (p.map (algebra_map T S)).roots := rfl\n\nlemma aroots_map (p : T[X]) (S) (A) [comm_ring S] [is_domain S] [algebra T S]\n  [comm_ring A] [is_domain A] [algebra S A] [algebra T A] [is_scalar_tower T S A] :\n(p.map (algebra_map T S)).aroots A = p.aroots A :=\nby rw [aroots_def, map_map, \u2190 is_scalar_tower.algebra_map_eq T S A]\n\nend polynomial\n\n\n\nsection gal_conj_classes\nvariables (F : Type*) [field F] (E : Type*) [field E] [algebra F E]\n\ndef is_gal_conj.setoid := mul_action.orbit_rel (E \u2243\u2090[F] E) E\ndef gal_conj_classes := mul_action.orbit_rel.quotient (E \u2243\u2090[F] E) E\n\nlocal attribute [instance] is_gal_conj.setoid\n\nvariable {E}\n\ndef is_gal_conj (x y : E) : Prop := (is_gal_conj.setoid F E).r x y\n\n-- need to fix the precedence\nnotation (name := is_gal_conj) x ` \u2248g[`:50 F `] ` y := is_gal_conj F x y\n\ninstance [decidable_eq E] [fintype (E \u2243\u2090[F] E)] (x y : E) : decidable (x \u2248g[F] y) :=\n  fintype.decidable_exists_fintype\n\ninstance [decidable_eq E] [fintype (E \u2243\u2090[F] E)] : decidable_eq (gal_conj_classes F E) :=\n@quotient.decidable_eq _ (is_gal_conj.setoid F E) (is_gal_conj.decidable F)\n\nnamespace is_gal_conj\n\ninstance : is_equiv E (is_gal_conj F) := quotient.has_equiv.equiv.is_equiv\n\n@[refl] lemma refl (x : E) : x \u2248g[F] x := refl x\n@[symm] lemma symm {x y : E} : (x \u2248g[F] y) \u2192 (y \u2248g[F] x) := symm\n@[trans] lemma trans {x y z : E} : (x \u2248g[F] y) \u2192 (y \u2248g[F] z) \u2192 (x \u2248g[F] z) := trans\n\nend is_gal_conj\n\nnamespace gal_conj_classes\n\ndef mk (x : E) : gal_conj_classes F E := \u27e6x\u27e7\n\ninstance : has_zero (gal_conj_classes F E) := \u27e8mk F 0\u27e9\n\nlemma zero_def : (0 : gal_conj_classes F E) = mk F 0 := rfl\n\nvariable {F}\n\nnoncomputable def out (c : gal_conj_classes F E) : E := c.out\n\n@[simp] theorem eq {x y : E} : mk F x = mk F y \u2194 x \u2248g[F] y := quotient.eq\n@[simp] theorem out_eq (q : gal_conj_classes F E) : mk F q.out = q := q.out_eq\ntheorem mk_out (x : E) : (mk F x).out \u2248 x := quotient.mk_out x\nlemma mk_eq_iff_out {x : E} {c : gal_conj_classes F E} :\n  mk F x = c \u2194 x \u2248g[F] c.out := quotient.mk_eq_iff_out\nlemma eq_mk_iff_out {c : gal_conj_classes F E} {x : E} :\n  c = mk F x \u2194 c.out \u2248g[F] x := quotient.eq_mk_iff_out\n@[simp] lemma out_equiv_out {c\u2081 c\u2082 : gal_conj_classes F E} :\n  (c\u2081.out \u2248g[F] c\u2082.out) \u2194 c\u2081 = c\u2082 := @quotient.out_equiv_out _ _ c\u2081 c\u2082\n\nlemma equiv_zero_iff (x : E) : (x \u2248g[F] 0) \u2194 x = 0 :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, by rw [h]\u27e9,\n  cases h with a ha,\n  simp_rw [\u2190 ha, alg_equiv.smul_def, map_zero],\nend\n\nlemma out_eq_zero_iff (c : gal_conj_classes F E) : c.out = 0 \u2194 c = 0 :=\nby rw [zero_def, eq_mk_iff_out, equiv_zero_iff]\n\nlemma zero_out : (0 : gal_conj_classes F E).out = 0 :=\n(out_eq_zero_iff 0).mpr rfl\n\nlemma mk_eq_zero_iff (x : E) : mk F x = 0 \u2194 x = 0 :=\nby rw [mk_eq_iff_out, zero_out, equiv_zero_iff]\n\nlemma mk_zero : mk F (0 : E) = 0 :=\n(mk_eq_zero_iff 0).mpr rfl\n\ndef orbit (c : gal_conj_classes F E) : set E := c.orbit\n\ninstance [decidable_eq E] [fintype (E \u2243\u2090[F] E)] (c : gal_conj_classes F E) :\n  fintype c.orbit :=\nquotient.rec_on_subsingleton' c (\u03bb a, set.fintype_range _)\n\nlemma mem_orbit {x : E} {c : gal_conj_classes F E} :\n  x \u2208 c.orbit \u2194 mk F x = c := mul_action.orbit_rel.quotient.mem_orbit\n\nlemma orbit_zero : (0 : gal_conj_classes F E).orbit = {0} :=\nby { ext, rw [mem_orbit, mk_eq_zero_iff], refl, }\n\ninstance : has_neg (gal_conj_classes F E) :=\n  \u27e8quotient.lift (\u03bb (x : E), mk F (-x)) begin\n    rintros _ y \u27e8f, rfl\u27e9, rw [eq],\n    use f, change f (-y) = -f y, rw [alg_equiv.map_neg],\nend\u27e9\n\nlemma mk_neg (x : E) : mk F (-x) = -mk F x := rfl\n\ninstance : has_involutive_neg (gal_conj_classes F E) :=\n{ neg_neg := \u03bb x, by rw [\u2190 out_eq x, \u2190 mk_neg, \u2190 mk_neg, neg_neg],\n  ..(infer_instance : has_neg (gal_conj_classes F E)), }\n\nlemma exist_mem_orbit_add_eq_zero (x y : gal_conj_classes F E) :\n  (\u2203 (a b : E), (a \u2208 x.orbit \u2227 b \u2208 y.orbit) \u2227 a + b = 0) \u2194 x = -y :=\nbegin\n  simp_rw [mem_orbit],\n  split,\n  { rintros \u27e8a, b, \u27e8rfl, rfl\u27e9, h\u27e9,\n    rw [\u2190 mk_neg, eq, add_eq_zero_iff_eq_neg.mp h], },\n  { rintro rfl,\n    refine \u27e8-y.out, y.out, _\u27e9,\n    simp_rw [mk_neg, out_eq, neg_add_self, eq_self_iff_true, true_and], },\nend\n\nvariable [is_separable F E]\n\nnoncomputable def minpoly : gal_conj_classes F E \u2192 F[X] :=\nquotient.lift (minpoly F) (\u03bb (a b : E) \u27e8f, h\u27e9, minpoly.eq_of_alg_hom_eq\n  f.symm.to_alg_hom f.symm.injective\n  (is_separable.is_integral F a) (h \u25b8 (f.symm_apply_apply b).symm))\n\nlemma minpoly_mk (x : E) : minpoly (mk F x) = _root_.minpoly F x := rfl\n\nlemma minpoly_out (c : gal_conj_classes F E) : _root_.minpoly F c.out = minpoly c :=\nby rw [\u2190 c.out_eq, minpoly_mk, c.out_eq]\n\nlemma minpoly.monic (c : gal_conj_classes F E) : (minpoly c).monic :=\nby { rw [\u2190 c.out_eq, minpoly_mk], exact minpoly.monic (is_separable.is_integral F _), }\n\nlemma minpoly.ne_zero (c : gal_conj_classes F E) : minpoly c \u2260 0 :=\nby { rw [\u2190 c.out_eq, minpoly_mk], exact minpoly.ne_zero (is_separable.is_integral F _), }\n\nlemma minpoly.irreducible (c : gal_conj_classes F E) : irreducible (minpoly c) :=\nby { rw [\u2190 c.out_eq, minpoly_mk], exact minpoly.irreducible (is_separable.is_integral F _), }\n\nlemma minpoly.splits [n : normal F E] (c : gal_conj_classes F E) :\n  splits (algebra_map F E) (minpoly c) :=\nby { rw [\u2190 c.out_eq, minpoly_mk], exact n.splits c.out, }\n\nlemma minpoly.separable (c : gal_conj_classes F E) : separable (minpoly c) :=\nby { rw [\u2190 c.out_eq, minpoly_mk], exact is_separable.separable F c.out, }\n\nlemma minpoly.inj [normal F E] {c d : gal_conj_classes F E} (h : minpoly c = minpoly d) : c = d :=\nbegin\n  let fc := intermediate_field.adjoin_root_equiv_adjoin F (is_separable.is_integral F c.out),\n  let fd := intermediate_field.adjoin_root_equiv_adjoin F (is_separable.is_integral F d.out),\n  let congr_f : adjoin_root (_root_.minpoly F c.out) \u2243\u2090[F] adjoin_root (_root_.minpoly F d.out),\n  { rw [minpoly_out, minpoly_out, h], },\n  have congr_f_apply : \u2200 x, congr_f x == x,\n  { intro x, change congr_f x == (alg_equiv.refl : _ \u2243\u2090[F] _) x,\n    dsimp only [congr_f],\n    refine fun_like.congr_heq _ heq.rfl _ _,\n    { simp_rw [eq_mpr_eq_cast, cast_cast],\n      refine cast_heq' _ (fun_like.ext_heq _ _ _ _ _),\n      any_goals { rw [minpoly_out, h], },\n      rintros x\u2081 x\u2082 rfl, refl, },\n    all_goals { rw [minpoly_out, minpoly_out, h], }, },\n  let f' := fc.symm.trans (congr_f.trans fd),\n  let f := f'.lift_normal E,\n  rw [\u2190 out_equiv_out],\n  refine \u27e8f.symm, _\u27e9,\n  dsimp only [f, alg_equiv.smul_def],\n  simp_rw [alg_equiv.symm_apply_eq, \u2190 intermediate_field.adjoin_simple.algebra_map_gen F c.out,\n    \u2190 intermediate_field.adjoin_simple.algebra_map_gen F d.out, alg_equiv.lift_normal_commutes],\n  apply congr_arg,\n  simp_rw [f', alg_equiv.trans_apply, \u2190 fd.symm_apply_eq, fc, fd,\n    intermediate_field.adjoin_root_equiv_adjoin_symm_apply_gen],\n  refine eq_of_heq (heq.trans _ (congr_f_apply _).symm),\n  rw [minpoly_out, minpoly_out, h],\nend\n\nlemma minpoly.injective [normal F E] : function.injective (@minpoly F _ E _ _ _) :=\n\u03bb x y, minpoly.inj\n\nlemma minpoly.nodup_aroots (c : gal_conj_classes F E) :\n  ((minpoly c).aroots E).nodup :=\nnodup_roots (minpoly.separable c).map\n\nlemma aeval_minpoly_iff [normal F E] (x : E) (c : gal_conj_classes F E) :\n  aeval x (minpoly c) = 0 \u2194 mk F x = c :=\nbegin\n  symmetry, split, { rintros rfl, exact minpoly.aeval _ _, },\n  intros h,\n  apply minpoly.inj,\n  rw [minpoly_mk, \u2190 minpoly.eq_of_irreducible (minpoly.irreducible c) h],\n  rw [(minpoly.monic c).leading_coeff, inv_one, map_one, mul_one],\nend\n\nlemma root_set_minpoly_eq_orbit [normal F E] (c : gal_conj_classes F E) :\n  (minpoly c).root_set E = c.orbit :=\nbegin\n  ext x, rw [mem_orbit],\n  simp_rw [mem_root_set, aeval_minpoly_iff x c],\n  simp [minpoly.ne_zero c],\nend\n\nlemma aroots_minpoly_eq_orbit_val [decidable_eq E] [fintype (E \u2243\u2090[F] E)] [normal F E]\n  (c : gal_conj_classes F E) : (minpoly c).aroots E = c.orbit.to_finset.1 :=\nbegin\n  simp_rw [\u2190 root_set_minpoly_eq_orbit, root_set_def, finset.to_finset_coe,\n    multiset.to_finset_val], symmetry, rw [multiset.dedup_eq_self],\n  exact nodup_roots ((separable_map _).mpr (minpoly.separable c)),\nend\n\nlemma orbit_eq_mk_aroots_minpoly [decidable_eq E] [fintype (E \u2243\u2090[F] E)] [normal F E]\n  (c : gal_conj_classes F E) :\n  c.orbit.to_finset = \u27e8(minpoly c).aroots E, minpoly.nodup_aroots c\u27e9 :=\nby simpa only [aroots_minpoly_eq_orbit_val]\n\nlemma minpoly.map_eq_prod [decidable_eq E] [fintype (E \u2243\u2090[F] E)] [normal F E]\n  (c : gal_conj_classes F E) :\n  (minpoly c).map (algebra_map F E) = \u220f x in c.orbit.to_finset, (X - C x) :=\nbegin\n  simp_rw [\u2190 root_set_minpoly_eq_orbit, finset.prod_eq_multiset_prod, root_set_def,\n    finset.to_finset_coe, multiset.to_finset_val],\n  rw [multiset.dedup_eq_self.mpr (nodup_roots _),\n    prod_multiset_X_sub_C_of_monic_of_roots_card_eq (monic.map _ _)],\n  { rw [splits_iff_card_roots.mp], rw [splits_id_iff_splits], exact minpoly.splits c, },\n  { exact minpoly.monic c, },\n  { exact (minpoly.separable c).map, },\nend\n/-\ndef class_of_roots_irreducible\n  {q : F[X]} (q_splits : q.splits (algebra_map F E))\n  (hq : _root_.irreducible q) : gal_conj_classes F E :=\n\u27e6root_of_splits (algebra_map F E) q_splits (degree_pos_of_irreducible hq).ne'\u27e7\n\nlemma minpoly_class_of_roots_irreducible_of_monic\n  {q : F[X]} (q_splits : splits (algebra_map F E) q)\n  (hq : _root_.irreducible q) (q_monic : q.monic) :\n  (class_of_roots_irreducible q_splits hq).minpoly = q :=\nbegin\n  dsimp only [class_of_roots_irreducible], rw [minpoly_mk],\n  exact (minpoly.eq_of_irreducible_of_monic hq (map_root_of_splits _ _ _) q_monic).symm,\nend\n\nlemma minpoly_class_of_roots_irreducible\n  {q : F[X]} (q_splits : splits (algebra_map F E) q)\n  (hq : _root_.irreducible q) :\n  (class_of_roots_irreducible q_splits hq).minpoly = q * C q.leading_coeff\u207b\u00b9 :=\nbegin\n  dsimp only [class_of_roots_irreducible], rw [minpoly_mk],\n  exact (minpoly.eq_of_irreducible hq (map_root_of_splits _ _ _)).symm,\nend\n\nlemma class_of_roots_irreducible_minpoly (c : gal_conj_classes F E)\n  (h\u2081 : c.minpoly.splits (algebra_map F E)) (h\u2082 : irreducible c.minpoly) :\n  class_of_roots_irreducible h\u2081 h\u2082 = c :=\nbegin\n  dsimp only [class_of_roots_irreducible], apply minpoly.inj, rw [minpoly_mk],\n  exact minpoly_class_of_roots_irreducible_of_monic _ _ (minpoly.monic c),\nend\n\nlemma root_set_C_mul (q : F[X]) {a : F} (a0 : a \u2260 0) :\n  q.root_set E = (C a * q).root_set E :=\nby { simp_rw [root_set, map_mul, map_C,\n  roots_C_mul _ (((algebra_map F E).map_ne_zero).mpr a0)], }\n\nlemma root_set_mul_C (q : F[X]) {a : F} (a0 : a \u2260 0) :\n  q.root_set E = (q * C a).root_set E :=\nby { rw [mul_comm], exact root_set_C_mul q a0, }\n\nlemma root_set_eq_orbit {q : F[X]} (q_splits : splits (algebra_map F E) q)\n  (hq : _root_.irreducible q) :\n  q.root_set E = (class_of_roots_irreducible q_splits hq).orbit :=\nbegin\n  rw [\u2190 root_set_minpoly_eq_orbit, minpoly_class_of_roots_irreducible, root_set_mul_C],\n  exact inv_ne_zero (leading_coeff_ne_zero.mpr hq.ne_zero),\nend\n\nlemma aroots_eq_orbit {q : F[X]} (q_splits : splits (algebra_map F E) q)\n  (hq : _root_.irreducible q) :\n  q.aroots E =\n    (class_of_roots_irreducible q_splits hq).orbit.to_finset.1 :=\nbegin\n  simp_rw [\u2190 root_set_eq_orbit q_splits hq, root_set_def, to_finset_coe,\n    multiset.to_finset_val], symmetry, rw [multiset.dedup_eq_self],\n  exact nodup_roots ((separable_map _).mpr hq.separable),\nend\n\ndef classes_of_roots {q : F[X]} (q_splits : q.splits (algebra_map F E)) :\n  multiset (gal_conj_classes F E) :=\n(unique_factorization_monoid.normalized_factors q).pmap\n  (\u03bb (q : F[X]) (hq : q.splits (algebra_map F E) \u2227 irreducible q),\n    class_of_roots_irreducible hq.1 hq.2)\n  (\u03bb d hd,\n  begin\n    refine \u27e8_, unique_factorization_monoid.irreducible_of_normalized_factor _ hd\u27e9,\n    have d_dvd_q := unique_factorization_monoid.dvd_of_mem_normalized_factors hd,\n    refine splits_of_splits_of_dvd _ _ q_splits d_dvd_q,\n    rintros rfl, simpa [unique_factorization_monoid.normalized_factors_zero] using hd,\n  end)\n\nsection\n\nopen list multiset\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n\nvariables (a : \u03b1) (s t : multiset \u03b1) (f g : \u03b1 \u2192 multiset \u03b2)\n\nnamespace multiset\n\ntheorem pmap_congr' {p q : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {g : \u03a0 a, q a \u2192 \u03b2}\n  (s : multiset \u03b1) {H\u2081 H\u2082} :\n  (\u2200 (a \u2208 s) h\u2081 h\u2082, f a h\u2081 = g a h\u2082) \u2192 pmap f s H\u2081 = pmap g s H\u2082 := sorry\n\nlemma count_dedup (m : multiset \u03b1) (a : \u03b1) :\n  m.dedup.count a = if a \u2208 m then 1 else 0 :=\nby { rcases m, simp [count_dedup], }\n\n@[simp]\nlemma dedup_bind_dedup (m : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b3) :\n  (m.dedup.bind f).dedup = (m.bind f).dedup :=\nby { ext x, simp_rw [count_dedup, multiset.mem_bind, multiset.mem_dedup], }\n\n@[simp]\nlemma dedup_dedup (m : multiset \u03b1) :\n  m.dedup.dedup = m.dedup :=\nby { ext x, simp_rw [count_dedup, multiset.mem_dedup], }\n\n@[simp]\nlemma to_finset_dedup (m : multiset \u03b1) :\n  m.dedup.to_finset = m.to_finset :=\nby simp_rw [multiset.to_finset, dedup_dedup]\n\n@[simp]\nlemma to_finset_bind_dedup (m : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b3) :\n  (m.dedup.bind f).to_finset = (m.bind f).to_finset :=\nby simp_rw [multiset.to_finset, dedup_bind_dedup]\n\nend multiset\n\nend\n\nlemma classes_of_roots_zero :\n  classes_of_roots (splits_zero (algebra_map F E)) = 0 :=\nby simp_rw [classes_of_roots, unique_factorization_monoid.normalized_factors_zero,\n  multiset.pmap_zero]\n\nsection\n/-\n`monic_normalize` caused following instance defeq problem:\n`(\u03bb (a b : F), classical.prop_decidable (a = b)) = (\u03bb (a b : F), rat.decidable_eq a b)`\n`classical.prop_decidable` is in `monic_normalize`\n-/\nlemma monic_normalize' {K : Type*} [field K] [decidable_eq K] {p : K[X]}\n  (hp0 : p \u2260 0) : monic (normalize p) :=\nbegin\n  rw [ne.def, \u2190 leading_coeff_eq_zero, \u2190 ne.def, \u2190 is_unit_iff_ne_zero] at hp0,\n  rw [monic, leading_coeff_normalize, normalize_eq_one],\n  apply hp0,\nend\n\nvariables {\u03b1 : Type*}\nlocal infix ` ~\u1d64 ` : 50 := associated\nvariables [cancel_comm_monoid_with_zero \u03b1] [decidable_eq \u03b1] [normalization_monoid \u03b1]\nvariables [unique_factorization_monoid \u03b1]\n\nopen unique_factorization_monoid\nnamespace unique_factorization_monoid\n\nlemma associated_iff_normalized_factors_eq_normalized_factors {x y : \u03b1} (hx : x \u2260 0) (hy : y \u2260 0) :\n  x ~\u1d64 y \u2194 normalized_factors x = normalized_factors y :=\nbegin\n  split,\n  { intro h,\n    apply le_antisymm; rw [\u2190 dvd_iff_normalized_factors_le_normalized_factors],\n    all_goals { simp [*, h.dvd, h.symm.dvd], }, },\n  { intro h,\n    apply associated_of_dvd_dvd; rw [dvd_iff_normalized_factors_le_normalized_factors],\n    all_goals { simp [*, h.le, h.ge], }, },\nend\n\nend unique_factorization_monoid\n\nend\n\nlemma prod_classes_of_roots_map_minpoly_of_monic\n  {q : F[X]} (q_splits : splits (algebra_map F E) q) (q0 : q \u2260 0)\n  (q_monic : q.monic) :\n  ((classes_of_roots q_splits).map (\u03bb c, minpoly c)).prod = q :=\nbegin\n  simp_rw [classes_of_roots, multiset.map_pmap],\n  suffices :\n    (multiset.pmap (\u03bb (q : F[X]) (h : q.splits (algebra_map F E) \u2227 irreducible q),\n      q) (unique_factorization_monoid.normalized_factors q) _).prod = q,\n  { refine eq.trans _ this, swap, congr' 1,\n    rw [(\u03bb _ _ _ _ _, iff.rfl : \u2200 {\u03b1 \u03b2} {f g : \u03b1 \u2192 \u03b2} (a : \u03b1), f a = g a \u2194 f a = g a)], -- hack\n    -- I wonder if there is a better method\n    refine multiset.pmap_congr' _ (\u03bb d hd _ _, minpoly_class_of_roots_irreducible_of_monic _ _ _),\n    swap, rw [\u2190 unique_factorization_monoid.normalize_normalized_factor _ hd],\n    have : d \u2260 0 := (unique_factorization_monoid.irreducible_of_normalized_factor _ hd).ne_zero,\n    exact monic_normalize' this, },\n  rw [multiset.pmap_eq_map, multiset.map_id'],\n  refine eq_of_monic_of_associated _ q_monic\n    (unique_factorization_monoid.normalized_factors_prod q0),\n  refine monic_multiset_prod_of_monic _ _ (\u03bb d hd, monic_normalize' _),\n  exact (unique_factorization_monoid.irreducible_of_factor _ hd).ne_zero,\nend\n\nlemma prod_classes_of_roots_map_minpoly\n  {q : F[X]} (q_splits : splits (algebra_map F E) q) (q0 : q \u2260 0) :\n  ((classes_of_roots q_splits).map (\u03bb c, minpoly c)).prod = q * C q.leading_coeff\u207b\u00b9 :=\nbegin\n  rw [\u2190 @prod_classes_of_roots_map_minpoly_of_monic p (q * C q.leading_coeff\u207b\u00b9)], rotate 1,\n  { exact splits_mul _ q_splits (splits_C _ _), },\n  { refine mul_ne_zero q0 _, rwa [ne.def, C_eq_zero, inv_eq_zero, leading_coeff_eq_zero], },\n  { exact monic_mul_leading_coeff_inv q0, },\n  congr' 2, dsimp only [classes_of_roots],\n  have l0 := inv_ne_zero (leading_coeff_ne_zero.mpr q0),\n  have ql0 := mul_ne_zero q0 (C_eq_zero.not.mpr l0),\n  have : associated q (q * C q.leading_coeff\u207b\u00b9),\n  { refine associated_mul_unit_right _ _ _,\n    rwa [is_unit_C, is_unit_iff_ne_zero], },\n  simp_rw [(unique_factorization_monoid.associated_iff_normalized_factors_eq_normalized_factors\n    q0 ql0).mp this],\nend\n\nlemma leading_coeff_mul_prod_classes_of_roots_map_minpoly\n  {q : F[X]} (q_splits : splits (algebra_map F E) q) :\n  C q.leading_coeff * ((classes_of_roots q_splits).map (\u03bb c, minpoly c)).prod = q :=\nbegin\n  rcases eq_or_ne q 0 with rfl | q0, { rw [leading_coeff_zero, C_0, zero_mul], },\n  rw [prod_classes_of_roots_map_minpoly q_splits q0, mul_left_comm, \u2190 C_mul, mul_inv_cancel,\n    C_1, mul_one],\n  exact leading_coeff_ne_zero.mpr q0,\nend\n\nlemma aroots_eq_classes_of_roots_bind_orbit\n  {q : F[X]} (q_splits : splits (algebra_map F E) q) :\n  q.aroots E =\n    (classes_of_roots q_splits).bind (\u03bb c, c.orbit.to_finset.1) :=\nbegin\n  rcases eq_or_ne q 0 with rfl | q0,\n  { rw [classes_of_roots_zero, multiset.zero_bind, map_zero, roots_zero], },\n  conv_lhs { rw [\u2190 leading_coeff_mul_prod_classes_of_roots_map_minpoly q_splits], },\n  rw [map_mul, map_multiset_prod, map_C, roots_C_mul, roots_multiset_prod,\n    multiset.bind_map, multiset.bind_map],\n  simp_rw [roots_minpoly_eq_orbit_val],\n  { intros h, simp_rw [multiset.mem_map] at h,\n    obtain \u27e8_, \u27e8\u27e8y, _, rfl\u27e9, h\u27e9\u27e9 := h,\n    rw [map_eq_zero] at h,\n    exact minpoly.ne_zero y h, },\n  { rwa [ring_hom.map_ne_zero, leading_coeff_ne_zero], },\nend\n\nlemma root_set_eq_classes_of_roots_bUnion_orbit\n  {q : F[X]} (q_splits : splits (algebra_map F E) q) :\n  q.root_set E =\n    \u22c3 c \u2208 ((classes_of_roots q_splits).to_finset : set (gal_conj_classes F E)), (c : _).orbit :=\nbegin\n  refine eq.trans _ (eq.trans\n    (congr_arg (\u03bb (m : multiset E), (m.to_finset : set E))\n    (roots_eq_classes_of_roots_bind_orbit q_splits)) _),\n  { exact root_set_def _ _, },\n  convert_to _ = \u22c3 (c : gal_conj_classes F E) (H : c \u2208 \u2191((classes_of_roots q_splits).to_finset)),\n    (c.orbit.to_finset : set E),\n  { simp_rw [set.coe_to_finset], },\n  rw [\u2190 coe_bUnion, finset.bUnion, finset.coe_inj, multiset.to_finset_val],\n  convert (multiset.to_finset_bind_dedup _ _).symm,\nend\n-/\n\nend gal_conj_classes\n\nend gal_conj_classes", "meta": {"author": "negiizhao", "repo": "transcendental", "sha": "c43bc907731093078887571a0adb3d6bcb1dc8cf", "save_path": "github-repos/lean/negiizhao-transcendental", "path": "github-repos/lean/negiizhao-transcendental/transcendental-c43bc907731093078887571a0adb3d6bcb1dc8cf/src/gal_conj.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6224593171945416, "lm_q1q2_score": 0.38745561028127745}}
{"text": "/-\nCopyright (c) 2020 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Heather Macbeth, Fr\u00e9d\u00e9ric Dupuis\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.analysis.normed_space.hahn_banach\nimport Mathlib.analysis.normed_space.inner_product\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u v \n\nnamespace Mathlib\n\n/-!\n# The topological dual of a normed space\n\nIn this file we define the topological dual of a normed space, and the bounded linear map from\na normed space into its double dual.\n\nWe also prove that, for base field `\ud835\udd5c` with `[is_R_or_C \ud835\udd5c]`, this map is an isometry.\n\nWe then consider inner product spaces, with base field over `\u211d` (the corresponding results for `\u2102`\nwill require the definition of conjugate-linear maps). We define `to_dual_map`, a continuous linear\nmap from `E` to its dual, which maps an element `x` of the space to `\u03bb y, \u27eax, y\u27eb`. We check\n(`to_dual_map_isometry`) that this map is an isometry onto its image, and particular is injective.\nWe also define `to_dual'` as the function taking taking a vector to its dual for a base field `\ud835\udd5c`\nwith `[is_R_or_C \ud835\udd5c]`; this is a function and not a linear map.\n\nFinally, under the hypothesis of completeness (i.e., for Hilbert spaces), we prove the Fr\u00e9chet-Riesz\nrepresentation (`to_dual_map_eq_top`), which states the surjectivity: every element of the dual\nof a Hilbert space `E` has the form `\u03bb u, \u27eax, u\u27eb` for some `x : E`.  This permits the map\n`to_dual_map` to be upgraded to an (isometric) continuous linear equivalence, `to_dual`, between a\nHilbert space and its dual.\n\n## References\n\n* [M. Einsiedler and T. Ward, *Functional Analysis, Spectral Theory, and Applications*]\n  [EinsiedlerWard2017]\n\n## Tags\n\ndual, Fr\u00e9chet-Riesz\n-/\n\nnamespace normed_space\n\n\n/-- The topological dual of a normed space `E`. -/\ndef dual (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2) [normed_group E]\n    [normed_space \ud835\udd5c E] :=\n  continuous_linear_map \ud835\udd5c E \ud835\udd5c\n\nprotected instance dual.inhabited (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2)\n    [normed_group E] [normed_space \ud835\udd5c E] : Inhabited (dual \ud835\udd5c E) :=\n  { default := 0 }\n\n/-- The inclusion of a normed space in its double (topological) dual. -/\ndef inclusion_in_double_dual' (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2)\n    [normed_group E] [normed_space \ud835\udd5c E] (x : E) : dual \ud835\udd5c (dual \ud835\udd5c E) :=\n  linear_map.mk_continuous (linear_map.mk (fun (f : dual \ud835\udd5c E) => coe_fn f x) sorry sorry) (norm x)\n    sorry\n\n@[simp] theorem dual_def (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2) [normed_group E]\n    [normed_space \ud835\udd5c E] (x : E) (f : dual \ud835\udd5c E) :\n    coe_fn (inclusion_in_double_dual' \ud835\udd5c E x) f = coe_fn f x :=\n  rfl\n\ntheorem double_dual_bound (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2)\n    [normed_group E] [normed_space \ud835\udd5c E] (x : E) : norm (inclusion_in_double_dual' \ud835\udd5c E x) \u2264 norm x :=\n  sorry\n\n/-- The inclusion of a normed space in its double (topological) dual, considered\n   as a bounded linear map. -/\ndef inclusion_in_double_dual (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] (E : Type u_2)\n    [normed_group E] [normed_space \ud835\udd5c E] : continuous_linear_map \ud835\udd5c E (dual \ud835\udd5c (dual \ud835\udd5c E)) :=\n  linear_map.mk_continuous\n    (linear_map.mk (fun (x : E) => inclusion_in_double_dual' \ud835\udd5c E x) sorry sorry) 1 sorry\n\n/-- If one controls the norm of every `f x`, then one controls the norm of `x`.\n    Compare `continuous_linear_map.op_norm_le_bound`. -/\ntheorem norm_le_dual_bound {\ud835\udd5c : Type v} [is_R_or_C \ud835\udd5c] {E : Type u} [normed_group E]\n    [normed_space \ud835\udd5c E] (x : E) {M : \u211d} (hMp : 0 \u2264 M)\n    (hM : \u2200 (f : dual \ud835\udd5c E), norm (coe_fn f x) \u2264 M * norm f) : norm x \u2264 M :=\n  sorry\n\n/-- The inclusion of a normed space in its double dual is an isometry onto its image.-/\ntheorem inclusion_in_double_dual_isometry {\ud835\udd5c : Type v} [is_R_or_C \ud835\udd5c] {E : Type u} [normed_group E]\n    [normed_space \ud835\udd5c E] (x : E) : norm (coe_fn (inclusion_in_double_dual \ud835\udd5c E) x) = norm x :=\n  sorry\n\nend normed_space\n\n\nnamespace inner_product_space\n\n\n/--\nGiven some `x` in an inner product space, we can define its dual as the continuous linear map\n`\u03bb y, \u27eax, y\u27eb`. Consider using `to_dual` or `to_dual_map` instead in the real case.\n-/\ndef to_dual' (\ud835\udd5c : Type u_1) {E : Type u_2} [is_R_or_C \ud835\udd5c] [inner_product_space \ud835\udd5c E] :\n    E \u2192+ normed_space.dual \ud835\udd5c E :=\n  add_monoid_hom.mk\n    (fun (x : E) =>\n      linear_map.mk_continuous (linear_map.mk (fun (y : E) => inner x y) sorry sorry) (norm x)\n        sorry)\n    sorry sorry\n\n@[simp] theorem to_dual'_apply (\ud835\udd5c : Type u_1) {E : Type u_2} [is_R_or_C \ud835\udd5c] [inner_product_space \ud835\udd5c E]\n    {x : E} {y : E} : coe_fn (coe_fn (to_dual' \ud835\udd5c) x) y = inner x y :=\n  rfl\n\n/-- In an inner product space, the norm of the dual of a vector `x` is `\u2225x\u2225` -/\n@[simp] theorem norm_to_dual'_apply (\ud835\udd5c : Type u_1) {E : Type u_2} [is_R_or_C \ud835\udd5c]\n    [inner_product_space \ud835\udd5c E] (x : E) : norm (coe_fn (to_dual' \ud835\udd5c) x) = norm x :=\n  sorry\n\ntheorem to_dual'_isometry (\ud835\udd5c : Type u_1) (E : Type u_2) [is_R_or_C \ud835\udd5c] [inner_product_space \ud835\udd5c E] :\n    isometry \u21d1(to_dual' \ud835\udd5c) :=\n  add_monoid_hom.isometry_of_norm (to_dual' \ud835\udd5c) (norm_to_dual'_apply \ud835\udd5c)\n\n/--\nFr\u00e9chet-Riesz representation: any `\u2113` in the dual of a Hilbert space `E` is of the form\n`\u03bb u, \u27eay, u\u27eb` for some `y : E`, i.e. `to_dual'` is surjective.\n-/\ntheorem to_dual'_surjective (\ud835\udd5c : Type u_1) (E : Type u_2) [is_R_or_C \ud835\udd5c] [inner_product_space \ud835\udd5c E]\n    [complete_space E] : function.surjective \u21d1(to_dual' \ud835\udd5c) :=\n  sorry\n\n/-- In a real inner product space `F`, the function that takes a vector `x` in `F` to its dual\n`\u03bb y, \u27eax, y\u27eb` is a continuous linear map. If the space is complete (i.e. is a Hilbert space),\nconsider using `to_dual` instead. -/\n-- TODO extend to `is_R_or_C` (requires a definition of conjugate linear maps)\n\ndef to_dual_map {F : Type u_1} [inner_product_space \u211d F] :\n    continuous_linear_map \u211d F (normed_space.dual \u211d F) :=\n  linear_map.mk_continuous (linear_map.mk \u21d1(to_dual' \u211d) sorry sorry) 1 sorry\n\n@[simp] theorem to_dual_map_apply {F : Type u_1} [inner_product_space \u211d F] {x : F} {y : F} :\n    coe_fn (coe_fn to_dual_map x) y = inner x y :=\n  rfl\n\n/-- In an inner product space, the norm of the dual of a vector `x` is `\u2225x\u2225` -/\n@[simp] theorem norm_to_dual_map_apply {F : Type u_1} [inner_product_space \u211d F] (x : F) :\n    norm (coe_fn to_dual_map x) = norm x :=\n  norm_to_dual'_apply \u211d x\n\ntheorem to_dual_map_isometry {F : Type u_1} [inner_product_space \u211d F] : isometry \u21d1to_dual_map :=\n  add_monoid_hom.isometry_of_norm (to_dual' \u211d) norm_to_dual_map_apply\n\ntheorem to_dual_map_injective {F : Type u_1} [inner_product_space \u211d F] :\n    function.injective \u21d1to_dual_map :=\n  isometry.injective to_dual_map_isometry\n\n@[simp] theorem ker_to_dual_map {F : Type u_1} [inner_product_space \u211d F] :\n    continuous_linear_map.ker to_dual_map = \u22a5 :=\n  iff.mpr linear_map.ker_eq_bot to_dual_map_injective\n\n@[simp] theorem to_dual_map_eq_iff_eq {F : Type u_1} [inner_product_space \u211d F] {x : F} {y : F} :\n    coe_fn to_dual_map x = coe_fn to_dual_map y \u2194 x = y :=\n  function.injective.eq_iff (iff.mp linear_map.ker_eq_bot ker_to_dual_map)\n\n/--\nFr\u00e9chet-Riesz representation: any `\u2113` in the dual of a real Hilbert space `F` is of the form\n`\u03bb u, \u27eay, u\u27eb` for some `y` in `F`.  See `inner_product_space.to_dual` for the continuous linear\nequivalence thus induced.\n-/\n-- TODO extend to `is_R_or_C` (requires a definition of conjugate linear maps)\n\ntheorem range_to_dual_map {F : Type u_1} [inner_product_space \u211d F] [complete_space F] :\n    continuous_linear_map.range to_dual_map = \u22a4 :=\n  iff.mpr linear_map.range_eq_top (to_dual'_surjective \u211d F)\n\n/--\nFr\u00e9chet-Riesz representation: If `F` is a Hilbert space, the function that takes a vector in `F` to\nits dual is a continuous linear equivalence.  -/\ndef to_dual {F : Type u_1} [inner_product_space \u211d F] [complete_space F] :\n    continuous_linear_equiv \u211d F (normed_space.dual \u211d F) :=\n  continuous_linear_equiv.of_isometry (continuous_linear_map.to_linear_map to_dual_map)\n    to_dual_map_isometry range_to_dual_map\n\n/--\nFr\u00e9chet-Riesz representation: If `F` is a Hilbert space, the function that takes a vector in `F` to\nits dual is an isometry.  -/\ndef isometric.to_dual {F : Type u_1} [inner_product_space \u211d F] [complete_space F] :\n    F \u2243\u1d62 normed_space.dual \u211d F :=\n  isometric.mk (linear_equiv.to_equiv (continuous_linear_equiv.to_linear_equiv to_dual))\n    (to_dual'_isometry \u211d F)\n\n@[simp] theorem to_dual_apply {F : Type u_1} [inner_product_space \u211d F] [complete_space F] {x : F}\n    {y : F} : coe_fn (coe_fn to_dual x) y = inner x y :=\n  rfl\n\n@[simp] theorem to_dual_eq_iff_eq {F : Type u_1} [inner_product_space \u211d F] [complete_space F]\n    {x : F} {y : F} : coe_fn to_dual x = coe_fn to_dual y \u2194 x = y :=\n  function.injective.eq_iff (continuous_linear_equiv.injective to_dual)\n\ntheorem to_dual_eq_iff_eq' {F : Type u_1} [inner_product_space \u211d F] [complete_space F] {x : F}\n    {x' : F} : (\u2200 (y : F), inner x y = inner x' y) \u2194 x = x' :=\n  sorry\n\n@[simp] theorem norm_to_dual_apply {F : Type u_1} [inner_product_space \u211d F] [complete_space F]\n    (x : F) : norm (coe_fn to_dual x) = norm x :=\n  norm_to_dual_map_apply x\n\n/-- In a Hilbert space, the norm of a vector in the dual space is the norm of its corresponding\nprimal vector. -/\ntheorem norm_to_dual_symm_apply {F : Type u_1} [inner_product_space \u211d F] [complete_space F]\n    (\u2113 : normed_space.dual \u211d F) : norm (coe_fn (continuous_linear_equiv.symm to_dual) \u2113) = norm \u2113 :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/normed_space/dual_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.38743339876627925}}
{"text": "import tag006T -- sheaves of types\nimport tag006N -- presheaves of rings\n\n-- this should really be done for abstract categories; we \"cheat\" here because\n-- equalizers in the category of commutative rings are the same as equalizers\n-- in the underlying category of sets: see tag0073 (lemma 6.9.2)\n\ndef is_sheaf_of_rings {\u03b1 : Type*} [T : topological_space \u03b1] \n  (PR : presheaf_of_rings \u03b1) : Prop :=\nis_sheaf_of_types PR.to_presheaf_of_types\n\n", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/src/tag0072.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.387422322852183}}
{"text": "import data.finset\n\nimport lib.core.rel3\nimport config\n\nvariables {\u03b1 : Type*} [linear_order \u03b1] (C : config \u03b1)\n\nstructure config.label (S : finset \u03b1) :=\n(slope : \u03b1 \u2192 \u03b1 \u2192 Prop)\n(decidable_slope : decidable_rel slope)\n-- The direction looks odd, but it is written in perspective of\n-- _where_ the edge ab is placed\n(extend_left : \u2200 {a b : \u03b1}, a \u2208 S \u2192 b \u2208 S \u2192 a < b \u2192 \u00acslope a b \u2192 \n  \u2200 {c : \u03b1}, c \u2208 S \u2192 b < c \u2192 C.cup3 a b c)\n(extend_right : \u2200 {a b : \u03b1}, a \u2208 S \u2192 b \u2208 S \u2192 a < b \u2192 slope a b \u2192\n  \u2200 {c : \u03b1}, c \u2208 S \u2192 c < a \u2192 C.cup3 c a b)\n\nattribute [instance] config.label.decidable_slope\n\ndef cap4_free_slope {S : finset \u03b1} (h : \u00acC.has_ncap 4 S) (a b : \u03b1) : Prop :=\n  \u2200 c : S, \u2191c < a \u2192 C.cup3 c a b\n\ninstance decidable_cap4_free_slope {S : finset \u03b1} (h : \u00acC.has_ncap 4 S) :\n  decidable_rel (cap4_free_slope C h) := \n\u03bb a b, by rw cap4_free_slope; simp; apply_instance\n\nvariable {C}\n\ndef cap4_free_label {S : finset \u03b1} (h : \u00acC.has_ncap 4 S) : C.label S :=\nbegin\n  use cap4_free_slope C h,\n  apply_instance,\n  { intros a b ha hb hab hn c hc hbc, \n    by_contra h', apply hn, intros d hd, \n    by_contra h'', apply h, use [[d, a, b, c]], simp [config.ncap], tauto },\n  { intros a b ha hb hab hy c hc hca,\n    exact hy \u27e8c, hc\u27e9 hca },\nend\n\nvariables {C} {S : finset \u03b1} {label : C.label S}\n\nprotected theorem config.cup.extend_left\n  {l : list \u03b1} (l_cup : C.cup l)\n  {a b : \u03b1} (s_ab : \u00aclabel.slope a b) \n  (ha : a \u2208 S) (hab : a < b) (l_in_S : l.in S)\n  (b_head_l : b \u2208 l.head') : C.cup (a :: l) :=\nbegin\n  cases l with b l,\n  { simp at b_head_l, tauto },\n  simp at b_head_l, subst b_head_l,\n  cases l with c l,\n  { simp, exact hab },\n  simp at l_in_S, simp,\n  refine \u27e8_, _, _\u27e9; try {tauto},\n  simp [config.cup] at l_cup,\n  apply label.extend_left; tauto,\nend \n\nprotected theorem config.cup.extend_right\n  {l : list \u03b1} (l_cup : C.cup l)\n  {a b : \u03b1} (s_ab : label.slope a b) \n  (hab : a < b) (hb : b \u2208 S) (l_in_S : l.in S)\n  (a_last_l : a \u2208 l.last') : C.cup (l ++ [b]) :=\nbegin\n  by_cases hl : 2 \u2264 l.length,\n  { rcases list.take_last2 hl with \u27e8c, a, l', eq_l\u27e9,\n    rw eq_l at a_last_l, simp at a_last_l, subst a_last_l,\n    rw eq_l, simp, rw \u2190eq_l,\n    refine \u27e8_, _, _\u27e9; try {tauto},\n    simp [config.cup] at l_cup,\n    rw eq_l at l_in_S, simp at l_in_S,\n    rw eq_l at l_cup, simp at l_cup,\n    apply label.extend_right; tauto },\n  cases l with p l, simp,\n  cases l with q l, simp at *, subst a_last_l; tauto,\n  exfalso, apply hl, exact le_add_self,\nend\n\nprotected theorem config.ncup.extend_left\n  {n : \u2115} {l : list \u03b1} (l_ncup : C.ncup n l)\n  {a b : \u03b1} (s_ab : \u00aclabel.slope a b) \n  (ha : a \u2208 S) (hab : a < b) (l_in_S : l.in S)\n  (b_head_l : b \u2208 l.head') : C.ncup (n+1) (a :: l) :=\nbegin\n  cases l_ncup with l_cup l_len, split,\n  apply l_cup.extend_left s_ab; try {assumption},\n  simp, assumption,\nend \n\nprotected theorem config.ncup.extend_right\n  {n : \u2115} {l : list \u03b1} (l_ncup : C.ncup n l)\n  {a b : \u03b1} (s_ab : label.slope a b) \n  (hab : a < b) (hb : b \u2208 S) (l_in_S : l.in S)\n  (a_last_l : a \u2208 l.last') : C.ncup (n+1) (l ++ [b]) :=\nbegin\n  cases l_ncup with l_cup l_len, split,\n  apply l_cup.extend_right s_ab; try {assumption},\n  simp, assumption,\nend\n\nvariable (label)\n\nopen order_dual\n\nprotected def config.label.mirror : \n  C.mirror.label S.mirror :=\n\u27e8 \u03bb a b, \u00ac(mirror2 label.slope a b), \n  \u03bb a b, @not.decidable _ (label.decidable_slope.mirror2 a b),\n  begin \n    intros a b a_in_S b_in_S hab hslope c c_in_S hbc,\n    simp [mirror2] at hslope,\n    simp [config.mirror, mirror3],\n    simp [finset.mirror] at a_in_S b_in_S c_in_S,\n    rcases a_in_S with \u27e8oa, \u27e8oa_in_S, oa_eq\u27e9\u27e9,\n    rcases b_in_S with \u27e8ob, \u27e8ob_in_S, ob_eq\u27e9\u27e9,\n    rcases c_in_S with \u27e8oc, \u27e8oc_in_S, oc_eq\u27e9\u27e9,\n    rw \u2190oa_eq at hab,\n    rw \u2190ob_eq at hab hbc,\n    rw \u2190oc_eq at hbc,\n    simp at hab hbc,\n    rw [\u2190oa_eq, \u2190ob_eq, \u2190oc_eq], simp,\n    rw [\u2190oa_eq, \u2190ob_eq] at hslope, simp at hslope,\n    apply label.extend_right; tauto,\n  end,\n  begin\n    intros a b a_in_S b_in_S hab hslope c c_in_S hca,\n    simp [mirror2] at hslope,\n    simp [config.mirror, mirror3],\n    simp [finset.mirror] at a_in_S b_in_S c_in_S,\n    rcases a_in_S with \u27e8oa, \u27e8oa_in_S, oa_eq\u27e9\u27e9,\n    rcases b_in_S with \u27e8ob, \u27e8ob_in_S, ob_eq\u27e9\u27e9,\n    rcases c_in_S with \u27e8oc, \u27e8oc_in_S, oc_eq\u27e9\u27e9,\n    rw \u2190oa_eq at hab hca,\n    rw \u2190ob_eq at hab,\n    rw \u2190oc_eq at hca,\n    simp at hab hca,\n    rw [\u2190oa_eq, \u2190ob_eq, \u2190oc_eq], simp,\n    rw [\u2190oa_eq, \u2190ob_eq] at hslope, simp at hslope,\n    apply label.extend_left; tauto,\n  end\u27e9\n\nvariable {label}\n\ndef mirror_slope {a b : \u03b1} :\n  \u00aclabel.mirror.slope (to_dual b) (to_dual a) \u2194 label.slope a b :=\nbegin\n  rw config.label.mirror, simp,\n  rw mirror2, simp,\nend", "meta": {"author": "jcpaik", "repo": "erdos-tuza-valtr", "sha": "7fceb6f4f7d73bc3a0a09f48426b0e9350bc82ef", "save_path": "github-repos/lean/jcpaik-erdos-tuza-valtr", "path": "github-repos/lean/jcpaik-erdos-tuza-valtr/erdos-tuza-valtr-7fceb6f4f7d73bc3a0a09f48426b0e9350bc82ef/src/etv/label.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.387422322852183}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.cone_category\nimport category_theory.adjunction\n\n/-!\n\n# Multi-(co)equalizers\n\nA *multiequalizer* is an equalizer of two morphisms between two products.\nSince both products and equalizers are limits, such an object is again a limit.\nThis file provides the diagram whose limit is indeed such an object.\nIn fact, it is well-known that any limit can be obtained as a multiequalizer.\nThe dual construction (multicoequalizers) is also provided.\n\n## Projects\n\nProve that a multiequalizer can be identified with\nan equalizer between products (and analogously for multicoequalizers).\n\nProve that the limit of any diagram is a multiequalizer (and similarly for colimits).\n\n-/\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses v u\n\n/-- The type underlying the multiequalizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multicospan {L R : Type v} (fst snd : R \u2192 L) : Type v\n| left : L \u2192 walking_multicospan\n| right : R \u2192 walking_multicospan\n\n/-- The type underlying the multiecoqualizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multispan {L R : Type v} (fst snd : L \u2192 R) : Type v\n| left : L \u2192 walking_multispan\n| right : R \u2192 walking_multispan\n\nnamespace walking_multicospan\n\nvariables {L R : Type v} {fst snd : R \u2192 L}\n\ninstance [inhabited L] : inhabited (walking_multicospan fst snd) :=\n\u27e8left (default _)\u27e9\n\n/-- Morphisms for `walking_multicospan`. -/\ninductive hom : \u03a0 (a b : walking_multicospan fst snd), Type v\n| id (A)  : hom A A\n| fst (b) : hom (left (fst b)) (right b)\n| snd (b) : hom (left (snd b)) (right b)\n\ninstance {a : walking_multicospan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multicospan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multicospan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst b) (hom.id X) := hom.fst b\n| _ _ _ (hom.snd b) (hom.id X) := hom.snd b\n\ninstance : small_category (walking_multicospan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multicospan\n\nnamespace walking_multispan\n\nvariables {L R : Type v} {fst snd : L \u2192 R}\n\ninstance [inhabited L] : inhabited (walking_multispan fst snd) :=\n\u27e8left (default _)\u27e9\n\n/-- Morphisms for `walking_multispan`. -/\ninductive hom : \u03a0 (a b : walking_multispan fst snd), Type v\n| id (A)  : hom A A\n| fst (a) : hom (left a) (right (fst a))\n| snd (a) : hom (left a) (right (snd a))\n\ninstance {a : walking_multispan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multispan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multispan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst a) (hom.id X) := hom.fst a\n| _ _ _ (hom.snd a) (hom.id X) := hom.snd a\n\ninstance : small_category (walking_multispan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multispan\n\n/-- This is a structure encapsulating the data necessary to define a `multicospan`. -/\n@[nolint has_inhabited_instance]\nstructure multicospan_index (C : Type u) [category.{v} C] :=\n(L R : Type v)\n(fst_to snd_to : R \u2192 L)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 b, left (fst_to b) \u27f6 right b)\n(snd : \u03a0 b, left (snd_to b) \u27f6 right b)\n\n/-- This is a structure encapsulating the data necessary to define a `multispan`. -/\n@[nolint has_inhabited_instance]\nstructure multispan_index (C : Type u) [category.{v} C] :=\n(L R : Type v)\n(fst_from snd_from : L \u2192 R)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 a, left a \u27f6 right (fst_from a))\n(snd : \u03a0 a, left a \u27f6 right (snd_from a))\n\nnamespace multicospan_index\n\nvariables {C : Type u} [category.{v} C] (I : multicospan_index C)\n\n/-- The multicospan associated to `I : multicospan_index`. -/\ndef multicospan : walking_multicospan I.fst_to I.snd_to \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multicospan.left a := I.left a\n  | walking_multicospan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multicospan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multicospan.hom.fst b := I.fst _\n  | _, _, walking_multicospan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multicospan_obj_left (a) :\n  I.multicospan.obj (walking_multicospan.left a) = I.left a := rfl\n\n@[simp] lemma multicospan_obj_right (b) :\n  I.multicospan.obj (walking_multicospan.right b) = I.right b := rfl\n\n@[simp] lemma multicospan_map_fst (b) :\n  I.multicospan.map (walking_multicospan.hom.fst b) = I.fst b := rfl\n\n@[simp] lemma multicospan_map_snd (b) :\n  I.multicospan.map (walking_multicospan.hom.snd b) = I.snd b := rfl\n\nvariables [has_product I.left] [has_product I.right]\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.fst`. -/\nnoncomputable\ndef fst_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.fst_to b) \u226b I.fst b)\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.snd`. -/\nnoncomputable\ndef snd_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.snd_to b) \u226b I.snd b)\n\n@[simp, reassoc]\nlemma fst_pi_map_\u03c0 (b) : I.fst_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.fst b :=\nby simp [fst_pi_map]\n\n@[simp, reassoc]\nlemma snd_pi_map_\u03c0 (b) : I.snd_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.snd b :=\nby simp [snd_pi_map]\n\n/--\nTaking the multiequalizer over the multicospan index is equivalent to taking the equalizer over\nthe two morphsims `\u220f I.left \u21c9 \u220f I.right`. This is the diagram of the latter.\n-/\n@[simps] protected noncomputable\ndef parallel_pair_diagram := parallel_pair I.fst_pi_map I.snd_pi_map\n\nend multicospan_index\n\nnamespace multispan_index\n\nvariables {C : Type u} [category.{v} C] (I : multispan_index C)\n\n/-- The multispan associated to `I : multispan_index`. -/\ndef multispan : walking_multispan I.fst_from I.snd_from \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multispan.left a := I.left a\n  | walking_multispan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multispan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multispan.hom.fst b := I.fst _\n  | _, _, walking_multispan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multispan_obj_left (a) :\n  I.multispan.obj (walking_multispan.left a) = I.left a := rfl\n\n@[simp] lemma multispan_obj_right (b) :\n  I.multispan.obj (walking_multispan.right b) = I.right b := rfl\n\n@[simp] lemma multispan_map_fst (a) :\n  I.multispan.map (walking_multispan.hom.fst a) = I.fst a := rfl\n\n@[simp] lemma multispan_map_snd (a) :\n  I.multispan.map (walking_multispan.hom.snd a) = I.snd a := rfl\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.fst`. -/\nnoncomputable\ndef fst_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.fst b \u226b sigma.\u03b9 _ (I.fst_from b))\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.snd`. -/\nnoncomputable\ndef snd_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.snd b \u226b sigma.\u03b9 _ (I.snd_from b))\n\n@[simp, reassoc]\nlemma \u03b9_fst_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.fst_sigma_map = I.fst b \u226b sigma.\u03b9 I.right _ :=\nby simp [fst_sigma_map]\n\n@[simp, reassoc]\nlemma \u03b9_snd_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.snd_sigma_map = I.snd b \u226b sigma.\u03b9 I.right _ :=\nby simp [snd_sigma_map]\n\n/--\nTaking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over\nthe two morphsims `\u2210 I.left \u21c9 \u2210 I.right`. This is the diagram of the latter.\n-/\nprotected noncomputable\nabbreviation parallel_pair_diagram := parallel_pair I.fst_sigma_map I.snd_sigma_map\n\nend multispan_index\n\nvariables {C : Type u} [category.{v} C]\n\n/-- A multifork is a cone over a multicospan. -/\n@[nolint has_inhabited_instance]\nabbreviation multifork (I : multicospan_index C) := cone I.multicospan\n\n/-- A multicofork is a cocone over a multispan. -/\n@[nolint has_inhabited_instance]\nabbreviation multicofork (I : multispan_index C) := cocone I.multispan\n\nnamespace multifork\n\nvariables {I : multicospan_index C} (K : multifork I)\n\n/-- The maps from the cone point of a multifork to the objects on the left. -/\ndef \u03b9 (a : I.L) : K.X \u27f6 I.left a :=\nK.\u03c0.app (walking_multicospan.left _)\n\n@[simp] lemma \u03b9_eq_app_left (a) : K.\u03b9 a = K.\u03c0.app (walking_multicospan.left _) := rfl\n\n@[simp] lemma app_left_fst (b) :\n  K.\u03c0.app (walking_multicospan.left (I.fst_to b)) \u226b I.fst b =\n    K.\u03c0.app (walking_multicospan.right b) :=\nby { rw \u2190 K.w (walking_multicospan.hom.fst b), refl }\n\n@[simp] lemma app_left_snd (b) :\n  K.\u03c0.app (walking_multicospan.left (I.snd_to b)) \u226b I.snd b =\n    K.\u03c0.app (walking_multicospan.right b) :=\nby { rw \u2190 K.w (walking_multicospan.hom.snd b), refl }\n\n/-- Construct a multifork using a collection `\u03b9` of morphisms. -/\n@[simps]\ndef of_\u03b9 (I : multicospan_index C) (P : C) (\u03b9 : \u03a0 a, P \u27f6 I.left a)\n  (w : \u2200 b, \u03b9 (I.fst_to b) \u226b I.fst b = \u03b9 (I.snd_to b) \u226b I.snd b) :\n  multifork I :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := \u03b9 _\n    | walking_multicospan.right b := \u03b9 (I.fst_to b) \u226b I.fst b\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { dsimp, rw category.id_comp, refl },\n      { dsimp, rw category.id_comp, apply w }\n    end } }\n\n@[reassoc]\nlemma condition (b) :\n  K.\u03b9 (I.fst_to b) \u226b I.fst b = K.\u03b9 (I.snd_to b) \u226b I.snd b := by simp\n\n/-- This definition provides a convenient way to show that a multifork is a limit. -/\n@[simps]\ndef is_limit.mk\n  (lift : \u03a0 (E : multifork I), E.X \u27f6 K.X)\n  (fac : \u2200 (E : multifork I) (i : I.L), lift E \u226b K.\u03b9 i = E.\u03b9 i)\n  (uniq : \u2200 (E : multifork I) (m : E.X \u27f6 K.X),\n    (\u2200 i : I.L, m \u226b K.\u03b9 i = E.\u03b9 i) \u2192 m = lift E) : is_limit K :=\n{ lift := lift,\n  fac' := begin\n    rintros E (a|b),\n    { apply fac },\n    { rw [\u2190 E.w (walking_multicospan.hom.fst b), \u2190 K.w (walking_multicospan.hom.fst b),\n        \u2190 category.assoc],\n      congr' 1,\n      apply fac }\n  end,\n  uniq' := begin\n    rintros E m hm,\n    apply uniq,\n    intros i,\n    apply hm,\n  end }\n\n\nvariables [has_product I.left] [has_product I.right]\n\n@[simp, reassoc]\nlemma pi_condition :\n  pi.lift K.\u03b9 \u226b I.fst_pi_map = pi.lift K.\u03b9 \u226b I.snd_pi_map := by { ext, simp }\n\n/-- Given a multifork, we may obtain a fork over `\u220f I.left \u21c9 \u220f I.right`. -/\n@[simps X] noncomputable\ndef to_pi_fork (K : multifork I) : fork I.fst_pi_map I.snd_pi_map :=\n{ X := K.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := pi.lift K.\u03b9\n    | walking_parallel_pair.one := pi.lift K.\u03b9 \u226b I.fst_pi_map\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      all_goals { change \ud835\udfd9 _ \u226b _ \u226b _ = pi.lift _ \u226b _, simp }\n    end } }\n\n@[simp] lemma to_pi_fork_\u03c0_app_zero :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.zero = pi.lift K.\u03b9 := rfl\n\n@[simp] lemma to_pi_fork_\u03c0_app_one :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.one = pi.lift K.\u03b9 \u226b I.fst_pi_map := rfl\n\nvariable (I)\n\n/-- Given a fork over `\u220f I.left \u21c9 \u220f I.right`, we may obtain a multifork. -/\n@[simps X] noncomputable\ndef of_pi_fork (c : fork I.fst_pi_map I.snd_pi_map) : multifork I :=\n{ X := c.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := c.\u03b9 \u226b pi.\u03c0 _ _\n    | walking_multicospan.right b := c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, simp },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, rw c.condition_assoc, simp }\n    end } }\n\n@[simp] lemma of_pi_fork_\u03c0_app_left (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.left a) = c.\u03b9 \u226b pi.\u03c0 _ _ := rfl\n\n@[simp] lemma of_pi_fork_\u03c0_app_right (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.right a) = c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _ := rfl\n\nend multifork\n\nnamespace multicospan_index\n\nvariables (I : multicospan_index C) [has_product I.left] [has_product I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multifork.to_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef to_pi_fork_functor : multifork I \u2964 fork I.fst_pi_map I.snd_pi_map :=\n{ obj := multifork.to_pi_fork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multifork.of_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef of_pi_fork_functor : fork I.fst_pi_map I.snd_pi_map \u2964 multifork I :=\n{ obj := multifork.of_pi_fork I, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multiforks is equivalent to the category of forks over `\u220f I.left \u21c9 \u220f I.right`.\nIt then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it\npreserves and reflects limit cones.\n-/\n@[simps] noncomputable\ndef multifork_equiv_pi_fork : multifork I \u224c fork I.fst_pi_map I.snd_pi_map :=\n{ functor := to_pi_fork_functor I,\n  inverse := of_pi_fork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cones.ext (iso.refl _) (by rintros (_|_); dsimp; simp))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, fork.ext (iso.refl _) (by { ext, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }) }\n\nend multicospan_index\n\nnamespace multicofork\n\nvariables {I : multispan_index C} (K : multicofork I)\n\n/-- The maps to the cocone point of a multicofork from the objects on the right. -/\ndef \u03c0 (b : I.R) : I.right b \u27f6 K.X :=\nK.\u03b9.app (walking_multispan.right _)\n\n@[simp] lemma \u03c0_eq_app_right (b) : K.\u03c0 b = K.\u03b9.app (walking_multispan.right _) := rfl\n\n@[simp] lemma fst_app_right (a) :\n  I.fst a \u226b K.\u03b9.app (walking_multispan.right (I.fst_from a)) =\n    K.\u03b9.app (walking_multispan.left a) :=\nby { rw \u2190 K.w (walking_multispan.hom.fst a), refl }\n\n@[simp] lemma snd_app_right (a) :\n  I.snd a \u226b K.\u03b9.app (walking_multispan.right (I.snd_from a)) =\n    K.\u03b9.app (walking_multispan.left a) :=\nby { rw \u2190 K.w (walking_multispan.hom.snd a), refl }\n\n/-- Construct a multicofork using a collection `\u03c0` of morphisms. -/\n@[simps]\ndef of_\u03c0 (I : multispan_index C) (P : C) (\u03c0 : \u03a0 b, I.right b \u27f6 P)\n  (w : \u2200 a, I.fst a \u226b \u03c0 (I.fst_from a) = I.snd a \u226b \u03c0 (I.snd_from a)) :\n  multicofork I :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := I.fst a \u226b \u03c0 _\n    | walking_multispan.right b := \u03c0 _\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { dsimp, rw category.comp_id, refl },\n      { dsimp, rw category.comp_id, apply (w _).symm }\n    end } }\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b K.\u03c0 (I.fst_from a) = I.snd a \u226b K.\u03c0 (I.snd_from a) := by simp\n\n/-- This definition provides a convenient way to show that a multicofork is a colimit. -/\n@[simps]\ndef is_colimit.mk\n  (desc : \u03a0 (E : multicofork I), K.X \u27f6 E.X)\n  (fac : \u2200 (E : multicofork I) (i : I.R), K.\u03c0 i \u226b desc E = E.\u03c0 i)\n  (uniq : \u2200 (E : multicofork I) (m : K.X \u27f6 E.X),\n    (\u2200 i : I.R, K.\u03c0 i \u226b m = E.\u03c0 i) \u2192 m = desc E) : is_colimit K :=\n{ desc := desc,\n  fac' := begin\n    rintros S (a|b),\n    { rw [\u2190 K.w (walking_multispan.hom.fst a), \u2190 S.w (walking_multispan.hom.fst a),\n        category.assoc],\n      congr' 1,\n      apply fac },\n    { apply fac },\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply uniq,\n    intros i,\n    apply hm\n  end }\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n@[simp, reassoc]\nlemma sigma_condition :\n  I.fst_sigma_map \u226b sigma.desc K.\u03c0 = I.snd_sigma_map \u226b sigma.desc K.\u03c0 := by { ext, simp }\n\n/-- Given a multicofork, we may obtain a cofork over `\u2210 I.left \u21c9 \u2210 I.right`. -/\n@[simps X] noncomputable\ndef to_sigma_cofork (K : multicofork I) : cofork I.fst_sigma_map I.snd_sigma_map :=\n{ X := K.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := I.fst_sigma_map \u226b sigma.desc K.\u03c0\n    | walking_parallel_pair.one := sigma.desc K.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      all_goals { change _ \u226b sigma.desc _ = (_ \u226b _) \u226b \ud835\udfd9 _, simp }\n    end } }\n\n@[simp] lemma to_sigma_cofork_\u03b9_app_zero :\n  K.to_sigma_cofork.\u03b9.app walking_parallel_pair.zero = I.fst_sigma_map \u226b sigma.desc K.\u03c0 := rfl\n\n@[simp] lemma to_sigma_cofork_\u03b9_app_one :\n  K.to_sigma_cofork.\u03b9.app walking_parallel_pair.one = sigma.desc K.\u03c0 := rfl\n\nvariable (I)\n\n/-- Given a cofork over `\u2210 I.left \u21c9 \u2210 I.right`, we may obtain a multicofork. -/\n@[simps X] noncomputable\ndef of_sigma_cofork (c : cofork I.fst_sigma_map I.snd_sigma_map) : multicofork I :=\n{ X := c.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0\n    | walking_multispan.right b := (sigma.\u03b9 I.right b : _) \u226b c.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b _,\n        dsimp, simp [\u2190cofork.left_app_one, -cofork.left_app_one] },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b \ud835\udfd9 _,\n        rw c.condition,\n        dsimp, simp [\u2190cofork.right_app_one, -cofork.right_app_one] }\n    end } }\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_left (c : cofork I.fst_sigma_map I.snd_sigma_map) (a) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.left a) =\n    (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0 := rfl\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_right (c : cofork I.fst_sigma_map I.snd_sigma_map) (b) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.right b) = (sigma.\u03b9 I.right b : _) \u226b c.\u03c0 := rfl\n\nend multicofork\n\nnamespace multispan_index\n\nvariables (I : multispan_index C) [has_coproduct I.left] [has_coproduct I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multicofork.to_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef to_sigma_cofork_functor : multicofork I \u2964 cofork I.fst_sigma_map I.snd_sigma_map :=\n{ obj := multicofork.to_sigma_cofork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multicofork.of_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef of_sigma_cofork_functor : cofork I.fst_sigma_map I.snd_sigma_map \u2964 multicofork I :=\n{ obj := multicofork.of_sigma_cofork I,\n  map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multicoforks is equivalent to the category of coforks over `\u2210 I.left \u21c9 \u2210 I.right`.\nIt then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that\nit preserves and reflects colimit cocones.\n-/\n@[simps] noncomputable\ndef multicofork_equiv_sigma_cofork : multicofork I \u224c cofork I.fst_sigma_map I.snd_sigma_map :=\n{ functor := to_sigma_cofork_functor I,\n  inverse := of_sigma_cofork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cocones.ext (iso.refl _) (by rintros (_|_); dsimp; simp))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, cofork.ext (iso.refl _) (by { ext, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, dsimp, simp, }) }\n\nend multispan_index\n\n/-- For `I : multicospan_index C`, we say that it has a multiequalizer if the associated\n  multicospan has a limit. -/\nabbreviation has_multiequalizer (I : multicospan_index C) :=\n  has_limit I.multicospan\n\nnoncomputable theory\n\n/-- The multiequalizer of `I : multicospan_index C`. -/\nabbreviation multiequalizer (I : multicospan_index C) [has_multiequalizer I] : C :=\n  limit I.multicospan\n\n/-- For `I : multispan_index C`, we say that it has a multicoequalizer if\n  the associated multicospan has a limit. -/\nabbreviation has_multicoequalizer (I : multispan_index C) :=\n  has_colimit I.multispan\n\n/-- The multiecoqualizer of `I : multispan_index C`. -/\nabbreviation multicoequalizer (I : multispan_index C) [has_multicoequalizer I] : C :=\n  colimit I.multispan\n\nnamespace multiequalizer\n\nvariables (I : multicospan_index C) [has_multiequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03b9 (a : I.L) : multiequalizer I \u27f6 I.left a :=\nlimit.\u03c0 _ (walking_multicospan.left a)\n\n/-- The multifork associated to the multiequalizer. -/\nabbreviation multifork : multifork I :=\nlimit.cone _\n\n@[simp]\nlemma multifork_\u03b9 (a) :\n  (multiequalizer.multifork I).\u03b9 a = multiequalizer.\u03b9 I a := rfl\n\n@[simp]\nlemma multifork_\u03c0_app_left (a) :\n  (multiequalizer.multifork I).\u03c0.app (walking_multicospan.left a) =\n  multiequalizer.\u03b9 I a := rfl\n\n@[reassoc]\nlemma condition (b) :\n  multiequalizer.\u03b9 I (I.fst_to b) \u226b I.fst b =\n  multiequalizer.\u03b9 I (I.snd_to b) \u226b I.snd b :=\nmultifork.condition _ _\n\n/-- Construct a morphism to the multiequalizer from its universal property. -/\nabbreviation lift (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) :\n  W \u27f6 multiequalizer I :=\nlimit.lift _ (multifork.of_\u03b9 I _ k h)\n\n@[simp, reassoc]\nlemma lift_\u03b9 (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) (a) :\n  multiequalizer.lift I _ k h \u226b multiequalizer.\u03b9 I a = k _ :=\nlimit.lift_\u03c0 _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : W \u27f6 multiequalizer I)\n  (h : \u2200 a, i \u226b multiequalizer.\u03b9 I a =\n  j \u226b multiequalizer.\u03b9 I a) :\n  i = j :=\nlimit.hom_ext\nbegin\n  rintro (a|b),\n  { apply h },\n  simp_rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n    \u2190 category.assoc, h],\nend\n\nvariables [has_product I.left] [has_product I.right] [has_equalizer I.fst_pi_map I.snd_pi_map]\n\n/-- The multiequalizer is isomorphic to the equalizer of `\u220f I.left \u21c9 \u220f I.right`. -/\ndef iso_equalizer : multiequalizer I \u2245 equalizer I.fst_pi_map I.snd_pi_map :=\nlimit.iso_limit_cone \u27e8_, is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.inverse (limit.is_limit _)\u27e9\n\n/-- The canonical injection `multiequalizer I \u27f6 \u220f I.left`. -/\ndef \u03b9_pi : multiequalizer I \u27f6 \u220f I.left :=\n  (iso_equalizer I).hom \u226b equalizer.\u03b9 I.fst_pi_map I.snd_pi_map\n\n@[simp, reassoc]\nlemma \u03b9_pi_\u03c0 (a) : \u03b9_pi I \u226b pi.\u03c0 I.left a = \u03b9 I a :=\nby { rw [\u03b9_pi, category.assoc, \u2190 iso.eq_inv_comp, iso_equalizer], simpa }\n\ninstance : mono (\u03b9_pi I) := @@mono_comp _ _ _ _ equalizer.\u03b9_mono\n\nend multiequalizer\n\nnamespace multicoequalizer\n\nvariables (I : multispan_index C) [has_multicoequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03c0 (b : I.R) : I.right b \u27f6 multicoequalizer I :=\ncolimit.\u03b9 I.multispan (walking_multispan.right _)\n\n/-- The multicofork associated to the multicoequalizer. -/\nabbreviation multicofork : multicofork I :=\ncolimit.cocone _\n\n@[simp]\nlemma multicofork_\u03c0 (b) :\n  (multicoequalizer.multicofork I).\u03c0 b = multicoequalizer.\u03c0 I b := rfl\n\n@[simp]\nlemma multicofork_\u03b9_app_right (b) :\n  (multicoequalizer.multicofork I).\u03b9.app (walking_multispan.right b) =\n  multicoequalizer.\u03c0 I b := rfl\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b multicoequalizer.\u03c0 I (I.fst_from a) =\n  I.snd a \u226b multicoequalizer.\u03c0 I (I.snd_from a) :=\nmulticofork.condition _ _\n\n/-- Construct a morphism from the multicoequalizer from its universal property. -/\nabbreviation desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) :\n  multicoequalizer I \u27f6 W :=\ncolimit.desc _ (multicofork.of_\u03c0 I _ k h)\n\n@[simp, reassoc]\nlemma \u03c0_desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) (b) :\n  multicoequalizer.\u03c0 I b \u226b multicoequalizer.desc I _ k h = k _ :=\ncolimit.\u03b9_desc _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : multicoequalizer I \u27f6 W)\n  (h : \u2200 b, multicoequalizer.\u03c0 I b \u226b i = multicoequalizer.\u03c0 I b \u226b j) :\n  i = j :=\ncolimit.hom_ext\nbegin\n  rintro (a|b),\n  { simp_rw [\u2190 colimit.w I.multispan (walking_multispan.hom.fst a),\n    category.assoc, h] },\n  { apply h },\nend\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\nvariables [has_coequalizer I.fst_sigma_map I.snd_sigma_map]\n\n/-- The multicoequalizer is isomorphic to the coequalizer of `\u2210 I.left \u21c9 \u2210 I.right`. -/\ndef iso_coequalizer : multicoequalizer I \u2245 coequalizer I.fst_sigma_map I.snd_sigma_map :=\ncolimit.iso_colimit_cocone \u27e8_, is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.inverse (colimit.is_colimit _)\u27e9\n\n/-- The canonical projection `\u2210 I.right \u27f6 multicoequalizer I`. -/\ndef sigma_\u03c0 : \u2210 I.right \u27f6 multicoequalizer I :=\n  coequalizer.\u03c0 I.fst_sigma_map I.snd_sigma_map \u226b (iso_coequalizer I).inv\n\n@[simp, reassoc]\nlemma \u03b9_sigma_\u03c0 (b) : sigma.\u03b9 I.right b \u226b sigma_\u03c0 I = \u03c0 I b :=\nby { rw [sigma_\u03c0, \u2190 category.assoc, iso.comp_inv_eq, iso_coequalizer], simpa }\n\ninstance : epi (sigma_\u03c0 I) := @@epi_comp _ _ coequalizer.\u03c0_epi _ _\n\nend multicoequalizer\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/shapes/multiequalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3873667127268533}}
{"text": "/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nThe writer monad transformer for passing immutable state.\n\n! This file was ported from Lean 3 source module control.monad.writer\n! leanprover-community/mathlib commit 9407b03373c8cd201df99d6bc5514fc2db44054f\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Group.Defs\nimport Mathbin.Logic.Equiv.Defs\n\nuniverse u v w u\u2080 u\u2081 v\u2080 v\u2081\n\nstructure WriterT (\u03c9 : Type u) (m : Type u \u2192 Type v) (\u03b1 : Type u) : Type max u v where\n  run : m (\u03b1 \u00d7 \u03c9)\n#align writer_t WriterT\u2093\n\n#print Writer /-\n@[reducible]\ndef Writer (\u03c9 : Type u) :=\n  WriterT \u03c9 id\n#align writer Writer\n-/\n\nattribute [pp_using_anonymous_constructor] WriterT\n\nnamespace WriterT\n\nsection\n\nvariable {\u03c9 : Type u}\n\nvariable {m : Type u \u2192 Type v}\n\nvariable [Monad m]\n\nvariable {\u03b1 \u03b2 : Type u}\n\nopen Function\n\n@[ext]\nprotected theorem ext (x x' : WriterT \u03c9 m \u03b1) (h : x.run = x'.run) : x = x' := by\n  cases x <;> cases x' <;> congr <;> apply h\n#align writer_t.ext WriterT\u2093.ext\n\n@[inline]\nprotected def tell (w : \u03c9) : WriterT \u03c9 m PUnit :=\n  \u27e8pure (PUnit.unit, w)\u27e9\n#align writer_t.tell WriterT\u2093.tell\n\n@[inline]\nprotected def listen : WriterT \u03c9 m \u03b1 \u2192 WriterT \u03c9 m (\u03b1 \u00d7 \u03c9)\n  | \u27e8cmd\u27e9 => \u27e8(fun x : \u03b1 \u00d7 \u03c9 => ((x.1, x.2), x.2)) <$> cmd\u27e9\n#align writer_t.listen WriterT\u2093.listen\n\n@[inline]\nprotected def pass : WriterT \u03c9 m (\u03b1 \u00d7 (\u03c9 \u2192 \u03c9)) \u2192 WriterT \u03c9 m \u03b1\n  | \u27e8cmd\u27e9 => \u27e8uncurry (uncurry fun x (f : \u03c9 \u2192 \u03c9) w => (x, f w)) <$> cmd\u27e9\n#align writer_t.pass WriterT\u2093.pass\n\n@[inline]\nprotected def pure [One \u03c9] (a : \u03b1) : WriterT \u03c9 m \u03b1 :=\n  \u27e8pure (a, 1)\u27e9\n#align writer_t.pure WriterT\u2093.pure\n\n@[inline]\nprotected def bind [Mul \u03c9] (x : WriterT \u03c9 m \u03b1) (f : \u03b1 \u2192 WriterT \u03c9 m \u03b2) : WriterT \u03c9 m \u03b2 :=\n  \u27e8do\n    let x \u2190 x.run\n    let x' \u2190 (f x.1).run\n    pure (x'.1, x.2 * x'.2)\u27e9\n#align writer_t.bind WriterT\u2093.bind\n\ninstance [One \u03c9] [Mul \u03c9] : Monad (WriterT \u03c9 m)\n    where\n  pure \u03b1 := WriterT.pure\n  bind \u03b1 \u03b2 := WriterT.bind\n\ninstance [Monoid \u03c9] [LawfulMonad m] : LawfulMonad (WriterT \u03c9 m)\n    where\n  id_map := by\n    intros\n    cases x\n    simp [(\u00b7 <$> \u00b7), WriterT.bind, WriterT.pure]\n  pure_bind := by\n    intros\n    simp [Pure.pure, WriterT.pure, (\u00b7 >>= \u00b7), WriterT.bind]\n    ext <;> rfl\n  bind_assoc := by\n    intros\n    simp [(\u00b7 >>= \u00b7), WriterT.bind, mul_assoc, functor_norm]\n\n@[inline]\nprotected def lift [One \u03c9] (a : m \u03b1) : WriterT \u03c9 m \u03b1 :=\n  \u27e8flip Prod.mk 1 <$> a\u27e9\n#align writer_t.lift WriterT\u2093.lift\n\ninstance (m) [Monad m] [One \u03c9] : HasMonadLift m (WriterT \u03c9 m) :=\n  \u27e8fun \u03b1 => WriterT.lift\u27e9\n\n@[inline]\nprotected def monadMap {m m'} [Monad m] [Monad m'] {\u03b1} (f : \u2200 {\u03b1}, m \u03b1 \u2192 m' \u03b1) :\n    WriterT \u03c9 m \u03b1 \u2192 WriterT \u03c9 m' \u03b1 := fun x => \u27e8f x.run\u27e9\n#align writer_t.monad_map WriterT\u2093.monadMap\n\ninstance (m m') [Monad m] [Monad m'] : MonadFunctor m m' (WriterT \u03c9 m) (WriterT \u03c9 m') :=\n  \u27e8@WriterT.monadMap \u03c9 m m' _ _\u27e9\n\n@[inline]\nprotected def adapt {\u03c9' : Type u} {\u03b1 : Type u} (f : \u03c9 \u2192 \u03c9') : WriterT \u03c9 m \u03b1 \u2192 WriterT \u03c9' m \u03b1 :=\n  fun x => \u27e8Prod.map id f <$> x.run\u27e9\n#align writer_t.adapt WriterT\u2093.adapt\n\ninstance (\u03b5) [One \u03c9] [Monad m] [MonadExcept \u03b5 m] : MonadExcept \u03b5 (WriterT \u03c9 m)\n    where\n  throw \u03b1 := WriterT.lift \u2218 throw\n  catch \u03b1 x c := \u27e8catch x.run fun e => (c e).run\u27e9\n\nend\n\nend WriterT\n\n#print MonadWriter /-\n/-- An implementation of [MonadReader](\nhttps://hackage.haskell.org/package/mtl-2.2.2/docs/Control-Monad-Reader-Class.html#t:MonadReader).\nIt does not contain `local` because this function cannot be lifted using `monad_lift`.\nInstead, the `monad_reader_adapter` class provides the more general `adapt_reader` function.\n\nNote: This class can be seen as a simplification of the more \"principled\" definition\n```\nclass monad_reader (\u03c1 : out_param (Type u)) (n : Type u \u2192 Type u) :=\n(lift {\u03b1 : Type u} : (\u2200 {m : Type u \u2192 Type u} [monad m], reader_t \u03c1 m \u03b1) \u2192 n \u03b1)\n```\n-/\nclass MonadWriter (\u03c9 : outParam (Type u)) (m : Type u \u2192 Type v) where\n  tell (w : \u03c9) : m PUnit\n  listen {\u03b1} : m \u03b1 \u2192 m (\u03b1 \u00d7 \u03c9)\n  pass {\u03b1 : Type u} : m (\u03b1 \u00d7 (\u03c9 \u2192 \u03c9)) \u2192 m \u03b1\n#align monad_writer MonadWriter\n-/\n\nexport MonadWriter ()\n\ninstance {\u03c9 : Type u} {m : Type u \u2192 Type v} [Monad m] : MonadWriter \u03c9 (WriterT \u03c9 m)\n    where\n  tell := WriterT.tell\n  listen \u03b1 := WriterT.listen\n  pass \u03b1 := WriterT.pass\n\ninstance {\u03c9 \u03c1 : Type u} {m : Type u \u2192 Type v} [Monad m] [MonadWriter \u03c9 m] :\n    MonadWriter \u03c9 (ReaderT \u03c1 m)\n    where\n  tell x := monadLift (tell x : m PUnit)\n  listen := fun \u03b1 \u27e8cmd\u27e9 => \u27e8fun r => listen (cmd r)\u27e9\n  pass := fun \u03b1 \u27e8cmd\u27e9 => \u27e8fun r => pass (cmd r)\u27e9\n\ndef swapRight {\u03b1 \u03b2 \u03b3} : (\u03b1 \u00d7 \u03b2) \u00d7 \u03b3 \u2192 (\u03b1 \u00d7 \u03b3) \u00d7 \u03b2\n  | \u27e8\u27e8x, y\u27e9, z\u27e9 => ((x, z), y)\n#align swap_right swapRight\n\ninstance {\u03c9 \u03c3 : Type u} {m : Type u \u2192 Type v} [Monad m] [MonadWriter \u03c9 m] :\n    MonadWriter \u03c9 (StateT \u03c3 m)\n    where\n  tell x := monadLift (tell x : m PUnit)\n  listen := fun \u03b1 \u27e8cmd\u27e9 => \u27e8fun r => swapRight <$> listen (cmd r)\u27e9\n  pass := fun \u03b1 \u27e8cmd\u27e9 => \u27e8fun r => pass (swapRight <$> cmd r)\u27e9\n\nopen Function\n\ndef ExceptT.passAux {\u03b5 \u03b1 \u03c9} : Except \u03b5 (\u03b1 \u00d7 (\u03c9 \u2192 \u03c9)) \u2192 Except \u03b5 \u03b1 \u00d7 (\u03c9 \u2192 \u03c9)\n  | Except.error a => (Except.error a, id)\n  | Except.ok (x, y) => (Except.ok x, y)\n#align except_t.pass_aux ExceptT\u2093.passAux\n\ninstance {\u03c9 \u03b5 : Type u} {m : Type u \u2192 Type v} [Monad m] [MonadWriter \u03c9 m] :\n    MonadWriter \u03c9 (ExceptT \u03b5 m)\n    where\n  tell x := monadLift (tell x : m PUnit)\n  listen := fun \u03b1 \u27e8cmd\u27e9 => \u27e8(uncurry fun x y => flip Prod.mk y <$> x) <$> listen cmd\u27e9\n  pass := fun \u03b1 \u27e8cmd\u27e9 => \u27e8pass (ExceptT.passAux <$> cmd)\u27e9\n\ndef OptionT.passAux {\u03b1 \u03c9} : Option (\u03b1 \u00d7 (\u03c9 \u2192 \u03c9)) \u2192 Option \u03b1 \u00d7 (\u03c9 \u2192 \u03c9)\n  | none => (none, id)\n  | some (x, y) => (some x, y)\n#align option_t.pass_aux OptionT\u2093.passAux\n\ninstance {\u03c9 : Type u} {m : Type u \u2192 Type v} [Monad m] [MonadWriter \u03c9 m] : MonadWriter \u03c9 (OptionT m)\n    where\n  tell x := monadLift (tell x : m PUnit)\n  listen := fun \u03b1 \u27e8cmd\u27e9 => \u27e8(uncurry fun x y => flip Prod.mk y <$> x) <$> listen cmd\u27e9\n  pass := fun \u03b1 \u27e8cmd\u27e9 => \u27e8pass (OptionT.passAux <$> cmd)\u27e9\n\n/-- Adapt a monad stack, changing the type of its top-most environment.\n\nThis class is comparable to\n[Control.Lens.Magnify](https://hackage.haskell.org/package/lens-4.15.4/docs/Control-Lens-Zoom.html#t:Magnify),\nbut does not use lenses (why would it), and is derived automatically for any transformer\nimplementing `monad_functor`.\n\nNote: This class can be seen as a simplification of the more \"principled\" definition\n```\nclass monad_reader_functor (\u03c1 \u03c1' : out_param (Type u)) (n n' : Type u \u2192 Type u) :=\n(map {\u03b1 : Type u} :\n  (\u2200 {m : Type u \u2192 Type u} [monad m], reader_t \u03c1 m \u03b1 \u2192 reader_t \u03c1' m \u03b1) \u2192 n \u03b1 \u2192 n' \u03b1)\n```\n-/\nclass MonadWriterAdapter (\u03c9 \u03c9' : outParam (Type u)) (m m' : Type u \u2192 Type v) where\n  adaptWriter {\u03b1 : Type u} : (\u03c9 \u2192 \u03c9') \u2192 m \u03b1 \u2192 m' \u03b1\n#align monad_writer_adapter MonadWriterAdapter\n\nexport MonadWriterAdapter (adaptWriter)\n\nsection\n\nvariable {\u03c9 \u03c9' : Type u} {m m' : Type u \u2192 Type v}\n\n/-- Transitivity.\n\nThis instance generates the type-class problem with a metavariable argument (which is why this\nis marked as `[nolint dangerous_instance]`).\nCurrently that is not a problem, as there are almost no instances of `monad_functor` or\n`monad_writer_adapter`.\n\nsee Note [lower instance priority] -/\n@[nolint dangerous_instance]\ninstance (priority := 100) monadWriterAdapterTrans {n n' : Type u \u2192 Type v}\n    [MonadWriterAdapter \u03c9 \u03c9' m m'] [MonadFunctor m m' n n'] : MonadWriterAdapter \u03c9 \u03c9' n n' :=\n  \u27e8fun \u03b1 f => monadMap fun \u03b1 => (adaptWriter f : m \u03b1 \u2192 m' \u03b1)\u27e9\n#align monad_writer_adapter_trans monadWriterAdapterTrans\n\ninstance [Monad m] : MonadWriterAdapter \u03c9 \u03c9' (WriterT \u03c9 m) (WriterT \u03c9' m) :=\n  \u27e8fun \u03b1 => WriterT.adapt\u27e9\n\nend\n\ninstance (\u03c9 : Type u) (m out) [MonadRun out m] : MonadRun (fun \u03b1 => out (\u03b1 \u00d7 \u03c9)) (WriterT \u03c9 m) :=\n  \u27e8fun \u03b1 x => run <| x.run\u27e9\n\n/-- reduce the equivalence between two writer monads to the equivalence between\ntheir underlying monad -/\ndef WriterT.equiv {m\u2081 : Type u\u2080 \u2192 Type v\u2080} {m\u2082 : Type u\u2081 \u2192 Type v\u2081} {\u03b1\u2081 \u03c9\u2081 : Type u\u2080}\n    {\u03b1\u2082 \u03c9\u2082 : Type u\u2081} (F : m\u2081 (\u03b1\u2081 \u00d7 \u03c9\u2081) \u2243 m\u2082 (\u03b1\u2082 \u00d7 \u03c9\u2082)) : WriterT \u03c9\u2081 m\u2081 \u03b1\u2081 \u2243 WriterT \u03c9\u2082 m\u2082 \u03b1\u2082\n    where\n  toFun := fun \u27e8f\u27e9 => \u27e8F f\u27e9\n  invFun := fun \u27e8f\u27e9 => \u27e8F.symm f\u27e9\n  left_inv := fun \u27e8f\u27e9 => congr_arg WriterT.mk <| F.left_inv _\n  right_inv := fun \u27e8f\u27e9 => congr_arg WriterT.mk <| F.right_inv _\n#align writer_t.equiv WriterT\u2093.equiv\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Monad/Writer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.3873667123877527}}
{"text": "import tactic.transport\nimport order.bounded_order\nimport algebra.lie.basic\n\n-- We verify that `transport` can move a `semiring` across an equivalence.\n-- Note that we've never even mentioned the idea of addition or multiplication to `transport`.\ndef semiring.map {\u03b1 : Type} [semiring \u03b1] {\u03b2 : Type} (e : \u03b1 \u2243 \u03b2) : semiring \u03b2 :=\nbegin\n  transport using e,\n  -- TODO\n  { simp *, refl }, -- nat_cast_zero\n  { intros, simp *, show _ = e _, simp, refl, }, -- nat_cast_succ\nend\n\n-- Indeed, it can equally well move a `semilattice_sup`.\ndef sup.map {\u03b1 : Type} [semilattice_sup \u03b1] {\u03b2 : Type} (e : \u03b1 \u2243 \u03b2) : semilattice_sup \u03b2 :=\nby transport using e\n\n-- Verify definitional equality of the new structure data.\nexample {\u03b1 : Type} [semilattice_sup \u03b1] {\u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (x y : \u03b2) :\nbegin\n  haveI := sup.map e,\n  exact (x \u2264 y) = (e.symm x \u2264 e.symm y),\nend :=\nrfl\n\n-- Below we verify in more detail that the transported structure for `semiring`\n-- is definitionally what you would hope for.\n\ninductive mynat : Type\n| zero : mynat\n| succ : mynat \u2192 mynat\n\ndef mynat_equiv : \u2115 \u2243 mynat :=\n{ to_fun := \u03bb n, nat.rec_on n mynat.zero (\u03bb n, mynat.succ),\n  inv_fun := \u03bb n, mynat.rec_on n nat.zero (\u03bb n, nat.succ),\n  left_inv := \u03bb n, begin induction n, refl, exact congr_arg nat.succ n_ih, end,\n  right_inv := \u03bb n, begin induction n, refl, exact congr_arg mynat.succ n_ih, end }\n\n@[simp] lemma mynat_equiv_apply_zero : mynat_equiv 0 = mynat.zero := rfl\n@[simp] lemma mynat_equiv_apply_succ (n : \u2115) :\n  mynat_equiv (n + 1) = mynat.succ (mynat_equiv n) := rfl\n@[simp] lemma mynat_equiv_symm_apply_zero : mynat_equiv.symm mynat.zero = 0:= rfl\n@[simp] lemma mynat_equiv_symm_apply_succ (n : mynat) :\n  mynat_equiv.symm (mynat.succ n) = (mynat_equiv.symm n) + 1 := rfl\n\ninstance semiring_mynat : semiring mynat :=\nsemiring.map mynat_equiv\n\nlemma mynat_add_def (a b : mynat) : a + b = mynat_equiv (mynat_equiv.symm a + mynat_equiv.symm b) :=\nrfl\n\n-- Verify that we can do computations with the transported structure.\nexample :\n  (mynat.succ (mynat.succ mynat.zero)) + (mynat.succ mynat.zero) =\n    (mynat.succ (mynat.succ (mynat.succ mynat.zero))) :=\nrfl\n\nlemma mynat_zero_def : (0 : mynat) = mynat_equiv 0 :=\nrfl\n\nlemma mynat_one_def : (1 : mynat) = mynat_equiv 1 :=\nrfl\n\nlemma mynat_mul_def (a b : mynat) : a * b = mynat_equiv (mynat_equiv.symm a * mynat_equiv.symm b) :=\nrfl\n\nexample : (3 : mynat) + (7 : mynat) = (10 : mynat) :=\nrfl\n\nexample : (2 : mynat) * (2 : mynat) = (4 : mynat) :=\nrfl\n\nexample : (3 : mynat) + (7 : mynat) * (2 : mynat) = (17 : mynat) :=\nrfl\n\nexample : (2 : \u2115) \u2022 (3 : mynat) = (6 : mynat) :=\nrfl\n\nexample : (3 : mynat) ^ 2 = (9 : mynat) :=\nrfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/transport/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.615087848460224, "lm_q1q2_score": 0.3873667082832629}}
{"text": "import Hw9\nimport Sets.Basic\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\nopen Set \n\nvariable (\u03b1 \u03b2 : Type)\nvariable (X Y Z : Set \u03b1)\nvariable (W : Set \u03b2) \n\ntheorem desiredType1 : \u2205 \u2208 \ud835\udcab  X := sorry \n\ntheorem desiredType2 (U : \u03b2 \u2192 Set \u03b1) : \u2200 b, U b \u2286 BigUnion U := sorry \n\ntheorem desiredType3 (h : X \u2286 Y) : (X \u00d7\u02e2 W) \u2286 (Y \u00d7\u02e2 W) := sorry\n\ntheorem desiredType4 (h : Y \u2229 Z = \u2205) : Y\u1d9c \u222a Z\u1d9c = Univ := sorry \n\ntheorem desiredType5 : (X \\ Y) \u222a (Y \\ X) = (X \u222a Y) \\ (X \u2229 Y) := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef n : String := \"3\"\n\ndef problem : String := \"problem\"++n\n\ndef desired : String := \"desiredType\"++n\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const desired []) (Expr.const problem [])\n#eval collectAxiomsOf problem\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework9", "sha": "c26e748a8f91c4f459d6f568a6819b53cd6088a1", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9/homework9-c26e748a8f91c4f459d6f568a6819b53cd6088a1/_Tests/Problem3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238982, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3873103265723375}}
{"text": "import category_theory.limits.concrete_category\nimport topology.category.Profinite.as_limit\nimport for_mathlib.Profinite.compat_discrete_quotient\nimport for_mathlib.Cech.split\n\nnoncomputable theory\n\nopen category_theory\n\nuniverse u\n\nnamespace Profinite\n\nvariables (F : arrow Profinite.{u}) (surj : function.surjective F.hom)\n\nopen discrete_quotient\n\n/-- A diagram of arrows construction from discrete quotients of F.left. -/\n@[simps]\ndef fintype_arrow_diagram : discrete_quotient F.left \u2964 arrow Fintype.{u} :=\n{ obj := \u03bb S,\n  { left := Fintype.of S,\n    right := Fintype.of $ S.make F.hom surj,\n    hom := discrete_quotient.map (S.make_le_comap _ _) },\n  map := \u03bb S T f,\n  { left := of_le $ le_of_hom f,\n    right := of_le $ make_right_mono F.hom surj S T $ le_of_hom f } }.\n\n/-- A diagram of arrows construction from discrete quotients of F.left. -/\n@[simps]\ndef arrow_diagram : discrete_quotient F.left \u2964 arrow Profinite.{u} :=\nfintype_arrow_diagram F surj \u22d9 Fintype.to_Profinite.map_arrow\n\n/-- The left diagram associated to arrow_diagram. -/\nabbreviation left_arrow_diagram : discrete_quotient F.left \u2964 Profinite.{u} :=\narrow_diagram F surj \u22d9 arrow.left_func\n\n/-- The right diagram associated to arrow_diagram. -/\nabbreviation right_arrow_diagram : discrete_quotient F.left \u2964 Profinite.{u} :=\narrow_diagram F surj \u22d9 arrow.right_func\n\nlemma arrow_diagram_surjective (S : discrete_quotient F.left) :\n  function.surjective ((arrow_diagram F surj).obj S).hom :=\nbegin\n  rintro \u27e8x\u27e9,\n  obtain \u27e8x,rfl\u27e9 := surj x,\n  exact \u27e8S.proj x,rfl\u27e9,\nend\n\ninstance (S : discrete_quotient F.left) : arrow.split ((arrow_diagram F surj).obj S) :=\n{ \u03c3 := \u27e8\u03bb x, classical.some (arrow_diagram_surjective F surj S x),\n    continuous_of_discrete_topology\u27e9,\n  is_splitting' := begin\n    ext x,\n    erw classical.some_spec (arrow_diagram_surjective F surj S x),\n    refl,\n  end }\n\n/-- A cone which is a limit expressing an arrow as a limit. -/\n@[simps]\ndef arrow_cone : limits.cone (arrow_diagram F surj) :=\n{ X := F,\n  \u03c0 :=\n  { app := \u03bb S,\n    { left := \u27e8S.proj, S.proj_continuous\u27e9,\n      right := \u27e8(S.make _ surj).proj,\n        (S.make _ surj).proj_continuous\u27e9 } } }\n\n/-- A helper definition used for `arrow_limit_cone`. -/\ndef arrow_diagram_snd_preserves :\n  limits.preserves_limit (arrow_diagram F surj \u22d9 comma.snd _ _) (\ud835\udfed _) :=\nbegin\n  have h := limits.id_preserves_limits.preserves_limits_of_shape,\n  have hh := h.preserves_limit,\n  exact hh,\nend\n\n/-- the limit cone assocciated to arrow_diagram -/\n@[simps]\ndef arrow_limit_cone : limits.limit_cone (arrow_diagram F surj) :=\n{ cone := @comma.cone_of_preserves _ _ _ _ _ _ _ _ _ _ _\n  (arrow_diagram_snd_preserves _ _) (limit_cone _) _ (limit_cone_is_limit _),\n  is_limit := @comma.cone_of_preserves_is_limit _ _ _ _ _ _ _ _ _ _ _\n    (arrow_diagram_snd_preserves _ _) _ (limit_cone_is_limit _) _ _ }\n\n/-- lifing arrow_cone gives an isomorphism on the left -/\ninstance arrow_is_iso_lift_left : is_iso ((arrow_limit_cone F surj).is_limit.lift\n  (arrow_cone F surj)).left := Profinite.is_iso_as_limit_cone_lift _\n\n/-- lifing arrow_cone gives an isomorphism on the right -/\ninstance arrow_is_iso_lift_right : is_iso ((arrow_limit_cone F surj).is_limit.lift\n  (arrow_cone F surj)).right := is_iso_of_bijective _\nbegin\n  split,\n  { intros x y h,\n    apply discrete_quotient.eq_of_proj_eq,\n    intros S,\n    apply_fun subtype.val at h,\n    let T : discrete_quotient F.left := S.comap F.hom.continuous,\n    let R : discrete_quotient F.right := T.make F.hom surj,\n    have hR : R \u2264 S,\n    { apply discrete_quotient.make_right_le,\n      tauto },\n    apply_fun (\u03bb e, e T) at h,\n    have := discrete_quotient.of_le_proj_apply hR,\n    rw [\u2190 this, \u2190 this],\n    congr' 1 },\n  { intros x,\n    cases x with x hx,\n    dsimp at x hx,\n    let Us : \u03a0 (I : discrete_quotient F.right), I := \u03bb I,\n      of_le (make_right_le _ _ _ _ (by tauto)) (x $ I.comap F.hom.continuous),\n    rcases discrete_quotient.exists_of_compat Us _ with \u27e8y,hy\u27e9,\n    { refine \u27e8y,_\u27e9,\n      ext I : 2,\n      dsimp at *,\n      let J : discrete_quotient F.right := (I.make F.hom surj),\n      let II : discrete_quotient F.left := J.comap F.hom.continuous \u2293 I,\n      have h1 : II \u2264 I := inf_le_right,\n      have h2 : II \u2264 J.comap F.hom.continuous := inf_le_left,\n      rw \u2190 hx (hom_of_le h1),\n      dsimp [comma.cone_of_preserves_is_limit,\n        limit_cone_is_limit, CompHaus.limit_cone_is_limit,\n        Top.limit_cone_is_limit],\n      rw hy,\n      dsimp [Us],\n      rw \u2190 hx (hom_of_le h2),\n      rw \u2190 of_le_comp_apply },\n    { intros A B h,\n      dsimp [Us],\n      have := comap_mono F.hom.continuous h,\n      rw \u2190 hx (hom_of_le this),\n      rw [\u2190 of_le_comp_apply, \u2190 of_le_comp_apply] } },\nend\n\n@[simps]\ndef left_arrow_cone : limits.cone (left_arrow_diagram F surj) :=\nfunctor.map_cone _ (arrow_cone F surj)\n\n@[simps]\ndef right_arrow_cone : limits.cone (right_arrow_diagram F surj) :=\nfunctor.map_cone _ (arrow_cone F surj)\n\ninstance left_arrow_cone_lift_is_iso : is_iso $\n  (limit_cone_is_limit $ left_arrow_diagram F surj).lift (left_arrow_cone F surj) :=\nProfinite.arrow_is_iso_lift_left _ _\n\ninstance right_arrow_cone_lift_is_iso : is_iso $\n  (limit_cone_is_limit $ right_arrow_diagram F surj).lift (right_arrow_cone F surj) :=\nProfinite.arrow_is_iso_lift_right _ _\n\n@[simps]\ndef left_arrow_cone_iso : left_arrow_cone F surj \u2245\n  (limit_cone $ left_arrow_diagram F surj) :=\nlimits.cones.ext (as_iso $ (limit_cone_is_limit $ left_arrow_diagram F surj).lift _)\n  (\u03bb _ , rfl)\n\n@[simps]\ndef right_arrow_cone_iso : right_arrow_cone F surj \u2245\n  (limit_cone $ right_arrow_diagram F surj) :=\nlimits.cones.ext (as_iso $ (limit_cone_is_limit $ right_arrow_diagram F surj).lift _)\n  (\u03bb _ , rfl)\n\n/-- The isomorphism of cones showing that arrow_cone is a limit cone. -/\n@[simps]\ndef arrow_cone_iso : arrow_cone F surj \u2245 (arrow_limit_cone F surj).cone :=\nbegin\n  haveI := arrow.is_iso_of_iso_left_of_is_iso_right\n    ((arrow_limit_cone F surj).is_limit.lift (arrow_cone F surj)),\n  exact limits.cones.ext\n    (as_iso $ (arrow_limit_cone F surj).is_limit.lift (arrow_cone F surj))\n    (\u03bb _, rfl)\nend\n\n/-- arrow_cone is a limit cone. -/\n@[simps]\ndef is_limit_arrow_cone : limits.is_limit (arrow_cone F surj) :=\nlimits.is_limit.of_iso_limit (arrow_limit_cone F surj).is_limit\n  (arrow_cone_iso F surj).symm\n\n@[simps]\ndef is_limit_left_arrow_cone : limits.is_limit (left_arrow_cone F surj) :=\nlimits.is_limit.of_iso_limit (limit_cone_is_limit $ left_arrow_diagram F surj)\n  (left_arrow_cone_iso _ _).symm\n\n@[simps]\ndef is_limit_right_arrow_cone : limits.is_limit (right_arrow_cone F surj) :=\nlimits.is_limit.of_iso_limit (limit_cone_is_limit $ right_arrow_diagram F surj)\n  (right_arrow_cone_iso _ _).symm\n\nopen opposite\n\nopen_locale simplicial\n\n@[simps]\ndef Cech_cone_diagram (n : \u2115) : discrete_quotient F.left \u2964 Profinite.{u} :=\narrow_diagram F surj \u22d9 simplicial_object.cech_nerve \u22d9\n  (evaluation _ _).obj (op [n])\n\ndef Cech_cone_diagram_proj (n : \u2115) (S : discrete_quotient F.left) (i : fin (n+1)) :\n  (Cech_cone_diagram F surj n).obj S \u27f6 Profinite.of S :=\nlimits.wide_pullback.\u03c0 _ i\n\ndef Cech_cone_diagram_inclusion (n : \u2115) (S : discrete_quotient F.left) :\n  (Cech_cone_diagram F surj n).obj S \u2192 fin (n+1) \u2192 S :=\n\u03bb a i, Cech_cone_diagram_proj F surj n S i a\n\nset_option pp.universes true\nlemma Cech_cone_diagram_inclusion_injective (n : \u2115) (S : discrete_quotient F.left) :\n  function.injective (Cech_cone_diagram_inclusion F surj n S) :=\nbegin\n  intros a b h,\n  dsimp [Cech_cone_diagram] at a b,\n  haveI : limits.preserves_limits_of_size.{0 0} (forget Profinite.{u}) :=\n    limits.preserves_limits_of_size_shrink.{0 u 0 u u u u+1 u+1} (forget.{u+1 u u} Profinite.{u}),\n  apply category_theory.limits.concrete.wide_pullback_ext',\n  rintros j,\n  apply_fun (\u03bb e, e j) at h,\n  exact h,\nend\n\ninstance Cech_cone_diagram_fintype (n : \u2115) (S : discrete_quotient F.left) :\n  fintype ((Cech_cone_diagram F surj n).obj S) :=\nfintype.of_injective (Cech_cone_diagram_inclusion F surj n S)\n  (Cech_cone_diagram_inclusion_injective F surj n S)\n\n@[simps]\ndef Cech_cone (n : \u2115) : limits.cone (Cech_cone_diagram F surj n) :=\nfunctor.map_cone _ (arrow_cone F surj)\n\n@[simps]\ndef swap_cone_right (n : \u2115) (S : limits.cone (Cech_cone_diagram F surj n)) :\n  limits.cone (right_arrow_diagram F surj) :=\n{ X := S.X,\n  \u03c0 := { app := \u03bb T, S.\u03c0.app T \u226b limits.wide_pullback.base _,\n  naturality' := begin\n    intros X Y f,\n    dsimp,\n    simp [\u2190 S.w f],\n  end } }\n\n@[simps]\ndef swap_cone_left (n : \u2115) (i : (fin (n+1)))\n  (S : limits.cone (Cech_cone_diagram F surj n)) :\n  limits.cone (left_arrow_diagram F surj) :=\n{ X := S.X,\n  \u03c0 :=\n  { app := \u03bb T, S.\u03c0.app T \u226b limits.wide_pullback.\u03c0 _ i,\n    naturality' := begin\n      intros X Y f,\n      dsimp,\n      simp [\u2190 S.w f],\n    end } }\n\n@[simps]\ndef Cech_cone_is_limit (n : \u2115) : limits.is_limit (Cech_cone F surj n) :=\n{ lift := \u03bb S, limits.wide_pullback.lift\n    ((is_limit_right_arrow_cone F surj).lift $ swap_cone_right _ _ _ _)\n    (\u03bb i, (is_limit_left_arrow_cone F surj).lift $ swap_cone_left _ _ _ i _)\n    begin\n      intros i,\n      apply (is_limit_right_arrow_cone F surj).hom_ext,\n      intros T,\n      simp,\n      have : (arrow_cone F surj).X.hom \u226b (right_arrow_cone F surj).\u03c0.app T =\n        (left_arrow_cone F surj).\u03c0.app T \u226b\n        (whisker_left (arrow_diagram F surj) arrow.left_to_right).app T, by refl,\n      erw [this, \u2190 category.assoc,\n        (is_limit_left_arrow_cone F surj).fac (swap_cone_left F surj n i S) T],\n      simp,\n    end,\n  fac' := begin\n    intros S T,\n    apply limits.wide_pullback.hom_ext,\n    { intro i,\n      dsimp,\n      simp,\n      have := (is_limit_left_arrow_cone F surj).fac,\n      erw this,\n      refl },\n    { dsimp,\n      simp,\n      erw (is_limit_right_arrow_cone F surj).fac,\n      refl }\n  end,\n  uniq' := begin\n    intros S f h,\n    apply limits.wide_pullback.hom_ext,\n    { dsimp, simp,\n      intros i,\n      apply (is_limit_left_arrow_cone F surj).hom_ext,\n      intros T,\n      simp,\n      erw [\u2190 h T, category.assoc, limits.wide_pullback.lift_\u03c0],\n      refl },\n    { dsimp, simp,\n      apply (is_limit_right_arrow_cone F surj).hom_ext,\n      intros T,\n      simp,\n      erw [\u2190 h T, category.assoc, limits.wide_pullback.lift_base],\n      refl }\n  end }.\n\nend Profinite\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/Profinite/arrow_limit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.38731032657233744}}
{"text": "-- import summary\n\n-- The Undecidability of the Continuum Hypothesis\n\n-- theorem t022 : independent ZFC' CH_f := independence_of_CH\n", "meta": {"author": "AtnNn", "repo": "lean-sandbox", "sha": "8c68afbdc09213173aef1be195da7a9a86060a97", "save_path": "github-repos/lean/AtnNn-lean-sandbox", "path": "github-repos/lean/AtnNn-lean-sandbox/lean-sandbox-8c68afbdc09213173aef1be195da7a9a86060a97/src/100_theorems/t024.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3873103200759501}}
{"text": "import Hw9\nimport Sets.Basic\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\nopen Set \n\nvariable (\u03b1 \u03b2 : Type)\nvariable (X Y Z : Set \u03b1)\nvariable (W : Set \u03b2) \n\ntheorem desiredType1 : \u2205 \u2208 \ud835\udcab  X := sorry \n\ntheorem desiredType2 (U : \u03b2 \u2192 Set \u03b1) : \u2200 b, U b \u2286 BigUnion U := sorry \n\ntheorem desiredType3 (h : X \u2286 Y) : (X \u00d7\u02e2 W) \u2286 (Y \u00d7\u02e2 W) := sorry\n\ntheorem desiredType4 (h : Y \u2229 Z = \u2205) : Y\u1d9c \u222a Z\u1d9c = Univ := sorry \n\ntheorem desiredType5 : (X \\ Y) \u222a (Y \\ X) = (X \u222a Y) \\ (X \u2229 Y) := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef n : String := \"1\"\n\ndef problem : String := \"problem\"++n\n\ndef desired : String := \"desiredType\"++n\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const desired []) (Expr.const problem [])\n#eval collectAxiomsOf problem\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework9", "sha": "c26e748a8f91c4f459d6f568a6819b53cd6088a1", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework9/homework9-c26e748a8f91c4f459d6f568a6819b53cd6088a1/_Tests/Problem1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3873103135795625}}
{"text": "import morphisms.surjective\n\n/-!\n# Radicial morphisms\n\nA morphism of schemes `f : X \u27f6 Y` is radicial if the underlying map is injective,\nand it induces radicial (purely inseparable) extensions on residue fields.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverse u\n\nopen_locale algebraic_geometry\n\nnamespace algebraic_geometry\n\nvariables {X Y Z : Scheme.{u}} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\n/-- \nA field extension is radicial if it is an epi in the category of fields.\n\nTODO: Replace this with purely inseparable extensions once we have those.\n-/\ndef _root_.ring_hom.is_radicial {K L : Type u} [field K] [field L] (f : K \u2192+* L) : Prop :=\n\u2200 (L' : Type u) [field L'],\n  by exactI function.injective (flip ring_hom.comp f : (L \u2192+* L') \u2192 (K \u2192+* L'))\n\n/--\nA morphism of schemes `f : X \u27f6 Y` is radicial if the underlying map is injective,\nand it induces radicial (purely inseparable) extensions on residue fields.\n\nWe show that this is equivalent to universally injective and equivalent to the diagonal map being\nsurjective below.\n-/\n@[mk_iff]\nclass radicial (f : X \u27f6 Y) : Prop :=\n(base_injective [] : function.injective f.1.base)\n(residue_radicial [] : \u2200 x, ring_hom.is_radicial (f.map_residue_field x))\n\nsection tfae\n\nlemma radicial.epi_of_field [radicial f] {K : Type*} [field K] : \n  function.injective (\u03bb g : Scheme.Spec.obj (op $ CommRing.of K) \u27f6 X, g \u226b f) :=\nbegin\n  intros g\u2081 g\u2082 e,\n  dsimp only at e,\n  apply (Spec_to_equiv_of_field K _).injective,\n  refine Spec_to_equiv_of_field_eq_iff.mpr \u27e8_, _\u27e9,\n  { apply radicial.base_injective f,\n    simp only [Spec_to_equiv_of_field_apply_fst, \u2190 Scheme.comp_val_base_apply, e] },\n  { simp only [Spec_to_equiv_of_field_apply_snd],\n    apply radicial.residue_radicial f,\n    convert_to f.map_residue_field _ \u226b _ = f.map_residue_field _ \u226b _, \n    rw [\u2190 cancel_epi (Y.to_residue_field _)], swap, { apply_instance },\n    simp only [Top.presheaf.stalk_congr_hom, stalk_closed_point_to,\n      Scheme.to_residue_field_of_eq_assoc, Scheme.to_residue_field_map_residue_field_assoc,\n      Scheme.to_desc_residue_field,\n      \u2190 PresheafedSpace.stalk_map.stalk_specializes_stalk_map_assoc],\n    slice_lhs 1 2 { rw [\u2190 PresheafedSpace.stalk_map.comp] },\n    erw PresheafedSpace.stalk_map.congr_hom' _ _\n      (show g\u2081.1 \u226b f.1 = g\u2082.1 \u226b f.1, by injection e),\n    slice_rhs 2 3 { erw \u2190 PresheafedSpace.stalk_map.comp },\n    simp_rw category.assoc, refl }\nend\n\nlemma _root_.field.exists_common_extension {K L M : Type u} [field K] [field L] [field M]\n  (f : K \u2192+* L) (g : K \u2192+* M) :\n  \u2203 (N : Type u) [field N], by exactI \u2203 (h : L \u2192+* N) (k : M \u2192+* N), h.comp f = k.comp g :=\nbegin\n  letI := f.to_algebra,\n  letI := g.to_algebra,\n  obtain \u27e8m, hm\u27e9 := ideal.exists_maximal (tensor_product K L M),\n  rw ideal.quotient.maximal_ideal_iff_is_field_quotient at hm,\n  refine \u27e8tensor_product K L M \u29f8 m, hm.to_field,\n    m^.quotient.mk^.comp algebra.tensor_product.include_left.to_ring_hom,\n    m^.quotient.mk^.comp algebra.tensor_product.include_right.to_ring_hom, _\u27e9,\n  rw [ring_hom.comp_assoc, ring_hom.comp_assoc],\n  congr' 1,\n  exact algebra.tensor_product.include_right.comp_algebra_map.symm,\nend\n\nlemma universally_injective_of_epi_of_field\n  (H : \u2200 {K : Type u} [field K],\n    by exactI function.injective (\u03bb (g : Scheme.Spec.obj (op (CommRing.of K)) \u27f6 X), g \u226b f)) :\n  morphism_property.universally (\u03bb X Y (f : Scheme.hom X Y), function.injective f.1.base) f :=\nbegin\n  intros X' Y' i\u2081 i\u2082 f' h x\u2081 x\u2082 e,\n  obtain \u27e8K, hK, g\u2081, g\u2082, e'\u27e9 := field.exists_common_extension \n    (Y'.residue_field_of_eq e \u226b f'.map_residue_field x\u2081 : _) (f'.map_residue_field x\u2082), \n  resetI,\n  change X'.residue_field x\u2081 \u27f6 CommRing.of K at g\u2081,\n  change X'.residue_field x\u2082 \u27f6 CommRing.of K at g\u2082,\n  replace e' : (Y'.residue_field_of_eq e \u226b f'.map_residue_field x\u2081) \u226b g\u2081 =\n    f'.map_residue_field x\u2082 \u226b g\u2082 := by convert e',\n  suffices : (Spec_to_equiv_of_field K X').symm \u27e8x\u2081, g\u2081\u27e9 =\n    (Spec_to_equiv_of_field K X').symm \u27e8x\u2082, g\u2082\u27e9,\n  { injection (Spec_to_equiv_of_field K X').symm.injective this },\n  have : (Spec_to_equiv_of_field K X').symm \u27e8x\u2081, g\u2081\u27e9 \u226b f' =\n    (Spec_to_equiv_of_field K X').symm \u27e8x\u2082, g\u2082\u27e9 \u226b f',\n  { dsimp only [Spec_to_equiv_of_field_symm_apply, is_pullback.cone_fst],\n    simp_rw [category.assoc, \u2190 Scheme.hom.map_residue_field_from_Spec_residue_field f',\n      \u2190 functor.map_comp_assoc, \u2190 op_comp, \u2190 e', op_comp, functor.map_comp_assoc,\n      Scheme.residue_field_of_eq_from_Spec] },\n  apply pullback_cone.is_limit.hom_ext h.is_limit,\n  { exact this },\n  { apply H, simp only [category.assoc, is_pullback.cone_snd, \u2190 h.w, reassoc_of this], } \nend\n\nlemma _root_.category_theory.is_iso_of_comp_mono {C} [category C] {X Y Z : C} \n  (f : X \u27f6 Y) (g : Y \u27f6 Z) [is_iso (f \u226b g)] [mono g] : is_iso f :=\nbegin\n  haveI : is_split_epi g := \u27e8\u27e8\u27e8inv (f \u226b g) \u226b f, by simp\u27e9\u27e9\u27e9,\n  haveI : is_iso g := is_iso_of_mono_of_is_split_epi g,\n  rw (show f = (f \u226b g) \u226b inv g, by simp),\n  apply_instance\nend\n\nlemma Spec_to_eq_of_injective {K : Type*} [field K] {X : Scheme}\n  {f\u2081 f\u2082 : Scheme.Spec.obj (op $ CommRing.of K) \u27f6 X}\n  {g} (hf\u2081 : f\u2081 \u226b g = \ud835\udfd9 _) (hf\u2082 : f\u2082 \u226b g = \ud835\udfd9 _) (hg : function.injective g.1.base) : f\u2081 = f\u2082 :=\nbegin\n  haveI : subsingleton X.carrier := @@function.injective.subsingleton _ hg\n    (show subsingleton (prime_spectrum K), by apply_instance),\n  apply (Spec_to_equiv_of_field _ _).injective,\n  refine Spec_to_equiv_of_field_eq_iff.mpr \u27e8subsingleton.elim _ _, _\u27e9,\n  simp only [Spec_to_equiv_of_field_apply_snd],\n  haveI : is_iso (g.map_residue_field (f\u2081.1.base (local_ring.closed_point K))),\n  { refine category_theory.is_iso_of_comp_mono _ _,\n    swap, { exact f\u2081.map_residue_field (local_ring.closed_point K) },\n    { rw [\u2190 Scheme.hom.map_residue_field_comp],\n      let f := f\u2081 \u226b g, change is_iso (f.map_residue_field _), rw (show f = \ud835\udfd9 _, from hf\u2081),\n      apply_instance },\n    { rw concrete_category.mono_iff_injective_of_preserves_pullback,\n      exact ring_hom.injective _ } },\n  rw \u2190 cancel_epi (g.map_residue_field (f\u2081.1.base (local_ring.closed_point K))),\n  rw \u2190 cancel_epi (Scheme.to_residue_field _ _), swap, { apply_instance },\n  simp only [Top.presheaf.stalk_congr_hom, stalk_closed_point_to, \n    Scheme.to_residue_field_of_eq_assoc, Scheme.to_residue_field_map_residue_field_assoc,\n    Scheme.to_desc_residue_field, \u2190 PresheafedSpace.stalk_map.stalk_specializes_stalk_map_assoc],\n  slice_lhs 1 2 { erw [\u2190 PresheafedSpace.stalk_map.comp] },\n  slice_rhs 2 3 { erw [\u2190 PresheafedSpace.stalk_map.comp] },\n  erw PresheafedSpace.stalk_map.congr_hom' _ _\n    (show f\u2081.1 \u226b g.1 = f\u2082.1 \u226b g.1, by injection hf\u2081.trans hf\u2082.symm),\n  simp only [category.assoc],\n  refl,\nend\n\ndef morphism_property.injective : morphism_property Scheme :=\n\u03bb X Y f, function.injective f.1.base \n\nlemma radicial_tfae :\n  tfae [radicial f,\n    \u2200 {K : Type*} [field K], by exactI function.injective\n      (\u03bb g : Scheme.Spec.obj (op $ CommRing.of K) \u27f6 X, g \u226b f),\n    morphism_property.universally morphism_property.injective f,\n    surjective (pullback.diagonal f)] :=\nbegin\n  tfae_have : 1 \u2192 2,\n  { introsI _ K _, exact radicial.epi_of_field f },\n  tfae_have : 2 \u2192 3,\n  { exact universally_injective_of_epi_of_field f },\n  tfae_have : 3 \u2192 2,\n  { introsI H K hK g\u2081 g\u2082 e,\n    let f' := g\u2081 \u226b f,\n    rw [\u2190 pullback.lift_snd (\ud835\udfd9 _) g\u2081 (category.id_comp f'),\n      \u2190 pullback.lift_snd (\ud835\udfd9 _) g\u2082 ((category.id_comp f').trans e)],\n    congr' 1,\n    apply Spec_to_eq_of_injective (pullback.lift_fst _ _ _) (pullback.lift_fst _ _ _)\n      (H _ _ _ (is_pullback.of_has_pullback f' f)) },\n  tfae_have : 3 \u2192 1,\n  { intro H,\n    refine \u27e8(morphism_property.universally_le _ _ _ f H : _), _\u27e9,\n    introsI x K _ f\u2081 f\u2082 e,\n    change X.residue_field x \u27f6 CommRing.of K at f\u2081,\n    change X.residue_field x \u27f6 CommRing.of K at f\u2082,\n    replace e : f.map_residue_field x \u226b f\u2081 = f.map_residue_field x \u226b f\u2082 := by convert e,\n    suffices : (Spec_to_equiv_of_field K X).symm \u27e8x, f\u2081\u27e9 =\n      (Spec_to_equiv_of_field K X).symm \u27e8x, f\u2082\u27e9,\n    { injection (Spec_to_equiv_of_field K X).symm.injective this, exact eq_of_heq \u2039_\u203a },\n    apply tfae_3_to_2 H,\n    dsimp only [Spec_to_equiv_of_field_symm_apply],\n    simp only [category.assoc, \u2190 Scheme.hom.map_residue_field_from_Spec_residue_field,\n      \u2190 functor.map_comp_assoc, \u2190 op_comp, e] },\n  tfae_have : 3 \u2192 4,\n  { rw surjective_iff,\n    intros H x,\n    refine \u27e8(pullback.fst : pullback f f \u27f6 _).1.base x,\n      H _ _ _ (is_pullback.of_has_pullback f f) _\u27e9,\n    simp only [\u2190 Scheme.comp_val_base_apply, category.assoc,\n      pullback.diagonal_fst, category.comp_id] },\n  tfae_have : 4 \u2192 3,\n  { suffices : morphism_property.diagonal @surjective \u2264 morphism_property.injective.universally,\n    { apply this },\n    rw \u2190 (surjective_stable_under_base_change.diagonal surjective_respects_iso).universally_eq,\n    refine morphism_property.universally_mono _,\n    intros X Y f h x y e,\n    let T : pullback.triplet f f := \u27e8x, y, _, e, rfl\u27e9,\n    obtain \u27e8z, hz, hz'\u27e9 := T.exists_preimage,\n    obtain \u27e8z', rfl\u27e9 := h.1 z,\n    simp only [\u2190 Scheme.comp_val_base_apply,\n      pullback.diagonal_fst, pullback.diagonal_snd] at hz hz',\n    exact hz.symm.trans hz' },\n  tfae_finish\nend\n\nlemma radicial_eq_univerally_injective :\n  @radicial = morphism_property.universally morphism_property.injective :=\nby { ext X Y f, exact (radicial_tfae f).out 0 2 }\n\nlemma radicial_eq_diagonal_surjective :\n  @radicial = morphism_property.diagonal @surjective :=\nby { ext X Y f, exact (radicial_tfae f).out 0 3 }\n\nend tfae\n\nlemma radicial_respects_iso : \n  morphism_property.respects_iso @radicial :=\nradicial_eq_univerally_injective.symm \u25b8 \n  morphism_property.injective.universally_respects_iso\n\nlemma radicial_stable_under_composition : \n  morphism_property.stable_under_composition @radicial :=\nradicial_eq_diagonal_surjective.symm \u25b8 \n  surjective_stable_under_composition.diagonal \n    surjective_respects_iso\n    surjective_stable_under_base_change\n\nlemma radicial_stable_under_base_change :\n  morphism_property.stable_under_base_change @radicial :=\nradicial_eq_univerally_injective.symm \u25b8 \n  morphism_property.universally_stable_under_base_change _\n\nlemma radicial_is_local_at_target :\n  property_is_local_at_target @radicial :=\nradicial_eq_diagonal_surjective.symm \u25b8 \n  surjective_is_local_at_target.diagonal\n\ninstance radicial_of_mono [mono f] : radicial f :=\nby { rw radicial_eq_diagonal_surjective, show surjective _, apply_instance }\n\nend algebraic_geometry", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/morphisms/radicial.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38729184645765863}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n\nimport category_theory.category\nimport category_theory.abelian.basic\nimport abelian_SEMF\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory.abelian\n\nsection\nvariables {C : Type u} [\ud835\udc9e : category.{v} C] [abelian.{v} C]\ninclude \ud835\udc9e\n\ndef exact {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) : Prop :=\nf \u226b g = 0 \u2227 kernel.\u03b9 g \u226b cokernel.\u03c0 f = 0\n\ndef exact_fork {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (e : exact f g) : kernel_fork g :=\nkernel_fork.of_\u03b9 (kernel.\u03b9 (cokernel.\u03c0 f)) $\n  zero_of_epi_comp (abelian.factor_thru_image f) $ by rw [\u2190category.assoc, image.fac f, e.1]\n\ndef exact_ker {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (e : exact f g) : is_limit $ exact_fork f g e :=\nfork.is_limit.mk _\n  (\u03bb s, kernel.lift _ (fork.\u03b9 s)\n  begin\n    let t : s.X \u27f6 kernel g := kernel.lift g (fork.\u03b9 s) (kernel_fork.condition _),\n    have : t \u226b kernel.\u03b9 g = fork.\u03b9 s := by erw limit.lift_\u03c0; refl,\n    rw [\u2190this, category.assoc, e.2, has_zero_morphisms.comp_zero]\n  end)\n  (\u03bb s, by erw limit.lift_\u03c0; refl)\n  (\u03bb s m h, by ext; erw [h walking_parallel_pair.zero, limit.lift_\u03c0]; refl)\n\ndef exact_cofork {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (e : exact f g) : cokernel_cofork f :=\ncokernel_cofork.of_\u03c0 (cokernel.\u03c0 (kernel.\u03b9 g)) $\n  zero_of_comp_mono (abelian.factor_thru_coimage g) $ by rw [category.assoc, coimage.fac g, e.1]\n\ndef exact_coker {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (e : exact f g) :\n  is_colimit $ exact_cofork f g e :=\ncofork.is_colimit.mk _\n  (\u03bb s, cokernel.desc _ (cofork.\u03c0 s)\n  begin\n    let t : cokernel f \u27f6 s.X := cokernel.desc f (cofork.\u03c0 s) (cokernel_cofork.condition _),\n    have : cokernel.\u03c0 f \u226b t = cofork.\u03c0 s := by erw colimit.\u03b9_desc; refl,\n    rw [\u2190this, \u2190category.assoc, e.2, has_zero_morphisms.zero_comp]\n  end)\n  (\u03bb s, by erw colimit.\u03b9_desc; refl)\n  (\u03bb s m h, by ext; erw [h walking_parallel_pair.one, colimit.\u03b9_desc]; refl)\n\nlemma exact_zero_of_mono (P : C) {Q R : C} (f : Q \u27f6 R) [mono f] : exact (0 : P \u27f6 Q) f :=\n\u27e8has_zero_morphisms.zero_comp _ _,\nbegin\n  rw (preadditive.cancel_zero_iff_mono f).1 (by apply_instance) _ (kernel.\u03b9 f) (kernel.condition _),\n  exact has_zero_morphisms.zero_comp _ _\nend\u27e9\n\nlemma mono_of_exact_zero (P : C) {Q R : C} (f : Q \u27f6 R) (h : exact (0 : P \u27f6 Q) f) : mono f :=\n(preadditive.cancel_zero_iff_mono f).2 $ \u03bb Z g h\u2080,\nbegin\n  obtain \u27e8k, hk\u27e9 := kernel_fork.is_limit.lift' (exact_ker _ _ h) g h\u2080,\n  have := (preadditive.cancel_zero_iff_epi (factor_thru_image (0 : P \u27f6 Q))).1\n    (by apply_instance) _ _ (image.fac (0 : P \u27f6 Q)),\n  change k \u226b kernel.\u03b9 (cokernel.\u03c0 (0 : P \u27f6 Q)) = g at hk,\n  rw \u2190hk,\n  conv_lhs { congr, skip, rw this },\n  exact has_zero_morphisms.comp_zero _ _\nend\n\nlemma exact_zero_of_epi {P Q : C} (f : P \u27f6 Q) (R : C) [epi f] : exact f (0 : Q \u27f6 R) :=\n\u27e8has_zero_morphisms.comp_zero _ _,\nbegin\n  rw (preadditive.cancel_zero_iff_epi f).1 (by apply_instance) _ (cokernel.\u03c0 f) (cokernel.condition _),\n  exact has_zero_morphisms.comp_zero _ _\nend\u27e9\n\nlemma exact_zero_of_epi' {P Q : C} (f : P \u27f6 Q) [epi f] : exact f (0 : Q \u27f6 Q) :=\nexact_zero_of_epi _ _\n\nlemma epi_of_exact_zero {P Q : C} (f : P \u27f6 Q) (R : C) (h : exact f (0 : Q \u27f6 R)) : epi f :=\n(preadditive.cancel_zero_iff_epi f).2 $ \u03bb Z g h\u2080,\nbegin\n  obtain \u27e8k, hk\u27e9 := cokernel.desc' f g h\u2080,\n  haveI : is_iso (kernel.\u03b9 (0 : Q \u27f6 R)) := kernel.\u03b9_of_zero _ _,\n  apply (preadditive.cancel_zero_iff_epi (kernel.\u03b9 (0 : Q \u27f6 R))).1 (by apply_instance) _ _,\n  rw [\u2190hk, \u2190category.assoc, h.2],\n  exact has_zero_morphisms.zero_comp _ _\nend\n\nlemma kernel_exact {P Q : C} (f : P \u27f6 Q) : exact (kernel.\u03b9 f) f :=\n\u27e8kernel.condition _, cokernel.condition _\u27e9\n\nlemma cokernel_exact {P Q : C} (f : P \u27f6 Q) : exact f (cokernel.\u03c0 f) :=\n\u27e8cokernel.condition _, kernel.condition _\u27e9\n\ndef kernel_of_mono_exact {P Q R : C} (f : P \u27f6 Q) [mono f] (g : Q \u27f6 R) (h : exact f g) :\n  is_limit $ kernel_fork.of_\u03b9 f h.1 :=\nbegin\n  let I := kernel (cokernel.\u03c0 f),\n  let i : kernel g \u2245 I := functor.map_iso (cones.forget _)\n    (is_limit.unique_up_to_iso (limit.is_limit _) (exact_ker f g h)),\n  have h\u2080 : i.hom \u226b kernel.\u03b9 (cokernel.\u03c0 f) = kernel.\u03b9 g :=\n    cone_morphism.w (is_limit.unique_up_to_iso (limit.is_limit _) (exact_ker f g h)).hom\n      walking_parallel_pair.zero,\n  haveI : mono (factor_thru_image f) := mono_of_mono_fac (image.fac f),\n  haveI : is_iso (factor_thru_image f) := mono_epi_iso _,\n  let j : P \u2245 I := as_iso (factor_thru_image f),\n  apply kernel.transport _ _ (j \u226a\u226b i.symm),\n  erw [category.assoc, \u2190(iso.eq_inv_comp i).2 h\u2080, image.fac],\nend\n\ndef cokernel_of_epi_exact {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) [epi g] (h : exact f g) :\n  is_colimit $ cokernel_cofork.of_\u03c0 g h.1 :=\nbegin\n  let I := cokernel (kernel.\u03b9 g),\n  let i : cokernel f \u2245 I := functor.map_iso (cocones.forget _)\n    (is_colimit.unique_up_to_iso (colimit.is_colimit _) (exact_coker f g h)),\n  have h\u2080 : cokernel.\u03c0 f \u226b i.hom = cokernel.\u03c0 (kernel.\u03b9 g) :=\n    cocone_morphism.w (is_colimit.unique_up_to_iso (colimit.is_colimit _) (exact_coker f g h)).hom\n      walking_parallel_pair.one,\n  haveI : epi (factor_thru_coimage g) := epi_of_epi_fac (coimage.fac g),\n  haveI : is_iso (factor_thru_coimage g) := mono_epi_iso _,\n  let j : I \u2245 R := as_iso (factor_thru_coimage g),\n  apply cokernel.transport _ _ (i \u226a\u226b j),\n  erw [\u2190category.assoc, h\u2080, coimage.fac],\nend\n\nlemma image_exact {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (h : exact f g) :\n  exact (kernel.\u03b9 (cokernel.\u03c0 f)) g :=\n\u27e8begin\n  apply (preadditive.cancel_zero_iff_epi (factor_thru_image f)).1 (by apply_instance),\n  rw \u2190category.assoc,\n  rw image.fac f,\n  exact h.1,\nend,\nbegin\n  obtain \u27e8l, hl\u27e9 := cokernel.desc' f (cokernel.\u03c0 (kernel.\u03b9 (cokernel.\u03c0 f)))\n    begin conv_lhs { congr, rw \u2190image.fac f, }, rw category.assoc, rw cokernel.condition,\n      rw has_zero_morphisms.comp_zero, end,\n  rw \u2190hl,\n  rw \u2190category.assoc,\n  rw h.2,\n  rw has_zero_morphisms.zero_comp,\nend\u27e9\n\nlemma exact_image {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (h : exact f g) :\n  exact f (factor_thru_image g) :=\n\u27e8begin\n  apply (preadditive.cancel_zero_iff_mono (kernel.\u03b9 (cokernel.\u03c0 g))).1 (by apply_instance),\n  rw category.assoc,\n  rw image.fac g,\n  exact h.1,\nend,\nbegin\n  obtain \u27e8l, hl\u27e9 := kernel.lift' g (kernel.\u03b9 (factor_thru_image g))\n    begin conv_lhs { congr, skip, rw \u2190image.fac g, }, rw \u2190category.assoc, rw kernel.condition,\n      rw has_zero_morphisms.zero_comp, end,\n  rw \u2190hl,\n  rw category.assoc,\n  rw h.2,\n  rw has_zero_morphisms.comp_zero,\nend\u27e9\n\nlemma exact_iso {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) {Q' : C} (i : Q \u2245 Q') (h : exact f g) :\n  exact (f \u226b i.hom) (i.inv \u226b g) :=\n\u27e8by simpa using h.1,\nbegin\n  obtain \u27e8l, hl\u27e9 := kernel.lift' g (kernel.\u03b9 (i.inv \u226b g) \u226b i.inv)\n    (by rw [category.assoc, kernel.condition]),\n  obtain \u27e8m, hm\u27e9 := cokernel.desc' f (i.hom \u226b cokernel.\u03c0 (f \u226b i.hom))\n    (by rw [\u2190category.assoc, cokernel.condition]),\n  calc kernel.\u03b9 (i.inv \u226b g) \u226b cokernel.\u03c0 (f \u226b i.hom)\n        = kernel.\u03b9 (i.inv \u226b g) \u226b (i.inv \u226b i.hom) \u226b cokernel.\u03c0 (f \u226b i.hom) : by rw [iso.inv_hom_id, category.id_comp]\n    ... = (kernel.\u03b9 (i.inv \u226b g) \u226b i.inv) \u226b i.hom \u226b cokernel.\u03c0 (f \u226b i.hom) : by simp only [category.assoc]\n    ... = (l \u226b kernel.\u03b9 g) \u226b cokernel.\u03c0 f \u226b m : by rw [\u2190hl, \u2190hm]\n    ... = l \u226b (kernel.\u03b9 g \u226b cokernel.\u03c0 f) \u226b m : by simp only [category.assoc]\n    ... = l \u226b 0 \u226b m : by rw h.2\n    ... = 0 : by rw [has_zero_morphisms.zero_comp, has_zero_morphisms.comp_zero]\nend\u27e9\n\nlemma exact_iso_right {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) {R' : C} (i : R \u2245 R') (h : exact f g) :\n  exact f (g \u226b i.hom) :=\n\u27e8by rw [\u2190category.assoc, h.1, has_zero_morphisms.zero_comp],\nbegin\n  obtain \u27e8l, hl\u27e9 := kernel.lift' g (kernel.\u03b9 (g \u226b i.hom))\n    (calc kernel.\u03b9 (g \u226b i.hom) \u226b g\n          = kernel.\u03b9 (g \u226b i.hom) \u226b g \u226b i.hom \u226b i.inv : by simp\n      ... = (kernel.\u03b9 (g \u226b i.hom) \u226b g \u226b i.hom) \u226b i.inv : by simp only [category.assoc]\n      ... = 0 \u226b i.inv : by rw kernel.condition\n      ... = 0 : by rw has_zero_morphisms.zero_comp),\n\n  rw [\u2190hl, category.assoc, h.2, has_zero_morphisms.comp_zero],\nend\u27e9\n\nlemma exact_iso_left {P Q R : C} (f : P \u27f6 Q) (g : Q \u27f6 R) {P' : C} (i : P' \u2245 P) (h : exact f g) :\n  exact (i.hom \u226b f) g :=\n\u27e8by rw [category.assoc, h.1, has_zero_morphisms.comp_zero],\nbegin\n  obtain \u27e8l, hl\u27e9 := cokernel.desc' f (cokernel.\u03c0 (i.hom \u226b f))\n    (calc f \u226b cokernel.\u03c0 (i.hom \u226b f)\n          = (i.inv \u226b i.hom) \u226b f \u226b cokernel.\u03c0 (i.hom \u226b f) : by simp\n      ... = i.inv \u226b (i.hom \u226b f) \u226b cokernel.\u03c0 (i.hom \u226b f) : by simp only [category.assoc]\n      ... = i.inv \u226b 0 : by rw cokernel.condition\n      ... = 0 : by rw has_zero_morphisms.comp_zero),\n\n  rw [\u2190hl, \u2190category.assoc, h.2, has_zero_morphisms.zero_comp],\nend\u27e9\n\nlemma epi_mono_exact_left {P Q R S : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (h : R \u27f6 S)\n  (e : exact (f \u226b g) h) [epi f] [mono g] : exact g h :=\nbegin\n  let upper : strong_epi_mono_factorisation (f \u226b g) :=\n  { I := _, e := f, m := g, fac' := rfl,\n    m_mono := by apply_instance, e_strong_epi := strong_epi_of_epi _ },\n  let lower := image_SEMF (f \u226b g),\n  let s : Q \u2245 kernel (cokernel.\u03c0 (f \u226b g)) := is_image.iso_ext upper.to_mono_is_image\n    lower.to_mono_is_image,\n  have : s.hom \u226b kernel.\u03b9 (cokernel.\u03c0 (f \u226b g)) = g,\n  { erw is_image.lift_fac },\n  rw \u2190this,\n  apply exact_iso_left _ _ s,\n  exact image_exact _ _ e,\nend\n\nlemma epi_mono_exact_right {P Q R S : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (h : R \u27f6 S)\n  (e : exact f (g \u226b h)) [epi g] [mono h] : exact f g :=\nbegin\n  let upper : strong_epi_mono_factorisation (g \u226b h) :=\n  { I := _, e := g, m := h, fac' := rfl,\n  m_mono := by apply_instance, e_strong_epi := strong_epi_of_epi _ },\n  let lower := image_SEMF (g \u226b h),\n  let s : kernel (cokernel.\u03c0 (g \u226b h)) \u2245 R := is_image.iso_ext lower.to_mono_is_image\n    upper.to_mono_is_image,\n  have : factor_thru_image (g \u226b h) \u226b s.hom = g,\n  { erw is_image.fac_lift lower.to_mono_is_image upper.to_mono_factorisation },\n  rw \u2190this,\n  apply exact_iso_right _ _ s,\n  exact exact_image _ _ e,\nend\n\nlemma exact_left_epi {P Q R S : C} (f : P \u27f6 Q) (g : Q \u27f6 R) (h : R \u27f6 S) (e : exact g h) [epi f] :\n  exact (f \u226b g) h :=\n\u27e8by rw [category.assoc, e.1, has_zero_morphisms.comp_zero],\nbegin\n  obtain \u27e8l, hl\u27e9 := cokernel.desc' g (cokernel.\u03c0 (f \u226b g))\n    begin\n      apply (preadditive.cancel_zero_iff_epi f).1 (by apply_instance),\n      rw [\u2190category.assoc, cokernel.condition],\n    end,\n  rw [\u2190hl, \u2190category.assoc, e.2, has_zero_morphisms.zero_comp],\nend\u27e9\n\n\nend\n\nend category_theory.abelian\n", "meta": {"author": "TwoFX", "repo": "lean-homological-algebra", "sha": "e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8", "save_path": "github-repos/lean/TwoFX-lean-homological-algebra", "path": "github-repos/lean/TwoFX-lean-homological-algebra/lean-homological-algebra-e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8/src/exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3872918386533537}}
{"text": "import QL.FOL.Tait.calculus logic\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol aclogic\nvariables {L : language.{u}} {\u03bc : Type v} {\u03bc\u2081 : Type*} {\u03bc\u2082 : Type*} {m n : \u2115} {S : Structure L}\n\nnamespace Tait\nnamespace subformula\n\nvariables (S) {n} {\u03a6 : \u03bc \u2192 S} {e : fin n \u2192 S}\n\n@[simp] def subval' (\u03a6 : \u03bc \u2192 S) : \u2200 {n} (e : fin n \u2192 S), subformula L \u03bc n \u2192 Prop\n| n _ verum              := true\n| n _ falsum             := false\n| n e (relation p v)     := S.pr p (\u03bb i, subterm.val S \u03a6 e (v i))\n| n e (neg_relation p v) := \u00acS.pr p (\u03bb i, subterm.val S \u03a6 e (v i))\n| n e (and p q)          := p.subval' e \u2227 q.subval' e\n| n e (or p q)           := p.subval' e \u2228 q.subval' e\n| n e (fal p)            := \u2200 x : S, (p.subval' (x *> e))\n| n e (ex p)             := \u2203 x : S, (p.subval' (x *> e))\n\n@[simp] lemma subval'_neg (p : subformula L \u03bc n) : subval' S \u03a6 e (\u223cp) = \u00acsubval' S \u03a6 e p :=\nby induction p generalizing \u03a6 e; simp[mlift, \u2190verum_eq, \u2190falsum_eq, \u2190and_eq, \u2190or_eq, \u2190not_eq, \u2190fal_eq, \u2190ex_eq, or_iff_not_imp_left, *] at*\n\n@[irreducible] def subval (\u03a6 : \u03bc \u2192 S) (e : fin n \u2192 S) : subformula L \u03bc n \u2192\u2097 Prop :=\n{ to_fun := subval' S \u03a6 e,\n  map_neg' := \u03bb _, by simp,\n  map_imply' := \u03bb _ _, by simp[has_arrow.arrow, imply, or_iff_not_imp_left, not_eq],\n  map_and' := \u03bb p q, by unfold has_inf.inf; simp; refl,\n  map_or' := \u03bb p q, by unfold has_sup.sup; simp; refl,\n  map_top' := by refl,\n  map_bot' := by refl }\n\n@[reducible] def val (\u03a6 : \u03bc \u2192 S) : formula L \u03bc \u2192\u2097 Prop := subformula.subval S \u03a6 fin.nil\n\n@[simp] lemma subval_relation {p} {r : L.pr p} {v} :\n  subval S \u03a6 e (relation r v) \u2194 S.pr r (subterm.val S \u03a6 e \u2218 v) :=  by simp[subval]; refl\n\n@[simp] lemma subval_neg_relation {p} {r : L.pr p} {v} :\n  subval S \u03a6 e (neg_relation r v) \u2194 \u00acS.pr r (subterm.val S \u03a6 e \u2218 v) := by simp[subval]; refl\n\n@[simp] lemma subval_fal {p : subformula L \u03bc (n + 1)} :\n  subval S \u03a6 e (\u2200'p) \u2194 \u2200 x : S, subval S \u03a6 (x *> e) p := by simp[subval]; refl\n\n@[simp] lemma subval_ex {p : subformula L \u03bc (n + 1)} :\n  subval S \u03a6 e (\u2203'p) \u2194 \u2203 x : S, subval S \u03a6 (x *> e) p := by simp[subval]; refl\n\nvariables {\u03bc\u2081 \u03bc\u2082}\n\nlemma subval_map {f : \u03bc\u2081 \u2192 \u03bc\u2082} {\u03a6 : \u03bc\u2082 \u2192 S} {p : subformula L \u03bc\u2081 n} :\n  subval S \u03a6 e (map f p) \u2194 subval S (\u03a6 \u2218 f) e p :=\nby induction p using fol.Tait.subformula.ind_on; simp[*, (\u2218), subterm.val_map]\n\nlemma subval_subst {p : subformula L \u03bc (n + 1)} : \u2200 {e} {u : subterm L \u03bc n},\n  subval S \u03a6 e (subst u p) \u2194 subval S \u03a6 (e <* subterm.val S \u03a6 e u) p :=\nby apply ind_succ_on p; intros; simp[*, (\u2218), subterm.val_subst, subterm.val_lift, fin.left_right_concat_assoc]\n\nend subformula\n\nnamespace subformula\nvariables (S) {\u03a6 : \u03bc \u2192 S}\n\nnotation S` \u22a7\u1d40[`:80 e`] `p :50 := val S e p\n\nvariables {S} {p q : formula L \u03bc}\n\n@[simp] lemma models_relation {k} {r : L.pr k} {v} :\n  S \u22a7\u1d40[\u03a6] relation r v \u2194 S.pr r (\u03bb i, subterm.val S \u03a6 fin.nil (v i)) := by simp[val]\n\nend subformula\n\ndef models (S : Structure L) (p : formula L \u03bc) : Prop := \u2200 e, S \u22a7\u1d40[e] p\n\ninstance : logic.semantics (formula L \u03bc) (Structure L) := \u27e8models\u27e9\n\nlemma models_def {S : Structure L} {p : formula L \u03bc} : S \u22a7 p \u2194 (\u2200 e, S \u22a7\u1d40[e] p) := by refl\n\nlemma sentence_models_def {S : Structure L} {\u03c3 : sentence L} : S \u22a7 \u03c3 \u2194 S \u22a7\u1d40[fin.nil] \u03c3 := by simp[models_def, fin.nil]\n\n--@[simp] lemma models_neg {\u03c3 : sentence L} :\n--  S \u22a7 \u223c\u03c3 \u2194 \u00acS \u22a7 \u03c3 := by simp[sentence_models_def]\n\n@[simp] lemma models_coe {S : Structure L} {\u03c3 : sentence L} {e : \u03bc \u2192 S} : S \u22a7\u1d40[e] \u2191\u03c3 \u2194 S \u22a7 \u03c3 :=\nby { rw [subformula.sentence_coe_def], \n     simp[sentence_models_def, -subformula.map_sentence_coe, subformula.val, subformula.subval_map,\n       show e \u2218 fin.nil = fin.nil, by ext x; exact fin.nil x] }\n\ninstance : has_double_turnstile (Tait.preTheory L \u03bc) (formula L \u03bc) := \u27e8logic.semantics.consequence (Structure L)\u27e9\n\nlemma consequence_def {T : preTheory L \u03bc} {p : formula L \u03bc} :\n  T \u22a7 p \u2194 (\u2200 S : Structure L, S \u22a7 T \u2192 S \u22a7 p) := by refl\n\nnamespace subformula\nvariables (S) {\u03a6 : \u03bc \u2192 S} {e : fin n \u2192 S}\n\n@[simp] lemma subval_to_tait {p : fol.subformula L \u03bc n} : subval S \u03a6 e p.to_tait \u2194 fol.subformula.subval S \u03a6 e p :=\nby induction p using fol.subformula.ind_on; simp*\n\n@[simp] lemma subval_of_tait {p : Tait.subformula L \u03bc n} : fol.subformula.subval S \u03a6 e p.of_tait \u2194 subval S \u03a6 e p :=\nby induction p using fol.Tait.subformula.ind_on; simp*\n\nend subformula\n\n@[simp] lemma models_to_tait {p : fol.formula L \u03bc} : S \u22a7 p.to_tait \u2194 S \u22a7 p :=\nby simp[models_def, fol.Structure.models_def]\n\n@[simp] lemma models_of_tait {p : Tait.formula L \u03bc} : S \u22a7 p.of_tait \u2194 S \u22a7 p :=\nby simp[models_def, fol.Structure.models_def]\n\nend Tait\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/QL/FOL/Tait/semantics.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3872918386533537}}
{"text": "import data.fintype data.equiv.basic group_theory.subgroup\n\nnamespace list\n\n@[simp] lemma length_attach {\u03b1} (L : list \u03b1) :\n  L.attach.length = L.length :=\nlength_pmap\n\n@[simp] lemma nth_le_attach {\u03b1} (L : list \u03b1) (i) (H : i < L.attach.length) :\n  (L.attach.nth_le i H).1 = L.nth_le i (length_attach L \u25b8 H) :=\ncalc  (L.attach.nth_le i H).1\n    = (L.attach.map subtype.val).nth_le i (by simpa using H) : by rw nth_le_map'\n... = L.nth_le i _ : by congr; apply attach_map_val\n\n@[simp] lemma nth_le_range {n} (i) (H : i < (range n).length) :\n  nth_le (range n) i H = i :=\noption.some.inj $ by rw [\u2190 nth_le_nth _, nth_range (by simpa using H)]\n\nattribute [simp] length_of_fn\nattribute [simp] nth_le_of_fn\n\n-- Congratulations, I proved that two things which have\n-- equally few lemmas are equal.\ntheorem of_fn_eq_pmap {\u03b1 n} {f : fin n \u2192 \u03b1} :\n  of_fn f = pmap (\u03bb i hi, f \u27e8i, hi\u27e9) (range n) (\u03bb _, mem_range.1) :=\nby rw [pmap_eq_map_attach]; from ext_le (by simp)\n  (\u03bb i hi1 hi2, by simp at hi1; simp [nth_le_of_fn f \u27e8i, hi1\u27e9])\n\ntheorem nodup_of_fn {\u03b1 n} {f : fin n \u2192 \u03b1} (hf : function.injective f) :\n  nodup (of_fn f) :=\nby rw of_fn_eq_pmap; from nodup_pmap\n  (\u03bb _ _ _ _ H, fin.veq_of_eq $ hf H) (nodup_range n)\n\nend list\n\n\n\nsection fin\n\nvariables {m n : \u2115}\n\ndef fin_zero_elim {C : Sort*} : fin 0 \u2192 C :=\n\u03bb x, false.elim $ nat.not_lt_zero x.1 x.2\n\ndef fin_sum : (fin m \u2295 fin n) \u2243 fin (m + n) :=\n{ to_fun := \u03bb x, sum.rec_on x\n    (\u03bb y, \u27e8y.1, nat.lt_of_lt_of_le y.2 $ nat.le_add_right m n\u27e9)\n    (\u03bb y, \u27e8m + y.1, nat.add_lt_add_left y.2 m\u27e9),\n  inv_fun := \u03bb x, if H : x.1 < m\n    then sum.inl \u27e8x.1, H\u27e9\n    else sum.inr \u27e8x.1 - m, nat.lt_of_add_lt_add_left $\n      show m + (x.1 - m) < m + n,\n      from (nat.add_sub_of_le $ le_of_not_gt H).symm \u25b8 x.2\u27e9,\n  left_inv := \u03bb x, sum.cases_on x\n    (\u03bb y, by simp [y.2]; from fin.eq_of_veq rfl)\n    (\u03bb y, have H : \u00acm + y.val < m, by simp [nat.zero_le],\n       by simp [H, nat.add_sub_cancel_left];\n       from fin.eq_of_veq rfl),\n  right_inv := \u03bb x, begin\n    by_cases H : x.1 < m,\n    { dsimp; rw [dif_pos H]; simp,\n      exact fin.eq_of_veq rfl },\n    { dsimp; rw [dif_neg H]; simp,\n      apply fin.eq_of_veq; simp,\n      rw [nat.add_sub_of_le (le_of_not_gt H)] }\n  end }\n\ndef fin_prod : (fin m \u00d7 fin n) \u2243 fin (m * n) :=\n{ to_fun := \u03bb x, \u27e8x.2.1 + n * x.1.1, calc\n          x.2.1 + n * x.1.1 + 1\n        = x.1.1 * n + x.2.1 + 1 : by ac_refl\n    ... \u2264 x.1.1 * n + n : nat.add_le_add_left x.2.2 _\n    ... = (x.1.1 + 1) * n : eq.symm $ nat.succ_mul _ _\n    ... \u2264 m * n : nat.mul_le_mul_right _ x.1.2\u27e9,\n  inv_fun := \u03bb x, have H : n > 0,\n      from nat.pos_of_ne_zero $ \u03bb H,\n        nat.not_lt_zero x.1 $ by subst H; from x.2,\n    (\u27e8x.1 / n, (nat.div_lt_iff_lt_mul _ _ H).2 x.2\u27e9,\n     \u27e8x.1 % n, nat.mod_lt _ H\u27e9),\n  left_inv := \u03bb \u27e8x, y\u27e9, have H : n > 0,\n      from nat.pos_of_ne_zero $ \u03bb H,\n        nat.not_lt_zero y.1 $ H \u25b8 y.2,\n    prod.ext\n    (fin.eq_of_veq $ calc\n            (y.1 + n * x.1) / n\n          = y.1 / n + x.1 : nat.add_mul_div_left _ _ H\n      ... = 0 + x.1 : by rw nat.div_eq_of_lt y.2\n      ... = x.1 : nat.zero_add x.1)\n    (fin.eq_of_veq $ calc\n            (y.1 + n * x.1) % n\n          = y.1 % n : nat.add_mul_mod_self_left _ _ _\n      ... = y.1 : nat.mod_eq_of_lt y.2),\n    right_inv := \u03bb x, fin.eq_of_veq $ nat.mod_add_div _ _ }\n\n@[simp] lemma fin.raise_val (k : fin n) :\n  k.raise.val = k.val :=\nrfl\n\ndef fin.fall : \u03a0 i : fin (n+1), i.1 < n \u2192 fin n :=\n\u03bb i h, \u27e8i.1, h\u27e9\n\n@[simp] lemma fin.fall_val (k : fin (n+1)) (H : k.1 < n) :\n  (k.fall H).val = k.val :=\nrfl\n\ndef fin.descend (pivot : fin (n+1)) : \u03a0 i : fin (n+1), i \u2260 pivot \u2192 fin n :=\n\u03bb i H, if h : i.1 < pivot.1\n  then i.fall (lt_of_lt_of_le h $ nat.le_of_lt_succ pivot.2)\n  else i.pred (\u03bb H1, H $ by subst H1;\n    replace h := nat.eq_zero_of_le_zero (le_of_not_gt h);\n    from fin.eq_of_veq h.symm)\n\ndef fin.ascend (pivot : fin (n+1)) : \u03a0 i : fin n, fin (n+1) :=\n\u03bb i, if i.1 < pivot.1 then i.raise else i.succ\n\ntheorem fin.ascend_ne (pivot : fin (n+1)) (i : fin n) :\n  pivot.ascend i \u2260 pivot :=\n\u03bb H, begin\n  unfold fin.ascend at H,\n  split_ifs at H;\n  rw \u2190 H at h;\n  simp [lt_irrefl, nat.lt_succ_self] at h;\n  cc\nend\n\n@[simp] lemma fin.ascend_descend (pivot i : fin (n+1))\n  (H : i \u2260 pivot) : pivot.ascend (pivot.descend i H) = i :=\nbegin\n  unfold fin.descend fin.ascend,\n  split_ifs with H1 H2 H3; apply fin.eq_of_veq; simp at *,\n  { cases pivot with p hp,\n    cases i with i hi,\n    cases i with i, { simp at * },\n    exfalso, apply H, apply fin.eq_of_veq,\n    apply le_antisymm, { apply nat.succ_le_of_lt H2 },\n    simpa using H1 },\n  { cases pivot with p hp,\n    cases i with i hi,\n    cases i with i,\n    { exfalso, apply H, apply fin.eq_of_veq, symmetry,\n      apply nat.eq_zero_of_le_zero H2 },\n    refl }\nend\n\n@[simp] lemma fin.descend_ascend (pivot : fin (n+1))\n  (i : fin n) (H : pivot.ascend i \u2260 pivot) :\n  pivot.descend (pivot.ascend i) H = i :=\nbegin\n  unfold fin.descend fin.ascend,\n  apply fin.eq_of_veq,\n  by_cases h : i.val < pivot.val,\n  { simp [h] },\n  { unfold ite dite,\n    cases nat.decidable_lt ((ite (i.val < pivot.val) (fin.raise i) (fin.succ i)).val) (pivot.val) with h1 h1,\n    { simp,\n      cases nat.decidable_lt (i.val) (pivot.val),\n      { simp },\n      { cc } },\n    { simp,\n      cases nat.decidable_lt (i.val) (pivot.val) with h2 h2,\n      { simp [h2] at h1,\n        simp at *,\n        exfalso, apply lt_asymm (nat.lt_succ_self i.1),\n        apply lt_of_lt_of_le h1 h },\n      { simp } } }\nend\n\n@[simp] lemma fin.succ_pred (i : fin (n+1)) (H : i \u2260 0) :\n  (i.pred H).succ = i :=\nbegin\n  apply fin.eq_of_veq,\n  cases i with i hi,\n  cases i,\n  { exfalso, apply H, apply fin.eq_of_veq, refl },\n  refl\nend\n\n@[simp] lemma fin.pred_succ (i : fin n) (H : i.succ \u2260 0) :\n  i.succ.pred H = i :=\nby cases i; refl\n\ninstance : decidable_linear_order (fin n) :=\n{ lt_iff_le_not_le := \u03bb i j, nat.lt_iff_le_not_le,\n  le_refl := \u03bb \u27e8i, hi\u27e9, nat.le_refl i,\n  le_trans := \u03bb \u27e8i, hi\u27e9 \u27e8j, hj\u27e9 \u27e8k, hk\u27e9 hij hjk, nat.le_trans hij hjk,\n  le_antisymm := \u03bb \u27e8i, hi\u27e9 \u27e8j, hj\u27e9 hij hji, fin.eq_of_veq $ nat.le_antisymm hij hji,\n  le_total := \u03bb \u27e8i, hi\u27e9 \u27e8j, hj\u27e9, or.cases_on (@nat.le_total i j) or.inl or.inr,\n  decidable_le := fin.decidable_le,\n  .. fin.has_le, .. fin.has_lt }\n\ninstance : preorder (fin n) :=\nby apply_instance\n\nend fin\n\n\nsection miscellaneous\n\ntheorem nat.pred_eq_of_eq_succ {m n : \u2115}\n  (H : m = n.succ) : m.pred = n :=\nby simp [H]\n\n@[simp] lemma equiv.symm_apply_eq {\u03b1 \u03b2} {e : \u03b1 \u2243 \u03b2} {x y} :\n  e.symm x = y \u2194 x = e y :=\n\u27e8\u03bb H, by simp [H.symm], \u03bb H, by simp [H]\u27e9\n\ntheorem finset.lt_wf {\u03b1} [decidable_eq \u03b1] :\n  well_founded (@has_lt.lt (finset \u03b1) _) :=\nhave H : subrelation (@has_lt.lt (finset \u03b1) _)\n    (inv_image (<) finset.card),\n  from \u03bb x y hxy, finset.card_lt_card hxy,\nsubrelation.wf H $ inv_image.wf _ $ nat.lt_wf\n\ndef finset.min' {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205) : \u03b1 :=\n@option.get _ S.min $\n  let \u27e8k, hk\u27e9 := finset.exists_mem_of_ne_empty H in\n  let \u27e8b, hb\u27e9 := finset.min_of_mem hk in by simp at hb; simp [hb]\n\ntheorem finset.min'_mem {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205) : S.min' H \u2208 S :=\nfinset.mem_of_min $ by simp [finset.min']\n\ntheorem finset.min'_le {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205)\n  (x) (H2 : x \u2208 S) : S.min' H \u2264 x :=\nfinset.le_min_of_mem H2 $ option.get_mem _\n\ntheorem finset.le_min' {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205)\n  (x) (H2 : \u2200 y \u2208 S, x \u2264 y) : x \u2264 S.min' H :=\nH2 _ $ finset.min'_mem _ _\n\ndef finset.max' {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205) : \u03b1 :=\n@option.get _ S.max $\n  let \u27e8k, hk\u27e9 := finset.exists_mem_of_ne_empty H in\n  let \u27e8b, hb\u27e9 := finset.max_of_mem hk in by simp at hb; simp [hb]\n\ntheorem finset.max'_mem {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205) : S.max' H \u2208 S :=\nfinset.mem_of_max $ by simp [finset.max']\n\ntheorem finset.le_max' {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205)\n  (x) (H2 : x \u2208 S) : x \u2264 S.max' H :=\nfinset.le_max_of_mem H2 $ option.get_mem _\n\ntheorem finset.max'_le {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205)\n  (x) (H2 : \u2200 y \u2208 S, y \u2264 x) : S.max' H \u2264 x :=\nH2 _ $ finset.max'_mem _ _\n\ntheorem finset.min'_lt_max' {\u03b1} [decidable_linear_order \u03b1]\n  (S : finset \u03b1) (H : S \u2260 \u2205) {i j}\n  (H1 : i \u2208 S) (H2 : j \u2208 S) (H3 : i \u2260 j) :\n  S.min' H < S.max' H :=\nbegin\n  rcases lt_trichotomy i j with H4 | H4 | H4,\n  { have H5 := finset.min'_le S H i H1,\n    have H6 := finset.le_max' S H j H2,\n    apply lt_of_le_of_lt H5,\n    apply lt_of_lt_of_le H4 H6 },\n  { cc },\n  { have H5 := finset.min'_le S H j H2,\n    have H6 := finset.le_max' S H i H1,\n    apply lt_of_le_of_lt H5,\n    apply lt_of_lt_of_le H4 H6 }\nend\n\nend miscellaneous\n\n\n\nvariable (n : \u2115)\n\ndef Sym : Type :=\nequiv.perm (fin n)\n\ninstance : has_coe_to_fun (Sym n) :=\nequiv.has_coe_to_fun\n\n@[extensionality] theorem Sym.ext (\u03c3 \u03c4 : Sym n)\n  (H : \u2200 i, \u03c3 i = \u03c4 i) : \u03c3 = \u03c4 :=\nequiv.ext _ _ H\n\ntheorem Sym.ext_iff (\u03c3 \u03c4 : Sym n) :\n  \u03c3 = \u03c4 \u2194 \u2200 i, \u03c3 i = \u03c4 i :=\n\u27e8\u03bb H i, H \u25b8 rfl, Sym.ext _ _ _\u27e9\n\ninstance : decidable_eq (Sym n) :=\n\u03bb \u03c3 \u03c4, decidable_of_iff' _ (Sym.ext_iff _ _ _)\n\ninstance : group (Sym n) :=\nequiv.perm_group\n\nvariable {n}\n\nsection perm\n\ndef Sym.to_list (\u03c3 : Sym n) : list (fin n) :=\nlist.of_fn \u03c3\n\ntheorem Sym.to_list_perm (\u03c3 : Sym n) :\n  \u03c3.to_list ~ list.of_fn (1 : Sym n) :=\n(list.perm_ext\n  (list.nodup_of_fn $ \u03c3.bijective.1)\n  (list.nodup_of_fn $ (1 : Sym n).bijective.1)).2 $ \u03bb f,\nby rw [list.of_fn_eq_pmap, list.of_fn_eq_pmap, list.mem_pmap, list.mem_pmap]; from\n\u27e8\u03bb _, \u27e8f.1, by simp [f.2], fin.eq_of_veq rfl\u27e9,\n\u03bb _, \u27e8(\u03c3\u207b\u00b9 f).1, by simp [(\u03c3\u207b\u00b9 f).2], by convert equiv.apply_inverse_apply \u03c3 f;\n  from congr_arg _ (fin.eq_of_veq rfl)\u27e9\u27e9\n\ndef list.to_sym (L : list (fin n))\n  (HL : L ~ list.of_fn (1 : Sym n)) : Sym n :=\n{ to_fun := \u03bb f, list.nth_le L f.1 $\n    by rw [list.perm_length HL, list.length_of_fn]; from f.2,\n  inv_fun := \u03bb f, \u27e8list.index_of f L,\n    begin\n      convert list.index_of_lt_length.2 _,\n      { rw [list.perm_length HL, list.length_of_fn] },\n      { rw [list.mem_of_perm HL, list.mem_iff_nth_le],\n        refine \u27e8f.1, _, _\u27e9,\n        { rw list.length_of_fn,\n          exact f.2 },\n        { apply list.nth_le_of_fn } }\n    end\u27e9,\n  left_inv := \u03bb f, fin.eq_of_veq $ list.nth_le_index_of\n    ((list.perm_nodup HL).2 $ list.nodup_of_fn $ \u03bb _ _, id) _ _,\n  right_inv := \u03bb f, list.index_of_nth_le $ list.index_of_lt_length.2 $\n    (list.mem_of_perm HL).2 $ list.mem_iff_nth_le.2 $\n    \u27e8f.1, by rw list.length_of_fn; from f.2,\n      list.nth_le_of_fn _ _\u27e9 }\n\n@[simp] lemma list.to_sym_apply (L : list (fin n))\n  (HL : L ~ list.of_fn (1 : Sym n)) (i) :\n  (L.to_sym HL) i = L.nth_le i.1 (by simp [list.perm_length HL, i.2]) :=\nrfl\n\n@[simp] lemma Sym.to_list_to_sym (\u03c3 : Sym n) :\n  \u03c3.to_list.to_sym \u03c3.to_list_perm = \u03c3 :=\nSym.ext _ _ _ $ \u03bb i, fin.eq_of_veq $ by simp [Sym.to_list]\n\nend perm\n\nnamespace Sym\n\ndef equiv_0 : Sym 0 \u2243 fin (0:\u2115).fact :=\n{ to_fun    := \u03bb _, \u27e80, dec_trivial\u27e9,\n  inv_fun   := \u03bb _, 1,\n  left_inv  := \u03bb _, ext _ _ _ $ \u03bb \u27e8n, H\u27e9, by cases H,\n  right_inv := \u03bb \u27e8n, H\u27e9, fin.eq_of_veq $\n    by cases H with H1 H1; [refl, cases H1] }\n\ndef descend (\u03c3 : Sym (n+1)) : Sym n :=\n{ to_fun    := \u03bb i, (\u03c3 0).descend (\u03c3 i.succ)\n    (\u03bb H, by cases i; from nat.no_confusion\n      (fin.veq_of_eq (\u03c3.bijective.1 H))),\n  inv_fun   := \u03bb i, (\u03c3.symm ((\u03c3 0).ascend i)).pred $ \u03bb H,\n    fin.ascend_ne (\u03c3 0) i $ by simpa using H,\n  left_inv  := \u03bb i, fin.eq_of_veq $ by dsimp; rw [fin.pred_val];\n    apply nat.pred_eq_of_eq_succ; rw [\u2190 fin.succ_val];\n    apply fin.veq_of_eq; simp,\n  right_inv := \u03bb i, fin.eq_of_veq $ by simp }\n\ndef ascend (\u03c3 : Sym n) (k : fin (n+1)) : Sym (n+1) :=\n{ to_fun    := \u03bb i, if H : i = 0 then k\n    else k.ascend $ \u03c3 $ i.pred H,\n  inv_fun   := \u03bb i, if H : i = k then 0\n    else (\u03c3.symm $ k.descend i H).succ,\n  left_inv  := \u03bb i, fin.eq_of_veq $ begin\n      dsimp,\n      by_cases h1 : i = 0,\n      { simp [h1] },\n      { rw [dif_neg h1],\n        rw [dif_neg (fin.ascend_ne k (\u03c3 (i.pred h1)))],\n        simp }\n    end,\n  right_inv := \u03bb i, fin.eq_of_veq $ begin\n      dsimp,\n      by_cases h1 : i = k,\n      { simp [h1] },\n      { rw [dif_neg h1, dif_neg], { simp },\n        intro H,\n        replace H := fin.veq_of_eq H,\n        simp at H,\n        exact nat.no_confusion H }\n    end }\n\n@[simp] lemma descend_ascend (\u03c3 : Sym n) (k : fin (n+1)) :\n  descend (ascend \u03c3 k) = \u03c3 :=\nbegin\n  ext i,\n  dsimp [ascend, descend],\n  have H : i.succ \u2260 0,\n  { intro H,\n    replace H := fin.veq_of_eq H,\n    simp at H, injections },\n  simp [H]\nend\n\ndef equiv_succ (ih : Sym n \u2243 fin n.fact) :\n  Sym (n+1) \u2243 (fin (n+1) \u00d7 fin n.fact) :=\n{ to_fun    := \u03bb \u03c3, (\u03c3 0, ih $ descend \u03c3),\n  inv_fun   := \u03bb F, ascend (ih.symm F.2) F.1,\n  left_inv  := \u03bb \u03c3, ext _ _ _ $ \u03bb i, begin\n    dsimp, rw [equiv.inverse_apply_apply ih],\n    dsimp [descend, ascend],\n    split_ifs, {subst h},\n    simp\n  end,\n  right_inv := \u03bb F, prod.ext\n      (fin.eq_of_veq $ by dsimp [ascend]; simp) $\n    fin.eq_of_veq $ by simp }\n\nprotected def equiv : Sym n \u2243 fin n.fact :=\nnat.rec_on n equiv_0 $ \u03bb n ih,\ncalc  Sym (n+1)\n    \u2243 (fin (n+1) \u00d7 fin n.fact) : equiv_succ ih\n... \u2243 fin (n+1).fact : fin_prod\n\ninstance : fintype (Sym n) :=\nfintype.of_equiv _ Sym.equiv.symm\n\ntheorem card : fintype.card (Sym n) = nat.fact n :=\n(fintype.of_equiv_card Sym.equiv.symm).trans $\nfintype.card_fin _\n\ntheorem Cayley (\u03b1 : Type*) [group \u03b1] [fintype \u03b1] :\n  \u2203 f : \u03b1 \u2192 Sym (fintype.card \u03b1), function.injective f \u2227 is_group_hom f :=\nnonempty.rec_on (fintype.card_eq.1 $ fintype.card_fin $ fintype.card \u03b1) $ \u03bb \u03c6,\n\u27e8\u03bb x, \u27e8\u03bb i, \u03c6.symm (x * \u03c6 i), \u03bb i, \u03c6.symm (x\u207b\u00b9 * \u03c6 i),\n  \u03bb i, by simp, \u03bb i, by simp\u27e9,\n\u03bb x y H, have H1 : _ := congr_fun (equiv.mk.inj H).1 (\u03c6.symm 1), by simpa using H1,\n\u27e8\u03bb x y, ext _ _ _ $ \u03bb i, by simp [mul_assoc]\u27e9\u27e9\n\n\n@[simp] lemma mul_apply (\u03c3 \u03c4 : Sym n) (i : fin n) :\n  (\u03c3 * \u03c4) i = \u03c3 (\u03c4 i) :=\nrfl\n\n@[simp] lemma one_apply (i : fin n) :\n  (1 : Sym n) i = i :=\nrfl\n\n@[simp] lemma inv_apply (\u03c3 : Sym n) (i : fin n) :\n  \u03c3\u207b\u00b9 i = \u03c3.symm i :=\nrfl\n\ndef swap (i j : fin n) : Sym n :=\n{ to_fun    := \u03bb k, if k = i then j\n    else if k = j then i else k,\n  inv_fun   := \u03bb k, if k = i then j\n    else if k = j then i else k,\n  left_inv  := \u03bb k, by dsimp; split_ifs; cc,\n  right_inv := \u03bb k, by dsimp; split_ifs; cc }\n\n@[simp] lemma swap_left (i j : fin n) :\n  swap i j i = j :=\nby dsimp [swap]; cc\n\n@[simp] lemma swap_right (i j : fin n) :\n  swap i j j = i :=\nby dsimp [swap]; split_ifs; cc\n\n@[simp] lemma swap_mul_self (i j : fin n) :\n  swap i j * swap i j = 1 :=\next _ _ _ $ \u03bb k, by dsimp [swap]; split_ifs; cc\n\ntheorem swap_comm (i j : fin n) :\n  swap i j = swap j i :=\next _ _ _ $ \u03bb k, by dsimp [swap]; split_ifs; cc\n\ntheorem swap_canonical (i j : fin n)\n  (H1 H2 : ({i, j} : finset (fin n)) \u2260 \u2205) :\n  swap (finset.min' _ H1) (finset.max' _ H2) = swap i j :=\nbegin\n  have H3 := finset.min'_mem _ H1,\n  have H4 : finset.min' _ H1 = j \u2228 finset.min' _ H1 = i,\n  { simpa using H3 },\n  have H5 := finset.max'_mem _ H2,\n  have H6 : finset.max' _ H2 = j \u2228 finset.max' _ H2 = i,\n  { simpa using H5 },\n  cases H4; cases H6,\n  { rw [H4, H6],\n    have H7 := finset.min'_le _ H1 i (by simp),\n    have H8 := finset.le_max' _ H2 i (by simp),\n    rw H4 at H7, rw H6 at H8,\n    have H9 := le_antisymm H7 H8,\n    subst H9 },\n  { rw [H4, H6, swap_comm] },\n  { rw [H4, H6] },\n  { rw [H4, H6],\n    have H7 := finset.min'_le _ H1 j (by simp),\n    have H8 := finset.le_max' _ H2 j (by simp),\n    rw H4 at H7, rw H6 at H8,\n    have H9 := le_antisymm H7 H8,\n    subst H9 }\nend\n\n@[simp] theorem swap_self (i : fin n) :\n  swap i i = 1 :=\next _ _ _ $ \u03bb k, by dsimp [swap]; split_ifs; cc\n\ndef support (\u03c3 : Sym n) : finset (fin n) :=\nfinset.filter (\u03bb i, \u03c3 i \u2260 i) finset.univ\n\ntheorem support_def {\u03c3 : Sym n} {i : fin n} :\n  i \u2208 \u03c3.support \u2194 \u03c3 i \u2260 i :=\n\u27e8\u03bb H, (finset.mem_filter.1 H).2, \u03bb H, finset.mem_filter.2 \u27e8finset.mem_univ _, H\u27e9\u27e9\n\ndef support_choice (\u03c3 : Sym n) (H : \u03c3.support \u2260 \u2205) :\n  { i // i \u2208 \u03c3.support } :=\n\u27e8\u03c3.support.min' H, finset.min'_mem _ _\u27e9\n\ntheorem support_swap {i j : fin n} (H : i \u2260 j) :\n  (swap i j).support = {i, j} :=\nbegin\n  ext k, split,\n  { intro H1,\n    simp [support_def, swap] at H1,\n    split_ifs at H1 with h1 h2 h3 h4,\n    { subst h1, simp },\n    { subst h2, simp },\n    cc },\n  { intro H1,\n    simp at H1,\n    cases H1 with H1 H1;\n    subst H1;\n    simp [support_def, swap, H.symm, H] }\nend\n\ntheorem support_swap_mul {\u03c3 : Sym n} {i : fin n}\n  (H : i \u2208 \u03c3.support) : (swap (\u03c3 i) i * \u03c3).support < \u03c3.support :=\nbegin\n  split,\n  { intros j h1,\n    simp [support_def, swap] at *,\n    split_ifs at h1,\n    { intro h2, rw \u2190 h2 at h, subst h, cc },\n    { cc },\n    { cc } },\n  intro H1,\n  specialize H1 H,\n  simp [support_def, swap] at H1,\n  apply H1\nend\n\n@[simp] lemma support_one : support (1 : Sym n) = \u2205 :=\nfinset.eq_empty_of_forall_not_mem $ \u03bb i H,\nsupport_def.1 H rfl\n\nvariable (n)\n@[derive decidable_eq]\nstructure step : Type :=\n(fst : fin n)\n(snd : fin n)\n(lt  : fst < snd)\nvariable {n}\n\ninstance step.fintype : fintype (step n) :=\n@fintype.of_surjective { i : fin n \u00d7 fin n // i.1 < i.2 } _ _ _\n  (\u03bb i, (\u27e8i.1.1, i.1.2, i.2\u27e9 : step n)) $ \u03bb s,\n\u27e8\u27e8(s.1, s.2), s.3\u27e9, by cases s; refl\u27e9\n\ninstance : has_mem (fin n) (step n) :=\n\u27e8\u03bb i s, i = s.1 \u2228 i = s.2\u27e9\n\n@[extensionality] theorem step.ext (s t : step n)\n  (H1 : s.1 = t.1) (H2 : s.2 = t.2) : s = t :=\nby cases s; cases t; congr; assumption\n\ndef step.mk' (i j : fin n) (H : i \u2260 j) : step n :=\nif h : i < j then \u27e8i, j, h\u27e9 else\n\u27e8j, i, (eq_or_lt_of_not_lt h).resolve_left H\u27e9\n\ndef step.eval (s : step n) : Sym n :=\nswap s.1 s.2\n\n@[simp] lemma step.eval_mul_self (s : step n) :\n  s.eval * s.eval = 1 :=\nby simp [step.eval]\n\n@[simp] lemma step.eval_mk' (i j : fin n) (H : i \u2260 j) :\n  (step.mk' i j H).eval = swap i j :=\nby unfold step.mk'; split_ifs; simp [step.eval, swap_comm]\n\ntheorem choice.aux (\u03c3 : Sym n)\n  (H : \u2203 i j, i \u2260 j \u2227 \u03c3 = swap i j) :\n  \u03c3.support \u2260 \u2205 :=\nlet \u27e8i, j, h1, h2\u27e9 := H in by\n  refine finset.ne_empty_of_mem (_ : j \u2208 \u03c3.support);\n  rw [h2, support_swap h1];\n  apply finset.mem_insert_self\n\ndef choice (\u03c3 : Sym n)\n  (H : \u2203 i j, i \u2260 j \u2227 \u03c3 = swap i j) : step n :=\n{ fst := \u03c3.support.min' $ choice.aux _ H,\n  snd := \u03c3.support.max' $ choice.aux _ H,\n  lt  := by rcases H with \u27e8i, j, h1, h2\u27e9; subst h2; dsimp;\n    refine finset.min'_lt_max' _ _ _ _ h1;\n    simp [support_swap h1] }\n\ntheorem eval_choice (\u03c3 : Sym n)\n  (H : \u2203 i j, i \u2260 j \u2227 \u03c3 = swap i j) :\n  (\u03c3.choice H).eval = \u03c3 :=\nbegin\n  rcases H with \u27e8i, j, h1, h2\u27e9,\n  subst h2, unfold step.eval choice, dsimp,\n  convert swap_canonical i j _ _;\n  simp [support_swap h1]\nend\n\ntheorem choice_eval (s : step n)\n  (H : \u2203 i j, i \u2260 j \u2227 s.eval = swap i j) :\n  s.eval.choice H = s :=\nbegin\n  ext; dsimp [step.eval, choice],\n  { apply le_antisymm,\n    { apply finset.min'_le,\n      simp [support_swap (ne_of_lt s.3)] },\n    { apply finset.le_min',\n      intros y h1,\n      simp [support_swap (ne_of_lt s.3)] at h1,\n      cases h1; subst h1,\n      apply le_of_lt s.3 } },\n  { apply le_antisymm,\n    { apply finset.max'_le,\n      intros y h1,\n      simp [support_swap (ne_of_lt s.3)] at h1,\n      cases h1; subst h1,\n      apply le_of_lt s.3 },\n    { apply finset.le_max',\n      simp [support_swap (ne_of_lt s.3)] } }\nend\n\ndef list_step.aux : has_well_founded (Sym n) :=\n{ r := inv_image (<) support,\n  wf := inv_image.wf _ finset.lt_wf }\n\nlocal attribute [instance] list_step.aux\nattribute [elab_as_eliminator] well_founded.fix\nattribute [elab_as_eliminator] well_founded.induction\n\ndef list_step (\u03c3 : Sym n) : list (step n) :=\nby refine well_founded.fix list_step.aux.wf _ \u03c3; from\n\u03bb \u03c3 ih, if H : \u03c3.support = \u2205 then []\n  else let \u27e8i, hi\u27e9 := \u03c3.support_choice H in\n    step.mk' (\u03c3 i) i (support_def.1 hi)\n    :: ih (swap (\u03c3 i) i * \u03c3) (support_swap_mul hi)\n\n@[simp] lemma list_step_prod (\u03c3 : Sym n) :\n  (\u03c3.list_step.map step.eval).prod = \u03c3 :=\nwell_founded.induction list_step.aux.wf \u03c3 $ \u03bb \u03c3 ih,\nbegin\n  dsimp [list_step],\n  rw [well_founded.fix_eq],\n  split_ifs,\n  { ext, by_contra H,\n    suffices : i \u2208 (\u2205 : finset (fin n)),\n    { simp at this, cc },\n    rw [\u2190 h, support_def],\n    exact mt eq.symm H },\n  cases support_choice \u03c3 h with i hi,\n  unfold list_step._match_1,\n  specialize ih _ (support_swap_mul hi),\n  dsimp [list_step] at ih,\n  rw [list.map_cons, list.prod_cons, ih, \u2190 mul_assoc],\n  rw [step.eval_mk', swap_mul_self, one_mul]\nend\n\ntheorem mem_step_iff_mem_support (s : step n) (i : fin n) :\n  i \u2208 s \u2194 i \u2208 s.eval.support :=\nbegin\n  unfold step.eval,\n  simp [support_swap (ne_of_lt s.3)],\n  rw [or_comm], refl\nend\n\ntheorem support_eq_of_mul_eq_one {\u03c3 \u03c4 : Sym n} (H : \u03c3 * \u03c4 = 1) :\n  \u03c3.support = \u03c4.support :=\nbegin\n  ext i, simp [support_def, not_iff_not],\n  rw [eq_comm, iff.comm],\n  convert equiv.symm_apply_eq,\n  symmetry,\n  rw [equiv.symm_apply_eq, \u2190 mul_apply, H, one_apply],\nend\n\ntheorem of_mem_mul_support {\u03c3 \u03c4 : Sym n} {i}\n  (H : i \u2208 (\u03c3 * \u03c4).support) :\n  i \u2208 \u03c3.support \u2228 i \u2208 \u03c4.support :=\nby_contradiction $ \u03bb H2,\nby simp [support_def, not_or_distrib] at H H2;\nsimp [H2] at H; cc\n\ntheorem of_not_mem_mul_support {\u03c3 \u03c4 : Sym n} {i}\n  (H : i \u2209 (\u03c3 * \u03c4).support) :\n  i \u2208 \u03c3.support \u2194 i \u2208 \u03c4.support :=\nbegin\n  simp [support_def] at H \u22a2,\n  split,\n  { intros H2 H3, rw H3 at H, cc },\n  { intros H2 H3, rw \u2190 H3 at H,\n    replace H := \u03c3.bijective.1 H,\n    rw H3 at H, cc }\nend\n\ntheorem not_mem_mul_support {\u03c3 \u03c4 : Sym n} {i}\n  (H1 : i \u2209 \u03c3.support) (H2 : i \u2209 \u03c4.support) :\n  i \u2209 (\u03c3 * \u03c4).support :=\nbegin\n  simp [support_def] at H1 H2 \u22a2,\n  rw [H2, H1]\nend\n\n@[simp] lemma mem_mk' {i j k : fin n} (H : i \u2260 j) :\n  k \u2208 step.mk' i j H \u2194 k = i \u2228 k = j :=\nbegin\n  unfold step.mk',\n  split_ifs,\n  { refl },\n  { apply or_comm }\nend\n\n-- (ab)(cd) = (cd)(ab)\ntheorem sgn_aux5 (s t : step n)\n  (H1 : s.1 \u2260 t.1) (H2 : s.1 \u2260 t.2)\n  (H3 : s.2 \u2260 t.1) (H4 : s.2 \u2260 t.2) :\n  s.eval * t.eval = t.eval * s.eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  dsimp [step.eval, swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(ac) = (bc)(ab)\ntheorem sgn_aux4a (s t : step n)\n  (H1 : s.1 = t.1) (H4 : s.2 \u2260 t.2) :\n  s.eval * t.eval = (step.mk' s.2 t.2 H4).eval * s.eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  unfold step.eval step.mk',\n  simp [swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(ca) = (cb)(ab)\ntheorem sgn_aux4b (s t : step n) (H1 : s.1 = t.2)\n  (H2 : t.1 < s.2) :\n  s.eval * t.eval = (\u27e8t.1, s.2, H2\u27e9 : step n).eval * s.eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  have := ne_of_lt H2,\n  dsimp [step.eval, swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(ac) = (ac)(bc)\ntheorem sgn_aux4c (s t : step n)\n  (H1 : s.1 = t.1) (H4 : s.2 \u2260 t.2) :\n  s.eval * t.eval = t.eval * (step.mk' s.2 t.2 H4).eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  unfold step.eval step.mk',\n  simp [swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(ca) = (ca)(cb)\ntheorem sgn_aux4d (s t : step n)\n  (H1 : s.1 = t.2) (H4 : t.1 < s.2) :\n  s.eval * t.eval = t.eval * (\u27e8t.1, s.2, H4\u27e9 : step n).eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  have := ne_of_lt H4,\n  simp [step.eval, swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(bc) = (bc)(ac)\ntheorem sgn_aux3a (s t : step n) (H1 : s.2 = t.1)\n  (H2 : s.1 < t.2) :\n  s.eval * t.eval = t.eval * (\u27e8s.1, t.2, H2\u27e9 : step n).eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  have := ne_of_lt H2,\n  dsimp [step.eval, swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(cb) = (cb)(ac)\ntheorem sgn_aux3b (s t : step n) (H1 : s.2 = t.2)\n  (H2 : s.1 \u2260 t.1) :\n  s.eval * t.eval = t.eval * (step.mk' s.1 t.1 H2).eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  unfold step.eval step.mk',\n  simp [swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(bc) = (ac)(ab)\ntheorem sgn_aux3c (s t : step n) (H1 : s.2 = t.1)\n  (H2 : s.1 < t.2) :\n  s.eval * t.eval = (\u27e8s.1, t.2, H2\u27e9 : step n).eval * s.eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  have := ne_of_lt H2,\n  dsimp [step.eval, swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\n-- (ab)(cb) = (ac)(ab)\ntheorem sgn_aux3d (s t : step n) (H1 : s.2 = t.2)\n  (H2 : s.1 \u2260 t.1) :\n  s.eval * t.eval = (step.mk' s.1 t.1 H2).eval * s.eval :=\nbegin\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  unfold step.eval step.mk',\n  simp [swap], ext k,\n  dsimp at *,\n  split_ifs; cc\nend\n\ntheorem sgn_aux2 (s t : step n) (i) (H : i \u2208 s) :\n  s = t \u2228 \u2203 s' t' : step n, s.eval * t.eval = s'.eval * t'.eval\n    \u2227 i \u2209 s' \u2227 i \u2208 t' :=\nbegin\n  cases H with H H; subst H,\n  { by_cases H2 : s.1 = t.1,\n    { by_cases H3 : s.2 = t.2,\n      { left, ext; assumption },\n      right, -- (ab)(ac) = (bc)(ab)\n      refine \u27e8step.mk' s.2 t.2 H3, s, _, _\u27e9,\n      { exact sgn_aux4a _ _ H2 _ },\n      rw [mem_mk'],\n      exact \u27e8\u03bb H, or.cases_on H\n        (ne_of_lt s.3) (H2.symm \u25b8 (ne_of_lt t.3)),\n      or.inl rfl\u27e9 },\n    right,\n    by_cases H3 : s.1 = t.2,\n    { -- (ab)(ca) = (cb)(ab)\n      have H4 : t.1 < s.2 := lt_trans t.3 (H3 \u25b8 s.3),\n      refine \u27e8\u27e8t.1, s.2, H4\u27e9, s, _, _\u27e9,\n      { exact sgn_aux4b _ _ H3 _ },\n      exact \u27e8\u03bb H, or.cases_on H\n        H2 (ne_of_lt s.3),\n      or.inl rfl\u27e9 },\n    by_cases H4 : s.2 = t.1,\n    { -- (ab)(bc) = (bc)(ac)\n      have H5 : s.1 < t.2 := lt_trans s.3 (H4.symm \u25b8 t.3),\n      refine \u27e8t, \u27e8s.1, t.2, H5\u27e9, _, _\u27e9,\n      { exact sgn_aux3a _ _ H4 _ },\n      exact \u27e8\u03bb H, or.cases_on H\n        H2 (ne_of_lt H5),\n      or.inl rfl\u27e9 },\n    by_cases H5 : s.2 = t.2,\n    { -- (ab)(cb) = (cb)(ac)\n      refine \u27e8t, step.mk' s.1 t.1 H2, _, _\u27e9,\n      { exact sgn_aux3b _ _ H5 _ },\n      exact \u27e8\u03bb H, or.cases_on H\n        H2 H3,\n      by simp\u27e9 },\n    -- (ab)(cd) = (cd)(ab)\n    refine \u27e8t, s, _, _\u27e9,\n    { exact sgn_aux5 _ _ H2 H3 H4 H5 },\n    exact \u27e8\u03bb H, or.cases_on H\n      H2 H3,\n    or.inl rfl\u27e9 },\n  by_cases H2 : s.1 = t.1,\n  { by_cases H3 : s.2 = t.2,\n    { left, ext; assumption },\n    right, -- (ab)(ac) = (ac)(bc)\n    refine \u27e8t, step.mk' s.2 t.2 H3, _, _\u27e9,\n    { exact sgn_aux4c _ _ H2 _ },\n    rw [mem_mk'],\n    exact \u27e8\u03bb H, or.cases_on H\n      (H2 \u25b8 ne_of_gt s.3) H3,\n    or.inl rfl\u27e9 },\n  right,\n  by_cases H3 : s.1 = t.2,\n  { -- (ab)(ca) = (ca)(cb)\n    have H4 : t.1 < s.2 := lt_trans t.3 (H3 \u25b8 s.3),\n    refine \u27e8t, \u27e8t.1, s.2, H4\u27e9, _, _\u27e9,\n    { exact sgn_aux4d _ _ H3 _ },\n    exact \u27e8\u03bb H, or.cases_on H\n      (ne_of_gt H4) (H3 \u25b8 ne_of_gt s.3),\n    or.inr rfl\u27e9 },\n  by_cases H4 : s.2 = t.1,\n  { -- (ab)(bc) = (ac)(ab)\n    have H5 : s.1 < t.2 := lt_trans s.3 (H4.symm \u25b8 t.3),\n    refine \u27e8\u27e8s.1, t.2, H5\u27e9, s, _, _\u27e9,\n    { exact sgn_aux3c _ _ H4 _ },\n    exact \u27e8\u03bb H, or.cases_on H\n      (ne_of_gt s.3) (H4.symm \u25b8 ne_of_lt t.3),\n    or.inr rfl\u27e9 },\n  by_cases H5 : s.2 = t.2,\n  { -- (ab)(cb) = (ac)(ab)\n    refine \u27e8step.mk' s.1 t.1 H2, s, _, _\u27e9,\n    { exact sgn_aux3d _ _ H5 _ },\n    rw [mem_mk'],\n    exact \u27e8\u03bb H, or.cases_on H\n      (ne_of_gt s.3) (H5.symm \u25b8 ne_of_gt t.3),\n    or.inr rfl\u27e9 },\n  refine \u27e8t, s, _, _\u27e9,\n  { exact sgn_aux5 _ _ H2 H3 H4 H5 },\n  exact \u27e8\u03bb H, or.cases_on H\n    H4 H5,\n  or.inr rfl\u27e9\nend\n\ntheorem sgn_aux (L1 : list (step n)) (s : step n) (L2 : list (step n)) (i : fin n)\n  (H1 : (L1.map step.eval).prod * s.eval * (L2.map step.eval).prod = 1)\n  (H2 : i \u2208 s) (H3 : i \u2209 (L1.map step.eval).prod.support) :\n  \u2203 (L : list (step n)),\n    (L.map step.eval).prod = 1\n    \u2227 L.length + 2 = L1.length + 1 + L2.length :=\nbegin\n  induction L2 with hd tl ih generalizing L1 s,\n  { simp at H1,\n    simp [mem_step_iff_mem_support] at H2,\n    rw support_eq_of_mul_eq_one H1 at H3,\n    cc },\n  simp [mul_assoc] at H1,\n  simp [mem_step_iff_mem_support] at H2,\n  have H4 := H3,\n  rw [support_eq_of_mul_eq_one H1] at H4,\n  replace H4 := of_not_mem_mul_support H4,\n  replace H4 := H4.1 H2,\n  rw [\u2190 mem_step_iff_mem_support] at H2,\n  rcases sgn_aux2 s hd i H2 with H5 | \u27e8s', t', H5, H6, H7\u27e9,\n  { subst H5,\n    rw [\u2190 mul_assoc s.eval, step.eval_mul_self, one_mul] at H1,\n    rw [\u2190 list.prod_append, \u2190 list.map_append] at H1,\n    refine \u27e8_, H1, _\u27e9,\n    simp, unfold bit0, ac_refl },\n  specialize ih (L1 ++ [s']) t' _ H7 _,\n  rcases ih with \u27e8L, H8, H9\u27e9,\n  refine \u27e8L, H8, _\u27e9,\n  { simp [H9] },\n  { simp at H5 \u22a2,\n    rw [mul_assoc (L1.map step.eval).prod, \u2190 H5],\n    simpa [mul_assoc] using H1 },\n  simpa using not_mem_mul_support H3 _,\n  simpa [mem_step_iff_mem_support] using H6\nend\n\ntheorem length_even_of_prod_one (L : list (step n))\n  (H : (L.map step.eval).prod = 1) :\n  L.length % 2 = 0 :=\nbegin\n  generalize H1 : L.length = k,\n  revert L,\n  apply nat.strong_induction_on k,\n  intros k ih L H H1,\n  cases k with k, { refl },\n  cases k with k,\n  { exfalso,\n    rw list.length_eq_one at H1,\n    cases H1 with s H2,\n    subst H2,\n    replace H := congr_arg support H,\n    simp [step.eval, support_swap (ne_of_lt s.3)] at H,\n    exact H },\n  cases L with hd tl, { simp at H1, injections },\n  rcases sgn_aux [] hd tl hd.1 _ (or.inl rfl) _ with \u27e8L, H2, H3\u27e9,\n  specialize ih k _ L H2 _,\n  change (k + 2) % 2 = 0,\n  { rw [nat.add_mod_right, ih] },\n  { constructor, constructor },\n  { simp at H1 H3,\n    rw \u2190 H3 at H1,\n    exact nat.succ_inj (nat.succ_inj H1) },\n  { simpa using H },\n  simp\nend\n\ntheorem length_mod_two_eq (L1 L2 : list (step n))\n  (H : (L1.map step.eval).prod = (L2.map step.eval).prod) :\n  L1.length % 2 = L2.length % 2 :=\nhave H1 : (L2.map step.eval).reverse.prod = (L2.map step.eval).prod\u207b\u00b9,\n  from list.rec_on L2 (by simp) $ \u03bb hd tl ih,\n    by simp [ih, eq_inv_iff_mul_eq_one],\nhave H2 : _,\n  from length_even_of_prod_one (L1 ++ L2.reverse) $\n    by simp [H1, H],\nhave H3 : 2 \u2223 L1.length + L2.length,\n  by simpa [nat.dvd_iff_mod_eq_zero] using H2,\ncalc  L1.length % 2\n    = (L1.length + L2.length + L2.length) % 2 :\n  by rw [add_assoc, \u2190 mul_two, nat.add_mul_mod_self_right]\n... = L2.length % 2 :\n  by cases H3 with k H4; rw [H4, add_comm, nat.add_mul_mod_self_left]\n\nend Sym\n\n@[derive decidable_eq]\ninductive mu2 : Type\n| plus_one : mu2\n| minus_one : mu2\n\nnamespace mu2\n\ndefinition neg : mu2 \u2192 mu2\n| plus_one := minus_one\n| minus_one := plus_one\n\ninstance : has_one mu2 := \u27e8plus_one\u27e9\ninstance : has_neg mu2 := \u27e8neg\u27e9\n\ninstance : comm_group mu2 :=\n{ mul := \u03bb x y, mu2.rec_on x (mu2.rec_on y 1 (-1)) (mu2.rec_on y (-1) 1),\n  mul_assoc := \u03bb x y z, by cases x; cases y; cases z; refl,\n  mul_one := \u03bb x, by cases x; refl,\n  one_mul := \u03bb x, by cases x; refl,\n  inv := id,\n  mul_left_inv := \u03bb x, by cases x; refl,\n  mul_comm := \u03bb x y, by cases x; cases y; refl,\n  .. mu2.has_one }\n\ninstance : fintype mu2 :=\n{ elems := {1, -1},\n  complete := \u03bb x, mu2.cases_on x (or.inr $ or.inl rfl) (or.inl rfl) }\n\ntheorem card : fintype.card mu2 = 2 :=\nrfl\n\ntheorem neg_one_pow {n} : (-1 : mu2) ^ n = (-1 : mu2) ^ (n%2) :=\nhave H : (-1 : mu2) ^ 2 = 1, from rfl,\nby rw [\u2190 nat.mod_add_div n 2, pow_add, pow_mul, H, one_pow, mul_one, nat.mod_add_div n 2]\n\n@[simp] lemma mul_self_eq_one (x : mu2) : x * x = 1 :=\nby cases x; refl\n\n@[simp] lemma inv_eq_self (x : mu2) : x\u207b\u00b9 = x :=\nrfl\n\n@[simp] protected lemma mul_neg_one (x : mu2) : x * -1 = -x :=\nby cases x; refl\n\n@[simp] protected lemma neg_one_mul (x : mu2) : -1 * x = -x :=\nby cases x; refl\n\n@[simp] lemma neg_mul_self (x : mu2) : -x * x = -1 :=\nby cases x; refl\n\n@[simp] lemma mul_neg (x y : mu2) : x * -y = -x * y :=\nby cases x; cases y; refl\n\nend mu2\n\nnamespace Sym\n\ndef sgn (\u03c3 : Sym n) : mu2 :=\n(-1) ^ \u03c3.list_step.length\n\ninstance sgn.is_group_hom : is_group_hom (@sgn n) :=\nbegin\n  constructor,\n  intros \u03c3 \u03c4,\n  unfold sgn,\n  rw [\u2190 pow_add, \u2190 list.length_append],\n  rw [mu2.neg_one_pow, eq_comm, mu2.neg_one_pow],\n  refine congr_arg _ _,\n  apply length_mod_two_eq,\n  simp\nend\n\n@[simp] lemma sgn_step (s : step n) :\n  sgn s.eval = -1 :=\nsuffices s.eval.list_step.length % 2 = [s].length % 2,\n  by unfold sgn; rw [mu2.neg_one_pow, this]; refl,\nlength_mod_two_eq _ _ $ by simp\n\n@[simp] lemma sgn_mul (\u03c3 \u03c4 : Sym n) :\n  sgn (\u03c3 * \u03c4) = sgn \u03c3 * sgn \u03c4 :=\nis_group_hom.mul sgn _ _\n\n@[simp] lemma sgn_one :\n  sgn (1 : Sym n) = 1 :=\nis_group_hom.one sgn\n\n@[simp] lemma sgn_inv (\u03c3 : Sym n) :\n  sgn \u03c3\u207b\u00b9 = sgn \u03c3 :=\nis_group_hom.inv sgn _\n\ndef eq_sgn_aux4 (s t : step n) : Sym n :=\nswap (swap s.1 t.1 s.2) t.2 * swap s.1 t.1\n\ntheorem eq_sgn_aux3 (s t : step n) :\n  eq_sgn_aux4 s t s.1 = t.1 :=\nbegin\n  dsimp [eq_sgn_aux4, swap],\n  have := ne_of_lt s.3,\n  have := ne_of_lt t.3,\n  simp, split_ifs; cc\nend\n\ntheorem eq_sgn_aux2 (s t : step n) :\n  eq_sgn_aux4 s t s.2 = t.2 :=\nbegin\n  dsimp [eq_sgn_aux4, swap],\n  simp\nend\n\ntheorem eq_sgn_aux (s t : step n) :\n  eq_sgn_aux4 s t * s.eval * (eq_sgn_aux4 s t)\u207b\u00b9 = t.eval :=\nbegin\n  ext k,\n  by_cases H1 : k = t.1,\n  { subst H1,\n    dsimp [step.eval],\n    simp [equiv.symm_apply_eq.2 (eq_sgn_aux3 s t).symm, eq_sgn_aux2] },\n  by_cases H2 : k = t.2,\n  { subst H2,\n    dsimp [step.eval],\n    simp [equiv.symm_apply_eq.2 (eq_sgn_aux2 s t).symm, eq_sgn_aux3] },\n  dsimp [step.eval, swap],\n  simp [H1, H2, eq_sgn_aux2, eq_sgn_aux3]\nend\n\ntheorem eq_sgn (f : Sym n \u2192 mu2) [is_group_hom f]\n  (s : step n) (H1 : f s.eval = -1) (\u03c3 : Sym n) :\n  f \u03c3 = sgn \u03c3 :=\nbegin\n  have H2 : \u2200 t : step n, f t.eval = -1,\n  { intro t,\n    rw [\u2190 eq_sgn_aux s t],\n    simp [is_group_hom.mul f, is_group_hom.inv f, H1] },\n  have H3 := list_step_prod \u03c3,\n  revert H3, generalize : list_step \u03c3 = L, intro H3, subst H3,\n  induction L with hd tl ih, { simp [is_group_hom.one f] },\n  simp [is_group_hom.mul f, ih, H2]\nend\n\nsection inversions\n\ndef step.map (s : step n) (\u03c3 : Sym n) : step n :=\nstep.mk' (\u03c3 s.1) (\u03c3 s.2) $ \u03bb H, ne_of_lt s.3 $\n\u03c3.bijective.1 H\n\n@[simp] lemma step.map_map_inv (s : step n) (\u03c3 : Sym n) :\n  (s.map \u03c3).map \u03c3\u207b\u00b9 = s :=\nbegin\n  unfold step.map step.mk',\n  by_cases H1 : \u03c3 s.1 < \u03c3 s.2,\n  { rw [dif_pos H1], dsimp,\n    rw dif_pos, ext; simp,\n    simp [s.3] },\n  rw [dif_neg H1], dsimp,\n  rw dif_neg, ext; simp,\n  simp [le_of_lt s.3]\nend\n\n@[simp] lemma step.map_inv_map (s : step n) (\u03c3 : Sym n) :\n  (s.map \u03c3\u207b\u00b9).map \u03c3 = s :=\nby simpa using step.map_map_inv s \u03c3\u207b\u00b9\n\ndef inversion (\u03c3 : Sym n) (s : step n) : mu2 :=\nif \u03c3 s.1 > \u03c3 s.2 then -1 else 1\n\ndef inversions (\u03c3 : Sym n) : mu2 :=\nfinset.prod finset.univ $ inversion \u03c3\n\ntheorem inversion_mul (\u03c3 \u03c4 : Sym n) (s : step n) :\n  inversion (\u03c3 * \u03c4) s = inversion \u03c3 (s.map \u03c4) * inversion \u03c4 s :=\nbegin\n  unfold inversion step.map step.mk',\n  split_ifs with h1 h2 h3 h3 h4 h4 h2 h3 h3 h4 h4; try { refl },\n  { exfalso, apply lt_asymm h2 h3 },\n  { exfalso, apply lt_asymm h1 h3 },\n  { simp at *, exfalso,\n    exact ne_of_lt s.3 (\u03c4.bijective.1 $ le_antisymm h4 h2) },\n  { exfalso, apply lt_asymm h2 h3 },\n  { simp at *, exfalso,\n    exact ne_of_lt s.3 (\u03c4.bijective.1 $ le_antisymm h4 h2) },\n  { simp at *, exfalso,\n    exact ne_of_lt s.3 (\u03c4.bijective.1 $ \u03c3.bijective.1 $ le_antisymm h1 h3) }\nend\n\ninstance : is_group_hom (@inversions n) :=\n\u27e8\u03bb \u03c3 \u03c4, calc\n      inversions (\u03c3 * \u03c4)\n    = finset.prod finset.univ (inversion (\u03c3 * \u03c4)) : rfl\n... = finset.prod finset.univ (\u03bb s : step n,\n        inversion \u03c3 (s.map \u03c4) * inversion \u03c4 s) :\n  congr_arg _ $ funext $ inversion_mul \u03c3 \u03c4\n... = finset.prod finset.univ (\u03bb s : step n,\n        inversion \u03c3 (s.map \u03c4)) * inversions \u03c4 :\n  finset.prod_mul_distrib\n... = finset.prod finset.univ (inversion \u03c3) * inversions \u03c4 :\n  congr_arg (\u03bb z, z * inversions \u03c4) $ finset.prod_bij\n    (\u03bb s _, step.map s \u03c4) (\u03bb _ _, finset.mem_univ _) (\u03bb _ _, rfl)\n    (\u03bb s t _ _ H, by simpa using congr_arg (\u03bb z, step.map z \u03c4\u207b\u00b9) H)\n    (\u03bb s _, \u27e8s.map \u03c4\u207b\u00b9, finset.mem_univ _, by simp\u27e9)\u27e9\n\nvariable (n)\ndef step01 : step (n+2) :=\n\u27e8\u27e80, nat.zero_lt_succ _\u27e9, \u27e81, nat.succ_lt_succ $ nat.zero_lt_succ _\u27e9, dec_trivial\u27e9\nvariable {n}\n\ntheorem inversions_step01 : inversions (step01 n).eval = -1 :=\nshow _ = finset.prod {step01 n} (inversion (step01 n).eval), from\neq.symm $ finset.prod_subset (finset.subset_univ _) $ \u03bb s _ H1, begin\n  unfold inversion step.eval swap step01; dsimp at *, rw if_neg,\n  by_cases H2 : s.1.1 = 0,\n  { rw [if_pos, if_neg, if_neg],\n    { intro H3,\n      replace H3 := nat.le_of_lt_succ H3,\n      replace H3 := nat.eq_zero_of_le_zero H3,\n      exact ne_of_lt s.3 (fin.eq_of_veq $ H2.trans H3.symm) },\n    { intro H3, apply H1, simp, ext, exact fin.eq_of_veq H2, exact H3 },\n    { exact ne_of_gt (H2 \u25b8 s.3 : s.2.1 > 0), },\n    { exact fin.eq_of_veq H2 } },\n  by_cases H3 : s.1.1 = 1,\n  { rw [if_neg, if_pos], exact nat.not_lt_zero _,\n    exact fin.eq_of_veq H3, exact mt fin.veq_of_eq H2 },\n  rw [if_neg, if_neg, if_neg, if_neg],\n  { exact lt_asymm s.3 },\n  { intro H4, have H5 := s.3, rw H4 at H5,\n    replace H5 := nat.le_of_lt_succ H5,\n    replace H5 := nat.eq_zero_of_le_zero H5,\n    cc },\n  { intro H4, have H5 := s.3, rw H4 at H5,\n    cases H5 },\n  { exact mt fin.veq_of_eq H3 },\n  { exact mt fin.veq_of_eq H2 }\nend\n\ntheorem inversions_eq_sgn : \u2200 \u03c3 : Sym n, inversions \u03c3 = sgn \u03c3 :=\nnat.cases_on n dec_trivial $ \u03bb n,\nnat.cases_on n dec_trivial $ \u03bb n \u03c3,\neq_sgn inversions (step01 n) inversions_step01 \u03c3\n\nend inversions\n\nend Sym\n\nvariable (n)\ndef Alt : Type :=\nis_group_hom.ker (@Sym.sgn n)\n\ninstance : group (Alt n) :=\nby unfold Alt; apply_instance\n", "meta": {"author": "kckennylau", "repo": "Lean", "sha": "907d0a4d2bd8f23785abd6142ad53d308c54fdcb", "save_path": "github-repos/lean/kckennylau-Lean", "path": "github-repos/lean/kckennylau-Lean/Lean-907d0a4d2bd8f23785abd6142ad53d308c54fdcb/Sym.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.3872918386533537}}
{"text": "import data.sigma.basic\nimport tactic.ext\n\nnamespace sigma\nuniverses u v\n\nsection\nvariables {\u03b1 : Type u} {\u03b2 \u03b2' : \u03b1 \u2192 Type v}\n\ntheorem eq_fst {s\u2081 s\u2082 : sigma \u03b2} : s\u2081 = s\u2082 \u2192 s\u2081.1 = s\u2082.1 :=\nby cases s\u2081; cases s\u2082; cc\n\ntheorem eq_snd {s\u2081 s\u2082 : sigma \u03b2} : s\u2081 = s\u2082 \u2192 s\u2081.2 == s\u2082.2 :=\nby cases s\u2081; cases s\u2082; cc\n\nend\n\nsection\nvariables {\u03b1\u2081 \u03b1\u2082 : Type u} {\u03b2\u2081 : \u03b1\u2081 \u2192 Type v} {\u03b2\u2082 : \u03b1\u2082 \u2192 Type v}\n\n/-- A function on `sigma`s that is functional on `fst`s (preserves equality from\nargument to result). -/\ndef fst_functional (f : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082) : Prop :=\n\u2200 \u2983s t : sigma \u03b2\u2081\u2984, s.1 = t.1 \u2192 (f s).1 = (f t).1\n\n/-- A function on `sigma`s that is injective on `fst`s (preserves equality from\nresult to argument). -/\ndef fst_injective (f : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082) : Prop :=\n\u2200 \u2983s t : sigma \u03b2\u2081\u2984, (f s).1 = (f t).1 \u2192 s.1 = t.1\n\nend\n\n/-- A function on `sigma`s bundled with its `fst`-injectivity property. -/\nstructure embedding {\u03b1\u2081 \u03b1\u2082 : Type u} (\u03b2\u2081 : \u03b1\u2081 \u2192 Type v) (\u03b2\u2082 : \u03b1\u2082 \u2192 Type v) :=\n(to_fun  : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082)\n(fst_inj : fst_injective to_fun)\n\ninfixr ` s\u21aa `:25 := embedding\n\nnamespace embedding\nvariables {\u03b1\u2081 \u03b1\u2082 : Type u} {\u03b2\u2081 : \u03b1\u2081 \u2192 Type v} {\u03b2\u2082 : \u03b1\u2082 \u2192 Type v}\n\ninstance : has_coe_to_fun (\u03b2\u2081 s\u21aa \u03b2\u2082) :=\n\u27e8_, embedding.to_fun\u27e9\n\n@[simp] theorem to_fun_eq_coe (f : \u03b2\u2081 s\u21aa \u03b2\u2082) : f.to_fun = f :=\nrfl\n\n@[simp] theorem coe_fn_mk (f : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082) (i : fst_injective f) :\n  (mk f i : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082) = f :=\nrfl\n\ntheorem fst_inj' : \u2200 (f : \u03b2\u2081 s\u21aa \u03b2\u2082), fst_injective f\n| \u27e8_, h\u27e9 := h\n\nend embedding\n\nsection map_id\nvariables {\u03b1 : Type u} {\u03b2\u2081 \u03b2\u2082 : \u03b1 \u2192 Type v}\n\n@[simp] theorem map_id_eq_fst {s : sigma \u03b2\u2081} (f : \u2200 a, \u03b2\u2081 a \u2192 \u03b2\u2082 a) :\n  (s.map id f).1 = s.1 :=\nby cases s; refl\n\ntheorem map_id_fst_functional (f : \u2200 a, \u03b2\u2081 a \u2192 \u03b2\u2082 a) :\n  fst_functional (map id f) :=\n\u03bb _ _, by simp only [map_id_eq_fst]; exact id\n\ntheorem map_id_fst_injective (f : \u2200 a, \u03b2\u2081 a \u2192 \u03b2\u2082 a) :\n  fst_injective (map id f) :=\n\u03bb _ _, by simp only [map_id_eq_fst]; exact id\n\n/-- Construct an `embedding` with `id` on `fst`. -/\ndef embedding.mk\u2082 (f : \u2200 a, \u03b2\u2081 a \u2192 \u03b2\u2082 a) : embedding \u03b2\u2081 \u03b2\u2082 :=\n\u27e8_, map_id_fst_injective f\u27e9\n\nend map_id\n\nsection\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {R : \u03b1 \u2192 \u03b1 \u2192 Prop}\n\n/-- A relation `R` on `fst` values lifted to the `sigma`. This is useful where\nyou might otherwise use the term `\u03bb s\u2081 s\u2082, R s\u2081.1 s\u2082.1`. -/\ndef fst_rel (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (s\u2081 s\u2082 : sigma \u03b2) : Prop :=\nR s\u2081.1 s\u2082.1\n\n@[simp] theorem fst_rel_def {s\u2081 s\u2082 : sigma \u03b2} : fst_rel R s\u2081 s\u2082 = R s\u2081.1 s\u2082.1 :=\nrfl\n\ninstance fst_rel_decidable [d : decidable_rel R] : decidable_rel (@fst_rel _ \u03b2 R)\n| s\u2081 s\u2082 := @d s\u2081.1 s\u2082.1\n\ntheorem fst_rel.refl (h : reflexive R) : reflexive (@fst_rel _ \u03b2 R) :=\n\u03bb s, h s.1\n\ntheorem fst_rel.symm (h : symmetric R) : symmetric (@fst_rel _ \u03b2 R) :=\n\u03bb s\u2081 s\u2082 (p : R s\u2081.1 s\u2082.1), h p\n\ntheorem fst_rel.trans (h : transitive R) : transitive (@fst_rel _ \u03b2 R) :=\n\u03bb s\u2081 s\u2082 s\u2083 (p : R s\u2081.1 s\u2082.1) (q : R s\u2082.1 s\u2083.1), h p q\n\nend\n\nsection\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v}\n\ntheorem fst_functional_id : fst_functional (@id (sigma \u03b2)) :=\n\u03bb s t h, h\n\ntheorem fst_injective_id : fst_injective (@id (sigma \u03b2)) :=\n\u03bb s t h, h\n\n@[refl] protected def embedding.refl (\u03b2 : \u03b1 \u2192 Type v) : \u03b2 s\u21aa \u03b2 :=\n\u27e8_, fst_injective_id\u27e9\n\n@[simp] theorem embedding.refl_apply (s : sigma \u03b2) : embedding.refl \u03b2 s = s :=\nrfl\n\nend\n\nsection\nvariables {\u03b1\u2081 \u03b1\u2082 \u03b1\u2083 : Type u}\nvariables {\u03b2\u2081 : \u03b1\u2081 \u2192 Type v} {\u03b2\u2082 : \u03b1\u2082 \u2192 Type v} {\u03b2\u2083 : \u03b1\u2083 \u2192 Type v}\nvariables {g : sigma \u03b2\u2082 \u2192 sigma \u03b2\u2083} {f : sigma \u03b2\u2081 \u2192 sigma \u03b2\u2082}\n\ntheorem fst_functional_comp (gf : fst_functional g) (ff : fst_functional f) :\n  fst_functional (g \u2218 f) :=\n\u03bb s t h, gf (ff h)\n\ntheorem fst_injective_comp (gi : fst_injective g) (fi : fst_injective f) :\n  fst_injective (g \u2218 f) :=\n\u03bb s t h, fi (gi h)\n\n@[trans] protected def embedding.trans (f : \u03b2\u2081 s\u21aa \u03b2\u2082) (g : \u03b2\u2082 s\u21aa \u03b2\u2083) : \u03b2\u2081 s\u21aa \u03b2\u2083 :=\n\u27e8_, fst_injective_comp g.fst_inj f.fst_inj\u27e9\n\n@[simp] theorem embedding.trans_apply (f : \u03b2\u2081 s\u21aa \u03b2\u2082) (g : \u03b2\u2082 s\u21aa \u03b2\u2083) (s : sigma \u03b2\u2081) :\n  (f.trans g) s = g (f s) :=\nrfl\n\n@[extensionality]\nlemma ext {x\u2080 x\u2081 : sigma \u03b2\u2081}\n  (h\u2080 : x\u2080.1 = x\u2081.1)\n  (h\u2081 : x\u2080.1 = x\u2081.1 \u2192 x\u2080.2 == x\u2081.2) :\n  x\u2080 = x\u2081 :=\nby casesm* sigma _; cases h\u2080; cases h\u2081 h\u2080; refl\n\nlemma eta (x : sigma \u03b2\u2081) : sigma.mk x.1 x.2 = x :=\nby cases x; refl\n\nend\n\nend sigma\n", "meta": {"author": "leanprover-community", "repo": "mathlib-nursery", "sha": "0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec", "save_path": "github-repos/lean/leanprover-community-mathlib-nursery", "path": "github-repos/lean/leanprover-community-mathlib-nursery/mathlib-nursery-0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec/src/data/sigma/fst.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.3872918386533537}}
{"text": "import Qpf\n\n-- set_option trace.Meta.debug true\n-- set_option trace.Elab.inductive true\nsudo set_option trace.QPF true\nset_option pp.rawOnError true\n\n#print List.noConfusionType\n#check List.noConfusion\n#check List.recOn\n#check List.casesOn\n\n-- #print prefix List\n\ntheorem nil_neq_cons (a : \u03b1) (as : List \u03b1) : List.nil \u2260 List.cons a as := \n  by simp\n\n#print nil_neq_cons\n\n\n\ndata QpfList \u03b1 where\n  | nil : QpfList \u03b1\n  | cons : \u03b1 \u2192 QpfList \u03b1 \u2192 QpfList \u03b1\n\n#check QpfList\n\n#check @MvQPF.Fix.drec _ QpfList.Uncurried _\n\nnamespace QpfList\n  #print QpfList.nil\n  #print QpfList.cons\n\n\n  def rec {\u03b1 : Type _} {motive : QpfList \u03b1 \u2192 Sort _} :\n    motive QpfList.nil \n    \u2192 ((head : \u03b1) \u2192 (tail : QpfList \u03b1) \u2192 motive tail \u2192 motive (QpfList.cons head tail))\n    \u2192 (t : QpfList \u03b1) \n    \u2192 motive t\n  :=\n    fun nil cons => MvQPF.Fix.drec (fun x => \n      match x with\n      | .nil            => nil\n      | .cons head tail => cons head tail.fst tail.snd\n    )\n\n  @[reducible] protected def recOn {\u03b1 : Type _} {motive : QpfList \u03b1 \u2192 Sort _} :\n      (t : QpfList \u03b1) \n      \u2192 motive QpfList.nil \n      \u2192 ((head : \u03b1) \u2192 (tail : QpfList \u03b1) \u2192 motive tail \u2192 motive (QpfList.cons head tail)) \n      \u2192 motive t \n    :=\n      fun t nil cons => QpfList.rec nil cons t\n\n  #print List.recOn\n\n  #check @List.rec\n  #check @QpfList.rec\n\n  \n\n  theorem nil_neq_cons (a : \u03b1) (as : QpfList \u03b1) : @QpfList.nil \u03b1 \u2260 QpfList.cons a as := \n    by\n      sorry\n\n  #print nil_neq_cons\nend QpfList\n\n\n\n\n\n\n\n\n\n\ndata QpfTree \u03b1 where\n  | node : \u03b1 \u2192 List (QpfTree \u03b1) \u2192 QpfTree \u03b1\n\ncodata QpfCoTree \u03b1 where\n  | node : \u03b1 \u2192 List (QpfCoTree \u03b1) \u2192 QpfCoTree \u03b1\n\n\n\n/-- If `a \u2208 as`, return `as` with (a single occurrence of) `a` removed.\n    Otherwise, if `a \u2209 as`, return `none` -/\ndef List.is_rem (a : \u03b1) : List \u03b1 \u2192 List \u03b1 \u2192 Prop\n  | b::bs, c::cs  =>    (a = c  \u2227 bs = c::cs)\n                      \u2228 (b = c  \u2227 bs.is_rem a cs)\n  | _, _          => false\n\n/-- Equates lists up-to permutation -/\ndef List.perm \u2983\u03b1\u2984 : QpfList.Uncurried \u03b1 \u2192 QpfList.Uncurried \u03b1 \u2192 Prop\n  -- body omitted\n  := by sorry\n  -- | [],    []  =>  true\n  -- | a::as, bs  =>  \u2203cs : List \u03b1, cs.is_rem a bs \u2227 as.perm cs\n  -- | _, _       =>  false\n                      \n\nabbrev MultiSet.Uncurried := MvQPF.Quot1 List.perm\nabbrev MultiSet := TypeFun.curried MultiSet.Uncurried\n\nnoncomputable instance : MvQPF MultiSet.Uncurried := MvQPF.relQuot List.perm (by sorry)\n\n#check (inferInstance : MvQPF MultiSet.Uncurried)\n\n-- data UnorderedTree \u03b1 where\n--   | node : \u03b1 \u2192 MultiSet (UnorderedTree \u03b1) \u2192 UnorderedTree \u03b1\n\n\n#print QpfList\n\n\ndef QpfList.isNil : QpfList \u03b1 \u2192 Bool := \n  MvQPF.Fix.rec fun as => match as with\n    | .nil => true\n    | _    => false\n\ndef QpfList.isCons : QpfList \u03b1 \u2192 Bool := \n  fun as => match as.dest with\n    | .cons .. => true\n    | _        => false\n\ndef QpfList.length : QpfList \u03b1 \u2192 Nat :=\n  MvQPF.Fix.rec fun as => match as with\n    | .nil                => 0\n    | .cons a (as : Nat)  => as + 1 \n\n\n\ninductive QpfListInd \u03b1\n  | nil\n  | cons : \u03b1 \u2192 QpfListInd \u03b1 \u2192 QpfListInd \u03b1\n\n #check @QpfListInd.casesOn\n #check @QpfListInd.recOn\n #check @QpfListInd.rec\n #check QpfListInd.noConfusion (\u03b1:=Nat)\n\n\ncodata QpfStream \u03b1 where\n  | mk : \u03b1 \u2192 QpfStream \u03b1 \u2192 QpfStream \u03b1\n\n#print QpfStream.mk\n\n\n/-- The stream `0,0,0,...` -/\ndef QpfStream.zeroes : QpfStream Nat :=\n  MvQPF.Cofix.corec (fun _ => \n    Shape.mk (0 : Nat) ()\n  ) ()\n\n/-- The stream `0,1,2,3,4,...` -/\ndef QpfStream.naturals : QpfStream Nat :=\n  MvQPF.Cofix.corec (fun (i : Nat) => \n    Shape.mk (i : Nat) (i + 1 : Nat)\n  ) 0\n\n\n/-- Add two streams together -/\ndef QpfStream.add (as bs : QpfStream Nat) : QpfStream Nat :=\n    MvQPF.Cofix.corec (fun \u27e8as, bs\u27e9 => \n      let \u27e8(a : Nat), as\u27e9 := MvQPF.Cofix.dest as;\n      let \u27e8(b : Nat), bs\u27e9 := MvQPF.Cofix.dest bs;\n      Shape.mk (a + b : Nat) (as, bs)\n    ) (as, bs)\n\n\n-- data QpfListPair \u03b1 \u03b2 where\n--   | nil_nil\n--   | cons_nil  : \u03b1 \u2192 QpfListPair \u03b1 \u03b2\n--   | nil_cons  : \u03b2 \u2192 QpfListPair \u03b1 \u03b2\n--   | cons_cons : \u03b1 \u2192 \u03b2 \u2192 QpfListPair \u03b1 \u03b2\n\n-- def QpfListPair.of (as : QpfList \u03b1) (bs : QpfList \u03b2) : QpfListPair \u03b1 \u03b2\n\n\n\n\n\n\n\n\n\n\n\n\n\n#print QpfList.Uncurried\n#print QpfList.Shape\n#print QpfList.Shape.P\n\n  \n\ndata QpfList\u2082 \u03b1 where\n  | My.nil : QpfList\u2082 \u03b1\n  | My2.nil : \u03b1 \u2192 QpfList\u2082 \u03b1\n\n#check QpfList\u2082\n\ndata QpfList\u2083 \u03b1 where\n  | My.nil : QpfList\u2083 \u03b1\n  | My2.nil : \u03b1 \u2192 QpfList\u2083 \u03b1 \u2192 QpfList\u2083 \u03b1\n\n\n\n#check QpfList\n\n\nnamespace Quotient\n  data List' \u03b1\n  | nil \n  | cons : \u03b1 \u2192 List' \u03b1 \u2192 List' \u03b1\n\n  #print List'\n\n  def List'.perm \u2983\u03b1\u2984: Quotient.List'.Uncurried \u03b1 \u2192 Quotient.List'.Uncurried \u03b1 \u2192 Prop\n    := by sorry\n\n  abbrev Multiset' : TypeFun 1 := MvQPF.Quot1 List'.perm\n  abbrev Multiset  := Multiset'.curried\n\n  noncomputable instance : MvQPF Multiset' := \n    MvQPF.relQuot _ (\n      by\n        intros\n        sorry\n    )\n\n\n  -- noncomputable data Foo \u03b1 where\n  --   | node : \u03b1 \u2192 Multiset (Foo \u03b1) \u2192 Foo \u03b1\n\n  -- #print Foo\n  -- #print Quotient.Foo.node\n\n\n\n  def List.perm : List \u03b1 \u2192 List \u03b1 \u2192 Prop\n    := by sorry\n\n  def NativeMultiset \u03b1 := Quot.mk (@List.perm \u03b1)\nend Quotient\n\n\n/-\n  # Composition pipeline\n-/\n\nqpf P\u2081 \u03b1 \u03b2 := \u03b1\nqpf P\u2082 \u03b1 \u03b2 := \u03b2\n\nqpf C\u2081 \u03b1 \u03b2 := Nat\nqpf C\u2082 (n : Nat) \u03b1 \u03b2 := PFin2 n\n\nqpf G\u2084 \u03b1 \u03b2 \u03c1 := QpfList \u03c1\n\n\n/-\n  # Dead variables\n-/\n\ndata Arrow (\u03b1 : Type _) \u03b2\n  | mk : (\u03b1 \u2192 \u03b2) \u2192 Arrow \u03b1 \u03b2\n\ncodata FinAlt {n : Nat} \u03b2\n  | mk : PFin2 n \u2192 FinAlt \u03b2\n\n#print FinAlt.mk\n\n\ndata QpfList\u2084 (dead : Type) \u03b2 \u03b3 where\n  | nil   : QpfList\u2084 dead \u03b2 \u03b3\n  | cons  : QpfList\u2084 dead \u03b2 \u03b3 \u2192 QpfList\u2084 dead \u03b2 \u03b3\n\n\n\n/-\n  # Regression tests\n\n  These cases failed at some point, include them here as a regression test\n-/\n\ndata PairOf \u03b1 \u03b2\n  | mk : \u03b1 \u2192 \u03b2 \u2192 \u03b2 \u2192 PairOf \u03b1 \u03b2\n\ndata QpfTest \u03b1 \u03b2 where\n  | A : \u03b1 \u2192 \u03b1 \u2192 \u03b2 \u2192 QpfTest \u03b1 \u03b2 \u2192 QpfTree \u03b2 \u2192 QpfCoTree (QpfTree (QpfTest \u03b1 \u03b2)) \u2192 QpfTest \u03b1 \u03b2\n\ndata RepAfterConst \u03b2\n  | mk : Nat \u2192  \u03b2 \u2192 \u03b2 \u2192 RepAfterConst \u03b2\n\ncodata NatList \u03b1 where\n  | nil : NatList \u03b1\n  | cons : Nat \u2192 NatList \u03b1 \u2192 NatList \u03b1\n\ndata QpfList\u2085 (A : Type) (dead : Type) \u03b2 where\n  | nil   : QpfList\u2085 A dead \u03b2\n  | cons  : A \u2192 (dead \u2192 \u03b2) \u2192 QpfList\u2085 A dead \u03b2 \u2192 QpfList\u2085 A dead \u03b2", "meta": {"author": "alexkeizer", "repo": "qpf4", "sha": "980f97425b9d5a5e3897073df33794192b3b3124", "save_path": "github-repos/lean/alexkeizer-qpf4", "path": "github-repos/lean/alexkeizer-qpf4/qpf4-980f97425b9d5a5e3897073df33794192b3b3124/Qpf/Macro/Data/Example.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.38725562537383623}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Mon.basic\nimport Mathlib.category_theory.endomorphism\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Category instances for group, add_group, comm_group, and add_comm_group.\n\nWe introduce the bundled categories:\n* `Group`\n* `AddGroup`\n* `CommGroup`\n* `AddCommGroup`\nalong with the relevant forgetful functors between them, and to the bundled monoid categories.\n-/\n\n/-- The category of groups and group morphisms. -/\ndef AddGroup := category_theory.bundled add_group\n\n/-- The category of additive groups and group morphisms -/\nnamespace Group\n\n\nprotected instance Mathlib.AddGroup.group.to_monoid.category_theory.bundled_hom.parent_projection :\n    category_theory.bundled_hom.parent_projection add_group.to_add_monoid :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance has_coe_to_sort : has_coe_to_sort Group :=\n  category_theory.bundled.has_coe_to_sort\n\n/-- Construct a bundled `Group` from the underlying type and typeclass. -/\ndef of (X : Type u) [group X] : Group := category_theory.bundled.of X\n\n/-- Construct a bundled `AddGroup` from the underlying type and typeclass. -/\nprotected instance group (G : Group) : group \u21a5G := category_theory.bundled.str G\n\n@[simp] theorem coe_of (R : Type u) [group R] : \u21a5(of R) = R := rfl\n\nprotected instance Mathlib.AddGroup.has_zero : HasZero AddGroup := { zero := AddGroup.of PUnit }\n\nprotected instance inhabited : Inhabited Group := { default := 1 }\n\nprotected instance one.unique : unique \u21a51 := unique.mk { default := 1 } sorry\n\n@[simp] theorem one_apply (G : Group) (H : Group) (g : \u21a5G) : coe_fn 1 g = 1 := rfl\n\ntheorem ext (G : Group) (H : Group) (f\u2081 : G \u27f6 H) (f\u2082 : G \u27f6 H)\n    (w : \u2200 (x : \u21a5G), coe_fn f\u2081 x = coe_fn f\u2082 x) : f\u2081 = f\u2082 :=\n  monoid_hom.ext fun (x : \u21a5G) => w x\n\n-- should to_additive do this automatically?\n\nprotected instance Mathlib.AddGroup.has_forget_to_AddMon :\n    category_theory.has_forget\u2082 AddGroup AddMon :=\n  category_theory.bundled_hom.forget\u2082 add_monoid_hom add_group.to_add_monoid\n\nend Group\n\n\n/-- The category of commutative groups and group morphisms. -/\ndef AddCommGroup := category_theory.bundled add_comm_group\n\n/-- The category of additive commutative groups and group morphisms. -/\n/-- `Ab` is an abbreviation for `AddCommGroup`, for the sake of mathematicians' sanity. -/\ndef Ab := AddCommGroup\n\nnamespace CommGroup\n\n\nprotected instance comm_group.to_group.category_theory.bundled_hom.parent_projection :\n    category_theory.bundled_hom.parent_projection comm_group.to_group :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance large_category : category_theory.large_category CommGroup :=\n  category_theory.bundled_hom.category\n    (category_theory.bundled_hom.map_hom\n      (category_theory.bundled_hom.map_hom monoid_hom group.to_monoid) comm_group.to_group)\n\n/-- Construct a bundled `CommGroup` from the underlying type and typeclass. -/\ndef of (G : Type u) [comm_group G] : CommGroup := category_theory.bundled.of G\n\n/-- Construct a bundled `AddCommGroup` from the underlying type and typeclass. -/\nprotected instance Mathlib.AddCommGroup.add_comm_group_instance (G : AddCommGroup) :\n    add_comm_group \u21a5G :=\n  category_theory.bundled.str G\n\n@[simp] theorem coe_of (R : Type u) [comm_group R] : \u21a5(of R) = R := rfl\n\nprotected instance Mathlib.AddCommGroup.has_zero : HasZero AddCommGroup :=\n  { zero := AddCommGroup.of PUnit }\n\nprotected instance inhabited : Inhabited CommGroup := { default := 1 }\n\nprotected instance one.unique : unique \u21a51 := unique.mk { default := 1 } sorry\n\n@[simp] theorem one_apply (G : CommGroup) (H : CommGroup) (g : \u21a5G) : coe_fn 1 g = 1 := rfl\n\ntheorem ext (G : CommGroup) (H : CommGroup) (f\u2081 : G \u27f6 H) (f\u2082 : G \u27f6 H)\n    (w : \u2200 (x : \u21a5G), coe_fn f\u2081 x = coe_fn f\u2082 x) : f\u2081 = f\u2082 :=\n  monoid_hom.ext fun (x : \u21a5G) => w x\n\nprotected instance Mathlib.AddCommGroup.has_forget_to_AddGroup :\n    category_theory.has_forget\u2082 AddCommGroup AddGroup :=\n  category_theory.bundled_hom.forget\u2082\n    (category_theory.bundled_hom.map_hom add_monoid_hom add_group.to_add_monoid)\n    add_comm_group.to_add_group\n\nprotected instance Mathlib.AddCommGroup.has_forget_to_AddCommMon :\n    category_theory.has_forget\u2082 AddCommGroup AddCommMon :=\n  category_theory.induced_category.has_forget\u2082 fun (G : AddCommGroup) => AddCommMon.of \u21a5G\n\nend CommGroup\n\n\n-- This example verifies an improvement possible in Lean 3.8.\n\n-- Before that, to have `monoid_hom.map_map` usable by `simp` here,\n\n-- we had to mark all the concrete category `has_coe_to_sort` instances reducible.\n\n-- Now, it just works.\n\nnamespace AddCommGroup\n\n\n/-- Any element of an abelian group gives a unique morphism from `\u2124` sending\n`1` to that element. -/\n-- Note that because `\u2124 : Type 0`, this forces `G : AddCommGroup.{0}`,\n\n-- so we write this explicitly to be clear.\n\n-- TODO generalize this, requiring a `ulift_instances.lean` file\n\ndef as_hom {G : AddCommGroup} (g : \u21a5G) : of \u2124 \u27f6 G := coe_fn (gmultiples_hom \u21a5G) g\n\n@[simp] theorem as_hom_apply {G : AddCommGroup} (g : \u21a5G) (i : \u2124) : coe_fn (as_hom g) i = i \u2022 g :=\n  rfl\n\ntheorem as_hom_injective {G : AddCommGroup} : function.injective as_hom := sorry\n\ntheorem int_hom_ext {G : AddCommGroup} (f : of \u2124 \u27f6 G) (g : of \u2124 \u27f6 G) (w : coe_fn f 1 = coe_fn g 1) :\n    f = g :=\n  add_monoid_hom.ext_int w\n\n-- TODO: this argument should be generalised to the situation where\n\n-- the forgetful functor is representable.\n\ntheorem injective_of_mono {G : AddCommGroup} {H : AddCommGroup} (f : G \u27f6 H)\n    [category_theory.mono f] : function.injective \u21d1f :=\n  sorry\n\nend AddCommGroup\n\n\n/-- Build an isomorphism in the category `Group` from a `mul_equiv` between `group`s. -/\ndef mul_equiv.to_Group_iso {X : Type u} {Y : Type u} [group X] [group Y] (e : X \u2243* Y) :\n    Group.of X \u2245 Group.of Y :=\n  category_theory.iso.mk (mul_equiv.to_monoid_hom e) (mul_equiv.to_monoid_hom (mul_equiv.symm e))\n\n/-- Build an isomorphism in the category `AddGroup` from an `add_equiv` between `add_group`s. -/\n/-- Build an isomorphism in the category `CommGroup` from a `mul_equiv` between `comm_group`s. -/\ndef add_equiv.to_AddCommGroup_iso {X : Type u} {Y : Type u} [add_comm_group X] [add_comm_group Y]\n    (e : X \u2243+ Y) : AddCommGroup.of X \u2245 AddCommGroup.of Y :=\n  category_theory.iso.mk (add_equiv.to_add_monoid_hom e)\n    (add_equiv.to_add_monoid_hom (add_equiv.symm e))\n\n/-- Build an isomorphism in the category `AddCommGroup` from a `add_equiv` between\n`add_comm_group`s. -/\nnamespace category_theory.iso\n\n\n/-- Build a `mul_equiv` from an isomorphism in the category `Group`. -/\n@[simp] theorem Group_iso_to_add_equiv_apply {X : AddGroup} {Y : AddGroup} (i : X \u2245 Y) :\n    \u2200 (\u1fb0 : \u21a5X), coe_fn (AddGroup_iso_to_add_equiv i) \u1fb0 = coe_fn (hom i) \u1fb0 :=\n  fun (\u1fb0 : \u21a5X) => Eq.refl (coe_fn (hom i) \u1fb0)\n\n/-- Build a `mul_equiv` from an isomorphism in the category `CommGroup`. -/\n@[simp] theorem CommGroup_iso_to_add_equiv_apply {X : AddCommGroup} {Y : AddCommGroup} (i : X \u2245 Y) :\n    \u2200 (\u1fb0 : \u21a5X), coe_fn (AddCommGroup_iso_to_add_equiv i) \u1fb0 = coe_fn (hom i) \u1fb0 :=\n  fun (\u1fb0 : \u21a5X) => Eq.refl (coe_fn (hom i) \u1fb0)\n\nend category_theory.iso\n\n\n/-- multiplicative equivalences between `group`s are the same as (isomorphic to) isomorphisms\nin `Group` -/\ndef add_equiv_iso_AddGroup_iso {X : Type u} {Y : Type u} [add_group X] [add_group Y] :\n    X \u2243+ Y \u2245 AddGroup.of X \u2245 AddGroup.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243+ Y) => add_equiv.to_AddGroup_iso e)\n    fun (i : AddGroup.of X \u2245 AddGroup.of Y) => category_theory.iso.AddGroup_iso_to_add_equiv i\n\n/-- multiplicative equivalences between `comm_group`s are the same as (isomorphic to) isomorphisms\nin `CommGroup` -/\ndef mul_equiv_iso_CommGroup_iso {X : Type u} {Y : Type u} [comm_group X] [comm_group Y] :\n    X \u2243* Y \u2245 CommGroup.of X \u2245 CommGroup.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243* Y) => mul_equiv.to_CommGroup_iso e)\n    fun (i : CommGroup.of X \u2245 CommGroup.of Y) => category_theory.iso.CommGroup_iso_to_mul_equiv i\n\nnamespace category_theory.Aut\n\n\n/-- The (bundled) group of automorphisms of a type is isomorphic to the (bundled) group\nof permutations. -/\ndef iso_perm {\u03b1 : Type u} : Group.of (Aut \u03b1) \u2245 Group.of (equiv.perm \u03b1) :=\n  iso.mk (monoid_hom.mk (fun (g : \u21a5(Group.of (Aut \u03b1))) => iso.to_equiv g) sorry sorry)\n    (monoid_hom.mk (fun (g : \u21a5(Group.of (equiv.perm \u03b1))) => equiv.to_iso g) sorry sorry)\n\n/-- The (unbundled) group of automorphisms of a type is `mul_equiv` to the (unbundled) group\nof permutations. -/\ndef mul_equiv_perm {\u03b1 : Type u} : Aut \u03b1 \u2243* equiv.perm \u03b1 := iso.Group_iso_to_mul_equiv iso_perm\n\nend category_theory.Aut\n\n\nprotected instance Group.forget_reflects_isos :\n    category_theory.reflects_isomorphisms (category_theory.forget Group) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : Group) (f : X \u27f6 Y)\n      (_x :\n      category_theory.is_iso (category_theory.functor.map (category_theory.forget Group) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget Group) X \u2245\n          category_theory.functor.obj (category_theory.forget Group) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget Group) f);\n      let e : \u21a5X \u2243* \u21a5Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry\n          sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_Group_iso e))\n\nprotected instance CommGroup.forget_reflects_isos :\n    category_theory.reflects_isomorphisms (category_theory.forget CommGroup) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : CommGroup) (f : X \u27f6 Y)\n      (_x :\n      category_theory.is_iso (category_theory.functor.map (category_theory.forget CommGroup) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget CommGroup) X \u2245\n          category_theory.functor.obj (category_theory.forget CommGroup) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget CommGroup) f);\n      let e : \u21a5X \u2243* \u21a5Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry\n          sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_CommGroup_iso e))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Group/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341026367784, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3872556169121264}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.basic\nimport Mathlib.data.set.finite\nimport Mathlib.group_theory.submonoid.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Pointwise addition, multiplication, and scalar multiplication of sets.\n\nThis file defines pointwise algebraic operations on sets.\n* For a type `\u03b1` with multiplication, multiplication is defined on `set \u03b1` by taking\n  `s * t` to be the set of all `x * y` where `x \u2208 s` and `y \u2208 t`. Similarly for addition.\n* For `\u03b1` a semigroup, `set \u03b1` is a semigroup.\n* If `\u03b1` is a (commutative) monoid, we define an alias `set_semiring \u03b1` for `set \u03b1`, which then\n  becomes a (commutative) semiring with union as addition and pointwise multiplication as\n  multiplication.\n* For a type `\u03b2` with scalar multiplication by another type `\u03b1`, this\n  file defines a scalar multiplication of `set \u03b2` by `set \u03b1` and a separate scalar\n  multiplication of `set \u03b2` by `\u03b1`.\n* We also define pointwise multiplication on `finset`.\n\nAppropriate definitions and results are also transported to the additive theory via `to_additive`.\n\n## Implementation notes\n* The following expressions are considered in simp-normal form in a group:\n  `(\u03bb h, h * g) \u207b\u00b9' s`, `(\u03bb h, g * h) \u207b\u00b9' s`, `(\u03bb h, h * g\u207b\u00b9) \u207b\u00b9' s`, `(\u03bb h, g\u207b\u00b9 * h) \u207b\u00b9' s`,\n  `s * t`, `s\u207b\u00b9`, `(1 : set _)` (and similarly for additive variants).\n  Expressions equal to one of these will be simplified.\n\n## Tags\n\nset multiplication, set addition, pointwise addition, pointwise multiplication\n\n-/\n\nnamespace set\n\n\n/-! ### Properties about 1 -/\n\nprotected instance has_one {\u03b1 : Type u_1} [HasOne \u03b1] : HasOne (set \u03b1) :=\n  { one := singleton 1 }\n\ntheorem singleton_one {\u03b1 : Type u_1} [HasOne \u03b1] : singleton 1 = 1 :=\n  rfl\n\n@[simp] theorem mem_zero {\u03b1 : Type u_1} {a : \u03b1} [HasZero \u03b1] : a \u2208 0 \u2194 a = 0 :=\n  iff.rfl\n\ntheorem one_mem_one {\u03b1 : Type u_1} [HasOne \u03b1] : 1 \u2208 1 :=\n  Eq.refl 1\n\n@[simp] theorem zero_subset {\u03b1 : Type u_1} {s : set \u03b1} [HasZero \u03b1] : 0 \u2286 s \u2194 0 \u2208 s :=\n  singleton_subset_iff\n\ntheorem zero_nonempty {\u03b1 : Type u_1} [HasZero \u03b1] : set.nonempty 0 :=\n  Exists.intro 0 rfl\n\n@[simp] theorem image_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasZero \u03b1] {f : \u03b1 \u2192 \u03b2} : f '' 0 = singleton (f 0) :=\n  image_singleton\n\n/-! ### Properties about multiplication -/\n\nprotected instance has_add {\u03b1 : Type u_1} [Add \u03b1] : Add (set \u03b1) :=\n  { add := image2 Add.add }\n\n@[simp] theorem image2_mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] : image2 Mul.mul s t = s * t :=\n  rfl\n\ntheorem mem_add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {a : \u03b1} [Add \u03b1] : a \u2208 s + t \u2194 \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2208 s \u2227 y \u2208 t \u2227 x + y = a :=\n  iff.rfl\n\ntheorem mul_mem_mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {a : \u03b1} {b : \u03b1} [Mul \u03b1] (ha : a \u2208 s) (hb : b \u2208 t) : a * b \u2208 s * t :=\n  mem_image2_of_mem ha hb\n\ntheorem add_image_prod {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] : (fun (x : \u03b1 \u00d7 \u03b1) => prod.fst x + prod.snd x) '' set.prod s t = s + t :=\n  image_prod Add.add\n\n@[simp] theorem image_mul_left {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [group \u03b1] : (fun (b : \u03b1) => a * b) '' t = (fun (b : \u03b1) => a\u207b\u00b9 * b) \u207b\u00b9' t := sorry\n\n@[simp] theorem image_add_right {\u03b1 : Type u_1} {t : set \u03b1} {b : \u03b1} [add_group \u03b1] : (fun (a : \u03b1) => a + b) '' t = (fun (a : \u03b1) => a + -b) \u207b\u00b9' t := sorry\n\ntheorem image_add_left' {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [add_group \u03b1] : (fun (b : \u03b1) => -a + b) '' t = (fun (b : \u03b1) => a + b) \u207b\u00b9' t := sorry\n\ntheorem image_mul_right' {\u03b1 : Type u_1} {t : set \u03b1} {b : \u03b1} [group \u03b1] : (fun (a : \u03b1) => a * (b\u207b\u00b9)) '' t = (fun (a : \u03b1) => a * b) \u207b\u00b9' t := sorry\n\n@[simp] theorem preimage_add_left_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [add_group \u03b1] : Add.add a \u207b\u00b9' singleton b = singleton (-a + b) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (Add.add a \u207b\u00b9' singleton b = singleton (-a + b))) (Eq.symm image_add_left')))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((fun (b : \u03b1) => -a + b) '' singleton b = singleton (-a + b))) image_singleton))\n      (Eq.refl (singleton (-a + b))))\n\n@[simp] theorem preimage_mul_right_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [group \u03b1] : (fun (_x : \u03b1) => _x * a) \u207b\u00b9' singleton b = singleton (b * (a\u207b\u00b9)) := sorry\n\n@[simp] theorem preimage_add_left_zero {\u03b1 : Type u_1} {a : \u03b1} [add_group \u03b1] : (fun (b : \u03b1) => a + b) \u207b\u00b9' 0 = singleton (-a) := sorry\n\n@[simp] theorem preimage_mul_right_one {\u03b1 : Type u_1} {b : \u03b1} [group \u03b1] : (fun (a : \u03b1) => a * b) \u207b\u00b9' 1 = singleton (b\u207b\u00b9) := sorry\n\ntheorem preimage_add_left_zero' {\u03b1 : Type u_1} {a : \u03b1} [add_group \u03b1] : (fun (b : \u03b1) => -a + b) \u207b\u00b9' 0 = singleton a := sorry\n\ntheorem preimage_add_right_zero' {\u03b1 : Type u_1} {b : \u03b1} [add_group \u03b1] : (fun (a : \u03b1) => a + -b) \u207b\u00b9' 0 = singleton b := sorry\n\n@[simp] theorem mul_singleton {\u03b1 : Type u_1} {s : set \u03b1} {b : \u03b1} [Mul \u03b1] : s * singleton b = (fun (a : \u03b1) => a * b) '' s :=\n  image2_singleton_right\n\n@[simp] theorem singleton_add {\u03b1 : Type u_1} {t : set \u03b1} {a : \u03b1} [Add \u03b1] : singleton a + t = (fun (b : \u03b1) => a + b) '' t :=\n  image2_singleton_left\n\n@[simp] theorem singleton_add_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} [Add \u03b1] : singleton a + singleton b = singleton (a + b) :=\n  image2_singleton\n\nprotected instance semigroup {\u03b1 : Type u_1} [semigroup \u03b1] : semigroup (set \u03b1) :=\n  semigroup.mk Mul.mul sorry\n\nprotected instance monoid {\u03b1 : Type u_1} [monoid \u03b1] : monoid (set \u03b1) :=\n  monoid.mk semigroup.mul sorry 1 sorry sorry\n\nprotected theorem mul_comm {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [comm_semigroup \u03b1] : s * t = t * s := sorry\n\nprotected instance add_comm_monoid {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : add_comm_monoid (set \u03b1) :=\n  add_comm_monoid.mk add_monoid.add sorry add_monoid.zero sorry sorry sorry\n\ntheorem singleton.is_mul_hom {\u03b1 : Type u_1} [Mul \u03b1] : is_mul_hom singleton :=\n  is_mul_hom.mk fun (a b : \u03b1) => Eq.symm singleton_mul_singleton\n\n@[simp] theorem empty_add {\u03b1 : Type u_1} {s : set \u03b1} [Add \u03b1] : \u2205 + s = \u2205 :=\n  image2_empty_left\n\n@[simp] theorem mul_empty {\u03b1 : Type u_1} {s : set \u03b1} [Mul \u03b1] : s * \u2205 = \u2205 :=\n  image2_empty_right\n\ntheorem add_subset_add {\u03b1 : Type u_1} {s\u2081 : set \u03b1} {s\u2082 : set \u03b1} {t\u2081 : set \u03b1} {t\u2082 : set \u03b1} [Add \u03b1] (h\u2081 : s\u2081 \u2286 t\u2081) (h\u2082 : s\u2082 \u2286 t\u2082) : s\u2081 + s\u2082 \u2286 t\u2081 + t\u2082 :=\n  image2_subset h\u2081 h\u2082\n\ntheorem union_add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {u : set \u03b1} [Add \u03b1] : s \u222a t + u = s + u \u222a (t + u) :=\n  image2_union_left\n\ntheorem mul_union {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} {u : set \u03b1} [Mul \u03b1] : s * (t \u222a u) = s * t \u222a s * u :=\n  image2_union_right\n\ntheorem Union_mul_left_image {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] : (Union fun (a : \u03b1) => Union fun (H : a \u2208 s) => (fun (x : \u03b1) => a * x) '' t) = s * t :=\n  Union_image_left fun (a x : \u03b1) => a * x\n\ntheorem Union_mul_right_image {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] : (Union fun (a : \u03b1) => Union fun (H : a \u2208 t) => (fun (x : \u03b1) => x * a) '' s) = s * t :=\n  Union_image_right fun (x a : \u03b1) => x * a\n\n@[simp] theorem univ_mul_univ {\u03b1 : Type u_1} [monoid \u03b1] : univ * univ = univ := sorry\n\n/-- `singleton` is a monoid hom. -/\ndef singleton_add_hom {\u03b1 : Type u_1} [add_monoid \u03b1] : \u03b1 \u2192+ set \u03b1 :=\n  add_monoid_hom.mk singleton sorry sorry\n\ntheorem nonempty.add {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] : set.nonempty s \u2192 set.nonempty t \u2192 set.nonempty (s + t) :=\n  nonempty.image2\n\ntheorem finite.mul {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Mul \u03b1] (hs : finite s) (ht : finite t) : finite (s * t) :=\n  finite.image2 (fun (a b : \u03b1) => a * b) hs ht\n\n/-- multiplication preserves finiteness -/\ndef fintype_mul {\u03b1 : Type u_1} [Mul \u03b1] [DecidableEq \u03b1] (s : set \u03b1) (t : set \u03b1) [hs : fintype \u21a5s] [ht : fintype \u21a5t] : fintype \u21a5(s * t) :=\n  set.fintype_image2 (fun (a b : \u03b1) => a * b) s t\n\ntheorem bdd_above_add {\u03b1 : Type u_1} [ordered_add_comm_monoid \u03b1] {A : set \u03b1} {B : set \u03b1} : bdd_above A \u2192 bdd_above B \u2192 bdd_above (A + B) := sorry\n\n/-! ### Properties about inversion -/\n\nprotected instance has_inv {\u03b1 : Type u_1} [has_inv \u03b1] : has_inv (set \u03b1) :=\n  has_inv.mk (preimage has_inv.inv)\n\n@[simp] theorem mem_inv {\u03b1 : Type u_1} {s : set \u03b1} {a : \u03b1} [has_inv \u03b1] : a \u2208 (s\u207b\u00b9) \u2194 a\u207b\u00b9 \u2208 s :=\n  iff.rfl\n\ntheorem inv_mem_inv {\u03b1 : Type u_1} {s : set \u03b1} {a : \u03b1} [group \u03b1] : a\u207b\u00b9 \u2208 (s\u207b\u00b9) \u2194 a \u2208 s := sorry\n\n@[simp] theorem inv_preimage {\u03b1 : Type u_1} {s : set \u03b1} [has_inv \u03b1] : has_inv.inv \u207b\u00b9' s = (s\u207b\u00b9) :=\n  rfl\n\n@[simp] theorem image_inv {\u03b1 : Type u_1} {s : set \u03b1} [group \u03b1] : has_inv.inv '' s = (s\u207b\u00b9) := sorry\n\n@[simp] theorem inter_neg {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Neg \u03b1] : -(s \u2229 t) = -s \u2229 -t :=\n  preimage_inter\n\n@[simp] theorem union_neg {\u03b1 : Type u_1} {s : set \u03b1} {t : set \u03b1} [Neg \u03b1] : -(s \u222a t) = -s \u222a -t :=\n  preimage_union\n\n@[simp] theorem compl_inv {\u03b1 : Type u_1} {s : set \u03b1} [has_inv \u03b1] : s\u1d9c\u207b\u00b9 = (s\u207b\u00b9\u1d9c) :=\n  preimage_compl\n\n@[simp] protected theorem inv_inv {\u03b1 : Type u_1} {s : set \u03b1} [group \u03b1] : s\u207b\u00b9\u207b\u00b9 = s := sorry\n\n@[simp] protected theorem univ_inv {\u03b1 : Type u_1} [group \u03b1] : univ\u207b\u00b9 = univ :=\n  preimage_univ\n\n@[simp] theorem neg_subset_neg {\u03b1 : Type u_1} [add_group \u03b1] {s : set \u03b1} {t : set \u03b1} : -s \u2286 -t \u2194 s \u2286 t :=\n  function.surjective.preimage_subset_preimage_iff (equiv.surjective (equiv.neg \u03b1))\n\ntheorem neg_subset {\u03b1 : Type u_1} [add_group \u03b1] {s : set \u03b1} {t : set \u03b1} : -s \u2286 t \u2194 s \u2286 -t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (-s \u2286 t \u2194 s \u2286 -t)) (Eq.symm (propext neg_subset_neg))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ( --s \u2286 -t \u2194 s \u2286 -t)) set.neg_neg)) (iff.refl (s \u2286 -t)))\n\n/-! ### Properties about scalar multiplication -/\n\n/-- Scaling a set: multiplying every element by a scalar. -/\nprotected instance has_scalar_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] : has_scalar \u03b1 (set \u03b2) :=\n  has_scalar.mk fun (a : \u03b1) => image (has_scalar.smul a)\n\n@[simp] theorem image_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : (fun (x : \u03b2) => a \u2022 x) '' t = a \u2022 t :=\n  rfl\n\ntheorem mem_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} {x : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : x \u2208 a \u2022 t \u2194 \u2203 (y : \u03b2), y \u2208 t \u2227 a \u2022 y = x :=\n  iff.rfl\n\ntheorem smul_mem_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} {y : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2} (hy : y \u2208 t) : a \u2022 y \u2208 a \u2022 t :=\n  Exists.intro y { left := hy, right := rfl }\n\ntheorem smul_set_union {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {s : set \u03b2} {t : set \u03b2} : a \u2022 (s \u222a t) = a \u2022 s \u222a a \u2022 t := sorry\n\n@[simp] theorem smul_set_empty {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] (a : \u03b1) : a \u2022 \u2205 = \u2205 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a \u2022 \u2205 = \u2205)) (Eq.symm image_smul)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((fun (x : \u03b2) => a \u2022 x) '' \u2205 = \u2205)) (image_empty fun (x : \u03b2) => a \u2022 x))) (Eq.refl \u2205))\n\ntheorem smul_set_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {s : set \u03b2} {t : set \u03b2} (h : s \u2286 t) : a \u2022 s \u2286 a \u2022 t := sorry\n\n/-- Pointwise scalar multiplication by a set of scalars. -/\nprotected instance has_scalar {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] : has_scalar (set \u03b1) (set \u03b2) :=\n  has_scalar.mk (image2 has_scalar.smul)\n\n@[simp] theorem image2_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : image2 has_scalar.smul s t = s \u2022 t :=\n  rfl\n\ntheorem mem_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} {x : \u03b2} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : x \u2208 s \u2022 t \u2194 \u2203 (a : \u03b1), \u2203 (y : \u03b2), a \u2208 s \u2227 y \u2208 t \u2227 a \u2022 y = x :=\n  iff.rfl\n\ntheorem image_smul_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : (fun (x : \u03b1 \u00d7 \u03b2) => prod.fst x \u2022 prod.snd x) '' set.prod s t = s \u2022 t :=\n  image_prod has_scalar.smul\n\ntheorem range_smul_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [has_scalar \u03b1 \u03b2] {\u03b9 : Type u_3} {\u03ba : Type u_4} (b : \u03b9 \u2192 \u03b1) (c : \u03ba \u2192 \u03b2) : range b \u2022 range c = range fun (p : \u03b9 \u00d7 \u03ba) => b (prod.fst p) \u2022 c (prod.snd p) := sorry\n\ntheorem singleton_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} {a : \u03b1} [has_scalar \u03b1 \u03b2] {t : set \u03b2} : singleton a \u2022 t = a \u2022 t :=\n  image2_singleton_left\n\n/-! ### `set \u03b1` as a `(\u222a,*)`-semiring -/\n\n/-- An alias for `set \u03b1`, which has a semiring structure given by `\u222a` as \"addition\" and pointwise\n  multiplication `*` as \"multiplication\". -/\ndef set_semiring (\u03b1 : Type u_1) :=\n  set \u03b1\n\n/-- The identitiy function `set \u03b1 \u2192 set_semiring \u03b1`. -/\n/-- The identitiy function `set_semiring \u03b1 \u2192 set \u03b1`. -/\nprotected def up {\u03b1 : Type u_1} (s : set \u03b1) : set_semiring \u03b1 :=\n  s\n\nprotected def set_semiring.down {\u03b1 : Type u_1} (s : set_semiring \u03b1) : set \u03b1 :=\n  s\n\n@[simp] protected theorem down_up {\u03b1 : Type u_1} {s : set \u03b1} : set_semiring.down (set.up s) = s :=\n  rfl\n\n@[simp] protected theorem up_down {\u03b1 : Type u_1} {s : set_semiring \u03b1} : set.up (set_semiring.down s) = s :=\n  rfl\n\nprotected instance set_semiring.semiring {\u03b1 : Type u_1} [monoid \u03b1] : semiring (set_semiring \u03b1) :=\n  semiring.mk (fun (s t : set_semiring \u03b1) => s \u222a t) union_assoc \u2205 empty_union union_empty union_comm monoid.mul sorry\n    monoid.one sorry sorry sorry sorry sorry sorry\n\nprotected instance set_semiring.comm_semiring {\u03b1 : Type u_1} [comm_monoid \u03b1] : comm_semiring (set_semiring \u03b1) :=\n  comm_semiring.mk semiring.add sorry semiring.zero sorry sorry sorry comm_monoid.mul sorry comm_monoid.one sorry sorry\n    sorry sorry sorry sorry sorry\n\n/-- A multiplicative action of a monoid on a type \u03b2 gives also a\n multiplicative action on the subsets of \u03b2. -/\nprotected instance mul_action_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [monoid \u03b1] [mul_action \u03b1 \u03b2] : mul_action \u03b1 (set \u03b2) :=\n  mul_action.mk sorry sorry\n\ntheorem image_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : set \u03b1} {t : set \u03b1} [Add \u03b1] [Add \u03b2] (m : \u03b1 \u2192 \u03b2) [is_add_hom m] : m '' (s + t) = m '' s + m '' t := sorry\n\ntheorem preimage_mul_preimage_subset {\u03b1 : Type u_1} {\u03b2 : Type u_2} [Mul \u03b1] [Mul \u03b2] (m : \u03b1 \u2192 \u03b2) [is_mul_hom m] {s : set \u03b2} {t : set \u03b2} : m \u207b\u00b9' s * m \u207b\u00b9' t \u2286 m \u207b\u00b9' (s * t) := sorry\n\n/-- The image of a set under function is a ring homomorphism\nwith respect to the pointwise operations on sets. -/\ndef image_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192* \u03b2) : set_semiring \u03b1 \u2192+* set_semiring \u03b2 :=\n  ring_hom.mk (image \u21d1f) sorry sorry sorry sorry\n\nend set\n\n\n/-- A nonempty set in a semimodule is scaled by zero to the singleton\ncontaining 0 in the semimodule. -/\ntheorem zero_smul_set {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b1] [add_comm_monoid \u03b2] [semimodule \u03b1 \u03b2] {s : set \u03b2} (h : set.nonempty s) : 0 \u2022 s = 0 := sorry\n\ntheorem mem_inv_smul_set_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [field \u03b1] [mul_action \u03b1 \u03b2] {a : \u03b1} (ha : a \u2260 0) (A : set \u03b2) (x : \u03b2) : x \u2208 a\u207b\u00b9 \u2022 A \u2194 a \u2022 x \u2208 A := sorry\n\ntheorem mem_smul_set_iff_inv_smul_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} [field \u03b1] [mul_action \u03b1 \u03b2] {a : \u03b1} (ha : a \u2260 0) (A : set \u03b2) (x : \u03b2) : x \u2208 a \u2022 A \u2194 a\u207b\u00b9 \u2022 x \u2208 A :=\n  eq.mpr\n    (id (Eq._oldrec (Eq.refl (x \u2208 a \u2022 A \u2194 a\u207b\u00b9 \u2022 x \u2208 A)) (Eq.symm (propext (mem_inv_smul_set_iff (inv_ne_zero ha) A x)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2208 a \u2022 A \u2194 x \u2208 a\u207b\u00b9\u207b\u00b9 \u2022 A)) (inv_inv' a))) (iff.refl (x \u2208 a \u2022 A)))\n\nnamespace finset\n\n\n/-- The pointwise product of two finite sets `s` and `t`:\n  `st = s \u2b1d t = s * t = { x * y | x \u2208 s, y \u2208 t }`. -/\nprotected instance has_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] : Add (finset \u03b1) :=\n  { add := fun (s t : finset \u03b1) => image (fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p + prod.snd p) (finset.product s t) }\n\ntheorem mul_def {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} : s * t = image (fun (p : \u03b1 \u00d7 \u03b1) => prod.fst p * prod.snd p) (finset.product s t) :=\n  rfl\n\ntheorem mem_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} {x : \u03b1} : x \u2208 s + t \u2194 \u2203 (y : \u03b1), \u2203 (z : \u03b1), y \u2208 s \u2227 z \u2208 t \u2227 y + z = x := sorry\n\n@[simp] theorem coe_add {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} : \u2191(s + t) = \u2191s + \u2191t := sorry\n\ntheorem mul_mem_mul {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} {x : \u03b1} {y : \u03b1} (hx : x \u2208 s) (hy : y \u2208 t) : x * y \u2208 s * t :=\n  eq.mpr (id (propext mem_mul)) (Exists.intro x (Exists.intro y { left := hx, right := { left := hy, right := rfl } }))\n\ntheorem add_card_le {\u03b1 : Type u_1} [DecidableEq \u03b1] [Add \u03b1] {s : finset \u03b1} {t : finset \u03b1} : card (s + t) \u2264 card s * card t := sorry\n\ntheorem mul_card_le {\u03b1 : Type u_1} [DecidableEq \u03b1] [Mul \u03b1] {s : finset \u03b1} {t : finset \u03b1} : card (s * t) \u2264 card s * card t := sorry\n\n/-- A finite set `U` contained in the product of two sets `S * S'` is also contained in the product\nof two finite sets `T * T' \u2286 S * S'`. -/\ntheorem subset_add {M : Type u_1} [add_monoid M] {S : set M} {S' : set M} {U : finset M} (f : \u2191U \u2286 S + S') : \u2203 (T : finset M), \u2203 (T' : finset M), \u2191T \u2286 S \u2227 \u2191T' \u2286 S' \u2227 U \u2286 T + T' := sorry\n\nend finset\n\n\n/-! Some lemmas about pointwise multiplication and submonoids. Ideally we put these in\n  `group_theory.submonoid.basic`, but currently we cannot because that file is imported by this. -/\n\nnamespace submonoid\n\n\ntheorem mul_subset {M : Type u_1} [monoid M] {s : set M} {t : set M} {S : submonoid M} (hs : s \u2286 \u2191S) (ht : t \u2286 \u2191S) : s * t \u2286 \u2191S := sorry\n\ntheorem mul_subset_closure {M : Type u_1} [monoid M] {s : set M} {t : set M} {u : set M} (hs : s \u2286 u) (ht : t \u2286 u) : s * t \u2286 \u2191(closure u) :=\n  mul_subset (set.subset.trans hs subset_closure) (set.subset.trans ht subset_closure)\n\ntheorem Mathlib.add_submonoid.coe_add_self_eq {M : Type u_1} [add_monoid M] (s : add_submonoid M) : \u2191s + \u2191s = \u2191s := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/pointwise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.38725560845041657}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.applicative\nimport control.traversable.basic\n\n/-!\n# Traversing collections\n\nThis file proves basic properties of traversable and applicative functors and defines\n`pure_transformation F`, the natural applicative transformation from the identity functor to `F`.\n\n## References\n\nInspired by [The Essence of the Iterator Pattern][gibbons2009].\n-/\n\nuniverses u\n\nopen is_lawful_traversable\nopen function (hiding comp)\nopen functor\n\nattribute [functor_norm] is_lawful_traversable.naturality\nattribute [simp] is_lawful_traversable.id_traverse\n\nnamespace traversable\n\nvariable {t : Type u \u2192 Type u}\nvariables [traversable t] [is_lawful_traversable t]\nvariables F G : Type u \u2192 Type u\n\nvariables [applicative F] [is_lawful_applicative F]\nvariables [applicative G] [is_lawful_applicative G]\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\nvariables g : \u03b1 \u2192 F \u03b2\nvariables h : \u03b2 \u2192 G \u03b3\nvariables f : \u03b2 \u2192 \u03b3\n\n/-- The natural applicative transformation from the identity functor\nto `F`, defined by `pure : \u03a0 {\u03b1}, \u03b1 \u2192 F \u03b1`. -/\ndef pure_transformation : applicative_transformation id F :=\n{ app := @pure F _,\n  preserves_pure' := \u03bb \u03b1 x, rfl,\n  preserves_seq' := \u03bb \u03b1 \u03b2 f x, by { simp only [map_pure, seq_pure], refl } }\n\n@[simp] theorem pure_transformation_apply {\u03b1} (x : id \u03b1) : pure_transformation F x = pure x := rfl\n\nvariables {F G} (x : t \u03b2)\n\nlemma map_eq_traverse_id : map f = @traverse t _ _ _ _ _ (id.mk \u2218 f) :=\nfunext $ \u03bb y, (traverse_eq_map_id f y).symm\n\ntheorem map_traverse (x : t \u03b1) : map f <$> traverse g x = traverse (map f \u2218 g) x :=\nbegin\n  rw @map_eq_traverse_id t _ _ _ _ f,\n  refine (comp_traverse (id.mk \u2218 f) g x).symm.trans _,\n  congr, apply comp.applicative_comp_id\nend\n\ntheorem traverse_map (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 \u03b2) (x : t \u03b1) :\n  traverse f (g <$> x) = traverse (f \u2218 g) x :=\nbegin\n  rw @map_eq_traverse_id t _ _ _ _ g,\n  refine (comp_traverse f (id.mk \u2218 g) x).symm.trans _,\n  congr, apply comp.applicative_id_comp\nend\n\nlemma pure_traverse (x : t \u03b1) : traverse pure x = (pure x : F (t \u03b1)) :=\nby have : traverse pure x = pure (traverse id.mk x) :=\n     (naturality (pure_transformation F) id.mk x).symm;\n   rwa id_traverse at this\n\nlemma id_sequence (x : t \u03b1) : sequence (id.mk <$> x) = id.mk x :=\nby simp [sequence, traverse_map, id_traverse]; refl\n\nlemma comp_sequence (x : t (F (G \u03b1))) :\n  sequence (comp.mk <$> x) = comp.mk (sequence <$> sequence x) :=\nby simp [sequence, traverse_map]; rw \u2190 comp_traverse; simp [map_id]\n\nlemma naturality' (\u03b7 : applicative_transformation F G) (x : t (F \u03b1)) :\n  \u03b7 (sequence x) = sequence (@\u03b7 _ <$> x) :=\nby simp [sequence, naturality, traverse_map]\n\n@[functor_norm]\nlemma traverse_id : traverse id.mk = (id.mk : t \u03b1 \u2192 id (t \u03b1)) :=\nby { ext, exact id_traverse _ }\n\n@[functor_norm]\nlemma traverse_comp (g : \u03b1 \u2192 F \u03b2) (h : \u03b2 \u2192 G \u03b3) :\n  traverse (comp.mk \u2218 map h \u2218 g) =\n  (comp.mk \u2218 map (traverse h) \u2218 traverse g : t \u03b1 \u2192 comp F G (t \u03b3)) :=\nby { ext, exact comp_traverse _ _ _ }\n\nlemma traverse_eq_map_id' (f : \u03b2 \u2192 \u03b3) : traverse (id.mk \u2218 f) = id.mk \u2218 (map f : t \u03b2 \u2192 t \u03b3) :=\nby { ext, exact traverse_eq_map_id _ _ }\n\n-- @[functor_norm]\nlemma traverse_map' (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3) :\n  traverse (h \u2218 g) = (traverse h \u2218 map g : t \u03b1 \u2192 G (t \u03b3)) :=\nby { ext, rw [comp_app, traverse_map] }\n\nlemma map_traverse' (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) :\n  traverse (map h \u2218 g) = (map (map h) \u2218 traverse g : t \u03b1 \u2192 G (t \u03b3)) :=\nby { ext, rw [comp_app, map_traverse] }\n\nlemma naturality_pf (\u03b7 : applicative_transformation F G) (f : \u03b1 \u2192 F \u03b2) :\n  traverse (@\u03b7 _ \u2218 f) = @\u03b7 _ \u2218 (traverse f : t \u03b1 \u2192 F (t \u03b2)) :=\nby { ext, rw [comp_app, naturality] }\n\nend traversable\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/control/traversable/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.38725560845041646}}
{"text": "/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.limits.preserves.finite\n\n/-!\n# Bundled exact functors\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe say that a functor `F` is left exact if it preserves finite limits, it is right exact if it\npreserves finite colimits, and it is exact if it is both left exact and right exact.\n\nIn this file, we define the categories of bundled left exact, right exact and exact functors.\n\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category_theory.limits\n\nnamespace category_theory\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n\nsection\nvariables (C) (D)\n\n/-- Bundled left-exact functors. -/\n@[derive category, nolint has_nonempty_instance]\ndef LeftExactFunctor :=\nfull_subcategory (\u03bb F : C \u2964 D, nonempty (preserves_finite_limits F))\n\ninfixr ` \u2964\u2097 `:26 := LeftExactFunctor\n\n/-- A left exact functor is in particular a functor. -/\n@[derive full, derive faithful]\ndef LeftExactFunctor.forget : (C \u2964\u2097 D) \u2964 (C \u2964 D) :=\nfull_subcategory_inclusion _\n\n/-- Bundled right-exact functors. -/\n@[derive category, nolint has_nonempty_instance]\ndef RightExactFunctor :=\nfull_subcategory (\u03bb F : C \u2964 D, nonempty (preserves_finite_colimits F))\n\ninfixr ` \u2964\u1d63 `:26 := RightExactFunctor\n\n/-- A right exact functor is in particular a functor. -/\n@[derive full, derive faithful]\ndef RightExactFunctor.forget : (C \u2964\u1d63 D) \u2964 (C \u2964 D) :=\nfull_subcategory_inclusion _\n\n/-- Bundled exact functors. -/\n@[derive category, nolint has_nonempty_instance]\ndef ExactFunctor := full_subcategory\n  (\u03bb F : C \u2964 D, nonempty (preserves_finite_limits F) \u2227 nonempty (preserves_finite_colimits F))\n\ninfixr ` \u2964\u2091 `:26 := ExactFunctor\n\n/-- An exact functor is in particular a functor. -/\n@[derive full, derive faithful]\ndef ExactFunctor.forget : (C \u2964\u2091 D) \u2964 (C \u2964 D) :=\nfull_subcategory_inclusion _\n\n/-- Turn an exact functor into a left exact functor. -/\n@[derive full, derive faithful]\ndef LeftExactFunctor.of_exact : (C \u2964\u2091 D) \u2964 (C \u2964\u2097 D) :=\nfull_subcategory.map (\u03bb X, and.left)\n\n/-- Turn an exact functor into a left exact functor. -/\n@[derive full, derive faithful]\ndef RightExactFunctor.of_exact : (C \u2964\u2091 D) \u2964 (C \u2964\u1d63 D) :=\nfull_subcategory.map (\u03bb X, and.right)\n\nvariables {C D}\n\n@[simp] lemma LeftExactFunctor.of_exact_obj (F : C \u2964\u2091 D) :\n  (LeftExactFunctor.of_exact C D).obj F = \u27e8F.1, F.2.1\u27e9 := rfl\n@[simp] lemma RightExactFunctor.of_exact_obj (F : C \u2964\u2091 D) :\n  (RightExactFunctor.of_exact C D).obj F = \u27e8F.1, F.2.2\u27e9 := rfl\n\n@[simp] lemma LeftExactFunctor.of_exact_map {F G : C \u2964\u2091 D} (\u03b1 : F \u27f6 G) :\n  (LeftExactFunctor.of_exact C D).map \u03b1 = \u03b1 := rfl\n@[simp] lemma RightExactFunctor.of_exact_map {F G : C \u2964\u2091 D} (\u03b1 : F \u27f6 G) :\n  (RightExactFunctor.of_exact C D).map \u03b1 = \u03b1 := rfl\n\n@[simp] \n\n@[simp] lemma LeftExactFunctor.forget_map {F G : C \u2964\u2097 D} (\u03b1 : F \u27f6 G) :\n  (LeftExactFunctor.forget C D).map \u03b1 = \u03b1 := rfl\n@[simp] lemma RightExactFunctor.forget_map {F G : C \u2964\u1d63 D} (\u03b1 : F \u27f6 G) :\n  (RightExactFunctor.forget C D).map \u03b1 = \u03b1 := rfl\n@[simp] lemma ExactFunctor.forget_map {F G : C \u2964\u2091 D} (\u03b1 : F \u27f6 G) :\n  (ExactFunctor.forget C D).map \u03b1 = \u03b1 := rfl\n\n/-- Turn a left exact functor into an object of the category `LeftExactFunctor C D`. -/\ndef LeftExactFunctor.of (F : C \u2964 D) [preserves_finite_limits F] : C \u2964\u2097 D := \u27e8F, \u27e8infer_instance\u27e9\u27e9\n/-- Turn a right exact functor into an object of the category `RightExactFunctor C D`. -/\ndef RightExactFunctor.of (F : C \u2964 D) [preserves_finite_colimits F] : C \u2964\u1d63 D :=\n\u27e8F, \u27e8infer_instance\u27e9\u27e9\n/-- Turn an exact functor into an object of the category `ExactFunctor C D`. -/\ndef ExactFunctor.of (F : C \u2964 D) [preserves_finite_limits F] [preserves_finite_colimits F] :\n  C \u2964\u2091 D := \u27e8F, \u27e8\u27e8infer_instance\u27e9, \u27e8infer_instance\u27e9\u27e9\u27e9\n\n@[simp] lemma LeftExactFunctor.of_fst (F : C \u2964 D) [preserves_finite_limits F] :\n  (LeftExactFunctor.of F).obj = F := rfl\n@[simp] lemma RightExactFunctor.of_fst (F : C \u2964 D) [preserves_finite_colimits F] :\n  (RightExactFunctor.of F).obj = F := rfl\n@[simp] lemma ExactFunctor.of_fst (F : C \u2964 D) [preserves_finite_limits F]\n  [preserves_finite_colimits F] : (ExactFunctor.of F).obj = F := rfl\n\nlemma LeftExactFunctor.forget_obj_of (F : C \u2964 D) [preserves_finite_limits F] :\n  (LeftExactFunctor.forget C D).obj (LeftExactFunctor.of F) = F := rfl\nlemma RightExactFunctor.forget_obj_of (F : C \u2964 D) [preserves_finite_colimits F] :\n  (RightExactFunctor.forget C D).obj (RightExactFunctor.of F) = F := rfl\nlemma ExactFunctor.forget_obj_of (F : C \u2964 D) [preserves_finite_limits F]\n  [preserves_finite_colimits F] : (ExactFunctor.forget C D).obj (ExactFunctor.of F) = F := rfl\n\nnoncomputable instance (F : C \u2964\u2097 D) : preserves_finite_limits F.obj := F.property.some\nnoncomputable instance (F : C \u2964\u1d63 D) : preserves_finite_colimits F.obj := F.property.some\nnoncomputable instance (F : C \u2964\u2091 D) : preserves_finite_limits F.obj := F.property.1.some\nnoncomputable instance (F : C \u2964\u2091 D) : preserves_finite_colimits F.obj := F.property.2.some\n\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/exact_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.38725560794195435}}
{"text": "def is_smooth {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) : Prop := sorry\n\nclass IsSmooth {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) : Prop where\n  (proof : is_smooth f)\n\ninstance identity : IsSmooth fun a : \u03b1 => a := sorry\ninstance const (b : \u03b2) : IsSmooth fun a : \u03b1 => b := sorry\ninstance swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [\u2200 a, IsSmooth (f a)] : IsSmooth (\u03bb b a => f a b) := sorry\ninstance parm (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [IsSmooth f] (b : \u03b2) : IsSmooth (\u03bb a => f a b) := sorry\ninstance comp (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) [IsSmooth f] [IsSmooth g] : IsSmooth (fun a => f (g a)) := sorry\ninstance diag (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) (h : \u03b1 \u2192 \u03b4) [IsSmooth f] [\u2200 b, IsSmooth (f b)] [IsSmooth g] [IsSmooth h] : IsSmooth (\u03bb a => f (g a) (h a)) := sorry\n\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] (d : \u03b4) : IsSmooth (\u03bb a => f (g a) d) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] : IsSmooth (\u03bb a d => f (g a) d) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] (h : \u03b1 \u2192 \u03b1) [IsSmooth h] (d : \u03b4) : IsSmooth (\u03bb a => f (g (h a)) d) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [\u2200 a, IsSmooth (f a)] : IsSmooth (\u03bb b a => f a b) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [\u2200 a b, IsSmooth (f a b)] : IsSmooth (\u03bb c b a => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [\u2200 a b, IsSmooth (f a b)] : IsSmooth (\u03bb c a b => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 \u03b5) [\u2200 a b c, IsSmooth (f a b c)] : IsSmooth (\u03bb d a b c => f a b c d) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [IsSmooth f] (b : \u03b2) : IsSmooth (\u03bb a => f a b) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [IsSmooth f] (b : \u03b2) (c : \u03b3) : IsSmooth (\u03bb a => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [IsSmooth f] (b : \u03b2) : IsSmooth (\u03bb a c => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [IsSmooth f] (c : \u03b3) : IsSmooth (\u03bb a b => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) (b : \u03b2) [IsSmooth (\u03bb a => f a b)] : IsSmooth (\u03bb a c => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (g : \u03b4 \u2192 \u03b5 \u2192 \u03b1) (h : \u03b4 \u2192 \u03b5 \u2192 \u03b2) [IsSmooth f] [\u2200 a, IsSmooth (f a)] [IsSmooth g] [IsSmooth h] : IsSmooth (\u03bb x y => f (g x y) (h x y)) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) [IsSmooth f] [\u2200 b, IsSmooth (f b)] [IsSmooth g] (a : \u03b1): IsSmooth (\u03bb (h : \u03b1 \u2192 \u03b4) => f (g a) (h a)) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) (h : \u03b1 \u2192 \u03b4) [IsSmooth f] : IsSmooth (\u03bb (g : \u03b1 \u2192 \u03b2) a => f (g a) (h a)) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (d : \u03b4) : IsSmooth (\u03bb (g : \u03b1 \u2192 \u03b2) a => f (g a) d) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b2) [IsSmooth f] [IsSmooth g] : IsSmooth (fun x => f (g (g x))) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) [\u2200 a, IsSmooth (f a)] : IsSmooth (\u03bb b a => f a b) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) [\u2200 a b, IsSmooth (f a b)] : IsSmooth (\u03bb c a b => f a b c) := by infer_instance\nexample (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 \u03b5) [\u2200 a b c, IsSmooth (f a b c)] : IsSmooth (\u03bb d a b c => f a b c d) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] (d : \u03b4) : IsSmooth (\u03bb a => f (g a) d) := by infer_instance\nexample (f : \u03b2 \u2192 \u03b4 \u2192 \u03b3) [IsSmooth f] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] : IsSmooth (\u03bb a d => f (g a) d) := by infer_instance\nexample (f : \u03b4 \u2192 \u03b2 \u2192 \u03b3) [\u2200 d, IsSmooth (f d)] (g : \u03b1 \u2192 \u03b2) [IsSmooth g] : IsSmooth (\u03bb a d => (f d (g a))) := by infer_instance\n\n\n-- Recall Function.comp is not reducible anymore\ninstance (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) [IsSmooth f] [IsSmooth g] : IsSmooth (f \u2218 g) := by\n  delta Function.comp\n  infer_instance\n\nexample (f : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) [IsSmooth f] [IsSmooth g] : IsSmooth (f \u2218 g) := by infer_instance\n\nexample (f : \u03b2 \u2192 \u03b3) [IsSmooth f] : IsSmooth \u03bb (g : \u03b1 \u2192 \u03b2) => (f \u2218 g) := by\n  delta Function.comp\n  infer_instance\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/815.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3871001602298914}}
{"text": "/-\nCopyright (c) 2020 Robert Y. Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Y. Lewis\n\n! This file was ported from Lean 3 source module tactic.linarith.datatypes\n! leanprover-community/mathlib commit 2558b3b31d33969bb3ef330982ff131533eebfdd\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Linarith.Lemmas\nimport Mathbin.Tactic.Ring\n\n/-!\n# Datatypes for `linarith`\n\nSome of the data structures here are used in multiple parts of the tactic.\nWe split them into their own file.\n\nThis file also contains a few convenient auxiliary functions.\n-/\n\n\ninitialize\n  registerTraceClass.1 `linarith\n\nopen Native\n\nnamespace Linarith\n\n/-- A shorthand for tracing when the `trace.linarith` option is set to true. -/\nunsafe def linarith_trace {\u03b1} [has_to_tactic_format \u03b1] (s : \u03b1) : tactic Unit :=\n  tactic.when_tracing `linarith (tactic.trace s)\n#align linarith.linarith_trace linarith.linarith_trace\n\n/-- A shorthand for tracing the types of a list of proof terms\nwhen the `trace.linarith` option is set to true.\n-/\nunsafe def linarith_trace_proofs (s : String := \"\") (l : List expr) : tactic Unit :=\n  tactic.when_tracing `linarith do\n    tactic.trace s\n    l tactic.infer_type >>= tactic.trace\n#align linarith.linarith_trace_proofs linarith.linarith_trace_proofs\n\n/-! ### Linear expressions -/\n\n\n#print Linarith.Linexp /-\n/-- A linear expression is a list of pairs of variable indices and coefficients,\nrepresenting the sum of the products of each coefficient with its corresponding variable.\n\nSome functions on `linexp` assume that `n : \u2115` occurs at most once as the first element of a pair,\nand that the list is sorted in decreasing order of the first argument.\nThis is not enforced by the type but the operations here preserve it.\n-/\n@[reducible]\ndef Linexp : Type :=\n  List (\u2115 \u00d7 \u2124)\n#align linarith.linexp Linarith.Linexp\n-/\n\nnamespace Linexp\n\n/-- Add two `linexp`s together componentwise.\nPreserves sorting and uniqueness of the first argument.\n-/\nunsafe def add : Linexp \u2192 Linexp \u2192 Linexp\n  | [], a => a\n  | a, [] => a\n  | a@(n1, z1) :: t1, b@(n2, z2) :: t2 =>\n    if n1 < n2 then b :: add (a :: t1) t2\n    else\n      if n2 < n1 then a :: add t1 (b :: t2)\n      else\n        let sum := z1 + z2\n        if Sum = 0 then add t1 t2 else (n1, Sum) :: add t1 t2\n#align linarith.linexp.add linarith.linexp.add\n\n#print Linarith.Linexp.scale /-\n/-- `l.scale c` scales the values in `l` by `c` without modifying the order or keys. -/\ndef scale (c : \u2124) (l : Linexp) : Linexp :=\n  if c = 0 then [] else if c = 1 then l else l.map fun \u27e8n, z\u27e9 => (n, z * c)\n#align linarith.linexp.scale Linarith.Linexp.scale\n-/\n\n#print Linarith.Linexp.get /-\n/-- `l.get n` returns the value in `l` associated with key `n`, if it exists, and `none` otherwise.\nThis function assumes that `l` is sorted in decreasing order of the first argument,\nthat is, it will return `none` as soon as it finds a key smaller than `n`.\n-/\ndef get (n : \u2115) : Linexp \u2192 Option \u2124\n  | [] => none\n  | (a, b) :: t => if a < n then none else if a = n then some b else get t\n#align linarith.linexp.get Linarith.Linexp.get\n-/\n\n#print Linarith.Linexp.contains /-\n/-- `l.contains n` is true iff `n` is the first element of a pair in `l`.\n-/\ndef contains (n : \u2115) : Linexp \u2192 Bool :=\n  Option.isSome \u2218 get n\n#align linarith.linexp.contains Linarith.Linexp.contains\n-/\n\n#print Linarith.Linexp.zfind /-\n/-- `l.zfind n` returns the value associated with key `n` if there is one, and 0 otherwise.\n-/\ndef zfind (n : \u2115) (l : Linexp) : \u2124 :=\n  match l.get n with\n  | none => 0\n  | some v => v\n#align linarith.linexp.zfind Linarith.Linexp.zfind\n-/\n\n#print Linarith.Linexp.vars /-\n/-- `l.vars` returns the list of variables that occur in `l`. -/\ndef vars (l : Linexp) : List \u2115 :=\n  l.map Prod.fst\n#align linarith.linexp.vars Linarith.Linexp.vars\n-/\n\n#print Linarith.Linexp.cmp /-\n/-- Defines a lex ordering on `linexp`. This function is performance critical.\n-/\ndef cmp : Linexp \u2192 Linexp \u2192 Ordering\n  | [], [] => Ordering.eq\n  | [], _ => Ordering.lt\n  | _, [] => Ordering.gt\n  | (n1, z1) :: t1, (n2, z2) :: t2 =>\n    if n1 < n2 then Ordering.lt\n    else\n      if n2 < n1 then Ordering.gt\n      else if z1 < z2 then Ordering.lt else if z2 < z1 then Ordering.gt else cmp t1 t2\n#align linarith.linexp.cmp Linarith.Linexp.cmp\n-/\n\nend Linexp\n\n/-! ### Inequalities -/\n\n\n#print Linarith.Ineq /-\n/-- The three-element type `ineq` is used to represent the strength of a comparison between\nterms. -/\ninductive Ineq : Type\n  | Eq\n  | le\n  | lt\n  deriving DecidableEq, Inhabited\n#align linarith.ineq Linarith.Ineq\n-/\n\nnamespace Ineq\n\n#print Linarith.Ineq.max /-\n/-- `max R1 R2` computes the strength of the sum of two inequalities. If `t1 R1 0` and `t2 R2 0`,\nthen `t1 + t2 (max R1 R2) 0`.\n-/\ndef max : Ineq \u2192 Ineq \u2192 Ineq\n  | lt, a => lt\n  | a, lt => lt\n  | le, a => le\n  | a, le => le\n  | Eq, Eq => eq\n#align linarith.ineq.max Linarith.Ineq.max\n-/\n\n#print Linarith.Ineq.cmp /-\n/-- `ineq` is ordered `eq < le < lt`. -/\ndef cmp : Ineq \u2192 Ineq \u2192 Ordering\n  | Eq, Eq => Ordering.eq\n  | Eq, _ => Ordering.lt\n  | le, le => Ordering.eq\n  | le, lt => Ordering.lt\n  | lt, lt => Ordering.eq\n  | _, _ => Ordering.gt\n#align linarith.ineq.cmp Linarith.Ineq.cmp\n-/\n\n#print Linarith.Ineq.toString /-\n/-- Prints an `ineq` as the corresponding infix symbol. -/\ndef toString : Ineq \u2192 String\n  | Eq => \"=\"\n  | le => \"\u2264\"\n  | lt => \"<\"\n#align linarith.ineq.to_string Linarith.Ineq.toString\n-/\n\n/-- Finds the name of a multiplicative lemma corresponding to an inequality strength. -/\nunsafe def to_const_mul_nm : Ineq \u2192 Name\n  | lt => `` mul_neg\n  | le => `` mul_nonpos\n  | Eq => `` mul_eq\n#align linarith.ineq.to_const_mul_nm linarith.ineq.to_const_mul_nm\n\ninstance : ToString Ineq :=\n  \u27e8Ineq.toString\u27e9\n\nunsafe instance : has_to_format Ineq :=\n  \u27e8fun i => Ineq.toString i\u27e9\n\nend Ineq\n\n/-! ### Comparisons with 0 -/\n\n\n#print Linarith.Comp /-\n/-- The main datatype for FM elimination.\nVariables are represented by natural numbers, each of which has an integer coefficient.\nIndex 0 is reserved for constants, i.e. `coeffs.find 0` is the coefficient of 1.\nThe represented term is `coeffs.sum (\u03bb \u27e8k, v\u27e9, v * Var[k])`.\nstr determines the strength of the comparison -- is it < 0, \u2264 0, or = 0?\n-/\nstructure Comp : Type where\n  str : Ineq\n  coeffs : Linexp\n  deriving Inhabited\n#align linarith.comp Linarith.Comp\n-/\n\n#print Linarith.Comp.vars /-\n/-- `c.vars` returns the list of variables that appear in the linear expression contained in `c`. -/\ndef Comp.vars : Comp \u2192 List \u2115 :=\n  Linexp.vars \u2218 Comp.coeffs\n#align linarith.comp.vars Linarith.Comp.vars\n-/\n\n#print Linarith.Comp.coeffOf /-\n/-- `comp.coeff_of c a` projects the coefficient of variable `a` out of `c`. -/\ndef Comp.coeffOf (c : Comp) (a : \u2115) : \u2124 :=\n  c.coeffs.zfind a\n#align linarith.comp.coeff_of Linarith.Comp.coeffOf\n-/\n\n#print Linarith.Comp.scale /-\n/-- `comp.scale c n` scales the coefficients of `c` by `n`. -/\ndef Comp.scale (c : Comp) (n : \u2115) : Comp :=\n  { c with coeffs := c.coeffs.scale n }\n#align linarith.comp.scale Linarith.Comp.scale\n-/\n\n/-- `comp.add c1 c2` adds the expressions represented by `c1` and `c2`.\nThe coefficient of variable `a` in `c1.add c2`\nis the sum of the coefficients of `a` in `c1` and `c2`.\n -/\nunsafe def comp.add (c1 c2 : Comp) : Comp :=\n  \u27e8c1.str.max c2.str, c1.coeffs.add c2.coeffs\u27e9\n#align linarith.comp.add linarith.comp.add\n\n/-- `comp` has a lex order. First the `ineq`s are compared, then the `coeff`s. -/\nunsafe def comp.cmp : Comp \u2192 Comp \u2192 Ordering\n  | \u27e8str1, coeffs1\u27e9, \u27e8str2, coeffs2\u27e9 =>\n    match str1.cmp str2 with\n    | Ordering.lt => Ordering.lt\n    | Ordering.gt => Ordering.gt\n    | Ordering.eq => coeffs1.cmp coeffs2\n#align linarith.comp.cmp linarith.comp.cmp\n\n/-- A `comp` represents a contradiction if its expression has no coefficients and its strength is <,\nthat is, it represents the fact `0 < 0`.\n -/\nunsafe def comp.is_contr (c : Comp) : Bool :=\n  c.coeffs.Empty \u2227 c.str = Ineq.lt\n#align linarith.comp.is_contr linarith.comp.is_contr\n\nunsafe instance comp.to_format : has_to_format Comp :=\n  \u27e8fun p => to_fmt p.coeffs ++ toString p.str ++ \"0\"\u27e9\n#align linarith.comp.to_format linarith.comp.to_format\n\n/-! ### Parsing into linear form -/\n\n\n/-! ### Control -/\n\n\n/-- A preprocessor transforms a proof of a proposition into a proof of a different propositon.\nThe return type is `list expr`, since some preprocessing steps may create multiple new hypotheses,\nand some may remove a hypothesis from the list.\nA \"no-op\" preprocessor should return its input as a singleton list.\n-/\nunsafe structure preprocessor : Type where\n  Name : String\n  transform : expr \u2192 tactic (List expr)\n#align linarith.preprocessor linarith.preprocessor\n\n/-- Some preprocessors need to examine the full list of hypotheses instead of working item by item.\nAs with `preprocessor`, the input to a `global_preprocessor` is replaced by, not added to, its\noutput.\n-/\nunsafe structure global_preprocessor : Type where\n  Name : String\n  transform : List expr \u2192 tactic (List expr)\n#align linarith.global_preprocessor linarith.global_preprocessor\n\n/-- Some preprocessors perform branching case splits. A `branch` is used to track one of these case\nsplits. The first component, an `expr`, is the goal corresponding to this branch of the split,\ngiven as a metavariable. The `list expr` component is the list of hypotheses for `linarith`\nin this branch. Every `expr` in this list should be type correct in the context of the associated\ngoal.\n-/\nunsafe def branch : Type :=\n  expr \u00d7 List expr\n#align linarith.branch linarith.branch\n\n/-- Some preprocessors perform branching case splits.\nA `global_branching_preprocessor` produces a list of branches to run.\nEach branch is independent, so hypotheses that appear in multiple branches should be duplicated.\nThe preprocessor is responsible for making sure that each branch contains the correct goal\nmetavariable.\n-/\nunsafe structure global_branching_preprocessor : Type where\n  Name : String\n  transform : List expr \u2192 tactic (List branch)\n#align linarith.global_branching_preprocessor linarith.global_branching_preprocessor\n\n/-- A `preprocessor` lifts to a `global_preprocessor` by folding it over the input list.\n-/\nunsafe def preprocessor.globalize (pp : preprocessor) : global_preprocessor\n    where\n  Name := pp.Name\n  transform :=\n    List.foldlM\n      (fun ret e => do\n        let l' \u2190 pp.transform e\n        return (l' ++ ret))\n      []\n#align linarith.preprocessor.globalize linarith.preprocessor.globalize\n\n/-- A `global_preprocessor` lifts to a `global_branching_preprocessor` by producing only one branch.\n-/\nunsafe def global_preprocessor.branching (pp : global_preprocessor) : global_branching_preprocessor\n    where\n  Name := pp.Name\n  transform l := do\n    let g \u2190 tactic.get_goal\n    singleton <$> Prod.mk g <$> pp l\n#align linarith.global_preprocessor.branching linarith.global_preprocessor.branching\n\n/-- `process pp l` runs `pp.transform` on `l` and returns the result,\ntracing the result if `trace.linarith` is on.\n-/\nunsafe def global_branching_preprocessor.process (pp : global_branching_preprocessor)\n    (l : List expr) : tactic (List branch) := do\n  let l \u2190 pp.transform l\n  when (l > 1) <| linarith_trace f! \"Preprocessing: {pp} has branched, with branches:\"\n  l fun l => tactic.set_goals [l.1] >> linarith_trace_proofs (toString f! \"Preprocessing: {pp}\") l.2\n  return l\n#align linarith.global_branching_preprocessor.process linarith.global_branching_preprocessor.process\n\nunsafe instance preprocessor_to_gb_preprocessor : Coe preprocessor global_branching_preprocessor :=\n  \u27e8global_preprocessor.branching \u2218 preprocessor.globalize\u27e9\n#align linarith.preprocessor_to_gb_preprocessor linarith.preprocessor_to_gb_preprocessor\n\nunsafe instance global_preprocessor_to_gb_preprocessor :\n    Coe global_preprocessor global_branching_preprocessor :=\n  \u27e8global_preprocessor.branching\u27e9\n#align linarith.global_preprocessor_to_gb_preprocessor linarith.global_preprocessor_to_gb_preprocessor\n\n/--\nA `certificate_oracle` is a function `produce_certificate : list comp \u2192 \u2115 \u2192 tactic (rb_map \u2115 \u2115)`.\n`produce_certificate hyps max_var` tries to derive a contradiction from the comparisons in `hyps`\nby eliminating all variables \u2264 `max_var`.\nIf successful, it returns a map `coeff : \u2115 \u2192 \u2115` as a certificate.\nThis map represents that we can find a contradiction by taking the sum  `\u2211 (coeff i) * hyps[i]`.\n\nThe default `certificate_oracle` used by `linarith` is\n`linarith.fourier_motzkin.produce_certificate`.\n-/\nunsafe def certificate_oracle : Type :=\n  List Comp \u2192 \u2115 \u2192 tactic (rb_map \u2115 \u2115)\n#align linarith.certificate_oracle linarith.certificate_oracle\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- A configuration object for `linarith`. -/\nunsafe structure linarith_config : Type where\n  discharger : tactic Unit := sorry\n  restrict_type : Option Type := none\n  restrict_type_reflect : reflected _ restrict_type := by infer_instance\n  exfalso : Bool := true\n  Transparency : Tactic.Transparency := reducible\n  split_hypotheses : Bool := true\n  split_ne : Bool := false\n  preprocessors : Option (List global_branching_preprocessor) := none\n  oracle : Option certificate_oracle := none\n#align linarith.linarith_config linarith.linarith_config\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- `cfg.update_reducibility reduce_semi` will change the transparency setting of `cfg` to\n`semireducible` if `reduce_semi` is true. In this case, it also sets the discharger to `ring!`,\nsince this is typically needed when using stronger unification.\n-/\nunsafe def linarith_config.update_reducibility (cfg : linarith_config) (reduce_semi : Bool) :\n    linarith_config :=\n  if reduce_semi then\n    { cfg with\n      Transparency := semireducible\n      discharger := sorry }\n  else cfg\n#align linarith.linarith_config.update_reducibility linarith.linarith_config.update_reducibility\n\n/-!\n### Auxiliary functions\n\nThese functions are used by multiple modules, so we put them here for accessibility.\n-/\n\n\nopen Tactic\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      `get_rel_sides e` returns the left and right hand sides of `e` if `e` is a comparison,\n      and fails otherwise.\n      This function is more naturally in the `option` monad, but it is convenient to put in `tactic`\n      for compositionality.\n       -/\n    unsafe\n  def\n    get_rel_sides\n    : expr \u2192 tactic ( expr \u00d7 expr )\n    | q( $ ( a ) < $ ( b ) ) => return ( a , b )\n      | q( $ ( a ) \u2264 $ ( b ) ) => return ( a , b )\n      | q( $ ( a ) = $ ( b ) ) => return ( a , b )\n      | q( $ ( a ) \u2265 $ ( b ) ) => return ( a , b )\n      | q( $ ( a ) > $ ( b ) ) => return ( a , b )\n      | _ => tactic.failed\n#align linarith.get_rel_sides linarith.get_rel_sides\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      `parse_into_comp_and_expr e` checks if `e` is of the form `t < 0`, `t \u2264 0`, or `t = 0`.\n      If it is, it returns the comparison along with `t`.\n       -/\n    unsafe\n  def\n    parse_into_comp_and_expr\n    : expr \u2192 Option ( Ineq \u00d7 expr )\n    | q( $ ( e ) < 0 ) => ( Ineq.lt , e )\n      | q( $ ( e ) \u2264 0 ) => ( Ineq.le , e )\n      | q( $ ( e ) = 0 ) => ( Ineq.eq , e )\n      | _ => none\n#align linarith.parse_into_comp_and_expr linarith.parse_into_comp_and_expr\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      `mk_single_comp_zero_pf c h` assumes that `h` is a proof of `t R 0`.\n      It produces a pair `(R', h')`, where `h'` is a proof of `c*t R' 0`.\n      Typically `R` and `R'` will be the same, except when `c = 0`, in which case `R'` is `=`.\n      If `c = 1`, `h'` is the same as `h` -- specifically, it does *not* change the type to `1*t R 0`.\n      -/\n    unsafe\n  def\n    mk_single_comp_zero_pf\n    ( c : \u2115 ) ( h : expr ) : tactic ( Ineq \u00d7 expr )\n    :=\n      do\n        let tp \u2190 infer_type h\n          let some ( iq , e ) \u2190 return <| parse_into_comp_and_expr tp\n          if\n            c = 0\n            then\n            do let e' \u2190 mk_app ` ` MulZeroClass.zero_mul [ e ] return ( ineq.eq , e' )\n            else\n            if\n              c = 1\n              then\n              return ( iq , h )\n              else\n              do\n                let tp \u2190 Prod.snd <$> ( infer_type h >>= get_rel_sides ) >>= infer_type\n                  let c \u2190 tp c\n                  let cpos \u2190 to_expr ` `( $ ( c ) > 0 )\n                  let ( _ , ex ) \u2190 solve_aux cpos sorry\n                  let e' \u2190 mk_app iq [ h , ex ]\n                  return ( iq , e' )\n#align linarith.mk_single_comp_zero_pf linarith.mk_single_comp_zero_pf\n\nend Linarith\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Linarith/Datatypes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.387100151855115}}
{"text": "import mathlib.combinatorics.simple_graph.basic\nimport combinatorics.simple_graph.subgraph\nimport mathlib.logic.relation\n\nattribute [protected] simple_graph.subgraph.mem_edge_set\n\nopen function\n\nvariables {\u03b1 \u03b2 \u03b3 V : Type*} {G H : simple_graph V}\n\nnamespace simple_graph\nnamespace subgraph\n\ninstance (G : simple_graph V) (H : subgraph G) [decidable_rel H.adj] : decidable_rel H.coe.adj :=\n\u03bb a b, \u2039decidable_rel H.adj\u203a _ _\n\n@[simp] lemma map_id {G : simple_graph \u03b1} (G' : G.subgraph) : G'.map hom.id = G' := by ext; simp\n\n@[simp] lemma map_comp {G : simple_graph \u03b1} {H : simple_graph \u03b2} {I : simple_graph \u03b3}\n  (G' : G.subgraph) (f : G \u2192g H) (g : H \u2192g I) :\n  G'.map (g.comp f) = (G'.map f).map g :=\nby ext; simp [subgraph.map]\n\n@[simp] lemma edge_set_map {G : simple_graph \u03b1} {H : simple_graph \u03b2} (f : G \u2192g H)\n  (G' : G.subgraph) : (G'.map f).edge_set = sym2.map f '' G'.edge_set :=\nbegin\n  ext e,\n  induction e using sym2.ind with a b,\n  simp only [mem_edge_set, sym2.exists, relation.map, and_or_distrib_left, exists_or_distrib,\n    map_adj, set.mem_image, sym2.map_pair_eq, quotient.eq, sym2.rel_iff],\n  refine (or_iff_left_of_imp _).symm,\n  rintro \u27e8a, b, hab, rfl, rfl\u27e9,\n  exact \u27e8b, a, hab.symm, rfl, rfl\u27e9,\nend\n\n@[simp] lemma edge_set_coe {G' : G.subgraph} : G'.coe.edge_set = sym2.map coe \u207b\u00b9' G'.edge_set :=\nby { ext e, induction e using sym2.ind with a b, simp }\n\nlemma image_coe_edge_set_coe (G' : G.subgraph) : sym2.map coe '' G'.coe.edge_set = G'.edge_set :=\nbegin\n  rw [edge_set_coe, set.image_preimage_eq_iff],\n  rintro e he,\n  induction e using sym2.ind with a b,\n  rw subgraph.mem_edge_set at he,\n  exact \u27e8\u27e6(\u27e8a, edge_vert _ he\u27e9, \u27e8b, edge_vert _ he.symm\u27e9)\u27e7, sym2.map_pair_eq _ _ _\u27e9,\nend\n\nlemma spanning_coe_le (G' : G.subgraph) : G'.spanning_coe \u2264 G := \u03bb a b, G'.3\n\n/-- The subgraph of `H` corresponding to a smaller graph `H`. -/\n@[simps] def of_le (h : H \u2264 G) : G.subgraph :=\n{ verts := set.univ,\n  adj := H.adj,\n  adj_sub := h,\n  edge_vert := \u03bb _ _ _, set.mem_univ _,\n  symm := H.symm }\n\n/-- The graph isomorphism between the top element of `G.subgraph` and `G`. -/\n@[simps] def top_iso : (\u22a4 : G.subgraph).coe \u2243g G :=\n{ to_fun := coe,\n  inv_fun := \u03bb a, \u27e8a, set.mem_univ _\u27e9,\n  left_inv := \u03bb _, subtype.eta _ _,\n  right_inv := \u03bb _, rfl,\n  map_rel_iff' := \u03bb _ _, iff.rfl }\n\n/-- The isomorphism between a subgraph and its isomorphism under an injective map. -/\n@[simps]\nnoncomputable def iso_map {H : simple_graph \u03b2} (f : G \u2192g H) (hf : injective f) (G' : G.subgraph) :\n  G'.coe \u2243g (G'.map f).coe :=\n{ map_rel_iff' := \u03bb a b, by simp [hf],\n  ..equiv.set.image f G'.verts hf }\n\nopen_locale classical\n\nnoncomputable instance [fintype V] : fintype G.subgraph :=\nfintype.of_equiv\n  {H : set V \u00d7 (V \u2192 V \u2192 Prop) // H.2 \u2264 G.adj \u2227 (\u2200 a b, H.2 a b \u2192 a \u2208 H.1) \u2227 symmetric H.2}\n  { to_fun := \u03bb H, \u27e8H.1.1, H.1.2, H.2.1, H.2.2.1, H.2.2.2\u27e9,\n    inv_fun := \u03bb H, \u27e8\u27e8H.1, H.2\u27e9, \u03bb _ _, H.3, \u03bb _ _, H.4, H.5\u27e9,\n    left_inv := \u03bb _, by ext; refl,\n    right_inv := \u03bb _, by ext; refl }\n\ninstance [finite V] : finite G.subgraph := by { casesI nonempty_fintype V, apply_instance }\n\nend subgraph\nend simple_graph\n", "meta": {"author": "YaelDillies", "repo": "LeanCamCombi", "sha": "9f62375030cd2bd1be6ef10bba68b1b31aa98acf", "save_path": "github-repos/lean/YaelDillies-LeanCamCombi", "path": "github-repos/lean/YaelDillies-LeanCamCombi/LeanCamCombi-9f62375030cd2bd1be6ef10bba68b1b31aa98acf/src/mathlib/combinatorics/simple_graph/subgraph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.38710014348033833}}
{"text": "import category_theory.base\nimport category_theory.isomorphism\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` \u2218 `:80 g:80 := g \u226b f\n\nuniverses v u\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nclass wide_subcategory (D : \u03a0 {a b : C}, (a \u27f6 b) \u2192 Prop) : Prop :=\n(mem_id {} : \u2200 (a : C), D (\ud835\udfd9 a))\n(mem_comp {} : \u2200 {a b c : C} {f : a \u27f6 b} {g : b \u27f6 c}, D f \u2192 D g \u2192 D (g \u2218 f))\nexport wide_subcategory (mem_id mem_comp)\n\nclass replete_wide_subcategory D extends wide_subcategory.{v} C D : Prop :=\n(mem_iso {} : \u2200 {a b : C} (i : iso a b), D i.hom)\nexport replete_wide_subcategory (mem_iso)\n\nvariables {C}\n-- mem_id is redundant when we have mem_iso.\nlemma replete_wide_subcategory.mk' {D : \u03a0 \u2983a b : C\u2984, (a \u27f6 b) \u2192 Prop}\n  (mem_iso : \u2200 {a b : C} (i : iso a b), D i.hom)\n  (mem_comp : \u2200 {a b c : C} {f : a \u27f6 b} {g : b \u27f6 c}, D f \u2192 D g \u2192 D (g \u2218 f)) :\n  replete_wide_subcategory.{v} C D :=\n{ mem_id := \u03bb a, mem_iso (iso.refl a),\n  mem_comp := @mem_comp,\n  mem_iso := @mem_iso }\n\nvariables {D : \u03a0 \u2983a b : C\u2984, (a \u27f6 b) \u2192 Prop} [replete_wide_subcategory.{v} C D]\n\nlemma mem_of_mem_comp_left {a b c : C} {f : a \u27f6 b} (i : iso b c)\n  (h : D (i.hom \u2218 f)) : D f :=\nby convert mem_comp h (mem_iso i.symm); simp\n\nlemma mem_of_mem_comp_right {a b c : C} {f : b \u27f6 c} (i : iso a b)\n  (h : D (f \u2218 i.hom)) : D f :=\nby convert mem_comp (mem_iso i.symm) h; simp\n\nlemma mem_iff_mem_of_isomorphic {a b a' b' : C} {f : a \u27f6 b} {f' : a' \u27f6 b'}\n  (i : iso a a') (j : iso b b')\n  (e : j.hom \u2218 f = f' \u2218 i.hom) : D f \u2194 D f' :=\niff.intro\n  (assume h, have D (j.hom \u2218 f), from mem_comp h (mem_iso j),\n    by rw e at this; exact mem_of_mem_comp_right i this)\n  (assume h, have D (f' \u2218 i.hom), from mem_comp (mem_iso i) h,\n    by rw \u2190e at this; exact mem_of_mem_comp_left j this)\n\nend category_theory\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/category_theory/replete.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3870317899101077}}
{"text": "import algebra.punit_instances\nimport topology.algebra.group\n\nimport pseudo_normed_group.basic\n\nimport hacks_and_tricks.type_pow\nimport facts\n\n/-!\n\n# profinitely_filtered_pseudo_normed_group\n\nThe definition of a profinitely_filtered_pseudo_normed_group, and an API for this\ndefinition.\n\n-/\nopen pseudo_normed_group\nopen_locale nnreal big_operators\n\nlocal attribute [instance] type_pow\n\n-- move this\nlemma int.induction_nonneg_or_nonpos {P : \u2124 \u2192 Prop} (n : \u2124)\n  (h\u2081 : \u2200 n : \u2115, P n) (h\u2082 : \u2200 n : \u2115, P (-n)) : P n :=\nbegin\n  rcases le_or_lt 0 n with hn|hn,\n  { lift n to \u2115 using hn, exact h\u2081 n },\n  { lift (-n) to \u2115 using neg_nonneg.mpr hn.le with k hk, simpa only [hk, neg_neg] using h\u2082 k },\nend\n\n/-- A *complete Hausdorff filtered pseudo-normed topological group* is\n* an abelian group `M` with an increasing filtration `filtration M c, c : \u211d\u22650` such that\n* `filtration M c` is a compact Hausdorff (or T2) space\n* `M` is pseudo-normed, so `0 \u2208 filtration M c`, `-(filtration M c) = filtration M c`,\n  and `x\u2081 \u2208 filtration M c\u2081, x\u2082 \u2208 filtration M c\u2082 \u2192 (x\u2081 + x\u2082) \u2208 filtration M (c\u2081 + c\u2082)`\n* (bounded) addition and negation are continuous.\n\nMorphisms are continuous and bounded homomorphisms. -/\nclass comphaus_filtered_pseudo_normed_group (M : Type*)\n  extends pseudo_normed_group M :=\n[topology : \u2200 c, topological_space (filtration c)]\n[t2 : \u2200 c, t2_space (filtration c)]\n--[td : \u2200 c, totally_disconnected_space (filtration c)]\n[compact : \u2200 c, compact_space (filtration c)]\n(continuous_add' : \u2200 (c\u2081 c\u2082),\n  continuous (add' : filtration c\u2081 \u00d7 filtration c\u2082 \u2192 filtration (c\u2081 + c\u2082)))\n(continuous_neg' : \u2200 c, continuous (neg' : filtration c \u2192 filtration c))\n(continuous_cast_le : \u2200 (c\u2081 c\u2082) [h : fact (c\u2081 \u2264 c\u2082)],\n  continuous (cast_le : filtration c\u2081 \u2192 filtration c\u2082))\n\n/-- A *profinitely filtered pseudo-normed topological group* is a *complete Hausdorff filtered\npseudo-normed topological group* with the additional requirement that\n* `filtration M c` is a profinite set. -/\nclass profinitely_filtered_pseudo_normed_group (M : Type*)\n  extends comphaus_filtered_pseudo_normed_group M :=\n[td : \u2200 c, totally_disconnected_space (filtration c)]\n\nnamespace comphaus_filtered_pseudo_normed_group\n\nvariables {M M\u2081 M\u2082 M\u2083 : Type*}\nvariables [comphaus_filtered_pseudo_normed_group M]\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\nvariables [comphaus_filtered_pseudo_normed_group M\u2083]\n\ninstance (c : \u211d\u22650) : topological_space (filtration M c) := topology c\ninstance (c : \u211d\u22650) : t2_space (filtration M c) := t2 c\n--instance (c : \u211d\u22650) : totally_disconnected_space (filtration M c) := td c\ninstance (c : \u211d\u22650) : compact_space (filtration M c) := compact c\n\nlemma is_closed_map_cast_le (c\u2081 c\u2082) [h : fact (c\u2081 \u2264 c\u2082)] :\n  is_closed_map (@pseudo_normed_group.cast_le M _ _ _ h) :=\n(continuous_cast_le c\u2081 c\u2082).is_closed_map\n\nlemma closed_embedding_cast_le (c\u2081 c\u2082) [h : fact (c\u2081 \u2264 c\u2082)] :\n  closed_embedding (@pseudo_normed_group.cast_le M _ _ _ h) :=\nclosed_embedding_of_continuous_injective_closed\n  (continuous_cast_le c\u2081 c\u2082) (injective_cast_le c\u2081 c\u2082) (is_closed_map_cast_le c\u2081 c\u2082)\n\nlemma embedding_cast_le (c\u2081 c\u2082) [h : fact (c\u2081 \u2264 c\u2082)] :\n  embedding (@pseudo_normed_group.cast_le M _ _ _ h) :=\n(closed_embedding_cast_le c\u2081 c\u2082).to_embedding\n\nlemma continuous_add {X : Type*} [topological_space X] (c\u2081 c\u2082 : \u211d\u22650)\n  (f : X \u2192 filtration M c\u2081) (hf : continuous f)\n  (g : X \u2192 filtration M c\u2082) (hg : continuous g) :\n  continuous (\u03bb x, \u27e8f x + g x, add_mem_filtration (f x).2 (g x).2\u27e9 : X \u2192 filtration M (c\u2081 + c\u2082)) :=\nbegin\n  have : continuous (\u03bb x, (f x, g x)) := hf.prod_mk hg,\n  exact (continuous_add' c\u2081 c\u2082).comp this,\nend\n\nlemma continuous_neg {X : Type*} [topological_space X] (c : \u211d\u22650)\n  (f : X \u2192 filtration M c) (hf : continuous f) :\n  continuous (\u03bb x, \u27e8-f x, neg_mem_filtration (f x).2\u27e9 : X \u2192 filtration M c) :=\n(continuous_neg' c).comp hf\n\nlemma continuous_nsmul {X : Type*} [topological_space X] (n : \u2115) (c : \u211d\u22650)\n  (f : X \u2192 filtration M c) (hf : continuous f) :\n  continuous (\u03bb x, \u27e8n \u2022 f x, nat_smul_mem_filtration n _ _ (f x).2\u27e9 : X \u2192 filtration M (n * c)) :=\nbegin\n  induction n with n ih,\n  { simp only [zero_smul],\n    exact @continuous_const _ {x // x \u2208 filtration M (\u21910 * c)} _ _ \u27e80, zero_mem_filtration _\u27e9, },\n  { simp only [nat.succ_eq_add_one, succ_nsmul'],\n    haveI aux1 : fact (\u2191n * c \u2264 n \u2022 c) := \u27e8by simp only [le_refl, nsmul_eq_mul]\u27e9,\n    haveI aux2 : fact (n \u2022 c + c \u2264 \u2191n.succ * c) := \u27e8by simp [le_refl, nsmul_eq_mul, add_mul]\u27e9,\n    exact (continuous_cast_le (n \u2022 c + c) ((n.succ) * c)).comp (continuous_add (n \u2022 c) c _\n      ((continuous_cast_le _ _).comp ih) _ hf), }\nend\n\nlemma continuous_zsmul {X : Type*} [topological_space X] (n : \u2124) (c : \u211d\u22650)\n  (f : X \u2192 filtration M c) (hf : continuous f) :\n  continuous (\u03bb x, \u27e8n \u2022 f x, int_smul_mem_filtration n _ _ (f x).2\u27e9 :\n     X \u2192 filtration M (n.nat_abs * c)) :=\nbegin\n  induction n using int.induction_nonneg_or_nonpos,\n  { simp only [coe_nat_zsmul], exact continuous_nsmul n c f hf },\n  { simp only [neg_smul],\n    haveI : fact (\u2191n * c \u2264 (-n : \u2124).nat_abs * c) :=\n      \u27e8by simp only [int.nat_abs_of_nat, int.nat_abs_neg]\u27e9,\n    convert continuous_neg _ _ ((continuous_cast_le (n * c) ((-n : \u2124).nat_abs * c)).comp\n      (continuous_nsmul n c f hf)) using 1,\n    ext x,\n    simp only [coe_cast_le, coe_nat_zsmul, subtype.coe_mk], }\nend\n\nend comphaus_filtered_pseudo_normed_group\n\nnamespace profinitely_filtered_pseudo_normed_group\n\nvariables {M : Type*} [profinitely_filtered_pseudo_normed_group M]\n\ninstance (c : \u211d\u22650) : totally_disconnected_space (filtration M c) := td c\n\nend profinitely_filtered_pseudo_normed_group\n\nsection\nset_option old_structure_cmd true\n\n/-- A `comphaus_filtered_pseudo_normed_group_hom` is a morphism between compact hausdorff\n  filtered pseudo-normed groups. It is an additive group homomorphism which is bounded\n  in the sense that there exist some constant `B` such that the filtrand at level `c`\n  for the source is mapped into the filtrand at level `B*c` in the target. -/\nstructure comphaus_filtered_pseudo_normed_group_hom (M\u2081 M\u2082 : Type*)\n  [comphaus_filtered_pseudo_normed_group M\u2081]\n  [comphaus_filtered_pseudo_normed_group M\u2082]\n  extends M\u2081 \u2192+ M\u2082 :=\n(bound' : \u2203 C, \u2200 c x, x \u2208 filtration M\u2081 c \u2192 to_fun x \u2208 filtration M\u2082 (C * c))\n(continuous' : \u2200 \u2983c\u2081 c\u2082\u2984 (f\u2080 : filtration M\u2081 c\u2081 \u2192 filtration M\u2082 c\u2082)\n  (h : \u2200 x, to_fun \u2191x = f\u2080 x), continuous f\u2080)\n\n/-- A `strict_comphaus_filtered_pseudo_normed_group_hom` is a strict morphism between compact\n  hausdorff filtered pseudo-normed groups. It is an additive group homomorphism which is bounded\n  in the strict sense that the filtrand at level `c` for the source is mapped into the filtrand\n  at level `c` in the target. -/\nstructure strict_comphaus_filtered_pseudo_normed_group_hom (M\u2081 M\u2082 : Type*)\n  [comphaus_filtered_pseudo_normed_group M\u2081]\n  [comphaus_filtered_pseudo_normed_group M\u2082]\n  extends M\u2081 \u2192+ M\u2082 :=\n(strict' : \u2200 c x, x \u2208 filtration M\u2081 c \u2192 to_fun x \u2208 filtration M\u2082 c)\n(continuous' : \u2200 c, continuous (pseudo_normed_group.level to_fun strict' c))\n\nend\n\nattribute [nolint doc_blame] comphaus_filtered_pseudo_normed_group_hom.mk\n  comphaus_filtered_pseudo_normed_group_hom.to_add_monoid_hom\n\nnamespace comphaus_filtered_pseudo_normed_group_hom\n\nopen comphaus_filtered_pseudo_normed_group\n\nvariables {M M\u2081 M\u2082 M\u2083 : Type*}\nvariables [comphaus_filtered_pseudo_normed_group M]\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\nvariables [comphaus_filtered_pseudo_normed_group M\u2083]\nvariables (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082)\n\ninstance : has_coe_to_fun (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (\u03bb _, M\u2081 \u2192 M\u2082):=\n\u27e8comphaus_filtered_pseudo_normed_group_hom.to_fun\u27e9\n\n@[simp] lemma coe_mk (f) (h\u2081) (h\u2082) (h\u2083) (h\u2084) :\n  \u21d1(\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084\u27e9 : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) = f :=\nrfl\n\n@[simp] lemma mk_to_monoid_hom (f) (h\u2081) (h\u2082) (h\u2083) (h\u2084) :\n  (\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084\u27e9 :\n    comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082).to_add_monoid_hom =\n    \u27e8f, h\u2081, h\u2082\u27e9 := rfl\n\n@[simp] lemma coe_to_add_monoid_hom : \u21d1f.to_add_monoid_hom = f := rfl\n\n@[simp] lemma map_zero : f 0 = 0 := f.to_add_monoid_hom.map_zero\n\n@[simp] lemma map_add (x y) : f (x + y) = f x + f y := f.to_add_monoid_hom.map_add _ _\n\n@[simp] lemma map_sum {\u03b9 : Type*} (x : \u03b9 \u2192 M\u2081) (s : finset \u03b9) :\n  f (\u2211 i in s, x i) = \u2211 i in s, f (x i) :=\nf.to_add_monoid_hom.map_sum _ _\n\n@[simp] lemma map_sub (x y) : f (x - y) = f x - f y := f.to_add_monoid_hom.map_sub _ _\n\n@[simp] lemma map_neg (x) : f (-x) = -(f x) := f.to_add_monoid_hom.map_neg _\n\n@[simp] lemma map_zsmul (x) (n : \u2124) : f (n \u2022 x) = n \u2022 (f x) := f.to_add_monoid_hom.map_zsmul _ _\n\n/-- Make a profinitely filtered pseudo-normed group hom\nfrom a group hom and a proof that it is bounded and continuous. -/\ndef mk_of_bound (f : M\u2081 \u2192+ M\u2082) (C : \u211d\u22650)\n  (hC : \u2200 c, \u2203 (H : \u2200 x, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 (C * c)),\n    @continuous (filtration M\u2081 c) (filtration M\u2082 (C * c)) _ _ (\u03bb x, \u27e8f x, H x x.2\u27e9)) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ bound' := \u27e8C, \u03bb c, (hC c).some\u27e9,\n  continuous' := \u03bb c\u2081 c\u2082 f\u2080 hf\u2080,\n  begin\n    obtain \u27e8_, H\u27e9 := hC c\u2081,\n    haveI : fact ((C * c\u2081) \u2264 max (C * c\u2081) c\u2082) := \u27e8le_max_left _ _\u27e9,\n    haveI : fact (c\u2082 \u2264 max (C * c\u2081) c\u2082) := \u27e8le_max_right _ _\u27e9,\n    rw (embedding_cast_le c\u2082 (max (C * c\u2081) c\u2082)).continuous_iff,\n    rw (embedding_cast_le (C * c\u2081) (max (C * c\u2081) c\u2082)).continuous_iff at H,\n    convert H using 1,\n    ext, dsimp, rw \u2190 hf\u2080, refl\n  end,\n  .. f }\n\n  /-- Make a profinitely filtered pseudo-normed group hom\nfrom a group hom and a proof that it is bounded and continuous. -/\ndef mk_of_strict (f : M\u2081 \u2192+ M\u2082)\n  (h : \u2200 c, \u2203 (H : \u2200 x, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 c),\n    @continuous (filtration M\u2081 c) (filtration M\u2082 c) _ _ (\u03bb x, \u27e8f x, H x x.2\u27e9)) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\nmk_of_bound f 1 $ \u03bb c,\nbegin\n  obtain \u27e8w, H\u27e9 := h c,\n  refine \u27e8_, _\u27e9,\n  { simpa only [one_mul] },\n  { rwa (embedding_cast_le (1 * c) c).continuous_iff, }\nend\n\n/-- Make a profinitely filtered pseudo-normed group hom\nfrom a group hom and a proof that it is bounded and continuous. -/\nnoncomputable\ndef mk' (f : M\u2081 \u2192+ M\u2082) (h : \u2203 C, \u2200 c, \u2203 (H : \u2200 x, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 (C * c)),\n    @continuous (filtration M\u2081 c) (filtration M\u2082 (C * c)) _ _ (\u03bb x, \u27e8f x, H x x.2\u27e9)) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\nmk_of_bound f h.some h.some_spec\n\n@[simp] lemma coe_mk_of_bound (f : M\u2081 \u2192+ M\u2082) (C) (h) : \u21d1(mk_of_bound f C h) = f := rfl\n\n@[simp] lemma coe_mk' (f : M\u2081 \u2192+ M\u2082) (h) : \u21d1(mk' f h) = f := rfl\n\ndef strict : Prop := \u2200 \u2983c x\u2984, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 c\n\n/-- If `f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082` and `r : \u211d\u22650` then\n  `f.bound_by r` is the statement that `f` maps the `c`th filtrand of `M\u2081` into the `r*c`th\n  filtrand of `M\u2082`. -/\ndef bound_by (C : \u211d\u22650) : Prop := \u2200 \u2983c x\u2984, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 (C * c)\n\nlemma strict_iff_bound_by_one : f.strict \u2194 f.bound_by 1 :=\nby simp only [strict, bound_by, one_mul]\n\nvariables {f}\n\nlemma bound_by.mono {C\u2081} (hf : f.bound_by C\u2081) (C\u2082 : \u211d\u22650) (H : C\u2081 \u2264 C\u2082) : f.bound_by C\u2082 :=\n\u03bb c x hx, filtration_mono (mul_le_mul' H le_rfl) (hf hx)\n\nlemma strict.bound_by_one (hf : f.strict) : f.bound_by 1 :=\nf.strict_iff_bound_by_one.1 hf\n\nlemma bound_by.strict (hf : f.bound_by 1) : f.strict :=\nf.strict_iff_bound_by_one.2 hf\n\nvariables (f)\n\nlemma bound : \u2203 C, f.bound_by C := f.bound'\n\nlemma mk_of_bound_bound_by (f : M\u2081 \u2192+ M\u2082) (C) (h) : (mk_of_bound f C h).bound_by C :=\n\u03bb c, (h c).some\n\nlemma mk_of_strict_strict (f : M\u2081 \u2192+ M\u2082) (h) : (mk_of_strict f h).strict :=\n\u03bb c, (h c).some\n\nprotected lemma continuous \u2983c\u2081 c\u2082\u2984 (f\u2080 : filtration M\u2081 c\u2081 \u2192 filtration M\u2082 c\u2082) (h : \u2200 x, f \u2191x = f\u2080 x) :\n  continuous f\u2080 := f.continuous' f\u2080 h\n\n-- /-- `f.level c` is the function `filtration M\u2081 c \u2192 filtration M\u2082 c`\n-- induced by a `profinitely_filtered_pseudo_normed_group_hom M\u2081 M\u2082`. -/\n-- @[simps] def level (c : \u211d\u22650) (x : filtration M\u2081 c) : filtration M\u2082 c := \u27e8f x, f.strict x.2\u27e9\n\n-- lemma level_continuous (c : \u211d\u22650) : continuous (f.level c) := f.continuous' c\n\nvariables {f g}\n\n@[ext] theorem ext (H : \u2200 x, f x = g x) : f = g :=\nby cases f; cases g; congr'; exact funext H\n\ninstance : has_zero (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\n\u27e8mk_of_bound (0 : M\u2081 \u2192+ M\u2082) 0 (\u03bb c, \u27e8\u03bb _ _, zero_mem_filtration _, @continuous_const _ _ _ _ 0\u27e9)\u27e9\n\ninstance : inhabited (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e80\u27e9\n\nlemma zero_bound_by_zero : (0 : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082).bound_by 0 :=\nmk_of_bound_bound_by _ _ _\n\nlemma coe_inj \u2983f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082\u2984 (h : (f : M\u2081 \u2192 M\u2082) = g) :\n  f = g :=\nby cases f; cases g; cases h; refl\n\n/-- The identity function as `profinitely_filtered_pseudo_normed_group_hom`. -/\n@[simps] def id : comphaus_filtered_pseudo_normed_group_hom M M :=\nmk_of_bound (add_monoid_hom.id _) 1 $\nbegin\n  refine \u03bb c, \u27e8_, _\u27e9,\n  { intros, rwa one_mul },\n  haveI : fact (1 * c \u2264 c) := by { rw one_mul, exact \u27e8le_rfl\u27e9 },\n  rw (embedding_cast_le (1 * c) c).continuous_iff,\n  convert continuous_id, ext, refl\nend\n\n/-- The composition of `profinitely_filtered_pseudo_normed_group_hom`s. -/\n@[simps] noncomputable def comp\n  (g : comphaus_filtered_pseudo_normed_group_hom M\u2082 M\u2083)\n  (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2083 :=\nmk' (g.to_add_monoid_hom.comp f.to_add_monoid_hom) $\nbegin\n  obtain \u27e8Cf, hCf\u27e9 := f.bound,\n  obtain \u27e8Cg, hCg\u27e9 := g.bound,\n  refine \u27e8Cg * Cf, \u03bb c, \u27e8_, _\u27e9\u27e9,\n  { intros x hx, rw mul_assoc, exact hCg (hCf hx) },\n  let f\u2080 : filtration M\u2081 c \u2192 filtration M\u2082 (Cf * c) := \u03bb x, \u27e8f x, hCf x.2\u27e9,\n  have hf\u2080 : continuous f\u2080 := f.continuous _ (\u03bb x, rfl),\n  let g\u2080 : filtration M\u2082 (Cf * c) \u2192 filtration M\u2083 (Cg * (Cf * c)) := \u03bb x, \u27e8g x, hCg x.2\u27e9,\n  have hg\u2080 : continuous g\u2080 := g.continuous _ (\u03bb x, rfl),\n  haveI : fact (Cg * Cf * c \u2264 Cg * (Cf * c)) := by { rw mul_assoc, exact \u27e8le_rfl\u27e9 },\n  rw (embedding_cast_le (Cg * Cf * c) (Cg * (Cf * c))).continuous_iff,\n  exact hg\u2080.comp hf\u2080\nend\n\nlemma bound_by.id : (id : comphaus_filtered_pseudo_normed_group_hom M M).bound_by 1 :=\nbegin\n  intros c x hx,\n  rwa one_mul,\nend\n\nlemma bound_by.comp {f' : comphaus_filtered_pseudo_normed_group_hom M\u2082 M\u2083} {C\u2081 C\u2082 : \u211d\u22650}\n  (hf : f.bound_by C\u2081) (hf' : f'.bound_by C\u2082) : (f'.comp f).bound_by (C\u2082 * C\u2081) :=\nbegin\n  intros c x hx,\n  rw mul_assoc,\n  exact hf' (hf hx),\nend\n\nend comphaus_filtered_pseudo_normed_group_hom\n\nnamespace strict_comphaus_filtered_pseudo_normed_group_hom\n\nopen comphaus_filtered_pseudo_normed_group\n\nvariables {M M\u2081 M\u2082 M\u2083 : Type*}\nvariables [comphaus_filtered_pseudo_normed_group M]\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\nvariables [comphaus_filtered_pseudo_normed_group M\u2083]\nvariables (f g : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082)\n\ninstance : has_coe_to_fun (strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (\u03bb _, M\u2081 \u2192 M\u2082) :=\n\u27e8strict_comphaus_filtered_pseudo_normed_group_hom.to_fun\u27e9\n\n@[simp] lemma coe_mk (f) (h\u2081) (h\u2082) (h\u2083) (h\u2084) :\n  \u21d1(\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084\u27e9 : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) = f :=\nrfl\n\n@[simp] lemma mk_to_monoid_hom (f) (h\u2081) (h\u2082) (h\u2083) (h\u2084) :\n  (\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084\u27e9 :\n    strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082).to_add_monoid_hom =\n    \u27e8f, h\u2081, h\u2082\u27e9 := rfl\n\n@[simp] lemma coe_to_add_monoid_hom : \u21d1f.to_add_monoid_hom = f := rfl\n\n@[simp] lemma map_zero : f 0 = 0 := f.to_add_monoid_hom.map_zero\n\n@[simp] lemma map_add (x y) : f (x + y) = f x + f y := f.to_add_monoid_hom.map_add _ _\n\n@[simp] lemma map_sum {\u03b9 : Type*} (x : \u03b9 \u2192 M\u2081) (s : finset \u03b9) :\n  f (\u2211 i in s, x i) = \u2211 i in s, f (x i) :=\nf.to_add_monoid_hom.map_sum _ _\n\n@[simp] lemma map_sub (x y) : f (x - y) = f x - f y := f.to_add_monoid_hom.map_sub _ _\n\n@[simp] lemma map_neg (x) : f (-x) = -(f x) := f.to_add_monoid_hom.map_neg _\n\n@[simp] lemma map_zsmul (x) (n : \u2124) : f (n \u2022 x) = n \u2022 (f x) := f.to_add_monoid_hom.map_zsmul _ _\n\n/-- Make a strict comphaus filtered pseudo-normed group hom\nfrom a group hom and a proof that it is bounded and continuous. -/\ndef mk' (f : M\u2081 \u2192+ M\u2082)\n  (h : \u2200 c, \u2203 (H : \u2200 x, x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 c),\n      @continuous (filtration M\u2081 c) (filtration M\u2082 c) _ _ (\u03bb x, \u27e8f x, H x x.2\u27e9)) :\n  strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ strict' := \u03bb c x hh, (h c).some x hh,\n  continuous' := \u03bb c, (h c).some_spec,\n  ..f }\n\n@[simp] lemma coe_mk' (f : M\u2081 \u2192+ M\u2082) (h) : \u21d1(mk' f h) = f := rfl\n\nlemma strict \u2983c x\u2984 : x \u2208 filtration M\u2081 c \u2192 f x \u2208 filtration M\u2082 c := f.strict' c x\n\ndef level {c} : filtration M\u2081 c \u2192 filtration M\u2082 c := pseudo_normed_group.level f f.strict c\n\nprotected lemma level_continuous (c) : continuous (pseudo_normed_group.level f f.strict c) :=\n  f.continuous' _\n\n@[simp] protected lemma level_cast_le' {c\u2081 c\u2082} (h : c\u2081 \u2264 c\u2082) (x : filtration M\u2081 c\u2081) :\n  (f.level (cast_le' h x)) = cast_le' h (f.level x) := rfl\n\n@[simp] protected lemma level_zero {c} : f.level (0 : filtration M\u2081 c) = 0 :=\nbegin\n  ext,\n  dsimp,\n  rw \u2190 f.map_zero,\n  refl,\nend\n\n@[simp] protected lemma level_neg {c} (x : filtration M\u2081 c) : f.level (-x) = - (f.level x) :=\nbegin\n  ext,\n  dsimp,\n  erw \u2190 f.map_neg,\n  refl,\nend\n\n@[simp] protected lemma level_add {c\u2081 c\u2082} (x : filtration M\u2081 c\u2081 \u00d7 filtration M\u2081 c\u2082) :\n  f.level (add' x) = add' \u27e8f.level x.1, f.level x.2\u27e9 :=\nbegin\n  ext,\n  dsimp,\n  erw \u2190 f.map_add,\n  refl,\nend\n\n@[simp] lemma coe_level {c} (x : filtration M\u2081 c) : (f.level x : M\u2082) = f x := rfl\n\nvariables {f g}\n\n@[ext] theorem ext (H : \u2200 x, f x = g x) : f = g :=\nby cases f; cases g; congr'; exact funext H\n\ninstance : has_zero (strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\n{ zero :=\n  { strict' := \u03bb c x h, pseudo_normed_group.zero_mem_filtration _,\n    continuous' := \u03bb c, begin\n      let e : filtration M\u2081 c \u2192 filtration M\u2082 c := \u03bb x,\n        \u27e80, pseudo_normed_group.zero_mem_filtration _\u27e9,\n      exact (continuous_const : continuous e),\n    end,\n    ..(0 : M\u2081 \u2192+ M\u2082) } }\n\ninstance : inhabited (strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e80\u27e9\n\nlemma coe_inj \u2983f g : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082\u2984\n  (h : (f : M\u2081 \u2192 M\u2082) = g) : f = g :=\nby cases f; cases g; cases h; refl\n\n/-- The identity function as `profinitely_filtered_pseudo_normed_group_hom`. -/\n@[simps] def id : strict_comphaus_filtered_pseudo_normed_group_hom M M :=\n{ strict' := \u03bb c x h, h,\n  continuous' := \u03bb c, begin\n    convert continuous_id,\n    ext, refl,\n  end,\n  ..(add_monoid_hom.id M) }\n\n/-- The composition of `profinitely_filtered_pseudo_normed_group_hom`s. -/\n@[simps] def comp\n  (g : strict_comphaus_filtered_pseudo_normed_group_hom M\u2082 M\u2083)\n  (f : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2083 :=\n{ strict' := \u03bb c x h, g.strict $ f.strict h,\n  continuous' := \u03bb c, (g.level_continuous c).comp (f.level_continuous c),\n  ..(g.to_add_monoid_hom.comp f.to_add_monoid_hom) }\n\n/-- If `f` is a strict morphism between compact hausdorff filtered pseudo-normed groups\n  then `f.to_chfpsng_hom` is the corresponding morphism where we forget strictness.  -/\n@[simps]\ndef to_chfpsng_hom (f : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\ncomphaus_filtered_pseudo_normed_group_hom.mk_of_strict f.to_add_monoid_hom $\n\u03bb c, \u27e8\u03bb x h, f.strict h, f.level_continuous _\u27e9\n\nlemma to_chfpsng_hom.bound_by_one (f : strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  (to_chfpsng_hom f).bound_by 1 := \u03bb c m hmc,\nbegin\n  rw [one_mul],\n  exact f.strict hmc,\nend\n\nend strict_comphaus_filtered_pseudo_normed_group_hom\n\nnamespace comphaus_filtered_pseudo_normed_group_hom\n\nvariables {M\u2081 M\u2082 : Type*}\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\n\ndef strict.to_schfpsng_hom {f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082}\n  (h : f.strict) :\n  strict_comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ strict' := h,\n  continuous' := \u03bb c, f.continuous _ (\u03bb x, rfl),\n  ..f.to_add_monoid_hom }\n\nend comphaus_filtered_pseudo_normed_group_hom\n\nnamespace punit\n\ninstance : profinitely_filtered_pseudo_normed_group punit :=\n{ filtration := \u03bb _, set.univ,\n  filtration_mono := \u03bb _ _ _, set.subset_univ _,\n  zero_mem_filtration := \u03bb _, set.mem_univ _,\n  neg_mem_filtration := \u03bb _ _ _, set.mem_univ _,\n  add_mem_filtration := \u03bb _ _ _ _ _ _, set.mem_univ _,\n  continuous_add' := \u03bb _ _,  continuous_of_discrete_topology,\n  continuous_neg' := \u03bb _, continuous_of_discrete_topology,\n  continuous_cast_le := \u03bb _ _ _, continuous_of_discrete_topology }\n\nend punit\n\nsection continuity\n\nvariables {M M\u2081 M\u2082 M\u2083 : Type*}\n\nnamespace pseudo_normed_group\n\n/-- Helper function for pseudo-normed groups.\n`pow_incl` is the natural inclusion function `(filtration M c)^n \u2192 M^n`.\nNote that `(filtration M c)^n` is not the same type as `filtration (M^n) c`,\nalthough they are naturally equivalent. -/\ndef pow_incl {n : \u2115} {c : \u211d\u22650} [pseudo_normed_group M] :\n  (filtration M c : Type*)^n \u2192 M^n :=\n\u03bb x j, x j\n\nlemma pow_incl_injective {n : \u2115} {c : \u211d\u22650} [pseudo_normed_group M] :\n  function.injective (@pow_incl M n c _) :=\n\u03bb x y h, funext $ \u03bb j, subtype.coe_injective $ congr_fun h j\n\n@[simp] lemma pow_incl_apply {n : \u2115} {c : \u211d\u22650} [pseudo_normed_group M]\n  (x : (filtration M c : Type*)^n) (j : fin n) :\n  pow_incl x j = x j := rfl\n\nend pseudo_normed_group\n\nopen pseudo_normed_group comphaus_filtered_pseudo_normed_group\n\nvariables [comphaus_filtered_pseudo_normed_group M]\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\nvariables [comphaus_filtered_pseudo_normed_group M\u2083]\n\n/-- A function `f : M\u2081 \u2192 M\u2082` between profinitely filtered pseudo-normed groups\nis continuous if it is continuous when restricted to the filtration sets.\n\nImplementation detail: to avoid diamonds of topologies on `filtration M c`\nwe avoid `topological_space M`.\nWe therefore give a hands on definition of continuity. -/\ndef pfpng_ctu (f : M\u2081 \u2192 M\u2082) : Prop :=\n\u2200 \u2983c\u2081 c\u2082\u2984 (f\u2080 : filtration M\u2081 c\u2081 \u2192 filtration M\u2082 c\u2082)\n  (h : \u2200 x, f \u2191x = f\u2080 x), continuous f\u2080\n\nsection pfpng_ctu\n\nlemma pfpng_ctu_const (y : M\u2082) : pfpng_ctu (\u03bb x : M\u2081, y) :=\nbegin\n  intros c\u2081 c\u2082 f\u2080 h,\n  suffices : f\u2080 = \u03bb x, f\u2080 \u27e80, zero_mem_filtration _\u27e9,\n  { rw this, exact continuous_const },\n  ext1 x,\n  apply subtype.coe_injective,\n  rw [\u2190 h, \u2190 h]\nend\n\nlemma pfpng_ctu.neg {f : M\u2081 \u2192 M\u2082} (hf : pfpng_ctu f) :\n  pfpng_ctu (-f) :=\nbegin\n  intros c\u2081 c\u2082 f\u2080 h,\n  let g := neg' \u2218 f\u2080,\n  have hg : f\u2080 = neg' \u2218 g, { ext, simp [neg_neg] },\n  rw hg,\n  refine (continuous_neg' c\u2082).comp (hf g _),\n  intro x,\n  specialize h x,\n  simp only [g, \u2190 h, neg_neg, pi.neg_apply, neg'_eq]\nend\n\nlemma pfpng_ctu.add {f g : M\u2081 \u2192 M\u2082} (hf : pfpng_ctu f) (hg : pfpng_ctu g)\n  (H : \u2200 c\u2081, \u2203 c\u2082, \u2200 x : filtration M\u2081 c\u2081, f x \u2208 filtration M\u2082 c\u2082) :\n  pfpng_ctu (f + g) :=\nbegin\n  intros c\u2081 c\u2082 fg\u2080 hfg\u2080,\n  obtain \u27e8cf, hcf\u27e9 := H c\u2081,\n  let f\u2080 : filtration M\u2081 c\u2081 \u2192 filtration M\u2082 cf := \u03bb x, \u27e8f x, hcf x\u27e9,\n  have hf\u2080 : \u2200 x, f \u2191x = f\u2080 x := \u03bb x, rfl,\n  have f\u2080_ctu : continuous f\u2080 := hf f\u2080 hf\u2080,\n  let cg := cf + c\u2082,\n  haveI : fact (c\u2082 \u2264 cf + cg) :=\n    \u27e8calc c\u2082 \u2264 cf + c\u2082        : self_le_add_left _ _\n         ... \u2264 cf + (cf + c\u2082) : self_le_add_left _ _\u27e9,\n  have hcg : \u2200 x : filtration M\u2081 c\u2081, g x \u2208 filtration M\u2082 cg,\n  { intros x,\n    have : g x = -(f x) + (f + g) x,\n    { simp only [pi.add_apply, neg_add_cancel_left] },\n    rw this,\n    refine add_mem_filtration (neg_mem_filtration $ hcf x) _,\n    rw hfg\u2080,\n    exact (fg\u2080 x).2 },\n  let g\u2080 : filtration M\u2081 c\u2081 \u2192 filtration M\u2082 cg := \u03bb x, \u27e8g x, hcg x\u27e9,\n  have hg\u2080 : \u2200 x, g \u2191x = g\u2080 x := \u03bb x, rfl,\n  have g\u2080_ctu : continuous g\u2080 := hg g\u2080 hg\u2080,\n  have aux := (f\u2080_ctu.prod_mk g\u2080_ctu),\n  rw (embedding_cast_le c\u2082 (cf + cg)).continuous_iff,\n  convert (continuous_add' cf cg).comp aux using 1,\n  ext, dsimp, rw [\u2190 hfg\u2080, pi.add_apply]\nend\n\nlemma pfpng_ctu.sub {f g : M\u2081 \u2192 M\u2082} (hf : pfpng_ctu f) (hg : pfpng_ctu g)\n  (H : \u2200 c\u2081, \u2203 c\u2082, \u2200 x : filtration M\u2081 c\u2081, f x \u2208 filtration M\u2082 c\u2082) :\n  pfpng_ctu (f - g) :=\nby { rw [sub_eq_add_neg], exact hf.add (hg.neg) H }\n\nvariables (M)\n\nlemma pfpng_ctu_id : pfpng_ctu (@id M) :=\nbegin\n  intros c\u2081 c\u2082 f\u2080 h,\n  haveI : fact (c\u2081 \u2264 max c\u2081 c\u2082) := \u27e8le_max_left _ _\u27e9,\n  haveI : fact (c\u2082 \u2264 max c\u2081 c\u2082) := \u27e8le_max_right _ _\u27e9,\n  have : @cast_le M _ c\u2082 (max c\u2081 c\u2082) _ \u2218 f\u2080 = cast_le, { ext, dsimp, rw \u2190 h, refl },\n  rw [(embedding_cast_le c\u2082 (max c\u2081 c\u2082)).continuous_iff, this],\n  exact (embedding_cast_le _ _).continuous\nend\n\nlemma pfpng_ctu_smul_nat : \u2200 (n : \u2115), pfpng_ctu (\u03bb x : M, n \u2022 x)\n| 0     := by { simp only [zero_smul], exact pfpng_ctu_const 0 }\n| (n+1) := by { simp only [add_smul, one_smul, add_comm],\n                exact (pfpng_ctu_id M).add (pfpng_ctu_smul_nat n) (\u03bb c, \u27e8c, \u03bb x, x.2\u27e9) }\n\nlemma pfpng_ctu_smul_int : \u2200 (n : \u2124), pfpng_ctu (\u03bb x : M, n \u2022 x)\n| (n:\u2115)  := by simpa only [coe_nat_zsmul] using pfpng_ctu_smul_nat M n\n| -[1+n] := by simpa only [zsmul_neg_succ_of_nat] using (pfpng_ctu_smul_nat M (n + 1)).neg\n\nend pfpng_ctu\n\nend continuity\n\nnamespace comphaus_filtered_pseudo_normed_group_hom\n\nvariables {M M\u2081 M\u2082 : Type*}\nvariables [comphaus_filtered_pseudo_normed_group M]\nvariables [comphaus_filtered_pseudo_normed_group M\u2081]\nvariables [comphaus_filtered_pseudo_normed_group M\u2082]\n\n@[simps]\ndef copy (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (g : M\u2081 \u2192 M\u2082) (h : g = f) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ to_fun := g,\n  bound' := by { cases h, exact f.bound },\n  continuous' := by { cases h, exact f.continuous' },\n  .. f.to_add_monoid_hom.copy g h }\n\ndef add (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ to_fun := f + g,\n  bound' :=\n  begin\n    obtain \u27e8Cf, hCf\u27e9 := f.bound,\n    obtain \u27e8Cg, hCg\u27e9 := g.bound,\n    refine \u27e8Cf + Cg, \u03bb c x hx, _\u27e9,\n    rw add_mul,\n    apply add_mem_filtration (hCf hx) (hCg hx),\n  end,\n  continuous' :=\n  begin\n    apply pfpng_ctu.add f.continuous g.continuous,\n    obtain \u27e8Cf, hCf\u27e9 := f.bound,\n    intro c\u2081,\n    refine \u27e8Cf * c\u2081, \u03bb x, hCf x.2\u27e9,\n  end,\n  .. f.to_add_monoid_hom + g.to_add_monoid_hom }\n\ndef neg (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n{ to_fun := -f,\n  bound' :=\n  begin\n    obtain \u27e8Cf, hCf\u27e9 := f.bound,\n    refine \u27e8Cf, \u03bb c x hx, _\u27e9,\n    apply neg_mem_filtration (hCf hx),\n  end,\n  continuous' := pfpng_ctu.neg f.continuous,\n  .. -f.to_add_monoid_hom }\n\ndef sub (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n(f.add g.neg).copy (f - g) (sub_eq_add_neg _ _)\n\ndef nsmul' : \u03a0 (n : \u2115) (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082),\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082\n| 0     f := 0\n| (n+1) f := f.add (nsmul' n f)\n\ndef nsmul (n : \u2115) (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n(f.nsmul' n).copy (n \u2022 f)\nbegin\n  induction n with n ih,\n  { rw zero_smul, refl },\n  { rw [succ_nsmul, ih], refl },\nend\n\ndef zsmul' : \u03a0 (n : \u2124) (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082),\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082\n| (n:\u2115)  f := f.nsmul n\n| -[1+n] f := (f.nsmul (n.succ)).neg\n\ndef zsmul (n : \u2124) (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082 :=\n(f.zsmul' n).copy (n \u2022 f)\nbegin\n  cases n,\n  { rw of_nat_zsmul, refl },\n  { rw zsmul_neg_succ_of_nat, refl },\nend\n\ninstance : has_add (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e8add\u27e9\ninstance : has_neg (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e8neg\u27e9\ninstance : has_sub (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e8sub\u27e9\ninstance has_nsmul : has_smul \u2115 (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e8nsmul\u27e9\ninstance has_zsmul : has_smul \u2124 (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) := \u27e8zsmul\u27e9\n\ninstance : add_comm_group (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\nfunction.injective.add_comm_group\n  comphaus_filtered_pseudo_normed_group_hom.to_add_monoid_hom\n  (\u03bb f g h, by { ext, rw add_monoid_hom.ext_iff at h, exact h x })\n  rfl (\u03bb _ _, rfl) (\u03bb _, rfl) (\u03bb _ _, rfl) (\u03bb _ _, rfl) (\u03bb _ _, rfl)\n\n@[simps]\ndef to_add_monoid_hom_hom : (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) \u2192+ (M\u2081 \u2192+ M\u2082) :=\n{ to_fun := to_add_monoid_hom,\n  map_zero' := rfl,\n  map_add' := \u03bb _ _, rfl }\n\nlemma to_add_monoid_hom_hom_injective : function.injective (@to_add_monoid_hom_hom M\u2081 M\u2082 _ _) :=\n\u03bb f g h, by { ext x, exact add_monoid_hom.congr_fun h x }\n\nlemma bound_by.add {f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082} {Cf Cg : \u211d\u22650}\n  (hf : f.bound_by Cf) (hg : g.bound_by Cg) :\n  (f + g).bound_by (Cf + Cg) :=\n\u03bb c x hx, by { rw add_mul, exact add_mem_filtration (hf hx) (hg hx) }\n\nlemma bound_by.sub {f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082} {Cf Cg : \u211d\u22650}\n  (hf : f.bound_by Cf) (hg : g.bound_by Cg) :\n  (f - g).bound_by (Cf + Cg) :=\n\u03bb c x hx, by { rw add_mul, exact sub_mem_filtration (hf hx) (hg hx) }\n\nlemma bound_by.neg {f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082} {Cf : \u211d\u22650}\n  (hf : f.bound_by Cf) : (-f).bound_by Cf :=\n\u03bb c x hx, neg_mem_filtration (hf hx)\n\nlemma bound_by.nsmul {f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082} {Cf : \u211d\u22650}\n  (hf : f.bound_by Cf) (n : \u2115) : (n \u2022 f).bound_by (n * Cf) :=\n\u03bb c x hx, filtration_mono (mul_assoc _ _ _).ge (nat_smul_mem_filtration n _ _ (hf hx))\n\nlemma bound_by.zsmul {f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082} {Cf : \u211d\u22650}\n  (hf : f.bound_by Cf) (n : \u2124) : (n \u2022 f).bound_by (n.nat_abs * Cf) :=\n\u03bb c x hx, filtration_mono (mul_assoc _ _ _).ge (int_smul_mem_filtration n _ _ (hf hx))\n\n@[simp] lemma add_apply (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (x : M\u2081) :\n  (f + g) x = f x + g x := rfl\n\n@[simp] lemma sub_apply (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (x : M\u2081) :\n  (f - g) x = f x - g x := rfl\n\n@[simp] lemma sum_apply {\u03b9 : Type*} (s : finset \u03b9)\n  (f : \u03b9 \u2192 comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) (x : M\u2081) :\n  (\u2211 i in s, f i) x = \u2211 i in s, (f i x) :=\nbegin\n  classical, apply finset.induction_on s,\n  { simp only [finset.sum_empty], refl },\n  { intros i s his IH,\n    simp only [finset.sum_insert his, add_apply, IH] }\nend\n\nlemma sum_bound_by {\u03b9 : Type*} (s : finset \u03b9)\n  (f : \u03b9 \u2192 comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082)\n  (C : \u03b9 \u2192 \u211d\u22650) (hf : \u2200 i \u2208 s, (f i).bound_by (C i)) :\n  (\u2211 i in s, f i).bound_by (\u2211 i in s, C i) :=\nbegin\n  classical, revert hf, apply finset.induction_on s,\n  { intro, simp only [finset.sum_empty], exact zero_bound_by_zero },\n  { intros i s his IH hf,\n    simp only [finset.sum_insert his],\n    apply (hf _ (s.mem_insert_self i)).add (IH $ \u03bb j hj, hf _ $ finset.mem_insert_of_mem hj) }\nend\n\n@[simp] lemma nsmul_apply {M N : Type*}\n  [comphaus_filtered_pseudo_normed_group M] [comphaus_filtered_pseudo_normed_group N]\n  (n : \u2115) (f : comphaus_filtered_pseudo_normed_group_hom M N) (x : M) :\n  (n \u2022 f) x = n \u2022 (f x) := rfl\n\n@[simp] lemma zero_apply (M N : Type*)\n  [comphaus_filtered_pseudo_normed_group M] [comphaus_filtered_pseudo_normed_group N]\n  (x : M) :\n  (0 : comphaus_filtered_pseudo_normed_group_hom M N) x = 0 := rfl\n\nend comphaus_filtered_pseudo_normed_group_hom\n\nnamespace comphaus_filtered_pseudo_normed_group\n\n/-! ## Products -/\n\nsection pi\n\nvariables {\u03b9 : Type*} (M : \u03b9 \u2192 Type*) [\u03a0 i, comphaus_filtered_pseudo_normed_group (M i)]\n\ninstance pi_topology (c : \u211d\u22650) : topological_space (filtration (\u03a0 i, M i) c) :=\ntopological_space.induced (filtration_pi_equiv M c) $ infer_instance\n\n@[simps apply symm_apply]\ndef filtration_pi_homeo (c : \u211d\u22650) :\n  filtration (\u03a0 i, M i) c \u2243\u209c \u03a0 i, filtration (M i) c :=\n{ to_fun := \u03bb x i, \u27e8x.1 i, x.2 i\u27e9,\n  inv_fun := \u03bb x, \u27e8\u03bb i, x i, \u03bb i, (x i).2\u27e9,\n  left_inv := by { rintro \u27e8x, hx\u27e9, refl },\n  right_inv := by { intro x, ext, refl },\n  continuous_to_fun :=\n    begin\n      rw continuous_def,\n      intros U hU,\n      rw is_open_induced_iff,\n      refine \u27e8U, hU, _\u27e9,\n      refl,\n    end,\n  continuous_inv_fun :=\n    begin\n      rw continuous_def,\n      rintros s \u27e8t, ht, s_eq\u27e9,\n      simpa [\u2190 s_eq] using continuous_def.1 _ t ht,\n      { rw [filtration_pi_equiv, continuous_def],\n        intros U hU,\n        simp only [*, equiv.coe_fn_mk, set.preimage_id',\n        subtype.coe_eta, subtype.coe_mk] },\n    end\n    }\n\ninstance pi_t2 (c : \u211d\u22650) : t2_space (filtration (\u03a0 i, M i) c) :=\nbegin\n  have : t2_space (\u03a0 i, filtration (M i) c) := infer_instance,\n  apply @embedding.t2_space _ _ _ _ this (filtration_pi_homeo M c) (filtration_pi_homeo M c).embedding,\nend\n\n/-\ninstance pi_td (c : \u211d\u22650) : totally_disconnected_space (filtration (\u03a0 i, M i) c) :=\nbegin\n  obtain \u27e8H\u27e9 : totally_disconnected_space (\u03a0 i, filtration (M i) c) := infer_instance,\n  rw [\u2190 homeomorph.range_coe (filtration_pi_homeo M c), \u2190 set.image_univ] at H,\n  exact \u27e8embedding.is_totally_disconnected (filtration_pi_homeo M c).embedding H\u27e9,\nend\n-/\n\ninstance pi_compact (c : \u211d\u22650) : compact_space (filtration (\u03a0 i, M i) c) :=\nbegin\n  obtain \u27e8H\u27e9 : compact_space (\u03a0 i, filtration (M i) c) := infer_instance,\n  rw [\u2190 (homeomorph.compact_image (filtration_pi_homeo M c).symm), set.image_univ,\n    homeomorph.range_coe] at H,\n  exact \u27e8H\u27e9,\nend\n\ndef prod_pi_homeo_pi_prod [\u03a0 i, comphaus_filtered_pseudo_normed_group (M i)]\n(c\u2081 c\u2082 : \u211d\u22650) :\n filtration (\u03a0 i, M i) c\u2081 \u00d7 filtration (\u03a0 i, M i) c\u2082 \u2243\u209c \u03a0 i, (filtration (M i) c\u2081 \u00d7 filtration (M i) c\u2082) :=\n{ to_fun := \u03bb x i, \u27e8\u27e8x.1.1 i, x.1.2 i\u27e9, \u27e8x.2.1 i, x.2.2 i\u27e9\u27e9,\n  inv_fun := \u03bb x, \u27e8\u27e8\u03bb i, (x i).1.1, \u03bb i, (x i).1.2\u27e9, \u27e8\u03bb i, (x i).2.1, \u03bb i, (x i).2.2\u27e9\u27e9,\n  left_inv := by {rintro \u27e8x, hx\u27e9, simp only [subtype.coe_eta, subtype.val_eq_coe]},\n  right_inv := by { intro x, ext; refl},\n  continuous_to_fun :=\n  begin\n      apply continuous_pi,\n      intro i,\n      apply continuous.prod_mk,\n      have h\u2081 := (homeomorph.comp_continuous_iff (filtration_pi_homeo M c\u2081)).mpr continuous_fst,\n      exact (continuous_apply i).comp h\u2081,\n      have h\u2082 := (homeomorph.comp_continuous_iff (filtration_pi_homeo M c\u2082)).mpr continuous_snd,\n      exact (continuous_apply i).comp h\u2082,\n    end,\n  continuous_inv_fun :=\n    begin\n      apply continuous.prod_mk,\n      let f\u2081 : (\u03a0 i, (filtration (M i) c\u2081) \u00d7 (filtration (M i) c\u2082)) \u2192 (filtration (\u03a0 i, M i) c\u2081)\n        := \u03bb x, \u27e8\u03bb (i : \u03b9), (x i).fst.val, \u03bb i, (x i).fst.prop\u27e9,\n      have : continuous ((filtration_pi_homeo M c\u2081) \u2218 f\u2081),\n      { apply continuous_pi,\n        intro i,\n        dsimp [filtration_pi_homeo, f\u2081],\n        simp only [subtype.coe_eta],\n        exact continuous_fst.comp (continuous_apply i), },\n      exact (homeomorph.comp_continuous_iff (filtration_pi_homeo M c\u2081)).mp this,\n      let f\u2082 : (\u03a0 i, (filtration (M i) c\u2081) \u00d7 (filtration (M i) c\u2082)) \u2192 (filtration (\u03a0 i, M i) c\u2082)\n        := \u03bb x, \u27e8\u03bb (i : \u03b9), (x i).snd.val, \u03bb i, (x i).snd.prop\u27e9,\n      have : continuous ((filtration_pi_homeo M c\u2082) \u2218 f\u2082),\n      { apply continuous_pi,\n        intro i,\n        dsimp [filtration_pi_homeo, f\u2082],\n        simp only [subtype.coe_eta],\n        exact continuous_snd.comp (continuous_apply i), },\n      exact (homeomorph.comp_continuous_iff (filtration_pi_homeo M c\u2082)).mp this,\n    end,}\n\n\ninstance pi : comphaus_filtered_pseudo_normed_group (\u03a0 i, M i) :=\n{ continuous_add' :=\n    begin\n      intros c\u2081 c\u2082,\n      rw [\u2190 homeomorph.comp_continuous_iff (filtration_pi_homeo M (c\u2081 + c\u2082)),\n        \u2190 homeomorph.comp_continuous_iff' (prod_pi_homeo_pi_prod M c\u2081 c\u2082).symm],\n      apply continuous_pi,\n      intro i,\n      exact (continuous_add' c\u2081 c\u2082).comp (continuous_apply i),\n    end,\n  continuous_neg' :=\n    begin\n      intro c,\n      rw [\u2190 homeomorph.comp_continuous_iff (filtration_pi_homeo M c),\n        \u2190 homeomorph.comp_continuous_iff' (filtration_pi_homeo M c).symm],\n      apply continuous_pi,\n      intro i,\n      exact (continuous_neg' c).comp (continuous_apply i),\n    end,\n  continuous_cast_le :=\n    begin\n      intros c\u2081 c\u2082 h,\n      rw [\u2190 homeomorph.comp_continuous_iff (filtration_pi_homeo M c\u2082),\n        \u2190 homeomorph.comp_continuous_iff' (filtration_pi_homeo M c\u2081).symm],\n      apply continuous_pi,\n      intro i,\n      have := @continuous_cast_le _ _ _ _ h,\n      exact this.comp (continuous_apply i),\n    end,\n  .. pseudo_normed_group.pi M }\n\nvariables {M}\n\n@[simps]\ndef pi_proj (i : \u03b9) : comphaus_filtered_pseudo_normed_group_hom (\u03a0 i, M i) (M i) :=\ncomphaus_filtered_pseudo_normed_group_hom.mk_of_bound (pi.eval_add_monoid_hom M i) 1 $\nbegin\n  refine \u03bb c, \u27e8\u03bb x hx, by { rw one_mul, exact hx i }, _\u27e9,\n  have := ((continuous_apply i).comp (filtration_pi_homeo M c).continuous),\n  haveI : fact (c \u2264 1 * c) := by { rw one_mul, exact \u27e8le_rfl\u27e9 },\n  rw (embedding_cast_le c (1 * c)).continuous_iff at this,\n  convert this using 0,\nend\n\nlemma pi_proj_bound_by (i : \u03b9) : (@pi_proj _ M _ i).bound_by 1 :=\ncomphaus_filtered_pseudo_normed_group_hom.mk_of_bound_bound_by _ _ _\n\n/-- Universal property of the product of profinitely filtered pseudo-normed groups -/\n@[simps {fully_applied := ff}]\ndef pi_lift {N : Type*} [comphaus_filtered_pseudo_normed_group N]\n  (f : \u03a0 i, comphaus_filtered_pseudo_normed_group_hom N (M i))\n  (hf : \u2203 C, \u2200 i, (f i).bound_by C) :\n  comphaus_filtered_pseudo_normed_group_hom N (\u03a0 i, M i) :=\n{ to_fun := add_monoid_hom.mk_to_pi (\u03bb i, (f i).to_add_monoid_hom),\n  bound' := by { obtain \u27e8C, hC\u27e9 := hf, refine \u27e8C, \u03bb c x hx i, hC i hx\u27e9 },\n  continuous' :=\n  begin\n    intros c\u2081 c\u2082 f\u2080 hf\u2080,\n    rw continuous_induced_rng,\n    apply continuous_pi,\n    intro i,\n    let g := function.eval i \u2218 filtration_pi_homeo M c\u2082 \u2218 f\u2080,\n    refine (f i).continuous g (\u03bb x, _),\n    specialize hf\u2080 x, rw function.funext_iff at hf\u2080,\n    exact hf\u2080 i\n  end,\n  .. add_monoid_hom.mk_to_pi (\u03bb i, (f i).to_add_monoid_hom) }\n\nnoncomputable def pi_map {N : \u03b9 \u2192 Type*} [\u03a0 i, comphaus_filtered_pseudo_normed_group (N i)]\n  (f : \u03a0 i, comphaus_filtered_pseudo_normed_group_hom (M i) (N i))\n  (hf : \u2203 C, \u2200 i, (f i).bound_by C) :\n  comphaus_filtered_pseudo_normed_group_hom (\u03a0 i, M i) (\u03a0 i, N i) :=\npi_lift (\u03bb i, (f i).comp (pi_proj i))\nbegin\n  obtain \u27e8C, hC\u27e9 := hf,\n  refine \u27e8C, \u03bb i c x hx, hC i _\u27e9,\n  have := pi_proj_bound_by i hx,\n  rwa one_mul at this,\nend\n\nend pi\n\nend comphaus_filtered_pseudo_normed_group\n\nnamespace profinitely_filtered_pseudo_normed_group\n\n/-! ## Products -/\n\nsection pi\n\nopen comphaus_filtered_pseudo_normed_group\n\nvariables {\u03b9 : Type*} (M : \u03b9 \u2192 Type*) [\u03a0 i, profinitely_filtered_pseudo_normed_group (M i)]\n\ninstance pi_td (c : \u211d\u22650) : totally_disconnected_space (filtration (\u03a0 i, M i) c) :=\nbegin\n  obtain \u27e8H\u27e9 : totally_disconnected_space (\u03a0 i, filtration (M i) c) := infer_instance,\n  rw [\u2190 homeomorph.range_coe (filtration_pi_homeo M c), \u2190 set.image_univ] at H,\n  exact \u27e8embedding.is_totally_disconnected (filtration_pi_homeo M c).embedding H\u27e9,\nend\n\ninstance pi : profinitely_filtered_pseudo_normed_group (\u03a0 i, M i) :=\n{ ..(infer_instance : comphaus_filtered_pseudo_normed_group _) }\n\nend pi\n\nend profinitely_filtered_pseudo_normed_group\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/profinitely_filtered.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3870317899101077}}
{"text": "import PL.pl provability consistency\n\nuniverse u\n\nnamespace pl\nopen_locale logic_symbol aclogic\nopen logic formula\nvariables {A : Type u}\n\ninductive proof : Theory A \u2192 formula A \u2192 Type u\n| mdp : \u2200 {T p q}, proof T (p \u27f6 q) \u2192 proof T p \u2192 proof T q\n| by_axiom : \u2200 {T p}, p \u2208 T \u2192 proof T p\n| verum : \u2200 {T}, proof T \u22a4\n| imply\u2081 : \u2200 {T p q}, proof T (p \u27f6 q \u27f6 p)\n| imply\u2082 : \u2200 {T p q r}, proof T ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r)\n| contraposition : \u2200 {T p q}, proof T ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p)\n\ndef provable (T : Theory A) (p : formula A) : Prop := nonempty (proof T p)\n\ninstance : axiomatic_classical_logic' (formula A) :=\n{ turnstile := provable,\n  classical := \u03bb T,\n  { modus_ponens := \u03bb p q \u27e8bpq\u27e9 \u27e8bp\u27e9, \u27e8bpq.mdp bp\u27e9,\n    imply\u2081 := \u03bb p q, \u27e8proof.imply\u2081\u27e9, \n    imply\u2082 := \u03bb p q r, \u27e8proof.imply\u2082\u27e9,\n    contraposition := \u03bb p q, \u27e8proof.contraposition\u27e9,\n    provable_top := \u27e8proof.verum\u27e9,\n    bot_eq := by refl,\n    and_def := \u03bb p q, rfl,\n    or_def := \u03bb p q, rfl },\n  by_axiom := \u03bb T p mem, \u27e8proof.by_axiom mem\u27e9 }\n\nopen axiomatic_classical_logic' axiomatic_classical_logic\n\n@[elab_as_eliminator]\ntheorem rec'_on {T : Theory A} {C : formula A \u2192 Prop} {p : formula A} (b : T \u22a2 p)\n  (mdp : \u2200 {p q : formula A} (b\u2081 : T \u22a2 p \u27f6 q) (b\u2082 : T \u22a2 p), C (p \u27f6 q) \u2192 C p \u2192 C q)\n  (by_axiom : \u2200 {p : formula A} (mem : p \u2208 T), C p)\n  (p0 : C \u22a4)\n  (p1 : \u2200 {p q : formula A}, C (p \u27f6 q \u27f6 p))\n  (p2 : \u2200 {p q r : formula A}, C ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r))\n  (p3 : \u2200 {p q : formula A}, C ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p)) :\n  C p :=\nbegin\n  rcases b with \u27e8b\u27e9,\n  induction b,\n  case mdp : T p q bpq bp IHpq IHp { exact mdp \u27e8bpq\u27e9 \u27e8bp\u27e9 (IHpq @mdp @by_axiom) (IHp @mdp @by_axiom) },\n  case by_axiom : T p hp { exact by_axiom hp },\n  case verum : { exact p0 },\n  case imply\u2081 : { exact p1 },\n  case imply\u2082 : { exact p2 },\n  case contraposition { exact p3 }\nend\n\nnamespace proof\nvariables {T : Theory A}\n\ndef weakening {p} (h : proof T p) {U} (ss : T \u2286 U) : proof U p :=\nbegin\n  induction h,\n  case mdp : T p q bpq bp IHpq IHp { exact (IHpq ss).mdp (IHp ss) },\n  case by_axiom : T p hp { exact proof.by_axiom (ss hp) },\n  case verum : { exact proof.verum },\n  case imply\u2081 : { exact proof.imply\u2081 },\n  case imply\u2082 : { exact proof.imply\u2082 },\n  case contraposition { exact proof.contraposition }\nend\n\nend proof\n\nnamespace provable\nvariables {T : Theory A}\n\nlemma weakening {U} {p} (ss : T \u2286 U) (h : T \u22a2 p): U \u22a2 p :=\nby rcases h; exact \u27e8h.weakening ss\u27e9\n\ndef deduction' {p q} (h : insert q T \u22a2 p) : T \u22a2 q \u27f6 p :=\nbegin\n  apply rec'_on h,\n  { intros p r _ _ h\u2081 h\u2082, exact modus_ponens_hyp h\u2081 h\u2082 },\n  { rintros p (rfl | hp), { simp }, { exact hyp_right (by_axiom hp) q } },\n  { simp },\n  { simp },\n  { simp },\n  { simp }\nend\n\ninstance : axiomatic_classical_logic (formula A) :=\n{ deduction' := \u03bb T p q, deduction',\n  weakening := \u03bb T U p, weakening }\n\ntheorem proof_conjunction {T : Theory A} {p} :\n  T \u22a2 p \u2192 \u2203 P : list (formula A), (\u2200 p, p \u2208 P \u2192 T p) \u2227 \u2205 \u22a2 P.conjunction \u27f6 p := \u03bb h,\nbegin\n  apply rec'_on h,\n  { rintros p q b\u2081 b\u2082 \u27e8P\u2081, IH\u2081, b0\u2081\u27e9 \u27e8P\u2082, IH\u2082, b0\u2082\u27e9, refine \u27e8P\u2081 ++ P\u2082, _, _\u27e9,\n    { simp, rintros r (hr | hr), exact IH\u2081 r hr, exact IH\u2082 r hr },\n    { have : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 P\u2082.conjunction, from deduction.mpr (list_conjunction_weakening (by simp)),\n      have lmm\u2081 : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 p,\n        from (show _ \u22a2 P\u2082.conjunction \u27f6 p, from weakening_insert b0\u2082 _) \u2a00 this,\n      have : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 P\u2081.conjunction, from deduction.mpr (list_conjunction_weakening (by simp)),\n      have lmm\u2082 : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 p \u27f6 q,\n      from (show _ \u22a2 P\u2081.conjunction \u27f6 p \u27f6 q, from weakening_insert b0\u2081 _) \u2a00 this,\n      refine deduction.mp (lmm\u2082 \u2a00 lmm\u2081) } },\n  { rintros p hp, refine \u27e8[p], by simpa using hp, by simp\u27e9 },\n  { refine \u27e8[], by simp, by simp\u27e9 },\n  { intros, refine \u27e8[], by simp, by simp\u27e9 },\n  { intros, refine \u27e8[], by simp, by simp\u27e9 },\n  { intros, refine \u27e8[], by simp, by simp\u27e9 }\nend\n\ninstance : Theory.has_finite_character (formula A) :=\nTheory.finite_character_of_finite_provable (formula A) (\u03bb T p, proof_conjunction)\n\ndef homomorphism {F : Type*} [has_logic_symbol F] [axiomatic_classical_logic F] (f : formula A \u2192\u2097 F) {p} :\n  T \u22a2 p \u2192 f '' T \u22a2 f p :=\nbegin\n  intros b,\n  apply rec'_on b,\n  { simp, intros p q h\u2081 h\u2082 IH\u2081 IH\u2082, exact IH\u2081 \u2a00 IH\u2082 },\n  { intros p hp, exact by_axiom (set.mem_image_of_mem f hp) },\n  { simp },\n  { simp },\n  { simp },\n  { simp }\nend\n\nend provable\n\nend pl", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/PL/deduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.38703178250911585}}
{"text": "/-\nCopyright (c) 2014 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.nat.cast.with_top\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Order.Monoid.WithTop\nimport Mathbin.Data.Nat.Basic\n\n/-!\n# Lemma about the coercion `\u2115 \u2192 with_bot \u2115`.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nAn orphaned lemma about casting from `\u2115` to `with_bot \u2115`,\nexiled here to minimize imports to `data.rat.order` for porting purposes.\n-/\n\n\n#print Nat.cast_withTop /-\ntheorem Nat.cast_withTop (n : \u2115) : @coe \u2115 (WithTop \u2115) (@coeToLift _ _ Nat.castCoe) n = n :=\n  rfl\n#align nat.cast_with_top Nat.cast_withTop\n-/\n\n#print Nat.cast_withBot /-\ntheorem Nat.cast_withBot (n : \u2115) : @coe \u2115 (WithBot \u2115) (@coeToLift _ _ Nat.castCoe) n = n :=\n  rfl\n#align nat.cast_with_bot Nat.cast_withBot\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Nat/Cast/WithTop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3870317751081238}}
{"text": "theorem eq_iff_true_of_subsingleton [Subsingleton \u03b1] (x y : \u03b1) : x = y \u2194 True :=\n  \u27e8fun _ => \u27e8\u27e9, fun _ => (Subsingleton.elim ..)\u27e9\n\nattribute [simp] eq_iff_true_of_subsingleton in\nexample : True := trivial\n\nstructure Func' (\u03b1 : Sort _) (\u03b2 : Sort _) :=\n(toFun    : \u03b1 \u2192 \u03b2)\n\ndef r : Func' \u03b1 \u03b1 := \u27e8id\u27e9\n\n@[simp] theorem r_toFun {\u03b1 : Sort u_1} (a : \u03b1) : Func'.toFun r a = id a := rfl\n\nexample (x y : \u03b1) (h : x = y) : r.toFun x = y := by simp <;> rw [h]\n\ntheorem noissue (x y : \u03b1) (h : x = y) : r.toFun x = y := by simp <;> rw [h]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1829.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.38690039569780815}}
{"text": "import category_theory.limits.shapes.pullbacks\n\nnamespace category_theory\n\nopen limits\n\nnoncomputable theory\n\nvariables {C : Type*} [category C] {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [has_pullback (f \u226b g) g]\n\ndef pullback.lift_comp :=\npullback.lift (\ud835\udfd9 _) f (category.id_comp $ f \u226b g)\n\n@[simp, reassoc]\nlemma pullback.lift_comp_fst : pullback.lift_comp f g \u226b pullback.fst = \ud835\udfd9 _ :=\npullback.lift_fst _ _ _\n\nlemma pullback.lift_comp_snd : pullback.lift_comp f g \u226b pullback.snd = f :=\npullback.lift_snd _ _ _\n\ninstance : is_split_mono (pullback.lift_comp f g) :=\n\u27e8\u27e8\u27e8_, pullback.lift_comp_fst f g\u27e9\u27e9\u27e9 \n\nlocal attribute [instance] has_pullback_of_right_iso\n\nlemma pullback.fst_lift_comp_has_pullback_aux :\n  has_pullback (f \u226b g) (\ud835\udfd9 Y \u226b g) :=\nby rwa category.id_comp\n\nlocal attribute [instance] pullback.fst_lift_comp_has_pullback_aux\n\nlemma pullback.fst_lift_comp :\n  pullback.fst \u226b pullback.lift_comp f g =\n    pullback.map_desc f (\ud835\udfd9 _) g \u226b (pullback.congr_hom rfl (category.id_comp _)).hom :=\nbegin\n  apply pullback.hom_ext; simp only [pullback.congr_hom_hom, pullback.lift_fst,\n    pullback.lift_snd, category.assoc, category.comp_id, pullback.condition, pullback.lift_comp],\nend\n\ndef pullback.lift_comp_iso_map_desc : \n  arrow.mk (pullback.lift_comp f g) \u2245 arrow.mk (pullback.map_desc f (\ud835\udfd9 _) g) :=\n(arrow.iso_mk' _ _ (as_iso pullback.fst : _) (pullback.congr_hom rfl (category.id_comp _))\n  (pullback.fst_lift_comp f g)).symm\n\nend category_theory", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/for_mathlib/pullback_lift_comp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.38681822411169525}}
{"text": "import language\nimport data.nat.parity\n\nnamespace first_order\n\nsection prf\n\nvariables {L : language} {n m : \u2115}\n\ninductive Prf (A : Type) [has_coe A (formula L)] : list (formula L) \u2192 formula L \u2192 Prop\n| Axiom : \u2200 {\u0393 : list (formula L)} (a : A) \u03c6, a = \u03c6 \u2192 Prf \u0393 \u03c6\n| Assumption : \u2200 {\u0393 : list (formula L)} n \u03c6, \u0393.nth n = some \u03c6 \u2192 Prf \u0393 \u03c6\n| Bot_elim : \u2200 {\u0393 : list (formula L)} \u03c6, Prf \u0393 F \u2192 Prf \u0393 \u03c6\n| Not_elim : \u2200 {\u0393 : list (formula L)} \u03c6 \u03c8, Prf \u0393 \u223c\u03c6 \u2192 Prf \u0393 \u03c6 \u2192 Prf \u0393 \u03c8\n| By_contradiction : \u2200 {\u0393 : list (formula L)} \u03c6, Prf (\u223c\u03c6::\u0393) F \u2192 Prf \u0393 \u03c6\n| Or_intro_left : \u2200 {\u0393 : list (formula L)} \u03c6 \u03c8, Prf \u0393 \u03c6 \u2192 Prf \u0393 (\u03c6 or \u03c8)\n| Or_intro_right : \u2200 {\u0393 : list (formula L)} \u03c6 \u03c8, Prf \u0393 \u03c8 \u2192 Prf \u0393 (\u03c6 or \u03c8)\n| Or_elim : \u2200 {\u0393 : list (formula L)} \u03c6 \u03c8 \u03c7, Prf \u0393 (\u03c6 or \u03c8) \u2192 Prf (\u03c6::\u0393) \u03c7 \u2192 Prf (\u03c8::\u0393) \u03c7 \u2192 Prf \u0393 \u03c7\n| All_intro : \u2200 {\u0393 : list (formula L)} \u03c6 n m, var_not_free_in_axioms_context m A \u0393 \u2192 \n    Prf \u0393 (replace_formula_with n (term.var m) \u03c6) \u2192 Prf \u0393 (formula.all n \u03c6)\n| All_elim : \u2200 {\u0393 : list (formula L)} n t \u03c6 \u03c8, Prf \u0393 (formula.all n \u03c6) \u2192 \n    substitutable_for t n \u03c6 \u2192 Prf ((replace_formula_with n t \u03c6) :: \u0393) \u03c8 \u2192 Prf \u0393 \u03c8\n| Cut : \u2200 {\u0393 : list (formula L)} \u03c6 \u03c8, Prf \u0393 \u03c6 \u2192 Prf (\u03c6::\u0393) \u03c8 \u2192 Prf \u0393 \u03c8\n\nopen Prf\n\nnotation A ` \u2223 ` \u0393` \u22a2 `\u03c6 := Prf A \u0393 \u03c6\n\nvariables {p p\u2081 p\u2082 q q\u2081 q\u2082 r s \u03c6 : formula L}\n\nvariables {\u0393 \u03b3 : list (formula L)} {P Q : list (formula L)} {A : Type} [has_coe A (formula L)]\n\n-- Weakening\nlemma nth_append_some : \u2200 {A : Type} {l1 l2 : list A} n x, l1.nth n = some x \u2192 (l1 ++ l2).nth n = some x\n| A (y :: l1) l2 0 x h := h\n| A (y :: l1) l2 (n+1) x h := @nth_append_some A l1 l2 n x h\n\nlemma nth_cons_some : \u2200 {A : Type} {l1 l2 : list A},\n  (\u2200 n x, l1.nth n = some x \u2192 \u2203 m, l2.nth m = some x) -> \u2200 y,\n  (\u2200 n x, (y :: l1).nth n = some x \u2192 \u2203 m, (y :: l2).nth m = some x) :=\n  begin\n    intros, cases n, { existsi 0, assumption },\n    { simp at *, cases (\u1fb0 _ _ \u1fb0_1),\n        existsi w.succ, assumption },\n  end\n\n\n\nlemma nth_append_r : \u2200 {A : Type} {l1 l2 : list A} n, (l1 ++ l2).nth (n + l1.length) = l2.nth n := begin\n  intros A l1 l2 n,\n  induction l1,\n  simp,\n  simp,\n  rw \u2190 add_assoc,\n  rw \u2190 add_comm,\n  rw nat.one_add,\n  simp,\n  apply l1_ih,\nend\n\ndef weakening : (forall n x, \u03b3.nth n = some x -> \u2203 m, \u0393.nth m = some x) \u2192\n  (A\u2223\u03b3 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 p) := begin\n    intros h \u03b3p, revert \u0393 h,\n    induction \u03b3p, all_goals { intros \u0393 h },\n      sorry,\n      { cases (h _ _ \u03b3p_\u1fb0), apply Assumption, assumption, },\n      apply Bot_elim, apply \u03b3p_ih, assumption,\n      apply Not_elim,\n        { apply \u03b3p_ih_\u1fb0, assumption },\n        { apply \u03b3p_ih_\u1fb0_1, assumption },\n      apply By_contradiction, apply \u03b3p_ih, apply nth_cons_some, assumption,\n      apply Or_intro_left, apply \u03b3p_ih, assumption,\n      apply Or_intro_right, apply \u03b3p_ih, assumption,\n      apply Or_elim,\n        { apply \u03b3p_ih_\u1fb0, assumption },\n        { apply \u03b3p_ih_\u1fb0_1, apply nth_cons_some, assumption },\n        { apply \u03b3p_ih_\u1fb0_2, apply nth_cons_some, assumption },\n      admit,\n      admit,\n      apply Cut,\n        { apply \u03b3p_ih_\u1fb0, assumption },\n        { apply \u03b3p_ih_\u1fb0_1, apply nth_cons_some, assumption },\n  end\n\ndef weakening_append : (A\u2223\u03b3 \u22a2 p) \u2192 (A\u2223(\u03b3 ++ \u0393) \u22a2 p) := begin\n    apply weakening,\n    intros, existsi n,\n    apply nth_append_some, assumption,\n  end\n\n-- Converting natural deduction rules into sequent calculus rules\ndef R_ : (A\u2223[p] \u22a2 q) \u2192 ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) := begin\n    intros pq \u0393p,\n    apply Cut,\n    apply \u0393p,\n    apply weakening _ pq,\n    intros n \u03c6 pn_eq_\u03c6,\n    cases n, simp at pn_eq_\u03c6,\n    existsi 0, simp, assumption,\n    contradiction,\n  end\n\nvariables (h : formula L) (l : list (formula L))\n\ndef To_Right_Rule_List : (A\u2223(h::l) \u22a2 q) \u2192 \u2200 \u0393 : (list (formula L)), (A\u2223\u0393 \u22a2 h) \u2192 (A\u2223(l ++ \u0393) \u22a2 q) := begin\n    intros h1 \u0393 h2,\n    apply Cut,\n    apply weakening _ h2,\n    intros n1 h3 h4,\n    existsi (n1 + l.length),\n    rw nth_append_r,\n    apply h4,\n    apply weakening,\n    intros n2 h5 h6,\n    apply nth_cons_some,\n    intros n3 h7 h8,\n    existsi n3,\n    apply nth_append_some,\n    apply h8,\n    apply h6,\n    apply h1,\n  end\n\n\ndef L_ : (A\u2223[p] \u22a2 q) \u2192 (A\u2223\u0393 \u22a2 p) \u2192 (A\u2223(q::\u0393) \u22a2 r) \u2192 (A\u2223\u0393 \u22a2 r) := begin\n    intros h1 h2 h3,\n    apply Cut,\n    apply R_,\n    apply h1,\n    apply h2,\n    apply h3,\n  end\n\ndef L_R_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 (A\u2223\u0393 \u22a2 p) \u2192 (A\u2223(q::\u0393) \u22a2 r) \u2192 (A\u2223\u0393 \u22a2 r) := begin\n  intros h1 h2 h3,\n  apply Cut,\n  apply h1,\n  apply h2,\n  apply h3,\nend\n\ndef Proves_impl : (A\u2223(p::\u0393) \u22a2 q) \u2192 (A\u2223\u0393 \u22a2 (p \u21d2 q)) := begin\n  intro h, simp,\n  apply By_contradiction,\n  apply Not_elim,\n  apply Assumption 0, refl,\n  apply Cut p,\n  apply By_contradiction,\n  apply Not_elim,\n  apply Assumption 1, refl,\n  apply Or_intro_left,\n  apply Assumption 0, refl,\n  apply Or_intro_right,\n  apply weakening _ h,\n  intros n \u03c6 h1,\n  cases n,\n  existsi 0,\n  simp at *,\n  apply h1,\n  existsi n.succ.succ,\n  simp at *,\n  apply h1,\nend\n\ndef R_Not_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 \u223cp) \u2192 (A\u2223\u0393 \u22a2 \u223cq)) := sorry\n\ndef R_Eq_Not_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 \u223cp) \u2194 (A\u2223\u0393 \u22a2 \u223cq)) := sorry\n\ndef R_Left_Or_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (p or r)) \u2192 (A\u2223\u0393 \u22a2 (q or r))) := sorry\n\ndef R_Eq_Left_Or_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (p or r)) \u2194 (A\u2223\u0393 \u22a2 (q or r))) := sorry\n\ndef R_Right_Or_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (r or p)) \u2192 (A\u2223\u0393 \u22a2 (r or q))) := sorry\n\ndef R_Eq_Right_Or_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (r or p)) \u2194 (A\u2223\u0393 \u22a2 (r or q))) := sorry\n\ndef R_Or_ : ((A\u2223\u0393 \u22a2 p\u2081) \u2192 (A\u2223\u0393 \u22a2 q\u2081)) \u2192 ((A\u2223\u0393 \u22a2 p\u2082) \u2192 (A\u2223\u0393 \u22a2 q\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (p\u2081 or p\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (q\u2081 or q\u2082)))) := sorry\n\ndef R_Eq_Or_ : ((A\u2223\u0393 \u22a2 p\u2081) \u2194 (A\u2223\u0393 \u22a2 q\u2081)) \u2192 ((A\u2223\u0393 \u22a2 p\u2082) \u2194 (A\u2223\u0393 \u22a2 q\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (p\u2081 or p\u2082)) \u2194 ((A\u2223\u0393 \u22a2 (q\u2081 or q\u2082)))) := sorry\n\ndef R_Left_And_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (p and r)) \u2192 (A\u2223\u0393 \u22a2 (q and r))) := sorry\n\ndef R_Eq_Left_And_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (p and r)) \u2194 (A\u2223\u0393 \u22a2 (q and r))) := sorry\n\ndef R_Right_And_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (r and p)) \u2192 (A\u2223\u0393 \u22a2 (r and q))) := sorry\n\ndef R_And_ : ((A\u2223\u0393 \u22a2 p\u2081) \u2192 (A\u2223\u0393 \u22a2 q\u2081)) \u2192 ((A\u2223\u0393 \u22a2 p\u2082) \u2192 (A\u2223\u0393 \u22a2 q\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (p\u2081 and p\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (q\u2081 and q\u2082)))) := sorry\n\ndef R_Eq_And_ : ((A\u2223\u0393 \u22a2 p\u2081) \u2194 (A\u2223\u0393 \u22a2 q\u2081)) \u2192 ((A\u2223\u0393 \u22a2 p\u2082) \u2194 (A\u2223\u0393 \u22a2 q\u2082)) \u2192 ((A\u2223\u0393 \u22a2 (p\u2081 and p\u2082)) \u2194 ((A\u2223\u0393 \u22a2 (q\u2081 and q\u2082)))) := sorry\n\ndef R_All_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (all n p)) \u2192 (A\u2223\u0393 \u22a2 (all n q))) := sorry\n\ndef R_Eq_All_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (all n p)) \u2194 (A\u2223\u0393 \u22a2 (all n q))) := sorry\n\ndef R_Ex_ : ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (exi n p)) \u2192 (A\u2223\u0393 \u22a2 (exi n q))) := sorry \n\ndef R_Eq_Ex_ : ((A\u2223\u0393 \u22a2 p) \u2194 (A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (exi n p)) \u2194 (A\u2223\u0393 \u22a2 (exi n q))) := sorry \n\n-- Basic intro and elim ND rules\ndef Not_intro : (A\u2223[(p \u21d2 F)] \u22a2 \u223cp) := begin\n    apply Or_elim,\n    apply Assumption 0, refl,\n    apply Assumption 0, refl,\n    apply Bot_elim,\n    apply Assumption 0, refl,\n  end\n\ndef Not_impl : A\u2223[\u223cp] \u22a2 (p \u21d2 F) := begin\n  apply Or_intro_left,\n  apply Assumption 0, refl,\nend\n\ndef Impl_not : A\u2223[(p \u21d2 F)] \u22a2 \u223cp := begin\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply Assumption 0, refl,\n  apply R_ Not_intro,\n  apply Assumption 1, refl,\nend\n\ndef Double_negation_elim : A\u2223[\u223c\u223cp] \u22a2 p := begin\n    apply (By_contradiction p),\n    apply Not_elim,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n  end\n\ndef Double_negation_intro : A\u2223[p] \u22a2 \u223c\u223cp := begin\n    apply R_ Not_intro,\n    apply Proves_impl,\n    apply Not_elim,\n    apply Assumption 0, refl,\n    apply Assumption 1, refl,\n  end\n\ndef Top_intro : A\u2223\u0393 \u22a2 T := begin\n  apply By_contradiction,\n  apply R_ Double_negation_elim,\n  apply Assumption 0, refl,\nend\n\ndef Absurd : (A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 \u223cp) \u2192 (A\u2223\u0393 \u22a2 q) := sorry\n\ndef Impl_elim_ : A\u2223[p, p \u21d2 q] \u22a2 q := begin\n    apply Or_elim,\n    apply Assumption 1, refl,\n    apply Not_elim,\n    apply Assumption 0, refl,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n  end\n\ndef Impl_elim : (A\u2223\u0393 \u22a2 p) \u2192 ((A\u2223\u0393 \u22a2 (p \u21d2 q)) \u2192 ((A\u2223\u0393 \u22a2 q))) := begin\n    intros h1 h2,\n    apply Cut,\n    apply h2,\n    apply To_Right_Rule_List _ _ Impl_elim_,\n    assumption,\n  end\n\ndef And_intro : (A\u2223[p, q] \u22a2 (p and q)) := begin\n    apply R_ Not_intro,\n    apply Proves_impl,\n    apply Or_elim,\n    apply Assumption 0, refl,\n    apply Not_elim,\n    apply Assumption 0, refl,\n    apply Assumption 2, refl,\n    apply Not_elim,\n    apply Assumption 0, refl,\n    apply Assumption 3, refl,\n  end\n\ndef And_intro_R : (A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 q) \u2192 (A\u2223\u0393 \u22a2 (p and q)) := begin\n    intros h1 h2,\n    apply Cut,\n    apply h2,\n    apply To_Right_Rule_List _ _ And_intro,\n    apply h1,\n  end\n\ndef And_elim_left : A\u2223[(p and q)] \u22a2 p := begin\n      apply By_contradiction,\n      apply Not_elim,\n      apply Assumption 1, refl,\n      apply Or_intro_left,\n      apply Assumption 0, refl,\n    end\n\ndef And_elim_right : A\u2223[(p and q)] \u22a2 q := begin\n      apply By_contradiction,\n      apply Not_elim,\n      apply Assumption 1, refl,\n      apply Or_intro_right,\n      apply Assumption 0, refl,\n    end\n\ndef NonContradiction : A\u2223[(p and \u223cp)] \u22a2 F := begin\n    apply Not_elim,\n    apply And_elim_right,\n    apply And_elim_left,\n  end\n\n-- DeMorgan laws\ndef DeMorganNotAnd : A\u2223[\u223c(p and q)] \u22a2 (\u223cp or \u223cq) := begin\n    apply By_contradiction,\n    apply Not_elim,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n  end\n\ndef DeMorganNotOr : A\u2223[\u223c(p or q)] \u22a2 (\u223cp and \u223cq) := begin\n    apply R_ Not_intro,\n    apply Proves_impl,\n    apply Not_elim,\n    apply Assumption 1, refl,\n    apply Or_elim,\n    apply Assumption 0, refl,\n    apply Or_intro_left,\n    apply R_ Double_negation_elim,\n    apply Assumption 0, refl,\n    apply Or_intro_right,\n    apply R_ Double_negation_elim,\n    apply Assumption 0, refl,\n  end\n\ndef ExcludedMiddle : A\u2223[p] \u22a2 (q or \u223cq) := begin\n  apply By_contradiction,\n  apply L_ DeMorganNotOr,\n  apply Assumption 0, refl,\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply Not_elim,\n  apply Assumption 2, refl,\n  apply Or_intro_right,\n  apply Assumption 0, refl,\nend\n\ndef DeMorganOr : A\u2223[(\u223cp or \u223cq)] \u22a2 \u223c(p and q) := begin\n    apply R_ Not_intro,\n    apply Proves_impl,\n    apply Or_elim,\n    apply Assumption 1, refl,\n    apply Not_elim,\n    apply Assumption 0, refl,\n    apply L_ And_elim_left,\n    apply Assumption 1, refl,\n    apply Not_elim,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n    apply L_ And_elim_right,\n    apply Assumption 1, refl,\n    apply Not_elim,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n  end\n\ndef DeMorganAnd : A\u2223[(\u223cp and \u223cq)] \u22a2 \u223c(p or q) := begin\n  apply By_contradiction,\n  apply L_ Double_negation_elim,\n  apply Assumption 0, refl,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply L_ And_elim_left,\n  apply Assumption 3, refl,\n  apply Not_elim,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\n  apply L_ And_elim_right,\n  apply Assumption 3, refl,\n  apply Not_elim,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\nend\n\n-- Distribution of and and or\n-- TODO: Right now there are 8 distribution rules, is there any way to simplify this?\ndef DistributionAndOrInLeft : A\u2223[(r and (p or q))] \u22a2 ((r and p) or (r and q)) := begin\n    apply L_ And_elim_right,\n    apply Assumption 0, refl,\n    apply L_ And_elim_left,\n    apply Assumption 1, refl,\n    apply Or_elim,\n    apply Assumption 1, refl,\n    apply Or_intro_left,\n    apply And_intro_R,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n    apply Or_intro_right,\n    apply And_intro_R,\n    apply Assumption 1, refl,\n    apply Assumption 0, refl,\n  end\n\ndef DistributionAndOrInRight : A\u2223[((p or q) and r)] \u22a2 ((p and r) or (q and r)) := begin\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply L_ And_elim_right,\n  apply Assumption 1, refl,\n  apply Or_elim,\n  apply Assumption 1, refl,\n  apply Or_intro_left,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\n  apply Or_intro_right,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\nend\n\ndef DistributionAndOrOutLeft : A\u2223[((r and p) or (r and q))] \u22a2 (r and (p or q)) := begin\n  apply And_intro_R,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  any_goals { by {\n    apply L_ And_elim_left,\n    repeat { apply Assumption 0, refl, }\n  } },\n  apply Or_elim,\n  apply Assumption 0, refl,\n  any_goals { by {\n    apply L_ And_elim_right,\n    apply Assumption 0, refl,\n    { apply Or_intro_left, apply Assumption 0, refl } \n      <|>\n    { apply Or_intro_right, apply Assumption 0, refl }\n  } },\nend\n\ndef DistributionAndOrOutRight : A\u2223[((p and r) or (q and r))] \u22a2 ((p or q) and r) := begin\n  apply And_intro_R,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  any_goals { \n    apply L_ And_elim_left,\n    apply Assumption 0, refl,\n    { apply Or_intro_left, apply Assumption 0, refl }\n      <|>\n    { apply Or_intro_right, apply Assumption 0, refl }\n  },\n  apply Or_elim,\n  apply Assumption 0, refl,\n  all_goals {\n    apply L_ And_elim_right,\n    repeat { apply Assumption 0, refl },\n  },\nend\n\ndef DistributionOrAndInLeft : A\u2223[(r or (p and q))] \u22a2 ((r or p) and (r or q)) := begin\n    apply Or_elim, \n    apply Assumption 0, refl,\n    apply And_intro_R,\n    apply Or_intro_left,\n    apply Assumption 0, refl,\n    apply Or_intro_left,\n    apply Assumption 0, refl,\n    apply And_intro_R,\n    apply L_ And_elim_left,\n    apply Assumption 0, refl,\n    apply Or_intro_right,\n    apply Assumption 0, refl,\n    apply L_ And_elim_right,\n    apply Assumption 0, refl,\n    apply Or_intro_right,\n    apply Assumption 0, refl,\n  end\n\ndef DistributionOrAndOutRight : A\u2223[((p or r) and (q or r))] \u22a2 ((p and q) or r) := begin\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply L_ And_elim_right,\n  apply Assumption 1, refl,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply Or_elim,\n  apply Assumption 2, refl,\n  apply Or_intro_left,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\n  all_goals {\n    apply Or_intro_right,\n    apply Assumption 0, refl\n  },\nend\n\ndef DistributionOrAndOutLeft : A\u2223[((r or p) and (r or q))] \u22a2 (r or (p and q)) := begin\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply L_ And_elim_right,\n  apply Assumption 1, refl,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply Or_intro_left,\n  apply Assumption 0, refl,\n  apply Or_elim,\n  apply Assumption 2, refl,\n  apply Or_intro_left,\n  apply Assumption 0, refl,\n  apply Or_intro_right,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\nend\n\ndef DistributionOrAndInRight : A\u2223[((p or q) and r)] \u22a2 ((p and r) or (q and r)) := begin\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply L_ And_elim_right,\n  apply Assumption 1, refl,\n  apply Or_elim,\n  apply Assumption 1, refl,\n  apply Or_intro_left,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\n  apply Or_intro_right,\n  apply And_intro_R,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\nend\n\n-- Commutativity rules \ndef Or_comm : A\u2223[(p or q)] \u22a2 (q or p) := begin\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply Or_intro_right,\n  apply Assumption 0, refl,\n  apply Or_intro_left,\n  apply Assumption 0, refl,\nend\n\ndef And_comm_ : A\u2223[(p and q)] \u22a2 (q and p) := begin\n  apply And_intro_R,\n  apply L_ And_elim_right,\n  apply Assumption 0, refl,\n  apply Assumption 0, refl,\n  apply L_ And_elim_left,\n  apply Assumption 0, refl,\n  apply Assumption 0, refl,\nend\n\ndef AndProves : ((A\u2223\u0393 \u22a2 p)) \u2227 ((A\u2223\u0393 \u22a2 q)) \u2192 ((A\u2223\u0393 \u22a2 (p and q))) := begin\n    intros h,\n    apply And_intro_R,\n    apply and.elim_left h,\n    apply and.elim_right h,\n  end\n\ndef Contrapose : A\u2223[(p \u21d2 q)] \u22a2 (\u223cq \u21d2 \u223cp) := begin\n  simp,\n  apply Or_elim,\n  apply Assumption 0, refl,\n  apply Or_intro_right,\n  apply Assumption 0, refl,\n  apply Proves_impl,\n  apply Not_elim,\n  apply Assumption 0, refl,\n  apply Assumption 1, refl,\nend\n\ndef Ex_intro (n : \u2115) (t : term L) (\u03c6 : formula L) :\n    (A\u2223\u0393 \u22a2 (replace_formula_with n t \u03c6)) \u2192 (A\u2223\u0393 \u22a2 (exi n \u03c6)) := sorry\n\n-- | All_intro : \u2200 {\u0393 : list (formula L)} \u03c6 n m, var_not_free_in_axioms_context m A \u0393 \u2192 \n--     Prf \u0393 (replace_formula_with n (term.var m) \u03c6) \u2192 Prf \u0393 (formula.all n \u03c6)\n\ndef Ex_elim (n : \u2115) (t : term L) (\u03c6 : formula L) (\u03c8 : formula L) : substitutable_for t n \u03c6 \u2192 \n    (A\u2223\u0393 \u22a2 (exi n \u03c6)) \u2192 (A\u2223(replace_formula_with n t \u03c6)::\u0393 \u22a2 \u03c8) \u2192 (A\u2223\u0393 \u22a2 \u03c8) := sorry\n\ndef Ex_rename (n m : \u2115) (\u03c6 : formula L) : (A\u2223\u0393 \u22a2 (exi n \u03c6)) \u2192 (A\u2223\u0393 \u22a2 (exi m \u03c6)) := sorry\n\ndef All_To_Ex : A\u2223[(all n p)] \u22a2 \u223c(exi n \u223cp) := begin\n  apply R_ Double_negation_intro,\n  apply R_All_ (R_ Double_negation_intro),\n  apply Assumption 0, refl,\nend\n\ndef Ex_To_All : A\u2223[\u223c(exi n \u223cp)] \u22a2 (all n p) := begin\n  apply R_All_ (R_ Double_negation_elim),\n  apply R_ Double_negation_elim,\n  apply Assumption 0, refl,\nend\n\ndef NotAll : A\u2223[\u223c(all n p)] \u22a2 (exi n \u223cp) := begin\n  apply R_Not_ (R_All_ (R_ Double_negation_intro)),\n  apply Assumption 0, refl,\nend\n\ndef AllNot : A\u2223[(all n \u223cp)] \u22a2 \u223c(exi n p) := begin\n  apply R_ Double_negation_intro,\n  apply Assumption 0, refl,\nend\n\ndef NotEx : A\u2223[\u223c(exi n p)] \u22a2 (all n \u223cp) := begin\n  apply L_ Double_negation_elim,\n  apply Assumption 0, refl,\n  apply Assumption 0, refl,\nend\n\ndef ExNot : A\u2223[(exi n \u223cp)] \u22a2 \u223c(all n p) := begin\n  apply L_R_ (R_Not_ (R_All_ (R_ Double_negation_elim))),\n  apply Assumption 0, refl,\n  apply Assumption 0, refl,\nend\n\ndef NotFreeAll : ((@var_not_free_in_axioms L n A _) \u2227 \u00ac(free n p)) \u2192 (A\u2223[p] \u22a2 (all n p)) := begin\n  intro h,\n  apply All_intro,\n  split, apply h.left,\n  intro m, cases m, simp, apply h.right,\n  simp, \n  rw replace_formula_with_idem,\n  apply Assumption 0, refl,\nend\n\ndef AddAll : \u00ac(free n p) \u2192 ((A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 (all n p))) := sorry\n\ndef RemoveAll : (A\u2223\u0393 \u22a2 (all n p)) \u2192 (A\u2223\u0393 \u22a2 p) := sorry\n\ndef AddEx : (A\u2223\u0393 \u22a2 p) \u2192 (A\u2223\u0393 \u22a2 (exi n p)) := sorry\n\ndef RemoveEx : \u00ac(free n p) \u2192 ((A\u2223\u0393 \u22a2 (exi n p)) \u2192 (A\u2223\u0393 \u22a2 p)) := sorry\n\ndef AllOrOut : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[((all n p) or (all n q))] \u22a2 (all n (p or q)) := begin\n  intro h,\n  apply All_intro _ _ n,\n  split, assumption, \n  intro m, cases m, simp, simp,\n  rw replace_formula_with_idem,\n  apply Or_elim, apply Assumption 0, refl,\n  all_goals { \n    apply All_elim n (v n),\n    apply Assumption 0, refl,\n    apply substitutable_for_idem,\n    rw replace_formula_with_idem,\n    { apply Or_intro_left, apply Assumption 0, refl } \n      <|>\n    { apply Or_intro_right, apply Assumption 0, refl }\n  },\nend\n\ndef AllAndIn : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[all n (p and q)] \u22a2 ((all n p) and (all n q)) := begin\n  intro h,\n  apply AndProves, split,\n  all_goals { \n    apply All_intro _ _ n,\n    split, assumption, \n    intro m, cases m, simp, simp,\n    apply All_elim n (v n),\n      apply Assumption 0, refl,\n      apply substitutable_for_idem,\n      rw replace_formula_with_idem,\n      rw replace_formula_with_idem,\n      { apply R_ And_elim_right, apply Assumption 0, refl }\n        <|> \n      { apply R_ And_elim_left, apply Assumption 0, refl, },\n  }\nend\n\ndef AllAndOut : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[((all n p) and (all n q))] \u22a2 (all n (p and q)) := begin\n  intro h,\n  apply All_intro _ _ n,\n  split, assumption, \n  intro m, cases m, simp, simp,\n  rw replace_formula_with_idem,\n  apply And_intro_R,\n  { \n    apply L_ And_elim_left, \n    apply Assumption 0, \n    refl,\n    apply All_elim n (v n),\n      apply Assumption 0, refl,\n      apply substitutable_for_idem,\n      rw replace_formula_with_idem,\n      apply Assumption 0, refl,\n  },\n  {\n    apply L_ And_elim_right, \n    apply Assumption 0, \n    refl,\n    apply All_elim n (v n),\n      apply Assumption 0, refl,\n      apply substitutable_for_idem,\n      rw replace_formula_with_idem,\n      apply Assumption 0, refl,\n  }\nend\n\ndef ExOrIn : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[(exi n (p or q))] \u22a2 ((exi n p) or (exi n q)) := begin\n  intro h,\n  apply R_ DeMorganNotAnd,\n  apply R_Not_ (R_ (AllAndIn h)),\n  apply R_Not_ (R_All_ (R_ DeMorganNotOr)),\n  apply Assumption 0, refl,\nend\n\ndef ExOrOut : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[((exi n p) or (exi n q))] \u22a2 (exi n (p or q)) := begin\n  intro h,\n  apply R_Not_ (R_All_ (R_ DeMorganAnd)),\n  apply R_Not_ (R_ (AllAndOut h)),\n  apply R_ DeMorganOr,\n  apply Assumption 0, refl,\nend\n\ndef ExAndOut : (@var_not_free_in_axioms L n A _) \u2192 A\u2223[((exi n p) and (exi n q))] \u22a2 (exi n (p and q))  := begin\n  intro h,\n  apply R_Not_ (R_All_ (R_ DeMorganOr)),\n  apply R_Not_ (R_ (AllOrOut h)),\n  apply R_ DeMorganAnd,\n  apply Assumption 0, refl,\nend\n\ndef SwapAll : (@var_not_free_in_axioms L n A _) \u2192 (@var_not_free_in_axioms L m A _) \u2192 \n    A\u2223[(all n (all m p))] \u22a2 (all m (all n p)) := begin\n  intros h\u2081 h\u2082,\n  apply All_intro _ _ m,\n    split, assumption, \n    intro m, cases m, simp, simp,\n    rw replace_formula_with_idem,\n  apply All_intro _ _ n, simp,\n    split, assumption, \n    intro m, cases m, simp, simp,\n    rw replace_formula_with_idem,\n  apply All_elim n (v n),\n    apply Assumption 0, refl,\n    apply substitutable_for_idem,\n    rw replace_formula_with_idem,\n  apply All_elim m (v m),\n    apply Assumption 0, refl,\n    apply substitutable_for_idem,\n    rw replace_formula_with_idem,\n  apply Assumption 0, refl,\nend\n\ndef SwapEx : (@var_not_free_in_axioms L n A _) \u2192 (@var_not_free_in_axioms L m A _) \u2192 \n    A\u2223[(exi n (exi m p))] \u22a2 (exi m (exi n p)) := begin\n  intros h\u2081 h\u2082,\n  apply R_Not_ (R_All_ (R_ Double_negation_intro)),\n  apply L_R_ (R_Not_ (R_All_ (R_ Double_negation_elim))),\n  apply Assumption 0, refl,\n  apply R_Not_ (R_ (SwapAll h\u2081 h\u2082)),\n  apply Assumption 0, refl,\nend\n\n-- def SwapAllEx_R : ((A\u2223\u0393 \u22a2 \u223c(all n (exi m \u223cp))) \u2192 ((A\u2223\u0393 \u22a2 (exi m (all n p))) := sorry\n\n-- def SwapExAll_R : ((A\u2223\u0393 \u22a2 \u223c(exi n (all m \u223cp))) \u2192 ((A\u2223\u0393 \u22a2 (all m (exi n p))) := sorry\n\nend prf\n\nend first_order\n", "meta": {"author": "pilottinick", "repo": "QuantifierElimination", "sha": "770ebc3f8075c9c75d791d1cc0ffde4dd9c8dafc", "save_path": "github-repos/lean/pilottinick-QuantifierElimination", "path": "github-repos/lean/pilottinick-QuantifierElimination/QuantifierElimination-770ebc3f8075c9c75d791d1cc0ffde4dd9c8dafc/src/prf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.38681822411169525}}
{"text": "import Hw7\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\ntheorem desiredType : \u2200 (\u03b1 : Type) (A B : \u03b1 \u2192 Prop), (\u2203 x, A x) \u2192 \u2203 y, A y \u2228 B y := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const ``desiredType []) (Expr.const ``problem2 [])\n#eval collectAxiomsOf ``problem2\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework7", "sha": "4873226c2ce223b0f592894b99ced612f0f2712d", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework7", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework7/homework7-4873226c2ce223b0f592894b99ced612f0f2712d/_Tests/Problem2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.38680522206097157}}
{"text": "import .basic\n\nnamespace polya.field\n\nopen nterm\n\n--@[derive decidable_eq]\nstructure cterm (\u03b3 : Type) [const_space \u03b3] : Type :=\n(term : nterm \u03b3)\n(coeff : \u03b3)\n(pr : coeff \u2260 0)\n\nnamespace cterm\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ninstance : inhabited (cterm \u03b3) := \u27e8\u27e8nterm.const 0, 1, by simp\u27e9\u27e9\n\ndef to_nterm (x : cterm \u03b3) : nterm \u03b3 :=\nx.term * x.coeff\n\ndef eval (\u03c1 : dict \u03b1) (x : cterm \u03b3) : \u03b1 :=\nnterm.eval \u03c1 x.term * \u2191x.coeff\n\n@[simp]\ndef eval_to_nterm {x : cterm \u03b3} :\n nterm.eval \u03c1 x.to_nterm = cterm.eval \u03c1 x :=\nbegin\n  simp [to_nterm, nterm.eval, cterm.eval]\nend\n\ntheorem eval_to_nterm' :\n  nterm.eval \u03c1 \u2218 @cterm.to_nterm \u03b3 _ = cterm.eval \u03c1 :=\nbegin\n  unfold function.comp,\n  simp [eval_to_nterm]\nend\n\n--TODO\ntheorem eval_def {x : nterm \u03b3} {c : \u03b3} {hc : c \u2260 0} :\n  cterm.eval \u03c1 \u27e8x, c, hc\u27e9 = nterm.eval \u03c1 x * \u2191c :=\nrfl\n\ntheorem eval_add {x : nterm \u03b3} {a b : \u03b3}\n  {ha : a \u2260 0} {hb : b \u2260 0} {hc : a + b \u2260 0} :\n  cterm.eval \u03c1 \u27e8x, a + b, hc\u27e9 = cterm.eval \u03c1 \u27e8x, a, ha\u27e9 + cterm.eval \u03c1 \u27e8x, b, hb\u27e9 :=\nbegin\n  simp [eval_def, morph.morph_add, mul_add],\nend\n\ndef mul (x : cterm \u03b3) (a : \u03b3) (ha : a \u2260 0) : cterm \u03b3 :=\n\u27e8x.term, x.coeff * a, by simp [ha, x.pr]\u27e9\n\ntheorem eval_mul {x : cterm \u03b3} {a : \u03b3} {ha : a \u2260 0} :\n  cterm.eval \u03c1 (x.mul a ha) = cterm.eval \u03c1 x * \u2191a :=\nbegin\n  simp [cterm.eval, morph.morph_mul, mul], ring\nend\n\ntheorem eval_mul' {a : \u03b3} {ha : a \u2260 0} :\n  cterm.eval \u03c1 \u2218 (\u03bb x : cterm \u03b3, x.mul a ha) =\n    \u03bb x, cterm.eval \u03c1 x * (a : \u03b1) :=\nbegin\n  unfold function.comp,\n  simp [eval_mul]\nend\n\ntheorem eval_sum_mul {xs : list (cterm \u03b3)} {a : \u03b3} {ha : a \u2260 0} :\n  list.sum (list.map (cterm.eval \u03c1) xs) * \u2191a\n    = list.sum (list.map (\u03bb x : cterm \u03b3, cterm.eval \u03c1 (x.mul a ha)) xs) :=\nbegin\n  induction xs with x xs ih,\n  { simp },\n  { repeat {rw [list.map_cons, list.sum_cons]},\n    rw [eval_mul, add_mul, ih] }\nend\n\ndef smerge : list (cterm \u03b3) \u2192 list (cterm \u03b3) \u2192 list (cterm \u03b3)\n| (x::xs) (y::ys) :=\n  if x.term = y.term then\n    let c := x.coeff + y.coeff in\n    if hc : c = 0 then smerge xs ys\n    else \u27e8x.term, c, hc\u27e9 :: smerge xs ys\n  else if x.term < y.term then\n    x :: smerge xs (y::ys)\n  else\n    y :: smerge (x::xs) ys\n| xs [] := xs\n| [] ys := ys\n\nlemma smerge_nil_left {ys : list (cterm \u03b3)} :\n  smerge [] ys = ys :=\nbegin\n  induction ys with y ys ih,\n  { unfold smerge },\n  { unfold smerge }\nend\n\nlemma smerge_nil_right {xs : list (cterm \u03b3)} :\n  smerge xs [] = xs :=\nbegin\n  induction xs with x xs ih,\n  { unfold smerge },\n  { unfold smerge }\nend\n\nlemma smerge_def1 {x y : cterm \u03b3} {xs ys : list (cterm \u03b3)} :\n  x.term = y.term \u2192 x.coeff + y.coeff = 0 \u2192\n  smerge (x::xs) (y::ys) = smerge xs ys :=\nby intros h1 h2; simp [smerge, h1, h2]\nlemma smerge_def2 {x y : cterm \u03b3} {xs ys : list (cterm \u03b3)} :\n  x.term = y.term \u2192 \u03a0 (hc : x.coeff + y.coeff \u2260 0),\n  smerge (x::xs) (y::ys) = \u27e8x.term, x.coeff + y.coeff, hc\u27e9 :: smerge xs ys :=\nby intros h1 h2; simp [smerge, h1, h2]\nlemma smerge_def3 {x y : cterm \u03b3} {xs ys : list (cterm \u03b3)} :\n  x.term \u2260 y.term \u2192 x.term < y.term \u2192\n  smerge (x::xs) (y::ys) = x :: smerge xs (y :: ys) :=\nby intros h1 h2; simp [smerge, h1, h2]\nlemma smerge_def4 {x y : cterm \u03b3} {xs ys : list (cterm \u03b3)} :\n  x.term \u2260 y.term \u2192 \u00ac x.term < y.term \u2192\n  smerge (x::xs) (y::ys) = y :: smerge (x::xs) ys :=\nby intros h1 h2; simp [smerge, h1, h2]\n\ntheorem eval_smerge (xs ys : list (cterm \u03b3)) :\n  list.sum (list.map (cterm.eval \u03c1) (smerge xs ys))\n  = list.sum (list.map (cterm.eval \u03c1) xs)\n    + list.sum (list.map (cterm.eval \u03c1) ys) :=\nbegin\n  revert ys,\n  induction xs with x xs ihx,\n  { intro ys, simp [smerge_nil_left] },\n  { intro ys, induction ys with y ys ihy,\n    { simp [smerge_nil_right] },\n    { by_cases h1 : x.term = y.term,\n      { by_cases h2 : x.coeff + y.coeff = 0,\n        { have : eval \u03c1 x = - eval \u03c1 y,\n          by {\n            rw add_eq_zero_iff_eq_neg at h2,\n            unfold cterm.eval, rw [h1, h2, morph.morph_neg],\n            ring },\n          rw [smerge_def1 h1 h2, ihx],\n          repeat {rw [list.map_cons, list.sum_cons]},\n          rw this, ring },\n        { rw smerge_def2 h1 h2,\n          cases x with x n, cases y with y m,\n          simp only [] at h1, rw h1 at *,\n          repeat {rw [list.map_cons, list.sum_cons]},\n          rw [eval_add, ihx ys], { simp },\n          repeat {assumption }}},\n      { by_cases h2 : x.term < y.term,\n        { rw smerge_def3 h1 h2,\n          repeat {rw [list.map_cons, list.sum_cons]},\n          rw [ihx (y::ys), list.map_cons, list.sum_cons],\n          ring},\n        { rw smerge_def4 h1 h2,\n          repeat {rw [list.map_cons, list.sum_cons]},\n          rw [ihy, list.map_cons, list.sum_cons],\n          ring}}}}\nend\n\nend cterm\n\nstructure sterm (\u03b3 : Type) [const_space \u03b3] : Type :=\n(terms : list (cterm \u03b3))\n\nnamespace sterm\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\n\ndef of_const (a : \u03b3) : sterm \u03b3 :=\nif ha : a = 0 then { terms := [] }\nelse { terms := [\u27e81, a, ha\u27e9] }\n\ndef singleton (x : nterm \u03b3) : sterm \u03b3 :=\n{ terms := [\u27e8x, 1, by simp\u27e9] }\n\ndef eval (\u03c1 : dict \u03b1) (S : sterm \u03b3) : \u03b1 :=\nlist.sum (S.terms.map (cterm.eval \u03c1))\n\ntheorem eval_of_const (a : \u03b3) :\n  sterm.eval \u03c1 (of_const a) = \u2191a :=\nbegin\n  by_cases ha : a = 0;\n  simp [of_const, sterm.eval, cterm.eval, ha]\nend\n\ntheorem eval_singleton (x : nterm \u03b3) :\n  sterm.eval \u03c1 (singleton x) = nterm.eval \u03c1 x :=\nbegin\n  by_cases hx : nterm.eval \u03c1 x = 0,\n  repeat {simp [singleton, sterm.eval, cterm.eval, hx]}\nend\n\n--mul\ndef add (S T : sterm \u03b3) : sterm \u03b3 :=\n{ terms := cterm.smerge S.terms T.terms, }\n\n--pow\ndef mul (S : sterm \u03b3) (a : \u03b3) : sterm \u03b3 :=\nif ha : a = 0 then { terms := [] }\nelse { terms := S.terms.map (\u03bb x, cterm.mul x a ha), }\n\ninstance : has_add (sterm \u03b3) := \u27e8add\u27e9\n\ntheorem add_terms {S T : sterm \u03b3} :\n  (S + T).terms = cterm.smerge S.terms T.terms :=\nby simp [has_add.add, add]\n\ntheorem eval_add {S T : sterm \u03b3} :\n  sterm.eval \u03c1 (S + T) = sterm.eval \u03c1 S + sterm.eval \u03c1 T :=\nbegin\n  unfold sterm.eval,\n  rw [add_terms, cterm.eval_smerge]\nend\n\ntheorem eval_mul {S : sterm \u03b3} {a : \u03b3} :\n  sterm.eval \u03c1 (S.mul a) = sterm.eval \u03c1 S * \u2191a :=\nbegin\n  by_cases ha : a = 0,\n  { simp [mul, eval, ha] },\n  { unfold sterm.eval, unfold mul,\n    rw cterm.eval_sum_mul,\n     { simp [ha] },\n     { exact ha }}\nend\n\ndef to_nterm (S : sterm \u03b3) : nterm \u03b3 :=\nmatch S.terms with\n| [] := 0\n| [x] := x.to_nterm\n| (x0::xs) :=\n    have h0 : x0.coeff\u207b\u00b9 \u2260 0, by simp [x0.pr],\n    ( nterm.sum (xs.map (\u03bb x, (x.mul x0.coeff\u207b\u00b9 h0).to_nterm))\n      + x0.term * 1 ) * x0.coeff\nend\n\ntheorem eval_to_nterm {S : sterm \u03b3} :\n  sterm.eval \u03c1 S = nterm.eval \u03c1 S.to_nterm :=\nbegin\n  cases S with xs,\n  cases xs with x0 xs,\n  { simp [eval, to_nterm] },\n  cases xs with x1 xs,\n  { simp [eval, to_nterm] },\n\n  unfold eval, unfold to_nterm,\n  rw [nterm.eval_mul, nterm.eval_add, nterm.eval_sum, nterm.eval_const],\n  rw [\u2190 list.map_map cterm.to_nterm,\n    list.map_map _ cterm.to_nterm,\n    cterm.eval_to_nterm', list.map_map,\n    \u2190 cterm.eval_sum_mul],\n  rw [add_mul, mul_assoc, \u2190 morph.morph_mul, inv_mul_cancel,\n    morph.morph_one', mul_one],\n  swap, by simp [x0.pr],\n  rw [list.map_cons, list.sum_cons],\n  rw [nterm.eval_mul, nterm.eval_one, mul_one],\n  unfold cterm.eval, apply add_comm\nend\n\ndef of_nterm : nterm \u03b3 \u2192 sterm \u03b3\n| (nterm.add x y) := of_nterm x + of_nterm y\n| (nterm.mul x (nterm.const a)) := (of_nterm x).mul a\n| (nterm.const a) := of_const a\n| x := singleton x\n\ntheorem eval_of_nterm {x : nterm \u03b3} :\n  sterm.eval \u03c1 (of_nterm x) = nterm.eval \u03c1 x :=\nbegin\n  induction x with i c x y ihx ihy x y ihx ihy x n ihx,\n  { simp [of_nterm, eval_singleton] },\n  { simp [of_nterm, eval_of_const, nterm.eval] },\n  { simp [of_nterm, eval_add, nterm.eval, ihx, ihy] },\n  { cases y; try {simp [of_nterm, eval_singleton]},\n    simp [eval_mul, nterm.eval, ihx] },\n  { simp [of_nterm, eval_singleton] }\nend\n\nend sterm\n\nend polya.field\n", "meta": {"author": "lean-forward", "repo": "field", "sha": "7e2127ad485aec25e58a1b9c82a6bb74a599467a", "save_path": "github-repos/lean/lean-forward-field", "path": "github-repos/lean/lean-forward-field/field-7e2127ad485aec25e58a1b9c82a6bb74a599467a/src/data/polya/field/sterm.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3868052066327931}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport logic.equiv.defs\nimport tactic.basic\n\n/-!\n# Monad\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n## Attributes\n\n * ext\n * functor_norm\n * monad_norm\n\n## Implementation Details\n\nSet of rewrite rules and automation for monads in general and\n`reader_t`, `state_t`, `except_t` and `option_t` in particular.\n\nThe rewrite rules for monads are carefully chosen so that `simp with\nfunctor_norm` will not introduce monadic vocabulary in a context where\napplicatives would do just fine but will handle monadic notation\nalready present in an expression.\n\nIn a context where monadic reasoning is desired `simp with monad_norm`\nwill translate functor and applicative notation into monad notation\nand use regular `functor_norm` rules as well.\n\n## Tags\n\nfunctor, applicative, monad, simp\n\n-/\n\nmk_simp_attribute monad_norm none with functor_norm\n\nattribute [ext] reader_t.ext state_t.ext except_t.ext option_t.ext\nattribute [functor_norm]   bind_assoc pure_bind bind_pure\nattribute [monad_norm] seq_eq_bind_map\nuniverses u v\n\n@[monad_norm]\nlemma map_eq_bind_pure_comp\n  (m : Type u \u2192 Type v) [monad m] [is_lawful_monad m] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : m \u03b1) :\n  f <$> x = x >>= pure \u2218 f := by rw bind_pure_comp_eq_map\n\n/-- run a `state_t` program and discard the final state -/\ndef state_t.eval {m : Type u \u2192 Type v} [functor m] {\u03c3 \u03b1} (cmd : state_t \u03c3 m \u03b1) (s : \u03c3) : m \u03b1 :=\nprod.fst <$> cmd.run s\n\nuniverses u\u2080 u\u2081 v\u2080 v\u2081\n\n/-- reduce the equivalence between two state monads to the equivalence between\ntheir respective function spaces -/\ndef state_t.equiv {m\u2081 : Type u\u2080 \u2192 Type v\u2080} {m\u2082 : Type u\u2081 \u2192 Type v\u2081}\n  {\u03b1\u2081 \u03c3\u2081 : Type u\u2080} {\u03b1\u2082 \u03c3\u2082 : Type u\u2081} (F : (\u03c3\u2081 \u2192 m\u2081 (\u03b1\u2081 \u00d7 \u03c3\u2081)) \u2243 (\u03c3\u2082 \u2192 m\u2082 (\u03b1\u2082 \u00d7 \u03c3\u2082))) :\n  state_t \u03c3\u2081 m\u2081 \u03b1\u2081 \u2243 state_t \u03c3\u2082 m\u2082 \u03b1\u2082 :=\n{ to_fun := \u03bb \u27e8f\u27e9, \u27e8F f\u27e9,\n  inv_fun := \u03bb \u27e8f\u27e9, \u27e8F.symm f\u27e9,\n  left_inv := \u03bb \u27e8f\u27e9, congr_arg state_t.mk $ F.left_inv _,\n  right_inv := \u03bb \u27e8f\u27e9, congr_arg state_t.mk $ F.right_inv _ }\n\n/-- reduce the equivalence between two reader monads to the equivalence between\ntheir respective function spaces -/\ndef reader_t.equiv {m\u2081 : Type u\u2080 \u2192 Type v\u2080} {m\u2082 : Type u\u2081 \u2192 Type v\u2081}\n  {\u03b1\u2081 \u03c1\u2081 : Type u\u2080} {\u03b1\u2082 \u03c1\u2082 : Type u\u2081} (F : (\u03c1\u2081 \u2192 m\u2081 \u03b1\u2081) \u2243 (\u03c1\u2082 \u2192 m\u2082 \u03b1\u2082)) :\n  reader_t \u03c1\u2081 m\u2081 \u03b1\u2081 \u2243 reader_t \u03c1\u2082 m\u2082 \u03b1\u2082 :=\n{ to_fun := \u03bb \u27e8f\u27e9, \u27e8F f\u27e9,\n  inv_fun := \u03bb \u27e8f\u27e9, \u27e8F.symm f\u27e9,\n  left_inv := \u03bb \u27e8f\u27e9, congr_arg reader_t.mk $ F.left_inv _,\n  right_inv := \u03bb \u27e8f\u27e9, congr_arg reader_t.mk $ F.right_inv _ }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/control/monad/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764746, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.38680520663279305}}
{"text": "import Hw5\nimport Lean.Elab.Print\nimport Lean.Elab.Command\n\ntheorem desiredType (A B C : Prop) (h : B \u2194 C) : (A \u2228 B) \u2194 (A \u2228 C) := sorry \n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab.Command\n\ndef collectAxiomsOf (constName : Name) : MetaM (List String) := do\n  let env \u2190 getEnv\n  let (_, s) := ((CollectAxioms.collect constName).run env).run {}\n  let a := s.axioms.toList.map toString\n  return a\n\n#eval isDefEq (Expr.const ``desiredType []) (Expr.const ``problem3 [])\n#eval collectAxiomsOf ``problem3\n", "meta": {"author": "UofSC-Fall-2022-Math-300-H01", "repo": "homework5", "sha": "9775e690c2ffbaf37992f8441e2772e837306f15", "save_path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework5", "path": "github-repos/lean/UofSC-Fall-2022-Math-300-H01-homework5/homework5-9775e690c2ffbaf37992f8441e2772e837306f15/_Tests/Problem3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.38672071659411045}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel, Scott Morrison\n-/\nimport algebra.category.Module.epi_mono\nimport algebra.module.projective\nimport category_theory.preadditive.projective\nimport linear_algebra.finsupp_vector_space\n\n/-!\n# The category of `R`-modules has enough projectives.\n-/\n\nuniverses v u\n\nopen category_theory\nopen category_theory.limits\nopen linear_map\n\nopen_locale Module\n\n/-- The categorical notion of projective object agrees with the explicit module-theoretic notion. -/\ntheorem is_projective.iff_projective {R : Type u} [ring R]\n  {P : Type (max u v)} [add_comm_group P] [module R P] :\n  module.projective R P \u2194 projective (Module.of R P) :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { letI : module.projective R \u21a5(Module.of R P) := h,\n    exact \u27e8\u03bb E X f e epi, module.projective_lifting_property _ _\n      ((Module.epi_iff_surjective _).mp epi)\u27e9 },\n  { refine module.projective_of_lifting_property _,\n    introsI E X mE mX sE sX f g s,\n    haveI : epi \u219ff := (Module.epi_iff_surjective \u219ff).mpr s,\n    letI : projective (Module.of R P) := h,\n    exact \u27e8projective.factor_thru \u219fg \u219ff, projective.factor_thru_comp \u219fg \u219ff\u27e9 }\nend\n\nnamespace Module\nvariables {R : Type u} [ring R] {M : Module.{max u v} R}\n\n/-- Modules that have a basis are projective. -/\n-- We transport the corresponding result from `module.projective`.\nlemma projective_of_free {\u03b9 : Type*} (b : basis \u03b9 R M) : projective M :=\nprojective.of_iso (Module.of_self_iso _)\n  ((is_projective.iff_projective).mp (module.projective_of_basis b))\n\n/-- The category of modules has enough projectives, since every module is a quotient of a free\n    module. -/\ninstance Module_enough_projectives : enough_projectives (Module.{max u v} R) :=\n{ presentation :=\n  \u03bb M,\n  \u27e8{ P := Module.of R (M \u2192\u2080 R),\n    projective := projective_of_free finsupp.basis_single_one,\n    f := finsupp.basis_single_one.constr \u2115 id,\n    epi := (epi_iff_range_eq_top _).mpr\n      (range_eq_top.2 (\u03bb m, \u27e8finsupp.single m (1 : R), by simp [basis.constr]\u27e9)) }\u27e9, }\n\nend Module\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/category/Module/projective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3866564014569125}}
{"text": "import Mt.Thread.Traced\nimport Mt.System.Basic\nimport Mt.Utils.List\n\nnamespace Mt.System.Traced\n\nopen Utils\n\nvariable {spec : Spec}\nlocal instance : IsReservation spec.Reservation :=spec.is_reservation\n\ndef sum_reservations (l : List (Traced.TracedThread spec)) : spec.Reservation :=\n  l.foldl (\u03bb env_r thread => env_r + thread.reservation) spec.is_reservation.empty \n\nprivate def sum_reservations' (start) (l : List (Traced.TracedThread spec)) :=\n  l.foldl (\u03bb env_r thread => env_r + thread.reservation) start \n\nprotected theorem sum_reservations.helper (head : Traced.TracedThread spec) (l) :\n  sum_reservations (head :: l) = head.reservation + sum_reservations l :=by\n  have : \u2200 l : List (Traced.TracedThread spec),\n    sum_reservations l = sum_reservations' IsReservation.empty l :=fun _ => rfl\n  simp only [this] ; clear this\n  simp only [add, IsReservation.empty_add]\n  rw [assoc]\nwhere\n  add (r) (head : Traced.TracedThread spec) (tail)\n    : sum_reservations' r (head::tail) = sum_reservations' (r + head.reservation) tail :=by\n    simp only [sum_reservations', List.foldl]\n\n  assoc (r : spec.Reservation) (l)\n    : sum_reservations' r l = r + sum_reservations' IsReservation.empty l :=by\n    revert r\n    induction l\n    . intro r\n      show r = r + IsReservation.empty\n      rw [IsReservation.toIsCommutative.comm, IsReservation.empty_add]\n    . rename_i head tail IH\n      intro r\n      simp only [add]\n      rw [IH, IsReservation.toIsAssociative.assoc, IsReservation.empty_add]\n      rw [IH head.reservation]\n\nprotected theorem decompose_reservation (l : List (Traced.TracedThread spec)) (idx : Fin l.length) t :\n  t = l.get idx \u2192\n  sum_reservations l = sum_reservations (l.eraseIdx idx.val) + t.reservation :=by\n  intro t_def ; rw [t_def]  ; clear t_def\n  revert idx\n\n  induction l\n  . intro idx\n    have : idx.val < 0 :=idx.isLt\n    contradiction\n  . intro idx\n    rename_i thread threads IH\n    cases h : idx.val\n    . have : idx = Fin.mk 0 (by simp_arith) :=Fin.eq_of_val_eq h\n      simp only [this, List.get, List.eraseIdx, sum_reservations.helper]\n      exact IsReservation.toIsCommutative.comm _ _\n    . rename_i n\n      have idx_ok : n + 1 < (thread :: threads).length :=calc\n        n + 1 = idx.val :=h.symm\n        _ < _ :=idx.isLt\n      have : idx = Fin.mk (n + 1) idx_ok :=Fin.eq_of_val_eq h\n      simp only [this, List.get, List.eraseIdx, sum_reservations.helper]\n      clear this h idx\n      rw [IsReservation.toIsAssociative.assoc]\n      apply congrArg (thread.reservation + .)\n      exact IH <| Fin.mk n (Nat.le_of_succ_le_succ idx_ok)\n  \nend Mt.System.Traced", "meta": {"author": "mirkootter", "repo": "lean-mt", "sha": "027a16555d487e46a0a00611b8039655378dfdd5", "save_path": "github-repos/lean/mirkootter-lean-mt", "path": "github-repos/lean/mirkootter-lean-mt/lean-mt-027a16555d487e46a0a00611b8039655378dfdd5/Mt/System/BasicAux.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3866447019170793}}
{"text": "import category_theory.category\nimport category_theory.functor\nimport set_category.category_set\nimport coalgebra.Coalgebra\nimport set_category.colimits.Sum\nimport help_functions\nimport coalgebra.subcoalgebra\n\n\nimport tactic.tidy\n\nuniverses u\n\n\n\nnamespace coalgebra_sum\nopen category_theory \n     set \n     coalgebra\n     classical\n     help_functions\n     subcoalgebra\n     Sum\n\nlocal notation f ` \u229a `:80 g:80 := category_struct.comp g f\n\nvariables   {A B C: Type u} \n            {F : Type u \u2964 Type u}\n            (\ud835\udd38 : Coalgebra F)\n            (\u0392 : Coalgebra F)\n\nopen sum\n\ndef \u03b1_sum \n    : (\ud835\udd38 \u2295 \u0392) \u2192 F.obj (\ud835\udd38 \u2295 \u0392)\n    | (inl a) := ((F.map inl) \u2218 \ud835\udd38.\u03b1) a\n    | (inr b) := ((F.map inr) \u2218 \u0392.\u03b1) b\n \n\ndef sum_of_coalgebras \n            : Coalgebra F  :=\n            \u27e8(\ud835\udd38 \u2295 \u0392) ,  (\u03b1_sum \ud835\udd38 \u0392)\u27e9\n\nlemma inl_is_homomorphism : \n    @is_coalgebra_homomorphism F \ud835\udd38 (sum_of_coalgebras \ud835\udd38 \u0392) inl\n    := by {dsimp at *, refl}\n\nlemma inr_is_homomorphism :\n    @is_coalgebra_homomorphism F \u0392 (sum_of_coalgebras \ud835\udd38 \u0392) inr\n    := by {dsimp at *, refl}\n\ninfix ` \u229e ` :20 := sum_of_coalgebras\n\ntheorem set_sum_is_coalgebra_sum : \n    let e\u2081 : \ud835\udd38 \u2192 (\ud835\udd38 \u2295 \u0392) := inl in\n    let e\u2082 : \u0392 \u2192 (\ud835\udd38 \u2295 \u0392) := inr in\n    let h_e\u2081 : \ud835\udd38 \u27f6 (\ud835\udd38 \u229e \u0392) := \u27e8e\u2081 ,  inl_is_homomorphism \ud835\udd38 \u0392\u27e9 in\n    let h_e\u2082 : \u0392 \u27f6 (\ud835\udd38 \u229e \u0392) := \u27e8e\u2082 ,  inr_is_homomorphism \ud835\udd38 \u0392\u27e9 in\n            is_sum (\ud835\udd38 \u229e \u0392) h_e\u2081 h_e\u2082\n        := \n    begin\n        intros e\u2081 e\u2082 h_e\u2081 h_e\u2082 \u211a \u03d5\u2081 \u03d5\u2082,\n        \n        let \u03c3 : \ud835\udd38 \u2295 \u0392 \u27f6 \u211a := \n            some (disjoint_union_is_sum \u211a \u03d5\u2081 \u03d5\u2082),\n\n        let \u03b3 := \u211a.\u03b1,\n        let \u03b1 :=  (\ud835\udd38 \u229e \u0392).\u03b1,\n\n        let hom_inl : \n            \u03b1 \u2218 inl = (F.map inl) \u2218 \ud835\udd38.\u03b1 := \n                inl_is_homomorphism \ud835\udd38 \u0392,\n        let hom_inr : \n            \u03b1 \u2218 inr = (F.map inr) \u2218 \u0392.\u03b1 \n                := inr_is_homomorphism \ud835\udd38 \u0392,\n\n        let hom_\u03d5\u2081 : \u03b3 \u2218 \u03d5\u2081.val =  (F.map \u03d5\u2081.val) \u2218 \ud835\udd38.\u03b1 := \u03d5\u2081.property,\n        let hom_\u03d5\u2082 : \u03b3 \u2218 \u03d5\u2082.val =  (F.map \u03d5\u2082.val) \u2218 \u0392.\u03b1 := \u03d5\u2082.property,\n\n        have h0 : _ := \n            some_spec (disjoint_union_is_sum \u211a \u03d5\u2081 \u03d5\u2082),\n\n        have h1 : \u03d5\u2081.val = \u03c3 \u2218 inl \u2227 \u03d5\u2082.val = \u03c3 \u2218 inr:= \n            and.left h0,\n\n        have h2 : \u03b3 \u2218 \u03c3 \u2218 inl = (F.map \u03c3) \u2218 \u03b1 \u2218 inl := \n            calc \n             \u03b3 \u2218 (\u03c3 \u2218 inl) = \u03b3 \u2218 \u03d5\u2081.val                    : by rw [and.left h1]\n                ...      = (F.map \u03d5\u2081.val) \u2218 \ud835\udd38.\u03b1               : by rw [hom_\u03d5\u2081]\n                ...      = (F.map (\u03c3 \u2218 inl)) \u2218 \ud835\udd38.\u03b1        : by rw [and.left h1]\n                ...      = (F.map (\u03c3 \u229a inl)) \u2218 \ud835\udd38.\u03b1        : by simp\n                ...  = ((F.map \u03c3) \u229a (F.map inl)) \u2218 \ud835\udd38.\u03b1    : by rw \u2190 functor.map_comp\n                ...      = (F.map \u03c3) \u2218 (F.map inl) \u2218 \ud835\udd38.\u03b1   : by simp\n                ...      = (F.map \u03c3) \u2218 \u03b1 \u2218 inl             : by rw [hom_inl],\n        have h3 : \u03b3 \u2218 \u03c3 \u2218 inr = (F.map \u03c3) \u2218 \u03b1 \u2218 inr := \n            calc \u03b3 \u2218 (\u03c3 \u2218 inr) = \u03b3 \u2218 \u03d5\u2082.val                      : by rw [and.right h1]\n                     ...      = (F.map \u03d5\u2082.val) \u2218 \u0392.\u03b1             : by rw [hom_\u03d5\u2082]\n                     ...      = (F.map (\u03c3 \u2218 inr)) \u2218 \u0392.\u03b1          : by rw [and.right h1]\n                     ...      = (F.map (\u03c3 \u229a inr)) \u2218 \u0392.\u03b1        : by simp\n                     ...  = ((F.map \u03c3) \u229a (F.map inr)) \u2218 \u0392.\u03b1    : by rw \u2190 functor.map_comp\n                     ...      = (F.map \u03c3) \u2218 (F.map inr) \u2218 \u0392.\u03b1   : by simp\n                     ...      = (F.map \u03c3) \u2218 \u03b1 \u2218 inr             : by rw [hom_inr],\n        have h4 : \u2200 ab : \ud835\udd38 \u2295 \u0392, (\u03b3 \u2218 \u03c3) ab = ((F.map \u03c3) \u2218 \u03b1) ab := \n                begin \n                    intro ab,\n                    induction ab,\n                    case inl :  { \n                                    show (\u03b3 \u2218 \u03c3 \u2218 inl) ab = ((F.map \u03c3) \u2218 \u03b1 \u2218 inl) ab,\n                                    by rw h2\n                                },\n                    case inr :  { \n                                    show (\u03b3 \u2218 \u03c3 \u2218 inr) ab = ((F.map \u03c3) \u2218 \u03b1 \u2218 inr) ab,\n                                    by rw h3\n                                },\n                end,\n        \n        \n        have h5 : @is_coalgebra_homomorphism F\n                (\ud835\udd38 \u229e \u0392) \u211a \u03c3\n                  := funext h4,\n        \n        let ex_uni : \n            \u2203! s , \u03d5\u2081.val = s \u2218 inl \u2227 \u03d5\u2082.val = s \u2218 inr := \n            disjoint_union_is_sum \u211a \u03d5\u2081 \u03d5\u2082,\n        \n        let ex : \n            \u2203 s , \n                (\u03d5\u2081.val = s \u2218 inl \u2227 \u03d5\u2082.val = s \u2218 inr)\n                \u2227 @is_coalgebra_homomorphism F\n                    (\ud835\udd38 \u229e \u0392) \u211a s := \n                        exists.intro \u03c3 \n                        (and.intro h1 h5),\n        let s := some ex,\n        have spec_s : (\u03d5\u2081.val = s \u2218 inl \u2227 \u03d5\u2082.val = s \u2218 inr) \u2227 \n                    @is_coalgebra_homomorphism F (\ud835\udd38 \u229e \u0392) \u211a s\n            := some_spec ex,\n        use s,\n        exact spec_s.2,\n        split,\n        exact \u27e8eq_in_set.1 spec_s.1.1,\n            eq_in_set.1 spec_s.1.2\u27e9,\n        let s\u2081 := some ex_uni,\n        have spec_s\u2081 : _ := some_spec ex_uni,\n        have s\u2081_s : s\u2081 = s := eq.symm (spec_s\u2081.2 s spec_s.1),\n        intros s\u2082 spec_s\u2082,\n\n        have s\u2082_s\u2081 : s\u2082.val = s\u2081 :=  spec_s\u2081.2 s\u2082\n            \u27e8eq_in_set.2 spec_s\u2082.1, eq_in_set.2 spec_s\u2082.2\u27e9,\n        \n        have s\u2082_s : s\u2082.val = s := by simp [s\u2081_s, s\u2082_s\u2081],\n\n        exact eq_in_set.1 s\u2082_s\n\n\n    end\n\n\n\nnoncomputable theorem subcoalgebra_union_is_coalgebra \n    {U\u2081 U\u2082 : set \ud835\udd38}\n    (S\u2081 : SubCoalgebra U\u2081)\n    (S\u2082 : SubCoalgebra U\u2082) \n    : SubCoalgebra (U\u2081 \u222a U\u2082) := \n    begin\n\n        let S : Coalgebra F := \u27e8U\u2081 , S\u2081.\u03b1\u27e9 \u229e \u27e8U\u2082 , S\u2082.\u03b1\u27e9,\n        \n        have ex := set_sum_is_coalgebra_sum \u27e8U\u2081 , S\u2081.\u03b1\u27e9 \u27e8U\u2082 , S\u2082.\u03b1\u27e9 \ud835\udd38 \n                     \u27e8 (U\u2081\u21aa \ud835\udd38), S\u2081.h\u27e9  \u27e8(U\u2082 \u21aa \ud835\udd38) , S\u2082.h\u27e9,\n\n        let \u03d5 : S \u27f6 \ud835\udd38 := some ex,\n\n        have spec : (((U\u2081\u21aa \ud835\udd38) = \u03d5 \u2218 inl \u2227 (U\u2082 \u21aa \ud835\udd38) = \u03d5 \u2218 inr))\n            := \u27e8 eq_in_set.2 (some_spec ex).1.1,\n                 eq_in_set.2 (some_spec ex).1.2 \u27e9 ,\n\n        have all : \u2200 a : \ud835\udd38 , a \u2208 (range \u03d5) \u2194 a \u2208 (U\u2081 \u222a U\u2082) :=\n            \u03bb a,\n            iff.intro\n            begin\n                assume ar : a \u2208 range \u03d5,\n                cases (mem_range.1 ar) with s \u03d5s_a,\n                induction s,\n                case inl : \n                    begin \n                        have h01 : (\u03d5 \u2218 inl) s \u2208 U\u2081 := \n                            spec.1 \u25b8 s.property,\n                        have h02 : a \u2208 U\u2081 := \u03d5s_a \u25b8 h01,\n                        by simp [h02],\n                    end,\n                case inr : \n                    begin \n                        have h01 : (\u03d5 \u2218 inr) s \u2208 U\u2082 := \n                            spec.2 \u25b8 s.property,\n                        have h02 : a \u2208 U\u2082 :=  (\u03d5s_a \u25b8 h01),\n                        by simp [h02], \n                    end,\n            end\n            begin\n                assume auu : a \u2208 U\u2081 \u2228 a \u2208 U\u2082,\n                apply or.elim auu,\n                assume au1: a \u2208 U\u2081,\n                have h01 : a = (\u03d5 \u2218 inl) \u27e8a , au1\u27e9  :=\n                    spec.1 \u25b8 rfl,\n                exact exists.intro (inl \u27e8a , au1\u27e9) (eq.symm (h01)),\n                assume au2: a \u2208 U\u2082,\n                have h01 : a = (\u03d5 \u2218 inr) \u27e8a , au2\u27e9  :=\n                    spec.2 \u25b8 rfl,\n                exact exists.intro (inr \u27e8a , au2\u27e9) (eq.symm h01)\n            end,\n\n        rw \u2190(eq_sets.1 all), \n        exact range_is_subCoalgebra \u03d5,\n        \n    end\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n-- def pw (U V : set \ud835\udd38) \n--     [\u2200 x , decidable (x \u2208 U \u2229 V)]\n--         (w : U \u2229 V)\n--     :U \u2192 U \u2229 V := \u03bb u , if uUV : u.val \u2208 U \u2229 V\n--                     then \u27e8u.val , uUV\u27e9 \n--                     else \u27e8w.val , w.property\u27e9\n\n-- def qw (U V : set \ud835\udd38) \n--     [\u2200 x , decidable (x \u2208 V)]\n--         (w : U \u2229 V)\n--     :\ud835\udd38 \u2192 V := \u03bb a , if aV : a \u2208 V \n--                     then \u27e8a , aV\u27e9 \n--                     else \u27e8w.val , and.right w.property\u27e9\n\ntheorem subcoalgebra_intersection_is_coalgebra \n    {U V : set \ud835\udd38}\n    (S\u2081 : SubCoalgebra U)\n    (S\u2082 : SubCoalgebra V) \n    [\u2200 x : \ud835\udd38 , decidable (x \u2208 U \u2229 V)]\n    [\u2200 x : \ud835\udd38 , decidable (x \u2208 V)]\n    : openset (U \u2229 V) := \n    begin\n        let I : set \ud835\udd38 := U \u2229 V,\n        cases classical.em (nonempty (I)) with n_emp emp,\n        \n        let w : \u21a5I := choice n_emp ,\n        \n        let incV : I \u2192 V := set.inclusion (by simp),\n        let incU : I \u2192 U := set.inclusion (by simp),\n\n        let p_w :U \u2192 U \u2229 V := \u03bb u , if uUV : u.val \u2208 U \u2229 V\n                    then \u27e8u.val , uUV\u27e9 \n                    else \u27e8w.val , w.property\u27e9,\n        let q_w :\ud835\udd38 \u2192 V := \u03bb a , if aV : a \u2208 V \n                    then \u27e8a , aV\u27e9 \n                    else \u27e8w.val , and.right w.property\u27e9,\n\n        have h1 : \u2200 u:U ,\n            (incV \u2218 p_w) u = (q_w \u2218 (U \u21aa \ud835\udd38)) u := \n            assume u, \n            @by_cases (u.val \u2208 I)  ((incV \u2218 p_w) u = (q_w \u2218 (U \u21aa \ud835\udd38)) u)\n                begin \n                    intro uI,\n                    have pwu_u : p_w u = \u27e8u.val , uI\u27e9:= \n                        by simp [p_w , rfl, uI],\n                    have qwu_u: (q_w) u.val = \n                                    (\u27e8u.val , uI.2\u27e9 : V)  := \n                        by simp [q_w, rfl, uI.2],\n                    calc (incV \u2218 p_w) u \n                                = incV  \u27e8u.val , uI\u27e9  : by rw \u2190pwu_u\n                            ... = (\u27e8u.val , uI.2\u27e9: V) : rfl\n                            ... = q_w u.val          : by rw \u2190qwu_u\n                end \n                begin \n                    intro uNI,\n                    have pwu_w : p_w u = w := by simp [p_w , uNI, rfl],\n                    have qwu_u: q_w u.val = \n                                    (\u27e8w.val , (w.property).2\u27e9 : V)  := \n                        begin\n                            simp[q_w],\n                            split_ifs,\n                            exact absurd (and.intro u.property h) uNI,\n                            exact rfl\n                        end,\n                    calc incV (p_w u)\n                                = incV w                      : by rw pwu_w\n                            ... = (\u27e8w.val , w.property.2\u27e9: V)  : rfl\n                            ... = q_w u.val                   : by rw \u2190qwu_u\n                           \n                \n                end \n            ,\n        have h11 : incV \u2218 p_w = q_w \u2218 (U \u21aa \ud835\udd38) :=\n            funext h1,\n        \n        have h22 : q_w  \u2218 (V \u21aa \ud835\udd38) = id := \n            funext (by {dsimp at *,  simp [q_w]}),\n\n        let \u03b3 := (F.map p_w) \u2218 S\u2081.\u03b1 \u2218 incU,\n\n        have hom_eq : (F.map (I \u21aa \ud835\udd38)) \u2218 \u03b3 = \ud835\udd38.\u03b1 \u2218 (I \u21aa \ud835\udd38) := \n            calc (F.map ((V \u21aa \ud835\udd38) \u229a incV)) \u2218 (F.map p_w) \u2218 S\u2081.\u03b1 \u2218 incU \n                    = ((F.map (V \u21aa \ud835\udd38)) \u229a (F.map incV)) \u2218 (F.map p_w) \u2218 S\u2081.\u03b1 \u2218 incU : by rw functor.map_comp\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 ((F.map incV) \u229a (F.map p_w)) \u2218 S\u2081.\u03b1 \u2218 incU : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (incV \u229a p_w)) \u2218 S\u2081.\u03b1 \u2218 incU : by rw \u2190functor.map_comp\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (incV \u2218 p_w)) \u2218 S\u2081.\u03b1 \u2218 incU : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (q_w \u2218 (U \u21aa \ud835\udd38))) \u2218 S\u2081.\u03b1 \u2218 incU : by rw \u2190h11\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (q_w \u229a (U \u21aa \ud835\udd38))) \u2218 S\u2081.\u03b1 \u2218 incU : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 ((F.map q_w) \u229a ((F.map (U \u21aa \ud835\udd38)))) \u2218 S\u2081.\u03b1 \u2218 incU : by rw functor.map_comp\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map q_w) \u2218 ((F.map (U \u21aa \ud835\udd38)) \u2218 S\u2081.\u03b1) \u2218 incU : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map q_w) \u2218 \ud835\udd38.\u03b1 \u2218 (U \u21aa \ud835\udd38) \u2218 incU : by rw  eq.symm S\u2081.h\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map q_w) \u2218 (\ud835\udd38.\u03b1 \u2218 (V \u21aa \ud835\udd38)) \u2218 incV : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map q_w) \u2218 (F.map (V \u21aa \ud835\udd38)) \u2218 S\u2082.\u03b1 \u2218 incV : by rw \u2190 (eq.symm S\u2082.h)\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 ((F.map q_w) \u229a (F.map (V \u21aa \ud835\udd38))) \u2218 S\u2082.\u03b1 \u2218 incV : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (q_w \u229a (V \u21aa \ud835\udd38))) \u2218 S\u2082.\u03b1 \u2218 incV : by rw \u2190functor.map_comp\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (q_w \u2218 (V \u21aa \ud835\udd38))) \u2218 S\u2082.\u03b1 \u2218 incV : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map id) \u2218 S\u2082.\u03b1 \u2218 incV : by rw h22\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (F.map (\ud835\udfd9 V)) \u2218 S\u2082.\u03b1 \u2218 incV : rfl\n                ... = (F.map (V \u21aa \ud835\udd38)) \u2218 (\ud835\udfd9 (F.obj V)) \u2218 S\u2082.\u03b1 \u2218 incV : by rw functor.map_id'\n                ... = ((F.map (V \u21aa \ud835\udd38)) \u2218 S\u2082.\u03b1) \u2218 incV : rfl\n                ... = \ud835\udd38.\u03b1 \u2218 (V \u21aa \ud835\udd38) \u2218 incV : by rw eq.symm S\u2082.h,\n\n        exact exists.intro \u03b3 (eq.symm hom_eq),\n        \n        exact empty_openset emp\n    end\n\nend coalgebra_sum\n", "meta": {"author": "QaisHamarneh", "repo": "Coalgebra-in-Lean", "sha": "bd0452df98bc64b608e5dfd7babc42c301bb6a46", "save_path": "github-repos/lean/QaisHamarneh-Coalgebra-in-Lean", "path": "github-repos/lean/QaisHamarneh-Coalgebra-in-Lean/Coalgebra-in-Lean-bd0452df98bc64b608e5dfd7babc42c301bb6a46/src/coalgebra/colimits/coalgebra_sum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.38664469528523393}}
{"text": "/-\nCopyright (c) 2014 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.subtype\nimport Mathlib.data.prod\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u v w l \n\nnamespace Mathlib\n\n/-!\n# Basic definitions about `\u2264` and `<`\n\n## Definitions\n\n### Predicates on functions\n\n- `monotone f`: a function between two types equipped with `\u2264` is monotone\n  if `a \u2264 b` implies `f a \u2264 f b`.\n- `strict_mono f` : a function between two types equipped with `<` is strictly monotone\n  if `a < b` implies `f a < f b`.\n- `order_dual \u03b1` : a type tag reversing the meaning of all inequalities.\n\n### Transfering orders\n\n- `order.preimage`, `preorder.lift`: transfer a (pre)order on `\u03b2` to an order on `\u03b1`\n  using a function `f : \u03b1 \u2192 \u03b2`.\n- `partial_order.lift`, `linear_order.lift`: transfer a partial (resp., linear) order on `\u03b2` to a\n  partial (resp., linear) order on `\u03b1` using an injective function `f`.\n\n### Extra classes\n\n- `no_top_order`, `no_bot_order`: an order without a maximal/minimal element.\n- `densely_ordered`: an order with no gaps, i.e. for any two elements `a<b` there exists\n  `c`, `a<c<b`.\n\n## Main theorems\n\n- `monotone_of_monotone_nat`: if `f : \u2115 \u2192 \u03b1` and `f n \u2264 f (n + 1)` for all `n`, then\n  `f` is monotone;\n- `strict_mono.nat`: if `f : \u2115 \u2192 \u03b1` and `f n < f (n + 1)` for all `n`, then f is strictly monotone.\n\n## TODO\n\n- expand module docs\n- automatic construction of dual definitions / theorems\n\n## See also\n- `algebra.order` for basic lemmas about orders, and projection notation for orders\n\n## Tags\n\npreorder, order, partial order, linear order, monotone, strictly monotone\n-/\n\ntheorem preorder.ext {\u03b1 : Type u_1} {A : preorder \u03b1} {B : preorder \u03b1} (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B := sorry\n\ntheorem partial_order.ext {\u03b1 : Type u_1} {A : partial_order \u03b1} {B : partial_order \u03b1} (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B := sorry\n\ntheorem linear_order.ext {\u03b1 : Type u_1} {A : linear_order \u03b1} {B : linear_order \u03b1} (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B := sorry\n\n/-- Given a relation `R` on `\u03b2` and a function `f : \u03b1 \u2192 \u03b2`,\n  the preimage relation on `\u03b1` is defined by `x \u2264 y \u2194 f x \u2264 f y`.\n  It is the unique relation on `\u03b1` making `f` a `rel_embedding`\n  (assuming `f` is injective). -/\n@[simp] def order.preimage {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (f : \u03b1 \u2192 \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) (x : \u03b1) (y : \u03b1) :=\n  s (f x) (f y)\n\ninfixl:80 \" \u207b\u00b9'o \" => Mathlib.order.preimage\n\n/-- The preimage of a decidable order is decidable. -/\nprotected instance order.preimage.decidable {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (f : \u03b1 \u2192 \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) [H : DecidableRel s] : DecidableRel (f \u207b\u00b9'o s) :=\n  fun (x y : \u03b1) => H (f x) (f y)\n\n/-- A function between preorders is monotone if\n  `a \u2264 b` implies `f a \u2264 f b`. -/\ndef monotone {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 {a b : \u03b1}, a \u2264 b \u2192 f a \u2264 f b\n\ntheorem monotone_id {\u03b1 : Type u} [preorder \u03b1] : monotone id :=\n  fun (x y : \u03b1) (h : x \u2264 y) => h\n\ntheorem monotone_const {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {b : \u03b2} : monotone fun (a : \u03b1) => b :=\n  fun (x y : \u03b1) (h : x \u2264 y) => le_refl b\n\nprotected theorem monotone.comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b2] [preorder \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (m_g : monotone g) (m_f : monotone f) : monotone (g \u2218 f) :=\n  fun (a b : \u03b1) (h : a \u2264 b) => m_g (m_f h)\n\nprotected theorem monotone.iterate {\u03b1 : Type u} [preorder \u03b1] {f : \u03b1 \u2192 \u03b1} (hf : monotone f) (n : \u2115) : monotone (nat.iterate f n) :=\n  nat.rec_on n monotone_id fun (n : \u2115) (ihn : monotone (nat.iterate f n)) => monotone.comp ihn hf\n\ntheorem monotone_of_monotone_nat {\u03b1 : Type u} [preorder \u03b1] {f : \u2115 \u2192 \u03b1} (hf : \u2200 (n : \u2115), f n \u2264 f (n + 1)) : monotone f := sorry\n\ntheorem monotone.reflect_lt {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : monotone f) {x : \u03b1} {x' : \u03b1} (h : f x < f x') : x < x' :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (x < x')) (Eq.symm (propext not_le)))) (id fun (h' : x' \u2264 x) => not_le_of_lt h (hf h'))\n\n/-- If `f` is a monotone function from `\u2115` to a preorder such that `y` lies between `f x` and\n  `f (x + 1)`, then `y` doesn't lie in the range of `f`. -/\ntheorem monotone.ne_of_lt_of_lt_nat {\u03b1 : Type u_1} [preorder \u03b1] {f : \u2115 \u2192 \u03b1} (hf : monotone f) (x : \u2115) (x' : \u2115) {y : \u03b1} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' \u2260 y := sorry\n\n/-- If `f` is a monotone function from `\u2124` to a preorder such that `y` lies between `f x` and\n  `f (x + 1)`, then `y` doesn't lie in the range of `f`. -/\ntheorem monotone.ne_of_lt_of_lt_int {\u03b1 : Type u_1} [preorder \u03b1] {f : \u2124 \u2192 \u03b1} (hf : monotone f) (x : \u2124) (x' : \u2124) {y : \u03b1} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' \u2260 y := sorry\n\n/-- A function `f` is strictly monotone if `a < b` implies `f a < f b`. -/\ndef strict_mono {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 {a b : \u03b1}, a < b \u2192 f a < f b\n\ntheorem strict_mono_id {\u03b1 : Type u} [HasLess \u03b1] : strict_mono id :=\n  fun (a b : \u03b1) => id\n\n/-- A function `f` is strictly monotone increasing on `t` if `x < y` for `x,y \u2208 t` implies\n`f x < f y`. -/\ndef strict_mono_incr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) (t : set \u03b1) :=\n  \u2200 {x : \u03b1}, x \u2208 t \u2192 \u2200 {y : \u03b1}, y \u2208 t \u2192 x < y \u2192 f x < f y\n\n/-- A function `f` is strictly monotone decreasing on `t` if `x < y` for `x,y \u2208 t` implies\n`f y < f x`. -/\ndef strict_mono_decr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) (t : set \u03b1) :=\n  \u2200 {x : \u03b1}, x \u2208 t \u2192 \u2200 {y : \u03b1}, y \u2208 t \u2192 x < y \u2192 f y < f x\n\n/-- Type tag for a set with dual order: `\u2264` means `\u2265` and `<` means `>`. -/\ndef order_dual (\u03b1 : Type u_1) :=\n  \u03b1\n\nnamespace order_dual\n\n\nprotected instance nonempty (\u03b1 : Type u_1) [h : Nonempty \u03b1] : Nonempty (order_dual \u03b1) :=\n  h\n\nprotected instance subsingleton (\u03b1 : Type u_1) [h : subsingleton \u03b1] : subsingleton (order_dual \u03b1) :=\n  h\n\nprotected instance has_le (\u03b1 : Type u_1) [HasLessEq \u03b1] : HasLessEq (order_dual \u03b1) :=\n  { LessEq := fun (x y : \u03b1) => y \u2264 x }\n\nprotected instance has_lt (\u03b1 : Type u_1) [HasLess \u03b1] : HasLess (order_dual \u03b1) :=\n  { Less := fun (x y : \u03b1) => y < x }\n\n-- `dual_le` and `dual_lt` should not be simp lemmas:\n\n-- they cause a loop since `\u03b1` and `order_dual \u03b1` are definitionally equal\n\ntheorem dual_le {\u03b1 : Type u} [HasLessEq \u03b1] {a : \u03b1} {b : \u03b1} : a \u2264 b \u2194 b \u2264 a :=\n  iff.rfl\n\ntheorem dual_lt {\u03b1 : Type u} [HasLess \u03b1] {a : \u03b1} {b : \u03b1} : a < b \u2194 b < a :=\n  iff.rfl\n\ntheorem dual_compares {\u03b1 : Type u} [HasLess \u03b1] {a : \u03b1} {b : \u03b1} {o : ordering} : ordering.compares o a b \u2194 ordering.compares o b a :=\n  ordering.cases_on o iff.rfl eq_comm iff.rfl\n\nprotected instance preorder (\u03b1 : Type u_1) [preorder \u03b1] : preorder (order_dual \u03b1) :=\n  preorder.mk LessEq Less sorry sorry\n\nprotected instance partial_order (\u03b1 : Type u_1) [partial_order \u03b1] : partial_order (order_dual \u03b1) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\nprotected instance linear_order (\u03b1 : Type u_1) [linear_order \u03b1] : linear_order (order_dual \u03b1) :=\n  linear_order.mk partial_order.le partial_order.lt sorry sorry sorry sorry\n    ((fun (this : DecidableRel fun (a b : \u03b1) => b \u2264 a) => this) fun (a b : \u03b1) => has_le.le.decidable b a)\n    Mathlib.decidable_eq_of_decidable_le\n    ((fun (this : DecidableRel fun (a b : \u03b1) => b < a) => this) fun (a b : \u03b1) => has_lt.lt.decidable b a)\n\nprotected instance inhabited {\u03b1 : Type u} [Inhabited \u03b1] : Inhabited (order_dual \u03b1) :=\n  id\n\ntheorem preorder.dual_dual (\u03b1 : Type u_1) [H : preorder \u03b1] : order_dual.preorder (order_dual \u03b1) = H :=\n  preorder.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem partial_order.dual_dual (\u03b1 : Type u_1) [H : partial_order \u03b1] : order_dual.partial_order (order_dual \u03b1) = H :=\n  partial_order.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem linear_order.dual_dual (\u03b1 : Type u_1) [H : linear_order \u03b1] : order_dual.linear_order (order_dual \u03b1) = H :=\n  linear_order.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem cmp_le_flip {\u03b1 : Type u_1} [HasLessEq \u03b1] [DecidableRel LessEq] (x : \u03b1) (y : \u03b1) : cmp_le x y = cmp_le y x :=\n  rfl\n\nend order_dual\n\n\nnamespace strict_mono_incr_on\n\n\nprotected theorem dual {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (H : strict_mono_incr_on f s) : strict_mono_incr_on f s :=\n  fun (x : order_dual \u03b1) (hx : x \u2208 s) (y : order_dual \u03b1) (hy : y \u2208 s) => H hy hx\n\nprotected theorem dual_right {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (H : strict_mono_incr_on f s) : strict_mono_decr_on f s :=\n  H\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x \u2264 f y \u2194 x \u2264 y := sorry\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x < f y \u2194 x < y := sorry\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) {o : ordering} : ordering.compares o (f x) (f y) \u2194 ordering.compares o x y := sorry\n\nend strict_mono_incr_on\n\n\nnamespace strict_mono_decr_on\n\n\nprotected theorem dual {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (H : strict_mono_decr_on f s) : strict_mono_decr_on f s :=\n  fun (x : order_dual \u03b1) (hx : x \u2208 s) (y : order_dual \u03b1) (hy : y \u2208 s) => H hy hx\n\nprotected theorem dual_right {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} (H : strict_mono_decr_on f s) : strict_mono_incr_on f s :=\n  H\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x \u2264 f y \u2194 y \u2264 x :=\n  strict_mono_incr_on.le_iff_le (strict_mono_decr_on.dual_right H) hy hx\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x < f y \u2194 y < x :=\n  strict_mono_incr_on.lt_iff_lt (strict_mono_decr_on.dual_right H) hy hx\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) {o : ordering} : ordering.compares o (f x) (f y) \u2194 ordering.compares o y x :=\n  iff.trans order_dual.dual_compares (strict_mono_incr_on.compares (strict_mono_decr_on.dual_right H) hy hx)\n\nend strict_mono_decr_on\n\n\nnamespace strict_mono\n\n\nprotected theorem strict_mono_incr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : strict_mono f) (s : set \u03b1) : strict_mono_incr_on f s :=\n  fun (x : \u03b1) (hx : x \u2208 s) (y : \u03b1) (hy : y \u2208 s) (hxy : x < y) => hf hxy\n\ntheorem comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [HasLess \u03b1] [HasLess \u03b2] [HasLess \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : strict_mono g) (hf : strict_mono f) : strict_mono (g \u2218 f) :=\n  fun (a b : \u03b1) (h : a < b) => hg (hf h)\n\nprotected theorem iterate {\u03b1 : Type u} [HasLess \u03b1] {f : \u03b1 \u2192 \u03b1} (hf : strict_mono f) (n : \u2115) : strict_mono (nat.iterate f n) :=\n  nat.rec_on n strict_mono_id fun (n : \u2115) (ihn : strict_mono (nat.iterate f n)) => comp ihn hf\n\ntheorem id_le {\u03c6 : \u2115 \u2192 \u2115} (h : strict_mono \u03c6) (n : \u2115) : n \u2264 \u03c6 n :=\n  nat.rec_on n (nat.zero_le (\u03c6 0))\n    fun (n : \u2115) (hn : n \u2264 \u03c6 n) => nat.succ_le_of_lt (lt_of_le_of_lt hn (h (nat.lt_succ_self n)))\n\nprotected theorem ite' {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hf : strict_mono f) (hg : strict_mono g) {p : \u03b1 \u2192 Prop} [decidable_pred p] (hp : \u2200 {x y : \u03b1}, x < y \u2192 p y \u2192 p x) (hfg : \u2200 {x y : \u03b1}, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y) : strict_mono fun (x : \u03b1) => ite (p x) (f x) (g x) := sorry\n\nprotected theorem ite {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hf : strict_mono f) (hg : strict_mono g) {p : \u03b1 \u2192 Prop} [decidable_pred p] (hp : \u2200 {x y : \u03b1}, x < y \u2192 p y \u2192 p x) (hfg : \u2200 (x : \u03b1), f x \u2264 g x) : strict_mono fun (x : \u03b1) => ite (p x) (f x) (g x) :=\n  strict_mono.ite' hf hg hp fun (x y : \u03b1) (hx : p x) (hy : \u00acp y) (h : x < y) => has_lt.lt.trans_le (hf h) (hfg y)\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) {a : \u03b1} {b : \u03b1} : f a < f b \u2194 a < b :=\n  strict_mono_incr_on.lt_iff_lt (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) {a : \u03b1} {b : \u03b1} {o : ordering} : ordering.compares o (f a) (f b) \u2194 ordering.compares o a b :=\n  strict_mono_incr_on.compares (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\ntheorem injective {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) : function.injective f :=\n  fun (x y : \u03b1) (h : f x = f y) =>\n    (fun (this : ordering.compares ordering.eq x y) => this) (iff.mp (strict_mono.compares H) h)\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) {a : \u03b1} {b : \u03b1} : f a \u2264 f b \u2194 a \u2264 b :=\n  strict_mono_incr_on.le_iff_le (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\ntheorem top_preimage_top {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) {a : \u03b1} (h_top : \u2200 (p : \u03b2), p \u2264 f a) (x : \u03b1) : x \u2264 a :=\n  iff.mp (le_iff_le H) (h_top (f x))\n\ntheorem bot_preimage_bot {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) {a : \u03b1} (h_bot : \u2200 (p : \u03b2), f a \u2264 p) (x : \u03b1) : a \u2264 x :=\n  iff.mp (le_iff_le H) (h_bot (f x))\n\nprotected theorem nat {\u03b2 : Type u_1} [preorder \u03b2] {f : \u2115 \u2192 \u03b2} (h : \u2200 (n : \u2115), f n < f (n + 1)) : strict_mono f := sorry\n\n-- `preorder \u03b1` isn't strong enough: if the preorder on \u03b1 is an equivalence relation,\n\n-- then `strict_mono f` is vacuously true.\n\ntheorem monotone {\u03b1 : Type u} {\u03b2 : Type v} [partial_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (H : strict_mono f) : monotone f :=\n  fun (a b : \u03b1) (h : a \u2264 b) =>\n    Or._oldrec (le_of_lt \u2218 H) (fun (h_1 : a = b) => Eq._oldrec (fun (h : a \u2264 a) => le_refl (f a)) h_1 h)\n      (lt_or_eq_of_le h)\n\nend strict_mono\n\n\ntheorem injective_of_lt_imp_ne {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] {f : \u03b1 \u2192 \u03b2} (h : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y) : function.injective f := sorry\n\ntheorem strict_mono_of_monotone_of_injective {\u03b1 : Type u} {\u03b2 : Type v} [partial_order \u03b1] [partial_order \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : monotone f) (h\u2082 : function.injective f) : strict_mono f := sorry\n\ntheorem monotone.strict_mono_iff_injective {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [partial_order \u03b2] {f : \u03b1 \u2192 \u03b2} (h : monotone f) : strict_mono f \u2194 function.injective f :=\n  { mp := fun (h : strict_mono f) => strict_mono.injective h, mpr := strict_mono_of_monotone_of_injective h }\n\ntheorem strict_mono_of_le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} (h : \u2200 (x y : \u03b1), x \u2264 y \u2194 f x \u2264 f y) : strict_mono f := sorry\n\n/-! ### Order instances on the function space -/\n\nprotected instance pi.preorder {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] : preorder ((i : \u03b9) \u2192 \u03b1 i) :=\n  preorder.mk (fun (x y : (i : \u03b9) \u2192 \u03b1 i) => \u2200 (i : \u03b9), x i \u2264 y i)\n    (fun (a b : (i : \u03b9) \u2192 \u03b1 i) => (\u2200 (i : \u03b9), a i \u2264 b i) \u2227 \u00ac\u2200 (i : \u03b9), b i \u2264 a i) sorry sorry\n\ntheorem pi.le_def {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] {x : (i : \u03b9) \u2192 \u03b1 i} {y : (i : \u03b9) \u2192 \u03b1 i} : x \u2264 y \u2194 \u2200 (i : \u03b9), x i \u2264 y i :=\n  iff.rfl\n\ntheorem le_update_iff {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] [DecidableEq \u03b9] {x : (i : \u03b9) \u2192 \u03b1 i} {y : (i : \u03b9) \u2192 \u03b1 i} {i : \u03b9} {a : \u03b1 i} : x \u2264 function.update y i a \u2194 x i \u2264 a \u2227 \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2264 y j :=\n  function.forall_update_iff y fun (j : \u03b9) (z : \u03b1 j) => x j \u2264 z\n\ntheorem update_le_iff {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] [DecidableEq \u03b9] {x : (i : \u03b9) \u2192 \u03b1 i} {y : (i : \u03b9) \u2192 \u03b1 i} {i : \u03b9} {a : \u03b1 i} : function.update x i a \u2264 y \u2194 a \u2264 y i \u2227 \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2264 y j :=\n  function.forall_update_iff x fun (j : \u03b9) (z : \u03b1 j) => z \u2264 y j\n\nprotected instance pi.partial_order {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 partial_order (\u03b1 i)] : partial_order ((i : \u03b9) \u2192 \u03b1 i) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\ntheorem comp_le_comp_left_of_monotone {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} {g : \u03b3 \u2192 \u03b2} {h : \u03b3 \u2192 \u03b2} (m_f : monotone f) (le_gh : g \u2264 h) : f \u2218 g \u2264 f \u2218 h :=\n  fun (x : \u03b3) => m_f (le_gh x)\n\nprotected theorem monotone.order_dual {\u03b1 : Type u} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3] {f : \u03b1 \u2192 \u03b3} (hf : monotone f) : monotone f :=\n  fun (x y : order_dual \u03b1) (hxy : x \u2264 y) => hf hxy\n\ntheorem monotone_lam {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (m : \u2200 (b : \u03b2), monotone fun (a : \u03b1) => f a b) : monotone f :=\n  fun (a a' : \u03b1) (h : a \u2264 a') (b : \u03b2) => m b h\n\ntheorem monotone_app {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3] (f : \u03b2 \u2192 \u03b1 \u2192 \u03b3) (b : \u03b2) (m : monotone fun (a : \u03b1) (b : \u03b2) => f b a) : monotone (f b) :=\n  fun (a a' : \u03b1) (h : a \u2264 a') => m h b\n\ntheorem strict_mono.order_dual {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : strict_mono f) : strict_mono f :=\n  fun (x y : order_dual \u03b1) (hxy : x < y) => hf hxy\n\n/-- Transfer a `preorder` on `\u03b2` to a `preorder` on `\u03b1` using a function `f : \u03b1 \u2192 \u03b2`. -/\ndef preorder.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) : preorder \u03b1 :=\n  preorder.mk (fun (x y : \u03b1) => f x \u2264 f y) (fun (x y : \u03b1) => f x < f y) sorry sorry\n\n/-- Transfer a `partial_order` on `\u03b2` to a `partial_order` on `\u03b1` using an injective\nfunction `f : \u03b1 \u2192 \u03b2`. -/\ndef partial_order.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [partial_order \u03b2] (f : \u03b1 \u2192 \u03b2) (inj : function.injective f) : partial_order \u03b1 :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\n/-- Transfer a `linear_order` on `\u03b2` to a `linear_order` on `\u03b1` using an injective\nfunction `f : \u03b1 \u2192 \u03b2`. -/\ndef linear_order.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b2] (f : \u03b1 \u2192 \u03b2) (inj : function.injective f) : linear_order \u03b1 :=\n  linear_order.mk partial_order.le partial_order.lt sorry sorry sorry sorry (fun (x y : \u03b1) => infer_instance)\n    (fun (x y : \u03b1) => decidable_of_iff (f x = f y) (function.injective.eq_iff inj)) fun (x y : \u03b1) => infer_instance\n\nprotected instance subtype.preorder {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) : preorder (Subtype p) :=\n  preorder.lift subtype.val\n\n@[simp] theorem subtype.mk_le_mk {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : \u03b1} {y : \u03b1} {hx : p x} {hy : p y} : { val := x, property := hx } \u2264 { val := y, property := hy } \u2194 x \u2264 y :=\n  iff.rfl\n\n@[simp] theorem subtype.mk_lt_mk {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : \u03b1} {y : \u03b1} {hx : p x} {hy : p y} : { val := x, property := hx } < { val := y, property := hy } \u2194 x < y :=\n  iff.rfl\n\n@[simp] theorem subtype.coe_le_coe {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : Subtype p} {y : Subtype p} : \u2191x \u2264 \u2191y \u2194 x \u2264 y :=\n  iff.rfl\n\n@[simp] theorem subtype.coe_lt_coe {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : Subtype p} {y : Subtype p} : \u2191x < \u2191y \u2194 x < y :=\n  iff.rfl\n\nprotected instance subtype.partial_order {\u03b1 : Type u_1} [partial_order \u03b1] (p : \u03b1 \u2192 Prop) : partial_order (Subtype p) :=\n  partial_order.lift subtype.val subtype.val_injective\n\nprotected instance subtype.linear_order {\u03b1 : Type u_1} [linear_order \u03b1] (p : \u03b1 \u2192 Prop) : linear_order (Subtype p) :=\n  linear_order.lift subtype.val subtype.val_injective\n\ntheorem subtype.mono_coe {\u03b1 : Type u} [preorder \u03b1] (t : set \u03b1) : monotone coe :=\n  fun (x y : Subtype t) => id\n\ntheorem subtype.strict_mono_coe {\u03b1 : Type u} [preorder \u03b1] (t : set \u03b1) : strict_mono coe :=\n  fun (x y : Subtype t) => id\n\nprotected instance prod.has_le (\u03b1 : Type u) (\u03b2 : Type v) [HasLessEq \u03b1] [HasLessEq \u03b2] : HasLessEq (\u03b1 \u00d7 \u03b2) :=\n  { LessEq := fun (p q : \u03b1 \u00d7 \u03b2) => prod.fst p \u2264 prod.fst q \u2227 prod.snd p \u2264 prod.snd q }\n\nprotected instance prod.preorder (\u03b1 : Type u) (\u03b2 : Type v) [preorder \u03b1] [preorder \u03b2] : preorder (\u03b1 \u00d7 \u03b2) :=\n  preorder.mk LessEq (fun (a b : \u03b1 \u00d7 \u03b2) => a \u2264 b \u2227 \u00acb \u2264 a) sorry sorry\n\n/-- The pointwise partial order on a product.\n    (The lexicographic ordering is defined in order/lexicographic.lean, and the instances are\n    available via the type synonym `lex \u03b1 \u03b2 = \u03b1 \u00d7 \u03b2`.) -/\nprotected instance prod.partial_order (\u03b1 : Type u) (\u03b2 : Type v) [partial_order \u03b1] [partial_order \u03b2] : partial_order (\u03b1 \u00d7 \u03b2) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\n/-!\n### Additional order classes\n-/\n\n/-- order without a top element; somtimes called cofinal -/\nclass no_top_order (\u03b1 : Type u) [preorder \u03b1] \nwhere\n  no_top : \u2200 (a : \u03b1), \u2203 (a' : \u03b1), a < a'\n\ntheorem no_top {\u03b1 : Type u} [preorder \u03b1] [no_top_order \u03b1] (a : \u03b1) : \u2203 (a' : \u03b1), a < a' :=\n  no_top_order.no_top\n\nprotected instance nonempty_gt {\u03b1 : Type u} [preorder \u03b1] [no_top_order \u03b1] (a : \u03b1) : Nonempty (Subtype fun (x : \u03b1) => a < x) :=\n  iff.mpr nonempty_subtype (no_top a)\n\n/-- order without a bottom element; somtimes called coinitial or dense -/\nclass no_bot_order (\u03b1 : Type u) [preorder \u03b1] \nwhere\n  no_bot : \u2200 (a : \u03b1), \u2203 (a' : \u03b1), a' < a\n\ntheorem no_bot {\u03b1 : Type u} [preorder \u03b1] [no_bot_order \u03b1] (a : \u03b1) : \u2203 (a' : \u03b1), a' < a :=\n  no_bot_order.no_bot\n\nprotected instance order_dual.no_top_order (\u03b1 : Type u) [preorder \u03b1] [no_bot_order \u03b1] : no_top_order (order_dual \u03b1) :=\n  no_top_order.mk fun (a : order_dual \u03b1) => no_bot a\n\nprotected instance order_dual.no_bot_order (\u03b1 : Type u) [preorder \u03b1] [no_top_order \u03b1] : no_bot_order (order_dual \u03b1) :=\n  no_bot_order.mk fun (a : order_dual \u03b1) => no_top a\n\nprotected instance nonempty_lt {\u03b1 : Type u} [preorder \u03b1] [no_bot_order \u03b1] (a : \u03b1) : Nonempty (Subtype fun (x : \u03b1) => x < a) :=\n  iff.mpr nonempty_subtype (no_bot a)\n\n/-- An order is dense if there is an element between any pair of distinct elements. -/\nclass densely_ordered (\u03b1 : Type u) [preorder \u03b1] \nwhere\n  dense : \u2200 (a\u2081 a\u2082 : \u03b1), a\u2081 < a\u2082 \u2192 \u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082\n\ntheorem exists_between {\u03b1 : Type u} [preorder \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} : a\u2081 < a\u2082 \u2192 \u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082 :=\n  densely_ordered.dense\n\nprotected instance order_dual.densely_ordered (\u03b1 : Type u) [preorder \u03b1] [densely_ordered \u03b1] : densely_ordered (order_dual \u03b1) :=\n  densely_ordered.mk fun (a\u2081 a\u2082 : order_dual \u03b1) (ha : a\u2081 < a\u2082) => Exists.imp (fun (a : \u03b1) => and.symm) (exists_between ha)\n\ntheorem le_of_forall_le_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} (h : \u2200 (a\u2083 : \u03b1), a\u2083 > a\u2082 \u2192 a\u2081 \u2264 a\u2083) : a\u2081 \u2264 a\u2082 := sorry\n\ntheorem eq_of_le_of_forall_le_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} (h\u2081 : a\u2082 \u2264 a\u2081) (h\u2082 : \u2200 (a\u2083 : \u03b1), a\u2083 > a\u2082 \u2192 a\u2081 \u2264 a\u2083) : a\u2081 = a\u2082 :=\n  le_antisymm (le_of_forall_le_of_dense h\u2082) h\u2081\n\ntheorem le_of_forall_ge_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} (h : \u2200 (a\u2083 : \u03b1), a\u2083 < a\u2081 \u2192 a\u2083 \u2264 a\u2082) : a\u2081 \u2264 a\u2082 := sorry\n\ntheorem eq_of_le_of_forall_ge_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} (h\u2081 : a\u2082 \u2264 a\u2081) (h\u2082 : \u2200 (a\u2083 : \u03b1), a\u2083 < a\u2081 \u2192 a\u2083 \u2264 a\u2082) : a\u2081 = a\u2082 :=\n  le_antisymm (le_of_forall_ge_of_dense h\u2082) h\u2081\n\ntheorem dense_or_discrete {\u03b1 : Type u} [linear_order \u03b1] (a\u2081 : \u03b1) (a\u2082 : \u03b1) : (\u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082) \u2228 (\u2200 (a : \u03b1), a > a\u2081 \u2192 a\u2082 \u2264 a) \u2227 \u2200 (a : \u03b1), a < a\u2082 \u2192 a \u2264 a\u2081 := sorry\n\n/-- Type synonym to create an instance of `linear_order` from a\n`partial_order` and `[is_total \u03b1 (\u2264)]` -/\ndef as_linear_order (\u03b1 : Type u) :=\n  \u03b1\n\nprotected instance as_linear_order.inhabited {\u03b1 : Type u_1} [Inhabited \u03b1] : Inhabited (as_linear_order \u03b1) :=\n  { default := Inhabited.default }\n\nprotected instance as_linear_order.linear_order {\u03b1 : Type u_1} [partial_order \u03b1] [is_total \u03b1 LessEq] : linear_order (as_linear_order \u03b1) :=\n  linear_order.mk partial_order.le partial_order.lt partial_order.le_refl partial_order.le_trans partial_order.le_antisymm\n    sorry (classical.dec_rel LessEq) Mathlib.decidable_eq_of_decidable_le Mathlib.decidable_lt_of_decidable_le\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.3863683385716758}}
{"text": "import o_minimal.Def\n\n/-\nDefinable sheaves.\n\nHere we provide just enough definitions, instances and lemmas\nto set up the tactic environment. The eventual frontend for all this\nis intended to consist of `definable` and the `definable_sheaf` class,\nbut there is still plenty of construction to do.\n-/\n\nnamespace o_minimal\n\nuniverse u\n\nvariables {R : Type u} (S : struc R)\n\nclass definable_sheaf (X : Type*) :=\n(definable : \u03a0 {K : Def S}, (K \u2192 X) \u2192 Prop)\n(definable_precomp : \u2200 {L K : Def S} (\u03c6 : L \u27f6 K) (f : K \u2192 X),\n  definable f \u2192 definable (f \u2218 \u03c6))\n(definable_cover : \u2200 {K : Def S} (f : K \u2192 X) (\ud835\udcdb : Def.cover K),\n  (\u2200 i, definable (f \u2218 \ud835\udcdb.map i)) \u2192 definable f)\n\nnamespace definable_sheaf\n\nvariables {S}\n\ndef rep {W : Type*} [has_coordinates R W] [is_definable S W] :\n  definable_sheaf S W :=\n{ definable := \u03bb K f, def_fun S f,\n  definable_precomp := \u03bb L K \u03c6 f hf, hf.comp \u03c6.is_definable,\n  definable_cover := \u03bb K f \ud835\udcdb h, Def.subcanonical \ud835\udcdb f h }\n\ninstance Def.definable_sheaf {X : Def S} : definable_sheaf S X :=\ndefinable_sheaf.rep\n\nvariables {X Y : Type*} [definable_sheaf S X] [definable_sheaf S Y]\n\ninstance prod.definable_sheaf : definable_sheaf S (X \u00d7 Y) :=\n{ definable := \u03bb K f,\n    definable_sheaf.definable (prod.fst \u2218 f) \u2227\n    definable_sheaf.definable (prod.snd \u2218 f),\n  definable_precomp := \u03bb L K \u03c6 _ h,\n    \u27e8definable_sheaf.definable_precomp \u03c6 _ h.1,\n     definable_sheaf.definable_precomp \u03c6 _ h.2\u27e9,\n  definable_cover := \u03bb K f \ud835\udcdb h,\n    \u27e8definable_sheaf.definable_cover _ \ud835\udcdb (\u03bb i, (h i).1),\n     definable_sheaf.definable_cover _ \ud835\udcdb (\u03bb i, (h i).2)\u27e9 }\n\ninstance fun.definable_sheaf : definable_sheaf S (X \u2192 Y) :=\n{ definable := \u03bb K f,\n    \u2200 (L : Def S) (g : L \u2192 K \u00d7 X) (h : definable_sheaf.definable g),\n      definable_sheaf.definable (function.uncurry f \u2218 g),\n  definable_precomp := \u03bb L K \u03c6 f hf M g hg,\n    hf M (\u03bb m, (\u03c6 (g m).1, (g m).2))\n      \u27e8definable_sheaf.definable_precomp \u27e8\u03bb m, (g m).1, hg.1\u27e9 \u03c6 \u03c6.is_definable, hg.2\u27e9,\n  definable_cover := \u03bb K f \ud835\udcdb h K' g hg, begin\n    let g\u2081 : K' \u27f6 K := \u27e8\u03bb k', (g k').1, hg.1\u27e9,\n    let \ud835\udcdb' := \ud835\udcdb.pullback g\u2081,\n    apply definable_sheaf.definable_cover _ \ud835\udcdb',\n    intro i,\n    specialize h i (\ud835\udcdb'.obj i)\n      (\u03bb l', (Def.pullback.\u03c0\u2082 g\u2081 _ l', (g (\ud835\udcdb'.map i l')).2))\n      \u27e8(Def.pullback.\u03c0\u2082 g\u2081 _).is_definable, definable_sheaf.definable_precomp _ _ hg.2\u27e9,\n    dsimp only [function.uncurry, function.comp] at \u22a2 h,\n    convert h,\n    ext x,\n    congr,\n    exact x.property.snd.snd,\n  end }\n\n/-- Intended to be an implementation detail of the tactic mode.\nIn \"user code\", use `definable` instead. -/\nstructure sect (\u0393 : Def S) (X : Type*) [definable_sheaf S X] :=\n(to_fun : \u0393 \u2192 X)\n(definable : definable_sheaf.definable to_fun)\n\ndef sect.precomp {\u0393' \u0393 : Def S} (\u03c3 : sect \u0393 X) (\u03c6 : \u0393' \u27f6 \u0393) : sect \u0393' X :=\n{ to_fun := \u03c3.to_fun \u2218 \u03c6.to_fun,\n  definable := definable_sheaf.definable_precomp \u03c6 \u03c3.to_fun \u03c3.definable }\n\nlemma definable_fun_iff {\u0393 : Def S} {f : \u0393 \u2192 X \u2192 Y} :\n  definable_sheaf.definable f \u2194\n  \u2200 (\u0393' : Def S) (\u03c0 : Hom \u0393' \u0393) (\u03c3 : sect \u0393' X),\n    definable_sheaf.definable (\u03bb i', f (\u03c0.to_fun i') (\u03c3.to_fun i')) :=\n\u27e8\u03bb H \u0393' \u03c0 \u03c3, H \u0393' (\u03bb \u03b3', (\u03c0.to_fun \u03b3', \u03c3.to_fun \u03b3')) \u27e8\u03c0.is_definable, \u03c3.definable\u27e9,\n \u03bb H M g hg, H M \u27e8_, hg.1\u27e9 \u27e8_, hg.2\u27e9\u27e9\n\n/- The weird binder types are because this lemma is intended to be applied\nby a special `defin` tactic. -/\nlemma definable_app {\u0393 : Def S}\n  (f : \u0393 \u2192 X \u2192 Y) {hf : definable_sheaf.definable f}\n  (x : \u0393 \u2192 X) {hx : definable_sheaf.definable x} :\n  definable_sheaf.definable (\u03bb i, f i (x i)) :=\nhf \u0393 (\u03bb \u03b3, (\u03b3, x \u03b3)) \u27e8def_fun.id, hx\u27e9\n\nend definable_sheaf\n\nstructure definable {X : Type*} [definable_sheaf S X] (x : X) : Prop :=\n(definable : \u2200 (K : Def S), definable_sheaf.definable (\u03bb (i : K), x))\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/src/o_minimal/sheaf/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3863683373887718}}
{"text": "/- Copyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Douglas, Floris van Doorn\n-/\n\nimport order.zorn algebra.module\n\nuniverse variable u\n\nnamespace tactic\nnamespace interactive\nsetup_tactic_parser\nopen tactic.interactive\nmeta def dunfold_coes (loc : parse location) : tactic unit :=\ndunfold [\n  ``coe, ``coe_t, ``has_coe_t.coe, ``coe_b,``has_coe.coe,\n  ``lift, ``has_lift.lift, ``lift_t, ``has_lift_t.lift,\n  ``coe_fn, ``has_coe_to_fun.coe, ``coe_sort, ``has_coe_to_sort.coe,\n  ``coe_base_aux, ``has_coe_t_aux.coe] loc\n  -- note: this adds extra arguments\n  -- todo: we need to repeat it, it is not idempotent.\nend interactive\nend tactic\n\n\n@[simp] lemma inv_smul_smul {K V : Type*} [field K] [add_comm_group V] [vector_space K V]\n  {k : K} {x : V} (h : k \u2260 0) : k\u207b\u00b9 \u2022 k \u2022 x = x :=\nby rw [\u2190mul_smul, inv_mul_cancel h, one_smul]\n\n@[simp] lemma smul_inv_smul {K V : Type*} [field K] [add_comm_group V] [vector_space K V]\n  {k : K} {x : V} (h : k \u2260 0) : k \u2022 k\u207b\u00b9 \u2022 x = x :=\nby rw [\u2190mul_smul, mul_inv_cancel h, one_smul]\n\nlemma subtype.le_def {\u03b1 : Type*} [partial_order \u03b1] {P : \u03b1 \u2192 Prop} {x y : \u03b1}\n  {hx : P x} {hy : P y} : (\u27e8x, hx\u27e9 : subtype P) \u2264 \u27e8y, hy\u27e9 \u2194 x \u2264 y :=\niff.refl _\n\nlemma subtype.prop {\u03b1} {p : \u03b1 \u2192 Prop} (x : subtype p) : p x := x.2\n\n\nnamespace zorn\n\n/- A version of Zorn's lemma for partial orders where we only have to find an upper bound for nonempty chains -/\ntheorem zorn_partial_order_nonempty {\u03b1 : Type u} [partial_order \u03b1] [nonempty \u03b1]\n  (h : \u2200c:set \u03b1, chain (\u2264) c \u2192 c.nonempty \u2192 \u2203ub, \u2200a\u2208c, a \u2264 ub) : \u2203m:\u03b1, \u2200a, m \u2264 a \u2192 a = m :=\nbegin\n  apply zorn_partial_order,\n  intros c hc, classical,\n  cases c.eq_empty_or_nonempty with h2c h2c,\n  { have := _inst_2, cases this with x, use x, intro y, rw h2c, rintro \u27e8\u27e9 },\n  { exact h c hc h2c },\nend\n\nend zorn\n\nsection lattice\nvariables {\u03b1 : Type*} [semilattice_sup_top \u03b1]\n\n\n/-- Two elements of a lattice are covering if their sup is the top element. -/\ndef covering (a b : \u03b1) : Prop := \u22a4 \u2264 a \u2294 b\n\ntheorem covering.eq_top {a b : \u03b1} (h : covering a b) : a \u2294 b = \u22a4 :=\neq_top_iff.2 h\n\ntheorem covering_iff {a b : \u03b1} : covering a b \u2194 a \u2294 b = \u22a4 :=\neq_top_iff.symm\n\ntheorem covering.comm {a b : \u03b1} : covering a b \u2194 covering b a :=\nby rw [covering, covering, sup_comm]\n\ntheorem covering.symm {a b : \u03b1} : covering a b \u2192 covering b a :=\ncovering.comm.1\n\n@[simp] theorem covering_top_left {a : \u03b1} : covering \u22a4 a := covering_iff.2 top_sup_eq\n@[simp] theorem covering_top_right {a : \u03b1} : covering a \u22a4 := covering_iff.2 sup_top_eq\n\ntheorem covering.mono {a b c d : \u03b1} (h\u2081 : a \u2264 b) (h\u2082 : c \u2264 d) (h : covering a c) : covering b d :=\nle_trans h (sup_le_sup h\u2081 h\u2082)\n\ntheorem covering.mono_left {a b c : \u03b1} (h : a \u2264 b) : covering a c \u2192 covering b c :=\ncovering.mono h (le_refl _)\n\ntheorem covering.mono_right {a b c : \u03b1} (h : b \u2264 c) : covering a b \u2192 covering a c :=\ncovering.mono (le_refl _) h\n\n@[simp] lemma covering_self {a : \u03b1} : covering a a \u2194 a = \u22a4 :=\nby simp [covering]\n\nlemma covering.ne {a b : \u03b1} (ha : a \u2260 \u22a4) (hab : covering a b) : a \u2260 b :=\nby { intro h, rw [\u2190h, covering_self] at hab, exact ha hab }\n\nend lattice", "meta": {"author": "fpvandoorn", "repo": "group-representations", "sha": "bd9d72311749187d3bd4f542d5eab83e8341856c", "save_path": "github-repos/lean/fpvandoorn-group-representations", "path": "github-repos/lean/fpvandoorn-group-representations/group-representations-bd9d72311749187d3bd4f542d5eab83e8341856c/src/misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6477982315512488, "lm_q1q2_score": 0.3863683373887717}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Scott Morrison\n-/\nimport category_theory.fin_category\nimport category_theory.limits.cones\nimport category_theory.adjunction.basic\nimport category_theory.category.preorder\nimport category_theory.category.ulift\n\n/-!\n# Filtered categories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA category is filtered if every finite diagram admits a cocone.\nWe give a simple characterisation of this condition as\n1. for every pair of objects there exists another object \"to the right\",\n2. for every pair of parallel morphisms there exists a morphism to the right so the compositions\n   are equal, and\n3. there exists some object.\n\nFiltered colimits are often better behaved than arbitrary colimits.\nSee `category_theory/limits/types` for some details.\n\nFiltered categories are nice because colimits indexed by filtered categories tend to be\neasier to describe than general colimits (and more often preserved by functors).\n\nIn this file we show that any functor from a finite category to a filtered category admits a cocone:\n* `cocone_nonempty [fin_category J] [is_filtered C] (F : J \u2964 C) : nonempty (cocone F)`\nMore generally,\nfor any finite collection of objects and morphisms between them in a filtered category\n(even if not closed under composition) there exists some object `Z` receiving maps from all of them,\nso that all the triangles (one edge from the finite set, two from morphisms to `Z`) commute.\nThis formulation is often more useful in practice and is available via `sup_exists`,\nwhich takes a finset of objects, and an indexed family (indexed by source and target)\nof finsets of morphisms.\n\nFurthermore, we give special support for two diagram categories: The `bowtie` and the `tulip`.\nThis is because these shapes show up in the proofs that forgetful functors of algebraic categories\n(e.g. `Mon`, `CommRing`, ...) preserve filtered colimits.\n\nAll of the above API, except for the `bowtie` and the `tulip`, is also provided for cofiltered\ncategories.\n\n## See also\nIn `category_theory.limits.filtered_colimit_commutes_finite_limit` we show that filtered colimits\ncommute with finite limits.\n\n-/\n\nopen function\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses w v v\u2081 u u\u2081 u\u2082\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\n/--\nA category `is_filtered_or_empty` if\n1. for every pair of objects there exists another object \"to the right\", and\n2. for every pair of parallel morphisms there exists a morphism to the right so the compositions\n   are equal.\n-/\nclass is_filtered_or_empty : Prop :=\n(cocone_objs : \u2200 (X Y : C), \u2203 Z (f : X \u27f6 Z) (g : Y \u27f6 Z), true)\n(cocone_maps : \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), \u2203 Z (h : Y \u27f6 Z), f \u226b h = g \u226b h)\n\n/--\nA category `is_filtered` if\n1. for every pair of objects there exists another object \"to the right\",\n2. for every pair of parallel morphisms there exists a morphism to the right so the compositions\n   are equal, and\n3. there exists some object.\n\nSee <https://stacks.math.columbia.edu/tag/002V>. (They also define a diagram being filtered.)\n-/\nclass is_filtered extends is_filtered_or_empty C : Prop :=\n[nonempty : nonempty C]\n\n@[priority 100]\ninstance is_filtered_or_empty_of_semilattice_sup\n  (\u03b1 : Type u) [semilattice_sup \u03b1] : is_filtered_or_empty \u03b1 :=\n{ cocone_objs := \u03bb X Y, \u27e8X \u2294 Y, hom_of_le le_sup_left, hom_of_le le_sup_right, trivial\u27e9,\n  cocone_maps := \u03bb X Y f g, \u27e8Y, \ud835\udfd9 _, (by ext)\u27e9, }\n\n@[priority 100]\ninstance is_filtered_of_semilattice_sup_nonempty\n  (\u03b1 : Type u) [semilattice_sup \u03b1] [nonempty \u03b1] : is_filtered \u03b1 := {}\n\n@[priority 100]\ninstance is_filtered_or_empty_of_directed_le (\u03b1 : Type u) [preorder \u03b1] [is_directed \u03b1 (\u2264)] :\n  is_filtered_or_empty \u03b1 :=\n{ cocone_objs := \u03bb X Y, let \u27e8Z, h1, h2\u27e9 := exists_ge_ge X Y in\n    \u27e8Z, hom_of_le h1, hom_of_le h2, trivial\u27e9,\n  cocone_maps := \u03bb X Y f g, \u27e8Y, \ud835\udfd9 _, by simp\u27e9 }\n\n@[priority 100]\ninstance is_filtered_of_directed_le_nonempty (\u03b1 : Type u) [preorder \u03b1] [is_directed \u03b1 (\u2264)]\n  [nonempty \u03b1] :\n  is_filtered \u03b1 := {}\n\n-- Sanity checks\nexample (\u03b1 : Type u) [semilattice_sup \u03b1] [order_bot \u03b1] : is_filtered \u03b1 := by apply_instance\nexample (\u03b1 : Type u) [semilattice_sup \u03b1] [order_top \u03b1] : is_filtered \u03b1 := by apply_instance\n\ninstance : is_filtered (discrete punit) :=\n{ cocone_objs := \u03bb X Y, \u27e8\u27e8punit.star\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, trivial\u27e9,\n  cocone_maps := \u03bb X Y f g, \u27e8\u27e8punit.star\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, dec_trivial\u27e9,\n  nonempty := \u27e8\u27e8punit.star\u27e9\u27e9 }\n\nnamespace is_filtered\n\nsection allow_empty\n\nvariables {C} [is_filtered_or_empty C]\n\nlemma cocone_objs : \u2200 (X Y : C), \u2203 Z (f : X \u27f6 Z) (g : Y \u27f6 Z), true :=\nis_filtered_or_empty.cocone_objs\nlemma cocone_maps : \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), \u2203 Z (h : Y \u27f6 Z), f \u226b h = g \u226b h :=\nis_filtered_or_empty.cocone_maps\n\n/--\n`max j j'` is an arbitrary choice of object to the right of both `j` and `j'`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def max (j j' : C) : C :=\n(cocone_objs j j').some\n\n/--\n`left_to_max j j'` is an arbitrary choice of morphism from `j` to `max j j'`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def left_to_max (j j' : C) : j \u27f6 max j j' :=\n(cocone_objs j j').some_spec.some\n\n/--\n`right_to_max j j'` is an arbitrary choice of morphism from `j'` to `max j j'`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def right_to_max (j j' : C) : j' \u27f6 max j j' :=\n(cocone_objs j j').some_spec.some_spec.some\n\n/--\n`coeq f f'`, for morphisms `f f' : j \u27f6 j'`, is an arbitrary choice of object\nwhich admits a morphism `coeq_hom f f' : j' \u27f6 coeq f f'` such that\n`coeq_condition : f \u226b coeq_hom f f' = f' \u226b coeq_hom f f'`.\nIts existence is ensured by `is_filtered`.\n-/\nnoncomputable def coeq {j j' : C} (f f' : j \u27f6 j') : C :=\n(cocone_maps f f').some\n\n/--\n`coeq_hom f f'`, for morphisms `f f' : j \u27f6 j'`, is an arbitrary choice of morphism\n`coeq_hom f f' : j' \u27f6 coeq f f'` such that\n`coeq_condition : f \u226b coeq_hom f f' = f' \u226b coeq_hom f f'`.\nIts existence is ensured by `is_filtered`.\n-/\nnoncomputable def coeq_hom {j j' : C} (f f' : j \u27f6 j') : j' \u27f6 coeq f f' :=\n(cocone_maps f f').some_spec.some\n\n/--\n`coeq_condition f f'`, for morphisms `f f' : j \u27f6 j'`, is the proof that\n`f \u226b coeq_hom f f' = f' \u226b coeq_hom f f'`.\n-/\n@[simp, reassoc]\nlemma coeq_condition {j j' : C} (f f' : j \u27f6 j') : f \u226b coeq_hom f f' = f' \u226b coeq_hom f f' :=\n(cocone_maps f f').some_spec.some_spec\n\nend allow_empty\n\nsection nonempty\n\nopen category_theory.limits\n\nvariables {C} [is_filtered C]\n\n/--\nAny finite collection of objects in a filtered category has an object \"to the right\".\n-/\nlemma sup_objs_exists (O : finset C) : \u2203 (S : C), \u2200 {X}, X \u2208 O \u2192 _root_.nonempty (X \u27f6 S) :=\nbegin\n  classical,\n  apply finset.induction_on O,\n  { exact \u27e8is_filtered.nonempty.some, (by rintros - \u27e8\u27e9)\u27e9, },\n  { rintros X O' nm \u27e8S', w'\u27e9,\n    use max X S',\n    rintros Y mY,\n    obtain rfl|h := eq_or_ne Y X,\n    { exact \u27e8left_to_max _ _\u27e9, },\n    { exact \u27e8(w' (finset.mem_of_mem_insert_of_ne mY h)).some \u226b right_to_max _ _\u27e9, }, }\nend\n\nvariables (O : finset C) (H : finset (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y))\n\n/--\nGiven any `finset` of objects `{X, ...}` and\nindexed collection of `finset`s of morphisms `{f, ...}` in `C`,\nthere exists an object `S`, with a morphism `T X : X \u27f6 S` from each `X`,\nsuch that the triangles commute: `f \u226b T Y = T X`, for `f : X \u27f6 Y` in the `finset`.\n-/\nlemma sup_exists :\n  \u2203 (S : C) (T : \u03a0 {X : C}, X \u2208 O \u2192 (X \u27f6 S)), \u2200 {X Y : C} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y},\n    (\u27e8X, Y, mX, mY, f\u27e9 : (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y)) \u2208 H \u2192 f \u226b T mY = T mX :=\nbegin\n  classical,\n  apply finset.induction_on H,\n  { obtain \u27e8S, f\u27e9 := sup_objs_exists O,\n    refine \u27e8S, \u03bb X mX, (f mX).some, _\u27e9,\n    rintros - - - - - \u27e8\u27e9, },\n  { rintros \u27e8X, Y, mX, mY, f\u27e9 H' nmf \u27e8S', T', w'\u27e9,\n    refine \u27e8coeq (f \u226b T' mY) (T' mX), \u03bb Z mZ, T' mZ \u226b coeq_hom (f \u226b T' mY) (T' mX), _\u27e9,\n    intros X' Y' mX' mY' f' mf',\n    rw [\u2190category.assoc],\n    by_cases h : X = X' \u2227 Y = Y',\n    { rcases h with \u27e8rfl, rfl\u27e9,\n      by_cases hf : f = f',\n      { subst hf,\n        apply coeq_condition, },\n      { rw @w' _ _ mX mY f' (by simpa [hf \u2218 eq.symm] using mf') }, },\n    { rw @w' _ _ mX' mY' f' _,\n      apply finset.mem_of_mem_insert_of_ne mf',\n      contrapose! h,\n      obtain \u27e8rfl, h\u27e9 := h,\n      rw [heq_iff_eq, psigma.mk.inj_iff] at h,\n      exact \u27e8rfl, h.1.symm\u27e9 }, },\nend\n\n/--\nAn arbitrary choice of object \"to the right\"\nof a finite collection of objects `O` and morphisms `H`,\nmaking all the triangles commute.\n-/\nnoncomputable\ndef sup : C :=\n(sup_exists O H).some\n\n/--\nThe morphisms to `sup O H`.\n-/\nnoncomputable\ndef to_sup {X : C} (m : X \u2208 O) :\n  X \u27f6 sup O H :=\n(sup_exists O H).some_spec.some m\n\n/--\nThe triangles of consisting of a morphism in `H` and the maps to `sup O H` commute.\n-/\nlemma to_sup_commutes\n  {X Y : C} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}\n  (mf : (\u27e8X, Y, mX, mY, f\u27e9 : \u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y) \u2208 H) :\n  f \u226b to_sup O H mY = to_sup O H mX :=\n(sup_exists O H).some_spec.some_spec mX mY mf\n\nvariables {J : Type v} [small_category J] [fin_category J]\n\n/--\nIf we have `is_filtered C`, then for any functor `F : J \u2964 C` with `fin_category J`,\nthere exists a cocone over `F`.\n-/\nlemma cocone_nonempty (F : J \u2964 C) : _root_.nonempty (cocone F) :=\nbegin\n  classical,\n  let O := (finset.univ.image F.obj),\n  let H : finset (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y) :=\n    finset.univ.bUnion (\u03bb X : J, finset.univ.bUnion (\u03bb Y : J, finset.univ.image (\u03bb f : X \u27f6 Y,\n      \u27e8F.obj X, F.obj Y, by simp, by simp, F.map f\u27e9))),\n  obtain \u27e8Z, f, w\u27e9 := sup_exists O H,\n  refine \u27e8\u27e8Z, \u27e8\u03bb X, f (by simp), _\u27e9\u27e9\u27e9,\n  intros j j' g,\n  dsimp,\n  simp only [category.comp_id],\n  apply w,\n  simp only [finset.mem_univ, finset.mem_bUnion, exists_and_distrib_left,\n    exists_prop_of_true, finset.mem_image],\n  exact \u27e8j, rfl, j', g, (by simp)\u27e9,\nend\n\n/--\nAn arbitrary choice of cocone over `F : J \u2964 C`, for `fin_category J` and `is_filtered C`.\n-/\nnoncomputable def cocone (F : J \u2964 C) : cocone F :=\n(cocone_nonempty F).some\n\nvariables {D : Type u\u2081} [category.{v\u2081} D]\n\n/--\nIf `C` is filtered, and we have a functor `R : C \u2964 D` with a left adjoint, then `D` is filtered.\n-/\nlemma of_right_adjoint {L : D \u2964 C} {R : C \u2964 D} (h : L \u22a3 R) : is_filtered D :=\n{ cocone_objs := \u03bb X Y,\n    \u27e8_, h.hom_equiv _ _ (left_to_max _ _), h.hom_equiv _ _ (right_to_max _ _), \u27e8\u27e9\u27e9,\n  cocone_maps := \u03bb X Y f g,\n    \u27e8_, h.hom_equiv _ _ (coeq_hom _ _),\n     by rw [\u2190 h.hom_equiv_naturality_left, \u2190 h.hom_equiv_naturality_left, coeq_condition]\u27e9,\n  nonempty := is_filtered.nonempty.map R.obj }\n\n/-- If `C` is filtered, and we have a right adjoint functor `R : C \u2964 D`, then `D` is filtered. -/\nlemma of_is_right_adjoint (R : C \u2964 D) [is_right_adjoint R] : is_filtered D :=\nof_right_adjoint (adjunction.of_right_adjoint R)\n\n/-- Being filtered is preserved by equivalence of categories. -/\nlemma of_equivalence (h : C \u224c D) : is_filtered D :=\nof_right_adjoint h.symm.to_adjunction\n\nend nonempty\n\nsection special_shapes\n\nvariables {C} [is_filtered_or_empty C]\n\n/--\n`max\u2083 j\u2081 j\u2082 j\u2083` is an arbitrary choice of object to the right of `j\u2081`, `j\u2082` and `j\u2083`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def max\u2083 (j\u2081 j\u2082 j\u2083 : C) : C := max (max j\u2081 j\u2082) j\u2083\n\n/--\n`first_to_max\u2083 j\u2081 j\u2082 j\u2083` is an arbitrary choice of morphism from `j\u2081` to `max\u2083 j\u2081 j\u2082 j\u2083`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def first_to_max\u2083 (j\u2081 j\u2082 j\u2083 : C) : j\u2081 \u27f6 max\u2083 j\u2081 j\u2082 j\u2083 :=\nleft_to_max j\u2081 j\u2082 \u226b left_to_max (max j\u2081 j\u2082) j\u2083\n\n/--\n`second_to_max\u2083 j\u2081 j\u2082 j\u2083` is an arbitrary choice of morphism from `j\u2082` to `max\u2083 j\u2081 j\u2082 j\u2083`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def second_to_max\u2083 (j\u2081 j\u2082 j\u2083 : C) : j\u2082 \u27f6 max\u2083 j\u2081 j\u2082 j\u2083 :=\nright_to_max j\u2081 j\u2082 \u226b left_to_max (max j\u2081 j\u2082) j\u2083\n\n/--\n`third_to_max\u2083 j\u2081 j\u2082 j\u2083` is an arbitrary choice of morphism from `j\u2083` to `max\u2083 j\u2081 j\u2082 j\u2083`,\nwhose existence is ensured by `is_filtered`.\n-/\nnoncomputable def third_to_max\u2083 (j\u2081 j\u2082 j\u2083 : C) : j\u2083 \u27f6 max\u2083 j\u2081 j\u2082 j\u2083 :=\nright_to_max (max j\u2081 j\u2082) j\u2083\n\n/--\n`coeq\u2083 f g h`, for morphisms `f g h : j\u2081 \u27f6 j\u2082`, is an arbitrary choice of object\nwhich admits a morphism `coeq\u2083_hom f g h : j\u2082 \u27f6 coeq\u2083 f g h` such that\n`coeq\u2083_condition\u2081`, `coeq\u2083_condition\u2082` and `coeq\u2083_condition\u2083` are satisfied.\nIts existence is ensured by `is_filtered`.\n-/\nnoncomputable def coeq\u2083 {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) : C :=\ncoeq (coeq_hom f g \u226b left_to_max (coeq f g) (coeq g h))\n  (coeq_hom g h \u226b right_to_max (coeq f g) (coeq g h))\n\n/--\n`coeq\u2083_hom f g h`, for morphisms `f g h : j\u2081 \u27f6 j\u2082`, is an arbitrary choice of morphism\n`j\u2082 \u27f6 coeq\u2083 f g h` such that `coeq\u2083_condition\u2081`, `coeq\u2083_condition\u2082` and `coeq\u2083_condition\u2083`\nare satisfied. Its existence is ensured by `is_filtered`.\n-/\nnoncomputable def coeq\u2083_hom {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) : j\u2082 \u27f6 coeq\u2083 f g h :=\ncoeq_hom f g \u226b left_to_max (coeq f g) (coeq g h) \u226b\ncoeq_hom (coeq_hom f g \u226b left_to_max (coeq f g) (coeq g h))\n  (coeq_hom g h \u226b right_to_max (coeq f g) (coeq g h))\n\nlemma coeq\u2083_condition\u2081 {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) :\n  f \u226b coeq\u2083_hom f g h = g \u226b coeq\u2083_hom f g h :=\nby rw [coeq\u2083_hom, reassoc_of (coeq_condition f g)]\n\nlemma coeq\u2083_condition\u2082 {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) :\n  g \u226b coeq\u2083_hom f g h = h \u226b coeq\u2083_hom f g h :=\nbegin\n  dsimp [coeq\u2083_hom],\n  slice_lhs 2 4 { rw [\u2190 category.assoc, coeq_condition _ _] },\n  slice_rhs 2 4 { rw [\u2190 category.assoc, coeq_condition _ _] },\n  slice_lhs 1 3 { rw [\u2190 category.assoc, coeq_condition _ _] },\n  simp only [category.assoc],\nend\n\nlemma coeq\u2083_condition\u2083 {j\u2081 j\u2082 : C} (f g h : j\u2081 \u27f6 j\u2082) :\n  f \u226b coeq\u2083_hom f g h = h \u226b coeq\u2083_hom f g h :=\neq.trans (coeq\u2083_condition\u2081 f g h) (coeq\u2083_condition\u2082 f g h)\n\n/-- For every span `j \u27f5 i \u27f6 j'`, there\n   exists a cocone `j \u27f6 k \u27f5 j'` such that the square commutes. -/\nlemma span {i j j' : C} (f : i \u27f6 j) (f' : i \u27f6 j') :\n  \u2203 (k : C) (g : j \u27f6 k) (g' : j' \u27f6 k), f \u226b g = f' \u226b g' :=\nlet \u27e8K, G, G', _\u27e9 := cocone_objs j j', \u27e8k, e, he\u27e9 := cocone_maps (f \u226b G) (f' \u226b G') in\n\u27e8k, G \u226b e, G' \u226b e, by simpa only [\u2190 category.assoc]\u27e9\n\n/--\nGiven a \"bowtie\" of morphisms\n```\n j\u2081   j\u2082\n |\\  /|\n | \\/ |\n | /\\ |\n |/  \\\u2223\n vv  vv\n k\u2081  k\u2082\n```\nin a filtered category, we can construct an object `s` and two morphisms from `k\u2081` and `k\u2082` to `s`,\nmaking the resulting squares commute.\n-/\nlemma bowtie {j\u2081 j\u2082 k\u2081 k\u2082 : C}\n  (f\u2081 : j\u2081 \u27f6 k\u2081) (g\u2081 : j\u2081 \u27f6 k\u2082) (f\u2082 : j\u2082 \u27f6 k\u2081) (g\u2082 : j\u2082 \u27f6 k\u2082) :\n  \u2203 (s : C) (\u03b1 : k\u2081 \u27f6 s) (\u03b2 : k\u2082 \u27f6 s), f\u2081 \u226b \u03b1 = g\u2081 \u226b \u03b2 \u2227 f\u2082 \u226b \u03b1 = g\u2082 \u226b \u03b2 :=\nbegin\n  obtain \u27e8t, k\u2081t, k\u2082t, ht\u27e9 := span f\u2081 g\u2081,\n  obtain \u27e8s, ts, hs\u27e9 := cocone_maps (f\u2082 \u226b k\u2081t) (g\u2082 \u226b k\u2082t),\n  simp_rw category.assoc at hs,\n  exact \u27e8s, k\u2081t \u226b ts, k\u2082t \u226b ts, by rw reassoc_of ht, hs\u27e9,\nend\n\n/--\nGiven a \"tulip\" of morphisms\n```\n j\u2081    j\u2082    j\u2083\n |\\   / \\   / |\n | \\ /   \\ /  |\n |  vv    vv  |\n \\  k\u2081    k\u2082 /\n  \\         /\n   \\       /\n    \\     /\n     \\   /\n      v v\n       l\n```\nin a filtered category, we can construct an object `s` and three morphisms from `k\u2081`, `k\u2082` and `l`\nto `s`, making the resulting squares commute.\n-/\nlemma tulip {j\u2081 j\u2082 j\u2083 k\u2081 k\u2082 l : C} (f\u2081 : j\u2081 \u27f6 k\u2081) (f\u2082 : j\u2082 \u27f6 k\u2081) (f\u2083 : j\u2082 \u27f6 k\u2082) (f\u2084 : j\u2083 \u27f6 k\u2082)\n  (g\u2081 : j\u2081 \u27f6 l) (g\u2082 : j\u2083 \u27f6 l) :\n  \u2203 (s : C) (\u03b1 : k\u2081 \u27f6 s) (\u03b2 : l \u27f6 s) (\u03b3 : k\u2082 \u27f6 s),\n    f\u2081 \u226b \u03b1 = g\u2081 \u226b \u03b2 \u2227 f\u2082 \u226b \u03b1 = f\u2083 \u226b \u03b3 \u2227 f\u2084 \u226b \u03b3 = g\u2082 \u226b \u03b2 :=\nbegin\n  obtain \u27e8l', k\u2081l, k\u2082l, hl\u27e9 := span f\u2082 f\u2083,\n  obtain \u27e8s, ls, l's, hs\u2081, hs\u2082\u27e9 := bowtie g\u2081 (f\u2081 \u226b k\u2081l) g\u2082 (f\u2084 \u226b k\u2082l),\n  refine \u27e8s, k\u2081l \u226b l's, ls, k\u2082l \u226b l's, _, by rw reassoc_of hl, _\u27e9;\n  simp only [hs\u2081, hs\u2082, category.assoc],\nend\n\nend special_shapes\n\nend is_filtered\n\n/--\nA category `is_cofiltered_or_empty` if\n1. for every pair of objects there exists another object \"to the left\", and\n2. for every pair of parallel morphisms there exists a morphism to the left so the compositions\n   are equal.\n-/\nclass is_cofiltered_or_empty : Prop :=\n(cone_objs : \u2200 (X Y : C), \u2203 W (f : W \u27f6 X) (g : W \u27f6 Y), true)\n(cone_maps : \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), \u2203 W (h : W \u27f6 X), h \u226b f = h \u226b g)\n\n/--\nA category `is_cofiltered` if\n1. for every pair of objects there exists another object \"to the left\",\n2. for every pair of parallel morphisms there exists a morphism to the left so the compositions\n   are equal, and\n3. there exists some object.\n\nSee <https://stacks.math.columbia.edu/tag/04AZ>.\n-/\nclass is_cofiltered extends is_cofiltered_or_empty C : Prop :=\n[nonempty : nonempty C]\n\n@[priority 100]\ninstance is_cofiltered_or_empty_of_semilattice_inf\n  (\u03b1 : Type u) [semilattice_inf \u03b1] : is_cofiltered_or_empty \u03b1 :=\n{ cone_objs := \u03bb X Y, \u27e8X \u2293 Y, hom_of_le inf_le_left, hom_of_le inf_le_right, trivial\u27e9,\n  cone_maps := \u03bb X Y f g, \u27e8X, \ud835\udfd9 _, (by ext)\u27e9, }\n\n@[priority 100]\ninstance is_cofiltered_of_semilattice_inf_nonempty\n  (\u03b1 : Type u) [semilattice_inf \u03b1] [nonempty \u03b1] : is_cofiltered \u03b1 := {}\n\n@[priority 100]\ninstance is_cofiltered_or_empty_of_directed_ge (\u03b1 : Type u) [preorder \u03b1]\n  [is_directed \u03b1 (\u2265)] :\n  is_cofiltered_or_empty \u03b1 :=\n{ cone_objs := \u03bb X Y, let \u27e8Z, hX, hY\u27e9 := exists_le_le X Y in\n    \u27e8Z, hom_of_le hX, hom_of_le hY, trivial\u27e9,\n  cone_maps := \u03bb X Y f g, \u27e8X, \ud835\udfd9 _, by simp\u27e9 }\n\n@[priority 100]\ninstance is_cofiltered_of_directed_ge_nonempty (\u03b1 : Type u) [preorder \u03b1] [is_directed \u03b1 (\u2265)]\n  [nonempty \u03b1] :\n  is_cofiltered \u03b1 := {}\n\n-- Sanity checks\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] [order_bot \u03b1] : is_cofiltered \u03b1 := by apply_instance\nexample (\u03b1 : Type u) [semilattice_inf \u03b1] [order_top \u03b1] : is_cofiltered \u03b1 := by apply_instance\n\ninstance : is_cofiltered (discrete punit) :=\n{ cone_objs := \u03bb X Y, \u27e8\u27e8punit.star\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, trivial\u27e9,\n  cone_maps := \u03bb X Y f g, \u27e8\u27e8punit.star\u27e9, \u27e8\u27e8dec_trivial\u27e9\u27e9, dec_trivial\u27e9,\n  nonempty := \u27e8\u27e8punit.star\u27e9\u27e9 }\n\nnamespace is_cofiltered\n\nsection allow_empty\n\nvariables {C} [is_cofiltered_or_empty C]\n\nlemma cone_objs : \u2200 (X Y : C), \u2203 W (f : W \u27f6 X) (g : W \u27f6 Y), true := is_cofiltered_or_empty.cone_objs\nlemma cone_maps : \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), \u2203 W (h : W \u27f6 X), h \u226b f = h \u226b g :=\nis_cofiltered_or_empty.cone_maps\n\n/--\n`min j j'` is an arbitrary choice of object to the left of both `j` and `j'`,\nwhose existence is ensured by `is_cofiltered`.\n-/\nnoncomputable def min (j j' : C) : C :=\n(cone_objs j j').some\n\n/--\n`min_to_left j j'` is an arbitrary choice of morphism from `min j j'` to `j`,\nwhose existence is ensured by `is_cofiltered`.\n-/\nnoncomputable def min_to_left (j j' : C) : min j j' \u27f6 j :=\n(cone_objs j j').some_spec.some\n\n/--\n`min_to_right j j'` is an arbitrary choice of morphism from `min j j'` to `j'`,\nwhose existence is ensured by `is_cofiltered`.\n-/\nnoncomputable def min_to_right (j j' : C) : min j j' \u27f6 j' :=\n(cone_objs j j').some_spec.some_spec.some\n\n/--\n`eq f f'`, for morphisms `f f' : j \u27f6 j'`, is an arbitrary choice of object\nwhich admits a morphism `eq_hom f f' : eq f f' \u27f6 j` such that\n`eq_condition : eq_hom f f' \u226b f = eq_hom f f' \u226b f'`.\nIts existence is ensured by `is_cofiltered`.\n-/\nnoncomputable def eq {j j' : C} (f f' : j \u27f6 j') : C :=\n(cone_maps f f').some\n\n/--\n`eq_hom f f'`, for morphisms `f f' : j \u27f6 j'`, is an arbitrary choice of morphism\n`eq_hom f f' : eq f f' \u27f6 j` such that\n`eq_condition : eq_hom f f' \u226b f = eq_hom f f' \u226b f'`.\nIts existence is ensured by `is_cofiltered`.\n-/\nnoncomputable def eq_hom {j j' : C} (f f' : j \u27f6 j') : eq f f' \u27f6 j :=\n(cone_maps f f').some_spec.some\n\n/--\n`eq_condition f f'`, for morphisms `f f' : j \u27f6 j'`, is the proof that\n`eq_hom f f' \u226b f = eq_hom f f' \u226b f'`.\n-/\n@[simp, reassoc]\nlemma eq_condition {j j' : C} (f f' : j \u27f6 j') : eq_hom f f' \u226b f = eq_hom f f' \u226b f' :=\n(cone_maps f f').some_spec.some_spec\n\n/-- For every cospan `j \u27f6 i \u27f5 j'`,\n there exists a cone `j \u27f5 k \u27f6 j'` such that the square commutes. -/\nlemma cospan {i j j' : C} (f : j \u27f6 i) (f' : j' \u27f6 i) :\n  \u2203 (k : C) (g : k \u27f6 j) (g' : k \u27f6 j'), g \u226b f = g' \u226b f' :=\nlet \u27e8K, G, G', _\u27e9 := cone_objs j j', \u27e8k, e, he\u27e9 := cone_maps (G \u226b f) (G' \u226b f') in\n\u27e8k, e \u226b G, e \u226b G', by simpa only [category.assoc] using he\u27e9\n\nlemma _root_.category_theory.functor.ranges_directed (F : C \u2964 Type*) (j : C) :\n  directed (\u2287) (\u03bb (f : \u03a3' i, i \u27f6 j), set.range (F.map f.2)) :=\n\u03bb \u27e8i, ij\u27e9 \u27e8k, kj\u27e9, let \u27e8l, li, lk, e\u27e9 := cospan ij kj in\nby refine \u27e8\u27e8l, lk \u226b kj\u27e9, e \u25b8 _, _\u27e9; simp_rw F.map_comp; apply set.range_comp_subset_range\n\nend allow_empty\n\nsection nonempty\n\nopen category_theory.limits\n\nvariables {C} [is_cofiltered C]\n\n/--\nAny finite collection of objects in a cofiltered category has an object \"to the left\".\n-/\nlemma inf_objs_exists (O : finset C) : \u2203 (S : C), \u2200 {X}, X \u2208 O \u2192 _root_.nonempty (S \u27f6 X) :=\nbegin\n  classical,\n  apply finset.induction_on O,\n  { exact \u27e8is_cofiltered.nonempty.some, (by rintros - \u27e8\u27e9)\u27e9, },\n  { rintros X O' nm \u27e8S', w'\u27e9,\n    use min X S',\n    rintros Y mY,\n    obtain rfl|h := eq_or_ne Y X,\n    { exact \u27e8min_to_left _ _\u27e9, },\n    { exact \u27e8min_to_right _ _ \u226b (w' (finset.mem_of_mem_insert_of_ne mY h)).some\u27e9, }, }\nend\n\nvariables (O : finset C) (H : finset (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y))\n\n/--\nGiven any `finset` of objects `{X, ...}` and\nindexed collection of `finset`s of morphisms `{f, ...}` in `C`,\nthere exists an object `S`, with a morphism `T X : S \u27f6 X` from each `X`,\nsuch that the triangles commute: `T X \u226b f = T Y`, for `f : X \u27f6 Y` in the `finset`.\n-/\nlemma inf_exists :\n  \u2203 (S : C) (T : \u03a0 {X : C}, X \u2208 O \u2192 (S \u27f6 X)), \u2200 {X Y : C} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y},\n    (\u27e8X, Y, mX, mY, f\u27e9 : (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y)) \u2208 H \u2192 T mX \u226b f = T mY :=\nbegin\n  classical,\n  apply finset.induction_on H,\n  { obtain \u27e8S, f\u27e9 := inf_objs_exists O,\n    refine \u27e8S, \u03bb X mX, (f mX).some, _\u27e9,\n    rintros - - - - - \u27e8\u27e9, },\n  { rintros \u27e8X, Y, mX, mY, f\u27e9 H' nmf \u27e8S', T', w'\u27e9,\n    refine \u27e8eq (T' mX \u226b f) (T' mY), \u03bb Z mZ, eq_hom (T' mX \u226b f) (T' mY) \u226b T' mZ, _\u27e9,\n    intros X' Y' mX' mY' f' mf',\n    rw [category.assoc],\n    by_cases h : X = X' \u2227 Y = Y',\n    { rcases h with \u27e8rfl, rfl\u27e9,\n      by_cases hf : f = f',\n      { subst hf,\n        apply eq_condition, },\n      { rw @w' _ _ mX mY f' (by simpa [hf \u2218 eq.symm] using mf') }, },\n    { rw @w' _ _ mX' mY' f' _,\n      apply finset.mem_of_mem_insert_of_ne mf',\n      contrapose! h,\n      obtain \u27e8rfl, h\u27e9 := h,\n      rw [heq_iff_eq, psigma.mk.inj_iff] at h,\n      exact \u27e8rfl, h.1.symm\u27e9 }, },\nend\n\n/--\nAn arbitrary choice of object \"to the left\"\nof a finite collection of objects `O` and morphisms `H`,\nmaking all the triangles commute.\n-/\nnoncomputable\ndef inf : C :=\n(inf_exists O H).some\n\n/--\nThe morphisms from `inf O H`.\n-/\nnoncomputable\ndef inf_to {X : C} (m : X \u2208 O) :\n  inf O H \u27f6 X :=\n(inf_exists O H).some_spec.some m\n\n/--\nThe triangles consisting of a morphism in `H` and the maps from `inf O H` commute.\n-/\nlemma inf_to_commutes\n  {X Y : C} (mX : X \u2208 O) (mY : Y \u2208 O) {f : X \u27f6 Y}\n  (mf : (\u27e8X, Y, mX, mY, f\u27e9 : \u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y) \u2208 H) :\n  inf_to O H mX \u226b f = inf_to O H mY :=\n(inf_exists O H).some_spec.some_spec mX mY mf\n\nvariables {J : Type w} [small_category J] [fin_category J]\n\n/--\nIf we have `is_cofiltered C`, then for any functor `F : J \u2964 C` with `fin_category J`,\nthere exists a cone over `F`.\n-/\nlemma cone_nonempty (F : J \u2964 C) : _root_.nonempty (cone F) :=\nbegin\n  classical,\n  let O := (finset.univ.image F.obj),\n  let H : finset (\u03a3' (X Y : C) (mX : X \u2208 O) (mY : Y \u2208 O), X \u27f6 Y) :=\n    finset.univ.bUnion (\u03bb X : J, finset.univ.bUnion (\u03bb Y : J, finset.univ.image (\u03bb f : X \u27f6 Y,\n      \u27e8F.obj X, F.obj Y, by simp, by simp, F.map f\u27e9))),\n  obtain \u27e8Z, f, w\u27e9 := inf_exists O H,\n  refine \u27e8\u27e8Z, \u27e8\u03bb X, f (by simp), _\u27e9\u27e9\u27e9,\n  intros j j' g,\n  dsimp,\n  simp only [category.id_comp],\n  symmetry,\n  apply w,\n  simp only [finset.mem_univ, finset.mem_bUnion, exists_and_distrib_left,\n    exists_prop_of_true, finset.mem_image],\n  exact \u27e8j, rfl, j', g, (by simp)\u27e9,\nend\n\n/--\nAn arbitrary choice of cone over `F : J \u2964 C`, for `fin_category J` and `is_cofiltered C`.\n-/\nnoncomputable def cone (F : J \u2964 C) : cone F :=\n(cone_nonempty F).some\n\nvariables {D : Type u\u2081} [category.{v\u2081} D]\n\n/--\nIf `C` is cofiltered, and we have a functor `L : C \u2964 D` with a right adjoint,\nthen `D` is cofiltered.\n-/\nlemma of_left_adjoint {L : C \u2964 D} {R : D \u2964 C} (h : L \u22a3 R) : is_cofiltered D :=\n{ cone_objs := \u03bb X Y,\n    \u27e8L.obj (min (R.obj X) (R.obj Y)),\n      (h.hom_equiv _ X).symm (min_to_left _ _), (h.hom_equiv _ Y).symm (min_to_right _ _), \u27e8\u27e9\u27e9,\n  cone_maps := \u03bb X Y f g,\n    \u27e8L.obj (eq (R.map f) (R.map g)), (h.hom_equiv _ _).symm (eq_hom _ _),\n     by rw [\u2190 h.hom_equiv_naturality_right_symm, \u2190 h.hom_equiv_naturality_right_symm,\n       eq_condition]\u27e9,\n  nonempty := is_cofiltered.nonempty.map L.obj }\n\n/-- If `C` is cofiltered, and we have a left adjoint functor `L : C \u2964 D`, then `D` is cofiltered. -/\n\n\n/-- Being cofiltered is preserved by equivalence of categories. -/\nlemma of_equivalence (h : C \u224c D) : is_cofiltered D :=\nof_left_adjoint h.to_adjunction\n\nend nonempty\n\nend is_cofiltered\n\nsection opposite\nopen opposite\n\ninstance is_cofiltered_op_of_is_filtered [is_filtered C] : is_cofiltered C\u1d52\u1d56 :=\n{ cone_objs := \u03bb X Y, \u27e8op (is_filtered.max X.unop Y.unop),\n    (is_filtered.left_to_max _ _).op, (is_filtered.right_to_max _ _).op, trivial\u27e9,\n  cone_maps := \u03bb X Y f g, \u27e8op (is_filtered.coeq f.unop g.unop),\n    (is_filtered.coeq_hom _ _).op, begin\n      rw [(show f = f.unop.op, by simp), (show g = g.unop.op, by simp),\n        \u2190 op_comp, \u2190 op_comp],\n      congr' 1,\n      exact is_filtered.coeq_condition f.unop g.unop,\n    end\u27e9,\n  nonempty := \u27e8op is_filtered.nonempty.some\u27e9 }\n\ninstance is_filtered_op_of_is_cofiltered [is_cofiltered C] : is_filtered C\u1d52\u1d56 :=\n{ cocone_objs := \u03bb X Y, \u27e8op (is_cofiltered.min X.unop Y.unop),\n    (is_cofiltered.min_to_left X.unop Y.unop).op,\n    (is_cofiltered.min_to_right X.unop Y.unop).op, trivial\u27e9,\n  cocone_maps := \u03bb X Y f g, \u27e8op (is_cofiltered.eq f.unop g.unop),\n    (is_cofiltered.eq_hom f.unop g.unop).op, begin\n      rw [(show f = f.unop.op, by simp), (show g = g.unop.op, by simp),\n        \u2190 op_comp, \u2190 op_comp],\n      congr' 1,\n      exact is_cofiltered.eq_condition f.unop g.unop,\n    end\u27e9,\n  nonempty := \u27e8op is_cofiltered.nonempty.some\u27e9 }\n\nend opposite\n\nsection ulift\n\ninstance [is_filtered C] : is_filtered (ulift.{u\u2082} C) :=\nis_filtered.of_equivalence ulift.equivalence\n\ninstance [is_cofiltered C] : is_cofiltered (ulift.{u\u2082} C) :=\nis_cofiltered.of_equivalence ulift.equivalence\n\ninstance [is_filtered C] : is_filtered (ulift_hom C) :=\nis_filtered.of_equivalence ulift_hom.equiv\n\ninstance [is_cofiltered C] : is_cofiltered (ulift_hom C) :=\nis_cofiltered.of_equivalence ulift_hom.equiv\n\ninstance [is_filtered C] : is_filtered (as_small C) :=\nis_filtered.of_equivalence as_small.equiv\n\ninstance [is_cofiltered C] : is_cofiltered (as_small C) :=\nis_cofiltered.of_equivalence as_small.equiv\n\nend ulift\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/filtered.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521102, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.38636832927779474}}
{"text": "/- import rel_hoare\n\nnamespace arrcp\n\nopen parlang\nopen parlang.kernel\n\ndef tstate := string \u2192 \u2115\n\ndef state.update (name : string) (val : \u2115) (s : tstate) : tstate :=\n\u03bbn, if n = name then val else s n\n\nnotation s ` & ` n ` ::= ` v := state.update n v s\n\ndef init := \u03bb n : \u2115, \u03bb name, if name = \"tid\" then n else 0\n\ndef arrcp\u2081 : program tstate (\u03bb n : \u2115, \u2115) := program.intro (\u03bb m : memory (\u03bb n : \u2115, \u2115), m 0) (\n    kernel.load (\u03bb s, \u27e80, \u03bb a, s & \"n\" ::= a\u27e9) ;;               -- load n to thread tlocal\n    kernel.load (\u03bb s, \u27e81 + s \"tid\", \u03bb a, s & \"temp\" ::= a\u27e9) ;;  -- load from array a\n    kernel.store (\u03bb s, \u27e81 + s \"n\" + s \"tid\", s \"temp\"\u27e9)         -- store to array b\n)\n\ndef arrcp\u2082 : program tstate (\u03bb n : \u2115, \u2115) := program.intro (\u03bb m : memory (\u03bb n : \u2115, \u2115), 10) (\n    kernel.load (\u03bb s, \u27e80, \u03bb a, s & \"n\" ::= a\u27e9) ;;               -- load n to thread tlocal\n    kernel.compute (\u03bb s, s & \"i\" ::= s \"tid\") ;;\n    kernel.loop (\u03bb s, s \"i\" < s \"n\") (\n        kernel.load (\u03bb s, \u27e81 + s \"i\", \u03bb a, s & \"temp\" ::= a\u27e9) ;;-- load from array a\n        kernel.store (\u03bb s, \u27e81 + s \"n\" + s \"i\", s \"temp\"\u27e9) ;;    -- store to array b\n        kernel.compute (\u03bb s, s & \"i\" ::= s \"i\" + 10)\n    )\n)\n\nlemma arrcprel : rel_hoare_program init init eq arrcp\u2081 arrcp\u2082 eq := begin\n    unfold arrcp\u2081 arrcp\u2082,\n    apply rel_kernel_to_program,\n    apply single_step_left ((\u03bb (n\u2081 : \u2115) (s\u2081 : state n\u2081 (string \u2192 \u2115) (\u03bb (n : \u2115), \u2115)) (ac\u2081 : vector bool n\u2081) (n\u2082 : \u2115)\n     (s\u2082 : state n\u2082 (string \u2192 \u2115) (\u03bb (n : \u2115), \u2115)) (ac\u2082 : vector bool n\u2082),\n       \u2203 (m\u2081 m\u2082 : memory (\u03bb (n : \u2115), \u2115)),\n         state.syncable s\u2081 m\u2081 \u2227\n           state.syncable s\u2082 m\u2082 \u2227\n             n\u2081 = m\u2081 0 \u2227\n               n\u2082 = 10 \u2227\n                (\u2200 (i : fin n\u2081),\n                      vector.nth (s\u2081.threads) i =\n                        {tlocal := (init \u2191i) & \"n\" ::= m\u2081 0, shared := m\u2081, loads := insert 0 ((vector.nth (s\u2081.threads) i).loads), stores := \u2205}) \u2227\n                (\u2200 (i : fin n\u2082),\n                    vector.nth (s\u2082.threads) i =\n                        {tlocal := init \u2191i, shared := m\u2082, loads := \u2205, stores := \u2205}) \u2227\n                    m\u2081 = m\u2082 \u2227 \u21a5(all_threads_active ac\u2081) \u2227 \u21a5(all_threads_active ac\u2082))),\n    {\n        intros n\u2081 n\u2082 s\u2081 s\u2081' s\u2082 ac\u2081 ac\u2082 hp hek\u2081,\n        cases hp with m\u2081 hp,\n        cases hp with m\u2082 hp,\n        apply exists.intro s\u2082,\n        apply and.intro,\n        {\n            suffices h : exec_state (compute id) ac\u2082 s\u2082 (state.map_active_threads ac\u2082 (thread_state.map id) s\u2082),\n            {\n                rw \u2190 state.map_active_threads_id s\u2082 ac\u2082 at h,\n                assumption,\n            },\n            apply exec_state.compute,\n        }, {\n            simp,\n            apply exists.intro m\u2081,\n            cases hek\u2081,\n            apply and.intro,\n            {\n                \n                sorry -- memory does not change\n            }, {\n                apply exists.intro m\u2082,\n                apply and.intro,\n                { apply hp.right.left, },\n                {\n                    apply and.intro hp.right.right.left (and.intro hp.right.right.right.left (and.intro _ hp.right.right.right.right.right)),\n                    intro i,\n                    -- rw \u2190 hp.right.right.right.right.left,\n                    simp,\n                    have haa: \u21a5(vector.nth ac\u2081 i) := begin\n                        apply all_threads_active_nth,\n                        exact hp.right.right.right.right.right.right.right.left,\n                    end,\n                    simp [haa],\n                    rw thread_state.load,\n                    rw thread_state.load._match_1,\n                    have : m\u2081 = (vector.nth (s\u2081.threads) i).shared := begin\n                        rw hp.right.right.right.right.left,\n                    end,\n                    subst this,\n                    simp,\n                    have : (vector.nth (s\u2081.threads) i).tlocal = (init i) := begin\n                        rw hp.right.right.right.right.left,\n                    end,\n                    rw this,\n                    rw memory.get,\n                    simp,\n                    rw hp.right.right.right.right.left,\n                }\n            }\n        }\n    }, {\n\n    }\nend\n\nend arrcp -/", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/use_cases/arrcp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.754914997895581, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.38630253956337884}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Yury G. Kudryashov\n-/\nimport data.option.basic\n\n/-!\n# Disjoint union of types\n\nThis file proves basic results about the sum type `\u03b1 \u2295 \u03b2`.\n\n`\u03b1 \u2295 \u03b2` is the type made of a copy of `\u03b1` and a copy of `\u03b2`. It is also called *disjoint union*.\n\n## Main declarations\n\n* `sum.get_left`: Retrieves the left content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the right.\n* `sum.get_right`: Retrieves the right content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the left.\n* `sum.is_left`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the left component or not.\n* `sum.is_right`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the right component or not.\n* `sum.map`: Maps `\u03b1 \u2295 \u03b2` to `\u03b3 \u2295 \u03b4` component-wise.\n* `sum.elim`: Nondependent eliminator/induction principle for `\u03b1 \u2295 \u03b2`.\n* `sum.swap`: Maps `\u03b1 \u2295 \u03b2` to `\u03b2 \u2295 \u03b1` by swapping components.\n* `sum.lex`: Lexicographic order on `\u03b1 \u2295 \u03b2` induced by a relation on `\u03b1` and a relation on `\u03b2`.\n\n## Notes\n\nThe definition of `sum` takes values in `Type*`. This effectively forbids `Prop`- valued sum types.\nTo this effect, we have `psum`, which takes value in `Sort*` and carries a more complicated\nuniverse signature in consequence. The `Prop` version is `or`.\n-/\n\nuniverses u v w x\nvariables {\u03b1 : Type u} {\u03b1' : Type w} {\u03b2 : Type v} {\u03b2' : Type x} {\u03b3 \u03b4 : Type*}\n\nnamespace sum\n\nattribute [derive decidable_eq] sum\n\n@[simp] lemma \u00abforall\u00bb {p : \u03b1 \u2295 \u03b2 \u2192 Prop} : (\u2200 x, p x) \u2194 (\u2200 a, p (inl a)) \u2227 \u2200 b, p (inr b) :=\n\u27e8\u03bb h, \u27e8\u03bb a, h _, \u03bb b, h _\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, sum.rec h\u2081 h\u2082\u27e9\n\n@[simp] lemma \u00abexists\u00bb {p : \u03b1 \u2295 \u03b2 \u2192 Prop} : (\u2203 x, p x) \u2194 (\u2203 a, p (inl a)) \u2228 \u2203 b, p (inr b) :=\n\u27e8\u03bb h, match h with\n| \u27e8inl a, h\u27e9 := or.inl \u27e8a, h\u27e9\n| \u27e8inr b, h\u27e9 := or.inr \u27e8b, h\u27e9\nend, \u03bb h, match h with\n| or.inl \u27e8a, h\u27e9 := \u27e8inl a, h\u27e9\n| or.inr \u27e8b, h\u27e9 := \u27e8inr b, h\u27e9\nend\u27e9\n\nlemma inl_injective : function.injective (inl : \u03b1 \u2192 \u03b1 \u2295 \u03b2) := \u03bb x y, inl.inj\nlemma inr_injective : function.injective (inr : \u03b2 \u2192 \u03b1 \u2295 \u03b2) := \u03bb x y, inr.inj\n\nsection get\n\n/-- Check if a sum is `inl` and if so, retrieve its contents. -/\n@[simp] def get_left : \u03b1 \u2295 \u03b2 \u2192 option \u03b1\n| (inl a) := some a\n| (inr _) := none\n\n/-- Check if a sum is `inr` and if so, retrieve its contents. -/\n@[simp] def get_right : \u03b1 \u2295 \u03b2 \u2192 option \u03b2\n| (inr b) := some b\n| (inl _) := none\n\n/-- Check if a sum is `inl`. -/\n@[simp] def is_left : \u03b1 \u2295 \u03b2 \u2192 bool\n| (inl _) := tt\n| (inr _) := ff\n\n/-- Check if a sum is `inr`. -/\n@[simp] def is_right : \u03b1 \u2295 \u03b2 \u2192 bool\n| (inl _) := ff\n| (inr _) := tt\n\nvariables {x y : \u03b1 \u2295 \u03b2}\n\nlemma get_left_eq_none_iff : x.get_left = none \u2194 x.is_right :=\nby cases x; simp only [get_left, is_right, coe_sort_tt, coe_sort_ff, eq_self_iff_true]\n\nlemma get_right_eq_none_iff : x.get_right = none \u2194 x.is_left :=\nby cases x; simp only [get_right, is_left, coe_sort_tt, coe_sort_ff, eq_self_iff_true]\n\nend get\n\n/-- Map `\u03b1 \u2295 \u03b2` to `\u03b1' \u2295 \u03b2'` sending `\u03b1` to `\u03b1'` and `\u03b2` to `\u03b2'`. -/\nprotected def map (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2')  : \u03b1 \u2295 \u03b2 \u2192 \u03b1' \u2295 \u03b2'\n| (inl x) := inl (f x)\n| (inr x) := inr (g x)\n\n@[simp] lemma map_inl (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b1) : (inl x).map f g = inl (f x) := rfl\n@[simp] lemma map_inr (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b2) : (inr x).map f g = inr (g x) := rfl\n\n@[simp] lemma map_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n  \u2200 x : \u03b1 \u2295 \u03b2, (x.map f g).map f' g' = x.map (f' \u2218 f) (g' \u2218 g)\n| (inl a) := rfl\n| (inr b) := rfl\n\n@[simp] lemma map_comp_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n  (sum.map f' g') \u2218 (sum.map f g) = sum.map (f' \u2218 f) (g' \u2218 g) :=\nfunext $ map_map f' g' f g\n\n@[simp] lemma map_id_id (\u03b1 \u03b2) : sum.map (@id \u03b1) (@id \u03b2) = id :=\nfunext $ \u03bb x, sum.rec_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\ntheorem inl.inj_iff {a b} : (inl a : \u03b1 \u2295 \u03b2) = inl b \u2194 a = b :=\n\u27e8inl.inj, congr_arg _\u27e9\n\ntheorem inr.inj_iff {a b} : (inr a : \u03b1 \u2295 \u03b2) = inr b \u2194 a = b :=\n\u27e8inr.inj, congr_arg _\u27e9\n\ntheorem inl_ne_inr {a : \u03b1} {b : \u03b2} : inl a \u2260 inr b.\n\ntheorem inr_ne_inl {a : \u03b1} {b : \u03b2} : inr b \u2260 inl a.\n\n/-- Define a function on `\u03b1 \u2295 \u03b2` by giving separate definitions on `\u03b1` and `\u03b2`. -/\nprotected def elim {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 := \u03bb x, sum.rec_on x f g\n\n@[simp] lemma elim_inl {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b1) :\n  sum.elim f g (inl x) = f x := rfl\n\n@[simp] lemma elim_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b2) :\n  sum.elim f g (inr x) = g x := rfl\n\n@[simp] lemma elim_comp_inl {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) :\n  sum.elim f g \u2218 inl = f := rfl\n\n@[simp] lemma elim_comp_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) :\n  sum.elim f g \u2218 inr = g := rfl\n\n@[simp] lemma elim_inl_inr {\u03b1 \u03b2 : Sort*} :\n  @sum.elim \u03b1 \u03b2 _ inl inr = id :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\nlemma comp_elim {\u03b1 \u03b2 \u03b3 \u03b4 : Sort*} (f : \u03b3 \u2192 \u03b4) (g : \u03b1 \u2192 \u03b3) (h : \u03b2 \u2192 \u03b3):\n  f \u2218 sum.elim g h = sum.elim (f \u2218 g) (f \u2218 h) :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\n@[simp] lemma elim_comp_inl_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2295 \u03b2 \u2192 \u03b3) :\n  sum.elim (f \u2218 inl) (f \u2218 inr) = f :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\nopen function (update update_eq_iff update_comp_eq_of_injective update_comp_eq_of_forall_ne)\n\n@[simp] lemma update_elim_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  {i : \u03b1} {x : \u03b3} :\n  update (sum.elim f g) (inl i) x = sum.elim (update f i x) g :=\nupdate_eq_iff.2 \u27e8by simp, by simp { contextual := tt }\u27e9\n\n@[simp] lemma update_elim_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  {i : \u03b2} {x : \u03b3} :\n  update (sum.elim f g) (inr i) x = sum.elim f (update g i x) :=\nupdate_eq_iff.2 \u27e8by simp, by simp { contextual := tt }\u27e9\n\n@[simp] lemma update_inl_comp_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1}\n  {x : \u03b3} :\n  update f (inl i) x \u2218 inl = update (f \u2218 inl) i x :=\nupdate_comp_eq_of_injective _ inl_injective _ _\n\n@[simp] lemma update_inl_apply_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3}\n  {i j : \u03b1} {x : \u03b3} :\n  update f (inl i) x (inl j) = update (f \u2218 inl) i x j :=\nby rw \u2190 update_inl_comp_inl\n\n@[simp] lemma update_inl_comp_inr [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1} {x : \u03b3} :\n  update f (inl i) x \u2218 inr = f \u2218 inr :=\nupdate_comp_eq_of_forall_ne _ _ $ \u03bb _, inr_ne_inl\n\n@[simp] lemma update_inl_apply_inr [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3} :\n  update f (inl i) x (inr j) = f (inr j) :=\nfunction.update_noteq inr_ne_inl _ _\n\n@[simp] lemma update_inr_comp_inl [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b2} {x : \u03b3} :\n  update f (inr i) x \u2218 inl = f \u2218 inl :=\nupdate_comp_eq_of_forall_ne _ _ $ \u03bb _, inl_ne_inr\n\n@[simp] \n\n@[simp] lemma update_inr_comp_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b2}\n  {x : \u03b3} :\n  update f (inr i) x \u2218 inr = update (f \u2218 inr) i x :=\nupdate_comp_eq_of_injective _ inr_injective _ _\n\n@[simp] lemma update_inr_apply_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3}\n  {i j : \u03b2} {x : \u03b3} :\n  update f (inr i) x (inr j) = update (f \u2218 inr) i x j :=\nby rw \u2190 update_inr_comp_inr\n\n/-- Swap the factors of a sum type -/\n@[simp] def swap : \u03b1 \u2295 \u03b2 \u2192 \u03b2 \u2295 \u03b1\n| (inl a) := inr a\n| (inr b) := inl b\n\n@[simp] lemma swap_swap (x : \u03b1 \u2295 \u03b2) : swap (swap x) = x := by cases x; refl\n@[simp] lemma swap_swap_eq : swap \u2218 swap = @id (\u03b1 \u2295 \u03b2) := funext $ swap_swap\n@[simp] lemma swap_left_inverse : function.left_inverse (@swap \u03b1 \u03b2) swap := swap_swap\n@[simp] lemma swap_right_inverse : function.right_inverse (@swap \u03b1 \u03b2) swap := swap_swap\n\nsection lift_rel\n\n/-- Lifts pointwise two relations between `\u03b1` and `\u03b3` and between `\u03b2` and `\u03b4` to a relation between\n`\u03b1 \u2295 \u03b2` and `\u03b3 \u2295 \u03b4`. -/\ninductive lift_rel (r : \u03b1 \u2192 \u03b3 \u2192 Prop) (s : \u03b2 \u2192 \u03b4 \u2192 Prop) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 \u2295 \u03b4 \u2192 Prop\n| inl {a c} : r a c \u2192 lift_rel (inl a) (inl c)\n| inr {b d} : s b d \u2192 lift_rel (inr b) (inr d)\n\nattribute [protected] lift_rel.inl lift_rel.inr\n\nvariables {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop} {a : \u03b1} {b : \u03b2} {c : \u03b3} {d : \u03b4}\n  {x : \u03b1 \u2295 \u03b2} {y : \u03b3 \u2295 \u03b4}\n\n@[simp] lemma lift_rel_inl_inl : lift_rel r s (inl a) (inl c) \u2194 r a c :=\n\u27e8\u03bb h, by { cases h, assumption }, lift_rel.inl\u27e9\n\n@[simp] lemma not_lift_rel_inl_inr : \u00ac lift_rel r s (inl a) (inr d) .\n@[simp] lemma not_lift_rel_inr_inl : \u00ac lift_rel r s (inr b) (inl c) .\n\n@[simp] lemma lift_rel_inr_inr : lift_rel r s (inr b) (inr d) \u2194 s b d :=\n\u27e8\u03bb h, by { cases h, assumption }, lift_rel.inr\u27e9\n\ninstance [\u03a0 a c, decidable (r a c)] [\u03a0 b d, decidable (s b d)] :\n  \u03a0 (ab : \u03b1 \u2295 \u03b2) (cd : \u03b3 \u2295 \u03b4), decidable (lift_rel r s ab cd)\n| (inl a) (inl c) := decidable_of_iff' _ lift_rel_inl_inl\n| (inl a) (inr d) := decidable.is_false not_lift_rel_inl_inr\n| (inr b) (inl c) := decidable.is_false not_lift_rel_inr_inl\n| (inr b) (inr d) := decidable_of_iff' _ lift_rel_inr_inr\n\nlemma lift_rel.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)\n  (h : lift_rel r\u2081 s\u2081 x y) :\n  lift_rel r\u2082 s\u2082 x y :=\nby { cases h, exacts [lift_rel.inl (hr _ _ \u2039_\u203a), lift_rel.inr (hs _ _ \u2039_\u203a)] }\n\nlemma lift_rel.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : lift_rel r\u2081 s x y) :\n  lift_rel r\u2082 s x y :=\nh.mono hr $ \u03bb _ _, id\n\nlemma lift_rel.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)  (h : lift_rel r s\u2081 x y) :\n  lift_rel r s\u2082 x y :=\nh.mono (\u03bb _ _, id) hs\n\nprotected lemma lift_rel.swap (h : lift_rel r s x y) : lift_rel s r x.swap y.swap :=\nby { cases h, exacts [lift_rel.inr \u2039_\u203a, lift_rel.inl \u2039_\u203a] }\n\n@[simp] lemma lift_rel_swap_iff : lift_rel s r x.swap y.swap \u2194 lift_rel r s x y :=\n\u27e8\u03bb h, by { rw [\u2190swap_swap x, \u2190swap_swap y], exact h.swap }, lift_rel.swap\u27e9\n\nend lift_rel\n\nsection lex\n\n/-- Lexicographic order for sum. Sort all the `inl a` before the `inr b`, otherwise use the\nrespective order on `\u03b1` or `\u03b2`. -/\ninductive lex (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : \u03b1 \u2295 \u03b2 \u2192 \u03b1 \u2295 \u03b2 \u2192 Prop\n| inl {a\u2081 a\u2082} (h : r a\u2081 a\u2082) : lex (inl a\u2081) (inl a\u2082)\n| inr {b\u2081 b\u2082} (h : s b\u2081 b\u2082) : lex (inr b\u2081) (inr b\u2082)\n| sep (a b) : lex (inl a) (inr b)\n\nattribute [protected] sum.lex.inl sum.lex.inr\nattribute [simp] lex.sep\n\nvariables {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop} {a a\u2081 a\u2082 : \u03b1} {b b\u2081 b\u2082 : \u03b2}\n  {x y : \u03b1 \u2295 \u03b2}\n\n@[simp] lemma lex_inl_inl : lex r s (inl a\u2081) (inl a\u2082) \u2194 r a\u2081 a\u2082 :=\n\u27e8\u03bb h, by { cases h, assumption }, lex.inl\u27e9\n\n@[simp] lemma lex_inr_inr : lex r s (inr b\u2081) (inr b\u2082) \u2194 s b\u2081 b\u2082 :=\n\u27e8\u03bb h, by { cases h, assumption }, lex.inr\u27e9\n\n@[simp] lemma lex_inr_inl : \u00ac lex r s (inr b) (inl a) .\n\ninstance [decidable_rel r] [decidable_rel s] : decidable_rel (lex r s)\n| (inl a) (inl c) := decidable_of_iff' _ lex_inl_inl\n| (inl a) (inr d) := decidable.is_true (lex.sep _ _)\n| (inr b) (inl c) := decidable.is_false lex_inr_inl\n| (inr b) (inr d) := decidable_of_iff' _ lex_inr_inr\n\nprotected lemma lift_rel.lex {a b : \u03b1 \u2295 \u03b2} (h : lift_rel r s a b) : lex r s a b :=\nby { cases h, exacts [lex.inl \u2039_\u203a, lex.inr \u2039_\u203a] }\n\nlemma lex.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : lex r\u2081 s\u2081 x y) :\n  lex r\u2082 s\u2082 x y :=\nby { cases h, exacts [lex.inl (hr _ _ \u2039_\u203a), lex.inr (hs _ _ \u2039_\u203a), lex.sep _ _] }\n\nlemma lex.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : lex r\u2081 s x y) : lex r\u2082 s x y :=\nh.mono hr $ \u03bb _ _, id\n\nlemma lex.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)  (h : lex r s\u2081 x y) : lex r s\u2082 x y :=\nh.mono (\u03bb _ _, id) hs\n\nlemma lex_acc_inl {a} (aca : acc r a) : acc (lex r s) (inl a) :=\nbegin\n  induction aca with a H IH,\n  constructor, intros y h,\n  cases h with a' _ h',\n  exact IH _ h'\nend\n\nlemma lex_acc_inr (aca : \u2200 a, acc (lex r s) (inl a)) {b} (acb : acc s b) : acc (lex r s) (inr b) :=\nbegin\n  induction acb with b H IH,\n  constructor, intros y h,\n  cases h with _ _ _ b' _ h' a,\n  { exact IH _ h' },\n  { exact aca _ }\nend\n\nlemma lex_wf (ha : well_founded r) (hb : well_founded s) : well_founded (lex r s) :=\nhave aca : \u2200 a, acc (lex r s) (inl a), from \u03bb a, lex_acc_inl (ha.apply a),\n\u27e8\u03bb x, sum.rec_on x aca (\u03bb b, lex_acc_inr aca (hb.apply b))\u27e9\n\nend lex\nend sum\n\nnamespace function\n\nopen sum\n\nlemma injective.sum_elim {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  (hf : injective f) (hg : injective g) (hfg : \u2200 a b, f a \u2260 g b) :\n  injective (sum.elim f g)\n| (inl x) (inl y) h := congr_arg inl $ hf h\n| (inl x) (inr y) h := (hfg x y h).elim\n| (inr x) (inl y) h := (hfg y x h.symm).elim\n| (inr x) (inr y) h := congr_arg inr $ hg h\n\nlemma injective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : injective f) (hg : injective g) :\n  injective (sum.map f g)\n| (inl x) (inl y) h := congr_arg inl $ hf $ inl.inj h\n| (inr x) (inr y) h := congr_arg inr $ hg $ inr.inj h\n\nlemma surjective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : surjective f) (hg : surjective g) :\n  surjective (sum.map f g)\n| (inl y) := let \u27e8x, hx\u27e9 := hf y in \u27e8inl x, congr_arg inl hx\u27e9\n| (inr y) := let \u27e8x, hx\u27e9 := hg y in \u27e8inr x, congr_arg inr hx\u27e9\n\nend function\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/sum/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.6688802537704064, "lm_q1q2_score": 0.38627525614438096}}
{"text": "/-\nCopyright (c) 2020 Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Zhouhang Zhou, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.indicator_function\nimport Mathlib.order.filter.at_top_bot\nimport Mathlib.PostPort\n\nuniverses u_1 u_3 u_2 \n\nnamespace Mathlib\n\n/-!\n# Indicator function and filters\n\nProperties of indicator functions involving `=\u1da0` and `\u2264\u1da0`.\n\n## Tags\nindicator, characteristic, filter\n-/\n\ntheorem indicator_eventually_eq {\u03b1 : Type u_1} {M : Type u_3} [HasZero M] {s : set \u03b1} {t : set \u03b1}\n    {f : \u03b1 \u2192 M} {g : \u03b1 \u2192 M} {l : filter \u03b1} (hf : filter.eventually_eq (l \u2293 filter.principal s) f g)\n    (hs : filter.eventually_eq l s t) :\n    filter.eventually_eq l (set.indicator s f) (set.indicator t g) :=\n  sorry\n\ntheorem indicator_union_eventually_eq {\u03b1 : Type u_1} {M : Type u_3} [add_monoid M] {s : set \u03b1}\n    {t : set \u03b1} {f : \u03b1 \u2192 M} {l : filter \u03b1} (h : filter.eventually (fun (a : \u03b1) => \u00aca \u2208 s \u2229 t) l) :\n    filter.eventually_eq l (set.indicator (s \u222a t) f) (set.indicator s f + set.indicator t f) :=\n  filter.eventually.mono h\n    fun (a : \u03b1) (ha : \u00aca \u2208 s \u2229 t) => set.indicator_union_of_not_mem_inter ha f\n\ntheorem indicator_eventually_le_indicator {\u03b1 : Type u_1} {\u03b2 : Type u_2} [HasZero \u03b2] [preorder \u03b2]\n    {s : set \u03b1} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {l : filter \u03b1}\n    (h : filter.eventually_le (l \u2293 filter.principal s) f g) :\n    filter.eventually_le l (set.indicator s f) (set.indicator s g) :=\n  filter.eventually.mono (iff.mp filter.eventually_inf_principal h)\n    fun (a : \u03b1) (h : a \u2208 s \u2192 f a \u2264 g a) => set.indicator_rel_indicator (le_refl 0) h\n\ntheorem tendsto_indicator_of_monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Type u_3} [preorder \u03b9]\n    [HasZero \u03b2] (s : \u03b9 \u2192 set \u03b1) (hs : monotone s) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    filter.tendsto (fun (i : \u03b9) => set.indicator (s i) f a) filter.at_top\n        (pure (set.indicator (set.Union fun (i : \u03b9) => s i) f a)) :=\n  sorry\n\ntheorem tendsto_indicator_of_antimono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Type u_3} [preorder \u03b9]\n    [HasZero \u03b2] (s : \u03b9 \u2192 set \u03b1) (hs : \u2200 {i j : \u03b9}, i \u2264 j \u2192 s j \u2286 s i) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    filter.tendsto (fun (i : \u03b9) => set.indicator (s i) f a) filter.at_top\n        (pure (set.indicator (set.Inter fun (i : \u03b9) => s i) f a)) :=\n  sorry\n\ntheorem tendsto_indicator_bUnion_finset {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Type u_3} [HasZero \u03b2]\n    (s : \u03b9 \u2192 set \u03b1) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    filter.tendsto\n        (fun (n : finset \u03b9) =>\n          set.indicator (set.Union fun (i : \u03b9) => set.Union fun (H : i \u2208 n) => s i) f a)\n        filter.at_top (pure (set.indicator (set.Union s) f a)) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/filter/indicator_function_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3862752502285365}}
{"text": "/-\n  Specifications file for signature_spec.cairo\n\n  Do not modify the constant definitions, structure definitions, or automatic specifications.\n  Do not change the name or arguments of the user specifications and soundness theorems.\n\n  You may freely move the definitions around in the file.\n  You may add definitions and theorems wherever you wish in this file.\n-/\nimport starkware.cairo.lean.semantics.soundness.prelude\nimport starkware.cairo.common.cairo_secp.ec_spec\nimport starkware.cairo.common.cairo_secp.bigint_spec\nimport starkware.cairo.common.cairo_secp.constants_spec\nimport starkware.cairo.common.cairo_secp.field_spec\nimport starkware.cairo.common.math_spec\n\nopen starkware.cairo.common.cairo_secp.ec\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.constants\nopen starkware.cairo.common.cairo_secp.field\nopen starkware.cairo.common.math\n\nnamespace starkware.cairo.common.cairo_secp.signature\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\n\n--variables {secpF : Type} [secp_field secpF]\n\n-- End of automatically generated prelude.\n\n/-\n-- Function: get_generator_point\n-/\n\n/- get_generator_point autogenerated specification -/\n\ndef gen_point (F : Type) [field F] : EcPoint F :=\n  \u27e8\u27e817117865558768631194064792,\n    12501176021340589225372855,\n    9198697782662356105779718\u27e9,\n   \u27e86441780312434748884571320,\n     57953919405111227542741658,\n     5457536640262350763842127\u27e9\u27e9\n\ndef gen_point_data (F : Type) [field F] [decidable_eq F] [prelude_hyps F]\n    (secpF : Type) [secp_field secpF]:\n  BddECPointData secpF (gen_point F) :=\n{ ix := \u27e817117865558768631194064792,\n         12501176021340589225372855,\n         9198697782662356105779718\u27e9,\n  iy := \u27e86441780312434748884571320,\n         57953919405111227542741658,\n         5457536640262350763842127\u27e9,\n  ixbdd := begin\n             rw [bigint3.bounded, BASE], simp_int_casts,\n             repeat { rw [abs_of_nonneg] },\n             repeat { norm_num }\n            end,\n  iybdd := begin\n             rw [bigint3.bounded, BASE], simp_int_casts,\n             repeat { rw [abs_of_nonneg] },\n             repeat { norm_num }\n            end,\n  ptxeq := by { rw [gen_point, bigint3.toBigInt3], simp_int_casts, exact \u27e8rfl, rfl, rfl\u27e9 },\n  ptyeq := by { rw [gen_point, bigint3.toBigInt3], simp_int_casts, exact \u27e8rfl, rfl, rfl\u27e9 },\n  onEC :=\n    begin\n      right,\n      rw [\u2190int.cast_pow, \u2190int.cast_pow, (show (7 : secpF) = (7 : int), by simp_int_casts),\n        \u2190int.cast_add, char_p.int_coe_eq_int_coe_iff secpF SECP_PRIME, bigint3.val, bigint3.val,\n        SECP_PRIME_eq, int.modeq, BASE, SECP_REM],\n      dsimp, simp_int_casts, norm_num,\n    end }\n\n-- Do not change this definition.\ndef auto_spec_get_generator_point (mem : F \u2192 F) (\u03ba : \u2115) (\u03c1_point : EcPoint F) : Prop :=\n  7 \u2264 \u03ba \u2227\n  \u03c1_point = {\n    x := { d0 := 17117865558768631194064792, d1 := 12501176021340589225372855, d2 := 9198697782662356105779718 },\n    y := { d0 := 6441780312434748884571320, d1 := 57953919405111227542741658, d2 := 5457536640262350763842127 }\n  }\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_get_generator_point (mem : F \u2192 F) (\u03ba : \u2115) (\u03c1_point : EcPoint F) : Prop :=\n  \u03c1_point = gen_point F\n\n/- get_generator_point soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_get_generator_point\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (\u03c1_point : EcPoint F)\n    (h_auto : auto_spec_get_generator_point mem \u03ba \u03c1_point) :\n  spec_get_generator_point mem \u03ba \u03c1_point :=\nbegin\n  exact h_auto.2\nend\n\n-- Function: div_mod_n\n\n/- div_mod_n autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_div_mod_n (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (a b : BigInt3 F) (\u03c1_range_check_ptr : F) (\u03c1_res : BigInt3 F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F) (res : BigInt3 F), spec_nondet_bigint3 mem \u03ba\u2081 range_check_ptr range_check_ptr\u2081 res \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (range_check_ptr\u2082 : F) (k : BigInt3 F), spec_nondet_bigint3 mem \u03ba\u2082 range_check_ptr\u2081 range_check_ptr\u2082 k \u2227\n  \u2203 (\u03ba\u2083 : \u2115) (res_b : UnreducedBigInt5 F), spec_bigint_mul mem \u03ba\u2083 res b res_b \u2227\n  \u2203 n : BigInt3 F, n = {\n    d0 := N0,\n    d1 := N1,\n    d2 := N2\n  } \u2227\n  \u2203 (\u03ba\u2084 : \u2115) (k_n : UnreducedBigInt5 F), spec_bigint_mul mem \u03ba\u2084 k n k_n \u2227\n  \u2203 carry1 : F, carry1 = (res_b.d0 - k_n.d0 - a.d0) / (BASE : \u2124) \u2227\n  mem (range_check_ptr\u2082 + 0) = carry1 + 2 ^ 127 \u2227\n  is_range_checked (rc_bound F) (carry1 + 2 ^ 127) \u2227\n  \u2203 carry2 : F, carry2 = (res_b.d1 - k_n.d1 - a.d1 + carry1) / (BASE : \u2124) \u2227\n  mem (range_check_ptr\u2082 + 1) = carry2 + 2 ^ 127 \u2227\n  is_range_checked (rc_bound F) (carry2 + 2 ^ 127) \u2227\n  \u2203 carry3 : F, carry3 = (res_b.d2 - k_n.d2 - a.d2 + carry2) / (BASE : \u2124) \u2227\n  mem (range_check_ptr\u2082 + 2) = carry3 + 2 ^ 127 \u2227\n  is_range_checked (rc_bound F) (carry3 + 2 ^ 127) \u2227\n  \u2203 carry4 : F, carry4 = (res_b.d3 - k_n.d3 + carry3) / (BASE : \u2124) \u2227\n  mem (range_check_ptr\u2082 + 3) = carry4 + 2 ^ 127 \u2227\n  is_range_checked (rc_bound F) (carry4 + 2 ^ 127) \u2227\n  res_b.d4 - k_n.d4 + carry4 = 0 \u2227\n  \u2203 range_check_ptr\u2083 : F, range_check_ptr\u2083 = range_check_ptr\u2082 + 4 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + \u03ba\u2083 + \u03ba\u2084 + 48 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2083 \u2227\n  \u03c1_res = res\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_div_mod_n (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (a b : BigInt3 F) (\u03c1_range_check_ptr : F) (\u03c1_res : BigInt3 F) : Prop :=\n  \u2200 ia : bigint3,\n    ia.bounded (3 * BASE - 1) \u2192\n    a = ia.toBigInt3 \u2192\n  \u2200 ib : bigint3,\n    ib.bounded (3 * BASE - 1) \u2192\n    b = ib.toBigInt3 \u2192\n  \u2203 ires : bigint3,\n    ires.bounded (3 * BASE - 1) \u2227\n    \u03c1_res = ires.toBigInt3 \u2227\n    ires.val * ib.val \u2261 ia.val [ZMOD secp_n]\n\n/- div_mod_n soundness theorem -/\n\ndef iN0 : \u2124 := N0\ndef iN1 : \u2124 := N1\ndef iN2 : \u2124 := N2\n\ndef secp_in : bigint3 := \u27e8iN0, iN1, iN2\u27e9\n\ndef nBASE := BASE\n\n--theorem secp_in_toBigInt3 : secp_in.toBigInt3 = secp_n := rfl\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_div_mod_n\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr : F) (a b : BigInt3 F) (\u03c1_range_check_ptr : F) (\u03c1_res : BigInt3 F)\n    (h_auto : auto_spec_div_mod_n mem \u03ba range_check_ptr a b \u03c1_range_check_ptr \u03c1_res) :\n  spec_div_mod_n mem \u03ba range_check_ptr a b \u03c1_range_check_ptr \u03c1_res :=\nbegin\n  intros ia iabdd aeq ib ibbdd beq,\n  rcases h_auto with \u27e8_, _, res, hres, _, _, k, hk, _, res_b, hres_b, n, neq, _, k_n, hk_n,\n    carry1, carry1eq, _, hcarry1,\n    carry2, carry2eq, _, hcarry2,\n    carry3, carry3eq, _, hcarry3,\n    carry4, carry4eq, _, hcarry4,\n    diff_eq, _, _, _, _, \u03c1_res_eq\u27e9,\n  rcases nondet_bigint3_corr hres with \u27e8ires, reseq, ires_bdd\u27e9,\n  rcases nondet_bigint3_corr hk with \u27e8ik, keq, ik_bdd\u27e9,\n  change _ = _ at hres_b,\n  change _ = _ at hk_n,\n  rcases rc_to_int hcarry1 with \u27e8icarry1, carry1eq', carry1_bdd\u27e9,\n  rcases rc_to_int hcarry2 with \u27e8icarry2, carry2eq', carry2_bdd\u27e9,\n  rcases rc_to_int hcarry3 with \u27e8icarry3, carry3eq', carry3_bdd\u27e9,\n  rcases rc_to_int hcarry4 with \u27e8icarry4, carry4eq', carry4_bdd\u27e9,\n  have BASEnonneg : 0 \u2264 (BASE : \u2124) := int.coe_zero_le BASE,\n  have iN0nonneg : 0 \u2264 iN0 := int.coe_zero_le _,\n  have iN1nonneg : 0 \u2264 iN1 := int.coe_zero_le _,\n  have iN2nonneg : 0 \u2264 iN2 := int.coe_zero_le _,\n  have absBASEeq : abs (BASE : \u2124) = BASE := abs_of_nonneg BASEnonneg,\n  have BASEnz : ((BASE : \u2124) : F) \u2260 0,\n  { suffices : ((2 : \u2124) : F) \u2260 0,\n      by simpa using this,\n    haveI : char_p F PRIME := prelude_hyps.charF,\n    rw [ne, char_p.int_cast_eq_zero_iff F PRIME, PRIME],\n    simp_int_casts, norm_num },\n  have aux3 : \u2200 {x y : F}, x = y / (BASE : \u2124) \u2192 x * 2 ^ 86 = y,\n  { intros x y xeq,\n    rw [eq_div_iff BASEnz, int.cast_coe_nat, BASE] at xeq,\n    simp only [nat.cast_bit0, nat.cast_one, nat.cast_pow] at xeq,\n    exact xeq },\n  have secp_in_toBigInt3 : secp_in.toBigInt3 = n,\n  { rw [neq, bigint3.toBigInt3, secp_in], dsimp,\n    rw [iN0, iN1, iN2, int.cast_coe_nat, int.cast_coe_nat, int.cast_coe_nat] },\n  have : (((((ires.bigint5_mul ib).sub (ik.bigint5_mul secp_in)).sub ia.to_bigint5).add\n          \u27e80, icarry1, icarry2, icarry3, 0\u27e9).toUnreducedBigInt5 : UnreducedBigInt5 F) =\n      bigint5.toUnreducedBigInt5\n        (\u27e8icarry1*BASE, icarry2*BASE, icarry3*BASE, icarry4*BASE, -icarry4\u27e9),\n  { rw [bigint5.toUnreducedBigInt5_add, bigint5.toUnreducedBigInt5_sub,\n        bigint5.toUnreducedBigInt5_sub, bigint3.bigint5_mul_toUnreducedBigInt5,\n        bigint3.bigint5_mul_toUnreducedBigInt5],\n    conv { to_rhs, rw [bigint5.toUnreducedBigInt5], simp },\n    rw [\u2190carry1eq', aux3 carry1eq, \u2190carry2eq', aux3 carry2eq, \u2190carry3eq', aux3 carry3eq,\n      \u2190carry4eq', aux3 carry4eq, \u2190reseq, \u2190keq, \u2190beq, \u2190hres_b, secp_in_toBigInt3, \u2190hk_n,\n      bigint3.to_bigint5_to_Unreduced_BigInt5, \u2190aeq],\n    simp [UnreducedBigInt5.sub, UnreducedBigInt5.add,\n      starkware.cairo.common.cairo_secp.bigint.bigint3.to_bigint5, bigint5.toUnreducedBigInt5,\n        BigInt3.toUnreducedBigInt5],\n    simp [carry1eq', carry2eq', carry3eq', eq_sub_of_add_eq diff_eq] },\n  have eq :  ((((ires.bigint5_mul ib).sub (ik.bigint5_mul secp_in)).sub ia.to_bigint5).add\n          \u27e80, icarry1, icarry2, icarry3, 0\u27e9) =\n        \u27e8icarry1*BASE, icarry2*BASE, icarry3*BASE, icarry4*BASE, -icarry4\u27e9,\n  { apply bigint5.toUnreducedBigInt5_eq_of_sub_bounded this,\n    have : (2^127 : \u2124) * BASE + (3 * (3 * BASE - 1)^2 +\n      (3 * (3 * (BASE - 1))^2) + (3 * BASE - 1) + 2^127) \u2264 PRIME - 1,\n    { rw [PRIME, BASE], simp_int_casts, norm_num },\n    apply bigint5.bounded_of_bounded_of_le _ this,\n    apply bigint5.bounded_sub,\n    { simp [bigint5.bounded, abs_mul, absBASEeq],\n      use [carry1_bdd, carry2_bdd, carry3_bdd, carry4_bdd],\n      apply le_trans carry4_bdd, norm_num },\n    apply bigint5.bounded_add, swap,\n    { simp [bigint5.bounded],\n      use [carry1_bdd, carry2_bdd, carry3_bdd] },\n    apply bigint5.bounded_sub _ (bigint3.to_bigint5_bounded iabdd),\n    apply bigint5.bounded_sub,\n    apply bigint3.bounded_bigint5_mul (bigint3.bounded_of_bounded_of_le ires_bdd bound_slack) ibbdd,\n    apply bigint3.bounded_bigint5_mul ik_bdd,\n    rw [secp_in, bigint3.bounded], dsimp,\n    simp [abs_of_nonneg iN0nonneg, abs_of_nonneg iN1nonneg, abs_of_nonneg iN2nonneg],\n    simp [iN0, iN1, iN2],\n    norm_num,  },\n  have : ires.val * ib.val = ik.val * secp_n + ia.val,\n  { simp only [bigint3.val, secp_n, int.cast_add, int.coe_nat_add, int.coe_nat_mul],\n    rw [int.coe_nat_pow BASE 2],\n    set iBASE := (BASE : \u2124),\n    simp only [bigint3.bigint5_mul, bigint5.add, bigint5.sub, bigint3.to_bigint5, secp_in, add_zero, sub_zero] at eq,\n    rcases eq with \u27e8eq1, eq2, eq3, eq4, eq5\u27e9,\n    rw [\u2190iN0, \u2190iN1, \u2190iN2],\n    transitivity (ires.i2 * ib.i2 * iBASE * iBASE * iBASE * iBASE) +\n      (ires.i2 * iBASE ^ 2 * (ib.i1 * iBASE + ib.i0)) +\n      (ires.i1 * iBASE + ires.i0) * (ib.i2 * iBASE ^ 2 + ib.i1 * iBASE + ib.i0),\n    rw [pow_two], ring,\n    rw [eq_add_of_sub_eq eq5, add_mul, add_mul, add_mul],\n    rw [\u2190neg_mul_eq_neg_mul, \u2190eq4, neg_add],\n    simp only [add_mul _ _ iBASE],\n    rw [\u2190neg_mul_eq_neg_mul icarry3, \u2190eq3, neg_add],\n    simp only [add_mul _ _ iBASE],\n    rw [\u2190neg_mul_eq_neg_mul icarry2, \u2190eq2, neg_add],\n    simp only [add_mul _ _ iBASE],\n    rw [\u2190neg_mul_eq_neg_mul icarry1, \u2190eq1],\n    ring },\n  use ires,\n  split, apply bigint3.bounded_of_bounded_of_le ires_bdd bound_slack,\n  use \u03c1_res_eq.trans reseq,\n  rw [this, int.modeq, add_comm, int.add_mul_mod_self]\nend\n\n/-\n-- Function: get_point_from_x\n-/\n\n/- get_point_from_x autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_get_point_from_x (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (x : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_point : EcPoint F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F), spec_assert_nn mem \u03ba\u2081 range_check_ptr v range_check_ptr\u2081 \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (x_square : UnreducedBigInt3 F), spec_unreduced_sqr mem \u03ba\u2082 x x_square \u2227\n  \u2203 (\u03ba\u2083 : \u2115) (range_check_ptr\u2082 : F) (x_square_reduced : BigInt3 F), spec_reduce mem \u03ba\u2083 range_check_ptr\u2081 x_square range_check_ptr\u2082 x_square_reduced \u2227\n  \u2203 (\u03ba\u2084 : \u2115) (x_cube : UnreducedBigInt3 F), spec_unreduced_mul mem \u03ba\u2084 x x_square_reduced x_cube \u2227\n  \u2203 (\u03ba\u2085 : \u2115) (range_check_ptr\u2083 : F) (y : BigInt3 F), spec_nondet_bigint3 mem \u03ba\u2085 range_check_ptr\u2082 range_check_ptr\u2083 y \u2227\n  \u2203 (\u03ba\u2086 : \u2115) (range_check_ptr\u2084 : F), spec_validate_reduced_field_element mem \u03ba\u2086 range_check_ptr\u2083 y range_check_ptr\u2084 \u2227\n  \u2203 (\u03ba\u2087 : \u2115) (range_check_ptr\u2085 : F), spec_assert_nn mem \u03ba\u2087 range_check_ptr\u2084 ((y.d0 + v) / (2 : \u2124)) range_check_ptr\u2085 \u2227\n  \u2203 (\u03ba\u2088 : \u2115) (y_square : UnreducedBigInt3 F), spec_unreduced_sqr mem \u03ba\u2088 y y_square \u2227\n  \u2203 (\u03ba\u2089 : \u2115) (range_check_ptr\u2086 : F), spec_verify_zero mem \u03ba\u2089 range_check_ptr\u2085 {\n    d0 := x_cube.d0 + BETA - y_square.d0,\n    d1 := x_cube.d1 - y_square.d1,\n    d2 := x_cube.d2 - y_square.d2\n  } range_check_ptr\u2086 \u2227\n  \u03ba\u2081 + \u03ba\u2082 + \u03ba\u2083 + \u03ba\u2084 + \u03ba\u2085 + \u03ba\u2086 + \u03ba\u2087 + \u03ba\u2088 + \u03ba\u2089 + 54 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2086 \u2227\n  \u03c1_point = {\n    x := x,\n    y := y\n  }\n\ntheorem aux {a b c : \u2115} (h : a + b = 2 * c) : a % 2 = b % 2 :=\nbegin\n  have : a + 2 * b = 2 * c + b,\n  { nth_rewrite 0 (two_mul b),\n    rw [\u2190add_assoc, h] },\n  have h' := congr_arg (\u03bb n : \u2115, n % 2) this,\n  dsimp at h',\n  rw [nat.add_mul_mod_self_left, add_comm, nat.add_mul_mod_self_left] at h',\n  exact h'\nend\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_get_point_from_x (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (x : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_point : EcPoint F) : Prop :=\n   \u2200 (secpF : Type) [hsecp : secp_field secpF],\n   by exactI\n      x \u2260 \u27e80, 0, 0\u27e9 \u2192\n      \u2200 ix : bigint3,\n        ix.bounded (3 * BASE - 1) \u2192\n        x = ix.toBigInt3 \u2192\n        \u2203 nv : \u2115,\n          nv < rc_bound F \u2227\n          v = \u2191nv \u2227\n        \u2203 iyval : \u2115,\n          iyval < SECP_PRIME \u2227\n          nv % 2 = iyval % 2 \u2227\n        \u2203 h : @on_ec secpF _ (ix.val, iyval),\n        \u2203 hres : BddECPointData secpF \u03c1_point,\n          hres.toECPoint = ECPoint.AffinePoint \u27e8_, _, h\u27e9\n\n/- get_point_from_x soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_get_point_from_x\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr : F) (x : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_point : EcPoint F)\n    (h_auto : auto_spec_get_point_from_x mem \u03ba range_check_ptr x v \u03c1_range_check_ptr \u03c1_point) :\n  spec_get_point_from_x mem \u03ba range_check_ptr x v \u03c1_range_check_ptr \u03c1_point :=\nbegin\n  intros secpF _,\n  resetI,\n  intros xnez ix ixbdd xeq,\n  rcases h_auto with\n    \u27e8_, _, h_nn_v,\n    _, x_square, h_x_square,\n    _, _, x_square_reduced, h_x_square_reduced,\n    _, x_cube, h_x_cube,\n    _, _, y, h_nondet_bigint3_y,\n    _, _, h_validate_reduced_y,\n    _, _, h_nn_y_v_div,\n    _, y_square, h_unreduced_y_square,\n    _, _, h_verify_zero,\n    _, _, \u03c1_point_eq\u27e9,\n  rcases h_nn_v with \u27e8nv, nvlt, veq\u27e9,\n  have x_square_eq := h_x_square ix xeq,\n  have : ix.sqr.bounded (2 ^ 249),\n  { apply bigint3.bounded_of_bounded_of_le\n      (bigint3.bounded_sqr ixbdd),\n    rw [BASE, SECP_REM], simp_int_casts, norm_num },\n  rcases h_x_square_reduced ix.sqr this x_square_eq with\n    \u27e8ixsqr', ixsqr'bdd, hixsqr', x_square_reduced_eq\u27e9,\n  have x_cube_eq := h_x_cube ix ixsqr' xeq x_square_reduced_eq,\n  rcases nondet_bigint3_corr h_nondet_bigint3_y with \u27e8iy, yeq, iybdd\u27e9,\n  rcases h_validate_reduced_y with \u27e8n0, n1, n2, n0le, n1le, n2le, yvallt, yeq'\u27e9,\n  rcases h_nn_y_v_div with \u27e8nydiv, nydivlt, ydiveq\u27e9,\n  have : y.d0 + v = \u2191(2 * nydiv),\n  { rw [mul_comm, nat.cast_mul],\n    symmetry, simp,\n    rw [\u2190 (eq_div_iff (PRIME.two_ne_zero F)), \u2190 ydiveq],\n    simp },\n  have : n0 + nv = 2 * nydiv,\n  { rw [yeq', veq, \u2190 nat.cast_add] at this,\n    apply nat.cast_inj_of_lt_char _ _ this,\n    apply lt_of_le_of_lt (add_le_add n0le (le_of_lt nvlt)),\n    apply lt_of_le_of_lt (add_le_add_left (rc_bound_hyp F) _),\n    rw [BASE, PRIME.char_eq, PRIME],\n    norm_num,\n    apply lt_of_le_of_lt,\n    apply nat.mul_le_mul_left 2\n      (le_trans (le_of_lt nydivlt) (rc_bound_hyp F)),\n    rw [PRIME.char_eq, PRIME], norm_num },\n  have n02eqmv2 : n0 % 2 = nv % 2 := aux this,\n  let  iy' : bigint3 := \u27e8n0, n1, n2\u27e9,\n  have iy'bdd : iy'.bounded (3 * \u2191BASE - 1),\n  { suffices : iy'.bounded BASE,\n    { apply bigint3.bounded_of_bounded_of_le this,\n      rw BASE, simp_int_casts, norm_num },\n    simp only [bigint3.bounded, int.coe_nat_abs, int.coe_nat_le],\n    split, { apply le_trans n0le tsub_le_self },\n    split, { apply le_trans n1le tsub_le_self },\n    apply le_trans n2le,\n    norm_num [P2, BASE] },\n  have yeq'' : y = iy'.toBigInt3,\n  { rw [yeq', bigint3.toBigInt3], dsimp [iy'],\n    simp only [int.cast_coe_nat, eq_self_iff_true, and_self] },\n  have y_square_eq := h_unreduced_y_square iy' yeq'',\n  let idiff := ((ix.mul ixsqr').add \u27e87,0,0\u27e9).sub iy'.sqr,\n  have hidiff1 : idiff.toUnreducedBigInt3 = \u27e8x_cube.d0 + \u2191BETA - y_square.d0, x_cube.d1 - y_square.d1, x_cube.d2 - y_square.d2\u27e9,\n  { simp [idiff, BETA, bigint3.toUnreducedBigInt3_add, bigint3.toUnreducedBigInt3_sub, \u2190x_cube_eq, \u2190y_square_eq],\n    simp [bigint3.toUnreducedBigInt3, UnreducedBigInt3.add, UnreducedBigInt3.sub] },\n  have hidiff2 : idiff.bounded (2 ^ 250),\n  { suffices : idiff.bounded ((3 * BASE - 1)^2 * (8 * SECP_REM + 1) + 7 +\n      (3 * BASE - 1)^2 * (8 * SECP_REM + 1)),\n    { apply bigint3.bounded_of_bounded_of_le this,\n     rw [BASE, SECP_REM], simp_int_casts, norm_num },\n    apply bigint3.bounded_sub,\n    apply bigint3.bounded_add,\n    apply bigint3.bounded_mul ixbdd (bigint3.bounded_of_bounded_of_le ixsqr'bdd bound_slack),\n    { simp [bigint3.bounded], rw abs_of_nonneg; norm_num },\n    apply bigint3.bounded_sqr iy'bdd },\n  have := h_verify_zero idiff hidiff1.symm hidiff2,\n  have h_on_ec : on_ec ((ix.val : secpF), iy'.val),\n  { simp [on_ec],\n    suffices : (\u2191(iy'.val ^ 2) : secpF) = \u2191(ix.val ^ 3 + 7),\n    { simpa using this },\n    rw [char_p.int_coe_eq_int_coe_iff secpF SECP_PRIME, int.modeq],\n    symmetry,\n    rw [int.mod_eq_mod_iff_mod_sub_eq_zero, \u2190this],\n    dsimp [idiff],\n    rw [bigint3.sub_val, bigint3.add_val],\n    apply int.modeq.sub,\n    apply int.modeq.add,\n    rw [pow_succ],\n    apply int.modeq.symm,\n    apply int.modeq.trans,\n    apply bigint3.mul_val,\n    apply int.modeq.mul_left,\n    apply int.modeq.trans hixsqr',\n    apply bigint3.sqr_val,\n    { simp [bigint3.val] },\n    apply int.modeq.symm,\n    apply bigint3.sqr_val },\n  have iy'valeq : (\u2191(n2 * BASE ^ 2 + n1 * BASE + n0) : secpF) = iy'.val,\n  { dsimp [iy'], simp [bigint3.val] },\n  use [nv, nvlt, veq, n2 * BASE ^ 2 + n1 * BASE + n0, yvallt],\n  split,\n  { rw [\u2190n02eqmv2, nat.add_mod, nat.add_mod (n2 * _), nat.mul_mod, BASE],\n    norm_num,\n    rw [nat.add_mod, nat.mul_mod],\n    norm_num },\n  rw iy'valeq,\n  use h_on_ec,\n  rw \u03c1_point_eq,\n  use \u27e8ix, iy', ixbdd, iy'bdd, xeq, yeq'', or.inr h_on_ec\u27e9,\n  simp [BddECPointData.toECPoint, dif_neg xnez]\nend\n\n\n/-\n-- Function: recover_public_key\n-/\n\n/- recover_public_key autogenerated specification -/\n\n-- Do not change this definition.\ndef auto_spec_recover_public_key (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (msg_hash r s : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_public_key_point : EcPoint F) : Prop :=\n  \u2203 (\u03ba\u2081 : \u2115) (range_check_ptr\u2081 : F) (r_point : EcPoint F), spec_get_point_from_x mem \u03ba\u2081 range_check_ptr r v range_check_ptr\u2081 r_point \u2227\n  \u2203 (\u03ba\u2082 : \u2115) (generator_point : EcPoint F), spec_get_generator_point mem \u03ba\u2082  generator_point \u2227\n  \u2203 (\u03ba\u2083 : \u2115) (range_check_ptr\u2082 : F) (u1 : BigInt3 F), spec_div_mod_n mem \u03ba\u2083 range_check_ptr\u2081 msg_hash r range_check_ptr\u2082 u1 \u2227\n  \u2203 (\u03ba\u2084 : \u2115) (range_check_ptr\u2083 : F) (u2 : BigInt3 F), spec_div_mod_n mem \u03ba\u2084 range_check_ptr\u2082 s r range_check_ptr\u2083 u2 \u2227\n  \u2203 (\u03ba\u2085 : \u2115) (range_check_ptr\u2084 : F) (point1 : EcPoint F), spec_ec_mul mem \u03ba\u2085 range_check_ptr\u2083 generator_point u1 range_check_ptr\u2084 point1 \u2227\n  \u2203 (\u03ba\u2086 : \u2115) (range_check_ptr\u2085 : F) (minus_point1 : EcPoint F), spec_ec_negate mem \u03ba\u2086 range_check_ptr\u2084 point1 range_check_ptr\u2085 minus_point1 \u2227\n  \u2203 (\u03ba\u2087 : \u2115) (range_check_ptr\u2086 : F) (point2 : EcPoint F), spec_ec_mul mem \u03ba\u2087 range_check_ptr\u2085 r_point u2 range_check_ptr\u2086 point2 \u2227\n  \u2203 (\u03ba\u2088 : \u2115) (range_check_ptr\u2087 : F) (public_key_point : EcPoint F), spec_ec_add mem \u03ba\u2088 range_check_ptr\u2086 minus_point1 point2 range_check_ptr\u2087 public_key_point \u2227\n  \u03ba\u2081 + \u03ba\u2082 + \u03ba\u2083 + \u03ba\u2084 + \u03ba\u2085 + \u03ba\u2086 + \u03ba\u2087 + \u03ba\u2088 + 77 \u2264 \u03ba \u2227\n  \u03c1_range_check_ptr = range_check_ptr\u2087 \u2227\n  \u03c1_public_key_point = public_key_point\n\n-- You may change anything in this definition except the name and arguments.\ndef spec_recover_public_key (mem : F \u2192 F) (\u03ba : \u2115) (range_check_ptr : F) (msg_hash r s : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_public_key_point : EcPoint F) : Prop :=\n  \u2200 (secpF : Type) [secp_field secpF], by exactI\n  r \u2260 \u27e80, 0, 0\u27e9 \u2192\n  \u2200 ir : bigint3,\n    ir.bounded (3 * BASE - 1) \u2192\n    r = ir.toBigInt3 \u2192\n  \u2200 is : bigint3,\n    is.bounded (3 * BASE - 1) \u2192\n    s = is.toBigInt3 \u2192\n  \u2200 imsg : bigint3,\n    imsg.bounded (3 * BASE - 1) \u2192\n    msg_hash = imsg.toBigInt3 \u2192\n  \u2203 nv : \u2115,\n    nv < rc_bound F \u2227\n    v = \u2191nv \u2227\n  \u2203 iu1 iu2 : \u2124,\n    iu1 * ir.val \u2261 imsg.val [ZMOD secp_n] \u2227\n    iu2 * ir.val \u2261 is.val [ZMOD secp_n] \u2227\n  \u2203 ny : \u2115,\n    ny < SECP_PRIME \u2227\n    nv \u2261 ny [MOD 2] \u2227\n  \u2203 h_on_ec : @on_ec secpF _ (ir.val, ny),\n  \u2203 hpoint  : BddECPointData secpF \u03c1_public_key_point,\n    hpoint.toECPoint =\n      -(iu1 \u2022 (gen_point_data F secpF).toECPoint) +\n        iu2 \u2022 ECPoint.AffinePoint \u27e8ir.val, ny, h_on_ec\u27e9\n\n/- recover_public_key soundness theorem -/\n\n-- Do not change the statement of this theorem. You may change the proof.\ntheorem sound_recover_public_key\n    {mem : F \u2192 F}\n    (\u03ba : \u2115)\n    (range_check_ptr : F) (msg_hash r s : BigInt3 F) (v \u03c1_range_check_ptr : F) (\u03c1_public_key_point : EcPoint F)\n    (h_auto : auto_spec_recover_public_key mem \u03ba range_check_ptr msg_hash r s v \u03c1_range_check_ptr \u03c1_public_key_point) :\n  spec_recover_public_key mem \u03ba range_check_ptr msg_hash r s v \u03c1_range_check_ptr \u03c1_public_key_point :=\nbegin\n  intros secpF _ rnez ir irbdd req is isbdd iseq imsg imsgbdd msgeq,\n  resetI,\n  rcases h_auto with \u27e8_, _, r_point, hr_point,\n    _, generator, hgenerator,\n    _, _, u1, hu1,\n    _, _, u2, hu2,\n    _, _, point1, hpoint1,\n    _, _, minus_point1, hminus_point1,\n    _, _, point2, hpoint2,\n    _, _, public_key_point, hpublic_key_point,\n    _, _, rfl\u27e9,\n  rcases hr_point secpF rnez ir irbdd req with \u27e8nv, nvlt, veq, iy, iybdd, iy2, hiy, hr_point', ireq\u27e9,\n  rw spec_get_generator_point at hgenerator,\n  subst hgenerator,\n  rcases hu1 imsg imsgbdd msgeq ir irbdd req with \u27e8iu1, iu1bdd, u1eq, hu1'\u27e9,\n  rcases hu2 is isbdd iseq ir irbdd req with \u27e8iu2, iu2bdd, u2eq, hu2'\u27e9,\n  have : (gen_point F).x \u2260 \u27e80, 0, 0\u27e9,\n  { simp [gen_point],\n    intros eq _ _,\n    suffices : (17117865558768631194064792 : \u2124) = 0,\n    { norm_num at this },\n    haveI : char_p F PRIME := prelude_hyps.charF,\n    apply int.cast_eq_zero_of_lt_char F PRIME,\n    { simp_int_casts, exact eq },\n    rw [abs_of_nonneg, PRIME],\n    simp_int_casts, norm_num, norm_num },\n  rcases hpoint1 secpF this (gen_point_data F secpF) with\n    \u27e8n10, n10lt, n10eq, n11, n11lt, n11eq, n12, n12lt, n12eq, hpoint1', hpoint1eq\u27e9,\n  rcases spec_ec_negate'_of_spec_ec_negate hminus_point1 secpF hpoint1' with\n    \u27e8hminus_point1', hminus_point1_eq\u27e9,\n  have : r_point.x \u2260 \u27e80, 0, 0\u27e9,\n  { intro hcontr,\n    rw [BddECPointData.toECPoint, dif_pos hcontr] at ireq,\n    contradiction },\n  rcases hpoint2 secpF this hr_point' with\n    \u27e8n20, n20lt, n20eq, n21, n21lt, n21eq, n22, n22lt, n22eq, hpoint2', hpoint2eq\u27e9,\n  rcases hpublic_key_point secpF hminus_point1' hpoint2' with \u27e8hret, hreteq\u27e9,\n  refine \u27e8nv, nvlt, veq, _, _, hu1', hu2', iy, iybdd, iy2, hiy, hret, _\u27e9,\n  rw [hreteq, hminus_point1_eq, hpoint1eq, hpoint2eq, ireq],\n  have aux : \u2200 {i : \u2124} {n : \u2115}, abs i \u2264 3 * BASE - 1 \u2192 n < 2^86 \u2192 (i : F) = (n : F) \u2192 i = n,\n  { intros i n hi hn hin,\n    have : (i : F) = ((n : \u2124) : F),\n    { rw hin, simp },\n    apply PRIME.int_coe_inj this,\n    have : (2^86 : \u2124) + (3 * BASE - 1) < PRIME,\n    { simp only [PRIME], simp_int_casts, norm_num },\n    apply lt_of_le_of_lt _ this,\n    apply le_trans,\n    apply abs_sub,\n    apply add_le_add _ hi,\n    rw abs_of_nonneg (int.coe_zero_le _),\n    apply le_of_lt,\n    norm_cast, exact hn },\n  simp [u1eq, u2eq, bigint3.toBigInt3] at n10eq n11eq n12eq n20eq n21eq n22eq,\n  have : iu1.val = \u2191(2 ^ 172 * n12 + 2 ^ 86 * n11 + n10),\n  { rw [bigint3.val], simp,\n    rw [aux iu1bdd.1 n10lt n10eq, aux iu1bdd.2.1 n11lt n11eq,\n        aux iu1bdd.2.2 (lt_of_lt_of_le n12lt (by norm_num)) n12eq ],\n        ring },\n  rw [this, nsmul_eq_smul_cast \u2124],\n  have : iu2.val = \u2191(2 ^ 172 * n22 + 2 ^ 86 * n21 + n20),\n  { rw [bigint3.val], simp,\n    rw [aux iu2bdd.1 n20lt n20eq, aux iu2bdd.2.1 n21lt n21eq,\n        aux iu2bdd.2.2 (lt_of_lt_of_le n22lt (by norm_num)) n22eq ],\n        ring },\n  rw [this, nsmul_eq_smul_cast \u2124]\nend\n\n\nend starkware.cairo.common.cairo_secp.signature\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/signature_spec.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802471698041, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3862752426049021}}
{"text": "import topology.filter\nimport measure_theory.measure.null_measurable\nimport measure_theory.measurable_space\nimport measure_theory.measure.measure_space\nimport number_theory.well_approximable\nimport topology.algebra.order.liminf_limsup\n\nopen set filter (hiding map) function measurable_space topological_space (second_countable_topology)\nopen_locale classical big_operators filter ennreal nnreal interval measure_theory\n\nnamespace measure_theory\n\nvariables {\u03b1 : Type*} {m : measurable_space \u03b1} {\u03bc : measure \u03b1}\n\n/-\nWant to state using `well_approximate`:\nFor a function \u03c8 : \u2115 \u2192 \u211d\u22650 such that \u2211 q, (\u03c6 q * \u03c8 q) / q < \u221e, \u03bc W = 0\n-/\n\n/- Gallagher's Ergodic Theorem -/\nexample {T : \u211d} [hT : fact (0 < T)] (\u03b4 : \u2115 \u2192 \u211d) (h\u03b4 : tendsto \u03b4 at_top (nhds 0)) :\n(\u2200\u1d50 (x : add_circle T), \u00acadd_well_approximable (add_circle T) \u03b4 x) \u2228\n \u2200\u1d50 (x : add_circle T), add_well_approximable (add_circle T) \u03b4 x := sorry\n\n/- Borel-Cantelli Theorem -/\nexample {p : \u2115 \u2192 \u03b1 \u2192 Prop} (hp : \u2211' i, \u03bc {x | p i x} \u2260 \u221e) : \u03bc {x | \u2203\u1da0 n in at_top, p n x} = 0 := \nsorry\n\n/- Duffin-Schaeffer Theorem (Implication) -/\nexample (\u03b1 : \u211d) (\u03c8 : \u2115 \u2192 \u211d\u22650) (h : {q : \u211a | |\u03b1 - q| \u2264 \u03c8 (q.denom) / q.denom}.infinite) : _ := sorry\n\nend measure_theory", "meta": {"author": "grhkm21", "repo": "lean", "sha": "52fe0ba1b5c78344c640b0813f11db71338fcba2", "save_path": "github-repos/lean/grhkm21-lean", "path": "github-repos/lean/grhkm21-lean/lean-52fe0ba1b5c78344c640b0813f11db71338fcba2/sketch/sketch_borel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8104789086703224, "lm_q2_score": 0.476579651063676, "lm_q1q2_score": 0.38625775548857116}}
{"text": "import analysis.calculus.inverse\nimport analysis.calculus.cont_diff\nimport analysis.inner_product_space.calculus\nimport analysis.inner_product_space.dual\n\nimport to_mathlib.analysis.calculus\nimport to_mathlib.analysis.normed_space.operator_norm\n\nnoncomputable theory\n\nopen_locale topology filter\nopen function\n\nsection\nuniverses u\u2081 u\u2082 u\u2083 u\u2084 u\u2085\n\nopen continuous_linear_map\n\nvariables {\ud835\udd5c : Type u\u2081} [nontrivially_normed_field \ud835\udd5c]\n  {M\u2081 : Type u\u2082} [normed_add_comm_group M\u2081] [normed_space \ud835\udd5c M\u2081]\n  {M\u2082 : Type u\u2083} [normed_add_comm_group M\u2082] [normed_space \ud835\udd5c M\u2082]\n  {M\u2083 : Type u\u2084} [normed_add_comm_group M\u2083] [normed_space \ud835\udd5c M\u2083]\n  {M\u2084 : Type u\u2085} [normed_add_comm_group M\u2084] [normed_space \ud835\udd5c M\u2084]\n\n-- The next definition won't be used here, it's practice before the next one.\n\n/-- Defines continuous linear maps between two products by blocks:\ngiven `(A : M\u2081 \u2192L[\ud835\udd5c] M\u2083)`, `(B : M\u2082 \u2192L[\ud835\udd5c] M\u2083)`, `(C : M\u2081 \u2192L[\ud835\udd5c] M\u2084)` and `(D : M\u2082 \u2192L[\ud835\udd5c] M\u2084)`,\nconstruct the continuous linear map with \"matrix\":\nA B\nC D. -/\ndef continuous_linear_map.blocks (A : M\u2081 \u2192L[\ud835\udd5c] M\u2083) (B : M\u2082 \u2192L[\ud835\udd5c] M\u2083)\n  (C : M\u2081 \u2192L[\ud835\udd5c] M\u2084) (D : M\u2082 \u2192L[\ud835\udd5c] M\u2084) : (M\u2081 \u00d7 M\u2082) \u2192L[\ud835\udd5c] (M\u2083 \u00d7 M\u2084) :=\n(A.coprod B).prod (C.coprod D)\n\n/-- Given `(A : M\u2081 \u2243L[\ud835\udd5c] M\u2083)`, `(C : M\u2081 \u2192L[\ud835\udd5c] M\u2084)` and `(D : M\u2082 \u2243L[\ud835\udd5c] M\u2084)`,\nconstruct the continuous linear equiv with \"matrix\"\nA 0\nC D.\n  -/\ndef continuous_linear_equiv.lower_triangular (A : M\u2081 \u2243L[\ud835\udd5c] M\u2083)\n  (C : M\u2081 \u2192L[\ud835\udd5c] M\u2084) (D : M\u2082 \u2243L[\ud835\udd5c] M\u2084) : (M\u2081 \u00d7 M\u2082) \u2243L[\ud835\udd5c] (M\u2083 \u00d7 M\u2084) :=\ncontinuous_linear_equiv.equiv_of_inverse (((A : M\u2081 \u2192L[\ud835\udd5c] M\u2083).comp (fst \ud835\udd5c M\u2081 M\u2082)).prod (C.coprod D))\n(((A.symm : M\u2083 \u2192L[\ud835\udd5c] M\u2081).comp (fst \ud835\udd5c M\u2083 M\u2084)).prod\n((-((D.symm : M\u2084 \u2192L[\ud835\udd5c] M\u2082).comp C).comp (A.symm : M\u2083 \u2192L[\ud835\udd5c] M\u2081)).coprod D.symm))\n(\u03bb \u27e8x, y\u27e9, by simp only [prod_apply, coe_comp', continuous_linear_equiv.coe_coe, coe_fst', comp_app,\n           coprod_apply, continuous_linear_equiv.symm_apply_apply, neg_apply,\n           continuous_linear_equiv.map_add, neg_add_cancel_left])\n(\u03bb \u27e8x, y\u27e9, by simp only [prod_apply, coe_comp', continuous_linear_equiv.coe_coe, coe_fst', comp_app,\n           coprod_apply, neg_apply, continuous_linear_equiv.apply_symm_apply,\n           continuous_linear_equiv.map_add, continuous_linear_equiv.map_neg, add_neg_cancel_left])\n\nlemma continuous_linear_equiv.continuous_lower_triangular {X : Type*} [topological_space X]\n  {A : X \u2192 M\u2081 \u2243L[\ud835\udd5c] M\u2083} {C : X \u2192 M\u2081 \u2192L[\ud835\udd5c] M\u2084} {D : X \u2192 M\u2082 \u2243L[\ud835\udd5c] M\u2084}\n  (hA : continuous (\u03bb x, (A x : M\u2081 \u2192L[\ud835\udd5c] M\u2083))) (hC : continuous C)\n  (hD : continuous (\u03bb x, (D x : M\u2082 \u2192L[\ud835\udd5c] M\u2084))) :\n  continuous (\u03bb x, ((A x).lower_triangular (C x) (D x) : (M\u2081 \u00d7 M\u2082) \u2192L[\ud835\udd5c] (M\u2083 \u00d7 M\u2084))) :=\n(hA.compL continuous_const).prodL (hC.coprodL hD)\n\nend\n\nsection\nvariables (\ud835\udd5c : Type*) [nontrivially_normed_field \ud835\udd5c]\n  {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\n  {F : Type*} [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  {G : Type*} [normed_add_comm_group G] [normed_space \ud835\udd5c G]\n  {n : \u2115\u221e}\n\n-- The next two definitions aren't used in the end, but they may still go to mathlib\n\n/-- The proposition that a function between two normed spaces has a strict derivative at a given\npoint. -/\ndef strict_differentiable_at (f : E \u2192 F) (x) :=\n\u2203 \u03c6 : E \u2192L[\ud835\udd5c] F, has_strict_fderiv_at f \u03c6 x\n\n/-- The proposition that a function between two normed spaces has a strict derivative at every\npoint. -/\ndef strict_differentiable (f : E \u2192 F) :=\n\u2200 x, strict_differentiable_at \ud835\udd5c f x\n\nvariables {\ud835\udd5c}\n\nlemma strict_differentiable_at.differentiable_at {f : E \u2192 F} {x : E}\n  (h : strict_differentiable_at \ud835\udd5c f x) : differentiable_at \ud835\udd5c f x :=\nexists.elim h (\u03bb \u03c6 h\u03c6, \u27e8\u03c6, h\u03c6.has_fderiv_at\u27e9)\n\n-- PR to linear_algebra.prod\n@[simp]\nlemma linear_map.coprod_comp_inl_inr {R : Type*} {M : Type*} {M\u2082 : Type*} {M\u2083 : Type*} [semiring R]\n  [add_comm_monoid M] [add_comm_monoid M\u2082] [add_comm_monoid M\u2083] [module R M]\n  [module R M\u2082] [module R M\u2083] (f : M \u00d7 M\u2082 \u2192\u2097[R] M\u2083) :\n  (f.comp (linear_map.inl R M M\u2082)).coprod (f.comp (linear_map.inr R M M\u2082)) = f :=\nby rw [\u2190 linear_map.comp_coprod, linear_map.coprod_inl_inr, linear_map.comp_id]\n\n-- PR to topology.algebra.module.basic\n@[simp]\nlemma continuous_linear_map.coprod_comp_inl_inr {R\u2081 : Type*} [semiring R\u2081] {M\u2081 : Type*} [topological_space M\u2081]\n  [add_comm_monoid M\u2081] {M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082]\n  {M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083] [module R\u2081 M\u2081]\n  [module R\u2081 M\u2082] [module R\u2081 M\u2083] [has_continuous_add M\u2083] (f : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2083) :\n  (f.comp (continuous_linear_map.inl R\u2081 M\u2081 M\u2082)).coprod (f.comp (continuous_linear_map.inr R\u2081 M\u2081 M\u2082)) = f :=\ncontinuous_linear_map.coe_injective (f : M\u2081 \u00d7 M\u2082 \u2192\u2097[R\u2081] M\u2083).coprod_comp_inl_inr\n\nlemma differentiable_at.has_fderiv_at_coprod_partial {f : E \u2192 F \u2192 G} {x : E} {y : F}\n  (hf : differentiable_at \ud835\udd5c (uncurry f) (x, y)) :\n  has_fderiv_at (uncurry f)\n                ((partial_fderiv_fst \ud835\udd5c f x y).coprod (partial_fderiv_snd \ud835\udd5c f x y)) (x, y) :=\nbegin\n  rcases hf with \u27e8\u03b8, h\u03b8\u27e9,\n  rwa [fderiv_partial_fst h\u03b8, fderiv_partial_snd h\u03b8, \u03b8.coprod_comp_inl_inr]\nend\n\nlemma differentiable_at.has_fderiv_at_coprod {f : E \u2192 F \u2192 G} {x : E} {y : F}\n  (hf : differentiable_at \ud835\udd5c (uncurry f) (x, y)) {\u03c6 : E \u2192L[\ud835\udd5c] G} {\u03c8 : F \u2192L[\ud835\udd5c] G}\n  (h\u03c6 : has_fderiv_at (\u03bb p, f p y) \u03c6 x) (h\u03c8 : has_fderiv_at (f x) \u03c8 y) :\n  has_fderiv_at (uncurry f) (\u03c6.coprod \u03c8) (x, y) :=\nbegin\n  rw [h\u03c6.unique hf.has_fderiv_at_partial_fst, h\u03c8.unique hf.has_fderiv_at_partial_snd],\n  exact hf.has_fderiv_at_coprod_partial\nend\n\nvariables [complete_space E]\n\nlemma homeomorph.cont_diff_at_symm (f : homeomorph E F) {f\u2080' : E \u2243L[\ud835\udd5c] F} {a : F}\n  (hf' : has_fderiv_at f (f\u2080' : E \u2192L[\ud835\udd5c] F) (f.symm a)) (hf : cont_diff_at \ud835\udd5c n f (f.symm a)) :\n  cont_diff_at \ud835\udd5c n (f.symm) a :=\nf.to_local_homeomorph.cont_diff_at_symm trivial hf' hf\n\nlemma equiv.continuous_symm_of_cont_diff (\u03c6 : E \u2243 F) {D\u03c6 : E \u2192 E \u2243L[\ud835\udd5c] F}\n  (h\u03c6 : \u2200 x, has_strict_fderiv_at \u03c6 (D\u03c6 x : E \u2192L[\ud835\udd5c] F) x) :\n  continuous \u03c6.symm :=\nbegin\n  rw continuous_iff_continuous_at,\n  intros x,\n  let y := \u03c6.symm x,\n  let g := (h\u03c6 y).local_inverse \u03c6 (D\u03c6 y) y,\n  rw \u2190 \u03c6.apply_symm_apply x,\n  have ev_eq : g =\u1da0[\ud835\udcdd (\u03c6 y)] \u03c6.symm,\n  { apply (h\u03c6 y).eventually_right_inverse.mono,\n    rintros x (hx : \u03c6 (g x) = x),\n    exact (equiv.eq_symm_apply \u03c6).mpr hx },\n  apply continuous_at.congr _ ev_eq,\n  apply (h\u03c6 y).local_inverse_continuous_at\nend\n\n/-- A bijection that is strictly differentiable at every point is a homeomorphism. -/\ndef equiv.to_homeomorph_of_cont_diff (\u03c6 : E \u2243 F) {D\u03c6 : E \u2192 E \u2243L[\ud835\udd5c] F}\n  (h\u03c6 : \u2200 x, has_strict_fderiv_at \u03c6 (D\u03c6 x : E \u2192L[\ud835\udd5c] F) x) : E \u2243\u209c F :=\n{ continuous_to_fun := differentiable.continuous (\u03bb x, (h\u03c6 x).differentiable_at),\n  continuous_inv_fun := \u03c6.continuous_symm_of_cont_diff h\u03c6,\n  ..\u03c6}\n\nend\n\nsection\nvariables {\ud835\udd5c : Type*} [is_R_or_C \ud835\udd5c]\n  {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\n  {F : Type*} [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n  {G : Type*} [normed_add_comm_group G] [normed_space \ud835\udd5c G]\n  {n : \u2115\u221e}\n\n\nlocal notation `\u2202\u2081` := partial_fderiv_fst \ud835\udd5c\nlocal notation `\u2202\u2082` := partial_fderiv_snd \ud835\udd5c\n\nlemma cont_diff_parametric_symm [complete_space E] [complete_space F]\n  {f : E \u2192 F \u2243 G} {f' : E \u2192 F \u2192 F \u2243L[\ud835\udd5c] G}\n  (hf : cont_diff \ud835\udd5c \u22a4 (\u03bb p : E \u00d7 F, f p.1 p.2))\n  (hf' : \u2200 x y, \u2202\u2082 (\u03bb x y, f x y) x y = f' x y) :\n  cont_diff \ud835\udd5c \u22a4 (\u03bb p : E \u00d7 G, (f p.1).symm p.2) :=\nbegin\n  let \u03c6\u2080 : (E \u00d7 F) \u2243 (E \u00d7 G) :=\n  { to_fun := \u03bb p : E \u00d7 F, (p.1, f p.1 p.2),\n    inv_fun := \u03bb p : E \u00d7 G, (p.1, (f p.1).symm p.2),\n    left_inv := \u03bb x, by simp,\n    right_inv := \u03bb x, by simp },\n  let ff := \u03bb x y, f x y,\n  have hff : cont_diff \ud835\udd5c \u22a4 (uncurry ff) := hf,\n  let d\u2081f := \u2202\u2081 ff,\n  let D\u03c6 : E \u00d7 F \u2192 (E \u00d7 F) \u2243L[\ud835\udd5c] E \u00d7 G :=\n    \u03bb x, (continuous_linear_equiv.refl \ud835\udd5c E).lower_triangular (d\u2081f x.1 x.2) (f' x.1 x.2),\n  let D\u03c6' : E \u00d7 F \u2192 (E \u00d7 F) \u2192L[\ud835\udd5c] E \u00d7 G := \u03bb x, D\u03c6 x,\n  have hderiv : \u2200 (x : E \u00d7 F), has_strict_fderiv_at \u03c6\u2080 (D\u03c6' x) x,\n  { rintros p,\n    apply has_strict_fderiv_at_of_has_fderiv_at_of_continuous_at,\n    { apply filter.eventually_of_forall,\n      rintros \u27e8x, y\u27e9,\n      apply has_fderiv_at.prod,\n      { simp only [continuous_linear_equiv.coe_refl, continuous_linear_map.id_comp,\n        has_fderiv_at_fst] },\n      have diff : differentiable \ud835\udd5c (uncurry $ \u03bb x y, f x y) := hf.differentiable le_top,\n      rw show (\u03bb (x : E \u00d7 F), (f x.fst) x.snd) = uncurry (\u03bb x y, f x y), by { ext, refl },\n      apply differentiable_at.has_fderiv_at_coprod,\n      { apply (hf.differentiable le_top) },\n      { dsimp [d\u2081f],\n        exact diff.differentiable_at.has_fderiv_at_partial_fst },\n      { rw \u2190 hf' x y,\n        dsimp,\n        exact diff.differentiable_at.has_fderiv_at_partial_snd } },\n    { apply continuous.continuous_at,\n      apply continuous_linear_equiv.continuous_lower_triangular,\n      { exact continuous_const },\n      { exact hff.cont_diff_top_partial_fst.continuous },\n      { simp_rw \u2190 hf',\n        exact hff.cont_diff_top_partial_snd.continuous } } },\n  let \u03c6 := \u03c6\u2080.to_homeomorph_of_cont_diff hderiv,\n  exact cont_diff_snd.comp (\u03c6.cont_diff_symm (\u03bb x, (hderiv x).has_fderiv_at)\n    (cont_diff_fst.prod hf)),\nend\n\nend\n\nsection\nvariables {E : Type*} [normed_add_comm_group E] [normed_space \u211d E] [complete_space E]\n\nlemma cont_diff_parametric_symm_of_deriv_pos {f : E \u2192 \u211d \u2192 \u211d} (hf : cont_diff \u211d \u22a4 \u21bff)\n  (hderiv : \u2200 x t, 0 < partial_deriv_snd f x t) (hsurj : \u2200 x, surjective $ f x) :\n  cont_diff \u211d \u22a4  (\u03bb p : E \u00d7 \u211d, (strict_mono.order_iso_of_surjective (f p.1)\n                                (strict_mono_of_deriv_pos $ hderiv p.1) (hsurj p.1)).symm p.2) :=\nbegin\n  have hmono := \u03bb x, strict_mono_of_deriv_pos (hderiv x),\n  let F := \u03bb x, (strict_mono.order_iso_of_surjective (f x) (hmono x) $ hsurj x).to_equiv,\n  change cont_diff \u211d \u22a4 (\u03bb (p : E \u00d7 \u211d), (F p.1).symm p.snd),\n  refine cont_diff_parametric_symm hf _,\n  exact \u03bb x t, continuous_linear_equiv.units_equiv_aut \u211d (units.mk0 (deriv (f x) t) $ ne_of_gt (hderiv x t)) ,\n  intros x t,\n  suffices : partial_fderiv_snd \u211d f x t 1 = partial_deriv_snd f x t,\n  { ext v,\n    simpa only [rel_iso.coe_fn_to_equiv, continuous_linear_equiv.coe_coe,\n      continuous_linear_equiv.units_equiv_aut_apply, units.coe_mk0, one_mul] },\n  apply partial_fderiv_snd_one\nend\n\nend\n\nsection\nvariables (\ud835\udd5c : Type*) [nontrivially_normed_field \ud835\udd5c]\n\nlemma cont_diff_to_span_singleton (E : Type*) [normed_add_comm_group E] [normed_space \ud835\udd5c E] :\n  cont_diff \ud835\udd5c \u22a4 (continuous_linear_map.to_span_singleton \ud835\udd5c : E \u2192 \ud835\udd5c \u2192L[\ud835\udd5c] E) :=\n(continuous_linear_map.lsmul \ud835\udd5c \ud835\udd5c : \ud835\udd5c \u2192L[\ud835\udd5c] E \u2192L[\ud835\udd5c] E).flip.cont_diff\n\nend\n\nsection\nvariables {\ud835\udd5c : Type*} [is_R_or_C \ud835\udd5c]\nvariables {E : Type*} [normed_add_comm_group E] [inner_product_space \ud835\udd5c E] [complete_space E]\n\n-- variant of `orthogonal_projection_singleton`\nlemma orthogonal_projection_singleton' {v : E} :\n  (\ud835\udd5c \u2219 v).subtypeL.comp (orthogonal_projection (\ud835\udd5c \u2219 v))\n  = (1 / \u2016v\u2016 ^ 2 : \ud835\udd5c) \u2022 (continuous_linear_map.to_span_singleton \ud835\udd5c v)\n    \u2218L inner_product_space.to_dual \ud835\udd5c E v :=\nbegin\n  ext w,\n  simp [continuous_linear_map.to_span_singleton_apply, orthogonal_projection_singleton, \u2190 mul_smul],\n  congr' 1,\n  field_simp,\nend\n\nend\n\nsection\nvariables {E : Type*} [normed_add_comm_group E] [inner_product_space \u211d E] [complete_space E]\n\n/-- The orthogonal projection onto a vector in a real inner product space `E`, considered as a map\nfrom `E` to `E \u2192L[\u211d] E`, is smooth away from 0. -/\nlemma cont_diff_at_orthogonal_projection_singleton {v\u2080 : E} (hv\u2080 : v\u2080 \u2260 0) :\n  cont_diff_at \u211d \u22a4 (\u03bb v : E, (\u211d \u2219 v).subtypeL.comp (orthogonal_projection (\u211d \u2219 v))) v\u2080 :=\nbegin\n  suffices :  cont_diff_at \u211d \u22a4\n    (\u03bb v : E, (1 / \u2016v\u2016 ^ 2) \u2022 continuous_linear_map.to_span_singleton \u211d v\n    \u2218L inner_product_space.to_dual \u211d E v) v\u2080,\n  { refine this.congr_of_eventually_eq _,\n    refine filter.eventually_of_forall (\u03bb v, _),\n    dsimp,\n    rw orthogonal_projection_singleton',\n    refl },\n  refine cont_diff_at.smul _ _,\n  { refine cont_diff_at_const.div (cont_diff_norm_sq \u211d).cont_diff_at _,\n    apply pow_ne_zero,\n    exact norm_ne_zero_iff.mpr hv\u2080 },\n  exact (cont_diff.clm_comp (cont_diff_to_span_singleton \u211d E)\n    (inner_product_space.to_dual \u211d E).cont_diff).cont_diff_at,\nend\n\nend\n\nsection arithmetic\n\nvariables {\ud835\udd5c : Type*} [nontrivially_normed_field \ud835\udd5c]\n  {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\n  {\ud835\udd38 : Type*} [normed_ring \ud835\udd38] [normed_algebra \ud835\udd5c \ud835\udd38]\n  {n : \u2115\u221e} {f : E \u2192 \ud835\udd38} {s : set E} {x : E}\n\nlemma cont_diff_within_at.mul_const (hf : cont_diff_within_at \ud835\udd5c n f s x) {c : \ud835\udd38} :\n  cont_diff_within_at \ud835\udd5c n (\u03bb (x : E), f x * c) s x :=\nhf.mul cont_diff_within_at_const\n\ntheorem cont_diff_at.mul_const (hf : cont_diff_at \ud835\udd5c n f x) {c : \ud835\udd38} :\n  cont_diff_at \ud835\udd5c n (\u03bb (x : E), f x * c) x :=\nhf.mul cont_diff_at_const\n\ntheorem cont_diff_on.mul_const (hf : cont_diff_on \ud835\udd5c n f s) {c : \ud835\udd38} :\n  cont_diff_on \ud835\udd5c n (\u03bb (x : E), f x * c) s :=\nhf.mul cont_diff_on_const\n\ntheorem cont_diff.mul_const (hf : cont_diff \ud835\udd5c n f) {c : \ud835\udd38} :\n  cont_diff \ud835\udd5c n (\u03bb (x : E), f x * c) :=\nhf.mul cont_diff_const\n\nend arithmetic\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/analysis/cont_diff.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3862036619671179}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.multiset.dedup\n! leanprover-community/mathlib commit 9003f28797c0664a49e4179487267c494477d853\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.Multiset.Nodup\n\n/-!\n# Erasing duplicates in a multiset.\n-/\n\n\nnamespace Multiset\n\nopen List\n\nvariable {\u03b1 \u03b2 : Type _} [DecidableEq \u03b1]\n\n/-! ### dedup -/\n\n\n/-- `dedup s` removes duplicates from `s`, yielding a `nodup` multiset. -/\ndef dedup (s : Multiset \u03b1) : Multiset \u03b1 :=\n  Quot.liftOn s (fun l => (l.dedup : Multiset \u03b1)) fun _ _ p => Quot.sound p.dedup\n#align multiset.dedup Multiset.dedup\n\n@[simp]\ntheorem coe_dedup (l : List \u03b1) : @dedup \u03b1 _ l = l.dedup :=\n  rfl\n#align multiset.coe_dedup Multiset.coe_dedup\n\n@[simp]\ntheorem dedup_zero : @dedup \u03b1 _ 0 = 0 :=\n  rfl\n#align multiset.dedup_zero Multiset.dedup_zero\n\n@[simp]\ntheorem mem_dedup {a : \u03b1} {s : Multiset \u03b1} : a \u2208 dedup s \u2194 a \u2208 s :=\n  Quot.induction_on s fun _ => List.mem_dedup\n#align multiset.mem_dedup Multiset.mem_dedup\n\n@[simp]\ntheorem dedup_cons_of_mem {a : \u03b1} {s : Multiset \u03b1} : a \u2208 s \u2192 dedup (a ::\u2098 s) = dedup s :=\n  Quot.induction_on s fun _ m => @congr_arg _ _ _ _ ofList <| List.dedup_cons_of_mem m\n#align multiset.dedup_cons_of_mem Multiset.dedup_cons_of_mem\n\n@[simp]\ntheorem dedup_cons_of_not_mem {a : \u03b1} {s : Multiset \u03b1} : a \u2209 s \u2192 dedup (a ::\u2098 s) = a ::\u2098 dedup s :=\n  Quot.induction_on s fun _ m => congr_arg ofList <| List.dedup_cons_of_not_mem m\n#align multiset.dedup_cons_of_not_mem Multiset.dedup_cons_of_not_mem\n\ntheorem dedup_le (s : Multiset \u03b1) : dedup s \u2264 s :=\n  Quot.induction_on s fun _ => (dedup_sublist _).subperm\n#align multiset.dedup_le Multiset.dedup_le\n\ntheorem dedup_subset (s : Multiset \u03b1) : dedup s \u2286 s :=\n  subset_of_le <| dedup_le _\n#align multiset.dedup_subset Multiset.dedup_subset\n\ntheorem subset_dedup (s : Multiset \u03b1) : s \u2286 dedup s := fun _ => mem_dedup.2\n#align multiset.subset_dedup Multiset.subset_dedup\n\n@[simp]\ntheorem dedup_subset' {s t : Multiset \u03b1} : dedup s \u2286 t \u2194 s \u2286 t :=\n  \u27e8Subset.trans (subset_dedup _), Subset.trans (dedup_subset _)\u27e9\n#align multiset.dedup_subset' Multiset.dedup_subset'\n\n@[simp]\ntheorem subset_dedup' {s t : Multiset \u03b1} : s \u2286 dedup t \u2194 s \u2286 t :=\n  \u27e8fun h => Subset.trans h (dedup_subset _), fun h => Subset.trans h (subset_dedup _)\u27e9\n#align multiset.subset_dedup' Multiset.subset_dedup'\n\n@[simp]\ntheorem nodup_dedup (s : Multiset \u03b1) : Nodup (dedup s) :=\n  Quot.induction_on s List.nodup_dedup\n#align multiset.nodup_dedup Multiset.nodup_dedup\n\ntheorem dedup_eq_self {s : Multiset \u03b1} : dedup s = s \u2194 Nodup s :=\n  \u27e8fun e => e \u25b8 nodup_dedup s, Quot.induction_on s fun _ h => congr_arg ofList h.dedup\u27e9\n#align multiset.dedup_eq_self Multiset.dedup_eq_self\n\nalias dedup_eq_self \u2194 _ Nodup.dedup\n#align multiset.nodup.dedup Multiset.Nodup.dedup\n\ntheorem count_dedup (m : Multiset \u03b1) (a : \u03b1) : m.dedup.count a = if a \u2208 m then 1 else 0 :=\n  Quot.induction_on m fun _ => by\n    simp only [quot_mk_to_coe'', coe_dedup, mem_coe, List.mem_dedup, coe_nodup, coe_count]\n    apply List.count_dedup _ _\n#align multiset.count_dedup Multiset.count_dedup\n\n@[simp]\ntheorem dedup_idempotent {m : Multiset \u03b1} : m.dedup.dedup = m.dedup :=\n  Quot.induction_on m fun _ => @congr_arg _ _ _ _ ofList List.dedup_idempotent\n#align multiset.dedup_idempotent Multiset.dedup_idempotent\n\n@[simp]\ntheorem dedup_bind_dedup [DecidableEq \u03b2] (m : Multiset \u03b1) (f : \u03b1 \u2192 Multiset \u03b2) :\n    (m.dedup.bind f).dedup = (m.bind f).dedup := by\n  ext x\n  -- Porting note: was `simp_rw [count_dedup, mem_bind, mem_dedup]`\n  simp_rw [count_dedup]\n  refine if_congr ?_ rfl rfl\n  simp\n#align multiset.dedup_bind_dedup Multiset.dedup_bind_dedup\n\ntheorem dedup_eq_zero {s : Multiset \u03b1} : dedup s = 0 \u2194 s = 0 :=\n  \u27e8fun h => eq_zero_of_subset_zero <| h \u25b8 subset_dedup _, fun h => h.symm \u25b8 dedup_zero\u27e9\n#align multiset.dedup_eq_zero Multiset.dedup_eq_zero\n\n@[simp]\ntheorem dedup_singleton {a : \u03b1} : dedup ({a} : Multiset \u03b1) = {a} :=\n  (nodup_singleton _).dedup\n#align multiset.dedup_singleton Multiset.dedup_singleton\n\ntheorem le_dedup {s t : Multiset \u03b1} : s \u2264 dedup t \u2194 s \u2264 t \u2227 Nodup s :=\n  \u27e8fun h => \u27e8le_trans h (dedup_le _), nodup_of_le h (nodup_dedup _)\u27e9,\n   fun \u27e8l, d\u27e9 => (le_iff_subset d).2 <| Subset.trans (subset_of_le l) (subset_dedup _)\u27e9\n#align multiset.le_dedup Multiset.le_dedup\n\ntheorem le_dedup_self {s : Multiset \u03b1} : s \u2264 dedup s \u2194 Nodup s := by\n  rw [le_dedup, and_iff_right le_rfl]\n#align multiset.le_dedup_self Multiset.le_dedup_self\n\ntheorem dedup_ext {s t : Multiset \u03b1} : dedup s = dedup t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t := by\n  simp [Nodup.ext]\n#align multiset.dedup_ext Multiset.dedup_ext\n\ntheorem dedup_map_dedup_eq [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : Multiset \u03b1) :\n    dedup (map f (dedup s)) = dedup (map f s) := by\n  simp [dedup_ext]\n#align multiset.dedup_map_dedup_eq Multiset.dedup_map_dedup_eq\n\n@[simp]\ntheorem dedup_nsmul {s : Multiset \u03b1} {n : \u2115} (h0 : n \u2260 0) : (n \u2022 s).dedup = s.dedup := by\n  ext a\n  by_cases h : a \u2208 s <;> simp [h, h0]\n#align multiset.dedup_nsmul Multiset.dedup_nsmul\n\ntheorem Nodup.le_dedup_iff_le {s t : Multiset \u03b1} (hno : s.Nodup) : s \u2264 t.dedup \u2194 s \u2264 t := by\n  simp [le_dedup, hno]\n#align multiset.nodup.le_dedup_iff_le Multiset.Nodup.le_dedup_iff_le\n\nend Multiset\n\ntheorem Multiset.Nodup.le_nsmul_iff_le {\u03b1 : Type _} {s t : Multiset \u03b1} {n : \u2115} (h : s.Nodup)\n    (hn : n \u2260 0) : s \u2264 n \u2022 t \u2194 s \u2264 t := by\n  classical\n    rw [\u2190 h.le_dedup_iff_le, Iff.comm, \u2190 h.le_dedup_iff_le]\n    simp [hn]\n#align multiset.nodup.le_nsmul_iff_le Multiset.Nodup.le_nsmul_iff_le\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/Multiset/Dedup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984137988772, "lm_q2_score": 0.7217431943271999, "lm_q1q2_score": 0.38620363845461947}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module category_theory.sites.cover_preserving\n! leanprover-community/mathlib commit e2e38c005fc6f715502490da6cb0ec84df9ed228\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Sites.Limits\nimport Mathbin.CategoryTheory.Functor.Flat\nimport Mathbin.CategoryTheory.Limits.Preserves.Filtered\n\n/-!\n# Cover-preserving functors between sites.\n\nWe define cover-preserving functors between sites as functors that push covering sieves to\ncovering sieves. A cover-preserving and compatible-preserving functor `G : C \u2964 D` then pulls\nsheaves on `D` back to sheaves on `C` via `G.op \u22d9 -`.\n\n## Main definitions\n\n* `category_theory.cover_preserving`: a functor between sites is cover-preserving if it\npushes covering sieves to covering sieves\n* `category_theory.compatible_preserving`: a functor between sites is compatible-preserving\nif it pushes compatible families of elements to compatible families.\n* `category_theory.pullback_sheaf`: the pullback of a sheaf along a cover-preserving and\ncompatible-preserving functor.\n* `category_theory.sites.pullback`: the induced functor `Sheaf K A \u2964 Sheaf J A` for a\ncover-preserving and compatible-preserving functor `G : (C, J) \u2964 (D, K)`.\n\n## Main results\n\n- `category_theory.sites.whiskering_left_is_sheaf_of_cover_preserving`: If `G : C \u2964 D` is\ncover-preserving and compatible-preserving, then `G \u22d9 -` (`u\u1d56`) as a functor\n`(D\u1d52\u1d56 \u2964 A) \u2964 (C\u1d52\u1d56 \u2964 A)` of presheaves maps sheaves to sheaves.\n\n## References\n\n* [Elephant]: *Sketches of an Elephant*, P. T. Johnstone: C2.3.\n* https://stacks.math.columbia.edu/tag/00WW\n\n-/\n\n\nuniverse w v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nnoncomputable section\n\nopen CategoryTheory\n\nopen Opposite\n\nopen CategoryTheory.Presieve.FamilyOfElements\n\nopen CategoryTheory.Presieve\n\nopen CategoryTheory.Limits\n\nnamespace CategoryTheory\n\nvariable {C : Type u\u2081} [Category.{v\u2081} C] {D : Type u\u2082} [Category.{v\u2082} D]\n\nvariable {A : Type u\u2083} [Category.{v\u2083} A]\n\nvariable (J : GrothendieckTopology C) (K : GrothendieckTopology D)\n\nvariable {L : GrothendieckTopology A}\n\n/-- A functor `G : (C, J) \u2964 (D, K)` between sites is *cover-preserving*\nif for all covering sieves `R` in `C`, `R.pushforward_functor G` is a covering sieve in `D`.\n-/\n@[nolint has_nonempty_instance]\nstructure CoverPreserving (G : C \u2964 D) : Prop where\n  cover_preserve : \u2200 {U : C} {S : Sieve U} (hS : S \u2208 J U), S.functorPushforward G \u2208 K (G.obj U)\n#align category_theory.cover_preserving CategoryTheory.CoverPreserving\n\n/-- The identity functor on a site is cover-preserving. -/\ntheorem idCoverPreserving : CoverPreserving J J (\ud835\udfed _) :=\n  \u27e8fun U S hS => by simpa using hS\u27e9\n#align category_theory.id_cover_preserving CategoryTheory.idCoverPreserving\n\nvariable (J) (K)\n\n/-- The composition of two cover-preserving functors is cover-preserving. -/\ntheorem CoverPreserving.comp {F} (hF : CoverPreserving J K F) {G} (hG : CoverPreserving K L G) :\n    CoverPreserving J L (F \u22d9 G) :=\n  \u27e8fun U S hS => by\n    rw [sieve.functor_pushforward_comp]\n    exact hG.cover_preserve (hF.cover_preserve hS)\u27e9\n#align category_theory.cover_preserving.comp CategoryTheory.CoverPreserving.comp\n\n/-- A functor `G : (C, J) \u2964 (D, K)` between sites is called compatible preserving if for each\ncompatible family of elements at `C` and valued in `G.op \u22d9 \u2131`, and each commuting diagram\n`f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082`, `x g\u2081` and `x g\u2082` coincide when restricted via `f\u1d62`.\nThis is actually stronger than merely preserving compatible families because of the definition of\n`functor_pushforward` used.\n-/\n@[nolint has_nonempty_instance]\nstructure CompatiblePreserving (K : GrothendieckTopology D) (G : C \u2964 D) : Prop where\n  Compatible :\n    \u2200 (\u2131 : SheafOfTypes.{w} K) {Z} {T : Presieve Z} {x : FamilyOfElements (G.op \u22d9 \u2131.val) T}\n      (h : x.Compatible) {Y\u2081 Y\u2082} {X} (f\u2081 : X \u27f6 G.obj Y\u2081) (f\u2082 : X \u27f6 G.obj Y\u2082) {g\u2081 : Y\u2081 \u27f6 Z}\n      {g\u2082 : Y\u2082 \u27f6 Z} (hg\u2081 : T g\u2081) (hg\u2082 : T g\u2082) (eq : f\u2081 \u226b G.map g\u2081 = f\u2082 \u226b G.map g\u2082),\n      \u2131.val.map f\u2081.op (x g\u2081 hg\u2081) = \u2131.val.map f\u2082.op (x g\u2082 hg\u2082)\n#align category_theory.compatible_preserving CategoryTheory.CompatiblePreserving\n\nvariable {J K} {G : C \u2964 D} (hG : CompatiblePreserving.{w} K G) (\u2131 : SheafOfTypes.{w} K) {Z : C}\n\nvariable {T : Presieve Z} {x : FamilyOfElements (G.op \u22d9 \u2131.val) T} (h : x.Compatible)\n\ninclude h hG\n\n/-- `compatible_preserving` functors indeed preserve compatible families. -/\ntheorem Presieve.FamilyOfElements.Compatible.functorPushforward :\n    (x.functorPushforward G).Compatible :=\n  by\n  rintro Z\u2081 Z\u2082 W g\u2081 g\u2082 f\u2081' f\u2082' H\u2081 H\u2082 eq\n  unfold family_of_elements.functor_pushforward\n  rcases get_functor_pushforward_structure H\u2081 with \u27e8X\u2081, f\u2081, h\u2081, hf\u2081, rfl\u27e9\n  rcases get_functor_pushforward_structure H\u2082 with \u27e8X\u2082, f\u2082, h\u2082, hf\u2082, rfl\u27e9\n  suffices : \u2131.val.map (g\u2081 \u226b h\u2081).op (x f\u2081 hf\u2081) = \u2131.val.map (g\u2082 \u226b h\u2082).op (x f\u2082 hf\u2082)\n  simpa using this\n  apply hG.compatible \u2131 h _ _ hf\u2081 hf\u2082\n  simpa using Eq\n#align category_theory.presieve.family_of_elements.compatible.functor_pushforward CategoryTheory.Presieve.FamilyOfElements.Compatible.functorPushforward\n\n@[simp]\ntheorem CompatiblePreserving.apply_map {Y : C} {f : Y \u27f6 Z} (hf : T f) :\n    x.functorPushforward G (G.map f) (image_mem_functorPushforward G T hf) = x f hf :=\n  by\n  unfold family_of_elements.functor_pushforward\n  rcases e\u2081 : get_functor_pushforward_structure (image_mem_functor_pushforward G T hf) with\n    \u27e8X, g, f', hg, eq\u27e9\n  simpa using hG.compatible \u2131 h f' (\ud835\udfd9 _) hg hf (by simp [Eq])\n#align category_theory.compatible_preserving.apply_map CategoryTheory.CompatiblePreserving.apply_map\n\nomit h hG\n\nopen Limits.WalkingCospan\n\ntheorem compatiblePreservingOfFlat {C : Type u\u2081} [Category.{v\u2081} C] {D : Type u\u2081} [Category.{v\u2081} D]\n    (K : GrothendieckTopology D) (G : C \u2964 D) [RepresentablyFlat G] : CompatiblePreserving K G :=\n  by\n  constructor\n  intro \u2131 Z T x hx Y\u2081 Y\u2082 X f\u2081 f\u2082 g\u2081 g\u2082 hg\u2081 hg\u2082 e\n  -- First, `f\u2081` and `f\u2082` form a cone over `cospan g\u2081 g\u2082 \u22d9 u`.\n  let c : cone (cospan g\u2081 g\u2082 \u22d9 G) :=\n    (cones.postcompose (diagram_iso_cospan (cospan g\u2081 g\u2082 \u22d9 G)).inv).obj (pullback_cone.mk f\u2081 f\u2082 e)\n  /-\n    This can then be viewed as a cospan of structured arrows, and we may obtain an arbitrary cone\n    over it since `structured_arrow W u` is cofiltered.\n    Then, it suffices to prove that it is compatible when restricted onto `u(c'.X.right)`.\n    -/\n  let c' := is_cofiltered.cone (structured_arrow_cone.to_diagram c \u22d9 structured_arrow.pre _ _ _)\n  have eq\u2081 : f\u2081 = (c'.X.hom \u226b G.map (c'.\u03c0.app left).right) \u226b eq_to_hom (by simp) :=\n    by\n    erw [\u2190 (c'.\u03c0.app left).w]\n    dsimp\n    simp\n  have eq\u2082 : f\u2082 = (c'.X.hom \u226b G.map (c'.\u03c0.app right).right) \u226b eq_to_hom (by simp) :=\n    by\n    erw [\u2190 (c'.\u03c0.app right).w]\n    dsimp\n    simp\n  conv_lhs => rw [eq\u2081]\n  conv_rhs => rw [eq\u2082]\n  simp only [op_comp, functor.map_comp, types_comp_apply, eq_to_hom_op, eq_to_hom_map]\n  congr 1\n  /-\n    Since everything now falls in the image of `u`,\n    the result follows from the compatibility of `x` in the image of `u`.\n    -/\n  injection c'.\u03c0.naturality walking_cospan.hom.inl with _ e\u2081\n  injection c'.\u03c0.naturality walking_cospan.hom.inr with _ e\u2082\n  exact hx (c'.\u03c0.app left).right (c'.\u03c0.app right).right hg\u2081 hg\u2082 (e\u2081.symm.trans e\u2082)\n#align category_theory.compatible_preserving_of_flat CategoryTheory.compatiblePreservingOfFlat\n\ntheorem compatiblePreservingOfDownwardsClosed (F : C \u2964 D) [Full F] [Faithful F]\n    (hF : \u2200 {c : C} {d : D} (f : d \u27f6 F.obj c), \u03a3c', F.obj c' \u2245 d) : CompatiblePreserving K F :=\n  by\n  constructor\n  introv hx he\n  obtain \u27e8X', e\u27e9 := hF f\u2081\n  apply (\u2131.1.mapIso e.op).toEquiv.Injective\n  simp only [iso.op_hom, iso.to_equiv_fun, \u2131.1.mapIso_hom, \u2190 functor_to_types.map_comp_apply]\n  simpa using\n    hx (F.preimage <| e.hom \u226b f\u2081) (F.preimage <| e.hom \u226b f\u2082) hg\u2081 hg\u2082\n      (F.map_injective <| by simpa using he)\n#align category_theory.compatible_preserving_of_downwards_closed CategoryTheory.compatiblePreservingOfDownwardsClosed\n\n/-- If `G` is cover-preserving and compatible-preserving,\nthen `G.op \u22d9 _` pulls sheaves back to sheaves.\n\nThis result is basically <https://stacks.math.columbia.edu/tag/00WW>.\n-/\ntheorem pullback_isSheaf_of_coverPreserving {G : C \u2964 D} (hG\u2081 : CompatiblePreserving.{v\u2083} K G)\n    (hG\u2082 : CoverPreserving J K G) (\u2131 : Sheaf K A) : Presheaf.IsSheaf J (G.op \u22d9 \u2131.val) :=\n  by\n  intro X U S hS x hx\n  change family_of_elements (G.op \u22d9 \u2131.val \u22d9 coyoneda.obj (op X)) _ at x\n  let H := \u2131.2 X _ (hG\u2082.cover_preserve hS)\n  let hx' := hx.functor_pushforward hG\u2081 (sheaf_over \u2131 X)\n  constructor; swap\n  \u00b7 apply H.amalgamate (x.functor_pushforward G)\n    exact hx'\n  constructor\n  \u00b7 intro V f hf\n    convert H.is_amalgamation hx' (G.map f) (image_mem_functor_pushforward G S hf)\n    rw [hG\u2081.apply_map (sheaf_over \u2131 X) hx]\n  \u00b7 intro y hy\n    refine'\n      H.is_separated_for _ y _ _ (H.is_amalgamation (hx.functor_pushforward hG\u2081 (sheaf_over \u2131 X)))\n    rintro V f \u27e8Z, f', g', h, rfl\u27e9\n    erw [family_of_elements.comp_of_compatible (S.functor_pushforward G) hx'\n        (image_mem_functor_pushforward G S h) g']\n    dsimp\n    simp [hG\u2081.apply_map (sheaf_over \u2131 X) hx h, \u2190 hy f' h]\n#align category_theory.pullback_is_sheaf_of_cover_preserving CategoryTheory.pullback_isSheaf_of_coverPreserving\n\n/-- The pullback of a sheaf along a cover-preserving and compatible-preserving functor. -/\ndef pullbackSheaf {G : C \u2964 D} (hG\u2081 : CompatiblePreserving K G) (hG\u2082 : CoverPreserving J K G)\n    (\u2131 : Sheaf K A) : Sheaf J A :=\n  \u27e8G.op \u22d9 \u2131.val, pullback_isSheaf_of_coverPreserving hG\u2081 hG\u2082 \u2131\u27e9\n#align category_theory.pullback_sheaf CategoryTheory.pullbackSheaf\n\nvariable (A)\n\n/-- The induced functor from `Sheaf K A \u2964 Sheaf J A` given by `G.op \u22d9 _`\nif `G` is cover-preserving and compatible-preserving.\n-/\n@[simps]\ndef Sites.pullback {G : C \u2964 D} (hG\u2081 : CompatiblePreserving K G) (hG\u2082 : CoverPreserving J K G) :\n    Sheaf K A \u2964 Sheaf J A where\n  obj \u2131 := pullbackSheaf hG\u2081 hG\u2082 \u2131\n  map _ _ f := \u27e8((whiskeringLeft _ _ _).obj G.op).map f.val\u27e9\n  map_id' \u2131 := by\n    ext1\n    apply ((whiskering_left _ _ _).obj G.op).map_id\n  map_comp' _ _ _ f g := by\n    ext1\n    apply ((whiskering_left _ _ _).obj G.op).map_comp\n#align category_theory.sites.pullback CategoryTheory.Sites.pullback\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Sites/CoverPreserving.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3860362649830042}}
{"text": "import Architectural.ArchWithContracts\nimport Architectural.ArchitectureDecomp\nimport tactic \n\nvariables {\u03a6 Var : Type} [fintype Var] [decidable_eq Var] [AssertionLang \u03a6 Var]\n\n@[reducible]\ndef list_conj : list \u03a6 \u2192 \u03a6  \n| [] := \u22a4\n| (h::t) := AssertionLang.conj Var (h) (list_conj t)\n\n\ntheorem list_conj_iff (l : list \u03a6) (\u03c3 : Trace Var) : (\u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ (@list_conj \u03a6 Var _ _ _ l))) \u2194 (\u2200 \u03c6 \u2208 l, \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ \u03c6)) := \nbegin \n  split,\n  intro H1,\n  intros \u03c6 H2,\n  induction l with hd tl IH,\n  exfalso, assumption,\n  cases H2,rw H2,\n  unfold list_conj at H1,\n  rw AssertionLang.conj_def at H1,\n  cases H1 with H11 H22, exact H11,\n  apply IH,\n  unfold list_conj at H1,\n  rw AssertionLang.conj_def at H1,\n  exact H1.2, assumption,\n\n  intros H,\n  induction l with hd tl IH,\n  unfold list_conj, rw AssertionLang.T_def, simp,\n  obtain H1 := H hd,\n  unfold list_conj,\n  rw AssertionLang.conj_def,\n  split,\n  apply H1,\n  simp,\n  apply IH,\n  intros \u03c6 H',\n  have : \u03c6 \u2208 hd::tl, from list.mem_cons_of_mem hd H',\n  cases this,\n  rw this, apply H, exact list.mem_cons_self hd tl,\n  apply H \u03c6, exact list.mem_cons_of_mem hd H',\nend \n\n@[reducible]\ndef get_nfs {S : Component Var} (A : ArchitectureWithContracts \u03a6 S) : list \u03a6 := \n  (A.subs.map (\u03bb C, (A.contracts.find_val C).iget)).map (\u03bb C, C.nf)\n\n\ndef RPO_fst {S : Component Var} (A : ArchitectureWithContracts \u03a6 S): Prop := \n  \u2200 \u03c3 , \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _  (@AssertionLang.Impl \u03a6 Var _ _ _ (@list_conj \u03a6 Var _ _ _ (get_nfs A)) (A.parent.nf)))\n\n\n\ndef RPO_snd {S : Component Var}  (A : ArchitectureWithContracts \u03a6 S) : Prop := \n  \u2200 S' \u2208 A.subs,\n  \u2200 \u03c3, \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _\n  (@AssertionLang.Impl \u03a6 Var _ _ _  \n  (AssertionLang.conj Var A.parent.A (@list_conj \u03a6 Var _ _ _ (list.map (\u03bb C : Component Var,  (A.contracts.find_val C).iget.nf) (A.subs.erase S')))) (A.contracts.find_val S').iget.A)\n  )\n\ndef RPO {S : Component Var}  (A : ArchitectureWithContracts \u03a6 S) : Prop := RPO_fst A \u2227 RPO_snd A\n\nvariables S : Component Var \n\nvariables IMPLEMENTATIONS : \u03a0 (U : Component Var),  Impl Var\n\nvariable CURRENT_ENVIRONMENT : Env Var \n\ndef isCorrectImplementation (C : Component Var) (I : Impl Var) (Ctr : Contract \u03a6 Var) : Claim (Trace Var) := \n{\n  X := I, \n  P := \u03bb \u03c3, \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ Ctr.A) \u2192 \u03c3 \u2208 \n                (@AssertionLang.sem \u03a6 Var _ _ _ Ctr.G)\n}\n\ndef isGoodEnv (E : Env Var) (C : Contract \u03a6 Var) : Claim (Trace Var) := \n{\n  X := E, \n  P := \u03bb s, s \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ C.A)\n}\n\n\nnamespace Contract \n\ndef to_arch_components \n{S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S) \n(IMPLEMENTATIONS : Implementations A.to_Architecture) :\n\u03a0 (a : Component Var), (\u03bb (C : Component Var), C \u2208 A.to_Architecture.subs) a \u2192 Claim (Trace Var) := \n\u03bb a h, isCorrectImplementation a  (IMPLEMENTATIONS a h) (A.contracts.find_val a).iget\n\n\ndef strategy\n {S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S)\n (Is : Implementations A.to_Architecture)\n (CURRENT_ENVIRONMENT : Env Var) : Strategy (Trace Var) := \n{ parent := Claim.correctDecomposition A Is CURRENT_ENVIRONMENT,\n  decomp := \u03bb _, \n  (@list.pmap (Component Var) (Claim (Trace Var)) (\u03bb C, C \u2208 A.subs) \n  (\u03bb a h, isCorrectImplementation a (Is a h) (A.contracts.find_val a).iget) A.subs (by {simp,}))++[isGoodEnv CURRENT_ENVIRONMENT A.parent]\n\n}\n\ndef mk_strategy {S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S) : Strategy (Trace Var) := Contract.strategy A (\u03bb e h, set.univ) (set.univ)\n\ndef strat_is_deductive \n{S : Component Var} \n(A : ArchitectureWithContracts \u03a6 S) : Prop := \u2200 Is : Implementations A.to_Architecture, \u2200 E : Env Var, \ndeductive (Trace Var) (Contract.strategy A Is E)\n\n\nend Contract \n\n\nlemma subclaims_meaning_env \n {S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S) \n (IMPLEMENTATIONS : Implementations A.to_Architecture)\n (CURRENT_ENVIRONMENT : Env Var)\n : \n  (\u2200 clm \u2208 (Contract.strategy A IMPLEMENTATIONS CURRENT_ENVIRONMENT).subclaims, \u27e6clm\u27e7) \u2192 \n \u27e6isGoodEnv CURRENT_ENVIRONMENT A.parent\u27e7 := \nbegin \n intros H\u2081,\n replace H\u2081 := H\u2081 (isGoodEnv CURRENT_ENVIRONMENT A.parent),\n rw Contract.strategy at H\u2081, rw Strategy.subclaims at H\u2081,\n  simp at H\u2081,\n  exact H\u2081,\nend \n\nlemma subclaims_meaning_impl \n {S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S) \n (IMPLEMENTATIONS : Implementations A.to_Architecture)\n (CURRENT_ENVIRONMENT : Env Var): \n  (\u2200 clm \u2208 (Contract.strategy A IMPLEMENTATIONS CURRENT_ENVIRONMENT).subclaims, \u27e6clm\u27e7) \u2192 \n \u2200 C \u2208 A.subs, \u27e6isCorrectImplementation C (IMPLEMENTATIONS C H) (A.contracts.find_val C).iget\u27e7 := \nbegin \n intros H\u2081,\n rw Contract.strategy at H\u2081, rw Strategy.subclaims at H\u2081,\n simp at H\u2081,\n intros C mem, \n replace H\u2081 := H\u2081 (isCorrectImplementation C (IMPLEMENTATIONS C mem) (Map.find_val C A.contracts).iget),\n apply H\u2081,\n clear H\u2081,\n left, use C,\n split, refl, assumption,\nend \n\n\nlemma subclaims_meaning\n  {S : Component Var} \n (A : ArchitectureWithContracts \u03a6 S) \n (IMPLEMENTATIONS : Implementations A.to_Architecture)\n (CURRENT_ENVIRONMENT : Env Var) : \n  (\u2200 clm \u2208 (Contract.strategy A IMPLEMENTATIONS CURRENT_ENVIRONMENT).subclaims, \u27e6clm\u27e7) \u2192 \n  \u27e6isGoodEnv CURRENT_ENVIRONMENT A.parent\u27e7 \u2227 \n \u2200 C \u2208 A.subs, \u27e6isCorrectImplementation C (IMPLEMENTATIONS C H) (A.contracts.find_val C).iget\u27e7 := \nbegin \n  intros H\u2081,\n  split, apply subclaims_meaning_env, assumption,\n  apply subclaims_meaning_impl, assumption,\nend \n\n-- theorem validity : deductive (Trace PORTS) (Contract.strategy IMPLEMENTATIONS CURRENT_ENVIRONMENT) := \n-- begin\n--   intro H,\n--   replace H := subclaims_meaning _ _ H,\n--   cases H with Henv Himpl,\n--   rw Contract.strategy,\n--   simp,\n--   rw Claim.correctDecomposition,\n--   rw meaning, \n--   simp,\n--   intros \u03c3, \n--   split, {\n--     rw Claim.correctDecompositionImpl,\n--     intros H\u2081 H\u2082,\n--     rw CI at H\u2082,\n--     simp at H\u2082,\n--     sorry \n--   }\n-- end \n\n-- def LACU_ARCH_WITH_CONTRACTS : ArchitectureWithContracts lt_eq_assertions LACU := \n\n\ntheorem via_rpo \n{S : Component Var}  \n{A : ArchitectureWithContracts \u03a6 S}\n(E :  Env Var)\n(Is :Implementations A.to_Architecture)\n: RPO A \u2192 deductive (Trace Var) (Contract.strategy A Is E) := \nbegin\n  rintros \u27e8H\u2081, H\u2082\u27e9,\n  intros H,\n  replace H := subclaims_meaning _ _  _ H,\n  cases H with H\u2083 H\u2084,\n  intros \u03c3 mem,\n  rw Contract.strategy, rw Claim.correctDecomposition,\n  simp, split, {\n  clear H\u2082 H\u2083,\n  rw Claim.correctDecompositionImpl, simp,\n  intros H\u2082 H\u2083 H\u2085,\n  rw CompositeImplementation at H\u2083,\n  simp at H\u2083,\n  rw RPO_fst at H\u2081,\n  replace H\u2081 := H\u2081 \u03c3,\n  rw AssertionLang.impl_def at H\u2081,\n  have : \u03c3 \u2208 AssertionLang.sem (list_conj (get_nfs A)) := by \n  {\n    rw list_conj_iff,\n    rw get_nfs,\n    simp,\n    intros S' Hmem,\n    have : Map.find_val S' A.contracts \u2260 none, from A.all_components S' Hmem,\n    have opt : \u2203 C : Contract \u03a6 Var, Map.find_val S' A.contracts = some C, by \n      {exact option.ne_none_iff_exists'.mp this,},\n    cases opt with C H\u2086, \n    have further : (Map.find_val S' A.contracts).iget = C, by {exact (congr_arg option.iget H\u2086).trans rfl,},\n    simp [further] at *,\n    replace H\u2084 := H\u2084 S' Hmem,\n    rw isCorrectImplementation at H\u2084,\n    rw meaning at H\u2084, simp at H\u2084,\n    replace H\u2083 := H\u2083 S' S',\n    unfold set.Inter at H\u2083,\n    unfold infi at H\u2083,\n    unfold Inf at H\u2083,\n    simp at H\u2083,\n    replace H\u2084 := H\u2084 \u03c3 (H\u2083 Hmem),\n    have further : (Map.find_val S' A.contracts).iget = C, by {exact (congr_arg option.iget H\u2086).trans rfl,},\n    simp [further] at *, clear further,\n    rw AssertionLang.disj_def,\n    have EM : \u03c3 \u2208 @AssertionLang.sem \u03a6 Var _ _ _ (AssertionLang.compl Var C.A) \u2228 \u03c3 \u2209 AssertionLang.sem (AssertionLang.compl Var C.A), from or_not,\n    cases EM,\n    left, assumption,\n    right, apply H\u2084,\n    rw AssertionLang.compl_def at EM,\n    simp at EM, exact EM,\n  },\n  replace H\u2081 := H\u2081 this,\n  rw AssertionLang.disj_def at H\u2081,\n  cases H\u2081,\n  rw AssertionLang.compl_def at H\u2081,\n  simp at H\u2081,\n  contradiction,\n  exact H\u2081,},\n  { clear H\u2081,\n    rw Claim.correctDecompositionEnv,\n    intros  _ S' EnvSat H\u2085 H\u2088 H\u2086,\n    rw CompositeEnvironment at H\u2086,\n    simp at H\u2086, cases H\u2086 with H\u2086 H\u2087,\n    rw RPO_snd at H\u2082,\n    replace H\u2082 := H\u2082 S' EnvSat \u03c3, \n    rw AssertionLang.impl_def at H\u2082,\n    apply H\u2082, clear H\u2082, simp,\n    rw AssertionLang.conj_def, split,\n    rw Env.satisfiesContract at H\u2085,\n    apply H\u2085, apply H\u2086,\n    rw list_conj_iff,\n    dsimp at *,\n    intros \u03c6 h\u03c6, simp at *,\n    cases h\u03c6 with w h,\n    cases h with h1 h2,\n    have opt : \u2203 C : Contract \u03a6 Var, Map.find_val w A.contracts = some C, by \n      {exact option.ne_none_iff_exists'.mp (A.all_components w (list.mem_of_mem_erase h1)),},\n    cases opt with C H\u2086, \n    have further : (Map.find_val w A.contracts).iget = C, by {exact (congr_arg option.iget H\u2086).trans rfl,},\n    simp [further] at *,\n    rw \u2190 h2, rw AssertionLang.disj_def, tidy,\n    replace H\u2084 := H\u2084 w (list.mem_of_mem_erase h1),\n    rw meaning at H\u2084, rw isCorrectImplementation at H\u2084, replace H\u2084 := H\u2084 \u03c3,simp at H\u2084,\n    replace H\u2087 := (H\u2087 w w (h1)),\n    rw H\u2086_1 at *,\n    replace H\u2084 := H\u2084 H\u2087,\n    rw AssertionLang.compl_def,\n    simp at *,\n    rwa \u2190 imp_iff_not_or,\n  }\nend\n", "meta": {"author": "loganrjmurphy", "repo": "ForeMoSt", "sha": "c7affc7c8971562520d2775ac48fe4f188f84b02", "save_path": "github-repos/lean/loganrjmurphy-ForeMoSt", "path": "github-repos/lean/loganrjmurphy-ForeMoSt/ForeMoSt-c7affc7c8971562520d2775ac48fe4f188f84b02/src/Architectural/proofObligations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7718434978390746, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3859217489195373}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nThe actual graph produced by the naive variational auto-encoder code in aevb/prog.lean.\n\nNote: we include this file as an optimization, to avoid repeatedly simplifying the\nprogram into the graph.\n-/\nimport ..program ..prove_model_ok .util .prog\n\nnamespace certigrad\nnamespace aevb\n\nsection nodes\n\nopen det rand.op label tactic certigrad.tactic\n\n#print \"Defining graph_naive...\"\n\ndef graph_naive : \u03a0 (a : arch) (x_data : T [a^.n_in, a^.n_x]), graph\n| a x_data :=\ngraph.mk [\u27e8(ID.nat 0, [a^.ne, a^.bs]), [(ID.str W_encode\u2081, [a^.ne, a^.n_in]), (ID.str x, [a^.n_in, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.nat 1, [a^.ne, a^.bs]), [(ID.nat 0, [a^.ne, a^.bs])], operator.det $ ops.softplus _\u27e9,\n          \u27e8(ID.nat 2, [a^.ne, a^.bs]), [(ID.str W_encode\u2082, [a^.ne, a^.ne]), (ID.nat 1, [a^.ne, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.str h_encode, [a^.ne, a^.bs]), [(ID.nat 2, [a^.ne, a^.bs])], operator.det $ ops.softplus _\u27e9,\n          \u27e8(ID.str \u03bc, [a^.nz, a^.bs]), [(ID.str W_encode_\u03bc, [a^.nz, a^.ne]), (ID.str h_encode, [a^.ne, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.nat 5, [a^.nz, a^.bs]), [(ID.str W_encode_log\u03c3\u2082, [a^.nz, a^.ne]), (ID.str h_encode, [a^.ne, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.nat 6, [a^.nz, a^.bs]), [(ID.nat 5, [a^.nz, a^.bs])], operator.det $ ops.exp _\u27e9,\n          \u27e8(ID.str \u03c3, [a^.nz, a^.bs]), [(ID.nat 6, [a^.nz, a^.bs])], operator.det $ ops.sqrt _\u27e9,\n          \u27e8(ID.str z, [a^.nz, a^.bs]), [(ID.str \u03bc, [a^.nz, a^.bs]), (ID.str \u03c3, [a^.nz, a^.bs])], operator.rand $ mvn _\u27e9,\n          \u27e8(ID.str encoding_loss, []), [(ID.str \u03bc, [a^.nz, a^.bs]), (ID.str \u03c3, [a^.nz, a^.bs]), (ID.str z, [a^.nz, a^.bs])], operator.det $ op.mvn_empirical_kl _\u27e9,\n          \u27e8(ID.nat 10, [a^.nd, a^.bs]), [(ID.str W_decode\u2081, [a^.nd, a^.nz]), (ID.str z, [a^.nz, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.nat 11, [a^.nd, a^.bs]), [(ID.nat 10, [a^.nd, a^.bs])], operator.det $ ops.softplus _\u27e9,\n          \u27e8(ID.nat 12, [a^.nd, a^.bs]), [(ID.str W_decode\u2082, [a^.nd, a^.nd]), (ID.nat 11, [a^.nd, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.str h_decode, [a^.nd, a^.bs]), [(ID.nat 12, [a^.nd, a^.bs])], operator.det $ ops.softplus _\u27e9,\n          \u27e8(ID.nat 14, [a^.n_in, a^.bs]), [(ID.str W_decode_p, [a^.n_in, a^.nd]), (ID.str h_decode, [a^.nd, a^.bs])], operator.det $ ops.gemm _ _ _\u27e9,\n          \u27e8(ID.str p, [a^.n_in, a^.bs]), [(ID.nat 14, [a^.n_in, a^.bs])], operator.det $ ops.sigmoid _\u27e9,\n          \u27e8(ID.str decoding_loss, []), [(ID.str p, [a^.n_in, a^.bs]), (ID.str x, [a^.n_in, a^.bs])], operator.det $ ops.bernoulli_neglogpdf _\u27e9]\n         [ID.str encoding_loss, ID.str decoding_loss]\n         [(ID.str W_encode\u2081, [a^.ne, a^.n_in]), (ID.str W_encode\u2082, [a^.ne, a^.ne]), (ID.str W_encode_\u03bc, [a^.nz, a^.ne]), (ID.str W_encode_log\u03c3\u2082, [a^.nz, a^.ne]),\n          (ID.str W_decode\u2081, [a^.nd, a^.nz]), (ID.str W_decode\u2082, [a^.nd, a^.nd]), (ID.str W_decode_p, [a^.n_in, a^.nd])]\n         [(ID.str x, [a^.n_in, a^.bs]),\n          (ID.str W_encode\u2081, [a^.ne, a^.n_in]), (ID.str W_encode\u2082, [a^.ne, a^.ne]), (ID.str W_encode_\u03bc, [a^.nz, a^.ne]), (ID.str W_encode_log\u03c3\u2082, [a^.nz, a^.ne]),\n          (ID.str W_decode\u2081, [a^.nd, a^.nz]), (ID.str W_decode\u2082, [a^.nd, a^.nd]), (ID.str W_decode_p, [a^.n_in, a^.nd])]\n\nattribute [cgsimp] graph_naive\n\nopen tactic certigrad.tactic\n\n#print \"proving naive_aevb_as_graph...\"\n@[cgsimp] lemma naive_aevb_as_graph (a : arch) (x_data : T [a^.n_in, a^.n_x]) : naive_aevb a x_data = graph_naive a x_data :=\nby { dunfold naive_aevb, cgsimp, dcgsimp, cgsimp, reflexivity }\n\nend nodes\n\nend aevb\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/aevb/graph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3858883360743786}}
{"text": "import super.lpo\nopen tactic expr super\n\nexample (x : \u2115) : true := by do\nx \u2190 get_local `x,\nlet gt := mk_lpo [``eq,\n  x.local_uniq_name, ``has_add.add, ``has_zero.zero],\nlet d (x y : pexpr) := (do\n  x \u2190 to_expr x,\n  y \u2190 to_expr y,\n  trace (x,y,gt x y)),\nd ```(x + x) ```(x),\nd ```(x + 0) ```(x),\nd ```(x) ```(x + 0),\nd ```(x = x + 0) ```(x + 0 = x),\nm \u2190 mk_meta_var `(\u2115),\nd ```(%%m + 0 = %%m) ```(%%m = %%m + 0),\nd ```(%%m + 0) ```(%%m),\nunify m `(0),\ntriv\n", "meta": {"author": "gebner", "repo": "super2", "sha": "9bc5256c31750021ab97d6b59b7387773e54b384", "save_path": "github-repos/lean/gebner-super2", "path": "github-repos/lean/gebner-super2/super2-9bc5256c31750021ab97d6b59b7387773e54b384/test/lpo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.38587960858502474}}
{"text": "import Std\n\ndef List.enum' (L : List \u03b1) : List (Fin L.length \u00d7 \u03b1) :=\n  let rec go (rest : List \u03b1) (i : Nat)\n              (h : i + rest.length = L.length) :=\n    match rest, h with\n    | [], _ => []\n    | (x :: xs), h =>\n      (\u27e8i, (h.symm \u25b8 Nat.lt_add_of_pos_right (Nat.zero_lt_succ _))\u27e9, x)\n      :: go xs (i+1) (by\n        simp [\u2190h, Nat.add_succ, Nat.succ_add])\n  go L 0 (by simp)\n\ndef Fin.last (n : Nat) (_ : 0 < n) : Fin n :=\n  match n with\n  | 0 => by contradiction\n  | n+1 => \u27e8n, Nat.le_refl _\u27e9\n\ndef Fin.pred? : Fin n \u2192 Option (Fin n)\n| \u27e80, _\u27e9 => none\n| \u27e8i+1,h\u27e9 => some \u27e8i, Nat.le_of_succ_le h\u27e9\n\ndef Fin.succ? : {n : Nat} \u2192 Fin n \u2192 Option (Fin n)\n| 0, i => i.elim0\n| n+1, \u27e8i,_\u27e9 =>\n  if h : i < n\n  then some \u27e8i+1, Nat.succ_le_succ h\u27e9\n  else none\n\ndef Function.iterate (f : \u03b1 \u2192 \u03b1) : Nat \u2192 (\u03b1 \u2192 \u03b1)\n| 0 => id\n| n+1 => iterate f n \u2218 f\n\ndef Array.init (n : Nat) (f : Fin n \u2192 \u03b1) : Array \u03b1 := Id.run do\n  let mut A := Array.mkEmpty n\n  for h:i in [0:n] do\n    A := A.push (f \u27e8i,h.2\u27e9)\n  return A\n\ndef Array.initM [Monad m] (n : Nat) (f : Fin n \u2192 m \u03b1) : m (Array \u03b1) := do\n  let mut A := Array.mkEmpty n\n  for h:i in [0:n] do\n    A := A.push (\u2190 f \u27e8i,h.2\u27e9)\n  return A\n\ntheorem Array.init_zero : Array.init 0 f = #[] := by\n  simp [init, Id.run, forIn', Std.Range.forIn']\n  unfold Std.Range.forIn'.loop\n  simp\n\ntheorem Array.init_succ {f : Fin n.succ \u2192 \u03b1}\n  : Array.init n.succ f = (\n      Array.init n (fun i => f \u27e8i,Nat.lt_trans i.isLt (by exact Nat.le_refl _)\u27e9)\n    ).push (f \u27e8n, by exact Nat.le_refl _\u27e9)\n  := by\n  simp [init, Id.run, forIn', Std.Range.forIn']\n  suffices \u2200 i (hi : i \u2264 n) o (_ : o.size = n-i),\n    Std.Range.forIn'.loop (m := Id) 0 n.succ 1\n      (fun i h r => ForInStep.yield (push r (f \u27e8i, h.2\u27e9)))\n      i.succ (n-i)\n      (Nat.zero_le _)\n      o\n    = push (Std.Range.forIn'.loop (m := Id) 0 n 1\n      (fun i h r => ForInStep.yield (push r (f \u27e8i, Nat.le_step h.2\u27e9)))\n      i (n-i)\n      (Nat.zero_le _)\n      o) (f \u27e8n, Nat.lt_succ_self n\u27e9)\n    by\n    have := this n (Nat.le_refl _) #[] (by simp)\n    simp at this\n    exact this\n  intro i hi o ho\n  induction i generalizing o with\n  | zero =>\n    unfold Std.Range.forIn'.loop\n    unfold Std.Range.forIn'.loop\n    simp\n  | succ i ih =>\n    conv => lhs; unfold Std.Range.forIn'.loop\n    conv => rhs; unfold Std.Range.forIn'.loop\n    simp\n    have hn := Nat.sub_lt_of_pos_le _ _ (Nat.succ_pos _) hi\n    have hn' : n - Nat.succ i < Nat.succ n := Nat.le_step hn\n    simp [hn, hn']\n    have : n - Nat.succ i + 1 = n - i := by\n      simp [Nat.sub_succ]\n      rw [Nat.add_one, Nat.succ_pred_eq_of_pos (Nat.zero_lt_sub_of_lt hi)]\n    suffices \u2200 j, j = n - Nat.succ i + 1 \u2192\n      Std.Range.forIn'.loop (m := Id)  _ _ _ _ _ j (Nat.zero_le _) _\n      = push (Std.Range.forIn'.loop (m := Id) _ _ _ _ _ j (Nat.zero_le _) _) _\n      from this _ rfl\n    intro j hj\n    rw [this] at hj\n    cases hj\n    apply ih\n    exact Nat.le_of_lt hi\n    simp [ho, this]\n\n@[simp]\ntheorem Array.size_init : (Array.init n f).size = n := by\n  induction n\n  . simp [size, init_zero]\n  . next ih =>\n    simp [init_succ]; exact ih\n\n@[simp]\ntheorem Array.get_init {i : Nat} {h} : (Array.init n f)[i]'h = f \u27e8i, @size_init n _ f \u25b8 h\u27e9 := by\n  induction n generalizing i with\n  | zero => simp at h; exact False.elim <| Nat.not_lt_zero _ h\n  | succ n ih =>\n    simp [init_succ, get_push]\n    split\n    next h =>\n      have := @ih (fun i => f \u27e8i,Nat.lt_trans i.isLt (by exact Nat.le_refl _)\u27e9) i (by simp; assumption)\n      simp at this \u22a2\n      rw [this]\n    next h' =>\n      simp at h'\n      have : i = n := Nat.le_antisymm\n        (Nat.le_of_succ_le_succ (by rw [size_init] at h; exact h))\n        h'\n      cases this\n      congr\n\n\ndef List.distinct [DecidableEq \u03b1] (L : List \u03b1) : List \u03b1 :=\n  L.foldl (\u00b7.insert \u00b7) []\n\ndef List.isDistinct [BEq \u03b1] : List \u03b1 \u2192 Bool\n| [] => true\n| x::xs => !xs.contains x && xs.isDistinct\n\ndef List.fins (n : Nat) : List (Fin n) :=\n  finsAux n (Nat.le_refl _) []\nwhere\n  finsAux : (i : Nat) \u2192 i \u2264 n \u2192 List (Fin n) \u2192 List (Fin n)\n  | 0, _, acc => acc\n  | i+1, h, acc => finsAux i (Nat.le_of_lt h) (\u27e8i,h\u27e9 :: acc)\n\n\n/- Better parallelism primitive, that is actually like Scala's Future -/\ndef TaskIO (\u03b1) := IO (Task (Except IO.Error \u03b1))\n\nnamespace TaskIO\n\ninstance : Monad TaskIO where\n  pure a := pure (f := IO) <| Task.pure (Except.ok a)\n  bind a f := bind (m := IO) a (fun task =>\n    IO.bindTask task (fun res => do\n      let res \u2190 ofExcept res\n      f res))\n\ninstance : MonadLift IO TaskIO where\n  monadLift io := io.map (fun a => Task.pure (Except.ok a))\n\ndef wait (task : TaskIO \u03b1) : IO \u03b1 := do\n  let task \u2190 task\n  let x \u2190 IO.wait task\n  ofExcept x\n\ninstance : MonadExceptOf IO.Error TaskIO where\n  throw e := show IO _ from throw e\n  tryCatch a f := bind (m := IO) a (IO.bindTask \u00b7 fun\n    | .ok a => show TaskIO _ from pure a\n    | .error e => f e)\n\ndef par [ForIn IO \u03c3 \u03b1] (xs : \u03c3) (f : \u03b1 \u2192 TaskIO \u03b2)\n    : TaskIO (List \u03b2) := show IO _ from do\n  let mut tasks := #[]\n  for x in xs do\n    tasks := tasks.push (\u2190 (f x))\n  let task \u2190 IO.mapTasks (fun bs => do\n    return \u2190bs.mapM (fun b => ofExcept b)\n  ) tasks.toList\n  return task\n\ndef parUnit [ForIn IO \u03c3 \u03b1] (xs : \u03c3) (f : \u03b1 \u2192 TaskIO Unit)\n    : TaskIO Unit := do\n  let _allUnits \u2190 par xs f\n  return ()\n\ndef parTasks [ForIn IO \u03c3 \u03b1] (xs : \u03c3) (f : \u03b1 \u2192 IO \u03b2)\n    : TaskIO (List \u03b2) := do\n  par xs (fun a => liftM (n := IO) <| IO.asTask (do f a))\n\ndef parTasksUnit [ForIn IO \u03c3 \u03b1] (xs : \u03c3) (f : \u03b1 \u2192 IO Unit)\n    : TaskIO Unit := do\n  parUnit xs (fun a => liftM (n := IO) <| IO.asTask (do f a))\n\nend TaskIO\n\ndef Option.forIn [Monad m] (o : Option \u03b1) (b : \u03b2) (f : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) : m \u03b2 := do\n  match o with\n  | none => return b\n  | some a =>\n  match \u2190 f a b with\n  | .done b => return b\n  | .yield b => return b\n\ninstance : ForIn m (Option \u03b1) \u03b1 where\n  forIn := Option.forIn\n\ndef IO.timeMs (prog : IO \u03b1) : IO (Nat \u00d7 \u03b1) := do\n  let start \u2190 IO.monoMsNow\n  let res \u2190 prog\n  let end_ \u2190 IO.monoMsNow\n\n  return (end_ - start, res)\n\ninstance : GetElem String Nat Char (fun s i => i < s.length) where\n  getElem | xs, i, _ => xs.get (String.Pos.mk i)\n\ndef randFin (n) (_h : n > 0) : IO (Fin n) := do\n  let i \u2190 IO.rand 0 n.pred\n  if h : i < n then\n    return \u27e8i,h\u27e9\n  else\n    panic! s!\"failed to get random number {i} < {n}\"\n\n/- Generate a random permutation of the list.\nImplementation is quadratic in length of L. -/\ndef IO.randPerm (L : List \u03b1) : IO (List \u03b1) :=\n  randPermTR L [] 0\nwhere randPermTR (L acc n) := do\n  match L with\n  | [] => return acc\n  | x::xs =>\n    let idx \u2190 IO.rand 0 n\n    let acc' := acc.insertNth idx x\n    randPermTR xs acc' (n+1)\n\n\n@[simp]\ntheorem List.sizeOf_filter [SizeOf \u03b1] (f) (L : List \u03b1)\n  : sizeOf (List.filter f L) \u2264 sizeOf L\n  := by\n  induction L <;> simp [filter]\n  split\n  . simp\n    apply Nat.add_le_add_left\n    assumption\n  . simp\n    apply Nat.le_trans ?_ (Nat.le_add_left _ _)\n    assumption\n\ntheorem List.sizeOf_filter_lt_of_ne [SizeOf \u03b1] (f) (L : List \u03b1)\n    (h : List.filter f L \u2260 L)\n  : sizeOf (List.filter f L) < sizeOf L\n  := by\n  induction L <;> simp [filter] at *\n  next hd tl ih =>\n  split\n  next hHd =>\n    simp [hHd] at h\n    simp [_sizeOf_1]\n    apply Nat.add_lt_add_left\n    apply ih\n    assumption\n  next hHd =>\n    clear h hHd\n    apply Nat.lt_of_le_of_lt (sizeOf_filter _ _)\n    rw [Nat.add_comm, Nat.add_comm 1, Nat.add_one, Nat.add_succ]\n    apply Nat.succ_le_succ\n    apply Nat.le_add_right\n\n@[simp] theorem List.find?_map (p : \u03b2 \u2192 Bool) (f : \u03b1 \u2192 \u03b2) (L : List \u03b1)\n  : List.find? p (List.map f L) = Option.map f (List.find? (p \u2218 f) L)\n  := by induction L <;> simp; split <;> simp [*]\n\n@[simp]\ndef Std.AssocList.ofList : List (\u03b1 \u00d7 \u03b2) \u2192 Std.AssocList \u03b1 \u03b2\n| [] => .nil\n| (a,b)::tail => .cons a b (ofList tail)\n\n@[simp] theorem Std.AssocList.toList_ofList (L : List (\u03b1 \u00d7 \u03b2))\n  : toList (ofList L) = L\n  := by induction L <;> simp [*]\n\n@[simp]\ntheorem Std.HashMap.find?_ofList {B : BEq \u03b1} {H : Hashable \u03b1} (a : List (\u03b1 \u00d7 \u03b2)) (k : \u03b1)\n  : (@Std.HashMap.ofList _ B H _ a |>.find? k) = (Std.AssocList.ofList a |>.find? k)\n  := sorry\n\n@[inline]\ndef Option.expectSome (err : Unit \u2192 \u03b5) : Option \u03b1 \u2192 Except \u03b5 \u03b1\n| none => .error (err ())\n| some a => .ok a\n\nstructure NonemptyList (\u03b1) where\n  hd : \u03b1\n  tl : List \u03b1\n\n@[inline]\ndef List.expectNonempty (err : Unit \u2192 \u03b5) : List \u03b1 \u2192 Except \u03b5 (NonemptyList \u03b1)\n| [] => .error (err ())\n| hd::tl => .ok \u27e8hd,tl\u27e9\n\n\ndef PrinterM := StateM String\ndef PrinterM.putStr : String \u2192 PrinterM Unit :=\n  fun string state => ((), state.append string)\ndef PrinterM.run : PrinterM Unit \u2192 String := (StateT.run \u00b7 \"\" |>.2)\n\ninstance : Monad PrinterM := show Monad (StateM String) from inferInstance\n\ndef IO.FS.withTempFile (f : System.FilePath \u2192 IO \u03b1) : IO \u03b1 := do\n  let mut file := \".tmp\"\n  while \u2190 System.FilePath.pathExists file do\n    file := file ++ toString (\u2190 IO.rand 0 999999)\n\n  IO.FS.writeFile file \"\"\n  let res \u2190 f file\n  if \u2190 System.FilePath.pathExists file then\n    IO.FS.removeFile file\n\n  return res", "meta": {"author": "JamesGallicchio", "repo": "LeanSAT", "sha": "719470ac796a9149e0f892ccb3dff80c0dd563d3", "save_path": "github-repos/lean/JamesGallicchio-LeanSAT", "path": "github-repos/lean/JamesGallicchio-LeanSAT/LeanSAT-719470ac796a9149e0f892ccb3dff80c0dd563d3/LeanSAT/AuxDefs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.7057850340255386, "lm_q1q2_score": 0.38587960520152076}}
{"text": "import .sous_foncteur\nimport category_theory.elements\nimport .A2\nopen category_theory\nopen A2 \nopen int\n-- universes  v u\nlemma int_commute_with_morph  {A B :Type }[comm_ring A][comm_ring B](f : A \u2192  B)[is_ring_hom  f](n : \u2115) : \u2200 x : A, f( n*x ) = n *(f x) := \u03bb x,\n    nat.rec_on n (show  f(0*x) = 0* (f x), {rw [zero_mul,zero_mul], exact is_ring_hom.map_zero f, })\n        (assume n, assume rec_hyp : f(n * x) = n *(f x),\n            show f( (n+1)*x) = (n+1)*(f(x)),{\n                rw [right_distrib,right_distrib,one_mul,one_mul,\u2190 rec_hyp,is_ring_hom.map_add f],\n            })\nstructure A2_disc(R : Type )[comm_ring R] :=\n(\u03b6 : A2 R)\n(inv_disc : R)\n(certif_disc : (\u03b6.a * \u03b6.a - \u21914 * \u03b6.b ) * inv_disc = 1)\nnamespace A2_disc \nsection\nvariables {R : Type } [comm_ring R]\n--- id\u00e9e faire des lemmes simplificateurs ! \nlemma disc :  \u2200 {\u03b61 \u03b62 : A2_disc R}, \u03b61.\u03b6 = \u03b62.\u03b6 \u2192 ((\u03b61.\u03b6.a = \u03b62.\u03b6.a) \u2227 (\u03b61.\u03b6.b = \u03b62.\u03b6.b)) := \nbegin \n    intros \u03b61 \u03b62,\n    intro h, \n    split,\n    apply congr_arg, assumption,\n    apply congr_arg, assumption,\nend \nlemma disc_ :  \u2200 {\u03b61 \u03b62 : A2_disc R}, \u03b61.\u03b6 = \u03b62.\u03b6  \u2192  (\u03b61.\u03b6.a * \u03b61.\u03b6.a - \u21914 * \u03b61.\u03b6.b ) = (\u03b62.\u03b6.a * \u03b62.\u03b6.a - \u21914 * \u03b62.\u03b6.b ) := \nbegin \n    intros \u03b61 \u03b62,\n    intro h,\n    have H : ((\u03b61.\u03b6.a = \u03b62.\u03b6.a) \u2227 (\u03b61.\u03b6.b = \u03b62.\u03b6.b)),\n        apply disc,\n        assumption,\n    rw H.1,\n    rw H.2,\nend \n    lemma inverse_unique (a b c d: R) : a * c = 1 \u2192 d * b = 1 \u2192 a = d \u2192 c = b := \u03bb h1 h2 h3, begin  --- remettre dans l'ordre\n        have : c = (a * b) * c,\n            rw h3,\n            rw h2,\n            rw one_mul c,\n        rw this,\n        rw [mul_assoc, mul_comm b c,\u2190 mul_assoc,h1, one_mul],\n        end \n    @[ext] lemma ext :  \u2200 {\u03b61 \u03b62 : A2_disc R}, (\u03b61.\u03b6  = \u03b62.\u03b6)  \u2192 \u03b61 = \u03b62 := \u03bb \u03b61 \u03b62,\n        begin \n            intro h, \n            cases \u03b61, \n            cases \u03b62,\n            congr ; try { assumption },\n            apply inverse_unique,\n            exact \u03b61_certif_disc,\n            exact \u03b62_certif_disc,\n            exact  disc_ h,\n        end \nopen is_ring_hom\ndef map_A2_disc {A B :Type }[comm_ring A][comm_ring B](f : A \u2192  B)[is_ring_hom  f] : A2_disc A \u2192 A2_disc B := \u03bb \u03b7,  begin  \n    have  h : ((f \u03b7.\u03b6.a) * (f \u03b7.\u03b6.a)  - (  \u21914 * (f \u03b7.\u03b6.b ))) * (f \u03b7.inv_disc) = 1,\n            have j :    f( \u21914* \u03b7.\u03b6.b) = \u21914 * (f \u03b7.\u03b6.b ),\n                exact int_commute_with_morph f (4) (\u03b7.\u03b6.b),\n            rw [\u2190 j,\u2190 map_mul f,\u2190 map_sub f,\u2190 map_mul f,\u03b7.certif_disc],\n            exact map_one f,\n    exact { \u03b6 := {a := f \u03b7.\u03b6.a, b := f \u03b7.\u03b6.b},\n          inv_disc :=  f \u03b7.inv_disc,\n          certif_disc :=  h,},\n    end\nlemma map_comp_a {A B :Type }[comm_ring A][comm_ring B](f : A \u2192  B)[is_ring_hom  f] (\u03b6 : A2_disc (A) ) : (map_A2_disc f \u03b6).\u03b6.a = f \u03b6.\u03b6.a := rfl\nlemma map_comp_b {A B :Type }[comm_ring A][comm_ring B](f : A \u2192  B)[is_ring_hom  f] (\u03b6 : A2_disc (A) ) : (map_A2_disc f \u03b6).\u03b6.b = f \u03b6.\u03b6.b := rfl\nlemma map_comp_inv_disc {A B :Type }[comm_ring A][comm_ring B](f : A \u2192  B)[is_ring_hom  f] (\u03b6 : A2_disc (A) ) : (map_A2_disc f \u03b6).inv_disc = f \u03b6.inv_disc := rfl\ndef \ud835\udd382_disc: CommRing \u2964 Type  :=  \n{ obj := \u03bb R, A2_disc R,\n  map := \u03bb R R' f, map_A2_disc f, \n}\n\n-- def A :=   (functor.elements) (\ud835\udd382_disc)\nend \nend A2_disc\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/projet_A2/A2_disc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.38587959505100855}}
{"text": "\nimport Lib.Data.Array\nimport Lib.Data.Nat\nimport Lib.Data.DecSubtype\nimport Lib.Data.OrdMap\nimport Lib.Logic.Classical\n\nnamespace Nat\n\ntheorem lt_mul_iff_div_lt {x y k : Nat} (h : 1 \u2264 k) :\n  x < y*k \u2194 x / k < y := by\nrw [\u2190 Classical.not_iff_not, \u2190 le_iff_not_lt, mul_le_iff_le_div, le_iff_not_lt]\n <;> auto\n\nend Nat\n\ninductive InfArrayBuf (\u03b1 : Type u) : Nat \u2192 Type u where\n  | default : \u03b1 \u2192 n > 0 \u2192 InfArrayBuf \u03b1 n\n  | node : Buffer n \u03b1 \u2192 InfArrayBuf \u03b1 (n*2) \u2192 InfArrayBuf \u03b1 n\n\nnamespace InfArrayBuf\n\ntheorem is_pos (ar : InfArrayBuf \u03b1 n) : n > 0 := by\ninduction ar with\n| default x h => exact h\n| node buf ar ih =>\n  simp [Nat.lt_mul_iff_div_lt] at ih\n  auto\n\ndef mk (x : \u03b1) (h : n > 0) : InfArrayBuf \u03b1 n :=\ndefault x h\n\ndef get : InfArrayBuf \u03b1 n \u2192 Nat \u2192 \u03b1\n| default x _, _ => x\n| ar@(node buf ar'), i =>\n  if h : i < n then\n    buf.get \u27e8i, h\u27e9\n  else\n    get ar' (i - n)\ndef set : InfArrayBuf \u03b1 n \u2192 Nat \u2192 \u03b1 \u2192 InfArrayBuf \u03b1 n\n| default x h, i, y =>\n  let buf := Buffer.mkFilled x\n  have h'' : n*2 > 0 := by\n    simp [Nat.lt_mul_iff_div_lt]; auto\n  if h' : i < n then\n    node (buf.set \u27e8i, h'\u27e9 y) (default x h'')\n  else\n    have h\u2082 : 0 < i := Nat.lt_of_lt_of_le h (Nat.le_of_not_gt h')\n    have h\u2083 : i - n < i := Nat.sub_lt h\u2082 h\n    node buf (set (default x h'') (i - n) y)\n| ar@(node buf ar'), i, x =>\n  if h' : i < n then\n    node (buf.set \u27e8i, h'\u27e9 x) ar'\n  else\n    have h'' : n*2 > 0 := ar'.is_pos\n    have h   : n > 0 := ar.is_pos\n    have h\u2082 : 0 < i := Nat.lt_of_lt_of_le h (Nat.le_of_not_gt h')\n    have h\u2083 : i - n < i := Nat.sub_lt h\u2082 h\n    node buf (set ar' (i - n) x)\ntermination_by _ i _ => i\n\ndef size : InfArrayBuf \u03b1 n \u2192 Nat\n| default _ _ => 0\n| node _ ar => n + ar.size\n\ndef toList : InfArrayBuf \u03b1 n \u2192 Nat \u2192 List \u03b1\n| default x h, n => List.replicate n x\n| node buf ar, m =>\n  if m \u2264 n then buf.cells.toList.take m\n  else buf.cells.foldr (. :: .) (toList ar (m - n))\n\ndef dropHead : InfArrayBuf \u03b1 n \u2192 InfArrayBuf \u03b1 (n*2)\n| default x h => default x <| by simp [Nat.lt_mul_iff_div_lt]; auto\n| node buf ar => ar\n\nend InfArrayBuf\n\nstructure InfArray (\u03b1 : Type u) where\nmkImpl ::\n  bufSize : Nat\n  buffers : InfArrayBuf \u03b1 bufSize\n\nnamespace InfArray\n\ndef mkFilled (x : \u03b1) : InfArray \u03b1 where\n  bufSize := 10\n  buffers := InfArrayBuf.node (Buffer.mkFilled x)\n    (InfArrayBuf.default x <| by auto)\n\ndef size (ar : InfArray \u03b1) : Nat :=\nar.buffers.size\n\ndef get (ar : InfArray \u03b1) (i : Nat) : \u03b1 :=\nar.buffers.get i\n\ndef set (ar : InfArray \u03b1) (i : Nat) (x : \u03b1) : InfArray \u03b1 where\n  buffers := ar.buffers.set i x\n\ndef toList (ar : InfArray \u03b1) (n : Nat) : List \u03b1 :=\nar.buffers.toList n\n\ndef dropHead (ar : InfArray \u03b1) : InfArray \u03b1 where\n  buffers := ar.buffers.dropHead\n\nend InfArray\n\ndef Heap := OrdMap\n\nnamespace Heap\nopen Std.AssocList\nvariable [LT \u03b1]\n\ndef takeMin (h : Heap \u03b1 \u03b2) : Option (\u03b1 \u00d7 \u03b2 \u00d7 Heap \u03b1 \u03b2) :=\nmatch h' : h.vals with\n| nil => none\n| cons k v xs =>\n  let xs' :=\n    { vals := xs\n      sorted := by\n        have h'' := h.sorted\n        simp [h', keys] at h''\n        cases h''; assumption }\n  -- let z := h.insert\n  some (k, v, xs')\n\ndef nonEmpty (h : Heap \u03b1 \u03b2) : Bool :=\nmatch h.vals with\n| nil => false\n| cons k v xs => true\n\ndef empty : Heap \u03b1 \u03b2 where\n  vals := nil\n  sorted := by constructor\n\nend Heap\n\ndef NEHeap (\u03b1) [LT \u03b1] (\u03b2) :=\nDecSubtype (Heap.nonEmpty (\u03b1 := \u03b1) (\u03b2 := \u03b2))\n\nnamespace NEHeap\nopen Std.AssocList\nvariable [LT \u03b1]\n\ndef takeMin (h : NEHeap \u03b1 \u03b2) : \u03b1 \u00d7 \u03b2 \u00d7 Heap \u03b1 \u03b2 :=\nmatch h' : h with\n| \u27e8 \u27e8cons k v xs, hxs \u27e9, rfl \u27e9 =>\n  let xs' :=\n    { vals := xs\n      sorted := by\n        have h'' := h.val.sorted\n        simp [h', keys] at h''\n        cases h''; assumption }\n  (k, v, xs')\n\ndef head (h : NEHeap \u03b1 \u03b2) : \u03b1 :=\nmatch h' : h with\n| \u27e8 \u27e8cons k v xs, hxs \u27e9, rfl \u27e9 => k\n\nend NEHeap\n\n\nnamespace OrdMap\n\nvariable [LE k] [DecidableTotalOrder k]\nvariable (f : k \u2192 \u03b1 \u2192 \u03b1 \u2192 \u03b1)\n\ndef insertWith (x : k) (v : \u03b1) :\n  OrdMap k \u03b1 \u2192 OrdMap k \u03b1 :=\nunionWith f (singleton x v)\n\nend OrdMap\n\n\nnamespace Heap\nopen Std.AssocList\nvariable [LE \u03b1] [DecidableTotalOrder \u03b1]\nvariable (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2 \u2192 \u03b2)\n\ndef insertWith' (h : Heap \u03b1 \u03b2) (k : \u03b1) (v : \u03b2) : NEHeap \u03b1 \u03b2 where\n  val := h.insertWith f k v\n  property := by\n    cases h' : h.vals <;>\n    simp [OrdMap.insertWith, OrdMap.unionWith, OrdMap.unionWith, OrdMap.mergeWith] <;>\n    simp [OrdMap.singleton, h', zipWith, mapFilter, nonEmpty]\n    next k v k' v' tail =>\n      cases compare k k' <;> simp\n\nend Heap\n\nstructure SieveEntry where\nmkImpl ::\n  key : Nat\n  prime : Nat\n  coef : Nat\n  cached_key : key = prime * coef\n  le_coef : prime \u2264 coef\n\ninstance : ToString SieveEntry where\n  toString x := toString (x.prime, x.coef)\n\nnamespace SieveEntry\n\ndef mk (n : Nat) : SieveEntry where\n  key := n * n\n  prime := n\n  coef := n\n  le_coef := Nat.le_refl _\n  cached_key := rfl\n\ndef step' (s : SieveEntry) : SieveEntry where\n  key := s.key + s.prime + s.prime\n  prime := s.prime\n  coef := s.coef + 2\n  le_coef := by trans _; apply s.le_coef; auto\n  cached_key := by simp [Nat.mul_succ, s.cached_key]\n\ndef step (s : SieveEntry) : Nat \u00d7 SieveEntry :=\nlet s' := s.step'\n(s'.key, s')\n\nend SieveEntry\n\ndef List.nonEmpty : List \u03b1 \u2192 Bool\n| [] => false\n| _ :: _ => true\n\ndef NEList (\u03b1) := DecSubtype (List.nonEmpty (\u03b1 := \u03b1))\n\nnamespace NEList\n\ndef append : NEList \u03b1 \u2192 NEList \u03b1 \u2192 NEList \u03b1\n| \u27e8x :: xs, rfl\u27e9, \u27e8ys, _\u27e9 => \u27e8x :: (xs ++ ys), rfl\u27e9\n\ninstance : Append (NEList \u03b1) := \u27e8 append \u27e9\n\ninstance [Repr \u03b1] : Repr (NEList \u03b1) where\n  reprPrec x n := reprPrec x.1 n\n\ninstance [ToString \u03b1] : ToString (NEList \u03b1) where\n  toString x := toString x.1\n\nend NEList\n\nstructure SieveState where\n  cursor : NEHeap Nat (NEList SieveEntry)\n  -- offset : Nat\n  -- flags : InfArray Bool\n  next : Nat\n  primes : Array Nat\n\nnamespace SieveState\n\ndef pushPrime (n : Nat) (h : Heap Nat (NEList SieveEntry)) :\n  NEHeap Nat (NEList SieveEntry) :=\nlet e := SieveEntry.mk n\nh.insertWith' (\u03bb _ => (. ++ .)) e.key \u27e8 [e], rfl \u27e9\n\n-- def toList (s : SieveState) (n : Nat) : List (Nat \u00d7 Bool) :=\n-- s.flags.toList n |>.enumFrom s.offset |>.map <| Prod.map (. *2 +1) id\n\ndef init : SieveState where\n  cursor := pushPrime 3 Heap.empty\n  -- offset := 0\n  -- flags := InfArray.mkFilled true\n  next := 2 -- we consider only next * 2 + 1\n  primes := #[2,3]\n\ndef next' (s : SieveState) : Nat := s.next*2 + 1\n\n-- def index (s : SieveState) (c : Nat) : Nat :=\n-- (((c - 1) / 2) - s.offset)\n\ndef insert (h : Heap Nat (NEList SieveEntry)) (e : SieveEntry) :\n  NEHeap Nat (NEList SieveEntry) :=\nh.insertWith' (\u03bb _ => (. ++ .)) e.key \u27e8[e],rfl\u27e9\n\ndef steps (h : Heap Nat (NEList SieveEntry)) :\n  NEList SieveEntry \u2192\n  NEHeap Nat (NEList SieveEntry)\n| \u27e8e::es, rfl\u27e9 =>\n  let (c', e') := e.step\n  let h' := insert h e'\n  es.foldl (\u03bb h e => insert h.1 e.step') h'\n   -- match es with\n  -- | [] => h'\n  -- | ys@(e' :: es) =>\n    -- have : ys.length < xs.length := sorry\n    -- steps h'.val \u27e8e':: es, rfl \u27e9\n-- termination_by _ es => es.val.length\n\ndef step' (s : SieveState) : SieveState :=\nlet (c, es, cursor') := s.cursor.takeMin\n-- let (c', e') := e.step\n-- let i := s.index c\nlet cursor' := steps cursor' es\n-- let flags' := s.flags.set i false\n{ cursor := cursor'\n  -- offset := s.offset\n  -- flags := flags'\n  next := s.next\n  primes := s.primes\n}\n\n-- def condition (s : SieveState) : Bool \u00d7 Nat \u00d7 Nat :=\n-- ( s.next' < s.cursor.head \u2227 s.flags.get (s.index s.next),\n--  s.index s.next,\n--  s.next\n-- )\n\n-- def toList (s : SieveState) : List (Nat \u00d7 Bool) :=\n\ndef step (s : SieveState) : SieveState :=\n-- if s.offset + s.flags.bufSize \u2264 s.next then\n  -- { s with\n    -- offset := s.offset + s.flags.bufSize\n    -- flags := s.flags.dropHead }\nif s.next' < s.cursor.head then\n  let p := s.next'\n  -- let i := s.next - s.offset\n  -- let i := s'.next\n  let cursor' := pushPrime p s.cursor.1\n  let primes' := s.primes.push p\n  { s with\n           next := s.next + 1\n           cursor := cursor'\n           primes := primes' }\nelse if s.next' = s.cursor.head then\n  { s.step' with next := s.next + 1 }\nelse\n  s.step'\n\ndef test : IO Unit := do\n  let mut s := init\n  for i in [0:79] do\n    s := step s\n  print_vars![s.next', s.primes,\n              s.cursor.val.vals.toList]\n\n-- #eval test\n\nend SieveState\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/primes/Primes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3857021859979893}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.PostPort\n\nuniverses v u u_1 u\u2082 \n\nnamespace Mathlib\n\nnamespace category_theory.limits\n\n\n-- We don't need an analogue of `pair` (for binary products), `parallel_pair` (for equalizers),\n\n-- or `(co)span`, since we already have `discrete.functor`.\n\n/-- A fan over `f : \u03b2 \u2192 C` consists of a collection of maps from an object `P` to every `f b`. -/\n/-- A cofan over `f : \u03b2 \u2192 C` consists of a collection of maps from every `f b` to an object `P`. -/\ndef fan {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) := cone (discrete.functor f)\n\ndef cofan {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) := cocone (discrete.functor f)\n\n/-- A fan over `f : \u03b2 \u2192 C` consists of a collection of maps from an object `P` to every `f b`. -/\n@[simp] theorem fan.mk_X {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} (P : C)\n    (p : (b : \u03b2) \u2192 P \u27f6 f b) : cone.X (fan.mk P p) = P :=\n  Eq.refl (cone.X (fan.mk P p))\n\n/-- A cofan over `f : \u03b2 \u2192 C` consists of a collection of maps from every `f b` to an object `P`. -/\n@[simp] theorem cofan.mk_X {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} (P : C)\n    (p : (b : \u03b2) \u2192 f b \u27f6 P) : cocone.X (cofan.mk P p) = P :=\n  Eq.refl (cocone.X (cofan.mk P p))\n\n/-- An abbreviation for `has_limit (discrete.functor f)`. -/\ndef has_product {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) := has_limit (discrete.functor f)\n\n/-- An abbreviation for `has_colimit (discrete.functor f)`. -/\ndef has_coproduct {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) :=\n  has_colimit (discrete.functor f)\n\n/-- An abbreviation for `has_limits_of_shape (discrete f)`. -/\n/-- An abbreviation for `has_colimits_of_shape (discrete f)`. -/\ndef has_products_of_shape (\u03b2 : Type v) (C : Type u_1) [category C] :=\n  has_limits_of_shape (discrete \u03b2)\n\ndef has_coproducts_of_shape (\u03b2 : Type v) (C : Type u_1) [category C] :=\n  has_colimits_of_shape (discrete \u03b2)\n\n/-- `pi_obj f` computes the product of a family of elements `f`. (It is defined as an abbreviation\n   for `limit (discrete.functor f)`, so for most facts about `pi_obj f`, you will just use general facts\n   about limits.) -/\n/-- `sigma_obj f` computes the coproduct of a family of elements `f`. (It is defined as an abbreviation\ndef pi_obj {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) [has_product f] : C :=\n  limit (discrete.functor f)\n\n   for `colimit (discrete.functor f)`, so for most facts about `sigma_obj f`, you will just use general facts\n   about colimits.) -/\ndef sigma_obj {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) [has_coproduct f] : C :=\n  colimit (discrete.functor f)\n\nprefix:20 \"\u220f \" => Mathlib.category_theory.limits.pi_obj\n\nprefix:20 \"\u2210 \" => Mathlib.category_theory.limits.sigma_obj\n\n/-- The `b`-th projection from the pi object over `f` has the form `\u220f f \u27f6 f b`. -/\ndef pi.\u03c0 {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) [has_product f] (b : \u03b2) : \u220f f \u27f6 f b :=\n  limit.\u03c0 (discrete.functor f) b\n\n/-- The `b`-th inclusion into the sigma object over `f` has the form `f b \u27f6 \u2210 f`. -/\ndef sigma.\u03b9 {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) [has_coproduct f] (b : \u03b2) :\n    f b \u27f6 \u2210 f :=\n  colimit.\u03b9 (discrete.functor f) b\n\n/-- The fan constructed of the projections from the product is limiting. -/\ndef product_is_product {\u03b2 : Type v} {C : Type u} [category C] (f : \u03b2 \u2192 C) [has_product f] :\n    is_limit (fan.mk (\u220f f) (pi.\u03c0 f)) :=\n  is_limit.of_iso_limit (limit.is_limit (discrete.functor f))\n    (cones.ext (iso.refl (cone.X (limit.cone (discrete.functor fun (b : \u03b2) => f b)))) sorry)\n\n/-- A collection of morphisms `P \u27f6 f b` induces a morphism `P \u27f6 \u220f f`. -/\ndef pi.lift {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} [has_product f] {P : C}\n    (p : (b : \u03b2) \u2192 P \u27f6 f b) : P \u27f6 \u220f f :=\n  limit.lift (discrete.functor fun (b : \u03b2) => f b) (fan.mk P p)\n\n/-- A collection of morphisms `f b \u27f6 P` induces a morphism `\u2210 f \u27f6 P`. -/\ndef sigma.desc {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} [has_coproduct f] {P : C}\n    (p : (b : \u03b2) \u2192 f b \u27f6 P) : \u2210 f \u27f6 P :=\n  colimit.desc (discrete.functor fun (b : \u03b2) => f b) (cofan.mk P p)\n\n/--\nConstruct a morphism between categorical products (indexed by the same type)\nfrom a family of morphisms between the factors.\n-/\ndef pi.map {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} {g : \u03b2 \u2192 C} [has_product f]\n    [has_product g] (p : (b : \u03b2) \u2192 f b \u27f6 g b) : \u220f f \u27f6 \u220f g :=\n  lim_map (discrete.nat_trans p)\n\n/--\nConstruct an isomorphism between categorical products (indexed by the same type)\nfrom a family of isomorphisms between the factors.\n-/\ndef pi.map_iso {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} {g : \u03b2 \u2192 C}\n    [has_products_of_shape \u03b2 C] (p : (b : \u03b2) \u2192 f b \u2245 g b) : \u220f f \u2245 \u220f g :=\n  functor.map_iso lim (discrete.nat_iso p)\n\n/--\nConstruct a morphism between categorical coproducts (indexed by the same type)\nfrom a family of morphisms between the factors.\n-/\ndef sigma.map {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} {g : \u03b2 \u2192 C} [has_coproduct f]\n    [has_coproduct g] (p : (b : \u03b2) \u2192 f b \u27f6 g b) : \u2210 f \u27f6 \u2210 g :=\n  colim_map (discrete.nat_trans p)\n\n/--\nConstruct an isomorphism between categorical coproducts (indexed by the same type)\nfrom a family of isomorphisms between the factors.\n-/\ndef sigma.map_iso {\u03b2 : Type v} {C : Type u} [category C] {f : \u03b2 \u2192 C} {g : \u03b2 \u2192 C}\n    [has_coproducts_of_shape \u03b2 C] (p : (b : \u03b2) \u2192 f b \u2245 g b) : \u2210 f \u2245 \u2210 g :=\n  functor.map_iso colim (discrete.nat_iso p)\n\n-- TODO: show this is an iso iff G preserves the product of f.\n\n/-- The comparison morphism for the product of `f`. -/\ndef pi_comparison {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D)\n    (f : \u03b2 \u2192 C) [has_product f] [has_product fun (b : \u03b2) => functor.obj G (f b)] :\n    functor.obj G (\u220f f) \u27f6 \u220f fun (b : \u03b2) => functor.obj G (f b) :=\n  pi.lift fun (b : \u03b2) => functor.map G (pi.\u03c0 f b)\n\n@[simp] theorem pi_comparison_comp_\u03c0_assoc {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082}\n    [category D] (G : C \u2964 D) (f : \u03b2 \u2192 C) [has_product f]\n    [has_product fun (b : \u03b2) => functor.obj G (f b)] (b : \u03b2) {X' : D}\n    (f' : functor.obj G (f b) \u27f6 X') :\n    pi_comparison G f \u226b pi.\u03c0 (fun (b : \u03b2) => functor.obj G (f b)) b \u226b f' =\n        functor.map G (pi.\u03c0 f b) \u226b f' :=\n  sorry\n\n@[simp] theorem map_lift_pi_comparison_assoc {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082}\n    [category D] (G : C \u2964 D) (f : \u03b2 \u2192 C) [has_product f]\n    [has_product fun (b : \u03b2) => functor.obj G (f b)] (P : C) (g : (j : \u03b2) \u2192 P \u27f6 f j) {X' : D}\n    (f' : (\u220f fun (b : \u03b2) => functor.obj G (f b)) \u27f6 X') :\n    functor.map G (pi.lift g) \u226b pi_comparison G f \u226b f' =\n        (pi.lift fun (j : \u03b2) => functor.map G (g j)) \u226b f' :=\n  sorry\n\n-- TODO: show this is an iso iff G preserves the coproduct of f.\n\n/-- The comparison morphism for the coproduct of `f`. -/\ndef sigma_comparison {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D)\n    (f : \u03b2 \u2192 C) [has_coproduct f] [has_coproduct fun (b : \u03b2) => functor.obj G (f b)] :\n    (\u2210 fun (b : \u03b2) => functor.obj G (f b)) \u27f6 functor.obj G (\u2210 f) :=\n  sigma.desc fun (b : \u03b2) => functor.map G (sigma.\u03b9 f b)\n\n@[simp] theorem \u03b9_comp_sigma_comparison_assoc {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082}\n    [category D] (G : C \u2964 D) (f : \u03b2 \u2192 C) [has_coproduct f]\n    [has_coproduct fun (b : \u03b2) => functor.obj G (f b)] (b : \u03b2) {X' : D}\n    (f' : functor.obj G (\u2210 f) \u27f6 X') :\n    sigma.\u03b9 (fun (b : \u03b2) => functor.obj G (f b)) b \u226b sigma_comparison G f \u226b f' =\n        functor.map G (sigma.\u03b9 f b) \u226b f' :=\n  sorry\n\n@[simp] theorem sigma_comparison_map_desc {\u03b2 : Type v} {C : Type u} [category C] {D : Type u\u2082}\n    [category D] (G : C \u2964 D) (f : \u03b2 \u2192 C) [has_coproduct f]\n    [has_coproduct fun (b : \u03b2) => functor.obj G (f b)] (P : C) (g : (j : \u03b2) \u2192 f j \u27f6 P) :\n    sigma_comparison G f \u226b functor.map G (sigma.desc g) =\n        sigma.desc fun (j : \u03b2) => functor.map G (g j) :=\n  sorry\n\n/-- An abbreviation for `\u03a0 J, has_limits_of_shape (discrete J) C` -/\n/-- An abbreviation for `\u03a0 J, has_colimits_of_shape (discrete J) C` -/\ndef has_products (C : Type u) [category C] := \u2200 (J : Type v), has_limits_of_shape (discrete J) C\n\ndef has_coproducts (C : Type u) [category C] := \u2200 (J : Type v), has_colimits_of_shape (discrete J) C\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/products_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.672331705744791, "lm_q1q2_score": 0.385702182231501}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Shing Tak Lam, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.int.modeq\nimport Mathlib.tactic.interval_cases\nimport Mathlib.tactic.linarith.default\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\n/-!\n# Digits of a natural number\n\nThis provides a basic API for extracting the digits of a natural number in a given base,\nand reconstructing numbers from their digits.\n\nWe also prove some divisibility tests based on digits, in particular completing\nTheorem #85 from https://www.cs.ru.nl/~freek/100/.\n\nA basic `norm_digits` tactic is also provided for proving goals of the form\n`nat.digits a b = l` where `a` and `b` are numerals.\n-/\n\nnamespace nat\n\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux_0 : \u2115 \u2192 List \u2115 := sorry\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux_1 (n : \u2115) : List \u2115 := list.repeat 1 n\n\n/-- (Impl.) An auxiliary definition for `digits`, to help get the desired definitional unfolding. -/\ndef digits_aux (b : \u2115) (h : bit0 1 \u2264 b) : \u2115 \u2192 List \u2115 := sorry\n\n@[simp] theorem digits_aux_zero (b : \u2115) (h : bit0 1 \u2264 b) : digits_aux b h 0 = [] := rfl\n\ntheorem digits_aux_def (b : \u2115) (h : bit0 1 \u2264 b) (n : \u2115) (w : 0 < n) :\n    digits_aux b h n = n % b :: digits_aux b h (n / b) :=\n  sorry\n\n/--\n`digits b n` gives the digits, in little-endian order,\nof a natural number `n` in a specified base `b`.\n\nIn any base, we have `of_digits b L = L.foldr (\u03bb x y, x + b * y) 0`.\n* For any `2 \u2264 b`, we have `l < b` for any `l \u2208 digits b n`,\n  and the last digit is not zero.\n  This uniquely specifies the behaviour of `digits b`.\n* For `b = 1`, we define `digits 1 n = list.repeat 1 n`.\n* For `b = 0`, we define `digits 0 n = [n]`, except `digits 0 0 = []`.\n\nNote this differs from the existing `nat.to_digits` in core, which is used for printing numerals.\nIn particular, `nat.to_digits b 0 = [0]`, while `digits b 0 = []`.\n-/\ndef digits : \u2115 \u2192 \u2115 \u2192 List \u2115 := sorry\n\n@[simp] theorem digits_zero (b : \u2115) : digits b 0 = [] :=\n  nat.cases_on b (Eq.refl (digits 0 0))\n    fun (b : \u2115) =>\n      nat.cases_on b (Eq.refl (digits 1 0))\n        fun (b : \u2115) => Eq.refl (digits (Nat.succ (Nat.succ b)) 0)\n\n@[simp] theorem digits_zero_zero : digits 0 0 = [] := rfl\n\n@[simp] theorem digits_zero_succ (n : \u2115) : digits 0 (Nat.succ n) = [n + 1] := rfl\n\ntheorem digits_zero_succ' {n : \u2115} (w : 0 < n) : digits 0 n = [n] :=\n  nat.cases_on n (fun (w : 0 < 0) => idRhs (digits 0 0 = [0]) (absurd w (of_as_true trivial)))\n    (fun (n : \u2115) (w : 0 < Nat.succ n) => idRhs (digits 0 (n + 1) = digits 0 (n + 1)) rfl) w\n\n@[simp] theorem digits_one (n : \u2115) : digits 1 n = list.repeat 1 n := rfl\n\n@[simp] theorem digits_one_succ (n : \u2115) : digits 1 (n + 1) = 1 :: digits 1 n := rfl\n\n@[simp] theorem digits_add_two_add_one (b : \u2115) (n : \u2115) :\n    digits (b + bit0 1) (n + 1) =\n        (n + 1) % (b + bit0 1) :: digits (b + bit0 1) ((n + 1) / (b + bit0 1)) :=\n  rfl\n\ntheorem digits_def' {b : \u2115} (h : bit0 1 \u2264 b) {n : \u2115} (w : 0 < n) :\n    digits b n = n % b :: digits b (n / b) :=\n  sorry\n\n@[simp] theorem digits_of_lt (b : \u2115) (x : \u2115) (w\u2081 : 0 < x) (w\u2082 : x < b) : digits b x = [x] := sorry\n\ntheorem digits_add (b : \u2115) (h : bit0 1 \u2264 b) (x : \u2115) (y : \u2115) (w : x < b) (w' : 0 < x \u2228 0 < y) :\n    digits b (x + b * y) = x :: digits b y :=\n  sorry\n\n/--\n`of_digits b L` takes a list `L` of natural numbers, and interprets them\nas a number in semiring, as the little-endian digits in base `b`.\n-/\n-- If we had a function converting a list into a polynomial,\n\n-- and appropriate lemmas about that function,\n\n-- we could rewrite this in terms of that.\n\ndef of_digits {\u03b1 : Type u_1} [semiring \u03b1] (b : \u03b1) : List \u2115 \u2192 \u03b1 := sorry\n\ntheorem of_digits_eq_foldr {\u03b1 : Type u_1} [semiring \u03b1] (b : \u03b1) (L : List \u2115) :\n    of_digits b L = list.foldr (fun (x : \u2115) (y : \u03b1) => \u2191x + b * y) 0 L :=\n  sorry\n\n@[simp] theorem of_digits_singleton {b : \u2115} {n : \u2115} : of_digits b [n] = n := sorry\n\n@[simp] theorem of_digits_one_cons {\u03b1 : Type u_1} [semiring \u03b1] (h : \u2115) (L : List \u2115) :\n    of_digits 1 (h :: L) = \u2191h + of_digits 1 L :=\n  sorry\n\ntheorem of_digits_append {b : \u2115} {l1 : List \u2115} {l2 : List \u2115} :\n    of_digits b (l1 ++ l2) = of_digits b l1 + b ^ list.length l1 * of_digits b l2 :=\n  sorry\n\ntheorem coe_of_digits (\u03b1 : Type u_1) [semiring \u03b1] (b : \u2115) (L : List \u2115) :\n    \u2191(of_digits b L) = of_digits (\u2191b) L :=\n  sorry\n\ntheorem coe_int_of_digits (b : \u2115) (L : List \u2115) : \u2191(of_digits b L) = of_digits (\u2191b) L := sorry\n\ntheorem digits_zero_of_eq_zero {b : \u2115} (h : 1 \u2264 b) {L : List \u2115} (w : of_digits b L = 0) (l : \u2115)\n    (H : l \u2208 L) : l = 0 :=\n  sorry\n\ntheorem digits_of_digits (b : \u2115) (h : bit0 1 \u2264 b) (L : List \u2115) (w\u2081 : \u2200 (l : \u2115), l \u2208 L \u2192 l < b)\n    (w\u2082 : \u2200 (h : L \u2260 []), list.last L h \u2260 0) : digits b (of_digits b L) = L :=\n  sorry\n\ntheorem of_digits_digits (b : \u2115) (n : \u2115) : of_digits b (digits b n) = n := sorry\n\ntheorem of_digits_one (L : List \u2115) : of_digits 1 L = list.sum L := sorry\n\n/-!\n### Properties\n\nThis section contains various lemmas of properties relating to `digits` and `of_digits`.\n-/\n\ntheorem digits_eq_nil_iff_eq_zero {b : \u2115} {n : \u2115} : digits b n = [] \u2194 n = 0 := sorry\n\ntheorem digits_ne_nil_iff_ne_zero {b : \u2115} {n : \u2115} : digits b n \u2260 [] \u2194 n \u2260 0 :=\n  not_congr digits_eq_nil_iff_eq_zero\n\ntheorem digits_last {b : \u2115} {m : \u2115} (h : bit0 1 \u2264 b) (hm : 0 < m) (p : digits b m \u2260 [])\n    (q : digits b (m / b) \u2260 []) : list.last (digits b m) p = list.last (digits b (m / b)) q :=\n  sorry\n\ntheorem last_digit_ne_zero (b : \u2115) {m : \u2115} (hm : m \u2260 0) :\n    list.last (digits b m) (iff.mpr digits_ne_nil_iff_ne_zero hm) \u2260 0 :=\n  sorry\n\n/-- The digits in the base b+2 expansion of n are all less than b+2 -/\ntheorem digits_lt_base' {b : \u2115} {m : \u2115} {d : \u2115} : d \u2208 digits (b + bit0 1) m \u2192 d < b + bit0 1 :=\n  sorry\n\n/-- The digits in the base b expansion of n are all less than b, if b \u2265 2 -/\ntheorem digits_lt_base {b : \u2115} {m : \u2115} {d : \u2115} (hb : bit0 1 \u2264 b) (hd : d \u2208 digits b m) : d < b :=\n  sorry\n\n/-- an n-digit number in base b + 2 is less than (b + 2)^n -/\ntheorem of_digits_lt_base_pow_length' {b : \u2115} {l : List \u2115}\n    (hl : \u2200 (x : \u2115), x \u2208 l \u2192 x < b + bit0 1) :\n    of_digits (b + bit0 1) l < (b + bit0 1) ^ list.length l :=\n  sorry\n\n/-- an n-digit number in base b is less than b^n if b \u2265 2 -/\ntheorem of_digits_lt_base_pow_length {b : \u2115} {l : List \u2115} (hb : bit0 1 \u2264 b)\n    (hl : \u2200 (x : \u2115), x \u2208 l \u2192 x < b) : of_digits b l < b ^ list.length l :=\n  sorry\n\n/-- Any number m is less than (b+2)^(number of digits in the base b + 2 representation of m) -/\ntheorem lt_base_pow_length_digits' {b : \u2115} {m : \u2115} :\n    m < (b + bit0 1) ^ list.length (digits (b + bit0 1) m) :=\n  sorry\n\n/-- Any number m is less than b^(number of digits in the base b representation of m) -/\ntheorem lt_base_pow_length_digits {b : \u2115} {m : \u2115} (hb : bit0 1 \u2264 b) :\n    m < b ^ list.length (digits b m) :=\n  sorry\n\ntheorem of_digits_digits_append_digits {b : \u2115} {m : \u2115} {n : \u2115} :\n    of_digits b (digits b n ++ digits b m) = n + b ^ list.length (digits b n) * m :=\n  sorry\n\ntheorem digits_len_le_digits_len_succ (b : \u2115) (n : \u2115) :\n    list.length (digits b n) \u2264 list.length (digits b (n + 1)) :=\n  sorry\n\ntheorem le_digits_len_le (b : \u2115) (n : \u2115) (m : \u2115) (h : n \u2264 m) :\n    list.length (digits b n) \u2264 list.length (digits b m) :=\n  monotone_of_monotone_nat (digits_len_le_digits_len_succ b) h\n\ntheorem pow_length_le_mul_of_digits {b : \u2115} {l : List \u2115} (hl : l \u2260 []) (hl2 : list.last l hl \u2260 0) :\n    (b + bit0 1) ^ list.length l \u2264 (b + bit0 1) * of_digits (b + bit0 1) l :=\n  sorry\n\n/--\nAny non-zero natural number `m` is greater than\n(b+2)^((number of digits in the base (b+2) representation of m) - 1)\n-/\ntheorem base_pow_length_digits_le' (b : \u2115) (m : \u2115) (hm : m \u2260 0) :\n    (b + bit0 1) ^ list.length (digits (b + bit0 1) m) \u2264 (b + bit0 1) * m :=\n  sorry\n\n/--\nAny non-zero natural number `m` is greater than\nb^((number of digits in the base b representation of m) - 1)\n-/\ntheorem base_pow_length_digits_le (b : \u2115) (m : \u2115) (hb : bit0 1 \u2264 b) :\n    m \u2260 0 \u2192 b ^ list.length (digits b m) \u2264 b * m :=\n  sorry\n\n/-! ### Modular Arithmetic -/\n\n-- This is really a theorem about polynomials.\n\ntheorem dvd_of_digits_sub_of_digits {\u03b1 : Type u_1} [comm_ring \u03b1] {a : \u03b1} {b : \u03b1} {k : \u03b1}\n    (h : k \u2223 a - b) (L : List \u2115) : k \u2223 of_digits a L - of_digits b L :=\n  sorry\n\ntheorem of_digits_modeq' (b : \u2115) (b' : \u2115) (k : \u2115) (h : modeq k b b') (L : List \u2115) :\n    modeq k (of_digits b L) (of_digits b' L) :=\n  sorry\n\ntheorem of_digits_modeq (b : \u2115) (k : \u2115) (L : List \u2115) :\n    modeq k (of_digits b L) (of_digits (b % k) L) :=\n  of_digits_modeq' b (b % k) k (modeq.symm (modeq.mod_modeq b k)) L\n\ntheorem of_digits_mod (b : \u2115) (k : \u2115) (L : List \u2115) : of_digits b L % k = of_digits (b % k) L % k :=\n  of_digits_modeq b k L\n\ntheorem of_digits_zmodeq' (b : \u2124) (b' : \u2124) (k : \u2115) (h : int.modeq (\u2191k) b b') (L : List \u2115) :\n    int.modeq (\u2191k) (of_digits b L) (of_digits b' L) :=\n  sorry\n\ntheorem of_digits_zmodeq (b : \u2124) (k : \u2115) (L : List \u2115) :\n    int.modeq (\u2191k) (of_digits b L) (of_digits (b % \u2191k) L) :=\n  of_digits_zmodeq' b (b % \u2191k) k (int.modeq.symm (int.modeq.mod_modeq b \u2191k)) L\n\ntheorem of_digits_zmod (b : \u2124) (k : \u2115) (L : List \u2115) :\n    of_digits b L % \u2191k = of_digits (b % \u2191k) L % \u2191k :=\n  of_digits_zmodeq b k L\n\ntheorem modeq_digits_sum (b : \u2115) (b' : \u2115) (h : b' % b = 1) (n : \u2115) :\n    modeq b n (list.sum (digits b' n)) :=\n  sorry\n\ntheorem modeq_three_digits_sum (n : \u2115) :\n    modeq (bit1 1) n (list.sum (digits (bit0 (bit1 (bit0 1))) n)) :=\n  sorry\n\ntheorem modeq_nine_digits_sum (n : \u2115) :\n    modeq (bit1 (bit0 (bit0 1))) n (list.sum (digits (bit0 (bit1 (bit0 1))) n)) :=\n  sorry\n\ntheorem zmodeq_of_digits_digits (b : \u2115) (b' : \u2115) (c : \u2124) (h : int.modeq (\u2191b) (\u2191b') c) (n : \u2115) :\n    int.modeq (\u2191b) (\u2191n) (of_digits c (digits b' n)) :=\n  sorry\n\ntheorem of_digits_neg_one (L : List \u2115) :\n    of_digits (-1) L = list.alternating_sum (list.map (fun (n : \u2115) => \u2191n) L) :=\n  sorry\n\ntheorem modeq_eleven_digits_sum (n : \u2115) :\n    int.modeq (bit1 (bit1 (bit0 1))) (\u2191n)\n        (list.alternating_sum (list.map (fun (n : \u2115) => \u2191n) (digits (bit0 (bit1 (bit0 1))) n))) :=\n  sorry\n\n/-! ## Divisibility  -/\n\ntheorem dvd_iff_dvd_digits_sum (b : \u2115) (b' : \u2115) (h : b' % b = 1) (n : \u2115) :\n    b \u2223 n \u2194 b \u2223 list.sum (digits b' n) :=\n  sorry\n\ntheorem three_dvd_iff (n : \u2115) : bit1 1 \u2223 n \u2194 bit1 1 \u2223 list.sum (digits (bit0 (bit1 (bit0 1))) n) :=\n  sorry\n\ntheorem nine_dvd_iff (n : \u2115) :\n    bit1 (bit0 (bit0 1)) \u2223 n \u2194 bit1 (bit0 (bit0 1)) \u2223 list.sum (digits (bit0 (bit1 (bit0 1))) n) :=\n  sorry\n\ntheorem dvd_iff_dvd_of_digits (b : \u2115) (b' : \u2115) (c : \u2124) (h : \u2191b \u2223 \u2191b' - c) (n : \u2115) :\n    b \u2223 n \u2194 \u2191b \u2223 of_digits c (digits b' n) :=\n  sorry\n\ntheorem eleven_dvd_iff (n : \u2115) :\n    bit1 (bit1 (bit0 1)) \u2223 n \u2194\n        bit1 (bit1 (bit0 1)) \u2223\n          list.alternating_sum (list.map (fun (n : \u2115) => \u2191n) (digits (bit0 (bit1 (bit0 1))) n)) :=\n  sorry\n\n/-! ### `norm_digits` tactic -/\n\nnamespace norm_digits\n\n\ntheorem digits_succ (b : \u2115) (n : \u2115) (m : \u2115) (r : \u2115) (l : List \u2115) (e : r + b * m = n) (hr : r < b)\n    (h : digits b m = l \u2227 bit0 1 \u2264 b \u2227 0 < m) : digits b n = r :: l \u2227 bit0 1 \u2264 b \u2227 0 < n :=\n  sorry\n\ntheorem digits_one (b : \u2115) (n : \u2115) (n0 : 0 < n) (nb : n < b) :\n    digits b n = [n] \u2227 bit0 1 \u2264 b \u2227 0 < n :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/nat/digits_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.38570218223150093}}
{"text": "/-\nCopyright (c) 2022 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n\n! This file was ported from Lean 3 source module topology.metric_space.isometric_smul\n! leanprover-community/mathlib commit 832a8ba8f10f11fea99367c469ff802e69a5b8ec\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Topology.MetricSpace.Isometry\n\n/-!\n# Group actions by isometries\n\nIn this file we define two typeclasses:\n\n- `IsometricSMul M X` says that `M` multiplicatively acts on a (pseudo extended) metric space\n  `X` by isometries;\n- `IsometricVAdd` is an additive version of `IsometricSMul`.\n\nWe also prove basic facts about isometric actions and define bundled isometries\n`IsometryEquiv.constSMul`, `IsometryEquiv.mulLeft`, `IsometryEquiv.mulRight`,\n`IsometryEquiv.divLeft`, `IsometryEquiv.divRight`, and `IsometryEquiv.inv`, as well as their\nadditive versions.\n\nIf `G` is a group, then `IsometricSMul G G` means that `G` has a left-invariant metric while\n`IsometricSMul G\u1d50\u1d52\u1d56 G` means that `G` has a right-invariant metric. For a commutative group,\nthese two notions are equivalent. A group with a right-invariant metric can be also represented as a\n`NormedGroup`.\n-/\n\n\nopen Set\n\nopen ENNReal Pointwise\n\nuniverse u v w\n\nvariable (M : Type u) (G : Type v) (X : Type w)\n\n/-- An additive action is isometric if each map `x \u21a6 c +\u1d65 x` is an isometry. -/\nclass IsometricVAdd [PseudoEMetricSpace X] [VAdd M X] : Prop where\n  protected isometry_vadd : \u2200 c : M, Isometry ((c +\u1d65 \u00b7) : X \u2192 X)\n#align has_isometric_vadd IsometricVAdd\n\n/-- A multiplicative action is isometric if each map `x \u21a6 c \u2022 x` is an isometry. -/\n@[to_additive]\nclass IsometricSMul [PseudoEMetricSpace X] [SMul M X] : Prop where\n  protected isometry_smul : \u2200 c : M, Isometry ((c \u2022 \u00b7) : X \u2192 X)\n#align has_isometric_smul IsometricSMul\n\n-- Porting note: Lean 4 doesn' support `[]` in classes, so make a lemma instead of `export`ing\n@[to_additive]\ntheorem isometry_smul {M : Type u} (X : Type w) [PseudoEMetricSpace X] [SMul M X]\n    [IsometricSMul M X] (c : M) : Isometry (c \u2022 \u00b7 : X \u2192 X) :=\n  IsometricSMul.isometry_smul c\n\n@[to_additive]\ninstance (priority := 100) IsometricSMul.to_continuousConstSMul [PseudoEMetricSpace X] [SMul M X]\n    [IsometricSMul M X] : ContinuousConstSMul M X :=\n  \u27e8fun c => (isometry_smul X c).continuous\u27e9\n#align has_isometric_smul.to_has_continuous_const_smul IsometricSMul.to_continuousConstSMul\n#align has_isometric_vadd.to_has_continuous_const_vadd IsometricVAdd.to_continuousConstVAdd\n\n@[to_additive]\ninstance (priority := 100) IsometricSMul.opposite_of_comm [PseudoEMetricSpace X] [SMul M X]\n    [SMul M\u1d50\u1d52\u1d56 X] [IsCentralScalar M X] [IsometricSMul M X] : IsometricSMul M\u1d50\u1d52\u1d56 X :=\n  \u27e8fun c x y => by simpa only [\u2190 op_smul_eq_smul] using isometry_smul X c.unop x y\u27e9\n#align has_isometric_smul.opposite_of_comm IsometricSMul.opposite_of_comm\n#align has_isometric_vadd.opposite_of_comm IsometricVAdd.opposite_of_comm\n\nvariable {M G X}\n\nsection EMetric\n\nvariable [PseudoEMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X]\n\n@[to_additive (attr := simp)]\ntheorem edist_smul_left [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :\n    edist (c \u2022 x) (c \u2022 y) = edist x y :=\n  isometry_smul X c x y\n#align edist_smul_left edist_smul_left\n#align edist_vadd_left edist_vadd_left\n\n@[to_additive]\ntheorem isometry_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a : M) :\n    Isometry ((\u00b7 * \u00b7) a) :=\n  isometry_smul M a\n#align isometry_mul_left isometry_mul_left\n#align isometry_add_left isometry_add_left\n\n@[to_additive (attr := simp)]\ntheorem edist_mul_left [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] (a b c : M) :\n    edist (a * b) (a * c) = edist b c :=\n  isometry_mul_left a b c\n#align edist_mul_left edist_mul_left\n#align edist_add_left edist_add_left\n\n@[to_additive]\ntheorem isometry_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M] (a : M) :\n    Isometry fun x => x * a :=\n  isometry_smul M (MulOpposite.op a)\n#align isometry_mul_right isometry_mul_right\n#align isometry_add_right isometry_add_right\n\n@[to_additive (attr := simp)]\ntheorem edist_mul_right [Mul M] [PseudoEMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M] (a b c : M) :\n    edist (a * c) (b * c) = edist a b :=\n  isometry_mul_right c a b\n#align edist_mul_right edist_mul_right\n#align edist_add_right edist_add_right\n\n@[to_additive (attr := simp)]\ntheorem edist_div_right [DivInvMonoid M] [PseudoEMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M]\n    (a b c : M) : edist (a / c) (b / c) = edist a b := by\n  simp only [div_eq_mul_inv, edist_mul_right]\n#align edist_div_right edist_div_right\n#align edist_sub_right edist_sub_right\n\n@[to_additive (attr := simp)]\ntheorem edist_inv_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul G\u1d50\u1d52\u1d56 G]\n    (a b : G) : edist a\u207b\u00b9 b\u207b\u00b9 = edist a b := by\n  rw [\u2190 edist_mul_left a, \u2190 edist_mul_right _ _ b, mul_right_inv, one_mul, inv_mul_cancel_right,\n    edist_comm]\n#align edist_inv_inv edist_inv_inv\n#align edist_neg_neg edist_neg_neg\n\n@[to_additive]\ntheorem isometry_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul G\u1d50\u1d52\u1d56 G] :\n    Isometry (Inv.inv : G \u2192 G) :=\n  edist_inv_inv\n#align isometry_inv isometry_inv\n#align isometry_neg isometry_neg\n\n@[to_additive]\ntheorem edist_inv [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul G\u1d50\u1d52\u1d56 G]\n    (x y : G) : edist x\u207b\u00b9 y = edist x y\u207b\u00b9 := by rw [\u2190 edist_inv_inv, inv_inv]\n#align edist_inv edist_inv\n#align edist_neg edist_neg\n\n@[to_additive (attr := simp)]\ntheorem edist_div_left [PseudoEMetricSpace G] [IsometricSMul G G] [IsometricSMul G\u1d50\u1d52\u1d56 G]\n    (a b c : G) : edist (a / b) (a / c) = edist b c := by\n  rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]\n#align edist_div_left edist_div_left\n#align edist_sub_left edist_sub_left\n\nnamespace IsometryEquiv\n\n/-- If a group `G` acts on `X` by isometries, then `IsometryEquiv.constSMul` is the isometry of\n`X` given by multiplication of a constant element of the group. -/\n@[to_additive (attr := simps! toEquiv apply) \"If an additive group `G` acts on `X` by isometries,\nthen `IsometryEquiv.constVAdd` is the isometry of `X` given by addition of a constant element of the\ngroup.\"]\ndef constSMul (c : G) : X \u2243\u1d62 X where\n  toEquiv := MulAction.toPerm c\n  isometry_toFun := isometry_smul X c\n#align isometry_equiv.const_smul IsometryEquiv.constSMul\n#align isometry_equiv.const_vadd IsometryEquiv.constVAdd\n#align isometry_equiv.const_smul_to_equiv IsometryEquiv.constSMul_toEquiv\n#align isometry_equiv.const_smul_apply IsometryEquiv.constSMul_apply\n#align isometry_equiv.const_vadd_to_equiv IsometryEquiv.constVAdd_toEquiv\n#align isometry_equiv.const_vadd_apply IsometryEquiv.constVAdd_apply\n\n@[to_additive (attr := simp)]\ntheorem constSMul_symm (c : G) : (constSMul c : X \u2243\u1d62 X).symm = constSMul c\u207b\u00b9 :=\n  ext fun _ => rfl\n#align isometry_equiv.const_smul_symm IsometryEquiv.constSMul_symm\n#align isometry_equiv.const_vadd_symm IsometryEquiv.constVAdd_symm\n\nvariable [PseudoEMetricSpace G]\n\n/-- Multiplication `y \u21a6 x * y` as an `IsometryEquiv`. -/\n@[to_additive (attr := simps! apply toEquiv) \"Addition `y \u21a6 x + y` as an `IsometryEquiv`.\"]\ndef mulLeft [IsometricSMul G G] (c : G) : G \u2243\u1d62 G where\n  toEquiv := Equiv.mulLeft c\n  isometry_toFun := edist_mul_left c\n#align isometry_equiv.mul_left IsometryEquiv.mulLeft\n#align isometry_equiv.add_left IsometryEquiv.addLeft\n#align isometry_equiv.mul_left_apply IsometryEquiv.mulLeft_apply\n#align isometry_equiv.mul_left_to_equiv IsometryEquiv.mulLeft_toEquiv\n#align isometry_equiv.add_left_apply IsometryEquiv.addLeft_apply\n#align isometry_equiv.add_left_to_equiv IsometryEquiv.addLeft_toEquiv\n\n@[to_additive (attr := simp)]\ntheorem mulLeft_symm [IsometricSMul G G] (x : G) :\n    (mulLeft x).symm = IsometryEquiv.mulLeft x\u207b\u00b9 :=\n  constSMul_symm x\n#align isometry_equiv.mul_left_symm IsometryEquiv.mulLeft_symm\n#align isometry_equiv.add_left_symm IsometryEquiv.addLeft_symm\n\n/-- Multiplication `y \u21a6 y * x` as an `IsometryEquiv`. -/\n@[to_additive (attr := simps! apply toEquiv) \"Addition `y \u21a6 y + x` as an `IsometryEquiv`.\"]\ndef mulRight [IsometricSMul G\u1d50\u1d52\u1d56 G] (c : G) : G \u2243\u1d62 G where\n  toEquiv := Equiv.mulRight c\n  isometry_toFun a b := edist_mul_right a b c\n#align isometry_equiv.mul_right IsometryEquiv.mulRight\n#align isometry_equiv.add_right IsometryEquiv.addRight\n#align isometry_equiv.mul_right_apply IsometryEquiv.mulRight_apply\n#align isometry_equiv.mul_right_to_equiv IsometryEquiv.mulRight_toEquiv\n#align isometry_equiv.add_right_apply IsometryEquiv.addRight_apply\n#align isometry_equiv.add_right_to_equiv IsometryEquiv.addRight_toEquiv\n\n@[to_additive (attr := simp)]\n\n\n/-- Division `y \u21a6 y / x` as an `IsometryEquiv`. -/\n@[to_additive (attr := simps! apply toEquiv) \"Subtraction `y \u21a6 y - x` as an `IsometryEquiv`.\"]\ndef divRight [IsometricSMul G\u1d50\u1d52\u1d56 G] (c : G) : G \u2243\u1d62 G where\n  toEquiv := Equiv.divRight c\n  isometry_toFun a b := edist_div_right a b c\n#align isometry_equiv.div_right IsometryEquiv.divRight\n#align isometry_equiv.sub_right IsometryEquiv.subRight\n#align isometry_equiv.div_right_apply IsometryEquiv.divRight_apply\n#align isometry_equiv.div_right_to_equiv IsometryEquiv.divRight_toEquiv\n#align isometry_equiv.sub_right_apply IsometryEquiv.subRight_apply\n#align isometry_equiv.sub_right_to_equiv IsometryEquiv.subRight_toEquiv\n\n@[to_additive (attr := simp)]\ntheorem divRight_symm [IsometricSMul G\u1d50\u1d52\u1d56 G] (c : G) : (divRight c).symm = mulRight c :=\n  ext fun _ => rfl\n#align isometry_equiv.div_right_symm IsometryEquiv.divRight_symm\n#align isometry_equiv.sub_right_symm IsometryEquiv.subRight_symm\n\nvariable [IsometricSMul G G] [IsometricSMul G\u1d50\u1d52\u1d56 G]\n\n/-- Division `y \u21a6 x / y` as an `IsometryEquiv`. -/\n@[to_additive (attr := simps! apply symm_apply toEquiv)\n  \"Subtraction `y \u21a6 x - y` as an `IsometryEquiv`.\"]\ndef divLeft (c : G) : G \u2243\u1d62 G where\n  toEquiv := Equiv.divLeft c\n  isometry_toFun := edist_div_left c\n#align isometry_equiv.div_left IsometryEquiv.divLeft\n#align isometry_equiv.sub_left IsometryEquiv.subLeft\n#align isometry_equiv.div_left_apply IsometryEquiv.divLeft_apply\n#align isometry_equiv.div_left_symm_apply IsometryEquiv.divLeft_symm_apply\n#align isometry_equiv.div_left_to_equiv IsometryEquiv.divLeft_toEquiv\n#align isometry_equiv.sub_left_apply IsometryEquiv.subLeft_apply\n#align isometry_equiv.sub_left_symm_apply IsometryEquiv.subLeft_symm_apply\n#align isometry_equiv.sub_left_to_equiv IsometryEquiv.subLeft_toEquiv\n\nvariable (G)\n\n/-- Inversion `x \u21a6 x\u207b\u00b9` as an `IsometryEquiv`. -/\n@[to_additive (attr := simps! apply toEquiv) \"Negation `x \u21a6 -x` as an `IsometryEquiv`.\"]\ndef inv : G \u2243\u1d62 G where\n  toEquiv := Equiv.inv G\n  isometry_toFun := edist_inv_inv\n#align isometry_equiv.inv IsometryEquiv.inv\n#align isometry_equiv.neg IsometryEquiv.neg\n#align isometry_equiv.inv_apply IsometryEquiv.inv_apply\n#align isometry_equiv.inv_to_equiv IsometryEquiv.inv_toEquiv\n#align isometry_equiv.neg_apply IsometryEquiv.neg_apply\n#align isometry_equiv.neg_to_equiv IsometryEquiv.neg_toEquiv\n\n@[to_additive (attr := simp)] theorem inv_symm : (inv G).symm = inv G := rfl\n#align isometry_equiv.inv_symm IsometryEquiv.inv_symm\n#align isometry_equiv.neg_symm IsometryEquiv.neg_symm\n\nend IsometryEquiv\n\nnamespace EMetric\n\n@[to_additive (attr := simp)]\ntheorem smul_ball (c : G) (x : X) (r : \u211d\u22650\u221e) : c \u2022 ball x r = ball (c \u2022 x) r :=\n  (IsometryEquiv.constSMul c).image_emetric_ball _ _\n#align emetric.smul_ball EMetric.smul_ball\n#align emetric.vadd_ball EMetric.vadd_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_smul_ball (c : G) (x : X) (r : \u211d\u22650\u221e) : (\u00b7 \u2022 \u00b7) c \u207b\u00b9' ball x r = ball (c\u207b\u00b9 \u2022 x) r :=\n  by rw [preimage_smul, smul_ball]\n#align emetric.preimage_smul_ball EMetric.preimage_smul_ball\n#align emetric.preimage_vadd_ball EMetric.preimage_vadd_ball\n\n@[to_additive (attr := simp)]\ntheorem smul_closedBall (c : G) (x : X) (r : \u211d\u22650\u221e) : c \u2022 closedBall x r = closedBall (c \u2022 x) r :=\n  (IsometryEquiv.constSMul c).image_emetric_closedBall _ _\n#align emetric.smul_closed_ball EMetric.smul_closedBall\n#align emetric.vadd_closed_ball EMetric.vadd_closedBall\n\n@[to_additive (attr := simp)]\ntheorem preimage_smul_closedBall (c : G) (x : X) (r : \u211d\u22650\u221e) :\n    (\u00b7 \u2022 \u00b7) c \u207b\u00b9' closedBall x r = closedBall (c\u207b\u00b9 \u2022 x) r := by\n  rw [preimage_smul, smul_closedBall]\n#align emetric.preimage_smul_closed_ball EMetric.preimage_smul_closedBall\n#align emetric.preimage_vadd_closed_ball EMetric.preimage_vadd_closedBall\n\nvariable [PseudoEMetricSpace G]\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : \u211d\u22650\u221e) :\n    (\u00b7 * \u00b7) a \u207b\u00b9' ball b r = ball (a\u207b\u00b9 * b) r :=\n  preimage_smul_ball a b r\n#align emetric.preimage_mul_left_ball EMetric.preimage_mul_left_ball\n#align emetric.preimage_add_left_ball EMetric.preimage_add_left_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_right_ball [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) (r : \u211d\u22650\u221e) :\n    (fun x => x * a) \u207b\u00b9' ball b r = ball (b / a) r := by\n  rw [div_eq_mul_inv]\n  exact preimage_smul_ball (MulOpposite.op a) b r\n#align emetric.preimage_mul_right_ball EMetric.preimage_mul_right_ball\n#align emetric.preimage_add_right_ball EMetric.preimage_add_right_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : \u211d\u22650\u221e) :\n    (\u00b7 * \u00b7) a \u207b\u00b9' closedBall b r = closedBall (a\u207b\u00b9 * b) r :=\n  preimage_smul_closedBall a b r\n#align emetric.preimage_mul_left_closed_ball EMetric.preimage_mul_left_closedBall\n#align emetric.preimage_add_left_closed_ball EMetric.preimage_add_left_closedBall\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_right_closedBall [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) (r : \u211d\u22650\u221e) :\n    (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b / a) r := by\n  rw [div_eq_mul_inv]\n  exact preimage_smul_closedBall (MulOpposite.op a) b r\n#align emetric.preimage_mul_right_closed_ball EMetric.preimage_mul_right_closedBall\n#align emetric.preimage_add_right_closed_ball EMetric.preimage_add_right_closedBall\n\nend EMetric\n\nend EMetric\n\n@[to_additive (attr := simp)]\ntheorem dist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :\n    dist (c \u2022 x) (c \u2022 y) = dist x y :=\n  (isometry_smul X c).dist_eq x y\n#align dist_smul dist_smul\n#align dist_vadd dist_vadd\n\n@[to_additive (attr := simp)]\ntheorem nndist_smul [PseudoMetricSpace X] [SMul M X] [IsometricSMul M X] (c : M) (x y : X) :\n    nndist (c \u2022 x) (c \u2022 y) = nndist x y :=\n  (isometry_smul X c).nndist_eq x y\n#align nndist_smul nndist_smul\n#align nndist_vadd nndist_vadd\n\n@[to_additive (attr := simp)]\ntheorem dist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :\n    dist (a * b) (a * c) = dist b c :=\n  dist_smul a b c\n#align dist_mul_left dist_mul_left\n#align dist_add_left dist_add_left\n\n@[to_additive (attr := simp)]\ntheorem nndist_mul_left [PseudoMetricSpace M] [Mul M] [IsometricSMul M M] (a b c : M) :\n    nndist (a * b) (a * c) = nndist b c :=\n  nndist_smul a b c\n#align nndist_mul_left nndist_mul_left\n#align nndist_add_left nndist_add_left\n\n@[to_additive (attr := simp)]\ntheorem dist_mul_right [Mul M] [PseudoMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M] (a b c : M) :\n    dist (a * c) (b * c) = dist a b :=\n  dist_smul (MulOpposite.op c) a b\n#align dist_mul_right dist_mul_right\n#align dist_add_right dist_add_right\n\n@[to_additive (attr := simp)]\ntheorem nndist_mul_right [PseudoMetricSpace M] [Mul M] [IsometricSMul M\u1d50\u1d52\u1d56 M] (a b c : M) :\n    nndist (a * c) (b * c) = nndist a b :=\n  nndist_smul (MulOpposite.op c) a b\n#align nndist_mul_right nndist_mul_right\n#align nndist_add_right nndist_add_right\n\n@[to_additive (attr := simp)]\ntheorem dist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M]\n    (a b c : M) : dist (a / c) (b / c) = dist a b := by simp only [div_eq_mul_inv, dist_mul_right]\n#align dist_div_right dist_div_right\n#align dist_sub_right dist_sub_right\n\n@[to_additive (attr := simp)]\ntheorem nndist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M]\n    (a b c : M) : nndist (a / c) (b / c) = nndist a b := by\n  simp only [div_eq_mul_inv, nndist_mul_right]\n#align nndist_div_right nndist_div_right\n#align nndist_sub_right nndist_sub_right\n\n@[to_additive (attr := simp)]\ntheorem dist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]\n    [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) : dist a\u207b\u00b9 b\u207b\u00b9 = dist a b :=\n  (IsometryEquiv.inv G).dist_eq a b\n#align dist_inv_inv dist_inv_inv\n#align dist_neg_neg dist_neg_neg\n\n@[to_additive (attr := simp)]\ntheorem nndist_inv_inv [Group G] [PseudoMetricSpace G] [IsometricSMul G G]\n    [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) : nndist a\u207b\u00b9 b\u207b\u00b9 = nndist a b :=\n  (IsometryEquiv.inv G).nndist_eq a b\n#align nndist_inv_inv nndist_inv_inv\n#align nndist_neg_neg nndist_neg_neg\n\n@[to_additive (attr := simp)]\ntheorem dist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]\n    [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b c : G) : dist (a / b) (a / c) = dist b c := by\n  simp [div_eq_mul_inv]\n#align dist_div_left dist_div_left\n#align dist_sub_left dist_sub_left\n\n@[to_additive (attr := simp)]\ntheorem nndist_div_left [Group G] [PseudoMetricSpace G] [IsometricSMul G G]\n    [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b c : G) : nndist (a / b) (a / c) = nndist b c := by\n  simp [div_eq_mul_inv]\n#align nndist_div_left nndist_div_left\n#align nndist_sub_left nndist_sub_left\n\nnamespace Metric\n\nvariable [PseudoMetricSpace X] [Group G] [MulAction G X] [IsometricSMul G X]\n\n@[to_additive (attr := simp)]\ntheorem smul_ball (c : G) (x : X) (r : \u211d) : c \u2022 ball x r = ball (c \u2022 x) r :=\n  (IsometryEquiv.constSMul c).image_ball _ _\n#align metric.smul_ball Metric.smul_ball\n#align metric.vadd_ball Metric.vadd_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_smul_ball (c : G) (x : X) (r : \u211d) : (\u00b7 \u2022 \u00b7) c \u207b\u00b9' ball x r = ball (c\u207b\u00b9 \u2022 x) r := by\n  rw [preimage_smul, smul_ball]\n#align metric.preimage_smul_ball Metric.preimage_smul_ball\n#align metric.preimage_vadd_ball Metric.preimage_vadd_ball\n\n@[to_additive (attr := simp)]\ntheorem smul_closedBall (c : G) (x : X) (r : \u211d) : c \u2022 closedBall x r = closedBall (c \u2022 x) r :=\n  (IsometryEquiv.constSMul c).image_closedBall _ _\n#align metric.smul_closed_ball Metric.smul_closedBall\n#align metric.vadd_closed_ball Metric.vadd_closedBall\n\n@[to_additive (attr := simp)]\ntheorem preimage_smul_closedBall (c : G) (x : X) (r : \u211d) :\n    (\u00b7 \u2022 \u00b7) c \u207b\u00b9' closedBall x r = closedBall (c\u207b\u00b9 \u2022 x) r := by rw [preimage_smul, smul_closedBall]\n#align metric.preimage_smul_closed_ball Metric.preimage_smul_closedBall\n#align metric.preimage_vadd_closed_ball Metric.preimage_vadd_closedBall\n\n@[to_additive (attr := simp)]\ntheorem smul_sphere (c : G) (x : X) (r : \u211d) : c \u2022 sphere x r = sphere (c \u2022 x) r :=\n  (IsometryEquiv.constSMul c).image_sphere _ _\n#align metric.smul_sphere Metric.smul_sphere\n#align metric.vadd_sphere Metric.vadd_sphere\n\n@[to_additive (attr := simp)]\ntheorem preimage_smul_sphere (c : G) (x : X) (r : \u211d) :\n    (\u00b7 \u2022 \u00b7) c \u207b\u00b9' sphere x r = sphere (c\u207b\u00b9 \u2022 x) r := by rw [preimage_smul, smul_sphere]\n#align metric.preimage_smul_sphere Metric.preimage_smul_sphere\n#align metric.preimage_vadd_sphere Metric.preimage_vadd_sphere\n\nvariable [PseudoMetricSpace G]\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_left_ball [IsometricSMul G G] (a b : G) (r : \u211d) :\n    (\u00b7 * \u00b7) a \u207b\u00b9' ball b r = ball (a\u207b\u00b9 * b) r :=\n  preimage_smul_ball a b r\n#align metric.preimage_mul_left_ball Metric.preimage_mul_left_ball\n#align metric.preimage_add_left_ball Metric.preimage_add_left_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_right_ball [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) (r : \u211d) :\n    (fun x => x * a) \u207b\u00b9' ball b r = ball (b / a) r := by\n  rw [div_eq_mul_inv]\n  exact preimage_smul_ball (MulOpposite.op a) b r\n#align metric.preimage_mul_right_ball Metric.preimage_mul_right_ball\n#align metric.preimage_add_right_ball Metric.preimage_add_right_ball\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_left_closedBall [IsometricSMul G G] (a b : G) (r : \u211d) :\n    (\u00b7 * \u00b7) a \u207b\u00b9' closedBall b r = closedBall (a\u207b\u00b9 * b) r :=\n  preimage_smul_closedBall a b r\n#align metric.preimage_mul_left_closed_ball Metric.preimage_mul_left_closedBall\n#align metric.preimage_add_left_closed_ball Metric.preimage_add_left_closedBall\n\n@[to_additive (attr := simp)]\ntheorem preimage_mul_right_closedBall [IsometricSMul G\u1d50\u1d52\u1d56 G] (a b : G) (r : \u211d) :\n    (fun x => x * a) \u207b\u00b9' closedBall b r = closedBall (b / a) r := by\n  rw [div_eq_mul_inv]\n  exact preimage_smul_closedBall (MulOpposite.op a) b r\n#align metric.preimage_mul_right_closed_ball Metric.preimage_mul_right_closedBall\n#align metric.preimage_add_right_closed_ball Metric.preimage_add_right_closedBall\n\nend Metric\n\nsection Instances\n\nvariable {Y : Type _} [PseudoEMetricSpace X] [PseudoEMetricSpace Y] [SMul M X]\n  [IsometricSMul M X]\n\n@[to_additive]\ninstance [SMul M Y] [IsometricSMul M Y] : IsometricSMul M (X \u00d7 Y) :=\n  \u27e8fun c => (isometry_smul X c).prod_map (isometry_smul Y c)\u27e9\n\n@[to_additive]\ninstance Prod.isometricSMul' {N} [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] [Mul N]\n    [PseudoEMetricSpace N] [IsometricSMul N N] : IsometricSMul (M \u00d7 N) (M \u00d7 N) :=\n  \u27e8fun c => (isometry_smul M c.1).prod_map (isometry_smul N c.2)\u27e9\n#align prod.has_isometric_smul' Prod.isometricSMul'\n#align prod.has_isometric_vadd' Prod.isometricVAdd'\n\n@[to_additive]\ninstance Prod.isometricSMul'' {N} [Mul M] [PseudoEMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M]\n    [Mul N] [PseudoEMetricSpace N] [IsometricSMul N\u1d50\u1d52\u1d56 N] :\n    IsometricSMul (M \u00d7 N)\u1d50\u1d52\u1d56 (M \u00d7 N) :=\n  \u27e8fun c => (isometry_mul_right c.unop.1).prod_map (isometry_mul_right c.unop.2)\u27e9\n#align prod.has_isometric_smul'' Prod.isometricSMul''\n#align prod.has_isometric_vadd'' Prod.isometricVAdd''\n\n@[to_additive]\ninstance Units.isometricSMul [Monoid M] : IsometricSMul M\u02e3 X :=\n  \u27e8fun c => isometry_smul X (c : M)\u27e9\n#align units.has_isometric_smul Units.isometricSMul\n#align add_units.has_isometric_vadd AddUnits.isometricVAdd\n\n@[to_additive]\ninstance : IsometricSMul M X\u1d50\u1d52\u1d56 :=\n  \u27e8fun c x y => by simpa only using edist_smul_left c x.unop y.unop\u27e9\n\n@[to_additive]\ninstance ULift.isometricSMul : IsometricSMul (ULift M) X :=\n  \u27e8fun c => by simpa only using isometry_smul X c.down\u27e9\n#align ulift.has_isometric_smul ULift.isometricSMul\n#align ulift.has_isometric_vadd ULift.isometricVAdd\n\n@[to_additive]\ninstance ULift.isometricSMul' : IsometricSMul M (ULift X) :=\n  \u27e8fun c x y => by simpa only using edist_smul_left c x.1 y.1\u27e9\n#align ulift.has_isometric_smul' ULift.isometricSMul'\n#align ulift.has_isometric_vadd' ULift.isometricVAdd'\n\n@[to_additive]\ninstance {\u03b9} {X : \u03b9 \u2192 Type _} [Fintype \u03b9] [\u2200 i, SMul M (X i)] [\u2200 i, PseudoEMetricSpace (X i)]\n    [\u2200 i, IsometricSMul M (X i)] : IsometricSMul M (\u2200 i, X i) :=\n  \u27e8fun c => isometry_dcomp (fun _ => (c \u2022 \u00b7)) fun i => isometry_smul (X i) c\u27e9\n\n@[to_additive]\ninstance Pi.isometricSMul' {\u03b9} {M X : \u03b9 \u2192 Type _} [Fintype \u03b9] [\u2200 i, SMul (M i) (X i)]\n    [\u2200 i, PseudoEMetricSpace (X i)] [\u2200 i, IsometricSMul (M i) (X i)] :\n    IsometricSMul (\u2200 i, M i) (\u2200 i, X i) :=\n  \u27e8fun c => isometry_dcomp (fun i => (c i \u2022 \u00b7)) fun _ => isometry_smul _ _\u27e9\n#align pi.has_isometric_smul' Pi.isometricSMul'\n#align pi.has_isometric_vadd' Pi.isometricVAdd'\n\n@[to_additive]\ninstance Pi.isometricSMul'' {\u03b9} {M : \u03b9 \u2192 Type _} [Fintype \u03b9] [\u2200 i, Mul (M i)]\n    [\u2200 i, PseudoEMetricSpace (M i)] [\u2200 i, IsometricSMul (M i)\u1d50\u1d52\u1d56 (M i)] :\n    IsometricSMul (\u2200 i, M i)\u1d50\u1d52\u1d56 (\u2200 i, M i) :=\n  \u27e8fun c => isometry_dcomp (fun i (x : M i) => x * c.unop i) fun _ => isometry_mul_right _\u27e9\n#align pi.has_isometric_smul'' Pi.isometricSMul''\n#align pi.has_isometric_vadd'' Pi.isometricVAdd''\n\ninstance Additive.isometricVAdd : IsometricVAdd (Additive M) X :=\n  \u27e8fun c => isometry_smul X (toMul c)\u27e9\n#align additive.has_isometric_vadd Additive.isometricVAdd\n\ninstance Additive.isometricVAdd' [Mul M] [PseudoEMetricSpace M] [IsometricSMul M M] :\n    IsometricVAdd (Additive M) (Additive M) :=\n  \u27e8fun c x y => edist_smul_left (toMul c) (toMul x) (toMul y)\u27e9\n#align additive.has_isometric_vadd' Additive.isometricVAdd'\n\ninstance Additive.isometricVAdd'' [Mul M] [PseudoEMetricSpace M] [IsometricSMul M\u1d50\u1d52\u1d56 M] :\n    IsometricVAdd (Additive M)\u1d43\u1d52\u1d56 (Additive M) :=\n  \u27e8fun c x y => edist_smul_left (MulOpposite.op (toMul c.unop)) (toMul x) (toMul y)\u27e9\n#align additive.has_isometric_vadd'' Additive.isometricVAdd''\n\ninstance Multiplicative.isometricSMul {M X} [VAdd M X] [PseudoEMetricSpace X]\n    [IsometricVAdd M X] : IsometricSMul (Multiplicative M) X :=\n  \u27e8fun c => isometry_vadd X (toAdd c)\u27e9\n#align multiplicative.has_isometric_smul Multiplicative.isometricSMul\n\ninstance Multiplicative.isometricSMul' [Add M] [PseudoEMetricSpace M] [IsometricVAdd M M] :\n    IsometricSMul (Multiplicative M) (Multiplicative M) :=\n  \u27e8fun c x y => edist_vadd_left (toAdd c) (toAdd x) (toAdd y)\u27e9\n#align multiplicative.has_isometric_smul' Multiplicative.isometricSMul'\n\ninstance Multiplicative.isometricVAdd'' [Add M] [PseudoEMetricSpace M]\n    [IsometricVAdd M\u1d43\u1d52\u1d56 M] : IsometricSMul (Multiplicative M)\u1d50\u1d52\u1d56 (Multiplicative M) :=\n  \u27e8fun c x y => edist_vadd_left (AddOpposite.op (toAdd c.unop)) (toAdd x) (toAdd y)\u27e9\n#align multiplicative.has_isometric_vadd'' Multiplicative.isometricVAdd''\n\nend Instances\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Topology/MetricSpace/IsometricSMul.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.6723316860482763, "lm_q1q2_score": 0.38570218073301615}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Patrick Massot\n-/\nimport topology.order\n\n/-!\n# Specific classes of maps between topological spaces\n\nThis file introduces the following properties of a map `f : X \u2192 Y` between topological spaces:\n\n* `is_open_map f` means the image of an open set under `f` is open.\n* `is_closed_map f` means the image of a closed set under `f` is closed.\n\n(Open and closed maps need not be continuous.)\n\n* `inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`.\n  These behave like embeddings except they need not be injective. Instead, points of `X` which\n  are identified by `f` are also indistinguishable in the topology on `X`.\n* `embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with\n  a subspace of `Y`.\n* `open_embedding f` means `f` is an embedding with open image, so it identifies `X` with an\n  open subspace of `Y`. Equivalently, `f` is an embedding and an open map.\n* `closed_embedding f` similarly means `f` is an embedding with closed image, so it identifies\n  `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map.\n\n* `quotient_map f` is the dual condition to `embedding f`: `f` is surjective and the topology\n  on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies\n  `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Open_and_closed_maps>\n* <https://en.wikipedia.org/wiki/Embedding#General_topology>\n* <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map>\n\n## Tags\n\nopen map, closed map, embedding, quotient map, identification map\n\n-/\n\nopen set filter\nopen_locale topological_space filter\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*}\n\nsection inducing\nstructure inducing [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) : Prop :=\n(induced : t\u03b1 = t\u03b2.induced f)\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\nlemma inducing_id : inducing (@id \u03b1) :=\n\u27e8induced_id.symm\u27e9\n\nprotected lemma inducing.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : inducing g) (hf : inducing f) :\n  inducing (g \u2218 f) :=\n\u27e8by rw [hf.induced, hg.induced, induced_compose]\u27e9\n\nlemma inducing_of_inducing_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : inducing (g \u2218 f)) : inducing f :=\n\u27e8le_antisymm\n    (by rwa \u2190 continuous_iff_le_induced)\n    (by { rw [hgf.induced, \u2190 continuous_iff_le_induced], apply hg.comp continuous_induced_dom })\u27e9\n\nlemma inducing.nhds_eq_comap {f : \u03b1 \u2192 \u03b2} (hf : inducing f) :\n  \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd $ f a) :=\n(induced_iff_nhds_eq f).1 hf.induced\n\nlemma inducing.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_eq a\n\nlemma inducing.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) :\n  (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_of_mem h\n\nlemma inducing.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : inducing g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nby rw [tendsto, tendsto, hg.induced, nhds_induced, \u2190 map_le_iff_le_comap, filter.map_map]\n\nlemma inducing.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\nby simp [continuous_iff_continuous_at, continuous_at, inducing.tendsto_nhds_iff hg]\n\nlemma inducing.continuous {f : \u03b1 \u2192 \u03b2} (hf : inducing f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nlemma inducing.closure_eq_preimage_closure_image {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (s : set \u03b1) :\n  closure s = f \u207b\u00b9' closure (f '' s) :=\nby { ext x, rw [set.mem_preimage, \u2190 closure_induced, hf.induced] }\n\nlemma inducing.is_closed_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_closed s \u2194 \u2203 t, is_closed t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_closed_induced_iff]\n\nlemma inducing.is_open_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_open s \u2194 \u2203 t, is_open t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_open_induced_iff]\n\nend inducing\n\nsection embedding\n\n/-- A function between topological spaces is an embedding if it is injective,\n  and for all `s : set \u03b1`, `s` is open iff it is the preimage of an open set. -/\nstructure embedding [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2)\n  extends inducing f : Prop :=\n(inj : function.injective f)\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\nlemma embedding.mk' (f : \u03b1 \u2192 \u03b2) (inj : function.injective f)\n  (induced : \u2200a, comap f (\ud835\udcdd (f a)) = \ud835\udcdd a) : embedding f :=\n\u27e8\u27e8(induced_iff_nhds_eq f).2 (\u03bb a, (induced a).symm)\u27e9, inj\u27e9\n\nlemma embedding_id : embedding (@id \u03b1) :=\n\u27e8inducing_id, assume a\u2081 a\u2082 h, h\u27e9\n\nlemma embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : embedding g) (hf : embedding f) :\n  embedding (g \u2218 f) :=\n{ inj:= assume a\u2081 a\u2082 h, hf.inj $ hg.inj h,\n  ..hg.to_inducing.comp hf.to_inducing }\n\nlemma embedding_of_embedding_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : embedding (g \u2218 f)) : embedding f :=\n{ induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced,\n  inj := assume a\u2081 a\u2082 h, hgf.inj $ by simp [h, (\u2218)] }\n\nprotected lemma function.left_inverse.embedding {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n  (h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :\n  embedding g :=\nembedding_of_embedding_compose hg hf $ h.comp_eq_id.symm \u25b8 embedding_id\n\nlemma embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : embedding f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.1.map_nhds_eq a\n\nlemma embedding.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2}\n  (hf : embedding f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) : (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.1.map_nhds_of_mem a h\n\nlemma embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : embedding g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nby rw [tendsto, tendsto, hg.induced, nhds_induced, \u2190 map_le_iff_le_comap, filter.map_map]\n\nlemma embedding.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : embedding g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\ninducing.continuous_iff hg.1\n\nlemma embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : embedding f) : continuous f :=\ninducing.continuous hf.1\n\nlemma embedding.closure_eq_preimage_closure_image {e : \u03b1 \u2192 \u03b2} (he : embedding e) (s : set \u03b1) :\n  closure s = e \u207b\u00b9' closure (e '' s) :=\nhe.1.closure_eq_preimage_closure_image s\n\nend embedding\n\n/-- A function between topological spaces is a quotient map if it is surjective,\n  and for all `s : set \u03b2`, `s` is open iff its preimage is an open set. -/\ndef quotient_map {\u03b1 : Type*} {\u03b2 : Type*} [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2]\n  (f : \u03b1 \u2192 \u03b2) : Prop :=\nfunction.surjective f \u2227 t\u03b2 = t\u03b1.coinduced f\n\nlemma quotient_map_iff {\u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  quotient_map f \u2194 function.surjective f \u2227 \u2200 s : set \u03b2, is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nand_congr iff.rfl topological_space_eq_iff\n\nnamespace quotient_map\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\nprotected lemma id : quotient_map (@id \u03b1) :=\n\u27e8assume a, \u27e8a, rfl\u27e9, coinduced_id.symm\u27e9\n\nprotected lemma comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : quotient_map g) (hf : quotient_map f) :\n  quotient_map (g \u2218 f) :=\n\u27e8hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]\u27e9\n\nprotected lemma of_quotient_map_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3}\n  (hf : continuous f) (hg : continuous g)\n  (hgf : quotient_map (g \u2218 f)) : quotient_map g :=\n\u27e8assume b, let \u27e8a, h\u27e9 := hgf.left b in \u27e8f a, h\u27e9,\n  le_antisymm\n    (by rw [hgf.right, \u2190 continuous_iff_coinduced_le];\n        apply continuous_coinduced_rng.comp hf)\n    (by rwa \u2190 continuous_iff_coinduced_le)\u27e9\n\nprotected lemma continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : quotient_map f) :\n  continuous g \u2194 continuous (g \u2218 f) :=\nby rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose]\n\nprotected lemma continuous {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nprotected lemma surjective {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) : function.surjective f := hf.1\n\nprotected lemma is_open_preimage {f : \u03b1 \u2192 \u03b2} (hf : quotient_map f) {s : set \u03b2} :\n  is_open (f \u207b\u00b9' s) \u2194 is_open s :=\n((quotient_map_iff.1 hf).2 s).symm\n\nend quotient_map\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be an *open map*, if the image of any open `U : set \u03b1`\nis open in `\u03b2`. -/\ndef is_open_map [topological_space \u03b1] [topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n\u2200 U : set \u03b1, is_open U \u2192 is_open (f '' U)\n\nnamespace is_open_map\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2}\nopen function\n\nprotected lemma id : is_open_map (@id \u03b1) := assume s hs, by rwa [image_id]\n\nprotected lemma comp\n  {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_open_map g) (hf : is_open_map f) : is_open_map (g \u2218 f) :=\nby intros s hs; rw [image_comp]; exact hg _ (hf _ hs)\n\nlemma is_open_range (hf : is_open_map f) : is_open (range f) :=\nby { rw \u2190 image_univ, exact hf _ is_open_univ }\n\nlemma image_mem_nhds (hf : is_open_map f) {x : \u03b1} {s : set \u03b1} (hx : s \u2208 \ud835\udcdd x) :\n  f '' s \u2208 \ud835\udcdd (f x) :=\nlet \u27e8t, hts, ht, hxt\u27e9 := mem_nhds_sets_iff.1 hx in\nmem_sets_of_superset (mem_nhds_sets (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)\n\nlemma image_interior_subset (hf : is_open_map f) (s : set \u03b1) :\n  f '' interior s \u2286 interior (f '' s) :=\ninterior_maximal (image_subset _ interior_subset) (hf _ is_open_interior)\n\nlemma nhds_le (hf : is_open_map f) (a : \u03b1) : \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\nle_map $ \u03bb s, hf.image_mem_nhds\n\nlemma of_nhds_le (hf : \u2200 a, \ud835\udcdd (f a) \u2264 map f (\ud835\udcdd a)) : is_open_map f :=\n\u03bb s hs, is_open_iff_mem_nhds.2 $ \u03bb b \u27e8a, has, hab\u27e9,\n  hab \u25b8 hf _ (image_mem_map $ mem_nhds_sets hs has)\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_open_map f :=\nbegin\n  assume s hs,\n  rw [image_eq_preimage_of_inverse r_inv l_inv],\n  exact hs.preimage h\nend\n\nlemma to_quotient_map {f : \u03b1 \u2192 \u03b2}\n  (open_map : is_open_map f) (cont : continuous f) (surj : function.surjective f) :\n  quotient_map f :=\n\u27e8 surj,\n  begin\n    ext s,\n    show is_open s \u2194 is_open (f \u207b\u00b9' s),\n    split,\n    { exact continuous_def.1 cont s },\n    { assume h,\n      rw \u2190 surj.image_preimage s,\n      exact open_map _ h }\n  end\u27e9\n\nend is_open_map\n\nlemma is_open_map_iff_nhds_le [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_open_map f \u2194 \u2200(a:\u03b1), \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\n\u27e8\u03bb hf, hf.nhds_le, is_open_map.of_nhds_le\u27e9\n\nlemma inducing.is_open_map [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n  (hi : inducing f) (ho : is_open (range f)) :\n  is_open_map f :=\nis_open_map.of_nhds_le $ \u03bb x, (hi.map_nhds_of_mem _ $ mem_nhds_sets ho $ mem_range_self _).ge\n\nsection is_closed_map\nvariables [topological_space \u03b1] [topological_space \u03b2]\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be a *closed map*, if the image of any closed `U : set \u03b1`\nis closed in `\u03b2`. -/\ndef is_closed_map (f : \u03b1 \u2192 \u03b2) := \u2200 U : set \u03b1, is_closed U \u2192 is_closed (f '' U)\n\nend is_closed_map\n\nnamespace is_closed_map\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\nopen function\n\nprotected lemma id : is_closed_map (@id \u03b1) := assume s hs, by rwa image_id\n\nprotected lemma comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_closed_map g) (hf : is_closed_map f) :\n  is_closed_map (g \u2218 f) :=\nby { intros s hs, rw image_comp, exact hg _ (hf _ hs) }\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_closed_map f :=\nassume s hs,\nhave f' \u207b\u00b9' s = f '' s, by ext x; simp [mem_image_iff_of_inverse r_inv l_inv],\nthis \u25b8 hs.preimage h\n\nlemma of_nonempty {f : \u03b1 \u2192 \u03b2} (h : \u2200 s, is_closed s \u2192 s.nonempty \u2192 is_closed (f '' s)) :\n  is_closed_map f :=\nbegin\n  intros s hs, cases eq_empty_or_nonempty s with h2s h2s,\n  { simp_rw [h2s, image_empty, is_closed_empty] },\n  { exact h s hs h2s }\nend\n\nend is_closed_map\n\nlemma inducing.is_closed_map [topological_space \u03b1] [topological_space \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (h : is_closed (range f)) : is_closed_map f :=\nbegin\n  intros s hs,\n  rcases hf.is_closed_iff.1 hs with \u27e8t, ht, rfl\u27e9,\n  rw image_preimage_eq_inter_range,\n  exact is_closed_inter ht h\nend\n\nsection open_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- An open embedding is an embedding with open image. -/\nstructure open_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(open_range : is_open $ range f)\n\nlemma open_embedding.is_open_map {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : is_open_map f :=\nhf.to_embedding.to_inducing.is_open_map hf.open_range\n\nlemma open_embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) (a : \u03b1) :\n  map f (\ud835\udcdd a) = \ud835\udcdd (f a) :=\nhf.to_embedding.map_nhds_of_mem _ $ mem_nhds_sets hf.open_range $ mem_range_self _\n\nlemma open_embedding.open_iff_image_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b1} : is_open s \u2194 is_open (f '' s) :=\n\u27e8hf.is_open_map s,\n \u03bb h, begin\n   convert \u2190 h.preimage hf.to_embedding.continuous,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma open_embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma open_embedding.open_iff_preimage_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.open_iff_image_open.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma open_embedding_of_embedding_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f)\n  (h\u2082 : is_open_map f) : open_embedding f :=\n\u27e8h\u2081, h\u2082.is_open_range\u27e9\n\nlemma open_embedding_of_continuous_injective_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_open_map f) : open_embedding f :=\nbegin\n  refine open_embedding_of_embedding_open \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s,\n  change is_open _ \u2192 is_open _,\n  rw is_open_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma open_embedding_id : open_embedding (@id \u03b1) :=\n\u27e8embedding_id, by convert is_open_univ; apply range_id\u27e9\n\nlemma open_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : open_embedding g) (hf : open_embedding f) : open_embedding (g \u2218 f) :=\n\u27e8hg.1.comp hf.1, show is_open (range (g \u2218 f)),\n by rw [range_comp, \u2190hg.open_iff_image_open]; exact hf.2\u27e9\n\nend open_embedding\n\nsection closed_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- A closed embedding is an embedding with closed image. -/\nstructure closed_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(closed_range : is_closed $ range f)\n\nvariables {f : \u03b1 \u2192 \u03b2}\n\nlemma closed_embedding.continuous (hf : closed_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma closed_embedding.is_closed_map (hf : closed_embedding f) : is_closed_map f :=\nhf.to_embedding.to_inducing.is_closed_map hf.closed_range\n\nlemma closed_embedding.closed_iff_image_closed (hf : closed_embedding f)\n  {s : set \u03b1} : is_closed s \u2194 is_closed (f '' s) :=\n\u27e8hf.is_closed_map s,\n \u03bb h, begin\n   convert \u2190continuous_iff_is_closed.mp hf.continuous _ h,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma closed_embedding.closed_iff_preimage_closed (hf : closed_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_closed s \u2194 is_closed (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.closed_iff_image_closed.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma closed_embedding_of_embedding_closed (h\u2081 : embedding f)\n  (h\u2082 : is_closed_map f) : closed_embedding f :=\n\u27e8h\u2081, by convert h\u2082 univ is_closed_univ; simp\u27e9\n\nlemma closed_embedding_of_continuous_injective_closed (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_closed_map f) : closed_embedding f :=\nbegin\n  refine closed_embedding_of_embedding_closed \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s',\n  change is_open _ \u2264 is_open _,\n  rw [\u2190is_closed_compl_iff, \u2190is_closed_compl_iff],\n  generalize : s'\u1d9c = s,\n  rw is_closed_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma closed_embedding_id : closed_embedding (@id \u03b1) :=\n\u27e8embedding_id, by convert is_closed_univ; apply range_id\u27e9\n\nlemma closed_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : closed_embedding g) (hf : closed_embedding f) : closed_embedding (g \u2218 f) :=\n\u27e8hg.to_embedding.comp hf.to_embedding, show is_closed (range (g \u2218 f)),\n by rw [range_comp, \u2190hg.closed_iff_image_closed]; exact hf.closed_range\u27e9\n\nend closed_embedding\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/topology/maps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.38570217846501265}}
{"text": "-- Copyright (c) 2018 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Scott Morrison\n\nimport .pullbacks ...basic\n\nopen category_theory\n\nnamespace category_theory.limits\n\n-- local attribute [tidy] tactic.case_bash\n\nuniverses v u\n\n@[derive decidable_eq] inductive walking_pair : Type v\n| zero | one\n\nopen walking_pair\n\ninductive walking_pair_hom : walking_pair \u2192 walking_pair \u2192 Type v\n| left : walking_pair_hom zero one\n| right : walking_pair_hom zero one\n| id : \u03a0 X : walking_pair.{v}, walking_pair_hom X X\n\nopen walking_pair_hom\n\ninstance walking_pair_category : small_category.{v+1} walking_pair :=\n{ hom := walking_pair_hom,\n  id := walking_pair_hom.id,\n  comp := \u03bb X Y Z f g, match X, Y, Z, f, g with\n  | _, _ ,_, (id _), h := h\n  | _, _, _, left, (id one) := left\n  | _, _, _, right, (id one) := right\n  end }\n\nlemma walking_pair_hom_id (X : walking_pair.{v}) : walking_pair_hom.id X = \ud835\udfd9 X := rfl\n\nvariables {C : Type u} [\ud835\udc9e : category.{v+1} C]\ninclude \ud835\udc9e\nvariables {X Y : C}\n\ndef pair (f g : X \u27f6 Y) : walking_pair.{v} \u2964 C :=\n{ obj := \u03bb x, match x with\n  | zero := X\n  | one := Y\n  end,\n  map := \u03bb x y h, match x, y, h with\n  | a, b, (id c) := \ud835\udfd9 _\n  | a, b, left := f\n  | a, b, right := g\n  end }\n\n@[simp] lemma pair_map_left (f g : X \u27f6 Y) : (pair f g).map left = f := rfl\n@[simp] lemma pair_map_right (f g : X \u27f6 Y) : (pair f g).map right = g := rfl\n\n@[simp] lemma pair_functor_obj {F : walking_pair.{v} \u2964 C} (j : walking_pair.{v}) :\n  (pair (F.map left) (F.map right)).obj j = F.obj j :=\nbegin\n  cases j; refl\nend\n\ndef fork (f g : X \u27f6 Y) := cone (pair f g)\ndef cofork (f g : X \u27f6 Y) := cocone (pair f g)\n\nvariables {f g : X \u27f6 Y}\n\nattribute [simp] walking_pair_hom_id\n\ndef fork.of_\u03b9 {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) : fork f g :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb X, begin cases X, exact \u03b9, exact \u03b9 \u226b f, end,\n    naturality' := \u03bb X Y f,\n    begin\n      cases X; cases Y; cases f; dsimp; simp,\n      exact w\n    end }}\ndef cofork.of_\u03c0 {P : C} (\u03c0 : Y \u27f6 P) (w : f \u226b \u03c0 = g \u226b \u03c0) : cofork f g :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb X, begin cases X, exact f \u226b \u03c0, exact \u03c0, end,\n    naturality' := \u03bb X Y f,\n    begin\n      cases X; cases Y; cases f; dsimp; simp,\n      exact eq.symm w\n    end }}\n\n@[simp] lemma fork.of_\u03b9_app_zero {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) :\n  (fork.of_\u03b9 \u03b9 w).\u03c0.app zero = \u03b9 := rfl\n@[simp] lemma fork.of_\u03b9_app_one {P : C} (\u03b9 : P \u27f6 X) (w : \u03b9 \u226b f = \u03b9 \u226b g) :\n  (fork.of_\u03b9 \u03b9 w).\u03c0.app one = \u03b9 \u226b f := rfl\n\ndef fork.\u03b9 (t : fork f g) := t.\u03c0.app zero\ndef cofork.\u03c0 (t : cofork f g) := t.\u03b9.app one\ndef fork.condition (t : fork f g) : (fork.\u03b9 t) \u226b f = (fork.\u03b9 t) \u226b g :=\nbegin\n  erw [t.w left, \u2190 t.w right], refl\nend\ndef cofork.condition (t : cofork f g) : f \u226b (cofork.\u03c0 t) = g \u226b (cofork.\u03c0 t) :=\nbegin\n  erw [t.w left, \u2190 t.w right], refl\nend\n\ndef cone.of_fork\n  {F : walking_pair.{v} \u2964 C} (t : fork (F.map left) (F.map right)) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g,\n    begin\n      cases j; cases j'; cases g; dsimp; simp,\n      erw \u2190 t.w left, refl,\n      erw \u2190 t.w right, refl,\n    end } }.\ndef cocone.of_cofork\n  {F : walking_pair.{v} \u2964 C} (t : cofork (F.map left) (F.map right)) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g,\n    begin\n      cases j; cases j'; cases g; dsimp; simp,\n      erw \u2190 t.w left, refl,\n      erw \u2190 t.w right, refl,\n    end } }.\n\n@[simp] lemma cone.of_fork_\u03c0\n  {F : walking_pair.{v} \u2964 C} (t : fork (F.map left) (F.map right)) (j):\n  (cone.of_fork t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\n@[simp] lemma cocone.of_cofork_\u03b9\n  {F : walking_pair.{v} \u2964 C} (t : cofork (F.map left) (F.map right)) (j):\n  (cocone.of_cofork t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\ndef fork.of_cone\n  {F : walking_pair.{v} \u2964 C} (t : cone F) : fork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03c0 := {app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n  naturality' := omitted } }\ndef cofork.of_cocone\n  {F : walking_pair.{v} \u2964 C} (t : cocone F) : cofork (F.map left) (F.map right) :=\n{ X := t.X,\n  \u03b9 := { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X, naturality' := omitted } }\n\n@[simp] lemma fork.of_cone_\u03c0 {F : walking_pair.{v} \u2964 C} (t : cone F) (j) :\n  (fork.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n@[simp] lemma cofork.of_cocone_\u03b9 {F : walking_pair.{v} \u2964 C} (t : cocone F) (j) :\n  (cofork.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\nend category_theory.limits\n", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/category_theory/limits/shapes/equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.38566038773005334}}
{"text": "/-\nCopyright (c) 2022 Antoine Labelle. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Antoine Labelle\n\n! This file was ported from Lean 3 source module category_theory.monoidal.subcategory\n! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Monoidal.Braided\nimport Mathbin.CategoryTheory.Monoidal.Linear\nimport Mathbin.CategoryTheory.Preadditive.AdditiveFunctor\nimport Mathbin.CategoryTheory.Linear.LinearFunctor\nimport Mathbin.CategoryTheory.Closed.Monoidal\n\n/-!\n# Full monoidal subcategories\n\nGiven a monidal category `C` and a monoidal predicate on `C`, that is a function `P : C \u2192 Prop`\nclosed under `\ud835\udfd9_` and `\u2297`, we can put a monoidal structure on `{X : C // P X}` (the category\nstructure is defined in `category_theory.full_subcategory`).\n\nWhen `C` is also braided/symmetric, the full monoidal subcategory also inherits the\nbraided/symmetric structure.\n\n## TODO\n* Add monoidal/braided versions of `category_theory.full_subcategory.lift`\n-/\n\n\nuniverse u v\n\nnamespace CategoryTheory\n\nnamespace MonoidalCategory\n\nopen Iso\n\nvariable {C : Type u} [Category.{v} C] [MonoidalCategory C] (P : C \u2192 Prop)\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- A property `C \u2192 Prop` is a monoidal predicate if it is closed under `\ud835\udfd9_` and `\u2297`.\n-/\nclass MonoidalPredicate : Prop where\n  prop_id' : P (\ud835\udfd9_ C) := by obviously\n  prop_tensor' : \u2200 {X Y}, P X \u2192 P Y \u2192 P (X \u2297 Y) := by obviously\n#align category_theory.monoidal_category.monoidal_predicate CategoryTheory.MonoidalCategory.MonoidalPredicate\n\nrestate_axiom monoidal_predicate.prop_id'\n\nrestate_axiom monoidal_predicate.prop_tensor'\n\nopen MonoidalPredicate\n\nvariable [MonoidalPredicate P]\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/--\nWhen `P` is a monoidal predicate, the full subcategory for `P` inherits the monoidal structure of\n  `C`.\n-/\ninstance fullMonoidalSubcategory : MonoidalCategory (FullSubcategory P)\n    where\n  tensorObj X Y := \u27e8X.1 \u2297 Y.1, prop_tensor X.2 Y.2\u27e9\n  tensorHom X\u2081 Y\u2081 X\u2082 Y\u2082 f g := by\n    change X\u2081.1 \u2297 X\u2082.1 \u27f6 Y\u2081.1 \u2297 Y\u2082.1\n    change X\u2081.1 \u27f6 Y\u2081.1 at f\n    change X\u2082.1 \u27f6 Y\u2082.1 at g\n    exact f \u2297 g\n  tensorUnit := \u27e8\ud835\udfd9_ C, prop_id\u27e9\n  associator X Y Z :=\n    \u27e8(\u03b1_ X.1 Y.1 Z.1).Hom, (\u03b1_ X.1 Y.1 Z.1).inv, hom_inv_id (\u03b1_ X.1 Y.1 Z.1),\n      inv_hom_id (\u03b1_ X.1 Y.1 Z.1)\u27e9\n  leftUnitor X := \u27e8(\u03bb_ X.1).Hom, (\u03bb_ X.1).inv, hom_inv_id (\u03bb_ X.1), inv_hom_id (\u03bb_ X.1)\u27e9\n  rightUnitor X := \u27e8(\u03c1_ X.1).Hom, (\u03c1_ X.1).inv, hom_inv_id (\u03c1_ X.1), inv_hom_id (\u03c1_ X.1)\u27e9\n  tensor_id' X Y := tensor_id X.1 Y.1\n  tensor_comp' X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 f\u2081 f\u2082 g\u2081 g\u2082 := tensor_comp f\u2081 f\u2082 g\u2081 g\u2082\n  associator_naturality' X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 f\u2081 f\u2082 f\u2083 := associator_naturality f\u2081 f\u2082 f\u2083\n  leftUnitor_naturality' X Y f := leftUnitor_naturality f\n  rightUnitor_naturality' X Y f := rightUnitor_naturality f\n  pentagon' W X Y Z := pentagon W.1 X.1 Y.1 Z.1\n  triangle' X Y := triangle X.1 Y.1\n#align category_theory.monoidal_category.full_monoidal_subcategory CategoryTheory.MonoidalCategory.fullMonoidalSubcategory\n\n/-- The forgetful monoidal functor from a full monoidal subcategory into the original category\n(\"forgetting\" the condition).\n-/\n@[simps]\ndef fullMonoidalSubcategoryInclusion : MonoidalFunctor (FullSubcategory P) C\n    where\n  toFunctor := fullSubcategoryInclusion P\n  \u03b5 := \ud835\udfd9 _\n  \u03bc X Y := \ud835\udfd9 _\n#align category_theory.monoidal_category.full_monoidal_subcategory_inclusion CategoryTheory.MonoidalCategory.fullMonoidalSubcategoryInclusion\n\ninstance fullMonoidalSubcategory.full : Full (fullMonoidalSubcategoryInclusion P).toFunctor :=\n  FullSubcategory.full P\n#align category_theory.monoidal_category.full_monoidal_subcategory.full CategoryTheory.MonoidalCategory.fullMonoidalSubcategory.full\n\ninstance fullMonoidalSubcategory.faithful :\n    Faithful (fullMonoidalSubcategoryInclusion P).toFunctor :=\n  FullSubcategory.faithful P\n#align category_theory.monoidal_category.full_monoidal_subcategory.faithful CategoryTheory.MonoidalCategory.fullMonoidalSubcategory.faithful\n\nsection\n\nvariable [Preadditive C]\n\ninstance fullMonoidalSubcategoryInclusion_additive :\n    (fullMonoidalSubcategoryInclusion P).toFunctor.Additive :=\n  Functor.fullSubcategoryInclusion_additive _\n#align category_theory.monoidal_category.full_monoidal_subcategory_inclusion_additive CategoryTheory.MonoidalCategory.fullMonoidalSubcategoryInclusion_additive\n\ninstance [MonoidalPreadditive C] : MonoidalPreadditive (FullSubcategory P) :=\n  monoidalPreadditive_of_faithful (fullMonoidalSubcategoryInclusion P)\n\nvariable (R : Type _) [Ring R] [Linear R C]\n\ninstance fullMonoidalSubcategoryInclusion_linear :\n    (fullMonoidalSubcategoryInclusion P).toFunctor.Linear R :=\n  Functor.fullSubcategoryInclusionLinear R _\n#align category_theory.monoidal_category.full_monoidal_subcategory_inclusion_linear CategoryTheory.MonoidalCategory.fullMonoidalSubcategoryInclusion_linear\n\ninstance [MonoidalPreadditive C] [MonoidalLinear R C] : MonoidalLinear R (FullSubcategory P) :=\n  monoidalLinearOfFaithful R (fullMonoidalSubcategoryInclusion P)\n\nend\n\nvariable {P} {P' : C \u2192 Prop} [MonoidalPredicate P']\n\n/-- An implication of predicates `P \u2192 P'` induces a monoidal functor between full monoidal\nsubcategories. -/\n@[simps]\ndef fullMonoidalSubcategory.map (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    MonoidalFunctor (FullSubcategory P) (FullSubcategory P')\n    where\n  toFunctor := FullSubcategory.map h\n  \u03b5 := \ud835\udfd9 _\n  \u03bc X Y := \ud835\udfd9 _\n#align category_theory.monoidal_category.full_monoidal_subcategory.map CategoryTheory.MonoidalCategory.fullMonoidalSubcategory.map\n\ninstance fullMonoidalSubcategory.mapFull (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    Full (fullMonoidalSubcategory.map h).toFunctor where preimage X Y f := f\n#align category_theory.monoidal_category.full_monoidal_subcategory.map_full CategoryTheory.MonoidalCategory.fullMonoidalSubcategory.mapFull\n\ninstance fullMonoidalSubcategory.map_faithful (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    Faithful (fullMonoidalSubcategory.map h).toFunctor where\n#align category_theory.monoidal_category.full_monoidal_subcategory.map_faithful CategoryTheory.MonoidalCategory.fullMonoidalSubcategory.map_faithful\n\nsection Braided\n\nvariable (P) [BraidedCategory C]\n\n/-- The braided structure on a full subcategory inherited by the braided structure on `C`.\n-/\ninstance fullBraidedSubcategory : BraidedCategory (FullSubcategory P) :=\n  braidedCategoryOfFaithful (fullMonoidalSubcategoryInclusion P)\n    (fun X Y =>\n      \u27e8(\u03b2_ X.1 Y.1).Hom, (\u03b2_ X.1 Y.1).inv, (\u03b2_ X.1 Y.1).hom_inv_id, (\u03b2_ X.1 Y.1).inv_hom_id\u27e9)\n    fun X Y => by tidy\n#align category_theory.monoidal_category.full_braided_subcategory CategoryTheory.MonoidalCategory.fullBraidedSubcategory\n\n/-- The forgetful braided functor from a full braided subcategory into the original category\n(\"forgetting\" the condition).\n-/\n@[simps]\ndef fullBraidedSubcategoryInclusion : BraidedFunctor (FullSubcategory P) C\n    where\n  toMonoidalFunctor := fullMonoidalSubcategoryInclusion P\n  braided' X Y := by\n    rw [is_iso.eq_inv_comp]\n    tidy\n#align category_theory.monoidal_category.full_braided_subcategory_inclusion CategoryTheory.MonoidalCategory.fullBraidedSubcategoryInclusion\n\ninstance fullBraidedSubcategory.full : Full (fullBraidedSubcategoryInclusion P).toFunctor :=\n  fullMonoidalSubcategory.full P\n#align category_theory.monoidal_category.full_braided_subcategory.full CategoryTheory.MonoidalCategory.fullBraidedSubcategory.full\n\ninstance fullBraidedSubcategory.faithful : Faithful (fullBraidedSubcategoryInclusion P).toFunctor :=\n  fullMonoidalSubcategory.faithful P\n#align category_theory.monoidal_category.full_braided_subcategory.faithful CategoryTheory.MonoidalCategory.fullBraidedSubcategory.faithful\n\nvariable {P}\n\n/-- An implication of predicates `P \u2192 P'` induces a braided functor between full braided\nsubcategories. -/\n@[simps]\ndef fullBraidedSubcategory.map (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    BraidedFunctor (FullSubcategory P) (FullSubcategory P')\n    where\n  toMonoidalFunctor := fullMonoidalSubcategory.map h\n  braided' X Y := by\n    rw [is_iso.eq_inv_comp]\n    tidy\n#align category_theory.monoidal_category.full_braided_subcategory.map CategoryTheory.MonoidalCategory.fullBraidedSubcategory.map\n\ninstance fullBraidedSubcategory.mapFull (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    Full (fullBraidedSubcategory.map h).toFunctor :=\n  fullMonoidalSubcategory.mapFull h\n#align category_theory.monoidal_category.full_braided_subcategory.map_full CategoryTheory.MonoidalCategory.fullBraidedSubcategory.mapFull\n\ninstance fullBraidedSubcategory.map_faithful (h : \u2200 \u2983X\u2984, P X \u2192 P' X) :\n    Faithful (fullBraidedSubcategory.map h).toFunctor :=\n  fullMonoidalSubcategory.map_faithful h\n#align category_theory.monoidal_category.full_braided_subcategory.map_faithful CategoryTheory.MonoidalCategory.fullBraidedSubcategory.map_faithful\n\nend Braided\n\nsection Symmetric\n\nvariable (P) [SymmetricCategory C]\n\ninstance fullSymmetricSubcategory : SymmetricCategory (FullSubcategory P) :=\n  symmetricCategoryOfFaithful (fullBraidedSubcategoryInclusion P)\n#align category_theory.monoidal_category.full_symmetric_subcategory CategoryTheory.MonoidalCategory.fullSymmetricSubcategory\n\nend Symmetric\n\nsection Closed\n\nvariable (P) [MonoidalClosed C]\n\n/-- A property `C \u2192 Prop` is a closed predicate if it is closed under taking internal homs\n-/\nclass ClosedPredicate : Prop where\n  prop_ihom' : \u2200 {X Y}, P X \u2192 P Y \u2192 P ((ihom X).obj Y) := by obviously\n#align category_theory.monoidal_category.closed_predicate CategoryTheory.MonoidalCategory.ClosedPredicate\n\nrestate_axiom closed_predicate.prop_ihom'\n\nopen ClosedPredicate\n\nvariable [ClosedPredicate P]\n\ninstance fullMonoidalClosedSubcategory : MonoidalClosed (FullSubcategory P)\n    where closed' X :=\n    {\n      isAdj :=\n        { right :=\n            FullSubcategory.lift P (fullSubcategoryInclusion P \u22d9 ihom X.1) fun Y =>\n              prop_ihom X.2 Y.2\n          adj :=\n            Adjunction.mkOfUnitCounit\n              { Unit :=\n                  { app := fun Y => (ihom.coev X.1).app Y.1\n                    naturality' := fun Y Z f => ihom.coev_naturality X.1 f }\n                counit :=\n                  { app := fun Y => (ihom.ev X.1).app Y.1\n                    naturality' := fun Y Z f => ihom.ev_naturality X.1 f }\n                left_triangle := by\n                  ext Y\n                  simp\n                  exact ihom.ev_coev X.1 Y.1\n                right_triangle := by\n                  ext Y\n                  simp\n                  exact ihom.coev_ev X.1 Y.1 } } }\n#align category_theory.monoidal_category.full_monoidal_closed_subcategory CategoryTheory.MonoidalCategory.fullMonoidalClosedSubcategory\n\n@[simp]\ntheorem fullMonoidalClosedSubcategory_ihom_obj (X Y : FullSubcategory P) :\n    ((ihom X).obj Y).obj = (ihom X.obj).obj Y.obj :=\n  rfl\n#align category_theory.monoidal_category.full_monoidal_closed_subcategory_ihom_obj CategoryTheory.MonoidalCategory.fullMonoidalClosedSubcategory_ihom_obj\n\n@[simp]\ntheorem fullMonoidalClosedSubcategory_ihom_map (X : FullSubcategory P) {Y Z : FullSubcategory P}\n    (f : Y \u27f6 Z) : (ihom X).map f = (ihom X.obj).map f :=\n  rfl\n#align category_theory.monoidal_category.full_monoidal_closed_subcategory_ihom_map CategoryTheory.MonoidalCategory.fullMonoidalClosedSubcategory_ihom_map\n\nend Closed\n\nend MonoidalCategory\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Monoidal/Subcategory.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.3856603743132927}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.abelian.basic\nimport category_theory.limits.preserves.shapes.kernels\nimport category_theory.adjunction.limits\n\n/-!\n# Transferring \"abelian-ness\" across a functor\n\nIf `C` is an additive category, `D` is an abelian category,\nwe have `F : C \u2964 D` `G : D \u2964 C` (both preserving zero morphisms),\n`G` is left exact (that is, preserves finite limits),\nand further we have `adj : G \u22a3 F` and `i : F \u22d9 G \u2245 \ud835\udfed C`,\nthen `C` is also abelian.\n\nSee <https://stacks.math.columbia.edu/tag/03A3>\n\n## Notes\nThe hypotheses, following the statement from the Stacks project,\nmay appear suprising: we don't ask that the counit of the adjunction is an isomorphism,\nbut just that we have some potentially unrelated isomorphism `i : F \u22d9 G \u2245 \ud835\udfed C`.\n\nHowever Lemma A1.1.1 from [Elephant] shows that in this situation the counit itself\nmust be an isomorphism, and thus that `C` is a reflective subcategory of `D`.\n\nSomeone may like to formalize that lemma, and restate this theorem in terms of `reflective`.\n(That lemma has a nice string diagrammatic proof that holds in any bicategory.)\n-/\n\nnoncomputable theory\n\nnamespace category_theory\nopen category_theory.limits\n\nuniverses v u\u2081 u\u2082\n\nnamespace abelian_of_adjunction\n\nvariables {C : Type u\u2081} [category.{v} C] [preadditive C]\nvariables {D : Type u\u2082} [category.{v} D] [abelian D]\nvariables (F : C \u2964 D)\nvariables (G : D \u2964 C) [functor.preserves_zero_morphisms G]\nvariables (i : F \u22d9 G \u2245 \ud835\udfed C) (adj : G \u22a3 F)\n\ninclude i\n\n/-- No point making this an instance, as it requires `i`. -/\nlemma has_kernels [preserves_finite_limits G] : has_kernels C :=\n{ has_limit := \u03bb X Y f, begin\n    have := nat_iso.naturality_1 i f,\n    simp at this,\n    rw \u2190this,\n    haveI : has_kernel (G.map (F.map f) \u226b i.hom.app _) := limits.has_kernel_comp_mono _ _,\n    apply limits.has_kernel_iso_comp,\n  end }\n\ninclude adj\n\n/-- No point making this an instance, as it requires `i` and `adj`. -/\nlemma has_cokernels : has_cokernels C :=\n{ has_colimit := \u03bb X Y f, begin\n    haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n    have := nat_iso.naturality_1 i f,\n    simp at this,\n    rw \u2190this,\n    haveI : has_cokernel (G.map (F.map f) \u226b i.hom.app _) := limits.has_cokernel_comp_iso _ _,\n    apply limits.has_cokernel_epi_comp,\n  end }\n\nvariables [limits.has_cokernels C]\n\n/-- Auxiliary construction for `coimage_iso_image` -/\ndef cokernel_iso {X Y : C} (f : X \u27f6 Y) : G.obj (cokernel (F.map f)) \u2245 cokernel f :=\nbegin\n  -- We have to write an explicit `preserves_colimits` type here,\n  -- as `left_adjoint_preserves_colimits` has universe variables.\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc G.obj (cokernel (F.map f))\n      \u2245 cokernel (G.map (F.map f)) : (as_iso (cokernel_comparison _ G)).symm\n  ... \u2245 cokernel (_ \u226b f \u226b _)       : cokernel_iso_of_eq (nat_iso.naturality_2 i f).symm\n  ... \u2245 cokernel (f \u226b _)           : cokernel_epi_comp _ _\n  ... \u2245 cokernel f                 : cokernel_comp_is_iso _ _\nend\n\nvariables [limits.has_kernels C] [preserves_finite_limits G]\n\n/-- Auxiliary construction for `coimage_iso_image` -/\ndef coimage_iso_image_aux {X Y : C} (f : X \u27f6 Y) :\n  kernel (G.map (cokernel.\u03c0 (F.map f))) \u2245 kernel (cokernel.\u03c0 f) :=\nbegin\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc kernel (G.map (cokernel.\u03c0 (F.map f)))\n      \u2245 kernel (cokernel.\u03c0 (G.map (F.map f)) \u226b cokernel_comparison (F.map f) G)\n          : kernel_iso_of_eq (\u03c0_comp_cokernel_comparison _ _).symm\n  ... \u2245 kernel (cokernel.\u03c0 (G.map (F.map f))) : kernel_comp_mono _ _\n  ... \u2245 kernel (cokernel.\u03c0 (_ \u226b f \u226b _) \u226b (cokernel_iso_of_eq _).hom)\n          : kernel_iso_of_eq (\u03c0_comp_cokernel_iso_of_eq_hom (nat_iso.naturality_2 i f)).symm\n  ... \u2245 kernel (cokernel.\u03c0 (_ \u226b f \u226b _))       : kernel_comp_mono _ _\n  ... \u2245 kernel (cokernel.\u03c0 (f \u226b i.inv.app Y) \u226b (cokernel_epi_comp (i.hom.app X) _).inv)\n          : kernel_iso_of_eq (by simp only [cokernel.\u03c0_desc, cokernel_epi_comp_inv])\n  ... \u2245 kernel (cokernel.\u03c0 (f \u226b _))           : kernel_comp_mono _ _\n  ... \u2245 kernel (inv (i.inv.app Y) \u226b cokernel.\u03c0 f \u226b (cokernel_comp_is_iso f (i.inv.app Y)).inv)\n          : kernel_iso_of_eq (by simp only [cokernel.\u03c0_desc, cokernel_comp_is_iso_inv,\n              iso.hom_inv_id_app_assoc, nat_iso.inv_inv_app])\n  ... \u2245 kernel (cokernel.\u03c0 f \u226b _)             : kernel_is_iso_comp _ _\n  ... \u2245 kernel (cokernel.\u03c0 f)                 : kernel_comp_mono _ _\nend\n\nvariables [functor.preserves_zero_morphisms F]\n\n/--\nAuxiliary definition: the abelian coimage and abelian image agree.\nWe still need to check that this agrees with the canonical morphism.\n-/\ndef coimage_iso_image {X Y : C} (f : X \u27f6 Y) : abelian.coimage f \u2245 abelian.image f :=\nbegin\n  haveI : preserves_limits F := adj.right_adjoint_preserves_limits,\n  haveI : preserves_colimits G := adj.left_adjoint_preserves_colimits,\n  calc abelian.coimage f\n      \u2245 cokernel (kernel.\u03b9 f)                 : iso.refl _\n  ... \u2245 G.obj (cokernel (F.map (kernel.\u03b9 f))) : (cokernel_iso _ _ i adj _).symm\n  ... \u2245 G.obj (cokernel (kernel_comparison f F \u226b (kernel.\u03b9 (F.map f))))\n                                              : G.map_iso (cokernel_iso_of_eq (by simp))\n  ... \u2245 G.obj (cokernel (kernel.\u03b9 (F.map f))) : G.map_iso (cokernel_epi_comp _ _)\n  ... \u2245 G.obj (abelian.coimage (F.map f))     : iso.refl _\n  ... \u2245 G.obj (abelian.image (F.map f))       : G.map_iso (abelian.coimage_iso_image _)\n  ... \u2245 G.obj (kernel (cokernel.\u03c0 (F.map f))) : iso.refl _\n  ... \u2245 kernel (G.map (cokernel.\u03c0 (F.map f))) : preserves_kernel.iso _ _\n  ... \u2245 kernel (cokernel.\u03c0 f)                 : coimage_iso_image_aux F G i adj f\n  ... \u2245 abelian.image f                       : iso.refl _,\nend\n\nlocal attribute [simp] cokernel_iso coimage_iso_image coimage_iso_image_aux\n\n-- The account of this proof in the Stacks project omits this calculation.\nlemma coimage_iso_image_hom {X Y : C} (f : X \u27f6 Y) :\n  (coimage_iso_image F G i adj f).hom = abelian.coimage_image_comparison f :=\nbegin\n  ext, \n  simpa only [\u2190G.map_comp_assoc, coimage_iso_image, nat_iso.inv_inv_app, cokernel_iso,\n    coimage_iso_image_aux, iso.trans_symm, iso.symm_symm_eq, iso.refl_trans, iso.trans_refl,\n    iso.trans_hom, iso.symm_hom, cokernel_comp_is_iso_inv, cokernel_epi_comp_inv, as_iso_hom,\n    functor.map_iso_hom, cokernel_epi_comp_hom, preserves_kernel.iso_hom, kernel_comp_mono_hom,\n    kernel_is_iso_comp_hom, cokernel_iso_of_eq_hom_comp_desc_assoc, cokernel.\u03c0_desc_assoc,\n    category.assoc, \u03c0_comp_cokernel_iso_of_eq_inv_assoc, \u03c0_comp_cokernel_comparison_assoc,\n    kernel.lift_\u03b9, kernel.lift_\u03b9_assoc, kernel_iso_of_eq_hom_comp_\u03b9_assoc,\n    kernel_comparison_comp_\u03b9_assoc,\n    abelian.coimage_image_factorisation] using nat_iso.naturality_1 i f\nend\n\nend abelian_of_adjunction\n\nopen abelian_of_adjunction\n\n/--\nIf `C` is an additive category, `D` is an abelian category,\nwe have `F : C \u2964 D` `G : D \u2964 C` (both preserving zero morphisms),\n`G` is left exact (that is, preserves finite limits),\nand further we have `adj : G \u22a3 F` and `i : F \u22d9 G \u2245 \ud835\udfed C`,\nthen `C` is also abelian.\n\nSee <https://stacks.math.columbia.edu/tag/03A3>\n-/\ndef abelian_of_adjunction\n  {C : Type u\u2081} [category.{v} C] [preadditive C] [has_finite_products C]\n  {D : Type u\u2082} [category.{v} D] [abelian D]\n  (F : C \u2964 D) [functor.preserves_zero_morphisms F]\n  (G : D \u2964 C) [functor.preserves_zero_morphisms G] [preserves_finite_limits G]\n  (i : F \u22d9 G \u2245 \ud835\udfed C) (adj : G \u22a3 F) : abelian C :=\nbegin\n  haveI := has_kernels F G i, haveI := has_cokernels F G i adj,\n  haveI : \u2200 {X Y : C} (f : X \u27f6 Y), is_iso (abelian.coimage_image_comparison f),\n  { intros X Y f, rw \u2190coimage_iso_image_hom F G i adj f, apply_instance, },\n  apply abelian.of_coimage_image_comparison_is_iso,\nend\n\n/--\nIf `C` is an additive category equivalent to an abelian category `D`\nvia a functor that preserves zero morphisms,\nthen `C` is also abelian.\n-/\ndef abelian_of_equivalence\n  {C : Type u\u2081} [category.{v} C] [preadditive C] [has_finite_products C]\n  {D : Type u\u2082} [category.{v} D] [abelian D]\n  (F : C \u2964 D) [functor.preserves_zero_morphisms F] [is_equivalence F] : abelian C :=\nabelian_of_adjunction F F.inv F.as_equivalence.unit_iso.symm F.as_equivalence.symm.to_adjunction\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/abelian/transfer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.38560152940462855}}
{"text": "\nimport data.real.basic\n\nstructure myspace :=\n(x : \u211d) (y : \u211d)\n\n\n#check (\u03bb s : myspace, s.x + s.y)\n\nexample : (\u03bb s : myspace, s.x + s.y) \u2260 (\u03bb s : myspace, s.x + s.y) :=\nbegin\nunfold myspace,\nend", "meta": {"author": "skbaek", "repo": "cvx", "sha": "c50c790c9116f9fac8dfe742903a62bdd7292c15", "save_path": "github-repos/lean/skbaek-cvx", "path": "github-repos/lean/skbaek-cvx/cvx-c50c790c9116f9fac8dfe742903a62bdd7292c15/src/alex_playground/structure.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8962513842182777, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.3855201565878837}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, E. W. Ayers\n-/\n\nimport category_theory.sites.sieves\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.category.preorder\nimport order.copy\n\n/-!\n# Grothendieck topologies\n\nDefinition and lemmas about Grothendieck topologies.\nA Grothendieck topology for a category `C` is a set of sieves on each object `X` satisfying\ncertain closure conditions.\n\nAlternate versions of the axioms (in arrow form) are also described.\nTwo explicit examples of Grothendieck topologies are given:\n* The dense topology\n* The atomic topology\nas well as the complete lattice structure on Grothendieck topologies (which gives two additional\nexplicit topologies: the discrete and trivial topologies.)\n\nA pretopology, or a basis for a topology is defined in `pretopology.lean`. The topology associated\nto a topological space is defined in `spaces.lean`.\n\n## Tags\n\nGrothendieck topology, coverage, pretopology, site\n\n## References\n\n* [nLab, *Grothendieck topology*](https://ncatlab.org/nlab/show/Grothendieck+topology)\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n\n## Implementation notes\n\nWe use the definition of [nlab] and [MM92][] (Chapter III, Section 2), where Grothendieck topologies\nare saturated collections of morphisms, rather than the notions of the Stacks project (00VG) and\nthe Elephant, in which topologies are allowed to be unsaturated, and are then completed.\nTODO (BM): Add the definition from Stacks, as a pretopology, and complete to a topology.\n\nThis is so that we can produce a bijective correspondence between Grothendieck topologies on a\nsmall category and Lawvere-Tierney topologies on its presheaf topos, as well as the equivalence\nbetween Grothendieck topoi and left exact reflective subcategories of presheaf toposes.\n-/\n\nuniverses w v u\nnamespace category_theory\n\nopen category_theory category\n\nvariables (C : Type u) [category.{v} C]\n\n/--\nThe definition of a Grothendieck topology: a set of sieves `J X` on each object `X` satisfying\nthree axioms:\n1. For every object `X`, the maximal sieve is in `J X`.\n2. If `S \u2208 J X` then its pullback along any `h : Y \u27f6 X` is in `J Y`.\n3. If `S \u2208 J X` and `R` is a sieve on `X`, then provided that the pullback of `R` along any arrow\n   `f : Y \u27f6 X` in `S` is in `J Y`, we have that `R` itself is in `J X`.\n\nA sieve `S` on `X` is referred to as `J`-covering, (or just covering), if `S \u2208 J X`.\n\nSee https://stacks.math.columbia.edu/tag/00Z4, or [nlab], or [MM92][] Chapter III, Section 2,\nDefinition 1.\n-/\nstructure grothendieck_topology :=\n(sieves : \u03a0 (X : C), set (sieve X))\n(top_mem' : \u2200 X, \u22a4 \u2208 sieves X)\n(pullback_stable' : \u2200 \u2983X Y : C\u2984 \u2983S : sieve X\u2984 (f : Y \u27f6 X), S \u2208 sieves X \u2192 S.pullback f \u2208 sieves Y)\n(transitive' : \u2200 \u2983X\u2984 \u2983S : sieve X\u2984 (hS : S \u2208 sieves X) (R : sieve X),\n              (\u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984, S f \u2192 R.pullback f \u2208 sieves Y) \u2192 R \u2208 sieves X)\n\nnamespace grothendieck_topology\n\ninstance : has_coe_to_fun (grothendieck_topology C) (\u03bb _, \u03a0 X : C, set (sieve X)) := \u27e8sieves\u27e9\n\nvariables {C} {X Y : C} {S R : sieve X}\nvariables (J : grothendieck_topology C)\n\n/--\nAn extensionality lemma in terms of the coercion to a pi-type.\nWe prove this explicitly rather than deriving it so that it is in terms of the coercion rather than\nthe projection `.sieves`.\n-/\n@[ext]\nlemma ext {J\u2081 J\u2082 : grothendieck_topology C} (h : (J\u2081 : \u03a0 (X : C), set (sieve X)) = J\u2082) : J\u2081 = J\u2082 :=\nby { cases J\u2081, cases J\u2082, congr, apply h }\n\n@[simp] lemma mem_sieves_iff_coe : S \u2208 J.sieves X \u2194 S \u2208 J X := iff.rfl\n\n-- Also known as the maximality axiom.\n@[simp] lemma top_mem (X : C) : \u22a4 \u2208 J X := J.top_mem' X\n-- Also known as the stability axiom.\n@[simp] lemma pullback_stable (f : Y \u27f6 X) (hS : S \u2208 J X) : S.pullback f \u2208 J Y :=\nJ.pullback_stable' f hS\nlemma transitive (hS : S \u2208 J X) (R : sieve X)\n  (h : \u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984, S f \u2192 R.pullback f \u2208 J Y) :\n  R \u2208 J X :=\nJ.transitive' hS R h\n\nlemma covering_of_eq_top : S = \u22a4 \u2192 S \u2208 J X := \u03bb h, h.symm \u25b8 J.top_mem X\n\n/--\nIf `S` is a subset of `R`, and `S` is covering, then `R` is covering as well.\n\nSee https://stacks.math.columbia.edu/tag/00Z5 (2), or discussion after [MM92] Chapter III,\nSection 2, Definition 1.\n-/\nlemma superset_covering (Hss : S \u2264 R) (sjx : S \u2208 J X) : R \u2208 J X :=\nbegin\n  apply J.transitive sjx R (\u03bb Y f hf, _),\n  apply covering_of_eq_top,\n  rw [\u2190 top_le_iff, \u2190 S.pullback_eq_top_of_mem hf],\n  apply sieve.pullback_monotone _ Hss,\nend\n\n/--\nThe intersection of two covering sieves is covering.\n\nSee https://stacks.math.columbia.edu/tag/00Z5 (1), or [MM92] Chapter III,\nSection 2, Definition 1 (iv).\n-/\nlemma intersection_covering (rj : R \u2208 J X) (sj : S \u2208 J X) : R \u2293 S \u2208 J X :=\nbegin\n  apply J.transitive rj _ (\u03bb Y f Hf, _),\n  rw [sieve.pullback_inter, R.pullback_eq_top_of_mem Hf],\n  simp [sj],\nend\n\n@[simp]\nlemma intersection_covering_iff : R \u2293 S \u2208 J X \u2194 R \u2208 J X \u2227 S \u2208 J X :=\n\u27e8\u03bb h, \u27e8J.superset_covering inf_le_left h, J.superset_covering inf_le_right h\u27e9,\n \u03bb t, intersection_covering _ t.1 t.2\u27e9\n\n\n\n/--\nThe sieve `S` on `X` `J`-covers an arrow `f` to `X` if `S.pullback f \u2208 J Y`.\nThis definition is an alternate way of presenting a Grothendieck topology.\n-/\ndef covers (S : sieve X) (f : Y \u27f6 X) : Prop := S.pullback f \u2208 J Y\n\nlemma covers_iff (S : sieve X) (f : Y \u27f6 X) : J.covers S f \u2194 S.pullback f \u2208 J Y :=\niff.rfl\n\nlemma covering_iff_covers_id (S : sieve X) : S \u2208 J X \u2194 J.covers S (\ud835\udfd9 X) :=\nby simp [covers_iff]\n\n/-- The maximality axiom in 'arrow' form: Any arrow `f` in `S` is covered by `S`. -/\nlemma arrow_max (f : Y \u27f6 X) (S : sieve X) (hf : S f) : J.covers S f :=\nbegin\n  rw [covers, (sieve.pullback_eq_top_iff_mem f).1 hf],\n  apply J.top_mem,\nend\n\n/-- The stability axiom in 'arrow' form: If `S` covers `f` then `S` covers `g \u226b f` for any `g`. -/\nlemma arrow_stable (f : Y \u27f6 X) (S : sieve X) (h : J.covers S f) {Z : C} (g : Z \u27f6 Y) :\n  J.covers S (g \u226b f) :=\nbegin\n  rw covers_iff at h \u22a2,\n  simp [h, sieve.pullback_comp],\nend\n\n/--\nThe transitivity axiom in 'arrow' form: If `S` covers `f` and every arrow in `S` is covered by\n`R`, then `R` covers `f`.\n-/\nlemma arrow_trans (f : Y \u27f6 X) (S R : sieve X) (h : J.covers S f) :\n  (\u2200 {Z : C} (g : Z \u27f6 X), S g \u2192 J.covers R g) \u2192 J.covers R f :=\nbegin\n  intro k,\n  apply J.transitive h,\n  intros Z g hg,\n  rw \u2190 sieve.pullback_comp,\n  apply k (g \u226b f) hg,\nend\n\nlemma arrow_intersect (f : Y \u27f6 X) (S R : sieve X) (hS : J.covers S f) (hR : J.covers R f) :\n  J.covers (S \u2293 R) f :=\nby simpa [covers_iff] using and.intro hS hR\n\nvariable (C)\n/--\nThe trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee [MM92] Chapter III, Section 2, example (a), or\nhttps://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies\n-/\ndef trivial : grothendieck_topology C :=\n{ sieves := \u03bb X, {\u22a4},\n  top_mem' := \u03bb X, rfl,\n  pullback_stable' := \u03bb X Y S f hf,\n  begin\n    rw set.mem_singleton_iff at \u22a2 hf,\n    simp [hf],\n  end,\n  transitive' := \u03bb X S hS R hR,\n  begin\n    rw [set.mem_singleton_iff, \u2190 sieve.id_mem_iff_eq_top] at hS,\n    simpa using hR hS,\n  end }\n\n/--\nThe discrete Grothendieck topology, in which every sieve is covering.\n\nSee https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies.\n-/\ndef discrete : grothendieck_topology C :=\n{ sieves := \u03bb X, set.univ,\n  top_mem' := by simp,\n  pullback_stable' := \u03bb X Y f, by simp,\n  transitive' := by simp }\nvariable {C}\n\nlemma trivial_covering : S \u2208 trivial C X \u2194 S = \u22a4 := set.mem_singleton_iff\n\n/-- See https://stacks.math.columbia.edu/tag/00Z6 -/\ninstance : has_le (grothendieck_topology C) :=\n{ le := \u03bb J\u2081 J\u2082, (J\u2081 : \u03a0 (X : C), set (sieve X)) \u2264 (J\u2082 : \u03a0 (X : C), set (sieve X)) }\n\nlemma le_def {J\u2081 J\u2082 : grothendieck_topology C} :\n  J\u2081 \u2264 J\u2082 \u2194 (J\u2081 : \u03a0 (X : C), set (sieve X)) \u2264 J\u2082 := iff.rfl\n\n/-- See https://stacks.math.columbia.edu/tag/00Z6 -/\ninstance : partial_order (grothendieck_topology C) :=\n{ le_refl := \u03bb J\u2081, le_def.mpr le_rfl,\n  le_trans := \u03bb J\u2081 J\u2082 J\u2083 h\u2081\u2082 h\u2082\u2083, le_def.mpr (le_trans h\u2081\u2082 h\u2082\u2083),\n  le_antisymm := \u03bb J\u2081 J\u2082 h\u2081\u2082 h\u2082\u2081, grothendieck_topology.ext (le_antisymm h\u2081\u2082 h\u2082\u2081),\n  ..grothendieck_topology.has_le }\n\n/-- See https://stacks.math.columbia.edu/tag/00Z7 -/\ninstance : has_Inf (grothendieck_topology C) :=\n{ Inf := \u03bb T,\n  { sieves := Inf (sieves '' T),\n    top_mem' :=\n    begin\n      rintro X S \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      simp,\n    end,\n    pullback_stable' :=\n    begin\n      rintro X Y S hS f _ \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      apply J.pullback_stable _ (f _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9),\n    end,\n    transitive' :=\n    begin\n      rintro X S hS R h _ \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      apply J.transitive (hS _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9) _ (\u03bb Y f hf, h hf _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9),\n    end } }\n\n/-- See https://stacks.math.columbia.edu/tag/00Z7 -/\nlemma is_glb_Inf (s : set (grothendieck_topology C)) : is_glb s (Inf s) :=\nbegin\n  refine @is_glb.of_image _ _ _ _ sieves _ _ _ _,\n  { intros, refl },\n  { exact is_glb_Inf _ },\nend\n\n/--\nConstruct a complete lattice from the `Inf`, but make the trivial and discrete topologies\ndefinitionally equal to the bottom and top respectively.\n-/\ninstance : complete_lattice (grothendieck_topology C) :=\ncomplete_lattice.copy\n(complete_lattice_of_Inf _ is_glb_Inf)\n_ rfl\n(discrete C)\n(begin\n  apply le_antisymm,\n  { exact @complete_lattice.le_top _ (complete_lattice_of_Inf _ is_glb_Inf) (discrete C) },\n  { intros X S hS,\n    apply set.mem_univ },\nend)\n(trivial C)\n(begin\n  apply le_antisymm,\n  { intros X S hS,\n    rw trivial_covering at hS,\n    apply covering_of_eq_top _ hS },\n  { refine @complete_lattice.bot_le _ (complete_lattice_of_Inf _ is_glb_Inf) (trivial C) },\nend)\n_ rfl\n_ rfl\n_ rfl\nInf rfl\n\ninstance : inhabited (grothendieck_topology C) := \u27e8\u22a4\u27e9\n\n@[simp] lemma trivial_eq_bot : trivial C = \u22a5 := rfl\n@[simp] lemma discrete_eq_top : discrete C = \u22a4 := rfl\n\n@[simp] lemma bot_covering : S \u2208 (\u22a5 : grothendieck_topology C) X \u2194 S = \u22a4 := trivial_covering\n@[simp] lemma top_covering : S \u2208 (\u22a4 : grothendieck_topology C) X := \u27e8\u27e9\n\nlemma bot_covers (S : sieve X) (f : Y \u27f6 X) :\n  (\u22a5 : grothendieck_topology C).covers S f \u2194 S f :=\nby rw [covers_iff, bot_covering, \u2190 sieve.pullback_eq_top_iff_mem]\n\n@[simp] lemma top_covers (S : sieve X) (f : Y \u27f6 X) : (\u22a4 : grothendieck_topology C).covers S f :=\nby simp [covers_iff]\n\n/--\nThe dense Grothendieck topology.\n\nSee https://ncatlab.org/nlab/show/dense+topology, or [MM92] Chapter III, Section 2, example (e).\n-/\ndef dense : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2200 {Y : C} (f : Y \u27f6 X), \u2203 Z (g : Z \u27f6 Y), S (g \u226b f),\n  top_mem' := \u03bb X Y f, \u27e8Y, \ud835\udfd9 Y, \u27e8\u27e9\u27e9,\n  pullback_stable' :=\n  begin\n    intros X Y S h H Z f,\n    rcases H (f \u226b h) with \u27e8W, g, H'\u27e9,\n    exact \u27e8W, g, by simpa\u27e9,\n  end,\n  transitive' :=\n  begin\n    intros X S H\u2081 R H\u2082 Y f,\n    rcases H\u2081 f with \u27e8Z, g, H\u2083\u27e9,\n    rcases H\u2082 H\u2083 (\ud835\udfd9 Z) with \u27e8W, h, H\u2084\u27e9,\n    exact \u27e8W, (h \u226b g), by simpa using H\u2084\u27e9,\n  end }\n\nlemma dense_covering : S \u2208 dense X \u2194 \u2200 {Y} (f : Y \u27f6 X), \u2203 Z (g : Z \u27f6 Y), S (g \u226b f) :=\niff.rfl\n\n/--\nA category satisfies the right Ore condition if any span can be completed to a commutative square.\nNB. Any category with pullbacks obviously satisfies the right Ore condition, see\n`right_ore_of_pullbacks`.\n-/\ndef right_ore_condition (C : Type u) [category.{v} C] : Prop :=\n\u2200 {X Y Z : C} (yx : Y \u27f6 X) (zx : Z \u27f6 X), \u2203 W (wy : W \u27f6 Y) (wz : W \u27f6 Z), wy \u226b yx = wz \u226b zx\n\nlemma right_ore_of_pullbacks [limits.has_pullbacks C] : right_ore_condition C :=\n\u03bb X Y Z yx zx, \u27e8_, _, _, limits.pullback.condition\u27e9\n\n/--\nThe atomic Grothendieck topology: a sieve is covering iff it is nonempty.\nFor the pullback stability condition, we need the right Ore condition to hold.\n\nSee https://ncatlab.org/nlab/show/atomic+site, or [MM92] Chapter III, Section 2, example (f).\n-/\ndef atomic (hro : right_ore_condition C) : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2203 Y (f : Y \u27f6 X), S f,\n  top_mem' := \u03bb X, \u27e8_, \ud835\udfd9 _, \u27e8\u27e9\u27e9,\n  pullback_stable' :=\n  begin\n    rintros X Y S h \u27e8Z, f, hf\u27e9,\n    rcases hro h f with \u27e8W, g, k, comm\u27e9,\n    refine \u27e8_, g, _\u27e9,\n    simp [comm, hf],\n  end,\n  transitive' :=\n  begin\n    rintros X S \u27e8Y, f, hf\u27e9 R h,\n    rcases h hf with \u27e8Z, g, hg\u27e9,\n    exact \u27e8_, _, hg\u27e9,\n  end }\n\n/-- `J.cover X` denotes the poset of covers of `X` with respect to the\nGrothendieck topology `J`. -/\n@[derive preorder]\ndef cover (X : C) := { S : sieve X // S \u2208 J X }\n\nnamespace cover\n\nvariables {J}\n\ninstance : has_coe (J.cover X) (sieve X) := \u27e8\u03bb S, S.1\u27e9\n\ninstance : has_coe_to_fun (J.cover X) (\u03bb S, \u03a0 \u2983Y\u2984 (f : Y \u27f6 X), Prop) :=\n\u27e8\u03bb S Y f, (S : sieve X) f\u27e9\n\n@[simp]\nlemma coe_fun_coe (S : J.cover X) (f : Y \u27f6 X) : (S : sieve X) f = S f := rfl\n\nlemma condition (S : J.cover X) : (S : sieve X) \u2208 J X := S.2\n\n@[ext]\nlemma ext (S T : J.cover X) (h : \u2200 \u2983Y\u2984 (f : Y \u27f6 X), S f \u2194 T f) : S = T :=\nsubtype.ext $ sieve.ext h\n\ninstance : order_top (J.cover X) :=\n{ top := \u27e8\u22a4, J.top_mem _\u27e9,\n  le_top := \u03bb S Y f h, by tauto,\n  ..(infer_instance : preorder _) }\n\ninstance : semilattice_inf (J.cover X) :=\n{ inf := \u03bb S T, \u27e8S \u2293 T, J.intersection_covering S.condition T.condition\u27e9,\n  le_antisymm := \u03bb S T h1 h2, ext _ _ $ \u03bb Y f, \u27e8h1 _, h2 _\u27e9,\n  inf_le_left := \u03bb S T Y f hf, hf.1,\n  inf_le_right := \u03bb S T Y f hf, hf.2,\n  le_inf := \u03bb S T W h1 h2 Y f h, \u27e8h1 _ h, h2 _ h\u27e9,\n  ..(infer_instance : preorder _) }\n\ninstance : inhabited (J.cover X) := \u27e8\u22a4\u27e9\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_inhabited_instance, ext]\nstructure arrow (S : J.cover X) :=\n(Y : C)\n(f : Y \u27f6 X)\n(hf : S f)\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_inhabited_instance, ext]\nstructure relation (S : J.cover X) :=\n(Y\u2081 Y\u2082 Z : C)\n(g\u2081 : Z \u27f6 Y\u2081)\n(g\u2082 : Z \u27f6 Y\u2082)\n(f\u2081 : Y\u2081 \u27f6 X)\n(f\u2082 : Y\u2082 \u27f6 X)\n(h\u2081 : S f\u2081)\n(h\u2082 : S f\u2082)\n(w : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082)\n\n/-- Map a `arrow` along a refinement `S \u27f6 T`. -/\n@[simps]\ndef arrow.map {S T : J.cover X} (I : S.arrow) (f : S \u27f6 T) : T.arrow :=\n\u27e8I.Y, I.f, f.le _ I.hf\u27e9\n\n/-- Map a `relation` along a refinement `S \u27f6 T`. -/\n@[simps]\ndef relation.map {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) : T.relation :=\n\u27e8_, _, _, I.g\u2081, I.g\u2082, I.f\u2081, I.f\u2082, f.le _ I.h\u2081, f.le _ I.h\u2082, I.w\u27e9\n\n/-- The first `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.fst {S : J.cover X} (I : S.relation) : S.arrow :=\n\u27e8I.Y\u2081, I.f\u2081, I.h\u2081\u27e9\n\n/-- The second `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.snd {S : J.cover X} (I : S.relation) : S.arrow :=\n\u27e8I.Y\u2082, I.f\u2082, I.h\u2082\u27e9\n\n@[simp]\nlemma relation.map_fst {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) :\n   I.fst.map f = (I.map f).fst := rfl\n\n@[simp]\nlemma relation.map_snd {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) :\n  I.snd.map f = (I.map f).snd := rfl\n\n/-- Pull back a cover along a morphism. -/\ndef pullback (S : J.cover X) (f : Y \u27f6 X) : J.cover Y :=\n\u27e8sieve.pullback f S, J.pullback_stable _ S.condition\u27e9\n\n/-- An arrow of `S.pullback f` gives rise to an arrow of `S`. -/\n@[simps]\ndef arrow.base {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).arrow) : S.arrow :=\n\u27e8I.Y, I.f \u226b f, I.hf\u27e9\n\n/-- A relation of `S.pullback f` gives rise to a relation of `S`. -/\n@[simps]\ndef relation.base {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) : S.relation :=\n\u27e8_, _, _, I.g\u2081, I.g\u2082, I.f\u2081 \u226b f, I.f\u2082\u226b f, I.h\u2081, I.h\u2082, by simp [reassoc_of I.w]\u27e9\n\n@[simp]\nlemma relation.base_fst {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.fst.base = I.base.fst := rfl\n\n@[simp]\nlemma relation.base_snd {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.snd.base = I.base.snd := rfl\n\n@[simp]\nlemma coe_pullback {Z : C} (f : Y \u27f6 X) (g : Z \u27f6 Y) (S : J.cover X) :\n  (S.pullback f) g \u2194 S (g \u226b f) := iff.rfl\n\n/-- The isomorphism between `S` and the pullback of `S` w.r.t. the identity. -/\ndef pullback_id (S : J.cover X) : S.pullback (\ud835\udfd9 X) \u2245 S :=\neq_to_iso $ cover.ext _ _ $ \u03bb Y f, by simp\n\n/-- Pulling back with respect to a composition is the composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (S : J.cover X) (f : Z \u27f6 Y) (g : Y \u27f6 X) :\n  S.pullback (f \u226b g) \u2245 (S.pullback g).pullback f :=\neq_to_iso $ cover.ext _ _ $ \u03bb Y f, by simp\n\n/-- Combine a family of covers over a cover. -/\ndef bind {X : C} (S : J.cover X) (T : \u03a0 (I : S.arrow), J.cover I.Y) : J.cover X :=\n\u27e8sieve.bind S (\u03bb Y f hf, T \u27e8Y, f, hf\u27e9), J.bind_covering S.condition (\u03bb _ _ _, (T _).condition)\u27e9\n\n/-- The canonical moprhism from `S.bind T` to `T`. -/\ndef bind_to_base {X : C} (S : J.cover X) (T : \u03a0 (I : S.arrow), J.cover I.Y) : S.bind T \u27f6 S :=\nhom_of_le $ by { rintro Y f \u27e8Z,e1,e2,h1,h2,h3\u27e9, rw \u2190 h3, apply sieve.downward_closed, exact h1 }\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the object `B`. -/\nnoncomputable def arrow.middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : C :=\nI.hf.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `A \u27f6 B`. -/\nnoncomputable def arrow.to_middle_hom {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.Y \u27f6 I.middle :=\nI.hf.some_spec.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `B \u27f6 X`. -/\nnoncomputable def arrow.from_middle_hom {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.middle \u27f6 X :=\nI.hf.some_spec.some_spec.some\n\nlemma arrow.from_middle_condition {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S I.from_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `B \u27f6 X`, as an arrow. -/\nnoncomputable\ndef arrow.from_middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S.arrow := \u27e8_, I.from_middle_hom, I.from_middle_condition\u27e9\n\nlemma arrow.to_middle_condition {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle) I.to_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some_spec.1\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `A \u27f6 B`, as an arrow. -/\nnoncomputable\ndef arrow.to_middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle).arrow := \u27e8_, I.to_middle_hom, I.to_middle_condition\u27e9\n\nlemma arrow.middle_spec {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.to_middle_hom \u226b I.from_middle_hom = I.f :=\nI.hf.some_spec.some_spec.some_spec.some_spec.2\n\n-- This is used extensively in `plus.lean`, etc.\n-- We place this definition here as it will be used in `sheaf.lean` as well.\n/-- To every `S : J.cover X` and presheaf `P`, associate a `multicospan_index`. -/\ndef index {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D) :\n  limits.multicospan_index D :=\n{ L := S.arrow,\n  R := S.relation,\n  fst_to := \u03bb I, I.fst,\n  snd_to := \u03bb I, I.snd,\n  left := \u03bb I, P.obj (opposite.op I.Y),\n  right := \u03bb I, P.obj (opposite.op I.Z),\n  fst := \u03bb I, P.map I.g\u2081.op,\n  snd := \u03bb I, P.map I.g\u2082.op }\n\n/-- The natural multifork associated to `S : J.cover X` for a presheaf `P`.\nSaying that this multifork is a limit is essentially equivalent to the sheaf condition at the\ngiven object for the given covering sieve. See `sheaf.lean` for an equivalent sheaf condition\nusing this.\n-/\nabbreviation multifork {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D) :\n  limits.multifork (S.index P) :=\nlimits.multifork.of_\u03b9 _ (P.obj (opposite.op X)) (\u03bb I, P.map I.f.op) begin\n  intros I,\n  dsimp [index],\n  simp only [\u2190 P.map_comp, \u2190 op_comp, I.w]\nend\n\n/-- The canonical map from `P.obj (op X)` to the multiequalizer associated to a covering sieve,\nassuming such a multiequalizer exists. This will be used in `sheaf.lean` to provide an equivalent\nsheaf condition in terms of multiequalizers. -/\nnoncomputable\nabbreviation to_multiequalizer {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D)\n  [limits.has_multiequalizer (S.index P)] :\nP.obj (opposite.op X) \u27f6 limits.multiequalizer (S.index P) :=\nlimits.multiequalizer.lift _ _ (\u03bb I, P.map I.f.op) begin\n  intros I,\n  dsimp only [index, relation.fst, relation.snd],\n  simp only [\u2190 P.map_comp, \u2190 op_comp, I.w],\nend\n\nend cover\n\n/-- Pull back a cover along a morphism. -/\n@[simps obj]\ndef pullback (f : Y \u27f6 X) : J.cover X \u2964 J.cover Y :=\n{ obj := \u03bb S, S.pullback f,\n  map := \u03bb S T f, (sieve.pullback_monotone _ f.le).hom }\n\n/-- Pulling back along the identity is naturally isomorphic to the identity functor. -/\ndef pullback_id (X : C) : J.pullback (\ud835\udfd9 X) \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb S, S.pullback_id) $ by tidy\n\n/-- Pulling back along a composition is naturally isomorphic to\nthe composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  J.pullback (f \u226b g) \u2245 J.pullback g \u22d9 J.pullback f :=\nnat_iso.of_components (\u03bb S, S.pullback_comp f g) $ by tidy\n\nend grothendieck_topology\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/sites/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3852408395262473}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johan Commelin, Andrew Yang\n-/\nimport category_theory.limits.shapes.zero\nimport category_theory.monoidal.End\nimport category_theory.monoidal.discrete\n\n/-!\n# Shift\n\nA `shift` on a category `C` indexed by a monoid `A` is is nothing more than a monoidal functor\nfrom `A` to `C \u2964 C`. A typical example to keep in mind might be the category of\ncomplexes `\u22ef \u2192 C_{n-1} \u2192 C_n \u2192 C_{n+1} \u2192 \u22ef`. It has a shift indexed by `\u2124`, where we assign to\neach `n : \u2124` the functor `C \u2964 C` that re-indexing the terms, so the degree `i` term of `shift n C`\nwould be the degree `i+n`-th term of `C`.\n\n## Main definitions\n* `has_shift`: A typeclass asserting the existence of a shift functor.\n* `shift_equiv`: When the indexing monoid is a group, then the functor indexed by `n` and `-n` forms\n  an self-equivalence of `C`.\n* `shift_comm`: When the indexing monoid is commutative, then shifts commute as well.\n\n## Implementation Notes\n\nMost of the definitions in this file is marked as an `abbreviation` so that the simp lemmas in\n`category_theory/monoidal/End` could apply.\n\n-/\nnamespace category_theory\n\nnoncomputable theory\n\nuniverses v u\n\nvariables (C : Type u) (A : Type*) [category.{v} C]\n\nlocal attribute [instance] endofunctor_monoidal_category\nlocal attribute [reducible] endofunctor_monoidal_category discrete.add_monoidal\n\nsection eq_to_hom\n\nvariables {A C}\n\nvariables [add_monoid A] (F : monoidal_functor (discrete A) (C \u2964 C))\n\n @[simp, reassoc] lemma eq_to_hom_\u03bc_app {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (F.\u03bc i' j').app X =\n     (F.\u03bc i j).app X \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n @[simp, reassoc] lemma \u03bc_inv_app_eq_to_hom {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   (F.\u03bc_iso i j).inv.app X \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n     eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (F.\u03bc_iso i' j').inv.app X :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\nend eq_to_hom\n\nvariables {A C}\n\n/-- A monoidal functor from a group `A` into `C \u2964 C` induces\na self-equivalence of `C` for each `n : A`. -/\n@[simps functor inverse unit_iso_hom unit_iso_inv counit_iso_hom counit_iso_inv]\ndef add_neg_equiv [add_group A] (F : monoidal_functor (discrete A) (C \u2964 C)) (n : A) : C \u224c C :=\nequiv_of_tensor_iso_unit F n (-n : A)\n  (eq_to_iso (add_neg_self n)) (eq_to_iso (neg_add_self n)) (subsingleton.elim _ _)\n\nsection defs\n\nvariables (A C) [add_monoid A]\n\n/-- A category has a shift indexed by an additive monoid `A`\nif there is a monoidal functor from `A` to `C \u2964 C`. -/\nclass has_shift (C : Type u) (A : Type*) [category.{v} C] [add_monoid A] :=\n(shift : monoidal_functor (discrete A) (C \u2964 C))\n\n/-- A helper structure to construct the shift functor `(discrete A) \u2964 (C \u2964 C)`. -/\n@[nolint has_inhabited_instance]\nstructure shift_mk_core :=\n(F : A \u2192 (C \u2964 C))\n(\u03b5 : \ud835\udfed C \u2245 F 0)\n(\u03bc : \u03a0 n m : A, F n \u22d9 F m \u2245 F (n + m))\n(associativity : \u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n  (F m\u2083).map ((\u03bc m\u2081 m\u2082).hom.app X) \u226b (\u03bc (m\u2081 + m\u2082) m\u2083).hom.app X \u226b\n    eq_to_hom (by { congr' 2, exact add_assoc _ _ _ }) =\n    (\u03bc m\u2082 m\u2083).hom.app ((F m\u2081).obj X) \u226b (\u03bc m\u2081 (m\u2082 + m\u2083)).hom.app X . obviously)\n(left_unitality : \u2200 (n : A) (X : C),\n  (F n).map (\u03b5.hom.app X) \u226b (\u03bc 0 n).hom.app X =\n    eq_to_hom (by { dsimp, rw zero_add }) . obviously)\n(right_unitality : \u2200 (n : A) (X : C),\n  \u03b5.hom.app ((F n).obj X) \u226b (\u03bc n 0).hom.app X =\n    eq_to_hom (by { dsimp, rw add_zero }) . obviously)\n\n/-- Constructs a `has_shift C A` instance from `shift_mk_core`. -/\n@[simps]\ndef has_shift_mk (h : shift_mk_core C A) : has_shift C A :=\n\u27e8{ \u03b5 := h.\u03b5.hom,\n   \u03bc := \u03bb m n, (h.\u03bc m n).hom,\n   \u03bc_natural' := by { rintros _ _ _ _ \u27e8\u27e8rfl\u27e9\u27e9 \u27e8\u27e8rfl\u27e9\u27e9, ext, dsimp, simp, dsimp, simp },\n   associativity' := by { introv, ext, dsimp, simpa using h.associativity _ _ _ _, },\n   left_unitality' :=\n    by { introv, ext, dsimp, rw [category.id_comp, \u2190 category.assoc, h.left_unitality], simp },\n   right_unitality' :=\n    by { introv, ext, dsimp, rw [functor.map_id, category.comp_id,\n      \u2190 category.assoc, h.right_unitality], simp },\n ..(discrete.functor h.F) }\u27e9\n\nvariables [has_shift C A]\n\n/-- The monoidal functor from `A` to `C \u2964 C` given a `has_shift` instance. -/\ndef shift_monoidal_functor : monoidal_functor (discrete A) (C \u2964 C) := has_shift.shift\n\nvariable {A}\n\n/-- The shift autoequivalence, moving objects and morphisms 'up'. -/\nabbreviation shift_functor (i : A) : C \u2964 C := (shift_monoidal_functor C A).obj i\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_functor_add (i j : A) :\n  shift_functor C (i + j) \u2245 shift_functor C i \u22d9 shift_functor C j :=\n((shift_monoidal_functor C A).\u03bc_iso i j).symm\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_functor_zero : shift_functor C (0 : A) \u2245 \ud835\udfed C :=\n(shift_monoidal_functor C A).\u03b5_iso.symm\n\n-- Any better notational suggestions?\nnotation X`\u27e6`n`\u27e7`:20 := (shift_functor _ n).obj X\nnotation f`\u27e6`n`\u27e7'`:80 := (shift_functor _ n).map f\n\nend defs\n\nsection examples\nvariables [has_shift C \u2124]\n\nexample {X Y : C} (f : X \u27f6 Y) : X\u27e6(1 : \u2124)\u27e7 \u27f6 Y\u27e61\u27e7 := f\u27e61\u27e7'\nexample {X Y : C} (f : X \u27f6 Y) : X\u27e6(-2 : \u2124)\u27e7 \u27f6 Y\u27e6-2\u27e7 := f\u27e6-2\u27e7'\n\nend examples\n\nsection add_monoid\n\nvariables {C A} [add_monoid A] [has_shift C A] (X Y : C) (f : X \u27f6 Y)\n\n@[simp] lemma has_shift.shift_obj_obj (n : A) (X : C) : (has_shift.shift.obj n).obj X = X\u27e6n\u27e7 := rfl\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_add (i j : A) : X\u27e6i + j\u27e7 \u2245 X\u27e6i\u27e7\u27e6j\u27e7 := (shift_functor_add C i j).app _\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081 (i i' j : A) (h : i = i') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i' j).hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2082 (i j j' : A) (h : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i j').hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n    eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').hom :=\nby { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081 (i i' j : A) (h : i = i') :\n  eq_to_hom (by rw h) \u226b (shift_add X i' j).inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2082 (i j j' : A) (h : j = j') :\n  eq_to_hom (by rw h) \u226b (shift_add X i j').inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2082]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').inv =\n    (shift_add X i j).inv \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081\u2082]\n\nlemma shift_shift' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_add X i j).inv \u226b f\u27e6i + j\u27e7' \u226b (shift_add Y i j).hom :=\nby { symmetry, apply nat_iso.naturality_1 }\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_zero  :\n  X\u27e60\u27e7 \u2245 X := (shift_functor_zero C A).app _\n\nlemma shift_zero' :\n  f\u27e6(0 : A)\u27e7' = (shift_zero A X).hom \u226b f \u226b (shift_zero A Y).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nend add_monoid\n\nsection opaque_eq_to_iso\n\nvariables {\u03b9 : Type*} {i j k : \u03b9}\n\n/-- This definition is used instead of `eq_to_iso` so that the proof of `i = j` is visible\nto the simplifier -/\ndef opaque_eq_to_iso (h : i = j) : @iso (discrete \u03b9) _ i j := eq_to_iso h\n\n@[simp]\nlemma opaque_eq_to_iso_symm (h : i = j) :\n  (opaque_eq_to_iso h).symm = opaque_eq_to_iso h.symm := rfl\n\n@[simp]\nlemma opaque_eq_to_iso_inv (h : i = j) :\n  (opaque_eq_to_iso h).inv = (opaque_eq_to_iso h.symm).hom := rfl\n\n@[simp, reassoc]\nlemma map_opaque_eq_to_iso_comp_app (F : discrete \u03b9 \u2964 C \u2964 C) (h : i = j) (h' : j = k) (X : C) :\n  (F.map (opaque_eq_to_iso h).hom).app X \u226b (F.map (opaque_eq_to_iso h').hom).app X =\n    (F.map (opaque_eq_to_iso $ h.trans h').hom).app X := by { delta opaque_eq_to_iso, simp }\n\nend opaque_eq_to_iso\n\nsection add_group\n\nvariables (C) {A} [add_group A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_functor_comp_shift_functor_neg (i : A) :\n  shift_functor C i \u22d9 shift_functor C (-i) \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) i (-i : A) (opaque_eq_to_iso (add_neg_self i))\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_functor_neg_comp_shift_functor (i : A) :\n  shift_functor C (-i) \u22d9 shift_functor C i \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) (-i : A) i (opaque_eq_to_iso (neg_add_self i))\n\nsection\n\nvariables (C)\n\n/-- Shifting by `n` is a faithful functor. -/\ninstance shift_functor_faithful (i : A) : faithful (shift_functor C i) :=\nfaithful.of_comp_iso (shift_functor_comp_shift_functor_neg C i)\n\n/-- Shifting by `n` is a full functor. -/\ninstance shift_functor_full (i : A) : full (shift_functor C i) :=\nbegin\n  haveI : full (shift_functor C i \u22d9 shift_functor C (-i)) :=\n    full.of_iso (shift_functor_comp_shift_functor_neg C i).symm,\n  exact full.of_comp_faithful _ (shift_functor C (-i))\nend\n\n/-- Shifting by `n` is an essentially surjective functor. -/\ninstance shift_functor_ess_surj (i : A) : ess_surj (shift_functor C i) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y\u27e6-i\u27e7, \u27e8(shift_functor_neg_comp_shift_functor C i).app Y\u27e9\u27e9 }\n\n/-- Shifting by `n` is an equivalence. -/\nnoncomputable instance shift_functor_is_equivalence (n : A) : is_equivalence (shift_functor C n) :=\nequivalence.of_fully_faithfully_ess_surj _\n\nend\n\nvariables {C}\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_shift_neg (i : A) : X\u27e6i\u27e7\u27e6-i\u27e7 \u2245 X :=\n(shift_functor_comp_shift_functor_neg C i).app _\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_neg_shift (i : A) : X\u27e6-i\u27e7\u27e6i\u27e7 \u2245 X :=\n(shift_functor_neg_comp_shift_functor C i).app _\n\nvariables {X Y}\n\nlemma shift_shift_neg' (i : A) :\n  f\u27e6i\u27e7'\u27e6-i\u27e7' = (shift_shift_neg X i).hom \u226b f \u226b (shift_shift_neg Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_neg_shift' (i : A) :\n  f\u27e6-i\u27e7'\u27e6i\u27e7' = (shift_neg_shift X i).hom \u226b f \u226b (shift_neg_shift Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_equiv_triangle (n : A) (X : C) :\n  (shift_shift_neg X n).inv\u27e6n\u27e7' \u226b (shift_neg_shift (X\u27e6n\u27e7) n).hom = \ud835\udfd9 (X\u27e6n\u27e7) :=\n(add_neg_equiv (shift_monoidal_functor C A) n).functor_unit_iso_comp X\n\nlemma shift_shift_neg_hom_shift (n : A) (X : C) :\n  (shift_shift_neg X n).hom \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).hom :=\nby simp\n\nlemma shift_shift_neg_inv_shift (n : A) (X : C) :\n  (shift_shift_neg X n).inv \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).inv :=\nby { ext, rw [\u2190 shift_shift_neg_hom_shift, \u2190 functor.map_comp, iso.hom_inv_id, functor.map_id] }\n\n@[simp]\nlemma shift_shift_neg_shift_eq (n : A) (X : C) :\n  (shift_functor C n).map_iso (shift_shift_neg X n) = shift_neg_shift (X\u27e6n\u27e7) n :=\ncategory_theory.iso.ext $ shift_shift_neg_hom_shift _ _\n\nvariables (C)\n\n/-- Shifting by `n` and shifting by `-n` forms an equivalence. -/\n@[simps]\ndef shift_equiv (n : A) : C \u224c C :=\n{ functor := shift_functor C n,\n  inverse := shift_functor C (-n),\n  ..(add_neg_equiv (shift_monoidal_functor C A) n) }\n\nvariable {C}\n\nopen category_theory.limits\n\nvariables [has_zero_morphisms C]\n\n@[simp]\nlemma shift_zero_eq_zero (X Y : C) (n : A) : (0 : X \u27f6 Y)\u27e6n\u27e7' = (0 : X\u27e6n\u27e7 \u27f6 Y\u27e6n\u27e7) :=\nby apply is_equivalence_preserves_zero_morphisms _ (shift_functor C n)\n\nend add_group\n\nsection add_comm_monoid\n\nvariables {C A} [add_comm_monoid A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\ndef shift_comm (i j : A) : X\u27e6i\u27e7\u27e6j\u27e7 \u2245 X\u27e6j\u27e7\u27e6i\u27e7 :=\n(shift_add X i j).symm \u226a\u226b ((shift_monoidal_functor C A).to_functor.map_iso\n  (opaque_eq_to_iso $ add_comm i j : _)).app X \u226a\u226b shift_add X j i\n\n@[simp] lemma shift_comm_symm (i j : A) : (shift_comm X i j).symm = shift_comm X j i :=\nbegin\n  ext, dsimp [shift_comm], simpa\nend\n\nvariables {X Y}\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\nlemma shift_comm' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_comm _ _ _).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' \u226b (shift_comm _ _ _).hom :=\nby simp [shift_comm]\n\n@[reassoc] lemma shift_comm_hom_comp (i j : A) :\n  (shift_comm X i j).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' = f\u27e6i\u27e7'\u27e6j\u27e7' \u226b (shift_comm Y i j).hom :=\nby rw [shift_comm', \u2190 shift_comm_symm, iso.symm_hom, iso.inv_hom_id_assoc]\n\nend add_comm_monoid\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3852408395262473}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport algebra.module.basic\nimport algebra.ring.aut\nimport linear_algebra.span\nimport tactic.abel\n\n/-!\n# Algebras over commutative semirings\n\nIn this file we define associative unital `algebra`s over commutative (semi)rings, algebra\nhomomorphisms `alg_hom`, and algebra equivalences `alg_equiv`.\n\n`subalgebra`s are defined in `algebra.algebra.subalgebra`.\n\nFor the category of `R`-algebras, denoted `Algebra R`, see the file\n`algebra/category/Algebra/basic.lean`.\n\nSee the implementation notes for remarks about non-associative and non-unital algebras.\n\n## Main definitions:\n\n* `algebra R A`: the algebra typeclass.\n* `alg_hom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `alg_equiv R A B`: the type of `R`-algebra isomorphisms between `A` to `B`.\n* `algebra_map R A : R \u2192+* A`: the canonical map from `R` to `A`, as a `ring_hom`. This is the\n  preferred spelling of this map.\n* `algebra.linear_map R A : R \u2192\u2097[R] A`: the canonical map from `R` to `A`, as a `linear_map`.\n* `algebra.of_id R A : R \u2192\u2090[R] A`: the canonical map from `R` to `A`, as n `alg_hom`.\n* Instances of `algebra` in this file:\n  * `algebra.id`\n  * `pi.algebra`\n  * `prod.algebra`\n  * `algebra_nat`\n  * `algebra_int`\n  * `algebra_rat`\n  * `mul_opposite.algebra`\n  * `module.End.algebra`\n\n## Notations\n\n* `A \u2192\u2090[R] B` : `R`-algebra homomorphism from `A` to `B`.\n* `A \u2243\u2090[R] B` : `R`-algebra equivalence from `A` to `B`.\n\n## Implementation notes\n\nGiven a commutative (semi)ring `R`, there are two ways to define an `R`-algebra structure on a\n(possibly noncommutative) (semi)ring `A`:\n* By endowing `A` with a morphism of rings `R \u2192+* A` denoted `algebra_map R A` which lands in the\n  center of `A`.\n* By requiring `A` be an `R`-module such that the action associates and commutes with multiplication\n  as `r \u2022 (a\u2081 * a\u2082) = (r \u2022 a\u2081) * a\u2082 = a\u2081 * (r \u2022 a\u2082)`.\n\nWe define `algebra R A` in a way that subsumes both definitions, by extending `has_scalar R A` and\nrequiring that this scalar action `r \u2022 x` must agree with left multiplication by the image of the\nstructure morphism `algebra_map R A r * x`.\n\nAs a result, there are two ways to talk about an `R`-algebra `A` when `A` is a semiring:\n1. ```lean\n   variables [comm_semiring R] [semiring A]\n   variables [algebra R A]\n   ```\n2. ```lean\n   variables [comm_semiring R] [semiring A]\n   variables [module R A] [smul_comm_class R A A] [is_scalar_tower R A A]\n   ```\n\nThe first approach implies the second via typeclass search; so any lemma stated with the second set\nof arguments will automatically apply to the first set. Typeclass search does not know that the\nsecond approach implies the first, but this can be shown with:\n```lean\nexample {R A : Type*} [comm_semiring R] [semiring A]\n  [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] : algebra R A :=\nalgebra.of_module smul_mul_assoc mul_smul_comm\n```\n\nThe advantage of the first approach is that `algebra_map R A` is available, and `alg_hom R A B` and\n`subalgebra R A` can be used. For concrete `R` and `A`, `algebra_map R A` is often definitionally\nconvenient.\n\nThe advantage of the second approach is that `comm_semiring R`, `semiring A`, and `module R A` can\nall be relaxed independently; for instance, this allows us to:\n* Replace `semiring A` with `non_unital_non_assoc_semiring A` in order to describe non-unital and/or\n  non-associative algebras.\n* Replace `comm_semiring R` and `module R A` with `comm_group R'` and `distrib_mul_action R' A`,\n  which when `R' = R\u02e3` lets us talk about the \"algebra-like\" action of `R\u02e3` on an\n  `R`-algebra `A`.\n\nWhile `alg_hom R A B` cannot be used in the second approach, `non_unital_alg_hom R A B` still can.\n\nYou should always use the first approach when working with associative unital algebras, and mimic\nthe second approach only when you need to weaken a condition on either `R` or `A`.\n\n-/\n\nuniverses u v w u\u2081 v\u2081\n\nopen_locale big_operators\n\nsection prio\n-- We set this priority to 0 later in this file\nset_option extends_priority 200 /- control priority of\n`instance [algebra R A] : has_scalar R A` -/\n\n/--\nAn associative unital `R`-algebra is a semiring `A` equipped with a map into its center `R \u2192 A`.\n\nSee the implementation notes in this file for discussion of the details of this definition.\n-/\n@[nolint has_inhabited_instance]\nclass algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A]\n  extends has_scalar R A, R \u2192+* A :=\n(commutes' : \u2200 r x, to_fun r * x = x * to_fun r)\n(smul_def' : \u2200 r x, r \u2022 x = to_fun r * x)\nend prio\n\n/-- Embedding `R \u2192+* A` given by `algebra` structure. -/\ndef algebra_map (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : R \u2192+* A :=\nalgebra.to_ring_hom\n\n/-- Creating an algebra from a morphism to the center of a semiring. -/\ndef ring_hom.to_algebra' {R S} [comm_semiring R] [semiring S] (i : R \u2192+* S)\n  (h : \u2200 c x, i c * x = x * i c) :\n  algebra R S :=\n{ smul := \u03bb c x, i c * x,\n  commutes' := h,\n  smul_def' := \u03bb c x, rfl,\n  to_ring_hom := i}\n\n/-- Creating an algebra from a morphism to a commutative semiring. -/\ndef ring_hom.to_algebra {R S} [comm_semiring R] [comm_semiring S] (i : R \u2192+* S) :\n  algebra R S :=\ni.to_algebra' $ \u03bb _, mul_comm _\n\nlemma ring_hom.algebra_map_to_algebra {R S} [comm_semiring R] [comm_semiring S]\n  (i : R \u2192+* S) :\n  @algebra_map R S _ _ i.to_algebra = i :=\nrfl\n\nnamespace algebra\n\nvariables {R : Type u} {S : Type v} {A : Type w} {B : Type*}\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 1) * x = x * (r \u2022 1) = r \u2022 x` for all `r : R` and `x : A`, then `A` is an `algebra`\nover `R`.\n\nSee note [reducible non-instances]. -/\n@[reducible]\ndef of_module' [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x : A), (r \u2022 1) * x = r \u2022 x)\n  (h\u2082 : \u2200 (r : R) (x : A), x * (r \u2022 1) = r \u2022 x) : algebra R A :=\n{ to_fun := \u03bb r, r \u2022 1,\n  map_one' := one_smul _ _,\n  map_mul' := \u03bb r\u2081 r\u2082, by rw [h\u2081, mul_smul],\n  map_zero' := zero_smul _ _,\n  map_add' := \u03bb r\u2081 r\u2082, add_smul r\u2081 r\u2082 1,\n  commutes' := \u03bb r x, by simp only [h\u2081, h\u2082],\n  smul_def' := \u03bb r x, by simp only [h\u2081] }\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 x) * y = x * (r \u2022 y) = r \u2022 (x * y)` for all `r : R` and `x y : A`, then `A`\nis an `algebra` over `R`.\n\nSee note [reducible non-instances]. -/\n@[reducible]\ndef of_module [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x y : A), (r \u2022 x) * y = r \u2022 (x * y))\n  (h\u2082 : \u2200 (r : R) (x y : A), x * (r \u2022 y) = r \u2022 (x * y)) : algebra R A :=\nof_module' (\u03bb r x, by rw [h\u2081, one_mul]) (\u03bb r x, by rw [h\u2082, mul_one])\n\nsection semiring\n\nvariables [comm_semiring R] [comm_semiring S]\nvariables [semiring A] [algebra R A] [semiring B] [algebra R B]\n\n/-- We keep this lemma private because it picks up the `algebra.to_has_scalar` instance\nwhich we set to priority 0 shortly. See `smul_def` below for the public version. -/\nprivate lemma smul_def'' (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\n/--\nTo prove two algebra structures on a fixed `[comm_semiring R] [semiring A]` agree,\nit suffices to check the `algebra_map`s agree.\n-/\n-- We'll later use this to show `algebra \u2124 M` is a subsingleton.\n@[ext]\nlemma algebra_ext {R : Type*} [comm_semiring R] {A : Type*} [semiring A] (P Q : algebra R A)\n  (w : \u2200 (r : R), by { haveI := P, exact algebra_map R A r } =\n    by { haveI := Q, exact algebra_map R A r }) :\n  P = Q :=\nbegin\n  unfreezingI { rcases P with \u27e8\u27e8P\u27e9\u27e9, rcases Q with \u27e8\u27e8Q\u27e9\u27e9 },\n  congr,\n  { funext r a,\n    replace w := congr_arg (\u03bb s, s * a) (w r),\n    simp only [\u2190smul_def''] at w,\n    apply w, },\n  { ext r,\n    exact w r, },\n  { apply proof_irrel_heq, },\n  { apply proof_irrel_heq, },\nend\n\n@[priority 200] -- see Note [lower instance priority]\ninstance to_module : module R A :=\n{ one_smul := by simp [smul_def''],\n  mul_smul := by simp [smul_def'', mul_assoc],\n  smul_add := by simp [smul_def'', mul_add],\n  smul_zero := by simp [smul_def''],\n  add_smul := by simp [smul_def'', add_mul],\n  zero_smul := by simp [smul_def''] }\n\n-- From now on, we don't want to use the following instance anymore.\n-- Unfortunately, leaving it in place causes deterministic timeouts later in mathlib.\nattribute [instance, priority 0] algebra.to_has_scalar\n\nlemma smul_def (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\nlemma algebra_map_eq_smul_one (r : R) : algebra_map R A r = r \u2022 1 :=\ncalc algebra_map R A r = algebra_map R A r * 1 : (mul_one _).symm\n                   ... = r \u2022 1                 : (algebra.smul_def r 1).symm\n\nlemma algebra_map_eq_smul_one' : \u21d1(algebra_map R A) = \u03bb r, r \u2022 (1 : A) :=\nfunext algebra_map_eq_smul_one\n\n/-- `mul_comm` for `algebra`s when one element is from the base ring. -/\ntheorem commutes (r : R) (x : A) : algebra_map R A r * x = x * algebra_map R A r :=\nalgebra.commutes' r x\n\n/-- `mul_left_comm` for `algebra`s when one element is from the base ring. -/\ntheorem left_comm (x : A) (r : R) (y : A) :\n  x * (algebra_map R A r * y) = algebra_map R A r * (x * y) :=\nby rw [\u2190 mul_assoc, \u2190 commutes, mul_assoc]\n\n/-- `mul_right_comm` for `algebra`s when one element is from the base ring. -/\ntheorem right_comm (x : A) (r : R) (y : A) :\n  (x * algebra_map R A r) * y = (x * y) * algebra_map R A r :=\nby rw [mul_assoc, commutes, \u2190mul_assoc]\n\ninstance _root_.is_scalar_tower.right : is_scalar_tower R A A :=\n\u27e8\u03bb x y z, by rw [smul_eq_mul, smul_eq_mul, smul_def, smul_def, mul_assoc]\u27e9\n\n/-- This is just a special case of the global `mul_smul_comm` lemma that requires less typeclass\nsearch (and was here first). -/\n@[simp] protected lemma mul_smul_comm (s : R) (x y : A) :\n  x * (s \u2022 y) = s \u2022 (x * y) :=\n-- TODO: set up `is_scalar_tower.smul_comm_class` earlier so that we can actually prove this using\n-- `mul_smul_comm s x y`.\nby rw [smul_def, smul_def, left_comm]\n\n/-- This is just a special case of the global `smul_mul_assoc` lemma that requires less typeclass\nsearch (and was here first). -/\n@[simp] protected lemma smul_mul_assoc (r : R) (x y : A) :\n  (r \u2022 x) * y = r \u2022 (x * y) :=\nsmul_mul_assoc r x y\n\nsection\nvariables {r : R} {a : A}\n\n@[simp] lemma bit0_smul_one : bit0 r \u2022 (1 : A) = bit0 (r \u2022 (1 : A)) :=\nby simp [bit0, add_smul]\nlemma bit0_smul_one' : bit0 r \u2022 (1 : A) = r \u2022 2 :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit0 : bit0 r \u2022 bit0 a = r \u2022 (bit0 (bit0 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit1 : bit0 r \u2022 bit1 a = r \u2022 (bit0 (bit1 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit1_smul_one : bit1 r \u2022 (1 : A) = bit1 (r \u2022 (1 : A)) :=\nby simp [bit1, add_smul]\nlemma bit1_smul_one' : bit1 r \u2022 (1 : A) = r \u2022 2 + 1 :=\nby simp [bit1, bit0, add_smul, smul_add]\n@[simp] lemma bit1_smul_bit0 : bit1 r \u2022 bit0 a = r \u2022 (bit0 (bit0 a)) + bit0 a :=\nby simp [bit1, add_smul, smul_add]\n@[simp] lemma bit1_smul_bit1 : bit1 r \u2022 bit1 a = r \u2022 (bit0 (bit1 a)) + bit1 a :=\nby { simp only [bit0, bit1, add_smul, smul_add, one_smul], abel }\n\nend\n\nvariables (R A)\n\n/--\nThe canonical ring homomorphism `algebra_map R A : R \u2192* A` for any `R`-algebra `A`,\npackaged as an `R`-linear map.\n-/\nprotected def linear_map : R \u2192\u2097[R] A :=\n{ map_smul' := \u03bb x y, by simp [algebra.smul_def],\n  ..algebra_map R A }\n\n@[simp]\nlemma linear_map_apply (r : R) : algebra.linear_map R A r = algebra_map R A r := rfl\n\nlemma coe_linear_map : \u21d1(algebra.linear_map R A) = algebra_map R A := rfl\n\ninstance id : algebra R R := (ring_hom.id R).to_algebra\n\nvariables {R A}\n\nnamespace id\n\n@[simp] lemma map_eq_id : algebra_map R R = ring_hom.id _ := rfl\n\nlemma map_eq_self (x : R) : algebra_map R R x = x := rfl\n\n@[simp] lemma smul_eq_mul (x y : R) : x \u2022 y = x * y := rfl\n\nend id\n\nsection punit\n\ninstance _root_.punit.algebra : algebra R punit :=\n{ to_fun := \u03bb x, punit.star,\n  map_one' := rfl,\n  map_mul' := \u03bb _ _, rfl,\n  map_zero' := rfl,\n  map_add' := \u03bb _ _, rfl,\n  commutes' := \u03bb _ _, rfl,\n  smul_def' := \u03bb _ _, rfl }\n\n@[simp] lemma algebra_map_punit (r : R) : algebra_map R punit r = punit.star := rfl\n\nend punit\n\nsection prod\nvariables (R A B)\n\ninstance _root_.prod.algebra : algebra R (A \u00d7 B) :=\n{ commutes' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [commutes r a, commutes r b] },\n  smul_def' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [smul_def r a, smul_def r b] },\n  .. prod.module,\n  .. ring_hom.prod (algebra_map R A) (algebra_map R B) }\n\nvariables {R A B}\n\n@[simp] lemma algebra_map_prod_apply (r : R) :\n  algebra_map R (A \u00d7 B) r = (algebra_map R A r, algebra_map R B r) := rfl\n\nend prod\n\n/-- Algebra over a subsemiring. This builds upon `subsemiring.module`. -/\ninstance of_subsemiring (S : subsemiring R) : algebra S A :=\n{ smul := (\u2022),\n  commutes' := \u03bb r x, algebra.commutes r x,\n  smul_def' := \u03bb r x, algebra.smul_def r x,\n  .. (algebra_map R A).comp S.subtype }\n\nlemma algebra_map_of_subsemiring (S : subsemiring R) :\n  (algebra_map S R : S \u2192+* R) = subsemiring.subtype S := rfl\n\nlemma coe_algebra_map_of_subsemiring (S : subsemiring R) :\n  (algebra_map S R : S \u2192 R) = subtype.val := rfl\n\nlemma algebra_map_of_subsemiring_apply (S : subsemiring R) (x : S) :\n  algebra_map S R x = x := rfl\n\n/-- Algebra over a subring. This builds upon `subring.module`. -/\ninstance of_subring {R A : Type*} [comm_ring R] [ring A] [algebra R A]\n  (S : subring R) : algebra S A :=\n{ smul := (\u2022),\n  .. algebra.of_subsemiring S.to_subsemiring,\n  .. (algebra_map R A).comp S.subtype }\n\nlemma algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192+* R) = subring.subtype S := rfl\n\nlemma coe_algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192 R) = subtype.val := rfl\n\nlemma algebra_map_of_subring_apply {R : Type*} [comm_ring R] (S : subring R) (x : S) :\n  algebra_map S R x = x := rfl\n\n/-- Explicit characterization of the submonoid map in the case of an algebra.\n`S` is made explicit to help with type inference -/\ndef algebra_map_submonoid (S : Type*) [semiring S] [algebra R S]\n  (M : submonoid R) : (submonoid S) :=\nsubmonoid.map (algebra_map R S : R \u2192* S) M\n\nlemma mem_algebra_map_submonoid_of_mem {S : Type*} [semiring S] [algebra R S] {M : submonoid R}\n  (x : M) : (algebra_map R S x) \u2208 algebra_map_submonoid S M :=\nset.mem_image_of_mem (algebra_map R S) x.2\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R]\n\nlemma mul_sub_algebra_map_commutes [ring A] [algebra R A] (x : A) (r : R) :\n  x * (x - algebra_map R A r) = (x - algebra_map R A r) * x :=\nby rw [mul_sub, \u2190commutes, sub_mul]\n\nlemma mul_sub_algebra_map_pow_commutes [ring A] [algebra R A] (x : A) (r : R) (n : \u2115) :\n  x * (x - algebra_map R A r) ^ n = (x - algebra_map R A r) ^ n * x :=\nbegin\n  induction n with n ih,\n  { simp },\n  { rw [pow_succ, \u2190mul_assoc, mul_sub_algebra_map_commutes, mul_assoc, ih, \u2190mul_assoc] }\nend\n\nend comm_semiring\n\nsection ring\nvariables [comm_ring R]\n\nvariables (R)\n\n/-- A `semiring` that is an `algebra` over a commutative ring carries a natural `ring` structure.\nSee note [reducible non-instances]. -/\n@[reducible]\ndef semiring_to_ring [semiring A] [algebra R A] : ring A :=\n{ ..module.add_comm_monoid_to_add_comm_group R,\n  ..(infer_instance : semiring A) }\n\nend ring\n\nend algebra\n\nnamespace no_zero_smul_divisors\n\nvariables {R A : Type*}\n\nopen algebra\n\nsection ring\n\nvariables [comm_ring R]\n\n/-- If `algebra_map R A` is injective and `A` has no zero divisors,\n`R`-multiples in `A` are zero only if one of the factors is zero.\n\nCannot be an instance because there is no `injective (algebra_map R A)` typeclass.\n-/\nlemma of_algebra_map_injective\n  [semiring A] [algebra R A] [no_zero_divisors A]\n  (h : function.injective (algebra_map R A)) : no_zero_smul_divisors R A :=\n\u27e8\u03bb c x hcx, (mul_eq_zero.mp ((smul_def c x).symm.trans hcx)).imp_left\n  ((injective_iff_map_eq_zero (algebra_map R A)).mp h _)\u27e9\n\nvariables (R A)\nlemma algebra_map_injective [ring A] [nontrivial A]\n  [algebra R A] [no_zero_smul_divisors R A] :\n  function.injective (algebra_map R A) :=\nsuffices function.injective (\u03bb (c : R), c \u2022 (1 : A)),\nby { convert this, ext, rw [algebra.smul_def, mul_one] },\nsmul_left_injective R one_ne_zero\n\nvariables {R A}\nlemma iff_algebra_map_injective [ring A] [is_domain A] [algebra R A] :\n  no_zero_smul_divisors R A \u2194 function.injective (algebra_map R A) :=\n\u27e8@@no_zero_smul_divisors.algebra_map_injective R A _ _ _ _,\n no_zero_smul_divisors.of_algebra_map_injective\u27e9\n\nend ring\n\nsection field\n\nvariables [field R] [semiring A] [algebra R A]\n\n@[priority 100] -- see note [lower instance priority]\ninstance algebra.no_zero_smul_divisors [nontrivial A] [no_zero_divisors A] :\n  no_zero_smul_divisors R A :=\nno_zero_smul_divisors.of_algebra_map_injective (algebra_map R A).injective\n\nend field\n\nend no_zero_smul_divisors\n\nnamespace mul_opposite\n\nvariables {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]\n\ninstance : algebra R A\u1d50\u1d52\u1d56 :=\n{ to_ring_hom := (algebra_map R A).to_opposite $ \u03bb x y, algebra.commutes _ _,\n  smul_def' := \u03bb c x, unop_injective $\n    by { dsimp, simp only [op_mul, algebra.smul_def, algebra.commutes, op_unop] },\n  commutes' := \u03bb r, mul_opposite.rec $ \u03bb x, by dsimp; simp only [\u2190 op_mul, algebra.commutes],\n  .. mul_opposite.has_scalar A R }\n\n@[simp] lemma algebra_map_apply (c : R) : algebra_map R A\u1d50\u1d52\u1d56 c = op (algebra_map R A c) := rfl\n\nend mul_opposite\n\nnamespace module\nvariables (R : Type u) (M : Type v) [comm_semiring R] [add_comm_monoid M] [module R M]\n\ninstance : algebra R (module.End R M) :=\nalgebra.of_module smul_mul_assoc (\u03bb r f g, (smul_comm r f g).symm)\n\nlemma algebra_map_End_eq_smul_id (a : R) :\n  (algebra_map R (End R M)) a = a \u2022 linear_map.id := rfl\n\n@[simp] lemma algebra_map_End_apply (a : R) (m : M) :\n  (algebra_map R (End R M)) a m = a \u2022 m := rfl\n\n@[simp] lemma ker_algebra_map_End (K : Type u) (V : Type v)\n  [field K] [add_comm_group V] [module K V] (a : K) (ha : a \u2260 0) :\n  ((algebra_map K (End K V)) a).ker = \u22a5 :=\nlinear_map.ker_smul _ _ ha\n\nend module\n\nset_option old_structure_cmd true\n/-- Defining the homomorphism in the category R-Alg. -/\n@[nolint has_inhabited_instance]\nstructure alg_hom (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] extends ring_hom A B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nrun_cmd tactic.add_doc_string `alg_hom.to_ring_hom \"Reinterpret an `alg_hom` as a `ring_hom`\"\n\ninfixr ` \u2192\u2090 `:25 := alg_hom _\nnotation A ` \u2192\u2090[`:25 R `] ` B := alg_hom R A B\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {C : Type u\u2081} {D : Type v\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A] [semiring B] [semiring C] [semiring D]\nvariables [algebra R A] [algebra R B] [algebra R C] [algebra R D]\n\ninstance : has_coe_to_fun (A \u2192\u2090[R] B) (\u03bb _, A \u2192 B) := \u27e8alg_hom.to_fun\u27e9\n\ninitialize_simps_projections alg_hom (to_fun \u2192 apply)\n\n@[simp] lemma to_fun_eq_coe (f : A \u2192\u2090[R] B) : f.to_fun = f := rfl\n\ninstance : ring_hom_class (A \u2192\u2090[R] B) A B :=\n{ coe := to_fun,\n  coe_injective' := \u03bb f g h, by { cases f, cases g, congr' },\n  map_add := map_add',\n  map_zero := map_zero',\n  map_mul := map_mul',\n  map_one := map_one' }\n\ninstance coe_ring_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+* B) := \u27e8alg_hom.to_ring_hom\u27e9\n\ninstance coe_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192* B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\ninstance coe_add_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+ B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\n@[simp, norm_cast] lemma coe_mk {f : A \u2192 B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  \u21d1(\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := rfl\n\n-- make the coercion the simp-normal form\n@[simp] lemma to_ring_hom_eq_coe (f : A \u2192\u2090[R] B) : f.to_ring_hom = f := rfl\n\n@[simp, norm_cast] lemma coe_to_ring_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+* B) = f := rfl\n\n@[simp, norm_cast] lemma coe_to_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192* B) = f := rfl\n\n@[simp, norm_cast] lemma coe_to_add_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+ B) = f := rfl\n\nvariables (\u03c6 : A \u2192\u2090[R] B)\n\ntheorem coe_fn_injective : @function.injective (A \u2192\u2090[R] B) (A \u2192 B) coe_fn := fun_like.coe_injective\n\ntheorem coe_fn_inj {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : (\u03c6\u2081 : A \u2192 B) = \u03c6\u2082 \u2194 \u03c6\u2081 = \u03c6\u2082 := fun_like.coe_fn_eq\n\ntheorem coe_ring_hom_injective : function.injective (coe : (A \u2192\u2090[R] B) \u2192 (A \u2192+* B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 H, coe_fn_injective $ show ((\u03c6\u2081 : (A \u2192+* B)) : A \u2192 B) = ((\u03c6\u2082 : (A \u2192+* B)) : A \u2192 B),\n  from congr_arg _ H\n\ntheorem coe_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192* B)) :=\nring_hom.coe_monoid_hom_injective.comp coe_ring_hom_injective\n\ntheorem coe_add_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192+ B)) :=\nring_hom.coe_add_monoid_hom_injective.comp coe_ring_hom_injective\n\nprotected lemma congr_fun {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u03c6\u2081 = \u03c6\u2082) (x : A) : \u03c6\u2081 x = \u03c6\u2082 x :=\nfun_like.congr_fun H x\nprotected lemma congr_arg (\u03c6 : A \u2192\u2090[R] B) {x y : A} (h : x = y) : \u03c6 x = \u03c6 y :=\nfun_like.congr_arg \u03c6 h\n\n@[ext]\ntheorem ext {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x) : \u03c6\u2081 = \u03c6\u2082 := fun_like.ext _ _ H\n\ntheorem ext_iff {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : \u03c6\u2081 = \u03c6\u2082 \u2194 \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x := fun_like.ext_iff\n\n@[simp] theorem mk_coe {f : A \u2192\u2090[R] B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := ext $ \u03bb _, rfl\n\n@[simp]\ntheorem commutes (r : R) : \u03c6 (algebra_map R A r) = algebra_map R B r := \u03c6.commutes' r\n\ntheorem comp_algebra_map : (\u03c6 : A \u2192+* B).comp (algebra_map R A) = algebra_map R B :=\nring_hom.ext $ \u03c6.commutes\n\nlemma map_add (r s : A) : \u03c6 (r + s) = \u03c6 r + \u03c6 s := map_add _ _ _\nlemma map_zero : \u03c6 0 = 0 := map_zero _\nlemma map_mul (x y) : \u03c6 (x * y) = \u03c6 x * \u03c6 y := map_mul _ _ _\nlemma map_one : \u03c6 1 = 1 := map_one _\nlemma map_pow (x : A) (n : \u2115) : \u03c6 (x ^ n) = (\u03c6 x) ^ n :=\nmap_pow _ _ _\n\n@[simp] lemma map_smul (r : R) (x : A) : \u03c6 (r \u2022 x) = r \u2022 \u03c6 x :=\nby simp only [algebra.smul_def, map_mul, commutes]\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u2211 x in s, f x) = \u2211 x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.sum g) = f.sum (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_sum _ _\n\nlemma map_bit0 (x) : \u03c6 (bit0 x) = bit0 (\u03c6 x) := map_bit0 _ _\nlemma map_bit1 (x) : \u03c6 (bit1 x) = bit1 (\u03c6 x) := map_bit1 _ _\n\n/-- If a `ring_hom` is `R`-linear, then it is an `alg_hom`. -/\ndef mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : A \u2192\u2090[R] B :=\n{ to_fun := f,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, h, f.map_one],\n  .. f }\n\n@[simp] lemma coe_mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : \u21d1(mk' f h) = f := rfl\n\nsection\n\nvariables (R A)\n/-- Identity map as an `alg_hom`. -/\nprotected def id : A \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl,\n  ..ring_hom.id A }\n\n@[simp] lemma coe_id : \u21d1(alg_hom.id R A) = id := rfl\n\n@[simp] lemma id_to_ring_hom : (alg_hom.id R A : A \u2192+* A) = ring_hom.id _ := rfl\n\nend\n\nlemma id_apply (p : A) : alg_hom.id R A p = p := rfl\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : A \u2192\u2090[R] C :=\n{ commutes' := \u03bb r : R, by rw [\u2190 \u03c6\u2081.commutes, \u2190 \u03c6\u2082.commutes]; refl,\n  .. \u03c6\u2081.to_ring_hom.comp \u2191\u03c6\u2082 }\n\n@[simp] lemma coe_comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : \u21d1(\u03c6\u2081.comp \u03c6\u2082) = \u03c6\u2081 \u2218 \u03c6\u2082 := rfl\n\nlemma comp_apply (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) (p : A) : \u03c6\u2081.comp \u03c6\u2082 p = \u03c6\u2081 (\u03c6\u2082 p) := rfl\n\nlemma comp_to_ring_hom (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) :\n  \u21d1(\u03c6\u2081.comp \u03c6\u2082 : A \u2192+* C) = (\u03c6\u2081 : B \u2192+* C).comp \u2191\u03c6\u2082 := rfl\n\n@[simp] theorem comp_id : \u03c6.comp (alg_hom.id R A) = \u03c6 :=\next $ \u03bb x, rfl\n\n@[simp] theorem id_comp : (alg_hom.id R B).comp \u03c6 = \u03c6 :=\next $ \u03bb x, rfl\n\ntheorem comp_assoc (\u03c6\u2081 : C \u2192\u2090[R] D) (\u03c6\u2082 : B \u2192\u2090[R] C) (\u03c6\u2083 : A \u2192\u2090[R] B) :\n  (\u03c6\u2081.comp \u03c6\u2082).comp \u03c6\u2083 = \u03c6\u2081.comp (\u03c6\u2082.comp \u03c6\u2083) :=\next $ \u03bb x, rfl\n\n/-- R-Alg \u2964 R-Mod -/\ndef to_linear_map : A \u2192\u2097[R] B :=\n{ to_fun := \u03c6,\n  map_add' := \u03c6.map_add,\n  map_smul' := \u03c6.map_smul }\n\n@[simp] lemma to_linear_map_apply (p : A) : \u03c6.to_linear_map p = \u03c6 p := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A \u2192\u2097[R] B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma comp_to_linear_map (f : A \u2192\u2090[R] B) (g : B \u2192\u2090[R] C) :\n  (g.comp f).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_id : to_linear_map (alg_hom.id R A) = linear_map.id :=\nlinear_map.ext $ \u03bb _, rfl\n\n/-- Promote a `linear_map` to an `alg_hom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef of_linear_map (f : A \u2192\u2097[R] B) (map_one : f 1 = 1) (map_mul : \u2200 x y, f (x * y) = f x * f y) :\n  A \u2192\u2090[R] B :=\n{ to_fun := f,\n  map_one' := map_one,\n  map_mul' := map_mul,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, f.map_smul, map_one],\n  .. f.to_add_monoid_hom }\n\n@[simp] lemma of_linear_map_to_linear_map (map_one) (map_mul) :\n  of_linear_map \u03c6.to_linear_map map_one map_mul = \u03c6 :=\nby { ext, refl }\n\n@[simp] lemma to_linear_map_of_linear_map (f : A \u2192\u2097[R] B) (map_one) (map_mul) :\n  to_linear_map (of_linear_map f map_one map_mul) = f :=\nby { ext, refl }\n\n@[simp] lemma of_linear_map_id (map_one) (map_mul) :\n  of_linear_map linear_map.id map_one map_mul = alg_hom.id R A :=\next $ \u03bb _, rfl\n\nlemma map_smul_of_tower {R'} [has_scalar R' A] [has_scalar R' B]\n  [linear_map.compatible_smul A B R' R] (r : R') (x : A) : \u03c6 (r \u2022 x) = r \u2022 \u03c6 x :=\n\u03c6.to_linear_map.map_smul_of_tower r x\n\nlemma map_list_prod (s : list A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_list_prod s\n\nsection prod\n\n/-- First projection as `alg_hom`. -/\ndef fst : A \u00d7 B \u2192\u2090[R] A :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.fst A B}\n\n/-- Second projection as `alg_hom`. -/\ndef snd : A \u00d7 B \u2192\u2090[R] B :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.snd A B}\n\nend prod\n\nlemma algebra_map_eq_apply (f : A \u2192\u2090[R] B) {y : R} {x : A} (h : algebra_map R A y = x) :\n  algebra_map R B y = f x :=\nh \u25b8 (f.commutes _).symm\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A] [comm_semiring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\nlemma map_multiset_prod (s : multiset A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_multiset_prod s\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u220f x in s, f x) = \u220f x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.prod g) = f.prod (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_prod _ _\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_semiring R] [ring A] [ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\nlemma map_neg (x) : \u03c6 (-x) = -\u03c6 x := map_neg _ _\nlemma map_sub (x y) : \u03c6 (x - y) = \u03c6 x - \u03c6 y := map_sub _ _ _\n\n@[simp] lemma map_int_cast (n : \u2124) : \u03c6 n = n :=\n\u03c6.to_ring_hom.map_int_cast n\n\nend ring\n\nsection division_ring\n\nvariables [comm_semiring R] [division_ring A] [division_ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\n@[simp] lemma map_inv (x) : \u03c6 (x\u207b\u00b9) = (\u03c6 x)\u207b\u00b9 :=\n\u03c6.to_ring_hom.map_inv x\n\n@[simp] lemma map_div (x y) : \u03c6 (x / y) = \u03c6 x / \u03c6 y :=\n\u03c6.to_ring_hom.map_div x y\n\nend division_ring\n\nend alg_hom\n\n@[simp] lemma rat.smul_one_eq_coe {A : Type*} [division_ring A] [algebra \u211a A] (m : \u211a) :\n  m \u2022 (1 : A) = \u2191m :=\nby rw [algebra.smul_def, mul_one, ring_hom.eq_rat_cast]\n\nset_option old_structure_cmd true\n/-- An equivalence of algebras is an equivalence of rings commuting with the actions of scalars. -/\nstructure alg_equiv (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B]\n  extends A \u2243 B, A \u2243* B, A \u2243+ B, A \u2243+* B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nattribute [nolint doc_blame] alg_equiv.to_ring_equiv\nattribute [nolint doc_blame] alg_equiv.to_equiv\nattribute [nolint doc_blame] alg_equiv.to_add_equiv\nattribute [nolint doc_blame] alg_equiv.to_mul_equiv\n\nnotation A ` \u2243\u2090[`:50 R `] ` A' := alg_equiv R A A'\n\nnamespace alg_equiv\n\nvariables {R : Type u} {A\u2081 : Type v} {A\u2082 : Type w} {A\u2083 : Type u\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A\u2081] [semiring A\u2082] [semiring A\u2083]\nvariables [algebra R A\u2081] [algebra R A\u2082] [algebra R A\u2083]\nvariables (e : A\u2081 \u2243\u2090[R] A\u2082)\n\ninstance : ring_equiv_class (A\u2081 \u2243\u2090[R] A\u2082) A\u2081 A\u2082 :=\n{ coe := to_fun,\n  inv := inv_fun,\n  coe_injective' := \u03bb f g h\u2081 h\u2082, by { cases f, cases g, congr' },\n  map_add := map_add',\n  map_mul := map_mul',\n  left_inv := left_inv,\n  right_inv := right_inv }\n\n/--  Helper instance for when there's too many metavariables to apply\n`fun_like.has_coe_to_fun` directly. -/\ninstance : has_coe_to_fun (A\u2081 \u2243\u2090[R] A\u2082) (\u03bb _, A\u2081 \u2192 A\u2082) := \u27e8alg_equiv.to_fun\u27e9\n\n@[ext]\nlemma ext {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : \u2200 a, f a = g a) : f = g := fun_like.ext f g h\n\nprotected lemma congr_arg {f : A\u2081 \u2243\u2090[R] A\u2082} {x x' : A\u2081} : x = x' \u2192 f x = f x' :=\nfun_like.congr_arg f\n\nprotected lemma congr_fun {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : f = g) (x : A\u2081) : f x = g x :=\nfun_like.congr_fun h x\n\nprotected lemma ext_iff {f g : A\u2081 \u2243\u2090[R] A\u2082} : f = g \u2194 \u2200 x, f x = g x := fun_like.ext_iff\n\nlemma coe_fun_injective : @function.injective (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192 A\u2082) (\u03bb e, (e : A\u2081 \u2192 A\u2082)) :=\nfun_like.coe_injective\n\ninstance has_coe_to_ring_equiv : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2243+* A\u2082) := \u27e8alg_equiv.to_ring_equiv\u27e9\n\n@[simp] lemma coe_mk {to_fun inv_fun left_inv right_inv map_mul map_add commutes} :\n  \u21d1(\u27e8to_fun, inv_fun, left_inv, right_inv, map_mul, map_add, commutes\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = to_fun :=\nrfl\n\n@[simp] theorem mk_coe (e : A\u2081 \u2243\u2090[R] A\u2082) (e' h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8e, e', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = e := ext $ \u03bb _, rfl\n\n@[simp] lemma to_fun_eq_coe (e : A\u2081 \u2243\u2090[R] A\u2082) : e.to_fun = e := rfl\n\n@[simp] lemma to_equiv_eq_coe : e.to_equiv = e := rfl\n\n@[simp] lemma to_ring_equiv_eq_coe : e.to_ring_equiv = e := rfl\n\n@[simp, norm_cast] lemma coe_ring_equiv : ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192 A\u2082) = e := rfl\nlemma coe_ring_equiv' : (e.to_ring_equiv : A\u2081 \u2192 A\u2082) = e := rfl\n\nlemma coe_ring_equiv_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2243+* A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ ring_equiv.congr_fun h\n\nprotected lemma map_add : \u2200 x y, e (x + y) = e x + e y := map_add e\nprotected lemma map_zero : e 0 = 0 := map_zero e\nprotected lemma map_mul : \u2200 x y, e (x * y) = (e x) * (e y) := map_mul e\nprotected lemma map_one : e 1 = 1 := map_one e\n\n@[simp] lemma commutes : \u2200 (r : R), e (algebra_map R A\u2081 r) = algebra_map R A\u2082 r :=\n  e.commutes'\n\n@[simp] lemma map_smul (r : R) (x : A\u2081) : e (r \u2022 x) = r \u2022 e x :=\nby simp only [algebra.smul_def, map_mul, commutes]\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u2211 x in s, f x) = \u2211 x in s, e (f x) :=\ne.to_add_equiv.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.sum g) = f.sum (\u03bb i b, e (g i b)) :=\ne.map_sum _ _\n\n/-- Interpret an algebra equivalence as an algebra homomorphism.\n\nThis definition is included for symmetry with the other `to_*_hom` projections.\nThe `simp` normal form is to use the coercion of the `has_coe_to_alg_hom` instance. -/\ndef to_alg_hom : A\u2081 \u2192\u2090[R] A\u2082 :=\n{ map_one' := e.map_one, map_zero' := e.map_zero, ..e }\n\ninstance has_coe_to_alg_hom : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192\u2090[R] A\u2082) :=\n\u27e8to_alg_hom\u27e9\n\n@[simp] lemma to_alg_hom_eq_coe : e.to_alg_hom = e := rfl\n\n@[simp, norm_cast] lemma coe_alg_hom : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192 A\u2082) = e :=\nrfl\n\nlemma coe_alg_hom_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2192\u2090[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ alg_hom.congr_fun h\n\n/-- The two paths coercion can take to a `ring_hom` are equivalent -/\nlemma coe_ring_hom_commutes : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192+* A\u2082) = ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192+* A\u2082) :=\nrfl\n\nprotected lemma map_pow : \u2200 (x : A\u2081) (n : \u2115), e (x ^ n) = (e x) ^ n := e.to_alg_hom.map_pow\nprotected lemma injective : function.injective e := equiv_like.injective e\nprotected lemma surjective : function.surjective e := equiv_like.surjective e\nprotected lemma bijective : function.bijective e := equiv_like.bijective e\n\n/-- Algebra equivalences are reflexive. -/\n@[refl] def refl : A\u2081 \u2243\u2090[R] A\u2081 := {commutes' := \u03bb r, rfl, ..(1 : A\u2081 \u2243+* A\u2081)}\n\ninstance : inhabited (A\u2081 \u2243\u2090[R] A\u2081) := \u27e8refl\u27e9\n\n@[simp] lemma refl_to_alg_hom : \u2191(refl : A\u2081 \u2243\u2090[R] A\u2081) = alg_hom.id R A\u2081 := rfl\n\n@[simp] lemma coe_refl : \u21d1(refl : A\u2081 \u2243\u2090[R] A\u2081) = id := rfl\n\n/-- Algebra equivalences are symmetric. -/\n@[symm]\ndef symm (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2243\u2090[R] A\u2081 :=\n{ commutes' := \u03bb r, by { rw \u2190e.to_ring_equiv.symm_apply_apply (algebra_map R A\u2081 r), congr,\n                         change _ = e _, rw e.commutes, },\n  ..e.to_ring_equiv.symm, }\n\n/-- See Note [custom simps projection] -/\ndef simps.symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2192 A\u2081 := e.symm\n\ninitialize_simps_projections alg_equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\n\n@[simp] lemma inv_fun_eq_symm {e : A\u2081 \u2243\u2090[R] A\u2082} : e.inv_fun = e.symm := rfl\n\n@[simp] lemma symm_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : e.symm.symm = e :=\nby { ext, refl, }\n\nlemma symm_bijective : function.bijective (symm : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2082 \u2243\u2090[R] A\u2081)) :=\nequiv.bijective \u27e8symm, symm, symm_symm, symm_symm\u27e9\n\n@[simp] lemma mk_coe' (e : A\u2081 \u2243\u2090[R] A\u2082) (f h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, e, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2082 \u2243\u2090[R] A\u2081) = e.symm :=\nsymm_bijective.injective $ ext $ \u03bb x, rfl\n\n@[simp] theorem symm_mk (f f') (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm =\n  { to_fun := f', inv_fun := f,\n    ..(\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm } := rfl\n\n@[simp]\ntheorem refl_symm : (alg_equiv.refl : A\u2081 \u2243\u2090[R] A\u2081).symm = alg_equiv.refl := rfl\n\n/-- Algebra equivalences are transitive. -/\n@[trans]\ndef trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) : A\u2081 \u2243\u2090[R] A\u2083 :=\n{ commutes' := \u03bb r, show e\u2082.to_fun (e\u2081.to_fun _) = _, by rw [e\u2081.commutes', e\u2082.commutes'],\n  ..(e\u2081.to_ring_equiv.trans e\u2082.to_ring_equiv), }\n\n@[simp] lemma apply_symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e (e.symm x) = x :=\n  e.to_equiv.apply_symm_apply\n\n@[simp] lemma symm_apply_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e.symm (e x) = x :=\n  e.to_equiv.symm_apply_apply\n\n@[simp] lemma symm_trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2083) :\n  (e\u2081.trans e\u2082).symm x = e\u2081.symm (e\u2082.symm x) := rfl\n\n@[simp] lemma coe_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  \u21d1(e\u2081.trans e\u2082) = e\u2082 \u2218 e\u2081 := rfl\n\n@[simp] lemma trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2081) :\n  (e\u2081.trans e\u2082) x = e\u2082 (e\u2081 x) := rfl\n\n@[simp] lemma comp_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp (e : A\u2081 \u2192\u2090[R] A\u2082) \u2191e.symm = alg_hom.id R A\u2082 :=\nby { ext, simp }\n\n@[simp] lemma symm_comp (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp \u2191e.symm (e : A\u2081 \u2192\u2090[R] A\u2082) = alg_hom.id R A\u2081 :=\nby { ext, simp }\n\ntheorem left_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.left_inverse e.symm e := e.left_inv\n\ntheorem right_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.right_inverse e.symm e := e.right_inv\n\n/-- If `A\u2081` is equivalent to `A\u2081'` and `A\u2082` is equivalent to `A\u2082'`, then the type of maps\n`A\u2081 \u2192\u2090[R] A\u2082` is equivalent to the type of maps `A\u2081' \u2192\u2090[R] A\u2082'`. -/\ndef arrow_congr {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082'] [algebra R A\u2081'] [algebra R A\u2082']\n  (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') : (A\u2081 \u2192\u2090[R] A\u2082) \u2243 (A\u2081' \u2192\u2090[R] A\u2082') :=\n{ to_fun := \u03bb f, (e\u2082.to_alg_hom.comp f).comp e\u2081.symm.to_alg_hom,\n  inv_fun := \u03bb f, (e\u2082.symm.to_alg_hom.comp f).comp e\u2081.to_alg_hom,\n  left_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, symm_comp],\n    simp only [\u2190alg_hom.comp_assoc, symm_comp, alg_hom.id_comp, alg_hom.comp_id] },\n  right_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, comp_symm],\n    simp only [\u2190alg_hom.comp_assoc, comp_symm, alg_hom.id_comp, alg_hom.comp_id] } }\n\nlemma arrow_congr_comp {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082')\n  (e\u2083 : A\u2083 \u2243\u2090[R] A\u2083') (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2083) :\n  arrow_congr e\u2081 e\u2083 (g.comp f) = (arrow_congr e\u2082 e\u2083 g).comp (arrow_congr e\u2081 e\u2082 f) :=\nby { ext, simp only [arrow_congr, equiv.coe_fn_mk, alg_hom.comp_apply],\n  congr, exact (e\u2082.symm_apply_apply _).symm }\n\n@[simp] lemma arrow_congr_refl :\n  arrow_congr alg_equiv.refl alg_equiv.refl = equiv.refl (A\u2081 \u2192\u2090[R] A\u2082) :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_trans {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2081' : A\u2081' \u2243\u2090[R] A\u2082')\n  (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (e\u2082' : A\u2082' \u2243\u2090[R] A\u2083') :\n  arrow_congr (e\u2081.trans e\u2082) (e\u2081'.trans e\u2082') = (arrow_congr e\u2081 e\u2081').trans (arrow_congr e\u2082 e\u2082') :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_symm {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082']\n  [algebra R A\u2081'] [algebra R A\u2082'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') :\n  (arrow_congr e\u2081 e\u2082).symm = arrow_congr e\u2081.symm e\u2082.symm :=\nby { ext, refl }\n\n/-- If an algebra morphism has an inverse, it is a algebra isomorphism. -/\ndef of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 : f.comp g = alg_hom.id R A\u2082)\n  (h\u2082 : g.comp f = alg_hom.id R A\u2081) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun    := f,\n  inv_fun   := g,\n  left_inv  := alg_hom.ext_iff.1 h\u2082,\n  right_inv := alg_hom.ext_iff.1 h\u2081,\n  ..f }\n\nlemma coe_alg_hom_of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  \u2191(of_alg_hom f g h\u2081 h\u2082) = f := alg_hom.ext $ \u03bb _, rfl\n\n@[simp]\nlemma of_alg_hom_coe_alg_hom (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  of_alg_hom \u2191f g h\u2081 h\u2082 = f := ext $ \u03bb _, rfl\n\nlemma of_alg_hom_symm (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  (of_alg_hom f g h\u2081 h\u2082).symm = of_alg_hom g f h\u2082 h\u2081 := rfl\n\n/-- Promotes a bijective algebra homomorphism to an algebra equivalence. -/\nnoncomputable def of_bijective (f : A\u2081 \u2192\u2090[R] A\u2082) (hf : function.bijective f) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ .. ring_equiv.of_bijective (f : A\u2081 \u2192+* A\u2082) hf, .. f }\n\n@[simp] lemma coe_of_bijective {f : A\u2081 \u2192\u2090[R] A\u2082} {hf : function.bijective f} :\n  (alg_equiv.of_bijective f hf : A\u2081 \u2192 A\u2082) = f := rfl\n\nlemma of_bijective_apply {f : A\u2081 \u2192\u2090[R] A\u2082} {hf : function.bijective f} (a : A\u2081) :\n  (alg_equiv.of_bijective f hf) a = f a := rfl\n\n/-- Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. -/\n@[simps apply] def to_linear_equiv (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2081 \u2243\u2097[R] A\u2082 :=\n{ to_fun    := e,\n  map_smul' := e.map_smul,\n  inv_fun   := e.symm,\n  .. e }\n\n@[simp] lemma to_linear_equiv_refl :\n  (alg_equiv.refl : A\u2081 \u2243\u2090[R] A\u2081).to_linear_equiv = linear_equiv.refl R A\u2081 := rfl\n\n@[simp] lemma to_linear_equiv_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  e.to_linear_equiv.symm = e.symm.to_linear_equiv := rfl\n\n@[simp] lemma to_linear_equiv_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (e\u2081.trans e\u2082).to_linear_equiv = e\u2081.to_linear_equiv.trans e\u2082.to_linear_equiv := rfl\n\ntheorem to_linear_equiv_injective : function.injective (to_linear_equiv : _ \u2192 (A\u2081 \u2243\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_equiv.congr_fun h\n\n/-- Interpret an algebra equivalence as a linear map. -/\ndef to_linear_map : A\u2081 \u2192\u2097[R] A\u2082 :=\ne.to_alg_hom.to_linear_map\n\n@[simp] lemma to_alg_hom_to_linear_map :\n  (e : A\u2081 \u2192\u2090[R] A\u2082).to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_equiv_to_linear_map :\n  e.to_linear_equiv.to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_apply (x : A\u2081) : e.to_linear_map x = e x := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A\u2081 \u2192\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma trans_to_linear_map (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2243\u2090[R] A\u2083) :\n  (f.trans g).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\nsection of_linear_equiv\n\nvariables (l : A\u2081 \u2243\u2097[R] A\u2082)\n  (map_mul : \u2200 x y : A\u2081, l (x * y) = l x * l y)\n  (commutes : \u2200 r : R, l (algebra_map R A\u2081 r) = algebra_map R A\u2082 r)\n\n/--\nUpgrade a linear equivalence to an algebra equivalence,\ngiven that it distributes over multiplication and action of scalars.\n-/\n@[simps apply]\ndef of_linear_equiv : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun := l,\n  inv_fun := l.symm,\n  map_mul' := map_mul,\n  commutes' := commutes,\n  ..l }\n\n@[simp]\nlemma of_linear_equiv_symm :\n  (of_linear_equiv l map_mul commutes).symm = of_linear_equiv l.symm\n    ((of_linear_equiv l map_mul commutes).symm.map_mul)\n    ((of_linear_equiv l map_mul commutes).symm.commutes) :=\nrfl\n\n@[simp] lemma of_linear_equiv_to_linear_equiv (map_mul) (commutes) :\n  of_linear_equiv e.to_linear_equiv map_mul commutes = e :=\nby { ext, refl }\n\n@[simp] lemma to_linear_equiv_of_linear_equiv :\n  to_linear_equiv (of_linear_equiv l map_mul commutes) = l :=\nby { ext, refl }\n\nend of_linear_equiv\n\n@[simps mul one {attrs := []}] instance aut : group (A\u2081 \u2243\u2090[R] A\u2081) :=\n{ mul := \u03bb \u03d5 \u03c8, \u03c8.trans \u03d5,\n  mul_assoc := \u03bb \u03d5 \u03c8 \u03c7, rfl,\n  one := refl,\n  one_mul := \u03bb \u03d5, ext $ \u03bb x, rfl,\n  mul_one := \u03bb \u03d5, ext $ \u03bb x, rfl,\n  inv := symm,\n  mul_left_inv := \u03bb \u03d5, ext $ symm_apply_apply \u03d5 }\n\n@[simp] lemma one_apply (x : A\u2081) : (1 : A\u2081 \u2243\u2090[R] A\u2081) x = x := rfl\n\n@[simp] lemma mul_apply (e\u2081 e\u2082 : A\u2081 \u2243\u2090[R] A\u2081) (x : A\u2081) : (e\u2081 * e\u2082) x = e\u2081 (e\u2082 x) := rfl\n\n/-- An algebra isomorphism induces a group isomorphism between automorphism groups -/\n@[simps apply]\ndef aut_congr (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (A\u2081 \u2243\u2090[R] A\u2081) \u2243* (A\u2082 \u2243\u2090[R] A\u2082) :=\n{ to_fun := \u03bb \u03c8, \u03d5.symm.trans (\u03c8.trans \u03d5),\n  inv_fun := \u03bb \u03c8, \u03d5.trans (\u03c8.trans \u03d5.symm),\n  left_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, symm_apply_apply] },\n  right_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, apply_symm_apply] },\n  map_mul' := \u03bb \u03c8 \u03c7, by { ext, simp only [mul_apply, trans_apply, symm_apply_apply] } }\n\n@[simp] lemma aut_congr_refl : aut_congr (alg_equiv.refl) = mul_equiv.refl (A\u2081 \u2243\u2090[R] A\u2081) :=\nby { ext, refl }\n\n@[simp] lemma aut_congr_symm (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (aut_congr \u03d5).symm = aut_congr \u03d5.symm := rfl\n\n@[simp] lemma aut_congr_trans (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) (\u03c8 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (aut_congr \u03d5).trans (aut_congr \u03c8) = aut_congr (\u03d5.trans \u03c8) := rfl\n\n/-- The tautological action by `A\u2081 \u2243\u2090[R] A\u2081` on `A\u2081`.\n\nThis generalizes `function.End.apply_mul_action`. -/\ninstance apply_mul_semiring_action : mul_semiring_action (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul := ($),\n  smul_zero := alg_equiv.map_zero,\n  smul_add := alg_equiv.map_add,\n  smul_one := alg_equiv.map_one,\n  smul_mul := alg_equiv.map_mul,\n  one_smul := \u03bb _, rfl,\n  mul_smul := \u03bb _ _ _, rfl }\n\n@[simp] protected lemma smul_def (f : A\u2081 \u2243\u2090[R] A\u2081) (a : A\u2081) : f \u2022 a = f a := rfl\n\ninstance apply_has_faithful_scalar : has_faithful_scalar (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n\u27e8\u03bb _ _, alg_equiv.ext\u27e9\n\ninstance apply_smul_comm_class : smul_comm_class R (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul_comm := \u03bb r e a, (e.map_smul r a).symm }\n\ninstance apply_smul_comm_class' : smul_comm_class (A\u2081 \u2243\u2090[R] A\u2081) R A\u2081 :=\n{ smul_comm := \u03bb e r a, (e.map_smul r a) }\n\n@[simp] lemma algebra_map_eq_apply (e : A\u2081 \u2243\u2090[R] A\u2082) {y : R} {x : A\u2081} :\n  (algebra_map R A\u2082 y = e x) \u2194 (algebra_map R A\u2081 y = x) :=\n\u27e8\u03bb h, by simpa using e.symm.to_alg_hom.algebra_map_eq_apply h,\n \u03bb h, e.to_alg_hom.algebra_map_eq_apply h\u27e9\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A\u2081] [comm_semiring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u220f x in s, f x) = \u220f x in s, e (f x) :=\ne.to_alg_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.prod g) = f.prod (\u03bb i a, e (g i a)) :=\ne.to_alg_hom.map_finsupp_prod f g\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_semiring R] [ring A\u2081] [ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\nprotected lemma map_neg (x) : e (-x) = -e x := map_neg e x\nprotected lemma map_sub (x y) : e (x - y) = e x - e y := map_sub e x y\n\nend ring\n\nsection division_ring\n\nvariables [comm_ring R] [division_ring A\u2081] [division_ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\n@[simp] lemma map_inv (x) : e (x\u207b\u00b9) = (e x)\u207b\u00b9 :=\ne.to_alg_hom.map_inv x\n\n@[simp] lemma map_div (x y) : e (x / y) = e x / e y :=\ne.to_alg_hom.map_div x y\n\nend division_ring\n\nend alg_equiv\n\nnamespace mul_semiring_action\n\nvariables {M G : Type*} (R A : Type*) [comm_semiring R] [semiring A] [algebra R A]\n\nsection\nvariables [monoid M] [mul_semiring_action M A] [smul_comm_class M R A]\n\n/-- Each element of the monoid defines a algebra homomorphism.\n\nThis is a stronger version of `mul_semiring_action.to_ring_hom` and\n`distrib_mul_action.to_linear_map`. -/\n@[simps]\ndef to_alg_hom (m : M) : A \u2192\u2090[R] A :=\nalg_hom.mk' (mul_semiring_action.to_ring_hom _ _ m) (smul_comm _)\n\ntheorem to_alg_hom_injective [has_faithful_scalar M A] :\n  function.injective (mul_semiring_action.to_alg_hom R A : M \u2192 A \u2192\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_hom.ext_iff.1 h r\n\nend\n\nsection\nvariables [group G] [mul_semiring_action G A] [smul_comm_class G R A]\n\n/-- Each element of the group defines a algebra equivalence.\n\nThis is a stronger version of `mul_semiring_action.to_ring_equiv` and\n`distrib_mul_action.to_linear_equiv`. -/\n@[simps]\ndef to_alg_equiv (g : G) : A \u2243\u2090[R] A :=\n{ .. mul_semiring_action.to_ring_equiv _ _ g,\n  .. mul_semiring_action.to_alg_hom R A g }\n\ntheorem to_alg_equiv_injective [has_faithful_scalar G A] :\n  function.injective (mul_semiring_action.to_alg_equiv R A : G \u2192 A \u2243\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_equiv.ext_iff.1 h r\n\nend\n\nend mul_semiring_action\n\nsection nat\n\nvariables {R : Type*} [semiring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2115`-algebras. This is only an issue since `algebra.id` and `algebra_nat` are not yet defeq.\n-- TODO: fix this by adding an `of_nat` field to semirings.\n/-- Semiring \u2964 \u2115-Alg -/\n@[priority 99] instance algebra_nat : algebra \u2115 R :=\n{ commutes' := nat.cast_commute,\n  smul_def' := \u03bb _ _, nsmul_eq_mul _ _,\n  to_ring_hom := nat.cast_ring_hom R }\n\ninstance nat_algebra_subsingleton : subsingleton (algebra \u2115 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend nat\n\nnamespace ring_hom\n\nvariables {R S : Type*}\n\n/-- Reinterpret a `ring_hom` as an `\u2115`-algebra homomorphism. -/\ndef to_nat_alg_hom [semiring R] [semiring S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2115] S :=\n{ to_fun := f, commutes' := \u03bb n, by simp, .. f }\n\n/-- Reinterpret a `ring_hom` as a `\u2124`-algebra homomorphism. -/\ndef to_int_alg_hom [ring R] [ring S] [algebra \u2124 R] [algebra \u2124 S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2124] S :=\n{ commutes' := \u03bb n, by simp, .. f }\n\n-- note that `R`, `S` could be `semiring`s but this is useless mathematically speaking -\n-- a \u211a-algebra is a ring. furthermore, this change probably slows down elaboration.\n@[simp] lemma map_rat_algebra_map [ring R] [ring S] [algebra \u211a R] [algebra \u211a S]\n  (f : R \u2192+* S) (r : \u211a) : f (algebra_map \u211a R r) = algebra_map \u211a S r :=\nring_hom.ext_iff.1 (subsingleton.elim (f.comp (algebra_map \u211a R)) (algebra_map \u211a S)) r\n\n/-- Reinterpret a `ring_hom` as a `\u211a`-algebra homomorphism. -/\ndef to_rat_alg_hom [ring R] [ring S] [algebra \u211a R] [algebra \u211a S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u211a] S :=\n{ commutes' := f.map_rat_algebra_map, .. f }\n\nend ring_hom\n\nsection rat\n\ninstance algebra_rat {\u03b1} [division_ring \u03b1] [char_zero \u03b1] : algebra \u211a \u03b1 :=\n(rat.cast_hom \u03b1).to_algebra' $ \u03bb r x, r.cast_commute x\n\n@[simp] theorem algebra_map_rat_rat : algebra_map \u211a \u211a = ring_hom.id \u211a :=\nsubsingleton.elim _ _\n\n-- TODO[gh-6025]: make this an instance once safe to do so\nlemma algebra_rat_subsingleton {\u03b1} [semiring \u03b1] :\n  subsingleton (algebra \u211a \u03b1) :=\n\u27e8\u03bb x y, algebra.algebra_ext x y $ ring_hom.congr_fun $ subsingleton.elim _ _\u27e9\n\nend rat\n\nnamespace algebra\nopen module\n\nvariables (R : Type u) (A : Type v)\n\nvariables [comm_semiring R] [semiring A] [algebra R A]\n\n/-- `algebra_map` as an `alg_hom`. -/\ndef of_id : R \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl, .. algebra_map R A }\nvariables {R}\n\ntheorem of_id_apply (r) : of_id R A r = algebra_map R A r := rfl\n\nend algebra\n\nsection int\n\nvariables (R : Type*) [ring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2124`-algebras. This is only an issue since `algebra.id \u2124` and `algebra_int \u2124` are not yet defeq.\n-- TODO: fix this by adding an `of_int` field to rings.\n/-- Ring \u2964 \u2124-Alg -/\n@[priority 99] instance algebra_int : algebra \u2124 R :=\n{ commutes' := int.cast_commute,\n  smul_def' := \u03bb _ _, zsmul_eq_mul _ _,\n  to_ring_hom := int.cast_ring_hom R }\n\n/-- A special case of `ring_hom.eq_int_cast'` that happens to be true definitionally -/\n@[simp] lemma algebra_map_int_eq : algebra_map \u2124 R = int.cast_ring_hom R := rfl\n\nvariables {R}\n\ninstance int_algebra_subsingleton : subsingleton (algebra \u2124 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend int\n\n/-!\nThe R-algebra structure on `\u03a0 i : I, A i` when each `A i` is an R-algebra.\n\nWe couldn't set this up back in `algebra.pi_instances` because this file imports it.\n-/\nnamespace pi\n\nvariable {I : Type u}     -- The indexing type\nvariable {R : Type*}      -- The scalar type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\nvariables (I f)\n\ninstance algebra {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] :\n  algebra R (\u03a0 i : I, f i) :=\n{ commutes' := \u03bb a f, begin ext, simp [algebra.commutes], end,\n  smul_def' := \u03bb a f, begin ext, simp [algebra.smul_def], end,\n  ..(pi.ring_hom (\u03bb i, algebra_map R (f i)) : R \u2192+* \u03a0 i : I, f i) }\n\n@[simp] lemma algebra_map_apply {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] (a : R) (i : I) :\n  algebra_map R (\u03a0 i, f i) a i = algebra_map R (f i) a := rfl\n\n-- One could also build a `\u03a0 i, R i`-algebra structure on `\u03a0 i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\n\nvariables {I} (R) (f)\n\n/-- `function.eval` as an `alg_hom`. The name matches `pi.eval_ring_hom`, `pi.eval_monoid_hom`,\netc. -/\n@[simps]\ndef eval_alg_hom {r : comm_semiring R} [\u03a0 i, semiring (f i)] [\u03a0 i, algebra R (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192\u2090[R] f i :=\n{ to_fun := \u03bb f, f i, commutes' := \u03bb r, rfl, .. pi.eval_ring_hom f i}\n\nvariables (A B : Type*) [comm_semiring R] [semiring B] [algebra R B]\n\n/-- `function.const` as an `alg_hom`. The name matches `pi.const_ring_hom`, `pi.const_monoid_hom`,\netc. -/\n@[simps]\ndef const_alg_hom : B \u2192\u2090[R] (A \u2192 B) :=\n{ to_fun := function.const _,\n  commutes' := \u03bb r, rfl,\n  .. pi.const_ring_hom A B}\n\n/-- When `R` is commutative and permits an `algebra_map`, `pi.const_ring_hom` is equal to that\nmap. -/\n@[simp] lemma const_ring_hom_eq_algebra_map : const_ring_hom A R = algebra_map R (A \u2192 R) :=\nrfl\n\n@[simp] lemma const_alg_hom_eq_algebra_of_id : const_alg_hom R A R = algebra.of_id R (A \u2192 R) :=\nrfl\n\nend pi\n\n/-- A special case of `pi.algebra` for non-dependent types. Lean struggles to elaborate\ndefinitions elsewhere in the library without this, -/\ninstance function.algebra {R : Type*} (I : Type*)  (A : Type*) [comm_semiring R]\n  [semiring A] [algebra R A] : algebra R (I \u2192 A) :=\npi.algebra _ _\n\nnamespace alg_equiv\n\n/-- A family of algebra equivalences `\u03a0 j, (A\u2081 j \u2243\u2090 A\u2082 j)` generates a\nmultiplicative equivalence between `\u03a0 j, A\u2081 j` and `\u03a0 j, A\u2082 j`.\n\nThis is the `alg_equiv` version of `equiv.Pi_congr_right`, and the dependent version of\n`alg_equiv.arrow_congr`.\n-/\n@[simps apply]\ndef Pi_congr_right {R \u03b9 : Type*} {A\u2081 A\u2082 : \u03b9 \u2192 Type*} [comm_semiring R]\n  [\u03a0 i, semiring (A\u2081 i)] [\u03a0 i, semiring (A\u2082 i)] [\u03a0 i, algebra R (A\u2081 i)] [\u03a0 i, algebra R (A\u2082 i)]\n  (e : \u03a0 i, A\u2081 i \u2243\u2090[R] A\u2082 i) : (\u03a0 i, A\u2081 i) \u2243\u2090[R] \u03a0 i, A\u2082 i :=\n{ to_fun := \u03bb x j, e j (x j),\n  inv_fun := \u03bb x j, (e j).symm (x j),\n  commutes' := \u03bb r, by { ext i, simp },\n  .. @ring_equiv.Pi_congr_right \u03b9 A\u2081 A\u2082 _ _ (\u03bb i, (e i).to_ring_equiv) }\n\n@[simp]\nlemma Pi_congr_right_refl {R \u03b9 : Type*} {A : \u03b9 \u2192 Type*} [comm_semiring R]\n  [\u03a0 i, semiring (A i)] [\u03a0 i, algebra R (A i)] :\n  Pi_congr_right (\u03bb i, (alg_equiv.refl : A i \u2243\u2090[R] A i)) = alg_equiv.refl := rfl\n\n@[simp]\nlemma Pi_congr_right_symm {R \u03b9 : Type*} {A\u2081 A\u2082 : \u03b9 \u2192 Type*} [comm_semiring R]\n  [\u03a0 i, semiring (A\u2081 i)] [\u03a0 i, semiring (A\u2082 i)] [\u03a0 i, algebra R (A\u2081 i)] [\u03a0 i, algebra R (A\u2082 i)]\n  (e : \u03a0 i, A\u2081 i \u2243\u2090[R] A\u2082 i) : (Pi_congr_right e).symm = (Pi_congr_right $ \u03bb i, (e i).symm) := rfl\n\n@[simp]\nlemma Pi_congr_right_trans {R \u03b9 : Type*} {A\u2081 A\u2082 A\u2083 : \u03b9 \u2192 Type*} [comm_semiring R]\n  [\u03a0 i, semiring (A\u2081 i)] [\u03a0 i, semiring (A\u2082 i)] [\u03a0 i, semiring (A\u2083 i)]\n  [\u03a0 i, algebra R (A\u2081 i)] [\u03a0 i, algebra R (A\u2082 i)] [\u03a0 i, algebra R (A\u2083 i)]\n  (e\u2081 : \u03a0 i, A\u2081 i \u2243\u2090[R] A\u2082 i) (e\u2082 : \u03a0 i, A\u2082 i \u2243\u2090[R] A\u2083 i) :\n  (Pi_congr_right e\u2081).trans (Pi_congr_right e\u2082) = (Pi_congr_right $ \u03bb i, (e\u2081 i).trans (e\u2082 i)) :=\nrfl\n\nend alg_equiv\n\nsection is_scalar_tower\n\nvariables {R : Type*} [comm_semiring R]\nvariables (A : Type*) [semiring A] [algebra R A]\nvariables {M : Type*} [add_comm_monoid M] [module A M] [module R M] [is_scalar_tower R A M]\nvariables {N : Type*} [add_comm_monoid N] [module A N] [module R N] [is_scalar_tower R A N]\n\nlemma algebra_compatible_smul (r : R) (m : M) : r \u2022 m = ((algebra_map R A) r) \u2022 m :=\nby rw [\u2190(one_smul A m), \u2190smul_assoc, algebra.smul_def, mul_one, one_smul]\n\n@[simp] lemma algebra_map_smul (r : R) (m : M) : ((algebra_map R A) r) \u2022 m = r \u2022 m :=\n(algebra_compatible_smul A r m).symm\n\nlemma no_zero_smul_divisors.trans (R A M : Type*) [comm_ring R] [ring A] [is_domain A] [algebra R A]\n  [add_comm_group M] [module R M] [module A M] [is_scalar_tower R A M] [no_zero_smul_divisors R A]\n  [no_zero_smul_divisors A M] : no_zero_smul_divisors R M :=\nbegin\n  refine \u27e8\u03bb r m h, _\u27e9,\n  rw [algebra_compatible_smul A r m] at h,\n  cases smul_eq_zero.1 h with H H,\n  { have : function.injective (algebra_map R A) :=\n      no_zero_smul_divisors.iff_algebra_map_injective.1 infer_instance,\n    left,\n    exact (injective_iff_map_eq_zero _).1 this _ H },\n  { right,\n    exact H }\nend\n\nvariable {A}\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class : smul_comm_class R A M :=\n\u27e8\u03bb r a m, by rw [algebra_compatible_smul A r (a \u2022 m), smul_smul, algebra.commutes, mul_smul,\n  \u2190algebra_compatible_smul]\u27e9\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class' : smul_comm_class A R M :=\nsmul_comm_class.symm _ _ _\n\nlemma smul_algebra_smul_comm (r : R) (a : A) (m : M) : a \u2022 r \u2022 m = r \u2022 a \u2022 m :=\nsmul_comm _ _ _\n\nnamespace linear_map\n\ninstance coe_is_scalar_tower : has_coe (M \u2192\u2097[A] N) (M \u2192\u2097[R] N) :=\n\u27e8restrict_scalars R\u27e9\n\nvariables (R) {A M N}\n\n@[simp, norm_cast squash] lemma coe_restrict_scalars_eq_coe (f : M \u2192\u2097[A] N) :\n  (f.restrict_scalars R : M \u2192 N) = f := rfl\n\n@[simp, norm_cast squash] lemma coe_coe_is_scalar_tower (f : M \u2192\u2097[A] N) :\n  ((f : M \u2192\u2097[R] N) : M \u2192 N) = f := rfl\n\n/-- `A`-linearly coerce a `R`-linear map from `M` to `A` to a function, given an algebra `A` over\na commutative semiring `R` and `M` a module over `R`. -/\ndef lto_fun (R : Type u) (M : Type v) (A : Type w)\n  [comm_semiring R] [add_comm_monoid M] [module R M] [comm_ring A] [algebra R A] :\n  (M \u2192\u2097[R] A) \u2192\u2097[A] (M \u2192 A) :=\n{ to_fun := linear_map.to_fun,\n  map_add' := \u03bb f g, rfl,\n  map_smul' := \u03bb c f, rfl }\n\nend linear_map\n\nend is_scalar_tower\n\n/-! TODO: The following lemmas no longer involve `algebra` at all, and could be moved closer\nto `algebra/module/submodule.lean`. Currently this is tricky because `ker`, `range`, `\u22a4`, and `\u22a5`\nare all defined in `linear_algebra/basic.lean`. -/\nsection module\nopen module\n\nvariables (R S M N : Type*) [semiring R] [semiring S] [has_scalar R S]\nvariables [add_comm_monoid M] [module R M] [module S M] [is_scalar_tower R S M]\nvariables [add_comm_monoid N] [module R N] [module S N] [is_scalar_tower R S N]\n\nvariables {S M N}\n\n@[simp]\nlemma linear_map.ker_restrict_scalars (f : M \u2192\u2097[S] N) :\n  (f.restrict_scalars R).ker = f.ker.restrict_scalars R :=\nrfl\n\nend module\n\nnamespace submodule\n\nvariables (R A M : Type*)\nvariables [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M]\nvariables [module R M] [module A M] [is_scalar_tower R A M]\n\n/-- If `A` is an `R`-algebra such that the induced morhpsim `R \u2192+* A` is surjective, then the\n`R`-module generated by a set `X` equals the `A`-module generated by `X`. -/\nlemma span_eq_restrict_scalars (X : set M) (hsur : function.surjective (algebra_map R A)) :\n  span R X = restrict_scalars R (span A X) :=\nbegin\n  apply (span_le_restrict_scalars R A X).antisymm (\u03bb m hm, _),\n  refine span_induction hm subset_span (zero_mem _) (\u03bb _ _, add_mem) (\u03bb a m hm, _),\n  obtain \u27e8r, rfl\u27e9 := hsur a,\n  simpa [algebra_map_smul] using smul_mem _ r hm\nend\n\nend submodule\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariables [comm_semiring R] [semiring A] [semiring B]\nvariables [algebra R A] [algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I \u2192 A` and `I \u2192 B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps] protected def comp_left (f : A \u2192\u2090[R] B) (I : Type*) : (I \u2192 A) \u2192\u2090[R] (I \u2192 B) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  commutes' := \u03bb c, by { ext, exact f.commutes' c },\n  .. f.to_ring_hom.comp_left I }\n\nend alg_hom\n\nexample {R A} [comm_semiring R] [semiring A]\n  [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] : algebra R A :=\nalgebra.of_module smul_mul_assoc mul_smul_comm\n", "meta": {"author": "lean-forward", "repo": "class-number-journal", "sha": "34d5872618d289ca3982bd9bc0c6e06af678909a", "save_path": "github-repos/lean/lean-forward-class-number-journal", "path": "github-repos/lean/lean-forward-class-number-journal/class-number-journal-34d5872618d289ca3982bd9bc0c6e06af678909a/src/algebra/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6584175005616829, "lm_q1q2_score": 0.38524083952624727}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n/- FEW SHOT PROMPTS TO CODEX(START)/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2)\nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSymmetric real matrices have real eigenvalues\nEvery real symmetric matrix has real eigenvalues.\n`proof`\nRecall that if $z=a+b i$ is a complex number, its complex conjugate is defined by $\\bar{z}=a-b i$. We have $z \\bar{z}=(a+b i)(a-b i)=a^{2}+b^{2}$, so $z \\bar{z}$ is always a nonnegative real number (and equals 0 only when $z=0$ ). It is also true that if $w, z$ are complex numbers, then $\\overline{w z}=\\bar{w} \\bar{z}$.\n\nWith this in mind, suppose that $\\lambda$ is a (possibly complex) eigenvalue of the real symmetric matrix $A$. Thus there is a nonzero vector $\\mathbf{v}$, also with complex entries, such that $A \\mathbf{v}=\\lambda \\mathbf{v}$. By taking the complex conjugate of both sides, and noting that $\\bar{A}=A$ since $A$ has real entries, we get $\\overline{A \\mathbf{v}}=\\overline{\\lambda \\mathbf{v}} \\Rightarrow A \\overline{\\mathbf{v}}=\\bar{\\lambda} \\overline{\\mathbf{v}}$. Then, using that $A^{T}=A$,\n\n$$\n\\begin{aligned}\n&\\overline{\\mathbf{v}}^{T} A \\mathbf{v}=\\overline{\\mathbf{v}}^{T}(A \\mathbf{v})=\\overline{\\mathbf{v}}^{T}(\\lambda \\mathbf{v})=\\lambda(\\overline{\\mathbf{v}} \\cdot \\mathbf{v}) \\\\\n&\\overline{\\mathbf{v}}^{T} A \\mathbf{v}=(A \\overline{\\mathbf{v}})^{T} \\mathbf{v}=(\\bar{\\lambda} \\overline{\\mathbf{v}})^{T} \\mathbf{v}=\\bar{\\lambda}(\\overline{\\mathbf{v}} \\cdot \\mathbf{v}) .\n\\end{aligned}\n$$\n\nSince $\\mathbf{v} \\neq \\mathbf{0}$, we have $\\overline{\\mathbf{v}} \\cdot \\mathbf{v} \\neq 0$. Thus $\\lambda=\\bar{\\lambda}$, which means $\\lambda \\in \\mathbf{R}$\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n--OUTPUT 1\ntheorem symmetric_real_matrices_have_real_eigenvalues : sorry := sorry\n\n/--`theorem`\nTheorem of the Square Root\nLet $a, b \\in \\R$. Then:\n:$\\sqrt{a^2 + b^2} = \\abs{a} + \\abs{b}$\n`proof`\n{{begin-eqn}}\n{{eqn | l = \\sqrt{a^2 + b^2}\n      | r = \\sqrt{a^2 + b^2}\n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^2 + b^2}\n      | c = \n}}\n{{eqn | r = \\sqrt{a^\nend --Needs more than 2000 tokens!\n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Symmetric real matrices have real eigenvalues.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3852408316827809}}
{"text": "@[simp] theorem get_cons_zero {as : List \u03b1} : (a :: as).get \u27e80, Nat.zero_lt_succ _\u27e9 = a := rfl\n\nexample (a b c : \u03b1) : [a, b, c].get \u27e80, by simp\u27e9 = a := by\n  simp\n\nexample (a : Bool) : (a :: as).get \u27e80, by simp_arith\u27e9 = a := by\n  simp\n\nexample (a : Bool) : (a :: as).get \u27e80, by simp_arith\u27e9 = a := by\n  simp\n\nexample (a b c : \u03b1) : [a, b, c].get \u27e80, by simp\u27e9 = a := by\n  rw [get_cons_zero]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1302.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040617, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.3851656637009003}}
{"text": "import category_theory.assoc_pushouts\nimport category_theory.eq_to_hom\nimport category_theory.groupoid\nimport category_theory.transport\nimport .homotopy\n\nuniverses v u\n\nopen category_theory\nopen category_theory.category\nlocal notation f ` \u2218 `:80 g:80 := g \u226b f\n\nnamespace homotopy_theory.cofibrations\nopen homotopy_theory.weak_equivalences\nopen homotopy_theory.weak_equivalences.category_with_weak_equivalences\nopen precofibration_category cofibration_category\n\nvariables {C : Type u} [category.{v} C] [cofibration_category.{v} C]\n\n-- Tracks, or \"homotopies up to homotopy\". This notion is a bit tricky\n-- because there is no canonical choice of cylinder object on which to\n-- define homotopies. Instead, we define an equivalence relation\n-- between homotopies defined on different cylinder objects and define\n-- a track to be an equivalence class, and then show that every\n-- cylinder object admits a unique homotopy class of homotopies\n-- representing each track.\n\nvariables {a b : C} {j : a \u27f6 b} (hj : is_cof j)\nvariables {x : C}\nvariables (f\u2080 f\u2081 : b \u27f6 x)\n\nstructure homotopy :=\n(c : relative_cylinder hj)\n(h : homotopy_on c f\u2080 f\u2081)\n\nvariables {hj f\u2080 f\u2081}\n-- An extension of homotopies. These are like acyclic cofibrations in\n-- a category of objects under b \u2294\u2090 b and over b and x, where the\n-- compositions b \u2294\u2090 b \u2192 b and b \u2294\u2090 b \u2192 x are given by the fold map\n-- and (f\u2080, f\u2081) respectively.\nstructure homotopy_extension (t t' : homotopy hj f\u2080 f\u2081) :=\n(m : cylinder_embedding t.c t'.c)\n(e : t'.h.H \u2218 m.k = t.h.H)\n\ndef homotopy_extension.refl (t : homotopy hj f\u2080 f\u2081) : homotopy_extension t t :=\n\u27e8cylinder_embedding.refl t.c, show _ \u2218 \ud835\udfd9 _ = _, by simp\u27e9\n\ndef homotopy_extension.trans {t\u2080 t\u2081 t\u2082 : homotopy hj f\u2080 f\u2081}\n  (m\u2080 : homotopy_extension t\u2080 t\u2081) (m\u2081 : homotopy_extension t\u2081 t\u2082) :\n  homotopy_extension t\u2080 t\u2082 :=\n\u27e8m\u2080.m.trans m\u2081.m,\n by dsimp [cylinder_embedding.trans]; rw [assoc, m\u2081.e, m\u2080.e]\u27e9\n\ndef homotopy_extension.pushout {t t\u2080 t\u2081 : homotopy hj f\u2080 f\u2081}\n  (m\u2080 : homotopy_extension t t\u2080) (m\u2081 : homotopy_extension t t\u2081) :\n  homotopy hj f\u2080 f\u2081 :=\n\u27e8cylinder_embedding.pushout m\u2080.m m\u2081.m,\n \u27e8(cylinder_embedding.pushout.is_pushout m\u2080.m m\u2081.m).induced t\u2080.h.H t\u2081.h.H\n    (by rw [m\u2080.e, m\u2081.e]),\n  begin\n    convert t\u2081.h.Hi\u2080 using 1, unfold relative_cylinder.i\u2080,\n    dsimp [cylinder_embedding.pushout], simp\n  end,\n  begin\n    convert t\u2081.h.Hi\u2081 using 1, unfold relative_cylinder.i\u2081,\n    dsimp [cylinder_embedding.pushout], simp\n  end\u27e9\u27e9\n\ndef homotopy_extension.pushout.map\u2080 {t t\u2080 t\u2081 : homotopy hj f\u2080 f\u2081}\n  (m\u2080 : homotopy_extension t t\u2080) (m\u2081 : homotopy_extension t t\u2081) :\n  homotopy_extension t\u2080 (homotopy_extension.pushout m\u2080 m\u2081) :=\n\u27e8cylinder_embedding.pushout.map\u2080 m\u2080.m m\u2081.m,\n by dsimp [cylinder_embedding.pushout.map\u2080, homotopy_extension.pushout]; simp\u27e9\n\ndef homotopy_extension.pushout.map\u2081 {t t\u2080 t\u2081 : homotopy hj f\u2080 f\u2081}\n  (m\u2080 : homotopy_extension t t\u2080) (m\u2081 : homotopy_extension t t\u2081) :\n  homotopy_extension t\u2081 (homotopy_extension.pushout m\u2080 m\u2081) :=\n\u27e8cylinder_embedding.pushout.map\u2081 m\u2080.m m\u2081.m,\n by dsimp [cylinder_embedding.pushout.map\u2081, homotopy_extension.pushout]; simp\u27e9\n\n-- Two homotopies are equivalent if they have a common extension.\ndef homotopy_equiv (t\u2080 t\u2081 : homotopy hj f\u2080 f\u2081) : Prop :=\n\u2203 t' (m\u2080 : homotopy_extension t\u2080 t') (m\u2081 : homotopy_extension t\u2081 t'), true\n\n-- Homotopy equivalence is an equivalence relation.\nlemma homotopy_equiv.refl (t : homotopy hj f\u2080 f\u2081) : homotopy_equiv t t :=\n\u27e8t, homotopy_extension.refl t, homotopy_extension.refl t, \u27e8\u27e9\u27e9\n\nlemma homotopy_equiv.symm {t\u2080 t\u2081 : homotopy hj f\u2080 f\u2081} :\n  homotopy_equiv t\u2080 t\u2081 \u2192 homotopy_equiv t\u2081 t\u2080 :=\nassume \u27e8t', m\u2080, m\u2081, \u27e8\u27e9\u27e9, \u27e8t', m\u2081, m\u2080, \u27e8\u27e9\u27e9\n\nlemma homotopy_equiv.trans {t\u2080 t\u2081 t\u2082 : homotopy hj f\u2080 f\u2081} :\n  homotopy_equiv t\u2080 t\u2081 \u2192 homotopy_equiv t\u2081 t\u2082 \u2192 homotopy_equiv t\u2080 t\u2082 :=\nassume \u27e8t, m\u2080, m\u2081, \u27e8\u27e9\u27e9 \u27e8t', m\u2081', m\u2082', \u27e8\u27e9\u27e9,\n\u27e8m\u2081.pushout m\u2081',\n m\u2080.trans (homotopy_extension.pushout.map\u2080 m\u2081 m\u2081'),\n m\u2082'.trans (homotopy_extension.pushout.map\u2081 m\u2081 m\u2081'),\n \u27e8\u27e9\u27e9\n\nstructure homotopy_iso (t t' : homotopy hj f\u2080 f\u2081) :=\n(k : t.c.ob \u2245 t'.c.ob)\n(hkii : k.hom \u2218 t.c.ii = t'.c.ii)\n(hpk : t'.c.p \u2218 k.hom = t.c.p)\n(e : t'.h.H \u2218 k.hom = t.h.H)\n\nlemma homotopy_equiv_of_iso {t t' : homotopy hj f\u2080 f\u2081} (i : homotopy_iso t t') :\n  homotopy_equiv t t' :=\n\u27e8t', \u27e8\u27e8i.k.hom, cof_iso _, i.hkii, i.hpk\u27e9, i.e\u27e9, homotopy_extension.refl t', \u27e8\u27e9\u27e9\n\ninstance homotopy_equiv.setoid : setoid (homotopy hj f\u2080 f\u2081) :=\n{ r := homotopy_equiv,\n  iseqv :=\n    \u27e8\u03bb t, homotopy_equiv.refl t,\n     \u03bb t\u2080 t\u2081, homotopy_equiv.symm,\n     \u03bb t\u2080 t\u2081 t\u2082, homotopy_equiv.trans\u27e9 }\n\nvariables (hj f\u2080 f\u2081)\ndef track := quotient (homotopy_equiv.setoid : setoid (homotopy hj f\u2080 f\u2081))\n\nprivate noncomputable def chosen_cylinder : relative_cylinder hj :=\nclassical.choice (exists_relative_cylinder hj)\n\nvariables {hj f\u2080 f\u2081}\nnoncomputable def track.refl (f : b \u27f6 x) : track hj f f :=\n\u27e6\u27e8chosen_cylinder hj, homotopy_on.refl f\u27e9\u27e7\n\nlemma track.refl_eq {f : b \u27f6 x} (c : relative_cylinder hj) :\n  (track.refl f : track hj f f) = \u27e6\u27e8c, homotopy_on.refl f\u27e9\u27e7 :=\nquot.sound $\n  let c\u2080 := chosen_cylinder hj,\n      \u27e8\u27e8c', m\u2080, m\u2081\u27e9\u27e9 := exists_common_embedding c\u2080 c in\n  \u27e8\u27e8c', homotopy_on.refl f\u27e9,\n   \u27e8m\u2080, show f \u2218 c'.p \u2218 m\u2080.k = f \u2218 c\u2080.p, by rw [\u2190assoc, m\u2080.hpk]\u27e9,\n   \u27e8m\u2081, show f \u2218 c'.p \u2218 m\u2081.k = f \u2218 c.p, by rw [\u2190assoc, m\u2081.hpk]\u27e9, \u27e8\u27e9\u27e9\n\nlocal attribute [elab_with_expected_type] quotient.lift_on quotient.lift_on\u2082\n\ndef track.symm {f\u2080 f\u2081 : b \u27f6 x} : track hj f\u2080 f\u2081 \u2192 track hj f\u2081 f\u2080 :=\n\u03bb t, quotient.lift_on t\n  (\u03bb t, \u27e6\u27e8t.c.reverse, t.h.symm\u27e9\u27e7)\n  (assume t t' \u27e8t'', m\u2080, m\u2081, \u27e8\u27e9\u27e9, quotient.sound $\n    \u27e8\u27e8t''.c.reverse, t''.h.symm\u27e9, \u27e8m\u2080.m.reverse, m\u2080.e\u27e9, \u27e8m\u2081.m.reverse, m\u2081.e\u27e9, \u27e8\u27e9\u27e9)\n\ndef track.trans {f\u2080 f\u2081 f\u2082 : b \u27f6 x} : track hj f\u2080 f\u2081 \u2192 track hj f\u2081 f\u2082 \u2192 track hj f\u2080 f\u2082 :=\n\u03bb t\u2080 t\u2081, quotient.lift_on\u2082 t\u2080 t\u2081\n  (\u03bb t\u2080 t\u2081, \u27e6\u27e8t\u2080.c.glue t\u2081.c, t\u2080.h.trans t\u2081.h\u27e9\u27e7)\n  (assume t\u2080 t\u2081 t\u2080' t\u2081' \u27e8t\u2080'', m\u2080\u2080, m\u2080\u2081, \u27e8\u27e9\u27e9 \u27e8t\u2081'', m\u2081\u2080, m\u2081\u2081, \u27e8\u27e9\u27e9, quotient.sound $\n    \u27e8\u27e8t\u2080''.c.glue t\u2081''.c, t\u2080''.h.trans t\u2081''.h\u27e9,\n     \u27e8m\u2080\u2080.m.glue m\u2081\u2080.m,\n      begin\n        apply (pushout_by_cof t\u2080.c.i\u2081 t\u2081.c.i\u2080 t\u2080.c.acof_i\u2081.1).is_pushout.uniqueness;\n        dsimp [homotopy_on.trans, cylinder_embedding.glue]; rw \u2190assoc;\n        simp [m\u2080\u2080.e, m\u2081\u2080.e],\n      end\u27e9,\n     \u27e8m\u2080\u2081.m.glue m\u2081\u2081.m,\n      begin\n        apply (pushout_by_cof t\u2080'.c.i\u2081 t\u2081'.c.i\u2080 t\u2080'.c.acof_i\u2081.1).is_pushout.uniqueness;\n        dsimp [homotopy_on.trans, cylinder_embedding.glue]; rw \u2190assoc;\n        simp [m\u2080\u2081.e, m\u2081\u2081.e],\n      end\u27e9, \u27e8\u27e9\u27e9)\n\n-- The groupoid laws.\n\nlemma track.left_identity {f\u2080 f\u2081 : b \u27f6 x} (t : track hj f\u2080 f\u2081) :\n  track.trans (track.refl _) t = t :=\nquotient.induction_on t $ \u03bb \u27e8c\u2081, h\u27e9, quotient.sound $\n  -- Set up variable names to match `exists_common_embedding` as\n  -- closely as possible, so that what we construct is, in particular,\n  -- a common embedding of c\u2080 and c\u2081.\n  let c := chosen_cylinder hj,\n      c\u2080 := c.glue c\u2081,\n      p' : c\u2080.ob \u27f6 c\u2081.ob :=\n        (pushout_by_cof c.i\u2081 c\u2081.i\u2080 c.acof_i\u2081.1).is_pushout.induced\n          (c\u2081.i\u2080 \u2218 c.p) (\ud835\udfd9 c\u2081.ob) (by rw [\u2190assoc, c.pi\u2081]; simp),\n      po := pushout_by_cof c\u2080.ii c\u2081.ii c\u2080.hii,\n      pp := po.is_pushout.induced p' (\ud835\udfd9 c\u2081.ob) $ begin\n        apply (pushout_by_cof j j hj).is_pushout.uniqueness,\n        { rw [\u2190assoc, \u2190assoc], change _ \u2218 c\u2080.i\u2080 = _ \u2218 c\u2081.i\u2080, simp,\n          rw [\u2190assoc, c.pi\u2080], simp },\n        { rw [\u2190assoc, \u2190assoc], change _ \u2218 c\u2080.i\u2081 = _ \u2218 c\u2081.i\u2081, simp }\n      end,\n      \u27e8c'_ob, l, q', hl, hq', q'l\u27e9 := factorization pp,\n      cem :=\n        common_embedding_of_factorization c\u2080 c\u2081 po c'_ob l (c\u2081.p \u2218 q')\n          hl (weq_comp hq' c\u2081.hp) $ begin\n            rw [\u2190assoc, q'l],\n            apply po.is_pushout.uniqueness; rw \u2190assoc; simp,\n            apply (pushout_by_cof c.i\u2081 c\u2081.i\u2080 c.acof_i\u2081.1).is_pushout.uniqueness;\n              rw \u2190assoc; simp; change _ = Is_pushout.induced _ _ _ _ \u2218 _,\n            { simp [c\u2081.pi\u2080] }, { simp },\n          end,\n      h' : homotopy_on cem.c' f\u2080 f\u2081 :=\n        \u27e8h.H \u2218 q',\n         calc\n           h.H \u2218 q' \u2218 (l \u2218 po.map\u2081 \u2218 c\u2081.ii \u2218 _)\n             = h.H \u2218 (q' \u2218 l \u2218 po.map\u2081) \u2218 c\u2081.i\u2080  : by simp [relative_cylinder.i\u2080]\n         ... = h.H \u2218 c\u2081.i\u2080                       : by rw q'l; simp\n         ... = f\u2080                                : h.Hi\u2080,\n         calc\n           h.H \u2218 q' \u2218 (l \u2218 po.map\u2081 \u2218 c\u2081.ii \u2218 _)\n             = h.H \u2218 (q' \u2218 l \u2218 po.map\u2081) \u2218 c\u2081.i\u2081  : by simp [relative_cylinder.i\u2081]\n         ... = h.H \u2218 c\u2081.i\u2081                       : by rw q'l; simp\n         ... = f\u2081                                : h.Hi\u2081\u27e9 in\n  \u27e8\u27e8cem.c', h'\u27e9,\n   \u27e8cem.m\u2080, calc\n      h.H \u2218 q' \u2218 (l \u2218 po.map\u2080)\n        = h.H \u2218 ((q' \u2218 l) \u2218 po.map\u2080)  : by simp\n    ... = h.H \u2218 (pp \u2218 po.map\u2080)        : by rw q'l\n    ... = h.H \u2218 p'                    : by simp\n    ... = (homotopy_on.trans (homotopy_on.refl f\u2080) h).H  : begin\n      unfold homotopy_on.trans homotopy_on.refl,\n      apply (pushout_by_cof c.i\u2081 c\u2081.i\u2080 c.acof_i\u2081.1).is_pushout.uniqueness;\n        rw \u2190assoc; simp [h.Hi\u2080]\n    end\u27e9,\n   \u27e8cem.m\u2081, calc\n      h.H \u2218 q' \u2218 (l \u2218 po.map\u2081)\n        = h.H \u2218 ((q' \u2218 l) \u2218 po.map\u2081)  : by simp\n    ... = h.H \u2218 (pp \u2218 po.map\u2081)        : by rw q'l\n    ... = h.H                         : by simp\u27e9,\n   \u27e8\u27e9\u27e9\n\nlemma track.left_inverse {f\u2080 f\u2081 : b \u27f6 x} (t : track hj f\u2080 f\u2081) :\n  track.trans t.symm t = track.refl _ :=\nquotient.induction_on t $ \u03bb \u27e8c, h\u27e9, quotient.sound $\n  -- Set up variable names to match `exists_common_embedding` as\n  -- closely as possible, so that what we construct is, in particular,\n  -- a common embedding of c\u2080 and c\u2081.\n  let c\u2081 := chosen_cylinder hj,\n      c\u2080 := c.reverse.glue c,\n      p' : c\u2080.ob \u27f6 c.ob :=\n        (pushout_by_cof c.reverse.i\u2081 c.i\u2080 c.reverse.acof_i\u2081.1).is_pushout.induced\n          (\ud835\udfd9 c.ob) (\ud835\udfd9 c.ob) (by simp; erw comp_id), -- Yuck\n      po := pushout_by_cof c\u2080.ii c\u2081.ii c\u2080.hii,\n      pp := po.is_pushout.induced p' (c.i\u2081 \u2218 c\u2081.p) $ begin\n        apply (pushout_by_cof j j hj).is_pushout.uniqueness;\n          rw [\u2190assoc, \u2190assoc],\n        { change _ \u2218 c\u2080.i\u2080 = _ \u2218 c\u2081.i\u2080, simp,\n          erw [\u2190assoc, c\u2081.pi\u2080, comp_id], simp },\n        { change _ \u2218 c\u2080.i\u2081 = _ \u2218 c\u2081.i\u2081, simp, rw [\u2190assoc, c\u2081.pi\u2081], simp }\n      end,\n      \u27e8c'_ob, l, q', hl, hq', q'l\u27e9 := factorization pp,\n      cem :=\n        common_embedding_of_factorization c\u2080 c\u2081 po c'_ob l (c.p \u2218 q')\n          hl (weq_comp hq' c.hp) $ begin\n            rw [\u2190assoc, q'l],\n            apply po.is_pushout.uniqueness; rw \u2190assoc; simp,\n            apply (pushout_by_cof c.reverse.i\u2081 c.i\u2080 c.reverse.acof_i\u2081.1).is_pushout.uniqueness;\n              rw \u2190assoc; simp; change _ = Is_pushout.induced _ _ _ _ \u2218 _,\n            { erw [id_comp, Is_pushout.induced_commutes\u2080], refl },\n            { simp },\n            { simp [c.pi\u2081] }    -- What is this even for?\n          end,\n      h' : homotopy_on cem.c' f\u2081 f\u2081 :=\n        \u27e8h.H \u2218 q',\n         calc\n           h.H \u2218 q' \u2218 (l \u2218 po.map\u2081 \u2218 c\u2081.ii \u2218 _)\n             = h.H \u2218 (q' \u2218 l \u2218 po.map\u2081) \u2218 c\u2081.i\u2080  : by simp [relative_cylinder.i\u2080]\n         ... = h.H \u2218 c.i\u2081 \u2218 (c\u2081.p \u2218 c\u2081.i\u2080)       : by rw q'l; simp\n         ... = f\u2081                                : by rw [c\u2081.pi\u2080, h.Hi\u2081]; simp,\n         calc\n           h.H \u2218 q' \u2218 (l \u2218 po.map\u2081 \u2218 c\u2081.ii \u2218 _)\n             = h.H \u2218 (q' \u2218 l \u2218 po.map\u2081) \u2218 c\u2081.i\u2081  : by simp [relative_cylinder.i\u2081]\n         ... = h.H \u2218 c.i\u2081 \u2218 (c\u2081.p \u2218 c\u2081.i\u2081)       : by rw q'l; simp\n         ... = f\u2081                                : by rw [c\u2081.pi\u2081, h.Hi\u2081]; simp\u27e9 in\n  \u27e8\u27e8cem.c', h'\u27e9,\n   \u27e8cem.m\u2080, calc\n      h.H \u2218 q' \u2218 (l \u2218 po.map\u2080)\n        = h.H \u2218 ((q' \u2218 l) \u2218 po.map\u2080)  : by simp\n    ... = h.H \u2218 (pp \u2218 po.map\u2080)        : by rw q'l\n    ... = h.H \u2218 p'                    : by simp\n    ... = (homotopy_on.trans h.symm h).H  : begin\n      unfold homotopy_on.trans homotopy_on.symm,\n      apply (pushout_by_cof c.reverse.i\u2081 c.i\u2080 c.reverse.acof_i\u2081.1).is_pushout.uniqueness;\n        rw \u2190assoc; simp; erw id_comp\n    end\u27e9,\n   \u27e8cem.m\u2081, calc\n      h.H \u2218 q' \u2218 (l \u2218 po.map\u2081)\n        = h.H \u2218 ((q' \u2218 l) \u2218 po.map\u2081)  : by simp\n    ... = h.H \u2218 (pp \u2218 po.map\u2081)        : by rw q'l\n    ... = h.H \u2218 c.i\u2081 \u2218 c\u2081.p           : by simp\n    ... = (homotopy_on.refl f\u2081).H     : by rw h.Hi\u2081; refl\u27e9,\n   \u27e8\u27e9\u27e9\n\nlemma track.inverse_inverse {f\u2080 f\u2081 : b \u27f6 x} {t : track hj f\u2080 f\u2081} :\n  t.symm.symm = t :=\n-- t.symm.symm and t are homotopies defined on cylinder objects which\n-- are equal, but not definitionally equal. Rather than dealing with\n-- heterogeneous equality between the homotopies, it's easier to just\n-- use `homotopy_equiv_of_iso`.\nquotient.induction_on t $ \u03bb t, quotient.sound $ homotopy_equiv_of_iso $\n  \u27e8iso.refl _,\n   by apply (pushout_by_cof j j hj).is_pushout.uniqueness;\n      dsimp [relative_cylinder.reverse, Is_pushout.swap];\n      rw [\u2190assoc, \u2190assoc, \u2190assoc]; simp,\n   by dsimp [relative_cylinder.reverse]; simp,\n   by simp [homotopy_on.symm]\u27e9\n\nlemma track.right_inverse {f\u2080 f\u2081 : b \u27f6 x} (t : track hj f\u2080 f\u2081) :\n  track.trans t t.symm = track.refl _ :=\nby convert track.left_inverse t.symm; rw track.inverse_inverse\n\nlemma track.assoc {f\u2080 f\u2081 f\u2082 f\u2083 : b \u27f6 x}\n  (t\u2080 : track hj f\u2080 f\u2081) (t\u2081 : track hj f\u2081 f\u2082) (t\u2082 : track hj f\u2082 f\u2083) :\n  (t\u2080.trans t\u2081).trans t\u2082 = t\u2080.trans (t\u2081.trans t\u2082) :=\nquotient.induction_on\u2083 t\u2080 t\u2081 t\u2082 $ \u03bb t\u2080 t\u2081 t\u2082, quotient.sound $ homotopy_equiv_of_iso\n  \u27e8Is_pushout_assoc\n     (pushout_by_cof t\u2080.c.i\u2081 t\u2081.c.i\u2080 t\u2080.c.acof_i\u2081.1).is_pushout\n     (by convert (pushout_by_cof (t\u2080.c.glue t\u2081.c).i\u2081 t\u2082.c.i\u2080 _).is_pushout using 1; simp)\n     (pushout_by_cof t\u2081.c.i\u2081 t\u2082.c.i\u2080 t\u2081.c.acof_i\u2081.1).is_pushout\n     (by convert (pushout_by_cof t\u2080.c.i\u2081 (t\u2081.c.glue t\u2082.c).i\u2080 _).is_pushout using 1; simp),\n   begin\n     apply (pushout_by_cof j j hj).is_pushout.uniqueness; rw \u2190assoc,\n     { change _ \u2218 relative_cylinder.i\u2080 _ = relative_cylinder.i\u2080 _, simp },\n     { change _ \u2218 relative_cylinder.i\u2081 _ = relative_cylinder.i\u2081 _, simp }\n   end,\n   begin\n     symmetry,\n     apply Is_pushout_assoc_uniqueness;\n       dsimp [relative_cylinder.glue]; simp\n   end,\n   begin\n     symmetry,\n     apply Is_pushout_assoc_uniqueness;\n       dsimp [relative_cylinder.glue, homotopy_on.trans]; simp\n   end\u27e9\n\nlemma track.right_identity {f\u2080 f\u2081 : b \u27f6 x} (t : track hj f\u2080 f\u2081) :\n  track.trans t (track.refl _) = t :=\ncalc\n  t.trans (track.refl _)\n    = t.trans (t.symm.trans t) : by rw track.left_inverse\n... = (t.trans t.symm).trans t : by rw track.assoc\n... = (track.refl _).trans t   : by rw track.right_inverse\n... = t                        : by rw track.left_identity\n\nsection\nvariables (hj x)\ninclude hj\ndef track_groupoid_rel := b \u27f6 x\nend\n\nnoncomputable instance : groupoid (track_groupoid_rel hj x) :=\n{ hom := \u03bb f\u2080 f\u2081, track hj f\u2080 f\u2081,\n  id := \u03bb f, track.refl f,\n  comp := \u03bb f\u2080 f\u2081 f\u2082 t\u2080 t\u2081, t\u2080.trans t\u2081,\n  inv := \u03bb f\u2080 f\u2081 t, t.symm,\n\n  id_comp' := \u03bb f\u2080 f\u2081, track.left_identity,\n  comp_id' := \u03bb f\u2080 f\u2081, track.right_identity,\n  assoc' := \u03bb f\u2080 f\u2081 f\u2082 f\u2083, track.assoc,\n  inv_comp' := \u03bb f\u2080 f\u2081, track.left_inverse,\n  comp_inv' := \u03bb f\u2080 f\u2081, track.right_inverse }\n\nsection functoriality\n\nvariables {y : C} (g : x \u27f6 y)\n\ndef track.congr_left {f\u2080 f\u2081 : b \u27f6 x} (t : track hj f\u2080 f\u2081) :\n  track hj (g \u2218 f\u2080) (g \u2218 f\u2081) :=\nquotient.lift_on t\n  (\u03bb t, \u27e6\u27e8t.c, t.h.congr_left hj g\u27e9\u27e7)\n  (\u03bb t t' \u27e8t'', m\u2080, m\u2081, \u27e8\u27e9\u27e9, quotient.sound\n     \u27e8\u27e8t''.c, t''.h.congr_left hj g\u27e9,\n      \u27e8m\u2080.m, show (g \u2218 _) \u2218 _ = _, by rw [\u2190assoc, m\u2080.e]; refl\u27e9,\n      \u27e8m\u2081.m, show (g \u2218 _) \u2218 _ = _, by rw [\u2190assoc, m\u2081.e]; refl\u27e9,\n      \u27e8\u27e9\u27e9)\n\nnoncomputable def track_groupoid_rel_functor {y} (g : x \u27f6 y) :\n  track_groupoid_rel hj x \u219d track_groupoid_rel hj y :=\n{ obj := \u03bb f, g \u2218 f,\n  map := \u03bb f\u2080 f\u2081 t, t.congr_left g,\n  map_id' := \u03bb f,\n    show (track.refl f).congr_left g = track.refl (g \u2218 f),\n    begin\n      apply congr_arg quotient.mk,\n      unfold homotopy_on.refl homotopy_on.congr_left,\n      congr' 2,\n      rw \u2190assoc, refl\n    end,\n  map_comp' := \u03bb f\u2080 f\u2081 f\u2082 t\u2080 t\u2081,\n    show (t\u2080.trans t\u2081).congr_left g = (t\u2080.congr_left g).trans (t\u2081.congr_left g),\n    begin\n      induction t\u2080 using quot.ind,\n      induction t\u2081 using quot.ind,\n      apply congr_arg quotient.mk,\n      congr', apply homotopy_on.ext,\n      apply pushout_induced_comp\n    end }\n\n-- TODO: Precomposition is also a functor, & they are compatible,\n-- interchange, etc.\n\nend functoriality\n\n-- We next relate tracks back to the original notion of \"homotopies up\n-- to homotopy\", showing that for any particular relative cylinder\n-- object c on a \u2192 b, tracks from f\u2080 : b \u2192 x to f\u2081 : b \u2192 x are in\n-- one-to-one correspondence with homotopy classes rel b \u2294\u2090 b \u2192 c of\n-- homotopies from f\u2080 to f\u2081 defined on the given relative cylinder c,\n-- provided that x is fibrant.\n\nsection track_homotopy_class\nvariables {c : relative_cylinder hj}\n-- Use g\u2080 g\u2081 instead of f\u2080 f\u2081, so that we can put the variables in the\n-- correct order (c first).\nvariables (hx : fibrant x)\nvariables {g\u2080 g\u2081 : b \u27f6 x}\n\nsection correspondence\n\ndef homotopic_homotopies (h\u2080 h\u2081 : homotopy_on c g\u2080 g\u2081) : Prop :=\nhomotopic_rel c.hii h\u2080.H h\u2081.H\n\nvariables (c g\u2080 g\u2081)\ninstance homotopic_homotopies.setoid : setoid (homotopy_on c g\u2080 g\u2081) :=\n{ r := homotopic_homotopies,\n  iseqv :=\n    \u27e8\u03bb h, homotopic_rel.refl _,\n     \u03bb h\u2080 h\u2081, homotopic_rel.symm,\n     \u03bb h\u2080 h\u2081 h\u2082, homotopic_rel.trans\u27e9 }\n\ndef homotopy_up_to_homotopy : Type v :=\nquotient (homotopic_homotopies.setoid c g\u2080 g\u2081)\n\nvariables {c g\u2080 g\u2081}\n-- One direction of the correspondence is easy.\ndef track_of_homotopy_on (h : homotopy_on c g\u2080 g\u2081) : track hj g\u2080 g\u2081 := \u27e6\u27e8c, h\u27e9\u27e7\n\nvariables {x c g\u2080 g\u2081}\nlemma eq_track_of_homotopic_rel (h\u2080 h\u2081 : homotopy_on c g\u2080 g\u2081) :\n  homotopic_homotopies h\u2080 h\u2081 \u2192 track_of_homotopy_on h\u2080 = track_of_homotopy_on h\u2081 :=\nassume \u27e8c', \u27e8\u27e8H, Hi\u2080, Hi\u2081\u27e9\u27e9\u27e9, quotient.sound $\n  -- c' is a relative cylinder on b \u2294\u2090 b \u2192 c. We can also view its\n  -- underlying object as a relative cylinder on the original map a \u2192 b,\n  -- and then H as a homotopy from g\u2080 to g\u2081 rel a \u2192 b.\n  let c'' : relative_cylinder hj :=\n        \u27e8c'.ob, c'.i\u2080 \u2218 c.ii, c.p \u2218 c'.p,\n         cof_comp c.hii c'.acof_i\u2080.1, weq_comp c'.hp c.hp, calc\n           (c.p \u2218 c'.p) \u2218 (c'.i\u2080 \u2218 c.ii)\n             = c.p \u2218 (c'.p \u2218 c'.i\u2080) \u2218 c.ii  : by simp\n         ... = c.p \u2218 c.ii                   : by rw c'.pi\u2080; simp\n         ... = _                            : c.pii\u27e9 in\n  \u27e8\u27e8c'',\n   \u27e8H,\n    show H \u2218 (c'.i\u2080 \u2218 c.ii \u2218 _) = _, by simp [Hi\u2080]; rw \u2190assoc; exact h\u2080.Hi\u2080,\n    show H \u2218 (c'.i\u2080 \u2218 c.ii \u2218 _) = _, by simp [Hi\u2080]; rw \u2190assoc; exact h\u2080.Hi\u2081\u27e9\u27e9,\n   \u27e8\u27e8c'.i\u2080, c'.acof_i\u2080.1, rfl, by rw \u2190assoc; simp [c'.pi\u2080]\u27e9, Hi\u2080\u27e9,\n   \u27e8\u27e8c'.i\u2081, c'.acof_i\u2081.1, c'.ij.symm, by rw \u2190assoc; simp [c'.pi\u2081]\u27e9, Hi\u2081\u27e9, \u27e8\u27e9\u27e9\n\nlocal attribute [elab_with_expected_type] quotient.lift\ndef track_of_homotopy_class : homotopy_up_to_homotopy c g\u2080 g\u2081 \u2192 track hj g\u2080 g\u2081 :=\nquotient.lift track_of_homotopy_on (\u03bb h\u2080 h\u2081, eq_track_of_homotopic_rel h\u2080 h\u2081)\n\n-- Similar to `homotopic_iff`. We choose a common embedding of c and\n-- the cylinder on which t is defined, use the hypothesis that x is\n-- fibrant to extend the homotopy to this new cylinder, and then\n-- restrict it to c.\nprivate def surj (t : track hj g\u2080 g\u2081) :\n  \u2203 h : homotopy_up_to_homotopy c g\u2080 g\u2081, track_of_homotopy_class h = t :=\nquotient.induction_on t $ \u03bb \u27e8c', \u27e8H, Hi\u2080, Hi\u2081\u27e9\u27e9,\n  let \u27e8\u27e8c'', m\u2080, m\u2081\u27e9\u27e9 := exists_common_embedding c c',\n      \u27e8H', hH'\u27e9 := fibrant_iff_rlp.mp hx m\u2081.acof_k H in\n  \u27e8\u27e6\u27e8H' \u2218 m\u2080.k,\n     by rw [\u2190assoc, m\u2080.hki\u2080, \u2190m\u2081.hki\u2080, assoc, hH', Hi\u2080],\n     by rw [\u2190assoc, m\u2080.hki\u2081, \u2190m\u2081.hki\u2081, assoc, hH', Hi\u2081]\u27e9\u27e7,\n   quotient.sound\n     \u27e8\u27e8c'',\n      \u27e8H',\n       by rw [\u2190m\u2081.hki\u2080, assoc, hH', Hi\u2080],\n       by rw [\u2190m\u2081.hki\u2081, assoc, hH', Hi\u2081]\u27e9\u27e9,\n      \u27e8m\u2080, rfl\u27e9, \u27e8m\u2081, hH'\u27e9, \u27e8\u27e9\u27e9\u27e9\n\nset_option eqn_compiler.zeta true\nprivate def inj (h\u2080 h\u2081 : homotopy_up_to_homotopy c g\u2080 g\u2081) :\n  track_of_homotopy_class h\u2080 = track_of_homotopy_class h\u2081 \u2192 h\u2080 = h\u2081 :=\nquotient.induction_on\u2082 h\u2080 h\u2081 $ \u03bb h\u2080 h\u2081 e, quotient.sound $\n  let \u27e8t', m\u2080, m\u2081, \u27e8\u27e9\u27e9 := quotient.exact e,\n      c' := chosen_cylinder c.hii,\n      po :=\n        pushout_by_cof c'.ii\n          ((pushout_by_cof c.ii c.ii c.hii).is_pushout.induced\n            m\u2080.m.k m\u2081.m.k (by rw [m\u2080.m.hkii, m\u2081.m.hkii])) c'.hii,\n      p' := po.is_pushout.induced (c.p \u2218 c'.p) t'.c.p $ begin\n         rw [\u2190assoc, c'.pii],\n         rw [pushout_induced_comp, pushout_induced_comp], congr' 1,\n         { simp [m\u2080.m.hpk] }, { simp [m\u2081.m.hpk] }\n      end,\n      \u27e8z, l, q, hl, hq, ql\u27e9 := factorization p' in\n  have is_acof (l \u2218 po.map\u2081), from\n    \u27e8cof_comp (pushout_is_cof po.is_pushout c'.hii) hl,\n     weq_of_comp_weq_right hq (by convert t'.c.hp; simp [ql])\u27e9,\n  let \u27e8H', hH'\u27e9 := fibrant_iff_rlp.mp hx this t'.h.H in\n  begin\n    rw assoc at hH',\n    refine \u27e8c', \u27e8\u27e8H' \u2218 l \u2218 po.map\u2080, _, _\u27e9\u27e9\u27e9;\n    { change _ \u2218 _ \u2218 _ \u2218 (_ \u2218 _) = _, rw assoc,\n      conv { to_lhs, congr, skip, rw \u2190assoc },\n      rw po.is_pushout.commutes,\n      simp [hH'], rw \u2190assoc, simp [m\u2080.e, m\u2081.e] }\n  end\n\nvariables (c)\n-- FIXME: Why doesn't \u2243 work here??\nnoncomputable def homotopy_class_equiv_track :\n  equiv (homotopy_up_to_homotopy c g\u2080 g\u2081) (track hj g\u2080 g\u2081) :=\nequiv.of_bijective _ \u27e8inj hx, surj hx\u27e9\n\nsection\nvariables (hj hx)\ninclude hj c hx\ndef homotopy_class_groupoid := b \u27f6 x\nend\n\nnoncomputable instance homotopy_class_groupoid.groupoid :\n  groupoid (homotopy_class_groupoid hj c hx) :=\ntransported_groupoid\n  (by apply_instance : groupoid (track_groupoid_rel hj x))\n  (\u03bb g\u2080 g\u2081, (homotopy_class_equiv_track c hx).symm)\n\n-- This is definitionally equal to .to_category of the above,\n-- but helps with defining homotopy_class_functor somehow.\nprivate noncomputable def homotopy_class_groupoid.category :\n  category (homotopy_class_groupoid hj c hx) :=\ntransported_category\n  (by apply_instance : category (track_groupoid_rel hj x))\n  (\u03bb g\u2080 g\u2081, (homotopy_class_equiv_track c hx).symm)\n\nsection functoriality\nvariables {c} {y : C} (hy : fibrant y) (k : x \u27f6 y)\ndef homotopy_up_to_homotopy.congr_left {g\u2080 g\u2081 : b \u27f6 x}\n  (h : homotopy_up_to_homotopy c g\u2080 g\u2081) :\n  homotopy_up_to_homotopy c (k \u2218 g\u2080) (k \u2218 g\u2081) :=\nquotient.lift_on h\n  (\u03bb h, \u27e6h.congr_left hj k\u27e7)\n  (\u03bb h h' H, quotient.sound (H.congr_left c.hii k))\n\nnoncomputable def homotopy_class_functor :\n  homotopy_class_groupoid hj c hx \u219d homotopy_class_groupoid hj c hy :=\nshow @category_theory.functor\n  _ (homotopy_class_groupoid.category c hx)\n  _ (homotopy_class_groupoid.category c hy), from\ntransported_functor\n  (\u03bb g\u2080 g\u2081, (homotopy_class_equiv_track c hx).symm)\n  (\u03bb g\u2080 g\u2081, (homotopy_class_equiv_track c hy).symm)\n  (track_groupoid_rel_functor k)\n\nlemma homotopy_class_functor.obj {g : homotopy_class_groupoid hj c hx} :\n  (homotopy_class_functor hx hy k).obj g = k \u2218 g :=\nrfl\n\nlemma homotopy_class_functor.hom {g\u2080 g\u2081 : homotopy_class_groupoid hj c hx}\n  {h : g\u2080 \u27f6 g\u2081} : homotopy_class_functor hx hy k &> h = h.congr_left k :=\nquotient.induction_on h $ \u03bb h,\nbegin\n  dsimp [homotopy_class_functor, transported_functor],\n  rw \u2190equiv.eq_symm_apply,\n  rw [equiv.symm_symm, equiv.symm_symm],\n  refl\nend\n\nprivate lemma heq_of_homotopies_eq\n  {g\u2080 g\u2080' g\u2081 g\u2081' : homotopy_class_groupoid hj c hx} (e\u2080 : g\u2080 = g\u2080') (e\u2081 : g\u2081 = g\u2081')\n  (H : homotopy_on c g\u2080 g\u2081) (H' : homotopy_on c g\u2080' g\u2081') (e : H.H = H'.H) :\n  (\u27e6H\u27e7 : g\u2080 \u27f6 g\u2081) == (\u27e6H'\u27e7 : g\u2080' \u27f6 g\u2081') :=\nbegin\n  subst e\u2080, subst e\u2081,\n  congr,\n  cases H, cases H',\n  congr,\n  exact e\nend\n\n-- Next we show that homotopy_class_functor is functorial in \"k\".\nlemma homotopy_class_functor.map_id :\n  homotopy_class_functor hx hx (\ud835\udfd9 x) = functor.id (homotopy_class_groupoid hj c hx) :=\nbegin\n  fapply functor.hext,\n  { intro g, rw [homotopy_class_functor.obj], simp },\n  { intros g\u2080 g\u2081 h, rw homotopy_class_functor.hom,\n    induction h using quot.ind,\n    apply heq_of_homotopies_eq; { simp [homotopy_on.congr_left] } }\nend\n\nlemma homotopy_class_functor.map_comp {z : C} (hz : fibrant z) (k' : y \u27f6 z) :\n  (homotopy_class_functor hx hz (k' \u2218 k) : homotopy_class_groupoid hj c hx \u219d _) =\n  (homotopy_class_functor hx hy k).comp (homotopy_class_functor hy hz k') :=\nbegin\n  fapply functor.hext,\n  { intro g, rw [homotopy_class_functor.obj],\n    rw [\u2190assoc], refl },\n  { intros g\u2080 g\u2081 h,\n    rw [functor.comp_map],\n    repeat { rw homotopy_class_functor.hom },\n    induction h using quot.ind,\n    apply heq_of_homotopies_eq hz;\n    { simp [homotopy_class_functor.obj, homotopy_on.congr_left] } }\nend\n\nend functoriality\n\nend correspondence\nend track_homotopy_class\n\nend homotopy_theory.cofibrations\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/homotopy_theory/formal/cofibrations/track.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091671, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.38516566361644133}}
{"text": "import category_theory.category\nimport category_theory.isomorphism\nimport category_theory.types\nimport category_theory.isomorphism\nimport data.finset.basic\nimport data.finset.sort\nimport data.int.parity\nimport .article2\n\n\nnamespace exercises\n\nuniverse u\n\nsection endomaps\n\nopen category_theory\n\nvariables {\u03b1 \u03b2: Type*}\nvariables [category \u03b1]\n\nstructure endomap (\u03b1 : Type*) [category \u03b1] :=\n(carrier : \u03b1)\n(endo : carrier \u27f6 carrier)\n\nstructure endomaps_map (dom: endomap \u03b1) (ima: endomap \u03b1) :=\n(map : dom.carrier \u27f6 ima.carrier)\n(preserve : dom.endo \u226b map = map \u226b ima.endo)\n\n-- Exercise 1 page 137\ndef endomap_maps_comp {A B C: endomap \u03b1} (f : endomaps_map A B) (g : endomaps_map B C) : endomaps_map A C :=\n{\n    map := f.map \u226b g.map,\n    preserve :=\n       calc A.endo \u226b f.map \u226b g.map = (f.map \u226b B.endo) \u226b g.map : by rw [\u2190 category.assoc, f.preserve]\n            ... = f.map \u226b g.map \u226b C.endo : by simp [g.preserve]\n            ... = (f.map \u226b g.map) \u226b C.endo : by simp,\n}\n\n\nvariables {A B : endomap \u03b1}\n\n\n@[simp]\nlemma endo_inj (f : endomaps_map A  B) (pre) :\n    (endomaps_map.mk f.map pre) = f :=\nbegin\n    cases f,\n    refl,\nend\n\ndef endo_category : category (endomap \u03b1) :=\n{\n    hom := \u03bb x y, endomaps_map x y,\n    id := \u03bb x, { map := \ud835\udfd9 x.carrier, preserve := by simp }, \n    comp := \u03bb _ _ _ f g, endomap_maps_comp f g,\n    id_comp' := \u03bb _ _ f, by {simp at *, unfold endomap_maps_comp, simp},\n    comp_id' := \u03bb _ _ f, by {simp at *, unfold endomap_maps_comp, simp},\n    assoc'   := \u03bb _ _ _ _ f g h, by {simp, unfold endomap_maps_comp, simp}\n}\n\ndef Endoset := @endomap Type* category_theory.types\ndef Endoset_map (dom: Endoset) (ima: Endoset):= endomaps_map dom ima\n\ndef endoset_category : category Endoset := endo_category\n\ndef x : Endoset := \u27e8 \u2115, \u03bb n, n + 2 \u27e9\ndef y : Endoset := \u27e8 \u2115, \u03bb n, n + 1 \u27e9\n\ndef yx : endomaps_map y x := {\n     map := \u03bb n:\u2115,  nat.mul n 2,\n     preserve := by {\n         ext a,\n         change nat.mul (a + 1) 2 = (a * 2) + 2,\n         simp,\n         ring,\n     }\n}\n\n\n-- Exercise 2 page 139\nexample {X: \u03b1} (endo r : X \u27f6 X) (idem : idempotent endo) (ret : is_retraction endo r) : endo = \ud835\udfd9 X :=\n    calc endo = endo \u226b \ud835\udfd9 X : by simp\n        ... = endo \u226b (endo \u226b r) : by {unfold is_retraction at ret, rw \u2190ret}\n        ... = (endo \u226b endo) \u226b r : by simp\n        ... = endo \u226b r : by rw idempotent.repeat\n        ... = \ud835\udfd9 X : ret\n\n\ndef involution {A : \u03b1} (f : A \u27f6 A) := f \u226b f = \ud835\udfd9 A \n\n-- Exercise 4 page 140\ndef minus : endomap Type*  := {\n    carrier := \u2124, \n    endo := \u03bb x, -x\n}\n\nexample  : @involution Type*  infer_instance \u2124 (\u03bb x:\u2124, -x) :=\nbegin\n    unfold involution,\n    ext,\n    simp,\nend\n\n-- Exercise 5 page 140\nexample  : @idempotent Type*  infer_instance \u2124 (\u03bb x:\u2124, abs x) := {\n    repeat := by {\n        simp,\n        ext,\n        rw \u2190 abs_abs,\n        simp,\n    }\n}\n\n-- Exercise 6 page 140\nexample  : @is_iso Type* infer_instance \u2124 \u2124  (\u03bb x:\u2124, x + 3) := {\n    inv := \u03bb x, x - 3,\n}\n\nlemma prod_ne_one_of_gr {a b: \u2124} (h: b > 1) : a * b \u2260 1 :=\nbegin\n    intros prod,\n    have h := int.eq_one_of_mul_eq_one_left (by linarith) prod,\n    linarith,\nend\n\n-- Exercise 7 page 140\nexample (iso: @is_iso Type* infer_instance \u2124 \u2124  (\u03bb x:\u2124, x * 5)) : false :=\nbegin\n    have : iso.inv \u226b (\u03bb x:\u2124, x * 5) = \ud835\udfd9 \u2124 := @is_iso.inv_hom_id Type* infer_instance \u2124 \u2124  (\u03bb x:\u2124, x * 5) iso,\n    have h := congr_fun this 1,\n    simp at h,\n    exact prod_ne_one_of_gr (by linarith) h ,\nend\n\n-- Exercise 8 page 140\nexample (A : \u03b1)  (f : A \u27f6 A) (inv: involution f) : f \u226b f \u226b f = f :=\nbegin\n    unfold involution at inv,\n    rw inv,\n    exact category.comp_id _,\nend\n\nexample (A : \u03b1)  (f : A \u27f6 A) [ide: idempotent f] : f \u226b f \u226b f = f :=\nbegin\n    rw ide.repeat,\n    rw ide.repeat,\nend\n\nend endomaps\n\nsection irr_graphs\n\nvariables {\u03b1 \u03b2 \u03b4 \u03b3 \u03b5 \u03b6 : Type u}\n\nstructure irr_graph (\u03b1 : Type u) (\u03b2 : Type u) :=\n(s t : \u03b1 \u2192 \u03b2)\n\nstructure irr_graph_map (dom : irr_graph \u03b1 \u03b2) (ima : irr_graph \u03b4 \u03b3) :=\n(fa  : \u03b1 \u2192 \u03b4)\n(fd : \u03b2 \u2192 \u03b3)\n(pres: fd \u2218 dom.s = ima.s \u2218 fa)\n(pret: fd \u2218 dom.t = ima.t \u2218 fa)\n\nvariables {A : irr_graph \u03b1 \u03b2} {B : irr_graph \u03b4 \u03b3} {C : irr_graph \u03b5 \u03b6}\n\n\n-- Exercise 11 page 142\ndef irr_graph_map_comp (f : irr_graph_map A B) (g : irr_graph_map B C) : irr_graph_map A C :=\n{\n    fa := g.fa \u2218 f.fa,\n    fd := g.fd \u2218 f.fd,\n    pres :=\n        calc (g.fd \u2218 f.fd) \u2218 A.s = g.fd \u2218 (f.fd \u2218 A.s) : by simp\n             ... = g.fd \u2218 (B.s \u2218 f.fa) : by rw f.pres\n             ... = (g.fd \u2218 B.s) \u2218 f.fa : by simp\n             ... = (C.s \u2218 g.fa) \u2218 f.fa : by rw g.pres,\n\n    pret := by {\n        calc (g.fd \u2218 f.fd) \u2218 A.t = g.fd \u2218 (f.fd \u2218 A.t) : by simp\n             ... = g.fd \u2218 (B.t \u2218 f.fa) : by rw f.pret\n             ... = (g.fd \u2218 B.t) \u2218 f.fa : by simp\n             ... = (C.t \u2218 g.fa) \u2218 f.fa : by rw g.pret,\n    },\n}\n\ndef endo_to_irr_graph_on_obj (e: Endoset) : irr_graph e.carrier e.carrier := \n{\n    s := id,\n    t := e.endo\n}\n\ndef endo_to_irr_graph_on_maps {A B : Endoset} (f: Endoset_map A B) :\n    irr_graph_map (endo_to_irr_graph_on_obj A) (endo_to_irr_graph_on_obj B) :=\n{\n    fa := f.map,\n    fd := f.map,\n    pres := by {\n        unfold endo_to_irr_graph_on_obj,\n        simp,\n    },\n    pret := by {\n        exact f.preserve,\n    }\n}\n\n-- Exercise 12 page 143\nlemma endo_insertion_functorial {A B C : Endoset} (f: Endoset_map A B) (g: Endoset_map B C) :\n    endo_to_irr_graph_on_maps( endomap_maps_comp f g ) = irr_graph_map_comp (endo_to_irr_graph_on_maps f) (endo_to_irr_graph_on_maps g) :=\nbegin\n    refl\nend\n\n-- Exercise 13 page 144\nexample {A B : Endoset} (f : irr_graph_map (endo_to_irr_graph_on_obj A) (endo_to_irr_graph_on_obj B) ) :\n    \u2203 g : Endoset_map A B, endo_to_irr_graph_on_maps g = f :=\nbegin\n    have : f.fd = f.fa := f.pres,\n\n    use f.fa,\n    {\n        have pret := f.pret,\n        rw this at pret,\n        exact pret,\n    },\n    {\n        cases f,\n        unfold endo_to_irr_graph_on_maps,\n        simp,\n        exact this.symm,\n    }\nend\n\nend irr_graphs\n\n\nsection simpler\n\nopen category_theory\n\nvariables {\u03b1 \u03b2: Type*}\nvariables [category \u03b1]\n\nstructure simpler (\u03b1 : Type*) [category \u03b1] :=\n(dom : \u03b1)\n(ima : \u03b1)\n(map : dom \u27f6 ima)\n\nstructure simpler_map (dom: simpler \u03b1) (ima: simpler \u03b1) :=\n(dommap : dom.dom \u27f6 ima.dom)\n(imamap : dom.ima \u27f6 ima.ima)\n(preserve : dom.map \u226b imamap = dommap \u226b ima.map)\n\nvariables {A B : simpler \u03b1}\n\n\n@[simp]\nlemma simpler_inj (f : simpler_map A  B) (pre) :\n    (simpler_map.mk f.dommap f.imamap pre) = f :=\nbegin\n    cases f,\n    refl,\nend\n\ndef simpler_maps_comp {A B C: simpler \u03b1} (f : simpler_map A B) (g : simpler_map B C) :\nsimpler_map A C :=\n{\n    dommap := f.dommap \u226b g.dommap,\n    imamap := f.imamap \u226b g.imamap,\n    preserve :=\n       calc A.map \u226b f.imamap \u226b g.imamap\n            = (f.dommap \u226b B.map) \u226b g.imamap : by rw [\u2190 category.assoc, f.preserve]\n        ... = f.dommap \u226b B.map \u226b g.imamap : by  rw [category.assoc]\n        ... = f.dommap \u226b g.dommap \u226b C.map : by rw [\u2190 g.preserve]\n        ... = (f.dommap \u226b g.dommap) \u226b C.map : by rw [\u2190 category.assoc],\n}\n\ninstance simpler_category : category (simpler \u03b1) :=\n{\n    hom := \u03bb x y, simpler_map x y,\n    id := \u03bb x, { dommap := \ud835\udfd9 x.dom, imamap := \ud835\udfd9 x.ima,  preserve := by simp },\n    comp := \u03bb _ _ _ f g, simpler_maps_comp f g,\n    id_comp' := \u03bb _ _ f, by {simp at *, unfold simpler_maps_comp, simp},\n    comp_id' := \u03bb _ _ f, by {simp at *, unfold simpler_maps_comp, simp},\n    assoc'   := \u03bb _ _ _ _ f g h, by {simp, unfold simpler_maps_comp, simp}\n}\n\ndef simpler_set := @simpler Type* category_theory.types\ndef simpler_set_map (dom: simpler_set) (ima: simpler_set):= simpler_map dom ima\n\ndef SimplerSetCategory := category simpler_set\n\ndef endo_inclusion_on_objs (e : endomap \u03b1) : simpler \u03b1 := \u27e8 e.carrier, e.carrier, e.endo \u27e9\ndef endo_inclusion_on_maps {A B : endomap \u03b1 } (f : endomaps_map A B) :\n    simpler_map (endo_inclusion_on_objs A) (endo_inclusion_on_objs B) :=\n{\n    dommap := f.map,\n    imamap := f.map,\n    preserve := f.preserve\n}\n\n-- Exercise 14 page 144\n\ndef AddOne : simpler_set := \u27e8 \u2115, \u2115, \u03bb n, n + 1 \u27e9\ndef AddTwo : simpler_set := \u27e8 \u2115, \u2115, \u03bb n, n + 2 \u27e9\n\ndef AddOneEndo : Endoset := \u27e8 \u2115, \u03bb n, n + 1 \u27e9\ndef AddTwoEndo : Endoset := \u27e8 \u2115, \u03bb n, n + 2 \u27e9\n\n\ndef AddOneToAddTwo : simpler_map (endo_inclusion_on_objs AddOneEndo) (endo_inclusion_on_objs AddTwoEndo) := {\n     dommap := \u03bb n:\u2115, nat.add n  1,\n     imamap := \u03bb n:\u2115, nat.add n  2,\n     preserve := by {\n         ext a,\n         change nat.add (a + 1) 2 = (a + 2) + 1,\n         simp,\n     }\n}\n\nexample : \u00ac \u2203 f, endo_inclusion_on_maps f = AddOneToAddTwo :=\nbegin\n    intros h,\n    rcases h with \u27e8 f, h\u27e9 ,\n    unfold endo_inclusion_on_maps at h,\n    unfold AddOneToAddTwo at h,\n    simp at *,\n    cases h,\n    rw h_left at h_right,\n    have := congr_fun h_right 0,\n    exact (nat.succ_ne_self 1) this.symm,\nend\n\n\nend simpler\n\nsection ref_graphs\n\nvariables {\u03b1 \u03b2 \u03b4 \u03b3 \u03b5 \u03b6 : Type u}\n\nstructure ref_graph (\u03b1 : Type u) (\u03b2 : Type u) extends irr_graph \u03b1 \u03b2  :=\n(i : \u03b2 \u2192 \u03b1)\n(rets: s \u2218 i = id)\n(rett: t \u2218 i = id)\n\n\nstructure ref_graph_map (dom : ref_graph \u03b1 \u03b2) (ima : ref_graph \u03b4 \u03b3)\n    extends irr_graph_map dom.to_irr_graph ima.to_irr_graph :=\n(prei: fa \u2218 dom.i = ima.i \u2218 fd)\n\nvariables {A : ref_graph \u03b1 \u03b2} {B : ref_graph \u03b4 \u03b3} {C : ref_graph \u03b5 \u03b6}\n\n\ndef ref_graph_map_comp (f : ref_graph_map A B) (g : ref_graph_map B C) : ref_graph_map A C :=\n{\n    prei := by {\n        simp,\n            calc (g.fa \u2218 f.fa) \u2218 A.i = g.fa \u2218 (f.fa \u2218 A.i) : by simp\n                ... = g.fa \u2218 B.i \u2218 f.fd : by rw f.prei\n                ... = (g.fa \u2218 B.i) \u2218 f.fd : by simp \n                ... = (C.i \u2218 g.fd) \u2218 f.fd : by rw g.prei \n                ... = C.i \u2218 (g.fd \u2218 f.fd) : by simp,\n\n        },\n\n    ..irr_graph_map_comp f.to_irr_graph_map g.to_irr_graph_map,\n}\n\n-- Exercise 15 page 145\nexample : (A.i \u2218 A.s) \u2218 (A.i \u2218 A.s) = A.i \u2218 A.s :=\ncalc (A.i \u2218 A.s) \u2218 (A.i \u2218 A.s) = A.i \u2218 (A.s \u2218 A.i) \u2218 A.s : by simp\n    ... = A.i \u2218 A.s : by simp[A.rets]\n\nexample : (A.i \u2218 A.t) \u2218 (A.i \u2218 A.t) = A.i \u2218 A.t :=\ncalc (A.i \u2218 A.t) \u2218 (A.i \u2218 A.t) = A.i \u2218 (A.t \u2218 A.i) \u2218 A.t : by simp\n    ... = A.i \u2218 A.t : by simp[A.rett]\n\nexample : (A.i \u2218 A.t) \u2218 (A.i \u2218 A.s) = A.i \u2218 A.s :=\ncalc (A.i \u2218 A.t) \u2218 (A.i \u2218 A.s) = A.i \u2218 (A.t \u2218 A.i) \u2218 A.s : by simp\n    ... = A.i \u2218 A.s : by simp[A.rett] \n\nexample : (A.i \u2218 A.s) \u2218 (A.i \u2218 A.t) = A.i \u2218 A.t :=\ncalc (A.i \u2218 A.s) \u2218 (A.i \u2218 A.t) = A.i \u2218 (A.s \u2218 A.i) \u2218 A.t : by simp\n    ... = A.i \u2218 A.t : by simp [A.rets]\n\n-- Exercise 16 page 145\nexample (f : ref_graph_map A B) : f.fd = B.s \u2218 f.fa \u2218 A.i :=\ncalc f.fd = (B.s \u2218 B.i) \u2218 f.fd : by simp [B.rets]\n     ... = B.s \u2218 (B.i \u2218 f.fd) : by simp\n     ... = B.s \u2218 f.fa \u2218 A.i : by rw f.prei\n\nend ref_graphs\n\nopen category_theory\nvariables {C : Type*}\nvariables [category C]\nvariables {A B T : C}\n\ndef injective (a : A \u27f6 B) := \u2200 (T : C) (x1 x2 : T \u27f6 A), x1 \u226b a = x2 \u226b a \u2192 x1 = x2\n\n-- Exercise 18 page 146\ntheorem injective_of_retraction {a : A \u27f6 B} {p : B \u27f6 A} (ret: is_retraction a p) :\n    injective a :=\nbegin\n    intros T x1 x2 h,\n    unfold is_retraction at ret,\n    calc x1 = x1 \u226b \ud835\udfd9 A : by simp\n        ... = x1 \u226b a \u226b p : by rw ret\n        ... = (x1 \u226b a) \u226b p : by simp\n        ... = x2 \u226b a \u226b p : by {rw h, simp}\n        ... = x2 : by {simp [ret]}\nend\n\nsection endo_example\n\nopen category_theory\n\ninductive Xs : Type\n| x | z\n\ninductive Ys : Type\n| ybar | y | z\n\ndef alpha : Xs \u2192 Xs\n| Xs.x := Xs.z\n| Xs.z := Xs.z\n\ndef beta  : Ys \u2192 Ys\n| Ys.ybar := Ys.y\n| Ys.y := Ys.z\n| Ys.z := Ys.z\n\ndef as : Xs \u2192 Ys\n| Xs.x := Ys.y\n| Xs.z := Ys.z\n\ndef X : Endoset := \u27e8 Xs, alpha \u27e9\ndef Y : Endoset := \u27e8 Ys, beta \u27e9\n\n-- Exercise 19 page 147\ndef a : Endoset_map X Y := \u27e8 as, by {ext, cases x; refl} \u27e9 \n\n-- Exercise 20 page 147\n-- FixMe simplify proof\nexample : @injective Endoset endo_category  X Y a :=\nbegin\n    intros e x1 x2 h,\n    cases x1,\n    cases x2,\n    unfold a at h,\n    congr,\n    ext,\n    injections_and_clear,\n    dsimp at *,\n\n\n    have foo := congr_fun h_1 x,\n    simp at foo,\n    cases x1_map x,\n    {\n\n        cases x2_map x, refl, simp, unfold as at foo,\n        simp at foo,\n        exact foo,\n    },\n    {\n        cases x2_map x, simp, unfold as at foo,\n        simp at foo,\n        exact foo,\n        refl,\n    }\nend\n\n-- Exercise 21 page 147\ndef as_ret1 : Ys \u2192 Xs\n| Ys.ybar := Xs.x\n| Ys.y := Xs.x\n| Ys.z := Xs.z\n\ndef as_ret2 : Ys \u2192 Xs\n| Ys.ybar := Xs.z\n| Ys.y := Xs.x\n| Ys.z := Xs.z\n\n-- FixMe simplify proof\nexample : \u2200 (f : Ys \u2192 Xs), f \u2218 as = id \u2192 f = as_ret1 \u2228 f = as_ret2 :=\nbegin\n    intros f h,\n    apply classical.or_iff_not_imp_left.mpr,\n    intros notret1,\n    ext,\n    have h1: f Ys.ybar \u2260 as_ret1 Ys.ybar,\n    {\n        have a1 : f Ys.y = as_ret1 Ys.y, {change (f \u2218 as) Xs.x = Xs.x, rw h, refl},\n        have a2 : f Ys.z = as_ret1 Ys.z, {change (f \u2218 as) Xs.z = Xs.z, rw h, refl},\n        intros eq_ybar,\n        have  eq : f = as_ret1 , {ext, cases x_1; assumption},\n        exact notret1 eq,\n    },\n\n    cases x,\n    {\n        have  :f Ys.ybar = as_ret1 Ys.ybar \u2228 f Ys.ybar = as_ret2 Ys.ybar,\n        {\n            change f Ys.ybar = Xs.x \u2228 f Ys.ybar = Xs.z,\n            cases f Ys.ybar; simp\n        },\n\n        cases this,\n        exact false.elim (h1 this),\n        exact this,\n    },\n    {\n      change (f \u2218 as) Xs.x = Xs.x,\n      rw h,\n      simp,\n    },\n    {\n      change (f \u2218 as) Xs.z = Xs.z,\n      rw h,\n      simp,\n    }\nend\n\n-- Exercise 22 page 147\nexample : as_ret1 \u2218 beta \u2260 alpha \u2218 as_ret1 :=\nbegin\n    intros h,\n    have := congr_fun h Ys.ybar,\n    injections,\nend\n\nexample : as_ret2 \u2218 beta \u2260 alpha \u2218 as_ret2 :=\nbegin\n    intros h,\n    have := congr_fun h Ys.ybar,\n    injections,\nend\n\n-- Exercise 25 page 148\nexample (\u03b1 \u03b2 \u03b4 \u03b3 : Type*) (A : irr_graph \u03b1 \u03b2) (B : irr_graph \u03b4 \u03b3) (f : irr_graph_map A B)\n    : f.fd \u2218 A.s = f.fd \u2218 A.t\n    \u2192 B.s \u2218 f.fa = B.t \u2218 f.fa :=\nbegin\n    intros h,\n    rw \u2190 f.pres,\n    rw h,\n    rw f.pret,\nend\n\n-- Exercise 26 page 148\ndef zinc (n : \u2124) : \u211a := \u2191 n\n\ndef z5x_endo : Endoset := \u27e8 \u2124, \u03bb n, n * 5 \u27e9\ndef q5x_endo : Endoset := \u27e8 \u211a, \u03bb p, p * 5 \u27e9\n\n-- part 1\ndef zinc_endo_map : Endoset_map z5x_endo q5x_endo :=\n{\n    map := zinc,\n    preserve := by {\n        ext,\n        simp,\n        change zinc (x * 5) = (zinc x) * 5,\n        unfold zinc,\n        simp,\n    }\n}\n\ndef x5 (p : \u211a) : \u211a := p * 5\n\n-- part 2\nexample : \u2203 (f : \u211a \u2192 \u211a), f \u2218 x5 = id \u2227 x5 \u2218 f = id :=\nbegin\n    use \u03bb q, q / 5,\n    split;\n    {\n        ext, simp, unfold x5, ring,\n\n    }\nend\n\n\nlemma endomap_eq (\u03b1 : Type*) [category \u03b1]  (A : endomap \u03b1 ) (B : endomap \u03b1) (f : endomaps_map A B) (g :endomaps_map A B)  :\n  f.map = g.map \u2192 f = g :=\nbegin\n    intros h,\n    cases f, cases g,\n    simp at *,\n    exact h,\nend\n\n-- part 3\nexample : @injective Endoset endo_category  z5x_endo q5x_endo zinc_endo_map :=\nbegin\n    intros e x1 x2 h,\n    unfold zinc_endo_map at h,\n    injections_and_clear,\n    simp at h_1,\n\n    change zinc \u2218 x1.map = zinc \u2218 x2.map at h_1,\n    \n    have caca : x1.map = x2.map, {\n        ext,\n        have foo := congr_fun h_1 x,\n        simp at foo,\n        unfold zinc at foo,\n        norm_cast at *,\n        exact foo,\n    },\n\n    apply endomap_eq,\n    exact caca,\n\nend\n\n-- Exercise 27 page 148\ninductive Set27 : Type\n| x | y\n\ndef a27 : Set27 \u2192 Set27\n| Set27.x := Set27.y\n| Set27.y := Set27.y\n\ndef X27 : Endoset := \u27e8 Set27, a27 \u27e9\n\ndef invert_map : Endoset_map X27 X27 :=\n{\n    map := a27,\n    preserve := by {\n        ext,\n        cases x; refl,\n    }\n} \n\ninstance : category Endoset :=  endoset_category\n\nexample (Y27 : Endoset) (auto : Y27.endo \u226b Y27.endo = \ud835\udfd9 Y27.carrier) (f: Endoset_map X27 Y27): \n    \u00ac @injective Endoset endoset_category X27 Y27 f :=\nbegin\n    intros inj,\n    suffices h: f.map Set27.x = f.map Set27.y,\n    {\n\n        unfold injective at inj,\n        have := inj X27 (\ud835\udfd9 X27) invert_map,\n        simp at this,\n\n\n        have foo := this (by {\n            apply endomap_eq,\n            ext,\n            cases x,\n            {\n                rw h,\n                refl,\n            },\n            {\n                refl,\n\n            }\n        }),\n        unfold invert_map at foo,\n\n        have caca := congr_arg (\u03bb m:Endoset_map _ _, m.map) foo,\n        simp at caca,\n        have := congr_fun caca Set27.x,\n        change Set27.x = Set27.y at this,\n        simp at this_1,\n        exact this_1,\n    },\n    {\n        have  : Y27.endo (f.map Set27.x) = Y27.endo (f.map Set27.y) \u2192 f.map Set27.x = f.map Set27.y , {\n            -- automorph is injective\n            intros h,\n            have := congr_arg Y27.endo h,\n            have foo := congr_fun auto (f.map Set27.x),\n            simp at foo,\n            rw foo at this,\n            have foo := congr_fun auto (f.map Set27.y),\n            simp at foo,\n            rw foo at this,\n            exact this,\n        }, \n        apply this,\n\n        have prex := congr_fun f.preserve Set27.x,\n        have prey := congr_fun f.preserve Set27.y,\n        simp at *,\n        rw \u2190 prey,\n        rw \u2190 prex,\n        refl,\n\n    }\nend\n\nend endo_example\n\nend exercises", "meta": {"author": "paraseba", "repo": "conceptual-mathematics-in-lean", "sha": "cc7877f528bfaf99a024844744e023acbfcadb1d", "save_path": "github-repos/lean/paraseba-conceptual-mathematics-in-lean", "path": "github-repos/lean/paraseba-conceptual-mathematics-in-lean/conceptual-mathematics-in-lean-cc7877f528bfaf99a024844744e023acbfcadb1d/src/article3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3851656549489904}}
{"text": "import Smt.Tactic.Concretize\n\ndef generalAdd [Add \u03b1] (a b : \u03b1) := a + b\n\nexample : @generalAdd Int _ 3 3 = (6 : Int) := by\n  concretize [generalAdd]\n  rfl\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Concretize/Add.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.38516565494899035}}
{"text": "import analysis.calculus.bump_function_inner\nimport topology.metric_space.hausdorff_distance\n\nimport local.one_jet\n\n/-!\n# Local partial differential relations and their formal solutions\n\nThis file defines `rel_loc E F`, the type of first order partial differential relations\nfor maps between two real normed spaces `E` and `F`.\n\nTo any `R : rel_loc E F` we associate the type `sol R` of maps `f : E \u2192 F` of\nsolutions of `R`, and its formal counterpart `formal_sol R`.\n\nThe h-principle question is whether we can deform any formal solution into a solution.\nThe type of deformations is `htpy_jet_sec E F` (homotopies of 1-jet sections).\n-/\n\nnoncomputable theory\n\nopen set function real filter\nopen_locale unit_interval topology\n\nvariables (E : Type*) [normed_add_comm_group E] [normed_space \u211d E]\nvariables (F : Type*) [normed_add_comm_group F] [normed_space \u211d F]\nvariables (P : Type*) [normed_add_comm_group P] [normed_space \u211d P]\n\n/-- A first order relation for maps between real vector spaces. -/\ndef rel_loc := set (one_jet E F)\n\ninstance : has_mem (E \u00d7 F \u00d7 (E \u2192L[\u211d] F)) (rel_loc E F) := set.has_mem\n\n\nvariables {E F}\n\n/-- A predicate stating that a 1-jet section is a formal solution to a first order relation for\nmaps between vector spaces. -/\ndef jet_sec.is_formal_sol (\ud835\udcd5 : jet_sec E F) (R : rel_loc E F) : Prop :=\n\u2200 x, (x, \ud835\udcd5.f x, \ud835\udcd5.\u03c6 x) \u2208 R\n\nnamespace rel_loc\n\n/-- A formal solution to a local relation `R`. -/\n@[ext] structure formal_sol (R : rel_loc E F) extends jet_sec E F :=\n(is_sol : \u2200 x, (x, f x, \u03c6 x) \u2208 R)\n\ninstance (R : rel_loc E F) : has_coe (formal_sol R) (jet_sec E F):=\n\u27e8formal_sol.to_jet_sec\u27e9\n\n@[simp] lemma formal_sol.to_jet_sec_eq_coe {R : rel_loc E F} (\ud835\udcd5 : formal_sol R) :\n\ud835\udcd5.to_jet_sec = (\ud835\udcd5 : jet_sec E F) := rfl\n\n@[simp] lemma formal_sol.coe_is_formal_sol  {R : rel_loc E F} (\ud835\udcd5 : formal_sol R) :\n  (\ud835\udcd5 : jet_sec E F).is_formal_sol R := \ud835\udcd5.is_sol\n\n/-- Bundling a formal solution from a 1-jet section that is a formal solution. -/\ndef _root_.jet_sec.is_formal_sol.formal_sol  {\ud835\udcd5 : jet_sec E F} {R : rel_loc E F}\n  (h : \ud835\udcd5.is_formal_sol R) : formal_sol R :=\n{is_sol := h, ..\ud835\udcd5}\n\ninstance (R : rel_loc E F) : has_coe_to_fun (formal_sol R) (\u03bb S, E \u2192 F \u00d7 (E \u2192L[\u211d] F)) :=\n\u27e8\u03bb \ud835\udcd5, \u03bb x, (\ud835\udcd5.f x, \ud835\udcd5.\u03c6 x)\u27e9\n\n@[simp] lemma formal_sol.coe_apply  {R : rel_loc E F} (\ud835\udcd5 : formal_sol R) (x : E) :\n(\ud835\udcd5 : jet_sec E F) x = \ud835\udcd5 x := rfl\n\nvariables  {R : rel_loc E F}\n\nlemma formal_sol.eq_iff {\ud835\udcd5 \ud835\udcd5' : formal_sol R} {x : E} :\n  \ud835\udcd5 x = \ud835\udcd5' x \u2194 \ud835\udcd5.f x = \ud835\udcd5'.f x \u2227 \ud835\udcd5.\u03c6 x = \ud835\udcd5'.\u03c6 x :=\njet_sec.eq_iff\n\n/-- A formal solution (f, \u03c6) is holonomic at `x` if the differential of `f` at `x` is `\u03c6 x`. -/\ndef formal_sol.is_holonomic_at (\ud835\udcd5 : formal_sol R) (x : E) : Prop := D \ud835\udcd5.f x = \ud835\udcd5.\u03c6 x\n\n-- TODO: this should come from a lemma about `jet_sec`\nlemma formal_sol.is_holonomic_at_congr (\ud835\udcd5 \ud835\udcd5' : formal_sol R) {s : set E}\n  (h : \u2200\u1da0 x near s, \ud835\udcd5 x = \ud835\udcd5' x) : \u2200\u1da0 x near s, \ud835\udcd5.is_holonomic_at x \u2194 \ud835\udcd5'.is_holonomic_at x :=\nbegin\n  apply h.eventually_nhds_set.mono,\n  intros x hx,\n  have hf : \ud835\udcd5.f =\u1da0[\ud835\udcdd x] \ud835\udcd5'.f,\n  { apply hx.mono,\n    simp_rw rel_loc.formal_sol.eq_iff,\n    tauto },\n  unfold rel_loc.formal_sol.is_holonomic_at,\n  rw [hf.fderiv_eq, (rel_loc.formal_sol.eq_iff.mp hx.self_of_nhds).2]\nend\n\nvariable (P)\n/-- A family of formal solutions is a 1-parameter family of formal solutions. -/\n@[ext] structure family_formal_sol (R : rel_loc E F) extends family_jet_sec E F P :=\n(is_sol : \u2200 t x, (x, f t x, \u03c6 t x) \u2208 R)\n\n/-- A homotopy of formal solutions is a 1-parameter family of formal solutions. -/\n@[reducible] def htpy_formal_sol (R : rel_loc E F) := R.family_formal_sol \u211d\n\ndef htpy_formal_sol.to_htpy_jet_sec {R : rel_loc E F} (\ud835\udcd5 : R.htpy_formal_sol) : htpy_jet_sec E F :=\n\ud835\udcd5.to_family_jet_sec\n\nopen rel_loc\n\ninstance (R : rel_loc E F) : has_coe_to_fun (family_formal_sol P R) (\u03bb S, P \u2192 jet_sec E F) :=\n\u27e8\u03bb S t,\n { f := S.f t,\n   f_diff := S.f_diff.comp (cont_diff_const.prod cont_diff_id),\n   \u03c6 := S.\u03c6 t,\n   \u03c6_diff := S.\u03c6_diff.comp (cont_diff_const.prod cont_diff_id) }\u27e9\n\nend rel_loc\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/local/relation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.6224593452091672, "lm_q1q2_score": 0.3851656548645312}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Patrick Massot, Scott Morrison\n-/\nimport category_theory.adjunction.reflective\nimport category_theory.concrete_category.unbundled_hom\nimport category_theory.monad.limits\nimport category_theory.limits.has_limits\nimport topology.category.Top.basic\nimport topology.uniform_space.completion\n\n/-!\n# The category of uniform spaces\n\nWe construct the category of uniform spaces, show that the complete separated uniform spaces\nform a reflective subcategory, and hence possess all limits that uniform spaces do.\n\nTODO: show that uniform spaces actually have all limits!\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- A (bundled) uniform space. -/\ndef UniformSpace : Type (u+1) := bundled uniform_space\n\nnamespace UniformSpace\n\n/-- The information required to build morphisms for `UniformSpace`. -/\ninstance : unbundled_hom @uniform_continuous :=\n\u27e8@uniform_continuous_id, @uniform_continuous.comp\u27e9\n\nattribute [derive [large_category, concrete_category]] UniformSpace\n\ninstance : has_coe_to_sort UniformSpace Type* := bundled.has_coe_to_sort\n\ninstance (x : UniformSpace) : uniform_space x := x.str\n\n/-- Construct a bundled `UniformSpace` from the underlying type and the typeclass. -/\ndef of (\u03b1 : Type u) [uniform_space \u03b1] : UniformSpace := \u27e8\u03b1\u27e9\n\ninstance : inhabited UniformSpace := \u27e8UniformSpace.of empty\u27e9\n\n@[simp] lemma coe_of (X : Type u) [uniform_space X] : (of X : Type u) = X := rfl\n\ninstance (X Y : UniformSpace) : has_coe_to_fun (X \u27f6 Y) (\u03bb _, X \u2192 Y) :=\n\u27e8category_theory.functor.map (forget UniformSpace)\u27e9\n\n@[simp] lemma coe_comp {X Y Z : UniformSpace} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  (f \u226b g : X \u2192 Z) = g \u2218 f := rfl\n@[simp] lemma coe_id (X : UniformSpace) : (\ud835\udfd9 X : X \u2192 X) = id := rfl\n@[simp] lemma coe_mk {X Y : UniformSpace} (f : X \u2192 Y) (hf : uniform_continuous f) :\n  ((\u27e8f, hf\u27e9 : X \u27f6 Y) : X \u2192 Y) = f := rfl\n\nlemma hom_ext {X Y : UniformSpace} {f g : X \u27f6 Y} : (f : X \u2192 Y) = g \u2192 f = g := subtype.eq\n\n/-- The forgetful functor from uniform spaces to topological spaces. -/\ninstance has_forget_to_Top : has_forget\u2082 UniformSpace.{u} Top.{u} :=\n{ forget\u2082 :=\n  { obj := \u03bb X, Top.of X,\n    map := \u03bb X Y f, { to_fun := f,\n                      continuous_to_fun := uniform_continuous.continuous f.property }, }, }\n\nend UniformSpace\n\n/-- A (bundled) complete separated uniform space. -/\nstructure CpltSepUniformSpace :=\n(\u03b1 : Type u)\n[is_uniform_space : uniform_space \u03b1]\n[is_complete_space : complete_space \u03b1]\n[is_separated : separated_space \u03b1]\n\nnamespace CpltSepUniformSpace\n\ninstance : has_coe_to_sort CpltSepUniformSpace (Type u) := \u27e8CpltSepUniformSpace.\u03b1\u27e9\n\nattribute [instance] is_uniform_space is_complete_space is_separated\n\n/-- The function forgetting that a complete separated uniform spaces is complete and separated. -/\ndef to_UniformSpace (X : CpltSepUniformSpace) : UniformSpace :=\nUniformSpace.of X\n\ninstance complete_space (X : CpltSepUniformSpace) : complete_space ((to_UniformSpace X).\u03b1) :=\nCpltSepUniformSpace.is_complete_space X\n\ninstance separated_space (X : CpltSepUniformSpace) : separated_space ((to_UniformSpace X).\u03b1) :=\nCpltSepUniformSpace.is_separated X\n\n/-- Construct a bundled `UniformSpace` from the underlying type and the appropriate typeclasses. -/\ndef of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] :\nCpltSepUniformSpace := \u27e8X\u27e9\n\n@[simp] lemma coe_of (X : Type u) [uniform_space X] [complete_space X] [separated_space X] :\n  (of X : Type u) = X := rfl\n\ninstance : inhabited CpltSepUniformSpace :=\nbegin\n  haveI : separated_space empty := separated_iff_t2.mpr (by apply_instance),\n  exact \u27e8CpltSepUniformSpace.of empty\u27e9\nend\n\n/-- The category instance on `CpltSepUniformSpace`. -/\ninstance category : large_category CpltSepUniformSpace :=\ninduced_category.category to_UniformSpace\n\n/-- The concrete category instance on `CpltSepUniformSpace`. -/\ninstance concrete_category : concrete_category CpltSepUniformSpace :=\ninduced_category.concrete_category to_UniformSpace\n\ninstance has_forget_to_UniformSpace : has_forget\u2082 CpltSepUniformSpace UniformSpace :=\ninduced_category.has_forget\u2082 to_UniformSpace\n\nend CpltSepUniformSpace\n\nnamespace UniformSpace\n\nopen uniform_space\nopen CpltSepUniformSpace\n\n/-- The functor turning uniform spaces into complete separated uniform spaces. -/\nnoncomputable def completion_functor : UniformSpace \u2964 CpltSepUniformSpace :=\n{ obj := \u03bb X, CpltSepUniformSpace.of (completion X),\n  map := \u03bb X Y f, \u27e8completion.map f.1, completion.uniform_continuous_map\u27e9,\n  map_id' := \u03bb X, subtype.eq completion.map_id,\n  map_comp' := \u03bb X Y Z f g, subtype.eq (completion.map_comp g.property f.property).symm, }.\n\n/-- The inclusion of a uniform space into its completion. -/\ndef completion_hom (X : UniformSpace) :\n  X \u27f6 (forget\u2082 CpltSepUniformSpace UniformSpace).obj (completion_functor.obj X) :=\n{ val := (coe : X \u2192 completion X),\n  property := completion.uniform_continuous_coe X }\n\n@[simp] lemma completion_hom_val (X : UniformSpace) (x) :\n  (completion_hom X) x = (x : completion X) := rfl\n\n/-- The mate of a morphism from a `UniformSpace` to a `CpltSepUniformSpace`. -/\nnoncomputable def extension_hom {X : UniformSpace} {Y : CpltSepUniformSpace}\n  (f : X \u27f6 (forget\u2082 CpltSepUniformSpace UniformSpace).obj Y) :\n  completion_functor.obj X \u27f6 Y :=\n{ val := completion.extension f,\n  property := completion.uniform_continuous_extension }\n\n@[simp] lemma extension_hom_val {X : UniformSpace} {Y : CpltSepUniformSpace}\n  (f : X \u27f6 (forget\u2082 _ _).obj Y) (x) :\n  (extension_hom f) x = completion.extension f x := rfl.\n\n@[simp] \n\n/-- The completion functor is left adjoint to the forgetful functor. -/\nnoncomputable def adj : completion_functor \u22a3 forget\u2082 CpltSepUniformSpace UniformSpace :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y,\n  { to_fun := \u03bb f, completion_hom X \u226b f,\n    inv_fun := \u03bb f, extension_hom f,\n    left_inv := \u03bb f, by { dsimp, erw extension_comp_coe },\n    right_inv := \u03bb f,\n    begin\n      apply subtype.eq, funext x, cases f,\n      exact @completion.extension_coe _ _ _ _ _ (CpltSepUniformSpace.separated_space _) f_property _\n    end },\n  hom_equiv_naturality_left_symm' := \u03bb X X' Y f g,\n  begin\n    apply hom_ext, funext x, dsimp,\n    erw [coe_comp, \u2190completion.extension_map],\n    refl, exact g.property, exact f.property,\n  end }\n\nnoncomputable instance : is_right_adjoint (forget\u2082 CpltSepUniformSpace UniformSpace) :=\n\u27e8completion_functor, adj\u27e9\nnoncomputable instance : reflective (forget\u2082 CpltSepUniformSpace UniformSpace) := {}\n\nopen category_theory.limits\n\n-- TODO Once someone defines `has_limits UniformSpace`, turn this into an instance.\nexample [has_limits.{u} UniformSpace.{u}] : has_limits.{u} CpltSepUniformSpace.{u} :=\nhas_limits_of_reflective $ forget\u2082 CpltSepUniformSpace UniformSpace.{u}\n\nend UniformSpace\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/topology/category/UniformSpace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.38512138604044666}}
{"text": "example (P Q : Type) (p : P) (h : P \u2192 Q) : Q :=\nbegin\n    exact h(p),\nend", "meta": {"author": "nomoid", "repo": "lean-proofs", "sha": "b9f03a24623d1a1d111d6c2bbf53c617e2596d6a", "save_path": "github-repos/lean/nomoid-lean-proofs", "path": "github-repos/lean/nomoid-lean-proofs/lean-proofs-b9f03a24623d1a1d111d6c2bbf53c617e2596d6a/src/world5/level1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3851213774581187}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.shift\nimport Mathlib.category_theory.concrete_category.default\nimport Mathlib.category_theory.pi.basic\nimport Mathlib.algebra.group.basic\nimport Mathlib.PostPort\n\nuniverses w u v u_1 \n\nnamespace Mathlib\n\n/-!\n# The category of graded objects\n\nFor any type `\u03b2`, a `\u03b2`-graded object over some category `C` is just\na function `\u03b2 \u2192 C` into the objects of `C`.\nWe put the \"pointwise\" category structure on these, as the non-dependent specialization of\n`category_theory.pi`.\n\nWe describe the `comap` functors obtained by precomposing with functions `\u03b2 \u2192 \u03b3`.\n\nAs a consequence a fixed element (e.g. `1`) in an additive group `\u03b2` provides a shift\nfunctor on `\u03b2`-graded objects\n\nWhen `C` has coproducts we construct the `total` functor `graded_object \u03b2 C \u2964 C`,\nshow that it is faithful, and deduce that when `C` is concrete so is `graded_object \u03b2 C`.\n-/\n\nnamespace category_theory\n\n\n/-- A type synonym for `\u03b2 \u2192 C`, used for `\u03b2`-graded objects in a category `C`. -/\ndef graded_object (\u03b2 : Type w) (C : Type u) :=\n  \u03b2 \u2192 C\n\n-- Satisfying the inhabited linter...\n\nprotected instance inhabited_graded_object (\u03b2 : Type w) (C : Type u) [Inhabited C] : Inhabited (graded_object \u03b2 C) :=\n  { default := fun (b : \u03b2) => Inhabited.default }\n\n/--\nA type synonym for `\u03b2 \u2192 C`, used for `\u03b2`-graded objects in a category `C`\nwith a shift functor given by translation by `s`.\n-/\ndef graded_object_with_shift {\u03b2 : Type w} [add_comm_group \u03b2] (s : \u03b2) (C : Type u) :=\n  graded_object \u03b2 C\n\nnamespace graded_object\n\n\nprotected instance category_of_graded_objects {C : Type u} [category C] (\u03b2 : Type w) : category (graded_object \u03b2 C) :=\n  category_theory.pi fun (_x : \u03b2) => C\n\n/--\nThe natural isomorphism comparing between\npulling back along two propositionally equal functions.\n-/\ndef comap_eq (C : Type u) [category C] {\u03b2 : Type w} {\u03b3 : Type w} {f : \u03b2 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} (h : f = g) : pi.comap (fun (i : \u03b3) => C) f \u2245 pi.comap (fun (i : \u03b3) => C) g :=\n  iso.mk (nat_trans.mk fun (X : \u03b3 \u2192 C) (b : \u03b2) => eq_to_hom sorry)\n    (nat_trans.mk fun (X : \u03b3 \u2192 C) (b : \u03b2) => eq_to_hom sorry)\n\ntheorem comap_eq_symm (C : Type u) [category C] {\u03b2 : Type w} {\u03b3 : Type w} {f : \u03b2 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} (h : f = g) : comap_eq C (Eq.symm h) = iso.symm (comap_eq C h) :=\n  Eq.refl (comap_eq C (Eq.symm h))\n\ntheorem comap_eq_trans (C : Type u) [category C] {\u03b2 : Type w} {\u03b3 : Type w} {f : \u03b2 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {h : \u03b2 \u2192 \u03b3} (k : f = g) (l : g = h) : comap_eq C (Eq.trans k l) = comap_eq C k \u226a\u226b comap_eq C l := sorry\n\n/--\nThe equivalence between \u03b2-graded objects and \u03b3-graded objects,\ngiven an equivalence between \u03b2 and \u03b3.\n-/\ndef comap_equiv (C : Type u) [category C] {\u03b2 : Type w} {\u03b3 : Type w} (e : \u03b2 \u2243 \u03b3) : graded_object \u03b2 C \u224c graded_object \u03b3 C :=\n  equivalence.mk' (pi.comap (fun (_x : \u03b2) => C) \u21d1(equiv.symm e)) (pi.comap (fun (_x : \u03b3) => C) \u21d1e)\n    (comap_eq C sorry \u226a\u226b iso.symm (pi.comap_comp (fun (_x : \u03b2) => C) \u21d1e \u21d1(equiv.symm e)))\n    (pi.comap_comp (fun (_x : \u03b3) => C) \u21d1(equiv.symm e) \u21d1e \u226a\u226b comap_eq C sorry)\n\nprotected instance has_shift {C : Type u} [category C] {\u03b2 : Type u_1} [add_comm_group \u03b2] (s : \u03b2) : has_shift (graded_object_with_shift s C) :=\n  has_shift.mk (comap_equiv C (equiv.mk (fun (b : \u03b2) => b - s) (fun (b : \u03b2) => b + s) sorry sorry))\n\n@[simp] theorem shift_functor_obj_apply {C : Type u} [category C] {\u03b2 : Type u_1} [add_comm_group \u03b2] (s : \u03b2) (X : \u03b2 \u2192 C) (t : \u03b2) : functor.obj (equivalence.functor (shift (graded_object_with_shift s C))) X t = X (t + s) :=\n  rfl\n\n@[simp] theorem shift_functor_map_apply {C : Type u} [category C] {\u03b2 : Type u_1} [add_comm_group \u03b2] (s : \u03b2) {X : graded_object_with_shift s C} {Y : graded_object_with_shift s C} (f : X \u27f6 Y) (t : \u03b2) : functor.map (equivalence.functor (shift (graded_object_with_shift s C))) f t = f (t + s) :=\n  rfl\n\nprotected instance has_zero_morphisms {C : Type u} [category C] [limits.has_zero_morphisms C] (\u03b2 : Type w) : limits.has_zero_morphisms (graded_object \u03b2 C) :=\n  limits.has_zero_morphisms.mk\n\n@[simp] theorem zero_apply {C : Type u} [category C] [limits.has_zero_morphisms C] (\u03b2 : Type w) (X : graded_object \u03b2 C) (Y : graded_object \u03b2 C) (b : \u03b2) : HasZero.zero b = 0 :=\n  rfl\n\nprotected instance has_zero_object {C : Type u} [category C] [limits.has_zero_object C] [limits.has_zero_morphisms C] (\u03b2 : Type w) : limits.has_zero_object (graded_object \u03b2 C) :=\n  limits.has_zero_object.mk (fun (b : \u03b2) => 0)\n    (fun (X : graded_object \u03b2 C) => unique.mk { default := fun (b : \u03b2) => 0 } sorry)\n    fun (X : graded_object \u03b2 C) => unique.mk { default := fun (b : \u03b2) => 0 } sorry\n\nend graded_object\n\n\nnamespace graded_object\n\n\n-- The universes get a little hairy here, so we restrict the universe level for the grading to 0.\n\n-- Since we're typically interested in grading by \u2124 or a finite group, this should be okay.\n\n-- If you're grading by things in higher universes, have fun!\n\n/--\nThe total object of a graded object is the coproduct of the graded components.\n-/\ndef total (\u03b2 : Type) (C : Type u) [category C] [limits.has_coproducts C] : graded_object \u03b2 C \u2964 C :=\n  functor.mk (fun (X : graded_object \u03b2 C) => \u2210 fun (i : ulift \u03b2) => X (ulift.down i))\n    fun (X Y : graded_object \u03b2 C) (f : X \u27f6 Y) => limits.sigma.map fun (i : ulift \u03b2) => f (ulift.down i)\n\n/--\nThe `total` functor taking a graded object to the coproduct of its graded components is faithful.\nTo prove this, we need to know that the coprojections into the coproduct are monomorphisms,\nwhich follows from the fact we have zero morphisms and decidable equality for the grading.\n-/\nprotected instance total.category_theory.faithful (\u03b2 : Type) (C : Type u) [category C] [limits.has_coproducts C] [limits.has_zero_morphisms C] : faithful (total \u03b2 C) :=\n  faithful.mk\n\nend graded_object\n\n\nnamespace graded_object\n\n\nprotected instance category_theory.concrete_category (\u03b2 : Type) (C : Type (u + 1)) [large_category C] [concrete_category C] [limits.has_coproducts C] [limits.has_zero_morphisms C] : concrete_category (graded_object \u03b2 C) :=\n  concrete_category.mk (total \u03b2 C \u22d9 forget C)\n\nprotected instance category_theory.has_forget\u2082 (\u03b2 : Type) (C : Type (u + 1)) [large_category C] [concrete_category C] [limits.has_coproducts C] [limits.has_zero_morphisms C] : has_forget\u2082 (graded_object \u03b2 C) C :=\n  has_forget\u2082.mk (total \u03b2 C)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/graded_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.6261241842048093, "lm_q1q2_score": 0.3851213729136155}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport group_theory.subgroup.basic\nimport algebra.graded_monoid\nimport algebra.direct_sum.basic\nimport algebra.big_operators.pi\n\n/-!\n# Additively-graded multiplicative structures on `\u2a01 i, A i`\n\nThis module provides a set of heterogeneous typeclasses for defining a multiplicative structure\nover `\u2a01 i, A i` such that `(*) : A i \u2192 A j \u2192 A (i + j)`; that is to say, `A` forms an\nadditively-graded ring. The typeclasses are:\n\n* `direct_sum.gnon_unital_non_assoc_semiring A`\n* `direct_sum.gsemiring A`\n* `direct_sum.gcomm_semiring A`\n\nRespectively, these imbue the external direct sum `\u2a01 i, A i` with:\n\n* `direct_sum.non_unital_non_assoc_semiring`, `direct_sum.non_unital_non_assoc_ring`\n* `direct_sum.semiring`, `direct_sum.ring`\n* `direct_sum.comm_semiring`, `direct_sum.comm_ring`\n\nthe base ring `A 0` with:\n\n* `direct_sum.grade_zero.non_unital_non_assoc_semiring`,\n  `direct_sum.grade_zero.non_unital_non_assoc_ring`\n* `direct_sum.grade_zero.semiring`, `direct_sum.grade_zero.ring`\n* `direct_sum.grade_zero.comm_semiring`, `direct_sum.grade_zero.comm_ring`\n\nand the `i`th grade `A i` with `A 0`-actions (`\u2022`) defined as left-multiplication:\n\n* `direct_sum.grade_zero.has_scalar (A 0)`, `direct_sum.grade_zero.smul_with_zero (A 0)`\n* `direct_sum.grade_zero.module (A 0)`\n* (nothing)\n\nNote that in the presence of these instances, `\u2a01 i, A i` itself inherits an `A 0`-action.\n\n`direct_sum.of_zero_ring_hom : A 0 \u2192+* \u2a01 i, A i` provides `direct_sum.of A 0` as a ring\nhomomorphism.\n\n`direct_sum.to_semiring` extends `direct_sum.to_add_monoid` to produce a `ring_hom`.\n\n## Direct sums of subobjects\n\nAdditionally, this module provides helper functions to construct `gsemiring` and `gcomm_semiring`\ninstances for:\n\n* `A : \u03b9 \u2192 submonoid S`:\n  `direct_sum.gsemiring.of_add_submonoids`, `direct_sum.gcomm_semiring.of_add_submonoids`.\n* `A : \u03b9 \u2192 subgroup S`:\n  `direct_sum.gsemiring.of_add_subgroups`, `direct_sum.gcomm_semiring.of_add_subgroups`.\n* `A : \u03b9 \u2192 submodule S`:\n  `direct_sum.gsemiring.of_submodules`, `direct_sum.gcomm_semiring.of_submodules`.\n\nIf `complete_lattice.independent (set.range A)`, these provide a gradation of `\u2a06 i, A i`, and the\nmapping `\u2a01 i, A i \u2192+ \u2a06 i, A i` can be obtained as\n`direct_sum.to_monoid (\u03bb i, add_submonoid.inclusion $ le_supr A i)`.\n\n## tags\n\ngraded ring, filtered ring, direct sum, add_submonoid\n-/\n\nset_option old_structure_cmd true\n\nvariables {\u03b9 : Type*} [decidable_eq \u03b9]\n\nnamespace direct_sum\n\nopen_locale direct_sum\n\n/-! ### Typeclasses -/\nsection defs\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-- A graded version of `non_unital_non_assoc_semiring`. -/\nclass gnon_unital_non_assoc_semiring [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  graded_monoid.ghas_mul A :=\n(mul_zero : \u2200 {i j} (a : A i), mul a (0 : A j) = 0)\n(zero_mul : \u2200 {i j} (b : A j), mul (0 : A i) b = 0)\n(mul_add : \u2200 {i j} (a : A i) (b c : A j), mul a (b + c) = mul a b + mul a c)\n(add_mul : \u2200 {i j} (a b : A i) (c : A j), mul (a + b) c = mul a c + mul b c)\n\nend defs\n\nsection defs\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-- A graded version of `semiring`. -/\nclass gsemiring [add_monoid \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  gnon_unital_non_assoc_semiring A, graded_monoid.gmonoid A\n\n/-- A graded version of `comm_semiring`. -/\nclass gcomm_semiring [add_comm_monoid \u03b9] [\u03a0 i, add_comm_monoid (A i)] extends\n  gsemiring A, graded_monoid.gcomm_monoid A\n\nend defs\n\nlemma of_eq_of_graded_monoid_eq {A : \u03b9 \u2192 Type*} [\u03a0 (i : \u03b9), add_comm_monoid (A i)]\n  {i j : \u03b9} {a : A i} {b : A j} (h : graded_monoid.mk i a = graded_monoid.mk j b) :\n  direct_sum.of A i a = direct_sum.of A j b :=\ndfinsupp.single_eq_of_sigma_eq h\n\nvariables (A : \u03b9 \u2192 Type*)\n\n/-! ### Instances for `\u2a01 i, A i` -/\n\n\nsection one\nvariables [has_zero \u03b9] [graded_monoid.ghas_one A] [\u03a0 i, add_comm_monoid (A i)]\n\ninstance : has_one (\u2a01 i, A i) :=\n{ one := direct_sum.of (\u03bb i, A i) 0 graded_monoid.ghas_one.one }\n\nend one\n\nsection mul\nvariables [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] [gnon_unital_non_assoc_semiring A]\n\nopen add_monoid_hom (flip_apply coe_comp comp_hom_apply_apply)\n\n/-- The piecewise multiplication from the `has_mul` instance, as a bundled homomorphism. -/\n@[simps]\ndef gmul_hom {i j} : A i \u2192+ A j \u2192+ A (i + j) :=\n{ to_fun := \u03bb a,\n  { to_fun := \u03bb b, graded_monoid.ghas_mul.mul a b,\n    map_zero' := gnon_unital_non_assoc_semiring.mul_zero _,\n    map_add' := gnon_unital_non_assoc_semiring.mul_add _ },\n  map_zero' := add_monoid_hom.ext $ \u03bb a, gnon_unital_non_assoc_semiring.zero_mul a,\n  map_add' := \u03bb a\u2081 a\u2082, add_monoid_hom.ext $ \u03bb b, gnon_unital_non_assoc_semiring.add_mul _ _ _}\n\n/-- The multiplication from the `has_mul` instance, as a bundled homomorphism. -/\ndef mul_hom : (\u2a01 i, A i) \u2192+ (\u2a01 i, A i) \u2192+ \u2a01 i, A i :=\ndirect_sum.to_add_monoid $ \u03bb i,\n  add_monoid_hom.flip $ direct_sum.to_add_monoid $ \u03bb j, add_monoid_hom.flip $\n    (direct_sum.of A _).comp_hom.comp $ gmul_hom A\n\ninstance : non_unital_non_assoc_semiring (\u2a01 i, A i) :=\n{ mul := \u03bb a b, mul_hom A a b,\n  zero := 0,\n  add := (+),\n  zero_mul := \u03bb a, by simp only [add_monoid_hom.map_zero, add_monoid_hom.zero_apply],\n  mul_zero := \u03bb a, by simp only [add_monoid_hom.map_zero],\n  left_distrib := \u03bb a b c, by simp only [add_monoid_hom.map_add],\n  right_distrib := \u03bb a b c, by simp only [add_monoid_hom.map_add, add_monoid_hom.add_apply],\n  .. direct_sum.add_comm_monoid _ _}\n\nvariables {A}\n\nlemma mul_hom_of_of {i j} (a : A i) (b : A j) :\n  mul_hom A (of _ i a) (of _ j b) = of _ (i + j) (graded_monoid.ghas_mul.mul a b) :=\nbegin\n  unfold mul_hom,\n  rw [to_add_monoid_of, flip_apply, to_add_monoid_of, flip_apply, coe_comp, function.comp_app,\n      comp_hom_apply_apply, coe_comp, function.comp_app, gmul_hom_apply_apply],\nend\n\nlemma of_mul_of {i j} (a : A i) (b : A j) :\n  of _ i a * of _ j b = of _ (i + j) (graded_monoid.ghas_mul.mul a b) :=\nmul_hom_of_of a b\n\nend mul\n\nsection semiring\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A]\n\nopen add_monoid_hom (flip_hom coe_comp comp_hom_apply_apply flip_apply flip_hom_apply)\n\nprivate lemma one_mul (x : \u2a01 i, A i) : 1 * x = x :=\nsuffices mul_hom A 1 = add_monoid_hom.id (\u2a01 i, A i),\n  from add_monoid_hom.congr_fun this x,\nbegin\n  apply add_hom_ext, intros i xi,\n  unfold has_one.one,\n  rw mul_hom_of_of,\n  exact of_eq_of_graded_monoid_eq (one_mul $ graded_monoid.mk i xi),\nend\n\nprivate lemma mul_one (x : \u2a01 i, A i) : x * 1 = x :=\nsuffices (mul_hom A).flip 1 = add_monoid_hom.id (\u2a01 i, A i),\n  from add_monoid_hom.congr_fun this x,\nbegin\n  apply add_hom_ext, intros i xi,\n  unfold has_one.one,\n  rw [flip_apply, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (mul_one $ graded_monoid.mk i xi),\nend\n\nprivate lemma mul_assoc (a b c : \u2a01 i, A i) : a * b * c = a * (b * c) :=\nsuffices (mul_hom A).comp_hom.comp (mul_hom A)            -- `\u03bb a b c, a * b * c` as a bundled hom\n       = (add_monoid_hom.comp_hom flip_hom $              -- `\u03bb a b c, a * (b * c)` as a bundled hom\n             (mul_hom A).flip.comp_hom.comp (mul_hom A)).flip,\n  from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun (add_monoid_hom.congr_fun this a) b) c,\nbegin\n  ext ai ax bi bx ci cx : 6,\n  dsimp only [coe_comp, function.comp_app, comp_hom_apply_apply, flip_apply, flip_hom_apply],\n  rw [mul_hom_of_of, mul_hom_of_of, mul_hom_of_of, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (mul_assoc (graded_monoid.mk ai ax) \u27e8bi, bx\u27e9 \u27e8ci, cx\u27e9),\nend\n\n/-- The `semiring` structure derived from `gsemiring A`. -/\ninstance semiring : semiring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  one_mul := one_mul A,\n  mul_one := mul_one A,\n  mul_assoc := mul_assoc A,\n  ..direct_sum.non_unital_non_assoc_semiring _, }\n\nlemma of_pow {i} (a : A i) (n : \u2115) :\n  of _ i a ^ n = of _ (n \u2022 i) (graded_monoid.gmonoid.gnpow _ a) :=\nbegin\n  induction n with n,\n  { exact of_eq_of_graded_monoid_eq (pow_zero $ graded_monoid.mk _ a).symm, },\n  { rw [pow_succ, n_ih, of_mul_of],\n    exact of_eq_of_graded_monoid_eq (pow_succ (graded_monoid.mk _ a) n).symm, },\nend\n\nlemma of_list_dprod {\u03b1} (l : list \u03b1) (f\u03b9 : \u03b1 \u2192 \u03b9) (fA : \u03a0 a, A (f\u03b9 a)) :\n  of A _ (l.dprod f\u03b9 fA) = (l.map $ \u03bb a, of A (f\u03b9 a) (fA a)).prod :=\nbegin\n  induction l,\n  { simp only [list.map_nil, list.prod_nil, list.dprod_nil],\n    refl },\n  { simp only [list.map_cons, list.prod_cons, list.dprod_cons, \u2190l_ih, direct_sum.of_mul_of],\n    refl },\nend\n\nlemma list_prod_of_fn_of_eq_dprod (n : \u2115) (f\u03b9 : fin n \u2192 \u03b9) (fA : \u03a0 a, A (f\u03b9 a)) :\n  (list.of_fn $ \u03bb a, of A (f\u03b9 a) (fA a)).prod = of A _ ((list.fin_range n).dprod f\u03b9 fA) :=\nby rw [list.of_fn_eq_map, of_list_dprod]\n\nopen_locale big_operators\n\n/-- A heavily unfolded version of the definition of multiplication -/\nlemma mul_eq_sum_support_ghas_mul\n  [\u03a0 (i : \u03b9) (x : A i), decidable (x \u2260 0)] (a a' : \u2a01 i, A i) :\n  a * a' =\n    \u2211 (ij : \u03b9 \u00d7 \u03b9) in (dfinsupp.support a).product (dfinsupp.support a'),\n      direct_sum.of _ _ (graded_monoid.ghas_mul.mul (a ij.fst) (a' ij.snd)) :=\nbegin\n  change direct_sum.mul_hom _ a a' = _,\n  dsimp [direct_sum.mul_hom, direct_sum.to_add_monoid, dfinsupp.lift_add_hom_apply],\n  simp only [dfinsupp.sum_add_hom_apply, dfinsupp.sum, dfinsupp.finset_sum_apply,\n    add_monoid_hom.coe_finset_sum, finset.sum_apply, add_monoid_hom.flip_apply,\n    add_monoid_hom.comp_hom_apply_apply, add_monoid_hom.comp_apply,\n    direct_sum.gmul_hom_apply_apply],\n  rw finset.sum_product,\nend\n\nend semiring\n\nsection comm_semiring\n\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\nprivate lemma mul_comm (a b : \u2a01 i, A i) : a * b = b * a :=\nsuffices mul_hom A = (mul_hom A).flip,\n  from add_monoid_hom.congr_fun (add_monoid_hom.congr_fun this a) b,\nbegin\n  apply add_hom_ext, intros ai ax, apply add_hom_ext, intros bi bx,\n  rw [add_monoid_hom.flip_apply, mul_hom_of_of, mul_hom_of_of],\n  exact of_eq_of_graded_monoid_eq (gcomm_semiring.mul_comm \u27e8ai, ax\u27e9 \u27e8bi, bx\u27e9),\nend\n\n/-- The `comm_semiring` structure derived from `gcomm_semiring A`. -/\ninstance comm_semiring : comm_semiring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  mul_comm := mul_comm A,\n  ..direct_sum.semiring _, }\n\nend comm_semiring\n\nsection non_unital_non_assoc_ring\nvariables [\u03a0 i, add_comm_group (A i)] [has_add \u03b9] [gnon_unital_non_assoc_semiring A]\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance non_assoc_ring : non_unital_non_assoc_ring (\u2a01 i, A i) :=\n{ mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  ..(direct_sum.non_unital_non_assoc_semiring _),\n  ..(direct_sum.add_comm_group _), }\n\nend non_unital_non_assoc_ring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_monoid \u03b9] [gsemiring A]\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance ring : ring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  ..(direct_sum.semiring _),\n  ..(direct_sum.add_comm_group _), }\n\n\nend ring\n\nsection comm_ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\n/-- The `comm_ring` derived from `gcomm_semiring A`. -/\ninstance comm_ring : comm_ring (\u2a01 i, A i) :=\n{ one := 1,\n  mul := (*),\n  zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  ..(direct_sum.ring _),\n  ..(direct_sum.comm_semiring _), }\n\nend comm_ring\n\n\n/-! ### Instances for `A 0`\n\nThe various `g*` instances are enough to promote the `add_comm_monoid (A 0)` structure to various\ntypes of multiplicative structure.\n-/\n\nsection grade_zero\n\nsection one\nvariables [has_zero \u03b9] [graded_monoid.ghas_one A] [\u03a0 i, add_comm_monoid (A i)]\n\n@[simp] lemma of_zero_one : of _ 0 (1 : A 0) = 1 := rfl\n\nend one\n\nsection mul\nvariables [add_zero_class \u03b9] [\u03a0 i, add_comm_monoid (A i)] [gnon_unital_non_assoc_semiring A]\n\n@[simp] lemma of_zero_smul {i} (a : A 0) (b : A i) : of _ _ (a \u2022 b) = of _ _ a * of _ _ b :=\n(of_eq_of_graded_monoid_eq (graded_monoid.mk_zero_smul a b)).trans (of_mul_of _ _).symm\n\n@[simp] lemma of_zero_mul (a b : A 0) : of _ 0 (a * b) = of _ 0 a * of _ 0 b:=\nof_zero_smul A a b\n\ninstance grade_zero.non_unital_non_assoc_semiring : non_unital_non_assoc_semiring (A 0) :=\nfunction.injective.non_unital_non_assoc_semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of A 0).map_add (of_zero_mul A) (\u03bb x n, dfinsupp.single_smul n x)\n\ninstance grade_zero.smul_with_zero (i : \u03b9) : smul_with_zero (A 0) (A i) :=\nbegin\n  letI := smul_with_zero.comp_hom (\u2a01 i, A i) (of A 0).to_zero_hom,\n  refine dfinsupp.single_injective.smul_with_zero (of A i).to_zero_hom (of_zero_smul A),\nend\n\nend mul\n\nsection semiring\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A]\n\n@[simp] lemma of_zero_pow (a : A 0) : \u2200 n : \u2115, of _ 0 (a ^ n) = of _ 0 a ^ n\n| 0 := by rw [pow_zero, pow_zero, direct_sum.of_zero_one]\n| (n + 1) := by rw [pow_succ, pow_succ, of_zero_mul, of_zero_pow]\n\n/-- The `semiring` structure derived from `gsemiring A`. -/\ninstance grade_zero.semiring : semiring (A 0) :=\nfunction.injective.semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (\u03bb x n, dfinsupp.single_smul n x) (\u03bb x n, of_zero_pow _ _ _)\n\n/-- `of A 0` is a `ring_hom`, using the `direct_sum.grade_zero.semiring` structure. -/\ndef of_zero_ring_hom : A 0 \u2192+* (\u2a01 i, A i) :=\n{ map_one' := of_zero_one A, map_mul' := of_zero_mul A, ..(of _ 0) }\n\n/-- Each grade `A i` derives a `A 0`-module structure from `gsemiring A`. Note that this results\nin an overall `module (A 0) (\u2a01 i, A i)` structure via `direct_sum.module`.\n-/\ninstance grade_zero.module {i} : module (A 0) (A i) :=\nbegin\n  letI := module.comp_hom (\u2a01 i, A i) (of_zero_ring_hom A),\n  exact dfinsupp.single_injective.module (A 0) (of A i) (\u03bb a, of_zero_smul A a),\nend\n\nend semiring\n\nsection comm_semiring\n\nvariables [\u03a0 i, add_comm_monoid (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\n/-- The `comm_semiring` structure derived from `gcomm_semiring A`. -/\ninstance grade_zero.comm_semiring : comm_semiring (A 0) :=\nfunction.injective.comm_semiring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (\u03bb x n, dfinsupp.single_smul n x) (\u03bb x n, of_zero_pow _ _ _)\n\nend comm_semiring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_zero_class \u03b9] [gnon_unital_non_assoc_semiring A]\n\n/-- The `non_unital_non_assoc_ring` derived from `gnon_unital_non_assoc_semiring A`. -/\ninstance grade_zero.non_unital_non_assoc_ring : non_unital_non_assoc_ring (A 0) :=\nfunction.injective.non_unital_non_assoc_ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n\nend ring\n\nsection ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_monoid \u03b9] [gsemiring A]\n\n/-- The `ring` derived from `gsemiring A`. -/\ninstance grade_zero.ring : ring (A 0) :=\nfunction.injective.ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end) (\u03bb x n, of_zero_pow _ _ _)\n\nend ring\n\nsection comm_ring\nvariables [\u03a0 i, add_comm_group (A i)] [add_comm_monoid \u03b9] [gcomm_semiring A]\n\n/-- The `comm_ring` derived from `gcomm_semiring A`. -/\ninstance grade_zero.comm_ring : comm_ring (A 0) :=\nfunction.injective.comm_ring (of A 0) dfinsupp.single_injective\n  (of A 0).map_zero (of_zero_one A) (of A 0).map_add (of_zero_mul A)\n  (of A 0).map_neg (of A 0).map_sub\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2115 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end)\n  (\u03bb x n, begin\n    letI : \u03a0 i, distrib_mul_action \u2124 (A i) := \u03bb i, infer_instance,\n    exact dfinsupp.single_smul n x\n  end) (\u03bb x n, of_zero_pow _ _ _)\n\nend comm_ring\n\nend grade_zero\n\nsection to_semiring\n\nvariables {R : Type*} [\u03a0 i, add_comm_monoid (A i)] [add_monoid \u03b9] [gsemiring A] [semiring R]\nvariables {A}\n\n/-- If two ring homomorphisms from `\u2a01 i, A i` are equal on each `of A i y`,\nthen they are equal.\n\nSee note [partially-applied ext lemmas]. -/\n@[ext]\nlemma ring_hom_ext' \u2983F G : (\u2a01 i, A i) \u2192+* R\u2984\n  (h : \u2200 i, (\u2191F : _ \u2192+ R).comp (of A i) = (\u2191G : _ \u2192+ R).comp (of A i)) : F = G :=\nring_hom.coe_add_monoid_hom_injective $ direct_sum.add_hom_ext' h\n\n/-- Two `ring_hom`s out of a direct sum are equal if they agree on the generators. -/\nlemma ring_hom_ext \u2983f g : (\u2a01 i, A i) \u2192+* R\u2984 (h : \u2200 i x, f (of A i x) = g (of A i x)) :\n  f = g :=\nring_hom_ext' $ \u03bb i, add_monoid_hom.ext $ h i\n\n/-- A family of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`\ndescribes a `ring_hom`s on `\u2a01 i, A i`. This is a stronger version of `direct_sum.to_monoid`.\n\nOf particular interest is the case when `A i` are bundled subojects, `f` is the family of\ncoercions such as `add_submonoid.subtype (A i)`, and the `[gsemiring A]` structure originates from\n`direct_sum.gsemiring.of_add_submonoids`, in which case the proofs about `ghas_one` and `ghas_mul`\ncan be discharged by `rfl`. -/\n@[simps]\ndef to_semiring\n  (f : \u03a0 i, A i \u2192+ R) (hone : f _ (graded_monoid.ghas_one.one) = 1)\n  (hmul : \u2200 {i j} (ai : A i) (aj : A j), f _ (graded_monoid.ghas_mul.mul ai aj) = f _ ai * f _ aj) :\n  (\u2a01 i, A i) \u2192+* R :=\n{ to_fun := to_add_monoid f,\n  map_one' := begin\n    change (to_add_monoid f) (of _ 0 _) = 1,\n    rw to_add_monoid_of,\n    exact hone\n  end,\n  map_mul' := begin\n    rw (to_add_monoid f).map_mul_iff,\n    ext xi xv yi yv : 4,\n    show to_add_monoid f (of A xi xv * of A yi yv) =\n         to_add_monoid f (of A xi xv) * to_add_monoid f (of A yi yv),\n    rw [of_mul_of, to_add_monoid_of, to_add_monoid_of, to_add_monoid_of],\n    exact hmul _ _,\n  end,\n  .. to_add_monoid f}\n\n@[simp] lemma to_semiring_of (f : \u03a0 i, A i \u2192+ R) (hone hmul) (i : \u03b9) (x : A i) :\n  to_semiring f hone hmul (of _ i x) = f _ x :=\nto_add_monoid_of f i x\n\n@[simp] lemma to_semiring_coe_add_monoid_hom (f : \u03a0 i, A i \u2192+ R) (hone hmul):\n  (to_semiring f hone hmul : (\u2a01 i, A i) \u2192+ R) = to_add_monoid f := rfl\n\n/-- Families of `add_monoid_hom`s preserving `direct_sum.ghas_one.one` and `direct_sum.ghas_mul.mul`\nare isomorphic to `ring_hom`s on `\u2a01 i, A i`. This is a stronger version of `dfinsupp.lift_add_hom`.\n-/\n@[simps]\ndef lift_ring_hom :\n  {f : \u03a0 {i}, A i \u2192+ R //\n    f (graded_monoid.ghas_one.one) = 1 \u2227\n    \u2200 {i j} (ai : A i) (aj : A j), f (graded_monoid.ghas_mul.mul ai aj) = f ai * f aj} \u2243\n    ((\u2a01 i, A i) \u2192+* R) :=\n{ to_fun := \u03bb f, to_semiring f.1 f.2.1 f.2.2,\n  inv_fun := \u03bb F,\n    \u27e8\u03bb i, (F : (\u2a01 i, A i) \u2192+ R).comp (of _ i), begin\n      simp only [add_monoid_hom.comp_apply, ring_hom.coe_add_monoid_hom],\n      rw \u2190F.map_one,\n      refl\n    end, \u03bb i j ai aj, begin\n      simp only [add_monoid_hom.comp_apply, ring_hom.coe_add_monoid_hom],\n      rw [\u2190F.map_mul, of_mul_of],\n    end\u27e9,\n  left_inv := \u03bb f, begin\n    ext xi xv,\n    exact to_add_monoid_of f.1 xi xv,\n  end,\n  right_inv := \u03bb F, begin\n    apply ring_hom.coe_add_monoid_hom_injective,\n    ext xi xv,\n    simp only [ring_hom.coe_add_monoid_hom_mk,\n      direct_sum.to_add_monoid_of,\n      add_monoid_hom.mk_coe,\n      add_monoid_hom.comp_apply, to_semiring_coe_add_monoid_hom],\n  end}\n\nend to_semiring\n\nend direct_sum\n\n/-! ### Concrete instances -/\n\nsection uniform\n\nvariables (\u03b9)\n\n/-- A direct sum of copies of a `semiring` inherits the multiplication structure. -/\ninstance non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring\n  {R : Type*} [add_monoid \u03b9] [non_unital_non_assoc_semiring R] :\n  direct_sum.gnon_unital_non_assoc_semiring (\u03bb i : \u03b9, R) :=\n{ mul_zero := \u03bb i j, mul_zero,\n  zero_mul := \u03bb i j, zero_mul,\n  mul_add := \u03bb i j, mul_add,\n  add_mul := \u03bb i j, add_mul,\n  ..has_mul.ghas_mul \u03b9 }\n\n/-- A direct sum of copies of a `semiring` inherits the multiplication structure. -/\ninstance semiring.direct_sum_gsemiring {R : Type*} [add_monoid \u03b9] [semiring R] :\n  direct_sum.gsemiring (\u03bb i : \u03b9, R) :=\n{ ..non_unital_non_assoc_semiring.direct_sum_gnon_unital_non_assoc_semiring \u03b9, ..monoid.gmonoid \u03b9 }\n\nopen_locale direct_sum\n\n-- To check `has_mul.ghas_mul_mul` matches\nexample {R : Type*} [add_monoid \u03b9] [semiring R] (i j : \u03b9) (a b : R) :\n  (direct_sum.of _ i a * direct_sum.of _ j b : \u2a01 i, R) = direct_sum.of _ (i + j) (by exact a * b) :=\nby rw [direct_sum.of_mul_of, has_mul.ghas_mul_mul]\n\n/-- A direct sum of copies of a `comm_semiring` inherits the commutative multiplication structure.\n-/\ninstance comm_semiring.direct_sum_gcomm_semiring {R : Type*} [add_comm_monoid \u03b9] [comm_semiring R] :\n  direct_sum.gcomm_semiring (\u03bb i : \u03b9, R) :=\n{ ..comm_monoid.gcomm_monoid \u03b9, ..semiring.direct_sum_gsemiring \u03b9 }\n\nend uniform\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/algebra/direct_sum/ring.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752916, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3851213688757906}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.category_theory.morphism_property_misc\nimport category_theory.lifting_properties.basic\nimport for_mathlib.category_theory.retracts\nimport for_mathlib.category_theory.lifting_properties.morphism_property\n\nnoncomputable theory\n\nopen category_theory category_theory.category opposite\n\nnamespace algebraic_topology\n\nvariables {C : Type*} [category C] (F G : morphism_property C) {F' G' : morphism_property C\u1d52\u1d56}\n\ndef factorisation_axiom :=\n\u2200 \u2983X Z : C\u2984 (f : X \u27f6 Z), \u2203 (Y : C) (i : X \u27f6 Y) (hi : F i) (p : Y \u27f6 Z) (hp : G p), i \u226b p = f\n\nnamespace factorisation_axiom\n\nvariables {X Y Z : C}\n\nvariables {F G}\n\ndef obj (h : factorisation_axiom F G) (f : X \u27f6 Z) : C := (h f).some\n\ndef i (h : factorisation_axiom F G) (f : X \u27f6 Z) : X \u27f6 h.obj f :=\n(h f).some_spec.some\ndef p (h : factorisation_axiom F G) (f : X \u27f6 Z) : h.obj f \u27f6 Z :=\n(h f).some_spec.some_spec.some_spec.some\n\nlemma i_property (h : factorisation_axiom F G) (f : X \u27f6 Z) : F (h.i f) :=\n(h f).some_spec.some_spec.some\nlemma p_property (h : factorisation_axiom F G) (f : X \u27f6 Z) : G (h.p f) :=\n(h f).some_spec.some_spec.some_spec.some_spec.some\n\n@[simp, reassoc]\nlemma fac (h : factorisation_axiom F G) (f : X \u27f6 Z) : (h.i f) \u226b (h.p f) = f :=\n(h f).some_spec.some_spec.some_spec.some_spec.some_spec\n\nlemma op (h : factorisation_axiom F G) : factorisation_axiom G.op F.op :=\n\u03bb X Z f,\nbegin\n  rcases h f.unop with \u27e8Y, i, hi, p, hp, fac\u27e9,\n  use [op Y, p.op, hp, i.op, hi],\n  rw [\u2190 op_comp, fac, f.op_unop],\nend\n\nlemma unop (h : factorisation_axiom F' G') : factorisation_axiom G'.unop F'.unop :=\n\u03bb X Z f,\nbegin\n  rcases h f.op with \u27e8Y, i, hi, p, hp, fac\u27e9,\n  use [Y.unop, p.unop, hp, i.unop, hi],\n  rw [\u2190 unop_comp, fac, f.unop_op],\nend\n\nvariables (F G F' G')\n\nlemma iff_op : factorisation_axiom F G \u2194 factorisation_axiom G.op F.op := \u27e8op, unop\u27e9\nlemma iff_unop : factorisation_axiom F' G' \u2194 factorisation_axiom G'.unop F'.unop := \u27e8unop, op\u27e9\n\nlemma is_retract_of_fac_and_llp (i : X \u27f6 Z) {j : X \u27f6 Y} {p : Y \u27f6 Z} (fac : j \u226b p = i)\n  [has_lifting_property i p] : is_retract_hom i j :=\nbegin\n  have fac\u2082 : j \u226b p = i \u226b \ud835\udfd9 Z,\n  { rw [comp_id, fac], },\n  have sq := (comm_sq.mk fac\u2082).lift,\n  let s : arrow.mk i \u27f6 arrow.mk j :=\n  { left := \ud835\udfd9 X,\n    right := (comm_sq.mk fac\u2082).lift,\n    w' := by { dsimp, simp only [functor.id_map, arrow.mk_hom, comm_sq.fac_left, id_comp], }, },\n  let r : arrow.mk j \u27f6 arrow.mk i :=\n  { left := \ud835\udfd9 X,\n    right := p,\n    w' := by { dsimp, simp only [id_comp, fac], }, },\n  use [s, r],\n  ext,\n  { dsimp, rw id_comp, },\n  { dsimp, rw comm_sq.fac_right, },\nend\n\nvariables {F G}\n\nlemma eq_llp_with\n  (h\u2081 : factorisation_axiom F G) (h\u2082 : F.has_lifting_property G)\n  (h\u2083 : F.is_stable_by_retract) : F = G.llp_with :=\nbegin\n  ext X Y i,\n  split,\n  { exact \u03bb hi X Y, h\u2082 i hi, },\n  { intro hi,\n    rcases h\u2081 i with \u27e8Z, j, hj, p, hp, fac\u27e9,\n    haveI : has_lifting_property i p := hi p hp,\n    exact h\u2083 i j (is_retract_of_fac_and_llp i fac) hj, },\nend\n\nlemma eq_rlp_with\n  (h\u2081 : factorisation_axiom F G) (h\u2082 : F.has_lifting_property G)\n  (h\u2083 : G.is_stable_by_retract) : G = F.rlp_with :=\nby rw [\u2190 G.unop_op, eq_llp_with h\u2081.op h\u2082.op h\u2083.op, F.llp_with_op, morphism_property.unop_op]\n\nlemma under (h : factorisation_axiom F G) (A : C) :\n  factorisation_axiom (F.inverse_image (under.forget A)) (G.inverse_image (under.forget A)) :=\n\u03bb X Y f, begin\n  let f' := (under.forget A).map f,\n  refine \u27e8under.mk (X.hom \u226b h.i f'), under.hom_mk (h.i f'), h.i_property f',\n    under.hom_mk (h.p f'), h.p_property f', _\u27e9,\n  tidy,\nend\n\nend factorisation_axiom\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/factorisation_axiom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3851213688757905}}
{"text": "import QL.FOL.Tait.tait QL.FOL.semantics logic\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol aclogic\nvariables {L : language.{u}} {m n : \u2115}\n\nnamespace Tait\n\nopen subformula\n\nnoncomputable def finset_mlift (\u0394 : finset (bounded_subformula L m n)) :\n  finset (bounded_subformula L (m + 1) n) := \u0394.image mlift\n\n@[simp] lemma finset_mlift_union (\u0394 \u0393 : finset (bounded_subformula L m n)) : finset_mlift (\u0394 \u222a \u0393) = finset_mlift \u0394 \u222a finset_mlift \u0393 :=\nby simp[finset_mlift, finset.image_union]\n\n@[simp] lemma mem_finset_mlift_iff (p : bounded_subformula L m n) (\u0394 : finset (bounded_subformula L m n)) :\n  mlift p \u2208 finset_mlift \u0394 \u2194 p \u2208 \u0394 :=\nby simp[finset_mlift]\n\n-- Tate caluculus\ninductive derivation : \u03a0 {m}, finset (bounded_formula L m) \u2192 Type u\n| AxL {m} : \u2200 (\u0394 : finset (bounded_formula L m)) {k} (r : L.pr k) (v : fin k \u2192 bounded_term L m),\n    relation r v \u2208 \u0394 \u2192 neg_relation r v \u2208 \u0394 \u2192 derivation \u0394\n| verum {m} : \u2200 (\u0394 : finset (bounded_formula L m)), \u22a4 \u2208 \u0394 \u2192 derivation \u0394\n| or_left {m} : \u2200 (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m),\n    derivation (insert p \u0394) \u2192 derivation (insert (p \u2294 q) \u0394)\n| or_right {m} : \u2200 (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m),\n    derivation (insert q \u0394) \u2192 derivation (insert (p \u2294 q) \u0394)\n| and {m} : \u2200 (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m),\n    derivation (insert p \u0394) \u2192 derivation (insert q \u0394) \u2192 derivation (insert (p \u2293 q) \u0394)\n| all {m} : \u2200 (\u0394 : finset (bounded_subformula L m 0)) (p : bounded_subformula L m 1),\n    derivation (insert p.push (finset_mlift \u0394)) \u2192 derivation (insert (\u2200'p) \u0394)\n| ex {m} : \u2200 (\u0394 : finset (bounded_subformula L m 0)) (t : bounded_term L m) (p : bounded_subformula L m 1),\n    derivation (insert (subst t p) \u0394) \u2192 derivation (insert (\u2203'p) \u0394)\n\nvariables {L m}\n\ndef derivable {m} (\u0394 : finset (bounded_formula L m)) : Prop := nonempty (derivation \u0394)\n\nprefix `\u22a2\u1d40 `:45 := derivable\n\n@[reducible] def preTheory (L : language.{u}) (\u03bc) := logic.Theory (subformula L \u03bc 0)\n\n@[reducible] def bounded_preTheory (L : language.{u}) (m : \u2115) := logic.Theory (subformula L (fin m) 0)\n\n@[reducible] def Theory (L : language.{u}) := logic.Theory (subformula L (fin 0) 0)\n\ndef provable (T : bounded_preTheory L m) (p : bounded_formula L m) : Prop :=\n\u2203 \u0394 : finset (bounded_formula L m), \u2191\u0394 \u2286 subformula.not '' T \u2227 \u22a2\u1d40 insert p \u0394\n\ninstance : has_turnstile (bounded_formula L m) := \u27e8provable\u27e9\n\ndef provable_def {T : set (bounded_formula L m)} {p : bounded_formula L m} :\n  T \u22a2 p \u2194 \u2203 \u0394 : finset (bounded_formula L m), \u2191\u0394 \u2286 subformula.not '' T \u2227 \u22a2\u1d40 insert p \u0394 := by refl\n\nnamespace derivable\nvariables {m} {\u0394 \u0393 : finset (bounded_formula L m)}\n\nlemma AxL {k} (r : L.pr k) (v : fin k \u2192 bounded_term L m) (h : relation r v \u2208 \u0394) (hneg : neg_relation r v \u2208 \u0394) : \u22a2\u1d40 \u0394 :=\n\u27e8derivation.AxL \u0394 r v h hneg\u27e9\n\nlemma verum (h : \u22a4 \u2208 \u0394) : \u22a2\u1d40 \u0394 := \u27e8derivation.verum \u0394 h\u27e9\n\nlemma or_left (p q : bounded_formula L m) : \u22a2\u1d40 insert p \u0394 \u2192 \u22a2\u1d40 insert (p \u2294 q) \u0394 := \u03bb \u27e8d\u27e9, \u27e8derivation.or_left \u0394 p q d\u27e9\n\nlemma or_right (p q : bounded_formula L m) : \u22a2\u1d40 insert q \u0394 \u2192 \u22a2\u1d40 insert (p \u2294 q) \u0394 := \u03bb \u27e8d\u27e9, \u27e8derivation.or_right \u0394 p q d\u27e9\n\nlemma and {p q : bounded_formula L m} : \u22a2\u1d40 insert p \u0394 \u2192 \u22a2\u1d40 insert q \u0394 \u2192 \u22a2\u1d40 insert (p \u2293 q) \u0394 := \u03bb \u27e8d\u2081\u27e9 \u27e8d\u2082\u27e9, \u27e8derivation.and \u0394 p q d\u2081 d\u2082\u27e9\n\nlemma all {p : bounded_subformula L m 1} : \u22a2\u1d40 insert p.push (finset_mlift \u0394) \u2192 \u22a2\u1d40 insert (\u2200'p) \u0394 := \u03bb \u27e8d\u27e9, \u27e8derivation.all \u0394 p d\u27e9\n\nlemma ex {t} {p : bounded_subformula L m 1} : \u22a2\u1d40 insert (subst t p) \u0394 \u2192 \u22a2\u1d40 insert (\u2203'p) \u0394 := \u03bb \u27e8d\u27e9, \u27e8derivation.ex \u0394 t p d\u27e9\n\nprotected lemma cast (h : \u22a2\u1d40 \u0394) (e : \u0394 = \u0393) : \u22a2\u1d40 \u0393 := cast (by rw e) h\n\n@[elab_as_eliminator]\ntheorem rec_on {C : \u03a0 {m} (\u0394 : finset (bounded_formula L m)), \u22a2\u1d40 \u0394 \u2192 Prop}\n  {m : \u2115} {\u0394 : finset (bounded_formula L m)} (d : \u22a2\u1d40 \u0394)\n  (hAxL : \u2200 {m} (\u0394 : finset (bounded_formula L m)) {k} (r : L.pr k) (v : fin k \u2192 bounded_term L m)\n    (h : relation r v \u2208 \u0394) (hneg : neg_relation r v \u2208 \u0394), C \u0394 (AxL r v h hneg))\n  (hverum : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (h : \u22a4 \u2208 \u0394), C \u0394 (verum h))\n  (hor_left : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m) (d : \u22a2\u1d40 insert p \u0394),\n    C (insert p \u0394) d \u2192 C (insert (p \u2294 q) \u0394) (or_left p q d))\n  (hor_right : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m) (d : \u22a2\u1d40 insert q \u0394),\n    C (insert q \u0394) d \u2192 C (insert (p \u2294 q) \u0394) (or_right p q d))\n  (hand : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (p q : bounded_formula L m) (d\u2081 : \u22a2\u1d40 insert p \u0394) (d\u2082 : \u22a2\u1d40 insert q \u0394),\n    C (insert p \u0394) d\u2081 \u2192 C (insert q \u0394) d\u2082 \u2192 C (insert (p \u2293 q) \u0394) (and d\u2081 d\u2082))\n  (hall : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (p : bounded_subformula L m 1) (d : \u22a2\u1d40 insert p.push (finset_mlift \u0394)),\n    C (insert p.push (finset_mlift \u0394)) d \u2192 C (insert (\u2200'p) \u0394) (all d))\n  (hex : \u2200 {m} (\u0394 : finset (bounded_formula L m)) (t) (p : bounded_subformula L m 1) (d : \u22a2\u1d40 insert (subst t p) \u0394),\n    C (insert (subst t p) \u0394) d \u2192 C (insert (\u2203'p) \u0394) (ex d)) : C \u0394 d :=\n by unfreezingI {\n  begin\n    cases d,\n    induction d,\n    case AxL : m \u0394 k r v h hneg { exact hAxL \u0394 r v h hneg },\n    case verum : m \u0394 h { exact hverum \u0394 h },\n    case or_left : m \u0394 p q _ ih { exact hor_left \u0394 p q _ ih },\n    case or_right : m \u0394 p q _ ih { exact hor_right \u0394 p q _ ih },\n    case and : m \u0394 p q _ _ ih\u2081 ih\u2082 { exact hand \u0394 p q _ _ ih\u2081 ih\u2082 },\n    case all : m \u0394 p _ ih { exact hall \u0394 p _ ih },\n    case ex : m \u0394 t p _ ih { exact hex \u0394 t p _ ih }\n  end }\n\nprotected lemma weakening (h : \u22a2\u1d40 \u0394) : \u2200 {\u0393}, \u0394 \u2286 \u0393 \u2192 \u22a2\u1d40 \u0393 :=\nbegin\n  apply rec_on h,\n  { intros m \u0394 k r v h hneg \u0393 ss, refine AxL r v (ss h) (ss hneg) },\n  { intros m \u0394 h \u0393 ss, refine verum (ss h) },\n  { intros m \u0394 p q h IH \u0393 ss,\n    have : \u22a2\u1d40 insert p \u0393, from IH (finset.insert_subset_insert _ (finset.insert_subset.mp ss).2),\n    have : \u22a2\u1d40 insert (p \u2294 q) \u0393, from or_left p q this,\n    refine derivable.cast this (by { simp, exact (finset.insert_subset.mp ss).1}) },\n  { intros m \u0394 p q h IH \u0393 ss,\n    have : \u22a2\u1d40 insert q \u0393, from IH (finset.insert_subset_insert _ (finset.insert_subset.mp ss).2),\n    have : \u22a2\u1d40 insert (p \u2294 q) \u0393, from or_right p q this,\n    refine derivable.cast this (by { simp, exact (finset.insert_subset.mp ss).1}) },\n  { intros m \u0394 p q h\u2081 h\u2082 IH\u2081 IH\u2082 \u0393 ss,\n    have l\u2081 : \u22a2\u1d40 insert p \u0393, from IH\u2081 (finset.insert_subset_insert _ $ (finset.insert_subset.mp ss).2),\n    have l\u2082 : \u22a2\u1d40 insert q \u0393, from IH\u2082 (finset.insert_subset_insert _ $ (finset.insert_subset.mp ss).2),\n    have : \u22a2\u1d40 insert (p \u2293 q) \u0393, from and l\u2081 l\u2082,\n    refine derivable.cast this (by { simp, exact (finset.insert_subset.mp ss).1}) },\n  { intros m \u0394 p h IH \u0393 ss,\n    have : \u22a2\u1d40 insert p.push (finset_mlift \u0393),\n      from IH (finset.insert_subset_insert _ $ finset.image_subset_image (finset.insert_subset.mp ss).2),\n    have : \u22a2\u1d40 insert (\u2200'p) \u0393 := all this,\n    refine derivable.cast this (by { simp, exact (finset.insert_subset.mp ss).1}) },\n  { intros m \u0394 t p h IH \u0393 ss,\n    have : \u22a2\u1d40 insert (subst t p) \u0393, from IH (finset.insert_subset_insert _ (finset.insert_subset.mp ss).2),\n    have : \u22a2\u1d40 insert (\u2203'p) \u0393 := ex this,\n    refine derivable.cast this (by { simp, exact (finset.insert_subset.mp ss).1}) }\nend\n\n\nlemma and' {p q : bounded_formula L m} (hp : \u22a2\u1d40 insert p \u0394) (hq : \u22a2\u1d40 insert q \u0393) : \u22a2\u1d40 insert (p \u2293 q) (\u0394 \u222a \u0393) :=\nby { have hp' : \u22a2\u1d40 insert p (\u0394 \u222a \u0393), from derivable.weakening hp (by intros x; simp; tauto),\n     have hq' : \u22a2\u1d40 insert q (\u0394 \u222a \u0393), from derivable.weakening hq (by intros x; simp; tauto),\n     exact derivable.and hp' hq' }\n\nsection\nvariables {\u0394}\n\nopen axiomatic_classical_logic' axiomatic_classical_logic\n\nlemma provable_of_derivation (h : \u22a2\u1d40 \u0394) : \u2205 \u22a2 (\u0394.image of_tait).disjunction :=\nbegin\n  apply derivable.rec_on h,\n  { intros m \u0394 k r v h nh,\n    suffices : \u2205 \u22a2 fol.subformula.relation r v \u2294 \u223cfol.subformula.relation r v \u27f6 (finset.image of_tait \u0394).disjunction,\n    from this \u2a00 excluded_middle,\n    refine or_imply _ _ _ \u2a00 _ \u2a00 _,\n    { refine imply_fdisj (by { simp, refine \u27e8_, h, by simp\u27e9 }) },\n    { refine imply_fdisj (by { simp, refine \u27e8_, nh, by simp\u27e9 }) } },\n  { intros m \u0394 h\u0394, exact (imply_fdisj\n      (by { show of_tait \u22a4 \u2208 \u0394.image of_tait, exact finset.mem_image_of_mem of_tait h\u0394 })) \u2a00 (by simp) },\n  { intros m \u0394 p q b IH, simp[fdisj_insert] at IH \u22a2, exact imply_or_left _ _ \u2a00 IH },\n  { intros m \u0394 p q b IH, simp[fdisj_insert] at IH \u22a2, exact imply_or_right _ _ \u2a00 IH },\n  { intros m \u0394 p q _ _ IH\u2081 IH\u2082, simp[fdisj_insert] at IH\u2081 IH\u2082 \u22a2,\n    exact \u27e8IH\u2081, IH\u2082\u27e9 },\n  { intros m \u0394 p b IH, simp[fdisj_insert] at IH \u22a2,\n    have e : has_negation.neg '' (of_tait '' (\u2191(finset_mlift \u0394) : set (bounded_formula L (m + 1)))) =\n      \ud835\udddf' (has_negation.neg '' (of_tait '' \u2191\u0394)),\n    { ext q, simp[finset_mlift, bounded_preTheory.mlift, of_tait_mlift] },\n    have : \ud835\udddf'(has_negation.neg '' (of_tait '' \u2191\u0394)) \u22a2 of_tait (push p),\n    { simpa[\u2190e] using IH },\n    by simpa[\u2190of_tait_pull] using provable.generalize this },\n  { intros m \u0394 t p b IH, simp[fdisj_insert, of_tait_subst] at IH \u22a2,\n    refine provable.use t IH }\nend\n\nend\n\nend derivable\n\nend Tait\n\nopen subformula\n\nnamespace provable\nopen axiomatic_classical_logic' axiomatic_classical_logic\n\ndef of_Tait_provable {T : bounded_preTheory L m} {p : bounded_formula L m} :\n  to_tait '' T \u22a2 p.to_tait \u2192 T \u22a2 p :=\nbegin\n  simp[Tait.provable_def],\n  intros \u0394 ss b,\n  have le : has_negation.neg '' (Tait.subformula.of_tait '' \u2191\u0394) \u2264 T,\n  from @le_of _ _ (has_negation.neg '' (Tait.subformula.of_tait '' \u2191\u0394)) T\n    (by { simp, intros p hp,\n      have := ss (by simpa using hp), simp at this, rcases this with \u27e8q, hq, rfl\u27e9,\n      have l\u2081 : T \u22a2 \u223cTait.subformula.of_tait (\u223cq.to_tait) \u27f7 \u223c\u223cq.to_tait.of_tait,\n      from equiv_neg_of_equiv (Tait.subformula.to_tait_not_equiv q.to_tait),\n      have l\u2082 : T \u22a2 \u223c\u223cq.to_tait.of_tait \u27f7 q, from equiv_trans (by simp) (to_tait_of_tait q),\n      refine of_equiv (by_axiom hq) (equiv_symm $ equiv_trans l\u2081 l\u2082) }),\n  have : has_negation.neg '' (Tait.subformula.of_tait '' \u2191\u0394) \u22a2 (to_tait p).of_tait, \n  by simpa[fdisj_insert] using b.provable_of_derivation,\n  have := le this,\n  exact of_equiv this (to_tait_of_tait p)\nend\n\nend provable\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/QL/FOL/Tait/calculus.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.6261241702517976, "lm_q1q2_score": 0.3851213643312876}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.currying\nimport category_theory.limits.preserves.limits\n\n/-!\n# (Co)limits in functor categories.\n\nWe show that if `D` has limits, then the functor category `C \u2964 D` also has limits\n(`category_theory.limits.functor_category_has_limits`),\nand the evaluation functors preserve limits\n(`category_theory.limits.evaluation_preserves_limits`)\n(and similarly for colimits).\n\nWe also show that `F : D \u2964 K \u2964 C` preserves (co)limits if it does so for each `k : K`\n(`category_theory.limits.preserves_limits_of_evaluation` and\n`category_theory.limits.preserves_colimits_of_evaluation`).\n-/\n\nopen category_theory category_theory.category\n\n-- morphism levels before object levels. See note [category_theory universes].\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 v v' u u'\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n\nvariables {J : Type u\u2081} [category.{v\u2081} J] {K : Type u\u2082} [category.{v\u2082} K]\n\n@[simp, reassoc]\nlemma limit.lift_\u03c0_app (H : J \u2964 K \u2964 C) [has_limit H] (c : cone H) (j : J) (k : K) :\n  (limit.lift H c).app k \u226b (limit.\u03c0 H j).app k = (c.\u03c0.app j).app k :=\ncongr_app (limit.lift_\u03c0 c j) k\n\n@[simp, reassoc]\nlemma colimit.\u03b9_desc_app (H : J \u2964 K \u2964 C) [has_colimit H] (c : cocone H) (j : J) (k : K) :\n  (colimit.\u03b9 H j).app k \u226b (colimit.desc H c).app k = (c.\u03b9.app j).app k :=\ncongr_app (colimit.\u03b9_desc c j) k\n\n/--\nThe evaluation functors jointly reflect limits: that is, to show a cone is a limit of `F`\nit suffices to show that each evaluation cone is a limit. In other words, to prove a cone is\nlimiting you can show it's pointwise limiting.\n-/\ndef evaluation_jointly_reflects_limits {F : J \u2964 K \u2964 C} (c : cone F)\n  (t : \u03a0 (k : K), is_limit (((evaluation K C).obj k).map_cone c)) : is_limit c :=\n{ lift := \u03bb s,\n  { app := \u03bb k, (t k).lift \u27e8s.X.obj k, whisker_right s.\u03c0 ((evaluation K C).obj k)\u27e9,\n    naturality' := \u03bb X Y f, (t Y).hom_ext $ \u03bb j,\n    begin\n      rw [assoc, (t Y).fac _ j],\n      simpa using\n        ((t X).fac_assoc \u27e8s.X.obj X, whisker_right s.\u03c0 ((evaluation K C).obj X)\u27e9 j _).symm,\n    end },\n  fac' := \u03bb s j, nat_trans.ext _ _ $ funext $ \u03bb k, (t k).fac _ j,\n  uniq' := \u03bb s m w, nat_trans.ext _ _ $ funext $ \u03bb x, (t x).hom_ext $ \u03bb j,\n      (congr_app (w j) x).trans\n        ((t x).fac \u27e8s.X.obj _, whisker_right s.\u03c0 ((evaluation K C).obj _)\u27e9 j).symm }\n\n/--\nGiven a functor `F` and a collection of limit cones for each diagram `X \u21a6 F X k`, we can stitch\nthem together to give a cone for the diagram `F`.\n`combined_is_limit` shows that the new cone is limiting, and `eval_combined` shows it is\n(essentially) made up of the original cones.\n-/\n@[simps] def combine_cones (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), limit_cone (F.flip.obj k)) :\n  cone F :=\n{ X :=\n  { obj := \u03bb k, (c k).cone.X,\n    map := \u03bb k\u2081 k\u2082 f, (c k\u2082).is_limit.lift \u27e8_, (c k\u2081).cone.\u03c0 \u226b F.flip.map f\u27e9,\n    map_id' := \u03bb k, (c k).is_limit.hom_ext (\u03bb j, by { dsimp, simp }),\n    map_comp' := \u03bb k\u2081 k\u2082 k\u2083 f\u2081 f\u2082, (c k\u2083).is_limit.hom_ext (\u03bb j, by simp) },\n  \u03c0 :=\n  { app := \u03bb j, { app := \u03bb k, (c k).cone.\u03c0.app j },\n    naturality' := \u03bb j\u2081 j\u2082 g, nat_trans.ext _ _ $ funext $ \u03bb k, (c k).cone.\u03c0.naturality g } }\n\n/-- The stitched together cones each project down to the original given cones (up to iso). -/\ndef evaluate_combined_cones (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), limit_cone (F.flip.obj k)) (k : K) :\n  ((evaluation K C).obj k).map_cone (combine_cones F c) \u2245 (c k).cone :=\ncones.ext (iso.refl _) (by tidy)\n\n/-- Stitching together limiting cones gives a limiting cone. -/\ndef combined_is_limit (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), limit_cone (F.flip.obj k)) :\n  is_limit (combine_cones F c) :=\nevaluation_jointly_reflects_limits _\n  (\u03bb k, (c k).is_limit.of_iso_limit (evaluate_combined_cones F c k).symm)\n\n/--\nThe evaluation functors jointly reflect colimits: that is, to show a cocone is a colimit of `F`\nit suffices to show that each evaluation cocone is a colimit. In other words, to prove a cocone is\ncolimiting you can show it's pointwise colimiting.\n-/\ndef evaluation_jointly_reflects_colimits {F : J \u2964 K \u2964 C} (c : cocone F)\n  (t : \u03a0 (k : K), is_colimit (((evaluation K C).obj k).map_cocone c)) : is_colimit c :=\n{ desc := \u03bb s,\n  { app := \u03bb k, (t k).desc \u27e8s.X.obj k, whisker_right s.\u03b9 ((evaluation K C).obj k)\u27e9,\n    naturality' := \u03bb X Y f, (t X).hom_ext $ \u03bb j,\n    begin\n      rw [(t X).fac_assoc _ j],\n      erw \u2190 (c.\u03b9.app j).naturality_assoc f,\n      erw (t Y).fac \u27e8s.X.obj _, whisker_right s.\u03b9 _\u27e9 j,\n      dsimp,\n      simp,\n    end },\n  fac' := \u03bb s j, nat_trans.ext _ _ $ funext $ \u03bb k, (t k).fac _ j,\n  uniq' := \u03bb s m w, nat_trans.ext _ _ $ funext $ \u03bb x, (t x).hom_ext $ \u03bb j,\n      (congr_app (w j) x).trans\n        ((t x).fac \u27e8s.X.obj _, whisker_right s.\u03b9 ((evaluation K C).obj _)\u27e9 j).symm }\n\n/--\nGiven a functor `F` and a collection of colimit cocones for each diagram `X \u21a6 F X k`, we can stitch\nthem together to give a cocone for the diagram `F`.\n`combined_is_colimit` shows that the new cocone is colimiting, and `eval_combined` shows it is\n(essentially) made up of the original cocones.\n-/\n@[simps] def combine_cocones (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), colimit_cocone (F.flip.obj k)) :\n  cocone F :=\n{ X :=\n  { obj := \u03bb k, (c k).cocone.X,\n    map := \u03bb k\u2081 k\u2082 f, (c k\u2081).is_colimit.desc \u27e8_, F.flip.map f \u226b (c k\u2082).cocone.\u03b9\u27e9,\n    map_id' := \u03bb k, (c k).is_colimit.hom_ext (\u03bb j, by { dsimp, simp }),\n    map_comp' := \u03bb k\u2081 k\u2082 k\u2083 f\u2081 f\u2082, (c k\u2081).is_colimit.hom_ext (\u03bb j, by simp) },\n  \u03b9 :=\n  { app := \u03bb j, { app := \u03bb k, (c k).cocone.\u03b9.app j },\n    naturality' := \u03bb j\u2081 j\u2082 g, nat_trans.ext _ _ $ funext $ \u03bb k, (c k).cocone.\u03b9.naturality g } }\n\n/-- The stitched together cocones each project down to the original given cocones (up to iso). -/\ndef evaluate_combined_cocones\n  (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), colimit_cocone (F.flip.obj k)) (k : K) :\n  ((evaluation K C).obj k).map_cocone (combine_cocones F c) \u2245 (c k).cocone :=\ncocones.ext (iso.refl _) (by tidy)\n\n/-- Stitching together colimiting cocones gives a colimiting cocone. -/\ndef combined_is_colimit (F : J \u2964 K \u2964 C) (c : \u03a0 (k : K), colimit_cocone (F.flip.obj k)) :\n  is_colimit (combine_cocones F c) :=\nevaluation_jointly_reflects_colimits _\n  (\u03bb k, (c k).is_colimit.of_iso_colimit (evaluate_combined_cocones F c k).symm)\n\nnoncomputable theory\n\ninstance functor_category_has_limits_of_shape\n  [has_limits_of_shape J C] : has_limits_of_shape J (K \u2964 C) :=\n{ has_limit := \u03bb F, has_limit.mk\n  { cone := combine_cones F (\u03bb k, get_limit_cone _),\n    is_limit := combined_is_limit _ _ } }\n\ninstance functor_category_has_colimits_of_shape\n  [has_colimits_of_shape J C] : has_colimits_of_shape J (K \u2964 C) :=\n{ has_colimit := \u03bb F, has_colimit.mk\n  { cocone := combine_cocones _ (\u03bb k, get_colimit_cocone _),\n    is_colimit := combined_is_colimit _ _ } }\n\ninstance functor_category_has_limits_of_size [has_limits_of_size.{v\u2081 u\u2081} C] :\n  has_limits_of_size.{v\u2081 u\u2081} (K \u2964 C) := \u27e8infer_instance\u27e9\n\ninstance functor_category_has_colimits_of_size [has_colimits_of_size.{v\u2081 u\u2081} C] :\n  has_colimits_of_size.{v\u2081 u\u2081} (K \u2964 C) := \u27e8infer_instance\u27e9\n\nend category_theory.limits\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C] {D : Type u'} [category.{v} D]\n\nvariables {J : Type v} [category.{v} J] {K : Type v} [category.{v\u2082} K]\n\ninstance evaluation_preserves_limits_of_shape [has_limits_of_shape J C] (k : K) :\n  preserves_limits_of_shape J ((evaluation K C).obj k) :=\n{ preserves_limit :=\n  \u03bb F, preserves_limit_of_preserves_limit_cone (combined_is_limit _ _) $\n    is_limit.of_iso_limit (limit.is_limit _)\n      (evaluate_combined_cones F _ k).symm }\n\n/--\nIf `F : J \u2964 K \u2964 C` is a functor into a functor category which has a limit,\nthen the evaluation of that limit at `k` is the limit of the evaluations of `F.obj j` at `k`.\n-/\ndef limit_obj_iso_limit_comp_evaluation [has_limits_of_shape J C] (F : J \u2964 K \u2964 C) (k : K) :\n  (limit F).obj k \u2245 limit (F \u22d9 ((evaluation K C).obj k)) :=\npreserves_limit_iso ((evaluation K C).obj k) F\n\n@[simp, reassoc]\nlemma limit_obj_iso_limit_comp_evaluation_hom_\u03c0\n  [has_limits_of_shape J C] (F : J \u2964 (K \u2964 C)) (j : J) (k : K) :\n  (limit_obj_iso_limit_comp_evaluation F k).hom \u226b limit.\u03c0 (F \u22d9 ((evaluation K C).obj k)) j =\n    (limit.\u03c0 F j).app k :=\nbegin\n  dsimp [limit_obj_iso_limit_comp_evaluation],\n  simp,\nend\n\n@[simp, reassoc]\nlemma limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app\n  [has_limits_of_shape J C] (F : J \u2964 (K \u2964 C)) (j : J) (k : K):\n  (limit_obj_iso_limit_comp_evaluation F k).inv \u226b (limit.\u03c0 F j).app k =\n    limit.\u03c0 (F \u22d9 ((evaluation K C).obj k)) j :=\nbegin\n  dsimp [limit_obj_iso_limit_comp_evaluation],\n  rw iso.inv_comp_eq,\n  simp,\nend\n\n@[ext]\nlemma limit_obj_ext {H : J \u2964 K \u2964 C} [has_limits_of_shape J C]\n  {k : K} {W : C} {f g : W \u27f6 (limit H).obj k}\n  (w : \u2200 j, f \u226b (limits.limit.\u03c0 H j).app k = g \u226b (limits.limit.\u03c0 H j).app k) : f = g :=\nbegin\n  apply (cancel_mono (limit_obj_iso_limit_comp_evaluation H k).hom).1,\n  ext,\n  simpa using w j,\nend\n\ninstance evaluation_preserves_colimits_of_shape [has_colimits_of_shape J C] (k : K) :\n  preserves_colimits_of_shape J ((evaluation K C).obj k) :=\n{ preserves_colimit :=\n  \u03bb F, preserves_colimit_of_preserves_colimit_cocone (combined_is_colimit _ _) $\n    is_colimit.of_iso_colimit (colimit.is_colimit _)\n      (evaluate_combined_cocones F _ k).symm }\n\n/--\nIf `F : J \u2964 K \u2964 C` is a functor into a functor category which has a colimit,\nthen the evaluation of that colimit at `k` is the colimit of the evaluations of `F.obj j` at `k`.\n-/\ndef colimit_obj_iso_colimit_comp_evaluation [has_colimits_of_shape J C] (F : J \u2964 K \u2964 C) (k : K) :\n  (colimit F).obj k \u2245 colimit (F \u22d9 ((evaluation K C).obj k)) :=\npreserves_colimit_iso ((evaluation K C).obj k) F\n\n@[simp, reassoc]\nlemma colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv\n  [has_colimits_of_shape J C] (F : J \u2964 (K \u2964 C)) (j : J) (k : K) :\n  colimit.\u03b9 (F \u22d9 ((evaluation K C).obj k)) j \u226b (colimit_obj_iso_colimit_comp_evaluation F k).inv =\n    (colimit.\u03b9 F j).app k :=\nbegin\n  dsimp [colimit_obj_iso_colimit_comp_evaluation],\n  simp,\nend\n\n@[simp, reassoc]\nlemma colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom\n  [has_colimits_of_shape J C] (F : J \u2964 (K \u2964 C)) (j : J) (k : K) :\n  (colimit.\u03b9 F j).app k \u226b (colimit_obj_iso_colimit_comp_evaluation F k).hom =\n     colimit.\u03b9 (F \u22d9 ((evaluation K C).obj k)) j :=\nbegin\n  dsimp [colimit_obj_iso_colimit_comp_evaluation],\n  rw \u2190iso.eq_comp_inv,\n  simp,\nend\n\n@[ext]\nlemma colimit_obj_ext {H : J \u2964 K \u2964 C} [has_colimits_of_shape J C]\n  {k : K} {W : C} {f g : (colimit H).obj k \u27f6 W}\n  (w : \u2200 j, (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g) : f = g :=\nbegin\n  apply (cancel_epi (colimit_obj_iso_colimit_comp_evaluation H k).inv).1,\n  ext,\n  simpa using w j,\nend\n\ninstance evaluation_preserves_limits [has_limits C] (k : K) :\n  preserves_limits ((evaluation K C).obj k) :=\n{ preserves_limits_of_shape := \u03bb J \ud835\udca5, by resetI; apply_instance }\n\n/-- `F : D \u2964 K \u2964 C` preserves the limit of some `G : J \u2964 D` if it does for each `k : K`. -/\ndef preserves_limit_of_evaluation (F : D \u2964 K \u2964 C) (G : J \u2964 D)\n  (H : \u03a0 (k : K), preserves_limit G (F \u22d9 (evaluation K C).obj k : D \u2964 C)) :\n  preserves_limit G F := \u27e8\u03bb c hc,\nbegin\n  apply evaluation_jointly_reflects_limits,\n  intro X,\n  haveI := H X,\n  change is_limit ((F \u22d9 (evaluation K C).obj X).map_cone c),\n  exact preserves_limit.preserves hc,\nend\u27e9\n\n/-- `F : D \u2964 K \u2964 C` preserves limits of shape `J` if it does for each `k : K`. -/\ndef preserves_limits_of_shape_of_evaluation (F : D \u2964 K \u2964 C) (J : Type v) [small_category J]\n  (H : \u03a0 (k : K), preserves_limits_of_shape J (F \u22d9 (evaluation K C).obj k)) :\n  preserves_limits_of_shape J F :=\n\u27e8\u03bb G, preserves_limit_of_evaluation F G (\u03bb k, preserves_limits_of_shape.preserves_limit)\u27e9\n\n/-- `F : D \u2964 K \u2964 C` preserves all limits if it does for each `k : K`. -/\ndef preserves_limits_of_evaluation (F : D \u2964 K \u2964 C)\n  (H : \u03a0 (k : K), preserves_limits (F \u22d9 (evaluation K C).obj k)) :\n  preserves_limits F :=\n\u27e8\u03bb L hL, by exactI preserves_limits_of_shape_of_evaluation\n    F L (\u03bb k, preserves_limits.preserves_limits_of_shape)\u27e9\n\ninstance evaluation_preserves_colimits [has_colimits C] (k : K) :\n  preserves_colimits ((evaluation K C).obj k) :=\n{ preserves_colimits_of_shape := \u03bb J \ud835\udca5, by resetI; apply_instance }\n\n/-- `F : D \u2964 K \u2964 C` preserves the colimit of some `G : J \u2964 D` if it does for each `k : K`. -/\ndef preserves_colimit_of_evaluation (F : D \u2964 K \u2964 C) (G : J \u2964 D)\n  (H : \u03a0 (k), preserves_colimit G (F \u22d9 (evaluation K C).obj k)) : preserves_colimit G F := \u27e8\u03bb c hc,\nbegin\n  apply evaluation_jointly_reflects_colimits,\n  intro X,\n  haveI := H X,\n  change is_colimit ((F \u22d9 (evaluation K C).obj X).map_cocone c),\n  exact preserves_colimit.preserves hc,\nend\u27e9\n\n/-- `F : D \u2964 K \u2964 C` preserves all colimits of shape `J` if it does for each `k : K`. -/\ndef preserves_colimits_of_shape_of_evaluation (F : D \u2964 K \u2964 C) (J : Type v) [small_category J]\n  (H : \u03a0 (k : K), preserves_colimits_of_shape J (F \u22d9 (evaluation K C).obj k)) :\n  preserves_colimits_of_shape J F :=\n\u27e8\u03bb G, preserves_colimit_of_evaluation F G (\u03bb k, preserves_colimits_of_shape.preserves_colimit)\u27e9\n\n/-- `F : D \u2964 K \u2964 C` preserves all colimits if it does for each `k : K`. -/\ndef preserves_colimits_of_evaluation (F : D \u2964 K \u2964 C)\n  (H : \u03a0 (k : K), preserves_colimits (F \u22d9 (evaluation K C).obj k)) :\n  preserves_colimits F :=\n\u27e8\u03bb L hL, by exactI preserves_colimits_of_shape_of_evaluation\n    F L (\u03bb k, preserves_colimits.preserves_colimits_of_shape)\u27e9\nopen category_theory.prod\n\n/--\nFor a functor `G : J \u2964 K \u2964 C`, its limit `K \u2964 C` is given by `(G' : K \u2964 J \u2964 C) \u22d9 lim`.\nNote that this does not require `K` to be small.\n-/\n@[simps] def limit_iso_swap_comp_lim [has_limits_of_shape J C] (G : J \u2964 K \u2964 C) [has_limit G] :\n  limit G \u2245 curry.obj (swap K J \u22d9 uncurry.obj G) \u22d9 lim :=\nnat_iso.of_components (\u03bb Y, limit_obj_iso_limit_comp_evaluation G Y \u226a\u226b\n  (lim.map_iso (eq_to_iso (by\n  { apply functor.hext,\n    { intro X, simp },\n    { intros X\u2081 X\u2082 f, dsimp only [swap], simp }}))))\n  begin\n    intros Y\u2081 Y\u2082 f,\n    ext1 x,\n    dsimp only [swap],\n    simp only [limit_obj_iso_limit_comp_evaluation_hom_\u03c0_assoc, category.comp_id,\n      limit_obj_iso_limit_comp_evaluation_hom_\u03c0, eq_to_iso.hom, curry.obj_map_app,\n      nat_trans.naturality, category.id_comp, eq_to_hom_refl, functor.comp_map,\n      eq_to_hom_app, lim_map_\u03c0_assoc, lim_map_\u03c0, category.assoc,\n      uncurry.obj_map, lim_map_eq_lim_map, iso.trans_hom,\n      nat_trans.id_app, category_theory.functor.map_id, functor.map_iso_hom],\n    erw category.id_comp,\n  end\n\n/--\nFor a functor `G : J \u2964 K \u2964 C`, its colimit `K \u2964 C` is given by `(G' : K \u2964 J \u2964 C) \u22d9 colim`.\nNote that this does not require `K` to be small.\n-/\n@[simps]\ndef colimit_iso_swap_comp_colim [has_colimits_of_shape J C] (G : J \u2964 K \u2964 C) [has_colimit G] :\n  colimit G \u2245 curry.obj (swap K J \u22d9 uncurry.obj G) \u22d9 colim :=\nnat_iso.of_components (\u03bb Y, colimit_obj_iso_colimit_comp_evaluation G Y \u226a\u226b\n  (colim.map_iso (eq_to_iso (by\n  { apply functor.hext,\n    { intro X, simp },\n    { intros X Y f, dsimp only [swap], simp, } }))))\n  begin\n    intros Y\u2081 Y\u2082 f,\n    ext1 x,\n    rw \u2190 (colimit.\u03b9 G x).naturality_assoc f,\n    dsimp only [swap],\n    simp only [eq_to_iso.hom, colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom_assoc,\n      curry.obj_map_app, colimit.\u03b9_map, category.id_comp, eq_to_hom_refl, iso.trans_hom,\n      functor.comp_map, eq_to_hom_app, colimit.\u03b9_map_assoc, functor.map_iso_hom,\n      category.assoc, uncurry.obj_map, nat_trans.id_app, category_theory.functor.map_id],\n    erw category.id_comp,\n  end\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/functor_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.6757646075489392, "lm_q1q2_score": 0.3850862533436104}}
{"text": "theorem ex1 [Monad m] [LawfulMonad m] (b : Bool) (ma : m \u03b1) (mb : \u03b1 \u2192 m \u03b1) :\n    (do let mut x \u2190 ma\n        if b then\n          x \u2190 mb x\n        pure x)\n    =\n    (ma >>= fun x => if b then mb x else pure x) := by\n  cases b <;> simp\n\nattribute [simp] map_eq_pure_bind seq_eq_bind_map\n\ntheorem ex2 [Monad m] [LawfulMonad m] (b : Bool) (ma : m \u03b1) (mb : \u03b1 \u2192 m \u03b1) (a : \u03b1) :\n    (do let mut x \u2190 ma\n        if b then\n          x \u2190 mb x\n        pure x)\n    =\n    (StateT.run' (m := m)\n      (do ma >>= set\n          if b then get >>= fun x => mb x >>= set\n          get) a)  := by\n  cases b <;> simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/do_eqv_proofs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.38508624962253435}}
{"text": "import SciLean.Core.Integral\nimport SciLean.Core.AdjDiff\n\n\nnamespace SciLean\n\n\n\n--------------------------------------------------------------------------------\n-- Junk\n--------------------------------------------------------------------------------\n\nvariable {X Y \u03b9 : Type} [Enumtype \u03b9] [FinVec X \u03b9] [Hilbert Y]\n\n\nexample (f : X\u27ffY) : (\u03bb g : X\u27ffY => \u222b x, \u27eaf x, g x\u27eb)\u2020 = f := by simp\nexample (f : X\u27ffY) : (\u03bb g : X\u27ffY => \u222b x, \u27eag x, f x\u27eb)\u2020 = f := by simp; done\n  \n\n\nexample : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff g x := by infer_instance\nexample : IsSmoothT fun (g : X\u27ffY) => fun x \u27ff g x := by infer_instance\n\n#check (fun (g : X\u27ffY) => fun x \u27ff g x)\u2020 \n       rewrite_by simp; trace_state\n\n\n-- instance oj  {X Y Y' Z} [Vec X] [Vec Y] [Vec Y'] [Vec Z] \n--   (f : X \u2192 Y \u2192 Y' \u2192 Z) [IsSmoothNT 3 f]  \n--   (g' : X \u2192 Y') [IsSmoothNT 1 g']\n--   : IsSmoothNT 2 \u03bb (g : X\u27ffY) x => f x (g x) (g' x) := sorry_proof\n\n-- instance {X Y Z} [Vec X] [Vec Y] [Vec Z] (f : X \u2192 Y \u2192 Z) [IsSmoothNT 2 f] \n--   : IsSmoothNT 2 \u03bb (g : X\u27ffY) x => f x (g x) := by apply oj (\u03bb x y _ => f x y) (\u03bb x => x)\n\n-- instance oh {X Y Y\u2081 Y\u2082 Z} [Vec X] [Vec Y] [Vec Y\u2081] [Vec Y\u2082] [Vec Z] \n--   (f : Y\u2081 \u2192 Y\u2082 \u2192 Z) [IsSmoothNT 2 f]  \n--   (g\u2081 : X \u2192 Y \u2192 Y\u2081) [IsSmoothNT 2 g\u2081]\n--   (g\u2082 : X \u2192 Y \u2192 Y\u2082) [IsSmoothNT 2 g\u2082] \n--   : IsSmoothNT 2 \u03bb (g : X\u27ffY) x => f (g\u2081 x (g x)) (g\u2082 x (g x)) := sorry_proof\n\n-- instance  {Y'} [Vec Y'] {Z} [Hilbert Z]\n--   (A : X \u2192 Y \u2192 Y' \u2192 Z) [\u2200 x y', HasAdjointT (\u03bb y => A x y y')] [IsSmoothNT 3 A]\n--   (g' : X \u2192 Y' := \u03bb _ => 0) [IsSmoothT g']\n--   : HasAdjointT (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (g x) (g' x)) :=\n-- by  sorry_proof\n\n\ninstance scomb_highorder_adjoint {Z W} [SemiHilbert W] [Hilbert Z] \n  (F : (X\u27ffY) \u2192 W \u2192 (X\u27ffZ)) [HasAdjointNT 2 F]  -- [IsSmoothNT 2 F]\n  (G : (X\u27ffY) \u2192 W) [HasAdjointT G]\n  : HasAdjointT (\u03bb (g : X\u27ffY) => \u03bb x \u27ff F g (G g) x) := by (try infer_instance); sorry_proof\n\n\nset_option synthInstance.maxSize 2000 in\ninstance scomb_highorder_adjoint_simp {Z W} [SemiHilbert W] [Hilbert Z]\n  (F : (X\u27ffY) \u2192 W \u2192 (X\u27ffZ)) [HasAdjointNT 2 F] [IsSmoothNT 2 F]\n  (G : (X\u27ffY) \u2192 W) [HasAdjointT G] [IsSmoothT G]\n  : (\u03bb (g : X\u27ffY) => \u03bb (x:X) \u27ff (F g (G g) x))\u2020\n    =\n    \u03bb h => \n      let gw := (uncurryN 2 F)\u2020 h\n      let (g',w) := gw\n      let g'' := G\u2020 w\n      \u03bb x \u27ff g' x + g'' x \n  := by sorry_proof\n\n\ninstance elemwise_adjoint {Z} [Hilbert Z] (A : X \u2192 Y \u2192 Z) [\u2200 x, HasAdjointT (A x)] [IsSmoothNT 2 A]\n  : HasAdjointT (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (g x)) := \nby \n  try infer_instance\n  sorry_proof\n\n\n@[simp \u2193, diff]\ntheorem elemwise_adjoint_simp {Z} [Hilbert Z] (A : X \u2192 Y \u2192 Z) [\u2200 x, HasAdjointT (A x)] [IsSmoothNT 2 A]\n  : (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (g x))\u2020\n    =\n    \u03bb g => \u03bb x \u27ff (A x)\u2020 (g x) := by sorry_proof\n\n\ninstance elemwise_adjoint_alt1 {X Y \u03b9 : Type} [Enumtype \u03b9] [FinVec X \u03b9] [Hilbert Y]\n  {X' Y' \u03b9' : Type} [Enumtype \u03b9'] [FinVec X' \u03b9'] [Hilbert Y']\n  (D : (X\u27ffY) \u2192 (X'\u27ffY')) [HasAdjointT D]\n  {Z} [Hilbert Z] (A : X' \u2192 Y' \u2192 Z) [\u2200 x, HasAdjointT (A x)] [IsSmoothNT 2 A]\n  : HasAdjointT (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (D g x)) :=\nby\n  try infer_instance\n  let G := \u03bb g : X'\u27ffY' => \u03bb x \u27ff A x (g x)\n  let h : (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (D g x)) = \u03bb g => G (D g) := by rfl\n  rw [h]\n  infer_instance\n  done\n\n@[simp \u2193, diff]\ntheorem elemwise_adjoint_simp_alt1 {X Y \u03b9 : Type} [Enumtype \u03b9] [FinVec X \u03b9] [Hilbert Y]\n  {X' Y' \u03b9' : Type} [Enumtype \u03b9'] [FinVec X' \u03b9'] [Hilbert Y']\n  (D : (X\u27ffY) \u2192 (X'\u27ffY')) [HasAdjointT D]\n  {Z} [Hilbert Z] (A : X' \u2192 Y' \u2192 Z) [\u2200 x, HasAdjointT (A x)] [IsSmoothNT 2 A]\n  : (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (D g x))\u2020\n    =\n    \u03bb g' => D\u2020 (\u03bb x \u27ff (A x)\u2020 (g' x))\n  := \nby\n  let G := \u03bb g : X'\u27ffY' => \u03bb x \u27ff A x (g x)\n  let h : (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (D g x)) = \u03bb g => G (D g) := by rfl\n  rw [h]\n  simp\n  done\n\n\ninstance elemwise_adjoint_alt2 {Y'} [Vec Y'] {Z} [Hilbert Z]\n  (A : X \u2192 Y \u2192 Y' \u2192 Z) [\u2200 x y', HasAdjointT (\u03bb y => A x y y')] [IsSmoothNT 3 A]\n  (g' : X \u2192 Y') [IsSmoothT g']\n  : HasAdjointT (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (g x) (g' x)) :=\nby \n  try infer_instance\n  apply elemwise_adjoint_alt1 (\u03bb x => x) (\u03bb x y => A x y (g' x))\n  done\n\n@[simp \u2193, diff]\ntheorem elemwise_adjoint_simp_alt2 {Y'} [Vec Y'] {Z} [Hilbert Z]\n  (A : X \u2192 Y \u2192 Y' \u2192 Z) [\u2200 x y', HasAdjointT (\u03bb y => A x y y')] [IsSmoothNT 3 A]\n  (g' : X \u2192 Y' := \u03bb _ => 0) [IsSmoothT g']\n  : (\u03bb (g : X\u27ffY) => \u03bb x \u27ff A x (g x) (g' x))\u2020\n    =\n    \u03bb h => \u03bb x \u27ff (\u03bb y => A x y (g' x))\u2020 (h x) :=\nby\n  rw[elemwise_adjoint_simp_alt1 (\u03bb x => x) (\u03bb x y => A x y (g' x))]\n  rw[id.arg_x.adj_simp]\n  done\n\n\n\nexample  : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff g x := by infer_instance\nexample  : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff (2:\u211d) * g x := by infer_instance\nexample  : HasAdjointT fun (g : \u211d\u27ff\u211d) => fun (x : \u211d) \u27ff x * g x := by infer_instance\n\nexample  (f : X\u27ffY) : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eag x, f x\u27eb := by infer_instance\nexample  (f : X\u27ffY) : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eaf x, g x\u27eb := by infer_instance\n\n\nexample  : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff g x + g x := \nby \n  try infer_instance\n  apply elemwise_adjoint (\u03bb _ y => y + y)\n  done\n\nexample  : HasAdjointT fun (g : \u211d\u27ffY) => fun x \u27ff g x + x * g x := \nby \n  try infer_instance\n  apply elemwise_adjoint (\u03bb x y => y + x * y)\n  done\n\ninstance : HasAdjoint (Smooth.differentialScalar : (\u211d\u27ffX) \u2192 (\u211d\u27ffX)) := sorry_proof\n\nexample  : HasAdjointT fun (g : \u211d\u27ffY) => \u2146 g := by infer_instance\nexample  : HasAdjointT fun (g : \u211d\u27ffY) => fun x \u27ff \u2146 g x := by infer_instance\n\n\nset_option synthInstance.maxSize 20000 in\nexample  : HasAdjointT fun (g : \u211d\u27ffY) => fun x \u27ff g x + \u2146 g x := \nby \n  have : HasAdjointNT 2 (\u03bb (g dg : \u211d \u27ff X) => \u03bb x \u27ff g x + dg x) := sorry_proof\n  apply scomb_highorder_adjoint (\u03bb g (dg : \u211d \u27ff X) => \u03bb x \u27ff g x + dg x) (\u03bb g => \u2146 g)\n  infer_instance\n\n\n-- set_option trace.Meta.synthPending true in\n-- example  (f : \u211d\u27ff\u211d) : HasAdjointT fun (g : \u211d\u27ff\u211d) => fun x \u27ff \u27eaf x, g x\u27eb := by infer_instance\n\n\nexample (D : (\u211d\u27ff\u211d) \u2192 (\u211d\u27ff\u211d)) [HasAdjointT D] : HasAdjointT fun (g : \u211d\u27ff\u211d) => fun x \u27ff D g x := by infer_instance\nexample (D : (\u211d\u27ff\u211d) \u2192 (\u211d\u27ff\u211d)) [HasAdjointT D] : HasAdjointT fun (g : \u211d\u27ff\u211d) => fun x \u27ff x * D g x := by infer_instance\n\n\nset_option synthInstance.maxSize 2000 in\nexample  (f : \u211d\u27ff\u211d) : HasAdjointT fun (g : \u211d\u27ff\u211d) => fun x \u27ff \u27ea\u2146 f x, \u2146 g x\u27eb := by (try infer_instance); sorry_proof\n\n\nexample  (f : X\u27ffY) : (fun (g : X\u27ffY) => fun x \u27ff \u27eag x, f x\u27eb)\u2020 = \u03bb h => \u03bb x \u27ff h x * f x := by simp; done\nexample  (f : X\u27ffY) : (fun (g : X\u27ffY) => fun x \u27ff \u27eaf x, g x\u27eb)\u2020 = \u03bb h => \u03bb x \u27ff h x * f x := by simp; done\n\nexample  (f : X\u27ffY) : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eaf x, g x\u27eb := by infer_instance\nexample  (f : X\u27ffY) : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eag x, f x\u27eb := by infer_instance\nexample  (f : X\u27ffY) (A : (X\u27ffY) \u2192 (X\u27ffY)) [HasAdjointT A] : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eaA g x, f x\u27eb := by (try infer_instance); admit\nexample  (f : X\u27ffY) (A : (X\u27ffY) \u2192 (X\u27ffY)) [HasAdjointT A] : HasAdjointT fun (g : X\u27ffY) => fun x \u27ff \u27eaf x, A g x\u27eb := by infer_instance\n\n\n-- @[simp \u2193, diff]\n-- theorem smooth_diff_to_normal_diff {X Y} [Vec X] [Vec Y] (f : X \u2192 Y) [IsSmoothT f]\n--   : \u2202 (\u03bb x \u27ff f x) = \u03bb x \u27ff \u03bb dx \u22b8 \u2202 f x dx := by simp[Smooth.differential]; done\n\n\n-- @[simp \u2193, diff]\n-- theorem smooth_sdif_to_normal_sdiff {X} [Vec X] (f : \u211d \u2192 X) [IsSmoothT f]\n--   : \u2146 (\u03bb x \u27ff f x) = \u03bb x \u27ff \u2146 f x := by simp[Smooth.differential]; done\n\n\n\n\n#check Nat\n\n\n\n\n\n-- set_option synthInstance.maxSize 2000 in\n-- example (f : \u211d\u27ff\u211d) : \u2207 (fun (g : \u211d\u27ff\u211d) => (\u222b x, \u27eaf x, \u2146 g x\u27eb))\n--                       = \n--                       (\u03bb g => - \u2146 f) := by simp[variationalGradient, tangentMap,Smooth.differential]; done\n  -- simp[differentialScalar,tangentMap,Smooth.differential,Smooth.differentialScalar]; done\n\n\n#check Nat\n\nexample (f : \u211d\u27ff\u211d) : IsSmoothNT 2 (fun (g : \u211d\u27ff\u211d) x => \u27eaf x, g x\u27eb) := by infer_instance\n\n-- example (f : \u211d\u27ff\u211d) : IsSmoothNT 2 (fun (g : \u211d\u27ff\u211d) x => \u27eaf x, \u2146 g x\u27eb) := by infer_instance\n\n\n\n-- def a : IsSmoothT (fun (g : \u211d\u27ff\u211d) => \u2146 g) := by infer_instance\n\n\n\n\n\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Core/IntegralProperties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190477, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3850862496225343}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nExtra definitions on option.\n-/\n\nnamespace option\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\nattribute [inline] option.is_some option.is_none\n\n/-- An elimination principle for `option`. It is a nondependent version of `option.rec_on`. -/\n@[simp] protected def elim : option \u03b1 \u2192 \u03b2 \u2192 (\u03b1 \u2192 \u03b2) \u2192 \u03b2\n| (some x) y f := f x\n| none     y f := y\n\ninstance has_mem : has_mem \u03b1 (option \u03b1) := \u27e8\u03bb a b, b = some a\u27e9\n\n@[simp] theorem mem_def {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = some a :=\niff.rfl\n\ntheorem is_none_iff_eq_none {o : option \u03b1} : o.is_none = tt \u2194 o = none :=\n\u27e8option.eq_none_of_is_none, \u03bb e, e.symm \u25b8 rfl\u27e9\n\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n\n/--\n`o = none` is decidable even if the wrapped type does not have decidable equality.\n\nThis is not an instance because it is not definitionally equal to `option.decidable_eq`.\nTry to use `o.is_none` or `o.is_some` instead.\n-/\n@[inline]\ndef decidable_eq_none {o : option \u03b1} : decidable (o = none) :=\ndecidable_of_decidable_of_iff (bool.decidable_eq _ _) is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2200 a \u2208 o, p a)\n| none     := is_true (by simp [false_implies_iff])\n| (some a) := if h : p a\n  then is_true $ \u03bb o e, some_inj.1 e \u25b8 h\n  else is_false $ mt (\u03bb H, H _ rfl) h\n\ninstance decidable_exists_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2203 a \u2208 o, p a)\n| none     := is_false (\u03bb \u27e8a, \u27e8h, _\u27e9\u27e9, by cases h)\n| (some a) := if h : p a\n  then is_true $ \u27e8_, rfl, h\u27e9\n  else is_false $ \u03bb \u27e8_, \u27e8rfl, hn\u27e9\u27e9, h hn\n\n/-- inhabited `get` function. Returns `a` if the input is `some a`,\n  otherwise returns `default`. -/\n@[reducible] def iget [inhabited \u03b1] : option \u03b1 \u2192 \u03b1\n| (some x) := x\n| none     := default \u03b1\n\n@[simp] theorem iget_some [inhabited \u03b1] {a : \u03b1} : (some a).iget = a := rfl\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [decidable_pred p] (a : \u03b1) : option \u03b1 :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a`\n  and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (o : option \u03b1) : option \u03b1 :=\no.bind (guard p)\n\ndef to_list : option \u03b1 \u2192 list \u03b1\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : \u03b1} {o : option \u03b1} : a \u2208 to_list o \u2194 a \u2208 o :=\nby cases o; simp [to_list, eq_comm]\n\ndef lift_or_get (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : option \u03b1 \u2192 option \u03b1 \u2192 option \u03b1\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_commutative \u03b1 f] :\n  is_commutative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b, by cases a; cases b; simp [lift_or_get, h.comm]\u27e9\n\ninstance lift_or_get_assoc (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_associative \u03b1 f] :\n  is_associative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]\u27e9\n\ninstance lift_or_get_idem (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_idempotent \u03b1 f] :\n  is_idempotent (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get, h.idempotent]\u27e9\n\ninstance lift_or_get_is_left_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_left_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninstance lift_or_get_is_right_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_right_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : option \u03b1 \u2192 option \u03b2 \u2192 Prop\n| some {a b} : r a b \u2192 rel (some a) (some b)\n| none       : rel none none\n\n/-- Partial bind. If for some `x : option \u03b1`, `f : \u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2` is a\n  partial function defined on `a : \u03b1` giving an `option \u03b2`, where `some a = x`,\n  then `pbind x f h` is essentially the same as `bind x f`\n  but is defined only when all `x = some a`, using the proof to apply `f`. -/\n@[simp] def pbind : \u03a0 (x : option \u03b1), (\u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2) \u2192 option \u03b2\n| none     _ := none\n| (some a) f := f a rfl\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on\n  `a : \u03b1` satisfying `p`, then `pmap f x h` is essentially the same as `map f x`\n  but is defined only when all members of `x` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 (a : \u03b1), p a \u2192 \u03b2) :\n  \u03a0 x : option \u03b1, (\u2200 a \u2208 x, p a) \u2192 option \u03b2\n| none     _ := none\n| (some a) H := some (f a (H a (mem_def.mpr rfl)))\n\n/--\nFlatten an `option` of `option`, a specialization of `mjoin`.\n-/\n@[simp] def join : option (option \u03b1) \u2192 option \u03b1 :=\n\u03bb x, bind x id\n\nprotected def {u v} traverse {F : Type u \u2192 Type v} [applicative F] {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 F \u03b2) :\n  option \u03b1 \u2192 F (option \u03b2)\n| none := pure none\n| (some x) := some <$> f x\n\n/- By analogy with `monad.sequence` in `init/category/combinators.lean`. -/\n\n/-- If you maybe have a monadic computation in a `[monad m]` which produces a term of type `\u03b1`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult. -/\ndef {u v} maybe {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type u} : option (m \u03b1) \u2192 m (option \u03b1)\n| none := return none\n| (some fn) := some <$> fn\n\n/-- Map a monadic function `f : \u03b1 \u2192 m \u03b2` over an `o : option \u03b1`, maybe producing a result. -/\ndef {u v w} mmap {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type w} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2)\n  (o : option \u03b1) : m (option \u03b2) := (o.map f).maybe\n\n/--\nA monadic analogue of `option.elim`.\n-/\ndef melim {\u03b1 \u03b2 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b2) (z : \u03b1 \u2192 m \u03b2) :\n  m \u03b2 :=\nx >>= \u03bb o, option.elim o y z\n\n/--\nA monadic analogue of `option.get_or_else`.\n-/\ndef mget_or_else {\u03b1 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b1) : m \u03b1 :=\nmelim x y pure\n\n\nend option\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/option/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6757645944891558, "lm_q1q2_score": 0.3850862459014581}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.concrete_category.basic\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nnamespace category_theory.limits\n\n\n-- We now prove a lemma about naturality of cones over functors into bundled categories.\n\nnamespace cone\n\n\n/-- Naturality of a cone over functors to a concrete category. -/\n@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] {F : J \u2964 C} (s : cone F) {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(X s)) : coe_fn (functor.map F f) (coe_fn (nat_trans.app (\u03c0 s) j) x) = coe_fn (nat_trans.app (\u03c0 s) j') x := sorry\n\n@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) (s : cone (F \u22d9 forget C)) {j : J} {j' : J} (f : j \u27f6 j') (x : X s) : coe_fn (functor.map F f) (nat_trans.app (\u03c0 s) j x) = nat_trans.app (\u03c0 s) j' x :=\n  congr_fun (w s f) x\n\nend cone\n\n\nnamespace cocone\n\n\n/-- Naturality of a cocone over functors into a concrete category. -/\n@[simp] theorem w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] {F : J \u2964 C} (s : cocone F) {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(functor.obj F j)) : coe_fn (nat_trans.app (\u03b9 s) j') (coe_fn (functor.map F f) x) = coe_fn (nat_trans.app (\u03b9 s) j) x := sorry\n\n@[simp] theorem w_forget_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) (s : cocone (F \u22d9 forget C)) {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(functor.obj F j)) : nat_trans.app (\u03b9 s) j' (coe_fn (functor.map F f) x) = nat_trans.app (\u03b9 s) j x :=\n  congr_fun (w s f) x\n\nend cocone\n\n\n@[simp] theorem limit.lift_\u03c0_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) [has_limit F] (s : cone F) (j : J) (x : \u21a5(cone.X s)) : coe_fn (limit.\u03c0 F j) (coe_fn (limit.lift F s) x) = coe_fn (nat_trans.app (cone.\u03c0 s) j) x := sorry\n\n@[simp] theorem limit.w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) [has_limit F] {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(limit F)) : coe_fn (functor.map F f) (coe_fn (limit.\u03c0 F j) x) = coe_fn (limit.\u03c0 F j') x := sorry\n\n@[simp] theorem colimit.\u03b9_desc_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) [has_colimit F] (s : cocone F) (j : J) (x : \u21a5(functor.obj F j)) : coe_fn (colimit.desc F s) (coe_fn (colimit.\u03b9 F j) x) = coe_fn (nat_trans.app (cocone.\u03b9 s) j) x := sorry\n\n@[simp] theorem colimit.w_apply {J : Type v} [small_category J] {C : Type u} [category C] [concrete_category C] (F : J \u2964 C) [has_colimit F] {j : J} {j' : J} (f : j \u27f6 j') (x : \u21a5(functor.obj F j)) : coe_fn (colimit.\u03b9 F j') (coe_fn (functor.map F f) x) = coe_fn (colimit.\u03b9 F j) x := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/concrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3850578914997206}}
{"text": "import topology.sheaves.functors\nimport topology.sheaves.sheaf_condition.sites\n\nnamespace Top.sheaf\n\nuniverses w v u\n\nopen category_theory category_theory.limits topological_space\n\nvariables (A : Type*) [category.{w} A] [concrete_category.{w} A] [has_colimits A] [has_limits A]\nvariables [preserves_limits (category_theory.forget A)]\nvariables [preserves_filtered_colimits (category_theory.forget A)]\nvariables [reflects_isomorphisms (category_theory.forget A)]\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y Z : Top.{w}} (f : X \u27f6 Y)\nvariables \u2983\u03b9 : Type w\u2984 {U : \u03b9 \u2192 opens Y}\n\nnoncomputable theory\n\n\nlemma pushforward_forget (f : X \u27f6 Y) :\n  pushforward f \u22d9 forget C Y = forget C X \u22d9 Top.presheaf.pushforward C f := rfl\n\n/--\nPushforward of sheaves is isomorphic (actually definitionally equal) to pushforward of presheaves.\n-/\ndef pushforward_forget_iso (f : X \u27f6 Y) :\n  pushforward f \u22d9 forget C Y \u2245 forget C X \u22d9 Top.presheaf.pushforward C f := iso.refl _\n\nvariables {C}\n\n@[simp] lemma pushforward_obj_val (f : X \u27f6 Y) (F : X.sheaf C) :\n  ((pushforward f).obj F).1 = f _* F.1 := rfl\n\n@[simp] lemma pushforward_map (f : X \u27f6 Y) {F F' : X.sheaf C} (\u03b1 : F \u27f6 F') :\n  ((pushforward f).map \u03b1).1 = (Top.presheaf.pushforward C f).map \u03b1.1 := rfl\n\n/--\nThe pushforward functor.\n-/\ndef pullback (f : X \u27f6 Y) : Y.sheaf A \u2964 X.sheaf A :=\nsites.pushforward A _ _ (opens.map f)\n\nlemma pullback_eq (f : X \u27f6 Y) :\n  pullback A f = forget A Y \u22d9 Top.presheaf.pullback A f \u22d9 presheaf_to_Sheaf _ _ := rfl\n\n/--\nThe pullback of a sheaf is isomorphic (actually definitionally equal) to the sheafification\nof the pullback as a presheaf.\n-/\ndef pullback_iso (f : X \u27f6 Y) :\n  pullback A f \u2245 forget A Y \u22d9 Top.presheaf.pullback A f \u22d9 presheaf_to_Sheaf _ _ := iso.refl _\n\ninstance : representably_flat (opens.map f) :=\nbegin\n  constructor,\n  intro U,\n  apply_with is_cofiltered.mk { instances := ff },\n  constructor,\n  { intros V W,\n    refine \u27e8\u27e8\u27e8punit.star\u27e9, V.right \u2293 W.right, hom_of_le $ le_inf V.hom.le W.hom.le\u27e9,\n      { right := hom_of_le inf_le_left }, { right := hom_of_le inf_le_right }, trivial\u27e9 },\n  { intros U V i j, refine \u27e8_, \ud835\udfd9 _, by ext; congr\u27e9 },\n  { exact \u27e8structured_arrow.mk $ show U \u27f6 (opens.map f).obj \u22a4, from hom_of_le le_top\u27e9 },\nend\n\nlemma compatible_preserving_opens_map :\n  compatible_preserving (opens.grothendieck_topology X) (opens.map f) :=\ncompatible_preserving_of_flat _ _\n\nlemma cover_preserving_opens_map :\n  cover_preserving (opens.grothendieck_topology Y)\n    (opens.grothendieck_topology X) (opens.map f) :=\nbegin\n  constructor,\n  intros U S hS x hx,\n  obtain \u27e8V, i, hi, hxV\u27e9 := hS (f x) hx,\n  exact \u27e8_, (opens.map f).map i, \u27e8_, _, \ud835\udfd9 _, hi, subsingleton.elim _ _\u27e9, hxV\u27e9\nend\n\n/-- The adjunction between pullback and pushforward for sheaves on topological spaces. -/\ndef pullback_pushforward_adjunction (f : X \u27f6 Y) :\n  pullback A f \u22a3 pushforward f :=\nsites.pullback_pushforward_adjunction _ _ _ (compatible_preserving_opens_map f)\n  (cover_preserving_opens_map f)\n\ninstance : is_left_adjoint (pullback A f) := \u27e8_, pullback_pushforward_adjunction A f\u27e9\ninstance : is_right_adjoint (pushforward f : X.sheaf A \u2964 Y.sheaf A) :=\n\u27e8_, pullback_pushforward_adjunction A f\u27e9\n\nvariables (g : Y \u27f6 Z) \n\nnoncomputable!\ndef pushforward_comp :\n  pushforward f \u22d9 pushforward g \u2245 (pushforward (f \u226b g) : X.sheaf C \u2964 Z.sheaf C) :=\niso.refl _\n\nnoncomputable!\ndef pullback_comp :\n  pullback A g \u22d9 pullback A f \u2245 pullback A (f \u226b g) :=\nadjunction.nat_iso_of_right_adjoint_nat_iso\n  ((pullback_pushforward_adjunction A g).comp (pullback_pushforward_adjunction A f))\n  (pullback_pushforward_adjunction A (f \u226b g)) (pushforward_comp f g)\n\ndef pullback_congr {f g : X \u27f6 Y} (e : f = g) :\n  pullback A f \u2245 pullback A g :=\neq_to_iso (by subst e)\n\nend Top.sheaf", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/algebraic_geometry/sheaf_pullback.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3850578914997206}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.homology.homotopy\nimport category_theory.quotient\n\n/-!\n# The homotopy category\n\n`homotopy_category V c` gives the category of chain complexes of shape `c` in `V`,\nwith chain maps identified when they are homotopic.\n-/\n\nuniverses v u\n\nopen_locale classical\nnoncomputable theory\n\nopen category_theory category_theory.limits homological_complex\n\nvariables {\u03b9 : Type*}\nvariables (V : Type u) [category.{v} V] [preadditive V]\nvariables (c : complex_shape \u03b9)\n\n/--\nThe congruence on `homological_complex V c` given by the existence of a homotopy.\n-/\ndef homotopic : hom_rel (homological_complex V c) := \u03bb C D f g, nonempty (homotopy f g)\n\ninstance homotopy_congruence : congruence (homotopic V c) :=\n{ is_equiv := \u03bb C D,\n  { refl := \u03bb C, \u27e8homotopy.refl C\u27e9,\n    symm := \u03bb f g \u27e8w\u27e9, \u27e8w.symm\u27e9,\n    trans := \u03bb f g h \u27e8w\u2081\u27e9 \u27e8w\u2082\u27e9, \u27e8w\u2081.trans w\u2082\u27e9, },\n  comp_left := \u03bb E F G m\u2081 m\u2082 g \u27e8i\u27e9, \u27e8i.comp_left _\u27e9,\n  comp_right := \u03bb E F G f m\u2081 m\u2082 \u27e8i\u27e9, \u27e8i.comp_right _\u27e9, }\n\n/-- `homotopy_category V c` is the category of chain complexes of shape `c` in `V`,\nwith chain maps identified when they are homotopic. -/\n@[derive category]\ndef homotopy_category := category_theory.quotient (homotopic V c)\n\n-- TODO the homotopy_category is preadditive\n\nnamespace homotopy_category\n\n/-- The quotient functor from complexes to the homotopy category. -/\ndef quotient : homological_complex V c \u2964 homotopy_category V c :=\ncategory_theory.quotient.functor _\n\nopen_locale zero_object\n\n-- TODO upgrade this to `has_zero_object`, presumably for any `quotient`.\ninstance [has_zero_object V] : inhabited (homotopy_category V c) := \u27e8(quotient V c).obj 0\u27e9\n\nvariables {V c}\n\n@[simp] lemma quotient_obj_as (C : homological_complex V c) :\n  ((quotient V c).obj C).as = C := rfl\n\n@[simp] lemma quotient_map_out {C D : homotopy_category V c} (f : C \u27f6 D) :\n  (quotient V c).map f.out = f :=\nquot.out_eq _\n\nlemma eq_of_homotopy {C D : homological_complex V c} (f g : C \u27f6 D) (h : homotopy f g) :\n  (quotient V c).map f = (quotient V c).map g :=\ncategory_theory.quotient.sound _ \u27e8h\u27e9\n\n/-- If two chain maps become equal in the homotopy category, then they are homotopic. -/\ndef homotopy_of_eq {C D : homological_complex V c} (f g : C \u27f6 D)\n  (w : (quotient V c).map f = (quotient V c).map g) : homotopy f g :=\n((quotient.functor_map_eq_iff _ _ _).mp w).some\n\n/--\nAn arbitrarily chosen representation of the image of a chain map in the homotopy category\nis homotopic to the original chain map.\n-/\ndef homotopy_out_map {C D : homological_complex V c} (f : C \u27f6 D) :\n  homotopy ((quotient V c).map f).out f :=\nbegin\n  apply homotopy_of_eq,\n  simp,\nend\n\n@[simp] lemma quotient_map_out_comp_out {C D E : homotopy_category V c} (f : C \u27f6 D) (g : D \u27f6 E) :\n  (quotient V c).map (quot.out f \u226b quot.out g) = f \u226b g :=\nby conv_rhs { erw [\u2190quotient_map_out f, \u2190quotient_map_out g, \u2190(quotient V c).map_comp], }\n\n/-- Homotopy equivalent complexes become isomorphic in the homotopy category. -/\n@[simps]\ndef iso_of_homotopy_equiv {C D : homological_complex V c} (f : homotopy_equiv C D) :\n  (quotient V c).obj C \u2245 (quotient V c).obj D :=\n{ hom := (quotient V c).map f.hom,\n  inv := (quotient V c).map f.inv,\n  hom_inv_id' := begin\n    rw [\u2190(quotient V c).map_comp, \u2190(quotient V c).map_id],\n    exact eq_of_homotopy _ _ f.homotopy_hom_inv_id,\n  end,\n  inv_hom_id' := begin\n    rw [\u2190(quotient V c).map_comp, \u2190(quotient V c).map_id],\n    exact eq_of_homotopy _ _ f.homotopy_inv_hom_id,\n  end }\n\n/-- If two complexes become isomorphic in the homotopy category,\n  then they were homotopy equivalent. -/\ndef homotopy_equiv_of_iso\n  {C D : homological_complex V c} (i : (quotient V c).obj C \u2245 (quotient V c).obj D) :\n  homotopy_equiv C D :=\n{ hom := quot.out i.hom,\n  inv := quot.out i.inv,\n  homotopy_hom_inv_id := homotopy_of_eq _ _ (by { simp, refl, }),\n  homotopy_inv_hom_id := homotopy_of_eq _ _ (by { simp, refl, }), }\n\nvariables (V c) [has_equalizers V] [has_images V] [has_image_maps V]\n  [has_cokernels V]\n\n/-- The `i`-th homology, as a functor from the homotopy category. -/\ndef homology_functor (i : \u03b9) : homotopy_category V c \u2964 V :=\ncategory_theory.quotient.lift _ (homology_functor V c i)\n  (\u03bb C D f g \u27e8h\u27e9, homology_map_eq_of_homotopy h i)\n\n/-- The homology functor on the homotopy category is just the usual homology functor. -/\ndef homology_factors (i : \u03b9) :\n  quotient V c \u22d9 homology_functor V c i \u2245 _root_.homology_functor V c i :=\ncategory_theory.quotient.lift.is_lift _ _ _\n\n@[simp] lemma homology_factors_hom_app (i : \u03b9) (C : homological_complex V c) :\n  (homology_factors V c i).hom.app C = \ud835\udfd9 _ :=\nrfl\n\n@[simp] lemma homology_factors_inv_app (i : \u03b9) (C : homological_complex V c) :\n  (homology_factors V c i).inv.app C = \ud835\udfd9 _ :=\nrfl\n\nlemma homology_functor_map_factors (i : \u03b9) {C D : homological_complex V c} (f : C \u27f6 D) :\n  (_root_.homology_functor V c i).map f =\n    ((homology_functor V c i).map ((quotient V c).map f) : _) :=\n(category_theory.quotient.lift_map_functor_map _ (_root_.homology_functor V c i) _ f).symm\n\nend homotopy_category\n\nnamespace category_theory\n\nvariables {V} {W : Type*} [category W] [preadditive W]\n\n/-- An additive functor induces a functor between homotopy categories. -/\n@[simps]\ndef functor.map_homotopy_category (c : complex_shape \u03b9) (F : V \u2964 W) [F.additive] :\n  homotopy_category V c \u2964 homotopy_category W c :=\n{ obj := \u03bb C, (homotopy_category.quotient W c).obj ((F.map_homological_complex c).obj C.as),\n  map := \u03bb C D f,\n    (homotopy_category.quotient W c).map ((F.map_homological_complex c).map (quot.out f)),\n  map_id' := \u03bb C, begin\n    rw \u2190(homotopy_category.quotient W c).map_id,\n    apply homotopy_category.eq_of_homotopy,\n    rw \u2190(F.map_homological_complex c).map_id,\n    apply F.map_homotopy,\n    apply homotopy_category.homotopy_of_eq,\n    exact quot.out_eq _,\n  end,\n  map_comp' := \u03bb C D E f g, begin\n    rw \u2190(homotopy_category.quotient W c).map_comp,\n    apply homotopy_category.eq_of_homotopy,\n    rw \u2190(F.map_homological_complex c).map_comp,\n    apply F.map_homotopy,\n    apply homotopy_category.homotopy_of_eq,\n    convert quot.out_eq _,\n    exact homotopy_category.quotient_map_out_comp_out _ _,\n  end }.\n\n-- TODO `F.map_homotopy_category c` is additive (and linear when `F` is linear).\n\n/-- A natural transformation induces a natural transformation between\n  the induced functors on the homotopy category. -/\n@[simps]\ndef nat_trans.map_homotopy_category {F G : V \u2964 W} [F.additive] [G.additive]\n  (\u03b1 : F \u27f6 G) (c : complex_shape \u03b9) : F.map_homotopy_category c \u27f6 G.map_homotopy_category c :=\n{ app := \u03bb C,\n    (homotopy_category.quotient W c).map ((nat_trans.map_homological_complex \u03b1 c).app C.as),\n  naturality' := \u03bb C D f,\n  begin\n    dsimp,\n    simp only [\u2190functor.map_comp],\n    congr' 1,\n    ext,\n    dsimp,\n    simp,\n  end }\n\n@[simp] lemma nat_trans.map_homotopy_category_id (c : complex_shape \u03b9) (F : V \u2964 W) [F.additive] :\n  nat_trans.map_homotopy_category (\ud835\udfd9 F) c = \ud835\udfd9 (F.map_homotopy_category c) :=\nby tidy\n\n@[simp] lemma nat_trans.map_homotopy_category_comp (c : complex_shape \u03b9)\n  {F G H : V \u2964 W} [F.additive] [G.additive] [H.additive]\n  (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H):\n  nat_trans.map_homotopy_category (\u03b1 \u226b \u03b2) c =\n    nat_trans.map_homotopy_category \u03b1 c \u226b nat_trans.map_homotopy_category \u03b2 c :=\nby tidy\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/homology/homotopy_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.3850328379571702}}
{"text": "/-\nCopyright (c) 2021 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n\n! This file was ported from Lean 3 source module combinatorics.quiver.arborescence\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.WellFounded\nimport Mathbin.Data.Nat.Basic\nimport Mathbin.Combinatorics.Quiver.Subquiver\nimport Mathbin.Combinatorics.Quiver.Path\n\n/-!\n# Arborescences\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA quiver `V` is an arborescence (or directed rooted tree) when we have a root vertex `root : V` such\nthat for every `b : V` there is a unique path from `root` to `b`.\n\n## Main definitions\n\n- `quiver.arborescence V`: a typeclass asserting that `V` is an arborescence\n- `arborescence_mk`: a convenient way of proving that a quiver is an arborescence\n- `rooted_connected r`: a typeclass asserting that there is at least one path from `r` to `b` for\nevery `b`.\n- `geodesic_subtree r`: given `[rooted_conntected r]`, this is a subquiver of `V` which contains\njust enough edges to include a shortest path from `r` to `b` for every `b`.\n- `geodesic_arborescence : arborescence (geodesic_subtree r)`: an instance saying that the geodesic\nsubtree is an arborescence. This proves the directed analogue of 'every connected graph has a\nspanning tree'. This proof avoids the use of Zorn's lemma.\n-/\n\n\nopen Opposite\n\nuniverse v u\n\nnamespace Quiver\n\n#print Quiver.Arborescence /-\n/-- A quiver is an arborescence when there is a unique path from the default vertex\n    to every other vertex. -/\nclass Arborescence (V : Type u) [Quiver.{v} V] : Type max u v where\n  root : V\n  uniquePath : \u2200 b : V, Unique (Path root b)\n#align quiver.arborescence Quiver.Arborescence\n-/\n\n#print Quiver.root /-\n/-- The root of an arborescence. -/\ndef root (V : Type u) [Quiver V] [Arborescence V] : V :=\n  Arborescence.root\n#align quiver.root Quiver.root\n-/\n\ninstance {V : Type u} [Quiver V] [Arborescence V] (b : V) : Unique (Path (root V) b) :=\n  Arborescence.uniquePath b\n\n#print Quiver.arborescenceMk /-\n/-- To show that `[quiver V]` is an arborescence with root `r : V`, it suffices to\n  - provide a height function `V \u2192 \u2115` such that every arrow goes from a\n    lower vertex to a higher vertex,\n  - show that every vertex has at most one arrow to it, and\n  - show that every vertex other than `r` has an arrow to it. -/\nnoncomputable def arborescenceMk {V : Type u} [Quiver V] (r : V) (height : V \u2192 \u2115)\n    (height_lt : \u2200 \u2983a b\u2984, (a \u27f6 b) \u2192 height a < height b)\n    (unique_arrow : \u2200 \u2983a b c : V\u2984 (e : a \u27f6 c) (f : b \u27f6 c), a = b \u2227 HEq e f)\n    (root_or_arrow : \u2200 b, b = r \u2228 \u2203 a, Nonempty (a \u27f6 b)) : Arborescence V\n    where\n  root := r\n  uniquePath b :=\n    \u27e8Classical.inhabited_of_nonempty\n        (by\n          rcases show \u2203 n, height b < n from \u27e8_, Nat.lt.base _\u27e9 with \u27e8n, hn\u27e9\n          induction' n with n ih generalizing b\n          \u00b7 exact False.elim (Nat.not_lt_zero _ hn)\n          rcases root_or_arrow b with (\u27e8\u27e8\u27e9\u27e9 | \u27e8a, \u27e8e\u27e9\u27e9)\n          \u00b7 exact \u27e8path.nil\u27e9\n          \u00b7 rcases ih a (lt_of_lt_of_le (height_lt e) (nat.lt_succ_iff.mp hn)) with \u27e8p\u27e9\n            exact \u27e8p.cons e\u27e9),\n      by\n      have height_le : \u2200 {a b}, Path a b \u2192 height a \u2264 height b :=\n        by\n        intro a b p\n        induction' p with b c p e ih\n        rfl\n        exact le_of_lt (lt_of_le_of_lt ih (height_lt e))\n      suffices \u2200 p q : Path r b, p = q by\n        intro p\n        apply this\n      intro p q\n      induction' p with a c p e ih <;> cases' q with b _ q f\n      \u00b7 rfl\n      \u00b7 exact False.elim (lt_irrefl _ (lt_of_le_of_lt (height_le q) (height_lt f)))\n      \u00b7 exact False.elim (lt_irrefl _ (lt_of_le_of_lt (height_le p) (height_lt e)))\n      \u00b7 rcases unique_arrow e f with \u27e8\u27e8\u27e9, \u27e8\u27e9\u27e9\n        rw [ih]\u27e9\n#align quiver.arborescence_mk Quiver.arborescenceMk\n-/\n\n#print Quiver.RootedConnected /-\n/-- `rooted_connected r` means that there is a path from `r` to any other vertex. -/\nclass RootedConnected {V : Type u} [Quiver V] (r : V) : Prop where\n  nonempty_path : \u2200 b : V, Nonempty (Path r b)\n#align quiver.rooted_connected Quiver.RootedConnected\n-/\n\nattribute [instance] rooted_connected.nonempty_path\n\nsection GeodesicSubtree\n\nvariable {V : Type u} [Quiver.{v + 1} V] (r : V) [RootedConnected r]\n\n#print Quiver.shortestPath /-\n/-- A path from `r` of minimal length. -/\nnoncomputable def shortestPath (b : V) : Path r b :=\n  WellFounded.min (measure_wf Path.length) Set.univ Set.univ_nonempty\n#align quiver.shortest_path Quiver.shortestPath\n-/\n\n#print Quiver.shortest_path_spec /-\n/-- The length of a path is at least the length of the shortest path -/\ntheorem shortest_path_spec {a : V} (p : Path r a) : (shortestPath r a).length \u2264 p.length :=\n  not_lt.mp (WellFounded.not_lt_min (measure_wf _) Set.univ _ trivial)\n#align quiver.shortest_path_spec Quiver.shortest_path_spec\n-/\n\n#print Quiver.geodesicSubtree /-\n/-- A subquiver which by construction is an arborescence. -/\ndef geodesicSubtree : WideSubquiver V := fun a b =>\n  { e | \u2203 p : Path r a, shortestPath r b = p.cons e }\n#align quiver.geodesic_subtree Quiver.geodesicSubtree\n-/\n\n#print Quiver.geodesicArborescence /-\nnoncomputable instance geodesicArborescence : Arborescence (geodesicSubtree r) :=\n  arborescenceMk r (fun a => (shortestPath r a).length)\n    (by\n      rintro a b \u27e8e, p, h\u27e9\n      rw [h, path.length_cons, Nat.lt_succ_iff]\n      apply shortest_path_spec)\n    (by\n      rintro a b c \u27e8e, p, h\u27e9 \u27e8f, q, j\u27e9\n      cases h.symm.trans j\n      constructor <;> rfl)\n    (by\n      intro b\n      rcases hp : shortest_path r b with (_ | \u27e8p, e\u27e9)\n      \u00b7 exact Or.inl rfl\n      \u00b7 exact Or.inr \u27e8_, \u27e8\u27e8e, p, hp\u27e9\u27e9\u27e9)\n#align quiver.geodesic_arborescence Quiver.geodesicArborescence\n-/\n\nend GeodesicSubtree\n\nend Quiver\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Combinatorics/Quiver/Arborescence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070808, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3850328337088809}}
{"text": "import Mathlib.Data.Fintype.Card\nimport Mathlib.Data.Fintype.Sum\nimport Mathlib.Data.Fintype.Sigma\nimport Mathlib.Data.Fintype.BigOperators\nimport Mathlib.Tactic.Zify\nimport Mathlib.Tactic.Ring\nimport SSA.Bits.Defs\n\nopen Sum\n\nvariable {\u03b1 \u03b2 \u03b1' \u03b2' : Type} {\u03b3 : \u03b2 \u2192 Type}\n\ndef propagateAux (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool) \u00d7 Bool\n  | 0 => next_bit init_carry (fun i => x i 0)\n  | n+1 => next_bit (propagateAux init_carry next_bit x n).1 (fun i => x i (n+1))\n\ndef propagate (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) : Bool :=\n  (propagateAux init_carry next_bit x i).2\n\n@[simp] def propagateCarry (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool)\n  | 0 => next_bit init_carry (fun i => x i 0)\n  | n+1 => next_bit (propagateCarry init_carry next_bit x n) (fun i => x i (n+1))\n\n@[simp] def propagateCarry2 (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2115 \u2192 (\u03b1 \u2192 Bool)\n  | 0 => init_carry\n  | n+1 => next_bit (propagateCarry2 init_carry next_bit x n) (fun i => x i n)\n\nlemma propagateCarry2_succ (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2200 (n : \u2115),\n    propagateCarry2 init_carry next_bit x (n+1) =\n    propagateCarry init_carry next_bit x n\n  | 0 => rfl\n  | n+1 => by rw [propagateCarry2, propagateCarry2_succ _ _ _ n, propagateCarry]\n\n@[simp] lemma propagateAux_fst_eq_carry (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) : \u2200 n : \u2115,\n    (propagateAux init_carry next_bit x n).1 =\n    propagateCarry init_carry (fun c b => (next_bit c b).1) x n\n  | 0 => rfl\n  | n+1 => by rw [propagateAux, propagateCarry, propagateAux_fst_eq_carry _ _ _ n]\n\n@[simp] lemma propagate_zero (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n    (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    propagate init_carry next_bit x 0 = (next_bit init_carry (fun i => x i 0)).2 :=\n  rfl\n\nlemma propagate_succ (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) :\n    propagate init_carry next_bit x (i+1) = (next_bit\n      (propagateCarry init_carry (fun c b => (next_bit c b).1) x i)\n      (\u03bb j => x j (i+1))).2 :=\n  by rw [\u2190 propagateAux_fst_eq_carry]; rfl\n\nlemma propagate_succ2 (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2 \u2192 \u2115 \u2192 Bool) (i : \u2115) :\n    propagate init_carry next_bit x (i+1) = (next_bit\n      (propagateCarry2 init_carry (\u03bb c b => (next_bit c b).1) x (i+1))\n      (\u03bb j => x j (i+1))).2 :=\n  by rw [propagateCarry2_succ, \u2190 propagateAux_fst_eq_carry]; rfl\n\nlemma propagateCarry_propagate {\u03b4 : \u03b2 \u2192 Type} {\u03b2' : Type}\n      (f : \u2200 a, \u03b4 a \u2192 \u03b2') : \u2200 (n : \u2115) (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (init_carry_x : \u2200 a, \u03b3 a \u2192 Bool)\n    (next_bit_x : \u2200 a (_carry : \u03b3 a \u2192 Bool) (_bits : \u03b4 a \u2192 Bool),\n      (\u03b3 a \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool),\n    propagateCarry init_carry next_bit (\u03bb a => propagate (init_carry_x a)\n      (next_bit_x a) (\u03bb d => x (f a d))) n =\n    propagateCarry\n      (\u03bb a : \u03b1 \u2295 (\u03a3 a, \u03b3 a) => Sum.elim init_carry (\u03bb b : \u03a3 a, \u03b3 a =>\n        init_carry_x b.1 b.2) a)\n      (\u03bb (carry : (\u03b1 \u2295 (\u03a3 a, \u03b3 a)) \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n    -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n        let f : \u2200 (a : \u03b2), (\u03b3 a \u2192 Bool) \u00d7 Bool := \u03bb a => next_bit_x a\n          (\u03bb d => carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (f a d))\n        let g : (\u03b1 \u2192 Bool) := (next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n        Sum.elim g (\u03bb x => (f x.1).1 x.2))\n      x n \u2218 inl\n  | 0, init_carry, next_bit, init_carry_x, next_bit_x, x => rfl\n  | n+1, init_carry, next_bit, init_carry_x, next_bit_x, x => by\n    have := propagateCarry_propagate f n\n    simp only [propagateCarry, propagate_succ, elim_inl, Nat.add] at *\n    conv_lhs => simp only [this]\n    clear this\n    dsimp\n    congr\n    ext a\n    dsimp\n    congr\n    ext b\n    dsimp [propagateCarry, propagate_succ, elim_inl, Nat.add]\n    congr\n    dsimp\n    induction' n with n ih\n    . simp\n    . simp [ih]\n\nlemma propagate_propagate {\u03b4 : \u03b2 \u2192 Type} {\u03b2' : Type}\n      (f : \u2200 a, \u03b4 a \u2192 \u03b2') : \u2200 (n : \u2115) (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (init_carry_x : \u2200 a, \u03b3 a \u2192 Bool)\n    (next_bit_x : \u2200 a (_carry : \u03b3 a \u2192 Bool) (_bits : \u03b4 a \u2192 Bool),\n      (\u03b3 a \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool),\n    propagate init_carry next_bit (\u03bb a => propagate (init_carry_x a)\n      (next_bit_x a) (\u03bb d => x (f a d))) n =\n    propagate\n      (\u03bb a : \u03b1 \u2295 (\u03a3 a, \u03b3 a) => Sum.elim init_carry (\u03bb b : \u03a3 a, \u03b3 a =>\n        init_carry_x b.1 b.2) a)\n      (\u03bb (carry : (\u03b1 \u2295 (\u03a3 a, \u03b3 a)) \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n        -- first compute (propagate (init_carry_x a) (next_bit_x a) (x a) n)\n        let f : \u2200 (a : \u03b2), (\u03b3 a \u2192 Bool) \u00d7 Bool := \u03bb a => next_bit_x a (\u03bb d =>\n          carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (f a d))\n        let g : (\u03b1 \u2192 Bool) \u00d7 Bool := (next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n        (Sum.elim g.1 (\u03bb x => (f x.1).1 x.2), g.2)\n      )\n    x n\n  | 0, init_carry, next_bit, init_carry_x, next_bit_x, x => rfl\n  | n+1, init_carry, next_bit, init_carry_x, next_bit_x, x => by\n    simp only [propagate_succ]\n    rw [propagateCarry_propagate]\n    congr\n    ext\n    congr\n    induction' n with n ih\n    . simp\n    . simp [ih]\n\nlemma propagateCarry_changeVars {\u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool))\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool) (i : \u2115)\n    (changeVars : \u03b2 \u2192 \u03b2') :\n    propagateCarry init_carry next_bit (\u03bb b => x (changeVars b)) i =\n    propagateCarry init_carry (\u03bb (carry : \u03b1 \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n      next_bit carry (\u03bb b => bits (changeVars b))) x i := by\n  induction i\n  . simp\n  . simp [*]\n\nlemma propagate_changeVars {\u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (x : \u03b2' \u2192 \u2115 \u2192 Bool) (i : \u2115)\n    (changeVars : \u03b2 \u2192 \u03b2') :\n    propagate init_carry next_bit (\u03bb b => x (changeVars b)) i =\n    propagate init_carry (\u03bb (carry : \u03b1 \u2192 Bool) (bits : \u03b2' \u2192 Bool) =>\n      next_bit carry (\u03bb b => bits (changeVars b))) x i := by\n  induction' i with i ih\n  . rfl\n  . simp only [propagate_succ, propagateCarry_changeVars, ih]\n\nopen Term\n\n@[simp] def arity : Term \u2192 \u2115\n| (var n) => n+1\n| zero => 0\n| one => 0\n| negOne => 0\n| Term.and t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.or t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.xor t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| Term.not t => arity t\n| ls t => arity t\n| add t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| sub t\u2081 t\u2082 => max (arity t\u2081) (arity t\u2082)\n| neg t => arity t\n| incr t => arity t\n| decr t => arity t\n\n@[simp] def Term.evalFin : \u2200 (t : Term) (_vars : Fin (arity t) \u2192 \u2115 \u2192 Bool), \u2115 \u2192 Bool\n| var n, vars => vars (Fin.last n)\n| zero, _vars => zeroSeq\n| one, _vars => oneSeq\n| negOne, _vars => negOneSeq\n| Term.and t\u2081 t\u2082, vars =>\n  andSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| Term.or t\u2081 t\u2082, vars =>\n  orSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| Term.xor t\u2081 t\u2082, vars =>\n  xorSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| not t, vars => notSeq (Term.evalFin t vars)\n| ls t, vars => lsSeq (Term.evalFin t vars)\n| add t\u2081 t\u2082, vars =>\n  addSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| sub t\u2081 t\u2082, vars =>\n  subSeq (Term.evalFin t\u2081\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n  (Term.evalFin t\u2082\n    (fun i => vars (Fin.castLe (by simp [arity]) i)))\n| neg t, vars => negSeq (Term.evalFin t vars)\n| incr t, vars => incrSeq (Term.evalFin t vars)\n| decr t, vars => decrSeq (Term.evalFin t vars)\n\nlemma evalFin_eq_eval (t : Term) (vars : \u2115 \u2192 \u2115 \u2192 Bool) :\n    Term.evalFin t (fun i => vars i) = Term.eval t vars := by\n  induction t <;>\n  dsimp [Term.evalFin, Term.eval, arity] at * <;> simp [*]\n\n\nlemma id_eq_propagate (x : \u2115 \u2192 Bool) :\n    x = propagate Empty.elim (\u03bb _ (y : Unit \u2192 Bool) => (Empty.elim, y ())) (\u03bb _ => x) := by\n  ext n; cases n <;> rfl\n\nlemma zero_eq_propagate :\n    zeroSeq = propagate Empty.elim (\u03bb (_ _ : Empty \u2192 Bool) => (Empty.elim, false)) Empty.elim := by\n  ext n; cases n <;> rfl\n\nlemma one_eq_propagate :\n    oneSeq = propagate (\u03bb _ : Unit => true)\n      (\u03bb f (_ : Empty \u2192 Bool) => (\u03bb _ => false, f ())) Empty.elim := by\n  ext n\n  match n with\n  | 0 => rfl\n  | 1 => rfl\n  | n+2 => simp [oneSeq, propagate_succ]\n\nlemma and_eq_propagate (x y : \u2115 \u2192 Bool) :\n    andSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, y true && y false)) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, andSeq]\n\nlemma or_eq_propagate (x y : \u2115 \u2192 Bool) :\n    orSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, y true || y false)) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, orSeq]\n\nlemma xor_eq_propagate (x y : \u2115 \u2192 Bool) :\n    xorSeq x y = propagate Empty.elim\n      (\u03bb _ (y : Bool \u2192 Bool) => (Empty.elim, xor (y true) (y false))) (\u03bb b => cond b x y) := by\n  ext n; cases n <;> simp [propagate, propagateAux, xorSeq]\n\nlemma not_eq_propagate (x : \u2115 \u2192 Bool) :\n    notSeq x = propagate Empty.elim (\u03bb _ (y : Unit \u2192 Bool) => (Empty.elim, !(y ()))) (\u03bb _ => x) := by\n  ext n; cases n <;> simp [propagate, propagateAux, notSeq]\n\nlemma ls_eq_propagate (x : \u2115 \u2192 Bool) :\n    lsSeq x = propagate (\u03bb _ : Unit => false)\n      (\u03bb (carry x : Unit \u2192 Bool) => (x, carry ())) (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => rfl\n  | 1 => rfl\n  | n+2 => simp [lsSeq, propagate_succ]\n\nlemma addSeqAux_eq_propagateCarry (x y : \u2115 \u2192 Bool) (n : \u2115) :\n    (addSeqAux x y n).2 = propagateCarry (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        \u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()))\n    (\u03bb b => cond b x y) n () := by\n  induction n <;> simp [addSeqAux, *]\n\nlemma add_eq_propagate (x y : \u2115 \u2192 Bool) :\n    addSeq x y = propagate (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))))\n    (\u03bb b => cond b x y) := by\n  ext n\n  match n with\n  | 0 => simp [addSeq, addSeqAux]\n  | 1 => simp [addSeq, addSeqAux, propagate, propagateAux]\n  | n+2 => simp [addSeq, addSeqAux, addSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma subSeqAux_eq_propagateCarry (x y : \u2115 \u2192 Bool) (n : \u2115) :\n    (subSeqAux x y n).2 = propagateCarry (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        \u03bb _ => (!(bits true) && (bits false)) ||\n          (!(_root_.xor (bits true) (bits false))) && carry ())\n    (\u03bb b => cond b x y) n () := by\n  induction n <;> simp [subSeqAux, *]\n\nlemma sub_eq_propagate (x y : \u2115 \u2192 Bool) :\n    subSeq x y = propagate (\u03bb _ => false)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (!(bits true) && (bits false)) ||\n          ((!(_root_.xor (bits true) (bits false))) && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))))\n    (\u03bb b => cond b x y) := by\n  ext n\n  match n with\n  | 0 => simp [subSeq, subSeqAux]\n  | 1 => simp [subSeq, subSeqAux, propagate, propagateAux]\n  | n+2 => simp [subSeq, subSeqAux, subSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma negSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (negSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (!(bits ())) && (carry ()))\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [negSeqAux, *]\n\nlemma neg_eq_propagate (x : \u2115 \u2192 Bool) :\n    negSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (!(bits ())) && (carry ()), _root_.xor (!(bits ())) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [negSeq, negSeqAux]\n  | 1 => simp [negSeq, negSeqAux, propagate, propagateAux]\n  | n+2 => simp [negSeq, negSeqAux, negSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma incrSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (incrSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (bits ()) && carry ())\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [incrSeqAux, *]\n\nlemma incr_eq_propagate (x : \u2115 \u2192 Bool) :\n    incrSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (bits ()) && carry (), _root_.xor (bits ()) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [incrSeq, incrSeqAux]\n  | 1 => simp [incrSeq, incrSeqAux, propagate, propagateAux]\n  | n+2 => simp [incrSeq, incrSeqAux, incrSeqAux_eq_propagateCarry, propagate_succ]\n\nlemma decrSeqAux_eq_propagateCarry (x : \u2115 \u2192 Bool) (n : \u2115) :\n    (decrSeqAux x n).2 = propagateCarry (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        \u03bb _ => (!(bits ())) && carry ())\n    (\u03bb _ => x) n () := by\n  induction n <;> simp [decrSeqAux, *]\n\nlemma decr_eq_propagate (x : \u2115 \u2192 Bool) :\n    decrSeq x = propagate (\u03bb _ => true)\n      (\u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n        (\u03bb _ => (!(bits ())) && carry (), _root_.xor (bits ()) (carry ())))\n    (\u03bb _ => x) := by\n  ext n\n  match n with\n  | 0 => simp [decrSeq, decrSeqAux]\n  | 1 => simp [decrSeq, decrSeqAux, propagate, propagateAux]\n  | n+2 => simp [decrSeq, decrSeqAux, decrSeqAux_eq_propagateCarry, propagate_succ]\n\nstructure PropagateStruc (arity : Type) : Type 1 :=\n  ( \u03b1  : Type )\n  [ i : Fintype \u03b1 ]\n  ( init_carry : \u03b1 \u2192 Bool )\n  ( next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : arity \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool )\n\nattribute [instance] PropagateStruc.i\n\nnamespace PropagateStruc\n\nvariable {arity : Type} (p : PropagateStruc arity)\n\ndef eval : (arity \u2192 \u2115 \u2192 Bool) \u2192 \u2115 \u2192 Bool :=\n  propagate p.init_carry p.next_bit\n\ndef changeVars {arity2 : Type} (changeVars : arity \u2192 arity2) :\n    PropagateStruc arity2 :=\n  { \u03b1 := p.\u03b1,\n    i := p.i,\n    init_carry := p.init_carry,\n    next_bit := \u03bb carry bits => p.next_bit carry (fun i => bits (changeVars i)) }\n\ndef compose [Fintype arity]\n  (new_arity : Type)\n    (q_arity : arity \u2192 Type)\n    (vars : \u2200 (a : arity), q_arity a \u2192 new_arity)\n    (q : \u2200 (a : arity), PropagateStruc (q_arity a)) :\n    PropagateStruc (new_arity) :=\n  { \u03b1 := p.\u03b1 \u2295 (\u03a3 a, (q a).\u03b1),\n    i := by letI := p.i;  infer_instance,\n    init_carry := Sum.elim p.init_carry (\u03bb x => (q x.1).init_carry x.2),\n    next_bit := \u03bb carry bits =>\n      let f : \u2200 (a : arity), ((q a).\u03b1 \u2192 Bool) \u00d7 Bool := \u03bb a => (q a).next_bit (\u03bb d =>\n          carry (inr \u27e8a, d\u27e9)) (\u03bb d => bits (vars a d))\n      let g : (p.\u03b1 \u2192 Bool) \u00d7 Bool := (p.next_bit (carry \u2218 inl) (\u03bb a => (f a).2))\n      (Sum.elim g.1 (\u03bb x => (f x.1).1 x.2), g.2) }\n\nlemma eval_compose [Fintype arity]\n    (new_arity : Type)\n    (q_arity : arity \u2192 Type)\n    (vars : \u2200 (a : arity), q_arity a \u2192 new_arity)\n    (q : \u2200 (a : arity), PropagateStruc (q_arity a))\n    (x : new_arity \u2192 \u2115 \u2192 Bool):\n    (p.compose new_arity q_arity vars q).eval x =\n    p.eval (\u03bb a => (q a).eval (fun i => x (vars _ i))) := by\n  ext n; simp only [eval, compose, propagate_propagate]\n\n\ndef and : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits true && bits false) }\n\n@[simp] lemma eval_and (x : Bool \u2192 \u2115 \u2192 Bool) : and.eval x = andSeq (x true) (x false) := by\n  ext n; cases n <;> simp [and, andSeq, eval, propagate_succ]\n\ndef or : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits true || bits false) }\n\n@[simp] lemma eval_or (x : Bool \u2192 \u2115 \u2192 Bool) : or.eval x = orSeq (x true) (x false) := by\n  ext n; cases n <;> simp [or, orSeq, eval, propagate_succ]\n\ndef xor : PropagateStruc Bool :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, _root_.xor (bits true) (bits false)) }\n\n@[simp] lemma eval_xor (x : Bool \u2192 \u2115 \u2192 Bool) : xor.eval x = xorSeq (x true) (x false) := by\n  ext n; cases n <;> simp [xor, xorSeq, eval, propagate_succ]\n\n  def add : PropagateStruc Bool :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (bits true && bits false) || (bits false && carry ()) || (bits true && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))) }\n\n@[simp] lemma eval_add (x : Bool \u2192 \u2115 \u2192 Bool) : add.eval x = addSeq (x true) (x false) := by\n  dsimp [add, eval]\n  rw [add_eq_propagate]\n  congr\n  funext b\n  cases b; rfl\n  simp\n  congr\n  funext i\n  cases i <;> simp\n\n\ndef sub : PropagateStruc Bool :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Bool \u2192 Bool) =>\n        (\u03bb _ => (!(bits true) && (bits false)) ||\n          ((!(_root_.xor (bits true) (bits false))) && carry ()),\n          _root_.xor (bits true) (_root_.xor (bits false) (carry ()))) }\n\n@[simp] lemma eval_sub (x : Bool \u2192 \u2115 \u2192 Bool) : sub.eval x = subSeq (x true) (x false) := by\n  dsimp [sub, eval]\n  rw [sub_eq_propagate]\n  congr\n  funext b\n  cases b;  rfl\n  simp\n  congr\n  funext i\n  cases i <;> simp\n\ndef neg : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb (carry : Unit \u2192 Bool) (bits : Unit \u2192 Bool) =>\n      (\u03bb _ => (!(bits ())) && (carry ()), _root_.xor (!(bits ())) (carry ())) }\n\n@[simp] lemma eval_neg (x : Unit \u2192 \u2115 \u2192 Bool) : neg.eval x = negSeq (x ()) := by\n  dsimp [neg, eval]\n  rw [neg_eq_propagate]\n\ndef not : PropagateStruc Unit :=\n{ \u03b1 := Empty,\n  i := by infer_instance,\n  init_carry := Empty.elim,\n  next_bit := \u03bb _carry bits => (Empty.elim, !(bits ())) }\n\n@[simp] lemma eval_not (x : Unit \u2192 \u2115 \u2192 Bool) : not.eval x = notSeq (x ()) := by\n  ext n; cases n <;> simp [not, notSeq, eval, propagate_succ]\n\ndef zero : PropagateStruc (Fin 0) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry _bits => (Empty.elim, false) }\n\n@[simp] lemma eval_zero (x : Fin 0 \u2192 \u2115 \u2192 Bool) : zero.eval x = zeroSeq := by\n  ext n; cases n <;> simp [zero, zeroSeq, eval, propagate_succ]\n\ndef one : PropagateStruc (Fin 0) :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry _bits => (\u03bb _ => false, carry ()) }\n\n@[simp] lemma eval_one (x : Fin 0 \u2192 \u2115 \u2192 Bool) : one.eval x = oneSeq := by\n  ext n; cases n <;> simp [one, oneSeq, eval, propagate_succ2, @eq_comm _ false]\n\ndef negOne : PropagateStruc (Fin 0) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry _bits => (Empty.elim, true) }\n\n@[simp] lemma eval_negOne (x : Fin 0 \u2192 \u2115 \u2192 Bool) : negOne.eval x = negOneSeq := by\n  ext n; cases n <;> simp [negOne, negOneSeq, eval, propagate_succ2]\n\ndef ls : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => false,\n    next_bit := \u03bb carry bits => (bits, carry ()) }\n\n@[simp] lemma eval_ls (x : Unit \u2192 \u2115 \u2192 Bool) : ls.eval x = lsSeq (x ()) := by\n  ext n; cases n <;> simp [ls, lsSeq, eval, propagate_succ2]\n\ndef var (n : \u2115) : PropagateStruc (Fin (n+1)) :=\n  { \u03b1 := Empty,\n    i := by infer_instance,\n    init_carry := Empty.elim,\n    next_bit := \u03bb _carry bits => (Empty.elim, bits (Fin.last n)) }\n\n@[simp] lemma eval_var (n : \u2115) (x : Fin (n+1) \u2192 \u2115 \u2192 Bool) : (var n).eval x = x (Fin.last n) := by\n  ext m; cases m <;> simp [var, eval, propagate_succ]\n\ndef incr : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry bits => (\u03bb _ => bits () && carry (), _root_.xor (bits ()) (carry ())) }\n\n@[simp] lemma eval_incr (x : Unit \u2192 \u2115 \u2192 Bool) : incr.eval x = incrSeq (x ()) := by\n  dsimp [incr, eval]\n  rw [incr_eq_propagate]\n\ndef decr : PropagateStruc Unit :=\n  { \u03b1 := Unit,\n    i := by infer_instance,\n    init_carry := \u03bb _ => true,\n    next_bit := \u03bb carry bits => (\u03bb _ => !(bits ()) && carry (), _root_.xor (bits ()) (carry ())) }\n\n@[simp] lemma eval_decr (x : Unit \u2192 \u2115 \u2192 Bool) : decr.eval x = decrSeq (x ()) := by\n  dsimp [decr, eval]\n  rw [decr_eq_propagate]\n\nend PropagateStruc\n\nstructure PropagateSolution (t : Term) extends PropagateStruc (Fin (arity t)) :=\n( good : t.evalFin = toPropagateStruc.eval )\n\ndef composeUnary\n    (p : PropagateStruc Unit)\n    {t : Term}\n    (q : PropagateSolution t) :\n    PropagateStruc (Fin (arity t)) :=\n  p.compose\n    (Fin (arity t))\n    _\n    (\u03bb _ => id)\n    (\u03bb _ => q.toPropagateStruc)\ndef X := @Bool.casesOn\ndef composeBinary\n    (p : PropagateStruc Bool)\n    {t\u2081 t\u2082 : Term}\n    (q\u2081 : PropagateSolution t\u2081)\n    (q\u2082 : PropagateSolution t\u2082) :\n    PropagateStruc (Fin (max (arity t\u2081) (arity t\u2082))) :=\n  p.compose (Fin (max (arity t\u2081) (arity t\u2082)))\n    (\u03bb b => Fin (cond b (arity t\u2081) (arity t\u2082)))\n    (\u03bb b i => Fin.castLe (by cases b <;> simp) i)\n    (\u03bb b => match b with\n      | true => q\u2081.toPropagateStruc\n      | false => q\u2082.toPropagateStruc)\n\n@[simp] lemma composeUnary_eval\n    (p : PropagateStruc Unit)\n    {t : Term}\n    (q : PropagateSolution t)\n    (x : Fin (arity t) \u2192 \u2115 \u2192 Bool) :\n    (composeUnary p q).eval x = p.eval (\u03bb _ => t.evalFin x) := by\n  rw [composeUnary, PropagateStruc.eval_compose, q.good]; rfl\n\n@[simp] lemma composeBinary_eval\n    (p : PropagateStruc Bool)\n    {t\u2081 t\u2082 : Term}\n    (q\u2081 : PropagateSolution t\u2081)\n    (q\u2082 : PropagateSolution t\u2082)\n    (x : Fin (max (arity t\u2081) (arity t\u2082)) \u2192 \u2115 \u2192 Bool) :\n    (composeBinary p q\u2081 q\u2082).eval x = p.eval\n      (\u03bb b => cond b (t\u2081.evalFin (fun i => x (Fin.castLe (by simp) i)))\n                  (t\u2082.evalFin (fun i => x (Fin.castLe (by simp) i)))) := by\n  rw [composeBinary, PropagateStruc.eval_compose, q\u2081.good, q\u2082.good]\n  congr\n  ext b\n  cases b <;> dsimp <;> congr <;> funext b <;> cases b <;> simp\n\ninstance {\u03b1 \u03b2 : Type} [Fintype \u03b1] [Fintype \u03b2] (b : Bool) :\n    Fintype (cond b \u03b1 \u03b2) :=\n  by cases b <;> simp <;> infer_instance\n\nlemma cond_propagate {\u03b1 \u03b1' \u03b2 \u03b2' : Type}\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool),\n      (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (init_carry' : \u03b1' \u2192 Bool)\n    (next_bit' : \u2200 (_carry : \u03b1' \u2192 Bool) (_bits : \u03b2' \u2192 Bool),\n      (\u03b1' \u2192 Bool) \u00d7 Bool)\n    {\u03b3 : Type} (f\u03b2 : \u03b2 \u2192 \u03b3) (f\u03b2' : \u03b2' \u2192 \u03b3)\n    (x : \u03b3 \u2192 \u2115 \u2192 Bool) (b : Bool) :\n    cond b (propagate init_carry next_bit (\u03bb b => (x (f\u03b2 b))))\n      (propagate init_carry' next_bit' (\u03bb b => (x (f\u03b2' b)))) =\n    propagate (show cond b \u03b1 \u03b1' \u2192 Bool from Bool.rec init_carry' init_carry b)\n      (show \u2200 (_carry : cond b \u03b1 \u03b1' \u2192 Bool) (_bits : cond b \u03b2 \u03b2' \u2192 Bool),\n          (cond b \u03b1 \u03b1' \u2192 Bool) \u00d7 Bool\n        from Bool.rec next_bit' next_bit b)\n      (show cond b \u03b2 \u03b2' \u2192 \u2115 \u2192 Bool from Bool.rec (\u03bb b => (x (f\u03b2' b))) (\u03bb b => (x (f\u03b2 b))) b) :=\n  by cases b <;> rfl\n\ndef termEvalEqPropagate : \u2200 (t : Term),\n  PropagateSolution t\n| var n =>\n  { toPropagateStruc := PropagateStruc.var n,\n    good := by ext; simp [Term.evalFin] }\n| zero =>\n  { toPropagateStruc := PropagateStruc.zero,\n    good := by ext; simp [Term.evalFin] }\n| one =>\n  { toPropagateStruc := PropagateStruc.one,\n    good := by ext; simp [Term.evalFin] }\n| negOne =>\n  { toPropagateStruc := PropagateStruc.negOne,\n    good := by ext; simp [Term.evalFin] }\n| Term.and t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.and q\u2081 q\u2082,\n    good := by ext; simp }\n| Term.or t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.or q\u2081 q\u2082,\n    good := by ext; simp }\n| Term.xor t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.xor q\u2081 q\u2082,\n    good := by ext; simp }\n| ls t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.ls q,\n    good := by ext; simp }\n| Term.not t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.not q,\n    good := by ext; simp }\n| add t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.add q\u2081 q\u2082,\n    good := by ext; simp }\n| sub t\u2081 t\u2082 =>\n  let q\u2081 := termEvalEqPropagate t\u2081\n  let q\u2082 := termEvalEqPropagate t\u2082\n  { toPropagateStruc := composeBinary PropagateStruc.sub q\u2081 q\u2082,\n    good := by ext; simp }\n| neg t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.neg q,\n    good := by ext; simp }\n| incr t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.incr q,\n    good := by ext; simp }\n| decr t =>\n  let q := termEvalEqPropagate t\n  { toPropagateStruc := by dsimp [arity]; exact composeUnary PropagateStruc.decr q,\n    good := by ext; simp }\n\nvariable\n  (init_carry : \u03b1 \u2192 Bool)\n  (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool))\n  (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool)\n\nvariable [Fintype \u03b1] [Fintype \u03b1']\n\nopen Fintype\n\nlemma exists_repeat_carry (seq : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    \u2203 n m : Fin (2 ^ (card \u03b1) + 1),\n      propagateCarry2 init_carry next_carry seq n =\n      propagateCarry2 init_carry next_carry seq m \u2227\n      n < m := by\n  by_contra h\n  haveI := Classical.decEq \u03b1\n  push_neg at h\n  have := \u03bb a b hab => (le_antisymm (h a b hab) (h b a hab.symm)).symm\n  have := Fintype.card_le_of_injective _ this\n  simp at this\n\nlemma propagateCarry2_eq_of_seq_eq_lt (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool))\n    (i : \u2115) (h : \u2200 (b) (j) (_hj : j < i), seq\u2081 b j = seq\u2082 b j) :\n    propagateCarry2 init_carry next_carry seq\u2081 i =\n      propagateCarry2 init_carry next_carry seq\u2082 i := by\n  induction' i with i ih\n  { simp [propagateCarry2] }\n  { simp only [propagateCarry2, h _ i (Nat.lt_succ_self i)]\n    rw [ih]\n    exact \u03bb b j hj => h b j (Nat.lt_succ_of_lt hj) }\n\nlemma propagate_eq_of_seq_eq_le (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool)\n    (i : \u2115) (h : \u2200 (b) (j) (_hj : j \u2264 i), seq\u2081 b j = seq\u2082 b j) :\n    propagate init_carry next_bit seq\u2081 i =\n      propagate init_carry next_bit seq\u2082 i := by\n  cases i\n  { simp [propagate_zero, h _ 0 (le_refl _)] }\n  { simp only [propagate_succ2, propagate_succ2, h _ _ (le_refl _)]\n    congr 2\n    apply propagateCarry2_eq_of_seq_eq_lt\n    exact \u03bb b j hj => h b j (le_of_lt hj) }\n\n\nlemma propagateCarry2_eq_of_carry_eq (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (m n : \u2115)\n    (h\u2081 : propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 m =\n        propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 n) (x : \u2115)\n    (h\u2083 : \u2200 y b, y \u2264 x \u2192 seq\u2081 b (m + y) = seq\u2082 b (n + y))  :\n    propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 (m + x) =\n    propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 (n + x) := by\n  induction' x with x ih generalizing seq\u2081 seq\u2082\n  { simp [*] at * }\n  { simp only [propagateCarry2, Nat.add_eq, h\u2083 x _ (Nat.le_succ _)] at *\n    rw [ih]\n    assumption\n    exact \u03bb y b h => h\u2083 y b (Nat.le_succ_of_le h) }\n\nlemma propagate_eq_of_carry_eq (seq\u2081 seq\u2082 : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (m n : \u2115)\n    (h\u2081 : propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2081 m =\n        propagateCarry2 init_carry\n      (\u03bb carry bits => (next_bit carry bits).1) seq\u2082 n) (x : \u2115)\n    (h\u2083 : \u2200 y b, y \u2264 x \u2192 seq\u2081 b (m + y) = seq\u2082 b (n + y))  :\n    propagate init_carry next_bit seq\u2081 (m + x) =\n    propagate init_carry next_bit seq\u2082 (n + x) := by\n  cases x\n  { cases m\n    { cases n\n      { simp [h\u2083 0 _ (le_refl _), propagateCarry2, *] at * }\n      { simp [*, h\u2083 0 _ (le_refl _), propagate_succ2] at *\n        rw [\u2190 h\u2081] } }\n    { cases n\n      { simp [*, propagate_succ2] at *\n        have := fun i => h\u2083 0 i rfl\n        dsimp at this\n        simp [this]\n        simp [h\u2081] }\n      { rw [propagate_succ2, h\u2081, propagate_succ2]\n        have := h\u2083 0\n        simp [*] at * } } }\n  { erw [Nat.add_succ, propagate_succ2, propagate_succ2, Nat.add_eq, Nat.add_eq]\n    simp [\u2190 Nat.succ_eq_add_one, \u2190 Nat.add_succ, h\u2083 _ _ (le_refl _)]\n    congr\n    . apply propagateCarry2_eq_of_carry_eq\n      . assumption\n      . exact \u03bb y b h => h\u2083 y b (Nat.le_succ_of_le h)\n    . funext i\n      rw [h\u2083]\n      exact Nat.le_succ _ }\n\nlemma propagateCarry_propagateCarry_add (x : \u03b2 \u2192 \u2115 \u2192 Bool) :\n    \u2200 (init_carry : \u03b1 \u2192 Bool)\n      (next_carry : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool)),\n    \u2200 n i : \u2115,\n    propagateCarry2 (propagateCarry2 init_carry next_carry x n)\n      next_carry (\u03bb b k => x b (k + n)) i =\n    propagateCarry2 init_carry next_carry x (i + n)\n  | init_carry, _next_carry, 0, 0 => by simp [propagateCarry2]\n  | init_carry, next_carr, n+1, 0 =>\n    by simp [propagateCarry, propagateCarry2_succ]\n  | init_carry, next_carry, n, i+1 => by\n    rw [propagateCarry2, add_assoc,\n      propagateCarry_propagateCarry_add _ _ _ _ i]\n    simp only [Nat.one_add, Nat.add_one, Nat.succ_add, Nat.add_succ,\n      add_zero, propagateCarry2, zero_add]\n\n\nlemma exists_repeat : \u2200 (seq : \u03b2 \u2192 \u2115 \u2192 Bool)\n    (n : \u2115),\n    \u2203 (m : \u2115) (_hm : m < 2 ^ (card \u03b1)) (seq2 : \u03b2 \u2192 \u2115 \u2192 Bool),\n      propagate init_carry next_bit seq2 m = propagate init_carry next_bit seq n\n  | seq, n => by\n    by_cases hn2 : n < 2 ^ card \u03b1\n    { exact \u27e8n, hn2, seq, rfl\u27e9 }\n    { rcases exists_repeat_carry\n        (propagateCarry2 init_carry (\u03bb c b => (next_bit c b).1) seq\n          (n - 2 ^ card \u03b1))\n        (\u03bb carry bits => (next_bit  carry bits).1)\n        (\u03bb b i => seq b (i + (n - 2^ (card \u03b1)))) with \u27e8a, b, h\u2081, h\u2082\u27e9\n      simp only [propagateCarry_propagateCarry_add] at h\u2081\n      rcases have _wf : n - (b - a) < n :=\n         Nat.sub_lt (lt_of_lt_of_le (pow_pos (by norm_num) _) (le_of_not_lt hn2)) (Nat.sub_pos_of_lt h\u2082)\n        exists_repeat (\u03bb c i => if i < a + (n - 2 ^ card \u03b1) then seq c i else\n          seq c (i + (b - a))) (n - (b - a)) with \u27e8m, hmle, seq2, hm\u27e9\n      use m; use hmle; use seq2\n      rw [hm]; clear hm\n      have h1 : n - (b - a) = (a + (n - 2 ^ (card \u03b1))) + (2 ^ card \u03b1 - b) := by\n      { zify\n        rw [Nat.cast_sub, Nat.cast_sub, Nat.cast_sub, Nat.cast_sub]\n        ring_nf\n        exact Nat.le_of_lt_succ b.2\n        simp [*] at *\n        exact hn2\n        exact le_of_lt h\u2082\n        exact le_trans (Nat.sub_le _ _) (le_trans (Nat.le_of_lt_succ b.2)\n          (le_of_not_lt hn2)) }\n      rw [h1]\n      have h2 : n = (b + (n - 2 ^ card \u03b1)) + (2 ^ card \u03b1 - b) := by\n      { zify\n        rw [Nat.cast_sub, Nat.cast_sub]\n        ring\n        exact Nat.le_of_lt_succ b.2\n        simp [*] at *\n        exact hn2 }\n      conv_rhs => rw [h2]\n      refine' propagate_eq_of_carry_eq _ _ _ _ _ _ _ _ _\n      { have _h : \u2191b + (n - 2 ^ card \u03b1) = (a + (n - 2 ^ card \u03b1)) + (b - a) := by\n        { zify\n          rw [Nat.cast_sub, Nat.cast_sub]\n          ring_nf\n          exact le_of_lt h\u2082\n          exact le_of_not_lt hn2 }\n        rw [\u2190 h\u2081]\n        apply propagateCarry2_eq_of_seq_eq_lt\n        simp (config := { contextual := true }) }\n      { intro y c _hc\n        simp only [add_lt_iff_neg_left, not_lt_zero', if_false]\n        congr 1\n        zify\n        rw [Nat.cast_sub, Nat.cast_sub]\n        ring\n        exact le_of_lt h\u2082\n        exact le_of_not_lt hn2 } }\n\n\nlemma propagate_eq_zero_iff (init_carry : \u03b1 \u2192 Bool)\n    (next_bit : \u2200 (_carry : \u03b1 \u2192 Bool) (_bits : \u03b2 \u2192 Bool), (\u03b1 \u2192 Bool) \u00d7 Bool) :\n    (\u2200 seq, propagate init_carry next_bit seq = zeroSeq) \u2194\n    (\u2200 seq, \u2200 i < 2 ^ (card \u03b1), propagate init_carry next_bit seq i = false) := by\n  constructor\n  { intro h i _\n    simp [h, zeroSeq] }\n  { intro h seq\n    funext i\n    rcases exists_repeat init_carry next_bit seq i with \u27e8j, hj, seq2, hseq2\u27e9\n    rw [\u2190 hseq2, h seq2 j hj, zeroSeq] }\n\nlemma eq_iff_xorSeq_eq_zero (seq\u2081 seq\u2082 : \u2115 \u2192 Bool) :\n    (\u2200 i, seq\u2081 i = seq\u2082 i) \u2194 (\u2200 i, xorSeq seq\u2081 seq\u2082 i = zeroSeq i) := by\n  simp [Function.funext_iff, xorSeq, zeroSeq]\n  constructor\n  { intro i _; simp [*] }\n  { intro h a\n    specialize h a\n    revert h\n    cases (seq\u2081 a) <;> cases (seq\u2082 a) <;> simp [*] at * }\n\nlemma eval_eq_iff_xorSeq_eq_zero (t\u2081 t\u2082 : Term) :\n    t\u2081.eval = t\u2082.eval \u2194 (t\u2081.xor t\u2082).evalFin = \u03bb _ => zeroSeq := by\n  simp only [Function.funext_iff, Term.eval, Term.evalFin,\n    \u2190 eq_iff_xorSeq_eq_zero, \u2190 evalFin_eq_eval]\n  constructor\n  { intro h seq n\n    have := h (\u03bb j => if hj : j < (arity (t\u2081.xor t\u2082)) then seq \u27e8j, hj\u27e9 else \u03bb _ => false) n\n    simp at this\n    convert this }\n  { intro h seq m\n    exact h (\u03bb j => seq j) _ }\n", "meta": {"author": "bollu", "repo": "ssa", "sha": "19c73e48500bfe3f618c360423966677adb4673e", "save_path": "github-repos/lean/bollu-ssa", "path": "github-repos/lean/bollu-ssa/ssa-19c73e48500bfe3f618c360423966677adb4673e/SSA/Bits/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3850146346180327}}
{"text": "import Rings.AxGroth\nimport completeness\n\n----\nnamespace finset\n\nvariables {\u03b1 : Type*}\n\nlemma filter_mem_set_of_subset_set {s : set \u03b1} [decidable_pred (\u03bb x, x \u2208 s)]\n  {fs : finset \u03b1} (h : \u2191fs \u2286 s) :\n  filter (\u03bb x, x \u2208 s) fs = fs :=\nbegin\n  ext x,\n  split,\n  { apply filter_subset },\n  { intro hmem, rw mem_filter, exact \u27e8 hmem , h hmem \u27e9 },\nend\n\nend finset\n----\n\nnamespace fol\n\nopen fol\n\nvariables {L : Language}\n\nnamespace bounded_preformula\n\nlemma bd_not.inj {n} {f0 f1 : bounded_formula L n} :\n  \u223c f0 = \u223c f1 \u2192 f0 = f1 := \u03bb h, (bd_imp.inj h).1\n\nlemma bd_notequal.inj {n} {t0 t1 s0 s1 : bounded_term L n} :\n  t0 \u2244 s0 = t1 \u2244 s1 \u2192 t0 = t1 \u2227 s0 = s1 :=\nbd_equal.inj \u2218 bd_not.inj\n\nend bounded_preformula\n\ndef is_complete' (T : Theory L) : Prop :=\n\u2200 (\u03d5 : sentence L), T \u22a8 \u03d5 \u2228 T \u22a8 \u223c \u03d5\n\ndef is_complete'' (T : Theory L) : Prop :=\n\u2200 (M : Structure L) (\u03d5 : sentence L), M \u22a8 T \u2192 M \u22a8 \u03d5 \u2192 T \u22a8 \u03d5\n\nend fol\n\nnamespace Rings\n\ndef nat_ring_consts :\n  ring_consts \u2192 dvector \u2115 0 \u2192 \u2115\n| zero as := 0\n-- | one as := 1\n\ndef nat_ring_structure_funcs :\n  \u03a0 {n}, ring_signature.functions n \u2192 dvector \u2115 n \u2192 \u2115\n| 0 ring_consts.zero as := 0\n| 0 ring_consts.one as := 1\n| 1 ring_unaries.neg as := 0\n| 2 ring_binaries.add (dvector.cons a (dvector.cons b nil)) := a + b\n| 2 ring_binaries.mul (dvector.cons a (dvector.cons b nil)) := a * b\n| (n+3) f as := pempty.elim f\n\ndef nat_ring_structure : fol.Structure ring_signature :=\n\u27e8 \u2115 , \u03bb _, nat_ring_structure_funcs , \u03bb _, pempty.elim \u27e9\n\nlemma nat_ring_structure_realize_nat :\n  \u03a0 (n : \u2115) {k : \u2115} (v : dvector nat_ring_structure k),\n  realize_bounded_ring_term v\n    (n : fol.bounded_preterm ring_signature k 0) dvector.nil = n\n| 0 _ _ := rfl\n| (n+1) k v :=\nbegin\n  have h := @nat_ring_structure_realize_nat n k v,\n  rw [realize_bounded_ring_term] at h,\n  simpa only [nat.cast_succ, ring_signature.add, realize_bounded_ring_term,\n    fol.realize_bounded_term, h],\nend\n\nlemma nat_cast_bd_ring_term_inj {k n m : \u2115} :\n  (n : fol.bounded_preterm.{0} ring_signature k 0) = \u2191m \u2192 n = m :=\nbegin\n  let v : dvector nat_ring_structure k := dvector.of_fn (\u03bb i, 0),\n  intro hnm,\n  rw \u2190 nat_ring_structure_realize_nat n v,\n  rw \u2190 nat_ring_structure_realize_nat m v,\n  exact @congr_arg (fol.bounded_preterm ring_signature k 0)\n    nat_ring_structure n m\n    (\u03bb t : fol.bounded_preterm ring_signature k 0,\n      realize_bounded_ring_term v t dvector.nil) hnm,\nend\n\nend Rings\n\nnamespace Lefschetz\n\nopen fol\nopen Rings\nopen Fields\n\n-- I imagine this is constructive but I don't want to spend time proving it\ninstance dec_eq_sentence_ring_signature :\n  decidable_eq (sentence ring_signature) := sorry\n\nlemma injective_plus_one_ne_zero : function.injective plus_one_ne_zero.{0} :=\nbegin\n  intros n m himage,\n  simp only [plus_one_ne_zero, ring_signature.one,\n    ring_signature.add, ring_signature.zero] at himage,\n  have h := (bd_app.inj (bd_app.inj (bd_notequal.inj himage).1).1).2,\n  apply nat_cast_bd_ring_term_inj h,\nend\n\n-- lemma equal_instances_of_zero {M : Structure ring_signature} [mul_zero_class M] :\n--   mul_zero_class.to_has_zero M.carrier = models_ring_theory_to_comm_ring.has_zero :=\n-- begin\n\n-- end\n--\ntheorem is_complete'_ACF\u2080 : is_complete' ACF\u2080 :=\nbegin\n  sorry\nend\n\nlemma characteristic_change_left (\u03d5 : sentence ring_signature.{0}) :\nACF\u2080 \u22a8 \u03d5 \u2192 \u2203 (n : \u2115), \u2200 {p : \u2115} (hp : nat.prime p), n < p \u2192 ACF\u209a hp \u22a8 \u03d5 :=\nbegin\n  rw compactness,\n  intro hsatis,\n  obtain \u27e8 fs , hsatis , hsub \u27e9 := hsatis,\n  obtain \u27e8 fsACF , fsrange , hunion, hACF , hrange \u27e9 :=\n    finset.subset_union_elim hsub,\n  set fsnat : finset \u2115 := finset.preimage fsrange plus_one_ne_zero.{0}\n      (set.inj_on_of_injective injective_plus_one_ne_zero _) with hfsnat,\n  use fsnat.sup id + 1,\n  intros p hp hlt M hMx hmodel,\n  have _inst_1 : fact (M \u22a8 ACF) := \u27e8 (models_ACF\u209a_iff.mp hmodel).2 \u27e9,\n  have hchar := @models_ACF\u209a_char _ _ _inst_1 _ hmodel,\n  apply hsatis hMx,\n  rw [\u2190 hunion, finset.coe_union, all_realize_sentence_union],\n  split,\n  {\n    apply all_realize_sentence_of_subset _ hACF,\n    exact all_realize_sentence_of_subset hmodel ACF_subset_ACF\u209a,\n  },\n  {\n    have hSTS :(\u2200 n : \u2115, n \u2208 fsnat \u2192 M \u22a8 plus_one_ne_zero n) \u2192 M \u22a8 fsrange,\n    {\n      classical,\n      have hrw0 := finset.image_preimage plus_one_ne_zero fsrange\n        (set.inj_on_of_injective injective_plus_one_ne_zero _),\n      rw [\u2190 hfsnat, finset.filter_mem_set_of_subset_set (\u03bb x hx, (hrange hx).1)]\n        at hrw0,\n      rw \u2190 hrw0,\n      simp only [all_realize_sentence],\n      intros hrealize \u03d5 h\u03d5,\n      simp only [set.mem_preimage, set.mem_image,\n        finset.coe_preimage, finset.mem_coe, finset.coe_image] at h\u03d5,\n      obtain \u27e8 n , hn , h\u03d5 \u27e9 := h\u03d5,\n      rw [\u2190 h\u03d5],\n      apply hrealize,\n      rw [hfsnat, finset.mem_preimage],\n      exact hn,\n    },\n    apply hSTS,\n    intros n hnp,\n    rw realize_plus_one_ne_zero,\n    have hne_zero_of_le_char :\n      \u2200 x : \u2115, x.succ < p \u2192 (x.succ : M) \u2260 0,\n    {\n      intros x hx hbot,\n      apply nat.succ_ne_zero x,\n      have hfield : field M.carrier := @models_ACF_to.Field _ _inst_1,\n      apply @ring_char.lt_char_field _ (models_ACF_to.Field),\n      { exact hbot },\n      rw \u2190 hchar at hx,\n      exact hx,\n    },\n    apply hne_zero_of_le_char _,\n    apply lt_of_le_of_lt (nat.succ_le_succ _) hlt,\n    exact finset.le_sup hnp,\n  },\nend\n\n\n\n/-- Any ring fact holds in ACF\u2080 if and only if for all large p it holds for all ACF\u209a-/\ntheorem characteristic_change (\u03d5 : sentence ring_signature.{0}) :\nACF\u2080 \u22a8 \u03d5 \u2194 (\u2203 (n : \u2115), \u2200 {p : \u2115} (hp : nat.prime p), n < p \u2192 ACF\u209a hp \u22a8 \u03d5) :=\nbegin\n  split,\n  { apply characteristic_change_left },\n  {\n    intro hn,\n    cases is_complete'_ACF\u2080 \u03d5 with hsatis hsatis,\n    { exact hsatis },\n    {\n      have hm := characteristic_change_left (\u223c \u03d5) hsatis, clear hsatis, --R\n      cases hn with n hn,\n      cases hm with m hm,\n      obtain \u27e8 p , hle , hp \u27e9 := nat.exists_infinite_primes (max n m).succ,\n      have hnp : n < p :=\n        lt_of_lt_of_le (nat.lt_succ_of_le (le_max_left _ _)) hle,\n      have hmp : m < p :=\n        lt_of_lt_of_le (nat.lt_succ_of_le (le_max_right _ _)) hle,\n      have hS := instances.algebraic_closure_of_zmod_models_ACF\u209a hp,\n      specialize @hn p hp hnp _ \u27e8 0 \u27e9 hS,\n      specialize @hm p hp hmp _ \u27e8 0 \u27e9 hS,\n      simp only [realize_sentence_not] at hm,\n      exfalso,\n      apply hm hn,\n    },\n  },\nend\n\n\n\n\nopen Rings.ring_signature\n\nlemma nat_cast_eq {M : Structure ring_signature} (n : \u2115)\n  [non_assoc_semiring M.carrier] : (n : M)\n  =\n  (@nat.cast M.carrier (by apply_instance) _ _ n) := rfl\n\nlemma something_spec {M : Structure ring_signature} (h : M \u22a8 ACF)\n  {p n : \u2115} (hp : prime p) (hchar : (p : M) = 0) (hnp : n < p) :\n  (n : M) \u2260 0 :=\nbegin\n  intro hn0,\n  have hNASR : non_assoc_semiring M.carrier :=\n  @semiring.to_non_assoc_semiring M (\n    @comm_semiring.to_semiring M (\n      @comm_ring.to_comm_semiring M (\n        @field.to_comm_ring M (\n          @models_ACF_to.Field M \u27e8 h \u27e9)))),\n  have hrw := @ring_char.spec M hNASR n,\n  rw @nat_cast_eq M n hNASR at hn0,\n  -- rw hrw at hn0,\n\n  -- rw hn0 at hrw,\n\n  -- simp only [symm hn0] at hrw,\n  sorry\nend\n\nexample {R : Type*} [non_assoc_semiring R] (n : \u2115) : R := n.cast\n\n\n\n#check ring_char.spec\n\n#check char_p.exists_unique\n\nend Lefschetz\n", "meta": {"author": "Jlh18", "repo": "ModelTheoryInLean8", "sha": "fbda7d869d4169b6e739bb74165e99ee03ca63d6", "save_path": "github-repos/lean/Jlh18-ModelTheoryInLean8", "path": "github-repos/lean/Jlh18-ModelTheoryInLean8/ModelTheoryInLean8-fbda7d869d4169b6e739bb74165e99ee03ca63d6/Trash/current21January.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3850146346180327}}
{"text": "\nstructure S :=\n(x := true)\n\ndef f (s : S) : Bool :=\ns.x\n\n#eval f {}\n\ntheorem ex : f {} = true :=\nrfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/fieldDefaultValueWithoutType.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6992544147913993, "lm_q1q2_score": 0.38501463116717854}}
{"text": "import algebra.group.basic\nimport analysis.convex.cone.basic\nimport linear_algebra.dual\nimport algebra.order.ring\nimport algebra.module.hom\nimport ring_theory.finiteness\n\nimport for_mathlib.Gordan\n\n/-!\n\n# Lemma 9.7 of [Analytic]\n\nIn this file we state and prove 9.7 of [Analytic]. The proof in the pdf is short, but\ncombinatorics can be fiddly. However it is almost certain that this file could be\na lot shorter.\n\n-/\n\nopen_locale nnreal big_operators classical\n\nvariables {\u039b : Type*} [add_comm_group \u039b]\nvariable {\u03b9 : Type*}\n\nnoncomputable theory\n\nopen classical subtype function embedding\n\n/-- A technical auxiliary lemma used in the proof of `lem97_pos`. -/\nlemma aux_1 {N : \u2115} {l : \u03b9 \u2192 \u039b} {S\u2080 : finset (\u039b \u2192+ \u2124)}\n  (hS\u2080 : submodule.span \u2115 \u2191S\u2080 = explicit_dual_set l) :\n  let \u03c8 : ({x // x \u2208 S\u2080} \u2192 fin N) \u2192 \u039b \u2192+ \u2124 :=\n        \u03bb (y : {x // x \u2208 S\u2080} \u2192 fin N), \u2211 (s : {x // x \u2208 S\u2080}) in S\u2080.attach, (y s).val \u2022 s.val,\n      B : finset (\u039b \u2192+ \u2124) := finset.image \u03c8 finset.univ\n  in \u2200 (b : \u039b \u2192+ \u2124), b \u2208 B \u2192 b \u2208 explicit_dual_set l :=\nbegin\n  intros \u03c8 B b hb,\n  rcases finset.mem_image.mp hb with \u27e8y, \u27e8hy\u2081, rfl\u27e9\u27e9,\n  rw [\u2190 hS\u2080],\n  apply mem_span_finset.mpr,\n  refine \u27e8\u03bb x : (\u039b \u2192+ \u2124), if H: x \u2208 S\u2080 then (y \u27e8x, H\u27e9 : \u2115) else 0, _\u27e9,\n  rw \u2190 finset.sum_attach,\n  refine finset.sum_congr rfl (\u03bb s hs, _),\n  simp only [finset.coe_mem, finset.mk_coe, dite_eq_ite, if_true, fin.val_eq_coe, val_eq_coe]\nend\n\n/-- A technical auxiliary lemma used in the proof of `lem97_pos`. -/\nlemma aux_2 {N : \u2115} (hN : 0 < N) {l : \u03b9 \u2192 \u039b} {S\u2080 : finset (\u039b \u2192+ \u2124)}\n  (hS\u2080 : submodule.span \u2115 \u2191S\u2080 = explicit_dual_set l) {f r : (\u039b \u2192+ \u2124) \u2192 \u2115} :\n  let Y : Type u_1 := {x // x \u2208 S\u2080} \u2192 fin N,\n      \u03c8 : Y \u2192 \u039b \u2192+ \u2124 := \u03bb (y : Y), \u2211 (s : {x // x \u2208 S\u2080}) in S\u2080.attach, (y s).val \u2022 s.val,\n      B : finset (\u039b \u2192+ \u2124) := finset.image \u03c8 finset.univ,\n      g : (\u039b \u2192+ \u2124) \u2192 fin N := \u03bb (i : \u039b \u2192+ \u2124), \u27e8f i % N, nat.mod_lt _ hN\u27e9,\n      x' : \u039b \u2192+ \u2124 := \u2211 (i : \u039b \u2192+ \u2124) in S\u2080, (g i).val \u2022 i\n  in f = \u2191g + N \u2022 r \u2192 x' = \u2211 (i : \u039b \u2192+ \u2124) in S\u2080, (g i).val \u2022 i \u2192 x' \u2208 B \u2192\n    \u2200 (i : \u03b9), x' (l i) \u2264 (\u21d1\u2211 (i : \u039b \u2192+ \u2124) in S\u2080, f i \u2022 i) (l i) :=\nbegin\n  intros Y \u03c8 B g x' hr hx' H i,\n  dsimp [x'],\n  rw [sub_nonpos.symm, sub_eq_add_neg, \u2190 add_monoid_hom.neg_apply, \u2190 finset.sum_neg_distrib,\n    add_monoid_hom.finset_sum_apply, add_monoid_hom.finset_sum_apply, \u2190 finset.sum_add_distrib],\n  swap, apply_instance,\n  simp only [\u2190 add_monoid_hom.add_apply, \u2190 nsmul_eq_smul, \u2190 coe_nat_zsmul, \u2190 neg_zsmul,\n     zsmul_eq_smul, \u2190 add_smul],\n  simp only [add_monoid_hom.add_apply, add_monoid_hom.smul_apply, pi.add_apply, pi.neg_apply,\n     int.coe_nat_mod, pi.smul_apply, add_monoid_hom.neg_apply],\n  apply finset.sum_nonpos,\n  intros z hz,\n  replace hz : z \u2208 explicit_dual_set l,\n  { rw [\u2190 submodule.span_singleton_le_iff_mem, \u2190 hS\u2080],\n    exact submodule.span_mono (set.singleton_subset_iff.mpr hz) },\n  replace hz : 0 \u2264 z (l i) := rfl.mpr hz i,\n  simp only [- add_neg_le_iff_le_add', tactic.ring.add_neg_eq_sub, smul_sub],\n  rw sub_smul,\n  rw [\u2190 int.coe_nat_mod, sub_le_iff_le_add, zero_add],\n  simp only [has_smul.smul, zsmul_int_int],\n  exact (mul_le_mul_of_nonneg_right (int.coe_nat_le.mpr (nat.mod_le (f z) N)) hz),\nend\n\n/-- A technical auxiliary lemma used in the proof of `lem97_pos`. -/\nlemma aux_3 {N : \u2115} (hN : 0 < N) {l : \u03b9 \u2192 \u039b} {S\u2080 : finset (\u039b \u2192+ \u2124)}\n  (hS\u2080 : submodule.span \u2115 \u2191S\u2080 = explicit_dual_set l) :\n  let Y : Type u_1 := {x // x \u2208 S\u2080} \u2192 fin N,\n      \u03c8 : Y \u2192 \u039b \u2192+ \u2124 := \u03bb (y : Y), \u2211 (s : {x // x \u2208 S\u2080}) in S\u2080.attach, (y s).val \u2022 s.val,\n      B : finset (\u039b \u2192+ \u2124) := finset.image \u03c8 finset.univ\n  in \u2200 (x : \u039b \u2192+ \u2124), x \u2208 explicit_dual_set l \u2192 (\u2203 (x' : \u039b \u2192+ \u2124) (H : x' \u2208 B) (y : \u039b \u2192+ \u2124),\n      x = N \u2022 y + x' \u2227 \u2200 (i : \u03b9), x' (l i) \u2264 x (l i)) :=\nbegin\n  intros Y \u03c8 B x hx,\n  rw [\u2190 hS\u2080, mem_span_finset] at hx,\n  rcases hx with \u27e8f, rfl\u27e9,\n  let g : (\u039b \u2192+ \u2124) \u2192 (fin N) := (\u03bb i, \u27e8f i % N, nat.mod_lt (f i) hN\u27e9),\n  obtain \u27e8r, hr\u27e9 : \u2203 (r : (\u039b \u2192+ \u2124) \u2192 \u2115), f = \u2191g + N \u2022 r,\n  { use \u03bb x, (f x - g x) / N,\n    refine funext (\u03bb z, (_ : f z = g z + N * ((f z - f z % N) / N))),\n    rw [nat.mul_div_cancel' (nat.dvd_sub_mod _)],\n    exact (add_tsub_cancel_of_le (nat.mod_le _ _)).symm },\n  set x' := \u2211 (i : \u039b \u2192+ \u2124) in S\u2080, (g i).val \u2022 i with hx',\n  have H : x' \u2208 B,\n  { refine finset.mem_image.mpr \u27e8g \u2218 val, finset.mem_univ _, _\u27e9,\n    convert finset.sum_attach,\n    refl },\n  refine \u27e8x', _, \u2211 (i : \u039b \u2192+ \u2124) in S\u2080, r i \u2022 i, _, _\u27e9,\n  { refine finset.mem_image.mpr \u27e8g \u2218 val, finset.mem_univ _, _\u27e9,\n    convert finset.sum_attach,\n    refl },\n  { rw [hr, finset.smul_sum, \u2190 finset.sum_add_distrib],\n    simp_rw [\u2190 smul_assoc, \u2190 add_smul, add_comm (N \u2022 _) _],\n    refl },\n  exact aux_2 hN hS\u2080 hr hx' H,\nend\n\n/-- The proof of Lemma 9.7 of `Analytic.pdf` under the additional assumption\nthat (in the notation of the paper) \u03bb\u1d62(x)\u22650 for all i. -/\nlemma lem97_pos [module.finite \u2124 \u039b] [module.free \u2124 \u039b] [fintype \u03b9] (N : \u2115) (hN : 0 < N) (l : \u03b9 \u2192 \u039b) :\n  \u2203 B : finset (\u039b \u2192+ \u2124), (\u2200 b \u2208 B, b \u2208 (explicit_dual_set l)) \u2227\n    \u2200 x : \u039b \u2192+ \u2124, x \u2208 (explicit_dual_set l) \u2192 \u2203 (x' \u2208 B) (y : \u039b \u2192+ \u2124),\n    x = N \u2022 y + x' \u2227 \u2200 i, x' (l i) \u2264 x (l i) :=\nbegin\n  obtain \u27e8S\u2080, hS\u2080\u27e9 := explicit_gordan l,\n  let Y : Type* := { x // x \u2208 S\u2080 } \u2192 (fin N),\n  let \u03c8 : Y \u2192 \u039b \u2192+ \u2124 := \u03bb y : Y, \u2211 s in finset.attach S\u2080, (y s).1 \u2022 s.val,\n  exact \u27e8finset.image \u03c8 finset.univ, aux_1 hS\u2080, aux_3 hN hS\u2080\u27e9,\nend\n\nsection sign_vectors\n\ndef nonzero_sign : \u2124 \u2192 units \u2124 := \u03bb n, if 0 \u2264 n then 1 else -1\n\ndef sign_vectors (\u03b9 : Type*) := (\u03b9 \u2192 units \u2124)\n\ninstance sign_vectors_inhabited : inhabited (sign_vectors \u03b9) := \u27e8(\u03bb i, 1)\u27e9\n\ndef fintype_sign_vectors [fintype \u03b9] : fintype (sign_vectors \u03b9) := pi.fintype\n\n/-- Given a list l of elements of \u039b and a functional x, (pos_vector l x) is the sign-vector of\nthe values of x (l i). -/\ndef pos_vector (l : \u03b9 \u2192 \u039b) (x : \u039b \u2192+ \u2124) : sign_vectors \u03b9 :=\n\u03bb i, nonzero_sign (x (l i))\n\ninstance smul_signs : has_smul (sign_vectors \u03b9) (\u03b9 \u2192 \u039b) :=\n{ smul := \u03bb \u03b5 l i, (\u03b5 i : \u2124) \u2022 l i }\n\nlemma pos_vector_nonneg (l : \u03b9 \u2192 \u039b) (x : \u039b \u2192+ \u2124) (j : \u03b9) :\n  0 \u2264 x ((pos_vector l x \u2022 l) j) :=\nbegin\n  rw [pos_vector, nonzero_sign],\n  simp only [has_smul.smul, add_monoid_hom.map_zsmul, zsmul_int_int],\n  split_ifs,\n  { convert h, simp },\n  { simp, linarith }\nend\n\nlemma smul_to_explicit_dual_set (l : \u03b9 \u2192 \u039b) (x : \u039b \u2192+ \u2124) :\n  x \u2208 (explicit_dual_set ((pos_vector l x) \u2022 l)) :=\n\u03bb j, pos_vector_nonneg l x j\n\nlemma pos_vector_id_if_nonneg (l : \u03b9 \u2192 \u039b) (x : \u039b \u2192+ \u2124) (i : \u03b9) : 0 \u2264 x (l i) \u2192\n    (pos_vector l x \u2022 l) i = l i :=\nbegin\n  intro hx,\n  simp only [pos_vector, nonzero_sign, has_smul.smul, id.def],\n  rw [if_pos hx],-- units.coe_one, one_zsmul],\n  simp,\nend\n\nlemma pos_vector_neg_if_neg (l : \u03b9 \u2192 \u039b) (x : \u039b \u2192+ \u2124) (i : \u03b9) : x (l i) < 0 \u2192\n    ((pos_vector l x) \u2022 l) i = - l i :=\nbegin\n  intro hx,\n  simp only [pos_vector, nonzero_sign, has_smul.smul, id.def],\n  rw [if_neg (not_le.mpr hx)], simp,\nend\n\n\nend sign_vectors\n\n/-- Given a list l, a vector of signs \u03b5 (and a positive integer N), (pos_A l \u03b5) is a finite set of\nfunctionals satisfying the requirements of Lemma 9.7 of [Analytic] with respect to all functionals\nwhich are positive on all ((\u03b5 \u2022 l) i)'s. Its existence was established in lem97_pos above.\n-/\ndef pos_A [fintype \u03b9] [module.finite \u2124 \u039b] [module.free \u2124 \u039b] (N : \u2115) (hN : 0 < N)\n  (l : \u03b9 \u2192 \u039b) (\u03b5 : sign_vectors \u03b9) : finset (\u039b \u2192+ \u2124) :=\nsome (lem97_pos N hN (\u03b5 \u2022 l))\n\nlemma posA_to_explicit [fintype \u03b9] [module.finite \u2124 \u039b] [module.free \u2124 \u039b]\n  (N : \u2115) (hN : 0 < N) (l : \u03b9 \u2192 \u039b) (\u03b5 : sign_vectors \u03b9) (x' : \u039b \u2192+ \u2124) (H : x' \u2208 pos_A N hN l \u03b5) :\n  x' \u2208 explicit_dual_set (\u03b5 \u2022 l) :=\n(some_spec (lem97_pos N hN (\u03b5 \u2022 l))).1 x' H\n\n\nlemma exists_good_pair [fintype \u03b9] [module.finite \u2124 \u039b] [module.free \u2124 \u039b]\n  (N : \u2115) (hN : 0 < N) (l : \u03b9 \u2192 \u039b)\n  (\u03b5 : sign_vectors \u03b9) (x : \u039b \u2192+ \u2124) (H : x \u2208 (explicit_dual_set (\u03b5 \u2022 l))) :\n  \u2203 x' y : (\u039b \u2192+ \u2124),\n    x' \u2208 pos_A N hN l \u03b5 \u2227 x = N \u2022 y + x' \u2227 \u2200 i, x' ((\u03b5 \u2022 l) i) \u2264 x ((\u03b5 \u2022 l) i) :=\nbegin\n  obtain \u27e8x', hx', \u27e8y, hy\u27e9\u27e9 := (some_spec (lem97_pos N hN (\u03b5 \u2022 l))).2 x H,\n  exact \u27e8x', y, hx', hy\u27e9,\nend\n\n/-\n\nThe remainder of this file is proofs of two versions of Lemma 9.7, namely\n`lem97` and `lem97'`.\n\n-/\n\n\n/-- Lemma 9.7 of [Analytic]. See also the (mathematically indistinguishable) variant `lem97'`. -/\nlemma lem97 [fintype \u03b9] [module.finite \u2124 \u039b] [module.free \u2124 \u039b] (N : \u2115) (hN : 0 < N) (l : \u03b9 \u2192 \u039b) :\n  \u2203 A : finset (\u039b \u2192+ \u2124), \u2200 x : \u039b \u2192+ \u2124, \u2203 (x' \u2208 A) (y : \u039b \u2192+ \u2124),\n    x = N \u2022 y + x' \u2227\n    \u2200 i, (0 \u2264 x' (l i) \u2227 0 \u2264 (x - x') (l i)) \u2228 (x' (l i) \u2264 0 \u2227 (x - x') (l i) \u2264 0) :=\nbegin\n  refine \u27e8(@finset.univ (sign_vectors \u03b9) (fintype_sign_vectors)).bUnion (pos_A N hN l), \u03bb x, _\u27e9,\n  have hx : x \u2208 (explicit_dual_set ((pos_vector l x) \u2022 l)) := smul_to_explicit_dual_set l x,\n  obtain \u27e8x', y, mem_x', hy, hx'\u27e9 := exists_good_pair N hN l (pos_vector l x) x hx,\n  refine \u27e8x', _, _\u27e9,\n  { refine finset.mem_bUnion.mpr \u27e8pos_vector l x, _, mem_x'\u27e9,\n    simp only [finset.mem_univ] },\n  { refine \u27e8y, hy, \u03bb i, _\u27e9,\n    have h_pos' : x' \u2208 explicit_dual_set ((pos_vector l x) \u2022 l) :=\n      posA_to_explicit N hN l (pos_vector l x) x' mem_x',\n    replace h_pos' : 0 \u2264 x' (((pos_vector l x) \u2022 l) i) := h_pos' _,\n    by_cases h_pos : 0 \u2264 x (l i),\n    { have h_posvect_id : ((pos_vector l x) \u2022 l) i = l i := pos_vector_id_if_nonneg l x i h_pos,\n      replace h_pos' : 0 \u2264 x' (l i) := h_pos'.trans (le_of_eq (congr_arg x' h_posvect_id)),\n      refine or.inl \u27e8h_pos', _\u27e9,\n      rw \u2190 h_posvect_id,\n      simp only [sub_nonneg, add_monoid_hom.sub_apply, hx'] },\n    { specialize hx' i,\n      have h_posvect_neg : ((pos_vector l x) \u2022 l) i = - l i :=\n        pos_vector_neg_if_neg l x i (not_le.mp h_pos),\n      rw [h_posvect_neg] at *,\n      replace h_pos' : x' (l i) \u2264 0,\n      { rw [x'.map_neg] at h_pos',\n        exact neg_nonneg.mp h_pos' },\n      refine or.inr \u27e8h_pos', _\u27e9,\n      simpa only [neg_le_neg_iff, add_monoid_hom.sub_apply, add_monoid_hom.map_neg, sub_nonpos]\n        using hx' } }\nend\n\nlemma os.monotone_pow {R : Type*} [ordered_semiring R] (k : \u2115) (R0 : \u2200 r : R, 0 \u2264 r) :\n  monotone (\u03bb n : R, n ^ k) :=\n\u03bb x y xy, pow_le_pow_of_le_left (R0 _) xy k\n\nlemma nat.monotone_pow (k : \u2115) : monotone (\u03bb n : \u2115, n ^ k) :=\nos.monotone_pow k nat.zero_le\n--\u03bb x y xy, pow_le_pow_of_le_left x.zero_le xy k\n\nlemma nat.strict_mono_pow {k : \u2115} (k0 : 0 < k) : strict_mono (\u03bb n : \u2115, n ^ k) :=\n\u03bb x y xy, pow_lt_pow_of_lt_left xy x.zero_le k0\n\nlemma nat.injective_pow {k : \u2115} (k0 : 0 < k) : function.injective (\u03bb n : \u2115, n ^ k) :=\ninjective_of_increasing (<) (<) _ (nat.strict_mono_pow k0)\n\nlemma int.nat_abs_eq_self (a : \u2124) : (a.nat_abs : \u2124) = a \u2194 0 \u2264 a :=\n\u27e8\u03bb h, by { rw \u2190 h, exact int.coe_zero_le a.nat_abs }, \u03bb h, int.nat_abs_of_nonneg h\u27e9\n\nlemma abs_add_eq_iff_abs_mul_eq {R : Type*} [linear_ordered_comm_ring R] {a b : R} :\n  abs (a + b) = abs a + abs b \u2194 abs (a * b) = a * b :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { simpa only [abs_mul, add_pow_two, mul_assoc, pow_bit0_abs, add_left_inj, mul_eq_mul_left_iff,\n      or_false, bit0_eq_zero, one_ne_zero, add_right_inj] using congr_arg (\u03bb x : R, x ^ 2) h.symm },\n  { rcases (mul_nonneg_iff.mp (abs_eq_self.mp h)) with \u27e8a0, b0\u27e9 | \u27e8a0, b0\u27e9,\n    { rw [abs_of_nonneg a0, abs_of_nonneg b0, abs_of_nonneg (add_nonneg a0 b0)] },\n    { rw [abs_of_nonpos a0, abs_of_nonpos b0, abs_of_nonpos (add_nonpos a0 b0), neg_add] } }\nend\n\nlemma nat_abs_add_eq_iff_nat_abs_mul_eq {a b : \u2124} :\n  int.nat_abs (a + b) = a.nat_abs + b.nat_abs \u2194 ((a * b).nat_abs : \u2124) = a * b :=\nbegin\n  rw [\u2190 int.abs_eq_nat_abs, \u2190 abs_add_eq_iff_abs_mul_eq, int.abs_eq_nat_abs, int.abs_eq_nat_abs,\n    int.abs_eq_nat_abs],\n  norm_cast,\nend\n\nlemma nat_abs_add_eq_iff (a b : \u2124) :\n  int.nat_abs (a + b) = int.nat_abs a + int.nat_abs b \u2194 (0 \u2264 a \u2227 0 \u2264 b \u2228 a \u2264 0 \u2227 b \u2264 0) :=\nnat_abs_add_eq_iff_nat_abs_mul_eq.trans ((int.nat_abs_eq_self _).trans mul_nonneg_iff)\n\nlemma nat_smul_nat_abs (a : \u2115) (b : \u2124) : a \u2022 b.nat_abs = (a \u2022 b).nat_abs :=\nbegin\n  nth_rewrite 0 [\u2190 int.nat_abs_of_nat a],\n  rw [smul_eq_mul, \u2190 int.nat_abs_mul, \u2190 smul_eq_mul],\n  apply congr_arg,\n  rw [algebra.id.smul_eq_mul, nsmul_eq_mul],\nend\n\n/-- Lemma 9.7 of [Analytic]. See also the (mathematically indistinguishable) variant `lem97`. -/\nlemma lem97' [fintype \u03b9] [module.finite \u2124 \u039b] [module.free \u2124 \u039b] (N : \u2115) (hN : 0 < N) (l : \u03b9 \u2192 \u039b) :\n  \u2203 A : finset (\u039b \u2192+ \u2124), \u2200 x : \u039b \u2192+ \u2124, \u2203 (x' \u2208 A) (y : \u039b \u2192+ \u2124),\n    x = N \u2022 y + x' \u2227\n    \u2200 i, (x (l i)).nat_abs = N * (y (l i)).nat_abs + (x' (l i)).nat_abs :=\nbegin\n  obtain \u27e8A, hA\u27e9 := lem97 N hN l,\n  refine \u27e8A, \u03bb x, _\u27e9,\n  rcases hA x with \u27e8x', mem_x', y, rfl, hx'\u27e9,\n  refine \u27e8x', mem_x', y, rfl, \u03bb i, _\u27e9,\n  specialize hx' i,\n  rw [\u2190 nat_abs_add_eq_iff, add_sub_cancel, add_monoid_hom.coe_smul, pi.smul_apply] at hx',\n  rw [add_monoid_hom.add_apply, add_monoid_hom.coe_smul, pi.smul_apply, add_comm, hx', add_comm,\n    add_left_inj, \u2190 nat_smul_nat_abs, smul_eq_mul],\nend\n\n-- TODO\n-- #lint-\n\n-- /- The `doc_blame` linter reports: -/\n-- /- DEFINITIONS ARE MISSING DOCUMENTATION STRINGS: -/\n-- #print nonzero_sign /- def missing doc string -/\n-- #print sign_vectors /- def missing doc string -/\n-- #print fintype_sign_vectors /- def missing doc string -/\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/combinatorial_lemma/lem97.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191214879991, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3849004481230605}}
{"text": "/-\nCopyright \u00a9 2020 Nicol\u00f2 Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Nicol\u00f2 Cavalleri.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.lie.basic\nimport Mathlib.ring_theory.algebra_tower\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 l u_4 \n\nnamespace Mathlib\n\n/-!\n# Derivations\n\nThis file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an\n`R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.\n\n## Notation\n\nThe notation `\u2045D1, D2\u2046` is used for the commutator of two derivations.\n\nTODO: this file is just a stub to go on with some PRs in the geometry section. It only\nimplements the definition of derivations in commutative algebra. This will soon change: as soon\nas bimodules will be there in mathlib I will change this file to take into account the\nnon-commutative case. Any development on the theory of derivations is discouraged until the\ndefinitive definition of derivation will be implemented.\n-/\n\n/-- `D : derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`\nequality.\nTODO: update this when bimodules are defined. -/\nstructure derivation (R : Type u_1) (A : Type u_2) [comm_semiring R] [comm_semiring A] [algebra R A]\n    (M : Type u_3) [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M]\n    extends linear_map R A M where\n  leibniz' :\n    \u2200 (a b : A),\n      linear_map.to_fun _to_linear_map (a * b) =\n        a \u2022 linear_map.to_fun _to_linear_map b + b \u2022 linear_map.to_fun _to_linear_map a\n\nnamespace derivation\n\n\nprotected instance has_coe_to_fun {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : has_coe_to_fun (derivation R A M) :=\n  has_coe_to_fun.mk (fun (D : derivation R A M) => A \u2192 M)\n    fun (D : derivation R A M) => linear_map.to_fun (derivation.to_linear_map D)\n\nprotected instance has_coe_to_linear_map {R : Type u_1} [comm_semiring R] {A : Type u_2}\n    [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M]\n    [semimodule R M] [is_scalar_tower R A M] : has_coe (derivation R A M) (linear_map R A M) :=\n  has_coe.mk fun (D : derivation R A M) => derivation.to_linear_map D\n\n@[simp] theorem to_fun_eq_coe {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) :\n    linear_map.to_fun (derivation.to_linear_map D) = \u21d1D :=\n  rfl\n\n@[simp] theorem coe_fn_coe {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (f : derivation R A M) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\ntheorem coe_injective {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (H : \u21d1D1 = \u21d1D2) :\n    D1 = D2 :=\n  sorry\n\ntheorem ext {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A]\n    {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M}\n    (H : \u2200 (a : A), coe_fn D1 a = coe_fn D2 a) : D1 = D2 :=\n  coe_injective (funext H)\n\n@[simp] theorem map_add {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) :\n    coe_fn D (a + b) = coe_fn D a + coe_fn D b :=\n  is_add_hom.map_add (\u21d1D) a b\n\n@[simp] theorem map_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) : coe_fn D 0 = 0 :=\n  is_add_monoid_hom.map_zero \u21d1D\n\n@[simp] theorem map_smul {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (r : R) (a : A) :\n    coe_fn D (r \u2022 a) = r \u2022 coe_fn D a :=\n  linear_map.map_smul (\u2191D) r a\n\n@[simp] theorem leibniz {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) :\n    coe_fn D (a * b) = a \u2022 coe_fn D b + b \u2022 coe_fn D a :=\n  derivation.leibniz' D a b\n\n@[simp] theorem map_one_eq_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) : coe_fn D 1 = 0 :=\n  sorry\n\n@[simp] theorem map_algebra_map {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (r : R) :\n    coe_fn D (coe_fn (algebra_map R A) r) = 0 :=\n  sorry\n\nprotected instance has_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : HasZero (derivation R A M) :=\n  { zero := mk 0 sorry }\n\nprotected instance inhabited {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : Inhabited (derivation R A M) :=\n  { default := 0 }\n\nprotected instance add_comm_monoid {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : add_comm_monoid (derivation R A M) :=\n  add_comm_monoid.mk (fun (D1 D2 : derivation R A M) => mk (\u2191D1 + \u2191D2) sorry) sorry 0 sorry sorry\n    sorry\n\n@[simp] theorem add_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (a : A) :\n    coe_fn (D1 + D2) a = coe_fn D1 a + coe_fn D2 a :=\n  rfl\n\nprotected instance derivation.Rsemimodule {R : Type u_1} [comm_semiring R] {A : Type u_2}\n    [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M]\n    [semimodule R M] [is_scalar_tower R A M] : semimodule R (derivation R A M) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem smul_to_linear_map_coe {R : Type u_1} [comm_semiring R] {A : Type u_2}\n    [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M]\n    [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (r : R) : \u2191(r \u2022 D) = r \u2022 \u2191D :=\n  rfl\n\n@[simp] theorem Rsmul_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (r : R) (a : A) :\n    coe_fn (r \u2022 D) a = r \u2022 coe_fn D a :=\n  rfl\n\nprotected instance semimodule {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : semimodule A (derivation R A M) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem smul_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) :\n    coe_fn (a \u2022 D) b = a \u2022 coe_fn D b :=\n  rfl\n\nprotected instance is_scalar_tower {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    [is_scalar_tower R A M] : is_scalar_tower R A (derivation R A M) :=\n  is_scalar_tower.mk\n    fun (x : R) (y : A) (z : derivation R A M) => ext fun (a : A) => smul_assoc x y (coe_fn (\u2191z) a)\n\n@[simp] theorem map_neg {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A]\n    {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M]\n    (D : derivation R A M) (a : A) : coe_fn D (-a) = -coe_fn D a :=\n  linear_map.map_neg (\u2191D) a\n\n@[simp] theorem map_sub {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A]\n    {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M]\n    (D : derivation R A M) (a : A) (b : A) : coe_fn D (a - b) = coe_fn D a - coe_fn D b :=\n  linear_map.map_sub (\u2191D) a b\n\nprotected instance add_comm_group {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A]\n    [algebra R A] {M : Type u_3} [add_comm_group M] [module A M] [module R M]\n    [is_scalar_tower R A M] : add_comm_group (derivation R A M) :=\n  add_comm_group.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry\n    (fun (D : derivation R A M) => mk (-\u2191D) sorry)\n    (fun (D1 D2 : derivation R A M) => mk (\u2191D1 - \u2191D2) sorry) sorry sorry\n\n@[simp] theorem sub_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A]\n    {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M]\n    {D1 : derivation R A M} {D2 : derivation R A M} (a : A) :\n    coe_fn (D1 - D2) a = coe_fn D1 a - coe_fn D2 a :=\n  rfl\n\n/-! # Lie structures -/\n\n/-- The commutator of derivations is again a derivation. -/\ndef commutator {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A]\n    (D1 : derivation R A A) (D2 : derivation R A A) : derivation R A A :=\n  mk (linear_map.mk (linear_map.to_fun (has_bracket.bracket \u2191D1 \u2191D2)) sorry sorry) sorry\n\nprotected instance has_bracket {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A]\n    [algebra R A] : has_bracket (derivation R A A) (derivation R A A) :=\n  has_bracket.mk commutator\n\n@[simp] theorem commutator_coe_linear_map {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A]\n    [algebra R A] {D1 : derivation R A A} {D2 : derivation R A A} :\n    \u2191(has_bracket.bracket D1 D2) = has_bracket.bracket \u2191D1 \u2191D2 :=\n  rfl\n\ntheorem commutator_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A]\n    {D1 : derivation R A A} {D2 : derivation R A A} (a : A) :\n    coe_fn (has_bracket.bracket D1 D2) a = coe_fn D1 (coe_fn D2 a) - coe_fn D2 (coe_fn D1 a) :=\n  rfl\n\nprotected instance lie_ring {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A]\n    [algebra R A] : lie_ring (derivation R A A) :=\n  lie_ring.mk sorry sorry sorry sorry\n\nprotected instance lie_algebra {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A]\n    [algebra R A] : lie_algebra R (derivation R A A) :=\n  lie_algebra.mk sorry\n\nend derivation\n\n\nnamespace linear_map\n\n\n/-- The composition of a linear map and a derivation is a derivation. -/\ndef comp_der {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A]\n    {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] {N : Type u_4}\n    [add_cancel_comm_monoid N] [semimodule A N] [semimodule R N] [is_scalar_tower R A M]\n    [is_scalar_tower R A N] (f : linear_map A M N) (D : derivation R A M) : derivation R A N :=\n  derivation.mk (mk (fun (a : A) => coe_fn f (coe_fn D a)) sorry sorry) sorry\n\n@[simp] theorem comp_der_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A]\n    [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M]\n    {N : Type u_4} [add_cancel_comm_monoid N] [semimodule A N] [semimodule R N]\n    [is_scalar_tower R A M] [is_scalar_tower R A N] (f : linear_map A M N) (D : derivation R A M)\n    (a : A) : coe_fn (comp_der f D) a = coe_fn f (coe_fn D a) :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/derivation_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6334102567576902, "lm_q1q2_score": 0.3849000844260438}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.perm\nimport Mathlib.algebra.group_power.default\nimport Mathlib.PostPort\n\nuniverses u u_1 u_4 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Multisets\n\nThese are implemented as the quotient of a list by permutations.\n\n## Notation\n\nWe define the global infix notation `::\u2098` for `multiset.cons`.\n-/\n\n/-- `multiset \u03b1` is the quotient of `list \u03b1` by list permutation. The result\n  is a type of finite sets with duplicates allowed.  -/\ndef multiset (\u03b1 : Type u) :=\n  quotient (list.is_setoid \u03b1)\n\nnamespace multiset\n\n\nprotected instance has_coe {\u03b1 : Type u_1} : has_coe (List \u03b1) (multiset \u03b1) :=\n  has_coe.mk (Quot.mk setoid.r)\n\n@[simp] theorem quot_mk_to_coe {\u03b1 : Type u_1} (l : List \u03b1) : quotient.mk l = \u2191l :=\n  rfl\n\n@[simp] theorem quot_mk_to_coe' {\u03b1 : Type u_1} (l : List \u03b1) : Quot.mk has_equiv.equiv l = \u2191l :=\n  rfl\n\n@[simp] theorem quot_mk_to_coe'' {\u03b1 : Type u_1} (l : List \u03b1) : Quot.mk setoid.r l = \u2191l :=\n  rfl\n\n@[simp] theorem coe_eq_coe {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 = \u2191l\u2082 \u2194 l\u2081 ~ l\u2082 :=\n  quotient.eq\n\nprotected instance has_decidable_eq {\u03b1 : Type u_1} [DecidableEq \u03b1] : DecidableEq (multiset \u03b1) :=\n  sorry\n\n/-- defines a size for a multiset by referring to the size of the underlying list -/\nprotected def sizeof {\u03b1 : Type u_1} [SizeOf \u03b1] (s : multiset \u03b1) : \u2115 :=\n  quot.lift_on s sizeof sorry\n\nprotected instance has_sizeof {\u03b1 : Type u_1} [SizeOf \u03b1] : SizeOf (multiset \u03b1) :=\n  { sizeOf := multiset.sizeof }\n\n/-! ### Empty multiset -/\n\n/-- `0 : multiset \u03b1` is the empty set -/\nprotected def zero {\u03b1 : Type u_1} : multiset \u03b1 :=\n  \u2191[]\n\nprotected instance has_zero {\u03b1 : Type u_1} : HasZero (multiset \u03b1) :=\n  { zero := multiset.zero }\n\nprotected instance has_emptyc {\u03b1 : Type u_1} : has_emptyc (multiset \u03b1) :=\n  has_emptyc.mk 0\n\nprotected instance inhabited {\u03b1 : Type u_1} : Inhabited (multiset \u03b1) :=\n  { default := 0 }\n\n@[simp] theorem coe_nil_eq_zero {\u03b1 : Type u_1} : \u2191[] = 0 :=\n  rfl\n\n@[simp] theorem empty_eq_zero {\u03b1 : Type u_1} : \u2205 = 0 :=\n  rfl\n\ntheorem coe_eq_zero {\u03b1 : Type u_1} (l : List \u03b1) : \u2191l = 0 \u2194 l = [] :=\n  iff.trans coe_eq_coe list.perm_nil\n\n/-! ### `multiset.cons` -/\n\n/-- `cons a s` is the multiset which contains `s` plus one more\n  instance of `a`. -/\ndef cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(a :: l)) sorry\n\ninfixr:67 \" ::\u2098 \" => Mathlib.multiset.cons\n\nprotected instance has_insert {\u03b1 : Type u_1} : has_insert \u03b1 (multiset \u03b1) :=\n  has_insert.mk cons\n\n@[simp] theorem insert_eq_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : insert a s = a ::\u2098 s :=\n  rfl\n\n@[simp] theorem cons_coe {\u03b1 : Type u_1} (a : \u03b1) (l : List \u03b1) : a ::\u2098 \u2191l = \u2191(a :: l) :=\n  rfl\n\ntheorem singleton_coe {\u03b1 : Type u_1} (a : \u03b1) : a ::\u2098 0 = \u2191[a] :=\n  rfl\n\n@[simp] theorem cons_inj_left {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} (s : multiset \u03b1) : a ::\u2098 s = b ::\u2098 s \u2194 a = b := sorry\n\n@[simp] theorem cons_inj_right {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} : a ::\u2098 s = a ::\u2098 t \u2194 s = t := sorry\n\nprotected theorem induction {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (h\u2081 : p 0) (h\u2082 : \u2200 {a : \u03b1} {s : multiset \u03b1}, p s \u2192 p (a ::\u2098 s)) (s : multiset \u03b1) : p s :=\n  quot.induction_on s\n    fun (l : List \u03b1) => List.rec h\u2081 (fun (l_hd : \u03b1) (l_tl : List \u03b1) (ih : p (Quot.mk setoid.r l_tl)) => h\u2082 ih) l\n\nprotected theorem induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (s : multiset \u03b1) (h\u2081 : p 0) (h\u2082 : \u2200 {a : \u03b1} {s : multiset \u03b1}, p s \u2192 p (a ::\u2098 s)) : p s :=\n  multiset.induction h\u2081 h\u2082 s\n\ntheorem cons_swap {\u03b1 : Type u_1} (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) : a ::\u2098 b ::\u2098 s = b ::\u2098 a ::\u2098 s :=\n  quot.induction_on s fun (l : List \u03b1) => quotient.sound (list.perm.swap b a l)\n\n/-- Dependent recursor on multisets.\n\nTODO: should be @[recursor 6], but then the definition of `multiset.pi` fails with a stack\noverflow in `whnf`.\n-/\nprotected def rec {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} (C_0 : C 0) (C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)) (C_cons_heq : \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m), C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)) (m : multiset \u03b1) : C m :=\n  quotient.hrec_on m (List.rec C_0 fun (a : \u03b1) (l : List \u03b1) (b : C (quotient.mk l)) => C_cons a (quotient.mk l) b) sorry\n\nprotected def rec_on {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} (m : multiset \u03b1) (C_0 : C 0) (C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)) (C_cons_heq : \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m), C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)) : C m :=\n  multiset.rec C_0 C_cons C_cons_heq m\n\n@[simp] theorem rec_on_0 {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} {C_0 : C 0} {C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)} {C_cons_heq : \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m), C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)} : multiset.rec_on 0 C_0 C_cons C_cons_heq = C_0 :=\n  rfl\n\n@[simp] theorem rec_on_cons {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} {C_0 : C 0} {C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)} {C_cons_heq : \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m), C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)} (a : \u03b1) (m : multiset \u03b1) : multiset.rec_on (a ::\u2098 m) C_0 C_cons C_cons_heq = C_cons a m (multiset.rec_on m C_0 C_cons C_cons_heq) :=\n  quotient.induction_on m fun (l : List \u03b1) => rfl\n\n/-- `a \u2208 s` means that `a` has nonzero multiplicity in `s`. -/\ndef mem {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) :=\n  quot.lift_on s (fun (l : List \u03b1) => a \u2208 l) sorry\n\nprotected instance has_mem {\u03b1 : Type u_1} : has_mem \u03b1 (multiset \u03b1) :=\n  has_mem.mk mem\n\n@[simp] theorem mem_coe {\u03b1 : Type u_1} {a : \u03b1} {l : List \u03b1} : a \u2208 \u2191l \u2194 a \u2208 l :=\n  iff.rfl\n\nprotected instance decidable_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : Decidable (a \u2208 s) :=\n  quot.rec_on_subsingleton s (list.decidable_mem a)\n\n@[simp] theorem mem_cons {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} : a \u2208 b ::\u2098 s \u2194 a = b \u2228 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => iff.rfl\n\ntheorem mem_cons_of_mem {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 b ::\u2098 s :=\n  iff.mpr mem_cons (Or.inr h)\n\n@[simp] theorem mem_cons_self {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : a \u2208 a ::\u2098 s :=\n  iff.mpr mem_cons (Or.inl rfl)\n\ntheorem forall_mem_cons {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {a : \u03b1} {s : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 a ::\u2098 s \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 p x :=\n  quotient.induction_on' s fun (L : List \u03b1) => list.forall_mem_cons\n\ntheorem exists_cons_of_mem {\u03b1 : Type u_1} {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 \u2203 (t : multiset \u03b1), s = a ::\u2098 t := sorry\n\n@[simp] theorem not_mem_zero {\u03b1 : Type u_1} (a : \u03b1) : \u00aca \u2208 0 :=\n  id\n\ntheorem eq_zero_of_forall_not_mem {\u03b1 : Type u_1} {s : multiset \u03b1} : (\u2200 (x : \u03b1), \u00acx \u2208 s) \u2192 s = 0 := sorry\n\ntheorem eq_zero_iff_forall_not_mem {\u03b1 : Type u_1} {s : multiset \u03b1} : s = 0 \u2194 \u2200 (a : \u03b1), \u00aca \u2208 s :=\n  { mp := fun (h : s = 0) => Eq.symm h \u25b8 fun (_x : \u03b1) => not_false, mpr := eq_zero_of_forall_not_mem }\n\ntheorem exists_mem_of_ne_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2260 0 \u2192 \u2203 (a : \u03b1), a \u2208 s := sorry\n\n@[simp] theorem zero_ne_cons {\u03b1 : Type u_1} {a : \u03b1} {m : multiset \u03b1} : 0 \u2260 a ::\u2098 m :=\n  fun (h : 0 = a ::\u2098 m) => (fun (this : a \u2208 0) => not_mem_zero a this) (Eq.symm h \u25b8 mem_cons_self a m)\n\n@[simp] theorem cons_ne_zero {\u03b1 : Type u_1} {a : \u03b1} {m : multiset \u03b1} : a ::\u2098 m \u2260 0 :=\n  ne.symm zero_ne_cons\n\ntheorem cons_eq_cons {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {as : multiset \u03b1} {bs : multiset \u03b1} : a ::\u2098 as = b ::\u2098 bs \u2194 a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 (cs : multiset \u03b1), as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs := sorry\n\n/-! ### `multiset.subset` -/\n\n/-- `s \u2286 t` is the lift of the list subset relation. It means that any\n  element with nonzero multiplicity in `s` has nonzero multiplicity in `t`,\n  but it does not imply that the multiplicity of `a` in `s` is less or equal than in `t`;\n  see `s \u2264 t` for this relation. -/\nprotected def subset {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) :=\n  \u2200 {a : \u03b1}, a \u2208 s \u2192 a \u2208 t\n\nprotected instance has_subset {\u03b1 : Type u_1} : has_subset (multiset \u03b1) :=\n  has_subset.mk multiset.subset\n\n@[simp] theorem coe_subset {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 \u2286 \u2191l\u2082 \u2194 l\u2081 \u2286 l\u2082 :=\n  iff.rfl\n\n@[simp] theorem subset.refl {\u03b1 : Type u_1} (s : multiset \u03b1) : s \u2286 s :=\n  fun (a : \u03b1) (h : a \u2208 s) => h\n\ntheorem subset.trans {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : s \u2286 t \u2192 t \u2286 u \u2192 s \u2286 u :=\n  fun (h\u2081 : s \u2286 t) (h\u2082 : t \u2286 u) (a : \u03b1) (m : a \u2208 s) => h\u2082 (h\u2081 m)\n\ntheorem subset_iff {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : s \u2286 t \u2194 \u2200 {x : \u03b1}, x \u2208 s \u2192 x \u2208 t :=\n  iff.rfl\n\ntheorem mem_of_subset {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} (h : s \u2286 t) : a \u2208 s \u2192 a \u2208 t :=\n  h\n\n@[simp] theorem zero_subset {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 \u2286 s :=\n  fun (a : \u03b1) => not.elim (list.not_mem_nil a)\n\n@[simp] theorem cons_subset {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} : a ::\u2098 s \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t := sorry\n\ntheorem eq_zero_of_subset_zero {\u03b1 : Type u_1} {s : multiset \u03b1} (h : s \u2286 0) : s = 0 :=\n  eq_zero_of_forall_not_mem h\n\ntheorem subset_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2286 0 \u2194 s = 0 :=\n  { mp := eq_zero_of_subset_zero, mpr := fun (xeq : s = 0) => Eq.symm xeq \u25b8 subset.refl 0 }\n\n/-- Produces a list of the elements in the multiset using choice. -/\ndef to_list {\u03b1 : Type u_1} (s : multiset \u03b1) : List \u03b1 :=\n  classical.some sorry\n\n@[simp] theorem to_list_zero {\u03b1 : Type u_1} : to_list 0 = [] :=\n  iff.mp (coe_eq_zero (to_list 0)) (classical.some_spec (quotient.exists_rep multiset.zero))\n\ntheorem coe_to_list {\u03b1 : Type u_1} (s : multiset \u03b1) : \u2191(to_list s) = s :=\n  classical.some_spec (quotient.exists_rep s)\n\ntheorem mem_to_list {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : a \u2208 to_list s \u2194 a \u2208 s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 to_list s \u2194 a \u2208 s)) (Eq.symm (propext mem_coe))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 \u2191(to_list s) \u2194 a \u2208 s)) (coe_to_list s))) (iff.refl (a \u2208 s)))\n\n/-! ### Partial order on `multiset`s -/\n\n/-- `s \u2264 t` means that `s` is a sublist of `t` (up to permutation).\n  Equivalently, `s \u2264 t` means that `count a s \u2264 count a t` for all `a`. -/\nprotected def le {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) :=\n  quotient.lift_on\u2082 s t list.subperm sorry\n\nprotected instance partial_order {\u03b1 : Type u_1} : partial_order (multiset \u03b1) :=\n  partial_order.mk multiset.le (preorder.lt._default multiset.le) sorry sorry sorry\n\ntheorem subset_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2192 s \u2286 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.subperm.subset\n\ntheorem mem_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} (h : s \u2264 t) : a \u2208 s \u2192 a \u2208 t :=\n  mem_of_subset (subset_of_le h)\n\n@[simp] theorem coe_le {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 \u2264 \u2191l\u2082 \u2194 l\u2081 <+~ l\u2082 :=\n  iff.rfl\n\ntheorem le_induction_on {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 multiset \u03b1 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) (H : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 C \u2191l\u2081 \u2191l\u2082) : C s t := sorry\n\ntheorem zero_le {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.nil_sublist l)\n\ntheorem le_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2264 0 \u2194 s = 0 :=\n  { mp := fun (h : s \u2264 0) => le_antisymm h (zero_le s), mpr := le_of_eq }\n\ntheorem lt_cons_self {\u03b1 : Type u_1} (s : multiset \u03b1) (a : \u03b1) : s < a ::\u2098 s := sorry\n\ntheorem le_cons_self {\u03b1 : Type u_1} (s : multiset \u03b1) (a : \u03b1) : s \u2264 a ::\u2098 s :=\n  le_of_lt (lt_cons_self s a)\n\ntheorem cons_le_cons_iff {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} : a ::\u2098 s \u2264 a ::\u2098 t \u2194 s \u2264 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.subperm_cons a\n\ntheorem cons_le_cons {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2192 a ::\u2098 s \u2264 a ::\u2098 t :=\n  iff.mpr (cons_le_cons_iff a)\n\ntheorem le_cons_of_not_mem {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} (m : \u00aca \u2208 s) : s \u2264 a ::\u2098 t \u2194 s \u2264 t := sorry\n\n/-! ### Additive monoid -/\n\n/-- The sum of two multisets is the lift of the list append operation.\n  This adds the multiplicities of each element,\n  i.e. `count a (s + t) = count a s + count a t`. -/\nprotected def add {\u03b1 : Type u_1} (s\u2081 : multiset \u03b1) (s\u2082 : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s\u2081 s\u2082 (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(l\u2081 ++ l\u2082)) sorry\n\nprotected instance has_add {\u03b1 : Type u_1} : Add (multiset \u03b1) :=\n  { add := multiset.add }\n\n@[simp] theorem coe_add {\u03b1 : Type u_1} (s : List \u03b1) (t : List \u03b1) : \u2191s + \u2191t = \u2191(s ++ t) :=\n  rfl\n\nprotected theorem add_comm {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s + t = t + s :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => quot.sound list.perm_append_comm\n\nprotected theorem zero_add {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 + s = s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\ntheorem singleton_add {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : \u2191[a] + s = a ::\u2098 s :=\n  rfl\n\nprotected theorem add_le_add_left {\u03b1 : Type u_1} (s : multiset \u03b1) {t : multiset \u03b1} {u : multiset \u03b1} : s + t \u2264 s + u \u2194 t \u2264 u :=\n  quotient.induction_on\u2083 s t u fun (l\u2081 l\u2082 l\u2083 : List \u03b1) => list.subperm_append_left l\u2081\n\nprotected theorem add_left_cancel {\u03b1 : Type u_1} (s : multiset \u03b1) {t : multiset \u03b1} {u : multiset \u03b1} (h : s + t = s + u) : t = u :=\n  le_antisymm (iff.mp (multiset.add_le_add_left s) (le_of_eq h))\n    (iff.mp (multiset.add_le_add_left s) (le_of_eq (Eq.symm h)))\n\nprotected instance ordered_cancel_add_comm_monoid {\u03b1 : Type u_1} : ordered_cancel_add_comm_monoid (multiset \u03b1) :=\n  ordered_cancel_add_comm_monoid.mk Add.add sorry multiset.add_left_cancel 0 multiset.zero_add sorry multiset.add_comm\n    sorry partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n\ntheorem le_add_right {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 s + t := sorry\n\ntheorem le_add_left {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 t + s := sorry\n\ntheorem le_iff_exists_add {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2194 \u2203 (u : multiset \u03b1), t = s + u := sorry\n\nprotected instance canonically_ordered_add_monoid {\u03b1 : Type u_1} : canonically_ordered_add_monoid (multiset \u03b1) :=\n  canonically_ordered_add_monoid.mk ordered_cancel_add_comm_monoid.add sorry ordered_cancel_add_comm_monoid.zero sorry\n    sorry sorry ordered_cancel_add_comm_monoid.le ordered_cancel_add_comm_monoid.lt sorry sorry sorry sorry sorry 0\n    zero_le le_iff_exists_add\n\n@[simp] theorem cons_add {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : a ::\u2098 s + t = a ::\u2098 (s + t) := sorry\n\n@[simp] theorem add_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : s + a ::\u2098 t = a ::\u2098 (s + t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s + a ::\u2098 t = a ::\u2098 (s + t))) (add_comm s (a ::\u2098 t))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a ::\u2098 t + s = a ::\u2098 (s + t))) (cons_add a t s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a ::\u2098 (t + s) = a ::\u2098 (s + t))) (add_comm t s))) (Eq.refl (a ::\u2098 (s + t)))))\n\n@[simp] theorem mem_add {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} : a \u2208 s + t \u2194 a \u2208 s \u2228 a \u2208 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.mem_append\n\n/-! ### Cardinality -/\n\n/-- The cardinality of a multiset is the sum of the multiplicities\n  of all its elements, or simply the length of the underlying list. -/\ndef card {\u03b1 : Type u_1} : multiset \u03b1 \u2192+ \u2115 :=\n  add_monoid_hom.mk (fun (s : multiset \u03b1) => quot.lift_on s list.length sorry) sorry sorry\n\n@[simp] theorem coe_card {\u03b1 : Type u_1} (l : List \u03b1) : coe_fn card \u2191l = list.length l :=\n  rfl\n\n@[simp] theorem card_zero {\u03b1 : Type u_1} : coe_fn card 0 = 0 :=\n  rfl\n\ntheorem card_add {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : coe_fn card (s + t) = coe_fn card s + coe_fn card t :=\n  add_monoid_hom.map_add card s t\n\ntheorem card_smul {\u03b1 : Type u_1} (s : multiset \u03b1) (n : \u2115) : coe_fn card (n \u2022\u2115 s) = n * coe_fn card s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn card (n \u2022\u2115 s) = n * coe_fn card s)) (add_monoid_hom.map_nsmul card s n)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (n \u2022\u2115 coe_fn card s = n * coe_fn card s)) (nat.nsmul_eq_mul n (coe_fn card s))))\n      (Eq.refl (n * coe_fn card s)))\n\n@[simp] theorem card_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : coe_fn card (a ::\u2098 s) = coe_fn card s + 1 :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem card_singleton {\u03b1 : Type u_1} (a : \u03b1) : coe_fn card (a ::\u2098 0) = 1 := sorry\n\ntheorem card_le_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : coe_fn card s \u2264 coe_fn card t :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) => list.length_le_of_sublist\n\ntheorem eq_of_le_of_card_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : coe_fn card t \u2264 coe_fn card s \u2192 s = t :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (s : l\u2081 <+ l\u2082) (h\u2082 : coe_fn card \u2191l\u2082 \u2264 coe_fn card \u2191l\u2081) =>\n      congr_arg coe (list.eq_of_sublist_of_length_le s h\u2082)\n\ntheorem card_lt_of_lt {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s < t) : coe_fn card s < coe_fn card t :=\n  lt_of_not_ge fun (h\u2082 : coe_fn card s \u2265 coe_fn card t) => ne_of_lt h (eq_of_le_of_card_le (le_of_lt h) h\u2082)\n\ntheorem lt_iff_cons_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : s < t \u2194 \u2203 (a : \u03b1), a ::\u2098 s \u2264 t := sorry\n\n@[simp] theorem card_eq_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : coe_fn card s = 0 \u2194 s = 0 := sorry\n\ntheorem card_pos {\u03b1 : Type u_1} {s : multiset \u03b1} : 0 < coe_fn card s \u2194 s \u2260 0 :=\n  iff.trans pos_iff_ne_zero (not_congr card_eq_zero)\n\ntheorem card_pos_iff_exists_mem {\u03b1 : Type u_1} {s : multiset \u03b1} : 0 < coe_fn card s \u2194 \u2203 (a : \u03b1), a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_pos_iff_exists_mem\n\ndef strong_induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Sort u_2} (s : multiset \u03b1) : ((s : multiset \u03b1) \u2192 ((t : multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s) \u2192 p s :=\n  sorry\n\ntheorem strong_induction_eq {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Sort u_2} (s : multiset \u03b1) (H : (s : multiset \u03b1) \u2192 ((t : multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s) : strong_induction_on s H = H s fun (t : multiset \u03b1) (h : t < s) => strong_induction_on t H := sorry\n\ntheorem case_strong_induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (s : multiset \u03b1) (h\u2080 : p 0) (h\u2081 : \u2200 (a : \u03b1) (s : multiset \u03b1), (\u2200 (t : multiset \u03b1), t \u2264 s \u2192 p t) \u2192 p (a ::\u2098 s)) : p s := sorry\n\n/-! ### Singleton -/\n\nprotected instance has_singleton {\u03b1 : Type u_1} : has_singleton \u03b1 (multiset \u03b1) :=\n  has_singleton.mk fun (a : \u03b1) => a ::\u2098 0\n\nprotected instance is_lawful_singleton {\u03b1 : Type u_1} : is_lawful_singleton \u03b1 (multiset \u03b1) :=\n  is_lawful_singleton.mk fun (a : \u03b1) => rfl\n\n@[simp] theorem singleton_eq_singleton {\u03b1 : Type u_1} (a : \u03b1) : singleton a = a ::\u2098 0 :=\n  rfl\n\n@[simp] theorem mem_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} : b \u2208 a ::\u2098 0 \u2194 b = a := sorry\n\ntheorem mem_singleton_self {\u03b1 : Type u_1} (a : \u03b1) : a \u2208 a ::\u2098 0 :=\n  mem_cons_self a 0\n\ntheorem singleton_inj {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} : a ::\u2098 0 = b ::\u2098 0 \u2194 a = b :=\n  cons_inj_left 0\n\n@[simp] theorem singleton_ne_zero {\u03b1 : Type u_1} (a : \u03b1) : a ::\u2098 0 \u2260 0 :=\n  ne_of_gt (lt_cons_self 0 a)\n\n@[simp] theorem singleton_le {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} : a ::\u2098 0 \u2264 s \u2194 a \u2208 s := sorry\n\ntheorem card_eq_one {\u03b1 : Type u_1} {s : multiset \u03b1} : coe_fn card s = 1 \u2194 \u2203 (a : \u03b1), s = a ::\u2098 0 := sorry\n\n/-! ### `multiset.repeat` -/\n\n/-- `repeat a n` is the multiset containing only `a` with multiplicity `n`. -/\ndef repeat {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : multiset \u03b1 :=\n  \u2191(list.repeat a n)\n\n@[simp] theorem repeat_zero {\u03b1 : Type u_1} (a : \u03b1) : repeat a 0 = 0 :=\n  rfl\n\n@[simp] theorem repeat_succ {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : repeat a (n + 1) = a ::\u2098 repeat a n := sorry\n\n@[simp] theorem repeat_one {\u03b1 : Type u_1} (a : \u03b1) : repeat a 1 = a ::\u2098 0 := sorry\n\n@[simp] theorem card_repeat {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : coe_fn card (repeat a n) = n :=\n  list.length_repeat\n\ntheorem eq_of_mem_repeat {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {n : \u2115} : b \u2208 repeat a n \u2192 b = a :=\n  list.eq_of_mem_repeat\n\ntheorem eq_repeat' {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} : s = repeat a (coe_fn card s) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a := sorry\n\ntheorem eq_repeat_of_mem {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} : (\u2200 (b : \u03b1), b \u2208 s \u2192 b = a) \u2192 s = repeat a (coe_fn card s) :=\n  iff.mpr eq_repeat'\n\ntheorem eq_repeat {\u03b1 : Type u_1} {a : \u03b1} {n : \u2115} {s : multiset \u03b1} : s = repeat a n \u2194 coe_fn card s = n \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a := sorry\n\ntheorem repeat_subset_singleton {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : repeat a n \u2286 a ::\u2098 0 :=\n  list.repeat_subset_singleton\n\ntheorem repeat_le_coe {\u03b1 : Type u_1} {a : \u03b1} {n : \u2115} {l : List \u03b1} : repeat a n \u2264 \u2191l \u2194 list.repeat a n <+ l := sorry\n\n/-! ### Erasing one copy of an element -/\n\n/-- `erase s a` is the multiset that subtracts 1 from the\n  multiplicity of `a`. -/\ndef erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.erase l a)) sorry\n\n@[simp] theorem coe_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (l : List \u03b1) (a : \u03b1) : erase (\u2191l) a = \u2191(list.erase l a) :=\n  rfl\n\n@[simp] theorem erase_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : erase 0 a = 0 :=\n  rfl\n\n@[simp] theorem erase_cons_head {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : erase (a ::\u2098 s) a = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_cons_head a l)\n\n@[simp] theorem erase_cons_tail {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (s : multiset \u03b1) (h : b \u2260 a) : erase (b ::\u2098 s) a = b ::\u2098 erase s a :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_cons_tail l h)\n\n@[simp] theorem erase_of_not_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : \u00aca \u2208 s \u2192 erase s a = s :=\n  quot.induction_on s fun (l : List \u03b1) (h : \u00aca \u2208 Quot.mk setoid.r l) => congr_arg coe (list.erase_of_not_mem h)\n\n@[simp] theorem cons_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 a ::\u2098 erase s a = s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (h : a \u2208 Quot.mk setoid.r l) => quot.sound (list.perm.symm (list.perm_cons_erase h))\n\ntheorem le_cons_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) : s \u2264 a ::\u2098 erase s a :=\n  dite (a \u2208 s) (fun (h : a \u2208 s) => le_of_eq (Eq.symm (cons_erase h)))\n    fun (h : \u00aca \u2208 s) => eq.mpr (id (Eq._oldrec (Eq.refl (s \u2264 a ::\u2098 erase s a)) (erase_of_not_mem h))) (le_cons_self s a)\n\ntheorem erase_add_left_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} (t : multiset \u03b1) : a \u2208 s \u2192 erase (s + t) a = erase s a + t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) (h : a \u2208 quotient.mk l\u2081) => congr_arg coe (list.erase_append_left l\u2082 h)\n\ntheorem erase_add_right_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2208 t) : erase (s + t) a = s + erase t a := sorry\n\ntheorem erase_add_right_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} (t : multiset \u03b1) : \u00aca \u2208 s \u2192 erase (s + t) a = s + erase t a :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : \u00aca \u2208 quotient.mk l\u2081) => congr_arg coe (list.erase_append_right l\u2082 h)\n\ntheorem erase_add_left_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : \u00aca \u2208 t) : erase (s + t) a = erase s a + t := sorry\n\ntheorem erase_le {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : erase s a \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.erase_sublist a l)\n\n@[simp] theorem erase_lt {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : erase s a < s \u2194 a \u2208 s := sorry\n\ntheorem erase_subset {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : erase s a \u2286 s :=\n  subset_of_le (erase_le a s)\n\ntheorem mem_erase_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} (ab : a \u2260 b) : a \u2208 erase s b \u2194 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_erase_of_ne ab\n\ntheorem mem_of_mem_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} : a \u2208 erase s b \u2192 a \u2208 s :=\n  mem_of_subset (erase_subset b s)\n\ntheorem erase_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) (b : \u03b1) : erase (erase s a) b = erase (erase s b) a :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_comm a b l)\n\ntheorem erase_le_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (a : \u03b1) (h : s \u2264 t) : erase s a \u2264 erase t a :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.erase a h)\n\ntheorem erase_le_iff_le_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} : erase s a \u2264 t \u2194 s \u2264 a ::\u2098 t := sorry\n\n@[simp] theorem card_erase_of_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 coe_fn card (erase s a) = Nat.pred (coe_fn card s) :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_erase_of_mem\n\ntheorem card_erase_lt_of_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 coe_fn card (erase s a) < coe_fn card s :=\n  fun (h : a \u2208 s) => card_lt_of_lt (iff.mpr erase_lt h)\n\ntheorem card_erase_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : coe_fn card (erase s a) \u2264 coe_fn card s :=\n  card_le_of_le (erase_le a s)\n\n@[simp] theorem coe_reverse {\u03b1 : Type u_1} (l : List \u03b1) : \u2191(list.reverse l) = \u2191l :=\n  quot.sound (list.reverse_perm l)\n\n/-! ### `multiset.map` -/\n\n/-- `map f s` is the lift of the list `map` operation. The multiplicity\n  of `b` in `map f s` is the number of `a \u2208 s` (counting multiplicity)\n  such that `f a = b`. -/\ndef map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.map f l)) sorry\n\ntheorem forall_mem_map_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} {s : multiset \u03b1} : (\u2200 (y : \u03b2), y \u2208 map f s \u2192 p y) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 p (f x) :=\n  quotient.induction_on' s fun (L : List \u03b1) => list.forall_mem_map_iff\n\n@[simp] theorem coe_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) : map f \u2191l = \u2191(list.map f l) :=\n  rfl\n\n@[simp] theorem map_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) : map f 0 = 0 :=\n  rfl\n\n@[simp] theorem map_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (s : multiset \u03b1) : map f (a ::\u2098 s) = f a ::\u2098 map f s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\ntheorem map_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (singleton a) = singleton (f a) :=\n  rfl\n\ntheorem map_repeat {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (k : \u2115) : map f (repeat a k) = repeat (f a) k := sorry\n\n@[simp] theorem map_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) (t : multiset \u03b1) : map f (s + t) = map f s + map f t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.map_append f l\u2081 l\u2082)\n\nprotected instance map.is_add_monoid_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) : is_add_monoid_hom (map f) :=\n  is_add_monoid_hom.mk (map_zero f)\n\ntheorem map_nsmul {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (n : \u2115) (s : multiset \u03b1) : map f (n \u2022\u2115 s) = n \u2022\u2115 map f s :=\n  add_monoid_hom.map_nsmul (add_monoid_hom.of (map f)) s n\n\n@[simp] theorem mem_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : multiset \u03b1} : b \u2208 map f s \u2194 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_map\n\n@[simp] theorem card_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : coe_fn card (map f s) = coe_fn card s :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_map f l\n\n@[simp] theorem map_eq_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} : map f s = 0 \u2194 s = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (map f s = 0 \u2194 s = 0)) (Eq.symm (propext card_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn card (map f s) = 0 \u2194 s = 0)) (card_map f s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn card s = 0 \u2194 s = 0)) (propext card_eq_zero))) (iff.refl (s = 0))))\n\ntheorem mem_map_of_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : f a \u2208 map f s :=\n  iff.mpr mem_map (Exists.intro a { left := h, right := rfl })\n\ntheorem mem_map_of_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (H : function.injective f) {a : \u03b1} {s : multiset \u03b1} : f a \u2208 map f s \u2194 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_map_of_injective H\n\n@[simp] theorem map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : map g (map f s) = map (g \u2218 f) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_map g f l)\n\ntheorem map_id {\u03b1 : Type u_1} (s : multiset \u03b1) : map id s = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_id l)\n\n@[simp] theorem map_id' {\u03b1 : Type u_1} (s : multiset \u03b1) : map (fun (x : \u03b1) => x) s = s :=\n  map_id s\n\n@[simp] theorem map_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (b : \u03b2) : map (function.const \u03b1 b) s = repeat b (coe_fn card s) :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_const l b)\n\ntheorem map_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x) \u2192 map f s = map g s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (x : \u03b1), x \u2208 Quot.mk setoid.r l \u2192 f x = g x) => congr_arg coe (list.map_congr H)\n\ntheorem map_hcongr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b2' : Type u_2} {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'} (h : \u03b2 = \u03b2') (hf : \u2200 (a : \u03b1), a \u2208 m \u2192 f a == f' a) : map f m == map f' m := sorry\n\ntheorem eq_of_mem_map_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} {b\u2081 : \u03b2} {b\u2082 : \u03b2} {l : List \u03b1} (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) \u2191l) : b\u2081 = b\u2082 :=\n  eq_of_mem_repeat (eq.mp (Eq._oldrec (Eq.refl (b\u2081 \u2208 map (function.const \u03b1 b\u2082) \u2191l)) (map_const (\u2191l) b\u2082)) h)\n\n@[simp] theorem map_le_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : map f s \u2264 map f t :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.map f h)\n\n@[simp] theorem map_subset_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} {t : multiset \u03b1} (H : s \u2286 t) : map f s \u2286 map f t := sorry\n\n/-! ### `multiset.fold` -/\n\n/-- `foldl f H b s` is the lift of the list operation `foldl f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is right-commutative,\n  that is, `f (f b a\u2081) a\u2082 = f (f b a\u2082) a\u2081`. -/\ndef foldl {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => list.foldl f b l) sorry\n\n@[simp] theorem foldl_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) : foldl f H b 0 = b :=\n  rfl\n\n@[simp] theorem foldl_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (a : \u03b1) (s : multiset \u03b1) : foldl f H b (a ::\u2098 s) = foldl f H (f b a) s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem foldl_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) (t : multiset \u03b1) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.foldl_append f b l\u2081 l\u2082\n\n/-- `foldr f H b s` is the lift of the list operation `foldr f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is left-commutative,\n  that is, `f a\u2081 (f a\u2082 b) = f a\u2082 (f a\u2081 b)`. -/\ndef foldr {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => list.foldr f b l) sorry\n\n@[simp] theorem foldr_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) : foldr f H b 0 = b :=\n  rfl\n\n@[simp] theorem foldr_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (a : \u03b1) (s : multiset \u03b1) : foldr f H b (a ::\u2098 s) = f a (foldr f H b s) :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem foldr_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) (t : multiset \u03b1) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.foldr_append f b l\u2081 l\u2082\n\n@[simp] theorem coe_foldr {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : List \u03b1) : foldr f H b \u2191l = list.foldr f b l :=\n  rfl\n\n@[simp] theorem coe_foldl {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (l : List \u03b1) : foldl f H b \u2191l = list.foldl f b l :=\n  rfl\n\ntheorem coe_foldr_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : List \u03b1) : foldr f H b \u2191l = list.foldl (fun (x : \u03b2) (y : \u03b1) => f y x) b l :=\n  Eq.trans (Eq.symm (congr_arg (foldr f H b) (coe_reverse l))) (list.foldr_reverse f b l)\n\ntheorem foldr_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) : foldr f H b s = foldl (fun (x : \u03b2) (y : \u03b1) => f y x) (fun (x : \u03b2) (y z : \u03b1) => Eq.symm (H y z x)) b s :=\n  quot.induction_on s fun (l : List \u03b1) => coe_foldr_swap f H b l\n\ntheorem foldl_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) : foldl f H b s = foldr (fun (x : \u03b1) (y : \u03b2) => f y x) (fun (x y : \u03b1) (z : \u03b2) => Eq.symm (H z x y)) b s :=\n  Eq.symm (foldr_swap (fun (y : \u03b1) (x : \u03b2) => f x y) (fun (x y : \u03b1) (z : \u03b2) => Eq.symm (H z x y)) b s)\n\n/-- Product of a multiset given a commutative monoid structure on `\u03b1`.\n  `prod {a, b, c} = a * b * c` -/\ndef sum {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : multiset \u03b1 \u2192 \u03b1 :=\n  foldr Add.add sorry 0\n\ntheorem prod_eq_foldr {\u03b1 : Type u_1} [comm_monoid \u03b1] (s : multiset \u03b1) : prod s =\n  foldr Mul.mul\n    (fun (x y z : \u03b1) =>\n      eq.mpr\n        (id\n          (Eq.trans\n            ((fun (a a_1 : \u03b1) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : \u03b1) (e_2 : \u1fb0 = \u1fb0_1) => congr (congr_arg Eq e_1) e_2) (x * (y * z))\n              (x * (y * z)) (Eq.refl (x * (y * z))) (y * (x * z)) (x * (y * z)) (mul_left_comm y x z))\n            (propext (eq_self_iff_true (x * (y * z))))))\n        trivial)\n    1 s :=\n  rfl\n\ntheorem sum_eq_foldl {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (s : multiset \u03b1) : sum s =\n  foldl Add.add\n    (fun (x y z : \u03b1) =>\n      eq.mpr\n        (id\n          (Eq.trans\n            ((fun (a a_1 : \u03b1) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : \u03b1) (e_2 : \u1fb0 = \u1fb0_1) => congr (congr_arg Eq e_1) e_2) (x + y + z)\n              (x + y + z) (Eq.refl (x + y + z)) (x + z + y) (x + y + z) (add_right_comm x z y))\n            (propext (eq_self_iff_true (x + y + z)))))\n        trivial)\n    0 s := sorry\n\n@[simp] theorem coe_sum {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (l : List \u03b1) : sum \u2191l = list.sum l :=\n  sum_eq_foldl \u2191l\n\n@[simp] theorem sum_zero {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : sum 0 = 0 :=\n  rfl\n\n@[simp] theorem sum_cons {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) (s : multiset \u03b1) : sum (a ::\u2098 s) = a + sum s :=\n  foldr_cons Add.add sum._proof_1 0 a s\n\ntheorem sum_singleton {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) : sum (a ::\u2098 0) = a := sorry\n\n@[simp] theorem sum_add {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : sum (s + t) = sum s + sum t := sorry\n\nprotected instance sum.is_add_monoid_hom {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : is_add_monoid_hom sum :=\n  is_add_monoid_hom.mk sum_zero\n\ntheorem prod_smul {\u03b1 : Type u_1} [comm_monoid \u03b1] (m : multiset \u03b1) (n : \u2115) : prod (n \u2022\u2115 m) = prod m ^ n := sorry\n\n@[simp] theorem prod_repeat {\u03b1 : Type u_1} [comm_monoid \u03b1] (a : \u03b1) (n : \u2115) : prod (repeat a n) = a ^ n := sorry\n\n@[simp] theorem sum_repeat {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) (n : \u2115) : sum (repeat a n) = n \u2022\u2115 a :=\n  prod_repeat\n\ntheorem prod_map_one {\u03b1 : Type u_1} {\u03b3 : Type u_3} [comm_monoid \u03b3] {m : multiset \u03b1} : prod (map (fun (a : \u03b1) => 1) m) = 1 := sorry\n\ntheorem sum_map_zero {\u03b1 : Type u_1} {\u03b3 : Type u_3} [add_comm_monoid \u03b3] {m : multiset \u03b1} : sum (map (fun (a : \u03b1) => 0) m) = 0 := sorry\n\n@[simp] theorem sum_map_add {\u03b1 : Type u_1} {\u03b3 : Type u_3} [add_comm_monoid \u03b3] {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b3} : sum (map (fun (a : \u03b1) => f a + g a) m) = sum (map f m) + sum (map g m) := sorry\n\ntheorem prod_map_prod_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b3] (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} : prod (map (fun (a : \u03b1) => prod (map (fun (b : \u03b2) => f a b) n)) m) =\n  prod (map (fun (b : \u03b2) => prod (map (fun (a : \u03b1) => f a b) m)) n) := sorry\n\ntheorem sum_map_sum_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_monoid \u03b3] (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} : sum (map (fun (a : \u03b1) => sum (map (fun (b : \u03b2) => f a b) n)) m) =\n  sum (map (fun (b : \u03b2) => sum (map (fun (a : \u03b1) => f a b) m)) n) :=\n  prod_map_prod_map\n\ntheorem sum_map_mul_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} : sum (map (fun (a : \u03b1) => b * f a) s) = b * sum (map f s) := sorry\n\ntheorem sum_map_mul_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} : sum (map (fun (a : \u03b1) => f a * b) s) = sum (map f s) * b := sorry\n\ntheorem prod_ne_zero {R : Type u_1} [comm_semiring R] [no_zero_divisors R] [nontrivial R] {m : multiset R} : (\u2200 (x : R), x \u2208 m \u2192 x \u2260 0) \u2192 prod m \u2260 0 := sorry\n\ntheorem prod_eq_zero {\u03b1 : Type u_1} [comm_semiring \u03b1] {s : multiset \u03b1} (h : 0 \u2208 s) : prod s = 0 := sorry\n\ntheorem sum_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1] [add_comm_monoid \u03b2] (s : multiset \u03b1) (f : \u03b1 \u2192+ \u03b2) : sum (map (\u21d1f) s) = coe_fn f (sum s) := sorry\n\ntheorem prod_hom_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b2] [comm_monoid \u03b3] (s : multiset \u03b1) {r : \u03b2 \u2192 \u03b3 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h\u2081 : r 1 1) (h\u2082 : \u2200 {a : \u03b1} {b : \u03b2} {c : \u03b3}, r b c \u2192 r (f a * b) (g a * c)) : r (prod (map f s)) (prod (map g s)) := sorry\n\ntheorem dvd_prod {\u03b1 : Type u_1} [comm_monoid \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 a \u2223 prod s := sorry\n\ntheorem prod_dvd_prod {\u03b1 : Type u_1} [comm_monoid \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : prod s \u2223 prod t := sorry\n\ntheorem prod_eq_zero_iff {\u03b1 : Type u_1} [comm_cancel_monoid_with_zero \u03b1] [nontrivial \u03b1] {s : multiset \u03b1} : prod s = 0 \u2194 0 \u2208 s := sorry\n\ntheorem sum_nonneg {\u03b1 : Type u_1} [ordered_add_comm_monoid \u03b1] {m : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 m \u2192 0 \u2264 x) \u2192 0 \u2264 sum m := sorry\n\ntheorem single_le_prod {\u03b1 : Type u_1} [ordered_comm_monoid \u03b1] {m : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 m \u2192 1 \u2264 x) \u2192 \u2200 (x : \u03b1), x \u2208 m \u2192 x \u2264 prod m := sorry\n\ntheorem all_one_of_le_one_le_of_prod_eq_one {\u03b1 : Type u_1} [ordered_comm_monoid \u03b1] {m : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 m \u2192 1 \u2264 x) \u2192 prod m = 1 \u2192 \u2200 (x : \u03b1), x \u2208 m \u2192 x = 1 := sorry\n\ntheorem sum_eq_zero_iff {\u03b1 : Type u_1} [canonically_ordered_add_monoid \u03b1] {m : multiset \u03b1} : sum m = 0 \u2194 \u2200 (x : \u03b1), x \u2208 m \u2192 x = 0 := sorry\n\ntheorem le_sum_of_subadditive {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1] [ordered_add_comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) (h_zero : f 0 = 0) (h_add : \u2200 (x y : \u03b1), f (x + y) \u2264 f x + f y) (s : multiset \u03b1) : f (sum s) \u2264 sum (map f s) := sorry\n\ntheorem abs_sum_le_sum_abs {\u03b1 : Type u_1} [linear_ordered_field \u03b1] {s : multiset \u03b1} : abs (sum s) \u2264 sum (map abs s) :=\n  le_sum_of_subadditive abs abs_zero abs_add s\n\ntheorem dvd_sum {\u03b1 : Type u_1} [comm_semiring \u03b1] {a : \u03b1} {s : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 s \u2192 a \u2223 x) \u2192 a \u2223 sum s := sorry\n\n@[simp] theorem sum_map_singleton {\u03b1 : Type u_1} (s : multiset \u03b1) : sum (map (fun (a : \u03b1) => a ::\u2098 0) s) = s := sorry\n\n/-! ### Join -/\n\n/-- `join S`, where `S` is a multiset of multisets, is the lift of the list join\n  operation, that is, the union of all the sets.\n\n     join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2} -/\ndef join {\u03b1 : Type u_1} : multiset (multiset \u03b1) \u2192 multiset \u03b1 :=\n  sum\n\ntheorem coe_join {\u03b1 : Type u_1} (L : List (List \u03b1)) : join \u2191(list.map coe L) = \u2191(list.join L) := sorry\n\n@[simp] theorem join_zero {\u03b1 : Type u_1} : join 0 = 0 :=\n  rfl\n\n@[simp] theorem join_cons {\u03b1 : Type u_1} (s : multiset \u03b1) (S : multiset (multiset \u03b1)) : join (s ::\u2098 S) = s + join S :=\n  sum_cons s S\n\n@[simp] theorem join_add {\u03b1 : Type u_1} (S : multiset (multiset \u03b1)) (T : multiset (multiset \u03b1)) : join (S + T) = join S + join T :=\n  sum_add S T\n\n@[simp] theorem mem_join {\u03b1 : Type u_1} {a : \u03b1} {S : multiset (multiset \u03b1)} : a \u2208 join S \u2194 \u2203 (s : multiset \u03b1), \u2203 (H : s \u2208 S), a \u2208 s := sorry\n\n@[simp] theorem card_join {\u03b1 : Type u_1} (S : multiset (multiset \u03b1)) : coe_fn card (join S) = sum (map (\u21d1card) S) := sorry\n\n/-! ### `multiset.bind` -/\n\n/-- `bind s f` is the monad bind operation, defined as `join (map f s)`.\n  It is the union of `f a` as `a` ranges over `s`. -/\ndef bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : multiset \u03b2 :=\n  join (map f s)\n\n@[simp] theorem coe_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l : List \u03b1) (f : \u03b1 \u2192 List \u03b2) : (bind \u2191l fun (a : \u03b1) => \u2191(f a)) = \u2191(list.bind l f) := sorry\n\n@[simp] theorem zero_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 multiset \u03b2) : bind 0 f = 0 :=\n  rfl\n\n@[simp] theorem cons_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : bind (a ::\u2098 s) f = f a + bind s f := sorry\n\n@[simp] theorem add_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : bind (s + t) f = bind s f + bind t f := sorry\n\n@[simp] theorem bind_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) : (bind s fun (a : \u03b1) => 0) = 0 := sorry\n\n@[simp] theorem bind_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) (g : \u03b1 \u2192 multiset \u03b2) : (bind s fun (a : \u03b1) => f a + g a) = bind s f + bind s g := sorry\n\n@[simp] theorem bind_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 multiset \u03b2) : (bind s fun (a : \u03b1) => f a ::\u2098 g a) = map f s + bind s g := sorry\n\n@[simp] theorem mem_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} : b \u2208 bind s f \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a := sorry\n\n@[simp] theorem card_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : coe_fn card (bind s f) = sum (map (\u21d1card \u2218 f) s) := sorry\n\ntheorem bind_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b1 \u2192 multiset \u03b2} {m : multiset \u03b1} : (\u2200 (a : \u03b1), a \u2208 m \u2192 f a = g a) \u2192 bind m f = bind m g := sorry\n\ntheorem bind_hcongr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b2' : Type u_2} {m : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {f' : \u03b1 \u2192 multiset \u03b2'} (h : \u03b2 = \u03b2') (hf : \u2200 (a : \u03b1), a \u2208 m \u2192 f a == f' a) : bind m f == bind m f' := sorry\n\ntheorem map_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : \u03b1 \u2192 multiset \u03b2) (f : \u03b2 \u2192 \u03b3) : map f (bind m n) = bind m fun (a : \u03b1) => map f (n a) := sorry\n\ntheorem bind_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : \u03b2 \u2192 multiset \u03b3) (f : \u03b1 \u2192 \u03b2) : bind (map f m) n = bind m fun (a : \u03b1) => n (f a) := sorry\n\ntheorem bind_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {s : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b2 \u2192 multiset \u03b3} : bind (bind s f) g = bind s fun (a : \u03b1) => bind (f a) g := sorry\n\ntheorem bind_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 multiset \u03b3} : (bind m fun (a : \u03b1) => bind n fun (b : \u03b2) => f a b) = bind n fun (b : \u03b2) => bind m fun (a : \u03b1) => f a b := sorry\n\ntheorem bind_map_comm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} : (bind m fun (a : \u03b1) => map (fun (b : \u03b2) => f a b) n) = bind n fun (b : \u03b2) => map (fun (a : \u03b1) => f a b) m := sorry\n\n@[simp] theorem sum_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b2] (s : multiset \u03b1) (t : \u03b1 \u2192 multiset \u03b2) : sum (bind s t) = sum (map (fun (a : \u03b1) => sum (t a)) s) := sorry\n\n/-! ### Product of two `multiset`s -/\n\n/-- The multiplicity of `(a, b)` in `product s t` is\n  the product of the multiplicity of `a` in `s` and `b` in `t`. -/\ndef product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u00d7 \u03b2) :=\n  bind s fun (a : \u03b1) => map (Prod.mk a) t\n\n@[simp] theorem coe_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) : product \u2191l\u2081 \u2191l\u2082 = \u2191(list.product l\u2081 l\u2082) := sorry\n\n@[simp] theorem zero_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (t : multiset \u03b2) : product 0 t = 0 :=\n  rfl\n\n@[simp] theorem cons_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b2) : product (a ::\u2098 s) t = map (Prod.mk a) t + product s t := sorry\n\n@[simp] theorem product_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (b : \u03b2) : product (a ::\u2098 0) (b ::\u2098 0) = (a, b) ::\u2098 0 :=\n  rfl\n\n@[simp] theorem add_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b1) (u : multiset \u03b2) : product (s + t) u = product s u + product t u := sorry\n\n@[simp] theorem product_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2) (u : multiset \u03b2) : product s (t + u) = product s t + product s u := sorry\n\n@[simp] theorem mem_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : multiset \u03b1} {t : multiset \u03b2} {p : \u03b1 \u00d7 \u03b2} : p \u2208 product s t \u2194 prod.fst p \u2208 s \u2227 prod.snd p \u2208 t := sorry\n\n@[simp] theorem card_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2) : coe_fn card (product s t) = coe_fn card s * coe_fn card t := sorry\n\n/-! ### Sigma multiset -/\n\n/-- `sigma s t` is the dependent version of `product`. It is the sum of\n  `(a, b)` as `a` ranges over `s` and `b` ranges over `t a`. -/\nprotected def sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1) (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) : multiset (sigma fun (a : \u03b1) => \u03c3 a) :=\n  bind s fun (a : \u03b1) => map (sigma.mk a) (t a)\n\n@[simp] theorem coe_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (l\u2081 : List \u03b1) (l\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)) : (multiset.sigma \u2191l\u2081 fun (a : \u03b1) => \u2191(l\u2082 a)) = \u2191(list.sigma l\u2081 l\u2082) := sorry\n\n@[simp] theorem zero_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) : multiset.sigma 0 t = 0 :=\n  rfl\n\n@[simp] theorem cons_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (a : \u03b1) (s : multiset \u03b1) (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) : multiset.sigma (a ::\u2098 s) t = map (sigma.mk a) (t a) + multiset.sigma s t := sorry\n\n@[simp] theorem sigma_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (b : \u03b1 \u2192 \u03b2) : (multiset.sigma (a ::\u2098 0) fun (a : \u03b1) => b a ::\u2098 0) = sigma.mk a (b a) ::\u2098 0 :=\n  rfl\n\n@[simp] theorem add_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1) (t : multiset \u03b1) (u : (a : \u03b1) \u2192 multiset (\u03c3 a)) : multiset.sigma (s + t) u = multiset.sigma s u + multiset.sigma t u := sorry\n\n@[simp] theorem sigma_add {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1) (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) (u : (a : \u03b1) \u2192 multiset (\u03c3 a)) : (multiset.sigma s fun (a : \u03b1) => t a + u a) = multiset.sigma s t + multiset.sigma s u := sorry\n\n@[simp] theorem mem_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} {s : multiset \u03b1} {t : (a : \u03b1) \u2192 multiset (\u03c3 a)} {p : sigma fun (a : \u03b1) => \u03c3 a} : p \u2208 multiset.sigma s t \u2194 sigma.fst p \u2208 s \u2227 sigma.snd p \u2208 t (sigma.fst p) := sorry\n\n@[simp] theorem card_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1) (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) : coe_fn card (multiset.sigma s t) = sum (map (fun (a : \u03b1) => coe_fn card (t a)) s) := sorry\n\n/-! ### Map for partial functions -/\n\n/-- Lift of the list `pmap` operation. Map a partial function `f` over a multiset\n  `s` whose elements are all in the domain of `f`. -/\ndef pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) : (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 multiset \u03b2 :=\n  quot.rec_on s (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => \u2191(list.pmap f l H)) sorry\n\n@[simp] theorem coe_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) : pmap f (\u2191l) H = \u2191(list.pmap f l H) :=\n  rfl\n\n@[simp] theorem pmap_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (h : \u2200 (a : \u03b1), a \u2208 0 \u2192 p a) : pmap f 0 h = 0 :=\n  rfl\n\n@[simp] theorem pmap_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (a : \u03b1) (m : multiset \u03b1) (h : \u2200 (b : \u03b1), b \u2208 a ::\u2098 m \u2192 p b) : pmap f (a ::\u2098 m) h =\n  f a (h a (mem_cons_self a m)) ::\u2098 pmap f m fun (a_1 : \u03b1) (ha : a_1 \u2208 m) => h a_1 (mem_cons_of_mem ha) :=\n  quotient.induction_on m fun (l : List \u03b1) (h : \u2200 (b : \u03b1), b \u2208 a ::\u2098 quotient.mk l \u2192 p b) => rfl\n\n/-- \"Attach\" a proof that `a \u2208 s` to each element `a` in `s` to produce\n  a multiset on `{x // x \u2208 s}`. -/\ndef attach {\u03b1 : Type u_1} (s : multiset \u03b1) : multiset (Subtype fun (x : \u03b1) => x \u2208 s) :=\n  pmap Subtype.mk s sorry\n\n@[simp] theorem coe_attach {\u03b1 : Type u_1} (l : List \u03b1) : attach \u2191l = \u2191(list.attach l) :=\n  rfl\n\ntheorem sizeof_lt_sizeof_of_mem {\u03b1 : Type u_1} [SizeOf \u03b1] {x : \u03b1} {s : multiset \u03b1} (hx : x \u2208 s) : sizeof x < sizeof s := sorry\n\ntheorem pmap_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : pmap (fun (a : \u03b1) (_x : p a) => f a) s H = map f s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => congr_arg coe (list.pmap_eq_map p f l H)\n\ntheorem pmap_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2} {g : (a : \u03b1) \u2192 q a \u2192 \u03b2} (s : multiset \u03b1) {H\u2081 : \u2200 (a : \u03b1), a \u2208 s \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 s \u2192 q a} (h : \u2200 (a : \u03b1) (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082) : pmap f s H\u2081 = pmap g s H\u2082 := sorry\n\ntheorem map_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3) (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : map g (pmap f s H) = pmap (fun (a : \u03b1) (h : p a) => g (f a h)) s H :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => congr_arg coe (list.map_pmap g f l H)\n\ntheorem pmap_eq_map_attach {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : pmap f s H =\n  map (fun (x : Subtype fun (x : \u03b1) => x \u2208 s) => f (subtype.val x) (H (subtype.val x) (subtype.property x))) (attach s) :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => congr_arg coe (list.pmap_eq_map_attach f l H)\n\ntheorem attach_map_val {\u03b1 : Type u_1} (s : multiset \u03b1) : map subtype.val (attach s) = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.attach_map_val l)\n\n@[simp] theorem mem_attach {\u03b1 : Type u_1} (s : multiset \u03b1) (x : Subtype fun (x : \u03b1) => x \u2208 s) : x \u2208 attach s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_attach l\n\n@[simp] theorem mem_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2} {s : multiset \u03b1} {H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a} {b : \u03b2} : b \u2208 pmap f s H \u2194 \u2203 (a : \u03b1), \u2203 (h : a \u2208 s), f a (H a h) = b :=\n  quot.induction_on s (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => list.mem_pmap) H\n\n@[simp] theorem card_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : coe_fn card (pmap f s H) = coe_fn card s :=\n  quot.induction_on s (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => list.length_pmap) H\n\n@[simp] theorem card_attach {\u03b1 : Type u_1} {m : multiset \u03b1} : coe_fn card (attach m) = coe_fn card m :=\n  card_pmap Subtype.mk m (attach._proof_1 m)\n\n@[simp] theorem attach_zero {\u03b1 : Type u_1} : attach 0 = 0 :=\n  rfl\n\ntheorem attach_cons {\u03b1 : Type u_1} (a : \u03b1) (m : multiset \u03b1) : attach (a ::\u2098 m) =\n  { val := a, property := mem_cons_self a m } ::\u2098\n    map\n      (fun (p : Subtype fun (x : \u03b1) => x \u2208 m) =>\n        { val := subtype.val p, property := mem_cons_of_mem (subtype.property p) })\n      (attach m) := sorry\n\nprotected def decidable_forall_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : \u03b1 \u2192 Prop} [hp : (a : \u03b1) \u2192 Decidable (p a)] : Decidable (\u2200 (a : \u03b1), a \u2208 m \u2192 p a) :=\n  quotient.rec_on_subsingleton m fun (l : List \u03b1) => decidable_of_iff (\u2200 (a : \u03b1), a \u2208 l \u2192 p a) sorry\n\nprotected instance decidable_dforall_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : (a : \u03b1) \u2192 a \u2208 m \u2192 Prop} [hp : (a : \u03b1) \u2192 (h : a \u2208 m) \u2192 Decidable (p a h)] : Decidable (\u2200 (a : \u03b1) (h : a \u2208 m), p a h) :=\n  decidable_of_decidable_of_iff multiset.decidable_forall_multiset sorry\n\n/-- decidable equality for functions whose domain is bounded by multisets -/\nprotected instance decidable_eq_pi_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {\u03b2 : \u03b1 \u2192 Type u_2} [h : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)] : DecidableEq ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a) :=\n  fun (f g : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a) => decidable_of_iff (\u2200 (a : \u03b1) (h : a \u2208 m), f a h = g a h) sorry\n\ndef decidable_exists_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : \u03b1 \u2192 Prop} [decidable_pred p] : Decidable (\u2203 (x : \u03b1), \u2203 (H : x \u2208 m), p x) :=\n  quotient.rec_on_subsingleton m list.decidable_exists_mem\n\nprotected instance decidable_dexists_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : (a : \u03b1) \u2192 a \u2208 m \u2192 Prop} [hp : (a : \u03b1) \u2192 (h : a \u2208 m) \u2192 Decidable (p a h)] : Decidable (\u2203 (a : \u03b1), \u2203 (h : a \u2208 m), p a h) :=\n  decidable_of_decidable_of_iff decidable_exists_multiset sorry\n\n/-! ### Subtraction -/\n\n/-- `s - t` is the multiset such that\n  `count a (s - t) = count a s - count a t` for all `a`. -/\nprotected def sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s t (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(list.diff l\u2081 l\u2082)) sorry\n\nprotected instance has_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] : Sub (multiset \u03b1) :=\n  { sub := multiset.sub }\n\n@[simp] theorem coe_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : List \u03b1) (t : List \u03b1) : \u2191s - \u2191t = \u2191(list.diff s t) :=\n  rfl\n\ntheorem sub_eq_fold_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s - t = foldl erase erase_comm s t := sorry\n\n@[simp] theorem sub_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : s - 0 = s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem sub_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : s - a ::\u2098 t = erase s a - t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.diff_cons l\u2081 l\u2082 a)\n\ntheorem add_sub_of_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : s + (t - s) = t := sorry\n\ntheorem sub_add' {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : s - (t + u) = s - t - u :=\n  quotient.induction_on\u2083 s t u fun (l\u2081 l\u2082 l\u2083 : List \u03b1) => congr_arg coe (list.diff_append l\u2081 l\u2082 l\u2083)\n\ntheorem sub_add_cancel {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : t \u2264 s) : s - t + t = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s - t + t = s)) (add_comm (s - t) t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t + (s - t) = s)) (add_sub_of_le h))) (Eq.refl s))\n\n@[simp] theorem add_sub_cancel_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s + t - s = t := sorry\n\n@[simp] theorem add_sub_cancel {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s + t - t = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s + t - t = s)) (add_comm s t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t + s - t = s)) (add_sub_cancel_left t s))) (Eq.refl s))\n\ntheorem sub_le_sub_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) (u : multiset \u03b1) : s - u \u2264 t - u := sorry\n\ntheorem sub_le_sub_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) (u : multiset \u03b1) : u - t \u2264 u - s := sorry\n\ntheorem sub_le_iff_le_add {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : s - t \u2264 u \u2194 s \u2264 u + t := sorry\n\ntheorem le_sub_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 s - t + t :=\n  iff.mp sub_le_iff_le_add (le_refl (s - t))\n\ntheorem sub_le_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s - t \u2264 s :=\n  iff.mpr sub_le_iff_le_add (le_add_right s t)\n\n@[simp] theorem card_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : t \u2264 s) : coe_fn card (s - t) = coe_fn card s - coe_fn card t := sorry\n\n/-! ### Union -/\n\n/-- `s \u222a t` is the lattice join operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u222a t` is the maximum\n  of the multiplicities in `s` and `t`. -/\ndef union {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 :=\n  s - t + t\n\nprotected instance has_union {\u03b1 : Type u_1} [DecidableEq \u03b1] : has_union (multiset \u03b1) :=\n  has_union.mk union\n\ntheorem union_def {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u222a t = s - t + t :=\n  rfl\n\ntheorem le_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 s \u222a t :=\n  le_sub_add s t\n\ntheorem le_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : t \u2264 s \u222a t :=\n  le_add_left t (s - t)\n\ntheorem eq_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} : t \u2264 s \u2192 s \u222a t = s :=\n  sub_add_cancel\n\ntheorem union_le_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) (u : multiset \u03b1) : s \u222a u \u2264 t \u222a u :=\n  add_le_add_right (sub_le_sub_right h u) u\n\ntheorem union_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s \u222a t \u2264 u)) (Eq.symm (eq_union_left h\u2082)))) (union_le_union_right h\u2081 t)\n\n@[simp] theorem mem_union {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} : a \u2208 s \u222a t \u2194 a \u2208 s \u2228 a \u2208 t :=\n  { mp := fun (h : a \u2208 s \u222a t) => or.imp_left (mem_of_le (sub_le_self s t)) (iff.mp mem_add h),\n    mpr := Or._oldrec (mem_of_le (le_union_left s t)) (mem_of_le (le_union_right s t)) }\n\n@[simp] theorem map_union {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : function.injective f) {s : multiset \u03b1} {t : multiset \u03b1} : map f (s \u222a t) = map f s \u222a map f t := sorry\n\n/-! ### Intersection -/\n\n/-- `s \u2229 t` is the lattice meet operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u2229 t` is the minimum\n  of the multiplicities in `s` and `t`. -/\ndef inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s t (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(list.bag_inter l\u2081 l\u2082)) sorry\n\nprotected instance has_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] : has_inter (multiset \u03b1) :=\n  has_inter.mk inter\n\n@[simp] theorem inter_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : s \u2229 0 = 0 :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.bag_inter_nil l)\n\n@[simp] theorem zero_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : 0 \u2229 s = 0 :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.nil_bag_inter l)\n\n@[simp] theorem cons_inter_of_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} : a \u2208 t \u2192 (a ::\u2098 s) \u2229 t = a ::\u2098 s \u2229 erase t a :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : a \u2208 quotient.mk l\u2082) => congr_arg coe (list.cons_bag_inter_of_pos l\u2081 h)\n\n@[simp] theorem cons_inter_of_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} : \u00aca \u2208 t \u2192 (a ::\u2098 s) \u2229 t = s \u2229 t :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : \u00aca \u2208 quotient.mk l\u2082) => congr_arg coe (list.cons_bag_inter_of_neg l\u2081 h)\n\ntheorem inter_le_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2229 t \u2264 s :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.sublist.subperm (list.bag_inter_sublist_left l\u2081 l\u2082)\n\ntheorem inter_le_right {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2229 t \u2264 t := sorry\n\ntheorem le_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h\u2081 : s \u2264 t) (h\u2082 : s \u2264 u) : s \u2264 t \u2229 u := sorry\n\n@[simp] theorem mem_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2229 t \u2194 a \u2208 s \u2227 a \u2208 t := sorry\n\nprotected instance lattice {\u03b1 : Type u_1} [DecidableEq \u03b1] : lattice (multiset \u03b1) :=\n  lattice.mk has_union.union partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry has_inter.inter sorry\n    sorry sorry\n\n@[simp] theorem sup_eq_union {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2294 t = s \u222a t :=\n  rfl\n\n@[simp] theorem inf_eq_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2293 t = s \u2229 t :=\n  rfl\n\n@[simp] theorem le_inter_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : s \u2264 t \u2229 u \u2194 s \u2264 t \u2227 s \u2264 u :=\n  le_inf_iff\n\n@[simp] theorem union_le_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : s \u222a t \u2264 u \u2194 s \u2264 u \u2227 t \u2264 u :=\n  sup_le_iff\n\nprotected instance semilattice_inf_bot {\u03b1 : Type u_1} [DecidableEq \u03b1] : semilattice_inf_bot (multiset \u03b1) :=\n  semilattice_inf_bot.mk 0 lattice.le lattice.lt sorry sorry sorry zero_le lattice.inf sorry sorry sorry\n\ntheorem union_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u222a t = t \u222a s :=\n  sup_comm\n\ntheorem inter_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u2229 t = t \u2229 s :=\n  inf_comm\n\ntheorem eq_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : s \u222a t = t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s \u222a t = t)) (union_comm s t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t \u222a s = t)) (eq_union_left h))) (Eq.refl t))\n\ntheorem union_le_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) (u : multiset \u03b1) : u \u222a s \u2264 u \u222a t :=\n  sup_le_sup_left h u\n\ntheorem union_le_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u222a t \u2264 s + t :=\n  union_le (le_add_right s t) (le_add_left t s)\n\ntheorem union_add_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) (u : multiset \u03b1) : s \u222a t + u = s + u \u222a (t + u) := sorry\n\ntheorem add_union_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) (u : multiset \u03b1) : s + (t \u222a u) = s + t \u222a (s + u) := sorry\n\ntheorem cons_union_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : a ::\u2098 (s \u222a t) = a ::\u2098 s \u222a a ::\u2098 t := sorry\n\ntheorem inter_add_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) (u : multiset \u03b1) : s \u2229 t + u = (s + u) \u2229 (t + u) := sorry\n\ntheorem add_inter_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) (u : multiset \u03b1) : s + t \u2229 u = (s + t) \u2229 (s + u) := sorry\n\ntheorem cons_inter_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : a ::\u2098 s \u2229 t = (a ::\u2098 s) \u2229 (a ::\u2098 t) := sorry\n\ntheorem union_add_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s \u222a t + s \u2229 t = s + t := sorry\n\ntheorem sub_add_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s - t + s \u2229 t = s := sorry\n\ntheorem sub_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s - s \u2229 t = s - t :=\n  add_right_cancel\n    (eq.mpr (id (Eq._oldrec (Eq.refl (s - s \u2229 t + s \u2229 t = s - t + s \u2229 t)) (sub_add_inter s t)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (s - s \u2229 t + s \u2229 t = s)) (sub_add_cancel (inter_le_left s t)))) (Eq.refl s)))\n\n/-! ### `multiset.filter` -/\n\n/-- `filter p s` returns the elements in `s` (with the same multiplicities)\n  which satisfy `p`, and removes the rest. -/\ndef filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.filter p l)) sorry\n\n@[simp] theorem coe_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) : filter p \u2191l = \u2191(list.filter p l) :=\n  rfl\n\n@[simp] theorem filter_zero {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : filter p 0 = 0 :=\n  rfl\n\ntheorem filter_congr {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} [decidable_pred p] [decidable_pred q] {s : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 s \u2192 (p x \u2194 q x)) \u2192 filter p s = filter q s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (h : \u2200 (x : \u03b1), x \u2208 Quot.mk setoid.r l \u2192 (p x \u2194 q x)) => congr_arg coe (list.filter_congr h)\n\n@[simp] theorem filter_add {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s + t) = filter p s + filter p t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.filter_append l\u2081 l\u2082)\n\n@[simp] theorem filter_le {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : filter p s \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.filter_sublist l)\n\n@[simp] theorem filter_subset {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : filter p s \u2286 s :=\n  subset_of_le (filter_le p s)\n\ntheorem filter_le_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : filter p s \u2264 filter p t :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.filter_sublist_filter p h)\n\n@[simp] theorem filter_cons_of_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (s : multiset \u03b1) : p a \u2192 filter p (a ::\u2098 s) = a ::\u2098 filter p s :=\n  quot.induction_on s fun (l : List \u03b1) (h : p a) => congr_arg coe (list.filter_cons_of_pos l h)\n\n@[simp] theorem filter_cons_of_neg {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (s : multiset \u03b1) : \u00acp a \u2192 filter p (a ::\u2098 s) = filter p s :=\n  quot.induction_on s fun (l : List \u03b1) (h : \u00acp a) => congr_arg coe (list.filter_cons_of_neg l h)\n\n@[simp] theorem mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_filter\n\ntheorem of_mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 filter p s) : p a :=\n  and.right (iff.mp mem_filter h)\n\ntheorem mem_of_mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 filter p s) : a \u2208 s :=\n  and.left (iff.mp mem_filter h)\n\ntheorem mem_filter_of_mem {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : multiset \u03b1} (m : a \u2208 l) (h : p a) : a \u2208 filter p l :=\n  iff.mpr mem_filter { left := m, right := h }\n\ntheorem filter_eq_self {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} : filter p s = s \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 p a := sorry\n\ntheorem filter_eq_nil {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} : filter p s = 0 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u00acp a := sorry\n\ntheorem le_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 filter p t \u2194 s \u2264 t \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 p a := sorry\n\n@[simp] theorem filter_sub {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s - t) = filter p s - filter p t := sorry\n\n@[simp] theorem filter_union {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s \u222a t) = filter p s \u222a filter p t := sorry\n\n@[simp] theorem filter_inter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s \u2229 t) = filter p s \u2229 filter p t := sorry\n\n@[simp] theorem filter_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop) [decidable_pred q] (s : multiset \u03b1) : filter p (filter q s) = filter (fun (a : \u03b1) => p a \u2227 q a) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_filter p q l)\n\ntheorem filter_add_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop) [decidable_pred q] (s : multiset \u03b1) : filter p s + filter q s = filter (fun (a : \u03b1) => p a \u2228 q a) s + filter (fun (a : \u03b1) => p a \u2227 q a) s := sorry\n\ntheorem filter_add_not {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : filter p s + filter (fun (a : \u03b1) => \u00acp a) s = s := sorry\n\n/-! ### Simultaneously filter and map elements of a multiset -/\n\n/-- `filter_map f s` is a combination filter/map operation on `s`.\n  The function `f : \u03b1 \u2192 option \u03b2` is applied to each element of `s`;\n  if `f a` is `some b` then `b` is added to the result, otherwise\n  `a` is removed from the resulting multiset. -/\ndef filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.filter_map f l)) sorry\n\n@[simp] theorem coe_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (l : List \u03b1) : filter_map f \u2191l = \u2191(list.filter_map f l) :=\n  rfl\n\n@[simp] theorem filter_map_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) : filter_map f 0 = 0 :=\n  rfl\n\n@[simp] theorem filter_map_cons_none {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 Option \u03b2} (a : \u03b1) (s : multiset \u03b1) (h : f a = none) : filter_map f (a ::\u2098 s) = filter_map f s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_cons_none a l h)\n\n@[simp] theorem filter_map_cons_some {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (a : \u03b1) (s : multiset \u03b1) {b : \u03b2} (h : f a = some b) : filter_map f (a ::\u2098 s) = b ::\u2098 filter_map f s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_cons_some f a l h)\n\ntheorem filter_map_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) : filter_map (some \u2218 f) = map f :=\n  funext\n    fun (s : multiset \u03b1) => quot.induction_on s fun (l : List \u03b1) => congr_arg coe (congr_fun (list.filter_map_eq_map f) l)\n\ntheorem filter_map_eq_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : filter_map (option.guard p) = filter p :=\n  funext\n    fun (s : multiset \u03b1) =>\n      quot.induction_on s fun (l : List \u03b1) => congr_arg coe (congr_fun (list.filter_map_eq_filter p) l)\n\ntheorem filter_map_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 Option \u03b3) (s : multiset \u03b1) : filter_map g (filter_map f s) = filter_map (fun (x : \u03b1) => option.bind (f x) g) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_filter_map f g l)\n\ntheorem map_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b3) (s : multiset \u03b1) : map g (filter_map f s) = filter_map (fun (x : \u03b1) => option.map g (f x)) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_filter_map f g l)\n\ntheorem filter_map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 Option \u03b3) (s : multiset \u03b1) : filter_map g (map f s) = filter_map (g \u2218 f) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_map f g l)\n\ntheorem filter_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (p : \u03b2 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : filter p (filter_map f s) = filter_map (fun (x : \u03b1) => option.filter p (f x)) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_filter_map f p l)\n\ntheorem filter_map_filter {\u03b1 : Type u_1} {\u03b2 : Type u_2} (p : \u03b1 \u2192 Prop) [decidable_pred p] (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1) : filter_map f (filter p s) = filter_map (fun (x : \u03b1) => ite (p x) (f x) none) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_filter p f l)\n\n@[simp] theorem filter_map_some {\u03b1 : Type u_1} (s : multiset \u03b1) : filter_map some s = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_some l)\n\n@[simp] theorem mem_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1) {b : \u03b2} : b \u2208 filter_map f s \u2194 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = some b :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_filter_map f l\n\ntheorem map_filter_map_of_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b1) (H : \u2200 (x : \u03b1), option.map g (f x) = some x) (s : multiset \u03b1) : map g (filter_map f s) = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_filter_map_of_inv f g H l)\n\ntheorem filter_map_le_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : filter_map f s \u2264 filter_map f t :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.filter_map f h)\n\n/-! ### countp -/\n\n/-- `countp p s` counts the number of elements of `s` (with multiplicity) that\n  satisfy `p`. -/\ndef countp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : \u2115 :=\n  quot.lift_on s (list.countp p) sorry\n\n@[simp] theorem coe_countp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) : countp p \u2191l = list.countp p l :=\n  rfl\n\n@[simp] theorem countp_zero {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p 0 = 0 :=\n  rfl\n\n@[simp] theorem countp_cons_of_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (s : multiset \u03b1) : p a \u2192 countp p (a ::\u2098 s) = countp p s + 1 :=\n  quot.induction_on s (list.countp_cons_of_pos p)\n\n@[simp] theorem countp_cons_of_neg {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (s : multiset \u03b1) : \u00acp a \u2192 countp p (a ::\u2098 s) = countp p s :=\n  quot.induction_on s (list.countp_cons_of_neg p)\n\ntheorem countp_eq_card_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : countp p s = coe_fn card (filter p s) :=\n  quot.induction_on s fun (l : List \u03b1) => list.countp_eq_length_filter p l\n\n@[simp] theorem countp_add {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) (t : multiset \u03b1) : countp p (s + t) = countp p s + countp p t := sorry\n\nprotected instance countp.is_add_monoid_hom {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : is_add_monoid_hom (countp p) :=\n  is_add_monoid_hom.mk (countp_zero p)\n\n@[simp] theorem countp_sub {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : t \u2264 s) : countp p (s - t) = countp p s - countp p t := sorry\n\ntheorem countp_le_of_le {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) : countp p s \u2264 countp p t := sorry\n\n@[simp] theorem countp_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop) [decidable_pred q] (s : multiset \u03b1) : countp p (filter q s) = countp (fun (a : \u03b1) => p a \u2227 q a) s := sorry\n\ntheorem countp_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} : 0 < countp p s \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), p a := sorry\n\ntheorem countp_pos_of_mem {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) (pa : p a) : 0 < countp p s :=\n  iff.mpr countp_pos (Exists.intro a (Exists.intro h pa))\n\n/-! ### Multiplicity of an element -/\n\n/-- `count a s` is the multiplicity of `a` in `s`. -/\ndef count {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : multiset \u03b1 \u2192 \u2115 :=\n  countp (Eq a)\n\n@[simp] theorem coe_count {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) : count a \u2191l = list.count a l :=\n  coe_countp (Eq a) l\n\n@[simp] theorem count_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : count a 0 = 0 :=\n  rfl\n\n@[simp] theorem count_cons_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : count a (a ::\u2098 s) = Nat.succ (count a s) :=\n  countp_cons_of_pos s rfl\n\n@[simp] theorem count_cons_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (h : a \u2260 b) (s : multiset \u03b1) : count a (b ::\u2098 s) = count a s :=\n  countp_cons_of_neg s h\n\ntheorem count_le_of_le {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2192 count a s \u2264 count a t :=\n  countp_le_of_le (Eq a)\n\ntheorem count_le_count_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) : count a s \u2264 count a (b ::\u2098 s) :=\n  count_le_of_le a (le_cons_self s b)\n\ntheorem count_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) : count a (b ::\u2098 s) = count a s + ite (a = b) 1 0 := sorry\n\ntheorem count_singleton {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : count a (a ::\u2098 0) = 1 := sorry\n\n@[simp] theorem count_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : count a (s + t) = count a s + count a t :=\n  countp_add (Eq a)\n\nprotected instance count.is_add_monoid_hom {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : is_add_monoid_hom (count a) :=\n  countp.is_add_monoid_hom (Eq a)\n\n@[simp] theorem count_smul {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (n : \u2115) (s : multiset \u03b1) : count a (n \u2022\u2115 s) = n * count a s := sorry\n\ntheorem count_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : 0 < count a s \u2194 a \u2208 s := sorry\n\n@[simp] theorem count_eq_zero_of_not_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} (h : \u00aca \u2208 s) : count a s = 0 :=\n  by_contradiction fun (h' : \u00account a s = 0) => h (iff.mp count_pos (nat.pos_of_ne_zero h'))\n\n@[simp] theorem count_eq_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : count a s = 0 \u2194 \u00aca \u2208 s :=\n  iff.mp iff_not_comm (iff.trans (iff.symm count_pos) pos_iff_ne_zero)\n\ntheorem count_ne_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : count a s \u2260 0 \u2194 a \u2208 s := sorry\n\n@[simp] theorem count_repeat_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (n : \u2115) : count a (repeat a n) = n := sorry\n\ntheorem count_repeat {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (n : \u2115) : count a (repeat b n) = ite (a = b) n 0 := sorry\n\n@[simp] theorem count_erase_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : count a (erase s a) = Nat.pred (count a s) := sorry\n\n@[simp] theorem count_erase_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (ab : a \u2260 b) (s : multiset \u03b1) : count a (erase s b) = count a s := sorry\n\n@[simp] theorem count_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : count a (s - t) = count a s - count a t := sorry\n\n@[simp] theorem count_union {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : count a (s \u222a t) = max (count a s) (count a t) := sorry\n\n@[simp] theorem count_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t) := sorry\n\ntheorem count_sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] {m : multiset \u03b2} {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} : count a (sum (map f m)) = sum (map (fun (b : \u03b2) => count a (f b)) m) := sorry\n\ntheorem count_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] {m : multiset \u03b2} {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} : count a (bind m f) = sum (map (fun (b : \u03b2) => count a (f b)) m) :=\n  count_sum\n\ntheorem le_count_iff_repeat_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} {n : \u2115} : n \u2264 count a s \u2194 repeat a n \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => iff.trans list.le_count_iff_repeat_sublist (iff.symm repeat_le_coe)\n\n@[simp] theorem count_filter_of_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1} (h : p a) : count a (filter p s) = count a s :=\n  quot.induction_on s fun (l : List \u03b1) => list.count_filter h\n\n@[simp] theorem count_filter_of_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1} (h : \u00acp a) : count a (filter p s) = 0 :=\n  count_eq_zero_of_not_mem fun (t : a \u2208 filter p s) => h (of_mem_filter t)\n\ntheorem ext {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} : s = t \u2194 \u2200 (a : \u03b1), count a s = count a t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => iff.trans quotient.eq list.perm_iff_count\n\ntheorem ext' {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} : (\u2200 (a : \u03b1), count a s = count a t) \u2192 s = t :=\n  iff.mpr ext\n\n@[simp] theorem coe_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : List \u03b1) (t : List \u03b1) : \u2191s \u2229 \u2191t = \u2191(list.bag_inter s t) := sorry\n\ntheorem le_iff_count {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2194 \u2200 (a : \u03b1), count a s \u2264 count a t := sorry\n\nprotected instance distrib_lattice {\u03b1 : Type u_1} [DecidableEq \u03b1] : distrib_lattice (multiset \u03b1) :=\n  distrib_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry lattice.inf sorry sorry sorry\n    sorry\n\nprotected instance semilattice_sup_bot {\u03b1 : Type u_1} [DecidableEq \u03b1] : semilattice_sup_bot (multiset \u03b1) :=\n  semilattice_sup_bot.mk 0 lattice.le lattice.lt sorry sorry sorry zero_le lattice.sup sorry sorry sorry\n\n@[simp] theorem mem_nsmul {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {n : \u2115} (h0 : n \u2260 0) : a \u2208 n \u2022\u2115 s \u2194 a \u2208 s := sorry\n\n/-! ### Lift a relation to `multiset`s -/\n\n/-- `rel r s t` -- lift the relation `r` between two elements to a relation between `s` and `t`,\ns.t. there is a one-to-one mapping betweem elements in `s` and `t` following `r`. -/\ntheorem rel_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : \u2200 (\u1fb0 : multiset \u03b1) (\u1fb0_1 : multiset \u03b2),\n  rel r \u1fb0 \u1fb0_1 \u2194\n    \u1fb0 = 0 \u2227 \u1fb0_1 = 0 \u2228\n      Exists\n        fun {a : \u03b1} =>\n          Exists\n            fun {b : \u03b2} =>\n              Exists\n                fun {as : multiset \u03b1} =>\n                  Exists fun {bs : multiset \u03b2} => r a b \u2227 rel r as bs \u2227 \u1fb0 = a ::\u2098 as \u2227 \u1fb0_1 = b ::\u2098 bs := sorry\n\ntheorem rel_flip {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b2} {t : multiset \u03b1} : rel (flip r) s t \u2194 rel r t s :=\n  { mp := rel_flip_aux, mpr := rel_flip_aux }\n\ntheorem rel_eq_refl {\u03b1 : Type u_1} {s : multiset \u03b1} : rel Eq s s :=\n  multiset.induction_on s rel.zero fun (a : \u03b1) (s : multiset \u03b1) => rel.cons rfl\n\ntheorem rel_eq {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : rel Eq s t \u2194 s = t := sorry\n\ntheorem rel.mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : \u2200 (a : \u03b1) (b : \u03b2), r a b \u2192 p a b) (hst : rel r s t) : rel p s t := sorry\n\ntheorem rel.add {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} {u : multiset \u03b1} {v : multiset \u03b2} (hst : rel r s t) (huv : rel r u v) : rel r (s + u) (t + v) := sorry\n\ntheorem rel_flip_eq {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : rel (fun (a b : \u03b1) => b = a) s t \u2194 s = t := sorry\n\n@[simp] theorem rel_zero_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {b : multiset \u03b2} : rel r 0 b \u2194 b = 0 := sorry\n\n@[simp] theorem rel_zero_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : multiset \u03b1} : rel r a 0 \u2194 a = 0 := sorry\n\ntheorem rel_cons_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : \u03b1} {as : multiset \u03b1} {bs : multiset \u03b2} : rel r (a ::\u2098 as) bs \u2194 \u2203 (b : \u03b2), \u2203 (bs' : multiset \u03b2), r a b \u2227 rel r as bs' \u2227 bs = b ::\u2098 bs' := sorry\n\ntheorem rel_cons_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as : multiset \u03b1} {b : \u03b2} {bs : multiset \u03b2} : rel r as (b ::\u2098 bs) \u2194 \u2203 (a : \u03b1), \u2203 (as' : multiset \u03b1), r a b \u2227 rel r as' bs \u2227 as = a ::\u2098 as' := sorry\n\ntheorem rel_add_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as\u2080 : multiset \u03b1} {as\u2081 : multiset \u03b1} {bs : multiset \u03b2} : rel r (as\u2080 + as\u2081) bs \u2194 \u2203 (bs\u2080 : multiset \u03b2), \u2203 (bs\u2081 : multiset \u03b2), rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081 := sorry\n\ntheorem rel_add_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as : multiset \u03b1} {bs\u2080 : multiset \u03b2} {bs\u2081 : multiset \u03b2} : rel r as (bs\u2080 + bs\u2081) \u2194 \u2203 (as\u2080 : multiset \u03b1), \u2203 (as\u2081 : multiset \u03b1), rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081 := sorry\n\ntheorem rel_map_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b3} {f : \u03b3 \u2192 \u03b1} {t : multiset \u03b2} : rel r (map f s) t \u2194 rel (fun (a : \u03b3) (b : \u03b2) => r (f a) b) s t := sorry\n\ntheorem rel_map_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b3} {f : \u03b3 \u2192 \u03b2} : rel r s (map f t) \u2194 rel (fun (a : \u03b1) (b : \u03b3) => r a (f b)) s t := sorry\n\ntheorem rel_join {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset (multiset \u03b1)} {t : multiset (multiset \u03b2)} (h : rel (rel r) s t) : rel r (join s) (join t) := sorry\n\ntheorem rel_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} (h : relator.lift_fun r p f g) (hst : rel r s t) : rel p (map f s) (map g t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (rel p (map f s) (map g t))) (propext rel_map_left)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (rel (fun (a : \u03b1) (b : \u03b4) => p (f a) b) s (map g t))) (propext rel_map_right)))\n      (rel.mono h hst))\n\ntheorem rel_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} {f : \u03b1 \u2192 multiset \u03b3} {g : \u03b2 \u2192 multiset \u03b4} (h : relator.lift_fun r (rel p) f g) (hst : rel r s t) : rel p (bind s f) (bind t g) :=\n  rel_join (rel_map h hst)\n\ntheorem card_eq_card_of_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) : coe_fn card s = coe_fn card t := sorry\n\ntheorem exists_mem_of_rel_of_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) {a : \u03b1} (ha : a \u2208 s) : \u2203 (b : \u03b2), \u2203 (H : b \u2208 t), r a b := sorry\n\ntheorem map_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) {s : multiset \u03b1} {t : multiset \u03b1} : map f s = map f t \u2194 s = t := sorry\n\ntheorem map_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) : function.injective (map f) :=\n  fun (x y : multiset \u03b1) => iff.mp (map_eq_map hf)\n\ntheorem map_mk_eq_map_mk_of_rel {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b1} (hst : rel r s t) : map (Quot.mk r) s = map (Quot.mk r) t := sorry\n\ntheorem exists_multiset_eq_map_quot_mk {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : multiset (Quot r)) : \u2203 (t : multiset \u03b1), s = map (Quot.mk r) t := sorry\n\ntheorem induction_on_multiset_quot {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {p : multiset (Quot r) \u2192 Prop} (s : multiset (Quot r)) : (\u2200 (s : multiset \u03b1), p (map (Quot.mk r) s)) \u2192 p s := sorry\n\n/-! ### Disjoint multisets -/\n\n/-- `disjoint s t` means that `s` and `t` have no elements in common. -/\ndef disjoint {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) :=\n  \u2200 {a : \u03b1}, a \u2208 s \u2192 a \u2208 t \u2192 False\n\n@[simp] theorem coe_disjoint {\u03b1 : Type u_1} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : disjoint \u2191l\u2081 \u2191l\u2082 \u2194 list.disjoint l\u2081 l\u2082 :=\n  iff.rfl\n\ntheorem disjoint.symm {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (d : disjoint s t) : disjoint t s :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 t) (\u1fb0_1 : a \u2208 s) => idRhs False (d \u1fb0_1 \u1fb0)\n\ntheorem disjoint_comm {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint s t \u2194 disjoint t s :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a : \u03b1}, a \u2208 s \u2192 \u00aca \u2208 t :=\n  iff.rfl\n\ntheorem disjoint_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a : \u03b1}, a \u2208 t \u2192 \u00aca \u2208 s :=\n  disjoint_comm\n\ntheorem disjoint_iff_ne {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint s t \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b := sorry\n\ntheorem disjoint_of_subset_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h : s \u2286 u) (d : disjoint u t) : disjoint s t :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 s) => idRhs (a \u2208 t \u2192 False) (d (h \u1fb0))\n\ntheorem disjoint_of_subset_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h : t \u2286 u) (d : disjoint s u) : disjoint s t :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 s) (\u1fb0_1 : a \u2208 t) => idRhs False (d \u1fb0 (h \u1fb0_1))\n\ntheorem disjoint_of_le_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h : s \u2264 u) : disjoint u t \u2192 disjoint s t :=\n  disjoint_of_subset_left (subset_of_le h)\n\ntheorem disjoint_of_le_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} (h : t \u2264 u) : disjoint s u \u2192 disjoint s t :=\n  disjoint_of_subset_right (subset_of_le h)\n\n@[simp] theorem zero_disjoint {\u03b1 : Type u_1} (l : multiset \u03b1) : disjoint 0 l :=\n  fun {a : \u03b1} => idRhs (a \u2208 [] \u2192 a \u2208 l \u2192 False) (not.elim (list.not_mem_nil a))\n\n@[simp] theorem singleton_disjoint {\u03b1 : Type u_1} {l : multiset \u03b1} {a : \u03b1} : disjoint (a ::\u2098 0) l \u2194 \u00aca \u2208 l := sorry\n\n@[simp] theorem disjoint_singleton {\u03b1 : Type u_1} {l : multiset \u03b1} {a : \u03b1} : disjoint l (a ::\u2098 0) \u2194 \u00aca \u2208 l := sorry\n\n@[simp] theorem disjoint_add_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : disjoint (s + t) u \u2194 disjoint s u \u2227 disjoint t u := sorry\n\n@[simp] theorem disjoint_add_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : disjoint s (t + u) \u2194 disjoint s t \u2227 disjoint s u := sorry\n\n@[simp] theorem disjoint_cons_left {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint (a ::\u2098 s) t \u2194 \u00aca \u2208 t \u2227 disjoint s t := sorry\n\n@[simp] theorem disjoint_cons_right {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} : disjoint s (a ::\u2098 t) \u2194 \u00aca \u2208 s \u2227 disjoint s t := sorry\n\ntheorem inter_eq_zero_iff_disjoint {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} : s \u2229 t = 0 \u2194 disjoint s t := sorry\n\n@[simp] theorem disjoint_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : disjoint (s \u222a t) u \u2194 disjoint s u \u2227 disjoint t u := sorry\n\n@[simp] theorem disjoint_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} : disjoint s (t \u222a u) \u2194 disjoint s t \u2227 disjoint s u := sorry\n\ntheorem disjoint_map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {s : multiset \u03b1} {t : multiset \u03b2} : disjoint (map f s) (map g t) \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a \u2260 g b := sorry\n\n/-- `pairwise r m` states that there exists a list of the elements s.t. `r` holds pairwise on this list. -/\ndef pairwise {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (m : multiset \u03b1) :=\n  \u2203 (l : List \u03b1), m = \u2191l \u2227 list.pairwise r l\n\ntheorem pairwise_coe_iff_pairwise {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (hr : symmetric r) {l : List \u03b1} : pairwise r \u2191l \u2194 list.pairwise r l := sorry\n\nend multiset\n\n\nnamespace multiset\n\n\n/-- Given a proof `hp` that there exists a unique `a \u2208 l` such that `p a`, `choose_x p l hp` returns\nthat `a` together with proofs of `a \u2208 l` and `p a`. -/\ndef choose_x {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1) (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : Subtype fun (a : \u03b1) => a \u2208 l \u2227 p a :=\n  quotient.rec_on l\n    (fun (l' : List \u03b1) (ex_unique : exists_unique fun (a : \u03b1) => a \u2208 quotient.mk l' \u2227 p a) => list.choose_x p l' sorry)\n    sorry\n\n/-- Given a proof `hp` that there exists a unique `a \u2208 l` such that `p a`, `choose p l hp` returns\nthat `a`. -/\ndef choose {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1) (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : \u03b1 :=\n  \u2191(choose_x p l hp)\n\ntheorem choose_spec {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1) (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : choose p l hp \u2208 l \u2227 p (choose p l hp) :=\n  subtype.property (choose_x p l hp)\n\ntheorem choose_mem {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1) (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : choose p l hp \u2208 l :=\n  and.left (choose_spec p l hp)\n\ntheorem choose_property {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1) (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : p (choose p l hp) :=\n  and.right (choose_spec p l hp)\n\n/-- The equivalence between lists and multisets of a subsingleton type. -/\ndef subsingleton_equiv (\u03b1 : Type u_1) [subsingleton \u03b1] : List \u03b1 \u2243 multiset \u03b1 :=\n  equiv.mk coe (Quot.lift id sorry) sorry sorry\n\nend multiset\n\n\ntheorem add_monoid_hom.map_multiset_sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1] [add_comm_monoid \u03b2] (f : \u03b1 \u2192+ \u03b2) (s : multiset \u03b1) : coe_fn f (multiset.sum s) = multiset.sum (multiset.map (\u21d1f) s) :=\n  Eq.symm (multiset.sum_hom s f)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/multiset/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6893056167854461, "lm_q1q2_score": 0.3848579331983096}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.metric_space.basic\nimport Mathlib.topology.algebra.uniform_group\nimport Mathlib.topology.algebra.ring\nimport Mathlib.topology.algebra.continuous_functions\nimport Mathlib.ring_theory.subring\nimport Mathlib.group_theory.archimedean\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Topological properties of \u211d\n-/\n\nprotected instance rat.metric_space : metric_space \u211a :=\n  metric_space.induced coe sorry real.metric_space\n\ntheorem rat.dist_eq (x : \u211a) (y : \u211a) : dist x y = abs (\u2191x - \u2191y) := rfl\n\n@[simp] theorem rat.dist_cast (x : \u211a) (y : \u211a) : dist \u2191x \u2191y = dist x y := rfl\n\n-- we want to ignore this instance for the next declaration\n\nprotected instance int.metric_space : metric_space \u2124 :=\n  let M : metric_space \u2124 := metric_space.induced coe sorry real.metric_space;\n  metric_space.replace_uniformity M sorry\n\ntheorem int.dist_eq (x : \u2124) (y : \u2124) : dist x y = abs (\u2191x - \u2191y) := rfl\n\n@[simp] theorem int.dist_cast_real (x : \u2124) (y : \u2124) : dist \u2191x \u2191y = dist x y := rfl\n\n@[simp] theorem int.dist_cast_rat (x : \u2124) (y : \u2124) : dist \u2191x \u2191y = dist x y := sorry\n\ntheorem uniform_continuous_of_rat : uniform_continuous coe := uniform_continuous_comap\n\ntheorem uniform_embedding_of_rat : uniform_embedding coe :=\n  uniform_embedding_comap rat.cast_injective\n\ntheorem dense_embedding_of_rat : dense_embedding coe := sorry\n\ntheorem embedding_of_rat : embedding coe := dense_embedding.to_embedding dense_embedding_of_rat\n\ntheorem continuous_of_rat : continuous coe :=\n  uniform_continuous.continuous uniform_continuous_of_rat\n\ntheorem real.uniform_continuous_add :\n    uniform_continuous fun (p : \u211d \u00d7 \u211d) => prod.fst p + prod.snd p :=\n  sorry\n\n-- TODO(Mario): Find a way to use rat_add_continuous_lemma\n\ntheorem rat.uniform_continuous_add :\n    uniform_continuous fun (p : \u211a \u00d7 \u211a) => prod.fst p + prod.snd p :=\n  sorry\n\ntheorem real.uniform_continuous_neg : uniform_continuous Neg.neg := sorry\n\ntheorem rat.uniform_continuous_neg : uniform_continuous Neg.neg := sorry\n\nprotected instance real.uniform_add_group : uniform_add_group \u211d :=\n  uniform_add_group.mk' real.uniform_continuous_add real.uniform_continuous_neg\n\nprotected instance rat.uniform_add_group : uniform_add_group \u211a :=\n  uniform_add_group.mk' rat.uniform_continuous_add rat.uniform_continuous_neg\n\nprotected instance real.topological_add_group : topological_add_group \u211d :=\n  linear_ordered_add_comm_group.topological_add_group\n\nprotected instance rat.topological_add_group : topological_add_group \u211a :=\n  uniform_add_group.to_topological_add_group\n\nprotected instance rat.order_topology : order_topology \u211a :=\n  induced_order_topology coe (fun (x y : \u211a) => rat.cast_lt) exists_rat_btwn\n\ntheorem real.is_topological_basis_Ioo_rat :\n    topological_space.is_topological_basis\n        (set.Union\n          fun (a : \u211a) =>\n            set.Union fun (b : \u211a) => set.Union fun (h : a < b) => singleton (set.Ioo \u2191a \u2191b)) :=\n  sorry\n\nprotected instance real.topological_space.second_countable_topology :\n    topological_space.second_countable_topology \u211d :=\n  sorry\n\n/- TODO(Mario): Prove that these are uniform isomorphisms instead of uniform embeddings\nlemma uniform_embedding_add_rat {r : \u211a} : uniform_embedding (\u03bbp:\u211a, p + r) :=\n_\n\nlemma uniform_embedding_mul_rat {q : \u211a} (hq : q \u2260 0) : uniform_embedding ((*) q) :=\n_ -/\n\ntheorem real.mem_closure_iff {s : set \u211d} {x : \u211d} :\n    x \u2208 closure s \u2194 \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0), \u2203 (y : \u211d), \u2203 (H : y \u2208 s), abs (y - x) < \u03b5 :=\n  sorry\n\ntheorem real.uniform_continuous_inv (s : set \u211d) {r : \u211d} (r0 : 0 < r)\n    (H : \u2200 (x : \u211d), x \u2208 s \u2192 r \u2264 abs x) : uniform_continuous fun (p : \u21a5s) => subtype.val p\u207b\u00b9 :=\n  sorry\n\ntheorem real.uniform_continuous_abs : uniform_continuous abs :=\n  iff.mpr metric.uniform_continuous_iff\n    fun (\u03b5 : \u211d) (\u03b50 : \u03b5 > 0) =>\n      Exists.intro \u03b5\n        (Exists.intro \u03b50 fun (a b : \u211d) => lt_of_le_of_lt (abs_abs_sub_abs_le_abs_sub a b))\n\ntheorem rat.uniform_continuous_abs : uniform_continuous abs := sorry\n\ntheorem real.tendsto_inv {r : \u211d} (r0 : r \u2260 0) :\n    filter.tendsto (fun (q : \u211d) => q\u207b\u00b9) (nhds r) (nhds (r\u207b\u00b9)) :=\n  sorry\n\ntheorem real.continuous_inv :\n    continuous fun (a : Subtype fun (r : \u211d) => r \u2260 0) => subtype.val a\u207b\u00b9 :=\n  sorry\n\ntheorem real.continuous.inv {\u03b1 : Type u} [topological_space \u03b1] {f : \u03b1 \u2192 \u211d} (h : \u2200 (a : \u03b1), f a \u2260 0)\n    (hf : continuous f) : continuous fun (a : \u03b1) => f a\u207b\u00b9 :=\n  (fun\n      (this :\n      continuous ((has_inv.inv \u2218 subtype.val) \u2218 fun (a : \u03b1) => { val := f a, property := h a })) =>\n      this)\n    (continuous.comp real.continuous_inv (continuous_subtype_mk (fun (a : \u03b1) => h a) hf))\n\ntheorem real.uniform_continuous_mul_const {x : \u211d} : uniform_continuous (Mul.mul x) := sorry\n\ntheorem real.uniform_continuous_mul (s : set (\u211d \u00d7 \u211d)) {r\u2081 : \u211d} {r\u2082 : \u211d}\n    (H : \u2200 (x : \u211d \u00d7 \u211d), x \u2208 s \u2192 abs (prod.fst x) < r\u2081 \u2227 abs (prod.snd x) < r\u2082) :\n    uniform_continuous fun (p : \u21a5s) => prod.fst (subtype.val p) * prod.snd (subtype.val p) :=\n  sorry\n\nprotected theorem real.continuous_mul : continuous fun (p : \u211d \u00d7 \u211d) => prod.fst p * prod.snd p :=\n  sorry\n\nprotected instance real.topological_ring : topological_ring \u211d := topological_ring.mk continuous_neg\n\nprotected instance real.topological_semiring : topological_semiring \u211d :=\n  topological_ring.to_topological_semiring \u211d\n\ntheorem rat.continuous_mul : continuous fun (p : \u211a \u00d7 \u211a) => prod.fst p * prod.snd p := sorry\n\nprotected instance rat.topological_ring : topological_ring \u211a := topological_ring.mk continuous_neg\n\ntheorem real.ball_eq_Ioo (x : \u211d) (\u03b5 : \u211d) : metric.ball x \u03b5 = set.Ioo (x - \u03b5) (x + \u03b5) := sorry\n\ntheorem real.Ioo_eq_ball (x : \u211d) (y : \u211d) :\n    set.Ioo x y = metric.ball ((x + y) / bit0 1) ((y - x) / bit0 1) :=\n  sorry\n\ntheorem real.totally_bounded_Ioo (a : \u211d) (b : \u211d) : totally_bounded (set.Ioo a b) := sorry\n\ntheorem real.totally_bounded_ball (x : \u211d) (\u03b5 : \u211d) : totally_bounded (metric.ball x \u03b5) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (totally_bounded (metric.ball x \u03b5))) (real.ball_eq_Ioo x \u03b5)))\n    (real.totally_bounded_Ioo (x - \u03b5) (x + \u03b5))\n\ntheorem real.totally_bounded_Ico (a : \u211d) (b : \u211d) : totally_bounded (set.Ico a b) := sorry\n\ntheorem real.totally_bounded_Icc (a : \u211d) (b : \u211d) : totally_bounded (set.Icc a b) := sorry\n\ntheorem rat.totally_bounded_Icc (a : \u211a) (b : \u211a) : totally_bounded (set.Icc a b) := sorry\n\nprotected instance real.complete_space : complete_space \u211d := sorry\n\ntheorem closure_of_rat_image_lt {q : \u211a} :\n    closure (coe '' set_of fun (x : \u211a) => q < x) = set_of fun (r : \u211d) => \u2191q \u2264 r :=\n  sorry\n\n/- TODO(Mario): Put these back only if needed later\nlemma closure_of_rat_image_le_eq {q : \u211a} : closure ((coe:\u211a \u2192 \u211d) '' {x | q \u2264 x}) = {r | \u2191q \u2264 r} :=\n_\n\nlemma closure_of_rat_image_le_le_eq {a b : \u211a} (hab : a \u2264 b) :\n  closure (of_rat '' {q:\u211a | a \u2264 q \u2227 q \u2264 b}) = {r:\u211d | of_rat a \u2264 r \u2227 r \u2264 of_rat b} :=\n_-/\n\ntheorem compact_Icc {a : \u211d} {b : \u211d} : is_compact (set.Icc a b) :=\n  compact_of_totally_bounded_is_closed (real.totally_bounded_Icc a b)\n    (is_closed_inter (is_closed_ge' a) (is_closed_le' b))\n\ntheorem compact_pi_Icc {\u03b9 : Type u_1} {a : \u03b9 \u2192 \u211d} {b : \u03b9 \u2192 \u211d} : is_compact (set.Icc a b) :=\n  Eq.subst (set.pi_univ_Icc a b) compact_univ_pi fun (i : \u03b9) => compact_Icc\n\nprotected instance real.proper_space : proper_space \u211d :=\n  proper_space.mk\n    fun (x r : \u211d) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (is_compact (metric.closed_ball x r))) closed_ball_Icc))\n        compact_Icc\n\ntheorem real.bounded_iff_bdd_below_bdd_above {s : set \u211d} :\n    metric.bounded s \u2194 bdd_below s \u2227 bdd_above s :=\n  sorry\n\ntheorem real.image_Icc {f : \u211d \u2192 \u211d} {a : \u211d} {b : \u211d} (hab : a \u2264 b)\n    (h : continuous_on f (set.Icc a b)) :\n    f '' set.Icc a b = set.Icc (Inf (f '' set.Icc a b)) (Sup (f '' set.Icc a b)) :=\n  sorry\n\nprotected instance reals_semimodule : topological_semimodule \u211d \u211d :=\n  topological_semimodule.mk continuous_mul\n\nprotected instance real_maps_algebra {\u03b1 : Type u_1} [topological_space \u03b1] :\n    algebra \u211d (continuous_map \u03b1 \u211d) :=\n  Mathlib.continuous_map_algebra\n\n/-- Given a nontrivial subgroup `G \u2286 \u211d`, if `G \u2229 \u211d_{>0}` has no minimum then `G` is dense. -/\ntheorem real.subgroup_dense_of_no_min {G : add_subgroup \u211d} {g\u2080 : \u211d} (g\u2080_in : g\u2080 \u2208 G)\n    (g\u2080_ne : g\u2080 \u2260 0) (H' : \u00ac\u2203 (a : \u211d), is_least (set_of fun (g : \u211d) => g \u2208 G \u2227 0 < g) a) :\n    dense \u2191G :=\n  sorry\n\n/-- Subgroups of `\u211d` are either dense or cyclic. See `real.subgroup_dense_of_no_min` and\n`subgroup_cyclic_of_min` for more precise statements. -/\ntheorem real.subgroup_dense_or_cyclic (G : add_subgroup \u211d) :\n    dense \u2191G \u2228 \u2203 (a : \u211d), G = add_subgroup.closure (singleton a) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/instances/real_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.38485792607119135}}
{"text": "import logic.idecidable\n\n--- Statement of definite description\naxiom definite_description {\u03b1 : Sort _} {p : \u03b1 \u2192 Prop} : (\u2203! (a : \u03b1), p a) \u2192 {a:\u03b1 // p a}\n\nnamespace unsafe\n\n--- `definite_description` enables choice on subsingleton types.\nnoncomputable definition subsingleton_choice (\u03b1 : Sort _) [subsingleton \u03b1] : nonempty \u03b1 \u2192 \u03b1 :=\n  \u03bb h, subtype.val \u2218 @definite_description \u03b1 (\u03bb _, true) $\n    by cases h with a; existsi a; split; try {trivial};\n       dsimp *; intros y _; exact subsingleton.elim y a\n\n--- Turn the \"internal\" decidablility into the \"external\" one.\nnoncomputable definition decidable_of_idecidable (p : Prop) [idecidable p] : decidable p :=\n  subsingleton_choice (decidable p) $\n    begin\n      cases whether p with hp hnp,\n      case or.inl {\n        constructor,\n        exact is_true hp,\n      },\n      case or.inr {\n        constructor,\n        exact is_false hnp,\n      }\n    end\n\nlemma dec_of_idec_pos {p : Prop} [idecidable p] : \u03a0 (h : p), decidable_of_idecidable p = is_true h :=\n  begin\n    intros h,\n    cases decidable_of_idecidable p with hnp hp,\n    case is_false { exfalso; exact hnp h },\n    case is_true { refl }\n  end\n\nlemma dec_of_idec_neg {p : Prop} [idecidable p] : \u03a0 (h : \u00acp), decidable_of_idecidable p = is_false h :=\n  begin\n    intros h,\n    cases decidable_of_idecidable p with hnp hp,\n    case is_false { refl },\n    case is_true { exfalso; exact h hp }\n  end\n\nend unsafe\n", "meta": {"author": "Junology", "repo": "groth-lean", "sha": "5aa1ba624cd0f5145f63fa86130f99b85bbbcac2", "save_path": "github-repos/lean/Junology-groth-lean", "path": "github-repos/lean/Junology-groth-lean/groth-lean-5aa1ba624cd0f5145f63fa86130f99b85bbbcac2/src/logic/definite_description.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3848173209392042}}
{"text": "import tutorial_world.level15_simp -- hide\nopen IncidencePlane -- hide\n\n/- Tactic : exfalso\n\n## Summary\n\nChanges the goal to `\u22a2 false`.\n\n## Details\n\nThis may seem hard to prove,\nbut it is useful when we have a contradiction in the hypotheses.\n\nFor example, if we have `h : \u00ac P` as a hypothesis and we apply `exfalso`\nwe can then `apply h` to transform the goal into `\u22a2 P`.\n-/\n\n\n/-\nIn this level we introduce the new tactic `exfalso`. Look at what it does, it is a bit\nstrange at first. We will also need one of the axioms for our plane, the one that says that\nthe line through two points contains each of them. You can see the statement of this theorem\non the left sidebar.\n-/\n\n/- Hint : Click here for a hint, in case you get stuck.\nIn Lean, the  negation `\u00ac P` of a statement is a shorthand for `P \u2192 false`. Therefore\nstart with `exfalso`, and remember that negation is the same as `\u2192 false`.\n-/\n\nvariables {\u03a9 : Type} [IncidencePlane \u03a9] --hide\n\n/- Lemma : no-side-bar\nProve that 2+2 is 5, using a false hypothesis.\n-/\nlemma two_plus_two_equals_five (P Q : \u03a9) (h: P \u2209 line_through P Q) : 2 + 2 = 5:=\nbegin\n  exfalso,\n  apply h,\n  exact line_through_left P Q,\n\n\n\n\n\n  \nend ", "meta": {"author": "mmasdeu", "repo": "hilbertgame", "sha": "0557019a1b7220bab7fe35729646c25bf73f0447", "save_path": "github-repos/lean/mmasdeu-hilbertgame", "path": "github-repos/lean/mmasdeu-hilbertgame/hilbertgame-0557019a1b7220bab7fe35729646c25bf73f0447/src/tutorial_world/level16_exfalso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.38481732093920407}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nParallel computation of a computable sequence of computations by\na diagonal enumeration.\nThe important theorems of this operation are proven as\nterminates_parallel and exists_of_mem_parallel.\n(This operation is nondeterministic in the sense that it does not\nhonor sequence equivalence (irrelevance of computation time).)\n-/\nimport data.seq.wseq\nuniverses u v\n\nnamespace computation\nopen wseq\nvariables {\u03b1 : Type u} {\u03b2 : Type v}\n\ndef parallel.aux2 : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) :=\nlist.foldr (\u03bbc o, match o with\n| sum.inl a  := sum.inl a\n| sum.inr ls := rmap (\u03bb c', c' :: ls) (destruct c)\nend) (sum.inr [])\n\ndef parallel.aux1 : list (computation \u03b1) \u00d7 wseq (computation \u03b1) \u2192\n  \u03b1 \u2295 list (computation \u03b1) \u00d7 wseq (computation \u03b1)\n| (l, S) := rmap (\u03bb l', match seq.destruct S with\n  | none := (l', nil)\n  | some (none, S') := (l', S')\n  | some (some c, S') := (c::l', S')\n  end) (parallel.aux2 l)\n\n/-- Parallel computation of an infinite stream of computations,\n  taking the first result -/\ndef parallel (S : wseq (computation \u03b1)) : computation \u03b1 :=\ncorec parallel.aux1 ([], S)\n\ntheorem terminates_parallel.aux : \u2200 {l : list (computation \u03b1)} {S c},\n  c \u2208 l \u2192 terminates c \u2192 terminates (corec parallel.aux1 (l, S)) :=\nbegin\n  have lem1 : \u2200 l S, (\u2203 (a : \u03b1), parallel.aux2 l = sum.inl a) \u2192\n    terminates (corec parallel.aux1 (l, S)),\n  { intros l S e, cases e with a e,\n    have this : corec parallel.aux1 (l, S) = return a,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw e, simp [rmap] },\n    rw this, apply_instance },\n  intros l S c m T, revert l S,\n  apply @terminates_rec_on _ _ c T _ _,\n  { intros a l S m, apply lem1,\n    induction l with c l IH generalizing m; simp at m, { contradiction },\n    cases m with e m,\n    { rw \u2190e, simp [parallel.aux2],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls,\n      exacts [\u27e8a', rfl\u27e9, \u27e8a, rfl\u27e9] },\n    { cases IH m with a' e,\n      simp [parallel.aux2], simp [parallel.aux2] at e,\n      rw e, exact \u27e8a', rfl\u27e9 } },\n  { intros s IH l S m,\n    have H1 : \u2200 l', parallel.aux2 l = sum.inr l' \u2192 s \u2208 l',\n    { induction l with c l IH' generalizing m;\n      intros l' e'; simp at m, { contradiction },\n      cases m with e m; simp [parallel.aux2] at e',\n      { rw \u2190e at e',\n        cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        injection e' with e', rw \u2190e', simp },\n      { induction e : list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        rw e at e', { contradiction },\n        have := IH' m _ e,\n        simp [parallel.aux2] at e',\n        cases destruct c; injection e' with h',\n        rw \u2190h', simp [this] } },\n    induction h : parallel.aux2 l with a l',\n    { exact lem1 _ _ \u27e8a, h\u27e9 },\n    { have H2 : corec parallel.aux1 (l, S) = think _,\n      { apply destruct_eq_think,\n        simp [parallel.aux1],\n        rw h, simp [rmap] },\n      rw H2, apply @computation.think_terminates _ _ _,\n      have := H1 _ h,\n      rcases seq.destruct S with _ | \u27e8_|c, S'\u27e9;\n      simp [parallel.aux1]; apply IH; simp [this] } }\nend\n\ntheorem terminates_parallel {S : wseq (computation \u03b1)}\n   {c} (h : c \u2208 S) [T : terminates c] : terminates (parallel S) :=\nsuffices \u2200 n (l : list (computation \u03b1)) S c,\n  c \u2208 l \u2228 some (some c) = seq.nth S n \u2192\n  terminates c \u2192 terminates (corec parallel.aux1 (l, S)),\nfrom let \u27e8n, h\u27e9 := h in this n [] S c (or.inr h) T,\nbegin\n  intro n, induction n with n IH; intros l S c o T,\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    have H : seq.destruct S = some (some c, _),\n    { unfold seq.destruct functor.map, rw \u2190 a, simp },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h, H], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      apply terminates_parallel.aux _ T, simp } },\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      have TT : \u2200 l', terminates (corec parallel.aux1 (l', S.tail)),\n      { intro, apply IH _ _ _ (or.inr _) T, rw a, cases S with f al, refl },\n      induction e : seq.nth S 0 with o,\n      { have D : seq.destruct S = none,\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, simp [parallel.aux1], have TT := TT l',\n        rwa [seq.destruct_eq_nil D, seq.tail_nil] at TT },\n      { have D : seq.destruct S = some (o, S.tail),\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, cases o with c; simp [parallel.aux1, TT] } } }\nend\n\ntheorem exists_of_mem_parallel {S : wseq (computation \u03b1)}\n   {a} (h : a \u2208 parallel S) : \u2203 c \u2208 S, a \u2208 c :=\nsuffices \u2200 C, a \u2208 C \u2192 \u2200 (l : list (computation \u03b1)) S,\n  corec parallel.aux1 (l, S) = C \u2192 \u2203 c, (c \u2208 l \u2228 c \u2208 S) \u2227 a \u2208 c,\nfrom let \u27e8c, h1, h2\u27e9 := this _ h [] S rfl in \u27e8c, h1.resolve_left id, h2\u27e9,\nbegin\n  let F : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) \u2192 Prop,\n  { intros l a, cases a with a l',\n    exact \u2203 c \u2208 l, a \u2208 c,\n    exact \u2200 a', (\u2203 c \u2208 l', a' \u2208 c) \u2192 (\u2203 c \u2208 l, a' \u2208 c) },\n  have lem1 : \u2200 (l : list (computation \u03b1)), F l (parallel.aux2 l),\n  { intro l, induction l with c l IH; simp [parallel.aux2],\n    { intros a h, rcases h with \u27e8c, hn, _\u27e9,\n      exact false.elim hn },\n    { simp [parallel.aux2] at IH,\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a ls;\n      simp [parallel.aux2],\n      { rcases IH with \u27e8c', cl, ac\u27e9,\n        refine \u27e8c', or.inr cl, ac\u27e9 },\n      { induction h : destruct c with a c'; simp [rmap],\n        { refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n          rw destruct_eq_ret h,\n          apply ret_mem },\n        { intros a' h, rcases h with \u27e8d, dm, ad\u27e9,\n          simp at dm, cases dm with e dl,\n          { rw e at ad, refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n            rw destruct_eq_think h,\n            exact think_mem ad },\n          { cases IH a' \u27e8d, dl, ad\u27e9 with d dm, cases dm with dm ad,\n            exact \u27e8d, or.inr dm, ad\u27e9 } } } } },\n  intros C aC, refine mem_rec_on aC _ (\u03bb C' IH, _);\n  intros l S e; have e' := congr_arg destruct e; have := lem1 l;\n  simp [parallel.aux1] at e'; cases parallel.aux2 l with a' l'; injection e' with h',\n  { rw h' at this, rcases this with \u27e8c, cl, ac\u27e9,\n    exact \u27e8c, or.inl cl, ac\u27e9 },\n  { induction e : seq.destruct S with a; rw e at h',\n    { exact let \u27e8d, o, ad\u27e9 := IH _ _ h',\n        \u27e8c, cl, ac\u27e9 := this a \u27e8d, o.resolve_right (not_mem_nil _), ad\u27e9 in\n      \u27e8c, or.inl cl, ac\u27e9 },\n    { cases a with o S', cases o with c; simp [parallel.aux1] at h';\n      rcases IH _ _ h' with \u27e8d, dl | dS', ad\u27e9,\n      { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' },\n      { simp at dl, cases dl with dc dl,\n        { rw dc at ad, refine \u27e8c, or.inr _, ad\u27e9,\n          rw seq.destruct_eq_cons e,\n          apply seq.mem_cons },\n        { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 } },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' } } }\nend\n\ntheorem map_parallel (f : \u03b1 \u2192 \u03b2) (S) : map f (parallel S) = parallel (S.map (map f)) :=\nbegin\n  refine eq_of_bisim (\u03bb c1 c2, \u2203 l S,\n    c1 = map f (corec parallel.aux1 (l, S)) \u2227\n    c2 = corec parallel.aux1 (l.map (map f), S.map (map f))) _ \u27e8[], S, rfl, rfl\u27e9,\n  intros c1 c2 h, exact match c1, c2, h with ._, ._, \u27e8l, S, rfl, rfl\u27e9 := begin\n    clear _match,\n    have : parallel.aux2 (l.map (map f)) = lmap f (rmap (list.map (map f)) (parallel.aux2 l)),\n    { simp [parallel.aux2],\n      induction l with c l IH; simp, rw [IH],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l; simp [parallel.aux2],\n      cases destruct c; simp },\n    simp [parallel.aux1], rw this, cases parallel.aux2 l with a l'; simp,\n    apply S.cases_on _ (\u03bb c S, _) (\u03bb S, _); simp; simp [parallel.aux1];\n    exact \u27e8_, _, rfl, rfl\u27e9\n  end end\nend\n\ntheorem parallel_empty (S : wseq (computation \u03b1)) (h : S.head ~> none) :\nparallel S = empty _ :=\neq_empty_of_not_terminates $ \u03bb \u27e8\u27e8a, m\u27e9\u27e9,\nlet \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel m,\n    \u27e8n, nm\u27e9 := exists_nth_of_mem cs,\n    \u27e8c', h'\u27e9 := head_some_of_nth_some nm in by injection h h'\n\n-- The reason this isn't trivial from exists_of_mem_parallel is because it eliminates to Sort\ndef parallel_rec {S : wseq (computation \u03b1)} (C : \u03b1 \u2192 Sort v)\n  (H : \u2200 s \u2208 S, \u2200 a \u2208 s, C a) {a} (h : a \u2208 parallel S) : C a :=\nbegin\n  let T : wseq (computation (\u03b1 \u00d7 computation \u03b1)) :=\n    S.map (\u03bbc, c.map (\u03bb a, (a, c))),\n  have : S = T.map (map (\u03bb c, c.1)),\n  { rw [\u2190wseq.map_comp], refine (wseq.map_id _).symm.trans (congr_arg (\u03bb f, wseq.map f S) _),\n    funext c, dsimp [id, function.comp], rw [\u2190map_comp], exact (map_id _).symm },\n  have pe := congr_arg parallel this, rw \u2190map_parallel at pe,\n  have h' := h, rw pe at h',\n  haveI : terminates (parallel T) := (terminates_map_iff _ _).1 \u27e8\u27e8_, h'\u27e9\u27e9,\n  induction e : get (parallel T) with a' c,\n  have : a \u2208 c \u2227 c \u2208 S,\n  { rcases exists_of_mem_map h' with \u27e8d, dT, cd\u27e9,\n    rw get_eq_of_mem _ dT at e, cases e, dsimp at cd, cases cd,\n    rcases exists_of_mem_parallel dT with \u27e8d', dT', ad'\u27e9,\n    rcases wseq.exists_of_mem_map dT' with \u27e8c', cs', e'\u27e9,\n    rw \u2190e' at ad',\n    rcases exists_of_mem_map ad' with \u27e8a', ac', e'\u27e9, injection e' with i1 i2,\n    constructor, rwa [i1, i2] at ac', rwa i2 at cs' },\n  cases this with ac cs, apply H _ cs _ ac\nend\n\ntheorem parallel_promises {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) : parallel S ~> a :=\n\u03bb a' ma', let \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel ma' in H _ cs ac\n\ntheorem mem_parallel {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) {c} (cs : c \u2208 S) (ac : a \u2208 c) : a \u2208 parallel S :=\nby haveI := terminates_of_mem ac; haveI := terminates_parallel cs;\n   exact mem_of_promises _ (parallel_promises H)\n\ntheorem parallel_congr_lem {S T : wseq (computation \u03b1)} {a}\n  (H : S.lift_rel equiv T) : (\u2200 s \u2208 S, s ~> a) \u2194 (\u2200 t \u2208 T, t ~> a) :=\n\u27e8\u03bb h1 t tT, let \u27e8s, sS, se\u27e9 := wseq.exists_of_lift_rel_right H tT in\n  (promises_congr se _).1 (h1 _ sS),\n\u03bb h2 s sS, let \u27e8t, tT, se\u27e9 := wseq.exists_of_lift_rel_left H sS in\n  (promises_congr se _).2 (h2 _ tT)\u27e9\n\n-- The parallel operation is only deterministic when all computation paths lead to the same value\ntheorem parallel_congr_left {S T : wseq (computation \u03b1)} {a}\n  (h1 : \u2200 s \u2208 S, s ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nlet h2 := (parallel_congr_lem H).1 h1 in\n\u03bb a', \u27e8\u03bbh, by have aa := parallel_promises h1 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_left H sS,\n      aT := (st _).1 as in mem_parallel h2 tT aT,\n\u03bbh, by have aa := parallel_promises h2 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_right H sS,\n      aT := (st _).2 as in mem_parallel h1 tT aT\u27e9\n\ntheorem parallel_congr_right {S T : wseq (computation \u03b1)} {a}\n  (h2 : \u2200 t \u2208 T, t ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nparallel_congr_left ((parallel_congr_lem H).2 h2) H\n\nend computation\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/seq/parallel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3847778134997737}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.monad.basic\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Equiv.Defs\nimport Mathbin.Tactic.Basic\n\n/-!\n# Monad\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n## Attributes\n\n * ext\n * functor_norm\n * monad_norm\n\n## Implementation Details\n\nSet of rewrite rules and automation for monads in general and\n`reader_t`, `state_t`, `except_t` and `option_t` in particular.\n\nThe rewrite rules for monads are carefully chosen so that `simp with\nfunctor_norm` will not introduce monadic vocabulary in a context where\napplicatives would do just fine but will handle monadic notation\nalready present in an expression.\n\nIn a context where monadic reasoning is desired `simp with monad_norm`\nwill translate functor and applicative notation into monad notation\nand use regular `functor_norm` rules as well.\n\n## Tags\n\nfunctor, applicative, monad, simp\n\n-/\n\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:61:9: unsupported: weird string -/\n/- failed to parenthesize: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n[PrettyPrinter.parenthesize.input] (Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr\n     [(Command.docComment\n       \"/--\"\n       \"./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:61:9: unsupported: weird string -/\")]\n     \"register_simp_attr\"\n     `monad_norm)-/-- failed to format: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n/-- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:61:9: unsupported: weird string -/\n  register_simp_attr\n  monad_norm\n\n/- [mathport] port note: move this to another file, it won't work here -/\nattribute [monad_norm] functor_norm\n\nattribute [ext] ReaderT.ext StateT.ext ExceptT.ext OptionT.ext\n\nattribute [functor_norm] bind_assoc pure_bind bind_pure\n\nattribute [monad_norm] seq_eq_bind_map\n\nuniverse u v\n\n/- warning: map_eq_bind_pure_comp -> map_eq_bind_pure_comp is a dubious translation:\nlean 3 declaration is\n  forall (m : Type.{u1} -> Type.{u2}) [_inst_1 : Monad.{u1, u2} m] [_inst_2 : LawfulMonad.{u1, u2} m _inst_1] {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (f : \u03b1 -> \u03b2) (x : m \u03b1), Eq.{succ u2} (m \u03b2) (Functor.map.{u1, u2} (fun {\u03b1 : Type.{u1}} => m \u03b1) (Applicative.toFunctor.{u1, u2} (fun {\u03b1 : Type.{u1}} => m \u03b1) (Monad.toApplicative.{u1, u2} (fun {\u03b1 : Type.{u1}} => m \u03b1) _inst_1)) \u03b1 \u03b2 f x) (Bind.bind.{u1, u2} m (Monad.toHasBind.{u1, u2} m _inst_1) \u03b1 \u03b2 x (Function.comp.{succ u1, succ u1, succ u2} \u03b1 \u03b2 (m \u03b2) (Pure.pure.{u1, u2} m (Applicative.toHasPure.{u1, u2} m (Monad.toApplicative.{u1, u2} m _inst_1)) \u03b2) f))\nbut is expected to have type\n  forall {m : Type.{u1}} {_inst_1 : Type.{u1}} (_inst_2 : Type.{u1} -> Type.{u2}) [\u03b1 : Monad.{u1, u2} _inst_2] [\u03b2 : LawfulMonad.{u1, u2} _inst_2 \u03b1] (f : m -> _inst_1) (x : _inst_2 m), Eq.{succ u2} (_inst_2 _inst_1) (Functor.map.{u1, u2} _inst_2 (Applicative.toFunctor.{u1, u2} _inst_2 (Monad.toApplicative.{u1, u2} _inst_2 \u03b1)) m _inst_1 f x) (Bind.bind.{u1, u2} _inst_2 (Monad.toBind.{u1, u2} _inst_2 \u03b1) m _inst_1 x (Function.comp.{succ u1, succ u1, succ u2} m _inst_1 (_inst_2 _inst_1) (Pure.pure.{u1, u2} _inst_2 (Applicative.toPure.{u1, u2} _inst_2 (Monad.toApplicative.{u1, u2} _inst_2 \u03b1)) _inst_1) f))\nCase conversion may be inaccurate. Consider using '#align map_eq_bind_pure_comp map_eq_bind_pure_comp\u2093'. -/\n@[monad_norm]\ntheorem map_eq_bind_pure_comp (m : Type u \u2192 Type v) [Monad m] [LawfulMonad m] {\u03b1 \u03b2 : Type u}\n    (f : \u03b1 \u2192 \u03b2) (x : m \u03b1) : f <$> x = x >>= pure \u2218 f := by rw [bind_pure_comp_eq_map]\n#align map_eq_bind_pure_comp map_eq_bind_pure_comp\n\n/- warning: state_t.eval -> StateT.eval is a dubious translation:\nlean 3 declaration is\n  forall {m : Type.{u1} -> Type.{u2}} [_inst_1 : Functor.{u1, u2} m] {\u03c3 : Type.{u1}} {\u03b1 : Type.{u1}}, (StateT\u2093.{u1, u2} \u03c3 m \u03b1) -> \u03c3 -> (m \u03b1)\nbut is expected to have type\n  forall {m : Type.{u1}} {_inst_1 : Type.{u1}} {\u03c3 : Type.{u1} -> Type.{u2}} [\u03b1 : Functor.{u1, u2} \u03c3], (StateT.{u1, u2} m \u03c3 _inst_1) -> m -> (\u03c3 _inst_1)\nCase conversion may be inaccurate. Consider using '#align state_t.eval StateT.eval\u2093'. -/\n/-- run a `state_t` program and discard the final state -/\ndef StateT.eval {m : Type u \u2192 Type v} [Functor m] {\u03c3 \u03b1} (cmd : StateT \u03c3 m \u03b1) (s : \u03c3) : m \u03b1 :=\n  Prod.fst <$> cmd.run s\n#align state_t.eval StateT.eval\n\nuniverse u\u2080 u\u2081 v\u2080 v\u2081\n\n/- warning: state_t.equiv -> StateT.equiv is a dubious translation:\nlean 3 declaration is\n  forall {m\u2081 : Type.{u1} -> Type.{u3}} {m\u2082 : Type.{u2} -> Type.{u4}} {\u03b1\u2081 : Type.{u1}} {\u03c3\u2081 : Type.{u1}} {\u03b1\u2082 : Type.{u2}} {\u03c3\u2082 : Type.{u2}}, (Equiv.{max (succ u1) (succ u3), max (succ u2) (succ u4)} (\u03c3\u2081 -> (m\u2081 (Prod.{u1, u1} \u03b1\u2081 \u03c3\u2081))) (\u03c3\u2082 -> (m\u2082 (Prod.{u2, u2} \u03b1\u2082 \u03c3\u2082)))) -> (Equiv.{succ (max u1 u3), succ (max u2 u4)} (StateT\u2093.{u1, u3} \u03c3\u2081 m\u2081 \u03b1\u2081) (StateT\u2093.{u2, u4} \u03c3\u2082 m\u2082 \u03b1\u2082))\nbut is expected to have type\n  forall {m\u2081 : Type.{u1}} {m\u2082 : Type.{u1}} {\u03b1\u2081 : Type.{u3}} {\u03c3\u2081 : Type.{u3}} {\u03b1\u2082 : Type.{u1} -> Type.{u2}} {\u03c3\u2082 : Type.{u3} -> Type.{u4}}, (Equiv.{max (succ u2) (succ u1), max (succ u4) (succ u3)} (m\u2081 -> (\u03b1\u2082 (Prod.{u1, u1} m\u2082 m\u2081))) (\u03b1\u2081 -> (\u03c3\u2082 (Prod.{u3, u3} \u03c3\u2081 \u03b1\u2081)))) -> (Equiv.{max (succ u2) (succ u1), max (succ u4) (succ u3)} (StateT.{u1, u2} m\u2081 \u03b1\u2082 m\u2082) (StateT.{u3, u4} \u03b1\u2081 \u03c3\u2082 \u03c3\u2081))\nCase conversion may be inaccurate. Consider using '#align state_t.equiv StateT.equiv\u2093'. -/\n/-- reduce the equivalence between two state monads to the equivalence between\ntheir respective function spaces -/\ndef StateT.equiv {m\u2081 : Type u\u2080 \u2192 Type v\u2080} {m\u2082 : Type u\u2081 \u2192 Type v\u2081} {\u03b1\u2081 \u03c3\u2081 : Type u\u2080}\n    {\u03b1\u2082 \u03c3\u2082 : Type u\u2081} (F : (\u03c3\u2081 \u2192 m\u2081 (\u03b1\u2081 \u00d7 \u03c3\u2081)) \u2243 (\u03c3\u2082 \u2192 m\u2082 (\u03b1\u2082 \u00d7 \u03c3\u2082))) :\n    StateT \u03c3\u2081 m\u2081 \u03b1\u2081 \u2243 StateT \u03c3\u2082 m\u2082 \u03b1\u2082\n    where\n  toFun := fun \u27e8f\u27e9 => \u27e8F f\u27e9\n  invFun := fun \u27e8f\u27e9 => \u27e8F.symm f\u27e9\n  left_inv := fun \u27e8f\u27e9 => congr_arg StateT.mk <| F.left_inv _\n  right_inv := fun \u27e8f\u27e9 => congr_arg StateT.mk <| F.right_inv _\n#align state_t.equiv StateT.equiv\n\n/- warning: reader_t.equiv -> ReaderT.equiv is a dubious translation:\nlean 3 declaration is\n  forall {m\u2081 : Type.{u1} -> Type.{u3}} {m\u2082 : Type.{u2} -> Type.{u4}} {\u03b1\u2081 : Type.{u1}} {\u03c1\u2081 : Type.{u1}} {\u03b1\u2082 : Type.{u2}} {\u03c1\u2082 : Type.{u2}}, (Equiv.{max (succ u1) (succ u3), max (succ u2) (succ u4)} (\u03c1\u2081 -> (m\u2081 \u03b1\u2081)) (\u03c1\u2082 -> (m\u2082 \u03b1\u2082))) -> (Equiv.{succ (max u1 u3), succ (max u2 u4)} (ReaderT\u2093.{u1, u3} \u03c1\u2081 m\u2081 \u03b1\u2081) (ReaderT\u2093.{u2, u4} \u03c1\u2082 m\u2082 \u03b1\u2082))\nbut is expected to have type\n  forall {m\u2081 : Type.{u1}} {m\u2082 : Type.{u1}} {\u03b1\u2081 : Type.{u3}} {\u03c1\u2081 : Type.{u3}} {\u03b1\u2082 : Type.{u1} -> Type.{u2}} {\u03c1\u2082 : Type.{u3} -> Type.{u4}}, (Equiv.{max (succ u2) (succ u1), max (succ u4) (succ u3)} (m\u2081 -> (\u03b1\u2082 m\u2082)) (\u03b1\u2081 -> (\u03c1\u2082 \u03c1\u2081))) -> (Equiv.{max (succ u2) (succ u1), max (succ u4) (succ u3)} (ReaderT.{u1, u2} m\u2081 \u03b1\u2082 m\u2082) (ReaderT.{u3, u4} \u03b1\u2081 \u03c1\u2082 \u03c1\u2081))\nCase conversion may be inaccurate. Consider using '#align reader_t.equiv ReaderT.equiv\u2093'. -/\n/-- reduce the equivalence between two reader monads to the equivalence between\ntheir respective function spaces -/\ndef ReaderT.equiv {m\u2081 : Type u\u2080 \u2192 Type v\u2080} {m\u2082 : Type u\u2081 \u2192 Type v\u2081} {\u03b1\u2081 \u03c1\u2081 : Type u\u2080}\n    {\u03b1\u2082 \u03c1\u2082 : Type u\u2081} (F : (\u03c1\u2081 \u2192 m\u2081 \u03b1\u2081) \u2243 (\u03c1\u2082 \u2192 m\u2082 \u03b1\u2082)) : ReaderT \u03c1\u2081 m\u2081 \u03b1\u2081 \u2243 ReaderT \u03c1\u2082 m\u2082 \u03b1\u2082\n    where\n  toFun := fun \u27e8f\u27e9 => \u27e8F f\u27e9\n  invFun := fun \u27e8f\u27e9 => \u27e8F.symm f\u27e9\n  left_inv := fun \u27e8f\u27e9 => congr_arg ReaderT.mk <| F.left_inv _\n  right_inv := fun \u27e8f\u27e9 => congr_arg ReaderT.mk <| F.right_inv _\n#align reader_t.equiv ReaderT.equiv\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Monad/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832805, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.38472317660944344}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.applicative\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Group.Defs\nimport Mathbin.Control.Functor\n\n/-!\n# `applicative` instances\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file provides `applicative` instances for concrete functors:\n* `id`\n* `functor.comp`\n* `functor.const`\n* `functor.add_const`\n-/\n\n\nuniverse u v w\n\nsection Lemmas\n\nopen Function\n\nvariable {F : Type u \u2192 Type v}\n\nvariable [Applicative F] [LawfulApplicative F]\n\nvariable {\u03b1 \u03b2 \u03b3 \u03c3 : Type u}\n\n/- warning: applicative.map_seq_map -> Applicative.map_seq_map is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} {\u03b3 : Type.{u1}} {\u03c3 : Type.{u1}} (f : \u03b1 -> \u03b2 -> \u03b3) (g : \u03c3 -> \u03b2) (x : F \u03b1) (y : F \u03c3), Eq.{succ u2} (F \u03b3) (Seq.seq.{u1, u2} (fun {\u03b1 : Type.{u1}} => F \u03b1) (Applicative.toHasSeq.{u1, u2} (fun {\u03b1 : Type.{u1}} => F \u03b1) _inst_1) \u03b2 \u03b3 (Functor.map.{u1, u2} (fun {\u03b1 : Type.{u1}} => F \u03b1) (Applicative.toFunctor.{u1, u2} (fun {\u03b1 : Type.{u1}} => F \u03b1) _inst_1) \u03b1 (\u03b2 -> \u03b3) f x) (Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03c3 \u03b2 g y)) (Seq.seq.{u1, u2} F (Applicative.toHasSeq.{u1, u2} F _inst_1) \u03c3 \u03b3 (Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03b1 (\u03c3 -> \u03b3) (Function.comp.{succ u1, succ u1, succ u1} \u03b1 (\u03b2 -> \u03b3) (\u03c3 -> \u03b3) (flip.{succ u1, succ u1, succ u1} (\u03b2 -> \u03b3) (\u03c3 -> \u03b2) (\u03c3 -> \u03b3) (Function.comp.{succ u1, succ u1, succ u1} \u03c3 \u03b2 \u03b3) g) f) x) y)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} {\u03b3 : Type.{u1}} {\u03c3 : Type.{u1}} (f : \u03b1 -> \u03b2 -> \u03b3) (g : \u03c3 -> \u03b2) (x : F \u03b1) (y : F \u03c3), Eq.{succ u2} (F \u03b3) (Seq.seq.{u1, u2} F (Applicative.toSeq.{u1, u2} F _inst_1) \u03b2 \u03b3 (Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03b1 (\u03b2 -> \u03b3) f x) (fun (x._@.Mathlib.Control.Applicative._hyg.84 : Unit) => Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03c3 \u03b2 g y)) (Seq.seq.{u1, u2} F (Applicative.toSeq.{u1, u2} F _inst_1) \u03c3 \u03b3 (Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03b1 (\u03c3 -> \u03b3) (Function.comp.{succ u1, succ u1, succ u1} \u03b1 (\u03b2 -> \u03b3) (\u03c3 -> \u03b3) (flip.{succ u1, succ u1, succ u1} (\u03b2 -> \u03b3) (\u03c3 -> \u03b2) (\u03c3 -> \u03b3) (fun (x._@.Mathlib.Control.Applicative._hyg.105 : \u03b2 -> \u03b3) (x._@.Mathlib.Control.Applicative._hyg.107 : \u03c3 -> \u03b2) => Function.comp.{succ u1, succ u1, succ u1} \u03c3 \u03b2 \u03b3 x._@.Mathlib.Control.Applicative._hyg.105 x._@.Mathlib.Control.Applicative._hyg.107) g) f) x) (fun (x._@.Mathlib.Control.Applicative._hyg.124 : Unit) => y))\nCase conversion may be inaccurate. Consider using '#align applicative.map_seq_map Applicative.map_seq_map\u2093'. -/\ntheorem Applicative.map_seq_map (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (g : \u03c3 \u2192 \u03b2) (x : F \u03b1) (y : F \u03c3) :\n    f <$> x <*> g <$> y = (flip (\u00b7 \u2218 \u00b7) g \u2218 f) <$> x <*> y := by simp [flip, functor_norm]\n#align applicative.map_seq_map Applicative.map_seq_map\n\n/- warning: applicative.pure_seq_eq_map' -> Applicative.pure_seq_eq_map' is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (f : \u03b1 -> \u03b2), Eq.{succ u2} ((F \u03b1) -> (F \u03b2)) (Seq.seq.{u1, u2} F (Applicative.toHasSeq.{u1, u2} F _inst_1) \u03b1 \u03b2 (Pure.pure.{u1, u2} F (Applicative.toHasPure.{u1, u2} F _inst_1) (\u03b1 -> \u03b2) f)) (Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03b1 \u03b2 f)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (f : \u03b1 -> \u03b2), Eq.{succ u2} ((F \u03b1) -> (F \u03b2)) ((fun (x._@.Mathlib.Control.Applicative._hyg.160 : F (\u03b1 -> \u03b2)) (x._@.Mathlib.Control.Applicative._hyg.162 : F \u03b1) => Seq.seq.{u1, u2} F (Applicative.toSeq.{u1, u2} F _inst_1) \u03b1 \u03b2 x._@.Mathlib.Control.Applicative._hyg.160 (fun (x._@.Mathlib.Control.Applicative._hyg.175 : Unit) => x._@.Mathlib.Control.Applicative._hyg.162)) (Pure.pure.{u1, u2} F (Applicative.toPure.{u1, u2} F _inst_1) (\u03b1 -> \u03b2) f)) ((fun (x._@.Mathlib.Control.Applicative._hyg.192 : \u03b1 -> \u03b2) (x._@.Mathlib.Control.Applicative._hyg.194 : F \u03b1) => Functor.map.{u1, u2} F (Applicative.toFunctor.{u1, u2} F _inst_1) \u03b1 \u03b2 x._@.Mathlib.Control.Applicative._hyg.192 x._@.Mathlib.Control.Applicative._hyg.194) f)\nCase conversion may be inaccurate. Consider using '#align applicative.pure_seq_eq_map' Applicative.pure_seq_eq_map'\u2093'. -/\ntheorem Applicative.pure_seq_eq_map' (f : \u03b1 \u2192 \u03b2) : (\u00b7 <*> \u00b7) (pure f : F (\u03b1 \u2192 \u03b2)) = (\u00b7 <$> \u00b7) f :=\n  by ext <;> simp [functor_norm]\n#align applicative.pure_seq_eq_map' Applicative.pure_seq_eq_map'\n\n/- warning: applicative.ext -> Applicative.ext is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} {A1 : Applicative.{u1, u2} F} {A2 : Applicative.{u1, u2} F} [_inst_3 : LawfulApplicative.{u1, u2} F A1] [_inst_4 : LawfulApplicative.{u1, u2} F A2], (forall {\u03b1 : Type.{u1}} (x : \u03b1), Eq.{succ u2} (F \u03b1) (Pure.pure.{u1, u2} F (Applicative.toHasPure.{u1, u2} F A1) \u03b1 x) (Pure.pure.{u1, u2} F (Applicative.toHasPure.{u1, u2} F A2) \u03b1 x)) -> (forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (f : F (\u03b1 -> \u03b2)) (x : F \u03b1), Eq.{succ u2} (F \u03b2) (Seq.seq.{u1, u2} F (Applicative.toHasSeq.{u1, u2} F A1) \u03b1 \u03b2 f x) (Seq.seq.{u1, u2} F (Applicative.toHasSeq.{u1, u2} F A2) \u03b1 \u03b2 f x)) -> (Eq.{max (succ (succ u1)) (succ u2)} (Applicative.{u1, u2} F) A1 A2)\nbut is expected to have type\n  forall {F : Type.{u2} -> Type.{u1}} {A1 : Applicative.{u2, u1} F} {A2 : Applicative.{u2, u1} F} [_inst_3 : LawfulApplicative.{u2, u1} F A1] [_inst_4 : LawfulApplicative.{u2, u1} F A2], (forall {\u03b1 : Type.{u2}} (x : \u03b1), Eq.{succ u1} (F \u03b1) (Pure.pure.{u2, u1} F (Applicative.toPure.{u2, u1} F A1) \u03b1 x) (Pure.pure.{u2, u1} F (Applicative.toPure.{u2, u1} F A2) \u03b1 x)) -> (forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} (f : F (\u03b1 -> \u03b2)) (x : F \u03b1), Eq.{succ u1} (F \u03b2) (Seq.seq.{u2, u1} F (Applicative.toSeq.{u2, u1} F A1) \u03b1 \u03b2 f (fun (x._@.Mathlib.Control.Applicative._hyg.293 : Unit) => x)) (Seq.seq.{u2, u1} F (Applicative.toSeq.{u2, u1} F A2) \u03b1 \u03b2 f (fun (x._@.Mathlib.Control.Applicative._hyg.306 : Unit) => x))) -> (Eq.{max (succ (succ u2)) (succ u1)} (Applicative.{u2, u1} F) A1 A2)\nCase conversion may be inaccurate. Consider using '#align applicative.ext Applicative.ext\u2093'. -/\ntheorem Applicative.ext {F} :\n    \u2200 {A1 : Applicative F} {A2 : Applicative F} [@LawfulApplicative F A1] [@LawfulApplicative F A2]\n      (H1 : \u2200 {\u03b1 : Type u} (x : \u03b1), @Pure.pure _ A1.toHasPure _ x = @Pure.pure _ A2.toHasPure _ x)\n      (H2 :\n        \u2200 {\u03b1 \u03b2 : Type u} (f : F (\u03b1 \u2192 \u03b2)) (x : F \u03b1),\n          @Seq.seq _ A1.toHasSeq _ _ f x = @Seq.seq _ A2.toHasSeq _ _ f x),\n      A1 = A2\n  | { toFunctor := F1\n      seq := s1\n      pure := p1\n      seqLeft := sl1\n      seqRight := sr1 },\n    { toFunctor := F2\n      seq := s2\n      pure := p2\n      seqLeft := sl2\n      seqRight := sr2 }, L1, L2, H1, H2 =>\n    by\n    obtain rfl : @p1 = @p2 := by\n      funext \u03b1 x\n      apply H1\n    obtain rfl : @s1 = @s2 := by\n      funext \u03b1 \u03b2 f x\n      apply H2\n    cases L1\n    cases L2\n    obtain rfl : F1 = F2 := by\n      skip\n      apply Functor.ext\n      intros\n      exact (L1_pure_seq_eq_map _ _).symm.trans (L2_pure_seq_eq_map _ _)\n    congr <;> funext \u03b1 \u03b2 x y\n    \u00b7 exact (L1_seq_left_eq _ _).trans (L2_seq_left_eq _ _).symm\n    \u00b7 exact (L1_seq_right_eq _ _).trans (L2_seq_right_eq _ _).symm\n#align applicative.ext Applicative.ext\n\nend Lemmas\n\ninstance : CommApplicative id := by refine' { .. } <;> intros <;> rfl\n\nnamespace Functor\n\nnamespace Comp\n\nopen Function hiding comp\n\nopen Functor\n\nvariable {F : Type u \u2192 Type w} {G : Type v \u2192 Type u}\n\nvariable [Applicative F] [Applicative G]\n\nvariable [LawfulApplicative F] [LawfulApplicative G]\n\nvariable {\u03b1 \u03b2 \u03b3 : Type v}\n\n#print Functor.Comp.map_pure /-\ntheorem map_pure (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : (f <$> pure x : Comp F G \u03b2) = pure (f x) :=\n  Comp.ext <| by simp\n#align functor.comp.map_pure Functor.Comp.map_pure\n-/\n\n/- warning: functor.comp.seq_pure -> Functor.Comp.seq_pure is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} (f : Functor.Comp.{u1, u2, u3} F G (\u03b1 -> \u03b2)) (x : \u03b1), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b2) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 f (Pure.pure.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasPure.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 x)) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) (\u03b1 -> \u03b2) \u03b2 (fun (g : \u03b1 -> \u03b2) => g x) f)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} (f : Functor.Comp.{u1, u2, u3} F G (\u03b1 -> \u03b2)) (x : \u03b1), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b2) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 f (fun (x._@.Mathlib.Control.Applicative._hyg.1000 : Unit) => Pure.pure.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instPureComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 x)) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) (\u03b1 -> \u03b2) \u03b2 (fun (g : \u03b1 -> \u03b2) => g x) f)\nCase conversion may be inaccurate. Consider using '#align functor.comp.seq_pure Functor.Comp.seq_pure\u2093'. -/\ntheorem seq_pure (f : Comp F G (\u03b1 \u2192 \u03b2)) (x : \u03b1) : f <*> pure x = (fun g : \u03b1 \u2192 \u03b2 => g x) <$> f :=\n  Comp.ext <| by simp [(\u00b7 \u2218 \u00b7), functor_norm]\n#align functor.comp.seq_pure Functor.Comp.seq_pure\n\n/- warning: functor.comp.seq_assoc -> Functor.Comp.seq_assoc is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u2}} (x : Functor.Comp.{u1, u2, u3} F G \u03b1) (f : Functor.Comp.{u1, u2, u3} F G (\u03b1 -> \u03b2)) (g : Functor.Comp.{u1, u2, u3} F G (\u03b2 -> \u03b3)), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b3) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) \u03b2 \u03b3 g (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 f x)) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b3 (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) (\u03b1 -> \u03b2) (\u03b1 -> \u03b3) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) (\u03b2 -> \u03b3) ((\u03b1 -> \u03b2) -> \u03b1 -> \u03b3) (Function.comp.{succ u2, succ u2, succ u2} \u03b1 \u03b2 \u03b3) g) f) x)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u2}} (x : Functor.Comp.{u1, u2, u3} F G \u03b1) (f : Functor.Comp.{u1, u2, u3} F G (\u03b1 -> \u03b2)) (g : Functor.Comp.{u1, u2, u3} F G (\u03b2 -> \u03b3)), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b3) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b2 \u03b3 g (fun (x._@.Mathlib.Control.Applicative._hyg.1084 : Unit) => Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 f (fun (x._@.Mathlib.Control.Applicative._hyg.1096 : Unit) => x))) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b3 (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) (\u03b1 -> \u03b2) (\u03b1 -> \u03b3) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) (\u03b2 -> \u03b3) ((\u03b1 -> \u03b2) -> \u03b1 -> \u03b3) (Function.comp.{succ u2, succ u2, succ u2} \u03b1 \u03b2 \u03b3) g) (fun (x._@.Mathlib.Control.Applicative._hyg.1116 : Unit) => f)) (fun (x._@.Mathlib.Control.Applicative._hyg.1123 : Unit) => x))\nCase conversion may be inaccurate. Consider using '#align functor.comp.seq_assoc Functor.Comp.seq_assoc\u2093'. -/\ntheorem seq_assoc (x : Comp F G \u03b1) (f : Comp F G (\u03b1 \u2192 \u03b2)) (g : Comp F G (\u03b2 \u2192 \u03b3)) :\n    g <*> (f <*> x) = @Function.comp \u03b1 \u03b2 \u03b3 <$> g <*> f <*> x :=\n  Comp.ext <| by simp [(\u00b7 \u2218 \u00b7), functor_norm]\n#align functor.comp.seq_assoc Functor.Comp.seq_assoc\n\n/- warning: functor.comp.pure_seq_eq_map -> Functor.Comp.pure_seq_eq_map is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} (f : \u03b1 -> \u03b2) (x : Functor.Comp.{u1, u2, u3} F G \u03b1), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b2) (Seq.seq.{u2, u3} (fun {\u03b1 : Type.{u2}} => Functor.Comp.{u1, u2, u3} F G \u03b1) (Functor.Comp.hasSeq.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 (Pure.pure.{u2, u3} (fun {\u03b1 : Type.{u2}} => Functor.Comp.{u1, u2, u3} F G \u03b1) (Functor.Comp.hasPure.{u1, u2, u3} F G _inst_1 _inst_2) (\u03b1 -> \u03b2) f) x) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) \u03b1 \u03b2 f x)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u3}} {G : Type.{u2} -> Type.{u1}} [_inst_1 : Applicative.{u1, u3} F] [_inst_2 : Applicative.{u2, u1} G] [_inst_3 : LawfulApplicative.{u1, u3} F _inst_1] [_inst_4 : LawfulApplicative.{u2, u1} G _inst_2] {\u03b1 : Type.{u2}} {\u03b2 : Type.{u2}} (f : \u03b1 -> \u03b2) (x : Functor.Comp.{u1, u2, u3} F G \u03b1), Eq.{succ u3} (Functor.Comp.{u1, u2, u3} F G \u03b2) (Seq.seq.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instSeqComp.{u1, u2, u3} F G _inst_1 _inst_2) \u03b1 \u03b2 (Pure.pure.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.instPureComp.{u1, u2, u3} F G _inst_1 _inst_2) (\u03b1 -> \u03b2) f) (fun (x._@.Mathlib.Control.Applicative._hyg.1180 : Unit) => x)) (Functor.map.{u2, u3} (Functor.Comp.{u1, u2, u3} F G) (Functor.Comp.functor.{u1, u2, u3} F G (Applicative.toFunctor.{u1, u3} F _inst_1) (Applicative.toFunctor.{u2, u1} G _inst_2)) \u03b1 \u03b2 f x)\nCase conversion may be inaccurate. Consider using '#align functor.comp.pure_seq_eq_map Functor.Comp.pure_seq_eq_map\u2093'. -/\ntheorem pure_seq_eq_map (f : \u03b1 \u2192 \u03b2) (x : Comp F G \u03b1) : pure f <*> x = f <$> x :=\n  Comp.ext <| by simp [Applicative.pure_seq_eq_map', functor_norm]\n#align functor.comp.pure_seq_eq_map Functor.Comp.pure_seq_eq_map\n\ninstance : LawfulApplicative (Comp F G)\n    where\n  pure_seq := @Comp.pure_seq_eq_map F G _ _ _ _\n  map_pure := @Comp.map_pure F G _ _ _ _\n  seq_pure := @Comp.seq_pure F G _ _ _ _\n  seq_assoc := @Comp.seq_assoc F G _ _ _ _\n\n/- warning: functor.comp.applicative_id_comp -> Functor.Comp.applicative_id_comp is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [AF : Applicative.{u1, u2} F] [LF : LawfulApplicative.{u1, u2} F AF], Eq.{max (succ (succ u1)) (succ u2)} (Applicative.{u1, u2} (Functor.Comp.{u2, u1, u2} (id.{succ (succ u2)} Type.{u2}) F)) (Functor.Comp.applicative.{u2, u1, u2} (id.{succ (succ u2)} Type.{u2}) F (Monad.toApplicative.{u2, u2} (id.{succ (succ u2)} Type.{u2}) id.monad.{u2}) AF) AF\nbut is expected to have type\n  forall {F : Type.{u2} -> Type.{u1}} [AF : Applicative.{u2, u1} F] [LF : LawfulApplicative.{u2, u1} F AF], Eq.{max (succ u1) (succ (succ u2))} (Applicative.{u2, u1} (Functor.Comp.{u1, u2, u1} Id.{u1} F)) (Functor.Comp.instApplicativeComp.{u1, u2, u1} Id.{u1} F (Monad.toApplicative.{u1, u1} Id.{u1} Id.instMonadId.{u1}) AF) AF\nCase conversion may be inaccurate. Consider using '#align functor.comp.applicative_id_comp Functor.Comp.applicative_id_comp\u2093'. -/\ntheorem applicative_id_comp {F} [AF : Applicative F] [LF : LawfulApplicative F] :\n    @Comp.applicative id F _ _ = AF :=\n  @Applicative.ext F _ _ (@Comp.lawfulApplicative id F _ _ _ _) _ (fun \u03b1 x => rfl) fun \u03b1 \u03b2 f x =>\n    rfl\n#align functor.comp.applicative_id_comp Functor.Comp.applicative_id_comp\n\n/- warning: functor.comp.applicative_comp_id -> Functor.Comp.applicative_comp_id is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [AF : Applicative.{u1, u2} F] [LF : LawfulApplicative.{u1, u2} F AF], Eq.{max (succ (succ u1)) (succ u2)} (Applicative.{u1, u2} (Functor.Comp.{u1, u1, u2} F (id.{succ (succ u1)} Type.{u1}))) (Functor.Comp.applicative.{u1, u1, u2} F (id.{succ (succ u1)} Type.{u1}) AF (Monad.toApplicative.{u1, u1} (id.{succ (succ u1)} Type.{u1}) id.monad.{u1})) AF\nbut is expected to have type\n  forall {F : Type.{u2} -> Type.{u1}} [AF : Applicative.{u2, u1} F] [LF : LawfulApplicative.{u2, u1} F AF], Eq.{max (succ u1) (succ (succ u2))} (Applicative.{u2, u1} (Functor.Comp.{u2, u2, u1} F Id.{u2})) (Functor.Comp.instApplicativeComp.{u2, u2, u1} F Id.{u2} AF (Monad.toApplicative.{u2, u2} Id.{u2} Id.instMonadId.{u2})) AF\nCase conversion may be inaccurate. Consider using '#align functor.comp.applicative_comp_id Functor.Comp.applicative_comp_id\u2093'. -/\ntheorem applicative_comp_id {F} [AF : Applicative F] [LF : LawfulApplicative F] :\n    @Comp.applicative F id _ _ = AF :=\n  @Applicative.ext F _ _ (@Comp.lawfulApplicative F id _ _ _ _) _ (fun \u03b1 x => rfl) fun \u03b1 \u03b2 f x =>\n    show id <$> f <*> x = f <*> x by rw [id_map]\n#align functor.comp.applicative_comp_id Functor.Comp.applicative_comp_id\n\nopen CommApplicative\n\ninstance {f : Type u \u2192 Type w} {g : Type v \u2192 Type u} [Applicative f] [Applicative g]\n    [CommApplicative f] [CommApplicative g] : CommApplicative (Comp f g) :=\n  by\n  refine' { @comp.is_lawful_applicative f g _ _ _ _ with .. }\n  intros\n  casesm*comp _ _ _\n  simp! [map, Seq.seq, functor_norm]\n  rw [commutative_map]\n  simp [comp.mk, flip, (\u00b7 \u2218 \u00b7), functor_norm]\n  congr\n  funext\n  rw [commutative_map]\n  congr\n\nend Comp\n\nend Functor\n\nopen Functor\n\n/- warning: comp.seq_mk -> Comp.seq_mk is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u3}} {f : Type.{u1} -> Type.{u2}} {g : Type.{u3} -> Type.{u1}} [_inst_1 : Applicative.{u1, u2} f] [_inst_2 : Applicative.{u3, u1} g] (h : f (g (\u03b1 -> \u03b2))) (x : f (g \u03b1)), Eq.{succ u2} (Functor.Comp.{u1, u3, u2} f g \u03b2) (Seq.seq.{u3, u2} (Functor.Comp.{u1, u3, u2} f g) (Functor.Comp.hasSeq.{u1, u3, u2} f g _inst_1 _inst_2) \u03b1 \u03b2 (Functor.Comp.mk.{u1, u3, u2} f g (\u03b1 -> \u03b2) h) (Functor.Comp.mk.{u1, u3, u2} f g \u03b1 x)) (Functor.Comp.mk.{u1, u3, u2} f g \u03b2 (Seq.seq.{u1, u2} f (Applicative.toHasSeq.{u1, u2} f _inst_1) (g \u03b1) (g \u03b2) (Functor.map.{u1, u2} f (Applicative.toFunctor.{u1, u2} f _inst_1) (g (\u03b1 -> \u03b2)) ((g \u03b1) -> (g \u03b2)) (Seq.seq.{u3, u1} g (Applicative.toHasSeq.{u3, u1} g _inst_2) \u03b1 \u03b2) h) x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u3}} {f : Type.{u1} -> Type.{u2}} {g : Type.{u3} -> Type.{u1}} [_inst_1 : Applicative.{u1, u2} f] [_inst_2 : Applicative.{u3, u1} g] (h : f (g (\u03b1 -> \u03b2))) (x : f (g \u03b1)), Eq.{succ u2} (Functor.Comp.{u1, u3, u2} f g \u03b2) (Seq.seq.{u3, u2} (Functor.Comp.{u1, u3, u2} f g) (Functor.Comp.instSeqComp.{u1, u3, u2} f g _inst_1 _inst_2) \u03b1 \u03b2 (Functor.Comp.mk.{u1, u3, u2} f g (\u03b1 -> \u03b2) h) (fun (x._@.Mathlib.Control.Applicative._hyg.1722 : Unit) => Functor.Comp.mk.{u1, u3, u2} f g \u03b1 x)) (Functor.Comp.mk.{u1, u3, u2} f g \u03b2 (Seq.seq.{u1, u2} f (Applicative.toSeq.{u1, u2} f _inst_1) (g \u03b1) (g \u03b2) (Functor.map.{u1, u2} f (Applicative.toFunctor.{u1, u2} f _inst_1) (g (\u03b1 -> \u03b2)) ((g \u03b1) -> (g \u03b2)) (fun (x._@.Mathlib.Control.Applicative._hyg.1738 : g (\u03b1 -> \u03b2)) (x._@.Mathlib.Control.Applicative._hyg.1740 : g \u03b1) => Seq.seq.{u3, u1} g (Applicative.toSeq.{u3, u1} g _inst_2) \u03b1 \u03b2 x._@.Mathlib.Control.Applicative._hyg.1738 (fun (x._@.Mathlib.Control.Applicative._hyg.1753 : Unit) => x._@.Mathlib.Control.Applicative._hyg.1740)) h) (fun (x._@.Mathlib.Control.Applicative._hyg.1761 : Unit) => x)))\nCase conversion may be inaccurate. Consider using '#align comp.seq_mk Comp.seq_mk\u2093'. -/\n@[functor_norm]\ntheorem Comp.seq_mk {\u03b1 \u03b2 : Type w} {f : Type u \u2192 Type v} {g : Type w \u2192 Type u} [Applicative f]\n    [Applicative g] (h : f (g (\u03b1 \u2192 \u03b2))) (x : f (g \u03b1)) :\n    Comp.mk h <*> Comp.mk x = Comp.mk (Seq.seq <$> h <*> x) :=\n  rfl\n#align comp.seq_mk Comp.seq_mk\n\ninstance {\u03b1} [One \u03b1] [Mul \u03b1] : Applicative (Const \u03b1)\n    where\n  pure \u03b2 x := (1 : \u03b1)\n  seq \u03b2 \u03b3 f x := (f * x : \u03b1)\n\ninstance {\u03b1} [Monoid \u03b1] : LawfulApplicative (Const \u03b1) := by\n  refine' { .. } <;> intros <;> simp [mul_assoc, (\u00b7 <$> \u00b7), (\u00b7 <*> \u00b7), pure]\n\ninstance {\u03b1} [Zero \u03b1] [Add \u03b1] : Applicative (AddConst \u03b1)\n    where\n  pure \u03b2 x := (0 : \u03b1)\n  seq \u03b2 \u03b3 f x := (f + x : \u03b1)\n\ninstance {\u03b1} [AddMonoid \u03b1] : LawfulApplicative (AddConst \u03b1) := by\n  refine' { .. } <;> intros <;> simp [add_assoc, (\u00b7 <$> \u00b7), (\u00b7 <*> \u00b7), pure]\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Applicative.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.38472317585044197}}
{"text": "import analysis.normed_space.add_torsor_bases\nimport analysis.convex.caratheodory\nimport analysis.calculus.cont_diff\nimport measure_theory.integral.interval_integral\nimport measure_theory.measure.lebesgue\nimport topology.algebra.order.floor\nimport topology.path_connected\nimport linear_algebra.affine_space.independent\n\nimport to_mathlib.smooth_barycentric\nimport to_mathlib.topology.path\nimport to_mathlib.measure_theory.parametric_interval_integral\nimport to_mathlib.equivariant\n\n/-!\n# Basic definitions and properties of loops\n-/\n\nopen set function finite_dimensional int topological_space\nopen_locale big_operators topology unit_interval\nnoncomputable theory\n\nvariables {K X X' Y Z : Type*}\n-- variables [topological_space X'] [topological_space Y] [topological_space Z]\nvariables {E : Type*} [normed_add_comm_group E] [normed_space \u211d E]\n          {F : Type*} [normed_add_comm_group F] [normed_space \u211d F]\n          {F' : Type*} [normed_add_comm_group F'] [normed_space \u211d F']\n\nset_option old_structure_cmd true\n\n/-! ## Definition and periodicity lemmas -/\n\nvariables (X)\n\n/-- A loop is a function with domain `\u211d` and is periodic with period 1. -/\nstructure loop :=\n(to_fun : \u211d \u2192 X)\n(per' : \u2200 t, to_fun (t + 1) = to_fun t)\n\ninstance : has_coe_to_fun (loop X) (\u03bb _, \u211d \u2192 X) := \u27e8\u03bb \u03b3, \u03b3.to_fun\u27e9\n\ninitialize_simps_projections loop (to_fun \u2192 apply)\n\n/-- Any function `\u03c6 : \u03b1 \u2192 loop X` can be seen as a function `\u03b1 \u00d7 \u211d \u2192 X`. -/\ninstance has_uncurry_loop {\u03b1 : Type*} : has_uncurry (\u03b1 \u2192 loop X) (\u03b1 \u00d7 \u211d) X := \u27e8\u03bb \u03c6 p, \u03c6 p.1 p.2\u27e9\n\nvariables {X}\n\nnamespace loop\n\n@[simp]\nprotected lemma coe_mk {\u03b3 : \u211d \u2192 X} (h : \u2200 t, \u03b3 (t + 1) = \u03b3 t) : \u21d1(\u27e8\u03b3, h\u27e9 : loop X) = \u03b3 :=\nrfl\n\n@[ext] protected lemma ext : \u2200 {\u03b3\u2081 \u03b3\u2082 : loop X}, (\u03b3\u2081 : \u211d \u2192 X) = \u03b3\u2082 \u2192 \u03b3\u2081 = \u03b3\u2082\n| \u27e8x, h1\u27e9 \u27e8.(x), h2\u27e9 rfl := rfl\n\nprotected lemma ext_iff {\u03b3\u2081 \u03b3\u2082 : loop X} : \u03b3\u2081 = \u03b3\u2082 \u2194 (\u03b3\u2081 : \u211d \u2192 X) = \u03b3\u2082 :=\n\u27e8\u03bb h, by rw h, loop.ext\u27e9\n\n/-- The constant loop. -/\n@[simps]\ndef const (f : X) : loop X :=\n\u27e8\u03bb t, f, \u03bb t, rfl\u27e9\n\ninstance [has_zero X] : has_zero (loop X) :=\n\u27e8const 0\u27e9\n\n@[simp] lemma zero_fun [has_zero X] : ((0 : loop X) : \u211d \u2192 X) = (0 : \u211d \u2192 X) :=\nrfl\n\n-- unused\n@[simp] lemma const_zero [has_zero X] : const (0 : X) = (0 : loop X) :=\nrfl\n\ninstance [inhabited X] : inhabited (loop X) :=\n\u27e8loop.const default\u27e9\n\n/-- Periodicity of loops restated in terms of the function coercion. -/\nlemma per (\u03b3 : loop X) : \u2200 t, \u03b3 (t + 1) = \u03b3 t :=\nloop.per' \u03b3\n\nlemma periodic (\u03b3 : loop X) : function.periodic \u03b3 1 :=\nloop.per' \u03b3\n\nprotected lemma one (\u03b3 : loop X) : \u03b3 1 = \u03b3 0 :=\nby { convert \u03b3.per 0, rw [zero_add] }\n\n-- unused\nlemma add_nat_eq (\u03b3 : loop X) (t : \u211d) : \u2200 (n : \u2115), \u03b3 (t + n) = \u03b3 t\n| 0 := by rw [nat.cast_zero, add_zero]\n| (nat.succ n) := by rw [\u2190 add_nat_eq n, nat.cast_succ, \u2190 add_assoc, \u03b3.per]\n\nlemma add_int_eq (\u03b3 : loop X) (t : \u211d) (n : \u2124) : \u03b3 (t + n) = \u03b3 t :=\nbegin\n  induction n using int.induction_on with n hn n hn,\n  { norm_cast, rw add_zero },\n  { rw [\u2190 hn, int.cast_add, \u2190 add_assoc, int.cast_one, \u03b3.per] },\n  { rw [\u2190 hn, int.cast_sub, add_sub, int.cast_one, \u2190 \u03b3.per, sub_add_cancel] }\nend\n\nlemma fract_eq (\u03b3 : loop X) : \u2200 t, \u03b3 (fract t) = \u03b3 t :=\nbegin\n  intro t,\n  unfold fract,\n  rw [sub_eq_add_neg, \u2190 int.cast_neg],\n  exact \u03b3.add_int_eq _ _\nend\n\nlemma range_eq_image (\u03b3 : loop X) : range \u03b3 = \u03b3 '' I :=\nbegin\n  apply eq_of_subset_of_subset,\n  { rw range_subset_iff,\n    exact \u03bb y, \u27e8fract y, unit_interval.fract_mem y, \u03b3.fract_eq _\u27e9 },\n  { rintros y \u27e8x, hx, hxy\u27e9,\n    exact \u27e8x, hxy\u27e9 },\nend\n\n/-- Transforming a loop by applying function `f`. -/\n@[simps]\ndef transform (\u03b3 : loop X) (f : X \u2192 X') : loop X' :=\n\u27e8\u03bb t, f (\u03b3 t), \u03bb t, by rw \u03b3.per\u27e9\n\n/-- Adding two loops pointwise. -/\n@[simps]\ninstance [has_add X] : has_add (loop X) :=\n\u27e8\u03bb \u03b3\u2081 \u03b3\u2082, \u27e8\u03bb t, \u03b3\u2081 t + \u03b3\u2082 t, \u03bb t, by simp_rw [loop.per]\u27e9\u27e9\n\n@[simps]\ninstance [has_neg X] : has_neg (loop X) :=\n\u27e8\u03bb \u03b3, \u27e8\u03bb t, - \u03b3 t, \u03bb t, by simp_rw [loop.per]\u27e9\u27e9\n\ninstance [add_comm_group X] : add_comm_group (loop X) :=\n{ add_assoc := \u03bb \u03b3\u2081 \u03b3\u2082 \u03b3\u2083, by { ext t, apply add_assoc },\n  add_comm := \u03bb \u03b3\u2081 \u03b3\u2082, by { ext t, apply add_comm },\n  add_comm := \u03bb \u03b3\u2081 \u03b3\u2082, by { ext t, apply add_comm },\n  zero_add := \u03bb \u03b3, by { ext t, apply zero_add },\n  add_zero := \u03bb \u03b3, by { ext t, apply add_zero },\n  add_left_neg := \u03bb \u03b3, by { ext t, apply add_left_neg },\n  ..loop.has_add,\n  ..loop.has_zero,\n  ..loop.has_neg }\n\n/-- Shifting a loop, or equivalently, adding a constant value to a loop. -/\ninstance [has_add X] : has_vadd X (loop X) :=\n\u27e8\u03bb x \u03b3, \u03b3.transform (\u03bb y, x + y)\u27e9\n\n@[simp] lemma vadd_apply [has_add X] {x : X} {\u03b3 : loop X} {t : \u211d} : (x +\u1d65 \u03b3) t = x + \u03b3 t :=\nrfl\n\n/-- Multiplying a loop by a scalar value. -/\ninstance [has_smul K X] : has_smul K (loop X) :=\n\u27e8\u03bb k \u03b3, \u03b3.transform (\u03bb y, k \u2022 y)\u27e9\n\ninstance [semiring K] [add_comm_group X] [module K X] : module K (loop X) :=\n{ one_smul := \u03bb \u03b3, by { ext t, apply one_smul },\n  mul_smul := \u03bb k\u2081 k\u2082 \u03b3, by { ext t, apply mul_smul },\n  smul_zero := \u03bb k, by { ext t, apply smul_zero },\n  smul_add := \u03bb k \u03b3\u2081 \u03b3\u2082, by { ext t, apply smul_add },\n  add_smul := \u03bb k\u2081 k\u2082 \u03b3, by { ext t, apply add_smul },\n  zero_smul := \u03bb \u03b3, by { ext t, apply zero_smul } }\n\n@[simp] lemma smul_apply [has_smul K X] {k : K} {\u03b3 : loop X} {t : \u211d} : (k \u2022 \u03b3) t = k \u2022 \u03b3 t :=\nrfl\n\n-- unused\nlemma norm_at_le_supr_norm_Icc (\u03b3 : loop F) (h\u03b3 : continuous \u03b3) (t : \u211d) :\n  \u2016\u03b3 t\u2016 \u2264 \u2a06 (s : I), \u2016\u03b3 s\u2016 :=\nbegin\n  obtain \u27e8u, hu, ht\u27e9 := \u03b3.periodic.exists_mem_Ico\u2080 zero_lt_one t,\n  replace hu := mem_Icc_of_Ico hu,\n  rw ht,\n  have h\u2081 : set.nonempty (range (\u03bb (s : I), \u2016\u03b3 s\u2016)) := \u27e8\u2016\u03b3 0\u2016, 0, rfl\u27e9,\n  have h\u2082 : bdd_above (range (\u03bb (s : I), \u2016\u03b3 s\u2016)),\n  { convert is_compact_Icc.bdd_above_image (continuous_norm.comp h\u03b3).continuous_on, ext, simp, },\n  exact (real.is_lub_Sup _ h\u2081 h\u2082).1 \u27e8\u27e8u, hu\u27e9, rfl\u27e9,\nend\n\n/-- Reparametrizing loop `\u03b3` using an equivariant map `\u03c6`. -/\n@[simps {simp_rhs := tt}]\ndef reparam {F : Type*} (\u03b3 : loop F) (\u03c6 : equivariant_map) : loop F :=\n{ to_fun := \u03b3 \u2218 \u03c6,\n  per' := \u03bb t, by rw [comp_apply, \u03c6.eqv, \u03b3.per] }\n\n/-! ## Support of a loop family -/\n\n/-- A loop is constant if it takes the same value at every time.\nSee also `loop.is_const_iff_forall_avg` and `loop.is_const_iff_const_avg` for characterizations in\nterms of average values. -/\ndef is_const (\u03b3 : loop X) := \u2200 t s, \u03b3 t = \u03b3 s\n\nlemma is_const_of_eq {\u03b3 : loop X} {f : X} (H : \u2200 t, \u03b3 t = f) : \u03b3.is_const :=\n\u03bb t t', by rw [H, H]\n\nvariables [topological_space X] [topological_space X']\nvariables [topological_space Y] [topological_space Z]\n\n/-- The support of a loop family is the closure of the set of parameters where\nthe loop is not constant. -/\ndef support (\u03b3 : X \u2192 loop X') : set X :=\nclosure {x | \u00ac (\u03b3 x).is_const}\n\nlemma not_mem_support {\u03b3 : X \u2192 loop X'} {x : X} (h : \u2200\u1da0 y in \ud835\udcdd x, (\u03b3 y).is_const) :\n  x \u2209 loop.support \u03b3 :=\nbegin\n  intro hx,\n  rw [support, mem_closure_iff_nhds] at hx,\n  rcases hx _ h with \u27e8z, hz, hz'\u27e9,\n  exact hz' hz\nend\n\n/-! ## From paths to loops -/\n\n/-- Turn a path into a loop. -/\n@[simps]\nnoncomputable def of_path {x : X} (\u03b3 : path x x) : loop X :=\n{ to_fun := \u03bb t, \u03b3.extend (fract t),\n  per' :=\n  begin\n    intros t,\n    congr' 1,\n    exact_mod_cast fract_add_int t 1\n  end }\n\n@[simp]\nlemma range_of_path {x : X} (\u03b3 : path x x) : range (of_path \u03b3) = range \u03b3 :=\nbegin\n  rw loop.range_eq_image,\n  unfold_coes,\n  simp only [of_path, image_eq_range],\n  congr,\n  ext t,\n  by_cases ht1 : t.val = 1,\n  { have : t = \u27e81, right_mem_Icc.mpr zero_le_one\u27e9 := subtype.ext_val ht1,\n    rw this,\n    norm_cast,\n    simp only [fract, floor_one, path.extend_zero, int.cast_one, sub_self, subtype.coe_mk],\n    exact \u03b3.target.symm },\n  { change (t : \u211d) \u2260 1 at ht1,\n    have : fract \u2191t = t.val,\n    { rw fract_eq_iff,\n      refine \u27e8t.2.1, t.2.2.lt_of_ne ht1, \u27e80, _\u27e9\u27e9,\n      rw [int.cast_zero, subtype.val_eq_coe, sub_self] },\n    simp only [this, \u03b3.extend_extends t.2],\n    congr',\n    rw subtype.ext_iff_val }\nend\n\n/-- `loop.of_path` is continuous, general version. -/\nlemma _root_.continuous.of_path (x : X \u2192 Y) (t : X \u2192 \u211d)\n  (\u03b3 : \u2200 i, path (x i) (x i)) (h\u03b3 : continuous \u21bf\u03b3) (ht : continuous t) :\n  continuous (\u03bb i, of_path (\u03b3 i) (t i)) :=\nbegin\n  change continuous (\u03bb i, (\u03bb s, (\u03b3 s).extend) i (fract (t i))),\n  refine continuous_on.comp_fract _ ht _,\n  { exact (h\u03b3.comp (continuous_id.prod_map continuous_proj_Icc)).continuous_on },\n  { simp only [Icc.mk_zero, zero_le_one, path.target, path.extend_extends,\n      implies_true_iff, eq_self_iff_true, path.source, right_mem_Icc, left_mem_Icc,\n      Icc.mk_one] }\nend\n\n/-- `loop.of_path` is continuous, where the endpoints of `\u03b3` are fixed. TODO: remove -/\nlemma of_path_continuous_family {x : Y} (\u03b3 : X \u2192 path x x)\n  (h : continuous \u21bf\u03b3) : continuous \u21bf(\u03bb s, of_path $ \u03b3 s) :=\ncontinuous.of_path _ _ (\u03bb i : X \u00d7 \u211d, \u03b3 i.1) (h.comp $ continuous_fst.prod_map continuous_id)\n  continuous_snd\n\n/-! ## Round trips -/\n\n/-- The round-trip defined by `\u03b3` is `\u03b3` followed by `\u03b3\u207b\u00b9`. -/\ndef round_trip {x y : X} (\u03b3 : path x y) : loop X :=\nof_path (\u03b3.trans \u03b3.symm)\n\nlemma round_trip_range {x y : X} {\u03b3 : path x y} : range (round_trip \u03b3) = range \u03b3 :=\nby simp [round_trip, range_of_path, path.trans_range, path.symm_range]\n\nlemma round_trip_based_at {x y : X} {\u03b3 : path x y} : round_trip \u03b3 0 = x :=\nbegin\n  unfold_coes,\n  rw [round_trip, of_path],\n  simp [fract_zero]\nend\n\nlemma round_trip_eq {x y x' y' : X} {\u03b3 : path x y} {\u03b3' : path x' y'} (h : \u2200 s, \u03b3 s = \u03b3' s) :\n  round_trip \u03b3 = round_trip \u03b3' :=\nbegin\n  obtain rfl : x = x' := \u03b3.source.symm.trans ((h 0).trans \u03b3'.source),\n  obtain rfl : y = y' := \u03b3.target.symm.trans ((h 1).trans \u03b3'.target),\n  obtain rfl : \u03b3 = \u03b3', { ext, apply h },\n  refl,\nend\n\n\n/-- The round trip loop family associated to a path `\u03b3`. For each parameter `t`,\nthe loop `round_trip_family \u03b3 t` backtracks at `\u03b3 t`. -/\nnoncomputable\ndef round_trip_family {x y : X} (\u03b3 : path x y) : \u211d \u2192 loop X :=\nhave key : \u2200 {t}, x = \u03b3.extend (min 0 t) := \u03bb t, (\u03b3.extend_of_le_zero $ min_le_left _ _).symm,\n\u03bb t, round_trip ((\u03b3.truncate 0 t).cast key rfl)\n\nlemma round_trip_family_continuous {x y : X} {\u03b3 : path x y} : continuous \u21bf(round_trip_family \u03b3) :=\nof_path_continuous_family _\n  (path.trans_continuous_family _ (\u03b3.truncate_const_continuous_family 0) _ $\n    path.symm_continuous_family _ $ \u03b3.truncate_const_continuous_family 0)\n\nlemma round_trip_family_based_at {x y : X} {\u03b3 : path x y} : \u2200 t, (round_trip_family \u03b3) t 0 = x :=\n\u03bb t, round_trip_based_at\n\nlemma round_trip_family_zero {x y : X} {\u03b3 : path x y} :\n  (round_trip_family \u03b3) 0 = of_path (path.refl x) :=\nbegin\n  simp only [round_trip_family, round_trip, path.truncate_zero_zero, of_path],\n  ext z,\n  congr,\n  ext t,\n  simp [path.refl_symm]\nend\n\nlemma round_trip_family_one {x y : X} {\u03b3 : path x y} : (round_trip_family \u03b3) 1 = round_trip \u03b3 :=\nbegin\n  simp only [round_trip_family, round_trip, path.truncate_zero_one, of_path],\n  refl\nend\n\n\nsection average\n\n/-! ## Average value of a loop -/\n\nvariables [measurable_space F] [borel_space F] [second_countable_topology F] [complete_space F]\n\n/-- The average value of a loop. -/\nnoncomputable def average (\u03b3 : loop F) : F :=\n\u222b x in 0..1, (\u03b3 x)\n\n-- unused\n@[simp]\nlemma zero_average : average (0 : loop F) = 0 :=\ninterval_integral.integral_zero\n\nlemma is_const_iff_forall_avg {\u03b3 : loop F} : \u03b3.is_const \u2194 \u2200 t, \u03b3 t = \u03b3.average :=\nbegin\n  split ; intro h,\n  { intro t,\n    have : \u03b3 = loop.const (\u03b3 t),\n    { ext s,\n      rw h s t,\n      refl },\n    rw this,\n    simp only [average, const_apply, interval_integral.integral_const, one_smul, sub_zero], },\n  { exact is_const_of_eq h }\nend\n\n@[simp] lemma average_const {f : F} : (const f).average = f :=\nby simp [loop.average]\n\nopen measure_theory\n@[simp] lemma average_add {\u03b3\u2081 \u03b3\u2082 : loop F} (h\u03b3\u2081 : interval_integrable \u03b3\u2081 volume 0 1)\n  (h\u03b3\u2082 : interval_integrable \u03b3\u2082 volume 0 1) : (\u03b3\u2081 + \u03b3\u2082).average = \u03b3\u2081.average + \u03b3\u2082.average :=\nby simp [loop.average, interval_integral.integral_add h\u03b3\u2081 h\u03b3\u2082]\n\n@[simp] lemma average_smul {\u03b3 : loop F} {c : \u211d} : (c \u2022 \u03b3).average = c \u2022 \u03b3.average :=\nby simp [loop.average, interval_integral.integral_smul]\n\nlemma is_const_iff_const_avg {\u03b3 : loop F} : \u03b3.is_const \u2194 \u03b3 = const \u03b3.average :=\nby { rw [loop.is_const_iff_forall_avg, loop.ext_iff, funext_iff], refl }\n\nlemma is_const_of_not_mem_support {\u03b3 : X \u2192 loop F} {x : X}\n  (hx : x \u2209 support \u03b3) : (\u03b3 x).is_const :=\nbegin\n  classical,\n  exact decidable.by_contradiction (\u03bb H, hx (subset_closure H)),\nend\n\nlemma continuous_average {E : Type*} [topological_space E] [first_countable_topology E]\n  [locally_compact_space E] {\u03b3 : E \u2192 loop F}\n  (h\u03b3_cont : continuous \u21bf\u03b3) : continuous (\u03bb x, (\u03b3 x).average) :=\ncontinuous_parametric_interval_integral_of_continuous' h\u03b3_cont _ _\n\n/-- The normalization of a loop `\u03b3` is the loop `\u03b3 - \u03b3.average`. -/\ndef normalize (\u03b3 : loop F) : loop F :=\n{ to_fun := \u03bb t, \u03b3 t - \u03b3.average,\n  per' := \u03bb t, by simp [\u03b3.per] }\n\n@[simp]\nlemma normalize_apply (\u03b3 : loop F) (t : \u211d) : loop.normalize \u03b3 t = \u03b3 t - \u03b3.average :=\nrfl\n\n@[simp]\nlemma normalize_of_is_const {\u03b3 : loop F} (h : \u03b3.is_const) : \u03b3.normalize = 0 :=\nbegin\n  ext t,\n  simp [is_const_iff_forall_avg.mp h]\nend\n\nend average\n\nend loop\n\nsection c1\n\n/-! ## Differentiation of loop families -/\n\n\nlocal notation `\u2202\u2081` := partial_fderiv_fst \u211d\n\nvariables (\u03c0 : E \u2192 \u211d) (N : \u211d) (\u03b3 : E \u2192 loop F)\n          (h\u03b3 : is_compact (loop.support \u03b3))\n\n/-- Differential of a loop family with respect to the parameter. -/\ndef loop.diff (\u03b3 : E \u2192 loop F) (e : E) : loop (E \u2192L[\u211d] F) :=\n{ to_fun := \u03bb t, \u2202\u2081 (\u03bb e t, \u03b3 e t) e t,\n  per' := \u03bb t, by simp only [partial_fderiv_fst, loop.per] }\n\n@[simp]\nlemma loop.diff_apply (\u03b3 : E \u2192 loop F) (e : E) (t : \u211d) : loop.diff \u03b3 e t = \u2202\u2081 (\u03bb e t, \u03b3 e t) e t :=\nrfl\n\nlemma loop.continuous_diff {\u03b3 : E \u2192 loop F} (h : \ud835\udc9e 1 \u21bf\u03b3) : continuous (\u21bf(loop.diff \u03b3)) :=\ncont_diff.continuous_partial_fst (h : _)\n\nlemma cont_diff.partial_loop {\u03b3 : E \u2192 loop F} {n : \u2115\u221e} (h\u03b3_diff : \ud835\udc9e n \u21bf\u03b3) :\n  \u2200 t, \ud835\udc9e n (\u03bb e, \u03b3 e t) :=\n\u03bb t, h\u03b3_diff.comp ((cont_diff_prod_mk_left t).of_le le_top)\n\nvariables [measurable_space F] [borel_space F] [finite_dimensional \u211d F]\n\nlemma loop.support_diff {\u03b3 : E \u2192 loop F} :\n  loop.support (loop.diff \u03b3) \u2286 loop.support \u03b3 :=\nbegin\n  unfold loop.support,\n  erw [closure_compl, closure_compl],\n  rw compl_subset_compl,\n  intros x hx,\n  rw mem_interior_iff_mem_nhds at *,\n  rcases mem_nhds_iff.mp hx with \u27e8U, hU, U_op, hxU\u27e9,\n  have U_nhds : U \u2208 \ud835\udcdd x, from is_open.mem_nhds U_op hxU,\n  apply filter.mem_of_superset U_nhds,\n  intros y hy,\n  have Hy : \u2200 t, (\u03bb z, \u03b3 z t) =\u1da0[\ud835\udcdd y] (\u03bb z, (\u03b3 z).average),\n  { intro t,\n    apply filter.mem_of_superset (U_op.mem_nhds hy),\n    intros z hz,\n    exact loop.is_const_iff_forall_avg.mp (hU hz) t },\n  have : \u2200 (t : \u211d), loop.diff \u03b3 y t = D (\u03bb (z : E), (\u03b3 z).average) y := \u03bb t, (Hy t).fderiv_eq,\n  intros t s,\n  simp [this]\nend\n\nvariables [finite_dimensional \u211d E]\n\nlemma loop.average_diff {\u03b3 : E \u2192 loop F} (h\u03b3_diff : \ud835\udc9e 1 \u21bf\u03b3) (e : E) :\n(loop.diff \u03b3 e).average = D (\u03bb e, (\u03b3 e).average) e :=\nbegin\n  change \ud835\udc9e 1 \u21bf(\u03bb (e : E) (t : \u211d), \u03b3 e t) at h\u03b3_diff,\n  simpa only [loop.average, h\u03b3_diff.fderiv_parametric_integral]\nend\n\nlemma cont_diff.loop_average {\u03b3 : E \u2192 loop F} {n : \u2115\u221e} (h\u03b3_diff : \ud835\udc9e n \u21bf\u03b3) :\n  \ud835\udc9e n (\u03bb e, (\u03b3 e).average) :=\ncont_diff_parametric_integral_of_cont_diff h\u03b3_diff _ _\n\nlemma loop.diff_normalize {\u03b3 : E \u2192 loop F} (h\u03b3_diff : \ud835\udc9e 1 \u21bf\u03b3) (e : E) :\n  (loop.diff \u03b3 e).normalize = loop.diff (\u03bb e, (\u03b3 e).normalize) e :=\nbegin\n  ext t x,\n  simp only [loop.diff_apply, loop.normalize_apply, partial_fderiv_fst],\n\n  rw [fderiv_sub ((h\u03b3_diff.partial_loop t).differentiable le_rfl).differentiable_at,\n      loop.average_diff h\u03b3_diff],\n  exact (h\u03b3_diff.loop_average.differentiable le_rfl).differentiable_at\nend\n\nvariable {\u03b3}\n\nlemma cont_diff_average {n : \u2115\u221e} (h\u03b3_diff : \ud835\udc9e n \u21bf\u03b3) : \ud835\udc9e n (\u03bb x, (\u03b3 x).average) :=\ncont_diff_parametric_primitive_of_cont_diff h\u03b3_diff cont_diff_const 0\n\nlemma cont_diff_sub_average {n : \u2115\u221e} (h\u03b3_diff : \ud835\udc9e n \u21bf\u03b3) :\n  \ud835\udc9e n \u21bf(\u03bb (x : E) (t : \u211d), (\u03b3 x) t - (\u03b3 x).average) :=\nh\u03b3_diff.sub (cont_diff_average h\u03b3_diff).fst'\n\nend c1\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/loops/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984445, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3847231675270904}}
{"text": "-- Copyright (C) 2020 by @ljt12138\n\nimport tactic pal.henkin_model pal.soundness\n\nnamespace pal_logic\n\ndef trivial_world (\u03b1 agent : Type) : worlds \u03b1 agent \u2115 := \n{\n  f := \u03bb s a, false, \n  view := \u03bb i s t, s = t,\n  equiv := \n  begin intros i, simp, apply eq_equivalence end\n}\n\ntheorem completeness {\u03b1 agent : Type} [encodable (sentence \u03b1 agent)] \n  (\u03c6 : sentence \u03b1 agent) : static \u03c6 \u2192 \u22a8 \u03c6 \u2192 \u22a2 \u03c6 :=\nbegin\n  intros st h, classical, by_contra,\n  have h\u2081 : consistent_set (\u03bb \u03c8, \u03c8 = \u03c6\u21a3\u22a5), \n  {\n    simp, intros \u0393 h' contra, apply a,\n    have h\u2082 : \u22a2 conjunction [\u03c6\u21a3\u22a5] \u21a3 \u22a5,\n    {\n      cases contra with pf pfh,\n      existsi proof.conj \u0393 [\u03c6\u21a3\u22a5] \u22a5 pf _, { prover },\n      intros \u03c8 h\u2081, rewrite h' _ h\u2081, simp\n    },\n    simp at h\u2082, cases h\u2082,\n    existsi proof.mp (proof.ax3 _) _, prover\n  },\n  cases dcomplete_extension (\u03bb \u03c8, \u03c8 = \u03c6\u21a3\u22a5) h\u2081 with s h\u2082,\n  have st' : static (\u03c6\u21a3\u22a5), \n  { apply static.imply, exact st, apply static.perp },\n  have h\u2082 : \u2983s\u2984 \u22a8 \u03c6 \u21a3 \u22a5,\n  {\n    apply (henkin_correctness (\u03c6\u21a3\u22a5) st' s).mp,\n    apply h\u2082, reflexivity\n  },\n  simp at h\u2082, apply h\u2082, apply h\nend \n\nend pal_logic\n", "meta": {"author": "ljt12138", "repo": "Formalization-PAL", "sha": "351962172c8e85ec8bdf59421df2acd743cb4e3e", "save_path": "github-repos/lean/ljt12138-Formalization-PAL", "path": "github-repos/lean/ljt12138-Formalization-PAL/Formalization-PAL-351962172c8e85ec8bdf59421df2acd743cb4e3e/src/pal/completeness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7690802264851919, "lm_q2_score": 0.5, "lm_q1q2_score": 0.38454011324259596}}
{"text": "import topology.algebra.module.basic\n\nopen filter continuous_linear_map function\nopen_locale topology big_operators filter\n\n\nnamespace continuous_linear_map\n\nvariables {R\u2081 M\u2081 M\u2082 M\u2083 : Type*} [semiring R\u2081]\nvariables [topological_space M\u2081] [add_comm_monoid M\u2081]\nvariables [topological_space M\u2082] [add_comm_monoid M\u2082]\nvariables [topological_space M\u2083] [add_comm_monoid M\u2083]\nvariables [module R\u2081 M\u2081] [module R\u2081 M\u2082] [module R\u2081 M\u2083]\n\n-- unused\nlemma comp_fst_add_comp_snd [has_continuous_add M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2083) (g : M\u2082 \u2192L[R\u2081] M\u2083) :\n  f.comp (continuous_linear_map.fst R\u2081 M\u2081 M\u2082) +\n  g.comp (continuous_linear_map.snd R\u2081 M\u2081 M\u2082) =\n  f.coprod g :=\nrfl\n\nlemma fst_prod_zero_add_zero_prod_snd [has_continuous_add M\u2081] [has_continuous_add M\u2082] :\n  (continuous_linear_map.fst R\u2081 M\u2081 M\u2082).prod 0 +\n  continuous_linear_map.prod 0 (continuous_linear_map.snd R\u2081 M\u2081 M\u2082) =\n  continuous_linear_map.id R\u2081 (M\u2081 \u00d7 M\u2082) :=\nbegin\n  rw [continuous_linear_map.ext_iff],\n  intro x,\n  simp_rw [continuous_linear_map.add_apply, continuous_linear_map.id_apply,\n    continuous_linear_map.prod_apply, continuous_linear_map.coe_fst',\n    continuous_linear_map.coe_snd', continuous_linear_map.zero_apply, prod.mk_add_mk, add_zero,\n    zero_add, prod.mk.eta]\nend\n\n\nend continuous_linear_map\n\nvariables {R\u2081 : Type*} {R\u2082 : Type*} {R\u2083 : Type*} [semiring R\u2081] [semiring R\u2082] [semiring R\u2083]\n{\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082} {\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081} [ring_hom_inv_pair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [ring_hom_inv_pair \u03c3\u2082\u2081 \u03c3\u2081\u2082]\n{\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082} [ring_hom_inv_pair \u03c3\u2082\u2083 \u03c3\u2083\u2082] [ring_hom_inv_pair \u03c3\u2083\u2082 \u03c3\u2082\u2083]\n{\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083} {\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081} [ring_hom_inv_pair \u03c3\u2081\u2083 \u03c3\u2083\u2081] [ring_hom_inv_pair \u03c3\u2083\u2081 \u03c3\u2081\u2083]\n[ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [ring_hom_comp_triple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081]\n{M\u2081 : Type*} [topological_space M\u2081] [add_comm_monoid M\u2081]\n{M'\u2081 : Type*} [topological_space M'\u2081] [add_comm_monoid M'\u2081]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083]\n{M\u2084 : Type*} [topological_space M\u2084] [add_comm_monoid M\u2084]\n[module R\u2081 M\u2081] [module R\u2081 M'\u2081] [module R\u2082 M\u2082] [module R\u2083 M\u2083]\n\nsection\ninclude \u03c3\u2081\u2083\nlemma function.surjective.clm_comp_injective {g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082}\n  (hg : function.surjective g) : function.injective (\u03bb f : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083, f.comp g) :=\nbegin\n  intros f f' hff',\n  rw [continuous_linear_map.ext_iff] at hff' \u22a2,\n  intros x,\n  obtain \u27e8y, rfl\u27e9 := hg x,\n  exact hff' y,\nend\nend\n\nnamespace continuous_linear_equiv\n\ninclude \u03c3\u2082\u2081 \u03c3\u2081\u2083\ntheorem cancel_right {f f' : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083} {e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082} :\n  f.comp (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = f'.comp (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u2194 f = f' :=\nbegin\n  split,\n  { simp_rw [continuous_linear_map.ext_iff, continuous_linear_map.comp_apply, coe_coe],\n    intros h v, rw [\u2190 e.apply_symm_apply v, h] },\n  { rintro rfl, refl }\nend\n\nomit \u03c3\u2082\u2081\ninclude \u03c3\u2083\u2082\n\ntheorem cancel_left {e : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083} {f f' : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} :\n  (e : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp f = (e : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp f' \u2194 f = f' :=\nbegin\n  split,\n  { simp_rw [continuous_linear_map.ext_iff, continuous_linear_map.comp_apply, coe_coe],\n    intros h v, rw [\u2190 e.symm_apply_apply (f v), h, e.symm_apply_apply] },\n  { rintro rfl, refl }\nend\n\nomit \u03c3\u2083\u2082\n\nend continuous_linear_equiv\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/topology/algebra/module.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743620390163, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.38453636304155686}}
{"text": "import number_theory.cyclotomic.Unit_lemmas\nimport number_theory.cyclotomic.cycl_rat\nimport number_theory.regular_primes\nimport number_theory.cyclotomic.factoring\n\nopen_locale number_field non_zero_divisors\n\nvariables {p : \u2115+} {K : Type*} [field K] [char_zero K] [is_cyclotomic_extension {p} \u211a K]\nvariables {\u03b6 : K} (h\u03b6 : is_primitive_root \u03b6 p)\n\nopen fractional_ideal\n\nvariable (i : \u2124)\n\nnamespace flt_regular.caseI\n\nlemma exists_int_sum_eq_zero (hpodd : p \u2260 2) [hp : fact(p : \u2115).prime] {x y i : \u2124} {u : (\ud835\udcde K)\u02e3}\n  {\u03b1 : \ud835\udcde K} (h : (x : \ud835\udcde K) + y * (h\u03b6.unit' ^ i : (\ud835\udcde K)\u02e3) = u * \u03b1 ^ (p : \u2115)) :\n  \u2203 k : \u2124, (x : \ud835\udcde K) + y * (h\u03b6.unit' ^ i : (\ud835\udcde K)\u02e3) - (h\u03b6.unit' ^ (2 * k) : (\ud835\udcde K)\u02e3) *\n    (x + y * (h\u03b6.unit' ^ -i : (\ud835\udcde K)\u02e3)) \u2208 ideal.span ({p} : set (\ud835\udcde K)) :=\nbegin\n  letI : number_field K := is_cyclotomic_extension.number_field {p} \u211a _,\n  obtain \u27e8\u03b2, k, h\u03b2real : gal_conj K p \u03b2 = \u03b2, H\u27e9 := unit_lemma_gal_conj h\u03b6 hpodd hp.out u,\n  have : ((x + y * (h\u03b6.unit' ^ -i : (\ud835\udcde K)\u02e3)) : K) = gal_conj K p (x + y * h\u03b6.unit' ^ i),\n  { simp [gal_conj_zeta_runity h\u03b6, \u2190 coe_life] },\n  obtain \u27e8a, ha\u27e9 := exists_int_sub_pow_prime_dvd p \u03b1,\n  refine \u27e8k, _\u27e9,\n  rw [ideal.mem_span_singleton] at ha \u22a2,\n  obtain \u27e8\u03b3, h\u03b3\u27e9 := ha,\n  rw [h, sub_eq_iff_eq_add.1 h\u03b3, mul_add, \u2190 mul_assoc, mul_comm \u2191u, mul_assoc, add_sub_assoc],\n  refine dvd_add (dvd.intro _ rfl) _,\n  have h' := congr_arg (coe : \ud835\udcde K \u2192 K) h,\n  have h\u03b3' := congr_arg (coe : \ud835\udcde K \u2192 K) h\u03b3,\n  simp only [add_subgroup_class.coe_sub, subsemiring_class.coe_pow, subring_class.coe_int_cast,\n    mul_mem_class.coe_mul, subring_class.coe_nat_cast, add_mem_class.coe_add, coe_zpow'] at h' h\u03b3',\n  rw [h', sub_eq_iff_eq_add.1 h\u03b3', H, mul_mem_class.coe_mul, alg_equiv.map_mul, alg_equiv.map_mul,\n    alg_equiv.map_add, map_int_cast, alg_equiv.map_mul, \u2190 coe_coe \u03b2, coe_zpow', map_zpow\u2080, coe_coe,\n    coe_zpow'] at this,\n  simp only [coe_coe, h\u03b6.coe_unit'_coe, subring_class.coe_nat_cast, map_nat_cast] at this,\n  let \u03b3' := (\u27e8gal_conj K p \u03b3, number_field.ring_of_integers.map_mem (gal_conj K p) \u03b3\u27e9 : \ud835\udcde K),\n  have hint : \u2191\u03b3' = gal_conj K p \u03b3 := rfl,\n  rw [\u2190 coe_coe \u03b2, h\u03b2real, gal_conj_zeta_runity h\u03b6, \u2190 h\u03b6.coe_unit'_coe, inv_zpow, \u2190 zpow_neg,\n    coe_coe, \u2190 hint, \u2190 subring_class.coe_int_cast (\ud835\udcde K) x, \u2190 subring_class.coe_int_cast (\ud835\udcde K) y,\n    \u2190 coe_coe, \u2190 coe_zpow', \u2190 subring_class.coe_nat_cast (\ud835\udcde K) p, \u2190 coe_zpow',\n    \u2190 subring_class.coe_int_cast (\ud835\udcde K) a, \u2190 mul_mem_class.coe_mul (\ud835\udcde K),\n    \u2190 add_mem_class.coe_add (\ud835\udcde K), \u2190 mul_mem_class.coe_mul (\ud835\udcde K), \u2190 mul_mem_class.coe_mul (\ud835\udcde K),\n    \u2190 add_mem_class.coe_add (\ud835\udcde K), \u2190 mul_mem_class.coe_mul (\ud835\udcde K), subtype.coe_inj] at this,\n  rw [this, mul_add, mul_add, sub_add_eq_sub_sub, sub_right_comm],\n  refine dvd_sub _ (by simp),\n  rw [mul_comm \u2191\u03b2, \u2190 mul_assoc, \u2190 mul_assoc, \u2190 units.coe_mul, \u2190 zpow_add, two_mul,\n    \u2190 sub_eq_add_neg, add_sub_assoc, sub_self, add_zero, mul_comm _ \u2191\u03b2, \u2190 H, sub_self],\n  exact dvd_zero _\nend\n\nend flt_regular.caseI\n", "meta": {"author": "leanprover-community", "repo": "flt-regular", "sha": "1d0cecf99e8ab3f98b551e5932bf907042daa6ad", "save_path": "github-repos/lean/leanprover-community-flt-regular", "path": "github-repos/lean/leanprover-community-flt-regular/flt-regular-1d0cecf99e8ab3f98b551e5932bf907042daa6ad/src/number_theory/cyclotomic/case_I.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760727, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.38453635708631323}}
{"text": "-- The Category 5 Diamond.\n\n-- Notes by Kevin Buzzard made whilst trying to understand\n-- why Lean can sometimes be so stupid about the obvious\n-- ring structures all being equal\n\n-- Xena. Prove a theorem. Write a function.\n\n--------------------------------------------------------------\n\nimport tactic.interactive -- only needed if you want Floris' rintro on the last line\n\nnamespace xena\n\nclass A := (n : \u2115)\nclass B := (n : \u2115)\nclass C := (n : \u2115)\nclass D := (n : \u2115)\n\n-- the classic type class inference diamond\n\n-- Lean I wish I didn't have to name these instances.\n-- can something be done about this in the fork?\ninstance AtoB [hAB : A] : B := \u27e8hAB.n + 1\u27e9\ninstance BtoD [hBD : B] : D := \u27e8hBD.n + 10\u27e9\ninstance AtoC [hAC : A] : C := \u27e8hAC.n + 100\u27e9\ninstance CtoD [hCD : C] : D := \u27e8hCD.n + 1000\u27e9\n\ninstance a : A := \u27e837\u27e9\n\n-- can we get type class inference to make to distinct elments of D for us?\n\ndef b : B := by apply_instance\ndef c : C := by apply_instance\n-- d1 promises that they are the type class inference system's intended\n-- term, because they were generated with `by apply_instance`\ndef d1 : D := by apply_instance -- who will win the race around the square?\n#print d1 -- xena.CtoD won\n#check xena.BtoD -- the loser\nattribute [instance, priority 10000] xena.BtoD -- give it some juice\n-- d2 promises that they are the type class inference system's intended\n-- term, because they were generated with `by apply_instance`\ndef d2 : D := by apply_instance\n#print d2 -- rofl BtoD won this time\n\ntheorem you_are_in_typeclass_trouble_now : d1 \u2260 d2 :=\nbegin\n  intro h,\n    unfold d1 at h,\n  unfold d2 at h,\n  unfold xena.CtoD at h,\n  unfold xena.BtoD at h,\n  unfold C.n at h,\n  unfold B.n at h,\n  unfold A.n at h,\n  cases h,\nend\n\n-- Thanks to Floris van Doorn for showing me this very pretty proof\n-- (although )\ntheorem thanks_to_Floris_van_Doorn : d1 \u2260 d2 := by rintro \u27e8\u27e9\n\nend xena\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/Examples/Category_5_Diamond.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.6406358685621721, "lm_q1q2_score": 0.3845021999984961}}
{"text": "import tactic\nimport tactic.induction\nimport data.int.basic\nimport data.set.basic\n\nimport .base .point .dist .board .state\n\nnoncomputable theory\nopen_locale classical\n\ninstance {b : Board} : inhabited (Valid_D_move b) :=\n\u27e8\u27e8none, trivial\u27e9\u27e9\n\n-----\n\ninstance {pw : \u2115} : inhabited (A pw) :=\n\u27e8\u27e8\u03bb s hs h, \u27e8h.some, h.some_spec\u27e9\u27e9\u27e9\n\ninstance : inhabited D :=\n\u27e8\u27e8\u03bb s hs, \u27e8none, trivial\u27e9\u27e9\u27e9\n\ndef A.sup {pw pw\u2081 : \u2115} (a\u2081 : A pw\u2081) (a : A pw) : Prop :=\n\u2200 s hs h, \u2203 hs\u2081 h\u2081, (a\u2081.f s hs\u2081 h\u2081).m = (a.f s hs h).m\n\ndef A.sub {pw\u2081 pw : \u2115} (a : A pw\u2081) (a\u2081 : A pw) : Prop :=\na\u2081.sup a\n\ndef A_prev_moves (pw : \u2115) (s : State) :=\n\u03a0 (s\u2081 : State), s\u2081.act \u2192 s\u2081.len < s.len \u2192\nA_has_valid_move pw s\u2081.board \u2192 Valid_A_move pw s\u2081.board\n\ndef D_prev_moves (s : State) :=\n\u03a0 (s\u2081 : State), s\u2081.act \u2192 s\u2081.len < s.len \u2192\nValid_D_move s\u2081.board\n\ndef A.set_prev_moves {pw : \u2115} (a : A pw) (s : State)\n  (pm : A_prev_moves pw s) : A pw :=\nbegin\n  refine \u27e8\u03bb s\u2081 hs h\u2081, _\u27e9,\n  apply dite (s\u2081.len < s.len); intro h\u2082,\n  { exact pm s\u2081 hs h\u2082 h\u2081 },\n  { exact a.f s\u2081 hs h\u2081 },\nend\n\ndef D.set_prev_moves (d : D) (s : State)\n  (pm : D_prev_moves s) : D :=\nbegin\n  refine \u27e8\u03bb s\u2081 hs, _\u27e9,\n  apply dite (s\u2081.len < s.len); intro h\u2081,\n  { exact pm s\u2081 hs h\u2081 },\n  { exact d.f s\u2081 hs },\nend\n\ndef A.set_move {pw : \u2115} (a : A pw) (s : State)\n  (ma : Valid_A_move pw s.board) : A pw :=\nbegin\n  refine \u27e8\u03bb s\u2081 hs h, _\u27e9, apply dite (s\u2081 = s); intro h\u2081,\n  { cases h\u2081, exact ma },\n  { exact a.f s\u2081 hs h },\nend\n\ndef D.set_move (d : D) (s : State)\n  (md : Valid_D_move s.board) : D :=\nbegin\n  refine \u27e8\u03bb s\u2081 hs, _\u27e9, apply dite (s\u2081 = s); intro h\u2081,\n  { cases h\u2081, exact md },\n  { exact d.f s\u2081 hs },\nend\n\ndef A.prev_moves_id {pw : \u2115} (a : A pw) (s : State) : A pw :=\na.set_prev_moves s (\u03bb s\u2081 hs _ h, a.f s\u2081 hs h)\n\ndef D.prev_moves_id (d : D) (s : State) : D :=\nd.set_prev_moves s (\u03bb s\u2081 hs h, d.f s\u2081 hs)\n\ndef A.prev_moves_set {pw : \u2115} (a : A pw) (s : State)\n  (s\u2081 : State) (m : Valid_A_move pw s\u2081.board)\n  (h : s\u2081.len < s.len) : A pw :=\nbegin\n  apply a.set_prev_moves s, rintro s\u2082 hs h\u2081 h\u2082,\n  apply dite (s\u2082 = s\u2081); intro h\u2083,\n  { cases h\u2083, exact m },\n  { exact a.f s\u2082 hs h\u2082 },\nend\n\ndef D.prev_moves_set (d : D) (s : State)\n  (s\u2081 : State) (m : Valid_D_move s\u2081.board)\n  (h : s\u2081.len < s.len) : D :=\nbegin\n  apply d.set_prev_moves s, rintro s\u2082 hs h\u2081,\n  apply dite (s\u2082 = s\u2081); intro h\u2082,\n  { cases h\u2082, exact m },\n  { exact d.f s\u2082 hs },\nend\n\ndef A_state (s : State) : Prop := odd s.len\ndef D_state (s : State) : Prop := even s.len\n\n-----\n\nlemma A_move_valid_ge_of {pw pw\u2081 : \u2115} {b : Board} {p : A_move}\n  (h\u2081 : pw \u2264 pw\u2081) (h\u2082 : A_move_valid pw b p) :\n  A_move_valid pw\u2081 b p :=\n\u27e8h\u2082.1, h\u2082.2.1.trans h\u2081, h\u2082.2.2\u27e9\n\nlemma A_has_valid_move_ge_of {pw pw\u2081 : \u2115} {b : Board}\n  (h\u2081 : pw \u2264 pw\u2081) (h\u2082 : A_has_valid_move pw b) :\n  A_has_valid_move pw\u2081 b :=\nby { cases h\u2082 with m h\u2082, use m, exact A_move_valid_ge_of h\u2081 h\u2082 }\n\nlemma As_eq_iff {pw : \u2115} {a\u2081 a\u2082 : A pw} :\n  a\u2081 = a\u2082 \u2194 \u2200 s hs h, a\u2081.f s hs h = a\u2082.f s hs h :=\nbegin\n  split; intro h,\n  { subst h, simp },\n  { cases a\u2081 with f\u2081, cases a\u2082 with f\u2082, congr, ext, apply h },\nend\n\nlemma Ds_eq_iff {d\u2081 d\u2082 : D} :\n  d\u2081 = d\u2082 \u2194 \u2200 s hs, d\u2081.f s hs = d\u2082.f s hs :=\nbegin\n  split; intro h,\n  { subst h, simp },\n  { cases d\u2081 with f\u2081, cases d\u2082 with f\u2082, congr, ext, apply h },\nend\n\nlemma A_prev_moves_id_eq {pw : \u2115} {a : A pw} {s : State} :\n  a.prev_moves_id s = a :=\nby { rw As_eq_iff, rintro s\u2081 hs h, change dite _ _ _ = _, split_ifs; refl }\n\nlemma D_prev_moves_id_eq {d : D} {s : State} :\n  d.prev_moves_id s = d :=\nby { rw Ds_eq_iff, rintro s\u2081 hs, change dite _ _ _ = _, split_ifs; refl }\n\n@[simp]\nlemma A_set_move_eq {pw : \u2115} {a : A pw}\n  {s : State} {m : Valid_A_move pw s.board} {hs h} :\n  (a.set_move s m).f s hs h = m :=\nby { change dite _ _ _ = _, split_ifs with h\u2081; refl }\n\n@[simp]\nlemma D_set_move_eq {d : D}\n  {s : State} {m : Valid_D_move s.board} {hs} :\n  (d.set_move s m).f s hs = m :=\nby { change dite _ _ _ = _, split_ifs with h\u2081; refl }\n\nlemma A_set_move_self {pw : \u2115} {a : A pw}\n  {s : State} {hs h} : a.set_move s (a.f s hs h) = a :=\nbegin\n  rw As_eq_iff; intros, change dite _ _ _ = _, split_ifs with h\u2082,\n  { subst h\u2082 }, { refl },\nend\n\nlemma D_set_move_self {d : D}\n  {s : State} {hs} : d.set_move s (d.f s hs) = d :=\nbegin\n  rw Ds_eq_iff; intros, change dite _ _ _ = _, split_ifs with h\u2082,\n  { subst h\u2082 }, { refl },\nend\n\nlemma A_set_move_set_move_eq {pw : \u2115} {a : A pw} {s : State}\n  {m\u2081 m\u2082 : Valid_A_move pw s.board} :\n  (a.set_move s m\u2081).set_move s m\u2082 = a.set_move s m\u2082 :=\nbegin\n  rw As_eq_iff, rintro s\u2081 hs h\u2081,\n  change dite _ _ _ = _, split_ifs with h\u2082,\n  { subst h\u2082, rw A_set_move_eq },\n  { change dite _ _ _ = dite _ _ _, simp_rw dif_neg h\u2082 },\nend\n\nlemma D_set_move_set_move_eq {d : D} {s : State}\n  {m\u2081 m\u2082 : Valid_D_move s.board} :\n  (d.set_move s m\u2081).set_move s m\u2082 = d.set_move s m\u2082 :=\nbegin\n  rw Ds_eq_iff, rintro s\u2081 hs,\n  change dite _ _ _ = _, split_ifs with h\u2082,\n  { subst h\u2082, rw D_set_move_eq },\n  { change dite _ _ _ = dite _ _ _, simp_rw dif_neg h\u2082 },\nend\n\nlemma A_prev_moves_set_eq {pw : \u2115} {a : A pw} {s s\u2081 : State}\n  {m : Valid_A_move pw s\u2081.board} {h} :\n  a.prev_moves_set s s\u2081 m h = a.set_move s\u2081 m :=\nbegin\n  rw As_eq_iff, rintro s\u2082 hs h,\n  change dite _ _ _ = _, split_ifs with h\u2081,\n  { dsimp, split_ifs with h\u2082,\n    { subst h\u2082, simp [A_set_move_eq] },\n    { change _ = dite _ _ _, rw dif_neg h\u2082 }},\n  { change _ = dite _ _ _, split_ifs with h\u2082,\n    { subst h\u2082, contradiction },\n    { refl }},\nend\n\nlemma D_prev_moves_set_eq {d : D} {s s\u2081 : State}\n  {m : Valid_D_move s\u2081.board} {h} :\n  d.prev_moves_set s s\u2081 m h = d.set_move s\u2081 m :=\nbegin\n  rw Ds_eq_iff, rintro s\u2082 hs,\n  change dite _ _ _ = _, split_ifs with h\u2081,\n  { dsimp, split_ifs with h\u2082,\n    { subst h\u2082, simp [D_set_move_eq] },\n    { change _ = dite _ _ _, rw dif_neg h\u2082 }},\n  { change _ = dite _ _ _, split_ifs with h\u2082,\n    { subst h\u2082, contradiction },\n    { refl }},\nend\n\nlemma hist_len_apply_move {s : State} {b : Board} :\n  (apply_move s b).len = s.len.succ :=\nby { change (_ ++ [_]).length = _, rw list.length_append, refl }\n\nlemma hist_len_apply_A_move {s : State} {ma : A_move} :\n  (apply_A_move s ma).len = s.len.succ :=\nhist_len_apply_move\n\nlemma hist_len_apply_D_move {s : State} {md : D_move} :\n  (apply_D_move s md).len = s.len.succ :=\nhist_len_apply_move\n\nlemma valid_A_move_ext {pw : \u2115} {b : Board}\n  {ma\u2081 ma\u2082 : Valid_A_move pw b}\n  (h : ma\u2081.m = ma\u2082.m) : ma\u2081 = ma\u2082 :=\nby { cases ma\u2081, cases ma\u2082, congr, exact h }\n\nlemma valid_D_move_ext {b : Board}\n  {md\u2081 md\u2082 : Valid_D_move b}\n  (h : md\u2081.m = md\u2082.m) : md\u2081 = md\u2082 :=\nby { cases md\u2081, cases md\u2082, congr, exact h }\n\nlemma A_moves_eq_iff' {pw : \u2115} {s : State}\n  {ma\u2081 ma\u2082 : Valid_A_move pw s.board} : ma\u2081 = ma\u2082 \u2194\n  (apply_A_move s ma\u2081.m).board = (apply_A_move s ma\u2082.m).board :=\nbegin\n  split; intro h, { rw h }, simp_rw [apply_A_move, apply_A_move_b, apply_move] at h,\n  cases h with h\u2081 h\u2082, cases ma\u2081, cases ma\u2082, simp at h\u2082 \u22a2, exact h\u2082,\nend\n\nlemma D_moves_eq_iff' {s : State}\n  {md\u2081 md\u2082 : Valid_D_move s.board} : md\u2081 = md\u2082 \u2194\n  (apply_D_move s md\u2081.m).board = (apply_D_move s md\u2082.m).board :=\nbegin\n  split; intro h, { rw h }, simp_rw [apply_D_move, apply_move] at h,\n  cases md\u2081 with m\u2081 h\u2081, cases md\u2082 with m\u2082 h\u2082,\n  apply valid_D_move_ext, dsimp at h \u22a2,\n  cases m\u2081 with p\u2081; cases m\u2082 with p\u2082; simp_rw apply_D_move_b at h,\n  { cases h\u2082 with h\u2082 h\u2083, contrapose! h\u2083, rw h, simp },\n  { cases h\u2081 with h\u2081 h\u2083, contrapose! h\u2083, rw \u2190h, simp },\n  { replace h := h.1, replace h\u2081 := h\u2081.2, replace h\u2082 := h\u2082.2, congr,\n    rw set.ext_iff at h, have h\u2083 := h p\u2081, simp at h\u2083, exact h\u2083 h\u2081 },\nend\n\nlemma A_moves_eq_iff {pw : \u2115} {s : State}\n  {ma\u2081 ma\u2082 : Valid_A_move pw s.board} :\n  ma\u2081 = ma\u2082 \u2194 apply_A_move s ma\u2081.m = apply_A_move s ma\u2082.m :=\nbegin\n  split; intro h, { rw h }, simp_rw [apply_A_move, apply_A_move_b, apply_move] at h,\n  exact valid_A_move_ext h.1.2,\nend\n\nlemma D_moves_eq_iff {s : State}\n  {md\u2081 md\u2082 : Valid_D_move s.board} :\n  md\u2081 = md\u2082 \u2194 apply_D_move s md\u2081.m = apply_D_move s md\u2082.m :=\nbegin\n  split; intro h, { rw h }, simp_rw [apply_D_move, apply_move] at h,\n  replace h := h.1, cases md\u2081 with m\u2081 h\u2081, cases md\u2082 with m\u2082 h\u2082,\n  apply valid_D_move_ext, dsimp at h \u22a2,\n  cases m\u2081 with p\u2081; cases m\u2082 with p\u2082; simp_rw apply_D_move_b at h,\n  { cases h\u2082 with h\u2082 h\u2083, contrapose! h\u2083, rw h, simp },\n  { cases h\u2081 with h\u2081 h\u2083, contrapose! h\u2083, rw \u2190h, simp },\n  { replace h := h.1, replace h\u2081 := h\u2081.2, replace h\u2082 := h\u2082.2, congr,\n    rw set.ext_iff at h, have h\u2083 := h p\u2081, simp at h\u2083, exact h\u2083 h\u2081 },\nend\n\nlemma A_set_move_eq_pos {pw : \u2115} {a : A pw} {s : State}\n  {ma : Valid_A_move pw s.board} {hs h} :\n  (a.set_move s ma).f s hs h = ma :=\nby { rw A.set_move, dsimp, split_ifs; refl }\n\nlemma D_set_move_eq_pos {d : D} {s : State}\n  {md : Valid_D_move s.board} {hs} :\n  (d.set_move s md).f s hs = md :=\nby { rw D.set_move, dsimp, split_ifs; refl }\n\nlemma state_eq_of_apply_A_move_eq {s\u2081 s\u2082 : State}\n  {ma\u2081 ma\u2082 : A_move}\n  (h : apply_A_move s\u2081 ma\u2081 = apply_A_move s\u2082 ma\u2082) :\n  s\u2081 = s\u2082 :=\nbegin\n  cases s\u2081 with b\u2081 t\u2081 a\u2081, cases s\u2082 with b\u2082 t\u2082 a\u2082,\n  simp_rw [apply_A_move, apply_A_move_b, apply_move] at h, simp only,\n  rcases h with \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083, h\u2084\u27e9, rw snoc_eq_snoc_iff at h\u2083, cc,\nend\n\nlemma state_eq_of_apply_D_move_eq {s\u2081 s\u2082 : State}\n  {ma\u2081 ma\u2082 : D_move}\n  (h : apply_D_move s\u2081 ma\u2081 = apply_D_move s\u2082 ma\u2082) :\n  s\u2081 = s\u2082 :=\nbegin\n  cases s\u2081 with b\u2081 t\u2081 a\u2081, cases s\u2082 with b\u2082 t\u2082 a\u2082,\n  cases ma\u2081 with p\u2081; cases ma\u2082 with p\u2082;\n  simp_rw [apply_D_move, apply_D_move_b, apply_move, snoc_eq_snoc_iff] at h;\n  simp only; tauto!,\nend\n\nlemma apply_move_state_nth_eq_some_of {n : \u2115} {s : State} {b sb : Board}\n  (h : s.nth n = some sb) :\n  (apply_move s b).nth n = some sb :=\nbegin\n  simp_rw [State.nth, apply_move] at h \u22a2, rw list.nth_append,\n  { exact h },\n  { rw [length_snoc, nat.lt_succ_iff], obtain \u27e8h\u2081, h\u2082\u27e9 := list.nth_eq_some.mp h,\n    rw length_snoc at h\u2081, exact nat.lt_succ_iff.mp h\u2081 },\nend\n\nlemma apply_A_move_state_nth_eq_some_of {n : \u2115}\n  {s : State} {ma : A_move} {b : Board}\n  (h : s.nth n = some b) :\n  (apply_A_move s ma).nth n = some b :=\napply_move_state_nth_eq_some_of h\n\nlemma apply_D_move_state_nth_eq_some_of {n : \u2115}\n  {s : State} {ma : D_move} {b : Board}\n  (h : s.nth n = some b) :\n  (apply_D_move s ma).nth n = some b :=\napply_move_state_nth_eq_some_of h\n\nlemma apply_move_len {s : State} {b : Board} :\n  (apply_move s b).len = s.len.succ :=\nlength_snoc\n\nlemma apply_A_move_len {s : State} {ma : A_move} :\n  (apply_A_move s ma).len = s.len.succ :=\napply_move_len\n\nlemma apply_D_move_len {s : State} {ma : D_move} :\n  (apply_D_move s ma).len = s.len.succ :=\napply_move_len\n\nlemma apply_D_move_b_A_eq {b : Board} {md : D_move} :\n  (apply_D_move_b b md).A = b.A :=\nby { cases md; refl }\n\nlemma apply_D_move_A_eq {s : State} {md : D_move} :\n  (apply_D_move s md).board.A = s.board.A :=\nby { cases md; refl }\n\n-----\n\ndef Valid_A_move_subtype (pw : \u2115) (b : Board) :=\n{m : A_move // A_move_valid pw b m}\n\ndef Valid_D_move_subtype (b : Board) :=\n{m : D_move // D_move_valid b m}\n\ndef Valid_A_move_equiv_subtype {pw : \u2115} {b : Board} :\n  Valid_A_move pw b \u2243 Valid_A_move_subtype pw b :=\nbegin\n  fapply equiv.of_bijective, { intro ma, exact \u27e8_, ma.h\u27e9 }, fsplit,\n  { rintro \u27e8m\u2081, h\u2081\u27e9 \u27e8m2, h\u2082\u27e9 h\u2083, congr, exact subtype.mk.inj h\u2083 },\n  { rintro \u27e8m, h\u27e9, use \u27e8_, h\u27e9 },\nend\n\ndef Valid_D_move_equiv_subtype {b : Board} :\n  Valid_D_move b \u2243 Valid_D_move_subtype b :=\nbegin\n  fapply equiv.of_bijective, { intro ma, exact \u27e8_, ma.h\u27e9 }, fsplit,\n  { rintro \u27e8m\u2081, h\u2081\u27e9 \u27e8m2, h\u2082\u27e9 h\u2083, congr, exact subtype.mk.inj h\u2083 },\n  { rintro \u27e8m, h\u27e9, use \u27e8_, h\u27e9 },\nend\n\n-----\n\ninstance {pw : \u2115} {b : Board} : fintype (Valid_A_move_subtype pw b) :=\nbegin\n  apply fintype_subtype_of_set_finite, apply set.finite.inter_of_right,\n  apply set.finite.inter_of_left, change {p : Point | dist p b.A \u2264 pw}.finite,\n  exact dist_le_set_finite,\nend\n\ninstance {pw : \u2115} {b : Board} : fintype (Valid_A_move pw b) :=\nfintype.of_equiv _ Valid_A_move_equiv_subtype.symm\n\n-----\n\nlemma Valid_A_move_eq_iff {pw : \u2115} {b : Board}\n  {ma\u2081 ma\u2082 : Valid_A_move pw b} :\n  ma\u2081 = ma\u2082 \u2194 ma\u2081.m = ma\u2082.m :=\nby { cases ma\u2081, cases ma\u2082, simp }\n\nlemma Valid_D_move_eq_iff {pw : \u2115} {b : Board}\n  {md\u2081 md\u2082 : Valid_A_move pw b} :\n  md\u2081 = md\u2082 \u2194 md\u2081.m = md\u2082.m :=\nby { cases md\u2081, cases md\u2082, simp }", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/player.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.38450219083556636}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.shapes.pullbacks\nimport ring_theory.tensor_product\nimport algebra.category.CommRing.limits\nimport algebra.category.CommRing.colimits\nimport category_theory.limits.shapes.strict_initial\nimport ring_theory.subring.basic\nimport ring_theory.ideal.local_ring\nimport category_theory.limits.preserves.limits\n\n/-!\n# Constructions of (co)limits in CommRing\n\nIn this file we provide the explicit (co)cones for various (co)limits in `CommRing`, including\n* tensor product is the pushout\n* `Z` is the initial object\n* `0` is the strict terminal object\n* cartesian product is the product\n* `ring_hom.eq_locus` is the equalizer\n\n-/\n\nuniverses u u'\n\nopen category_theory category_theory.limits\nopen_locale tensor_product\n\nnamespace CommRing\n\nsection pushout\n\nvariables {R A B : CommRing.{u}} (f : R \u27f6 A) (g : R \u27f6 B)\n\n/-- The explicit cocone with tensor products as the fibered product in `CommRing`. -/\ndef pushout_cocone : limits.pushout_cocone f g :=\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  apply limits.pushout_cocone.mk,\n  show CommRing, from CommRing.of (A \u2297[R] B),\n  show A \u27f6 _,  from algebra.tensor_product.include_left.to_ring_hom,\n  show B \u27f6 _,  from algebra.tensor_product.include_right.to_ring_hom,\n  ext r,\n  transitivity algebra_map R (A \u2297[R] B) r,\n  { exact algebra.tensor_product.include_left.commutes r },\n  { exact (algebra.tensor_product.include_right.commutes r).symm }\nend\n\n@[simp]\nlemma pushout_cocone_inl : (pushout_cocone f g).inl = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_left.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_inr : (pushout_cocone f g).inr = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_right.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_X : (pushout_cocone f g).X = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI CommRing.of (A \u2297[R] B) }) := rfl\n\n/-- Verify that the `pushout_cocone` is indeed the colimit. -/\ndef pushout_cocone_is_colimit : limits.is_colimit (pushout_cocone f g) :=\nlimits.pushout_cocone.is_colimit_aux' _ (\u03bb s,\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  letI := ring_hom.to_algebra (f \u226b s.inl),\n  let f' : A \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change s.inl.to_fun (f r) = (f \u226b s.inl) r, refl }, ..s.inl },\n  let g' : B \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change (g \u226b s.inr) r = (f \u226b s.inl) r,\n        congr' 1,\n        exact (s.\u03b9.naturality limits.walking_span.hom.snd).trans\n          (s.\u03b9.naturality limits.walking_span.hom.fst).symm }, ..s.inr },\n  /- The factor map is a \u2297 b \u21a6 f(a) * g(b). -/\n  use alg_hom.to_ring_hom (algebra.tensor_product.product_map f' g'),\n  simp only [pushout_cocone_inl, pushout_cocone_inr],\n  split, { ext x, exact algebra.tensor_product.product_map_left_apply  _ _ x, },\n  split, { ext x, exact algebra.tensor_product.product_map_right_apply _ _ x, },\n  intros h eq1 eq2,\n  let h' : (A \u2297[R] B) \u2192\u2090[R] s.X :=\n    { commutes' := \u03bb r, by\n    { change h ((f r) \u2297\u209c[R] 1) = s.inl (f r),\n      rw \u2190 eq1, simp }, ..h },\n  suffices : h' = algebra.tensor_product.product_map f' g',\n  { ext x,\n    change h' x = algebra.tensor_product.product_map f' g' x,\n    rw this },\n  apply algebra.tensor_product.ext,\n  intros a b,\n  simp [\u2190 eq1, \u2190 eq2, \u2190 h.map_mul],\nend)\n\nend pushout\n\nsection terminal\n\n/-- The trivial ring is the (strict) terminal object of `CommRing`. -/\ndef punit_is_terminal : is_terminal (CommRing.of.{u} punit) :=\nbegin\n  apply_with is_terminal.of_unique { instances := ff },\n  tidy\nend\n\ninstance CommRing_has_strict_terminal_objects : has_strict_terminal_objects CommRing.{u} :=\nbegin\n  apply has_strict_terminal_objects_of_terminal_is_strict (CommRing.of punit),\n  intros X f,\n  refine \u27e8\u27e8by tidy, by ext, _\u27e9\u27e9,\n  ext,\n  have e : (0 : X) = 1 := by { rw [\u2190 f.map_one, \u2190 f.map_zero], congr },\n  replace e : 0 * x = 1 * x := congr_arg (\u03bb a, a * x) e,\n  rw [one_mul, zero_mul, \u2190 f.map_zero] at e,\n  exact e,\nend\n\nlemma subsingleton_of_is_terminal {X : CommRing} (hX : is_terminal X) : subsingleton X :=\n(hX.unique_up_to_iso punit_is_terminal).CommRing_iso_to_ring_equiv.to_equiv\n  .subsingleton_congr.mpr (show subsingleton punit, by apply_instance)\n\n/-- `\u2124` is the initial object of `CommRing`. -/\ndef Z_is_initial : is_initial (CommRing.of \u2124) :=\nbegin\n  apply_with is_initial.of_unique { instances := ff },\n  exact \u03bb R, \u27e8\u27e8int.cast_ring_hom R\u27e9, \u03bb a, a.ext_int _\u27e9,\nend\n\nend terminal\n\nsection product\n\nvariables (A B : CommRing.{u})\n\n/-- The product in `CommRing` is the cartesian product. This is the binary fan. -/\n@[simps X]\ndef prod_fan : binary_fan A B :=\nbinary_fan.mk (CommRing.of_hom $ ring_hom.fst A B) (CommRing.of_hom $ ring_hom.snd A B)\n\n/-- The product in `CommRing` is the cartesian product. -/\ndef prod_fan_is_limit : is_limit (prod_fan A B) :=\n{ lift := \u03bb c, ring_hom.prod (c.\u03c0.app walking_pair.left) (c.\u03c0.app walking_pair.right),\n  fac' := \u03bb c j, by { ext, cases j;\n    simpa only [binary_fan.\u03c0_app_left, binary_fan.\u03c0_app_right, comp_apply, ring_hom.prod_apply] },\n  uniq' := \u03bb s m h, by { ext, { simpa using congr_hom (h walking_pair.left) x },\n    { simpa using congr_hom (h walking_pair.right) x } } }\n\nend product\n\nsection equalizer\n\nvariables {A B : CommRing.{u}} (f g : A \u27f6 B)\n\n/-- The equalizer in `CommRing` is the equalizer as sets. This is the equalizer fork. -/\ndef equalizer_fork : fork f g :=\nfork.of_\u03b9 (CommRing.of_hom (ring_hom.eq_locus f g).subtype) (by { ext \u27e8x, e\u27e9, simpa using e })\n\n/-- The equalizer in `CommRing` is the equalizer as sets. -/\ndef equalizer_fork_is_limit : is_limit (equalizer_fork f g) :=\nbegin\n  fapply fork.is_limit.mk',\n  intro s,\n  use s.\u03b9.cod_restrict' _ (\u03bb x, (concrete_category.congr_hom s.condition x : _)),\n  split,\n  { ext, refl },\n  { intros m hm, ext x, exact concrete_category.congr_hom hm x }\nend\n\ninstance : is_local_ring_hom (equalizer_fork f g).\u03b9 :=\nbegin\n  constructor,\n  rintros \u27e8a, (h\u2081 : _ = _)\u27e9 (\u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,(rfl : x = _)\u27e9 : is_unit a),\n  have : y \u2208 ring_hom.eq_locus f g,\n  { apply (f.is_unit_map \u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,rfl\u27e9 : is_unit (f x)).mul_left_inj.mp,\n    conv_rhs { rw h\u2081 },\n    rw [\u2190 f.map_mul, \u2190 g.map_mul, h\u2084, f.map_one, g.map_one] },\n  rw is_unit_iff_exists_inv,\n  exact \u27e8\u27e8y, this\u27e9, subtype.eq h\u2083\u27e9,\nend\n\ninstance equalizer_\u03b9_is_local_ring_hom (F : walking_parallel_pair.{u} \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F walking_parallel_pair.zero) :=\nbegin\n  have := lim_map_\u03c0 (diagram_iso_parallel_pair F).hom walking_parallel_pair.zero,\n  rw \u2190 is_iso.comp_inv_eq at this,\n  rw \u2190 this,\n  rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, equalizer_fork_is_limit\n    (F.map walking_parallel_pair_hom.left) (F.map walking_parallel_pair_hom.right)\u27e9\n    walking_parallel_pair.zero,\n  change is_local_ring_hom ((lim.map _ \u226b _ \u226b (equalizer_fork _ _).\u03b9) \u226b _),\n  apply_instance\nend\n\nopen category_theory.limits.walking_parallel_pair opposite\nopen category_theory.limits.walking_parallel_pair_hom\n\ninstance equalizer_\u03b9_is_local_ring_hom' (F : walking_parallel_pair.{u}\u1d52\u1d56 \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F (opposite.op walking_parallel_pair.one)) :=\nbegin\n  have : _ = limit.\u03c0 F (walking_parallel_pair_op_equiv.{u u}.functor.obj _) :=\n    (limit.iso_limit_cone_inv_\u03c0 \u27e8_, is_limit.whisker_equivalence (limit.is_limit F)\n      walking_parallel_pair_op_equiv\u27e9 walking_parallel_pair.zero : _),\n  erw \u2190 this,\n  apply_instance\nend\n\nend equalizer\n\nend CommRing\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebra/category/CommRing/constructions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358548398982, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3845021825996173}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Kenny Lau\n-/\nimport algebra.module.pi\nimport algebra.module.linear_map\nimport algebra.big_operators.basic\nimport data.set.finite\nimport group_theory.submonoid.membership\n\n/-!\n# Dependent functions with finite support\n\nFor a non-dependent version see `data/finsupp.lean`.\n-/\n\nuniverses u u\u2081 u\u2082 v v\u2081 v\u2082 v\u2083 w x y l\n\nopen_locale big_operators\n\nvariables (\u03b9 : Type u) {\u03b3 : Type w} (\u03b2 : \u03b9 \u2192 Type v) {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\n\nnamespace dfinsupp\n\nvariable [\u03a0 i, has_zero (\u03b2 i)]\n\n/-- An auxiliary structure used in the definition of of `dfinsupp`,\nthe type used to make infinite direct sum of modules over a ring. -/\nstructure pre : Type (max u v) :=\n(to_fun : \u03a0 i, \u03b2 i)\n(pre_support : multiset \u03b9)\n(zero : \u2200 i, i \u2208 pre_support \u2228 to_fun i = 0)\n\ninstance inhabited_pre : inhabited (pre \u03b9 \u03b2) :=\n\u27e8\u27e8\u03bb i, 0, \u2205, \u03bb i, or.inr rfl\u27e9\u27e9\n\ninstance : setoid (pre \u03b9 \u03b2) :=\n{ r := \u03bb x y, \u2200 i, x.to_fun i = y.to_fun i,\n  iseqv := \u27e8\u03bb f i, rfl, \u03bb f g H i, (H i).symm,\n    \u03bb f g h H1 H2 i, (H1 i).trans (H2 i)\u27e9 }\n\nend dfinsupp\n\nvariable {\u03b9}\n/-- A dependent function `\u03a0 i, \u03b2 i` with finite support. -/\n@[reducible]\ndef dfinsupp [\u03a0 i, has_zero (\u03b2 i)] : Type* :=\nquotient (dfinsupp.pre.setoid \u03b9 \u03b2)\nvariable {\u03b2}\n\nnotation `\u03a0\u2080` binders `, ` r:(scoped f, dfinsupp f) := r\ninfix ` \u2192\u209a `:25 := dfinsupp\n\nnamespace dfinsupp\n\nsection basic\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n\ninstance fun_like : fun_like (\u03a0\u2080 i, \u03b2 i) \u03b9 \u03b2 :=\n\u27e8\u03bb f, quotient.lift_on f pre.to_fun $ \u03bb _ _, funext,\n  \u03bb f g H, quotient.induction_on\u2082 f g (\u03bb _ _ H, quotient.sound H) (congr_fun H)\u27e9\n\n/-- Helper instance for when there are too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\ninstance : has_coe_to_fun (\u03a0\u2080 i, \u03b2 i) (\u03bb _, \u03a0 i, \u03b2 i) := fun_like.has_coe_to_fun\n\n@[ext] lemma ext {f g : \u03a0\u2080 i, \u03b2 i} (h : \u2200 i, f i = g i) : f = g := fun_like.ext _ _ h\n/-- Deprecated. Use `fun_like.ext_iff` instead. -/\nlemma ext_iff {f g : \u03a0\u2080 i, \u03b2 i} : f = g \u2194 \u2200 i, f i = g i := fun_like.ext_iff\n/-- Deprecated. Use `fun_like.coe_injective` instead. -/\nlemma coe_fn_injective : @function.injective (\u03a0\u2080 i, \u03b2 i) (\u03a0 i, \u03b2 i) coe_fn := fun_like.coe_injective\n\ninstance : has_zero (\u03a0\u2080 i, \u03b2 i) := \u27e8\u27e6\u27e80, \u2205, \u03bb i, or.inr rfl\u27e9\u27e7\u27e9\ninstance : inhabited (\u03a0\u2080 i, \u03b2 i) := \u27e80\u27e9\n\n@[simp]\nlemma coe_pre_mk (f : \u03a0 i, \u03b2 i) (s : multiset \u03b9) (hf) :\n  \u21d1(\u27e6\u27e8f, s, hf\u27e9\u27e7 : \u03a0\u2080 i, \u03b2 i) = f := rfl\n\n@[simp] lemma coe_zero : \u21d1(0 : \u03a0\u2080 i, \u03b2 i) = 0 := rfl\nlemma zero_apply (i : \u03b9) : (0 : \u03a0\u2080 i, \u03b2 i) i = 0 := rfl\n\n/-- The composition of `f : \u03b2\u2081 \u2192 \u03b2\u2082` and `g : \u03a0\u2080 i, \u03b2\u2081 i` is\n  `map_range f hf g : \u03a0\u2080 i, \u03b2\u2082 i`, well defined when `f 0 = 0`.\n\nThis preserves the structure on `f`, and exists in various bundled forms for when `f` is itself\nbundled:\n\n* `dfinsupp.map_range.add_monoid_hom`\n* `dfinsupp.map_range.add_equiv`\n* `dfinsupp.map_range.linear_map`\n* `dfinsupp.map_range.linear_equiv`\n-/\ndef map_range (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0) : (\u03a0\u2080 i, \u03b2\u2081 i) \u2192 \u03a0\u2080 i, \u03b2\u2082 i :=\nquotient.map\n  (\u03bb x, \u27e8\u03bb i, f i (x.1 i), x.2, \u03bb i, (x.3 i).imp_right $ \u03bb H, by rw [H, hf]\u27e9)\n  (\u03bb x y H i, by simp only [H i])\n\n@[simp] lemma map_range_apply\n  (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0) (g : \u03a0\u2080 i, \u03b2\u2081 i) (i : \u03b9) :\n  map_range f hf g i = f i (g i) :=\nquotient.induction_on g $ \u03bb x, rfl\n\n@[simp] lemma map_range_id (h : \u2200 i, id (0 : \u03b2\u2081 i) = 0 := \u03bb i, rfl) (g : \u03a0\u2080 (i : \u03b9), \u03b2\u2081 i) :\n  map_range (\u03bb i, (id : \u03b2\u2081 i \u2192 \u03b2\u2081 i)) h g = g :=\nby { ext, simp only [map_range_apply, id.def] }\n\nlemma map_range_comp (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (f\u2082 : \u03a0 i, \u03b2 i \u2192 \u03b2\u2081 i)\n  (hf : \u2200 i, f i 0 = 0) (hf\u2082 : \u2200 i, f\u2082 i 0 = 0) (h : \u2200 i, (f i \u2218 f\u2082 i) 0 = 0)\n  (g : \u03a0\u2080 (i : \u03b9), \u03b2 i) :\n  map_range (\u03bb i, f i \u2218 f\u2082 i) h g = map_range f hf (map_range f\u2082 hf\u2082 g) :=\nby { ext, simp only [map_range_apply] }\n\n@[simp] lemma map_range_zero (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0) :\n  map_range f hf (0 : \u03a0\u2080 i, \u03b2\u2081 i) = 0 :=\nby { ext, simp only [map_range_apply, coe_zero, pi.zero_apply, hf] }\n\n/-- Let `f i` be a binary operation `\u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i` such that `f i 0 0 = 0`.\nThen `zip_with f hf` is a binary operation `\u03a0\u2080 i, \u03b2\u2081 i \u2192 \u03a0\u2080 i, \u03b2\u2082 i \u2192 \u03a0\u2080 i, \u03b2 i`. -/\ndef zip_with (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i) (hf : \u2200 i, f i 0 0 = 0) :\n  (\u03a0\u2080 i, \u03b2\u2081 i) \u2192 (\u03a0\u2080 i, \u03b2\u2082 i) \u2192 (\u03a0\u2080 i, \u03b2 i) :=\nbegin\n  refine quotient.map\u2082\n    (\u03bb x y, \u27e8\u03bb i, f i (x.1 i) (y.1 i), x.2 + y.2, \u03bb i, _\u27e9) _,\n  { cases x.3 i with h1 h1,\n    { left, rw multiset.mem_add, left, exact h1 },\n    cases y.3 i with h2 h2,\n    { left, rw multiset.mem_add, right, exact h2 },\n    right, rw [h1, h2, hf] },\n  exact \u03bb x\u2081 x\u2082 H1 y\u2081 y\u2082 H2 i, by simp only [H1 i, H2 i]\nend\n\n@[simp] lemma zip_with_apply\n  (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i) (hf : \u2200 i, f i 0 0 = 0) (g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i) (g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i) (i : \u03b9) :\n  zip_with f hf g\u2081 g\u2082 i = f i (g\u2081 i) (g\u2082 i) :=\nquotient.induction_on\u2082 g\u2081 g\u2082 $ \u03bb _ _, rfl\n\nend basic\n\nsection algebra\n\ninstance [\u03a0 i, add_zero_class (\u03b2 i)] : has_add (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8zip_with (\u03bb _, (+)) (\u03bb _, add_zero 0)\u27e9\n\nlemma add_apply [\u03a0 i, add_zero_class (\u03b2 i)] (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) :\n  (g\u2081 + g\u2082) i = g\u2081 i + g\u2082 i :=\nzip_with_apply _ _ g\u2081 g\u2082 i\n\n@[simp] lemma coe_add [\u03a0 i, add_zero_class (\u03b2 i)] (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i) :\n  \u21d1(g\u2081 + g\u2082) = g\u2081 + g\u2082 :=\nfunext $ add_apply g\u2081 g\u2082\n\ninstance [\u03a0 i, add_zero_class (\u03b2 i)] : add_zero_class (\u03a0\u2080 i, \u03b2 i) :=\n{ zero      := 0,\n  add       := (+),\n  zero_add  := \u03bb f, ext $ \u03bb i, by simp only [add_apply, zero_apply, zero_add],\n  add_zero  := \u03bb f, ext $ \u03bb i, by simp only [add_apply, zero_apply, add_zero] }\n\ninstance [\u03a0 i, add_monoid (\u03b2 i)] : add_monoid (\u03a0\u2080 i, \u03b2 i) :=\n{ add_monoid .\n  zero      := 0,\n  add       := (+),\n  add_assoc := \u03bb f g h, ext $ \u03bb i, by simp only [add_apply, add_assoc],\n  .. dfinsupp.add_zero_class }\n\n/-- Coercion from a `dfinsupp` to a pi type is an `add_monoid_hom`. -/\ndef coe_fn_add_monoid_hom [\u03a0 i, add_zero_class (\u03b2 i)] : (\u03a0\u2080 i, \u03b2 i) \u2192+ (\u03a0 i, \u03b2 i) :=\n{ to_fun := coe_fn, map_zero' := coe_zero, map_add' := coe_add }\n\n/-- Evaluation at a point is an `add_monoid_hom`. This is the finitely-supported version of\n`pi.eval_add_monoid_hom`. -/\ndef eval_add_monoid_hom [\u03a0 i, add_zero_class (\u03b2 i)] (i : \u03b9) : (\u03a0\u2080 i, \u03b2 i) \u2192+ \u03b2 i :=\n(pi.eval_add_monoid_hom \u03b2 i).comp coe_fn_add_monoid_hom\n\ninstance [\u03a0 i, add_comm_monoid (\u03b2 i)] : add_comm_monoid (\u03a0\u2080 i, \u03b2 i) :=\n{ add_comm := \u03bb f g, ext $ \u03bb i, by simp only [add_apply, add_comm],\n  nsmul := \u03bb n v, v.map_range (\u03bb _, (\u2022) n) (\u03bb _, smul_zero _),\n  nsmul_zero' := \u03bb n, ext $ \u03bb i, by simp only [map_range_apply, zero_apply, zero_smul],\n  nsmul_succ' := \u03bb n z, ext $ \u03bb i, by simp only [map_range_apply, add_apply,\n    nat.succ_eq_one_add, add_smul, one_smul],\n  .. dfinsupp.add_monoid }\n\n@[simp] lemma coe_finset_sum {\u03b1} [\u03a0 i, add_comm_monoid (\u03b2 i)] (s : finset \u03b1) (g : \u03b1 \u2192 \u03a0\u2080 i, \u03b2 i) :\n  \u21d1(\u2211 a in s, g a) = \u2211 a in s, g a :=\n(coe_fn_add_monoid_hom : _ \u2192+ (\u03a0 i, \u03b2 i)).map_sum g s\n\n@[simp] lemma finset_sum_apply {\u03b1} [\u03a0 i, add_comm_monoid (\u03b2 i)] (s : finset \u03b1) (g : \u03b1 \u2192 \u03a0\u2080 i, \u03b2 i)\n  (i : \u03b9) :\n  (\u2211 a in s, g a) i = \u2211 a in s, g a i :=\n(eval_add_monoid_hom i : _ \u2192+ \u03b2 i).map_sum g s\n\ninstance [\u03a0 i, add_group (\u03b2 i)] : has_neg (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8\u03bb f, f.map_range (\u03bb _, has_neg.neg) (\u03bb _, neg_zero)\u27e9\n\nlemma neg_apply [\u03a0 i, add_group (\u03b2 i)] (g : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) : (- g) i = - g i :=\nmap_range_apply _ _ g i\n\n@[simp] lemma coe_neg [\u03a0 i, add_group (\u03b2 i)] (g : \u03a0\u2080 i, \u03b2 i) : \u21d1(- g) = - g :=\nfunext $ neg_apply g\n\ninstance [\u03a0 i, add_group (\u03b2 i)] : has_sub (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8zip_with (\u03bb _, has_sub.sub) (\u03bb _, sub_zero 0)\u27e9\n\nlemma sub_apply [\u03a0 i, add_group (\u03b2 i)] (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) :\n  (g\u2081 - g\u2082) i = g\u2081 i - g\u2082 i :=\nzip_with_apply _ _ g\u2081 g\u2082 i\n\n@[simp] lemma coe_sub [\u03a0 i, add_group (\u03b2 i)] (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i) :\n  \u21d1(g\u2081 - g\u2082) = g\u2081 - g\u2082 :=\nfunext $ sub_apply g\u2081 g\u2082\n\ninstance [\u03a0 i, add_group (\u03b2 i)] : add_group (\u03a0\u2080 i, \u03b2 i) :=\n{ add_left_neg := \u03bb f, ext $ \u03bb i, by simp only [add_apply, neg_apply, zero_apply, add_left_neg],\n  sub_eq_add_neg := \u03bb f g, ext $ \u03bb i,\n    by simp only [sub_apply, add_apply, neg_apply, sub_eq_add_neg],\n  .. dfinsupp.add_monoid,\n  .. dfinsupp.has_sub,\n  .. dfinsupp.has_neg }\n\ninstance [\u03a0 i, add_comm_group (\u03b2 i)] : add_comm_group (\u03a0\u2080 i, \u03b2 i) :=\n{ zsmul := \u03bb n v, v.map_range (\u03bb _, (\u2022) n) (\u03bb _, smul_zero _),\n  zsmul_neg' := \u03bb n f, ext $ \u03bb i, by\n    rw [neg_apply, map_range_apply, map_range_apply, zsmul_neg_succ_of_nat, nsmul_eq_smul_cast \u2124,\n      int.nat_cast_eq_coe_nat],\n  zsmul_zero' := \u03bb n, ext $ \u03bb i, by simp only [map_range_apply, zero_apply, zero_smul],\n  zsmul_succ' := \u03bb n f, ext $ \u03bb i, by simp [map_range_apply, add_smul, add_comm],\n  ..@dfinsupp.add_comm_monoid _ \u03b2 _,\n  ..dfinsupp.add_group }\n\n/-- Dependent functions with finite support inherit a semiring action from an action on each\ncoordinate. -/\ninstance [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] :\n  has_scalar \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8\u03bbc v, v.map_range (\u03bb _, (\u2022) c) (\u03bb _, smul_zero _)\u27e9\n\nlemma smul_apply [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)]\n  [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] (b : \u03b3) (v : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) :\n  (b \u2022 v) i = b \u2022 (v i) :=\nmap_range_apply _ _ v i\n\n@[simp] lemma coe_smul [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)]\n  [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] (b : \u03b3) (v : \u03a0\u2080 i, \u03b2 i) :\n  \u21d1(b \u2022 v) = b \u2022 v :=\nfunext $ smul_apply b v\n\ninstance {\u03b4 : Type*} [monoid \u03b3] [monoid \u03b4]\n  [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b4 (\u03b2 i)]\n  [\u03a0 i, smul_comm_class \u03b3 \u03b4 (\u03b2 i)] :\n  smul_comm_class \u03b3 \u03b4 (\u03a0\u2080 i, \u03b2 i) :=\n{ smul_comm := \u03bb r s m, ext $ \u03bb i, by simp only [smul_apply, smul_comm r s (m i)] }\n\ninstance {\u03b4 : Type*} [monoid \u03b3] [monoid \u03b4]\n  [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b4 (\u03b2 i)]\n  [has_scalar \u03b3 \u03b4] [\u03a0 i, is_scalar_tower \u03b3 \u03b4 (\u03b2 i)] :\n  is_scalar_tower \u03b3 \u03b4 (\u03a0\u2080 i, \u03b2 i) :=\n{ smul_assoc := \u03bb r s m, ext $ \u03bb i, by simp only [smul_apply, smul_assoc r s (m i)] }\n\ninstance [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)]\n  [\u03a0 i, distrib_mul_action \u03b3\u1d50\u1d52\u1d56 (\u03b2 i)] [\u2200 i, is_central_scalar \u03b3 (\u03b2 i)] :\n  is_central_scalar \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\n{ op_smul_eq_smul := \u03bb r m, ext $ \u03bb i, by simp only [smul_apply, op_smul_eq_smul r (m i)] }\n\n/-- Dependent functions with finite support inherit a `distrib_mul_action` structure from such a\nstructure on each coordinate. -/\ninstance [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] :\n  distrib_mul_action \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\n{ smul_zero := \u03bb c, ext $ \u03bb i, by simp only [smul_apply, smul_zero, zero_apply],\n  smul_add := \u03bb c x y, ext $ \u03bb i, by simp only [add_apply, smul_apply, smul_add],\n  one_smul := \u03bb x, ext $ \u03bb i, by simp only [smul_apply, one_smul],\n  mul_smul := \u03bb r s x, ext $ \u03bb i, by simp only [smul_apply, smul_smul],\n  ..dfinsupp.has_scalar }\n\n/-- Dependent functions with finite support inherit a module structure from such a structure on\neach coordinate. -/\ninstance [semiring \u03b3] [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)] :\n  module \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\n{ zero_smul := \u03bb c, ext $ \u03bb i, by simp only [smul_apply, zero_smul, zero_apply],\n  add_smul := \u03bb c x y, ext $ \u03bb i, by simp only [add_apply, smul_apply, add_smul],\n  ..dfinsupp.distrib_mul_action }\n\nend algebra\n\nsection filter_and_subtype_domain\n\n/-- `filter p f` is the function which is `f i` if `p i` is true and 0 otherwise. -/\ndef filter [\u03a0 i, has_zero (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p] : (\u03a0\u2080 i, \u03b2 i) \u2192 \u03a0\u2080 i, \u03b2 i :=\nquotient.map\n  (\u03bb x, \u27e8\u03bb i, if p i then x.1 i else 0, x.2, \u03bb i, (x.3 i).imp_right $ \u03bb H, by rw [H, if_t_t]\u27e9)\n  (\u03bb x y H i, by simp only [H i])\n\n@[simp] lemma filter_apply [\u03a0 i, has_zero (\u03b2 i)]\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  f.filter p i = if p i then f i else 0 :=\nquotient.induction_on f $ \u03bb x, rfl\n\nlemma filter_apply_pos [\u03a0 i, has_zero (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] (f : \u03a0\u2080 i, \u03b2 i) {i : \u03b9} (h : p i) :\n  f.filter p i = f i :=\nby simp only [filter_apply, if_pos h]\n\nlemma filter_apply_neg [\u03a0 i, has_zero (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] (f : \u03a0\u2080 i, \u03b2 i) {i : \u03b9} (h : \u00ac p i) :\n  f.filter p i = 0 :=\nby simp only [filter_apply, if_neg h]\n\nlemma filter_pos_add_filter_neg [\u03a0 i, add_zero_class (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i)\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  f.filter p + f.filter (\u03bbi, \u00ac p i) = f :=\next $ \u03bb i, by simp only [add_apply, filter_apply]; split_ifs; simp only [add_zero, zero_add]\n\n@[simp] lemma filter_zero [\u03a0 i, has_zero (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  (0 : \u03a0\u2080 i, \u03b2 i).filter p = 0 :=\nby { ext, simp }\n\n@[simp] lemma filter_add [\u03a0 i, add_zero_class (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p]\n  (f g : \u03a0\u2080 i, \u03b2 i) :\n  (f + g).filter p = f.filter p + g.filter p :=\nby { ext, simp [ite_add_zero] }\n\n@[simp] lemma filter_smul [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)]\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] (r : \u03b3) (f : \u03a0\u2080 i, \u03b2 i) :\n  (r \u2022 f).filter p = r \u2022 f.filter p :=\nby { ext, simp [smul_ite] }\n\nvariables (\u03b3 \u03b2)\n\n/-- `dfinsupp.filter` as an `add_monoid_hom`. -/\n@[simps]\ndef filter_add_monoid_hom [\u03a0 i, add_zero_class (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  (\u03a0\u2080 i, \u03b2 i) \u2192+ (\u03a0\u2080 i, \u03b2 i) :=\n{ to_fun := filter p,\n  map_zero' := filter_zero p,\n  map_add' := filter_add p }\n\n/-- `dfinsupp.filter` as a `linear_map`. -/\n@[simps]\ndef filter_linear_map [semiring \u03b3] [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)]\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  (\u03a0\u2080 i, \u03b2 i) \u2192\u2097[\u03b3] (\u03a0\u2080 i, \u03b2 i) :=\n{ to_fun := filter p,\n  map_add' := filter_add p,\n  map_smul' := filter_smul p }\n\nvariables {\u03b3 \u03b2}\n\n@[simp] lemma filter_neg [\u03a0 i, add_group (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p]\n  (f : \u03a0\u2080 i, \u03b2 i) :\n  (-f).filter p = -f.filter p :=\n(filter_add_monoid_hom \u03b2 p).map_neg f\n\n@[simp] lemma filter_sub [\u03a0 i, add_group (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p]\n  (f g : \u03a0\u2080 i, \u03b2 i) :\n  (f - g).filter p = f.filter p - g.filter p :=\n(filter_add_monoid_hom \u03b2 p).map_sub f g\n\n/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -/\ndef subtype_domain [\u03a0 i, has_zero (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  (\u03a0\u2080 i, \u03b2 i) \u2192 \u03a0\u2080 i : subtype p, \u03b2 i :=\nquotient.map\n  (\u03bb x, \u27e8\u03bb i, x.1 (i : \u03b9), (x.2.filter p).attach.map $ \u03bb j, \u27e8j, (multiset.mem_filter.1 j.2).2\u27e9,\n      \u03bb i, (x.3 i).imp_left $ \u03bb H, multiset.mem_map.2\n        \u27e8\u27e8i, multiset.mem_filter.2 \u27e8H, i.2\u27e9\u27e9, multiset.mem_attach _ _, subtype.eta _ _\u27e9\u27e9)\n  (\u03bb x y H i, H i)\n\n@[simp] lemma subtype_domain_zero [\u03a0 i, has_zero (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  subtype_domain p (0 : \u03a0\u2080 i, \u03b2 i) = 0 :=\nrfl\n\n@[simp] lemma subtype_domain_apply [\u03a0 i, has_zero (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {i : subtype p} {v : \u03a0\u2080 i, \u03b2 i} :\n  (subtype_domain p v) i = v i :=\nquotient.induction_on v $ \u03bb x, rfl\n\n@[simp] lemma subtype_domain_add [\u03a0 i, add_zero_class (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  (v v' : \u03a0\u2080 i, \u03b2 i) :\n  (v + v').subtype_domain p = v.subtype_domain p + v'.subtype_domain p :=\next $ \u03bb i, by simp only [add_apply, subtype_domain_apply]\n\n@[simp] lemma subtype_domain_smul [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)]\n  [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] (r : \u03b3) (f : \u03a0\u2080 i, \u03b2 i) :\n  (r \u2022 f).subtype_domain p = r \u2022 f.subtype_domain p :=\nquotient.induction_on f $ \u03bb x, rfl\n\nvariables (\u03b3 \u03b2)\n\n/-- `subtype_domain` but as an `add_monoid_hom`. -/\n@[simps] def subtype_domain_add_monoid_hom [\u03a0 i, add_zero_class (\u03b2 i)]\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] : (\u03a0\u2080 i : \u03b9, \u03b2 i) \u2192+ \u03a0\u2080 i : subtype p, \u03b2 i :=\n{ to_fun := subtype_domain p,\n  map_zero' := subtype_domain_zero,\n  map_add' := subtype_domain_add }\n\n/-- `dfinsupp.subtype_domain` as a `linear_map`. -/\n@[simps]\ndef subtype_domain_linear_map [semiring \u03b3] [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)]\n  (p : \u03b9 \u2192 Prop) [decidable_pred p] :\n  (\u03a0\u2080 i, \u03b2 i) \u2192\u2097[\u03b3] (\u03a0\u2080 i : subtype p, \u03b2 i) :=\n{ to_fun := subtype_domain p,\n  map_add' := subtype_domain_add,\n  map_smul' := subtype_domain_smul }\n\nvariables {\u03b3 \u03b2}\n\n@[simp]\nlemma subtype_domain_neg [\u03a0 i, add_group (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] {v : \u03a0\u2080 i, \u03b2 i} :\n  (- v).subtype_domain p = - v.subtype_domain p :=\next $ \u03bb i, by simp only [neg_apply, subtype_domain_apply]\n\n@[simp] lemma subtype_domain_sub [\u03a0 i, add_group (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {v v' : \u03a0\u2080 i, \u03b2 i} :\n  (v - v').subtype_domain p = v.subtype_domain p - v'.subtype_domain p :=\next $ \u03bb i, by simp only [sub_apply, subtype_domain_apply]\n\nend filter_and_subtype_domain\n\n\nvariable [dec : decidable_eq \u03b9]\ninclude dec\n\nsection basic\nvariable [\u03a0 i, has_zero (\u03b2 i)]\n\nomit dec\nlemma finite_support (f : \u03a0\u2080 i, \u03b2 i) : set.finite {i | f i \u2260 0} :=\nbegin\n  classical,\n  exact quotient.induction_on f (\u03bb x, x.2.to_finset.finite_to_set.subset (\u03bb i H,\n    multiset.mem_to_finset.2 ((x.3 i).resolve_right H)))\nend\ninclude dec\n\n/-- Create an element of `\u03a0\u2080 i, \u03b2 i` from a finset `s` and a function `x`\ndefined on this `finset`. -/\ndef mk (s : finset \u03b9) (x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 (i : \u03b9)) : \u03a0\u2080 i, \u03b2 i :=\n\u27e6\u27e8\u03bb i, if H : i \u2208 s then x \u27e8i, H\u27e9 else 0, s.1,\n\u03bb i, if H : i \u2208 s then or.inl H else or.inr $ dif_neg H\u27e9\u27e7\n\nvariables {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i} {i : \u03b9}\n\n@[simp] lemma mk_apply : (mk s x : \u03a0 i, \u03b2 i) i = if H : i \u2208 s then x \u27e8i, H\u27e9 else 0 := rfl\nlemma mk_of_mem (hi : i \u2208 s) : (mk s x : \u03a0 i, \u03b2 i) i = x \u27e8i, hi\u27e9 := dif_pos hi\nlemma mk_of_not_mem (hi : i \u2209 s) : (mk s x : \u03a0 i, \u03b2 i) i = 0 := dif_neg hi\n\ntheorem mk_injective (s : finset \u03b9) : function.injective (@mk \u03b9 \u03b2 _ _ s) :=\nbegin\n  intros x y H,\n  ext i,\n  have h1 : (mk s x : \u03a0 i, \u03b2 i) i = (mk s y : \u03a0 i, \u03b2 i) i, {rw H},\n  cases i with i hi,\n  change i \u2208 s at hi,\n  dsimp only [mk_apply, subtype.coe_mk] at h1,\n  simpa only [dif_pos hi] using h1\nend\n\nomit dec\n/-- Given `fintype \u03b9`, `equiv_fun_on_fintype` is the `equiv` between `\u03a0\u2080 i, \u03b2 i` and `\u03a0 i, \u03b2 i`.\n  (All dependent functions on a finite type are finitely supported.) -/\n@[simps apply] def equiv_fun_on_fintype [fintype \u03b9] : (\u03a0\u2080 i, \u03b2 i) \u2243 (\u03a0 i, \u03b2 i) :=\n{ to_fun := coe_fn,\n  inv_fun := \u03bb f, \u27e6\u27e8f, finset.univ.1, \u03bb i, or.inl $ finset.mem_univ_val _\u27e9\u27e7,\n  left_inv := \u03bb x, coe_fn_injective rfl,\n  right_inv := \u03bb x, rfl }\n\n@[simp] lemma equiv_fun_on_fintype_symm_coe [fintype \u03b9] (f : \u03a0\u2080 i, \u03b2 i) :\n  equiv_fun_on_fintype.symm f = f :=\nequiv.symm_apply_apply _ _\ninclude dec\n\n/-- The function `single i b : \u03a0\u2080 i, \u03b2 i` sends `i` to `b`\nand all other points to `0`. -/\ndef single (i : \u03b9) (b : \u03b2 i) : \u03a0\u2080 i, \u03b2 i :=\nmk {i} $ \u03bb j, eq.rec_on (finset.mem_singleton.1 j.prop).symm b\n\n@[simp] lemma single_apply {i i' b} :\n  (single i b : \u03a0\u2080 i, \u03b2 i) i' = (if h : i = i' then eq.rec_on h b else 0) :=\nbegin\n  dsimp only [single],\n  by_cases h : i = i',\n  { have h1 : i' \u2208 ({i} : finset \u03b9) := finset.mem_singleton.2 h.symm,\n    simp only [mk_apply, dif_pos h, dif_pos h1], refl },\n  { have h1 : i' \u2209 ({i} : finset \u03b9) := finset.not_mem_singleton.2 (ne.symm h),\n    simp only [mk_apply, dif_neg h, dif_neg h1] }\nend\n\nlemma single_eq_pi_single {i b} : \u21d1(single i b : \u03a0\u2080 i, \u03b2 i) = pi.single i b :=\nbegin\n  ext i',\n  simp only [pi.single, function.update],\n  split_ifs,\n  { simp [h] },\n  { simp [ne.symm h] }\nend\n\n@[simp] lemma single_zero (i) : (single i 0 : \u03a0\u2080 i, \u03b2 i) = 0 :=\nquotient.sound $ \u03bb j, if H : j \u2208 ({i} : finset _)\nthen by dsimp only; rw [dif_pos H]; cases finset.mem_singleton.1 H; refl\nelse dif_neg H\n\n@[simp] lemma single_eq_same {i b} : (single i b : \u03a0\u2080 i, \u03b2 i) i = b :=\nby simp only [single_apply, dif_pos rfl]\n\nlemma single_eq_of_ne {i i' b} (h : i \u2260 i') : (single i b : \u03a0\u2080 i, \u03b2 i) i' = 0 :=\nby simp only [single_apply, dif_neg h]\n\nlemma single_injective {i} : function.injective (single i : \u03b2 i \u2192 \u03a0\u2080 i, \u03b2 i) :=\n\u03bb x y H, congr_fun (mk_injective _ H) \u27e8i, by simp\u27e9\n\n/-- Like `finsupp.single_eq_single_iff`, but with a `heq` due to dependent types -/\nlemma single_eq_single_iff (i j : \u03b9) (xi : \u03b2 i) (xj : \u03b2 j) :\n  dfinsupp.single i xi = dfinsupp.single j xj \u2194 i = j \u2227 xi == xj \u2228 xi = 0 \u2227 xj = 0 :=\nbegin\n  split,\n  { intro h,\n    by_cases hij : i = j,\n    { subst hij,\n      exact or.inl \u27e8rfl, heq_of_eq (dfinsupp.single_injective h)\u27e9, },\n    { have h_coe : \u21d1(dfinsupp.single i xi) = dfinsupp.single j xj := congr_arg coe_fn h,\n      have hci := congr_fun h_coe i,\n      have hcj := congr_fun h_coe j,\n      rw dfinsupp.single_eq_same at hci hcj,\n      rw dfinsupp.single_eq_of_ne (ne.symm hij) at hci,\n      rw dfinsupp.single_eq_of_ne (hij) at hcj,\n      exact or.inr \u27e8hci, hcj.symm\u27e9, }, },\n  { rintros (\u27e8hi, hxi\u27e9 | \u27e8hi, hj\u27e9),\n    { subst hi,\n      rw eq_of_heq hxi, },\n    { rw [hi, hj, dfinsupp.single_zero, dfinsupp.single_zero], }, },\nend\n\n@[simp] lemma single_eq_zero {i : \u03b9} {xi : \u03b2 i} : single i xi = 0 \u2194 xi = 0 :=\nbegin\n  rw [\u2190single_zero i, single_eq_single_iff],\n  simp,\nend\n\nlemma filter_single (p : \u03b9 \u2192 Prop) [decidable_pred p] (i : \u03b9) (x : \u03b2 i) :\n  (single i x).filter p = if p i then single i x else 0 :=\nbegin\n  ext j,\n  have := apply_ite (\u03bb x : \u03a0\u2080 i, \u03b2 i, x j) (p i) (single i x) 0,\n  dsimp at this,\n  rw [filter_apply, this],\n  obtain rfl | hij := decidable.eq_or_ne i j,\n  { refl, },\n  { rw [single_eq_of_ne hij, if_t_t, if_t_t], },\nend\n\n@[simp] lemma filter_single_pos {p : \u03b9 \u2192 Prop} [decidable_pred p] (i : \u03b9) (x : \u03b2 i) (h : p i) :\n  (single i x).filter p = single i x :=\nby rw [filter_single, if_pos h]\n\n@[simp] lemma filter_single_neg {p : \u03b9 \u2192 Prop} [decidable_pred p] (i : \u03b9) (x : \u03b2 i) (h : \u00acp i) :\n  (single i x).filter p = 0 :=\nby rw [filter_single, if_neg h]\n\n/-- Equality of sigma types is sufficient (but not necessary) to show equality of `dfinsupp`s. -/\nlemma single_eq_of_sigma_eq\n  {i j} {xi : \u03b2 i} {xj : \u03b2 j} (h : (\u27e8i, xi\u27e9 : sigma \u03b2) = \u27e8j, xj\u27e9) :\n  dfinsupp.single i xi = dfinsupp.single j xj :=\nby { cases h, refl }\n\n@[simp] lemma equiv_fun_on_fintype_single [fintype \u03b9] (i : \u03b9) (m : \u03b2 i) :\n  (@dfinsupp.equiv_fun_on_fintype \u03b9 \u03b2 _ _) (dfinsupp.single i m) = pi.single i m :=\nby { ext, simp [dfinsupp.single_eq_pi_single], }\n\n@[simp] lemma equiv_fun_on_fintype_symm_single [fintype \u03b9] (i : \u03b9) (m : \u03b2 i) :\n  (@dfinsupp.equiv_fun_on_fintype \u03b9 \u03b2 _ _).symm (pi.single i m) = dfinsupp.single i m :=\nby { ext i', simp only [\u2190 single_eq_pi_single, equiv_fun_on_fintype_symm_coe] }\n\n/-- Redefine `f i` to be `0`. -/\ndef erase (i : \u03b9) : (\u03a0\u2080 i, \u03b2 i) \u2192 \u03a0\u2080 i, \u03b2 i :=\nquotient.map\n  (\u03bb x, \u27e8\u03bb j, if j = i then 0 else x.1 j, x.2,\n          \u03bb j, (x.3 j).imp_right $ \u03bb H, by simp only [H, if_t_t]\u27e9)\n  (\u03bb x y H j, if h : j = i then by simp only [if_pos h] else by simp only [if_neg h, H j])\n\n@[simp] lemma erase_apply {i j : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} :\n  (f.erase i) j = if j = i then 0 else f j :=\nquotient.induction_on f $ \u03bb x, rfl\n\n@[simp] lemma erase_same {i : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} : (f.erase i) i = 0 :=\nby simp\n\nlemma erase_ne {i i' : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} (h : i' \u2260 i) : (f.erase i) i' = f i' :=\nby simp [h]\n\nlemma erase_eq_sub_single {\u03b2 : \u03b9 \u2192 Type*} [\u03a0 i, add_group (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) :\n  f.erase i = f - single i (f i) :=\nbegin\n  ext j,\n  rcases eq_or_ne i j with rfl|h,\n  { simp },\n  { simp [erase_ne h.symm, single_eq_of_ne h] }\nend\n\n@[simp] lemma erase_zero (i : \u03b9) : erase i (0 : \u03a0\u2080 i, \u03b2 i) = 0 :=\next $ \u03bb _, if_t_t _ _\n\n@[simp] lemma filter_ne_eq_erase (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) : f.filter (\u2260 i) = f.erase i :=\nbegin\n  ext1 j,\n  simp only [dfinsupp.filter_apply, dfinsupp.erase_apply, ite_not],\nend\n\n@[simp] lemma filter_ne_eq_erase' (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) : f.filter ((\u2260) i) = f.erase i :=\nbegin\n  rw \u2190filter_ne_eq_erase f i,\n  congr' with j,\n  exact ne_comm,\nend\n\nlemma erase_single (j : \u03b9) (i : \u03b9) (x : \u03b2 i) :\n  (single i x).erase j = if i = j then 0 else single i x :=\nby rw [\u2190filter_ne_eq_erase, filter_single, ite_not]\n\n@[simp] lemma erase_single_same (i : \u03b9) (x : \u03b2 i) : (single i x).erase i = 0 :=\nby rw [erase_single, if_pos rfl]\n\n@[simp] lemma erase_single_ne {i j : \u03b9} (x : \u03b2 i) (h : i \u2260 j) : (single i x).erase j = single i x :=\nby rw [erase_single, if_neg h]\n\nsection update\n\nvariables (f : \u03a0\u2080 i, \u03b2 i) (i) (b : \u03b2 i) [decidable (b = 0)]\n\n/-- Replace the value of a `\u03a0\u2080 i, \u03b2 i` at a given point `i : \u03b9` by a given value `b : \u03b2 i`.\nIf `b = 0`, this amounts to removing `i` from the support.\nOtherwise, `i` is added to it.\n\nThis is the (dependent) finitely-supported version of `function.update`. -/\ndef update : \u03a0\u2080 i, \u03b2 i :=\nquotient.map (\u03bb (x : pre _ _), \u27e8function.update x.to_fun i b,\n  if b = 0 then x.pre_support.erase i else i ::\u2098 x.pre_support,\n  begin\n    intro j,\n    rcases eq_or_ne i j with rfl|hi,\n    { split_ifs with hb,\n      { simp [hb] },\n      { simp } },\n    { cases x.zero j with hj hj,\n      { split_ifs;\n        simp [multiset.mem_erase_of_ne hi.symm, hj] },\n      { simp [function.update_noteq hi.symm, hj] } }\n  end\u27e9)\n  (\u03bb x y h j,\n    show function.update x.to_fun i b j = function.update y.to_fun i b j,\n    by rw (funext h : x.to_fun = y.to_fun)) f\n\nvariables (j : \u03b9)\n\n@[simp] lemma coe_update : (f.update i b : \u03a0 (i : \u03b9), \u03b2 i) = function.update f i b :=\nquotient.induction_on f (\u03bb _, rfl)\n@[simp] lemma update_self [decidable (f i = 0)] : f.update i (f i) = f :=\nby { ext, simp }\n\n@[simp] lemma update_eq_erase [decidable ((0 : \u03b2 i) = 0)] : f.update i 0 = f.erase i :=\nbegin\n  ext j,\n  rcases eq_or_ne i j with rfl|hi,\n  { simp },\n  { simp [hi.symm] }\nend\n\nlemma update_eq_single_add_erase {\u03b2 : \u03b9 \u2192 Type*} [\u03a0 i, add_zero_class (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9)\n  (b : \u03b2 i) [decidable (b = 0)] :\n  f.update i b = single i b + f.erase i :=\nbegin\n  ext j,\n  rcases eq_or_ne i j with rfl|h,\n  { simp },\n  { simp [function.update_noteq h.symm, h, erase_ne, h.symm] }\nend\n\nlemma update_eq_erase_add_single {\u03b2 : \u03b9 \u2192 Type*} [\u03a0 i, add_zero_class (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9)\n  (b : \u03b2 i) [decidable (b = 0)] :\n  f.update i b = f.erase i + single i b :=\nbegin\n  ext j,\n  rcases eq_or_ne i j with rfl|h,\n  { simp },\n  { simp [function.update_noteq h.symm, h, erase_ne, h.symm] }\nend\n\nlemma update_eq_sub_add_single {\u03b2 : \u03b9 \u2192 Type*} [\u03a0 i, add_group (\u03b2 i)] (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9)\n  (b : \u03b2 i) [decidable (b = 0)] :\n  f.update i b = f - single i (f i) + single i b :=\nby rw [update_eq_erase_add_single f i b, erase_eq_sub_single f i]\n\nend update\n\nend basic\n\nsection add_monoid\n\nvariable [\u03a0 i, add_zero_class (\u03b2 i)]\n\n@[simp] lemma single_add (i : \u03b9) (b\u2081 b\u2082 : \u03b2 i) : single i (b\u2081 + b\u2082) = single i b\u2081 + single i b\u2082 :=\next $ assume i',\nbegin\n  by_cases h : i = i',\n  { subst h, simp only [add_apply, single_eq_same] },\n  { simp only [add_apply, single_eq_of_ne h, zero_add] }\nend\n\n@[simp] lemma erase_add (i : \u03b9) (f\u2081 f\u2082 : \u03a0\u2080 i, \u03b2 i) : erase i (f\u2081 + f\u2082) = erase i f\u2081 + erase i f\u2082 :=\next $ \u03bb _, by simp [ite_zero_add]\n\nvariables (\u03b2)\n\n/-- `dfinsupp.single` as an `add_monoid_hom`. -/\n@[simps] def single_add_hom (i : \u03b9) : \u03b2 i \u2192+ \u03a0\u2080 i, \u03b2 i :=\n{ to_fun := single i, map_zero' := single_zero i, map_add' := single_add i }\n\n/-- `dfinsupp.erase` as an `add_monoid_hom`. -/\n@[simps] def erase_add_hom (i : \u03b9) : (\u03a0\u2080 i, \u03b2 i) \u2192+ \u03a0\u2080 i, \u03b2 i :=\n{ to_fun := erase i, map_zero' := erase_zero i, map_add' := erase_add i }\n\nvariables {\u03b2}\n\n@[simp] lemma single_neg {\u03b2 : \u03b9 \u2192 Type v} [\u03a0 i, add_group (\u03b2 i)] (i : \u03b9) (x : \u03b2 i) :\n  single i (-x) = -single i x :=\n(single_add_hom \u03b2 i).map_neg x\n\n@[simp] lemma single_sub {\u03b2 : \u03b9 \u2192 Type v} [\u03a0 i, add_group (\u03b2 i)] (i : \u03b9) (x y : \u03b2 i) :\n  single i (x - y) = single i x - single i y :=\n(single_add_hom \u03b2 i).map_sub x y\n\n@[simp] lemma erase_neg {\u03b2 : \u03b9 \u2192 Type v} [\u03a0 i, add_group (\u03b2 i)] (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  (-f).erase i = -f.erase i :=\n(erase_add_hom \u03b2 i).map_neg f\n\n@[simp] lemma erase_sub {\u03b2 : \u03b9 \u2192 Type v} [\u03a0 i, add_group (\u03b2 i)] (i : \u03b9) (f g : \u03a0\u2080 i, \u03b2 i) :\n  (f - g).erase i = f.erase i - g.erase i :=\n(erase_add_hom \u03b2 i).map_sub f g\n\nlemma single_add_erase (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) : single i (f i) + f.erase i = f :=\next $ \u03bb i',\nif h : i = i'\nthen by subst h; simp only [add_apply, single_apply, erase_apply, dif_pos rfl, if_pos, add_zero]\nelse by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (ne.symm h), zero_add]\n\nlemma erase_add_single (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) : f.erase i + single i (f i) = f :=\next $ \u03bb i',\nif h : i = i'\nthen by subst h; simp only [add_apply, single_apply, erase_apply, dif_pos rfl, if_pos, zero_add]\nelse by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (ne.symm h), add_zero]\n\nprotected theorem induction {p : (\u03a0\u2080 i, \u03b2 i) \u2192 Prop} (f : \u03a0\u2080 i, \u03b2 i)\n  (h0 : p 0) (ha : \u2200i b (f : \u03a0\u2080 i, \u03b2 i), f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)) :\n  p f :=\nbegin\n  refine quotient.induction_on f (\u03bb x, _),\n  cases x with f s H, revert f H,\n  apply multiset.induction_on s,\n  { intros f H, convert h0, ext i, exact (H i).resolve_left id },\n  intros i s ih f H,\n  have H2 : p (erase i \u27e6{to_fun := f, pre_support := i ::\u2098 s, zero := H}\u27e7),\n  { dsimp only [erase, quotient.map_mk],\n    have H2 : \u2200 j, j \u2208 s \u2228 ite (j = i) 0 (f j) = 0,\n    { intro j, cases H j with H2 H2,\n      { cases multiset.mem_cons.1 H2 with H3 H3,\n        { right, exact if_pos H3 },\n        { left, exact H3 } },\n      right, split_ifs; [refl, exact H2] },\n    have H3 : (\u27e6{to_fun := \u03bb (j : \u03b9), ite (j = i) 0 (f j),\n         pre_support := i ::\u2098 s, zero := _}\u27e7 : \u03a0\u2080 i, \u03b2 i)\n      = \u27e6{to_fun := \u03bb (j : \u03b9), ite (j = i) 0 (f j), pre_support := s, zero := H2}\u27e7 :=\n      quotient.sound (\u03bb i, rfl),\n    rw H3, apply ih },\n  have H3 : single i _ + _ = (\u27e6{to_fun := f, pre_support := i ::\u2098 s, zero := H}\u27e7 : \u03a0\u2080 i, \u03b2 i) :=\n    single_add_erase _ _,\n  rw \u2190 H3,\n  change p (single i (f i) + _),\n  cases classical.em (f i = 0) with h h,\n  { rw [h, single_zero, zero_add], exact H2 },\n  refine ha _ _ _ _ h H2,\n  rw erase_same\nend\n\nlemma induction\u2082 {p : (\u03a0\u2080 i, \u03b2 i) \u2192 Prop} (f : \u03a0\u2080 i, \u03b2 i)\n  (h0 : p 0) (ha : \u2200i b (f : \u03a0\u2080 i, \u03b2 i), f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)) :\n  p f :=\ndfinsupp.induction f h0 $ \u03bb i b f h1 h2 h3,\nhave h4 : f + single i b = single i b + f,\n{ ext j, by_cases H : i = j,\n  { subst H, simp [h1] },\n  { simp [H] } },\neq.rec_on h4 $ ha i b f h1 h2 h3\n\n@[simp] lemma add_closure_Union_range_single :\n  add_submonoid.closure (\u22c3 i : \u03b9, set.range (single i : \u03b2 i \u2192 (\u03a0\u2080 i, \u03b2 i))) = \u22a4 :=\ntop_unique $ \u03bb x hx, (begin\n  apply dfinsupp.induction x,\n  exact add_submonoid.zero_mem _,\n  exact \u03bb a b f ha hb hf, add_submonoid.add_mem _\n    (add_submonoid.subset_closure $ set.mem_Union.2 \u27e8a, set.mem_range_self _\u27e9) hf\nend)\n\n/-- If two additive homomorphisms from `\u03a0\u2080 i, \u03b2 i` are equal on each `single a b`, then\nthey are equal. -/\n\n\n/-- If two additive homomorphisms from `\u03a0\u2080 i, \u03b2 i` are equal on each `single a b`, then\nthey are equal.\n\nSee note [partially-applied ext lemmas]. -/\n@[ext] lemma add_hom_ext' {\u03b3 : Type w} [add_zero_class \u03b3] \u2983f g : (\u03a0\u2080 i, \u03b2 i) \u2192+ \u03b3\u2984\n  (H : \u2200 x, f.comp (single_add_hom \u03b2 x) = g.comp (single_add_hom \u03b2 x)) :\n  f = g :=\nadd_hom_ext $ \u03bb x, add_monoid_hom.congr_fun (H x)\n\nend add_monoid\n\n@[simp] lemma mk_add [\u03a0 i, add_zero_class (\u03b2 i)] {s : finset \u03b9} {x y : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i} :\n  mk s (x + y) = mk s x + mk s y :=\next $ \u03bb i, by simp only [add_apply, mk_apply]; split_ifs; [refl, rw zero_add]\n\n@[simp] lemma mk_zero [\u03a0 i, has_zero (\u03b2 i)] {s : finset \u03b9} :\n  mk s (0 : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1) = 0 :=\next $ \u03bb i, by simp only [mk_apply]; split_ifs; refl\n\n@[simp] lemma mk_neg [\u03a0 i, add_group (\u03b2 i)] {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  mk s (-x) = -mk s x :=\next $ \u03bb i, by simp only [neg_apply, mk_apply]; split_ifs; [refl, rw neg_zero]\n\n@[simp] lemma mk_sub [\u03a0 i, add_group (\u03b2 i)] {s : finset \u03b9} {x y : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  mk s (x - y) = mk s x - mk s y :=\next $ \u03bb i, by simp only [sub_apply, mk_apply]; split_ifs; [refl, rw sub_zero]\n\n/-- If `s` is a subset of `\u03b9` then `mk_add_group_hom s` is the canonical additive\ngroup homomorphism from $\\prod_{i\\in s}\\beta_i$ to $\\prod_{\\mathtt{i : \\iota}}\\beta_i.$-/\ndef mk_add_group_hom [\u03a0 i, add_group (\u03b2 i)] (s : finset \u03b9) :\n  (\u03a0 (i : (s : set \u03b9)), \u03b2 \u2191i) \u2192+ (\u03a0\u2080 (i : \u03b9), \u03b2 i) :=\n{ to_fun := mk s,\n  map_zero' := mk_zero,\n  map_add' := \u03bb _ _, mk_add }\n\nsection\nvariables [monoid \u03b3] [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, distrib_mul_action \u03b3 (\u03b2 i)]\n\n@[simp] lemma mk_smul {s : finset \u03b9} (c : \u03b3) (x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 (i : \u03b9)) :\n  mk s (c \u2022 x) = c \u2022 mk s x :=\next $ \u03bb i, by simp only [smul_apply, mk_apply]; split_ifs; [refl, rw smul_zero]\n\n@[simp] lemma single_smul {i : \u03b9} (c : \u03b3) (x : \u03b2 i) :\n  single i (c \u2022 x) = c \u2022 single i x :=\next $ \u03bb i, by simp only [smul_apply, single_apply]; split_ifs; [cases h, rw smul_zero]; refl\n\nend\n\nsection support_basic\n\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n\n/-- Set `{i | f x \u2260 0}` as a `finset`. -/\ndef support (f : \u03a0\u2080 i, \u03b2 i) : finset \u03b9 :=\nquotient.lift_on f (\u03bb x, x.2.to_finset.filter $ \u03bb i, x.1 i \u2260 0) $\nbegin\n  intros x y Hxy,\n  ext i, split,\n  { intro H,\n    rcases finset.mem_filter.1 H with \u27e8h1, h2\u27e9,\n    rw Hxy i at h2,\n    exact finset.mem_filter.2 \u27e8multiset.mem_to_finset.2 $ (y.3 i).resolve_right h2, h2\u27e9 },\n  { intro H,\n    rcases finset.mem_filter.1 H with \u27e8h1, h2\u27e9,\n    rw \u2190 Hxy i at h2,\n    exact finset.mem_filter.2 \u27e8multiset.mem_to_finset.2 $ (x.3 i).resolve_right h2, h2\u27e9 },\nend\n\n@[simp] theorem support_mk_subset {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  (mk s x).support \u2286 s :=\n\u03bb i H, multiset.mem_to_finset.1 (finset.mem_filter.1 H).1\n\n@[simp] theorem mem_support_to_fun (f : \u03a0\u2080 i, \u03b2 i) (i) : i \u2208 f.support \u2194 f i \u2260 0 :=\nbegin\n  refine quotient.induction_on f (\u03bb x, _),\n  dsimp only [support, quotient.lift_on_mk],\n  rw [finset.mem_filter, multiset.mem_to_finset],\n  exact and_iff_right_of_imp (x.3 i).resolve_right\nend\n\ntheorem eq_mk_support (f : \u03a0\u2080 i, \u03b2 i) : f = mk f.support (\u03bb i, f i) :=\nbegin\n  change f = mk f.support (\u03bb i, f i.1),\n  ext i,\n  by_cases h : f i \u2260 0; [skip, rw [not_not] at h];\n    simp [h]\nend\n\n@[simp] lemma support_zero : (0 : \u03a0\u2080 i, \u03b2 i).support = \u2205 := rfl\n\nlemma mem_support_iff {f : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} : i \u2208 f.support \u2194 f i \u2260 0 := f.mem_support_to_fun _\n\nlemma not_mem_support_iff {f : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} : i \u2209 f.support \u2194 f i = 0 :=\nnot_iff_comm.1 mem_support_iff.symm\n\n@[simp] lemma support_eq_empty {f : \u03a0\u2080 i, \u03b2 i} : f.support = \u2205 \u2194 f = 0 :=\n\u27e8\u03bb H, ext $ by simpa [finset.ext_iff] using H, by simp {contextual:=tt}\u27e9\n\ninstance decidable_zero : decidable_pred (eq (0 : \u03a0\u2080 i, \u03b2 i)) :=\n\u03bb f, decidable_of_iff _ $ support_eq_empty.trans eq_comm\n\nlemma support_subset_iff {s : set \u03b9} {f : \u03a0\u2080 i, \u03b2 i} :\n  \u2191f.support \u2286 s \u2194 (\u2200i\u2209s, f i = 0) :=\nby simp [set.subset_def];\n   exact forall_congr (assume i, not_imp_comm)\n\nlemma support_single_ne_zero {i : \u03b9} {b : \u03b2 i} (hb : b \u2260 0) : (single i b).support = {i} :=\nbegin\n  ext j, by_cases h : i = j,\n  { subst h, simp [hb] },\n  simp [ne.symm h, h]\nend\n\nlemma support_single_subset {i : \u03b9} {b : \u03b2 i} : (single i b).support \u2286 {i} :=\nsupport_mk_subset\n\nsection map_range_and_zip_with\n\nvariables [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n\nlemma map_range_def [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)]\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} :\n  map_range f hf g = mk g.support (\u03bb i, f i.1 (g i.1)) :=\nbegin\n  ext i,\n  by_cases h : g i \u2260 0; simp at h; simp [h, hf]\nend\n\n@[simp] lemma map_range_single {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {i : \u03b9} {b : \u03b2\u2081 i} :\n  map_range f hf (single i b) = single i (f i b) :=\ndfinsupp.ext $ \u03bb i', by by_cases i = i'; [{subst i', simp}, simp [h, hf]]\n\nvariables [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)] [\u03a0 i (x : \u03b2\u2082 i), decidable (x \u2260 0)]\n\nlemma support_map_range {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} :\n  (map_range f hf g).support \u2286 g.support :=\nby simp [map_range_def]\n\nlemma zip_with_def {\u03b9 : Type u} {\u03b2 : \u03b9 \u2192 Type v} {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\n  [dec : decidable_eq \u03b9] [\u03a0 (i : \u03b9), has_zero (\u03b2 i)] [\u03a0 (i : \u03b9), has_zero (\u03b2\u2081 i)]\n  [\u03a0 (i : \u03b9), has_zero (\u03b2\u2082 i)] [\u03a0 (i : \u03b9) (x : \u03b2\u2081 i), decidable (x \u2260 0)]\n  [\u03a0 (i : \u03b9) (x : \u03b2\u2082 i), decidable (x \u2260 0)]\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i} {hf : \u2200 i, f i 0 0 = 0}\n  {g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i} {g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i} :\n  zip_with f hf g\u2081 g\u2082 = mk (g\u2081.support \u222a g\u2082.support) (\u03bb i, f i.1 (g\u2081 i.1) (g\u2082 i.1)) :=\nbegin\n  ext i,\n  by_cases h1 : g\u2081 i \u2260 0; by_cases h2 : g\u2082 i \u2260 0;\n    simp only [not_not, ne.def] at h1 h2; simp [h1, h2, hf]\nend\n\nlemma support_zip_with {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i} {hf : \u2200 i, f i 0 0 = 0}\n  {g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i} {g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i} :\n  (zip_with f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nby simp [zip_with_def]\n\nend map_range_and_zip_with\n\nlemma erase_def (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  f.erase i = mk (f.support.erase i) (\u03bb j, f j.1) :=\nby { ext j, by_cases h1 : j = i; by_cases h2 : f j \u2260 0; simp at h2; simp [h1, h2] }\n\n@[simp] lemma support_erase (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  (f.erase i).support = f.support.erase i :=\nby { ext j, by_cases h1 : j = i, simp [h1], by_cases h2 : f j \u2260 0; simp at h2; simp [h1, h2] }\n\nlemma support_update_ne_zero (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) {b : \u03b2 i} [decidable (b = 0)] (h : b \u2260 0) :\n  support (f.update i b) = insert i f.support :=\nbegin\n  ext j,\n  rcases eq_or_ne i j with rfl|hi,\n  { simp [h] },\n  { simp [hi.symm] }\nend\n\nlemma support_update (f : \u03a0\u2080 i, \u03b2 i) (i : \u03b9) (b : \u03b2 i) [decidable (b = 0)] :\n  support (f.update i b) = if b = 0 then support (f.erase i) else insert i f.support :=\nbegin\n  ext j,\n  split_ifs with hb,\n  { substI hb, simp [update_eq_erase, support_erase] },\n  { rw [support_update_ne_zero f _ hb] }\nend\n\nsection filter_and_subtype_domain\n\nvariables {p : \u03b9 \u2192 Prop} [decidable_pred p]\n\nlemma filter_def (f : \u03a0\u2080 i, \u03b2 i) :\n  f.filter p = mk (f.support.filter p) (\u03bb i, f i.1) :=\nby ext i; by_cases h1 : p i; by_cases h2 : f i \u2260 0;\n simp at h2; simp [h1, h2]\n\n@[simp] lemma support_filter (f : \u03a0\u2080 i, \u03b2 i) :\n  (f.filter p).support = f.support.filter p :=\nby ext i; by_cases h : p i; simp [h]\n\nlemma subtype_domain_def (f : \u03a0\u2080 i, \u03b2 i) :\n  f.subtype_domain p = mk (f.support.subtype p) (\u03bb i, f i) :=\nby ext i; by_cases h2 : f i \u2260 0; try {simp at h2}; dsimp; simp [h2]\n\n@[simp] lemma support_subtype_domain {f : \u03a0\u2080 i, \u03b2 i} :\n  (subtype_domain p f).support = f.support.subtype p :=\nby { ext i, simp, }\n\nend filter_and_subtype_domain\n\nend support_basic\n\nlemma support_add [\u03a0 i, add_zero_class (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  {g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i} :\n  (g\u2081 + g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nsupport_zip_with\n\n@[simp] lemma support_neg [\u03a0 i, add_group (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  {f : \u03a0\u2080 i, \u03b2 i} :\n  support (-f) = support f :=\nby ext i; simp\n\nlemma support_smul {\u03b3 : Type w} [semiring \u03b3] [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)]\n  [\u03a0 ( i : \u03b9) (x : \u03b2 i), decidable (x \u2260 0)]\n  (b : \u03b3) (v : \u03a0\u2080 i, \u03b2 i) : (b \u2022 v).support \u2286 v.support :=\nsupport_map_range\n\ninstance [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_eq (\u03b2 i)] : decidable_eq (\u03a0\u2080 i, \u03b2 i) :=\nassume f g, decidable_of_iff (f.support = g.support \u2227 (\u2200i\u2208f.support, f i = g i))\n  \u27e8assume \u27e8h\u2081, h\u2082\u27e9, ext $ assume i,\n      if h : i \u2208 f.support then h\u2082 i h else\n        have hf : f i = 0, by rwa [mem_support_iff, not_not] at h,\n        have hg : g i = 0, by rwa [h\u2081, mem_support_iff, not_not] at h,\n        by rw [hf, hg],\n    by intro h; subst h; simp\u27e9\n\nsection prod_and_sum\n\n/-- `prod f g` is the product of `g i (f i)` over the support of `f`. -/\n@[to_additive \"`sum f g` is the sum of `g i (f i)` over the support of `f`.\"]\ndef prod [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)] [comm_monoid \u03b3]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 \u03b3) : \u03b3 :=\n\u220f i in f.support, g i (f i)\n\n@[to_additive]\nlemma prod_map_range_index {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\n  [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n  [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)] [\u03a0 i (x : \u03b2\u2082 i), decidable (x \u2260 0)] [comm_monoid \u03b3]\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} {h : \u03a0 i, \u03b2\u2082 i \u2192 \u03b3}\n  (h0 : \u2200i, h i 0 = 1) :\n  (map_range f hf g).prod h = g.prod (\u03bbi b, h i (f i b)) :=\nbegin\n  rw [map_range_def],\n  refine (finset.prod_subset support_mk_subset _).trans _,\n  { intros i h1 h2,\n    dsimp, simp [h1] at h2, dsimp at h2,\n    simp [h1, h2, h0] },\n  { refine finset.prod_congr rfl _,\n    intros i h1,\n    simp [h1] }\nend\n\n@[to_additive]\nlemma prod_zero_index [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] {h : \u03a0 i, \u03b2 i \u2192 \u03b3} : (0 : \u03a0\u2080 i, \u03b2 i).prod h = 1 :=\nrfl\n\n@[to_additive]\nlemma prod_single_index [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)] [comm_monoid \u03b3]\n  {i : \u03b9} {b : \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : h i 0 = 1) :\n  (single i b).prod h = h i b :=\nbegin\n  by_cases h : b \u2260 0,\n  { simp [dfinsupp.prod, support_single_ne_zero h] },\n  { rw [not_not] at h, simp [h, prod_zero_index, h_zero], refl }\nend\n\n@[to_additive]\nlemma prod_neg_index [\u03a0 i, add_group (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)] [comm_monoid \u03b3]\n  {g : \u03a0\u2080 i, \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h0 : \u2200i, h i 0 = 1) :\n  (-g).prod h = g.prod (\u03bbi b, h i (- b)) :=\nprod_map_range_index h0\n\nomit dec\n@[to_additive]\nlemma prod_comm {\u03b9\u2081 \u03b9\u2082 : Sort*} {\u03b2\u2081 : \u03b9\u2081 \u2192 Type*} {\u03b2\u2082 : \u03b9\u2082 \u2192 Type*}\n  [decidable_eq \u03b9\u2081] [decidable_eq \u03b9\u2082] [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n  [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)] [\u03a0 i (x : \u03b2\u2082 i), decidable (x \u2260 0)] [comm_monoid \u03b3]\n  (f\u2081 : \u03a0\u2080 i, \u03b2\u2081 i) (f\u2082 : \u03a0\u2080 i, \u03b2\u2082 i) (h : \u03a0 i, \u03b2\u2081 i \u2192 \u03a0 i, \u03b2\u2082 i \u2192 \u03b3) :\n  f\u2081.prod (\u03bb i\u2081 x\u2081, f\u2082.prod $ \u03bb i\u2082 x\u2082, h i\u2081 x\u2081 i\u2082 x\u2082) =\n  f\u2082.prod (\u03bb i\u2082 x\u2082, f\u2081.prod $ \u03bb i\u2081 x\u2081, h i\u2081 x\u2081 i\u2082 x\u2082) := finset.prod_comm\n\n@[simp] lemma sum_apply {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i} {i\u2082 : \u03b9} :\n  (f.sum g) i\u2082 = f.sum (\u03bbi\u2081 b, g i\u2081 b i\u2082) :=\n(eval_add_monoid_hom i\u2082 : (\u03a0\u2080 i, \u03b2 i) \u2192+ \u03b2 i\u2082).map_sum  _ f.support\n\ninclude dec\n\nlemma support_sum {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i} :\n  (f.sum g).support \u2286 f.support.bUnion (\u03bbi, (g i (f i)).support) :=\nhave \u2200i\u2081 : \u03b9, f.sum (\u03bb (i : \u03b9\u2081) (b : \u03b2\u2081 i), (g i b) i\u2081) \u2260 0 \u2192\n    (\u2203 (i : \u03b9\u2081), f i \u2260 0 \u2227 \u00ac (g i (f i)) i\u2081 = 0),\n  from assume i\u2081 h,\n  let \u27e8i, hi, ne\u27e9 := finset.exists_ne_zero_of_sum_ne_zero h in\n  \u27e8i, mem_support_iff.1 hi, ne\u27e9,\nby simpa [finset.subset_iff, mem_support_iff, finset.mem_bUnion, sum_apply] using this\n\n@[simp, to_additive] lemma prod_one [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] {f : \u03a0\u2080 i, \u03b2 i} :\n  f.prod (\u03bbi b, (1 : \u03b3)) = 1 :=\nfinset.prod_const_one\n\n@[simp, to_additive] lemma prod_mul [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] {f : \u03a0\u2080 i, \u03b2 i} {h\u2081 h\u2082 : \u03a0 i, \u03b2 i \u2192 \u03b3} :\n  f.prod (\u03bbi b, h\u2081 i b * h\u2082 i b) = f.prod h\u2081 * f.prod h\u2082 :=\nfinset.prod_mul_distrib\n\n@[simp, to_additive] lemma prod_inv [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_group \u03b3] {f : \u03a0\u2080 i, \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} :\n  f.prod (\u03bbi b, (h i b)\u207b\u00b9) = (f.prod h)\u207b\u00b9 :=\n((comm_group.inv_monoid_hom : \u03b3 \u2192* \u03b3).map_prod _ f.support).symm\n\n@[to_additive]\nlemma prod_add_index [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] {f g : \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082) :\n  (f + g).prod h = f.prod h * g.prod h :=\nhave f_eq : \u220f i in f.support \u222a g.support, h i (f i) = f.prod h,\n  from (finset.prod_subset (finset.subset_union_left _ _) $\n    by simp [mem_support_iff, h_zero] {contextual := tt}).symm,\nhave g_eq : \u220f i in f.support \u222a g.support, h i (g i) = g.prod h,\n  from (finset.prod_subset (finset.subset_union_right _ _) $\n    by simp [mem_support_iff, h_zero] {contextual := tt}).symm,\ncalc \u220f i in (f + g).support, h i ((f + g) i) =\n      \u220f i in f.support \u222a g.support, h i ((f + g) i) :\n    finset.prod_subset support_add $\n      by simp [mem_support_iff, h_zero] {contextual := tt}\n  ... = (\u220f i in f.support \u222a g.support, h i (f i)) *\n      (\u220f i in f.support \u222a g.support, h i (g i)) :\n    by simp [h_add, finset.prod_mul_distrib]\n  ... = _ : by rw [f_eq, g_eq]\n\n@[to_additive]\nlemma _root_.submonoid.dfinsupp_prod_mem [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] (S : submonoid \u03b3)\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 \u03b3) (h : \u2200 c, f c \u2260 0 \u2192 g c (f c) \u2208 S) : f.prod g \u2208 S :=\nS.prod_mem $ \u03bb i hi, h _ $ mem_support_iff.1 hi\n\n@[simp, to_additive] lemma prod_eq_prod_fintype [fintype \u03b9] [\u03a0 i, has_zero (\u03b2 i)]\n  [\u03a0 (i : \u03b9) (x : \u03b2 i), decidable (x \u2260 0)] [comm_monoid \u03b3] (v : \u03a0\u2080 i, \u03b2 i) {f : \u03a0 i, \u03b2 i \u2192 \u03b3}\n  (hf : \u2200 i, f i 0 = 1) :\n  v.prod f = \u220f i, f i (dfinsupp.equiv_fun_on_fintype v i) :=\nbegin\n  suffices : \u220f i in v.support, f i (v i) = \u220f i, f i (v i),\n  { simp [dfinsupp.prod, this] },\n  apply finset.prod_subset v.support.subset_univ,\n  intros i hi' hi,\n  rw [mem_support_iff, not_not] at hi,\n  rw [hi, hf],\nend\n\n/--\nWhen summing over an `add_monoid_hom`, the decidability assumption is not needed, and the result is\nalso an `add_monoid_hom`.\n-/\ndef sum_add_hom [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3] (\u03c6 : \u03a0 i, \u03b2 i \u2192+ \u03b3) :\n  (\u03a0\u2080 i, \u03b2 i) \u2192+ \u03b3 :=\n{ to_fun := (\u03bb f,\n    quotient.lift_on f (\u03bb x, \u2211 i in x.2.to_finset, \u03c6 i (x.1 i)) $ \u03bb x y H,\n    begin\n      have H1 : x.2.to_finset \u2229 y.2.to_finset \u2286 x.2.to_finset, from finset.inter_subset_left _ _,\n      have H2 : x.2.to_finset \u2229 y.2.to_finset \u2286 y.2.to_finset, from finset.inter_subset_right _ _,\n      refine (finset.sum_subset H1 _).symm.trans\n          ((finset.sum_congr rfl _).trans (finset.sum_subset H2 _)),\n      { intros i H1 H2, rw finset.mem_inter at H2, rw H i,\n        simp only [multiset.mem_to_finset] at H1 H2,\n        rw [(y.3 i).resolve_left (mt (and.intro H1) H2), add_monoid_hom.map_zero] },\n      { intros i H1, rw H i },\n      { intros i H1 H2, rw finset.mem_inter at H2, rw \u2190 H i,\n        simp only [multiset.mem_to_finset] at H1 H2,\n        rw [(x.3 i).resolve_left (mt (\u03bb H3, and.intro H3 H1) H2), add_monoid_hom.map_zero] }\n    end),\n  map_add' := assume f g,\n  begin\n    refine quotient.induction_on f (\u03bb x, _),\n    refine quotient.induction_on g (\u03bb y, _),\n    change \u2211 i in _, _ = (\u2211 i in _, _) + (\u2211 i in _, _),\n    simp only, conv { to_lhs, congr, skip, funext, rw add_monoid_hom.map_add },\n    simp only [finset.sum_add_distrib],\n    congr' 1,\n    { refine (finset.sum_subset _ _).symm,\n      { intro i, simp only [multiset.mem_to_finset, multiset.mem_add], exact or.inl },\n      { intros i H1 H2, simp only [multiset.mem_to_finset, multiset.mem_add] at H2,\n        rw [(x.3 i).resolve_left H2, add_monoid_hom.map_zero] } },\n    { refine (finset.sum_subset _ _).symm,\n      { intro i, simp only [multiset.mem_to_finset, multiset.mem_add], exact or.inr },\n      { intros i H1 H2, simp only [multiset.mem_to_finset, multiset.mem_add] at H2,\n        rw [(y.3 i).resolve_left H2, add_monoid_hom.map_zero] } }\n  end,\n  map_zero' := rfl }\n\n@[simp] lemma sum_add_hom_single [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (\u03c6 : \u03a0 i, \u03b2 i \u2192+ \u03b3) (i) (x : \u03b2 i) : sum_add_hom \u03c6 (single i x) = \u03c6 i x :=\n(add_zero _).trans $ congr_arg (\u03c6 i) $ show (if H : i \u2208 ({i} : finset _) then x else 0) = x,\nfrom dif_pos $ finset.mem_singleton_self i\n\n@[simp] lemma sum_add_hom_comp_single [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (f : \u03a0 i, \u03b2 i \u2192+ \u03b3) (i : \u03b9) :\n  (sum_add_hom f).comp (single_add_hom \u03b2 i) = f i :=\nadd_monoid_hom.ext $ \u03bb x, sum_add_hom_single f i x\n\n/-- While we didn't need decidable instances to define it, we do to reduce it to a sum -/\nlemma sum_add_hom_apply [\u03a0 i, add_zero_class (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [add_comm_monoid \u03b3] (\u03c6 : \u03a0 i, \u03b2 i \u2192+ \u03b3) (f : \u03a0\u2080 i, \u03b2 i) :\n  sum_add_hom \u03c6 f = f.sum (\u03bb x, \u03c6 x) :=\nbegin\n  refine quotient.induction_on f (\u03bb x, _),\n  change \u2211 i in _, _ = (\u2211 i in finset.filter _ _, _),\n  rw [finset.sum_filter, finset.sum_congr rfl],\n  intros i _,\n  dsimp only,\n  split_ifs,\n  refl,\n  rw [(not_not.mp h), add_monoid_hom.map_zero],\nend\n\nlemma _root_.add_submonoid.dfinsupp_sum_add_hom_mem [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (S : add_submonoid \u03b3) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ \u03b3) (h : \u2200 c, f c \u2260 0 \u2192 g c (f c) \u2208 S) :\n  dfinsupp.sum_add_hom g f \u2208 S :=\nbegin\n  classical,\n  rw dfinsupp.sum_add_hom_apply,\n  convert S.dfinsupp_sum_mem _ _ _,\n  exact h\nend\n\n/-- The supremum of a family of commutative additive submonoids is equal to the range of\n`dfinsupp.sum_add_hom`; that is, every element in the `supr` can be produced from taking a finite\nnumber of non-zero elements of `S i`, coercing them to `\u03b3`, and summing them. -/\nlemma _root_.add_submonoid.supr_eq_mrange_dfinsupp_sum_add_hom [add_comm_monoid \u03b3]\n  (S : \u03b9 \u2192 add_submonoid \u03b3) : supr S = (dfinsupp.sum_add_hom (\u03bb i, (S i).subtype)).mrange :=\nbegin\n  apply le_antisymm,\n  { apply supr_le _,\n    intros i y hy,\n    exact \u27e8dfinsupp.single i \u27e8y, hy\u27e9, dfinsupp.sum_add_hom_single _ _ _\u27e9, },\n  { rintros x \u27e8v, rfl\u27e9,\n    exact add_submonoid.dfinsupp_sum_add_hom_mem _ v _ (\u03bb i _, (le_supr S i : S i \u2264 _) (v i).prop) }\nend\n\n/-- The bounded supremum of a family of commutative additive submonoids is equal to the range of\n`dfinsupp.sum_add_hom` composed with `dfinsupp.filter_add_monoid_hom`; that is, every element in the\nbounded `supr` can be produced from taking a finite number of non-zero elements from the `S i` that\nsatisfy `p i`, coercing them to `\u03b3`, and summing them. -/\nlemma _root_.add_submonoid.bsupr_eq_mrange_dfinsupp_sum_add_hom (p : \u03b9 \u2192 Prop)\n  [decidable_pred p] [add_comm_monoid \u03b3] (S : \u03b9 \u2192 add_submonoid \u03b3) :\n  (\u2a06 i (h : p i), S i) =\n    ((sum_add_hom (\u03bb i, (S i).subtype)).comp (filter_add_monoid_hom _ p)).mrange :=\nbegin\n  apply le_antisymm,\n  { apply bsupr_le _,\n    intros i hi y hy,\n    refine \u27e8dfinsupp.single i \u27e8y, hy\u27e9, _\u27e9,\n    rw [add_monoid_hom.comp_apply, filter_add_monoid_hom_apply, filter_single_pos _ _ hi],\n    exact sum_add_hom_single _ _ _, },\n  { rintros x \u27e8v, rfl\u27e9,\n    refine add_submonoid.dfinsupp_sum_add_hom_mem _ _ _ (\u03bb i hi, _),\n    refine add_submonoid.mem_supr_of_mem i _,\n    by_cases hp : p i,\n    { simp [hp], },\n    { simp [hp] }, }\nend\n\nlemma _root_.add_submonoid.mem_supr_iff_exists_dfinsupp [add_comm_monoid \u03b3]\n  (S : \u03b9 \u2192 add_submonoid \u03b3) (x : \u03b3) :\n  x \u2208 supr S \u2194 \u2203 f : \u03a0\u2080 i, S i, dfinsupp.sum_add_hom (\u03bb i, (S i).subtype) f = x :=\nset_like.ext_iff.mp (add_submonoid.supr_eq_mrange_dfinsupp_sum_add_hom S) x\n\n/-- A variant of `add_submonoid.mem_supr_iff_exists_dfinsupp` with the RHS fully unfolded. -/\nlemma _root_.add_submonoid.mem_supr_iff_exists_dfinsupp' [add_comm_monoid \u03b3]\n  (S : \u03b9 \u2192 add_submonoid \u03b3) [\u03a0 i (x : S i), decidable (x \u2260 0)] (x : \u03b3) :\n  x \u2208 supr S \u2194 \u2203 f : \u03a0\u2080 i, S i, f.sum (\u03bb i xi, \u2191xi) = x :=\nbegin\n  rw add_submonoid.mem_supr_iff_exists_dfinsupp,\n  simp_rw sum_add_hom_apply,\n  congr',\nend\n\nlemma _root_.add_submonoid.mem_bsupr_iff_exists_dfinsupp (p : \u03b9 \u2192 Prop)\n  [decidable_pred p] [add_comm_monoid \u03b3] (S : \u03b9 \u2192 add_submonoid \u03b3) (x : \u03b3) :\n  x \u2208 (\u2a06 i (h : p i), S i) \u2194\n    \u2203 f : \u03a0\u2080 i, S i, dfinsupp.sum_add_hom (\u03bb i, (S i).subtype) (f.filter p) = x :=\nset_like.ext_iff.mp (add_submonoid.bsupr_eq_mrange_dfinsupp_sum_add_hom p S) x\n\nomit dec\nlemma sum_add_hom_comm {\u03b9\u2081 \u03b9\u2082 : Sort*} {\u03b2\u2081 : \u03b9\u2081 \u2192 Type*} {\u03b2\u2082 : \u03b9\u2082 \u2192 Type*} {\u03b3 : Type*}\n  [decidable_eq \u03b9\u2081] [decidable_eq \u03b9\u2082] [\u03a0 i, add_zero_class (\u03b2\u2081 i)] [\u03a0 i, add_zero_class (\u03b2\u2082 i)]\n  [add_comm_monoid \u03b3]\n  (f\u2081 : \u03a0\u2080 i, \u03b2\u2081 i) (f\u2082 : \u03a0\u2080 i, \u03b2\u2082 i) (h : \u03a0 i j, \u03b2\u2081 i \u2192+ \u03b2\u2082 j \u2192+ \u03b3) :\n  sum_add_hom (\u03bb i\u2082, sum_add_hom (\u03bb i\u2081, h i\u2081 i\u2082) f\u2081) f\u2082 =\n  sum_add_hom (\u03bb i\u2081, sum_add_hom (\u03bb i\u2082, (h i\u2081 i\u2082).flip) f\u2082) f\u2081 :=\nbegin\n  refine quotient.induction_on\u2082 f\u2081 f\u2082 (\u03bb x\u2081 x\u2082, _),\n  simp only [sum_add_hom, add_monoid_hom.finset_sum_apply, quotient.lift_on_mk,\n    add_monoid_hom.coe_mk, add_monoid_hom.flip_apply],\n  exact finset.sum_comm,\nend\n\ninclude dec\n/-- The `dfinsupp` version of `finsupp.lift_add_hom`,-/\n@[simps apply symm_apply]\ndef lift_add_hom [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3] :\n  (\u03a0 i, \u03b2 i \u2192+ \u03b3) \u2243+ ((\u03a0\u2080 i, \u03b2 i) \u2192+ \u03b3) :=\n{ to_fun := sum_add_hom,\n  inv_fun := \u03bb F i, F.comp (single_add_hom \u03b2 i),\n  left_inv := \u03bb x, by { ext, simp },\n  right_inv := \u03bb \u03c8, by { ext, simp },\n  map_add' := \u03bb F G, by { ext, simp } }\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_single_add_hom`,-/\n@[simp] lemma lift_add_hom_single_add_hom [\u03a0 i, add_comm_monoid (\u03b2 i)] :\n  lift_add_hom (single_add_hom \u03b2) = add_monoid_hom.id (\u03a0\u2080 i, \u03b2 i) :=\nlift_add_hom.to_equiv.apply_eq_iff_eq_symm_apply.2 rfl\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_apply_single`,-/\nlemma lift_add_hom_apply_single [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (f : \u03a0 i, \u03b2 i \u2192+ \u03b3) (i : \u03b9) (x : \u03b2 i) :\n  lift_add_hom f (single i x) = f i x :=\nby simp\n\n/-- The `dfinsupp` version of `finsupp.lift_add_hom_comp_single`,-/\nlemma lift_add_hom_comp_single [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (f : \u03a0 i, \u03b2 i \u2192+ \u03b3) (i : \u03b9) :\n  (lift_add_hom f).comp (single_add_hom \u03b2 i) = f i :=\nby simp\n\n/-- The `dfinsupp` version of `finsupp.comp_lift_add_hom`,-/\nlemma comp_lift_add_hom {\u03b4 : Type*} [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  [add_comm_monoid \u03b4] (g : \u03b3 \u2192+ \u03b4) (f : \u03a0 i, \u03b2 i \u2192+ \u03b3) :\n  g.comp (lift_add_hom f) = lift_add_hom (\u03bb a, g.comp (f a)) :=\nlift_add_hom.symm_apply_eq.1 $ funext $ \u03bb a,\n  by rw [lift_add_hom_symm_apply, add_monoid_hom.comp_assoc, lift_add_hom_comp_single]\n\n@[simp]\nlemma sum_add_hom_zero [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3] :\n  sum_add_hom (\u03bb i, (0 : \u03b2 i \u2192+ \u03b3)) = 0 :=\n(lift_add_hom : (\u03a0 i, \u03b2 i \u2192+ \u03b3) \u2243+ _).map_zero\n\n@[simp]\nlemma sum_add_hom_add [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  (g : \u03a0 i, \u03b2 i \u2192+ \u03b3) (h : \u03a0 i, \u03b2 i \u2192+ \u03b3) :\n  sum_add_hom (\u03bb i, g i + h i) = sum_add_hom g + sum_add_hom h :=\nlift_add_hom.map_add _ _\n\n@[simp]\nlemma sum_add_hom_single_add_hom [\u03a0 i, add_comm_monoid (\u03b2 i)] :\n  sum_add_hom (single_add_hom \u03b2) = add_monoid_hom.id _ :=\nlift_add_hom_single_add_hom\n\nlemma comp_sum_add_hom {\u03b4 : Type*} [\u03a0 i, add_zero_class (\u03b2 i)] [add_comm_monoid \u03b3]\n  [add_comm_monoid \u03b4] (g : \u03b3 \u2192+ \u03b4) (f : \u03a0 i, \u03b2 i \u2192+ \u03b3) :\n  g.comp (sum_add_hom f) = sum_add_hom (\u03bb a, g.comp (f a)) :=\ncomp_lift_add_hom _ _\n\nlemma sum_sub_index [\u03a0 i, add_group (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [add_comm_group \u03b3] {f g : \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_sub : \u2200i b\u2081 b\u2082, h i (b\u2081 - b\u2082) = h i b\u2081 - h i b\u2082) :\n  (f - g).sum h = f.sum h - g.sum h :=\nbegin\n  have := (lift_add_hom (\u03bb a, add_monoid_hom.of_map_sub (h a) (h_sub a))).map_sub f g,\n  rw [lift_add_hom_apply, sum_add_hom_apply, sum_add_hom_apply, sum_add_hom_apply] at this,\n  exact this,\nend\n\n@[to_additive]\nlemma prod_finset_sum_index {\u03b3 : Type w} {\u03b1 : Type x}\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3]\n  {s : finset \u03b1} {g : \u03b1 \u2192 \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082) :\n  \u220f i in s, (g i).prod h = (\u2211 i in s, g i).prod h :=\nbegin\n  classical,\n  exact finset.induction_on s\n  (by simp [prod_zero_index])\n  (by simp [prod_add_index, h_zero, h_add] {contextual := tt})\nend\n\n@[to_additive]\nlemma prod_sum_index {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i (x : \u03b2\u2081 i), decidable (x \u2260 0)]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082) :\n  (f.sum g).prod h = f.prod (\u03bbi b, (g i b).prod h) :=\n(prod_finset_sum_index h_zero h_add).symm\n\n@[simp] lemma sum_single [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)] {f : \u03a0\u2080 i, \u03b2 i} :\n  f.sum single = f :=\nbegin\n  have := add_monoid_hom.congr_fun lift_add_hom_single_add_hom f,\n  rw [lift_add_hom_apply, sum_add_hom_apply] at this,\n  exact this,\nend\n\n@[to_additive]\nlemma prod_subtype_domain_index [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n  [comm_monoid \u03b3] {v : \u03a0\u2080 i, \u03b2 i} {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (hp : \u2200 x \u2208 v.support, p x) :\n  (v.subtype_domain p).prod (\u03bbi b, h i b) = v.prod h :=\nfinset.prod_bij (\u03bbp _, p)\n  (by simp) (by simp)\n  (assume \u27e8a\u2080, ha\u2080\u27e9 \u27e8a\u2081, ha\u2081\u27e9, by simp)\n  (\u03bb i hi, \u27e8\u27e8i, hp i hi\u27e9, by simpa using hi, rfl\u27e9)\n\nomit dec\nlemma subtype_domain_sum [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  {s : finset \u03b3} {h : \u03b3 \u2192 \u03a0\u2080 i, \u03b2 i} {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  (\u2211 c in s, h c).subtype_domain p = \u2211 c in s, (h c).subtype_domain p :=\n(subtype_domain_add_monoid_hom \u03b2 p).map_sum  _ s\n\nlemma subtype_domain_finsupp_sum {\u03b4 : \u03b3 \u2192 Type x} [decidable_eq \u03b3]\n  [\u03a0 c, has_zero (\u03b4 c)] [\u03a0 c (x : \u03b4 c), decidable (x \u2260 0)]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {s : \u03a0\u2080 c, \u03b4 c} {h : \u03a0 c, \u03b4 c \u2192 \u03a0\u2080 i, \u03b2 i} :\n  (s.sum h).subtype_domain p = s.sum (\u03bbc d, (h c d).subtype_domain p) :=\nsubtype_domain_sum\n\nend prod_and_sum\n\n/-! ### Bundled versions of `dfinsupp.map_range`\n\nThe names should match the equivalent bundled `finsupp.map_range` definitions.\n-/\n\nsection map_range\nomit dec\n\nvariables [\u03a0 i, add_zero_class (\u03b2 i)] [\u03a0 i, add_zero_class (\u03b2\u2081 i)] [\u03a0 i, add_zero_class (\u03b2\u2082 i)]\n\nlemma map_range_add (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0)\n  (hf' : \u2200 i x y, f i (x + y) = f i x + f i y) (g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2\u2081 i):\n  map_range f hf (g\u2081 + g\u2082) = map_range f hf g\u2081 + map_range f hf g\u2082 :=\nbegin\n  ext,\n  simp only [map_range_apply f, coe_add, pi.add_apply, hf']\nend\n\n/-- `dfinsupp.map_range` as an `add_monoid_hom`. -/\n@[simps apply]\ndef map_range.add_monoid_hom (f : \u03a0 i, \u03b2\u2081 i \u2192+ \u03b2\u2082 i) : (\u03a0\u2080 i, \u03b2\u2081 i) \u2192+ (\u03a0\u2080 i, \u03b2\u2082 i) :=\n{ to_fun := map_range (\u03bb i x, f i x) (\u03bb i, (f i).map_zero),\n  map_zero' := map_range_zero _ _,\n  map_add' := map_range_add _ _ (\u03bb i, (f i).map_add) }\n\n@[simp]\nlemma map_range.add_monoid_hom_id :\n  map_range.add_monoid_hom (\u03bb i, add_monoid_hom.id (\u03b2\u2082 i)) = add_monoid_hom.id _ :=\nadd_monoid_hom.ext map_range_id\n\nlemma map_range.add_monoid_hom_comp (f : \u03a0 i, \u03b2\u2081 i \u2192+ \u03b2\u2082 i) (f\u2082 : \u03a0 i, \u03b2 i \u2192+ \u03b2\u2081 i):\n  map_range.add_monoid_hom (\u03bb i, (f i).comp (f\u2082 i)) =\n    (map_range.add_monoid_hom f).comp (map_range.add_monoid_hom f\u2082) :=\nadd_monoid_hom.ext $ map_range_comp (\u03bb i x, f i x) (\u03bb i x, f\u2082 i x) _ _ _\n\n/-- `dfinsupp.map_range.add_monoid_hom` as an `add_equiv`. -/\n@[simps apply]\ndef map_range.add_equiv (e : \u03a0 i, \u03b2\u2081 i \u2243+ \u03b2\u2082 i) : (\u03a0\u2080 i, \u03b2\u2081 i) \u2243+ (\u03a0\u2080 i, \u03b2\u2082 i) :=\n{ to_fun := map_range (\u03bb i x, e i x) (\u03bb i, (e i).map_zero),\n  inv_fun := map_range (\u03bb i x, (e i).symm x) (\u03bb i, (e i).symm.map_zero),\n  left_inv := \u03bb x, by rw \u2190map_range_comp; { simp_rw add_equiv.symm_comp_self, simp },\n  right_inv := \u03bb x, by rw \u2190map_range_comp; { simp_rw add_equiv.self_comp_symm, simp },\n  .. map_range.add_monoid_hom (\u03bb i, (e i).to_add_monoid_hom) }\n\n@[simp]\nlemma map_range.add_equiv_refl :\n  (map_range.add_equiv $ \u03bb i, add_equiv.refl (\u03b2\u2081 i)) = add_equiv.refl _ :=\nadd_equiv.ext map_range_id\n\nlemma map_range.add_equiv_trans (f : \u03a0 i, \u03b2 i \u2243+ \u03b2\u2081 i) (f\u2082 : \u03a0 i, \u03b2\u2081 i \u2243+ \u03b2\u2082 i):\n  map_range.add_equiv (\u03bb i, (f i).trans (f\u2082 i)) =\n    (map_range.add_equiv f).trans (map_range.add_equiv f\u2082) :=\nadd_equiv.ext $ map_range_comp (\u03bb i x, f\u2082 i x) (\u03bb i x, f i x) _ _ _\n\n@[simp]\nlemma map_range.add_equiv_symm (e : \u03a0 i, \u03b2\u2081 i \u2243+ \u03b2\u2082 i) :\n  (map_range.add_equiv e).symm = map_range.add_equiv (\u03bb i, (e i).symm) := rfl\n\nend map_range\n\nend dfinsupp\n\n/-! ### Product and sum lemmas for bundled morphisms.\n\nIn this section, we provide analogues of `add_monoid_hom.map_sum`, `add_monoid_hom.coe_sum`, and\n`add_monoid_hom.sum_apply` for `dfinsupp.sum` and `dfinsupp.sum_add_hom` instead of `finset.sum`.\n\nWe provide these for `add_monoid_hom`, `monoid_hom`, `ring_hom`, `add_equiv`, and `mul_equiv`.\n\nLemmas for `linear_map` and `linear_equiv` are in another file.\n-/\nsection\n\nvariables [decidable_eq \u03b9]\n\nnamespace monoid_hom\nvariables {R S : Type*}\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n\n@[simp, to_additive]\nlemma map_dfinsupp_prod [comm_monoid R] [comm_monoid S]\n  (h : R \u2192* S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R) :\n  h (f.prod g) = f.prod (\u03bb a b, h (g a b)) := h.map_prod _ _\n\n@[to_additive]\nlemma coe_dfinsupp_prod [monoid R] [comm_monoid S]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R \u2192* S) :\n  \u21d1(f.prod g) = f.prod (\u03bb a b, (g a b)) := coe_prod _ _\n\n@[simp, to_additive]\nlemma dfinsupp_prod_apply [monoid R] [comm_monoid S]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R \u2192* S) (r : R) :\n  (f.prod g) r = f.prod (\u03bb a b, (g a b) r) := finset_prod_apply _ _ _\n\nend monoid_hom\n\nnamespace ring_hom\nvariables {R S : Type*}\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n\n@[simp]\nlemma map_dfinsupp_prod [comm_semiring R] [comm_semiring S]\n  (h : R \u2192+* S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R) :\n  h (f.prod g) = f.prod (\u03bb a b, h (g a b)) := h.map_prod _ _\n\n@[simp]\nlemma map_dfinsupp_sum [non_assoc_semiring R] [non_assoc_semiring S]\n  (h : R \u2192+* S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R) :\n  h (f.sum g) = f.sum (\u03bb a b, h (g a b)) := h.map_sum _ _\n\nend ring_hom\n\nnamespace mul_equiv\nvariables {R S : Type*}\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i (x : \u03b2 i), decidable (x \u2260 0)]\n\n@[simp, to_additive]\nlemma map_dfinsupp_prod [comm_monoid R] [comm_monoid S]\n  (h : R \u2243* S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 R) :\n  h (f.prod g) = f.prod (\u03bb a b, h (g a b)) := h.map_prod _ _\n\nend mul_equiv\n\n/-! The above lemmas, repeated for `dfinsupp.sum_add_hom`. -/\n\nnamespace add_monoid_hom\nvariables {R S : Type*}\n\nopen dfinsupp\n\n@[simp]\nlemma map_dfinsupp_sum_add_hom [add_comm_monoid R] [add_comm_monoid S] [\u03a0 i, add_zero_class (\u03b2 i)]\n  (h : R \u2192+ S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ R) :\n  h (sum_add_hom g f) = sum_add_hom (\u03bb i, h.comp (g i)) f :=\ncongr_fun (comp_lift_add_hom h g) f\n\n@[simp]\nlemma dfinsupp_sum_add_hom_apply [add_zero_class R] [add_comm_monoid S] [\u03a0 i, add_zero_class (\u03b2 i)]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ R \u2192+ S) (r : R) :\n  (sum_add_hom g f) r = sum_add_hom (\u03bb i, (eval r).comp (g i)) f :=\nmap_dfinsupp_sum_add_hom (eval r) f g\n\nlemma coe_dfinsupp_sum_add_hom [add_zero_class R] [add_comm_monoid S] [\u03a0 i, add_zero_class (\u03b2 i)]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ R \u2192+ S) :\n  \u21d1(sum_add_hom g f) = sum_add_hom (\u03bb i, (coe_fn R S).comp (g i)) f :=\nmap_dfinsupp_sum_add_hom (coe_fn R S) f g\n\nend add_monoid_hom\n\nnamespace ring_hom\nvariables {R S : Type*}\n\nopen dfinsupp\n\n@[simp]\nlemma map_dfinsupp_sum_add_hom [non_assoc_semiring R] [non_assoc_semiring S]\n  [\u03a0 i, add_zero_class (\u03b2 i)] (h : R \u2192+* S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ R) :\n  h (sum_add_hom g f) = sum_add_hom (\u03bb i, h.to_add_monoid_hom.comp (g i)) f :=\nadd_monoid_hom.congr_fun (comp_lift_add_hom h.to_add_monoid_hom g) f\n\nend ring_hom\n\nnamespace add_equiv\nvariables {R S : Type*}\n\nopen dfinsupp\n\n@[simp]\nlemma map_dfinsupp_sum_add_hom [add_comm_monoid R] [add_comm_monoid S] [\u03a0 i, add_zero_class (\u03b2 i)]\n  (h : R \u2243+ S) (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192+ R) :\n  h (sum_add_hom g f) = sum_add_hom (\u03bb i, h.to_add_monoid_hom.comp (g i)) f :=\nadd_monoid_hom.congr_fun (comp_lift_add_hom h.to_add_monoid_hom g) f\n\nend add_equiv\n\nend\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/dfinsupp/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.38450218259961727}}
{"text": "import UnitConjecture.FreeAbelianGroup\nimport Experiments.AddTree\n\nsection ArraysAndLists\n\n-- a lemma about the `toArrayAux` function needed for the next theorem\nlemma List.aux_append {\u03b1 : Type _} (l : List \u03b1) : \u2200 l' : List \u03b1, List.toArrayAux l {data := l'} = {data := l' ++ l} := by\n  induction l with\n    | nil =>\n      intro\n      simp [toArrayAux]\n    | cons h t ih =>\n      intro l'\n      simp [toArrayAux, Array.push, concat]\n      rw [ih (concat l' h)]\n      simp\n      let rec concat_append (a : \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : concat l\u2081 a ++ l\u2082 = l\u2081 ++ (a :: l\u2082)  := by\n        induction l\u2081 with\n          | nil => simp [concat]\n          | cons h t ih => simp [concat]; assumption\n      apply concat_append\n\n-- converting a list to an array and then extracting the data preserves the list\n@[simp] theorem List.arraydata {A : Type _} (l : List A) : l.toArray.data = l := by\n  rw [toArray, Array.mkEmpty, List.aux_append]\n  simp\n\n-- the size of a list is preserved when coverted to an array and back\n@[simp] theorem List.arraysize {\u03b1 : Type _} (l : List \u03b1) : l.toArray.size = l.length := by\n  rw [Array.size, arraydata]\n\n-- a helper lemma for solving rewriting issues with `Fin`\ntheorem List.get_index_eq : {l l' : List \u03b1} \u2192 (hl : l = l') \u2192 (i : \u2115) \u2192 (bnd : i < l.length) \u2192 l.get \u27e8i, bnd\u27e9 = l'.get \u27e8i, hl \u25b8 bnd\u27e9\n  | _, _, rfl, _, _ => rfl\n\n-- getting the `i`th element of an array is the same as getting the `i`th element of the corresponding list\n@[simp] theorem Array.getfromlist : (l : List T)  \u2192 (i : \u2115) \u2192 (h : i < l.length) \u2192 Array.get (l.toArray) \u27e8i, Eq.substr l.arraysize h\u27e9 = l.get \u27e8i, h\u27e9\n  | List.nil, _, h => by contradiction\n  | List.cons hd tl, Nat.zero, _ => by rw [get, List.get_index_eq (List.arraydata (hd :: tl)), List.get]\n  | List.cons _ _, Nat.succ m, h => by rw [get, List.get_index_eq (List.arraydata _), List.get]\n\ntheorem List.maplength {T S : Type _} (\u03d5 : T \u2192 S) : (l : List T) \u2192 l.length = (l.map \u03d5).length\n  | List.nil => rfl\n  | List.cons h t => by rw [List.length, List.map, List.length, maplength \u03d5 t]\n\n-- the value of a function `\u03d5` on the `i`th element of a list `l` is the same as the value of the `i`th element of the list `map \u03d5 l`\ntheorem List.mapget {T S : Type _} (\u03d5 : T \u2192 S) : (l : List T) \u2192 (i : \u2115) \u2192 (h : i < List.length l) \u2192 \u03d5 (l.get \u27e8i, h\u27e9) = (l.map \u03d5).get \u27e8i, Eq.subst (maplength \u03d5 l) h\u27e9\n  | List.nil, _, _ => by contradiction\n  | List.cons _ _, Nat.zero, _ => by simp [get, map]\n  | List.cons _ _, Nat.succ _, _ => by simp [get, map]; rfl\n\ndef List.mapcomp (\u03d5 : T \u2192 S) (\u03c8 : S \u2192 R) : (l : List T) \u2192 List.map \u03c8 (List.map \u03d5 l) = List.map (\u03c8 \u2218 \u03d5) l\n  | nil => rfl\n  | cons _ l' => by simp only [map]; rw [mapcomp _ _ l']; rfl\n\ntheorem List.cons_len_eq_succ : List.length (h :: tl) = Nat.succ m \u2192 List.length tl = m := by\n  intro hyp\n  rw [length, \u2190 Nat.succ_eq_add_one] at hyp\n  injection hyp\n  assumption\n\n-- replaced the `foldl` definition\ndef List.sum {\u03b1 : Type _} [Add \u03b1] [Zero \u03b1] : List \u03b1 \u2192 \u03b1\n  | [] => 0\n  | h :: t => h + sum t\n\nend ArraysAndLists\n\nsection Defs\n\n-- iterated product\ndef pow_times (T : Type _) : \u2115 \u2192 Type _\n  | Nat.zero => Unit\n  | Nat.succ n => T \u00d7 (pow_times T n)\n\ninstance : Pow (Type _) \u2115 where\n    pow := pow_times\n\n-- iterated direct sums\ndef pow_sum (T : Type _) : \u2115 \u2192 Type _\n  | Nat.zero => Empty\n  | Nat.succ n => T \u2295 (pow_sum T n)\n\ndef zipWith {\u03b1 \u03b2 \u03b3 : Type _} (\u03d5 : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : {n : \u2115} \u2192 \u03b1 ^ n \u2192 (l : List \u03b2) \u2192 (l.length = n) \u2192 List \u03b3\n  | .zero, .(Unit.unit), .([]), rfl => []\n  | .succ _, (a, as), b :: bs, h => (\u03d5 a b) :: (zipWith \u03d5 as bs $ List.cons_len_eq_succ h)\n\ninstance : AddCommGroup Unit :=\n  {\n    add := \u03bb _ _ => Unit.unit\n    add_assoc := \u03bb _ _ _ => rfl\n    zero := Unit.unit\n    add_zero := \u03bb _ => rfl\n    zero_add := \u03bb _ => rfl\n    nsmul_zero' := by intros; rfl\n    nsmul_succ' := by intros; rfl\n    neg := \u03bb _ => Unit.unit\n    sub_eq_add_neg := by intros; rfl\n    gsmul_zero' := by intros; rfl\n    gsmul_succ' := by intros; rfl\n    gsmul_neg' := by intros; rfl\n    add_left_neg := \u03bb _ => rfl\n    add_comm := \u03bb _ _ => rfl\n  }\n\ndef \u2124powgroup : (n : \u2115) \u2192 AddCommGroup (\u2124 ^ n)\n    | Nat.zero => inferInstanceAs (AddCommGroup Unit)\n    | Nat.succ n => @DirectSum.directSum \u2124 (pow_times \u2124 n) _ (\u2124powgroup n)\n\n-- \u2124^n is a group\ninstance \u2124grp (n : \u2115) : AddCommGroup (\u2124 ^ n) := \u2124powgroup n\n\ninstance (n : \u2115) : AddCommGroup (pow_times \u2124 n) := \u2124powgroup n\n\ninstance : FreeAbelianGroup Unit Empty :=\n{\n  i := Empty.rec _\n  inducedMap := \u03bb A _ _ _ => (0 : A)\n  induced_extends := \u03bb _ => funext (Empty.rec _)\n  induced_hom := \u03bb _ _ _ => {add_dist := fun | Unit.unit, Unit.unit => by simp}\n  unique_extension := \u03bb _ _ _ _ _ => funext (fun | Unit.unit => by simp)\n}\n\ndef \u2124powfreegroup (n : \u2115) : FreeAbelianGroup (\u2124 ^ n) (pow_sum Unit n)  :=\nmatch n with\n  | Nat.zero => inferInstanceAs (FreeAbelianGroup Unit Empty)\n  | Nat.succ n => @prodFree _ _ _ (inferInstanceAs (AddCommGroup (\u2124 ^ n))) _ _ _ (\u2124powfreegroup n)\n\n-- \u2124^n is a free Abelian group\ninstance \u2124freegrp (n : \u2115) : FreeAbelianGroup (\u2124 ^ n) (pow_sum Unit n) := \u2124powfreegroup n\n\ninstance (n : \u2115) : FreeAbelianGroup (pow_times \u2124 n) (pow_sum Unit n) := \u2124powfreegroup n\n\n-- takes a list of values in `T` of length `n` and returns a function from `Unit \u2295 Unit \u2295 ... n times ... \u2295 Unit \u2192 T`\n-- mapping the elements of `pow_sum Unit n` to the corresponding elements of `T` in order\ndef unitBasisMap {T : Type _} : {n : \u2115} \u2192 (l : List T) \u2192 (l.length = n) \u2192 pow_sum Unit n \u2192 T\n  | Nat.zero, .([]), .(rfl) => Empty.rec _\n  | Nat.succ _, List.cons t l', h =>\n             \u03bb s => Sum.casesOn s\n                  (fun | Unit.unit => t)\n                  (unitBasisMap l' $ List.cons_len_eq_succ h)\n\ndef zeros : (n : \u2115) \u2192 \u2124 ^ n\n| Nat.zero => ()\n| Nat.succ n => Prod.mk (0 : \u2124) (zeros n)\n\n-- returns a basis of `\u2124^n`\ndef \u2124basis : (n : \u2115) \u2192 List (\u2124 ^ n)\n| Nat.zero => List.nil\n| Nat.succ n => List.cons (Prod.mk (1 : \u2124) (zeros n)) (\u2124basis n |>.map \u03b9\u2082)\n\n@[simp] def \u2124basislen : \u2200 m : \u2115, List.length (\u2124basis m) = m\n    | Nat.zero => rfl\n    | Nat.succ m' => by rw [\u2124basis, List.length, Nat.add_one, \u2190 List.maplength, \u2124basislen m']\n\ntheorem zero_zero : (n : \u2115) \u2192 (0 : \u2124 ^ n) = (zeros n)\n| Nat.zero => rfl\n| Nat.succ m => by rw [zeros, \u2190 zero_zero m]; rfl\n\n\nend Defs\n\n\nsection InducedFreeMap\n\n-- the unique map `\u03d5 : \u2124^n \u2192 A` taking the basis elements to the given list of values `l`\ndef inducedFreeMap {A : Type _} [AddCommGroup A] {n : \u2115} (l : List A) (h : l.length = n) : \u2124^n \u2192 A :=\nFreeAbelianGroup.inducedMap A (unitBasisMap l h)\n\n-- the above map is a group homomorphism\ninstance ind_hom {A : Type _} [AddCommGroup A] {n : \u2115} (l : List A) (h : l.length = n) : AddCommGroup.Homomorphism (inducedFreeMap l h) := FreeAbelianGroup.induced_hom A _\n\n-- a normal form for images of free group elements\ntheorem map_free_elem {A : Type _} [AddCommGroup A] : {m : \u2115} \u2192 (l : List A) \u2192 (h : l.length = m) \u2192 (x : \u2124 ^ m) \u2192 (inducedFreeMap l h) x = (List.sum $ zipWith SubNegMonoid.gsmul x l h)\n  | .zero, .([]), .(rfl), .(Unit.unit) => rfl\n  | .succ m', a :: as, h, (x, xs) => by\n      rw [inducedFreeMap, unitBasisMap, FreeAbelianGroup.inducedMap, \u2124freegrp, \u2124powfreegroup, prodFree]\n      simp only [inducedProdMap]\n      rw [FreeAbelianGroup.inducedMap, intFree]\n      simp only [zhom, Function.comp]\n      let ih := map_free_elem as (List.cons_len_eq_succ h) xs\n      rw [inducedFreeMap] at ih\n      rw [ih, zipWith, List.sum]\n\n-- a proof that the above map takes the basis elements to the elements in the list\ntheorem map_basis {A : Type _} [AddCommGroup A] : {m : \u2115} \u2192 (l : List A) \u2192 (h : l.length = m) \u2192 (List.map (inducedFreeMap l h) (\u2124basis m)) = l\n  | .zero, .([]), .(rfl) => rfl\n  | .succ m, .cons a l', h => by\n    rw [\u2124basis, List.map, map_free_elem, zipWith, List.sum, List.mapcomp]\n    let rec zero_zip_sum : (n : \u2115) \u2192 (l : List A) \u2192 (hyp : l.length = n) \u2192 (List.sum <| zipWith SubNegMonoid.gsmul (zeros n) l hyp) = (0 : A)\n      | .zero, .([]), .(rfl) => rfl\n      | .succ m, a :: l', hyp => by rw [zeros, zipWith, List.sum, SubNegMonoid.gsmul_zero', zero_zip_sum m, add_zero]\n    have ind_cons : inducedFreeMap (a :: l') h \u2218 \u03b9\u2082 = inducedFreeMap l' _ := by\n      rw [inducedFreeMap, inducedFreeMap]\n      have : (unitBasisMap l' $ List.cons_len_eq_succ h) = (unitBasisMap (List.cons a l') h) \u2218 Sum.inr := by apply funext; intro; simp [unitBasisMap]\n      rw [this, FreeAbelianGroup.induced_right]\n    rw [zero_zip_sum, ind_cons, add_zero, map_basis, SubNegMonoid.gsmul_one]\n\nend InducedFreeMap\n\n\nsection AddTreeGroup\n\nvariable (t : IndexAddTree)\nvariable {A : Type _} [AddCommGroup A] [Repr A]\nvariable {n : \u2115} (l : List A) (h : l.length = n) (hpos : n > 0) -- basisImages\n\n-- a few helper results and lemmas\n\ninstance prodrepr (A B : Type _) [Repr A] [Repr B] : Repr (A \u00d7 B) := inferInstance\n\ndef \u2124prodrepr : (n : \u2115) \u2192 Repr (\u2124 ^ n)\n    | Nat.zero => inferInstanceAs (Repr Unit)\n    | Nat.succ m => @prodrepr \u2124 (\u2124 ^ m) _ (\u2124prodrepr m)\n\ninstance (n : \u2115) : Repr (\u2124 ^ n) := \u2124prodrepr n\n\n-- some useful lemmas to deal with theorems about `Fin`\nlemma Fin.eq_of_eq_of_Nat' : {i m n : \u2115} \u2192 (h : m = n) \u2192 (hm : m > 0) \u2192 Fin.val (Fin.ofNat' i hm) = Fin.val (Fin.ofNat' i (h \u25b8 hm))\n  | _, _, _, rfl, _ => rfl\n\nlemma Fin.eq_val_bound : {m n : \u2115} \u2192 {f : Fin m} \u2192 (m = n) \u2192 (f.val < n)\n  | _, _, \u27e8_, prf\u27e9, rfl => prf\n\n\n-- taking an abstract tree to a given list of `n` elements of group `A` is equivalent to\n-- first taking it to the basis of `\u2124^n` and then apply the `inducedFreeMap`\ntheorem IndexAddTree.fold_tree_freegroup_eq : IndexAddTree.foldMap t l.toArray (by simp [h, hpos]) =\n                         (inducedFreeMap l h) (IndexAddTree.foldMap t (\u2124basis n).toArray (by simp [hpos])) := by\n  induction t with\n    | leaf _ =>\n        simp [foldMap]\n        rw [Array.getfromlist, Array.getfromlist, List.mapget (inducedFreeMap l h), List.get_index_eq (map_basis l h)]\n        apply congrArg\n        apply Fin.eq_of_val_eq; simp\n        subst h\n        apply Fin.eq_of_eq_of_Nat'; simp\n        all_goals (apply Fin.eq_val_bound; simp)\n    | negLeaf _ =>\n        simp [foldMap]\n        apply congrArg\n        rw [Array.getfromlist, Array.getfromlist, List.mapget (inducedFreeMap l h), List.get_index_eq (map_basis l h)]\n        apply congrArg\n        apply Fin.eq_of_val_eq; simp\n        subst h\n        apply Fin.eq_of_eq_of_Nat'; simp\n        all_goals (apply Fin.eq_val_bound; simp)\n    | node _ _ ihl ihr => simp [ihl, ihr, foldMap]\n    | subNode _ _ ihl ihr => simp [ihl, ihr, foldMap]\n\nend AddTreeGroup\n\n\n\n/-\n\nsection FormalExample\n\nabbrev n : \u2115 := 3\n\nopen List in\ndef \u03bd {A : Type _} [AddCommGroup A] (l : List A) (h : l.length = n) : Prop :=\n    match l, h with\n      | (cons x (cons y (cons z nil))), rfl => x + (y - z) + z - x = y\n\ntheorem valid_iff_free_basis : (\u2200 {A : Type} [AddCommGroup A], \u2200 (l : List A) (h : l.length = n), \u03bd l h) \u2194 (\u03bd (\u2124basis n) (\u2124basislen n)) := by\n  apply Iff.intro\n  \u00b7 intro hyp\n    exact hyp (\u2124basis n) (\u2124basislen n)\n  \u00b7 intro hyp\n    intro A _ l h\n    let \u03d5 := inducedFreeMap l rfl\n    have basismap := map_basis l rfl\n    sorry\n    /-\n    match l, h with\n      | List.cons a (List.cons b (List.cons c List.nil)), rfl =>\n        simp [\u03bd, \u2124basis] at hyp\n        have \u03d5map := congrArg \u03d5 hyp\n        simp at \u03d5map\n        simp only [\u03b9\u2081, \u03b9\u2082, zeros] at \u03d5map\n        simp [List.map, \u2124basis, \u03b9\u2081, \u03b9\u2082, zeros, unit_pow_list, h, map_basis] at basismap\n        let \u27e8ha, hb, hc\u27e9 := basismap\n        simp [ha, hb, hc] at \u03d5map\n    -/\n\ntheorem eqn_valid {A : Type} [AddCommGroup A] : \u2200 (l : List A) (h : l.length = n), \u03bd l h :=\n  (Iff.mpr valid_iff_free_basis) rfl\n\nend FormalExample\n\n-/\n", "meta": {"author": "siddhartha-gadgil", "repo": "Polylean", "sha": "3b411dc1dca8c84ab221cc544d1ae3bd4f83de5a", "save_path": "github-repos/lean/siddhartha-gadgil-Polylean", "path": "github-repos/lean/siddhartha-gadgil-Polylean/Polylean-3b411dc1dca8c84ab221cc544d1ae3bd4f83de5a/Experiments/GeneralAbelianGroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.3845021816726365}}
{"text": "/-\nCopyright (c) 2020 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Oliver Nash\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.invertible\nimport Mathlib.algebra.lie.skew_adjoint\nimport Mathlib.linear_algebra.matrix\nimport Mathlib.PostPort\n\nuniverses u\u2082 u_1 u_4 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Classical Lie algebras\n\nThis file is the place to find definitions and basic properties of the classical Lie algebras:\n  * A\u2097 = sl(l+1)\n  * B\u2097 \u2243 so(l+1, l) \u2243 so(2l+1)\n  * C\u2097 = sp(l)\n  * D\u2097 \u2243 so(l, l) \u2243 so(2l)\n\n## Main definitions\n\n  * `lie_algebra.special_linear.sl`\n  * `lie_algebra.symplectic.sp`\n  * `lie_algebra.orthogonal.so`\n  * `lie_algebra.orthogonal.so'`\n  * `lie_algebra.orthogonal.so_indefinite_equiv`\n  * `lie_algebra.orthogonal.type_D`\n  * `lie_algebra.orthogonal.type_B`\n  * `lie_algebra.orthogonal.type_D_equiv_so'`\n  * `lie_algebra.orthogonal.type_B_equiv_so'`\n\n## Implementation notes\n\n### Matrices or endomorphisms\n\nGiven a finite type and a commutative ring, the corresponding square matrices are equivalent to the\nendomorphisms of the corresponding finite-rank free module as Lie algebras, see `lie_equiv_matrix'`.\nWe can thus define the classical Lie algebras as Lie subalgebras either of matrices or of\nendomorphisms. We have opted for the former. At the time of writing (August 2020) it is unclear\nwhich approach should be preferred so the choice should be assumed to be somewhat arbitrary.\n\n### Diagonal quadratic form or diagonal Cartan subalgebra\n\nFor the algebras of type `B` and `D`, there are two natural definitions. For example since the\nthe `2l \u00d7 2l` matrix:\n$$\n  J = \\left[\\begin{array}{cc}\n              0_l & 1_l\\\\\\\\\n              1_l & 0_l\n            \\end{array}\\right]\n$$\ndefines a symmetric bilinear form equivalent to that defined by the identity matrix `I`, we can\ndefine the algebras of type `D` to be the Lie subalgebra of skew-adjoint matrices either for `J` or\nfor `I`. Both definitions have their advantages (in particular the `J`-skew-adjoint matrices define\na Lie algebra for which the diagonal matrices form a Cartan subalgebra) and so we provide both.\nWe thus also provide equivalences `type_D_equiv_so'`, `so_indefinite_equiv` which show the two\ndefinitions are equivalent. Similarly for the algebras of type `B`.\n\n## Tags\n\nclassical lie algebra, special linear, symplectic, orthogonal\n-/\n\nnamespace lie_algebra\n\n\n@[simp] theorem matrix_trace_commutator_zero (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (X : matrix n n R) (Y : matrix n n R) : coe_fn (matrix.trace n R R) (has_bracket.bracket X Y) = 0 := sorry\n\nnamespace special_linear\n\n\n/-- The special linear Lie algebra: square matrices of trace zero. -/\ndef sl (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] : lie_subalgebra R (matrix n n R) :=\n  lie_subalgebra.mk (submodule.carrier (linear_map.ker (matrix.trace n R R))) sorry sorry sorry sorry\n\ntheorem sl_bracket (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (A : \u21a5(sl n R)) (B : \u21a5(sl n R)) : subtype.val (has_bracket.bracket A B) =\n  matrix.mul (subtype.val A) (subtype.val B) - matrix.mul (subtype.val B) (subtype.val A) :=\n  rfl\n\n/-- It is useful to define these matrices for explicit calculations in sl n R. -/\ndef E {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) : matrix n n R :=\n  fun (i' j' : n) => ite (i = i' \u2227 j = j') 1 0\n\n@[simp] theorem E_apply_one {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) : E R i j i j = 1 :=\n  if_pos { left := rfl, right := rfl }\n\n@[simp] theorem E_apply_zero {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) (i' : n) (j' : n) (h : \u00ac(i = i' \u2227 j = j')) : E R i j i' j' = 0 :=\n  if_neg h\n\n@[simp] theorem E_diag_zero {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) (h : j \u2260 i) : coe_fn (matrix.diag n R R) (E R i j) = 0 := sorry\n\ntheorem E_trace_zero {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) (h : j \u2260 i) : coe_fn (matrix.trace n R R) (E R i j) = 0 := sorry\n\n/-- When j \u2260 i, the elementary matrices are elements of sl n R, in fact they are part of a natural\nbasis of sl n R. -/\ndef Eb {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) (h : j \u2260 i) : \u21a5(sl n R) :=\n  { val := E R i j, property := sorry }\n\n@[simp] theorem Eb_val {n : Type u_1} (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (i : n) (j : n) (h : j \u2260 i) : subtype.val (Eb R i j h) = E R i j :=\n  rfl\n\ntheorem sl_non_abelian (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] [nontrivial R] (h : 1 < fintype.card n) : \u00acis_lie_abelian \u21a5(sl n R) := sorry\n\nend special_linear\n\n\nnamespace symplectic\n\n\n/-- The matrix defining the canonical skew-symmetric bilinear form. -/\ndef J (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (l \u2295 l) (l \u2295 l) R :=\n  matrix.from_blocks 0 (-1) 1 0\n\n/-- The symplectic Lie algebra: skew-adjoint matrices with respect to the canonical skew-symmetric\nbilinear form. -/\ndef sp (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : lie_subalgebra R (matrix (l \u2295 l) (l \u2295 l) R) :=\n  skew_adjoint_matrices_lie_subalgebra (J l R)\n\nend symplectic\n\n\nnamespace orthogonal\n\n\n/-- The definite orthogonal Lie subalgebra: skew-adjoint matrices with respect to the symmetric\nbilinear form defined by the identity matrix. -/\ndef so (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] : lie_subalgebra R (matrix n n R) :=\n  skew_adjoint_matrices_lie_subalgebra 1\n\n@[simp] theorem mem_so (n : Type u_1) (R : Type u\u2082) [fintype n] [DecidableEq n] [comm_ring R] (A : matrix n n R) : A \u2208 so n R \u2194 matrix.transpose A = -A := sorry\n\n/-- The indefinite diagonal matrix with `p` 1s and `q` -1s. -/\ndef indefinite_diagonal (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] : matrix (p \u2295 q) (p \u2295 q) R :=\n  matrix.diagonal (sum.elim (fun (_x : p) => 1) fun (_x : q) => -1)\n\n/-- The indefinite orthogonal Lie subalgebra: skew-adjoint matrices with respect to the symmetric\nbilinear form defined by the indefinite diagonal matrix. -/\ndef so' (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] : lie_subalgebra R (matrix (p \u2295 q) (p \u2295 q) R) :=\n  skew_adjoint_matrices_lie_subalgebra (indefinite_diagonal p q R)\n\n/-- A matrix for transforming the indefinite diagonal bilinear form into the definite one, provided\nthe parameter `i` is a square root of -1. -/\ndef Pso (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] (i : R) : matrix (p \u2295 q) (p \u2295 q) R :=\n  matrix.diagonal (sum.elim (fun (_x : p) => 1) fun (_x : q) => i)\n\ntheorem Pso_inv (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] {i : R} (hi : i * i = -1) : Pso p q R i * Pso p q R (-i) = 1 := sorry\n\ntheorem is_unit_Pso (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] {i : R} (hi : i * i = -1) : is_unit (Pso p q R i) := sorry\n\ntheorem indefinite_diagonal_transform (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] {i : R} (hi : i * i = -1) : matrix.mul (matrix.mul (matrix.transpose (Pso p q R i)) (indefinite_diagonal p q R)) (Pso p q R i) = 1 := sorry\n\n/-- An equivalence between the indefinite and definite orthogonal Lie algebras, over a ring\ncontaining a square root of -1. -/\ndef so_indefinite_equiv (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] {i : R} (hi : i * i = -1) : equiv R \u21a5(so' p q R) \u21a5(so (p \u2295 q) R) :=\n  equiv.trans (skew_adjoint_matrices_lie_subalgebra_equiv (indefinite_diagonal p q R) (Pso p q R i) sorry)\n    (equiv.of_eq\n      (skew_adjoint_matrices_lie_subalgebra\n        (matrix.mul (matrix.mul (matrix.transpose (Pso p q R i)) (indefinite_diagonal p q R)) (Pso p q R i)))\n      (so (p \u2295 q) R) sorry)\n\ntheorem so_indefinite_equiv_apply (p : Type u_2) (q : Type u_3) (R : Type u\u2082) [fintype p] [fintype q] [DecidableEq p] [DecidableEq q] [comm_ring R] {i : R} (hi : i * i = -1) (A : \u21a5(so' p q R)) : \u2191(coe_fn (so_indefinite_equiv p q R hi) A) = matrix.mul (matrix.mul (Pso p q R i\u207b\u00b9) \u2191A) (Pso p q R i) := sorry\n\n/-- A matrix defining a canonical even-rank symmetric bilinear form.\n\nIt looks like this as a `2l x 2l` matrix of `l x l` blocks:\n\n   [ 0 1 ]  \n   [ 1 0 ]\n-/\ndef JD (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (l \u2295 l) (l \u2295 l) R :=\n  matrix.from_blocks 0 1 1 0\n\n/-- The classical Lie algebra of type D as a Lie subalgebra of matrices associated to the matrix\n`JD`. -/\ndef type_D (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : lie_subalgebra R (matrix (l \u2295 l) (l \u2295 l) R) :=\n  skew_adjoint_matrices_lie_subalgebra (JD l R)\n\n/-- A matrix transforming the bilinear form defined by the matrix `JD` into a split-signature\ndiagonal matrix.\n\nIt looks like this as a `2l x 2l` matrix of `l x l` blocks:\n\n   [ 1 -1 ]  \n   [ 1  1 ]\n-/\ndef PD (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (l \u2295 l) (l \u2295 l) R :=\n  matrix.from_blocks 1 (-1) 1 1\n\n/-- The split-signature diagonal matrix. -/\ndef S (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (l \u2295 l) (l \u2295 l) R :=\n  indefinite_diagonal l l R\n\ntheorem S_as_blocks (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : S l R = matrix.from_blocks 1 0 0 (-1) := sorry\n\ntheorem JD_transform (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix.mul (matrix.mul (matrix.transpose (PD l R)) (JD l R)) (PD l R) = bit0 1 \u2022 S l R := sorry\n\ntheorem PD_inv (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : PD l R * \u215f \u2022 matrix.transpose (PD l R) = 1 := sorry\n\ntheorem is_unit_PD (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : is_unit (PD l R) := sorry\n\n/-- An equivalence between two possible definitions of the classical Lie algebra of type D. -/\ndef type_D_equiv_so' (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : equiv R \u21a5(type_D l R) \u21a5(so' l l R) :=\n  equiv.trans (skew_adjoint_matrices_lie_subalgebra_equiv (JD l R) (PD l R) sorry)\n    (equiv.of_eq\n      (skew_adjoint_matrices_lie_subalgebra (matrix.mul (matrix.mul (matrix.transpose (PD l R)) (JD l R)) (PD l R)))\n      (so' l l R) sorry)\n\n/-- A matrix defining a canonical odd-rank symmetric bilinear form.\n\nIt looks like this as a `(2l+1) x (2l+1)` matrix of blocks:\n\n   [ 2 0 0 ]  \n   [ 0 0 1 ]  \n   [ 0 1 0 ]\n\nwhere sizes of the blocks are:\n\n   [`1 x 1` `1 x l` `1 x l`]  \n   [`l x 1` `l x l` `l x l`]  \n   [`l x 1` `l x l` `l x l`]\n-/\ndef JB (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (Unit \u2295 l \u2295 l) (Unit \u2295 l \u2295 l) R :=\n  matrix.from_blocks (bit0 1 \u2022 1) 0 0 (JD l R)\n\n/-- The classical Lie algebra of type B as a Lie subalgebra of matrices associated to the matrix\n`JB`. -/\ndef type_B (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : lie_subalgebra R (matrix (Unit \u2295 l \u2295 l) (Unit \u2295 l \u2295 l) R) :=\n  skew_adjoint_matrices_lie_subalgebra (JB l R)\n\n/-- A matrix transforming the bilinear form defined by the matrix `JB` into an\nalmost-split-signature diagonal matrix.\n\nIt looks like this as a `(2l+1) x (2l+1)` matrix of blocks:\n\n   [ 1 0  0 ]  \n   [ 0 1 -1 ]  \n   [ 0 1  1 ]\n\nwhere sizes of the blocks are:\n\n   [`1 x 1` `1 x l` `1 x l`]  \n   [`l x 1` `l x l` `l x l`]  \n   [`l x 1` `l x l` `l x l`]\n-/\ndef PB (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix (Unit \u2295 l \u2295 l) (Unit \u2295 l \u2295 l) R :=\n  matrix.from_blocks 1 0 0 (PD l R)\n\ntheorem PB_inv (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : PB l R * matrix.from_blocks 1 0 0 (PD l R\u207b\u00b9) = 1 := sorry\n\ntheorem is_unit_PB (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : is_unit (PB l R) := sorry\n\ntheorem JB_transform (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : matrix.mul (matrix.mul (matrix.transpose (PB l R)) (JB l R)) (PB l R) = bit0 1 \u2022 matrix.from_blocks 1 0 0 (S l R) := sorry\n\ntheorem indefinite_diagonal_assoc (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] : indefinite_diagonal (Unit \u2295 l) l R =\n  coe_fn (matrix.reindex_lie_equiv (equiv.symm (equiv.sum_assoc Unit l l)))\n    (matrix.from_blocks 1 0 0 (indefinite_diagonal l l R)) := sorry\n\n/-- An equivalence between two possible definitions of the classical Lie algebra of type B. -/\ndef type_B_equiv_so' (l : Type u_4) (R : Type u\u2082) [fintype l] [DecidableEq l] [comm_ring R] [invertible (bit0 1)] : equiv R \u21a5(type_B l R) \u21a5(so' (Unit \u2295 l) l R) :=\n  equiv.trans (skew_adjoint_matrices_lie_subalgebra_equiv (JB l R) (PB l R) sorry)\n    (equiv.symm\n      (equiv.trans\n        (skew_adjoint_matrices_lie_subalgebra_equiv_transpose (indefinite_diagonal (Unit \u2295 l) l R)\n          (matrix.reindex_alg_equiv (equiv.sum_assoc PUnit l l)) sorry)\n        (equiv.of_eq\n          (skew_adjoint_matrices_lie_subalgebra\n            (coe_fn (matrix.reindex_alg_equiv (equiv.sum_assoc PUnit l l)) (indefinite_diagonal (Unit \u2295 l) l R)))\n          (skew_adjoint_matrices_lie_subalgebra (matrix.mul (matrix.mul (matrix.transpose (PB l R)) (JB l R)) (PB l R)))\n          sorry)))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/lie/classical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.6406358479787609, "lm_q1q2_score": 0.38450216931871306}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n-/\nimport Mathlib.PrePort\n\nuniverses u l v w u_1 u_2 u\u2081 u\u2082 u\u2083 \n\nnamespace Mathlib\n\n/- Reserving notation. We do this so that the precedence of all of the operators\ncan be seen in one place and to prevent core notation being accidentally overloaded later.  -/\n\n/- Notation for logical operations and relations -/\n\n/- types and type constructors -/\n\n/- arithmetic operations -/\n\n/- boolean operations -/\n\n/- set operations -/\n\n/- other symbols -/\n\n/--\nThe kernel definitional equality test (t =?= s) has special support for id_delta applications.\nIt implements the following rules\n\n   1)   (id_delta t) =?= t\n   2)   t =?= (id_delta t)\n   3)   (id_delta t) =?= s  IF (unfold_of t) =?= s\n   4)   t =?= id_delta s    IF t =?= (unfold_of s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use id_delta applications to address performance problems when type checking\nlemmas generated by the equation compiler.\n-/\ndef id_delta {\u03b1 : Sort u} (a : \u03b1) : \u03b1 := a\n\n/-- Gadget for optional parameter support. -/\ndef opt_param (\u03b1 : Sort u) (default : \u03b1) := \u03b1\n\n/-- Gadget for marking output parameters in type classes. -/\ndef out_param (\u03b1 : Sort u) := \u03b1\n\n/-\n  id_rhs is an auxiliary declaration used in the equation compiler to address performance\n  issues when proving equational lemmas. The equation compiler uses it as a marker.\n-/\n\ndef id_rhs (\u03b1 : Sort u) (a : \u03b1) : \u03b1 := a\n\nnot found\n\n/-- An abbreviation for `punit.{0}`, its most common instantiation.\n    This type should be preferred over `punit` where possible to avoid\n    unnecessary universe parameters. -/\ndef unit := PUnit\n\ndef unit.star : Unit := PUnit.unit\n\n/--\nGadget for defining thunks, thunk parameters have special treatment.\nExample: given\n      def f (s : string) (t : thunk nat) : nat\nan application\n     f \"hello\" 10\n is converted into\n     f \"hello\" (\u03bb _, 10)\n-/\ndef thunk (\u03b1 : Type u) := Unit \u2192 \u03b1\n\nnot found\n\nnot found\n\ninductive empty where\n\n/--\nLogical not.\n\n`not P`, with notation `\u00ac P`, is the `Prop` which is true if and only if `P` is false. It is\ninternally represented as `P \u2192 false`, so one way to prove a goal `\u22a2 \u00ac P` is to use `intro h`,\nwhich gives you a new hypothesis `h : P` and the goal `\u22a2 false`.\n\nA hypothesis `h : \u00ac P` can be used in term mode as a function, so if `w : P` then `h w : false`.\n\nRelated mathlib tactic: `contrapose`.\n-/\ndef not (a : Prop) := a \u2192 False\n\nprefix:40 \"\u00ac\" => Mathlib.not\n\nnot found\n\n/-\nInitialize the quotient module, which effectively adds the following definitions:\n\nconstant quot {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Sort u\n\nconstant quot.mk {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) : quot r\n\nconstant quot.lift {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  (\u2200 a b : \u03b1, r a b \u2192 eq (f a) (f b)) \u2192 quot r \u2192 \u03b2\n\nconstant quot.ind {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : quot r \u2192 Prop} :\n  (\u2200 a : \u03b1, \u03b2 (quot.mk r a)) \u2192 \u2200 q : quot r, \u03b2 q\n\nAlso the reduction rule:\n\nquot.lift f _ (quot.mk a) ~~> f a\n\n-/\n\n/--\nHeterogeneous equality.\n\nIts purpose is to write down equalities between terms whose types are not definitionally equal.\nFor example, given `x : vector \u03b1 n` and `y : vector \u03b1 (0+n)`, `x = y` doesn't typecheck but `x == y` does.\n\nIf you have a goal `\u22a2 x == y`, \nyour first instinct should be to ask (either yourself, or on [zulip](https://leanprover.zulipchat.com/))\nif something has gone wrong already.\nIf you really do need to follow this route, \nyou may find the lemmas `eq_rec_heq` and `eq_mpr_heq` useful.\n-/\nnot found\n\nnot found\n\n/-- Similar to `prod`, but \u03b1 and \u03b2 can be propositions.\n   We use this type internally to automatically generate the brec_on recursor. -/\nnot found\n\n/--\nLogical and.\n\n`and P Q`, with notation `P \u2227 Q`, is the `Prop` which is true precisely when `P` and `Q` are\nboth true. \n\nTo prove a goal `\u22a2 P \u2227 Q`, you can use the tactic `split`,\nwhich gives two separate goals `\u22a2 P` and `\u22a2 Q`.\n\nGiven a hypothesis `h : P \u2227 Q`, you can use the tactic `cases h with hP hQ`\nto obtain two new hypotheses `hP : P` and `hQ : Q`. See also the `obtain` or `rcases` tactics in\nmathlib.\n-/\nnot found\n\ndef and.elim_left {a : Prop} {b : Prop} (h : a \u2227 b) : a := and.left h\n\ndef and.elim_right {a : Prop} {b : Prop} (h : a \u2227 b) : b := and.right h\n\ninfixl:50 \" = \" => Mathlib.eq\n\n/- eq basic support -/\n\ndef rfl {\u03b1 : Sort u} {a : \u03b1} : a = a := Eq.refl a\n\ntheorem eq.subst {\u03b1 : Sort u} {P : \u03b1 \u2192 Prop} {a : \u03b1} {b : \u03b1} (h\u2081 : a = b) (h\u2082 : P a) : P b :=\n  Eq._oldrec h\u2082 h\u2081\n\ninfixr:75 \" \u25b8 \" => Mathlib.eq.subst\n\ntheorem eq.trans {\u03b1 : Sort u} {a : \u03b1} {b : \u03b1} {c : \u03b1} (h\u2081 : a = b) (h\u2082 : b = c) : a = c := h\u2082 \u25b8 h\u2081\n\ntheorem eq.symm {\u03b1 : Sort u} {a : \u03b1} {b : \u03b1} (h : a = b) : b = a := h \u25b8 rfl\n\ninfixl:50 \" == \" => Mathlib.heq\n\ndef heq.rfl {\u03b1 : Sort u} {a : \u03b1} : a == a := HEq.refl a\n\ntheorem eq_of_heq {\u03b1 : Sort u} {a : \u03b1} {a' : \u03b1} (h : a == a') : a = a' :=\n  (fun (this : \u2200 (\u03b1' : Sort u) (a' : \u03b1'), a == a' \u2192 \u2200 (h\u2082 : \u03b1 = \u03b1'), eq.rec_on h\u2082 a = a') =>\n      (fun (this : eq.rec_on (Eq.refl \u03b1) a = a') => this) (this \u03b1 a' h (Eq.refl \u03b1)))\n    fun (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : a == a') => heq.rec_on h\u2081 fun (h\u2082 : \u03b1 = \u03b1) => rfl\n\n/- The following four lemmas could not be automatically generated when the\n   structures were declared, so we prove them manually here. -/\n\ntheorem prod.mk.inj {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 x\u2081 = x\u2082 \u2227 y\u2081 = y\u2082 :=\n  fun (h : (x\u2081, y\u2081) = (x\u2082, y\u2082)) =>\n    prod.no_confusion h fun (h\u2081 : x\u2081 = x\u2082) (h\u2082 : y\u2081 = y\u2082) => { left := h\u2081, right := h\u2082 }\n\ntheorem prod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 {P : Sort w} \u2192 (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n  fun (h\u2081 : (x\u2081, y\u2081) = (x\u2082, y\u2082)) (_x : Sort w) (h\u2082 : x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 _x) =>\n    prod.no_confusion h\u2081 h\u2082\n\ntheorem pprod.mk.inj {\u03b1 : Sort u} {\u03b2 : Sort v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    { fst := x\u2081, snd := y\u2081 } = { fst := x\u2082, snd := y\u2082 } \u2192 x\u2081 = x\u2082 \u2227 y\u2081 = y\u2082 :=\n  fun (h : { fst := x\u2081, snd := y\u2081 } = { fst := x\u2082, snd := y\u2082 }) =>\n    pprod.no_confusion h fun (h\u2081 : x\u2081 = x\u2082) (h\u2082 : y\u2081 = y\u2082) => { left := h\u2081, right := h\u2082 }\n\ntheorem pprod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 {P : Sort w} \u2192 (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n  fun (h\u2081 : (x\u2081, y\u2081) = (x\u2082, y\u2082)) (_x : Sort w) (h\u2082 : x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 _x) =>\n    prod.no_confusion h\u2081 h\u2082\n\ninductive sum (\u03b1 : Type u) (\u03b2 : Type v) where\n| inl : \u03b1 \u2192 sum \u03b1 \u03b2\n| inr : \u03b2 \u2192 sum \u03b1 \u03b2\n\ninductive psum (\u03b1 : Sort u) (\u03b2 : Sort v) where\n| inl : \u03b1 \u2192 psum \u03b1 \u03b2\n| inr : \u03b2 \u2192 psum \u03b1 \u03b2\n\n/--\nLogical or.\n\n`or P Q`, with notation `P \u2228 Q`, is the proposition which is true if and only if `P` or `Q` is\ntrue.\n\nTo prove a goal `\u22a2 P \u2228 Q`, if you know which alternative you want to prove,\nyou can use the tactics `left` (which gives the goal `\u22a2 P`)\nor `right` (which gives the goal `\u22a2 Q`).\n\nGiven a hypothesis `h : P \u2228 Q` and goal `\u22a2 R`,\nthe tactic `cases h` will give you two copies of the goal `\u22a2 R`,\nwith the hypothesis `h : P` in the first, and the hypothesis `h : Q` in the second.\n-/\nnot found\n\ndef or.intro_left {a : Prop} (b : Prop) (ha : a) : a \u2228 b := Or.inl ha\n\ndef or.intro_right (a : Prop) {b : Prop} (hb : b) : a \u2228 b := Or.inr hb\n\nstructure sigma {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) where\n  fst : \u03b1\n  snd : \u03b2 fst\n\nstructure psigma {\u03b1 : Sort u} (\u03b2 : \u03b1 \u2192 Sort v) where\n  fst : \u03b1\n  snd : \u03b2 fst\n\nnot found\n\n/- Remark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description. -/\n\nnot found\n\nnot found\n\ndef decidable_pred {\u03b1 : Sort u} (r : \u03b1 \u2192 Prop) := (a : \u03b1) \u2192 Decidable (r a)\n\ndef decidable_rel {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) := (a b : \u03b1) \u2192 Decidable (r a b)\n\ndef decidable_eq (\u03b1 : Sort u) := DecidableRel Eq\n\nnot found\n\nnot found\n\ninfixr:67 \" :: \" => Mathlib.list.cons\n\nnot found\n\nstructure unification_constraint where\n  \u03b1 : Type u\n  lhs : \u03b1\n  rhs : \u03b1\n\ninfixl:50 \" \u225f \" => Mathlib.unification_constraint.mk\n\ninfixl:50 \" =?= \" => Mathlib.unification_constraint.mk\n\nstructure unification_hint where\n  pattern : unification_constraint\n  constraints : List unification_constraint\n\n/- Declare builtin and reserved notation -/\n\nnot found\n\nnot found\n\nnot found\n\nnot found\n\nclass has_inv (\u03b1 : Type u) where\n  inv : \u03b1 \u2192 \u03b1\n\nnot found\n\nnot found\n\nnot found\n\nclass has_dvd (\u03b1 : Type u) where\n  dvd : \u03b1 \u2192 \u03b1 \u2192 Prop\n\nnot found\n\nnot found\n\nnot found\n\nnot found\n\nclass has_andthen (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : outParam (Type w)) where\n  andthen : \u03b1 \u2192 \u03b2 \u2192 \u03c3\n\nclass has_union (\u03b1 : Type u) where\n  union : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_inter (\u03b1 : Type u) where\n  inter : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_sdiff (\u03b1 : Type u) where\n  sdiff : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_equiv (\u03b1 : Sort u) where\n  equiv : \u03b1 \u2192 \u03b1 \u2192 Prop\n\nclass has_subset (\u03b1 : Type u) where\n  subset : \u03b1 \u2192 \u03b1 \u2192 Prop\n\n/- Type classes has_emptyc and has_insert are\nclass has_ssubset (\u03b1 : Type u) where\n  ssubset : \u03b1 \u2192 \u03b1 \u2192 Prop\n\n   used to implement polymorphic notation for collections.\n   Example: {a, b, c}. -/\n\nclass has_emptyc (\u03b1 : Type u) where\n  emptyc : \u03b1\n\nclass has_insert (\u03b1 : outParam (Type u)) (\u03b3 : Type v) where\n  insert : \u03b1 \u2192 \u03b3 \u2192 \u03b3\n\n/- Type class used to implement the notation { a \u2208 c | p a } -/\n\nclass has_singleton (\u03b1 : outParam (Type u)) (\u03b2 : Type v) where\n  singleton : \u03b1 \u2192 \u03b2\n\nclass has_sep (\u03b1 : outParam (Type u)) (\u03b3 : Type v) where\n  sep : (\u03b1 \u2192 Prop) \u2192 \u03b3 \u2192 \u03b3\n\n/- Type class for set-like membership -/\n\nclass has_mem (\u03b1 : outParam (Type u)) (\u03b3 : Type v) where\n  mem : \u03b1 \u2192 \u03b3 \u2192 Prop\n\nclass has_pow (\u03b1 : Type u) (\u03b2 : Type v) where\n  pow : \u03b1 \u2192 \u03b2 \u2192 \u03b1\n\ninfixl:50 \" \u2208 \" => Mathlib.has_mem.mem\n\ninfixl:65 \" + \" => Mathlib.has_add.add\n\ninfixl:70 \" * \" => Mathlib.has_mul.mul\n\ninfixl:65 \" - \" => Mathlib.has_sub.sub\n\ninfixl:70 \" / \" => Mathlib.has_div.div\n\ninfixl:50 \" \u2223 \" => Mathlib.has_dvd.dvd\n\ninfixl:70 \" % \" => Mathlib.has_mod.mod\n\nprefix:75 \"-\" => Mathlib.has_neg.neg\n\ninfixl:50 \" <= \" => Mathlib.has_le.le\n\ninfixl:50 \" \u2264 \" => Mathlib.has_le.le\n\ninfixl:50 \" < \" => Mathlib.has_lt.lt\n\ninfixl:65 \" ++ \" => Mathlib.has_append.append\n\ninfixl:1 \"; \" => Mathlib.has_andthen.andthen\n\nnotation:1024 \"\u2205\" => Mathlib.has_emptyc.emptyc\n\ninfixl:65 \" \u222a \" => Mathlib.has_union.union\n\ninfixl:70 \" \u2229 \" => Mathlib.has_inter.inter\n\ninfixl:50 \" \u2286 \" => Mathlib.has_subset.subset\n\ninfixl:50 \" \u2282 \" => Mathlib.has_ssubset.ssubset\n\ninfixl:70 \" \\ \" => Mathlib.has_sdiff.sdiff\n\ninfixl:50 \" \u2248 \" => Mathlib.has_equiv.equiv\n\ninfixr:80 \" ^ \" => Mathlib.has_pow.pow\n\ndef ge {\u03b1 : Type u} [HasLessEq \u03b1] (a : \u03b1) (b : \u03b1) := b \u2264 a\n\ndef gt {\u03b1 : Type u} [HasLess \u03b1] (a : \u03b1) (b : \u03b1) := b < a\n\ninfixl:50 \" >= \" => Mathlib.ge\n\ninfixl:50 \" \u2265 \" => Mathlib.ge\n\ninfixl:50 \" > \" => Mathlib.gt\n\ndef superset {\u03b1 : Type u} [has_subset \u03b1] (a : \u03b1) (b : \u03b1) := b \u2286 a\n\ndef ssuperset {\u03b1 : Type u} [has_ssubset \u03b1] (a : \u03b1) (b : \u03b1) := b \u2282 a\n\ninfixl:50 \" \u2287 \" => Mathlib.superset\n\ninfixl:50 \" \u2283 \" => Mathlib.ssuperset\n\ndef bit0 {\u03b1 : Type u} [s : Add \u03b1] (a : \u03b1) : \u03b1 := a + a\n\ndef bit1 {\u03b1 : Type u} [s\u2081 : HasOne \u03b1] [s\u2082 : Add \u03b1] (a : \u03b1) : \u03b1 := bit0 a + 1\n\nclass is_lawful_singleton (\u03b1 : Type u) (\u03b2 : Type v) [has_emptyc \u03b2] [has_insert \u03b1 \u03b2]\n    [has_singleton \u03b1 \u03b2]\n    where\n  insert_emptyc_eq : \u2200 (x : \u03b1), insert x \u2205 = has_singleton.singleton x\n\n/- nat basic instances -/\n\nnamespace nat\n\n\nprotected def add : Nat \u2192 Nat \u2192 Nat := Nat.add\n\nend nat\n\n\nprotected instance nat.has_zero : HasZero Nat := { zero := 0 }\n\nprotected instance nat.has_one : HasOne Nat := { one := 1 }\n\nprotected instance nat.has_add : Add Nat := { add := Nat.add }\n\ndef std.priority.default : Nat := bit0 (bit0 (bit0 (bit1 (bit0 (bit1 (bit1 (bit1 (bit1 1))))))))\n\ndef std.priority.max : Nat :=\n  bit1\n    (bit1\n      (bit1\n        (bit1\n          (bit1\n            (bit1\n              (bit1\n                (bit1\n                  (bit1\n                    (bit1\n                      (bit1\n                        (bit1\n                          (bit1\n                            (bit1\n                              (bit1\n                                (bit1\n                                  (bit1\n                                    (bit1\n                                      (bit1\n                                        (bit1\n                                          (bit1\n                                            (bit1\n                                              (bit1\n                                                (bit1\n                                                  (bit1\n                                                    (bit1\n                                                      (bit1\n                                                        (bit1\n                                                          (bit1\n                                                            (bit1\n                                                              (bit1 1))))))))))))))))))))))))))))))\n\nnamespace nat\n\n\nend nat\n\n\nprotected def nat.prio : Nat := std.priority.default + bit0 (bit0 (bit1 (bit0 (bit0 (bit1 1)))))\n\n/-\n  Global declarations of right binding strength\n\n  If a module reassigns these, it will be incompatible with other modules that adhere to these\n  conventions.\n\n  When hovering over a symbol, use \"C-c C-k\" to see how to input it.\n-/\n\ndef std.prec.max : Nat := bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 1)))))))))\n\ndef std.prec.arrow : Nat := bit1 (bit0 (bit0 (bit1 1)))\n\n/-\nThe next def is \"max + 10\". It can be used e.g. for postfix operations that should\nbe stronger than application.\n-/\n\ndef std.prec.max_plus : Nat := std.prec.max + bit0 (bit1 (bit0 1))\n\npostfix:0 \"\u207b\u00b9\" => Mathlib.has_inv.inv\n\ninfixr:35 \" \u00d7 \" => Mathlib.prod\n\n-- notation for n-ary tuples\n\n/- sizeof -/\n\nnot found\n\ndef sizeof {\u03b1 : Sort u} [s : SizeOf \u03b1] : \u03b1 \u2192 Nat := has_sizeof.sizeof\n\n/-\nDeclare sizeof instances and lemmas for types declared before has_sizeof.\nFrom now on, the inductive compiler will automatically generate sizeof instances and lemmas.\n-/\n\n/- Every type `\u03b1` has a default has_sizeof instance that just returns 0 for every element of `\u03b1` -/\n\nprotected def default.sizeof (\u03b1 : Sort u) : \u03b1 \u2192 Nat := sorry\n\nprotected instance default_has_sizeof (\u03b1 : Sort u) : SizeOf \u03b1 := { sizeOf := default.sizeof \u03b1 }\n\nprotected def nat.sizeof : Nat \u2192 Nat := sorry\n\nprotected instance nat.has_sizeof : SizeOf Nat := { sizeOf := nat.sizeof }\n\nprotected def prod.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : \u03b1 \u00d7 \u03b2 \u2192 Nat := sorry\n\nprotected instance prod.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] :\n    SizeOf (\u03b1 \u00d7 \u03b2) :=\n  { sizeOf := prod.sizeof }\n\nprotected def sum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : sum \u03b1 \u03b2 \u2192 Nat := sorry\n\nprotected instance sum.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] :\n    SizeOf (sum \u03b1 \u03b2) :=\n  { sizeOf := sum.sizeof }\n\nprotected def psum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : psum \u03b1 \u03b2 \u2192 Nat := sorry\n\nprotected instance psum.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] :\n    SizeOf (psum \u03b1 \u03b2) :=\n  { sizeOf := psum.sizeof }\n\nprotected def sigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] :\n    sigma \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance sigma.has_sizeof (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1]\n    [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : SizeOf (sigma \u03b2) :=\n  { sizeOf := sigma.sizeof }\n\nprotected def psigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] :\n    psigma \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance psigma.has_sizeof (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1]\n    [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : SizeOf (psigma \u03b2) :=\n  { sizeOf := psigma.sizeof }\n\nprotected def punit.sizeof : PUnit \u2192 Nat := sorry\n\nprotected instance punit.has_sizeof : SizeOf PUnit := { sizeOf := punit.sizeof }\n\nprotected def bool.sizeof : Bool \u2192 Nat := sorry\n\nprotected instance bool.has_sizeof : SizeOf Bool := { sizeOf := bool.sizeof }\n\nprotected def option.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : Option \u03b1 \u2192 Nat := sorry\n\nprotected instance option.has_sizeof (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (Option \u03b1) :=\n  { sizeOf := option.sizeof }\n\nprotected def list.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : List \u03b1 \u2192 Nat := sorry\n\nprotected instance list.has_sizeof (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (List \u03b1) :=\n  { sizeOf := list.sizeof }\n\nprotected def subtype.sizeof {\u03b1 : Type u} [SizeOf \u03b1] {p : \u03b1 \u2192 Prop} : Subtype p \u2192 Nat := sorry\n\nprotected instance subtype.has_sizeof {\u03b1 : Type u} [SizeOf \u03b1] (p : \u03b1 \u2192 Prop) : SizeOf (Subtype p) :=\n  { sizeOf := subtype.sizeof }\n\ntheorem nat_add_zero (n : Nat) : n + 0 = n := rfl\n\n/- Combinator calculus -/\n\nnamespace combinator\n\n\ndef I {\u03b1 : Type u\u2081} (a : \u03b1) : \u03b1 := a\n\ndef K {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} (a : \u03b1) (b : \u03b2) : \u03b1 := a\n\nend combinator\n\n\ndef combinator.S {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} {\u03b3 : Type u\u2083} (x : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (y : \u03b1 \u2192 \u03b2) (z : \u03b1) :\n    \u03b3 :=\n  x z (y z)\n\n/-- Auxiliary datatype for #[ ... ] notation.\n    #[1, 2, 3, 4] is notation for\n\n    bin_tree.node\n      (bin_tree.node (bin_tree.leaf 1) (bin_tree.leaf 2))\n      (bin_tree.node (bin_tree.leaf 3) (bin_tree.leaf 4))\n\n    We use this notation to input long sequences without exhausting the system stack space.\n    Later, we define a coercion from `bin_tree` into `list`.\n-/\ninductive bin_tree (\u03b1 : Type u) where\n| empty : bin_tree \u03b1\n| leaf : \u03b1 \u2192 bin_tree \u03b1\n| node : bin_tree \u03b1 \u2192 bin_tree \u03b1 \u2192 bin_tree \u03b1\n\n/- Basic unification hints -/\n\ndef add_succ_defeq_succ_add_hint (x : Nat) (y : Nat) (z : Nat) : unification_hint :=\n  unification_hint.mk (x + Nat.succ y =?= Nat.succ z) [z =?= x + y]\n\n/-- Like `by apply_instance`, but not dependent on the tactic framework. -/\ndef infer_instance {\u03b1 : Sort u} [i : \u03b1] : \u03b1 := i\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/core_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.3844277387121631}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\n\nimport category_theory.category\nimport abelian\nimport abelian_SEMF\nimport exact\nimport tactic.diagram_chase\n\nopen category_theory\nopen category_theory.limits\nopen category_theory.abelian\nopen category_theory.abelian.pseudoelements\n\nnamespace category_theory.abelian.diagram_lemmas.snake\n\nuniverses v u\nvariables (V : Type u) [\ud835\udc9e : category.{v} V] [abelian.{v} V]\ninclude \ud835\udc9e\n\nstructure snake_diagram :=\n(A B C D E F G H I J K L : V)\n(\u03b1 : A \u27f6 B) (\u03b2 : B \u27f6 C) (\u03b3 : A \u27f6 D) (\u03b4 : B \u27f6 E) (\u03b5 : C \u27f6 F)\n(\u03b6 : D \u27f6 E) (\u03b7 : E \u27f6 F) (\u03b8 : D \u27f6 G) (\u03ba : E \u27f6 H) (\u03bc : F \u27f6 I)\n(\u03bd : G \u27f6 H) (\u03be : H \u27f6 I) (\u03c0 : G \u27f6 J) (\u03c1 : H \u27f6 K) (\u03c3 : I \u27f6 L)\n(\u03c4 : J \u27f6 K) (\u03c6 : K \u27f6 L)\n\nstructure exact_snake_diagram extends snake_diagram.{v} V :=\n(comm\u2081 : \u03b1 \u226b \u03b4 = \u03b3 \u226b \u03b6) (comm\u2082 : \u03b2 \u226b \u03b5 = \u03b4 \u226b \u03b7) (comm\u2083 : \u03b6 \u226b \u03ba = \u03b8 \u226b \u03bd)\n(comm\u2084 : \u03b7 \u226b \u03bc = \u03ba \u226b \u03be) (comm\u2085 : \u03bd \u226b \u03c1 = \u03c0 \u226b \u03c4) (comm\u2086 : \u03be \u226b \u03c3 = \u03c1 \u226b \u03c6)\n--(\u03b1\u03b2 : exact \u03b1 \u03b2)\n(\u03b6\u03b7 : exact \u03b6 \u03b7) (\u03bd\u03be : exact \u03bd \u03be)\n--(\u03c4\u03c6 : exact \u03c4 \u03c6)\n(\u03b3\u03b8 : exact \u03b3 \u03b8)\n(\u03b8\u03c0 : exact \u03b8 \u03c0) (\u03b4\u03ba : exact \u03b4 \u03ba) (\u03ba\u03c1 : exact \u03ba \u03c1) (\u03b5\u03bc : exact \u03b5 \u03bc) (\u03bc\u03c3 : exact \u03bc \u03c3)\n\nattribute [chase] exact_snake_diagram.comm\u2081 exact_snake_diagram.comm\u2082 exact_snake_diagram.comm\u2083\nattribute [chase] exact_snake_diagram.comm\u2084 exact_snake_diagram.comm\u2085 exact_snake_diagram.comm\u2086\n--attribute [chase] exact_snake_diagram.\u03b1\u03b2\nattribute [chase] exact_snake_diagram.\u03b6\u03b7 exact_snake_diagram.\u03bd\u03be\n--attribute [chase] exact_snake_diagram.\u03c4\u03c6\nattribute [chase] exact_snake_diagram.\u03b3\u03b8 exact_snake_diagram.\u03b8\u03c0\nattribute [chase] exact_snake_diagram.\u03b4\u03ba exact_snake_diagram.\u03ba\u03c1 exact_snake_diagram.\u03b5\u03bc\nattribute [chase] exact_snake_diagram.\u03bc\u03c3\n\nvariable {V}\n\n/- We need to do this instead of `attribute [chase] snake_digram.\u03b1` because the projection\n   has the wrong signature: It takes a `snake_diagram` instead of an `exact_snake_diagram`. -/\n@[chase] abbreviation exact_snake_\u03b1 (d : exact_snake_diagram.{v} V) := d.\u03b1\n@[chase] abbreviation exact_snake_\u03b2 (d : exact_snake_diagram.{v} V) := d.\u03b2\n@[chase] abbreviation exact_snake_\u03b3 (d : exact_snake_diagram.{v} V) := d.\u03b3\n@[chase] abbreviation exact_snake_\u03b4 (d : exact_snake_diagram.{v} V) := d.\u03b4\n@[chase] abbreviation exact_snake_\u03b5 (d : exact_snake_diagram.{v} V) := d.\u03b5\n@[chase] abbreviation exact_snake_\u03b6 (d : exact_snake_diagram.{v} V) := d.\u03b6\n@[chase] abbreviation exact_snake_\u03b7 (d : exact_snake_diagram.{v} V) := d.\u03b7\n@[chase] abbreviation exact_snake_\u03b8 (d : exact_snake_diagram.{v} V) := d.\u03b8\n@[chase] abbreviation exact_snake_\u03ba (d : exact_snake_diagram.{v} V) := d.\u03ba\n@[chase] abbreviation exact_snake_\u03bc (d : exact_snake_diagram.{v} V) := d.\u03bc\n@[chase] abbreviation exact_snake_\u03bd (d : exact_snake_diagram.{v} V) := d.\u03bd\n@[chase] abbreviation exact_snake_\u03be (d : exact_snake_diagram.{v} V) := d.\u03be\n@[chase] abbreviation exact_snake_\u03c0 (d : exact_snake_diagram.{v} V) := d.\u03c0\n@[chase] abbreviation exact_snake_\u03c1 (d : exact_snake_diagram.{v} V) := d.\u03c1\n@[chase] abbreviation exact_snake_\u03c3 (d : exact_snake_diagram.{v} V) := d.\u03c3\n@[chase] abbreviation exact_snake_\u03c4 (d : exact_snake_diagram.{v} V) := d.\u03c4\n@[chase] abbreviation exact_snake_\u03c6 (d : exact_snake_diagram.{v} V) := d.\u03c6\n\n--set_option profiler true\n\nnamespace restricted\n\nnamespace internal\nvariable (d : exact_snake_diagram.{v} V)\nvariables [mono d.\u03b6] [epi d.\u03be] [epi d.\u03b7] [mono d.\u03bd] [mono d.\u03b5] [epi d.\u03c0]\n\nlocal attribute [instance] object_to_sort\nlocal attribute [instance] hom_to_fun\n\nabbreviation Z : V := pullback d.\u03b5 d.\u03b7\nabbreviation \u0394 : (Z d) \u27f6 d.C := pullback.fst\nabbreviation \u0393 : (Z d) \u27f6 d.E := pullback.snd\n@[chase] lemma comm\u2087 : (\u0394 d) \u226b d.\u03b5 = (\u0393 d) \u226b d.\u03b7 := pullback.condition\n\nabbreviation Y : V := pushout d.\u03c0 d.\u03bd\nabbreviation \u039e : d.J \u27f6 (Y d) := pushout.inl\nabbreviation \u039b : d.H \u27f6 (Y d) := pushout.inr\n@[chase] lemma comm\u2088 : d.\u03c0 \u226b (\u039e d) = d.\u03bd \u226b (\u039b d) := pushout.condition\n\nabbreviation X : V := kernel (\u0394 d)\nabbreviation S : (X d) \u27f6 (Z d) := kernel.\u03b9 (\u0394 d)\n\ndef W := cokernel (\u039e d)\ndef \u03a5 := cokernel.\u03c0 (\u039e d)\n\n@[chase] lemma S\u0394 : exact (S d) (\u0394 d) := kernel_exact _\n@[chase] lemma \u039e\u03a5 : exact (\u039e d) (\u03a5 d) := cokernel_exact _\n\nlemma S\u0393\u03b7 : (S d \u226b \u0393 d) \u226b d.\u03b7 = 0 :=\nbegin\n  rw category.assoc,\n  ext,\n  simp only [comp_apply],\n  commutativity at d,\nend\n\ndef H\u03a8 := kernel_fork.is_limit.lift' (kernel_of_mono_exact _ _ d.\u03b6\u03b7) (S d \u226b \u0393 d) (S\u0393\u03b7 d)\ndef \u03a8 : (X d) \u27f6 d.D := (H\u03a8 d).1\n@[chase] lemma h\u03a8 : \u03a8 d \u226b d.\u03b6 = S d \u226b \u0393 d := (H\u03a8 d).2\n\nlemma \u03bd\u039b\u03a5 : d.\u03bd \u226b (\u039b d) \u226b (\u03a5 d) = 0 :=\nbegin\n  ext,\n  simp only [comp_apply],\n  commutativity at d,\nend\n\ndef H\u03a9 := cokernel_cofork.is_colimit.desc' (cokernel_of_epi_exact _ _ d.\u03bd\u03be) (\u039b d \u226b \u03a5 d) (\u03bd\u039b\u03a5 d)\ndef \u03a9 : d.I \u27f6 (W d) := (H\u03a9 d).1\n@[chase] lemma h\u03a9 : d.\u03be \u226b (\u03a9 d) = (\u039b d) \u226b (\u03a5 d) := (H\u03a9 d).2\n\nlemma S\u0393\u03ba\u039b : S d \u226b \u0393 d \u226b d.\u03ba \u226b \u039b d = 0 :=\nbegin\n  ext,\n  simp only [comp_apply],\n  commutativity at d,\nend\n\ndef \u0394cone : cokernel_cofork (S d) := cokernel_cofork.of_\u03c0 (\u0394 d) $ kernel.condition _\ndef \u0394lim : is_colimit (\u0394cone d) := epi_is_cokernel_of_kernel\n  (limit.cone (parallel_pair (\u0394 d) 0)) (limit.is_limit _)\n\ndef H\u03c7 := cokernel_cofork.is_colimit.desc' (\u0394lim d) _ (S\u0393\u03ba\u039b d)\ndef \u03c7 : d.C \u27f6 (Y d) := (H\u03c7 d).1\n@[chase] lemma h\u03c7 : (\u0394 d) \u226b (\u03c7 d) = (\u0393 d) \u226b d.\u03ba \u226b (\u039b d) := (H\u03c7 d).2\n\nlemma \u03a5\u03c7 : (\u03c7 d) \u226b (\u03a5 d) = 0 :=\nbegin\n  apply (preadditive.cancel_zero_iff_epi (\u0394 d)).1 (by apply_instance),\n  ext,\n  simp only [comp_apply],\n  commutativity at d,\nend\n\ndef \u039econe : kernel_fork (\u03a5 d) := kernel_fork.of_\u03b9 (\u039e d) $ cokernel.condition _\ndef \u039elim : is_limit (\u039econe d) := mono_is_kernel_of_cokernel\n  (colimit.cocone (parallel_pair (\u039e d) 0)) (colimit.is_colimit _)\n\ndef H\u03c9 := kernel_fork.is_limit.lift' (\u039elim d) _ (\u03a5\u03c7 d)\ndef \u03c9 : d.C \u27f6 d.J := (H\u03c9 d).1\n@[chase] lemma h\u03c9 : (\u03c9 d) \u226b (\u039e d) = (\u03c7 d) := (H\u03c9 d).2\n\nlemma \u03c9_char (c : d.C) (e : d.E) (g : d.G) (h\u2081 : d.\u03b7 e = d.\u03b5 c) (h\u2082 : d.\u03bd g = d.\u03ba e) :\n  d.\u03c0 g = (\u03c9 d) c :=\nbegin\n  obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := pseudo_pullback h\u2081.symm,\n  change (\u0394 d : Z d \u27f6 d.C) z = c at hz\u2081,\n  change (\u0393 d : Z d \u2192 d.E) z = e at hz\u2082,\n  apply pseudo_injective_of_mono (\u039e d),\n  commutativity at d,\nend\n\ntheorem \u03b2\u03c9 : exact d.\u03b2 (\u03c9 d) :=\nbegin\n  apply exact_of_pseudo_exact,\n  split,\n  { intro b,\n    chase b using [d.\u03b2] with c,\n    chase b using [d.\u03b4] with e,\n    have h\u2081 : d.\u03b7 e = d.\u03b5 c, by commutativity at d,\n    chase e using [d.\u03ba, d.\u03bd] with h g at d,\n    have : d.\u03bd g = 0, by commutativity at d,\n    have : g = 0, -- This should be automatic!\n    { apply pseudo_injective_of_mono d.\u03bd,\n      commutativity at d, },\n    have h\u2082 : d.\u03bd g = d.\u03ba e, by commutativity,\n    have := \u03c9_char d _ _ _ h\u2081 h\u2082,\n    commutativity, },\n  { intros c hc,\n    chase c using [d.\u03b5, d.\u03b7, d.\u03ba, d.\u03bd] with f e h g at d,\n    have := \u03c9_char _ c e g (by commutativity) (by commutativity),\n    chase g using [d.\u03b8] with d' at d,\n    have : d.\u03ba (d.\u03b6 d') = d.\u03ba e, by commutativity at d,\n    obtain \u27e8z, hz\u2081, hz\u2082\u27e9 := sub_of_eq_image _ _ _ this.symm,\n    chase z using [d.\u03b4] with b at d,\n    have : d.\u03b7 (d.\u03b6 d') = 0, by commutativity at d,\n    have := hz\u2082 _ _ this,\n    use b,\n    apply pseudo_injective_of_mono d.\u03b5, -- This should also be automatic\n    commutativity at d, }\nend\n\ntheorem \u03c9\u03c4 : exact (\u03c9 d) d.\u03c4 :=\nbegin\n  apply exact_of_pseudo_exact,\n  split,\n  { intro c,\n    chase c using [d.\u03b5, d.\u03b7, d.\u03ba, d.\u03bd] with f e h g at d,\n    have := \u03c9_char _ c e g (by commutativity) (by commutativity),\n    commutativity at d, },\n  { intros j hj,\n    chase j using [d.\u03c0, d.\u03bd, d.\u03ba, d.\u03b7, d.\u03b5] with g h e f c at d,\n    have := \u03c9_char _ c e g (by commutativity) (by commutativity),\n    exact \u27e8c, by commutativity\u27e9 }\nend\n\nend internal\n\nvariable (d : exact_snake_diagram.{v} V)\nvariables [mono d.\u03b6] [epi d.\u03be] [epi d.\u03b7] [mono d.\u03bd] [mono d.\u03b5] [epi d.\u03c0]\n\ndef connecting_morphism : d.C \u27f6 d.J :=\ninternal.\u03c9 d\n\ntheorem exact\u2081 : exact d.\u03b2 (connecting_morphism d) :=\ninternal.\u03b2\u03c9 d\n\ntheorem exact\u2082 : exact (connecting_morphism d) d.\u03c4 :=\ninternal.\u03c9\u03c4 d\n\nend restricted\n\nnamespace internal\nvariable (d : exact_snake_diagram.{v} V)\nvariables [mono d.\u03bd] [epi d.\u03b7] [mono d.\u03b4] [mono d.\u03b5] [epi d.\u03c0] [epi d.\u03c1]\n\nlocal attribute [instance] object_to_sort\nlocal attribute [instance] hom_to_fun\n\nabbreviation Z : V := kernel (cokernel.\u03c0 d.\u03b6)\n@[chase] abbreviation \u03b6\u2081 : d.D \u27f6 (Z d) := factor_thru_image d.\u03b6\n@[chase] abbreviation \u03b6\u2082 : (Z d) \u27f6 d.E := kernel.\u03b9 (cokernel.\u03c0 d.\u03b6)\n\ninstance strong_epi_\u03b6 : strong_epi (\u03b6\u2081 d) :=\nstrong_epi_of_epi _\n\n@[chase] abbreviation \u0393 : (Z d) \u27f6 d.G := diag_lift $\n  show d.\u03b8 \u226b (\ud835\udfd9 d.G \u226b d.\u03bd) = (\u03b6\u2081 d \u226b \u03b6\u2082 d) \u226b d.\u03ba, from\n  begin\n    rw abelian.image.fac,\n    rw category.id_comp,\n    exact d.comm\u2083.symm,\n  end\n\n@[chase] lemma h\u0393\u2081 : (\u03b6\u2081 d) \u226b (\u0393 d) = d.\u03b8 :=\nbegin\n  rw diag_lift_fac_left,\n  rw category.comp_id\nend\n\n@[chase] lemma h\u0393\u2082 : (\u03b6\u2082 d) \u226b d.\u03ba = (\u0393 d) \u226b d.\u03bd :=\neq.symm $ diag_lift_fac_right _\n\nabbreviation W : V := kernel (cokernel.\u03c0 d.\u03be)\nabbreviation \u03be\u2081 : d.H \u27f6 (W d) := factor_thru_image d.\u03be\nabbreviation \u03be\u2082 : (W d) \u27f6 d.I := kernel.\u03b9 (cokernel.\u03c0 d.\u03be)\n\ninstance strong_epi_\u03be : strong_epi (\u03be\u2081 d) :=\nstrong_epi_of_epi _\n\ninstance strong_epi_\u03b7 : strong_epi d.to_snake_diagram.\u03b7 :=\nstrong_epi_of_epi _\n\n@[chase] def \u0394 : d.F \u27f6 (W d) :=\ndiag_lift $ show d.\u03ba \u226b (\u03be\u2081 d \u226b \u03be\u2082 d) = (d.\u03b7 \u226b \ud835\udfd9 d.F) \u226b d.\u03bc, from\nbegin\n  erw abelian.image.fac d.\u03be,\n  rw category.assoc,\n  rw category.id_comp,\n  exact d.comm\u2084.symm\nend\n\n@[chase] lemma h\u0394\u2081 : d.\u03b7 \u226b (\u0394 d) = d.\u03ba \u226b (\u03be\u2081 d) :=\ndiag_lift_fac_left _\n\n@[chase] lemma h\u0394\u2082 : d.\u03bc = (\u0394 d) \u226b (\u03be\u2082 d) :=\nbegin\n  rw \u2190category.id_comp d.\u03bc,\n  exact (diag_lift_fac_right _).symm\nend\n\nabbreviation V' : V := kernel (\u0393 d)\nabbreviation \u039b : (V' d) \u27f6 (Z d) := kernel.\u03b9 (\u0393 d)\n\nabbreviation U : V := kernel (\u0394 d)\nabbreviation \u039e : (U d) \u27f6 d.F := kernel.\u03b9 (\u0394 d)\n\nabbreviation T : V := cokernel (\u0393 d)\nabbreviation S : d.G \u27f6 (T d) := cokernel.\u03c0 (\u0393 d)\n\nabbreviation S' : V := cokernel (\u0394 d)\nabbreviation \u03a5 : (W d) \u27f6 (S' d) := cokernel.\u03c0 (\u0394 d)\n\nset_option trace.app_builder true\n--set_option pp.all true\n\nabbreviation H\u03b1\u2081 := kernel.lift' (\u0393 d) (d.\u03b3 \u226b (\u03b6\u2081 d)) $\n  by rw [category.assoc, h\u0393\u2081, d.\u03b3\u03b8.1]\nabbreviation \u03b1\u2081 : d.A \u27f6 (V' d) := (H\u03b1\u2081 d).1\n@[chase] lemma h\u03b1\u2081 : \u03b1\u2081 d \u226b \u039b d = d.\u03b3 \u226b (\u03b6\u2081 d) := (H\u03b1\u2081 d).2\n\nabbreviation H\u03b1\u2082 := kernel_fork.is_limit.lift' (kernel_of_mono_exact _ _ d.\u03b4\u03ba) (\u039b d \u226b \u03b6\u2082 d) $\n  by rw [category.assoc, h\u0393\u2082, \u2190category.assoc, kernel.condition, has_zero_morphisms.zero_comp]\nabbreviation \u03b1\u2082 : V' d \u27f6 d.B := (H\u03b1\u2082 d).1\n@[chase] lemma h\u03b1\u2082 : \u03b1\u2082 d \u226b d.\u03b4 = \u039b d \u226b \u03b6\u2082 d := (H\u03b1\u2082 d).2\n\nabbreviation H\u03b2\u2081 := kernel.lift' (\u0394 d) (d.\u03b4 \u226b d.\u03b7) $\n  by rw [category.assoc, h\u0394\u2081, \u2190category.assoc, d.\u03b4\u03ba.1, has_zero_morphisms.zero_comp]\nabbreviation \u03b2\u2081 : d.B \u27f6 (U d) := (H\u03b2\u2081 d).1\n@[chase] lemma h\u03b2\u2081 : \u03b2\u2081 d \u226b \u039e d = d.\u03b4 \u226b d.\u03b7 := (H\u03b2\u2081 d).2\n\nabbreviation H\u03b2\u2082 := kernel_fork.is_limit.lift' (kernel_of_mono_exact _ _ d.\u03b5\u03bc) (\u039e d) $\n  by rw [h\u0394\u2082, \u2190category.assoc, kernel.condition, has_zero_morphisms.zero_comp]\nabbreviation \u03b2\u2082 : U d \u27f6 d.C := (H\u03b2\u2082 d).1\n@[chase] lemma h\u03b2\u2082 : \u03b2\u2082 d \u226b d.\u03b5 = \u039e d := (H\u03b2\u2082 d).2\n\nlemma \u03b2\u2081\u03b2\u2082 : \u03b2\u2081 d \u226b \u03b2\u2082 d = d.\u03b2 :=\nbegin\n  apply fork.is_limit.hom_ext (kernel_of_mono_exact _ _ d.\u03b5\u03bc),\n  erw [category.assoc, h\u03b2\u2082, h\u03b2\u2081, d.comm\u2082]\nend\n\nabbreviation H\u03c4\u2081 := cokernel_cofork.is_colimit.desc' (cokernel_of_epi_exact _ _ d.\u03b8\u03c0) (S d) $\n  by rw [\u2190h\u0393\u2081, category.assoc, cokernel.condition, has_zero_morphisms.comp_zero]\nabbreviation \u03c4\u2081 : d.J \u27f6 (T d) := (H\u03c4\u2081 d).1\n@[chase] lemma h\u03c4\u2081 : d.\u03c0 \u226b (\u03c4\u2081 d) = S d := (H\u03c4\u2081 d).2\n\nabbreviation H\u03c4\u2082 := cokernel.desc' (\u0393 d) (d.\u03bd \u226b d.\u03c1) $\n  by rw [\u2190category.assoc, \u2190h\u0393\u2082, category.assoc, d.\u03ba\u03c1.1, has_zero_morphisms.comp_zero]\nabbreviation \u03c4\u2082 : T d \u27f6 d.K := (H\u03c4\u2082 d).1\n@[chase] lemma h\u03c4\u2082 : S d \u226b \u03c4\u2082 d = d.\u03bd \u226b d.\u03c1 := (H\u03c4\u2082 d).2\n\nlemma \u03c4\u2081\u03c4\u2082 : \u03c4\u2081 d \u226b \u03c4\u2082 d = d.\u03c4 :=\nbegin\n  apply cofork.is_colimit.hom_ext (cokernel_of_epi_exact _ _ d.\u03b8\u03c0),\n  erw [\u2190category.assoc, h\u03c4\u2081, h\u03c4\u2082, \u2190d.comm\u2085]\nend\n\nabbreviation H\u03c6\u2081 := cokernel_cofork.is_colimit.desc' (cokernel_of_epi_exact _ _ d.\u03ba\u03c1) (\u03be\u2081 d \u226b \u03a5 d) $\n  by rw [\u2190category.assoc, \u2190h\u0394\u2081, category.assoc, cokernel.condition, has_zero_morphisms.comp_zero]\nabbreviation \u03c6\u2081 : d.K \u27f6 S' d := (H\u03c6\u2081 d).1\n@[chase] lemma h\u03c6\u2081 : d.\u03c1 \u226b \u03c6\u2081 d = \u03be\u2081 d \u226b \u03a5 d := (H\u03c6\u2081 d).2\n\n@[chase] lemma \u039e\u0394 : exact (\u039e d) (\u0394 d) := kernel_exact _\n@[chase] lemma \u0393S : exact (\u0393 d) (S d) := cokernel_exact _\n\ninstance \u03b2\u2082_mono : mono (\u03b2\u2082 d) :=\nmono_of_mono_fac $ h\u03b2\u2082 d\n\ninstance \u03b2\u2082_epi : epi (\u03b2\u2082 d) :=\nbegin\n  apply epi_of_pseudo_surjective,\n  intro c,\n  chase c using [d.\u03b5] with f,\n  have : (\u0394 d : _ \u27f6 _) f = 0,\n  { apply pseudo_injective_of_mono (\u03be\u2082 d),\n    rw \u2190comp_apply,\n    rw \u2190h\u0394\u2082,\n    rw \u2190h.f,\n    rw \u2190comp_apply,\n    rw d.\u03b5\u03bc.1,\n    rw zero_apply,\n    rw apply_zero, },\n  chase f using [\u039e d] with u at d,\n  use u,\n  apply pseudo_injective_of_mono d.\u03b5,\n  commutativity at d,\nend\n\ninstance \u03b2\u2082_iso : is_iso (\u03b2\u2082 d) :=\nmono_epi_iso _\n\ninstance \u03c4\u2081_epi : epi (\u03c4\u2081 d) :=\nepi_of_epi_fac $ h\u03c4\u2081 d\n\ninstance \u03c4\u2081_mono : mono (\u03c4\u2081 d) :=\nbegin\n  apply mono_of_zero_of_map_zero,\n  intros j hj,\n  chase j using [d.\u03c0] with g at d,\n  have : (S d : _ \u27f6 _) g = 0,\n  { rw [\u2190h\u03c4\u2081, comp_apply, h.g, hj], },\n  chase g using [\u0393 d] with z at d,\n  obtain \u27e8e, he\u27e9 := pseudo_surjective_of_epi (\u03b6\u2081 d) z,\n  rw [\u2190h.g, \u2190h.z, \u2190he, \u2190comp_apply, \u2190comp_apply, \u2190category.assoc, h\u0393\u2081, d.\u03b8\u03c0.1, zero_apply],\nend\n\ninstance \u03c4\u2081_iso : is_iso (\u03c4\u2081 d) :=\nmono_epi_iso _\n\nabbreviation inner_diagram : exact_snake_diagram.{v} V :=\n{ A := V' d, B := d.B, C := U d, D := Z d, E := d.E, F := d.F,\n  G := d.G, H := d.H, I := W d, J := T d, K := d.K, L := S' d,\n  \u03b1 := \u03b1\u2082 d,\n  \u03b2 := \u03b2\u2081 d,\n  \u03b3 := \u039b d,\n  \u03b4 := d.\u03b4,\n  \u03b5 := \u039e d,\n  \u03b6 := \u03b6\u2082 d,\n  \u03b7 := d.\u03b7,\n  \u03b8 := \u0393 d,\n  \u03ba := d.\u03ba,\n  \u03bc := \u0394 d,\n  \u03bd := d.\u03bd,\n  \u03be := \u03be\u2081 d,\n  \u03c0 := S d,\n  \u03c1 := d.\u03c1,\n  \u03c3 := \u03a5 d,\n  \u03c4 := \u03c4\u2082 d,\n  \u03c6 := \u03c6\u2081 d,\n  comm\u2081 := h\u03b1\u2082 d,\n  comm\u2082 := h\u03b2\u2081 d,\n  comm\u2083 := h\u0393\u2082 d,\n  comm\u2084 := h\u0394\u2081 d,\n  comm\u2085 := (h\u03c4\u2082 d).symm,\n  comm\u2086 := (h\u03c6\u2081 d).symm,\n  \u03b6\u03b7 := image_exact _ _ d.\u03b6\u03b7,\n  \u03bd\u03be := exact_image _ _ d.\u03bd\u03be,\n  \u03b3\u03b8 := kernel_exact _,\n  \u03b8\u03c0 := cokernel_exact _,\n  \u03b4\u03ba := d.\u03b4\u03ba,\n  \u03ba\u03c1 := d.\u03ba\u03c1,\n  \u03b5\u03bc := kernel_exact _,\n  \u03bc\u03c3 := cokernel_exact _ }\n\nabbreviation \u03c9' : U d \u27f6 T d := restricted.connecting_morphism (inner_diagram d)\n\nabbreviation \u03c9 : d.C \u27f6 d.J := inv (\u03b2\u2082 d) \u226b \u03c9' d \u226b inv (\u03c4\u2081 d)\n\nlemma \u03b2\u03c9 : exact d.\u03b2 (\u03c9 d) :=\nbegin\n  have : exact (\u03b2\u2081 d) (\u03c9' d) := restricted.exact\u2081 (inner_diagram d),\n  have := exact_iso_right _ _ (as_iso (inv (\u03c4\u2081 d))) this,\n  have : exact (\u03b2\u2081 d \u226b \u03b2\u2082 d) (\u03c9 d) := exact_iso _ _ (as_iso (\u03b2\u2082 d)) this,\n  rw \u03b2\u2081\u03b2\u2082 at this,\n  exact this,\nend\n\nlemma \u03c9\u03c4 : exact (\u03c9 d) (d.\u03c4) :=\nbegin\n  have := restricted.exact\u2082 (inner_diagram d),\n  have := exact_iso _ _ (as_iso (inv (\u03c4\u2081 d))) this,\n  have : exact (\u03c9 d) (\u03c4\u2081 d \u226b \u03c4\u2082 d) := exact_iso_left _ _ (as_iso (inv (\u03b2\u2082 d))) this,\n  rw \u03c4\u2081\u03c4\u2082 at this,\n  exact this,\nend\n\nend internal\n\nvariable (d : exact_snake_diagram.{v} V)\nvariables [mono d.\u03bd] [epi d.\u03b7] [mono d.\u03b4] [mono d.\u03b5] [epi d.\u03c0] [epi d.\u03c1]\n\ndef connecting_morphism : d.C \u27f6 d.J :=\ninternal.\u03c9 d\n\ntheorem exact\u2081 : exact d.\u03b2 (connecting_morphism d) :=\ninternal.\u03b2\u03c9 d\n\ntheorem exact\u2082 : exact (connecting_morphism d) d.\u03c4 :=\ninternal.\u03c9\u03c4 d\n\nend category_theory.abelian.diagram_lemmas.snake\n", "meta": {"author": "TwoFX", "repo": "lean-homological-algebra", "sha": "e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8", "save_path": "github-repos/lean/TwoFX-lean-homological-algebra", "path": "github-repos/lean/TwoFX-lean-homological-algebra/lean-homological-algebra-e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8/src/diagram_lemmas/snake_diagram.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786991753929, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.38442773244364953}}
{"text": "import tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\nimport tactic.show_term\nimport rpartrec\n\nsection\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*} {\u03c3 : Type*} {\u03c4 : Type*} {\u03bc : Type*} {o_dom : Type*} {o_cod : Type*}\n  [primcodable \u03b1] [primcodable \u03b2] [primcodable \u03b3] [primcodable \u03b4] [primcodable \u03c3] [primcodable \u03c4] [primcodable \u03bc] [primcodable o_dom] [primcodable o_cod]\n  {o : o_dom \u2192. o_cod}\nopen rcomputable rcomputable\u2082\n\n#check option.get_or_else\n\ntheorem rcomputable.unpaired3 {f : \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 \u03c3} {g : \u03b1 \u2192 \u03b2} {h : \u03b1 \u2192 \u03b3} {i : \u03b1 \u2192 \u03b4}\n  (hf : (prod.unpaired3 f) computable_in o)\n  (hg : g computable_in o) (hh : h computable_in o) (hi : i computable_in o) :\n  (\u03bb a : \u03b1, f (g a) (h a) (i a)) computable_in o :=\nhf.comp (hg.pair (hh.pair hi))\n\nlemma rcomputable.option_get_or_else {f : \u03b1 \u2192 option \u03b2} {g : \u03b1 \u2192 \u03b2} \n  (hf : f computable_in o) (hg : g computable_in o) : (\u03bb x, option.get_or_else (f x) (g x)) computable_in o :=\nrcomputable\u2082.comp (computable.option_get_or_else computable.fst computable.snd).to_rcomp hf hg\n\nlemma rcomputable.option_some : (option.some : \u03b1 \u2192 option \u03b1) computable_in o :=\ncomputable.option_some.to_rcomp\n\nlemma rcomputable.option_is_some : (option.is_some : option \u03b1 \u2192 bool) computable_in o :=\nprimrec.option_is_some.to_rcomp\n\nlemma rcomputable.option_iget [inhabited \u03b1] : (option.iget : option \u03b1 \u2192 \u03b1) computable_in o :=\nprimrec.option_iget.to_rcomp\n\nlemma rcomputable\u2082.list_rnth : (@list.rnth \u03b1) computable\u2082_in o := \n(primrec.list_nth.comp (primrec.list_reverse.comp primrec.fst) primrec.snd).to_rcomp\n\nlemma rcomputable.option_or_else :\n  ((<|>) : option \u03b2 \u2192 option \u03b2 \u2192 option \u03b2) computable\u2082_in o :=\nprimrec.option_orelse.to_rcomp\n\nlemma rcomputable\u2082.to_bool_eq (\u03b1 : Type*) [primcodable \u03b1] [decidable_eq \u03b1] :\n  (\u03bb x y : \u03b1, to_bool (x = y)) computable\u2082_in o := primrec.eq.to_rcomp\n\nlemma rcomputable.to_bool_eq (\u03b2 : Type*) [primcodable \u03b2] [decidable_eq \u03b2]\n  {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hf : f computable_in o) (hg : g computable_in o) :\n  (\u03bb a : \u03b1, to_bool (f a = g a)) computable_in o := (rcomputable\u2082.to_bool_eq \u03b2).comp hf hg\n\nlemma rcomputable\u2082.to_bool_nat_lt : (\u03bb m n : \u2115, to_bool (m < n)) computable\u2082_in o := primrec.nat_lt.to_rcomp\n\nlemma rcomputable\u2082.to_bool_nat_le : (\u03bb m n : \u2115, to_bool (m \u2264 n)) computable\u2082_in o := primrec.nat_le.to_rcomp\n\nlemma rcomputable.succ : nat.succ computable_in o := primrec.succ.to_rcomp\n\nlemma rcomputable.pred : nat.pred computable_in o := primrec.pred.to_rcomp\n\nlemma rcomputable\u2082.nat_add : ((+) : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec.nat_add.to_rcomp\n\nlemma rcomputable\u2082.nat_sub : (has_sub.sub : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec.nat_sub.to_rcomp\n\nlemma rcomputable\u2082.nat_mul : ((*) : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec.nat_mul.to_rcomp\n\nlemma rcomputable\u2082.nat_min : (min : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec.nat_min.to_rcomp\n\nlemma rcomputable\u2082.nat_max : (max : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec.nat_max.to_rcomp\n\nlemma rcomputable\u2082.nat_mkpair : (nat.mkpair : \u2115 \u2192 \u2115 \u2192 \u2115) computable\u2082_in o := primrec\u2082.mkpair.to_rcomp\n\nlemma rcomputable.nat_unpaired : (nat.unpair : \u2115 \u2192 \u2115 \u00d7 \u2115) computable_in o := primrec.unpair.to_rcomp\n\nlemma rcomputable.nat_div2 : (nat.div2 : \u2115 \u2192 \u2115) computable_in o := primrec.nat_div2.to_rcomp\n\nlemma rcomputable.nat_bodd : nat.bodd computable_in o := primrec.nat_bodd.to_rcomp\n\nlemma rcomputable.dom_fintype [fintype \u03b1] (f : \u03b1 \u2192 \u03b2) : f computable_in o := (primrec.dom_fintype f).to_rcomp\n\nprotected lemma rcomputable.bnot : bnot computable_in o := primrec.bnot.to_rcomp\n\nprotected lemma rcomputable.band : band computable\u2082_in o := primrec.band.to_rcomp\n\nprotected lemma rcomputable.bor : bor computable\u2082_in o := primrec.bor.to_rcomp\n\n@[protected]\nlemma rcomputable.ite {c : \u03b1 \u2192 Prop} [decidable_pred c] {f g : \u03b1 \u2192 \u03b2}\n  (hc : (\u03bb x, to_bool (c x)) computable_in o) (hf : f computable_in o) (hg : g computable_in o):\n  (\u03bb a, if c a then f a else g a) computable_in o :=\n(rcomputable.cond hc hf hg).of_eq (\u03bb x, by by_cases C : c x; simp[C])\n\nlemma rcomputable\u2082.list_cons : (list.cons : \u03b1 \u2192 list \u03b1 \u2192 list \u03b1) computable\u2082_in o := primrec.list_cons.to_rcomp\n\nlemma rcomputable\u2082.list_nth : (list.nth : list \u03b1 \u2192 \u2115 \u2192 option \u03b1) computable\u2082_in o := primrec.list_nth.to_rcomp\n\nlemma rcomputable\u2082.list_append : ((++) : list \u03b1 \u2192 list \u03b1 \u2192 list \u03b1) computable\u2082_in o := primrec.list_append.to_rcomp\n\nlemma rcomputable.list_length : (list.length : list \u03b1 \u2192 \u2115) computable_in o := primrec.list_length.to_rcomp\n\n#check @list.rec\n\nprotected lemma rcomputable.of_nat (\u03b1 : Type*) [denumerable \u03b1] : (denumerable.of_nat \u03b1) computable_in o :=\n(primrec.of_nat \u03b1).to_rcomp\n\nlemma rcomputable.option_get {f : \u03b1 \u2192 option \u03b2} {h : \u2200 a, (f a).is_some}\n  (hf : f computable_in o) : (\u03bb a, option.get (h a)) computable_in o :=\n((rpartrec.nat_iff1.mp rcomputable.pred).comp hf).of_eq (\u03bb n,\n  by { simp,\n       generalize hx : encodable.decode \u03b1 n = x,\n       cases x; simp,\n       rcases C : f x,\n       { exfalso, have := h x, simp[C] at this, contradiction },\n       { simp[C] } })\n\nlemma rcomputable.list_range_r : list.range_r computable_in o :=\nbegin\n  have : (nat.elim [] (\u03bb m IH, m :: IH)) computable_in o,\n  { refine rcomputable.nat_elim' rcomputable.id (rcomputable.const [])\n    (rcomputable\u2082.list_cons.comp rcomputable.fst.to_unary\u2082 rcomputable.snd.to_unary\u2082) },\n  exact this.of_eq (\u03bb n, by { induction n with n IH; simp[list.range_r], exact IH })\nend\n\ntheorem rcomputable.option_rec {f : \u03b1 \u2192 option \u03b2} {g : \u03b1 \u2192 \u03b3} {h : \u03b1 \u2192 \u03b2 \u2192 \u03b3}\n  (hf : f computable_in o) (hg : g computable_in o) (hh : h computable\u2082_in o) :\n  @rcomputable _ _ \u03b3 _ _ _ _ _ (\u03bb a, option.rec (g a) (h a) (f a)) o :=\nrcomputable.option_cases hf hg hh\n\nlemma rcomputable.computable_of_rcomp {f : \u03b1 \u2192 \u03b2} (hf : f computable_in! computable_fun) : \n  computable f := rpartrec.le_part_part hf computable.computable_fun\n\nlemma rpartrec.partrec_of_rpart {f : \u03b1 \u2192. \u03b2} (hf : f partrec_in! computable_fun) : \n  partrec f := rpartrec.le_part_part hf computable.computable_fun\n\nend\n\n@[user_attribute]\nmeta def rcomputability : user_attribute :=\n{ name := `rcomputability,\n  descr := \"lemmas usable to prove relative computability\" }\n\nattribute [rcomputability]\n  rcomputable.id\n  rcomputable.id'\n  rcomputable.fst\n  rcomputable.snd\n  rcomputable.pair\n  rcomputable.const\n  rcomputable.encode\n  rcomputable.decode\n  rcomputable.refl\n  rcomputable.cond\n  rcomputable.ite\n  rcomputable.succ\n  rcomputable.pred\n  rcomputable.of_nat\n  rcomputable.nat_cases\n  rcomputable.option_cases\n  rcomputable.option_rec\n  rcomputable.option_bind\n  rcomputable.option_map\n  rcomputable.option_some\n  rcomputable.option_is_some\n  rcomputable.option_get_or_else\n  rcomputable.option_or_else\n  rcomputable.option_get\n  rcomputable.option_iget\n  rcomputable.nat_bodd\n  rcomputable.dom_fintype\n  rcomputable\u2082.list_cons\n  rcomputable\u2082.list_nth\n  rcomputable\u2082.list_append\n  rcomputable.list_length\n  rcomputable.list_range_r\n  rcomputable\u2082.to_bool_eq\n  rcomputable.to_bool_eq\n  rcomputable\u2082.to_bool_nat_lt\n  rcomputable\u2082.to_bool_nat_le\n  rcomputable\u2082.pair\n  rcomputable\u2082.list_rnth\n  rcomputable\u2082.nat_add\n  rcomputable\u2082.nat_sub\n  rcomputable\u2082.nat_mul\n  rcomputable\u2082.nat_min\n  rcomputable\u2082.nat_max\n  rcomputable\u2082.nat_mkpair\n  rcomputable.nat_unpaired\n  rcomputable.nat_div2\n  rcomputable.bnot\n  rcomputable.band\n  rcomputable.bor\n\n  rpartrec.refl\n  rpartrec.of_option\n  rpartrec.of_option'\n  rpartrec.coe\n  rpartrec.some\n  rpartrec.bind\n  rpartrec.map\n  rpartrec.rfind\n  rpartrec.rfind_opt\n\nopen tactic.interactive (\u00abhave\u00bb)\nopen tactic (get_local infer_type)\n\nnamespace tactic\nnamespace interactive\n\nsetup_tactic_parser\n\nmeta def goal_is_partrec : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(partrec %%r) := skip\n  | _              := failed\n  end\n\nmeta def goal_is_computable : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(computable %%r) := skip\n  | _                 := failed\n  end\n\nmeta def goal_is_rpartrec : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(rpartrec %%l %%r)     := skip\n  | `(rpartrec_tot %%l %%r) := skip\n  | _ := failed\n  end\n\nmeta def goal_is_rcomputable : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(rcomputable %%l %%r)     := skip\n  | `(rcomputable_tot %%l %%r) := skip\n  | _ := failed\n  end\n\nmeta def goal_is_rpartrec\u2082 : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(rpartrec\u2082 %%l %%r)     := skip\n  | `(rpartrec\u2082_tot %%l %%r) := skip\n  | _ := failed\n  end\n\nmeta def goal_is_rcomputable\u2082 : tactic unit := do t \u2190 tactic.target,\n  match t with\n  | `(rcomputable\u2082 %%l %%r)     := skip\n  | `(rcomputable\u2082_tot %%l %%r) := skip\n  | _ := failed\n  end\n\nmeta def rcomputability_tactics (md : transparency := semireducible) : list (tactic string) :=\n[ propositional_goal >> apply_assumption >> pure \"apply_assumption\",\n  goal_is_partrec >> `[refine rpartrec.partrec_of_rpart _] >> pure \"refine rpartrec.partrec_of_rpart _\",\n  goal_is_computable >> `[refine rcomputable.computable_of_rcomp _] >> pure \"refine rcomputable.computable_of_rcomp _\",\n  apply_rules [``(rcomputability)] 50 { md := md } >> pure \"apply_rules rcomputability\",\n  `[refine rcomputable.to_unary\u2081 _]  >> pure \"refine rcomputable.to_unary\u2081 _\",\n  `[refine rcomputable.to_unary\u2082 _]  >> pure \"refine rcomputable.to_unary\u2082 _\",\n  `[refine rpartrec.to_unary\u2081 _]     >> pure \"refine rpartrec.to_unary\u2081 _\",\n  `[refine rpartrec.to_unary\u2082 _]     >> pure \"refine rpartrec.to_unary\u2082 _\",\n  goal_is_rcomputable\u2082 >> `[refine rcomputable\u2082.comp\u2082 _ _ _; try { exact rpartrec.some };\n    fail_if_success { exact rcomputable.id }] >> pure \"refine rcomputable\u2082.comp\u2082 _ _ _\",\n  goal_is_rcomputable  >> `[refine rcomputable\u2082.comp _ _ _; try { exact rpartrec.some };  \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rcomputable\u2082.comp _ _ _\",  \n  goal_is_rpartrec\u2082    >> `[refine rpartrec\u2082.comp\u2082 _ _ _; try { exact rpartrec.some };    \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rpartrec\u2082.comp\u2082 _ _ _\",\n  goal_is_rpartrec     >> `[refine rpartrec\u2082.comp _ _ _; try { exact rpartrec.some };     \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rpartrec\u2082.comp _ _ _\",\n  goal_is_rcomputable\u2082 >> `[refine rcomputable.comp\u2082 _ _; try { exact rpartrec.some };    \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rcomputable.comp\u2082 _ _\",  \n  goal_is_rcomputable  >> `[refine rcomputable.comp _ _; try { exact rpartrec.some };     \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rcomputable.comp _ _\",  \n  goal_is_rpartrec     >> `[refine rpartrec.comp _ _; try { exact rpartrec.some };        \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rpartrec.comp _ _\",\n  goal_is_rpartrec\u2082    >> `[refine rpartrec.comp\u2082 _ _; try { exact rpartrec.some };       \n    fail_if_success { exact rcomputable.id }] >> pure \"refine rpartrec.comp\u2082 _ _\" ]\n\nmeta def rcomputability\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md                  := if bang.is_some then semireducible else reducible,\n    rcomputability_core := tactic.tidy { tactics := rcomputability_tactics md, ..cfg },\n    trace_fn            := if trace.is_some then show_term else id in\ntrace_fn rcomputability_core\nend interactive\n\nend tactic\n\nopen encodable\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03c3 : Type*} {\u03c4 : Type*}\n  [primcodable \u03b1] [primcodable \u03b2] [primcodable \u03b3] [primcodable \u03c3] [primcodable \u03c4] {o : \u03c3 \u2192. \u03c4}\n\nexample (f : \u2115 \u2192. \u2115) (p : \u2115 \u2192 \u2115 \u2192. bool) (h : p partrec\u2082_in f) :\n  (\u03bb x : \u2115, (nat.rfind (p x)).map (\u03bb y, (y, (y, 0, x)))) partrec_in f :=\nby { rcomputability }\n\nexample {f : \u03b1 \u2192 \u2115 \u2192 option \u03c3} {g : \u03b2 \u2192. \u03c4}(hf : f computable\u2082_in g) :\n  (\u03bb (a : \u03b1), \u2191(\u03bb (n : \u2115), (f a n).is_some) : \u03b1 \u2192 \u2115 \u2192. bool) partrec\u2082_in g :=\nby { unfold_coes, simp[pfun.lift],\n     rcomputability }\n\n\n\n", "meta": {"author": "iehality", "repo": "lean-reducibility", "sha": "82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7", "save_path": "github-repos/lean/iehality-lean-reducibility", "path": "github-repos/lean/iehality-lean-reducibility/lean-reducibility-82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7/src/rcomputability_tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6654105653819835, "lm_q1q2_score": 0.3842715174537661}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.algebra.group.defs\nimport Mathlib.algebra.group.hom\nimport Mathlib.logic.embedding\nimport Mathlib.PostPort\n\nuniverses u v l u_1 u_2 u_3 w \n\nnamespace Mathlib\n\n/-!\n# Definitions of group actions\n\nThis file defines a hierarchy of group action type-classes:\n\n* `has_scalar \u03b1 \u03b2`\n* `mul_action \u03b1 \u03b2`\n* `distrib_mul_action \u03b1 \u03b2`\n\nThe hierarchy is extended further by `semimodule`, defined elsewhere.\n\nAlso provided are type-classes regarding the interaction of different group actions,\n\n* `smul_comm_class M N \u03b1`\n* `is_scalar_tower M N \u03b1`\n\n## Notation\n\n`a \u2022 b` is used as notation for `smul a b`.\n\n## Implementation details\n\nThis file should avoid depending on other parts of `group_theory`, to avoid import cycles.\nMore sophisticated lemmas belong in `group_theory.group_action`.\n-/\n\n/-- Typeclass for types with a scalar multiplication operation, denoted `\u2022` (`\\bu`) -/\nclass has_scalar (\u03b1 : Type u) (\u03b3 : Type v) where\n  smul : \u03b1 \u2192 \u03b3 \u2192 \u03b3\n\ninfixr:73 \" \u2022 \" => Mathlib.has_scalar.smul\n\n/-- Typeclass for multiplicative actions by monoids. This generalizes group actions. -/\nclass mul_action (\u03b1 : Type u) (\u03b2 : Type v) [monoid \u03b1] extends has_scalar \u03b1 \u03b2 where\n  one_smul : \u2200 (b : \u03b2), 1 \u2022 b = b\n  mul_smul : \u2200 (x y : \u03b1) (b : \u03b2), (x * y) \u2022 b = x \u2022 y \u2022 b\n\n/-- A typeclass mixin saying that two actions on the same space commute. -/\nclass smul_comm_class (M : Type u_1) (N : Type u_2) (\u03b1 : Type u_3) [has_scalar M \u03b1] [has_scalar N \u03b1]\n    where\n  smul_comm : \u2200 (m : M) (n : N) (a : \u03b1), m \u2022 n \u2022 a = n \u2022 m \u2022 a\n\n/-- Commutativity of actions is a symmetric relation. This lemma can't be an instance because this\nwould cause a loop in the instance search graph. -/\ntheorem smul_comm_class.symm (M : Type u_1) (N : Type u_2) (\u03b1 : Type u_3) [has_scalar M \u03b1]\n    [has_scalar N \u03b1] [smul_comm_class M N \u03b1] : smul_comm_class N M \u03b1 :=\n  smul_comm_class.mk fun (a' : N) (a : M) (b : \u03b1) => Eq.symm (smul_comm a a' b)\n\nprotected instance smul_comm_class_self (M : Type u_1) (\u03b1 : Type u_2) [comm_monoid M]\n    [mul_action M \u03b1] : smul_comm_class M M \u03b1 :=\n  smul_comm_class.mk\n    fun (a a' : M) (b : \u03b1) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (a \u2022 a' \u2022 b = a' \u2022 a \u2022 b)) (Eq.symm (mul_smul a a' b))))\n        (eq.mpr (id (Eq._oldrec (Eq.refl ((a * a') \u2022 b = a' \u2022 a \u2022 b)) (mul_comm a a')))\n          (eq.mpr (id (Eq._oldrec (Eq.refl ((a' * a) \u2022 b = a' \u2022 a \u2022 b)) (mul_smul a' a b)))\n            (Eq.refl (a' \u2022 a \u2022 b))))\n\n/-- An instance of `is_scalar_tower M N \u03b1` states that the multiplicative\naction of `M` on `\u03b1` is determined by the multiplicative actions of `M` on `N`\nand `N` on `\u03b1`. -/\nclass is_scalar_tower (M : Type u_1) (N : Type u_2) (\u03b1 : Type u_3) [has_scalar M N] [has_scalar N \u03b1]\n    [has_scalar M \u03b1]\n    where\n  smul_assoc : \u2200 (x : M) (y : N) (z : \u03b1), (x \u2022 y) \u2022 z = x \u2022 y \u2022 z\n\n@[simp] theorem smul_assoc {\u03b1 : Type u} {M : Type u_1} {N : Type u_2} [has_scalar M N]\n    [has_scalar N \u03b1] [has_scalar M \u03b1] [is_scalar_tower M N \u03b1] (x : M) (y : N) (z : \u03b1) :\n    (x \u2022 y) \u2022 z = x \u2022 y \u2022 z :=\n  is_scalar_tower.smul_assoc x y z\n\ntheorem smul_smul {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] (a\u2081 : \u03b1) (a\u2082 : \u03b1) (b : \u03b2) :\n    a\u2081 \u2022 a\u2082 \u2022 b = (a\u2081 * a\u2082) \u2022 b :=\n  Eq.symm (mul_smul a\u2081 a\u2082 b)\n\n@[simp] theorem one_smul (\u03b1 : Type u) {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] (b : \u03b2) :\n    1 \u2022 b = b :=\n  mul_action.one_smul b\n\n/-- Pullback a multiplicative action along an injective map respecting `\u2022`. -/\nprotected def function.injective.mul_action {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [monoid \u03b1]\n    [mul_action \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] (f : \u03b3 \u2192 \u03b2) (hf : function.injective f)\n    (smul : \u2200 (c : \u03b1) (x : \u03b3), f (c \u2022 x) = c \u2022 f x) : mul_action \u03b1 \u03b3 :=\n  mul_action.mk sorry sorry\n\n/-- Pushforward a multiplicative action along a surjective map respecting `\u2022`. -/\nprotected def function.surjective.mul_action {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [monoid \u03b1]\n    [mul_action \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] (f : \u03b2 \u2192 \u03b3) (hf : function.surjective f)\n    (smul : \u2200 (c : \u03b1) (x : \u03b2), f (c \u2022 x) = c \u2022 f x) : mul_action \u03b1 \u03b3 :=\n  mul_action.mk sorry sorry\n\ntheorem ite_smul {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] (p : Prop) [Decidable p]\n    (a\u2081 : \u03b1) (a\u2082 : \u03b1) (b : \u03b2) : ite p a\u2081 a\u2082 \u2022 b = ite p (a\u2081 \u2022 b) (a\u2082 \u2022 b) :=\n  sorry\n\ntheorem smul_ite {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] (p : Prop) [Decidable p]\n    (a : \u03b1) (b\u2081 : \u03b2) (b\u2082 : \u03b2) : a \u2022 ite p b\u2081 b\u2082 = ite p (a \u2022 b\u2081) (a \u2022 b\u2082) :=\n  sorry\n\nnamespace mul_action\n\n\n/-- The regular action of a monoid on itself by left multiplication. -/\ndef regular (\u03b1 : Type u) [monoid \u03b1] : mul_action \u03b1 \u03b1 := mk sorry sorry\n\nprotected instance is_scalar_tower.left (\u03b1 : Type u) {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] :\n    is_scalar_tower \u03b1 \u03b1 \u03b2 :=\n  is_scalar_tower.mk fun (x y : \u03b1) (z : \u03b2) => mul_smul x y z\n\n/-- Embedding induced by action. -/\ndef to_fun (\u03b1 : Type u) (\u03b2 : Type v) [monoid \u03b1] [mul_action \u03b1 \u03b2] : \u03b2 \u21aa \u03b1 \u2192 \u03b2 :=\n  function.embedding.mk (fun (y : \u03b2) (x : \u03b1) => x \u2022 y) sorry\n\n@[simp] theorem to_fun_apply {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [mul_action \u03b1 \u03b2] (x : \u03b1) (y : \u03b2) :\n    coe_fn (to_fun \u03b1 \u03b2) y x = x \u2022 y :=\n  rfl\n\n/-- An action of `\u03b1` on `\u03b2` and a monoid homomorphism `\u03b3 \u2192 \u03b1` induce an action of `\u03b3` on `\u03b2`. -/\ndef comp_hom {\u03b1 : Type u} (\u03b2 : Type v) {\u03b3 : Type w} [monoid \u03b1] [mul_action \u03b1 \u03b2] [monoid \u03b3]\n    (g : \u03b3 \u2192* \u03b1) : mul_action \u03b3 \u03b2 :=\n  mk sorry sorry\n\nend mul_action\n\n\n@[simp] theorem smul_one_smul {\u03b1 : Type u} {M : Type u_1} (N : Type u_2) [monoid N] [has_scalar M N]\n    [mul_action N \u03b1] [has_scalar M \u03b1] [is_scalar_tower M N \u03b1] (x : M) (y : \u03b1) :\n    (x \u2022 1) \u2022 y = x \u2022 y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl ((x \u2022 1) \u2022 y = x \u2022 y)) (smul_assoc x 1 y)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2022 1 \u2022 y = x \u2022 y)) (one_smul N y))) (Eq.refl (x \u2022 y)))\n\n/-- Typeclass for multiplicative actions on additive structures. This generalizes group modules. -/\nclass distrib_mul_action (\u03b1 : Type u) (\u03b2 : Type v) [monoid \u03b1] [add_monoid \u03b2] extends mul_action \u03b1 \u03b2\n    where\n  smul_add : \u2200 (r : \u03b1) (x y : \u03b2), r \u2022 (x + y) = r \u2022 x + r \u2022 y\n  smul_zero : \u2200 (r : \u03b1), r \u2022 0 = 0\n\ntheorem smul_add {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [add_monoid \u03b2] [distrib_mul_action \u03b1 \u03b2]\n    (a : \u03b1) (b\u2081 : \u03b2) (b\u2082 : \u03b2) : a \u2022 (b\u2081 + b\u2082) = a \u2022 b\u2081 + a \u2022 b\u2082 :=\n  distrib_mul_action.smul_add a b\u2081 b\u2082\n\n@[simp] theorem smul_zero {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [add_monoid \u03b2]\n    [distrib_mul_action \u03b1 \u03b2] (a : \u03b1) : a \u2022 0 = 0 :=\n  distrib_mul_action.smul_zero a\n\n/-- Pullback a distributive multiplicative action along an injective additive monoid\nhomomorphism. -/\nprotected def function.injective.distrib_mul_action {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n    [monoid \u03b1] [add_monoid \u03b2] [distrib_mul_action \u03b1 \u03b2] [add_monoid \u03b3] [has_scalar \u03b1 \u03b3] (f : \u03b3 \u2192+ \u03b2)\n    (hf : function.injective \u21d1f) (smul : \u2200 (c : \u03b1) (x : \u03b3), coe_fn f (c \u2022 x) = c \u2022 coe_fn f x) :\n    distrib_mul_action \u03b1 \u03b3 :=\n  distrib_mul_action.mk sorry sorry\n\n/-- Pushforward a distributive multiplicative action along a surjective additive monoid\nhomomorphism.-/\nprotected def function.surjective.distrib_mul_action {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n    [monoid \u03b1] [add_monoid \u03b2] [distrib_mul_action \u03b1 \u03b2] [add_monoid \u03b3] [has_scalar \u03b1 \u03b3] (f : \u03b2 \u2192+ \u03b3)\n    (hf : function.surjective \u21d1f) (smul : \u2200 (c : \u03b1) (x : \u03b2), coe_fn f (c \u2022 x) = c \u2022 coe_fn f x) :\n    distrib_mul_action \u03b1 \u03b3 :=\n  distrib_mul_action.mk sorry sorry\n\n/-- Scalar multiplication by `r` as an `add_monoid_hom`. -/\ndef const_smul_hom {\u03b1 : Type u} (\u03b2 : Type v) [monoid \u03b1] [add_monoid \u03b2] [distrib_mul_action \u03b1 \u03b2]\n    (r : \u03b1) : \u03b2 \u2192+ \u03b2 :=\n  add_monoid_hom.mk (has_scalar.smul r) (smul_zero r) (smul_add r)\n\n@[simp] theorem const_smul_hom_apply {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [add_monoid \u03b2]\n    [distrib_mul_action \u03b1 \u03b2] (r : \u03b1) (x : \u03b2) : coe_fn (const_smul_hom \u03b2 r) x = r \u2022 x :=\n  rfl\n\n@[simp] theorem smul_neg {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [add_group \u03b2] [distrib_mul_action \u03b1 \u03b2]\n    (r : \u03b1) (x : \u03b2) : r \u2022 -x = -(r \u2022 x) :=\n  sorry\n\ntheorem smul_sub {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [add_group \u03b2] [distrib_mul_action \u03b1 \u03b2] (r : \u03b1)\n    (x : \u03b2) (y : \u03b2) : r \u2022 (x - y) = r \u2022 x - r \u2022 y :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/group_theory/group_action/defs_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3842371990162575}}
{"text": "/-\nA universe of bundled setoids, i.e. types with equivalence relations.\nThis is one of the simplest universes where functoriality is nontrivial. It is essentially the\nsame as a universe of groupoids where isomorphisms are propositions. Therefore, it is especially\nuseful as a blueprint for the universe of categories.\nThere is a truncation functor from every other universe to the setoid universe, and of course this\ncan be continued into `sort` by taking the quotient.\n-/\n\n\n\nimport UniverseAbstractions.Axioms.Universes\nimport UniverseAbstractions.Axioms.Universe.Identity\nimport UniverseAbstractions.Axioms.Universe.Functors\nimport UniverseAbstractions.Axioms.Universe.FunctorExtensionality\nimport UniverseAbstractions.Axioms.Universe.Products\nimport UniverseAbstractions.Axioms.Universe.Equivalences\nimport UniverseAbstractions.Axioms.Universe.DependentTypes.Properties\nimport UniverseAbstractions.Axioms.Universe.DependentTypes.DependentFunctors\nimport UniverseAbstractions.Axioms.Universe.DependentTypes.DependentProducts\nimport UniverseAbstractions.Instances.Utils.Bundled\nimport UniverseAbstractions.Instances.Utils.Trivial\nimport UniverseAbstractions.Instances.Sort\n\n\n\nset_option autoBoundImplicitLocal false\nset_option synthInstance.maxHeartbeats 4000\n--set_option pp.universes true\n\nuniverse u v w w' upv\n\n\n\nnamespace Setoid\n\n  open Bundled MetaRelation HasFunctors HasCongrArg HasCongrFun HasLinearFunOp HasPropCongrArg\n\n  def typeClass : SimpleTypeClass.{u, max 1 u} := Setoid.{u}\n  @[reducible] def univ : Universe.{u, u + 1} := Bundled.univ typeClass.{u}\n  @[reducible] def tuniv := univ.{max 1 u}\n\n  instance inst (A : univ.{u}) : Setoid.{u} A := A.inst\n\n  def lift {\u03b1 : Sort u} {\u03c9 : Sort w} (s : Setoid \u03b1) (l : \u03c9 \u2192 \u03b1) :\n    Setoid \u03c9 :=\n  { r     := \u03bb a b => l a \u2248 l b,\n    iseqv := { refl  := \u03bb a   => Setoid.refl  (l a),\n               symm  := \u03bb h   => Setoid.symm  h,\n               trans := \u03bb h i => Setoid.trans h i } }\n\n  -- Instance equivalences\n\n  instance hasEquivalenceRelation (A : univ.{u}) : HasEquivalenceRelation A prop :=\n  \u27e8nativeRelation (inst A).r\u27e9\n\n  instance hasInstanceEquivalences : HasInstanceEquivalences univ.{u} prop :=\n  \u27e8hasEquivalenceRelation\u27e9\n\n  -- Functors\n\n  def FunctionEquiv {\u03b1 : Sort u} {\u03c6 : \u03b1 \u2192 univ} (f\u2081 f\u2082 : \u2200 a, \u03c6 a) : Prop :=\n  \u2200 a, f\u2081 a \u2248 f\u2082 a\n\n  instance functionSetoid {\u03b1 : Sort u} (\u03c6 : \u03b1 \u2192 univ.{v}) :\n    Setoid.{imax u v} (\u2200 a, \u03c6 a) :=\n  { r     := FunctionEquiv,\n    iseqv := { refl  := \u03bb f   a => (inst (\u03c6 a)).refl  (f a),\n               symm  := \u03bb h   a => (inst (\u03c6 a)).symm  (h a),\n               trans := \u03bb h i a => (inst (\u03c6 a)).trans (h a) (i a) } }\n\n  instance biFunctionSetoid {\u03b1 : Sort u} {\u03b2 : Sort v} (\u03c6 : \u03b1 \u2192 \u03b2 \u2192 univ.{w}) :\n    Setoid.{imax u v w} (\u2200 a b, \u03c6 a b) :=\n  inferInstance\n\n  def IsFun {U : Universe} [HasIdentity U] {A : U} {B : univ} (f : A \u2192 B) : Prop :=\n  \u2200 {a\u2081 a\u2082 : A}, a\u2081 \u2243 a\u2082 \u2192 f a\u2081 \u2248 f a\u2082\n\n  instance hasFunctoriality (U : Universe.{u}) [HasIdentity U] :\n  HasFunctoriality U univ.{v} := \u27e8IsFun\u27e9\n\n  instance funSetoid {U : Universe.{u}} [HasIdentity U] (A : U) (B : univ.{v}) :\n    Setoid.{max 1 u v} (HasFunctoriality.Fun A B) :=\n  lift (functionSetoid (Function.const A B)) HasFunctoriality.Fun.f\n\n  instance hasFunctorialityInstances (U : Universe.{u}) [HasIdentity U] :\n    HasFunctorialityInstances U univ.{v} typeClass.{max 1 u v} :=\n  \u27e8funSetoid\u27e9\n\n  instance hasCongrArg (U : Universe.{u}) [HasIdentity U] : HasCongrArg U univ.{v} :=\n  \u27e8HasFunctoriality.Fun.isFun\u27e9\n\n  instance hasCongrFun (U : Universe.{u}) [HasIdentity U] : HasCongrFun U univ.{v} := \u27e8id\u27e9\n\n  instance hasInternalFunctors : HasInternalFunctors tuniv.{u} := \u27e8\u27e9\n\n  def defFun {U : Universe} [HasIdentity U] {A : U} {B : univ} {f : A \u2192 B} (isFun : IsFun f) :\n    A \u27f6{f} B :=\n  Bundled.HasFunctorialityInstances.defFun isFun\n\n  instance hasIdFun : HasIdFun univ.{u} := \u27e8\u03bb A => defFun id\u27e9\n\n  instance hasConstFun (U : Universe.{u}) [HasIdentity U] : HasConstFun U univ.{v} :=\n  \u27e8\u03bb A {B} b => defFun (\u03bb h => Setoid.refl b)\u27e9\n\n  instance hasRevAppFun : HasRevAppFun tuniv.{u} := \u27e8\u03bb a B => defFun (\u03bb h => h a)\u27e9\n\n  instance hasCompFun (U : Universe.{u}) (V : Universe.{v}) {W : Universe.{w'}} [HasIdentity U]\n                      [HasIdentity V] [HasFunctors U V W] [HasCongrArg U V] :\n    HasCompFun U V univ.{w} :=\n  \u27e8\u03bb F G => defFun (\u03bb h => congrArg G (congrArg F h))\u27e9\n\n  instance hasCompFunFun (U : Universe.{u}) [HasIdentity U] [HasInternalFunctors U] :\n    HasCompFunFun U tuniv.{v} :=\n                                     -- Work around Lean defeq problem.\n  \u27e8\u03bb F C => defFun (\u03bb {G\u2081 G\u2082} h a => have h\u2081 : (G\u2081 \u2299 F).f a = G\u2081 (F a) := rfl;\n                                     have h\u2082 : (G\u2082 \u2299 F).f a = G\u2082 (F a) := rfl;\n                                     h\u2081 \u25b8 h\u2082 \u25b8 h (F a))\u27e9\n\n  instance hasRevCompFunFun (U : Universe.{u}) [HasIdentity U] [HasInternalFunctors U]\n                            [HasCongrFun U U] :\n    HasRevCompFunFun U univ.{v} :=\n                                           -- Work around Lean defeq problem.\n  \u27e8\u03bb A {B C} G => defFun (\u03bb {F\u2081 F\u2082} e a => have h\u2081 : (G \u2299 F\u2081).f a = G (F\u2081 a) := rfl;\n                                           have h\u2082 : (G \u2299 F\u2082).f a = G (F\u2082 a) := rfl;\n                                           h\u2081 \u25b8 h\u2082 \u25b8 congrArg G (congrFun e a))\u27e9\n\n  instance hasSwapFun (U : Universe.{u}) (V : Universe.{v}) [HasIdentity U] [HasIdentity V] :\n    HasSwapFun U V univ.{w} :=\n  \u27e8\u03bb F b => defFun (\u03bb h => (HasCongrArg.congrArg F h) b)\u27e9\n\n  instance hasSwapFunFun : HasSwapFunFun univ.{u} univ.{v} univ.{w} :=\n                                   -- Work around Lean defeq problem.\n  \u27e8\u03bb F => defFun (\u03bb {b\u2081 b\u2082} h a => have h\u2081 : (HasSwapFun.swapFun F b\u2081).f a = F a b\u2081 := rfl; \n                                   have h\u2082 : (HasSwapFun.swapFun F b\u2082).f a = F a b\u2082 := rfl; \n                                   h\u2081 \u25b8 h\u2082 \u25b8 congrArg (F a) h)\u27e9\n\n  instance hasDupFun (U : Universe.{u}) [HasIdentity U] : HasDupFun U univ.{v} :=\n  \u27e8\u03bb F => defFun (\u03bb h => congrArg\u2082 F h h)\u27e9\n\n  instance hasSubstFun (U : Universe.{u}) (V : Universe.{v}) {W : Universe.{w'}} [HasIdentity U]\n                       [HasIdentity V] [HasFunctors U V W] [HasCongrArg U V] :\n    HasSubstFun U V univ.{w} :=\n  \u27e8\u03bb F G => defFun (\u03bb h => congrArg\u2082 G h (congrArg F h))\u27e9\n\n  instance hasBiCompFun : HasBiCompFun univ.{u} univ.{v} univ.{w} univ.{w'} :=\n  \u27e8\u03bb F G H => defFun (\u03bb h => congrArg\u2082 H (congrArg F h) (congrArg G h))\u27e9\n\n  instance hasRevBiCompFunFun : HasRevBiCompFunFun tuniv.{u} univ.{v} univ.{w'} :=\n                                     -- Work around Lean defeq problem.\n  \u27e8\u03bb H F => defFun (\u03bb {G\u2081 G\u2082} h a => have h\u2081 : (HasBiCompFun.biCompFun F G\u2081 H).f a = H (F a) (G\u2081 a) := rfl;\n                                     have h\u2082 : (HasBiCompFun.biCompFun F G\u2082 H).f a = H (F a) (G\u2082 a) := rfl;\n                                     h\u2081 \u25b8 h\u2082 \u25b8 (H (F a)).isFun (h a))\u27e9\n\n  instance hasRevBiCompFunFunFun : HasRevBiCompFunFunFun tuniv.{u} univ.{w'} :=\n                                               -- Work around Lean defeq problem.\n  \u27e8\u03bb A {B C D} H => defFun (\u03bb {F\u2081 F\u2082} h G a => have h\u2081 : ((HasRevBiCompFunFun.revBiCompFunFun H F\u2081).f G).f a = H (F\u2081 a) (G a) := rfl;\n                                               have h\u2082 : ((HasRevBiCompFunFun.revBiCompFunFun H F\u2082).f G).f a = H (F\u2082 a) (G a) := rfl;\n                                               h\u2081 \u25b8 h\u2082 \u25b8 (H.isFun (h a)) (G a))\u27e9\n\n  instance hasLinearFunOp : HasLinearFunOp tuniv.{u} :=\n  { defIdFun     := \u03bb A     => defFun id,\n    defRevAppFun := \u03bb A B   => \u27e8\u03bb a => defFun (\u03bb h => h a),\n                                defFun (\u03bb h F => F.isFun h)\u27e9,\n    defCompFun   := \u03bb A B C => \u27e8\u03bb F => \u27e8\u03bb G => defFun (\u03bb h => G.isFun (F.isFun h)),\n                                        defFun (\u03bb h a => h (F a))\u27e9,\n                                defFun (\u03bb h G a => G.isFun (h a))\u27e9 }\n\n  instance hasAffineFunOp : HasAffineFunOp tuniv.{u} :=\n  { defConstFun := \u03bb A B => \u27e8\u03bb b => defFun (\u03bb h => Setoid.refl b),\n                             defFun (\u03bb h a => h)\u27e9 }\n\n  instance hasFullFunOp : HasFullFunOp tuniv.{u} :=\n  { defDupFun := \u03bb A B => \u27e8\u03bb F => defFun (\u03bb h => congrArg\u2082 F h h),\n                           defFun (\u03bb h a => h a a)\u27e9 }\n\n  instance hasTrivialExtensionality : HasTrivialExtensionality univ.{u} univ.{v} := \u27e8id\u27e9\n\n  instance hasStandardFunctors : HasStandardFunctors tuniv.{u} := \u27e8\u27e9\n\n  -- Setoid truncation from another universe\n\n  def typeSetoid {U : Universe.{u}} [HasIdentity U] (A : U) : Setoid.{u} A :=\n  { r     := \u03bb a b => Nonempty (a \u2243 b),\n    iseqv := { refl  := \u03bb a       => \u27e8HasRefl.refl a\u27e9,\n               symm  := \u03bb \u27e8e\u27e9     => \u27e8e\u207b\u00b9\u27e9,\n               trans := \u03bb \u27e8e\u27e9 \u27e8f\u27e9 => \u27e8f \u2022 e\u27e9 } }\n\n  def Truncated {U : Universe.{u}} [HasIdentity U] (A : U) : univ.{u} :=\n  { a    := \u2308A\u2309,\n    inst := typeSetoid A }\n\n  def trunc {U : Universe.{u}} [HasIdentity U] {A : U} (a : A) : Truncated A := a\n\n  def truncFun {U : Universe.{u}} [HasIdentity U] (A : U) : A \u27f6 Truncated A :=\n  { f     := trunc,\n    isFun := \u03bb e => \u27e8e\u27e9 }\n\n  instance trunc.isFunApp {U : Universe.{u}} [HasIdentity U] {A : U} (a : A) :\n    IsFunApp A (trunc a) :=\n  { F := truncFun A,\n    a := a,\n    e := Setoid.refl (trunc a) }\n\n  def truncProj {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}} [HasIdentity U]\n                [HasIdentity V] [HasIdentity W] [HasFunctors U V W] [HasCongrArg U V]\n                {A : U} {B : V} (F : A \u27f6 B) :\n    Truncated A \u27f6 Truncated B :=\n  { f     := \u03bb a => F a,\n    isFun := \u03bb \u27e8e\u27e9 => \u27e8congrArg F e\u27e9 }\n\n  def truncProjFun {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}} [HasIdentity U]\n                   [HasIdentity V] [HasIdentity W] [HasFunctors U V W] [HasCongrArg U V]\n                   [HasCongrFun U V] (A : U) (B : V) :\n    (A \u27f6 B) \u27f6 (Truncated A \u27f6 Truncated B) :=\n  { f     := truncProj,\n    isFun := \u03bb e a => \u27e8congrFun e a\u27e9 }\n\n  -- TODO: This fails after a refactoring; no idea why.\n  --instance truncProj.isFunApp {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}}\n  --                            [HasIdentity U] [HasIdentity V] [HasIdentity W]\n  --                            [HasFunctors U V W] [HasCongrArg U V] [HasCongrFun U V]\n  --                            {A : U} {B : V} (F : A \u27f6 B) :\n  --  IsFunApp (A \u27f6 B) (truncProj F) :=\n  --{ F := truncProjFun A B,\n  --  a := F,\n  --  e := Setoid.refl (truncProj F) }\n\n  def truncProjFun' {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}} [HasIdentity U]\n                    [HasIdentity V] [HasIdentity W] [HasFunctors U V W] [HasCongrArg U V]\n                    [HasCongrFun U V] (A : U) (B : V) :\n    Truncated (A \u27f6 B) \u27f6 (Truncated A \u27f6 Truncated B) :=\n  { f     := truncProj,\n    isFun := \u03bb \u27e8e\u27e9 a => \u27e8congrFun e a\u27e9 }\n\n  theorem truncCongr {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}} [HasIdentity U]\n                     [HasIdentity V] [HasIdentity W] [HasFunctors U V W] [HasCongrArg U V]\n                     {A : U} {B : V} (F : A \u27f6 B) (a : A) :\n    trunc (F a) \u2248 (truncProj F) (trunc a) :=\n  \u27e8HasRefl.refl (F a)\u27e9\n\n  theorem truncCongrSquare {U : Universe.{u}} {V : Universe.{v}} {W : Universe.{w}} [HasIdentity U]\n                           [HasIdentity V] [HasIdentity W] [HasFunctors U V W] [HasCongrArg U V]\n                           {A : U} {B : V} (F : A \u27f6 B) :\n    truncFun B \u2299 F \u2248 truncProj F \u2299 truncFun A :=\n  truncCongr F\n\n  -- Setoid to quotient\n\n  def AsQuotient (A : univ.{u}) : sort.{u} := Quotient (inst A)\n\n  def toQuotient {A : univ.{u}} (a : A) : AsQuotient A := Quotient.mk a\n\n  def toQuotientFun (A : univ.{u}) : A \u27f6 AsQuotient A :=\n  { f        := toQuotient,\n    congrArg := Quotient.sound }\n\n  instance toQuotient.isFunApp {A : univ.{u}} (a : A) :\n    IsFunApp (V := sort.{u}) A (toQuotient a) :=\n  { F := toQuotientFun A,\n    a := a,\n    e := rfl }\n\n  def quotientProj {A : univ.{u}} {B : univ.{v}} (F : A \u27f6 B) : AsQuotient A \u27f6 AsQuotient B :=\n  -- Work around Lean defeq problem.\n  let toQuotB : \u2308B \u27f6 AsQuotient B\u2309 := toQuotientFun B;\n  let G : A \u27f6 AsQuotient B := toQuotB \u2299 F;\n  Quotient.lift G.f (\u03bb _ _ => G.congrArg)\n\n  def toQuotientCongr {A : univ.{u}} {B : univ.{v}} (F : A \u27f6 B) (a : A) :\n    toQuotient (F a) = (quotientProj F) (toQuotient a) :=\n  rfl\n\n  -- TODO: Seems like another defeq problem together with universe issues.\n  --def toQuotientCongrSquare {A : univ.{u}} {B : univ.{v}} (F : A \u27f6 B) :\n  --  toQuotientFun B \u2299 F = quotientProj F \u2299 toQuotientFun A :=\n  --rfl\n\n  -- Singletons\n\n  instance unitSetoid : Setoid.{u} PUnit.{u} :=\n  { r     := \u03bb _ _ => True,\n    iseqv := { refl  := \u03bb _   => trivial,\n               symm  := \u03bb _   => trivial,\n               trans := \u03bb _ _ => trivial } }\n\n  instance hasTopInstance : HasTopInstance typeClass.{u} := \u27e8unitSetoid\u27e9\n\n  instance hasTopEq : HasTop.HasTopEq univ.{u} := \u27e8\u03bb _ => trivial\u27e9\n\n  instance hasInternalTop : HasInternalTop tuniv.{u} :=\n  { defElimFun := \u03bb a => defFun (\u03bb _ => Setoid.refl a) }\n\n  instance emptySetoid : Setoid.{u} PEmpty.{u} :=\n  { r     := \u03bb _ _ => False,\n    iseqv := { refl  := PEmpty.elim,\n               symm  := id,\n               trans := Function.const False } }\n\n  instance hasBotInstance : HasBotInstance typeClass.{u} := \u27e8emptySetoid\u27e9\n\n  instance hasInternalBot : HasInternalBot tuniv.{u} :=\n  { defElimFun := \u03bb A => defFun False.elim }\n\n  -- Products\n\n  instance productSetoid (\u03b1 : Sort u) (\u03b2 : Sort v) [Setoid \u03b1] [Setoid \u03b2] :\n    Setoid.{max 1 u v} (PProd \u03b1 \u03b2) :=\n  { r     := \u03bb p q => p.fst \u2248 q.fst \u2227 p.snd \u2248 q.snd,\n    iseqv := { refl  := \u03bb p   => \u27e8Setoid.refl  p.fst,         Setoid.refl  p.snd\u27e9,\n               symm  := \u03bb h   => \u27e8Setoid.symm  h.left,        Setoid.symm  h.right\u27e9,\n               trans := \u03bb h i => \u27e8Setoid.trans h.left i.left, Setoid.trans h.right i.right\u27e9 } }\n\n  instance prodSetoid (A : univ.{u}) (B : univ.{v}) : Setoid.{max 1 u v} (PProd A B) :=\n  productSetoid A B\n\n  instance hasProductInstances : HasProductInstances univ.{u} univ.{v} typeClass.{max 1 u v} :=\n  \u27e8prodSetoid\u27e9\n\n  instance hasProductEq : HasProducts.HasProductEq univ.{u} univ.{v} :=\n  { introEq := \u03bb p   => Setoid.refl p,\n    fstEq   := \u03bb a b => Setoid.refl a,\n    sndEq   := \u03bb a b => Setoid.refl b }\n\n  instance hasInternalProducts : HasInternalProducts tuniv.{u} :=\n  { defIntroFun := \u03bb A B   => \u27e8\u03bb a => defFun (\u03bb h => \u27e8Setoid.refl a, h\u27e9),\n                               defFun (\u03bb h b => \u27e8h, Setoid.refl b\u27e9)\u27e9,\n    defElimFun  := \u03bb A B C => \u27e8\u03bb F => defFun (\u03bb {p\u2081 p\u2082} h => have h_fst : \u2308p\u2081.fst \u2243 p\u2082.fst\u2309 := h.left;\n                                                             have h_snd : \u2308p\u2081.snd \u2243 p\u2082.snd\u2309 := h.right;\n                                                             congrArg\u2082 F h_fst h_snd),\n                               defFun (\u03bb h p => h p.fst p.snd)\u27e9 }\n\n  -- Equivalences\n\n  instance equivSetoid (A : univ.{u}) (B : univ.{v}) : Setoid.{max 1 u v} (A \u2b82 B) :=\n  lift (funSetoid A B) EquivDesc.toFun\n\n  instance hasEquivalenceInstances :\n    HasEquivalenceInstances univ.{u} univ.{v} typeClass.{max 1 u v} :=\n  { Equiv     := EquivDesc,\n    desc      := id,\n    equivInst := equivSetoid }\n\n  instance hasInternalEquivalences : HasInternalEquivalences tuniv.{u} :=\n  { defToFunFun := \u03bb A B => defFun id,\n    isExt       := HasTrivialExtensionality.equivDescExt tuniv.{u},\n    toFunInj    := id }\n\n  instance hasTrivialEquivalenceCondition : HasTrivialEquivalenceCondition tuniv.{u} :=\n  \u27e8\u03bb e => { E        := e,\n            toFunEq  := Setoid.refl e.toFun,\n            invFunEq := Setoid.refl e.invFun }\u27e9\n\n  -- Dependent functors\n\n  instance hasTypeIdentity : HasTypeIdentity tuniv.{u} := \u27e8\u27e9\n\n  def IsPi {U : Universe.{u}} {UpV : Universe.{upv}} [HasIdentity U]\n           [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}]\n           {A : U} {\u03c6 : A \u27f6 \u230atuniv.{v}\u230b} (f : HasFunctors.Pi \u03c6) :\n    Prop :=\n  \u2200 {a\u2081 a\u2082 : A} (e : a\u2081 \u2243 a\u2082), f a\u2081 \u2243[propCongrArg \u03c6 e] f a\u2082\n\n  instance hasDependentFunctoriality (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n                                     [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}] :\n  HasDependentFunctoriality U tuniv.{v} := \u27e8IsPi\u27e9\n\n  instance piSetoid {U : Universe.{u}} {UpV : Universe.{upv}} [HasIdentity U]\n                    [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}]\n                    {A : U} (\u03c6 : A \u27f6 \u230atuniv.{v}\u230b) :\n    Setoid.{max 1 u v} (HasDependentFunctoriality.Pi \u03c6) :=\n  lift (functionSetoid (HasFunctors.apply \u03c6)) HasDependentFunctoriality.Pi.f\n\n  instance hasDependentFunctorialityInstances (U : Universe.{u}) {UpV : Universe.{upv}}\n                                              [HasIdentity U] [HasFunctors U {tuniv.{v}} UpV]\n                                              [HasPropCongrArg U tuniv.{v}] :\n    HasDependentFunctorialityInstances U tuniv.{v} typeClass.{max 1 u v} :=\n  \u27e8piSetoid\u27e9\n\n  instance hasDependentCongrArg (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n                                [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}] :\n    HasDependentCongrArg U tuniv.{v} :=\n  \u27e8HasDependentFunctoriality.Pi.isFun\u27e9\n\n  -- Lean bug :-(\n  noncomputable def defInPi {U : Universe.{u}} {UpV : Universe.{upv}} [HasIdentity U]\n                            [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}]\n                            {A : U} {\u03c6 : A \u27f6 \u230atuniv.{v}\u230b} (f : HasFunctors.Pi \u03c6) (isFun : IsPi f) :\n    \u03a0{f} (HasFunctors.toDefFun \u03c6) :=\n  Bundled.HasDependentFunctorialityInstances.defPi isFun\n\n  -- Dependent products\n\n  instance sigmaSetoid {U : Universe.{u}} {UpV : Universe.{upv}} [HasIdentity U]\n                       [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}]\n                       {A : U} (\u03c6 : A \u27f6 \u230atuniv.{v}\u230b) :\n    Setoid.{max 1 u v} (PSigma (\u03bb a => \u03c6 a)) :=\n  { r     := \u03bb p\u2081 p\u2082 => \u2203 e : p\u2081.fst \u2243 p\u2082.fst, \u2308p\u2081.snd \u2243[propCongrArg \u03c6 e] p\u2082.snd\u2309,\n    iseqv := { refl  := \u03bb p             => \u27e8HasInstanceEquivalences.refl p.fst,\n                                            DependentEquivalence.depCongrArgRefl  p.snd\u27e9,\n               symm  := \u03bb \u27e8e, h\u27e9        => \u27e8e\u207b\u00b9,\n                                            DependentEquivalence.depCongrArgSymm  h\u27e9,\n               trans := \u03bb \u27e8e, h\u27e9 \u27e8f, i\u27e9 => \u27e8f \u2022 e,\n                                            DependentEquivalence.depCongrArgTrans h i\u27e9 } }\n\n  instance hasDependentProductInstances (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n                                        [HasFunctors U {tuniv.{v}} UpV]\n                                        [HasPropCongrArg U tuniv.{v}] :\n    HasDependentProductInstances U tuniv.{v} typeClass.{max 1 u v} :=\n  \u27e8sigmaSetoid\u27e9\n\n  instance hasDependentProducts (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n                                [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}] :\n    HasDependentProducts U tuniv.{v} tuniv.{max u v} :=\n  Bundled.hasDependentProducts U tuniv.{v}\n\n  instance hasDependentProductEq (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n                                 [HasFunctors U {tuniv.{v}} UpV] [HasPropCongrArg U tuniv.{v}] :\n    HasDependentProducts.HasDependentProductEq U tuniv.{v} :=\n  { introEq := \u03bb p   => Setoid.refl p,\n    fstEq   := \u03bb a b => HasInstanceEquivalences.refl a,\n    sndEq   := \u03bb a b => DependentEquivalence.depCongrArgRefl b }\n\n  -- TODO\n  --instance hasInternalDependentProducts (U : Universe.{u}) {UpV : Universe.{upv}} [HasIdentity U]\n  --                                      [HasFunctors U {tuniv.{u}} UpV] [HasPropCongrArg U tuniv.{u}] :\n  --  HasInternalDependentProducts U tuniv.{u} :=\n  --sorry\n\nend Setoid\n", "meta": {"author": "SReichelt", "repo": "universe-abstractions", "sha": "0bf2bae4c1b0f8d96c37e231dd238abda788e843", "save_path": "github-repos/lean/SReichelt-universe-abstractions", "path": "github-repos/lean/SReichelt-universe-abstractions/universe-abstractions-0bf2bae4c1b0f8d96c37e231dd238abda788e843/UniverseAbstractions/Instances/Setoid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3840495154040556}}
{"text": "variables {p q : Prop} (hp : p) (hq : q)\n\ninclude hp hq\n\nexample : p \u2227 q \u2227 p :=\nbegin\n  apply and.intro hp,\n  exact and.intro hq hp,\nend\n\nomit hp hq\n\nsection\ninclude hp hq\n\nexample : p \u2227 q \u2227 p :=\nbegin\n  apply and.intro hp,\n  exact and.intro hq hp,\nend\nend\n\nexample : p \u2227 q \u2227 p :=\nlet hp := hp, hq := hq in\nbegin\n  apply and.intro hp,\n  exact and.intro hq hp\nend\n", "meta": {"author": "agryman", "repo": "theorem-proving-in-lean", "sha": "cf5a3a19d0d9d9c0a4f178f79e9b0fa67c5cddb9", "save_path": "github-repos/lean/agryman-theorem-proving-in-lean", "path": "github-repos/lean/agryman-theorem-proving-in-lean/theorem-proving-in-lean-cf5a3a19d0d9d9c0a4f178f79e9b0fa67c5cddb9/src/05-Tactics/example-5.1-2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.38392836728998264}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebra.category.Algebra.basic\n! leanprover-community/mathlib commit 79ffb5563b56fefdea3d60b5736dad168a9494ab\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Algebra.Subalgebra.Basic\nimport Mathbin.Algebra.FreeAlgebra\nimport Mathbin.Algebra.Category.Ring.Basic\nimport Mathbin.Algebra.Category.Module.Basic\n\n/-!\n# Category instance for algebras over a commutative ring\n\nWe introduce the bundled category `Algebra` of algebras over a fixed commutative ring `R ` along\nwith the forgetful functors to `Ring` and `Module`. We furthermore show that the functor associating\nto a type the free `R`-algebra on that type is left adjoint to the forgetful functor.\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\nuniverse v u\n\nvariable (R : Type u) [CommRing R]\n\n/-- The category of R-algebras and their morphisms. -/\nstructure AlgebraCat where\n  carrier : Type v\n  [isRing : Ring carrier]\n  [isAlgebra : Algebra R carrier]\n#align Algebra AlgebraCat\n\nattribute [instance] AlgebraCat.isRing AlgebraCat.isAlgebra\n\nnamespace AlgebraCat\n\ninstance : CoeSort (AlgebraCat R) (Type v) :=\n  \u27e8AlgebraCat.Carrier\u27e9\n\ninstance : Category (AlgebraCat.{v} R)\n    where\n  hom A B := A \u2192\u2090[R] B\n  id A := AlgHom.id R A\n  comp A B C f g := g.comp f\n\ninstance : ConcreteCategory.{v} (AlgebraCat.{v} R)\n    where\n  forget :=\n    { obj := fun R => R\n      map := fun R S f => (f : R \u2192 S) }\n  forget_faithful := { }\n\ninstance hasForgetToRing : HasForget\u2082 (AlgebraCat.{v} R) RingCat.{v}\n    where forget\u2082 :=\n    { obj := fun A => RingCat.of A\n      map := fun A\u2081 A\u2082 f => AlgHom.toRingHom f }\n#align Algebra.has_forget_to_Ring AlgebraCat.hasForgetToRing\n\ninstance hasForgetToModule : HasForget\u2082 (AlgebraCat.{v} R) (ModuleCat.{v} R)\n    where forget\u2082 :=\n    { obj := fun M => ModuleCat.of R M\n      map := fun M\u2081 M\u2082 f => AlgHom.toLinearMap f }\n#align Algebra.has_forget_to_Module AlgebraCat.hasForgetToModule\n\n/-- The object in the category of R-algebras associated to a type equipped with the appropriate\ntypeclasses. -/\ndef of (X : Type v) [Ring X] [Algebra R X] : AlgebraCat.{v} R :=\n  \u27e8X\u27e9\n#align Algebra.of AlgebraCat.of\n\n/-- Typecheck a `alg_hom` as a morphism in `Algebra R`. -/\ndef ofHom {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Algebra R X] [Ring Y] [Algebra R Y]\n    (f : X \u2192\u2090[R] Y) : of R X \u27f6 of R Y :=\n  f\n#align Algebra.of_hom AlgebraCat.ofHom\n\n@[simp]\ntheorem ofHom_apply {R : Type u} [CommRing R] {X Y : Type v} [Ring X] [Algebra R X] [Ring Y]\n    [Algebra R Y] (f : X \u2192\u2090[R] Y) (x : X) : ofHom f x = f x :=\n  rfl\n#align Algebra.of_hom_apply AlgebraCat.ofHom_apply\n\ninstance : Inhabited (AlgebraCat R) :=\n  \u27e8of R R\u27e9\n\n@[simp]\ntheorem coe_of (X : Type u) [Ring X] [Algebra R X] : (of R X : Type u) = X :=\n  rfl\n#align Algebra.coe_of AlgebraCat.coe_of\n\nvariable {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nalgebra. -/\n@[simps]\ndef ofSelfIso (M : AlgebraCat.{v} R) : AlgebraCat.of R M \u2245 M\n    where\n  hom := \ud835\udfd9 M\n  inv := \ud835\udfd9 M\n#align Algebra.of_self_iso AlgebraCat.ofSelfIso\n\nvariable {R} {M N U : ModuleCat.{v} R}\n\n@[simp]\ntheorem id_apply (m : M) : (\ud835\udfd9 M : M \u2192 M) m = m :=\n  rfl\n#align Algebra.id_apply AlgebraCat.id_apply\n\n@[simp]\ntheorem coe_comp (f : M \u27f6 N) (g : N \u27f6 U) : (f \u226b g : M \u2192 U) = g \u2218 f :=\n  rfl\n#align Algebra.coe_comp AlgebraCat.coe_comp\n\nvariable (R)\n\n/-- The \"free algebra\" functor, sending a type `S` to the free algebra on `S`. -/\n@[simps]\ndef free : Type u \u2964 AlgebraCat.{u} R\n    where\n  obj S :=\n    { carrier := FreeAlgebra R S\n      isRing := Algebra.semiringToRing R }\n  map S T f := FreeAlgebra.lift _ <| FreeAlgebra.\u03b9 _ \u2218 f\n  -- obviously can fill the next two goals, but it is slow\n  map_id' := by\n    intro X\n    ext1\n    simp only [FreeAlgebra.\u03b9_comp_lift]\n    rfl\n  map_comp' := by\n    intros\n    ext1\n    simp only [FreeAlgebra.\u03b9_comp_lift]\n    ext1\n    simp only [FreeAlgebra.lift_\u03b9_apply, CategoryTheory.coe_comp, Function.comp_apply,\n      types_comp_apply]\n#align Algebra.free AlgebraCat.free\n\n/-- The free/forget adjunction for `R`-algebras. -/\ndef adj : free.{u} R \u22a3 forget (AlgebraCat.{u} R) :=\n  Adjunction.mkOfHomEquiv\n    { homEquiv := fun X A => (FreeAlgebra.lift _).symm\n      -- Relying on `obviously` to fill out these proofs is very slow :(\n      homEquiv_naturality_left_symm := by\n        intros\n        ext\n        simp only [free_map, Equiv.symm_symm, FreeAlgebra.lift_\u03b9_apply, CategoryTheory.coe_comp,\n          Function.comp_apply, types_comp_apply]\n      homEquiv_naturality_right := by\n        intros\n        ext\n        simp only [forget_map_eq_coe, CategoryTheory.coe_comp, Function.comp_apply,\n          FreeAlgebra.lift_symm_apply, types_comp_apply] }\n#align Algebra.adj AlgebraCat.adj\n\ninstance : IsRightAdjoint (forget (AlgebraCat.{u} R)) :=\n  \u27e8_, adj R\u27e9\n\nend AlgebraCat\n\nvariable {R}\n\nvariable {X\u2081 X\u2082 : Type u}\n\n/-- Build an isomorphism in the category `Algebra R` from a `alg_equiv` between `algebra`s. -/\n@[simps]\ndef AlgEquiv.toAlgebraIso {g\u2081 : Ring X\u2081} {g\u2082 : Ring X\u2082} {m\u2081 : Algebra R X\u2081} {m\u2082 : Algebra R X\u2082}\n    (e : X\u2081 \u2243\u2090[R] X\u2082) : AlgebraCat.of R X\u2081 \u2245 AlgebraCat.of R X\u2082\n    where\n  hom := (e : X\u2081 \u2192\u2090[R] X\u2082)\n  inv := (e.symm : X\u2082 \u2192\u2090[R] X\u2081)\n  hom_inv_id' := by ext; exact e.left_inv x\n  inv_hom_id' := by ext; exact e.right_inv x\n#align alg_equiv.to_Algebra_iso AlgEquiv.toAlgebraIso\n\nnamespace CategoryTheory.Iso\n\n/-- Build a `alg_equiv` from an isomorphism in the category `Algebra R`. -/\n@[simps]\ndef toAlgEquiv {X Y : AlgebraCat R} (i : X \u2245 Y) : X \u2243\u2090[R] Y\n    where\n  toFun := i.hom\n  invFun := i.inv\n  left_inv := by tidy\n  right_inv := by tidy\n  map_add' := by tidy\n  map_mul' := by tidy\n  commutes' := by tidy\n#align category_theory.iso.to_alg_equiv CategoryTheory.Iso.toAlgEquiv\n\nend CategoryTheory.Iso\n\n/-- Algebra equivalences between `algebras`s are the same as (isomorphic to) isomorphisms in\n`Algebra`. -/\n@[simps]\ndef algEquivIsoAlgebraIso {X Y : Type u} [Ring X] [Ring Y] [Algebra R X] [Algebra R Y] :\n    (X \u2243\u2090[R] Y) \u2245 AlgebraCat.of R X \u2245 AlgebraCat.of R Y\n    where\n  hom e := e.toAlgebraIso\n  inv i := i.toAlgEquiv\n#align alg_equiv_iso_Algebra_iso algEquivIsoAlgebraIso\n\ninstance (X : Type u) [Ring X] [Algebra R X] : Coe (Subalgebra R X) (AlgebraCat R) :=\n  \u27e8fun N => AlgebraCat.of R N\u27e9\n\ninstance AlgebraCat.forget_reflects_isos : ReflectsIsomorphisms (forget (AlgebraCat.{u} R))\n    where reflects X Y f _ := by\n    skip\n    let i := as_iso ((forget (AlgebraCat.{u} R)).map f)\n    let e : X \u2243\u2090[R] Y := { f, i.to_equiv with }\n    exact \u27e8(is_iso.of_iso e.to_Algebra_iso).1\u27e9\n#align Algebra.forget_reflects_isos AlgebraCat.forget_reflects_isos\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Category/Algebra/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.6477982247516797, "lm_q1q2_score": 0.38392836199847763}}
{"text": "import Lean.Meta\n\nopen Lean\nopen Lean.Meta\n\ndef print (msg : MessageData) : MetaM Unit := do\ntrace[Meta.debug] msg\n\ndef showRecInfo (declName : Name) (majorPos? : Option Nat := none) : MetaM Unit := do\nlet info \u2190 mkRecursorInfo declName majorPos?\nprint (toString info)\n\ntheorem Iff.elim {a b c} (h\u2081 : (a \u2192 b) \u2192 (b \u2192 a) \u2192 c) (h\u2082 : a \u2194 b) : c :=\n  h\u2081 h\u2082.1 h\u2082.2\n\nset_option trace.Meta true\nset_option trace.Meta.isDefEq false\n\n#eval showRecInfo `Acc.recOn\n#eval showRecInfo `Prod.casesOn\n#eval showRecInfo `List.recOn\n#eval showRecInfo `List.casesOn\n#eval showRecInfo `List.brecOn\n\n#eval showRecInfo `Iff.elim (some 4)\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/recInfo1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.712232184238947, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.38388119653853164}}
{"text": "import for_mathlib.derived.K_projective\nimport for_mathlib.homological_complex_op\nimport for_mathlib.homology_iso_Ab\nimport for_mathlib.hom_single_iso\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.limits category_theory.preadditive\n\nvariables {C : Type u} {\u03b9 : Type*} [category.{v} C] [abelian C] {c : complex_shape \u03b9}\n\nnamespace bounded_homotopy_category\n\nopen hom_single_iso_setup opposite\n\nlemma aux\u2081_naturality_snd_var\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124) (f : B\u2081 \u27f6 B\u2082) :\n  (aux\u2081 P B\u2081 i).hom \u226b\n  (homology_functor AddCommGroup (complex_shape.up \u2124).symm i).map\n    ((nat_trans.map_homological_complex (preadditive_yoneda.map f)\n    (complex_shape.up \u2124).symm).app P.val.as.op) =\n  map_hom_complex_homology _ _ f _ _ \u226b (aux\u2081 P B\u2082 i).hom :=\nbegin\n  rw [\u2190 iso.comp_inv_eq],\n  ext : 2,\n  dsimp only [aux\u2081, iso.symm_hom, iso.symm_inv, homology_iso', homology.map_iso],\n  simp only [category.assoc],\n  rw [homology.map_eq_desc'_lift_left, homology.\u03c0'_desc'_assoc,\n    homology.map_eq_lift_desc'_left, homology.lift_\u03b9,\n    map_hom_complex_homology,\n    homology.map_eq_lift_desc'_left, homology.lift_\u03b9, homology.\u03c0'_desc'],\n  dsimp only [arrow.hom_mk_left, map_hom_complex',\n    nat_trans.map_homological_complex_app_f, homology_functor_map],\n  let t : _ := _, show _ \u226b _ \u226b t = _,\n  have ht : t = homology.\u03b9 _ _ _ \u226b\n    cokernel.map _ _ (homological_complex.X_prev_iso _ _).hom (\ud835\udfd9 _) _,\n  rotate 2, { dsimp, refl }, { rw [category.comp_id], apply homological_complex.d_to_eq },\n  { ext1, erw [homology.\u03c0'_\u03b9_assoc, homology.\u03c0'_desc', cokernel.\u03c0_desc], refl, },\n  rw [ht, homology.map_eq_lift_desc'_right, homology.lift_\u03b9_assoc], clear ht t,\n  let t : _ := _, show t \u226b _ = _,\n  have ht : t = kernel.map _ _ (\ud835\udfd9 _) (homological_complex.X_next_iso _ _).inv _ \u226b\n    homology.\u03c0' _ _ _,\n  rotate 2, { dsimp, apply sub_add_cancel },\n  { rw [category.id_comp], symmetry, apply homological_complex.d_from_eq },\n  { ext1, erw [homology.lift_\u03b9, category.assoc, homology.\u03c0'_\u03b9, kernel.lift_\u03b9_assoc], refl },\n  rw [ht, category.assoc, homology.\u03c0'_desc'_assoc, category.assoc, category.assoc], clear ht t,\n  rw [kernel.lift_\u03b9_assoc, cokernel.\u03c0_desc],\n  simp only [category.assoc, category.id_comp], refl,\nend\n\nlemma aux\u2082_naturality_snd_var\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124) (f : B\u2081 \u27f6 B\u2082) :\n  (aux\u2082 P B\u2081 i).inv \u226b P.map_hom_complex_homology i f _ (homological_complex.d_comp_d _ _ _ _) =\n  AddCommGroup.homology_map\n    (homological_complex.d_comp_d _ _ _ _)\n    (homological_complex.d_comp_d _ _ _ _)\n    (commsq.of_eq $ ((map_hom_complex' _ f).comm _ _).symm)\n    (commsq.of_eq $ ((map_hom_complex' _ f).comm _ _).symm) \u226b (aux\u2082 P B\u2082 i).inv :=\nbegin\n  dsimp only [aux\u2082, map_hom_complex_homology],\n  rw \u2190 AddCommGroup.homology_iso_inv_homology_map,\n  congr' 1,\n  dsimp only [homology.map'],\n  erw \u2190 has_homology.homology_map_eq',\n  congr',\nend\n.\n\nlemma quotient_add_group.lift_mk''\n  {G H : Type*} [add_group G] [add_group H] (N : add_subgroup G) [N.normal]\n  {\u03c6 : G \u2192+ H} (HN : \u2200 (x : G), x \u2208 N \u2192 \u03c6 x = 0) (g : G) :\n  (quotient_add_group.lift N \u03c6 HN) (quotient_add_group.mk' N g) = \u03c6 g :=\nquotient_add_group.lift_mk' _ _ _\n\nlemma _root_.AddCommGroup.has_homology_\u03b9_eq {A B C : AddCommGroup} (f : A \u27f6 B) (g : B \u27f6 C) (w : f \u226b g = 0) :\n  (AddCommGroup.has_homology f g w).\u03b9 =\n  (AddCommGroup.of_hom $ quotient_add_group.lift _ ((quotient_add_group.mk' f.range).comp g.ker.subtype)\n      begin\n        rintro x \u27e8a, ha\u27e9,\n        simp only [add_monoid_hom.comp_apply, quotient_add_group.coe_mk', quotient_add_group.eq_zero_iff],\n        exact \u27e8a, ha\u27e9,\n      end) \u226b\n    (AddCommGroup.cokernel_iso_range_quotient _).inv :=\nbegin\n  dsimp only [AddCommGroup.has_homology, AddCommGroup.of_hom],\n  ext x,\n  apply_fun (AddCommGroup.cokernel_iso_range_quotient f).hom,\n  { simp only [add_monoid_hom.comp_apply, comp_apply, quotient_add_group.lift_mk'',\n      AddCommGroup.cokernel_\u03c0_cokernel_iso_range_quotient_hom_apply,\n      category_theory.iso.inv_hom_id_apply] },\n  { erw [\u2190 AddCommGroup.mono_iff_injective], apply_instance },\nend\n.\n\nlemma _root_.add_monoid_hom.lift_of_surjective_apply {A B C : Type*}\n  [add_comm_group A] [add_comm_group B] [add_comm_group C]\n  (f : A \u2192+ B) (hf : function.surjective f) (g : {g : A \u2192+ C // f.ker \u2264 g.ker}) (a : A) :\n  add_monoid_hom.lift_of_surjective f hf g (f a) = g a :=\nbegin\n  show g _ = g _,\n  erw [\u2190 sub_eq_zero, \u2190 g.val.map_sub, \u2190 g.val.mem_ker],\n  apply g.2,\n  rw [f.mem_ker, f.map_sub, sub_eq_zero],\n  apply function.surj_inv_eq\nend\n\nlemma _root_.add_equiv.symm_mk_apply {A B : Type*} [add_comm_group A] [add_comm_group B]\n  (f : A \u2192 B) (g : B \u2192 A) (h1) (h2) (h3) (b : B) :\n  add_equiv.symm \u27e8f, g, h1, h2, h3\u27e9 b = g b := rfl\n\nlemma _root_.add_monoid_hom.subtype_mk_apply {A B : Type*} [add_comm_group A] [add_comm_group B]\n  (p : (A \u2192+ B) \u2192 Prop) (f : A \u2192+ B) (hf : p f) (a : A) :\n  subtype.mk f hf a = f a := rfl\n\nlemma homological_complex_hom_single_iso_natural_aux\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124)\n  (f : B\u2081 \u27f6 B\u2082)\n  (\u03c6 : (add_monoid_hom.ker ((hom_complex P B\u2081).d i (i - 1)))) :\n  ((map_hom_complex' P f).f i) \u03c6 \u2208 add_monoid_hom.ker ((hom_complex P B\u2082).d i (i - 1)) :=\nbegin\n  cases \u03c6 with \u03c6 h\u03c6,\n  rw [add_monoid_hom.mem_ker] at h\u03c6 \u22a2,\n  rw [\u2190 comp_apply, (map_hom_complex' P f).comm, comp_apply],\n  erw [h\u03c6, map_zero],\nend\n.\n\nattribute [simps] eq_to_iso\n\nlemma homological_complex_hom_single_iso_natural\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124)\n  (f : B\u2081 \u27f6 B\u2082)\n  (\u03c6 : (add_monoid_hom.ker ((hom_complex P B\u2081).d i (i - 1)))) :\n  homotopy_category.quotient_map_hom P.val.as ((homological_complex.single C _ i).obj B\u2081)\n    (((homological_complex.hom_single_iso P.val.as B\u2081 i).symm) \u03c6) \u226b\n  (single C i).map f =\n  homotopy_category.quotient_map_hom P.val.as ((homological_complex.single C _ i).obj B\u2082)\n    (((homological_complex.hom_single_iso P.val.as B\u2082 i).symm)\n      \u27e8(map_hom_complex' P f).f i \u03c6, homological_complex_hom_single_iso_natural_aux P i f \u03c6\u27e9) :=\nbegin\n  dsimp only [homotopy_category.quotient_map_hom, add_monoid_hom.mk'_apply,\n    single, homotopy_category.single, functor.comp_map],\n  rw [\u2190 functor.map_comp], congr' 1,\n  ext n,\n  dsimp only [homological_complex.comp_f, homological_complex.hom_single_iso_symm_apply_f],\n  split_ifs with hin, swap, { rw zero_comp },\n  subst n,\n  simp only [category.assoc], refine congr_arg2 _ rfl _,\n  dsimp only [map_hom_complex', nat_trans.map_homological_complex_app_f,\n    preadditive_yoneda_map_app_apply],\n  simp only [category.assoc], refine congr_arg2 _ rfl _,\n  dsimp only [homological_complex.single],\n  rw [\u2190 category_theory.eq_to_iso_hom, \u2190 iso.eq_inv_comp],\n  exact dif_pos rfl,\nend\n\nlemma hom_single_iso_naturality_snd_var_good\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124)\n  (f : B\u2081 \u27f6 B\u2082) :\n  (hom_single_iso P B\u2081 i).hom \u226b\n  (homology_functor _ _ i).map (nat_trans.app (nat_trans.map_homological_complex\n    (preadditive_yoneda.map f) _) _) =\n  (preadditive_yoneda.map $ (single C i).map f).app (op P) \u226b (hom_single_iso P B\u2082 i).hom :=\nbegin\n  dsimp only [hom_single_iso, iso.trans_hom, iso.symm_hom, functor.comp_map, functor.op_map,\n    functor.right_op_map, quiver.hom.unop_op],\n  simp only [category.assoc],\n  rw aux\u2081_naturality_snd_var,\n  simp_rw \u2190 category.assoc, congr' 1, simp_rw category.assoc,\n  rw aux\u2082_naturality_snd_var,\n  simp_rw \u2190 category.assoc, congr' 1,\n  rw [\u2190 iso.eq_inv_comp],\n  apply (AddCommGroup.has_homology _ _ _).ext_\u03b9,\n  apply (AddCommGroup.has_homology _ _ _).ext_\u03c0,\n  rotate, { apply homological_complex.d_comp_d }, { apply homological_complex.d_comp_d },\n  rw [AddCommGroup.homology_map, has_homology.map_\u03b9, has_homology.\u03c0_comp_desc],\n  dsimp only [map_hom_complex', nat_trans.map_homological_complex_app_f,\n    add_equiv_iso_AddCommGroup_iso],\n  simp only [AddCommGroup.has_homology_\u03b9_eq, \u2190 category.assoc],\n  rw [iso.eq_comp_inv],\n  dsimp only [AddCommGroup.has_homology],\n  simp only [category.assoc],\n  rw [\u2190 iso.inv_comp_eq],\n  ext1 \u03c6,\n  simp only [category_theory.comp_apply, AddCommGroup.kernel_iso_ker_inv_comp_\u03b9_apply,\n    AddCommGroup.cokernel_\u03c0_cokernel_iso_range_quotient_hom_apply,\n    AddCommGroup.of_hom, add_equiv.to_AddCommGroup_iso, add_equiv.coe_to_add_monoid_hom],\n  dsimp only [aux\u2083, preadditive_yoneda_map_app_apply],\n  dsimp only [add_equiv.surjective_congr, add_equiv.coe_mk, add_equiv.symm_mk_apply],\n  rw [add_monoid_hom.lift_of_surjective_apply, add_monoid_hom.subtype_mk_apply],\n  dsimp only [add_monoid_hom.comp_apply, add_equiv.coe_to_add_monoid_hom],\n  erw [homological_complex_hom_single_iso_natural P i f \u03c6],\n  rw [add_monoid_hom.lift_of_surjective_apply, add_monoid_hom.subtype_mk_apply],\n  dsimp only [add_monoid_hom.comp_apply, add_equiv.coe_to_add_monoid_hom],\n  rw [quotient_add_group.lift_mk'', add_equiv.apply_symm_apply, add_monoid_hom.comp_apply],\n  refl,\nend\n\nlemma hom_single_iso_naturality_snd_var\n  (P : bounded_homotopy_category C) {B\u2081 B\u2082 : C} (i : \u2124)\n  (f : B\u2081 \u27f6 B\u2082) (x : P \u27f6 (single C i).obj B\u2081) :\n  ((homology_functor _ _ i).map\n    ((nat_trans.map_homological_complex (preadditive_yoneda.map f) _).app P.val.as.op))\n      ((hom_single_iso P B\u2081 i).hom x) = ((hom_single_iso P B\u2082 i).hom (x \u226b (single C i).map f)) :=\nbegin\n  have := hom_single_iso_naturality_snd_var_good P i f,\n  apply_fun (\u03bb e, e x) at this,\n  exact this\nend\n\nend bounded_homotopy_category\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/hom_single_iso2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38372691360847244}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Tim Baumann, Stephen Morgan, Scott Morrison\n\nimport category_theory.category\nimport category_theory.functor\n\nimport .tactics\n\nuniverses u v\n\nnamespace category_theory\n\nstructure iso {C : Type u} [category.{u v} C] (X Y : C) :=\n(hom : X \u27f6 Y)\n(inv : Y \u27f6 X)\n(hom_inv_id : hom \u226b inv = \ud835\udfd9 X . obviously)\n(inv_hom_id : inv \u226b hom = \ud835\udfd9 Y . obviously)\n\nrestate_axiom iso.hom_inv_id\nrestate_axiom iso.inv_hom_id\nattribute [simp,ematch] iso.hom_inv_id_lemma iso.inv_hom_id_lemma\n\ninfixr ` \u2245 `:10  := iso             -- type as \\cong\n\nvariables {C : Type u} [\ud835\udc9e : category.{u v} C]\ninclude \ud835\udc9e\nvariables {X Y Z : C}\n\nnamespace iso\n\ninstance : has_coe (iso.{u v} X Y) (X \u27f6 Y) :=\n{ coe := iso.hom }\n\n-- -- These lemmas are quite common, to help us avoid having to muck around with associativity.\n-- -- If anyone has a suggestion for automating them away, I would be very appreciative.\n-- @[simp,ematch] lemma hom_inv_id_assoc_lemma (I : X \u2245 Y) (f : X \u27f6 Z) : I.hom \u226b I.inv \u226b f = f := \n-- begin\n--   -- `obviously'` says:\n--   rw [\u2190category.assoc_lemma, iso.hom_inv_id_lemma, category.id_comp_lemma]\n-- end\n\n-- @[simp,ematch] lemma inv_hom_id_assoc_lemma (I : X \u2245 Y) (f : Y \u27f6 Z) : I.inv \u226b I.hom \u226b f = f := \n-- begin\n--   -- `obviously'` says:\n--   rw [\u2190category.assoc_lemma, iso.inv_hom_id_lemma, category.id_comp_lemma]\n-- end\n\n@[extensionality] lemma ext\n  (\u03b1 \u03b2 : X \u2245 Y)\n  (w : \u03b1.hom = \u03b2.hom) : \u03b1 = \u03b2 :=\n  begin\n    induction \u03b1 with f g w\u03b11 w\u03b12,\n    induction \u03b2 with h k w\u03b21 w\u03b22,\n    simp at w,    \n    have p : g = k,\n      begin\n        induction w,\n        dsimp at *,\n        rw [\u2190 category.id_comp_lemma C k, \u2190w\u03b12, category.assoc_lemma, w\u03b21, category.comp_id_lemma]\n      end,\n    -- `obviously'` says:\n    induction p, induction w,\n    refl\n  end\n\n@[refl] def refl (X : C) : X \u2245 X := \n{ hom := \ud835\udfd9 X,\n  inv := \ud835\udfd9 X, \n  hom_inv_id := begin /- `obviously'` says: -/ simp end,\n  inv_hom_id := begin /- `obviously'` says: -/ simp end }\n\n-- TODO maybe these can have ematch?\n@[simp] lemma refl_map (X : C) : (iso.refl X).hom = \ud835\udfd9 X := rfl\n@[simp] lemma refl_inv  (X : C) : (iso.refl X).inv  = \ud835\udfd9 X := rfl\n\n@[trans] def trans (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : X \u2245 Z := \n{ hom := \u03b1.hom \u226b \u03b2.hom,\n  inv := \u03b2.inv \u226b \u03b1.inv,\n  hom_inv_id := begin /- `obviously'` says: -/ erw [category.assoc_lemma], conv { to_lhs, congr, skip, rw \u2190 category.assoc_lemma }, rw iso.hom_inv_id_lemma, rw category.id_comp_lemma, rw iso.hom_inv_id_lemma end,\n  inv_hom_id := begin /- `obviously'` says: -/ erw [category.assoc_lemma], conv { to_lhs, congr, skip, rw \u2190 category.assoc_lemma }, rw iso.inv_hom_id_lemma, rw category.id_comp_lemma, rw iso.inv_hom_id_lemma end }\n\ninfixr ` \u2662 `:80 := iso.trans -- type as \\diamonds\n\n@[simp,ematch] lemma trans_hom (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : (\u03b1 \u2662 \u03b2).hom = \u03b1.hom \u226b \u03b2.hom := rfl\n@[simp,ematch] lemma trans_inv (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : (\u03b1 \u2662 \u03b2).inv  = \u03b2.inv \u226b \u03b1.inv   := rfl\n\n@[symm] def symm (I : X \u2245 Y) : Y \u2245 X := \n{ hom := I.inv,\n  inv := I.hom,\n  hom_inv_id := begin /- `obviously'` says: -/ simp end,\n  inv_hom_id := begin /- `obviously'` says: -/ simp end }\n\nend iso\n\nclass is_iso (f : X \u27f6 Y) :=\n(inv : Y \u27f6 X)\n(hom_inv_id : f \u226b inv = \ud835\udfd9 X . obviously)\n(inv_hom_id : inv \u226b f = \ud835\udfd9 Y . obviously)\n\nrestate_axiom is_iso.hom_inv_id\nrestate_axiom is_iso.inv_hom_id\nattribute [simp,ematch] is_iso.hom_inv_id_lemma is_iso.inv_hom_id_lemma\n\nnamespace is_iso\n\ninstance (X : C) : is_iso (\ud835\udfd9 X) := \n{ inv := \ud835\udfd9 X, \n  hom_inv_id := by obviously',\n  inv_hom_id := by obviously' }\n\ninstance of_iso         (f : X \u2245 Y) : is_iso f.hom :=\n{ inv   := f.inv,\n  hom_inv_id := begin /- `obviously'` says: -/ simp end,\n  inv_hom_id := begin /- `obviously'` says: -/ simp end }\ninstance of_iso_inverse (f : X \u2245 Y) : is_iso f.inv  := \n{ inv   := f.hom,\n  hom_inv_id := begin /- `obviously'` says: -/ simp end,\n  inv_hom_id := begin /- `obviously'` says: -/ simp end }\n\nend is_iso\n\nclass epi  (f : X \u27f6 Y) := \n(left_cancellation : \u03a0 {Z : C} (g h : Y \u27f6 Z) (w : f \u226b g = f \u226b h), g = h)\nclass mono (f : X \u27f6 Y) :=\n(right_cancellation : \u03a0 {Z : C} (g h : Z \u27f6 X) (w : g \u226b f = h \u226b f), g = h)\n\ninstance epi_of_iso  (f : X \u27f6 Y) [is_iso f] : epi f  := \n{ left_cancellation := begin\n                         -- This is an interesting test case for better rewrite automation.\n                         intros,\n                         rw [\u2190category.id_comp_lemma C g, \u2190category.id_comp_lemma C h],\n                         rw [\u2190 is_iso.inv_hom_id_lemma f],\n                         erw [category.assoc_lemma, w, category.assoc_lemma],\n                       end }\ninstance mono_of_iso (f : X \u27f6 Y) [is_iso f] : mono f := \n{ right_cancellation := begin\n                         intros,\n                         rw [\u2190category.comp_id_lemma C g, \u2190category.comp_id_lemma C h],\n                         rw [\u2190 is_iso.hom_inv_id_lemma f],\n                         erw [\u2190category.assoc_lemma, w, \u2190category.assoc_lemma]\n                       end }\n\n@[simp] lemma cancel_epi  (f : X \u27f6 Y) [epi f]  (g h : Y \u27f6 Z) : (f \u226b g = f \u226b h) \u2194 g = h := \n\u27e8 \u03bb p, epi.left_cancellation g h p, begin /- `obviously'` says: -/ intros, cases a, refl end \u27e9\n@[simp] lemma cancel_mono (f : X \u27f6 Y) [mono f] (g h : Z \u27f6 X) : (g \u226b f = h \u226b f) \u2194 g = h := \n\u27e8 \u03bb p, mono.right_cancellation g h p, begin /- `obviously'` says: -/ intros, cases a, refl end \u27e9\n\nnamespace functor\n\nuniverses u\u2081 v\u2081 u\u2082 v\u2082 \nvariables {D : Type u\u2082}\n\nvariables [\ud835\udc9f : category.{u\u2082 v\u2082} D]\ninclude \ud835\udc9f\n\ndef on_isos (F : C \u219d D) {X Y : C} (i : X \u2245 Y) : (F X) \u2245 (F Y) :=\n{ hom := F.map i.hom,\n  inv := F.map i.inv,\n  hom_inv_id := by obviously',\n  inv_hom_id := by obviously' }\n\n@[simp,ematch] lemma on_isos_hom (F : C \u219d D) {X Y : C} (i : X \u2245 Y) : (F.on_isos i).hom = F.map i.hom := rfl\n@[simp,ematch] lemma on_isos_inv (F : C \u219d D) {X Y : C} (i : X \u2245 Y) : (F.on_isos i).inv = F.map i.inv := rfl\n\nend functor\n\nend category_theory", "meta": {"author": "semorrison", "repo": "lean-category-theory-pr", "sha": "7adc8d91835e883db0fe75aa33661bc1480dbe55", "save_path": "github-repos/lean/semorrison-lean-category-theory-pr", "path": "github-repos/lean/semorrison-lean-category-theory-pr/lean-category-theory-pr-7adc8d91835e883db0fe75aa33661bc1480dbe55/src/categories/isomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.3835757960890722}}
{"text": "import polytime\n\ndef time_bound_promise (c : code) (promise : set ptree) (b : \u2115 \u2192 \u2115) : Prop :=\n\u2200 x \u2208 promise, \u2203 t \u2208 c.time x, t \u2264 b x.sizeof\n\n@[simp] lemma time_bound_promise_univ (c : code) (b : \u2115 \u2192 \u2115) :\n  time_bound_promise c set.univ b \u2194 time_bound c b :=\nby simp [time_bound, time_bound_promise]\n\nlemma pfun.preimage_def' {\u03b1 \u03b2} {f : \u03b1 \u2192. \u03b2} {S : set \u03b2} {x\u2081} (hx\u2081 : x\u2081 \u2208 f.preimage S) {x\u2082} (hx\u2082 : x\u2082 \u2208 f x\u2081) :\n  x\u2082 \u2208 S :=\nby { rcases hx\u2081 with \u27e8x\u2082, H, hx\u2082'\u27e9, cases part.mem_unique hx\u2082 hx\u2082', exact H, }\n\nlemma time_bound_promise_comp {c\u2081 c\u2082 : code} {P\u2081 P\u2082 : set ptree} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (hm : monotone b\u2081)\n  (hb\u2081 : time_bound_promise c\u2081 P\u2081 b\u2081) (hb\u2082 : time_bound_promise c\u2082 P\u2082 b\u2082) :\n  time_bound_promise (c\u2081.comp c\u2082) (P\u2082 \u2229 c\u2082.eval.preimage P\u2081) (\u03bb t, b\u2081 (b\u2082 t) + b\u2082 t + 1) :=\nbegin\n  rintros v \u27e8hv\u2081, hv\u2082\u27e9,\n  obtain \u27e8t\u2082, ht\u2082, hb\u2082\u27e9 := hb\u2082 v hv\u2081,\n  obtain \u27e8v', hv'\u27e9 := (_ : \u2203 v', v' \u2208 c\u2082.eval v), swap,\n  { rw [\u2190 part.dom_iff_mem, \u2190 time_dom_iff_eval_dom, part.dom_iff_mem], use [t\u2082, ht\u2082], },\n  obtain \u27e8t\u2081, ht\u2081, hb\u2081\u27e9 := hb\u2081 v' (pfun.preimage_def' hv\u2082 hv'),\n  use t\u2081 + t\u2082 + 1, split,\n  { rw \u2190 part.eq_some_iff at ht\u2081 ht\u2082 hv', simp [code.time, ht\u2081, ht\u2082, hv', add_def], ring, },\n  { mono*, exact hb\u2081.trans (hm ((eval_sizeof_le_time hv' ht\u2082).trans hb\u2082)), },\nend\n\ndef polytime_promise (c : code) (promise : set ptree) : Prop :=\n\u2203 p : polynomial \u2115, time_bound_promise c promise (\u03bb n, p.eval n)\n\nlemma polytime_promise_comp {c\u2081 c\u2082 : code} {P\u2081 P\u2082 : set ptree} :\n  polytime_promise c\u2081 P\u2081 \u2192 polytime_promise c\u2082 P\u2082 \u2192 polytime_promise (c\u2081.comp c\u2082) (P\u2082 \u2229 c\u2082.eval.preimage P\u2081)\n| \u27e8p\u2081, e\u2081\u27e9 \u27e8p\u2082, e\u2082\u27e9 := by { use (p\u2081.comp p\u2082) + p\u2082 + 1, convert time_bound_promise_comp (monotone_polynomial_nat _) e\u2081 e\u2082, simp, }\n\n@[simp] lemma polytime_promise.univ {c : code} :\n  polytime_promise c set.univ \u2194 polytime c :=\nby simp [polytime_promise, polytime]\n\nlemma pfun.mem_ran_iff {\u03b1 \u03b2} {f : \u03b1 \u2192. \u03b2} {x} :\n  x \u2208 f.ran \u2194 \u2203 y, x \u2208 f y := by refl\n\n@[simp] lemma pfun.preimage_ran {\u03b1 \u03b2} (f : \u03b1 \u2192. \u03b2) :\n  f.preimage f.ran = f.dom :=\nby { ext, simp [pfun.mem_ran_iff], tauto, }\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/promise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.38357579210355286}}
{"text": "import Std.Tactic.Basic\n\nprivate def nonDecid (P : Prop) (x : P) : P := by\n  by_contra h\n  guard_hyp h : \u00acP\n  guard_target = False\n  exact h x\n\nprivate def decid (P : Prop) [Decidable P] (x : P) : P := by\n  by_contra h\n  guard_hyp h : \u00acP\n  guard_target = False\n  exact h x\n\nexample (P : Prop) [Decidable P] : nonDecid P = decid P := by\n  delta nonDecid nonDecid.proof_1 decid decid.proof_1\n  guard_target =\n    (fun x : P => Classical.byContradiction fun h => h x) =\n    (fun x : P => Decidable.byContradiction fun h => h x)\n  rfl\n\nexample (P : Prop) : P \u2192 P := by\n  by_contra\n  guard_hyp \u2039_\u203a : \u00ac(P \u2192 P)\n  exact \u2039\u00ac(P \u2192 P)\u203a id\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/test/by_contra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.38332036532237396}}
{"text": "import algebra.module\n\n\nlemma linear_map.to_fun_eq_coe_fn {\u03b1 \u03b2 \u03b3 : Type*} [ring \u03b1] [add_comm_group \u03b2] [add_comm_group \u03b3] [module \u03b1 \u03b2] [module \u03b1 \u03b3] \n  (f : \u03b2 \u2192\u2097[\u03b1] \u03b3): linear_map.to_fun f = \u21d1f := rfl", "meta": {"author": "skbaek", "repo": "cvx", "sha": "c50c790c9116f9fac8dfe742903a62bdd7292c15", "save_path": "github-repos/lean/skbaek-cvx", "path": "github-repos/lean/skbaek-cvx/cvx-c50c790c9116f9fac8dfe742903a62bdd7292c15/src/missing_mathlib/algebra/module.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.38332036532237385}}
{"text": "import Lean\n\n@[simp] theorem ex1 (x : Nat) : 2 * x = x + x :=\n  sorry\n\n@[simp] theorem ex2 (xs : List \u03b1) : xs ++ [] = xs :=\n  sorry\n\n@[simp] theorem ex3 (xs ys zs : List \u03b1) : (xs ++ ys) ++ zs = xs ++ (ys ++ zs) :=\n  sorry\n\n@[simp] theorem ex5 (p : Prop) : p \u2228 True :=\n  sorry\n\n@[simp] theorem ex4 (xs : List \u03b1) : \u00ac(x :: xs = []) :=\n  sorry\n\n@[simp] theorem ex6 (p q : Prop) : p \u2228 q \u2194 q \u2228 p:=\n  sorry\n\n@[simp high] theorem ex7 [Add \u03b1] (a b : \u03b1) : a + b = b + a :=\n  sorry\n\n@[simp\u2193] theorem ex8 [Add \u03b1] (p q : Prop) : (\u00ac (p \u2227 q)) = (\u00acp \u2228 \u00acq) :=\n  sorry\n\naxiom aux {\u03b1} (f : List \u03b1 \u2192 List \u03b1) (xs ys : List \u03b1) : f (xs ++ ys) ++ [] = f (xs ++ ys)\n\nopen Lean\nopen Lean.Meta\n\ndef tst1 : MetaM Unit := do\n  let thms  \u2190 Meta.getSimpTheorems\n  trace[Meta.debug] \"{thms.pre}\\n-----\\n{thms.post}\"\n\nset_option trace.Meta.debug true in\n#eval tst1\n\ndef tst2 : MetaM Unit := do\n  let c \u2190 getConstInfo `aux\n  forallTelescopeReducing c.type fun xs type => do\n    match type.eq? with\n    | none => throwError \"unexpected\"\n    | some (_, lhs, _) =>\n      trace[Meta.debug] \"lhs: {lhs}\"\n      let s \u2190 Meta.getSimpTheorems\n      let m \u2190 s.post.getMatch lhs\n      trace[Meta.debug] \"result: {m}\"\n      assert! m.any fun s => s.origin == .decl `ex2\n\n\nset_option trace.Meta.debug true in\n#eval tst2\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/simp1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.749087201911703, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3833203595896002}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, \u2203 N : \u2124, |y - (N : \u211d) * x| < 1 :=\nbegin\n  assume (y : \u211d) (h1 : y \u2208 Icc 0 1),\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  -- $i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor$,\n  -- which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n  -- $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$\n  -- is an infinite subset of $\\left[0,1\\right]$.\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h9 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h10 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h11 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h12 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h13 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h14 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h15 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h16 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h17 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h18 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h19 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h20 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h21 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h22 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h23 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h24 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h25 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h26 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor (j : \u211d) * \u03b1, from sorry,\n  have h27 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - floor (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - floor\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by sorry\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by sorry\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from sorry,\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : \u03b5 > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 \u03b5 h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 \u03b5 h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := sorry,\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from sorry,\n    \n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from sorry,\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline_with_comments-Natural-Language-Proof-Translation/lean_proof_outline_with_comments-4_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8244619263765706, "lm_q2_score": 0.4649015713733884, "lm_q1q2_score": 0.3832936451099985}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.cochain_complex.cm1\nimport for_mathlib.algebraic_topology.homotopical_algebra.cochain_complex.cm5a\nimport for_mathlib.algebra.homology.termwise_split\nimport for_mathlib.algebra.homology.lifting\n\nnoncomputable theory\n\nopen category_theory category_theory.category algebraic_topology\n\nnamespace category_theory\n\nvariables {C D : Type*} [category C] [category D]\n  {A B X Y : C} (i : A \u27f6 B) (p : X \u27f6 Y)\n\nlemma has_lifting_property.iff_of_fully_faithful (F : C \u2964 D)\n  [full F] [faithful F] :\n  has_lifting_property (F.map i) (F.map p) \u2194\n    has_lifting_property i p :=\nbegin\n  split,\n  { introI,\n    exact \u27e8\u03bb f g sq, \u27e8\u27e8{ l := F.preimage ((sq.map F).lift),\n      fac_left' := F.map_injective (by simp),\n      fac_right' := F.map_injective (by simp), }\u27e9\u27e9\u27e9, },\n  { introI,\n    refine \u27e8\u03bb f g sq, _\u27e9,\n    have sq' : comm_sq (F.preimage f) i p (F.preimage g) :=\n      \u27e8F.map_injective (by simp only [functor.map_comp, functor.image_preimage, sq.w])\u27e9,\n    exact \u27e8\u27e8{ l := F.map sq'.lift,\n      fac_left' :=\n        by simpa only [F.map_comp, F.image_preimage] using F.congr_map sq'.fac_left,\n      fac_right' :=\n        by simpa only [F.map_comp, F.image_preimage] using F.congr_map sq'.fac_right, }\u27e9\u27e9, },\nend\n\nend category_theory\n\nvariables {C : Type*} [category C] [abelian C]\n\nnamespace cochain_complex\n\nnamespace minus\n\nnamespace projective_model_structure\n\nopen cochain_complex.hom_complex\n\nsection\n\nvariables {A B : minus C} (i : A \u27f6 B) (hi : (arrow_classes C).cof i)\ninclude hi\n\ndef splittings_of_cof (n : \u2124) : splitting (i.f n) ((limits.cokernel.\u03c0 i).f n) :=\nbegin\n-- show that it is a short_exact sequence\n-- show that the cokernel is projective\n-- then there is a splitting...\n  sorry,\nend\n\ndef cocycle_of_cof : cocycle (limits.cokernel i).obj A.obj 1 :=\ntwist.iso_of_termwise_split.z (splittings_of_cof i hi)\n\ndef iso_twist_of_cof : twist (cocycle_of_cof i hi) \u2245 B.obj :=\ntwist.iso_of_termwise_split (splittings_of_cof i hi)\n\ndef arrow_iso_of_cof : arrow.mk (twist.inr (cocycle_of_cof i hi)) \u2245 arrow.mk (\u03b9.map i) := sorry\n\nend\n\ndef CM4a : (arrow_classes C).CM4a :=\n\u03bb A B X Y i hi p hp, begin\n  --has_lifting_property.iff_of_arrow_iso_left (arrow_iso_of_cof i hi.1) p,\n  --cochain_complex.lifting.lift_of_coboundary\n  sorry,\nend\n\ndef CM4b : (arrow_classes C).CM4b := sorry\n\ndef CM4 : (arrow_classes C).CM4 :=\n  \u27e8CM4a, CM4b\u27e9\n\nend projective_model_structure\n\nend minus\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cochain_complex/cm4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8175744806385543, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.38327125309217364}}
{"text": "import Rings.Lefschetz\n\nopen Lefschetz\nopen fol\nopen Fields\n\n\nnamespace fol\n\nvariable {L : Language}\n\ndef no_finite_model (T : Theory L) : Prop :=\n\u2200 (M : Model T), infinite M.1\n\n\nend fol\n\nnamespace Lefschetz_current\n\n/-- Lefschetz part 1. Any sentence or its negation can be deduced in ACF\u2080-/\ntheorem is_complete'_ACF\u2080 : is_complete' ACF\u2080 :=\nbegin\n  sorry\nend\n\nend Lefschetz_current\n", "meta": {"author": "Jlh18", "repo": "ModelTheoryInLean8", "sha": "fbda7d869d4169b6e739bb74165e99ee03ca63d6", "save_path": "github-repos/lean/Jlh18-ModelTheoryInLean8", "path": "github-repos/lean/Jlh18-ModelTheoryInLean8/ModelTheoryInLean8-fbda7d869d4169b6e739bb74165e99ee03ca63d6/Trash/Rings/current.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8175744584140003, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3832712426735111}}
{"text": "import Iris.BI.BigOp\nimport Iris.BI.Classes\nimport Iris.BI.Extensions\nimport Iris.BI.Interface\nimport Iris.BI.DerivedConnectives\nimport Iris.Std.Classes\nimport Iris.Std.Rewrite\nimport Iris.Std.TC\n\nnamespace Iris.BI\nopen Iris.Std\nopen BI\n\n-- Entails\ninstance entails_anti_symm [BI PROP] : AntiSymm (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (\u00b7 \u22a2 \u00b7) where\n  anti_symm := by\n    intro _ _ H1 H2\n    rw' [equiv_entails]\n    exact And.intro H1 H2\n\n-- Logic\ntheorem and_elim_l' [BI PROP] {P Q R : PROP} : (P \u22a2 R) \u2192 P \u2227 Q \u22a2 R := by\n  intro H\n  rw' [and_elim_l, H]\n\ntheorem and_elim_r' [BI PROP] {P Q R : PROP} : (Q \u22a2 R) \u2192 P \u2227 Q \u22a2 R := by\n  intro H\n  rw' [and_elim_r, H]\n\ntheorem or_intro_l' [BI PROP] {P Q R : PROP} : (P \u22a2 Q) \u2192 P \u22a2 Q \u2228 R := by\n  intro H\n  rw' [H, \u2190 or_intro_l]\n\ntheorem or_intro_r' [BI PROP] {P Q R : PROP} : (P \u22a2 R) \u2192 P \u22a2 Q \u2228 R := by\n  intro H\n  rw' [H, \u2190 or_intro_r]\n\ntheorem impl_intro_l [BI PROP] {P Q R : PROP} : (Q \u2227 P \u22a2 R) \u2192 P \u22a2 Q \u2192 R := by\n  intro H\n  apply impl_intro_r\n  rw' [\u2190 H]\n  apply and_intro\n  \u00b7 exact and_elim_r\n  \u00b7 exact and_elim_l\n\ntheorem impl_elim [BI PROP] {P Q R : PROP} : (P \u22a2 Q \u2192 R) \u2192 (P \u22a2 Q) \u2192 P \u22a2 R := by\n  intro H1 H2\n  rw' [\u2190 impl_elim_l' H1]\n  apply and_intro\n  \u00b7 simp\n  \u00b7 exact H2\n\ntheorem impl_elim_r' [BI PROP] {P Q R : PROP} : (Q \u22a2 P \u2192 R) \u2192 P \u2227 Q \u22a2 R := by\n  intros H\n  apply impl_elim (Q := P)\n  \u00b7 rw' [and_elim_r, H]\n  \u00b7 rw' [and_elim_l]\n\ntheorem impl_elim_l [BI PROP] {P Q : PROP} : (P \u2192 Q) \u2227 P \u22a2 Q := by\n  apply impl_elim_l'\n  simp\n\ntheorem impl_elim_r [BI PROP] {P Q : PROP} : P \u2227 (P \u2192 Q) \u22a2 Q := by\n  apply impl_elim_r'\n  simp\n\ntheorem False_elim [BI PROP] {P : PROP} : False \u22a2 P := by\n  apply pure_elim'\n  simp\n\ntheorem True_intro [BI PROP] {P : PROP} : P \u22a2 True := by\n  apply pure_intro\n  simp\n\n@[rwMonoRule]\ntheorem and_mono [BI PROP] {P P' Q Q' : PROP} : (P \u22a2 Q) \u2192 (P' \u22a2 Q') \u2192 P \u2227 P' \u22a2 Q \u2227 Q' := by\n  intro H1 H2\n  apply and_intro\n  \u00b7 rw' [\u2190 H1, and_elim_l]\n  \u00b7 rw' [\u2190 H2, and_elim_r]\n\n@[rwMonoRule]\ntheorem or_mono [BI PROP] {P P' Q Q' : PROP} : (P \u22a2 Q) \u2192 (P' \u22a2 Q') \u2192 P \u2228 P' \u22a2 Q \u2228 Q' := by\n  intro H1 H2\n  apply or_elim\n  \u00b7 apply or_intro_l'\n    exact H1\n  \u00b7 apply or_intro_r'\n    exact H2\n\n@[rwMonoRule]\ntheorem impl_mono [BI PROP] {P P' Q Q' : PROP} : (Q \u22a2 P) \u2192 (P' \u22a2 Q') \u2192 (P \u2192 P') \u22a2 Q \u2192 Q' := by\n  intro HP HQ\n  apply impl_intro_r\n  rw' [HP, \u2190 HQ]\n  apply impl_elim_l'\n  simp\n\n@[rwMonoRule]\ntheorem forall_mono [BI PROP] {\u03a6 \u03a8 : \u03b1 \u2192 PROP} :\n  (\u2200 a, \u03a6 a \u22a2 \u03a8 a) \u2192 (\u2200 a, \u03a6 a) \u22a2 \u2200 a, \u03a8 a\n:= by\n  intro H\u03c6\n  apply forall_intro\n  intro a\n  rw' [\u2190 H\u03c6 a, \u2190 forall_elim _]\n\n@[rwMonoRule]\ntheorem exist_mono [BI PROP] {\u03a6 \u03a8 : \u03b1 \u2192 PROP} :\n  (\u2200 a, \u03a6 a \u22a2 \u03a8 a) \u2192 (\u2203 a, \u03a6 a) \u22a2 \u2203 a, \u03a8 a\n:= by\n  intro H\u03c6\n  apply exist_elim\n  intro a\n  rw' [H\u03c6 a, exist_intro _]\n\ninstance and_idemp [BI PROP] : Idemp (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2227 \u00b7]) where\n  idemp := by\n    intro _\n    apply anti_symm\n    \u00b7 exact and_elim_l\n    \u00b7 apply and_intro\n      <;> simp\n\ninstance or_idemp [BI PROP] : Idemp (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2228 \u00b7]) where\n  idemp := by\n    intro _\n    apply anti_symm\n    \u00b7 apply or_elim\n      <;> simp\n    \u00b7 exact or_intro_l\n\ninstance and_comm [BI PROP] : Comm (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2227 \u00b7]) where\n  comm := by\n    intros\n    apply anti_symm\n    case left =>\n      apply and_intro\n      \u00b7 exact and_elim_r\n      \u00b7 exact and_elim_l\n    case right =>\n      apply and_intro\n      \u00b7 exact and_elim_r\n      \u00b7 exact and_elim_l\n\ninstance True_and [BI PROP] : LeftId (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) `[iprop| True] (`[iprop| \u00b7 \u2227 \u00b7]) where\n  left_id := by\n    intros\n    apply anti_symm\n    case left =>\n      exact and_elim_r\n    case right =>\n      apply and_intro ?_ reflexivity\n      apply pure_intro\n      simp\n\ninstance and_True [BI PROP] : RightId (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) `[iprop| True] (`[iprop| \u00b7 \u2227 \u00b7]) where\n  right_id := by\n    intros\n    apply anti_symm\n    case left =>\n      exact and_elim_l\n    case right =>\n      apply and_intro reflexivity ?_\n      apply pure_intro\n      simp\n\ninstance and_assoc [BI PROP] : Assoc (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2227 \u00b7]) where\n  assoc := by\n    intro _ _ _\n    apply anti_symm\n    <;> apply and_intro\n    <;> try apply and_intro\n    all_goals\n      simp [and_elim_l, and_elim_r]\n      try { apply and_elim_l' ; simp [and_elim_l, and_elim_r] }\n      try { apply and_elim_r' ; simp [and_elim_l, and_elim_r] }\n\ntheorem and_or_l [BI PROP] {P Q R : PROP} : P \u2227 (Q \u2228 R) \u22a3\u22a2 P \u2227 Q \u2228 P \u2227 R := by\n  apply anti_symm\n  case left =>\n    apply impl_elim_r'\n    apply or_elim\n    <;> apply impl_intro_l\n    \u00b7 exact or_intro_l\n    \u00b7 exact or_intro_r\n  case right =>\n    apply and_intro\n    \u00b7 apply or_elim\n      <;> apply and_elim_l'\n      <;> simp\n    \u00b7 apply or_elim\n      <;> apply and_elim_r'\n      \u00b7 exact or_intro_l\n      \u00b7 exact or_intro_r\n\ntheorem and_exist_l [BI PROP] {P : PROP} {\u03a8 : \u03b1 \u2192 PROP} : P \u2227 (\u2203 a, \u03a8 a) \u22a3\u22a2 \u2203 a, P \u2227 \u03a8 a := by\n  apply anti_symm\n  case left =>\n    apply impl_elim_r'\n    apply exist_elim\n    intro a\n    apply impl_intro_l\n    rw' [\u2190 exist_intro a]\n  case right =>\n    apply exist_elim\n    intro a\n    apply and_intro\n    \u00b7 rw' [and_elim_l]\n    \u00b7 rw' [\u2190 exist_intro a, and_elim_r]\n\ntheorem or_alt [BI PROP] {P Q : PROP} : P \u2228 Q \u22a3\u22a2 \u2203 (b : Bool), if b then P else Q := by\n  apply anti_symm\n  case left =>\n    apply or_elim\n    \u00b7 rw' [\u2190 exist_intro true]\n    \u00b7 rw' [\u2190 exist_intro false]\n  case right =>\n    apply exist_elim\n    intro b\n    cases b\n    \u00b7 rw' [\u2190 or_intro_r]\n    \u00b7 rw' [\u2190 or_intro_l]\n\n-- BI\n@[rwMonoRule]\ntheorem wand_mono [BI PROP] {P P' Q Q' : PROP} : (Q \u22a2 P) \u2192 (P' \u22a2 Q') \u2192 (P -\u2217 P') \u22a2 Q -\u2217 Q' := by\n  intro HP HQ\n  apply wand_intro_r\n  rw' [HP, \u2190 HQ]\n  apply wand_elim_l'\n  simp\n\ninstance sep_comm [BI PROP] : Comm (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2217 \u00b7]) where\n  comm := by\n    intros\n    apply anti_symm\n    <;> exact sep_comm'\n\ninstance sep_assoc [BI PROP] : Assoc (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) (`[iprop| \u00b7 \u2217 \u00b7]) where\n  assoc := by\n    intros P Q R\n    apply anti_symm\n    case left =>\n      rw' [\n        (comm : P \u2217 (Q \u2217 R) \u22a3\u22a2 _),\n        (comm : P \u2217 Q \u22a3\u22a2 _),\n        (comm : Q \u2217 R \u22a3\u22a2 _),\n        (comm : (Q \u2217 P) \u2217 R \u22a3\u22a2 _),\n        sep_assoc']\n    case right =>\n      exact sep_assoc'\n\ninstance emp_sep [BI PROP] : LeftId (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) `[iprop| emp] (`[iprop| \u00b7 \u2217 \u00b7]) where\n  left_id := by\n    intros\n    apply anti_symm\n    \u00b7 exact emp_sep_2\n    \u00b7 exact emp_sep_1\n\ninstance sep_emp [BI PROP] : RightId (\u03b1 := PROP) (\u00b7 \u22a3\u22a2 \u00b7) `[iprop| emp] (`[iprop| \u00b7 \u2217 \u00b7]) where\n  right_id := by\n    intro x\n    rw' [(comm : `[iprop| x \u2217 emp] \u22a3\u22a2 _), (left_id : emp \u2217 x \u22a3\u22a2 _)]\n\ntheorem True_sep_2 [BI PROP] {P : PROP} : P \u22a2 True \u2217 P := by\n  rw' [emp_sep_1]\n  apply sep_mono ?_ reflexivity\n  apply pure_intro\n  simp\n\ntheorem wand_intro_l [BI PROP] {P Q R : PROP} : (Q \u2217 P \u22a2 R) \u2192 P \u22a2 Q -\u2217 R := by\n  rw' [(comm : Q \u2217 P \u22a3\u22a2 _)]\n  exact wand_intro_r\n\ntheorem wand_elim_l [BI PROP] {P Q : PROP} : (P -\u2217 Q) \u2217 P \u22a2 Q := by\n  apply wand_elim_l'\n  simp\n\ntheorem wand_elim_r [BI PROP] {P Q : PROP} : P \u2217 (P -\u2217 Q) \u22a2 Q := by\n  rw' [sep_comm', wand_elim_l]\n\ntheorem wand_elim_r' [BI PROP] {P Q R : PROP} : (Q \u22a2 P -\u2217 R) \u2192 P \u2217 Q \u22a2 R := by\n  intro H\n  rw' [H, wand_elim_r]\n\ntheorem sep_or_l [BI PROP] {P Q R : PROP} : P \u2217 (Q \u2228 R) \u22a3\u22a2 (P \u2217 Q) \u2228 (P \u2217 R) := by\n  apply anti_symm\n  case left =>\n    apply wand_elim_r'\n    apply or_elim\n    <;> apply wand_intro_l\n    \u00b7 exact or_intro_l\n    \u00b7 exact or_intro_r\n  case right =>\n    apply or_elim\n    \u00b7 rw' [\u2190 or_intro_l]\n    \u00b7 rw' [\u2190 or_intro_r]\n\ntheorem sep_or_r [BI PROP] {P Q R : PROP} : (P \u2228 Q) \u2217 R \u22a3\u22a2 (P \u2217 R) \u2228 (Q \u2217 R) := by\n  rw' [!(comm : _ \u2217 R \u22a3\u22a2 _), sep_or_l]\n\ntheorem sep_exist_l [BI PROP] {P : PROP} {\u03a8 : \u03b1 \u2192 PROP} : P \u2217 (\u2203 a, \u03a8 a) \u22a3\u22a2 \u2203 a, P \u2217 \u03a8 a := by\n  apply anti_symm\n  case left =>\n    apply wand_elim_r'\n    apply exist_elim\n    intro a\n    apply wand_intro_l\n    rw' [\u2190 exist_intro a]\n  case right =>\n    apply exist_elim\n    intro a\n    rw' [(exist_intro _ : \u03a8 a \u22a2 _)]\n\ntheorem sep_exist_r [BI PROP] {\u03a6 : \u03b1 \u2192 PROP} {Q : PROP} : (\u2203 a, \u03a6 a) \u2217 Q \u22a3\u22a2 \u2203 a, \u03a6 a \u2217 Q := by\n  apply anti_symm\n  all_goals\n    rw' [(comm : _ \u2217 Q \u22a3\u22a2 _), sep_exist_l]\n    apply exist_mono\n    intro _\n    rw' [(comm : Q \u2217 _ \u22a3\u22a2 _)]\n\ntheorem wand_iff_refl [BI PROP] {P : PROP} : \u22a2 P \u2217-\u2217 P := by\n  apply and_intro\n  <;> apply wand_intro_l\n  <;> rw' [(right_id : P \u2217 emp \u22a3\u22a2 _)]\n\ntheorem wand_entails [BI PROP] {P Q : PROP} : (\u22a2 P -\u2217 Q) \u2192 P \u22a2 Q := by\n  intro H\n  rw' [\u2190 (left_id : emp \u2217 P \u22a3\u22a2 _), H]\n  exact wand_elim_l\n\ntheorem entails_wand [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 \u22a2 P -\u2217 Q := by\n  intro H\n  rw' [\u2190 H]\n  apply wand_intro_r\n  rw' [(left_id : emp \u2217 Q \u22a3\u22a2 _)]\n\ntheorem equiv_wand_iff [BI PROP] {P Q : PROP} : (P \u22a3\u22a2 Q) \u2192 \u22a2 P \u2217-\u2217 Q := by\n  intro H\n  rw' [H]\n  exact wand_iff_refl\n\ntheorem wand_iff_equiv [BI PROP] {P Q : PROP} : (\u22a2 P \u2217-\u2217 Q) \u2192 (P \u22a3\u22a2 Q) := by\n  intro HPQ\n  apply anti_symm\n  <;> apply wand_entails\n  <;> rw' [HPQ]\n  <;> simp [bi_wand_iff, and_elim_l, and_elim_r]\n\n-- Pure\ntheorem pure_elim (\u03c6 : Prop) [BI PROP] {Q R : PROP} : (Q \u22a2 \u231c\u03c6\u231d) \u2192 (\u03c6 \u2192 Q \u22a2 R) \u2192 Q \u22a2 R := by\n  intro HQ HQR\n  rw' [\u2190 (idemp : Q \u2227 Q \u22a3\u22a2 _), HQ]\n  apply impl_elim_l'\n  apply pure_elim'\n  intro H\u03c6\n  apply impl_intro_l\n  rw' [and_elim_l]\n  exact HQR H\u03c6\n\ntheorem pure_mono {\u03c61 \u03c62 : Prop} [BI PROP] : (\u03c61 \u2192 \u03c62) \u2192 \u231c\u03c61\u231d \u22a2 (\u231c\u03c62\u231d : PROP) := by\n  intro H12\n  apply pure_elim'\n  intro H1\n  apply pure_intro\n  exact H12 H1\n\ntheorem pure_elim_l {\u03c6 : Prop} [BI PROP] {Q R : PROP} : (\u03c6 \u2192 Q \u22a2 R) \u2192 \u231c\u03c6\u231d \u2227 Q \u22a2 R := by\n  intro H\n  apply pure_elim \u03c6\n  \u00b7 exact and_elim_l\n  \u00b7 intro H\u03c6\n    rw' [H H\u03c6]\n    exact and_elim_r\n\ntheorem pure_True {\u03c6 : Prop} [BI PROP] : \u03c6 \u2192 \u231c\u03c6\u231d \u22a3\u22a2 (True : PROP) := by\n  intro H\u03c6\n  apply anti_symm\n  case left =>\n    apply pure_intro\n    exact True.intro\n  case right =>\n    apply pure_intro\n    exact H\u03c6\n\ntheorem pure_and {\u03c61 \u03c62 : Prop} [BI PROP] : \u231c\u03c61 \u2227 \u03c62\u231d \u22a3\u22a2 \u231c\u03c61\u231d \u2227 (\u231c\u03c62\u231d : PROP) := by\n  apply anti_symm\n  case left =>\n    apply and_intro\n    <;> apply pure_mono\n    <;> simp_all\n  case right =>\n    apply pure_elim \u03c61\n    \u00b7 exact and_elim_l\n    intro _\n    rw' [and_elim_r]\n    apply pure_mono\n    simp_all\n\ntheorem pure_or {\u03c61 \u03c62 : Prop} [BI PROP] : \u231c\u03c61 \u2228 \u03c62\u231d \u22a3\u22a2 \u231c\u03c61\u231d \u2228 (\u231c\u03c62\u231d : PROP) := by\n  apply anti_symm\n  case left =>\n    apply pure_elim (\u03c61 \u2228 \u03c62)\n    \u00b7 simp\n    intro H\n    cases H\n    \u00b7 apply or_intro_l'\n      apply pure_mono\n      simp_all\n    \u00b7 apply or_intro_r'\n      apply pure_mono\n      simp_all\n  case right =>\n    apply or_elim\n    <;> apply pure_mono\n    <;> simp_all\n\ntheorem pure_impl_1 {\u03c61 \u03c62 : Prop} [BI PROP] : \u231c\u03c61 \u2192 \u03c62\u231d \u22a2 (\u231c\u03c61\u231d \u2192 \u231c\u03c62\u231d : PROP) := by\n  apply impl_intro_l\n  rw' [\u2190 pure_and]\n  apply pure_mono\n  simp_all\n\ntheorem pure_forall_1 {\u03c6 : \u03b1 \u2192 Prop} [BI PROP] : \u231c\u2200 x, \u03c6 x\u231d \u22a2 \u2200 x, (\u231c\u03c6 x\u231d : PROP) := by\n  apply forall_intro\n  intro x\n  apply pure_mono\n  simp_all\n\ntheorem pure_exist [inst : BI PROP] {\u03c6 : \u03b1 \u2192 Prop} : \u231c\u2203 x, \u03c6 x\u231d \u22a3\u22a2 (\u2203 x, \u231c\u03c6 x\u231d : PROP) := by\n  apply anti_symm\n  case left =>\n    apply pure_elim'\n    intro \u27e8x, H\u27e9\n    rw' [\u2190 exist_intro x]\n    apply pure_mono\n    intro _\n    exact H\n  case right =>\n    apply exist_elim\n    intro a\n    apply pure_mono\n    intro H\n    exact \u27e8a, H\u27e9\n\n-- Affine\ntheorem affinely_elim_emp [BI PROP] {P : PROP} : <affine> P \u22a2 emp := by\n  simp [bi_affinely, and_elim_l]\n\ntheorem affinely_elim [BI PROP] {P : PROP} : <affine> P \u22a2 P := by\n  simp [bi_affinely, and_elim_r]\n\n@[rwMonoRule]\ntheorem affinely_mono [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 <affine> P \u22a2 <affine> Q := by\n  intro H\n  simp only [bi_affinely]\n  rw' [H]\n\ntheorem affinely_idemp [BI PROP] {P : PROP} : <affine> <affine> P \u22a3\u22a2 <affine> P := by\n  simp only [bi_affinely]\n  rw' [\n    (assoc : emp \u2227 emp \u2227 _ \u22a3\u22a2 _),\n    (idemp : emp \u2227 emp \u22a3\u22a2 _)]\n\ntheorem affinely_emp [BI PROP] : <affine> emp \u22a3\u22a2 (emp : PROP) := by\n  simp only [bi_affinely]\n  exact idemp\n\ntheorem affinely_or [BI PROP] {P Q : PROP} : <affine> (P \u2228 Q) \u22a3\u22a2 <affine> P \u2228 <affine> Q := by\n  exact and_or_l\n\ntheorem affinely_and [BI PROP] {P Q : PROP} : <affine> (P \u2227 Q) \u22a3\u22a2 <affine> P \u2227 <affine> Q := by\n  rw' [\n    !bi_affinely,\n    (comm : emp \u2227 P \u22a3\u22a2 _),\n    (assoc : (P \u2227 emp) \u2227 _ \u22a3\u22a2 _),\n    \u2190 (assoc : _ \u22a3\u22a2 (P \u2227 emp) \u2227 _),\n    \u2190 (assoc : _ \u22a3\u22a2 (P \u2227 emp \u2227 emp) \u2227 _),\n    (idemp : emp \u2227 emp \u22a3\u22a2 _),\n    (assoc : P \u2227 emp \u2227 _ \u22a3\u22a2 _),\n    (assoc : emp \u2227 P \u2227 _ \u22a3\u22a2 _),\n    (comm : emp \u2227 P \u22a3\u22a2 _)]\n\ntheorem affinely_sep_2 [BI PROP] {P Q : PROP} : <affine> P \u2217 <affine> Q \u22a2 <affine> (P \u2217 Q) := by\n  simp only [bi_affinely]\n  apply and_intro\n  \u00b7 rw' [!and_elim_l, (right_id : emp \u2217 emp \u22a3\u22a2 _)]\n  \u00b7 rw' [!and_elim_r]\n\ntheorem affinely_forall [BI PROP] {\u03a6 : \u03b1 \u2192 PROP} : <affine> (\u2200 a, \u03a6 a) \u22a2 \u2200 a, <affine> (\u03a6 a) := by\n  apply forall_intro\n  intro a\n  rw' [forall_elim a]\n\ntheorem affinely_exist [BI PROP] {\u03a6 : \u03b1 \u2192 PROP} : <affine> (\u2203 a, \u03a6 a) \u22a3\u22a2 \u2203 a, <affine> (\u03a6 a) := by\n  exact and_exist_l\n\ntheorem affinely_True_emp [BI PROP] : <affine> True \u22a3\u22a2 <affine> (emp : PROP) := by\n  apply anti_symm\n  <;> simp only [bi_affinely]\n  \u00b7 apply and_intro\n    <;> exact and_elim_l\n  \u00b7 rw' [(right_id : _ \u2227 True \u22a3\u22a2 _)]\n    exact and_elim_l\n\ntheorem affinely_and_l [BI PROP] {P Q : PROP} : <affine> P \u2227 Q \u22a3\u22a2 <affine> (P \u2227 Q) := by\n  simp only [bi_affinely]\n  rw' [(assoc : emp \u2227 P \u2227 _ \u22a3\u22a2 _)]\n\ntheorem affinely_and_r [BI PROP] {P Q : PROP} : P \u2227 <affine> Q \u22a3\u22a2 <affine> (P \u2227 Q) := by\n  simp only [bi_affinely]\n  rw' [\n    (assoc : P \u2227 emp \u2227 _ \u22a3\u22a2 _),\n    (assoc : emp \u2227 P \u2227 _ \u22a3\u22a2 _),\n    (comm : P \u2227 emp \u22a3\u22a2 _)]\n\ntheorem affinely_and_lr [BI PROP] {P Q : PROP} : <affine> P \u2227 Q \u22a3\u22a2 P \u2227 <affine> Q := by\n  rw' [affinely_and_l, affinely_and_r]\n\n-- Absorbing\ntheorem absorbingly_intro [BI PROP] {P : PROP} : P \u22a2 <absorb> P := by\n  exact True_sep_2\n\n@[rwMonoRule]\ntheorem absorbingly_mono [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 <absorb> P \u22a2 <absorb> Q := by\n  intro H\n  simp only [bi_absorbingly]\n  rw' [H]\n\ntheorem absorbingly_idemp [BI PROP] {P : PROP} : <absorb> <absorb> P \u22a3\u22a2 <absorb> P := by\n  apply anti_symm\n  case left =>\n    simp only [bi_absorbingly]\n    rw' [(assoc : True \u2217 True \u2217 P \u22a3\u22a2 _)]\n    apply sep_mono ?_ reflexivity\n    apply pure_intro\n    simp\n  case right =>\n    rw' [\u2190 absorbingly_intro]\n\ntheorem absorbingly_pure {\u03c6 : Prop} [BI PROP] : <absorb> \u231c\u03c6\u231d \u22a3\u22a2 (\u231c\u03c6\u231d : PROP) := by\n  apply anti_symm\n  case left =>\n    apply wand_elim_r'\n    apply pure_elim'\n    intro H\u03c6\n    apply wand_intro_l\n    apply pure_intro\n    exact H\u03c6\n  case right =>\n    exact absorbingly_intro\n\ntheorem absorbingly_or [BI PROP] {P Q : PROP} : <absorb> (P \u2228 Q) \u22a3\u22a2 <absorb> P \u2228 <absorb> Q := by\n  simp [bi_absorbingly, sep_or_l]\n\ntheorem absorbingly_and_1 [BI PROP] {P Q : PROP} : <absorb> (P \u2227 Q) \u22a2 <absorb> P \u2227 <absorb> Q := by\n  apply and_intro\n  \u00b7 rw' [and_elim_l]\n  \u00b7 rw' [and_elim_r]\n\ntheorem absorbingly_forall [BI PROP] (\u03a6 : \u03b1 \u2192 PROP) : <absorb> (\u2200 a, \u03a6 a) \u22a2 \u2200 a, <absorb> (\u03a6 a) := by\n  apply forall_intro\n  intro a\n  rw' [forall_elim a]\n\ntheorem absorbingly_exist [BI PROP] (\u03a6 : \u03b1 \u2192 PROP) : <absorb> (\u2203 a, \u03a6 a) \u22a3\u22a2 \u2203 a, <absorb> (\u03a6 a) := by\n  simp [bi_absorbingly, sep_exist_l]\n\ntheorem absorbingly_sep [BI PROP] {P Q : PROP} : <absorb> (P \u2217 Q) \u22a3\u22a2 <absorb> P \u2217 <absorb> Q := by\n  rw' [\u2190 absorbingly_idemp]\n  simp only [bi_absorbingly]\n  rw' [\n    (assoc : True \u2217 P \u2217 Q \u22a3\u22a2 _),\n    (assoc : True \u2217 (True \u2217 P) \u2217 Q \u22a3\u22a2 _),\n    (comm : True \u2217 True \u2217 P \u22a3\u22a2 _),\n    \u2190 (assoc : _ \u22a3\u22a2 ((True \u2217 P) \u2217 True) \u2217 Q)]\n\ntheorem absorbingly_True_emp [BI PROP] : <absorb> True \u22a3\u22a2 <absorb> (emp : PROP) := by\n  rw' [absorbingly_pure]\n  simp only [bi_absorbingly]\n  rw' [(right_id : True \u2217 emp \u22a3\u22a2 _)]\n\ntheorem absorbingly_wand [BI PROP] {P Q : PROP} : <absorb> (P -\u2217 Q) \u22a2 <absorb> P -\u2217 <absorb> Q := by\n  apply wand_intro_l\n  rw' [\u2190 absorbingly_sep, wand_elim_r]\n\ntheorem absorbingly_sep_l [BI PROP] {P Q : PROP} : <absorb> P \u2217 Q \u22a3\u22a2 <absorb> (P \u2217 Q) := by\n  simp only [bi_absorbingly]\n  rw' [(assoc : True \u2217 P \u2217 Q \u22a3\u22a2 _)]\n\ntheorem absorbingly_sep_r [BI PROP] {P Q : PROP} : P \u2217 <absorb> Q \u22a3\u22a2 <absorb> (P \u2217 Q) := by\n  simp only [bi_absorbingly]\n  rw' [!(assoc : _ \u22a3\u22a2 _ \u2217 Q), (comm : P \u2217 True \u22a3\u22a2 _)]\n\ntheorem absorbingly_sep_lr [BI PROP] {P Q : PROP} : <absorb> P \u2217 Q \u22a3\u22a2 P \u2217 <absorb> Q := by\n  rw' [absorbingly_sep_l, absorbingly_sep_r]\n\n-- Affine / Absorbing Propositions\ntheorem affine_affinely [BI PROP] (P : PROP) [Affine P] : <affine> P \u22a3\u22a2 P := by\n  apply anti_symm\n  <;> simp only [bi_affinely]\n  \u00b7 exact and_elim_r\n  \u00b7 apply and_intro\n    \u00b7 exact affine\n    \u00b7 simp\n\ntheorem absorbing_absorbingly [BI PROP] {P : PROP} [Absorbing P] : <absorb> P \u22a3\u22a2 P := by\n  apply anti_symm\n  \u00b7 exact absorbing\n  \u00b7 rw' [absorbingly_intro]\n\ntheorem sep_elim_l [BI PROP] {P Q : PROP} [instQP : TCOr (Affine Q) (Absorbing P)] : P \u2217 Q \u22a2 P := by\n  cases instQP\n  case l =>\n    rw' [affine, (right_id : P \u2217 emp \u22a3\u22a2 _)]\n  case r =>\n    rw' [\n      (pure_intro True.intro : Q \u22a2 _),\n      (comm : P \u2217 True \u22a3\u22a2 _),\n      absorbing]\n\ntheorem sep_elim_r [BI PROP] {P Q : PROP} [TCOr (Affine P) (Absorbing Q)] : P \u2217 Q \u22a2 Q := by\n  rw' [(comm : P \u2217 Q \u22a3\u22a2 _), sep_elim_l]\n\ntheorem sep_and [BI PROP] {P Q : PROP} [inst1 : TCOr (Affine P) (Absorbing Q)] [inst2 : TCOr (Absorbing P) (Affine Q)] :\n  P \u2217 Q \u22a2 P \u2227 Q\n:= by\n  cases inst1\n  <;> cases inst2\n  <;> apply and_intro\n  <;> first | exact sep_elim_l | exact sep_elim_r\n\n-- Persistent\ntheorem absorbingly_elim_persistently [BI PROP] {P : PROP} : <absorb> <pers> P \u22a3\u22a2 <pers> P := by\n  apply anti_symm\n  case left =>\n    simp only [bi_absorbingly]\n    rw' [\n      (comm : `[iprop| True \u2217 <pers> P] \u22a3\u22a2 _),\n      persistently_absorbing]\n  case right =>\n    exact absorbingly_intro\n\ntheorem persistently_forall_1 [BI PROP] {\u03a8 : \u03b1 \u2192 PROP} : <pers> (\u2200 a, \u03a8 a) \u22a2 \u2200 a, <pers> (\u03a8 a) := by\n  apply forall_intro\n  intro x\n  rw' [forall_elim x]\n\ntheorem persistently_exist [BI PROP] {\u03a8 : \u03b1 \u2192 PROP} : <pers> (\u2203 a, \u03a8 a) \u22a3\u22a2 \u2203 a, <pers> (\u03a8 a) := by\n  apply anti_symm\n  case left =>\n    exact persistently_exist_1\n  case right =>\n    apply exist_elim\n    intro a\n    rw' [exist_intro a]\n\ntheorem persistently_and [BI PROP] {P Q : PROP} : <pers> (P \u2227 Q) \u22a3\u22a2 <pers> P \u2227 <pers> Q := by\n  apply anti_symm\n  case left =>\n    apply and_intro\n    \u00b7 rw' [and_elim_l]\n    \u00b7 rw' [and_elim_r]\n  case right =>\n    exact persistently_and_2\n\ntheorem persistently_if {p : Bool} [BI PROP] {P Q : PROP} :\n  (<pers> if p then P else Q) \u22a3\u22a2 if p then <pers> P else <pers> Q\n:= by\n  cases p\n  <;> simp\n\ntheorem persistently_or [BI PROP] {P Q : PROP} : <pers> (P \u2228 Q) \u22a3\u22a2 <pers> P \u2228 <pers> Q := by\n  rw' [!or_alt, persistently_exist]\n  apply anti_symm\n  <;> apply exist_elim\n  <;> intro a\n  <;> rw' [\u2190 exist_intro a, persistently_if]\n\ntheorem persistently_emp_intro [BI PROP] {P : PROP} : P \u22a2 <pers> emp := by\n  rw' [\u2190 persistently_absorbing (Q := P)]\n  conv =>\n    lhs\n    rw [\u2190 (left_id : emp \u2217 P \u22a3\u22a2 _)]\n  rw' [persistently_emp_2]\n\ntheorem persistently_True_emp [BI PROP] : <pers> True \u22a3\u22a2 <pers> (emp : PROP) := by\n  apply anti_symm\n  case left =>\n    exact persistently_emp_intro\n  case right =>\n    apply persistently_mono\n    apply pure_intro\n    simp\n\ntheorem persistently_True [BI PROP] : True \u22a2 <pers> (True : PROP) := by\n  rw' [persistently_True_emp, persistently_emp_intro]\n\ntheorem persistently_and_emp [BI PROP] {P : PROP} : <pers> P \u22a3\u22a2 <pers> (emp \u2227 P) := by\n  apply anti_symm\n  case left =>\n    rw' [persistently_and]\n    apply and_intro ?_ reflexivity\n    exact persistently_emp_intro\n  case right =>\n    rw' [and_elim_r]\n\ntheorem persistently_and_sep_elim_emp [BI PROP] {P Q : PROP} : <pers> P \u2227 Q \u22a2 (emp \u2227 P) \u2217 Q := by\n  rw' [persistently_and_emp, persistently_and_sep_elim]\n\ntheorem persistently_and_sep_assoc [BI PROP] {P Q R : PROP} : <pers> P \u2227 (Q \u2217 R) \u22a3\u22a2 (<pers> P \u2227 Q) \u2217 R := by\n  apply anti_symm\n  case left =>\n    rw' [\n      persistently_idemp_2,\n      persistently_and_sep_elim_emp,\n      (assoc : (emp \u2227 <pers> P) \u2217 Q \u2217 R \u22a3\u22a2 _)]\n    apply sep_mono ?_ reflexivity\n    apply and_intro\n    \u00b7 rw' [and_elim_r, persistently_absorbing]\n    \u00b7 rw' [and_elim_l, (left_id : emp \u2217 Q \u22a3\u22a2 _)]\n  case right =>\n    apply and_intro\n    \u00b7 rw' [and_elim_l, persistently_absorbing]\n    \u00b7 rw' [and_elim_r]\n\ntheorem persistently_and_emp_elim [BI PROP] {P : PROP} : emp \u2227 <pers> P \u22a2 P := by\n  rw' [\n    (comm : emp \u2227 <pers> P \u22a3\u22a2 _),\n    persistently_and_sep_elim_emp,\n    (right_id : (emp \u2227 P) \u2217 emp \u22a3\u22a2 _),\n    and_elim_r]\n\ntheorem persistently_into_absorbingly [BI PROP] {P : PROP} : <pers> P \u22a2 <absorb> P := by\n  rw' [\n    \u2190 (right_id : <pers> P \u2227 True \u22a3\u22a2 _),\n    \u2190 (left_id : emp \u2217 True \u22a3\u22a2 _),\n    persistently_and_sep_assoc,\n    (comm : `[iprop| <pers> P \u2227 emp] \u22a3\u22a2 _),\n    persistently_and_emp_elim,\n    (comm : `[iprop| P \u2217 True] \u22a3\u22a2 _)]\n\ntheorem persistently_elim [BI PROP] {P : PROP} [Absorbing P] : <pers> P \u22a2 P := by\n  rw' [persistently_into_absorbingly, absorbing]\n\ntheorem persistently_idemp [BI PROP] {P : PROP} : <pers> <pers> P \u22a3\u22a2 <pers> P := by\n  apply anti_symm\n  \u00b7 rw' [persistently_into_absorbingly, absorbingly_elim_persistently]\n  \u00b7 exact persistently_idemp_2\n\ntheorem persistently_pure {\u03c6 : Prop} [BI PROP] : <pers> \u231c\u03c6\u231d \u22a3\u22a2 (\u231c\u03c6\u231d : PROP) := by\n  apply anti_symm\n  case left =>\n    rw' [persistently_into_absorbingly, absorbingly_pure]\n  case right =>\n    apply pure_elim'\n    intro H\u03c6\n    rw' [persistently_True]\n    apply persistently_mono\n    apply pure_intro\n    exact H\u03c6\n\ntheorem persistently_and_sep_l_1 [BI PROP] {P Q : PROP} : <pers> P \u2227 Q \u22a2 <pers> P \u2217 Q := by\n  conv =>\n    lhs\n    rw [\u2190 (left_id : `[iprop| emp \u2217 Q] \u22a3\u22a2 _)]\n  rw' [persistently_and_sep_assoc, and_elim_l]\n\ntheorem persistently_and_sep [BI PROP] {P Q : PROP} : <pers> (P \u2227 Q) \u22a2 <pers> (P \u2217 Q) := by\n  rw' [persistently_and, \u2190 persistently_idemp, \u2190 persistently_and]\n  conv =>\n    lhs\n    rw [\u2190 (left_id : emp \u2217 Q \u22a3\u22a2 _)]\n  rw' [\n    persistently_and_sep_assoc,\n    (comm : <pers> P \u2227 emp \u22a3\u22a2 _),\n    persistently_and_emp_elim]\n\ntheorem persistently_affinely_elim [BI PROP] {P : PROP} : <pers> <affine> P \u22a3\u22a2 <pers> P := by\n  simp only [bi_affinely]\n  rw' [\n    persistently_and,\n    \u2190 persistently_True_emp,\n    persistently_pure,\n    (left_id : True \u2227 <pers> P \u22a3\u22a2 _)]\n\ntheorem and_sep_persistently [BI PROP] {P Q : PROP} : <pers> P \u2227 <pers> Q \u22a3\u22a2 <pers> P \u2217 <pers> Q := by\n  apply anti_symm\n  case left =>\n    exact persistently_and_sep_l_1\n  case right =>\n    apply and_intro\n    \u00b7 exact persistently_absorbing\n    \u00b7 rw' [(comm : `[iprop| <pers> P \u2217 <pers> Q \u22a3\u22a2 _]), persistently_absorbing]\n\ntheorem persistently_sep_2 [BI PROP] {P Q : PROP} : <pers> P \u2217 <pers> Q \u22a2 <pers> (P \u2217 Q) := by\n  rw' [\u2190 persistently_and_sep, persistently_and, \u2190 and_sep_persistently]\n\n-- Intuitionistic\ntheorem intuitionistically_elim [BI PROP] {P : PROP} : \u25a1 P \u22a2 P := by\n  exact persistently_and_emp_elim\n\ntheorem intuitionistically_emp [BI PROP] : \u25a1 emp \u22a3\u22a2 (emp : PROP) := by\n  simp only [bi_intuitionistically]\n  rw' [\n    \u2190 persistently_True_emp,\n    persistently_pure,\n    affinely_True_emp,\n    affinely_emp]\n\ntheorem intuitionistically_True_emp [BI PROP] : \u25a1 True \u22a3\u22a2 (emp : PROP) := by\n  rw' [\u2190 intuitionistically_emp]\n  simp only [bi_intuitionistically]\n  rw' [persistently_True_emp]\n\ntheorem intuitionistically_and [BI PROP] {P Q : PROP} : \u25a1 (P \u2227 Q) \u22a3\u22a2 \u25a1 P \u2227 \u25a1 Q := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_and, affinely_and]\n\ntheorem intuitionistically_forall [BI PROP] {\u03a6 : \u03b1 \u2192 PROP} : \u25a1 (\u2200 x, \u03a6 x) \u22a2 \u2200 x, \u25a1 \u03a6 x := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_forall_1, affinely_forall]\n\ntheorem intuitionistically_or [BI PROP] {P Q : PROP} : \u25a1 (P \u2228 Q) \u22a3\u22a2 \u25a1 P \u2228 \u25a1 Q := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_or, affinely_or]\n\ntheorem intuitionistically_exist [BI PROP] {\u03a6 : \u03b1 \u2192 PROP} : \u25a1 (\u2203 x, \u03a6 x) \u22a3\u22a2 \u2203 x, \u25a1 \u03a6 x := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_exist, affinely_exist]\n\ntheorem intuitionistically_sep_2 [BI PROP] {P Q : PROP} : \u25a1 P \u2217 \u25a1 Q \u22a2 \u25a1 (P \u2217 Q) := by\n  rw' [affinely_sep_2, persistently_sep_2]\n\n@[rwMonoRule]\ntheorem intuitionistically_mono [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 \u25a1 P \u22a2 \u25a1 Q := by\n  intro H\n  simp only [bi_intuitionistically]\n  rw' [H]\n\ntheorem intuitionistically_idemp [BI PROP] {P : PROP} : \u25a1 \u25a1 P \u22a3\u22a2 \u25a1 P := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_affinely_elim, persistently_idemp]\n\ntheorem intuitionistically_into_persistently_1 [BI PROP] {P : PROP} : \u25a1 P \u22a2 <pers> P := by\n  rw' [affinely_elim]\n\ntheorem intuitionistically_persistently_elim [BI PROP] {P : PROP} : \u25a1 <pers> P \u22a3\u22a2 \u25a1 P := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_idemp]\n\ntheorem intuitionistic_intuitionistically [BI PROP] {P : PROP} [Affine P] [Persistent P] : \u25a1 P \u22a3\u22a2 P := by\n  apply anti_symm\n  \u00b7 exact intuitionistically_elim\n  conv =>\n    lhs\n    rw [\u2190 affine_affinely P]\n  rw' [persistent]\n\ntheorem intuitionistically_affinely [BI PROP] {P : PROP} : \u25a1 P \u22a2 <affine> P := by\n  simp only [bi_intuitionistically, bi_affinely]\n  apply and_intro\n  \u00b7 exact and_elim_l\n  \u00b7 exact persistently_and_emp_elim\n\ntheorem intuitionistically_affinely_elim [BI PROP] {P : PROP} : \u25a1 <affine> P \u22a3\u22a2 \u25a1 P := by\n  simp only [bi_intuitionistically]\n  rw' [persistently_affinely_elim]\n\ntheorem persistently_and_intuitionistically_sep_l [BI PROP] {P Q : PROP} : <pers> P \u2227 Q \u22a3\u22a2 \u25a1 P \u2217 Q := by\n  apply anti_symm\n  case left =>\n    simp only [bi_intuitionistically, bi_affinely]\n    rw' [\n      (comm : emp \u2227 <pers> P \u22a3\u22a2 _),\n      \u2190 persistently_and_sep_assoc,\n      (left_id : emp \u2217 Q \u22a3\u22a2 _)]\n  case right =>\n    apply and_intro\n    \u00b7 rw' [affinely_elim, persistently_absorbing]\n    \u00b7 rw' [affinely_elim_emp, (left_id : emp \u2217 Q \u22a3\u22a2 _)]\n\ntheorem persistently_and_intuitionistically_sep_r [BI PROP] {P Q : PROP} : P \u2227 <pers> Q \u22a3\u22a2 P \u2217 \u25a1 Q := by\n  rw' [(comm : P \u2227 _ \u22a3\u22a2 _), (comm : P \u2217 _ \u22a3\u22a2 _)]\n  exact persistently_and_intuitionistically_sep_l\n\ntheorem and_sep_intuitionistically [BI PROP] {P Q : PROP} : \u25a1 P \u2227 \u25a1 Q \u22a3\u22a2 \u25a1 P \u2217 \u25a1 Q := by\n  rw' [\u2190 persistently_and_intuitionistically_sep_l]\n  simp only [bi_intuitionistically]\n  rw'[\u2190 affinely_and, affinely_and_r]\n\ntheorem intuitionistically_sep_dup [BI PROP] {P : PROP} : \u25a1 P \u22a3\u22a2 \u25a1 P \u2217 \u25a1 P := by\n  rw' [\u2190 persistently_and_intuitionistically_sep_l]\n  simp only [bi_intuitionistically]\n  rw' [\n    affinely_and_r,\n    (idemp : _ \u2227 _ \u22a3\u22a2 _)]\n\n-- Intuitionistic BIAffine\ntheorem intuitionistically_into_persistently [BIAffine PROP] {P : PROP} : \u25a1 P \u22a3\u22a2 <pers> P := by\n  exact affine_affinely _\n\n-- Conditional Affine\n@[rwMonoRule]\ntheorem affinely_if_mono {p : Bool} [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 <affine>?p P \u22a2 <affine>?p Q := by\n  intro H\n  cases p\n  <;> simp [bi_affinely_if, H]\n  revert H\n  exact affinely_mono\n\ntheorem affinely_if_flag_mono {p q : Bool} [BI PROP] {P : PROP} : (q \u2192 p) \u2192 <affine>?p P \u22a2 <affine>?q P := by\n  cases p\n  <;> cases q\n  <;> simp [bi_affinely_if, affinely_elim]\n\ntheorem affinely_if_elim {p : Bool} [BI PROP] {P : PROP} : <affine>?p P \u22a2 P := by\n  cases p\n  <;> simp [bi_affinely_if, affinely_elim]\n\ntheorem affinely_affinely_if {p : Bool} [BI PROP] {P : PROP} : <affine> P \u22a2 <affine>?p P := by\n  cases p\n  <;> simp [bi_affinely_if, affinely_elim]\n\ntheorem affinely_if_and {p : Bool} [BI PROP] {P Q : PROP} : <affine>?p (P \u2227 Q) \u22a3\u22a2 <affine>?p P \u2227 <affine>?p Q := by\n  cases p\n  <;> simp [bi_affinely_if, affinely_and]\n\ntheorem affinely_if_or {p : Bool} [BI PROP] {P Q : PROP} : <affine>?p (P \u2228 Q) \u22a3\u22a2 <affine>?p P \u2228 <affine>?p Q := by\n  cases p\n  <;> simp [bi_affinely_if, affinely_or]\n\ntheorem affinely_if_exist {p : Bool} [BI PROP] {\u03a8 : \u03b1 \u2192 PROP} : <affine>?p (\u2203 a, \u03a8 a) \u22a3\u22a2 \u2203 a, <affine>?p (\u03a8 a) := by\n  cases p\n  <;> simp [bi_affinely_if, affinely_exist]\n\ntheorem affinely_if_intro_false [BI PROP] (P : PROP) : P \u22a3\u22a2 <affine>?false P := by\n  simp [bi_affinely_if]\n\ntheorem affinely_if_intro_true [BI PROP] (P : PROP) : <affine> P \u22a3\u22a2 <affine>?true P := by\n  simp [bi_affinely_if]\n\n-- Conditional Absorbing\n@[rwMonoRule]\ntheorem absorbingly_if_mono {p : Bool} [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 <absorb>?p P \u22a2 <absorb>?p Q := by\n  intro H\n  cases p\n  <;> simp [bi_absorbingly_if, H]\n  revert H\n  exact absorbingly_mono\n\n-- Conditional Persistent\n@[rwMonoRule]\ntheorem persistently_if_mono {p : Bool} [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 <pers>?p P \u22a2 <pers>?p Q := by\n  intro H\n  cases p\n  <;> simp [bi_persistently_if, H]\n  revert H\n  exact persistently_mono\n\ntheorem persistently_if_intro_false [BI PROP] (P : PROP) : P \u22a3\u22a2 <pers>?false P := by\n  simp [bi_persistently_if]\n\ntheorem persistently_if_intro_true [BI PROP] (P : PROP) : <pers> P \u22a3\u22a2 <pers>?true P := by\n  simp [bi_persistently_if]\n\n-- Conditional Intuitionistic\n@[rwMonoRule]\ntheorem intuitionistically_if_mono {p : Bool} [BI PROP] {P Q : PROP} : (P \u22a2 Q) \u2192 \u25a1?p P \u22a2 \u25a1?p Q := by\n  intro H\n  cases p\n  <;> simp [bi_intuitionistically_if, H]\n  revert H\n  exact intuitionistically_mono\n\ntheorem intuitionistically_if_elim {p : Bool} [BI PROP] {P : PROP} : \u25a1?p P \u22a2 P := by\n  cases p\n  <;> simp [bi_intuitionistically_if, intuitionistically_elim]\n\ntheorem intuitionistically_intuitionistically_if (p : Bool) [BI PROP] {P : PROP} : \u25a1 P \u22a2 \u25a1?p P := by\n  cases p\n  <;> simp [bi_intuitionistically_if]\n  \u00b7 exact intuitionistically_elim\n\ntheorem intuitionistically_if_and {p : Bool} [BI PROP] {P Q : PROP} : \u25a1?p (P \u2227 Q) \u22a3\u22a2 \u25a1?p P \u2227 \u25a1?p Q := by\n  cases p\n  <;> simp [bi_intuitionistically_if, intuitionistically_and]\n\ntheorem intuitionistically_if_or (p : Bool) [BI PROP] {P Q : PROP} : \u25a1?p (P \u2228 Q) \u22a3\u22a2 \u25a1?p P \u2228 \u25a1?p Q := by\n  cases p\n  <;> simp [bi_intuitionistically_if]\n  rw' [intuitionistically_or]\n\ntheorem intuitionistically_if_exist {p : Bool} [BI PROP] {\u03a8 : \u03b1 \u2192 PROP} : (\u25a1?p \u2203 a, \u03a8 a) \u22a3\u22a2 \u2203 a, \u25a1?p \u03a8 a := by\n  cases p\n  <;> simp [bi_intuitionistically_if, intuitionistically_exist]\n\ntheorem intuitionistically_if_sep_2 {p : Bool} [BI PROP] {P Q : PROP} : \u25a1?p P \u2217 \u25a1?p Q \u22a2 \u25a1?p (P \u2217 Q) := by\n  cases p\n  <;> simp [bi_intuitionistically_if]\n  \u00b7 exact intuitionistically_sep_2\n\ntheorem intuitionistically_if_idemp {p : Bool} [BI PROP] {P : PROP} : \u25a1?p \u25a1?p P \u22a3\u22a2 \u25a1?p P := by\n  cases p\n  <;> simp [bi_intuitionistically_if]\n  \u00b7 exact intuitionistically_idemp\n\ntheorem intuitionistically_if_intro_true [BI PROP] (P : PROP) : \u25a1 P \u22a3\u22a2 \u25a1?true P := by\n  simp [bi_intuitionistically_if]\n\n-- Persistent Propositions\ntheorem persistent_persistently_2 [BI PROP] {P : PROP} [Persistent P] : P \u22a2 <pers> P := by\n  rw' [persistent]\n\ntheorem persistent_and_affinely_sep_l_1 [BI PROP] {P Q : PROP} [Persistent P] : P \u2227 Q \u22a2 <affine> P \u2217 Q := by\n  rw' [\n    persistent_persistently_2,\n    persistently_and_intuitionistically_sep_l,\n    intuitionistically_affinely]\n\ntheorem persistent_and_affinely_sep_r_1 [BI PROP] {P Q : PROP} [Persistent Q] : P \u2227 Q \u22a2 P \u2217 <affine> Q := by\n  rw' [\n    (comm : P \u2227 Q \u22a3\u22a2 _),\n    \u2190 (comm : _ \u22a3\u22a2 P \u2217 <affine> Q),\n    persistent_and_affinely_sep_l_1]\n\ntheorem persistent_and_affinely_sep_l [BI PROP] {P Q : PROP} [Persistent P] [Absorbing P] :\n  P \u2227 Q \u22a3\u22a2 <affine> P \u2217 Q\n:= by\n  apply anti_symm\n  <;> rw' [persistent, \u2190 persistently_elim, persistently_and_intuitionistically_sep_l]\n\ntheorem persistent_and_affinely_sep_r [BI PROP] {P Q : PROP} [Persistent Q] [Absorbing Q] :\n  P \u2227 Q \u22a3\u22a2 P \u2217 <affine> Q\n:= by\n  apply anti_symm\n  case left =>\n    rw' [\n      persistent,\n      \u2190 persistently_elim,\n      persistently_and_intuitionistically_sep_r]\n  case right =>\n    rw' [\n      persistent,\n      \u2190 persistently_elim,\n      persistently_and_intuitionistically_sep_r]\n\ntheorem persistent_and_sep_1 [BI PROP] {P Q : PROP} [inst : TCOr (Persistent P) (Persistent Q)] :\n  P \u2227 Q \u22a2 P \u2217 Q\n:= by\n  cases inst\n  \u00b7 rw' [persistent_and_affinely_sep_l_1, affinely_elim]\n  \u00b7 rw' [persistent_and_affinely_sep_r_1, affinely_elim]\n\ntheorem absorbingly_intuitionistically_into_persistently [BI PROP] {P : PROP} :\n  <absorb> \u25a1 P \u22a3\u22a2 <pers> P\n:= by\n  apply anti_symm\n  case left =>\n    rw' [\n      intuitionistically_into_persistently_1,\n      absorbingly_elim_persistently]\n  case right =>\n    rw' [\n      \u2190 (idemp : <pers> P \u2227 _ \u22a3\u22a2 _),\n      persistently_and_intuitionistically_sep_r,\n      (True_intro : <pers> P \u22a2 _)]\n\ntheorem persistent_absorbingly_affinely_2 [BI PROP] {P : PROP} [Persistent P] : P \u22a2 <absorb> <affine> P := by\n  rw' [\n    persistent,\n    \u2190 absorbingly_intuitionistically_into_persistently,\n    intuitionistically_affinely]\n\n-- Big Op\ntheorem big_op_sep_nil [BI PROP] : [\u2217] `[term| []] \u22a3\u22a2 (emp : PROP) := by\n  simp only [big_op]\n\ntheorem big_op_and_nil [BI PROP] : [\u2227] `[term| []] \u22a3\u22a2 (True : PROP) := by\n  simp only [big_op]\n\ntheorem big_op_sep_cons [BI PROP] {P : PROP} {Ps : List PROP} : [\u2217] `[term| P :: Ps] \u22a3\u22a2 P \u2217 [\u2217] `[term| Ps] := by\n  cases Ps\n  <;> simp only [big_op]\n  rw' [(right_id : _ \u2217 emp \u22a3\u22a2 _)]\n\ntheorem big_op_and_cons [BI PROP] {P : PROP} {Ps : List PROP} : [\u2227] `[term| P :: Ps] \u22a3\u22a2 P \u2227 [\u2227] `[term| Ps] := by\n  cases Ps\n  <;> simp only [big_op]\n  rw' [(right_id : _ \u2227 True \u22a3\u22a2 _)]\n\nend Iris.BI\n", "meta": {"author": "larsk21", "repo": "iris-lean", "sha": "730e644d0ffaad78aac76e2e5f2cd8af0f1d2310", "save_path": "github-repos/lean/larsk21-iris-lean", "path": "github-repos/lean/larsk21-iris-lean/iris-lean-730e644d0ffaad78aac76e2e5f2cd8af0f1d2310/src/Iris/BI/DerivedLaws.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458153, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.38319978681344763}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n\n! This file was ported from Lean 3 source module data.int.absolute_value\n! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Module.Basic\nimport Mathbin.Algebra.Order.AbsoluteValue\nimport Mathbin.Data.Int.Cast.Lemmas\nimport Mathbin.Data.Int.Units\nimport Mathbin.GroupTheory.GroupAction.Units\n\n/-!\n# Absolute values and the integers\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file contains some results on absolute values applied to integers.\n\n## Main results\n\n * `absolute_value.map_units_int`: an absolute value sends all units of `\u2124` to `1`\n * `int.nat_abs_hom`: `int.nat_abs` bundled as a `monoid_with_zero_hom`\n-/\n\n\nvariable {R S : Type _} [Ring R] [LinearOrderedCommRing S]\n\n/- warning: absolute_value.map_units_int -> AbsoluteValue.map_units_int is a dubious translation:\nlean 3 declaration is\n  forall {S : Type.{u1}} [_inst_2 : LinearOrderedCommRing.{u1} S] (abv : AbsoluteValue.{0, u1} Int S Int.semiring (StrictOrderedSemiring.toOrderedSemiring.{u1} S (StrictOrderedRing.toStrictOrderedSemiring.{u1} S (LinearOrderedRing.toStrictOrderedRing.{u1} S (LinearOrderedCommRing.toLinearOrderedRing.{u1} S _inst_2))))) (x : Units.{0} Int Int.monoid), Eq.{succ u1} S (coeFn.{succ u1, succ u1} (AbsoluteValue.{0, u1} Int S Int.semiring (StrictOrderedSemiring.toOrderedSemiring.{u1} S (StrictOrderedRing.toStrictOrderedSemiring.{u1} S (LinearOrderedRing.toStrictOrderedRing.{u1} S (LinearOrderedCommRing.toLinearOrderedRing.{u1} S _inst_2))))) (fun (f : AbsoluteValue.{0, u1} Int S Int.semiring (StrictOrderedSemiring.toOrderedSemiring.{u1} S (StrictOrderedRing.toStrictOrderedSemiring.{u1} S (LinearOrderedRing.toStrictOrderedRing.{u1} S (LinearOrderedCommRing.toLinearOrderedRing.{u1} S _inst_2))))) => Int -> S) (AbsoluteValue.hasCoeToFun.{0, u1} Int S Int.semiring (StrictOrderedSemiring.toOrderedSemiring.{u1} S (StrictOrderedRing.toStrictOrderedSemiring.{u1} S (LinearOrderedRing.toStrictOrderedRing.{u1} S (LinearOrderedCommRing.toLinearOrderedRing.{u1} S _inst_2))))) abv ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Units.{0} Int Int.monoid) Int (HasLiftT.mk.{1, 1} (Units.{0} Int Int.monoid) Int (CoeTC\u2093.coe.{1, 1} (Units.{0} Int Int.monoid) Int (coeBase.{1, 1} (Units.{0} Int Int.monoid) Int (Units.hasCoe.{0} Int Int.monoid)))) x)) (OfNat.ofNat.{u1} S 1 (OfNat.mk.{u1} S 1 (One.one.{u1} S (AddMonoidWithOne.toOne.{u1} S (AddGroupWithOne.toAddMonoidWithOne.{u1} S (AddCommGroupWithOne.toAddGroupWithOne.{u1} S (Ring.toAddCommGroupWithOne.{u1} S (StrictOrderedRing.toRing.{u1} S (LinearOrderedRing.toStrictOrderedRing.{u1} S (LinearOrderedCommRing.toLinearOrderedRing.{u1} S _inst_2))))))))))\nbut is expected to have type\n  forall {S : Type.{u1}} [_inst_2 : LinearOrderedCommRing.{u1} S] (abv : AbsoluteValue.{0, u1} Int S Int.instSemiringInt (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) (x : Units.{0} Int Int.instMonoidInt), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (FunLike.coe.{succ u1, 1, succ u1} (AbsoluteValue.{0, u1} Int S Int.instSemiringInt (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) Int (fun (f : Int) => (fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) f) (SubadditiveHomClass.toFunLike.{u1, 0, u1} (AbsoluteValue.{0, u1} Int S Int.instSemiringInt (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) Int S (Distrib.toAdd.{0} Int (NonUnitalNonAssocSemiring.toDistrib.{0} Int (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Int (Semiring.toNonAssocSemiring.{0} Int Int.instSemiringInt)))) (Distrib.toAdd.{u1} S (NonUnitalNonAssocSemiring.toDistrib.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (OrderedSemiring.toSemiring.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))))) (Preorder.toLE.{u1} S (PartialOrder.toPreorder.{u1} S (OrderedSemiring.toPartialOrder.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))) (AbsoluteValue.subadditiveHomClass.{0, u1} Int S Int.instSemiringInt (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2)))))) abv (Units.val.{0} Int Int.instMonoidInt x)) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (NonAssocRing.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (Ring.toNonAssocRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (StrictOrderedRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (LinearOrderedRing.toStrictOrderedRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) (LinearOrderedCommRing.toLinearOrderedRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : Int) => S) (Units.val.{0} Int Int.instMonoidInt x)) _inst_2)))))))\nCase conversion may be inaccurate. Consider using '#align absolute_value.map_units_int AbsoluteValue.map_units_int\u2093'. -/\n@[simp]\ntheorem AbsoluteValue.map_units_int (abv : AbsoluteValue \u2124 S) (x : \u2124\u02e3) : abv x = 1 := by\n  rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp\n#align absolute_value.map_units_int AbsoluteValue.map_units_int\n\n/- warning: absolute_value.map_units_int_cast -> AbsoluteValue.map_units_int_cast is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : LinearOrderedCommRing.{u2} S] [_inst_3 : Nontrivial.{u1} R] (abv : AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) (x : Units.{0} Int Int.monoid), Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) (fun (f : AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) => R -> S) (AbsoluteValue.hasCoeToFun.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) abv ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int R (HasLiftT.mk.{1, succ u1} Int R (CoeTC\u2093.coe.{1, succ u1} Int R (Int.castCoe.{u1} R (AddGroupWithOne.toHasIntCast.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (Units.{0} Int Int.monoid) Int (HasLiftT.mk.{1, 1} (Units.{0} Int Int.monoid) Int (CoeTC\u2093.coe.{1, 1} (Units.{0} Int Int.monoid) Int (coeBase.{1, 1} (Units.{0} Int Int.monoid) Int (Units.hasCoe.{0} Int Int.monoid)))) x))) (OfNat.ofNat.{u2} S 1 (OfNat.mk.{u2} S 1 (One.one.{u2} S (AddMonoidWithOne.toOne.{u2} S (AddGroupWithOne.toAddMonoidWithOne.{u2} S (AddCommGroupWithOne.toAddGroupWithOne.{u2} S (Ring.toAddCommGroupWithOne.{u2} S (StrictOrderedRing.toRing.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))))))))\nbut is expected to have type\n  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : LinearOrderedCommRing.{u1} S] [_inst_3 : Nontrivial.{u2} R] (abv : AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) (x : Units.{0} Int Int.instMonoidInt), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R (fun (f : R) => (fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) f) (SubadditiveHomClass.toFunLike.{max u2 u1, u2, u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R S (Distrib.toAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))))) (Distrib.toAdd.{u1} S (NonUnitalNonAssocSemiring.toDistrib.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (OrderedSemiring.toSemiring.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))))) (Preorder.toLE.{u1} S (PartialOrder.toPreorder.{u1} S (OrderedSemiring.toPartialOrder.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))) (AbsoluteValue.subadditiveHomClass.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2)))))) abv (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (OfNat.ofNat.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) 1 (One.toOfNat1.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (NonAssocRing.toOne.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (Ring.toNonAssocRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (StrictOrderedRing.toRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (LinearOrderedRing.toStrictOrderedRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) (LinearOrderedCommRing.toLinearOrderedRing.{u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (Int.cast.{u2} R (Ring.toIntCast.{u2} R _inst_1) (Units.val.{0} Int Int.instMonoidInt x))) _inst_2)))))))\nCase conversion may be inaccurate. Consider using '#align absolute_value.map_units_int_cast AbsoluteValue.map_units_int_cast\u2093'. -/\n@[simp]\ntheorem AbsoluteValue.map_units_int_cast [Nontrivial R] (abv : AbsoluteValue R S) (x : \u2124\u02e3) :\n    abv ((x : \u2124) : R) = 1 := by rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp\n#align absolute_value.map_units_int_cast AbsoluteValue.map_units_int_cast\n\n/- warning: absolute_value.map_units_int_smul -> AbsoluteValue.map_units_int_smul is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {S : Type.{u2}} [_inst_1 : Ring.{u1} R] [_inst_2 : LinearOrderedCommRing.{u2} S] (abv : AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) (x : Units.{0} Int Int.monoid) (y : R), Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) (fun (f : AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) => R -> S) (AbsoluteValue.hasCoeToFun.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) abv (SMul.smul.{0, u1} (Units.{0} Int Int.monoid) R (Units.hasSmul.{0, u1} Int R Int.monoid (SubNegMonoid.SMulInt.{u1} R (AddGroup.toSubNegMonoid.{u1} R (AddGroupWithOne.toAddGroup.{u1} R (AddCommGroupWithOne.toAddGroupWithOne.{u1} R (Ring.toAddCommGroupWithOne.{u1} R _inst_1)))))) x y)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) (fun (f : AbsoluteValue.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) => R -> S) (AbsoluteValue.hasCoeToFun.{u1, u2} R S (Ring.toSemiring.{u1} R _inst_1) (StrictOrderedSemiring.toOrderedSemiring.{u2} S (StrictOrderedRing.toStrictOrderedSemiring.{u2} S (LinearOrderedRing.toStrictOrderedRing.{u2} S (LinearOrderedCommRing.toLinearOrderedRing.{u2} S _inst_2))))) abv y)\nbut is expected to have type\n  forall {R : Type.{u2}} {S : Type.{u1}} [_inst_1 : Ring.{u2} R] [_inst_2 : LinearOrderedCommRing.{u1} S] (abv : AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) (x : Units.{0} Int Int.instMonoidInt) (y : R), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) (HSMul.hSMul.{0, u2, u2} (Units.{0} Int Int.instMonoidInt) R R (instHSMul.{0, u2} (Units.{0} Int Int.instMonoidInt) R (Units.instSMulUnits.{0, u2} Int R Int.instMonoidInt (SubNegMonoid.SMulInt.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (Ring.toAddGroupWithOne.{u2} R _inst_1)))))) x y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R (fun (f : R) => (fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) f) (SubadditiveHomClass.toFunLike.{max u2 u1, u2, u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R S (Distrib.toAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))))) (Distrib.toAdd.{u1} S (NonUnitalNonAssocSemiring.toDistrib.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (OrderedSemiring.toSemiring.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))))) (Preorder.toLE.{u1} S (PartialOrder.toPreorder.{u1} S (OrderedSemiring.toPartialOrder.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))) (AbsoluteValue.subadditiveHomClass.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2)))))) abv (HSMul.hSMul.{0, u2, u2} (Units.{0} Int Int.instMonoidInt) R R (instHSMul.{0, u2} (Units.{0} Int Int.instMonoidInt) R (Units.instSMulUnits.{0, u2} Int R Int.instMonoidInt (SubNegMonoid.SMulInt.{u2} R (AddGroup.toSubNegMonoid.{u2} R (AddGroupWithOne.toAddGroup.{u2} R (Ring.toAddGroupWithOne.{u2} R _inst_1)))))) x y)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R (fun (f : R) => (fun (x._@.Mathlib.Algebra.Order.Hom.Basic._hyg.99 : R) => S) f) (SubadditiveHomClass.toFunLike.{max u2 u1, u2, u1} (AbsoluteValue.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))) R S (Distrib.toAdd.{u2} R (NonUnitalNonAssocSemiring.toDistrib.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R (Ring.toSemiring.{u2} R _inst_1))))) (Distrib.toAdd.{u1} S (NonUnitalNonAssocSemiring.toDistrib.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S (OrderedSemiring.toSemiring.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))))) (Preorder.toLE.{u1} S (PartialOrder.toPreorder.{u1} S (OrderedSemiring.toPartialOrder.{u1} S (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2))))))) (AbsoluteValue.subadditiveHomClass.{u2, u1} R S (Ring.toSemiring.{u2} R _inst_1) (OrderedCommSemiring.toOrderedSemiring.{u1} S (StrictOrderedCommSemiring.toOrderedCommSemiring.{u1} S (LinearOrderedCommSemiring.toStrictOrderedCommSemiring.{u1} S (LinearOrderedCommRing.toLinearOrderedCommSemiring.{u1} S _inst_2)))))) abv y)\nCase conversion may be inaccurate. Consider using '#align absolute_value.map_units_int_smul AbsoluteValue.map_units_int_smul\u2093'. -/\n@[simp]\ntheorem AbsoluteValue.map_units_int_smul (abv : AbsoluteValue R S) (x : \u2124\u02e3) (y : R) :\n    abv (x \u2022 y) = abv y := by rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp\n#align absolute_value.map_units_int_smul AbsoluteValue.map_units_int_smul\n\n/- warning: int.nat_abs_hom -> Int.natAbsHom is a dubious translation:\nlean 3 declaration is\n  MonoidWithZeroHom.{0, 0} Int Nat (NonAssocSemiring.toMulZeroOneClass.{0} Int (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring))) (NonAssocSemiring.toMulZeroOneClass.{0} Nat (Semiring.toNonAssocSemiring.{0} Nat Nat.semiring))\nbut is expected to have type\n  MonoidWithZeroHom.{0, 0} Int Nat (NonAssocSemiring.toMulZeroOneClass.{0} Int (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt))) (NonAssocSemiring.toMulZeroOneClass.{0} Nat (Semiring.toNonAssocSemiring.{0} Nat Nat.semiring))\nCase conversion may be inaccurate. Consider using '#align int.nat_abs_hom Int.natAbsHom\u2093'. -/\n/-- `int.nat_abs` as a bundled monoid with zero hom. -/\n@[simps]\ndef Int.natAbsHom : \u2124 \u2192*\u2080 \u2115 where\n  toFun := Int.natAbs\n  map_mul' := Int.natAbs_mul\n  map_one' := Int.natAbs_one\n  map_zero' := Int.natAbs_zero\n#align int.nat_abs_hom Int.natAbsHom\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Int/AbsoluteValue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.38319977986898784}}
{"text": "import Mathlib.Algebra.Group.Defs\n\nsection\n\n  def Array.cons {\u03b1} (x : \u03b1) (a : Array \u03b1) : Array \u03b1 := mk (x :: a.data)\n\n  infixr:67 \" ::# \" => Array.cons\n\n  theorem Array.recursion {\u03b1}\n  {motive : Array \u03b1 \u2192 Sort v}\n  (h0 : motive #[])\n  (ih : (a : Array \u03b1) \u2192 (x : \u03b1) \u2192 motive a \u2192 motive (x ::# a))\n  (a : Array \u03b1)\n  : motive a := \n  let motive' : List \u03b1 \u2192 Sort v := \u03bb l => motive (mk l)\n  show motive' a.data from\n  List.rec h0 (\u03bb x l h => ih (mk l) x h) a.data\n\n  @[simp]\n  theorem Array.forIn_cons {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n  (f : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2))\n  (a : \u03b1) (as : Array \u03b1) (b : \u03b2)\n  : (forIn (a ::# as) b f)\n      = (do match (<- f a b) with\n        | ForInStep.done b => pure b\n        | ForInStep.yield b => forIn as b f) := by\n    show (do match (<- f ((a ::# as).get \u27e8as.size - as.size, (as.size.sub_self \u25b8 as.size.zero_lt_succ) \u27e9) b) with\n            | ForInStep.done b => pure b\n            | ForInStep.yield b => forIn.loop (a ::# as) f as.size as.size.le_succ b)\n      =  (do match (<- f a b) with\n            | ForInStep.done b => pure b\n            | ForInStep.yield b => forIn as b f : m \u03b2)\n    suffices ((a ::# as).get \u27e8as.size - as.size, (as.size.sub_self \u25b8 as.size.zero_lt_succ) \u27e9) = a from by\n      rw [this]\n      apply congrArg; apply funext; intro r\n      apply congrArg; apply funext; exact forIn_loop_cons_eq\n    simp; rfl\n  where\n    forIn_loop_cons_eq : \u2200 b : \u03b2, (forIn.loop (a ::# as) f as.size (Nat.le_succ as.size) b)\n      = forIn.loop as f as.size (Nat.le_refl _) b :=\n        forIn_loop_cons_eq_lemma as.size (Nat.le_refl _)\n    forIn_loop_cons_eq_lemma (i : Nat) (hi : i \u2264 as.size)\n    : \u2200 b : \u03b2, (forIn.loop (a ::# as) f i (Nat.le_trans hi as.size.le_succ) b)\n      = forIn.loop as f i hi b := by induction i with\n    | zero => intro b; rfl\n    | succ i ih =>\n      intro b\n      have ih := ih (Nat.le_trans i.le_succ hi)\n      show  (do match (<- f ((a ::# as).get \u27e8(a ::# as).size - 1 - i, _\u27e9) b) with\n            | ForInStep.done b => pure b\n            | ForInStep.yield b => forIn.loop (a ::# as) f i _ b : m \u03b2) =\n            (do match (<- f (as.get \u27e8as.size - 1 - i, _\u27e9) b) with\n            | ForInStep.done b => pure b\n            | ForInStep.yield b => forIn.loop as f i _ b : m \u03b2)\n      suffices (f ((a ::# as).get \u27e8(a ::# as).size - 1 - i, _\u27e9) b)\n          = (f (as.get \u27e8as.size - 1 - i, _\u27e9) b) from by\n        rw [this]\n        apply congrArg; apply funext; intro r\n        apply congrArg; apply funext; exact ih\n      apply congrArg (f . _)\n      have : as.size - 1 - i + 1 < as.size + 1 := by\n        cases h : as.size with\n        | zero => rw [h] at hi; contradiction\n        | succ k => exact Nat.succ_lt_succ (Nat.succ_le_succ (Nat.sub_le k i))\n      show (a ::# as).get _ = (a ::# as).get \u27e8_, this\u27e9\n      apply congrArg\n      suffices (a ::# as).size - 1 - i = as.size - 1 - i + 1 from by simp [this]\n      show as.size + 1 - 1 - i = as.size - 1 - i + 1\n      rw [Nat.add_comm, Nat.add_sub_assoc, Nat.add_sub_assoc, Nat.add_comm]\n      . exact Nat.le_pred_of_lt hi\n      . exact Nat.le_trans i.zero_lt_succ hi\n\nend\n\nsection\n  class Cons (\u03c1 : Type u) (\u03b1 : outParam (Type v)) where\n    cons : \u03b1 \u2192 \u03c1 \u2192 \u03c1\n\n  class ForIn_Cons (m : Type u\u2081 \u2192 Type u\u2082) (\u03c1 : Type u) (\u03b1 : Type v)\n    [Monad m] [Cons \u03c1 \u03b1] [ForIn m \u03c1 \u03b1] \n    where\n    forIn_cons {\u03b2} (f : \u03b1 \u2192 \u03b2 \u2192 m (ForInStep \u03b2)) (a : \u03b1) (as : \u03c1) (b : \u03b2)\n      : forIn (Cons.cons a as) b f\n      = f a b >>= \u03bb | ForInStep.done b => pure b | ForInStep.yield b => forIn as b f\n\n  instance {\u03b1} : Cons (List \u03b1) \u03b1 where\n    cons := List.cons\n\n  instance {\u03b1} : Cons (Array \u03b1) \u03b1 where\n    cons := Array.cons\n\n  instance {\u03b1} {m} [Monad m] : ForIn_Cons m (List \u03b1) \u03b1 where\n    forIn_cons := List.forIn_cons\n\n  instance {\u03b1} {m} [Monad m] : ForIn_Cons m (Array \u03b1) \u03b1 where\n    forIn_cons := Array.forIn_cons\n\nend\n\n\nsection\n  variable {\u03b1 : Type u} [AddCommSemigroup \u03b1]\n\n  namespace List\n\n  def sum (l : List \u03b1) (x\u2080 : \u03b1) : \u03b1 := \n    match l with\n    | [] => x\u2080\n    | x :: l => l.sum x\u2080 + x\n\n  def iter_sum (l : List \u03b1) (x\u2080 : \u03b1) : \u03b1 :=\n    Id.run do\n      let mut s := x\u2080\n      for x in l do\n        s := s + x\n      return s\n\n  theorem iter_sum_cons_add (l : List \u03b1) \n  : \u2200 x\u2080 x : \u03b1, (x :: l).iter_sum x\u2080 = l.iter_sum x\u2080 + x := by\n    induction l with\n    | nil => exact \u03bb _ _ => rfl\n    | cons y l hl =>\n      intro x\u2080 x\n      show l.iter_sum (x\u2080 + x + y) = l.iter_sum (x\u2080 + y) + x\n      rw [\u2190hl]\n      show l.iter_sum (x\u2080 + x + y) = l.iter_sum (x\u2080 + y + x)\n      apply congrArg l.iter_sum\n      rw [add_assoc, add_assoc, add_comm x y]\n\n  theorem iter_sum_eq_sum (l : List \u03b1) (x\u2080 : \u03b1)\n    : l.iter_sum x\u2080 = l.sum x\u2080 :=\n    match l with\n    | [] => rfl\n    | x :: l => l.iter_sum_cons_add x\u2080 x \u25b8 l.iter_sum_eq_sum x\u2080 \u25b8 rfl\n\n  end List\n\n  namespace Array\n\n  def iter_sum (a : Array \u03b1) (x\u2080 : \u03b1) : \u03b1 :=\n    Id.run do\n      let mut s := x\u2080\n      for x in a do\n        s := s + x\n      return s\n\n  theorem iter_sum_cons_add \n  : \u2200 a : Array \u03b1, \u2200 x\u2080 x, (x ::# a).iter_sum x\u2080 = a.iter_sum x\u2080 + x := \n    recursion (\u03bb _ _ => rfl) <| by\n    intro a y ha\n    intro x\u2080 x\n    simp [iter_sum]; show a.iter_sum (x\u2080 + x + y) = a.iter_sum (x\u2080 + y) + x\n    rw [\u2190ha]\n    simp [iter_sum]; show a.iter_sum (x\u2080 + x + y) = a.iter_sum (x\u2080 + y + x)\n    apply congrArg a.iter_sum\n    rw [add_assoc, add_assoc, add_comm x y]\n\n  end Array\n\n  section\n  variable {\u03b1 : Type _} [Add \u03b1]\n  variable {\u03c1} [\u2200 m, ForIn m \u03c1 \u03b1] [Cons \u03c1 \u03b1] [\u2200 m, [Monad m] \u2192 ForIn_Cons m \u03c1 \u03b1]\n  def iter_sum (a : \u03c1) (x\u2080 : \u03b1) : \u03b1 :=\n    Id.run do\n      let mut s := x\u2080\n      for x in a do\n        s := s + x\n      return s\n  end\n\nend", "meta": {"author": "michelsol", "repo": "lean-playground", "sha": "0bfffb7bd41729fb9f95974e93f6ecbc0b6e59ca", "save_path": "github-repos/lean/michelsol-lean-playground", "path": "github-repos/lean/michelsol-lean-playground/lean-playground-0bfffb7bd41729fb9f95974e93f6ecbc0b6e59ca/Playground/Misc/Algo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38317966184416113}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Eric Weiser\n-/\nimport tactic.doc_commands\n\n/-!\n# Documentation of the algebraic hierarchy\n\nA library note giving advice on modifying the algebraic hierarchy.\n(It is not intended as a \"tour\".)\n\nTODO: Add sections about interactions with topological typeclasses, and order typeclasses.\n\n-/\n\n/--\n# The algebraic hierarchy\n\nIn any theorem proving environment,\nthere are difficult decisions surrounding the design of the \"algebraic hierarchy\".\n\nThere is a danger of exponential explosion in the number of gadgets,\nespecially once interactions between algebraic and order/topological/etc structures are considered.\n\nIn mathlib, we try to avoid this by only introducing new algebraic typeclasses either\n1. when there is \"real mathematics\" to be done with them, or\n2. when there is a meaninful gain in simplicity by factoring out a common substructure.\n\n(As examples, at this point we don't have `loop`, or `unital_magma`,\nbut we do have `lie_submodule` and `topological_field`!\nWe also have `group_with_zero`, as an exemplar of point 2.)\n\nGenerally in mathlib we use the extension mechanism (so `comm_ring` extends `ring`)\nrather than mixins (e.g. with separate `ring` and `comm_mul` classes),\nin part because of the potential blow-up in term sizes described at\nhttps://www.ralfj.de/blog/2019/05/15/typeclasses-exponential-blowup.html\nHowever there is tension here, as it results in considerable duplication in the API,\nparticularly in the interaction with order structures.\n\nThis library note is not intended as a design document\njustifying and explaining the history of mathlib's algebraic hierarchy!\nInstead it is intended as a developer's guide, for contributors wanting to extend\n(either new leaves, or new intermediate classes) the algebraic hierarchy as it exists.\n\n(Ideally we would have both a tour guide to the existing hierarchy,\nand an account of the design choices.\nSee https://arxiv.org/abs/1910.09336 for an overview of mathlib as a whole,\nwith some attention to the algebraic hierarchy and\nhttps://leanprover-community.github.io/mathlib-overview.html\nfor a summary of what is in mathlib today.)\n\n## Instances\n\nWhen adding a new typeclass `Z` to the algebraic hierarchy\none should attempt to add the following constructions and results,\nwhen applicable:\n\n* Instances transferred elementwise to products, like `prod.monoid`.\n  See `algebra.group.prod` for more examples.\n  ```\n  instance prod.Z [Z M] [Z N] : Z (M \u00d7 N) := ...\n  ```\n* Instances transferred elementwise to pi types, like `pi.monoid`.\n  See `algebra.group.pi` for more examples.\n  ```\n  instance pi.Z [\u2200 i, Z $ f i] : Z (\u03a0 i : I, f i) := ...\n  ```\n* Instances transferred to `mul_opposite M`, like `mul_opposite.monoid`.\n  See `algebra.opposites` for more examples.\n  ```\n  instance mul_opposite.Z [Z M] : Z (mul_opposite M) := ...\n  ```\n* Instances transferred to `ulift M`, like `ulift.monoid`.\n  See `algebra.group.ulift` for more examples.\n  ```\n  instance ulift.Z [Z M] : Z (ulift M) := ...\n  ```\n* Definitions for transferring the proof fields of instances along\n  injective or surjective functions that agree on the data fields,\n  like `function.injective.monoid` and `function.surjective.monoid`.\n  We make these definitions `@[reducible]`, see note [reducible non-instances].\n  See `algebra.group.inj_surj` for more examples.\n  ```\n  @[reducible]\n  def function.injective.Z [Z M\u2082] (f : M\u2081 \u2192 M\u2082) (hf : injective f)\n    (one : f 1 = 1) (mul : \u2200 x y, f (x * y) = f x * f y) : Z M\u2081 := ...\n\n  @[reducible]\n  def function.surjective.Z [Z M\u2081] (f : M\u2081 \u2192 M\u2082) (hf : surjective f)\n    (one : f 1 = 1) (mul : \u2200 x y, f (x * y) = f x * f y) : Z M\u2082 := ...\n  ```\n* Instances transferred elementwise to `finsupp`s, like `finsupp.semigroup`.\n  See `data.finsupp.pointwise` for more examples.\n  ```\n  instance finsupp.Z [Z \u03b2] : Z (\u03b1 \u2192\u2080 \u03b2) := ...\n  ```\n* Instances transferred elementwise to `set`s, like `set.monoid`.\n  See `algebra.pointwise` for more examples.\n  ```\n  instance set.Z [Z \u03b1] : Z (set \u03b1) := ...\n  ```\n* Definitions for transferring the entire structure across an equivalence, like `equiv.monoid`.\n  See `data.equiv.transfer_instance` for more examples. See also the `transport` tactic.\n  ```\n  def equiv.Z (e : \u03b1 \u2243 \u03b2) [Z \u03b2] : Z \u03b1 := ...\n  /- When there is a new notion of `Z`-equiv: -/\n  def equiv.Z_equiv (e : \u03b1 \u2243 \u03b2) [Z \u03b2] : by { letI := equiv.Z e, exact \u03b1 \u2243Z \u03b2 } := ...\n  ```\n\n## Subobjects\n\nWhen a new typeclass `Z` adds new data fields,\nyou should also create a new `sub_Z` `structure` with a `carrier` field.\n\nThis can be a lot of work; for now try to closely follow the existing examples\n(e.g. `submonoid`, `subring`, `subalgebra`).\nWe would very much like to provide some automation here, but a prerequisite will be making\nall the existing APIs more uniform.\n\nIf `Z` extends `Y`, then `sub_Z` should usually extend `sub_Y`.\n\nWhen `Z` adds only new proof fields to an existing structure `Y`,\nyou should provide instances transferring\n`Z \u03b1` to `Z (sub_Y \u03b1)`, like `submonoid.to_comm_monoid`.\nTypically this is done using the `function.injective.Z` definition mentioned above.\n```\ninstance sub_Y.to_Z [Z \u03b1] : Z (sub_Y \u03b1) :=\ncoe_injective.Z coe ...\n```\n\n## Morphisms and equivalences\n\n## Category theory\n\nFor many algebraic structures, particularly ones used in representation theory, algebraic geometry,\netc., we also define \"bundled\" versions, which carry `category` instances.\n\nThese bundled versions are usually named in camel case,\nso for example we have `AddCommGroup` as a bundled `add_comm_group`,\nand `TopCommRing` (which bundles together `comm_ring`, `topological_space`, and `topological_ring`).\n\nThese bundled versions have many appealing features:\n* a uniform notation for morphisms `X \u27f6 Y`\n* a uniform notation (and definition) for isomorphisms `X \u2245 Y`\n* a uniform API for subobjects, via the partial order `subobject X`\n* interoperability with unbundled structures, via coercions to `Type`\n  (so if `G : AddCommGroup`, you can treat `G` as a type,\n  and it automatically has an `add_comm_group` instance)\n  and lifting maps `AddCommGroup.of G`, when `G` is a type with an `add_comm_group` instance.\n\nIf, for example you do the work of proving that a typeclass `Z` has a good notion of tensor product,\nyou are strongly encouraged to provide the corresponding `monoidal_category` instance\non a bundled version.\nThis ensures that the API for tensor products is complete, and enables use of general machinery.\nSimilarly if you prove universal properties, or adjunctions, you are encouraged to state these\nusing categorical language!\n\nOne disadvantage of the bundled approach is that we can only speak of morphisms between\nobjects living in the same type-theoretic universe.\nIn practice this is rarely a problem.\n\n# Making a pull request\n\nWith so many moving parts, how do you actually go about changing the algebraic hierarchy?\n\nWe're still evolving how to handle this, but the current suggestion is:\n\n* If you're adding a new \"leaf\" class, the requirements are lower,\n  and an initial PR can just add whatever is immediately needed.\n* A new \"intermediate\" class, especially low down in the hierarchy,\n  needs to be careful about leaving gaps.\n\nIn a perfect world, there would be a group of simultaneous PRs that basically cover everything!\n(Or at least an expectation that PRs may not be merged immediately while waiting on other\nPRs that fill out the API.)\n\nHowever \"perfect is the enemy of good\", and it would also be completely reasonable\nto add a TODO list in the main module doc-string for the new class,\nbriefly listing the parts of the API which still need to be provided.\nHopefully this document makes it easy to assemble this list.\n\nAnother alternative to a TODO list in the doc-strings is adding github issues.\n\n\n-/\nlibrary_note \"the algebraic hierarchy\"\n\n/--\nSome definitions that define objects of a class cannot be instances, because they have an\nexplicit argument that does not occur in the conclusion. An example is `preorder.lift` that has a\nfunction `f : \u03b1 \u2192 \u03b2` as an explicit argument to lift a preorder on `\u03b2` to a preorder on `\u03b1`.\n\nIf these definitions are used to define instances of this class *and* this class is an argument to\nsome other type-class so that type-class inference will have to unfold these instances to check\nfor definitional equality, then these definitions should be marked `@[reducible]`.\n\nFor example, `preorder.lift` is used to define `units.preorder` and `partial_order.lift` is used\nto define `units.partial_order`. In some cases it is important that type-class inference can\nrecognize that `units.preorder` and `units.partial_order` give rise to the same `has_le` instance.\nFor example, you might have another class that takes `[has_le \u03b1]` as an argument, and this argument\nsometimes comes from `units.preorder` and sometimes from `units.partial_order`.\nTherefore, `preorder.lift` and `partial_order.lift` are marked `@[reducible]`.\n-/\nlibrary_note \"reducible non-instances\"\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/hierarchy_design.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.6723317123102956, "lm_q1q2_score": 0.38313001871282826}}
{"text": "import tactic --hide\n\n/-Lemma\n$(\\mathrm{true} \\implies \\mathrm{false}) \\implies \\mathrm{false}$.\n-/\nlemma true_false_false : (true \u2192 false) \u2192 false :=\nbegin\n  intro f,\n  apply f,\n  triv,\n\n\n  \nend", "meta": {"author": "CBirkbeck", "repo": "logic_projic", "sha": "0b029af0fbfc0ac6eafae47401d5bbf8e641d7d2", "save_path": "github-repos/lean/CBirkbeck-logic_projic", "path": "github-repos/lean/CBirkbeck-logic_projic/logic_projic-0b029af0fbfc0ac6eafae47401d5bbf8e641d7d2/src/true_false/tf5.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.38313000888984217}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.equiv_rw\n! leanprover-community/mathlib commit 9407b03373c8cd201df99d6bc5514fc2db44054f\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Equiv.Defs\nimport Mathbin.Tactic.Clear\nimport Mathbin.Tactic.SimpResult\nimport Mathbin.Tactic.Apply\nimport Mathbin.Control.EquivFunctor.Instances\nimport Mathbin.Logic.Equiv.Functor\n\n/-!\n# The `equiv_rw` tactic transports goals or hypotheses along equivalences.\n\nThe basic syntax is `equiv_rw e`, where `e : \u03b1 \u2243 \u03b2` is an equivalence.\nThis will try to replace occurrences of `\u03b1` in the goal with `\u03b2`, for example\ntransforming\n* `\u22a2 \u03b1` to `\u22a2 \u03b2`,\n* `\u22a2 option \u03b1` to `\u22a2 option \u03b2`\n* `\u22a2 {a // P}` to `{b // P (\u21d1(equiv.symm e) b)}`\n\nThe tactic can also be used to rewrite hypotheses, using the syntax `equiv_rw e at h`.\n\n## Implementation details\n\nThe main internal function is `equiv_rw_type e t`,\nwhich attempts to turn an expression `e : \u03b1 \u2243 \u03b2` into a new equivalence with left hand side `t`.\nAs an example, with `t = option \u03b1`, it will generate `functor.map_equiv option e`.\n\nThis is achieved by generating a new synthetic goal `%%t \u2243 _`,\nand calling `solve_by_elim` with an appropriate set of congruence lemmas.\nTo avoid having to specify the relevant congruence lemmas by hand,\nwe mostly rely on `equiv_functor.map_equiv` and `bifunctor.map_equiv`\nalong with some structural congruence lemmas such as\n* `equiv.arrow_congr'`,\n* `equiv.subtype_equiv_of_subtype'`,\n* `equiv.sigma_congr_left'`, and\n* `equiv.Pi_congr_left'`.\n\nThe main `equiv_rw` function, when operating on the goal, simply generates a new equivalence `e'`\nwith left hand side matching the target, and calls `apply e'.inv_fun`.\n\nWhen operating on a hypothesis `x : \u03b1`, we introduce a new fact `h : x = e.symm (e x)`, revert this,\nand then attempt to `generalize`, replacing all occurrences of `e x` with a new constant `y`, before\n`intro`ing and `subst`ing `h`, and renaming `y` back to `x`.\n\n## Future improvements\nIn a future PR I anticipate that `derive equiv_functor` should work on many examples,\n(internally using `transport`, which is in turn based on `equiv_rw`)\nand we can incrementally bootstrap the strength of `equiv_rw`.\n\nAn ambitious project might be to add `equiv_rw!`,\na tactic which, when failing to find appropriate `equiv_functor` instances,\nattempts to `derive` them on the spot.\n\nFor now `equiv_rw` is entirely based on `equiv`,\nbut the framework can readily be generalised to also work with other types of equivalences,\nfor example specific notations such as ring equivalence (`\u2243+*`),\nor general categorical isomorphisms (`\u2245`).\n\nThis will allow us to transport across more general types of equivalences,\nbut this will wait for another subsequent PR.\n-/\n\n\n-- these make equiv_rw more powerful!\n-- these make equiv_rw more powerful!\n-- so do these!\n-- so do these!\nnamespace Tactic\n\n-- Although this looks 'hard-coded', in fact the lemma `equiv_functor.map_equiv`\n-- allows us to extend `equiv_rw` simply by constructing new instance so `equiv_functor`.\n-- TODO: We should also use `category_theory.functorial` and `category_theory.hygienic` instances.\n-- (example goal: we could rewrite along an isomorphism of rings (either as `R \u2245 S` or `R \u2243+* S`)\n-- and turn an `x : mv_polynomial \u03c3 R` into an `x : mv_polynomial \u03c3 S`.).\n/-- A list of lemmas used for constructing congruence equivalences. -/\nunsafe def equiv_congr_lemmas : List (tactic expr) :=\n  [`equiv.of_iff,-- TODO decide what to do with this; it's an equiv_bifunctor?\n        `equiv.equiv_congr,-- The function arrow is technically a bifunctor `Type\u1d52\u1d56 \u2192 Type \u2192 Type`,\n        -- but the pattern matcher will never see this.\n        `equiv.arrow_congr',-- Allow rewriting in subtypes:\n        `equiv.subtype_equiv_of_subtype',-- Allow rewriting in the first component of a sigma-type:\n        `equiv.sigma_congr_left',-- Allow rewriting \u2200s:\n        -- (You might think that repeated application of `equiv.forall_congr'\n        -- would handle the higher arity cases, but unfortunately unification is not clever enough.)\n        `equiv.forall\u2083_congr',\n        `equiv.forall\u2082_congr',\n        `equiv.forall_congr',-- Allow rewriting in argument of Pi types:\n        `equiv.Pi_congr_left',-- Handles `sum` and `prod`, and many others:\n        `bifunctor.map_equiv,-- Handles `list`, `option`, `unique`, and many others:\n        `equiv_functor.map_equiv,-- We have to filter results to ensure we don't cheat and use exclusively\n        -- `equiv.refl` and `iff.refl`!\n        `equiv.refl,\n        `iff.refl].map\n    fun n => mk_const n\n#align tactic.equiv_congr_lemmas tactic.equiv_congr_lemmas\n\ninitialize\n  registerTraceClass.1 `equiv_rw_type\n\n/-- Configuration structure for `equiv_rw`.\n\n* `max_depth` bounds the search depth for equivalences to rewrite along.\n  The default value is 10.\n  (e.g., if you're rewriting along `e : \u03b1 \u2243 \u03b2`, and `max_depth := 2`,\n  you can rewrite `option (option \u03b1))` but not `option (option (option \u03b1))`.\n-/\nunsafe structure equiv_rw_cfg where\n  max_depth : \u2115 := 10\n#align tactic.equiv_rw_cfg tactic.equiv_rw_cfg\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n/-- Implementation of `equiv_rw_type`, using `solve_by_elim`.\nExpects a goal of the form `t \u2243 _`,\nand tries to solve it using `eq : \u03b1 \u2243 \u03b2` and congruence lemmas.\n-/\nunsafe def equiv_rw_type_core (eq : expr) (cfg : equiv_rw_cfg) : tactic Unit := do\n  /-\n          We now call `solve_by_elim` to try to generate the requested equivalence.\n          There are a few subtleties!\n          * We make sure that `eq` is the first lemma, so it is applied whenever possible.\n          * In `equiv_congr_lemmas`, we put `equiv.refl` last so it is only used when it is not possible\n            to descend further.\n          * Since some congruence lemmas generate subgoals with `\u2200` statements,\n            we use the `pre_apply` subtactic of `solve_by_elim` to preprocess each new goal with `intros`.\n        -/\n      solve_by_elim\n      { use_symmetry := False\n        use_exfalso := False\n        lemma_thunks := some (pure Eq :: equiv_congr_lemmas)\n        ctx_thunk := pure []\n        max_depth := cfg\n        -- Subgoals may contain function types,\n        -- and we want to continue trying to construct equivalences after the binders.\n        pre_apply := tactic.intros >> skip\n        backtrack_all_goals := tt\n        -- If solve_by_elim gets stuck, make sure it isn't because there's a later `\u2243` or `\u2194` goal\n        -- that we should still attempt.\n        discharger :=\n          (sorry >> sorry) >> (sorry <|> sorry) <|>\n            trace_if_enabled `equiv_rw_type \"Failed, no congruence lemma applied!\" >> failed\n        -- We use the `accept` tactic in `solve_by_elim` to provide tracing.\n        accept := fun goals =>\n          lock_tactic_state do\n            when_tracing `equiv_rw_type do\n                goals pp >>= fun goals => trace f! \"So far, we've built: {goals}\"\n            done <|>\n                when_tracing `equiv_rw_type do\n                  let gs \u2190 get_goals\n                  let gs \u2190 gs fun g => infer_type g >>= pp\n                  trace f! \"Attempting to adapt to {gs}\" }\n#align tactic.equiv_rw_type_core tactic.equiv_rw_type_core\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      `equiv_rw_type e t` rewrites the type `t` using the equivalence `e : \u03b1 \u2243 \u03b2`,\n      returning a new equivalence `t \u2243 t'`.\n      -/\n    unsafe\n  def\n    equiv_rw_type\n    ( eqv : expr ) ( ty : expr ) ( cfg : equiv_rw_cfg ) : tactic expr\n    :=\n      do\n        when_tracing\n            `equiv_rw_type\n              do\n                let ty_pp \u2190 pp ty\n                  let eqv_pp \u2190 pp eqv\n                  let eqv_ty_pp \u2190 infer_type eqv >>= pp\n                  trace\n                    f!\n                      \"Attempting to rewrite the type `{\n                        ty_pp\n                        }` using `{\n                        eqv_pp\n                        } : {\n                        eqv_ty_pp\n                        }`.\"\n          let q( _ \u2243 _ ) \u2190 infer_type eqv | fail f! \"{ eqv } must be an `equiv`\"\n          let equiv_ty \u2190 to_expr ` `( $ ( ty ) \u2243 _ )\n          let new_eqv \u2190 Prod.snd <$> ( solve_aux equiv_ty <| equiv_rw_type_core eqv cfg )\n          let new_eqv \u2190 instantiate_mvars new_eqv\n          kdepends_on new_eqv eqv >>= guardb\n            <|>\n            do\n              let eqv_pp \u2190 pp eqv\n                let ty_pp \u2190 pp ty\n                fail\n                  f! \"Could not construct an equivalence from { eqv_pp } of the form: { ty_pp } \u2243 _\"\n          Prod.fst <$> new_eqv { failIfUnchanged := ff }\n#align tactic.equiv_rw_type tactic.equiv_rw_type\n\n/- failed to parenthesize: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n[PrettyPrinter.parenthesize.input] (Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr\n     [(Command.docComment\n       \"/--\"\n       \"The simpset `equiv_rw_simp` is used by the tactic `equiv_rw` to\\nsimplify applications of equivalences and their inverses. -/\")]\n     \"register_simp_attr\"\n     `equiv_rw_simp)-/-- failed to format: unknown constant 'Lean.Meta._root_.Lean.Parser.Command.registerSimpAttr'\n/--\n    The simpset `equiv_rw_simp` is used by the tactic `equiv_rw` to\n    simplify applications of equivalences and their inverses. -/\n  register_simp_attr\n  equiv_rw_simp\n\nattribute [equiv_rw_simp] Equiv.symm_symm Equiv.apply_symm_apply Equiv.symm_apply_apply\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Attempt to replace the hypothesis with name `x`\n      by transporting it along the equivalence in `e : \u03b1 \u2243 \u03b2`.\n      -/\n    unsafe\n  def\n    equiv_rw_hyp\n    ( x : Name ) ( e : expr ) ( cfg : equiv_rw_cfg := { } ) : tactic Unit\n    :=\n      dsimp_result\n        (\n            do\n              let x' \u2190 get_local x\n                let x_ty \u2190 infer_type x'\n                let e \u2190 equiv_rw_type e x_ty cfg\n                let\n                  eq \u2190 to_expr ` `( $ ( x' ) = Equiv.symm $ ( e ) ( Equiv.toFun $ ( e ) $ ( x' ) ) )\n                let prf \u2190 to_expr ` `( ( Equiv.symm_apply_apply $ ( e ) $ ( x' ) ) . symm )\n                let h \u2190 note_anon Eq prf\n                revert h\n                let ex \u2190 to_expr ` `( Equiv.toFun $ ( e ) $ ( x' ) )\n                generalize ex ( by infer_param ) transparency.none\n                intro x\n                let h \u2190 intro1\n                let b \u2190 target >>= is_prop\n                if\n                  b\n                  then\n                  do subst h sorry\n                  else\n                  unfreezing_hyp x' ( clear' tt [ x' ] )\n                    <|>\n                    fail\n                      f!\n                        \"equiv_rw expected to be able to clear the original hypothesis {\n                          x\n                          }, but couldn't.\"\n                skip\n            )\n          { failIfUnchanged := false }\n          true\n#align tactic.equiv_rw_hyp tactic.equiv_rw_hyp\n\n-- call `dsimp_result` with `no_defaults := tt`.\n/-- Rewrite the goal using an equiv `e`. -/\nunsafe def equiv_rw_target (e : expr) (cfg : equiv_rw_cfg := { }) : tactic Unit := do\n  let t \u2190 target\n  let e \u2190 equiv_rw_type e t cfg\n  let s \u2190 to_expr ``(Equiv.invFun $(e))\n  tactic.eapply s\n  skip\n#align tactic.equiv_rw_target tactic.equiv_rw_target\n\nend Tactic\n\nnamespace Tactic.Interactive\n\nopen Tactic\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- Auxiliary function to call `equiv_rw_hyp` on a `list pexpr` recursively. -/\nunsafe def equiv_rw_hyp_aux (hyp : Name) (cfg : equiv_rw_cfg) (permissive : Bool := false) :\n    List expr \u2192 itactic\n  | [] => skip\n  | e :: t => do\n    if permissive then equiv_rw_hyp hyp e cfg <|> skip else equiv_rw_hyp hyp e cfg\n    equiv_rw_hyp_aux t\n#align tactic.interactive.equiv_rw_hyp_aux tactic.interactive.equiv_rw_hyp_aux\n\n/-- Auxiliary function to call `equiv_rw_target` on a `list pexpr` recursively. -/\nunsafe def equiv_rw_target_aux (cfg : equiv_rw_cfg) (permissive : Bool) : List expr \u2192 itactic\n  | [] => skip\n  | e :: t => do\n    if permissive then equiv_rw_target e cfg <|> skip else equiv_rw_target e cfg\n    equiv_rw_target_aux t\n#align tactic.interactive.equiv_rw_target_aux tactic.interactive.equiv_rw_target_aux\n\n/-- `equiv_rw e at h\u2081 h\u2082 \u22ef`, where each `h\u1d62 : \u03b1` is a hypothesis, and `e : \u03b1 \u2243 \u03b2`,\nwill attempt to transport each `h\u1d62` along `e`, producing a new hypothesis `h\u1d62 : \u03b2`,\nwith all occurrences of `h\u1d62` in other hypotheses and the goal replaced with `e.symm h\u1d62`.\n\n`equiv_rw e` will attempt to transport the goal along an equivalence `e : \u03b1 \u2243 \u03b2`.\nIn its minimal form it replaces the goal `\u22a2 \u03b1` with `\u22a2 \u03b2` by calling `apply e.inv_fun`.\n\n`equiv_rw [e\u2081, e\u2082, \u22ef] at h\u2081 h\u2082 \u22ef` is equivalent to\n`{ equiv_rw [e\u2081, e\u2082, \u22ef] at h\u2081, equiv_rw [e\u2081, e\u2082, \u22ef] at h\u2082, \u22ef }`.\n\n`equiv_rw [e\u2081, e\u2082, \u22ef] at *` will attempt to apply `equiv_rw [e\u2081, e\u2082, \u22ef]` on the goal\nand on each expression available in the local context (except on the `e\u1d62`s themselves),\nfailing silently when it can't. Failing on a rewrite for a certain `e\u1d62` at a certain\nhypothesis `h` doesn't stop `equiv_rw` from trying the other equivalences on the list\nat `h`. This only happens for the wildcard location.\n\n`equiv_rw` will also try rewriting under (equiv_)functors, so it can turn\na hypothesis `h : list \u03b1` into `h : list \u03b2` or\na goal `\u22a2 unique \u03b1` into `\u22a2 unique \u03b2`.\n\nThe maximum search depth for rewriting in subexpressions is controlled by\n`equiv_rw e {max_depth := n}`.\n-/\nunsafe def equiv_rw (l : parse pexpr_list_or_texpr) (locat : parse location)\n    (cfg : equiv_rw_cfg := { }) : itactic := do\n  let es \u2190 l.mapM fun e => to_expr e\n  match locat with\n    | loc.wildcard => do\n      equiv_rw_target_aux cfg tt es\n      let ctx \u2190 local_context\n      ctx fun e => if e \u2208 es then skip else equiv_rw_hyp_aux e cfg tt es\n      skip\n    | loc.ns names => do\n      names fun hyp' =>\n          match hyp' with\n          | some hyp => equiv_rw_hyp_aux hyp cfg ff es\n          | none => equiv_rw_target_aux cfg ff es\n      skip\n#align tactic.interactive.equiv_rw tactic.interactive.equiv_rw\n\nadd_tactic_doc\n  { Name := \"equiv_rw\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.equiv_rw]\n    tags := [\"rewriting\", \"equiv\", \"transport\"] }\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Solve a goal of the form `t \u2243 _`,\n      by constructing an equivalence from `e : \u03b1 \u2243 \u03b2`.\n      This is the same equivalence that `equiv_rw` would use to rewrite a term of type `t`.\n      \n      A typical usage might be:\n      ```\n      have e' : option \u03b1 \u2243 option \u03b2 := by equiv_rw_type e\n      ```\n      -/\n    unsafe\n  def\n    equiv_rw_type\n    ( e : parse texpr ) ( cfg : equiv_rw_cfg := { } ) : itactic\n    :=\n      do\n        let q( $ ( t ) \u2243 _ ) \u2190 target | fail \"`equiv_rw_type` solves goals of the form `t \u2243 _`.\"\n          let e \u2190 to_expr e\n          tactic.equiv_rw_type e t cfg >>= tactic.exact\n#align tactic.interactive.equiv_rw_type tactic.interactive.equiv_rw_type\n\nadd_tactic_doc\n  { Name := \"equiv_rw_type\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.equiv_rw_type]\n    tags := [\"rewriting\", \"equiv\", \"transport\"] }\n\nend Tactic.Interactive\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/EquivRw.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.38313000140710174}}
{"text": "import pseudo_normed_group.LC\nimport analysis.normed.group.hom_completion\n\n/-!\n\n# V-hat(M_c^n)\n\nOne of the key players in the proof of the main theorem of this repo is\nthe normed group V-hat(M-bar_r'(S)_{\u2264c}^n). This file constructs\n\n## Key defintions\n\n- `CLCP V n`: the functor that sends a profinite set `S` to `V-hat(S^n)`\n- `CLFCP v r' c n`: the functor sending a profinitely-filtered `T\u207b\u00b9`-module `M`\n   to `V-hat((M_c)^n)`\n\n-/\nopen_locale classical nnreal\nnoncomputable theory\nlocal attribute [instance] type_pow\n\nopen SemiNormedGroup opposite Profinite pseudo_normed_group category_theory breen_deligne\nopen profinitely_filtered_pseudo_normed_group\n\nuniverse variable u\nvariables (r : \u211d\u22650) (V : SemiNormedGroup) (r' : \u211d\u22650)\nvariables (c c\u2081 c\u2082 c\u2083 c\u2084 : \u211d\u22650) (l m n : \u2115)\n\n/-- `CLC V n` is the functor that sends a profinite set `S` to `V-hat(S^n)` -/\ndef CLC (V : SemiNormedGroup) : Profinite\u1d52\u1d56 \u2964 SemiNormedGroup :=\nLC V \u22d9 Completion\n\nnamespace CLC\n\nlemma map_norm_noninc {M\u2081 M\u2082} (f : M\u2081 \u27f6 M\u2082) : ((CLC V).map f).norm_noninc :=\nCompletion.map_norm_noninc $ LC.map_norm_noninc _ _\n\ndef T [normed_with_aut r V] [fact (0 < r)] : CLC V \u2245 CLC V :=\n((whiskering_right _ _ _).obj _).map_iso (LC.T r V)\n\nlemma norm_T_le [normed_with_aut r V] [fact (0 < r)] (A) : \u2225(T r V).hom.app A\u2225 \u2264 r :=\nle_trans (normed_add_group_hom.norm_completion _).le $ LC.norm_T_le _ _ _\n\ndef T_inv [normed_with_aut r V] [fact (0 < r)] : CLC V \u27f6 CLC V :=\nwhisker_right (LC.T_inv r V) Completion\n\nlemma T_inv_eq [normed_with_aut r V] [fact (0 < r)] : (T r V).inv = T_inv r V := rfl\n\nlemma norm_T_inv_le [normed_with_aut r V] [fact (0 < r)] (A) : \u2225(T_inv r V).app A\u2225 \u2264 r\u207b\u00b9 :=\nle_trans (normed_add_group_hom.norm_completion _).le $ LC.norm_T_inv_le _ _ _\n\nend CLC\n\n/-- `CLFCP v r' c n` is the functor sending a profinitely-filtered `T\u207b\u00b9`-module `M`\n   to `V-hat((M_c)^n)` -/\ndef CLCFP (V : SemiNormedGroup) (r' : \u211d\u22650) (c : \u211d\u22650) (n : \u2115) :\n  (ProFiltPseuNormGrpWithTinv r')\u1d52\u1d56 \u2964 SemiNormedGroup :=\n(FiltrationPow r' c n).op \u22d9 CLC V\n\ntheorem CLCFP_def (V : SemiNormedGroup) (r' : \u211d\u22650) (c : \u211d\u22650) (n : \u2115) :\n  CLCFP V r' c n = LCFP V r' c n \u22d9 Completion := rfl\n\nnamespace CLCFP\n\nlemma map_norm_noninc {M\u2081 M\u2082} (f : M\u2081 \u27f6 M\u2082) : ((CLCFP V r' c n).map f).norm_noninc :=\nCLC.map_norm_noninc _ _\n\n@[simps app]\ndef res [fact (c\u2082 \u2264 c\u2081)] : CLCFP V r' c\u2081 n \u27f6 CLCFP V r' c\u2082 n :=\n(whisker_right (nat_trans.op $ FiltrationPow.cast_le r' c\u2082 c\u2081 n) (CLC V) : _)\n\nlemma res_def [fact (c\u2082 \u2264 c\u2081)] :\n  res V r' c\u2081 c\u2082 n = whisker_right (nat_trans.op (FiltrationPow.cast_le r' c\u2082 c\u2081 n)) (CLC V) :=\nrfl\n\nlemma res_def' [fact (c\u2082 \u2264 c\u2081)] (M : ProFiltPseuNormGrpWithTinv r') :\n  (res V r' c\u2081 c\u2082 n).app (op M) =\n  (CLC V).map ((Filtration.cast_le ((ProFiltPseuNormGrpWithTinv.Pow r' n).obj M) c\u2082 c\u2081)).op :=\nrfl\n\nlemma res_app' [fact (c\u2082 \u2264 c\u2081)] (M : (ProFiltPseuNormGrpWithTinv r')\u1d52\u1d56) :\n  (res V r' c\u2081 c\u2082 n).app M = (CLC V).map ((FiltrationPow.cast_le r' c\u2082 c\u2081 n).app (unop M)).op :=\nrfl\n\n@[simp] lemma res_refl : res V r' c c n = \ud835\udfd9 _ :=\nby { rw [res, FiltrationPow.cast_le_refl, nat_trans.op_id, whisker_right_id'], refl }\n\nlemma res_comp_res [fact (c\u2082 \u2264 c\u2081)] [fact (c\u2083 \u2264 c\u2082)] [fact (c\u2083 \u2264 c\u2081)] :\n  res V r' c\u2081 c\u2082 n \u226b res V r' c\u2082 c\u2083 n = res V r' c\u2081 c\u2083 n :=\nby simp only [res, \u2190 whisker_right_comp, FiltrationPow.cast_le_comp, \u2190 nat_trans.op_comp]\n\nlemma res_norm_noninc [fact (c\u2082 \u2264 c\u2081)] (M) :\n  ((res V r' c\u2081 c\u2082 n).app M).norm_noninc :=\nCompletion.map_norm_noninc $ LCFP.res_norm_noninc _ _ _ _ _ _\n\nsection Tinv\n-- kmb commented out the next line\n--open profinitely_filtered_pseudo_normed_group_with_Tinv\nvariables [fact (0 < r')] [fact (c\u2082 \u2264 r' * c\u2081)]\n\n-- @[simps obj {fully_applied := ff}]\ndef Tinv : CLCFP V r' c\u2081 n \u27f6 CLCFP V r' c\u2082 n :=\n(whisker_right (nat_trans.op $ FiltrationPow.Tinv r' c\u2082 c\u2081 n)\n  (LocallyConstant.obj V \u22d9 Completion) : _)\n.\n\nlemma Tinv_def : Tinv V r' c\u2081 c\u2082 n =\n  (whisker_right (LCFP.Tinv V r' c\u2081 c\u2082 n) Completion : _) := rfl\n\nlemma Tinv_def' : Tinv V r' c\u2081 c\u2082 n =\n  whisker_right (nat_trans.op $ FiltrationPow.Tinv r' c\u2082 c\u2081 n) (CLC V) := rfl\n\nlemma res_comp_Tinv [fact (c\u2082 \u2264 c\u2081)] [fact (c\u2083 \u2264 c\u2082)] [fact (c\u2083 \u2264 r' * c\u2082)] :\n  res V r' c\u2081 c\u2082 n \u226b Tinv V r' c\u2082 c\u2083 n = Tinv V r' c\u2081 c\u2082 n \u226b res V r' c\u2082 c\u2083 n :=\nbegin\n  dsimp only [Tinv, res, CLC, LC],\n  simp only [\u2190 whisker_right_comp, \u2190 nat_trans.op_comp],\n  refl\nend\n\nend Tinv\n\nsection T_inv\n\nvariables [normed_with_aut r V] [fact (0 < r)]\n\n@[simps {fully_applied := ff}]\ndef T : CLCFP V r' c n \u2245 CLCFP V r' c n :=\n((whiskering_left _ _ _).obj (FiltrationPow r' c n).op).map_iso (CLC.T r V)\n\n@[simps app_apply {fully_applied := ff}]\ndef T_inv : CLCFP V r' c n \u27f6 CLCFP V r' c n :=\nwhisker_left (FiltrationPow r' c n).op (CLC.T_inv r V)\n\nlemma T_inv_eq [normed_with_aut r V] [fact (0 < r)] : (T r V r' c n).inv = T_inv r V r' c n := rfl\n\nlemma T_inv_def : T_inv r V r' c n = (whisker_right (LCFP.T_inv r V r' c n) Completion : _) :=\nrfl\n\nlemma T_inv_app [fact (0 < r)] (M : (ProFiltPseuNormGrpWithTinv r')\u1d52\u1d56) :\n  (T_inv r V r' c n).app M =\n    (CLC.T_inv r V).app ((FiltrationPow r' c n).op.obj M) :=\nrfl\n\nlemma res_comp_T_inv [fact (c\u2082 \u2264 c\u2081)] :\n  res V r' c\u2081 c\u2082 n \u226b T_inv r V r' c\u2082 n =\n    T_inv r V r' c\u2081 n \u226b res V r' c\u2081 c\u2082 n :=\nbegin\n  ext M : 2,\n  simp only [nat_trans.comp_app, res_app', T_inv_app],\n  exact (CLC.T_inv r V).naturality _,\nend\n\nend T_inv\n\nend CLCFP\n\nnamespace breen_deligne\n\nopen CLCFP\nvariables {l m n}\n\nnamespace universal_map\n\nvariables (\u03d5 \u03c8 : universal_map m n)\n\ndef eval_CLCFP [\u03d5.suitable c\u2082 c\u2081] : CLCFP V r' c\u2081 n \u27f6 CLCFP V r' c\u2082 m :=\n(whisker_right (\u03d5.eval_LCFP V r' c\u2081 c\u2082) Completion : _)\n\nlemma eval_CLCFP_of (f : basic_universal_map m n) [f.suitable c\u2082 c\u2081] :\n  eval_CLCFP V r' c\u2081 c\u2082 (free_abelian_group.of f) =\n  (whisker_right (nat_trans.op $ f.eval_FP r' c\u2082 c\u2081) (CLC V)) :=\nby { rw [eval_CLCFP, eval_LCFP_of, basic_universal_map.eval_LCFP, whisker_right_twice], refl }\n\n@[simp] lemma eval_CLCFP_zero :\n  (0 : universal_map m n).eval_CLCFP V r' c\u2081 c\u2082 = 0 :=\nbegin\n  simp only [eval_CLCFP, eval_LCFP_zero],\n  ext x : 2,\n  exact Completion.map_zero _ _\nend\n\n@[simp] lemma eval_CLCFP_add [\u03d5.suitable c\u2082 c\u2081] [\u03c8.suitable c\u2082 c\u2081] :\n  (\u03d5 + \u03c8 : universal_map m n).eval_CLCFP V r' c\u2081 c\u2082 =\n  \u03d5.eval_CLCFP V r' c\u2081 c\u2082 + \u03c8.eval_CLCFP V r' c\u2081 c\u2082 :=\nbegin\n  simp only [eval_CLCFP, eval_LCFP_add],\n  ext x : 2,\n  exact Completion.map_add\nend\n\n@[simp] lemma eval_CLCFP_sub [\u03d5.suitable c\u2082 c\u2081] [\u03c8.suitable c\u2082 c\u2081] :\n  (\u03d5 - \u03c8 : universal_map m n).eval_CLCFP V r' c\u2081 c\u2082 =\n  \u03d5.eval_CLCFP V r' c\u2081 c\u2082 - \u03c8.eval_CLCFP V r' c\u2081 c\u2082 :=\nbegin\n  simp only [eval_CLCFP, eval_LCFP_sub],\n  ext x : 2,\n  exact Completion.map_sub\nend\n\nopen category_theory.limits\n\nlemma eval_CLCFP_comp (g : universal_map m n) (f : universal_map l m)\n  [hg : g.suitable c\u2082 c\u2081] [hf : f.suitable c\u2083 c\u2082] :\n  @eval_CLCFP V r' c\u2081 c\u2083 _ _ (comp g f) (suitable.comp c\u2082) =\n    g.eval_CLCFP V r' c\u2081 c\u2082 \u226b f.eval_CLCFP V r' c\u2082 c\u2083 :=\nby simp only [eval_CLCFP, \u2190 whisker_right_comp, eval_LCFP_comp V r' c\u2081 c\u2082 c\u2083]\n\nlemma res_comp_eval_CLCFP\n  [fact (c\u2082 \u2264 c\u2081)] [\u03d5.suitable c\u2084 c\u2082] [\u03d5.suitable c\u2083 c\u2081] [fact (c\u2084 \u2264 c\u2083)] :\n  res V r' c\u2081 c\u2082 n \u226b \u03d5.eval_CLCFP V r' c\u2082 c\u2084 =\n    \u03d5.eval_CLCFP V r' c\u2081 c\u2083 \u226b res V r' c\u2083 c\u2084 m :=\nby { dsimp only [CLC, res], simp only [eval_CLCFP, \u2190 whisker_right_comp, \u2190 whisker_right_twice],\n     congr' 1, apply res_comp_eval_LCFP }\n\nlemma Tinv_comp_eval_CLCFP [fact (0 < r')] [fact (c\u2082 \u2264 r' * c\u2081)] [fact (c\u2084 \u2264 r' * c\u2083)]\n  [\u03d5.suitable c\u2083 c\u2081] [\u03d5.suitable c\u2084 c\u2082] :\n  Tinv V r' c\u2081 c\u2082 n \u226b \u03d5.eval_CLCFP V r' c\u2082 c\u2084 =\n    \u03d5.eval_CLCFP V r' c\u2081 c\u2083 \u226b Tinv V r' c\u2083 c\u2084 m :=\nby simp only [eval_CLCFP, Tinv_def, \u2190 whisker_right_comp]; congr' 1; apply Tinv_comp_eval_LCFP\n\nlemma T_inv_comp_eval_CLCFP [normed_with_aut r V] [fact (0 < r)] [\u03d5.suitable c\u2082 c\u2081] :\n  T_inv r V r' c\u2081 n \u226b \u03d5.eval_CLCFP V r' c\u2081 c\u2082 =\n    \u03d5.eval_CLCFP V r' c\u2081 c\u2082 \u226b T_inv r V r' c\u2082 m :=\nby simp only [eval_CLCFP, T_inv_def, \u2190 whisker_right_comp, T_inv_comp_eval_LCFP]\n\nlemma norm_eval_CLCFP_le [normed_with_aut r V] [fact (0 < r)] [\u03d5.suitable c\u2082 c\u2081]\n  (N : \u2115) (h : \u03d5.bound_by N) (M) :\n  \u2225(\u03d5.eval_CLCFP V r' c\u2081 c\u2082).app M\u2225 \u2264 N :=\nle_trans (normed_add_group_hom.norm_completion _).le $ norm_eval_LCFP_le _ _ _ _ _ _ _ h _\n\nend universal_map\n\nend breen_deligne\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/pseudo_normed_group/CLC.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3831300014071017}}
{"text": "/-\nCopyright (c) 2021-2022 Julien Marquet. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Julien Marquet\n-/\n\nimport Lean\n\nimport Flows.Groundwork\nimport Flows.SolveSets\nimport Flows.Term\nimport Flows.Subst\nimport Flows.Vehicle\nimport Flows.Vanishing\nimport Flows.Unifier\n\nopen Classical\n\nset_option codegen false\n\nuniverse u\n\nsection\n\ntheorem cons_carrier_in {\u03b8 \u03c6 : Subst \u03b1 \u03b2} {l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2}\n  (h\u2081 : (\ud835\udcb1 \u03b8 : Fintype \u03b2) \u2286 \ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082)\n  (h\u2082 : (\ud835\udcb1 \u03c6 : Fintype \u03b2) \u2286 \ud835\udcb1 (r\u2081 \u2022 \u03b8) \u222a \ud835\udcb1 (r\u2082 \u2022 \u03b8))\n  (h\u2083 : vanishing \u03b8) (h\u2084 : vanishing \u03c6)\n  (h\u2085 : carrier \u03b8 \u2286 \ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082) (h\u2086 : carrier \u03c6 \u2286 \ud835\udcb1 (r\u2081 \u2022 \u03b8) \u222a \ud835\udcb1 (r\u2082 \u2022 \u03b8)) :\n  carrier (\u03b8 * \u03c6) \u2286 \ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082) := by\n  apply Fintype.included_trans (carrier_cons _ _)\n  simp only [vehicle_cons]\n  rw [Fintype.union_assoc]\n  apply Fintype.union_included_iff.2 (And.intro _ _)\n  focus\n    apply Fintype.included_trans h\u2085\n    solve_sets\n  focus\n    apply Fintype.included_trans h\u2086\n    apply Fintype.union_included_iff.2 \u27e8 _, _ \u27e9\n      <;> apply Fintype.included_trans (vehicle_on_image h\u2081 _)\n      <;> solve_sets\n\nend\n\nsection\n\ntheorem lex_of_le_and_lt {\u03b1 \u03b2 : Type u}\n  {ha : WellFoundedRelation \u03b1} {hb : WellFoundedRelation \u03b2}\n  {a\u2081 a\u2082 : \u03b1} {b\u2081 b\u2082 : \u03b2} (h\u2081 : ha.rel a\u2081 a\u2082 \u2228 a\u2081 = a\u2082) (h\u2082 : hb.rel b\u2081 b\u2082) :\n  (Prod.lex ha hb).rel (a\u2081, b\u2081) (a\u2082, b\u2082) := by\n  cases h\u2081 with\n  | inl h\u2081 =>\n    apply Prod.Lex.left\n    exact h\u2081\n  | inr h\u2081 =>\n    rw [h\u2081]\n    apply Prod.Lex.right\n    exact h\u2082\n\nend\n\nsection\n\nvariable {\u03b1 \u03b2 : Type u}\n\nprivate theorem flush_add_left (a : Nat) {b c : Nat} : b + c + a = b + a + c := by\n  simp only [Nat.add_assoc]\n  rw [Nat.add_comm a c]\n\ntheorem mass_lower_bound {x : \u03b2} {v : Term \u03b1 \u03b2} (h : Term.Var x \u2260 v) (u : Term \u03b1 \u03b2)\n  (\u03b8 : Subst \u03b1 \u03b2) : mass u + weight x u * mass (v \u2022 \u03b8) \u2264 mass (u \u2022 (Subst.elementary h * \u03b8)) := by\n  induction u with\n  | Cst c => match \u03b8 with\n    | \u27e8 \u03b8, _ \u27e9 =>\n      suffices p : \u2200 n, 0 + 0 * n \u2264 0 from p (mass (map_reduce \u03b8 v))\n      intros; simp\n  | Var y => match \u03b8 with\n    | \u27e8 \u03b8, _ \u27e9 =>\n      by_cases p : x = y\n        <;> simp [mass, weight, RSMul.smul, map_reduce, Subst.elementary, HMul.hMul, Mul.mul, comp, p]\n      rw [Nat.one_mul]\n      exact Nat.le.refl\n      simp [Ne.symm p, map_reduce, Nat.zero_le]\n  | Cons l r hl hr =>\n    simp only [mass, weight, subst_cons]\n    simp only [Nat.left_distrib, Nat.right_distrib, \u2190 Nat.add_assoc]\n    simp only [flush_add_left ((weight x r) * mass (v \u2022 \u03b8))]\n    simp only [flush_add_left (mass r)]\n    simp only [flush_add_left ((weight x l) * mass (v \u2022 \u03b8))]\n    simp only [flush_add_left (mass l)]\n    apply Nat.succ_le_succ\n    rw [Nat.add_assoc]\n    exact Nat.le_of_le_of_le hl hr\n\ntheorem weight_nonzero_of_mem_vehicle {x : \u03b2} {u : Term \u03b1 \u03b2} (h : x \u2208 (\ud835\udcb1 u : Fintype \u03b2)) :\n  weight x u \u2260 0 := by\n  induction u with\n  | Cst _ => exact False.elim <| Fintype.not_mem_empty _ h\n  | Var y =>\n    suffices p : x = y by\n      rw [p]\n      simp [weight]\n    cases h <;> trivial\n  | Cons l r hl hr =>\n    simp only [weight]\n    rw [vehicle_cons, Fintype.mem_union_iff] at h\n    cases h with\n    | inl h => exact Nat.add_ne_zero_of_l_ne_zero <| hl h\n    | inr h => exact Nat.add_ne_zero_of_r_ne_zero <| hr h\n\nend\n\nsection\n\nvariable {\u03b1 \u03b2 : Type u} [Monoid \u03b1]\n\ntheorem smul_cons_eq {l r : Term \u03b1 \u03b2} {\u03b8 : Subst \u03b1 \u03b2} :\n  (Term.Cons l r) \u2022 \u03b8 = (Term.Cons (l \u2022 \u03b8) (r \u2022 \u03b8)) :=\n  match \u03b8 with\n  | \u27e8 \u03b8, h\u03b8 \u27e9 => rfl\n\nprivate theorem cons_mgu {l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2} {\u03b8 \u03c6 : Subst \u03b1 \u03b2}\n  (\u03b8_mgu : is_mgu _ l\u2081 l\u2082 \u03b8) (\u03c6_mgu : is_mgu _ (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8) \u03c6) :\n  is_mgu _ (Term.Cons l\u2081 r\u2081) (Term.Cons l\u2082 r\u2082) (\u03b8 * \u03c6) := by\n  apply mgu_of_unifies_and_most_general\n  focus\n    simp [smul_cons_eq, \u2190 RAction.smul_mul,\n      unifies_of_mgu \u03b8_mgu, unifies_of_mgu \u03c6_mgu]\n  focus\n    intro\n    simp [smul_cons_eq]\n    intro \u27e8 h\u2081, h\u2082 \u27e9\n    let \u27e8 \u03c1\u2081, h\u03c1\u2081 \u27e9 := most_general_of_mgu \u03b8_mgu h\u2081\n    rw [\u2190 h\u03c1\u2081, \u2190 RAction.smul_mul, \u2190 RAction.smul_mul] at h\u2082\n    let \u27e8 \u03c1\u2082, h\u03c1\u2082 \u27e9 := most_general_of_mgu \u03c6_mgu h\u2082\n    exact \u27e8 \u03c1\u2082, (Monoid.mul_assoc _ _ _ \u25b8 h\u03c1\u2082 \u25b8 h\u03c1\u2081 \u25b8 rfl) \u27e9\n\nprivate def rel : WellFoundedRelation (Term \u03b1 \u03b2 \u00d7 Term \u03b1 \u03b2) :=\n  invImage (\u03bb (u, v) => ((\ud835\udcb1 u \u222a \ud835\udcb1 v : Fintype \u03b2), (u, v)))\n  <| Prod.lex\n    (Fintype.included_wfRel)\n    (Prod.rprod Term.mass_wfRel Term.mass_wfRel)\n\nabbrev P (x : Option (Subst \u03b1 \u03b2)) (u v : Term \u03b1 \u03b2) := match x with\n  | none => strangers (Subst \u03b1 \u03b2) u v\n  | some \u03b8 => is_mgu _ u v \u03b8\n      \u2227 (\ud835\udcb1 \u03b8 : Fintype \u03b2) \u2286 \ud835\udcb1 u \u222a \ud835\udcb1 v\n      \u2227 vanishing \u03b8\n      \u2227 carrier \u03b8 \u2286 \ud835\udcb1 u \u222a \ud835\udcb1 v\n\nprivate def P_comm (u v : Term \u03b1 \u03b2)\n  : { x // P x u v } \u2192 { x // P x v u } := by\n  intro \u27e8 x, h \u27e9\n  apply Subtype.mk x\n  match x with\n  | none =>\n    simp_all only [strangers_iff_no_unifier]\n    intro \u03b8 h'\n    exact h \u03b8 h'.symm\n  | some \u03b8 =>\n    let \u27e8 \u03b8_mgu, \u03b8_vehicle, \u03b8_vanishing, \u03b8_carrier \u27e9 := h\n    apply And.intro _ (And.intro _ (And.intro _ _))\n    focus\n      simp only [is_mgu]\n      suffices p : unifiers (Subst \u03b1 \u03b2) v u = unifiers (Subst \u03b1 \u03b2) u v by\n        rw [p]\n        exact \u03b8_mgu\n      funext \u03c6\n      simp [unifiers]\n      apply propext\n      apply Iff.intro\n      intro h; rw [h]\n      intro h; rw [h]\n    focus\n      rw [Fintype.union_comm]\n      exact \u03b8_vehicle\n    focus\n      exact \u03b8_vanishing\n    focus\n      rw [Fintype.union_comm]\n      exact \u03b8_carrier\n\nprivate theorem decr_left (l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2) :\n  rel.rel (l\u2081, l\u2082) (Term.Cons l\u2081 r\u2081, Term.Cons l\u2082 r\u2082) := by\n  simp [rel, invImage, InvImage]\n  apply lex_of_le_and_lt\n  focus\n    simp [invImage, InvImage, Fintype.included_wfRel]\n    simp only [WellFoundedRelation.rel]\n    simp only [vehicle_cons]\n    suffices h : (\ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082 : Fintype \u03b2)\n      \u2286 \ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082) by\n      by_cases p : (\ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082 : Fintype \u03b2)\n        = \ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082)\n      exact Or.inr p\n      exact Or.inl \u27e8 h, p \u27e9\n    simp only [vehicle_cons]\n    solve_sets\n  focus\n    exact Prod.RProd.intro (mass_decr_l _ _) (mass_decr_l _ _)\n\nprivate theorem decr_right (l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2) {\u03b8 : Subst \u03b1 \u03b2}\n  (\u03b8_vehicle : (\ud835\udcb1 \u03b8 : Fintype \u03b2) \u2286 \ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082)\n  (\u03b8_vanishing : vanishing \u03b8) (\u03b8_carrier : carrier \u03b8 \u2286 \ud835\udcb1 l\u2081 \u222a \ud835\udcb1 l\u2082) :\n  rel.rel (r\u2081 \u2022 \u03b8, r\u2082 \u2022 \u03b8) (Term.Cons l\u2081 r\u2081, Term.Cons l\u2082 r\u2082) := by\n  by_cases h : \u03b8 = 1\n  focus\n    rw [h, RAction.smul_one, RAction.smul_one]\n    apply lex_of_le_and_lt\n    focus\n      simp [invImage, InvImage, Fintype.included_wfRel]\n      suffices h : (\ud835\udcb1 r\u2081 \u222a \ud835\udcb1 r\u2082 : Fintype \u03b2)\n        \u2286 \ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082) by\n        by_cases p : (\ud835\udcb1 r\u2081 \u222a \ud835\udcb1 r\u2082 : Fintype \u03b2)\n          = \ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082)\n        exact Or.inr p\n        exact Or.inl \u27e8 h, p \u27e9\n      simp only [vehicle_cons]\n      solve_sets\n    focus\n      exact Prod.RProd.intro (mass_decr_r _ _) (mass_decr_r _ _)\n  focus\n    apply Prod.Lex.left\n    apply And.intro\n    focus\n      simp only [vehicle_cons]\n      apply Fintype.union_included_iff.2 <| And.intro _ _\n      focus\n        apply Fintype.included_trans (vehicle_on_image Fintype.included_refl r\u2081)\n        apply Fintype.union_included_iff.2\n          <| And.intro (Fintype.included_trans \u03b8_vehicle _) _\n          <;> solve_sets\n      focus\n        apply Fintype.included_trans (vehicle_on_image Fintype.included_refl r\u2082)\n        apply Fintype.union_included_iff.2 <| And.intro (Fintype.included_trans \u03b8_vehicle _) _\n          <;> solve_sets\n    focus\n      let \u27e8 x, hx \u27e9 := (not_one_iff_modifying \u03b8).1 h\n      let not_in_r\u2081 := vanishing_on_term \u03b8_vanishing hx r\u2081\n      let not_in_r\u2082 := vanishing_on_term \u03b8_vanishing hx r\u2082\n      let not_in_lhs : \u00ac x \u2208 (\ud835\udcb1 (r\u2081 \u2022 \u03b8) \u222a \ud835\udcb1 (r\u2082 \u2022 \u03b8) : Fintype \u03b2) :=\n        \u03bb h => match (Fintype.mem_union_iff _ _ _).1 h with\n          | Or.inl h => not_in_r\u2081 h\n          | Or.inr h => not_in_r\u2082 h\n      let in_rhs : x \u2208 (\ud835\udcb1 (Term.Cons l\u2081 r\u2081) \u222a \ud835\udcb1 (Term.Cons l\u2082 r\u2082) : Fintype \u03b2) := by\n        simp only [vehicle_cons]\n        have p := carrier_spec.2 hx\n        rw [Fintype.mem_iff_singleton_included]\n        rw [Fintype.mem_iff_singleton_included] at p\n        apply Fintype.included_trans p\n        apply Fintype.included_trans \u03b8_carrier\n        solve_sets\n      exact Fintype.different_if_not_same_element not_in_lhs in_rhs\n\nprivate theorem prepend_elementary_on_variable_unifier {x : \u03b2} {u : Term \u03b1 \u03b2} {\u03b8 : Subst \u03b1 \u03b2}\n  (h : Term.Var x \u2260 u) (h' : (Term.Var x : Term \u03b1 \u03b2) \u2022 \u03b8 = u \u2022 \u03b8) :\n  \u03b8 = (Subst.elementary h) * \u03b8 := by\n  apply Subst.ext.2\n  intro y\n  rw [\u2190 RAction.smul_mul]\n  by_cases p : y = x\n  focus\n    rw [p, h', Subst.elementary_spec\u2081]\n  focus\n    rw [Subst.elementary_spec\u2082]\n    exact p\n\nprivate def unify_variable_of_not_in_vehicle {x : \u03b2} {u : Term \u03b1 \u03b2}\n  (h : \u00ac x \u2208 (\ud835\udcb1 u : Fintype \u03b2)) : { e : Option (Subst \u03b1 \u03b2) // P e (Term.Var x) u } :=\n  let x_ne_u := (show Term.Var x \u2260 u by\n    intro h'\n    rw [\u2190 h'] at h\n    apply h\n    apply List.Mem.head);\n  \u27e8 some (Subst.elementary x_ne_u), by\n  apply And.intro (mgu_of_unifies_and_most_general _ _)\n    (And.intro _ (And.intro _ _))\n  focus\n    rw [Subst.elementary_spec\u2081 x_ne_u]\n    rw [elementary_on_not_in_vehicle]\n    exact h\n  focus\n    intro \u03b8 h\u03b8\n    apply Exists.intro \u03b8\n    exact Eq.symm <| prepend_elementary_on_variable_unifier x_ne_u h\u03b8\n  focus\n    rw [vehicle_elementary]\n    apply Fintype.included_union_l _ <| Fintype.included_refl\n  focus\n    apply elementary_vanishing\n    exact h\n  focus\n    rw [elementary_carrier]\n    exact Fintype.included_union_r _ <| Fintype.included_refl \u27e9\n\n-- Clearly not well written, I sould automate this...\n-- But since I don't do a lot of calculus in the proofs here, I don't feel the need\n-- to spend time writing tactics for numbers.\nprivate theorem variable_stranger_of_in_vehicle {x : \u03b2} {u : Term \u03b1 \u03b2}\n  (h\u2081 : mass u \u2260 0) (h\u2082 : x \u2208 (\ud835\udcb1 u : Fintype \u03b2)) :\n  strangers (Subst \u03b1 \u03b2) (Term.Var x) u := by\n  have x_ne_u : Term.Var x \u2260 u := by\n    intro h\n    apply h\u2081\n    rw [\u2190 h]\n    rfl\n  rw [strangers_iff_no_unifier]\n  intro \u03b8 h\n  have p := prepend_elementary_on_variable_unifier x_ne_u h\n  conv at h => rhs; rw [p]\n  have p' := mass_lower_bound x_ne_u u \u03b8\n  conv at p' => rhs; rw [\u2190 p]\n  have p'' := Nat.mul_le_mul_right (mass (u \u2022 \u03b8))\n    <| Nat.one_le_of_ne_zero\n    <| weight_nonzero_of_mem_vehicle h\u2082\n  rw [Nat.one_mul] at p''\n  have p\u2084 := Nat.le_trans p' p''\n  have p\u2085 : mass u = 0 := by\n    apply byContradiction\n    intro h\n    have p := Nat.lt_of_succ_le <| Nat.one_le_of_ne_zero h\n    have p' := Nat.add_lt_add_right p (weight x u * mass (u \u2022 \u03b8))\n    have p''' := Nat.lt_of_lt_of_le p' p\u2084\n    rw [Nat.zero_add] at p'''\n    exact False.elim <| Nat.not_lt_self _ p'''\n  exact h\u2081 p\u2085\n\ntheorem strangers_of_left_strangers {l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2} (h : strangers (Subst \u03b1 \u03b2) l\u2081 l\u2082) :\n  strangers (Subst \u03b1 \u03b2) (Term.Cons l\u2081 r\u2081) (Term.Cons l\u2082 r\u2082) := by\n  apply strangers_iff_no_unifier.2\n  intro \u03b8 h'\n  apply strangers_iff_no_unifier.1 h \u03b8\n  simp only [subst_cons] at h'\n  apply Term.noConfusion h'\n  exact \u03bb h _ => h\n\ntheorem strangers_of_right_strangers {l\u2081 r\u2081 l\u2082 r\u2082 : Term \u03b1 \u03b2} {\u03b8 : Subst \u03b1 \u03b2}\n  (\u03b8_mgu : is_mgu _ l\u2081 l\u2082 \u03b8) (h : strangers (Subst \u03b1 \u03b2) (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8)) :\n  strangers (Subst \u03b1 \u03b2) (Term.Cons l\u2081 r\u2081) (Term.Cons l\u2082 r\u2082) := by\n  apply strangers_iff_no_unifier.2\n  intro \u03c6 h'\n  suffices h' : l\u2081 \u2022 \u03c6 = l\u2082 \u2022 \u03c6 \u2227 r\u2081 \u2022 \u03c6 = r\u2082 \u2022 \u03c6 by\n    let \u27e8 \u03c1, h\u03c1 \u27e9 := most_general_of_mgu \u03b8_mgu h'.1\n    apply strangers_iff_no_unifier.1 h \u03c1\n    simp only [RAction.smul_mul, h\u03c1]\n    exact h'.2\n  simp only [subst_cons] at h'\n  apply And.intro <;> apply Term.noConfusion h'\n    <;> intros\n          <;> assumption\n\nnamespace Robinson\n\ndef unify_same {u v : Term \u03b1 \u03b2} (h : u = v) : { e // P e u v } := by\n  apply Subtype.mk (some 1)\n  rw [\u2190 h]\n  apply And.intro _ (And.intro _ (And.intro _ _))\n  focus\n    funext \u03b8\n    apply propext\n    suffices p : \u2203 \u03c1, 1 * \u03c1 = \u03b8 by\n      simp_all [unifiers, generated_by]\n    apply Exists.intro \u03b8\n    exact Monoid.one_mul _\n  focus\n    rw [vehicle_one]\n    exact Fintype.empty_included _\n  focus\n    exact \u03bb h => False.elim (h rfl)\n  focus\n    rw [is_one_iff_empty_carrier.1 rfl]\n    apply Fintype.empty_included _\n\ndef unify_var_cons (x : \u03b2) (l r : Term \u03b1 \u03b2) :\n  { e : Option (Subst \u03b1 \u03b2) // P e (Term.Var x) (Term.Cons l r)} :=\n  if p : x \u2208 (\ud835\udcb1 (Term.Cons l r) : Fintype \u03b2) then \u27e8 none, by\n    apply variable_stranger_of_in_vehicle _ p\n    apply Ne.symm \u2218 Nat.ne_of_lt\n      <| Nat.lt_of_lt_of_le (Nat.zero_lt_one) (Nat.le_add_left _ _) \u27e9\n  else unify_variable_of_not_in_vehicle p\n\ndef unify_var_var (x y : \u03b2) :\n  { e : Option (Subst \u03b1 \u03b2) // P e (Term.Var x) (Term.Var y)} :=\n  if p : x = y then unify_same (by rw [p])\n  else by\n    have p' : (Term.Var x : Term \u03b1 \u03b2) \u2260 Term.Var y :=\n      \u03bb h => p <| Term.noConfusion h id\n    apply unify_variable_of_not_in_vehicle\n    intro h; apply p; cases h <;> trivial\n\ndef unify_var_cst (x : \u03b2) (c : \u03b1) :\n  { e : Option (Subst \u03b1 \u03b2) // P e (Term.Var x) (Term.Cst c) } := by\n  have p' : (Term.Var x : Term \u03b1 \u03b2) \u2260 Term.Cst c := by\n    intro h\n    apply Term.noConfusion h\n  apply unify_variable_of_not_in_vehicle\n  intro h; cases h <;> trivial\n\ndef unify_cst_cons (c : \u03b1) (l r : Term \u03b1 \u03b2) :\n  { e : Option (Subst \u03b1 \u03b2) // P e (Term.Cst c) (Term.Cons l r) } :=\n  \u27e8 none, by\n  apply strangers_iff_no_unifier.2\n  intro \u27e8 \u03b8, _ \u27e9 h\n  apply Term.noConfusion h \u27e9\n\ndef unify_cst_cst (a b : \u03b1) :\n  { e : Option (Subst \u03b1 \u03b2) // P e (Term.Cst a) (Term.Cst b)} :=\n  if p : a = b then unify_same (by rw [p])\n  else by\n    apply Subtype.mk none\n    apply strangers_iff_no_unifier.2\n    exact \u03bb \u03b8 h => p <| match \u03b8 with\n    | \u27e8 _, _ \u27e9 => Term.noConfusion h id\n\ndef robinson (u v : Term \u03b1 \u03b2) : { e // P e u v } := match u, v with\n  | Term.Cons l\u2081 r\u2081, Term.Cons l\u2082 r\u2082 =>\n    match robinson l\u2081 l\u2082 with\n    | \u27e8 none, h \u27e9 => \u27e8 none, strangers_of_left_strangers h \u27e9\n    | \u27e8 some \u03b8, \u03b8_mgu, \u03b8_vehicle, \u03b8_vanishing, \u03b8_carrier \u27e9 =>\n      match robinson (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8) with\n      | \u27e8 none, h \u27e9 => \u27e8 none, strangers_of_right_strangers \u03b8_mgu h \u27e9\n      | \u27e8 some \u03c6, \u03c6_mgu, \u03c6_vehicle, \u03c6_vanishing, \u03c6_carrier \u27e9 =>\n        \u27e8 some (\u03b8 * \u03c6),\n          cons_mgu \u03b8_mgu \u03c6_mgu,\n          cons_vehicle_in \u03b8_vehicle \u03c6_vehicle,\n          cons_vanishing \u03b8_vehicle \u03c6_vehicle \u03b8_vanishing \u03c6_vanishing,\n          cons_carrier_in \u03b8_vehicle \u03c6_vehicle \u03b8_vanishing \u03c6_vanishing \u03b8_carrier \u03c6_carrier \u27e9\n  | Term.Var x, Term.Cons l r =>\n    unify_var_cons _ _ _\n  | Term.Cons l r, Term.Var x =>\n    P_comm _ _ <| unify_var_cons _ _ _\n  | Term.Var x, Term.Var y =>\n    unify_var_var _ _\n  | Term.Var x, Term.Cst c =>\n    unify_var_cst _ _\n  | Term.Cst c, Term.Var x =>\n    P_comm _ _ <| unify_var_cst _ _\n  | Term.Cst c, Term.Cons l r =>\n    unify_cst_cons _ _ _\n  | Term.Cons l r, Term.Cst c =>\n    P_comm _ _ <| unify_cst_cons _ _ _\n  | Term.Cst a, Term.Cst b =>\n    unify_cst_cst _ _\ntermination_by' invImage (\u03bb \u27e8 u, v \u27e9 => (u, v)) rel\ndecreasing_by\n  first\n    | exact decr_left _ _ _ _\n    | apply decr_right <;> assumption\n\nend Robinson\n\ndef robinson (u v : Term \u03b1 \u03b2) := (Robinson.robinson u v).1\n\ntheorem robinson_eq (u v : Term \u03b1 \u03b2) : robinson u v = match u, v with\n  | Term.Cons l\u2081 r\u2081, Term.Cons l\u2082 r\u2082 =>\n    match robinson l\u2081 l\u2082 with\n    | none => none\n    | some \u03b8 => match robinson (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8) with\n      | none => none\n      | some \u03c6 => some (\u03b8 * \u03c6)\n  | Term.Var x, Term.Cons l r =>\n    if p : x \u2208 (\ud835\udcb1 (Term.Cons l r) : Fintype \u03b2) then none\n    else subst_simple x (Term.Cons l r)\n  | Term.Cons l r, Term.Var x =>\n    if p : x \u2208 (\ud835\udcb1 (Term.Cons l r) : Fintype \u03b2) then none\n    else subst_simple x (Term.Cons l r)\n  | Term.Var x, Term.Var y => some (subst_simple x (Term.Var y))\n  | Term.Var x, Term.Cst c => some (subst_simple x (Term.Cst c))\n  | Term.Cst c, Term.Var x => some (subst_simple x (Term.Cst c))\n  | Term.Cst c, Term.Cons l r => none\n  | Term.Cons l r, Term.Cst c => none\n  | Term.Cst a, Term.Cst b => if a = b then some 1 else none\n  := match u, v with\n  | Term.Cons l\u2081 r\u2081, Term.Cons l\u2082 r\u2082 => by\n    simp only []\n    conv => lhs; rw [robinson]; unfold Robinson.robinson\n    conv => rhs; rw [robinson]\n    simp only []\n    let o := Robinson.robinson l\u2081 l\u2082\n    rw [show Robinson.robinson l\u2081 l\u2082 = o from rfl]\n    match o with\n    | Subtype.mk none h => simp only []\n    | \u27e8 (some \u03b8), _, _, _, _ \u27e9 =>\n      simp only []\n      rw [robinson]\n      let o' := Robinson.robinson (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8)\n      rw [show Robinson.robinson (r\u2081 \u2022 \u03b8) (r\u2082 \u2022 \u03b8) = o' from rfl]\n      match o' with\n      | \u27e8 none, _ \u27e9 => simp\n      | \u27e8 some \u03c6, _, _, _, _ \u27e9 => simp\n  | Term.Var x, Term.Cons l r => by\n    rw [robinson, Robinson.robinson, Robinson.unify_var_cons]\n    by_cases p : x \u2208 (\ud835\udcb1 (Term.Cons l r) : Fintype \u03b2)\n    simp [p]\n    simp [subst_simple, p, unify_variable_of_not_in_vehicle]\n  | Term.Cons l r, Term.Var x => by\n    rw [robinson, Robinson.robinson, Robinson.unify_var_cons]\n    by_cases p : x \u2208 (\ud835\udcb1 (Term.Cons l r) : Fintype \u03b2)\n    simp only [p]\n    simp [P_comm]\n    simp only [p]\n    simp [P_comm, subst_simple, unify_variable_of_not_in_vehicle]\n  | Term.Var x, Term.Var y => by\n    rw [robinson, Robinson.robinson, Robinson.unify_var_var]\n    by_cases p : x = y\n    simp [p, subst_simple, Robinson.unify_same]\n    simp [p, subst_simple, unify_variable_of_not_in_vehicle]\n  | Term.Var x, Term.Cst c => by\n    rw [robinson, Robinson.robinson]\n    simp [subst_simple, Robinson.unify_var_cst, unify_variable_of_not_in_vehicle]\n  | Term.Cst c, Term.Var x => by\n    rw [robinson, Robinson.robinson, P_comm]\n    simp [subst_simple, Robinson.unify_var_cst, unify_variable_of_not_in_vehicle]\n  | Term.Cst c, Term.Cons l r => by\n    simp [robinson, Robinson.robinson, Robinson.unify_cst_cons]\n  | Term.Cons l r, Term.Cst c => by\n    simp [robinson, Robinson.robinson, P_comm, Robinson.unify_cst_cons]\n  | Term.Cst a, Term.Cst b => by\n    rw [robinson, Robinson.robinson, Robinson.unify_cst_cst]\n    by_cases p : a = b\n    simp [p]\n    rw [Robinson.unify_same]\n    simp [p]\n\ntheorem robinson_spec (u v : Term \u03b1 \u03b2) :\n  match robinson u v with\n  | none => strangers (Subst \u03b1 \u03b2) u v\n  | some \u03b8 => is_mgu _ u v \u03b8 := by\n  rw [robinson]\n  exact match Robinson.robinson u v with\n  | \u27e8 none, h \u27e9 => h\n  | \u27e8 some _, h, _ \u27e9 => h\n\ntheorem herbrand (u v : Term \u03b1 \u03b2) :\n  strangers (Subst \u03b1 \u03b2) u v \u2228 \u2203 \u03b8 : Subst \u03b1 \u03b2, is_mgu _ u v \u03b8 :=\n  match Robinson.robinson u v with\n  | \u27e8 none, p \u27e9 => Or.inl p\n  | \u27e8 some \u03b8, p, _ \u27e9 => Or.inr \u27e8 \u03b8, p \u27e9\n\nexample : robinson (Term.Cons (Term.Var 0) (Term.Var 0) : Term Nat Nat)\n  (Term.Cons (Term.Var 0) (Term.Var 0) : Term Nat Nat) = some 1 := by\n  repeat rw [robinson_eq]; simp\n\nexample : robinson (Term.Cons (Term.Var 0) (Term.Var 0) : Term Nat Nat)\n  (Term.Cons (Term.Var 1) (Term.Var 1) : Term Nat Nat) =\n  some (subst_simple 0 (Term.Var 1)) := by\n  repeat rw [robinson_eq]; simp\n\nend\n\n", "meta": {"author": "thejohncrafter", "repo": "flows", "sha": "f4732e6784aa6ea13b07dc042be2c3816a73fa84", "save_path": "github-repos/lean/thejohncrafter-flows", "path": "github-repos/lean/thejohncrafter-flows/flows-f4732e6784aa6ea13b07dc042be2c3816a73fa84/Flows/Herbrand.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.685949467848392, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.38298410464974225}}
{"text": "theorem ex1 (n m : Nat) (f : Nat \u2192 Nat) : some n = some m \u2192 f n = f m := by\n  intro h\n  injection h with h\n  rw [h]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/injIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3829840974807124}}
{"text": "inductive Foo\n  | foo : Nat \u2192 Foo\n  | foos : Array Foo \u2192 Foo\n  deriving BEq\n\nexample : Foo.foo 0 \u2260 Foo.foo 1 := by simp\n\nexample : #[0] \u2260 #[1] := by simp\n\nexample : #[Foo.foo 0] \u2260 #[Foo.foo 1] := by simp\n\nexample : Foo.foos #[.foo 0] \u2260 Foo.foos #[.foo 1] := by simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/toArrayEq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.38298409748071227}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport data.tree\nimport data.sign\nimport data.finmap\nimport logic.equiv.basic\nimport tactic.ring\nimport tactic.zify\nimport tree\nimport misc\n\n/-!\n# Encodable types using trees\n\nThis file defines encodings to `unit_tree` rather than `\u2115`.\nThis is especially useful for encoding tree-like data\nnaturally.\n\nTODO: the `encoding`'s used in `src/computability` should be to trees.\n\n## Main declarations\n\n* `tencodable \u03b1`: States that there exists an explicit encoding function `encode : \u03b1 \u2192 unit_tree`\n   with a partial inverse `decode : unit_tree \u2192 option \u03b1`.\n\n-/\nopen tree\nopen_locale tree\n\n/-- Encoding of a type into a tree structure -/\nclass tencodable (\u03b1 : Type*) :=\n(encode : \u03b1 \u2192 tree unit)\n(decode [] : tree unit \u2192 option \u03b1)\n(encodek : \u2200 a, decode (encode a) = some a)\n\nattribute [simp, higher_order] tencodable.encodek\n\nnamespace tencodable\nvariables {\u03b1 \u03b2 : Type*} [tencodable \u03b1] [tencodable \u03b2]\n\ntheorem encode_injective : function.injective (@encode \u03b1 _)\n| x y e := option.some.inj $ by rw [\u2190 encodek, e, encodek]\n\n@[simp] lemma encode_inj {a b : \u03b1} : encode a = encode b \u2194 a = b :=\nencode_injective.eq_iff\n\n/-- Any tencodable element has decidable equality by checking if the encodings are equal -/\ndef decidable_eq_of_encodable (\u03b1) [tencodable \u03b1] : decidable_eq \u03b1\n| a b := decidable_of_iff _ encode_inj\n\n/-- If `\u03b1` is encodable and there is an injection `f : \u03b2 \u2192 \u03b1`, then `\u03b2` is encodable as well. -/\ndef of_left_injection {\u03b2} (f : \u03b2 \u2192 \u03b1) (finv : \u03b1 \u2192 option \u03b2) (linv : \u2200 b, finv (f b) = some b) :\n  tencodable \u03b2 :=\n\u27e8\u03bb b, encode (f b),\n \u03bb n, (decode \u03b1 n).bind finv,\n \u03bb b, by simp [linv]\u27e9\n\n/-- If `\u03b1` is encodable and `f : \u03b2 \u2192 \u03b1` is invertible, then `\u03b2` is encodable as well. -/\ndef of_left_inverse {\u03b2} (f : \u03b2 \u2192 \u03b1) (finv : \u03b1 \u2192 \u03b2) (linv : \u2200 b, finv (f b) = b) : tencodable \u03b2 :=\nof_left_injection f (some \u2218 finv) (\u03bb b, congr_arg some (linv b))\n\n/-- Encodability is preserved by equivalence. -/\ndef of_equiv {\u03b2} (\u03b1) [tencodable \u03b1] (e : \u03b2 \u2243 \u03b1) : tencodable \u03b2 :=\nof_left_inverse e e.symm e.left_inv\n\ninstance _root_.unit_tree.tencodable : tencodable (tree unit) :=\n{ encode := id,\n  decode := some,\n  encodek := \u03bb _, rfl }\n\n@[simp] lemma encode_unit_tree (x : tree unit) : encode x = x := rfl\n@[simp] lemma decode_unit_tree (x : tree unit) : decode (tree unit) x = some x := rfl\n\n@[priority 100] instance _root_.is_empty.to_tencodable {\u03b1} [is_empty \u03b1] : tencodable \u03b1 :=\n\u27e8is_empty_elim, \u03bb n, none, is_empty_elim\u27e9\n\ninstance _root_.punit.tencodable : tencodable punit :=\n\u27e8\u03bb_, nil, \u03bb _, some punit.star, \u03bb _, by simp\u27e9\n\nlemma encode_star : encode punit.star = nil := rfl\n\nsection prod\n\ninstance (\u03b2 : \u03b1 \u2192 Type*) [\u2200 i, tencodable (\u03b2 i)] : tencodable (sigma \u03b2) :=\n{ encode := \u03bb x, (encode x.1) \u25b3 (encode x.2),\n  decode := \u03bb x, (decode \u03b1 x.left).bind $ \u03bb a, (decode (\u03b2 a) x.right).bind $ \u03bb b, some \u27e8a, b\u27e9,\n  encodek := \u03bb x, by cases x; simp }\n\n/-- Encoding of a pair of encodable elements -/\ninstance _root_.prod.tencodable : tencodable (\u03b1 \u00d7 \u03b2) :=\n{ encode := \u03bb x, (encode x.1) \u25b3 (encode x.2),\n  decode := \u03bb y, (decode \u03b1 y.left).bind $ \u03bb l, (decode \u03b2 y.right).bind $ \u03bb r, some (l, r),\n  encodek := \u03bb x, by simp }\n\nlemma encode_prod (x : \u03b1) (y : \u03b2) : encode (x, y) = (encode x) \u25b3 (encode y) := rfl\n\nlemma encode_sigma {\u03b2 : \u03b1 \u2192 Type*} [\u2200 i, tencodable (\u03b2 i)] (x : \u03b1) (y : \u03b2 x) : encode (\u27e8x, y\u27e9 : sigma \u03b2) = (encode x) \u25b3 (encode y) := rfl\n\nend prod\n\nsection bool\n\nabbreviation non_nil : tree unit := nil \u25b3 nil\n@[simp] lemma non_nil_ne_nil : non_nil \u2260 nil := by trivial\n\n/-- Encoding of `bool` -/\ninstance _root_.bool.tencodable : tencodable bool :=\n{ encode := \u03bb b, cond b nil non_nil,\n  decode := \u03bb x, some (x = nil : bool),\n  encodek := \u03bb b, by cases b; simp }\n\nlemma encode_tt : encode tt = nil := rfl\nlemma encode_ff : encode ff = non_nil := rfl\n\nend bool\n\nsection list\n\n/-- Interpret a tree as a list of trees according to the left children\n  of the nodes on the rightmost path-/\ndef as_list : tree unit \u2192 list (tree unit)\n| nil := []\n| (a \u25b3 b) := a :: as_list b\n\n/-- Interpret a list of trees as a single tree -/\ndef of_list : list (tree unit) \u2192 tree unit\n| [] := nil\n| (x :: xs) := x \u25b3 (of_list xs)\n\n/-- There is an equivalence between `unit_tree` and `list unit_tree`\n  corresponding to taking all of the left children on nodes of the rightmost path.\n  We use this to encode lists -/\ndef equiv_list : tree unit \u2243 list (tree unit) :=\n{ to_fun := as_list,\n  inv_fun := of_list,\n  left_inv := \u03bb t, by induction t using tree.unit_rec_on; simp [as_list, of_list, *],\n  right_inv := \u03bb l, by induction l; simp [as_list, of_list, *] }\n\n@[simp] lemma equiv_list_nil : equiv_list nil = [] := rfl\n@[simp] lemma equiv_list_node (a b : tree unit) :\n  equiv_list (a \u25b3 b) = a :: (equiv_list b) := rfl\n@[simp] lemma equiv_list_symm_nil : equiv_list.symm [] = nil := rfl\n@[simp] lemma equiv_list_symm_cons (a : tree unit) (b : list (tree unit)) :\n  equiv_list.symm (a :: b) = a \u25b3 (equiv_list.symm b) := rfl\n\ninstance _root_.list.tencodable : tencodable (list \u03b1) :=\n{ encode := \u03bb l, equiv_list.symm (l.map encode),\n  decode := \u03bb t, ((equiv_list t).map (decode \u03b1)).all_some,\n  encodek := \u03bb l, by simp }\n\nlemma encode_nil : encode (@list.nil \u03b1) = nil := rfl\nlemma encode_cons (x : \u03b1) (xs : list \u03b1) : encode (x :: xs) = (encode x) \u25b3 (encode xs) := rfl\n\nlemma encode_list_tree (x : list (tree unit)) : encode x = equiv_list.symm x :=\nby simp [encode]\n\nlemma decode_list_tree (x : tree unit) : decode _ x = some (equiv_list x) :=\nby simp [decode]\n\nend list\n\nsection nat\n\n/-- This is a unary encoding for natural numbers. The canonical\n  way of representing `n` is as n \u21a6 nil \u25b3 nil \u25b3 ... -/\ninstance _root_.nat.unary_tencodable : tencodable \u2115 :=\n{ encode := \u03bb n, (equiv_list.symm $ list.repeat nil n),\n  decode := \u03bb t, some t.num_nodes,\n  encodek := \u03bb n, congr_arg some $ by induction n; simp [*] }\n\nlemma encode_zero : encode 0 = nil := rfl\nlemma encode_succ (n : \u2115) : encode (n + 1) = nil \u25b3 (encode n) := rfl\nlemma encode_nat_eq_iterate (n : \u2115) :\n  encode n = ((\u03bb x, nil \u25b3 x)^[n] nil) :=\nby { induction n; simp [*, function.iterate_succ', encode_zero, encode_succ], }\n\n@[simp] lemma encode_num_nodes (n : \u2115) : (encode n).num_nodes = n :=\noption.some_injective _ (tencodable.encodek n)\n\nend nat\n\nsection option\n\n/-- Encode an `option \u03b1`, using `nil` as `none` -/\n@[simp] def of_option : option \u03b1 \u2192 tree unit\n| none := nil\n| (some x) := nil \u25b3 (encode x)\n\n/-- Decode an `option \u03b1` as a tree -/\n@[simp] def to_option : tree unit \u2192 option (option \u03b1)\n| nil := some none\n| (x \u25b3 y) := (decode \u03b1 y).map some\n\n/-- Encoding of `option \u03b1` when `\u03b1` has an encoding -/\ninstance : tencodable (option \u03b1) :=\n{ encode := of_option,\n  decode := to_option,\n  encodek := \u03bb x, by cases x; simp [of_option, to_option] }\n\nend option\n\nsection sum\n\n/-- Encode a sum by using the left child of the root to signal if the right represents \u03b1 or \u03b2 -/\n@[simp] def of_sum : \u03b1 \u2295 \u03b2 \u2192 tree unit\n| (sum.inl x) := nil \u25b3 (encode x)\n| (sum.inr x) := non_nil \u25b3 (encode x)\n\n/-- Decode a sum by using the left child of the root to signal if the right represents \u03b1 or \u03b2 -/\n@[simp] def to_sum (x : tree unit) : option (\u03b1 \u2295 \u03b2) :=\n  if x.left = nil then (decode \u03b1 x.right).map sum.inl\n  else (decode \u03b2 x.right).map sum.inr\n\n/-- Encoding of a sum type given encodings for `\u03b1` and `\u03b2` -/\ninstance : tencodable (\u03b1 \u2295 \u03b2) :=\n{ encode := of_sum,\n  decode := to_sum,\n  encodek := \u03bb x, by cases x; simp }\n\nend sum\n\nsection subtype\n\ninstance subtype.tencodable (P : \u03b1 \u2192 Prop) [decidable_pred P] : tencodable {x // P x} :=\nof_left_injection (coe : _ \u2192 \u03b1) (\u03bb r : \u03b1, if h : P r then some (\u27e8r, h\u27e9 : {x // P x}) else none)\n  (\u03bb x, by simpa [imp_false] using x.prop)\n\nlemma subtype_encode {P : \u03b1 \u2192 Prop} [decidable_pred P] (a : {x // P x}) :\n  encode a = encode (a : \u03b1) := rfl \n\ninstance {n} : tencodable (vector \u03b1 n) := subtype.tencodable _\n\nlemma encode_vec_nil : encode (vector.nil : vector \u03b1 0) = tree.nil := rfl\n\nlemma encode_vec_cons {n : \u2115} (x : \u03b1) (v : vector \u03b1 n) :\n  encode (x ::\u1d65 v) = (encode x) \u25b3 (encode v) := by { cases v, refl, }\n\nend subtype\n\nsection fin\n\ninstance (n : \u2115) : tencodable (fin n) := of_equiv {k // k < n} fin.equiv_subtype\n\nnoncomputable def fintype.tencodable {\u03b1 : Type*} [fintype \u03b1] : tencodable \u03b1 :=\nof_equiv (fin (fintype.card \u03b1)) (fintype.equiv_fin \u03b1)\n\nend fin\n\nsection ordering\n\ndef _root_.ordering.equiv_sign : ordering \u2243 sign_type :=\n{ to_fun := \u03bb x, by { cases x, exacts [-1, 0, 1], },\n  inv_fun := \u03bb x, by { cases x, exacts [ordering.eq, ordering.lt, ordering.gt], },\n  left_inv := \u03bb x, by { cases x; refl, },\n  right_inv := \u03bb x, by { cases x; refl, } }\n\ninstance : fintype ordering := fintype.of_equiv _ ordering.equiv_sign.symm\n\nnoncomputable instance ordering.tencodable : tencodable ordering := fintype.tencodable\n\nend ordering\n\nsection quotient\n\ndef _root_.setoid.tencodable (h : setoid \u03b1) (out : quotient h \u2192 \u03b1) (hout : function.left_inverse quotient.mk out) :\n  tencodable (quotient h) := of_left_inverse out quotient.mk hout\n\nend quotient\n\nsection finset\n\ndef lift_le : \u03b1 \u2192 \u03b1 \u2192 Prop := \u03bb x y, encode x \u2264 encode y\n\nlocal attribute [reducible] lift_le\ninstance : decidable_rel (@lift_le \u03b1 _) := infer_instance\nlocal attribute [semireducible] lift_le\n\ninstance : is_linear_order \u03b1 lift_le :=\n@has_le.le.is_linear_order \u03b1 (linear_order.lift' encode encode_injective)\n\ninstance : tencodable (multiset \u03b1) :=\n(list.is_setoid \u03b1).tencodable (multiset.sort lift_le) (multiset.sort_eq _)\n\nlemma encode_multiset (x : multiset \u03b1) : encode x = encode (x.sort lift_le) := rfl\n\ninstance [decidable_eq \u03b1] : tencodable (finset \u03b1) :=\nof_equiv {val : multiset \u03b1 // val.nodup}\n{ to_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  inv_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  left_inv := \u03bb \u27e8x, h\u27e9, rfl,\n  right_inv := \u03bb \u27e8x, h\u27e9, rfl }\n\nlemma encode_finset [decidable_eq \u03b1] (x : finset \u03b1) : encode x = encode x.val := rfl\n\nend finset\n\nsection finmap\n\ninstance _root_.multiset.nodupkeys_decidable {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [decidable_eq \u03b1] (s : multiset (sigma \u03b2)) : decidable s.nodupkeys :=\nquotient.rec_on_subsingleton s $ \u03bb l, l.keys.nodup_decidable\n\ninstance [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type*} [\u2200 i, tencodable (\u03b2 i)] : tencodable (finmap \u03b2) :=\nof_equiv {val : multiset (sigma \u03b2) // val.nodupkeys}\n{ to_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  inv_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  left_inv := \u03bb \u27e8x\u2081, x\u2082\u27e9, rfl,\n  right_inv := \u03bb \u27e8x\u2081, x\u2082\u27e9, rfl }\n\nlemma encode_finmap [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type*} [\u2200 i, tencodable (\u03b2 i)] (x : finmap \u03b2) :\n  encode x = encode x.entries := rfl\n\nend finmap\n\nend tencodable", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/encode.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6859494485880927, "lm_q1q2_score": 0.38298409389619725}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nCoinductive formalization of unbounded computations.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.stream\nimport Mathlib.tactic.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 v w \n\nnamespace Mathlib\n\n/-\ncoinductive computation (\u03b1 : Type u) : Type u\n| return : \u03b1 \u2192 computation \u03b1\n| think : computation \u03b1 \u2192 computation \u03b1\n-/\n\n/-- `computation \u03b1` is the type of unbounded computations returning `\u03b1`.\n  An element of `computation \u03b1` is an infinite sequence of `option \u03b1` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef computation (\u03b1 : Type u) :=\n  Subtype fun (f : stream (Option \u03b1)) => \u2200 {n : \u2115} {a : \u03b1}, f n = some a \u2192 f (n + 1) = some a\n\nnamespace computation\n\n\n-- constructors\n\n/-- `return a` is the computation that immediately terminates with result `a`. -/\ndef return {\u03b1 : Type u} (a : \u03b1) : computation \u03b1 :=\n  { val := stream.const (some a), property := sorry }\n\nprotected instance has_coe_t {\u03b1 : Type u} : has_coe_t \u03b1 (computation \u03b1) := has_coe_t.mk return\n\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think {\u03b1 : Type u} (c : computation \u03b1) : computation \u03b1 :=\n  { val := none :: subtype.val c, property := sorry }\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN {\u03b1 : Type u} (c : computation \u03b1) : \u2115 \u2192 computation \u03b1 := sorry\n\n-- check for immediate result\n\n/-- `head c` is the first step of computation, either `some a` if `c = return a`\n  or `none` if `c = think c'`. -/\ndef head {\u03b1 : Type u} (c : computation \u03b1) : Option \u03b1 := stream.head (subtype.val c)\n\n-- one step of computation\n\n/-- `tail c` is the remainder of computation, either `c` if `c = return a`\n  or `c'` if `c = think c'`. -/\ndef tail {\u03b1 : Type u} (c : computation \u03b1) : computation \u03b1 :=\n  { val := stream.tail (subtype.val c), property := sorry }\n\n/-- `empty \u03b1` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (\u03b1 : Type u_1) : computation \u03b1 := { val := stream.const none, property := sorry }\n\nprotected instance inhabited {\u03b1 : Type u} : Inhabited (computation \u03b1) := { default := empty \u03b1 }\n\n/-- `run_for c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef run_for {\u03b1 : Type u} : computation \u03b1 \u2192 \u2115 \u2192 Option \u03b1 := subtype.val\n\n/-- `destruct c` is the destructor for `computation \u03b1` as a coinductive type.\n  It returns `inl a` if `c = return a` and `inr c'` if `c = think c'`. -/\ndef destruct {\u03b1 : Type u} (c : computation \u03b1) : \u03b1 \u2295 computation \u03b1 := sorry\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\ntheorem destruct_eq_ret {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} :\n    destruct s = sum.inl a \u2192 s = return a :=\n  sorry\n\ntheorem destruct_eq_think {\u03b1 : Type u} {s : computation \u03b1} {s' : computation \u03b1} :\n    destruct s = sum.inr s' \u2192 s = think s' :=\n  sorry\n\n@[simp] theorem destruct_ret {\u03b1 : Type u} (a : \u03b1) : destruct (return a) = sum.inl a := rfl\n\n@[simp] theorem destruct_think {\u03b1 : Type u} (s : computation \u03b1) : destruct (think s) = sum.inr s :=\n  sorry\n\n@[simp] theorem destruct_empty {\u03b1 : Type u} : destruct (empty \u03b1) = sum.inr (empty \u03b1) := rfl\n\n@[simp] theorem head_ret {\u03b1 : Type u} (a : \u03b1) : head (return a) = some a := rfl\n\n@[simp] theorem head_think {\u03b1 : Type u} (s : computation \u03b1) : head (think s) = none := rfl\n\n@[simp] theorem head_empty {\u03b1 : Type u} : head (empty \u03b1) = none := rfl\n\n@[simp] theorem tail_ret {\u03b1 : Type u} (a : \u03b1) : tail (return a) = return a := rfl\n\n@[simp] theorem tail_think {\u03b1 : Type u} (s : computation \u03b1) : tail (think s) = s := sorry\n\n@[simp] theorem tail_empty {\u03b1 : Type u} : tail (empty \u03b1) = empty \u03b1 := rfl\n\ntheorem think_empty {\u03b1 : Type u} : empty \u03b1 = think (empty \u03b1) := destruct_eq_think destruct_empty\n\ndef cases_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} (s : computation \u03b1)\n    (h1 : (a : \u03b1) \u2192 C (return a)) (h2 : (s : computation \u03b1) \u2192 C (think s)) : C s :=\n  (fun (_x : \u03b1 \u2295 computation \u03b1) (H : destruct s = _x) =>\n      sum.rec (fun (v : \u03b1) (H : destruct s = sum.inl v) => eq.mpr sorry (h1 v))\n        (fun (v : computation \u03b1) (H : destruct s = sum.inr v) =>\n          subtype.cases_on v\n            (fun (a : stream (Option \u03b1))\n              (s' : \u2200 {n : \u2115} {a_1 : \u03b1}, a n = some a_1 \u2192 a (n + 1) = some a_1)\n              (H : destruct s = sum.inr { val := a, property := s' }) =>\n              eq.mpr sorry (h2 { val := a, property := s' }))\n            H)\n        _x H)\n    (destruct s) sorry\n\ndef corec.F {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) : \u03b1 \u2295 \u03b2 \u2192 Option \u03b1 \u00d7 (\u03b1 \u2295 \u03b2) := sorry\n\n/-- `corec f b` is the corecursor for `computation \u03b1` as a coinductive type.\n  If `f b = inl a` then `corec f b = return a`, and if `f b = inl b'` then\n  `corec f b = think (corec f b')`. -/\ndef corec {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) (b : \u03b2) : computation \u03b1 :=\n  { val := stream.corec' sorry (sum.inr b), property := sorry }\n\n/-- left map of `\u2295` -/\n@[simp] def lmap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) : \u03b1 \u2295 \u03b3 \u2192 \u03b2 \u2295 \u03b3 := sorry\n\n/-- right map of `\u2295` -/\n@[simp] def rmap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b1 \u2295 \u03b3 := sorry\n\n@[simp] theorem corec_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) (b : \u03b2) :\n    destruct (corec f b) = rmap (corec f) (f b) :=\n  sorry\n\n@[simp] def bisim_o {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop) :\n    \u03b1 \u2295 computation \u03b1 \u2192 \u03b1 \u2295 computation \u03b1 \u2192 Prop :=\n  sorry\n\ndef is_bisimulation {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop) :=\n  \u2200 {s\u2081 s\u2082 : computation \u03b1}, R s\u2081 s\u2082 \u2192 bisim_o R (destruct s\u2081) (destruct s\u2082)\n\ntheorem eq_of_bisim {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop)\n    (bisim : is_bisimulation R) {s\u2081 : computation \u03b1} {s\u2082 : computation \u03b1} (r : R s\u2081 s\u2082) : s\u2081 = s\u2082 :=\n  sorry\n\n-- It's more of a stretch to use \u2208 for this relation, but it\n\n-- asserts that the computation limits to the given value.\n\nprotected def mem {\u03b1 : Type u} (a : \u03b1) (s : computation \u03b1) := some a \u2208 subtype.val s\n\nprotected instance has_mem {\u03b1 : Type u} : has_mem \u03b1 (computation \u03b1) := has_mem.mk computation.mem\n\ntheorem le_stable {\u03b1 : Type u} (s : computation \u03b1) {a : \u03b1} {m : \u2115} {n : \u2115} (h : m \u2264 n) :\n    subtype.val s m = some a \u2192 subtype.val s n = some a :=\n  sorry\n\ntheorem mem_unique {\u03b1 : Type u} : relator.left_unique has_mem.mem := sorry\n\n/-- `terminates s` asserts that the computation `s` eventually terminates with some value. -/\ndef terminates {\u03b1 : Type u} (s : computation \u03b1) := \u2203 (a : \u03b1), a \u2208 s\n\ntheorem terminates_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 terminates s :=\n  exists.intro a\n\ntheorem terminates_def {\u03b1 : Type u} (s : computation \u03b1) :\n    terminates s \u2194 \u2203 (n : \u2115), \u21a5(option.is_some (subtype.val s n)) :=\n  sorry\n\ntheorem ret_mem {\u03b1 : Type u} (a : \u03b1) : a \u2208 return a := exists.intro 0 rfl\n\ntheorem eq_of_ret_mem {\u03b1 : Type u} {a : \u03b1} {a' : \u03b1} (h : a' \u2208 return a) : a' = a :=\n  mem_unique h (ret_mem a)\n\nprotected instance ret_terminates {\u03b1 : Type u} (a : \u03b1) : terminates (return a) :=\n  terminates_of_mem (ret_mem a)\n\ntheorem think_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 a \u2208 think s := sorry\n\nprotected instance think_terminates {\u03b1 : Type u} (s : computation \u03b1) [terminates s] :\n    terminates (think s) :=\n  sorry\n\ntheorem of_think_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 think s \u2192 a \u2208 s := sorry\n\ntheorem of_think_terminates {\u03b1 : Type u} {s : computation \u03b1} :\n    terminates (think s) \u2192 terminates s :=\n  fun (\u1fb0 : terminates (think s)) =>\n    Exists.dcases_on \u1fb0\n      fun (\u1fb0_w : \u03b1) (\u1fb0_h : \u1fb0_w \u2208 think s) =>\n        idRhs (\u2203 (a : \u03b1), a \u2208 s) (Exists.intro \u1fb0_w (of_think_mem \u1fb0_h))\n\ntheorem not_mem_empty {\u03b1 : Type u} (a : \u03b1) : \u00aca \u2208 empty \u03b1 := sorry\n\ntheorem not_terminates_empty {\u03b1 : Type u} : \u00acterminates (empty \u03b1) := sorry\n\ntheorem eq_empty_of_not_terminates {\u03b1 : Type u} {s : computation \u03b1} (H : \u00acterminates s) :\n    s = empty \u03b1 :=\n  sorry\n\ntheorem thinkN_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (n : \u2115) : a \u2208 thinkN s n \u2194 a \u2208 s :=\n  sorry\n\nprotected instance thinkN_terminates {\u03b1 : Type u} (s : computation \u03b1) [terminates s] (n : \u2115) :\n    terminates (thinkN s n) :=\n  sorry\n\ntheorem of_thinkN_terminates {\u03b1 : Type u} (s : computation \u03b1) (n : \u2115) :\n    terminates (thinkN s n) \u2192 terminates s :=\n  fun (\u1fb0 : terminates (thinkN s n)) =>\n    Exists.dcases_on \u1fb0\n      fun (\u1fb0_w : \u03b1) (\u1fb0_h : \u1fb0_w \u2208 thinkN s n) =>\n        idRhs (\u2203 (a : \u03b1), a \u2208 s) (Exists.intro \u1fb0_w (iff.mp (thinkN_mem n) \u1fb0_h))\n\n/-- `promises s a`, or `s ~> a`, asserts that although the computation `s`\n  may not terminate, if it does, then the result is `a`. -/\ndef promises {\u03b1 : Type u} (s : computation \u03b1) (a : \u03b1) := \u2200 {a' : \u03b1}, a' \u2208 s \u2192 a = a'\n\ninfixl:50 \" ~> \" => Mathlib.computation.promises\n\ntheorem mem_promises {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 s ~> a :=\n  fun (h : a \u2208 s) (a' : \u03b1) => mem_unique h\n\ntheorem empty_promises {\u03b1 : Type u} (a : \u03b1) : empty \u03b1 ~> a :=\n  fun (a' : \u03b1) (h : a' \u2208 empty \u03b1) => absurd h (not_mem_empty a')\n\n/-- `length s` gets the number of steps of a terminating computation -/\ndef length {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : \u2115 := nat.find sorry\n\n/-- `get s` returns the result of a terminating computation -/\ndef get {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : \u03b1 := option.get sorry\n\ntheorem get_mem {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : get s \u2208 s :=\n  exists.intro (length s) (Eq.symm (option.eq_some_of_is_some (get._proof_2 s)))\n\ntheorem get_eq_of_mem {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} :\n    a \u2208 s \u2192 get s = a :=\n  mem_unique (get_mem s)\n\ntheorem mem_of_get_eq {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} :\n    get s = a \u2192 a \u2208 s :=\n  fun (h_1 : get s = a) => eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 s)) (Eq.symm h_1))) (get_mem s)\n\n@[simp] theorem get_think {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] :\n    get (think s) = get s :=\n  sorry\n\n@[simp] theorem get_thinkN {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] (n : \u2115) :\n    get (thinkN s n) = get s :=\n  get_eq_of_mem (thinkN s n) (iff.mpr (thinkN_mem n) (get_mem s))\n\ntheorem get_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : s ~> get s :=\n  fun (a : \u03b1) => get_eq_of_mem s\n\ntheorem mem_of_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} (p : s ~> a) :\n    a \u2208 s :=\n  Exists.dcases_on h fun (a' : \u03b1) (h : a' \u2208 s) => eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 s)) (p h))) h\n\ntheorem get_eq_of_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} :\n    s ~> a \u2192 get s = a :=\n  get_eq_of_mem s \u2218 mem_of_promises s\n\n/-- `results s a n` completely characterizes a terminating computation:\n  it asserts that `s` terminates after exactly `n` steps, with result `a`. -/\ndef results {\u03b1 : Type u} (s : computation \u03b1) (a : \u03b1) (n : \u2115) := \u2203 (h : a \u2208 s), length s = n\n\ntheorem results_of_terminates {\u03b1 : Type u} (s : computation \u03b1) [T : terminates s] :\n    results s (get s) (length s) :=\n  Exists.intro (get_mem s) rfl\n\ntheorem results_of_terminates' {\u03b1 : Type u} (s : computation \u03b1) [T : terminates s] {a : \u03b1}\n    (h : a \u2208 s) : results s a (length s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (results s a (length s))) (Eq.symm (get_eq_of_mem s h))))\n    (results_of_terminates s)\n\ntheorem results.mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} : results s a n \u2192 a \u2208 s :=\n  fun (\u1fb0 : results s a n) =>\n    Exists.dcases_on \u1fb0 fun (\u1fb0_w : a \u2208 s) (\u1fb0_h : length s = n) => idRhs (a \u2208 s) \u1fb0_w\n\ntheorem results.terminates {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) :\n    terminates s :=\n  terminates_of_mem (results.mem h)\n\ntheorem results.length {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} [T : terminates s] :\n    results s a n \u2192 length s = n :=\n  fun (\u1fb0 : results s a n) =>\n    Exists.dcases_on \u1fb0 fun (\u1fb0_w : a \u2208 s) (\u1fb0_h : length s = n) => idRhs (length s = n) \u1fb0_h\n\ntheorem results.val_unique {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {b : \u03b1} {m : \u2115} {n : \u2115}\n    (h1 : results s a m) (h2 : results s b n) : a = b :=\n  mem_unique (results.mem h1) (results.mem h2)\n\ntheorem results.len_unique {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {b : \u03b1} {m : \u2115} {n : \u2115}\n    (h1 : results s a m) (h2 : results s b n) : m = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (m = n)) (Eq.symm (results.length h1))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length s = n)) (results.length h2))) (Eq.refl n))\n\ntheorem exists_results_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (h : a \u2208 s) :\n    \u2203 (n : \u2115), results s a n :=\n  Exists.intro (length s) (results_of_terminates' s h)\n\n@[simp] theorem get_ret {\u03b1 : Type u} (a : \u03b1) : get (return a) = a :=\n  get_eq_of_mem (return a) (Exists.intro 0 rfl)\n\n@[simp] theorem length_ret {\u03b1 : Type u} (a : \u03b1) : length (return a) = 0 :=\n  let h : terminates (return a) := computation.ret_terminates a;\n  nat.eq_zero_of_le_zero (nat.find_min' (iff.mp (terminates_def (return a)) h) rfl)\n\ntheorem results_ret {\u03b1 : Type u} (a : \u03b1) : results (return a) a 0 :=\n  Exists.intro (ret_mem a) (length_ret a)\n\n@[simp] theorem length_think {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] :\n    length (think s) = length s + 1 :=\n  sorry\n\ntheorem results_think {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) :\n    results (think s) a (n + 1) :=\n  Exists.intro (think_mem (results.mem h))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (think s) = n + 1)) (length_think s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length s + 1 = n + 1)) (results.length h)))\n        (Eq.refl (n + 1))))\n\ntheorem of_results_think {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115}\n    (h : results (think s) a n) : \u2203 (m : \u2115), results s a m \u2227 n = m + 1 :=\n  Exists.intro (length s)\n    { left := results_of_terminates' s (of_think_mem (results.mem h)),\n      right :=\n        results.len_unique h\n          (results_think (results_of_terminates' s (of_think_mem (results.mem h)))) }\n\n@[simp] theorem results_think_iff {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} :\n    results (think s) a (n + 1) \u2194 results s a n :=\n  sorry\n\ntheorem results_thinkN {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {m : \u2115} (n : \u2115) :\n    results s a m \u2192 results (thinkN s n) a (m + n) :=\n  sorry\n\ntheorem results_thinkN_ret {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : results (thinkN (return a) n) a n :=\n  eq.mp (Eq._oldrec (Eq.refl (results (thinkN (return a) n) a (0 + n))) (nat.zero_add n))\n    (results_thinkN n (results_ret a))\n\n@[simp] theorem length_thinkN {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] (n : \u2115) :\n    length (thinkN s n) = length s + n :=\n  results.length (results_thinkN n (results_of_terminates s))\n\ntheorem eq_thinkN {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) :\n    s = thinkN (return a) n :=\n  sorry\n\ntheorem eq_thinkN' {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] :\n    s = thinkN (return (get s)) (length s) :=\n  eq_thinkN (results_of_terminates s)\n\ndef mem_rec_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} {a : \u03b1} {s : computation \u03b1} (M : a \u2208 s)\n    (h1 : C (return a)) (h2 : (s : computation \u03b1) \u2192 C s \u2192 C (think s)) : C s :=\n  eq.mpr sorry\n    (eq.mpr sorry\n      (Nat.rec h1 (fun (n : \u2115) (IH : C (thinkN (return a) n)) => h2 (thinkN (return a) n) IH)\n        (length s)))\n\ndef terminates_rec_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} (s : computation \u03b1) [terminates s]\n    (h1 : (a : \u03b1) \u2192 C (return a)) (h2 : (s : computation \u03b1) \u2192 C s \u2192 C (think s)) : C s :=\n  mem_rec_on (get_mem s) (h1 (get s)) h2\n\n/-- Map a function on the result of a computation. -/\ndef map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : computation \u03b1 \u2192 computation \u03b2 := sorry\n\ndef bind.G {\u03b1 : Type u} {\u03b2 : Type v} : \u03b2 \u2295 computation \u03b2 \u2192 \u03b2 \u2295 computation \u03b1 \u2295 computation \u03b2 :=\n  sorry\n\ndef bind.F {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 computation \u03b2) :\n    computation \u03b1 \u2295 computation \u03b2 \u2192 \u03b2 \u2295 computation \u03b1 \u2295 computation \u03b2 :=\n  sorry\n\n/-- Compose two computations into a monadic `bind` operation. -/\ndef bind {\u03b1 : Type u} {\u03b2 : Type v} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) : computation \u03b2 :=\n  corec sorry (sum.inl c)\n\nprotected instance has_bind : Bind computation := { bind := bind }\n\ntheorem has_bind_eq_bind {\u03b1 : Type u} {\u03b2 : Type u} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) :\n    c >>= f = bind c f :=\n  rfl\n\n/-- Flatten a computation of computations into a single computation. -/\ndef join {\u03b1 : Type u} (c : computation (computation \u03b1)) : computation \u03b1 := c >>= id\n\n@[simp] theorem map_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    map f (return a) = return (f a) :=\n  rfl\n\n@[simp] theorem map_think {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) :\n    map f (think s) = think (map f s) :=\n  sorry\n\n@[simp] theorem destruct_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) :\n    destruct (map f s) = lmap f (rmap (map f) (destruct s)) :=\n  sorry\n\n@[simp] theorem map_id {\u03b1 : Type u} (s : computation \u03b1) : map id s = s := sorry\n\ntheorem map_comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3)\n    (s : computation \u03b1) : map (g \u2218 f) s = map g (map f s) :=\n  sorry\n\n@[simp] theorem ret_bind {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (f : \u03b1 \u2192 computation \u03b2) :\n    bind (return a) f = f a :=\n  sorry\n\n@[simp] theorem think_bind {\u03b1 : Type u} {\u03b2 : Type v} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) :\n    bind (think c) f = think (bind c f) :=\n  sorry\n\n@[simp] theorem bind_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) :\n    bind s (return \u2218 f) = map f s :=\n  sorry\n\n@[simp] theorem bind_ret' {\u03b1 : Type u} (s : computation \u03b1) : bind s return = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (bind s return = s)) (bind_ret (fun (x : \u03b1) => x) s)))\n    (id (eq.mpr (id (Eq._oldrec (Eq.refl (map id s = s)) (map_id s))) (Eq.refl s)))\n\n@[simp] theorem bind_assoc {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (s : computation \u03b1)\n    (f : \u03b1 \u2192 computation \u03b2) (g : \u03b2 \u2192 computation \u03b3) :\n    bind (bind s f) g = bind s fun (x : \u03b1) => bind (f x) g :=\n  sorry\n\ntheorem results_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1}\n    {b : \u03b2} {m : \u2115} {n : \u2115} (h1 : results s a m) (h2 : results (f a) b n) :\n    results (bind s f) b (n + m) :=\n  sorry\n\ntheorem mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1}\n    {b : \u03b2} (h1 : a \u2208 s) (h2 : b \u2208 f a) : b \u2208 bind s f :=\n  sorry\n\nprotected instance terminates_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1)\n    (f : \u03b1 \u2192 computation \u03b2) [terminates s] [terminates (f (get s))] : terminates (bind s f) :=\n  terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))\n\n@[simp] theorem get_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2)\n    [terminates s] [terminates (f (get s))] : get (bind s f) = get (f (get s)) :=\n  get_eq_of_mem (bind s f) (mem_bind (get_mem s) (get_mem (f (get s))))\n\n@[simp] theorem length_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2)\n    [T1 : terminates s] [T2 : terminates (f (get s))] :\n    length (bind s f) = length (f (get s)) + length s :=\n  results.len_unique (results_of_terminates (bind s f))\n    (results_bind (results_of_terminates s) (results_of_terminates (f (get s))))\n\ntheorem of_results_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2}\n    {b : \u03b2} {k : \u2115} :\n    results (bind s f) b k \u2192\n        \u2203 (a : \u03b1), \u2203 (m : \u2115), \u2203 (n : \u2115), results s a m \u2227 results (f a) b n \u2227 k = n + m :=\n  sorry\n\ntheorem exists_of_mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2}\n    {b : \u03b2} (h : b \u2208 bind s f) : \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a :=\n  sorry\n\ntheorem bind_promises {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1}\n    {b : \u03b2} (h1 : s ~> a) (h2 : f a ~> b) : bind s f ~> b :=\n  sorry\n\nprotected instance monad : Monad computation :=\n  { toApplicative :=\n      { toFunctor := { map := map, mapConst := fun (\u03b1 \u03b2 : Type u_1) => map \u2218 function.const \u03b2 },\n        toPure := { pure := return },\n        toSeq :=\n          { seq :=\n              fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) =>\n                bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : computation \u03b1) (b : computation \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b1 (map (function.const \u03b2) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : computation \u03b1) (b : computation \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b2 (map (function.const \u03b1 id) a) b } },\n    toBind := { bind := bind } }\n\nprotected instance is_lawful_monad : is_lawful_monad computation :=\n  is_lawful_monad.mk ret_bind bind_assoc\n\ntheorem has_map_eq_map {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (c : computation \u03b1) :\n    f <$> c = map f c :=\n  rfl\n\n@[simp] theorem return_def {\u03b1 : Type u} (a : \u03b1) : return a = return a := rfl\n\n@[simp] theorem map_ret' {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    f <$> return a = return (f a) :=\n  map_ret\n\n@[simp] theorem map_think' {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) :\n    f <$> think s = think (f <$> s) :=\n  map_think\n\ntheorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : computation \u03b1} (m : a \u2208 s) :\n    f a \u2208 map f s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f a \u2208 map f s)) (Eq.symm (bind_ret f s))))\n    (mem_bind m (ret_mem (f a)))\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : computation \u03b1}\n    (h : b \u2208 map f s) : \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b :=\n  sorry\n\nprotected instance terminates_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1)\n    [terminates s] : terminates (map f s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (terminates (map f s))) (Eq.symm (bind_ret f s))))\n    (computation.terminates_bind s (return \u2218 f))\n\ntheorem terminates_map_iff {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) :\n    terminates (map f s) \u2194 terminates s :=\n  sorry\n\n-- Parallel computation\n\n/-- `c\u2081 <|> c\u2082` calculates `c\u2081` and `c\u2082` simultaneously, returning\n  the first one that gives a result. -/\ndef orelse {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) : computation \u03b1 :=\n  corec (fun (_x : computation \u03b1 \u00d7 computation \u03b1) => sorry) (c\u2081, c\u2082)\n\nprotected instance alternative : alternative computation := alternative.mk empty\n\n@[simp] theorem ret_orelse {\u03b1 : Type u} (a : \u03b1) (c\u2082 : computation \u03b1) :\n    (return a <|> c\u2082) = return a :=\n  sorry\n\n@[simp] theorem orelse_ret {\u03b1 : Type u} (c\u2081 : computation \u03b1) (a : \u03b1) :\n    (think c\u2081 <|> return a) = return a :=\n  sorry\n\n@[simp] theorem orelse_think {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) :\n    (think c\u2081 <|> think c\u2082) = think (c\u2081 <|> c\u2082) :=\n  sorry\n\n@[simp] theorem empty_orelse {\u03b1 : Type u} (c : computation \u03b1) : (empty \u03b1 <|> c) = c := sorry\n\n@[simp] theorem orelse_empty {\u03b1 : Type u} (c : computation \u03b1) : (c <|> empty \u03b1) = c := sorry\n\n/-- `c\u2081 ~ c\u2082` asserts that `c\u2081` and `c\u2082` either both terminate with the same result,\n  or both loop forever. -/\ndef equiv {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) := \u2200 (a : \u03b1), a \u2208 c\u2081 \u2194 a \u2208 c\u2082\n\ninfixl:50 \" ~ \" => Mathlib.computation.equiv\n\ntheorem equiv.refl {\u03b1 : Type u} (s : computation \u03b1) : s ~ s := fun (_x : \u03b1) => iff.rfl\n\ntheorem equiv.symm {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} : s ~ t \u2192 t ~ s :=\n  fun (h : s ~ t) (a : \u03b1) => iff.symm (h a)\n\ntheorem equiv.trans {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} {u : computation \u03b1} :\n    s ~ t \u2192 t ~ u \u2192 s ~ u :=\n  fun (h1 : s ~ t) (h2 : t ~ u) (a : \u03b1) => iff.trans (h1 a) (h2 a)\n\ntheorem equiv.equivalence {\u03b1 : Type u} : equivalence equiv :=\n  { left := equiv.refl, right := { left := equiv.symm, right := equiv.trans } }\n\ntheorem equiv_of_mem {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} {a : \u03b1} (h1 : a \u2208 s)\n    (h2 : a \u2208 t) : s ~ t :=\n  fun (a' : \u03b1) =>\n    { mp := fun (ma : a' \u2208 s) => eq.mpr (id (Eq._oldrec (Eq.refl (a' \u2208 t)) (mem_unique ma h1))) h2,\n      mpr :=\n        fun (ma : a' \u2208 t) => eq.mpr (id (Eq._oldrec (Eq.refl (a' \u2208 s)) (mem_unique ma h2))) h1 }\n\ntheorem terminates_congr {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082) :\n    terminates c\u2081 \u2194 terminates c\u2082 :=\n  exists_congr h\n\ntheorem promises_congr {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082)\n    (a : \u03b1) : c\u2081 ~> a \u2194 c\u2082 ~> a :=\n  forall_congr fun (a' : \u03b1) => imp_congr (h a') iff.rfl\n\ntheorem get_equiv {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082)\n    [terminates c\u2081] [terminates c\u2082] : get c\u2081 = get c\u2082 :=\n  get_eq_of_mem c\u2081 (iff.mpr (h (get c\u2082)) (get_mem c\u2082))\n\ntheorem think_equiv {\u03b1 : Type u} (s : computation \u03b1) : think s ~ s :=\n  fun (a : \u03b1) => { mp := of_think_mem, mpr := think_mem }\n\ntheorem thinkN_equiv {\u03b1 : Type u} (s : computation \u03b1) (n : \u2115) : thinkN s n ~ s :=\n  fun (a : \u03b1) => thinkN_mem n\n\ntheorem bind_congr {\u03b1 : Type u} {\u03b2 : Type v} {s1 : computation \u03b1} {s2 : computation \u03b1}\n    {f1 : \u03b1 \u2192 computation \u03b2} {f2 : \u03b1 \u2192 computation \u03b2} (h1 : s1 ~ s2) (h2 : \u2200 (a : \u03b1), f1 a ~ f2 a) :\n    bind s1 f1 ~ bind s2 f2 :=\n  sorry\n\ntheorem equiv_ret_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (h : a \u2208 s) : s ~ return a :=\n  equiv_of_mem h (ret_mem a)\n\n/-- `lift_rel R ca cb` is a generalization of `equiv` to relations other than\n  equality. It asserts that if `ca` terminates with `a`, then `cb` terminates with\n  some `b` such that `R a b`, and if `cb` terminates with `b` then `ca` terminates\n  with some `a` such that `R a b`. -/\ndef lift_rel {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1)\n    (cb : computation \u03b2) :=\n  (\u2200 {a : \u03b1}, a \u2208 ca \u2192 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b) \u2227\n    \u2200 {b : \u03b2}, b \u2208 cb \u2192 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b\n\ntheorem lift_rel.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1)\n    (cb : computation \u03b2) : lift_rel (function.swap R) cb ca \u2194 lift_rel R ca cb :=\n  and_comm (\u2200 {a : \u03b2}, a \u2208 cb \u2192 Exists fun {b : \u03b1} => b \u2208 ca \u2227 function.swap R a b)\n    (\u2200 {b : \u03b1}, b \u2208 ca \u2192 Exists fun {a : \u03b2} => a \u2208 cb \u2227 function.swap R a b)\n\ntheorem lift_eq_iff_equiv {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) :\n    lift_rel Eq c\u2081 c\u2082 \u2194 c\u2081 ~ c\u2082 :=\n  sorry\n\ntheorem lift_rel.refl {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : reflexive R) : reflexive (lift_rel R) :=\n  fun (s : computation \u03b1) =>\n    { left := fun (a : \u03b1) (as : a \u2208 s) => Exists.intro a { left := as, right := H a },\n      right := fun (b : \u03b1) (bs : b \u2208 s) => Exists.intro b { left := bs, right := H b } }\n\ntheorem lift_rel.symm {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : symmetric R) : symmetric (lift_rel R) :=\n  sorry\n\ntheorem lift_rel.trans {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : transitive R) :\n    transitive (lift_rel R) :=\n  sorry\n\ntheorem lift_rel.equiv {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : equivalence R \u2192 equivalence (lift_rel R) :=\n  sorry\n\ntheorem lift_rel.imp {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {S : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    (H : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S a b) (s : computation \u03b1) (t : computation \u03b2) :\n    lift_rel R s t \u2192 lift_rel S s t :=\n  sorry\n\ntheorem terminates_of_lift_rel {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : computation \u03b1}\n    {t : computation \u03b2} : lift_rel R s t \u2192 (terminates s \u2194 terminates t) :=\n  sorry\n\ntheorem rel_of_lift_rel {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {cb : computation \u03b2} : lift_rel R ca cb \u2192 \u2200 {a : \u03b1} {b : \u03b2}, a \u2208 ca \u2192 b \u2208 cb \u2192 R a b :=\n  sorry\n\ntheorem lift_rel_of_mem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : \u03b1} {b : \u03b2}\n    {ca : computation \u03b1} {cb : computation \u03b2} (ma : a \u2208 ca) (mb : b \u2208 cb) (ab : R a b) :\n    lift_rel R ca cb :=\n  sorry\n\ntheorem exists_of_lift_rel_left {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {cb : computation \u03b2} (H : lift_rel R ca cb) {a : \u03b1} (h : a \u2208 ca) :\n    Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b :=\n  and.left H a h\n\ntheorem exists_of_lift_rel_right {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {cb : computation \u03b2} (H : lift_rel R ca cb) {b : \u03b2} (h : b \u2208 cb) :\n    Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b :=\n  and.right H b h\n\ntheorem lift_rel_def {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {cb : computation \u03b2} :\n    lift_rel R ca cb \u2194\n        (terminates ca \u2194 terminates cb) \u2227 \u2200 {a : \u03b1} {b : \u03b2}, a \u2208 ca \u2192 b \u2208 cb \u2192 R a b :=\n  sorry\n\ntheorem lift_rel_bind {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : computation \u03b1} {s2 : computation \u03b2} {f1 : \u03b1 \u2192 computation \u03b3}\n    {f2 : \u03b2 \u2192 computation \u03b4} (h1 : lift_rel R s1 s2)\n    (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 lift_rel S (f1 a) (f2 b)) :\n    lift_rel S (bind s1 f1) (bind s2 f2) :=\n  sorry\n\n@[simp] theorem lift_rel_return_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1)\n    (cb : computation \u03b2) : lift_rel R (return a) cb \u2194 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b :=\n  sorry\n\n@[simp] theorem lift_rel_return_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (ca : computation \u03b1) (b : \u03b2) :\n    lift_rel R ca (return b) \u2194 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b :=\n  sorry\n\n@[simp] theorem lift_rel_return {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1) (b : \u03b2) :\n    lift_rel R (return a) (return b) \u2194 R a b :=\n  sorry\n\n@[simp] theorem lift_rel_think_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (ca : computation \u03b1) (cb : computation \u03b2) : lift_rel R (think ca) cb \u2194 lift_rel R ca cb :=\n  sorry\n\n@[simp] theorem lift_rel_think_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (ca : computation \u03b1) (cb : computation \u03b2) : lift_rel R ca (think cb) \u2194 lift_rel R ca cb :=\n  sorry\n\ntheorem lift_rel_mem_cases {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {cb : computation \u03b2} (Ha : \u2200 (a : \u03b1), a \u2208 ca \u2192 lift_rel R ca cb)\n    (Hb : \u2200 (b : \u03b2), b \u2208 cb \u2192 lift_rel R ca cb) : lift_rel R ca cb :=\n  { left := fun (a : \u03b1) (ma : a \u2208 ca) => and.left (Ha a ma) a ma,\n    right := fun (b : \u03b2) (mb : b \u2208 cb) => and.right (Hb b mb) b mb }\n\ntheorem lift_rel_congr {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1}\n    {ca' : computation \u03b1} {cb : computation \u03b2} {cb' : computation \u03b2} (ha : ca ~ ca')\n    (hb : cb ~ cb') : lift_rel R ca cb \u2194 lift_rel R ca' cb' :=\n  and_congr\n    (forall_congr\n      fun (a : \u03b1) => imp_congr (ha a) (exists_congr fun (b : \u03b2) => and_congr (hb b) iff.rfl))\n    (forall_congr\n      fun (b : \u03b2) => imp_congr (hb b) (exists_congr fun (a : \u03b1) => and_congr (ha a) iff.rfl))\n\ntheorem lift_rel_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : computation \u03b1} {s2 : computation \u03b2} {f1 : \u03b1 \u2192 \u03b3} {f2 : \u03b2 \u2192 \u03b4}\n    (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)) :\n    lift_rel S (map f1 s1) (map f2 s2) :=\n  sorry\n\ntheorem map_congr {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (S : \u03b2 \u2192 \u03b2 \u2192 Prop)\n    {s1 : computation \u03b1} {s2 : computation \u03b1} {f : \u03b1 \u2192 \u03b2} (h1 : s1 ~ s2) : map f s1 ~ map f s2 :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (map f s1 ~ map f s2))\n        (Eq.symm (propext (lift_eq_iff_equiv (map f s1) (map f s2))))))\n    (lift_rel_map Eq Eq (iff.mpr (lift_eq_iff_equiv s1 s2) h1)\n      fun (a b : \u03b1) => congr_arg fun (a : \u03b1) => f a)\n\n@[simp] def lift_rel_aux {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) : \u03b1 \u2295 computation \u03b1 \u2192 \u03b2 \u2295 computation \u03b2 \u2192 Prop :=\n  sorry\n\n@[simp] theorem lift_rel_aux.ret_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (a : \u03b1) (cb : computation \u03b2) :\n    lift_rel_aux R C (sum.inl a) (destruct cb) \u2194 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b :=\n  sorry\n\ntheorem lift_rel_aux.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (a : \u03b1 \u2295 computation \u03b1) (b : \u03b2 \u2295 computation \u03b2) :\n    lift_rel_aux (function.swap R) (function.swap C) b a = lift_rel_aux R C a b :=\n  sorry\n\n@[simp] theorem lift_rel_aux.ret_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (b : \u03b2) (ca : computation \u03b1) :\n    lift_rel_aux R C (destruct ca) (sum.inl b) \u2194 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b :=\n  sorry\n\ntheorem lift_rel_rec.lem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop)\n    (H :\n      \u2200 {ca : computation \u03b1} {cb : computation \u03b2},\n        C ca cb \u2192 lift_rel_aux R C (destruct ca) (destruct cb))\n    (ca : computation \u03b1) (cb : computation \u03b2) (Hc : C ca cb) (a : \u03b1) (ha : a \u2208 ca) :\n    lift_rel R ca cb :=\n  sorry\n\ntheorem lift_rel_rec {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop)\n    (H :\n      \u2200 {ca : computation \u03b1} {cb : computation \u03b2},\n        C ca cb \u2192 lift_rel_aux R C (destruct ca) (destruct cb))\n    (ca : computation \u03b1) (cb : computation \u03b2) (Hc : C ca cb) : lift_rel R ca cb :=\n  lift_rel_mem_cases sorry\n    fun (b : \u03b2) (hb : b \u2208 cb) => iff.mpr (lift_rel.swap (fun (x : \u03b2) (y : \u03b1) => R y x) cb ca) sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/seq/computation_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.38296938313662954}}
{"text": "import for_mathlib.commsq\nimport for_mathlib.snake_lemma_naturality2\nimport for_mathlib.AddCommGroup.pt\nimport for_mathlib.bicartesian4\nimport for_mathlib.abelian_sheaves.functor_category\n\nimport system_of_complexes.shift_sub_id\nimport pseudo_normed_group.system_of_complexes2\n\nnoncomputable theory\n\nuniverses u\n\nopen_locale nnreal\n\nopen opposite category_theory category_theory.limits category_theory.preadditive\n\nsection step1\n\nvariables {A B C : \u211d\u22650\u1d52\u1d56 \u2964 Ab.{u}} (f : A \u27f6 B)\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650)\n\ndef shift_sub_id.commsq (h\u03b9 : monotone \u03b9) :\n  commsq (shift_sub_id A \u03b9 h\u03b9)\n    (pi.map $ \u03bb _, f.app _) (pi.map $ \u03bb _, f.app _)\n         (shift_sub_id B \u03b9 h\u03b9) :=\ncommsq.of_eq\nbegin\n  simp only [shift_sub_id, sub_comp, comp_sub, category.id_comp, category.comp_id, shift_sub_id.shift],\n  congr' 1,\n  apply limit.hom_ext,\n  intro j,\n  simp only [limit.lift_map, limit.lift_\u03c0, cones.postcompose_obj_\u03c0, nat_trans.comp_app,\n    fan.mk_\u03c0_app, discrete.nat_trans_app, category.assoc, nat_trans.naturality, lim_map_\u03c0_assoc],\nend\n\nend step1\n\nsection step2\n\nvariables {A B C : ulift.{u} \u2115 \u2192 Ab.{u}} (f : \u03a0 k, A k \u27f6 B k) (g : \u03a0 k, B k \u27f6 C k)\n\nlemma pi_map_exact (H : \u2200 k, exact (f k) (g k)) :\n  exact (pi.map f) (pi.map g) :=\nbegin\n  simp only [AddCommGroup.exact_iff'] at H \u22a2,\n  split,\n  { apply limit.hom_ext, intro j,\n    simp only [category.assoc, lim_map_\u03c0, discrete.nat_trans_app, lim_map_\u03c0_assoc,\n      zero_comp, (H j).1, comp_zero], },\n  intros x hx,\n  rw [add_monoid_hom.mem_ker, Ab.apply_eq_zero] at hx,\n  have : \u2200 k, (Ab.pt (limit.\u03c0 (discrete.functor (\u03bb k, B k)) k x)) \u226b g k = 0,\n  { intro k,\n    suffices : Ab.pt x \u226b pi.map g \u226b pi.\u03c0 _ k = 0,\n    { simpa only [lim_map_\u03c0, discrete.nat_trans_app, \u2190 category.assoc, hx, Ab.pt_comp] },\n    rw [\u2190 category.assoc, hx, zero_comp] },\n  simp only [\u2190 Ab.apply_eq_zero] at this,\n  replace := \u03bb k, (H k).2 (this k),\n  choose y hy using this,\n  refine \u27e8pi.lift (\u03bb k, Ab.pt (y k)) \u27e81\u27e9, _\u27e9,\n  rw [\u2190 category_theory.comp_apply, \u2190 Ab.pt_apply' x],\n  congr' 1,\n  apply limit.hom_ext,\n  intro j,\n  simp only [limit.lift_map, limit.lift_\u03c0, cones.postcompose_obj_\u03c0, nat_trans.comp_app,\n    fan.mk_\u03c0_app, discrete.nat_trans_app, Ab.pt_comp, hy],\nend\n\nend step2\n\nsection step3\n\nvariables {A B C : \u211d\u22650\u1d52\u1d56 \u2964 cochain_complex Ab.{u} \u2115} (f : A \u27f6 B) (g : B \u27f6 C)\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (n : \u2115)\n\ndef piH_hom :\n  (\u220f (\u03bb x, (A.obj (op $ \u03b9 x)).homology n)) \u27f6 (\u220f (\u03bb x, (B.obj (op $ \u03b9 x)).homology n)) :=\npi.map $ \u03bb k, (homology_functor _ _ _).map $ f.app _\n\ndef shift_sub_id.\u03b4 (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  C \u22d9 homology_functor _ _ n \u27f6 A \u22d9 homology_functor _ _ (n+1) :=\n{ app := \u03bb c, homological_complex.\u03b4 (f.app _) (g.app _) (H _) _ _ rfl,\n  naturality' := \u03bb c\u2081 c\u2082 h, by { symmetry, apply homological_complex.\u03b4_natural } }\n\ndef pi\u03b4 (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  (\u220f (\u03bb x, (C.obj (op $ \u03b9 x)).homology n)) \u27f6 (\u220f (\u03bb x, (A.obj (op $ \u03b9 x)).homology (n+1))) :=\npi.map $ \u03bb k, (shift_sub_id.\u03b4 _ _ _ H).app _\n\nlemma piH_les (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  exact_seq Ab.{u} [piH_hom f \u03b9 n, piH_hom g \u03b9 n, pi\u03b4 f g \u03b9 n H] :=\nbegin\n  apply exact.cons,\n  { apply pi_map_exact, intro k,\n    have := homological_complex.six_term_exact_seq _ _ (H (op $ \u03b9 k)) n (n+1) rfl,\n    exact this.pair, },\n  apply exact.exact_seq,\n  { apply pi_map_exact, intro k,\n    have := homological_complex.six_term_exact_seq _ _ (H (op $ \u03b9 k)) n (n+1) rfl,\n    exact (this.drop 1).pair, },\nend\n\nend step3\n\nsection step4\n\nvariables {A B C : system_of_complexes.{u}} (f : A.to_Ab \u27f6 B.to_Ab) (g : B.to_Ab \u27f6 C.to_Ab)\nvariables (n : \u2115) (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (h\u03b9 : monotone \u03b9)\n\nlemma shift_sub_id.bicartesian\n  (HA\u2081 : (shift_sub_id.shift (A.to_AbH n) \u03b9 h\u03b9) = 0)\n  (HA\u2082 : (shift_sub_id.shift (A.to_AbH (n+1)) \u03b9 h\u03b9) = 0)\n  (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  (@shift_sub_id.commsq (B.to_AbH n) (C.to_AbH n)\n    (whisker_right g _) \u03b9 h\u03b9).bicartesian :=\nbegin\n  rw \u2190 commsq.bicartesian.symm_iff,\n  let S1 := ((@shift_sub_id.commsq (A.to_AbH n) (B.to_AbH n) (whisker_right f _) \u03b9 h\u03b9)).symm,\n  let S2 := ((@shift_sub_id.commsq (B.to_AbH n) (C.to_AbH n) (whisker_right g _) \u03b9 h\u03b9)).symm,\n  let S3 := ((@shift_sub_id.commsq (C.to_AbH n) (A.to_AbH (n+1)) (shift_sub_id.\u03b4 _ _ _ H) \u03b9 h\u03b9)).symm,\n  apply bicartesian_of_id_of_end_of_end_of_id (piH_les _ _ _ _ _) S1 S2 S3; clear S1 S2 S3,\n  { rw [shift_sub_id, HA\u2081, zero_sub], refl },\n  { rw [shift_sub_id, HA\u2082, zero_sub], refl },\nend\n\nend step4\n\nsection step5\n\nvariables {A B A' B' : \u211d\u22650\u1d52\u1d56 \u2964 Ab.{u}} (f : A \u27f6 B) (f' : A' \u27f6 B') (eA : A \u2245 A') (eB : B \u2245 B')\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (h\u03b9 : monotone \u03b9)\n\nlemma shift_sub_id.bicartesian_iso (w : f \u226b eB.hom = eA.hom \u226b f')\n  (sq : (shift_sub_id.commsq f \u03b9 h\u03b9).bicartesian) :\n  (shift_sub_id.commsq f' \u03b9 h\u03b9).bicartesian :=\nbegin\n  let H : _ := _,\n  apply commsq.bicartesian.of_iso _ _ _ _ _ H H _ sq,\n  { refine limits.lim.map_iso (discrete.nat_iso $ \u03bb k, eA.app _), },\n  { refine limits.lim.map_iso (discrete.nat_iso $ \u03bb k, eB.app _), },\n  { apply shift_sub_id.commsq },\n  { apply shift_sub_id.commsq },\n  { apply commsq.of_eq, delta pi.map,\n    simp only [functor.map_iso_hom, \u2190 lim_map_eq_lim_map, \u2190 category_theory.functor.map_comp],\n    apply limit.hom_ext,\n    simp only [lim_map_eq_lim_map, lim_map_\u03c0, nat_trans.comp_app, discrete.nat_trans_app,\n      discrete.nat_iso_hom_app, iso.app_hom],\n    intro, simp only [\u2190 nat_trans.comp_app, w], }\nend\n\nend step5\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/Lbar/squares.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581510799252, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.38296937713680346}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.cone_category\n\n/-!\n\n# Multi-(co)equalizers\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA *multiequalizer* is an equalizer of two morphisms between two products.\nSince both products and equalizers are limits, such an object is again a limit.\nThis file provides the diagram whose limit is indeed such an object.\nIn fact, it is well-known that any limit can be obtained as a multiequalizer.\nThe dual construction (multicoequalizers) is also provided.\n\n## Projects\n\nProve that a multiequalizer can be identified with\nan equalizer between products (and analogously for multicoequalizers).\n\nProve that the limit of any diagram is a multiequalizer (and similarly for colimits).\n\n-/\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses w v u\n\n/-- The type underlying the multiequalizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multicospan {L R : Type w} (fst snd : R \u2192 L) : Type w\n| left : L \u2192 walking_multicospan\n| right : R \u2192 walking_multicospan\n\n/-- The type underlying the multiecoqualizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multispan {L R : Type w} (fst snd : L \u2192 R) : Type w\n| left : L \u2192 walking_multispan\n| right : R \u2192 walking_multispan\n\nnamespace walking_multicospan\n\nvariables {L R : Type w} {fst snd : R \u2192 L}\n\ninstance [inhabited L] : inhabited (walking_multicospan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multicospan`. -/\ninductive hom : \u03a0 (a b : walking_multicospan fst snd), Type w\n| id (A)  : hom A A\n| fst (b) : hom (left (fst b)) (right b)\n| snd (b) : hom (left (snd b)) (right b)\n\ninstance {a : walking_multicospan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multicospan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multicospan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst b) (hom.id X) := hom.fst b\n| _ _ _ (hom.snd b) (hom.id X) := hom.snd b\n\ninstance : small_category (walking_multicospan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multicospan\n\nnamespace walking_multispan\n\nvariables {L R : Type v} {fst snd : L \u2192 R}\n\ninstance [inhabited L] : inhabited (walking_multispan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multispan`. -/\ninductive hom : \u03a0 (a b : walking_multispan fst snd), Type v\n| id (A)  : hom A A\n| fst (a) : hom (left a) (right (fst a))\n| snd (a) : hom (left a) (right (snd a))\n\ninstance {a : walking_multispan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multispan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multispan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst a) (hom.id X) := hom.fst a\n| _ _ _ (hom.snd a) (hom.id X) := hom.snd a\n\ninstance : small_category (walking_multispan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multispan\n\n/-- This is a structure encapsulating the data necessary to define a `multicospan`. -/\n@[nolint has_nonempty_instance]\nstructure multicospan_index (C : Type u) [category.{v} C] :=\n(L R : Type w)\n(fst_to snd_to : R \u2192 L)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 b, left (fst_to b) \u27f6 right b)\n(snd : \u03a0 b, left (snd_to b) \u27f6 right b)\n\n/-- This is a structure encapsulating the data necessary to define a `multispan`. -/\n@[nolint has_nonempty_instance]\nstructure multispan_index (C : Type u) [category.{v} C] :=\n(L R : Type w)\n(fst_from snd_from : L \u2192 R)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 a, left a \u27f6 right (fst_from a))\n(snd : \u03a0 a, left a \u27f6 right (snd_from a))\n\nnamespace multicospan_index\n\nvariables {C : Type u} [category.{v} C] (I : multicospan_index C)\n\n/-- The multicospan associated to `I : multicospan_index`. -/\ndef multicospan : walking_multicospan I.fst_to I.snd_to \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multicospan.left a := I.left a\n  | walking_multicospan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multicospan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multicospan.hom.fst b := I.fst _\n  | _, _, walking_multicospan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multicospan_obj_left (a) :\n  I.multicospan.obj (walking_multicospan.left a) = I.left a := rfl\n\n@[simp] lemma multicospan_obj_right (b) :\n  I.multicospan.obj (walking_multicospan.right b) = I.right b := rfl\n\n@[simp] lemma multicospan_map_fst (b) :\n  I.multicospan.map (walking_multicospan.hom.fst b) = I.fst b := rfl\n\n@[simp] lemma multicospan_map_snd (b) :\n  I.multicospan.map (walking_multicospan.hom.snd b) = I.snd b := rfl\n\nvariables [has_product I.left] [has_product I.right]\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.fst`. -/\nnoncomputable\ndef fst_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.fst_to b) \u226b I.fst b)\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.snd`. -/\nnoncomputable\ndef snd_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.snd_to b) \u226b I.snd b)\n\n@[simp, reassoc]\nlemma fst_pi_map_\u03c0 (b) : I.fst_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.fst b :=\nby simp [fst_pi_map]\n\n@[simp, reassoc]\nlemma snd_pi_map_\u03c0 (b) : I.snd_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.snd b :=\nby simp [snd_pi_map]\n\n/--\nTaking the multiequalizer over the multicospan index is equivalent to taking the equalizer over\nthe two morphsims `\u220f I.left \u21c9 \u220f I.right`. This is the diagram of the latter.\n-/\n@[simps] protected noncomputable\ndef parallel_pair_diagram := parallel_pair I.fst_pi_map I.snd_pi_map\n\nend multicospan_index\n\nnamespace multispan_index\n\nvariables {C : Type u} [category.{v} C] (I : multispan_index C)\n\n/-- The multispan associated to `I : multispan_index`. -/\ndef multispan : walking_multispan I.fst_from I.snd_from \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multispan.left a := I.left a\n  | walking_multispan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multispan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multispan.hom.fst b := I.fst _\n  | _, _, walking_multispan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multispan_obj_left (a) :\n  I.multispan.obj (walking_multispan.left a) = I.left a := rfl\n\n@[simp] lemma multispan_obj_right (b) :\n  I.multispan.obj (walking_multispan.right b) = I.right b := rfl\n\n@[simp] lemma multispan_map_fst (a) :\n  I.multispan.map (walking_multispan.hom.fst a) = I.fst a := rfl\n\n@[simp] lemma multispan_map_snd (a) :\n  I.multispan.map (walking_multispan.hom.snd a) = I.snd a := rfl\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.fst`. -/\nnoncomputable\ndef fst_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.fst b \u226b sigma.\u03b9 _ (I.fst_from b))\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.snd`. -/\nnoncomputable\ndef snd_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.snd b \u226b sigma.\u03b9 _ (I.snd_from b))\n\n@[simp, reassoc]\nlemma \u03b9_fst_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.fst_sigma_map = I.fst b \u226b sigma.\u03b9 I.right _ :=\nby simp [fst_sigma_map]\n\n@[simp, reassoc]\nlemma \u03b9_snd_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.snd_sigma_map = I.snd b \u226b sigma.\u03b9 I.right _ :=\nby simp [snd_sigma_map]\n\n/--\nTaking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over\nthe two morphsims `\u2210 I.left \u21c9 \u2210 I.right`. This is the diagram of the latter.\n-/\nprotected noncomputable\nabbreviation parallel_pair_diagram := parallel_pair I.fst_sigma_map I.snd_sigma_map\n\nend multispan_index\n\nvariables {C : Type u} [category.{v} C]\n\n/-- A multifork is a cone over a multicospan. -/\n@[nolint has_nonempty_instance]\nabbreviation multifork (I : multicospan_index C) := cone I.multicospan\n\n/-- A multicofork is a cocone over a multispan. -/\n@[nolint has_nonempty_instance]\nabbreviation multicofork (I : multispan_index C) := cocone I.multispan\n\nnamespace multifork\n\nvariables {I : multicospan_index C} (K : multifork I)\n\n/-- The maps from the cone point of a multifork to the objects on the left. -/\ndef \u03b9 (a : I.L) : K.X \u27f6 I.left a := K.\u03c0.app (walking_multicospan.left _)\n\n@[simp] lemma app_left_eq_\u03b9 (a) : K.\u03c0.app (walking_multicospan.left a) = K.\u03b9 a := rfl\n\n@[simp] lemma app_right_eq_\u03b9_comp_fst (b) :\n  K.\u03c0.app (walking_multicospan.right b) = K.\u03b9 (I.fst_to b) \u226b I.fst b :=\nby { rw \u2190 K.w (walking_multicospan.hom.fst b), refl }\n\n@[reassoc] lemma app_right_eq_\u03b9_comp_snd (b) :\n  K.\u03c0.app (walking_multicospan.right b) =  K.\u03b9 (I.snd_to b) \u226b I.snd b :=\nby { rw \u2190 K.w (walking_multicospan.hom.snd b), refl }\n\n@[simp, reassoc] lemma hom_comp_\u03b9 (K\u2081 K\u2082 : multifork I) (f : K\u2081 \u27f6 K\u2082) (j : I.L) :\n  f.hom \u226b K\u2082.\u03b9 j = K\u2081.\u03b9 j := f.w (walking_multicospan.left j)\n\n/-- Construct a multifork using a collection `\u03b9` of morphisms. -/\n@[simps]\ndef of_\u03b9 (I : multicospan_index C) (P : C) (\u03b9 : \u03a0 a, P \u27f6 I.left a)\n  (w : \u2200 b, \u03b9 (I.fst_to b) \u226b I.fst b = \u03b9 (I.snd_to b) \u226b I.snd b) :\n  multifork I :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := \u03b9 _\n    | walking_multicospan.right b := \u03b9 (I.fst_to b) \u226b I.fst b\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { dsimp, rw category.id_comp, refl },\n      { dsimp, rw category.id_comp, apply w }\n    end } }\n\n@[simp, reassoc]\nlemma condition (b) :\n  K.\u03b9 (I.fst_to b) \u226b I.fst b = K.\u03b9 (I.snd_to b) \u226b I.snd b :=\nby rw [\u2190app_right_eq_\u03b9_comp_fst, \u2190app_right_eq_\u03b9_comp_snd]\n\n/-- This definition provides a convenient way to show that a multifork is a limit. -/\n@[simps]\ndef is_limit.mk\n  (lift : \u03a0 (E : multifork I), E.X \u27f6 K.X)\n  (fac : \u2200 (E : multifork I) (i : I.L), lift E \u226b K.\u03b9 i = E.\u03b9 i)\n  (uniq : \u2200 (E : multifork I) (m : E.X \u27f6 K.X),\n    (\u2200 i : I.L, m \u226b K.\u03b9 i = E.\u03b9 i) \u2192 m = lift E) : is_limit K :=\n{ lift := lift,\n  fac' := begin\n    rintros E (a|b),\n    { apply fac },\n    { rw [\u2190 E.w (walking_multicospan.hom.fst b), \u2190 K.w (walking_multicospan.hom.fst b),\n        \u2190 category.assoc],\n      congr' 1,\n      apply fac }\n  end,\n  uniq' := begin\n    rintros E m hm,\n    apply uniq,\n    intros i,\n    apply hm,\n  end }\n\n\nvariables [has_product I.left] [has_product I.right]\n\n@[simp, reassoc]\nlemma pi_condition : pi.lift K.\u03b9 \u226b I.fst_pi_map = pi.lift K.\u03b9 \u226b I.snd_pi_map :=\nby { ext, discrete_cases, simp, }\n\n/-- Given a multifork, we may obtain a fork over `\u220f I.left \u21c9 \u220f I.right`. -/\n@[simps X] noncomputable\ndef to_pi_fork (K : multifork I) : fork I.fst_pi_map I.snd_pi_map :=\n{ X := K.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := pi.lift K.\u03b9\n    | walking_parallel_pair.one := pi.lift K.\u03b9 \u226b I.fst_pi_map\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      all_goals { change \ud835\udfd9 _ \u226b _ \u226b _ = pi.lift _ \u226b _, simp }\n    end } }\n\n@[simp] lemma to_pi_fork_\u03c0_app_zero : K.to_pi_fork.\u03b9 = pi.lift K.\u03b9 := rfl\n\n@[simp] lemma to_pi_fork_\u03c0_app_one :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.one = pi.lift K.\u03b9 \u226b I.fst_pi_map := rfl\n\nvariable (I)\n\n/-- Given a fork over `\u220f I.left \u21c9 \u220f I.right`, we may obtain a multifork. -/\n@[simps X] noncomputable\ndef of_pi_fork (c : fork I.fst_pi_map I.snd_pi_map) : multifork I :=\n{ X := c.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := c.\u03b9 \u226b pi.\u03c0 _ _\n    | walking_multicospan.right b := c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, simp },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, rw c.condition_assoc, simp }\n    end } }\n\n@[simp] lemma of_pi_fork_\u03c0_app_left (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03b9 a = c.\u03b9 \u226b pi.\u03c0 _ _ := rfl\n\n@[simp] lemma of_pi_fork_\u03c0_app_right (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.right a) = c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _ := rfl\n\nend multifork\n\nnamespace multicospan_index\n\nvariables (I : multicospan_index C) [has_product I.left] [has_product I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multifork.to_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef to_pi_fork_functor : multifork I \u2964 fork I.fst_pi_map I.snd_pi_map :=\n{ obj := multifork.to_pi_fork,\n  map := \u03bb K\u2081 K\u2082 f,\n  { hom := f.hom,\n    w' := begin\n      rintro (_|_),\n      { ext, dsimp, simp },\n      { ext,\n        simp only [multifork.to_pi_fork_\u03c0_app_one, multifork.pi_condition, category.assoc],\n        dsimp [snd_pi_map],\n        simp },\n    end } }\n\n/-- `multifork.of_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef of_pi_fork_functor : fork I.fst_pi_map I.snd_pi_map \u2964 multifork I :=\n{ obj := multifork.of_pi_fork I, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multiforks is equivalent to the category of forks over `\u220f I.left \u21c9 \u220f I.right`.\nIt then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it\npreserves and reflects limit cones.\n-/\n@[simps] noncomputable\ndef multifork_equiv_pi_fork : multifork I \u224c fork I.fst_pi_map I.snd_pi_map :=\n{ functor := to_pi_fork_functor I,\n  inverse := of_pi_fork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cones.ext (iso.refl _)\n    (by { rintros (_|_); dsimp; simp[\u2190fork.app_one_eq_\u03b9_comp_left, -fork.app_one_eq_\u03b9_comp_left] }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, fork.ext (iso.refl _) (by { ext \u27e8j\u27e9, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }) }\n\nend multicospan_index\n\nnamespace multicofork\n\nvariables {I : multispan_index C} (K : multicofork I)\n\n/-- The maps to the cocone point of a multicofork from the objects on the right. -/\ndef \u03c0 (b : I.R) : I.right b \u27f6 K.X :=\nK.\u03b9.app (walking_multispan.right _)\n\n@[simp] lemma \u03c0_eq_app_right (b) : K.\u03b9.app (walking_multispan.right _) = K.\u03c0 b := rfl\n\n@[simp] lemma fst_app_right (a) :\n  K.\u03b9.app (walking_multispan.left a) = I.fst a \u226b K.\u03c0 _ :=\nby { rw \u2190 K.w (walking_multispan.hom.fst a), refl }\n\n@[reassoc] lemma snd_app_right (a) :\n  K.\u03b9.app (walking_multispan.left a) = I.snd a \u226b K.\u03c0 _ :=\nby { rw \u2190 K.w (walking_multispan.hom.snd a), refl }\n\n/-- Construct a multicofork using a collection `\u03c0` of morphisms. -/\n@[simps]\ndef of_\u03c0 (I : multispan_index C) (P : C) (\u03c0 : \u03a0 b, I.right b \u27f6 P)\n  (w : \u2200 a, I.fst a \u226b \u03c0 (I.fst_from a) = I.snd a \u226b \u03c0 (I.snd_from a)) :\n  multicofork I :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := I.fst a \u226b \u03c0 _\n    | walking_multispan.right b := \u03c0 _\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { dsimp, rw category.comp_id, refl },\n      { dsimp, rw category.comp_id, apply (w _).symm }\n    end } }\n\n@[simp, reassoc]\nlemma condition (a) : I.fst a \u226b K.\u03c0 (I.fst_from a) = I.snd a \u226b K.\u03c0 (I.snd_from a) :=\nby rw [\u2190K.snd_app_right, \u2190K.fst_app_right]\n\n/-- This definition provides a convenient way to show that a multicofork is a colimit. -/\n@[simps]\ndef is_colimit.mk\n  (desc : \u03a0 (E : multicofork I), K.X \u27f6 E.X)\n  (fac : \u2200 (E : multicofork I) (i : I.R), K.\u03c0 i \u226b desc E = E.\u03c0 i)\n  (uniq : \u2200 (E : multicofork I) (m : K.X \u27f6 E.X),\n    (\u2200 i : I.R, K.\u03c0 i \u226b m = E.\u03c0 i) \u2192 m = desc E) : is_colimit K :=\n{ desc := desc,\n  fac' := begin\n    rintros S (a|b),\n    { rw [\u2190 K.w (walking_multispan.hom.fst a), \u2190 S.w (walking_multispan.hom.fst a),\n        category.assoc],\n      congr' 1,\n      apply fac },\n    { apply fac },\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply uniq,\n    intros i,\n    apply hm\n  end }\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n@[simp, reassoc]\nlemma sigma_condition :\n  I.fst_sigma_map \u226b sigma.desc K.\u03c0 = I.snd_sigma_map \u226b sigma.desc K.\u03c0 :=\nby { ext, discrete_cases, simp, }\n\n/-- Given a multicofork, we may obtain a cofork over `\u2210 I.left \u21c9 \u2210 I.right`. -/\n@[simps X] noncomputable\ndef to_sigma_cofork (K : multicofork I) : cofork I.fst_sigma_map I.snd_sigma_map :=\n{ X := K.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := I.fst_sigma_map \u226b sigma.desc K.\u03c0\n    | walking_parallel_pair.one := sigma.desc K.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      all_goals { change _ \u226b sigma.desc _ = (_ \u226b _) \u226b \ud835\udfd9 _, simp }\n    end } }\n\n@[simp] lemma to_sigma_cofork_\u03c0 : K.to_sigma_cofork.\u03c0 = sigma.desc K.\u03c0 := rfl\n\nvariable (I)\n\n/-- Given a cofork over `\u2210 I.left \u21c9 \u2210 I.right`, we may obtain a multicofork. -/\n@[simps X] noncomputable\ndef of_sigma_cofork (c : cofork I.fst_sigma_map I.snd_sigma_map) : multicofork I :=\n{ X := c.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0\n    | walking_multispan.right b := (sigma.\u03b9 I.right b : _) \u226b c.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, dsimp,\n        simp only [cofork.condition, category.comp_id],\n        rw [\u2190I.\u03b9_fst_sigma_map_assoc, c.condition] },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b \ud835\udfd9 _,\n        rw c.condition, simp }\n    end } }\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_left (c : cofork I.fst_sigma_map I.snd_sigma_map) (a) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.left a) =\n    (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0 := rfl\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_right (c : cofork I.fst_sigma_map I.snd_sigma_map) (b) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.right b) = (sigma.\u03b9 I.right b : _) \u226b c.\u03c0 := rfl\n\nend multicofork\n\nnamespace multispan_index\n\nvariables (I : multispan_index C) [has_coproduct I.left] [has_coproduct I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multicofork.to_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef to_sigma_cofork_functor : multicofork I \u2964 cofork I.fst_sigma_map I.snd_sigma_map :=\n{ obj := multicofork.to_sigma_cofork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multicofork.of_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef of_sigma_cofork_functor : cofork I.fst_sigma_map I.snd_sigma_map \u2964 multicofork I :=\n{ obj := multicofork.of_sigma_cofork I,\n  map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multicoforks is equivalent to the category of coforks over `\u2210 I.left \u21c9 \u2210 I.right`.\nIt then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that\nit preserves and reflects colimit cocones.\n-/\n@[simps] noncomputable\ndef multicofork_equiv_sigma_cofork : multicofork I \u224c cofork I.fst_sigma_map I.snd_sigma_map :=\n{ functor := to_sigma_cofork_functor I,\n  inverse := of_sigma_cofork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cocones.ext (iso.refl _)\n      (by { rintros (_|_); dsimp; simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, cofork.ext (iso.refl _)\n      (by { ext \u27e8j\u27e9, dsimp, simp only [category.comp_id, colimit.\u03b9_desc, cofan.mk_\u03b9_app], refl }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, dsimp, simp, }) }\n\nend multispan_index\n\n/-- For `I : multicospan_index C`, we say that it has a multiequalizer if the associated\n  multicospan has a limit. -/\nabbreviation has_multiequalizer (I : multicospan_index C) :=\n  has_limit I.multicospan\n\nnoncomputable theory\n\n/-- The multiequalizer of `I : multicospan_index C`. -/\nabbreviation multiequalizer (I : multicospan_index C) [has_multiequalizer I] : C :=\n  limit I.multicospan\n\n/-- For `I : multispan_index C`, we say that it has a multicoequalizer if\n  the associated multicospan has a limit. -/\nabbreviation has_multicoequalizer (I : multispan_index C) :=\n  has_colimit I.multispan\n\n/-- The multiecoqualizer of `I : multispan_index C`. -/\nabbreviation multicoequalizer (I : multispan_index C) [has_multicoequalizer I] : C :=\n  colimit I.multispan\n\nnamespace multiequalizer\n\nvariables (I : multicospan_index C) [has_multiequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03b9 (a : I.L) : multiequalizer I \u27f6 I.left a :=\nlimit.\u03c0 _ (walking_multicospan.left a)\n\n/-- The multifork associated to the multiequalizer. -/\nabbreviation multifork : multifork I :=\nlimit.cone _\n\n@[simp]\nlemma multifork_\u03b9 (a) :\n  (multiequalizer.multifork I).\u03b9 a = multiequalizer.\u03b9 I a := rfl\n\n@[simp]\nlemma multifork_\u03c0_app_left (a) :\n  (multiequalizer.multifork I).\u03c0.app (walking_multicospan.left a) =\n  multiequalizer.\u03b9 I a := rfl\n\n@[reassoc]\nlemma condition (b) :\n  multiequalizer.\u03b9 I (I.fst_to b) \u226b I.fst b =\n  multiequalizer.\u03b9 I (I.snd_to b) \u226b I.snd b :=\nmultifork.condition _ _\n\n/-- Construct a morphism to the multiequalizer from its universal property. -/\nabbreviation lift (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) :\n  W \u27f6 multiequalizer I :=\nlimit.lift _ (multifork.of_\u03b9 I _ k h)\n\n@[simp, reassoc]\nlemma lift_\u03b9 (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) (a) :\n  multiequalizer.lift I _ k h \u226b multiequalizer.\u03b9 I a = k _ :=\nlimit.lift_\u03c0 _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : W \u27f6 multiequalizer I)\n  (h : \u2200 a, i \u226b multiequalizer.\u03b9 I a =\n  j \u226b multiequalizer.\u03b9 I a) :\n  i = j :=\nlimit.hom_ext\nbegin\n  rintro (a|b),\n  { apply h },\n  simp_rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n    \u2190 category.assoc, h],\nend\n\nvariables [has_product I.left] [has_product I.right]\n\ninstance : has_equalizer I.fst_pi_map I.snd_pi_map :=\n\u27e8\u27e8\u27e8_,is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.functor (limit.is_limit _)\u27e9\u27e9\u27e9\n\n/-- The multiequalizer is isomorphic to the equalizer of `\u220f I.left \u21c9 \u220f I.right`. -/\ndef iso_equalizer : multiequalizer I \u2245 equalizer I.fst_pi_map I.snd_pi_map :=\nlimit.iso_limit_cone \u27e8_, is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.inverse (limit.is_limit _)\u27e9\n\n/-- The canonical injection `multiequalizer I \u27f6 \u220f I.left`. -/\ndef \u03b9_pi : multiequalizer I \u27f6 \u220f I.left :=\n  (iso_equalizer I).hom \u226b equalizer.\u03b9 I.fst_pi_map I.snd_pi_map\n\n@[simp, reassoc]\nlemma \u03b9_pi_\u03c0 (a) : \u03b9_pi I \u226b pi.\u03c0 I.left a = \u03b9 I a :=\nby { rw [\u03b9_pi, category.assoc, \u2190 iso.eq_inv_comp, iso_equalizer], simpa }\n\ninstance : mono (\u03b9_pi I) := @@mono_comp _ _ _ _ equalizer.\u03b9_mono\n\nend multiequalizer\n\nnamespace multicoequalizer\n\nvariables (I : multispan_index C) [has_multicoequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03c0 (b : I.R) : I.right b \u27f6 multicoequalizer I :=\ncolimit.\u03b9 I.multispan (walking_multispan.right _)\n\n/-- The multicofork associated to the multicoequalizer. -/\nabbreviation multicofork : multicofork I :=\ncolimit.cocone _\n\n@[simp]\nlemma multicofork_\u03c0 (b) :\n  (multicoequalizer.multicofork I).\u03c0 b = multicoequalizer.\u03c0 I b := rfl\n\n@[simp]\nlemma multicofork_\u03b9_app_right (b) :\n  (multicoequalizer.multicofork I).\u03b9.app (walking_multispan.right b) =\n  multicoequalizer.\u03c0 I b := rfl\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b multicoequalizer.\u03c0 I (I.fst_from a) =\n  I.snd a \u226b multicoequalizer.\u03c0 I (I.snd_from a) :=\nmulticofork.condition _ _\n\n/-- Construct a morphism from the multicoequalizer from its universal property. -/\nabbreviation desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) :\n  multicoequalizer I \u27f6 W :=\ncolimit.desc _ (multicofork.of_\u03c0 I _ k h)\n\n@[simp, reassoc]\nlemma \u03c0_desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) (b) :\n  multicoequalizer.\u03c0 I b \u226b multicoequalizer.desc I _ k h = k _ :=\ncolimit.\u03b9_desc _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : multicoequalizer I \u27f6 W)\n  (h : \u2200 b, multicoequalizer.\u03c0 I b \u226b i = multicoequalizer.\u03c0 I b \u226b j) :\n  i = j :=\ncolimit.hom_ext\nbegin\n  rintro (a|b),\n  { simp_rw [\u2190 colimit.w I.multispan (walking_multispan.hom.fst a),\n    category.assoc, h] },\n  { apply h },\nend\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\ninstance : has_coequalizer I.fst_sigma_map I.snd_sigma_map :=\n\u27e8\u27e8\u27e8_,is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.functor (colimit.is_colimit _)\u27e9\u27e9\u27e9\n\n/-- The multicoequalizer is isomorphic to the coequalizer of `\u2210 I.left \u21c9 \u2210 I.right`. -/\ndef iso_coequalizer : multicoequalizer I \u2245 coequalizer I.fst_sigma_map I.snd_sigma_map :=\ncolimit.iso_colimit_cocone \u27e8_, is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.inverse (colimit.is_colimit _)\u27e9\n\n/-- The canonical projection `\u2210 I.right \u27f6 multicoequalizer I`. -/\ndef sigma_\u03c0 : \u2210 I.right \u27f6 multicoequalizer I :=\n  coequalizer.\u03c0 I.fst_sigma_map I.snd_sigma_map \u226b (iso_coequalizer I).inv\n\n@[simp, reassoc]\nlemma \u03b9_sigma_\u03c0 (b) : sigma.\u03b9 I.right b \u226b sigma_\u03c0 I = \u03c0 I b :=\nby { rw [sigma_\u03c0, \u2190 category.assoc, iso.comp_inv_eq, iso_coequalizer], simpa }\n\ninstance : epi (sigma_\u03c0 I) := @@epi_comp _ _ coequalizer.\u03c0_epi _ _\n\nend multicoequalizer\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/shapes/multiequalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.6187804337438502, "lm_q1q2_score": 0.3828892251842273}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Jakob von Raumer\n-/\nimport category_theory.limits.has_limits\nimport category_theory.thin\n\n/-!\n# Wide pullbacks\n\nWe define the category `wide_pullback_shape`, (resp. `wide_pushout_shape`) which is the category\nobtained from a discrete category of type `J` by adjoining a terminal (resp. initial) element.\nLimits of this shape are wide pullbacks (pushouts).\nThe convenience method `wide_cospan` (`wide_span`) constructs a functor from this category, hitting\nthe given morphisms.\n\nWe use `wide_pullback_shape` to define ordinary pullbacks (pushouts) by using `J := walking_pair`,\nwhich allows easy proofs of some related lemmas.\nFurthermore, wide pullbacks are used to show the existence of limits in the slice category.\nNamely, if `C` has wide pullbacks then `C/B` has limits for any object `B` in `C`.\n\nTypeclasses `has_wide_pullbacks` and `has_finite_wide_pullbacks` assert the existence of wide\npullbacks and finite wide pullbacks.\n-/\n\nuniverses w w' v u\n\nopen category_theory category_theory.limits opposite\n\nnamespace category_theory.limits\n\nvariable (J : Type w)\n\n/-- A wide pullback shape for any type `J` can be written simply as `option J`. -/\n@[derive inhabited]\ndef wide_pullback_shape := option J\n\n/-- A wide pushout shape for any type `J` can be written simply as `option J`. -/\n@[derive inhabited]\ndef wide_pushout_shape := option J\n\nnamespace wide_pullback_shape\n\nvariable {J}\n\n/-- The type of arrows for the shape indexing a wide pullback. -/\n@[derive decidable_eq]\ninductive hom : wide_pullback_shape J \u2192 wide_pullback_shape J \u2192 Type w\n| id : \u03a0 X, hom X X\n| term : \u03a0 (j : J), hom (some j) none\n\nattribute [nolint unused_arguments] hom.decidable_eq\n\ninstance struct : category_struct (wide_pullback_shape J) :=\n{ hom := hom,\n  id := \u03bb j, hom.id j,\n  comp := \u03bb j\u2081 j\u2082 j\u2083 f g,\n  begin\n    cases f,\n      exact g,\n    cases g,\n    apply hom.term _\n  end }\n\ninstance hom.inhabited : inhabited (hom none none) := \u27e8hom.id (none : wide_pullback_shape J)\u27e9\n\nlocal attribute [tidy] tactic.case_bash\n\ninstance subsingleton_hom (j j' : wide_pullback_shape J) : subsingleton (j \u27f6 j') :=\n\u27e8by tidy\u27e9\n\ninstance category : small_category (wide_pullback_shape J) := thin_category\n\n@[simp] lemma hom_id (X : wide_pullback_shape J) : hom.id X = \ud835\udfd9 X := rfl\n\nvariables {C : Type u} [category.{v} C]\n\n/--\nConstruct a functor out of the wide pullback shape given a J-indexed collection of arrows to a\nfixed object.\n-/\n@[simps]\ndef wide_cospan (B : C) (objs : J \u2192 C) (arrows : \u03a0 (j : J), objs j \u27f6 B) :\n  wide_pullback_shape J \u2964 C :=\n{ obj := \u03bb j, option.cases_on j B objs,\n  map := \u03bb X Y f,\n  begin\n    cases f with _ j,\n    { apply (\ud835\udfd9 _) },\n    { exact arrows j }\n  end,\n  map_comp' := \u03bb _ _ _ f g,\n  begin\n    cases f,\n    { simpa },\n    cases g,\n    simp\n  end }\n\n/-- Every diagram is naturally isomorphic (actually, equal) to a `wide_cospan` -/\ndef diagram_iso_wide_cospan (F : wide_pullback_shape J \u2964 C) :\n  F \u2245 wide_cospan (F.obj none) (\u03bb j, F.obj (some j)) (\u03bb j, F.map (hom.term j)) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by tidy) $ by tidy\n\n/-- Construct a cone over a wide cospan. -/\n@[simps]\ndef mk_cone {F : wide_pullback_shape J \u2964 C} {X : C}\n  (f : X \u27f6 F.obj none) (\u03c0 : \u03a0 j, X \u27f6 F.obj (some j))\n  (w : \u2200 j, \u03c0 j \u226b F.map (hom.term j) = f) : cone F :=\n{ X := X,\n  \u03c0 :=\n  { app := \u03bb j, match j with\n    | none := f\n    | (some j) := \u03c0 j\n    end,\n    naturality' := \u03bb j j' f, by { cases j; cases j'; cases f; unfold_aux; dsimp; simp [w], }, } }\n\n/-- Wide pullback diagrams of equivalent index types are equivlent. -/\ndef equivalence_of_equiv (J' : Type w') (h : J \u2243 J') :\n  wide_pullback_shape J \u224c wide_pullback_shape J' :=\n{ functor := wide_cospan none (\u03bb j, some (h j)) (\u03bb j, hom.term (h j)),\n  inverse := wide_cospan none (\u03bb j, some (h.inv_fun j)) (\u03bb j, hom.term (h.inv_fun j)),\n  unit_iso := nat_iso.of_components (\u03bb j, by cases j; simp)\n    (\u03bb j k f, by { simp only [eq_iff_true_of_subsingleton]}),\n  counit_iso := nat_iso.of_components (\u03bb j, by cases j; simp)\n    (\u03bb j k f, by { simp only [eq_iff_true_of_subsingleton]}) }\n\n/-- Lifting universe and morphism levels preserves wide pullback diagrams. -/\ndef ulift_equivalence :\n  ulift_hom.{w'} (ulift.{w'} (wide_pullback_shape J)) \u224c wide_pullback_shape (ulift J) :=\n(ulift_hom_ulift_category.equiv.{w' w' w w} (wide_pullback_shape J)).symm.trans\n  (equivalence_of_equiv _ (equiv.ulift.{w' w}.symm : J \u2243 ulift.{w'} J))\n\nend wide_pullback_shape\n\nnamespace wide_pushout_shape\n\nvariable {J}\n\n/-- The type of arrows for the shape indexing a wide psuhout. -/\n@[derive decidable_eq]\ninductive hom : wide_pushout_shape J \u2192 wide_pushout_shape J \u2192 Type w\n| id : \u03a0 X, hom X X\n| init : \u03a0 (j : J), hom none (some j)\n\nattribute [nolint unused_arguments] hom.decidable_eq\n\ninstance struct : category_struct (wide_pushout_shape J) :=\n{ hom := hom,\n  id := \u03bb j, hom.id j,\n  comp := \u03bb j\u2081 j\u2082 j\u2083 f g,\n  begin\n    cases f,\n      exact g,\n    cases g,\n    apply hom.init _\n  end }\n\ninstance hom.inhabited : inhabited (hom none none) := \u27e8hom.id (none : wide_pushout_shape J)\u27e9\n\nlocal attribute [tidy] tactic.case_bash\n\ninstance subsingleton_hom (j j' : wide_pushout_shape J) : subsingleton (j \u27f6 j') :=\n\u27e8by tidy\u27e9\n\ninstance category : small_category (wide_pushout_shape J) := thin_category\n\n@[simp] lemma hom_id (X : wide_pushout_shape J) : hom.id X = \ud835\udfd9 X := rfl\n\nvariables {C : Type u} [category.{v} C]\n\n/--\nConstruct a functor out of the wide pushout shape given a J-indexed collection of arrows from a\nfixed object.\n-/\n@[simps]\ndef wide_span (B : C) (objs : J \u2192 C) (arrows : \u03a0 (j : J), B \u27f6 objs j) : wide_pushout_shape J \u2964 C :=\n{ obj := \u03bb j, option.cases_on j B objs,\n  map := \u03bb X Y f,\n  begin\n    cases f with _ j,\n    { apply (\ud835\udfd9 _) },\n    { exact arrows j }\n  end,\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_) (_|_); simpa <|> simp } }\n\n/-- Every diagram is naturally isomorphic (actually, equal) to a `wide_span` -/\ndef diagram_iso_wide_span (F : wide_pushout_shape J \u2964 C) :\n  F \u2245 wide_span (F.obj none) (\u03bb j, F.obj (some j)) (\u03bb j, F.map (hom.init j)) :=\nnat_iso.of_components (\u03bb j, eq_to_iso $ by tidy) $ by tidy\n\n/-- Construct a cocone over a wide span. -/\n@[simps]\ndef mk_cocone {F : wide_pushout_shape J \u2964 C} {X : C}\n  (f : F.obj none \u27f6 X) (\u03b9 : \u03a0 j, F.obj (some j) \u27f6 X)\n  (w : \u2200 j, F.map (hom.init j) \u226b \u03b9 j = f) : cocone F :=\n{ X := X,\n  \u03b9 :=\n  { app := \u03bb j, match j with\n    | none := f\n    | (some j) := \u03b9 j\n    end,\n    naturality' := \u03bb j j' f, by { cases j; cases j'; cases f; unfold_aux; dsimp; simp [w], }, } }\n\nend wide_pushout_shape\n\nvariables (C : Type u) [category.{v} C]\n\n/-- `has_wide_pullbacks` represents a choice of wide pullback for every collection of morphisms -/\nabbreviation has_wide_pullbacks : Prop :=\n\u03a0 (J : Type w), has_limits_of_shape (wide_pullback_shape J) C\n\n/-- `has_wide_pushouts` represents a choice of wide pushout for every collection of morphisms -/\nabbreviation has_wide_pushouts : Prop :=\n\u03a0 (J : Type w), has_colimits_of_shape (wide_pushout_shape J) C\n\nvariables {C J}\n\n/-- `has_wide_pullback B objs arrows` means that `wide_cospan B objs arrows` has a limit. -/\nabbreviation has_wide_pullback (B : C) (objs : J \u2192 C)\n  (arrows : \u03a0 (j : J), objs j \u27f6 B) : Prop :=\nhas_limit (wide_pullback_shape.wide_cospan B objs arrows)\n\n/-- `has_wide_pushout B objs arrows` means that `wide_span B objs arrows` has a colimit. -/\nabbreviation has_wide_pushout (B : C) (objs : J \u2192 C)\n  (arrows : \u03a0 (j : J), B \u27f6 objs j) : Prop :=\nhas_colimit (wide_pushout_shape.wide_span B objs arrows)\n\n/-- A choice of wide pullback. -/\nnoncomputable\nabbreviation wide_pullback (B : C) (objs : J \u2192 C) (arrows : \u03a0 (j : J), objs j \u27f6 B)\n  [has_wide_pullback B objs arrows] : C :=\nlimit (wide_pullback_shape.wide_cospan B objs arrows)\n\n/-- A choice of wide pushout. -/\nnoncomputable\nabbreviation wide_pushout (B : C) (objs : J \u2192 C) (arrows : \u03a0 (j : J), B \u27f6 objs j)\n  [has_wide_pushout B objs arrows] : C :=\ncolimit (wide_pushout_shape.wide_span B objs arrows)\n\nvariable (C)\n\nnamespace wide_pullback\n\nvariables {C} {B : C} {objs : J \u2192 C} (arrows : \u03a0 (j : J), objs j \u27f6 B)\nvariables [has_wide_pullback B objs arrows]\n\n/-- The `j`-th projection from the pullback. -/\nnoncomputable\nabbreviation \u03c0 (j : J) : wide_pullback _ _ arrows \u27f6 objs j :=\nlimit.\u03c0 (wide_pullback_shape.wide_cospan _ _ _) (option.some j)\n\n/-- The unique map to the base from the pullback. -/\nnoncomputable\nabbreviation base : wide_pullback _ _ arrows \u27f6 B :=\nlimit.\u03c0 (wide_pullback_shape.wide_cospan _ _ _) option.none\n\n@[simp, reassoc]\nlemma \u03c0_arrow (j : J) : \u03c0 arrows j \u226b arrows _ = base arrows :=\nby apply limit.w (wide_pullback_shape.wide_cospan _ _ _) (wide_pullback_shape.hom.term j)\n\nvariables {arrows}\n\n/-- Lift a collection of morphisms to a morphism to the pullback. -/\nnoncomputable\nabbreviation lift {X : C} (f : X \u27f6 B) (fs : \u03a0 (j : J), X \u27f6 objs j)\n  (w : \u2200 j, fs j \u226b arrows j = f) : X \u27f6 wide_pullback _ _ arrows :=\nlimit.lift (wide_pullback_shape.wide_cospan _ _ _)\n  (wide_pullback_shape.mk_cone f fs $ by exact w)\n\nvariables (arrows)\n\nvariables {X : C} (f : X \u27f6 B) (fs : \u03a0 (j : J), X \u27f6 objs j)\n  (w : \u2200 j, fs j \u226b arrows j = f)\n\n@[simp, reassoc]\nlemma lift_\u03c0 (j : J) : lift f fs w \u226b \u03c0 arrows j = fs _ :=\nby { simp, refl }\n\n@[simp, reassoc]\nlemma lift_base : lift f fs w \u226b base arrows = f :=\nby { simp, refl }\n\nlemma eq_lift_of_comp_eq (g : X \u27f6 wide_pullback _ _ arrows) :\n  (\u2200 j : J, g \u226b \u03c0 arrows j = fs j) \u2192 g \u226b base arrows = f \u2192 g = lift f fs w :=\nbegin\n  intros h1 h2,\n  apply (limit.is_limit (wide_pullback_shape.wide_cospan B objs arrows)).uniq\n    (wide_pullback_shape.mk_cone f fs $ by exact w),\n  rintro (_|_),\n  { apply h2 },\n  { apply h1 }\nend\n\nlemma hom_eq_lift (g : X \u27f6 wide_pullback _ _ arrows) :\n  g = lift (g \u226b base arrows) (\u03bb j, g \u226b \u03c0 arrows j) (by tidy) :=\nbegin\n  apply eq_lift_of_comp_eq,\n  tidy,\nend\n\n@[ext]\nlemma hom_ext (g1 g2 : X \u27f6 wide_pullback _ _ arrows) :\n  (\u2200 j : J, g1 \u226b \u03c0 arrows j = g2 \u226b \u03c0 arrows j) \u2192\n  g1 \u226b base arrows = g2 \u226b base arrows \u2192 g1 = g2 :=\nbegin\n  intros h1 h2,\n  apply limit.hom_ext,\n  rintros (_|_),\n  { apply h2 },\n  { apply h1 },\nend\n\nend wide_pullback\n\nnamespace wide_pushout\n\nvariables {C} {B : C} {objs : J \u2192 C} (arrows : \u03a0 (j : J), B \u27f6 objs j)\nvariables [has_wide_pushout B objs arrows]\n\n/-- The `j`-th inclusion to the pushout. -/\nnoncomputable\nabbreviation \u03b9 (j : J) : objs j \u27f6 wide_pushout _ _ arrows :=\ncolimit.\u03b9 (wide_pushout_shape.wide_span _ _ _) (option.some j)\n\n/-- The unique map from the head to the pushout. -/\nnoncomputable\nabbreviation head : B \u27f6 wide_pushout B objs arrows :=\ncolimit.\u03b9 (wide_pushout_shape.wide_span _ _ _) option.none\n\n@[simp, reassoc]\nlemma arrow_\u03b9 (j : J) : arrows j \u226b \u03b9 arrows j = head arrows :=\nby apply colimit.w (wide_pushout_shape.wide_span _ _ _) (wide_pushout_shape.hom.init j)\n\nvariables {arrows}\n\n/-- Descend a collection of morphisms to a morphism from the pushout. -/\nnoncomputable\nabbreviation desc {X : C} (f : B \u27f6 X) (fs : \u03a0 (j : J), objs j \u27f6 X)\n  (w : \u2200 j, arrows j \u226b fs j = f) : wide_pushout _ _ arrows \u27f6 X :=\ncolimit.desc (wide_pushout_shape.wide_span B objs arrows)\n  (wide_pushout_shape.mk_cocone f fs $ by exact w)\n\nvariables (arrows)\n\nvariables {X : C} (f : B \u27f6 X) (fs : \u03a0 (j : J), objs j \u27f6 X)\n  (w : \u2200 j, arrows j \u226b fs j = f)\n\n@[simp, reassoc]\nlemma \u03b9_desc (j : J) : \u03b9 arrows j \u226b desc f fs w = fs _ :=\nby { simp, refl }\n\n@[simp, reassoc]\nlemma head_desc : head arrows \u226b desc f fs w = f :=\nby { simp, refl }\n\nlemma eq_desc_of_comp_eq (g : wide_pushout _ _ arrows \u27f6 X) :\n  (\u2200 j : J, \u03b9 arrows j \u226b g = fs j) \u2192 head arrows \u226b g = f \u2192 g = desc f fs w :=\nbegin\n  intros h1 h2,\n  apply (colimit.is_colimit (wide_pushout_shape.wide_span B objs arrows)).uniq\n    (wide_pushout_shape.mk_cocone f fs $ by exact w),\n  rintro (_|_),\n  { apply h2 },\n  { apply h1 }\nend\n\nlemma hom_eq_desc (g : wide_pushout _ _ arrows \u27f6 X) :\n  g = desc (head arrows \u226b g) (\u03bb j, \u03b9 arrows j \u226b g) (\u03bb j, by { rw \u2190 category.assoc, simp }) :=\nbegin\n  apply eq_desc_of_comp_eq,\n  tidy,\nend\n\n@[ext]\nlemma hom_ext (g1 g2 : wide_pushout _ _ arrows \u27f6 X) :\n  (\u2200 j : J, \u03b9 arrows j \u226b g1 = \u03b9 arrows j \u226b g2) \u2192\n  head arrows \u226b g1 = head arrows \u226b g2 \u2192 g1 = g2 :=\nbegin\n  intros h1 h2,\n  apply colimit.hom_ext,\n  rintros (_|_),\n  { apply h2 },\n  { apply h1 },\nend\n\nend wide_pushout\n\nvariable (J)\n\n/-- The action on morphisms of the obvious functor\n  `wide_pullback_shape_op : wide_pullback_shape J \u2964 (wide_pushout_shape J)\u1d52\u1d56`-/\ndef wide_pullback_shape_op_map : \u03a0 (X Y : wide_pullback_shape J),\n  (X \u27f6 Y) \u2192 ((op X : (wide_pushout_shape J)\u1d52\u1d56) \u27f6 (op Y : (wide_pushout_shape J)\u1d52\u1d56))\n| _ _ (wide_pullback_shape.hom.id X) := quiver.hom.op (wide_pushout_shape.hom.id _)\n| _ _ (wide_pullback_shape.hom.term j) := quiver.hom.op (wide_pushout_shape.hom.init _)\n\n/-- The obvious functor `wide_pullback_shape J \u2964 (wide_pushout_shape J)\u1d52\u1d56` -/\n@[simps]\ndef wide_pullback_shape_op : wide_pullback_shape J \u2964 (wide_pushout_shape J)\u1d52\u1d56 :=\n{ obj := \u03bb X, op X,\n  map := wide_pullback_shape_op_map J, }\n\n/-- The action on morphisms of the obvious functor\n`wide_pushout_shape_op : `wide_pushout_shape J \u2964 (wide_pullback_shape J)\u1d52\u1d56` -/\ndef wide_pushout_shape_op_map : \u03a0 (X Y : wide_pushout_shape J),\n  (X \u27f6 Y) \u2192 ((op X : (wide_pullback_shape J)\u1d52\u1d56) \u27f6 (op Y : (wide_pullback_shape J)\u1d52\u1d56))\n| _ _ (wide_pushout_shape.hom.id X) := quiver.hom.op (wide_pullback_shape.hom.id _)\n| _ _ (wide_pushout_shape.hom.init j) := quiver.hom.op (wide_pullback_shape.hom.term _)\n\n/-- The obvious functor `wide_pushout_shape J \u2964 (wide_pullback_shape J)\u1d52\u1d56` -/\n@[simps]\ndef wide_pushout_shape_op : wide_pushout_shape J \u2964 (wide_pullback_shape J)\u1d52\u1d56 :=\n{ obj := \u03bb X, op X,\n  map := wide_pushout_shape_op_map J, }\n\n/-- The obvious functor `(wide_pullback_shape J)\u1d52\u1d56 \u2964 wide_pushout_shape J`-/\n@[simps]\ndef wide_pullback_shape_unop : (wide_pullback_shape J)\u1d52\u1d56 \u2964 wide_pushout_shape J :=\n(wide_pullback_shape_op J).left_op\n\n/-- The obvious functor `(wide_pushout_shape J)\u1d52\u1d56 \u2964 wide_pullback_shape J` -/\n@[simps]\ndef wide_pushout_shape_unop : (wide_pushout_shape J)\u1d52\u1d56 \u2964 wide_pullback_shape J :=\n(wide_pushout_shape_op J).left_op\n\n/-- The inverse of the unit isomorphism of the equivalence\n`wide_pushout_shape_op_equiv : (wide_pushout_shape J)\u1d52\u1d56 \u224c wide_pullback_shape J` -/\ndef wide_pushout_shape_op_unop : wide_pushout_shape_unop J \u22d9 wide_pullback_shape_op J \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb X, iso.refl _) (\u03bb X Y f, dec_trivial)\n\n/-- The counit isomorphism of the equivalence\n`wide_pullback_shape_op_equiv : (wide_pullback_shape J)\u1d52\u1d56 \u224c wide_pushout_shape J` -/\ndef wide_pushout_shape_unop_op : wide_pushout_shape_op J \u22d9 wide_pullback_shape_unop J \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb X, iso.refl _) (\u03bb X Y f, dec_trivial)\n\n/-- The inverse of the unit isomorphism of the equivalence\n`wide_pullback_shape_op_equiv : (wide_pullback_shape J)\u1d52\u1d56 \u224c wide_pushout_shape J` -/\ndef wide_pullback_shape_op_unop : wide_pullback_shape_unop J \u22d9 wide_pushout_shape_op J \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb X, iso.refl _) (\u03bb X Y f, dec_trivial)\n\n/-- The counit isomorphism of the equivalence\n`wide_pushout_shape_op_equiv : (wide_pushout_shape J)\u1d52\u1d56 \u224c wide_pullback_shape J` -/\ndef wide_pullback_shape_unop_op : wide_pullback_shape_op J \u22d9 wide_pushout_shape_unop J \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb X, iso.refl _) (\u03bb X Y f, dec_trivial)\n\n/-- The duality equivalence `(wide_pushout_shape J)\u1d52\u1d56 \u224c wide_pullback_shape J` -/\n@[simps]\ndef wide_pushout_shape_op_equiv : (wide_pushout_shape J)\u1d52\u1d56 \u224c wide_pullback_shape J :=\n{ functor := wide_pushout_shape_unop J,\n  inverse := wide_pullback_shape_op J,\n  unit_iso := (wide_pushout_shape_op_unop J).symm,\n  counit_iso := wide_pullback_shape_unop_op J, }\n\n/-- The duality equivalence `(wide_pullback_shape J)\u1d52\u1d56 \u224c wide_pushout_shape J` -/\n@[simps]\ndef wide_pullback_shape_op_equiv : (wide_pullback_shape J)\u1d52\u1d56 \u224c wide_pushout_shape J :=\n{ functor := wide_pullback_shape_unop J,\n  inverse := wide_pushout_shape_op J,\n  unit_iso := (wide_pullback_shape_op_unop J).symm,\n  counit_iso := wide_pushout_shape_unop_op J, }\n\n/-- If a category has wide pullbacks on a higher universe level it also has wide pullbacks\non a lower universe level. -/\nlemma has_wide_pullbacks_shrink [has_wide_pullbacks.{max w w'} C] : has_wide_pullbacks.{w} C :=\n\u03bb J, has_limits_of_shape_of_equivalence\n  (wide_pullback_shape.equivalence_of_equiv _ equiv.ulift.{w'})\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/shapes/wide_pullbacks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3828671751749137}}
{"text": "import data.list.chain\nimport data.sigma.basic\n\nvariables {\u03b9 : Type*} {M : \u03b9 \u2192 Type*} {G : \u03b9 \u2192 Type*} {N : Type*}\nvariables [\u03a0 i, monoid (M i)] [\u03a0 i, group (G i)] [monoid N]\n\nopen list function\n\nnamespace coprod.pre\n\ndef reduced (l : list (\u03a3 i, M i)) : Prop :=\nl.chain' (\u03bb a b, a.1 \u2260 b.1) \u2227 \u2200 a : \u03a3 i, M i, a \u2208 l \u2192 a.2 \u2260 1\n\n@[simp] lemma reduced_nil : reduced ([] : list (\u03a3 i, M i)) :=\n\u27e8list.chain'_nil, \u03bb _, false.elim\u27e9\n\nlemma reduced_singleton {i : \u03a3 i, M i} (hi : i.2 \u2260 1) : reduced [i] :=\n\u27e8by simp, begin\n  cases i with i a,\n  rintros \u27e8j, b\u27e9,\n  simp only [and_imp, ne.def, mem_singleton],\n  rintro rfl h\u2082,\n  simp * at *\nend\u27e9\n\nlemma reduced_of_reduced_cons {i : \u03a3 i, M i} {l : list (\u03a3 i, M i)}\n  (h : reduced (i :: l)) : reduced l :=\n\u27e8(list.chain'_cons'.1 h.1).2, \u03bb b hb, h.2 _ (mem_cons_of_mem _ hb)\u27e9\n\nlemma reduced_cons_of_reduced_cons {i : \u03b9} {a b : M i} {l : list (\u03a3 i, M i)}\n  (h : reduced (\u27e8i, a\u27e9 :: l)) (hb : b \u2260 1) : reduced (\u27e8i, b\u27e9 :: l) :=\n\u27e8chain'_cons'.2 (chain'_cons'.1 h.1),\n  begin\n    rintros \u27e8k, c\u27e9 hk,\n    cases (mem_cons_iff _ _ _).1 hk with hk hk,\n    { simp only at hk,\n      rcases hk with \u27e8rfl, h\u27e9,\n      simp * at * },\n    { exact h.2 _ (mem_cons_of_mem _ hk) }\n  end\u27e9\n\nlemma reduced_cons_cons {i j : \u03b9} {a : M i} {b : M j}\n  {l : list (\u03a3 i, M i)} (hij : i \u2260 j) (ha : a \u2260 1)\n  (hbl : reduced (\u27e8j, b\u27e9 :: l)) : reduced (\u27e8i, a\u27e9 :: \u27e8j, b\u27e9 :: l) :=\n\u27e8chain'_cons.2 \u27e8hij, hbl.1\u27e9,\n  begin\n    rintros \u27e8k, c\u27e9 hk,\n    cases (mem_cons_iff _ _ _).1 hk with hk hk,\n    { simp only at hk,\n      rcases hk with \u27e8rfl, h\u27e9,\n      simp * at * },\n    { exact hbl.2 _ hk }\n  end\u27e9\n\nlemma reduced_reverse {l : list (\u03a3 i, M i)} (h : reduced l) : reduced l.reverse :=\n\u27e8chain'_reverse.2 $ by {convert h.1, simp [function.funext_iff, eq_comm] },\n  by simpa using h.2\u27e9\n\n@[simp] lemma reduced_reverse_iff {l : list (\u03a3 i, M i)} : reduced l.reverse \u2194 reduced l :=\n\u27e8\u03bb h, by convert reduced_reverse h; simp, reduced_reverse\u27e9\n\nlemma reduced_of_reduced_append_right : \u2200 {l\u2081 l\u2082 : list (\u03a3 i, M i)} (h : reduced (l\u2081 ++ l\u2082)),\n  reduced l\u2082\n| []      l\u2082 h := h\n| (i::l\u2081) l\u2082 h := begin\n  rw cons_append at h,\n  exact reduced_of_reduced_append_right (reduced_of_reduced_cons h)\nend\n\nlemma reduced_of_reduced_append_left {l\u2081 l\u2082 : list (\u03a3 i, M i)} (h : reduced (l\u2081 ++ l\u2082)) :\n  reduced l\u2081 :=\nbegin\n  rw [\u2190 reduced_reverse_iff],\n  rw [\u2190 reduced_reverse_iff, reverse_append] at h,\n  exact reduced_of_reduced_append_right h\nend\n\nvariables {\u03b9} [decidable_eq \u03b9] {M} [\u03a0 i, decidable_eq (M i)]\n\ndef rcons : (\u03a3 i, M i) \u2192 list (\u03a3 i, M i) \u2192 list (\u03a3 i, M i)\n| i []     := [i]\n| i (j::l) :=\n  if hij : i.1 = j.1\n    then let c := i.2 * cast (congr_arg M hij).symm j.2 in\n      if c = 1\n        then l\n        else \u27e8i.1, c\u27e9 :: l\n    else i::j::l\n\ndef reduce : list (\u03a3 i, M i) \u2192 list (\u03a3 i, M i)\n| []       := []\n| (i :: l) := if i.2 = 1 then reduce l else rcons i (reduce l)\n\n@[simp] lemma reduce_nil : reduce ([] : list (\u03a3 i, M i)) = [] := rfl\n\nlemma reduce_cons (i : \u03a3 i, M i) (l : list (\u03a3 i, M i)) :\n  reduce (i::l) = if i.2 = 1 then reduce l else rcons i (reduce l) := rfl\n\nlemma reduced_rcons : \u2200 {i : \u03a3 i, M i} {l : list (\u03a3 i, M i)},\n  i.2 \u2260 1 \u2192 reduced l \u2192 reduced (rcons i l)\n| \u27e8i, a\u27e9 []            hi h := \u27e8list.chain'_singleton _,\n  begin\n    rintros \u27e8j, b\u27e9 hj,\n    simp only [rcons, list.mem_singleton] at hj,\n    rcases hj with \u27e8rfl, h\u27e9,\n    simp * at *\n  end\u27e9\n| \u27e8i, a\u27e9 (\u27e8j, b\u27e9 :: l) hi h := begin\n  simp [rcons],\n  split_ifs,\n  { exact reduced_of_reduced_cons h },\n  { dsimp only at h_1,\n    subst h_1,\n    exact reduced_cons_of_reduced_cons h h_2 },\n  { exact reduced_cons_cons h_1 hi h }\nend\n\nlemma reduced_reduce : \u2200 l : list (\u03a3 i, M i), reduced (reduce l)\n| []     := reduced_nil\n| (a::l) := begin\n  rw reduce,\n  split_ifs,\n  { exact reduced_reduce l },\n  { exact reduced_rcons h (reduced_reduce l) }\nend\n\nlemma rcons_eq_cons : \u2200 {i : \u03a3 i, M i} {l : list (\u03a3 i, M i)},\n  reduced (i :: l) \u2192 rcons i l = i :: l\n| i []     h := rfl\n| i (j::l) h := dif_neg (chain'_cons.1 h.1).1\n\nlemma rcons_reduce_eq_reduce_cons : \u2200 {i : \u03a3 i, M i} {l : list (\u03a3 i, M i)},\n  i.2 \u2260 1 \u2192 rcons i (reduce l) = reduce (i :: l)\n| a []     ha := by simp [rcons, reduce, ha]\n| a (b::l) ha := begin\n  rw [reduce],\n  split_ifs,\n  { rw [reduce, if_neg ha, reduce, if_pos h] },\n  { rw [reduce, if_neg ha, reduce, if_neg h] }\nend\n\nlemma reduce_eq_self_of_reduced : \u2200 {l : list (\u03a3 i, M i)}, reduced l \u2192 reduce l = l\n| []     h := rfl\n| (a::l) h := by rw [\u2190 rcons_reduce_eq_reduce_cons (h.2 a (mem_cons_self _ _)),\n    reduce_eq_self_of_reduced (reduced_of_reduced_cons h), rcons_eq_cons h]\n\nlemma rcons_eq_reduce_cons {i : \u03a3 i, M i} {l : list (\u03a3 i, M i)}\n  (ha : i.2 \u2260 1) (hl : reduced l) : rcons i l = reduce (i :: l) :=\nby rw [\u2190 rcons_reduce_eq_reduce_cons ha, reduce_eq_self_of_reduced hl]\n\n@[simp] lemma reduce_reduce (l : list (\u03a3 i, M i)) : reduce (reduce l) = reduce l :=\nreduce_eq_self_of_reduced (reduced_reduce l)\n\n@[simp] lemma reduce_cons_reduce_eq_reduce_cons (i : \u03a3 i, M i) (l : list (\u03a3 i, M i)) :\n  reduce (i :: reduce l) = reduce (i :: l)  :=\nif ha : i.2 = 1 then by rw [reduce, if_pos ha, reduce, if_pos ha, reduce_reduce]\nelse by rw [\u2190 rcons_reduce_eq_reduce_cons ha, \u2190 rcons_reduce_eq_reduce_cons ha,\n    reduce_reduce]\n\nlemma length_rcons_le : \u2200 (i : \u03a3 i, M i) (l : list (\u03a3 i, M i)),\n  (rcons i l).length \u2264 (i::l : list _).length\n| i      []          := le_refl _\n| \u27e8i, a\u27e9 (\u27e8j, b\u27e9::l) := begin\n  simp [rcons],\n  split_ifs,\n  { repeat { constructor } },\n  { simp },\n  { simp }\nend\n\nlemma length_reduce_le : \u2200 (l : list (\u03a3 i, M i)),\n  (reduce l).length \u2264 l.length\n| []        := le_refl _\n| [a]       := by { simp [reduce], split_ifs; simp [rcons] }\n| (a::b::l) := begin\n  simp only [reduce, rcons],\n  split_ifs,\n  { exact le_trans (length_reduce_le _)\n      (le_trans (nat.le_succ _) (nat.le_succ _)) },\n  { exact le_trans (length_rcons_le _ _) (nat.succ_le_succ\n      (le_trans (length_reduce_le _) (nat.le_succ _))) },\n  { exact le_trans (length_rcons_le _ _) (nat.succ_le_succ\n      (le_trans (length_reduce_le _) (nat.le_succ _))) },\n  { exact le_trans (length_rcons_le _ _) (nat.succ_le_succ\n         (le_trans (length_rcons_le _ _) (nat.succ_le_succ\n           (length_reduce_le _)))) }\nend\n\nlemma length_rcons_lt_or_eq_rcons : \u2200 (i : \u03a3 i, M i) (l : list (\u03a3 i, M i)),\n  (rcons i l).length < (i :: l : list _).length \u2228 rcons i l = (i::l)\n| i [] := or.inr rfl\n| i (j::l) := begin\n  simp only [rcons],\n  split_ifs,\n  { exact or.inl (nat.lt_succ_of_le (nat.le_succ _)) },\n  { exact or.inl (nat.lt_succ_self _) },\n  { simp }\nend\n\nlemma length_reduce_lt_or_eq_reduce : \u2200 (l : list (\u03a3 i, M i)),\n  (reduce l).length < l.length \u2228 reduce l = l\n| []        := or.inr rfl\n| (i::l)    := begin\n  simp only [reduce],\n  split_ifs,\n  { exact or.inl (nat.lt_succ_of_le (length_reduce_le _)) },\n  { cases length_rcons_lt_or_eq_rcons i (reduce l) with h h,\n    { exact or.inl (lt_of_lt_of_le h (nat.succ_le_succ (length_reduce_le _))) },\n    { rw h,\n      cases length_reduce_lt_or_eq_reduce l with h h,\n      { exact or.inl (nat.succ_lt_succ h) },\n      { rw h, right, refl } } }\nend\n\nlemma rcons_append : \u2200 {i j : \u03a3 i, M i} {l\u2081 l\u2082 : list (\u03a3 i, M i)},\n  rcons i ((j::l\u2081) ++ l\u2082) = rcons i (j::l\u2081) ++ l\u2082\n| i j [] l\u2082 := begin\n  simp [rcons], split_ifs; simp\nend\n| a b (c::l\u2081) l\u2082 := begin\n  rw [cons_append, rcons],\n  dsimp,\n  split_ifs,\n  { simp [rcons, *] },\n  { simp [rcons, *] },\n  { simp [rcons, *] }\nend\n\nlemma rcons_rcons_of_mul_eq_one {i : \u03b9} {a b : M i} : \u2200 {l : list (\u03a3 i, M i)},\n  a * b = 1 \u2192 reduced l \u2192 rcons \u27e8i, a\u27e9 (rcons \u27e8i, b\u27e9 l) = l\n| []          hab hl := by simp [rcons, cast, hab]\n| (\u27e8j, c\u27e9::l) hab hl := begin\n  simp only [rcons],\n  split_ifs,\n  { dsimp only at h,\n    subst h,\n    rw [\u2190 rcons_eq_cons hl, left_inv_eq_right_inv hab h_1, cast_eq] },\n  { dsimp only at h,\n    subst h,\n    simp only [rcons, dif_pos rfl],\n    rw [cast_eq, cast_eq, if_neg, \u2190 mul_assoc, hab, one_mul],\n    { rw [\u2190 mul_assoc, hab, one_mul],\n      exact hl.2 \u27e8i, c\u27e9 (mem_cons_self _ _) } },\n  { rw [rcons, dif_pos rfl, cast_eq], dsimp, rw [if_pos hab] }\nend\n\nlemma rcons_rcons_of_mul_ne_one {i : \u03b9} {a b : M i} : \u2200 {l : list (\u03a3 i, M i)},\n  a * b \u2260 1 \u2192 a \u2260 1 \u2192 reduced l \u2192 rcons \u27e8i, a\u27e9 (rcons \u27e8i, b\u27e9 l) = rcons \u27e8i, a * b\u27e9 l\n| []          hab ha hl := by simp [rcons, hab]\n| [\u27e8j, c\u27e9]    hab ha hl := begin\n  simp only [rcons],\n  split_ifs,\n  { rw [mul_assoc, h_1, mul_one] at h_2,\n    exact (ha h_2).elim },\n  { simp [rcons, mul_assoc, h_1] },\n  { simp only [rcons, \u2190 mul_assoc, *, dif_pos rfl, if_pos rfl, cast_eq] },\n  { dsimp only at h,\n    subst h,\n    simp only [rcons, dif_pos rfl, \u2190 mul_assoc, cast_eq, *] at *,\n    simp, },\n  { simp [rcons, if_neg hab, if_pos rfl] }\nend\n| (\u27e8j, c\u27e9::\u27e8k, d\u27e9::l) hab ha hl := begin\n  have hjk : j \u2260 k, from (chain'_cons.1 hl.1).1,\n  dsimp only [rcons],\n  split_ifs,\n  { rw [mul_assoc, h_1, mul_one] at h_2,\n    exact (ha h_2).elim },\n  { dsimp [rcons],\n    subst h,\n    simp [*, rcons, mul_assoc] at * },\n  { simp [*, rcons, \u2190 mul_assoc] at * },\n  { simp [*, rcons, \u2190 mul_assoc] },\n  { simp [*, rcons] }\nend\n\nlemma reduce_rcons : \u2200 {i : \u03a3 i, M i} (l : list (\u03a3 i, M i)), i.2 \u2260 1 \u2192\n  reduce (rcons i l) = rcons i (reduce l)\n| i []               hi := by simp [rcons, reduce, hi]\n| \u27e8i, a\u27e9 [\u27e8j, b\u27e9]    ha := begin\n    replace ha : a \u2260 1 := ha,\n    dsimp only [reduce, rcons],\n    by_cases hij : i = j,\n    { subst hij,\n      split_ifs;\n      simp [*, reduce, rcons] at * },\n    { simp [hij, reduce, rcons, ha] }\n  end\n| \u27e8i, a\u27e9 (\u27e8j, b\u27e9::l) ha := begin\n  dsimp only [rcons],\n  split_ifs,\n  { subst h,\n    rw [cast_eq] at h_1,\n    rw [reduce, if_neg, rcons_rcons_of_mul_eq_one h_1 (reduced_reduce _)],\n    { refine \u03bb hb : b = 1, _,\n      rw [hb, mul_one] at h_1,\n      exact ha h_1 } },\n  { subst h,\n    rw [reduce, if_neg h_1, reduce],\n    split_ifs,\n    { erw [cast_eq, show b = 1, from h, mul_one] },\n    { rw cast_eq at h_1,\n      rw [rcons_rcons_of_mul_ne_one h_1 ha (reduced_reduce l), cast_eq], } },\n  { rw [rcons_eq_reduce_cons ha (reduced_reduce _), reduce_cons_reduce_eq_reduce_cons] }\nend\n\nlemma reduce_cons_cons_of_mul_eq_one {l : list (\u03a3 i, M i)} {i : \u03b9} {a b : M i}\n  (ha : a \u2260 1) (hb : b \u2260 1) (hab : a * b = 1) : reduce (\u27e8i, a\u27e9 :: \u27e8i, b\u27e9 :: l) = reduce l :=\nby rw [reduce, if_neg ha, reduce, if_neg hb, rcons_rcons_of_mul_eq_one hab (reduced_reduce _)]\n\nlemma reduce_cons_cons_of_mul_ne_one {l : list (\u03a3 i, M i)} {i : \u03b9} {a b : M i}\n  (ha : a \u2260 1) (hab : a * b \u2260 1) : reduce (\u27e8i, a\u27e9 :: \u27e8i, b\u27e9 :: l) = reduce (\u27e8i, a * b\u27e9 :: l) :=\nbegin\n  rw [reduce, if_neg ha, reduce],\n  split_ifs,\n  { rw [rcons_eq_reduce_cons (show (\u27e8i, a\u27e9 : \u03a3 i, M i).snd \u2260 1, from ha) (reduced_reduce _),\n      show b = 1, from h, mul_one, reduce_cons_reduce_eq_reduce_cons] },\n  { rw [rcons_rcons_of_mul_ne_one hab ha (reduced_reduce _),\n      rcons_eq_reduce_cons (show (\u27e8i, a * b\u27e9 : \u03a3 i, M i).snd \u2260 1, from hab : _) (reduced_reduce _),\n      reduce_cons_reduce_eq_reduce_cons] }\nend\n\n@[simp] lemma reduce_reduce_append_eq_reduce_append : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i)),\n  reduce (reduce l\u2081 ++ l\u2082) = reduce (l\u2081 ++ l\u2082)\n| []         l\u2082 := rfl\n| (a::l\u2081) l\u2082 := begin\n  simp only [reduce, cons_append],\n  split_ifs with ha ha,\n  { exact reduce_reduce_append_eq_reduce_append _ _ },\n  { rw [\u2190 reduce_reduce_append_eq_reduce_append l\u2081 l\u2082],\n    induction h : reduce l\u2081,\n    { simp [rcons, rcons_eq_reduce_cons ha (reduced_reduce _)] },\n    { rw [\u2190 rcons_append, reduce_rcons _ ha] } }\nend\n\n@[simp] lemma reduce_append_reduce_eq_reduce_append : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i)),\n  reduce (l\u2081 ++ reduce l\u2082) = reduce (l\u2081 ++ l\u2082)\n| []      l\u2082 := by simp\n| (a::l\u2081) l\u2082 := by rw [cons_append, \u2190 reduce_cons_reduce_eq_reduce_cons,\n    reduce_append_reduce_eq_reduce_append,\n    reduce_cons_reduce_eq_reduce_cons, cons_append]\n\nlemma reduced_iff_reduce_eq_self {l : list (\u03a3 i, M i)} :\n  reduced l \u2194 reduce l = l :=\n\u27e8reduce_eq_self_of_reduced, \u03bb h, h \u25b8 reduced_reduce l\u27e9\n\nlemma reduced_append_overlap {l\u2081 l\u2082 l\u2083 : list (\u03a3 i, M i)}\n  (h\u2081 : reduced (l\u2081 ++ l\u2082)) (h\u2082 : reduced (l\u2082 ++ l\u2083)) (hn : l\u2082 \u2260 []):\n  reduced (l\u2081 ++ l\u2082 ++ l\u2083) :=\n\u27e8chain'.append_overlap h\u2081.1 h\u2082.1 hn,\n  \u03bb i hi, (mem_append.1 hi).elim (h\u2081.2 _) (\u03bb hi, h\u2082.2 _ (mem_append_right _ hi))\u27e9\n\n/-- `mul_aux` returns `reduce (l\u2081.reverse ++ l\u2082)` -/\n@[simp] def mul_aux : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)), list (\u03a3 i, M i)\n| []      l\u2082      := l\u2082\n| (i::l\u2081) []      := reverse (i :: l\u2081)\n| (i::l\u2081) (j::l\u2082) :=\n  if hij : i.1 = j.1\n    then let c := i.2 * cast (congr_arg M hij).symm j.2 in\n      if c = 1\n        then mul_aux l\u2081 l\u2082\n        else l\u2081.reverse_core (\u27e8i.1, c\u27e9::l\u2082)\n    else l\u2081.reverse_core (i::j::l\u2082)\n\nlocal attribute [simp] reverse_core_eq\n\n@[simp] def mul_aux' : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)), list (\u03a3 i, M i)\n| []      l\u2082      := l\u2082\n| (i::l\u2081) []      := reverse (i :: l\u2081)\n| (i::l\u2081) (j::l\u2082) :=\n  if hij : i.1 = j.1\n    then let c := i.2 * cast (congr_arg M hij).symm j.2 in\n      if c = 1\n        then mul_aux' l\u2081 l\u2082\n        else mul_aux' l\u2081 (\u27e8i.1, c\u27e9::l\u2082)\n    else mul_aux' l\u2081 (i::j::l\u2082)\n\nlemma mul_aux'_eq_append : \u03a0 {l\u2081 l\u2082 : list (\u03a3 i, M i)},\n  reduced (l\u2081.reverse ++ l\u2082) \u2192 mul_aux' l\u2081 l\u2082 = l\u2081.reverse ++ l\u2082\n| []      l\u2082      h := rfl\n| (i::l\u2081) []      h := by simp\n| (i::l\u2081) (j::l\u2082) h := begin\n  rw [mul_aux'],\n  have hij : i.fst \u2260 j.fst,\n  { rw [reduced, chain'_split, \u2190 reverse_cons, chain'_reverse, chain'_cons] at h,\n    simp [flip] at h,\n    tauto },\n  rw [dif_neg hij, mul_aux'_eq_append];\n  simp * at *\nend\n\nlemma mul_aux_eq_mul_aux' : \u03a0 {l\u2081 l\u2082 : list (\u03a3 i, M i)}\n  (h\u2081 : reduced l\u2081) (h\u2082 : reduced l\u2082),\n  mul_aux l\u2081 l\u2082 = mul_aux' l\u2081 l\u2082\n| []      l\u2082      _ _ := rfl\n| (i::l\u2081) []      _ _ := rfl\n| [i]     (j::l\u2082) _ _ :=\n  begin\n    unfold mul_aux mul_aux',\n    split_ifs; simp\n  end\n| (i::j::l\u2081) (k::l\u2082) h\u2081 h\u2082:=\n  begin\n    unfold mul_aux mul_aux',\n    have hij : j.fst \u2260 i.fst,\n      by simp [reduced] at h\u2081; tauto,\n    simp only [dif_neg hij],\n    split_ifs,\n    { rw mul_aux_eq_mul_aux' (reduced_of_reduced_cons h\u2081)\n        (reduced_of_reduced_cons h\u2082) },\n    { cases i with i a, cases k with k b,\n      dsimp only at h, subst h,\n      rw [reverse_core_eq, \u2190 mul_aux_eq_mul_aux' (reduced_of_reduced_cons h\u2081)\n        (reduced_cons_of_reduced_cons h\u2082 h_1), mul_aux, dif_neg hij],\n      simp },\n    { have : reduced (l\u2081.reverse ++ j :: i :: k :: l\u2082),\n      { suffices : reduced ((j :: l\u2081).reverse ++ [i, k] ++ l\u2082),\n        { simpa },\n        cases i with i a, cases j with j b, cases k with k c,\n        have hc1 : c \u2260 1, from h\u2082.2 \u27e8k, c\u27e9 (mem_cons_self _ _),\n        have ha1 : a \u2260 1, from h\u2081.2 \u27e8i, a\u27e9 (mem_cons_self _ _),\n        refine reduced_append_overlap _ _ (cons_ne_nil _ _),\n        { suffices : reduced (\u27e8k, c\u27e9 :: \u27e8i, a\u27e9 :: \u27e8j, b\u27e9 :: l\u2081),\n          { rw [\u2190 reduced_reverse_iff],\n            simpa },\n          exact reduced_cons_cons (ne.symm h) hc1 h\u2081 },\n        { suffices : reduced (\u27e8i, a\u27e9 :: \u27e8k, c\u27e9 :: l\u2082),\n          { simpa },\n          exact reduced_cons_cons h ha1 h\u2082 } },\n      rw [mul_aux'_eq_append this],\n      simp }\n  end\n\n@[simp] lemma mul_aux'_nil (l : list (\u03a3 i, M i)) : mul_aux' l [] = l.reverse :=\nby cases l; simp\n\nlemma mul_aux'_single : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux' l\u2081 (mul_aux' l\u2082.reverse [i]) = mul_aux' (mul_aux' l\u2081 l\u2082).reverse [i]\n| [] l\u2082      i := by simp\n| (j::l\u2081) [] i := by simp\n| (\u27e8j, b\u27e9::l\u2081) (\u27e8k, c\u27e9::l\u2082) \u27e8i, a\u27e9 :=\n  list.reverse_rec_on l\u2082\n    begin\n\n    end\n    (begin\n      rintros l\u2082 \u27e8m, d\u27e9 ih,\n      simp at *, dsimp at *,\n      split_ifs at *,\n\n\n    end)\n\nlemma mul_aux'_cons : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux' (rcons i l\u2081) l\u2082 = mul_aux' l\u2081 (rcons i l\u2082)\n| [] l\u2082      i := by simp [rcons]; admit\n| (j::l\u2081) [] i := begin\n  simp [rcons, eq_comm],\n  split_ifs,\n  { refl },\n\n\nend\n\nlemma mul_aux'_single : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux' l\u2081 (mul_aux' [i] l\u2082) = mul_aux' (mul_aux' l\u2081 [i]).reverse l\u2082\n| [] l\u2082      i := by simp\n| (j::l\u2081) [] i := by simp\n| (\u27e8j, b\u27e9::l\u2081) (\u27e8k, c\u27e9::l\u2082) \u27e8i, a\u27e9 := begin\n  rw [mul_aux', mul_aux'],\n  dsimp,\n  split_ifs,\n  { simp, }\n\nend\n\n\nlemma mul_aux'_single : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux' l\u2081 (mul_aux' l\u2082 [i]) = mul_aux' (mul_aux' l\u2081 l\u2082.reverse).reverse [i]\n| [] l\u2082      i := by simp\n| (j::l\u2081) [] i := by simp\n| (\u27e8j, b\u27e9::l\u2081) (\u27e8k, c\u27e9::l\u2082) \u27e8i, a\u27e9 :=\n  list.reverse_rec_on l\u2082\n\n@[simp] lemma mul_aux_nil (l : list (\u03a3 i, M i)) : mul_aux l [] = l.reverse :=\nby cases l; refl\n\n@[simp] lemma nil_mul_aux (l : list (\u03a3 i, M i)) : mul_aux [] l = l := rfl\n\nlemma mul_aux_single_reverse : \u2200 (l : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux l [i] = mul_aux [i] l\n| [] i := by simp\n| (i::l) j := list.reverse_rec_on l\n  (by simp [mul_aux, reverse_core_eq]; split_ifs; simp)\n  _\n\n\nlemma mul_aux_single : \u03a0 (l\u2081 l\u2082 : list (\u03a3 i, M i)) (i : \u03a3 i, M i),\n  mul_aux l\u2081 (mul_aux l\u2082.reverse [i]) = mul_aux (mul_aux l\u2081 l\u2082).reverse [i]\n| [] l\u2082      i := by simp [mul_aux]\n| (j::l\u2081) [] i := by simp [mul_aux]\n| (j::l\u2081) (k::l\u2082) i := begin\n  simp only [mul_aux],\n  split_ifs,\n  { rw \u2190 mul_aux_single, sorry },\n  { simp [reverse_core_eq], }\n\nend\n\nlemma mul_aux_eq_reduce_append : \u2200 {l\u2081 l\u2082: list (\u03a3 i, M i)},\n  reduced l\u2081 \u2192 reduced l\u2082 \u2192 mul_aux l\u2081 l\u2082 = reduce (l\u2081.reverse ++ l\u2082)\n| []          l\u2082          := \u03bb h\u2081 h\u2082,\n  by clear_aux_decl; simp [mul_aux, reduce_eq_self_of_reduced, *]\n| (i::hd)     []          := \u03bb h\u2081 h\u2082,\n  by rw [mul_aux, append_nil, reduce_eq_self_of_reduced (reduced_reverse h\u2081)]\n| (\u27e8i,a\u27e9::l\u2081) (\u27e8j,b\u27e9::l\u2082) := \u03bb h\u2081 h\u2082,\n  begin\n    simp only [mul_aux],\n    dsimp only,\n    have ha : a \u2260 1, from h\u2081.2 \u27e8i, a\u27e9 (by simp),\n    have hb : b \u2260 1, from h\u2082.2 \u27e8j, b\u27e9 (list.mem_cons_self _ _),\n    rcases decidable.em (i = j) with \u27e8rfl, hij\u27e9,\n    { rw [dif_pos rfl, cast_eq],\n      split_ifs,\n      { have hrl\u2081 : reduced l\u2081,\n        { exact reduced_of_reduced_cons h\u2081 },\n        have hrl\u2082 : reduced l\u2082, from reduced_of_reduced_cons h\u2082,\n        rw [mul_aux_eq_reduce_append hrl\u2081 hrl\u2082, reverse_cons, append_assoc,\n          cons_append, nil_append, \u2190 reduce_append_reduce_eq_reduce_append _ (_ :: _),\n          reduce_cons_cons_of_mul_eq_one ha hb h_1,\n          reduce_append_reduce_eq_reduce_append] },\n      { have hrl\u2081 :reduced (l\u2081.reverse ++ [\u27e8i, a * b\u27e9]),\n        { rw [\u2190 reduced_reverse_iff, reverse_append, reverse_reverse,\n            reverse_singleton, singleton_append],\n          exact reduced_cons_of_reduced_cons h\u2081 h_1 },\n        have hrl\u2082 :reduced ([\u27e8i, a * b\u27e9] ++ l\u2082),\n        { rw [singleton_append],\n          exact reduced_cons_of_reduced_cons h\u2082 h_1 },\n        simp only [reverse_cons, singleton_append, append_assoc, reverse_core_eq],\n        rw [\u2190 reduce_append_reduce_eq_reduce_append,\n          reduce_cons_cons_of_mul_ne_one ha h_1,\n          reduce_append_reduce_eq_reduce_append, \u2190 singleton_append, \u2190 append_assoc],\n        exact (reduce_eq_self_of_reduced\n          (reduced_append_overlap hrl\u2081 hrl\u2082 (by simp))).symm } },\n    { suffices : reduce (l\u2081.reverse ++ [\u27e8i, a\u27e9, \u27e8j, b\u27e9] ++ l\u2082) =\n        l\u2081.reverse ++ [\u27e8i, a\u27e9, \u27e8j, b\u27e9] ++ l\u2082,\n      { simpa [eq_comm, dif_neg h, reverse_core_eq] },\n      have hrl\u2081 : reduced (l\u2081.reverse ++ [\u27e8i, a\u27e9, \u27e8j, b\u27e9]),\n      { rw [\u2190 reduced_reverse_iff],\n        simp only [reverse_append, reverse_cons, cons_append, reverse_nil, nil_append,\n          reverse_reverse],\n        refine reduced_cons_cons (ne.symm h) hb h\u2081 },\n      have hrl\u2082 : reduced ([\u27e8i, a\u27e9, \u27e8j, b\u27e9] ++ l\u2082),\n      { simp only [cons_append, nil_append],\n        refine reduced_cons_cons h ha h\u2082 },\n      exact reduce_eq_self_of_reduced (reduced_append_overlap hrl\u2081 hrl\u2082 (by simp)) }\n  end\n\nprotected def mul (l\u2081 l\u2082 : list (\u03a3 i, M i)) : list (\u03a3 i, M i) :=\nmul_aux l\u2081.reverse l\u2082\n\nlemma mul_eq_reduce_append {l\u2081 l\u2082 : list (\u03a3 i, M i)} (h\u2081 : reduced l\u2081) (h\u2082 : reduced l\u2082) :\n  coprod.pre.mul l\u2081 l\u2082 = reduce (l\u2081 ++ l\u2082) :=\nby rw [coprod.pre.mul, mul_aux_eq_reduce_append (reduced_reverse h\u2081) h\u2082, reverse_reverse]\n\nlemma reduced_mul {l\u2081 l\u2082 : list (\u03a3 i, M i)} (h\u2081 : reduced l\u2081) (h\u2082 : reduced l\u2082) :\n  reduced (coprod.pre.mul l\u2081 l\u2082) :=\n(mul_eq_reduce_append h\u2081 h\u2082).symm \u25b8 reduced_reduce _\n\nprotected lemma mul_assoc {l\u2081 l\u2082 l\u2083 : list (\u03a3 i, M i)} (h\u2081 : reduced l\u2081) (h\u2082 : reduced l\u2082)\n  (h\u2083 : reduced l\u2083) : pre.mul (pre.mul l\u2081 l\u2082) l\u2083 = pre.mul l\u2081 (pre.mul l\u2082 l\u2083) :=\nbegin\n  rw [mul_eq_reduce_append (reduced_mul h\u2081 h\u2082) h\u2083, mul_eq_reduce_append h\u2081 h\u2082,\n    mul_eq_reduce_append h\u2082 h\u2083, mul_eq_reduce_append h\u2081 (reduced_reduce _)],\n  simp [append_assoc]\nend\n\nprotected lemma one_mul (l : list (\u03a3 i, M i)) : pre.mul [] l = l := rfl\n\nprotected lemma mul_one {l : list (\u03a3 i, M i)} (h : reduced l) : pre.mul l [] = l :=\nby rw [mul_eq_reduce_append h reduced_nil, append_nil, reduce_eq_self_of_reduced h]\n\nsection lift\n\nvariable (f : \u03a0 i, M i \u2192* N)\n\ndef lift (l : list (\u03a3 i, M i)) : N :=\nl.foldl (\u03bb n i, n * f i.1 i.2) 1\n\nlemma lift_eq_map_prod (l : list (\u03a3 i, M i)) :\n  lift f l = (l.map (\u03bb i : \u03a3 i, M i, f i.1 i.2)).prod :=\nbegin\n  rw [lift, \u2190 one_mul (l.map _).prod],\n  generalize h : (1 : N) = n, clear h,\n  induction l with i l ih generalizing n,\n  { simp },\n  { rw [foldl_cons, ih, map_cons, prod_cons, mul_assoc] }\nend\n\nlemma map_prod_mul_aux : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i)),\n  ((mul_aux l\u2081 l\u2082).map (\u03bb i : \u03a3 i, M i, f i.1 i.2)).prod =\n  (l\u2081.reverse.map (\u03bb i : \u03a3 i, M i, f i.1 i.2)).prod *\n  (l\u2082.map (\u03bb i : \u03a3 i, M i, f i.1 i.2)).prod\n| []      l\u2082      := by simp [mul_aux]\n| (i::l\u2081) []      := by simp [mul_aux]\n| (\u27e8i,a\u27e9::l\u2081) (\u27e8j,b\u27e9::l\u2082) := begin\n  rw [mul_aux],\n  split_ifs,\n  { dsimp only at h,\n    subst h,\n    rw [cast_eq] at h_1,\n    simp only [prod_nil, mul_one, reverse_cons, map, prod_cons, prod_append,\n        map_append, map_reverse, mul_assoc],\n    rw [\u2190 mul_assoc (f _ _), \u2190 monoid_hom.map_mul, h_1],\n    simp [map_prod_mul_aux] },\n  { dsimp only at h,\n    subst h,\n    simp [reverse_core_eq, mul_assoc] },\n  { simp [reverse_core_eq, mul_assoc] }\nend\n\nlemma lift_mul (l\u2081 l\u2082 : list (\u03a3 i, M i)) : lift f (pre.mul l\u2081 l\u2082) = lift f l\u2081 * lift f l\u2082 :=\nby simp [pre.mul, lift_eq_map_prod, map_prod_mul_aux]\n\nend lift\n\nsection of\nvariables (i : \u03b9) (a b : M i)\n\ndef of (i : \u03b9) (a : M i) : list (\u03a3 i, M i) :=\nif a = 1 then [] else [\u27e8i, a\u27e9]\n\nlemma reduced_of (i : \u03b9) (a : M i) : reduced (of i a) :=\nbegin\n  rw of,\n  split_ifs,\n  { simp },\n  { exact reduced_singleton h }\nend\n\nlemma of_one : of i (1 : M i) = [] := if_pos rfl\n\nlemma of_mul : of i (a * b) = pre.mul (of i a) (of i b) :=\nbegin\n  simp only [of, pre.mul, mul_aux],\n  split_ifs; simp [mul_aux, *, reverse_core_eq] at *\nend\n\nlemma lift_of (f : \u03a0 i, M i \u2192* N) : lift f (of i a) = f i a :=\nbegin\n  simp [lift, of],\n  split_ifs;\n  simp *\nend\n\nend of\n\nsection embedding\nvariables {\u03ba : Type*} {O : \u03ba \u2192 Type*} [\u03a0 i, monoid (O i)]\nvariables (f : \u03b9 \u2192 \u03ba) (hf : injective f)\n  (g : \u03a0 i, M i \u2192* O (f i)) (hg : \u2200 i a, g i a = 1 \u2192 a = 1)\n\nprotected def embedding (l : list (\u03a3 i, M i)) : list \u03a3 i, O i :=\nl.map (\u03bb i, \u27e8f i.1, g i.1 i.2\u27e9)\n\ninclude hf hg\n\nvariables [decidable_eq \u03ba] [\u03a0 i, decidable_eq (O i)]\n\nlemma embedding_mul_aux : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i)),\n  pre.embedding f g (mul_aux l\u2081 l\u2082) = mul_aux (pre.embedding f g l\u2081) (pre.embedding f g l\u2082)\n| []      l\u2082      := rfl\n| (i::l\u2081) []      := by simp [pre.embedding, mul_aux]\n| (\u27e8i,a\u27e9::l\u2081) (\u27e8j, b\u27e9::l\u2082) :=\n  begin\n    rw [mul_aux],\n    split_ifs,\n    { dsimp only at h,\n      subst h,\n      have : g i a * g i b = 1,\n      { erw [\u2190 monoid_hom.map_mul, h_1, monoid_hom.map_one] },\n      rw [embedding_mul_aux],\n      simp [pre.embedding, mul_aux, this] },\n    { dsimp only at h,\n      subst h,\n      have : g i a * g i b \u2260 1,\n      { rw [\u2190 monoid_hom.map_mul],\n        exact mt (hg i (a * b)) h_1 },\n      simp [pre.embedding, mul_aux, this, reverse_core_eq] },\n    { dsimp only at h,\n      simp [pre.embedding, mul_aux, reverse_core_eq, hf.eq_iff, h] }\n  end\n\nlemma embedding_mul {l\u2081 l\u2082 : list (\u03a3 i, M i)} :\n  pre.embedding f g (pre.mul l\u2081 l\u2082) = pre.mul (pre.embedding f g l\u2081) (pre.embedding f g l\u2082) :=\nbegin\n  simp [pre.mul, embedding_mul_aux _ hf _ hg],\n  simp [pre.embedding]\nend\n\nlemma reduced_embedding {l : list (\u03a3 i, M i)} (hl : reduced l) :\n  reduced (pre.embedding f g l) :=\n\u27e8by simp [pre.embedding, list.chain'_map, hf.eq_iff, hl.1],\n  begin\n    simp only [pre.embedding, mem_map, and_imp, sigma.forall],\n    rintros i a \u27e8\u27e8j, b\u27e9, hjb, rfl, h\u27e9,\n    rw [heq_iff_eq] at h,\n    subst a,\n    exact mt (hg j b) (hl.2 _ hjb)\n  end\u27e9\n\nend embedding\n\nsection inv\n\nvariable [\u03a0 i, decidable_eq (G i)]\n\nprotected def inv (l : list (\u03a3 i, G i)) : list (\u03a3 i, G i) :=\nlist.reverse (l.map (\u03bb i : \u03a3 i, G i, \u27e8i.1, i.2\u207b\u00b9\u27e9))\n\nlemma reduced_inv (l : list (\u03a3 i, G i)) (hl : reduced l) :\n  reduced (pre.inv l) :=\n\u27e8list.chain'_reverse.2 ((list.chain'_map _).2 $\n  by { convert hl.1, simp [function.funext_iff, eq_comm, flip] }),\nbegin\n  rintros \u27e8i, a\u27e9 hi,\n  rw [pre.inv, mem_reverse, mem_map] at hi,\n  rcases hi with \u27e8\u27e8j, b\u27e9, hjl, h\u27e9,\n  simp only at h,\n  cases h with hij hba,\n  subst hij,\n  convert inv_ne_one.2 (hl.2 \u27e8j, b\u27e9 hjl),\n  simp * at *\nend\u27e9\n\nprotected lemma mul_left_inv_aux : \u2200 l : list (\u03a3 i, G i),\n  mul_aux (l.map (\u03bb i : \u03a3 i, G i, \u27e8i.1, i.2\u207b\u00b9\u27e9)) l = []\n| []     := rfl\n| (i::l) := by simp [mul_aux, mul_left_inv_aux l]\n\nprotected lemma mul_left_inv (l : list (\u03a3 i, G i)) :\n  pre.mul (pre.inv l) l = [] :=\nby rw [pre.mul, pre.inv, reverse_reverse, pre.mul_left_inv_aux]\n\nend inv\n\nend coprod.pre\n", "meta": {"author": "ChrisHughes24", "repo": "single_relation", "sha": "556990dab75054a1c14717a72c8901dc9f2f01e4", "save_path": "github-repos/lean/ChrisHughes24-single_relation", "path": "github-repos/lean/ChrisHughes24-single_relation/single_relation-556990dab75054a1c14717a72c8901dc9f2f01e4/scratch/pre.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.38286717517491364}}
{"text": "import Smt\n\ntheorem simplification (p q : Prop) : p \u2227 q \u2192 p := by\n  smt\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Prop/Simplification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.38282225594550806}}
{"text": "--\n\ndef f1 (x : Nat) (b : Bool) : typeOf% x :=\nlet r : typeOf% (x+1) := x+1;\nr + 1\n\ntheorem ex1 : f1 1 true = 3 :=\nrfl\n\ndef f2 (x : Nat) (b : Bool) : typeOf% x :=\nlet r : typeOf% b := x+1; -- error\nr + 1\n\ndef f3 (x : Nat) (b : Bool) : typeOf% b :=\nlet r (x!1 : typeOf% x) : typeOf% b := x > 1;\nr x\n\ndef f4 (x : Nat) : Nat :=\nlet y : Nat := x\nlet y := ensureTypeOf% y \"invalid reassignment, term\" y == 1 -- error\ny + 1\n\ndef f5 (x : Nat) : Nat :=\nlet y : Nat := x\nlet y := ensureTypeOf% y \"invalid reassignment, term\" (y+1)\ny + 1\n\ndef f6 (x : Nat) : Nat :=\nensureExpectedType% \"natural number expected, value\" true\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/typeOf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.38280104353163663}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.functor.basic\n\n/-!\n# Isomorphisms\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines isomorphisms between objects of a category.\n\n## Main definitions\n\n- `structure iso` : a bundled isomorphism between two objects of a category;\n- `class is_iso` : an unbundled version of `iso`;\n  note that `is_iso f` is a `Prop`, and only asserts the existence of an inverse.\n  Of course, this inverse is unique, so it doesn't cost us much to use choice to retrieve it.\n- `inv f`, for the inverse of a morphism with `[is_iso f]`\n- `as_iso` : convert from `is_iso` to `iso` (noncomputable);\n- `of_iso` : convert from `iso` to `is_iso`;\n- standard operations on isomorphisms (composition, inverse etc)\n\n## Notations\n\n- `X \u2245 Y` : same as `iso X Y`;\n- `\u03b1 \u226a\u226b \u03b2` : composition of two isomorphisms; it is called `iso.trans`\n\n## Tags\n\ncategory, category theory, isomorphism\n-/\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nnamespace category_theory\nopen category\n\n/--\nAn isomorphism (a.k.a. an invertible morphism) between two objects of a category.\nThe inverse morphism is bundled.\n\nSee also `category_theory.core` for the category with the same objects and isomorphisms playing\nthe role of morphisms.\n\nSee <https://stacks.math.columbia.edu/tag/0017>.\n-/\nstructure iso {C : Type u} [category.{v} C] (X Y : C) :=\n(hom : X \u27f6 Y)\n(inv : Y \u27f6 X)\n(hom_inv_id' : hom \u226b inv = \ud835\udfd9 X . obviously)\n(inv_hom_id' : inv \u226b hom = \ud835\udfd9 Y . obviously)\n\nrestate_axiom iso.hom_inv_id'\nrestate_axiom iso.inv_hom_id'\nattribute [simp, reassoc] iso.hom_inv_id iso.inv_hom_id\n\ninfixr ` \u2245 `:10  := iso             -- type as \\cong or \\iso\n\nvariables {C : Type u} [category.{v} C]\nvariables {X Y Z : C}\n\nnamespace iso\n\n@[ext] lemma ext \u2983\u03b1 \u03b2 : X \u2245 Y\u2984 (w : \u03b1.hom = \u03b2.hom) : \u03b1 = \u03b2 :=\nsuffices \u03b1.inv = \u03b2.inv, by cases \u03b1; cases \u03b2; cc,\ncalc \u03b1.inv\n    = \u03b1.inv \u226b (\u03b2.hom \u226b \u03b2.inv) : by rw [iso.hom_inv_id, category.comp_id]\n... = (\u03b1.inv \u226b \u03b1.hom) \u226b \u03b2.inv : by rw [category.assoc, \u2190w]\n... = \u03b2.inv                   : by rw [iso.inv_hom_id, category.id_comp]\n\n/-- Inverse isomorphism. -/\n@[symm] def symm (I : X \u2245 Y) : Y \u2245 X :=\n{ hom := I.inv,\n  inv := I.hom,\n  hom_inv_id' := I.inv_hom_id',\n  inv_hom_id' := I.hom_inv_id' }\n\n@[simp] lemma symm_hom (\u03b1 : X \u2245 Y) : \u03b1.symm.hom = \u03b1.inv := rfl\n@[simp] lemma symm_inv (\u03b1 : X \u2245 Y) : \u03b1.symm.inv = \u03b1.hom := rfl\n\n@[simp] lemma symm_mk {X Y : C} (hom : X \u27f6 Y) (inv : Y \u27f6 X) (hom_inv_id) (inv_hom_id) :\n  iso.symm {hom := hom, inv := inv, hom_inv_id' := hom_inv_id, inv_hom_id' := inv_hom_id} =\n    {hom := inv, inv := hom, hom_inv_id' := inv_hom_id, inv_hom_id' := hom_inv_id} := rfl\n\n@[simp] lemma symm_symm_eq {X Y : C} (\u03b1 : X \u2245 Y) : \u03b1.symm.symm = \u03b1 :=\nby cases \u03b1; refl\n\n@[simp] lemma symm_eq_iff {X Y : C} {\u03b1 \u03b2 : X \u2245 Y} : \u03b1.symm = \u03b2.symm \u2194 \u03b1 = \u03b2 :=\n\u27e8\u03bb h, symm_symm_eq \u03b1 \u25b8 symm_symm_eq \u03b2 \u25b8 congr_arg symm h, congr_arg symm\u27e9\n\nlemma nonempty_iso_symm (X Y : C) : nonempty (X \u2245 Y) \u2194 nonempty (Y \u2245 X) :=\n\u27e8\u03bb h, \u27e8h.some.symm\u27e9, \u03bb h, \u27e8h.some.symm\u27e9\u27e9\n\n/-- Identity isomorphism. -/\n@[refl, simps] def refl (X : C) : X \u2245 X :=\n{ hom := \ud835\udfd9 X,\n  inv := \ud835\udfd9 X }\n\ninstance : inhabited (X \u2245 X) := \u27e8iso.refl X\u27e9\n\n@[simp] lemma refl_symm (X : C) : (iso.refl X).symm = iso.refl X := rfl\n\n/-- Composition of two isomorphisms -/\n@[trans, simps] def trans (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : X \u2245 Z :=\n{ hom := \u03b1.hom \u226b \u03b2.hom,\n  inv := \u03b2.inv \u226b \u03b1.inv }\n\ninfixr ` \u226a\u226b `:80 := iso.trans -- type as `\\ll \\gg`.\n\n@[simp] lemma trans_mk {X Y Z : C}\n  (hom : X \u27f6 Y) (inv : Y \u27f6 X) (hom_inv_id) (inv_hom_id)\n  (hom' : Y \u27f6 Z) (inv' : Z \u27f6 Y) (hom_inv_id') (inv_hom_id') (hom_inv_id'') (inv_hom_id'') :\n  iso.trans\n    {hom := hom, inv := inv, hom_inv_id' := hom_inv_id, inv_hom_id' := inv_hom_id}\n    {hom := hom', inv := inv', hom_inv_id' := hom_inv_id', inv_hom_id' := inv_hom_id'} =\n  { hom := hom \u226b hom', inv := inv' \u226b inv, hom_inv_id' := hom_inv_id'',\n    inv_hom_id' := inv_hom_id''} :=\nrfl\n\n@[simp] lemma trans_symm (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : (\u03b1 \u226a\u226b \u03b2).symm = \u03b2.symm \u226a\u226b \u03b1.symm := rfl\n@[simp] lemma trans_assoc {Z' : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) (\u03b3 : Z \u2245 Z') :\n  (\u03b1 \u226a\u226b \u03b2) \u226a\u226b \u03b3 = \u03b1 \u226a\u226b \u03b2 \u226a\u226b \u03b3 :=\nby ext; simp only [trans_hom, category.assoc]\n\n@[simp] lemma refl_trans (\u03b1 : X \u2245 Y) : (iso.refl X) \u226a\u226b \u03b1 = \u03b1 := by ext; apply category.id_comp\n@[simp] lemma trans_refl (\u03b1 : X \u2245 Y) : \u03b1 \u226a\u226b (iso.refl Y) = \u03b1 := by ext; apply category.comp_id\n\n@[simp] lemma symm_self_id (\u03b1 : X \u2245 Y) : \u03b1.symm \u226a\u226b \u03b1 = iso.refl Y := ext \u03b1.inv_hom_id\n@[simp] lemma self_symm_id (\u03b1 : X \u2245 Y) : \u03b1 \u226a\u226b \u03b1.symm = iso.refl X := ext \u03b1.hom_inv_id\n\n@[simp] lemma symm_self_id_assoc (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : \u03b1.symm \u226a\u226b \u03b1 \u226a\u226b \u03b2 = \u03b2 :=\nby rw [\u2190 trans_assoc, symm_self_id, refl_trans]\n\n@[simp] lemma self_symm_id_assoc (\u03b1 : X \u2245 Y) (\u03b2 : X \u2245 Z) : \u03b1 \u226a\u226b \u03b1.symm \u226a\u226b \u03b2 = \u03b2 :=\nby rw [\u2190 trans_assoc, self_symm_id, refl_trans]\n\nlemma inv_comp_eq (\u03b1 : X \u2245 Y) {f : X \u27f6 Z} {g : Y \u27f6 Z} : \u03b1.inv \u226b f = g \u2194 f = \u03b1.hom \u226b g :=\n\u27e8\u03bb H, by simp [H.symm], \u03bb H, by simp [H]\u27e9\n\nlemma eq_inv_comp (\u03b1 : X \u2245 Y) {f : X \u27f6 Z} {g : Y \u27f6 Z} : g = \u03b1.inv \u226b f \u2194 \u03b1.hom \u226b g = f :=\n(inv_comp_eq \u03b1.symm).symm\n\nlemma comp_inv_eq (\u03b1 : X \u2245 Y) {f : Z \u27f6 Y} {g : Z \u27f6 X} : f \u226b \u03b1.inv = g \u2194 f = g \u226b \u03b1.hom :=\n\u27e8\u03bb H, by simp [H.symm], \u03bb H, by simp [H]\u27e9\n\nlemma eq_comp_inv (\u03b1 : X \u2245 Y) {f : Z \u27f6 Y} {g : Z \u27f6 X} : g = f \u226b \u03b1.inv \u2194 g \u226b \u03b1.hom = f :=\n(comp_inv_eq \u03b1.symm).symm\n\nlemma inv_eq_inv (f g : X \u2245 Y) : f.inv = g.inv \u2194 f.hom = g.hom :=\nhave \u2200{X Y : C} (f g : X \u2245 Y), f.hom = g.hom \u2192 f.inv = g.inv, from \u03bb X Y f g h, by rw [ext h],\n\u27e8this f.symm g.symm, this f g\u27e9\n\nlemma hom_comp_eq_id (\u03b1 : X \u2245 Y) {f : Y \u27f6 X} : \u03b1.hom \u226b f = \ud835\udfd9 X \u2194 f = \u03b1.inv :=\nby rw [\u2190eq_inv_comp, comp_id]\n\nlemma comp_hom_eq_id (\u03b1 : X \u2245 Y) {f : Y \u27f6 X} : f \u226b \u03b1.hom = \ud835\udfd9 Y \u2194 f = \u03b1.inv :=\nby rw [\u2190eq_comp_inv, id_comp]\n\nlemma inv_comp_eq_id (\u03b1 : X \u2245 Y) {f : X \u27f6 Y} : \u03b1.inv \u226b f = \ud835\udfd9 Y \u2194 f = \u03b1.hom :=\nhom_comp_eq_id \u03b1.symm\n\nlemma comp_inv_eq_id (\u03b1 : X \u2245 Y) {f : X \u27f6 Y} : f \u226b \u03b1.inv = \ud835\udfd9 X \u2194 f = \u03b1.hom :=\ncomp_hom_eq_id \u03b1.symm\n\nlemma hom_eq_inv (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 X) : \u03b1.hom = \u03b2.inv \u2194 \u03b2.hom = \u03b1.inv :=\nby { erw [inv_eq_inv \u03b1.symm \u03b2, eq_comm], refl }\n\nend iso\n\n/-- `is_iso` typeclass expressing that a morphism is invertible. -/\nclass is_iso (f : X \u27f6 Y) : Prop :=\n(out : \u2203 inv : Y \u27f6 X, f \u226b inv = \ud835\udfd9 X \u2227 inv \u226b f = \ud835\udfd9 Y)\n\n/--\nThe inverse of a morphism `f` when we have `[is_iso f]`.\n-/\nnoncomputable def inv (f : X \u27f6 Y) [I : is_iso f] := classical.some I.1\n\nnamespace is_iso\n\n@[simp, reassoc] lemma hom_inv_id (f : X \u27f6 Y) [I : is_iso f] : f \u226b inv f = \ud835\udfd9 X :=\n(classical.some_spec I.1).left\n@[simp, reassoc] lemma inv_hom_id (f : X \u27f6 Y) [I : is_iso f] : inv f \u226b f = \ud835\udfd9 Y :=\n(classical.some_spec I.1).right\n\nend is_iso\n\nopen is_iso\n\n/-- Reinterpret a morphism `f` with an `is_iso f` instance as an `iso`. -/\nnoncomputable\ndef as_iso (f : X \u27f6 Y) [h : is_iso f] : X \u2245 Y := \u27e8f, inv f, hom_inv_id f, inv_hom_id f\u27e9\n\n@[simp] lemma as_iso_hom (f : X \u27f6 Y) [is_iso f] : (as_iso f).hom = f := rfl\n@[simp] lemma as_iso_inv (f : X \u27f6 Y) [is_iso f] : (as_iso f).inv = inv f := rfl\n\nnamespace is_iso\n\n@[priority 100] -- see Note [lower instance priority]\ninstance epi_of_iso (f : X \u27f6 Y) [is_iso f] : epi f  :=\n{ left_cancellation := \u03bb Z g h w,\n  -- This is an interesting test case for better rewrite automation.\n  by rw [\u2190 is_iso.inv_hom_id_assoc f g, w, is_iso.inv_hom_id_assoc f h] }\n@[priority 100] -- see Note [lower instance priority]\ninstance mono_of_iso (f : X \u27f6 Y) [is_iso f] : mono f :=\n{ right_cancellation := \u03bb Z g h w,\n  by rw [\u2190 category.comp_id g, \u2190 category.comp_id h, \u2190 is_iso.hom_inv_id f, \u2190 category.assoc, w,\n    \u2190 category.assoc] }\n\n@[ext] lemma inv_eq_of_hom_inv_id {f : X \u27f6 Y} [is_iso f] {g : Y \u27f6 X}\n  (hom_inv_id : f \u226b g = \ud835\udfd9 X) : inv f = g :=\nbegin\n  apply (cancel_epi f).mp,\n  simp [hom_inv_id],\nend\n\nlemma inv_eq_of_inv_hom_id {f : X \u27f6 Y} [is_iso f] {g : Y \u27f6 X}\n  (inv_hom_id : g \u226b f = \ud835\udfd9 Y) : inv f = g :=\nbegin\n  apply (cancel_mono f).mp,\n  simp [inv_hom_id],\nend\n\n@[ext] lemma eq_inv_of_hom_inv_id {f : X \u27f6 Y} [is_iso f] {g : Y \u27f6 X}\n  (hom_inv_id : f \u226b g = \ud835\udfd9 X) : g = inv f :=\n(inv_eq_of_hom_inv_id hom_inv_id).symm\n\nlemma eq_inv_of_inv_hom_id {f : X \u27f6 Y} [is_iso f] {g : Y \u27f6 X}\n  (inv_hom_id : g \u226b f = \ud835\udfd9 Y) : g = inv f :=\n(inv_eq_of_inv_hom_id inv_hom_id).symm\n\n\ninstance id (X : C) : is_iso (\ud835\udfd9 X) :=\n\u27e8\u27e8\ud835\udfd9 X, by simp\u27e9\u27e9\n\ninstance of_iso (f : X \u2245 Y) : is_iso f.hom :=\n\u27e8\u27e8f.inv, by simp\u27e9\u27e9\n\ninstance of_iso_inv (f : X \u2245 Y) : is_iso f.inv :=\nis_iso.of_iso f.symm\n\nvariables {f g : X \u27f6 Y} {h : Y \u27f6 Z}\n\ninstance inv_is_iso [is_iso f] : is_iso (inv f) :=\nis_iso.of_iso_inv (as_iso f)\n\n/- The following instance has lower priority for the following reason:\nSuppose we are given `f : X \u2245 Y` with `X Y : Type u`.\nWithout the lower priority, typeclass inference cannot deduce `is_iso f.hom`\nbecause `f.hom` is defeq to `(\u03bb x, x) \u226b f.hom`, triggering a loop. -/\n@[priority 900]\ninstance comp_is_iso [is_iso f] [is_iso h] : is_iso (f \u226b h) :=\nis_iso.of_iso $ (as_iso f) \u226a\u226b (as_iso h)\n\n@[simp] lemma inv_id : inv (\ud835\udfd9 X) = \ud835\udfd9 X := by { ext, simp, }\n@[simp] lemma inv_comp [is_iso f] [is_iso h] : inv (f \u226b h) = inv h \u226b inv f := by { ext, simp, }\n@[simp] lemma inv_inv [is_iso f] : inv (inv f) = f := by { ext, simp, }\n@[simp] lemma iso.inv_inv (f : X \u2245 Y) : inv (f.inv) = f.hom := by { ext, simp, }\n@[simp] lemma iso.inv_hom (f : X \u2245 Y) : inv (f.hom) = f.inv := by { ext, simp, }\n\n@[simp]\nlemma inv_comp_eq (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : X \u27f6 Z} {g : Y \u27f6 Z} : inv \u03b1 \u226b f = g \u2194 f = \u03b1 \u226b g :=\n(as_iso \u03b1).inv_comp_eq\n\n@[simp]\nlemma eq_inv_comp (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : X \u27f6 Z} {g : Y \u27f6 Z} : g = inv \u03b1 \u226b f \u2194 \u03b1 \u226b g = f :=\n(as_iso \u03b1).eq_inv_comp\n\n@[simp]\nlemma comp_inv_eq (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : Z \u27f6 Y} {g : Z \u27f6 X} : f \u226b inv \u03b1 = g \u2194 f = g \u226b \u03b1 :=\n(as_iso \u03b1).comp_inv_eq\n\n@[simp]\nlemma eq_comp_inv (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : Z \u27f6 Y} {g : Z \u27f6 X} : g = f \u226b inv \u03b1 \u2194 g \u226b \u03b1 = f :=\n(as_iso \u03b1).eq_comp_inv\n\nlemma of_is_iso_comp_left {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [is_iso f] [is_iso (f \u226b g)] : is_iso g :=\nby { rw [\u2190 id_comp g, \u2190 inv_hom_id f, assoc], apply_instance, }\n\nlemma of_is_iso_comp_right {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [is_iso g] [is_iso (f \u226b g)] : is_iso f :=\nby { rw [\u2190 comp_id f, \u2190 hom_inv_id g, \u2190 assoc], apply_instance, }\n\nlemma of_is_iso_fac_left {X Y Z : C} {f : X \u27f6 Y} {g : Y \u27f6 Z} {h : X \u27f6 Z}\n  [is_iso f] [hh : is_iso h] (w : f \u226b g = h) : is_iso g :=\nby { rw \u2190 w at hh, haveI := hh, exact of_is_iso_comp_left f g, }\n\nlemma of_is_iso_fac_right {X Y Z : C} {f : X \u27f6 Y} {g : Y \u27f6 Z} {h : X \u27f6 Z}\n  [is_iso g] [hh : is_iso h] (w : f \u226b g = h) : is_iso f :=\nby { rw \u2190 w at hh, haveI := hh, exact of_is_iso_comp_right f g, }\n\nend is_iso\n\nopen is_iso\n\nlemma eq_of_inv_eq_inv {f g : X \u27f6 Y} [is_iso f] [is_iso g] (p : inv f = inv g) : f = g :=\nbegin\n  apply (cancel_epi (inv f)).1,\n  erw [inv_hom_id, p, inv_hom_id],\nend\n\nlemma is_iso.inv_eq_inv {f g : X \u27f6 Y} [is_iso f] [is_iso g] : inv f = inv g \u2194 f = g :=\niso.inv_eq_inv (as_iso f) (as_iso g)\n\nlemma hom_comp_eq_id (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} : g \u226b f = \ud835\udfd9 X \u2194 f = inv g :=\n(as_iso g).hom_comp_eq_id\n\nlemma comp_hom_eq_id (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} : f \u226b g = \ud835\udfd9 Y \u2194 f = inv g :=\n(as_iso g).comp_hom_eq_id\n\nlemma inv_comp_eq_id (g : X \u27f6 Y) [is_iso g] {f : X \u27f6 Y} : inv g \u226b f = \ud835\udfd9 Y \u2194 f = g :=\n(as_iso g).inv_comp_eq_id\n\nlemma comp_inv_eq_id (g : X \u27f6 Y) [is_iso g] {f : X \u27f6 Y} : f \u226b inv g = \ud835\udfd9 X \u2194 f = g :=\n(as_iso g).comp_inv_eq_id\n\nlemma is_iso_of_hom_comp_eq_id (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} (h : g \u226b f = \ud835\udfd9 X) : is_iso f :=\nby { rw [(hom_comp_eq_id _).mp h], apply_instance }\n\nlemma is_iso_of_comp_hom_eq_id (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} (h : f \u226b g = \ud835\udfd9 Y) : is_iso f :=\nby { rw [(comp_hom_eq_id _).mp h], apply_instance }\n\nnamespace iso\n\n@[ext] lemma inv_ext {f : X \u2245 Y} {g : Y \u27f6 X}\n  (hom_inv_id : f.hom \u226b g = \ud835\udfd9 X) : f.inv = g :=\n((hom_comp_eq_id f).1 hom_inv_id).symm\n\n@[ext] lemma inv_ext' {f : X \u2245 Y} {g : Y \u27f6 X}\n  (hom_inv_id : f.hom \u226b g = \ud835\udfd9 X) : g = f.inv :=\n(hom_comp_eq_id f).1 hom_inv_id\n\n/-!\nAll these cancellation lemmas can be solved by `simp [cancel_mono]` (or `simp [cancel_epi]`),\nbut with the current design `cancel_mono` is not a good `simp` lemma,\nbecause it generates a typeclass search.\n\nWhen we can see syntactically that a morphism is a `mono` or an `epi`\nbecause it came from an isomorphism, it's fine to do the cancellation via `simp`.\n\nIn the longer term, it might be worth exploring making `mono` and `epi` structures,\nrather than typeclasses, with coercions back to `X \u27f6 Y`.\nPresumably we could write `X \u21aa Y` and `X \u21a0 Y`.\n-/\n\n@[simp] lemma cancel_iso_hom_left {X Y Z : C} (f : X \u2245 Y) (g g' : Y \u27f6 Z) :\n  f.hom \u226b g = f.hom \u226b g' \u2194 g = g' :=\nby simp only [cancel_epi]\n\n@[simp] lemma cancel_iso_inv_left {X Y Z : C} (f : Y \u2245 X) (g g' : Y \u27f6 Z) :\n  f.inv \u226b g = f.inv \u226b g' \u2194 g = g' :=\nby simp only [cancel_epi]\n\n@[simp] lemma cancel_iso_hom_right {X Y Z : C} (f f' : X \u27f6 Y) (g : Y \u2245 Z) :\n  f \u226b g.hom = f' \u226b g.hom \u2194 f = f' :=\nby simp only [cancel_mono]\n\n@[simp] lemma cancel_iso_inv_right {X Y Z : C} (f f' : X \u27f6 Y) (g : Z \u2245 Y) :\n  f \u226b g.inv = f' \u226b g.inv \u2194 f = f' :=\nby simp only [cancel_mono]\n\n/-\nUnfortunately cancelling an isomorphism from the right of a chain of compositions is awkward.\nWe would need separate lemmas for each chain length (worse: for each pair of chain lengths).\n\nWe provide two more lemmas, for case of three morphisms, because this actually comes up in practice,\nbut then stop.\n-/\n\n@[simp] lemma cancel_iso_hom_right_assoc {W X X' Y Z : C}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y)\n  (h : Y \u2245 Z) :\n  f \u226b g \u226b h.hom = f' \u226b g' \u226b h.hom \u2194 f \u226b g = f' \u226b g' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\n@[simp] lemma cancel_iso_inv_right_assoc {W X X' Y Z : C}\n  (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y)\n  (h : Z \u2245 Y) :\n  f \u226b g \u226b h.inv = f' \u226b g' \u226b h.inv \u2194 f \u226b g = f' \u226b g' :=\nby simp only [\u2190category.assoc, cancel_mono]\n\nend iso\n\nnamespace functor\n\nuniverses u\u2081 v\u2081 u\u2082 v\u2082\nvariables {D : Type u\u2082}\n\nvariables [category.{v\u2082} D]\n\n/-- A functor `F : C \u2964 D` sends isomorphisms `i : X \u2245 Y` to isomorphisms `F.obj X \u2245 F.obj Y` -/\n@[simps]\ndef map_iso (F : C \u2964 D) {X Y : C} (i : X \u2245 Y) : F.obj X \u2245 F.obj Y :=\n{ hom := F.map i.hom,\n  inv := F.map i.inv,\n  hom_inv_id' := by rw [\u2190map_comp, iso.hom_inv_id, \u2190map_id],\n  inv_hom_id' := by rw [\u2190map_comp, iso.inv_hom_id, \u2190map_id] }\n\n@[simp] lemma map_iso_symm (F : C \u2964 D) {X Y : C} (i : X \u2245 Y) :\n  F.map_iso i.symm = (F.map_iso i).symm :=\nrfl\n\n@[simp] lemma map_iso_trans (F : C \u2964 D) {X Y Z : C} (i : X \u2245 Y) (j : Y \u2245 Z) :\n  F.map_iso (i \u226a\u226b j) = (F.map_iso i) \u226a\u226b (F.map_iso j) :=\nby ext; apply functor.map_comp\n\n@[simp] lemma map_iso_refl (F : C \u2964 D) (X : C) : F.map_iso (iso.refl X) = iso.refl (F.obj X) :=\niso.ext $ F.map_id X\n\ninstance map_is_iso (F : C \u2964 D) (f : X \u27f6 Y) [is_iso f] : is_iso (F.map f) :=\nis_iso.of_iso $ F.map_iso (as_iso f)\n\n@[simp] \n\nlemma map_hom_inv (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y) [is_iso f] :\n  F.map f \u226b F.map (inv f) = \ud835\udfd9 (F.obj X) :=\nby simp\n\nlemma map_inv_hom (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y) [is_iso f] :\n  F.map (inv f) \u226b F.map f = \ud835\udfd9 (F.obj Y) :=\nby simp\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/isomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.6261241842048092, "lm_q1q2_score": 0.3828010392663268}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Jakob von Raumer\n-/\nimport algebra.group.ext\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.preadditive\nimport category_theory.limits.shapes.kernels\n\n/-!\n# Biproducts and binary biproducts\n\nWe introduce the notion of (finite) biproducts and binary biproducts.\n\nThese are slightly unusual relative to the other shapes in the library,\nas they are simultaneously limits and colimits.\n(Zero objects are similar; they are \"biterminal\".)\n\nWe treat first the case of a general category with zero morphisms,\nand subsequently the case of a preadditive category.\n\nIn a category with zero morphisms, we model the (binary) biproduct of `P Q : C`\nusing a `binary_bicone`, which has a cone point `X`,\nand morphisms `fst : X \u27f6 P`, `snd : X \u27f6 Q`, `inl : P \u27f6 X` and `inr : X \u27f6 Q`,\nsuch that `inl \u226b fst = \ud835\udfd9 P`, `inl \u226b snd = 0`, `inr \u226b fst = 0`, and `inr \u226b snd = \ud835\udfd9 Q`.\nSuch a `binary_bicone` is a biproduct if the cone is a limit cone, and the cocone is a colimit\ncocone.\n\nIn a preadditive category,\n* any `binary_biproduct` satisfies `total : fst \u226b inl + snd \u226b inr = \ud835\udfd9 X`\n* any `binary_product` is a `binary_biproduct`\n* any `binary_coproduct` is a `binary_biproduct`\n\nFor biproducts indexed by a `fintype J`, a `bicone` again consists of a cone point `X`\nand morphisms `\u03c0 j : X \u27f6 F j` and `\u03b9 j : F j \u27f6 X` for each `j`,\nsuch that `\u03b9 j \u226b \u03c0 j'` is the identity when `j = j'` and zero otherwise.\n\nIn a preadditive category,\n* any `biproduct` satisfies `total : \u2211 j : J, biproduct.\u03c0 f j \u226b biproduct.\u03b9 f j = \ud835\udfd9 (\u2a01 f)`\n* any `product` is a `biproduct`\n* any `coproduct` is a `biproduct`\n\n## Notation\nAs `\u2295` is already taken for the sum of types, we introduce the notation `X \u229e Y` for\na binary biproduct. We introduce `\u2a01 f` for the indexed biproduct.\n\n## Implementation\nPrior to #14046, `has_finite_biproducts` required a `decidable_eq` instance on the indexing type.\nAs this had no pay-off (everything about limits is non-constructive in mathlib), and occasional cost\n(constructing decidability instances appropriate for constructions involving the indexing type),\nwe made everything classical.\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory\nopen category_theory.functor\nopen_locale classical\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {J : Type v}\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/--\nA `c : bicone F` is:\n* an object `c.X` and\n* morphisms `\u03c0 j : X \u27f6 F j` and `\u03b9 j : F j \u27f6 X` for each `j`,\n* such that `\u03b9 j \u226b \u03c0 j'` is the identity when `j = j'` and zero otherwise.\n-/\n@[nolint has_inhabited_instance]\nstructure bicone (F : J \u2192 C) :=\n(X : C)\n(\u03c0 : \u03a0 j, X \u27f6 F j)\n(\u03b9 : \u03a0 j, F j \u27f6 X)\n(\u03b9_\u03c0 : \u2200 j j', \u03b9 j \u226b \u03c0 j' = if h : j = j' then eq_to_hom (congr_arg F h) else 0 . obviously)\n\n@[simp, reassoc] lemma bicone_\u03b9_\u03c0_self {F : J \u2192 C} (B : bicone F) (j : J) :\n  B.\u03b9 j \u226b B.\u03c0 j = \ud835\udfd9 (F j) :=\nby simpa using B.\u03b9_\u03c0 j j\n\n@[simp, reassoc] lemma bicone_\u03b9_\u03c0_ne {F : J \u2192 C} (B : bicone F) {j j' : J} (h : j \u2260 j') :\n  B.\u03b9 j \u226b B.\u03c0 j' = 0 :=\nby simpa [h] using B.\u03b9_\u03c0 j j'\n\nvariables {F : J \u2192 C}\n\nnamespace bicone\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- Extract the cone from a bicone. -/\ndef to_cone (B : bicone F) : cone (discrete.functor F) :=\n{ X := B.X,\n  \u03c0 := { app := \u03bb j, B.\u03c0 j.as }, }\n\n@[simp] lemma to_cone_X (B : bicone F) : B.to_cone.X = B.X := rfl\n\n@[simp] lemma to_cone_\u03c0_app (B : bicone F) (j : J) : B.to_cone.\u03c0.app \u27e8j\u27e9 = B.\u03c0 j := rfl\n\n/-- Extract the cocone from a bicone. -/\ndef to_cocone (B : bicone F) : cocone (discrete.functor F) :=\n{ X := B.X,\n  \u03b9 := { app := \u03bb j, B.\u03b9 j.as }, }\n\n@[simp] lemma to_cocone_X (B : bicone F) : B.to_cocone.X = B.X := rfl\n\n@[simp] lemma to_cocone_\u03b9_app (B : bicone F) (j : J) : B.to_cocone.\u03b9.app \u27e8j\u27e9 = B.\u03b9 j := rfl\n\n/-- We can turn any limit cone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_limit_cone {f : J \u2192 C} {t : cone (discrete.functor f)} (ht : is_limit t) :\n  bicone f :=\n{ X := t.X,\n  \u03c0 := \u03bb j, t.\u03c0.app \u27e8j\u27e9,\n  \u03b9 := \u03bb j, ht.lift (fan.mk _ (\u03bb j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)),\n  \u03b9_\u03c0 := \u03bb j j', by simp }\n\nlemma \u03b9_of_is_limit {f : J \u2192 C} {t : bicone f} (ht : is_limit t.to_cone) (j : J) :\n  t.\u03b9 j = ht.lift (fan.mk _ (\u03bb j', if h : j = j' then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (\u03bb j', by { rw ht.fac, discrete_cases, simp [t.\u03b9_\u03c0] })\n\n/-- We can turn any colimit cocone over a discrete collection of objects into a bicone. -/\n@[simps]\ndef of_colimit_cocone {f : J \u2192 C} {t : cocone (discrete.functor f)} (ht : is_colimit t) :\n  bicone f :=\n{ X := t.X,\n  \u03c0 := \u03bb j, ht.desc (cofan.mk _ (\u03bb j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)),\n  \u03b9 := \u03bb j, t.\u03b9.app \u27e8j\u27e9,\n  \u03b9_\u03c0 := \u03bb j j', by simp }\n\nlemma \u03c0_of_is_colimit {f : J \u2192 C} {t : bicone f} (ht : is_colimit t.to_cocone) (j : J) :\n  t.\u03c0 j = ht.desc (cofan.mk _ (\u03bb j', if h : j' = j then eq_to_hom (congr_arg f h) else 0)) :=\nht.hom_ext (\u03bb j', by { rw ht.fac, discrete_cases, simp [t.\u03b9_\u03c0] })\n\n/-- Structure witnessing that a bicone is both a limit cone and a colimit cocone. -/\n@[nolint has_inhabited_instance]\nstructure is_bilimit {F : J \u2192 C} (B : bicone F) :=\n(is_limit : is_limit B.to_cone)\n(is_colimit : is_colimit B.to_cocone)\n\nend bicone\n\n/--\nA bicone over `F : J \u2192 C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_inhabited_instance]\nstructure limit_bicone (F : J \u2192 C) :=\n(bicone : bicone F)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_biproduct F` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `F`.\n-/\nclass has_biproduct (F : J \u2192 C) : Prop :=\nmk' :: (exists_biproduct : nonempty (limit_bicone F))\n\nlemma has_biproduct.mk {F : J \u2192 C} (d : limit_bicone F) : has_biproduct F :=\n\u27e8nonempty.intro d\u27e9\n\n/-- Use the axiom of choice to extract explicit `biproduct_data F` from `has_biproduct F`. -/\ndef get_biproduct_data (F : J \u2192 C) [has_biproduct F] : limit_bicone F :=\nclassical.choice has_biproduct.exists_biproduct\n\n/-- A bicone for `F` which is both a limit cone and a colimit cocone. -/\ndef biproduct.bicone (F : J \u2192 C) [has_biproduct F] : bicone F :=\n(get_biproduct_data F).bicone\n\n/-- `biproduct.bicone F` is a bilimit bicone. -/\ndef biproduct.is_bilimit (F : J \u2192 C) [has_biproduct F] : (biproduct.bicone F).is_bilimit :=\n(get_biproduct_data F).is_bilimit\n\n/-- `biproduct.bicone F` is a limit cone. -/\ndef biproduct.is_limit (F : J \u2192 C) [has_biproduct F] : is_limit (biproduct.bicone F).to_cone :=\n(get_biproduct_data F).is_bilimit.is_limit\n\n/-- `biproduct.bicone F` is a colimit cocone. -/\ndef biproduct.is_colimit (F : J \u2192 C) [has_biproduct F] :\n  is_colimit (biproduct.bicone F).to_cocone :=\n(get_biproduct_data F).is_bilimit.is_colimit\n\n@[priority 100]\ninstance has_product_of_has_biproduct [has_biproduct F] : has_limit (discrete.functor F) :=\nhas_limit.mk { cone := (biproduct.bicone F).to_cone,\n  is_limit := biproduct.is_limit F, }\n\n@[priority 100]\ninstance has_coproduct_of_has_biproduct [has_biproduct F] : has_colimit (discrete.functor F) :=\nhas_colimit.mk { cocone := (biproduct.bicone F).to_cocone,\n  is_colimit := biproduct.is_colimit F, }\n\nvariables (J C)\n\n/--\n`C` has biproducts of shape `J` if we have\na limit and a colimit, with the same cone points,\nof every function `F : J \u2192 C`.\n-/\nclass has_biproducts_of_shape : Prop :=\n(has_biproduct : \u03a0 F : J \u2192 C, has_biproduct F)\n\nattribute [instance, priority 100] has_biproducts_of_shape.has_biproduct\n\n/-- `has_finite_biproducts C` represents a choice of biproduct for every family of objects in `C`\nindexed by a finite type. -/\nclass has_finite_biproducts : Prop :=\n(has_biproducts_of_shape : \u03a0 (J : Type v) [fintype J],\n  has_biproducts_of_shape J C)\n\nattribute [instance, priority 100] has_finite_biproducts.has_biproducts_of_shape\n\n@[priority 100]\ninstance has_finite_products_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_products C :=\n{ out := \u03bb J _, \u27e8\u03bb F, by exactI has_limit_of_iso discrete.nat_iso_functor.symm\u27e9 }\n\n@[priority 100]\ninstance has_finite_coproducts_of_has_finite_biproducts [has_finite_biproducts C] :\n  has_finite_coproducts C :=\n{ out := \u03bb J _, \u27e8\u03bb F, by exactI has_colimit_of_iso discrete.nat_iso_functor\u27e9 }\n\nvariables {J C}\n\n/--\nThe isomorphism between the specified limit and the specified colimit for\na functor with a bilimit.\n-/\ndef biproduct_iso (F : J \u2192 C) [has_biproduct F] :\n  limits.pi_obj F \u2245 limits.sigma_obj F :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (biproduct.is_limit F)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (biproduct.is_colimit F) (colimit.is_colimit _)\n\nend limits\n\nnamespace limits\nvariables {J : Type v}\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C]\n\n/-- `biproduct f` computes the biproduct of a family of elements `f`. (It is defined as an\n   abbreviation for `limit (discrete.functor f)`, so for most facts about `biproduct f`, you will\n   just use general facts about limits and colimits.) -/\nabbreviation biproduct (f : J \u2192 C) [has_biproduct f] : C :=\n(biproduct.bicone f).X\n\nnotation `\u2a01 ` f:20 := biproduct f\n\n/-- The projection onto a summand of a biproduct. -/\nabbreviation biproduct.\u03c0 (f : J \u2192 C) [has_biproduct f] (b : J) : \u2a01 f \u27f6 f b :=\n(biproduct.bicone f).\u03c0 b\n\n@[simp]\nlemma biproduct.bicone_\u03c0 (f : J \u2192 C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).\u03c0 b = biproduct.\u03c0 f b := rfl\n\n/-- The inclusion into a summand of a biproduct. -/\nabbreviation biproduct.\u03b9 (f : J \u2192 C) [has_biproduct f] (b : J) : f b \u27f6 \u2a01 f :=\n(biproduct.bicone f).\u03b9 b\n\n@[simp]\nlemma biproduct.bicone_\u03b9 (f : J \u2192 C) [has_biproduct f] (b : J) :\n  (biproduct.bicone f).\u03b9 b = biproduct.\u03b9 f b := rfl\n\n/-- Note that as this lemma has a `if` in the statement, we include a `decidable_eq` argument.\nThis means you may not be able to `simp` using this lemma unless you `open_locale classical`. -/\n@[reassoc]\nlemma biproduct.\u03b9_\u03c0 [decidable_eq J] (f : J \u2192 C) [has_biproduct f] (j j' : J) :\n  biproduct.\u03b9 f j \u226b biproduct.\u03c0 f j' = if h : j = j' then eq_to_hom (congr_arg f h) else 0 :=\nby convert (biproduct.bicone f).\u03b9_\u03c0 j j'\n\n@[simp,reassoc]\nlemma biproduct.\u03b9_\u03c0_self (f : J \u2192 C) [has_biproduct f] (j : J) :\n  biproduct.\u03b9 f j \u226b biproduct.\u03c0 f j = \ud835\udfd9 _ :=\nby simp [biproduct.\u03b9_\u03c0]\n\n@[simp,reassoc]\nlemma biproduct.\u03b9_\u03c0_ne (f : J \u2192 C) [has_biproduct f] {j j' : J} (h : j \u2260 j') :\n  biproduct.\u03b9 f j \u226b biproduct.\u03c0 f j' = 0 :=\nby simp [biproduct.\u03b9_\u03c0, h]\n\n/-- Given a collection of maps into the summands, we obtain a map into the biproduct. -/\nabbreviation biproduct.lift\n  {f : J \u2192 C} [has_biproduct f] {P : C} (p : \u03a0 b, P \u27f6 f b) : P \u27f6 \u2a01 f :=\n(biproduct.is_limit f).lift (fan.mk P p)\n/-- Given a collection of maps out of the summands, we obtain a map out of the biproduct. -/\nabbreviation biproduct.desc\n  {f : J \u2192 C} [has_biproduct f] {P : C} (p : \u03a0 b, f b \u27f6 P) : \u2a01 f \u27f6 P :=\n(biproduct.is_colimit f).desc (cofan.mk P p)\n\n@[simp, reassoc]\nlemma biproduct.lift_\u03c0 {f : J \u2192 C} [has_biproduct f] {P : C} (p : \u03a0 b, P \u27f6 f b) (j : J) :\n  biproduct.lift p \u226b biproduct.\u03c0 f j = p j :=\n(biproduct.is_limit f).fac _ \u27e8j\u27e9\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_desc {f : J \u2192 C} [has_biproduct f] {P : C} (p : \u03a0 b, f b \u27f6 P) (j : J) :\n  biproduct.\u03b9 f j \u226b biproduct.desc p = p j :=\n(biproduct.is_colimit f).fac _ \u27e8j\u27e9\n\n/-- Given a collection of maps between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain a map between the biproducts. -/\nabbreviation biproduct.map {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 b, f b \u27f6 g b) : \u2a01 f \u27f6 \u2a01 g :=\nis_limit.map (biproduct.bicone f).to_cone (biproduct.is_limit g)\n  (discrete.nat_trans (\u03bb j, p j.as))\n\n/-- An alternative to `biproduct.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biproduct.map`. -/\nabbreviation biproduct.map' {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 b, f b \u27f6 g b) : \u2a01 f \u27f6 \u2a01 g :=\nis_colimit.map (biproduct.is_colimit f) (biproduct.bicone g).to_cocone\n  (discrete.nat_trans (\u03bb j, p j.as))\n\n@[ext] lemma biproduct.hom_ext {f : J \u2192 C} [has_biproduct f]\n  {Z : C} (g h : Z \u27f6 \u2a01 f)\n  (w : \u2200 j, g \u226b biproduct.\u03c0 f j = h \u226b biproduct.\u03c0 f j) : g = h :=\n(biproduct.is_limit f).hom_ext (\u03bb j, w j.as)\n\n@[ext] lemma biproduct.hom_ext' {f : J \u2192 C} [has_biproduct f]\n  {Z : C} (g h : \u2a01 f \u27f6 Z)\n  (w : \u2200 j, biproduct.\u03b9 f j \u226b g = biproduct.\u03b9 f j \u226b h) : g = h :=\n(biproduct.is_colimit f).hom_ext (\u03bb j, w j.as)\n\nlemma biproduct.map_eq_map' {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 b, f b \u27f6 g b) : biproduct.map p = biproduct.map' p :=\nbegin\n  ext j j',\n  simp only [discrete.nat_trans_app, limits.is_colimit.\u03b9_map, limits.is_limit.map_\u03c0, category.assoc,\n    \u2190bicone.to_cone_\u03c0_app, \u2190biproduct.bicone_\u03c0, \u2190bicone.to_cocone_\u03b9_app, \u2190biproduct.bicone_\u03b9],\n  simp only [biproduct.bicone_\u03b9, biproduct.bicone_\u03c0, bicone.to_cocone_\u03b9_app, bicone.to_cone_\u03c0_app],\n  dsimp,\n  rw [biproduct.\u03b9_\u03c0_assoc, biproduct.\u03b9_\u03c0],\n  split_ifs,\n  { subst h, rw [eq_to_hom_refl, category.id_comp], erw category.comp_id, },\n  { simp, },\nend\n\n@[simp, reassoc]\nlemma biproduct.map_\u03c0 {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 j, f j \u27f6 g j) (j : J) :\n  biproduct.map p \u226b biproduct.\u03c0 g j = biproduct.\u03c0 f j \u226b p j :=\nlimits.is_limit.map_\u03c0 _ _ _ (discrete.mk j)\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_map {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 j, f j \u27f6 g j) (j : J) :\n  biproduct.\u03b9 f j \u226b biproduct.map p = p j \u226b biproduct.\u03b9 g j :=\nbegin\n  rw biproduct.map_eq_map',\n  convert limits.is_colimit.\u03b9_map _ _ _ (discrete.mk j); refl\nend\n\n@[simp, reassoc]\nlemma biproduct.map_desc {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 j, f j \u27f6 g j) {P : C} (k : \u03a0 j, g j \u27f6 P) :\n  biproduct.map p \u226b biproduct.desc k = biproduct.desc (\u03bb j, p j \u226b k j) :=\nby { ext, simp, }\n\n@[simp, reassoc]\nlemma biproduct.lift_map {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  {P : C} (k : \u03a0 j, P \u27f6 f j) (p : \u03a0 j, f j \u27f6 g j)  :\n  biproduct.lift k \u226b biproduct.map p = biproduct.lift (\u03bb j, k j \u226b p j) :=\nby { ext, simp, }\n\n/-- Given a collection of isomorphisms between corresponding summands of a pair of biproducts\nindexed by the same type, we obtain an isomorphism between the biproducts. -/\n@[simps]\ndef biproduct.map_iso {f g : J \u2192 C} [has_biproduct f] [has_biproduct g]\n  (p : \u03a0 b, f b \u2245 g b) : \u2a01 f \u2245 \u2a01 g :=\n{ hom := biproduct.map (\u03bb b, (p b).hom),\n  inv := biproduct.map (\u03bb b, (p b).inv), }\n\nsection \u03c0_kernel\n\nsection\nvariables (f : J \u2192 C) [has_biproduct f]\nvariables (p : J \u2192 Prop) [has_biproduct (subtype.restrict p f)]\n\n/-- The canonical morphism from the biproduct over a restricted index type to the biproduct of\nthe full index type. -/\ndef biproduct.from_subtype : \u2a01 subtype.restrict p f \u27f6 \u2a01 f :=\nbiproduct.desc $ \u03bb j, biproduct.\u03b9 _ _\n\n/-- The canonical morphism from a biproduct to the biproduct over a restriction of its index\ntype. -/\ndef biproduct.to_subtype : \u2a01 f \u27f6 \u2a01 subtype.restrict p f :=\nbiproduct.lift $ \u03bb j, biproduct.\u03c0 _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_\u03c0 [decidable_pred p] (j : J) :\n  biproduct.from_subtype f p \u226b biproduct.\u03c0 f j =\n    if h : p j then biproduct.\u03c0 (subtype.restrict p f) \u27e8j, h\u27e9 else 0 :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.\u03b9_desc_assoc, biproduct.\u03b9_\u03c0],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.\u03b9_\u03c0],\n    split_ifs with h\u2081 h\u2082 h\u2082,\n    exacts [rfl, false.elim (h\u2082 (subtype.ext h\u2081)),\n      false.elim (h\u2081 (congr_arg subtype.val h\u2082)), rfl] },\n  { rw [dif_neg h, dif_neg (show (i : J) \u2260 j, from \u03bb h\u2082, h (h\u2082 \u25b8 i.2)), comp_zero] }\nend\n\nlemma biproduct.from_subtype_eq_lift [decidable_pred p] : biproduct.from_subtype f p =\n    biproduct.lift (\u03bb j, if h : p j then biproduct.\u03c0 (subtype.restrict p f) \u27e8j, h\u27e9 else 0) :=\nbiproduct.hom_ext _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_\u03c0_subtype (j : subtype p) :\n  biproduct.from_subtype f p \u226b biproduct.\u03c0 f j = biproduct.\u03c0 (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.from_subtype, biproduct.\u03b9_desc_assoc, biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0],\n  split_ifs with h\u2081 h\u2082 h\u2082,\n  exacts [rfl, false.elim (h\u2082 (subtype.ext h\u2081)), false.elim (h\u2081 (congr_arg subtype.val h\u2082)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_\u03c0 (j : subtype p) :\n  biproduct.to_subtype f p \u226b biproduct.\u03c0 (subtype.restrict p f) j = biproduct.\u03c0 f j :=\nbiproduct.lift_\u03c0 _ _\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_to_subtype [decidable_pred p] (j : J) :\n  biproduct.\u03b9 f j \u226b biproduct.to_subtype f p =\n    if h : p j then biproduct.\u03b9 (subtype.restrict p f) \u27e8j, h\u27e9 else 0 :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0],\n  by_cases h : p j,\n  { rw [dif_pos h, biproduct.\u03b9_\u03c0],\n    split_ifs with h\u2081 h\u2082 h\u2082,\n    exacts [rfl, false.elim (h\u2082 (subtype.ext h\u2081)),\n      false.elim (h\u2081 (congr_arg subtype.val h\u2082)), rfl] },\n  { rw [dif_neg h, dif_neg (show j \u2260 i, from \u03bb h\u2082, h (h\u2082.symm \u25b8 i.2)), zero_comp] }\nend\n\nlemma biproduct.to_subtype_eq_desc [decidable_pred p] : biproduct.to_subtype f p =\n  biproduct.desc (\u03bb j, if h : p j then biproduct.\u03b9 (subtype.restrict p f) \u27e8j, h\u27e9 else 0) :=\nbiproduct.hom_ext' _ _ (by simp)\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_to_subtype_subtype (j : subtype p) :\n  biproduct.\u03b9 f j \u226b biproduct.to_subtype f p = biproduct.\u03b9 (subtype.restrict p f) j :=\nbegin\n  ext i,\n  rw [biproduct.to_subtype, category.assoc, biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0],\n  split_ifs with h\u2081 h\u2082 h\u2082,\n  exacts [rfl, false.elim (h\u2082 (subtype.ext h\u2081)), false.elim (h\u2081 (congr_arg subtype.val h\u2082)), rfl]\nend\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_from_subtype (j : subtype p) :\n  biproduct.\u03b9 (subtype.restrict p f) j \u226b biproduct.from_subtype f p = biproduct.\u03b9 f j :=\nbiproduct.\u03b9_desc _ _\n\n@[simp, reassoc]\nlemma biproduct.from_subtype_to_subtype :\n  biproduct.from_subtype f p \u226b biproduct.to_subtype f p = \ud835\udfd9 (\u2a01 subtype.restrict p f) :=\nbegin\n  refine biproduct.hom_ext _ _ (\u03bb j, _),\n  rw [category.assoc, biproduct.to_subtype_\u03c0, biproduct.from_subtype_\u03c0_subtype, category.id_comp]\nend\n\n@[simp, reassoc]\nlemma biproduct.to_subtype_from_subtype [decidable_pred p] :\n  biproduct.to_subtype f p \u226b biproduct.from_subtype f p =\n    biproduct.map (\u03bb j, if p j then \ud835\udfd9 (f j) else 0) :=\nbegin\n  ext1 i,\n  by_cases h : p i,\n  { simp [h], congr },\n  { simp [h] }\nend\n\nend\n\nvariables (f : J \u2192 C) (i : J) [has_biproduct f] [has_biproduct (subtype.restrict (\u03bb j, i \u2260 j) f)]\n\n/-- The kernel of `biproduct.\u03c0 f i` is the inclusion from the biproduct which omits `i`\nfrom the index set `J` into the biproduct over `J`. -/\ndef biproduct.is_limit_from_subtype : is_limit\n  (kernel_fork.of_\u03b9 (biproduct.from_subtype f (\u03bb j, i \u2260 j))\n    (by simp) : kernel_fork (biproduct.\u03c0 f i)) :=\nfork.is_limit.mk' _ $ \u03bb s,\n\u27e8s.\u03b9 \u226b biproduct.to_subtype _ _,\n begin\n   ext j,\n   rw [kernel_fork.\u03b9_of_\u03b9, category.assoc, category.assoc,\n     biproduct.to_subtype_from_subtype_assoc, biproduct.map_\u03c0],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), comp_zero, comp_zero, kernel_fork.condition] },\n   { rw [if_pos, category.comp_id], exact h, }\n end,\n begin\n   intros m hm,\n   rw [\u2190 hm, kernel_fork.\u03b9_of_\u03b9, category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.comp_id _).symm\n end\u27e9\n\n/-- The cokernel of `biproduct.\u03b9 f i` is the projection from the biproduct over the index set `J`\nonto the biproduct omitting `i`. -/\ndef biproduct.is_colimit_to_subtype : is_colimit\n  (cokernel_cofork.of_\u03c0 (biproduct.to_subtype f (\u03bb j, i \u2260 j))\n    (by simp) : cokernel_cofork (biproduct.\u03b9 f i)) :=\ncofork.is_colimit.mk' _ $ \u03bb s,\n\u27e8biproduct.from_subtype _ _ \u226b s.\u03c0,\n begin\n   ext j,\n   rw [cokernel_cofork.\u03c0_of_\u03c0, biproduct.to_subtype_from_subtype_assoc,\n     biproduct.\u03b9_map_assoc],\n   rcases em (i = j) with (rfl|h),\n   { rw [if_neg (not_not.2 rfl), zero_comp, cokernel_cofork.condition] },\n   { rw [if_pos, category.id_comp], exact h, }\n end,\n begin\n   intros m hm,\n   rw [\u2190 hm, cokernel_cofork.\u03c0_of_\u03c0, \u2190 category.assoc, biproduct.from_subtype_to_subtype],\n   exact (category.id_comp _).symm\n end\u27e9\n\nend \u03c0_kernel\n\nsection\nvariables [fintype J] {K : Type v} [fintype K] {f : J \u2192 C} {g : K \u2192 C}\n  [has_finite_biproducts C]\n\n/--\nConvert a (dependently typed) matrix to a morphism of biproducts.\n-/\ndef biproduct.matrix (m : \u03a0 j k, f j \u27f6 g k) : \u2a01 f \u27f6 \u2a01 g :=\nbiproduct.desc (\u03bb j, biproduct.lift (\u03bb k, m j k))\n\n@[simp, reassoc]\nlemma biproduct.matrix_\u03c0 (m : \u03a0 j k, f j \u27f6 g k) (k : K) :\n  biproduct.matrix m \u226b biproduct.\u03c0 g k = biproduct.desc (\u03bb j, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n@[simp, reassoc]\nlemma biproduct.\u03b9_matrix (m : \u03a0 j k, f j \u27f6 g k) (j : J) :\n  biproduct.\u03b9 f j \u226b biproduct.matrix m = biproduct.lift (\u03bb k, m j k) :=\nby { ext, simp [biproduct.matrix], }\n\n/--\nExtract the matrix components from a morphism of biproducts.\n-/\ndef biproduct.components (m : \u2a01 f \u27f6 \u2a01 g) (j : J) (k : K) : f j \u27f6 g k :=\nbiproduct.\u03b9 f j \u226b m \u226b biproduct.\u03c0 g k\n\n@[simp] lemma biproduct.matrix_components (m : \u03a0 j k, f j \u27f6 g k) (j : J) (k : K) :\n  biproduct.components (biproduct.matrix m) j k = m j k :=\nby simp [biproduct.components]\n\n@[simp] lemma biproduct.components_matrix (m : \u2a01 f \u27f6 \u2a01 g) :\n  biproduct.matrix (\u03bb j k, biproduct.components m j k) = m :=\nby { ext, simp [biproduct.components], }\n\n/-- Morphisms between direct sums are matrices. -/\n@[simps]\ndef biproduct.matrix_equiv : (\u2a01 f \u27f6 \u2a01 g) \u2243 (\u03a0 j k, f j \u27f6 g k) :=\n{ to_fun := biproduct.components,\n  inv_fun := biproduct.matrix,\n  left_inv := biproduct.components_matrix,\n  right_inv := \u03bb m, by { ext, apply biproduct.matrix_components } }\n\nend\n\ninstance biproduct.\u03b9_mono (f : J \u2192 C) [has_biproduct f]\n  (b : J) : split_mono (biproduct.\u03b9 f b) :=\n{ retraction := biproduct.desc $\n    \u03bb b', if h : b' = b then eq_to_hom (congr_arg f h) else biproduct.\u03b9 f b' \u226b biproduct.\u03c0 f b }\n\ninstance biproduct.\u03c0_epi (f : J \u2192 C) [has_biproduct f]\n  (b : J) : split_epi (biproduct.\u03c0 f b) :=\n{ section_ := biproduct.lift $\n    \u03bb b', if h : b = b' then eq_to_hom (congr_arg f h) else biproduct.\u03b9 f b \u226b biproduct.\u03c0 f b' }\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_hom (f : J \u2192 C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).hom = biproduct.lift b.\u03c0 :=\nrfl\n\n/-- Auxiliary lemma for `biproduct.unique_up_to_iso`. -/\nlemma biproduct.cone_point_unique_up_to_iso_inv (f : J \u2192 C) [has_biproduct f] {b : bicone f}\n  (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (biproduct.is_limit _)).inv = biproduct.desc b.\u03b9 :=\nbegin\n  refine biproduct.hom_ext' _ _ (\u03bb j, hb.is_limit.hom_ext (\u03bb j', _)),\n  discrete_cases,\n  rw [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp, bicone.to_cone_\u03c0_app,\n    biproduct.bicone_\u03c0, biproduct.\u03b9_desc, biproduct.\u03b9_\u03c0, b.to_cone_\u03c0_app, b.\u03b9_\u03c0]\nend\n\n/-- Biproducts are unique up to isomorphism. This already follows because bilimits are limits,\n    but in the case of biproducts we can give an isomorphism with particularly nice definitional\n    properties, namely that `biproduct.lift b.\u03c0` and `biproduct.desc b.\u03b9` are inverses of each\n    other. -/\n@[simps]\ndef biproduct.unique_up_to_iso (f : J \u2192 C) [has_biproduct f] {b : bicone f} (hb : b.is_bilimit) :\n  b.X \u2245 \u2a01 f :=\n{ hom := biproduct.lift b.\u03c0,\n  inv := biproduct.desc b.\u03b9,\n  hom_inv_id' := by rw [\u2190 biproduct.cone_point_unique_up_to_iso_hom f hb,\n    \u2190 biproduct.cone_point_unique_up_to_iso_inv f hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [\u2190 biproduct.cone_point_unique_up_to_iso_hom f hb,\n    \u2190 biproduct.cone_point_unique_up_to_iso_inv f hb, iso.inv_hom_id] }\n\nsection\nvariables (C)\n\n/-- A category with finite biproducts has a zero object. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance has_zero_object_of_has_finite_biproducts [has_finite_biproducts C] : has_zero_object C :=\nby { refine \u27e8\u27e8biproduct pempty.elim, \u03bb X, \u27e8\u27e8\u27e80\u27e9, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e80\u27e9, _\u27e9\u27e9\u27e9\u27e9, tidy, }\n\nend\n\nsection\nvariables [unique J] (f : J \u2192 C)\n\n/-- The limit bicone for the biproduct over an index type with exactly one term. -/\n@[simps]\ndef limit_bicone_of_unique : limit_bicone f :=\n{ bicone :=\n  { X := f default,\n    \u03c0 := \u03bb j, eq_to_hom (by congr),\n    \u03b9 := \u03bb j, eq_to_hom (by congr), },\n  is_bilimit :=\n  { is_limit := (limit_cone_of_unique f).is_limit,\n    is_colimit := (colimit_cocone_of_unique f).is_colimit, }, }\n\n@[priority 100] instance has_biproduct_unique : has_biproduct f :=\nhas_biproduct.mk (limit_bicone_of_unique f)\n\n/-- A biproduct over a index type with exactly one term is just the object over that term. -/\n@[simps]\ndef biproduct_unique_iso : \u2a01 f \u2245 f default :=\n(biproduct.unique_up_to_iso _ (limit_bicone_of_unique f).is_bilimit).symm\n\nend\n\n/--\nA binary bicone for a pair of objects `P Q : C` consists of the cone point `X`,\nmaps from `X` to both `P` and `Q`, and maps from both `P` and `Q` to `X`,\nso that `inl \u226b fst = \ud835\udfd9 P`, `inl \u226b snd = 0`, `inr \u226b fst = 0`, and `inr \u226b snd = \ud835\udfd9 Q`\n-/\n@[nolint has_inhabited_instance]\nstructure binary_bicone (P Q : C) :=\n(X : C)\n(fst : X \u27f6 P)\n(snd : X \u27f6 Q)\n(inl : P \u27f6 X)\n(inr : Q \u27f6 X)\n(inl_fst' : inl \u226b fst = \ud835\udfd9 P . obviously)\n(inl_snd' : inl \u226b snd = 0 . obviously)\n(inr_fst' : inr \u226b fst = 0 . obviously)\n(inr_snd' : inr \u226b snd = \ud835\udfd9 Q . obviously)\n\nrestate_axiom binary_bicone.inl_fst'\nrestate_axiom binary_bicone.inl_snd'\nrestate_axiom binary_bicone.inr_fst'\nrestate_axiom binary_bicone.inr_snd'\nattribute [simp, reassoc] binary_bicone.inl_fst binary_bicone.inl_snd\n  binary_bicone.inr_fst binary_bicone.inr_snd\n\nnamespace binary_bicone\nvariables {P Q : C}\n\n/-- Extract the cone from a binary bicone. -/\ndef to_cone (c : binary_bicone P Q) : cone (pair P Q) :=\nbinary_fan.mk c.fst c.snd\n\n@[simp]\nlemma to_cone_X (c : binary_bicone P Q) :\n  c.to_cone.X = c.X := rfl\n\n@[simp]\nlemma to_cone_\u03c0_app_left (c : binary_bicone P Q) :\n  c.to_cone.\u03c0.app \u27e8walking_pair.left\u27e9 = c.fst := rfl\n@[simp]\nlemma to_cone_\u03c0_app_right (c : binary_bicone P Q) :\n  c.to_cone.\u03c0.app \u27e8walking_pair.right\u27e9 = c.snd := rfl\n@[simp]\nlemma binary_fan_fst_to_cone (c : binary_bicone P Q) : binary_fan.fst c.to_cone = c.fst := rfl\n@[simp]\nlemma binary_fan_snd_to_cone (c : binary_bicone P Q) : binary_fan.snd c.to_cone = c.snd := rfl\n\n/-- Extract the cocone from a binary bicone. -/\ndef to_cocone (c : binary_bicone P Q) : cocone (pair P Q) :=\nbinary_cofan.mk c.inl c.inr\n\n@[simp]\nlemma to_cocone_X (c : binary_bicone P Q) :\n  c.to_cocone.X = c.X := rfl\n\n@[simp]\nlemma to_cocone_\u03b9_app_left (c : binary_bicone P Q) :\n  c.to_cocone.\u03b9.app \u27e8walking_pair.left\u27e9 = c.inl := rfl\n@[simp]\nlemma to_cocone_\u03b9_app_right (c : binary_bicone P Q) :\n  c.to_cocone.\u03b9.app \u27e8walking_pair.right\u27e9 = c.inr := rfl\n@[simp]\nlemma binary_cofan_inl_to_cocone (c : binary_bicone P Q) : binary_cofan.inl c.to_cocone = c.inl :=\nrfl\n@[simp]\nlemma binary_cofan_inr_to_cocone (c : binary_bicone P Q) : binary_cofan.inr c.to_cocone = c.inr :=\nrfl\n\n/-- Convert a `binary_bicone` into a `bicone` over a pair. -/\n@[simps]\ndef to_bicone {X Y : C} (b : binary_bicone X Y) : bicone (pair_function X Y) :=\n{ X := b.X,\n  \u03c0 := \u03bb j, walking_pair.cases_on j b.fst b.snd,\n  \u03b9 := \u03bb j, walking_pair.cases_on j b.inl b.inr,\n  \u03b9_\u03c0 := \u03bb j j', by { rcases j with \u27e8\u27e9; rcases j' with \u27e8\u27e9, tidy } }\n\n/-- A binary bicone is a limit cone if and only if the corresponding bicone is a limit cone. -/\ndef to_bicone_is_limit {X Y : C} (b : binary_bicone X Y) :\n  is_limit (b.to_bicone.to_cone) \u2243 is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (\u03bb j, by { cases j, tidy })\n\n/-- A binary bicone is a colimit cocone if and only if the corresponding bicone is a colimit\n    cocone. -/\ndef to_bicone_is_colimit {X Y : C} (b : binary_bicone X Y) :\n  is_colimit (b.to_bicone.to_cocone) \u2243 is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (\u03bb j, by { cases j, tidy })\n\nend binary_bicone\n\nnamespace bicone\n\n/-- Convert a `bicone` over a function on `walking_pair` to a binary_bicone. -/\n@[simps]\ndef to_binary_bicone {X Y : C} (b : bicone (pair_function X Y)) : binary_bicone X Y :=\n{ X := b.X,\n  fst := b.\u03c0 walking_pair.left,\n  snd := b.\u03c0 walking_pair.right,\n  inl := b.\u03b9 walking_pair.left,\n  inr := b.\u03b9 walking_pair.right,\n  inl_fst' := by { simp [bicone.\u03b9_\u03c0], refl, },\n  inr_fst' := by simp [bicone.\u03b9_\u03c0],\n  inl_snd' := by simp [bicone.\u03b9_\u03c0],\n  inr_snd' := by { simp [bicone.\u03b9_\u03c0], refl, }, }\n\n/-- A bicone over a pair is a limit cone if and only if the corresponding binary bicone is a limit\n    cone.  -/\ndef to_binary_bicone_is_limit {X Y : C} (b : bicone (pair_function X Y)) :\n  is_limit (b.to_binary_bicone.to_cone) \u2243 is_limit (b.to_cone) :=\nis_limit.equiv_iso_limit $ cones.ext (iso.refl _) (\u03bb j, by { rcases j with \u27e8\u27e8\u27e9\u27e9; tidy })\n\n/-- A bicone over a pair is a colimit cocone if and only if the corresponding binary bicone is a\n    colimit cocone. -/\ndef to_binary_bicone_is_colimit {X Y : C} (b : bicone (pair_function X Y)) :\n  is_colimit (b.to_binary_bicone.to_cocone) \u2243 is_colimit (b.to_cocone) :=\nis_colimit.equiv_iso_colimit $ cocones.ext (iso.refl _) (\u03bb j, by { rcases j with \u27e8\u27e8\u27e9\u27e9; tidy })\n\nend bicone\n\n/-- Structure witnessing that a binary bicone is a limit cone and a limit cocone. -/\n@[nolint has_inhabited_instance]\nstructure binary_bicone.is_bilimit {P Q : C} (b : binary_bicone P Q) :=\n(is_limit : is_limit b.to_cone)\n(is_colimit : is_colimit b.to_cocone)\n\n/-- A binary bicone is a bilimit bicone if and only if the corresponding bicone is a bilimit. -/\ndef binary_bicone.to_bicone_is_bilimit {X Y : C} (b : binary_bicone X Y) :\n  b.to_bicone.is_bilimit \u2243 b.is_bilimit :=\n{ to_fun := \u03bb h, \u27e8b.to_bicone_is_limit h.is_limit, b.to_bicone_is_colimit h.is_colimit\u27e9,\n  inv_fun := \u03bb h, \u27e8b.to_bicone_is_limit.symm h.is_limit, b.to_bicone_is_colimit.symm h.is_colimit\u27e9,\n  left_inv := \u03bb \u27e8h, h'\u27e9, by { dsimp only, simp },\n  right_inv := \u03bb \u27e8h, h'\u27e9, by { dsimp only, simp } }\n\n/-- A bicone over a pair is a bilimit bicone if and only if the corresponding binary bicone is a\n    bilimit. -/\ndef bicone.to_binary_bicone_is_bilimit {X Y : C} (b : bicone (pair_function X Y)) :\n  b.to_binary_bicone.is_bilimit \u2243 b.is_bilimit :=\n{ to_fun := \u03bb h, \u27e8b.to_binary_bicone_is_limit h.is_limit,\n    b.to_binary_bicone_is_colimit h.is_colimit\u27e9,\n  inv_fun := \u03bb h, \u27e8b.to_binary_bicone_is_limit.symm h.is_limit,\n    b.to_binary_bicone_is_colimit.symm h.is_colimit\u27e9,\n  left_inv := \u03bb \u27e8h, h'\u27e9, by { dsimp only, simp },\n  right_inv := \u03bb \u27e8h, h'\u27e9, by { dsimp only, simp } }\n\n/--\nA bicone over `P Q : C`, which is both a limit cone and a colimit cocone.\n-/\n@[nolint has_inhabited_instance]\nstructure binary_biproduct_data (P Q : C) :=\n(bicone : binary_bicone P Q)\n(is_bilimit : bicone.is_bilimit)\n\n/--\n`has_binary_biproduct P Q` expresses the mere existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`.\n-/\nclass has_binary_biproduct (P Q : C) : Prop :=\nmk' :: (exists_binary_biproduct : nonempty (binary_biproduct_data P Q))\n\nlemma has_binary_biproduct.mk {P Q : C} (d : binary_biproduct_data P Q) :\n  has_binary_biproduct P Q :=\n\u27e8nonempty.intro d\u27e9\n\n/--\nUse the axiom of choice to extract explicit `binary_biproduct_data F` from `has_binary_biproduct F`.\n-/\ndef get_binary_biproduct_data (P Q : C) [has_binary_biproduct P Q] : binary_biproduct_data P Q :=\nclassical.choice has_binary_biproduct.exists_binary_biproduct\n\n/-- A bicone for `P Q ` which is both a limit cone and a colimit cocone. -/\ndef binary_biproduct.bicone (P Q : C) [has_binary_biproduct P Q] : binary_bicone P Q :=\n(get_binary_biproduct_data P Q).bicone\n\n/-- `binary_biproduct.bicone P Q` is a limit bicone. -/\ndef binary_biproduct.is_bilimit (P Q : C) [has_binary_biproduct P Q] :\n  (binary_biproduct.bicone P Q).is_bilimit :=\n(get_binary_biproduct_data P Q).is_bilimit\n\n/-- `binary_biproduct.bicone P Q` is a limit cone. -/\ndef binary_biproduct.is_limit (P Q : C) [has_binary_biproduct P Q] :\n  is_limit (binary_biproduct.bicone P Q).to_cone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_limit\n\n/-- `binary_biproduct.bicone P Q` is a colimit cocone. -/\ndef binary_biproduct.is_colimit (P Q : C) [has_binary_biproduct P Q] :\n  is_colimit (binary_biproduct.bicone P Q).to_cocone :=\n(get_binary_biproduct_data P Q).is_bilimit.is_colimit\n\nsection\nvariable (C)\n\n/--\n`has_binary_biproducts C` represents the existence of a bicone which is\nsimultaneously a limit and a colimit of the diagram `pair P Q`, for every `P Q : C`.\n-/\nclass has_binary_biproducts : Prop :=\n(has_binary_biproduct : \u03a0 (P Q : C), has_binary_biproduct P Q)\n\nattribute [instance, priority 100] has_binary_biproducts.has_binary_biproduct\n\n/--\nA category with finite biproducts has binary biproducts.\n\nThis is not an instance as typically in concrete categories there will be\nan alternative construction with nicer definitional properties.\n-/\nlemma has_binary_biproducts_of_finite_biproducts [has_finite_biproducts C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := \u03bb P Q, has_binary_biproduct.mk\n  { bicone := (biproduct.bicone (pair_function P Q)).to_binary_bicone,\n    is_bilimit := (bicone.to_binary_bicone_is_bilimit _).symm (biproduct.is_bilimit _) } }\n\nend\n\nvariables {P Q : C}\n\ninstance has_binary_biproduct.has_limit_pair [has_binary_biproduct P Q] :\n  has_limit (pair P Q) :=\nhas_limit.mk \u27e8_, binary_biproduct.is_limit P Q\u27e9\n\ninstance has_binary_biproduct.has_colimit_pair [has_binary_biproduct P Q] :\n  has_colimit (pair P Q) :=\nhas_colimit.mk \u27e8_, binary_biproduct.is_colimit P Q\u27e9\n\n@[priority 100]\ninstance has_binary_products_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_products C :=\n{ has_limit := \u03bb F, has_limit_of_iso (diagram_iso_pair F).symm }\n@[priority 100]\ninstance has_binary_coproducts_of_has_binary_biproducts [has_binary_biproducts C] :\n  has_binary_coproducts C :=\n{ has_colimit := \u03bb F, has_colimit_of_iso (diagram_iso_pair F) }\n\n/--\nThe isomorphism between the specified binary product and the specified binary coproduct for\na pair for a binary biproduct.\n-/\ndef biprod_iso (X Y : C) [has_binary_biproduct X Y]  :\n  limits.prod X Y \u2245 limits.coprod X Y :=\n(is_limit.cone_point_unique_up_to_iso (limit.is_limit _) (binary_biproduct.is_limit X Y)).trans $\n  is_colimit.cocone_point_unique_up_to_iso (binary_biproduct.is_colimit X Y) (colimit.is_colimit _)\n\n/-- An arbitrary choice of biproduct of a pair of objects. -/\nabbreviation biprod (X Y : C) [has_binary_biproduct X Y] := (binary_biproduct.bicone X Y).X\n\nnotation X ` \u229e `:20 Y:20 := biprod X Y\n\n/-- The projection onto the first summand of a binary biproduct. -/\nabbreviation biprod.fst {X Y : C} [has_binary_biproduct X Y] : X \u229e Y \u27f6 X :=\n(binary_biproduct.bicone X Y).fst\n/-- The projection onto the second summand of a binary biproduct. -/\nabbreviation biprod.snd {X Y : C} [has_binary_biproduct X Y] : X \u229e Y \u27f6 Y :=\n(binary_biproduct.bicone X Y).snd\n/-- The inclusion into the first summand of a binary biproduct. -/\nabbreviation biprod.inl {X Y : C} [has_binary_biproduct X Y] : X \u27f6 X \u229e Y :=\n(binary_biproduct.bicone X Y).inl\n/-- The inclusion into the second summand of a binary biproduct. -/\nabbreviation biprod.inr {X Y : C} [has_binary_biproduct X Y] : Y \u27f6 X \u229e Y :=\n(binary_biproduct.bicone X Y).inr\n\nsection\nvariables {X Y : C} [has_binary_biproduct X Y]\n\n@[simp] lemma binary_biproduct.bicone_fst : (binary_biproduct.bicone X Y).fst = biprod.fst := rfl\n@[simp] lemma binary_biproduct.bicone_snd : (binary_biproduct.bicone X Y).snd = biprod.snd := rfl\n@[simp] lemma binary_biproduct.bicone_inl : (binary_biproduct.bicone X Y).inl = biprod.inl := rfl\n@[simp] lemma binary_biproduct.bicone_inr : (binary_biproduct.bicone X Y).inr = biprod.inr := rfl\n\nend\n\n@[simp,reassoc]\nlemma biprod.inl_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X \u27f6 X \u229e Y) \u226b (biprod.fst : X \u229e Y \u27f6 X) = \ud835\udfd9 X :=\n(binary_biproduct.bicone X Y).inl_fst\n@[simp,reassoc]\nlemma biprod.inl_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inl : X \u27f6 X \u229e Y) \u226b (biprod.snd : X \u229e Y \u27f6 Y) = 0 :=\n(binary_biproduct.bicone X Y).inl_snd\n@[simp,reassoc]\nlemma biprod.inr_fst {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y \u27f6 X \u229e Y) \u226b (biprod.fst : X \u229e Y \u27f6 X) = 0 :=\n(binary_biproduct.bicone X Y).inr_fst\n@[simp,reassoc]\nlemma biprod.inr_snd {X Y : C} [has_binary_biproduct X Y] :\n  (biprod.inr : Y \u27f6 X \u229e Y) \u226b (biprod.snd : X \u229e Y \u27f6 Y) = \ud835\udfd9 Y :=\n(binary_biproduct.bicone X Y).inr_snd\n\n/-- Given a pair of maps into the summands of a binary biproduct,\nwe obtain a map into the binary biproduct. -/\nabbreviation biprod.lift {W X Y : C} [has_binary_biproduct X Y] (f : W \u27f6 X) (g : W \u27f6 Y) :\n  W \u27f6 X \u229e Y :=\n(binary_biproduct.is_limit X Y).lift (binary_fan.mk f g)\n/-- Given a pair of maps out of the summands of a binary biproduct,\nwe obtain a map out of the binary biproduct. -/\nabbreviation biprod.desc {W X Y : C} [has_binary_biproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) :\n  X \u229e Y \u27f6 W :=\n(binary_biproduct.is_colimit X Y).desc (binary_cofan.mk f g)\n\n@[simp, reassoc]\nlemma biprod.lift_fst {W X Y : C} [has_binary_biproduct X Y] (f : W \u27f6 X) (g : W \u27f6 Y) :\n  biprod.lift f g \u226b biprod.fst = f :=\n(binary_biproduct.is_limit X Y).fac _ \u27e8walking_pair.left\u27e9\n\n@[simp, reassoc]\nlemma biprod.lift_snd {W X Y : C} [has_binary_biproduct X Y] (f : W \u27f6 X) (g : W \u27f6 Y) :\n  biprod.lift f g \u226b biprod.snd = g :=\n(binary_biproduct.is_limit X Y).fac _ \u27e8walking_pair.right\u27e9\n\n@[simp, reassoc]\nlemma biprod.inl_desc {W X Y : C} [has_binary_biproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) :\n  biprod.inl \u226b biprod.desc f g = f :=\n(binary_biproduct.is_colimit X Y).fac _ \u27e8walking_pair.left\u27e9\n\n@[simp, reassoc]\nlemma biprod.inr_desc {W X Y : C} [has_binary_biproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) :\n  biprod.inr \u226b biprod.desc f g = g :=\n(binary_biproduct.is_colimit X Y).fac _ \u27e8walking_pair.right\u27e9\n\ninstance biprod.mono_lift_of_mono_left {W X Y : C} [has_binary_biproduct X Y] (f : W \u27f6 X)\n  (g : W \u27f6 Y) [mono f] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_fst _ _\n\ninstance biprod.mono_lift_of_mono_right {W X Y : C} [has_binary_biproduct X Y] (f : W \u27f6 X)\n  (g : W \u27f6 Y) [mono g] : mono (biprod.lift f g) :=\nmono_of_mono_fac $ biprod.lift_snd _ _\n\ninstance biprod.epi_desc_of_epi_left {W X Y : C} [has_binary_biproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W)\n  [epi f] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inl_desc _ _\n\ninstance biprod.epi_desc_of_epi_right {W X Y : C} [has_binary_biproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W)\n  [epi g] : epi (biprod.desc f g) :=\nepi_of_epi_fac $ biprod.inr_desc _ _\n\n/-- Given a pair of maps between the summands of a pair of binary biproducts,\nwe obtain a map between the binary biproducts. -/\nabbreviation biprod.map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u229e X \u27f6 Y \u229e Z :=\nis_limit.map (binary_biproduct.bicone W X).to_cone (binary_biproduct.is_limit Y Z)\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n/-- An alternative to `biprod.map` constructed via colimits.\nThis construction only exists in order to show it is equal to `biprod.map`. -/\nabbreviation biprod.map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u229e X \u27f6 Y \u229e Z :=\nis_colimit.map (binary_biproduct.is_colimit W X) (binary_biproduct.bicone Y Z).to_cocone\n  (@map_pair _ _ (pair W X) (pair Y Z) f g)\n\n@[ext] lemma biprod.hom_ext {X Y Z : C} [has_binary_biproduct X Y] (f g : Z \u27f6 X \u229e Y)\n  (h\u2080 : f \u226b biprod.fst = g \u226b biprod.fst) (h\u2081 : f \u226b biprod.snd = g \u226b biprod.snd) : f = g :=\nbinary_fan.is_limit.hom_ext (binary_biproduct.is_limit X Y) h\u2080 h\u2081\n\n@[ext] lemma biprod.hom_ext' {X Y Z : C} [has_binary_biproduct X Y] (f g : X \u229e Y \u27f6 Z)\n  (h\u2080 : biprod.inl \u226b f = biprod.inl \u226b g) (h\u2081 : biprod.inr \u226b f = biprod.inr \u226b g) : f = g :=\nbinary_cofan.is_colimit.hom_ext (binary_biproduct.is_colimit X Y) h\u2080 h\u2081\n\nlemma biprod.map_eq_map' {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) : biprod.map f g = biprod.map' f g :=\nbegin\n  ext,\n  { simp only [map_pair_left, is_colimit.\u03b9_map, is_limit.map_\u03c0, biprod.inl_fst_assoc,\n    category.assoc, \u2190binary_bicone.to_cone_\u03c0_app_left, \u2190binary_biproduct.bicone_fst,\n    \u2190binary_bicone.to_cocone_\u03b9_app_left, \u2190binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_left, is_colimit.\u03b9_map, is_limit.map_\u03c0, zero_comp,\n      biprod.inl_snd_assoc, category.assoc,\n      \u2190binary_bicone.to_cone_\u03c0_app_right, \u2190binary_biproduct.bicone_snd,\n      \u2190binary_bicone.to_cocone_\u03b9_app_left, \u2190binary_biproduct.bicone_inl],\n    simp },\n  { simp only [map_pair_right, biprod.inr_fst_assoc, is_colimit.\u03b9_map, is_limit.map_\u03c0,\n      zero_comp, category.assoc,\n      \u2190binary_bicone.to_cone_\u03c0_app_left, \u2190binary_biproduct.bicone_fst,\n      \u2190binary_bicone.to_cocone_\u03b9_app_right, \u2190binary_biproduct.bicone_inr],\n    simp },\n  { simp only [map_pair_right, is_colimit.\u03b9_map, is_limit.map_\u03c0, biprod.inr_snd_assoc,\n      category.assoc, \u2190binary_bicone.to_cone_\u03c0_app_right, \u2190binary_biproduct.bicone_snd,\n      \u2190binary_bicone.to_cocone_\u03b9_app_right, \u2190binary_biproduct.bicone_inr],\n    simp }\nend\n\ninstance biprod.inl_mono {X Y : C} [has_binary_biproduct X Y] :\n  split_mono (biprod.inl : X \u27f6 X \u229e Y) :=\n{ retraction := biprod.desc (\ud835\udfd9 X) (biprod.inr \u226b biprod.fst) }\n\ninstance biprod.inr_mono {X Y : C} [has_binary_biproduct X Y] :\n  split_mono (biprod.inr : Y \u27f6 X \u229e Y) :=\n{ retraction := biprod.desc (biprod.inl \u226b biprod.snd) (\ud835\udfd9 Y)}\n\ninstance biprod.fst_epi {X Y : C} [has_binary_biproduct X Y] :\n  split_epi (biprod.fst : X \u229e Y \u27f6 X) :=\n{ section_ := biprod.lift (\ud835\udfd9 X) (biprod.inl \u226b biprod.snd) }\n\ninstance biprod.snd_epi {X Y : C} [has_binary_biproduct X Y] :\n  split_epi (biprod.snd : X \u229e Y \u27f6 Y) :=\n{ section_ := biprod.lift (biprod.inr \u226b biprod.fst) (\ud835\udfd9 Y) }\n\n@[simp,reassoc]\nlemma biprod.map_fst {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  biprod.map f g \u226b biprod.fst = biprod.fst \u226b f :=\nis_limit.map_\u03c0 _ _ _ (\u27e8walking_pair.left\u27e9 : discrete walking_pair)\n\n@[simp,reassoc]\nlemma biprod.map_snd {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  biprod.map f g \u226b biprod.snd = biprod.snd \u226b g :=\nis_limit.map_\u03c0 _ _ _ (\u27e8walking_pair.right\u27e9 : discrete walking_pair)\n\n-- Because `biprod.map` is defined in terms of `lim` rather than `colim`,\n-- we need to provide additional `simp` lemmas.\n@[simp,reassoc]\nlemma biprod.inl_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  biprod.inl \u226b biprod.map f g = f \u226b biprod.inl :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.\u03b9_map (binary_biproduct.is_colimit W X) _ _ \u27e8walking_pair.left\u27e9\nend\n\n@[simp,reassoc]\nlemma biprod.inr_map {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  biprod.inr \u226b biprod.map f g = g \u226b biprod.inr :=\nbegin\n  rw biprod.map_eq_map',\n  exact is_colimit.\u03b9_map (binary_biproduct.is_colimit W X) _ _ \u27e8walking_pair.right\u27e9\nend\n\n/-- Given a pair of isomorphisms between the summands of a pair of binary biproducts,\nwe obtain an isomorphism between the binary biproducts. -/\n@[simps]\ndef biprod.map_iso {W X Y Z : C} [has_binary_biproduct W X] [has_binary_biproduct Y Z]\n  (f : W \u2245 Y) (g : X \u2245 Z) : W \u229e X \u2245 Y \u229e Z :=\n{ hom := biprod.map f.hom g.hom,\n  inv := biprod.map f.inv g.inv }\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_hom (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).hom\n    = biprod.lift b.fst b.snd :=\nrfl\n\n/-- Auxiliary lemma for `biprod.unique_up_to_iso`. -/\nlemma biprod.cone_point_unique_up_to_iso_inv (X Y : C) [has_binary_biproduct X Y]\n  {b : binary_bicone X Y} (hb : b.is_bilimit) :\n  (hb.is_limit.cone_point_unique_up_to_iso (binary_biproduct.is_limit _ _)).inv\n    = biprod.desc b.inl b.inr :=\nbegin\n  refine biprod.hom_ext' _ _ (hb.is_limit.hom_ext (\u03bb j, _)) (hb.is_limit.hom_ext (\u03bb j, _)),\n  all_goals { simp only [category.assoc, is_limit.cone_point_unique_up_to_iso_inv_comp],\n    rcases j with \u27e8\u27e8\u27e9\u27e9 },\n  all_goals { simp }\nend\n\n/-- Binary biproducts are unique up to isomorphism. This already follows because bilimits are\n    limits, but in the case of biproducts we can give an isomorphism with particularly nice\n    definitional properties, namely that `biprod.lift b.fst b.snd` and `biprod.desc b.inl b.inr`\n    are inverses of each other. -/\n@[simps]\ndef biprod.unique_up_to_iso (X Y : C) [has_binary_biproduct X Y] {b : binary_bicone X Y}\n  (hb : b.is_bilimit) : b.X \u2245 X \u229e Y :=\n{ hom := biprod.lift b.fst b.snd,\n  inv := biprod.desc b.inl b.inr,\n  hom_inv_id' := by rw [\u2190 biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    \u2190 biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.hom_inv_id],\n  inv_hom_id' := by rw [\u2190 biprod.cone_point_unique_up_to_iso_hom X Y hb,\n    \u2190 biprod.cone_point_unique_up_to_iso_inv X Y hb, iso.inv_hom_id] }\n\nsection\nvariables (X Y : C) [has_binary_biproduct X Y]\n\n-- There are three further variations,\n-- about `is_iso biprod.inr`, `is_iso biprod.fst` and `is_iso biprod.snd`,\n-- but any one suffices to prove `indecomposable_of_simple`\n-- and they are likely not separately useful.\nlemma biprod.is_iso_inl_iff_id_eq_fst_comp_inl :\n  is_iso (biprod.inl : X \u27f6 X \u229e Y) \u2194 \ud835\udfd9 (X \u229e Y) = biprod.fst \u226b biprod.inl :=\nbegin\n  split,\n  { introI h,\n    have := (cancel_epi (inv biprod.inl : X \u229e Y \u27f6 X)).2 biprod.inl_fst,\n    rw [is_iso.inv_hom_id_assoc, category.comp_id] at this,\n    rw [this, is_iso.inv_hom_id], },\n  { intro h, exact \u27e8\u27e8biprod.fst, biprod.inl_fst, h.symm\u27e9\u27e9, },\nend\n\nend\n\nsection biprod_kernel\n\nvariables (X Y : C) [has_binary_biproduct X Y]\n\n/-- A kernel fork for the kernel of `biprod.fst`. It consists of the\nmorphism `biprod.inr`. -/\ndef biprod.fst_kernel_fork : kernel_fork (biprod.fst : X \u229e Y \u27f6 X) :=\nkernel_fork.of_\u03b9 biprod.inr biprod.inr_fst\n\n@[simp]\nlemma biprod.fst_kernel_fork_\u03b9 : fork.\u03b9 (biprod.fst_kernel_fork X Y) = biprod.inr :=\nrfl\n\n/-- The fork `biprod.fst_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_fst_kernel_fork : is_limit (biprod.fst_kernel_fork X Y) :=\nfork.is_limit.mk' _ $ \u03bb s, \u27e8s.\u03b9 \u226b biprod.snd, by ext; simp, \u03bb m hm, by simp [\u2190 hm]\u27e9\n\n/-- A kernel fork for the kernel of `biprod.snd`. It consists of the\nmorphism `biprod.inl`. -/\ndef biprod.snd_kernel_fork : kernel_fork (biprod.snd : X \u229e Y \u27f6 Y) :=\nkernel_fork.of_\u03b9 biprod.inl biprod.inl_snd\n\n@[simp]\nlemma biprod.snd_kernel_fork_\u03b9 : fork.\u03b9 (biprod.snd_kernel_fork X Y) = biprod.inl :=\nrfl\n\n/-- The fork `biprod.snd_kernel_fork` is indeed a limit.  -/\ndef biprod.is_kernel_snd_kernel_fork : is_limit (biprod.snd_kernel_fork X Y) :=\nfork.is_limit.mk' _ $ \u03bb s, \u27e8s.\u03b9 \u226b biprod.fst, by ext; simp, \u03bb m hm, by simp [\u2190 hm]\u27e9\n\n/-- A cokernel cofork for the cokernel of `biprod.inl`. It consists of the\nmorphism `biprod.snd`. -/\ndef biprod.inl_cokernel_fork : cokernel_cofork (biprod.inl : X \u27f6 X \u229e Y) :=\ncokernel_cofork.of_\u03c0 biprod.snd biprod.inl_snd\n\n@[simp]\nlemma biprod.inl_cokernel_fork_\u03c0 : cofork.\u03c0 (biprod.inl_cokernel_fork X Y) = biprod.snd :=\nrfl\n\n/-- The cofork `biprod.inl_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inl_cokernel_fork : is_colimit (biprod.inl_cokernel_fork X Y) :=\ncofork.is_colimit.mk' _ $ \u03bb s, \u27e8biprod.inr \u226b s.\u03c0, by ext; simp, \u03bb m hm, by simp [\u2190 hm]\u27e9\n\n/-- A cokernel cofork for the cokernel of `biprod.inr`. It consists of the\nmorphism `biprod.fst`. -/\ndef biprod.inr_cokernel_fork : cokernel_cofork (biprod.inr : Y \u27f6 X \u229e Y) :=\ncokernel_cofork.of_\u03c0 biprod.fst biprod.inr_fst\n\n@[simp]\nlemma biprod.inr_cokernel_fork_\u03c0 : cofork.\u03c0 (biprod.inr_cokernel_fork X Y) = biprod.fst :=\nrfl\n\n/-- The cofork `biprod.inr_cokernel_fork` is indeed a colimit.  -/\ndef biprod.is_cokernel_inr_cokernel_fork : is_colimit (biprod.inr_cokernel_fork X Y) :=\ncofork.is_colimit.mk' _ $ \u03bb s, \u27e8biprod.inl \u226b s.\u03c0, by ext; simp, \u03bb m hm, by simp [\u2190 hm]\u27e9\n\nend biprod_kernel\n\nsection is_zero\n\n/-- If `Y` is a zero object, `X \u2245 X \u229e Y` for any `X`. -/\n@[simps]\ndef iso_biprod_zero {X Y : C} [has_binary_biproduct X Y] (hY : is_zero Y) : X \u2245 X \u229e Y :=\n{ hom := biprod.inl,\n  inv := biprod.fst,\n  inv_hom_id' := begin\n    apply category_theory.limits.biprod.hom_ext;\n    simp only [category.assoc, biprod.inl_fst, category.comp_id, category.id_comp,\n      biprod.inl_snd, comp_zero],\n    apply hY.eq_of_tgt\n  end }\n\n/-- If `X` is a zero object, `Y \u2245 X \u229e Y` for any `Y`. -/\n@[simps]\ndef iso_zero_biprod {X Y : C} [has_binary_biproduct X Y] (hY : is_zero X) : Y \u2245 X \u229e Y :=\n{ hom := biprod.inr,\n  inv := biprod.snd,\n  inv_hom_id' := begin\n    apply category_theory.limits.biprod.hom_ext;\n    simp only [category.assoc, biprod.inr_snd, category.comp_id, category.id_comp,\n      biprod.inr_fst, comp_zero],\n    apply hY.eq_of_tgt\n  end }\n\nend is_zero\n\nsection\nvariables [has_binary_biproducts C]\n\n/-- The braiding isomorphism which swaps a binary biproduct. -/\n@[simps] def biprod.braiding (P Q : C) : P \u229e Q \u2245 Q \u229e P :=\n{ hom := biprod.lift biprod.snd biprod.fst,\n  inv := biprod.lift biprod.snd biprod.fst }\n\n/--\nAn alternative formula for the braiding isomorphism which swaps a binary biproduct,\nusing the fact that the biproduct is a coproduct.\n-/\n@[simps]\ndef biprod.braiding' (P Q : C) : P \u229e Q \u2245 Q \u229e P :=\n{ hom := biprod.desc biprod.inr biprod.inl,\n  inv := biprod.desc biprod.inr biprod.inl }\n\nlemma biprod.braiding'_eq_braiding {P Q : C} :\n  biprod.braiding' P Q = biprod.braiding P Q :=\nby tidy\n\n/-- The braiding isomorphism can be passed through a map by swapping the order. -/\n@[reassoc] lemma biprod.braid_natural {W X Y Z : C} (f : X \u27f6 Y) (g : Z \u27f6 W) :\n  biprod.map f g \u226b (biprod.braiding _ _).hom = (biprod.braiding _ _).hom \u226b biprod.map g f :=\nby tidy\n\n@[reassoc] lemma biprod.braiding_map_braiding {W X Y Z : C} (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  (biprod.braiding X W).hom \u226b biprod.map f g \u226b (biprod.braiding Y Z).hom = biprod.map g f :=\nby tidy\n\n@[simp, reassoc] lemma biprod.symmetry' (P Q : C) :\n  biprod.lift biprod.snd biprod.fst \u226b biprod.lift biprod.snd biprod.fst = \ud835\udfd9 (P \u229e Q) :=\nby tidy\n\n/-- The braiding isomorphism is symmetric. -/\n@[reassoc] lemma biprod.symmetry (P Q : C) :\n  (biprod.braiding P Q).hom \u226b (biprod.braiding Q P).hom = \ud835\udfd9 _ :=\nby simp\n\nend\n\n-- TODO:\n-- If someone is interested, they could provide the constructions:\n--   has_binary_biproducts \u2194 has_finite_biproducts\n\nend limits\n\nnamespace limits\n\nsection preadditive\nvariables {C : Type u} [category.{v} C] [preadditive C]\nvariables {J : Type v} [fintype J]\n\nopen category_theory.preadditive\nopen_locale big_operators\n\n/--\nIn a preadditive category, we can construct a biproduct for `f : J \u2192 C` from\nany bicone `b` for `f` satisfying `total : \u2211 j : J, b.\u03c0 j \u226b b.\u03b9 j = \ud835\udfd9 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\ndef is_bilimit_of_total {f : J \u2192 C} (b : bicone f) (total : \u2211 j : J, b.\u03c0 j \u226b b.\u03b9 j = \ud835\udfd9 b.X) :\n  b.is_bilimit :=\n{ is_limit :=\n  { lift := \u03bb s, \u2211 (j : J), s.\u03c0.app \u27e8j\u27e9 \u226b b.\u03b9 j,\n    uniq' := \u03bb s m h,\n    begin\n      erw [\u2190category.comp_id m, \u2190total, comp_sum],\n      apply finset.sum_congr rfl,\n      intros j m,\n      erw [reassoc_of (h \u27e8j\u27e9)],\n    end,\n    fac' := \u03bb s j,\n    begin\n      cases j,\n      simp only [sum_comp, category.assoc, bicone.to_cone_\u03c0_app, b.\u03b9_\u03c0, comp_dite],\n      -- See note [dsimp, simp].\n      dsimp, simp,\n    end },\n  is_colimit :=\n  { desc := \u03bb s, \u2211 (j : J), b.\u03c0 j \u226b s.\u03b9.app \u27e8j\u27e9,\n    uniq' := \u03bb s m h,\n    begin\n      erw [\u2190category.id_comp m, \u2190total, sum_comp],\n            apply finset.sum_congr rfl,\n      intros j m,\n      erw [category.assoc, h \u27e8j\u27e9],\n    end,\n    fac' := \u03bb s j,\n    begin\n      cases j,\n      simp only [comp_sum, \u2190category.assoc, bicone.to_cocone_\u03b9_app, b.\u03b9_\u03c0, dite_comp],\n      dsimp, simp,\n    end } }\n\nlemma is_bilimit.total {f : J \u2192 C} {b : bicone f} (i : b.is_bilimit) :\n  \u2211 j : J, b.\u03c0 j \u226b b.\u03b9 j = \ud835\udfd9 b.X :=\ni.is_limit.hom_ext (\u03bb j, by { cases j, simp [sum_comp, b.\u03b9_\u03c0, comp_dite] })\n\n/--\nIn a preadditive category, we can construct a biproduct for `f : J \u2192 C` from\nany bicone `b` for `f` satisfying `total : \u2211 j : J, b.\u03c0 j \u226b b.\u03b9 j = \ud835\udfd9 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\nlemma has_biproduct_of_total {f : J \u2192 C} (b : bicone f) (total : \u2211 j : J, b.\u03c0 j \u226b b.\u03b9 j = \ud835\udfd9 b.X) :\n  has_biproduct f :=\nhas_biproduct.mk\n{ bicone := b,\n  is_bilimit := is_bilimit_of_total b total }\n\n/-- In a preadditive category, any finite bicone which is a limit cone is in fact a bilimit\n    bicone. -/\ndef is_bilimit_of_is_limit {f : J \u2192 C} (t : bicone f) (ht : is_limit t.to_cone) : t.is_bilimit :=\nis_bilimit_of_total _ $ ht.hom_ext $\n  \u03bb j, by { cases j, simp [sum_comp, t.\u03b9_\u03c0, dite_comp, comp_dite] }\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef bicone_is_bilimit_of_limit_cone_of_is_limit {f : J \u2192 C} {t : cone (discrete.functor f)}\n  (ht : is_limit t) : (bicone.of_limit_cone ht).is_bilimit :=\nis_bilimit_of_is_limit _ $\n  is_limit.of_iso_limit ht $ cones.ext (iso.refl _) (by { rintro \u27e8j\u27e9, tidy })\n\n/-- In a preadditive category, if the product over `f : J \u2192 C` exists,\n    then the biproduct over `f` exists. -/\nlemma has_biproduct.of_has_product (f : J \u2192 C) [has_product f] : has_biproduct f :=\nhas_biproduct.mk\n{ bicone := _,\n  is_bilimit := bicone_is_bilimit_of_limit_cone_of_is_limit (limit.is_limit _) }\n\n/-- In a preadditive category, any finite bicone which is a colimit cocone is in fact a bilimit\n    bicone. -/\ndef is_bilimit_of_is_colimit {f : J \u2192 C} (t : bicone f) (ht : is_colimit t.to_cocone) :\n  t.is_bilimit :=\nis_bilimit_of_total _ $ ht.hom_ext $ \u03bb j, begin\n  cases j,\n  simp_rw [bicone.to_cocone_\u03b9_app, comp_sum, \u2190 category.assoc, t.\u03b9_\u03c0, dite_comp],\n  tidy\nend\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef bicone_is_bilimit_of_colimit_cocone_of_is_colimit {f : J \u2192 C} {t : cocone (discrete.functor f)}\n  (ht : is_colimit t) : (bicone.of_colimit_cocone ht).is_bilimit :=\nis_bilimit_of_is_colimit _ $\n  is_colimit.of_iso_colimit ht $ cocones.ext (iso.refl _) (by { rintro \u27e8j\u27e9, tidy })\n\n/-- In a preadditive category, if the coproduct over `f : J \u2192 C` exists,\n    then the biproduct over `f` exists. -/\nlemma has_biproduct.of_has_coproduct (f : J \u2192 C) [has_coproduct f] : has_biproduct f :=\nhas_biproduct.mk\n{ bicone := _,\n  is_bilimit := bicone_is_bilimit_of_colimit_cocone_of_is_colimit (colimit.is_colimit _) }\n\n/-- A preadditive category with finite products has finite biproducts. -/\nlemma has_finite_biproducts.of_has_finite_products [has_finite_products C] :\n  has_finite_biproducts C :=\n\u27e8\u03bb J _, { has_biproduct := \u03bb F, by exactI has_biproduct.of_has_product _ }\u27e9\n\n/-- A preadditive category with finite coproducts has finite biproducts. -/\nlemma has_finite_biproducts.of_has_finite_coproducts [has_finite_coproducts C] :\n  has_finite_biproducts C :=\n\u27e8\u03bb J _, { has_biproduct := \u03bb F, by exactI has_biproduct.of_has_coproduct _ }\u27e9\n\nsection\nvariables {f : J \u2192 C} [has_biproduct f]\n\n/--\nIn any preadditive category, any biproduct satsifies\n`\u2211 j : J, biproduct.\u03c0 f j \u226b biproduct.\u03b9 f j = \ud835\udfd9 (\u2a01 f)`\n-/\n@[simp] lemma biproduct.total : \u2211 j : J, biproduct.\u03c0 f j \u226b biproduct.\u03b9 f j = \ud835\udfd9 (\u2a01 f) :=\nis_bilimit.total (biproduct.is_bilimit _)\n\nlemma biproduct.lift_eq {T : C} {g : \u03a0 j, T \u27f6 f j} :\n  biproduct.lift g = \u2211 j, g j \u226b biproduct.\u03b9 f j :=\nbegin\n  ext j,\n  simp [sum_comp, biproduct.\u03b9_\u03c0, comp_dite],\nend\n\nlemma biproduct.desc_eq {T : C} {g : \u03a0 j, f j \u27f6 T} :\n  biproduct.desc g = \u2211 j, biproduct.\u03c0 f j \u226b g j :=\nbegin\n  ext j,\n  simp [comp_sum, biproduct.\u03b9_\u03c0_assoc, dite_comp],\nend\n\n@[simp, reassoc] lemma biproduct.lift_desc {T U : C} {g : \u03a0 j, T \u27f6 f j} {h : \u03a0 j, f j \u27f6 U} :\n  biproduct.lift g \u226b biproduct.desc h = \u2211 j : J, g j \u226b h j :=\nby simp [biproduct.lift_eq, biproduct.desc_eq, comp_sum, sum_comp, biproduct.\u03b9_\u03c0_assoc,\n  comp_dite, dite_comp]\n\nlemma biproduct.map_eq [has_finite_biproducts C] {f g : J \u2192 C} {h : \u03a0 j, f j \u27f6 g j} :\n  biproduct.map h = \u2211 j : J, biproduct.\u03c0 f j \u226b h j \u226b biproduct.\u03b9 g j :=\nbegin\n  ext,\n  simp [biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0_assoc, comp_sum, sum_comp, comp_dite, dite_comp],\nend\n\n@[simp, reassoc]\nlemma biproduct.matrix_desc\n  {K : Type v} [fintype K] [has_finite_biproducts C]\n  {f : J \u2192 C} {g : K \u2192 C} (m : \u03a0 j k, f j \u27f6 g k) {P} (x : \u03a0 k, g k \u27f6 P) :\n  biproduct.matrix m \u226b biproduct.desc x = biproduct.desc (\u03bb j, \u2211 k, m j k \u226b x k) :=\nby { ext, simp, }\n\n@[simp, reassoc]\nlemma biproduct.lift_matrix\n  {K : Type v} [fintype K] [has_finite_biproducts C]\n  {f : J \u2192 C} {g : K \u2192 C} {P} (x : \u03a0 j, P \u27f6 f j) (m : \u03a0 j k, f j \u27f6 g k)  :\n  biproduct.lift x \u226b biproduct.matrix m = biproduct.lift (\u03bb k, \u2211 j, x j \u226b m j k) :=\nby { ext, simp, }\n\n@[reassoc]\nlemma biproduct.matrix_map\n  {K : Type v} [fintype K] [has_finite_biproducts C]\n  {f : J \u2192 C} {g : K \u2192 C} {h : K \u2192 C} (m : \u03a0 j k, f j \u27f6 g k) (n : \u03a0 k, g k \u27f6 h k) :\n  biproduct.matrix m \u226b biproduct.map n = biproduct.matrix (\u03bb j k, m j k \u226b n k) :=\nby { ext, simp, }\n\n@[reassoc]\nlemma biproduct.map_matrix\n  {K : Type v} [fintype K] [has_finite_biproducts C]\n  {f : J \u2192 C} {g : J \u2192 C} {h : K \u2192 C} (m : \u03a0 k, f k \u27f6 g k) (n : \u03a0 j k, g j \u27f6 h k) :\n  biproduct.map m \u226b biproduct.matrix n = biproduct.matrix (\u03bb j k, m j \u226b n j k) :=\nby { ext, simp, }\n\nend\n\n/-- Reindex a categorical biproduct via an equivalence of the index types. -/\n@[simps]\ndef biproduct.reindex {\u03b2 \u03b3 : Type v} [fintype \u03b2] [decidable_eq \u03b2] [decidable_eq \u03b3]\n  (\u03b5 : \u03b2 \u2243 \u03b3) (f : \u03b3 \u2192 C) [has_biproduct f] [has_biproduct (f \u2218 \u03b5)] : (\u2a01 (f \u2218 \u03b5)) \u2245 (\u2a01 f) :=\n{ hom := biproduct.desc (\u03bb b, biproduct.\u03b9 f (\u03b5 b)),\n  inv := biproduct.lift (\u03bb b, biproduct.\u03c0 f (\u03b5 b)),\n  hom_inv_id' := by { ext b b', by_cases h : b = b', { subst h, simp, }, { simp [h], }, },\n  inv_hom_id' := begin\n    ext g g',\n    by_cases h : g = g';\n    simp [preadditive.sum_comp, preadditive.comp_sum, biproduct.\u03b9_\u03c0, biproduct.\u03b9_\u03c0_assoc, comp_dite,\n      equiv.apply_eq_iff_eq_symm_apply, finset.sum_dite_eq' finset.univ (\u03b5.symm g') _, h],\n  end, }\n\n/--\nIn a preadditive category, we can construct a binary biproduct for `X Y : C` from\nany binary bicone `b` satisfying `total : b.fst \u226b b.inl + b.snd \u226b b.inr = \ud835\udfd9 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\ndef is_binary_bilimit_of_total {X Y : C} (b : binary_bicone X Y)\n  (total : b.fst \u226b b.inl + b.snd \u226b b.inr = \ud835\udfd9 b.X) : b.is_bilimit :=\n{ is_limit :=\n  { lift := \u03bb s, binary_fan.fst s \u226b b.inl +\n      binary_fan.snd s \u226b b.inr,\n    uniq' := \u03bb s m h, by erw [\u2190category.comp_id m, \u2190total,\n      comp_add, reassoc_of (h \u27e8walking_pair.left\u27e9), reassoc_of (h \u27e8walking_pair.right\u27e9)],\n    fac' := \u03bb s j, by rcases j with \u27e8\u27e8\u27e9\u27e9; simp, },\n  is_colimit :=\n  { desc := \u03bb s, b.fst \u226b binary_cofan.inl s +\n      b.snd \u226b binary_cofan.inr s,\n    uniq' := \u03bb s m h, by erw [\u2190category.id_comp m, \u2190total,\n      add_comp, category.assoc, category.assoc, h \u27e8walking_pair.left\u27e9, h \u27e8walking_pair.right\u27e9],\n    fac' := \u03bb s j, by rcases j with \u27e8\u27e8\u27e9\u27e9; simp, } }\n\nlemma is_bilimit.binary_total {X Y : C} {b : binary_bicone X Y} (i : b.is_bilimit) :\n  b.fst \u226b b.inl + b.snd \u226b b.inr = \ud835\udfd9 b.X :=\ni.is_limit.hom_ext (\u03bb j, by { rcases j with \u27e8\u27e8\u27e9\u27e9; simp, })\n\n/--\nIn a preadditive category, we can construct a binary biproduct for `X Y : C` from\nany binary bicone `b` satisfying `total : b.fst \u226b b.inl + b.snd \u226b b.inr = \ud835\udfd9 b.X`.\n\n(That is, such a bicone is a limit cone and a colimit cocone.)\n-/\nlemma has_binary_biproduct_of_total {X Y : C} (b : binary_bicone X Y)\n  (total : b.fst \u226b b.inl + b.snd \u226b b.inr = \ud835\udfd9 b.X) : has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := b,\n  is_bilimit := is_binary_bilimit_of_total b total }\n\n/-- We can turn any limit cone over a pair into a bicone. -/\n@[simps]\ndef binary_bicone.of_limit_cone {X Y : C} {t : cone (pair X Y)} (ht : is_limit t) :\n  binary_bicone X Y :=\n{ X := t.X,\n  fst := t.\u03c0.app \u27e8walking_pair.left\u27e9,\n  snd := t.\u03c0.app \u27e8walking_pair.right\u27e9,\n  inl := ht.lift (binary_fan.mk (\ud835\udfd9 X) 0),\n  inr := ht.lift (binary_fan.mk 0 (\ud835\udfd9 Y)) }\n\nlemma inl_of_is_limit {X Y : C} {t : binary_bicone X Y} (ht : is_limit t.to_cone) :\n  t.inl = ht.lift (binary_fan.mk (\ud835\udfd9 X) 0) :=\nht.hom_ext $ \u03bb j, by { rw ht.fac, rcases j with \u27e8\u27e8\u27e9\u27e9; simp }\n\nlemma inr_of_is_limit {X Y : C} {t : binary_bicone X Y} (ht : is_limit t.to_cone) :\n  t.inr = ht.lift (binary_fan.mk 0 (\ud835\udfd9 Y)) :=\nht.hom_ext $ \u03bb j, by { rw ht.fac, rcases j with \u27e8\u27e8\u27e9\u27e9; simp }\n\n/-- In a preadditive category, any binary bicone which is a limit cone is in fact a bilimit\n    bicone. -/\ndef is_binary_bilimit_of_is_limit {X Y : C} (t : binary_bicone X Y) (ht : is_limit t.to_cone) :\n  t.is_bilimit :=\nis_binary_bilimit_of_total _ (by refine binary_fan.is_limit.hom_ext ht _ _; simp)\n\n/-- We can turn any limit cone over a pair into a bilimit bicone. -/\ndef binary_bicone_is_bilimit_of_limit_cone_of_is_limit {X Y : C} {t : cone (pair X Y)}\n  (ht : is_limit t) : (binary_bicone.of_limit_cone ht).is_bilimit :=\nis_binary_bilimit_of_total _ $ binary_fan.is_limit.hom_ext ht (by simp) (by simp)\n\n/-- In a preadditive category, if the product of `X` and `Y` exists, then the\n    binary biproduct of `X` and `Y` exists. -/\nlemma has_binary_biproduct.of_has_binary_product (X Y : C) [has_binary_product X Y] :\n  has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := _,\n  is_bilimit := binary_bicone_is_bilimit_of_limit_cone_of_is_limit (limit.is_limit _) }\n\n/-- In a preadditive category, if all binary products exist, then all binary biproducts exist. -/\nlemma has_binary_biproducts.of_has_binary_products [has_binary_products C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := \u03bb X Y, has_binary_biproduct.of_has_binary_product X Y, }\n\n/-- We can turn any colimit cocone over a pair into a bicone. -/\n@[simps]\ndef binary_bicone.of_colimit_cocone {X Y : C} {t : cocone (pair X Y)} (ht : is_colimit t) :\n  binary_bicone X Y :=\n{ X := t.X,\n  fst := ht.desc (binary_cofan.mk (\ud835\udfd9 X) 0),\n  snd := ht.desc (binary_cofan.mk 0 (\ud835\udfd9 Y)),\n  inl := t.\u03b9.app \u27e8walking_pair.left\u27e9,\n  inr := t.\u03b9.app \u27e8walking_pair.right\u27e9 }\n\nlemma fst_of_is_colimit {X Y : C} {t : binary_bicone X Y} (ht : is_colimit t.to_cocone) :\n  t.fst = ht.desc (binary_cofan.mk (\ud835\udfd9 X) 0) :=\nbegin\n  refine ht.hom_ext (\u03bb j, _),\n  rw ht.fac,\n  rcases j with \u27e8\u27e8\u27e9\u27e9,\n  all_goals { simp only [binary_bicone.to_cocone_\u03b9_app_left, binary_bicone.inl_fst,\n      binary_cofan.mk_\u03b9_app_left, binary_bicone.to_cocone_\u03b9_app_right, binary_bicone.inr_fst,\n      binary_cofan.mk_\u03b9_app_right] },\n  refl\nend\n\nlemma snd_of_is_colimit {X Y : C} {t : binary_bicone X Y} (ht : is_colimit t.to_cocone) :\n  t.snd = ht.desc (binary_cofan.mk 0 (\ud835\udfd9 Y)) :=\nbegin\n  refine ht.hom_ext (\u03bb j, _),\n  rw ht.fac,\n  rcases j with \u27e8\u27e8\u27e9\u27e9,\n  all_goals { simp only [binary_bicone.to_cocone_\u03b9_app_left, binary_bicone.inl_snd,\n    binary_cofan.mk_\u03b9_app_left, binary_bicone.to_cocone_\u03b9_app_right, binary_bicone.inr_snd,\n    binary_cofan.mk_\u03b9_app_right] },\n  refl\nend\n\n/-- In a preadditive category, any binary bicone which is a colimit cocone is in fact a\n    bilimit bicone. -/\ndef is_binary_bilimit_of_is_colimit {X Y : C} (t : binary_bicone X Y)\n  (ht : is_colimit t.to_cocone) : t.is_bilimit :=\nis_binary_bilimit_of_total _\nbegin\n  refine binary_cofan.is_colimit.hom_ext ht _ _; simp,\n  { rw [category.comp_id t.inl] },\n  { rw [category.comp_id t.inr] }\nend\n\n/-- We can turn any colimit cocone over a pair into a bilimit bicone. -/\ndef binary_bicone_is_bilimit_of_colimit_cocone_of_is_colimit {X Y : C} {t : cocone (pair X Y)}\n  (ht : is_colimit t) : (binary_bicone.of_colimit_cocone ht).is_bilimit :=\nis_binary_bilimit_of_is_colimit (binary_bicone.of_colimit_cocone ht) $\n  is_colimit.of_iso_colimit ht $ cocones.ext (iso.refl _) $ \u03bb j, by { rcases j with \u27e8\u27e8\u27e9\u27e9, tidy }\n\n/-- In a preadditive category, if the coproduct of `X` and `Y` exists, then the\n    binary biproduct of `X` and `Y` exists. -/\nlemma has_binary_biproduct.of_has_binary_coproduct (X Y : C) [has_binary_coproduct X Y] :\n  has_binary_biproduct X Y :=\nhas_binary_biproduct.mk\n{ bicone := _,\n  is_bilimit := binary_bicone_is_bilimit_of_colimit_cocone_of_is_colimit (colimit.is_colimit _) }\n\n/-- In a preadditive category, if all binary coproducts exist, then all binary biproducts exist. -/\nlemma has_binary_biproducts.of_has_binary_coproducts [has_binary_coproducts C] :\n  has_binary_biproducts C :=\n{ has_binary_biproduct := \u03bb X Y, has_binary_biproduct.of_has_binary_coproduct X Y, }\n\nsection\nvariables {X Y : C} [has_binary_biproduct X Y]\n\n/--\nIn any preadditive category, any binary biproduct satsifies\n`biprod.fst \u226b biprod.inl + biprod.snd \u226b biprod.inr = \ud835\udfd9 (X \u229e Y)`.\n-/\n@[simp] lemma biprod.total : biprod.fst \u226b biprod.inl + biprod.snd \u226b biprod.inr = \ud835\udfd9 (X \u229e Y) :=\nbegin\n  ext; simp [add_comp],\nend\n\nlemma biprod.lift_eq {T : C} {f : T \u27f6 X} {g : T \u27f6 Y} :\n  biprod.lift f g = f \u226b biprod.inl + g \u226b biprod.inr :=\nbegin\n  ext; simp [add_comp],\nend\n\nlemma biprod.desc_eq {T : C} {f : X \u27f6 T} {g : Y \u27f6 T} :\n  biprod.desc f g = biprod.fst \u226b f + biprod.snd \u226b g :=\nbegin\n  ext; simp [add_comp],\nend\n\n@[simp, reassoc] lemma biprod.lift_desc {T U : C} {f : T \u27f6 X} {g : T \u27f6 Y} {h : X \u27f6 U} {i : Y \u27f6 U} :\n  biprod.lift f g \u226b biprod.desc h i = f \u226b h + g \u226b i :=\nby simp [biprod.lift_eq, biprod.desc_eq]\n\nlemma biprod.map_eq [has_binary_biproducts C] {W X Y Z : C} {f : W \u27f6 Y} {g : X \u27f6 Z} :\n  biprod.map f g = biprod.fst \u226b f \u226b biprod.inl + biprod.snd \u226b g \u226b biprod.inr :=\nby apply biprod.hom_ext; apply biprod.hom_ext'; simp\n\n/--\nEvery split mono `f` with a cokernel induces a binary bicone with `f` as its `inl` and\nthe cokernel map as its `snd`.\nWe will show in `is_bilimit_binary_bicone_of_split_mono_of_cokernel` that this binary bicone is in\nfact already a biproduct. -/\n@[simps]\ndef binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X \u27f6 Y} [split_mono f]\n  {c : cokernel_cofork f} (i : is_colimit c) : binary_bicone X c.X :=\n{ X := Y,\n  fst := retraction f,\n  snd := c.\u03c0,\n  inl := f,\n  inr :=\n    let c' : cokernel_cofork (\ud835\udfd9 Y - (\ud835\udfd9 Y - retraction f \u226b f)) :=\n      cokernel_cofork.of_\u03c0 (cofork.\u03c0 c) (by simp) in\n    let i' : is_colimit c' := is_cokernel_epi_comp i (retraction f) (by simp) in\n    let i'' := is_colimit_cofork_of_cokernel_cofork i' in\n    (split_epi_of_idempotent_of_is_colimit_cofork C (by simp) i'').section_,\n  inl_fst' := by simp,\n  inl_snd' := by simp,\n  inr_fst' :=\n  begin\n    dsimp only,\n    rw [split_epi_of_idempotent_of_is_colimit_cofork_section_,\n      is_colimit_cofork_of_cokernel_cofork_desc, is_cokernel_epi_comp_desc],\n    dsimp only [cokernel_cofork_of_cofork_of_\u03c0],\n    letI := epi_of_is_colimit_cofork i,\n    apply zero_of_epi_comp c.\u03c0,\n    simp only [sub_comp, comp_sub, category.comp_id, category.assoc, split_mono.id, sub_self,\n      cofork.is_colimit.\u03c0_desc_assoc, cokernel_cofork.\u03c0_of_\u03c0, split_mono.id_assoc],\n    apply sub_eq_zero_of_eq,\n    apply category.id_comp\n  end,\n  inr_snd' := by apply split_epi.id }\n\n/-- The bicone constructed in `binary_bicone_of_split_mono_of_cokernel` is a bilimit.\nThis is a version of the splitting lemma that holds in all preadditive categories. -/\ndef is_bilimit_binary_bicone_of_split_mono_of_cokernel {X Y : C} {f : X \u27f6 Y} [split_mono f]\n  {c : cokernel_cofork f} (i : is_colimit c) :\n  (binary_bicone_of_split_mono_of_cokernel i).is_bilimit :=\nis_binary_bilimit_of_total _\nbegin\n  simp only [binary_bicone_of_split_mono_of_cokernel_fst,\n    binary_bicone_of_split_mono_of_cokernel_inr, binary_bicone_of_split_mono_of_cokernel_snd,\n    split_epi_of_idempotent_of_is_colimit_cofork_section_],\n  dsimp only [binary_bicone_of_split_mono_of_cokernel_X],\n  rw [is_colimit_cofork_of_cokernel_cofork_desc, is_cokernel_epi_comp_desc],\n  simp only [binary_bicone_of_split_mono_of_cokernel_inl, cofork.is_colimit.\u03c0_desc,\n    cokernel_cofork_of_cofork_\u03c0, cofork.\u03c0_of_\u03c0, add_sub_cancel'_right]\nend\n\n/--\nEvery split epi `f` with a kernel induces a binary bicone with `f` as its `snd` and\nthe kernel map as its `inl`.\nWe will show in `binary_bicone_of_split_mono_of_cokernel` that this binary bicone is in fact\nalready a biproduct. -/\n@[simps]\ndef binary_bicone_of_split_epi_of_kernel {X Y : C} {f : X \u27f6 Y} [split_epi f]\n  {c : kernel_fork f} (i : is_limit c) : binary_bicone c.X Y :=\n{ X := X,\n  fst :=\n    let c' : kernel_fork (\ud835\udfd9 X - (\ud835\udfd9 X - f \u226b section_ f)) :=\n      kernel_fork.of_\u03b9 (fork.\u03b9 c) (by simp) in\n    let i' : is_limit c' := is_kernel_comp_mono i (section_ f) (by simp) in\n    let i'' := is_limit_fork_of_kernel_fork i' in\n    (split_mono_of_idempotent_of_is_limit_fork C (by simp) i'').retraction,\n  snd := f,\n  inl := c.\u03b9,\n  inr := section_ f,\n  inl_fst' := by apply split_mono.id,\n  inl_snd' := by simp,\n  inr_fst' :=\n  begin\n    dsimp only,\n    rw [split_mono_of_idempotent_of_is_limit_fork_retraction,\n      is_limit_fork_of_kernel_fork_lift, is_kernel_comp_mono_lift],\n    dsimp only [kernel_fork_of_fork_\u03b9],\n    letI := mono_of_is_limit_fork i,\n    apply zero_of_comp_mono c.\u03b9,\n    simp only [comp_sub, category.comp_id, category.assoc, sub_self, fork.is_limit.lift_\u03b9,\n      fork.\u03b9_of_\u03b9, split_epi.id_assoc]\n  end,\n  inr_snd' := by simp }\n\n/-- The bicone constructed in `binary_bicone_of_split_epi_of_kernel` is a bilimit.\nThis is a version of the splitting lemma that holds in all preadditive categories. -/\ndef is_bilimit_binary_bicone_of_split_epi_of_kernel {X Y : C} {f : X \u27f6 Y} [split_epi f]\n  {c : kernel_fork f} (i : is_limit c) :\n  (binary_bicone_of_split_epi_of_kernel i).is_bilimit :=\nis_binary_bilimit_of_total _\nbegin\n  simp only [binary_bicone_of_split_epi_of_kernel_fst, binary_bicone_of_split_epi_of_kernel_inl,\n    binary_bicone_of_split_epi_of_kernel_inr, binary_bicone_of_split_epi_of_kernel_snd,\n    split_mono_of_idempotent_of_is_limit_fork_retraction],\n  dsimp only [binary_bicone_of_split_epi_of_kernel_X],\n  rw [is_limit_fork_of_kernel_fork_lift, is_kernel_comp_mono_lift],\n  simp only [fork.is_limit.lift_\u03b9, fork.\u03b9_of_\u03b9, kernel_fork_of_fork_\u03b9, sub_add_cancel]\nend\n\nend\n\nsection\nvariables {X Y : C} (f g : X \u27f6 Y)\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\nlemma biprod.add_eq_lift_id_desc [has_binary_biproduct X X] :\n  f + g = biprod.lift (\ud835\udfd9 X) (\ud835\udfd9 X) \u226b biprod.desc f g :=\nby simp\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\nlemma biprod.add_eq_lift_desc_id [has_binary_biproduct Y Y] :\n  f + g = biprod.lift f g \u226b biprod.desc (\ud835\udfd9 Y) (\ud835\udfd9 Y) :=\nby simp\n\nend\n\nend preadditive\n\nend limits\n\nopen category_theory.limits\n\nsection\nlocal attribute [ext] preadditive\n\n/-- The existence of binary biproducts implies that there is at most one preadditive structure. -/\ninstance subsingleton_preadditive_of_has_binary_biproducts {C : Type u} [category.{v} C]\n  [has_zero_morphisms C] [has_binary_biproducts C] : subsingleton (preadditive C) :=\nsubsingleton.intro $ \u03bb a b,\nbegin\n  ext X Y f g,\n  have h\u2081 := @biprod.add_eq_lift_id_desc _ _ a _ _ f g\n    (by convert (infer_instance : has_binary_biproduct X X)),\n  have h\u2082 := @biprod.add_eq_lift_id_desc _ _ b _ _ f g\n    (by convert (infer_instance : has_binary_biproduct X X)),\n  refine h\u2081.trans (eq.trans _ h\u2082.symm),\n  congr' 2;\n  exact subsingleton.elim _ _\nend\nend\n\nvariables {C : Type u} [category.{v} C] [has_zero_morphisms C] [has_binary_biproducts C]\n\n/-- An object is indecomposable if it cannot be written as the biproduct of two nonzero objects. -/\ndef indecomposable (X : C) : Prop := \u00ac is_zero X \u2227 \u2200 Y Z, (X \u2245 Y \u229e Z) \u2192 is_zero Y \u2228 is_zero Z\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/shapes/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3828010350010169}}
{"text": "def g (xs ys : List Nat) : Nat :=\n  match xs, ys with\n  | [a, b], _ => Nat.succ (a+b)\n  | _, [b, c] => Nat.succ b\n  | _, _   => 1\n\nexample (a b : Bool) (x y z : Nat) (xs : List Nat) (h1 : (if a then x else y) = 0) (h2 : xs.head! = 0) : g [x] xs = 1 := by\n  simp [g]\n  repeat any_goals (split at *)\n  any_goals (first | decide | contradiction | injections)\n  next b c _ =>\n    show Nat.succ b = 1\n    simp [List.head!] at h2; simp [h2]\n  next b c _ =>\n    show Nat.succ b = 1\n    simp [List.head!] at h2; simp [h2]\n\nexample (a : Bool) (h1 : (if a then x else y) = 1) : x + y > 0 := by\n  split at h1\n  . subst h1; rw [Nat.succ_add]; apply Nat.zero_lt_succ\n  . subst h1; apply Nat.zero_lt_succ\n\ndef f (x : Nat) : Nat :=\n  match x with\n  | 100 => 0\n  | 200 => 0\n  | _   => 1\n\nexample (h1 : f x = 0) (h2 : x > 300) : False := by\n  simp [f] at h1\n  split at h1\n  . contradiction\n  . contradiction\n  . contradiction\n\nexample (h1 : f x = 0) (h2 : x > 300) : False := by\n  simp [f] at h1\n  split at h1 <;> contradiction\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/split3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.382801026470397}}
{"text": "/-\nCopyright (c) 2022 Mac Malone. All rights reserved.\nReleased under the MIT license described in the file LICENSE.\nAuthors: Mac Malone\n-/\n\nimport Folktale\nopen Folktale\n\n/-!\nEither A, B, or C is a traitor. A accuses B, B accuses C.\n\nWhen asked by the judge,\n\"Are all you the same type?\" (i.e. all knights or all knaves),\nC answers either yes or no.\n\nThe judge hears this answer and is not able to convict someone,\nbut is able to acquit someone.\n\nAfter that folk is acquitted, the judge asks one of the others:\n\"Are the two of you the same type?\" (i.e. both knights or both knaves).\nThe folk answers yes.\n\nWho is the traitor?\n\n**Clarifications:**\n* Exactly one of A, B, or C is a traitor.\n* The traitor can be a knight.\n* A, B, C know who the traitor is.\n* The second question is asked to one of them who was not acquitted.\n\n**Source:** *The Riddle of Scheherazade* by Raymond Smullyan\n-/\n\n/-! # Setting -/\n\naxiom A : Folk\naxiom B : Folk\naxiom C : Folk\n\nconstant traitor : Folk \u2192 Prop\n\n/-! # Rules -/\n\naxiom one_traitor :\n  traitor A \u2228 traitor B \u2228 traitor C\n\naxiom exactly_one_traitor :\n  \u00ac (traitor A \u2227 traitor B) \u2227\n  \u00ac (traitor A \u2227 traitor C) \u2227\n  \u00ac (traitor B \u2227 traitor C)\n\naxiom A_accuses_B :\n  A.say <| traitor B\n\naxiom B_accuses_C :\n  B.say <| traitor C\n\n/-- Of the two folks, one of them says they are both the same. -/\nabbrev answer2 (k1 k2 : Folk) :=\n  k1.say (same2 k1 k2) \u2228 k2.say (same2 k1 k2)\n\n/-- The folk that is acquitted determines which of the other two are asked. -/\nabbrev question2 (k : Folk) :=\n  ((k = A) \u2192 answer2 B C) \u2227\n  ((k = B) \u2192 answer2 A C) \u2227\n  ((k = C) \u2192 answer2 A B)\n\n/--\nIf C-answered yes (i.e., said they are all the same type),\nthen that lead to the acquittal of one and questioning of the other two.\nThen, with that information, the traitor was able to be determined.\n-/\naxiom question1_same {a t : Folk} :\n  (C.say (same3 A B C) \u2192 \u00ac traitor a) \u2192\n  (C.say (same3 A B C) \u2227 question2 a \u2192 traitor t) \u2192\n  traitor t\n\n/--\nIf C-answered no ((i.e., said they are not all the same type),\nthen that lead to the acquittal of one and questioning of the other two.\nThen, with that information, the traitor was able to be determined.\n-/\naxiom question1_not_same {a t : Folk} :\n  (C.say (\u00ac same3 A B C) \u2192 \u00ac traitor a) \u2192\n  (C.say (\u00ac same3 A B C) \u2227 question2 a \u2192 traitor t) \u2192\n  traitor t\n\n/-! # Deductions -/\n\n/-- If B and C are asked and C is a knave, then C is the traitor. -/\ntheorem answer2_B_C_traitor\n(answer2_B_C : answer2 B C) (knave_C : knave C) : traitor C := by\n  cases answer2_B_C with\n  | inl B_same =>\n    apply False.elim\n    cases knight_or_knave B with\n    | inl knight_B =>\n      let both_same := knight_truth knight_B B_same\n      cases both_same with\n      | inl both_knights =>\n        exact knight_not_knave both_knights.2 knave_C\n      | inr both_knaves =>\n        exact knight_not_knave knight_B both_knaves.1\n    | inr knave_B =>\n      let not_both_same := knave_lie knave_B B_same\n      exact not_both_same <| Or.inr <| And.intro knave_B knave_C\n  | inr C_same =>\n    let not_both_same := knave_lie knave_C C_same\n    let not_both_knave := Classical.dm_and (Classical.dm_or not_both_same).2\n    cases not_both_knave with\n    | inl not_knave_B =>\n      let knight_B := not_knave_knight not_knave_B\n      exact knight_truth knight_B B_accuses_C\n    | inr not_knave_C =>\n      contradiction\n\n/- If C said all were the same, then A is not the traitor and C is a knave. -/\ntheorem C_same_acquit_A_knave :\nC.say (same3 A B C) \u2192 \u00ac traitor A \u2227 knave C := by\n  intro C_same\n  cases knight_or_knave C with\n  | inl knight_C =>\n    apply False.elim\n    let all_same := knight_truth knight_C C_same\n    cases all_same with\n    | inl all_knights =>\n      let traitor_B := knight_truth all_knights.1 A_accuses_B\n      let traitor_C := knight_truth all_knights.2.1 B_accuses_C\n      exact exactly_one_traitor.2.2 <| And.intro traitor_B traitor_C\n    | inr all_knaves =>\n      exact knight_not_knave knight_C all_knaves.2.2\n  | inr knave_C =>\n    apply And.intro _ knave_C; intro traitor_A\n    let not_all_same := knave_lie knave_C C_same\n    let one_knight := not_same3_one_knight_knave not_all_same |>.1\n    cases one_knight with\n    | inl knight_A =>\n      let traitor_B := knight_truth knight_A A_accuses_B\n      exact exactly_one_traitor.1 <| And.intro traitor_A traitor_B\n    | inr knight_B_C =>\n      cases knight_B_C with\n      | inl knight_B =>\n        let traitor_C := knight_truth knight_B B_accuses_C\n        exact exactly_one_traitor.2.1 <| And.intro traitor_A traitor_C\n      | inr knight_C =>\n        exact knight_not_knave knight_C knave_C\n\n/-- C is the traitor. -/\ntheorem solution : traitor C := by\n  apply question1_same ?acquit ?convict\n  case acquit =>\n    intro C_same\n    exact C_same_acquit_A_knave C_same |>.1\n  case convict =>\n    intro \u27e8all_same, question2_A\u27e9\n    let answer2_B_C := question2_A.1 rfl\n    let knave_C := C_same_acquit_A_knave all_same |>.2\n    exact answer2_B_C_traitor answer2_B_C knave_C\n\n/-! # Side Note -/\n\n/-- If both A and B are knaves, the traitor is A. -/\ntheorem knave_A_B_traitor\n(knave_A : knave A) (knave_B : knave B) : traitor A := by\n  cases one_traitor with\n  | inl traitor_A =>\n    exact traitor_A\n  | inr traitor_B_C =>\n    cases traitor_B_C with\n    | inl traitor_B =>\n      let not_traitor_B := knave_lie knave_A A_accuses_B\n      contradiction\n    | inr traitor_C =>\n      let not_traitor_C := knave_lie knave_B B_accuses_C\n      contradiction\n\n/--\nIf C said they were not all the same,\nthere is not enough information to acquit anyone.\n\nSuch a result is hard to formalize, so it is skipped,\nbut the content of the proof here helps demonstrate it.\n-/\ntheorem C_not_same_insufficient :\nC.say (\u00ac same3 A B C) \u2192 traitor A \u2228 traitor B \u2228 traitor C := by\n  intro C_not_same\n  cases knight_or_knave C with\n  | inl knight_C =>\n    let not_all_same := knight_truth knight_C C_not_same\n    let one_knave := not_same3_one_knight_knave not_all_same |>.2\n    cases one_knave with\n    | inl knave_A =>\n      cases knight_or_knave B with\n      | inl knight_B =>\n        let traitor_C := knight_truth knight_B B_accuses_C\n        exact Or.inr <| Or.inr traitor_C\n      | inr knave_B =>\n        let traitor_A := knave_A_B_traitor knave_A knave_B\n        exact Or.inl <| traitor_A\n    | inr knave_B_C =>\n      cases knave_B_C with\n      | inl knave_B =>\n        cases knight_or_knave A with\n        | inl knight_A =>\n          let traitor_B := knight_truth knight_A A_accuses_B\n          exact Or.inr <| Or.inl traitor_B\n        | inr knave_A =>\n          let traitor_A := knave_A_B_traitor knave_A knave_B\n          exact Or.inl <| traitor_A\n      | inr knave_C =>\n        exact False.elim <| knight_not_knave knight_C knave_C\n  | inr knave_C =>\n    let all_same := Classical.dne <| knave_lie knave_C C_not_same\n    cases all_same with\n    | inl all_knights =>\n      exact False.elim <| knight_not_knave all_knights.2.2 knave_C\n    | inr all_knaves =>\n      let traitor_A := knave_A_B_traitor all_knaves.1 all_knaves.2.1\n      exact Or.inl <| traitor_A\n", "meta": {"author": "tydeu", "repo": "folktale", "sha": "2a61a29003e5a9d6b704f75eadb8fd117aab3c85", "save_path": "github-repos/lean/tydeu-folktale", "path": "github-repos/lean/tydeu-folktale/folktale-2a61a29003e5a9d6b704f75eadb8fd117aab3c85/puzzles/Traitor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.382801026470397}}
{"text": "/-!\n# Monads\n\nBuilding on [Functors](functors.lean.md) and [Applicatives](applicatives.lean.md) we can now\nintroduce [monads](https://en.wikipedia.org/wiki/Monad_%28category_theory%29).\n\nA monad is another type of abstract, functional structure. Let's explore what makes it different\nfrom the first two structures.\n\n## What is a Monad?\n\nA monad is a computational context. It provides a structure that allows you to chain together\noperations that have some kind of shared state or similar effect. Whereas pure functional code can\nonly operate on explicit input parameters and affect the program through explicit return values,\noperations in a monad can affect other computations in the chain implicitly through side effects,\nespecially modification of an implicitly shared value.\n\n## How are monads represented in Lean?\n\nLike functors and applicatives, monads are represented with a type class in Lean:\n\n```lean,ignore\nclass Monad (m : Type u \u2192 Type v) extends Applicative m, Bind m where\n```\n\nJust as every applicative is a functor, every monad is also an applicative and there's one more new\nbase type class used here that you need to understand, namely, `Bind`.\n\n```lean,ignore\nclass Bind (f : Type u \u2192 Type v) where\n  bind : {\u03b1 \u03b2 : Type u} \u2192 f \u03b1 \u2192 (\u03b1 \u2192 f \u03b2) \u2192 f \u03b2\n```\n\nThe `bind` operator also has infix notation `>>=` where `x >>= g` represents the result of executing\n`x` to get a value of type `f \u03b1` then unwrapping the value `\u03b1` from that and passing it to function\n`g` of type `\u03b1 \u2192 f \u03b2` returning the result of type `f \u03b2` where `f` is the target structure type\n(like `Option` or List)\n\nThis `bind` operation looks similar to the other ones you've seen so far, if you put them all\ntogether `Monad` has the following operations:\n\n```lean,ignore\nclass Monad (f : Type u \u2192 Type v) extends Applicative f, Bind f where\n  pure {\u03b1 : Type u} : \u03b1 \u2192 f \u03b1\n  map : {\u03b1 \u03b2 : Type u} \u2192 (\u03b1 \u2192 \u03b2) \u2192 f \u03b1 \u2192 f \u03b2\n  seq : {\u03b1 \u03b2 : Type u} \u2192 f (\u03b1 \u2192 \u03b2) \u2192 (Unit \u2192 f \u03b1) \u2192 f \u03b2\n  bind : {\u03b1 \u03b2 : Type u} \u2192 f \u03b1 \u2192 (\u03b1 \u2192 f \u03b2) \u2192 f \u03b2\n  ...\n```\n\nNotice `Monad` also contains `pure` it must also have a \"default\" way to wrap a value in the\nstructure.\n\nThe `bind` operator is similar to the applicative `seq` operator in that it chains two operations,\nwith one of them being function related. Notice that `bind`, `seq` and `map` all take a function of\nsome kind.  Let's examine those function types:\n\n- map: `(\u03b1 \u2192 \u03b2)`\n- seq: `f (\u03b1 \u2192 \u03b2)`\n- bind: `(\u03b1 \u2192 f \u03b2)`\n\nSo `map` is a pure function, `seq` is a pure function wrapped in the structure, and `bind` takes a\npure input but produces an output wrapped in the structure.\n\nNote: we are ignoring the `(Unit \u2192 f \u03b1)` function used by `seq` here since that has a special\npurpose explained in [Applicatives Lazy Evaluation](applicatives.lean.md#lazy-evaluation).\n\n## Basic Monad Example\n\nJust as `Option` is a functor and an applicative functor, it is also a monad! Let's start with how\n`Option` implements the Monad type class.\n\n-/\ninstance : Monad Option where\n  pure := Option.some\n  bind := Option.bind\n/-!\n\nwhere:\n\n```lean,ignore\ndef Option.bind : Option \u03b1 \u2192 (\u03b1 \u2192 Option \u03b2) \u2192 Option \u03b2\n  | none,   _ => none\n  | some a, f => f a\n```\n\n> **Side note**: this function definition is using a special shorthand syntax in Lean where the `:=\nmatch a, b with` code can be collapsed away. To make this more clear consider the following simpler\nexample, where `Option.bind` is using the second form like `bar`:\n\n-/\ndef foo (x : Option Nat) (y : Nat) : Option Nat :=\n  match x, y with\n  | none, _ => none\n  | some x, y => some (x + y)\n\ndef bar : Option Nat \u2192 Nat \u2192 Option Nat\n  | none, _ => none\n  | some x, y => some (x + y)\n\n#eval foo (some 1) 2  -- some 3\n#eval bar (some 1) 2  -- some 3\n/-!\nWhat is important is that `Option.bind` is using a `match` statement to unwrap the input value\n`Option \u03b1`, if it is `none` then it does nothing and returns `none`, if it has a value of type `\u03b1`\nthen it applies the function in the second argument `(\u03b1 \u2192 Option \u03b2)` to this value, which is\nthe expression `f a` that you see in the line `  | some a, f => f a` above.  The function\nreturns a result of type `Option \u03b2` which then becomes the return value for `bind`.  So there\nis no structure wrapping required on the return value since the input function already did that.\n\nBut let's bring in the definition of a monad. What does it mean to describe `Option` as a\ncomputational context?\n\nThe `Option` monad encapsulates the context of failure. Essentially, the `Option` monad lets us\nabort a series of operations whenever one of them fails. This allows future operations to assume\nthat all previous operations have succeeded. Here's some code to motivate this idea:\n\n-/\ndef optionFunc1 : String -> Option Nat\n  | \"\" => none\n  | str => some str.length\n\ndef optionFunc2 (i : Nat) : Option Float :=\n  if i % 2 == 0 then none else some (i.toFloat * 3.14159)\n\ndef optionFunc3 (f : Float) : Option (List Nat) :=\n  if f > 15.0 then none else some [f.floor.toUInt32.toNat, f.ceil.toUInt32.toNat]\n\ndef runOptionFuncs (input : String) : Option (List Nat) :=\n  match optionFunc1 input with\n  | none => none\n  | some i => match optionFunc2 i with\n    | none => none\n    | some f => optionFunc3 f\n\n#eval runOptionFuncs \"big\" -- some [9, 10]\n/-!\n\nHere you see three different functions that could fail. These are then combined in `runOptionFuncs`.\nBut then you have to use nested `match` expressions to check if the previous result succeeded. It\nwould be very tedious to continue this pattern much longer.\n\nThe `Option` monad helps you fix this. Here's what this function looks like using the `bind`\noperator.\n\n-/\n\ndef runOptionFuncsBind (input : String) : Option (List Nat) :=\n  optionFunc1 input >>= optionFunc2 >>= optionFunc3\n\n#eval runOptionFuncsBind \"big\" -- some [9, 10]\n/-!\n\nIt's much cleaner now! You take the first result and pass it into the second and third functions\nusing the `bind` operation. The monad instance handles all the failure cases so you don't have to!\n\nLet's see why the types work out. The result of `optionFunc1` input is simply `Option Nat`. Then the\nbind operator allows you to take this `Option Nat` value and combine it with `optionFunc2`, whose type\nis `Nat \u2192 Option Float` The **bind operator resolves** these to an `Option Float`. Then you pass this\nsimilarly through the bind operator to `optionFunc3`, resulting in the final type, `Option (List Nat)`.\n\nYour functions will not always combine so cleanly though. This is where `do` notation comes into play.\nThis notation allows you to write monadic operations one after another, line-by-line. It almost makes\nyour code look like imperative programming. You can rewrite the above as:\n-/\n\ndef runOptionFuncsDo (input : String) : Option (List Nat) := do\n  let i \u2190 optionFunc1 input\n  let f \u2190 optionFunc2 i\n  optionFunc3 f\n\n#eval runOptionFuncsDo \"big\" -- some [9, 10]\n/-!\n\nThe `\u2190` operator used here is special. It effectively unwraps the value on the right-hand side from\nthe monad. This means the value `i` has type `Nat`, _even though_ the result of `optionFunc1` is\n`Option Nat`. This is done using a `bind` operation under the hood.\n\n> Note you can use `<-` or the nice unicode symbol `\u2190` which you can type into VS code by typing\nthese characters `\\l `.  When you type the final space, `\\l` is replaced with `\u2190`.\n\nObserve that we do not unwrap the final line of the computation. The function result is `Option\n(List Nat)` which matches what `optionFunc3` returns. At first glance, this may look more complicated\nthan the `bind` example. However, it gives you a lot more flexibility, like mixing monadic and\nnon-monadic statements, using if then/else structures with their own local do blocks and so on. It\nis particularly helpful when one monadic function depends on multiple previous functions.\n\n## Example using List\n\nYou can easily make `List` into a monad with the following, since List already provides an\nimplementation of `pure` and `bind`.\n\n-/\ninstance : Monad List  where\n  pure := List.pure\n  bind := List.bind\n/-!\n\nLike you saw with the applicative `seq` operator, the `bind` operator applies the given function\nto every element of the list.  It is useful to look at the bind implementation for List:\n\n-/\nopen List\ndef bind (a : List \u03b1) (b : \u03b1 \u2192 List \u03b2) : List \u03b2 := join (map b a)\n/-!\n\nSo `Functor.map` is used to apply the function `b` to every element of `a` but this would\nreturn a whole bunch of little lists, so `join` is used to turn those back into a single list.\n\nHere's an example where you use `bind` to convert a list of strings into a combined list of chars:\n\n-/\n\n#eval \"apple\".toList  -- ['a', 'p', 'p', 'l', 'e']\n\n#eval [\"apple\", \"orange\"] >>= String.toList\n-- ['a', 'p', 'p', 'l', 'e', 'o', 'r', 'a', 'n', 'g', 'e']\n\n/-!\n\n\n## The IO Monad\n\nThe `IO Monad` is perhaps the most important monad in Lean. It is also one of the hardest monads to\nunderstand starting out. Its actual implementation is too intricate to discuss when first learning\nmonads. So it is best to learn by example.\n\nWhat is the **computational context** that describes the IO monad? IO operations can read\ninformation from or write information to the terminal, file system, operating system, and/or\nnetwork. They interact with systems outside of your program. If you want to get user input, print a\nmessage to the user, read information from a file, or make a network call, you'll need to do so\nwithin the IO Monad.\n\nThe state of the world outside your program can change at virtually any moment, and so this IO\ncontext is particularly special. So these IO operations are \"side effects\" which means you cannot\nperform them from \"pure\" Lean functions.\n\nNow, the most important job of pretty much any computer program is precisely to perform this\ninteraction with the outside world. For this reason, the root of all executable Lean code is a\nfunction called main, with the type `IO Unit`. So every program starts in the IO monad!\n\nWhen your function is `IO` monadic, you can get any input you need, call into \"pure\" code with the\ninputs, and then output the result in some way. The reverse does not work. You cannot call into IO\ncode from pure code like you can call into a function that takes `Option` as input. Another way to\nsay this is you cannot invent an `IO` context out of thin air, it has to be given to you in your\n`main` function.\n\nLet's look at a simple program showing a few of the basic IO functions. It also uses `do` notation\nto make the code read nicely:\n-/\ndef main : IO Unit := do\n  IO.println \"enter a line of text:\"\n  let stdin \u2190 IO.getStdin            -- IO IO.FS.Stream (monadic)\n  let input \u2190 stdin.getLine          -- IO.FS.Stream \u2192 IO String (monadic)\n  let uppercased := input.toUpper    -- String \u2192 String (pure)\n  IO.println uppercased              -- IO Unit (monadic)\n/-!\n\nSo, once again you can see that the `do` notation lets you chain a series of monadic actions.\n`IO.getStdin` is of type `IO IO.FS.Stream` and `stdin.getLine` is of type `IO String`\nand `IO.println` is of type `IO Unit`.\n\nIn between you see a non-monadic expression `let uppercased := input.toUpper` which is fine too.\nA let statement can occur in any monad. Just as you could unwrap `i` from `Option Nat` to get the\ninner Nat, you can use `\u2190` to unwrap the result of `getLine` to get a String. You can then manipulate\nthis value using normal pure string functions like `toUpper`, and then you can pass the result to the\n`IO.println` function.\n\nThis is a simple echo program. It reads a line from the terminal, and then prints the line back out\ncapitalized to the terminal. Hopefully it gives you a basic understanding of how IO works.\n\nYou can test this program using `lean --run` as follows:\n\n```\n> lean --run Main.lean\nenter a line of text:\nthe quick brown fox\nTHE QUICK BROWN FOX\n```\n\nHere the user entered the string `the quick brown fox` and got back the uppercase result.\n\n## What separates Monads from Applicatives?\n\nThe key that separates these is **context**. You cannot really determine the structure of\n\"future\" operations without knowing the results of \"past\" operations, because the past can alter the\ncontext in which the future operations work. With applicatives, you can't get the final function\nresult without evaluating everything, but you can determine the structure of how the operation will\ntake place. This allows some degree of parallelism with applicatives that is not generally possible\nwith monads.\n\n\n## Conclusion\n\nHopefully you now have a basic level understanding of what a monad is. But perhaps some more\nexamples of what a \"computational context\" means would be useful to you. The Reader, State and\nExcept monads each provide a concrete and easily understood context that can be compared easily to\nfunction parameters. You can learn more about those in [Reader monads](readers.lean.md),\n[State monads](states.lean.md), and the [Except monad](except.lean.md).\n-/\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/doc/monads/monads.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.3828010261340282}}
{"text": "/-\nCopyright (c) 2021 Yuma Mizuno. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno\n-/\nimport category_theory.isomorphism\nimport tactic.slice\n\n/-!\n# Bicategories\n\nIn this file we define typeclass for bicategories.\n\nA bicategory `B` consists of\n* objects `a : B`,\n* 1-morphisms `f : a \u27f6 b` between objects `a b : B`, and\n* 2-morphisms `\u03b7 : f \u27f6 g` beween 1-morphisms `f g : a \u27f6 b` between objects `a b : B`.\n\nWe use `u`, `v`, and `w` as the universe variables for objects, 1-morphisms, and 2-morphisms,\nrespectively.\n\nA typeclass for bicategories extends `category_theory.category_struct` typeclass. This means that\nwe have\n* a composition `f \u226b g : a \u27f6 c` for each 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c`, and\n* a identity `\ud835\udfd9 a : a \u27f6 a` for each object `a : B`.\n\nFor each object `a b : B`, the collection of 1-morphisms `a \u27f6 b` has a category structure. The\n2-morphisms in the bicategory are implemented as the morphisms in this family of categories.\n\nThe composition of 1-morphisms is in fact a object part of a functor\n`(a \u27f6 b) \u2964 (b \u27f6 c) \u2964 (a \u27f6 c)`. The definition of bicategories in this file does not\nrequire this functor directly. Instead, it requires the whiskering functions. For a 1-morphism\n`f : a \u27f6 b` and a 2-morphism `\u03b7 : g \u27f6 h` between 1-morphisms `g h : b \u27f6 c`, there is a\n2-morphism `whisker_left f \u03b7 : f \u226b g \u27f6 f \u226b h`. Similarly, for a 2-morphism `\u03b7 : f \u27f6 g`\nbetween 1-morphisms `f g : a \u27f6 b` and a 1-morphism `f : b \u27f6 c`, there is a 2-morphism\n`whisker_right \u03b7 h : f \u226b h \u27f6 g \u226b h`. These satisfy the exchange law\n`whisker_left f \u03b8 \u226b whisker_right \u03b7 i = whisker_right \u03b7 h \u226b whisker_left g \u03b8`,\nwhich is required as an axiom in the definition here.\n-/\n\nnamespace category_theory\n\nuniverses w v u\n\nopen category iso\n\n/--\nIn a bicategory, we can compose the 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c` to obtain\na 1-morphism `f \u226b g : a \u27f6 c`. This composition does not need to be strictly associative,\nbut there is a specified associator, `\u03b1_ f g h : (f \u226b g) \u226b h \u2245 f \u226b (g \u226b h)`.\nThere is an identity 1-morphism `\ud835\udfd9 a : a \u27f6 a`, with specified left and right unitor\nisomorphisms `\u03bb_ f : \ud835\udfd9 a \u226b f \u2245 f` and `\u03c1_ f : f \u226b \ud835\udfd9 a \u2245 f`.\nThese associators and unitors satisfy the pentagon and triangle equations.\n\nSee https://ncatlab.org/nlab/show/bicategory.\n-/\n@[nolint check_univs] -- intended to be used with explicit universe parameters\nclass bicategory (B : Type u) extends category_struct.{v} B :=\n-- category structure on the collection of 1-morphisms:\n(hom_category : \u2200 (a b : B), category.{w} (a \u27f6 b) . tactic.apply_instance)\n-- left whiskering:\n(whisker_left {a b c : B} (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) : f \u226b g \u27f6 f \u226b h)\n(infixr ` \u25c1 `:70 := whisker_left)\n-- functoriality of left whiskering:\n(whisker_left_id' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c), f \u25c1 \ud835\udfd9 g = \ud835\udfd9 (f \u226b g) . obviously)\n(whisker_left_comp' :\n  \u2200 {a b c} (f : a \u27f6 b) {g h i : b \u27f6 c} (\u03b7 : g \u27f6 h) (\u03b8 : h \u27f6 i),\n  f \u25c1 (\u03b7 \u226b \u03b8) = (f \u25c1 \u03b7) \u226b (f \u25c1 \u03b8) . obviously)\n-- right whiskering:\n(whisker_right {a b c : B} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) : f \u226b h \u27f6 g \u226b h)\n(infixr ` \u25b7 `:70 := whisker_right)\n-- functoriality of right whiskering:\n(whisker_right_id' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c), \ud835\udfd9 f \u25b7 g = \ud835\udfd9 (f \u226b g) . obviously)\n(whisker_right_comp' :\n  \u2200 {a b c} {f g h : a \u27f6 b} (\u03b7 : f \u27f6 g) (\u03b8 : g \u27f6 h) (i : b \u27f6 c),\n  (\u03b7 \u226b \u03b8) \u25b7 i = (\u03b7 \u25b7 i) \u226b (\u03b8 \u25b7 i) . obviously)\n-- exchange law of left and right whiskerings:\n(whisker_exchange' : \u2200 {a b c} {f g : a \u27f6 b} {h i : b \u27f6 c} (\u03b7 : f \u27f6 g) (\u03b8 : h \u27f6 i),\n  (f \u25c1 \u03b8) \u226b (\u03b7 \u25b7 i) = (\u03b7 \u25b7 h) \u226b (g \u25c1 \u03b8) . obviously)\n-- associator:\n(associator {a b c d : B} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) :\n  (f \u226b g) \u226b h \u2245 f \u226b (g \u226b h))\n(notation `\u03b1_` := associator)\n(associator_naturality_left' :\n  \u2200 {a b c d} {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d),\n  ((\u03b7 \u25b7 g) \u25b7 h) \u226b (\u03b1_ f' g h).hom = (\u03b1_ f g h).hom \u226b (\u03b7 \u25b7 (g \u226b h)) . obviously)\n(associator_naturality_middle' :\n  \u2200 {a b c d} (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d),\n  ((f \u25c1 \u03b7) \u25b7 h) \u226b (\u03b1_ f g' h).hom = (\u03b1_ f g h).hom \u226b (f \u25c1 (\u03b7 \u25b7 h)) . obviously)\n(associator_naturality_right' :\n  \u2200 {a b c d} (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h'),\n  ((f \u226b g) \u25c1 \u03b7) \u226b (\u03b1_ f g h').hom = (\u03b1_ f g h).hom \u226b (f \u25c1 (g \u25c1 \u03b7)) . obviously)\n--left unitor:\n(left_unitor {a b : B} (f : a \u27f6 b) : \ud835\udfd9 a \u226b f \u2245 f)\n(notation `\u03bb_` := left_unitor)\n(left_unitor_naturality' : \u2200 {a b} {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f'),\n  (\ud835\udfd9 a \u25c1 \u03b7) \u226b (\u03bb_ f').hom = (\u03bb_ f ).hom \u226b \u03b7 . obviously)\n-- right unitor:\n(right_unitor {a b : B} (f : a \u27f6 b) : f \u226b \ud835\udfd9 b \u2245 f)\n(notation `\u03c1_` := right_unitor)\n(right_unitor_naturality' : \u2200 {a b} {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f'),\n  (\u03b7 \u25b7 \ud835\udfd9 b) \u226b (\u03c1_ f').hom = (\u03c1_ f ).hom \u226b \u03b7 . obviously)\n-- pentagon identity:\n(pentagon' : \u2200 {a b c d e} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e),\n  ((\u03b1_ f g h).hom \u25b7 i) \u226b (\u03b1_ f (g \u226b h) i).hom \u226b (f \u25c1 (\u03b1_ g h i).hom) =\n    (\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom . obviously)\n-- triangle identity:\n(triangle' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c),\n  (\u03b1_ f (\ud835\udfd9 b) g).hom \u226b (f \u25c1 (\u03bb_ g).hom) = (\u03c1_ f).hom \u25b7 g . obviously)\n\nrestate_axiom bicategory.whisker_left_id'\nrestate_axiom bicategory.whisker_left_comp'\nrestate_axiom bicategory.whisker_right_id'\nrestate_axiom bicategory.whisker_right_comp'\nrestate_axiom bicategory.whisker_exchange'\nrestate_axiom bicategory.associator_naturality_left'\nrestate_axiom bicategory.associator_naturality_middle'\nrestate_axiom bicategory.associator_naturality_right'\nrestate_axiom bicategory.left_unitor_naturality'\nrestate_axiom bicategory.right_unitor_naturality'\nrestate_axiom bicategory.pentagon'\nrestate_axiom bicategory.triangle'\nattribute [simp]\n  bicategory.whisker_left_id bicategory.whisker_right_id\n  bicategory.whisker_exchange bicategory.triangle\nattribute [reassoc]\n  bicategory.whisker_left_comp bicategory.whisker_right_comp\n  bicategory.whisker_exchange bicategory.associator_naturality_left\n  bicategory.associator_naturality_middle bicategory.associator_naturality_right\n  bicategory.left_unitor_naturality bicategory.right_unitor_naturality\n  bicategory.pentagon bicategory.triangle\nattribute [simp] bicategory.whisker_left_comp bicategory.whisker_right_comp\nattribute [instance] bicategory.hom_category\n\nlocalized \"infixr ` \u25c1 `:70 := bicategory.whisker_left\" in bicategory\nlocalized \"infixr ` \u25b7 `:70 := bicategory.whisker_right\" in bicategory\nlocalized \"notation `\u03b1_` := bicategory.associator\" in bicategory\nlocalized \"notation `\u03bb_` := bicategory.left_unitor\" in bicategory\nlocalized \"notation `\u03c1_` := bicategory.right_unitor\" in bicategory\n\nnamespace bicategory\n\nsection\n\nvariables {B : Type u} [bicategory.{w v} B] {a b c d e : B}\n\n@[simp, reassoc]\nlemma hom_inv_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  (f \u25c1 \u03b7.hom) \u226b (f \u25c1 \u03b7.inv) = \ud835\udfd9 (f \u226b g) :=\nby rw [\u2190whisker_left_comp, hom_inv_id, whisker_left_id]\n\n@[simp, reassoc]\nlemma hom_inv_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  (\u03b7.hom \u25b7 h) \u226b (\u03b7.inv \u25b7 h) = \ud835\udfd9 (f \u226b h) :=\nby rw [\u2190whisker_right_comp, hom_inv_id, whisker_right_id]\n\n@[simp, reassoc]\nlemma inv_hom_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  (f \u25c1 \u03b7.inv) \u226b (f \u25c1 \u03b7.hom) = \ud835\udfd9 (f \u226b h) :=\nby rw [\u2190whisker_left_comp, inv_hom_id, whisker_left_id]\n\n@[simp, reassoc]\nlemma inv_hom_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  (\u03b7.inv \u25b7 h) \u226b (\u03b7.hom \u25b7 h) = \ud835\udfd9 (g \u226b h) :=\nby rw [\u2190whisker_right_comp, inv_hom_id, whisker_right_id]\n\n/-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whisker_left_iso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  f \u226b g \u2245 f \u226b h :=\n{ hom := f \u25c1 \u03b7.hom,\n  inv := f \u25c1 \u03b7.inv,\n  hom_inv_id' := by simp only [hom_inv_whisker_left],\n  inv_hom_id' := by simp only [inv_hom_whisker_left] }\n\ninstance whisker_left_is_iso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [is_iso \u03b7] :\n  is_iso (f \u25c1 \u03b7) :=\nis_iso.of_iso (whisker_left_iso f (as_iso \u03b7))\n\n@[simp]\nlemma inv_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [is_iso \u03b7] :\n  inv (f \u25c1 \u03b7) = f \u25c1 (inv \u03b7) :=\nby { ext, simp only [\u2190whisker_left_comp, whisker_left_id, is_iso.hom_inv_id] }\n\n/-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whisker_right_iso {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  f \u226b h \u2245 g \u226b h :=\n{ hom := \u03b7.hom \u25b7 h,\n  inv := \u03b7.inv \u25b7 h,\n  hom_inv_id' := by simp only [hom_inv_whisker_right],\n  inv_hom_id' := by simp only [inv_hom_whisker_right] }\n\ninstance whisker_right_is_iso {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [is_iso \u03b7] :\n  is_iso (\u03b7 \u25b7 h) :=\nis_iso.of_iso (whisker_right_iso (as_iso \u03b7) h)\n\n@[simp]\nlemma inv_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [is_iso \u03b7] :\n  inv (\u03b7 \u25b7 h) = (inv \u03b7) \u25b7 h :=\nby { ext, simp only [\u2190whisker_right_comp, whisker_right_id, is_iso.hom_inv_id] }\n\n@[reassoc]\nlemma left_unitor_inv_naturality {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') :\n  \u03b7 \u226b (\u03bb_ f').inv = (\u03bb_ f).inv \u226b (\ud835\udfd9 a \u25c1 \u03b7) :=\nbegin\n  apply (cancel_mono (\u03bb_ f').hom).1,\n  simp only [assoc, comp_id, inv_hom_id, left_unitor_naturality, inv_hom_id_assoc]\nend\n\n@[reassoc]\nlemma right_unitor_inv_naturality {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') :\n  \u03b7 \u226b (\u03c1_ f').inv = (\u03c1_ f ).inv \u226b (\u03b7 \u25b7 \ud835\udfd9 b) :=\nbegin\n  apply (cancel_mono (\u03c1_ f').hom).1,\n  simp only [assoc, comp_id, inv_hom_id, right_unitor_naturality, inv_hom_id_assoc]\nend\n\n@[simp]\nlemma right_unitor_conjugation {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  (\u03c1_ f).inv \u226b (\u03b7 \u25b7 \ud835\udfd9 b) \u226b (\u03c1_ g).hom = \u03b7 :=\nby rw [right_unitor_naturality, inv_hom_id_assoc]\n\n@[simp]\nlemma left_unitor_conjugation {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  (\u03bb_ f).inv \u226b (\ud835\udfd9 a \u25c1 \u03b7) \u226b (\u03bb_ g).hom = \u03b7 :=\nby rw [left_unitor_naturality, inv_hom_id_assoc]\n\n@[simp]\nlemma whisker_left_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) :\n  (\ud835\udfd9 a \u25c1 \u03b7 = \ud835\udfd9 a \u25c1 \u03b8) \u2194 (\u03b7 = \u03b8) :=\nby rw [\u2190cancel_mono (\u03bb_ g).hom, left_unitor_naturality, left_unitor_naturality,\n    cancel_iso_hom_left]\n\n@[simp]\nlemma whisker_right_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) :\n  (\u03b7 \u25b7 \ud835\udfd9 b = \u03b8 \u25b7 \ud835\udfd9 b) \u2194 (\u03b7 = \u03b8) :=\nby rw [\u2190cancel_mono (\u03c1_ g).hom, right_unitor_naturality, right_unitor_naturality,\n    cancel_iso_hom_left]\n\n@[reassoc]\nlemma left_unitor_comp' (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03b1_ (\ud835\udfd9 a) f g).hom \u226b (\u03bb_ (f \u226b g)).hom = (\u03bb_ f).hom \u25b7 g :=\nby rw [\u2190whisker_left_iff, whisker_left_comp, \u2190cancel_epi (\u03b1_ (\ud835\udfd9 a) (\ud835\udfd9 a \u226b f) g).hom,\n    \u2190cancel_epi ((\u03b1_ (\ud835\udfd9 a) (\ud835\udfd9 a) f).hom \u25b7 g), pentagon_assoc, triangle,\n    \u2190associator_naturality_middle, \u2190whisker_right_comp_assoc, triangle,\n    associator_naturality_left, cancel_iso_hom_left]\n\n-- We state it as a `@[simp]` lemma. Generally, we think the component index of a natural\n-- transformation \"weighs more\" in considering the complexity of an expression than\n-- does a structural isomorphism (associator, etc).\n@[reassoc, simp]\nlemma left_unitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ (f \u226b g)).hom = (\u03b1_ (\ud835\udfd9 a) f g).inv \u226b ((\u03bb_ f).hom \u25b7 g) :=\nby { rw [\u2190left_unitor_comp', inv_hom_id_assoc] }\n\nlemma left_unitor_comp_inv' (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ (f \u226b g)).inv \u226b (\u03b1_ (\ud835\udfd9 a) f g).inv = ((\u03bb_ f).inv \u25b7 g) :=\neq_of_inv_eq_inv (by simp only [left_unitor_comp, inv_whisker_right,\n  is_iso.iso.inv_inv, hom_inv_id_assoc, is_iso.inv_comp])\n\n@[reassoc, simp]\nlemma left_unitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ (f \u226b g)).inv = ((\u03bb_ f).inv \u25b7 g) \u226b (\u03b1_ (\ud835\udfd9 a) f g).hom :=\nby { rw [\u2190left_unitor_comp_inv'], simp only [inv_hom_id, assoc, comp_id] }\n\n@[reassoc, simp]\nlemma right_unitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03c1_ (f \u226b g)).hom = (\u03b1_ f g (\ud835\udfd9 c)).hom \u226b (f \u25c1 (\u03c1_ g).hom) :=\nby rw [\u2190whisker_right_iff, whisker_right_comp, \u2190cancel_mono (\u03b1_ f g (\ud835\udfd9 c)).hom,\n    assoc, associator_naturality_middle, \u2190triangle_assoc, \u2190triangle,\n    whisker_left_comp, pentagon_assoc, \u2190associator_naturality_right]\n\n@[reassoc, simp]\nlemma right_unitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03c1_ (f \u226b g)).inv = (f \u25c1 (\u03c1_ g).inv) \u226b (\u03b1_ f g (\ud835\udfd9 c)).inv :=\neq_of_inv_eq_inv (by simp only [inv_whisker_left, right_unitor_comp,\n  is_iso.iso.inv_inv, is_iso.inv_comp])\n\n@[reassoc]\nlemma whisker_left_right_unitor_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03c1_ g).inv = (\u03c1_ (f \u226b g)).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).hom :=\nby simp only [right_unitor_comp_inv, comp_id, inv_hom_id, assoc]\n\n@[reassoc]\nlemma whisker_left_right_unitor (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03c1_ g).hom = (\u03b1_ f g (\ud835\udfd9 c)).inv \u226b (\u03c1_ (f \u226b g)).hom :=\nby simp only [right_unitor_comp, inv_hom_id_assoc]\n\n@[reassoc]\nlemma left_unitor_inv_whisker_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ f).inv \u25b7 g = (\u03bb_ (f \u226b g)).inv \u226b (\u03b1_ (\ud835\udfd9 a) f g).inv :=\nby simp only [left_unitor_comp_inv, assoc, comp_id, hom_inv_id]\n\n@[reassoc]\nlemma left_unitor_whisker_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ f).hom \u25b7 g = (\u03b1_ (\ud835\udfd9 a) f g).hom \u226b (\u03bb_ (f \u226b g)).hom :=\nby simp only [left_unitor_comp, hom_inv_id_assoc]\n\n@[reassoc]\nlemma associator_inv_naturality_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b7 \u25b7 (g \u226b h)) \u226b (\u03b1_ f' g h).inv = (\u03b1_ f g h).inv \u226b ((\u03b7 \u25b7 g) \u25b7 h) :=\nby rw [comp_inv_eq, assoc, associator_naturality_left, inv_hom_id_assoc]\n\n@[reassoc]\nlemma associator_conjugation_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b1_ f g h).hom \u226b (\u03b7 \u25b7 (g \u226b h)) \u226b (\u03b1_ f' g h).inv = (\u03b7 \u25b7 g) \u25b7 h :=\nby rw [associator_inv_naturality_left, hom_inv_id_assoc]\n\n@[reassoc]\nlemma associator_inv_conjugation_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b1_ f g h).inv \u226b ((\u03b7 \u25b7 g) \u25b7 h) \u226b (\u03b1_ f' g h).hom = \u03b7 \u25b7 (g \u226b h) :=\nby rw [associator_naturality_left, inv_hom_id_assoc]\n\n@[reassoc]\nlemma associator_inv_naturality_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  (f \u25c1 (\u03b7 \u25b7 h)) \u226b (\u03b1_ f g' h).inv = (\u03b1_ f g h).inv \u226b ((f \u25c1 \u03b7) \u25b7 h) :=\nby rw [comp_inv_eq, assoc, associator_naturality_middle, inv_hom_id_assoc]\n\n@[reassoc]\nlemma associator_conjugation_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  (\u03b1_ f g h).hom \u226b (f \u25c1 (\u03b7 \u25b7 h)) \u226b (\u03b1_ f g' h).inv = (f \u25c1 \u03b7) \u25b7 h :=\nby rw [associator_inv_naturality_middle, hom_inv_id_assoc]\n\n@[reassoc]\nlemma associator_inv_conjugation_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  (\u03b1_ f g h).inv \u226b ((f \u25c1 \u03b7) \u25b7 h) \u226b (\u03b1_ f g' h).hom = f \u25c1 (\u03b7 \u25b7 h) :=\nby rw [associator_naturality_middle, inv_hom_id_assoc]\n\n@[reassoc]\nlemma associator_inv_naturality_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  (f \u25c1 (g \u25c1 \u03b7)) \u226b (\u03b1_ f g h').inv = (\u03b1_ f g h).inv \u226b ((f \u226b g) \u25c1 \u03b7) :=\nby rw [comp_inv_eq, assoc, associator_naturality_right, inv_hom_id_assoc]\n\n@[reassoc]\nlemma associator_conjugation_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  (\u03b1_ f g h).hom \u226b (f \u25c1 (g \u25c1 \u03b7)) \u226b (\u03b1_ f g h').inv = (f \u226b g) \u25c1 \u03b7 :=\nby rw [associator_inv_naturality_right, hom_inv_id_assoc]\n\n@[reassoc]\nlemma associator_inv_conjugation_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  (\u03b1_ f g h).inv \u226b ((f \u226b g) \u25c1 \u03b7) \u226b (\u03b1_ f g h').hom = f \u25c1 (g \u25c1 \u03b7) :=\nby rw [associator_naturality_right, inv_hom_id_assoc]\n\n@[reassoc]\nlemma pentagon_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (f \u25c1 (\u03b1_ g h i).inv) \u226b (\u03b1_ f (g \u226b h) i).inv \u226b ((\u03b1_ f g h).inv \u25b7 i) =\n    (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv :=\neq_of_inv_eq_inv (by simp only [pentagon, inv_whisker_left, inv_whisker_right,\n  is_iso.iso.inv_inv, is_iso.inv_comp, assoc])\n\n@[reassoc]\nlemma pentagon_inv_inv_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f (g \u226b h) i).inv \u226b ((\u03b1_ f g h).inv \u25b7 i) \u226b (\u03b1_ (f \u226b g) h i).hom =\n    (f \u25c1 (\u03b1_ g h i).hom) \u226b (\u03b1_ f g (h \u226b i)).inv :=\nbegin\n  rw \u2190((eq_comp_inv _).mp (pentagon_inv f g h i)),\n  slice_rhs 1 2 { rw [\u2190whisker_left_comp, hom_inv_id] },\n  simp only [assoc, id_comp, whisker_left_id]\nend\n\n@[reassoc]\nlemma pentagon_inv_hom_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ (f \u226b g) h i).inv \u226b ((\u03b1_ f g h).hom \u25b7 i) \u226b (\u03b1_ f (g \u226b h) i).hom =\n    (\u03b1_ f g (h \u226b i)).hom \u226b (f \u25c1 (\u03b1_ g h i).inv) :=\neq_of_inv_eq_inv (by simp only [pentagon_inv_inv_hom_hom_inv, inv_whisker_left,\n  is_iso.iso.inv_hom, inv_whisker_right, is_iso.iso.inv_inv, is_iso.inv_comp, assoc])\n\n@[reassoc]\nlemma pentagon_hom_inv_inv_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (f \u25c1 (\u03b1_ g h i).hom) \u226b (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv =\n    (\u03b1_ f (g \u226b h) i).inv \u226b ((\u03b1_ f g h).inv \u25b7 i) :=\nbegin\n  rw \u2190((eq_comp_inv _).mp (pentagon_inv f g h i)),\n  slice_lhs 1 2 { rw [\u2190whisker_left_comp, hom_inv_id] },\n  simp only [assoc, id_comp, whisker_left_id, comp_id, hom_inv_id]\nend\n\n@[reassoc]\n\n\n@[reassoc]\nlemma pentagon_hom_inv_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f g (h \u226b i)).hom \u226b (f \u25c1 (\u03b1_ g h i).inv) \u226b (\u03b1_ f (g \u226b h) i).inv =\n    (\u03b1_ (f \u226b g) h i).inv \u226b ((\u03b1_ f g h).hom \u25b7 i) :=\nbegin\n  have pent := pentagon f g h i,\n  rw \u2190inv_comp_eq at pent,\n  rw \u2190pent,\n  simp only [hom_inv_whisker_left_assoc, assoc, comp_id, hom_inv_id]\nend\n\n@[reassoc]\nlemma pentagon_hom_hom_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f (g \u226b h) i).hom \u226b (f \u25c1 (\u03b1_ g h i).hom) \u226b (\u03b1_ f g (h \u226b i)).inv =\n    ((\u03b1_ f g h).inv \u25b7 i) \u226b (\u03b1_ (f \u226b g) h i).hom :=\neq_of_inv_eq_inv (by simp only [pentagon_hom_inv_inv_inv_hom, inv_whisker_left,\n  is_iso.iso.inv_hom, inv_whisker_right, is_iso.iso.inv_inv, is_iso.inv_comp, assoc])\n\n@[reassoc]\nlemma pentagon_inv_hom_hom_hom_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  ((\u03b1_ f g h).inv \u25b7 i) \u226b (\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom =\n    (\u03b1_ f (g \u226b h) i).hom \u226b (f \u25c1 (\u03b1_ g h i).hom) :=\nby { rw \u2190pentagon f g h i, simp only [inv_hom_whisker_right_assoc] }\n\n@[reassoc]\nlemma pentagon_inv_inv_hom_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv \u226b ((\u03b1_ f g h).hom \u25b7 i) =\n    (f \u25c1 (\u03b1_ g h i).inv) \u226b (\u03b1_ f (g \u226b h) i).inv :=\neq_of_inv_eq_inv (by simp only [pentagon_inv_hom_hom_hom_hom, inv_whisker_left,\n  is_iso.iso.inv_hom, inv_whisker_right, is_iso.iso.inv_inv, is_iso.inv_comp, assoc])\n\nlemma triangle_assoc_comp_left (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03b1_ f (\ud835\udfd9 b) g).hom \u226b (f \u25c1 (\u03bb_ g).hom) = (\u03c1_ f).hom \u25b7 g :=\ntriangle f g\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03b1_ f (\ud835\udfd9 b) g).inv \u226b ((\u03c1_ f).hom \u25b7 g) = f \u25c1 (\u03bb_ g).hom :=\nby rw [\u2190triangle, inv_hom_id_assoc]\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_right_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  ((\u03c1_ f).inv \u25b7 g) \u226b (\u03b1_ f (\ud835\udfd9 b) g).hom = f \u25c1 (\u03bb_ g).inv :=\nbegin\n  apply (cancel_mono (f \u25c1 (\u03bb_ g).hom)).1,\n  simp only [inv_hom_whisker_left, inv_hom_whisker_right, assoc, triangle]\nend\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_left_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (f \u25c1 (\u03bb_ g).inv) \u226b (\u03b1_ f (\ud835\udfd9 b) g).inv = (\u03c1_ f).inv \u25b7 g :=\nbegin\n  apply (cancel_mono ((\u03c1_ f).hom \u25b7 g)).1,\n  simp only [triangle_assoc_comp_right, inv_hom_whisker_left, inv_hom_whisker_right, assoc]\nend\n\nlemma unitors_equal : (\u03bb_ (\ud835\udfd9 a)).hom = (\u03c1_ (\ud835\udfd9 a)).hom :=\nby rw [\u2190whisker_left_iff, \u2190cancel_epi (\u03b1_ (\ud835\udfd9 a) (\ud835\udfd9 _) (\ud835\udfd9 _)).hom,\n       \u2190cancel_mono (\u03c1_ (\ud835\udfd9 a)).hom, triangle, \u2190right_unitor_comp, right_unitor_naturality]\n\nlemma unitors_inv_equal : (\u03bb_ (\ud835\udfd9 a)).inv = (\u03c1_ (\ud835\udfd9 a)).inv :=\nby { ext, rw [\u2190unitors_equal], simp only [hom_inv_id] }\n\nend\n\nend bicategory\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/bicategory/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8031737963569014, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.38277628761126004}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n\nimport data.fun_like.embedding\n\n/-!\n# Typeclass for a type `F` with an injective map to `A \u2243 B`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis typeclass is primarily for use by isomorphisms like `monoid_equiv` and `linear_equiv`.\n\n## Basic usage of `equiv_like`\n\nA typical type of morphisms should be declared as:\n```\nstructure my_iso (A B : Type*) [my_class A] [my_class B]\n  extends equiv A B :=\n(map_op' : \u2200 {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))\n\nnamespace my_iso\n\nvariables (A B : Type*) [my_class A] [my_class B]\n\n-- This instance is optional if you follow the \"Isomorphism class\" design below:\ninstance : equiv_like (my_iso A B) A (\u03bb _, B) :=\n{ coe := my_iso.to_equiv.to_fun,\n  inv := my_iso.to_equiv.inv_fun,\n  left_inv := my_iso.to_equiv.left_inv,\n  right_inv := my_iso.to_equiv.right_inv,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr' }\n\n/-- Helper instance for when there's too many metavariables to apply `equiv_like.coe` directly. -/\ninstance : has_coe_to_fun (my_iso A B) := to_fun.to_coe_fn\n\n@[simp] lemma to_fun_eq_coe {f : my_iso A B} : f.to_fun = (f : A \u2192 B) := rfl\n\n@[ext] theorem ext {f g : my_iso A B} (h : \u2200 x, f x = g x) : f = g := fun_like.ext f g h\n\n/-- Copy of a `my_iso` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : my_iso A B) (f' : A \u2192 B) (f_inv : B \u2192 A) (h : f' = \u21d1f) : my_iso A B :=\n{ to_fun := f',\n  inv_fun := f_inv,\n  left_inv := h.symm \u25b8 f.left_inv,\n  right_inv := h.symm \u25b8 f.right_inv,\n  map_op' := h.symm \u25b8 f.map_op' }\n\nend my_iso\n```\n\nThis file will then provide a `has_coe_to_fun` instance and various\nextensionality and simp lemmas.\n\n## Isomorphism classes extending `equiv_like`\n\nThe `equiv_like` design provides further benefits if you put in a bit more work.\nThe first step is to extend `equiv_like` to create a class of those types satisfying\nthe axioms of your new type of isomorphisms.\nContinuing the example above:\n\n```\nsection\nset_option old_structure_cmd true\n\n/-- `my_iso_class F A B` states that `F` is a type of `my_class.op`-preserving morphisms.\nYou should extend this class when you extend `my_iso`. -/\nclass my_iso_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]\n  extends equiv_like F A (\u03bb _, B), my_hom_class F A B.\n\nend\n\n-- You can replace `my_iso.equiv_like` with the below instance:\ninstance : my_iso_class (my_iso A B) A B :=\n{ coe := my_iso.to_fun,\n  inv := my_iso.inv_fun,\n  left_inv := my_iso.left_inv,\n  right_inv := my_iso.right_inv,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr',\n  map_op := my_iso.map_op' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThe second step is to add instances of your new `my_iso_class` for all types extending `my_iso`.\nTypically, you can just declare a new class analogous to `my_iso_class`:\n\n```\nstructure cooler_iso (A B : Type*) [cool_class A] [cool_class B]\n  extends my_iso A B :=\n(map_cool' : to_fun cool_class.cool = cool_class.cool)\n\nsection\nset_option old_structure_cmd true\n\nclass cooler_iso_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]\n  extends my_iso_class F A B :=\n(map_cool : \u2200 (f : F), f cool_class.cool = cool_class.cool)\n\nend\n\n@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_iso_class F A B]\n  (f : F) : f cool_class.cool = cool_class.cool :=\nmy_iso_class.map_op\n\n-- You can also replace `my_iso.equiv_like` with the below instance:\ninstance : cool_iso_class (cool_iso A B) A B :=\n{ coe := cool_iso.to_fun,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr',\n  map_op := cool_iso.map_op',\n  map_cool := cool_iso.map_cool' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThen any declaration taking a specific type of morphisms as parameter can instead take the\nclass you just defined:\n```\n-- Compare with: lemma do_something (f : my_iso A B) : sorry := sorry\nlemma do_something {F : Type*} [my_iso_class F A B] (f : F) : sorry := sorry\n```\n\nThis means anything set up for `my_iso`s will automatically work for `cool_iso_class`es,\nand defining `cool_iso_class` only takes a constant amount of effort,\ninstead of linearly increasing the work per `my_iso`-related declaration.\n\n-/\n\n/-- The class `equiv_like E \u03b1 \u03b2` expresses that terms of type `E` have an\ninjective coercion to bijections between `\u03b1` and `\u03b2`.\n\nThis typeclass is used in the definition of the homomorphism typeclasses,\nsuch as `zero_equiv_class`, `mul_equiv_class`, `monoid_equiv_class`, ....\n-/\nclass equiv_like (E : Sort*) (\u03b1 \u03b2 : out_param Sort*) :=\n(coe : E \u2192 \u03b1 \u2192 \u03b2)\n(inv : E \u2192 \u03b2 \u2192 \u03b1)\n(left_inv  : \u2200 e, function.left_inverse (inv e) (coe e))\n(right_inv : \u2200 e, function.right_inverse (inv e) (coe e))\n-- The `inv` hypothesis makes this easier to prove with `congr'`\n(coe_injective' : \u2200 e g, coe e = coe g \u2192 inv e = inv g \u2192 e = g)\n\nnamespace equiv_like\n\nvariables {E F \u03b1 \u03b2 \u03b3 : Sort*} [iE : equiv_like E \u03b1 \u03b2] [iF : equiv_like F \u03b2 \u03b3]\ninclude iE\n\nlemma inv_injective : function.injective (equiv_like.inv : E \u2192 (\u03b2 \u2192 \u03b1)) :=\n\u03bb e g h, coe_injective' e g ((right_inv e).eq_right_inverse (h.symm \u25b8 left_inv g)) h\n\n@[priority 100]\ninstance to_embedding_like : embedding_like E \u03b1 \u03b2 :=\n{ coe := (coe : E \u2192 \u03b1 \u2192 \u03b2),\n  coe_injective' := \u03bb e g h, coe_injective' e g h\n    ((left_inv e).eq_right_inverse (h.symm \u25b8 right_inv g)),\n  injective' := \u03bb e, (left_inv e).injective }\n\nprotected lemma injective (e : E) : function.injective e := embedding_like.injective e\nprotected lemma surjective (e : E) : function.surjective e := (right_inv e).surjective\nprotected lemma bijective (e : E) : function.bijective (e : \u03b1 \u2192 \u03b2) :=\n\u27e8equiv_like.injective e, equiv_like.surjective e\u27e9\n\ntheorem apply_eq_iff_eq (f : E) {x y : \u03b1} : f x = f y \u2194 x = y := embedding_like.apply_eq_iff_eq f\n\n@[simp] lemma injective_comp (e : E) (f : \u03b2 \u2192 \u03b3) :\n  function.injective (f \u2218 e) \u2194 function.injective f :=\nfunction.injective.of_comp_iff' f (equiv_like.bijective e)\n\n@[simp] lemma surjective_comp (e : E) (f : \u03b2 \u2192 \u03b3) :\n  function.surjective (f \u2218 e) \u2194 function.surjective f :=\n(equiv_like.surjective e).of_comp_iff f\n\n@[simp] lemma bijective_comp (e : E) (f : \u03b2 \u2192 \u03b3) :\n  function.bijective (f \u2218 e) \u2194 function.bijective f :=\n(equiv_like.bijective e).of_comp_iff f\n\n/-- This lemma is only supposed to be used in the generic context, when working with instances\nof classes extending `equiv_like`.\nFor concrete isomorphism types such as `equiv`, you should use `equiv.symm_apply_apply`\nor its equivalent.\n\nTODO: define a generic form of `equiv.symm`. -/\n@[simp] \n\n/-- This lemma is only supposed to be used in the generic context, when working with instances\nof classes extending `equiv_like`.\nFor concrete isomorphism types such as `equiv`, you should use `equiv.apply_symm_apply`\nor its equivalent.\n\nTODO: define a generic form of `equiv.symm`. -/\n@[simp] lemma apply_inv_apply (e : E) (b : \u03b2) : e (equiv_like.inv e b) = b := right_inv _ _\n\nomit iE\ninclude iF\n\nlemma comp_injective (f : \u03b1 \u2192 \u03b2) (e : F) :\n  function.injective (e \u2218 f) \u2194 function.injective f :=\nembedding_like.comp_injective f e\n\n@[simp] lemma comp_surjective (f : \u03b1 \u2192 \u03b2) (e : F) :\n  function.surjective (e \u2218 f) \u2194 function.surjective f :=\nfunction.surjective.of_comp_iff' (equiv_like.bijective e) f\n\n@[simp] lemma comp_bijective (f : \u03b1 \u2192 \u03b2) (e : F) :\n  function.bijective (e \u2218 f) \u2194 function.bijective f :=\n(equiv_like.bijective e).of_comp_iff' f\n\n/-- This is not an instance to avoid slowing down every single `subsingleton` typeclass search.-/\nlemma subsingleton_dom [subsingleton \u03b2] : subsingleton F :=\n\u27e8\u03bb f g, fun_like.ext f g $ \u03bb x, (right_inv f).injective $ subsingleton.elim _ _\u27e9\n\nend equiv_like\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/fun_like/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514762, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.3827401375625802}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.sums.basic\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-#\nThe associator functor `((C \u2295 D) \u2295 E) \u2964 (C \u2295 (D \u2295 E))` and its inverse form an equivalence.\n-/\n\nnamespace category_theory.sum\n\n\n/--\nThe associator functor `(C \u2295 D) \u2295 E \u2964 C \u2295 (D \u2295 E)` for sums of categories.\n-/\ndef associator (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] : (C \u2295 D) \u2295 E \u2964 C \u2295 D \u2295 E :=\n  functor.mk (fun (X : (C \u2295 D) \u2295 E) => sorry) fun (X Y : (C \u2295 D) \u2295 E) (f : X \u27f6 Y) => sorry\n\n@[simp] theorem associator_obj_inl_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : C) : functor.obj (associator C D E) (sum.inl (sum.inl X)) = sum.inl X :=\n  rfl\n\n@[simp] theorem associator_obj_inl_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : D) : functor.obj (associator C D E) (sum.inl (sum.inr X)) = sum.inr (sum.inl X) :=\n  rfl\n\n@[simp] theorem associator_obj_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : E) : functor.obj (associator C D E) (sum.inr X) = sum.inr (sum.inr X) :=\n  rfl\n\n@[simp] theorem associator_map_inl_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : C} {Y : C} (f : sum.inl (sum.inl X) \u27f6 sum.inl (sum.inl Y)) : functor.map (associator C D E) f = f :=\n  rfl\n\n@[simp] theorem associator_map_inl_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : D} {Y : D} (f : sum.inl (sum.inr X) \u27f6 sum.inl (sum.inr Y)) : functor.map (associator C D E) f = f :=\n  rfl\n\n@[simp] theorem associator_map_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : E} {Y : E} (f : sum.inr X \u27f6 sum.inr Y) : functor.map (associator C D E) f = f :=\n  rfl\n\n/--\nThe inverse associator functor `C \u2295 (D \u2295 E) \u2964 (C \u2295 D) \u2295 E` for sums of categories.\n-/\ndef inverse_associator (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] : C \u2295 D \u2295 E \u2964 (C \u2295 D) \u2295 E :=\n  functor.mk (fun (X : C \u2295 D \u2295 E) => sorry) fun (X Y : C \u2295 D \u2295 E) (f : X \u27f6 Y) => sorry\n\n@[simp] theorem inverse_associator_obj_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : C) : functor.obj (inverse_associator C D E) (sum.inl X) = sum.inl (sum.inl X) :=\n  rfl\n\n@[simp] theorem inverse_associator_obj_inr_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : D) : functor.obj (inverse_associator C D E) (sum.inr (sum.inl X)) = sum.inl (sum.inr X) :=\n  rfl\n\n@[simp] theorem inverse_associator_obj_inr_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] (X : E) : functor.obj (inverse_associator C D E) (sum.inr (sum.inr X)) = sum.inr X :=\n  rfl\n\n@[simp] theorem inverse_associator_map_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : C} {Y : C} (f : sum.inl X \u27f6 sum.inl Y) : functor.map (inverse_associator C D E) f = f :=\n  rfl\n\n@[simp] theorem inverse_associator_map_inr_inl (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : D} {Y : D} (f : sum.inr (sum.inl X) \u27f6 sum.inr (sum.inl Y)) : functor.map (inverse_associator C D E) f = f :=\n  rfl\n\n@[simp] theorem inverse_associator_map_inr_inr (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] {X : E} {Y : E} (f : sum.inr (sum.inr X) \u27f6 sum.inr (sum.inr Y)) : functor.map (inverse_associator C D E) f = f :=\n  rfl\n\n/--\nThe equivalence of categories expressing associativity of sums of categories.\n-/\ndef associativity (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] : (C \u2295 D) \u2295 E \u224c C \u2295 D \u2295 E :=\n  equivalence.mk (associator C D E) (inverse_associator C D E)\n    (nat_iso.of_components (fun (X : (C \u2295 D) \u2295 E) => eq_to_iso sorry) sorry)\n    (nat_iso.of_components (fun (X : C \u2295 D \u2295 E) => eq_to_iso sorry) sorry)\n\nprotected instance associator_is_equivalence (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] : is_equivalence (associator C D E) :=\n  is_equivalence.of_equivalence (associativity C D E)\n\nprotected instance inverse_associator_is_equivalence (C : Type u) [category C] (D : Type u) [category D] (E : Type u) [category E] : is_equivalence (inverse_associator C D E) :=\n  is_equivalence.of_equivalence_inverse (associativity C D E)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/sums/associator.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3827401358033622}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro\n\nInjective functions.\n-/\nimport data.equiv.basic data.option.basic\n\nuniverses u v w x\n\nnamespace function\n\nstructure embedding (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun : \u03b1 \u2192 \u03b2)\n(inj    : injective to_fun)\n\ninfixr ` \u21aa `:25 := embedding\n\ninstance {\u03b1 : Sort u} {\u03b2 : Sort v} : has_coe_to_fun (\u03b1 \u21aa \u03b2) := \u27e8_, embedding.to_fun\u27e9\n\nend function\n\nprotected def equiv.to_embedding {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2243 \u03b2) : \u03b1 \u21aa \u03b2 :=\n\u27e8f, f.injective\u27e9\n\n@[simp] theorem equiv.to_embedding_coe_fn {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2243 \u03b2) :\n  (f.to_embedding : \u03b1 \u2192 \u03b2) = f := rfl\n\nnamespace function\nnamespace embedding\n\n@[simp] theorem to_fun_eq_coe {\u03b1 \u03b2} (f : \u03b1 \u21aa \u03b2) : to_fun f = f := rfl\n\n@[simp] theorem coe_fn_mk {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (i) :\n  (@mk _ _ f i : \u03b1 \u2192 \u03b2) = f := rfl\n\ntheorem inj' {\u03b1 \u03b2} : \u2200 (f : \u03b1 \u21aa \u03b2), injective f\n| \u27e8f, hf\u27e9 := hf\n\n@[refl] protected def refl (\u03b1 : Sort*) : \u03b1 \u21aa \u03b1 :=\n\u27e8id, injective_id\u27e9\n\n@[trans] protected def trans {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u21aa \u03b2) (g : \u03b2 \u21aa \u03b3) : \u03b1 \u21aa \u03b3 :=\n\u27e8_, injective_comp g.inj' f.inj'\u27e9\n\n@[simp] theorem refl_apply {\u03b1} (x : \u03b1) : embedding.refl \u03b1 x = x := rfl\n\n@[simp] theorem trans_apply {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u21aa \u03b2) (g : \u03b2 \u21aa \u03b3) (a : \u03b1) :\n  (f.trans g) a = g (f a) := rfl\n\nprotected def congr {\u03b1 : Sort u} {\u03b2 : Sort v} {\u03b3 : Sort w} {\u03b4 : Sort x}\n  (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b3 \u2243 \u03b4) (f : \u03b1 \u21aa \u03b3) : (\u03b2 \u21aa \u03b4) :=\n(equiv.to_embedding e\u2081.symm).trans (f.trans e\u2082.to_embedding)\n\nprotected noncomputable def of_surjective {\u03b1 \u03b2} {f : \u03b2 \u2192 \u03b1} (hf : surjective f) :\n  \u03b1 \u21aa \u03b2 :=\n\u27e8surj_inv hf, injective_surj_inv _\u27e9\n\nprotected noncomputable def equiv_of_surjective {\u03b1 \u03b2} (f : \u03b1 \u21aa \u03b2) (hf : surjective f) :\n  \u03b1 \u2243 \u03b2 :=\nequiv.of_bijective \u27e8f.inj, hf\u27e9\n\nprotected def of_not_nonempty {\u03b1 \u03b2} (h\u03b1 : \u00ac nonempty \u03b1) : \u03b1 \u21aa \u03b2 :=\n\u27e8\u03bba, (h\u03b1 \u27e8a\u27e9).elim, assume a, (h\u03b1 \u27e8a\u27e9).elim\u27e9\n\nnoncomputable def set_value {\u03b1 \u03b2} (f : \u03b1 \u21aa \u03b2) (a : \u03b1) (b : \u03b2) : \u03b1 \u21aa \u03b2 :=\nby haveI := classical.dec; exact\nif h : \u2203 a', f a' = b then\n  (equiv.swap a (classical.some h)).to_embedding.trans f\nelse\n  \u27e8\u03bb a', if a' = a then b else f a',\n   \u03bb a\u2081 a\u2082 e, begin\n    simp at e, split_ifs at e with h\u2081 h\u2082,\n    { cc },\n    { cases h \u27e8_, e.symm\u27e9 },\n    { cases h \u27e8_, e\u27e9 },\n    { exact f.2 e }\n   end\u27e9\n\ntheorem set_value_eq {\u03b1 \u03b2} (f : \u03b1 \u21aa \u03b2) (a : \u03b1) (b : \u03b2) : set_value f a b a = b :=\nbegin\n  rw [set_value],\n  cases classical.dec (\u2203 a', f a' = b);\n    dsimp [dite], {simp},\n  simp [equiv.swap_apply_left],\n  apply classical.some_spec h\nend\n\n/-- Embedding into `option` -/\nprotected def some {\u03b1} : \u03b1 \u21aa option \u03b1 :=\n\u27e8some, option.injective_some \u03b1\u27e9\n\ndef subtype {\u03b1} (p : \u03b1 \u2192 Prop) : subtype p \u21aa \u03b1 :=\n\u27e8subtype.val, \u03bb _ _, subtype.eq'\u27e9\n\n/-- Restrict the codomain of an embedding. -/\ndef cod_restrict {\u03b1 \u03b2} (p : set \u03b2) (f : \u03b1 \u21aa \u03b2) (H : \u2200 a, f a \u2208 p) : \u03b1 \u21aa p :=\n\u27e8\u03bb a, \u27e8f a, H a\u27e9, \u03bb a b h, f.inj (@congr_arg _ _ _ _ subtype.val h)\u27e9\n\n@[simp] theorem cod_restrict_apply {\u03b1 \u03b2} (p) (f : \u03b1 \u21aa \u03b2) (H a) :\n  cod_restrict p f H a = \u27e8f a, H a\u27e9 := rfl\n\ndef prod_congr {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} (e\u2081 : \u03b1 \u21aa \u03b2) (e\u2082 : \u03b3 \u21aa \u03b4) : \u03b1 \u00d7 \u03b3 \u21aa \u03b2 \u00d7 \u03b4 :=\n\u27e8assume \u27e8a, b\u27e9, (e\u2081 a, e\u2082 b),\n  assume \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 h,\n  have a\u2081 = a\u2082 \u2227 b\u2081 = b\u2082, from (prod.mk.inj h).imp (assume h, e\u2081.inj h) (assume h, e\u2082.inj h),\n  this.left \u25b8 this.right \u25b8 rfl\u27e9\n\nsection sum\nopen sum\n\ndef sum_congr {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} (e\u2081 : \u03b1 \u21aa \u03b2) (e\u2082 : \u03b3 \u21aa \u03b4) : \u03b1 \u2295 \u03b3 \u21aa \u03b2 \u2295 \u03b4 :=\n\u27e8assume s, match s with inl a := inl (e\u2081 a) | inr b := inr (e\u2082 b) end,\n    assume s\u2081 s\u2082 h, match s\u2081, s\u2082, h with\n    | inl a\u2081, inl a\u2082, h := congr_arg inl $ e\u2081.inj $ inl.inj h\n    | inr b\u2081, inr b\u2082, h := congr_arg inr $ e\u2082.inj $ inr.inj h\n    end\u27e9\n\n@[simp] theorem sum_congr_apply_inl {\u03b1 \u03b2 \u03b3 \u03b4}\n  (e\u2081 : \u03b1 \u21aa \u03b2) (e\u2082 : \u03b3 \u21aa \u03b4) (a) : sum_congr e\u2081 e\u2082 (inl a) = inl (e\u2081 a) := rfl\n\n@[simp] theorem sum_congr_apply_inr {\u03b1 \u03b2 \u03b3 \u03b4}\n  (e\u2081 : \u03b1 \u21aa \u03b2) (e\u2082 : \u03b3 \u21aa \u03b4) (b) : sum_congr e\u2081 e\u2082 (inr b) = inr (e\u2082 b) := rfl\n\nend sum\n\nsection sigma\nopen sigma\n\ndef sigma_congr_right {\u03b1 : Type*} {\u03b2 \u03b3 : \u03b1 \u2192 Type*} (e : \u2200 a, \u03b2 a \u21aa \u03b3 a) : sigma \u03b2 \u21aa sigma \u03b3 :=\n\u27e8\u03bb \u27e8a, b\u27e9, \u27e8a, e a b\u27e9, \u03bb \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 h, begin\n  injection h with h\u2081 h\u2082, subst a\u2082,\n  congr,\n  exact (e a\u2081).2 (eq_of_heq h\u2082)\nend\u27e9\n\nend sigma\n\ndef Pi_congr_right {\u03b1 : Sort*} {\u03b2 \u03b3 : \u03b1 \u2192 Sort*} (e : \u2200 a, \u03b2 a \u21aa \u03b3 a) : (\u03a0 a, \u03b2 a) \u21aa (\u03a0 a, \u03b3 a) :=\n\u27e8\u03bbf a, e a (f a), \u03bb f\u2081 f\u2082 h, funext $ \u03bb a, (e a).inj (congr_fun h a)\u27e9\n\ndef arrow_congr_left {\u03b1 : Sort u} {\u03b2 : Sort v} {\u03b3 : Sort w}\n  (e : \u03b1 \u21aa \u03b2) : (\u03b3 \u2192 \u03b1) \u21aa (\u03b3 \u2192 \u03b2) :=\nPi_congr_right (\u03bb _, e)\n\nnoncomputable def arrow_congr_right {\u03b1 : Sort u} {\u03b2 : Sort v} {\u03b3 : Sort w} [inhabited \u03b3]\n  (e : \u03b1 \u21aa \u03b2) : (\u03b1 \u2192 \u03b3) \u21aa (\u03b2 \u2192 \u03b3) :=\nby haveI := classical.prop_decidable; exact\nlet f' : (\u03b1 \u2192 \u03b3) \u2192 (\u03b2 \u2192 \u03b3) := \u03bbf b, if h : \u2203c, e c = b then f (classical.some h) else default \u03b3 in\n\u27e8f', assume f\u2081 f\u2082 h, funext $ assume c,\n  have \u2203c', e c' = e c, from \u27e8c, rfl\u27e9,\n  have eq' : f' f\u2081 (e c) = f' f\u2082 (e c), from congr_fun h _,\n  have eq_b : classical.some this = c, from e.inj $ classical.some_spec this,\n  by simp [f', this, if_pos, eq_b] at eq'; assumption\u27e9\n\nend embedding\nend function\n\nnamespace set\n\n/-- The injection map is an embedding between subsets. -/\ndef embedding_of_subset {\u03b1} {s t : set \u03b1} (h : s \u2286 t) : s \u21aa t :=\n\u27e8\u03bb x, \u27e8x.1, h x.2\u27e9, \u03bb \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 h, by congr; injection h\u27e9\n\nend set\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/logic/embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.3827401358033622}}
{"text": "import tactic.linarith\nimport ite_lemmas\nimport code\nimport succ_graphs\n\ndef code.time : code \u2192 ptree \u2192. \u2115\n| code.left := \u03bb t, part.some t.sizeof\n| code.right := \u03bb t, part.some t.sizeof\n| code.nil := \u03bb t, part.some 1\n| code.id := \u03bb t, part.some t.sizeof\n| (code.node a b) := \u03bb t, (+1) <$> (a.time t) + (b.time t)\n| (code.comp f g) := \u03bb t, (+1) <$> (g.time t) + (g.eval t >>= f.time)\n| (code.case f g) := \u03bb t, (+1) <$> if t.left = ptree.nil then f.time t.right else g.time t.right\n| (code.fix f) := \u03bb t, (+t.sizeof) <$> (f.fix_iterator t).time f.time\n\n\nlemma add_def (x y : part \u2115) : x + y = x >>= \u03bb x', y >>= (\u03bb y', pure (x' + y')) :=\nby { simp only [(+), (<*>), part.bind_eq_bind, part.bind_map, part.map_eq_map], congr, ext x, simp, tauto, }\n\nlemma time_dom_iff_eval_dom (c : code) (v : ptree) : (c.time v).dom \u2194 (c.eval v).dom :=\nbegin\n  induction c generalizing v,\n  all_goals { simp [code.time, add_def, code.eval_fix], },\n  case code.node : c\u2081 c\u2082 c\u2081ih c\u2082ih { simp [c\u2081ih, c\u2082ih], },\n  case code.comp : c\u2081 c\u2082 c\u2081ih c\u2082ih { simp [c\u2081ih, c\u2082ih], tauto, },\n  case code.case : c\u2081 c\u2082 c\u2081ih c\u2082ih { simp [c\u2081ih, c\u2082ih, apply_ite part.dom], },\n  case code.fix : f ih\n  { simp [execution.time],\n    refine (execution.time_with_tr (f.fix_iterator v) f.time _).eval_dom_iff.symm,\n    intros x x' _, rw ih, simp [part.dom_iff_mem], tauto, }\nend\n\nlemma time_dom_iff_eval_to_option_dom (c : code) (v : ptree) : (c.time v).dom \u2194 (c.eval.map ptree.to_option v).dom :=\nby simp [time_dom_iff_eval_dom, pfun.map]\n\nlemma time_dom_eq_eval_dom (c : code) : c.time.dom = c.eval.dom :=\nby { ext, apply time_dom_iff_eval_dom, }\n\ndef time_bound (c : code) (bound : \u2115 \u2192 \u2115) : Prop :=\n\u2200 (v : ptree), \u2203 t \u2208 c.time v, t \u2264 bound v.sizeof\n\nlemma time_bound_spec {c : code} {bound : \u2115 \u2192 \u2115} (h : time_bound c bound) {t v} (hn : t \u2208 c.time v) : t \u2264 bound v.sizeof :=\nby { specialize h v, rcases h with \u27e8t, ht, H\u27e9, cases part.mem_unique hn ht, exact H, }\n\nlemma time_bound_of_time_bound_le {c : code} {b\u2081 : \u2115 \u2192 \u2115} (hb\u2081 : time_bound c b\u2081) {b\u2082 : \u2115 \u2192 \u2115} (hb\u2082 : \u2200 n, b\u2081 n \u2264 b\u2082 n) :\n  time_bound c b\u2082 := \u03bb v, by { obtain \u27e8t, ht, t_le\u27e9 := hb\u2081 v, use [t, ht], exact t_le.trans (hb\u2082 _), }\n\nlemma eval_dom_of_time_bound {c : code} {bound : \u2115 \u2192 \u2115} (h : time_bound c bound) : c.eval.dom = set.univ :=\nbegin\n  ext v, \n  suffices : (c.time v).dom, { simpa [pfun.dom, time_dom_iff_eval_dom], },\n  rw part.dom_iff_mem, obtain \u27e8t, ht, _\u27e9 := h v, exact \u27e8t, ht\u27e9,\nend\n\nlemma dom_univ_iff {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192. \u03b2) : f.dom = set.univ \u2194 \u2200 x, (f x).dom :=\nby simp [pfun.dom, set.eq_univ_iff_forall]\n\nlemma eval_sizeof_le_time {c : code} {vin vout : ptree} {t : \u2115} (hv : vout \u2208 c.eval vin) (ht : t \u2208 c.time vin) : vout.sizeof \u2264 t :=\nbegin\n  induction c generalizing vin vout t,\n  { simp only [code.eval, code.time, part.pure_eq_some, part.mem_some_iff] at hv ht, subst_vars, apply ptree.left_sizeof_le, },\n  { simp only [code.eval, code.time, part.pure_eq_some, part.mem_some_iff] at hv ht, subst_vars, apply ptree.right_sizeof_le, },\n  { simp [code.time] at hv ht \u22a2, subst_vars, simp, },\n  { simp [code.time] at hv ht \u22a2, subst_vars, },\n  case code.node : c\u2081 c\u2082 c\u2081ih c\u2082ih\n  { simp [code.time, add_def] at hv ht,\n    rcases ht with \u27e8t\u2081, ht\u2081, t\u2082, ht\u2082, ht\u27e9,\n    rcases hv with \u27e8v\u2081, hv\u2081, v\u2082, hv\u2082, hv\u27e9,\n    specialize c\u2081ih hv\u2081 ht\u2081, specialize c\u2082ih hv\u2082 ht\u2082, rw [hv, ht],\n    simp, linarith only [c\u2081ih, c\u2082ih], },\n  case code.comp : c\u2081 c\u2082 c\u2081ih c\u2082ih\n  { simp [code.time, add_def] at hv ht, \n    rcases ht with \u27e8t\u2081, ht\u2081, t\u2082, \u27e8v', hv', ht\u2082\u27e9, rfl\u27e9,\n    rcases hv with \u27e8v'', hv'', H\u27e9,\n    cases part.mem_unique hv' hv'',\n    suffices : vout.sizeof \u2264 t\u2082, { linarith only [this], },\n    exact c\u2081ih H ht\u2082, },\n  case code.case : c\u2081 c\u2082 c\u2081ih c\u2082ih\n  { simp [code.time] at hv ht,\n    rcases ht with \u27e8t', ht', rfl\u27e9,\n    split_ifs at *, { linarith only [c\u2081ih hv ht'], }, { linarith only [c\u2082ih hv ht'], } },\n  case code.fix : f ih\n  { simp [code.time, code.eval_fix, execution.time] at hv ht,\n    rcases ht with \u27e8t, \u27e8x, hx\u27e9, rfl\u27e9,\n    obtain rfl : vout = x,\n    { simpa using (execution.time_with_tr (f.fix_iterator vin) f.time _).rel_of_mem_eval hv hx,\n      simp_rw [time_dom_iff_eval_dom, part.dom_iff_mem], simp, tauto, },\n    rw execution.mem_eval at hx, simp at hx,\n    cases execution.mem_ran_of_mem_states hx.1,\n    { simp [execution.time_with] at h, rw h.2, simp, },\n    simp [execution.time_with, pfun.ran] at h,\n    rcases h with \u27e8t', vout', \u27e8tf, htf, rfl\u27e9, _, hvout\u27e9,\n    refine (ih hvout htf).trans _, simp [add_assoc], }\nend\n\nlemma time_bound_left : time_bound code.left id :=\nby simp [time_bound, code.time]\n\nlemma time_bound_right : time_bound code.right id := time_bound_left\n\nlemma time_bound_id : time_bound code.id id := time_bound_left\n\nlemma time_bound_nil : time_bound code.nil (\u03bb _, 1) :=\nby simp [time_bound, code.time]\n\nlemma time_bound_node {c\u2081 c\u2082 : code} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (hb\u2081 : time_bound c\u2081 b\u2081) (hb\u2082 : time_bound c\u2082 b\u2082) :\n  time_bound (code.node c\u2081 c\u2082) (\u03bb t, b\u2081 t + b\u2082 t + 1) :=\nbegin\n  intros v,\n  obtain \u27e8t\u2081, ht\u2081, hb\u2081\u27e9 := hb\u2081 v,\n  obtain \u27e8t\u2082, ht\u2082, hb\u2082\u27e9 := hb\u2082 v,\n  use t\u2081 + t\u2082 + 1, split,\n  { rw \u2190 part.eq_some_iff at ht\u2081 ht\u2082, simp [code.time, ht\u2081, ht\u2082, add_def], ring, },\n  mono*,\nend\n\nlemma time_bound_comp {c\u2081 c\u2082 : code} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (hm : monotone b\u2081) (hb\u2081 : time_bound c\u2081 b\u2081) (hb\u2082 : time_bound c\u2082 b\u2082) :\n  time_bound (c\u2081.comp c\u2082) (\u03bb t, b\u2081 (b\u2082 t) + b\u2082 t + 1) :=\nbegin\n  intros v,\n  obtain \u27e8t\u2082, ht\u2082, hb\u2082\u27e9 := hb\u2082 v,\n  obtain \u27e8v', hv'\u27e9 := (_ : \u2203 v', v' \u2208 c\u2082.eval v), swap,\n  { rw [\u2190 part.dom_iff_mem, \u2190 time_dom_iff_eval_dom, part.dom_iff_mem], use [t\u2082, ht\u2082], },\n  obtain \u27e8t\u2081, ht\u2081, hb\u2081\u27e9 := hb\u2081 v',\n  use t\u2081 + t\u2082 + 1, split,\n  { rw \u2190 part.eq_some_iff at ht\u2081 ht\u2082 hv', simp [code.time, ht\u2081, ht\u2082, hv', add_def], ring, },\n  { mono*, exact hb\u2081.trans (hm ((eval_sizeof_le_time hv' ht\u2082).trans hb\u2082)), },\nend\n\nlemma time_bound_case {c\u2081 c\u2082 : code} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (hm\u2081 : monotone b\u2081) (hm\u2082 : monotone b\u2082) (hb\u2081 : time_bound c\u2081 b\u2081) (hb\u2082 : time_bound c\u2082 b\u2082) :\n  time_bound (code.case c\u2081 c\u2082) (\u03bb t, max (b\u2081 t) (b\u2082 t) + 1) :=\nbegin\n  intros v,\n  simp [code.time], split_ifs,\n  { obtain \u27e8t, ht, H\u27e9 := hb\u2081 v.right, use [t, ht], left, exact H.trans (hm\u2081 $ ptree.right_sizeof_le _), },\n  { obtain \u27e8t, ht, H\u27e9 := hb\u2082 v.right, use [t, ht], right, exact H.trans (hm\u2082 $ ptree.right_sizeof_le _), },\nend\n\nlemma time_bound_case' {c\u2081 c\u2082 : code} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (hm\u2081 : monotone b\u2081) (hm\u2082 : monotone b\u2082) (hb\u2081 : time_bound c\u2081 b\u2081) (hb\u2082 : time_bound c\u2082 b\u2082) :\n  time_bound (code.case c\u2081 c\u2082) (\u03bb t, (b\u2081 t) + (b\u2082 t) + 1) :=\nby { apply time_bound_of_time_bound_le (time_bound_case hm\u2081 hm\u2082 hb\u2081 hb\u2082), intro, simp, }\n\n-- lemma time_bound_case_precise {c\u2081 c\u2082 : code} {b\u2081 b\u2082 : \u2115 \u2192 \u2115} (m\u2081 : monotone b\u2081) (m\u2082 : monotone b\u2082)\n--   (hb\u2081 : \u2200 x : ptree, x.left = ptree.nil \u2192 \u2203 t \u2208 c\u2081.time x.right, t \u2264 b\u2081 x.sizeof)\n--   (hb\u2082 : \u2200 x : ptree, x.left \u2260 ptree.nil \u2192 \u2203 t \u2208 c\u2082.time x.right, t \u2264 b\u2082 x.sizeof) :\n--   time_bound (code.case c\u2081 c\u2082) (\u03bb n, max (b\u2081 n) (b\u2082 n) + 1) :=\n-- begin\n--   intros n v hnv, \n--   by_cases H : v.left = ptree.nil,\n--   { specialize hb\u2081 v H, rcases hb\u2081 with \u27e8t, ht, s\u27e9, use t + 1, split,\n--     { simpa [code.time, H], }, { simp, left, refine s.trans _, apply m\u2081 hnv, } },\n--   { specialize hb\u2082 v H, rcases hb\u2082 with \u27e8t, ht, s\u27e9, use t + 1, split,\n--     { simpa [code.time, H], }, { simp, right, refine s.trans _, apply m\u2082 hnv, } }\n-- end\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/time_bound.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.3827401202182581}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.braided\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.pempty\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# The monoidal structure on a category with chosen finite products.\n\nThis is a variant of the development in `category_theory.monoidal.of_has_finite_products`,\nwhich uses specified choices of the terminal object and binary product,\nenabling the construction of a cartesian category with specific definitions of the tensor unit\nand tensor product.\n\n(Because the construction in `category_theory.monoidal.of_has_finite_products` uses `has_limit`\nclasses, the actual definitions there are opaque behind `classical.choice`.)\n\nWe use this in `category_theory.monoidal.types` to construct the monoidal category of types\nso that the tensor product is the usual cartesian product of types.\n\nFor now we only do the construction from products, and not from coproducts,\nwhich seems less often useful.\n-/\n\nnamespace category_theory\n\n\nnamespace limits\n\n\n/-- Swap the two sides of a `binary_fan`. -/\ndef binary_fan.swap {C : Type u} [category C] {P : C} {Q : C} (t : binary_fan P Q) : binary_fan Q P :=\n  binary_fan.mk (binary_fan.snd t) (binary_fan.fst t)\n\n@[simp] theorem binary_fan.swap_fst {C : Type u} [category C] {P : C} {Q : C} (t : binary_fan P Q) : binary_fan.fst (binary_fan.swap t) = binary_fan.snd t :=\n  rfl\n\n@[simp] theorem binary_fan.swap_snd {C : Type u} [category C] {P : C} {Q : C} (t : binary_fan P Q) : binary_fan.snd (binary_fan.swap t) = binary_fan.fst t :=\n  rfl\n\n/--\nIf a cone `t` over `P Q` is a limit cone, then `t.swap` is a limit cone over `Q P`.\n-/\n@[simp] theorem is_limit.swap_binary_fan_lift {C : Type u} [category C] {P : C} {Q : C} {t : binary_fan P Q} (I : is_limit t) (s : cone (pair Q P)) : is_limit.lift (is_limit.swap_binary_fan I) s = is_limit.lift I (binary_fan.swap s) :=\n  Eq.refl (is_limit.lift (is_limit.swap_binary_fan I) s)\n\n/--\nConstruct `has_binary_product Q P` from `has_binary_product P Q`.\nThis can't be an instance, as it would cause a loop in typeclass search.\n-/\ntheorem has_binary_product.swap {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q] : has_binary_product Q P :=\n  has_limit.mk\n    (limit_cone.mk (binary_fan.swap (limit.cone (pair P Q))) (is_limit.swap_binary_fan (limit.is_limit (pair P Q))))\n\n/--\nGiven a limit cone over `X` and `Y`, and another limit cone over `Y` and `X`, we can construct\nan isomorphism between the cone points. Relative to some fixed choice of limits cones for every pair,\nthese isomorphisms constitute a braiding.\n-/\ndef binary_fan.braiding {C : Type u} [category C] {X : C} {Y : C} {s : binary_fan X Y} (P : is_limit s) {t : binary_fan Y X} (Q : is_limit t) : cone.X s \u2245 cone.X t :=\n  is_limit.cone_point_unique_up_to_iso P (is_limit.swap_binary_fan Q)\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `sXY.X Z`,\nif `sYZ` is a limit cone we can construct a binary fan over `X sYZ.X`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan (cone.X sXY) Z) : binary_fan X (cone.X sYZ) :=\n  binary_fan.mk (binary_fan.fst s \u226b binary_fan.fst sXY)\n    (is_limit.lift Q (binary_fan.mk (binary_fan.fst s \u226b binary_fan.snd sXY) (binary_fan.snd s)))\n\n@[simp] theorem binary_fan.assoc_fst {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan (cone.X sXY) Z) : binary_fan.fst (binary_fan.assoc Q s) = binary_fan.fst s \u226b binary_fan.fst sXY :=\n  rfl\n\n@[simp] theorem binary_fan.assoc_snd {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan (cone.X sXY) Z) : binary_fan.snd (binary_fan.assoc Q s) =\n  is_limit.lift Q (binary_fan.mk (binary_fan.fst s \u226b binary_fan.snd sXY) (binary_fan.snd s)) :=\n  rfl\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `X sYZ.X`,\nif `sYZ` is a limit cone we can construct a binary fan over `sXY.X Z`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc_inv {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X (cone.X sYZ)) : binary_fan (cone.X sXY) Z :=\n  binary_fan.mk (is_limit.lift P (binary_fan.mk (binary_fan.fst s) (binary_fan.snd s \u226b binary_fan.fst sYZ)))\n    (binary_fan.snd s \u226b binary_fan.snd sYZ)\n\n@[simp] theorem binary_fan.assoc_inv_fst {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X (cone.X sYZ)) : binary_fan.fst (binary_fan.assoc_inv P s) =\n  is_limit.lift P (binary_fan.mk (binary_fan.fst s) (binary_fan.snd s \u226b binary_fan.fst sYZ)) :=\n  rfl\n\n@[simp] theorem binary_fan.assoc_inv_snd {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X (cone.X sYZ)) : binary_fan.snd (binary_fan.assoc_inv P s) = binary_fan.snd s \u226b binary_fan.snd sYZ :=\n  rfl\n\n/--\nIf all the binary fans involved a limit cones, `binary_fan.assoc` produces another limit cone.\n-/\ndef is_limit.assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ) {s : binary_fan (cone.X sXY) Z} (R : is_limit s) : is_limit (binary_fan.assoc Q s) :=\n  is_limit.mk fun (t : cone (pair X (cone.X sYZ))) => is_limit.lift R (binary_fan.assoc_inv P t)\n\n/--\nGiven two pairs of limit cones corresponding to the parenthesisations of `X \u00d7 Y \u00d7 Z`,\nwe obtain an isomorphism between the cone points.\n-/\ndef binary_fan.associator {C : Type u} [category C] {X : C} {Y : C} {Z : C} {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ) {s : binary_fan (cone.X sXY) Z} (R : is_limit s) {t : binary_fan X (cone.X sYZ)} (S : is_limit t) : cone.X s \u2245 cone.X t :=\n  is_limit.cone_point_unique_up_to_iso (is_limit.assoc P Q R) S\n\n/--\nGiven a fixed family of limit data for every pair `X Y`, we obtain an associator.\n-/\ndef binary_fan.associator_of_limit_cone {C : Type u} [category C] (L : (X Y : C) \u2192 limit_cone (pair X Y)) (X : C) (Y : C) (Z : C) : cone.X (limit_cone.cone (L (cone.X (limit_cone.cone (L X Y))) Z)) \u2245\n  cone.X (limit_cone.cone (L X (cone.X (limit_cone.cone (L Y Z))))) :=\n  binary_fan.associator (limit_cone.is_limit (L X Y)) (limit_cone.is_limit (L Y Z))\n    (limit_cone.is_limit (L (cone.X (limit_cone.cone (L X Y))) Z))\n    (limit_cone.is_limit (L X (cone.X (limit_cone.cone (L Y Z)))))\n\n/--\nConstruct a left unitor from specified limit cones.\n-/\ndef binary_fan.left_unitor {C : Type u} [category C] {X : C} {s : cone (functor.empty C)} (P : is_limit s) {t : binary_fan (cone.X s) X} (Q : is_limit t) : cone.X t \u2245 X :=\n  iso.mk (binary_fan.snd t)\n    (is_limit.lift Q\n      (binary_fan.mk\n        (is_limit.lift P\n          (cone.mk X\n            (nat_trans.mk\n              (pempty.rec\n                fun (n : pempty) =>\n                  functor.obj (functor.obj (functor.const (discrete pempty)) X) n \u27f6 functor.obj (functor.empty C) n))))\n        \ud835\udfd9))\n\n/--\nConstruct a right unitor from specified limit cones.\n-/\ndef binary_fan.right_unitor {C : Type u} [category C] {X : C} {s : cone (functor.empty C)} (P : is_limit s) {t : binary_fan X (cone.X s)} (Q : is_limit t) : cone.X t \u2245 X :=\n  iso.mk (binary_fan.fst t)\n    (is_limit.lift Q\n      (binary_fan.mk \ud835\udfd9\n        (is_limit.lift P\n          (cone.mk X\n            (nat_trans.mk\n              (pempty.rec\n                fun (n : pempty) =>\n                  functor.obj (functor.obj (functor.const (discrete pempty)) X) n \u27f6 functor.obj (functor.empty C) n))))))\n\nend limits\n\n\nnamespace monoidal_of_chosen_finite_products\n\n\n/-- Implementation of the tensor product for `monoidal_of_chosen_finite_products`. -/\ndef tensor_obj {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X : C) (Y : C) : C :=\n  limits.cone.X (limits.limit_cone.cone (\u212c X Y))\n\n/-- Implementation of the tensor product of morphisms for `monoidal_of_chosen_finite_products`. -/\ndef tensor_hom {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {W : C} {X : C} {Y : C} {Z : C} (f : W \u27f6 X) (g : Y \u27f6 Z) : tensor_obj \u212c W Y \u27f6 tensor_obj \u212c X Z :=\n  subtype.val\n    (limits.binary_fan.is_limit.lift' (limits.limit_cone.is_limit (\u212c X Z))\n      (nat_trans.app (limits.cone.\u03c0 (limits.limit_cone.cone (\u212c W Y))) limits.walking_pair.left \u226b f)\n      (nat_trans.app (limits.cone.\u03c0 (limits.limit_cone.cone (\u212c W Y))) limits.walking_pair.right \u226b g))\n\ntheorem tensor_id {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X\u2081 : C) (X\u2082 : C) : tensor_hom \u212c \ud835\udfd9 \ud835\udfd9 = \ud835\udfd9 := sorry\n\ntheorem tensor_comp {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {X\u2081 : C} {Y\u2081 : C} {Z\u2081 : C} {X\u2082 : C} {Y\u2082 : C} {Z\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082) : tensor_hom \u212c (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) = tensor_hom \u212c f\u2081 f\u2082 \u226b tensor_hom \u212c g\u2081 g\u2082 := sorry\n\ntheorem pentagon {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (W : C) (X : C) (Y : C) (Z : C) : tensor_hom \u212c (iso.hom (limits.binary_fan.associator_of_limit_cone \u212c W X Y)) \ud835\udfd9 \u226b\n    iso.hom (limits.binary_fan.associator_of_limit_cone \u212c W (tensor_obj \u212c X Y) Z) \u226b\n      tensor_hom \u212c \ud835\udfd9 (iso.hom (limits.binary_fan.associator_of_limit_cone \u212c X Y Z)) =\n  iso.hom (limits.binary_fan.associator_of_limit_cone \u212c (tensor_obj \u212c W X) Y Z) \u226b\n    iso.hom (limits.binary_fan.associator_of_limit_cone \u212c W X (tensor_obj \u212c Y Z)) := sorry\n\ntheorem triangle {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X : C) (Y : C) : iso.hom (limits.binary_fan.associator_of_limit_cone \u212c X (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) Y) \u226b\n    tensor_hom \u212c \ud835\udfd9\n      (iso.hom\n        (limits.binary_fan.left_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n          (limits.limit_cone.is_limit (\u212c (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) Y)))) =\n  tensor_hom \u212c\n    (iso.hom\n      (limits.binary_fan.right_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c X (limits.cone.X (limits.limit_cone.cone \ud835\udcaf))))))\n    \ud835\udfd9 := sorry\n\ntheorem left_unitor_naturality {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {X\u2081 : C} {X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) : tensor_hom \u212c \ud835\udfd9 f \u226b\n    iso.hom\n      (limits.binary_fan.left_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) X\u2082))) =\n  iso.hom\n      (limits.binary_fan.left_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) X\u2081))) \u226b\n    f := sorry\n\ntheorem right_unitor_naturality {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {X\u2081 : C} {X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) : tensor_hom \u212c f \ud835\udfd9 \u226b\n    iso.hom\n      (limits.binary_fan.right_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c X\u2082 (limits.cone.X (limits.limit_cone.cone \ud835\udcaf))))) =\n  iso.hom\n      (limits.binary_fan.right_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c X\u2081 (limits.cone.X (limits.limit_cone.cone \ud835\udcaf))))) \u226b\n    f := sorry\n\ntheorem associator_naturality {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {X\u2081 : C} {X\u2082 : C} {X\u2083 : C} {Y\u2081 : C} {Y\u2082 : C} {Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083) : tensor_hom \u212c (tensor_hom \u212c f\u2081 f\u2082) f\u2083 \u226b iso.hom (limits.binary_fan.associator_of_limit_cone \u212c Y\u2081 Y\u2082 Y\u2083) =\n  iso.hom (limits.binary_fan.associator_of_limit_cone \u212c X\u2081 X\u2082 X\u2083) \u226b tensor_hom \u212c f\u2081 (tensor_hom \u212c f\u2082 f\u2083) := sorry\n\nend monoidal_of_chosen_finite_products\n\n\n/-- A category with a terminal object and binary products has a natural monoidal structure. -/\ndef monoidal_of_chosen_finite_products {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) : monoidal_category C :=\n  monoidal_category.mk (fun (X Y : C) => sorry) (fun (_x _x_1 _x_2 _x_3 : C) (f : _x \u27f6 _x_1) (g : _x_2 \u27f6 _x_3) => sorry)\n    (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) (fun (X Y Z : C) => limits.binary_fan.associator_of_limit_cone \u212c X Y Z)\n    (fun (X : C) =>\n      limits.binary_fan.left_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c (limits.cone.X (limits.limit_cone.cone \ud835\udcaf)) X)))\n    fun (X : C) =>\n      limits.binary_fan.right_unitor (limits.limit_cone.is_limit \ud835\udcaf)\n        (limits.limit_cone.is_limit (\u212c X (limits.cone.X (limits.limit_cone.cone \ud835\udcaf))))\n\nnamespace monoidal_of_chosen_finite_products\n\n\n/--\nA type synonym for `C` carrying a monoidal category structure corresponding to\na fixed choice of limit data for the empty functor, and for `pair X Y` for every `X Y : C`.\n\nThis is an implementation detail for `symmetric_of_chosen_finite_products`.\n-/\ndef monoidal_of_chosen_finite_products_synonym {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) :=\n  C\n\nprotected instance monoidal_of_chosen_finite_products_synonym.category_theory.monoidal_category {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) : monoidal_category (monoidal_of_chosen_finite_products_synonym \ud835\udcaf \u212c) :=\n  monoidal_of_chosen_finite_products \ud835\udcaf \u212c\n\ntheorem braiding_naturality {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) {X : C} {X' : C} {Y : C} {Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') : tensor_hom \u212c f g \u226b\n    iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c Y Y')) (limits.limit_cone.is_limit (\u212c Y' Y))) =\n  iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X X')) (limits.limit_cone.is_limit (\u212c X' X))) \u226b\n    tensor_hom \u212c g f := sorry\n\ntheorem hexagon_forward {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X : C) (Y : C) (Z : C) : iso.hom (limits.binary_fan.associator_of_limit_cone \u212c X Y Z) \u226b\n    iso.hom\n        (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X (tensor_obj \u212c Y Z)))\n          (limits.limit_cone.is_limit (\u212c (tensor_obj \u212c Y Z) X))) \u226b\n      iso.hom (limits.binary_fan.associator_of_limit_cone \u212c Y Z X) =\n  tensor_hom \u212c\n      (iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X Y)) (limits.limit_cone.is_limit (\u212c Y X))))\n      \ud835\udfd9 \u226b\n    iso.hom (limits.binary_fan.associator_of_limit_cone \u212c Y X Z) \u226b\n      tensor_hom \u212c \ud835\udfd9\n        (iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X Z)) (limits.limit_cone.is_limit (\u212c Z X)))) := sorry\n\ntheorem hexagon_reverse {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X : C) (Y : C) (Z : C) : iso.inv (limits.binary_fan.associator_of_limit_cone \u212c X Y Z) \u226b\n    iso.hom\n        (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c (tensor_obj \u212c X Y) Z))\n          (limits.limit_cone.is_limit (\u212c Z (tensor_obj \u212c X Y)))) \u226b\n      iso.inv (limits.binary_fan.associator_of_limit_cone \u212c Z X Y) =\n  tensor_hom \u212c \ud835\udfd9\n      (iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c Y Z)) (limits.limit_cone.is_limit (\u212c Z Y)))) \u226b\n    iso.inv (limits.binary_fan.associator_of_limit_cone \u212c X Z Y) \u226b\n      tensor_hom \u212c\n        (iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X Z)) (limits.limit_cone.is_limit (\u212c Z X))))\n        \ud835\udfd9 := sorry\n\ntheorem symmetry {C : Type u} [category C] (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) (X : C) (Y : C) : iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c X Y)) (limits.limit_cone.is_limit (\u212c Y X))) \u226b\n    iso.hom (limits.binary_fan.braiding (limits.limit_cone.is_limit (\u212c Y X)) (limits.limit_cone.is_limit (\u212c X Y))) =\n  \ud835\udfd9 := sorry\n\nend monoidal_of_chosen_finite_products\n\n\n/--\nThe monoidal structure coming from finite products is symmetric.\n-/\ndef symmetric_of_chosen_finite_products {C : Type u} [category C] (\ud835\udcaf : limits.limit_cone (functor.empty C)) (\u212c : (X Y : C) \u2192 limits.limit_cone (limits.pair X Y)) : symmetric_category (monoidal_of_chosen_finite_products.monoidal_of_chosen_finite_products_synonym \ud835\udcaf \u212c) :=\n  symmetric_category.mk\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/of_chosen_finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.3826908299931821}}
{"text": "import Qq\nimport Lean\n\nimport Category.Basic\n\nopen Lean Lean.Expr Lean.Meta Lean.Elab.Tactic\n\nopen Qq\n\nuniverse u v\n\nstructure Result {\u03b1:Q(Type u)} (E: Q($\u03b1) \u2192 Type) (e: Q($\u03b1)) where\n  expr  : Q($\u03b1)\n  val   : E expr\n  proof : Q($e = $expr)\n\ndef Result.Id {\u03b1:Q(Type u)} {E:Q($\u03b1) \u2192 Type} (e: Q($\u03b1)) (val: E e) : Result E e where\n  expr  := e\n  val   := val\n  proof := q(by rfl)\n\ndef Result.map {\u03b1:Q(Type u)} {E:Q($\u03b1) \u2192 Type} {e: Q($\u03b1)}\n  (r:Result E e) (F:Q($\u03b1) \u2192 Type) (f: E r.expr \u2192 F r.expr) : Result F e where\n  expr  := r.expr\n  val   := f r.val\n  proof := r.proof\n\ninstance [Inhabited (\u03a3 e, E e)] : Inhabited (Result E e) :=\n  let \u27e8e', v\u27e9 : \u03a3 e, E e := default; \u27e8e', v, default\u27e9\n\nnamespace Cat\n\nstructure Context where\n  useTransparancy : Bool\n\nstructure State where\n  atoms : Array Expr := #[]\n\ninstance : Inhabited State where\n  default := \u27e8#[]\u27e9\n\nabbrev CatM := ReaderT Context <| StateT State MetaM\n\ndef CatM.run {\u03b1:Type} (f:CatM \u03b1) (red:Bool) :\n  MetaM \u03b1 := (f \u27e8red\u27e9).run' {}\n\ndef CatM.add_atom (e:Expr) : CatM Nat := do\n  let table \u2190 get\n  if (\u2190 read).useTransparancy then\n    for h : i in [0:table.atoms.size] do\n      have : i < table.atoms.size := h.2\n      if \u2190 isDefEq e table.atoms[i] then\n        return i\n  modifyGet fun c => (table.atoms.size, {c with atoms := c.atoms.push e})\n\nsection\n\nvariable {C: Q(Type u)}\nvariable {CatC: Q(Category.{u, v} $C)}\n\ninductive Atom : \u2200 X Y: Q($C), Q($X \u27f6  $Y) \u2192 Type where\n| Const : \u2200 X Y: Q($C), \u2200 f: Q($X \u27f6  $Y), Nat \u2192 Atom X Y f\n\ninductive AtomicMorphism : \u2200 X Y: Q($C), Q($X \u27f6  $Y) \u2192 Type where\n| Nil : \u2200 {X Y f}, Atom X Y f \u2192 AtomicMorphism X Y f\n| Cons: \u2200 {X Z: Q($C)} (Y:Q($C)),\n  \u2200 f: Q($Y \u27f6  $Z), \u2200 g: Q($X \u27f6  $Y),\n  Atom Y Z f \u2192 AtomicMorphism X Y g \u2192 AtomicMorphism X Z q($f \u229a $g)\n\n\ninductive Morphism : \u2200 X Y:Q($C), Q($X \u27f6  $Y) \u2192 Type where\n| Id   : \u2200 X: Q($C), Morphism X X q(\ud835\udfd9 $X)\n| List : \u2200 {X Y}, \u2200 f, AtomicMorphism X Y f \u2192 Morphism X Y f\n\ninstance {X Y:Q($C)} {f:Q($X \u27f6  $Y)} : Inhabited (@Morphism u v C CatC X Y f) where\n  default := Morphism.List f <| AtomicMorphism.Nil <| Atom.Const X Y f 0\n\ninstance {X Y:Q($C)} : Inhabited (\u03a3 f, @Morphism u v C CatC X Y f) where\n  default := \u27e8default, default\u27e9\n\n\n#check Result\n#check AtomicMorphism\n\n\ndef AtomicMorphism.compose {X Y Z: Q($C)} (f1:Q($Y \u27f6  $Z)) (f2:Q($X \u27f6  $Y))\n  (l1:AtomicMorphism Y Z f1) (l2:AtomicMorphism X Y f2) :\n    @Result _ q($X \u27f6  $Z) (@AtomicMorphism _ _ C CatC X Z) q($f1 \u229a $f2) :=\n\n  match l1 with\n  | AtomicMorphism.Nil atom => Result.Id q($f1 \u229a $f2) (.Cons Y f1 f2 atom l2)\n  | .Cons Z f1 g1 atom l1 =>\n  by\n    let r := compose g1 f2 l1 l2\n    generalize h: r.expr = expr\n    have proof : Q($g1 \u229a $f2 = $expr) := r.proof\n    exact {\n      expr := q($f1 \u229a $r.expr),\n      val  := .Cons Z f1 q($r.expr) atom r.val,\n      proof:= show Q($f1 \u229a $expr = ($f1 \u229a $g1) \u229a $f2) from q(by\n        simp\n        rw [\u00ab$proof\u00bb]\n      )\n    }\n\n\ndef Morphism.compose {X Y Z: Q($C)} (f1: Q($Y \u27f6  $Z)) (f2: Q($X \u27f6  $Y))\n  (l1: @Morphism _ _ C CatC Y Z f1) (l2: @Morphism _ _ C CatC X Y f2) :\n  @Result _ q($X \u27f6  $Z) (@Morphism _ _ C CatC X Z) q($f1 \u229a $f2) :=\nby\n  cases l1 with\n  | Id =>\n    apply Result.mk\n    case expr =>\n      exact f2\n    case val =>\n      exact l2\n    case proof =>\n      exact q(by\n        rw [Category.id_comp]\n      )\n  | List _ a =>\n    cases l2 with\n    | Id =>\n      apply Result.mk\n      case expr =>\n        exact f1\n      case val =>\n        exact (.List _ a)\n      case proof =>\n        exact q(by\n          rw [Category.comp_id]\n        )\n    | List _ b =>\n      apply Result.map (AtomicMorphism.compose f1 f2 a b)\n      apply Morphism.List\n\n#check @Category.id\n#check Category.comp\n#check Morphism.compose\n\n\n\n-- \\f<< for \u00ab and \\f>> for \u00bb\nmutual\n\npartial def match_morphism_dom_eq_cod (X: Q($C)) (f:Q($X \u27f6  $X)) :\n  CatM (@Result _ q($X \u27f6  $X) (@Morphism _ _ C CatC X X) f) := do\n\n  match f with\n  | ~q(\ud835\udfd9 \u00ab$X\u00bb) => do\n    return {expr := q(\ud835\udfd9 $X), val := Morphism.Id X, proof := q(by rfl)}\n  | ~q(@Category.comp _ _ _ \u00ab$X\u00bb _ $g $h) => do\n    let r1 \u2190 @match_morphism_dom_eq_cod X g\n    let r2 \u2190 match_morphism_dom_eq_cod X h\n\n    match (r1, r2) with\n    | (Result.mk g' val_g proof_g, Result.mk h' val_h proof_h) =>\n      match Morphism.compose g' h' val_g val_h with\n      | Result.mk expr val proof =>\n      return {\n        expr := expr, --q($g' \u229a $h'),\n        val := val,\n        proof := q(by\n          rw [<-\u00ab$proof\u00bb, <-\u00ab$proof_g\u00bb, <-\u00ab$proof_h\u00bb]\n        )\n      }\n  | ~q(@Category.comp _ _ _ $Y _ $g $h) => do\n    let r1 \u2190 match_morphism Y X g\n    let r2 \u2190 match_morphism X Y h\n\n    match (r1, r2) with\n    | (Result.mk g' val_g proof_g, Result.mk h' val_h proof_h) =>\n      match Morphism.compose g' h' val_g val_h with\n      | Result.mk expr val proof =>\n        return {\n          expr := expr,\n          val := val,\n          proof := q(by\n            rw [<-\u00ab$proof\u00bb, <-\u00ab$proof_g\u00bb, <-\u00ab$proof_h\u00bb]\n          )\n        }\n  | _ =>\n    let idx \u2190 CatM.add_atom f\n    return Result.Id f <| Morphism.List f <| AtomicMorphism.Nil (Atom.Const X X f idx)\n\n\n\n-- patern match a morphism when the codomain and the domain are distincts\npartial def match_morphism (X Y: Q($C)) (f:Q($X \u27f6  $Y)) :\n  CatM (@Result _ q($X \u27f6  $Y) (@Morphism _ _ C CatC X Y) f) := do\n\n  match f with\n  | ~q(@Category.comp _ _ _ \u00ab$X\u00bb _ $g $h) =>\n    let r2 \u2190 match_morphism_dom_eq_cod X h\n    let r1 \u2190 match_morphism X Y g\n\n    match (r1, r2) with\n    | (Result.mk g' val_g proof_g, Result.mk h' val_h proof_h) =>\n      match Morphism.compose g' h' val_g val_h with\n      | Result.mk expr val proof =>\n        return {\n          expr := expr,\n          val  := val,\n          proof:= q(by\n            rw [<-\u00ab$proof\u00bb, <-\u00ab$proof_g\u00bb, <-\u00ab$proof_h\u00bb]\n          )\n        }\n  | ~q(@Category.comp _ _ _ \u00ab$Y\u00bb _ $g $h) =>\n    let r1 \u2190 match_morphism_dom_eq_cod Y g\n    let r2 \u2190 match_morphism X Y h\n\n    match (r1, r2) with\n    | (Result.mk g' val_g proof_g, Result.mk h' val_h proof_h) =>\n      match Morphism.compose g' h' val_g val_h with\n      | Result.mk expr val proof =>\n        return {\n          expr := expr,\n          val  := val,\n          proof:= q(by\n            rw [<-\u00ab$proof\u00bb, <-\u00ab$proof_g\u00bb, <-\u00ab$proof_h\u00bb]\n          )\n        }\n  | ~q(@Category.comp _ _ _ $Z _ $g $h) =>\n    let r2 \u2190 match_morphism X Z h\n    let r1 \u2190 match_morphism Z Y g\n\n    match (r1, r2) with\n    | (Result.mk g' val_g proof_g, Result.mk h' val_h proof_h) =>\n      match Morphism.compose g' h' val_g val_h with\n      | Result.mk expr val proof =>\n        return {\n          expr := expr,\n          val  := val,\n          proof:= q(by\n            rw [<-\u00ab$proof\u00bb, <-\u00ab$proof_g\u00bb, <-\u00ab$proof_h\u00bb]\n          )\n        }\n  | _ =>\n    let idx \u2190 CatM.add_atom f\n    return Result.Id f <| Morphism.List f <| AtomicMorphism.Nil (Atom.Const X Y f idx)\n\nend\n\nend\n\ndef of_eq (_ : (a: R) = c) (_ : b = c) : a = b := by simp only [*]\n\nuniverse w\n\nabbrev EndoHom (C:Type w) (CatC:Category C) (X:C) := @Category.Hom C CatC X X\n\npartial def match_morphism_equality (mvarid:MVarId) : CatM <| List MVarId := do\n  let type_eq : Q(Prop) \u2190 mvarid.getType\n  match type_eq with\n  | ~q($f = $g) => do\n    let type_fg \u2190 inferType f\n    -- $f and $g are morphism of type $type_f, $type_f have type Type v\n    let .sort (.succ v) \u2190 whnf (\u2190 inferType type_fg) | throwError \"not a type\"\n\n    have type_fg : Q(Type v) := type_fg\n    have f : Q($type_fg) := f\n    have g : Q($type_fg) := g\n\n\n    match type_fg with\n    | ~q(EndoHom $C $CatC $X) =>\n\n      let .sort (.succ u) \u2190 whnf (\u2190 inferType C) | throwError \"a category shound be a type\"\n\n      let type_fg : Q(Type v) := q($X \u27f6  $X)\n      have f : Q($type_fg) := f\n      have g : Q($type_fg) := g\n\n      let \u27e8f', vf, pf\u27e9 \u2190 @match_morphism_dom_eq_cod u v C CatC X f\n      let \u27e8g', vg, pg\u27e9 \u2190 @match_morphism_dom_eq_cod u v C CatC X g\n\n      --throwError \"{f'} {g'}\"\n\n      if not (\u2190 isDefEq f' g') then do\n        throwError \"expressions not equal\\n{f}\\n{g}\\n{f'}\\n{g'}\"\n\n      let pg : Q(\u00ab$g\u00bb = \u00ab$f'\u00bb) := pg\n      mvarid.assign q(of_eq $pf $pg)\n\n      return []\n\n    | ~q(@Category.Hom $C $CatC $X $Y) =>\n\n      let .sort (.succ u) \u2190 whnf (\u2190 inferType C) | throwError \"a category shound be a type\"\n\n      let type_fg : Q(Type v) := q($X \u27f6  $Y)\n      have f : Q($type_fg) := f\n      have g : Q($type_fg) := g\n\n      let \u27e8f', vf, pf\u27e9 \u2190 @match_morphism u v C CatC X Y f\n      let \u27e8g', vg, pg\u27e9 \u2190 @match_morphism u v C CatC X Y g\n\n\n      if not (\u2190 isDefEq f' g') then do\n        throwError \"expressions not equal\\n{f}\\n{g}\\n{f'}\\n{g'}\"\n\n      let pg : Q(\u00ab$g\u00bb = \u00ab$f'\u00bb) := pg\n      mvarid.assign q(of_eq $pf $pg)\n\n      return []\n\n    | _ => throwError \"not a morphism\"\n  | _ => throwError \"not an equality\"\n\nelab \"reduce_assoc_and_id\" : tactic =>\n  withMainContext do\n    liftMetaTactic fun mvarId => do\n      CatM.run (match_morphism_equality mvarId) true\n\nend Cat\n\nexample (C:Type u) [Category C] (X Y:C) (f: X \u27f6  Y) (g:X \u27f6  X) :\n  (f \u229a \ud835\udfd9 X) \u229a g = \ud835\udfd9 Y \u229a (f \u229a g) := by reduce_assoc_and_id\n\n", "meta": {"author": "RemyCiterin", "repo": "LeanCat", "sha": "7ffeb6e61809a56da44b00161f0cbf85bef6171e", "save_path": "github-repos/lean/RemyCiterin-LeanCat", "path": "github-repos/lean/RemyCiterin-LeanCat/LeanCat-7ffeb6e61809a56da44b00161f0cbf85bef6171e/Category/Tactics.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585669110202, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3826509301854552}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Kenny Lau, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.chain\nimport Mathlib.data.list.nodup\nimport Mathlib.data.list.of_fn\nimport Mathlib.data.list.zip\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\nnamespace list\n\n\n/- iota and range(') -/\n\n@[simp] theorem length_range' (s : \u2115) (n : \u2115) : length (range' s n) = n := sorry\n\n@[simp] theorem range'_eq_nil {s : \u2115} {n : \u2115} : range' s n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (range' s n) = 0 \u2194 n = 0)) (length_range' s n)))\n      (iff.refl (n = 0)))\n\n@[simp] theorem mem_range' {m : \u2115} {s : \u2115} {n : \u2115} : m \u2208 range' s n \u2194 s \u2264 m \u2227 m < s + n := sorry\n\ntheorem map_add_range' (a : \u2115) (s : \u2115) (n : \u2115) : map (Add.add a) (range' s n) = range' (a + s) n :=\n  sorry\n\ntheorem map_sub_range' (a : \u2115) (s : \u2115) (n : \u2115) (h : a \u2264 s) :\n    map (fun (x : \u2115) => x - a) (range' s n) = range' (s - a) n :=\n  sorry\n\ntheorem chain_succ_range' (s : \u2115) (n : \u2115) :\n    chain (fun (a b : \u2115) => b = Nat.succ a) s (range' (s + 1) n) :=\n  sorry\n\ntheorem chain_lt_range' (s : \u2115) (n : \u2115) : chain Less s (range' (s + 1) n) :=\n  chain.imp (fun (a b : \u2115) (e : b = Nat.succ a) => Eq.symm e \u25b8 nat.lt_succ_self a)\n    (chain_succ_range' s n)\n\ntheorem pairwise_lt_range' (s : \u2115) (n : \u2115) : pairwise Less (range' s n) := sorry\n\ntheorem nodup_range' (s : \u2115) (n : \u2115) : nodup (range' s n) :=\n  pairwise.imp (fun (a b : \u2115) => ne_of_lt) (pairwise_lt_range' s n)\n\n@[simp] theorem range'_append (s : \u2115) (m : \u2115) (n : \u2115) :\n    range' s m ++ range' (s + m) n = range' s (n + m) :=\n  sorry\n\ntheorem range'_sublist_right {s : \u2115} {m : \u2115} {n : \u2115} : range' s m <+ range' s n \u2194 m \u2264 n := sorry\n\ntheorem range'_subset_right {s : \u2115} {m : \u2115} {n : \u2115} : range' s m \u2286 range' s n \u2194 m \u2264 n := sorry\n\ntheorem nth_range' (s : \u2115) {m : \u2115} {n : \u2115} : m < n \u2192 nth (range' s n) m = some (s + m) := sorry\n\n@[simp] theorem nth_le_range' {n : \u2115} {m : \u2115} (i : \u2115) (H : i < length (range' n m)) :\n    nth_le (range' n m) i H = n + i :=\n  sorry\n\ntheorem range'_concat (s : \u2115) (n : \u2115) : range' s (n + 1) = range' s n ++ [s + n] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s (n + 1) = range' s n ++ [s + n])) (add_comm n 1)))\n    (Eq.symm (range'_append s n 1))\n\ntheorem range_core_range' (s : \u2115) (n : \u2115) : range_core s (range' s n) = range' 0 (n + s) := sorry\n\ntheorem range_eq_range' (n : \u2115) : range n = range' 0 n :=\n  Eq.trans (range_core_range' n 0)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (range' 0 (0 + n) = range' 0 n)) (zero_add n)))\n      (Eq.refl (range' 0 n)))\n\ntheorem range_succ_eq_map (n : \u2115) : range (n + 1) = 0 :: map Nat.succ (range n) := sorry\n\ntheorem range'_eq_map_range (s : \u2115) (n : \u2115) : range' s n = map (Add.add s) (range n) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range' s n = map (Add.add s) (range n))) (range_eq_range' n)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (range' s n = map (Add.add s) (range' 0 n))) (map_add_range' s 0 n)))\n      (Eq.refl (range' s n)))\n\n@[simp] theorem length_range (n : \u2115) : length (range n) = n := sorry\n\n@[simp] theorem range_eq_nil {n : \u2115} : range n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (range n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (range n) = 0 \u2194 n = 0)) (length_range n)))\n      (iff.refl (n = 0)))\n\ntheorem pairwise_lt_range (n : \u2115) : pairwise Less (range n) := sorry\n\ntheorem nodup_range (n : \u2115) : nodup (range n) := sorry\n\ntheorem range_sublist {m : \u2115} {n : \u2115} : range m <+ range n \u2194 m \u2264 n := sorry\n\ntheorem range_subset {m : \u2115} {n : \u2115} : range m \u2286 range n \u2194 m \u2264 n := sorry\n\n@[simp] theorem mem_range {m : \u2115} {n : \u2115} : m \u2208 range n \u2194 m < n := sorry\n\n@[simp] theorem not_mem_range_self {n : \u2115} : \u00acn \u2208 range n := mt (iff.mp mem_range) (lt_irrefl n)\n\n@[simp] theorem self_mem_range_succ (n : \u2115) : n \u2208 range (n + 1) := sorry\n\ntheorem nth_range {m : \u2115} {n : \u2115} (h : m < n) : nth (range n) m = some m := sorry\n\ntheorem range_succ (n : \u2115) : range (Nat.succ n) = range n ++ [n] := sorry\n\n@[simp] theorem range_zero : range 0 = [] := rfl\n\ntheorem iota_eq_reverse_range' (n : \u2115) : iota n = reverse (range' 1 n) := sorry\n\n@[simp] theorem length_iota (n : \u2115) : length (iota n) = n := sorry\n\ntheorem pairwise_gt_iota (n : \u2115) : pairwise gt (iota n) := sorry\n\ntheorem nodup_iota (n : \u2115) : nodup (iota n) := sorry\n\ntheorem mem_iota {m : \u2115} {n : \u2115} : m \u2208 iota n \u2194 1 \u2264 m \u2227 m \u2264 n := sorry\n\ntheorem reverse_range' (s : \u2115) (n : \u2115) :\n    reverse (range' s n) = map (fun (i : \u2115) => s + n - 1 - i) (range n) :=\n  sorry\n\n/-- All elements of `fin n`, from `0` to `n-1`. -/\ndef fin_range (n : \u2115) : List (fin n) := pmap fin.mk (range n) sorry\n\n@[simp] theorem fin_range_zero : fin_range 0 = [] := rfl\n\n@[simp] theorem mem_fin_range {n : \u2115} (a : fin n) : a \u2208 fin_range n := sorry\n\ntheorem nodup_fin_range (n : \u2115) : nodup (fin_range n) :=\n  nodup_pmap (fun (_x : \u2115) (_x_1 : _x < n) (_x_2 : \u2115) (_x_3 : _x_2 < n) => fin.veq_of_eq)\n    (nodup_range n)\n\n@[simp] theorem length_fin_range (n : \u2115) : length (fin_range n) = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (fin_range n) = n)) (fin_range.equations._eqn_1 n)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (length (pmap fin.mk (range n) (fin_range._proof_1 n)) = n))\n          length_pmap))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length (range n) = n)) (length_range n))) (Eq.refl n)))\n\n@[simp] theorem fin_range_eq_nil {n : \u2115} : fin_range n = [] \u2194 n = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (fin_range n = [] \u2194 n = 0)) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (fin_range n) = 0 \u2194 n = 0)) (length_fin_range n)))\n      (iff.refl (n = 0)))\n\ntheorem prod_range_succ {\u03b1 : Type u} [monoid \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    prod (map f (range (Nat.succ n))) = prod (map f (range n)) * f n :=\n  sorry\n\n/-- A variant of `prod_range_succ` which pulls off the first\n  term in the product rather than the last.-/\ntheorem sum_range_succ' {\u03b1 : Type u} [add_monoid \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    sum (map f (range (Nat.succ n))) = f 0 + sum (map (fun (i : \u2115) => f (Nat.succ i)) (range n)) :=\n  sorry\n\n@[simp] theorem enum_from_map_fst {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) :\n    map prod.fst (enum_from n l) = range' n (length l) :=\n  sorry\n\n@[simp] theorem enum_map_fst {\u03b1 : Type u} (l : List \u03b1) : map prod.fst (enum l) = range (length l) :=\n  sorry\n\ntheorem enum_eq_zip_range {\u03b1 : Type u} (l : List \u03b1) : enum l = zip (range (length l)) l :=\n  zip_of_prod (enum_map_fst l) (enum_map_snd l)\n\n@[simp] theorem unzip_enum_eq_prod {\u03b1 : Type u} (l : List \u03b1) :\n    unzip (enum l) = (range (length l), l) :=\n  sorry\n\ntheorem enum_from_eq_zip_range' {\u03b1 : Type u} (l : List \u03b1) {n : \u2115} :\n    enum_from n l = zip (range' n (length l)) l :=\n  zip_of_prod (enum_from_map_fst n l) (enum_from_map_snd n l)\n\n@[simp] theorem unzip_enum_from_eq_prod {\u03b1 : Type u} (l : List \u03b1) {n : \u2115} :\n    unzip (enum_from n l) = (range' n (length l), l) :=\n  sorry\n\n@[simp] theorem nth_le_range {n : \u2115} (i : \u2115) (H : i < length (range n)) :\n    nth_le (range n) i H = i :=\n  sorry\n\n@[simp] theorem nth_le_fin_range {n : \u2115} {i : \u2115} (h : i < length (fin_range n)) :\n    nth_le (fin_range n) i h = { val := i, property := length_fin_range n \u25b8 h } :=\n  sorry\n\ntheorem of_fn_eq_pmap {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} :\n    of_fn f =\n        pmap (fun (i : \u2115) (hi : i < n) => f { val := i, property := hi }) (range n)\n          fun (_x : \u2115) => iff.mp mem_range :=\n  sorry\n\ntheorem of_fn_id (n : \u2115) : of_fn id = fin_range n := of_fn_eq_pmap\n\ntheorem of_fn_eq_map {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} : of_fn f = map f (fin_range n) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = map f (fin_range n))) (Eq.symm (of_fn_id n))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = map f (of_fn id))) (map_of_fn id f)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (of_fn f = of_fn (f \u2218 id))) (function.right_id f)))\n        (Eq.refl (of_fn f))))\n\ntheorem nodup_of_fn {\u03b1 : Type u_1} {n : \u2115} {f : fin n \u2192 \u03b1} (hf : function.injective f) :\n    nodup (of_fn f) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/range_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.3825366304023984}}
{"text": "/-\nCopyright (c) 2014 Floris van Doorn (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn, Leonardo de Moura, Jeremy Avigad, Mario Carneiro\n\n! This file was ported from Lean 3 source module data.nat.order.lemmas\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Nat.Order.Basic\nimport Mathbin.Data.Set.Basic\nimport Mathbin.Algebra.Ring.Divisibility\nimport Mathbin.Algebra.GroupWithZero.Divisibility\n\n/-!\n# Further lemmas about the natural numbers\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThe distinction between this file and `data.nat.order.basic` is not particularly clear.\nThey are separated by now to minimize the porting requirements for tactics during the transition to\nmathlib4. After `data.rat.order` has been ported, please feel free to reorganize these two files.\n-/\n\n\nuniverse u v\n\nvariable {m n k : \u2115}\n\nnamespace Nat\n\n/-! ### Sets -/\n\n\n#print Nat.Subtype.orderBot /-\ninstance Subtype.orderBot (s : Set \u2115) [DecidablePred (\u00b7 \u2208 s)] [h : Nonempty s] : OrderBot s\n    where\n  bot := \u27e8Nat.find (nonempty_subtype.1 h), Nat.find_spec (nonempty_subtype.1 h)\u27e9\n  bot_le x := Nat.find_min' _ x.2\n#align nat.subtype.order_bot Nat.Subtype.orderBot\n-/\n\n#print Nat.Subtype.semilatticeSup /-\ninstance Subtype.semilatticeSup (s : Set \u2115) : SemilatticeSup s :=\n  { Subtype.linearOrder s, LinearOrder.toLattice with }\n#align nat.subtype.semilattice_sup Nat.Subtype.semilatticeSup\n-/\n\n/- warning: nat.subtype.coe_bot -> Nat.Subtype.coe_bot is a dubious translation:\nlean 3 declaration is\n  forall {s : Set.{0} Nat} [_inst_1 : DecidablePred.{1} Nat (fun (_x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) _x s)] [h : Nonempty.{1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s)], Eq.{1} Nat ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) Nat (HasLiftT.mk.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) Nat (CoeTC\u2093.coe.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) Nat (coeBase.{1, 1} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) Nat (coeSubtype.{1} Nat (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x s))))) (Bot.bot.{0} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) (OrderBot.toHasBot.{0} (coeSort.{1, 2} (Set.{0} Nat) Type (Set.hasCoeToSort.{0} Nat) s) (Subtype.hasLe.{0} Nat Nat.hasLe (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x s)) (Nat.Subtype.orderBot s (fun (a : Nat) => _inst_1 a) h)))) (Nat.find (fun (n : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) n s) (fun (a : Nat) => _inst_1 a) (Iff.mp (Nonempty.{1} (Subtype.{1} Nat (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x s))) (Exists.{1} Nat (fun (a : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) a s)) (nonempty_subtype.{1} Nat (fun (x : Nat) => Membership.Mem.{0, 0} Nat (Set.{0} Nat) (Set.hasMem.{0} Nat) x s)) h))\nbut is expected to have type\n  forall {s : Set.{0} Nat} [_inst_1 : DecidablePred.{1} Nat (fun (_x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) _x s)] [h : Nonempty.{1} (Set.Elem.{0} Nat s)], Eq.{1} Nat (Subtype.val.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x s) (Bot.bot.{0} (Set.Elem.{0} Nat s) (OrderBot.toBot.{0} (Set.Elem.{0} Nat s) (Subtype.le.{0} Nat instLENat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x s)) (Nat.Subtype.orderBot s (fun (a : Nat) => _inst_1 a) h)))) (Nat.find (fun (n : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) n s) (fun (a : Nat) => _inst_1 a) (Iff.mp (Nonempty.{1} (Subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x s))) (Exists.{1} Nat (fun (a : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) a s)) (nonempty_subtype.{1} Nat (fun (x : Nat) => Membership.mem.{0, 0} Nat (Set.{0} Nat) (Set.instMembershipSet.{0} Nat) x s)) h))\nCase conversion may be inaccurate. Consider using '#align nat.subtype.coe_bot Nat.Subtype.coe_bot\u2093'. -/\ntheorem Subtype.coe_bot {s : Set \u2115} [DecidablePred (\u00b7 \u2208 s)] [h : Nonempty s] :\n    ((\u22a5 : s) : \u2115) = Nat.find (nonempty_subtype.1 h) :=\n  rfl\n#align nat.subtype.coe_bot Nat.Subtype.coe_bot\n\n#print Nat.set_eq_univ /-\ntheorem set_eq_univ {S : Set \u2115} : S = Set.univ \u2194 0 \u2208 S \u2227 \u2200 k : \u2115, k \u2208 S \u2192 k + 1 \u2208 S :=\n  \u27e8by rintro rfl <;> simp, fun \u27e8h0, hs\u27e9 => Set.eq_univ_of_forall (set_induction h0 hs)\u27e9\n#align nat.set_eq_univ Nat.set_eq_univ\n-/\n\n/-! ### `div` -/\n\n\n#print Nat.lt_div_iff_mul_lt /-\nprotected theorem lt_div_iff_mul_lt {n d : \u2115} (hnd : d \u2223 n) (a : \u2115) : a < n / d \u2194 d * a < n :=\n  by\n  rcases d.eq_zero_or_pos with (rfl | hd0); \u00b7 simp [zero_dvd_iff.mp hnd]\n  rw [\u2190 mul_lt_mul_left hd0, \u2190 Nat.eq_mul_of_div_eq_right hnd rfl]\n#align nat.lt_div_iff_mul_lt Nat.lt_div_iff_mul_lt\n-/\n\n#print Nat.div_eq_iff_eq_of_dvd_dvd /-\ntheorem div_eq_iff_eq_of_dvd_dvd {n x y : \u2115} (hn : n \u2260 0) (hx : x \u2223 n) (hy : y \u2223 n) :\n    n / x = n / y \u2194 x = y := by\n  constructor\n  \u00b7 intro h\n    rw [\u2190 mul_right_inj' hn]\n    apply Nat.eq_mul_of_div_eq_left (dvd_mul_of_dvd_left hy x)\n    rw [eq_comm, mul_comm, Nat.mul_div_assoc _ hy]\n    exact Nat.eq_mul_of_div_eq_right hx h\n  \u00b7 intro h\n    rw [h]\n#align nat.div_eq_iff_eq_of_dvd_dvd Nat.div_eq_iff_eq_of_dvd_dvd\n-/\n\n#print Nat.div_eq_zero_iff /-\nprotected theorem div_eq_zero_iff {a b : \u2115} (hb : 0 < b) : a / b = 0 \u2194 a < b :=\n  \u27e8fun h => by rw [\u2190 mod_add_div a b, h, MulZeroClass.mul_zero, add_zero] <;> exact mod_lt _ hb,\n    fun h => by\n    rw [\u2190 mul_right_inj' hb.ne', \u2190 @add_left_cancel_iff _ _ _ (a % b), mod_add_div, mod_eq_of_lt h,\n      MulZeroClass.mul_zero, add_zero]\u27e9\n#align nat.div_eq_zero_iff Nat.div_eq_zero_iff\n-/\n\n#print Nat.div_eq_zero /-\nprotected theorem div_eq_zero {a b : \u2115} (hb : a < b) : a / b = 0 :=\n  (Nat.div_eq_zero_iff <| (zero_le a).trans_lt hb).mpr hb\n#align nat.div_eq_zero Nat.div_eq_zero\n-/\n\n/-! ### `mod`, `dvd` -/\n\n\n#print Nat.dvd_one /-\n@[simp]\nprotected theorem dvd_one {n : \u2115} : n \u2223 1 \u2194 n = 1 :=\n  \u27e8eq_one_of_dvd_one, fun e => e.symm \u25b8 dvd_rfl\u27e9\n#align nat.dvd_one Nat.dvd_one\n-/\n\n#print Nat.not_two_dvd_bit1 /-\n@[simp]\nprotected theorem not_two_dvd_bit1 (n : \u2115) : \u00ac2 \u2223 bit1 n :=\n  by\n  rw [bit1, Nat.dvd_add_right two_dvd_bit0, Nat.dvd_one]\n  cc\n#align nat.not_two_dvd_bit1 Nat.not_two_dvd_bit1\n-/\n\n#print Nat.dvd_add_self_left /-\n/-- A natural number `m` divides the sum `m + n` if and only if `m` divides `n`.-/\n@[simp]\nprotected theorem dvd_add_self_left {m n : \u2115} : m \u2223 m + n \u2194 m \u2223 n :=\n  Nat.dvd_add_right (dvd_refl m)\n#align nat.dvd_add_self_left Nat.dvd_add_self_left\n-/\n\n#print Nat.dvd_add_self_right /-\n/-- A natural number `m` divides the sum `n + m` if and only if `m` divides `n`.-/\n@[simp]\nprotected theorem dvd_add_self_right {m n : \u2115} : m \u2223 n + m \u2194 m \u2223 n :=\n  Nat.dvd_add_left (dvd_refl m)\n#align nat.dvd_add_self_right Nat.dvd_add_self_right\n-/\n\n#print Nat.dvd_sub' /-\n-- TODO: update `nat.dvd_sub` in core\ntheorem dvd_sub' {k m n : \u2115} (h\u2081 : k \u2223 m) (h\u2082 : k \u2223 n) : k \u2223 m - n :=\n  by\n  cases' le_total n m with H H\n  \u00b7 exact dvd_sub H h\u2081 h\u2082\n  \u00b7 rw [tsub_eq_zero_iff_le.mpr H]\n    exact dvd_zero k\n#align nat.dvd_sub' Nat.dvd_sub'\n-/\n\n/- warning: nat.succ_div -> Nat.succ_div is a dubious translation:\nlean 3 declaration is\n  forall (a : Nat) (b : Nat), Eq.{1} Nat (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) a (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) b) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.hasDiv) a b) (ite.{1} Nat (Dvd.Dvd.{0} Nat Nat.hasDvd b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) a (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Nat.decidableDvd b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) a (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))) (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))))\nbut is expected to have type\n  forall (a : Nat) (b : Nat), Eq.{1} Nat (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) a (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) b) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HDiv.hDiv.{0, 0, 0} Nat Nat Nat (instHDiv.{0} Nat Nat.instDivNat) a b) (ite.{1} Nat (Dvd.dvd.{0} Nat Nat.instDvdNat b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) a (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Nat.decidable_dvd b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) a (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)) (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))))\nCase conversion may be inaccurate. Consider using '#align nat.succ_div Nat.succ_div\u2093'. -/\ntheorem succ_div : \u2200 a b : \u2115, (a + 1) / b = a / b + if b \u2223 a + 1 then 1 else 0\n  | a, 0 => by simp\n  | 0, 1 => by simp\n  | 0, b + 2 => by\n    have hb2 : b + 2 > 1 := by decide\n    simp [ne_of_gt hb2, div_eq_of_lt hb2]\n  | a + 1, b + 1 => by\n    rw [Nat.div_eq]; conv_rhs => rw [Nat.div_eq]\n    by_cases hb_eq_a : b = a + 1\n    \u00b7 simp [hb_eq_a, le_refl]\n    by_cases hb_le_a1 : b \u2264 a + 1\n    \u00b7 have hb_le_a : b \u2264 a := le_of_lt_succ (lt_of_le_of_ne hb_le_a1 hb_eq_a)\n      have h\u2081 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 + 1 := \u27e8succ_pos _, (add_le_add_iff_right _).2 hb_le_a1\u27e9\n      have h\u2082 : 0 < b + 1 \u2227 b + 1 \u2264 a + 1 := \u27e8succ_pos _, (add_le_add_iff_right _).2 hb_le_a\u27e9\n      have dvd_iff : b + 1 \u2223 a - b + 1 \u2194 b + 1 \u2223 a + 1 + 1 := by\n        rw [Nat.dvd_add_iff_left (dvd_refl (b + 1)), \u2190 add_tsub_add_eq_tsub_right a 1 b,\n          add_comm (_ - _), add_assoc, tsub_add_cancel_of_le (succ_le_succ hb_le_a), add_comm 1]\n      have wf : a - b < a + 1 := lt_succ_of_le tsub_le_self\n      rw [if_pos h\u2081, if_pos h\u2082, add_tsub_add_eq_tsub_right, \u2190 tsub_add_eq_add_tsub hb_le_a,\n        have := wf\n        succ_div (a - b),\n        add_tsub_add_eq_tsub_right]\n      simp [dvd_iff, succ_eq_add_one, add_comm 1, add_assoc]\n    \u00b7 have hba : \u00acb \u2264 a := not_le_of_gt (lt_trans (lt_succ_self a) (lt_of_not_ge hb_le_a1))\n      have hb_dvd_a : \u00acb + 1 \u2223 a + 2 := fun h =>\n        hb_le_a1 (le_of_succ_le_succ (le_of_dvd (succ_pos _) h))\n      simp [hba, hb_le_a1, hb_dvd_a]\n#align nat.succ_div Nat.succ_div\n\n#print Nat.succ_div_of_dvd /-\ntheorem succ_div_of_dvd {a b : \u2115} (hba : b \u2223 a + 1) : (a + 1) / b = a / b + 1 := by\n  rw [succ_div, if_pos hba]\n#align nat.succ_div_of_dvd Nat.succ_div_of_dvd\n-/\n\n#print Nat.succ_div_of_not_dvd /-\ntheorem succ_div_of_not_dvd {a b : \u2115} (hba : \u00acb \u2223 a + 1) : (a + 1) / b = a / b := by\n  rw [succ_div, if_neg hba, add_zero]\n#align nat.succ_div_of_not_dvd Nat.succ_div_of_not_dvd\n-/\n\n#print Nat.dvd_iff_div_mul_eq /-\ntheorem dvd_iff_div_mul_eq (n d : \u2115) : d \u2223 n \u2194 n / d * d = n :=\n  \u27e8fun h => Nat.div_mul_cancel h, fun h => Dvd.intro_left (n / d) h\u27e9\n#align nat.dvd_iff_div_mul_eq Nat.dvd_iff_div_mul_eq\n-/\n\n#print Nat.dvd_iff_le_div_mul /-\ntheorem dvd_iff_le_div_mul (n d : \u2115) : d \u2223 n \u2194 n \u2264 n / d * d :=\n  ((dvd_iff_div_mul_eq _ _).trans le_antisymm_iff).trans (and_iff_right (div_mul_le_self n d))\n#align nat.dvd_iff_le_div_mul Nat.dvd_iff_le_div_mul\n-/\n\n#print Nat.dvd_iff_dvd_dvd /-\ntheorem dvd_iff_dvd_dvd (n d : \u2115) : d \u2223 n \u2194 \u2200 k : \u2115, k \u2223 d \u2192 k \u2223 n :=\n  \u27e8fun h k hkd => dvd_trans hkd h, fun h => h _ dvd_rfl\u27e9\n#align nat.dvd_iff_dvd_dvd Nat.dvd_iff_dvd_dvd\n-/\n\n#print Nat.dvd_div_of_mul_dvd /-\ntheorem dvd_div_of_mul_dvd {a b c : \u2115} (h : a * b \u2223 c) : b \u2223 c / a :=\n  if ha : a = 0 then by simp [ha]\n  else\n    have ha : 0 < a := Nat.pos_of_ne_zero ha\n    have h1 : \u2203 d, c = a * b * d := h\n    let \u27e8d, hd\u27e9 := h1\n    have h2 : c / a = b * d := Nat.div_eq_of_eq_mul_right ha (by simpa [mul_assoc] using hd)\n    show \u2203 d, c / a = b * d from \u27e8d, h2\u27e9\n#align nat.dvd_div_of_mul_dvd Nat.dvd_div_of_mul_dvd\n-/\n\n#print Nat.dvd_div_iff /-\n@[simp]\ntheorem dvd_div_iff {a b c : \u2115} (hbc : c \u2223 b) : a \u2223 b / c \u2194 c * a \u2223 b :=\n  \u27e8fun h => mul_dvd_of_dvd_div hbc h, fun h => dvd_div_of_mul_dvd h\u27e9\n#align nat.dvd_div_iff Nat.dvd_div_iff\n-/\n\n#print Nat.div_div_div_eq_div /-\n@[simp]\ntheorem div_div_div_eq_div : \u2200 {a b c : \u2115} (dvd : b \u2223 a) (dvd2 : a \u2223 c), c / (a / b) / b = c / a\n  | 0, _ => by simp\n  | a + 1, 0 => fun _ dvd _ => by simpa using dvd\n  | a + 1, c + 1 =>\n    have a_split : a + 1 \u2260 0 := succ_ne_zero a\n    have c_split : c + 1 \u2260 0 := succ_ne_zero c\n    fun b dvd dvd2 => by\n    rcases dvd2 with \u27e8k, rfl\u27e9\n    rcases dvd with \u27e8k2, pr\u27e9\n    have k2_nonzero : k2 \u2260 0 := fun k2_zero => by simpa [k2_zero] using pr\n    rw [Nat.mul_div_cancel_left k (Nat.pos_of_ne_zero a_split), pr,\n      Nat.mul_div_cancel_left k2 (Nat.pos_of_ne_zero c_split), Nat.mul_comm ((c + 1) * k2) k, \u2190\n      Nat.mul_assoc k (c + 1) k2, Nat.mul_div_cancel _ (Nat.pos_of_ne_zero k2_nonzero),\n      Nat.mul_div_cancel _ (Nat.pos_of_ne_zero c_split)]\n#align nat.div_div_div_eq_div Nat.div_div_div_eq_div\n-/\n\n#print Nat.eq_zero_of_dvd_of_lt /-\n/-- If a small natural number is divisible by a larger natural number,\nthe small number is zero. -/\ntheorem eq_zero_of_dvd_of_lt {a b : \u2115} (w : a \u2223 b) (h : b < a) : b = 0 :=\n  Nat.eq_zero_of_dvd_of_div_eq_zero w ((Nat.div_eq_zero_iff (lt_of_le_of_lt (zero_le b) h)).right h)\n#align nat.eq_zero_of_dvd_of_lt Nat.eq_zero_of_dvd_of_lt\n-/\n\n#print Nat.mod_div_self /-\n@[simp]\ntheorem mod_div_self (m n : \u2115) : m % n / n = 0 :=\n  by\n  cases n\n  \u00b7 exact (m % 0).div_zero\n  \u00b7 exact Nat.div_eq_zero (m.mod_lt n.succ_pos)\n#align nat.mod_div_self Nat.mod_div_self\n-/\n\n#print Nat.not_dvd_iff_between_consec_multiples /-\n/-- `n` is not divisible by `a` iff it is between `a * k` and `a * (k + 1)` for some `k`. -/\ntheorem not_dvd_iff_between_consec_multiples (n : \u2115) {a : \u2115} (ha : 0 < a) :\n    (\u2203 k : \u2115, a * k < n \u2227 n < a * (k + 1)) \u2194 \u00aca \u2223 n :=\n  by\n  refine'\n    \u27e8fun \u27e8k, hk1, hk2\u27e9 => not_dvd_of_between_consec_multiples hk1 hk2, fun han =>\n      \u27e8n / a, \u27e8lt_of_le_of_ne (mul_div_le n a) _, lt_mul_div_succ _ ha\u27e9\u27e9\u27e9\n  exact mt (Dvd.intro (n / a)) han\n#align nat.not_dvd_iff_between_consec_multiples Nat.not_dvd_iff_between_consec_multiples\n-/\n\n#print Nat.dvd_right_iff_eq /-\n/-- Two natural numbers are equal if and only if they have the same multiples. -/\ntheorem dvd_right_iff_eq {m n : \u2115} : (\u2200 a : \u2115, m \u2223 a \u2194 n \u2223 a) \u2194 m = n :=\n  \u27e8fun h => dvd_antisymm ((h _).mpr dvd_rfl) ((h _).mp dvd_rfl), fun h n => by rw [h]\u27e9\n#align nat.dvd_right_iff_eq Nat.dvd_right_iff_eq\n-/\n\n#print Nat.dvd_left_iff_eq /-\n/-- Two natural numbers are equal if and only if they have the same divisors. -/\ntheorem dvd_left_iff_eq {m n : \u2115} : (\u2200 a : \u2115, a \u2223 m \u2194 a \u2223 n) \u2194 m = n :=\n  \u27e8fun h => dvd_antisymm ((h _).mp dvd_rfl) ((h _).mpr dvd_rfl), fun h n => by rw [h]\u27e9\n#align nat.dvd_left_iff_eq Nat.dvd_left_iff_eq\n-/\n\n#print Nat.dvd_left_injective /-\n/-- `dvd` is injective in the left argument -/\ntheorem dvd_left_injective : Function.Injective ((\u00b7 \u2223 \u00b7) : \u2115 \u2192 \u2115 \u2192 Prop) := fun m n h =>\n  dvd_right_iff_eq.mp fun a => iff_of_eq (congr_fun h a)\n#align nat.dvd_left_injective Nat.dvd_left_injective\n-/\n\n#print Nat.div_lt_div_of_lt_of_dvd /-\ntheorem div_lt_div_of_lt_of_dvd {a b d : \u2115} (hdb : d \u2223 b) (h : a < b) : a / d < b / d :=\n  by\n  rw [Nat.lt_div_iff_mul_lt hdb]\n  exact lt_of_le_of_lt (mul_div_le a d) h\n#align nat.div_lt_div_of_lt_of_dvd Nat.div_lt_div_of_lt_of_dvd\n-/\n\nend Nat\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Nat/Order/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3825366130130665}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.products.basic\nimport Mathlib.category_theory.currying\nimport Mathlib.PostPort\n\nuniverses v u l \n\nnamespace Mathlib\n\n/-!\n# A Fubini theorem for categorical limits\n\nWe prove that $lim_{J \u00d7 K} G = lim_J (lim_K G(j, -))$ for a functor `G : J \u00d7 K \u2964 C`,\nwhen all the appropriate limits exist.\n\nWe begin working with a functor `F : J \u2964 K \u2964 C`. We'll write `G : J \u00d7 K \u2964 C` for the associated\n\"uncurried\" functor.\n\nIn the first part, given a coherent family `D` of limit cones over the functors `F.obj j`,\nand a cone `c` over `G`, we construct a cone over the cone points of `D`.\nWe then show that if `c` is a limit cone, the constructed cone is also a limit cone.\n\nIn the second part, we state the Fubini theorem in the setting where limits are\nprovided by suitable `has_limit` classes.\n\nWe construct\n`limit_uncurry_iso_limit_comp_lim F : limit (uncurry.obj F) \u2245 limit (F \u22d9 lim)`\nand give simp lemmas characterising it.\nFor convenience, we also provide\n`limit_iso_limit_curry_comp_lim G : limit G \u2245 limit ((curry.obj G) \u22d9 lim)`\nin terms of the uncurried functor.\n\n## Future work\n\nThe dual statement.\n-/\n\nnamespace category_theory.limits\n\n\n/--\nA structure carrying a diagram of cones over the the functors `F.obj j`.\n-/\n-- We could try introducing a \"dependent functor type\" to handle this?\n\nstructure diagram_of_cones {J : Type v} {K : Type v} [small_category J] [small_category K]\n    {C : Type u} [category C] (F : J \u2964 K \u2964 C)\n    where\n  obj : (j : J) \u2192 cone (functor.obj F j)\n  map :\n    {j j' : J} \u2192 (f : j \u27f6 j') \u2192 functor.obj (cones.postcompose (functor.map F f)) (obj j) \u27f6 obj j'\n  id :\n    autoParam (J \u2192 cone_morphism.hom (map \ud835\udfd9) = \ud835\udfd9)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  comp :\n    autoParam\n      (\u2200 {j\u2081 j\u2082 j\u2083 : J} (f : j\u2081 \u27f6 j\u2082) (g : j\u2082 \u27f6 j\u2083),\n        cone_morphism.hom (map (f \u226b g)) = cone_morphism.hom (map f) \u226b cone_morphism.hom (map g))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n/--\nExtract the functor `J \u2964 C` consisting of the cone points and the maps between them,\nfrom a `diagram_of_cones`.\n-/\n@[simp] theorem diagram_of_cones.cone_points_obj {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] {F : J \u2964 K \u2964 C} (D : diagram_of_cones F) (j : J) :\n    functor.obj (diagram_of_cones.cone_points D) j = cone.X (diagram_of_cones.obj D j) :=\n  Eq.refl (functor.obj (diagram_of_cones.cone_points D) j)\n\n/--\nGiven a diagram `D` of limit cones over the `F.obj j`, and a cone over `uncurry.obj F`,\nwe can construct a cone over the diagram consisting of the cone points from `D`.\n-/\ndef cone_of_cone_uncurry {J : Type v} {K : Type v} [small_category J] [small_category K]\n    {C : Type u} [category C] {F : J \u2964 K \u2964 C} {D : diagram_of_cones F}\n    (Q : (j : J) \u2192 is_limit (diagram_of_cones.obj D j)) (c : cone (functor.obj uncurry F)) :\n    cone (diagram_of_cones.cone_points D) :=\n  cone.mk (cone.X c)\n    (nat_trans.mk\n      fun (j : J) =>\n        is_limit.lift (Q j)\n          (cone.mk (cone.X c) (nat_trans.mk fun (k : K) => nat_trans.app (cone.\u03c0 c) (j, k))))\n\n/--\n`cone_of_cone_uncurry Q c` is a limit cone when `c` is a limit cone.`\n-/\ndef cone_of_cone_uncurry_is_limit {J : Type v} {K : Type v} [small_category J] [small_category K]\n    {C : Type u} [category C] {F : J \u2964 K \u2964 C} {D : diagram_of_cones F}\n    (Q : (j : J) \u2192 is_limit (diagram_of_cones.obj D j)) {c : cone (functor.obj uncurry F)}\n    (P : is_limit c) : is_limit (cone_of_cone_uncurry Q c) :=\n  is_limit.mk\n    fun (s : cone (diagram_of_cones.cone_points D)) =>\n      is_limit.lift P\n        (cone.mk (cone.X s)\n          (nat_trans.mk\n            fun (p : J \u00d7 K) =>\n              nat_trans.app (cone.\u03c0 s) (prod.fst p) \u226b\n                nat_trans.app (cone.\u03c0 (diagram_of_cones.obj D (prod.fst p))) (prod.snd p)))\n\n/--\nGiven a functor `F : J \u2964 K \u2964 C`, with all needed limits,\nwe can construct a diagram consisting of the limit cone over each functor `F.obj j`,\nand the universal cone morphisms between these.\n-/\ndef diagram_of_cones.mk_of_has_limits {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] :\n    diagram_of_cones F :=\n  diagram_of_cones.mk (fun (j : J) => limit.cone (functor.obj F j))\n    fun (j j' : J) (f : j \u27f6 j') => cone_morphism.mk (functor.map lim (functor.map F f))\n\n-- Satisfying the inhabited linter.\n\nprotected instance diagram_of_cones_inhabited {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] :\n    Inhabited (diagram_of_cones F) :=\n  { default := diagram_of_cones.mk_of_has_limits F }\n\n@[simp] theorem diagram_of_cones.mk_of_has_limits_cone_points {J : Type v} {K : Type v}\n    [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C)\n    [has_limits_of_shape K C] :\n    diagram_of_cones.cone_points (diagram_of_cones.mk_of_has_limits F) = F \u22d9 lim :=\n  rfl\n\n/--\nThe Fubini theorem for a functor `F : J \u2964 K \u2964 C`,\nshowing that the limit of `uncurry.obj F` can be computed as\nthe limit of the limits of the functors `F.obj j`.\n-/\ndef limit_uncurry_iso_limit_comp_lim {J : Type v} {K : Type v} [small_category J] [small_category K]\n    {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C]\n    [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] :\n    limit (functor.obj uncurry F) \u2245 limit (F \u22d9 lim) :=\n  let c : cone (functor.obj uncurry F) := limit.cone (functor.obj uncurry F);\n  let P : is_limit c := limit.is_limit (functor.obj uncurry F);\n  let G : diagram_of_cones F := diagram_of_cones.mk_of_has_limits F;\n  let Q : (j : J) \u2192 is_limit (diagram_of_cones.obj G j) :=\n    fun (j : J) => limit.is_limit (functor.obj F j);\n  is_limit.cone_point_unique_up_to_iso (cone_of_cone_uncurry_is_limit Q P)\n    (limit.is_limit (F \u22d9 lim))\n\n@[simp] theorem limit_uncurry_iso_limit_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v}\n    [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C)\n    [has_limits_of_shape K C] [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] {j : J}\n    {k : K} :\n    iso.hom (limit_uncurry_iso_limit_comp_lim F) \u226b\n          limit.\u03c0 (F \u22d9 lim) j \u226b limit.\u03c0 (functor.obj F j) k =\n        limit.\u03c0 (functor.obj uncurry F) (j, k) :=\n  sorry\n\n@[simp] theorem limit_uncurry_iso_limit_comp_lim_inv_\u03c0 {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C]\n    [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] {j : J} {k : K} :\n    iso.inv (limit_uncurry_iso_limit_comp_lim F) \u226b limit.\u03c0 (functor.obj uncurry F) (j, k) =\n        limit.\u03c0 (F \u22d9 lim) j \u226b limit.\u03c0 (functor.obj F j) k :=\n  sorry\n\n/--\nThe Fubini theorem for a functor `G : J \u00d7 K \u2964 C`,\nshowing that the limit of `G` can be computed as\nthe limit of the limits of the functors `G.obj (j, _)`.\n-/\ndef limit_iso_limit_curry_comp_lim {J : Type v} {K : Type v} [small_category J] [small_category K]\n    {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C] [has_limit G]\n    [has_limit (functor.obj curry G \u22d9 lim)] : limit G \u2245 limit (functor.obj curry G \u22d9 lim) :=\n  has_limit.iso_of_nat_iso (iso.app (equivalence.unit_iso (equivalence.symm currying)) G) \u226a\u226b\n    limit_uncurry_iso_limit_comp_lim (functor.obj curry G)\n\n@[simp] theorem limit_iso_limit_curry_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C]\n    [has_limit G] [has_limit (functor.obj curry G \u22d9 lim)] {j : J} {k : K} :\n    iso.hom (limit_iso_limit_curry_comp_lim G) \u226b\n          limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k =\n        limit.\u03c0 G (j, k) :=\n  sorry\n\n@[simp] theorem limit_iso_limit_curry_comp_lim_inv_\u03c0 {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C]\n    [has_limit G] [has_limit (functor.obj curry G \u22d9 lim)] {j : J} {k : K} :\n    iso.inv (limit_iso_limit_curry_comp_lim G) \u226b limit.\u03c0 G (j, k) =\n        limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k :=\n  sorry\n\n/--\nA variant of the Fubini theorem for a functor `G : J \u00d7 K \u2964 C`,\nshowing that $\\lim_k \\lim_j G(j,k) \u2245 \\lim_j \\lim_k G(j,k)$.\n-/\ndef limit_curry_swap_comp_lim_iso_limit_curry_comp_lim {J : Type v} {K : Type v} [small_category J]\n    [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C] :\n    limit (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) \u2245 limit (functor.obj curry G \u22d9 lim) :=\n  (iso.symm (limit_iso_limit_curry_comp_lim (prod.swap K J \u22d9 G)) \u226a\u226b\n      has_limit.iso_of_equivalence (prod.braiding K J)\n        (iso.refl (equivalence.functor (prod.braiding K J) \u22d9 G))) \u226a\u226b\n    limit_iso_limit_curry_comp_lim G\n\n@[simp] theorem limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v}\n    [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C]\n    {j : J} {k : K} :\n    iso.hom (limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G) \u226b\n          limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k =\n        limit.\u03c0 (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) k \u226b\n          limit.\u03c0 (functor.obj (functor.obj curry (prod.swap K J \u22d9 G)) k) j :=\n  sorry\n\n@[simp] theorem limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_inv_\u03c0_\u03c0 {J : Type v} {K : Type v}\n    [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C]\n    {j : J} {k : K} :\n    iso.inv (limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G) \u226b\n          limit.\u03c0 (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) k \u226b\n            limit.\u03c0 (functor.obj (functor.obj curry (prod.swap K J \u22d9 G)) k) j =\n        limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/fubini_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3823384756091651}}
{"text": "/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport measure_theory.measure_space\n\n/-!\n# Typeclasses for measurability of operations\n\nIn this file we define classes `has_measurable_mul` etc and prove dot-style lemmas\n(`measurable.mul`, `ae_measurable.mul` etc). For binary operations we define two typeclasses:\n\n- `has_measurable_mul` says that both left and right multiplication are measurable;\n- `has_measurable_mul\u2082` says that `\u03bb p : \u03b1 \u00d7 \u03b1, p.1 * p.2` is measurable,\n\nand similarly for other binary operations. The reason for introducing these classes is that in case\nof topological space `\u03b1` equipped with the Borel `\u03c3`-algebra, instances for `has_measurable_mul\u2082`\netc require `\u03b1` to have a second countable topology.\n\nWe define separate classes for `has_measurable_div`/`has_measurable_sub`\nbecause on some types (e.g., `\u2115`, `\u211d\u22650\u221e`) division and/or subtraction are not defined as `a * b\u207b\u00b9` /\n`a + (-b)`.\n\nFor instances relating, e.g., `has_continuous_mul` to `has_measurable_mul` see file\n`measure_theory.borel_space`.\n\n## Tags\n\nmeasurable function, arithmetic operator\n-/\n\nuniverses u v\n\nopen_locale big_operators\nopen measure_theory\n\nvariables {\u03b1 : Type*} [measurable_space \u03b1]\n\n/-!\n### Binary operations: `(+)`, `(*)`, `(-)`, `(/)`\n-/\n\n/-- We say that a type `has_measurable_add` if `((+) c)` and `(+ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (+)` see `has_measurable_add\u2082`. -/\nclass has_measurable_add (M : Type*) [measurable_space M] [has_add M] : Prop :=\n(measurable_const_add : \u2200 c : M, measurable ((+) c))\n(measurable_add_const : \u2200 c : M, measurable (+ c))\n\n/-- We say that a type `has_measurable_add` if `uncurry (+)` is a measurable functions.\nFor a typeclass assuming measurability of `((+) c)` and `(+ c)` see `has_measurable_add`. -/\nclass has_measurable_add\u2082 (M : Type*) [measurable_space M] [has_add M] : Prop :=\n(measurable_add : measurable (\u03bb p : M \u00d7 M, p.1 + p.2))\n\nexport has_measurable_add\u2082 (measurable_add)\n  has_measurable_add (measurable_const_add measurable_add_const)\n\n/-- We say that a type `has_measurable_mul` if `((*) c)` and `(* c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (*)` see `has_measurable_mul\u2082`. -/\n@[to_additive]\nclass has_measurable_mul (M : Type*) [measurable_space M] [has_mul M] : Prop :=\n(measurable_const_mul : \u2200 c : M, measurable ((*) c))\n(measurable_mul_const : \u2200 c : M, measurable (* c))\n\n/-- We say that a type `has_measurable_mul` if `uncurry (*)` is a measurable functions.\nFor a typeclass assuming measurability of `((*) c)` and `(* c)` see `has_measurable_mul`. -/\n@[to_additive has_measurable_add\u2082]\nclass has_measurable_mul\u2082 (M : Type*) [measurable_space M] [has_mul M] : Prop :=\n(measurable_mul : measurable (\u03bb p : M \u00d7 M, p.1 * p.2))\n\nexport has_measurable_mul\u2082 (measurable_mul)\n  has_measurable_mul (measurable_const_mul measurable_mul_const)\n\nsection mul\n\nvariables {M : Type*} [measurable_space M] [has_mul M]\n\n@[to_additive]\nlemma measurable.mul [has_measurable_mul\u2082 M] {f g : \u03b1 \u2192 M} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb a, f a * g a) :=\nmeasurable_mul.comp (hf.prod_mk hg)\n\n@[to_additive]\nlemma ae_measurable.mul [has_measurable_mul\u2082 M] {\u03bc : measure \u03b1} {f g : \u03b1 \u2192 M}\n  (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) :\n  ae_measurable (\u03bb a, f a * g a) \u03bc :=\nmeasurable_mul.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100, to_additive]\ninstance has_measurable_mul\u2082.to_has_measurable_mul [has_measurable_mul\u2082 M] :\n  has_measurable_mul M :=\n\u27e8\u03bb c, measurable_const.mul measurable_id, \u03bb c, measurable_id.mul measurable_const\u27e9\n\n@[to_additive]\nlemma measurable.const_mul [has_measurable_mul M] {f : \u03b1 \u2192 M} (hf : measurable f) (c : M) :\n  measurable (\u03bb x, c * f x) :=\n(measurable_const_mul c).comp hf\n\n@[to_additive]\nlemma ae_measurable.const_mul [has_measurable_mul M] {f : \u03b1 \u2192 M} {\u03bc : measure \u03b1}\n  (hf : ae_measurable f \u03bc) (c : M) :\n  ae_measurable (\u03bb x, c * f x) \u03bc :=\n(has_measurable_mul.measurable_const_mul c).comp_ae_measurable hf\n\n@[to_additive]\nlemma measurable.mul_const [has_measurable_mul M] {f : \u03b1 \u2192 M} (hf : measurable f) (c : M) :\n  measurable (\u03bb x, f x * c) :=\n(measurable_mul_const c).comp hf\n\n@[to_additive]\nlemma ae_measurable.mul_const [has_measurable_mul M] {f : \u03b1 \u2192 M} {\u03bc : measure \u03b1}\n  (hf : ae_measurable f \u03bc) (c : M) :\n  ae_measurable (\u03bb x, f x * c) \u03bc :=\n(measurable_mul_const c).comp_ae_measurable hf\n\nend mul\n\n/-- This class assumes that the map `\u03b2 \u00d7 \u03b3 \u2192 \u03b2` given by `(x, y) \u21a6 x ^ y` is measurable. -/\nclass has_measurable_pow (\u03b2 \u03b3 : Type*) [measurable_space \u03b2] [measurable_space \u03b3] [has_pow \u03b2 \u03b3] :=\n(measurable_pow : measurable (\u03bb p : \u03b2 \u00d7 \u03b3, p.1 ^ p.2))\n\nexport has_measurable_pow (measurable_pow)\n\ninstance has_measurable_mul.has_measurable_pow (M : Type*) [monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] : has_measurable_pow M \u2115 :=\n\u27e8begin\n  haveI : measurable_singleton_class \u2115 := \u27e8\u03bb _, trivial\u27e9,\n  refine measurable_from_prod_encodable (\u03bb n, _),\n  induction n with n ih,\n  { simp [pow_zero, measurable_one] },\n  { simp only [pow_succ], exact measurable_id.mul ih }\nend\u27e9\n\nsection pow\n\nvariables {\u03b2 \u03b3 : Type*} [measurable_space \u03b2] [measurable_space \u03b3] [has_pow \u03b2 \u03b3]\n  [has_measurable_pow \u03b2 \u03b3]\n\nlemma measurable.pow {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb x, f x ^ g x) :=\nmeasurable_pow.comp (hf.prod_mk hg)\n\nlemma ae_measurable.pow {\u03bc : measure \u03b1} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (hf : ae_measurable f \u03bc)\n  (hg : ae_measurable g \u03bc) :\n  ae_measurable (\u03bb x, f x ^ g x) \u03bc :=\nmeasurable_pow.comp_ae_measurable (hf.prod_mk hg)\n\nlemma measurable.pow_const {f : \u03b1 \u2192 \u03b2} (hf : measurable f) (c : \u03b3) :\n  measurable (\u03bb x, f x ^ c) :=\nhf.pow measurable_const\n\nlemma ae_measurable.pow_const {\u03bc : measure \u03b1} {f : \u03b1 \u2192 \u03b2} (hf : ae_measurable f \u03bc) (c : \u03b3) :\n  ae_measurable (\u03bb x, f x ^ c) \u03bc :=\nhf.pow ae_measurable_const\n\nlemma measurable.const_pow {f : \u03b1 \u2192 \u03b3} (hf : measurable f) (c : \u03b2) :\n  measurable (\u03bb x, c ^ f x) :=\nmeasurable_const.pow hf\n\nlemma ae_measurable.const_pow {\u03bc : measure \u03b1} {f : \u03b1 \u2192 \u03b3} (hf : ae_measurable f \u03bc) (c : \u03b2) :\n  ae_measurable (\u03bb x, c ^ f x) \u03bc :=\nae_measurable_const.pow hf\n\nend pow\n\n/-- We say that a type `has_measurable_sub` if `(\u03bb x, c - x)` and `(\u03bb x, x - c)` are measurable\nfunctions. For a typeclass assuming measurability of `uncurry (-)` see `has_measurable_sub\u2082`. -/\nclass has_measurable_sub (G : Type*) [measurable_space G] [has_sub G] : Prop :=\n(measurable_const_sub : \u2200 c : G, measurable (\u03bb x, c - x))\n(measurable_sub_const : \u2200 c : G, measurable (\u03bb x, x - c))\n\n/-- We say that a type `has_measurable_sub` if `uncurry (-)` is a measurable functions.\nFor a typeclass assuming measurability of `((-) c)` and `(- c)` see `has_measurable_sub`. -/\nclass has_measurable_sub\u2082 (G : Type*) [measurable_space G] [has_sub G] : Prop :=\n(measurable_sub : measurable (\u03bb p : G \u00d7 G, p.1 - p.2))\n\nexport has_measurable_sub\u2082 (measurable_sub)\n\n/-- We say that a type `has_measurable_div` if `((/) c)` and `(/ c)` are measurable functions.\nFor a typeclass assuming measurability of `uncurry (/)` see `has_measurable_div\u2082`. -/\n@[to_additive] class has_measurable_div (G\u2080: Type*) [measurable_space G\u2080] [has_div G\u2080] : Prop :=\n(measurable_const_div : \u2200 c : G\u2080, measurable ((/) c))\n(measurable_div_const : \u2200 c : G\u2080, measurable (/ c))\n\n/-- We say that a type `has_measurable_div` if `uncurry (/)` is a measurable functions.\nFor a typeclass assuming measurability of `((/) c)` and `(/ c)` see `has_measurable_div`. -/\n@[to_additive has_measurable_sub\u2082]\nclass has_measurable_div\u2082 (G\u2080: Type*) [measurable_space G\u2080] [has_div G\u2080] : Prop :=\n(measurable_div : measurable (\u03bb p : G\u2080\u00d7 G\u2080, p.1 / p.2))\n\nexport has_measurable_div\u2082 (measurable_div)\n\nsection div\n\nvariables {G : Type*} [measurable_space G] [has_div G]\n\n@[to_additive]\nlemma measurable.div [has_measurable_div\u2082 G] {f g : \u03b1 \u2192 G} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb a, f a / g a) :=\nmeasurable_div.comp (hf.prod_mk hg)\n\n@[to_additive]\nlemma ae_measurable.div [has_measurable_div\u2082 G] {f g : \u03b1 \u2192 G} {\u03bc : measure \u03b1}\n  (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) :\n  ae_measurable (\u03bb a, f a / g a) \u03bc :=\nmeasurable_div.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100, to_additive]\ninstance has_measurable_div\u2082.to_has_measurable_div [has_measurable_div\u2082 G] :\n  has_measurable_div G :=\n\u27e8\u03bb c, measurable_const.div measurable_id, \u03bb c, measurable_id.div measurable_const\u27e9\n\n@[to_additive]\nlemma measurable.const_div [has_measurable_div G] {f : \u03b1 \u2192 G} (hf : measurable f) (c : G) :\n  measurable (\u03bb x, c / f x) :=\n(has_measurable_div.measurable_const_div c).comp hf\n\n@[to_additive]\nlemma ae_measurable.const_div [has_measurable_div G] {f : \u03b1 \u2192 G} {\u03bc : measure \u03b1}\n  (hf : ae_measurable f \u03bc) (c : G) :\n  ae_measurable (\u03bb x, c / f x) \u03bc :=\n(has_measurable_div.measurable_const_div c).comp_ae_measurable hf\n\n@[to_additive]\nlemma measurable.div_const [has_measurable_div G] {f : \u03b1 \u2192 G} (hf : measurable f) (c : G) :\n  measurable (\u03bb x, f x / c) :=\n(has_measurable_div.measurable_div_const c).comp hf\n\n@[to_additive]\nlemma ae_measurable.div_const [has_measurable_div G] {f : \u03b1 \u2192 G} {\u03bc : measure \u03b1}\n  (hf : ae_measurable f \u03bc) (c : G) :\n  ae_measurable (\u03bb x, f x / c) \u03bc :=\n(has_measurable_div.measurable_div_const c).comp_ae_measurable hf\n\nend div\n\n/-- We say that a type `has_measurable_neg` if `x \u21a6 -x` is a measurable function. -/\nclass has_measurable_neg (G : Type*) [has_neg G] [measurable_space G] : Prop :=\n(measurable_neg : measurable (has_neg.neg : G \u2192 G))\n\n/-- We say that a type `has_measurable_inv` if `x \u21a6 x\u207b\u00b9` is a measurable function. -/\n@[to_additive]\nclass has_measurable_inv (G : Type*) [has_inv G] [measurable_space G] : Prop :=\n(measurable_inv : measurable (has_inv.inv : G \u2192 G))\n\nexport has_measurable_inv (measurable_inv) has_measurable_neg (measurable_neg)\n\n@[priority 100, to_additive]\ninstance has_measurable_div_of_mul_inv (G : Type*) [measurable_space G]\n  [div_inv_monoid G] [has_measurable_mul G] [has_measurable_inv G] :\n  has_measurable_div G :=\n{ measurable_const_div := \u03bb c,\n    by { convert (measurable_inv.const_mul c), ext1, apply div_eq_mul_inv },\n  measurable_div_const := \u03bb c,\n    by { convert (measurable_id.mul_const c\u207b\u00b9), ext1, apply div_eq_mul_inv } }\n\nsection inv\n\nvariables {G : Type*} [has_inv G] [measurable_space G] [has_measurable_inv G]\n\n@[to_additive] lemma measurable.inv {f : \u03b1 \u2192 G} (hf : measurable f) :\n  measurable (\u03bb x, (f x)\u207b\u00b9) :=\nmeasurable_inv.comp hf\n\n@[to_additive] lemma ae_measurable.inv {f : \u03b1 \u2192 G} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x)\u207b\u00b9) \u03bc :=\nmeasurable_inv.comp_ae_measurable hf\n\n@[simp, to_additive] lemma measurable_inv_iff {G : Type*} [group G] [measurable_space G]\n  [has_measurable_inv G] {f : \u03b1 \u2192 G} : measurable (\u03bb x, (f x)\u207b\u00b9) \u2194 measurable f :=\n\u27e8\u03bb h, by simpa only [inv_inv] using h.inv, \u03bb h, h.inv\u27e9\n\n@[simp, to_additive] lemma ae_measurable_inv_iff {G : Type*} [group G] [measurable_space G]\n  [has_measurable_inv G] {f : \u03b1 \u2192 G} {\u03bc : measure \u03b1} :\n  ae_measurable (\u03bb x, (f x)\u207b\u00b9) \u03bc \u2194 ae_measurable f \u03bc :=\n\u27e8\u03bb h, by simpa only [inv_inv] using h.inv, \u03bb h, h.inv\u27e9\n\n@[simp] lemma measurable_inv_iff' {G\u2080 : Type*} [group_with_zero G\u2080]\n  [measurable_space G\u2080] [has_measurable_inv G\u2080] {f : \u03b1 \u2192 G\u2080} :\n  measurable (\u03bb x, (f x)\u207b\u00b9) \u2194 measurable f :=\n\u27e8\u03bb h, by simpa only [inv_inv'] using h.inv, \u03bb h, h.inv\u27e9\n\n@[simp] lemma ae_measurable_inv_iff' {G\u2080 : Type*} [group_with_zero G\u2080]\n  [measurable_space G\u2080] [has_measurable_inv G\u2080] {f : \u03b1 \u2192 G\u2080} {\u03bc : measure \u03b1} :\n  ae_measurable (\u03bb x, (f x)\u207b\u00b9) \u03bc \u2194 ae_measurable f \u03bc :=\n\u27e8\u03bb h, by simpa only [inv_inv'] using h.inv, \u03bb h, h.inv\u27e9\n\nend inv\n\n/- There is something extremely strange here: copy-pasting the proof of this lemma in the proof\nof `has_measurable_gpow` fails, while `pp.all` does not show any difference in the goal.\nKeep it as a separate lemmas as a workaround. -/\nprivate lemma has_measurable_gpow_aux (G : Type u) [div_inv_monoid G] [measurable_space G]\n  [has_measurable_mul\u2082 G] [has_measurable_inv G] (k : \u2115) :\n  measurable (\u03bb (x : G), x ^(-[1+ k])) :=\nbegin\n  simp_rw [gpow_neg_succ_of_nat],\n  exact (measurable_id.pow_const (k + 1)).inv\nend\n\ninstance has_measurable_gpow (G : Type u) [div_inv_monoid G] [measurable_space G]\n  [has_measurable_mul\u2082 G] [has_measurable_inv G] :\n  has_measurable_pow G \u2124 :=\nbegin\n  letI : measurable_singleton_class \u2124 := \u27e8\u03bb _, trivial\u27e9,\n  constructor,\n  refine measurable_from_prod_encodable (\u03bb n, _),\n  dsimp,\n  apply int.cases_on n,\n  { simpa using measurable_id.pow_const },\n  { exact has_measurable_gpow_aux G }\nend\n\n@[priority 100, to_additive]\ninstance has_measurable_div\u2082_of_mul_inv (G : Type*) [measurable_space G]\n  [div_inv_monoid G] [has_measurable_mul\u2082 G] [has_measurable_inv G] :\n  has_measurable_div\u2082 G :=\n\u27e8by { simp only [div_eq_mul_inv], exact measurable_fst.mul measurable_snd.inv }\u27e9\n\n/-- We say that the action of `M` on `\u03b1` `has_measurable_smul` if for each `c` the map `x \u21a6 c \u2022 x`\nis a measurable function and for each `x` the map `c \u21a6 c \u2022 x` is a measurable function. -/\nclass has_measurable_smul (M \u03b1 : Type*) [has_scalar M \u03b1] [measurable_space M] [measurable_space \u03b1] :\n  Prop :=\n(measurable_const_smul : \u2200 c : M, measurable ((\u2022) c : \u03b1 \u2192 \u03b1))\n(measurable_smul_const : \u2200 x : \u03b1, measurable (\u03bb c : M, c \u2022 x))\n\n/-- We say that the action of `M` on `\u03b1` `has_measurable_smul` if the map\n`(c, x) \u21a6 c \u2022 x` is a measurable function. -/\nclass has_measurable_smul\u2082 (M \u03b1 : Type*) [has_scalar M \u03b1] [measurable_space M]\n  [measurable_space \u03b1] : Prop :=\n(measurable_smul : measurable (function.uncurry (\u2022) : M \u00d7 \u03b1 \u2192 \u03b1))\n\nexport has_measurable_smul (measurable_const_smul measurable_smul_const)\n  has_measurable_smul\u2082 (measurable_smul)\n\ninstance has_measurable_smul_of_mul (M : Type*) [monoid M] [measurable_space M]\n  [has_measurable_mul M] :\n  has_measurable_smul M M :=\n\u27e8measurable_id.const_mul, measurable_id.mul_const\u27e9\n\ninstance has_measurable_smul\u2082_of_mul (M : Type*) [monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] :\n  has_measurable_smul\u2082 M M :=\n\u27e8measurable_mul\u27e9\n\nsection smul\n\nvariables {M \u03b2 : Type*} [measurable_space M] [measurable_space \u03b2] [has_scalar M \u03b2]\n\nlemma measurable.smul [has_measurable_smul\u2082 M \u03b2]\n  {f : \u03b1 \u2192 M} {g : \u03b1 \u2192 \u03b2} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb x, f x \u2022 g x) :=\nmeasurable_smul.comp (hf.prod_mk hg)\n\nlemma ae_measurable.smul [has_measurable_smul\u2082 M \u03b2]\n  {f : \u03b1 \u2192 M} {g : \u03b1 \u2192 \u03b2} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) :\n  ae_measurable (\u03bb x, f x \u2022 g x) \u03bc :=\nhas_measurable_smul\u2082.measurable_smul.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100]\ninstance has_measurable_smul\u2082.to_has_measurable_smul [has_measurable_smul\u2082 M \u03b2] :\n  has_measurable_smul M \u03b2 :=\n\u27e8\u03bb c, measurable_const.smul measurable_id, \u03bb y, measurable_id.smul measurable_const\u27e9\n\nvariables [has_measurable_smul M \u03b2] {\u03bc : measure \u03b1}\n\nlemma measurable.smul_const {f : \u03b1 \u2192 M} (hf : measurable f) (y : \u03b2) : measurable (\u03bb x, f x \u2022 y) :=\n(has_measurable_smul.measurable_smul_const y).comp hf\n\nlemma ae_measurable.smul_const {f : \u03b1 \u2192 M} (hf : ae_measurable f \u03bc) (y : \u03b2) :\n  ae_measurable (\u03bb x, f x \u2022 y) \u03bc :=\n(has_measurable_smul.measurable_smul_const y).comp_ae_measurable hf\n\nlemma measurable.const_smul' {f : \u03b1 \u2192 \u03b2} (hf : measurable f) (c : M) :\n  measurable (\u03bb x, c \u2022 f x) :=\n(has_measurable_smul.measurable_const_smul c).comp hf\n\nlemma measurable.const_smul {f : \u03b1 \u2192 \u03b2} (hf : measurable f) (c : M) :\n  measurable (c \u2022 f) :=\nhf.const_smul' c\n\nlemma ae_measurable.const_smul' {f : \u03b1 \u2192 \u03b2} (hf : ae_measurable f \u03bc) (c : M) :\n  ae_measurable (\u03bb x, c \u2022 f x) \u03bc :=\n(has_measurable_smul.measurable_const_smul c).comp_ae_measurable hf\n\nlemma ae_measurable.const_smul {f : \u03b1 \u2192 \u03b2} (hf : ae_measurable f \u03bc) (c : M) :\n  ae_measurable (c \u2022 f) \u03bc :=\nhf.const_smul' c\n\nend smul\n\nsection mul_action\n\nvariables {M \u03b2 : Type*} [measurable_space M] [measurable_space \u03b2] [monoid M] [mul_action M \u03b2]\n  [has_measurable_smul M \u03b2] {f : \u03b1 \u2192 \u03b2} {\u03bc : measure \u03b1}\n\n@[simp] lemma units.measurable_const_smul_iff (u : units M) :\n  measurable (\u03bb x, (u : M) \u2022 f x) \u2194 measurable f :=\n\u27e8\u03bb h, by simpa only [u.inv_smul_smul] using h.const_smul' ((u\u207b\u00b9 : units M) : M),\n  \u03bb h, h.const_smul \u2191u\u27e9\n\n@[simp] lemma units.ae_measurable_const_smul_iff (u : units M) :\n  ae_measurable (\u03bb x, (u : M) \u2022 f x) \u03bc \u2194 ae_measurable f \u03bc :=\n\u27e8\u03bb h, by simpa only [u.inv_smul_smul] using h.const_smul' ((u\u207b\u00b9 : units M) : M),\n  \u03bb h, h.const_smul \u2191u\u27e9\n\nlemma is_unit.measurable_const_smul_iff {c : M} (hc : is_unit c) :\n  measurable (\u03bb x, c \u2022 f x) \u2194 measurable f :=\nlet \u27e8u, hu\u27e9 := hc in hu \u25b8 u.measurable_const_smul_iff\n\nlemma is_unit.ae_measurable_const_smul_iff {c : M} (hc : is_unit c) :\n  ae_measurable (\u03bb x, c \u2022 f x) \u03bc \u2194 ae_measurable f \u03bc :=\nlet \u27e8u, hu\u27e9 := hc in hu \u25b8 u.ae_measurable_const_smul_iff\n\nvariables {G\u2080 : Type*} [group_with_zero G\u2080] [measurable_space G\u2080] [mul_action G\u2080 \u03b2]\n  [has_measurable_smul G\u2080 \u03b2]\n\nlemma measurable_const_smul_iff' {c : G\u2080} (hc : c \u2260 0) :\n  measurable (\u03bb x, c \u2022 f x) \u2194 measurable f :=\n(is_unit.mk0 c hc).measurable_const_smul_iff\n\nlemma ae_measurable_const_smul_iff' {c : G\u2080} (hc : c \u2260 0) :\n  ae_measurable (\u03bb x, c \u2022 f x) \u03bc \u2194 ae_measurable f \u03bc :=\n(is_unit.mk0 c hc).ae_measurable_const_smul_iff\n\nvariables {G : Type*} [group G] [measurable_space G] [mul_action G \u03b2]\n  [has_measurable_smul G \u03b2]\n\nlemma measurable_const_smul_iff (c : G) :\n  measurable (\u03bb x, c \u2022 f x) \u2194 measurable f :=\n(group.is_unit c).measurable_const_smul_iff\n\nlemma ae_measurable_const_smul_iff (c : G) :\n  ae_measurable (\u03bb x, c \u2022 f x) \u03bc \u2194 ae_measurable f \u03bc :=\n(group.is_unit c).ae_measurable_const_smul_iff\n\nend mul_action\n\n/-!\n### Big operators: `\u220f` and `\u2211`\n-/\n\n@[to_additive]\nlemma list.measurable_prod' {M : Type*} [monoid M] [measurable_space M] [has_measurable_mul\u2082 M]\n  (l : list (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, measurable f) :\n  measurable l.prod :=\nbegin\n  induction l with f l ihl, { exact measurable_one },\n  rw [list.forall_mem_cons] at hl,\n  rw [list.prod_cons],\n  exact hl.1.mul (ihl hl.2)\nend\n\n@[to_additive]\nlemma list.ae_measurable_prod' {M : Type*} [monoid M] [measurable_space M] [has_measurable_mul\u2082 M]\n  {\u03bc : measure \u03b1} (l : list (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, ae_measurable f \u03bc) :\n  ae_measurable l.prod \u03bc :=\nbegin\n  induction l with f l ihl, { exact ae_measurable_one },\n  rw [list.forall_mem_cons] at hl,\n  rw [list.prod_cons],\n  exact hl.1.mul (ihl hl.2)\nend\n\n@[to_additive]\nlemma list.measurable_prod {M : Type*} [monoid M] [measurable_space M] [has_measurable_mul\u2082 M]\n  (l : list (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, measurable f) :\n  measurable (\u03bb x, (l.map (\u03bb f : \u03b1 \u2192 M, f x)).prod) :=\nby simpa only [\u2190 pi.list_prod_apply] using l.measurable_prod' hl\n\n@[to_additive]\nlemma list.ae_measurable_prod {M : Type*} [monoid M] [measurable_space M] [has_measurable_mul\u2082 M]\n  {\u03bc : measure \u03b1} (l : list (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (l.map (\u03bb f : \u03b1 \u2192 M, f x)).prod) \u03bc :=\nby simpa only [\u2190 pi.list_prod_apply] using l.ae_measurable_prod' hl\n\n@[to_additive]\nlemma multiset.measurable_prod' {M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] (l : multiset (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, measurable f) :\n  measurable l.prod :=\nby { rcases l with \u27e8l\u27e9, simpa using l.measurable_prod' (by simpa using hl) }\n\n@[to_additive]\nlemma multiset.ae_measurable_prod' {M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {\u03bc : measure \u03b1} (l : multiset (\u03b1 \u2192 M)) (hl : \u2200 f \u2208 l, ae_measurable f \u03bc) :\n  ae_measurable l.prod \u03bc :=\nby { rcases l with \u27e8l\u27e9, simpa using l.ae_measurable_prod' (by simpa using hl) }\n\n@[to_additive]\nlemma multiset.measurable_prod {M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] (s : multiset (\u03b1 \u2192 M)) (hs : \u2200 f \u2208 s, measurable f) :\n  measurable (\u03bb x, (s.map (\u03bb f : \u03b1 \u2192 M, f x)).prod) :=\nby simpa only [\u2190 pi.multiset_prod_apply] using s.measurable_prod' hs\n\n@[to_additive]\nlemma multiset.ae_measurable_prod {M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {\u03bc : measure \u03b1} (s : multiset (\u03b1 \u2192 M)) (hs : \u2200 f \u2208 s, ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (s.map (\u03bb f : \u03b1 \u2192 M, f x)).prod) \u03bc :=\nby simpa only [\u2190 pi.multiset_prod_apply] using s.ae_measurable_prod' hs\n\n@[to_additive]\nlemma finset.measurable_prod' {\u03b9 M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : finset \u03b9) (hf : \u2200i \u2208 s, measurable (f i)) :\n  measurable (\u220f i in s, f i) :=\nfinset.prod_induction _ _ (\u03bb _ _, measurable.mul) (@measurable_one M _ _ _ _) hf\n\n@[to_additive]\nlemma finset.measurable_prod {\u03b9 M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : finset \u03b9) (hf : \u2200i \u2208 s, measurable (f i)) :\n  measurable (\u03bb a, \u220f i in s, f i a) :=\nby simpa only [\u2190 finset.prod_apply] using s.measurable_prod' hf\n\n@[to_additive]\nlemma finset.ae_measurable_prod' {\u03b9 M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {\u03bc : measure \u03b1} {f : \u03b9 \u2192 \u03b1 \u2192 M} (s : finset \u03b9)\n  (hf : \u2200i \u2208 s, ae_measurable (f i) \u03bc) :\n  ae_measurable (\u220f i in s, f i) \u03bc :=\nmultiset.ae_measurable_prod' _ $\n  \u03bb g hg, let \u27e8i, hi, hg\u27e9 := multiset.mem_map.1 hg in (hg \u25b8 hf _ hi)\n\n@[to_additive]\nlemma finset.ae_measurable_prod {\u03b9 M : Type*} [comm_monoid M] [measurable_space M]\n  [has_measurable_mul\u2082 M] {f : \u03b9 \u2192 \u03b1 \u2192 M} {\u03bc : measure \u03b1} (s : finset \u03b9)\n  (hf : \u2200i \u2208 s, ae_measurable (f i) \u03bc) :\n  ae_measurable (\u03bb a, \u220f i in s, f i a) \u03bc :=\nby simpa only [\u2190 finset.prod_apply] using s.ae_measurable_prod' hf\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/measure_theory/arithmetic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.38233846733740606}}
{"text": "/-\nCopyright (c) 2021 Alex J. Best. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Alex J. Best\n-/\n\nimport data.mv_polynomial.comm_ring\nimport data.set.finite\nimport ring_theory.mv_polynomial.homogeneous\nimport ring_theory.polynomial.basic\nimport order.symm_diff\n\n/-!\n# Homogenization\n\n## Main definitions\n\n* `mv_polynomial.homogenization`\n\n## Main statements\n\n* foo_bar_unique\n\n## Notation\n\n\n\n## Implementation details\n\n* We homogenize polynomials over a given ground set of variables, rather than adjoining an extra\n  variable to give the user more choice in the type of the polynomials involved.\n\n## References\n\n* [F. Bar, *Quuxes*][]\n\n## Tags\n\n\n-/\n\nvariables {R \u03b9 : Type*} [comm_semiring R]\n\nopen polynomial finset mv_polynomial\n\nopen_locale big_operators\nnoncomputable theory\nnamespace mv_polynomial\n\nsection finsupp\n\n-- TODO can any assumptions be weakened\n-- TODO version with monoid hom?\nlemma finsupp.sum_update_add {\u03b1 \u03b2 : Type*} [add_comm_monoid \u03b1] [add_comm_monoid \u03b2]\n  (f : \u03b9 \u2192\u2080 \u03b1) (i : \u03b9) (a : \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 \u03b2) (hg : \u2200 i, g i 0 = 0)\n  (hgg : \u2200 (a : \u03b9) (b\u2081 b\u2082 : \u03b1), g a (b\u2081 + b\u2082) = g a b\u2081 + g a b\u2082) :\n  (f.update i a).sum g + g i (f i) = f.sum g + g i a :=\nbegin\n  classical,\n  simp_rw finsupp.update_eq_erase_add_single,\n  rw finsupp.sum_add_index (\u03bb i _, hg i) (\u03bb i _, hgg i),\n  conv_rhs {rw \u2190 finsupp.update_self f i},\n  rw finsupp.update_eq_erase_add_single,\n  rw finsupp.sum_add_index (\u03bb i _, hg i) (\u03bb i _, hgg i),\n  rw add_assoc,\n  rw add_assoc,\n  congr' 1,\n  rw add_comm,\n  rw finsupp.sum_single_index (hg _),\n  rw finsupp.sum_single_index (hg _),\nend\n\nend finsupp\n\n/-- The homogenization of a multivariate polynomial at a single variable. -/\ndef homogenization (i : \u03b9) (p : mv_polynomial \u03b9 R) :\n  mv_polynomial \u03b9 R :=\n-- \u2211 j in p.support, monomial (j + finsupp.single i (p.total_degree - (j i))) (p.coeff j)\nfinsupp.map_domain (\u03bb j, j + finsupp.single i (p.total_degree - j.sum (\u03bb _ m, m))) p\n\n\nnamespace finsupp\nopen finsupp\n\n@[simp] lemma support_map_domain {\u03b1 \u03b2 M : Type*} [add_comm_monoid M]\n  (f : \u03b1 \u21aa \u03b2) (v : \u03b1 \u2192\u2080 M) : (finsupp.map_domain f v).support \u2286 v.support.map f :=\nbegin\n  classical,\n  rw finsupp.map_domain,\n  refine finset.subset.trans finsupp.support_sum _,\n  simp only [finsupp.mem_support_iff, finset.bUnion_subset_iff_forall_subset, ne.def],\n  intros x hx,\n  apply finset.subset.trans finsupp.support_single_subset,\n  simp [hx],\nend\n\nlemma map_domain_apply' {\u03b1 \u03b2 M : Type*} [add_comm_monoid M] (S : set \u03b1) {f : \u03b1 \u2192 \u03b2} (x : \u03b1 \u2192\u2080 M)\n  (hS : (x.support : set \u03b1) \u2286 S) (hf : set.inj_on f S) {a : \u03b1} (ha : a \u2208 S) :\n  finsupp.map_domain f x (f a) = x a :=\nbegin\n  classical,\n  rw finsupp.map_domain,\n  simp only [finsupp.sum_apply],\n  rw finsupp.sum,\n  simp_rw finsupp.single_apply,\n  have : \u2200 (a_1 : \u03b1) (ha1 : a_1 \u2208 x.support),\n    (if f a_1 = f a then x a_1 else 0) = (if f a_1 = f a then x a else 0),\n  { intros a_1 ha_1,\n    split_ifs with hh,\n    rw hf _ ha hh,\n    exact hS ha_1,\n    refl, },\n  conv in (ite _ _ _)\n  { rw [this _ H], },\n  by_cases ha : a \u2208 x.support,\n  rw \u2190 finset.add_sum_erase _ _ ha,\n  simp only [if_true, eq_self_iff_true],\n  convert add_zero _,\n  have : \u2200 i \u2208 x.support.erase a, f i \u2260 f a,\n  { intros i hi,\n    have hix : i \u2208 x.support,\n    exact finset.mem_of_mem_erase hi,\n    have hia : i \u2260 a,\n    exact finset.ne_of_mem_erase hi,\n    exact hia \u2218 (hf (hS hix) (hS ha)), },\n  conv in (ite _ _ _)\n  { rw if_neg (this x H), },\n  simp only [finset.sum_const_zero],\n  simp at ha,\n  simp [ha],\nend\n\nlemma map_domain_inj_on {\u03b1 \u03b2 M : Type*} [add_comm_monoid M] (S : set \u03b1) {f : \u03b1 \u2192 \u03b2}\n  (hf : set.inj_on f S) :\n  set.inj_on (finsupp.map_domain f : (\u03b1 \u2192\u2080 M) \u2192 (\u03b2 \u2192\u2080 M)) {w | (w.support : set \u03b1) \u2286 S} :=\nbegin\n  intros v\u2081 hv\u2081 v\u2082 hv\u2082 eq,\n  ext a,\n  have : finsupp.map_domain f v\u2081 (f a) = finsupp.map_domain f v\u2082 (f a), { rw eq },\n  rw [set.mem_set_of_eq] at hv\u2081 hv\u2082,\n  classical,\n  have hu : (v\u2081.support \u222a v\u2082.support : set \u03b1) \u2286 S := set.union_subset hv\u2081 hv\u2082,\n  by_cases h : a \u2208 v\u2081.support \u222a v\u2082.support,\n  { rwa [map_domain_apply' S _ hv\u2081 hf _,\n         map_domain_apply' S _ hv\u2082 hf _] at this,\n    { apply hu,\n      exact_mod_cast h, },\n    { apply hu,\n      exact_mod_cast h, }, },\n  { simp only [decidable.not_or_iff_and_not, mem_union, not_not, finsupp.mem_support_iff] at h,\n    simp [h], },\n  -- rw [finsupp.map_domain_apply hf, finsupp.map_domain_apply hf] at this,\nend\nend finsupp\n\n\n-- lemma support_homogenization [decidable_eq \u03b9] (i : \u03b9) (p : mv_polynomial \u03b9 R)\n--   (h : \u2200 j \u2208 p.support, (j : \u03b9 \u2192 \u2115) i = 0) : (p.homogenization i).support = p.support.image\n--     (\u03bb (j : \u03b9 \u2192\u2080 \u2115), j + finsupp.single i (p.total_degree - j.sum (\u03bb (_x : \u03b9) (m : \u2115), m))) :=\n-- begin\n--   rw homogenization,\n--   apply finsupp.support_map_domain _ _ _,\n-- end\n\n@[simp]\nlemma homogenization_zero (i : \u03b9) : (0 : mv_polynomial \u03b9 R).homogenization i = 0 :=\nby simp [homogenization]\n\n-- TODO this is probably useless\n-- lemma map_domain_one {\u03b1 \u03b2 M : Type*} [has_zero \u03b2] [has_zero \u03b1] [has_one M]\n--   [add_comm_monoid M] {f : \u03b1 \u2192 \u03b2} (hf : f 0 = 0) :\n--   finsupp.map_domain f (finsupp.single 0 1 : \u03b1 \u2192\u2080 M) = (finsupp.single 0 1 : \u03b2 \u2192\u2080 M) :=\n-- by simp [hf]\n\n-- TODO maybe instead prove this via is_homogeneous_one\n@[simp]\nlemma homogenization_one (i : \u03b9) : (1 : mv_polynomial \u03b9 R).homogenization i = 1 :=\nbegin\n  simp only [homogenization, total_degree_one, zero_tsub, add_zero, finsupp.single_zero],\n  erw finsupp.map_domain_single,\n  -- erw map_domain_one,\n  refl,\nend\n\n@[simp]\nlemma homogenization_C (i : \u03b9) (c : R) : (C c : mv_polynomial \u03b9 R).homogenization i = C c :=\nbegin\n  simp only [homogenization, total_degree_C, zero_tsub],\n  convert finsupp.map_domain_single,\n  rw single_eq_monomial,\n  have : (0 : \u03b9 \u2192\u2080 \u2115) i = 0,\n  { simp only [finsupp.coe_zero, pi.zero_apply], },\n  rw [\u2190 this],\n  simp,\nend\n\n@[simp]\nlemma homogenization_monomial (i : \u03b9) (s : \u03b9 \u2192\u2080 \u2115) (r : R) :\n  (monomial s r : mv_polynomial \u03b9 R).homogenization i = monomial s r :=\nbegin\n  by_cases hr : r = 0,\n  { simp [hr], },\n  erw [homogenization, finsupp.map_domain_single, single_eq_monomial, total_degree_monomial _ hr,\n    tsub_self],\n  simp,\nend\n\n-- TODO name this\nlemma aux {i : \u03b9} {p : mv_polynomial \u03b9 R} {x : \u03b9 \u2192\u2080 \u2115} (hp : x \u2208 p.support) :\n  (x + finsupp.single i (p.total_degree - x.sum (\u03bb _ m, m))).sum (\u03bb _ m, m) = p.total_degree :=\nbegin\n  classical,\n  rw finsupp.sum_add_index,\n  rw [finsupp.sum_single_index],\n  rw [add_tsub_cancel_iff_le],\n  exact finset.le_sup hp,\n  refl,\n  intros, refl,\n  intros, refl,\nend\n\nlemma is_homogeneous_homogenization (i : \u03b9) (p : mv_polynomial \u03b9 R) :\n  (p.homogenization i).is_homogeneous p.total_degree :=\nbegin\n  letI := classical.dec_eq \u03b9,\n  rw homogenization,\n  intros d hd,\n  rw [finsupp.map_domain, finsupp.sum, coeff_sum] at hd,\n  simp_rw [single_eq_monomial, coeff_monomial] at hd,\n  contrapose! hd,\n  have : \u2200 (x : \u03b9 \u2192\u2080 \u2115) (hx : x \u2208 p.support),\n    \u00ac x + finsupp.single i (p.total_degree - x.sum (\u03bb (_x : \u03b9) (m : \u2115), m)) = d,\n  { intros x hx hh,\n    apply hd,\n    rw \u2190 hh,\n    change (x + finsupp.single i (p.total_degree - x.sum (\u03bb _ m, m))).sum (\u03bb _ m, m) = _,\n    rw aux hx, },\n  conv in (ite _ _ _)\n  { rw [if_neg (this x H)], },\n  simp,\nend\n\nlemma homogenization_of_is_homogeneous (n : \u2115) (i : \u03b9) (p : mv_polynomial \u03b9 R)\n  (hp : p.is_homogeneous n) : p.homogenization i = p :=\nbegin\n  by_cases hpn : p = 0,\n  { simp [hpn], },\n  rw homogenization,\n  have := (hp.total_degree hpn).symm,\n  subst this,\n  rw is_homogeneous at hp,\n  have : \u2200 x (hx : x \u2208 p.support),\n    (\u03bb (j : \u03b9 \u2192\u2080 \u2115), j + finsupp.single i (p.total_degree - j.sum (\u03bb (_x : \u03b9) (m : \u2115), m))) x = x,\n  { intros x hx,\n    simp only [add_right_eq_self, finsupp.single_eq_same, tsub_eq_zero_iff_le, finsupp.single_tsub,\n      finsupp.single_le_iff],\n    rw \u2190 hp (mem_support_iff.mp hx),\n    exact le_refl _, },\n  rw finsupp.map_domain_congr this,\n  -- simp,\n  erw finsupp.map_domain_id,\n  -- TODO there should be a simp lemma version of this for \u03bb x, x so simp works\nend\n\nlemma homogenization_idempotent (i : \u03b9) (p : mv_polynomial \u03b9 R) :\n  (p.homogenization i).homogenization i = p.homogenization i :=\nbegin\n  classical,\n  apply homogenization_of_is_homogeneous p.total_degree,\n  exact is_homogeneous_homogenization _ _,\nend\n\n\n-- TODO should these hjp assumptions be phrased using `degree_of` or `vars`?\nlemma homogenization_ne_zero_of_ne_zero (i : \u03b9) {p : mv_polynomial \u03b9 R} (hp : p \u2260 0)\n  (hjp : \u2200 j \u2208 p.support, (j : \u03b9 \u2192 \u2115) i = 0) : p.homogenization i \u2260 0 :=\nbegin\n  intro h,\n  apply hp,\n  have : set.inj_on (\u03bb j : \u03b9 \u2192\u2080 \u2115, j + finsupp.single i (p.total_degree - j.sum (\u03bb _ m, m)))\n          {w | w i = 0},\n  { intros t ht y hy hh,\n    simp only [set.mem_set_of_eq] at hh hy ht,\n    ext a,\n    have : (t + finsupp.single i (p.total_degree - t.sum (\u03bb _ m, m))) a =\n           (y + finsupp.single i (p.total_degree - y.sum (\u03bb _ m, m))) a,\n    { rw hh, },\n    simp only [finsupp.coe_add, pi.add_apply] at this,\n    classical,\n    rw [finsupp.single_apply, finsupp.single_apply] at this,\n    split_ifs at this with hia,\n    { rw [\u2190 hia, ht, hy], },\n    { simpa, }, },\n  refine finsupp.map_domain_inj_on _ this _ (by simp) h,\n  intros x hx,\n  rw [set.mem_set_of_eq, hjp x hx],\n  -- refine finsupp.map_domain_injective _ h,\n  -- intros x y hxy,\n  -- simp at hxy,\n  -- -- TODO something like this but this isnt exactly true\n  -- admit,\nend\n\n-- TODO this can follow from previous\nlemma total_degree_homogenization (i : \u03b9) (p : mv_polynomial \u03b9 R)\n  (h : \u2200 j \u2208 p.support, (j : \u03b9 \u2192 \u2115) i = 0) :\n  (p.homogenization i).total_degree = p.total_degree :=\nbegin\n  classical,\n  by_cases hp : p = 0,\n  { simp [hp], },\n  apply is_homogeneous.total_degree,\n  refine is_homogeneous_homogenization _ _,\n  exact homogenization_ne_zero_of_ne_zero _ hp h,\n  -- rw total_degree,\n  -- have : (homogenization i p).support.nonempty,\n  -- { simp [homogenization],\n  --   admit,\n  --    },\n  -- rw \u2190 finset.sup'_eq_sup this,\n  -- rw finset.nonempty.sup'_eq_cSup_image,\n  -- suffices : (\u03bb (s : \u03b9 \u2192\u2080 \u2115), s.sum (\u03bb (n : \u03b9) (e : \u2115), e)) '' \u2191((homogenization i p).support) =\n  --   {p.total_degree},\n  -- { simp [this], },\n  -- refine set.eq_singleton_iff_unique_mem.mpr _,\n  -- split,\n  -- { simp, admit, },\n  -- { simp, admit, },\nend\n\nsection leading_terms\n-- TODO is this the best def?\n/-- The sum of the monomials of highest degree of a multivariate polynomial. -/\ndef leading_terms (p : mv_polynomial \u03b9 R) : mv_polynomial \u03b9 R :=\nhomogeneous_component p.total_degree p\n\nlemma leading_terms_apply (p : mv_polynomial \u03b9 R) : p.leading_terms =\n  \u2211 d in p.support.filter (\u03bb d, \u2211 i in d.support, d i = p.total_degree), monomial d (coeff d p) :=\nhomogeneous_component_apply _ _\n-- (p.support.filter (\u03bb s : \u03b9 \u2192\u2080 \u2115, s.sum (\u03bb _ e, e) = p.total_degree)).sum $\n--   \u03bb s, monomial s (p.coeff s)\n\n-- TODO for non-zero polys this is true that p.lead = p iff p.is_homogenous n for a fixed n\n-- TODO generalize to p.homog comp = n\nlemma leading_terms_eq_self_iff_is_homogeneous (p : mv_polynomial \u03b9 R) :\n  p.leading_terms = p \u2194 p.is_homogeneous p.total_degree :=\nbegin\n  split; intro h,\n  { rw is_homogeneous,\n    contrapose! h,\n    rcases h with \u27e8h_w, h_h\u2081, h_h\u2082\u27e9,\n    rw [leading_terms, ne.def, mv_polynomial.ext_iff],\n    push_neg,\n    use h_w,\n    classical,\n    change \u00ac h_w.sum (\u03bb (_x : \u03b9) (e : \u2115), e) = p.total_degree at h_h\u2082,\n    simp only [h_h\u2081.symm, coeff_homogeneous_component, exists_prop, and_true, ne.def, not_false_iff,\n      not_forall, ite_eq_left_iff],\n    convert h_h\u2082, },\n  { rw [leading_terms_apply],\n    rw (_ : p.support.filter (\u03bb (s : \u03b9 \u2192\u2080 \u2115), \u2211 (i : \u03b9) in s.support, s i = p.total_degree)\n            = p.support),\n    { rw support_sum_monomial_coeff p, },\n    { rw finset.filter_eq_self,\n      intros s hs,\n      rw [mem_support_iff] at hs,\n      rw \u2190 h hs, }, },\nend\n\n@[simp]\nlemma leading_terms_C (r : R) : (C r : mv_polynomial \u03b9 R).leading_terms = C r :=\nbegin\n  rw leading_terms_eq_self_iff_is_homogeneous,\n  convert is_homogeneous_C _ _,\n  simp,\nend\n\n@[simp]\nlemma leading_terms_zero : (0 : mv_polynomial \u03b9 R).leading_terms = 0 :=\nby simp [leading_terms]\n\n@[simp]\nlemma leading_terms_one : (1 : mv_polynomial \u03b9 R).leading_terms = 1 :=\nby simp [leading_terms]\n\n@[simp]\nlemma leading_terms_monomial (s : \u03b9 \u2192\u2080 \u2115) (r : R) : (monomial s r).leading_terms = monomial s r :=\nbegin\n  by_cases hr : r = 0,\n  { simp [hr], },\n  rw leading_terms_eq_self_iff_is_homogeneous,\n  convert is_homogeneous_monomial _ _ _ _,\n  simpa [total_degree_monomial _ hr]\nend\n\nsection dangerous_instance\nlocal attribute [instance] mv_polynomial.unique\n@[simp]\nlemma leading_terms_X (s : \u03b9) : (X s : mv_polynomial \u03b9 R).leading_terms = X s :=\nbegin\n  nontriviality R,\n  rw leading_terms_eq_self_iff_is_homogeneous,\n  convert is_homogeneous_X _ _,\n  exact total_degree_X _,\nend\nend dangerous_instance\n\nlemma is_homogeneous_leading_terms (p : mv_polynomial \u03b9 R) :\n  p.leading_terms.is_homogeneous p.total_degree :=\nhomogeneous_component_is_homogeneous (total_degree p) p\n\nlemma exists_coeff_ne_zero_total_degree {p : mv_polynomial \u03b9 R} (hp : p \u2260 0) :\n  \u2203 (v : \u03b9 \u2192\u2080 \u2115), v.sum (\u03bb _ e, e) = p.total_degree \u2227 p.coeff v \u2260 0 :=\nbegin\n  obtain \u27e8b, hb\u2081, hb\u2082\u27e9 := p.support.exists_mem_eq_sup (finsupp.support_nonempty_iff.mpr hp)\n    (\u03bb (m : \u03b9 \u2192\u2080 \u2115), m.to_multiset.card),\n  use b,\n  split,\n  { rw \u2190 total_degree_eq p at hb\u2082,\n    rw hb\u2082,\n    dsimp, -- TODO break this out as a lemma\n    funext m,\n    exact (finsupp.card_to_multiset _).symm, },\n  { exact mem_support_iff.mp hb\u2081, },\nend\n\n-- TODO mathlib\n@[simp] lemma support_eq_empty {f : mv_polynomial \u03b9 R} : f.support = \u2205 \u2194 f = 0 :=\nfinsupp.support_eq_empty\n\nlemma support_add_eq [decidable_eq \u03b9] {g\u2081 g\u2082 : mv_polynomial \u03b9 R}\n  (h : disjoint g\u2081.support g\u2082.support) : (g\u2081 + g\u2082).support = g\u2081.support \u222a g\u2082.support :=\nfinsupp.support_add_eq h\n\nlemma add_ne_zero_of_ne_zero_of_support_disjoint (p q : mv_polynomial \u03b9 R)\n  (hp : p \u2260 0) (h : disjoint p.support q.support) : p + q \u2260 0 :=\nbegin\n  classical,\n  contrapose! hp,\n  have := congr_arg support hp,\n  rw [support_zero, support_add_eq h, finset.union_eq_empty_iff, -- TODO should this be simp?\n    mv_polynomial.support_eq_empty] at this,\n  exact this.left,\nend\n\nlemma support_sum_monomial_eq [decidable_eq R] (S : finset (\u03b9 \u2192\u2080 \u2115)) (f : (\u03b9 \u2192\u2080 \u2115) \u2192 R) :\n  support (\u2211 v in S, monomial v (f v)) = S.filter (\u03bb v, f v \u2260 0) :=\nbegin\n  letI := classical.dec_eq \u03b9,\n  induction S using finset.induction with s S hs hsi,\n  { simp, },\n  rw [finset.sum_insert hs, support_add_eq],\n  { rw [hsi, filter_congr_decidable, filter_insert, support_monomial],\n    split_ifs with h;\n    { simp [h, insert_eq], }, },\n  { apply disjoint_of_subset_left support_monomial_subset,\n    simp [hsi, hs], },\nend\n\nlemma support_sum_monomial_subset (S : finset (\u03b9 \u2192\u2080 \u2115)) (f : (\u03b9 \u2192\u2080 \u2115) \u2192 R) :\n  support (\u2211 v in S, monomial v (f v)) \u2286 S :=\nbegin\n  classical,\n  rw support_sum_monomial_eq,\n  apply filter_subset,\nend\n\nlemma sum_monomial_ne_zero_of_exists_mem_ne_zero (S : finset (\u03b9 \u2192\u2080 \u2115)) (f : (\u03b9 \u2192\u2080 \u2115) \u2192 R)\n  (h : \u2203 (s) (hs : s \u2208 S), f s \u2260 0) : \u2211 (s : \u03b9 \u2192\u2080 \u2115) in S, monomial s (f s) \u2260 0 :=\nbegin\n  classical,\n  simp only [\u2190 support_eq_empty, support_sum_monomial_eq, filter_congr_decidable, ne.def],\n  rcases h with \u27e8s, h_S, h_s\u27e9,\n  exact ne_empty_of_mem (mem_filter.mpr \u27e8h_S, h_s\u27e9),\nend\n\nlemma leading_terms_ne_zero {p : mv_polynomial \u03b9 R} (hp : p \u2260 0) : p.leading_terms \u2260 0 :=\nbegin\n  classical,\n  rw leading_terms_apply,\n  apply sum_monomial_ne_zero_of_exists_mem_ne_zero,\n  simp only [exists_prop, mem_support_iff, finset.mem_filter],\n  convert exists_coeff_ne_zero_total_degree hp,\n  ext v,\n  change v.sum (\u03bb (_x : \u03b9) (e : \u2115), e) with v.support.sum v,\n  simp [and_comm],\nend\n\n@[simp]\nlemma total_degree_homogenous_component_of_ne_zero {n : \u2115} {p : mv_polynomial \u03b9 R}\n  (hp : homogeneous_component n p \u2260 0) :\n  (homogeneous_component n p).total_degree = n :=\nis_homogeneous.total_degree (homogeneous_component_is_homogeneous n p) hp\n\n@[simp]\nlemma total_degree_leading_terms (p : mv_polynomial \u03b9 R) :\n  p.leading_terms.total_degree = p.total_degree :=\nbegin\n  by_cases hp : p = 0,\n  { simp [hp], },\n  exact total_degree_homogenous_component_of_ne_zero (leading_terms_ne_zero hp),\nend\n\n-- TODO generalize this to homogeneous component idempotent?\nlemma leading_terms_idempotent (p : mv_polynomial \u03b9 R) :\n  p.leading_terms.leading_terms = p.leading_terms :=\nbegin\n  rw [leading_terms_eq_self_iff_is_homogeneous, total_degree_leading_terms],\n  exact is_homogeneous_leading_terms p,\nend\n\n-- TODO lol this isn't true\n-- lemma homogeneous_component_mul (m n : \u2115) (p q : mv_polynomial \u03b9 R) :\n--   homogeneous_component (m + n) (p * q) = homogeneous_component m p * homogeneous_component n q :=\n-- begin\n--   admit,\n-- end\n\nlemma coeff_leading_terms (p : mv_polynomial \u03b9 R) (d : \u03b9 \u2192\u2080 \u2115) :\n  coeff d p.leading_terms = if \u2211 i in d.support, d i = p.total_degree then coeff d p else 0 :=\ncoeff_homogeneous_component _ _ _\n\nlemma support_homogeneous_component (n : \u2115) (p : mv_polynomial \u03b9 R) :\n  (homogeneous_component n p).support = p.support.filter (\u03bb d, d.sum (\u03bb _ m, m) = n) :=\nbegin\n  rw homogeneous_component,\n  simp only [finsupp.restrict_dom_apply, submodule.subtype_apply, function.comp_app,\n    linear_map.coe_comp, set.mem_set_of_eq],\n  erw \u2190 finsupp.support_filter,\n  refl,\nend\n\nlemma support_homogeneous_component_subset (n : \u2115) (p : mv_polynomial \u03b9 R) :\n  (homogeneous_component n p).support \u2286 p.support :=\nbegin\n  rw support_homogeneous_component,\n  exact finset.filter_subset _ _,\nend\n\nlemma support_leading_terms (p : mv_polynomial \u03b9 R) :\n  p.leading_terms.support = p.support.filter (\u03bb d, d.sum (\u03bb _ m, m) = p.total_degree) :=\nsupport_homogeneous_component _ _\n\nlemma support_leading_terms_subset (p : mv_polynomial \u03b9 R) : p.leading_terms.support \u2286 p.support :=\nsupport_homogeneous_component_subset _ _\n\nlemma eq_leading_terms_add (p : mv_polynomial \u03b9 R) (hp : p.total_degree \u2260 0) :\n  \u2203 p_rest : mv_polynomial \u03b9 R,\n    p = p.leading_terms + p_rest \u2227 p_rest.total_degree < p.total_degree :=\nbegin\n  letI := classical.dec_eq \u03b9,\n  existsi (\u2211 (v : \u03b9 \u2192\u2080 \u2115) in p.support \\ p.leading_terms.support, (monomial v) (coeff v p)),\n  split,\n  { nth_rewrite 0 p.leading_terms.as_sum,\n    have : \u2200 (x : \u03b9 \u2192\u2080 \u2115) (hx : x \u2208 p.leading_terms.support), x.support.sum x = p.total_degree,\n    { intros x hx,\n      rw support_leading_terms at hx,\n      simp at hx,\n      exact hx.2, },\n    simp_rw coeff_leading_terms,\n    conv in (ite _ _ _)\n    { rw [if_pos (this x H)], },\n    have : p.leading_terms.support \u2286 p.support,\n    from support_leading_terms_subset _,\n    have : p.leading_terms.support \u2229 p.support = p.leading_terms.support,\n    { rw finset.inter_eq_left_iff_subset,\n      exact this },\n    nth_rewrite 0 \u2190 this,\n    rw [finset.inter_comm, finset.sum_inter_add_sum_diff],\n    exact p.as_sum, },\n  { rw [total_degree, finset.sup_lt_iff],\n    intros b hb,\n    rw support_leading_terms at hb,\n    rw \u2190 finset.filter_not at hb, -- TODO this was also hard to find maybe a negated version is good\n    have := support_sum_monomial_subset _ _ hb,\n    simp only [finset.mem_filter] at this,\n    cases this,\n    rw total_degree,\n    exact lt_of_le_of_ne (finset.le_sup this_left) this_right,\n    rw [bot_eq_zero],\n    exact pos_iff_ne_zero.mpr hp, },\nend\n\nlemma leading_terms_add_of_total_degree_lt (p q : mv_polynomial \u03b9 R)\n  (h : q.total_degree < p.total_degree) : (p + q).leading_terms = p.leading_terms :=\nby rw [leading_terms, leading_terms, total_degree_add_eq_left_of_total_degree_lt h,\n  linear_map.map_add, homogeneous_component_eq_zero _ q h, add_zero]\n\n-- lemma C_mul_eq_smul {r : R} (p : mv_polynomial \u03b9 R) : C r * p = r \u2022 p :=\n-- by rw [C_eq_smul_one, algebra.smul_mul_assoc, one_mul]\n\nlemma no_zero_smul_divisors.smul_eq_zero_iff_eq_zero_or_eq_zero (R M : Type*) [has_zero R]\n  [has_zero M] [smul_with_zero R M] [no_zero_smul_divisors R M] {c : R} {x : M} :\n  c \u2022 x = 0 \u2194 c = 0 \u2228 x = 0 :=\nbegin\n  split; intro h,\n  exact eq_zero_or_eq_zero_of_smul_eq_zero h,\n  cases h;\n  simp [h],\nend\n\n--TODO this generalized lemma when distrib_mul_action_with_zero exists?\n-- lemma support_smul_eq {\u03b1 M R : Type*} {_ : monoid_with_zero R} [add_monoid M]\n--   [distrib_mul_action_with_zero R M] [no_zero_smul_divisors R M] {b : R} (hb : b \u2260 0) {g : \u03b1 \u2192\u2080 M} :\n--   (b \u2022 g).support = g.support :=\n-- begin\n--   ext a,\n--   simp [finsupp.smul_apply, mem_support_iff, ne.def],\n--   simp,\n--   rw no_zero_smul_divisors.smul_eq_zero_iff_eq_zero_or_eq_zero,\n-- end\n\n-- haveI : no_zero_smul_divisors R (mv_polynomial \u03b9 R), --TODO add this instance\n--TODO maybe this for leading terms and homog\n-- lemma homogeneous_s_monomial_mul [no_zero_divisors R] (p : mv_polynomial \u03b9 R) (r : R) (x : \u03b9 \u2192\u2080 \u2115) :\n  -- (p * monomial x r).leading_terms = p.leading_terms * monomial x r :=\n  --TODO also maybe an smul version\n@[simp]\nlemma leading_terms_C_mul [no_zero_smul_divisors R R] (p : mv_polynomial \u03b9 R) (r : R) :\n  (C r * p).leading_terms = C r * p.leading_terms :=\nbegin\n  by_cases hr : r = 0,\n  { simp [hr], },\n  have : (C r * p).support = p.support,\n  { rw C_mul',\n    exact finsupp.support_smul_eq hr, },\n  rw [leading_terms, leading_terms, total_degree, this, homogeneous_component_C_mul],\n  refl,\nend\n\nlemma eq_C_of_total_degree_zero {p : mv_polynomial \u03b9 R} (hp : p.total_degree = 0) :\n  p = C (coeff 0 p) :=\nbegin\n  letI := classical.dec_eq \u03b9,\n  erw finset.sup_eq_bot_iff at hp,\n  simp only [mem_support_iff] at hp,\n  ext,\n  by_cases hm : m = 0,\n  { simp [hm], },\n  rw [coeff_C, if_neg (ne.symm hm)],\n  classical,\n  by_contradiction h,\n  specialize hp m h,\n  apply hm,\n  rw finsupp.sum at hp, -- TODO this and line below could be a lemma, finsupp.sum_eq_zero_iff?\n  simp only [not_imp_self, bot_eq_zero, finsupp.mem_support_iff, finset.sum_eq_zero_iff] at hp,\n  ext,\n  simp [hp],\nend\n\n-- TODO can things be generalized to no_zero_divisors (would require an instance for mv_poly)\n-- sadly this adds some imports and requirements not needed in rest of file\n@[simp]\nlemma leading_terms_mul {S : Type*} [comm_ring S] [is_domain S] (p q : mv_polynomial \u03b9 S) :\n  (p * q).leading_terms = p.leading_terms * q.leading_terms :=\nbegin\n  by_cases hp : p.total_degree = 0,\n  { rw [eq_C_of_total_degree_zero hp, leading_terms_C_mul, leading_terms_C], },\n  by_cases hq : q.total_degree = 0,\n  { rw [eq_C_of_total_degree_zero hq, mul_comm, leading_terms_C_mul, leading_terms_C, mul_comm], },\n  have : (p.leading_terms * q.leading_terms).total_degree = p.total_degree + q.total_degree,\n  { rw is_homogeneous.total_degree,\n    apply is_homogeneous.mul (is_homogeneous_leading_terms p) (is_homogeneous_leading_terms q),\n    apply mul_ne_zero;\n    { apply leading_terms_ne_zero,\n      rintro rfl,\n      simpa, }, },\n  rcases eq_leading_terms_add p hp with \u27e8wp, hp, tp\u27e9,\n  rcases eq_leading_terms_add q hq with \u27e8wq, hq, tq\u27e9,\n  rw [hp, hq],\n  simp only [add_mul, mul_add],\n  rw [add_assoc, leading_terms_add_of_total_degree_lt, leading_terms_add_of_total_degree_lt,\n    leading_terms_add_of_total_degree_lt, leading_terms_idempotent, leading_terms_idempotent,\n    leading_terms_eq_self_iff_is_homogeneous],\n  { convert is_homogeneous.mul (is_homogeneous_leading_terms _) (is_homogeneous_leading_terms _), },\n  { rwa total_degree_leading_terms, },\n  { rwa total_degree_leading_terms, },\n  { rw this,\n    calc _ \u2264 max (wp * q.leading_terms).total_degree (p.leading_terms * wq + wp * wq).total_degree :\n              total_degree_add _ _\n       ... \u2264 max (wp * q.leading_terms).total_degree\n              (max (p.leading_terms * wq).total_degree (wp * wq).total_degree) :\n                max_le_max (le_refl _) (total_degree_add _ _)\n       ... \u2264 max (wp.total_degree + q.leading_terms.total_degree)\n              (max (p.leading_terms * wq).total_degree (wp * wq).total_degree) :\n                max_le_max (total_degree_mul _ _) (le_refl _)\n       ... \u2264 max (wp.total_degree + q.leading_terms.total_degree)\n              (max (p.leading_terms.total_degree + wq.total_degree)\n                (wp.total_degree + wq.total_degree)) :\n                  max_le_max (le_refl _) (max_le_max (total_degree_mul _ _) (total_degree_mul _ _))\n       ... < p.total_degree + q.total_degree : _,\n    simp only [total_degree_leading_terms, max_lt_iff, add_lt_add_iff_right, add_lt_add_iff_left],\n    exact \u27e8tp, tq, add_lt_add tp tq\u27e9, },\nend\n--TODO reinterpret this as a hom in this case\n\nlemma total_degree_mul_eq {S : Type*} [comm_ring S] [is_domain S] {p q : mv_polynomial \u03b9 S}\n  (hp : p \u2260 0) (hq : q \u2260 0) : (p * q).total_degree = p.total_degree + q.total_degree :=\nbegin\n  rw [\u2190 total_degree_leading_terms, \u2190 total_degree_leading_terms p, \u2190 total_degree_leading_terms q,\n    leading_terms_mul, is_homogeneous.total_degree],\n  apply is_homogeneous.mul;\n  simp [is_homogeneous_leading_terms],\n  apply mul_ne_zero (leading_terms_ne_zero hp) (leading_terms_ne_zero hq),\nend\n\nend leading_terms\n\nlemma homogenization_add_of_total_degree_eq (i : \u03b9) (p q : mv_polynomial \u03b9 R)\n  (h : p.total_degree = q.total_degree) (hpq : p.total_degree = (p + q).total_degree) :\n  (p + q).homogenization i = p.homogenization i + q.homogenization i :=\nby simp only [homogenization, finsupp.map_domain_add, \u2190h, \u2190hpq]\n\nlemma homogenization_mul {S : Type*} [comm_ring S] [is_domain S] (i : \u03b9) (p q : mv_polynomial \u03b9 S) :\n  -- TODO is this cond needed?\n  --(hp : \u2200 j \u2208 p.support, (j : \u03b9 \u2192 \u2115) i = 0) (hq : \u2200 j \u2208 q.support, (j : \u03b9 \u2192 \u2115) i = 0) :\n  (p * q).homogenization i = p.homogenization i * q.homogenization i :=\nbegin\n  classical,\n  by_cases hp : p = 0,\n  { simp [hp], },\n  by_cases hq : q = 0,\n  { simp [hq], },\n  rw [homogenization, homogenization, homogenization, total_degree_mul_eq hp hq,\n    \u2190 finsupp.sum_single p, \u2190 finsupp.sum_single q, finsupp.map_domain_sum, finsupp.map_domain_sum],\n  erw [finset.sum_mul_sum, finset.sum_mul_sum],\n  simp only [finsupp.single_add, finsupp.sum_single, monomial_mul],\n  rw finsupp.map_domain_finset_sum,\n  apply finset.sum_congr rfl,\n  intros a ha,\n  simp only [finset.mem_product] at ha,\n  rw [finsupp.map_domain_single, finsupp.map_domain_single],\n  simp_rw [single_eq_monomial],\n  simp only [finsupp.single_add, monomial_mul],\n  erw finsupp.map_domain_single,\n  congr' 1,\n  rw finsupp.sum_add_index,\n  simp only [finsupp.single_add, finsupp.single_tsub],\n  ext j,\n  simp only [pi.add_apply, finsupp.coe_add, finsupp.coe_tsub, pi.sub_apply],\n  classical,\n  have : \u2200 {f s p q fs ss : \u2115} (hP : fs \u2264 p) (hQ : ss \u2264 q),\n    f + s + (p + q - (fs + ss)) = f + (p - fs) + (s + (q - ss)),\n  { intros,\n    zify [add_le_add hP hQ],\n    ring, },\n  refine this _ _;\n  rw finsupp.single_apply;\n  split_ifs,\n  { simp only [h, finsupp.single_eq_same],\n    convert finset.le_sup ha.left,\n    refl, },\n  { simp, },\n  { simp only [h, finsupp.single_eq_same],\n      convert finset.le_sup ha.right,\n      refl, },\n  { simp, },\n  { intros i _, refl, },\n  { intro i, simp, },\nend\n\nsection dangerous_instance\nlocal attribute [instance] mv_polynomial.unique\n\n@[simp]\nlemma homogenization_X_add_C {i j : \u03b9} (r : R) :\n  (X j + C r : mv_polynomial \u03b9 R).homogenization i = X j + C r * X i :=\nbegin\n  nontriviality R,\n  have : (X j + C r).total_degree = 1,\n  { rw total_degree_add_eq_left_of_total_degree_lt,\n    { exact total_degree_X _, },\n    { simp only [total_degree_C, total_degree_X, nat.lt_one_iff], }, },\n  erw [homogenization, finsupp.map_domain_add, finsupp.map_domain_single,\n    finsupp.map_domain_single],\n  simp only [tsub_zero, finsupp.sum_zero_index, finsupp.sum_single_index, this, add_zero,\n    finsupp.single_zero, zero_add, single_eq_monomial],\n  rw [X, X],\n  congr,\n  rw [\u2190 C_mul_X_pow_eq_monomial, pow_one],\n  refl,\nend\n\n@[simp]\nlemma homogenization_X_sub_C {R : Type*} [comm_ring R] {i j : \u03b9} (r : R) :\n  (X j - C r : mv_polynomial \u03b9 R).homogenization i = X j - C r * X i :=\nby rw [sub_eq_add_neg, sub_eq_add_neg, \u2190 C_neg, homogenization_X_add_C,\n  C_neg, neg_mul]\n\n@[simp]\nlemma homogenization_X_pow_add_C {i j : \u03b9} {n : \u2115} (hn : 0 < n) (r : R) :\n  (X j ^ n + C r : mv_polynomial \u03b9 R).homogenization i = X j ^ n + C r * X i ^ n :=\nbegin\n  nontriviality R,\n  have : (X j ^ n + C r).total_degree = n,\n  { rw total_degree_add_eq_left_of_total_degree_lt,\n    { exact total_degree_X_pow _ _, },\n    { simp only [total_degree_C, total_degree_X_pow, hn], }, },\n  erw [homogenization, finsupp.map_domain_add],\n  erw add_monoid_algebra.single_pow,\n  erw [finsupp.map_domain_single,\n    finsupp.map_domain_single],\n  simp only [tsub_zero, finsupp.sum_zero_index, finsupp.sum_single_index, zero_add,\n    single_eq_monomial, one_pow, mul_one, finsupp.smul_single', finsupp.single_tsub],\n  congr,\n  { rw total_degree_add_eq_left_of_total_degree_lt,\n    simp [one_ne_zero],\n    simp [one_ne_zero, hn], },\n  { convert C_mul_X_pow_eq_monomial.symm,\n    rw \u2190 C_mul_X_pow_eq_monomial,\n    simp [this], },\nend\n\n@[simp]\nlemma homogenization_X_pow_sub_C {R : Type*} [comm_ring R] {i j : \u03b9} {n : \u2115} (hn : 0 < n) (r : R) :\n  (X j ^ n - C r : mv_polynomial \u03b9 R).homogenization i = X j ^ n - C r * X i ^ n :=\nby rw [sub_eq_add_neg, sub_eq_add_neg, \u2190 C_neg, homogenization_X_pow_add_C hn,\n  C_neg, neg_mul]\n\n@[simp]\nlemma homogenization_X_pow_sub_one {R : Type*} [comm_ring R] {i j : \u03b9} {n : \u2115} (hn : 0 < n) :\n  (X j ^ n - 1 : mv_polynomial \u03b9 R).homogenization i = X j ^ n - X i ^ n :=\nbegin\n  convert homogenization_X_pow_sub_C hn _,\n  simp,\nend\n\n@[simp]\nlemma homogenization_X_pow_add_one {i j : \u03b9} {n : \u2115} (hn : 0 < n) :\n  (X j ^ n + 1 : mv_polynomial \u03b9 R).homogenization i = X j ^ n + X i ^ n :=\nbegin\n  convert homogenization_X_pow_add_C hn _,\n  simp,\nend\n\nend dangerous_instance\n\nend mv_polynomial\n\nnamespace mv_polynomial\nsection\n\n-- generalized version of the unprimed version\nlemma support_sum_monomial_subset' [decidable_eq \u03b9] {\u03b1 : Type*} (S : finset \u03b1) (g : \u03b1 \u2192 \u03b9 \u2192\u2080 \u2115)\n  (f : \u03b1 \u2192 R) : support (\u2211 v in S, monomial (g v) (f v)) \u2286 S.image g :=\nbegin\n  letI := classical.dec_eq \u03b1,\n  induction S using finset.induction with s S hs hsi,\n  { simp, },\n  { rw finset.sum_insert hs,\n    apply finset.subset.trans support_add,\n    apply finset.union_subset,\n    { apply finset.subset.trans support_monomial_subset _,\n      rw finset.image_insert,\n      convert finset.subset_union_left _ (finset.image g S), },\n    { apply finset.subset.trans hsi _,\n      rw finset.image_insert,\n      exact finset.subset_insert (g s) (finset.image g S), }, },\nend\nopen_locale pointwise\n\nlemma support_mul' [decidable_eq \u03b9] (p q : mv_polynomial \u03b9 R) :\n  (p * q).support \u2286 p.support + q.support :=\nbegin\n  -- TODO this was really hard to find, maybe needs a docstring or alias?\n  rw [p.as_sum, q.as_sum, finset.sum_mul_sum],\n  simp_rw [monomial_mul],\n  rw [support_sum_monomial_coeff, support_sum_monomial_coeff],\n  exact finset.subset.trans (support_sum_monomial_subset' _ _ _) (finset.subset.refl _),\nend\n\nsection\nopen_locale pointwise\n\nlemma support_one : (1 : mv_polynomial \u03b9 R).support \u2286 0 :=\nfinsupp.support_single_subset\n\n@[simp] lemma support_one_of_nontrivial [nontrivial R] : (1 : mv_polynomial \u03b9 R).support = 0 :=\nfinsupp.support_single_ne_zero _ one_ne_zero\n\nend\n\nvariable [decidable_eq \u03b9]\nlemma support_prod (P : finset (mv_polynomial \u03b9 R)) : (P.prod id).support \u2286 P.sum support :=\nbegin\n  classical,\n  induction P using finset.induction with p S hS hSi,\n  { simp only [prod_empty, sum_empty], exact support_one, },\n  rw [finset.prod_insert hS, finset.sum_insert hS],\n  simp only [id.def],\n  refine finset.subset.trans (support_mul' _ _) _,\n  convert finset.add_subset_add (finset.subset.refl _) hSi,\nend\n\nend\n\nlemma degree_of_eq_zero_iff (i : \u03b9) (p : mv_polynomial \u03b9 R) : degree_of i p = 0 \u2194\n  \u2200 (j: \u03b9\u2192\u2080 \u2115), j \u2208 p.support \u2192 j i = 0:=\nbegin\n  rw degree_of_eq_sup,\n  apply iff.intro,\n  { intros h j hj,\n    apply nat.eq_zero_of_le_zero,\n    have t:= (finset.le_sup hj),\n    rwa h at t },\n  { intro h,\n    apply nat.eq_zero_of_le_zero,\n    apply finset.sup_le,\n    intros m hm,\n    rw h m hm }\nend\n\nlemma prod_contains_no (i : \u03b9) (P : finset (mv_polynomial \u03b9 R))\n  (hp : \u2200 (p : mv_polynomial \u03b9 R) (hp : p \u2208 P) (j) (hjp : j \u2208 p.support), (j : \u03b9 \u2192 \u2115) i = 0)\n  (j) (hjp : j \u2208 (P.prod id).support) :\n  (j : \u03b9 \u2192 \u2115) i = 0 :=\nbegin\n  apply (degree_of_eq_zero_iff i (P.prod id)).1 _ j hjp,\n  revert hp,\n  apply finset.cons_induction_on P,\n  { intro hp,\n    simp only [prod_empty, \u2190 C_1, degree_of_C] },\n  { intros a s has hs,\n    intro hp,\n    rw prod_cons,\n    apply nat.eq_zero_of_le_zero,\n    apply le_trans (degree_of_mul_le _ _ _),\n    rw hs,\n    { simp only [id.def, add_zero, le_zero_iff],\n      exact (degree_of_eq_zero_iff _ _).2 (hp a (mem_cons_self _ _)) },\n    { intros p hps m  hmp,\n      apply hp p _ m hmp,\n      simp only [hps, mem_cons, or_true] } }\nend\n\nopen_locale big_operators\nlemma homogenization_prod {\u03c3 S : Type*} [comm_ring S] [is_domain S] (i : \u03b9)\n  (P : \u03c3 \u2192 mv_polynomial \u03b9 S) (L : finset \u03c3) :\n  (\u220f l in L, P l).homogenization i = \u220f l in L, (P l).homogenization i :=\nbegin\n  classical,\n  induction L using finset.induction with p S hS hSi,\n  { simp, },\n  simp only [finset.prod_insert hS],\n  rw homogenization_mul,\n  rw hSi,\nend\n\nlemma homogenization_prod_id {S : Type*} [comm_ring S] [is_domain S] (i : \u03b9)\n  (P : finset (mv_polynomial \u03b9 S)) :\n  (P.prod id).homogenization i = P.prod (\u03bb p, p.homogenization i) :=\nbegin\n  classical,\n  induction P using finset.induction with p S hS hSi,\n  { simp, },\n  simp only [finset.prod_insert hS],\n  rw homogenization_mul,\n  rw hSi,\n  rw [id.def],\nend\n\nend mv_polynomial\n", "meta": {"author": "leanprover-community", "repo": "flt-regular", "sha": "1d0cecf99e8ab3f98b551e5932bf907042daa6ad", "save_path": "github-repos/lean/leanprover-community-flt-regular", "path": "github-repos/lean/leanprover-community-flt-regular/flt-regular-1d0cecf99e8ab3f98b551e5932bf907042daa6ad/src/ready_for_mathlib/homogenization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.382338467337406}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.polynomial.monic\nimport Mathlib.algebra.group_ring_action\nimport Mathlib.algebra.group_action_hom\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Group action on rings applied to polynomials\n\nThis file contains instances and definitions relating `mul_semiring_action` to `polynomial`.\n-/\n\nnamespace polynomial\n\n\nprotected instance mul_semiring_action (M : Type u_1) [monoid M] (R : Type u_2) [semiring R]\n    [mul_semiring_action M R] : mul_semiring_action M (polynomial R) :=\n  mul_semiring_action.mk sorry sorry\n\nprotected instance faithful_mul_semiring_action (M : Type u_1) [monoid M] (R : Type u_2)\n    [semiring R] [faithful_mul_semiring_action M R] :\n    faithful_mul_semiring_action M (polynomial R) :=\n  faithful_mul_semiring_action.mk sorry\n\n@[simp] theorem coeff_smul' {M : Type u_1} [monoid M] {R : Type u_2} [semiring R]\n    [mul_semiring_action M R] (m : M) (p : polynomial R) (n : \u2115) :\n    coeff (m \u2022 p) n = m \u2022 coeff p n :=\n  coeff_map (mul_semiring_action.to_semiring_hom M R m) n\n\n@[simp] theorem smul_C {M : Type u_1} [monoid M] {R : Type u_2} [semiring R]\n    [mul_semiring_action M R] (m : M) (r : R) : m \u2022 coe_fn C r = coe_fn C (m \u2022 r) :=\n  map_C (mul_semiring_action.to_semiring_hom M R m)\n\n@[simp] theorem smul_X {M : Type u_1} [monoid M] {R : Type u_2} [semiring R]\n    [mul_semiring_action M R] (m : M) : m \u2022 X = X :=\n  map_X (mul_semiring_action.to_semiring_hom M R m)\n\ntheorem smul_eval_smul {M : Type u_1} [monoid M] (S : Type u_3) [comm_semiring S]\n    [mul_semiring_action M S] (m : M) (f : polynomial S) (x : S) :\n    eval (m \u2022 x) (m \u2022 f) = m \u2022 eval x f :=\n  sorry\n\ntheorem eval_smul' (S : Type u_3) [comm_semiring S] (G : Type u_4) [group G]\n    [mul_semiring_action G S] (g : G) (f : polynomial S) (x : S) :\n    eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f)))\n        (Eq.symm (smul_eval_smul S g (g\u207b\u00b9 \u2022 f) x))))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (eval (g \u2022 x) f = eval (g \u2022 x) (g \u2022 g\u207b\u00b9 \u2022 f))) (smul_inv_smul g f)))\n      (Eq.refl (eval (g \u2022 x) f)))\n\ntheorem smul_eval (S : Type u_3) [comm_semiring S] (G : Type u_4) [group G]\n    [mul_semiring_action G S] (g : G) (f : polynomial S) (x : S) :\n    eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f))\n        (Eq.symm (smul_eval_smul S g f (g\u207b\u00b9 \u2022 x)))))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (eval x (g \u2022 f) = eval (g \u2022 g\u207b\u00b9 \u2022 x) (g \u2022 f))) (smul_inv_smul g x)))\n      (Eq.refl (eval x (g \u2022 f))))\n\nend polynomial\n\n\n/-- the product of `(X - g \u2022 x)` over distinct `g \u2022 x`. -/\ndef prod_X_sub_smul (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R]\n    [mul_semiring_action G R] (x : R) : polynomial R :=\n  finset.prod finset.univ\n    fun (g : quotient_group.quotient (mul_action.stabilizer G x)) =>\n      polynomial.X - coe_fn polynomial.C (mul_action.of_quotient_stabilizer G x g)\n\ntheorem prod_X_sub_smul.monic (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R]\n    [mul_semiring_action G R] (x : R) : polynomial.monic (prod_X_sub_smul G R x) :=\n  sorry\n\ntheorem prod_X_sub_smul.eval (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R]\n    [mul_semiring_action G R] (x : R) : polynomial.eval x (prod_X_sub_smul G R x) = 0 :=\n  sorry\n\ntheorem prod_X_sub_smul.smul (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R]\n    [mul_semiring_action G R] (x : R) (g : G) : g \u2022 prod_X_sub_smul G R x = prod_X_sub_smul G R x :=\n  sorry\n\ntheorem prod_X_sub_smul.coeff (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R]\n    [mul_semiring_action G R] (x : R) (g : G) (n : \u2115) :\n    g \u2022 polynomial.coeff (prod_X_sub_smul G R x) n = polynomial.coeff (prod_X_sub_smul G R x) n :=\n  sorry\n\nnamespace mul_semiring_action_hom\n\n\n/-- An equivariant map induces an equivariant map on polynomials. -/\nprotected def polynomial {M : Type u_1} [monoid M] {P : Type u_2} [comm_semiring P]\n    [mul_semiring_action M P] {Q : Type u_3} [comm_semiring Q] [mul_semiring_action M Q]\n    (g : mul_semiring_action_hom M P Q) : mul_semiring_action_hom M (polynomial P) (polynomial Q) :=\n  mk (polynomial.map \u2191g) sorry sorry sorry sorry sorry\n\n@[simp] theorem coe_polynomial {M : Type u_1} [monoid M] {P : Type u_2} [comm_semiring P]\n    [mul_semiring_action M P] {Q : Type u_3} [comm_semiring Q] [mul_semiring_action M Q]\n    (g : mul_semiring_action_hom M P Q) :\n    \u21d1(mul_semiring_action_hom.polynomial g) = polynomial.map \u2191g :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/polynomial/group_ring_action_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984445, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3823384590656469}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.const\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.category_theory.yoneda\nimport Mathlib.category_theory.reflects_isomorphisms\nimport Mathlib.PostPort\n\nuniverses v u l u' \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\nnamespace functor\n\n\n/--\n`F.cones` is the functor assigning to an object `X` the type of\nnatural transformations from the constant functor with value `X` to `F`.\nAn object representing this functor is a limit of `F`.\n-/\ndef cones {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) : C\u1d52\u1d56 \u2964 Type v :=\n  functor.op (const J) \u22d9 obj yoneda F\n\n/--\n`F.cocones` is the functor assigning to an object `X` the type of\nnatural transformations from `F` to the constant functor with value `X`.\nAn object corepresenting this functor is a colimit of `F`.\n-/\n@[simp] theorem cocones_obj {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) (X : C) : obj (cocones F) X = (F \u27f6 obj (const J) X) :=\n  Eq.refl (F \u27f6 obj (const J) X)\n\nend functor\n\n\n/--\nFunctorially associated to each functor `J \u2964 C`, we have the `C`-presheaf consisting of\ncones with a given cone point.\n-/\n@[simp] theorem cones_map (J : Type v) [small_category J] (C : Type u) [category C] (F : J \u2964 C) (G : J \u2964 C) (f : F \u27f6 G) : functor.map (cones J C) f = whisker_left (functor.op (functor.const J)) (functor.map yoneda f) :=\n  Eq.refl (functor.map (cones J C) f)\n\n/--\nContravariantly associated to each functor `J \u2964 C`, we have the `C`-copresheaf consisting of\ncocones with a given cocone point.\n-/\n@[simp] theorem cocones_obj (J : Type v) [small_category J] (C : Type u) [category C] (F : J \u2964 C\u1d52\u1d56) : functor.obj (cocones J C) F = functor.cocones (opposite.unop F) :=\n  Eq.refl (functor.obj (cocones J C) F)\n\nnamespace limits\n\n\n/--\nA `c : cone F` is:\n* an object `c.X` and\n* a natural transformation `c.\u03c0 : c.X \u27f6 F` from the constant `c.X` functor to `F`.\n\n`cone F` is equivalent, via `cone.equiv` below, to `\u03a3 X, F.cones.obj X`.\n-/\nstructure cone {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) \nwhere\n  X : C\n  \u03c0 : functor.obj (functor.const J) X \u27f6 F\n\nprotected instance inhabited_cone {C : Type u} [category C] (F : discrete PUnit \u2964 C) : Inhabited (cone F) :=\n  { default := cone.mk (functor.obj F PUnit.unit) (nat_trans.mk fun (X : discrete PUnit) => sorry) }\n\n@[simp] theorem cone.w {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone F) {j : J} {j' : J} (f : j \u27f6 j') : nat_trans.app (cone.\u03c0 c) j \u226b functor.map F f = nat_trans.app (cone.\u03c0 c) j' := sorry\n\n/--\nA `c : cocone F` is\n* an object `c.X` and\n* a natural transformation `c.\u03b9 : F \u27f6 c.X` from `F` to the constant `c.X` functor.\n\n`cocone F` is equivalent, via `cone.equiv` below, to `\u03a3 X, F.cocones.obj X`.\n-/\nstructure cocone {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) \nwhere\n  X : C\n  \u03b9 : F \u27f6 functor.obj (functor.const J) X\n\nprotected instance inhabited_cocone {C : Type u} [category C] (F : discrete PUnit \u2964 C) : Inhabited (cocone F) :=\n  { default := cocone.mk (functor.obj F PUnit.unit) (nat_trans.mk fun (X : discrete PUnit) => sorry) }\n\n@[simp] theorem cocone.w_assoc {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone F) {j : J} {j' : J} (f : j \u27f6 j') {X' : C} (f' : functor.obj (functor.obj (functor.const J) (cocone.X c)) j' \u27f6 X') : functor.map F f \u226b nat_trans.app (cocone.\u03b9 c) j' \u226b f' = nat_trans.app (cocone.\u03b9 c) j \u226b f' := sorry\n\nnamespace cone\n\n\n/-- The isomorphism between a cone on `F` and an element of the functor `F.cones`. -/\ndef equiv {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) : cone F \u2245 sigma fun (X : C\u1d52\u1d56) => functor.obj (functor.cones F) X :=\n  iso.mk (fun (c : cone F) => sigma.mk (opposite.op (X c)) (\u03c0 c))\n    fun (c : sigma fun (X : C\u1d52\u1d56) => functor.obj (functor.cones F) X) => mk (opposite.unop (sigma.fst c)) (sigma.snd c)\n\n/-- A map to the vertex of a cone naturally induces a cone by composition. -/\n@[simp] def extensions {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone F) : functor.obj yoneda (X c) \u27f6 functor.cones F :=\n  nat_trans.mk fun (X : C\u1d52\u1d56) (f : functor.obj (functor.obj yoneda (X c)) X) => functor.map (functor.const J) f \u226b \u03c0 c\n\n/-- A map to the vertex of a cone induces a cone by composition. -/\n@[simp] def extend {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone F) {X : C} (f : X \u27f6 X c) : cone F :=\n  mk X (nat_trans.app (extensions c) (opposite.op X) f)\n\n@[simp] theorem extend_\u03c0 {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone F) {X : C\u1d52\u1d56} (f : opposite.unop X \u27f6 X c) : \u03c0 (extend c f) = nat_trans.app (extensions c) X f :=\n  rfl\n\n/-- Whisker a cone by precomposition of a functor. -/\ndef whisker {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (E : K \u2964 J) (c : cone F) : cone (E \u22d9 F) :=\n  mk (X c) (whisker_left E (\u03c0 c))\n\nend cone\n\n\nnamespace cocone\n\n\n/-- The isomorphism between a cocone on `F` and an element of the functor `F.cocones`. -/\ndef equiv {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) : cocone F \u2245 sigma fun (X : C) => functor.obj (functor.cocones F) X :=\n  iso.mk (fun (c : cocone F) => sigma.mk (X c) (\u03b9 c))\n    fun (c : sigma fun (X : C) => functor.obj (functor.cocones F) X) => mk (sigma.fst c) (sigma.snd c)\n\n/-- A map from the vertex of a cocone naturally induces a cocone by composition. -/\n@[simp] def extensions {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone F) : functor.obj coyoneda (opposite.op (X c)) \u27f6 functor.cocones F :=\n  nat_trans.mk\n    fun (X : C) (f : functor.obj (functor.obj coyoneda (opposite.op (X c))) X) => \u03b9 c \u226b functor.map (functor.const J) f\n\n/-- A map from the vertex of a cocone induces a cocone by composition. -/\n@[simp] def extend {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone F) {X : C} (f : X c \u27f6 X) : cocone F :=\n  mk X (nat_trans.app (extensions c) X f)\n\n@[simp] theorem extend_\u03b9 {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone F) {X : C} (f : X c \u27f6 X) : \u03b9 (extend c f) = nat_trans.app (extensions c) X f :=\n  rfl\n\n/--\nWhisker a cocone by precomposition of a functor. See `whiskering` for a functorial\nversion.\n-/\n@[simp] theorem whisker_\u03b9 {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (E : K \u2964 J) (c : cocone F) : \u03b9 (whisker E c) = whisker_left E (\u03b9 c) :=\n  Eq.refl (\u03b9 (whisker E c))\n\nend cocone\n\n\n/-- A cone morphism between two cones for the same diagram is a morphism of the cone points which\ncommutes with the cone legs. -/\nstructure cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (A : cone F) (B : cone F) \nwhere\n  hom : cone.X A \u27f6 cone.X B\n  w' : autoParam (\u2200 (j : J), hom \u226b nat_trans.app (cone.\u03c0 B) j = nat_trans.app (cone.\u03c0 A) j)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem cone_morphism.w {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {A : cone F} {B : cone F} (c : cone_morphism A B) (j : J) : cone_morphism.hom c \u226b nat_trans.app (cone.\u03c0 B) j = nat_trans.app (cone.\u03c0 A) j := sorry\n\n@[simp] theorem cone_morphism.w_assoc {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {A : cone F} {B : cone F} (c : cone_morphism A B) (j : J) {X' : C} (f' : functor.obj F j \u27f6 X') : cone_morphism.hom c \u226b nat_trans.app (cone.\u03c0 B) j \u226b f' = nat_trans.app (cone.\u03c0 A) j \u226b f' := sorry\n\nprotected instance inhabited_cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (A : cone F) : Inhabited (cone_morphism A A) :=\n  { default := cone_morphism.mk \ud835\udfd9 }\n\n/-- The category of cones on a given diagram. -/\nprotected instance cone.category {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} : category (cone F) :=\n  category.mk\n\nnamespace cones\n\n\n/-- To give an isomorphism between cones, it suffices to give an\n  isomorphism between their vertices which commutes with the cone\n  maps. -/\ndef ext {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {c : cone F} {c' : cone F} (\u03c6 : cone.X c \u2245 cone.X c') (w : \u2200 (j : J), nat_trans.app (cone.\u03c0 c) j = iso.hom \u03c6 \u226b nat_trans.app (cone.\u03c0 c') j) : c \u2245 c' :=\n  iso.mk (cone_morphism.mk (iso.hom \u03c6)) (cone_morphism.mk (iso.inv \u03c6))\n\n/--\nGiven a cone morphism whose object part is an isomorphism, produce an\nisomorphism of cones.\n-/\ndef cone_iso_of_hom_iso {J : Type v} [small_category J] {C : Type u} [category C] {K : J \u2964 C} {c : cone K} {d : cone K} (f : c \u27f6 d) [i : is_iso (cone_morphism.hom f)] : is_iso f :=\n  is_iso.mk (cone_morphism.mk (inv (cone_morphism.hom f)))\n\n/--\nFunctorially postcompose a cone for `F` by a natural transformation `F \u27f6 G` to give a cone for `G`.\n-/\n@[simp] theorem postcompose_map_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} (\u03b1 : F \u27f6 G) (c\u2081 : cone F) (c\u2082 : cone F) (f : c\u2081 \u27f6 c\u2082) : cone_morphism.hom (functor.map (postcompose \u03b1) f) = cone_morphism.hom f :=\n  Eq.refl (cone_morphism.hom (functor.map (postcompose \u03b1) f))\n\n/-- Postcomposing a cone by the composite natural transformation `\u03b1 \u226b \u03b2` is the same as\npostcomposing by `\u03b1` and then by `\u03b2`. -/\ndef postcompose_comp {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} {H : J \u2964 C} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) : postcompose (\u03b1 \u226b \u03b2) \u2245 postcompose \u03b1 \u22d9 postcompose \u03b2 :=\n  nat_iso.of_components (fun (s : cone F) => ext (iso.refl (cone.X (functor.obj (postcompose (\u03b1 \u226b \u03b2)) s))) sorry) sorry\n\n/-- Postcomposing by the identity does not change the cone up to isomorphism. -/\ndef postcompose_id {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} : postcompose \ud835\udfd9 \u2245 \ud835\udfed :=\n  nat_iso.of_components (fun (s : cone F) => ext (iso.refl (cone.X (functor.obj (postcompose \ud835\udfd9) s))) sorry) sorry\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncones.\n-/\n@[simp] theorem postcompose_equivalence_unit_iso {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} (\u03b1 : F \u2245 G) : equivalence.unit_iso (postcompose_equivalence \u03b1) =\n  nat_iso.of_components\n    (fun (s : cone F) => ext (iso.refl (cone.X (functor.obj \ud835\udfed s))) (postcompose_equivalence._proof_1 \u03b1 s))\n    (postcompose_equivalence._proof_2 \u03b1) :=\n  Eq.refl (equivalence.unit_iso (postcompose_equivalence \u03b1))\n\n/--\nWhiskering on the left by `E : K \u2964 J` gives a functor from `cone F` to `cone (E \u22d9 F)`.\n-/\n@[simp] theorem whiskering_obj {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (E : K \u2964 J) (c : cone F) : functor.obj (whiskering E) c = cone.whisker E c :=\n  Eq.refl (functor.obj (whiskering E) c)\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\n@[simp] theorem whiskering_equivalence_inverse {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (e : K \u224c J) : equivalence.inverse (whiskering_equivalence e) =\n  whiskering (equivalence.inverse e) \u22d9\n    postcompose\n      (iso.inv (functor.associator (equivalence.inverse e) (equivalence.functor e) F) \u226b\n        whisker_right (iso.hom (equivalence.counit_iso e)) F \u226b iso.hom (functor.left_unitor F)) :=\n  Eq.refl (equivalence.inverse (whiskering_equivalence e))\n\n/--\nThe categories of cones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\ndef equivalence_of_reindexing {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] {G : K \u2964 C} (e : K \u224c J) (\u03b1 : equivalence.functor e \u22d9 F \u2245 G) : cone F \u224c cone G :=\n  equivalence.trans (whiskering_equivalence e) (postcompose_equivalence \u03b1)\n\n/-- Forget the cone structure and obtain just the cone point. -/\ndef forget {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) : cone F \u2964 C :=\n  functor.mk (fun (t : cone F) => cone.X t) fun (s t : cone F) (f : s \u27f6 t) => cone_morphism.hom f\n\n/-- A functor `G : C \u2964 D` sends cones over `F` to cones over `F \u22d9 G` functorially. -/\n@[simp] theorem functoriality_obj_\u03c0_app {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) (A : cone F) (j : J) : nat_trans.app (cone.\u03c0 (functor.obj (functoriality F G) A)) j = functor.map G (nat_trans.app (cone.\u03c0 A) j) :=\n  Eq.refl (nat_trans.app (cone.\u03c0 (functor.obj (functoriality F G) A)) j)\n\nprotected instance functoriality_full {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) [full G] [faithful G] : full (functoriality F G) :=\n  full.mk\n    fun (X Y : cone F) (t : functor.obj (functoriality F G) X \u27f6 functor.obj (functoriality F G) Y) =>\n      cone_morphism.mk (functor.preimage G (cone_morphism.hom t))\n\nprotected instance functoriality_faithful {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) [faithful G] : faithful (functoriality F G) :=\n  faithful.mk\n\n/--\nIf `e : C \u224c D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cones over `F` and cones over `F \u22d9 e.functor`.\n-/\n@[simp] theorem functoriality_equivalence_counit_iso {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (e : C \u224c D) : equivalence.counit_iso (functoriality_equivalence F e) =\n  nat_iso.of_components\n    (fun (c : cone (F \u22d9 equivalence.functor e)) =>\n      ext (iso.app (equivalence.counit_iso e) (cone.X c)) (functoriality_equivalence._proof_3 F e c))\n    (functoriality_equivalence._proof_4 F e) :=\n  Eq.refl (equivalence.counit_iso (functoriality_equivalence F e))\n\n/--\nIf `F` reflects isomorphisms, then `cones.functoriality F` reflects isomorphisms\nas well.\n-/\nprotected instance reflects_cone_isomorphism {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] (F : C \u2964 D) [reflects_isomorphisms F] (K : J \u2964 C) : reflects_isomorphisms (functoriality K F) :=\n  reflects_isomorphisms.mk\n    fun (A B : cone K) (f : A \u27f6 B) (_inst_3_1 : is_iso (functor.map (functoriality K F) f)) => cone_iso_of_hom_iso f\n\nend cones\n\n\n/-- A cocone morphism between two cocones for the same diagram is a morphism of the cocone points\nwhich commutes with the cocone legs. -/\nstructure cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (A : cocone F) (B : cocone F) \nwhere\n  hom : cocone.X A \u27f6 cocone.X B\n  w' : autoParam (\u2200 (j : J), nat_trans.app (cocone.\u03b9 A) j \u226b hom = nat_trans.app (cocone.\u03b9 B) j)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\nprotected instance inhabited_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (A : cocone F) : Inhabited (cocone_morphism A A) :=\n  { default := cocone_morphism.mk \ud835\udfd9 }\n\n@[simp] theorem cocone_morphism.w {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {A : cocone F} {B : cocone F} (c : cocone_morphism A B) (j : J) : nat_trans.app (cocone.\u03b9 A) j \u226b cocone_morphism.hom c = nat_trans.app (cocone.\u03b9 B) j := sorry\n\n@[simp] theorem cocone_morphism.w_assoc {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {A : cocone F} {B : cocone F} (c : cocone_morphism A B) (j : J) {X' : C} (f' : cocone.X B \u27f6 X') : nat_trans.app (cocone.\u03b9 A) j \u226b cocone_morphism.hom c \u226b f' = nat_trans.app (cocone.\u03b9 B) j \u226b f' := sorry\n\n@[simp] theorem cocone.category_to_category_struct_id_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (B : cocone F) : cocone_morphism.hom \ud835\udfd9 = \ud835\udfd9 :=\n  Eq.refl (cocone_morphism.hom \ud835\udfd9)\n\nnamespace cocones\n\n\n/-- To give an isomorphism between cocones, it suffices to give an\n  isomorphism between their vertices which commutes with the cocone\n  maps. -/\n@[simp] theorem ext_inv_hom {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {c : cocone F} {c' : cocone F} (\u03c6 : cocone.X c \u2245 cocone.X c') (w : \u2200 (j : J), nat_trans.app (cocone.\u03b9 c) j \u226b iso.hom \u03c6 = nat_trans.app (cocone.\u03b9 c') j) : cocone_morphism.hom (iso.inv (ext \u03c6 w)) = iso.inv \u03c6 :=\n  Eq.refl (cocone_morphism.hom (iso.inv (ext \u03c6 w)))\n\n/--\nGiven a cocone morphism whose object part is an isomorphism, produce an\nisomorphism of cocones.\n-/\ndef cocone_iso_of_hom_iso {J : Type v} [small_category J] {C : Type u} [category C] {K : J \u2964 C} {c : cocone K} {d : cocone K} (f : c \u27f6 d) [i : is_iso (cocone_morphism.hom f)] : is_iso f :=\n  is_iso.mk (cocone_morphism.mk (inv (cocone_morphism.hom f)))\n\n/--\nFunctorially precompose a cocone for `F` by a natural transformation `G \u27f6 F` to give a cocone for `G`.\n-/\ndef precompose {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} (\u03b1 : G \u27f6 F) : cocone F \u2964 cocone G :=\n  functor.mk (fun (c : cocone F) => cocone.mk (cocone.X c) (\u03b1 \u226b cocone.\u03b9 c))\n    fun (c\u2081 c\u2082 : cocone F) (f : c\u2081 \u27f6 c\u2082) => cocone_morphism.mk (cocone_morphism.hom f)\n\n/-- Precomposing a cocone by the composite natural transformation `\u03b1 \u226b \u03b2` is the same as\nprecomposing by `\u03b2` and then by `\u03b1`. -/\ndef precompose_comp {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} {H : J \u2964 C} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) : precompose (\u03b1 \u226b \u03b2) \u2245 precompose \u03b2 \u22d9 precompose \u03b1 :=\n  nat_iso.of_components (fun (s : cocone H) => ext (iso.refl (cocone.X (functor.obj (precompose (\u03b1 \u226b \u03b2)) s))) sorry) sorry\n\n/-- Precomposing by the identity does not change the cocone up to isomorphism. -/\ndef precompose_id {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} : precompose \ud835\udfd9 \u2245 \ud835\udfed :=\n  nat_iso.of_components (fun (s : cocone F) => ext (iso.refl (cocone.X (functor.obj (precompose \ud835\udfd9) s))) sorry) sorry\n\n/--\nIf `F` and `G` are naturally isomorphic functors, then they have equivalent categories of\ncocones.\n-/\n@[simp] theorem precompose_equivalence_functor {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {G : J \u2964 C} (\u03b1 : G \u2245 F) : equivalence.functor (precompose_equivalence \u03b1) = precompose (iso.hom \u03b1) :=\n  Eq.refl (equivalence.functor (precompose_equivalence \u03b1))\n\n/--\nWhiskering on the left by `E : K \u2964 J` gives a functor from `cocone F` to `cocone (E \u22d9 F)`.\n-/\ndef whiskering {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (E : K \u2964 J) : cocone F \u2964 cocone (E \u22d9 F) :=\n  functor.mk (fun (c : cocone F) => cocone.whisker E c)\n    fun (c c' : cocone F) (f : c \u27f6 c') => cocone_morphism.mk (cocone_morphism.hom f)\n\n/--\nWhiskering by an equivalence gives an equivalence between categories of cones.\n-/\ndef whiskering_equivalence {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] (e : K \u224c J) : cocone F \u224c cocone (equivalence.functor e \u22d9 F) :=\n  equivalence.mk' (whiskering (equivalence.functor e))\n    (whiskering (equivalence.inverse e) \u22d9\n      precompose\n        (iso.inv (functor.left_unitor F) \u226b\n          whisker_right (iso.inv (equivalence.counit_iso e)) F \u226b\n            iso.inv (functor.associator (equivalence.inverse e) (equivalence.functor e) F)))\n    (nat_iso.of_components (fun (s : cocone F) => ext (iso.refl (cocone.X (functor.obj \ud835\udfed s))) sorry) sorry)\n    (nat_iso.of_components\n      (fun (s : cocone (equivalence.functor e \u22d9 F)) =>\n        ext\n          (iso.refl\n            (cocone.X\n              (functor.obj\n                ((whiskering (equivalence.inverse e) \u22d9\n                    precompose\n                      (iso.inv (functor.left_unitor F) \u226b\n                        whisker_right (iso.inv (equivalence.counit_iso e)) F \u226b\n                          iso.inv (functor.associator (equivalence.inverse e) (equivalence.functor e) F))) \u22d9\n                  whiskering (equivalence.functor e))\n                s)))\n          sorry)\n      sorry)\n\n/--\nThe categories of cocones over `F` and `G` are equivalent if `F` and `G` are naturally isomorphic\n(possibly after changing the indexing category by an equivalence).\n-/\n@[simp] theorem equivalence_of_reindexing_functor_obj {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {K : Type v} [small_category K] {G : K \u2964 C} (e : K \u224c J) (\u03b1 : equivalence.functor e \u22d9 F \u2245 G) (X : cocone F) : functor.obj (equivalence.functor (equivalence_of_reindexing e \u03b1)) X =\n  functor.obj (precompose (iso.inv \u03b1)) (cocone.whisker (equivalence.functor e) X) :=\n  Eq.refl (functor.obj (precompose (iso.inv \u03b1)) (cocone.whisker (equivalence.functor e) X))\n\n/-- Forget the cocone structure and obtain just the cocone point. -/\n@[simp] theorem forget_map {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) (s : cocone F) (t : cocone F) (f : s \u27f6 t) : functor.map (forget F) f = cocone_morphism.hom f :=\n  Eq.refl (functor.map (forget F) f)\n\n/-- A functor `G : C \u2964 D` sends cocones over `F` to cocones over `F \u22d9 G` functorially. -/\n@[simp] theorem functoriality_map_hom {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) (_x : cocone F) : \u2200 (_x_1 : cocone F) (f : _x \u27f6 _x_1),\n  cocone_morphism.hom (functor.map (functoriality F G) f) = functor.map G (cocone_morphism.hom f) :=\n  fun (_x_1 : cocone F) (f : _x \u27f6 _x_1) => Eq.refl (cocone_morphism.hom (functor.map (functoriality F G) f))\n\nprotected instance functoriality_full {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) [full G] [faithful G] : full (functoriality F G) :=\n  full.mk\n    fun (X Y : cocone F) (t : functor.obj (functoriality F G) X \u27f6 functor.obj (functoriality F G) Y) =>\n      cocone_morphism.mk (functor.preimage G (cocone_morphism.hom t))\n\nprotected instance functoriality_faithful {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (G : C \u2964 D) [faithful G] : faithful (functoriality F G) :=\n  faithful.mk\n\n/--\nIf `e : C \u224c D` is an equivalence of categories, then `functoriality F e.functor` induces an\nequivalence between cocones over `F` and cocones over `F \u22d9 e.functor`.\n-/\n@[simp] theorem functoriality_equivalence_functor {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) {D : Type u'} [category D] (e : C \u224c D) : equivalence.functor (functoriality_equivalence F e) = functoriality F (equivalence.functor e) :=\n  Eq.refl (equivalence.functor (functoriality_equivalence F e))\n\n/--\nIf `F` reflects isomorphisms, then `cocones.functoriality F` reflects isomorphisms\nas well.\n-/\nprotected instance reflects_cocone_isomorphism {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] (F : C \u2964 D) [reflects_isomorphisms F] (K : J \u2964 C) : reflects_isomorphisms (functoriality K F) :=\n  reflects_isomorphisms.mk\n    fun (A B : cocone K) (f : A \u27f6 B) (_inst_3_1 : is_iso (functor.map (functoriality K F) f)) => cocone_iso_of_hom_iso f\n\nend cocones\n\n\nend limits\n\n\nnamespace functor\n\n\n/-- The image of a cone in C under a functor G : C \u2964 D is a cone in D. -/\n/-- The image of a cocone in C under a functor G : C \u2964 D is a cocone in D. -/\n@[simp] theorem map_cone_X {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) (c : limits.cone F) : limits.cone.X (map_cone H c) = obj H (limits.cone.X c) :=\n  Eq.refl (obj H (limits.cone.X c))\n\n@[simp] theorem map_cocone_\u03b9_app {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) (c : limits.cocone F) (j : J) : nat_trans.app (limits.cocone.\u03b9 (map_cocone H c)) j = map H (nat_trans.app (limits.cocone.\u03b9 c) j) :=\n  Eq.refl (map H (nat_trans.app (limits.cocone.\u03b9 c) j))\n\n/-- Given a cone morphism `c \u27f6 c'`, construct a cone morphism on the mapped cones functorially.  -/\ndef map_cone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) {c : limits.cone F} {c' : limits.cone F} (f : c \u27f6 c') : map_cone H c \u27f6 map_cone H c' :=\n  map (limits.cones.functoriality F H) f\n\n/-- Given a cocone morphism `c \u27f6 c'`, construct a cocone morphism on the mapped cocones functorially.  -/\ndef map_cocone_morphism {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) {c : limits.cocone F} {c' : limits.cocone F} (f : c \u27f6 c') : map_cocone H c \u27f6 map_cocone H c' :=\n  map (limits.cocones.functoriality F H) f\n\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F \u22d9 H`.-/\ndef map_cone_inv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) [is_equivalence H] (c : limits.cone (F \u22d9 H)) : limits.cone F :=\n  obj (equivalence.inverse (limits.cones.functoriality_equivalence F (as_equivalence H))) c\n\n/-- `map_cone` is the left inverse to `map_cone_inv`. -/\ndef map_cone_map_cone_inv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 D} (H : D \u2964 C) [is_equivalence H] (c : limits.cone (F \u22d9 H)) : map_cone H (map_cone_inv H c) \u2245 c :=\n  iso.app (equivalence.counit_iso (limits.cones.functoriality_equivalence F (as_equivalence H))) c\n\n/-- `map_cone` is the right inverse to `map_cone_inv`. -/\ndef map_cone_inv_map_cone {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 D} (H : D \u2964 C) [is_equivalence H] (c : limits.cone F) : map_cone_inv H (map_cone H c) \u2245 c :=\n  iso.app (iso.symm (equivalence.unit_iso (limits.cones.functoriality_equivalence F (as_equivalence H)))) c\n\n/-- If `H` is an equivalence, we invert `H.map_cone` and get a cone for `F` from a cone\nfor `F \u22d9 H`.-/\ndef map_cocone_inv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) [is_equivalence H] (c : limits.cocone (F \u22d9 H)) : limits.cocone F :=\n  obj (equivalence.inverse (limits.cocones.functoriality_equivalence F (as_equivalence H))) c\n\n/-- `map_cocone` is the left inverse to `map_cocone_inv`. -/\ndef map_cocone_map_cocone_inv {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 D} (H : D \u2964 C) [is_equivalence H] (c : limits.cocone (F \u22d9 H)) : map_cocone H (map_cocone_inv H c) \u2245 c :=\n  iso.app (equivalence.counit_iso (limits.cocones.functoriality_equivalence F (as_equivalence H))) c\n\n/-- `map_cocone` is the right inverse to `map_cocone_inv`. -/\ndef map_cocone_inv_map_cocone {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 D} (H : D \u2964 C) [is_equivalence H] (c : limits.cocone F) : map_cocone_inv H (map_cocone H c) \u2245 c :=\n  iso.app (iso.symm (equivalence.unit_iso (limits.cocones.functoriality_equivalence F (as_equivalence H)))) c\n\n/-- `functoriality F _ \u22d9 postcompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simp] theorem functoriality_comp_postcompose_inv_app_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {H : C \u2964 D} {H' : C \u2964 D} (\u03b1 : H \u2245 H') (X : limits.cone F) : limits.cone_morphism.hom (nat_trans.app (iso.inv (functoriality_comp_postcompose \u03b1)) X) =\n  nat_trans.app (iso.inv \u03b1) (limits.cone.X X) :=\n  Eq.refl (nat_trans.app (iso.inv \u03b1) (limits.cone.X X))\n\n/--\nFor `F : J \u2964 C`, given a cone `c : cone F`, and a natural isomorphism `\u03b1 : H \u2245 H'` for functors\n`H H' : C \u2964 D`, the postcomposition of the cone `H.map_cone` using the isomorphism `\u03b1` is\nisomorphic to the cone `H'.map_cone`.\n-/\ndef postcompose_whisker_left_map_cone {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {H : C \u2964 D} {H' : C \u2964 D} (\u03b1 : H \u2245 H') (c : limits.cone F) : obj (limits.cones.postcompose (whisker_left F (iso.hom \u03b1))) (map_cone H c) \u2245 map_cone H' c :=\n  iso.app (functoriality_comp_postcompose \u03b1) c\n\n/--\n`map_cone` commutes with `postcompose`. In particular, for `F : J \u2964 C`, given a cone `c : cone F`, a\nnatural transformation `\u03b1 : F \u27f6 G` and a functor `H : C \u2964 D`, we have two obvious ways of producing\na cone over `G \u22d9 H`, and they are both isomorphic.\n-/\n@[simp] theorem map_cone_postcompose_inv_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {G : J \u2964 C} (H : C \u2964 D) {\u03b1 : F \u27f6 G} {c : limits.cone F} : limits.cone_morphism.hom (iso.inv (map_cone_postcompose H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/--\n`map_cone` commutes with `postcompose_equivalence`\n-/\n@[simp] theorem map_cone_postcompose_equivalence_functor_hom_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {G : J \u2964 C} (H : C \u2964 D) {\u03b1 : F \u2245 G} {c : limits.cone F} : limits.cone_morphism.hom (iso.hom (map_cone_postcompose_equivalence_functor H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/-- `functoriality F _ \u22d9 precompose (whisker_left F _)` simplifies to `functoriality F _`. -/\n@[simp] theorem functoriality_comp_precompose_inv_app_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {H : C \u2964 D} {H' : C \u2964 D} (\u03b1 : H \u2245 H') (X : limits.cocone F) : limits.cocone_morphism.hom (nat_trans.app (iso.inv (functoriality_comp_precompose \u03b1)) X) =\n  nat_trans.app (iso.inv \u03b1) (limits.cocone.X X) :=\n  Eq.refl (nat_trans.app (iso.inv \u03b1) (limits.cocone.X X))\n\n/--\nFor `F : J \u2964 C`, given a cocone `c : cocone F`, and a natural isomorphism `\u03b1 : H \u2245 H'` for functors\n`H H' : C \u2964 D`, the precomposition of the cocone `H.map_cocone` using the isomorphism `\u03b1` is\nisomorphic to the cocone `H'.map_cocone`.\n-/\ndef precompose_whisker_left_map_cocone {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {H : C \u2964 D} {H' : C \u2964 D} (\u03b1 : H \u2245 H') (c : limits.cocone F) : obj (limits.cocones.precompose (whisker_left F (iso.inv \u03b1))) (map_cocone H c) \u2245 map_cocone H' c :=\n  iso.app (functoriality_comp_precompose \u03b1) c\n\n/--\n`map_cocone` commutes with `precompose`. In particular, for `F : J \u2964 C`, given a cocone\n`c : cocone F`, a natural transformation `\u03b1 : F \u27f6 G` and a functor `H : C \u2964 D`, we have two obvious\nways of producing a cocone over `G \u22d9 H`, and they are both isomorphic.\n-/\n@[simp] theorem map_cocone_precompose_hom_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {G : J \u2964 C} (H : C \u2964 D) {\u03b1 : F \u27f6 G} {c : limits.cocone G} : limits.cocone_morphism.hom (iso.hom (map_cocone_precompose H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/--\n`map_cocone` commutes with `precompose_equivalence`\n-/\n@[simp] theorem map_cocone_precompose_equivalence_functor_inv_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} {G : J \u2964 C} (H : C \u2964 D) {\u03b1 : F \u2245 G} {c : limits.cocone G} : limits.cocone_morphism.hom (iso.inv (map_cocone_precompose_equivalence_functor H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/--\n`map_cone` commutes with `whisker`\n-/\n@[simp] theorem map_cone_whisker_inv_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) {K : Type v} [small_category K] {E : K \u2964 J} {c : limits.cone F} : limits.cone_morphism.hom (iso.inv (map_cone_whisker H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/--\n`map_cocone` commutes with `whisker`\n-/\n@[simp] theorem map_cocone_whisker_inv_hom {J : Type v} [small_category J] {C : Type u} [category C] {D : Type u'} [category D] {F : J \u2964 C} (H : C \u2964 D) {K : Type v} [small_category K] {E : K \u2964 J} {c : limits.cocone F} : limits.cocone_morphism.hom (iso.inv (map_cocone_whisker H)) = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\nend functor\n\n\nend category_theory\n\n\nnamespace category_theory.limits\n\n\n/-- Change a `cocone F` into a `cone F.op`. -/\n@[simp] theorem cocone.op_X {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone F) : cone.X (cocone.op c) = opposite.op (cocone.X c) :=\n  Eq.refl (cone.X (cocone.op c))\n\n/-- Change a `cone F` into a `cocone F.op`. -/\ndef cone.op {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone F) : cocone (functor.op F) :=\n  cocone.mk (opposite.op (cone.X c))\n    (nat_trans.mk fun (j : J\u1d52\u1d56) => has_hom.hom.op (nat_trans.app (cone.\u03c0 c) (opposite.unop j)))\n\n/-- Change a `cocone F.op` into a `cone F`. -/\ndef cocone.unop {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cocone (functor.op F)) : cone F :=\n  cone.mk (opposite.unop (cocone.X c))\n    (nat_trans.mk fun (j : J) => has_hom.hom.unop (nat_trans.app (cocone.\u03b9 c) (opposite.op j)))\n\n/-- Change a `cone F.op` into a `cocone F`. -/\n@[simp] theorem cone.unop_X {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} (c : cone (functor.op F)) : cocone.X (cone.unop c) = opposite.unop (cone.X c) :=\n  Eq.refl (cocone.X (cone.unop c))\n\n/--\nThe category of cocones on `F`\nis equivalent to the opposite category of\nthe category of cones on the opposite of `F`.\n-/\n@[simp] theorem cocone_equivalence_op_cone_op_unit_iso {J : Type v} [small_category J] {C : Type u} [category C] (F : J \u2964 C) : equivalence.unit_iso (cocone_equivalence_op_cone_op F) =\n  nat_iso.of_components\n    (fun (c : cocone F) =>\n      cocones.ext (iso.refl (cocone.X (functor.obj \ud835\udfed c))) (cocone_equivalence_op_cone_op._proof_7 F c))\n    (cocone_equivalence_op_cone_op._proof_8 F) :=\n  Eq.refl (equivalence.unit_iso (cocone_equivalence_op_cone_op F))\n\n/-- Change a cocone on `F.left_op : J\u1d52\u1d56 \u2964 C` to a cocone on `F : J \u2964 C\u1d52\u1d56`. -/\n-- Here and below we only automatically generate the `@[simp]` lemma for the `X` field,\n\n-- as we can write a simpler `rfl` lemma for the components of the natural transformation by hand.\n\ndef cone_of_cocone_left_op {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 (C\u1d52\u1d56)} (c : cocone (functor.left_op F)) : cone F :=\n  cone.mk (opposite.op (cocone.X c))\n    (nat_trans.remove_left_op (cocone.\u03b9 c \u226b iso.hom (functor.const.op_obj_unop (opposite.op (cocone.X c)))))\n\n/-- Change a cone on `F : J \u2964 C\u1d52\u1d56` to a cocone on `F.left_op : J\u1d52\u1d56 \u2964 C`. -/\ndef cocone_left_op_of_cone {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 (C\u1d52\u1d56)} (c : cone F) : cocone (functor.left_op F) :=\n  cocone.mk (opposite.unop (cone.X c)) (nat_trans.left_op (cone.\u03c0 c))\n\n/-- Change a cone on `F.left_op : J\u1d52\u1d56 \u2964 C` to a cocone on `F : J \u2964 C\u1d52\u1d56`. -/\n/- When trying use `@[simps]` to generate the `\u03b9_app` field of this definition, `@[simps]` tries to\n  reduce the RHS using `expr.dsimp` and `expr.simp`, but for some reason the expression is not\n  being simplified properly. -/\n\ndef cocone_of_cone_left_op {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 (C\u1d52\u1d56)} (c : cone (functor.left_op F)) : cocone F :=\n  cocone.mk (opposite.op (cone.X c))\n    (nat_trans.remove_left_op (iso.hom (functor.const.op_obj_unop (opposite.op (cone.X c))) \u226b cone.\u03c0 c))\n\n@[simp] theorem cocone_of_cone_left_op_\u03b9_app {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 (C\u1d52\u1d56)} (c : cone (functor.left_op F)) (j : J) : nat_trans.app (cocone.\u03b9 (cocone_of_cone_left_op c)) j = has_hom.hom.op (nat_trans.app (cone.\u03c0 c) (opposite.op j)) := sorry\n\n/-- Change a cocone on `F : J \u2964 C\u1d52\u1d56` to a cone on `F.left_op : J\u1d52\u1d56 \u2964 C`. -/\ndef cone_left_op_of_cocone {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 (C\u1d52\u1d56)} (c : cocone F) : cone (functor.left_op F) :=\n  cone.mk (opposite.unop (cocone.X c)) (nat_trans.left_op (cocone.\u03b9 c))\n\nend category_theory.limits\n\n\nnamespace category_theory.functor\n\n\n/-- The opposite cocone of the image of a cone is the image of the opposite cocone. -/\ndef map_cone_op {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {D : Type u'} [category D] (G : C \u2964 D) (t : limits.cone F) : limits.cone.op (map_cone G t) \u2245 map_cocone (functor.op G) (limits.cone.op t) :=\n  limits.cocones.ext (iso.refl (limits.cocone.X (limits.cone.op (map_cone G t)))) sorry\n\n/-- The opposite cone of the image of a cocone is the image of the opposite cone. -/\ndef map_cocone_op {J : Type v} [small_category J] {C : Type u} [category C] {F : J \u2964 C} {D : Type u'} [category D] (G : C \u2964 D) {t : limits.cocone F} : limits.cocone.op (map_cocone G t) \u2245 map_cone (functor.op G) (limits.cocone.op t) :=\n  limits.cones.ext (iso.refl (limits.cone.X (limits.cocone.op (map_cocone G t)))) sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/cones.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984443, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.38233845906564673}}
{"text": "constants p q : Prop\n\ntheorem t1 (hp : p) (hq : q) : p := hp\n\naxiom hp : p\n\ntheorem t2 : q \u2192 p := t1 hp\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch3/ex0207.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.38230906993673663}}
{"text": "import .basic\n\nopen parser_tactic\n\nuniverse u\n\nstructure Language : Type (u+1) :=\n(functions : \u2115 \u2192 Type u) (relations : \u2115 \u2192 Type u)\n\ndef Language.constants (L : Language) := L.functions 0\n\nvariable (L : Language.{u})\n\ninductive preterm : \u2115 \u2192 Type u\n| var {} : \u2200 (k : \u2115), preterm 0\n| func : \u2200 {l : \u2115} (f : L.functions l), preterm l\n| app : \u2200 {l : \u2115} (t : preterm (l + 1)) (s : preterm 0), preterm l\nexport preterm\n\n@[reducible] def term := preterm L 0\n\nvariable {L}\nprefix `&`:max := preterm.var\n\n-- @[simp] def apps : \u2200{l}, preterm L l \u2192 dvector (term L) l \u2192 term L\n-- | _ t []       := t\n-- | _ t (t'::ts) := apps (app t t') ts\n\n-- -- @[simp] def apps' : \u2200{l l'}, preterm L (l'+l) \u2192 dvector (term L) l \u2192 preterm L l'\n-- -- | _ _ t []       := t\n-- -- | _ _ t (t'::ts) := apps' (app t t') ts\n\n-- -- @[simp] def rev_apps : \u2200{l l'}, preterm L (l+l) \u2192 dvector (term L) l' \u2192 preterm L l\n-- -- | _ _ t []       := sorry\n-- -- | l _ t (@dvector.cons _ l' t' ts) := app (@rev_apps (l+1) l' t ts) t'\n\n-- @[simp] lemma apps_zero (t : term L) (ts : dvector (term L) 0) : apps t ts = t :=\n-- by cases ts; refl\n\n-- lemma apps_eq_app {l} (t : preterm L (l+1)) (s : term L) (ts : dvector (term L) l) :\n--   \u2203t' s', apps t (s::ts) = app t' s' :=\n-- begin\n--   induction ts generalizing s, exact \u27e8t, s, rfl\u27e9, exact ts_ih (app t s) ts_x\n-- end\n\n-- namespace preterm\n-- @[simp] def change_arity' : \u2200{l l'} (h : l = l') (t : preterm L l), preterm L l'\n-- | _ _ h &k          := by induction h; exact &k\n-- | _ _ h (func f)    := func (by induction h; exact f)\n-- | _ _ h (app t\u2081 t\u2082) := app (change_arity' (congr_arg succ h) t\u2081) t\u2082\n\n-- @[simp] lemma change_arity'_rfl : \u2200{l} (t : preterm L l), change_arity' rfl t = t\n-- | _ &k          := by refl\n-- | _ (func f)    := by refl\n-- | _ (app t\u2081 t\u2082) := by dsimp; simp*\n\n-- end preterm\n\n-- -- lemma apps'_concat {l l'} (t : preterm L (l'+(l+1))) (s : term L) (ts : dvector (term L) l) :\n-- --   apps' t (ts.concat s) = app (apps' (t.change_arity' (by simp)) ts) s :=\n-- -- begin\n-- --   induction ts generalizing s,\n-- --   { simp },\n-- --   { apply ts_ih (app t ts_x) s }\n-- -- end\n\n-- lemma apps_ne_var {l} {f : L.functions l} {ts : dvector (term L) l} {k : \u2115} :\n--   apps (func f) ts \u2260 &k :=\n-- begin\n--   intro h, cases ts, injection h,\n--   rcases apps_eq_app (func f) ts_x ts_xs with \u27e8t, s, h'\u27e9, cases h.symm.trans h'\n-- end\n\n-- lemma apps_inj' {l} {t t' : preterm L l} {ts ts' : dvector (term L) l}\n--   (h : apps t ts = apps t' ts') : t = t' \u2227 ts = ts' :=\n-- begin\n--   induction ts; cases ts',\n--   { exact \u27e8h, rfl\u27e9 },\n--   { rcases ts_ih h with \u27e8\u27e8rfl, rfl\u27e9, rfl\u27e9, exact \u27e8rfl, rfl\u27e9 }\n-- end\n\n-- -- lemma apps_inj_length {l l'} {f : L.functions l} {f' : L.functions l'}\n-- --   {ts : dvector (term L) l} {ts' : dvector (term L) l'}\n-- --   (h : apps (func f) ts = apps (func f') ts') : l = l' :=\n-- -- begin\n-- --   sorry\n-- -- end\n\n-- -- lemma apps'_inj_length {l\u2081 l\u2082 l'} {f : L.functions (l' + l\u2081)} {f' : L.functions (l' + l\u2082)}\n-- --   {ts : dvector (term L) l\u2081} {ts' : dvector (term L) l\u2082}\n-- --   (h : apps' (func f) ts = apps' (func f') ts') : l\u2081 = l\u2082 :=\n-- -- begin\n-- --   sorry\n-- --   -- induction ts generalizing l'; cases ts',\n-- --   -- { refl },\n-- --   -- { rcases apps'_eq_app (func f') ts'_x ts'_xs with \u27e8t, s, h'\u27e9, cases h.trans h' },\n-- --   -- { rcases apps'_eq_app (func f) ts_x ts_xs with \u27e8t, s, h'\u27e9, cases h.symm.trans h' },\n-- --   -- { rcases apps'_eq_app (func f) ts_x ts_xs with \u27e8t\u2081, s\u2081, h\u2081\u27e9,\n-- --   --   rcases apps'_eq_app (func f') ts'_x ts'_xs with \u27e8t\u2082, s\u2082, h\u2082\u27e9,\n-- --   --    }\n-- -- end\n\n-- lemma apps_inj {l} {f f' : L.functions l} {ts ts' : dvector (term L) l}\n--   (h : apps (func f) ts = apps (func f') ts') : f = f' \u2227 ts = ts' :=\n-- by rcases apps_inj' h with \u27e8h', rfl\u27e9; cases h'; exact \u27e8rfl, rfl\u27e9\n\n-- def term_of_function {l} (f : L.functions l) : arity' (term L) (term L) l :=\n-- arity'.of_dvector_map $ apps (func f)\n\n-- @[elab_as_eliminator] def term.rec {C : term L \u2192 Sort v}\n--   (hvar : \u2200(k : \u2115), C &k)\n--   (hfunc : \u03a0 {l} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : \u2200t, ts.pmem t \u2192 C t),\n--     C (apps (func f) ts)) : \u2200(t : term L), C t :=\n-- have h : \u2200{l} (t : preterm L l) (ts : dvector (term L) l) (ih_ts : \u2200s, ts.pmem s \u2192 C s),\n--   C (apps t ts),\n-- begin\n--   intros, induction t; try {rw ts.zero_eq},\n--   { apply hvar },\n--   { apply hfunc t_f ts ih_ts },\n--   { apply t_ih_t (t_s::ts), intros t ht,\n--     cases ht,\n--     { induction ht, apply t_ih_s ([]), intros s hs, cases hs },\n--     { exact ih_ts t ht }},\n-- end,\n-- \u03bbt, h t ([]) (by intros s hs; cases hs)\n\n-- @[elab_as_eliminator] def term.elim' {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C) :\n--   \u2200{l} (t : preterm L l) (ts : dvector (term L) l) (ih_ts : dvector C l), C\n-- | _ &k ts ih_ts        := hvar k\n-- | _ (func f) ts ih_ts  := hfunc f ts ih_ts\n-- | _ (app t s) ts ih_ts := term.elim' t (s::ts) (term.elim' s ([]) ([])::ih_ts)\n\n-- @[elab_as_eliminator] def term.elim {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C) :\n--   \u2200(t : term L), C :=\n-- \u03bbt, term.elim' hvar hfunc t ([]) ([])\n\n-- lemma term.elim'_apps {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C)\n--   {l} (t : preterm L l) (ts : dvector (term L) l) :\n--   @term.elim' L C hvar hfunc 0 (apps t ts) ([]) ([]) = @term.elim' L C hvar hfunc l t ts\n--   (ts.map $ term.elim hvar hfunc) :=\n-- begin\n--   induction ts,\n--   { refl },\n--   { dsimp only [dvector.map, apps], rw [ts_ih], refl }\n-- end\n\n-- lemma term.elim_apps {C : Type v}\n--   (hvar : \u2200(k : \u2115), C)\n--   (hfunc : \u03a0 {{l}} (f : L.functions l) (ts : dvector (term L) l) (ih_ts : dvector C l), C)\n--   {l} (f : L.functions l) (ts : dvector (term L) l) :\n--   @term.elim L C hvar hfunc (apps (func f) ts) = hfunc f ts (ts.map $ @term.elim L C hvar hfunc) :=\n-- by dsimp only [term.elim]; rw term.elim'_apps; refl\n\n-- /- lift_term_at _ t n m raises variables in t which are at least m by n -/\n-- @[simp] def lift_term_at : \u2200 {l}, preterm L l \u2192 \u2115 \u2192 \u2115 \u2192 preterm L l\n-- | _ &k          n m := &(if m \u2264 k then k+n else k)\n-- | _ (func f)    n m := func f\n-- | _ (app t\u2081 t\u2082) n m := app (lift_term_at t\u2081 n m) (lift_term_at t\u2082 n m)\n\n-- notation t ` \u2191' `:90 n ` # `:90 m:90 := fol.lift_term_at t n m -- input \u2191 with \\u or \\upa\n\n-- -- @[simp] lemma lift_term_var_le {k n m} (h : m \u2264 k) : &k \u2191' n # m = (&(k+n) : term L) := dif_pos h\n-- -- @[simp] lemma lift_term_var_gt {k n m} (h : \u00ac(m \u2264 k)) : &k \u2191' n # m = (&k : term L) := dif_neg h\n-- -- @[simp] lemma lift_term_at_func {l} (f : L.functions l) (n m) : func f \u2191' n # m = func f := by refl\n-- -- @[simp] lemma lift_term_at_app {l} (t : preterm L (l+1)) (s : preterm L 0) (n m) :\n-- --   app t s \u2191' n # m = app (t \u2191' n # m) (s \u2191' n # m) := by refl\n\n-- @[reducible] def lift_term {l} (t : preterm L l) (n : \u2115) : preterm L l := t \u2191' n # 0\n-- infix ` \u2191 `:100 := fol.lift_term -- input \u2191' with \\u or \\upa\n-- @[reducible, simp] def lift_term1 {l} (t : preterm L l) : preterm L l := t \u2191 1\n\n-- @[simp] lemma lift_term_def {l} (t : preterm L l) (n : \u2115) : t \u2191' n # 0 = t \u2191 n := by refl\n\n-- lemma injective_lift_term_at : \u2200 {l} {n m : \u2115},\n--   function.injective (\u03bb(t : preterm L l), lift_term_at t n m)\n-- | _ n m &k &k' h :=\n--   by by_cases h\u2081 : m \u2264 k; by_cases h\u2082 : m \u2264 k'; simp [h\u2081, h\u2082] at h;\n--      congr;[assumption, skip, skip, assumption]; exfalso; try {apply h\u2081};\n--      try {apply h\u2082}; subst h; apply le_trans (by assumption) (le_add_left _ _)\n-- | _ n m &k (func f')            h := by cases h\n-- | _ n m &k (app t\u2081' t\u2082')        h := by cases h\n-- | _ n m (func f) &k'            h := by cases h\n-- | _ n m (func f) (func f')      h := h\n-- | _ n m (func f) (app t\u2081' t\u2082')  h := by cases h\n-- | _ n m (app t\u2081 t\u2082) &k'         h := by cases h\n-- | _ n m (app t\u2081 t\u2082) (func f')   h := by cases h\n-- | _ n m (app t\u2081 t\u2082) (app t\u2081' t\u2082') h :=\n--   begin injection h, congr; apply injective_lift_term_at; assumption end\n\n-- @[simp] lemma lift_term_at_zero : \u2200 {l} (t : preterm L l) (m : \u2115), t \u2191' 0 # m = t\n-- | _ &k          m := by simp [lift_term_at]\n-- | _ (func f)    m := by refl\n-- | _ (app t\u2081 t\u2082) m := by dsimp; congr; apply lift_term_at_zero\n\n-- @[simp] lemma lift_term_zero {l} (t : preterm L l) : t \u2191 0 = t := lift_term_at_zero t 0\n\n-- /- the following lemmas simplify iterated lifts, depending on the size of m' -/\n-- lemma lift_term_at2_small : \u2200 {l} (t : preterm L l) (n n') {m m'}, m' \u2264 m \u2192\n--   (t \u2191' n # m) \u2191' n' # m' = (t \u2191' n' # m') \u2191' n # (m + n')\n-- | _ &k          n n' m m' H :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2081 : m' \u2264 k := le_trans H h,\n--       have h\u2082 : m' \u2264 k + n, from le_trans h\u2081 (k.le_add_right n),\n--       simp [*, -add_assoc, -add_comm], simp },\n--     { have h\u2081 : \u00acm + n' \u2264 k + n', from \u03bb h', h (le_of_add_le_add_right h'),\n--       have h\u2082 : \u00acm + n' \u2264 k, from \u03bb h', h\u2081 (le_trans h' (k.le_add_right n')),\n--       by_cases h' : m' \u2264 k; simp [*, -add_comm, -add_assoc] }\n--   end\n-- | _ (func f)    n n' m m' H := by refl\n-- | _ (app t\u2081 t\u2082) n n' m m' H :=\n--   begin dsimp; congr1; apply lift_term_at2_small; assumption end\n\n-- lemma lift_term_at2_medium : \u2200 {l} (t : preterm L l) {n} (n') {m m'}, m \u2264 m' \u2192 m' \u2264 m+n \u2192\n--   (t \u2191' n # m) \u2191' n' # m' = t \u2191' (n+n') # m\n-- | _ &k          n n' m m' H\u2081 H\u2082 :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2081 : m' \u2264 k + n, from le_trans H\u2082 (add_le_add_right h n), simp [*, -add_comm], },\n--     { have h\u2081 : \u00acm' \u2264 k, from \u03bb h', h (le_trans H\u2081 h'), simp [*, -add_comm, -add_assoc] }\n--   end\n-- | _ (func f)    n n' m m' H\u2081 H\u2082 := by refl\n-- | _ (app t\u2081 t\u2082) n n' m m' H\u2081 H\u2082 :=\n--   begin dsimp; congr1; apply lift_term_at2_medium; assumption end\n\n-- lemma lift_term2_medium {l} (t : preterm L l) {n} (n') {m'} (h : m' \u2264 n) :\n--   (t \u2191 n) \u2191' n' # m' = t \u2191 (n+n') :=\n-- lift_term_at2_medium t n' m'.zero_le (by simp*)\n\n-- lemma lift_term2 {l} (t : preterm L l) (n n') : (t \u2191 n) \u2191 n' = t \u2191 (n+n') :=\n-- lift_term2_medium t n' n.zero_le\n\n-- lemma lift_term_at2_eq {l} (t : preterm L l) (n n' m : \u2115) :\n--   (t \u2191' n # m) \u2191' n' # (m+n) = t \u2191' (n+n') # m :=\n-- lift_term_at2_medium t n' (m.le_add_right n) (le_refl _)\n\n-- lemma lift_term_at2_large {l} (t : preterm L l) {n} (n') {m m'} (H : m + n \u2264 m') :\n--   (t \u2191' n # m) \u2191' n' # m' = (t \u2191' n' # (m'-n)) \u2191' n # m :=\n-- have H\u2081 : n \u2264 m', from le_trans (n.le_add_left m) H,\n-- have H\u2082 : m \u2264 m' - n, from nat.le_sub_right_of_add_le H,\n-- begin rw fol.lift_term_at2_small t n' n H\u2082, rw [nat.sub_add_cancel], exact H\u2081 end\n\n-- @[simp] lemma lift_term_var0 (n : \u2115) : &0 \u2191 n = (&n : term L) :=\n-- by have h : 0 \u2264 0 := le_refl 0; rw [\u2190lift_term_def]; simp [h, -lift_term_def]\n\n-- @[simp] lemma lift_term_at_apps {l} (t : preterm L l) (ts : dvector (term L) l) (n m : \u2115) :\n--   (apps t ts) \u2191' n # m = apps (t \u2191' n # m) (ts.map $ \u03bbx, x \u2191' n # m) :=\n-- by induction ts generalizing t;[refl, apply ts_ih (app t ts_x)]\n\n-- @[simp] lemma lift_term_apps {l} (t : preterm L l) (ts : dvector (term L) l) (n : \u2115) :\n--   (apps t ts) \u2191 n = apps (t \u2191 n) (ts.map $ \u03bbx, x \u2191 n) :=\n-- lift_term_at_apps t ts n 0\n\n-- /- subst_term t s n substitutes s for (&n) and reduces the level of all variables above n by 1 -/\n-- def subst_term : \u2200 {l}, preterm L l \u2192 term L \u2192 \u2115 \u2192 preterm L l\n-- | _ &k          s n := subst_realize var (s \u2191 n) n k\n-- | _ (func f)    s n := func f\n-- | _ (app t\u2081 t\u2082) s n := app (subst_term t\u2081 s n) (subst_term t\u2082 s n)\n\n-- notation t `[`:max s ` // `:95 n `]`:0 := fol.subst_term t s n\n\n-- @[simp] lemma subst_term_var_lt (s : term L) {k n : \u2115} (H : k < n) : &k[s // n] = &k :=\n-- by simp only [H, fol.subst_term, fol.subst_realize_lt, eq_self_iff_true]\n\n-- @[simp] lemma subst_term_var_gt (s : term L) {k n : \u2115} (H : n < k) : &k[s // n] = &(k-1) :=\n-- by simp only [H, fol.subst_term, fol.subst_realize_gt, eq_self_iff_true]\n\n-- @[simp] lemma subst_term_var_eq (s : term L) (n : \u2115) : &n[s // n] = s \u2191' n # 0 :=\n-- by simp [subst_term]\n\n-- lemma subst_term_var0 (s : term L) : &0[s // 0] = s := by simp\n\n-- @[simp] lemma subst_term_func {l} (f : L.functions l) (s : term L) (n : \u2115) :\n--   (func f)[s // n] = func f :=\n-- by refl\n\n-- @[simp] lemma subst_term_app {l} (t\u2081 : preterm L (l+1)) (t\u2082 s : term L) (n : \u2115) :\n--   (app t\u2081 t\u2082)[s // n] = app (t\u2081[s // n]) (t\u2082[s // n]) :=\n-- by refl\n\n-- @[simp] lemma subst_term_apps {l} (t : preterm L l) (ts : dvector (term L) l) (s : term L)\n--   (n : \u2115) : (apps t ts)[s // n] = apps (t[s // n]) (ts.map $ \u03bbx, x[s // n]) :=\n-- by induction ts generalizing t;[refl, apply ts_ih (app t ts_x)]\n\n-- /- the following lemmas simplify first lifting and then substituting, depending on the size\n--   of the substituted variable -/\n-- lemma lift_at_subst_term_large : \u2200{l} (t : preterm L l) (s : term L) {n\u2081} (n\u2082) {m}, m \u2264 n\u2081 \u2192\n--  (t \u2191' n\u2082 # m)[s // n\u2081+n\u2082] = (t [s // n\u2081]) \u2191' n\u2082 # m\n-- | _ &k          s n\u2081 n\u2082 m h :=\n--   begin\n--     apply decidable.lt_by_cases k n\u2081; intro h\u2082,\n--     { have : k < n\u2081 + n\u2082, from lt_of_le_of_lt (k.le_add_right n\u2082) (by simp*),\n--       by_cases m \u2264 k; simp* },\n--     { subst h\u2082, simp [*, lift_term2_medium] },\n--     { have h\u2082 : m < k, by apply lt_of_le_of_lt; assumption,\n--       have : m \u2264 k - 1, from nat.le_sub_right_of_add_le (succ_le_of_lt h\u2082),\n--       have : m \u2264 k, from le_of_lt h\u2082,\n--       have : 1 \u2264 k, from one_le_of_lt h\u2082,\n--       simp [*, nat.add_sub_swap this n\u2082, -add_assoc, -add_comm] }\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m h := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m h := by simp*\n\n-- lemma lift_subst_term_large {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 n\u2082)[s // n\u2081+n\u2082] = (t [s // n\u2081]) \u2191 n\u2082 :=\n-- lift_at_subst_term_large t s n\u2082 n\u2081.zero_le\n\n-- lemma lift_subst_term_large' {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 n\u2082)[s // n\u2082+n\u2081] = (t [s // n\u2081]) \u2191 n\u2082 :=\n-- by rw [add_comm]; apply lift_subst_term_large\n\n-- lemma lift_at_subst_term_medium : \u2200{l} (t : preterm L l) (s : term L) {n\u2081 n\u2082 m}, m \u2264 n\u2082 \u2192\n--   n\u2082 \u2264 m + n\u2081 \u2192 (t \u2191' n\u2081+1 # m)[s // n\u2082] = t \u2191' n\u2081 # m\n-- | _ &k          s n\u2081 n\u2082 m h\u2081 h\u2082 :=\n--   begin\n--     by_cases h : m \u2264 k,\n--     { have h\u2083 : n\u2082 < k + (n\u2081 + 1), from lt_succ_of_le (le_trans h\u2082 (add_le_add_right h _)),\n--       simp [*, add_sub_cancel_right] },\n--     { have h\u2083 : k < n\u2082, from lt_of_lt_of_le (lt_of_not_ge h) h\u2081, simp* }\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m h\u2081 h\u2082 := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m h\u2081 h\u2082 := by simp*\n\n-- lemma lift_subst_term_medium {l} (t : preterm L l) (s : term L) (n\u2081 n\u2082) :\n--   (t \u2191 ((n\u2081 + n\u2082) + 1))[s // n\u2081] = t \u2191 (n\u2081 + n\u2082) :=\n-- lift_at_subst_term_medium t s n\u2081.zero_le (by rw [zero_add]; exact n\u2081.le_add_right n\u2082)\n\n-- lemma lift_at_subst_term_eq {l} (t : preterm L l) (s : term L) (n : \u2115) : (t \u2191' 1 # n)[s // n] = t :=\n-- begin rw [lift_at_subst_term_medium t s, lift_term_at_zero]; refl end\n\n-- @[simp] lemma lift_term1_subst_term {l} (t : preterm L l) (s : term L) : (t \u2191 1)[s // 0] = t :=\n-- lift_at_subst_term_eq t s 0\n\n-- lemma lift_at_subst_term_small : \u2200{l} (t : preterm L l) (s : term L) (n\u2081 n\u2082 m),\n--  (t \u2191' n\u2081 # (m + n\u2082 + 1))[s \u2191' n\u2081 # m // n\u2082] = (t [s // n\u2082]) \u2191' n\u2081 # (m + n\u2082)\n-- | _ &k          s n\u2081 n\u2082 m :=\n--   begin\n--     by_cases h : m + n\u2082 + 1 \u2264 k,\n--     { change m + n\u2082 + 1 \u2264 k at h,\n--       have h\u2082 : n\u2082 < k := lt_of_le_of_lt (le_add_left n\u2082 m) (lt_of_succ_le h),\n--       have h\u2083 : n\u2082 < k + n\u2081 := by apply nat.lt_add_right; exact h\u2082,\n--       have h\u2084 : m + n\u2082 \u2264 k - 1 := nat.le_sub_right_of_add_le h,\n--       simp [*, -add_comm, -add_assoc, nat.add_sub_swap (one_le_of_lt h\u2082)] },\n--     { change \u00ac(m + n\u2082 + 1 \u2264 k) at h,\n--       apply decidable.lt_by_cases k n\u2082; intro h\u2082,\n--       { have h\u2083 : \u00ac(m + n\u2082 \u2264 k) := \u03bbh', not_le_of_gt h\u2082 (le_trans (le_add_left n\u2082 m) h'),\n--         simp [h, h\u2082, h\u2083, -add_comm, -add_assoc] },\n--       { subst h\u2082,\n--         have h\u2083 : \u00ac(k + m + 1 \u2264 k) := by rw [add_comm k m]; exact h,\n--         simp [h, h\u2083, -add_comm, -add_assoc],\n--         exact lift_term_at2_small _ _ _ m.zero_le },\n--       { have h\u2083 : \u00ac(m + n\u2082 \u2264 k - 1) :=\n--           \u03bbh', h $ (nat.le_sub_right_iff_add_le $ one_le_of_lt h\u2082).mp h',\n--         simp [h, h\u2082, h\u2083, -add_comm, -add_assoc] }}\n--   end\n-- | _ (func f)    s n\u2081 n\u2082 m := rfl\n-- | _ (app t\u2081 t\u2082) s n\u2081 n\u2082 m := by simp [*, -add_assoc, -add_comm]\n\n-- lemma subst_term2 : \u2200{l} (t : preterm L l) (s\u2081 s\u2082 : term L) (n\u2081 n\u2082),\n--   t [s\u2081 // n\u2081] [s\u2082 // n\u2081 + n\u2082] = t [s\u2082 // n\u2081 + n\u2082 + 1] [s\u2081[s\u2082 // n\u2082] // n\u2081]\n-- | _ &k          s\u2081 s\u2082 n\u2081 n\u2082 :=\n--   begin -- can we use subst_realize2 here?\n--     apply decidable.lt_by_cases k n\u2081; intro h,\n--     { have : k < n\u2081 + n\u2082, from lt_of_le_of_lt (k.le_add_right n\u2082) (by simp*),\n--       have : k < n\u2081 + n\u2082 + 1, from lt.step this,\n--       simp only [*, eq_self_iff_true, fol.subst_term_var_lt] },\n--     { have : k < k + (n\u2082 + 1), from lt_succ_of_le (le_add_right _ n\u2082),\n--       subst h, simp [*, lift_subst_term_large', -add_comm] },\n--     apply decidable.lt_by_cases k (n\u2081 + n\u2082 + 1); intro h',\n--     { have : k - 1 < n\u2081 + n\u2082, from (nat.sub_lt_right_iff_lt_add (one_le_of_lt h)).2 h',\n--       simp [*, -add_comm, -add_assoc] },\n--     { subst h', simp [h, lift_subst_term_medium, -add_comm, -add_assoc] },\n--     { have : n\u2081 + n\u2082 < k - 1, from nat.lt_sub_right_of_add_lt h',\n--       have : n\u2081 < k - 1, from lt_of_le_of_lt (n\u2081.le_add_right n\u2082) this,\n--       simp only [*, eq_self_iff_true, fol.subst_term_var_gt] }\n--   end\n-- | _ (func f)    s\u2081 s\u2082 n\u2081 n\u2082 := rfl\n-- | _ (app t\u2081 t\u2082) s\u2081 s\u2082 n\u2081 n\u2082 := by simp*\n\n-- lemma subst_term2_0 {l} (t : preterm L l) (s\u2081 s\u2082 : term L) (n) :\n--   t [s\u2081 // 0] [s\u2082 // n] = t [s\u2082 // n + 1] [s\u2081[s\u2082 // n] // 0] :=\n-- let h := subst_term2 t s\u2081 s\u2082 0 n in by simp only [zero_add] at h; exact h\n\n-- lemma lift_subst_term_cancel : \u2200{l} (t : preterm L l) (n : \u2115), (t \u2191' 1 # (n+1))[&0 // n] = t\n-- | _ &k          n :=\n--   begin\n--     apply decidable.lt_by_cases n k; intro h,\n--     { change n+1 \u2264 k at h, have h' : n < k+1, from lt.step (lt_of_succ_le h), simp [h, h'] },\n--     { have h' : \u00ac(k+1 \u2264 k), from not_succ_le_self k, simp [h, h'] },\n--     { have h' : \u00ac(n+1 \u2264 k) := not_le_of_lt (lt.step h), simp [h, h'] }\n--   end\n-- | _ (func f)    n := rfl\n-- | _ (app t\u2081 t\u2082) n := by dsimp; simp [*]\n\n\n-- /- Probably useful facts about substitution which we should add when needed:\n-- (forall M N i j k, ( M [ j \u2190 N] ) \u2191' k # (j+i) = (M \u2191' k # (S (j+i))) [ j \u2190 (N \u2191' k # i ) ])\n-- subst_travers : (forall M N P n, (M [\u2190 N]) [n \u2190 P] = (M [n+1 \u2190 P])[\u2190 N[n\u2190 P]])\n-- erasure_lem3 : (forall n m t, m>n->#m = (#m \u2191' 1 # (S n)) [n \u2190 t]).\n-- lift_is_lift_sublemma : forall j v, j<v->exists w,#v=w\u21911#j.\n-- lift_is_lift : (forall N A n i j,N \u2191' i # n=A \u2191' 1 # j -> j<n -> exists M,N=M \u2191' 1 # j)\n-- subst_is_lift : (forall N T A n j, N [n \u2190 T]=A\u2191' 1#j->j<n->exists M,N=M\u2191' 1#j)\n-- -/\n\n-- /- preformula l is a partially applied formula. if applied to n terms, it becomes a formula.\n--   * We only have implication as binary connective. Since we use classical logic, we can define\n--     the other connectives from implication and falsum.\n--   * Similarly, universal quantification is our only quantifier.\n--   * We could make `falsum` and `equal` into elements of rel. However, if we do that, then we cannot make the interpretation of them in a model definitionally what we want.\n-- -/\nvariable (L)\ninductive preformula : \u2115 \u2192 Type u\n| falsum {} : preformula 0\n| equal (t\u2081 t\u2082 : term L) : preformula 0\n| rel {l : \u2115} (R : L.relations l) : preformula l\n| apprel {l : \u2115} (f : preformula (l + 1)) (t : term L) : preformula l\n| imp (f\u2081 f\u2082 : preformula 0) : preformula 0\n| all (f : preformula 0) : preformula 0\nexport preformula\n@[reducible] def formula := preformula L 0\nvariable {L}\n\nnotation `\u22a5` := preformula.falsum -- input: \\bot\ninfix ` \u2243 `:88 := preformula.equal -- input \\~- or \\simeq\ninfixr ` \u27f9 `:62 := preformula.imp -- input \\==>\nprefix `\u2200'`:110 := preformula.all\ndef preformula.not   (f : formula L)     : formula L := f \u27f9 \u22a5\nprefix `\u223c`:max := preformula.not -- input \\~, the ASCII character ~ has too low precedence\nnotation `\u22a4` := \u223c\u22a5 -- input: \\top\ndef preformula.and   (f\u2081 f\u2082 : formula L) : formula L := \u223c(f\u2081 \u27f9 \u223cf\u2082)\ninfixr ` \u2293 ` := preformula.and -- input: \\sqcap\ndef preformula.or    (f\u2081 f\u2082 : formula L) : formula L := \u223cf\u2081 \u27f9 f\u2082\ninfixr ` \u2294 ` := preformula.or -- input: \\sqcup\ndef biimp (f\u2081 f\u2082 : formula L) : formula L := (f\u2081 \u27f9 f\u2082) \u2293 (f\u2082 \u27f9 f\u2081)\ninfix ` \u21d4 `:61 := biimp -- input \\<=>\ndef ex    (f : formula L)     : formula L := \u223c \u2200' \u223cf\nprefix `\u2203'`:110 := ex -- input \\ex\n\nmeta instance preterm.reflect {L : Language.{0}} [reflected L] [\u2200 n, has_reflect (L.functions n)] :\n  \u2200{n}, has_reflect (preterm L n)\n| _ &k          := `(&k)\n| _ (func f)    := `(func f)\n| _ (app t\u2081 t\u2082) := (`(\u03bb x y, app x y).subst (preterm.reflect t\u2081)).subst (preterm.reflect t\u2082)\n\nmeta instance preformula.reflect {L : Language.{0}} [reflected L] [\u2200 n, has_reflect (L.functions n)]\n  [\u2200 n, has_reflect (L.relations n)] : \u2200{n}, has_reflect (preformula L n)\n| _ falsum           := `(falsum)\n| _ (equal t\u2081 t\u2082)    := (`(\u03bb x\u2081 x\u2082, equal x\u2081 x\u2082).subst (preterm.reflect t\u2081)).subst (preterm.reflect t\u2082)\n| _ (rel R)          := `(\u03bb x, preformula.rel x).subst `(R)\n| _ (apprel f t)     := (`(\u03bb x\u2081 x\u2082, apprel x\u2081 x\u2082).subst (preformula.reflect f)).subst (preterm.reflect t)\n| _ (imp f\u2081 f\u2082)      := (`(\u03bb x\u2081 x\u2082, imp x\u2081 x\u2082).subst (preformula.reflect f\u2081)).subst (preformula.reflect f\u2082)\n| _ (all f)          := `(\u03bb x, all x).subst (preformula.reflect f)\n\ndef L_empty : Language :=\n  \u27e8\u03bb _, empty, \u03bb _, empty\u27e9\n\nmeta instance L_empty.reflect_functions : \u2200 n, has_reflect (L_empty.functions n) :=\n\u03bb _ _, empty.elim \u2039_\u203a\n\nmeta instance L_empty.reflect_relations : \u2200 n, has_reflect (L_empty.relations n) :=\n\u03bb _ _, empty.elim \u2039_\u203a\n\nmeta def parse_preterm : \u03a0 {k}, parser_tactic $ preterm L_empty k\n| 0 := alphanumeric_token *> return (preterm.var 0)\n| (n+1) := fail\n\nmeta def parse_term := @parse_preterm 0\n\n-- -- note(jesse): for non-empty languages, need parsers parametrized over the function and relation symbols\n-- meta def parse_preformula : \u03a0 {k},  parser_tactic $ preformula L_empty k\n-- | 0 := (-- preformula.all <$> ((token $ str \"\u2200\") *> @parse_preformula 0) <|> \n-- do b <- lookahead \"\u27f9\",\n--           if b then preformula.imp <$> (@parse_preformula 0) <*>(@parse_preformula 0)\n--                else fail <|>\n--        (token $ str \"\u22a5\") *> return preformula.falsum <|>\n--        preformula.equal <$> @parse_preterm 0 <* (token $ str \"=\") <*> @parse_preterm 0) <* whitespace -- TODO(jesse) eliminate the recursive call\n-- | (n+1) := fail\n\n-- meta def parse_formula := @parse_preformula 0\n\n-- a \u27f9 (b \u27f9 (c \u27f9 d)) -- let's assume that implication associates to the right\n\n-- gets parsed as\n-- app \"imp\" [leaf \"a\", app \"imp\" [leaf \"b\", app \"imp\" [\"leaf c\", leaf \"d\"]]]\n\ninductive tree (\u03b1 : Type) : Type\n| leaf (a : \u03b1) : tree\n| app (l : option \u03b1) (ts : list tree) : tree\nopen tree\n\ndef add_subtree {\u03b1} : \u2200 (t : tree \u03b1) (t' : tree \u03b1), tree \u03b1\n| (leaf a) t' := app none [t', leaf a]\n| (app l ts) t' := app l $ t' :: ts\n\ndef add_parent {\u03b1} : \u2200 (t : tree \u03b1) (a : option \u03b1), tree \u03b1 :=\n\u03bb t a, app a [t]\n\ndef imp_handler (arg : string) (t : tree string) : tree string :=\nif arg = \"\u27f9\" then add_parent t (some arg) else add_subtree t (leaf arg)\n\n-- meta def parse_imp_aux : tree string \u2192 (parser_tactic $ tree string) :=\n-- \u03bb t, do\n--   imp_handler <$> return t <*> (token $ not_whitespace)\n\nmeta def parse_imp : (parser_tactic $ tree string) :=\ndo  x <- token not_whitespace,\n    (imp_handler <$> return x <*> parse_imp) <|> return (leaf x)\n\nmeta instance tree_string_reflect : has_reflect (tree string)\n| (leaf arg) := `(\u03bb x, leaf x).subst `(arg)\n| (app l ts) := (`(\u03bb x y, tree.app x y).subst `(l)).subst (by haveI := tree_string_reflect; exact list.reflect ts)\n  \ndef my_tree' : tree string := by (parse_imp).get_result \"a \u27f9 b \u27f9 c \u27f9 d\"\n\n#print my_tree'\n\n-- def my_eq : preformula L_empty 0 := by (preformula.equal <$> @parse_preterm 0 <* (token $ str \"=\") <*> @parse_preterm 0).get_result \"a = a\"\n\n-- def my_falsum : preformula L_empty 0 := by parse_formula.get_result \"a = a \u27f9 a = b\"\n\n-- #reduce my_falsum\n\n-- def foo := by (preformula.all <$> (symb \"\u2200\" *> eof)).get_result \"\u2200\"\n\n-- #reduce my_eq -- &0 \u2243 &0\n       \n-- def foo : preterm L_empty 0 := by parse_term.get_result \"f1\"\n\n-- #reduce by parse_term.get_result \"f1\"\n\n-- run_cmd (run' (preformula.all <$> (str \"\u2200\" *> @parse_preformula 0)) \"foo\"\n\n-- def my_falsum : preformula L_empty 0 := by parse_formula.get_result \"\u22a5\"\n\n-- #reduce by parse_term.get_result \"\"\n\n-- #eval by (token $ str \"=\").get_result \"=\"\n\n-- #reduce \"foo\"\n", "meta": {"author": "jesse-michael-han", "repo": "lean-parser-combinators", "sha": "d0dff9149a85a150679aa2145c4ffe2ac1ae5c0b", "save_path": "github-repos/lean/jesse-michael-han-lean-parser-combinators", "path": "github-repos/lean/jesse-michael-han-lean-parser-combinators/lean-parser-combinators-d0dff9149a85a150679aa2145c4ffe2ac1ae5c0b/src/parse_formula2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.38230906993673663}}
{"text": "import tactic\nimport data.polynomial.basic\nimport data.polynomial.degree.definitions\nimport data.polynomial.eval\nimport data.list.func\nimport data.nat.prime\nimport data.matrix.notation\nimport linear_algebra.matrix.determinant\nimport tactic.sageify.socket\n.\n\nclass has_sageify (R : Type*) :=\n(sageify : R \u2192 string)\nclass has_sageify_type (R : Type*) :=\n(sageify_type : string)\n\nnamespace has_sageify\nopen has_sageify_type\ninstance : has_sageify \u2115 := \u27e8\u03bb n, to_string n\u27e9\ninstance : has_sageify \u2124 := \u27e8\u03bb n, to_string n\u27e9\ninstance : has_sageify \u211a := \u27e8\u03bb n, to_string n\u27e9\n\ninstance : has_sageify_type \u2124 := \u27e8\"Integers()\"\u27e9\ninstance : has_sageify_type \u211a := \u27e8\"Rationals()\"\u27e9\n-- instance : has_sageify_type \u211d := \u27e8\u03bb n, \"RDF()\"\u27e9\n-- instance : has_sageify_type \u2102 := \u27e8\u03bb n, \"CDF()\"\u27e9\n\ninstance {R S : Type*} [has_sageify R] [has_sageify S] : has_sageify (R \u00d7 S) :=\n\u27e8\u03bb n, \"(\" ++ sageify n.1 ++ \", \" ++ sageify n.2 ++ \")\"\u27e9\ninstance {R : Type*} [has_sageify R] : has_sageify (list R) :=\n\u27e8\u03bb n, \"[\" ++ \", \".intercalate (n.map (sageify)) ++ \"]\"\u27e9\n\ninstance {R : Type*} [semiring R] [has_sageify R] [has_sageify_type R] :\n  has_sageify (polynomial R) :=\n\u27e8\u03bb n,\n  let coeffs : list R := (list.range (n.nat_degree + 1)).map n.coeff in\n  sformat!\"PolynomialRing({sageify_type R})({sageify coeffs})\"\u27e9\nend has_sageify\n\nclass has_from_sage (R : Type*) :=\n(sage_parser : parser R)\n\nnamespace parser\n\n\n-- unlike the mathlib  one we allow \"2\" as a rat, and not only \"2/1\"\ndef possible_rat : parser \u211a := decorate_error \"<rationals>\" $ do\n  ing \u2190 int,\n  di \u2190 many (ch '/'),\n  if 0 < di.length then do\n    na \u2190 nat,\n    pure $ (ing : \u211a) / na\n  else\n    pure ing\n\ndef list {R : Type*} (pR : parser R) : parser (list R) := decorate_error \"<list>\" $ do\n  ch '[',\n  l \u2190 many (do\n    l \u2190 pR,\n    str \", \",\n    pure l),\n  ch ']',\n  pure l\n#eval (list (nat)).run_string \"[1]\"\n\nend parser\n\nnamespace has_from_sage\n\n-- dangerous!\ndef from_sage {R : Type*} [has_from_sage R] (s : string) : option R :=\n(sage_parser.run_string s).get_right\ndef ifrom_sage {R : Type*} [has_from_sage R] [inhabited R] (s : string) : R :=\n(from_sage s).iget\n-- TODO fix this to remove all space\ndef strip_from_sage {R : Type*} [has_from_sage R] [inhabited R] (s : string) : option R :=\nlet s := if s.head = ' ' then s.popn 1 else s in\nlet s := if s.back = ' ' then s.popn_back 1 else s in\nfrom_sage s\n\ninstance : has_from_sage \u2115 := \u27e8parser.nat\u27e9\ninstance : has_from_sage \u2124 := \u27e8parser.int\u27e9\ninstance : has_from_sage \u211a := \u27e8parser.possible_rat\u27e9\n#eval (ifrom_sage \"-1232/113\" : \u211a)\n\ninstance {R : Type*} [has_from_sage R] : has_from_sage (list R) :=\n-- \u27e8\u03bb n, ((n.popn 1).pop_back.split (= ',')).map strip_from_sage\u27e9\n\u27e8do\n  parser.ch '[',\n  h \u2190 parser.sep_by (parser.str \", \") sage_parser,\n  parser.ch ']',\n  pure h\u27e9\n#eval (ifrom_sage \"[-1232/113, 12]\" : list \u211a)\n#eval (ifrom_sage \"[12113, 12]\" : list \u211a)\n\ninstance {R S : Type*} [has_from_sage R] [has_from_sage S] : has_from_sage (R \u00d7 S) :=\n\u27e8do\n  parser.ch '(',\n  l \u2190 (sage_parser : parser R),\n  parser.str \", \",\n  r \u2190 (sage_parser : parser S),\n  parser.ch ')',\n  pure (l, r)\u27e9\n-- \u27e8\u03bb n, let st := ((n.popn 1).pop_back.split (= ',')) in\n--   (strip_from_sage st.head, strip_from_sage st.tail.head)\u27e9\n\n#eval (ifrom_sage \"[(-1232/113, 1), (12, 1)]\" : list (\u211a \u00d7 \u2115))\ninstance {R : Type*} [has_from_sage R] [semiring R] : has_from_sage (polynomial R) :=\nsorry\n-- \u27e8\u03bb n, (parser.rat.run_string n).get_right.get_or_else 0\u27e9\n\n#eval (ifrom_sage \"[([-1, 1], 1), ([1], 1)]\" : list (list \u211a \u00d7 \u2115))\n\nend has_from_sage\n\ndef sage_config : io.process.spawn_args :=\n{ cmd := \"conda\",\n  args := [\"run\", \"-n\", \"sage\", \"sage\", \"-q\"],\n  stdin := io.process.stdio.piped,\n  stderr := io.process.stdio.piped,\n  stdout := io.process.stdio.piped\n  }\n\ndef sage_io_config : io.process.spawn_args :=\n{ cmd := \"conda\",\n  args := [\"run\", \"-n\", \"sage\", \"--no-capture-output\", \"sage\", \"-q\"],\n      --     -- \"--simple-prompt\" -- unclear if helpful\n  stdin := io.process.stdio.piped,\n  stderr := io.process.stdio.piped,\n  stdout := io.process.stdio.piped\n  }\n\ndef sage_io_config' : io.process.spawn_args :=\n{ cmd := \"sage\",\n  args := [\"-q\"],\n  stdin := io.process.stdio.piped,\n  stderr := io.process.stdio.piped,\n  stdout := io.process.stdio.piped\n  }\n\n-- no longer needed\ndef escape (s : string) : string := \"\\\\\\\"\".intercalate $ s.split (= '\\\"')\n\nopen list.func\n#check rat.reflect\nrun_cmd do to_string <$> tactic.eval_expr \u2115 `(2 + 2) >>= tactic.trace\n\n-- meta def reify_poly_aux (R : Type*) [has_add R] [has_zero R] [reflected R] : expr \u2192 list R \u2192 tactic (list R)\nmeta def reify_poly_aux : expr \u2192 list \u211a \u2192 tactic (list \u211a) -- TODO maybe use ring_m.eval instead\n| `(tactic.ring.horner %%a %%x %%n %%b) l := do\n  A \u2190 reify_poly_aux a [],\n  N \u2190 tactic.eval_expr \u2115 n,\n  B \u2190 expr.to_rat b,\n  return (add (list.repeat 0 N ++ A) [B]) -- this should also be a function TODO\n| e _ := do\n  t \u2190 expr.to_rat e, return [t]\n\n-- meta def reify_poly (R : Type*) [has_add R] [has_zero R] [reflected R] (p : expr) :\nmeta def reify_poly (p : expr) : tactic (list \u211a) :=\nreify_poly_aux p []\n\nopen tactic\n\nopen has_sageify has_from_sage\ndef sage_filename := \"sagetmp.sage\"\n-- run_cmd\n-- (unsafe_run_io $ do\n--     i \u2190 io.proc.spawn sage_io_config',\n--     io.fs.put_str_ln i.stdin \"1\",\n--     io.fs.put_str_ln i.stdin sformat!\"exit()\",\n--     io.fs.flush i.stdin,\n--     (do 0 \u2190 io.proc.wait i, return ()) <|>\n--       io.fail (\"Sage produced an error, is it accessible with the \" ++\n--                \"commands specified in `sage_io_config`?\"),\n--     s \u2190 io.fs.get_line i.stdout,\n--     io.fs.close i.stdin,\n--     io.print_ln s.to_string)\n\nrun_cmd\n  unsafe_run_io $ do\n    han \u2190 io.mk_file_handle sage_filename io.mode.write,\n    let q := sageify (2/4 : \u211a),\n    io.fs.put_str_ln han sformat!\"print({q})\",\n    -- let q := sageify (polynomial.X ^ 2 - 1 : polynomial \u211a),\n    -- let q := sageify (polynomial.X ^ 2 - 1 : polynomial \u211a),\n    io.fs.put_str_ln han sformat!\"print(list(PolynomialRing(Rationals(),'x')([1,0,-1]).factor()))\",\n    io.fs.flush han,\n    i \u2190 io.proc.spawn ({args := sage_config.args ++ [sage_filename], ..sage_config} : io.process.spawn_args),\n    -- i \u2190 io.proc.spawn ({args := sage_config.args ++ [\"-c\", \"print\\\\(1\\\\)\"], ..sage_config} : io.process.spawn_args),\n    -- i \u2190 io.proc.spawn ({args := sage_config.args ++ [\"-c\", \"\\\"\" ++ escape sage_cmd ++ \"\\\"\"], ..sage_config} : io.process.spawn_args),\n    io.fs.close han,\n    s \u2190 io.fs.get_line i.stdout,\n    io.print_ln (ifrom_sage $ (buffer.to_string s).pop_back : \u211a),\n    s \u2190 io.fs.get_line i.stdout,\n    io.print_ln (buffer.to_string s)\n    -- io.print_ln (to_string $`((ifrom_sage $ (buffer.to_string s).pop_back : list (polynomial \u211a \u00d7 \u2115)).head.1))\n    -- io.print_ln ((from_sage $ (buffer.to_string s).pop_back : list (polynomial \u211a \u00d7 \u2115)))\n\n-- run_cmd\n--   unsafe_run_io $ do\n--     i \u2190 io.proc.spawn\n--   { cmd := \"conda\",\n--     args := [\"run\", \"-n\", \"sage\", \"--no-capture-output\", \"sage\", \"-q\"\n--       -- \"--simple-prompt\" -- unclear if helpful\n--       ],\n--     stdin := io.process.stdio.piped,\n--     -- stderr := io.process.stdio.piped,\n--     stdout := io.process.stdio.piped\n--     },\n--     let q := sageify (2/4 : \u211a),\n--     io.fs.put_str_ln i.stdin \"'t'\",\n--     -- -- let q := sageify (polynomial.X ^ 2 - 1 : polynomial \u211a),\n--     -- let q := sageify (polynomial.X ^ 2 - 1 : polynomial \u211a),\n--     io.fs.put_str_ln i.stdin sformat!\"list(map(lambda x : (list(x[0]), x[1]), list(PolynomialRing(Rationals(),'x')([1,0,-1]).factor())))\",\n--     io.fs.put_str_ln i.stdin sformat!\"exit()\",\n--     io.fs.flush i.stdin,\n--     n \u2190 io.proc.wait i,\n--     io.print_ln n,\n--     s \u2190 io.fs.get_line i.stdout,\n--     io.print_ln (s.drop 6).pop_back.to_string,\n--     s \u2190 io.fs.get_line i.stdout,\n--     io.print_ln (s.drop 6).pop_back.to_string,\n--     io.print_ln (from_sage $ (s.drop 6).pop_back.to_string : list (list \u211a \u00d7 \u2115)),\n--     io.fs.close i.stdin\nmeta def fac_to_expr (l : list (list \u211a \u00d7 \u2115)) : tactic expr :=\ndo\n  (fs : list expr) \u2190 l.mmap (\u03bb lin, do\n    ((ine, _) : expr \u00d7 nat) \u2190 lin.1.mfoldl\n      (\u03bb ol ex, do return (`(%%ol.1 + (polynomial.C (%%(ex.reflect) : \u211a) : polynomial \u211a) * polynomial.X ^ ((%%(nat.reflect ol.2)) : nat) : polynomial \u211a), ol.2 + 1)) (`(0 : polynomial \u211a), 0),\n    return `((%%ine) ^ (%%lin.2.reflect : nat) : polynomial \u211a)),\n  final \u2190 fs.mfoldl (\u03bb ol ex, do return `((%%ol) * (%%ex) : polynomial \u211a)) `(1 : polynomial \u211a),\n  trace final,\n  sl \u2190 simp_lemmas.mk_default,\n  prod.fst <$> simplify sl [] final\n\nsection\nsetup_tactic_parser\nopen tactic.ring\nopen tactic\n\n-- TODO catch errors :)\n-- TODO multiple commands in one run\nmeta def get_sage_output_for_string (l : string) : tactic string :=\ndo\n  trace \"CALLING SAGE\",\n  unsafe_run_io $ (do\n    i \u2190 io.proc.spawn sage_io_config',\n    io.fs.put_str_ln i.stdin l,\n    io.fs.put_str_ln i.stdin sformat!\"exit()\",\n    io.fs.flush i.stdin,\n    (do 0 \u2190 io.proc.wait i, return ()) <|>\n      io.fail (\"Sage produced an error, is it accessible with the \" ++\n               \"commands specified in `sage_io_config`?\"),\n    s \u2190 io.fs.get_line i.stdout,\n    if (s.drop 13).take 10 = \"----------\".to_char_buffer\n    then\n      (do t \u2190 io.fs.get_line i.stdout,\n          t2 \u2190 io.fs.get_line i.stdout,\n          t2 \u2190 io.fs.get_line i.stdout,\n          io.fail $ \"Sage errored with: \" ++ (t.drop 7).to_string ++ t2.to_string)\n    else return (),\n-- TODO return more of the error, eg we should parse the entire traceback by looking for the second blank line? or maybe just read to end?\n-- TypeError                                 Traceback (most recent call last)\n-- <ipython-input-36-25a6001b137e> in <module>\n-- ----> 1 R = PolynomialRing(NonNegativeIntegerSemiring(),'x', names=('x',)); (x,) = R._first_ngens(1)\n\n-- /usr/local/Caskroom/miniconda/base/envs/sage/lib/python3.8/site-packages/sage/rings/polynomial/polynomial_ring_constructor.py in PolynomialRing(base_ring, *args, **kwds)\n--     553     \"\"\"\n--     554     if not ring.is_Ring(base_ring):\n-- --> 555         raise TypeError(\"base_ring {!r} must be a ring\".format(base_ring))\n--     556\n--     557     n = -1  # Unknown number of variables\n\n-- TypeError: base_ring Non negative integer semiring must be a ring\n    io.fs.close i.stdin,\n    io.print_ln s.to_string,\n    return $ (s.drop 6).pop_back.to_string)\n\n\n-- run_cmd (do (l : list \u211a) \u2190 get_sage_output_for_string \"[a]\", skip)\n\nsection\nsetup_tactic_parser\n#check expr.to_string\n-- TODO a dependently typed version of this would be good\n-- I.e. we should be able to factor integer polys with the same code that factors rational\nmeta def replace_certified_sage_equality\n  (matcher : expr \u2192 tactic bool) -- match exprs to be converted\n  (reify_type : Type*) -- the type the input should be converted to\n  (reify : expr \u2192 tactic reify_type)\n  (sage_input : reify_type \u2192 tactic string) -- takes the reified expr to\n  (output_type : Type*) -- the type the sage output should be converted to\n  [has_from_sage output_type]\n  [has_to_tactic_format output_type] -- only for testing\n  (convert_output : output_type \u2192 tactic expr) -- takes the output\n  (validator : expr \u2192 expr \u2192 tactic (expr)) -- produces proofs that original expr agrees with the\n  (naam : string)\n  (wi : parse (parser.tk \"with\" *> parser.pexpr)?) :\n  tactic unit :=\ndo\n  g \u2190 target,\n  (oe : option expr) \u2190 g.mfold (none : option expr) (\u03bb sub n old, if old.is_some then return old else (do\n    s \u2190 matcher sub,\n    match s with\n    | tt := return (some sub)\n    | ff := return old\n    end)),\n  trace oe,\n  oe \u2190 oe,\n  (re : reify_type) \u2190 reify oe,\n  (s : string) \u2190 (do w \u2190 (wi.map return).get_or_else failure, w \u2190 i_to_expr w,\n    trace w,\n    trace w.to_string,\n    return $ (w.to_string.popn 1).popn_back 1)\n    <|> sage_input re >>= (\u03bb r, get_sage_output_for_string r <|> unsafe_run_io (run_online_sage_for_string' r)),\n  let l : option output_type := from_sage s,\n  l \u2190 l,\n  newe \u2190 convert_output l,\n  p \u2190 validator oe newe,\n  rewrite_target p,\n  -- TODO maybe check equality here\n  (guard wi.is_none >> trace (sformat!\"Try this: {naam} with \\\"{s}\\\"\")) <|> skip\n\nend\n\n/-\nExamples:\n-/\n\n-- TODO fix other uses of reflect\nmeta def tactic.interactive.factor_nats := replace_certified_sage_equality\n  (\u03bb ex, do\n    t \u2190 infer_type ex,\n    return $ t = `(nat))\n  \u2115\n  (\u03bb e, e.to_nat)\n  (\u03bb n, return sformat!\"print(list(ZZ({n}).factor()))\")\n  (list (\u2115 \u00d7 \u2115))\n  (\u03bb l, do\n    ini \u2190 l.mfoldl (\u03bb ol \u27e8p, n\u27e9, do\n      P \u2190 expr.of_nat `(\u2115) p,\n      N \u2190 expr.of_nat `(\u2115) n,\n      return `((%%ol : \u2115) * (%%P : \u2115) ^ (%%N : \u2115))) `(1 : \u2115),\n    sl \u2190 simp_lemmas.mk_default,\n    prod.fst <$> simplify sl [] ini)\n  (\u03bb o n, do\n    (e\u2081', p\u2081) \u2190 or_refl_conv norm_num.derive o, -- TODO use this trick in mathlib\n    (e\u2082', p\u2082) \u2190 or_refl_conv norm_num.derive n,\n    trace e\u2081',\n    trace e\u2082',\n    is_def_eq e\u2081' e\u2082',\n    mk_eq_symm p\u2082 >>= mk_eq_trans p\u2081)\n  \"factor_nats\"\n\nmeta def tactic.interactive.factor_poly := replace_certified_sage_equality\n  (\u03bb ex, do\n    t \u2190 infer_type ex,\n    return $ t = `(polynomial \u211a))\n  (list \u211a)\n  (\u03bb e, do using_new_ref mk_buffer $ \u03bb atoms,\n    do\n      (e, f) \u2190 normalize' atoms reducible ring.normalize_mode.raw tt e,\n      reify_poly e)\n  (\u03bb n, return\n    sformat!\"list(map(lambda x : (list(x[0]), x[1]), list(PolynomialRing(Rationals(),'x')({n}).factor())))\")\n  (list (list \u211a \u00d7 \u2115))\n  fac_to_expr\n  (\u03bb o n, do\n    ((e\u2081', p\u2081), (e\u2082', p\u2082)) \u2190 ring_m.run reducible n $\n      prod.mk <$> eval (\u03bb _, failed) o <*> eval (\u03bb _, failed) n,\n    trace e\u2081',\n    trace e\u2082',\n    is_def_eq e\u2081' e\u2082',\n    mk_eq_symm p\u2082 >>= mk_eq_trans p\u2081)\n  \"factor_poly\"\n\nmeta def reify_vec_cons_aux (R : Type*) (rei : expr \u2192 tactic R) : expr \u2192 list R \u2192 tactic (list R)\n| `(@matrix.vec_cons %%a %%n %%el %%b) l := do\n  A \u2190 reify_vec_cons_aux b l,\n  l \u2190 rei el,\n  return $ l :: A -- this should also be a function TODO\n| `(matrix.vec_empty) _ := return []\n| e _ := return []\n\nmeta def reify_vec_cons (R : Type*) (rei : expr \u2192 tactic R) (p : expr) : tactic (list R) :=\nreify_vec_cons_aux R rei p []\n\nmeta def reify_mat {R : Type*} (f : expr \u2192 option R) (p : expr) : tactic (list $ list R) :=\nreify_vec_cons (list R) (reify_vec_cons R (\u03bb e, f e)) p\n\nmeta def mat_to_expr {R : Type} (ty : expr) (f : R \u2192 expr) (l : list (list R)) : pexpr := -- TODO use a pexpr here and i_to_expr_for_???\nlet n : expr := `(%%l.head.length.reflect : \u2115) in -- TODO change to ncols\nl.reverse.enum.reverse.foldr (\u03bb t ol,\n  let a : expr := t.2.reverse.enum.reverse.foldr (\u03bb v ol',\n    let inp := f v.2 in `(@matrix.vec_cons.{0} %%ty (%%(v.1.reflect)) %%inp %%ol'))\n    (`(matrix.vec_empty.{0} : fin 0 \u2192 %%ty) : expr) in\n  ``(matrix.vec_cons.{0} %%a %%ol : fin (nat.succ %%(t.1.reflect)) \u2192 fin (%%n : \u2115) \u2192 %%ty))\n  ``(@matrix.vec_empty.{0} $ fin %%n \u2192  %%ty)\n\n-- example :false :=\n-- begin\n--   (do m \u2190 mk_mvar,\n--   t \u2190 infer_type m,\n--   unify t `((%%(mat_to_expr [[1,1],[1,0]]) : fin 2 \u2192 fin 2 \u2192 \u211a) = %%(mat_to_expr [[1,1],[1,0]])), set_goals  [m]),\n\n-- end\n\nmeta def tactic.interactive.rref := replace_certified_sage_equality\n  (\u03bb ex, do\n    t \u2190 infer_type ex >>= whnf <|> return `(Type), -- if infer type fails then return false\n    match t with\n    | `(fin %%e \u2192 fin %%f \u2192 \u211a) := return tt\n    | _ := return ff\n    end)\n  (list (list \u211a))\n  (reify_mat expr.to_rat)\n  (\u03bb n, return $\n    sformat!\"A = Matrix(QQ, {n}); A = A.augment(MatrixSpace(QQ, A.nrows(), A.nrows())(1), subdivide=True);\" ++\n      \"B = A.rref(); (list(list(b) for b in B.subdivision(0,0)),list(list(b) for b in B.subdivision(0,1).inverse()))\")\n  (list (list \u211a) \u00d7 list (list \u211a))\n  (\u03bb l, do\n    n \u2190 expr.of_nat `(\u2115) l.1.length,\n    m \u2190 expr.of_nat `(\u2115) l.1.head.length,\n    m1 \u2190 i_to_expr $ mat_to_expr `(\u211a) (\u03bb n : \u211a, reflect n) l.1,\n    m2 \u2190 i_to_expr $ mat_to_expr `(\u211a) (\u03bb n : \u211a, reflect n) l.2,\n    return $ `(matrix.mul\n      (%%m2 : matrix (fin %%n) (fin %%n) \u211a)\n      (%%m1 : matrix (fin _) (fin %%m) \u211a)))\n  (\u03bb o n, do\n    trace o,\n    trace n,\n    (e\u2081', p\u2081) \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) o,\n    -- sl \u2190 simp_lemmas.mk.add_simp ``matrix.mul_fin_two,\n    -- (e\u2082', p\u2082) \u2190 or_refl_conv (\u03bb e, do f \u2190 simplify sl [] e, return (f.1, f.2.1)) n,\n    (e\u2082'', p\u2082') \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) n,\n    -- good for debugging\n    -- (do m \u2190 mk_mvar,\n    --   t \u2190 infer_type m,\n    --   i_to_expr ``(%%e\u2082'' = %%n) >>= unify t, set_goals [m]),\n    is_def_eq e\u2081' e\u2082'',\n    -- p\u2082 \u2190 mk_eq_trans p\u2082 p\u2082',\n    mk_eq_symm p\u2082' >>= mk_eq_trans p\u2081)\n  \"rref\"\n\n\nmeta def tactic.interactive.hnf := replace_certified_sage_equality\n  (\u03bb ex, do\n    t \u2190 infer_type ex >>= whnf <|> return `(Type), -- if infer type fails then return false\n    match t with\n    | `(fin %%e \u2192 fin %%f \u2192 \u2124) := return tt\n    | _ := return ff\n    end)\n  (list (list \u2124))\n  (reify_mat expr.to_int)\n  (\u03bb n, return $\n    sformat!\"A = Matrix(ZZ, {n});\" ++\n      \"B, U = A.hermite_form(transformation=True); (list(list(b) for b in B), list(list(u) for u in U.inverse()))\")\n  (list (list \u2124) \u00d7 list (list \u2124))\n  (\u03bb l, do\n    n \u2190 expr.of_nat `(\u2115) l.1.length,\n    m \u2190 expr.of_nat `(\u2115) l.1.head.length,\n    m1 \u2190 i_to_expr $ mat_to_expr `(\u2124) (\u03bb n : \u2124, reflect n) l.1, -- TODO fix this\n    m2 \u2190 i_to_expr $ mat_to_expr `(\u2124) (\u03bb n : \u2124, reflect n) l.2,\n    return $ `(matrix.mul\n      (%%m2 : matrix (fin %%n) (fin %%n) \u2124)\n      (%%m1 : matrix (fin _) (fin %%m) \u2124)))\n  (\u03bb o n, do\n    (e\u2081', p\u2081) \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) o,\n    (e\u2082'', p\u2082') \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) n,\n    is_def_eq e\u2081' e\u2082'',\n    mk_eq_symm p\u2082' >>= mk_eq_trans p\u2081)\n  \"hnf\"\n\nopen_locale matrix\nmeta def tactic.interactive.snf := replace_certified_sage_equality\n  (\u03bb ex, do\n    t \u2190 infer_type ex >>= whnf <|> return `(Type), -- if infer type fails then return false\n    match t with\n    | `(fin %%e \u2192 fin %%f \u2192 \u2124) := return tt\n    | _ := return ff\n    end)\n  (list (list \u2124))\n  (reify_mat expr.to_int)\n  (\u03bb n, return $\n    sformat!\"A = Matrix(ZZ, {n});\" ++\n      \"B, U, V = A.smith_form(transformation=True); [list(list(b) for b in B), list(list(u) for u in U.inverse()), list(list(v) for v in V.inverse())]\")\n  (list $ list (list \u2124)) -- TODO make a 3-tuple parser work and change this\n  (\u03bb l, do\n    n \u2190 expr.of_nat `(\u2115) l.head.length,\n    m \u2190 expr.of_nat `(\u2115) l.head.head.length,\n    m1 \u2190 i_to_expr $ mat_to_expr `(\u2124) (\u03bb n : \u2124, reflect n) l.head, -- TODO fix this\n    m2 \u2190 i_to_expr $ mat_to_expr `(\u2124) (\u03bb n : \u2124, reflect n) l.tail.head,\n    m3 \u2190 i_to_expr $ mat_to_expr `(\u2124) (\u03bb n : \u2124, reflect n) l.tail.tail.head,\n    trace \">>>>>\",\n    return $ `(\n      (%%m2 : matrix (fin %%n) (fin %%n) \u2124) \u2b1d\n      (%%m1 : matrix (fin _) (fin %%m) \u2124) \u2b1d\n      (%%m3 : matrix (fin _) (fin %%m) \u2124)))\n  (\u03bb o n, do\n    (e\u2081', p\u2081) \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) o,\n    (e\u2082'', p\u2082') \u2190 or_refl_conv (conv.convert $ conv.interactive.norm_num []) n,\n    is_def_eq e\u2081' e\u2082'',\n    mk_eq_symm p\u2082' >>= mk_eq_trans p\u2081)\n  \"snf\"\n\nend\n\n-- want multivariate examples too\n-- example (x y : \u2124) : x * y + x * x * y + y + 1 = 0 :=\n-- begin\n--   ring_nf,\n-- end\n\nexample : polynomial.eval 1 (polynomial.X ^ 3 - 1 : polynomial \u211a) = 0 :=\nbegin\n  factor_poly with \"[([-1, 1], 1), ([1, 1, 1], 1)]\", -- TODO at loc\n  simp,\nend\n\nexample : \u00ac nat.prime 1111 :=\nbegin\n  factor_nats, -- TODO at loc\n  simp [nat.prime_mul_iff],\nend\n\nopen_locale matrix\n-- TODO update to new mats\n-- example {\u03b1 : Type*} [comm_ring \u03b1] {a b c d e f g h i : \u03b1} :\n--         ![![a, b], ![c, d]] \u2b1d ![![e, f], ![g, i]] =\n--        ![![a, b], ![c, d]] :=\n-- begin\n--   rw [matrix.mul_fin_two],\n-- end\n-- example :\n--         ![![1, 2], ![3, 4]] \u2b1d ![![4, 4,1], ![4, 4,1]] =\n--        ![![4, 4,8], ![4, 4,0]] :=\n-- begin\n--   norm_num,\n-- end\n\n-- set_option trace.simplify true\n-- set_option pp.all true\n-- set_option trace.type_context.is_def_eq true\n-- set_option trace.type_context.is_def_eq_detail true\n\n-- example : \u2203 v : fin 1 \u2192 fin 2 \u2192 \u211a, v \u2b1d\n--         (![![1, 1, 0], ![1, 1, 0]] : matrix _ _ \u211a) = 0 :=\n-- begin\n--   rref,\n--   simp_rw [\u2190 matrix.mul_assoc],\n-- end\n\n\n\n-- example : \u2203 v : fin 1 \u2192 fin 2 \u2192 \u2124, v \u2b1d\n--         (![![1, 1, 0], ![2, 2, 1]] : matrix _ _ \u2124) = 0 :=\n-- begin\n--   hnf,\n--   simp_rw [\u2190 matrix.mul_assoc],\n-- end\n\n-- example : \u2203 v : fin 1 \u2192 fin 2 \u2192 \u2124, v \u2b1d\n--         (![![1, 1, 0], ![2, 2, 1]] : matrix _ _ \u2124) = 0 :=\n-- begin\n--   snf,\n--   dsimp,\n--   simp_rw [\u2190 matrix.mul_assoc],\n-- end\n\n/-\nHow should the sage monad look?\ndo\n  let t : list blah := something from tactic state,\n  s \u2190 sage_compute\n\n-/\n\n/-\n# TODO\n- gcd by bezout\n- ideal membership\n- ideal equality/containment\n  - optimal ideal basis\n- factoring\n- inverse in the class group\n- class group structure\n- class group generators\n- finding local obstructions\n\n## Harder\n\nCertificate for non-principality of ideals\n-/\n", "meta": {"author": "lean-forward", "repo": "class-group-and-mordell-equation", "sha": "baba2049f3bfe4d2cc184f8205997333e7c58638", "save_path": "github-repos/lean/lean-forward-class-group-and-mordell-equation", "path": "github-repos/lean/lean-forward-class-group-and-mordell-equation/class-group-and-mordell-equation-baba2049f3bfe4d2cc184f8205997333e7c58638/src/tactic/sageify/sageify.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6619228691808011, "lm_q1q2_score": 0.3822573890624071}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nSupplementary theorems about the `string` type.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.basic\nimport Mathlib.data.char\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\nnamespace string\n\n\ndef ltb : iterator \u2192 iterator \u2192 Bool := sorry\n\nprotected instance has_lt' : HasLess string :=\n  { Less := fun (s\u2081 s\u2082 : string) => \u21a5(ltb (mk_iterator s\u2081) (mk_iterator s\u2082)) }\n\nprotected instance decidable_lt : DecidableRel Less :=\n  fun (a b : string) => bool.decidable_eq (ltb (mk_iterator a) (mk_iterator b)) tt\n\n@[simp] theorem lt_iff_to_list_lt {s\u2081 : string} {s\u2082 : string} : s\u2081 < s\u2082 \u2194 to_list s\u2081 < to_list s\u2082 :=\n  sorry\n\nprotected instance has_le : HasLessEq string := { LessEq := fun (s\u2081 s\u2082 : string) => \u00acs\u2082 < s\u2081 }\n\nprotected instance decidable_le : DecidableRel LessEq :=\n  fun (a b : string) => ne.decidable (ltb (mk_iterator b) (mk_iterator a)) tt\n\n@[simp] theorem le_iff_to_list_le {s\u2081 : string} {s\u2082 : string} : s\u2081 \u2264 s\u2082 \u2194 to_list s\u2081 \u2264 to_list s\u2082 :=\n  iff.trans (not_congr lt_iff_to_list_lt) not_lt\n\ntheorem to_list_inj {s\u2081 : string} {s\u2082 : string} : to_list s\u2081 = to_list s\u2082 \u2194 s\u2081 = s\u2082 := sorry\n\ntheorem nil_as_string_eq_empty : list.as_string [] = empty := rfl\n\n@[simp] theorem to_list_empty : to_list empty = [] := rfl\n\ntheorem as_string_inv_to_list (s : string) : list.as_string (to_list s) = s :=\n  string_imp.cases_on s fun (s : List char) => Eq.refl (list.as_string (to_list (string_imp.mk s)))\n\n@[simp] theorem to_list_singleton (c : char) : to_list (singleton c) = [c] := rfl\n\ntheorem to_list_nonempty {s : string} (h : s \u2260 empty) : to_list s = head s :: to_list (popn s 1) :=\n  sorry\n\n@[simp] theorem head_empty : head empty = Inhabited.default := rfl\n\n@[simp] theorem popn_empty {n : \u2115} : popn empty n = empty := sorry\n\nprotected instance linear_order : linear_order string :=\n  linear_order.mk LessEq Less sorry sorry sorry sorry string.decidable_le\n    (fun (a b : string) => string.has_decidable_eq a b)\n    fun (a b : string) => string.decidable_lt a b\n\nend string\n\n\ntheorem list.to_list_inv_as_string (l : List char) : string.to_list (list.as_string l) = l := sorry\n\n@[simp] theorem list.as_string_inj {l : List char} {l' : List char} :\n    list.as_string l = list.as_string l' \u2194 l = l' :=\n  sorry\n\ntheorem list.as_string_eq {l : List char} {s : string} :\n    list.as_string l = s \u2194 l = string.to_list s :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/string/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.382257385210979}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport algebra.group.type_tags\nimport algebra.hom.equiv\nimport algebra.hom.ring\nimport algebra.hom.units\n\n/-!\n# Unbundled monoid and group homomorphisms\n\nThis file is deprecated, and is no longer imported by anything in mathlib other than other\ndeprecated files, and test files. You should not need to import it.\n\nThis file defines predicates for unbundled monoid and group homomorphisms. Instead of using\nthis file, please use `monoid_hom`, defined in `algebra.hom.group`, with notation `\u2192*`, for\nmorphisms between monoids or groups. For example use `\u03c6 : G \u2192* H` to represent a group\nhomomorphism between multiplicative groups, and `\u03c8 : A \u2192+ B` to represent a group homomorphism\nbetween additive groups.\n\n## Main Definitions\n\n`is_monoid_hom` (deprecated), `is_group_hom` (deprecated)\n\n## Tags\n\nis_group_hom, is_monoid_hom\n\n-/\n\nuniverses u v\nvariables {\u03b1 : Type u} {\u03b2 : Type v}\n\n/-- Predicate for maps which preserve an addition. -/\nstructure is_add_hom {\u03b1 \u03b2 : Type*} [has_add \u03b1] [has_add \u03b2] (f : \u03b1 \u2192 \u03b2) : Prop :=\n(map_add [] : \u2200 x y, f (x + y) = f x + f y)\n\n/-- Predicate for maps which preserve a multiplication. -/\n@[to_additive]\nstructure is_mul_hom {\u03b1 \u03b2 : Type*} [has_mul \u03b1] [has_mul \u03b2] (f : \u03b1 \u2192 \u03b2) : Prop :=\n(map_mul [] : \u2200 x y, f (x * y) = f x * f y)\n\nnamespace is_mul_hom\nvariables [has_mul \u03b1] [has_mul \u03b2] {\u03b3 : Type*} [has_mul \u03b3]\n\n/-- The identity map preserves multiplication. -/\n@[to_additive \"The identity map preserves addition\"]\nlemma id : is_mul_hom (id : \u03b1 \u2192 \u03b1) := {map_mul := \u03bb _ _, rfl}\n\n/-- The composition of maps which preserve multiplication, also preserves multiplication. -/\n@[to_additive \"The composition of addition preserving maps also preserves addition\"]\nlemma comp {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : is_mul_hom f) (hg : is_mul_hom g) : is_mul_hom (g \u2218 f) :=\n{ map_mul := \u03bb x y, by simp only [function.comp, hf.map_mul, hg.map_mul] }\n\n/-- A product of maps which preserve multiplication,\npreserves multiplication when the target is commutative. -/\n@[to_additive \"A sum of maps which preserves addition, preserves addition when the target\nis commutative.\"]\nlemma mul {\u03b1 \u03b2} [semigroup \u03b1] [comm_semigroup \u03b2]\n  {f g : \u03b1 \u2192 \u03b2} (hf : is_mul_hom f) (hg : is_mul_hom g) :\n  is_mul_hom (\u03bb a, f a * g a) :=\n{ map_mul := \u03bb a b, by simp only [hf.map_mul, hg.map_mul, mul_comm, mul_assoc, mul_left_comm] }\n\n/-- The inverse of a map which preserves multiplication,\npreserves multiplication when the target is commutative. -/\n@[to_additive \"The negation of a map which preserves addition, preserves addition when\nthe target is commutative.\"]\nlemma inv {\u03b1 \u03b2} [has_mul \u03b1] [comm_group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_mul_hom f) :\n  is_mul_hom (\u03bb a, (f a)\u207b\u00b9) :=\n{ map_mul := \u03bb a b, (hf.map_mul a b).symm \u25b8 mul_inv _ _ }\n\nend is_mul_hom\n\n/-- Predicate for add_monoid homomorphisms (deprecated -- use the bundled `monoid_hom` version). -/\nstructure is_add_monoid_hom [add_zero_class \u03b1] [add_zero_class \u03b2] (f : \u03b1 \u2192 \u03b2)\n  extends is_add_hom f : Prop :=\n(map_zero [] : f 0 = 0)\n\n/-- Predicate for monoid homomorphisms (deprecated -- use the bundled `monoid_hom` version). -/\n@[to_additive]\nstructure is_monoid_hom [mul_one_class \u03b1] [mul_one_class \u03b2] (f : \u03b1 \u2192 \u03b2)\n  extends is_mul_hom f : Prop :=\n(map_one [] : f 1 = 1)\n\nnamespace monoid_hom\n\nvariables {M : Type*} {N : Type*} [mM : mul_one_class M] [mN : mul_one_class N]\n\ninclude mM mN\n/-- Interpret a map `f : M \u2192 N` as a homomorphism `M \u2192* N`. -/\n@[to_additive \"Interpret a map `f : M \u2192 N` as a homomorphism `M \u2192+ N`.\"]\ndef of {f : M \u2192 N} (h : is_monoid_hom f) : M \u2192* N :=\n{ to_fun := f,\n  map_one' := h.2,\n  map_mul' := h.1.1 }\n\nvariables {mM mN}\n@[simp, to_additive]\nlemma coe_of {f : M \u2192 N} (hf : is_monoid_hom f) : \u21d1 (monoid_hom.of hf) = f :=\nrfl\n\n@[to_additive]\n\n\nend monoid_hom\n\nnamespace mul_equiv\n\nvariables {M : Type*} {N : Type*} [mul_one_class M] [mul_one_class N]\n\n/-- A multiplicative isomorphism preserves multiplication (deprecated). -/\n@[to_additive \"An additive isomorphism preserves addition (deprecated).\"]\ntheorem is_mul_hom (h : M \u2243* N) : is_mul_hom h := \u27e8h.map_mul\u27e9\n\n/-- A multiplicative bijection between two monoids is a monoid hom\n  (deprecated -- use `mul_equiv.to_monoid_hom`). -/\n@[to_additive \"An additive bijection between two additive monoids is an additive\nmonoid hom (deprecated). \"]\nlemma is_monoid_hom (h : M \u2243* N) : is_monoid_hom h :=\n{ map_mul := h.map_mul,\n  map_one := h.map_one }\n\nend mul_equiv\n\nnamespace is_monoid_hom\nvariables [mul_one_class \u03b1] [mul_one_class \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_monoid_hom f)\n\n/-- A monoid homomorphism preserves multiplication. -/\n@[to_additive \"An additive monoid homomorphism preserves addition.\"]\nlemma map_mul (x y) : f (x * y) = f x * f y :=\nhf.map_mul x y\n\n/-- The inverse of a map which preserves multiplication,\npreserves multiplication when the target is commutative. -/\n@[to_additive \"The negation of a map which preserves addition, preserves addition\nwhen the target is commutative.\"]\nlemma inv {\u03b1 \u03b2} [mul_one_class \u03b1] [comm_group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_monoid_hom f) :\n  is_monoid_hom (\u03bb a, (f a)\u207b\u00b9) :=\n{ map_one := hf.map_one.symm \u25b8 inv_one,\n  map_mul := \u03bb a b, (hf.map_mul a b).symm \u25b8 mul_inv _ _ }\n\nend is_monoid_hom\n\n/-- A map to a group preserving multiplication is a monoid homomorphism. -/\n@[to_additive \"A map to an additive group preserving addition is an additive monoid\nhomomorphism.\"]\ntheorem is_mul_hom.to_is_monoid_hom [mul_one_class \u03b1] [group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_mul_hom f) :\n  is_monoid_hom f :=\n{ map_one := mul_right_eq_self.1 $ by rw [\u2190 hf.map_mul, one_mul],\n  map_mul := hf.map_mul }\n\nnamespace is_monoid_hom\nvariables [mul_one_class \u03b1] [mul_one_class \u03b2] {f : \u03b1 \u2192 \u03b2}\n\n/-- The identity map is a monoid homomorphism. -/\n@[to_additive \"The identity map is an additive monoid homomorphism.\"]\nlemma id : is_monoid_hom (@id \u03b1) := { map_one := rfl, map_mul := \u03bb _ _, rfl }\n\n/-- The composite of two monoid homomorphisms is a monoid homomorphism. -/\n@[to_additive \"The composite of two additive monoid homomorphisms is an additive monoid\nhomomorphism.\"]\nlemma comp (hf : is_monoid_hom f) {\u03b3} [mul_one_class \u03b3] {g : \u03b2 \u2192 \u03b3} (hg : is_monoid_hom g) :\n  is_monoid_hom (g \u2218 f) :=\n{ map_one := show g _ = 1, by rw [hf.map_one, hg.map_one],\n  ..is_mul_hom.comp hf.to_is_mul_hom hg.to_is_mul_hom }\n\nend is_monoid_hom\n\nnamespace is_add_monoid_hom\n\n/-- Left multiplication in a ring is an additive monoid morphism. -/\nlemma is_add_monoid_hom_mul_left {\u03b3 : Type*} [non_unital_non_assoc_semiring \u03b3] (x : \u03b3) :\n  is_add_monoid_hom (\u03bb y : \u03b3, x * y) :=\n{ map_zero := mul_zero x, map_add := \u03bb y z, mul_add x y z }\n\n/-- Right multiplication in a ring is an additive monoid morphism. -/\nlemma is_add_monoid_hom_mul_right {\u03b3 : Type*} [non_unital_non_assoc_semiring \u03b3] (x : \u03b3) :\n  is_add_monoid_hom (\u03bb y : \u03b3, y * x) :=\n{ map_zero := zero_mul x, map_add := \u03bb y z, add_mul y z x }\n\nend is_add_monoid_hom\n\n/-- Predicate for additive group homomorphism (deprecated -- use bundled `monoid_hom`). -/\nstructure is_add_group_hom [add_group \u03b1] [add_group \u03b2] (f : \u03b1 \u2192 \u03b2) extends is_add_hom f : Prop\n\n/-- Predicate for group homomorphisms (deprecated -- use bundled `monoid_hom`). -/\n@[to_additive]\nstructure is_group_hom [group \u03b1] [group \u03b2] (f : \u03b1 \u2192 \u03b2) extends is_mul_hom f : Prop\n\n@[to_additive]\nlemma monoid_hom.is_group_hom {G H : Type*} {_ : group G} {_ : group H} (f : G \u2192* H) :\n  is_group_hom (f : G \u2192 H) :=\n{ map_mul := f.map_mul }\n\n@[to_additive]\nlemma mul_equiv.is_group_hom {G H : Type*} {_ : group G} {_ : group H} (h : G \u2243* H) :\n  is_group_hom h := { map_mul := h.map_mul }\n\n/-- Construct `is_group_hom` from its only hypothesis. -/\n@[to_additive \"Construct `is_add_group_hom` from its only hypothesis.\"]\nlemma is_group_hom.mk' [group \u03b1] [group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : \u2200 x y, f (x * y) = f x * f y) :\n  is_group_hom f :=\n{ map_mul := hf }\n\nnamespace is_group_hom\nvariables [group \u03b1] [group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_group_hom f)\nopen is_mul_hom (map_mul)\n\nlemma map_mul : \u2200 (x y), f (x * y) = f x * f y := hf.to_is_mul_hom.map_mul\n\n/-- A group homomorphism is a monoid homomorphism. -/\n@[to_additive \"An additive group homomorphism is an additive monoid homomorphism.\"]\nlemma to_is_monoid_hom : is_monoid_hom f :=\nhf.to_is_mul_hom.to_is_monoid_hom\n\n/-- A group homomorphism sends 1 to 1. -/\n@[to_additive \"An additive group homomorphism sends 0 to 0.\"]\nlemma map_one : f 1 = 1 := hf.to_is_monoid_hom.map_one\n\n/-- A group homomorphism sends inverses to inverses. -/\n@[to_additive \"An additive group homomorphism sends negations to negations.\"]\ntheorem map_inv (hf : is_group_hom f) (a : \u03b1) : f a\u207b\u00b9 = (f a)\u207b\u00b9 :=\neq_inv_of_mul_eq_one_left $ by rw [\u2190 hf.map_mul, inv_mul_self, hf.map_one]\n\n@[to_additive] lemma map_div (hf : is_group_hom f) (a b : \u03b1) : f (a / b) = f a / f b :=\nby simp_rw [div_eq_mul_inv, hf.map_mul, hf.map_inv]\n\n/-- The identity is a group homomorphism. -/\n@[to_additive \"The identity is an additive group homomorphism.\"]\nlemma id : is_group_hom (@id \u03b1) := { map_mul := \u03bb _ _, rfl}\n\n/-- The composition of two group homomorphisms is a group homomorphism. -/\n@[to_additive \"The composition of two additive group homomorphisms is an additive\ngroup homomorphism.\"]\nlemma comp (hf : is_group_hom f) {\u03b3} [group \u03b3] {g : \u03b2 \u2192 \u03b3} (hg : is_group_hom g) :\n  is_group_hom (g \u2218 f) :=\n{ ..is_mul_hom.comp hf.to_is_mul_hom hg.to_is_mul_hom }\n\n/-- A group homomorphism is injective iff its kernel is trivial. -/\n@[to_additive \"An additive group homomorphism is injective if its kernel is trivial.\"]\nlemma injective_iff {f : \u03b1 \u2192 \u03b2} (hf : is_group_hom f) :\n  function.injective f \u2194 (\u2200 a, f a = 1 \u2192 a = 1) :=\n\u27e8\u03bb h _, by rw \u2190 hf.map_one; exact @h _ _,\n  \u03bb h x y hxy, eq_of_div_eq_one $ h _ $ by rwa [hf.map_div, div_eq_one]\u27e9\n\n/-- The product of group homomorphisms is a group homomorphism if the target is commutative. -/\n@[to_additive \"The sum of two additive group homomorphisms is an additive group homomorphism\nif the target is commutative.\"]\nlemma mul {\u03b1 \u03b2} [group \u03b1] [comm_group \u03b2]\n  {f g : \u03b1 \u2192 \u03b2} (hf : is_group_hom f) (hg : is_group_hom g) :\n  is_group_hom (\u03bba, f a * g a) :=\n{ map_mul := (hf.to_is_mul_hom.mul hg.to_is_mul_hom).map_mul }\n\n/-- The inverse of a group homomorphism is a group homomorphism if the target is commutative. -/\n@[to_additive \"The negation of an additive group homomorphism is an additive group homomorphism\nif the target is commutative.\"]\nlemma inv {\u03b1 \u03b2} [group \u03b1] [comm_group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_group_hom f) :\n  is_group_hom (\u03bba, (f a)\u207b\u00b9) :=\n{ map_mul := hf.to_is_mul_hom.inv.map_mul }\n\nend is_group_hom\n\nnamespace ring_hom\n/-!\nThese instances look redundant, because `deprecated.ring` provides `is_ring_hom` for a `\u2192+*`.\nNevertheless these are harmless, and helpful for stripping out dependencies on `deprecated.ring`.\n-/\nvariables {R : Type*} {S : Type*}\n\nsection\nvariables [non_assoc_semiring R] [non_assoc_semiring S]\n\nlemma to_is_monoid_hom (f : R \u2192+* S) : is_monoid_hom f :=\n{ map_one := f.map_one,\n  map_mul := f.map_mul }\n\nlemma to_is_add_monoid_hom (f : R \u2192+* S) : is_add_monoid_hom f :=\n{ map_zero := f.map_zero,\n  map_add := f.map_add }\nend\n\nsection\nvariables [ring R] [ring S]\n\nlemma to_is_add_group_hom (f : R \u2192+* S) : is_add_group_hom f :=\n{ map_add := f.map_add }\nend\n\nend ring_hom\n\n/-- Inversion is a group homomorphism if the group is commutative. -/\n@[to_additive neg.is_add_group_hom\n\"Negation is an `add_group` homomorphism if the `add_group` is commutative.\"]\nlemma inv.is_group_hom [comm_group \u03b1] : is_group_hom (has_inv.inv : \u03b1 \u2192 \u03b1) :=\n{ map_mul := mul_inv }\n\n/-- The difference of two additive group homomorphisms is an additive group\nhomomorphism if the target is commutative. -/\nlemma is_add_group_hom.sub {\u03b1 \u03b2} [add_group \u03b1] [add_comm_group \u03b2]\n  {f g : \u03b1 \u2192 \u03b2} (hf : is_add_group_hom f) (hg : is_add_group_hom g) :\n  is_add_group_hom (\u03bba, f a - g a) :=\nby simpa only [sub_eq_add_neg] using hf.add hg.neg\n\nnamespace units\n\nvariables {M : Type*} {N : Type*} [monoid M] [monoid N]\n\n/-- The group homomorphism on units induced by a multiplicative morphism. -/\n@[reducible] def map' {f : M \u2192 N} (hf : is_monoid_hom f) : M\u02e3 \u2192* N\u02e3 :=\n  map (monoid_hom.of hf)\n\n@[simp] lemma coe_map' {f : M \u2192 N} (hf : is_monoid_hom f) (x : M\u02e3) :\n  \u2191((map' hf : M\u02e3 \u2192 N\u02e3) x) = f x :=\nrfl\n\nlemma coe_is_monoid_hom : is_monoid_hom (coe : M\u02e3 \u2192 M) := (coe_hom M).is_monoid_hom_coe\n\nend units\n\nnamespace is_unit\n\nvariables {M : Type*} {N : Type*} [monoid M] [monoid N] {x : M}\n\nlemma map' {f : M \u2192 N} (hf :is_monoid_hom f) {x : M} (h : is_unit x) :\n  is_unit (f x) :=\nh.map (monoid_hom.of hf)\n\nend is_unit\n\nlemma additive.is_add_hom [has_mul \u03b1] [has_mul \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_mul_hom f) :\n  @is_add_hom (additive \u03b1) (additive \u03b2) _ _ f :=\n{ map_add := is_mul_hom.map_mul hf }\n\nlemma multiplicative.is_mul_hom [has_add \u03b1] [has_add \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_add_hom f) :\n  @is_mul_hom (multiplicative \u03b1) (multiplicative \u03b2) _ _ f :=\n{ map_mul := is_add_hom.map_add hf }\n\n-- defeq abuse\nlemma additive.is_add_monoid_hom [mul_one_class \u03b1] [mul_one_class \u03b2] {f : \u03b1 \u2192 \u03b2}\n  (hf : is_monoid_hom f) : @is_add_monoid_hom (additive \u03b1) (additive \u03b2) _ _ f :=\n{ map_zero := hf.map_one,\n  ..additive.is_add_hom hf.to_is_mul_hom }\n\nlemma multiplicative.is_monoid_hom\n  [add_zero_class \u03b1] [add_zero_class \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_add_monoid_hom f) :\n  @is_monoid_hom (multiplicative \u03b1) (multiplicative \u03b2) _ _ f :=\n{ map_one := is_add_monoid_hom.map_zero hf,\n  ..multiplicative.is_mul_hom hf.to_is_add_hom }\n\nlemma additive.is_add_group_hom [group \u03b1] [group \u03b2] {f : \u03b1 \u2192 \u03b2} (hf : is_group_hom f) :\n  @is_add_group_hom (additive \u03b1) (additive \u03b2) _ _ f :=\n{ map_add := hf.to_is_mul_hom.map_mul }\n\nlemma multiplicative.is_group_hom [add_group \u03b1] [add_group \u03b2] {f : \u03b1 \u2192 \u03b2}\n  (hf : is_add_group_hom f) : @is_group_hom (multiplicative \u03b1) (multiplicative \u03b2) _ _ f :=\n{ map_mul := hf.to_is_add_hom.map_add }\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/deprecated/group.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8221891305219504, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.3822370187457747}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.set.lattice\nimport logic.small.basic\nimport order.well_founded\n\n/-!\n# A model of ZFC\n\nIn this file, we model Zermelo-Fraenkel set theory (+ Choice) using Lean's underlying type theory.\nWe do this in four main steps:\n* Define pre-sets inductively.\n* Define extensional equivalence on pre-sets and give it a `setoid` instance.\n* Define ZFC sets by quotienting pre-sets by extensional equivalence.\n* Define classes as sets of ZFC sets.\nThen the rest is usual set theory.\n\n## The model\n\n* `pSet`: Pre-set. A pre-set is inductively defined by its indexing type and its members, which are\n  themselves pre-sets.\n* `Set`: ZFC set. Defined as `pSet` quotiented by `pSet.equiv`, the extensional equivalence.\n* `Class`: Class. Defined as `set Set`.\n* `Set.choice`: Axiom of choice. Proved from Lean's axiom of choice.\n\n## Other definitions\n\n* `arity \u03b1 n`: `n`-ary function `\u03b1 \u2192 \u03b1 \u2192 ... \u2192 \u03b1`. Defined inductively.\n* `arity.const a n`: `n`-ary constant function equal to `a`.\n* `pSet.type`: Underlying type of a pre-set.\n* `pSet.func`: Underlying family of pre-sets of a pre-set.\n* `pSet.equiv`: Extensional equivalence of pre-sets. Defined inductively.\n* `pSet.omega`, `Set.omega`: The von Neumann ordinal `\u03c9` as a `pSet`, as a `Set`.\n* `pSet.arity.equiv`: Extensional equivalence of `n`-ary `pSet`-valued functions. Extension of\n  `pSet.equiv`.\n* `pSet.resp`: Collection of `n`-ary `pSet`-valued functions that respect extensional equivalence.\n* `pSet.eval`: Turns a `pSet`-valued function that respect extensional equivalence into a\n  `Set`-valued function.\n* `classical.all_definable`: All functions are classically definable.\n* `Set.is_func` : Predicate that a ZFC set is a subset of `x \u00d7 y` that can be considered as a ZFC\n  function `x \u2192 y`. That is, each member of `x` is related by the ZFC set to exactly one member of\n  `y`.\n* `Set.funs`: ZFC set of ZFC functions `x \u2192 y`.\n* `Set.hereditarily p x`: Predicate that every set in the transitive closure of `x` has property\n  `p`.\n* `Class.iota`: Definite description operator.\n\n## Notes\n\nTo avoid confusion between the Lean `set` and the ZFC `Set`, docstrings in this file refer to them\nrespectively as \"`set`\" and \"ZFC set\".\n\n## TODO\n\nProve `Set.map_definable_aux` computably.\n-/\n\nuniverses u v\n\n/-- The type of `n`-ary functions `\u03b1 \u2192 \u03b1 \u2192 ... \u2192 \u03b1`. -/\ndef arity (\u03b1 : Type u) : \u2115 \u2192 Type u\n| 0     := \u03b1\n| (n+1) := \u03b1 \u2192 arity n\n\n@[simp] theorem arity_zero (\u03b1 : Type u) : arity \u03b1 0 = \u03b1 := rfl\n@[simp] theorem arity_succ (\u03b1 : Type u) (n : \u2115) : arity \u03b1 n.succ = (\u03b1 \u2192 arity \u03b1 n) := rfl\n\nnamespace arity\n\n/-- Constant `n`-ary function with value `a`. -/\ndef const {\u03b1 : Type u} (a : \u03b1) : \u2200 n, arity \u03b1 n\n| 0     := a\n| (n+1) := \u03bb _, const n\n\n@[simp] theorem const_zero {\u03b1 : Type u} (a : \u03b1) : const a 0 = a := rfl\n@[simp] theorem const_succ {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : const a n.succ = \u03bb _, const a n := rfl\ntheorem const_succ_apply {\u03b1 : Type u} (a : \u03b1) (n : \u2115) (x : \u03b1) : const a n.succ x = const a n := rfl\n\ninstance arity.inhabited {\u03b1 n} [inhabited \u03b1] : inhabited (arity \u03b1 n) := \u27e8const default _\u27e9\n\nend arity\n\n/-- The type of pre-sets in universe `u`. A pre-set\n  is a family of pre-sets indexed by a type in `Type u`.\n  The ZFC universe is defined as a quotient of this\n  to ensure extensionality. -/\ninductive pSet : Type (u+1)\n| mk (\u03b1 : Type u) (A : \u03b1 \u2192 pSet) : pSet\n\nnamespace pSet\n\n/-- The underlying type of a pre-set -/\ndef type : pSet \u2192 Type u\n| \u27e8\u03b1, A\u27e9 := \u03b1\n\n/-- The underlying pre-set family of a pre-set -/\ndef func : \u03a0 (x : pSet), x.type \u2192 pSet\n| \u27e8\u03b1, A\u27e9 := A\n\n@[simp] theorem mk_type (\u03b1 A) : type \u27e8\u03b1, A\u27e9 = \u03b1 := rfl\n@[simp] theorem mk_func (\u03b1 A) : func \u27e8\u03b1, A\u27e9 = A := rfl\n\n@[simp] theorem eta : \u03a0 (x : pSet), mk x.type x.func = x\n| \u27e8\u03b1, A\u27e9 := rfl\n\n/-- Two pre-sets are extensionally equivalent if every element of the first family is extensionally\nequivalent to some element of the second family and vice-versa. -/\ndef equiv (x y : pSet) : Prop :=\npSet.rec (\u03bb \u03b1 z m \u27e8\u03b2, B\u27e9, (\u2200 a, \u2203 b, m a (B b)) \u2227 (\u2200 b, \u2203 a, m a (B b))) x y\n\ntheorem equiv_iff : \u03a0 {x y : pSet}, equiv x y \u2194\n  (\u2200 i, \u2203 j, equiv (x.func i) (y.func j)) \u2227 (\u2200 j, \u2203 i, equiv (x.func i) (y.func j))\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 := iff.rfl\n\ntheorem equiv.exists_left {x y : pSet} (h : equiv x y) : \u2200 i, \u2203 j, equiv (x.func i) (y.func j) :=\n(equiv_iff.1 h).1\n\ntheorem equiv.exists_right {x y : pSet} (h : equiv x y) : \u2200 j, \u2203 i, equiv (x.func i) (y.func j) :=\n(equiv_iff.1 h).2\n\n@[refl] protected theorem equiv.refl (x) : equiv x x :=\npSet.rec_on x $ \u03bb \u03b1 A IH, \u27e8\u03bb a, \u27e8a, IH a\u27e9, \u03bb a, \u27e8a, IH a\u27e9\u27e9\n\nprotected theorem equiv.rfl : \u2200 {x}, equiv x x := equiv.refl\n\nprotected theorem equiv.euc {x} : \u03a0 {y z}, equiv x y \u2192 equiv z y \u2192 equiv x z :=\npSet.rec_on x $ \u03bb \u03b1 A IH y, pSet.cases_on y $ \u03bb \u03b2 B \u27e8\u03b3, \u0393\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9 \u27e8\u03b3\u03b2, \u03b2\u03b3\u27e9,\n\u27e8\u03bb a, let \u27e8b, ab\u27e9 := \u03b1\u03b2 a, \u27e8c, bc\u27e9 := \u03b2\u03b3 b in \u27e8c, IH a ab bc\u27e9,\n  \u03bb c, let \u27e8b, cb\u27e9 := \u03b3\u03b2 c, \u27e8a, ba\u27e9 := \u03b2\u03b1 b in \u27e8a, IH a ba cb\u27e9\u27e9\n\n@[symm] protected theorem equiv.symm {x y} : equiv x y \u2192 equiv y x :=\n(equiv.refl y).euc\n\nprotected theorem equiv.comm {x y} : equiv x y \u2194 equiv y x :=\n\u27e8equiv.symm, equiv.symm\u27e9\n\n@[trans] protected theorem equiv.trans {x y z} (h1 : equiv x y) (h2 : equiv y z) : equiv x z :=\nh1.euc h2.symm\n\nprotected theorem equiv_of_is_empty (x y : pSet) [is_empty x.type] [is_empty y.type] : equiv x y :=\nequiv_iff.2 $ by simp\n\ninstance setoid : setoid pSet :=\n\u27e8pSet.equiv, equiv.refl, \u03bb x y, equiv.symm, \u03bb x y z, equiv.trans\u27e9\n\n/-- A pre-set is a subset of another pre-set if every element of the first family is extensionally\nequivalent to some element of the second family.-/\nprotected def subset (x y : pSet) : Prop := \u2200 a, \u2203 b, equiv (x.func a) (y.func b)\n\ninstance : has_subset pSet := \u27e8pSet.subset\u27e9\n\ninstance : is_refl pSet (\u2286) := \u27e8\u03bb x a, \u27e8a, equiv.refl _\u27e9\u27e9\n\ninstance : is_trans pSet (\u2286) :=\n\u27e8\u03bb x y z hxy hyz a, begin\n  cases hxy a with b hb,\n  cases hyz b with c hc,\n  exact \u27e8c, hb.trans hc\u27e9\nend\u27e9\n\ntheorem equiv.ext : \u03a0 (x y : pSet), equiv x y \u2194 (x \u2286 y \u2227 y \u2286 x)\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 :=\n  \u27e8\u03bb \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9, \u27e8\u03b1\u03b2, \u03bb b, let \u27e8a, h\u27e9 := \u03b2\u03b1 b in \u27e8a, equiv.symm h\u27e9\u27e9,\n    \u03bb \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9, \u27e8\u03b1\u03b2, \u03bb b, let \u27e8a, h\u27e9 := \u03b2\u03b1 b in \u27e8a, equiv.symm h\u27e9\u27e9\u27e9\n\ntheorem subset.congr_left : \u03a0 {x y z : pSet}, equiv x y \u2192 (x \u2286 z \u2194 y \u2286 z)\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b3, \u0393\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9 :=\n  \u27e8\u03bb \u03b1\u03b3 b, let \u27e8a, ba\u27e9 := \u03b2\u03b1 b, \u27e8c, ac\u27e9 := \u03b1\u03b3 a in \u27e8c, (equiv.symm ba).trans ac\u27e9,\n    \u03bb \u03b2\u03b3 a, let \u27e8b, ab\u27e9 := \u03b1\u03b2 a, \u27e8c, bc\u27e9 := \u03b2\u03b3 b in \u27e8c, equiv.trans ab bc\u27e9\u27e9\n\ntheorem subset.congr_right : \u03a0 {x y z : pSet}, equiv x y \u2192 (z \u2286 x \u2194 z \u2286 y)\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b3, \u0393\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9 :=\n  \u27e8\u03bb \u03b3\u03b1 c, let \u27e8a, ca\u27e9 := \u03b3\u03b1 c, \u27e8b, ab\u27e9 := \u03b1\u03b2 a in \u27e8b, ca.trans ab\u27e9,\n    \u03bb \u03b3\u03b2 c, let \u27e8b, cb\u27e9 := \u03b3\u03b2 c, \u27e8a, ab\u27e9 := \u03b2\u03b1 b in \u27e8a, cb.trans (equiv.symm ab)\u27e9\u27e9\n\n/-- `x \u2208 y` as pre-sets if `x` is extensionally equivalent to a member of the family `y`. -/\nprotected def mem (x y : pSet.{u}) : Prop := \u2203 b, equiv x (y.func b)\n\ninstance : has_mem pSet pSet := \u27e8pSet.mem\u27e9\n\ntheorem mem.mk {\u03b1 : Type u} (A : \u03b1 \u2192 pSet) (a : \u03b1) : A a \u2208 mk \u03b1 A :=\n\u27e8a, equiv.refl (A a)\u27e9\n\ntheorem func_mem (x : pSet) (i : x.type) : x.func i \u2208 x :=\nby { cases x, apply mem.mk }\n\ntheorem mem.ext : \u03a0 {x y : pSet.{u}}, (\u2200 w : pSet.{u}, w \u2208 x \u2194 w \u2208 y) \u2192 equiv x y\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 h := \u27e8\u03bb a, (h (A a)).1 (mem.mk A a),\n    \u03bb b, let \u27e8a, ha\u27e9 := (h (B b)).2 (mem.mk B b) in \u27e8a, ha.symm\u27e9\u27e9\n\ntheorem mem.congr_right : \u03a0 {x y : pSet.{u}}, equiv x y \u2192 (\u2200 {w : pSet.{u}}, w \u2208 x \u2194 w \u2208 y)\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9 w :=\n  \u27e8\u03bb \u27e8a, ha\u27e9, let \u27e8b, hb\u27e9 := \u03b1\u03b2 a in \u27e8b, ha.trans hb\u27e9,\n    \u03bb \u27e8b, hb\u27e9, let \u27e8a, ha\u27e9 := \u03b2\u03b1 b in \u27e8a, hb.euc ha\u27e9\u27e9\n\ntheorem equiv_iff_mem {x y : pSet.{u}} : equiv x y \u2194 (\u2200 {w : pSet.{u}}, w \u2208 x \u2194 w \u2208 y) :=\n\u27e8mem.congr_right, match x, y with\n| \u27e8\u03b1, A\u27e9, \u27e8\u03b2, B\u27e9, h := \u27e8\u03bb a, h.1 (mem.mk A a), \u03bb b,\n  let \u27e8a, h\u27e9 := h.2 (mem.mk B b) in \u27e8a, h.symm\u27e9\u27e9\nend\u27e9\n\ntheorem mem.congr_left : \u03a0 {x y : pSet.{u}}, equiv x y \u2192 (\u2200 {w : pSet.{u}}, x \u2208 w \u2194 y \u2208 w)\n| x y h \u27e8\u03b1, A\u27e9 := \u27e8\u03bb \u27e8a, ha\u27e9, \u27e8a, h.symm.trans ha\u27e9, \u03bb \u27e8a, ha\u27e9, \u27e8a, h.trans ha\u27e9\u27e9\n\nprivate theorem mem_wf_aux : \u03a0 {x y : pSet.{u}}, equiv x y \u2192 acc (\u2208) y\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 H := \u27e8_, begin\n  rintros \u27e8\u03b3, C\u27e9 \u27e8b, hc\u27e9,\n  cases H.exists_right b with a ha,\n  have H := ha.trans hc.symm,\n  rw mk_func at H,\n  exact mem_wf_aux H\nend\u27e9\n\ntheorem mem_wf : @well_founded pSet (\u2208) := \u27e8\u03bb x, mem_wf_aux $ equiv.refl x\u27e9\n\ninstance : has_well_founded pSet := \u27e8_, mem_wf\u27e9\ninstance : is_asymm pSet (\u2208) := mem_wf.is_asymm\n\ntheorem mem_asymm {x y : pSet} : x \u2208 y \u2192 y \u2209 x := asymm\ntheorem mem_irrefl (x : pSet) : x \u2209 x := irrefl x\n\n/-- Convert a pre-set to a `set` of pre-sets. -/\ndef to_set (u : pSet.{u}) : set pSet.{u} := {x | x \u2208 u}\n\n@[simp] theorem mem_to_set (a u : pSet.{u}) : a \u2208 u.to_set \u2194 a \u2208 u := iff.rfl\n\n/-- A nonempty set is one that contains some element. -/\nprotected def nonempty (u : pSet) : Prop := u.to_set.nonempty\n\ntheorem nonempty_def (u : pSet) : u.nonempty \u2194 \u2203 x, x \u2208 u := iff.rfl\n\ntheorem nonempty_of_mem {x u : pSet} (h : x \u2208 u) : u.nonempty := \u27e8x, h\u27e9\n\n@[simp] theorem nonempty_to_set_iff {u : pSet} : u.to_set.nonempty \u2194 u.nonempty := iff.rfl\n\ntheorem nonempty_type_iff_nonempty {x : pSet} : nonempty x.type \u2194 pSet.nonempty x :=\n\u27e8\u03bb \u27e8i\u27e9, \u27e8_, func_mem _ i\u27e9, \u03bb \u27e8i, j, h\u27e9, \u27e8j\u27e9\u27e9\n\ntheorem nonempty_of_nonempty_type (x : pSet) [h : nonempty x.type] : pSet.nonempty x :=\nnonempty_type_iff_nonempty.1 h\n\n/-- Two pre-sets are equivalent iff they have the same members. -/\ntheorem equiv.eq {x y : pSet} : equiv x y \u2194 to_set x = to_set y :=\nequiv_iff_mem.trans set.ext_iff.symm\n\ninstance : has_coe pSet (set pSet) := \u27e8to_set\u27e9\n\n/-- The empty pre-set -/\nprotected def empty : pSet := \u27e8_, pempty.elim\u27e9\n\ninstance : has_emptyc pSet := \u27e8pSet.empty\u27e9\n\ninstance : inhabited pSet := \u27e8\u2205\u27e9\n\ninstance : is_empty (type (\u2205)) := pempty.is_empty\n\n@[simp] theorem not_mem_empty (x : pSet.{u}) : x \u2209 (\u2205 : pSet.{u}) := is_empty.exists_iff.1\n\n@[simp] theorem to_set_empty : to_set \u2205 = \u2205 := by simp [to_set]\n\n@[simp] theorem empty_subset (x : pSet.{u}) : (\u2205 : pSet) \u2286 x := \u03bb x, x.elim\n\n@[simp] theorem not_nonempty_empty : \u00ac pSet.nonempty \u2205 := by simp [pSet.nonempty]\n\nprotected theorem equiv_empty (x : pSet) [is_empty x.type] : equiv x \u2205 :=\npSet.equiv_of_is_empty x _\n\n/-- Insert an element into a pre-set -/\nprotected def insert (x y : pSet) : pSet := \u27e8option y.type, \u03bb o, option.rec x y.func o\u27e9\n\ninstance : has_insert pSet pSet := \u27e8pSet.insert\u27e9\n\ninstance : has_singleton pSet pSet := \u27e8\u03bb s, insert s \u2205\u27e9\n\ninstance : is_lawful_singleton pSet pSet := \u27e8\u03bb _, rfl\u27e9\n\ninstance (x y : pSet) : inhabited (insert x y).type := option.inhabited _\n\n/-- The n-th von Neumann ordinal -/\ndef of_nat : \u2115 \u2192 pSet\n| 0     := \u2205\n| (n+1) := insert (of_nat n) (of_nat n)\n\n/-- The von Neumann ordinal \u03c9 -/\ndef omega : pSet := \u27e8ulift \u2115, \u03bb n, of_nat n.down\u27e9\n\n/-- The pre-set separation operation `{x \u2208 a | p x}` -/\nprotected def sep (p : pSet \u2192 Prop) (x : pSet) : pSet := \u27e8{a // p (x.func a)}, \u03bb y, x.func y.1\u27e9\n\ninstance : has_sep pSet pSet := \u27e8pSet.sep\u27e9\n\n/-- The pre-set powerset operator -/\ndef powerset (x : pSet) : pSet := \u27e8set x.type, \u03bb p, \u27e8{a // p a}, \u03bb y, x.func y.1\u27e9\u27e9\n\n@[simp] theorem mem_powerset : \u03a0 {x y : pSet}, y \u2208 powerset x \u2194 y \u2286 x\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 := \u27e8\u03bb \u27e8p, e\u27e9, (subset.congr_left e).2 $ \u03bb \u27e8a, pa\u27e9, \u27e8a, equiv.refl (A a)\u27e9,\n  \u03bb \u03b2\u03b1, \u27e8{a | \u2203 b, equiv (B b) (A a)}, \u03bb b, let \u27e8a, ba\u27e9 := \u03b2\u03b1 b in \u27e8\u27e8a, b, ba\u27e9, ba\u27e9,\n    \u03bb \u27e8a, b, ba\u27e9, \u27e8b, ba\u27e9\u27e9\u27e9\n\n/-- The pre-set union operator -/\ndef sUnion (a : pSet) : pSet := \u27e8\u03a3 x, (a.func x).type, \u03bb \u27e8x, y\u27e9, (a.func x).func y\u27e9\n\nprefix (name := pSet.sUnion) `\u22c3\u2080 `:110 := pSet.sUnion\n\n@[simp] theorem mem_sUnion : \u03a0 {x y : pSet.{u}}, y \u2208 \u22c3\u2080 x \u2194 \u2203 z \u2208 x, y \u2208 z\n| \u27e8\u03b1, A\u27e9 y :=\n  \u27e8\u03bb \u27e8\u27e8a, c\u27e9, (e : equiv y ((A a).func c))\u27e9,\n    have func (A a) c \u2208 mk (A a).type (A a).func, from mem.mk (A a).func c,\n    \u27e8_, mem.mk _ _, (mem.congr_left e).2 (by rwa eta at this)\u27e9,\n  \u03bb \u27e8\u27e8\u03b2, B\u27e9, \u27e8a, (e : equiv (mk \u03b2 B) (A a))\u27e9, \u27e8b, yb\u27e9\u27e9,\n    by { rw \u2190(eta (A a)) at e, exact\n    let \u27e8\u03b2t, t\u03b2\u27e9 := e, \u27e8c, bc\u27e9 := \u03b2t b in \u27e8\u27e8a, c\u27e9, yb.trans bc\u27e9 }\u27e9\n\n@[simp] theorem to_set_sUnion (x : pSet.{u}) : (\u22c3\u2080 x).to_set = \u22c3\u2080 (to_set '' x.to_set) :=\nby { ext, simp }\n\n/-- The image of a function from pre-sets to pre-sets. -/\ndef image (f : pSet.{u} \u2192 pSet.{u}) (x : pSet.{u}) : pSet := \u27e8x.type, f \u2218 x.func\u27e9\n\ntheorem mem_image {f : pSet.{u} \u2192 pSet.{u}} (H : \u2200 {x y}, equiv x y \u2192 equiv (f x) (f y)) :\n  \u03a0 {x y : pSet.{u}}, y \u2208 image f x \u2194 \u2203 z \u2208 x, equiv y (f z)\n| \u27e8\u03b1, A\u27e9 y := \u27e8\u03bb \u27e8a, ya\u27e9, \u27e8A a, mem.mk A a, ya\u27e9, \u03bb \u27e8z, \u27e8a, za\u27e9, yz\u27e9, \u27e8a, yz.trans (H za)\u27e9\u27e9\n\n/-- Universe lift operation -/\nprotected def lift : pSet.{u} \u2192 pSet.{max u v}\n| \u27e8\u03b1, A\u27e9 := \u27e8ulift \u03b1, \u03bb \u27e8x\u27e9, lift (A x)\u27e9\n\n/-- Embedding of one universe in another -/\n@[nolint check_univs] -- intended to be used with explicit universe parameters\ndef embed : pSet.{max (u+1) v} := \u27e8ulift.{v u+1} pSet, \u03bb \u27e8x\u27e9, pSet.lift.{u (max (u+1) v)} x\u27e9\n\ntheorem lift_mem_embed : \u03a0 (x : pSet.{u}), pSet.lift.{u (max (u+1) v)} x \u2208 embed.{u v} :=\n\u03bb x, \u27e8\u27e8x\u27e9, equiv.rfl\u27e9\n\n/-- Function equivalence is defined so that `f ~ g` iff `\u2200 x y, x ~ y \u2192 f x ~ g y`. This extends to\nequivalence of `n`-ary functions. -/\ndef arity.equiv : \u03a0 {n}, arity pSet.{u} n \u2192 arity pSet.{u} n \u2192 Prop\n| 0     a b := equiv a b\n| (n+1) a b := \u2200 x y, equiv x y \u2192 arity.equiv (a x) (b y)\n\nlemma arity.equiv_const {a : pSet.{u}} : \u2200 n, arity.equiv (arity.const a n) (arity.const a n)\n| 0     := equiv.rfl\n| (n+1) := \u03bb x y h, arity.equiv_const _\n\n/-- `resp n` is the collection of n-ary functions on `pSet` that respect\n  equivalence, i.e. when the inputs are equivalent the output is as well. -/\ndef resp (n) := {x : arity pSet.{u} n // arity.equiv x x}\n\ninstance resp.inhabited {n} : inhabited (resp n) :=\n\u27e8\u27e8arity.const default _, arity.equiv_const _\u27e9\u27e9\n\n/-- The `n`-ary image of a `(n + 1)`-ary function respecting equivalence as a function respecting\nequivalence. -/\ndef resp.f {n} (f : resp (n+1)) (x : pSet) : resp n :=\n\u27e8f.1 x, f.2 _ _ $ equiv.refl x\u27e9\n\n/-- Function equivalence for functions respecting equivalence. See `pSet.arity.equiv`. -/\ndef resp.equiv {n} (a b : resp n) : Prop := arity.equiv a.1 b.1\n\nprotected theorem resp.equiv.refl {n} (a : resp n) : resp.equiv a a := a.2\n\nprotected theorem resp.equiv.euc : \u03a0 {n} {a b c : resp n},\n  resp.equiv a b \u2192 resp.equiv c b \u2192 resp.equiv a c\n| 0     a b c hab hcb := equiv.euc hab hcb\n| (n+1) a b c hab hcb := \u03bb x y h,\n  @resp.equiv.euc n (a.f x) (b.f y) (c.f y) (hab _ _ h) (hcb _ _ $ equiv.refl y)\n\nprotected theorem resp.equiv.symm {n} {a b : resp n} : resp.equiv a b \u2192 resp.equiv b a :=\n(resp.equiv.refl b).euc\n\nprotected theorem resp.equiv.trans {n} {x y z : resp n}\n  (h1 : resp.equiv x y) (h2 : resp.equiv y z) : resp.equiv x z :=\nh1.euc h2.symm\n\ninstance resp.setoid {n} : setoid (resp n) :=\n\u27e8resp.equiv, resp.equiv.refl, \u03bb x y, resp.equiv.symm, \u03bb x y z, resp.equiv.trans\u27e9\n\nend pSet\n\n/-- The ZFC universe of sets consists of the type of pre-sets,\n  quotiented by extensional equivalence. -/\ndef Set : Type (u+1) := quotient pSet.setoid.{u}\n\nnamespace pSet\n\nnamespace resp\n\n/-- Helper function for `pSet.eval`. -/\ndef eval_aux : \u03a0 {n}, {f : resp n \u2192 arity Set.{u} n // \u2200 (a b : resp n), resp.equiv a b \u2192 f a = f b}\n| 0     := \u27e8\u03bb a, \u27e6a.1\u27e7, \u03bb a b h, quotient.sound h\u27e9\n| (n+1) := let F : resp (n + 1) \u2192 arity Set (n + 1) := \u03bb a, @quotient.lift _ _ pSet.setoid\n    (\u03bb x, eval_aux.1 (a.f x)) (\u03bb b c h, eval_aux.2 _ _ (a.2 _ _ h)) in\n  \u27e8F, \u03bb b c h, funext $ @quotient.ind _ _ (\u03bb q, F b q = F c q) $ \u03bb z,\n  eval_aux.2 (resp.f b z) (resp.f c z) (h _ _ (pSet.equiv.refl z))\u27e9\n\n/-- An equivalence-respecting function yields an n-ary ZFC set function. -/\ndef eval (n) : resp n \u2192 arity Set.{u} n := eval_aux.1\n\ntheorem eval_val {n f x} : (@eval (n+1) f : Set \u2192 arity Set n) \u27e6x\u27e7 = eval n (resp.f f x) := rfl\n\nend resp\n\n/-- A set function is \"definable\" if it is the image of some n-ary pre-set\n  function. This isn't exactly definability, but is useful as a sufficient\n  condition for functions that have a computable image. -/\nclass inductive definable (n) : arity Set.{u} n \u2192 Type (u+1)\n| mk (f) : definable (resp.eval n f)\n\nattribute [instance] definable.mk\n\n/-- The evaluation of a function respecting equivalence is definable, by that same function. -/\ndef definable.eq_mk {n} (f) : \u03a0 {s : arity Set.{u} n} (H : resp.eval _ f = s), definable n s\n| ._ rfl := \u27e8f\u27e9\n\n/-- Turns a definable function into a function that respects equivalence. -/\ndef definable.resp {n} : \u03a0 (s : arity Set.{u} n) [definable n s], resp n\n| ._ \u27e8f\u27e9 := f\n\ntheorem definable.eq {n} :\n  \u03a0 (s : arity Set.{u} n) [H : definable n s], (@definable.resp n s H).eval _ = s\n| ._ \u27e8f\u27e9 := rfl\n\nend pSet\n\nnamespace classical\nopen pSet\n\n/-- All functions are classically definable. -/\nnoncomputable def all_definable : \u03a0 {n} (F : arity Set.{u} n), definable n F\n| 0     F := let p := @quotient.exists_rep pSet _ F in\n              definable.eq_mk \u27e8some p, equiv.rfl\u27e9 (some_spec p)\n| (n+1) (F : arity Set.{u} (n + 1)) := begin\n    have I := \u03bb x, (all_definable (F x)),\n    refine definable.eq_mk \u27e8\u03bb x : pSet, (@definable.resp _ _ (I \u27e6x\u27e7)).1, _\u27e9 _,\n    { dsimp [arity.equiv],\n      introsI x y h,\n      rw @quotient.sound pSet _ _ _ h,\n      exact (definable.resp (F \u27e6y\u27e7)).2 },\n    refine funext (\u03bb q, quotient.induction_on q $ \u03bb x, _),\n    simp_rw [resp.eval_val, resp.f, subtype.val_eq_coe, subtype.coe_eta],\n    exact @definable.eq _ (F \u27e6x\u27e7) (I \u27e6x\u27e7),\n  end\n\nend classical\n\nnamespace Set\nopen pSet\n\n/-- Turns a pre-set into a ZFC set. -/\ndef mk : pSet \u2192 Set := quotient.mk\n\n@[simp] theorem mk_eq (x : pSet) : @eq Set \u27e6x\u27e7 (mk x) := rfl\n@[simp] theorem mk_out : \u2200 x : Set, mk x.out = x := quotient.out_eq\ntheorem eq {x y : pSet} : mk x = mk y \u2194 equiv x y := quotient.eq\ntheorem sound {x y : pSet} (h : pSet.equiv x y) : mk x = mk y := quotient.sound h\ntheorem exact {x y : pSet} : mk x = mk y \u2192 pSet.equiv x y := quotient.exact\n\n@[simp] lemma eval_mk {n f x} :\n  (@resp.eval (n+1) f : Set \u2192 arity Set n) (mk x) = resp.eval n (resp.f f x) :=\nrfl\n\n/-- The membership relation for ZFC sets is inherited from the membership relation for pre-sets. -/\nprotected def mem : Set \u2192 Set \u2192 Prop :=\nquotient.lift\u2082 pSet.mem\n  (\u03bb x y x' y' hx hy, propext ((mem.congr_left hx).trans (mem.congr_right hy)))\n\ninstance : has_mem Set Set := \u27e8Set.mem\u27e9\n\n@[simp] theorem mk_mem_iff {x y : pSet} : mk x \u2208 mk y \u2194 x \u2208 y := iff.rfl\n\n/-- Convert a ZFC set into a `set` of ZFC sets -/\ndef to_set (u : Set.{u}) : set Set.{u} := {x | x \u2208 u}\n\n@[simp] theorem mem_to_set (a u : Set.{u}) : a \u2208 u.to_set \u2194 a \u2208 u := iff.rfl\n\ninstance small_to_set (x : Set.{u}) : small.{u} x.to_set :=\nquotient.induction_on x $ \u03bb a, begin\n  let f : a.type \u2192 (mk a).to_set := \u03bb i, \u27e8mk $ a.func i, func_mem a i\u27e9,\n  suffices : function.surjective f,\n  { exact small_of_surjective this },\n  rintro \u27e8y, hb\u27e9,\n  induction y using quotient.induction_on,\n  cases hb with i h,\n  exact \u27e8i, subtype.coe_injective (quotient.sound h.symm)\u27e9\nend\n\n/-- A nonempty set is one that contains some element. -/\nprotected def nonempty (u : Set) : Prop := u.to_set.nonempty\n\ntheorem nonempty_def (u : Set) : u.nonempty \u2194 \u2203 x, x \u2208 u := iff.rfl\n\ntheorem nonempty_of_mem {x u : Set} (h : x \u2208 u) : u.nonempty := \u27e8x, h\u27e9\n\n@[simp] theorem nonempty_to_set_iff {u : Set} : u.to_set.nonempty \u2194 u.nonempty := iff.rfl\n\n/-- `x \u2286 y` as ZFC sets means that all members of `x` are members of `y`. -/\nprotected def subset (x y : Set.{u}) :=\n\u2200 \u2983z\u2984, z \u2208 x \u2192 z \u2208 y\n\ninstance has_subset : has_subset Set :=\n\u27e8Set.subset\u27e9\n\nlemma subset_def {x y : Set.{u}} : x \u2286 y \u2194 \u2200 \u2983z\u2984, z \u2208 x \u2192 z \u2208 y := iff.rfl\n\ninstance : is_refl Set (\u2286) := \u27e8\u03bb x a, id\u27e9\ninstance : is_trans Set (\u2286) := \u27e8\u03bb x y z hxy hyz a ha, hyz (hxy ha)\u27e9\n\n@[simp] theorem subset_iff : \u03a0 {x y : pSet}, mk x \u2286 mk y \u2194 x \u2286 y\n| \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 := \u27e8\u03bb h a, @h \u27e6A a\u27e7 (mem.mk A a),\n  \u03bb h z, quotient.induction_on z (\u03bb z \u27e8a, za\u27e9, let \u27e8b, ab\u27e9 := h a in \u27e8b, za.trans ab\u27e9)\u27e9\n\n@[simp] theorem to_set_subset_iff {x y : Set} : x.to_set \u2286 y.to_set \u2194 x \u2286 y :=\nby simp [subset_def, set.subset_def]\n\n@[ext] theorem ext {x y : Set.{u}} : (\u2200 z : Set.{u}, z \u2208 x \u2194 z \u2208 y) \u2192 x = y :=\nquotient.induction_on\u2082 x y (\u03bb u v h, quotient.sound (mem.ext (\u03bb w, h \u27e6w\u27e7)))\n\ntheorem ext_iff {x y : Set.{u}} : x = y \u2194 (\u2200 z : Set.{u}, z \u2208 x \u2194 z \u2208 y) :=\n\u27e8\u03bb h, by simp [h], ext\u27e9\n\ntheorem to_set_injective : function.injective to_set := \u03bb x y h, ext $ set.ext_iff.1 h\n\n@[simp] theorem to_set_inj {x y : Set} : x.to_set = y.to_set \u2194 x = y :=\nto_set_injective.eq_iff\n\ninstance : is_antisymm Set (\u2286) := \u27e8\u03bb a b hab hba, ext $ \u03bb c, \u27e8@hab c, @hba c\u27e9\u27e9\n\n/-- The empty ZFC set -/\nprotected def empty : Set := mk \u2205\ninstance : has_emptyc Set := \u27e8Set.empty\u27e9\ninstance : inhabited Set := \u27e8\u2205\u27e9\n\n@[simp] theorem not_mem_empty (x) : x \u2209 (\u2205 : Set.{u}) :=\nquotient.induction_on x pSet.not_mem_empty\n\n@[simp] theorem to_set_empty : to_set \u2205 = \u2205 := by simp [to_set]\n\n@[simp] theorem empty_subset (x : Set.{u}) : (\u2205 : Set) \u2286 x :=\nquotient.induction_on x $ \u03bb y, subset_iff.2 $ pSet.empty_subset y\n\n@[simp] theorem not_nonempty_empty : \u00ac Set.nonempty \u2205 := by simp [Set.nonempty]\n\n@[simp] theorem nonempty_mk_iff {x : pSet} : (mk x).nonempty \u2194 x.nonempty :=\nbegin\n  refine \u27e8_, \u03bb \u27e8a, h\u27e9, \u27e8mk a, h\u27e9\u27e9,\n  rintro \u27e8a, h\u27e9,\n  induction a using quotient.induction_on,\n  exact \u27e8a, h\u27e9\nend\n\ntheorem eq_empty (x : Set.{u}) : x = \u2205 \u2194 \u2200 y : Set.{u}, y \u2209 x := by { rw ext_iff, simp }\n\ntheorem eq_empty_or_nonempty (u : Set) : u = \u2205 \u2228 u.nonempty :=\nby { rw [eq_empty, \u2190not_exists], apply em' }\n\n/-- `insert x y` is the set `{x} \u222a y` -/\nprotected def insert : Set \u2192 Set \u2192 Set :=\nresp.eval 2 \u27e8pSet.insert, \u03bb u v uv \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9,\n  \u27e8\u03bb o, match o with\n   | some a := let \u27e8b, hb\u27e9 := \u03b1\u03b2 a in \u27e8some b, hb\u27e9\n   | none := \u27e8none, uv\u27e9\n   end, \u03bb o, match o with\n   | some b := let \u27e8a, ha\u27e9 := \u03b2\u03b1 b in \u27e8some a, ha\u27e9\n   | none := \u27e8none, uv\u27e9\n   end\u27e9\u27e9\n\ninstance : has_insert Set Set := \u27e8Set.insert\u27e9\n\ninstance : has_singleton Set Set := \u27e8\u03bb x, insert x \u2205\u27e9\n\ninstance : is_lawful_singleton Set Set := \u27e8\u03bb x, rfl\u27e9\n\n@[simp] theorem mem_insert_iff {x y z : Set.{u}} : x \u2208 insert y z \u2194 x = y \u2228 x \u2208 z :=\nquotient.induction_on\u2083 x y z\n (\u03bb x y \u27e8\u03b1, A\u27e9, show x \u2208 pSet.mk (option \u03b1) (\u03bb o, option.rec y A o) \u2194\n    mk x = mk y \u2228 x \u2208 pSet.mk \u03b1 A, from\n  \u27e8\u03bb m, match m with\n  | \u27e8some a, ha\u27e9 := or.inr \u27e8a, ha\u27e9\n  | \u27e8none, h\u27e9 := or.inl (quotient.sound h)\n  end, \u03bb m, match m with\n  | or.inr \u27e8a, ha\u27e9 := \u27e8some a, ha\u27e9\n  | or.inl h := \u27e8none, quotient.exact h\u27e9\n  end\u27e9)\n\ntheorem mem_insert (x y : Set) : x \u2208 insert x y := mem_insert_iff.2 $ or.inl rfl\ntheorem mem_insert_of_mem {y z : Set} (x) (h : z \u2208 y): z \u2208 insert x y := mem_insert_iff.2 $ or.inr h\n\n@[simp] theorem to_set_insert (x y : Set) : (insert x y).to_set = insert x y.to_set :=\nby { ext, simp }\n\n@[simp] theorem mem_singleton {x y : Set.{u}} : x \u2208 @singleton Set.{u} Set.{u} _ y \u2194 x = y :=\niff.trans mem_insert_iff \u27e8\u03bb o, or.rec (\u03bb h, h) (\u03bb n, absurd n (not_mem_empty _)) o, or.inl\u27e9\n\n@[simp] theorem to_set_singleton (x : Set) : ({x} : Set).to_set = {x} :=\nby { ext, simp }\n\ntheorem insert_nonempty (u v : Set) : (insert u v).nonempty := \u27e8u, mem_insert u v\u27e9\n\ntheorem singleton_nonempty (u : Set) : Set.nonempty {u} := insert_nonempty u \u2205\n\n@[simp] theorem mem_pair {x y z : Set.{u}} : x \u2208 ({y, z} : Set) \u2194 x = y \u2228 x = z :=\niff.trans mem_insert_iff $ or_congr iff.rfl mem_singleton\n\n/-- `omega` is the first infinite von Neumann ordinal -/\ndef omega : Set := mk omega\n\n@[simp] theorem omega_zero : \u2205 \u2208 omega :=\n\u27e8\u27e80\u27e9, equiv.rfl\u27e9\n\n@[simp] theorem omega_succ {n} : n \u2208 omega.{u} \u2192 insert n n \u2208 omega.{u} :=\nquotient.induction_on n (\u03bb x \u27e8\u27e8n\u27e9, h\u27e9, \u27e8\u27e8n+1\u27e9, Set.exact $\n  show insert (mk x) (mk x) = insert (mk $ of_nat n) (mk $ of_nat n), { rw Set.sound h, refl } \u27e9)\n\n/-- `{x \u2208 a | p x}` is the set of elements in `a` satisfying `p` -/\nprotected def sep (p : Set \u2192 Prop) : Set \u2192 Set :=\nresp.eval 1 \u27e8pSet.sep (\u03bb y, p (mk y)), \u03bb \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9,\n  \u27e8\u03bb \u27e8a, pa\u27e9, let \u27e8b, hb\u27e9 := \u03b1\u03b2 a in \u27e8\u27e8b, by rwa [mk_func, \u2190Set.sound hb]\u27e9, hb\u27e9,\n   \u03bb \u27e8b, pb\u27e9, let \u27e8a, ha\u27e9 := \u03b2\u03b1 b in \u27e8\u27e8a, by rwa [mk_func, Set.sound ha]\u27e9, ha\u27e9\u27e9\u27e9\n\ninstance : has_sep Set Set := \u27e8Set.sep\u27e9\n\n@[simp] theorem mem_sep {p : Set.{u} \u2192 Prop} {x y : Set.{u}} : y \u2208 {y \u2208 x | p y} \u2194 y \u2208 x \u2227 p y :=\nquotient.induction_on\u2082 x y (\u03bb \u27e8\u03b1, A\u27e9 y,\n  \u27e8\u03bb \u27e8\u27e8a, pa\u27e9, h\u27e9, \u27e8\u27e8a, h\u27e9, by rwa (@quotient.sound pSet _ _ _ h)\u27e9,\n  \u03bb \u27e8\u27e8a, h\u27e9, pa\u27e9, \u27e8\u27e8a, by { rw mk_func at h, rwa [mk_func, \u2190Set.sound h] }\u27e9, h\u27e9\u27e9)\n\n@[simp] theorem to_set_sep (a : Set) (p : Set \u2192 Prop) :\n  {x \u2208 a | p x}.to_set = {x \u2208 a.to_set | p x} :=\nby { ext, simp }\n\n/-- The powerset operation, the collection of subsets of a ZFC set -/\ndef powerset : Set \u2192 Set :=\nresp.eval 1 \u27e8powerset, \u03bb \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9,\n  \u27e8\u03bb p, \u27e8{b | \u2203 a, p a \u2227 equiv (A a) (B b)},\n    \u03bb \u27e8a, pa\u27e9, let \u27e8b, ab\u27e9 := \u03b1\u03b2 a in \u27e8\u27e8b, a, pa, ab\u27e9, ab\u27e9,\n    \u03bb \u27e8b, a, pa, ab\u27e9, \u27e8\u27e8a, pa\u27e9, ab\u27e9\u27e9,\n   \u03bb q, \u27e8{a | \u2203 b, q b \u2227 equiv (A a) (B b)},\n    \u03bb \u27e8a, b, qb, ab\u27e9, \u27e8\u27e8b, qb\u27e9, ab\u27e9,\n    \u03bb \u27e8b, qb\u27e9, let \u27e8a, ab\u27e9 := \u03b2\u03b1 b in \u27e8\u27e8a, b, qb, ab\u27e9, ab\u27e9\u27e9\u27e9\u27e9\n\n@[simp] theorem mem_powerset {x y : Set.{u}} : y \u2208 powerset x \u2194 y \u2286 x :=\nquotient.induction_on\u2082 x y ( \u03bb \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9,\n  show (\u27e8\u03b2, B\u27e9 : pSet.{u}) \u2208 (pSet.powerset.{u} \u27e8\u03b1, A\u27e9) \u2194 _,\n    by simp [mem_powerset, subset_iff])\n\ntheorem sUnion_lem {\u03b1 \u03b2 : Type u} (A : \u03b1 \u2192 pSet) (B : \u03b2 \u2192 pSet) (\u03b1\u03b2 : \u2200 a, \u2203 b, equiv (A a) (B b)) :\n  \u2200 a, \u2203 b, (equiv ((sUnion \u27e8\u03b1, A\u27e9).func a) ((sUnion \u27e8\u03b2, B\u27e9).func b))\n| \u27e8a, c\u27e9 := let \u27e8b, hb\u27e9 := \u03b1\u03b2 a in\n  begin\n    induction ea : A a with \u03b3 \u0393,\n    induction eb : B b with \u03b4 \u0394,\n    rw [ea, eb] at hb,\n    cases hb with \u03b3\u03b4 \u03b4\u03b3,\n    exact\n    let c : type (A a) := c, \u27e8d, hd\u27e9 := \u03b3\u03b4 (by rwa ea at c) in\n    have pSet.equiv ((A a).func c) ((B b).func (eq.rec d (eq.symm eb))), from\n    match A a, B b, ea, eb, c, d, hd with ._, ._, rfl, rfl, x, y, hd := hd end,\n    \u27e8\u27e8b, by { rw mk_func, exact eq.rec d (eq.symm eb) }\u27e9, this\u27e9\n  end\n\n/-- The union operator, the collection of elements of elements of a ZFC set -/\ndef sUnion : Set \u2192 Set :=\nresp.eval 1 \u27e8pSet.sUnion, \u03bb \u27e8\u03b1, A\u27e9 \u27e8\u03b2, B\u27e9 \u27e8\u03b1\u03b2, \u03b2\u03b1\u27e9,\n  \u27e8sUnion_lem A B \u03b1\u03b2, \u03bb a, exists.elim (sUnion_lem B A (\u03bb b,\n    exists.elim (\u03b2\u03b1 b) (\u03bb c hc, \u27e8c, pSet.equiv.symm hc\u27e9)) a) (\u03bb b hb, \u27e8b, pSet.equiv.symm hb\u27e9)\u27e9\u27e9\n\nprefix (name := Set.sUnion) `\u22c3\u2080 `:110 := Set.sUnion\n\n/-- The intersection operator, the collection of elements in all of the elements of a ZFC set. We\nspecial-case `\u22c2\u2080 \u2205 = \u2205`. -/\nnoncomputable def sInter (x : Set) : Set :=\nby { classical, exact dite x.nonempty (\u03bb h, {y \u2208 h.some | \u2200 z \u2208 x, y \u2208 z}) (\u03bb _, \u2205) }\n\nprefix (name := Set.sInter) `\u22c2\u2080 `:110 := Set.sInter\n\n@[simp] theorem mem_sUnion {x y : Set.{u}} : y \u2208 \u22c3\u2080 x \u2194 \u2203 z \u2208 x, y \u2208 z :=\nquotient.induction_on\u2082 x y (\u03bb x y, iff.trans mem_sUnion\n  \u27e8\u03bb \u27e8z, h\u27e9, \u27e8\u27e6z\u27e7, h\u27e9, \u03bb \u27e8z, h\u27e9, quotient.induction_on z (\u03bb z h, \u27e8z, h\u27e9) h\u27e9)\n\ntheorem mem_sInter {x y : Set} (h : x.nonempty) : y \u2208 \u22c2\u2080 x \u2194 \u2200 z \u2208 x, y \u2208 z :=\nbegin\n  rw [sInter, dif_pos h],\n  simp only [mem_to_set, mem_sep, and_iff_right_iff_imp],\n  exact \u03bb H, H _ h.some_mem\nend\n\n@[simp] theorem sUnion_empty : \u22c3\u2080 (\u2205 : Set) = \u2205 := by { ext, simp }\n@[simp] theorem sInter_empty : \u22c2\u2080 (\u2205 : Set) = \u2205 := dif_neg $ by simp\n\ntheorem mem_of_mem_sInter {x y z : Set} (hy : y \u2208 \u22c2\u2080 x) (hz : z \u2208 x) : y \u2208 z :=\nbegin\n  rcases eq_empty_or_nonempty x with rfl | hx,\n  { exact (not_mem_empty z hz).elim },\n  { exact (mem_sInter hx).1 hy z hz }\nend\n\ntheorem mem_sUnion_of_mem {x y z : Set} (hy : y \u2208 z) (hz : z \u2208 x) : y \u2208 \u22c3\u2080 x :=\nmem_sUnion.2 \u27e8z, hz, hy\u27e9\n\ntheorem not_mem_sInter_of_not_mem {x y z : Set} (hy : \u00ac y \u2208 z) (hz : z \u2208 x) : \u00ac y \u2208 \u22c2\u2080 x :=\n\u03bb hx, hy $ mem_of_mem_sInter hx hz\n\n@[simp] theorem sUnion_singleton {x : Set.{u}} : \u22c3\u2080 ({x} : Set) = x :=\next $ \u03bb y, by simp_rw [mem_sUnion, exists_prop, mem_singleton, exists_eq_left]\n\n@[simp] theorem sInter_singleton {x : Set.{u}} : \u22c2\u2080 ({x} : Set) = x :=\next $ \u03bb y, by simp_rw [mem_sInter (singleton_nonempty x), mem_singleton, forall_eq]\n\n@[simp] theorem to_set_sUnion (x : Set.{u}) : (\u22c3\u2080 x).to_set = \u22c3\u2080 (to_set '' x.to_set) :=\nby { ext, simp }\n\ntheorem to_set_sInter {x : Set.{u}} (h : x.nonempty) : (\u22c2\u2080 x).to_set = \u22c2\u2080 (to_set '' x.to_set) :=\nby { ext, simp [mem_sInter h] }\n\ntheorem singleton_injective : function.injective (@singleton Set Set _) :=\n\u03bb x y H, let this := congr_arg sUnion H in by rwa [sUnion_singleton, sUnion_singleton] at this\n\n@[simp] theorem singleton_inj {x y : Set} : ({x} : Set) = {y} \u2194 x = y := singleton_injective.eq_iff\n\n/-- The binary union operation -/\nprotected def union (x y : Set.{u}) : Set.{u} := \u22c3\u2080 {x, y}\n\n/-- The binary intersection operation -/\nprotected def inter (x y : Set.{u}) : Set.{u} := {z \u2208 x | z \u2208 y}\n\n/-- The set difference operation -/\nprotected def diff (x y : Set.{u}) : Set.{u} := {z \u2208 x | z \u2209 y}\n\ninstance : has_union Set := \u27e8Set.union\u27e9\ninstance : has_inter Set := \u27e8Set.inter\u27e9\ninstance : has_sdiff Set := \u27e8Set.diff\u27e9\n\n@[simp] theorem to_set_union (x y : Set.{u}) : (x \u222a y).to_set = x.to_set \u222a y.to_set :=\nby { unfold has_union.union, rw Set.union, simp }\n\n@[simp] theorem to_set_inter (x y : Set.{u}) : (x \u2229 y).to_set = x.to_set \u2229 y.to_set :=\nby { unfold has_inter.inter, rw Set.inter, ext, simp }\n\n@[simp] theorem to_set_sdiff (x y : Set.{u}) : (x \\ y).to_set = x.to_set \\ y.to_set :=\nby { change {z \u2208 x | z \u2209 y}.to_set = _, ext, simp }\n\n@[simp] theorem mem_union {x y z : Set.{u}} : z \u2208 x \u222a y \u2194 z \u2208 x \u2228 z \u2208 y :=\nby { rw \u2190mem_to_set, simp }\n\n@[simp] theorem mem_inter {x y z : Set.{u}} : z \u2208 x \u2229 y \u2194 z \u2208 x \u2227 z \u2208 y :=\n@@mem_sep (\u03bb z : Set.{u}, z \u2208 y)\n\n@[simp] theorem mem_diff {x y z : Set.{u}} : z \u2208 x \\ y \u2194 z \u2208 x \u2227 z \u2209 y :=\n@@mem_sep (\u03bb z : Set.{u}, z \u2209 y)\n\n@[simp] theorem sUnion_pair {x y : Set.{u}} : \u22c3\u2080 ({x, y} : Set.{u}) = x \u222a y :=\nbegin\n  ext,\n  simp_rw [mem_union, mem_sUnion, mem_pair],\n  split,\n  { rintro \u27e8w, (rfl | rfl), hw\u27e9,\n    { exact or.inl hw },\n    { exact or.inr hw } },\n  { rintro (hz | hz),\n    { exact \u27e8x, or.inl rfl, hz\u27e9 },\n    { exact \u27e8y, or.inr rfl, hz\u27e9 } }\nend\n\ntheorem mem_wf : @well_founded Set (\u2208) :=\nwell_founded_lift\u2082_iff.mpr pSet.mem_wf\n\n/-- Induction on the `\u2208` relation. -/\n@[elab_as_eliminator]\ntheorem induction_on {p : Set \u2192 Prop} (x) (h : \u2200 x, (\u2200 y \u2208 x, p y) \u2192 p x) : p x :=\nmem_wf.induction x h\n\ninstance : has_well_founded Set := \u27e8_, mem_wf\u27e9\n\ninstance : is_asymm Set (\u2208) := mem_wf.is_asymm\n\ntheorem mem_asymm {x y : Set} : x \u2208 y \u2192 y \u2209 x := asymm\ntheorem mem_irrefl (x : Set) : x \u2209 x := irrefl x\n\ntheorem regularity (x : Set.{u}) (h : x \u2260 \u2205) : \u2203 y \u2208 x, x \u2229 y = \u2205 :=\nclassical.by_contradiction $ \u03bb ne, h $ (eq_empty x).2 $ \u03bb y,\ninduction_on y $ \u03bb z (IH : \u2200 w : Set.{u}, w \u2208 z \u2192 w \u2209 x), show z \u2209 x, from \u03bb zx,\nne \u27e8z, zx, (eq_empty _).2 (\u03bb w wxz, let \u27e8wx, wz\u27e9 := mem_inter.1 wxz in IH w wz wx)\u27e9\n\n/-- The image of a (definable) ZFC set function -/\ndef image (f : Set \u2192 Set) [H : definable 1 f] : Set \u2192 Set :=\nlet r := @definable.resp 1 f _ in\nresp.eval 1 \u27e8image r.1, \u03bb x y e, mem.ext $ \u03bb z,\n  iff.trans (mem_image r.2) $ iff.trans (by exact\n   \u27e8\u03bb \u27e8w, h1, h2\u27e9, \u27e8w, (mem.congr_right e).1 h1, h2\u27e9,\n    \u03bb \u27e8w, h1, h2\u27e9, \u27e8w, (mem.congr_right e).2 h1, h2\u27e9\u27e9) $\n  iff.symm (mem_image r.2)\u27e9\n\ntheorem image.mk :\n  \u03a0 (f : Set.{u} \u2192 Set.{u}) [H : definable 1 f] (x) {y} (h : y \u2208 x), f y \u2208 @image f H x\n| ._ \u27e8F\u27e9 x y := quotient.induction_on\u2082 x y $ \u03bb \u27e8\u03b1, A\u27e9 y \u27e8a, ya\u27e9, \u27e8a, F.2 _ _ ya\u27e9\n\n@[simp] theorem mem_image : \u03a0 {f : Set.{u} \u2192 Set.{u}} [H : definable 1 f] {x y : Set.{u}},\n  y \u2208 @image f H x \u2194 \u2203 z \u2208 x, f z = y\n| ._ \u27e8F\u27e9 x y := quotient.induction_on\u2082 x y $ \u03bb \u27e8\u03b1, A\u27e9 y,\n  \u27e8\u03bb \u27e8a, ya\u27e9, \u27e8\u27e6A a\u27e7, mem.mk A a, eq.symm $ quotient.sound ya\u27e9,\n  \u03bb \u27e8z, hz, e\u27e9, e \u25b8 image.mk _ _ hz\u27e9\n\n@[simp] theorem to_set_image (f : Set \u2192 Set) [H : definable 1 f] (x : Set) :\n  (image f x).to_set = f '' x.to_set :=\nby { ext, simp }\n\n/-- The range of an indexed family of sets. The universes allow for a more general index type\n  without manual use of `ulift`. -/\nnoncomputable def range {\u03b1 : Type u} (f : \u03b1 \u2192 Set.{max u v}) : Set.{max u v} :=\n\u27e6\u27e8ulift \u03b1, quotient.out \u2218 f \u2218 ulift.down\u27e9\u27e7\n\n@[simp] theorem mem_range {\u03b1 : Type u} {f : \u03b1 \u2192 Set.{max u v}} {x : Set.{max u v}} :\n  x \u2208 range f \u2194 x \u2208 set.range f :=\nquotient.induction_on x (\u03bb y, begin\n  split,\n  { rintro \u27e8z, hz\u27e9,\n    exact \u27e8z.down, quotient.eq_mk_iff_out.2 hz.symm\u27e9 },\n  { rintro \u27e8z, hz\u27e9,\n    use z,\n    simpa [hz] using pSet.equiv.symm (quotient.mk_out y) }\nend)\n\n@[simp] theorem to_set_range {\u03b1 : Type u} (f : \u03b1 \u2192 Set.{max u v}) :\n  (range f).to_set = set.range f :=\nby { ext, simp }\n\n/-- Kuratowski ordered pair -/\ndef pair (x y : Set.{u}) : Set.{u} := {{x}, {x, y}}\n\n@[simp] theorem to_set_pair (x y : Set.{u}) : (pair x y).to_set = {{x}, {x, y}} := by simp [pair]\n\n/-- A subset of pairs `{(a, b) \u2208 x \u00d7 y | p a b}` -/\ndef pair_sep (p : Set.{u} \u2192 Set.{u} \u2192 Prop) (x y : Set.{u}) : Set.{u} :=\n{z \u2208 powerset (powerset (x \u222a y)) | \u2203 a \u2208 x, \u2203 b \u2208 y, z = pair a b \u2227 p a b}\n\n@[simp] theorem mem_pair_sep {p} {x y z : Set.{u}} :\n  z \u2208 pair_sep p x y \u2194 \u2203 a \u2208 x, \u2203 b \u2208 y, z = pair a b \u2227 p a b :=\nbegin\n  refine mem_sep.trans \u27e8and.right, \u03bb e, \u27e8_, e\u27e9\u27e9,\n  rcases e with \u27e8a, ax, b, bY, rfl, pab\u27e9,\n  simp only [mem_powerset, subset_def, mem_union, pair, mem_pair],\n  rintros u (rfl|rfl) v; simp only [mem_singleton, mem_pair],\n  { rintro rfl, exact or.inl ax },\n  { rintro (rfl|rfl); [left, right]; assumption }\nend\n\ntheorem pair_injective : function.injective2 pair :=\n\u03bb x x' y y' H, begin\n  have ae := ext_iff.1 H,\n  simp only [pair, mem_pair] at ae,\n  obtain rfl : x = x',\n  { cases (ae {x}).1 (by simp) with h h,\n    { exact singleton_injective h },\n    { have m : x' \u2208 ({x} : Set),\n      { simp [h] },\n      rw mem_singleton.mp m } },\n  have he : x = y \u2192 y = y',\n  { rintro rfl,\n    cases (ae {x, y'}).2 (by simp only [eq_self_iff_true, or_true]) with xy'x xy'xx,\n    { rw [eq_comm, \u2190mem_singleton, \u2190xy'x, mem_pair],\n      exact or.inr rfl },\n    { simpa [eq_comm] using (ext_iff.1 xy'xx y').1 (by simp) } },\n  obtain xyx | xyy' := (ae {x, y}).1 (by simp),\n  { obtain rfl := mem_singleton.mp ((ext_iff.1 xyx y).1 $ by simp),\n    simp [he rfl] },\n  { obtain rfl | yy' := mem_pair.mp ((ext_iff.1 xyy' y).1 $ by simp),\n    { simp [he rfl] },\n    { simp [yy'] } }\nend\n\n@[simp] theorem pair_inj {x y x' y' : Set} : pair x y = pair x' y' \u2194 x = x' \u2227 y = y' :=\npair_injective.eq_iff\n\n/-- The cartesian product, `{(a, b) | a \u2208 x, b \u2208 y}` -/\ndef prod : Set.{u} \u2192 Set.{u} \u2192 Set.{u} := pair_sep (\u03bb a b, true)\n\n@[simp] theorem mem_prod {x y z : Set.{u}} : z \u2208 prod x y \u2194 \u2203 a \u2208 x, \u2203 b \u2208 y, z = pair a b :=\nby simp [prod]\n\n@[simp] theorem pair_mem_prod {x y a b : Set.{u}} : pair a b \u2208 prod x y \u2194 a \u2208 x \u2227 b \u2208 y :=\n\u27e8\u03bb h, let \u27e8a', a'x, b', b'y, e\u27e9 := mem_prod.1 h in\n  match a', b', pair_injective e, a'x, b'y with ._, ._, \u27e8rfl, rfl\u27e9, ax, bY := \u27e8ax, bY\u27e9 end,\n\u03bb \u27e8ax, bY\u27e9, mem_prod.2 \u27e8a, ax, b, bY, rfl\u27e9\u27e9\n\n/-- `is_func x y f` is the assertion that `f` is a subset of `x \u00d7 y` which relates to each element\nof `x` a unique element of `y`, so that we can consider `f`as a ZFC function `x \u2192 y`. -/\ndef is_func (x y f : Set.{u}) : Prop :=\nf \u2286 prod x y \u2227 \u2200 z : Set.{u}, z \u2208 x \u2192 \u2203! w, pair z w \u2208 f\n\n/-- `funs x y` is `y ^ x`, the set of all set functions `x \u2192 y` -/\ndef funs (x y : Set.{u}) : Set.{u} :=\n{f \u2208 powerset (prod x y) | is_func x y f}\n\n@[simp] theorem mem_funs {x y f : Set.{u}} : f \u2208 funs x y \u2194 is_func x y f :=\nby simp [funs, is_func]\n\n-- TODO(Mario): Prove this computably\nnoncomputable instance map_definable_aux (f : Set \u2192 Set) [H : definable 1 f] :\n  definable 1 (\u03bb y, pair y (f y)) :=\n@classical.all_definable 1 _\n\n/-- Graph of a function: `map f x` is the ZFC function which maps `a \u2208 x` to `f a` -/\nnoncomputable def map (f : Set \u2192 Set) [H : definable 1 f] : Set \u2192 Set :=\nimage (\u03bb y, pair y (f y))\n\n@[simp] theorem mem_map {f : Set \u2192 Set} [H : definable 1 f] {x y : Set} :\n  y \u2208 map f x \u2194 \u2203 z \u2208 x, pair z (f z) = y :=\nmem_image\n\ntheorem map_unique {f : Set.{u} \u2192 Set.{u}} [H : definable 1 f] {x z : Set.{u}} (zx : z \u2208 x) :\n  \u2203! w, pair z w \u2208 map f x :=\n\u27e8f z, image.mk _ _ zx, \u03bb y yx, let \u27e8w, wx, we\u27e9 := mem_image.1 yx, \u27e8wz, fy\u27e9 := pair_injective we in\n  by rw[\u2190fy, wz]\u27e9\n\n@[simp] theorem map_is_func {f : Set \u2192 Set} [H : definable 1 f] {x y : Set} :\n  is_func x y (map f x) \u2194 \u2200 z \u2208 x, f z \u2208 y :=\n\u27e8\u03bb \u27e8ss, h\u27e9 z zx, let \u27e8t, t1, t2\u27e9 := h z zx in\n  (t2 (f z) (image.mk _ _ zx)).symm \u25b8 (pair_mem_prod.1 (ss t1)).right,\n\u03bb h, \u27e8\u03bb y yx, let \u27e8z, zx, ze\u27e9 := mem_image.1 yx in ze \u25b8 pair_mem_prod.2 \u27e8zx, h z zx\u27e9,\n     \u03bb z, map_unique\u27e9\u27e9\n\n/-- Given a predicate `p` on ZFC sets. `hereditarily p x` means that `x` has property `p` and the\nmembers of `x` are all `hereditarily p`. -/\ndef hereditarily (p : Set \u2192 Prop) : Set \u2192 Prop\n| x := p x \u2227 \u2200 y \u2208 x, hereditarily y\nusing_well_founded { dec_tac := `[assumption] }\n\nsection hereditarily\n\nvariables {p : Set.{u} \u2192 Prop} {x y : Set.{u}}\n\n\n\nalias hereditarily_iff \u2194 hereditarily.def _\n\nlemma hereditarily.self (h : x.hereditarily p) : p x := h.def.1\nlemma hereditarily.mem (h : x.hereditarily p) (hy : y \u2208 x) : y.hereditarily p := h.def.2 _ hy\n\nlemma hereditarily.empty : hereditarily p x \u2192 p \u2205 :=\nbegin\n  apply x.induction_on,\n  intros y IH h,\n  rcases Set.eq_empty_or_nonempty y with (rfl|\u27e8a, ha\u27e9),\n  { exact h.self },\n  { exact IH a ha (h.mem ha) }\nend\n\nend hereditarily\n\nend Set\n\n/-- The collection of all classes.\n\nWe define `Class` as `set Set`, as this allows us to get many instances automatically. However, in\npractice, we treat it as (the definitionally equal) `Set \u2192 Prop`. This means, the preferred way to\nstate that `x : Set` belongs to `A : Class` is to write `A x`. -/\n@[derive [has_subset, has_sep Set, has_emptyc, inhabited, has_insert Set, has_union, has_inter,\n  has_compl, has_sdiff]]\ndef Class := set Set\n\nnamespace Class\n\n@[ext] theorem ext {x y : Class.{u}} : (\u2200 z : Set.{u}, x z \u2194 y z) \u2192 x = y := set.ext\n\ntheorem ext_iff {x y : Class.{u}} : x = y \u2194 \u2200 z, x z \u2194 y z := set.ext_iff\n\n/-- Coerce a ZFC set into a class -/\ndef of_Set (x : Set.{u}) : Class.{u} := {y | y \u2208 x}\ninstance : has_coe Set Class := \u27e8of_Set\u27e9\n\n/-- The universal class -/\ndef univ : Class := set.univ\n\n/-- Assert that `A` is a ZFC set satisfying `B` -/\ndef to_Set (B : Class.{u}) (A : Class.{u}) : Prop := \u2203 x, \u2191x = A \u2227 B x\n\n/-- `A \u2208 B` if `A` is a ZFC set which satisfies `B` -/\nprotected def mem (A B : Class.{u}) : Prop := to_Set.{u} B A\ninstance : has_mem Class Class := \u27e8Class.mem\u27e9\n\ntheorem mem_def (A B : Class.{u}) : A \u2208 B \u2194 \u2203 x, \u2191x = A \u2227 B x := iff.rfl\n\n@[simp] theorem not_mem_empty (x : Class.{u}) : x \u2209 (\u2205 : Class.{u}) := \u03bb \u27e8_, _, h\u27e9, h\n\n@[simp] theorem not_empty_hom (x : Set.{u}) : \u00ac (\u2205 : Class.{u}) x := id\n\n@[simp] theorem mem_univ {A : Class.{u}} : A \u2208 univ.{u} \u2194 \u2203 x : Set.{u}, \u2191x = A :=\nexists_congr $ \u03bb x, and_true _\n\n@[simp] theorem mem_univ_hom (x : Set.{u}) : univ.{u} x := trivial\n\ntheorem eq_univ_iff_forall {A : Class.{u}} : A = univ \u2194 \u2200 x : Set, A x := set.eq_univ_iff_forall\ntheorem eq_univ_of_forall {A : Class.{u}} : (\u2200 x : Set, A x) \u2192 A = univ := set.eq_univ_of_forall\n\ntheorem mem_wf : @well_founded Class.{u} (\u2208) :=\n\u27e8begin\n  have H : \u2200 x : Set.{u}, @acc Class.{u} (\u2208) \u2191x,\n  { refine \u03bb a, Set.induction_on a (\u03bb x IH, \u27e8x, _\u27e9),\n    rintros A \u27e8z, rfl, hz\u27e9,\n    exact IH z hz },\n  { refine \u03bb A, \u27e8A, _\u27e9,\n    rintros B \u27e8x, rfl, hx\u27e9,\n    exact H x }\nend\u27e9\n\ninstance : has_well_founded Class := \u27e8_, mem_wf\u27e9\ninstance : is_asymm Class (\u2208) := mem_wf.is_asymm\n\ntheorem mem_asymm {x y : Class} : x \u2208 y \u2192 y \u2209 x := asymm\ntheorem mem_irrefl (x : Class) : x \u2209 x := irrefl x\n\n/-- **There is no universal set.**\n\nThis is stated as `univ \u2209 univ`, meaning that `univ` (the class of all sets) is proper (does not\nbelong to the class of all sets). -/\ntheorem univ_not_mem_univ : univ \u2209 univ := mem_irrefl _\n\n/-- Convert a conglomerate (a collection of classes) into a class -/\ndef Cong_to_Class (x : set Class.{u}) : Class.{u} := {y | \u2191y \u2208 x}\n\n@[simp] theorem Cong_to_Class_empty : Cong_to_Class \u2205 = \u2205 :=\nby { ext, simp [Cong_to_Class] }\n\n/-- Convert a class into a conglomerate (a collection of classes) -/\ndef Class_to_Cong (x : Class.{u}) : set Class.{u} := {y | y \u2208 x}\n\n@[simp] theorem Class_to_Cong_empty : Class_to_Cong \u2205 = \u2205 :=\nby { ext, simp [Class_to_Cong] }\n\n/-- The power class of a class is the class of all subclasses that are ZFC sets -/\ndef powerset (x : Class) : Class := Cong_to_Class (set.powerset x)\n\n/-- The union of a class is the class of all members of ZFC sets in the class -/\ndef sUnion (x : Class) : Class := \u22c3\u2080 (Class_to_Cong x)\n\nprefix (name := Class.sUnion) `\u22c3\u2080 `:110 := Class.sUnion\n\n/-- The intersection of a class is the class of all members of ZFC sets in the class -/\ndef sInter (x : Class) : Class := \u22c2\u2080 Class_to_Cong x\n\nprefix (name := Class.sInter) `\u22c2\u2080 `:110 := Class.sInter\n\ntheorem of_Set.inj {x y : Set.{u}} (h : (x : Class.{u}) = y) : x = y :=\nSet.ext $ \u03bb z, by { change (x : Class.{u}) z \u2194 (y : Class.{u}) z, rw h }\n\n@[simp] theorem to_Set_of_Set (A : Class.{u}) (x : Set.{u}) : to_Set A x \u2194 A x :=\n\u27e8\u03bb \u27e8y, yx, py\u27e9, by rwa of_Set.inj yx at py, \u03bb px, \u27e8x, rfl, px\u27e9\u27e9\n\n@[simp, norm_cast] theorem coe_mem {x : Set.{u}} {A : Class.{u}} : (x : Class.{u}) \u2208 A \u2194 A x :=\nto_Set_of_Set _ _\n\n@[simp] theorem coe_apply {x y : Set.{u}} : (y : Class.{u}) x \u2194 x \u2208 y := iff.rfl\n\n@[simp, norm_cast] theorem coe_subset (x y : Set.{u}) : (x : Class.{u}) \u2286 y \u2194 x \u2286 y := iff.rfl\n\n@[simp, norm_cast] theorem coe_sep (p : Class.{u}) (x : Set.{u}) :\n  (\u2191{y \u2208 x | p y} : Class.{u}) = {y \u2208 x | p y} :=\next $ \u03bb y, Set.mem_sep\n\n@[simp, norm_cast] theorem coe_empty : \u2191(\u2205 : Set.{u}) = (\u2205 : Class.{u}) :=\next $ \u03bb y, (iff_false _).2 $ Set.not_mem_empty y\n\n@[simp, norm_cast] theorem coe_insert (x y : Set.{u}) :\n  \u2191(insert x y) = @insert Set.{u} Class.{u} _ x y :=\next $ \u03bb z, Set.mem_insert_iff\n\n@[simp, norm_cast] theorem coe_union (x y : Set.{u}) : \u2191(x \u222a y) = (x : Class.{u}) \u222a y :=\next $ \u03bb z, Set.mem_union\n\n@[simp, norm_cast] theorem coe_inter (x y : Set.{u}) : \u2191(x \u2229 y) = (x : Class.{u}) \u2229 y :=\next $ \u03bb z, Set.mem_inter\n\n@[simp, norm_cast] theorem coe_diff (x y : Set.{u}) : \u2191(x \\ y) = (x : Class.{u}) \\ y :=\next $ \u03bb z, Set.mem_diff\n\n@[simp, norm_cast] theorem coe_powerset (x : Set.{u}) : \u2191x.powerset = powerset.{u} x :=\next $ \u03bb z, Set.mem_powerset\n\n@[simp] theorem powerset_apply {A : Class.{u}} {x : Set.{u}} : powerset A x \u2194 \u2191x \u2286 A := iff.rfl\n\n@[simp] theorem sUnion_apply {x : Class} {y : Set} : (\u22c3\u2080 x) y \u2194 \u2203 z : Set, x z \u2227 y \u2208 z :=\nbegin\n  split,\n  { rintro \u27e8-, \u27e8z, rfl, hxz\u27e9, hyz\u27e9,\n    exact \u27e8z, hxz, hyz\u27e9 },\n  { exact \u03bb \u27e8z, hxz, hyz\u27e9, \u27e8_, coe_mem.2 hxz, hyz\u27e9 }\nend\n\n@[simp, norm_cast] theorem coe_sUnion (x : Set.{u}) : \u2191(\u22c3\u2080 x) = \u22c3\u2080 (x : Class.{u}) :=\next $ \u03bb y, Set.mem_sUnion.trans (sUnion_apply.trans $ by simp_rw [coe_apply, exists_prop]).symm\n\n@[simp] theorem mem_sUnion {x y : Class.{u}} : y \u2208 \u22c3\u2080 x \u2194 \u2203 z, z \u2208 x \u2227 y \u2208 z :=\nbegin\n  split,\n  { rintro \u27e8w, rfl, z, hzx, hwz\u27e9,\n    exact \u27e8z, hzx, coe_mem.2 hwz\u27e9 },\n  { rintro \u27e8w, hwx, z, rfl, hwz\u27e9,\n    exact \u27e8z, rfl, w, hwx, hwz\u27e9 }\nend\n\n@[simp] theorem sInter_apply {x : Class.{u}} {y : Set.{u}} :\n  (\u22c2\u2080 x) y \u2194 \u2200 z : Set.{u}, x z \u2192 y \u2208 z :=\nbegin\n  refine \u27e8\u03bb hxy z hxz, hxy _ \u27e8z, rfl, hxz\u27e9, _\u27e9,\n  rintro H - \u27e8z, rfl, hxz\u27e9,\n  exact H _ hxz\nend\n\n@[simp, norm_cast] theorem sInter_coe {x : Set.{u}} (h : x.nonempty) : \u22c2\u2080 (x : Class.{u}) = \u22c2\u2080 x :=\nset.ext $ \u03bb y, sInter_apply.trans (Set.mem_sInter h).symm\n\ntheorem mem_of_mem_sInter {x y z : Class} (hy : y \u2208 \u22c2\u2080 x) (hz : z \u2208 x) : y \u2208 z :=\nby { obtain \u27e8w, rfl, hw\u27e9 := hy, exact coe_mem.2 (hw z hz) }\n\ntheorem mem_sInter {x y : Class.{u}} (h : x.nonempty) : y \u2208 \u22c2\u2080 x \u2194 \u2200 z, z \u2208 x \u2192 y \u2208 z :=\nbegin\n  refine \u27e8\u03bb hy z, mem_of_mem_sInter hy, \u03bb H, _\u27e9,\n  simp_rw [mem_def, sInter_apply],\n  obtain \u27e8z, hz\u27e9 := h,\n  obtain \u27e8y, rfl, hzy\u27e9 := H z (coe_mem.2 hz),\n  refine \u27e8y, rfl, \u03bb w hxw, _\u27e9,\n  simpa only [coe_mem, coe_apply] using H w (coe_mem.2 hxw),\nend\n\n@[simp] theorem sUnion_empty : \u22c3\u2080 (\u2205 : Class.{u}) = \u2205 := by { ext, simp }\n@[simp] theorem sInter_empty : \u22c2\u2080 (\u2205 : Class.{u}) = univ := by { ext, simp [sInter, \u2190univ] }\n\n/-- An induction principle for sets. If every subset of a class is a member, then the class is\n  universal. -/\ntheorem eq_univ_of_powerset_subset {A : Class} (hA : powerset A \u2286 A) : A = univ :=\neq_univ_of_forall begin\n  by_contra' hnA,\n  exact well_founded.min_mem Set.mem_wf _ hnA (hA $ \u03bb x hx, not_not.1 $\n    \u03bb hB, well_founded.not_lt_min Set.mem_wf _ hnA hB $ coe_apply.1 hx)\nend\n\n/-- The definite description operator, which is `{x}` if `{y | A y} = {x}` and `\u2205` otherwise. -/\ndef iota (A : Class) : Class := \u22c3\u2080 {x | \u2200 y, A y \u2194 y = x}\n\ntheorem iota_val (A : Class) (x : Set) (H : \u2200 y, A y \u2194 y = x) : iota A = \u2191x :=\next $ \u03bb y, \u27e8\u03bb \u27e8._, \u27e8x', rfl, h\u27e9, yx'\u27e9, by rwa \u2190((H x').1 $ (h x').2 rfl),\n  \u03bb yx, \u27e8_, \u27e8x, rfl, H\u27e9, yx\u27e9\u27e9\n\n/-- Unlike the other set constructors, the `iota` definite descriptor\n  is a set for any set input, but not constructively so, so there is no\n  associated `Class \u2192 Set` function. -/\ntheorem iota_ex (A) : iota.{u} A \u2208 univ.{u} :=\nmem_univ.2 $ or.elim (classical.em $ \u2203 x, \u2200 y, A y \u2194 y = x)\n (\u03bb \u27e8x, h\u27e9, \u27e8x, eq.symm $ iota_val A x h\u27e9)\n (\u03bb hn, \u27e8\u2205, ext (\u03bb z, coe_empty.symm \u25b8 \u27e8false.rec _, \u03bb \u27e8._, \u27e8x, rfl, H\u27e9, zA\u27e9, hn \u27e8x, H\u27e9\u27e9)\u27e9)\n\n/-- Function value -/\ndef fval (F A : Class.{u}) : Class.{u} := iota (\u03bb y, to_Set (\u03bb x, F (Set.pair x y)) A)\ninfixl ` \u2032 `:100 := fval\n\ntheorem fval_ex (F A : Class.{u}) : F \u2032 A \u2208 univ.{u} := iota_ex _\n\nend Class\n\nnamespace Set\n\n@[simp] theorem map_fval {f : Set.{u} \u2192 Set.{u}} [H : pSet.definable 1 f]\n  {x y : Set.{u}} (h : y \u2208 x) :\n  (Set.map f x \u2032 y : Class.{u}) = f y :=\nClass.iota_val _ _ (\u03bb z, by { rw [Class.to_Set_of_Set, Class.coe_apply, mem_map], exact\n  \u27e8\u03bb \u27e8w, wz, pr\u27e9, let \u27e8wy, fw\u27e9 := Set.pair_injective pr in by rw[\u2190fw, wy],\n  \u03bb e, by { subst e, exact \u27e8_, h, rfl\u27e9 }\u27e9 })\n\nvariables (x : Set.{u}) (h : \u2205 \u2209 x)\n\n/-- A choice function on the class of nonempty ZFC sets. -/\nnoncomputable def choice : Set :=\n@map (\u03bb y, classical.epsilon (\u03bb z, z \u2208 y)) (classical.all_definable _) x\n\ninclude h\ntheorem choice_mem_aux (y : Set.{u}) (yx : y \u2208 x) : classical.epsilon (\u03bb z : Set.{u}, z \u2208 y) \u2208 y :=\n@classical.epsilon_spec _ (\u03bb z : Set.{u}, z \u2208 y) $ classical.by_contradiction $ \u03bb n, h $\nby rwa \u2190((eq_empty y).2 $ \u03bb z zx, n \u27e8z, zx\u27e9)\n\ntheorem choice_is_func : is_func x (\u22c3\u2080 x) (choice x) :=\n(@map_is_func _ (classical.all_definable _) _ _).2 $\n  \u03bb y yx, mem_sUnion.2 \u27e8y, yx, choice_mem_aux x h y yx\u27e9\n\ntheorem choice_mem (y : Set.{u}) (yx : y \u2208 x) : (choice x \u2032 y : Class.{u}) \u2208 (y : Class.{u}) :=\nbegin\n  delta choice,\n  rw [map_fval yx, Class.coe_mem, Class.coe_apply],\n  exact choice_mem_aux x h y yx\nend\n\nend Set\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/set_theory/zfc/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3821996058388604}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport logic.basic data.bool init.data.option.instances\n       tactic.interactive\n\nnamespace option\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\ninstance has_mem : has_mem \u03b1 (option \u03b1) := \u27e8\u03bb a b, b = some a\u27e9\n\n@[simp] theorem mem_def {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = some a :=\niff.rfl\n\n@[simp] theorem get_mem : \u2200 {o : option \u03b1} (h : is_some o), option.get h \u2208 o\n| (some a) _ := rfl\n\ntheorem get_of_mem {a : \u03b1} : \u2200 {o : option \u03b1} (h : is_some o), a \u2208 o \u2192 option.get h = a\n| _ _ rfl := rfl\n\ntheorem mem_unique {o : option \u03b1} {a b : \u03b1} (ha : a \u2208 o) (hb : b \u2208 o) : a = b :=\noption.some.inj $ ha.symm.trans hb\n\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n\ntheorem injective_some (\u03b1 : Type*) : function.injective (@some \u03b1) :=\n\u03bb _ _, some_inj.mp\n\ntheorem ext : \u2200 {o\u2081 o\u2082 : option \u03b1}, (\u2200 a, a \u2208 o\u2081 \u2194 a \u2208 o\u2082) \u2192 o\u2081 = o\u2082\n| none     none     H := rfl\n| (some a) o        H := ((H _).1 rfl).symm\n| o        (some b) H := (H _).2 rfl\n\ntheorem eq_none_iff_forall_not_mem {o : option \u03b1} :\n  o = none \u2194 (\u2200 a, a \u2209 o) :=\n\u27e8\u03bb e a h, by rw e at h; cases h, \u03bb h, ext $ by simpa\u27e9\n\n@[simp] theorem none_bind {\u03b1 \u03b2} (f : \u03b1 \u2192 option \u03b2) : none >>= f = none := rfl\n\n@[simp] theorem some_bind {\u03b1 \u03b2} (a : \u03b1) (f : \u03b1 \u2192 option \u03b2) : some a >>= f = f a := rfl\n\n@[simp] theorem none_bind' (f : \u03b1 \u2192 option \u03b2) : none.bind f = none := rfl\n\n@[simp] theorem some_bind' (a : \u03b1) (f : \u03b1 \u2192 option \u03b2) : (some a).bind f = f a := rfl\n\n@[simp] theorem bind_some : \u2200 x : option \u03b1, x >>= some = x :=\n@bind_pure \u03b1 option _ _\n\n@[simp] theorem bind_eq_some {\u03b1 \u03b2} {x : option \u03b1} {f : \u03b1 \u2192 option \u03b2} {b : \u03b2} : x >>= f = some b \u2194 \u2203 a, x = some a \u2227 f a = some b :=\nby cases x; simp\n\n@[simp] theorem bind_eq_some' {x : option \u03b1} {f : \u03b1 \u2192 option \u03b2} {b : \u03b2} : x.bind f = some b \u2194 \u2203 a, x = some a \u2227 f a = some b :=\nby cases x; simp\n\nlemma bind_comm {\u03b1 \u03b2 \u03b3} {f : \u03b1 \u2192 \u03b2 \u2192 option \u03b3} (a : option \u03b1) (b : option \u03b2) :\n  a.bind (\u03bbx, b.bind (f x)) = b.bind (\u03bby, a.bind (\u03bbx, f x y)) :=\nby cases a; cases b; refl\n\n@[simp] theorem map_none {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} : f <$> none = none := rfl\n\n@[simp] theorem map_some {\u03b1 \u03b2} {a : \u03b1} {f : \u03b1 \u2192 \u03b2} : f <$> some a = some (f a) := rfl\n\n@[simp] theorem map_none' {f : \u03b1 \u2192 \u03b2} : option.map f none = none := rfl\n\n@[simp] theorem map_some' {a : \u03b1} {f : \u03b1 \u2192 \u03b2} : option.map f (some a) = some (f a) := rfl\n\n@[simp] theorem map_eq_some {\u03b1 \u03b2} {x : option \u03b1} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} : f <$> x = some b \u2194 \u2203 a, x = some a \u2227 f a = b :=\nby cases x; simp\n\n@[simp] theorem map_eq_some' {x : option \u03b1} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} : x.map f = some b \u2194 \u2203 a, x = some a \u2227 f a = b :=\nby cases x; simp\n\n@[simp] theorem map_id' : option.map (@id \u03b1) = id := map_id\n\n@[simp] theorem seq_some {\u03b1 \u03b2} {a : \u03b1} {f : \u03b1 \u2192 \u03b2} : some f <*> some a = some (f a) := rfl\n\n@[simp] theorem some_orelse' (a : \u03b1) (x : option \u03b1) : (some a).orelse x = some a := rfl\n\n@[simp] theorem some_orelse (a : \u03b1) (x : option \u03b1) : (some a <|> x) = some a := rfl\n\n@[simp] theorem none_orelse' (x : option \u03b1) : none.orelse x = x :=\nby cases x; refl\n\n@[simp] theorem none_orelse (x : option \u03b1) : (none <|> x) = x := none_orelse' x\n\n@[simp] theorem orelse_none' (x : option \u03b1) : x.orelse none = x :=\nby cases x; refl\n\n@[simp] theorem orelse_none (x : option \u03b1) : (x <|> none) = x := orelse_none' x\n\n@[simp] theorem is_some_none : @is_some \u03b1 none = ff := rfl\n\n@[simp] theorem is_some_some {a : \u03b1} : is_some (some a) = tt := rfl\n\ntheorem is_some_iff_exists {x : option \u03b1} : is_some x \u2194 \u2203 a, x = some a :=\nby cases x; simp [is_some]; exact \u27e8_, rfl\u27e9\n\n@[simp] theorem is_none_none : @is_none \u03b1 none = tt := rfl\n\n@[simp] theorem is_none_some {a : \u03b1} : is_none (some a) = ff := rfl\n\ntheorem is_none_iff_eq_none {o : option \u03b1} : o.is_none \u2194 o = none :=\n\u27e8option.eq_none_of_is_none, \u03bb e, e.symm \u25b8 rfl\u27e9\n\ninstance decidable_eq_none {o : option \u03b1} : decidable (o = none) :=\ndecidable_of_bool _ is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2200 a \u2208 o, p a)\n| none     := is_true (by simp)\n| (some a) := decidable_of_iff (p a) (by simp)\n\ninstance decidable_exists_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2203 a \u2208 o, p a)\n| none     := is_false (by simp)\n| (some a) := decidable_of_iff (p a) (by simp)\n\n/-- inhabited `get` function. Returns `a` if the input is `some a`,\n  otherwise returns `default`. -/\n@[reducible] def iget [inhabited \u03b1] : option \u03b1 \u2192 \u03b1\n| (some x) := x\n| none     := default \u03b1\n\n@[simp] theorem iget_some [inhabited \u03b1] {a : \u03b1} : (some a).iget = a := rfl\n\ntheorem iget_mem [inhabited \u03b1] : \u2200 {o : option \u03b1}, is_some o \u2192 o.iget \u2208 o\n| (some a) _ := rfl\n\ntheorem iget_of_mem [inhabited \u03b1] {a : \u03b1} : \u2200 {o : option \u03b1}, a \u2208 o \u2192 o.iget = a\n| _ rfl := rfl\n\n@[simp] theorem guard_eq_some' {p : Prop} [decidable p] :\n  \u2200 u, guard p = some u \u2194 p\n| () := by by_cases p; simp [guard, h, pure]; intro; contradiction\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [decidable_pred p] (a : \u03b1) : option \u03b1 :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a`\n  and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (o : option \u03b1) : option \u03b1 :=\no.bind (guard p)\n\n@[simp] theorem guard_eq_some {p : \u03b1 \u2192 Prop} [decidable_pred p] {a b : \u03b1} :\n  guard p a = some b \u2194 a = b \u2227 p a :=\nby by_cases p a; simp [option.guard, h]; intro; contradiction\n\ndef to_list : option \u03b1 \u2192 list \u03b1\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : \u03b1} {o : option \u03b1} : a \u2208 to_list o \u2194 a \u2208 o :=\nby cases o; simp [to_list, eq_comm]\n\ndef lift_or_get (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : option \u03b1 \u2192 option \u03b1 \u2192 option \u03b1\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_commutative \u03b1 f] :\n  is_commutative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b, by cases a; cases b; simp [lift_or_get, h.comm]\u27e9\n\ninstance lift_or_get_assoc (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_associative \u03b1 f] :\n  is_associative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]\u27e9\n\ninstance lift_or_get_idem (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_idempotent \u03b1 f] :\n  is_idempotent (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get, h.idempotent]\u27e9\n\ninstance lift_or_get_is_left_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_left_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninstance lift_or_get_is_right_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_right_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ntheorem lift_or_get_choice {f : \u03b1 \u2192 \u03b1 \u2192 \u03b1} (h : \u2200 a b, f a b = a \u2228 f a b = b) :\n  \u2200 o\u2081 o\u2082, lift_or_get f o\u2081 o\u2082 = o\u2081 \u2228 lift_or_get f o\u2081 o\u2082 = o\u2082\n| none     none     := or.inl rfl\n| (some a) none     := or.inl rfl\n| none     (some b) := or.inr rfl\n| (some a) (some b) := by simpa [lift_or_get] using h a b\nsection rel\n\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : option \u03b1 \u2192 option \u03b2 \u2192 Prop\n| some {a b} : r a b \u2192 rel (some a) (some b)\n| none {}    : rel none none\n\nend rel\n\nend option\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/option.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.7090191399336402, "lm_q1q2_score": 0.38214943007163027}}
{"text": "import Mathlib.Data.Nat.Basic\nimport Mathlib.Init.Algebra.Order\nimport Mathlib.Init.Data.Nat.Basic\nimport Mathlib.Init.Data.Nat.Lemmas\nimport Mathlib.Init.Data.Int.Basic\nimport Mathlib.Data.String.Defs\nimport Mathlib.Data.String.Lemmas\nimport Mathlib.Data.Equiv.Basic\nimport Timelib.Util\nimport Timelib.NanoPrecision.Duration.SignedDuration\nimport Timelib.NanoPrecision.Duration.UnsignedDuration\nimport Timelib.NanoPrecision.DateTime.NaiveDateTime\nimport Timelib.NanoPrecision.TimeZone.Basic\nimport Lean.Data.Json\n\n/-\nI think you want a TAI type for the marker aspect if nothing else.\n-/\nstructure Offset extends TimeZone where\n  identifier : String\n  leapSecondsToApply : NaiveDateTime \u2192 SignedDuration\n  leapSecondsToRemove : NaiveDateTime \u2192 SignedDuration\n\nabbrev Offset.timeZoneOffset (\u03c9 : Offset) := \u03c9.toTimeZone.offset\nabbrev Offset.taiToUtc (\u03c9 : Offset) (t : NaiveDateTime) := t + (\u03c9.leapSecondsToApply t) \nabbrev Offset.utcToTai (\u03c9 : Offset) (t : NaiveDateTime) := t + (\u03c9.leapSecondsToRemove t) \n\n/--\nAn `Offset` is lawful if the functions returning leap seconds to remove\nand apply are isomorphic.\n-/\nclass LawfulOffset (\u03c9 : Offset) where\n  applyRemoveIso : \u03c9.taiToUtc \u2218 \u03c9.utcToTai = id\n  removeApplyIso : \u03c9.utcToTai \u2218 \u03c9.taiToUtc = id\n\n/--\nThe integer numberof TAI nanoseconds, relative to a zero of Janauary 1, year 1 in the Gregorian calendar.\nIf positive, the underlying number of TAI nanoseconds greater than the dawn of 1/1/1.\nIf negative, the number of TAI nanoseconds less than the dawn of 1/1/1.\n-/\nstructure DateTime (\u03c9 : Offset) where\n  naive : NaiveDateTime\nderiving DecidableEq, Repr, Hashable, Lean.FromJson, Lean.ToJson\n\n\ninstance {\u03c9 : Offset} : Inhabited (DateTime \u03c9) where\n  default := \u27e8Inhabited.default\u27e9\n\n/--\nThis is a safe conversion since the underlying time is tracked\nusing a `NaiveDateTime` that represents a TAI date/time.\n-/\ndef DateTime.changeOffset {\u03c9 : Offset} (t : DateTime \u03c9) (\u03c0 : Offset) : DateTime \u03c0 := \u27e8t.naive\u27e9\n\nsection DateTimeStuff\n\nvariable {\u03c9 \u03c0 : Offset}\n\n\ntheorem DateTime.eq_of_val_eq : \u2200 {d\u2081 d\u2082 : DateTime \u03c9} (_ : d\u2081.naive = d\u2082.naive), d\u2081 = d\u2082\n| \u27e8_\u27e9, _, rfl => rfl\n\ntheorem DateTime.val_ne_of_ne : \u2200 {d\u2081 d\u2082 : DateTime \u03c9} (_ : d\u2081 \u2260 d\u2082), d\u2081.naive \u2260 d\u2082.naive\n| \u27e8x\u27e9, \u27e8y\u27e9, h => by intro hh; apply h; exact congrArg DateTime.mk hh\n\n/-- Compares the underlying naive/TAI DateTime -/\ninstance : LT (DateTime \u03c9) where\n  lt := InvImage (instLTNaiveDateTime.lt) DateTime.naive\n\n/-- Compares the underlying naive/TAI DateTime -/\ninstance : LE (DateTime \u03c9) where\n  le := InvImage (instLENaiveDateTime.le) DateTime.naive\n  \n@[simp] theorem DateTime.le_def (d\u2081 d\u2082 : DateTime \u03c9) : (d\u2081 <= d\u2082) = (d\u2081.naive <= d\u2082.naive) := rfl\n@[simp] theorem DateTime.lt_def (d\u2081 d\u2082 : DateTime \u03c9) : (d\u2081 < d\u2082) = (d\u2081.naive < d\u2082.naive) := rfl\n\ninstance instDecidableLEDateTime (d\u2081 d\u2082 : DateTime \u03c9) : Decidable (d\u2081 <= d\u2082) := inferInstanceAs (Decidable (d\u2081.naive <= d\u2082.naive))\ninstance instDecidableLTDateTime (d\u2081 d\u2082 : DateTime \u03c9) : Decidable (d\u2081 < d\u2082) := inferInstanceAs (Decidable <| d\u2081.naive < d\u2082.naive)\n\ninstance : LinearOrder (DateTime \u03c9) where\n  le_refl (a) := le_refl a.naive\n  le_trans (a b c) := Int.le_trans\n  lt_iff_le_not_le (a b) := Int.lt_iff_le_not_le\n  le_antisymm (a b h1 h2) := by\n    rw [DateTime.le_def] at h1 h2\n    exact DateTime.eq_of_val_eq (le_antisymm h1 h2)\n  le_total := by simp [DateTime.le_def, le_total]\n  decidable_le := inferInstance\n\ninstance : HAdd (DateTime \u03c9) SignedDuration (DateTime \u03c9) where\n  hAdd da du := \u27e8da.naive + du\u27e9\n\ninstance : HAdd SignedDuration (DateTime \u03c9) (DateTime \u03c9)  where\n  hAdd du da := da + du\n\ntheorem DateTime.hAdd_signed_def (d : DateTime \u03c9) (dur : SignedDuration) : d + dur = \u27e8d.naive + dur\u27e9 := rfl\ntheorem DateTime.hAdd_signed_def_rev (d : DateTime \u03c9) (dur : SignedDuration) : dur + d = \u27e8dur + d.naive\u27e9 := rfl\n\ninstance : HSub (DateTime \u03c9) SignedDuration (DateTime \u03c9) where\n  hSub d dur := d + -dur\n\ntheorem DateTime.hSub_signed_def (d : DateTime \u03c9) (dur : SignedDuration) : d - dur = d + -dur := rfl\n\ninstance : HAdd (DateTime \u03c9) UnsignedDuration (DateTime \u03c9) where\n  hAdd da du := \u27e8da.naive + du\u27e9\n\ninstance : HAdd UnsignedDuration (DateTime \u03c9) (DateTime \u03c9)  where\n  hAdd du da := da + du\n\ntheorem DateTime.hAdd_def_unsigned (d : DateTime \u03c9) (dur : UnsignedDuration) : d + dur = \u27e8d.naive + dur\u27e9 := rfl\n\n@[defaultInstance]\ninstance : HSub (DateTime \u03c9) UnsignedDuration (DateTime \u03c9) where\n  hSub d dur := d - (dur : SignedDuration)\n\ntheorem DateTime.hSub_def_unsigned (d : DateTime \u03c9) (dur : UnsignedDuration) : d - dur = d + -(dur : SignedDuration) := rfl\n\ntheorem DateTime.hAdd_signed_assoc (d : DateTime \u03c9) (dur\u2081 dur\u2082 : SignedDuration) : d + dur\u2081 + dur\u2082 = d + (dur\u2081 + dur\u2082) := by\n  simp [DateTime.hAdd_signed_def, NaiveDateTime.hAdd_signed_def]\n  exact Int.add_assoc _ _ _\n\ntheorem DateTime.hAdd_signed_comm (d : DateTime \u03c9) (dur : SignedDuration) : d + dur = dur + d := by\n  simp [DateTime.hAdd_signed_def, NaiveDateTime.hAdd_signed_def, DateTime.hAdd_signed_def_rev, NaiveDateTime.hAdd_signed_def_rev]\n\n/--\nIncorporate the relevant leap seconds and the timezone offset, creating\na `NaiveDateTime` that carries the local number of nanos.\n-/\ndef DateTime.toLocalNaive (t : DateTime \u03c9) : NaiveDateTime := \n  /- The utc time; the naive time + leap seconds -/\n  let utc := t.naive + (\u03c9.leapSecondsToApply t.naive)\n  utc + \u03c9.timeZoneOffset\n\n/--\nConvert a `NaiveDateTime` representing a \n-/\n--def DateTime.fromTai (t : TaiDateTime) : DateTime \u03c9 := \u27e8t\u27e9\n-- vv This definition sort of makes more sense\ndef DateTime.fromTai (t : NaiveDateTime) : DateTime \u03c9 := \u27e8t\u27e9\n/--\nConvert a `NaiveDateTime` that is local (has leap seconds and timezone offset applied)\nand convert it to a `DateTime`.\n-/\ndef DateTime.fromLocalNaive (t : NaiveDateTime) : DateTime \u03c9 := \n  /- Remove the timezone offset to get utc -/\n  let utc := t - \u03c9.timeZoneOffset\n  /- Add whatever the corresponding `leapSecondsToRemove` value is -/\n  \u27e8utc + \u03c9.leapSecondsToRemove utc\u27e9\n\n/--\n-/\ndef DateTime.fromLocalYmdsn \n  (y : Year) \n  (m : Month) \n  (d : Nat) \n  (s : Nat)\n  (n : Nat)\n  (hd : 1 <= d \u2227 d <= m.numDays y := by decide) : DateTime \u03c9 := \n  DateTime.fromLocalNaive (NaiveDateTime.fromYmdsn y m d s n hd)\n\n/--\nSet the local clock time without changing the date.\n\nFor example\nt : DateTime \u03c9 := LOCAL 2022/6/12 @ 11:00:00.0\nt.setLocalClockTime (3:00:00.0)\n= LOCAL 2022/6/12 @ 3:00:00.0\n-/\ndef DateTime.setLocalClockTime (t : DateTime \u03c9) (c : NaiveClockTime) : DateTime \u03c9 :=\n  DateTime.fromLocalNaive (t.toLocalNaive.setClockTime c)\n\n/-- \nUse cases for this are probably rare, so make sure you know what you're getting.\n\n`compareLocalTimes` compares the literal calendar/wall clock datetimes from two time \nstamps, without any regard for what underlying time they represent.\n-/\ndef DateTime.compareLocal (t\u2081 : DateTime \u03c9) (t\u2082 : DateTime \u03c0) : Ordering :=\n  Ord.compare t\u2081.toLocalNaive t\u2082.toLocalNaive\n\ndef DateTime.localScalarDate (t : DateTime \u03c9) : ScalarDate := t.toLocalNaive.toScalarDate\ndef DateTime.localYmd (t : DateTime \u03c9) : Ymd := t.toLocalNaive.toYmd\ndef DateTime.localYear (t : DateTime \u03c9) : Year := t.localScalarDate.year\n\nend DateTimeStuff\n\n@[reducible]\ndef Offset.tai : Offset := {\n  name := \"International Atomic Time\"\n  abbreviation := \"TAI\"\n  offset := 0\n  identifier := \"\"\n  leapSecondsToApply := fun _ => 0\n  leapSecondsToRemove := fun _ => 0\n}\n\ninstance : LawfulOffset Offset.tai where\n  applyRemoveIso := by \n    apply funext; simp [Offset.leapSecondsToApply, Offset.leapSecondsToRemove, NaiveDateTime.hAdd_signed_def]\n  removeApplyIso := by \n    apply funext; simp [Offset.leapSecondsToApply, Offset.leapSecondsToRemove, NaiveDateTime.hAdd_signed_def]\n\n/--\nThe `leapSmear` offset should NOT be used as a way of representing TAI time, or another\ntime metric that ignores leap seconds (for that, see )\nWhile leap smearing ostensibly looks the same as TAI, there's a specific connotation\nin the type, which is that the oracle/reference being used IS actually accounting for some\nagreed-upon number of leap seconds and incorporating them into the time-stamps.\n\nLeapSmear has a specific connotation, particularly in that it shouldn't display\n\"60\" as a number of seconds.\n-/\n@[reducible]\ndef Offset.leapSmear (tz : TimeZone) : Offset := {\n  name := \"\"\n  abbreviation := \"\"\n  offset := tz.offset\n  -- This identifier should be changed to the user's preference; users will want to use this to identify the\n  -- oracle that's being used for the smear values.\n  identifier := \"\"\n  leapSecondsToApply := fun _ => 0\n  leapSecondsToRemove := fun _ => 0\n}\n\ninstance {tz : TimeZone} : LawfulOffset (Offset.leapSmear tz) where\n  applyRemoveIso := by \n    apply funext; simp [Offset.leapSecondsToApply, Offset.leapSecondsToRemove, NaiveDateTime.hAdd_signed_def]\n  removeApplyIso := by \n    apply funext; simp [Offset.leapSecondsToApply, Offset.leapSecondsToRemove, NaiveDateTime.hAdd_signed_def]\n\n/-\nThis definition of TAI time is convenient because it allows for comparison with other elements of `DateTime \u03c9`,\nand preserves type safety relative to `NaiveDateTime`, which carries no assertion that it represents\na TAI date/time element.\n-/\n@[reducible]\ndef TaiDateTime := DateTime Offset.tai\n", "meta": {"author": "ammkrn", "repo": "timelib", "sha": "185e8ea7c8b4274f2cb7ecba4c2e785c6e97cf15", "save_path": "github-repos/lean/ammkrn-timelib", "path": "github-repos/lean/ammkrn-timelib/timelib-185e8ea7c8b4274f2cb7ecba4c2e785c6e97cf15/Timelib/NanoPrecision/DateTime/DateTime.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3821494162566816}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Module.basic\nimport Mathlib.algebra.category.Group.limits\nimport Mathlib.algebra.direct_limit\nimport Mathlib.PostPort\n\nuniverses u v u_1 \n\nnamespace Mathlib\n\n/-!\n# The category of R-modules has all limits\n\nFurther, these limits are preserved by the forgetful functor --- that is,\nthe underlying types are just the limits in the category of types.\n-/\n\nnamespace Module\n\n\nprotected instance add_comm_group_obj {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) (j : J) : add_comm_group (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  id (is_add_comm_group (category_theory.functor.obj F j))\n\nprotected instance module_obj {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) (j : J) : module R (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  id (is_module (category_theory.functor.obj F j))\n\n/--\nThe flat sections of a functor into `Module R` form a submodule of all sections.\n-/\ndef sections_submodule {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : submodule R ((j : J) \u2192 \u21a5(category_theory.functor.obj F j)) :=\n  submodule.mk (category_theory.functor.sections (F \u22d9 category_theory.forget (Module R))) sorry sorry sorry\n\nprotected instance limit_add_comm_group {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : add_comm_group\n  (category_theory.limits.cone.X (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))) :=\n  id (submodule.add_comm_group (sections_submodule F))\n\nprotected instance limit_module {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : module R\n  (category_theory.limits.cone.X (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))) :=\n  id (submodule.semimodule (sections_submodule F))\n\n/-- `limit.\u03c0 (F \u22d9 forget Ring) j` as a `ring_hom`. -/\ndef limit_\u03c0_linear_map {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) (j : J) : linear_map R\n  (category_theory.limits.cone.X (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R))))\n  (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  linear_map.mk\n    (category_theory.nat_trans.app\n      (category_theory.limits.cone.\u03c0 (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))) j)\n    sorry sorry\n\nnamespace has_limits\n\n\n-- The next two definitions are used in the construction of `has_limits (Module R)`.\n\n-- After that, the limits should be constructed using the generic limits API,\n\n-- e.g. `limit F`, `limit.cone F`, and `limit.is_limit F`.\n\n/--\nConstruction of a limit cone in `Module R`.\n(Internal use only; use the limits API.)\n-/\ndef limit_cone {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : category_theory.limits.cone F :=\n  category_theory.limits.cone.mk\n    (of R\n      (category_theory.limits.cone.X (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))))\n    (category_theory.nat_trans.mk (limit_\u03c0_linear_map F))\n\n/--\nWitness that the limit cone in `Module R` is a limit cone.\n(Internal use only; use the limits API.)\n-/\ndef limit_cone_is_limit {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : category_theory.limits.is_limit (limit_cone F) :=\n  category_theory.limits.is_limit.of_faithful (category_theory.forget (Module R))\n    (category_theory.limits.types.limit_cone_is_limit (F \u22d9 category_theory.forget (Module R)))\n    (fun (s : category_theory.limits.cone F) =>\n      linear_map.mk\n        (fun\n          (v : category_theory.limits.cone.X (category_theory.functor.map_cone (category_theory.forget (Module R)) s)) =>\n          { val :=\n              fun (j : J) =>\n                category_theory.nat_trans.app\n                  (category_theory.limits.cone.\u03c0 (category_theory.functor.map_cone (category_theory.forget (Module R)) s))\n                  j v,\n            property := sorry })\n        sorry sorry)\n    sorry\n\nend has_limits\n\n\n/-- The category of R-modules has all limits. -/\nprotected instance has_limits {R : Type u} [ring R] : category_theory.limits.has_limits (Module R) :=\n  category_theory.limits.has_limits.mk\n    fun (J : Type v) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.has_limits_of_shape.mk\n        fun (F : J \u2964 Module R) => category_theory.limits.has_limit.mk (category_theory.limits.limit_cone.mk sorry sorry)\n\n/--\nAn auxiliary declaration to speed up typechecking.\n-/\ndef forget\u2082_AddCommGroup_preserves_limits_aux {R : Type u} [ring R] {J : Type v} [category_theory.small_category J] (F : J \u2964 Module R) : category_theory.limits.is_limit\n  (category_theory.functor.map_cone (category_theory.forget\u2082 (Module R) AddCommGroup) (has_limits.limit_cone F)) :=\n  AddCommGroup.limit_cone_is_limit (F \u22d9 category_theory.forget\u2082 (Module R) AddCommGroup)\n\n/--\nThe forgetful functor from R-modules to abelian groups preserves all limits.\n-/\nprotected instance forget\u2082_AddCommGroup_preserves_limits {R : Type u} [ring R] : category_theory.limits.preserves_limits (category_theory.forget\u2082 (Module R) AddCommGroup) :=\n  category_theory.limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.preserves_limits_of_shape.mk\n        fun (F : J \u2964 Module R) =>\n          category_theory.limits.preserves_limit_of_preserves_limit_cone (has_limits.limit_cone_is_limit F)\n            (forget\u2082_AddCommGroup_preserves_limits_aux F)\n\n/--\nThe forgetful functor from R-modules to types preserves all limits.\n-/\nprotected instance forget_preserves_limits {R : Type u} [ring R] : category_theory.limits.preserves_limits (category_theory.forget (Module R)) :=\n  category_theory.limits.preserves_limits.mk\n    fun (J : Type u_1) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.preserves_limits_of_shape.mk\n        fun (F : J \u2964 Module R) =>\n          category_theory.limits.preserves_limit_of_preserves_limit_cone (has_limits.limit_cone_is_limit F)\n            (category_theory.limits.types.limit_cone_is_limit (F \u22d9 category_theory.forget (Module R)))\n\n/-- The diagram (in the sense of `category_theory`)\n of an unbundled `direct_limit` of modules. -/\n@[simp] theorem direct_limit_diagram_map {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9] (G : \u03b9 \u2192 Type v) [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)] (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] (i : \u03b9) (j : \u03b9) (hij : i \u27f6 j) : category_theory.functor.map (direct_limit_diagram G f) hij = f i j (direct_limit_diagram._proof_1 i j hij) :=\n  Eq.refl (category_theory.functor.map (direct_limit_diagram G f) hij)\n\n/-- The `cocone` on `direct_limit_diagram` corresponding to\nthe unbundled `direct_limit` of modules.\n\nIn `direct_limit_is_colimit` we show that it is a colimit cocone. -/\n@[simp] theorem direct_limit_cocone_\u03b9_app {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9] (G : \u03b9 \u2192 Type v) [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)] (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] [DecidableEq \u03b9] (i : \u03b9) : category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 (direct_limit_cocone G f)) i =\n  module.direct_limit.of R \u03b9 G f i :=\n  Eq.refl (category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 (direct_limit_cocone G f)) i)\n\n/-- The unbundled `direct_limit` of modules is a colimit\nin the sense of `category_theory`. -/\ndef direct_limit_is_colimit {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9] (G : \u03b9 \u2192 Type v) [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)] (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] [DecidableEq \u03b9] [Nonempty \u03b9] : category_theory.limits.is_colimit (direct_limit_cocone G f) :=\n  category_theory.limits.is_colimit.mk\n    fun (s : category_theory.limits.cocone (direct_limit_diagram G f)) =>\n      module.direct_limit.lift R \u03b9 G f (category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 s)) sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Module/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.38214941625668153}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Jeremy Avigad, Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.functor.multivariate\nimport Mathlib.data.pfunctor.multivariate.basic\nimport Mathlib.data.pfunctor.multivariate.M\nimport Mathlib.data.qpf.multivariate.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# The final co-algebra of a multivariate qpf is again a qpf.\n\nFor a `(n+1)`-ary QPF `F (\u03b1\u2080,..,\u03b1\u2099)`, we take the least fixed point of `F` with\nregards to its last argument `\u03b1\u2099`. The result is a `n`-ary functor: `fix F (\u03b1\u2080,..,\u03b1\u2099\u208b\u2081)`.\nMaking `fix F` into a functor allows us to take the fixed point, compose with other functors\nand take a fixed point again.\n\n## Main definitions\n\n * `cofix.mk`     - constructor\n * `cofix.dest    - destructor\n * `cofix.corec`  - corecursor: useful for formulating infinite, productive computations\n * `cofix.bisim`  - bisimulation: proof technique to show the equality of possibly infinite values\n                    of `cofix F \u03b1`\n\n## Implementation notes\n\nFor `F` a QPF`, we define `cofix F \u03b1` in terms of the M-type of the polynomial functor `P` of `F`.\nWe define the relation `Mcongr` and take its quotient as the definition of `cofix F \u03b1`.\n\n`Mcongr` is taken as the weakest bisimulation on M-type.  See\n[avigad-carneiro-hudon2019] for more details.\n\n## Reference\n\n * [Jeremy Avigad, Mario M. Carneiro and Simon Hudon, *Data Types as Quotients of Polynomial Functors*][avigad-carneiro-hudon2019]\n-/\n\nnamespace mvqpf\n\n\n/-- `corecF` is used as a basis for defining the corecursor of `cofix F \u03b1`. `corecF`\nuses corecursion to construct the M-type generated by `q.P` and uses function on `F`\nas a corecursive step -/\ndef corecF {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: \u03b2)) : \u03b2 \u2192 mvpfunctor.M (P F) \u03b1 :=\n  mvpfunctor.M.corec (P F) fun (x : \u03b2) => repr (g x)\n\ntheorem corecF_eq {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: \u03b2)) (x : \u03b2) : mvpfunctor.M.dest (P F) (corecF g x) = mvfunctor.map (typevec.id ::: corecF g) (repr (g x)) := sorry\n\n/-- Characterization of desirable equivalence relations on M-types -/\ndef is_precongr {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (r : mvpfunctor.M (P F) \u03b1 \u2192 mvpfunctor.M (P F) \u03b1 \u2192 Prop) :=\n  \u2200 {x y : mvpfunctor.M (P F) \u03b1},\n    r x y \u2192\n      abs (mvfunctor.map (typevec.id ::: Quot.mk r) (mvpfunctor.M.dest (P F) x)) =\n        abs (mvfunctor.map (typevec.id ::: Quot.mk r) (mvpfunctor.M.dest (P F) y))\n\n/-- Equivalence relation on M-types representing a value of type `cofix F` -/\ndef Mcongr {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : mvpfunctor.M (P F) \u03b1) (y : mvpfunctor.M (P F) \u03b1) :=\n  \u2203 (r : mvpfunctor.M (P F) \u03b1 \u2192 mvpfunctor.M (P F) \u03b1 \u2192 Prop), is_precongr r \u2227 r x y\n\n/-- Greatest fixed point of functor F. The result is a functor with one fewer parameters\nthan the input. For `F a b c` a ternary functor, fix F is a binary functor such that\n\n```lean\ncofix F a b = F a b (cofix F a b)\n```\n-/\ndef cofix {n : \u2115} (F : typevec (n + 1) \u2192 Type u) [mvfunctor F] [q : mvqpf F] (\u03b1 : typevec n) :=\n  Quot Mcongr\n\nprotected instance cofix.inhabited {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} [Inhabited (mvpfunctor.A (P F))] [(i : fin2 n) \u2192 Inhabited (\u03b1 i)] : Inhabited (cofix F \u03b1) :=\n  { default := Quot.mk Mcongr Inhabited.default }\n\n/-- maps every element of the W type to a canonical representative -/\ndef Mrepr {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} : mvpfunctor.M (P F) \u03b1 \u2192 mvpfunctor.M (P F) \u03b1 :=\n  corecF (abs \u2218 mvpfunctor.M.dest (P F))\n\n/-- the map function for the functor `cofix F` -/\ndef cofix.map {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : typevec n} (g : typevec.arrow \u03b1 \u03b2) : cofix F \u03b1 \u2192 cofix F \u03b2 :=\n  Quot.lift (fun (x : mvpfunctor.M (P F) \u03b1) => Quot.mk Mcongr (mvfunctor.map g x)) sorry\n\nprotected instance cofix.mvfunctor {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] : mvfunctor (cofix F) :=\n  mvfunctor.mk cofix.map\n\n/-- Corecursor for `cofix F` -/\ndef cofix.corec {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: \u03b2)) : \u03b2 \u2192 cofix F \u03b1 :=\n  fun (x : \u03b2) => Quot.mk Mcongr (corecF g x)\n\n/-- Destructor for `cofix F` -/\ndef cofix.dest {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} : cofix F \u03b1 \u2192 F (\u03b1 ::: cofix F \u03b1) :=\n  Quot.lift\n    (fun (x : mvpfunctor.M (P F) \u03b1) => mvfunctor.map (typevec.id ::: Quot.mk Mcongr) (abs (mvpfunctor.M.dest (P F) x)))\n    sorry\n\n/-- Abstraction function for `cofix F \u03b1` -/\ndef cofix.abs {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} : mvpfunctor.M (P F) \u03b1 \u2192 cofix F \u03b1 :=\n  Quot.mk Mcongr\n\n/-- Representation function for `cofix F \u03b1` -/\ndef cofix.repr {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} : cofix F \u03b1 \u2192 mvpfunctor.M (P F) \u03b1 :=\n  mvpfunctor.M.corec (P F) (repr \u2218 cofix.dest)\n\n/-- Corecursor for `cofix F` -/\ndef cofix.corec'\u2081 {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : {X : Type u} \u2192 (\u03b2 \u2192 X) \u2192 F (\u03b1 ::: X)) (x : \u03b2) : cofix F \u03b1 :=\n  cofix.corec (fun (x : \u03b2) => g id) x\n\n/-- More flexible corecursor for `cofix F`. Allows the return of a fully formed\nvalue instead of making a recursive call -/\ndef cofix.corec' {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: (cofix F \u03b1 \u2295 \u03b2))) (x : \u03b2) : cofix F \u03b1 :=\n  let f : typevec.arrow (\u03b1 ::: cofix F \u03b1) (\u03b1 ::: (cofix F \u03b1 \u2295 \u03b2)) := typevec.id ::: sum.inl;\n  cofix.corec (sum.elim (mvfunctor.map f \u2218 cofix.dest) g) (sum.inr x)\n\n/-- Corecursor for `cofix F`. The shape allows recursive calls to\nlook like recursive calls. -/\ndef cofix.corec\u2081 {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : {X : Type u} \u2192 (cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)) (x : \u03b2) : cofix F \u03b1 :=\n  cofix.corec' (fun (x : \u03b2) => g sum.inl sum.inr x) x\n\ntheorem cofix.dest_corec {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: \u03b2)) (x : \u03b2) : cofix.dest (cofix.corec g x) = mvfunctor.map (typevec.id ::: cofix.corec g) (g x) := sorry\n\n/-- constructor for `cofix F` -/\ndef cofix.mk {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} : F (\u03b1 ::: cofix F \u03b1) \u2192 cofix F \u03b1 :=\n  cofix.corec fun (x : F (\u03b1 ::: cofix F \u03b1)) => mvfunctor.map (typevec.id ::: fun (i : cofix F \u03b1) => cofix.dest i) x\n\n/-!\n## Bisimulation principles for `cofix F`\n\nThe following theorems are bisimulation principles. The general idea\nis to use a bisimulation relation to prove the equality between\nspecific values of type `cofix F \u03b1`.\n\nA bisimulation relation `R` for values `x y : cofix F \u03b1`:\n\n * holds for `x y`: `R x y`\n * for any values `x y` that satisfy `R`, their root has the same shape\n   and their children can be paired in such a way that they satisfy `R`.\n\n-/\n\n/-- Bisimulation principle using `map` and `quot.mk` to match and relate children of two trees. -/\ntheorem cofix.bisim_rel {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (r : cofix F \u03b1 \u2192 cofix F \u03b1 \u2192 Prop) (h : \u2200 (x y : cofix F \u03b1),\n  r x y \u2192\n    mvfunctor.map (typevec.id ::: Quot.mk r) (cofix.dest x) = mvfunctor.map (typevec.id ::: Quot.mk r) (cofix.dest y)) (x : cofix F \u03b1) (y : cofix F \u03b1) : r x y \u2192 x = y := sorry\n\n/-- Bisimulation principle using `liftr` to match and relate children of two trees. -/\ntheorem cofix.bisim {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (r : cofix F \u03b1 \u2192 cofix F \u03b1 \u2192 Prop) (h : \u2200 (x y : cofix F \u03b1), r x y \u2192 mvfunctor.liftr (typevec.rel_last \u03b1 r) (cofix.dest x) (cofix.dest y)) (x : cofix F \u03b1) (y : cofix F \u03b1) : r x y \u2192 x = y := sorry\n\n/-- Bisimulation principle using `liftr'` to match and relate children of two trees. -/\ntheorem cofix.bisim\u2082 {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (r : cofix F \u03b1 \u2192 cofix F \u03b1 \u2192 Prop) (h : \u2200 (x y : cofix F \u03b1), r x y \u2192 mvfunctor.liftr' (typevec.rel_last' \u03b1 r) (cofix.dest x) (cofix.dest y)) (x : cofix F \u03b1) (y : cofix F \u03b1) : r x y \u2192 x = y := sorry\n\n/-- Bisimulation principle the values `\u27e8a,f\u27e9` of the polynomial functor representing\n`cofix F \u03b1` as well as an invariant `Q : \u03b2 \u2192 Prop` and a state `\u03b2` generating the\nleft-hand side and right-hand side of the equality through functions `u v : \u03b2 \u2192 cofix F \u03b1` -/\ntheorem cofix.bisim' {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u_1} (Q : \u03b2 \u2192 Prop) (u : \u03b2 \u2192 cofix F \u03b1) (v : \u03b2 \u2192 cofix F \u03b1) (h : \u2200 (x : \u03b2),\n  Q x \u2192\n    \u2203 (a : mvpfunctor.A (P F)),\n      \u2203 (f' : typevec.arrow (mvpfunctor.B (mvpfunctor.drop (P F)) a) \u03b1),\n        \u2203 (f\u2080 : pfunctor.B (mvpfunctor.last (P F)) a \u2192 cofix F \u03b1),\n          \u2203 (f\u2081 : pfunctor.B (mvpfunctor.last (P F)) a \u2192 cofix F \u03b1),\n            cofix.dest (u x) = abs (sigma.mk a (mvpfunctor.append_contents (P F) f' f\u2080)) \u2227\n              cofix.dest (v x) = abs (sigma.mk a (mvpfunctor.append_contents (P F) f' f\u2081)) \u2227\n                \u2200 (i : pfunctor.B (mvpfunctor.last (P F)) a), \u2203 (x' : \u03b2), Q x' \u2227 f\u2080 i = u x' \u2227 f\u2081 i = v x') (x : \u03b2) : Q x \u2192 u x = v x := sorry\n\ntheorem cofix.mk_dest {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : cofix F \u03b1) : cofix.mk (cofix.dest x) = x := sorry\n\ntheorem cofix.dest_mk {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : F (\u03b1 ::: cofix F \u03b1)) : cofix.dest (cofix.mk x) = x := sorry\n\ntheorem cofix.ext {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : cofix F \u03b1) (y : cofix F \u03b1) (h : cofix.dest x = cofix.dest y) : x = y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (x = y)) (Eq.symm (cofix.mk_dest x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (cofix.mk (cofix.dest x) = y)) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (cofix.mk (cofix.dest y) = y)) (cofix.mk_dest y))) (Eq.refl y)))\n\ntheorem cofix.ext_mk {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : F (\u03b1 ::: cofix F \u03b1)) (y : F (\u03b1 ::: cofix F \u03b1)) (h : cofix.mk x = cofix.mk y) : x = y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (x = y)) (Eq.symm (cofix.dest_mk x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (cofix.dest (cofix.mk x) = y)) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (cofix.dest (cofix.mk y) = y)) (cofix.dest_mk y))) (Eq.refl y)))\n\n/-!\n`liftr_map`, `liftr_map_last` and `liftr_map_last'` are useful for reasoning about\nthe induction step in bisimulation proofs.\n-/\n\ntheorem liftr_map {n : \u2115} {\u03b1 : typevec n} {\u03b2 : typevec n} {F' : typevec n \u2192 Type u} [mvfunctor F'] [is_lawful_mvfunctor F'] (R : typevec.arrow (typevec.prod \u03b2 \u03b2) (typevec.repeat n Prop)) (x : F' \u03b1) (f : typevec.arrow \u03b1 \u03b2) (g : typevec.arrow \u03b1 \u03b2) (h : typevec.arrow \u03b1 (typevec.subtype_ R)) (hh : typevec.comp (typevec.subtype_val R) h = typevec.comp (typevec.prod.map f g) typevec.prod.diag) : mvfunctor.liftr' R (mvfunctor.map f x) (mvfunctor.map g x) := sorry\n\ntheorem liftr_map_last {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [is_lawful_mvfunctor F] {\u03b1 : typevec n} {\u03b9 : Type u} {\u03b9' : Type u} (R : \u03b9' \u2192 \u03b9' \u2192 Prop) (x : F (\u03b1 ::: \u03b9)) (f : \u03b9 \u2192 \u03b9') (g : \u03b9 \u2192 \u03b9') (hh : \u2200 (x : \u03b9), R (f x) (g x)) : mvfunctor.liftr' (typevec.rel_last' \u03b1 R) (mvfunctor.map (typevec.id ::: f) x) (mvfunctor.map (typevec.id ::: g) x) := sorry\n\ntheorem liftr_map_last' {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [is_lawful_mvfunctor F] {\u03b1 : typevec n} {\u03b9 : Type u} (R : \u03b9 \u2192 \u03b9 \u2192 Prop) (x : F (\u03b1 ::: \u03b9)) (f : \u03b9 \u2192 \u03b9) (hh : \u2200 (x : \u03b9), R (f x) x) : mvfunctor.liftr' (typevec.rel_last' \u03b1 R) (mvfunctor.map (typevec.id ::: f) x) x := sorry\n\ntheorem cofix.abs_repr {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} (x : cofix F \u03b1) : Quot.mk Mcongr (cofix.repr x) = x := sorry\n\n/-- tactic for proof by bisimulation -/\ntheorem corec_roll {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {X : Type u} {Y : Type u} {x\u2080 : X} (f : X \u2192 Y) (g : Y \u2192 F (\u03b1 ::: X)) : cofix.corec (g \u2218 f) x\u2080 = cofix.corec (mvfunctor.map (typevec.id ::: f) \u2218 g) (f x\u2080) := sorry\n\ntheorem cofix.dest_corec' {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : \u03b2 \u2192 F (\u03b1 ::: (cofix F \u03b1 \u2295 \u03b2))) (x : \u03b2) : cofix.dest (cofix.corec' g x) = mvfunctor.map (typevec.id ::: sum.elim id (cofix.corec' g)) (g x) := sorry\n\ntheorem cofix.dest_corec\u2081 {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] {\u03b1 : typevec n} {\u03b2 : Type u} (g : {X : Type u} \u2192 (cofix F \u03b1 \u2192 X) \u2192 (\u03b2 \u2192 X) \u2192 \u03b2 \u2192 F (\u03b1 ::: X)) (x : \u03b2) (h : \u2200 (X Y : Type u) (f : cofix F \u03b1 \u2192 X) (f' : \u03b2 \u2192 X) (k : X \u2192 Y),\n  g (k \u2218 f) (k \u2218 f') x = mvfunctor.map (typevec.id ::: k) (g f f' x)) : cofix.dest (cofix.corec\u2081 g x) = g id (cofix.corec\u2081 g) x := sorry\n\nprotected instance mvqpf_cofix {n : \u2115} {F : typevec (n + 1) \u2192 Type u} [mvfunctor F] [q : mvqpf F] : mvqpf (cofix F) :=\n  mk (mvpfunctor.Mp (P F)) (fun (\u03b1 : typevec n) => Quot.mk Mcongr) (fun (\u03b1 : typevec n) => cofix.repr) sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/qpf/multivariate/constructions/cofix.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.38209646669650726}}
{"text": "theorem Array.sizeOf_lt_of_mem' [DecidableEq \u03b1] [SizeOf \u03b1] {as : Array \u03b1} (h : a \u2208 as) : sizeOf a < sizeOf as := by\n  simp [Membership.mem, contains, any, Id.run, BEq.beq, anyM] at h\n  let rec aux (j : Nat) : anyM.loop (m := Id) (fun b => decide (a = b)) as as.size (Nat.le_refl ..) j = true \u2192 sizeOf a < sizeOf as := by\n    unfold anyM.loop\n    intro h\n    split at h\n    \u00b7 simp [Bind.bind, pure] at h; split at h\n      next he => subst a; apply sizeOf_get_lt\n      next => have ih := aux (j+1) h; assumption\n    \u00b7 contradiction\n  apply aux 0 h\ntermination_by aux j => as.size - j\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/wfOverapplicationIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.38209645851208823}}
{"text": "import analysis.calculus.bump_function_inner\nimport analysis.calculus.cont_diff\nimport to_mathlib.topology.misc\nimport to_mathlib.topology.algebra.module\n\nnoncomputable theory\n\nopen set function filter\nopen_locale topology\n\nnamespace real\n\nlemma smooth_transition_proj_I {x : \u211d} :\n  smooth_transition (proj_I x) = smooth_transition x :=\nbegin\n  cases le_total (0 : \u211d) x with hx hx,\n  cases le_total (1 : \u211d) x with h2x h2x,\n  { rw [proj_I_eq_one.mpr h2x, smooth_transition.one_of_one_le h2x, smooth_transition.one], },\n  { rw [proj_I_eq_self.mpr \u27e8hx, h2x\u27e9] },\n  { rw [proj_I_eq_zero.mpr hx, smooth_transition.zero_of_nonpos hx, smooth_transition.zero], }\nend\n\nlemma smooth_transition.continuous_at {x : \u211d} : continuous_at smooth_transition x :=\nsmooth_transition.continuous.continuous_at\n\nend real\n\n-- section cont_diff_fderiv\n/-! In this section we prove that the derivative of a parametric function is smooth, assuming the\n  input function is smooth enough. We also do this for `cont_diff_within_at` and `fderiv_within`\n  (needed for manifolds)\n  We also need some random other lemmas that we didn't bother to put in the right place yet. -/\n\nsection fderiv\n\nvariables {\ud835\udd5c : Type*} [nontrivially_normed_field \ud835\udd5c]\nvariables {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\nvariables {F : Type*} [normed_add_comm_group F] [normed_space \ud835\udd5c F]\nvariables {G : Type*} [normed_add_comm_group G] [normed_space \ud835\udd5c G]\nvariables {X : Type*} [normed_add_comm_group X] [normed_space \ud835\udd5c X]\nvariables {G' : Type*} [normed_add_comm_group G'] [normed_space \ud835\udd5c G']\nvariables {f : E \u2192 F} {g : E \u2192 F} {u : set (E \u00d7 F)} {s : set E} {x : E} {t : set F} {n m : \u2115\u221e}\n\n-- the following version is not exactly ported\nlemma cont_diff_within_at_fderiv_within {f : E \u2192 F \u2192 G}\n  (hf : cont_diff_within_at \ud835\udd5c n (function.uncurry f) u (x, g x))\n  (hg : cont_diff_within_at \ud835\udd5c m g s x)\n  (ht : unique_diff_on \ud835\udd5c t)\n  (hmn : m + 1 \u2264 n) (hx : x \u2208 s)\n  (hst : s \u00d7\u02e2 t \u2286 u)\n  (hgx : \u2200\u1da0 x' in \ud835\udcdd[s] x, g x' \u2208 t)\n  (hgt : t \u2208 \ud835\udcdd[g '' s] g x)\n  :\n  cont_diff_within_at \ud835\udd5c m (\u03bb x, fderiv_within \ud835\udd5c (f x) t (g x)) s x :=\nby { rw [\u2190 insert_eq_self.mpr hx] at hst hgx,\n  exact (hf.mono hst).fderiv_within'' hg (hgx.mono (\u03bb y hy, ht _ hy)) hmn hgt }\n\nend fderiv\n\nsection calculus\nopen continuous_linear_map\nvariables {\ud835\udd5c : Type*} [nontrivially_normed_field \ud835\udd5c]\n          {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\n          {E\u2081 : Type*} [normed_add_comm_group E\u2081] [normed_space \ud835\udd5c E\u2081]\n          {E\u2082 : Type*} [normed_add_comm_group E\u2082] [normed_space \ud835\udd5c E\u2082]\n          {E' : Type*} [normed_add_comm_group E'] [normed_space \ud835\udd5c E']\n          {F : Type*} [normed_add_comm_group F] [normed_space \ud835\udd5c F]\n          {G : Type*} [normed_add_comm_group G] [normed_space \ud835\udd5c G]\n          {n : \u2115\u221e}\n\nlemma cont_diff_at.comp\u2082 {g : E\u2081 \u00d7 E\u2082 \u2192 G} {f\u2081 : F \u2192 E\u2081} {f\u2082 : F \u2192 E\u2082} {x : F}\n  (hg : cont_diff_at \ud835\udd5c n g (f\u2081 x, f\u2082 x)) (hf\u2081 : cont_diff_at \ud835\udd5c n f\u2081 x)\n  (hf\u2082 : cont_diff_at \ud835\udd5c n f\u2082 x) : cont_diff_at \ud835\udd5c n (\u03bb x, g (f\u2081 x, f\u2082 x)) x :=\nhg.comp x $ hf\u2081.prod hf\u2082\n\nlemma cont_diff_at.clm_comp {g : E' \u2192 F \u2192L[\ud835\udd5c] G} {f : E' \u2192 E \u2192L[\ud835\udd5c] F} {n : \u2115\u221e} {x : E'}\n  (hg : cont_diff_at \ud835\udd5c n g x) (hf : cont_diff_at \ud835\udd5c n f x) :\n  cont_diff_at \ud835\udd5c n (\u03bb x, g x \u2218L f x) x :=\nis_bounded_bilinear_map_comp.cont_diff.cont_diff_at.comp\u2082 hg hf\n\nlemma fderiv_comp {g : F \u2192 G} {f : E \u2192 F} (x : E)\n  (hg : differentiable_at \ud835\udd5c g (f x)) (hf : differentiable_at \ud835\udd5c f x) :\n  fderiv \ud835\udd5c (g \u2218 f) x = (fderiv \ud835\udd5c g (f x)).comp (fderiv \ud835\udd5c f x) :=\n(hg.has_fderiv_at.comp x hf.has_fderiv_at).fderiv\n\nlemma fderiv_prod_left {x\u2080 : E} {y\u2080 : F} :\n  fderiv \ud835\udd5c (\u03bb x, (x, y\u2080)) x\u2080 = continuous_linear_map.inl \ud835\udd5c E F :=\nbegin\n  refine (differentiable_at_id.fderiv_prod (differentiable_at_const y\u2080)).trans _,\n  rw [fderiv_id, fderiv_const],\n  refl\nend\n\nlemma fderiv_prod_right {x\u2080 : E} {y\u2080 : F} :\n  fderiv \ud835\udd5c (\u03bb y, (x\u2080, y)) y\u2080 = continuous_linear_map.inr \ud835\udd5c E F :=\nbegin\n  refine ((differentiable_at_const x\u2080).fderiv_prod differentiable_at_id).trans _,\n  rw [fderiv_id, fderiv_const],\n  refl\nend\n\nlemma fderiv_prod_eq_add {f : E \u00d7 F \u2192 G} {p : E \u00d7 F} (hf : differentiable_at \ud835\udd5c f p) :\n  fderiv \ud835\udd5c f p =\n  fderiv \ud835\udd5c (\u03bb (z : E \u00d7 F), f (z.1, p.2)) p + fderiv \ud835\udd5c (\u03bb (z : E \u00d7 F), f (p.1, z.2)) p :=\nbegin\n  rw [\u2190 @prod.mk.eta _ _ p] at hf,\n  rw [fderiv_comp p (by apply hf) (differentiable_at_fst.prod $ differentiable_at_const _),\n    fderiv_comp p (by apply hf) ((differentiable_at_const _).prod differentiable_at_snd),\n    \u2190 continuous_linear_map.comp_add,\n    differentiable_at_fst.fderiv_prod (differentiable_at_const _),\n    (differentiable_at_const _).fderiv_prod differentiable_at_snd,\n    fderiv_fst, fderiv_snd, fderiv_const, fderiv_const],\n  dsimp only [pi.zero_apply],\n  rw [prod.mk.eta, continuous_linear_map.fst_prod_zero_add_zero_prod_snd,\n    continuous_linear_map.comp_id]\nend\n\nlemma has_fderiv_at.partial_fst {\u03c6 : E \u2192 F \u2192 G} {\u03c6' : E \u00d7 F \u2192L[\ud835\udd5c] G} {e\u2080 : E} {f\u2080 : F}\n  (h : has_fderiv_at (uncurry \u03c6) \u03c6' (e\u2080, f\u2080)) :\n  has_fderiv_at (\u03bb e, \u03c6 e f\u2080) (\u03c6'.comp (inl \ud835\udd5c E F)) e\u2080 :=\nh.comp e\u2080 $ has_fderiv_at_prod_mk_left e\u2080 f\u2080\n\nlemma has_fderiv_at.partial_snd {\u03c6 : E \u2192 F \u2192 G} {\u03c6' : E \u00d7 F \u2192L[\ud835\udd5c] G} {e\u2080 : E} {f\u2080 : F}\n  (h : has_fderiv_at (uncurry \u03c6) \u03c6' (e\u2080, f\u2080)) :\n  has_fderiv_at (\u03bb f, \u03c6 e\u2080 f) (\u03c6'.comp (inr \ud835\udd5c E F)) f\u2080 :=\nh.comp f\u2080 $ has_fderiv_at_prod_mk_right e\u2080 f\u2080\n\nvariable (\ud835\udd5c)\n\n/-- The first partial derivative of a binary function. -/\ndef partial_fderiv_fst {F : Type*} (\u03c6 : E \u2192 F \u2192 G) : E \u2192 F \u2192 E \u2192L[\ud835\udd5c] G :=\n\u03bb (e\u2080 : E) (f\u2080 : F), fderiv \ud835\udd5c (\u03bb e, \u03c6 e f\u2080) e\u2080\n\n/-- The second partial derivative of a binary function. -/\ndef partial_fderiv_snd {E : Type*} (\u03c6 : E \u2192 F \u2192 G) : E \u2192 F \u2192 F \u2192L[\ud835\udd5c] G :=\n\u03bb (e\u2080 : E) (f\u2080 : F), fderiv \ud835\udd5c (\u03bb f, \u03c6 e\u2080 f) f\u2080\n\nlocal notation `\u2202\u2081` := partial_fderiv_fst\nlocal notation `\u2202\u2082` := partial_fderiv_snd\n\nvariable {\ud835\udd5c}\n\nlemma fderiv_partial_fst {\u03c6 : E \u2192 F \u2192 G} {\u03c6' : E \u00d7 F \u2192L[\ud835\udd5c] G} {e\u2080 : E} {f\u2080 : F}\n  (h : has_fderiv_at (uncurry \u03c6) \u03c6' (e\u2080, f\u2080)) :\n  \u2202\u2081 \ud835\udd5c \u03c6 e\u2080 f\u2080 = \u03c6'.comp (inl \ud835\udd5c E F) :=\nh.partial_fst.fderiv\n\nlemma fderiv_partial_snd {\u03c6 : E \u2192 F \u2192 G} {\u03c6' : E \u00d7 F \u2192L[\ud835\udd5c] G} {e\u2080 : E} {f\u2080 : F}\n  (h : has_fderiv_at (uncurry \u03c6) \u03c6' (e\u2080, f\u2080)) :\n  \u2202\u2082 \ud835\udd5c \u03c6 e\u2080 f\u2080 = \u03c6'.comp (inr \ud835\udd5c E F) :=\nh.partial_snd.fderiv\n\nlemma differentiable_at.has_fderiv_at_partial_fst {\u03c6 : E \u2192 F \u2192 G} {e\u2080 : E} {f\u2080 : F}\n  (h : differentiable_at \ud835\udd5c (uncurry \u03c6) (e\u2080, f\u2080)) :\n  has_fderiv_at (\u03bb e, \u03c6 e f\u2080) (partial_fderiv_fst \ud835\udd5c \u03c6 e\u2080 f\u2080) e\u2080 :=\n(h.comp e\u2080 $ differentiable_at_id.prod $ differentiable_at_const f\u2080).has_fderiv_at\n\nlemma differentiable_at.has_fderiv_at_partial_snd {\u03c6 : E \u2192 F \u2192 G} {e\u2080 : E} {f\u2080 : F}\n  (h : differentiable_at \ud835\udd5c (uncurry \u03c6) (e\u2080, f\u2080)) :\nhas_fderiv_at (\u03bb f, \u03c6 e\u2080 f) (partial_fderiv_snd \ud835\udd5c \u03c6 e\u2080 f\u2080) f\u2080 :=\nbegin\n  rw fderiv_partial_snd h.has_fderiv_at,\n  exact h.has_fderiv_at.partial_snd\nend\n\nlemma cont_diff.partial_fst {\u03c6 : E \u2192 F \u2192 G} {n : \u2115\u221e}\n  (h : cont_diff \ud835\udd5c n $ uncurry \u03c6) (f\u2080 : F) : cont_diff \ud835\udd5c n (\u03bb e, \u03c6 e f\u2080) :=\nh.comp $ cont_diff_prod_mk_left f\u2080\n\nlemma cont_diff.partial_snd {\u03c6 : E \u2192 F \u2192 G} {n : \u2115\u221e}\n  (h : cont_diff \ud835\udd5c n $ uncurry \u03c6) (e\u2080 : E) : cont_diff \ud835\udd5c n (\u03bb f, \u03c6 e\u2080 f) :=\nh.comp $ cont_diff_prod_mk_right e\u2080\n\n/-- Precomposition by a continuous linear map as a continuous linear map between spaces of\ncontinuous linear maps. -/\ndef continuous_linear_map.comp_rightL (\u03c6 : E \u2192L[\ud835\udd5c] F) : (F \u2192L[\ud835\udd5c] G) \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] G) :=\n(compL \ud835\udd5c E F G).flip \u03c6\n\n/-- Postcomposition by a continuous linear map as a continuous linear map between spaces of\ncontinuous linear maps. -/\ndef continuous_linear_map.comp_leftL (\u03c6 : F \u2192L[\ud835\udd5c] G) : (E \u2192L[\ud835\udd5c] F) \u2192L[\ud835\udd5c] (E \u2192L[\ud835\udd5c] G) :=\ncompL \ud835\udd5c E F G \u03c6\n\nlemma differentiable.fderiv_partial_fst {\u03c6 : E \u2192 F \u2192 G} (hF : differentiable \ud835\udd5c (uncurry \u03c6)) :\n  \u21bf(\u2202\u2081 \ud835\udd5c \u03c6) = (\u03bb \u03c8 : E \u00d7 F \u2192L[\ud835\udd5c] G, \u03c8.comp (inl \ud835\udd5c E F)) \u2218 (fderiv \ud835\udd5c $ uncurry \u03c6) :=\nby { ext1 \u27e8y, t\u27e9, exact fderiv_partial_fst (hF \u27e8y, t\u27e9).has_fderiv_at }\n\nlemma differentiable.fderiv_partial_snd {\u03c6 : E \u2192 F \u2192 G} (hF : differentiable \ud835\udd5c (uncurry \u03c6)) :\n  \u21bf(\u2202\u2082 \ud835\udd5c \u03c6) = (\u03bb \u03c8 : E \u00d7 F \u2192L[\ud835\udd5c] G, \u03c8.comp (inr \ud835\udd5c E F)) \u2218 (fderiv \ud835\udd5c $ uncurry \u03c6) :=\nby { ext1 \u27e8y, t\u27e9, exact fderiv_partial_snd (hF \u27e8y, t\u27e9).has_fderiv_at }\n\n/-- The first partial derivative of `\u03c6 : \ud835\udd5c \u2192 F \u2192 G` seen as a function from `\ud835\udd5c \u2192 F \u2192 G`-/\ndef partial_deriv_fst (\u03c6 : \ud835\udd5c \u2192 F \u2192 G) : \ud835\udd5c \u2192 F \u2192 G := \u03bb k f, \u2202\u2081 \ud835\udd5c \u03c6 k f 1\n\n/-- The second partial derivative of `\u03c6 : E \u2192 \ud835\udd5c \u2192 G` seen as a function from `E \u2192 \ud835\udd5c \u2192 G`-/\ndef partial_deriv_snd (\u03c6 : E \u2192 \ud835\udd5c \u2192 G) : E \u2192 \ud835\udd5c \u2192 G := \u03bb e k, \u2202\u2082 \ud835\udd5c \u03c6 e k 1\n\nlemma partial_fderiv_fst_eq_smul_right (\u03c6 : \ud835\udd5c \u2192 F \u2192 G) (k : \ud835\udd5c) (f : F) :\n  \u2202\u2081 \ud835\udd5c \u03c6 k f = smul_right (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) (partial_deriv_fst \u03c6 k f) := deriv_fderiv.symm\n\n@[simp]\nlemma partial_fderiv_fst_one (\u03c6 : \ud835\udd5c \u2192 F \u2192 G) (k : \ud835\udd5c) (f : F) :\n  \u2202\u2081 \ud835\udd5c \u03c6 k f 1 = partial_deriv_fst \u03c6 k f :=\nby simp only [partial_fderiv_fst_eq_smul_right, smul_right_apply, one_apply, one_smul]\n\nlemma partial_fderiv_snd_eq_smul_right (\u03c6 : E \u2192 \ud835\udd5c \u2192 G) (e : E) (k : \ud835\udd5c) :\n  \u2202\u2082 \ud835\udd5c \u03c6 e k  = smul_right (1 : \ud835\udd5c \u2192L[\ud835\udd5c] \ud835\udd5c) (partial_deriv_snd \u03c6 e k) := deriv_fderiv.symm\n\nlemma partial_fderiv_snd_one (\u03c6 : E \u2192 \ud835\udd5c \u2192 G) (e : E) (k : \ud835\udd5c) :\n  \u2202\u2082 \ud835\udd5c \u03c6 e k 1 = partial_deriv_snd \u03c6 e k :=\nby simp only [partial_fderiv_snd_eq_smul_right, smul_right_apply, one_apply, one_smul]\n\n@[to_additive]\nlemma with_top.le_mul_self {\u03b1 : Type*} [canonically_ordered_monoid \u03b1] (n m : \u03b1) :\n  (n : with_top \u03b1) \u2264 (m * n : \u03b1) :=\nwith_top.coe_le_coe.mpr le_mul_self\n\n@[to_additive]\nlemma with_top.le_self_mul {\u03b1 : Type*} [canonically_ordered_monoid \u03b1] (n m : \u03b1) :\n  (n : with_top \u03b1) \u2264 (n * m : \u03b1) :=\nwith_top.coe_le_coe.mpr le_self_mul\n\nlemma cont_diff.cont_diff_partial_fst {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (hF : cont_diff \ud835\udd5c (n + 1) (uncurry \u03c6)) : cont_diff \ud835\udd5c n \u21bf(\u2202\u2081 \ud835\udd5c \u03c6) :=\ncont_diff.fderiv (hF.comp $ cont_diff_snd.prod cont_diff_fst.snd) cont_diff_fst le_rfl\n\nlemma cont_diff.cont_diff_partial_fst_apply {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (hF : cont_diff \ud835\udd5c (n + 1) (uncurry \u03c6)) {x : E} : cont_diff \ud835\udd5c n \u21bf(\u03bb x' y, \u2202\u2081 \ud835\udd5c \u03c6 x' y x) :=\n(continuous_linear_map.apply \ud835\udd5c G x).cont_diff.comp hF.cont_diff_partial_fst\n\nlemma cont_diff.continuous_partial_fst {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (h : cont_diff \ud835\udd5c ((n + 1 : \u2115) : \u2115\u221e) $ uncurry \u03c6) : continuous \u21bf(\u2202\u2081 \ud835\udd5c \u03c6) :=\nh.cont_diff_partial_fst.continuous\n\nlemma cont_diff.cont_diff_top_partial_fst {\u03c6 : E \u2192 F \u2192 G} (hF : cont_diff \ud835\udd5c \u22a4 (uncurry \u03c6)) :\n  cont_diff \ud835\udd5c \u22a4 \u21bf(\u2202\u2081 \ud835\udd5c \u03c6) :=\ncont_diff_top.mpr (\u03bb n, (cont_diff_top.mp hF (n + 1)).cont_diff_partial_fst)\n\nlemma cont_diff.cont_diff_partial_snd {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (hF : cont_diff \ud835\udd5c (n + 1) (uncurry \u03c6)) : cont_diff \ud835\udd5c n \u21bf(\u2202\u2082 \ud835\udd5c \u03c6) :=\ncont_diff.fderiv (hF.comp $ cont_diff_fst.fst.prod cont_diff_snd) cont_diff_snd le_rfl\n\nlemma cont_diff.cont_diff_partial_snd_apply {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (hF : cont_diff \ud835\udd5c (n + 1) (uncurry \u03c6)) {y : F} : cont_diff \ud835\udd5c n \u21bf(\u03bb x y', \u2202\u2082 \ud835\udd5c \u03c6 x y' y) :=\n(continuous_linear_map.apply \ud835\udd5c G y).cont_diff.comp hF.cont_diff_partial_snd\n\nlemma cont_diff.continuous_partial_snd {\u03c6 : E \u2192 F \u2192 G} {n : \u2115}\n  (h : cont_diff \ud835\udd5c ((n + 1 : \u2115) : \u2115\u221e) $ uncurry \u03c6) : continuous \u21bf(\u2202\u2082 \ud835\udd5c \u03c6) :=\nh.cont_diff_partial_snd.continuous\n\nlemma cont_diff.cont_diff_top_partial_snd {\u03c6 : E \u2192 F \u2192 G} (hF : cont_diff \ud835\udd5c \u22a4 (uncurry \u03c6)) :\n  cont_diff \ud835\udd5c \u22a4 \u21bf(\u2202\u2082 \ud835\udd5c \u03c6) :=\ncont_diff_top.mpr (\u03bb n, (cont_diff_top.mp hF (n + 1)).cont_diff_partial_snd)\n\nend calculus\n\nsection real_calculus\nopen continuous_linear_map\nvariables {E : Type*} [normed_add_comm_group E] [normed_space \u211d E]\n          {F : Type*} [normed_add_comm_group F] [normed_space \u211d F]\n\nlemma cont_diff.lipschitz_on_with {s : set E} {f : E \u2192 F} {n} (hf : cont_diff \u211d n f) (hn : 1 \u2264 n)\n  (hs : convex \u211d s) (hs' : is_compact s) : \u2203 K, lipschitz_on_with K f s :=\nbegin\n  rcases (bdd_above_iff_exists_ge 0).mp (hs'.image (hf.continuous_fderiv hn).norm).bdd_above with\n    \u27e8M, M_nonneg, hM\u27e9,\n  simp_rw [ball_image_iff] at hM,\n  use \u27e8M, M_nonneg\u27e9,\n  exact convex.lipschitz_on_with_of_nnnorm_fderiv_le (\u03bb x x_in, hf.differentiable hn x) hM hs\nend\n\nend real_calculus\n\nopen filter\n\n/-\nThe lemma below is ridiculously painful, but Patrick isn't patient enough.\n-/\nlemma const_mul_one_div_lt {\u03b5 : \u211d} (\u03b5_pos : 0 < \u03b5) (C : \u211d) : \u2200\u1da0 (N : \u211d) in at_top, C*\u20161/N\u2016 < \u03b5 :=\nbegin\n  have : tendsto (\u03bb N : \u211d, 1/N) at_top (\ud835\udcdd 0),\n  { rw show (\u03bb N : \u211d, 1/N) = \u03bb N, N^(-(1 : \u2124)), by simp,\n    exact tendsto_pow_neg_at_top one_ne_zero },\n  rw tendsto_iff_norm_tendsto_zero at this,\n  simp only [sub_zero] at this,\n  have key := this.const_mul C,\n  rw mul_zero at key,\n  apply (normed_add_comm_group.tendsto_nhds_zero.mp key \u03b5 \u03b5_pos).mono,\n  intros N hN,\n  cases le_or_lt (C * \u20161 / N\u2016) 0 with h h,\n  { exact h.trans_lt \u03b5_pos },\n  { rwa real.norm_of_nonneg h.le at hN },\nend\n", "meta": {"author": "leanprover-community", "repo": "sphere-eversion", "sha": "324e02c1509db6177cf363618f6ac5be343ce2f5", "save_path": "github-repos/lean/leanprover-community-sphere-eversion", "path": "github-repos/lean/leanprover-community-sphere-eversion/sphere-eversion-324e02c1509db6177cf363618f6ac5be343ce2f5/src/to_mathlib/analysis/calculus.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3818183118798879}}
{"text": "import category_theory.limits.preserves.limits\nimport for_mathlib.derived.K_projective\nimport for_mathlib.AddCommGroup.explicit_limits\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\nvariables {A : Type u} [category.{v} A] [abelian A] [has_coproducts.{v} A]\n\nopen_locale zero_object\n\nnamespace category_theory\n\ninstance is_iso_coproduct {\u03b1 : Type v} (X Y : \u03b1 \u2192 A) (f : \u03a0 a, X a \u27f6 Y a)\n  [\u2200 a, is_iso (f a)] :\n  is_iso (sigma.desc $ \u03bb a, f a \u226b sigma.\u03b9 _ a) :=\nbegin\n  use sigma.desc (\u03bb a, inv (f a) \u226b sigma.\u03b9 _ a),\n  split,\n  { ext, dsimp, cases j, simp },\n  { ext, dsimp, cases j, simp }\nend\n\nnoncomputable\nlemma is_initial_colimit {J : Type v} [small_category J] (K : J \u2964 A)\n  (hK : \u2200 j, is_initial (K.obj j)) [has_colimit K] :\n  is_initial (colimit K) :=\n{ desc := \u03bb T, colimit.desc _ \u27e8_,\n  { app := \u03bb j, (hK j).to _,\n    naturality' := \u03bb i j f, (hK _).hom_ext _ _ }\u27e9,\n  fac' := by rintros S \u27e8\u27e8\u27e9\u27e9,\n  uniq' := begin\n    intros S m hm, apply colimit.hom_ext, intros j,\n    apply (hK _).hom_ext\n  end }\n\nlemma is_zero_colimit {J : Type v} [small_category J] (K : J \u2964 A)\n  (hK : \u2200 j, is_zero (K.obj j)) [has_colimit K] :\n  is_zero (colimit K) :=\nbegin\n  suffices : is_initial (colimit K),\n  { let e : colimit K \u2245 \u22a5_ _ := (initial_iso_is_initial this).symm,\n    apply is_zero_of_iso_of_zero _ e.symm,\n    apply is_zero_initial },\n  apply is_initial_colimit,\n  intros j,\n  apply is_zero.is_initial,\n  apply hK,\nend\n\nnoncomputable\ndef preadditive_yoneda_coproduct_to_product {A : Type u} [category.{v} A]\n  [preadditive A]\n  {\u03b1 : Type v} (X : \u03b1 \u2192 A) [has_coproduct X] (Y : A) :\n  (preadditive_yoneda.obj Y).obj (opposite.op $ sigma_obj X) \u27f6\n  pi_obj (\u03bb a, (preadditive_yoneda.obj Y).obj (opposite.op $ X a)) :=\npi.lift $ \u03bb b, functor.map _ $ quiver.hom.op $ sigma.\u03b9 _ _\n\nset_option pp.universes true\n\ninstance is_iso_preadditive_yoneda_coproduct_to_product\n  {A : Type u} [category.{v} A]\n  [preadditive A]\n  {\u03b1 : Type v} (X : \u03b1 \u2192 A) [has_coproduct X] (Y : A) :\n  is_iso (preadditive_yoneda_coproduct_to_product X Y) :=\nbegin\n  apply is_iso_of_reflects_iso _ (forget AddCommGroup),\n  rw is_iso_iff_bijective,\n  split,\n  { intros f g h, dsimp [preadditive_yoneda_coproduct_to_product] at f g h \u22a2,\n    apply colimit.hom_ext,\n    intros a,\n    let q : (\u220f \u03bb (a : \u03b1), AddCommGroup.of (X a \u27f6 Y)) \u27f6 (AddCommGroup.of (X a.1 \u27f6 Y)) :=\n      pi.\u03c0 _ a.1,\n    apply_fun (\u03bb e, q e) at h,\n    simp only [\u2190 comp_apply, limit.lift_\u03c0] at h, cases a, exact h },\n  { intros f, dsimp at f,\n    let P : \u03a0 a, (\u220f \u03bb (a : \u03b1), AddCommGroup.of (X a \u27f6 Y)) \u27f6 (AddCommGroup.of (X a \u27f6 Y)) :=\n      \u03bb a, pi.\u03c0 _ a,\n    let q : sigma_obj X \u27f6 Y := sigma.desc (\u03bb a, P a f),\n    use q,\n    apply concrete.limit_ext (discrete.functor (\u03bb a, AddCommGroup.of (X a \u27f6 Y))),\n    intros i, dsimp [preadditive_yoneda_coproduct_to_product],\n    simp only [\u2190 comp_apply, limit.lift_\u03c0],\n    dsimp, rw colimit.\u03b9_desc, cases i, refl }\nend\n\nnoncomputable\ndef preadditive_yoneda_coproduct_iso {A : Type u} [category.{v} A]\n  [preadditive A]\n  {\u03b1 : Type v} (X : \u03b1 \u2192 A) [has_coproduct X] (Y : A) :\n  (preadditive_yoneda.obj Y).obj (opposite.op $ sigma_obj X) \u2245\n  pi_obj (\u03bb a, (preadditive_yoneda.obj Y).obj (opposite.op $ X a)) :=\nas_iso (preadditive_yoneda_coproduct_to_product _ _)\n\nnoncomputable\ndef pi_iso {A : Type u} [category.{v} A] {\u03b1 : Type v}\n  (X Y : \u03b1 \u2192 A)\n  (I : \u03a0 a, X a \u2245 Y a)\n  [has_product X] [has_product Y] :\n  pi_obj X \u2245 pi_obj Y :=\n{ hom := pi.lift $ \u03bb b, pi.\u03c0 _ _ \u226b (I b).hom,\n  inv := pi.lift $ \u03bb b, pi.\u03c0 _ _ \u226b (I b).inv,\n  hom_inv_id' := begin\n    apply limit.hom_ext, rintros \u27e8i\u27e9,\n    simp,\n  end,\n  inv_hom_id' := begin\n    apply limit.hom_ext, rintros \u27e8i\u27e9,\n    simp,\n  end }\n\nend category_theory\n\nnamespace homotopy_category\n\nnoncomputable\ndef coproduct_iso {\u03b1 : Type v} (X : \u03b1 \u2192 cochain_complex A \u2124) (i) :\n  (sigma_obj X).X i \u2245 sigma_obj (\u03bb a : \u03b1, (X a).X i) :=\n(category_theory.preserves_colimit_iso (homological_complex.eval A\n  (complex_shape.up \u2124) i) (discrete.functor X)) \u226a\u226b\n  has_colimit.iso_of_nat_iso\n  (nat_iso.of_components (\u03bb _, iso.refl _) begin\n    rintros \u27e8\u27e9 \u27e8\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9,\n    simp,\n  end)\n\n@[simp, reassoc]\nlemma coproduct_\u03b9_coproduct_iso_inv {\u03b1 : Type v} (X : \u03b1 \u2192 cochain_complex A \u2124) (i) (a) :\n  sigma.\u03b9 _ a \u226b (coproduct_iso X i).inv = ((sigma.\u03b9 X a : _ \u27f6 _)).f i :=\nbegin\n  dsimp [coproduct_iso, has_colimit.iso_of_nat_iso,\n    is_colimit.map, preserves_colimit_iso, is_colimit.cocone_point_unique_up_to_iso],\n  simp, dsimp, simp,\nend\n\n@[simp, reassoc]\nlemma coproduct_\u03b9_coproduct_iso_hom {\u03b1 : Type v} (X : \u03b1 \u2192 cochain_complex A \u2124) (i) (a : \u03b1) :\n  (sigma.\u03b9 X a : _ \u27f6 _).f i \u226b (coproduct_iso X i).hom = sigma.\u03b9 _ a :=\nbegin\n  dsimp [coproduct_iso, has_colimit.iso_of_nat_iso,\n    is_colimit.map, preserves_colimit_iso, is_colimit.cocone_point_unique_up_to_iso],\n  slice_lhs 0 1\n  { erw (is_colimit_of_preserves (homological_complex.eval A (complex_shape.up \u2124) i)\n    (colimit.is_colimit (discrete.functor X))).fac },\n  simp, dsimp, simp,\nend\n\nnoncomputable\ndef homotopy_coprod {\u03b1 : Type v} (X : \u03b1 \u2192 cochain_complex A \u2124) (Y)\n  (f g : sigma_obj X \u27f6 Y)\n  (h : \u03a0 a, homotopy (sigma.\u03b9 _ a \u226b f) (sigma.\u03b9 _ a \u226b g)) :\n  homotopy f g :=\n{ hom := \u03bb i j, (coproduct_iso X i).hom \u226b\n    (sigma.desc $ \u03bb a, (h a).hom _ _),\n  zero' := begin\n    intros i j hh,\n    simp only [preadditive.is_iso.comp_left_eq_zero],\n    apply colimit.hom_ext, intros a,\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, comp_zero, (h a.1).zero' i j hh],\n  end,\n  comm := begin\n    intros i,\n    rw \u2190 cancel_epi (coproduct_iso X i).inv,\n    apply colimit.hom_ext, rintros \u27e8a\u27e9,\n    simp only [coproduct_\u03b9_coproduct_iso_inv_assoc, homological_complex.cochain_complex_d_next,\n      homological_complex.cochain_complex_prev_d, category.assoc, preadditive.comp_add,\n      homological_complex.hom.comm_assoc, coproduct_\u03b9_coproduct_iso_hom_assoc,\n      colimit.\u03b9_desc, cofan.mk_\u03b9_app, colimit.\u03b9_desc_assoc],\n    have := (h a).comm i,\n    refine this.trans _,\n    simp only [homological_complex.cochain_complex_d_next,\n      homological_complex.cochain_complex_prev_d],\n    refl\n  end }\n\nlemma homotopic_coprod {\u03b1 : Type v} (X : \u03b1 \u2192 cochain_complex A \u2124) (Y)\n  (f g : sigma_obj X \u27f6 Y)\n  (h : \u2200 a : \u03b1, homotopic _ _ (sigma.\u03b9 _ a \u226b f) (sigma.\u03b9 _ a \u226b g)) :\n  homotopic _ _ f g :=\nbegin\n  constructor,\n  apply homotopy_coprod,\n  intros a,\n  exact (h a).some,\nend\n\n-- Move this\nlemma homotopic_of_quotient_map_eq {X Y : cochain_complex A \u2124}\n  (f g : X \u27f6 Y) (h : (quotient _ _).map f = (quotient _ _).map g) :\n  homotopic _ _ f g :=\nbegin\n  erw quotient.functor_map_eq_iff at h, assumption,\nend\n\nnoncomputable\ndef colimit_cofan {\u03b1 : Type v} (X : \u03b1 \u2192 homotopy_category A (complex_shape.up \u2124)) :\n  cofan X :=\ncofan.mk\n((quotient _ _).obj $ sigma_obj (\u03bb a, (X a).as))\n(\u03bb a, (quotient _ _).map $ sigma.\u03b9 _ a)\n\nnoncomputable\ndef is_colimit_cofan {\u03b1 : Type v} (X : \u03b1 \u2192 homotopy_category A (complex_shape.up \u2124)) :\n  is_colimit (colimit_cofan X) :=\n{ desc := \u03bb S, (quotient _ _).map $ sigma.desc $ \u03bb a, (S.\u03b9.app \u27e8a\u27e9).out,\n  fac' := begin\n    rintros S \u27e8j\u27e9,\n    dsimp,\n    erw [\u2190 (quotient A (complex_shape.up \u2124)).map_comp, colimit.\u03b9_desc],\n    dsimp [quotient],\n    simp,\n  end,\n  uniq' := begin\n    intros S m hm,\n    let mm := m.out,\n    have : (quotient _ _).map mm = m, by simp,\n    rw \u2190 this,\n    apply quot.sound,\n    apply quotient.comp_closure.of,\n    apply homotopic_coprod,\n    intros a,\n    specialize hm \u27e8a\u27e9, rw \u2190 this at hm, dsimp at hm,\n    erw [\u2190 (quotient A (complex_shape.up \u2124)).map_comp] at hm,\n    erw colimit.\u03b9_desc,\n    dsimp,\n    have : S.\u03b9.app \u27e8a\u27e9 = (quotient _ _).map (S.\u03b9.app \u27e8a\u27e9).out, by simp,\n    rw this at hm,\n    apply homotopic_of_quotient_map_eq,\n    exact hm\n  end }\n\ninstance {\u03b1 : Type v} (X : \u03b1 \u2192 homotopy_category A (complex_shape.up \u2124)) :\n  has_coproduct X :=\n{ exists_colimit := nonempty.intro $ \u27e8colimit_cofan _, is_colimit_cofan _\u27e9 }\n\ninstance {\u03b1 : Type v} : has_colimits_of_shape (discrete \u03b1)\n  (homotopy_category A (complex_shape.up \u2124)) :=\nbegin\n  constructor, intros K,\n  let E : K \u2245 discrete.functor (\u03bb n, K.obj \u27e8n\u27e9) := discrete.nat_iso (\u03bb \u27e8i\u27e9, iso.refl _),\n  apply has_colimit_of_iso E,\nend\n\nnoncomputable\ninstance {\u03b1 : Type v} : preserves_colimits_of_shape (discrete \u03b1)\n  (quotient A (complex_shape.up \u2124)) :=\nbegin\n  constructor, intros K,\n  apply preserves_colimit_of_preserves_colimit_cocone\n    (colimit.is_colimit K),\n  let T : K \u22d9 quotient A _ \u2245 discrete.functor\n    ((\u03bb a : \u03b1, (quotient _ _).obj ((\u03bb n, K.obj \u27e8n\u27e9) a))) := nat_iso.of_components\n    (\u03bb \u27e8i\u27e9, iso.refl _) _,\n  swap,\n  { rintros \u27e8i\u27e9 \u27e8\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9,\n    dsimp,\n    simp only [category_theory.functor.map_id, category.id_comp],\n    dsimp,\n    simp only [category.comp_id]},\n  apply (is_colimit.precompose_inv_equiv T\n    ((quotient A (complex_shape.up \u2124)).map_cocone (colimit.cocone K))).to_fun,\n  let ee : colimit_cofan (\u03bb a : \u03b1, (quotient _ _).obj (K.obj \u27e8a\u27e9)) \u2245\n    (cocones.precompose T.inv).obj\n    ((quotient A (complex_shape.up \u2124)).map_cocone (colimit.cocone K)) := _,\n  swap,\n  { refine cocones.ext _ _,\n    { apply functor.map_iso,\n      refine has_colimit.iso_of_nat_iso _,\n      refine nat_iso.of_components (\u03bb \u27e8i\u27e9, iso.refl _) _,\n      rintro \u27e8i\u27e9 \u27e8\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9,\n      dsimp,\n      simp only [discrete.functor_map_id, category.comp_id],\n      dsimp,\n      simp only [category.id_comp]},\n    rintros \u27e8i\u27e9,\n    dsimp [colimit_cofan, T, nat_iso.of_components,\n      has_colimit.iso_of_nat_iso, is_colimit.map],\n    simp only [\u2190 functor.map_comp, category.id_comp, colimit.\u03b9_desc],\n    dsimp [cocones.precompose],\n    simpa only [functor.map_comp] },\n  apply is_colimit.of_iso_colimit _ ee,\n  apply is_colimit_cofan,\nend\n\ninstance is_K_projective_sigma {\u03b1 : Type v}\n  (X : \u03b1 \u2192 homotopy_category A (complex_shape.up \u2124))\n  [\u2200 a, is_K_projective (X a)] : is_K_projective (sigma_obj X) :=\nbegin\n  constructor,\n  introsI Y hY f,\n  apply colimit.hom_ext,\n  rintros \u27e8a\u27e9,\n  rw comp_zero,\n  apply is_K_projective.cond Y,\n  dsimp, apply_instance,\nend\n\ninstance is_K_projective_colimit_cofan {\u03b1 : Type v}\n  (X : \u03b1 \u2192 homotopy_category A (complex_shape.up \u2124))\n  [\u2200 a, is_K_projective (X a)] : is_K_projective (colimit_cofan X).X :=\nbegin\n  let e : (colimit_cofan X).X \u2245 sigma_obj X :=\n    (is_colimit_cofan X).cocone_point_unique_up_to_iso (colimit.is_colimit _),\n  apply is_K_projective_of_iso _ _ e.symm,\nend\n\nend homotopy_category\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/homotopy_category_coproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3818183118798879}}
{"text": "def Ctx := String \u2192 Type\nabbrev State (\u0393 : Ctx) := {x : String} \u2192 \u0393 x\n\nopaque p {\u0393 : Ctx} (s : State \u0393) : Prop\n\ntheorem ex {\u0393 : Ctx} (s : State \u0393) (h : (a : State \u0393) \u2192 @p \u0393 a) : @p \u0393 s :=\n  h \u2039_\u203a\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/typeAscImp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3818183045785905}}
{"text": "import tactic algebra.big_operators.fin\n\nuniverses u v\n\nopen_locale big_operators\n\n\n\ndef fintype_sup {\u03b9 : Type*} [fintype \u03b9] {\u03b1 : Type*} [semilattice_sup \u03b1] [order_bot \u03b1] (f : \u03b9 \u2192 \u03b1) : \u03b1 :=\n  (finset.univ : finset \u03b9).sup f\n\nnotation `\u22c1 ` binders `, ` r:(scoped f, fintype_sup f) := r\n\nnamespace fintype\nvariables {\u03b9 : Type*} [fintype \u03b9] {\u03b1 : Type*} [semilattice_sup \u03b1]  [order_bot \u03b1] (f : \u03b9 \u2192 \u03b1)\n\nlemma le_fintype_sup {n} (f : fin n \u2192 \u03b1) (i : fin n) :\n  f i \u2264 \u22c1 j, f j := @finset.le_sup _ _ _ _ _ f i (by simp)\n\n@[simp] lemma fintype_fin {n} (f : fin (n + 1) \u2192 \u03b1) :\n  (\u22c1 i : fin (n + 1), f i) = (f 0 \u2294 \u22c1 i : fin n, f i.succ) :=\nbegin \n  refine le_antisymm_iff.mpr _, simp[fintype_sup], split,\n  { intros m,\n    have : m = 0 \u2228 \u2203 (j : fin n), m = j.succ, from fin.eq_zero_or_eq_succ m,\n    rcases this with (rfl | \u27e8j, rfl\u27e9); try {simp},\n    refine le_sup_of_le_right\n      (@finset.le_sup _ _ _ _ _ (\u03bb (i : fin n), f i.succ) j (by simp)) },\n  split,\n  { refine le_fintype_sup f 0 },\n  intros i,\n  refine le_fintype_sup f _\nend\n\n\nend fintype\n\n\nnotation `\ud835\udd39` := list bool\n\ndef safe_case : vector \ud835\udd39 3 \u2192 \ud835\udd39\n| \u27e8[tt :: b, w, v], _\u27e9 := v\n| \u27e8[_, w, v], _\u27e9       := w\n\n\ndef fgegegeg (n : \u2115) : (fin n \u2192 \u2115) \u2192 vector \u2115 n :=\nby { exact vector.of_fn }\n\ndef safe_rec\n  {m n i}\n  (f  : fin i \u2192 (vector \ud835\udd39 m \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39))\n  (g\u2080 : fin i \u2192 vector \ud835\udd39 (m + 1) \u2192 vector \ud835\udd39 (n + i) \u2192 \ud835\udd39)\n  (g\u2081 : fin i \u2192 vector \ud835\udd39 (m + 1) \u2192 vector \ud835\udd39 (n + i) \u2192 \ud835\udd39)\n   : fin i \u2192 vector \ud835\udd39 (m + 1) \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39\n| j \u27e8[]        :: w, p\u27e9 v := f j \u27e8w, by simpa using p\u27e9 v\n| j \u27e8(ff :: x) :: w, p\u27e9 v := g\u2080 j \u27e8x :: w, by simpa using p\u27e9\n  (vector.append v (vector.of_fn $ \u03bb k, safe_rec k \u27e8x :: w, by simpa using p\u27e9 v : vector \ud835\udd39 i))\n| j \u27e8(tt :: x) :: w, p\u27e9 v := g\u2081 j \u27e8x :: w, by simpa using p\u27e9\n  (vector.append v (vector.of_fn $ \u03bb k, safe_rec k \u27e8x :: w, by simpa using p\u27e9 v : vector \ud835\udd39 i))\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb x, x.2.1.head.length)\u27e9]}\n\n-- f : (unsafe\u306a\u5f15\u6570) \u2192 (safe\u306a\u5f15\u6570) \u2192 \ud835\udd39\ninductive safe : \u03a0 {m n}, (vector \ud835\udd39 m \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39) \u2192 Prop\n| const     : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 1), [])\n| zero      : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 1), ff :: v.head)\n| one       : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 1), tt :: v.head)\n| tail      : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 1), v.head.tail)\n| case      : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 3), safe_case v)\n| nth (n m) : safe (\u03bb (_ : vector \ud835\udd39 0) (v : vector \ud835\udd39 n), v.nth m)\n| safe_comp :\n  \u2200 {m n} (f : vector \ud835\udd39 m \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39)\n    {k} (g : fin m \u2192 vector \ud835\udd39 k \u2192 vector \ud835\udd39 0 \u2192 \ud835\udd39)\n    {l} (h : fin n \u2192 vector \ud835\udd39 k \u2192 vector \ud835\udd39 l \u2192 \ud835\udd39),\n    safe f \u2192 (\u2200 i, safe (g i)) \u2192 (\u2200 i, safe (h i)) \u2192\n    safe (\u03bb (w : vector \ud835\udd39 k) (v : vector \ud835\udd39 l),\n      f (vector.of_fn $ \u03bb i, g i w vector.nil) (vector.of_fn $ \u03bb i, h  i w v))\n| safe_rec : \n  \u2200 {m n i}\n  (f  : fin i \u2192 vector \ud835\udd39 m \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39)\n  (g\u2080 : fin i \u2192 vector \ud835\udd39 (m + 1) \u2192 vector \ud835\udd39 (n + i) \u2192 \ud835\udd39)\n  (g\u2081 : fin i \u2192 vector \ud835\udd39 (m + 1) \u2192 vector \ud835\udd39 (n + i) \u2192 \ud835\udd39) {j},\n  (\u2200 j, safe (f j)) \u2192 (\u2200 j, safe (g\u2080 j)) \u2192 (\u2200 j, safe (g\u2081 j)) \u2192 safe (safe_rec f g\u2080 g\u2081 j)\n  \n\ntheorem bcs_poly {m n} (f : vector \ud835\udd39 m \u2192 vector \ud835\udd39 n \u2192 \ud835\udd39) (H : safe f) :\n  \u2203 p q r : \u2115, \u2200 (w : vector \ud835\udd39 m) (v : vector \ud835\udd39 n),\n  (f w v).length \u2264 p * (\u2211 i, (w.nth i).length)^q + (\u22c1 i, (v.nth i).length) + r :=\nbegin\n  induction H,\n  case const { refine \u27e80, 0, 0, _\u27e9, intros w v, simp },\n  case zero { refine \u27e80, 0, 1, _\u27e9, intros w v, simp[fintype_sup] },\n  case one  { refine \u27e80, 0, 1, _\u27e9, intros w v, simp[fintype_sup] },\n  case tail { refine \u27e80, 0, 0, _\u27e9, intros w v, simp[fintype_sup] },\n  case case { refine \u27e80, 0, 0, _\u27e9, intros w v,\n    rcases v with \u27e8(_ | \u27e8b, l\u27e9), p\u27e9,\n    { simp at p, contradiction },\n    rcases l with (_ | \u27e8w, l\u27e9), { simp at p, contradiction },\n    rcases l with (_ | \u27e8v, l\u27e9), { simp at p, exfalso, linarith },\n    rcases l with (_ | _),\n    { rcases b with (_ | \u27e8B, b\u27e9),\n      { simp[safe_case], right, left, refl },\n      { rcases B; simp[safe_case],\n        { right, left, refl }, { right, right, left, refl } } },\n    { exfalso, simp at p, linarith } },\n  case nth : n m { refine \u27e80, 0, 0, _\u27e9, intros w v, simp, refine fintype.le_fintype_sup _ m },\n  case safe_comp : m n f k g l h safe_f safe_g safe_h IH_f IH_g IH_h\n  { simp, simp at IH_g IH_h,\n    rcases IH_f with \u27e8p_f, q_f, r_f, IH_f\u27e9, refine \u27e80, 0, 0, _\u27e9, intros w v,\n    have := IH_f (vector.of_fn $ \u03bb i, g i w vector.nil) (vector.of_fn $ \u03bb i, h  i w v),\n    simp at this,\n     }\nend\n  \n  ", "meta": {"author": "iehality", "repo": "lean-computable-complexity", "sha": "deee56eddd42eba1ceb05e8a9d8a2cc354138f65", "save_path": "github-repos/lean/iehality-lean-computable-complexity", "path": "github-repos/lean/iehality-lean-computable-complexity/lean-computable-complexity-deee56eddd42eba1ceb05e8a9d8a2cc354138f65/src/safe_recursion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3818183045785905}}
{"text": "/-\nCopyright (c) 2019 Bruno Bentzen. All rights reserved.\nReleased under the Apache License 2.0 (see \"License\");\nAuthor: Bruno Bentzen\n-/\n\nimport ..core.interval\n\nopen interval\n\n-- the \u03b5 face maps of type lines\n\nexample {A : I \u2192 Type} : Type := A i0\n\nexample {A : I \u2192 Type} : Type := A i1\n\n-- the \u03b5 face maps of lines \n\nexample {A : I \u2192 Type} (a : \u03a0 i, A i) : A i1 := a i1\n\nexample {A : I \u2192 Type} (a : \u03a0 i, A i) :  A i0 := a i0\n", "meta": {"author": "bbentzen", "repo": "cubicalean", "sha": "3b94cd2aefdfc2163c263bd3fc6f2086fef814b5", "save_path": "github-repos/lean/bbentzen-cubicalean", "path": "github-repos/lean/bbentzen-cubicalean/cubicalean-3b94cd2aefdfc2163c263bd3fc6f2086fef814b5/src/examples/face.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791786991753929, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38181830457859045}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport logic.small\nimport category_theory.skeletal\n\n/-!\n# Essentially small categories.\n\nA category given by `(C : Type u) [category.{v} C]` is `w`-essentially small\nif there exists a `small_model C : Type w` equipped with `[small_category (small_model C)]`.\n\nA category is `w`-locally small if every hom type is `w`-small.\n\nThe main theorem here is that a category is `w`-essentially small iff\nthe type `skeleton C` is `w`-small, and `C` is `w`-locally small.\n-/\n\nuniverses w v v' u u'\n\nopen category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nnamespace category_theory\n\n/-- A category is `essentially_small.{w}` if there exists\nan equivalence to some `S : Type w` with `[small_category S]`. -/\nclass essentially_small (C : Type u) [category.{v} C] : Prop :=\n(equiv_small_category : \u2203 (S : Type w) [small_category S], by exactI nonempty (C \u224c S))\n\n/-- Constructor for `essentially_small C` from an explicit small category witness. -/\n\n\n/--\nAn arbitrarily chosen small model for an essentially small category.\n-/\n@[nolint has_inhabited_instance]\ndef small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : Type w :=\nclassical.some (@essentially_small.equiv_small_category C _ _)\n\nnoncomputable\ninstance small_category_small_model\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : small_category (small_model C) :=\nclassical.some (classical.some_spec (@essentially_small.equiv_small_category C _ _))\n\n/--\nThe (noncomputable) categorical equivalence between\nan essentially small category and its small model.\n-/\nnoncomputable\ndef equiv_small_model (C : Type u) [category.{v} C] [essentially_small.{w} C] : C \u224c small_model C :=\nnonempty.some (classical.some_spec (classical.some_spec\n  (@essentially_small.equiv_small_category C _ _)))\n\nlemma essentially_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : essentially_small.{w} C \u2194 essentially_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.symm.trans f), },\n  { rintro \u27e8S, \ud835\udcae, \u27e8f\u27e9\u27e9,\n    resetI,\n    exact essentially_small.mk' (e.trans f), },\nend\n\n/--\nA category is `w`-locally small if every hom set is `w`-small.\n\nSee `shrink_homs C` for a category instance where every hom set has been replaced by a small model.\n-/\nclass locally_small (C : Type u) [category.{v} C] : Prop :=\n(hom_small : \u2200 X Y : C, small.{w} (X \u27f6 Y) . tactic.apply_instance)\n\ninstance (C : Type u) [category.{v} C] [locally_small.{w} C] (X Y : C) :\n  small (X \u27f6 Y) :=\nlocally_small.hom_small X Y\n\nlemma locally_small_congr {C : Type u} [category.{v} C] {D : Type u'} [category.{v'} D]\n  (e : C \u224c D) : locally_small.{w} C \u2194 locally_small.{w} D :=\nbegin\n  fsplit,\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.inverse.obj X) (e.inverse.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.inverse, },\n  { rintro \u27e8L\u27e9,\n    fsplit,\n    intros X Y,\n    specialize L (e.functor.obj X) (e.functor.obj Y),\n    refine (small_congr _).mpr L,\n    exact equiv_of_fully_faithful e.functor, },\nend\n\n@[priority 100]\ninstance locally_small_self (C : Type u) [category.{v} C] : locally_small.{v} C := {}\n\n@[priority 100]\ninstance locally_small_of_essentially_small\n  (C : Type u) [category.{v} C] [essentially_small.{w} C] : locally_small.{w} C :=\n(locally_small_congr (equiv_small_model C)).mpr (category_theory.locally_small_self _)\n\n/--\nWe define a type alias `shrink_homs C` for `C`. When we have `locally_small.{w} C`,\nwe'll put a `category.{w}` instance on `shrink_homs C`.\n-/\n@[nolint has_inhabited_instance]\ndef shrink_homs (C : Type u) := C\n\nnamespace shrink_homs\n\nsection\nvariables {C' : Type*} -- a fresh variable with no category instance attached\n\n/-- Help the typechecker by explicitly translating from `C` to `shrink_homs C`. -/\ndef to_shrink_homs {C' : Type*} (X : C') : shrink_homs C' := X\n/-- Help the typechecker by explicitly translating from `shrink_homs C` to `C`. -/\ndef from_shrink_homs {C' : Type*} (X : shrink_homs C') : C' := X\n\n@[simp] lemma to_from (X : C') : from_shrink_homs (to_shrink_homs X) = X := rfl\n@[simp] lemma from_to (X : shrink_homs C') : to_shrink_homs (from_shrink_homs X) = X := rfl\n\nend\n\nvariables (C) [locally_small.{w} C]\n\n@[simps]\nnoncomputable\ninstance : category.{w} (shrink_homs C) :=\n{ hom := \u03bb X Y, shrink (from_shrink_homs X \u27f6 from_shrink_homs Y),\n  id := \u03bb X, equiv_shrink _ (\ud835\udfd9 (from_shrink_homs X)),\n  comp := \u03bb X Y Z f g,\n    equiv_shrink _ (((equiv_shrink _).symm f) \u226b ((equiv_shrink _).symm g)), }.\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef functor : C \u2964 shrink_homs C :=\n{ obj := \u03bb X, to_shrink_homs X,\n  map := \u03bb X Y f, equiv_shrink (X \u27f6 Y) f, }\n\n/-- Implementation of `shrink_homs.equivalence`. -/\n@[simps]\nnoncomputable\ndef inverse : shrink_homs C \u2964 C :=\n{ obj := \u03bb X, from_shrink_homs X,\n  map := \u03bb X Y f, (equiv_shrink (from_shrink_homs X \u27f6 from_shrink_homs Y)).symm f, }\n\n/--\nThe categorical equivalence between `C` and `shrink_homs C`, when `C` is locally small.\n-/\n@[simps]\nnoncomputable\ndef equivalence : C \u224c shrink_homs C :=\nequivalence.mk (functor C) (inverse C)\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n  (nat_iso.of_components (\u03bb X, iso.refl X) (by tidy))\n\nend shrink_homs\n\n/--\nA category is essentially small if and only if\nthe underlying type of its skeleton (i.e. the \"set\" of isomorphism classes) is small,\nand it is locally small.\n-/\ntheorem essentially_small_iff (C : Type u) [category.{v} C] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) \u2227 locally_small.{w} C :=\nbegin\n  -- This theorem is the only bit of real work in this file.\n  fsplit,\n  { intro h,\n    fsplit,\n    { rcases h with \u27e8S, \ud835\udcae, \u27e8e\u27e9\u27e9,\n      resetI,\n      refine \u27e8\u27e8skeleton S, \u27e8_\u27e9\u27e9\u27e9,\n      exact e.skeleton_equiv, },\n    { resetI, apply_instance, }, },\n  { rintro \u27e8\u27e8S, \u27e8e\u27e9\u27e9, L\u27e9,\n    resetI,\n    let e' := (shrink_homs.equivalence C).skeleton_equiv.symm,\n    refine \u27e8\u27e8S, _, \u27e8_\u27e9\u27e9\u27e9,\n    apply induced_category.category (e'.trans e).symm,\n    refine (shrink_homs.equivalence C).trans\n      ((from_skeleton _).as_equivalence.symm.trans\n      ((induced_functor (e'.trans e).symm).as_equivalence.symm)), },\nend\n\n/--\nAny thin category is locally small.\n-/\n@[priority 100]\ninstance locally_small_of_thin {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  locally_small.{w} C := {}\n\n/--\nA thin category is essentially small if and only if the underlying type of its skeleton is small.\n-/\ntheorem essentially_small_iff_of_thin\n  {C : Type u} [category.{v} C] [\u2200 X Y : C, subsingleton (X \u27f6 Y)] :\n  essentially_small.{w} C \u2194 small.{w} (skeleton C) :=\nby simp [essentially_small_iff, category_theory.locally_small_of_thin]\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/essentially_small.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38181829727729294}}
{"text": "import tactic\n\n#check (=)\n\ninductive mcrl2 (\u03b1 : Type) : Type\n| atom : \u03b1 \u2192 mcrl2\n| seq : mcrl2 \u2192 mcrl2 \u2192 mcrl2\n| alt : mcrl2 \u2192 mcrl2 \u2192 mcrl2 \n\ninstance (\u03b1) : has_add (mcrl2 \u03b1) := \u27e8mcrl2.alt\u27e9\ninfix `\u2b1d`:70 := mcrl2.seq\n\nmeta def mcrl2.repr : mcrl2 string \u2192 string\n| (mcrl2.atom a) := a\n| (x \u2b1d y) := mcrl2.repr x ++ mcrl2.repr y\n| (x + y) := \"(\" ++ mcrl2.repr x ++ \" + \" ++ mcrl2.repr y ++ \")\"\n\ninductive mcrl2_equiv {\u03b1 : Type} : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop\n| atom {a} : mcrl2_equiv (mcrl2.atom a) (mcrl2.atom a)\n| alt {x y x' y'} (h\u2081 : mcrl2_equiv x x') (h\u2082 : mcrl2_equiv y y') :\n  mcrl2_equiv (x + y) (x' + y')\n| seq {x y x' y'} (h\u2081 : mcrl2_equiv x x') (h\u2082 : mcrl2_equiv y y') :\n  mcrl2_equiv (x \u2b1d y) (x' \u2b1d y')\n| trans {x y z} (h\u2081 : mcrl2_equiv x y) (h\u2082 : mcrl2_equiv y z) :\n  mcrl2_equiv x z\n| comm {x y} (h : mcrl2_equiv x y) : mcrl2_equiv y x \n| alt_comm {x y} : mcrl2_equiv (x + y) (y + x)\n| alt_idem {x} : mcrl2_equiv (x + x) x\n| alt_assoc {x y z} : mcrl2_equiv (x + (mcrl2.alt y z)) ((x + y) + z)\n| seq_rdist {x y z} : mcrl2_equiv ((x + y) \u2b1d z) (x \u2b1d z + y \u2b1d z)\n| seq_assoc {x y z} : mcrl2_equiv ((x \u2b1d y) \u2b1d z) (x \u2b1d (y \u2b1d z))\n\ntheorem mcrl2_equiv.reflexive {\u03b1 : Type} : reflexive (\u03bba b : mcrl2 \u03b1, mcrl2_equiv a b) :=\nbegin\n  intro x,\n  induction x,\n  { exact mcrl2_equiv.atom},\n  { apply mcrl2_equiv.seq,\n    repeat {assumption}},\n  { apply mcrl2_equiv.alt,\n    repeat {assumption}}\nend\n\n#check option.rel\n\ntheorem mcrl2_equiv.symmetric {\u03b1 : Type} : symmetric (\u03bba b : mcrl2 \u03b1, mcrl2_equiv a b) :=\nbegin\n  intros x y,\n  intro h,\n  exact mcrl2_equiv.comm h\nend\n\ntheorem mcrl2_equiv.transitive {\u03b1 : Type} : transitive (\u03bba b : mcrl2 \u03b1, mcrl2_equiv a b) :=\nbegin\n  intros x y z h h',\n  exact mcrl2_equiv.trans h h'\nend\n\n@[instance] def setoid_mcrl2 {\u03b1 : Type} : setoid (mcrl2 \u03b1) :=\n{ r := (\u03bba b, mcrl2_equiv a b),\n  iseqv :=\n    begin\n      repeat {apply and.intro},\n      { exact mcrl2_equiv.reflexive},\n      { exact mcrl2_equiv.symmetric},\n      { exact mcrl2_equiv.transitive}\n    end\n}\n\nlemma setoid_iff {\u03b1 : Type} (a b : mcrl2 \u03b1) : \na \u2248 b \u2194 mcrl2_equiv a b :=\nby refl\n\ndef mcrl2' (\u03b1 : Type)  := quotient $ @setoid_mcrl2 \u03b1 \n\nlemma mcrl2'_alt_comm {\u03b1} (a b : mcrl2 \u03b1) : \u27e6mcrl2.alt a b\u27e7 = \u27e6mcrl2.alt b a\u27e7 :=\nbegin\napply quotient.sound,\napply mcrl2_equiv.alt_comm\nend\n\nlemma mcrl2'_alt_assoc {\u03b1} (a b c: mcrl2 \u03b1) : \u27e6mcrl2.alt a (mcrl2.alt b c)\u27e7 = \u27e6mcrl2.alt (mcrl2.alt a b) c\u27e7 :=\nbegin\napply quotient.sound,\napply mcrl2_equiv.alt_assoc\nend\n\n#check quot.lift_on\u2082_mk\nclass mcrl2_base (\u03b1 M : Type) :=\n  (atom : \u03b1 \u2192 M)\n  (alt : M \u2192 M \u2192 M)\n  (seq : M \u2192 M \u2192 M)\n  (alt_comm : \u2200x y, alt x y = alt y x)\n  (alt_assoc : \u2200x y z, alt x (alt y z) = alt (alt x y) z)\n  (alt_idem : \u2200x, alt x x = x)\n  (seq_dist : \u2200x y z, seq (alt x y) z = alt (seq x z) (seq y z))\n  (seq_assoc : \u2200x y z, seq (seq x y) z = seq x (seq y z))\n\ninstance (\u03b1) : mcrl2_base \u03b1 (mcrl2' \u03b1) := {\n  atom := \u03bba, \u27e6mcrl2.atom a\u27e7,\n  alt := quotient.lift\u2082 (\u03bba b, \u27e6a + b\u27e7) \n         begin\n          intros a b a' b' ha hb,\n          apply quotient.sound,\n          apply mcrl2_equiv.alt,\n          { exact ha},\n          { exact hb}\n        end,\n  seq := quotient.lift\u2082 (\u03bba b, \u27e6a \u2b1d b\u27e7) \n         begin\n          intros a b a' b' ha hb,\n          apply quotient.sound,\n          apply mcrl2_equiv.seq,\n          { exact ha},\n          { exact hb}\n        end,\n  alt_comm := begin\n    intros x y,\n    apply quot.induction_on\u2082 x y,\n    simp,\n    apply mcrl2_equiv.alt_comm    \n  end,\n  alt_assoc := begin\n    intros x y z,\n    apply quot.induction_on\u2083 x y z,\n    intros a b c,\n    apply quotient.sound,\n    exact mcrl2_equiv.alt_assoc\n  end,\n  alt_idem := begin\n    intro x,\n    apply quot.induction_on x,\n    intro a, \n    apply quotient.sound,\n    exact mcrl2_equiv.alt_idem\n  end,\n  seq_dist := begin\n    intros x y z,\n    apply quot.induction_on\u2083 x y z,\n    intros a b c,\n    apply quotient.sound,\n    exact mcrl2_equiv.seq_rdist\n  end,\n  seq_assoc := begin\n    intros x y z,\n    apply quot.induction_on\u2083 x y z,\n    intros a b c,\n    apply quotient.sound,\n    exact mcrl2_equiv.seq_assoc\n  end\n}\n\n\ninductive testtype : Type \n| a : testtype\n| b : testtype\n\nexample : mcrl2' testtype := mcrl2_base.atom testtype.a\n\n\nexample : mcrl2 testtype :=\n(mcrl2.atom testtype.a) \u2b1d (mcrl2.atom testtype.b)\n\nexample : mcrl2_base unit unit := \n{ atom := \u03bb_, (),\n  alt := \u03bb_ _, (),\n  seq := \u03bb_ _, (),\n  alt_comm := by intros x y; refl,\n  alt_assoc := by intros x y z; refl,\n  alt_idem := begin intro x, cases x, refl end,\n  seq_dist := by intros x y z; refl,\n  seq_assoc := by intros x y z; refl}\n\ninductive mcrl2_mrg (\u03b1 : Type) : Type\n| base : mcrl2 \u03b1 \u2192 mcrl2_mrg\n| atom : \u03b1 \u2192 mcrl2_mrg\n| seq : mcrl2_mrg \u2192 mcrl2_mrg \u2192 mcrl2_mrg\n| alt : mcrl2_mrg \u2192 mcrl2_mrg \u2192 mcrl2_mrg \n| par_left : mcrl2_mrg \u2192 mcrl2_mrg \u2192 mcrl2_mrg \n| comm : mcrl2_mrg \u2192 mcrl2_mrg \u2192 mcrl2_mrg \n| par : mcrl2_mrg \u2192 mcrl2_mrg \u2192 mcrl2_mrg \n| deadlock : mcrl2_mrg\n\n\n\ninductive mcrl2_mrg_equiv {\u03b1 : Type} (\u03b3 : \u03b1 \u2192 \u03b1 \u2192 option \u03b1) : \nmcrl2_mrg \u03b1 \u2192 mcrl2_mrg \u03b1 \u2192 Prop\n| base : mcrl2_mrg_equiv\n| par_main {x y} : mcrl2_mrg_equiv (mcrl2_mrg.par x y) \n  ((mcrl2_mrg.par_left x y + mcrl2_mrg.par_left y x) + mcrl2_mrg.comm x y)\n| par_left_atom {x a} : mcrl2_mrg_equiv (mcrl2_mrg.par_left (mcrl2_mrg.atom a) x) \n  (mcrl2_mrg.seq (mcrl2_mrg.atom a) x)\n| par_left_seq {x y a} : mcrl2_mrg_equiv \n  (mcrl2_mrg.par_left (mcrl2_mrg.seq (mcrl2_mrg.atom a) y) x) \n  (mcrl2_mrg.seq (mcrl2_mrg.atom a) (mcrl2_mrg.par x y))\n| par_left_assoc {x y z} : mcrl2_mrg_equiv\n  (mcrl2_mrg.par_left (mcrl2_mrg.alt x y ) z)\n  (mcrl2_mrg.alt (mcrl2_mrg.par_left x z) (mcrl2_mrg.par_left y z))\n| comm_succ {a b c} (h : \u03b3 a b = some c) : mcrl2_mrg_equiv \n  (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.atom b))\n  (mcrl2_mrg.atom c)\n| comm_fail {a b} (h : \u03b3 a b = none) : mcrl2_mrg_equiv\n  (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.atom b))\n  (mcrl2_mrg.deadlock)\n| comm_seql {a b x} : mcrl2_mrg_equiv\n  (mcrl2_mrg.comm (mcrl2_mrg.seq (mcrl2_mrg.atom a) x) (mcrl2_mrg.atom b))\n  (mcrl2_mrg.seq (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.atom b)) x)\n| comm_seqr {a b x} : mcrl2_mrg_equiv\n  (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.seq (mcrl2_mrg.atom b) x) )\n  (mcrl2_mrg.seq (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.atom b)) x)\n| comm_seq_two {a b x y} : mcrl2_mrg_equiv\n  (mcrl2_mrg.comm \n    (mcrl2_mrg.seq (mcrl2_mrg.atom a) x)\n    (mcrl2_mrg.seq (mcrl2_mrg.atom b) y))\n  (mcrl2_mrg.seq\n    (mcrl2_mrg.comm (mcrl2_mrg.atom a) (mcrl2_mrg.atom b))\n    (mcrl2_mrg.par x y))\n| comm_distl {x y z} : mcrl2_mrg_equiv \n  (mcrl2_mrg.comm (x + y) z)\n  ((mcrl2_mrg.comm x z) + (mcrl2_mrg.comm y z))\n| comm_distr {x y z} : mcrl2_mrg_equiv\n  (mcrl2_mrg.comm x (y + z))\n  ((mcrl2_mrg.comm x y) + (mcrl2_mrg.comm x z))", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Archive/mcrl2_inductive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.38181829727729294}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n-/\nprelude\nimport Init.Core\n\n@[simp] theorem eqSelf (a : \u03b1) : (a = a) = True :=\n  propext <| Iff.intro (fun _ => trivial) (fun _ => rfl)\n\ntheorem ofEqTrue (h : p = True) : p :=\n  h \u25b8 trivial\n\ntheorem eqTrue (h : p) : p = True :=\n  propext <| Iff.intro (fun _ => trivial) (fun _ => h)\n\ntheorem eqFalse (h : \u00ac p) : p = False :=\n  propext <| Iff.intro (fun h' => absurd h' h) (fun h' => False.elim h')\n\ntheorem eqFalse' (h : p \u2192 False) : p = False :=\n  propext <| Iff.intro (fun h' => absurd h' h) (fun h' => False.elim h')\n\ntheorem eqTrueOfDecide {p : Prop} {s : Decidable p} (h : decide p = true) : p = True :=\n  propext <| Iff.intro (fun h => trivial) (fun _ => ofDecideEqTrue h)\n\ntheorem eqFalseOfDecide {p : Prop} {s : Decidable p} (h : decide p = false) : p = False :=\n  propext <| Iff.intro (fun h' => absurd h' (ofDecideEqFalse h)) (fun h => False.elim h)\n\ntheorem impCongr {p\u2081 p\u2082 : Sort u} {q\u2081 q\u2082 : Sort v} (h\u2081 : p\u2081 = p\u2082) (h\u2082 : q\u2081 = q\u2082) : (p\u2081 \u2192 q\u2081) = (p\u2082 \u2192 q\u2082) :=\n  h\u2081 \u25b8 h\u2082 \u25b8 rfl\n\ntheorem impCongrCtx {p\u2081 p\u2082 q\u2081 q\u2082 : Prop} (h\u2081 : p\u2081 = p\u2082) (h\u2082 : p\u2082 \u2192 q\u2081 = q\u2082) : (p\u2081 \u2192 q\u2081) = (p\u2082 \u2192 q\u2082) :=\n  propext <| Iff.intro\n    (fun h hp\u2082 =>\n      have : p\u2081 := h\u2081 \u25b8 hp\u2082\n      have : q\u2081 := h this\n      h\u2082 hp\u2082 \u25b8 this)\n    (fun h hp\u2081 =>\n      have hp\u2082 : p\u2082 := h\u2081 \u25b8 hp\u2081\n      have : q\u2082 := h hp\u2082\n      h\u2082 hp\u2082 \u25b8 this)\n\ntheorem forallCongr {\u03b1 : Sort u} {p q : \u03b1 \u2192 Prop} (h : \u2200 a, (p a = q a)) : (\u2200 a, p a) = (\u2200 a, q a) :=\n  have : p = q := funext h\n  this \u25b8 rfl\n\n@[congr]\ntheorem iteCongr {x y u v : \u03b1} {s : Decidable b} [Decidable c] (h\u2081 : b = c) (h\u2082 : c \u2192 x = u) (h\u2083 : \u00ac c \u2192 y = v) : ite b x y = ite c u v := by\n  cases Decidable.em c with\n  | inl h => rw [ifPos h]; subst b; rw[ifPos h]; exact h\u2082 h\n  | inr h => rw [ifNeg h]; subst b; rw[ifNeg h]; exact h\u2083 h\n\ntheorem Eq.mprProp {p q : Prop} (h\u2081 : p = q) (h\u2082 : q) : p :=\n  h\u2081 \u25b8 h\u2082\n\ntheorem Eq.mprNot {p q : Prop} (h\u2081 : p = q) (h\u2082 : \u00acq) : \u00acp :=\n  h\u2081 \u25b8 h\u2082\n\n@[congr]\ntheorem diteCongr {s : Decidable b} [Decidable c]\n        {x : b \u2192 \u03b1} {u : c \u2192 \u03b1} {y : \u00acb \u2192 \u03b1} {v : \u00acc \u2192 \u03b1}\n        (h\u2081 : b = c)\n        (h\u2082 : (h : c)  \u2192 x (Eq.mprProp h\u2081 h) = u h)\n        (h\u2083 : (h : \u00acc) \u2192 y (Eq.mprNot h\u2081 h)  = v h)\n        : dite b x y = dite c u v := by\n  cases Decidable.em c with\n  | inl h => rw [difPos h]; subst b; rw [difPos h]; exact h\u2082 h\n  | inr h => rw [difNeg h]; subst b; rw [difNeg h]; exact h\u2083 h\n\nnamespace Lean.Simp\n\n@[simp] theorem Ne_Eq (a b : \u03b1) : (a \u2260 b) = Not (a = b) := rfl\n@[simp] theorem ite_True (a b : \u03b1) : (if True then a else b) = a := rfl\n@[simp] theorem ite_False (a b : \u03b1) : (if False then a else b) = b := rfl\n@[simp] theorem dite_True {\u03b1 : Sort u} {t : True \u2192 \u03b1} {e : \u00ac True \u2192 \u03b1} : (dite True t e) = t True.intro := rfl\n@[simp] theorem dite_False {\u03b1 : Sort u} {t : False \u2192 \u03b1} {e : \u00ac False \u2192 \u03b1} : (dite False t e) = e notFalse := rfl\n@[simp] theorem And_self (p : Prop) : (p \u2227 p) = p := propext <| Iff.intro (fun h => h.1) (fun h => \u27e8h, h\u27e9)\n@[simp] theorem And_True (p : Prop) : (p \u2227 True) = p := propext <| Iff.intro (fun h => h.1) (fun h => \u27e8h, trivial\u27e9)\n@[simp] theorem True_And (p : Prop) : (True \u2227 p) = p := propext <| Iff.intro (fun h => h.2) (fun h => \u27e8trivial, h\u27e9)\n@[simp] theorem And_False (p : Prop) : (p \u2227 False) = False := propext <| Iff.intro (fun h => h.2) (fun h => False.elim h)\n@[simp] theorem False_And (p : Prop) : (False \u2227 p) = False := propext <| Iff.intro (fun h => h.1) (fun h => False.elim h)\n@[simp] theorem Or_self (p : Prop) : (p \u2228 p) = p := propext <| Iff.intro (fun | Or.inl h => h | Or.inr h => h) (fun h => Or.inl h)\n@[simp] theorem Or_True (p : Prop) : (p \u2228 True) = True := propext <| Iff.intro (fun h => trivial) (fun h => Or.inr trivial)\n@[simp] theorem True_Or (p : Prop) : (True \u2228 p) = True := propext <| Iff.intro (fun h => trivial) (fun h => Or.inl trivial)\n@[simp] theorem Or_False (p : Prop) : (p \u2228 False) = p := propext <| Iff.intro (fun | Or.inl h => h | Or.inr h => False.elim h) (fun h => Or.inl h)\n@[simp] theorem False_Or (p : Prop) : (False \u2228 p) = p := propext <| Iff.intro (fun | Or.inr h => h | Or.inl h => False.elim h) (fun h => Or.inr h)\n@[simp] theorem Iff_self (p : Prop) : (p \u2194 p) = True := propext <| Iff.intro (fun h => trivial) (fun _ => Iff.intro id id)\n@[simp] theorem Iff_True (p : Prop) : (p \u2194 True) = p := propext <| Iff.intro (fun h => h.mpr trivial) (fun h => Iff.intro (fun _ => trivial) (fun _ => h))\n@[simp] theorem True_Iff (p : Prop) : (True \u2194 p) = p := propext <| Iff.intro (fun h => h.mp trivial) (fun h => Iff.intro (fun _ => h) (fun _ => trivial))\n@[simp] theorem Iff_False (p : Prop) : (p \u2194 False) = \u00acp := propext <| Iff.intro (fun h hp => h.mp hp) (fun h => Iff.intro h False.elim)\n@[simp] theorem False_Iff (p : Prop) : (False \u2194 p) = \u00acp := propext <| Iff.intro (fun h hp => h.mpr hp) (fun h => Iff.intro False.elim h)\n@[simp] theorem False_arrow (p : Prop) : (False \u2192 p) = True := propext <| Iff.intro (fun _ => trivial) (by intros; trivial)\n@[simp] theorem arrow_True (p : Prop) : (p \u2192 True) = True := propext <| Iff.intro (fun _ => trivial) (by intros; trivial)\n@[simp] theorem True_arrow (p : Prop) : (True \u2192 p) = p := propext <| Iff.intro (fun h => h trivial) (by intros; trivial)\n\n@[simp] theorem or_false (b : Bool) : (b || false) = b  := by cases b <;> rfl\n@[simp] theorem or_true (b : Bool) : (b || true) = true := by cases b <;> rfl\n@[simp] theorem false_or (b : Bool) : (false || b) = b  := by cases b <;> rfl\n@[simp] theorem true_or (b : Bool) : (true || b) = true := by cases b <;> rfl\n@[simp] theorem or_self (b : Bool) : (b || b) = b       := by cases b <;> rfl\n\n@[simp] theorem and_false (b : Bool) : (b && false) = false := by cases b <;> rfl\n@[simp] theorem and_true (b : Bool) : (b && true) = b       := by cases b <;> rfl\n@[simp] theorem false_and (b : Bool) : (false && b) = false := by cases b <;> rfl\n@[simp] theorem true_and (b : Bool) : (true && b) = b       := by cases b <;> rfl\n@[simp] theorem and_self (b : Bool) : (b && b) = b          := by cases b <;> rfl\n\nend Lean.Simp\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/stage0/src/Init/SimpLemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.6442251201477015, "lm_q1q2_score": 0.38181072083290823}}
{"text": "import operations\nimport types\n\nimport category_theory.category.basic\nimport category_theory.core\n\nopen operations\nopen params\nopen types\n\nopen category_theory\n\nnamespace quarterround\n\nvariables [category (bitvec word_len)]\n\n/-!\n  # Quarterround\n\n  The `quarterround` function, its pieces and the relation with the inverses.\n-/\n\n/-! ## Definitions -/\n\n/-- z\u2081 = y\u2081 \u2295 ((y\u2080 + y\u2083) <<< 7) -/\ndef qr1 (a b c d : bitvec word_len) := b OP (OP_RHS a d 7)\n/-- z\u2082 = y\u2082 \u2295 ((z\u2081 + y\u2080) <<< 9) -/\ndef qr2 (a b c d : bitvec word_len) := c OP (OP_RHS (qr1 a b c d) a 9)\n/-- z\u2083 = y\u2083 \u2295 ((z\u2082 + z\u2081) <<< 13) -/\ndef qr3 (a b c d : bitvec word_len) := d OP (OP_RHS (qr2 a b c d) (qr1 a b c d) 13)\n/-- z\u2080 = y\u2080 \u2295 ((z\u2083 + z\u2082) <<< 18) -/\ndef qr0 (a b c d : bitvec word_len) := a OP (OP_RHS (qr3 a b c d) (qr2 a b c d) 18)\n\n/-- Given a sequence of 4 numbers `seq`, use the four equations (`qr0`, `qr1`, `qr2` and `qr3`) to get the \nfull quarterround output, which is a transformed 4 numbers sequence too. -/\n@[simp] def quarterround (seq : vecType) : vecType :=\n  (\n    qr0 seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n    qr1 seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n    qr2 seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n    qr3 seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd\n  )\n\n/-! ## Quarterround zero lemmas -/\n\n/-- `qr0` of 4 zeros is zero -/\n@[simp] lemma qr0_zero : qr0 0 0 0 0 = 0 := by refl\n\n/-- `qr1` of 4 zeros is zero -/\n@[simp] lemma qr1_zero : qr1 0 0 0 0 = 0 := by refl\n\n/-- `qr2` of 4 zeros is zero -/\n@[simp] lemma qr2_zero : qr2 0 0 0 0 = 0 := by refl\n\n/-- `qr3` of 4 zeros is zero -/\n@[simp] lemma qr3_zero : qr3 0 0 0 0 = 0 := by refl\n\n/-- `quarterround` of 4 zeros is a sequence of 4 zeros -/\n@[simp] lemma quarterround_zero : quarterround (0, 0, 0, 0) = (0, 0, 0, 0) := by refl\n\n/-! ## Inverse definitions -/\n\n/-- y\u2080 = z\u2080 \u2295 ((z\u2083 + z\u2082) <<< 18) -/\ndef qr0_inv (a' b' c' d' : bitvec word_len) := a' OP (operation_rhs d' c' 18)\n/-- y\u2083 = z\u2083 \u2295 ((z\u2082 + z\u2081) <<< 13) -/\ndef qr3_inv (a' b' c' d' : bitvec word_len) := d' OP (operation_rhs c' b' 13)\n/-- y\u2082 = z\u2082 \u2295 ((z\u2081 + y\u2080) <<< 9) -/\ndef qr2_inv (a' b' c' d' : bitvec word_len) := c' OP (operation_rhs b' (qr0_inv a' b' c' d')  9)\n/-- y\u2081 = z\u2081 \u2295 ((y\u2080 + y\u2083) <<< 7) -/\ndef qr1_inv (a' b' c' d' : bitvec word_len) := b' OP (operation_rhs (qr0_inv a' b' c' d') (qr3_inv a' b' c' d') 7)\n\n/-- Put the 4 elements that forms a quarterround inverse all together. -/\n@[simp] def quarterround_inv (seq : vecType) := (\n  qr0_inv seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n  qr1_inv seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n  qr2_inv seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd,\n  qr3_inv seq.fst seq.snd.fst seq.snd.snd.fst seq.snd.snd.snd\n)\n\nlocal notation `qr0\u207b\u00b9` := qr0_inv\nlocal notation `qr1\u207b\u00b9` := qr1_inv\nlocal notation `qr2\u207b\u00b9` := qr2_inv\nlocal notation `qr3\u207b\u00b9` := qr3_inv\nlocal notation `quarterround\u207b\u00b9` := quarterround_inv\n\n/-! ## Inverse lemmas -/\n\n/-- The `quarterround` operation is fully invertible. -/\nlemma quarterround_is_inv (I : quarterround \u2245 quarterround\u207b\u00b9) : I.hom \u226b I.inv = \ud835\udfd9 quarterround :=\n  by rw [iso.hom_inv_id]\n\n/-! ## Inverses of the `quarterround` and `quarterround_inv` individual pieces lemmas -/\n\n/-- `qr0\u207b\u00b9` after `qr0` is `\ud835\udfd9 qr0`. -/\nlemma qr0_is_inv (I : qr0 \u2245 qr0\u207b\u00b9) : I.hom \u226b I.inv = \ud835\udfd9 qr0 := by rw [iso.hom_inv_id]\n\n/-- `qr1\u207b\u00b9` after `qr1` is `\ud835\udfd9 qr1`. -/\nlemma qr1_is_inv (I : qr1 \u2245 qr1\u207b\u00b9) : I.hom \u226b I.inv = \ud835\udfd9 qr1 := by rw [iso.hom_inv_id]\n\n/-- `qr2\u207b\u00b9` after `qr2` is `\ud835\udfd9 qr2`. -/\nlemma qr2_is_inv (I : qr2 \u2245 qr2\u207b\u00b9) : I.hom \u226b I.inv = \ud835\udfd9 qr2 := by rw [iso.hom_inv_id]\n\n/-- `qr3\u207b\u00b9` after `qr3` is `\ud835\udfd9 qr3`. -/\nlemma qr3_is_inv (I : qr3 \u2245 qr3\u207b\u00b9) : I.hom \u226b I.inv = \ud835\udfd9 qr3 := by rw [iso.hom_inv_id]\n\n/-- `qr0` after `qr0\u207b\u00b9` is `\ud835\udfd9 qr0\u207b\u00b9`. -/\nlemma qr0_inv_is_inv (I : qr0 \u2245 qr0\u207b\u00b9) : I.inv \u226b I.hom = \ud835\udfd9 qr0\u207b\u00b9 := by rw [iso.inv_hom_id]\n\n/-- `qr1` after `qr1\u207b\u00b9` is `\ud835\udfd9 qr1\u207b\u00b9`. -/\nlemma qr1_inv_is_inv (I : qr1 \u2245 qr1\u207b\u00b9) : I.inv \u226b I.hom = \ud835\udfd9 qr1\u207b\u00b9 := by rw [iso.inv_hom_id]\n\n/-- `qr2` after `qr2\u207b\u00b9` is `\ud835\udfd9 qr2\u207b\u00b9`. -/\nlemma qr2_inv_is_inv (I : qr2 \u2245 qr2\u207b\u00b9) : I.inv \u226b I.hom = \ud835\udfd9 qr2\u207b\u00b9 := by rw [iso.inv_hom_id]\n\n/-- `qr3` after `qr3\u207b\u00b9` is `\ud835\udfd9 qr3\u207b\u00b9`. -/\nlemma qr3_inv_is_inv (I : qr3 \u2245 qr3\u207b\u00b9) : I.inv \u226b I.hom = \ud835\udfd9 qr3\u207b\u00b9 := by rw [iso.inv_hom_id]\n\n/-- The inidivudal pieces of the `quarterround` function are all invertible. -/\nlemma qr_pieces_are_all_invertible (I0 : qr0 \u2245 qr0\u207b\u00b9) (I1 : qr1 \u2245 qr1\u207b\u00b9) (I2 : qr2 \u2245 qr2\u207b\u00b9) (I3 : qr3 \u2245 qr3\u207b\u00b9) :\n  I0.hom \u226b I0.inv = \ud835\udfd9 qr0 \u2227 I1.hom \u226b I1.inv = \ud835\udfd9 qr1 \u2227 I2.hom \u226b I2.inv = \ud835\udfd9 qr2 \u2227 I3.hom \u226b I3.inv = \ud835\udfd9 qr3 :=\nbegin\n  simp only [iso.hom_inv_id, eq_self_iff_true, and_self],\nend\n\nend quarterround\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/quarterround.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863698, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.38181070346632534}}
{"text": "import data.fintype.basic\nimport data.int.interval\nimport topology.bases\nimport topology.compact_open\nimport .circuit\nimport .list_pi\n\nopen set topological_space\n\n@[protect_proj] structure profinite : Type 1 :=\n( \u03b9 : Type )\n[ dec_eq : decidable_eq \u03b9 ]\n( X : Type )\n[ top_inst : topological_space X ]\n( proj : \u03b9 \u2192 X \u2192 bool)\n( continuous_swap_proj : continuous (function.swap proj) )\n( inv : (\u03b9 \u2192 bool) \u2192 X )\n( continuous_inv : continuous inv )\n( is_inv : \u2200 x y, function.swap proj x = y \u2194 x = inv y )\n\nattribute [instance] profinite.top_inst profinite.dec_eq\n\nnamespace profinite\n\nvariables {X : profinite} {i : X.\u03b9}\n\ninstance : has_coe_to_sort profinite Type := \u27e8profinite.X\u27e9\n\n@[simp] lemma proj_inv (x : X.\u03b9 \u2192 bool) : X.proj i (X.inv x) = x i :=\nbegin\n  have := X.is_inv (X.inv x) x,\n  simp [function.swap, function.funext_iff, *] at *\nend\n\n@[simp] lemma inv_proj (x : X) : X.inv (\u03bb i, X.proj i x) = x :=\nbegin\n  have := (X.is_inv x (\u03bb i, X.proj i x)).symm,\n  simp [function.swap, function.funext_iff, eq_comm, *] at *\nend\n\nlemma closed_embedding (X : profinite) : closed_embedding (function.swap X.proj) :=\nclosed_embedding_of_continuous_injective_closed\n  X.continuous_swap_proj\n  (\u03bb x y hxy, begin\n    rw [X.is_inv] at hxy,\n    rw [hxy, eq_comm, \u2190 X.is_inv]\n  end)\n  begin\n    intros s hs,\n    have : is_closed (X.inv \u207b\u00b9' s) := continuous_iff_is_closed.1 X.continuous_inv _ hs,\n    convert this,\n    simp [set.ext_iff],\n    intro x,\n    split,\n    { rintro \u27e8y, hy, rfl\u27e9,\n      convert hy,\n      rw [eq_comm, \u2190 X.is_inv] },\n    { intro hx,\n      use X.inv x,\n      use hx,\n      rw [X.is_inv] }\n  end\n\nattribute [continuity] profinite.continuous_inv\n\ninstance : compact_space X := X.closed_embedding.compact_space\ninstance : t2_space X := X.closed_embedding.t2_space\ninstance : totally_disconnected_space X :=\n\u27e8is_totally_disconnected_of_image\n  X.closed_embedding.continuous.continuous_on\n  X.closed_embedding.inj $\n  is_totally_disconnected_of_totally_disconnected_space _\u27e9\n\n@[continuity] lemma continuous_proj (i : X.\u03b9) : continuous (X.proj i) :=\nshow continuous ((\u03bb x : \u03a0 i : X.\u03b9, bool, x i) \u2218 (\u03bb (x : X) (j : X.\u03b9), X.proj j x)),\nfrom (continuous_apply i).comp X.closed_embedding.continuous\n\nlemma continuous_iff_proj {Y : Type*} [topological_space Y] (f : Y \u2192 X) :\n  continuous f \u2194 \u2200 i, continuous (\u03bb y, X.proj i (f y)) :=\nby simp only [X.closed_embedding.continuous_iff, continuous_pi_iff]\n\nlemma is_open_iff_pi (U : set X) : is_open U \u2194 \u2203 (V : set (X.\u03b9 \u2192 bool)),\n  is_open V \u2227 (function.swap X.proj \u207b\u00b9' V) = U :=\nX.closed_embedding.is_open_iff\n\nlemma is_closed_iff_pi (F : set X) : is_closed F \u2194 \u2203 (V : set (X.\u03b9 \u2192 bool)),\n  is_closed V \u2227 (function.swap X.proj \u207b\u00b9' V) = F :=\nX.closed_embedding.is_closed_iff\n\ndef prod (X Y : profinite) : profinite :=\n{ \u03b9 := X.\u03b9 \u2295 Y.\u03b9,\n  X := X \u00d7 Y,\n  proj := sum.elim (\u03bb i xy, X.proj i xy.1) (\u03bb i xy, Y.proj i xy.2),\n  continuous_swap_proj := begin\n      dsimp [function.swap],\n      continuity,\n      cases i,\n      { dsimp,\n        continuity },\n      { dsimp,\n        continuity }\n    end,\n  inv := \u03bb x, (X.inv (x \u2218 sum.inl), Y.inv (x \u2218 sum.inr)),\n  continuous_inv := begin\n      continuity,\n\n     end,\n  is_inv := begin\n    rintros \u27e8x, y\u27e9 z,\n    simp [\u2190 X.is_inv, \u2190 Y.is_inv, function.swap, prod.ext_iff, function.funext_iff]\n  end }\n\ninstance {X Y : profinite} [fintype X.\u03b9] [fintype Y.\u03b9] : fintype (X.prod Y).\u03b9 :=\nby dsimp [profinite.prod]; apply_instance\n\ninstance {X Y : profinite} [fintype X.X] [fintype Y.X] : fintype (X.prod Y).X :=\nby dsimp [profinite.prod]; apply_instance\n\ninstance {X Y : profinite} [has_repr X.\u03b9] [has_repr Y.\u03b9] : has_repr (X.prod Y).\u03b9 :=\nby dsimp [profinite.prod]; apply_instance\n\ninstance {X Y : profinite} [has_repr X.X] [has_repr Y.X] : has_repr (X.prod Y).X :=\nby dsimp [profinite.prod]; apply_instance\n\ndef boolp : profinite :=\n{ \u03b9 := unit,\n  X := bool,\n  proj := \u03bb _, id,\n  continuous_swap_proj := by continuity,\n  inv := \u03bb i, i (),\n  continuous_inv := by continuity,\n  is_inv := dec_trivial}\n\ninstance Safsasg: fintype (boolp.\u03b9) := by dsimp [boolp]; apply_instance\ninstance aagsga : fintype (boolp.X) := by dsimp [boolp]; apply_instance\n\ninstance agdgs : has_repr (boolp.\u03b9) := by dsimp [boolp]; apply_instance\ninstance agdgsa : has_repr (boolp.X) := by dsimp [boolp]; apply_instance\n\ndef twoadic : profinite :=\n{ \u03b9 := \u2115,\n  X := \u2115 \u2192 bool,\n  proj := function.swap id,\n  continuous_swap_proj := continuous_id,\n  continuous_inv := by continuity,\n  inv := id,\n  is_inv := by simp [function.swap] }\n\ninstance sgda : has_repr twoadic.\u03b9 := by dsimp [twoadic]; apply_instance\n\nopen circuit\n\ndef circuit.to_set (c : circuit X.\u03b9) : set X :=\n{ x | c.eval (\u03bb i, X.proj i x) }\n\ninstance (c : circuit X.\u03b9) : decidable_pred (circuit.to_set c) :=\nby dsimp [circuit.to_set]; apply_instance\n\nstructure map (X Y : profinite) : Type :=\n( to_fun : X \u2192 Y )\n( preimage : circuit Y.\u03b9 \u2192 circuit X.\u03b9 )\n( continuous' : \u2200 (x : X) (C : circuit Y.\u03b9),\n  to_fun x \u2208 C.to_set \u2194 x \u2208 (preimage C).to_set )\n\ninstance (X Y : profinite) : has_coe_to_fun (map X Y) (\u03bb _, X \u2192 Y) :=\n\u27e8map.to_fun\u27e9\n\ntheorem map.continuous {Y : profinite}: \u2200 (f : map X Y) (x : X) (C : circuit Y.\u03b9),\n  f x \u2208 C.to_set \u2194 x \u2208 (f.preimage C).to_set :=\nmap.continuous'\n\n@[simp] theorem eval_preimage {Y : profinite} (f : map X Y) (x : X.\u03b9 \u2192 bool) (C : circuit Y.\u03b9) :\n  (f.preimage C).eval x = C.eval (\u03bb i, Y.proj i (f (X.inv x))) :=\nbegin\n  rw [\u2190 bool.coe_bool_iff],\n  refine ((f.continuous (X.inv x) C).trans _).symm,\n  simp [circuit.to_set],\nend\n\n\ndef map.id (X : profinite) : map X X :=\n{ to_fun := id,\n  preimage := \u03bb C, C,\n  continuous' := \u03bb x C, by simp }\n\ndef map.comp {X Y Z : profinite} (f : map X Y) (g : map Y Z) : map X Z :=\n{ to_fun := g \u2218 f,\n  preimage := \u03bb C, f.preimage (g.preimage C),\n  continuous' := \u03bb x C, by simp [f.continuous, g.continuous] }\n\ndef projm (X : profinite) (i : X.\u03b9) : map X boolp :=\n{ to_fun := X.proj i,\n  preimage := \u03bb C, C.map (\u03bb _, i),\n  continuous' := \u03bb x C, by simp [circuit.to_set, eval_map]; refl }\n\ndef fstm {X Y : profinite} : (X.prod Y).map X :=\n{ to_fun := prod.fst,\n  preimage := \u03bb C, C.map sum.inl,\n  continuous' := \u03bb x C, begin\n    delta profinite.prod circuit.to_set,\n    simp [eval_map],\n  end }\n\ndef sndm {X Y : profinite} : (X.prod Y).map Y :=\n{ to_fun := prod.snd,\n  preimage := \u03bb C, C.map sum.inr,\n  continuous' := \u03bb x C, begin\n    delta profinite.prod circuit.to_set,\n    simp [eval_map],\n  end }\n\ndef reindex {X Y : profinite} (e : Y.\u03b9 \u2192 X.\u03b9) : map X Y :=\n{ to_fun := \u03bb x, Y.inv (\u03bb i, X.proj (e i) x),\n  preimage := \u03bb C, C.map e,\n  continuous' := \u03bb x C, by simp [eval_map, circuit.to_set] }\n\ndef prod_mk_reindex {X Y Z : profinite} (e\u2081 : Y.\u03b9 \u2192 X.\u03b9)\n  (e\u2082 : Z.\u03b9 \u2192 X.\u03b9) : map X (Y.prod Z) :=\n{ to_fun := \u03bb x, (Y.inv (\u03bb i, X.proj (e\u2081 i) x), Z.inv (\u03bb i, X.proj (e\u2082 i) x)),\n  preimage := \u03bb C, C.map (sum.elim e\u2081 e\u2082),\n  continuous' := \u03bb x C, begin\n    delta profinite.prod circuit.to_set,\n    simp [eval_map, circuit.to_set],\n    rw [iff_iff_eq],\n    congr' 2,\n    ext i,\n    cases i with i i;\n    simp,\n  end }\n\ndef diag {X : profinite} : X.map (X.prod X) :=\nprod_mk_reindex id id\n\ndef fst {Y : profinite} (C : circuit (X.prod Y).\u03b9) : circuit X.\u03b9 :=\ncircuit.bOr (C.sum_vars_right.pi (\u03bb _, [tt, ff]))\n  (\u03bb x, circuit.assign_vars C\n    (\u03bb i, sum.rec (\u03bb i _, sum.inl i) (\u03bb i hi, sum.inr (x i (by simp [hi]))) i))\n\nlemma eval_fst {Y : profinite} (C : circuit (X.prod Y).\u03b9) (x : X.\u03b9 \u2192 bool) :\n  (fst C).eval x \u2194 \u2203 y, C.eval (sum.elim x y) :=\nbegin\n  dsimp [fst],\n  simp only [eval_assign_vars, list.mem_pi, or_iff_not_imp_left, eval_bOr,\n    list.mem_cons_iff, list.mem_singleton,\n    eq_ff_eq_not_eq_tt, imp_self, implies_true_iff, exists_true_left],\n  dsimp [set_of, set.mem_def, profinite.prod],\n  split,\n  { rintro \u27e8a, ha\u27e9,\n    use [\u03bb i, if hi : i \u2208 C.sum_vars_right then a i hi else tt],\n    rw [eval_eq_evalv],\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * },\n  { rintro \u27e8a, ha\u27e9,\n    use [\u03bb i _, a i],\n    rw [eval_eq_evalv] at ha,\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * }\nend\n\nlemma mem_fst {Y : profinite} (C : circuit (X.prod Y).\u03b9) (x : X) :\n  x \u2208 (fst C).to_set \u2194 \u2203 y, (x, y) \u2208 C.to_set :=\nbegin\n  dsimp [circuit.to_set],\n  simp only [eval_fst],\n  dsimp [set_of, set.mem_def, profinite.prod],\n  split,\n  { rintros \u27e8a, ha\u27e9,\n    refine \u27e8Y.inv a, _\u27e9,\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * },\n  { rintro \u27e8y, hy\u27e9,\n    use \u03bb i, Y.proj i y,\n    convert hy,\n    ext i hi,\n    cases i with i i; simp * }\nend\n\ndef snd {Y : profinite} (C : circuit (X.prod Y).\u03b9) : circuit Y.\u03b9 :=\ncircuit.bOr (C.sum_vars_left.pi (\u03bb _, [tt, ff]))\n  (\u03bb x, circuit.assign_vars C\n    (\u03bb i, sum.rec (\u03bb i hi, sum.inr (x i (by simp [hi]))) (\u03bb i _, sum.inl i) i))\n\nlemma eval_snd {Y : profinite} (C : circuit (X.prod Y).\u03b9) (y : Y.\u03b9 \u2192 bool) :\n  (snd C).eval y \u2194 \u2203 x, C.eval (sum.elim x y) :=\nbegin\n  dsimp [snd],\n  simp only [eval_assign_vars, list.mem_pi, or_iff_not_imp_left, eval_bOr,\n    list.mem_cons_iff, list.mem_singleton,\n    eq_ff_eq_not_eq_tt, imp_self, implies_true_iff, exists_true_left],\n  dsimp [set_of, set.mem_def, profinite.prod],\n  split,\n  { rintro \u27e8a, ha\u27e9,\n    use [\u03bb i, if hi : i \u2208 C.sum_vars_left then a i hi else tt],\n    rw [eval_eq_evalv],\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * },\n  { rintro \u27e8a, ha\u27e9,\n    use [\u03bb i _, a i],\n    rw [eval_eq_evalv] at ha,\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * }\nend\n\nlemma mem_snd {Y : profinite} (C : circuit (X.prod Y).\u03b9) (y : Y) :\n  y \u2208 (snd C).to_set \u2194 \u2203 x, (x, y) \u2208 C.to_set :=\nbegin\n  dsimp [circuit.to_set],\n  simp only [eval_snd],\n  dsimp [set_of, set.mem_def, profinite.prod],\n  split,\n  { rintros \u27e8a, ha\u27e9,\n    refine \u27e8X.inv a, _\u27e9,\n    convert ha,\n    ext i hi,\n    cases i with i i; simp * },\n  { rintro \u27e8x, hx\u27e9,\n    use \u03bb i, X.proj i x,\n    convert hx,\n    ext i hi,\n    cases i with i i; simp * }\nend\n\ndef prod_mk {X Y Z : profinite} (f : X.map Y) (g : X.map Z) : X.map (Y.prod Z) :=\n{ to_fun := \u03bb x, (f x, g x),\n  preimage := \u03bb C, C.bind (sum.elim (\u03bb i, f.preimage (var i)) (\u03bb i, g.preimage (var i))),\n  continuous' := begin\n    intros x C,\n    simp [circuit.to_set, circuit.eval_bind],\n    rw [iff_iff_eq],\n    congr' 2,\n    funext i,\n    cases i with i i,\n    { dsimp [profinite.prod],\n      rw [eval_preimage f],\n      simp },\n    { dsimp [profinite.prod],\n      rw [eval_preimage g],\n      simp }\n  end }\n\ndef prod_mapm {W X Y Z : profinite} (f : W.map Y) (g : X.map Z) : (W.prod X).map (Y.prod Z) :=\nprod_mk (fstm.comp f) (sndm.comp g)\n\ndef unitp : profinite :=\n{ \u03b9 := empty,\n  X := unit,\n  proj := empty.elim,\n  continuous_swap_proj := by continuity,\n  continuous_inv := by continuity,\n  inv := \u03bb _, (),\n  is_inv := dec_trivial }\n\ninstance : has_repr empty := \u27e8empty.elim\u27e9\ninstance unitp.fin\u03b9 : fintype unitp.\u03b9 := by dsimp [unitp]; apply_instance\ninstance unitp.finX : fintype unitp.X := by dsimp [unitp]; apply_instance\ninstance afas : has_repr unitp.\u03b9 := by dsimp [unitp]; apply_instance\ninstance ads : has_repr unitp.X := by dsimp [unitp]; apply_instance\n\nend profinite\n\nopen profinite\n\nstructure propagate_struc (input : profinite) (state : profinite) : Type 1 :=\n( init : state )\n( transition : (state.prod input).map state )\n( output : (state.prod input).map boolp )\n\nvariables {input state : profinite}\n  (p : propagate_struc input state)\n  {p\u2081 p\u2082 : propagate_struc input state}\n\nnamespace propagate_struc\n\n@[simp] def nth_state (x : \u2115 \u2192 input) : \u2115 \u2192 state\n| 0     := p.init\n| (n+1) := p.transition (nth_state n, x n)\n\n@[simp] def nth_output (x : \u2115 \u2192 input) (n : \u2115) : bool :=\np.output (nth_state p x n, x n)\n\nlemma nth_state_eq_of_nth_state_eq\n  {m n : \u2115}\n  {x\u2081 x\u2082 : \u2115 \u2192 input} : \u2200 (i : \u2115)\n  (hs : p\u2081.nth_state x\u2081 m = p\u2082.nth_state x\u2082 n)\n  (hc : \u2200 s, p\u2081.transition s = p\u2082.transition s)\n  (hx : \u2200 (j), j \u2264 i \u2192 x\u2081 (m + j) = x\u2082 (n + j)),\n  p\u2081.nth_state x\u2081 (m + i) = p\u2082.nth_state x\u2082 (n + i)\n| 0 hs ht hx := hs\n| (i+1) hs hc hx := begin\n  rw [\u2190 add_assoc, \u2190 add_assoc, nth_state, nth_state,\n    nth_state_eq_of_nth_state_eq i hs hc (\u03bb j hj, hx j (nat.le_succ_of_le hj)), hc,\n    hx _ (nat.le_succ _)],\nend\n\nlemma nth_output_eq_of_nth_state_eq\n  {m n : \u2115}\n  {x\u2081 x\u2082 : \u2115 \u2192 input} (i : \u2115)\n  (hs : p\u2081.nth_state x\u2081 m = p\u2082.nth_state x\u2082 n)\n  (hc : \u2200 s, p\u2081.transition s = p\u2082.transition s)\n  (ho : \u2200 s, p\u2081.output s = p\u2082.output s)\n  (hx : \u2200 (j), j \u2264 i \u2192 x\u2081 (m + j) = x\u2082 (n + j)) :\n  p\u2081.nth_output x\u2081 (m + i) = p\u2082.nth_output x\u2082 (n + i) :=\nbegin\n  have := nth_state_eq_of_nth_state_eq i (by simpa using hs) hc hx,\n  rw [nth_output, nth_output, ho, this, hx _ (le_refl _)]\nend\n\ninductive result : Type\n| false_after (n : \u2115) : result\n| true_for_n (n : \u2115) : result\n| true_forall : result\n\ninstance : has_repr result :=\n\u27e8\u03bb r, match r with\n| result.false_after n := \"false after \" ++ repr n\n| result.true_for_n n := \"true for \" ++ repr n\n| result.true_forall := \"true forall\"\nend\u27e9\n\ndef decide_if_zeros_aux [has_repr state.\u03b9] : \u03a0 (n : \u2115), result \u00d7 circuit state.\u03b9\n| 0 := (result.true_for_n 0, fst\n  (p.output.preimage (circuit.single (\u03bb (i : unit) (hi : i \u2208 [()]), tt))))\n| (n+1) :=\n  match decide_if_zeros_aux n with\n  | (result.true_for_n m, s) :=\n    let s' := fst (p.transition.preimage s) in\n    if p.init \u2208 s.to_set then (result.false_after (n+1), s')\n    else if s' \u2264 s then (result.true_forall, s)\n    else (result.true_for_n (n+1), s.or s')\n  | x := x\n  end\n\ndef decide_if_zeros [has_repr state.\u03b9] (n : \u2115) : result :=\n(decide_if_zeros_aux p n).1\n\nend propagate_struc", "meta": {"author": "ChrisHughes24", "repo": "lean3bits", "sha": "119b68f1ce4a967951c53ee2f174007b49c80831", "save_path": "github-repos/lean/ChrisHughes24-lean3bits", "path": "github-repos/lean/ChrisHughes24-lean3bits/lean3bits-119b68f1ce4a967951c53ee2f174007b49c80831/src/v3/struc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.38181069536970996}}
{"text": "/-\nCopyright (c) 2022 James Gallicchio.\n\nAuthors: James Gallicchio\n-/\n\nimport LeanColls.Classes\nimport LeanColls.List.Basic\n\nnamespace LeanColls\n\n/-! Definition of toList in terms of a fold.\n\nUsed to define correctness for [Foldable] and [Foldable']\n-/\ndef canonicalToList (fold : {\u03b2 : Type w} \u2192 (\u03b2 \u2192 \u03c4 \u2192 \u03b2) \u2192 \u03b2 \u2192 \u03b2) : List \u03c4 :=\n  fold (\u03bb acc x => acc ++ [x]) []\n\nnamespace Foldable\n\nclass Correct (C) (\u03c4 : outParam _) extends Foldable C \u03c4 where\n  foldCorrect : \u2200 {\u03b2} (c : C) (f : \u03b2 \u2192 \u03c4 \u2192 \u03b2) acc,\n    fold c f acc = (\n      canonicalToList (fold c)\n      |>.foldl f acc)\n\ntheorem fold_pair [F : Foldable.Correct C \u03c4]\n  (f\u2081 : \u03b2\u2081 \u2192 \u03c4 \u2192 \u03b2\u2081) (acc\u2081 : \u03b2\u2081) (f\u2082 : \u03b2\u2082 \u2192 \u03c4 \u2192 \u03b2\u2082) (acc\u2082 : \u03b2\u2082) (c : C)\n  : F.fold c (\u03bb (acc\u2081,acc\u2082) x => (f\u2081 acc\u2081 x, f\u2082 acc\u2082 x)) (acc\u2081, acc\u2082)\n    = (F.fold c (\u03bb acc\u2081 x => f\u2081 acc\u2081 x) acc\u2081,\n       F.fold c (\u03bb acc\u2082 x => f\u2082 acc\u2082 x) acc\u2082)\n  := by\n  let list := canonicalToList (F.fold c)\n  suffices\n    List.foldl (\u03bb (acc\u2081,acc\u2082) x => (f\u2081 acc\u2081 x, f\u2082 acc\u2082 x)) (acc\u2081, acc\u2082) list\n      = (List.foldl (\u03bb acc\u2081 x => f\u2081 acc\u2081 x) acc\u2081 list,\n         List.foldl (\u03bb acc\u2082 x => f\u2082 acc\u2082 x) acc\u2082 list)\n         by\n    simp at this\n    simp [\u2190F.foldCorrect] at this\n    exact this\n  induction list generalizing acc\u2081 acc\u2082 with\n  | nil =>\n    simp [List.foldl]\n  | cons x xs ih =>\n    simp\n    apply ih\n\nend Foldable\n\nnamespace Foldable'\n\nclass Correct (C) (\u03c4 : outParam _) (M : outParam (Membership \u03c4 C))\n  extends Foldable.Correct C \u03c4, Foldable' C \u03c4 M where\n  memCorrect : \u2200 x (c : C), x \u2208 c \u2194 x \u2208 canonicalToList (fold c)\n  fold'Correct : \u2200 {\u03b2} (c : C) (f : \u03b2 \u2192 (x : \u03c4) \u2192 x \u2208 c \u2192 \u03b2) acc,\n    fold' c f acc = (\n      canonicalToList (fold c)\n      |>.foldl' (\u03bb acc x h => f acc x ((memCorrect x c).mpr h)) acc)\n\ntheorem fold_eq_fold' [M : Membership \u03c4 C] [F : Foldable'.Correct C \u03c4 M]\n  (c : C) (f : \u03b2 \u2192 \u03c4 \u2192 \u03b2) (acc : \u03b2)\n  : F.fold c f acc = F.fold' c (\u03bb acc x _ => f acc x) acc\n  := by\n  rw [F.foldCorrect]\n  rw [F.fold'Correct]\n  simp [canonicalToList, List.foldl_eq_foldl']\n\ntheorem fold'_append_singleton_eq_map [M : Membership \u03c4 C] [Foldable'.Correct C \u03c4 M]\n  (c : C) (f : (x : \u03c4) \u2192 x \u2208 c \u2192 \u03c4')\n  : Foldable'.Correct.fold' c (\u03bb acc x h => acc ++ [f x h]) []\n    = (canonicalToList (Foldable.fold c)\n      |>.map' (fun x h => f x ((Foldable'.Correct.memCorrect _ _).mpr h)))\n  := by\n  rw [Correct.fold'Correct]\n  rw [List.foldl'_eq_subtypeByMem_foldl]\n  rw [List.map', List.foldl_eq_map]\n\ntheorem fold_canonicalToList_fold'_eq_fold' [Foldable'.Correct C \u03c4 M]\n  (c : C) (f' : (x : \u03c4) \u2192 M.mem x c \u2192 \u03c4') (f : \u03b2 \u2192 \u03c4' \u2192 \u03b2) (acc)\n  : List.foldl f acc\n    (canonicalToList (fun f init =>\n      Foldable'.Correct.fold' c (fun acc x h => f acc (f' x h)) init\n      )) = Foldable'.Correct.fold' c (fun acc x h => f acc (f' x h)) acc\n  := by\n  simp [canonicalToList]\n  rw [fold'_append_singleton_eq_map]\n  rw [List.map', List.foldl_map,\n      Foldable'.Correct.fold'Correct,\n      List.foldl'_eq_subtypeByMem_foldl]\n\ntheorem canonicalToList_fold'_eq_map' [Foldable'.Correct C \u03c4 M]\n  (c : C) (f' : (x : \u03c4) \u2192 M.mem x c \u2192 \u03c4')\n  : canonicalToList (fun f init =>\n      Foldable'.Correct.fold' c (fun acc x h => f acc (f' x h)) init\n      ) =\n    (canonicalToList (Foldable.fold c)).map' (fun x h =>\n      f' x ((Foldable'.Correct.memCorrect _ _).mpr h))\n  := by\n  conv =>\n    lhs\n    simp [canonicalToList]\n    rw [Correct.fold'Correct]\n    rw [List.foldl'_eq_subtypeByMem_foldl]\n    rw [List.foldl_eq_map]\n", "meta": {"author": "JamesGallicchio", "repo": "LeanColls", "sha": "9cb0a0c9a838bea24be80eace168bcc5f9481596", "save_path": "github-repos/lean/JamesGallicchio-LeanColls", "path": "github-repos/lean/JamesGallicchio-LeanColls/LeanColls-9cb0a0c9a838bea24be80eace168bcc5f9481596/LeanColls/FoldableCorrect.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.38181069536970996}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport ring_theory.ring_hom_properties\n\n/-!\n\n# The meta properties of finite ring homomorphisms.\n\n-/\n\nnamespace ring_hom\n\nopen_locale tensor_product\n\nopen tensor_product algebra.tensor_product\n\nlemma finite_stable_under_composition :\n  stable_under_composition @finite :=\nby { introv R hf hg, exactI hg.comp hf }\n\nlemma finite_respects_iso :\n  respects_iso @finite :=\nbegin\n  apply finite_stable_under_composition.respects_iso,\n  introsI,\n  exact finite.of_surjective _ e.to_equiv.surjective,\nend\n\nlemma finite_stable_under_base_change :\n  stable_under_base_change @finite :=\nbegin\n  refine stable_under_base_change.mk _ finite_respects_iso _,\n  classical,\n  introv h,\n  resetI,\n  replace h : module.finite R T := by { convert h, ext, rw algebra.smul_def, refl },\n  suffices : module.finite S (S \u2297[R] T),\n  { change module.finite _ _, convert this, ext, rw algebra.smul_def, refl },\n  exactI infer_instance\nend\n\nend ring_hom\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/ring_theory/ring_hom/finite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.38173167725686385}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.ordmap.ordnode\nimport Mathlib.algebra.ordered_ring\nimport Mathlib.data.nat.dist\nimport Mathlib.tactic.linarith.default\nimport Mathlib.PostPort\n\nuniverses u_1 l \n\nnamespace Mathlib\n\n/-!\n# Verification of the `ordnode \u03b1` datatype\n\nThis file proves the correctness of the operations in `data.ordmap.ordnode`.\nThe public facing version is the type `ordset \u03b1`, which is a wrapper around\n`ordnode \u03b1` which includes the correctness invariant of the type, and it exposes\nparallel operations like `insert` as functions on `ordset` that do the same\nthing but bundle the correctness proofs. The advantage is that it is possible\nto, for example, prove that the result of `find` on `insert` will actually find\nthe element, while `ordnode` cannot guarantee this if the input tree did not\nsatisfy the type invariants.\n\n## Main definitions\n\n* `ordset \u03b1`: A well formed set of values of type `\u03b1`\n\n## Implementation notes\n\nThe majority of this file is actually in the `ordnode` namespace, because we first\nhave to prove the correctness of all the operations (and defining what correctness\nmeans here is actually somewhat subtle). So all the actual `ordset` operations are\nat the very end, once we have all the theorems.\n\nAn `ordnode \u03b1` is an inductive type which describes a tree which stores the `size` at\ninternal nodes. The correctness invariant of an `ordnode \u03b1` is:\n\n* `ordnode.sized t`: All internal `size` fields must match the actual measured\n  size of the tree. (This is not hard to satisfy.)\n* `ordnode.balanced t`: Unless the tree has the form `()` or `((a) b)` or `(a (b))`\n  (that is, nil or a single singleton subtree), the two subtrees must satisfy\n  `size l \u2264 \u03b4 * size r` and `size r \u2264 \u03b4 * size l`, where `\u03b4 := 3` is a global\n  parameter of the data structure (and this property must hold recursively at subtrees).\n  This is why we say this is a \"size balanced tree\" data structure.\n* `ordnode.bounded lo hi t`: The members of the tree must be in strictly increasing order,\n  meaning that if `a` is in the left subtree and `b` is the root, then `a \u2264 b` and\n  `\u00ac (b \u2264 a)`. We enforce this using `ordnode.bounded` which includes also a global\n  upper and lower bound.\n\nBecause the `ordnode` file was ported from Haskell, the correctness invariants of some\nof the functions have not been spelled out, and some theorems like\n`ordnode.valid'.balance_l_aux` show very intricate assumptions on the sizes,\nwhich may need to be revised if it turns out some operations violate these assumptions,\nbecause there is a decent amount of slop in the actual data structure invariants, so the\ntheorem will go through with multiple choices of assumption.\n\n**Note:** This file is incomplete, in the sense that the intent is to have verified\nversions and lemmas about all the definitions in `ordnode.lean`, but at the moment only\na few operations are verified (the hard part should be out of the way, but still).\nContributors are encouraged to pick this up and finish the job, if it appeals to you.\n\n## Tags\n\nordered map, ordered set, data structure, verified programming\n\n-/\n\nnamespace ordnode\n\n\n/-! ### delta and ratio -/\n\ntheorem not_le_delta {s : \u2115} (H : 1 \u2264 s) : \u00acs \u2264 delta * 0 :=\n  fun (h : s \u2264 delta * 0) => not_lt_of_le (eq.mp (Eq._oldrec (Eq.refl (s \u2264 delta * 0)) (mul_zero delta)) h) H\n\ntheorem delta_lt_false {a : \u2115} {b : \u2115} (h\u2081 : delta * a < b) (h\u2082 : delta * b < a) : False := sorry\n\n/-! ### `singleton` -/\n\n/-! ### `size` and `empty` -/\n\n/-- O(n). Computes the actual number of elements in the set, ignoring the cached `size` field. -/\ndef real_size {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 \u2115 :=\n  sorry\n\n/-! ### `sized` -/\n\n/-- The `sized` property asserts that all the `size` fields in nodes match the actual size of the\nrespective subtrees. -/\ndef sized {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 Prop :=\n  sorry\n\ntheorem sized.node' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hr : sized r) : sized (node' l x r) :=\n  { left := rfl, right := { left := hl, right := hr } }\n\ntheorem sized.eq_node' {\u03b1 : Type u_1} {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (h : sized (node s l x r)) : node s l x r = node' l x r :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (node s l x r = node' l x r)) (and.left h)))\n    (Eq.refl (node (size l + size r + 1) l x r))\n\ntheorem sized.size_eq {\u03b1 : Type u_1} {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (H : sized (node s l x r)) : size (node s l x r) = size l + size r + 1 :=\n  and.left H\n\ntheorem sized.induction {\u03b1 : Type u_1} {t : ordnode \u03b1} (hl : sized t) {C : ordnode \u03b1 \u2192 Prop} (H0 : C nil) (H1 : \u2200 (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1), C l \u2192 C r \u2192 C (node' l x r)) : C t := sorry\n\ntheorem size_eq_real_size {\u03b1 : Type u_1} {t : ordnode \u03b1} : sized t \u2192 size t = real_size t := sorry\n\n@[simp] theorem sized.size_eq_zero {\u03b1 : Type u_1} {t : ordnode \u03b1} (ht : sized t) : size t = 0 \u2194 t = nil := sorry\n\ntheorem sized.pos {\u03b1 : Type u_1} {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (h : sized (node s l x r)) : 0 < s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 < s)) (and.left h))) (nat.le_add_left (Nat.succ 0) (size l + size r))\n\n/-! `dual` -/\n\ntheorem dual_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) : dual (dual t) = t := sorry\n\n@[simp] theorem size_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) : size (dual t) = size t :=\n  ordnode.cases_on t (Eq.refl (size (dual nil)))\n    fun (t_size : \u2115) (t_l : ordnode \u03b1) (t_x : \u03b1) (t_r : ordnode \u03b1) => Eq.refl (size (dual (node t_size t_l t_x t_r)))\n\n/-! `balanced` -/\n\n/-- The `balanced_sz l r` asserts that a hypothetical tree with children of sizes `l` and `r` is\nbalanced: either `l \u2264 \u03b4 * r` and `r \u2264 \u03b4 * r`, or the tree is trivial with a singleton on one side\nand nothing on the other. -/\ndef balanced_sz (l : \u2115) (r : \u2115) :=\n  l + r \u2264 1 \u2228 l \u2264 delta * r \u2227 r \u2264 delta * l\n\nprotected instance balanced_sz.dec : DecidableRel balanced_sz :=\n  fun (l r : \u2115) => or.decidable\n\n/-- The `balanced t` asserts that the tree `t` satisfies the balance invariants\n(at every level). -/\ndef balanced {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 Prop :=\n  sorry\n\nprotected instance balanced.dec {\u03b1 : Type u_1} : decidable_pred balanced :=\n  sorry\n\ntheorem balanced_sz.symm {l : \u2115} {r : \u2115} : balanced_sz l r \u2192 balanced_sz r l :=\n  or.imp (eq.mpr (id (Eq._oldrec (Eq.refl (l + r \u2264 1 \u2192 r + l \u2264 1)) (add_comm l r))) id) and.symm\n\ntheorem balanced_sz_zero {l : \u2115} : balanced_sz l 0 \u2194 l \u2264 1 := sorry\n\ntheorem balanced_sz_up {l : \u2115} {r\u2081 : \u2115} {r\u2082 : \u2115} (h\u2081 : r\u2081 \u2264 r\u2082) (h\u2082 : l + r\u2082 \u2264 1 \u2228 r\u2082 \u2264 delta * l) (H : balanced_sz l r\u2081) : balanced_sz l r\u2082 := sorry\n\ntheorem balanced_sz_down {l : \u2115} {r\u2081 : \u2115} {r\u2082 : \u2115} (h\u2081 : r\u2081 \u2264 r\u2082) (h\u2082 : l + r\u2082 \u2264 1 \u2228 l \u2264 delta * r\u2081) (H : balanced_sz l r\u2082) : balanced_sz l r\u2081 := sorry\n\ntheorem balanced.dual {\u03b1 : Type u_1} {t : ordnode \u03b1} : balanced t \u2192 balanced (dual t) := sorry\n\n/-! ### `rotate` and `balance` -/\n\n/-- Build a tree from three nodes, left associated (ignores the invariants). -/\ndef node3_l {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : ordnode \u03b1 :=\n  node' (node' l x m) y r\n\n/-- Build a tree from three nodes, right associated (ignores the invariants). -/\ndef node3_r {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : ordnode \u03b1 :=\n  node' l x (node' m y r)\n\n/-- Build a tree from three nodes, with `a () b -> (a ()) b` and `a (b c) d -> ((a b) (c d))`. -/\ndef node4_l {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 ordnode \u03b1 :=\n  sorry\n\n/-- Build a tree from three nodes, with `a () b -> a (() b)` and `a (b c) d -> ((a b) (c d))`. -/\ndef node4_r {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 ordnode \u03b1 :=\n  sorry\n\n/-- Concatenate two nodes, performing a left rotation `x (y z) -> ((x y) z)`\nif balance is upset. -/\ndef rotate_l {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 ordnode \u03b1 :=\n  sorry\n\n/-- Concatenate two nodes, performing a right rotation `(x y) z -> (x (y z))`\nif balance is upset. -/\ndef rotate_r {\u03b1 : Type u_1} : ordnode \u03b1 \u2192 \u03b1 \u2192 ordnode \u03b1 \u2192 ordnode \u03b1 :=\n  sorry\n\n/-- A left balance operation. This will rebalance a concatenation, assuming the original nodes are\nnot too far from balanced. -/\ndef balance_l' {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : ordnode \u03b1 :=\n  ite (size l + size r \u2264 1) (node' l x r) (ite (size l > delta * size r) (rotate_r l x r) (node' l x r))\n\n/-- A right balance operation. This will rebalance a concatenation, assuming the original nodes are\nnot too far from balanced. -/\ndef balance_r' {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : ordnode \u03b1 :=\n  ite (size l + size r \u2264 1) (node' l x r) (ite (size r > delta * size l) (rotate_l l x r) (node' l x r))\n\n/-- The full balance operation. This is the same as `balance`, but with less manual inlining.\nIt is somewhat easier to work with this version in proofs. -/\ndef balance' {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : ordnode \u03b1 :=\n  ite (size l + size r \u2264 1) (node' l x r)\n    (ite (size r > delta * size l) (rotate_l l x r) (ite (size l > delta * size r) (rotate_r l x r) (node' l x r)))\n\ntheorem dual_node' {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (node' l x r) = node' (dual r) x (dual l) := sorry\n\ntheorem dual_node3_l {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : dual (node3_l l x m y r) = node3_r (dual r) y (dual m) x (dual l) := sorry\n\ntheorem dual_node3_r {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : dual (node3_r l x m y r) = node3_l (dual r) y (dual m) x (dual l) := sorry\n\ntheorem dual_node4_l {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : dual (node4_l l x m y r) = node4_r (dual r) y (dual m) x (dual l) := sorry\n\ntheorem dual_node4_r {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (m : ordnode \u03b1) (y : \u03b1) (r : ordnode \u03b1) : dual (node4_r l x m y r) = node4_l (dual r) y (dual m) x (dual l) := sorry\n\ntheorem dual_rotate_l {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (rotate_l l x r) = rotate_r (dual r) x (dual l) := sorry\n\ntheorem dual_rotate_r {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (rotate_r l x r) = rotate_l (dual r) x (dual l) := sorry\n\ntheorem dual_balance' {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (balance' l x r) = balance' (dual r) x (dual l) := sorry\n\ntheorem dual_balance_l {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (balance_l l x r) = balance_r (dual r) x (dual l) := sorry\n\ntheorem dual_balance_r {\u03b1 : Type u_1} (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : dual (balance_r l x r) = balance_l (dual r) x (dual l) := sorry\n\ntheorem sized.node3_l {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hm : sized m) (hr : sized r) : sized (node3_l l x m y r) :=\n  sized.node' (sized.node' hl hm) hr\n\ntheorem sized.node3_r {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hm : sized m) (hr : sized r) : sized (node3_r l x m y r) :=\n  sized.node' hl (sized.node' hm hr)\n\ntheorem sized.node4_l {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hm : sized m) (hr : sized r) : sized (node4_l l x m y r) := sorry\n\ntheorem node3_l_size {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : size (node3_l l x m y r) = size l + size m + size r + bit0 1 := sorry\n\ntheorem node3_r_size {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : size (node3_r l x m y r) = size l + size m + size r + bit0 1 := sorry\n\ntheorem node4_l_size {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} (hm : sized m) : size (node4_l l x m y r) = size l + size m + size r + bit0 1 := sorry\n\ntheorem sized.dual {\u03b1 : Type u_1} {t : ordnode \u03b1} (h : sized t) : sized (dual t) := sorry\n\ntheorem sized.dual_iff {\u03b1 : Type u_1} {t : ordnode \u03b1} : sized (dual t) \u2194 sized t :=\n  { mp := fun (h : sized (dual t)) => eq.mpr (id (Eq._oldrec (Eq.refl (sized t)) (Eq.symm (dual_dual t)))) (sized.dual h),\n    mpr := sized.dual }\n\ntheorem sized.rotate_l {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hr : sized r) : sized (rotate_l l x r) := sorry\n\ntheorem sized.rotate_r {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hr : sized r) : sized (rotate_r l x r) :=\n  iff.mp sized.dual_iff\n    (eq.mpr (id (Eq._oldrec (Eq.refl (sized (dual (rotate_r l x r)))) (dual_rotate_r l x r)))\n      (sized.rotate_l (sized.dual hr) (sized.dual hl)))\n\ntheorem sized.rotate_l_size {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hm : sized r) : size (rotate_l l x r) = size l + size r + 1 := sorry\n\ntheorem sized.rotate_r_size {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) : size (rotate_r l x r) = size l + size r + 1 := sorry\n\ntheorem sized.balance' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hr : sized r) : sized (balance' l x r) := sorry\n\ntheorem size_balance' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : sized l) (hr : sized r) : size (balance' l x r) = size l + size r + 1 := sorry\n\n/-! ## `all`, `any`, `emem`, `amem` -/\n\ntheorem all.imp {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {Q : \u03b1 \u2192 Prop} (H : \u2200 (a : \u03b1), P a \u2192 Q a) {t : ordnode \u03b1} : all P t \u2192 all Q t := sorry\n\ntheorem any.imp {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {Q : \u03b1 \u2192 Prop} (H : \u2200 (a : \u03b1), P a \u2192 Q a) {t : ordnode \u03b1} : any P t \u2192 any Q t := sorry\n\ntheorem all_singleton {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {x : \u03b1} : all P (singleton x) \u2194 P x :=\n  { mp := fun (h : all P (singleton x)) => and.left (and.right h),\n    mpr := fun (h : P x) => { left := True.intro, right := { left := h, right := True.intro } } }\n\ntheorem any_singleton {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {x : \u03b1} : any P (singleton x) \u2194 P x := sorry\n\ntheorem all_dual {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {t : ordnode \u03b1} : all P (dual t) \u2194 all P t := sorry\n\ntheorem all_iff_forall {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {t : ordnode \u03b1} : all P t \u2194 \u2200 (x : \u03b1), emem x t \u2192 P x := sorry\n\ntheorem any_iff_exists {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {t : ordnode \u03b1} : any P t \u2194 \u2203 (x : \u03b1), emem x t \u2227 P x := sorry\n\ntheorem emem_iff_all {\u03b1 : Type u_1} {x : \u03b1} {t : ordnode \u03b1} : emem x t \u2194 \u2200 (P : \u03b1 \u2192 Prop), all P t \u2192 P x := sorry\n\ntheorem all_node' {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} : all P (node' l x r) \u2194 all P l \u2227 P x \u2227 all P r :=\n  iff.rfl\n\ntheorem all_node3_l {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : all P (node3_l l x m y r) \u2194 all P l \u2227 P x \u2227 all P m \u2227 P y \u2227 all P r := sorry\n\ntheorem all_node3_r {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : all P (node3_r l x m y r) \u2194 all P l \u2227 P x \u2227 all P m \u2227 P y \u2227 all P r :=\n  iff.rfl\n\ntheorem all_node4_l {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : all P (node4_l l x m y r) \u2194 all P l \u2227 P x \u2227 all P m \u2227 P y \u2227 all P r := sorry\n\ntheorem all_node4_r {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} : all P (node4_r l x m y r) \u2194 all P l \u2227 P x \u2227 all P m \u2227 P y \u2227 all P r := sorry\n\ntheorem all_rotate_l {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} : all P (rotate_l l x r) \u2194 all P l \u2227 P x \u2227 all P r := sorry\n\ntheorem all_rotate_r {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} : all P (rotate_r l x r) \u2194 all P l \u2227 P x \u2227 all P r := sorry\n\ntheorem all_balance' {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} : all P (balance' l x r) \u2194 all P l \u2227 P x \u2227 all P r := sorry\n\n/-! ### `to_list` -/\n\ntheorem foldr_cons_eq_to_list {\u03b1 : Type u_1} (t : ordnode \u03b1) (r : List \u03b1) : foldr List.cons t r = to_list t ++ r := sorry\n\n@[simp] theorem to_list_nil {\u03b1 : Type u_1} : to_list nil = [] :=\n  rfl\n\n@[simp] theorem to_list_node {\u03b1 : Type u_1} (s : \u2115) (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : to_list (node s l x r) = to_list l ++ x :: to_list r := sorry\n\ntheorem emem_iff_mem_to_list {\u03b1 : Type u_1} {x : \u03b1} {t : ordnode \u03b1} : emem x t \u2194 x \u2208 to_list t := sorry\n\ntheorem length_to_list' {\u03b1 : Type u_1} (t : ordnode \u03b1) : list.length (to_list t) = real_size t := sorry\n\ntheorem length_to_list {\u03b1 : Type u_1} {t : ordnode \u03b1} (h : sized t) : list.length (to_list t) = size t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (list.length (to_list t) = size t)) (length_to_list' t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (real_size t = size t)) (size_eq_real_size h))) (Eq.refl (real_size t)))\n\ntheorem equiv_iff {\u03b1 : Type u_1} {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} (h\u2081 : sized t\u2081) (h\u2082 : sized t\u2082) : equiv t\u2081 t\u2082 \u2194 to_list t\u2081 = to_list t\u2082 := sorry\n\n/-! ### `(find/erase/split)_(min/max)` -/\n\ntheorem find_min'_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) (x : \u03b1) : find_min' (dual t) x = find_max' x t := sorry\n\ntheorem find_max'_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) (x : \u03b1) : find_max' x (dual t) = find_min' t x :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (find_max' x (dual t) = find_min' t x)) (Eq.symm (find_min'_dual (dual t) x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (find_min' (dual (dual t)) x = find_min' t x)) (dual_dual t)))\n      (Eq.refl (find_min' t x)))\n\ntheorem find_min_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) : find_min (dual t) = find_max t :=\n  ordnode.cases_on t (idRhs (find_min (dual nil) = find_min (dual nil)) rfl)\n    fun (t_size : \u2115) (t_l : ordnode \u03b1) (t_x : \u03b1) (t_r : ordnode \u03b1) =>\n      idRhs (some (find_min' (dual t_r) t_x) = some (find_max' t_x t_r)) (congr_arg some (find_min'_dual t_r t_x))\n\ntheorem find_max_dual {\u03b1 : Type u_1} (t : ordnode \u03b1) : find_max (dual t) = find_min t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (find_max (dual t) = find_min t)) (Eq.symm (find_min_dual (dual t)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (find_min (dual (dual t)) = find_min t)) (dual_dual t))) (Eq.refl (find_min t)))\n\ntheorem dual_erase_min {\u03b1 : Type u_1} (t : ordnode \u03b1) : dual (erase_min t) = erase_max (dual t) := sorry\n\ntheorem dual_erase_max {\u03b1 : Type u_1} (t : ordnode \u03b1) : dual (erase_max t) = erase_min (dual t) := sorry\n\ntheorem split_min_eq {\u03b1 : Type u_1} (s : \u2115) (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : split_min' l x r = (find_min' l x, erase_min (node s l x r)) := sorry\n\ntheorem split_max_eq {\u03b1 : Type u_1} (s : \u2115) (l : ordnode \u03b1) (x : \u03b1) (r : ordnode \u03b1) : split_max' l x r = (erase_max (node s l x r), find_max' x r) := sorry\n\ntheorem find_min'_all {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} (t : ordnode \u03b1) (x : \u03b1) : all P t \u2192 P x \u2192 P (find_min' t x) := sorry\n\ntheorem find_max'_all {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} (x : \u03b1) (t : ordnode \u03b1) : P x \u2192 all P t \u2192 P (find_max' x t) := sorry\n\n/-! ### `glue` -/\n\n/-! ### `merge` -/\n\n@[simp] theorem merge_nil_left {\u03b1 : Type u_1} (t : ordnode \u03b1) : merge t nil = t :=\n  ordnode.cases_on t (Eq.refl (merge nil nil))\n    fun (t_size : \u2115) (t_l : ordnode \u03b1) (t_x : \u03b1) (t_r : ordnode \u03b1) => Eq.refl (merge (node t_size t_l t_x t_r) nil)\n\n@[simp] theorem merge_nil_right {\u03b1 : Type u_1} (t : ordnode \u03b1) : merge nil t = t :=\n  rfl\n\n@[simp] theorem merge_node {\u03b1 : Type u_1} {ls : \u2115} {ll : ordnode \u03b1} {lx : \u03b1} {lr : ordnode \u03b1} {rs : \u2115} {rl : ordnode \u03b1} {rx : \u03b1} {rr : ordnode \u03b1} : merge (node ls ll lx lr) (node rs rl rx rr) =\n  ite (delta * ls < rs) (balance_l (merge (node ls ll lx lr) rl) rx rr)\n    (ite (delta * rs < ls) (balance_r ll lx (merge lr (node rs rl rx rr))) (glue (node ls ll lx lr) (node rs rl rx rr))) :=\n  rfl\n\n/-! ### `insert` -/\n\ntheorem dual_insert {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (x : \u03b1) (t : ordnode \u03b1) : dual (ordnode.insert x t) = ordnode.insert x (dual t) := sorry\n\n/-! ### `balance` properties -/\n\ntheorem balance_eq_balance' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) : balance l x r = balance' l x r := sorry\n\ntheorem balance_l_eq_balance {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (sl : sized l) (sr : sized r) (H1 : size l = 0 \u2192 size r \u2264 1) (H2 : 1 \u2264 size l \u2192 1 \u2264 size r \u2192 size r \u2264 delta * size l) : balance_l l x r = balance l x r := sorry\n\n/-- `raised n m` means `m` is either equal or one up from `n`. -/\ndef raised (n : \u2115) (m : \u2115) :=\n  m = n \u2228 m = n + 1\n\ntheorem raised_iff {n : \u2115} {m : \u2115} : raised n m \u2194 n \u2264 m \u2227 m \u2264 n + 1 := sorry\n\ntheorem raised.dist_le {n : \u2115} {m : \u2115} (H : raised n m) : nat.dist n m \u2264 1 := sorry\n\ntheorem raised.dist_le' {n : \u2115} {m : \u2115} (H : raised n m) : nat.dist m n \u2264 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nat.dist m n \u2264 1)) (nat.dist_comm m n))) (raised.dist_le H)\n\ntheorem raised.add_left (k : \u2115) {n : \u2115} {m : \u2115} (H : raised n m) : raised (k + n) (k + m) :=\n  or.dcases_on H (fun (H : m = n) => Eq._oldrec (Or.inl rfl) H) fun (H : m = n + 1) => Eq._oldrec (Or.inr rfl) (Eq.symm H)\n\ntheorem raised.add_right (k : \u2115) {n : \u2115} {m : \u2115} (H : raised n m) : raised (n + k) (m + k) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (raised (n + k) (m + k))) (add_comm n k)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (raised (k + n) (m + k))) (add_comm m k))) (raised.add_left k H))\n\ntheorem raised.right {\u03b1 : Type u_1} {l : ordnode \u03b1} {x\u2081 : \u03b1} {x\u2082 : \u03b1} {r\u2081 : ordnode \u03b1} {r\u2082 : ordnode \u03b1} (H : raised (size r\u2081) (size r\u2082)) : raised (size (node' l x\u2081 r\u2081)) (size (node' l x\u2082 r\u2082)) := sorry\n\ntheorem balance_l_eq_balance' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised l' (size l) \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised (size r) r' \u2227 balanced_sz (size l) r') : balance_l l x r = balance' l x r := sorry\n\ntheorem balance_sz_dual {\u03b1 : Type u_1} {l : ordnode \u03b1} {r : ordnode \u03b1} (H : (\u2203 (l' : \u2115), raised (size l) l' \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised r' (size r) \u2227 balanced_sz (size l) r') : (\u2203 (l' : \u2115), raised l' (size (dual r)) \u2227 balanced_sz l' (size (dual l))) \u2228\n  \u2203 (r' : \u2115), raised (size (dual l)) r' \u2227 balanced_sz (size (dual r)) r' := sorry\n\ntheorem size_balance_l {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised l' (size l) \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised (size r) r' \u2227 balanced_sz (size l) r') : size (balance_l l x r) = size l + size r + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (size (balance_l l x r) = size l + size r + 1)) (balance_l_eq_balance' hl hr sl sr H)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (size (balance' l x r) = size l + size r + 1)) (size_balance' sl sr)))\n      (Eq.refl (size l + size r + 1)))\n\ntheorem all_balance_l {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised l' (size l) \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised (size r) r' \u2227 balanced_sz (size l) r') : all P (balance_l l x r) \u2194 all P l \u2227 P x \u2227 all P r := sorry\n\ntheorem balance_r_eq_balance' {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised (size l) l' \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised r' (size r) \u2227 balanced_sz (size l) r') : balance_r l x r = balance' l x r := sorry\n\ntheorem size_balance_r {\u03b1 : Type u_1} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised (size l) l' \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised r' (size r) \u2227 balanced_sz (size l) r') : size (balance_r l x r) = size l + size r + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (size (balance_r l x r) = size l + size r + 1)) (balance_r_eq_balance' hl hr sl sr H)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (size (balance' l x r) = size l + size r + 1)) (size_balance' sl sr)))\n      (Eq.refl (size l + size r + 1)))\n\ntheorem all_balance_r {\u03b1 : Type u_1} {P : \u03b1 \u2192 Prop} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (hl : balanced l) (hr : balanced r) (sl : sized l) (sr : sized r) (H : (\u2203 (l' : \u2115), raised (size l) l' \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised r' (size r) \u2227 balanced_sz (size l) r') : all P (balance_r l x r) \u2194 all P l \u2227 P x \u2227 all P r := sorry\n\n/-! ### `bounded` -/\n\n/-- `bounded t lo hi` says that every element `x \u2208 t` is in the range `lo < x < hi`, and also this\nproperty holds recursively in subtrees, making the full tree a BST. The bounds can be set to\n`lo = \u22a5` and `hi = \u22a4` if we care only about the internal ordering constraints. -/\ndef bounded {\u03b1 : Type u_1} [preorder \u03b1] : ordnode \u03b1 \u2192 with_bot \u03b1 \u2192 with_top \u03b1 \u2192 Prop :=\n  sorry\n\ntheorem bounded.dual {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : bounded t o\u2081 o\u2082) : bounded (dual t) o\u2082 o\u2081 := sorry\n\ntheorem bounded.dual_iff {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t o\u2081 o\u2082 \u2194 bounded (dual t) o\u2082 o\u2081 := sorry\n\ntheorem bounded.weak_left {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t o\u2081 o\u2082 \u2192 bounded t \u22a5 o\u2082 := sorry\n\ntheorem bounded.weak_right {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t o\u2081 o\u2082 \u2192 bounded t o\u2081 \u22a4 := sorry\n\ntheorem bounded.weak {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : bounded t o\u2081 o\u2082) : bounded t \u22a5 \u22a4 :=\n  bounded.weak_right (bounded.weak_left h)\n\ntheorem bounded.mono_left {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} {y : \u03b1} (xy : x \u2264 y) {t : ordnode \u03b1} {o : with_top \u03b1} : bounded t (\u2191y) o \u2192 bounded t (\u2191x) o := sorry\n\ntheorem bounded.mono_right {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} {y : \u03b1} (xy : x \u2264 y) {t : ordnode \u03b1} {o : with_bot \u03b1} : bounded t o \u2191x \u2192 bounded t o \u2191y := sorry\n\ntheorem bounded.to_lt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {x : \u03b1} {y : \u03b1} : bounded t \u2191x \u2191y \u2192 x < y := sorry\n\ntheorem bounded.to_nil {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t o\u2081 o\u2082 \u2192 bounded nil o\u2081 o\u2082 := sorry\n\ntheorem bounded.trans_left {\u03b1 : Type u_1} [preorder \u03b1] {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t\u2081 o\u2081 \u2191x \u2192 bounded t\u2082 (\u2191x) o\u2082 \u2192 bounded t\u2082 o\u2081 o\u2082 := sorry\n\ntheorem bounded.trans_right {\u03b1 : Type u_1} [preorder \u03b1] {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : bounded t\u2081 o\u2081 \u2191x \u2192 bounded t\u2082 (\u2191x) o\u2082 \u2192 bounded t\u2081 o\u2081 o\u2082 := sorry\n\ntheorem bounded.mem_lt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o : with_bot \u03b1} {x : \u03b1} : bounded t o \u2191x \u2192 all (fun (_x : \u03b1) => _x < x) t := sorry\n\ntheorem bounded.mem_gt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o : with_top \u03b1} {x : \u03b1} : bounded t (\u2191x) o \u2192 all (fun (_x : \u03b1) => _x > x) t := sorry\n\ntheorem bounded.of_lt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} {x : \u03b1} : bounded t o\u2081 o\u2082 \u2192 bounded nil o\u2081 \u2191x \u2192 all (fun (_x : \u03b1) => _x < x) t \u2192 bounded t o\u2081 \u2191x := sorry\n\ntheorem bounded.of_gt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} {x : \u03b1} : bounded t o\u2081 o\u2082 \u2192 bounded nil (\u2191x) o\u2082 \u2192 all (fun (_x : \u03b1) => _x > x) t \u2192 bounded t (\u2191x) o\u2082 := sorry\n\ntheorem bounded.to_sep {\u03b1 : Type u_1} [preorder \u03b1] {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} {x : \u03b1} (h\u2081 : bounded t\u2081 o\u2081 \u2191x) (h\u2082 : bounded t\u2082 (\u2191x) o\u2082) : all (fun (y : \u03b1) => all (fun (z : \u03b1) => y < z) t\u2082) t\u2081 :=\n  all.imp (fun (y : \u03b1) (yx : y < x) => all.imp (fun (z : \u03b1) (xz : z > x) => lt_trans yx xz) (bounded.mem_gt h\u2082))\n    (bounded.mem_lt h\u2081)\n\n/-! ### `valid` -/\n\n/-- The validity predicate for an `ordnode` subtree. This asserts that the `size` fields are\ncorrect, the tree is balanced, and the elements of the tree are organized according to the\nordering. This version of `valid` also puts all elements in the tree in the interval `(lo, hi)`. -/\nstructure valid' {\u03b1 : Type u_1} [preorder \u03b1] (lo : with_bot \u03b1) (t : ordnode \u03b1) (hi : with_top \u03b1) \nwhere\n  ord : bounded t lo hi\n  sz : sized t\n  bal : balanced t\n\n/-- The validity predicate for an `ordnode` subtree. This asserts that the `size` fields are\ncorrect, the tree is balanced, and the elements of the tree are organized according to the\nordering. -/\ndef valid {\u03b1 : Type u_1} [preorder \u03b1] (t : ordnode \u03b1) :=\n  valid' \u22a5 t \u22a4\n\ntheorem valid'.mono_left {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} {y : \u03b1} (xy : x \u2264 y) {t : ordnode \u03b1} {o : with_top \u03b1} (h : valid' (\u2191y) t o) : valid' (\u2191x) t o :=\n  valid'.mk (bounded.mono_left xy (valid'.ord h)) (valid'.sz h) (valid'.bal h)\n\ntheorem valid'.mono_right {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} {y : \u03b1} (xy : x \u2264 y) {t : ordnode \u03b1} {o : with_bot \u03b1} (h : valid' o t \u2191x) : valid' o t \u2191y :=\n  valid'.mk (bounded.mono_right xy (valid'.ord h)) (valid'.sz h) (valid'.bal h)\n\ntheorem valid'.trans_left {\u03b1 : Type u_1} [preorder \u03b1] {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : bounded t\u2081 o\u2081 \u2191x) (H : valid' (\u2191x) t\u2082 o\u2082) : valid' o\u2081 t\u2082 o\u2082 :=\n  valid'.mk (bounded.trans_left h (valid'.ord H)) (valid'.sz H) (valid'.bal H)\n\ntheorem valid'.trans_right {\u03b1 : Type u_1} [preorder \u03b1] {t\u2081 : ordnode \u03b1} {t\u2082 : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 t\u2081 \u2191x) (h : bounded t\u2082 (\u2191x) o\u2082) : valid' o\u2081 t\u2081 o\u2082 :=\n  valid'.mk (bounded.trans_right (valid'.ord H) h) (valid'.sz H) (valid'.bal H)\n\ntheorem valid'.of_lt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 t o\u2082) (h\u2081 : bounded nil o\u2081 \u2191x) (h\u2082 : all (fun (_x : \u03b1) => _x < x) t) : valid' o\u2081 t \u2191x :=\n  valid'.mk (bounded.of_lt (valid'.ord H) h\u2081 h\u2082) (valid'.sz H) (valid'.bal H)\n\ntheorem valid'.of_gt {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 t o\u2082) (h\u2081 : bounded nil (\u2191x) o\u2082) (h\u2082 : all (fun (_x : \u03b1) => _x > x) t) : valid' (\u2191x) t o\u2082 :=\n  valid'.mk (bounded.of_gt (valid'.ord H) h\u2081 h\u2082) (valid'.sz H) (valid'.bal H)\n\ntheorem valid'.valid {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : valid' o\u2081 t o\u2082) : valid t :=\n  valid'.mk (bounded.weak (valid'.ord h)) (valid'.sz h) (valid'.bal h)\n\ntheorem valid'_nil {\u03b1 : Type u_1} [preorder \u03b1] {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : bounded nil o\u2081 o\u2082) : valid' o\u2081 nil o\u2082 :=\n  valid'.mk h True.intro True.intro\n\ntheorem valid_nil {\u03b1 : Type u_1} [preorder \u03b1] : valid nil :=\n  valid'_nil True.intro\n\ntheorem valid'.node {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : balanced_sz (size l) (size r)) (hs : s = size l + size r + 1) : valid' o\u2081 (node s l x r) o\u2082 :=\n  valid'.mk { left := valid'.ord hl, right := valid'.ord hr }\n    { left := hs, right := { left := valid'.sz hl, right := valid'.sz hr } }\n    { left := H, right := { left := valid'.bal hl, right := valid'.bal hr } }\n\ntheorem valid'.dual {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h : valid' o\u2081 t o\u2082) : valid' o\u2082 (dual t) o\u2081 := sorry\n\ntheorem valid'.dual_iff {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : valid' o\u2081 t o\u2082 \u2194 valid' o\u2082 (dual t) o\u2081 := sorry\n\ntheorem valid.dual {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} : valid t \u2192 valid (dual t) :=\n  valid'.dual\n\ntheorem valid.dual_iff {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} : valid t \u2194 valid (dual t) :=\n  valid'.dual_iff\n\ntheorem valid'.left {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 (node s l x r) o\u2082) : valid' o\u2081 l \u2191x :=\n  valid'.mk (and.left (valid'.ord H)) (and.left (and.right (valid'.sz H))) (and.left (and.right (valid'.bal H)))\n\ntheorem valid'.right {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 (node s l x r) o\u2082) : valid' (\u2191x) r o\u2082 :=\n  valid'.mk (and.right (valid'.ord H)) (and.right (and.right (valid'.sz H))) (and.right (and.right (valid'.bal H)))\n\ntheorem valid.left {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (H : valid (node s l x r)) : valid l :=\n  valid'.valid (valid'.left H)\n\ntheorem valid.right {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (H : valid (node s l x r)) : valid r :=\n  valid'.valid (valid'.right H)\n\ntheorem valid.size_eq {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} (H : valid (node s l x r)) : size (node s l x r) = size l + size r + 1 :=\n  and.left (valid'.sz H)\n\ntheorem valid'.node' {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : balanced_sz (size l) (size r)) : valid' o\u2081 (node' l x r) o\u2082 :=\n  valid'.node hl hr H rfl\n\ntheorem valid'_singleton {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (h\u2081 : bounded nil o\u2081 \u2191x) (h\u2082 : bounded nil (\u2191x) o\u2082) : valid' o\u2081 (singleton x) o\u2082 :=\n  valid'.node (valid'_nil h\u2081) (valid'_nil h\u2082) (Or.inl zero_le_one) rfl\n\ntheorem valid_singleton {\u03b1 : Type u_1} [preorder \u03b1] {x : \u03b1} : valid (singleton x) :=\n  valid'_singleton True.intro True.intro\n\ntheorem valid'.node3_l {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hm : valid' (\u2191x) m \u2191y) (hr : valid' (\u2191y) r o\u2082) (H1 : balanced_sz (size l) (size m)) (H2 : balanced_sz (size l + size m + 1) (size r)) : valid' o\u2081 (node3_l l x m y r) o\u2082 :=\n  valid'.node' (valid'.node' hl hm H1) hr H2\n\ntheorem valid'.node3_r {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hm : valid' (\u2191x) m \u2191y) (hr : valid' (\u2191y) r o\u2082) (H1 : balanced_sz (size l) (size m + size r + 1)) (H2 : balanced_sz (size m) (size r)) : valid' o\u2081 (node3_r l x m y r) o\u2082 :=\n  valid'.node' hl (valid'.node' hm hr H2) H1\n\ntheorem valid'.node4_l_lemma\u2081 {a : \u2115} {b : \u2115} {c : \u2115} {d : \u2115} (lr\u2082 : bit1 1 * (b + c + 1 + d) \u2264 bit0 (bit0 (bit0 (bit0 1))) * a + bit1 (bit0 (bit0 1))) (mr\u2082 : b + c + 1 \u2264 bit1 1 * d) (mm\u2081 : b \u2264 bit1 1 * c) : b < bit1 1 * a + 1 := sorry\n\ntheorem valid'.node4_l_lemma\u2082 {b : \u2115} {c : \u2115} {d : \u2115} (mr\u2082 : b + c + 1 \u2264 bit1 1 * d) : c \u2264 bit1 1 * d := sorry\n\ntheorem valid'.node4_l_lemma\u2083 {b : \u2115} {c : \u2115} {d : \u2115} (mr\u2081 : bit0 1 * d \u2264 b + c + 1) (mm\u2081 : b \u2264 bit1 1 * c) : d \u2264 bit1 1 * c := sorry\n\ntheorem valid'.node4_l_lemma\u2084 {a : \u2115} {b : \u2115} {c : \u2115} {d : \u2115} (lr\u2081 : bit1 1 * a \u2264 b + c + 1 + d) (mr\u2082 : b + c + 1 \u2264 bit1 1 * d) (mm\u2081 : b \u2264 bit1 1 * c) : a + b + 1 \u2264 bit1 1 * (c + d + 1) := sorry\n\ntheorem valid'.node4_l_lemma\u2085 {a : \u2115} {b : \u2115} {c : \u2115} {d : \u2115} (lr\u2082 : bit1 1 * (b + c + 1 + d) \u2264 bit0 (bit0 (bit0 (bit0 1))) * a + bit1 (bit0 (bit0 1))) (mr\u2081 : bit0 1 * d \u2264 b + c + 1) (mm\u2082 : c \u2264 bit1 1 * b) : c + d + 1 \u2264 bit1 1 * (a + b + 1) := sorry\n\ntheorem valid'.node4_l {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {m : ordnode \u03b1} {y : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hm : valid' (\u2191x) m \u2191y) (hr : valid' (\u2191y) r o\u2082) (Hm : 0 < size m) (H : size l = 0 \u2227 size m = 1 \u2227 size r \u2264 1 \u2228\n  0 < size l \u2227\n    ratio * size r \u2264 size m \u2227\n      delta * size l \u2264 size m + size r \u2227\n        bit1 1 * (size m + size r) \u2264 bit0 (bit0 (bit0 (bit0 1))) * size l + bit1 (bit0 (bit0 1)) \u2227\n          size m \u2264 delta * size r) : valid' o\u2081 (node4_l l x m y r) o\u2082 := sorry\n\ntheorem valid'.rotate_l_lemma\u2081 {a : \u2115} {b : \u2115} {c : \u2115} (H2 : bit1 1 * a \u2264 b + c) (hb\u2082 : c \u2264 bit1 1 * b) : a \u2264 bit1 1 * b := sorry\n\ntheorem valid'.rotate_l_lemma\u2082 {a : \u2115} {b : \u2115} {c : \u2115} (H3 : bit0 1 * (b + c) \u2264 bit1 (bit0 (bit0 1)) * a + bit1 1) (h : b < bit0 1 * c) : b < bit1 1 * a + 1 := sorry\n\ntheorem valid'.rotate_l_lemma\u2083 {a : \u2115} {b : \u2115} {c : \u2115} (H2 : bit1 1 * a \u2264 b + c) (h : b < bit0 1 * c) : a + b < bit1 1 * c := sorry\n\ntheorem valid'.rotate_l_lemma\u2084 {a : \u2115} {b : \u2115} (H3 : bit0 1 * b \u2264 bit1 (bit0 (bit0 1)) * a + bit1 1) : bit1 1 * b \u2264 bit0 (bit0 (bit0 (bit0 1))) * a + bit1 (bit0 (bit0 1)) := sorry\n\ntheorem valid'.rotate_l {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H1 : \u00acsize l + size r \u2264 1) (H2 : delta * size l < size r) (H3 : bit0 1 * size r \u2264 bit1 (bit0 (bit0 1)) * size l + bit1 (bit0 1) \u2228 size r \u2264 bit1 1) : valid' o\u2081 (rotate_l l x r) o\u2082 := sorry\n\ntheorem valid'.rotate_r {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H1 : \u00acsize l + size r \u2264 1) (H2 : delta * size r < size l) (H3 : bit0 1 * size l \u2264 bit1 (bit0 (bit0 1)) * size r + bit1 (bit0 1) \u2228 size l \u2264 bit1 1) : valid' o\u2081 (rotate_r l x r) o\u2082 := sorry\n\ntheorem valid'.balance'_aux {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H\u2081 : bit0 1 * size r \u2264 bit1 (bit0 (bit0 1)) * size l + bit1 (bit0 1) \u2228 size r \u2264 bit1 1) (H\u2082 : bit0 1 * size l \u2264 bit1 (bit0 (bit0 1)) * size r + bit1 (bit0 1) \u2228 size l \u2264 bit1 1) : valid' o\u2081 (balance' l x r) o\u2082 := sorry\n\ntheorem valid'.balance'_lemma {\u03b1 : Type u_1} {l : ordnode \u03b1} {l' : \u2115} {r : ordnode \u03b1} {r' : \u2115} (H1 : balanced_sz l' r') (H2 : nat.dist (size l) l' \u2264 1 \u2227 size r = r' \u2228 nat.dist (size r) r' \u2264 1 \u2227 size l = l') : bit0 1 * size r \u2264 bit1 (bit0 (bit0 1)) * size l + bit1 (bit0 1) \u2228 size r \u2264 bit1 1 := sorry\n\ntheorem valid'.balance' {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : \u2203 (l' : \u2115),\n  \u2203 (r' : \u2115), balanced_sz l' r' \u2227 (nat.dist (size l) l' \u2264 1 \u2227 size r = r' \u2228 nat.dist (size r) r' \u2264 1 \u2227 size l = l')) : valid' o\u2081 (balance' l x r) o\u2082 := sorry\n\ntheorem valid'.balance {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : \u2203 (l' : \u2115),\n  \u2203 (r' : \u2115), balanced_sz l' r' \u2227 (nat.dist (size l) l' \u2264 1 \u2227 size r = r' \u2228 nat.dist (size r) r' \u2264 1 \u2227 size l = l')) : valid' o\u2081 (balance l x r) o\u2082 := sorry\n\ntheorem valid'.balance_l_aux {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H\u2081 : size l = 0 \u2192 size r \u2264 1) (H\u2082 : 1 \u2264 size l \u2192 1 \u2264 size r \u2192 size r \u2264 delta * size l) (H\u2083 : bit0 1 * size l \u2264 bit1 (bit0 (bit0 1)) * size r + bit1 (bit0 1) \u2228 size l \u2264 bit1 1) : valid' o\u2081 (balance_l l x r) o\u2082 := sorry\n\ntheorem valid'.balance_l {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : (\u2203 (l' : \u2115), raised l' (size l) \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised (size r) r' \u2227 balanced_sz (size l) r') : valid' o\u2081 (balance_l l x r) o\u2082 := sorry\n\ntheorem valid'.balance_r_aux {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H\u2081 : size r = 0 \u2192 size l \u2264 1) (H\u2082 : 1 \u2264 size r \u2192 1 \u2264 size l \u2192 size l \u2264 delta * size r) (H\u2083 : bit0 1 * size r \u2264 bit1 (bit0 (bit0 1)) * size l + bit1 (bit0 1) \u2228 size r \u2264 bit1 1) : valid' o\u2081 (balance_r l x r) o\u2082 := sorry\n\ntheorem valid'.balance_r {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) (H : (\u2203 (l' : \u2115), raised (size l) l' \u2227 balanced_sz l' (size r)) \u2228 \u2203 (r' : \u2115), raised r' (size r) \u2227 balanced_sz (size l) r') : valid' o\u2081 (balance_r l x r) o\u2082 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (valid' o\u2081 (balance_r l x r) o\u2082)) (propext valid'.dual_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (valid' o\u2082 (dual (balance_r l x r)) o\u2081)) (dual_balance_r l x r)))\n      (valid'.balance_l (valid'.dual hr) (valid'.dual hl) (balance_sz_dual H)))\n\ntheorem valid'.erase_max_aux {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 (node s l x r) o\u2082) : valid' o\u2081 (erase_max (node' l x r)) \u2191(find_max' x r) \u2227 size (node' l x r) = size (erase_max (node' l x r)) + 1 := sorry\n\ntheorem valid'.erase_min_aux {\u03b1 : Type u_1} [preorder \u03b1] {s : \u2115} {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (H : valid' o\u2081 (node s l x r) o\u2082) : valid' (\u2191(find_min' l x)) (erase_min (node' l x r)) o\u2082 \u2227 size (node' l x r) = size (erase_min (node' l x r)) + 1 := sorry\n\ntheorem erase_min.valid {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} (h : valid t) : valid (erase_min t) := sorry\n\ntheorem erase_max.valid {\u03b1 : Type u_1} [preorder \u03b1] {t : ordnode \u03b1} (h : valid t) : valid (erase_max t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (valid (erase_max t))) (propext valid.dual_iff)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (valid (dual (erase_max t)))) (dual_erase_max t))) (erase_min.valid (valid.dual h)))\n\ntheorem valid'.glue_aux {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l o\u2082) (hr : valid' o\u2081 r o\u2082) (sep : all (fun (x : \u03b1) => all (fun (y : \u03b1) => x < y) r) l) (bal : balanced_sz (size l) (size r)) : valid' o\u2081 (glue l r) o\u2082 \u2227 size (glue l r) = size l + size r := sorry\n\ntheorem valid'.glue {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {x : \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l \u2191x) (hr : valid' (\u2191x) r o\u2082) : balanced_sz (size l) (size r) \u2192 valid' o\u2081 (glue l r) o\u2082 \u2227 size (glue l r) = size l + size r :=\n  valid'.glue_aux (valid'.trans_right hl (valid'.ord hr)) (valid'.trans_left (valid'.ord hl) hr)\n    (bounded.to_sep (valid'.ord hl) (valid'.ord hr))\n\ntheorem valid'.merge_lemma {a : \u2115} {b : \u2115} {c : \u2115} (h\u2081 : bit1 1 * a < b + c + 1) (h\u2082 : b \u2264 bit1 1 * c) : bit0 1 * (a + b) \u2264 bit1 (bit0 (bit0 1)) * c + bit1 (bit0 1) := sorry\n\ntheorem valid'.merge_aux\u2081 {\u03b1 : Type u_1} [preorder \u03b1] {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} {ls : \u2115} {ll : ordnode \u03b1} {lx : \u03b1} {lr : ordnode \u03b1} {rs : \u2115} {rl : ordnode \u03b1} {rx : \u03b1} {rr : ordnode \u03b1} {t : ordnode \u03b1} (hl : valid' o\u2081 (node ls ll lx lr) o\u2082) (hr : valid' o\u2081 (node rs rl rx rr) o\u2082) (h : delta * ls < rs) (v : valid' o\u2081 t \u2191rx) (e : size t = ls + size rl) : valid' o\u2081 (balance_l t rx rr) o\u2082 \u2227 size (balance_l t rx rr) = ls + rs := sorry\n\ntheorem valid'.merge_aux {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {r : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} (hl : valid' o\u2081 l o\u2082) (hr : valid' o\u2081 r o\u2082) (sep : all (fun (x : \u03b1) => all (fun (y : \u03b1) => x < y) r) l) : valid' o\u2081 (merge l r) o\u2082 \u2227 size (merge l r) = size l + size r := sorry\n\ntheorem valid.merge {\u03b1 : Type u_1} [preorder \u03b1] {l : ordnode \u03b1} {r : ordnode \u03b1} (hl : valid l) (hr : valid r) (sep : all (fun (x : \u03b1) => all (fun (y : \u03b1) => x < y) r) l) : valid (merge l r) :=\n  and.left (valid'.merge_aux hl hr sep)\n\ntheorem insert_with.valid_aux {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (f : \u03b1 \u2192 \u03b1) (x : \u03b1) (hf : \u2200 (y : \u03b1), x \u2264 y \u2227 y \u2264 x \u2192 x \u2264 f y \u2227 f y \u2264 x) {t : ordnode \u03b1} {o\u2081 : with_bot \u03b1} {o\u2082 : with_top \u03b1} : valid' o\u2081 t o\u2082 \u2192\n  bounded nil o\u2081 \u2191x \u2192\n    bounded nil (\u2191x) o\u2082 \u2192 valid' o\u2081 (insert_with f x t) o\u2082 \u2227 raised (size t) (size (insert_with f x t)) := sorry\n\ntheorem insert_with.valid {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (f : \u03b1 \u2192 \u03b1) (x : \u03b1) (hf : \u2200 (y : \u03b1), x \u2264 y \u2227 y \u2264 x \u2192 x \u2264 f y \u2227 f y \u2264 x) {t : ordnode \u03b1} (h : valid t) : valid (insert_with f x t) :=\n  and.left (insert_with.valid_aux (fun (y : \u03b1) => f y) x hf h True.intro True.intro)\n\ntheorem insert_eq_insert_with {\u03b1 : Type u_1} [preorder \u03b1] [DecidableRel LessEq] (x : \u03b1) (t : ordnode \u03b1) : ordnode.insert x t = insert_with (fun (_x : \u03b1) => x) x t := sorry\n\ntheorem insert.valid {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (x : \u03b1) {t : ordnode \u03b1} (h : valid t) : valid (ordnode.insert x t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (valid (ordnode.insert x t))) (insert_eq_insert_with x t)))\n    (insert_with.valid (fun (_x : \u03b1) => x) x\n      (fun (_x : \u03b1) (_x : x \u2264 _x \u2227 _x \u2264 x) => { left := le_refl x, right := le_refl x }) h)\n\ntheorem insert'_eq_insert_with {\u03b1 : Type u_1} [preorder \u03b1] [DecidableRel LessEq] (x : \u03b1) (t : ordnode \u03b1) : insert' x t = insert_with id x t := sorry\n\ntheorem insert'.valid {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (x : \u03b1) {t : ordnode \u03b1} (h : valid t) : valid (insert' x t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (valid (insert' x t))) (insert'_eq_insert_with x t)))\n    (insert_with.valid id x (fun (_x : \u03b1) => id) h)\n\nend ordnode\n\n\n/-- An `ordset \u03b1` is a finite set of values, represented as a tree. The operations on this type\nmaintain that the tree is balanced and correctly stores subtree sizes at each level. The\ncorrectness property of the tree is baked into the type, so all operations on this type are correct\nby construction. -/\ndef ordset (\u03b1 : Type u_1) [preorder \u03b1] :=\n  Subtype fun (t : ordnode \u03b1) => ordnode.valid t\n\nnamespace ordset\n\n\n/-- O(1). The empty set. -/\ndef nil {\u03b1 : Type u_1} [preorder \u03b1] : ordset \u03b1 :=\n  { val := ordnode.nil, property := sorry }\n\n/-- O(1). Get the size of the set. -/\ndef size {\u03b1 : Type u_1} [preorder \u03b1] (s : ordset \u03b1) : \u2115 :=\n  ordnode.size (subtype.val s)\n\n/-- O(1). Construct a singleton set containing value `a`. -/\nprotected def singleton {\u03b1 : Type u_1} [preorder \u03b1] (a : \u03b1) : ordset \u03b1 :=\n  { val := singleton a, property := ordnode.valid_singleton }\n\nprotected instance has_emptyc {\u03b1 : Type u_1} [preorder \u03b1] : has_emptyc (ordset \u03b1) :=\n  has_emptyc.mk nil\n\nprotected instance inhabited {\u03b1 : Type u_1} [preorder \u03b1] : Inhabited (ordset \u03b1) :=\n  { default := nil }\n\nprotected instance has_singleton {\u03b1 : Type u_1} [preorder \u03b1] : has_singleton \u03b1 (ordset \u03b1) :=\n  has_singleton.mk ordset.singleton\n\n/-- O(1). Is the set empty? -/\ndef empty {\u03b1 : Type u_1} [preorder \u03b1] (s : ordset \u03b1) :=\n  s = \u2205\n\ntheorem empty_iff {\u03b1 : Type u_1} [preorder \u03b1] {s : ordset \u03b1} : s = \u2205 \u2194 \u21a5(ordnode.empty (subtype.val s)) := sorry\n\nprotected instance empty.decidable_pred {\u03b1 : Type u_1} [preorder \u03b1] : decidable_pred empty :=\n  fun (s : ordset \u03b1) => decidable_of_iff' (\u21a5(ordnode.empty (subtype.val s))) empty_iff\n\n/-- O(log n). Insert an element into the set, preserving balance and the BST property.\n  If an equivalent element is already in the set, this replaces it. -/\nprotected def insert {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (x : \u03b1) (s : ordset \u03b1) : ordset \u03b1 :=\n  { val := ordnode.insert x (subtype.val s), property := sorry }\n\nprotected instance has_insert {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] : has_insert \u03b1 (ordset \u03b1) :=\n  has_insert.mk ordset.insert\n\n/-- O(log n). Insert an element into the set, preserving balance and the BST property.\n  If an equivalent element is already in the set, the set is returned as is. -/\ndef insert' {\u03b1 : Type u_1} [preorder \u03b1] [is_total \u03b1 LessEq] [DecidableRel LessEq] (x : \u03b1) (s : ordset \u03b1) : ordset \u03b1 :=\n  { val := ordnode.insert' x (subtype.val s), property := sorry }\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/ordmap/ordset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7185944046238981, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3817240833901484}}
{"text": "import analysis.p_series\nimport number_theory.arithmetic_function\nimport algebra.squarefree\nimport data.list.intervals\nimport tactic\nimport measure_theory.integral.interval_integral\nimport analysis.special_functions.integrals\nimport defs\nimport summability\nimport general\n\nnoncomputable theory\nopen nat finset list finsupp set function filter measure_theory\nopen_locale topological_space interval big_operators filter ennreal asymptotics\n\nnamespace squarefree_sums\n\nlemma mem_Icc_Ico\n{a b c : \u211d}\n(hc : c \u2208 set.Icc a b)\n(hc' : c \u2260 b) :\nc \u2208 set.Ico a b :=\nbegin\n  simp, simp at hc,\n  simp [hc.left],\n  exact lt_of_le_of_ne hc.right hc',\nend\n\nlemma blech\n{a m : \u2115}\n{f : \u211d \u2192 \u211d}\n(ham : a \u2264 m)\n(hf_nonneg : \u2200 (b : \u211d), b \u2208 set.Ici (a : \u211d) \u2192 0 \u2264 f b)\n:\n{x : \u211d | x \u2208 set.Icc (m : \u211d) (\u2191m + 1) \u2192 0 \u2264 f x} = univ\n:=\nbegin\n  simp,\n  apply eq_univ_of_forall,\n  intros x,\n  simp,\n  intros hx hx',\n  have : x \u2208 set.Ici (a : \u211d),\n    simp,\n    calc (a : \u211d) \u2264 \u2191m : cast_le.mpr ham ... \u2264 x : hx,\n  exact hf_nonneg x this,\nend\n\nlemma tail_sum_le_tail_integral\n{a : \u2115}\n{l : \u211d}\n{f : \u211d \u2192 \u211d}\n(hf : tendsto (\u03bb (b : \u2115), \u222b (x : \u211d) in a..b, f x) at_top (\ud835\udcdd l))\n(hf_mono : antitone_on f (set.Ici (a : \u211d)))\n(hf_nonneg : \u2200 (b : \u211d), b \u2208 set.Ici (a : \u211d) \u2192 0 \u2264 f b)\n:\n(\u2211' (i : \u2115), (\u03bb (j : \u2115), ite (a + 1 \u2264 j) (f \u2191j) 0) i) \u2264 l :=\nbegin\n  by_cases h : summable (\u03bb (j : \u2115), ite (a + 1 \u2264 j) (f \u2191j) 0),\n  obtain \u27e8c, hc\u27e9 := h,\n  rw has_sum.tsum_eq hc,\n  rw has_sum_iff_tendsto_nat_of_nonneg at hc,\n  simp at hf,\n  refine le_of_tendsto_of_tendsto hc hf _,\n  rw [filter.eventually_le, eventually_at_top],\n  use a + 100,\n  intros n hn,\n  rw sum_ite,\n  simp,\n  have : filter (has_le.le (a + 1)) (finset.range n) = finset.Ico (a + 1) n,\n  {\n    ext d,\n    rw finset.mem_filter,\n    simp,\n    conv {to_lhs, rw and_comm},\n  },\n  rw this,\n  obtain \u27e8m, hm\u27e9 : \u2203m, n = m + 1, {\n    use n - 1,\n    exact (nat.sub_add_cancel (calc 1 \u2264 100 : by linarith ... \u2264 a + 100 : by linarith ... \u2264 n : hn)).symm,\n  },\n  have : a \u2264 m, {\n    have : a + 1 \u2264 m + 1, {\n      rw \u2190 hm,\n      calc a + 1 \u2264 a + 100 : by linarith ... \u2264 n : hn,\n    },\n    linarith,\n  },\n  rw hm,\n  transitivity,\n  refine antitone_sum_le_integral this _,\n  intros x hx y hy hxy,\n  exact hf_mono (mem_Icc_mem_Ici hx) (mem_Icc_mem_Ici hy) hxy,\n\n  have hf_mono_local: antitone_on f [(a : \u211d), \u2191m + 1], {\n    have : (n : \u211d) = (m : \u211d) + 1, simp [hm],\n    rw \u2190 this,\n    rw interval_eq_Icc (cast_le.mpr (calc a \u2264 a + 100 : by linarith ... \u2264 n : hn)),\n    intros x hx y hy hxy,\n    exact hf_mono (mem_Icc_mem_Ici hx) (mem_Icc_mem_Ici hy) hxy,\n  },\n  have uu: interval_integrable f real.measure_space.volume \u2191a (\u2191m + 1), {\n    exact antitone_on.interval_integrable hf_mono_local,\n  },\n\n  have hf_mono_local: antitone_on f [(a : \u211d), \u2191m], {\n    rw interval_eq_Icc (cast_le.mpr this),\n    intros x hx y hy hxy,\n    exact hf_mono (mem_Icc_mem_Ici hx) (mem_Icc_mem_Ici hy) hxy,\n  },\n  have ul: interval_integrable f real.measure_space.volume \u2191a \u2191m, {\n    exact antitone_on.interval_integrable hf_mono_local,\n  },\n\n  have hf_mono_local: antitone_on f [(m : \u211d), \u2191m + 1], {\n    rw interval_eq_Icc (calc (m : \u211d) \u2264 \u2191m + 1 : by simp),\n    intros x hx y hy hxy,\n    have ut : (m : \u211d) + 1 = \u2191(m + 1), simp,\n    rw ut at hx,\n    rw ut at hy,\n    exact hf_mono (mem_Icc_mem_Ici' hx this) (mem_Icc_mem_Ici' hy this) hxy,\n  },\n  have ur: interval_integrable f real.measure_space.volume \u2191m (\u2191m + 1), {\n    exact antitone_on.interval_integrable hf_mono_local,\n  },\n\n  have aa : interval_integral f \u2191a \u2191(m + 1) real.measure_space.volume = interval_integral f \u2191a \u2191m real.measure_space.volume + interval_integral f \u2191m \u2191(m + 1) real.measure_space.volume, {\n    symmetry,\n    refine interval_integral.integral_add_adjacent_intervals ul ur,\n  },\n  rw aa,\n  simp,\n  apply interval_integral.integral_nonneg_of_ae_restrict,\n  simp,\n  unfold filter.eventually_le,\n  simp,\n  rw filter.eventually_inf_principal,\n  rw filter.eventually_iff,\n  rw blech this hf_nonneg,\n  simp,\n  intros i,\n  by_cases hi : a + 1 \u2264 i,\n  simp [hi],\n  refine hf_nonneg i _,\n  simp,\n  calc a \u2264 a + 1 : le_succ a ... \u2264 i : hi,\n  simp [hi],\n\n  -- Now to the not summable case\n  rw not_summable_eq_zero h,\n  refine le_of_tendsto_of_tendsto (tendsto_const_nhds) hf _,\n  rw [filter.eventually_le, eventually_at_top],\n  use a + 1,\n  intros x hx,\n  apply interval_integral.integral_nonneg,\n  norm_cast, linarith,\n  intros u hu,\n  exact hf_nonneg u (mem_Icc_mem_Ici hu),\nend\n\ntheorem integral_tendsto_of_has_deriv_at {a b : \u211d} {f f' : \u211d \u2192 \u211d}\n  (hderiv : \u2200 x \u2208 Ici a, has_deriv_at f (f' x) x)\n  (hvanish : tendsto f at_top (\ud835\udcdd b))\n  (hint : \u2200 (b : \u211d), b \u2208 Ici a \u2192 interval_integrable f' volume a b) :\n  tendsto (\u03bb (b : \u211d), \u222b y in a..b, f' y) at_top (\ud835\udcdd (b - f a)) :=\nbegin\n  have hev : (\u03bb (x : \u211d), f x - f a) =\u1da0[at_top] (\u03bb (b : \u211d), \u222b y in a..b, f' y),\n  { rw [eventually_eq, eventually_at_top],\n    use a,\n    intros b hb,\n    have hderiv' : \u2200 x \u2208 [a, b], has_deriv_at f (f' x) x,\n    { intros x hx,\n      exact hderiv x (calc a = min a b : (min_eq_left hb.le).symm ... \u2264 x : hx.left), },\n    rw interval_integral.integral_eq_sub_of_has_deriv_at hderiv' (hint b hb.le), },\n  exact tendsto.congr' hev (filter.tendsto.sub_const (f a) hvanish),\nend\n\nlemma integral_rpow_tendsto_at_top (a r : \u211d) (ha : 0 < a) (hr : r < -1) :\ntendsto\n(\u03bb (y : \u211d), \u222b (x : \u211d) in a..y, x ^ r)\nat_top\n(\ud835\udcdd (-a ^ (r + 1) / (r + 1)))\n:=\nbegin\n  have : (\u03bb (y : \u211d), \u222b (x : \u211d) in a..y, x ^ r) =\u1da0[at_top] (\u03bb (y : \u211d), (y ^ (r + 1) / (r + 1)) - (a ^ (r + 1) / (r + 1))),\n  { rw [eventually_eq, eventually_at_top],\n    refine \u27e820, (\u03bb b hb, _)\u27e9,\n    rw integral_rpow,\n    { ring, },\n    { right,\n      split,\n      { linarith [hr], },\n      { exact not_mem_interval_of_lt ha (by linarith [hb.le]), }, }, },\n  rw tendsto_congr' this,\n  have : -a ^ (r + 1) / (r + 1) = 0 - (a ^ (r + 1) / (r + 1)), { ring, },\n  rw this,\n  apply tendsto.sub_const,\n  rw \u2190 zero_div (r + 1),\n  apply tendsto.div_const,\n  have hinf : tendsto (\u03bb (k : \u211d), k ^ -(r + 1)) at_top at_top,\n  { apply tendsto_rpow_at_top,\n    linarith [hr], },\n  have hev : (\u03bb (k : \u211d), k ^ -(r + 1)) =\u1da0[at_top] (\u03bb (k : \u211d), (k ^ (r + 1))\u207b\u00b9),\n  { rw [eventually_eq, eventually_at_top],\n    use 0,\n    intros b hb,\n    rw [\u2190real.inv_rpow hb.le, real.rpow_neg hb.le, \u2190real.inv_rpow hb.le], },\n  refine tendsto.congr _ (tendsto_inv_at_top_zero.comp (tendsto.congr' hev hinf)),\n  intros x,\n  simp only [comp_app, inv_inv],\nend\n\nend squarefree_sums\n", "meta": {"author": "khwilson", "repo": "squarefree_asymptotics", "sha": "b44adacc9ab77d48af7905ca33b83fc330857ac6", "save_path": "github-repos/lean/khwilson-squarefree_asymptotics", "path": "github-repos/lean/khwilson-squarefree_asymptotics/squarefree_asymptotics-b44adacc9ab77d48af7905ca33b83fc330857ac6/src/integral_facts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3817240769874595}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. \n  assume A : set (euclidean_space \u211d (fin n)),\n  assume h1 : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. \n  have h2 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_refinement A C \u2227 is_locally_finite C \u2227 is_open_cover C, from by auto [paracompact_space.paracompact_iff_locally_finite_open_refinement],\n  -- First, we define a collection of pen balls. \n  have h3 : \u2203 B0 : set (euclidean_space \u211d (fin n)), is_open B0, from by auto [is_open_univ],\n  -- Let $B_0 = \\phi$, \n  have h4 : \u2203 B0 : set (euclidean_space \u211d (fin n)), is_open B0 \u2227 B0 = \u2205, from by auto [is_open_univ],\n  -- and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0. \n  have h5 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm, from by auto [is_open_ball],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, \n  have h6 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_compact Bm, from by auto [is_open_ball, compact_iff_closed_inter_compact],\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ \n  have h7 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_compact Bm \u2227 is_finite (A \u2229 Bm), from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, \n  have h8 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_compact Bm \u2227 is_finite (A \u2229 Bm) \u2227 (A \u2229 Bm) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bm - 1))), from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). \n  have h9 : \u2200 m : \u2115, \u2203 Cm : set (euclidean_space \u211d (fin n)), is_open Cm \u2227 is_open_refinement A Cm \u2227 is_finite Cm, from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. \n  have h10 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open C \u2227 is_open_refinement A C \u2227 is_finite C, from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, \n  have h11 : \u2200 x : euclidean_space \u211d (fin n), \u2203 Bx : set (euclidean_space \u211d (fin n)), is_open Bx \u2227 is_compact Bx \u2227 is_finite (A \u2229 Bx) \u2227 (A \u2229 Bx) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bx - 1))), from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), \n  have h12 : \u2200 x : euclidean_space \u211d (fin n), \u2203 Bx : set (euclidean_space \u211d (fin n)), is_open Bx \u2227 is_compact Bx \u2227 is_finite (A \u2229 Bx) \u2227 (A \u2229 Bx) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bx - 1))) \u2227 \u2203 m : \u2115, x \u2208 Bm, from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- and so $x$ is an element of $\\mathcal{C}_m$. \n  have h13 : \u2200 x : euclidean_space \u211d (fin n), \u2203 Bx : set (euclidean_space \u211d (fin n)), is_open Bx \u2227 is_compact Bx \u2227 is_finite (A \u2229 Bx) \u2227 (A \u2229 Bx) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bx - 1))) \u2227 \u2203 m : \u2115, x \u2208 Bm \u2227 x \u2208 Cm, from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, \n  have h14 : \u2200 x : euclidean_space \u211d (fin n), \u2203 Bx : set (euclidean_space \u211d (fin n)), is_open Bx \u2227 is_compact Bx \u2227 is_finite (A \u2229 Bx) \u2227 (A \u2229 Bx) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bx - 1))) \u2227 \u2203 m : \u2115, x \u2208 Bm \u2227 x \u2208 Cm \u2227 is_locally_finite Bx, from by auto [is_open_ball, compact_iff_closed_inter_compact, compact_iff_finite_open_cover, h1],\n  -- neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, \n  have h15 : \u2200 x : euclidean_space \u211d (fin n), \u2203 Bx : set (euclidean_space \u211d (fin n)), is_open Bx \u2227 is_compact Bx \u2227 is_finite (A \u2229 Bx) \u2227 (A \u2229 Bx) \u2286 (A \u2229 (euclidean_space \u211d (fin n) \\ (Bx - 1))) \u2227 \u2203 m : \u2115, x \u2208 Bm \u2227 x \u2208 Cm \u2227 is_loc\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  have h1 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_cover C \u2227 is_locally_finite_open_refinement A C, from by auto using [use A],\n  -- First, we define a collection of pen balls.\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  have h2 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_ball Bm (0 : euclidean_space \u211d (fin n)) m, from by auto using [use {x | \u2203 i : fin n, x i \u2264 m}],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem,\n  have h3 : \u2200 m : \u2115, is_compact {x | \u2203 i : fin n, x i \u2264 m}, from by auto using [compact_iff_finite_open_cover, use {x | \u2203 i : fin n, x i \u2264 m}],\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  have h4 : \u2200 m : \u2115, \u2203 Cm : set (euclidean_space \u211d (fin n)), is_open_cover Cm \u2227 (\u2200 A \u2208 Cm, \u2203 B \u2208 A, (\u2200 x, x \u2208 B \u2192 x \u2208 {x | \u2203 i : fin n, x i \u2264 m}) \u2227 (\u2200 x, x \u2208 B \u2192 x \u2209 {x | \u2203 i : fin n, x i \u2264 (m-1)})), from by auto using [use {x | \u2203 i : fin n, x i \u2264 m}],\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  have h5 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_cover C \u2227 is_open_refinement A C, from by auto using [use {x | \u2203 i : fin n, x i \u2264 m}],\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$.\n  have h6 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, x \u2208 {x | \u2203 i : fin n, x i \u2264 m}, from by auto using [exists_nat_gt],\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$.\n  have h7 : \u2200 x : euclidean_space \u211d (fin n), is_locally_finite_open_refinement A {x | \u2203 i : fin n, x i \u2264 m}, from by auto using [use {x | \u2203 i : fin n, x i \u2264 m}],\n  -- So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  show paracompact_space (euclidean_space \u211d (fin n)), from by auto using [use {x | \u2203 i : fin n, x i \u2264 m}],\nend\n\n--OUTPUT 3\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n  -- we now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$\n  let C := {c : set (euclidean_space \u211d (fin n)) | \u2203 a : set (euclidean_space \u211d (fin n)), a \u2208 A \u2227 c \u2286 a},\n  let hC : is_open_refinement C A := by auto [is_open_refinement],\n  have hC1 : is_open_cover C, from by auto [is_open_cover],\n  have hC2 : is_locally_finite C, from by auto [is_locally_finite],\n  have hC3 : is_cover C, from by auto [is_cover],\n  -- so $\\mathcal{C}$ is an open refinement of $\\mathcal{A}$\n  show \u2203 C : set (euclidean_space \u211d (fin n)), is_open_refinement C A \u2227 is_locally_finite C \u2227 is_cover C, from by auto [hC, hC1, hC2, hC3],\nend\n\n--OUTPUT 4\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  have h1 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_cover C \u2227 is_locally_finite C \u2227 is_open_refinement A C \u2227 covers C, from by auto [paracompact_space.paracompact_iff_locally_finite_open_refinement],\n  -- First, we define a collection of pen balls.\n  have h2 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_ball Bm m \u2227 is_compact (closure Bm), from by auto [euclidean_space.is_open_ball, euclidean_space.is_compact_closure],\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  have h3 : \u2200 m : \u2115, \u2203 Bm : set (euclidean_space \u211d (fin n)), is_open Bm \u2227 is_ball Bm m \u2227 is_compact (closure Bm) \u2227 Bm \u2286 (\u22c3 (A \u2208 A), A), from by auto [hA, h2],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  have h4 : \u2200 m : \u2115, \u2203 Cm : set (euclidean_space \u211d (fin n)), is_open Cm \u2227 is_open_cover (Cm \u2229 (\u22c3 (A \u2208 A), A)) \u2227 is_open_refinement Cm A \u2227 Cm \u2286 (\u22c3 (A \u2208 A), A), from by auto [hA, h3],\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  have h5 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_refinement C A \u2227 C \u2286 (\u22c3 (A \u2208 A), A), from by auto [h4],\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$.\n  have h6 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_refinement C A \u2227 C \u2286 (\u22c3 (A \u2208 A), A) \u2227 covers C, from by auto [hA, h5],\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$.\n  have h7 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_refinement C A \u2227 is_locally_finite C \u2227 C \u2286 (\u22c3 (A \u2208 A), A) \u2227 covers C, from by auto [hA, h6],\n  -- So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  show paracompact_space (euclidean_space \u211d (fin n)), from by auto [paracompact_space.paracompact_iff_locally_finite_open_refinement, h7],\nend\n\n--OUTPUT 5\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. \n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  let C := {U : set (euclidean_space \u211d (fin n)) | \u2203 A \u2208 A, \u2203 V : set (euclidean_space \u211d (fin n)), A = V \u2227 is_open V \u2227 U \u2286 V},\n  have hC : is_open_cover C, from by auto [is_open_cover_iff],\n\n  -- First, we define a collection of pen balls. \n  let B0 : set (euclidean_space \u211d (fin n)) := \u2205,\n  have hB0 : is_open B0, from by auto [is_open_empty],\n  have hB0_1 : B0 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_empty],\n\n  let B1 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B0},\n  have hB1 : is_open B1, from by auto [is_open_ball],\n  have hB1_1 : B1 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B2 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B1},\n  have hB2 : is_open B2, from by auto [is_open_ball],\n  have hB2_1 : B2 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B3 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B2},\n  have hB3 : is_open B3, from by auto [is_open_ball],\n  have hB3_1 : B3 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B4 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B3},\n  have hB4 : is_open B4, from by auto [is_open_ball],\n  have hB4_1 : B4 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B5 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B4},\n  have hB5 : is_open B5, from by auto [is_open_ball],\n  have hB5_1 : B5 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B6 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B5},\n  have hB6 : is_open B6, from by auto [is_open_ball],\n  have hB6_1 : B6 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B7 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B6},\n  have hB7 : is_open B7, from by auto [is_open_ball],\n  have hB7_1 : B7 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B8 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B7},\n  have hB8 : is_open B8, from by auto [is_open_ball],\n  have hB8_1 : B8 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B9 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B8},\n  have hB9 : is_open B9, from by auto [is_open_ball],\n  have hB9_1 : B9 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B10 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B9},\n  have hB10 : is_open B10, from by auto [is_open_ball],\n  have hB10_1 : B10 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B11 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : euclidean_space \u211d (fin n)), \u2225x - y\u2225 < m \u2192 y \u2208 B10},\n  have hB11 : is_open B11, from by auto [is_open_ball],\n  have hB11_1 : B11 \u2286 (euclidean_space \u211d (fin n)), from by auto [set.subset_univ],\n\n  let B12 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2203 (m : \u211d), m > 0 \u2227 \u2200 (y : e\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$\n  -- First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  let B0 : set (euclidean_space \u211d (fin n)) := {x : euclidean_space \u211d (fin n) | \u2200 i : fin n, x.val i = 0},\n  have hB0 : is_open B0, from by auto [set.is_open_empty],\n  let Bm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, {x : euclidean_space \u211d (fin n) | \u2200 i : fin n, abs (x.val i) \u2264 m},\n  have hBm : \u2200 m : \u2115, is_open (Bm m), from by auto [set.is_open_ball],\n  have hBm_subset : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto [set.subset_iff, abs_le_add],\n\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  let Cm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, (\u22c2 (a : set (euclidean_space \u211d (fin n))) (h : a \u2208 A) (h2 : a \u2229 (Bm m) \u2260 \u2205), a) \u2229 (euclidean_space \u211d (fin n) \\ Bm (m-1)),\n  have hCm : \u2200 m : \u2115, is_open (Cm m), from by auto [set.is_open_inter, set.is_open_diff],\n  have hCm_subset : \u2200 m : \u2115, Cm m \u2286 Bm m, from by auto [set.subset_iff, abs_le_add],\n  have hCm_cover : \u2200 m : \u2115, Cm m \u2260 \u2205, from by auto [set.inter_nonempty_iff, set.diff_nonempty_iff, hA, hBm_subset],\n\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  let C : set (euclidean_space \u211d (fin n)) := \u22c3 (m : \u2115), Cm m,\n  have hC : is_open C, from by auto [set.is_open_union],\n  have hC_cover : C \u2260 \u2205, from by auto [set.nonempty_union],\n\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$.\n  have hC_cover_Rn : \u2200 x : euclidean_space \u211d (fin n), x \u2208 C, from by auto [set.mem_union, set.mem_inter, set.mem_diff, set.mem_inter_iff, set.mem_diff_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_inter_iff, set.mem_\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7401743505760728, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.38164863626437256}}
{"text": "import for_mathlib.truncation\nimport for_mathlib.Ext_quasi_iso\n\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.limits\n\nnamespace cochain_complex\nopen bounded_homotopy_category\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0] [enough_projectives \ud835\udcd0]\nvariables (C : cochain_complex \ud835\udcd0 \u2124)\n\nlemma Ext_\u03b9_succ_five_term_exact_seq (B : bounded_homotopy_category \ud835\udcd0) (i j : \u2124) :\n  let E := \u03bb n, ((Ext n).flip.obj B) in\n  exact_seq Ab.{v} $\n    [\n      (E j).map (bounded_homotopy_category.of_hom (truncation.to_imker C (i+1))).op\n    , (E j).map (bounded_homotopy_category.of_hom (truncation.\u03b9_succ C i)).op\n    , Ext_\u03b4 _ _ j B (truncation.short_exact_\u03b9_succ_to_imker C i)\n    , (E (j+1)).map (bounded_homotopy_category.of_hom (truncation.to_imker C (i+1))).op ] :=\nExt_five_term_exact_seq' _ _ j B (truncation.short_exact_\u03b9_succ_to_imker C i)\n\nend cochain_complex\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/truncation_Ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8418256432832333, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.3815674467812995}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-!\n# Extra definitions on `option`\n\nThis file defines more operations involving `option \u03b1`. Lemmas about them are located in other\nfiles under `data.option.`.\nOther basic operations on `option` are defined in the core library.\n-/\n\n\nnamespace option\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\nattribute [inline] option.is_some option.is_none\n\n/-- An elimination principle for `option`. It is a nondependent version of `option.rec_on`. -/\n@[simp] protected def elim : option \u03b1 \u2192 \u03b2 \u2192 (\u03b1 \u2192 \u03b2) \u2192 \u03b2\n| (some x) y f := f x\n| none     y f := y\n\ninstance has_mem : has_mem \u03b1 (option \u03b1) := \u27e8\u03bb a b, b = some a\u27e9\n\n@[simp] theorem mem_def {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = some a :=\niff.rfl\n\nlemma mem_iff {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = a := iff.rfl\n\ntheorem is_none_iff_eq_none {o : option \u03b1} : o.is_none = tt \u2194 o = none :=\n\u27e8option.eq_none_of_is_none, \u03bb e, e.symm \u25b8 rfl\u27e9\n\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n\nlemma mem_some_iff {\u03b1 : Type*} {a b : \u03b1} : a \u2208 some b \u2194 b = a :=\nby simp\n\n/--\n`o = none` is decidable even if the wrapped type does not have decidable equality.\n\nThis is not an instance because it is not definitionally equal to `option.decidable_eq`.\nTry to use `o.is_none` or `o.is_some` instead.\n-/\n@[inline]\ndef decidable_eq_none {o : option \u03b1} : decidable (o = none) :=\ndecidable_of_decidable_of_iff (bool.decidable_eq _ _) is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2200 a \u2208 o, p a)\n| none     := is_true (by simp [false_implies_iff])\n| (some a) := if h : p a\n  then is_true $ \u03bb o e, some_inj.1 e \u25b8 h\n  else is_false $ mt (\u03bb H, H _ rfl) h\n\ninstance decidable_exists_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2203 a \u2208 o, p a)\n| none     := is_false (\u03bb \u27e8a, \u27e8h, _\u27e9\u27e9, by cases h)\n| (some a) := if h : p a\n  then is_true $ \u27e8_, rfl, h\u27e9\n  else is_false $ \u03bb \u27e8_, \u27e8rfl, hn\u27e9\u27e9, h hn\n\n/-- Inhabited `get` function. Returns `a` if the input is `some a`, otherwise returns `default`. -/\n@[reducible] def iget [inhabited \u03b1] : option \u03b1 \u2192 \u03b1\n| (some x) := x\n| none     := default\n\n@[simp] theorem iget_some [inhabited \u03b1] {a : \u03b1} : (some a).iget = a := rfl\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [decidable_pred p] (a : \u03b1) : option \u03b1 :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a` and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (o : option \u03b1) : option \u03b1 :=\no.bind (guard p)\n\n/-- Cast of `option` to `list `. Returns `[a]` if the input is `some a`, and `[]` if it is\n`none`. -/\ndef to_list : option \u03b1 \u2192 list \u03b1\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : \u03b1} {o : option \u03b1} : a \u2208 to_list o \u2194 a \u2208 o :=\nby cases o; simp [to_list, eq_comm]\n\n/-- Two arguments failsafe function. Returns `f a b` if the inputs are `some a` and `some b`, and\n\"does nothing\" otherwise. -/\ndef lift_or_get (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : option \u03b1 \u2192 option \u03b1 \u2192 option \u03b1\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_commutative \u03b1 f] :\n  is_commutative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b, by cases a; cases b; simp [lift_or_get, h.comm]\u27e9\n\ninstance lift_or_get_assoc (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_associative \u03b1 f] :\n  is_associative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]\u27e9\n\ninstance lift_or_get_idem (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_idempotent \u03b1 f] :\n  is_idempotent (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get, h.idempotent]\u27e9\n\ninstance lift_or_get_is_left_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_left_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninstance lift_or_get_is_right_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_right_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\n/-- Lifts a relation `\u03b1 \u2192 \u03b2 \u2192 Prop` to a relation `option \u03b1 \u2192 option \u03b2 \u2192 Prop` by just adding\n`none ~ none`. -/\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : option \u03b1 \u2192 option \u03b2 \u2192 Prop\n/-- If `a ~ b`, then `some a ~ some b` -/\n| some {a b} : r a b \u2192 rel (some a) (some b)\n/-- `none ~ none` -/\n| none       : rel none none\n\n/-- Partial bind. If for some `x : option \u03b1`, `f : \u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2` is a\n  partial function defined on `a : \u03b1` giving an `option \u03b2`, where `some a = x`,\n  then `pbind x f h` is essentially the same as `bind x f`\n  but is defined only when all `x = some a`, using the proof to apply `f`. -/\n@[simp] def pbind : \u03a0 (x : option \u03b1), (\u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2) \u2192 option \u03b2\n| none     _ := none\n| (some a) f := f a rfl\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on `a : \u03b1` satisfying `p`,\nthen `pmap f x h` is essentially the same as `map f x` but is defined only when all members of `x`\nsatisfy `p`, using the proof to apply `f`. -/\n@[simp] def pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 (a : \u03b1), p a \u2192 \u03b2) :\n  \u03a0 x : option \u03b1, (\u2200 a \u2208 x, p a) \u2192 option \u03b2\n| none     _ := none\n| (some a) H := some (f a (H a (mem_def.mpr rfl)))\n\n/-- Flatten an `option` of `option`, a specialization of `mjoin`. -/\n@[simp] def join : option (option \u03b1) \u2192 option \u03b1 :=\n\u03bb x, bind x id\n\nprotected def {u v} traverse {F : Type u \u2192 Type v} [applicative F] {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 F \u03b2) :\n  option \u03b1 \u2192 F (option \u03b2)\n| none     := pure none\n| (some x) := some <$> f x\n\n/- By analogy with `monad.sequence` in `init/category/combinators.lean`. -/\n\n/-- If you maybe have a monadic computation in a `[monad m]` which produces a term of type `\u03b1`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult. -/\ndef {u v} maybe {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type u} : option (m \u03b1) \u2192 m (option \u03b1)\n| none      := return none\n| (some fn) := some <$> fn\n\n/-- Map a monadic function `f : \u03b1 \u2192 m \u03b2` over an `o : option \u03b1`, maybe producing a result. -/\ndef {u v w} mmap {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type w} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2)\n  (o : option \u03b1) : m (option \u03b2) := (o.map f).maybe\n\n/-- A monadic analogue of `option.elim`. -/\ndef melim {\u03b1 \u03b2 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b2) (z : \u03b1 \u2192 m \u03b2) :\n  m \u03b2 :=\nx >>= \u03bb o, option.elim o y z\n\n/-- A monadic analogue of `option.get_or_else`. -/\ndef mget_or_else {\u03b1 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b1) : m \u03b1 :=\nmelim x y pure\n\nend option\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/option/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.6477982043529715, "lm_q1q2_score": 0.3814813308222468}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.polynomial.monic\nimport Mathlib.algebra.group_ring_action\nimport Mathlib.algebra.group_action_hom\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Group action on rings applied to polynomials\n\nThis file contains instances and definitions relating `mul_semiring_action` to `polynomial`.\n-/\n\nnamespace polynomial\n\n\nprotected instance mul_semiring_action (M : Type u_1) [monoid M] (R : Type u_2) [semiring R] [mul_semiring_action M R] : mul_semiring_action M (polynomial R) :=\n  mul_semiring_action.mk sorry sorry\n\nprotected instance faithful_mul_semiring_action (M : Type u_1) [monoid M] (R : Type u_2) [semiring R] [faithful_mul_semiring_action M R] : faithful_mul_semiring_action M (polynomial R) :=\n  faithful_mul_semiring_action.mk sorry\n\n@[simp] theorem coeff_smul' {M : Type u_1} [monoid M] {R : Type u_2} [semiring R] [mul_semiring_action M R] (m : M) (p : polynomial R) (n : \u2115) : coeff (m \u2022 p) n = m \u2022 coeff p n :=\n  coeff_map (mul_semiring_action.to_semiring_hom M R m) n\n\n@[simp] theorem smul_C {M : Type u_1} [monoid M] {R : Type u_2} [semiring R] [mul_semiring_action M R] (m : M) (r : R) : m \u2022 coe_fn C r = coe_fn C (m \u2022 r) :=\n  map_C (mul_semiring_action.to_semiring_hom M R m)\n\n@[simp] theorem smul_X {M : Type u_1} [monoid M] {R : Type u_2} [semiring R] [mul_semiring_action M R] (m : M) : m \u2022 X = X :=\n  map_X (mul_semiring_action.to_semiring_hom M R m)\n\ntheorem smul_eval_smul {M : Type u_1} [monoid M] (S : Type u_3) [comm_semiring S] [mul_semiring_action M S] (m : M) (f : polynomial S) (x : S) : eval (m \u2022 x) (m \u2022 f) = m \u2022 eval x f := sorry\n\ntheorem eval_smul' (S : Type u_3) [comm_semiring S] (G : Type u_4) [group G] [mul_semiring_action G S] (g : G) (f : polynomial S) (x : S) : eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (eval (g \u2022 x) f = g \u2022 eval x (g\u207b\u00b9 \u2022 f))) (Eq.symm (smul_eval_smul S g (g\u207b\u00b9 \u2022 f) x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (eval (g \u2022 x) f = eval (g \u2022 x) (g \u2022 g\u207b\u00b9 \u2022 f))) (smul_inv_smul g f)))\n      (Eq.refl (eval (g \u2022 x) f)))\n\ntheorem smul_eval (S : Type u_3) [comm_semiring S] (G : Type u_4) [group G] [mul_semiring_action G S] (g : G) (f : polynomial S) (x : S) : eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (eval x (g \u2022 f) = g \u2022 eval (g\u207b\u00b9 \u2022 x) f)) (Eq.symm (smul_eval_smul S g f (g\u207b\u00b9 \u2022 x)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (eval x (g \u2022 f) = eval (g \u2022 g\u207b\u00b9 \u2022 x) (g \u2022 f))) (smul_inv_smul g x)))\n      (Eq.refl (eval x (g \u2022 f))))\n\nend polynomial\n\n\n/-- the product of `(X - g \u2022 x)` over distinct `g \u2022 x`. -/\ndef prod_X_sub_smul (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R] [mul_semiring_action G R] (x : R) : polynomial R :=\n  finset.prod finset.univ\n    fun (g : quotient_group.quotient (mul_action.stabilizer G x)) =>\n      polynomial.X - coe_fn polynomial.C (mul_action.of_quotient_stabilizer G x g)\n\ntheorem prod_X_sub_smul.monic (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R] [mul_semiring_action G R] (x : R) : polynomial.monic (prod_X_sub_smul G R x) := sorry\n\ntheorem prod_X_sub_smul.eval (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R] [mul_semiring_action G R] (x : R) : polynomial.eval x (prod_X_sub_smul G R x) = 0 := sorry\n\ntheorem prod_X_sub_smul.smul (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R] [mul_semiring_action G R] (x : R) (g : G) : g \u2022 prod_X_sub_smul G R x = prod_X_sub_smul G R x := sorry\n\ntheorem prod_X_sub_smul.coeff (G : Type u_2) [group G] [fintype G] (R : Type u_3) [comm_ring R] [mul_semiring_action G R] (x : R) (g : G) (n : \u2115) : g \u2022 polynomial.coeff (prod_X_sub_smul G R x) n = polynomial.coeff (prod_X_sub_smul G R x) n := sorry\n\nnamespace mul_semiring_action_hom\n\n\n/-- An equivariant map induces an equivariant map on polynomials. -/\nprotected def polynomial {M : Type u_1} [monoid M] {P : Type u_2} [comm_semiring P] [mul_semiring_action M P] {Q : Type u_3} [comm_semiring Q] [mul_semiring_action M Q] (g : mul_semiring_action_hom M P Q) : mul_semiring_action_hom M (polynomial P) (polynomial Q) :=\n  mk (polynomial.map \u2191g) sorry sorry sorry sorry sorry\n\n@[simp] theorem coe_polynomial {M : Type u_1} [monoid M] {P : Type u_2} [comm_semiring P] [mul_semiring_action M P] {Q : Type u_3} [comm_semiring Q] [mul_semiring_action M Q] (g : mul_semiring_action_hom M P Q) : \u21d1(mul_semiring_action_hom.polynomial g) = polynomial.map \u2191g :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/polynomial/group_ring_action.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.38148132948276886}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport data.set.basic\nimport tactic.monotonicity.basic\n\n/-!\n# Typeclass for types with a set-like extensionality property\n\nThe `has_mem` typeclass is used to let terms of a type have elements.\nMany instances of `has_mem` have a set-like extensionality property:\nthings are equal iff they have the same elements.  The `set_like`\ntypeclass provides a unified interface to define a `has_mem` that is\nextensional in this way.\n\nThe main use of `set_like` is for algebraic subobjects (such as\n`submonoid` and `submodule`), whose non-proof data consists only of a\ncarrier set.  In such a situation, the projection to the carrier set\nis injective.\n\nIn general, a type `A` is `set_like` with elements of type `B` if it\nhas an injective map to `set B`.  This module provides standard\nboilerplate for every `set_like`: a `coe_sort`, a `coe` to set, a\n`partial_order`, and various extensionality and simp lemmas.\n\nA typical subobject should be declared as:\n```\nstructure my_subobject (X : Type*) :=\n(carrier : set X)\n(op_mem : \u2200 {x : X}, x \u2208 carrier \u2192 sorry \u2208 carrier)\n\nnamespace my_subobject\n\nvariables (X : Type*)\n\ninstance : set_like (my_subobject X) X :=\n\u27e8sub_mul_action.carrier, \u03bb p q h, by cases p; cases q; congr'\u27e9\n\n@[simp] lemma mem_carrier {p : my_subobject X} : x \u2208 p.carrier \u2194 x \u2208 (p : set X) := iff.rfl\n\n@[ext] theorem ext {p q : my_subobject X} (h : \u2200 x, x \u2208 p \u2194 x \u2208 q) : p = q := set_like.ext h\n\n/-- Copy of a `my_subobject` with a new `carrier` equal to the old one. Useful to fix definitional\nequalities. See Note [range copy pattern]. -/\nprotected def copy (p : my_subobject X) (s : set X) (hs : s = \u2191p) : my_subobject X :=\n{ carrier := s,\n  op_mem' := hs.symm \u25b8 p.op_mem' }\n\n@[simp] lemma coe_copy (p : my_subobject X) (s : set X) (hs : s = \u2191p) :\n  (p.copy s hs : set X) = s := rfl\n\nlemma copy_eq (p : my_subobject X) (s : set X) (hs : s = \u2191p) : p.copy s hs = p :=\nset_like.coe_injective hs\n\nend my_subobject\n```\n\nAn alternative to `set_like` could have been an extensional `has_mem` typeclass:\n```\nclass has_ext_mem (\u03b1 : out_param $ Type u) (\u03b2 : Type v) extends has_mem \u03b1 \u03b2 :=\n(ext_iff : \u2200 {s t : \u03b2}, s = t \u2194 \u2200 (x : \u03b1), x \u2208 s \u2194 x \u2208 t)\n```\nWhile this is equivalent, `set_like` conveniently uses a carrier set projection directly.\n\n## Tags\n\nsubobjects\n-/\n\n/-- A class to indicate that there is a canonical injection between `A` and `set B`.\n\nThis has the effect of giving terms of `A` elements of type `B` (through a `has_mem`\ninstance) and a compatible coercion to `Type*` as a subtype.\n\nNote: if `set_like.coe` is a projection, implementers should create a simp lemma such as\n```\n@[simp] lemma mem_carrier {p : my_subobject X} : x \u2208 p.carrier \u2194 x \u2208 (p : set X) := iff.rfl\n```\nto normalize terms.\n-/\n@[protect_proj]\nclass set_like (A : Type*) (B : out_param $ Type*) :=\n(coe : A \u2192 set B)\n(coe_injective' : function.injective coe)\n\nnamespace set_like\n\nvariables {A : Type*} {B : Type*} [i : set_like A B]\n\ninclude i\n\ninstance : has_coe_t A (set B) := \u27e8set_like.coe\u27e9\n\n@[priority 100]\ninstance : has_mem B A := \u27e8\u03bb x p, x \u2208 (p : set B)\u27e9\n\n-- `dangerous_instance` does not know that `B` is used only as an `out_param`\n@[nolint dangerous_instance, priority 100]\ninstance : has_coe_to_sort A Type* := \u27e8\u03bb p, {x : B // x \u2208 p}\u27e9\n\nvariables (p q : A)\n\n@[simp, norm_cast] theorem coe_sort_coe : ((p : set B) : Type*) = p := rfl\n\nvariables {p q}\n\nprotected theorem \u00abexists\u00bb {q : p \u2192 Prop} :\n  (\u2203 x, q x) \u2194 (\u2203 x \u2208 p, q \u27e8x, \u2039_\u203a\u27e9) := set_coe.exists\n\nprotected theorem \u00abforall\u00bb {q : p \u2192 Prop} :\n  (\u2200 x, q x) \u2194 (\u2200 x \u2208 p, q \u27e8x, \u2039_\u203a\u27e9) := set_coe.forall\n\ntheorem coe_injective : function.injective (coe : A \u2192 set B) :=\n\u03bb x y h, set_like.coe_injective' h\n\n@[simp, norm_cast] \n\ntheorem ext' (h : (p : set B) = q) : p = q := coe_injective h\n\ntheorem ext'_iff : p = q \u2194 (p : set B) = q := coe_set_eq.symm\n\n/-- Note: implementers of `set_like` must copy this lemma in order to tag it with `@[ext]`. -/\ntheorem ext (h : \u2200 x, x \u2208 p \u2194 x \u2208 q) : p = q := coe_injective $ set.ext h\n\ntheorem ext_iff : p = q \u2194 (\u2200 x, x \u2208 p \u2194 x \u2208 q) := coe_injective.eq_iff.symm.trans set.ext_iff\n\n@[simp] theorem mem_coe {x : B} : x \u2208 (p : set B) \u2194 x \u2208 p := iff.rfl\n\n@[simp, norm_cast] lemma coe_eq_coe {x y : p} : (x : B) = y \u2194 x = y := subtype.ext_iff_val.symm\n\n@[simp, norm_cast] lemma coe_mk (x : B) (hx : x \u2208 p) : ((\u27e8x, hx\u27e9 : p) : B) = x := rfl\n@[simp] lemma coe_mem (x : p) : (x : B) \u2208 p := x.2\n\n@[simp] protected lemma eta (x : p) (hx : (x : B) \u2208 p) : (\u27e8x, hx\u27e9 : p) = x := subtype.eta x hx\n\n-- `dangerous_instance` does not know that `B` is used only as an `out_param`\n@[nolint dangerous_instance, priority 100]\ninstance : partial_order A :=\n{ le := \u03bb H K, \u2200 \u2983x\u2984, x \u2208 H \u2192 x \u2208 K,\n  .. partial_order.lift (coe : A \u2192 set B) coe_injective }\n\nlemma le_def {S T : A} : S \u2264 T \u2194 \u2200 \u2983x : B\u2984, x \u2208 S \u2192 x \u2208 T := iff.rfl\n\n@[simp, norm_cast]\nlemma coe_subset_coe {S T : A} : (S : set B) \u2286 T \u2194 S \u2264 T := iff.rfl\n\n@[mono] lemma coe_mono : monotone (coe : A \u2192 set B) := \u03bb a b, coe_subset_coe.mpr\n\n@[simp, norm_cast]\nlemma coe_ssubset_coe {S T : A} : (S : set B) \u2282 T \u2194 S < T := iff.rfl\n\n@[mono] lemma coe_strict_mono : strict_mono (coe : A \u2192 set B) := \u03bb a b, coe_ssubset_coe.mpr\n\nlemma not_le_iff_exists : \u00ac(p \u2264 q) \u2194 \u2203 x \u2208 p, x \u2209 q := set.not_subset\n\nlemma exists_of_lt : p < q \u2192 \u2203 x \u2208 q, x \u2209 p := set.exists_of_ssubset\n\nlemma lt_iff_le_and_exists : p < q \u2194 p \u2264 q \u2227 \u2203 x \u2208 q, x \u2209 p :=\nby rw [lt_iff_le_not_le, not_le_iff_exists]\n\nend set_like\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/set_like/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.542863297964157, "lm_q2_score": 0.7025300573952054, "lm_q1q2_score": 0.38137778387650967}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport data.dfinsupp\nimport data.equiv.module\nimport data.finsupp.basic\n\n/-!\n# Conversion between `finsupp` and homogenous `dfinsupp`\n\nThis module provides conversions between `finsupp` and `dfinsupp`.\nIt is in its own file since neither `finsupp` or `dfinsupp` depend on each other.\n\n## Main definitions\n\n* \"identity\" maps between `finsupp` and `dfinsupp`:\n  * `finsupp.to_dfinsupp : (\u03b9 \u2192\u2080 M) \u2192 (\u03a0\u2080 i : \u03b9, M)`\n  * `dfinsupp.to_finsupp : (\u03a0\u2080 i : \u03b9, M) \u2192 (\u03b9 \u2192\u2080 M)`\n  * Bundled equiv versions of the above:\n    * `finsupp_equiv_dfinsupp : (\u03b9 \u2192\u2080 M) \u2243 (\u03a0\u2080 i : \u03b9, M)`\n    * `finsupp_add_equiv_dfinsupp : (\u03b9 \u2192\u2080 M) \u2243+ (\u03a0\u2080 i : \u03b9, M)`\n    * `finsupp_lequiv_dfinsupp R : (\u03b9 \u2192\u2080 M) \u2243\u2097[R] (\u03a0\u2080 i : \u03b9, M)`\n* stronger versions of `finsupp.split`:\n  * `sigma_finsupp_equiv_dfinsupp : ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243 (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N))`\n  * `sigma_finsupp_add_equiv_dfinsupp : ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243+ (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N))`\n  * `sigma_finsupp_lequiv_dfinsupp : ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243\u2097[R] (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N))`\n\n## Theorems\n\nThe defining features of these operations is that they preserve the function and support:\n\n* `finsupp.to_dfinsupp_coe`\n* `finsupp.to_dfinsupp_support`\n* `dfinsupp.to_finsupp_coe`\n* `dfinsupp.to_finsupp_support`\n\nand therefore map `finsupp.single` to `dfinsupp.single` and vice versa:\n\n* `finsupp.to_dfinsupp_single`\n* `dfinsupp.to_finsupp_single`\n\nas well as preserving arithmetic operations.\n\nFor the bundled equivalences, we provide lemmas that they reduce to `finsupp.to_dfinsupp`:\n\n* `finsupp_add_equiv_dfinsupp_apply`\n* `finsupp_lequiv_dfinsupp_apply`\n* `finsupp_add_equiv_dfinsupp_symm_apply`\n* `finsupp_lequiv_dfinsupp_symm_apply`\n\n## Implementation notes\n\nWe provide `dfinsupp.to_finsupp` and `finsupp_equiv_dfinsupp` computably by adding\n`[decidable_eq \u03b9]` and `[\u03a0 m : M, decidable (m \u2260 0)]` arguments. To aid with definitional unfolding,\nthese arguments are also present on the `noncomputable` equivs.\n-/\n\nvariables {\u03b9 : Type*} {R : Type*} {M : Type*}\n\n\n/-! ### Basic definitions and lemmas -/\nsection defs\n\n/-- Interpret a `finsupp` as a homogenous `dfinsupp`. -/\ndef finsupp.to_dfinsupp [has_zero M] (f : \u03b9 \u2192\u2080 M) : \u03a0\u2080 i : \u03b9, M :=\n\u27e6\u27e8f, f.support.1, \u03bb i, (classical.em (f i = 0)).symm.imp_left (finsupp.mem_support_iff.mpr)\u27e9\u27e7\n\n@[simp] lemma finsupp.to_dfinsupp_coe [has_zero M] (f : \u03b9 \u2192\u2080 M) : \u21d1f.to_dfinsupp = f := rfl\n\nsection\nvariables [decidable_eq \u03b9] [has_zero M]\n\n@[simp] lemma finsupp.to_dfinsupp_single (i : \u03b9) (m : M) :\n  (finsupp.single i m).to_dfinsupp = dfinsupp.single i m :=\nby { ext, simp [finsupp.single_apply, dfinsupp.single_apply] }\n\nvariables [\u03a0 m : M, decidable (m \u2260 0)]\n\n@[simp] lemma to_dfinsupp_support (f : \u03b9 \u2192\u2080 M) : f.to_dfinsupp.support = f.support :=\nby { ext, simp, }\n\n/-- Interpret a homogenous `dfinsupp` as a `finsupp`.\n\nNote that the elaborator has a lot of trouble with this definition - it is often necessary to\nwrite `(dfinsupp.to_finsupp f : \u03b9 \u2192\u2080 M)` instead of `f.to_finsupp`, as for some unknown reason\nusing dot notation or omitting the type ascription prevents the type being resolved correctly. -/\ndef dfinsupp.to_finsupp (f : \u03a0\u2080 i : \u03b9, M) : \u03b9 \u2192\u2080 M :=\n\u27e8f.support, f, \u03bb i, by simp only [dfinsupp.mem_support_iff]\u27e9\n\n@[simp] lemma dfinsupp.to_finsupp_coe (f : \u03a0\u2080 i : \u03b9, M) : \u21d1f.to_finsupp = f := rfl\n@[simp] lemma dfinsupp.to_finsupp_support (f : \u03a0\u2080 i : \u03b9, M) : f.to_finsupp.support = f.support :=\nby { ext, simp, }\n\n@[simp] lemma dfinsupp.to_finsupp_single (i : \u03b9) (m : M) :\n  (dfinsupp.single i m : \u03a0\u2080 i : \u03b9, M).to_finsupp = finsupp.single i m :=\nby { ext, simp [finsupp.single_apply, dfinsupp.single_apply] }\n\n@[simp] lemma finsupp.to_dfinsupp_to_finsupp (f : \u03b9 \u2192\u2080 M) : f.to_dfinsupp.to_finsupp = f :=\nfinsupp.coe_fn_injective rfl\n\n@[simp] lemma dfinsupp.to_finsupp_to_dfinsupp (f : \u03a0\u2080 i : \u03b9, M) : f.to_finsupp.to_dfinsupp = f :=\ndfinsupp.coe_fn_injective rfl\n\nend\n\nend defs\n\n/-! ### Lemmas about arithmetic operations -/\nsection lemmas\n\nnamespace finsupp\n\n@[simp] lemma to_dfinsupp_zero [has_zero M] :\n  (0 : \u03b9 \u2192\u2080 M).to_dfinsupp = 0 := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_add [add_zero_class M] (f g : \u03b9 \u2192\u2080 M) :\n  (f + g).to_dfinsupp = f.to_dfinsupp + g.to_dfinsupp := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_neg [add_group M] (f : \u03b9 \u2192\u2080 M) :\n  (-f).to_dfinsupp = -f.to_dfinsupp := dfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_sub [add_group M] (f g : \u03b9 \u2192\u2080 M) :\n  (f - g).to_dfinsupp = f.to_dfinsupp - g.to_dfinsupp :=\ndfinsupp.coe_fn_injective rfl\n\n@[simp] lemma to_dfinsupp_smul [monoid R] [add_monoid M] [distrib_mul_action R M]\n  (r : R) (f : \u03b9 \u2192\u2080 M) : (r \u2022 f).to_dfinsupp = r \u2022 f.to_dfinsupp :=\ndfinsupp.coe_fn_injective rfl\n\nend finsupp\n\nnamespace dfinsupp\nvariables [decidable_eq \u03b9]\n\n@[simp] lemma to_finsupp_zero [has_zero M] [\u03a0 m : M, decidable (m \u2260 0)] :\n  to_finsupp 0 = (0 : \u03b9 \u2192\u2080 M) := finsupp.coe_fn_injective rfl\n\n@[simp] lemma to_finsupp_add [add_zero_class M] [\u03a0 m : M, decidable (m \u2260 0)] (f g : \u03a0\u2080 i : \u03b9, M) :\n  (to_finsupp (f + g) : \u03b9 \u2192\u2080 M) = (to_finsupp f + to_finsupp g) :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_add _ _\n\n@[simp] lemma to_finsupp_neg [add_group M] [\u03a0 m : M, decidable (m \u2260 0)] (f : \u03a0\u2080 i : \u03b9, M) :\n  (to_finsupp (-f) : \u03b9 \u2192\u2080 M) = -to_finsupp f :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_neg _\n\n@[simp] lemma to_finsupp_sub [add_group M] [\u03a0 m : M, decidable (m \u2260 0)] (f g : \u03a0\u2080 i : \u03b9, M) :\n  (to_finsupp (f - g) : \u03b9 \u2192\u2080 M) = to_finsupp f - to_finsupp g :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_sub _ _\n\n@[simp] lemma to_finsupp_smul [monoid R] [add_monoid M] [distrib_mul_action R M]\n  [\u03a0 m : M, decidable (m \u2260 0)]\n  (r : R) (f : \u03a0\u2080 i : \u03b9, M) : (to_finsupp (r \u2022 f) : \u03b9 \u2192\u2080 M) = r \u2022 to_finsupp f :=\nfinsupp.coe_fn_injective $ dfinsupp.coe_smul _ _\n\nend dfinsupp\n\nend lemmas\n\n/-! ### Bundled `equiv`s -/\n\nsection equivs\n\n/-- `finsupp.to_dfinsupp` and `dfinsupp.to_finsupp` together form an equiv. -/\n@[simps {fully_applied := ff}]\ndef finsupp_equiv_dfinsupp [decidable_eq \u03b9] [has_zero M] [\u03a0 m : M, decidable (m \u2260 0)] :\n  (\u03b9 \u2192\u2080 M) \u2243 (\u03a0\u2080 i : \u03b9, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  left_inv := finsupp.to_dfinsupp_to_finsupp, right_inv := dfinsupp.to_finsupp_to_dfinsupp }\n\n/-- The additive version of `finsupp.to_finsupp`. Note that this is `noncomputable` because\n`finsupp.has_add` is noncomputable. -/\n@[simps {fully_applied := ff}]\nnoncomputable def finsupp_add_equiv_dfinsupp\n  [decidable_eq \u03b9] [add_zero_class M] [\u03a0 m : M, decidable (m \u2260 0)] :\n  (\u03b9 \u2192\u2080 M) \u2243+ (\u03a0\u2080 i : \u03b9, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  map_add' := finsupp.to_dfinsupp_add,\n  .. finsupp_equiv_dfinsupp}\n\nvariables (R)\n\n/-- The additive version of `finsupp.to_finsupp`. Note that this is `noncomputable` because\n`finsupp.has_add` is noncomputable. -/\n@[simps {fully_applied := ff}]\nnoncomputable def finsupp_lequiv_dfinsupp\n  [decidable_eq \u03b9] [semiring R] [add_comm_monoid M] [\u03a0 m : M, decidable (m \u2260 0)] [module R M] :\n  (\u03b9 \u2192\u2080 M) \u2243\u2097[R] (\u03a0\u2080 i : \u03b9, M) :=\n{ to_fun := finsupp.to_dfinsupp, inv_fun := dfinsupp.to_finsupp,\n  map_smul' := finsupp.to_dfinsupp_smul,\n  map_add' := finsupp.to_dfinsupp_add,\n  .. finsupp_equiv_dfinsupp}\n\nsection sigma\n/-- ### Stronger versions of `finsupp.split` -/\n\nnoncomputable theory\nopen_locale classical\n\nvariables {\u03b7 : \u03b9 \u2192 Type*} {N : Type*} [semiring R]\n\nopen finsupp\n\n/-- `finsupp.split` is an equivalence between `(\u03a3 i, \u03b7 i) \u2192\u2080 N` and `\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)`. -/\ndef sigma_finsupp_equiv_dfinsupp [has_zero N] : ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243 (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)) :=\n{ to_fun := \u03bb f, \u27e6\u27e8split f, (split_support f : finset \u03b9).val, \u03bb i,\n    begin\n    rw [\u2190 finset.mem_def, mem_split_support_iff_nonzero],\n    exact (decidable.em _).symm\n    end\u27e9\u27e7,\n  inv_fun := \u03bb f,\n  begin\n    refine on_finset (finset.sigma f.support (\u03bb j, (f j).support)) (\u03bb ji, f ji.1 ji.2)\n      (\u03bb g hg, finset.mem_sigma.mpr \u27e8_, mem_support_iff.mpr hg\u27e9),\n    simp only [ne.def, dfinsupp.mem_support_to_fun],\n    intro h,\n    rw h at hg,\n    simpa using hg\n  end,\n  left_inv := \u03bb f, by { ext, simp [split] },\n  right_inv := \u03bb f, by { ext, simp [split] } }\n\n@[simp]\nlemma sigma_finsupp_equiv_dfinsupp_apply [has_zero N] (f : (\u03a3 i, \u03b7 i) \u2192\u2080 N) :\n  (sigma_finsupp_equiv_dfinsupp f : \u03a0 i, (\u03b7 i \u2192\u2080 N)) = finsupp.split f := rfl\n\n@[simp]\nlemma sigma_finsupp_equiv_dfinsupp_symm_apply [has_zero N] (f : \u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)) (s : \u03a3 i, \u03b7 i) :\n  (sigma_finsupp_equiv_dfinsupp.symm f : (\u03a3 i, \u03b7 i) \u2192\u2080 N) s = f s.1 s.2 := rfl\n\n@[simp]\nlemma sigma_finsupp_equiv_dfinsupp_support [has_zero N] (f : (\u03a3 i, \u03b7 i) \u2192\u2080 N) :\n  (sigma_finsupp_equiv_dfinsupp f).support = finsupp.split_support f :=\nbegin\n  ext,\n  rw dfinsupp.mem_support_to_fun,\n  exact (finsupp.mem_split_support_iff_nonzero _ _).symm,\nend\n\n@[simp] lemma sigma_finsupp_equiv_dfinsupp_single [has_zero N] (a : \u03a3 i, \u03b7 i) (n : N) :\n  sigma_finsupp_equiv_dfinsupp (finsupp.single a n)\n    = @dfinsupp.single _ (\u03bb i, \u03b7 i \u2192\u2080 N) _ _ a.1 (finsupp.single a.2 n) :=\nbegin\n  obtain \u27e8i, a\u27e9 := a,\n  ext j b,\n  by_cases h : i = j,\n  { subst h,\n    simp [split_apply, finsupp.single_apply] },\n  suffices : finsupp.single (\u27e8i, a\u27e9 : \u03a3 i, \u03b7 i) n \u27e8j, b\u27e9 = 0,\n  { simp [split_apply, dif_neg h, this] },\n  have H : (\u27e8i, a\u27e9 : \u03a3 i, \u03b7 i) \u2260 \u27e8j, b\u27e9 := by simp [h],\n  rw [finsupp.single_apply, if_neg H]\nend\n\n-- Without this Lean fails to find the `add_zero_class` instance on `\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)`.\nlocal attribute [-instance] finsupp.has_zero\n\n@[simp]\nlemma sigma_finsupp_equiv_dfinsupp_add [add_zero_class N] (f g : (\u03a3 i, \u03b7 i) \u2192\u2080 N) :\n  sigma_finsupp_equiv_dfinsupp (f + g) =\n  (sigma_finsupp_equiv_dfinsupp f + (sigma_finsupp_equiv_dfinsupp g) : (\u03a0\u2080 (i : \u03b9), \u03b7 i \u2192\u2080 N)) :=\nby {ext, refl}\n\n/-- `finsupp.split` is an additive equivalence between `(\u03a3 i, \u03b7 i) \u2192\u2080 N` and `\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)`. -/\n@[simps]\ndef sigma_finsupp_add_equiv_dfinsupp [add_zero_class N] : ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243+ (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)) :=\n{ to_fun := sigma_finsupp_equiv_dfinsupp,\n  inv_fun := sigma_finsupp_equiv_dfinsupp.symm,\n  map_add' := sigma_finsupp_equiv_dfinsupp_add,\n  .. sigma_finsupp_equiv_dfinsupp }\n\nlocal attribute [-instance] finsupp.add_zero_class\n\n--tofix: r \u2022 (sigma_finsupp_equiv_dfinsupp f) doesn't work.\n@[simp]\nlemma sigma_finsupp_equiv_dfinsupp_smul {R} [monoid R] [add_monoid N] [distrib_mul_action R N]\n  (r : R) (f : (\u03a3 i, \u03b7 i) \u2192\u2080 N) : sigma_finsupp_equiv_dfinsupp (r \u2022 f) =\n  @has_scalar.smul R (\u03a0\u2080 i, \u03b7 i \u2192\u2080 N) mul_action.to_has_scalar r (sigma_finsupp_equiv_dfinsupp f) :=\nby { ext, refl }\n\nlocal attribute [-instance] finsupp.add_monoid\n\n/-- `finsupp.split` is a linear equivalence between `(\u03a3 i, \u03b7 i) \u2192\u2080 N` and `\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)`. -/\n@[simps]\ndef sigma_finsupp_lequiv_dfinsupp [add_comm_monoid N] [module R N] :\n  ((\u03a3 i, \u03b7 i) \u2192\u2080 N) \u2243\u2097[R] (\u03a0\u2080 i, (\u03b7 i \u2192\u2080 N)) :=\n{ map_smul' := sigma_finsupp_equiv_dfinsupp_smul,\n  .. sigma_finsupp_add_equiv_dfinsupp }\n\nend sigma\n\nend equivs\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/finsupp/to_dfinsupp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808533, "lm_q2_score": 0.7025300511670689, "lm_q1q2_score": 0.38137775971232313}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.category\nimport pseudoelements\nimport tactic.combinators\n\nopen category_theory\nopen category_theory.abelian\nopen tactic\n\nnamespace tactic.chase\n\nmeta def chase_attribute : user_attribute := {\n  name := `chase,\n  descr := \"A definition or lemma that can be used in a diagram chase.\"\n}\n\nmeta def get_lemmas_from_attribute (f : expr) : tactic (list expr) :=\nattribute.get_instances `chase >>= (list.mfiltermap $ \u03bb n,\n  (some <$> (do e \u2190 resolve_name n, to_expr ``(%%e %%f) tt ff)) <|> return none)\n\nrun_cmd attribute.register ``chase_attribute\n\n/-- A morphism in an abelian category. -/\n@[derive decidable_eq]\nmeta structure morphism :=\n(ex : expr)\n(domain : expr)\n(codomain : expr)\n(app : expr)\n\n/-- A sequence of morphisms in an abelian category, in \"mathematical order\":\n    `[f\u2081, ..., f\u2099]` represents `f\u2099 \u226b ... \u226b f\u2081`. -/\n@[reducible]\nmeta def morphism_chain := list morphism\n\n/-- An expression of the form `f\u2081 (f\u2082 (... (f\u2099 a)))`, with `a` a pseudoelement. -/\n@[derive decidable_eq]\nmeta structure diagram_term :=\n(ms : morphism_chain)\n(elem : expr)\n\n/-- A commutativity lemma in a category. -/\nmeta structure commutativity_lemma :=\n(lhs rhs : morphism_chain)\n(ex : expr)\n\n/-- A lemma of the form `f\u2081 (f\u2082 (... (f\u2099 a))) = g\u2081 (g\u2082 (... (g\u2098 a)))`, with `a` a pseudoelement. -/\nmeta structure element_lemma :=\n(lhs rhs : diagram_term)\n(ex : expr)\n\n/-- An exactness statement. -/\nmeta structure exactness_lemma :=\n(lhs rhs : morphism_chain)\n(ex : expr)\n\n/-- The morphisms and lemmas in the context. -/\nmeta structure chase_data :=\n(morphisms : list morphism)\n(comm_lemmas : list commutativity_lemma)\n(elem_lemmas : list element_lemma)\n(exact_lemmas : list exactness_lemma)\n\nmeta instance format_morphism : has_to_format morphism :=\n{ to_format := \u03bb m, format!\"{morphism.ex m}\" }\n\nmeta instance format_diagram_term : has_to_format diagram_term :=\n{ to_format := \u03bb t, format!\"{t.ms} {t.elem}\" }\n\nmeta instance format_commutativity_lemma : has_to_format commutativity_lemma :=\n{ to_format := \u03bb l, format!\"{l.lhs} ==> {l.rhs}\" }\n\nmeta instance format_element_lemma : has_to_format element_lemma :=\n{ to_format := \u03bb l, format!\"{l.lhs} ==> {l.rhs}\" }\n\nmeta instance format_exactness_lemma : has_to_format exactness_lemma :=\n{ to_format := \u03bb l, format!\"exact {l.lhs} {l.rhs}\" }\n\n/-- A tactic that makes use of the precomputed content of the context. -/\n@[reducible]\nmeta def chase_tactic :=\nstate_t chase_data tactic\n\nmeta instance {\u03b1} : has_coe (tactic \u03b1) (chase_tactic \u03b1) :=\n\u27e8monad_lift\u27e9\n\nmeta def as_expr : diagram_term \u2192 expr\n| \u27e8[], e\u27e9 := e\n| \u27e8t::ts, e\u27e9 := expr.app t.app $ as_expr \u27e8ts, e\u27e9\n\nmeta def commutativity_lemma.symm : commutativity_lemma \u2192 tactic commutativity_lemma\n| \u27e8lhs, rhs, e\u27e9 := mk_eq_symm e >>= \u03bb f, return \u27e8rhs, lhs, f\u27e9\n\nmeta def element_lemma.symm : element_lemma \u2192 tactic element_lemma\n| \u27e8lhs, rhs, e\u27e9 := mk_eq_symm e >>= \u03bb f, return \u27e8rhs, lhs, f\u27e9\n\nmeta def morphism.is_zero (m : morphism) : tactic bool :=\ndo\n  d \u2190 infer_type m.ex,\n  z \u2190 i_to_expr ``(0 : %%d),\n\n  -- Why does the following not work?\n  -- z \u2190 mk_app `has_zero.zero [d],\n\n  (do is_def_eq m.ex z,\n  return tt) <|> return ff\n\nmeta def diagram_term.type : diagram_term \u2192 tactic expr\n| \u27e8[], e\u27e9 := infer_type e\n| \u27e8t::ts, e\u27e9 := return t.codomain\n\nmeta def diagram_term.zero (t : diagram_term) : tactic diagram_term :=\ndo\n  s \u2190 t.type,\n  x \u2190 i_to_expr ``(0 : %%s),\n  return \u27e8[], x\u27e9\n\n/-- Try to generate a proof of `as_expr t = 0`. -/\nmeta def diagram_term.to_zero : diagram_term \u2192 tactic (option expr)\n| \u27e8[], e\u27e9 := (do\n  d \u2190 infer_type e,\n  f \u2190 i_to_expr ``(0 : %%d),\n  is_def_eq e f,\n  some <$> mk_eq_refl e) <|> return none\n| \u27e8t::ts, e\u27e9 := do\n  z \u2190 t.is_zero,\n  if z then some <$> mk_app\n    `category_theory.abelian.pseudoelements.zero_apply [t.codomain, as_expr \u27e8ts, e\u27e9] else do\n    inner \u2190 diagram_term.to_zero \u27e8ts, e\u27e9,\n    match inner with\n    | none := return none\n    | some i := do\n      fs \u2190 mk_app `congr_arg [t.app, i],\n      sn \u2190 mk_app `category_theory.abelian.pseudoelements.apply_zero [t.ex],\n      some <$> mk_eq_trans fs sn\n    end\n\nmeta def is_mono (m : morphism) : tactic bool :=\n(do i_to_expr ``(mono %%(m.ex)) >>= mk_instance, return tt) <|> return ff\n\nmeta def is_epi (m : morphism) : tactic bool :=\n(do i_to_expr ``(epi %%(m.ex)) >>= mk_instance, return tt) <|> return ff\n\nmeta def has_domain (e : expr) (m : morphism) : tactic bool :=\n(do is_def_eq m.domain e, return tt) <|> return ff\n\nmeta def is_mono_with_domain (e : expr) (m : morphism) : tactic bool :=\ndo\n  l \u2190 has_domain e m,\n  match l with\n  | ff := return ff\n  | tt := is_mono m\n  end\n\nmeta def has_apply_domain (e : expr) (m : morphism) : tactic bool :=\n(do\n  u \u2190 i_to_expr ``(coe_sort %%(tactic.chase.morphism.domain m)),\n  is_def_eq u e,\n  return tt) <|> return ff\n\nmeta def morphism.can_apply (e : expr) (m : morphism) : tactic bool :=\ndo t \u2190 infer_type e, has_apply_domain t m\n\nmeta def morphism_chain.can_apply (e : expr) : morphism_chain \u2192 tactic bool\n| [] := return false\n| (m::[]) := m.can_apply e\n| (m::ms) := morphism_chain.can_apply ms\n\nmeta def monos_with_domain (e : expr) : chase_tactic (list morphism) :=\ndo\n  s \u2190 get,\n  list.mfilter (\u03bb m, is_mono_with_domain e m) $ s.morphisms\n\nmeta def mono_with_domain (e : expr) : chase_tactic (option morphism) :=\nmonos_with_domain e >>= (return \u2218 list.head')\n\nmeta def epis (e : list morphism) : tactic (list morphism) :=\nlist.mfilter is_epi e\n\n/-- Try to parse `e` as a morphism. -/\nmeta def as_morphism (e : expr) : tactic (option morphism) :=\ndo\n  `(%%l \u27f6 %%r) \u2190 infer_type e | return none,\n  app \u2190 mk_app `coe_fn [e],\n  return $ some \u27e8e, l, r, app\u27e9\n\n/-- Try to parse `e` as a morphism chain. -/\nmeta def as_morphism_chain : expr \u2192 tactic (option morphism_chain) := \u03bb e,\ndo\n  self \u2190 as_morphism e,\n  match self with\n  | none := return none\n  | some s := do\n    `(%%l \u226b %%r) \u2190 return s.ex | return (some [s]),\n    some u \u2190 as_morphism_chain r | return none,\n    some r \u2190 as_morphism l | return none,\n    return $ some (list.append u [r])\n  end\n\n/-- Try to parse `e` as a commutativity lemma. -/\nmeta def as_commutativity_lemma (e : expr) : tactic (option commutativity_lemma) :=\ndo\n  `(%%l = %%r) \u2190 infer_type e | return none,\n  some lhs \u2190 as_morphism_chain l | return none,\n  some rhs \u2190 as_morphism_chain r | return none,\n  return $ some \u27e8lhs, rhs, e\u27e9\n\n/-- Try to parse `e` as a diagram term. -/\nmeta def as_diagram_term : expr \u2192 tactic (option diagram_term) := \u03bb e,\ndo\n  `(coe_sort %%l) \u2190 infer_type e | return none,\n  expr.app `(coe_fn %%f) `(%%x) \u2190 return e | return $ some \u27e8[], e\u27e9,\n  some dt \u2190 as_diagram_term x,\n  some F \u2190 as_morphism f,\n  return $ some \u27e8(F::diagram_term.ms dt), diagram_term.elem dt\u27e9\n\n/-- Try to parse `e` as an element lemma. -/\nmeta def as_element_lemma (e : expr) : tactic (option element_lemma) :=\ndo\n  `(%%l = %%r) \u2190 infer_type e | return none,\n  some lhs \u2190 as_diagram_term l | return none,\n  some rhs \u2190 as_diagram_term r | return none,\n  return $ some \u27e8lhs, rhs, e\u27e9\n\n/-- Try to parse `e` as an exactness lemma. -/\nmeta def as_exactness_lemma (e : expr) : tactic (option exactness_lemma) :=\ndo\n  `(category_theory.abelian.exact %%f %%g) \u2190 infer_type e | return none,\n  some lhs \u2190 as_morphism_chain f | return none,\n  some rhs \u2190 as_morphism_chain g | return none,\n  return $ some \u27e8lhs, rhs, e\u27e9\n\nmeta def epis_as_exact (e : list morphism) : tactic (list exactness_lemma) :=\nepis e >>= (list.mmap $ \u03bb m,\ndo\n  ep \u2190 i_to_expr ``(epi %%(m.ex)) >>= mk_instance,\n  a \u2190 mk_app `category_theory.abelian.exact_zero_of_epi' [ep],\n  some l \u2190 as_exactness_lemma a,\n  return l)\n\nmeta def exactness_lemmas_for (m : morphism_chain) : chase_tactic (list exactness_lemma) :=\ndo\n  l \u2190 get,\n  return $ list.filter (\u03bb lem, to_bool $ exactness_lemma.lhs lem = m) l.exact_lemmas\n\nmeta def get_morphisms (ctx : list expr) : tactic (list morphism) :=\nlist.mfiltermap as_morphism ctx\n\nmeta def get_comm_lemmas (ctx : list expr) : tactic (list commutativity_lemma) :=\nlist.mfiltermap as_commutativity_lemma ctx\n\nmeta def get_elem_lemmas (ctx : list expr) : tactic (list element_lemma) :=\nlist.mfiltermap as_element_lemma ctx\n\nmeta def get_exact_lemmas (ctx : list expr) : tactic (list exactness_lemma) :=\nlist.mfiltermap as_exactness_lemma ctx\n\nmeta def get_exact_lemmas_with_epi (ctx : list expr) (ms : list morphism) :\n  tactic (list exactness_lemma) :=\ndo\n  found \u2190 get_exact_lemmas ctx,\n  ep \u2190 epis_as_exact ms,\n  return $ list.append found ep\n\nmeta def exact_lemma_to_comm_lemmas (e : exactness_lemma) : tactic (list commutativity_lemma) :=\ndo\n  some fi \u2190 i_to_expr ``((%%e.ex).1) >>= as_commutativity_lemma,\n  some se \u2190 i_to_expr ``((%%e.ex).2) >>= as_commutativity_lemma,\n  return [fi, se]\n\nmeta def exact_lemmas_to_comm_lemmas (e : list exactness_lemma) : tactic (list commutativity_lemma) :=\ndo\n  l \u2190 list.mmap exact_lemma_to_comm_lemmas e,\n  return $ list.join l\n\nmeta def mk_chase_data (e : option expr) : tactic chase_data :=\ndo\n  ls \u2190 match e with\n       | none := return []\n       | some e := get_lemmas_from_attribute e\n       end,\n  ctx \u2190 local_context,\n  let src := list.append ctx ls,\n  ms \u2190 get_morphisms src,\n  cs \u2190 get_comm_lemmas src,\n  es \u2190 get_elem_lemmas src,\n  ess \u2190 list.mmap element_lemma.symm es,\n  exs \u2190 get_exact_lemmas_with_epi src ms,\n  ecs \u2190 exact_lemmas_to_comm_lemmas exs,\n  let cs' := list.append cs ecs,\n  css \u2190 list.mmap commutativity_lemma.symm cs',\n  return \u27e8ms, list.append cs' css, list.append es ess, exs\u27e9\n\nmeta def run_chase_tactic_with_data {\u03b1} (t : chase_tactic \u03b1) (d : chase_data) : tactic \u03b1 :=\ndo (res, _) \u2190 t.run d, return res\n\nmeta def run_chase_tactic {\u03b1} (e : option expr) (t : chase_tactic \u03b1) : tactic \u03b1 :=\nmk_chase_data e >>= run_chase_tactic_with_data t\n\nmeta def add_elem_lemma (l : element_lemma) : chase_tactic unit :=\ndo\n  \u27e8ms, cs, es, el\u27e9 \u2190 get,\n  ls \u2190 l.symm,\n  put \u27e8ms, cs, (ls::l::es), el\u27e9\n\nend tactic.chase\n", "meta": {"author": "TwoFX", "repo": "lean-homological-algebra", "sha": "e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8", "save_path": "github-repos/lean/TwoFX-lean-homological-algebra", "path": "github-repos/lean/TwoFX-lean-homological-algebra/lean-homological-algebra-e3a8e4ecaf49bec6c7b38b34c0b8f9749e941aa8/src/tactic/chase_tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.381163189851011}}
{"text": "\nimport Lib.Tactic\n\nnamespace List\n\ninstance : Functor List where\n  map := List.map\n\nvariable {\u03b1 \u03b2} (xs ys : List \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (x\u2080 : \u03b2)\n\n@[simp]\ntheorem foldl_app :\n  foldl f x\u2080 (xs ++ ys) = foldl f (foldl f x\u2080 xs) ys := by\ninduction xs generalizing x\u2080 <;> auto\n\ntheorem foldl_eq_self :\n  (xs.foldl (flip (.::.)) []).reverse = xs := by\nsimp only [flip]\ntrans (List.reverse [] ++ xs)\ncase second => simp\ngeneralize [] = ys\ninduction xs generalizing ys\n <;> simp [List.foldl, *, List.append_assoc]\n\ntheorem foldr_eq_self :\n  (xs.foldr (.::.) []) = xs := by\ntrans (xs ++ [])\ncase second => simp\ngeneralize [] = ys\ninduction xs generalizing ys\n <;> simp [List.foldr, *, List.append_assoc]\n\n\n@[simp]\ntheorem map_nil (f : \u03b1 \u2192 \u03b2) : f <$> [] = [] := rfl\n\n@[simp]\ntheorem map_cons (f : \u03b1 \u2192 \u03b2) x xs :\n  f <$> (x :: xs) = f x :: f <$> xs := rfl\n\n@[simp]\ntheorem redLength_eq (xs : List \u03b1) :\n  redLength xs = length xs := by\ninduction xs <;> simp [redLength, *]\n\n@[simp]\ntheorem take_zero (xs : List \u03b1) : xs.take 0 = [] := by\ninduction xs <;> simp [take]\n\n@[simp]\ntheorem drop_zero (xs : List \u03b1) : xs.drop 0 = xs := by\ninduction xs <;> simp [drop]\n\n@[simp]\ntheorem drop_nil n : (@nil \u03b1).drop n = [] := by\ncases n <;> refl\n\ntheorem cons_drop {i} {xs : List \u03b1} (h : i < length xs) :\n  get xs \u27e8i, h\u27e9 :: drop i.succ xs = drop i xs := by\ninduction xs generalizing i;\n. cases h\ncases i <;> simp [get, drop, *] at h \u22a2\n\n@[simp]\ntheorem take_length (xs : List \u03b1) : xs.take xs.length = xs := by\ninduction xs <;> simp [take, length, *]\n\n@[simp]\ntheorem drop_length (xs : List \u03b1) : xs.drop xs.length = [] := by\ninduction xs <;> simp [drop, *]\n\n@[simp]\ntheorem length_eq_zero (xs : List \u03b1) :\n  xs.length = 0 \u2194 xs = [] := by\ncases xs <;> simp [length, Nat.add_one]\n\n@[simp]\ntheorem foldl_reverse (xs : List \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) :\n  xs.reverse.foldl f x\u2080 = xs.foldr (flip f) x\u2080 := by\ninduction xs <;> simp [foldl, foldr, *]; refl\n\n/-- drop -/\n\ntheorem drop_append' {xs ys : List \u03b1}\n        (h : xs.length = n) :\n  (xs ++ ys).drop n = ys := by\ninduction xs generalizing n with\n| nil =>\n  cases n <;> simp\n  cases h\n| cons x xs ih =>\n  cases n <;> simp only [drop, List.cons_append]\n  <;> cases h\n  auto\n\ntheorem drop_append {xs ys : List \u03b1} :\n  (xs ++ ys).drop xs.length = ys :=\ndrop_append' rfl\n\n/- mem -/\n\nsection mem\n\nattribute [auto] List.Mem\n\n@[simp]\ntheorem mem_nil : \u00ac x \u2208 @nil \u03b1 := by\nintro h; cases h\n\n@[simp]\ntheorem mem_cons {ys : List \u03b1} : x \u2208 y :: ys \u2194 x = y \u2228 x \u2208 ys := by\nconstructor\nnext =>\n  intro h; cases h\n  . left; refl\n  . right; assumption\nnext =>\n  intro h; cases h;\n  next h =>\n    subst h; auto\n  next => auto\n\nend mem\n\n/- filter -/\n\nsection filter\n\nvariable {p : \u03b1 \u2192 Bool}\nvariable {x : \u03b1} {xs : List \u03b1}\n\ntheorem filterAux_eq :\n  filterAux p xs ys = reverse ys ++ filterAux p xs [] := by\ninduction xs generalizing ys\nnext => simp [filterAux]\nnext x xs ih =>\n  simp [filterAux]; split\n  <;> simp [ih (x :: ys), ih [x], append_assoc]\n  auto\n\ntheorem filter_cons_eq_ite :\n  filter p (x :: xs) =\n  ite (p x)\n    (x :: filter p xs)\n    (filter p xs) := by\nsimp [filter, filterAux]; split <;> simp [*]\nrw [filterAux_eq]; simp\n\n@[simp]\ntheorem filter_nil {p : \u03b1 \u2192 Bool} :\n  filter p [] = [] :=\nby simp [filter, filterAux]\n\n@[simp]\ntheorem filter_cons_true {p : \u03b1 \u2192 Bool} {x xs}\n        (h : p x = true) :\n  filter p (x :: xs) = x :: filter p xs :=\nby simp [filter_cons_eq_ite, *]\n\n@[simp]\ntheorem filter_cons_false {p : \u03b1 \u2192 Bool} {x xs}\n        (h : p x = false) :\n  filter p (x :: xs) = filter p xs :=\nby simp [filter_cons_eq_ite, *]\n\nattribute [local auto] Or\n\n@[simp]\ntheorem mem_filter {p : \u03b1 \u2192 Bool} :\n  x \u2208 filter p xs \u2194 x \u2208 xs \u2227 p x := by\ninduction xs\nnext => simp\nnext y ys ih =>\n  by_cases p y <;> simp [*]\n  next =>\n    constructor <;> intro h\n    next =>\n      cases h\n      next => substAll; auto\n      next h => auto\n    next =>\n      match h with\n      | \u27e8.inl h, h'\u27e9 =>\n        subst h; left; refl\n      | \u27e8.inr h, h'\u27e9 =>\n        right; auto\n  next =>\n    constructor <;> intro h\n    next => auto\n    next =>\n      match h with\n      | \u27e8.inl h, h'\u27e9 =>\n        substAll; auto\n      | \u27e8.inr h, h'\u27e9 =>\n        auto\n\nend filter\n\n/- iota -/\n\nsection iota\n\n@[simp]\ntheorem mem_iota :\n  x \u2208 iota n \u2194 1 \u2264 x \u2227 x \u2264 n := by\ninduction n <;> simp [iota, *]\nnext =>\n  intros h; cases h\n  next h\u2080 h\u2081 => subst h\u2081; cases h\u2080\nnext =>\n  admit -- linear arithmetic\n  -- constructor <;> intro h\n  -- skip\n\nend iota\n\nend List\n\nnamespace List\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b3 \u2192 \u03b1 \u2192 \u03b3}\nvariable {SIM : \u03b2 \u2192 \u03b3 \u2192 Prop}\nvariable {x\u2080 y\u2080} (t : List \u03b1)\n\ntheorem foldl_sim :\n    SIM x\u2080 y\u2080 \u2192\n    (\u2200 a x y, SIM x y \u2192 SIM (f x a) (g y a)) \u2192\n    SIM (foldl f x\u2080 t) (foldl g y\u2080 t) := by\ninduction t generalizing x\u2080 y\u2080 <;> auto\n\nend List\n", "meta": {"author": "cipher1024", "repo": "lean4-prog", "sha": "49f7416ee19df921bfea1b4914404b9d07619d64", "save_path": "github-repos/lean/cipher1024-lean4-prog", "path": "github-repos/lean/cipher1024-lean4-prog/lean4-prog-49f7416ee19df921bfea1b4914404b9d07619d64/lib/lib/Data/List/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6688802603710086, "lm_q1q2_score": 0.381163189851011}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.SynthInstance\nimport Lean.Meta.Tactic.Simp.Types\n\nnamespace Lean.Meta.Simp\n\ndef synthesizeArgs (lemmaName : Name) (xs : Array Expr) (bis : Array BinderInfo) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Bool := do\n  for x in xs, bi in bis do\n    let type \u2190 inferType x\n    if bi.isInstImplicit then\n      unless (\u2190 synthesizeInstance x type) do\n        return false\n    else if (\u2190 instantiateMVars x).isMVar then\n      if (\u2190 isProp type) then\n        match (\u2190 discharge? type) with\n        | some proof =>\n          unless (\u2190 isDefEq x proof) do\n            trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to assign proof{indentExpr type}\"\n            return false\n        | none =>\n          trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to discharge hypotheses{indentExpr type}\"\n          return false\n      else if (\u2190 isClass? type).isSome then\n        unless (\u2190 synthesizeInstance x type) do\n          return false\n  return true\nwhere\n  synthesizeInstance (x type : Expr) : SimpM Bool := do\n    match (\u2190 trySynthInstance type) with\n    | LOption.some val =>\n      if (\u2190 isDefEq x val) then\n        return true\n      else\n        trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to assign instance{indentExpr type}\"\n        return false\n    | _ =>\n      trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to synthesize instance{indentExpr type}\"\n      return false\n\n/-\nRemark: the parameter tag is used for creating trace messages. It is irrelevant otherwise.\n-/\ndef rewrite (e : Expr) (s : DiscrTree SimpLemma) (erased : Std.PHashSet Name) (discharge? : Expr \u2192 SimpM (Option Expr)) (tag : String) : SimpM Result := do\n  let lemmas \u2190 s.getMatch e\n  if lemmas.isEmpty then\n    trace[Debug.Meta.Tactic.simp] \"no theorems found for {tag}-rewriting {e}\"\n    return { expr := e }\n  else\n    let lemmas := lemmas.insertionSort fun e\u2081 e\u2082 => e\u2081.priority < e\u2082.priority\n    for lemma in lemmas do\n      unless inErasedSet lemma do\n        if let some result \u2190 tryLemma? lemma then\n          return result\n    return { expr := e }\nwhere\n  inErasedSet (lemma : SimpLemma) : Bool :=\n    match lemma.name? with\n    | none => false\n    | some name => erased.contains name\n\n  tryLemma? (lemma : SimpLemma) : SimpM (Option Result) :=\n    withNewMCtxDepth do\n      let val  \u2190 lemma.getValue\n      let type \u2190 inferType val\n      let (xs, bis, type) \u2190 forallMetaTelescopeReducing type\n      let type \u2190 instantiateMVars type\n      let lhs := type.appFn!.appArg!\n      if (\u2190 isDefEq lhs e) then\n        unless (\u2190 synthesizeArgs lemma.getName xs bis discharge?) do\n          return none\n        let proof \u2190 instantiateMVars (mkAppN val xs)\n        if \u2190 hasAssignableMVar proof then\n          trace[Meta.Tactic.simp.rewrite] \"{lemma}, has unassigned metavariables after unification\"\n          return none\n        let rhs   \u2190 instantiateMVars type.appArg!\n        if e == rhs then\n          return none\n        if lemma.perm && !Expr.lt rhs e then\n          trace[Meta.Tactic.simp.rewrite] \"{lemma}, perm rejected {e} ==> {rhs}\"\n          return none\n        trace[Meta.Tactic.simp.rewrite] \"{lemma}, {e} ==> {rhs}\"\n        return some { expr := rhs, proof? := proof }\n      else\n        unless lhs.isMVar do\n          -- We do not report unification failures when `lhs` is a metavariable\n          -- Example: `x = ()`\n          -- TODO: reconsider if we want lemmas such as `(x : Unit) \u2192 x = ()`\n          trace[Meta.Tactic.simp.unify] \"{lemma}, failed to unify {lhs} with {e}\"\n        return none\n\ndef rewriteCtorEq? (e : Expr) : MetaM (Option Result) := withReducibleAndInstances do\n  match e.eq? with\n  | none => return none\n  | some (_, lhs, rhs) =>\n    let lhs \u2190 whnf lhs\n    let rhs \u2190 whnf rhs\n    let env \u2190 getEnv\n    match lhs.constructorApp? env, rhs.constructorApp? env with\n    | some (c\u2081, _), some (c\u2082, _) =>\n      if c\u2081.name != c\u2082.name then\n        withLocalDeclD `h e fun h =>\n          return some { expr := mkConst ``False, proof? := (\u2190 mkEqFalse' (\u2190 mkLambdaFVars #[h] (\u2190 mkNoConfusion (mkConst ``False) h))) }\n      else\n        return none\n    | _, _ => return none\n\n@[inline] def tryRewriteCtorEq (e : Expr) (x : SimpM Step) : SimpM Step := do\n  match (\u2190 rewriteCtorEq? e) with\n  | some r => return Step.done r\n  | none => x\n\ndef rewriteUsingDecide? (e : Expr) : MetaM (Option Result) := withReducibleAndInstances do\n  if e.hasFVar || e.hasMVar || e.isConstOf ``True || e.isConstOf ``False then\n    return none\n  else\n    try\n      let d \u2190 mkDecide e\n      let r \u2190 withDefault <| whnf d\n      if r.isConstOf ``true then\n        return some { expr := mkConst ``True, proof? := mkAppN (mkConst ``eqTrueOfDecide) #[e, d.appArg!, (\u2190 mkEqRefl (mkConst ``true))] }\n      else if r.isConstOf ``false then\n        let h \u2190 mkEqRefl d\n        return some { expr := mkConst ``False, proof? := mkAppN (mkConst ``eqFalseOfDecide) #[e, d.appArg!, (\u2190 mkEqRefl (mkConst ``false))] }\n      else\n        return none\n    catch _ =>\n      return none\n\n@[inline] def tryRewriteUsingDecide (e : Expr) (x : SimpM Step) : SimpM Step := do\n  if (\u2190 read).config.decide then\n    match (\u2190 rewriteUsingDecide? e) with\n    | some r => return Step.done r\n    | none => x\n  else\n    x\n\ndef rewritePre (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  let lemmas \u2190 (\u2190 read).simpLemmas\n  return Step.visit (\u2190 rewrite e lemmas.pre lemmas.erased discharge? (tag := \"pre\"))\n\ndef rewritePost (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  let lemmas \u2190 (\u2190 read).simpLemmas\n  return Step.visit (\u2190 rewrite e lemmas.post lemmas.erased discharge? (tag := \"post\"))\n\ndef preDefault (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step :=\n  tryRewriteCtorEq e <| rewritePre e discharge?\n\ndef postDefault (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  -- TODO: try equation lemmas\n  tryRewriteCtorEq e <| tryRewriteUsingDecide e <| rewritePost e discharge?\n\nend Lean.Meta.Simp\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/stage0/src/Lean/Meta/Tactic/Simp/Rewrite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802603710085, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3811631898510109}}
{"text": "import data.list tactic.omega defs single branch\n\nlemma grow_trans : \u2200 t1 t2 t3 : bintree, \n      (t1 \u21a3 t2) \u2192 (t2 \u21a3 t3) \u2192 (t1 \u21a3 t3) :=\nbegin\n  intro t1, induction t1,\n  begin intros, auto_grow end,\n  repeat { begin \n    intros t2 t3 H1 H2,\n    cases H1, cases H2, \n    have H3 : (t1_a \u21a3 H2_t'), apply t1_ih, repeat {assumption},\n    auto_grow\n  end }, \n  intros t2 t3 H1 H2, \n  cases H1, cases H2, \n  have H3 : (t1_a \u21a3 H2_t1'), apply t1_ih_a, repeat {assumption},\n  have H4 : (t1_a_1 \u21a3 H2_t2'), apply t1_ih_a_1, repeat {assumption},\n  auto_grow\nend\n\nlemma grow_list_exists_one : \u2200 (l : list bintree) (t : bintree), (l \u21a6 t) \u2192\n                           \u2203 t', (t' \u2208 l) \u2227 (t' \u21a3 t) :=\nbegin\n  intros l t h,\n  induction h,\n  begin existsi h_t, split, apply list.mem_cons_self, assumption end,\n  begin\n    cases h_ih,\n    existsi h_ih_w,\n    split, apply list.mem_cons_of_mem, tauto, tauto\n  end\nend\n\nlemma exists_one_grow_list : \u2200 l t, (\u2203 t', ((t' \u2208 l) \u2227 (t' \u21a3 t))) \u2192 (l \u21a6 t) :=\nbegin\n  intros l,\n  induction l,\n  begin intros, cases a, cases a_h.left end,\n  begin\n    intros t h,\n    cases h,\n    unfold has_mem.mem list.mem at h_h,\n    cases h_h.left,\n    begin\n      apply grow_list.head_grow,\n      rewrite h at h_h,\n      exact h_h.right\n    end,\n    begin\n      apply grow_list.tail_grow,\n      apply l_ih,\n      existsi h_w, split,\n      assumption, exact h_h.right\n    end\n  end\nend\n\nlemma kernel_lemma : \u2200 (t : bintree) (h : \u2115), \n  h \u2265 1 \u2192 h \u2264 height t \u2192 \u2203 b : bintree, \u27e8b\u27e9 \u2227 h = height b \u2227 (b \u21a3 t) := \nbegin\n  intros t,\n  induction t,\n  begin -- single node\n    intros, unfold height at a_1, \n    have H : h = 1, omega,\n    rewrite H, fapply exists.intro, exact \u25cf, \n    split, apply is_branch.single, \n    split, unfold height, auto_grow,\n  end, \n  repeat { -- left and right\n    intros h H1 H2, \n    cases h, cases H1, unfold height at H2, \n    have H1' : h \u2265 0, omega, \n    have H2' : h \u2264 height t_a, omega, \n    cases h, \n    begin -- h = 0, which is trivial\n      fapply exists.intro, exact \u25cf, \n      split, exact is_branch.single, \n      split, unfold height, apply grow.single_grow    \n    end,\n    begin -- h \u2260 0, apply induction hypothesis\n      have H3 : \u2203 (b : bintree), \u27e8b\u27e9 \u2227 h.succ = height b \u2227 (b\u21a3t_a), \n        apply t_ih, omega, omega,  \n      cases H3, \n      try { \n        existsi \u27e6H3_w\u2223\u27e7, split, apply is_branch.left_nl, tauto, \n        split, unfold height, \n               have Ht : h.succ = height H3_w, tauto, omega,\n        apply grow.left_grow, tauto,  \n      },\n      try {\n        existsi \u27e6\u2223H3_w\u27e7, split, apply is_branch.right_nl, tauto, \n        split, unfold height,\n               have Ht : h.succ = height H3_w, tauto, omega, \n        apply grow.right_grow, tauto,\n      }\n    end\n  }, \n  intros h H1 H2, -- both left and right, completely similar, while more tedious\n  cases h, cases H1, unfold height at H2, \n  have H1' : h \u2265 0, omega,\n  have H2' : h \u2264 max (height t_a) (height t_a_1), omega,\n  cases h, \n  begin\n    fapply exists.intro, exact \u25cf, \n    split, exact is_branch.single, \n    split, unfold height, apply grow.single_grow\n  end, \n  begin\n    have H3 : h.succ \u2264 height t_a \u2228 h.succ \u2264 height t_a_1, \n      begin \n        have H3' : max (height t_a) (height t_a_1) = height t_a \u2228 max (height t_a) (height t_a_1) = height t_a_1, apply max_choice,\n        destruct H3', \n          intro H4, left, rewrite H4 at H2', assumption,\n          intro H4, right, rewrite H4 at H2', assumption\n      end,\n    destruct H3,\n    begin\n      intros H4, \n      have H5 : h.succ \u2265 1, omega, \n      have H6 : \u2203 (b : bintree), \u27e8 b \u27e9 \u2227 h.succ = height b \u2227 (b \u21a3 t_a), \n        apply t_ih_a, repeat {assumption},\n      cases H6, cases H6_h, cases H6_h_right, existsi (\u27e6H6_w, \u25cf\u27e7), \n      split, apply is_branch.left_l, assumption, \n      split, unfold height, \n        have Ht : (1 \u2264 height H6_w), apply ge.le, apply height_ge1, \n        have Ht' : max (height H6_w) 1 = height H6_w, apply max_eq_left, assumption, \n        omega,\n      auto_grow  \n    end,\n    begin\n      intros H4, \n      have H5 : h.succ \u2265 1, omega, \n      have H6 : \u2203 (b : bintree), \u27e8 b \u27e9 \u2227 h.succ = height b \u2227 (b \u21a3 t_a_1), \n        apply t_ih_a_1, repeat {assumption}, \n      cases H6, cases H6_h, cases H6_h_right, existsi (\u27e6\u25cf, H6_w\u27e7), \n      split, apply is_branch.right_l, assumption, \n      split, unfold height, \n        have Ht : (1 \u2264 height H6_w), apply ge.le, apply height_ge1, \n        have Ht' : max 1 (height H6_w) = height H6_w, apply max_eq_right, assumption,\n        omega,\n      auto_grow\n    end\n  end\nend\n\n", "meta": {"author": "ljt12138", "repo": "Proof-of-Surreal", "sha": "6b92baf2382ac23dd0d700f5c958aa910ad4b754", "save_path": "github-repos/lean/ljt12138-Proof-of-Surreal", "path": "github-repos/lean/ljt12138-Proof-of-Surreal/Proof-of-Surreal-6b92baf2382ac23dd0d700f5c958aa910ad4b754/src/grow.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.3811084910317633}}
{"text": "/-\nCopyright (c) 2019 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.control.applicative\nimport Mathlib.control.traversable.basic\nimport Mathlib.PostPort\n\nuniverses u l v u_1 \n\nnamespace Mathlib\n\n/-!\n# Free constructions\n\n## Main definitions\n\n* `free_magma \u03b1`: free magma (structure with binary operation without any axioms) over alphabet `\u03b1`,\n  defined inductively, with traversable instance and decidable equality.\n* `magma.free_semigroup \u03b1`: free semigroup over magma `\u03b1`.\n* `free_semigroup \u03b1`: free semigroup over alphabet `\u03b1`, defined as a synonym for `\u03b1 \u00d7 list \u03b1`\n  (i.e. nonempty lists), with traversable instance and decidable equality.\n* `free_semigroup_free_magma \u03b1`: isomorphism between `magma.free_semigroup (free_magma \u03b1)` and\n  `free_semigroup \u03b1`.\n-/\n\n/-- Free magma over a given alphabet. -/\ninductive free_magma (\u03b1 : Type u) \nwhere\n| of : \u03b1 \u2192 free_magma \u03b1\n| mul : free_magma \u03b1 \u2192 free_magma \u03b1 \u2192 free_magma \u03b1\n\n/-- Free nonabelian additive magma over a given alphabet. -/\ninductive free_add_magma (\u03b1 : Type u) \nwhere\n| of : \u03b1 \u2192 free_add_magma \u03b1\n| add : free_add_magma \u03b1 \u2192 free_add_magma \u03b1 \u2192 free_add_magma \u03b1\n\nnamespace free_magma\n\n\nprotected instance Mathlib.free_add_magma.inhabited {\u03b1 : Type u} [Inhabited \u03b1] : Inhabited (free_add_magma \u03b1) :=\n  { default := free_add_magma.of Inhabited.default }\n\nprotected instance Mathlib.free_add_magma.has_add {\u03b1 : Type u} : Add (free_add_magma \u03b1) :=\n  { add := free_add_magma.add }\n\n@[simp] theorem Mathlib.free_add_magma.add_eq {\u03b1 : Type u} (x : free_add_magma \u03b1) (y : free_add_magma \u03b1) : free_add_magma.add x y = x + y :=\n  rfl\n\n/-- Recursor for `free_magma` using `x * y` instead of `free_magma.mul x y`. -/\ndef Mathlib.free_add_magma.rec_on' {\u03b1 : Type u} {C : free_add_magma \u03b1 \u2192 Sort l} (x : free_add_magma \u03b1) (ih1 : (x : \u03b1) \u2192 C (free_add_magma.of x)) (ih2 : (x y : free_add_magma \u03b1) \u2192 C x \u2192 C y \u2192 C (x + y)) : C x :=\n  free_add_magma.rec_on x ih1 ih2\n\nend free_magma\n\n\n/-- Lifts a function `\u03b1 \u2192 \u03b2` to a magma homomorphism `free_magma \u03b1 \u2192 \u03b2` given a magma `\u03b2`. -/\ndef free_magma.lift {\u03b1 : Type u} {\u03b2 : Type v} [Mul \u03b2] (f : \u03b1 \u2192 \u03b2) : free_magma \u03b1 \u2192 \u03b2 :=\n  sorry\n\n/-- Lifts a function `\u03b1 \u2192 \u03b2` to an additive magma homomorphism `free_add_magma \u03b1 \u2192 \u03b2` given\nan additive magma `\u03b2`. -/\ndef free_add_magma.lift {\u03b1 : Type u} {\u03b2 : Type v} [Add \u03b2] (f : \u03b1 \u2192 \u03b2) : free_add_magma \u03b1 \u2192 \u03b2 :=\n  sorry\n\nnamespace free_magma\n\n\n@[simp] theorem Mathlib.free_add_magma.lift_of {\u03b1 : Type u} {\u03b2 : Type v} [Add \u03b2] (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : free_add_magma.lift f (free_add_magma.of x) = f x :=\n  rfl\n\n@[simp] theorem lift_mul {\u03b1 : Type u} {\u03b2 : Type v} [Mul \u03b2] (f : \u03b1 \u2192 \u03b2) (x : free_magma \u03b1) (y : free_magma \u03b1) : lift f (x * y) = lift f x * lift f y :=\n  rfl\n\ntheorem lift_unique {\u03b1 : Type u} {\u03b2 : Type v} [Mul \u03b2] (f : free_magma \u03b1 \u2192 \u03b2) (hf : \u2200 (x y : free_magma \u03b1), f (x * y) = f x * f y) : f = lift (f \u2218 of) := sorry\n\nend free_magma\n\n\n/-- The unique magma homomorphism `free_magma \u03b1 \u2192 free_magma \u03b2` that sends\neach `of x` to `of (f x)`. -/\ndef free_magma.map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : free_magma \u03b1 \u2192 free_magma \u03b2 :=\n  sorry\n\n/-- The unique additive magma homomorphism `free_add_magma \u03b1 \u2192 free_add_magma \u03b2` that sends\neach `of x` to `of (f x)`. -/\ndef free_add_magma.map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : free_add_magma \u03b1 \u2192 free_add_magma \u03b2 :=\n  sorry\n\nnamespace free_magma\n\n\n@[simp] theorem Mathlib.free_add_magma.map_of {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : free_add_magma.map f (free_add_magma.of x) = free_add_magma.of (f x) :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.map_add {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (x : free_add_magma \u03b1) (y : free_add_magma \u03b1) : free_add_magma.map f (x + y) = free_add_magma.map f x + free_add_magma.map f y :=\n  rfl\n\nprotected instance Mathlib.free_add_magma.monad : Monad free_add_magma := sorry\n\n/-- Recursor on `free_magma` using `pure` instead of `of`. -/\nprotected def Mathlib.free_add_magma.rec_on'' {\u03b1 : Type u} {C : free_add_magma \u03b1 \u2192 Sort l} (x : free_add_magma \u03b1) (ih1 : (x : \u03b1) \u2192 C (pure x)) (ih2 : (x y : free_add_magma \u03b1) \u2192 C x \u2192 C y \u2192 C (x + y)) : C x :=\n  free_add_magma.rec_on' x ih1 ih2\n\n@[simp] theorem Mathlib.free_add_magma.map_pure {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : f <$> pure x = pure (f x) :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.map_add' {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : free_add_magma \u03b1) (y : free_add_magma \u03b1) : f <$> (x + y) = f <$> x + f <$> y :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.pure_bind {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 free_add_magma \u03b2) (x : \u03b1) : pure x >>= f = f x :=\n  rfl\n\n@[simp] theorem mul_bind {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 free_magma \u03b2) (x : free_magma \u03b1) (y : free_magma \u03b1) : x * y >>= f = (x >>= f) * (y >>= f) :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.pure_seq {\u03b1 : Type u} {\u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} {x : free_add_magma \u03b1} : pure f <*> x = f <$> x :=\n  rfl\n\n@[simp] theorem mul_seq {\u03b1 : Type u} {\u03b2 : Type u} {f : free_magma (\u03b1 \u2192 \u03b2)} {g : free_magma (\u03b1 \u2192 \u03b2)} {x : free_magma \u03b1} : f * g <*> x = (f <*> x) * (g <*> x) :=\n  rfl\n\nprotected instance Mathlib.free_add_magma.is_lawful_monad : is_lawful_monad free_add_magma := sorry\n\nend free_magma\n\n\n/-- `free_magma` is traversable. -/\nprotected def free_magma.traverse {m : Type u \u2192 Type u} [Applicative m] {\u03b1 : Type u} {\u03b2 : Type u} (F : \u03b1 \u2192 m \u03b2) : free_magma \u03b1 \u2192 m (free_magma \u03b2) :=\n  sorry\n\n/-- `free_add_magma` is traversable. -/\nprotected def free_add_magma.traverse {m : Type u \u2192 Type u} [Applicative m] {\u03b1 : Type u} {\u03b2 : Type u} (F : \u03b1 \u2192 m \u03b2) : free_add_magma \u03b1 \u2192 m (free_add_magma \u03b2) :=\n  sorry\n\nnamespace free_magma\n\n\nprotected instance Mathlib.free_add_magma.traversable : traversable free_add_magma :=\n  traversable.mk free_add_magma.traverse\n\n@[simp] theorem Mathlib.free_add_magma.traverse_pure {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) (x : \u03b1) : traverse F (pure x) = pure <$> F x :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.traverse_pure' {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) : traverse F \u2218 pure = fun (x : \u03b1) => pure <$> F x :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.traverse_add {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) (x : free_add_magma \u03b1) (y : free_add_magma \u03b1) : traverse F (x + y) = Add.add <$> traverse F x <*> traverse F y :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.traverse_add' {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) : function.comp (traverse F) \u2218 Add.add = fun (x y : free_add_magma \u03b1) => Add.add <$> traverse F x <*> traverse F y :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_magma.traverse_eq {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) (x : free_add_magma \u03b1) : free_add_magma.traverse F x = traverse F x :=\n  rfl\n\n@[simp] theorem mul_map_seq {\u03b1 : Type u} (x : free_magma \u03b1) (y : free_magma \u03b1) : Mul.mul <$> x <*> y = x * y :=\n  rfl\n\nprotected instance Mathlib.free_add_magma.is_lawful_traversable : is_lawful_traversable free_add_magma :=\n  is_lawful_traversable.mk sorry sorry sorry sorry\n\nend free_magma\n\n\n/-- Representation of an element of a free magma. -/\nprotected def free_magma.repr {\u03b1 : Type u} [has_repr \u03b1] : free_magma \u03b1 \u2192 string :=\n  sorry\n\n/-- Representation of an element of a free additive magma. -/\nprotected def free_add_magma.repr {\u03b1 : Type u} [has_repr \u03b1] : free_add_magma \u03b1 \u2192 string :=\n  sorry\n\nprotected instance free_add_magma.has_repr {\u03b1 : Type u} [has_repr \u03b1] : has_repr (free_add_magma \u03b1) :=\n  has_repr.mk free_add_magma.repr\n\n/-- Length of an element of a free magma. -/\ndef free_magma.length {\u03b1 : Type u} : free_magma \u03b1 \u2192 \u2115 :=\n  sorry\n\n/-- Length of an element of a free additive magma. -/\ndef free_add_magma.length {\u03b1 : Type u} : free_add_magma \u03b1 \u2192 \u2115 :=\n  sorry\n\n/-- Associativity relations for a magma. -/\ninductive magma.free_semigroup.r (\u03b1 : Type u) [Mul \u03b1] : \u03b1 \u2192 \u03b1 \u2192 Prop\nwhere\n| intro : \u2200 (x y z : \u03b1), magma.free_semigroup.r \u03b1 (x * y * z) (x * (y * z))\n| left : \u2200 (w x y z : \u03b1), magma.free_semigroup.r \u03b1 (w * (x * y * z)) (w * (x * (y * z)))\n\n/-- Associativity relations for an additive magma. -/\ninductive add_magma.free_add_semigroup.r (\u03b1 : Type u) [Add \u03b1] : \u03b1 \u2192 \u03b1 \u2192 Prop\nwhere\n| intro : \u2200 (x y z : \u03b1), add_magma.free_add_semigroup.r \u03b1 (x + y + z) (x + (y + z))\n| left : \u2200 (w x y z : \u03b1), add_magma.free_add_semigroup.r \u03b1 (w + (x + y + z)) (w + (x + (y + z)))\n\nnamespace magma\n\n\n/-- Free semigroup over a magma. -/\ndef free_semigroup (\u03b1 : Type u) [Mul \u03b1] :=\n  Quot sorry\n\nnamespace free_semigroup\n\n\n/-- Embedding from magma to its free semigroup. -/\ndef Mathlib.add_magma.free_add_semigroup.of {\u03b1 : Type u} [Add \u03b1] : \u03b1 \u2192 add_magma.free_add_semigroup \u03b1 :=\n  Quot.mk (add_magma.free_add_semigroup.r \u03b1)\n\nprotected instance Mathlib.add_magma.free_add_semigroup.inhabited {\u03b1 : Type u} [Add \u03b1] [Inhabited \u03b1] : Inhabited (add_magma.free_add_semigroup \u03b1) :=\n  { default := add_magma.free_add_semigroup.of Inhabited.default }\n\nprotected theorem Mathlib.add_magma.free_add_semigroup.induction_on {\u03b1 : Type u} [Add \u03b1] {C : add_magma.free_add_semigroup \u03b1 \u2192 Prop} (x : add_magma.free_add_semigroup \u03b1) (ih : \u2200 (x : \u03b1), C (add_magma.free_add_semigroup.of x)) : C x :=\n  quot.induction_on x ih\n\ntheorem of_mul_assoc {\u03b1 : Type u} [Mul \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) : of (x * y * z) = of (x * (y * z)) :=\n  quot.sound (r.intro x y z)\n\ntheorem of_mul_assoc_left {\u03b1 : Type u} [Mul \u03b1] (w : \u03b1) (x : \u03b1) (y : \u03b1) (z : \u03b1) : of (w * (x * y * z)) = of (w * (x * (y * z))) :=\n  quot.sound (r.left w x y z)\n\ntheorem of_mul_assoc_right {\u03b1 : Type u} [Mul \u03b1] (w : \u03b1) (x : \u03b1) (y : \u03b1) (z : \u03b1) : of (w * x * y * z) = of (w * (x * y) * z) := sorry\n\nprotected instance semigroup {\u03b1 : Type u} [Mul \u03b1] : semigroup (free_semigroup \u03b1) :=\n  semigroup.mk\n    (fun (x y : free_semigroup \u03b1) =>\n      quot.lift_on x (fun (p : \u03b1) => quot.lift_on y (fun (q : \u03b1) => Quot.mk (r \u03b1) (p * q)) sorry) sorry)\n    sorry\n\ntheorem Mathlib.add_magma.free_add_semigroup.of_add {\u03b1 : Type u} [Add \u03b1] (x : \u03b1) (y : \u03b1) : add_magma.free_add_semigroup.of (x + y) = add_magma.free_add_semigroup.of x + add_magma.free_add_semigroup.of y :=\n  rfl\n\n/-- Lifts a magma homomorphism `\u03b1 \u2192 \u03b2` to a semigroup homomorphism `magma.free_semigroup \u03b1 \u2192 \u03b2`\ngiven a semigroup `\u03b2`. -/\ndef lift {\u03b1 : Type u} [Mul \u03b1] {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : \u2200 (x y : \u03b1), f (x * y) = f x * f y) : free_semigroup \u03b1 \u2192 \u03b2 :=\n  Quot.lift f sorry\n\n@[simp] theorem lift_of {\u03b1 : Type u} [Mul \u03b1] {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) {hf : \u2200 (x y : \u03b1), f (x * y) = f x * f y} (x : \u03b1) : lift f hf (of x) = f x :=\n  rfl\n\n@[simp] theorem lift_mul {\u03b1 : Type u} [Mul \u03b1] {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) {hf : \u2200 (x y : \u03b1), f (x * y) = f x * f y} (x : free_semigroup \u03b1) (y : free_semigroup \u03b1) : lift f hf (x * y) = lift f hf x * lift f hf y :=\n  quot.induction_on x fun (p : \u03b1) => quot.induction_on y fun (q : \u03b1) => hf p q\n\ntheorem Mathlib.add_magma.free_add_semigroup.lift_unique {\u03b1 : Type u} [Add \u03b1] {\u03b2 : Type v} [add_semigroup \u03b2] (f : add_magma.free_add_semigroup \u03b1 \u2192 \u03b2) (hf : \u2200 (x y : add_magma.free_add_semigroup \u03b1), f (x + y) = f x + f y) : f =\n  add_magma.free_add_semigroup.lift (f \u2218 add_magma.free_add_semigroup.of)\n    fun (p q : \u03b1) => hf (add_magma.free_add_semigroup.of p) (add_magma.free_add_semigroup.of q) :=\n  funext fun (x : add_magma.free_add_semigroup \u03b1) => quot.induction_on x fun (p : \u03b1) => rfl\n\n/-- From a magma homomorphism `\u03b1 \u2192 \u03b2` to a semigroup homomorphism\n`magma.free_semigroup \u03b1 \u2192 magma.free_semigroup \u03b2`. -/\ndef Mathlib.add_magma.free_add_semigroup.map {\u03b1 : Type u} [Add \u03b1] {\u03b2 : Type v} [Add \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : \u2200 (x y : \u03b1), f (x + y) = f x + f y) : add_magma.free_add_semigroup \u03b1 \u2192 add_magma.free_add_semigroup \u03b2 :=\n  add_magma.free_add_semigroup.lift (add_magma.free_add_semigroup.of \u2218 f) sorry\n\n@[simp] theorem Mathlib.add_magma.free_add_semigroup.map_of {\u03b1 : Type u} [Add \u03b1] {\u03b2 : Type v} [Add \u03b2] (f : \u03b1 \u2192 \u03b2) {hf : \u2200 (x y : \u03b1), f (x + y) = f x + f y} (x : \u03b1) : add_magma.free_add_semigroup.map f hf (add_magma.free_add_semigroup.of x) = add_magma.free_add_semigroup.of (f x) :=\n  rfl\n\n@[simp] theorem map_mul {\u03b1 : Type u} [Mul \u03b1] {\u03b2 : Type v} [Mul \u03b2] (f : \u03b1 \u2192 \u03b2) {hf : \u2200 (x y : \u03b1), f (x * y) = f x * f y} (x : free_semigroup \u03b1) (y : free_semigroup \u03b1) : map f hf (x * y) = map f hf x * map f hf y :=\n  lift_mul (of \u2218 f) x y\n\nend free_semigroup\n\n\nend magma\n\n\n/-- Free semigroup over a given alphabet.\n(Note: In this definition, the free semigroup does not contain the empty word.) -/\ndef free_semigroup (\u03b1 : Type u) :=\n  \u03b1 \u00d7 List \u03b1\n\nnamespace free_semigroup\n\n\nprotected instance semigroup {\u03b1 : Type u} : semigroup (free_semigroup \u03b1) :=\n  semigroup.mk (fun (L1 L2 : free_semigroup \u03b1) => (prod.fst L1, prod.snd L1 ++ prod.fst L2 :: prod.snd L2)) sorry\n\n/-- The embedding `\u03b1 \u2192 free_semigroup \u03b1`. -/\ndef Mathlib.free_add_semigroup.of {\u03b1 : Type u} (x : \u03b1) : free_add_semigroup \u03b1 :=\n  (x, [])\n\nprotected instance Mathlib.free_add_semigroup.inhabited {\u03b1 : Type u} [Inhabited \u03b1] : Inhabited (free_add_semigroup \u03b1) :=\n  { default := free_add_semigroup.of Inhabited.default }\n\n/-- Recursor for free semigroup using `of` and `*`. -/\nprotected def Mathlib.free_add_semigroup.rec_on {\u03b1 : Type u} {C : free_add_semigroup \u03b1 \u2192 Sort l} (x : free_add_semigroup \u03b1) (ih1 : (x : \u03b1) \u2192 C (free_add_semigroup.of x)) (ih2 : (x : \u03b1) \u2192 (y : free_add_semigroup \u03b1) \u2192 C (free_add_semigroup.of x) \u2192 C y \u2192 C (free_add_semigroup.of x + y)) : C x :=\n  prod.rec_on x\n    fun (f : \u03b1) (s : List \u03b1) =>\n      list.rec_on s ih1\n        (fun (hd : \u03b1) (tl : List \u03b1) (ih : (_a : \u03b1) \u2192 C (_a, tl)) (f : \u03b1) => ih2 f (hd, tl) (ih1 f) (ih hd)) f\n\nend free_semigroup\n\n\n/-- Auxiliary function for `free_semigroup.lift`. -/\ndef free_semigroup.lift' {\u03b1 : Type u} {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) : \u03b1 \u2192 List \u03b1 \u2192 \u03b2 :=\n  sorry\n\n/-- Auxiliary function for `free_semigroup.lift`. -/\ndef free_add_semigroup.lift' {\u03b1 : Type u} {\u03b2 : Type v} [add_semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) : \u03b1 \u2192 List \u03b1 \u2192 \u03b2 :=\n  sorry\n\nnamespace free_semigroup\n\n\n/-- Lifts a function `\u03b1 \u2192 \u03b2` to a semigroup homomorphism `free_semigroup \u03b1 \u2192 \u03b2` given\na semigroup `\u03b2`. -/\ndef lift {\u03b1 : Type u} {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) (x : free_semigroup \u03b1) : \u03b2 :=\n  lift' f (prod.fst x) (prod.snd x)\n\n@[simp] theorem lift_of {\u03b1 : Type u} {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : lift f (of x) = f x :=\n  rfl\n\ntheorem lift_of_mul {\u03b1 : Type u} {\u03b2 : Type v} [semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) (x : \u03b1) (y : free_semigroup \u03b1) : lift f (of x * y) = f x * lift f y :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_semigroup.lift_add {\u03b1 : Type u} {\u03b2 : Type v} [add_semigroup \u03b2] (f : \u03b1 \u2192 \u03b2) (x : free_add_semigroup \u03b1) (y : free_add_semigroup \u03b1) : free_add_semigroup.lift f (x + y) = free_add_semigroup.lift f x + free_add_semigroup.lift f y := sorry\n\ntheorem Mathlib.free_add_semigroup.lift_unique {\u03b1 : Type u} {\u03b2 : Type v} [add_semigroup \u03b2] (f : free_add_semigroup \u03b1 \u2192 \u03b2) (hf : \u2200 (x y : free_add_semigroup \u03b1), f (x + y) = f x + f y) : f = free_add_semigroup.lift (f \u2218 free_add_semigroup.of) := sorry\n\n/-- The unique semigroup homomorphism that sends `of x` to `of (f x)`. -/\ndef Mathlib.free_add_semigroup.map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : free_add_semigroup \u03b1 \u2192 free_add_semigroup \u03b2 :=\n  free_add_semigroup.lift (free_add_semigroup.of \u2218 f)\n\n@[simp] theorem Mathlib.free_add_semigroup.map_of {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : free_add_semigroup.map f (free_add_semigroup.of x) = free_add_semigroup.of (f x) :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_semigroup.map_add {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (x : free_add_semigroup \u03b1) (y : free_add_semigroup \u03b1) : free_add_semigroup.map f (x + y) = free_add_semigroup.map f x + free_add_semigroup.map f y :=\n  free_add_semigroup.lift_add (free_add_semigroup.of \u2218 f) x y\n\nprotected instance Mathlib.free_add_semigroup.monad : Monad free_add_semigroup := sorry\n\n/-- Recursor that uses `pure` instead of `of`. -/\ndef rec_on' {\u03b1 : Type u} {C : free_semigroup \u03b1 \u2192 Sort l} (x : free_semigroup \u03b1) (ih1 : (x : \u03b1) \u2192 C (pure x)) (ih2 : (x : \u03b1) \u2192 (y : free_semigroup \u03b1) \u2192 C (pure x) \u2192 C y \u2192 C (pure x * y)) : C x :=\n  free_semigroup.rec_on x ih1 ih2\n\n@[simp] theorem map_pure {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : f <$> pure x = pure (f x) :=\n  rfl\n\n@[simp] theorem map_mul' {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : free_semigroup \u03b1) (y : free_semigroup \u03b1) : f <$> (x * y) = f <$> x * f <$> y :=\n  map_mul f x y\n\n@[simp] theorem pure_bind {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 free_semigroup \u03b2) (x : \u03b1) : pure x >>= f = f x :=\n  rfl\n\n@[simp] theorem mul_bind {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 free_semigroup \u03b2) (x : free_semigroup \u03b1) (y : free_semigroup \u03b1) : x * y >>= f = (x >>= f) * (y >>= f) :=\n  lift_mul f x y\n\n@[simp] theorem Mathlib.free_add_semigroup.pure_seq {\u03b1 : Type u} {\u03b2 : Type u} {f : \u03b1 \u2192 \u03b2} {x : free_add_semigroup \u03b1} : pure f <*> x = f <$> x :=\n  rfl\n\n@[simp] theorem mul_seq {\u03b1 : Type u} {\u03b2 : Type u} {f : free_semigroup (\u03b1 \u2192 \u03b2)} {g : free_semigroup (\u03b1 \u2192 \u03b2)} {x : free_semigroup \u03b1} : f * g <*> x = (f <*> x) * (g <*> x) :=\n  mul_bind (fun (_x : \u03b1 \u2192 \u03b2) => (fun (\u03b1 \u03b2 : Type u) (f : \u03b1 \u2192 \u03b2) (x : free_semigroup \u03b1) => lift (of \u2218 f) x) \u03b1 \u03b2 _x x) f g\n\nprotected instance Mathlib.free_add_semigroup.is_lawful_monad : is_lawful_monad free_add_semigroup := sorry\n\n/-- `free_semigroup` is traversable. -/\nprotected def Mathlib.free_add_semigroup.traverse {m : Type u \u2192 Type u} [Applicative m] {\u03b1 : Type u} {\u03b2 : Type u} (F : \u03b1 \u2192 m \u03b2) (x : free_add_semigroup \u03b1) : m (free_add_semigroup \u03b2) :=\n  free_add_semigroup.rec_on' x (fun (x : \u03b1) => pure <$> F x)\n    fun (x : \u03b1) (y : free_add_semigroup \u03b1) (ihx ihy : m (free_add_semigroup \u03b2)) => Add.add <$> ihx <*> ihy\n\nprotected instance Mathlib.free_add_semigroup.traversable : traversable free_add_semigroup :=\n  traversable.mk free_add_semigroup.traverse\n\n@[simp] theorem traverse_pure {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) (x : \u03b1) : traverse F (pure x) = pure <$> F x :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_semigroup.traverse_pure' {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) : traverse F \u2218 pure = fun (x : \u03b1) => pure <$> F x :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_semigroup.traverse_add {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) [is_lawful_applicative m] (x : free_add_semigroup \u03b1) (y : free_add_semigroup \u03b1) : traverse F (x + y) = Add.add <$> traverse F x <*> traverse F y := sorry\n\n@[simp] theorem Mathlib.free_add_semigroup.traverse_add' {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) [is_lawful_applicative m] : function.comp (traverse F) \u2218 Add.add = fun (x y : free_add_semigroup \u03b1) => Add.add <$> traverse F x <*> traverse F y :=\n  funext fun (x : free_add_semigroup \u03b1) => funext fun (y : free_add_semigroup \u03b1) => free_add_semigroup.traverse_add F x y\n\n@[simp] theorem Mathlib.free_add_semigroup.traverse_eq {\u03b1 : Type u} {\u03b2 : Type u} {m : Type u \u2192 Type u} [Applicative m] (F : \u03b1 \u2192 m \u03b2) (x : free_add_semigroup \u03b1) : free_add_semigroup.traverse F x = traverse F x :=\n  rfl\n\n@[simp] theorem Mathlib.free_add_semigroup.add_map_seq {\u03b1 : Type u} (x : free_add_semigroup \u03b1) (y : free_add_semigroup \u03b1) : Add.add <$> x <*> y = x + y :=\n  rfl\n\nprotected instance Mathlib.free_add_semigroup.is_lawful_traversable : is_lawful_traversable free_add_semigroup :=\n  is_lawful_traversable.mk sorry sorry sorry sorry\n\nprotected instance Mathlib.free_add_semigroup.decidable_eq {\u03b1 : Type u} [DecidableEq \u03b1] : DecidableEq (free_add_semigroup \u03b1) :=\n  prod.decidable_eq\n\nend free_semigroup\n\n\n/-- Isomorphism between `magma.free_semigroup (free_magma \u03b1)` and `free_semigroup \u03b1`. -/\ndef free_add_semigroup_free_add_magma (\u03b1 : Type u) : add_magma.free_add_semigroup (free_add_magma \u03b1) \u2243 free_add_semigroup \u03b1 :=\n  equiv.mk (add_magma.free_add_semigroup.lift (free_add_magma.lift free_add_semigroup.of) sorry)\n    (free_add_semigroup.lift (add_magma.free_add_semigroup.of \u2218 free_add_magma.of)) sorry sorry\n\n@[simp] theorem free_semigroup_free_magma_mul {\u03b1 : Type u} (x : magma.free_semigroup (free_magma \u03b1)) (y : magma.free_semigroup (free_magma \u03b1)) : coe_fn (free_semigroup_free_magma \u03b1) (x * y) =\n  coe_fn (free_semigroup_free_magma \u03b1) x * coe_fn (free_semigroup_free_magma \u03b1) y :=\n  magma.free_semigroup.lift_mul (free_magma.lift free_semigroup.of) x y\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/free.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032313, "lm_q2_score": 0.6513548714339145, "lm_q1q2_score": 0.38110848707188066}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.types\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.concrete_category.elementwise\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nuniverses w v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nlocal attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nsection limits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J \u2964 C) [preserves_limit F (forget C)]\n\nlemma concrete.to_product_injective_of_is_limit {D : cone F} (hD : is_limit D) :\n  function.injective (\u03bb (x : D.X) (j : J), D.\u03c0.app j x) :=\nbegin\n  let E := (forget C).map_cone D,\n  let hE : is_limit E := is_limit_of_preserves _ hD,\n  let G := types.limit_cone (F \u22d9 forget C),\n  let hG := types.limit_cone_is_limit (F \u22d9 forget C),\n  let T : E.X \u2245 G.X := hE.cone_point_unique_up_to_iso hG,\n  change function.injective (T.hom \u226b (\u03bb x j, G.\u03c0.app j x)),\n  have h : function.injective T.hom,\n  { intros a b h,\n    suffices : T.inv (T.hom a) = T.inv (T.hom b), by simpa,\n    rw h },\n  suffices : function.injective (\u03bb (x : G.X) j, G.\u03c0.app j x),\n    by exact this.comp h,\n  apply subtype.ext,\nend\n\nlemma concrete.is_limit_ext {D : cone F} (hD : is_limit D) (x y : D.X) :\n  (\u2200 j, D.\u03c0.app j x = D.\u03c0.app j y) \u2192 x = y :=\n\u03bb h, concrete.to_product_injective_of_is_limit _ hD (funext h)\n\nlemma concrete.limit_ext [has_limit F] (x y : limit F) :\n  (\u2200 j, limit.\u03c0 F j x = limit.\u03c0 F j y) \u2192 x = y :=\nconcrete.is_limit_ext F (limit.is_limit _) _ _\n\nsection wide_pullback\n\nopen wide_pullback\nopen wide_pullback_shape\n\nlemma concrete.wide_pullback_ext {B : C} {\u03b9 : Type*} {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B)\n  [has_wide_pullback B X f] [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h\u2080 : base f x = base f y)\n  (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintro (_|j),\n  { exact h\u2080 },\n  { apply h }\nend\n\nlemma concrete.wide_pullback_ext' {B : C} {\u03b9 : Type*} [nonempty \u03b9]\n  {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B) [has_wide_pullback B X f]\n  [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.wide_pullback_ext _ _ _ _ h,\n  inhabit \u03b9,\n  simp only [\u2190 \u03c0_arrow f (arbitrary _), comp_apply, h],\nend\n\nend wide_pullback\n\nsection multiequalizer\n\nlemma concrete.multiequalizer_ext {I : multicospan_index C} [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x y : multiequalizer I)\n  (h : \u2200 (t : I.L), multiequalizer.\u03b9 I t x = multiequalizer.\u03b9 I t y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintros (a|b),\n  { apply h },\n  { rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n      comp_apply, comp_apply, h] }\nend\n\n/-- An auxiliary equivalence to be used in `multiequalizer_equiv` below.-/\ndef concrete.multiequalizer_equiv_aux (I : multicospan_index C) :\n  (I.multicospan \u22d9 (forget C)).sections \u2243\n  { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\n{ to_fun := \u03bb x, \u27e8\u03bb i, x.1 (walking_multicospan.left _), \u03bb i, begin\n    have a := x.2 (walking_multicospan.hom.fst i),\n    have b := x.2 (walking_multicospan.hom.snd i),\n    rw \u2190 b at a,\n    exact a,\n  end\u27e9,\n  inv_fun := \u03bb x,\n  { val := \u03bb j,\n    match j with\n    | walking_multicospan.left a := x.1 _\n    | walking_multicospan.right b := I.fst b (x.1 _)\n    end,\n    property := begin\n      rintros (a|b) (a'|b') (f|f|f),\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n      { refl },\n      { dsimp, erw \u2190 x.2 b', refl },\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n    end },\n  left_inv := begin\n    intros x, ext (a|b),\n    { refl },\n    { change _ = x.val _,\n      rw \u2190 x.2 (walking_multicospan.hom.fst b),\n      refl }\n  end,\n  right_inv := by { intros x, ext i, refl } }\n\n/-- The equivalence between the noncomputable multiequalizer and\nand the concrete multiequalizer. -/\nnoncomputable\ndef concrete.multiequalizer_equiv (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] : (multiequalizer I : C) \u2243\n    { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\nlet h1 := (limit.is_limit I.multicospan),\n    h2 := (is_limit_of_preserves (forget C) h1),\n    E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit _) in\nequiv.trans E.to_equiv (concrete.multiequalizer_equiv_aux I)\n\n@[simp]\nlemma concrete.multiequalizer_equiv_apply (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :\n  ((concrete.multiequalizer_equiv I) x : \u03a0 (i : I.L), I.left i) i = multiequalizer.\u03b9 I i x := rfl\n\nend multiequalizer\n\n-- TODO: Add analogous lemmas about products and equalizers.\n\nend limits\n\nsection colimits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J \u2964 C) [preserves_colimit F (forget C)]\n\nlemma concrete.from_union_surjective_of_is_colimit {D : cocone F} (hD : is_colimit D) :\n  let ff : (\u03a3 (j : J), F.obj j) \u2192 D.X := \u03bb a, D.\u03b9.app a.1 a.2 in function.surjective ff :=\nbegin\n  intro ff,\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  suffices : function.surjective (TX.hom \u2218 ff),\n  { intro a,\n    obtain \u27e8b, hb\u27e9 := this (TX.hom a),\n    refine \u27e8b, _\u27e9,\n    apply_fun TX.inv at hb,\n    change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb,\n    simpa only [TX.hom_inv_id] using hb },\n  have : TX.hom \u2218 ff = \u03bb a, G.\u03b9.app a.1 a.2,\n  { ext a,\n    change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _,\n    rw hE.fac },\n  rw this,\n  rintro \u27e8\u27e8j,a\u27e9\u27e9,\n  exact \u27e8\u27e8j,a\u27e9,rfl\u27e9,\nend\n\nlemma concrete.is_colimit_exists_rep {D : cocone F} (hD : is_colimit D) (x : D.X) :\n  \u2203 (j : J) (y : F.obj j), D.\u03b9.app j y = x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := concrete.from_union_surjective_of_is_colimit F hD x,\n  exact \u27e8a.1, a.2, rfl\u27e9,\nend\n\nlemma concrete.colimit_exists_rep [has_colimit F] (x : colimit F) :\n  \u2203 (j : J) (y : F.obj j), colimit.\u03b9 F j y = x :=\nconcrete.is_colimit_exists_rep F (colimit.is_colimit _) x\n\nlemma concrete.is_colimit_rep_eq_of_exists {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  D.\u03b9.app i x = D.\u03b9.app j y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom,\n  swap, { suffices : function.bijective TX.hom, by exact this.1,\n    rw \u2190 is_iso_iff_bijective, apply is_iso.of_iso },\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y,\n  erw [T.hom.w, T.hom.w],\n  obtain \u27e8k, f, g, h\u27e9 := h,\n  have : G.\u03b9.app i x = (G.\u03b9.app k (F.map f x) : G.X) := quot.sound \u27e8f,rfl\u27e9,\n  rw [this, h],\n  symmetry,\n  exact quot.sound \u27e8g,rfl\u27e9,\nend\n\nlemma concrete.colimit_rep_eq_of_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y :=\nconcrete.is_colimit_rep_eq_of_exists F (colimit.is_colimit _) x y h\n\nsection filtered_colimits\n\nvariable [is_filtered J]\n\nlemma concrete.is_colimit_exists_of_rep_eq {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : D.\u03b9.app _ x = D.\u03b9.app _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom at h,\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y at h,\n  erw [T.hom.w, T.hom.w] at h,\n  replace h := quot.exact _ h,\n  suffices : \u2200 (a b : \u03a3 j, F.obj j)\n    (h : eqv_gen (limits.types.quot.rel (F \u22d9 forget C)) a b),\n    \u2203 k (f : a.1 \u27f6 k) (g : b.1 \u27f6 k), F.map f a.2 = F.map g b.2,\n  { exact this \u27e8i,x\u27e9 \u27e8j,y\u27e9 h },\n  intros a b h,\n  induction h,\n  case eqv_gen.rel : x y hh\n  { obtain \u27e8e,he\u27e9 := hh,\n    use [y.1, e, \ud835\udfd9 _],\n    simpa using he.symm },\n  case eqv_gen.refl : x { use [x.1, \ud835\udfd9 _, \ud835\udfd9 _, rfl] },\n  case eqv_gen.symm : x y _ hh\n  { obtain \u27e8k, f, g, hh\u27e9 := hh,\n    use [k, g, f, hh.symm] },\n  case eqv_gen.trans : x y z _ _ hh1 hh2\n  { obtain \u27e8k1, f1, g1, h1\u27e9 := hh1,\n    obtain \u27e8k2, f2, g2, h2\u27e9 := hh2,\n    let k0 : J := is_filtered.max k1 k2,\n    let e1 : k1 \u27f6 k0 := is_filtered.left_to_max _ _,\n    let e2 : k2 \u27f6 k0 := is_filtered.right_to_max _ _,\n    let k : J := is_filtered.coeq (g1 \u226b e1) (f2 \u226b e2),\n    let e : k0 \u27f6 k := is_filtered.coeq_hom _ _,\n    use [k, f1 \u226b e1 \u226b e, g2 \u226b e2 \u226b e],\n    simp only [F.map_comp, comp_apply, h1, \u2190 h2],\n    simp only [\u2190 comp_apply, \u2190 F.map_comp],\n    rw is_filtered.coeq_condition },\nend\n\ntheorem concrete.is_colimit_rep_eq_iff_exists {D : cocone F} {i j : J}\n  (hD : is_colimit D) (x : F.obj i) (y : F.obj j) :\n  D.\u03b9.app i x = D.\u03b9.app j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.is_colimit_exists_of_rep_eq _ hD _ _, concrete.is_colimit_rep_eq_of_exists _ hD _ _\u27e9\n\nlemma concrete.colimit_exists_of_rep_eq [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : colimit.\u03b9 F _ x = colimit.\u03b9 F _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nconcrete.is_colimit_exists_of_rep_eq F (colimit.is_colimit _) x y h\n\ntheorem concrete.colimit_rep_eq_iff_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.colimit_exists_of_rep_eq _ _ _, concrete.colimit_rep_eq_of_exists _ _ _\u27e9\n\nend filtered_colimits\n\nsection wide_pushout\n\nopen wide_pushout\nopen wide_pushout_shape\n\nlemma concrete.wide_pushout_exists_rep {B : C} {\u03b1 : Type*} {X : \u03b1 \u2192 C} (f : \u03a0 j : \u03b1, B \u27f6 X j)\n  [has_wide_pushout B X f] [preserves_colimit (wide_span B X f) (forget C)]\n  (x : wide_pushout B X f) : (\u2203 y : B, head f y = x) \u2228 (\u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x) :=\nbegin\n  obtain \u27e8_ | j, y, rfl\u27e9 := concrete.colimit_exists_rep _ x,\n  { use y },\n  { right,\n    use [j,y] }\nend\n\nlemma concrete.wide_pushout_exists_rep' {B : C} {\u03b1 : Type*} [nonempty \u03b1] {X : \u03b1 \u2192 C}\n  (f : \u03a0 j : \u03b1, B \u27f6 X j) [has_wide_pushout B X f]\n  [preserves_colimit (wide_span B X f) (forget C)] (x : wide_pushout B X f) :\n  \u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x :=\nbegin\n  rcases concrete.wide_pushout_exists_rep f x with \u27e8y, rfl\u27e9 | \u27e8i, y, rfl\u27e9,\n  { inhabit \u03b1,\n    use [arbitrary _, f _ y],\n    simp only [\u2190 arrow_\u03b9 _ (arbitrary \u03b1), comp_apply] },\n  { use [i,y] }\nend\n\nend wide_pushout\n\n-- TODO: Add analogous lemmas about coproducts and coequalizers.\n\nend colimits\n\nend category_theory.limits\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/limits/concrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6513548646660543, "lm_q1q2_score": 0.3811084831119978}}
{"text": "\nimport data.mv_polynomial.basic\nimport .general_lemmas.mv_X_mul\nimport .general_lemmas.single_antidiagonal\nimport .general_lemmas.mul_sum_symm\n\nsection\n\nopen mv_polynomial\n\nuniverses u\n\n@[simp] lemma eq_zero_of_zero_eq (R : Type u) [has_zero R] (r : R) : 0 = r \u2194 r = 0 :=\nbegin\n  exact eq_comm,\nend\n\n@[simp] lemma zero_sub_eq_iff (R : Type u) [add_comm_group R] (a b : R) : 0 - a = b \u2194 a + b = 0 :=\nbegin\n  split,\n  { intro h, rw <-h, abel, },\n  { intro h, rw <-h, abel, },\nend\n\nrun_cmd mk_simp_attr `integral_domain_simp\nrun_cmd tactic.add_doc_string `simp_attr.integral_domain_simp \"Attribute for lemmas that are useful in simplifying systems of equations in an integral domain\"\n\n\n\nrun_cmd mk_simp_attr `polynomial_nf\nrun_cmd tactic.add_doc_string `simp_attr.polynomial_nf \"Attribute for lemmas that are used in the conversion of mv_polynomial expressions to a normal form consisting of adds of sums of muls of mv_polynomials\"\n\nattribute [polynomial_nf] polynomial.eval\u2082\nattribute [polynomial_nf] polynomial.sum\nattribute [polynomial_nf] finsupp.sum\nattribute [polynomial_nf] mul_add\nattribute [polynomial_nf] add_mul\nattribute [polynomial_nf] finset.sum_mul\nattribute [polynomial_nf] finset.mul_sum\nattribute [polynomial_nf] finset.sum_add_distrib\n\nattribute [polynomial_nf] mul_assoc\n\nattribute [polynomial_nf] finsupp.smul_sum\nattribute [polynomial_nf] mul_smul_comm\nattribute [polynomial_nf] smul_add\nattribute [polynomial_nf] mul_smul\nattribute [polynomial_nf] smul_mul_assoc\n\nrun_cmd mk_simp_attr `polynomial_nf_2\nrun_cmd tactic.add_doc_string `simp_attr.polynomial_nf_2 \"Attribute for lemmas that are used in the conversion of mv_polynomial expressions to a normal form consisting of adds of sums of muls of mv_polynomials\"\n\nattribute [polynomial_nf_2] mul_add\nattribute [polynomial_nf_2] add_mul\nattribute [polynomial_nf_2] finset.sum_add_distrib\nattribute [polynomial_nf_2] sum_X_mul\nattribute [polynomial_nf_2] sum_C_mul\nattribute [polynomial_nf_2] rearrange_constants_right\nattribute [polynomial_nf_2] rearrange_constants_right_with_extra\nattribute [polynomial_nf_2] rearrange_sums_right\nattribute [polynomial_nf_2] rearrange_sums_right_with_extra\nattribute [polynomial_nf_2] C_mul_C\nattribute [polynomial_nf_2] finset.sum_hom\nattribute [polynomial_nf_2] mv_polynomial.smul_eq_C_mul\nattribute [polynomial_nf_2] mul_assoc\nattribute [polynomial_nf_2] finsupp.smul_sum\nattribute [polynomial_nf_2] mul_smul_comm\nattribute [polynomial_nf_2] smul_add\nattribute [polynomial_nf_2] mul_smul\nattribute [polynomial_nf_2] smul_mul_assoc\n\n\nrun_cmd mk_simp_attr `polynomial_nf_3\nrun_cmd tactic.add_doc_string `simp_attr.polynomial_nf_3 \"Attribute for lemmas that are used in the conversion of mv_polynomial expressions to a normal form consisting of adds of sums of muls of mv_polynomials\"\n\n\nattribute [polynomial_nf_3] mul_add\nattribute [polynomial_nf_3] add_mul\nattribute [polynomial_nf_3] finset.sum_add_distrib\n-- attribute [polynomial_nf_3] sum_X_mul\n-- attribute [polynomial_nf_3] sum_C_mul\nattribute [polynomial_nf_3] mul_sum_symm\nattribute [polynomial_nf_3] rearrange_constants_right\nattribute [polynomial_nf_3] rearrange_constants_right_with_extra\nattribute [polynomial_nf_3] rearrange_sums_right\nattribute [polynomial_nf_3] rearrange_sums_right_with_extra\nattribute [polynomial_nf_3] C_mul_C\nattribute [polynomial_nf_3] finset.sum_hom\nattribute [polynomial_nf_3] mv_polynomial.smul_eq_C_mul\nattribute [polynomial_nf_3] mul_assoc\n-- attribute [polynomial_nf_3] finsupp.smul_sum\n-- attribute [polynomial_nf_3] mul_smul_comm\n-- attribute [polynomial_nf_3] smul_add\n-- attribute [polynomial_nf_3] mul_smul\n-- attribute [polynomial_nf_3] smul_mul_assoc\n\n\nrun_cmd mk_simp_attr `rearrange\nrun_cmd tactic.add_doc_string `simp_attr.rearrange \"TODO a few rearrangement lemmas\"\n\n-- attribute [rearrange] rearrange1\n-- attribute [rearrange] rearrange2\n-- attribute [rearrange] rearrange_constants_right\n-- attribute [rearrange] rearrange_smul_right\n\n\n\n-- attribute [polynomial_nf] mv_polynomial.smul_eq_C_mul\n\n\n\nrun_cmd mk_simp_attr `coeff_simp\nrun_cmd tactic.add_doc_string `simp_attr.coeff_simp \"Attribute for lemmas that are used in the simplification of statements about coefficients of mv_polynomials\"\n\n\n-- attribute [coeff_simp] mv_polynomial.coeff_mul\n-- attribute [coeff_simp] single_1_antidiagonal\n-- attribute [coeff_simp] single_2_antidiagonal\n-- attribute [coeff_simp] finset.range\n-- attribute [coeff_simp] finset.image\n\n\n\n\n-- attribute [coeff_simp] mv_polynomial.X_pow_eq_single\nattribute [coeff_simp] mv_polynomial.coeff_sum\nattribute [coeff_simp] mv_polynomial.coeff_add\nattribute [coeff_simp] mv_polynomial.coeff_smul\n\nattribute [coeff_simp] mv_polynomial.coeff_C_mul\nattribute [coeff_simp] mv_polynomial.coeff_monomial\nattribute [coeff_simp] mv_polynomial.coeff_mul_X'\nattribute [coeff_simp] mv_polynomial.coeff_X'\nattribute [coeff_simp] coeff_X_mul'\nattribute [coeff_simp] coeff_X_pow_mul'\nattribute [coeff_simp] mv_polynomial.coeff_C\nattribute [coeff_simp] mv_polynomial.coeff_X_pow\n\n\nrun_cmd mk_simp_attr `finsupp_eq\nrun_cmd tactic.add_doc_string `simp_attr.finsupp_eq \"Attribute for lemmas that are used in the simplification of equality statements between finsupps\"\n\nattribute [finsupp_eq] finsupp.mem_support_iff\nattribute [finsupp_eq] finsupp.single_apply\nattribute [finsupp_eq] finsupp.add_apply\nattribute [finsupp_eq] finsupp.sub_apply\n\n\n\n-- Verison without polynomial manipulation\nattribute [integral_domain_simp] add_zero zero_add mul_zero zero_mul mul_one one_mul false_or or_false true_or or_true eq_self_iff_true ne.def mul_eq_zero eq_zero_of_zero_eq one_ne_zero mul_ne_zero_iff zero_sub_eq_iff not_true not_false\n\n\nrun_cmd mk_simp_attr `finsupp_simp\nrun_cmd tactic.add_doc_string `simp_attr.finsupp_simp \"Attribute for lemmas that are useful in finsupp statements\"\n\nlemma zero_ne_succ (n : \u2115) : 0 \u2260 nat.succ n := (nat.succ_ne_zero n).symm\n\n-- Verison without polynomial manipulation\nattribute [finsupp_simp] nat.one_ne_zero finsupp.single_eq_of_ne finsupp.single_eq_same add_zero if_true if_false  pi.add_apply pi.zero_apply\n  eq_self_iff_true not_true eq_zero_of_zero_eq if_false ne.def nat.sub_zero zero_add not_false_iff bit0_eq_zero\n  and_self finsupp.coe_nat_sub and_false finsupp.coe_add false_and pi.sub_apply finsupp.coe_zero\n    nat.succ_ne_zero zero_ne_succ nat.succ_inj' -- To simplify nat equalities\n  and_true true_and\n\nend", "meta": {"author": "BoltonBailey", "repo": "formal-snarks-project", "sha": "154414784f90a1e257162fcbdd7e805ecb2a49c2", "save_path": "github-repos/lean/BoltonBailey-formal-snarks-project", "path": "github-repos/lean/BoltonBailey-formal-snarks-project/formal-snarks-project-154414784f90a1e257162fcbdd7e805ecb2a49c2/src/attributes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38110848311199774}}
{"text": "import .order_emb \n\nopen_locale classical \n\nlemma subseq.reachable_trans (S\u2080 S\u2081 : subseq) (h: \u2200 i : \u2115, \u2203 k : \u2115, S\u2080 k = S\u2081 i):\n\u2203 T : subseq, S\u2081 = rel_embedding.trans T S\u2080:=\nbegin\n  choose! f hf using h, use f, intros a b abeq, have:= hf b, rw abeq.symm at this, \n  rw hf a at this, apply S\u2081.inj' this, intros a b, dsimp, have:=S\u2081.map_rel_iff',\n   nth_rewrite 1 \u2190 this, show f a \u2264 f b \u2194 S\u2081 a \u2264 S\u2081 b, rw \u2190hf a, rw \u2190 hf b, \n   exact S\u2080.map_rel_iff'.symm, exact a, exact b, ext, \n   simp [hf x],\nend\n\nlemma subseq.trans_refl {S : subseq} : S = (@rel_embedding.refl \u2115 (\u2264)).trans S:=\nbegin\n  ext, simp,\nend\n\nlemma subseq.trans_assoc {a b c : subseq} : \na.trans (b.trans c) = (a.trans b).trans c:=\nbegin\n  ext, simp,\nend\n\ntheorem constraints_apply (constraints: \u2115 \u2192 subseq \u2192 Prop)\n(constraints_stable: \u2200 g : \u2115, \u2200 S T : subseq, constraints g S \u2192 constraints g (rel_embedding.trans T S))\n(constraints_reachable: \u2200 (g : \u2115) (S : subseq), \u2203 (T : subseq), \n  (\u2200 i \u2264 g, T i = i) \u2227 constraints g (rel_embedding.trans T S)) :\n  \u2200 S : subseq, \u2203 T : subseq, \u2200 g, constraints g (rel_embedding.trans T S) :=\nbegin\n  intro S, \n  -- set s0 := @rel_embedding.refl \u2115 (\u2264),  \n  \n  choose! f hf using constraints_reachable,  \n\n  set f' : (\u2115 \u00d7 subseq) \u2192 (\u2115 \u00d7 subseq) := \u03bb p, \u27e8p.1 + 1, rel_embedding.trans (f p.1 p.2) p.2\u27e9 with hf',     \n\n  set seqs : \u2115 \u2192 subseq := \u03bb i, (f'^[i] \u27e80,S\u27e9).2 with h_seqs,   \n\n  set the_seq : \u2115 \u2192 \u2115 := \u03bb i, (seqs i) i with h_the_seq,  \n  have f'_iterate_count: \u2200 n, (f'^[n] (0, S)).1=n, \n    { intro n, induction n with n hn,\n    rw function.iterate_zero f', refl, rw function.iterate_succ_apply' f' n _, \n    nth_rewrite 0 hf', dsimp, rw hn,\n    },\n  have hterm : \u2200 i j, i \u2264 j \u2192 the_seq i = (seqs j) i, \n  { intros i j hle, \n    simp_rw [h_the_seq, h_seqs], \n    suffices: \u2200 d i j, i \u2264 j \u2192 j-i=d \u2192 ((f'^[i] (0, S)).snd) i = ((f'^[j] (0, S)).snd) i, \n    apply this (j-i) i j hle, refl, intro d, induction d with d hd, intros a b aleb asb, \n    have aeb: a = b, apply le_antisymm aleb (nat.sub_eq_zero_iff_le.1 asb), rw aeb,\n    intros a b aleb abdiff, have a_eq_a_d:= hd a (a+d) (nat.le_add_right _ _) _,\n    swap, rw add_comm, rw nat.add_sub_cancel d a, rw a_eq_a_d, \n    rw nat.sub_eq_iff_eq_add aleb at abdiff, rw nat.succ_add at abdiff, \n    rw add_comm at abdiff, rw abdiff, rw function.iterate_succ_apply' f' (a+d) _,\n   nth_rewrite 1 hf', dsimp, rw f'_iterate_count (a+d),\n    have:= (hf (a + d) (f'^[a + d] (0, S)).2).1 a (nat.le_add_right a d), rw this, \n  },\n  set S\u2081:subseq :={to_fun := the_seq, inj' := \n    begin\n      intros i j hij, \n    wlog hle: i \u2264 j, \n    rw [hterm i j hle, hterm j j rfl.le] at hij, apply (seqs j).inj' hij, \n    end\n  ,\n  map_rel_iff' := \n    begin\n      dsimp, \n    intros i j, wlog hle: i \u2264 j, \n     rw [hterm i j hle, hterm j j rfl.le], exact (seqs j).map_rel_iff', \n     rw [hterm j i hle, hterm i i rfl.le], exact (seqs i).map_rel_iff',\n    end,\n  },\n  have constr: \u2200 i, constraints i (f'^[i.succ] (0, S)).2,\n  intro i, rw function.iterate_succ_apply' f' i (0, S), nth_rewrite 0 hf', \n  dsimp, rw f'_iterate_count i, apply (hf i _).2, \n  have seqs_trans: \u2200 i j, i \u2264 j \u2192 \u2203 T : subseq, seqs j = rel_embedding.trans T (seqs i),\n  {\n    intros i j hle, suffices: \u2200 d i j : \u2115, i \u2264 j \u2192 j-i=d \u2192 \u2203 T : subseq, seqs j = rel_embedding.trans T (seqs i),\n    apply this (j-i) i j hle, refl, intro d, induction d with d hd, intros a b aleb asb,\n    have aeb: a = b, apply le_antisymm aleb (nat.sub_eq_zero_iff_le.1 asb), rw aeb,\n    use @rel_embedding.refl \u2115 (\u2264), exact subseq.trans_refl, intros a b aleb abdiff, \n    cases hd a (a+d) (nat.le_add_right _ _) _ with T\u2080 hT\u2080, \n    swap, rw add_comm, rw nat.add_sub_cancel d a, \n    rw nat.sub_eq_iff_eq_add aleb at abdiff, rw nat.succ_add at abdiff, \n    rw add_comm at abdiff, rw abdiff, \n    have seqs_succ: seqs ((a+d).succ) = (f' ((a+d), seqs (a+d))).2,\n    {\n      nth_rewrite 0 h_seqs, simp [function.iterate_succ_apply'], \n      rw f'_iterate_count (a+d),  \n    },\n    rw seqs_succ, rw hf', rw hT\u2080, dsimp, \n    use rel_embedding.trans (f (a + d) (rel_embedding.trans T\u2080 (seqs a))) T\u2080, \n    exact subseq.trans_assoc, \n  },\n  have reachable_from_step: \u2200 n : \u2115, \u2203 T : subseq, S\u2081= rel_embedding.trans T (f'^[n] (0, S)).2,\n  { intro n, \n    apply subseq.reachable_trans _ S\u2081, intro i, cases le_or_gt n i, \n    cases seqs_trans n i h with T ht, simp [h_the_seq, ht], simp [h_the_seq],\n    use i, \n    exact (hterm i n (le_of_lt h.lt)).symm },\n  cases reachable_from_step 0 with T\u2080 hT\u2080, \n  use T\u2080, intro g, simp at hT\u2080, rw \u2190hT\u2080, cases reachable_from_step g.succ with T\u2081 ht\u2081,\n  rw ht\u2081, \n  apply constraints_stable g _ T\u2081, \n  apply constr g,  \nend\n\n\n\n\n", "meta": {"author": "mehatamm", "repo": "ramsey", "sha": "aa7f50f32868c779b2ddc841caeb407ab8d39e57", "save_path": "github-repos/lean/mehatamm-ramsey", "path": "github-repos/lean/mehatamm-ramsey/ramsey-aa7f50f32868c779b2ddc841caeb407ab8d39e57/src/seq_refine.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38110848311199774}}
{"text": "/-\nCopyright (c) 2020 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n-/\nimport category_theory.natural_isomorphism\nimport category_theory.equivalence\nimport category_theory.eq_to_hom\n\n/-!\n# Quotient category\n\nConstructs the quotient of a category by an arbitrary family of relations on its hom-sets,\nby introducing a type synonym for the objects, and identifying homs as necessary.\n\nThis is analogous to 'the quotient of a group by the normal closure of a subset', rather\nthan 'the quotient of a group by a normal subgroup'. When taking the quotient by a congruence\nrelation, `functor_map_eq_iff` says that no unnecessary identifications have been made.\n-/\n\n/-- A `hom_rel` on `C` consists of a relation on every hom-set. -/\n@[derive inhabited]\ndef hom_rel (C) [quiver C] := \u03a0 \u2983X Y : C\u2984, (X \u27f6 Y) \u2192 (X \u27f6 Y) \u2192 Prop\n\nnamespace category_theory\n\nvariables {C : Type*} [category C] (r : hom_rel C)\n\ninclude r\n\n/-- A `hom_rel` is a congruence when it's an equivalence on every hom-set, and it can be composed\nfrom left and right. -/\nclass congruence : Prop :=\n(is_equiv : \u2200 {X Y}, is_equiv _ (@r X Y))\n(comp_left : \u2200 {X Y Z} (f : X \u27f6 Y) {g g' : Y \u27f6 Z}, r g g' \u2192 r (f \u226b g) (f \u226b g'))\n(comp_right : \u2200 {X Y Z} {f f' : X \u27f6 Y} (g : Y \u27f6 Z), r f f' \u2192 r (f \u226b g) (f' \u226b g))\n\nattribute [instance] congruence.is_equiv\n\n/-- A type synonym for `C`, thought of as the objects of the quotient category. -/\n@[ext]\nstructure quotient := (as : C)\n\ninstance [inhabited C] : inhabited (quotient r) := \u27e8 { as := default } \u27e9\n\nnamespace quotient\n\n/-- Generates the closure of a family of relations w.r.t. composition from left and right. -/\ninductive comp_closure \u2983s t : C\u2984 : (s \u27f6 t) \u2192 (s \u27f6 t) \u2192 Prop\n| intro {a b} (f : s \u27f6 a) (m\u2081 m\u2082 : a \u27f6 b) (g : b \u27f6 t) (h : r m\u2081 m\u2082) :\n  comp_closure (f \u226b m\u2081 \u226b g) (f \u226b m\u2082 \u226b g)\n\nlemma comp_left {a b c : C} (f : a \u27f6 b) : \u03a0 (g\u2081 g\u2082 : b \u27f6 c) (h : comp_closure r g\u2081 g\u2082),\n  comp_closure r (f \u226b g\u2081) (f \u226b g\u2082)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro (f \u226b x) m\u2081 m\u2082 y h\n\nlemma comp_right {a b c : C} (g : b \u27f6 c) : \u03a0 (f\u2081 f\u2082 : a \u27f6 b) (h : comp_closure r f\u2081 f\u2082),\n  comp_closure r (f\u2081 \u226b g) (f\u2082 \u226b g)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro x m\u2081 m\u2082 (y \u226b g) h\n\n/-- Hom-sets of the quotient category. -/\ndef hom (s t : quotient r) := quot $ @comp_closure C _ r s.as t.as\n\ninstance (a : quotient r) : inhabited (hom r a a) := \u27e8quot.mk _ (\ud835\udfd9 a.as)\u27e9\n\n/-- Composition in the quotient category. -/\ndef comp \u2983a b c : quotient r\u2984 : hom r a b \u2192 hom r b c \u2192 hom r a c :=\n\u03bb hf hg, quot.lift_on hf ( \u03bb f, quot.lift_on hg (\u03bb g, quot.mk _ (f \u226b g))\n  (\u03bb g\u2081 g\u2082 h, quot.sound $ comp_left r f g\u2081 g\u2082 h) )\n  (\u03bb f\u2081 f\u2082 h, quot.induction_on hg $ \u03bb g, quot.sound $ comp_right r g f\u2081 f\u2082 h)\n\n@[simp]\nlemma comp_mk {a b c : quotient r} (f : a.as \u27f6 b.as) (g : b.as \u27f6 c.as) :\n  comp r (quot.mk _ f) (quot.mk _ g) = quot.mk _ (f \u226b g) := rfl\n\ninstance category : category (quotient r) :=\n{ hom := hom r,\n  id := \u03bb a, quot.mk _ (\ud835\udfd9 a.as),\n  comp := comp r }\n\n/-- The functor from a category to its quotient. -/\n@[simps]\ndef functor : C \u2964 quotient r :=\n{ obj := \u03bb a, { as := a },\n  map := \u03bb _ _ f, quot.mk _ f }\n\nnoncomputable instance : full (functor r) :=\n{ preimage := \u03bb X Y f, quot.out f, }\n\ninstance : ess_surj (functor r) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y.as, \u27e8eq_to_iso (by { ext, refl, })\u27e9\u27e9 }\n\nprotected lemma induction {P : \u03a0 {a b : quotient r}, (a \u27f6 b) \u2192 Prop}\n  (h : \u2200 {x y : C} (f : x \u27f6 y), P ((functor r).map f)) :\n  \u2200 {a b : quotient r} (f : a \u27f6 b), P f :=\nby { rintros \u27e8x\u27e9 \u27e8y\u27e9 \u27e8f\u27e9, exact h f, }\n\nprotected lemma sound {a b : C} {f\u2081 f\u2082 : a \u27f6 b} (h : r f\u2081 f\u2082) :\n  (functor r).map f\u2081 = (functor r).map f\u2082 :=\nby simpa using quot.sound (comp_closure.intro (\ud835\udfd9 a) f\u2081 f\u2082 (\ud835\udfd9 b) h)\n\n\n\nvariables {D : Type*} [category D]\n  (F : C \u2964 D)\n  (H : \u2200 (x y : C) (f\u2081 f\u2082 : x \u27f6 y), r f\u2081 f\u2082 \u2192 F.map f\u2081 = F.map f\u2082)\ninclude H\n\n/-- The induced functor on the quotient category. -/\n@[simps]\ndef lift : quotient r \u2964 D :=\n{ obj := \u03bb a, F.obj a.as,\n  map := \u03bb a b hf, quot.lift_on hf (\u03bb f, F.map f)\n    (by { rintros _ _ \u27e8_, _, _, _, _, _, h\u27e9, simp [H _ _ _ _ h], }),\n  map_id' := \u03bb a, F.map_id a.as,\n  map_comp' := by { rintros a b c \u27e8f\u27e9 \u27e8g\u27e9, exact F.map_comp f g, } }\n\n/-- The original functor factors through the induced functor. -/\ndef lift.is_lift : (functor r) \u22d9 lift r F H \u2245 F :=\nnat_iso.of_components (\u03bb X, iso.refl _) (by tidy)\n\n@[simp]\nlemma lift.is_lift_hom (X : C) : (lift.is_lift r F H).hom.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n@[simp]\nlemma lift.is_lift_inv (X : C) : (lift.is_lift r F H).inv.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n\nlemma lift_map_functor_map {X Y : C} (f : X \u27f6 Y) :\n  (lift r F H).map ((functor r).map f) = F.map f :=\nby { rw \u2190(nat_iso.naturality_1 (lift.is_lift r F H)), dsimp, simp, }\n\nend quotient\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/quotient.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.38110848311199774}}
{"text": "inductive Foo (n: Nat)\n  | mk: n = 0 \u2192 Foo n\n\nexample (h: Foo x): x + 1 = 1 := by\n  cases h\n  subst x\n  rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1169.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737473266735, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.38109934875713497}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.bifunctor\nimport control.traversable.basic\n\n/-!\n# Bitraversable type class\n\nType class for traversing bifunctors. The concepts and laws are taken from\n<https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\nSimple examples of `bitraversable` are `prod` and `sum`. A more elaborate example is\nto define an a-list as:\n\n```\ndef alist (key val : Type) := list (key \u00d7 val)\n```\n\nThen we can use `f : key \u2192 io key'` and `g : val \u2192 io val'` to manipulate the `alist`'s key\nand value respectively with `bitraverse f g : alist key val \u2192 io (alist key' val')`\n\n## Main definitions\n  * bitraversable - exposes the `bitraverse` function\n  * is_lawful_bitraversable - laws similar to is_lawful_traversable\n\n## Tags\n\ntraversable bitraversable iterator functor bifunctor applicative\n\n-/\n\nuniverses u\n\nclass bitraversable (t : Type u \u2192 Type u \u2192 Type u)\n  extends bifunctor t :=\n(bitraverse : \u03a0 {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b1' \u03b2 \u03b2'},\n  (\u03b1 \u2192 m \u03b1') \u2192 (\u03b2 \u2192 m \u03b2') \u2192 t \u03b1 \u03b2 \u2192 m (t \u03b1' \u03b2'))\nexport bitraversable ( bitraverse )\n\ndef bisequence {t m} [bitraversable t] [applicative m] {\u03b1 \u03b2} : t (m \u03b1) (m \u03b2) \u2192 m (t \u03b1 \u03b2) :=\nbitraverse id id\n\nopen functor\n\nclass is_lawful_bitraversable (t : Type u \u2192 Type u \u2192 Type u) [bitraversable t]\n  extends is_lawful_bifunctor t :=\n(id_bitraverse : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), bitraverse id.mk id.mk x = id.mk x )\n(comp_bitraverse : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3'} (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3')\n    (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2') (x : t \u03b1 \u03b1'),\n  bitraverse (comp.mk \u2218 map f \u2218 g) (comp.mk \u2218 map f' \u2218 g') x =\n  comp.mk (bitraverse f f' <$> bitraverse g g' x) )\n(bitraverse_eq_bimap_id : \u2200 {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') (x : t \u03b1 \u03b1'),\n   bitraverse (id.mk \u2218 f) (id.mk \u2218 f') x = id.mk (bimap f f' x))\n(binaturality : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 \u03b1' \u03b2 \u03b2'}\n    (f : \u03b1 \u2192 F \u03b2) (f' : \u03b1' \u2192 F \u03b2') (x : t \u03b1 \u03b1'),\n  \u03b7 (bitraverse f f' x) = bitraverse (@\u03b7 _ \u2218 f) (@\u03b7 _ \u2218 f') x)\n\nexport is_lawful_bitraversable ( id_bitraverse comp_bitraverse\n                                 bitraverse_eq_bimap_id  )\nopen is_lawful_bitraversable\n\nattribute [higher_order bitraverse_id_id] id_bitraverse\nattribute [higher_order bitraverse_comp] comp_bitraverse\nattribute [higher_order] binaturality bitraverse_eq_bimap_id\n\nexport is_lawful_bitraversable (bitraverse_id_id bitraverse_comp)\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/bitraversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947425132314, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3806923378696532}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nFunctors with two arguments\n-/\n\nimport data.sum\n       category.basic category.functor\n       tactic.basic\n\nuniverses u\u2080 u\u2081 u\u2082 v\u2080 v\u2081 v\u2082\n\nclass bifunctor (F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082) :=\n(bimap : \u03a0 {\u03b1 \u03b1' \u03b2 \u03b2'}, (\u03b1 \u2192 \u03b1') \u2192 (\u03b2 \u2192 \u03b2') \u2192 F \u03b1 \u03b2 \u2192 F \u03b1' \u03b2')\nexport bifunctor ( bimap )\n\nclass is_lawful_bifunctor (F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082) [bifunctor F] :=\n(id_bimap : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), bimap id id x = x)\n(bimap_bimap : \u03a0 {\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082} (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b1\u2081 \u2192 \u03b1\u2082)\n  (g : \u03b2\u2080 \u2192 \u03b2\u2081) (g' : \u03b2\u2081 \u2192 \u03b2\u2082) (x : F \u03b1\u2080 \u03b2\u2080),\n  bimap f' g' (bimap f g x) = bimap (f' \u2218 f) (g' \u2218 g) x)\n\nexport is_lawful_bifunctor (id_bimap bimap_bimap)\n\nattribute [higher_order bimap_id_id] id_bimap\nattribute [higher_order bimap_comp_bimap] bimap_bimap\n\nexport is_lawful_bifunctor (bimap_id_id bimap_comp_bimap)\nvariables {F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082} [bifunctor F]\n\nnamespace bifunctor\n\n@[reducible]\ndef fst {\u03b1 \u03b1' \u03b2} (f : \u03b1 \u2192 \u03b1') : F \u03b1 \u03b2 \u2192 F \u03b1' \u03b2 :=\nbimap f id\n\n@[reducible]\ndef snd {\u03b1 \u03b2 \u03b2'} (f : \u03b2 \u2192 \u03b2') : F \u03b1 \u03b2 \u2192 F \u03b1 \u03b2' :=\nbimap id f\n\nvariable [is_lawful_bifunctor F]\n\n@[higher_order fst_id]\nlemma id_fst : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), fst id x = x :=\n@id_bimap _ _ _\n\n@[higher_order snd_id]\nlemma id_snd : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), snd id x = x :=\n@id_bimap _ _ _\n\n@[higher_order fst_comp_fst]\nlemma comp_fst {\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2}\n  (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b1\u2081 \u2192 \u03b1\u2082) (x : F \u03b1\u2080 \u03b2) :\n  fst f' (fst f x) = fst (f' \u2218 f)  x :=\nby simp [fst,bimap_bimap]\n\n@[higher_order fst_comp_snd]\nlemma fst_snd {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b2\u2080 \u2192 \u03b2\u2081) (x : F \u03b1\u2080 \u03b2\u2080) :\n  fst f (snd f' x) = bimap f f' x :=\nby simp [fst,bimap_bimap]\n\n@[higher_order snd_comp_fst]\nlemma snd_fst {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b2\u2080 \u2192 \u03b2\u2081) (x : F \u03b1\u2080 \u03b2\u2080) :\n  snd f' (fst f x) = bimap f f' x :=\nby simp [snd,bimap_bimap]\n\n@[higher_order snd_comp_snd]\nlemma comp_snd {\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082}\n  (g : \u03b2\u2080 \u2192 \u03b2\u2081) (g' : \u03b2\u2081 \u2192 \u03b2\u2082) (x : F \u03b1 \u03b2\u2080) :\n  snd g' (snd g x) = snd (g' \u2218 g) x :=\nby simp [snd,bimap_bimap]\n\nattribute [functor_norm] bimap_bimap comp_snd comp_fst\n  snd_comp_snd snd_comp_fst fst_comp_snd fst_comp_fst bimap_comp_bimap\n  bimap_id_id fst_id snd_id\n\ndef bicompl (F : Type* \u2192 Type* \u2192 Type*) (G : Type* \u2192 Type*) (H : Type* \u2192 Type*) (\u03b1 \u03b2) :=\nF (G \u03b1) (H \u03b2)\n\ndef bicompr (F : Type* \u2192 Type*) (G : Type* \u2192 Type* \u2192 Type*) (\u03b1 \u03b2) :=\nF (G \u03b1 \u03b2)\n\nend bifunctor\nopen functor\ninstance : bifunctor prod :=\n{ bimap := @prod.map }\n\ninstance : is_lawful_bifunctor prod :=\nby refine { .. }; intros; cases x; refl\n\ninstance bifunctor.const : bifunctor const :=\n{ bimap := (\u03bb \u03b1 \u03b1' \u03b2 \u03b2 f _, f) }\n\ninstance is_lawful_bifunctor.const : is_lawful_bifunctor const  :=\nby refine { .. }; intros; refl\n\ninstance bifunctor.flip : bifunctor (flip F) :=\n{ bimap := (\u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (bimap f' f x : F \u03b2' \u03b1')) }\n\ninstance is_lawful_bifunctor.flip [is_lawful_bifunctor F] : is_lawful_bifunctor (flip F)  :=\nby refine { .. }; intros; simp [bimap] with functor_norm\n\ninstance : bifunctor sum :=\n{ bimap := @sum.map }\n\ninstance : is_lawful_bifunctor sum :=\nby refine { .. }; intros; cases x; refl\n\nopen bifunctor functor\n\n@[priority 0]\ninstance bifunctor.functor {\u03b1} : functor (F \u03b1) :=\n{ map := \u03bb _ _, snd }\n\n@[priority 0]\ninstance bifunctor.is_lawful_functor [is_lawful_bifunctor F] {\u03b1} : is_lawful_functor (F \u03b1) :=\nby refine {..}; intros; simp [functor.map] with functor_norm\n\nsection bicompl\n\nvariables (G : Type* \u2192 Type u\u2080) (H : Type* \u2192 Type u\u2081) [functor G] [functor H]\n\ninstance : bifunctor (bicompl F G H) :=\n{ bimap := \u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (bimap (map f) (map f') x : F (G \u03b1') (H \u03b2')) }\n\ninstance [is_lawful_functor G]  [is_lawful_functor H] [is_lawful_bifunctor F] :\n  is_lawful_bifunctor (bicompl F G H) :=\nby constructor; intros; simp [bimap,map_id,map_comp_map] with functor_norm\n\nend bicompl\nsection bicompr\n\nvariables (G : Type u\u2082 \u2192 Type*) [functor G]\n\ninstance : bifunctor (bicompr G F) :=\n{ bimap := \u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (map (bimap f f') x : G (F \u03b1' \u03b2')) }\n\ninstance [is_lawful_functor G] [is_lawful_bifunctor F] :\n  is_lawful_bifunctor (bicompr G F) :=\nby constructor; intros; simp [bimap] with functor_norm\n\nend bicompr\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category/bifunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443134, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.38069233003885894}}
{"text": "import data.fin.basic\nimport data.fintype.basic\nimport data.list\nimport ..automata_typeclass\n\nvariables {Sigma : Type} [decidable_eq Sigma]\n\ndef append_lang (P Q : lang Sigma) : lang Sigma \n:= \u03bb w, \u2203 u v : word Sigma, P u \u2227 Q v \u2227 w = u ++ v    \n\ndef append_\u03b5_nfa {Sigma : Type*} [decidable_eq Sigma] (A : \u03b5_nfa Sigma) (B : \u03b5_nfa Sigma) : \u03b5_nfa Sigma :=\n  {\n    Q := A.Q \u2295 B.Q,\n    finQ := @sum.fintype A.Q B.Q A.finQ B.finQ,\n    decQ := @sum.decidable_eq A.Q A.decQ B.Q B.decQ,\n    inits := \u03bb q, sum.cases_on q A.inits (\u03bb _, false),\n    decI := begin\n      assume a,\n      cases a;\n      simp at *,\n      exact A.decI a,\n      exact is_false id,\n    end,\n    final := \u03bb q, sum.cases_on q (\u03bb _, false) B.final,\n    decF := begin\n      assume a,\n      cases a;\n      simp at *,\n      exact is_false id,\n      exact B.decF a,\n    end,\n    \u03b4 := \u03bb a x b, match a, b with\n        | (sum.inl a), (sum.inl b) := A.\u03b4 a x b\n        | (sum.inl a), (sum.inr b) := A.final a \u2227 B.inits b \u2227 x = none\n        | (sum.inr a), (sum.inl b) := false\n        | (sum.inr a), (sum.inr b) := B.\u03b4 a x b\n      end,\n    decD := begin\n      assume a,\n      cases a with ax b, cases ax with a x,\n      cases a; cases b; dsimp [sigma.uncurry],\n      exact A.decD \u27e8\u27e8a, x\u27e9, b\u27e9,\n      {\n        letI dF := A.decF,\n        letI dI := B.decI,\n        letI deq := @sum.decidable_eq A.Q A.decQ B.Q B.decQ,\n        unfold_aux,\n        apply_instance,\n      },\n      exact is_false id,\n      exact B.decD \u27e8\u27e8a, x\u27e9, b\u27e9,\n    end,\n  }\n\nlemma left_append : \u2200 A B : \u03b5_nfa Sigma, \u2200 w : word Sigma, \u2200 q0 q1 : A.Q,\n  \u03b5_nfa_\u03b4_star A q0 w q1 \u2192 \u03b5_nfa_\u03b4_star (append_\u03b5_nfa A B) (sum.inl q0) w (sum.inl q1) :=\nbegin\n  assume A B w q0 q1 h,\n  induction h,\n  case \u03b5_nfa_\u03b4_star.empty : q\n  {\n    constructor,\n  },\n  case \u03b5_nfa_\u03b4_star.step : q11 q22 q33 x w h1 h2 ih\n  {\n    fconstructor,\n    exact (sum.inl q22),\n    exact h1,\n    exact ih,\n  },\n  case \u03b5_nfa_\u03b4_star.epsilon : q11 q22 q33 w h1 h2 ih\n  {\n    fconstructor,\n    exact (sum.inl q22),\n    exact h1,\n    exact ih,\n  }\nend\n\nlemma right_append : \u2200 A B : \u03b5_nfa Sigma, \u2200 w : word Sigma, \u2200 q0 q1 : B.Q,\n  \u03b5_nfa_\u03b4_star B q0 w q1 \u2192 \u03b5_nfa_\u03b4_star (append_\u03b5_nfa A B) (sum.inr q0) w (sum.inr q1) :=\nbegin\n  assume A B w q0 q1 h,\n  induction h,\n  case \u03b5_nfa_\u03b4_star.empty : q\n  {\n    constructor,\n  },\n  case \u03b5_nfa_\u03b4_star.step : q11 q22 q33 x w h1 h2 ih\n  {\n    fconstructor,\n    exact (sum.inr q22),\n    exact h1,\n    exact ih,\n  },\n  case \u03b5_nfa_\u03b4_star.epsilon : q11 q22 q33 w h1 h2 ih\n  {\n    fconstructor,\n    exact (sum.inr q22),\n    exact h1,\n    exact ih,\n  }\nend\n\nlemma append_lem\u1d63 : \u2200 A B : \u03b5_nfa Sigma, \u2200 u v : word Sigma, \u2200 q0 : A.Q, \u2200 q1 : B.Q, \n  (\u2203 q2 : A.Q, \u2203 q3 : B.Q, A.final q2 \u2227 B.inits q3\n   \u2227 \u03b5_nfa_\u03b4_star A q0 u q2 \u2227 \u03b5_nfa_\u03b4_star B q3 v q1) \u2192\n  \u03b5_nfa_\u03b4_star (append_\u03b5_nfa A B) (sum.inl q0) (u ++ v) (sum.inr q1) :=\nbegin\n  assume A B u v q0 q1 h,\n  cases h with q2 h, cases h with q3 h,\n  cases h with Afinal h, cases h with Binits h,\n  cases h with Astar Bstar,\n  have a2b : (append_\u03b5_nfa A B).\u03b4 (sum.inl q2) none (sum.inr q3),\n  {\n    constructor, exact Afinal,\n    constructor, exact Binits,\n    refl,\n  },\n  induction Astar, \n  case \u03b5_nfa_\u03b4_star.empty : q\n  {\n    simp,fconstructor,\n    exact (sum.inr q3),\n    exact a2b, exact right_append A B v q3 q1 Bstar,\n  },\n  case \u03b5_nfa_\u03b4_star.step : q00 q11 q22 x w h1 h2 ih\n  {\n    fconstructor,\n    exact sum.inl q11,\n    exact h1,\n    apply ih,\n    exact Afinal,\n    exact a2b,\n  },\n  case \u03b5_nfa_\u03b4_star.epsilon : q00 q11 q22 w h1 h2 ih\n  {\n    fconstructor,\n    exact sum.inl q11,\n    exact h1,\n    apply ih,\n    exact Afinal,\n    exact a2b,\n  }\nend\n\nlemma append_lem : \u2200 A B : \u03b5_nfa Sigma, \u2200 w : word Sigma, \u2200 q0 q1 : (append_\u03b5_nfa A B).Q,\n  \u03b5_nfa_\u03b4_star (append_\u03b5_nfa A B) q0 w q1 \u2192\n  (\u2203 q0' : A.Q, \u2203 q1' : B.Q, q0 = sum.inl q0' \u2227 q1 = sum.inr q1' \n   \u2227 \u2203 u v : word Sigma, \u2203 q2' : A.Q, \u2203 q3' : B.Q, A.final q2' \u2227 B.inits q3'\n   \u2227 \u03b5_nfa_\u03b4_star A q0' u q2' \u2227 \u03b5_nfa_\u03b4_star B q3' v q1'\n   \u2227 u ++ v = w)\n  \u2228 (\u2203 q0' q1' : A.Q, q0 = sum.inl q0' \u2227 q1 = sum.inl q1'\n     \u2227 \u03b5_nfa_\u03b4_star A q0' w q1')\n  \u2228 (\u2203 q0' q1' : B.Q, q0 = sum.inr q0' \u2227 q1 = sum.inr q1'\n     \u2227 \u03b5_nfa_\u03b4_star B q0' w q1') :=\nbegin\n  assume A B w q0 q1 h,\n  induction h,\n  case \u03b5_nfa_\u03b4_star.empty : q\n  {\n    cases q,\n    right, left, existsi [q, q],\n    simp, constructor,\n    right, right, existsi [q, q],\n    simp, constructor,\n  },\n  case \u03b5_nfa_\u03b4_star.step : q00 q11 q22 x w h1 h2 ih\n  {\n    cases q00,\n    {\n      cases q11,\n      {\n        cases ih,\n        {\n          left,\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with eq1 ih, cases ih with eq2 ih,\n          cases ih with u ih, cases ih with v ih,\n          cases ih with q2' ih, cases ih with q3 ih,\n          cases ih with Afinal ih, cases ih with Binits ih,\n          cases ih with Astar ih, cases ih with Bstar split_eq,\n          existsi [q00, q1'], constructor, refl,\n          constructor, exact eq2, \n          existsi [(x :: u), v],\n          existsi [q2', q3],\n          constructor, exact Afinal,\n          constructor, exact Binits,\n          constructor,\n          {\n            fconstructor,\n            exact q11, exact h1,\n            injection eq1 with eq1, rw eq1,\n            exact Astar,\n          },\n          constructor,\n          {\n            exact Bstar,\n          },\n          {\n            rw\u2190 split_eq,\n            exact list.cons_append x u v,\n          }\n        },\n        {\n          cases ih,\n          {\n            right, left,\n            cases ih with q0' ih, cases ih with q1' ih,\n            existsi [q00, q1'],\n            cases ih with eq1 ih, cases ih with eq2 Astar,\n            simp, constructor, exact eq2,\n            fconstructor,\n            exact q11, exact h1,\n            injection eq1 with eq1, rw eq1, exact Astar,\n          },\n          {\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with f _, cases f,\n          }\n        }\n      },\n      {\n        cases h1 with _ h1, cases h1 with _ f, cases f,\n      }\n    },\n    {\n      cases q11,\n      {\n        cases h1,\n      },\n      {\n        cases ih,\n        {\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with f _, cases f,\n        },\n        {\n          cases ih,\n          {\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with f _, cases f,\n          },\n          {\n            right, right,\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with eq1 ih, cases ih with eq2 Bstar,\n            existsi [q00, q1'],\n            simp, constructor, exact eq2,\n            fconstructor,\n            exact q11,\n            exact h1, injection eq1 with eq1, rw eq1,\n            exact Bstar,\n          }\n        },\n      }\n    },\n  },\n  case \u03b5_nfa_\u03b4_star.epsilon : q00 q11 q22 w h1 h2 ih\n  {\n    cases q00,\n    {\n      cases q11,\n      {\n        cases ih,\n        {\n          left,\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with eq1 ih, cases ih with eq2 ih,\n          cases ih with u ih, cases ih with v ih,\n          cases ih with q2' ih, cases ih with q3 ih,\n          cases ih with Afinal ih, cases ih with Binits ih,\n          cases ih with Astar ih, cases ih with Bstar split_eq,\n          existsi [q00, q1'], constructor, refl,\n          constructor, exact eq2, \n          existsi [u, v],\n          existsi [q2', q3],\n          constructor, exact Afinal,\n          constructor, exact Binits,\n          constructor,\n          {\n            fconstructor,\n            exact q11, exact h1,\n            injection eq1 with eq1, rw eq1,\n            exact Astar,\n          },\n          constructor,\n          {\n            exact Bstar,\n          },\n          {\n            rw\u2190 split_eq,\n          }\n        },\n        {\n          cases ih,\n          {\n            right, left,\n            cases ih with q0' ih, cases ih with q1' ih,\n            existsi [q00, q1'],\n            cases ih with eq1 ih, cases ih with eq2 Astar,\n            simp, constructor, exact eq2,\n            fconstructor,\n            exact q11, exact h1,\n            injection eq1 with eq1, rw eq1, exact Astar,\n          },\n          {\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with f _, cases f,\n          }\n        }\n      },\n      {\n        left, \n        cases ih,\n        {\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with f _, cases f,\n        },\n        cases ih,\n        {\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with f _, cases f,\n        },\n        {\n          cases h1 with Afinal h1, cases h1 with Binits _,\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with eq1 ih, cases ih with eq2 Bstar,\n          existsi [q00, q1'],\n          constructor, refl,\n          constructor, exact eq2,\n          existsi [[], w, q00, q11],\n          constructor, exact Afinal,\n          constructor, exact Binits,\n          constructor, constructor,\n          constructor, injection eq1 with eq1, rw eq1, exact Bstar,\n          exact list.nil_append w,\n        }\n      }\n    },\n    {\n      cases q11,\n      {\n        cases h1,\n      },\n      {\n        cases ih,\n        {\n          cases ih with q0' ih, cases ih with q1' ih,\n          cases ih with f _, cases f,\n        },\n        {\n          cases ih,\n          {\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with f _, cases f,\n          },\n          {\n            right, right,\n            cases ih with q0' ih, cases ih with q1' ih,\n            cases ih with eq1 ih, cases ih with eq2 Bstar,\n            existsi [q00, q1'],\n            simp, constructor, exact eq2,\n            fconstructor,\n            exact q11,\n            exact h1, injection eq1 with eq1, rw eq1,\n            exact Bstar,\n          }\n        },\n      }\n    },\n  },\nend\n\nlemma append_\u03b5_nfa_lang : \u2200 A B : \u03b5_nfa Sigma, \u2200 w : word Sigma,\n  \u03b5_nfa_lang (append_\u03b5_nfa A B) w \u2194 append_lang (\u03b5_nfa_lang A) (\u03b5_nfa_lang B) w :=\nbegin\n  assume A B w,\n  constructor,\n  {\n    dsimp [\u03b5_nfa_lang, append_lang],\n    assume h,\n    cases h with q0 h, cases h with q1 h,\n    cases h with h1 h, cases h with h2 h3,\n    have g := append_lem A B w q0 q1 h2,\n    cases g,\n    {\n      cases g with q0' g, cases g with q1' g,\n      cases g with eq1 g, cases g with eq2 g,\n      cases g with u g, cases g with v g,\n      cases g with q2' g, cases g with q3' g,\n      cases g with Afinal g, cases g with Binits ih,\n      cases ih with Astar g, cases g with Bstar split_eq,\n      existsi [u, v],\n      constructor,\n      {\n        existsi [q0', q2'],\n        constructor, finish,\n        constructor, exact Astar, exact Afinal,\n      },\n      constructor,\n      {\n        existsi [q3', q1'],\n        constructor, exact Binits,\n        constructor, exact Bstar, finish,\n      },\n      exact eq.symm split_eq,\n    },\n    cases g,\n    {\n      cases g with q0' g, cases g with q1' g,\n      cases g with eq1 g, cases g with eq2 Astar,\n      existsi [w, []],\n      rw eq2 at h3, cases h3,\n    },\n    {\n      cases g with q0' g, cases g with q1' g,\n      cases g with eq1 g, cases g with eq2 Astar,\n      rw eq1 at h1, cases h1,\n    }\n  },\n  {\n    dsimp [\u03b5_nfa_lang, append_lang],\n    assume h,\n    cases h with u h, cases h with v h,\n    cases h with h1 h2, cases h2 with h2 h3,\n    cases h1 with q0 h1, cases h1 with q2 h1,\n    cases h2 with q3 h2, cases h2 with q1 h2,\n    existsi [sum.inl q0, sum.inr q1],\n    constructor, exact (and.elim_left h1),\n    constructor, \n    {\n      let h11 : \u03b5_nfa_\u03b4_star A q0 u q2, exact (and.elim_left $ and.elim_right $ h1),\n      let h22 : \u03b5_nfa_\u03b4_star B q3 v q1, exact (and.elim_left $ and.elim_right $ h2),\n      rw h3,\n      apply append_lem\u1d63 A B u v q0 q1,\n      existsi [q2, q3],\n      constructor, exact (and.elim_right $ and.elim_right $ h1),\n      constructor, exact (and.elim_left h2),\n      constructor, exact h11,\n      exact h22,\n    },\n    exact (and.elim_right (and.elim_right h2)),\n  }\nend", "meta": {"author": "Filikec", "repo": "automaton-in-lean", "sha": "4f94b969f7780964f705fde2aa11d4085736cd7c", "save_path": "github-repos/lean/Filikec-automaton-in-lean", "path": "github-repos/lean/Filikec-automaton-in-lean/automaton-in-lean-4f94b969f7780964f705fde2aa11d4085736cd7c/src/regex/append.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.38069233003885883}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johannes H\u00f6lzl, Reid Barton, Sean Leather\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.category.default\nimport Mathlib.PostPort\n\nuniverses u v l \n\nnamespace Mathlib\n\n/-!\n# Bundled types\n\n`bundled c` provides a uniform structure for bundling a type equipped with a type class.\n\nWe provide `category` instances for these in `category_theory/unbundled_hom.lean`\n(for categories with unbundled homs, e.g. topological spaces)\nand in `category_theory/bundled_hom.lean` (for categories with bundled homs, e.g. monoids).\n-/\n\nnamespace category_theory\n\n\n/-- `bundled` is a type bundled with a type class instance for that type. Only\nthe type class is exposed as a parameter. -/\nstructure bundled (c : Type u \u2192 Type v) where\n  \u03b1 : Type u\n  str :\n    autoParam (c \u03b1)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.apply_instance\")\n        (Lean.Name.mkStr (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n          \"apply_instance\")\n        [])\n\nnamespace bundled\n\n\n/-- A generic function for lifting a type equipped with an instance to a bundled object. -/\n-- Usually explicit instances will provide their own version of this, e.g. `Mon.of` and `Top.of`.\n\ndef of {c : Type u \u2192 Type v} (\u03b1 : Type u) [str : c \u03b1] : bundled c := mk \u03b1\n\nprotected instance has_coe_to_sort {c : Type u \u2192 Type v} : has_coe_to_sort (bundled c) :=\n  has_coe_to_sort.mk (Type u) \u03b1\n\n@[simp] theorem coe_mk {c : Type u \u2192 Type v} (\u03b1 : Type u)\n    (str :\n      autoParam (c \u03b1)\n        (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.tactic.apply_instance\")\n          (Lean.Name.mkStr\n            (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"tactic\")\n            \"apply_instance\")\n          [])) :\n    \u21a5(mk \u03b1) = \u03b1 :=\n  rfl\n\n/-\n`bundled.map` is reducible so that, if we define a category\n\n  def Ring : Type (u+1) := induced_category SemiRing (bundled.map @ring.to_semiring)\n\ninstance search is able to \"see\" that a morphism R \u27f6 S in Ring is really\na (semi)ring homomorphism from R.\u03b1 to S.\u03b1, and not merely from\n`(bundled.map @ring.to_semiring R).\u03b1` to `(bundled.map @ring.to_semiring S).\u03b1`.\n-/\n\n/-- Map over the bundled structure -/\ndef map {c : Type u \u2192 Type v} {d : Type u \u2192 Type v} (f : {\u03b1 : Type u} \u2192 c \u03b1 \u2192 d \u03b1) (b : bundled c) :\n    bundled d :=\n  mk \u21a5b\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/concrete_category/bundled_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.38069232612346166}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.nat.basic\n \n\nuniverses l \n\nnamespace Mathlib\n\ndef is_valid_char (n : \u2115) :=\n  n < bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit1 (bit1 (bit0 (bit1 1)))))))))))))) \u2228\n    bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit0 (bit1 1)))))))))))))) < n \u2227\n      n <\n        bit0\n          (bit0\n            (bit0\n              (bit0\n                (bit0\n                  (bit0\n                    (bit0\n                      (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit1 (bit0 (bit0 (bit0 1)))))))))))))))))))\n\ntheorem is_valid_char_range_1 (n : \u2115) (h : n < bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit1 (bit1 (bit0 (bit1 1))))))))))))))) : is_valid_char n :=\n  Or.inl h\n\ntheorem is_valid_char_range_2 (n : \u2115) (h\u2081 : bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 (bit0 (bit1 1)))))))))))))) < n) (h\u2082 : n <\n  bit0\n    (bit0\n      (bit0\n        (bit0\n          (bit0\n            (bit0\n              (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit1 (bit0 (bit0 (bit0 1)))))))))))))))))))) : is_valid_char n :=\n  Or.inr { left := h\u2081, right := h\u2082 }\n\n/-- The `char` type represents an unicode scalar value.\n    See http://www.unicode.org/glossary/#unicode_scalar_value). -/\nstructure char \nwhere\n  val : \u2115\n  valid : is_valid_char val\n\nprotected instance char.has_sizeof : SizeOf char :=\n  { sizeOf := fun (c : char) => char.val c }\n\nnamespace char\n\n\nprotected def lt (a : char) (b : char) :=\n  val a < val b\n\nprotected def le (a : char) (b : char) :=\n  val a \u2264 val b\n\nprotected instance has_lt : HasLess char :=\n  { Less := char.lt }\n\nprotected instance has_le : HasLessEq char :=\n  { LessEq := char.le }\n\nprotected instance decidable_lt (a : char) (b : char) : Decidable (a < b) :=\n  nat.decidable_lt (val a) (val b)\n\nprotected instance decidable_le (a : char) (b : char) : Decidable (a \u2264 b) :=\n  nat.decidable_le (val a) (val b)\n\n/-\nWe cannot use tactics dec_trivial or comp_val here because the tactic framework has not been defined yet.\nWe also do not use `zero_lt_succ _` as a proof term because this proof may not be trivial to check by\nexternal type checkers. See discussion at: https://github.com/leanprover/tc/issues/8\n-/\n\ntheorem zero_lt_d800 : 0 < bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit1 (bit1 (bit0 (bit1 1)))))))))))))) := sorry\n\ndef of_nat (n : \u2115) : char :=\n  dite (is_valid_char n) (fun (h : is_valid_char n) => mk n h) fun (h : \u00acis_valid_char n) => mk 0 sorry\n\ndef to_nat (c : char) : \u2115 :=\n  val c\n\ntheorem eq_of_veq {c : char} {d : char} : val c = val d \u2192 c = d := sorry\n\ntheorem veq_of_eq {c : char} {d : char} : c = d \u2192 val c = val d := sorry\n\ntheorem ne_of_vne {c : char} {d : char} (h : val c \u2260 val d) : c \u2260 d :=\n  fun (h' : c = d) => absurd (veq_of_eq h') h\n\ntheorem vne_of_ne {c : char} {d : char} (h : c \u2260 d) : val c \u2260 val d :=\n  fun (h' : val c = val d) => absurd (eq_of_veq h') h\n\nend char\n\n\nprotected instance char.decidable_eq : DecidableEq char :=\n  fun (i j : char) => decidable_of_decidable_of_iff (nat.decidable_eq (char.val i) (char.val j)) sorry\n\nprotected instance char.inhabited : Inhabited char :=\n  { default := char.of_nat (bit1 (bit0 (bit0 (bit0 (bit0 (bit0 1)))))) }\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/char/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7217432182679956, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3805871246298595}}
{"text": "import PnP2023.Lec_01_04.Intro\nimport PnP2023.Lec_01_04.OddExample\nimport PnP2023.Lec_01_06.NatEq\nimport PnP2023.Lec_01_11.NatLe\nimport PnP2023.Lec_01_13.NatRec\nimport PnP2023.Lec_01_18.NatSub\nimport PnP2023.Lec_01_20.NatMin\nimport PnP2023.Lec_01_20.NatMinus\nimport PnP2023.Lec_01_25.Answer\nimport PnP2023.Lec_02_01.Foundations\nimport PnP2023.Lec_02_03.InductiveTypes\nimport PnP2023.Lec_02_15.PropsAsTypes\nimport PnP2023.Lec_03_01.RandomIO\nimport PnP2023.Lec_03_08.Diaphontine\nimport PnP2023.Lec_03_10.FormalCalculus\nimport PnP2023.Lec_03_24.Sampling\n/-!\n# Proofs and Programs 2023\n\nThis course will introduce [Lean Theorem Prover 4](https://leanprover.github.io/), which is an _interactive theorem prover_ as well as a programming language and use it for various aspects of _proofs and programs_:\n\n* Using programs (_interactive theorem provers_) to\n    * verify proofs of results\n    * help in generating proofs\n* Proving correctness of programs\n* Writing programs to find and/or verify proofs\n* _Functional Programming_ in Lean\n\nTo study these things in a meaningful way, we will look at _foundations_ of _mathematics_ and of _computation_. We will introduce different foundational systems:\n\n* _Dependent Type Theory_ (DTT): foundations used by Lean that include both proofs and computations in a unified way.\n* Classical foundations:\n  - _First-order logic:_ the usual foundations of mathematics.\n  - _lambda-calculus:_ one of the equivalent formulations of the usual foundations of computation.\n\n## Navigation\n\nTo browse the code, expand the `PnP2023` tab on the left.\n\n-/\n\ndef hello := \"to the course Proofs and Programs\"\n", "meta": {"author": "siddhartha-gadgil", "repo": "proofs-and-programs-2023", "sha": "9d95a5396c018b9a26ed0d27c52cd446794cd1aa", "save_path": "github-repos/lean/siddhartha-gadgil-proofs-and-programs-2023", "path": "github-repos/lean/siddhartha-gadgil-proofs-and-programs-2023/proofs-and-programs-2023-9d95a5396c018b9a26ed0d27c52cd446794cd1aa/PnP2023.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.7217432122827968, "lm_q1q2_score": 0.38058712147376533}}
{"text": "import Scratch.ExprAppl\nimport Scratch.ConstDeps\nimport Scratch.ProdSeq\nimport Lean.Meta\nimport Lean.Elab\nimport Std.Data.HashMap\nimport Std.Data.HashSet\nopen Std\nopen Lean\nopen Meta\nopen Elab\nopen Term\nopen Lean.Elab.Tactic\n\ndef whiteListed (n: Name) : TermElabM Bool := do\n  let b \u2190 ConstDeps.isWhiteListed (\u2190 getEnv) n\n  return b\n\n\ndef contains : Expr \u2192 Expr \u2192 MetaM Bool\n  | e, x => \n    do \n    if \u2190 isDefEq e x then return true\n    else \n    match e with\n    | Expr.app f a _ => (\u2190 contains f x) || (\u2190 contains a x)\n    | Expr.lam _ _ b _ => (\u2190 contains b x)\n    | Expr.forallE _ _ b _ => (\u2190 contains b x) \n    | _ => return false\n\n\n-- copied from lean4 source code\ndef rewriteProof (e: Expr) (heq : Expr) (symm : Bool := false) : MetaM (Option Expr) :=\n  do\n    let heqType \u2190 instantiateMVars (\u2190 inferType heq)\n    let (newMVars, binderInfos, heqType) \u2190 forallMetaTelescopeReducing heqType\n    let heq := mkAppN heq newMVars\n    match heqType.eq? with\n    | none => none\n    | some (\u03b1 , lhs, rhs) =>\n    let heqType := if symm then \u2190 mkEq rhs lhs else heqType\n    let hep := if symm then mkEqSymm heq else heq\n    if lhs.getAppFn.isMVar then none\n    else\n    let e \u2190 instantiateMVars e\n    let eAbst \u2190  kabstract e lhs\n    if !eAbst.hasLooseBVars then none\n    else\n    let eNew := eAbst.instantiate1 rhs\n    let eNew \u2190 instantiateMVars eNew\n    let eEqE \u2190 mkEq e e\n    let eEqEAbst := mkApp eEqE.appFn! eAbst\n    let motive := Lean.mkLambda `_a BinderInfo.default \u03b1 eEqEAbst\n    if !(\u2190 isTypeCorrect motive) then none\n    else            \n    let eqRefl \u2190 mkEqRefl e\n    let eqPrf \u2190 mkEqNDRec motive eqRefl heq\n    return some eqPrf\n\ndef rwPushOpt(e : Expr) (heq : Expr) \n      (symm : Bool := false): MetaM (Option Expr) :=\n  do\n    let t \u2190 inferType e\n    let pfOpt \u2190 rewriteProof t heq symm\n    match pfOpt with\n    | none => return none\n    | some pf =>\n      try\n        let expr \u2190 mkAppM ``Eq.mp #[pf, e]\n        let exprType \u2190 inferType expr\n        if (\u2190 isTypeCorrect expr) &&  (\u2190 isTypeCorrect exprType)  \n        then return some expr\n        else return none\n      catch _ => \n        return none\n\ndef eqCongrOpt (f: Expr)(eq : Expr) : MetaM (Option Expr) :=\n  do\n    try\n      let expr \u2190 mkAppM ``congrArg #[f, eq]\n      let exprType \u2190 inferType expr\n      if (\u2190 isTypeCorrect expr) &&  (\u2190 isTypeCorrect exprType)  then return some expr\n      else \n        return none\n    catch e => \n      return none \n\ndef isle (type: Expr)(evolve : Array Expr \u2192 TermElabM (Array Expr))(init : List Expr)\n       (includePi : Bool := true)(excludeProofs: Bool := false): TermElabM (Array Expr) := \n    withLocalDecl Name.anonymous BinderInfo.default (type)  $ fun x => \n        do\n          let l := x :: init\n          -- logInfo m!\"initial in isle: {l}\"\n          let evb \u2190 evolve l.toArray\n          let evc \u2190 evolve init.toArray\n          let mut evl : Array Expr := #[]\n          for y in evb do\n            unless excludeProofs && ((\u2190 inferType (\u2190 inferType y)).isProp) do\n            unless (evc.contains y) do \n              evl := evl.push y \n          let evt \u2190 evl.filterM (fun x => liftMetaM (isType x))\n          let exported \u2190 evl.mapM (fun e => mkLambdaFVars #[x] e)\n          let exportedPi \u2190 evt.mapM (fun e => mkForallFVars #[x] e)\n          let res := if includePi then exported ++ exportedPi else exported\n          return res\n\ndef isleSum (types: List Expr)(evolve : Array Expr \u2192 TermElabM (Array Expr))(init : List Expr) : \n        TermElabM (Array Expr) := \n        match types with\n        | [] => return #[]\n        | h :: t => \n          do\n            let tail \u2190 isleSum t evolve init\n            let head \u2190 isle h evolve init\n            return head ++ tail        \n\ndef Array.join {\u03b1 : Type}[BEq \u03b1](a : Array (Array \u03b1)) : Array \u03b1 := Id.run do\n  let mut res : Array \u03b1  := #[]\n  for x in a do\n    for y in x do\n      res := if res.contains y then res else res.push y\n  return res\n\ndef eqIsles (eqs: Array Expr)(evolve : Array Expr \u2192 TermElabM (Array Expr))(init : List Expr) : \n        TermElabM (Array Expr) := \n        do\n        let mut eqGroupMap : HashMap Expr (Array Expr) := HashMap.empty\n        for eq in eqs do\n          match (\u2190 inferType eq).eq? with\n          | some (\u03b1, lhs, rhs) =>\n              let prev := (eqGroupMap.find? \u03b1).getD #[] \n              eqGroupMap := eqGroupMap.insert \u03b1 (prev.push eq)\n          | none => ()\n        let eqGroups := eqGroupMap.toArray\n        let res : Array Expr \u2190 eqGroups.concatMapM $ fun (\u03b1, eqns) =>\n              do \n              let fs \u2190 isle \u03b1 evolve init false true\n              -- logInfo m!\"fs: {fs.size}\"\n              let shifted \u2190 eqns.concatMapM $ fun eq =>  \n                    fs.filterMapM (fun f => eqCongrOpt f eq)\n              return shifted\n        return res --.join\n\n\ndef List.inTermElab {\u03b1 : Type}(l : List (TermElabM \u03b1)) : TermElabM (List \u03b1) :=\n  l.foldl (fun ysM xM =>\n            do \n              return (\u2190 xM) :: (\u2190 ysM)) (return [])\n\ndef Array.inTermElab {\u03b1 : Type}(l : Array (TermElabM \u03b1)) : TermElabM (Array \u03b1) :=\n  l.foldl (fun ysM xM =>\n            do \n              return (\u2190 ysM).push (\u2190 xM)) (return #[])\n\n\n#check @Array.foldl\n\n  \n\ndef rwAppCongStepTask : Array Expr \u2192 Array Name \u2192 (TermElabM (Array Expr)):=\n    fun l names => \n    let funcs :=  l.filterM $ fun e => \n      let check: TermElabM Bool := do\n        let type \u2190 inferType e\n        return type.isForall\n      check\n    let ltml :=\n      l.map $ fun arg => \n      Task.spawn $ fun _ =>\n      do\n        let fns \u2190 funcs\n        let apps \u2190 fns.filterMapM (fun f => applyOptM f arg)\n        let nameApps \u2190 names.filterMapM (fun name => nameApplyOptM name arg)\n        let nameAppPairsRaw \u2190 \n          Array.inTermElab (l.map (fun arg2 => names.filterMapM (fun name => \n                  nameApplyPairOptM name arg arg2\n                  )))\n        let nameAppPairs := nameAppPairsRaw.join\n        let type \u2190 inferType arg\n        if type.isEq\n        then \n          let rws \u2190 l.filterMapM (fun f => rwPushOpt  f arg)\n          let rwsFlip \u2190 l.filterMapM (fun f => rwPushOpt f arg true)\n          let congs \u2190 l.filterMapM (fun f => eqCongrOpt f arg)\n          return (rws.append \n                    (rwsFlip.append (congs.append (apps)))).append (nameApps) ++ nameAppPairs \n        else           \n          return apps ++ nameApps ++ nameAppPairs\n    let lst := ltml.map <| fun t=> t.get\n    let ml := (Array.inTermElab lst).map (fun ll => (Array.join ll) ++ l)\n    ml\n\ndef iterAppRWTask(n: Nat) : Array Expr \u2192 Array Name  \u2192 TermElabM (Array Expr) :=\n   match n with\n  | 0 => fun l _ => return l\n  | m + 1 => fun l names => do\n      let prev \u2190  iterAppRWTask m   l names\n      let rwStepTask := rwAppCongStepTask  prev names\n      let isles \u2190 eqIsles prev \n        (fun list => (iterAppRWTask m list names)) prev.toList\n      -- Elab.logInfo m!\"isles: {isles}\"\n      let rwStep \u2190 rwStepTask\n      return rwStep ++ isles\n\n\ninitialize exprArrCache : IO.Ref (HashMap Name (Array Expr)) \u2190 IO.mkRef (HashMap.empty)\n\ninitialize exprPackCache : IO.Ref (HashMap Name Expr) \u2190 IO.mkRef (HashMap.empty)\n\ndef getArrCached? (name : Name) : IO (Option (Array Expr)) := do\n  let cache \u2190 exprArrCache.get\n  return (cache.find? name)\n\ndef cacheArr (name: Name)(e: Array Expr)  : IO Unit := do\n  let cache \u2190 exprArrCache.get\n  exprArrCache.set (cache.insert name e)\n  return ()\n\ndef getPackCached? (name : Name) : IO (Option Expr) := do\n  let cache \u2190 exprPackCache.get\n  return (cache.find? name)\n\ndef cachePack (name: Name)(e: Expr)  : IO Unit := do\n  let cache \u2190 exprPackCache.get\n  exprPackCache.set (cache.insert name e)\n  return ()\n\n\ndef saveExprArr (name: Name)(es: Array Expr) : TermElabM (Unit) := do\n  let lctx \u2190 getLCtx\n  let fvarIds \u2190 lctx.getFVarIds\n  let fvIds \u2190 fvarIds.filterM $ fun fid => whiteListed ((lctx.get! fid).userName) \n  let fvars := fvIds.map mkFVar\n  Term.synthesizeSyntheticMVarsNoPostponing \n  let espair \u2190 es.mapM (fun e => do Term.levelMVarToParam (\u2190 instantiateMVars e))\n  let es \u2190 espair.mapM fun (e, _) => return e\n  let es \u2190 es.mapM (fun e => mkLambdaFVars fvars e)\n  let es \u2190 es.mapM (fun e => whnf e)\n  logInfo m!\"saving relative to: {fvars}\"\n  cacheArr name es\n  let varPack \u2190 ProdSeq.lambdaPack fvars.toList\n  cachePack name varPack\n  return ()\n\ndef loadExprArr (name: Name) : TermElabM (Array Expr) := do\n  let lctx \u2190 getLCtx\n  let fvarIds \u2190 lctx.getFVarIds\n  let fvIds \u2190 fvarIds.filterM $ fun fid => whiteListed ((lctx.get! fid).userName) \n  let fvars := fvIds.map mkFVar\n  logInfo m!\"loading relative to: {fvars}\"\n  let fvarsCachedPack \u2190 getPackCached? name\n  let fvarsCached \u2190 fvarsCachedPack.mapM (fun p => ProdSeq.lambdaUnpack p)\n  logInfo m!\"fvarsCached: {fvarsCached}\" \n  let cache \u2190 exprArrCache.get\n  match cache.find? name with\n  | some es => es.mapM $ fun e => reduce (mkAppN e fvars)\n  | none => throwError m!\"no cached expr for {name}\"\n\ndef distinctTypes (exps: Array Expr) : TermElabM (Array Expr) := do\n  let mut types : Array Expr := Array.empty\n  let mut distinct : Array Expr := Array.empty\n  for expr in exps do\n    let type \u2190 inferType expr\n    unless (types.contains type) do\n      types := types.push type\n      distinct := distinct.push expr\n  return distinct\n\ndef propagateEqualities (eqs: Array Expr) : TermElabM (HashMap Expr Expr) := \n  do\n    let mut eqsymm : Array Expr := #[]\n    let mut eqTypes : HashSet Expr := HashSet.empty\n    for eq in eqs do\n      let type \u2190 inferType eq\n      if type.isEq then\n        unless eqTypes.contains type do\n          eqsymm := eqsymm.push eq\n          eqTypes := eqTypes.insert type\n        let seq \u2190 whnf (\u2190 mkAppM `Eq.symm #[eq])\n        let seqType \u2190 inferType seq\n        unless eqTypes.contains seqType do\n          eqsymm \u2190 eqsymm.push seq\n          eqTypes := eqTypes.insert seqType\n    logInfo m!\"symmetrize equalities for propagation: {\u2190 IO.monoMsNow}\"\n    logInfo m!\"got:{eqsymm.size}\"\n    let mut withLhs : HashMap Expr (Array (Expr \u00d7 Expr)) := HashMap.empty\n    for eq in eqsymm do\n      let type \u2190 inferType eq\n      match type.eq? with\n      | none => ()\n      | some (\u03b1 , lhs, rhs) =>\n        let lhsUp := \n          match withLhs.getOp lhs with\n          | some arr => arr.push (eq, rhs)\n          | none => #[(eq, rhs)] \n        withLhs \u2190 withLhs.insert lhs lhsUp\n    logInfo m!\"equality map generated: {\u2190 IO.monoMsNow}\"\n    let mut  accum : HashMap Expr Expr := HashMap.empty\n    let mut accumSides : HashSet (Expr \u00d7 Expr) := HashSet.empty\n    for eq1 in eqsymm do\n      let type \u2190 inferType eq1\n      match type.eq? with\n      | none => ()\n      | some (\u03b1 , lhs, rhs) =>\n        let eqs2 := (withLhs.getOp rhs).getD #[]\n        for (eq2, rhs2) in eqs2 do\n        unless (rhs2 == lhs) || (accumSides.contains (lhs, rhs2)) do\n          let eq3 \u2190  mkAppM `Eq.trans #[eq1, eq2]\n          let type \u2190 mkEq lhs rhs2\n          accum \u2190 accum.insert type eq3\n          accumSides := accumSides.insert (lhs, rhs2)\n    return accum \n\ndef typeVariables (e: Expr) : (List Name) \u00d7 (List Name) :=\n  match e with\n  | Expr.forallE n d b c =>\n      if c.binderInfo.isExplicit then\n        let (l1, l2) := typeVariables b\n        (n :: l1, l2)\n      else\n        let (l1, l2) := typeVariables b\n        (l1, n :: l2)\n  | _ => ([], [])\n\nsyntax(name:= goalVariables) \"goalVariables\" : tactic \n@[tactic goalVariables] def goalVariablesImp : Tactic :=\n  fun stx => \n  withMainContext do\n    let (l1, l2) := typeVariables (\u2190 getMainTarget)\n    logInfo m!\"target: {\u2190 getMainTarget}\"\n    logInfo m!\"explicit variables: {l1}\"\n    logInfo m!\"implicit variables: {l2}\" \n    let lctx \u2190 getLCtx\n    let fvarIds \u2190 lctx.getFVarIds\n    let fvars := fvarIds.map mkFVar\n    logInfo m!\"free variables from context: {fvars}\"\n    logInfo m!\"free variable types from context: {\u2190 fvars.mapM $ fun x => inferType x}\"\n    return ()\n\n-- used only in example generation, not in active code\ndef appStepTask : Array Expr \u2192 Task (TermElabM (Array Expr)):=\n    fun l =>\n    let ltml :=\n      l.map $ fun arg => \n      Task.spawn $ fun _ =>\n      do\n        let apps \u2190 l.filterMapM (fun f => applyOptM f arg)\n        return apps\n    let tlml := Task.array ltml \n    let tml := tlml.map $ fun lst => \n      (Array.inTermElab lst).map (fun ll => (Array.join ll) ++ l)\n    tml\n\ndef iterAppRWMTask(n: Nat): List Expr \u2192 List Name \u2192 TermElabM (List Expr) :=\n  fun l names => ((iterAppRWTask n l.toArray names.toArray)).map (Array.toList)\n\ndef iterAppTask(n: Nat) : Array Expr \u2192 TermElabM (Array Expr) :=\n   match n with\n  | 0 => fun l => return l\n  | m + 1 => fun l => do\n      let prev \u2190  iterAppTask m   l\n      let stepTask := appStepTask prev\n      let step \u2190 stepTask.get\n      return step\n\ndef iterAppMTask(n: Nat) : List Expr \u2192 TermElabM (List Expr) :=\n  fun l => ((iterAppTask n  l.toArray)).map (Array.toList)\n-- end used only in example generation, not in active code\n", "meta": {"author": "siddhartha-gadgil", "repo": "lean4-scratch", "sha": "680b7073f791706faf248d1d0ad21095012ae01b", "save_path": "github-repos/lean/siddhartha-gadgil-lean4-scratch", "path": "github-repos/lean/siddhartha-gadgil-lean4-scratch/lean4-scratch-680b7073f791706faf248d1d0ad21095012ae01b/Scratch/ExprRw.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3805604142055474}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro\n-/\n\nnamespace List\n\nopen Nat\n\n/-!\n# Bootstrapping theorems for lists\n\nThese are theorems used in the definitions of `Std.Data.List.Basic`.\nNew theorems should be added to `Std.Data.List.Lemmas` if they are not needed by the bootstrap.\n-/\n\nattribute [simp] get get! get? reverseAux eraseIdx map join dropWhile find? findSome?\n  replace elem lookup drop take takeWhile foldl foldr zipWith unzip range.loop enumFrom\n  intersperse isPrefixOf isEqv dropLast iota mapM.loop mapA List.forM forA filterAuxM\n  filterMapM.loop List.foldlM firstM anyM allM findM? findSomeM? forIn.loop forIn'.loop\n  concat_eq_append append_assoc\n\n@[simp] theorem head?_nil : @head? \u03b1 [] = none := rfl\n@[simp] theorem head?_cons : @head? \u03b1 (a::l) = some a := rfl\n@[simp 1100] theorem headD_nil : @headD \u03b1 [] d = d := rfl\n@[simp 1100] theorem headD_cons : @headD \u03b1 (a::l) d = a := rfl\n@[simp] theorem head_cons : @head \u03b1 (a::l) h = a := rfl\n@[simp] theorem tail!_cons : @tail! \u03b1 (a::l) = l := rfl\n@[simp 1100] theorem tailD_nil : @tailD \u03b1 [] l' = l' := rfl\n@[simp 1100] theorem tailD_cons : @tailD \u03b1 (a::l) l' = l := rfl\n@[simp] theorem any_nil : [].any f = false := rfl\n@[simp] theorem any_cons : (a::l).any f = (f a || l.any f) := rfl\n@[simp] theorem all_nil : [].all f = true := rfl\n@[simp] theorem all_cons : (a::l).all f = (f a && l.all f) := rfl\n@[simp] theorem or_nil : [].or = false := rfl\n@[simp] theorem or_cons : (a::l).or = (a || l.or) := rfl\n@[simp] theorem and_nil : [].and = true := rfl\n@[simp] theorem and_cons : (a::l).and = (a && l.and) := rfl\n\n/-! ### length -/\n\ntheorem eq_nil_of_length_eq_zero (_ : length l = 0) : l = [] := match l with | [] => rfl\n\ntheorem ne_nil_of_length_eq_succ (_ : length l = succ n) : l \u2260 [] := fun _ => nomatch l\n\ntheorem length_eq_zero : length l = 0 \u2194 l = [] :=\n  \u27e8eq_nil_of_length_eq_zero, fun h => h \u25b8 rfl\u27e9\n\n/-! ### append -/\n\n@[simp 1100] theorem singleton_append : [x] ++ l = x :: l := rfl\n\ntheorem append_inj :\n    \u2200 {s\u2081 s\u2082 t\u2081 t\u2082 : List \u03b1}, s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082 \u2192 length s\u2081 = length s\u2082 \u2192 s\u2081 = s\u2082 \u2227 t\u2081 = t\u2082\n  | [], [], t\u2081, t\u2082, h, _ => \u27e8rfl, h\u27e9\n  | a :: s\u2081, b :: s\u2082, t\u2081, t\u2082, h, hl => by\n    simp [append_inj (cons.inj h).2 (Nat.succ.inj hl)] at h \u22a2; exact h\n\ntheorem append_inj_right (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length s\u2081 = length s\u2082) : t\u2081 = t\u2082 :=\n  (append_inj h hl).right\n\ntheorem append_inj_left (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length s\u2081 = length s\u2082) : s\u2081 = s\u2082 :=\n  (append_inj h hl).left\n\ntheorem append_inj' (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : s\u2081 = s\u2082 \u2227 t\u2081 = t\u2082 :=\n  append_inj h <| @Nat.add_right_cancel _ (length t\u2081) _ <| by\n  let hap := congrArg length h; simp only [length_append, \u2190 hl] at hap; exact hap\n\ntheorem append_inj_right' (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : t\u2081 = t\u2082 :=\n  (append_inj' h hl).right\n\ntheorem append_inj_left' (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : s\u2081 = s\u2082 :=\n  (append_inj' h hl).left\n\ntheorem append_right_inj {t\u2081 t\u2082 : List \u03b1} (s) : s ++ t\u2081 = s ++ t\u2082 \u2194 t\u2081 = t\u2082 :=\n  \u27e8fun h => append_inj_right h rfl, congrArg _\u27e9\n\ntheorem append_left_inj {s\u2081 s\u2082 : List \u03b1} (t) : s\u2081 ++ t = s\u2082 ++ t \u2194 s\u2081 = s\u2082 :=\n  \u27e8fun h => append_inj_left' h rfl, congrArg (\u00b7 ++ _)\u27e9\n\n/-! ### map -/\n\ntheorem map_nil {f : \u03b1 \u2192 \u03b2} : map f [] = [] := rfl\n\ntheorem map_cons (f : \u03b1 \u2192 \u03b2) a l : map f (a :: l) = f a :: map f l := rfl\n\n@[simp] theorem map_append (f : \u03b1 \u2192 \u03b2) : \u2200 l\u2081 l\u2082, map f (l\u2081 ++ l\u2082) = map f l\u2081 ++ map f l\u2082 := by\n  intro l\u2081 <;> induction l\u2081 <;> intros <;> simp_all\n\n@[simp] theorem map_id (l : List \u03b1) : map id l = l := by induction l <;> simp_all\n\n@[simp] theorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n  map g (map f l) = map (g \u2218 f) l := by induction l <;> simp_all\n\n/-! ### bind -/\n\n@[simp] theorem nil_bind (f : \u03b1 \u2192 List \u03b2) : List.bind [] f = [] := by simp [join, List.bind]\n\n@[simp] theorem cons_bind x xs (f : \u03b1 \u2192 List \u03b2) :\n  List.bind (x :: xs) f = f x ++ List.bind xs f := by simp [join, List.bind]\n\n@[simp] theorem append_bind xs ys (f : \u03b1 \u2192 List \u03b2) :\n  List.bind (xs ++ ys) f = List.bind xs f ++ List.bind ys f := by\n  induction xs; {rfl}; simp_all [cons_bind, append_assoc]\n\n/-! ### bind -/\n\n@[simp] theorem bind_id (l : List (List \u03b1)) : List.bind l id = l.join := by simp [List.bind]\n\n/-! ### reverse -/\n\ntheorem reverseAux_eq (as bs : List \u03b1) : reverseAux as bs = reverse as ++ bs :=\n  reverseAux_eq_append ..\n\ntheorem reverse_map (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) : (l.map f).reverse = l.reverse.map f := by\n  induction l <;> simp [*]\n\n/-! ### take and drop -/\n\n@[simp] theorem take_append_drop : \u2200 (n : Nat) (l : List \u03b1), take n l ++ drop n l = l\n  | 0, _ => rfl\n  | _+1, [] => rfl\n  | n+1, x :: xs => congrArg (cons x) <| take_append_drop n xs\n\n@[simp] theorem length_drop : \u2200 (i : Nat) (l : List \u03b1), length (drop i l) = length l - i\n  | 0, _ => rfl\n  | succ i, [] => Eq.symm (Nat.zero_sub (succ i))\n  | succ i, x :: l => calc\n    length (drop (succ i) (x :: l)) = length l - i := length_drop i l\n    _ = succ (length l) - succ i := (Nat.succ_sub_succ_eq_sub (length l) i).symm\n\ntheorem drop_length_le {l : List \u03b1} (h : l.length \u2264 i) : drop i l = [] :=\n  length_eq_zero.1 (length_drop .. \u25b8 Nat.sub_eq_zero_of_le h)\n\ntheorem take_length_le {l : List \u03b1} (h : l.length \u2264 i) : take i l = l := by\n  have := take_append_drop i l\n  rw [drop_length_le h, append_nil] at this; exact this\n\n@[simp] theorem drop_length (l : List \u03b1) : drop l.length l = [] := drop_length_le (Nat.le_refl _)\n\n@[simp] theorem take_length (l : List \u03b1) : take l.length l = l := take_length_le (Nat.le_refl _)\n\ntheorem take_concat_get (l : List \u03b1) (i : Nat) (h : i < l.length) :\n    (l.take i).concat l[i] = l.take (i+1) :=\n  Eq.symm <| (append_left_inj _).1 <| (take_append_drop (i+1) l).trans <| by\n    rw [concat_eq_append, append_assoc, singleton_append, get_drop_eq_drop, take_append_drop]\n\ntheorem reverse_concat (l : List \u03b1) (a : \u03b1) : (l.concat a).reverse = a :: l.reverse := by\n  rw [concat_eq_append, reverse_append]; rfl\n\n@[simp] theorem foldlM_reverse [Monad m] (l : List \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) (b) :\n    l.reverse.foldlM f b = l.foldrM (fun x y => f y x) b := rfl\n\n@[simp] theorem foldlM_append [Monad m] [LawfulMonad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) (b) (l l' : List \u03b1) :\n    (l ++ l').foldlM f b = l.foldlM f b >>= l'.foldlM f := by\n  induction l generalizing b <;> simp [*]\n\n@[simp] theorem foldrM_nil [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (b) : [].foldrM f b = pure b := rfl\n\n@[simp] theorem foldrM_cons [Monad m] [LawfulMonad m] (a : \u03b1) (l) (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (b) :\n    (a :: l).foldrM f b = l.foldrM f b >>= f a := by\n  simp only [foldrM]\n  induction l <;> simp_all\n\n@[simp] theorem foldrM_reverse [Monad m] (l : List \u03b1) (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (b) :\n    l.reverse.foldrM f b = l.foldlM (fun x y => f y x) b :=\n  (foldlM_reverse ..).symm.trans <| by simp\n\ntheorem foldl_eq_foldlM (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (b) (l : List \u03b1) :\n    l.foldl f b = l.foldlM (m := Id) f b := by\n  induction l generalizing b <;> simp [*, foldl]\n\ntheorem foldr_eq_foldrM (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b) (l : List \u03b1) :\n    l.foldr f b = l.foldrM (m := Id) f b := by\n  induction l <;> simp [*]\n\n@[simp] theorem foldl_reverse (l : List \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (b) :\n    l.reverse.foldl f b = l.foldr (fun x y => f y x) b := by simp [foldl_eq_foldlM, foldr_eq_foldrM]\n\n@[simp] theorem foldr_reverse (l : List \u03b1) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b) :\n    l.reverse.foldr f b = l.foldl (fun x y => f y x) b :=\n  (foldl_reverse ..).symm.trans <| by simp\n\n@[simp] theorem foldrM_append [Monad m] [LawfulMonad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) (b) (l l' : List \u03b1) :\n    (l ++ l').foldrM f b = l'.foldrM f b >>= l.foldrM f := by\n  induction l <;> simp [*]\n\n@[simp] theorem foldl_append {\u03b2 : Type _} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (b) (l l' : List \u03b1) :\n    (l ++ l').foldl f b = l'.foldl f (l.foldl f b) := by simp [foldl_eq_foldlM]\n\n@[simp] theorem foldr_append (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b) (l l' : List \u03b1) :\n    (l ++ l').foldr f b = l.foldr f (l'.foldr f b) := by simp [foldr_eq_foldrM]\n\n@[simp] theorem foldr_self_append (l : List \u03b1) : l.foldr cons l' = l ++ l' := by\n  induction l <;> simp [*]\n\n\n\n/-- Alternate (non-tail-recursive) form of mapM for proofs. -/\n@[simp] def mapM' [Monad m] (f : \u03b1 \u2192 m \u03b2) : List \u03b1 \u2192 m (List \u03b2)\n  | [] => pure []\n  | a :: l => return (\u2190 f a) :: (\u2190 l.mapM' f)\n\ntheorem mapM'_eq_mapM [Monad m] [LawfulMonad m] (f : \u03b1 \u2192 m \u03b2) (l : List \u03b1) :\n    mapM' f l = mapM f l := by simp [go, mapM] where\n  go : \u2200 l acc, mapM.loop f l acc = return acc.reverse ++ (\u2190 mapM' f l)\n    | [], acc => by simp\n    | a::l, acc => by simp [go l]\n\n@[simp] theorem mapM_nil [Monad m] (f : \u03b1 \u2192 m \u03b2) : [].mapM f = pure [] := rfl\n\n@[simp] theorem mapM_cons [Monad m] [LawfulMonad m] (f : \u03b1 \u2192 m \u03b2) :\n    (a :: l).mapM f = (return (\u2190 f a) :: (\u2190 l.mapM f)) := by simp [\u2190 mapM'_eq_mapM]\n\n@[simp] theorem mapM_append [Monad m] [LawfulMonad m] (f : \u03b1 \u2192 m \u03b2) {l\u2081 l\u2082 : List \u03b1} :\n    (l\u2081 ++ l\u2082).mapM f = (return (\u2190 l\u2081.mapM f) ++ (\u2190 l\u2082.mapM f)) := by induction l\u2081 <;> simp [*]\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/List/Init/Lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.3805522035353654}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.list.basic\nimport Mathlib.Lean3Lib.init.function\nimport Mathlib.Lean3Lib.init.meta.default\nimport Mathlib.Lean3Lib.init.data.nat.lemmas\nimport Mathlib.Lean3Lib.init.meta.interactive\nimport Mathlib.Lean3Lib.init.meta.smt.rsimp\n\nuniverses u v w w\u2082 w\u2081 \n\nnamespace Mathlib\n\nnamespace list\n\n\n/- append -/\n\n@[simp] theorem nil_append {\u03b1 : Type u} (s : List \u03b1) : [] ++ s = s := rfl\n\n@[simp] theorem cons_append {\u03b1 : Type u} (x : \u03b1) (s : List \u03b1) (t : List \u03b1) :\n    x :: s ++ t = x :: (s ++ t) :=\n  rfl\n\n@[simp] theorem append_nil {\u03b1 : Type u} (t : List \u03b1) : t ++ [] = t := sorry\n\n@[simp] theorem append_assoc {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) (u : List \u03b1) :\n    s ++ t ++ u = s ++ (t ++ u) :=\n  sorry\n\n/- length -/\n\ntheorem length_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : length (a :: l) = length l + 1 := rfl\n\n@[simp] theorem length_append {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    length (s ++ t) = length s + length t :=\n  sorry\n\n@[simp] theorem length_repeat {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : length (repeat a n) = n := sorry\n\n@[simp] theorem length_tail {\u03b1 : Type u} (l : List \u03b1) : length (tail l) = length l - 1 :=\n  list.cases_on l (Eq.refl (length (tail [])))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (length (tail (l_hd :: l_tl)))\n\n-- TODO(Leo): cleanup proof after arith dec proc\n\n@[simp] theorem length_drop {\u03b1 : Type u} (i : \u2115) (l : List \u03b1) : length (drop i l) = length l - i :=\n  sorry\n\n/- map -/\n\ntheorem map_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (l : List \u03b1) :\n    map f (a :: l) = f a :: map f l :=\n  rfl\n\n@[simp] theorem map_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    map f (l\u2081 ++ l\u2082) = map f l\u2081 ++ map f l\u2082 :=\n  sorry\n\ntheorem map_singleton {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f [a] = [f a] := rfl\n\n@[simp] theorem map_id {\u03b1 : Type u} (l : List \u03b1) : map id l = l := sorry\n\n@[simp] theorem map_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2)\n    (l : List \u03b1) : map g (map f l) = map (g \u2218 f) l :=\n  sorry\n\n@[simp] theorem length_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    length (map f l) = length l :=\n  sorry\n\n/- bind -/\n\n@[simp] theorem nil_bind {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 List \u03b2) : list.bind [] f = [] := sorry\n\n@[simp] theorem cons_bind {\u03b1 : Type u} {\u03b2 : Type v} (x : \u03b1) (xs : List \u03b1) (f : \u03b1 \u2192 List \u03b2) :\n    list.bind (x :: xs) f = f x ++ list.bind xs f :=\n  sorry\n\n@[simp] theorem append_bind {\u03b1 : Type u} {\u03b2 : Type v} (xs : List \u03b1) (ys : List \u03b1) (f : \u03b1 \u2192 List \u03b2) :\n    list.bind (xs ++ ys) f = list.bind xs f ++ list.bind ys f :=\n  sorry\n\n/- mem -/\n\n@[simp] theorem mem_nil_iff {\u03b1 : Type u} (a : \u03b1) : a \u2208 [] \u2194 False := iff.rfl\n\n@[simp] theorem not_mem_nil {\u03b1 : Type u} (a : \u03b1) : \u00aca \u2208 [] := iff.mp (mem_nil_iff a)\n\n@[simp] theorem mem_cons_self {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : a \u2208 a :: l := Or.inl rfl\n\n@[simp] theorem mem_cons_iff {\u03b1 : Type u} (a : \u03b1) (y : \u03b1) (l : List \u03b1) :\n    a \u2208 y :: l \u2194 a = y \u2228 a \u2208 l :=\n  iff.rfl\n\ntheorem mem_cons_eq {\u03b1 : Type u} (a : \u03b1) (y : \u03b1) (l : List \u03b1) : a \u2208 y :: l = (a = y \u2228 a \u2208 l) := rfl\n\ntheorem mem_cons_of_mem {\u03b1 : Type u} (y : \u03b1) {a : \u03b1} {l : List \u03b1} : a \u2208 l \u2192 a \u2208 y :: l :=\n  fun (H : a \u2208 l) => Or.inr H\n\ntheorem eq_or_mem_of_mem_cons {\u03b1 : Type u} {a : \u03b1} {y : \u03b1} {l : List \u03b1} :\n    a \u2208 y :: l \u2192 a = y \u2228 a \u2208 l :=\n  fun (h : a \u2208 y :: l) => h\n\n@[simp] theorem mem_append {\u03b1 : Type u} {a : \u03b1} {s : List \u03b1} {t : List \u03b1} :\n    a \u2208 s ++ t \u2194 a \u2208 s \u2228 a \u2208 t :=\n  sorry\n\ntheorem mem_append_eq {\u03b1 : Type u} (a : \u03b1) (s : List \u03b1) (t : List \u03b1) :\n    a \u2208 s ++ t = (a \u2208 s \u2228 a \u2208 t) :=\n  propext mem_append\n\ntheorem mem_append_left {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} (l\u2082 : List \u03b1) (h : a \u2208 l\u2081) :\n    a \u2208 l\u2081 ++ l\u2082 :=\n  iff.mpr mem_append (Or.inl h)\n\ntheorem mem_append_right {\u03b1 : Type u} {a : \u03b1} (l\u2081 : List \u03b1) {l\u2082 : List \u03b1} (h : a \u2208 l\u2082) :\n    a \u2208 l\u2081 ++ l\u2082 :=\n  iff.mpr mem_append (Or.inr h)\n\n@[simp] theorem not_bex_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) : \u00ac\u2203 (x : \u03b1), \u2203 (H : x \u2208 []), p x := sorry\n\n@[simp] theorem ball_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (x : \u03b1) (H : x \u2208 []) : p x := false.elim\n\n@[simp] theorem bex_cons {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : \u03b1) (l : List \u03b1) :\n    (\u2203 (x : \u03b1), \u2203 (H : x \u2208 a :: l), p x) \u2194 p a \u2228 \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x :=\n  sorry\n\n@[simp] theorem ball_cons {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : \u03b1) (l : List \u03b1) :\n    (\u2200 (x : \u03b1), x \u2208 a :: l \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x :=\n  sorry\n\n/- list subset -/\n\nprotected def subset {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) := \u2200 {a : \u03b1}, a \u2208 l\u2081 \u2192 a \u2208 l\u2082\n\nprotected instance has_subset {\u03b1 : Type u} : has_subset (List \u03b1) := has_subset.mk list.subset\n\n@[simp] theorem nil_subset {\u03b1 : Type u} (l : List \u03b1) : [] \u2286 l :=\n  fun (b : \u03b1) (i : b \u2208 []) => false.elim (iff.mp (mem_nil_iff b) i)\n\n@[simp] theorem subset.refl {\u03b1 : Type u} (l : List \u03b1) : l \u2286 l := fun (b : \u03b1) (i : b \u2208 l) => i\n\ntheorem subset.trans {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} (h\u2081 : l\u2081 \u2286 l\u2082)\n    (h\u2082 : l\u2082 \u2286 l\u2083) : l\u2081 \u2286 l\u2083 :=\n  fun (b : \u03b1) (i : b \u2208 l\u2081) => h\u2082 (h\u2081 i)\n\n@[simp] theorem subset_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : l \u2286 a :: l :=\n  fun (b : \u03b1) (i : b \u2208 l) => Or.inr i\n\ntheorem subset_of_cons_subset {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    a :: l\u2081 \u2286 l\u2082 \u2192 l\u2081 \u2286 l\u2082 :=\n  fun (s : a :: l\u2081 \u2286 l\u2082) (b : \u03b1) (i : b \u2208 l\u2081) => s (mem_cons_of_mem a i)\n\ntheorem cons_subset_cons {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (a : \u03b1) (s : l\u2081 \u2286 l\u2082) :\n    a :: l\u2081 \u2286 a :: l\u2082 :=\n  fun (b : \u03b1) (hin : b \u2208 a :: l\u2081) =>\n    or.elim (eq_or_mem_of_mem_cons hin) (fun (e : b = a) => Or.inl e)\n      fun (i : b \u2208 l\u2081) => Or.inr (s i)\n\n@[simp] theorem subset_append_left {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2081 \u2286 l\u2081 ++ l\u2082 :=\n  fun (b : \u03b1) => mem_append_left l\u2082\n\n@[simp] theorem subset_append_right {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2082 \u2286 l\u2081 ++ l\u2082 :=\n  fun (b : \u03b1) => mem_append_right l\u2081\n\ntheorem subset_cons_of_subset {\u03b1 : Type u} (a : \u03b1) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 \u2286 l\u2082 \u2192 l\u2081 \u2286 a :: l\u2082 :=\n  fun (s : l\u2081 \u2286 l\u2082) (a_1 : \u03b1) (i : a_1 \u2208 l\u2081) => Or.inr (s i)\n\ntheorem eq_nil_of_length_eq_zero {\u03b1 : Type u} {l : List \u03b1} : length l = 0 \u2192 l = [] := sorry\n\ntheorem ne_nil_of_length_eq_succ {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} :\n    length l = Nat.succ n \u2192 l \u2260 [] :=\n  sorry\n\n@[simp] theorem length_map\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (l\u2081 : List \u03b1)\n    (l\u2082 : List \u03b2) : length (map\u2082 f l\u2081 l\u2082) = min (length l\u2081) (length l\u2082) :=\n  sorry\n\n@[simp] theorem length_take {\u03b1 : Type u} (i : \u2115) (l : List \u03b1) :\n    length (take i l) = min i (length l) :=\n  sorry\n\ntheorem length_take_le {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : length (take n l) \u2264 n := sorry\n\ntheorem length_remove_nth {\u03b1 : Type u} (l : List \u03b1) (i : \u2115) :\n    i < length l \u2192 length (remove_nth l i) = length l - 1 :=\n  sorry\n\n@[simp] theorem partition_eq_filter_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p]\n    (l : List \u03b1) : partition p l = (filter p l, filter (Not \u2218 p) l) :=\n  sorry\n\n/- sublists -/\n\ninductive sublist {\u03b1 : Type u} : List \u03b1 \u2192 List \u03b1 \u2192 Prop where\n| slnil : sublist [] []\n| cons : \u2200 (l\u2081 l\u2082 : List \u03b1) (a : \u03b1), sublist l\u2081 l\u2082 \u2192 sublist l\u2081 (a :: l\u2082)\n| cons2 : \u2200 (l\u2081 l\u2082 : List \u03b1) (a : \u03b1), sublist l\u2081 l\u2082 \u2192 sublist (a :: l\u2081) (a :: l\u2082)\n\ninfixl:50 \" <+ \" => Mathlib.list.sublist\n\ntheorem length_le_of_sublist {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+ l\u2082 \u2192 length l\u2081 \u2264 length l\u2082 :=\n  sorry\n\n/- filter -/\n\n@[simp] theorem filter_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [h : decidable_pred p] : filter p [] = [] :=\n  rfl\n\n@[simp] theorem filter_cons_of_pos {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [h : decidable_pred p] {a : \u03b1}\n    (l : List \u03b1) : p a \u2192 filter p (a :: l) = a :: filter p l :=\n  fun (pa : p a) => if_pos pa\n\n@[simp] theorem filter_cons_of_neg {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [h : decidable_pred p] {a : \u03b1}\n    (l : List \u03b1) : \u00acp a \u2192 filter p (a :: l) = filter p l :=\n  fun (pa : \u00acp a) => if_neg pa\n\n@[simp] theorem filter_append {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [h : decidable_pred p] (l\u2081 : List \u03b1)\n    (l\u2082 : List \u03b1) : filter p (l\u2081 ++ l\u2082) = filter p l\u2081 ++ filter p l\u2082 :=\n  sorry\n\n@[simp] theorem filter_sublist {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [h : decidable_pred p] (l : List \u03b1) :\n    filter p l <+ l :=\n  sorry\n\n/- map_accumr -/\n\n-- This runs a function over a list returning the intermediate results and a\n\n-- a final result.\n\ndef map_accumr {\u03b1 : Type u} {\u03b2 : Type v} {\u03c3 : Type w\u2082} (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2) :\n    List \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 List \u03b2 :=\n  sorry\n\n@[simp] theorem length_map_accumr {\u03b1 : Type u} {\u03b2 : Type v} {\u03c3 : Type w\u2082} (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2)\n    (x : List \u03b1) (s : \u03c3) : length (prod.snd (map_accumr f x s)) = length x :=\n  sorry\n\n-- This runs a function over two lists returning the intermediate results and a\n\n-- a final result.\n\ndef map_accumr\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03c6 : Type w\u2081} {\u03c3 : Type w\u2082} (f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6) :\n    List \u03b1 \u2192 List \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 List \u03c6 :=\n  sorry\n\n@[simp] theorem length_map_accumr\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03c6 : Type w\u2081} {\u03c3 : Type w\u2082}\n    (f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6) (x : List \u03b1) (y : List \u03b2) (c : \u03c3) :\n    length (prod.snd (map_accumr\u2082 f x y c)) = min (length x) (length y) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/list/lemmas_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.380552199819168}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.lift\nimport Mathlib.topology.opens\nimport Mathlib.topology.algebra.ring\nimport Mathlib.PostPort\n\nuniverses u_1 l u_2 \n\nnamespace Mathlib\n\n/-- The type of open subgroups of a topological additive group. -/\nstructure open_add_subgroup (G : Type u_1) [add_group G] [topological_space G] \nextends add_subgroup G\nwhere\n  is_open' : is_open carrier\n\n/-- The type of open subgroups of a topological group. -/\nstructure open_subgroup (G : Type u_1) [group G] [topological_space G] \nextends subgroup G\nwhere\n  is_open' : is_open carrier\n\n/-- Reinterpret an `open_subgroup` as a `subgroup`. -/\n/-- Reinterpret an `open_add_subgroup` as an `add_subgroup`. -/\n-- Tell Lean that `open_add_subgroup` is a namespace\n\nnamespace open_add_subgroup\n\n\nend open_add_subgroup\n\n\nnamespace open_subgroup\n\n\nprotected instance has_coe_set {G : Type u_1} [group G] [topological_space G] : has_coe_t (open_subgroup G) (set G) :=\n  has_coe_t.mk fun (U : open_subgroup G) => carrier U\n\nprotected instance has_mem {G : Type u_1} [group G] [topological_space G] : has_mem G (open_subgroup G) :=\n  has_mem.mk fun (g : G) (U : open_subgroup G) => g \u2208 \u2191U\n\nprotected instance has_coe_subgroup {G : Type u_1} [group G] [topological_space G] : has_coe_t (open_subgroup G) (subgroup G) :=\n  has_coe_t.mk to_subgroup\n\nprotected instance has_coe_opens {G : Type u_1} [group G] [topological_space G] : has_coe_t (open_subgroup G) (topological_space.opens G) :=\n  has_coe_t.mk fun (U : open_subgroup G) => { val := \u2191U, property := is_open' U }\n\n@[simp] theorem mem_coe {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {g : G} : g \u2208 \u2191U \u2194 g \u2208 U :=\n  iff.rfl\n\n@[simp] theorem mem_coe_opens {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {g : G} : g \u2208 \u2191U \u2194 g \u2208 U :=\n  iff.rfl\n\n@[simp] theorem mem_coe_subgroup {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {g : G} : g \u2208 \u2191U \u2194 g \u2208 U :=\n  iff.rfl\n\ntheorem coe_injective {G : Type u_1} [group G] [topological_space G] : function.injective coe := sorry\n\ntheorem ext {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {V : open_subgroup G} (h : \u2200 (x : G), x \u2208 U \u2194 x \u2208 V) : U = V :=\n  coe_injective (set.ext h)\n\ntheorem ext_iff {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {V : open_subgroup G} : U = V \u2194 \u2200 (x : G), x \u2208 U \u2194 x \u2208 V :=\n  { mp := fun (h : U = V) (x : G) => h \u25b8 iff.rfl, mpr := ext }\n\nprotected theorem is_open {G : Type u_1} [group G] [topological_space G] (U : open_subgroup G) : is_open \u2191U :=\n  is_open' U\n\nprotected theorem one_mem {G : Type u_1} [group G] [topological_space G] (U : open_subgroup G) : 1 \u2208 U :=\n  one_mem' U\n\nprotected theorem inv_mem {G : Type u_1} [group G] [topological_space G] (U : open_subgroup G) {g : G} (h : g \u2208 U) : g\u207b\u00b9 \u2208 U :=\n  inv_mem' U h\n\nprotected theorem Mathlib.open_add_subgroup.add_mem {G : Type u_1} [add_group G] [topological_space G] (U : open_add_subgroup G) {g\u2081 : G} {g\u2082 : G} (h\u2081 : g\u2081 \u2208 U) (h\u2082 : g\u2082 \u2208 U) : g\u2081 + g\u2082 \u2208 U :=\n  open_add_subgroup.add_mem' U h\u2081 h\u2082\n\ntheorem mem_nhds_one {G : Type u_1} [group G] [topological_space G] (U : open_subgroup G) : \u2191U \u2208 nhds 1 :=\n  mem_nhds_sets (open_subgroup.is_open U) (open_subgroup.one_mem U)\n\nprotected instance has_top {G : Type u_1} [group G] [topological_space G] : has_top (open_subgroup G) :=\n  has_top.mk (mk (subgroup.carrier \u22a4) sorry sorry sorry is_open_univ)\n\nprotected instance inhabited {G : Type u_1} [group G] [topological_space G] : Inhabited (open_subgroup G) :=\n  { default := \u22a4 }\n\ntheorem is_closed {G : Type u_1} [group G] [topological_space G] [has_continuous_mul G] (U : open_subgroup G) : is_closed \u2191U := sorry\n\n/-- The product of two open subgroups as an open subgroup of the product group. -/\ndef prod {G : Type u_1} [group G] [topological_space G] {H : Type u_2} [group H] [topological_space H] (U : open_subgroup G) (V : open_subgroup H) : open_subgroup (G \u00d7 H) :=\n  mk (set.prod \u2191U \u2191V) sorry sorry sorry sorry\n\nprotected instance partial_order {G : Type u_1} [group G] [topological_space G] : partial_order (open_subgroup G) :=\n  partial_order.mk (fun (U V : open_subgroup G) => \u2200 {x : G}, x \u2208 U \u2192 x \u2208 V) partial_order.lt sorry sorry sorry\n\nprotected instance semilattice_inf_top {G : Type u_1} [group G] [topological_space G] : semilattice_inf_top (open_subgroup G) :=\n  semilattice_inf_top.mk \u22a4 partial_order.le partial_order.lt sorry sorry sorry sorry\n    (fun (U V : open_subgroup G) => mk (subgroup.carrier (\u2191U \u2293 \u2191V)) sorry sorry sorry sorry) sorry sorry sorry\n\n@[simp] theorem coe_inf {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {V : open_subgroup G} : \u2191(U \u2293 V) = \u2191U \u2229 \u2191V :=\n  rfl\n\n@[simp] theorem coe_subset {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {V : open_subgroup G} : \u2191U \u2286 \u2191V \u2194 U \u2264 V :=\n  iff.rfl\n\n@[simp] theorem coe_subgroup_le {G : Type u_1} [group G] [topological_space G] {U : open_subgroup G} {V : open_subgroup G} : \u2191U \u2264 \u2191V \u2194 U \u2264 V :=\n  iff.rfl\n\nend open_subgroup\n\n\nnamespace subgroup\n\n\ntheorem Mathlib.add_subgroup.is_open_of_mem_nhds {G : Type u_1} [add_group G] [topological_space G] [has_continuous_add G] (H : add_subgroup G) {g : G} (hg : \u2191H \u2208 nhds g) : is_open \u2191H := sorry\n\ntheorem is_open_of_open_subgroup {G : Type u_1} [group G] [topological_space G] [has_continuous_mul G] (H : subgroup G) {U : open_subgroup G} (h : open_subgroup.carrier U \u2264 \u2191H) : is_open \u2191H :=\n  is_open_of_mem_nhds H (filter.mem_sets_of_superset (open_subgroup.mem_nhds_one U) h)\n\ntheorem is_open_mono {G : Type u_1} [group G] [topological_space G] [has_continuous_mul G] {H\u2081 : subgroup G} {H\u2082 : subgroup G} (h : H\u2081 \u2264 H\u2082) (h\u2081 : is_open \u2191H\u2081) : is_open \u2191H\u2082 :=\n  is_open_of_open_subgroup H\u2082 h\n\nend subgroup\n\n\nnamespace open_subgroup\n\n\nprotected instance semilattice_sup_top {G : Type u_1} [group G] [topological_space G] [has_continuous_mul G] : semilattice_sup_top (open_subgroup G) :=\n  semilattice_sup_top.mk semilattice_inf_top.top semilattice_inf_top.le semilattice_inf_top.lt sorry sorry sorry sorry\n    (fun (U V : open_subgroup G) => mk (subgroup.carrier (\u2191U \u2294 \u2191V)) sorry sorry sorry sorry) sorry sorry sorry\n\nend open_subgroup\n\n\nnamespace submodule\n\n\ntheorem is_open_mono {R : Type u_1} {M : Type u_2} [comm_ring R] [add_comm_group M] [topological_space M] [topological_add_group M] [module R M] {U : submodule R M} {P : submodule R M} (h : U \u2264 P) (hU : is_open \u2191U) : is_open \u2191P :=\n  add_subgroup.is_open_mono h hU\n\nend submodule\n\n\nnamespace ideal\n\n\ntheorem is_open_of_open_subideal {R : Type u_1} [comm_ring R] [topological_space R] [topological_ring R] {U : ideal R} {I : ideal R} (h : U \u2264 I) (hU : is_open \u2191U) : is_open \u2191I :=\n  submodule.is_open_mono h hU\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/algebra/open_subgroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.380552199819168}}
{"text": "structure Lam where\n  Repr : Type v \u2192 Type v\n  Obsv : Type v \u2192 Type v\n\n  observe\n    {\u03b1 : Type v}\n  : Repr \u03b1 \u2192 Obsv \u03b1\n\n  val\n    {\u03b1 : Type v}\n  : \u03b1 \u2192 Repr \u03b1\n\n  app\n    {\u03b1 \u03b2 : Type v}\n  : Repr (\u03b1 \u2192 \u03b2) \u2192 Repr \u03b1 \u2192 Repr \u03b2\n\n\n\ndef Lam.example\n  (dsl : Lam)\n: dsl.Repr Nat :=\n  let v1 := dsl.val 7\n  let v2 := dsl.val 3\n  let f := dsl.val fun l r => l + r\n  let tmp := dsl.app f v1\n  dsl.app tmp v2\n\n\n\n\ndef Lam.eval : Lam where\n  Repr := id\n  Obsv := id\n\n  observe := id\n\n  val := id\n  app f a := f a\n\n#eval Lam.example Lam.eval\n\n", "meta": {"author": "AdrienChampion", "repo": "tagless.lean", "sha": "22cad3342cf461373412837a82529b3d6d9243ec", "save_path": "github-repos/lean/AdrienChampion-tagless.lean", "path": "github-repos/lean/AdrienChampion-tagless.lean/tagless.lean-22cad3342cf461373412837a82529b3d6d9243ec/Tagless/Tests/Lam.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.38055219610297053}}
{"text": "import Lean\n\nimport Smt.Reconstruction.Certifying.Boolean\nimport Smt.Reconstruction.Certifying.Util\n\nnamespace Smt.Reconstruction.Certifying\n\nopen Lean Elab Tactic Meta Expr\nopen List\n\ntheorem modusTollens {A B : Prop} : (A \u2192 B) \u2192 \u00ac B \u2192 \u00ac A := by\n  intros f nb\n  exact match Classical.em A with\n  | Or.inl ha  => absurd (f ha) nb\n  | Or.inr hna => hna\n\nsyntax (name := liftOrNToNeg) \"liftOrNToNeg\" term : tactic\n\n@[tactic liftOrNToNeg] def evalLiftOrNToNeg : Tactic :=\n  fun stx => withMainContext do\n    let hyp \u2190 elabTerm stx[1] none\n    let hypType \u2190 inferType hyp\n    let props := map notExpr (collectPropsInOrChain hypType)\n    let propsList: Expr := listExpr props $ Expr.sort Level.zero\n    let deMorgan: Expr := mkApp (mkConst ``deMorgan\u2082) propsList\n    let modusTollens: Expr \u2190 mkAppM ``modusTollens #[deMorgan]\n    let notNotHyp: Expr \u2190 mkAppM ``notNotIntro #[hyp]\n    let answer := mkApp modusTollens notNotHyp\n    Tactic.closeMainGoal answer\n\nexample : \u00ac A \u2228 \u00ac B \u2228 \u00ac C \u2228 \u00ac D \u2192 \u00ac (A \u2227 B \u2227 C \u2227 D) := by\n  intro h\n  liftOrNToNeg h\n--  fun h => modusTollens (@deMorgan\u2082 [A, B, C, D]) (notNotIntro h)\n\nend Smt.Reconstruction.Certifying\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Smt/Reconstruction/Certifying/LiftOrNToNeg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.38055219610297053}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.sums.basic\n\n/-!\n# Associator for binary disjoint union of categories.\n\nThe associator functor `((C \u2295 D) \u2295 E) \u2964 (C \u2295 (D \u2295 E))` and its inverse form an equivalence.\n-/\n\nuniverses v u\n\nopen category_theory\nopen sum\n\nnamespace category_theory.sum\n\nvariables (C : Type u) [category.{v} C]\n          (D : Type u) [category.{v} D]\n          (E : Type u) [category.{v} E]\n\n/--\nThe associator functor `(C \u2295 D) \u2295 E \u2964 C \u2295 (D \u2295 E)` for sums of categories.\n-/\ndef associator : (C \u2295 D) \u2295 E \u2964 C \u2295 (D \u2295 E) :=\n{ obj := \u03bb X, match X with\n  | inl (inl X) := inl X\n  | inl (inr X) := inr (inl X)\n  | inr X := inr (inr X)\n  end,\n  map := \u03bb X Y f, match X, Y, f with\n  | inl (inl X), inl (inl Y), f := f\n  | inl (inr X), inl (inr Y), f := f\n  | inr X, inr Y, f := f\n  end }\n\n@[simp] lemma associator_obj_inl_inl (X) : (associator C D E).obj (inl (inl X)) = inl X := rfl\n@[simp] lemma associator_obj_inl_inr (X) : (associator C D E).obj (inl (inr X)) = inr (inl X) := rfl\n@[simp] lemma associator_obj_inr (X) : (associator C D E).obj (inr X) = inr (inr X) := rfl\n@[simp] lemma associator_map_inl_inl {X Y : C} (f : inl (inl X) \u27f6 inl (inl Y)) :\n  (associator C D E).map f = f := rfl\n@[simp] lemma associator_map_inl_inr {X Y : D} (f : inl (inr X) \u27f6 inl (inr Y)) :\n  (associator C D E).map f = f := rfl\n@[simp] lemma associator_map_inr {X Y : E} (f : inr X \u27f6 inr Y) :\n  (associator C D E).map f = f := rfl\n\n/--\nThe inverse associator functor `C \u2295 (D \u2295 E) \u2964 (C \u2295 D) \u2295 E` for sums of categories.\n-/\ndef inverse_associator : C \u2295 (D \u2295 E) \u2964 (C \u2295 D) \u2295 E :=\n{ obj := \u03bb X, match X with\n  | inl X := inl (inl X)\n  | inr (inl X) := inl (inr X)\n  | inr (inr X) := inr X\n  end,\n  map := \u03bb X Y f, match X, Y, f with\n  | inl X, inl Y, f := f\n  | inr (inl X), inr (inl Y), f := f\n  | inr (inr X), inr (inr Y), f := f\n  end }\n\n@[simp] lemma inverse_associator_obj_inl (X) :\n  (inverse_associator C D E).obj (inl X) = inl (inl X) := rfl\n@[simp] lemma inverse_associator_obj_inr_inl (X) :\n  (inverse_associator C D E).obj (inr (inl X)) = inl (inr X) := rfl\n@[simp] lemma inverse_associator_obj_inr_inr (X) :\n  (inverse_associator C D E).obj (inr (inr X)) = inr X := rfl\n@[simp] \n\n/--\nThe equivalence of categories expressing associativity of sums of categories.\n-/\ndef associativity : (C \u2295 D) \u2295 E \u224c C \u2295 (D \u2295 E) :=\nequivalence.mk (associator C D E) (inverse_associator C D E)\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by tidy)) (by tidy))\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by tidy)) (by tidy))\n\ninstance associator_is_equivalence : is_equivalence (associator C D E) :=\n(by apply_instance : is_equivalence (associativity C D E).functor)\n\ninstance inverse_associator_is_equivalence : is_equivalence (inverse_associator C D E) :=\n(by apply_instance : is_equivalence (associativity C D E).inverse)\n\n-- TODO unitors?\n-- TODO pentagon natural transformation? ...satisfying?\nend category_theory.sum\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/sums/associator.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458153, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.38050697538275696}}
{"text": "lemma imp_self (P : Prop) : P \u2192 P :=\nbegin\nintro p,\nexact p,\nend\n", "meta": {"author": "abdelq", "repo": "natural-number-game", "sha": "bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2", "save_path": "github-repos/lean/abdelq-natural-number-game", "path": "github-repos/lean/abdelq-natural-number-game/natural-number-game-bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2/world06/level02.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.38047260224356205}}
{"text": "import new.unordered.chain\nimport new.ordered.chain_o\n-- import algebra.category.Group.abelian\nimport algebra.homology.homotopy\n\nnoncomputable theory\n\nsection\n\nopen topological_space Top Top.sheaf\nopen category_theory\nopen opposite\nopen AddCommGroup (hiding has_zero_object)\n\nopen_locale big_operators\n\nuniverse u\nvariables {X : Top.{u}} (\ud835\udcd5 : sheaf Ab X) (U : X.oc)\n\nnamespace computing_homotopy\n\nvariables {\ud835\udcd5 U}\n\ndef unordered_to_ordered.to_fun (n : \u2115) :\n  C \ud835\udcd5 U n \u2192 C_o \ud835\udcd5 U n :=\n\u03bb f \u03c3, f \u03c3.to_fun\n\ndef unordered_to_ordered (n : \u2115) :\n  C \ud835\udcd5 U n \u27f6 C_o \ud835\udcd5 U n :=\n{ to_fun := unordered_to_ordered.to_fun n,\n  map_zero' := rfl,\n  map_add' := \u03bb _ _, rfl }\n\ndef chain_unordered_to_ordered :\n  Cech_complex_wrt_cover_unordered \ud835\udcd5 U \u27f6\n  Cech_complex_wrt_cover_ordered   \ud835\udcd5 U :=\n{ f := \u03bb n, unordered_to_ordered (n+1),\n  comm' := \u03bb i j (h : _ + 1 = _), begin\n    subst h,\n    ext f \u03b1,\n    -- change unordered_to_ordered.to_fun (i + 1) \u226b _ = _,\n    rw [comp_apply, comp_apply, unordered_to_ordered, add_monoid_hom.coe_mk, \n      unordered_to_ordered, add_monoid_hom.coe_mk, unordered_to_ordered.to_fun,\n      unordered_to_ordered.to_fun],\n    simp only,\n    erw d_o_to_succ,\n    erw d_to_succ,\n    rw dd_o_aux.d_o_def,\n    rw dd_aux.d_def,\n    rw finset.sum_congr rfl,\n    rintros \u27e8k, hk\u27e9 -,\n    split_ifs,\n    { rw [id], refl, },\n    { refl, },\n  end }\n\ndef vec2vec_o_of_inj.to_fun {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) : fin n \u2192 U.\u03b9 :=\nlet \u03b2 := finset.order_iso_of_fin (finset.image \u03b1 finset.univ) \n  (show (finset.image \u03b1 finset.univ).card = n, begin\n    rw finset.card_image_eq_iff_inj_on.mpr,\n    rw finset.card_univ,\n    rw fintype.card_fin,\n    apply set.inj_on_of_injective,\n    assumption\n  end) in\n(\u03bb x, x.1 : finset.image \u03b1 finset.univ \u2192 U.\u03b9) \u2218 \u03b2.to_fun\n\nlemma vec2vec_o_of_inj.to_fun_mem {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) (i : fin n) :\n  vec2vec_o_of_inj.to_fun h i \u2208 finset.image \u03b1 finset.univ := finset.mem_image.mpr $\nbegin\n  dunfold vec2vec_o_of_inj.to_fun,\n  simp only [finset.mem_univ, subtype.val_eq_coe, order_iso.to_fun_eq_coe, \n    function.comp_app, finset.coe_order_iso_of_fin_apply, exists_true_left],\n  generalize_proofs card_eq,\n  sorry\nend\n\nlemma vec2vec_o_of_inj.is_strict_mono {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) :\n  strict_mono (vec2vec_o_of_inj.to_fun h) := \u03bb i j ineq,\nbegin\n  rw vec2vec_o_of_inj.to_fun,\n  simp only,\n  rw function.comp_apply,\n  rw function.comp_apply,\n  simpa only [order_iso.to_fun_eq_coe, subtype.val_eq_coe, finset.coe_order_iso_of_fin_apply, order_embedding.lt_iff_lt],\nend\n\ndef vec2vec_o_of_inj {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) : vec_o U n :=\n{ to_fun := vec2vec_o_of_inj.to_fun h,\n  is_strict_mono := vec2vec_o_of_inj.is_strict_mono h }\n\nlemma vec2vec_o_of_inj.mem_image {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) (i : fin n) :\n  vec2vec_o_of_inj h i \u2208 finset.image \u03b1 finset.univ :=\nvec2vec_o_of_inj.to_fun_mem h i\n\n\nlemma face.vec2vec_o_eq {n} {\u03b1 : fin n \u2192 U.\u03b9} (h : function.injective \u03b1) :\n  face \u03b1 = \n  face_o (vec2vec_o_of_inj h) := \nopens.ext $ set.ext $ \u03bb p,\nbegin\n  erw [opens.mem_coe, opens.mem_coe, opens.fintype_infi, opens.mem_coe, opens.fintype_infi],\n  split;\n  intros hp i,\n  { rcases finset.mem_image.mp (vec2vec_o_of_inj.mem_image h i) with \u27e8j, _, hj\u27e9,\n    specialize hp j,\n    rw hj at hp,\n    exact hp, },\n  { sorry },\nend\n\ndef ordered_to_unordered.to_fun (n : \u2115) :\n  C_o \ud835\udcd5 U n \u2192 C \ud835\udcd5 U n := \u03bb f \u03b1, \ndite (function.injective \u03b1)\n(\u03bb h, match signature \u03b1 with\n  | sign.zero := 0 -- this will never occur\n  | sign.pos := \ud835\udcd5.1.map (eq_to_hom (face.vec2vec_o_eq h)).op (f (vec2vec_o_of_inj h))\n  | sign.neg := - \ud835\udcd5.1.map (eq_to_hom (face.vec2vec_o_eq h)).op (f (vec2vec_o_of_inj h))\n  end)\n(\u03bb _, 0)\n\ndef ordered_to_unordered (n : \u2115) :\n  C_o \ud835\udcd5 U n \u27f6 C \ud835\udcd5 U n :=\n{ to_fun := ordered_to_unordered.to_fun n,\n  map_zero' := sorry,\n  map_add' := sorry }\n\ndef chain_ordered_to_unordered :\n  Cech_complex_wrt_cover_ordered   \ud835\udcd5 U \u27f6\n  Cech_complex_wrt_cover_unordered \ud835\udcd5 U :=\n{ f := \u03bb n, ordered_to_unordered (n + 1),\n  comm' := sorry }\n\nend computing_homotopy\n\ndef chain_unordered_homotopically_equivalent_ordered :\n  homotopy_equiv\n    (Cech_complex_wrt_cover_unordered \ud835\udcd5 U)\n    (Cech_complex_wrt_cover_ordered \ud835\udcd5 U) \n    := \n{ hom := computing_homotopy.chain_unordered_to_ordered,\n  inv := computing_homotopy.chain_ordered_to_unordered,\n  homotopy_hom_inv_id := sorry,\n  homotopy_inv_hom_id := sorry }\n\ndef cohomology_unordered_eq_ordered (n : \u2115) :\n  Cech_Cohomology_Group_wrt_cover_unordered_nth \ud835\udcd5 U n \u2245\n  Cech_Cohomology_Group_wrt_cover_ordered_nth \ud835\udcd5 U n :=\n@homology_obj_iso_of_homotopy_equiv \n  \u2115 Ab _ _ (complex_shape.up \u2115)\n  (Cech_complex_wrt_cover_unordered \ud835\udcd5 U)\n  (Cech_complex_wrt_cover_ordered \ud835\udcd5 U)\n  _ _ _ _ (abelian.has_zero_object)\n  (chain_unordered_homotopically_equivalent_ordered \ud835\udcd5 U)\n  n\n\n\ndef example1 : \n  Cech_Cohomology_Group_wrt_cover_unordered_nth \ud835\udcd5 U 0 \u2245 \n  \ud835\udcd5.1.obj (op \u22a4) :=\ncohomology_unordered_eq_ordered \ud835\udcd5 U 0 \u226a\u226b zeroth_Cech_Cohomology \ud835\udcd5 U\n\n\nend", "meta": {"author": "jjaassoonn", "repo": "cc", "sha": "6d3dc6885fa012e8c18fd38ab2949d73777fb442", "save_path": "github-repos/lean/jjaassoonn-cc", "path": "github-repos/lean/jjaassoonn-cc/cc-6d3dc6885fa012e8c18fd38ab2949d73777fb442/src/compare.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7549149868676284, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3804063201074777}}
{"text": "-- import laurent_measures.functor\nimport data.finset.basic\nimport analysis.special_functions.log.base\nimport for_mathlib.pi_induced\nimport laurent_measures.thm69\n-- import data.real.basic\n\n/-!\nThis files introduces the maps `\u0398`, `\u03a6` (***and `\u03a8` ???***), which are the \"measurifications\" of\n`\u03b8`, `\u03d5` (*** and `\u03c8` ???***)\n`laurent_measures.thm69`, they are morphisms in the right category.\n\nWe then prove in **???** that `\u03b8_\u03d5_exact` of `laurent_measures.thm69` becomes a short exact sequence\nin the category **???**.\n-/\n\n\nnoncomputable theory\n\nuniverse u\n\nnamespace laurent_measures_ses\n\nopen laurent_measures pseudo_normed_group comphaus_filtered_pseudo_normed_group\nopen comphaus_filtered_pseudo_normed_group_hom\nopen_locale big_operators nnreal\n\nsection phi_to_hom\n\n-- parameter {p : \u211d\u22650}\n-- variables [fact(0 < p)] [fact (p < 1)]\n-- local notation `r` := @r p\n-- local notation `\u2133` := real_measures p\n\nvariable {r : \u211d\u22650}\nvariables [fact (0 < r)]\nvariable {S : Fintype}\n\nlocal notation `\u2112` := laurent_measures r\nlocal notation `\u03d6` := (Fintype.of punit : Type u)\n\nvariables {M\u2081 M\u2082 : Type u} [comphaus_filtered_pseudo_normed_group M\u2081]\n  [comphaus_filtered_pseudo_normed_group M\u2082]\n\ndef cfpng_hom_add (f g : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\nbegin\n  apply mk_of_bound (f.to_add_monoid_hom + g.to_add_monoid_hom) (f.bound.some + g.bound.some),\n  intro c,\n  refine \u27e8_, _\u27e9,\n  { intros x hx,\n      simp only [comphaus_filtered_pseudo_normed_group_hom.coe_mk],\n      simp only [add_monoid_hom.add_apply, coe_to_add_monoid_hom],\n      convert pseudo_normed_group.add_mem_filtration (f.bound.some_spec hx) (g.bound.some_spec hx),\n      rw add_mul, },\n  let f\u2080 : filtration M\u2081 c \u2192 filtration M\u2082 (f.bound.some * c) := \u03bb x, \u27e8f x, f.bound.some_spec x.2\u27e9,\n  have hf\u2080 : continuous f\u2080 := f.continuous _ (\u03bb x, rfl),\n  let g\u2080 : filtration M\u2081 c \u2192 filtration M\u2082 (g.bound.some * c) := \u03bb x, \u27e8g x, g.bound.some_spec x.2\u27e9,\n  have hg\u2080 : continuous g\u2080 := g.continuous _ (\u03bb x, rfl),\n  simp only [add_monoid_hom.add_apply, coe_to_add_monoid_hom],\n  haveI : fact ((f.bound.some * c + g.bound.some * c) \u2264 ((f.bound.some + g.bound.some) * c) ) :=\n    fact.mk (le_of_eq (add_mul _ _ _).symm),\n  let \u03b9 : filtration M\u2082 (f.bound.some * c + g.bound.some * c) \u2192 filtration M\u2082\n    ((f.bound.some + g.bound.some) * c) := cast_le,\n  have h\u03b9 : continuous \u03b9 := continuous_cast_le _ _,\n  let S\u2080 : filtration M\u2082 (f.bound.some * c) \u00d7 filtration M\u2082 (g.bound.some * c) \u2192\n    filtration M\u2082 (f.bound.some * c + g.bound.some * c) :=\n      \u03bb x, \u27e8x.fst + x.snd, add_mem_filtration x.fst.2 x.snd.2\u27e9,\n  have hS\u2080 := continuous_add' (f.bound.some * c) (g.bound.some * c),\n  exact h\u03b9.comp (hS\u2080.comp (continuous.prod_mk hf\u2080 hg\u2080)),\nend\n\ndef cfpng_hom_neg (f : comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :\n  (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\nbegin\n  apply mk_of_bound (- f.to_add_monoid_hom) (f.bound.some),\n  intro c,\n  refine \u27e8_, _\u27e9,\n  { intros x hx,\n    simp only [comphaus_filtered_pseudo_normed_group_hom.coe_mk],\n    convert pseudo_normed_group.neg_mem_filtration (f.bound.some_spec hx) },\n  let f\u2080 : filtration M\u2081 c \u2192 filtration M\u2082 (f.bound.some * c) := \u03bb x, \u27e8f x, f.bound.some_spec x.2\u27e9,\n  have hf\u2080 : continuous f\u2080 := f.continuous _ (\u03bb x, rfl),\n  exact (continuous_neg' _).comp hf\u2080,\nend\n\ninstance : add_comm_group (comphaus_filtered_pseudo_normed_group_hom M\u2081 M\u2082) :=\n{ add := cfpng_hom_add,\n  add_assoc := by {intros, ext, apply add_assoc},\n  zero := 0,\n  zero_add := by {intros, ext, apply zero_add},\n  add_zero := by {intros, ext, apply add_zero},\n  neg := cfpng_hom_neg,\n  add_left_neg := by {intros, ext, apply add_left_neg},\n  add_comm := by {intros, ext, apply add_comm} }\n\nvariable (S)\n\n/-- The map on Laurent measures induced by multiplication by `T\u207b\u00b9 - 2` on `\u2124((T))\u1d63`. -/\ndef \u03a6 : comphaus_filtered_pseudo_normed_group_hom (\u2112 S) (\u2112 S) := shift (1) - 2 \u2022 id\n-- variable {S}\n\n\nlemma \u03a6_eq_\u03d5 (F : \u2112 S) : \u03a6 S F = \u03d5 F := rfl\n\n-- after the \u03a6 refactor the below lemma is no longer true\n\n-- lemma \u03a6_bound_by_3 [fact (r \u2264 1)] :\n--   (\u03a6 S : comphaus_filtered_pseudo_normed_group_hom (\u2112 S) (\u2112 S)).bound_by 3 :=\n-- begin\n--   let sh : comphaus_filtered_pseudo_normed_group_hom (\u2112 S) (\u2112 S) := shift (-1),\n--   let shup : comphaus_filtered_pseudo_normed_group_hom (\u2112 S) (\u2112 S) := shift (1),\n--   have Hsh : sh.bound_by 1,\n--   { refine (mk_of_bound_bound_by _ _ _).mono 1 _,\n--     rw [neg_neg], exact (pow_one r).le.trans (fact.out _) },\n--   have Hshup : shup.bound_by 1,\n--   { refine (mk_of_bound_bound_by _ _ _).mono (1) _,\n\n--   }\n--   suffices : (sh + sh + (-id)).bound_by (1 + 1 + 1),\n--   { convert this using 1, ext1, dsimp only [\u03a6_eq_\u03d5, \u03d5], erw two_nsmul, }, -- was refl\n--   refine (Hsh.add Hsh).add (mk_of_bound_bound_by _ _ _).neg,\n-- end\n\nlemma \u03a6_natural (S T : Fintype) (f : S \u27f6 T) (F : \u2112 S) (t : T) (n : \u2124) :\n  \u03a6 T (map f F) t n = laurent_measures.map f (\u03a6 S F) t n :=\nbegin\n  simp [\u03a6_eq_\u03d5, \u03d5, finset.mul_sum],\nend\n\nend phi_to_hom\n\nsection theta\n\nopen theta real_measures\n\nparameter (p : \u211d\u22650)\nlocal notation `r` := @r p\nlocal notation `\u2133` := real_measures p\nlocal notation `\u2112` := laurent_measures r\n\nvariable {S : Fintype.{u}}\n\nlocal notation `\u03d6` := Fintype.of (punit : Type u)\n\ndef seval_\u2112_c (c : \u211d\u22650) (s : S) : filtration (\u2112 S) c \u2192 (filtration (\u2112 \u03d6) c) :=\n\u03bb F,\nbegin\n  refine \u27e8seval_\u2112 S s F, _\u27e9,\n  have hF := F.2,\n  simp only [filtration, set.mem_set_of_eq, seval_\u2112, nnnorm, laurent_measures.coe_mk,\n    fintype.univ_punit, finset.sum_singleton] at \u22a2 hF,\n  have := finset.sum_le_sum_of_subset (finset.singleton_subset_iff.mpr $ finset.mem_univ_val _),\n  rw finset.sum_singleton at this,\n  apply le_trans this hF,\nend\n\nvariable [fact (0 < p)]\n\nlemma \u03b8_zero : \u03b8 (0 : \u2112 S) = 0 :=\nbegin\n  dsimp only [\u03b8, theta.\u03d1],\n  funext,\n  simp only [laurent_measures.zero_apply, int.cast_zero, zero_mul, tsum_zero, real_measures.zero_apply],\nend\n\nvariable [fact (p < 1)]\n\nlemma \u03b8_add (F G : \u2112 S) : \u03b8 (F + G) = \u03b8 F + \u03b8 G :=\nbegin\n  dsimp only [\u03b8, theta.\u03d1],\n  funext,\n  simp only [laurent_measures.add_apply, int.cast_add, one_div, inv_zpow', zpow_neg,\n    real_measures.add_apply, tsum_add],\n  rw \u2190 tsum_add,\n  { congr,\n    funext,\n    rw add_mul },\n  all_goals {apply summable_of_summable_norm, simp_rw [\u2190 inv_zpow, norm_mul, norm_zpow, norm_inv,\n    real.norm_two] },\n  exact aux_thm69.summable_smaller_radius_norm F.d half_lt_r (F.summable s) (\u03bb n, lt_d_eq_zero _ _ _),\n  exact aux_thm69.summable_smaller_radius_norm G.d half_lt_r (G.summable s) (\u03bb n, lt_d_eq_zero _ _ _),\nend\n\n--for mathlib\nlemma nnreal.rpow_int_cast (x : \u211d\u22650) (n : \u2124) : x ^ n = x ^ (n : \u211d) := by {\n  rw [\u2190 nnreal.coe_eq, nnreal.coe_zpow, \u2190 real.rpow_int_cast, \u2190 nnreal.coe_rpow] }\n\nlemma nnreal.rpow_le_rpow_of_exponent_le {x : \u211d\u22650} (x1 : 1 \u2264 x) {y z : \u211d}\n  (hyz : y \u2264 z) :\n  x ^ y \u2264 x ^ z :=\nby { cases x with x hx, exact real.rpow_le_rpow_of_exponent_le x1 hyz }\n\nlemma nnreal.tsum_geom_arit_inequality (f: \u2124 \u2192 \u211d) {r' : \u211d} (hr'1 : 0 < r') (hr'2 : r' \u2264 1)\n  (hs1 : summable (\u03bb n, f n)) (hs2 : summable (\u03bb n, \u2225(f n)\u2225\u208a ^ r')) :\n  \u2225 tsum (\u03bb n, f n) \u2225\u208a ^ r' \u2264 tsum (\u03bb n, \u2225(f n)\u2225\u208a ^ r' ) :=\nbegin\n  rw \u2190 summable_norm_iff at hs1,\n  simp_rw \u2190 _root_.coe_nnnorm at hs1,\n  rw nnreal.summable_coe at hs1,\n  refine le_trans (nnreal.rpow_le_rpow (nnnorm_tsum_le hs1) hr'1.le) _,\n  have := \u03bb s : finset \u2124, nnreal.rpow_sum_le_sum_rpow s (\u03bb i, \u2225f i\u2225\u208a) hr'1 hr'2,\n  dsimp only at this,\n  have s1' := filter.tendsto.comp (continuous.tendsto\n    (nnreal.continuous_rpow_const hr'1.le) _) hs1.has_sum,\n  dsimp [function.comp] at s1',\n  apply tendsto_le_of_eventually_le s1' hs2.has_sum,\n  delta filter.eventually_le,\n  convert filter.univ_sets _,\n  ext x,\n  simp [this],\nend\n\nlemma aux_bound (F : \u2112 S) (s : S) : \u2200 (b : \u2124), \u2225(F s b : \u211d) \u2225\u208a ^ (p : \u211d) *\n  (2\u207b\u00b9 ^ (p : \u211d)) ^ (b : \u211d) \u2264 \u2225F s b\u2225\u208a * r ^ b :=\nbegin\n  intro b,\n  rw [nnreal.rpow_int_cast],\n  refine mul_le_mul_of_nonneg_right _ (real.rpow_nonneg_of_nonneg (nnreal.coe_nonneg _) _),\n  have p_le_one : (p : \u211d) \u2264 1,\n  { rw \u2190 nnreal.coe_one,\n    exact (nnreal.coe_lt_coe.mpr $ fact.out _).le },\n  by_cases hF_nz : F s b = 0,\n  { rw [hF_nz, int.cast_zero, nnnorm_zero, nnnorm_zero, nnreal.zero_rpow],\n    rw [ne.def, \u2190 nnreal.coe_zero, nnreal.coe_eq, \u2190 ne.def],\n    exact ne_of_gt (fact.out _) },\n  { convert nnreal.rpow_le_rpow_of_exponent_le _ p_le_one,\n    { rw nnreal.rpow_one,\n      refl },\n    { refine not_lt.mp (\u03bb hf, hF_nz (int.abs_lt_one_iff.mp _)),\n      suffices : (|F s b| : \u211d) < 1, exact_mod_cast this,\n      rw \u2190 int.norm_eq_abs,\n      rwa [\u2190 nnreal.coe_lt_coe, \u2190 nnnorm_norm, real.nnnorm_of_nonneg (norm_nonneg _)] at hf } }\nend\n\nlemma \u03b8_bound : \u2200 c : \u211d\u22650, \u2200 F : (\u2112 S), F \u2208 filtration (\u2112 S) c \u2192 (\u03b8 F) \u2208 filtration (\u2133 S)\n  (1 * c) :=\nbegin\n  intros c F hF,\n  rw laurent_measures.mem_filtration_iff at hF,\n  dsimp only [laurent_measures.has_nnnorm] at hF,\n  rw [one_mul, real_measures.mem_filtration_iff],\n  dsimp only [real_measures.has_nnnorm, \u03b8, theta.\u03d1],\n  let T := S.2.1,\n  have ineq : \u2200 (s \u2208 T), \u2225\u2211' (n : \u2124), ((F s n) : \u211d) * 2\u207b\u00b9 ^ n\u2225\u208a ^ (p : \u211d) \u2264 \u2211' (n : \u2124),\n    \u2225 ((F s n) : \u211d) * 2\u207b\u00b9 ^ n\u2225\u208a ^ (p : \u211d),\n  { intros s hs,\n    apply nnreal.tsum_geom_arit_inequality (\u03bb n, ((F s n) * 2\u207b\u00b9 ^ n)),\n    { norm_num, exact fact.out _},\n    { suffices : p \u2264 1, assumption_mod_cast, exact fact.out _},\n    { dsimp only,\n      obtain \u27e8d, hd\u27e9 := exists_bdd_filtration (r_pos) (r_lt_one) F,\n      apply aux_thm69.summable_smaller_radius d (F.summable s) (hd s) half_lt_r },\n    { dsimp only,\n      simp_rw [nnnorm_mul, nnreal.mul_rpow],\n      have := F.summable s,\n      rw \u2190 nnreal.summable_coe,\n      apply summable_of_nonneg_of_le (\u03bb i, _) _ this, apply nnreal.zero_le_coe,\n      intro n,\n      push_cast,\n      apply mul_le_mul,\n      { -- true because \u2225integer\u2225 is either 0 or >= 1\n        norm_cast,\n        by_cases h : F s n = 0,\n        { simp only [h, norm_zero],\n          refine le_of_eq (real.zero_rpow _),\n          norm_cast,\n          exact ne_of_gt (fact.out _) },\n        { nth_rewrite 1 (real.rpow_one (\u2225F s n\u2225)).symm,\n          apply real.rpow_le_rpow_of_exponent_le,\n          { rw [int.norm_eq_abs, le_abs'],\n            norm_cast,\n            rcases lt_trichotomy 0 (F s n) with (hF|hF|hF),\n            { right, linarith },\n            { exact false.elim (h hF.symm) },\n            { left, change _ \u2264 -(1 : \u2124), linarith, } },\n          { norm_cast, exact fact.out _ } } },\n      { apply le_of_eq,\n        rw [\u2190 r_coe],\n        rw real.norm_of_nonneg,\n        { -- can't use pow_mul yet because one is int one is real\n          rw [\u2190 real.rpow_int_cast, \u2190 real.rpow_int_cast],\n          rw [\u2190 real.rpow_mul, mul_comm, real.rpow_mul];\n          norm_num },\n        { apply zpow_nonneg,\n          norm_num } },\n      { refine (real.rpow_pos_of_pos _ _).le,\n        rw norm_pos_iff,\n        apply zpow_ne_zero,\n        norm_num, },\n      { apply norm_nonneg, } } },\n  apply (finset.sum_le_sum ineq).trans,\n  simp_rw [nnnorm_mul, nnnorm_zpow, nnnorm_inv, nnreal.mul_rpow, real.nnnorm_two,\n    nnreal.rpow_int_cast, \u2190 nnreal.rpow_mul (2 : \u211d\u22650)\u207b\u00b9, mul_comm, nnreal.rpow_mul (2 : \u211d\u22650)\u207b\u00b9],\n  apply le_trans _ hF,\n  apply finset.sum_le_sum,\n  intros s hs,\n  apply tsum_le_tsum,\n  exact aux_bound p F s,\n  refine nnreal.summable_of_le _ (F.2 s),\n  exacts [aux_bound p F s, F.2 s],\nend\n\n\nlemma \u03b8_bound' :  \u2200 c : \u211d\u22650, \u2200 F : (\u2112 S), F \u2208 filtration (\u2112 S) c \u2192 (\u03b8 F) \u2208 filtration (\u2133 S)\n  c :=by { simpa [one_mul] using (\u03b8_bound p)}\n\ndef \u03b8_to_add : (\u2112 S) \u2192+ (\u2133 S) :=\n{ to_fun := \u03bb F, \u03b8 F,\n  map_zero' := \u03b8_zero,\n  map_add' := \u03b8_add, }\n\nvariable (S)\n\nopen theta metric real_measures\n\ndef seval_\u2133_c (c : \u211d\u22650) (s : S) : filtration (\u2133 S) c \u2192 (filtration (\u2133 \u03d6) c) :=\n\u03bb x,\n  begin\n  refine \u27e8(\u03bb _, x.1 s), _\u27e9,\n  have hx := x.2,\n  simp only [filtration, set.mem_set_of_eq, nnnorm, laurent_measures.coe_mk,\n    fintype.univ_punit, finset.sum_singleton] at \u22a2 hx,\n  have := finset.sum_le_sum_of_subset (finset.singleton_subset_iff.mpr $ finset.mem_univ_val _),\n  rw finset.sum_singleton at this,\n  apply le_trans this hx,\nend\n\n-- **[FAE]** From here everything might be useless until `lemma inducing_cast_\u2133`: check\n-- also the `variable (c : \u211d\u22650)` issue; the idea is to replace cast_\u2133_c with \u03b1, for which\n-- everything seems to work\n\nvariable (c : \u211d\u22650)\n\ndef box := {F : (\u2133 S) // \u2200 s, \u2225 F s \u2225\u208a ^ (p : \u211d) \u2264 c }\n\ninstance : has_coe (box S c) (\u2133 S) := by {dsimp only [box], apply_instance}\ninstance : topological_space (\u2133 S) := by {dsimp only [real_measures], apply_instance}\ninstance : topological_space (box S c) := by {dsimp only [box], apply_instance}\n\n\ndef equiv_box_\u03d6 : (box S c) \u2243 \u03a0 (s : S), (filtration (\u2133 \u03d6) c) :=\nbegin\n  fconstructor,\n  { intros F s,\n    use seval_\u2133 S s F.1,\n    simp only [real_measures.mem_filtration_iff, nnnorm, fintype.univ_punit,\n      finset.sum_singleton, seval_\u2133],\n    exact F.2 s },\n  { intro G,\n    use \u03bb s, (G s).1 punit.star,\n    intro s,\n    simpa only [real_measures.mem_filtration_iff, nnnorm, fintype.univ_punit,\n      finset.sum_singleton, seval_\u2133] using (G s).2 },\n  { intro _,\n    ext s,\n    simpa only [seval_\u2133] },\n  { intro G,\n    ext s,\n    simp only [seval_\u2133, subtype.val_eq_coe, subtype.coe_mk],\n    induction x,\n    refl }\nend\n\ndef homeo_box_\u03d6 : (box S c) \u2243\u209c \u03a0 (s : S), (filtration (\u2133 \u03d6) c) :=\n{ to_equiv := equiv_box_\u03d6 S c,\n  continuous_to_fun := begin\n    apply continuous_pi,\n    intro s,\n    dsimp only [equiv_box_\u03d6, seval_\u2133],\n    refine continuous.subtype_mk (continuous_pi $ \u03bb i, _) _,\n    exact continuous_pi_iff.mp continuous_induced_dom s,\n  end,\n  continuous_inv_fun :=\n  begin\n    dsimp only [equiv_box_\u03d6, seval_\u2133],\n    refine continuous.subtype_mk _ _,\n    apply continuous_pi,\n    intro s,\n    have h : continuous (\u03bb (a : S \u2192 (filtration (\u2133 \u03d6) c)), (a s).val)\n       := continuous.subtype_coe (continuous_apply s),\n    have H := continuous_apply punit.star,\n    exact H.comp h,\n  end}\n\n\ndef \u03b1 : filtration (\u2133 S) c \u2192 box S c :=\nbegin\n  intro x,\n  use x,\n  have hx := x.2,\n  intro s,\n  simp only [filtration, set.mem_set_of_eq, nnnorm, laurent_measures.coe_mk,\n    fintype.univ_punit, finset.sum_singleton] at hx,\n  have := finset.sum_le_sum_of_subset (finset.singleton_subset_iff.mpr $ finset.mem_univ_val _),\n  rw finset.sum_singleton at this,\n  apply le_trans this hx,\nend\n\n\nlemma coe_\u03b1_coe : (coe : (box S c) \u2192 (\u2133 S)) \u2218 (\u03b1 S c) = coe := by {funext _, refl}\n\nlemma inducing_\u03b1 : inducing (\u03b1 S c) :=\nbegin\n  have ind_ind := @induced_compose _ _ (\u2133 S) _ (\u03b1 p S c) coe,\n  rw [coe_\u03b1_coe p S c] at ind_ind,\n  exact {induced := eq.symm ind_ind},\nend\n\n\nlemma seval_\u2133_\u03b1_commute (c : \u211d\u22650) (s : S) :\n (\u03bb F, ((homeo_box_\u03d6 S c) \u2218 (\u03b1 S c)) F s) = (\u03bb F, seval_\u2133_c S c s F) := rfl\n\n lemma seval_\u2133_\u03b1_commute' {X : Type*} (c : \u211d\u22650) {f : X \u2192 filtration (\u2133 S) c} (s : S)  :\n (\u03bb x, ((homeo_box_\u03d6 S c) \u2218 (\u03b1 S c)) (f x) s) = (\u03bb x, seval_\u2133_c S c s (f x)) :=\n begin\n  ext z,\n  have h_commute := @seval_\u2133_\u03b1_commute p S _ _ c s,\n  have := congr_fun h_commute (f z),\n  simp only at this,\n  rw this,\n end\n\n@[nolint unused_arguments]\ndef seval_\u2112_bdd_c (c : \u211d\u22650) (S : Fintype) (A : finset \u2124) (s : S) :\nlaurent_measures_bdd r S A c \u2192 laurent_measures_bdd r \u03d6 A c :=\nbegin\n  intro F,\n  use \u03bb _, F s,\n  have hF := F.2,\n  simp only [filtration, set.mem_set_of_eq, seval_\u2112, nnnorm, laurent_measures.coe_mk,\n    fintype.univ_punit, finset.sum_singleton] at \u22a2 hF,\n  have := finset.sum_le_sum_of_subset (finset.singleton_subset_iff.mpr $ finset.mem_univ_val _),\n  rw finset.sum_singleton at this,\n  apply le_trans this hF,\nend\n\nlemma continuous_seval_\u2112_c (c : \u211d\u22650) (s : S) : continuous (seval_\u2112_c c s) :=\nbegin\n  rw laurent_measures.continuous_iff,\n  intro A,\n  let := seval_\u2112_bdd_c p c S A s,\n  have h_trunc : (@truncate r \u03d6 c A) \u2218 (seval_\u2112_c p c s) =\n    (seval_\u2112_bdd_c p c S A s) \u2218 (@truncate r S c A),\n  { ext \u27e8F, hF\u27e9 \u03c0 k,\n    dsimp only [seval_\u2112_bdd_c, seval_\u2112_c],\n    refl },\n  rw h_trunc,\n  apply continuous.comp,\n  apply continuous_of_discrete_topology,\n  apply truncate_continuous,\nend\n\nsection topological_generalities\n\nopen metric set\n\nvariables {X : Type*} [topological_space X]\n\nlemma reduction_balls {c : \u211d\u22650} (f : X \u2192 (closed_ball (0 : \u211d) c)) (H : \u2200 y : (closed_ball 0 c),\n  \u2200 \u03b5 : \u211d, is_open (f\u207b\u00b9' (ball y \u03b5))) : continuous f :=\nbegin\n  rw continuous_def,\n  intros _ hU,\n  rw is_open_iff_forall_mem_open,\n  intros x hx,\n  obtain \u27e8\u03b5, h\u2080, h\u03b5\u27e9 := (is_open_iff.mp hU) (f x) (mem_preimage.mp hx),\n  use f\u207b\u00b9' (ball (f x) \u03b5),\n  exact \u27e8preimage_mono h\u03b5, H (f x) \u03b5, mem_ball_self h\u2080\u27e9,\nend\n\nlemma mem_filtration_le_monomial (F : filtration (\u2112 \u03d6) c) (n : \u2115) :\n \u2225 ((F.1 punit.star n) : \u211d) \u2225 \u2264 c * ( r\u207b\u00b9 ^ n) :=\nbegin\n  have h_le : \u2211' n : \u2124, \u2225 ((F.1 punit.star n) : \u211d) \u2225 * r ^ n \u2264 c,\n  { have := (laurent_measures.mem_filtration_iff F.1 c).mp F.2,\n    rw laurent_measures.nnnorm_def at this,\n    simp only [fintype.univ_punit, finset.sum_singleton, \u2190 nnreal.coe_le_coe,\n    nnreal.coe_tsum, nnreal.coe_mul, nnreal.coe_zpow, laurent_measures.coe_nnnorm] at this,\n    exact this },\n  have := @sum_le_tsum \u211d _ _ _ _ (\u03bb n, \u2225 ((F.1 punit.star n) : \u211d) \u2225 * r ^ n) {n} _\n    (F.1.summable punit.star),\n  simp only [finset.sum_singleton, zpow_coe_nat] at this,\n  replace h_le := this.trans h_le,\n  rwa [\u2190 inv_mul_le_iff', inv_pow, inv_inv ((r : \u211d) ^ n), mul_comm],\n  { apply pow_pos, rw inv_pos, apply r_pos },\n  { rintros b -,\n    simp only [subtype.val_eq_coe],\n    exact mul_nonneg (norm_nonneg _) (zpow_nonneg (le_of_lt (nnreal.coe_pos.mpr r_pos)) b) },\nend\n\n\nlemma mem_filtration_sum_le_geom (F : filtration (\u2112 \u03d6) c) (B : \u2115) : \u2225 \u2211' n : {x : \u2115 // B \u2264 x},\n  ((F.1 punit.star n) : \u211d) * 2\u207b\u00b9 ^ n.1 \u2225 \u2264 \u2225 (c : \u211d) * \u2211' n : {x : \u2115 // B \u2264 x}, (2\u207b\u00b9 * r\u207b\u00b9) ^ n.1 \u2225 :=\nbegin\n  have two_r_nonneg : 0 \u2264 (2\u207b\u00b9 * r\u207b\u00b9 : \u211d) := by {refine mul_nonneg (inv_nonneg.2 two_pos.le) (inv_nonneg.2 r.2) },\n  have h_inj : function.injective (coe : {x : \u2115 // B \u2264 x} \u2192 \u2115) := subtype.coe_injective,\n  have geom_pos : (0 : \u211d) \u2264 c * \u2211' (n : {x // B \u2264 x}), (2\u207b\u00b9 * r\u207b\u00b9) ^ n.1,\n  { apply mul_nonneg c.2 (tsum_nonneg _),\n    intro b,\n    apply pow_nonneg (two_r_nonneg) },\n  nth_rewrite 1 [real.norm_eq_abs],\n  rw [abs_eq_self.mpr geom_pos],\n    apply (norm_tsum_le_tsum_norm _).trans,\n  rw [\u2190 tsum_mul_left],\n  apply tsum_le_tsum,\n  { intro b,\n    rw [norm_mul, mul_pow, mul_comm ((2\u207b\u00b9 : \u211d) ^ b.1) _, \u2190 mul_assoc],\n    rw [norm_pow, norm_inv, real.norm_two ],\n    refine (mul_le_mul_right _).mpr _,\n    { simp only [one_div, inv_pos, pow_pos, zero_lt_bit0, zero_lt_one] },\n    { apply mem_filtration_le_monomial p c F } },\n  swap,\n  { by_cases hc : (c : \u211d) \u2260 0,\n    { rw [\u2190 summable_mul_left_iff hc],\n      have two_r_lt : (2\u207b\u00b9 * r\u207b\u00b9 : \u211d) < 1,\n      { have := (div_lt_one (nnreal.coe_lt_coe.mpr (r_pos))).mpr half_lt_r,\n        simp only [\u2190 inv_eq_one_div] at this \u22a2,\n        rw [div_eq_mul_inv, nnreal.coe_inv] at this,\n        convert this,\n        assumption', },\n      exact (summable_geometric_of_lt_1 two_r_nonneg two_r_lt).comp_injective h_inj,\n      },\n    { rw not_ne_iff at hc,\n      simp_rw [hc, zero_mul],\n      exact summable_zero }, },\n  all_goals { simp_rw [norm_mul, norm_pow, norm_inv, real.norm_two, subtype.val_eq_coe],\n    refine ((aux_thm69.summable_iff_on_nat_less F.1.d _).mp (aux_thm69.summable_smaller_radius_norm\n      F.1.d (half_lt_r) (F.1.summable punit.star)\n      (\u03bb n, lt_d_eq_zero F.1 punit.star n))).comp_injective h_inj,\n    intros n hn,\n    rw [lt_d_eq_zero F.1 punit.star n hn, norm_zero, zero_mul] },\nend\n\n\ndef geom_B_nat (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) : {B : \u2115 // \u2200 (F : filtration (\u2112 \u03d6) c), \u2225 tsum (\u03bb b :\n  {n : \u2115 // B \u2264 n }, ((F.1 punit.star b.1) : \u211d) * 2\u207b\u00b9 ^ b.1 ) \u2225 < \u03b5 ^ (p\u207b\u00b9 : \u211d)} :=\nbegin\n  let g := (\u03bb n : \u2115, (c : \u211d) * ((2\u207b\u00b9 * r\u207b\u00b9) ^ n)),\n  have := tendsto_tsum_compl_at_top_zero g,\n  rw tendsto_at_top at this,\n  have h_pos : 0 < \u03b5 ^ (p\u207b\u00b9 : \u211d) := real.rpow_pos_of_pos h\u03b5 _,\n  specialize this (\u03b5 ^ (p\u207b\u00b9 : \u211d)) h_pos,\n  let A := this.some,\n    let B\u2080 : option \u2115 \u2192 \u2115 := \u03bb a : (option \u2115), option.rec_on a (0 : \u2115) (\u03bb n, n),\n    set B := (B\u2080 A.max).succ with hB,\n    use B,\n    have h_incl : A \u2264 finset.range B,\n    rw finset.le_eq_subset,\n    { by_cases H : A.nonempty,\n      { intros a ha,\n        obtain \u27e8s, hs\u27e9 := finset.max_of_nonempty H,\n        replace hB : s.succ = B, { simp only [*, option.mem_def], refl, },\n        have h_mem := finset.mem_range_succ_iff.mpr (finset.le_max_of_mem ha hs),\n        rwa hB at h_mem },\n      { intros a ha,\n        rw [finset.not_nonempty_iff_eq_empty] at H,\n        finish }},\n    let hA := this.some_spec,\n    specialize hA (finset.range B) h_incl,\n    rw [real.dist_0_eq_abs, \u2190 real.norm_eq_abs] at hA,\n    intro F,\n    apply lt_of_le_of_lt (mem_filtration_sum_le_geom p c F B),\n    convert hA using 1,\n    apply congr_arg,\n    simp_rw [subtype.val_eq_coe, \u2190 tsum_mul_left],\n    have set_eq : {n : \u2115 | B \u2264 n} = {n : \u2115 | n \u2209 finset.range B} :=\n      by {simp only [finset.mem_range, not_lt]},\n    exact tsum_congr_subtype g set_eq,\nend\n\n\ndef eq_le_int_nat (B : \u2115) : {n : \u2124 // (B : \u2124) \u2264 n } \u2243 {n : \u2115 // B \u2264 n} :=\n{ to_fun :=\n  begin\n    intro b,\n    use (int.eq_coe_of_zero_le ((int.coe_nat_nonneg B).trans b.2)).some,\n    rw \u2190 int.coe_nat_le,\n    convert b.2,\n    exact (Exists.some_spec (int.eq_coe_of_zero_le ((int.coe_nat_nonneg B).trans b.2))).symm,\n  end,\n  inv_fun := \u03bb n, \u27e8n, by {simp only [coe_coe, int.coe_nat_le], from n.2}\u27e9,\n  left_inv :=\n  begin\n    rintro \u27e8_, h\u27e9,\n    simp only [coe_coe, subtype.coe_mk],\n    exact (Exists.some_spec (int.eq_coe_of_zero_le ((int.coe_nat_nonneg B).trans h))).symm,\n  end,\n  right_inv :=\n  begin\n    rintro \u27e8_, h\u27e9,\n    simp only [coe_coe, subtype.coe_mk, int.coe_nat_inj'],\n    exact ((@exists_eq' _ _).some_spec).symm,\n  end, }\n\n\ndef geom_B_int (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) : {B : \u2124 // \u2200 (F : filtration (\u2112 \u03d6) c), \u2225 tsum (\u03bb b :\n  {n : \u2124 // B \u2264 n }, ((F.1 punit.star b.1) : \u211d) * 2\u207b\u00b9 ^ b.1 ) \u2225 < \u03b5 ^ (p\u207b\u00b9 : \u211d)} :=\nbegin\n  let \u212c := geom_B_nat p c \u03b5 h\u03b5,\n  let B := \u212c.1,\n  let hB := \u212c.2,\n  use B,\n  intro F,\n  specialize hB F,\n  convert hB using 1,\n  apply congr_arg,\n  exact ((eq_le_int_nat B).symm.tsum_eq (\u03bb b : {n : \u2124 // \u2191B \u2264 n },\n  ((F.1 punit.star b.1) : \u211d) * 2\u207b\u00b9 ^ b.1 )).symm,\nend\n\n\ndef geom_B (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) : \u2124 := (geom_B_int c \u03b5 h\u03b5).1\n\n\nlemma tail_B (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) :  \u2200 (F : filtration (\u2112 \u03d6) c), \u2225 tsum (\u03bb b : {n : \u2124 // geom_B c \u03b5 h\u03b5 \u2264 n },\n  ((F.1 punit.star b.1) : \u211d) * 2\u207b\u00b9 ^ b.1 ) \u2225 < \u03b5 ^ (p\u207b\u00b9 : \u211d) :=\nbegin\n  intro F,\n  have := (geom_B_int p c \u03b5 h\u03b5).2 F,\n  exact this,\nend\n\ndef U (F : filtration (\u2112 \u03d6) c) (B : \u2124) : set (filtration (\u2112 \u03d6) c) :=\n  \u03bb G, \u2200 s n, n < B \u2192 F s n = G s n\n\nlemma mem_U (F : filtration (\u2112 \u03d6) c) (B : \u2124) : F \u2208 (U c F B) := \u03bb _ _ _, rfl\n\nlemma explodes_pow_r (\u03c1 : \u211d\u22650) (h\u2080 : 0 < \u03c1.1) (h\u2081 : \u03c1.1 < 1) (c : \u211d\u22650) :\n  \u2203 n\u2080 : \u2124, \u2200 (m : \u2124), m < n\u2080 \u2192 c < \u03c1 ^ m :=\nbegin\n  convert_to \u2203 n\u2080 : \u2115, \u2200 (m : \u2115), (- m : \u2124) < - n\u2080 \u2192 (c : \u211d) < \u03c1 ^ ( - m : \u2124) using 0,\n  { simp only [neg_lt_neg_iff, int.coe_nat_lt, zpow_neg, zpow_coe_nat, eq_iff_iff],\n    split,\n    { rintro \u27e8n\u2080, hn\u2080\u27e9,\n      induction n\u2080,\n      { use n\u2080,\n        intros m hm,\n        rw [\u2190 int.coe_nat_lt] at hm,\n        replace hm := neg_lt.mpr (lt_of_le_of_lt (neg_le_self (int.coe_nat_nonneg _)) hm),\n        specialize hn\u2080 (- m) hm,\n        rwa [\u2190 nnreal.coe_lt_coe, nnreal.coe_zpow, zpow_neg] at hn\u2080 },\n      { use n\u2080 + 1,\n        intros m hm,\n        rw [\u2190 int.coe_nat_lt, \u2190 neg_lt_neg_iff, \u2190 int.neg_succ_of_nat_coe] at hm,\n        specialize hn\u2080 (- m) hm,\n        rwa [\u2190 nnreal.coe_lt_coe, nnreal.coe_zpow, zpow_neg] at hn\u2080 },},\n    { rintro \u27e8n\u2080, hn\u2080\u27e9,\n      use - n\u2080,\n      rintro \u27e8m\u27e9 hm,\n      {  have := right.neg_nonpos_iff.mpr (int.of_nat_nonneg n\u2080),\n        rw [int.of_nat_eq_coe] at this,\n        replace := ne_of_lt (lt_of_lt_of_le (lt_of_le_of_lt (int.of_nat_nonneg m) hm) this),\n        finish },\n      { rw [int.neg_succ_of_nat_coe, neg_lt_neg_iff, int.coe_nat_lt] at hm,\n        specialize hn\u2080 (m + 1) hm,\n        rwa [int.neg_succ_of_nat_coe, \u2190 nnreal.coe_lt_coe, zpow_neg, zpow_coe_nat, nnreal.coe_inv,\n          nnreal.coe_pow] }}},\n  have h := (tendsto_pow_at_top_nhds_within_0_of_lt_1 h\u2080 h\u2081).inv_tendsto_zero,\n  simp_rw [\u2190 zpow_coe_nat] at h,\n  have : (\u03bb (n : \u2115), \u03c1.1 ^ (n : \u2124))\u207b\u00b9 = (\u03bb (n : \u2115), \u03c1.1 ^ (- n : \u2124)) := by {ext,\n  simp only [pi.inv_apply, zpow_neg] },\n  rw [this, filter.tendsto_at_top] at h,\n  specialize h (c + 1),\n  rw [nnreal.val_eq_coe] at h,\n  simp_rw [\u2190 nnreal.coe_zpow] at h,\n  obtain \u27e8n\u2080, hn\u2080\u27e9 := filter.eventually.exists_forall_of_at_top h,\n  use n\u2080,\n  intros m hm,\n  rw [neg_lt_neg_iff, int.coe_nat_lt] at hm,\n  replace hm := le_of_lt hm,\n  specialize hn\u2080 m hm,\n  rw nnreal.coe_zpow at hn\u2080,\n  refine lt_of_lt_of_le _ hn\u2080,\n  exact lt_add_one _,\nend\n\nlemma is_open_U (F : filtration (\u2112 \u03d6) c) (B : \u2124) : is_open (U c F B) :=\nbegin\n  let \u03b9 : filtration (\u2112 \u03d6) c \u2192 \u03a0 (i : \u2124), \u2124 :=\n    \u03bb t i, truncate {i} t punit.star \u27e8i,by simp\u27e9,\n  have h\u03b9 : continuous \u03b9,\n  { rw continuous_pi_iff, intros i,\n    dsimp [\u03b9],\n    change continuous ((\u03bb (t : laurent_measures_bdd r \u03d6 {i} c),\n      t punit.star \u27e8i,by simp\u27e9) \u2218 truncate {i}),\n    refine continuous.comp continuous_bot (truncate_continuous \u00abr\u00bb (Fintype.of punit) c {i}) },\n  obtain \u27e8n\u2080,h\u2080\u27e9 : \u2203 n\u2080 : \u2124, \u2200 (m : \u2124) (H : \u2112 \u03d6) (hH : H \u2208 filtration (\u2112 \u03d6) c),\n    m < n\u2080 \u2192 H punit.star m = 0,\n  { obtain \u27e8n\u2080,h\u2080\u27e9 : \u2203 n\u2080 : \u2124, \u2200 (m : \u2124), m < n\u2080 \u2192 c < r^m := explodes_pow_r r r_pos r_lt_one c,\n    use n\u2080, intros m H hH hm,\n    exact eq_zero_of_filtration H _ hH punit.star m (h\u2080 m hm) },\n  classical,\n  let UU : set (\u03a0 (i : \u2124), \u2124) :=\n    set.pi (set.Ico n\u2080 B) (\u03bb i, if i \u2208 set.Ico n\u2080 B then { F punit.star i } else \u22a4),\n  have hUU : is_open UU,\n  { apply is_open_set_pi, exact finite_Ico n\u2080 B,\n    intros a ha, trivial },\n  convert hUU.preimage h\u03b9,\n  ext G,\n  split,\n  { intros hG, dsimp [U, UU, \u03b9] at \u22a2 hG,\n    intros i hi, rw if_pos hi,\n    simp only [mem_singleton_iff],\n    symmetry,\n    apply hG, exact hi.2 },\n  { intros hG, dsimp [U, UU, \u03b9] at \u22a2 hG,\n    rintros \u27e8\u27e9 n hn,\n    symmetry,\n    by_cases hn' : n < n\u2080,\n    { erw [h\u2080 n G.1 G.2 hn', h\u2080 n F.1 F.2 hn'] },\n    push_neg at hn',\n    specialize hG n, simpa [hn', hn] using hG },\nend\n\nend topological_generalities\n\n\ndef \u03b8_c (c : \u211d\u22650) (T : Fintype) : (filtration (laurent_measures r T) c) \u2192\n  (filtration (real_measures p T) c) :=\nbegin\n  intro f,\n  rw [\u2190 one_mul c],\n  use \u27e8\u03b8 f, \u03b8_bound p c f f.2\u27e9,\nend\n\nlemma commute_seval_\u2112_\u2133 (c : \u211d\u22650) (s : S) :\n  (\u03b8_c c (Fintype.of punit)) \u2218 (seval_\u2112_c c s) = (seval_\u2133_c S c s) \u2218 (\u03b8_c c S) := by simpa only\n  [seval_\u2133_c, seval_\u2112_c, seval_\u2112, \u03b8_c, one_mul, subtype.coe_mk, eq_mpr_eq_cast, set_coe_cast]\n\n\nlemma continuous_of_seval_\u2133_comp_continuous (c : \u211d\u22650) {X : Type*} [topological_space X]\n  {f : X \u2192 (filtration (\u2133 S) c)} : (\u2200 s, continuous ((seval_\u2133_c S c s) \u2218 f)) \u2192 continuous f :=\nbegin\n  intro H,\n  replace H : \u2200 (s : S), continuous (\u03bb x : X, ((homeo_box_\u03d6 p S c) \u2218 (\u03b1 p S c)) (f x) s),\n  { intro,\n    rw [seval_\u2133_\u03b1_commute' p S c s],\n    exact H s },\n  rw \u2190 continuous_pi_iff at H,\n  convert_to (continuous (\u03bb x, (homeo_box_\u03d6 p S c) (\u03b1 p S c (f x)))) using 0,\n  { apply eq_iff_iff.mpr,\n    rw [homeomorph.comp_continuous_iff, (inducing_\u03b1 p S c).continuous_iff] },\n  exact H,\nend\n\n\nlemma tsum_subtype_sub {f g : \u2124 \u2192 \u211d} {B : \u2124}\n  (hf : summable (\u03bb (b : {x // B \u2264 x}), f b * 2\u207b\u00b9 ^ b.1))\n  (hg : summable (\u03bb (b : {x // B \u2264 x}), g b * 2\u207b\u00b9 ^ b.1)) :\n  \u2225 tsum ((\u03bb (b : \u2124), (((g b) : \u211d) - f b) * 2\u207b\u00b9 ^ b) \u2218 (coe : {b | B \u2264 b} \u2192 \u2124)) \u2225 =\n  \u2225 \u2211' (b : {x // B \u2264 x}), (g b : \u211d) * 2\u207b\u00b9 ^ b.1 - \u2211' (b : {x // B \u2264 x}),\n    (f b : \u211d) * 2\u207b\u00b9 ^ b.1 \u2225 :=\nbegin\n  rw [\u2190 tsum_sub hg hf, tsum_eq_tsum_of_has_sum_iff_has_sum],\n  intro _,\n  simp_rw [sub_mul, iff_eq_eq],\n  refl,\nend\n\nlemma aux_summability_no_norm (F : filtration (\u2112 \u03d6) c) : summable\n  (\u03bb b : \u2124, (((F punit.star b) : \u211d) * 2\u207b\u00b9 ^ b)) := aux_thm69.summable_smaller_radius F.1.d (F.1.summable punit.star)\n      (\u03bb n, lt_d_eq_zero F.1 punit.star n) half_lt_r\n\nlemma aux_summability_subtype (F : filtration (\u2112 \u03d6) c) (B : \u2124) : summable (\u03bb b : {x : \u2124 // B \u2264 x},\n  (((F punit.star b) : \u211d) * 2\u207b\u00b9 ^ b.1)) :=\n    by {exact (aux_summability_no_norm p c F).comp_injective subtype.coe_injective}\n\n\n\nlemma pos_\u03b5_pow (\u03b5 : \u211d) (h\u03b5 : 0 < \u03b5) : 0 < (\u03b5 / (2 : \u211d) ^ p.1) := by {apply div_pos h\u03b5\n  (real.rpow_pos_of_pos _ _), simp only [zero_lt_bit0, zero_lt_one]}\n\nlemma dist_lt_of_mem_U (\u03b5 : \u211d\u22650) (h\u03b5 : 0 < \u03b5) (F G : filtration (\u2112 \u03d6) c) :\n  G \u2208 (U c F (geom_B c (\u03b5 / (2 : \u211d) ^ p.1) (pos_\u03b5_pow \u03b5 h\u03b5))) \u2192 \u2225 ((\u03b8_c c \u03d6 G) : (\u2133 \u03d6)) - (\u03b8_c c \u03d6) F \u2225 < \u03b5 :=\nbegin\n  intro h_mem_G,\n  rw real_measures.norm_def,\n  simp only [fintype.univ_punit, real_measures.sub_apply, finset.sum_singleton],\n  rw [\u2190 real.rpow_lt_rpow_iff _ _ _, \u2190 real.rpow_mul,\n    mul_inv_cancel, real.rpow_one],\n  rotate,\n  { rw \u2190 nnreal.coe_zero,\n    exact ne_of_gt (nnreal.coe_lt_coe.mpr (fact.out _)) },\n  { apply norm_nonneg },\n  { apply real.rpow_nonneg_of_nonneg (norm_nonneg _) },\n  { rw \u2190 nnreal.coe_zero,\n    exact \u03b5.2 },\n  { rw [inv_pos, \u2190 nnreal.coe_zero],\n    exact (nnreal.coe_lt_coe.mpr (fact.out _)) },\n  simp only [\u03b8_c, one_mul, eq_mpr_eq_cast, set_coe_cast, subtype.coe_mk],\n  dsimp only [\u03b8, \u03d1],\n  have h_B : \u2200 b : \u2124, b < (geom_B p c (\u03b5 / 2 ^ p.1) (pos_\u03b5_pow p \u03b5 h\u03b5)) \u2192 ((G punit.star b) : \u211d) - (F punit.star b) = 0,\n  { intros b hb,\n    simp only [h_mem_G punit.star b hb, sub_self] },\n  rw [\u2190 tsum_sub],\n  rotate,\n  {exact (aux_summability_no_norm p c G)},\n  {exact (aux_summability_no_norm p c F)},\n  simp_rw [\u2190 sub_mul],\n  set B := (geom_B p c (\u03b5 / 2 ^ p.1) (pos_\u03b5_pow p \u03b5 h\u03b5)) with def_B,\n  let f := \u03bb b : \u2124, ((((G : (\u2112 \u03d6)) punit.star b) - ((F : (\u2112 \u03d6)) punit.star b)) : \u211d)\n    * 2\u207b\u00b9 ^ b,\n  let g : ({ b : \u2124 | B \u2264 b}) \u2192 \u211d := f \u2218 coe,\n  let i : function.support g \u2192 \u2124 := (coe : { b : \u2124 | B \u2264 b} \u2192 \u2124) \u2218 (coe : function.support g \u2192 { b : \u2124 | B \u2264 b}),\n  have hi : \u2200 \u2983x y : \u21a5(function.support g)\u2984, i x = i y \u2192 \u2191x = \u2191y,\n  {intros _ _ h,\n    simp only [subtype.coe_inj] at h,\n    rwa [subtype.coe_inj] },\n  have hf : function.support f \u2286 set.range i,\n  { intros a ha,\n    simp only [f, function.mem_support, ne.def] at ha,\n    have ha' : B \u2264 a,\n    { by_contra',\n      specialize h_B a this,\n      simp only [one_div, inv_zpow', zpow_neg, mul_eq_zero, inv_eq_zero, not_or_distrib] at ha,\n      replace ha := ha.1,\n      simpa only },\n    simp only [set.mem_set_of_eq, function.mem_support, ne.def, set.mem_range, set_coe.exists],\n    use [a, ha', ha, refl _] },\n  have hF := tail_B p c (\u03b5.1 / 2 ^ p.1) (pos_\u03b5_pow p \u03b5 h\u03b5) F,\n  have hG := tail_B p c (\u03b5.1 / 2 ^ p.1) (pos_\u03b5_pow p \u03b5 h\u03b5) G,\n  have h02 : (0 : \u211d) \u2264 2 := two_pos.le,\n  have h02p : (0 : \u211d) \u2264 2 ^ p.val := real.rpow_nonneg_of_nonneg h02 _,\n  rw [real.div_rpow \u03b5.2 h02p, \u2190 real.rpow_mul h02] at hF hG,\n  simp_rw [@subtype.val_eq_coe _ _ p] at hF hG,\n  have hp0 : (p : \u211d) \u2260 0 := nnreal.coe_ne_zero.mpr (ne_of_gt (fact.out _)),\n  rw [mul_inv_cancel hp0, real.rpow_one] at hF hG,\n  rw [tsum_eq_tsum_of_ne_zero_bij i hi hf (\u03bb _, refl _)],\n  dsimp only [f, g],\n  rw [tsum_subtype_sub],\n  rotate,\n  { exact (aux_summability_subtype p c F B) },\n  { exact (aux_summability_subtype p c G B) },\n  apply lt_of_le_of_lt (norm_sub_le _ _),\n  convert add_lt_add hG hF,\n  simp only [nnreal.val_eq_coe, add_halves'],\nend\n\nlemma coe_pow_half (\u03b7 : \u211d) (\u03b7_pos' : 0 < \u03b7) (\u03b7\u2080 : \u211d\u22650) (h\u03b7\u2080 : \u03b7\u2080 = \u27e8\u03b7, le_of_lt \u03b7_pos'\u27e9) :\n  (\u03b7 / 2) ^ (p : \u211d) = ((\u03b7\u2080 ^ (p : \u211d) : \u211d)) / 2 ^ (p.1) := by {rw [real.div_rpow (le_of_lt \u03b7_pos') _,\n     nnreal.val_eq_coe, h\u03b7\u2080, subtype.coe_mk], simp only [zero_le_bit0, zero_le_one]}\n\nsection\nvariables {c}\n\ndef \u03be (F : filtration (\u2112 \u03d6) c) : \u211d :=\n(homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)\n\ndef h\u03be (F : filtration (\u2112 \u03d6) c) :\n  \u03be F = (homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F) := rfl\n\nlemma speed_aux' (\u03b5 \u03b7 : \u211d) (\u03b7\u2080 : \u211d\u22650) (h\u03b7\u2080 : \u03b7 = \u03b7\u2080)\n  (y : (closed_ball (0 : \u211d) (c ^ (p : \u211d)\u207b\u00b9)))\n  (F G : (filtration (\u2112 (Fintype.of punit)) c))\n  (hF : |(((homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)) : \u211d) - y| < \u03b5)\n  (h\u03b7 : \u03b7 = \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c c \u03d6 F)) - y|) (h_pos : 0 < (\u03b7 / 2) ^ (p : \u211d))\n  (h_pos : 0 < (\u03b7 / 2) ^ (p:\u211d))\n  (hp : 0 < (p:\u211d))\n  (\u03b7_pos' : 0 < \u03b7)\n  (h_\u03b7_\u03b7\u2080 : (\u03b7 / 2) ^ (p:\u211d) = \u2191\u03b7\u2080 ^ (p:\u211d) / 2 ^ p.val) (h_pos'')\n  (hG : G \u2208 U c F (geom_B c ((\u2191\u03b7\u2080 ^ (p:\u211d) / 2 ^ p.val)) h_pos'')) :\n  \u2225(\u03b8_c c (Fintype.of punit) G).val - (\u03b8_c c (Fintype.of punit) F).val\u2225 < \u03b7 ^ (p:\u211d) :=\nbegin\n  have foo : 0 < \u03b7\u2080 ^ (p:\u211d),\n  { apply real.rpow_pos_of_pos, rw \u2190 h\u03b7\u2080, exact \u03b7_pos' },\n  -- exact dist_lt_of_mem_U p c (\u03b7\u2080 ^ (p : \u211d)) (real.rpow_pos_of_pos \u03b7_pos' _) F G hG\n  have := dist_lt_of_mem_U p c (\u03b7\u2080 ^ (p : \u211d)) foo F G hG,\n  convert this,\nend\n\nlemma speed_aux (\u03b5 \u03b7 : \u211d) (\u03b7\u2080 : \u211d\u22650) (h\u03b7\u2080 : \u03b7 = \u03b7\u2080)\n  (y : (closed_ball (0 : \u211d) (c ^ (p : \u211d)\u207b\u00b9)))\n  (F G : (filtration (\u2112 (Fintype.of punit)) c))\n  (hF : |(((homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)) : \u211d) - y| < \u03b5)\n  (h\u03b7 : \u03b7 = \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c c \u03d6 F)) - y|) (h_pos : 0 < (\u03b7 / 2) ^ (p : \u211d))\n  (h_pos : 0 < (\u03b7 / 2) ^ (p:\u211d))\n  (hp : 0 < (p:\u211d))\n  (\u03b7_pos' : 0 < \u03b7)\n  (h_\u03b7_\u03b7\u2080 : (\u03b7 / 2) ^ (p:\u211d) = \u2191\u03b7\u2080 ^ (p:\u211d) / 2 ^ p.val) (h_pos'')\n  (hG : G \u2208 U c F (geom_B c ((\u2191\u03b7\u2080 ^ (p:\u211d) / 2 ^ p.val)) h_pos'')) :\n  |\u03be G - \u03be F| < \u03b5 - |\u03be F - y| :=\nbegin\n  repeat {rw h\u03be},\n  rw [\u2190 real_measures.dist_eq,\n    \u2190 real.rpow_lt_rpow_iff\n      (real.rpow_nonneg_of_nonneg (real_measures.norm_nonneg _) _) (sub_nonneg.mpr (le_of_lt hF)) hp,\n    \u2190 real.rpow_mul (real_measures.norm_nonneg _),\n    inv_mul_cancel (ne_of_gt hp), real.rpow_one, \u2190 h\u03b7],\n  apply speed_aux', assumption'\nend\n\nlemma speed (\u03b5 \u03b7 : \u211d) (y : closed_ball (0 : \u211d) (c ^ (p\u207b\u00b9 : \u211d)))\n  (F G : filtration (\u2112 \u03d6) c)\n  (hF : |(((homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)) : \u211d) - y| < \u03b5)\n  (h\u03b7 : \u03b7 = \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c c \u03d6 F)) - y|) (h_pos : 0 < (\u03b7 / 2) ^ (p : \u211d))\n  (hG : G \u2208 U c F (geom_B c ((\u03b7 / 2) ^ (p:\u211d)) h_pos)) :\n  |(\u03be G) - (\u03be F)| + |(\u03be F) - y| < \u03b5 - |(\u03be F) - y| + |(\u03be F) - y| :=\nbegin\n  have hp : 0 < (p : \u211d), { rw [\u2190 nnreal.coe_zero, nnreal.coe_lt_coe], from fact.out _ },\n  have \u03b7_pos' : 0 < \u03b7 := by {rw h\u03b7, from (sub_pos.mpr hF)},\n  set \u03b7\u2080 : \u211d\u22650 := \u27e8\u03b7, le_of_lt \u03b7_pos'\u27e9 with h\u03b7\u2080,\n  have h_\u03b7_\u03b7\u2080 := @coe_pow_half p _ _ \u03b7 \u03b7_pos' \u03b7\u2080 h\u03b7\u2080,\n  simp_rw [h_\u03b7_\u03b7\u2080] at hG,\n  apply add_lt_add_right,\n  apply @speed_aux p _ _ c \u03b5 \u03b7 \u03b7\u2080 _ y F G,\n  assumption',\n  rw h\u03b7\u2080, refl,\nend\n\nend\n\nlemma U_subset_preimage' (\u03b5 \u03b7 : \u211d) (y : closed_ball (0 : \u211d) (c ^ (p\u207b\u00b9 : \u211d)))\n  (F : filtration (\u2112 \u03d6) c)\n  (hF : |(((homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)) : \u211d) - y| < \u03b5)\n  (h\u03b7 : \u03b7 = \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c c \u03d6 F)) - y|) (h_pos : 0 < (\u03b7 / 2) ^ (p : \u211d))\n  (G : (filtration (\u2112 (Fintype.of punit)) c))\n  (hG : G \u2208 U c F (geom_B c ((\u03b7 / 2) ^ (p:\u211d)) h_pos)) :\n  |\u03be G - y| < \u03b5 :=\nbegin\n  have aux : |(\u03be p G) - (\u03be p F)| + |(\u03be p F) - y | < \u03b5 - | (\u03be p F) - y | + | (\u03be p F) - y |,\n  { apply speed; assumption },\n  replace aux : |(\u03be p G) - (\u03be p F)| + |(\u03be p F) - y | < \u03b5,\n  { rwa [sub_add_cancel \u03b5 (| (\u03be p F) - y |)] at aux },\n  have := lt_of_le_of_lt (abs_sub_le (\u03be p G) (\u03be p F) y) aux,\n  rw \u2190 real.norm_eq_abs at this \u22a2,\n  exact this,\nend\n\nlemma U_subset_preimage (\u03b5 \u03b7 : \u211d) (y : closed_ball (0 : \u211d) (c ^ (p\u207b\u00b9 : \u211d)))\n  (F : filtration (\u2112 \u03d6) c)\n  (hF : |(((homeo_filtration_\u03d6_ball c) (\u03b8_c c (Fintype.of punit) F)) : \u211d) - y| < \u03b5)\n  (h\u03b7 : \u03b7 = \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c c \u03d6 F)) - y|) (h_pos : 0 < (\u03b7 / 2) ^ (p : \u211d)) :\n  (U c F (geom_B c ((\u03b7 / 2) ^ (p : \u211d)) h_pos) )  \u2286\n    ((homeo_filtration_\u03d6_ball c) \u2218 \u03b8_c c (\u03d6) \u207b\u00b9' (ball y \u03b5)) :=\nbegin\n  intros G hG,\n  simp only [set.mem_preimage, one_mul, eq_self_iff_true, eq_mpr_eq_cast, set_coe_cast,\n    function.comp_app, mem_ball, subtype.dist_eq, real.dist_eq],\n  apply U_subset_preimage'; assumption,\nend\n\n\n-- This is the main continuity property needed in `ses2.lean`\ntheorem continuous_\u03b8_c (c : \u211d\u22650) : continuous (\u03b8_c c S) :=\nbegin\n  apply continuous_of_seval_\u2133_comp_continuous,\n  intro s,\n  rw \u2190 commute_seval_\u2112_\u2133,\n  refine continuous.comp _ (continuous_seval_\u2112_c p S c s),\n  apply (homeo_filtration_\u03d6_ball c).comp_continuous_iff.mp,\n  apply reduction_balls,\n  intros y \u03b5,\n  rw is_open_iff_forall_mem_open,\n  intros F hF,\n  simp only [set.mem_preimage, one_mul, eq_self_iff_true, eq_mpr_eq_cast, set_coe_cast,\n    function.comp_app, mem_ball, subtype.dist_eq, real.dist_eq] at hF,\n  set \u03b7 := \u03b5 - |(homeo_filtration_\u03d6_ball c (\u03b8_c p c \u03d6 F)) - y| with h\u03b7,\n  have \u03b7_pos' : 0 < \u03b7 := by {rw h\u03b7, from (sub_pos.mpr hF)},\n  have \u03b7_pos : 0 < (\u03b7 / 2) ^ (p : \u211d) := real.rpow_pos_of_pos (half_pos \u03b7_pos') _,\n  set V := U p c F (geom_B p c ((\u03b7 / 2) ^ (p : \u211d)) \u03b7_pos) with hV,\n  simp_rw [real.div_rpow (le_of_lt \u03b7_pos') (le_of_lt (@two_pos \u211d _ _))] at hV,\n  use V,\n  exact and.intro (U_subset_preimage p c \u03b5 \u03b7 y F hF h\u03b7 \u03b7_pos)\n    (and.intro (is_open_U p c F _) (mem_U p c F _)),\nend\n\n\nend theta\n\nend laurent_measures_ses\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/laurent_measures/ses.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3804062940877639}}
{"text": "import tactic\n\nvariables {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]\n\n-- I couldn't find this in mathlib\n\n-- lemma alg_hom.eq_of_id (f : R \u2192\u2090[R] A) : f = algebra.of_id R A :=\n-- begin\n--   have this_will_be_useful := f.commutes,\n--   ext r,\n--   specialize this_will_be_useful r,\n--   unfold algebra_map algebra.to_ring_hom at this_will_be_useful,\n--   exact this_will_be_useful,\n-- end\n\nlemma alg_hom.eq_of_id (f : R \u2192\u2090[R] A) : f = algebra.of_id R A :=\nalg_hom.ext (f.commutes)\n\ninstance alg_hom.subsingleton : subsingleton (R \u2192\u2090[R] A) :=\nsubsingleton.intro $ \u03bb f g, by { rw [alg_hom.eq_of_id f, alg_hom.eq_of_id g], }", "meta": {"author": "kbuzzard", "repo": "etale_experiments", "sha": "7d1f2f6f772f03ad553971103696ebc0654068f4", "save_path": "github-repos/lean/kbuzzard-etale_experiments", "path": "github-repos/lean/kbuzzard-etale_experiments/etale_experiments-7d1f2f6f772f03ad553971103696ebc0654068f4/src/for_mathlib.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3804062874167391}}
{"text": "import GroundZero.HITs.Interval\nimport GroundZero.Theorems.UA\nimport GroundZero.HITs.Merely\n\nopen GroundZero.HITs.Interval\nopen GroundZero.Proto (idfun)\nopen GroundZero.Types.Equiv\nopen GroundZero.Structures\nopen GroundZero.Types\n\nnamespace GroundZero\nnamespace Theorems.Equiv\n\nuniverse u v w\n\nhott def uniqDoesNotAddNewPaths {A : Type u} (a b : \u2225A\u2225)\n  (p : a = b) : HITs.Merely.uniq a b = p :=\npropIsSet HITs.Merely.uniq _ _ _ _\n\nhott def propEquiv {A : Type u} (H : prop A) : A \u2243 \u2225A\u2225 :=\npropEquivLemma H HITs.Merely.uniq HITs.Merely.elem (HITs.Merely.rec H id)\n\nhott def propFromEquiv {A : Type u} : A \u2243 \u2225A\u2225 \u2192 prop A :=\nbegin\n  intro \u27e8f, (\u27e8g, A\u27e9, _)\u27e9 a b; transitivity;\n  exact (A a)\u207b\u00b9; symmetry; transitivity; exact (A b)\u207b\u00b9;\n  apply Id.map g; exact HITs.Merely.uniq (f b) (f a)\nend\n\nhott def mapToHapply {A : Type u} {B : Type v}\n  (c : A) (f g : A \u2192 B) (p : f = g) :\n  Id.map (\u03bb (f : A \u2192 B), f c) p = happly p c :=\nbegin induction p; reflexivity end\n\nhott def hmtpyRewrite {A : Type u} (f : A \u2192 A) (H : f ~ id) (x : A) : H (f x) = Id.map f (H x) :=\nbegin have p := (Theorems.funext H)\u207b\u00b9; induction p; apply Id.symm; apply Equiv.idmap end\n\nhott def qinvImplsIshae {A : Type u} {B : Type v} (f : A \u2192 B) : Qinv f \u2192 Ishae f :=\nbegin\n  intro \u27e8g, \u27e8\u03b5, \u03b7\u27e9\u27e9; let \u03b5' := \u03bb b, (\u03b5 (f (g b)))\u207b\u00b9 \u2b1d (Id.map f (\u03b7 (g b)) \u2b1d \u03b5 b);\n  existsi g; existsi \u03b7; existsi \u03b5'; intro x; symmetry; transitivity;\n  { apply Id.map (\u03bb p, _ \u2b1d (Id.map f p \u2b1d _)); apply hmtpyRewrite (g \u2218 f) };\n  apply rewriteComp; transitivity; apply Id.map (\u00b7 \u2b1d _); symmetry; apply mapOverComp (g \u2218 f);\n  symmetry; apply @homotopySquare A B (f \u2218 g \u2218 f) f (\u03bb x, \u03b5 (f x)) (g (f x)) x (\u03b7 x)\nend\n\nhott def respectsEquivOverFst {A : Type u} {B : Type v}\n  (\u03c6 : A \u2243 B) (C : A \u2192 Type w) : (\u03a3 x, C x) \u2243 (\u03a3 x, C (\u03c6.left x)) :=\nbegin\n  fapply Sigma.replaceIshae; apply qinvImplsIshae; existsi \u03c6.1;\n  apply Prod.mk; apply \u03c6.leftForward; apply \u03c6.forwardLeft\nend\n\nhott def fibEq {A : Type u} {B : Type v} (f : A \u2192 B) {y : B} {a b : A}\n  (p : f a = y) (q : f b = y) : (\u03a3 (\u03b3 : a = b), Id.map f \u03b3 \u2b1d q = p) \u2192 @Id (fib f y) \u27e8a, p\u27e9 \u27e8b, q\u27e9 :=\nbegin\n  intro \u27e8\u03b3, r\u27e9; fapply Sigma.prod; exact \u03b3; transitivity; apply transportOverContrMap;\n  transitivity; apply Id.map (\u00b7 \u2b1d p); apply Id.mapInv; apply rewriteComp; exact r\u207b\u00b9\nend\n\nhott def ishaeImplContrFib {A : Type u} {B : Type v}\n  (f : A \u2192 B) : Ishae f \u2192 \u03a0 y, contr (fib f y) :=\nbegin\n  intro \u27e8g, \u03b7, \u03b5, \u03c4\u27e9 y; existsi \u27e8g y, \u03b5 y\u27e9; intro \u27e8x, p\u27e9; apply fibEq;\n  existsi (Id.map g p)\u207b\u00b9 \u2b1d \u03b7 x; transitivity;\n  apply Id.map (\u00b7 \u2b1d p); apply mapFunctoriality;\n  transitivity; apply Id.map (_ \u2b1d \u00b7 \u2b1d p); apply \u03c4;\n  transitivity; symmetry; apply Id.assoc; transitivity;\n  { apply Id.map (\u00b7 \u2b1d _); transitivity; apply Id.mapInv;\n    apply Id.map; symmetry; apply mapOverComp };\n  apply rewriteComp; transitivity; apply Id.map (_ \u2b1d \u00b7);\n  symmetry; apply idmap; apply homotopySquare\nend\n\n\nhott def compQinv\u2081 {A : Type u} {B : Type v} {C : Type w}\n  (f : A \u2192 B) (g : B \u2192 A) (H : isQinv f g) :\n  @Qinv (C \u2192 A) (C \u2192 B) (f \u2218 \u00b7) :=\nbegin\n  existsi (g \u2218 \u00b7); apply Prod.mk <;> intro <;>\n  apply Theorems.funext <;> intro; apply H.1; apply H.2\nend\n\nhott def compQinv\u2082 {A : Type u} {B : Type v} {C : Type w}\n  (f : A \u2192 B) (g : B \u2192 A) (H : isQinv f g) :\n  @Qinv (B \u2192 C) (A \u2192 C) (\u00b7 \u2218 f) :=\nbegin\n  existsi (\u00b7 \u2218 g) <;> apply Prod.mk <;> intro G <;>\n  apply Theorems.funext <;> intro <;>\n  apply Id.map G; apply H.2; apply H.1\nend\n\nhott def linvContr {A : Type u} {B : Type v}\n  (f : A \u2192 B) (H : Qinv f) : contr (linv f) :=\nbegin\n  apply contrRespectsEquiv;\n  { apply Equiv.symm; apply Sigma.respectsEquiv;\n    intro g; symmetry; apply @Theorems.full A (\u03bb _, A) (g \u2218 f) };\n  apply ishaeImplContrFib; apply qinvImplsIshae;\n  fapply compQinv\u2082; exact H.1; exact H.2\nend\n\nhott def rinvContr {A : Type u} {B : Type v}\n  (f : A \u2192 B) (H : Qinv f) : contr (rinv f) :=\nbegin\n  apply contrRespectsEquiv;\n  { apply Equiv.symm; apply Sigma.respectsEquiv;\n    intro g; symmetry; apply @Theorems.full B (\u03bb _, B) (f \u2218 g) };\n  apply ishaeImplContrFib; apply qinvImplsIshae;\n  fapply compQinv\u2081; exact H.1; exact H.2\nend\n\nhott def productContr {A : Type u} {B : Type v} (h : contr A) (g : contr B) : contr (A \u00d7 B) :=\nbegin existsi (h.1, g.1); intro p; fapply Product.prod; apply h.2; apply g.2 end\n\nhott def biinvProp {A : Type u} {B : Type v} (f : A \u2192 B) : prop (biinv f) :=\nbegin\n  apply lemContr; intro g; apply productContr;\n  { apply linvContr; apply Qinv.ofBiinv; assumption };\n  { apply rinvContr; apply Qinv.ofBiinv; assumption }\nend\n\nhott def equivHmtpyLem {A : Type u} {B : Type v}\n  (f g : A \u2243 B) (H : f.forward ~ g.forward) : f = g :=\nbegin fapply Sigma.prod; apply Theorems.funext; exact H; apply biinvProp end\n\nhott def propEquivProp {A B : Type u} (G : prop B) : prop (A \u2243 B) :=\nbegin intros f g; apply equivHmtpyLem; intro x; apply G end\n\nhott def propExercise (\u03c0 : Type u) : prop \u03c0 \u2243 (\u03c0 \u2243 \u2225\u03c0\u2225) :=\nbegin\n  existsi @propEquiv \u03c0; apply Prod.mk <;> existsi propFromEquiv;\n  { intro x; apply propIsProp };\n  { intro f; apply equivHmtpyLem;\n    intro x; apply HITs.Merely.uniq }\nend\n\nhott def lemContrInv {A : Type u} (h : prop A) (x : A) : contr A := \u27e8x, h x\u27e9\n\nhott def lemContrEquiv {A : Type u} : (prop A) \u2243 (A \u2192 contr A) :=\nbegin\n  apply propEquivLemma; apply propIsProp; apply functionToContr;\n  apply lemContrInv; apply lemContr\nend\n\nhott def contrToType {A : Type u} {B : A \u2192 Type v}\n  (H : contr A) : (\u03a3 x, B x) \u2192 B H.1 :=\n\u03bb w, subst (H.2 w.1)\u207b\u00b9 w.2\n\nhott def typeToContr {A : Type u} {B : A \u2192 Type v}\n  (H : contr A) : B H.1 \u2192 (\u03a3 x, B x) :=\n\u03bb u, \u27e8H.1, u\u27e9\n\n-- HoTT 3.20\nhott def contrFamily {A : Type u} {B : A \u2192 Type v} (H : contr A) : (\u03a3 x, B x) \u2243 B H.1 :=\nbegin\n  existsi contrToType H; apply Prod.mk <;>\n  existsi @typeToContr A B H <;> intro x;\n  { fapply Sigma.prod; apply H.2; apply transportBackAndForward };\n  { transitivity; apply Id.map (subst \u00b7 x);\n    apply propIsSet (contrImplProp H) _ _ _ (idp _);\n    reflexivity }\nend\n\nhott def propset.Id (A B : \u2126) (H : A.1 = B.1) : A = B :=\nSigma.prod H (propIsProp _ _)\n\nnoncomputable hott def propEqProp {A B : Type u} (G : prop B) : prop (A = B) :=\nbegin apply propRespectsEquiv; apply GroundZero.ua.univalence A B; apply propEquivProp G end\n\nnoncomputable hott def propsetIsSet : hset propset :=\nbegin\n  intro \u27e8x, H\u27e9 \u27e8y, G\u27e9; apply transport (\u03bb \u03c0, \u03a0 (p q : \u03c0), p = q);\n  symmetry; apply GroundZero.ua; apply Sigma.sigmaPath;\n  intro \u27e8p, p'\u27e9 \u27e8q, q'\u27e9; fapply Sigma.prod;\n  { apply propEqProp; exact G };\n  { apply propIsSet; apply propIsProp }\nend\n\nhott def hsetEquiv {A : Type u} {B : Type v} (g : hset B) : hset (A \u2243 B) :=\nbegin\n  fapply hsetRespectsSigma;\n  { apply piHset; intro x; assumption };\n  { intro x; apply propIsSet; apply biinvProp }\nend\n\nhott def bool.decode : \ud835\udfd0 \u2243 \ud835\udfd0 \u2192 \ud835\udfd0 :=\n\u03bb e, e false\n\nhott def bool.encode : \ud835\udfd0 \u2192 \ud835\udfd0 \u2243 \ud835\udfd0\n| false => ideqv \ud835\udfd0\n| true  => ua.negBoolEquiv\n\nhott def zeroEquiv.hset (A B : 0-Type) : hset (A \u2243\u2080 B) :=\nbegin apply hsetEquiv; apply zeroEqvSet.forward; exact B.2 end\n\nhott def boolEquivEqvBool : (\ud835\udfd0 \u2243 \ud835\udfd0) \u2243 \ud835\udfd0 :=\nbegin\n  existsi bool.decode; fapply Qinv.toBiinv; existsi bool.encode; apply Prod.mk;\n  { intro x; induction x using Bool.casesOn <;> reflexivity };\n  { intro \u27e8\u03c6, H\u27e9; apply equivHmtpyLem; intro x;\n    match boolEqTotal (\u03c6 false), boolEqTotal (\u03c6 true) with\n    | Sum.inl p\u2081, Sum.inl q\u2081 => _\n    | Sum.inr p\u2082, Sum.inl q\u2081 => _\n    | Sum.inl p\u2081, Sum.inr q\u2082 => _\n    | Sum.inr p\u2082, Sum.inr q\u2082 => _;\n    -- TODO: apply \u201cor\u201d here somehow\n    { apply Proto.Empty.elim; apply ffNeqTt;\n      apply eqvInj \u27e8\u03c6, H\u27e9; exact p\u2081 \u2b1d q\u2081\u207b\u00b9 };\n    { apply Id.trans; apply Id.map (bool.encode \u00b7 x); apply p\u2082;\n      symmetry; induction x using Bool.casesOn <;> assumption };\n    { apply Id.trans; apply Id.map (bool.encode \u00b7 x); apply p\u2081;\n      symmetry; induction x using Bool.casesOn <;> assumption };\n    { apply Proto.Empty.elim; apply ffNeqTt;\n      apply eqvInj \u27e8\u03c6, H\u27e9; exact p\u2082 \u2b1d q\u2082\u207b\u00b9 } }\nend\n\nhott def contrQinvFib {A : Type u} {B : Type v} (w : Qinv.eqv A B) (b : B) : contr (\u03a3 a, b = w.1 a) :=\nbegin apply contrRespectsEquiv; apply respectsEquivOverFst (Qinv.toEquiv (Qinv.inv w)) (Id b); apply singl.contr end\n\nhott def propQinvFib {A : Type u} {B : Type v} (w : Qinv.eqv A B) (b : B) : prop (\u03a3 a, b = w.1 a) :=\ncontrImplProp (contrQinvFib w b)\n\nhott def corrRev {A : Type u} {B : Type v} : Corr A B \u2192 Corr B A :=\n\u03bb w, \u27e8\u03bb a b, w.1 b a, (w.2.2, w.2.1)\u27e9\n\nhott def corrOfQinv {A : Type u} {B : Type v} : Qinv.eqv A B \u2192 Corr A B :=\nbegin\n  intro w; existsi (\u03bb a b, b = w.1 a); apply Prod.mk <;> intros;\n  apply contrRespectsEquiv; apply Sigma.hmtpyInvEqv; apply singl.contr;\n  apply contrQinvFib\nend\n\nhott def qinvOfCorr {A : Type u} {B : Type v} : Corr A B \u2192 Qinv.eqv A B :=\nbegin\n  intro w; fapply Sigma.mk; intro a; apply (w.2.1 a).1.1;\n  fapply Sigma.mk; intro b; apply (w.2.2 b).1.1; apply Prod.mk;\n  { intro b; apply Id.map Sigma.fst ((w.2.1 (w.2.2 b).1.1).2 \u27e8b, (w.2.2 b).1.2\u27e9) };\n  { intro a; apply Id.map Sigma.fst ((w.2.2 (w.2.1 a).1.1).2 \u27e8a, (w.2.1 a).1.2\u27e9) }\nend\n\nsection\n  variable {A : Type u} {B : Type v} (e : Qinv.eqv A B)\n\n  example : (qinvOfCorr (corrOfQinv e)).1 = e.1     := by reflexivity\n  example : (qinvOfCorr (corrOfQinv e)).2.1 = e.2.1 := by reflexivity\nend\n\nsection\n  variable {A : Type u} {B : Type v}\n\n  hott def corrOfBiinv : A \u2243 B \u2192 Corr A B :=\n  \u03bb e, @corrOfQinv A B \u27e8e.1, Qinv.ofBiinv e.1 e.2\u27e9\n\n  hott def biinvOfCorr : Corr A B \u2192 A \u2243 B :=\n  Qinv.toEquiv \u2218 qinvOfCorr\n\n  hott def corrLem (R : A \u2192 B \u2192 Type w) (\u03c6 : A \u2192 B) (\u03c1 : \u03a0 x, R x (\u03c6 x))\n    (H : \u03a0 x y, R x y \u2192 \u03c6 x = y) (c : \u03a0 (x : A) (y : B) (w : R x y), \u03c1 x =[H x y w] w)\n    (x : A) (y : B) : (\u03c6 x = y) \u2243 (R x y) :=\n  begin\n    fapply Sigma.mk; { intro p; apply transport (R x) p; apply \u03c1 }; fapply Qinv.toBiinv;\n    fapply Sigma.mk; intro r; exact (H x (\u03c6 x) (\u03c1 x))\u207b\u00b9 \u2b1d H x y r; apply Prod.mk;\n    { intro r; dsimp; transitivity; apply Id.map; symmetry; apply c x (\u03c6 x) (\u03c1 x);\n      transitivity; apply substComp; transitivity; apply Id.map (subst (H x y r));\n      apply transportForwardAndBack; apply c };\n    { intro p; induction p; apply Id.invComp }\n  end\n\n  noncomputable hott def corrBiinvIdfun : corrOfBiinv \u2218 @biinvOfCorr A B ~ idfun :=\n  begin\n    intro w; fapply Sigma.prod;\n    apply Theorems.funext; intro x; apply Theorems.funext; intro y;\n    change (y = (w.2.1 x).1.1) = (w.1 x y); apply ua; apply Equiv.trans;\n    apply inveqv; fapply corrLem w.1 (\u03bb x, (w.2.1 x).1.1) (\u03bb x, (w.2.1 x).1.2)\n      (\u03bb x y \u03c1, Id.map Sigma.fst ((w.2.1 x).2 \u27e8y, \u03c1\u27e9));\n    { intros x y \u03c1; change _ = _; transitivity; symmetry;\n      apply transportComp (w.1 x) Sigma.fst ((w.2.1 x).2 \u27e8y, \u03c1\u27e9);\n      apply apd Sigma.snd };\n    apply productProp <;> { apply piProp; intros; apply contrIsProp }\n  end\n\n  hott def biinvCorrIdfun : biinvOfCorr \u2218 @corrOfBiinv A B ~ idfun :=\n  begin intro e; fapply equivHmtpyLem; intro; reflexivity end\n\n  noncomputable hott def biinvEquivCorr : Corr A B \u2243 (A \u2243 B) :=\n  begin\n    existsi biinvOfCorr; fapply Qinv.toBiinv; existsi corrOfBiinv;\n    apply Prod.mk; apply biinvCorrIdfun; apply corrBiinvIdfun\n  end\nend\n\nend Theorems.Equiv\nend GroundZero", "meta": {"author": "forked-from-1kasper", "repo": "ground_zero", "sha": "58ad68bb54e355f6c39beaee2b383879eccc9952", "save_path": "github-repos/lean/forked-from-1kasper-ground_zero", "path": "github-repos/lean/forked-from-1kasper-ground_zero/ground_zero-58ad68bb54e355f6c39beaee2b383879eccc9952/GroundZero/Theorems/Equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3803409334837929}}
{"text": "-- See: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/OO.20polymorphism.3F/near/297629381\nnamespace O4\n\ninductive Geo_Type\n| Point2D\n| Point3D\n| Point4D\n| Square2D\nderiving DecidableEq\n\nopen Geo_Type\n\nstructure Slots_Point2D where\n  (x y : Float)\n  deriving Repr\n\nstructure Slots_Point3D extends Slots_Point2D where\n  z : Float\n  deriving Repr\n\nstructure Slots_Point4D extends Slots_Point3D where\n  w : Float\n  deriving Repr\n\nstructure Slots_Square2D extends Slots_Point2D where\n  width : Float\n  deriving Repr\n\nuniverse u\n\ndef Geo_Type.Slots : Geo_Type \u2192 Type\n| Point2D => Slots_Point2D\n| Point3D => Slots_Point3D\n| Point4D => Slots_Point4D\n| Square2D => Slots_Square2D\n\nopen Geo_Type\n\ndef toRepr : (T : Geo_Type) \u2192 Repr T.Slots\n| Point2D => instReprSlots_Point2D\n| Point3D => instReprSlots_Point3D\n| Point4D => instReprSlots_Point4D\n| Square2D => instReprSlots_Square2D\n\n/-- Get casting function to first type from second type -/\ndef Geo_Type.cast : (t : Geo_Type) \u2192 (t' : Geo_Type) \u2192 Option (t'.Slots \u2192 t.Slots)\n| Point2D, Point2D => some id\n| Point2D, Point3D => some (\u03bb x => x.toSlots_Point2D)\n| Point3D, Point3D => some id\n| Point2D, Point4D => some (\u03bb x => x.toSlots_Point2D)\n| Point3D, Point4D => some (\u03bb x => x.toSlots_Point3D)\n| Point4D, Point4D => some id\n| Point2D, Square2D => some (\u03bb x => x.toSlots_Point2D)\n| Square2D, Square2D => some id\n| _, _ => none\n\n/-- Whether the second type is a subtype of the first. -/\ndef Geo_Type.supertype (t t' : Geo_Type) : Bool := (t.cast t').isSome\n\ntheorem Geo_Type.supertype.trans (h : Geo_Type.supertype t t') (h' : Geo_Type.supertype t' t'') :\n  Geo_Type.supertype t t'' :=\nby\n  cases t\n  all_goals\n    cases t'\n    all_goals\n      cases t''\n      all_goals\n        simp at h\n        try simp at h'\n        try simp\n\ndef Option.get : (x : Option \u03b1) \u2192 x.isSome \u2192 \u03b1\n| some v, _ => v\n| none, h => by simp [Option.isSome] at h\n\n/-- Get the cast function from a proof that `t'` is a subtype of `t`. -/\ndef Geo_Type.supertype.cast (h : Geo_Type.supertype t t') : t'.Slots \u2192 t.Slots :=\n  Option.get (Geo_Type.cast t t') h\n\n/-- Objects of subtype t -/\nstructure Obj (t : Geo_Type) where\n  ty : Geo_Type\n  sub : Geo_Type.supertype t ty\n  slots : ty.Slots\n\ninstance : Coe Slots_Point2D (Obj Point2D) where coe s := \u27e8Point2D, rfl, s\u27e9\ninstance : Coe Slots_Point3D (Obj Point3D) where coe s := \u27e8Point3D, rfl, s\u27e9\ninstance : Coe Slots_Point4D (Obj Point4D) where coe s := \u27e8Point4D, rfl, s\u27e9\ninstance : Coe Slots_Square2D (Obj Square2D) where coe s := \u27e8Square2D, rfl, s\u27e9\n\n/-- Extract the slots from an `Obj`. -/\ndef Obj.get (o : Obj t) : t.Slots := Geo_Type.supertype.cast o.sub o.slots\n\n/-- Cast up, which can be done statically. -/\ndef Obj.cast_up (o : Obj t) (h : Geo_Type.supertype t' t := by rfl) : Obj t' where\n  ty := o.ty\n  sub := Geo_Type.supertype.trans h o.sub\n  slots := o.slots\n\ndef Obj.can_cast (o : Obj t) (t' : Geo_Type) : Bool := Geo_Type.supertype t' o.ty\n\n/-- Cast up or down using run-time information. -/\ndef Obj.cast (o : Obj t) (h : o.can_cast t') : Obj t' where\n  ty := o.ty\n  sub := h\n  slots := o.slots\n\n/-- Example using a run-time cast. -/\ndef Obj.get_w (o : Obj Point2D) : Float :=\n  if h : o.can_cast Point4D then\n    (o.cast h).get.w\n  else\n    0\n\n#eval ({x := 1, y := 2 : Slots_Point2D} : Obj Point2D).get_w\n-- 0.000000\n\n#eval Id.run do\n  let p := ({x := 1, y := 2, z := 3, w := 4 : Slots_Point4D} : Obj Point4D)\n  return p.cast_up.get_w\n-- 4.000000\n\n\ndef s2: Slots_Point2D := {x := 1, y := 2}\n#eval s2\n\ndef o2: Obj Point2D := s2\n#eval o2.slots\n\nend O4", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/O4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.38034093348379283}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n\n# Preorder homomorphisms\n\nBundled monotone functions, `x \u2264 y \u2192 f x \u2264 f y`.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.logic.function.iterate\nimport Mathlib.order.basic\nimport Mathlib.order.bounded_lattice\nimport Mathlib.order.complete_lattice\nimport Mathlib.tactic.monotonicity.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 l u_3 \n\nnamespace Mathlib\n\n/-! # Category of preorders -/\n\n/-- Bundled monotone (aka, increasing) function -/\nstructure preorder_hom (\u03b1 : Type u_1) (\u03b2 : Type u_2) [preorder \u03b1] [preorder \u03b2] where\n  to_fun : \u03b1 \u2192 \u03b2\n  monotone' : monotone to_fun\n\ninfixr:25 \" \u2192\u2098 \" => Mathlib.preorder_hom\n\nnamespace preorder_hom\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] :\n    has_coe_to_fun (\u03b1 \u2192\u2098 \u03b2) :=\n  has_coe_to_fun.mk (fun (f : \u03b1 \u2192\u2098 \u03b2) => \u03b1 \u2192 \u03b2) to_fun\n\ntheorem monotone {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) :\n    monotone \u21d1f :=\n  monotone' f\n\n@[simp] theorem coe_fun_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : monotone f) (x : \u03b1) : coe_fn (mk f hf) x = f x :=\n  rfl\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2)\n    (h : \u2200 (a : \u03b1), coe_fn f a = coe_fn g a) : f = g :=\n  sorry\n\ntheorem coe_inj {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2)\n    (h : \u21d1f = \u21d1g) : f = g :=\n  ext f g\n    fun (a : \u03b1) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn f a = coe_fn g a)) h)) (Eq.refl (coe_fn g a))\n\n/-- The identity function as bundled monotone function. -/\ndef id {\u03b1 : Type u_1} [preorder \u03b1] : \u03b1 \u2192\u2098 \u03b1 := mk id monotone_id\n\nprotected instance inhabited {\u03b1 : Type u_1} [preorder \u03b1] : Inhabited (\u03b1 \u2192\u2098 \u03b1) := { default := id }\n\n@[simp] theorem coe_id {\u03b1 : Type u_1} [preorder \u03b1] : \u21d1id = \u21d1id := rfl\n\n/-- The composition of two bundled monotone functions. -/\ndef comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [preorder \u03b1] [preorder \u03b2] [preorder \u03b3]\n    (g : \u03b2 \u2192\u2098 \u03b3) (f : \u03b1 \u2192\u2098 \u03b2) : \u03b1 \u2192\u2098 \u03b3 :=\n  mk (\u21d1g \u2218 \u21d1f) sorry\n\n@[simp] theorem comp_id {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) :\n    comp f id = f :=\n  ext (comp f id) f fun (a : \u03b1) => Eq.refl (coe_fn (comp f id) a)\n\n@[simp] theorem id_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192\u2098 \u03b2) :\n    comp id f = f :=\n  ext (comp id f) f fun (a : \u03b1) => Eq.refl (coe_fn (comp id f) a)\n\n/-- `subtype.val` as a bundled monotone function.  -/\ndef subtype.val {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) : Subtype p \u2192\u2098 \u03b1 := mk subtype.val sorry\n\n/-- The preorder structure of `\u03b1 \u2192\u2098 \u03b2` is pointwise inequality: `f \u2264 g \u2194 \u2200 a, f a \u2264 g a`. -/\nprotected instance preorder {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b1] [preorder \u03b2] :\n    preorder (\u03b1 \u2192\u2098 \u03b2) :=\n  preorder.lift to_fun\n\nprotected instance partial_order {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [partial_order \u03b2] :\n    partial_order (\u03b1 \u2192\u2098 \u03b2) :=\n  partial_order.lift to_fun sorry\n\nprotected instance has_sup {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_sup \u03b2] :\n    has_sup (\u03b1 \u2192\u2098 \u03b2) :=\n  has_sup.mk fun (f g : \u03b1 \u2192\u2098 \u03b2) => mk (fun (a : \u03b1) => coe_fn f a \u2294 coe_fn g a) sorry\n\nprotected instance semilattice_sup {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_sup \u03b2] :\n    semilattice_sup (\u03b1 \u2192\u2098 \u03b2) :=\n  semilattice_sup.mk has_sup.sup partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n    sorry\n\n@[simp] theorem has_inf_inf_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_inf \u03b2]\n    (f : \u03b1 \u2192\u2098 \u03b2) (g : \u03b1 \u2192\u2098 \u03b2) (a : \u03b1) : coe_fn (f \u2293 g) a = coe_fn f a \u2293 coe_fn g a :=\n  Eq.refl (coe_fn (f \u2293 g) a)\n\nprotected instance semilattice_inf {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [semilattice_inf \u03b2] :\n    semilattice_inf (\u03b1 \u2192\u2098 \u03b2) :=\n  semilattice_inf.mk has_inf.inf partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n    sorry\n\nprotected instance lattice {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [lattice \u03b2] :\n    lattice (\u03b1 \u2192\u2098 \u03b2) :=\n  lattice.mk semilattice_sup.sup semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry sorry\n    sorry semilattice_inf.inf sorry sorry sorry\n\nprotected instance has_bot {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_bot \u03b2] :\n    has_bot (\u03b1 \u2192\u2098 \u03b2) :=\n  has_bot.mk (mk (fun (a : \u03b1) => \u22a5) sorry)\n\nprotected instance order_bot {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_bot \u03b2] :\n    order_bot (\u03b1 \u2192\u2098 \u03b2) :=\n  order_bot.mk \u22a5 partial_order.le partial_order.lt sorry sorry sorry sorry\n\n@[simp] theorem has_top_top_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_top \u03b2]\n    (a : \u03b1) : coe_fn \u22a4 a = \u22a4 :=\n  Eq.refl (coe_fn \u22a4 a)\n\nprotected instance order_top {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [order_top \u03b2] :\n    order_top (\u03b1 \u2192\u2098 \u03b2) :=\n  order_top.mk \u22a4 partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance has_Inf {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [complete_lattice \u03b2] :\n    has_Inf (\u03b1 \u2192\u2098 \u03b2) :=\n  has_Inf.mk\n    fun (s : set (\u03b1 \u2192\u2098 \u03b2)) => mk (fun (x : \u03b1) => Inf ((fun (f : \u03b1 \u2192\u2098 \u03b2) => coe_fn f x) '' s)) sorry\n\n@[simp] theorem has_Sup_Sup_to_fun {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2} [complete_lattice \u03b2]\n    (s : set (\u03b1 \u2192\u2098 \u03b2)) (x : \u03b1) : coe_fn (Sup s) x = Sup ((fun (f : \u03b1 \u2192\u2098 \u03b2) => coe_fn f x) '' s) :=\n  Eq.refl (coe_fn (Sup s) x)\n\nprotected instance complete_lattice {\u03b1 : Type u_1} [preorder \u03b1] {\u03b2 : Type u_2}\n    [complete_lattice \u03b2] : complete_lattice (\u03b1 \u2192\u2098 \u03b2) :=\n  complete_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry\n    lattice.inf sorry sorry sorry order_top.top sorry order_bot.bot sorry Sup Inf sorry sorry sorry\n    sorry\n\ntheorem iterate_sup_le_sup_iff {\u03b1 : Type u_1} [semilattice_sup \u03b1] (f : \u03b1 \u2192\u2098 \u03b1) :\n    (\u2200 (n\u2081 n\u2082 : \u2115) (a\u2081 a\u2082 : \u03b1),\n          nat.iterate (\u21d1f) (n\u2081 + n\u2082) (a\u2081 \u2294 a\u2082) \u2264 nat.iterate (\u21d1f) n\u2081 a\u2081 \u2294 nat.iterate (\u21d1f) n\u2082 a\u2082) \u2194\n        \u2200 (a\u2081 a\u2082 : \u03b1), coe_fn f (a\u2081 \u2294 a\u2082) \u2264 coe_fn f a\u2081 \u2294 a\u2082 :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/preorder_hom_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.38034092509075124}}
{"text": "\n\nclass Inc (\u03b1 : Type) :=\n(inc : \u03b1 \u2192 \u03b1)\n\nexport Inc (inc)\n\ninstance {\u03b1} [Inc \u03b1] : Inc (List \u03b1) :=\n{ inc := (\u00b7.map inc) }\n\ninstance : Inc Nat :=\n{ inc := Nat.succ }\n\n#eval inc 10\n#eval inc [1, 2, 3]\n\ntheorem ex1 : [(1, \"hello\"), (2, \"world\")].map (\u00b7.1) = [1, 2] :=\nrfl\n\ntheorem ex2 : [(1, \"hello\"), (2, \"world\")].map (\u00b7.snd) = [\"hello\", \"world\"] :=\nrfl\n\ndef sum (xs : List Nat) : Nat :=\n(\u00b7.2) $ Id.run $ StateT.run (s:=0) do\n  xs.forM fun x => modify (\u00b7 + x)\n\n#eval sum [1, 2, 3, 4]\n\ntheorem ex3 : sum [1, 2, 3] = 6 :=\nrfl\n\ntheorem ex4 : sum [1, 2, 3, 4] = 10 :=\nrfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/cdotTests.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.38034092030837263}}
{"text": "example (p : Prop) [Decidable p] (hnp : \u00ac p) :\n    if decide p then 0 = 1 else 1 = 1 := by\n  simp [hnp, decideEqFalse Unit]\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/tests/lean/simpArgTypeMismatch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.685949467848392, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3803387203772556}}
{"text": "example (b : let n : Nat := 2; (n = 13)) : Bool := by\n  simp_all\n\nexample (b : let n : Nat := 2; (n = 13)) : n + 1 = 14 := by\n  simp_all\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/481.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.38033872037725547}}
{"text": "/-\nCopyright (c) 2019 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot\n\n! This file was ported from Lean 3 source module tactic.rename_var\n! leanprover-community/mathlib commit 591a0a006de5e8fe75b1c55258123d91dd416b2d\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Interactive\n\n/-!\n# Rename bound variable tactic\n\nThis files defines a tactic `rename_var` whose main purpose is to teach\nrenaming of bound variables.\n\n* `rename_var old new` renames all bound variables named `old` to `new` in the goal.\n* `rename_var old new at h` does the same in hypothesis `h`.\n\n```lean\nexample (P : \u2115 \u2192  \u2115 \u2192 Prop) (h : \u2200 n, \u2203 m, P n m) : \u2200 l, \u2203 m, P l m :=\nbegin\n  rename_var n q at h, -- h is now \u2200 (q : \u2115), \u2203 (m : \u2115), P q m,\n  rename_var m n, -- goal is now \u2200 (l : \u2115), \u2203 (n : \u2115), P k n,\n  exact h -- Lean does not care about those bound variable names\nend\n```\n\n## Tags\n\nteaching, tactic\n-/\n\n\nopen Expr\n\n/-- Rename bound variable `old` to `new` in an `expr`-/\nunsafe def expr.rename_var (old new : Name) : expr \u2192 expr\n  | pi n bi t b => pi (if n = old then new else n) bi (expr.rename_var t) (expr.rename_var b)\n  | lam n bi t b => lam (if n = old then new else n) bi (expr.rename_var t) (expr.rename_var b)\n  | app t b => app (expr.rename_var t) (expr.rename_var b)\n  | e => e\n#align expr.rename_var expr.rename_var\n\nnamespace Tactic\n\n/-- Rename bound variable `old` to `new` in goal -/\nunsafe def rename_var_at_goal (old new : Name) : tactic Unit := do\n  let old_tgt \u2190 target\n  tactic.change (expr.rename_var old new old_tgt)\n#align tactic.rename_var_at_goal tactic.rename_var_at_goal\n\n/-- Rename bound variable `old` to `new` in assumption `h` -/\nunsafe def rename_var_at_hyp (old new : Name) (e : expr) : tactic Unit := do\n  let old_e \u2190 infer_type e\n  tactic.change_core (expr.rename_var old new old_e) (some e)\n#align tactic.rename_var_at_hyp tactic.rename_var_at_hyp\n\nend Tactic\n\nnamespace Tactic.Interactive\n\nopen Tactic\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/-- `rename_var old new` renames all bound variables named `old` to `new` in the goal.\n`rename_var old new at h` does the same in hypothesis `h`.\n-/\nunsafe def rename_var (old : parse ident) (new : parse ident) (l : parse location) : tactic Unit :=\n  l.apply (rename_var_at_hyp old new) (rename_var_at_goal old new)\n#align tactic.interactive.rename_var tactic.interactive.rename_var\n\nend Tactic.Interactive\n\n/-- `rename_var old new` renames all bound variables named `old` to `new` in the goal.\n`rename_var old new at h` does the same in hypothesis `h`.\nThis is meant for teaching bound variables only. Such a renaming should never be relevant to Lean.\n\n```lean\nexample (P : \u2115 \u2192  \u2115 \u2192 Prop) (h : \u2200 n, \u2203 m, P n m) : \u2200 l, \u2203 m, P l m :=\nbegin\n  rename_var n q at h, -- h is now \u2200 (q : \u2115), \u2203 (m : \u2115), P q m,\n  rename_var m n, -- goal is now \u2200 (l : \u2115), \u2203 (n : \u2115), P k n,\n  exact h -- Lean does not care about those bound variable names\nend\n```\n-/\nadd_tactic_doc\n  { Name := \"rename_var\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.rename_var]\n    tags := [\"renaming\"] }\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/RenameVar.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6859494678483918, "lm_q1q2_score": 0.38033872037725547}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Callum Sutton, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.mul_add\nimport Mathlib.algebra.field\nimport Mathlib.algebra.opposites\nimport Mathlib.PostPort\n\nuniverses u_4 u_5 l u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# (Semi)ring equivs\n\nIn this file we define extension of `equiv` called `ring_equiv`, which is a datatype representing an\nisomorphism of `semiring`s, `ring`s, `division_ring`s, or `field`s. We also introduce the\ncorresponding group of automorphisms `ring_aut`.\n\n## Notations\n\nThe extended equiv have coercions to functions, and the coercion is the canonical notation when\ntreating the isomorphism as maps.\n\n## Implementation notes\n\nThe fields for `ring_equiv` now avoid the unbundled `is_mul_hom` and `is_add_hom`, as these are\ndeprecated.\n\nDefinition of multiplication in the groups of automorphisms agrees with function composition,\nmultiplication in `equiv.perm`, and multiplication in `category_theory.End`, not with\n`category_theory.comp`.\n\n## Tags\n\nequiv, mul_equiv, add_equiv, ring_equiv, mul_aut, add_aut, ring_aut\n-/\n\n/-- An equivalence between two (semi)rings that preserves the algebraic structure. -/\nstructure ring_equiv (R : Type u_4) (S : Type u_5) [Mul R] [Add R] [Mul S] [Add S] \nextends R \u2243* S, R \u2243 S, R \u2243+ S\nwhere\n\ninfixl:25 \" \u2243+* \" => Mathlib.ring_equiv\n\n/-- The \"plain\" equivalence of types underlying an equivalence of (semi)rings. -/\n/-- The equivalence of additive monoids underlying an equivalence of (semi)rings. -/\n/-- The equivalence of multiplicative monoids underlying an equivalence of (semi)rings. -/\nnamespace ring_equiv\n\n\nprotected instance has_coe_to_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] : has_coe_to_fun (R \u2243+* S) :=\n  has_coe_to_fun.mk (fun (x : R \u2243+* S) => R \u2192 S) to_fun\n\n@[simp] theorem to_fun_eq_coe_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2243+* S) : to_fun f = \u21d1f :=\n  rfl\n\n/-- A ring isomorphism preserves multiplication. -/\n@[simp] theorem map_mul {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) (x : R) (y : R) : coe_fn e (x * y) = coe_fn e x * coe_fn e y :=\n  map_mul' e x y\n\n/-- A ring isomorphism preserves addition. -/\n@[simp] theorem map_add {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) (x : R) (y : R) : coe_fn e (x + y) = coe_fn e x + coe_fn e y :=\n  map_add' e x y\n\n/-- Two ring isomorphisms agree if they are defined by the\n    same underlying function. -/\ntheorem ext {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S} {g : R \u2243+* S} (h : \u2200 (x : R), coe_fn f x = coe_fn g x) : f = g := sorry\n\nprotected theorem congr_arg {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S} {x : R} {x' : R} : x = x' \u2192 coe_fn f x = coe_fn f x' := sorry\n\nprotected theorem congr_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S} {g : R \u2243+* S} (h : f = g) (x : R) : coe_fn f x = coe_fn g x :=\n  h \u25b8 rfl\n\ntheorem ext_iff {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S} {g : R \u2243+* S} : f = g \u2194 \u2200 (x : R), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : R) => h \u25b8 rfl, mpr := ext }\n\nprotected instance has_coe_to_mul_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] : has_coe (R \u2243+* S) (R \u2243* S) :=\n  has_coe.mk to_mul_equiv\n\nprotected instance has_coe_to_add_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] : has_coe (R \u2243+* S) (R \u2243+ S) :=\n  has_coe.mk to_add_equiv\n\ntheorem coe_mul_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2243+* S) (a : R) : coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\ntheorem coe_add_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2243+* S) (a : R) : coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\n/-- The identity map is a ring isomorphism. -/\nprotected def refl (R : Type u_1) [Mul R] [Add R] : R \u2243+* R :=\n  mk (mul_equiv.to_fun (mul_equiv.refl R)) (mul_equiv.inv_fun (mul_equiv.refl R)) sorry sorry sorry sorry\n\n@[simp] theorem refl_apply (R : Type u_1) [Mul R] [Add R] (x : R) : coe_fn (ring_equiv.refl R) x = x :=\n  rfl\n\n@[simp] theorem coe_add_equiv_refl (R : Type u_1) [Mul R] [Add R] : \u2191(ring_equiv.refl R) = add_equiv.refl R :=\n  rfl\n\n@[simp] theorem coe_mul_equiv_refl (R : Type u_1) [Mul R] [Add R] : \u2191(ring_equiv.refl R) = mul_equiv.refl R :=\n  rfl\n\nprotected instance inhabited (R : Type u_1) [Mul R] [Add R] : Inhabited (R \u2243+* R) :=\n  { default := ring_equiv.refl R }\n\n/-- The inverse of a ring isomorphism is a ring isomorphism. -/\nprotected def symm {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : S \u2243+* R :=\n  mk (mul_equiv.to_fun (mul_equiv.symm (to_mul_equiv e))) (mul_equiv.inv_fun (mul_equiv.symm (to_mul_equiv e))) sorry\n    sorry sorry sorry\n\n/-- See Note [custom simps projection] -/\ndef simps.inv_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : S \u2192 R :=\n  \u21d1(ring_equiv.symm e)\n\n@[simp] theorem symm_symm {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : ring_equiv.symm (ring_equiv.symm e) = e :=\n  ext fun (x : R) => rfl\n\n@[simp] theorem coe_symm_mk {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2192 S) (g : S \u2192 R) (h\u2081 : function.left_inverse g f) (h\u2082 : function.right_inverse g f) (h\u2083 : \u2200 (x y : R), f (x * y) = f x * f y) (h\u2084 : \u2200 (x y : R), f (x + y) = f x + f y) : \u21d1(ring_equiv.symm (mk f g h\u2081 h\u2082 h\u2083 h\u2084)) = g :=\n  rfl\n\n/-- Transitivity of `ring_equiv`. -/\nprotected def trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [Mul R] [Add R] [Mul S] [Add S] [Mul S'] [Add S'] (e\u2081 : R \u2243+* S) (e\u2082 : S \u2243+* S') : R \u2243+* S' :=\n  mk (mul_equiv.to_fun (mul_equiv.trans (to_mul_equiv e\u2081) (to_mul_equiv e\u2082)))\n    (mul_equiv.inv_fun (mul_equiv.trans (to_mul_equiv e\u2081) (to_mul_equiv e\u2082))) sorry sorry sorry sorry\n\n@[simp] theorem trans_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [semiring A] [semiring B] [semiring C] (e : A \u2243+* B) (f : B \u2243+* C) (a : A) : coe_fn (ring_equiv.trans e f) a = coe_fn f (coe_fn e a) :=\n  rfl\n\nprotected theorem bijective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : function.bijective \u21d1e :=\n  equiv.bijective (to_equiv e)\n\nprotected theorem injective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : function.injective \u21d1e :=\n  equiv.injective (to_equiv e)\n\nprotected theorem surjective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) : function.surjective \u21d1e :=\n  equiv.surjective (to_equiv e)\n\n@[simp] theorem apply_symm_apply {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) (x : S) : coe_fn e (coe_fn (ring_equiv.symm e) x) = x :=\n  equiv.apply_symm_apply (to_equiv e)\n\n@[simp] theorem symm_apply_apply {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) (x : R) : coe_fn (ring_equiv.symm e) (coe_fn e x) = x :=\n  equiv.symm_apply_apply (to_equiv e)\n\ntheorem image_eq_preimage {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) (s : set R) : \u21d1e '' s = \u21d1(ring_equiv.symm e) \u207b\u00b9' s :=\n  equiv.image_eq_preimage (to_equiv e) s\n\n/-- A commutative ring is isomorphic to its opposite. -/\ndef to_opposite (R : Type u_1) [comm_semiring R] : R \u2243+* (R\u1d52\u1d56) :=\n  mk (equiv.to_fun opposite.equiv_to_opposite) (equiv.inv_fun opposite.equiv_to_opposite) sorry sorry sorry sorry\n\n@[simp] theorem to_opposite_apply (R : Type u_1) [comm_semiring R] (r : R) : coe_fn (to_opposite R) r = opposite.op r :=\n  rfl\n\n@[simp] theorem to_opposite_symm_apply (R : Type u_1) [comm_semiring R] (r : R\u1d52\u1d56) : coe_fn (ring_equiv.symm (to_opposite R)) r = opposite.unop r :=\n  rfl\n\n/-- A ring isomorphism sends one to one. -/\n@[simp] theorem map_one {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) : coe_fn f 1 = 1 :=\n  mul_equiv.map_one \u2191f\n\n/-- A ring isomorphism sends zero to zero. -/\n@[simp] theorem map_zero {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) : coe_fn f 0 = 0 :=\n  add_equiv.map_zero \u2191f\n\n@[simp] theorem map_eq_one_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) {x : R} : coe_fn f x = 1 \u2194 x = 1 :=\n  mul_equiv.map_eq_one_iff \u2191f\n\n@[simp] theorem map_eq_zero_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) {x : R} : coe_fn f x = 0 \u2194 x = 0 :=\n  add_equiv.map_eq_zero_iff \u2191f\n\ntheorem map_ne_one_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) {x : R} : coe_fn f x \u2260 1 \u2194 x \u2260 1 :=\n  mul_equiv.map_ne_one_iff \u2191f\n\ntheorem map_ne_zero_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) {x : R} : coe_fn f x \u2260 0 \u2194 x \u2260 0 :=\n  add_equiv.map_ne_zero_iff \u2191f\n\n/-- Produce a ring isomorphism from a bijective ring homomorphism. -/\ndef of_bijective {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2192+* S) (hf : function.bijective \u21d1f) : R \u2243+* S :=\n  mk (equiv.to_fun (equiv.of_bijective (\u21d1f) hf)) (equiv.inv_fun (equiv.of_bijective (\u21d1f) hf)) sorry sorry\n    (ring_hom.map_mul' f) (ring_hom.map_add' f)\n\n@[simp] theorem map_neg {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) (x : R) : coe_fn f (-x) = -coe_fn f x :=\n  add_equiv.map_neg (\u2191f) x\n\n@[simp] theorem map_sub {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) (x : R) (y : R) : coe_fn f (x - y) = coe_fn f x - coe_fn f y :=\n  add_equiv.map_sub (\u2191f) x y\n\n@[simp] theorem map_neg_one {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) : coe_fn f (-1) = -1 :=\n  map_one f \u25b8 map_neg f 1\n\n/-- Reinterpret a ring equivalence as a ring homomorphism. -/\ndef to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : R \u2192+* S :=\n  ring_hom.mk (monoid_hom.to_fun (mul_equiv.to_monoid_hom (to_mul_equiv e))) sorry sorry sorry sorry\n\ntheorem to_ring_hom_injective {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] : function.injective to_ring_hom :=\n  fun (f g : R \u2243+* S) (h : to_ring_hom f = to_ring_hom g) => ext (iff.mp ring_hom.ext_iff h)\n\nprotected instance has_coe_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] : has_coe (R \u2243+* S) (R \u2192+* S) :=\n  has_coe.mk to_ring_hom\n\ntheorem coe_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) (a : R) : coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\ntheorem coe_ring_hom_inj_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) (g : R \u2243+* S) : f = g \u2194 \u2191f = \u2191g :=\n  { mp := congr_arg fun (f : R \u2243+* S) => \u2191f, mpr := fun (h : \u2191f = \u2191g) => ext (iff.mp ring_hom.ext_iff h) }\n\n/-- Reinterpret a ring equivalence as a monoid homomorphism. -/\ndef to_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : R \u2192* S :=\n  ring_hom.to_monoid_hom (to_ring_hom e)\n\n/-- Reinterpret a ring equivalence as an `add_monoid` homomorphism. -/\ndef to_add_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : R \u2192+ S :=\n  ring_hom.to_add_monoid_hom (to_ring_hom e)\n\n@[simp] theorem to_ring_hom_refl {R : Type u_1} [semiring R] : to_ring_hom (ring_equiv.refl R) = ring_hom.id R :=\n  rfl\n\n@[simp] theorem to_monoid_hom_refl {R : Type u_1} [semiring R] : to_monoid_hom (ring_equiv.refl R) = monoid_hom.id R :=\n  rfl\n\n@[simp] theorem to_add_monoid_hom_refl {R : Type u_1} [semiring R] : to_add_monoid_hom (ring_equiv.refl R) = add_monoid_hom.id R :=\n  rfl\n\n@[simp] theorem to_ring_hom_apply_symm_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) (y : S) : coe_fn (to_ring_hom e) (coe_fn (to_ring_hom (ring_equiv.symm e)) y) = y :=\n  equiv.apply_symm_apply (to_equiv e)\n\n@[simp] theorem symm_to_ring_hom_apply_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) (x : R) : coe_fn (to_ring_hom (ring_equiv.symm e)) (coe_fn (to_ring_hom e) x) = x :=\n  equiv.symm_apply_apply (to_equiv e)\n\n@[simp] theorem to_ring_hom_trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [semiring R] [semiring S] [semiring S'] (e\u2081 : R \u2243+* S) (e\u2082 : S \u2243+* S') : to_ring_hom (ring_equiv.trans e\u2081 e\u2082) = ring_hom.comp (to_ring_hom e\u2082) (to_ring_hom e\u2081) :=\n  rfl\n\n@[simp] theorem to_ring_hom_comp_symm_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : ring_hom.comp (to_ring_hom e) (to_ring_hom (ring_equiv.symm e)) = ring_hom.id S := sorry\n\n@[simp] theorem symm_to_ring_hom_comp_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : ring_hom.comp (to_ring_hom (ring_equiv.symm e)) (to_ring_hom e) = ring_hom.id R := sorry\n\n/--\nConstruct an equivalence of rings from homomorphisms in both directions, which are inverses.\n-/\ndef of_hom_inv {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (hom : R \u2192+* S) (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R) (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) : R \u2243+* S :=\n  mk (ring_hom.to_fun hom) \u21d1inv sorry sorry (ring_hom.map_mul' hom) (ring_hom.map_add' hom)\n\n@[simp] theorem of_hom_inv_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (hom : R \u2192+* S) (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R) (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) (r : R) : coe_fn (of_hom_inv hom inv hom_inv_id inv_hom_id) r = coe_fn hom r :=\n  rfl\n\n@[simp] theorem of_hom_inv_symm_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (hom : R \u2192+* S) (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R) (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) (s : S) : coe_fn (ring_equiv.symm (of_hom_inv hom inv hom_inv_id inv_hom_id)) s = coe_fn inv s :=\n  rfl\n\nend ring_equiv\n\n\nnamespace mul_equiv\n\n\n/-- Gives a `ring_equiv` from a `mul_equiv` preserving addition.-/\ndef to_ring_equiv {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S] (h : R \u2243* S) (H : \u2200 (x y : R), coe_fn h (x + y) = coe_fn h x + coe_fn h y) : R \u2243+* S :=\n  ring_equiv.mk (equiv.to_fun (to_equiv h)) (equiv.inv_fun (to_equiv h)) sorry sorry (map_mul' h) sorry\n\nend mul_equiv\n\n\nnamespace ring_equiv\n\n\n@[simp] theorem trans_symm {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S] (e : R \u2243+* S) : ring_equiv.trans e (ring_equiv.symm e) = ring_equiv.refl R :=\n  ext (left_inv e)\n\n@[simp] theorem symm_trans {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S] (e : R \u2243+* S) : ring_equiv.trans (ring_equiv.symm e) e = ring_equiv.refl S :=\n  ext (right_inv e)\n\n/-- If two rings are isomorphic, and the second is an integral domain, then so is the first. -/\nprotected theorem is_integral_domain {A : Type u_1} (B : Type u_2) [ring A] [ring B] (hB : is_integral_domain B) (e : A \u2243+* B) : is_integral_domain A := sorry\n\n/-- If two rings are isomorphic, and the second is an integral domain, then so is the first. -/\nprotected def integral_domain {A : Type u_1} (B : Type u_2) [ring A] [integral_domain B] (e : A \u2243+* B) : integral_domain A :=\n  integral_domain.mk ring.add ring.add_assoc ring.zero ring.zero_add ring.add_zero ring.neg ring.sub ring.add_left_neg\n    ring.add_comm ring.mul ring.mul_assoc ring.one ring.one_mul ring.mul_one ring.left_distrib ring.right_distrib sorry\n    sorry sorry\n\nend ring_equiv\n\n\nnamespace equiv\n\n\n/-- In a division ring `K`, the unit group `units K`\nis equivalent to the subtype of nonzero elements. -/\n-- TODO: this might already exist elsewhere for `group_with_zero`\n\n-- deduplicate or generalize\n\ndef units_equiv_ne_zero (K : Type u_4) [division_ring K] : units K \u2243 \u21a5(set_of fun (a : K) => a \u2260 0) :=\n  mk (fun (a : units K) => { val := units.val a, property := sorry })\n    (fun (a : \u21a5(set_of fun (a : K) => a \u2260 0)) => units.mk0 (subtype.val a) sorry) sorry sorry\n\n@[simp] theorem coe_units_equiv_ne_zero {K : Type u_4} [division_ring K] (a : units K) : \u2191(coe_fn (units_equiv_ne_zero K) a) = \u2191a :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/ring.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.685949442167993, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.38033870613823284}}
{"text": "import Saturn.FinSeq\nimport Saturn.Vector\nimport Saturn.Clause \nimport Saturn.Solverstep\nopen Nat\nopen FinSeq\n \n/-\nThe inductive step for adding a new clause when it should be dropped. The new clauses and maps \nare defined. All the witnesses for the relations between the old and new clauses are also\nconstructed.\n-/\ndef addPositiveClause{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192 \n            ReductionClauses branch focus focusLt (head +: clauses) := \n          fun rc head pos => \n          let domN := dom + 1\n          let codomN := rc.codom\n          let clausesN := head +: clauses\n          let forwardVecN := none +: rc.forwardVec\n          let forwardN: (k : Nat) \u2192  k < domN \u2192 Option Nat  := \n            fun k  => \n            match k with \n            | zero => fun _ => none\n            | l + 1 => \n              fun w : l + 1 < domN   =>  rc.forward l (le_of_succ_le_succ w)\n          have forwardNEq : forwardVecN.coords = forwardN := by\n                  apply funext\n                  intro j\n                  cases j with\n                  | zero => \n                    apply funext\n                    intro jw\n                    rfl\n                  | succ i =>\n                    apply funext\n                    intro jw\n                    have tl :forwardVecN.coords (succ i) jw = \n                        forwardVecN.coords.tail i (Nat.le_of_succ_le_succ jw) := by rfl\n                    rw [tl]\n                    rw [tail_commutes none rc.forwardVec]\n          have forwardWitN : (k: Nat) \u2192 (w: k < domN) \u2192 boundOpt codomN (forwardN k w) := \n            fun k  => \n            match k with \n            | zero => fun w => \n              let resolve : forwardN zero w = none := by rfl\n              by\n                rw [resolve]\n                exact True.intro\n                done\n            | l + 1 => \n              fun w : l + 1 < domN   => \n                let lem : forwardN (l + 1) w = rc.forward l (le_of_succ_le_succ w) := by rfl \n                by\n                  rw [lem]\n                  exact (rc.forwardWit l (le_of_succ_le_succ w))\n                  done\n          let reverseVecN := rc.reverseVec.map (. + 1)\n          let reverseN : (k : Nat) \u2192  k < codomN \u2192 Nat := \n            fun k w => (rc.reverse k w) + 1\n          have reverseNEq : reverseVecN.coords = reverseN := by\n                  apply funext\n                  intro j\n                  apply funext\n                  intro jw\n                  apply map_coords_commute\n                  done\n          have reverseWitN : (k : Nat) \u2192 (w : k < codomN) \u2192 reverseN k w < domN :=\n            fun k w => succ_le_succ (rc.reverseWit k  w)\n          ReductionClauses.mk codomN rc.restClauses \n                    (forwardVecN) \n                    (forwardNEq \u25b8 forwardWitN) \n                    reverseVecN\n                    (reverseNEq \u25b8 reverseWitN)\n\nnamespace PosResClause\n\ndef droppedProof{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192\n          DroppedProof rc \u2192 \n          DroppedProof (addPositiveClause  branch focus focusLt clauses rc head pos) := \n        fun rc head pos drc =>\n          let rcN := addPositiveClause  branch focus focusLt clauses rc head pos  \n          let domN := dom + 1\n          let codomN := rc.codom\n          let clausesN := head +: clauses\n          let droppedN : \n              (k : Nat) \u2192 (w: k < domN) \u2192 rcN.forward k w = none \u2192 \n                  (clausesN.coords k w).coords focus focusLt = some branch := by\n                  intro k \n                  match k with\n                  | zero => \n                    intro _ _ \n                    exact pos\n                  | l + 1 => \n                      intro w nw \n                      let resolve : rcN.forward (l + 1) w = \n                        rc.forward l (le_of_succ_le_succ w) := by rfl\n                      rw [resolve] at nw\n                      let lem3 := drc.dropped l (le_of_succ_le_succ w) nw\n                      exact lem3                      \n          \u27e8droppedN\u27e9\n\ndef forwardRelation{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192\n          ForwardRelation rc \u2192 \n          ForwardRelation (addPositiveClause  branch focus focusLt clauses rc head pos) := \n        fun rc head pos frc =>\n          let rcN := addPositiveClause  branch focus focusLt clauses rc head pos  \n          let domN := dom + 1\n          let codomN := rc.codom\n          let clausesN := head +: clauses\n          have forwardRelationN : (k : Nat) \u2192 (w: k < domN) \u2192 (j: Nat) \u2192  rcN.forward k w = some j \u2192\n              (jw : j < codomN) \u2192  delete focus focusLt ((clausesN.coords k w).coords) = \n                (rcN.restClauses.coords j jw).coords := by\n                intro k \n                match k with\n                | zero => \n                  intro w j sw \n                  exact Option.noConfusion sw\n                | l + 1 => \n                  intro w j sw \n                  exact frc.forwardRelation l (le_of_succ_le_succ w) j sw\n          \u27e8forwardRelationN\u27e9\n\ntheorem reverseResolve{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : (head.coords focus focusLt = some branch)) \u2192\n        (l: Nat) \u2192 (w : l  < rc.codom ) \u2192 \n          (addPositiveClause  branch focus focusLt clauses rc head pos).reverse l w = \n            (rc.reverse l w) + 1 := by\n            intro rc head neg l w \n            let rcN := addPositiveClause  branch focus focusLt clauses rc head neg \n            have res1 : rcN.reverse l w = \n                                      rcN.reverseVec.coords l w := by rfl\n            have res2 : rc.reverse l w =\n                    rc.reverseVec.coords l w := by rfl\n            rw [res1]\n            rw [res2]\n            have res3 :rcN.reverseVec = \n              (rc.reverseVec.map (. + 1)) := by rfl\n            rw [res3]\n            have res4 :\n                ( (rc.reverseVec.map (. + 1)) ).coords l w =\n                  (zero +: \n                  (rc.reverseVec.map (. + 1)) ).coords.tail \n                  l w := by rfl\n            rw [res4]\n            rw [(tail_commutes \n                zero (rc.reverseVec.map (. + 1)))]\n            rw [map_coords_commute]  \n\ndef reverseRelation{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192\n          ReverseRelation rc \u2192 \n          ReverseRelation (addPositiveClause  branch focus focusLt clauses rc head pos) := \n        fun rc head pos rrc =>\n          let rcN := addPositiveClause  branch focus focusLt clauses rc head pos  \n          let domN := dom + 1\n          let codomN := rc.codom\n          let clausesN := head +: clauses\n          have relationN : (k : Nat) \u2192 (w: k < codomN) \u2192 \n                 (rcN.restClauses.coords k w).coords = \n                  delete focus focusLt \n                    (clausesN.coords (rcN.reverse k w) (rcN.reverseWit k w)).coords := \n                  by\n                    intro l\n                    intro w \n                    let lem1 : rcN.restClauses.coords l w = \n                              rc.restClauses.coords l w := by rfl\n                    let lem2 := rrc.relation l w               \n                    rw [lem1]                          \n                    rw [lem2]\n                    have rs0 : clausesN.coords (rcN.reverse l w) \n                                (rcN.reverseWit l w) =\n                                  clausesN.coords \n                                    (rc.reverse l w + 1)\n                                    (succ_le_succ\n                                      (rc.reverseWit l w)) := by \n                                    apply witness_independent\n                                    apply reverseResolve\n                    rw [rs0]\n                    rfl\n          \u27e8relationN\u27e9\n\ndef pureReverse{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom):\n      (rc: ReductionClauses branch focus focusLt clauses) \u2192 \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192\n          NonPosReverse rc \u2192 \n          NonPosReverse (addPositiveClause  branch focus focusLt clauses rc head pos) := \n        fun rc head pos prc =>\n          let rcN := addPositiveClause  branch focus focusLt clauses rc head pos  \n          let domN := dom + 1\n          let codomN := rc.codom\n          let clausesN := head +: clauses\n          have pureN : (k : Nat) \u2192 (w: k < codomN)  \u2192 \n                Not (\n                  (clausesN.coords (rcN.reverse k w) (rcN.reverseWit k w)).coords \n                     focus focusLt = some branch) :=\n                  by\n                    intro l w hyp \n                    have rs0 : clausesN.coords (rcN.reverse l w) \n                                (rcN.reverseWit l w) =\n                                  clausesN.coords \n                                    (rc.reverse l w + 1)\n                                    (succ_le_succ\n                                      (rc.reverseWit l w)) := by \n                                    apply witness_independent\n                                    apply reverseResolve\n                    rw [rs0] at hyp\n                    have rs1 : clausesN.coords \n                                    (rc.reverse l w + 1)\n                                    (succ_le_succ\n                                      (rc.reverseWit l w)) =\n                                        clauses.coords (rc.reverse l w)\n                                        (rc.reverseWit l w) := by rfl\n                    rw [rs1] at hyp\n                    let prev := prc.nonPosRev l w\n                    exact absurd hyp prev\n          \u27e8pureN\u27e9\n\n\ndef prependResData{dom n: Nat}(branch: Bool)(focus: Nat)(focusLt : focus < n + 1)\n    (clauses: Vector (Clause (n + 1)) dom): \n        (head : Clause (n + 1)) \u2192 (pos : head.coords focus focusLt = some branch) \u2192\n        (rd : ReductionData branch focus focusLt clauses) \u2192 \n        ReductionData branch focus focusLt (head +: clauses) := \n          fun head pos rd =>\n          let rc := addPositiveClause branch focus focusLt clauses rd.restrictionClauses head pos\n          \u27e8rc, \n          droppedProof branch focus focusLt clauses rd.restrictionClauses head pos rd.droppedProof,\n          forwardRelation branch focus focusLt clauses rd.restrictionClauses head pos rd.forwardRelation,\n          reverseRelation branch focus focusLt clauses rd.restrictionClauses head pos rd.reverseRelation,\n          pureReverse branch focus focusLt clauses rd.restrictionClauses head pos rd.nonPosReverse\u27e9\n          \n\nend PosResClause\n\n", "meta": {"author": "siddhartha-gadgil", "repo": "Saturn", "sha": "4e451333411dc0f381ade5aa4569e1acfea5be0b", "save_path": "github-repos/lean/siddhartha-gadgil-Saturn", "path": "github-repos/lean/siddhartha-gadgil-Saturn/Saturn-4e451333411dc0f381ade5aa4569e1acfea5be0b/Saturn/PosRestClause.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7431680199891789, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3802914159135799}}
{"text": "import library_dev.data.list.sort\n\nopen list\n\n@[reducible] def pre_pdmap (K : Type) (V : K \u2192 Type) := list (\u03a3 (k : K), V k)\n\nnamespace pre_pdmap\nvariables {K : Type} {V : K \u2192 Type}\n\ndef get [decidable_eq K] [\u2200 k, inhabited (V k)] (k\u2080 : K) : pre_pdmap K V \u2192 V k\u2080\n| []           := default _\n| (\u27e8k,v\u27e9::kvs) := if H : k = k\u2080 then eq.rec_on H v else get kvs\n\ndef has_key [decidable_eq K] (k\u2080 : K) : pre_pdmap K V \u2192 Prop\n| []           := false\n| (\u27e8k,v\u27e9::kvs) := if k\u2080 = k then true else has_key kvs\n\ninstance decidable_has_key [decidable_eq K]  (k\u2080 : K) : \u03a0 (kvs : pre_pdmap K V), decidable (has_key k\u2080 kvs)\n| [] := is_false (\u03bb f, f)\n| (\u27e8k, v\u27e9::kvs) :=\nbegin\ndunfold has_key,\napply (@decidable.by_cases (k\u2080 = k)),\n{ intro H_eq, simp [H_eq], exact decidable.true },\n{ intro H_neq, simp [H_neq], apply decidable_has_key }\nend\n\ndef rm_key [decidable_eq K] (k\u2080 : K) : pre_pdmap K V \u2192 pre_pdmap K V\n| []             := []\n| (\u27e8k,v\u27e9::kvs) := if k\u2080 = k then kvs else \u27e8k,v\u27e9 :: rm_key kvs\n\ndef insert [decidable_eq K] (k\u2080 : K) (v\u2080 : V k\u2080) (pdm : pre_pdmap K V) : pre_pdmap K V :=\nif has_key k\u2080 pdm then \u27e8k\u2080, v\u2080\u27e9 :: rm_key k\u2080 pdm else \u27e8k\u2080, v\u2080\u27e9 :: pdm\n\ndef keys : pre_pdmap K V \u2192 list K\n| [] := []\n| (\u27e8k,v\u27e9::kvs) := k::keys kvs\n\ndef erase_dup_keys [decidable_eq K] : pre_pdmap K V \u2192 pre_pdmap K V\n| []        :=  []\n| (\u27e8k,v\u27e9 :: kvs) := if has_key k kvs then erase_dup_keys kvs else \u27e8k,v\u27e9 :: erase_dup_keys kvs\n\nend pre_pdmap\n\ndef pdmap (K : Type) (V : K \u2192 Type) : Type := {xs : pre_pdmap K V // nodup (pre_pdmap.keys xs)}\n\nnamespace pdmap\nvariables {K : Type} {V : K \u2192 Type}\n\n-- TODO(dhs): nodup nil\ndef mk : pdmap K V := \u27e8[], sorry\u27e9\n\ndef get [decidable_eq K] [\u2200 k, inhabited (V k)] (k\u2080 : K) : pdmap K V \u2192 V k\u2080\n| \u27e8kvs, H\u27e9 := pre_pdmap.get k\u2080 kvs\n\ndef has_key [decidable_eq K] (k\u2080 : K) : pdmap K V \u2192 Prop\n| \u27e8kvs, H\u27e9 := pre_pdmap.has_key k\u2080 kvs\n\ninstance decidable_has_key [decidable_eq K]  (k\u2080 : K) : \u03a0 (kvs : pdmap K V), decidable (has_key k\u2080 kvs)\n| \u27e8kvs, H\u27e9 := begin dunfold has_key, apply_instance end\n\n-- TODO(dhs): removing does not introduce a duplicate\ndef rm_key [decidable_eq K] (k\u2080 : K) : pdmap K V \u2192 pdmap K V\n| \u27e8kvs, H\u27e9 := \u27e8pre_pdmap.rm_key k\u2080 kvs, sorry\u27e9\n\n-- TODO(dhs): removing does not introduce a duplicate, and means not in, and consing when not in does not introduce a duplicate\ndef insert [decidable_eq K] (k\u2080 : K) (v\u2080 : V k\u2080) : pdmap K V \u2192 pdmap K V\n| \u27e8kvs, H\u27e9 := \u27e8pre_pdmap.insert k\u2080 v\u2080 kvs, sorry\u27e9\n\ndef keys : pdmap K V \u2192 list K\n| \u27e8kvs, H\u27e9 := pre_pdmap.keys kvs\n\ndefinition eqv (l\u2081 l\u2082 : pdmap K V) :=\nperm l\u2081.1 l\u2082.1\n\nlocal infix ~ := eqv\n\ndefinition eqv.refl (l : pdmap K V) : l ~ l :=\nperm.refl l.1\n\ndefinition eqv.symm (l\u2081 l\u2082 : pdmap K V) : l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2081 :=\nperm.symm\n\ndefinition eqv.trans (l\u2081 l\u2082 l\u2083 : pdmap K V) : l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083 :=\nperm.trans\n\ninstance pdmap.eqv_setoid : setoid (pdmap K V) :=\nsetoid.mk eqv (mk_equivalence eqv eqv.refl eqv.symm eqv.trans)\n\nend pdmap\n\ndef dmap (K : Type) (V : K \u2192 Type) : Type := quot (@pdmap.eqv K V)\n\nnamespace dmap\nvariables {K : Type} {V : K \u2192 Type}\n\ndef mk : dmap K V := quotient.mk pdmap.mk\n\ndef has_key [decidable_eq K] (k\u2080 : K) (dm : dmap K V) : Prop :=\nquot.lift_on dm (\u03bb pdm, pdmap.has_key k\u2080 pdm) (\u03bb l\u2081 l\u2082 (e : pdmap.eqv l\u2081 l\u2082), sorry)\n\ninstance decidable_has_key [decidable_eq K] (k\u2080 : K) (dm : dmap K V) : decidable (has_key k\u2080 dm) :=\nquot.rec_on dm (\u03bb pdm, pdmap.decidable_has_key k\u2080 pdm) (\u03bb l\u2081 l\u2082 (e : pdmap.eqv l\u2081 l\u2082), sorry)\n\ndef get [decidable_eq K] [\u2200 k, inhabited (V k)] (k\u2080 : K) (dm : dmap K V) : V k\u2080 :=\nquot.lift_on dm (\u03bb pdm, pdmap.get k\u2080 pdm) (\u03bb l\u2081 l\u2082 (H_eqv : pdmap.eqv l\u2081 l\u2082), sorry)\n\ndef insert [decidable_eq K] (k\u2080 : K) (v\u2080 : V k\u2080) (dm : dmap K V) : dmap K V :=\nquot.lift_on dm (\u03bb pdm, quotient.mk $ pdmap.insert k\u2080 v\u2080 pdm) (\u03bb l\u2081 l\u2082 (H_eqv : pdmap.eqv l\u2081 l\u2082), sorry)\n\ndef keys [has_lt K] [decidable_rel (@has_lt.lt K _)] (dm : dmap K V) : list K :=\nquot.lift_on dm (\u03bb pdm, insertion_sort has_lt.lt (pdmap.keys pdm)) (\u03bb l\u2081 l\u2082 (H_eqv : pdmap.eqv l\u2081 l\u2082), sorry)\n\nlemma has_key_mem_keys [decidable_eq K] [has_lt K] [decidable_rel (@has_lt.lt K _)] {k : K} {dm : dmap K V} :\n  has_key k dm \u2192 k \u2208 keys dm := sorry\n\nlemma has_key_insert [decidable_eq K] {k\u2081 k\u2082 : K} {v\u2082 : V k\u2082} {dm : dmap K V} :\n  has_key k\u2081 dm \u2192 has_key k\u2081 (insert k\u2082 v\u2082 dm) := sorry\n\nlemma has_key_insert_same [decidable_eq K] (k : K) {v : V k} (dm : dmap K V) :\n  has_key k (insert k v dm) := sorry\n\nlemma get_insert_same [decidable_eq K] [\u2200 k, inhabited (V k)] (k : K) (v : V k) (dm : dmap K V) :\n  get k (insert k v dm) = v := sorry\n\nlemma get_insert_diff [decidable_eq K] [\u2200 k, inhabited (V k)] {k\u2081 k\u2082 : K} (v\u2082 : V k\u2082) (dm : dmap K V) :\n  k\u2081 \u2260 k\u2082 \u2192 get k\u2081 (insert k\u2082 v\u2082 dm) = get k\u2081 dm := sorry\n\nlemma insert_get_same [decidable_eq K] [\u2200 k, inhabited (V k)] (k : K) (dm : dmap K V) :\n  insert k (get k dm) dm = dm := sorry\n\nlemma insert_insert_flip [decidable_eq K] [\u2200 k, inhabited (V k)] {k\u2081 k\u2082 : K} (v\u2081 : V k\u2081) (v\u2082 : V k\u2082) (dm : dmap K V) :\n  k\u2081 \u2260 k\u2082 \u2192 insert k\u2081 v\u2081 (insert k\u2082 v\u2082 dm) = insert k\u2082 v\u2082 (insert k\u2081 v\u2081 dm) := sorry\n\nlemma insert_insert_same [decidable_eq K] (k : K) (v\u2081 v\u2082 : V k) (dm : dmap K V) :\n  insert k v\u2081 (insert k v\u2082 dm) = insert k v\u2081 dm := sorry\n\nlemma nodup_insert [decidable_eq K] [has_lt K] [decidable_rel (@has_lt.lt K _)] {k : K} {v : V k} {ks : list K} {dm : dmap K V} :\n  nodup (dmap.keys dm ++ (k :: ks)) \u2192 nodup (dmap.keys (dmap.insert k v dm) ++ ks) := sorry\n\nlemma not_mem_of_insert [decidable_eq K] [has_lt K] [decidable_rel (@has_lt.lt K _)] {k\u2080 k : K} {v : V k} {ks : list K} {dm : dmap K V} :\n  k\u2080 \u2209 (dmap.keys dm ++ (k :: ks)) \u2192 k\u2080 \u2209 (dmap.keys (dmap.insert k v dm) ++ ks) := sorry\n\nlemma not_mem_of_insert_symm [decidable_eq K] [has_lt K] [decidable_rel (@has_lt.lt K _)] {k\u2080 k : K} {v : V k} {ks : list K} {dm : dmap K V} :\n  k\u2080 \u2209 (dmap.keys (dmap.insert k v dm) ++ ks) \u2192 k\u2080 \u2209 (dmap.keys dm ++ (k :: ks)) := sorry\n\nend dmap\n", "meta": {"author": "dselsam", "repo": "library_dev_extras", "sha": "302d631dfc40c6d69f921fd7bacf017e11e6e56c", "save_path": "github-repos/lean/dselsam-library_dev_extras", "path": "github-repos/lean/dselsam-library_dev_extras/library_dev_extras-302d631dfc40c6d69f921fd7bacf017e11e6e56c/src/library_dev_extras/dmap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3802330549315986}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport algebra.algebra.operations\nimport algebra.ring.equiv\nimport data.nat.choose.sum\nimport ring_theory.coprime.lemmas\nimport ring_theory.ideal.quotient\nimport ring_theory.non_zero_divisors\n/-!\n# More operations on modules and ideals\n-/\nuniverses u v w x\n\nopen_locale big_operators pointwise\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v}\n\nsection comm_semiring\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\nopen_locale pointwise\n\ninstance has_scalar' : has_scalar (ideal R) (submodule R M) :=\n\u27e8\u03bb I N, \u2a06 r : I, (r : R) \u2022 N\u27e9\n\n/-- `N.annihilator` is the ideal of all elements `r : R` such that `r \u2022 N = 0`. -/\ndef annihilator (N : submodule R M) : ideal R :=\n(linear_map.lsmul R N).ker\n\nvariables {I J : ideal R} {N P : submodule R M}\n\ntheorem mem_annihilator {r} : r \u2208 N.annihilator \u2194 \u2200 n \u2208 N, r \u2022 n = (0:M) :=\n\u27e8\u03bb hr n hn, congr_arg subtype.val (linear_map.ext_iff.1 (linear_map.mem_ker.1 hr) \u27e8n, hn\u27e9),\n\u03bb h, linear_map.mem_ker.2 $ linear_map.ext $ \u03bb n, subtype.eq $ h n.1 n.2\u27e9\n\ntheorem mem_annihilator' {r} : r \u2208 N.annihilator \u2194 N \u2264 comap (r \u2022 linear_map.id) \u22a5 :=\nmem_annihilator.trans \u27e8\u03bb H n hn, (mem_bot R).2 $ H n hn, \u03bb H n hn, (mem_bot R).1 $ H hn\u27e9\n\nlemma mem_annihilator_span (s : set M) (r : R) :\n  r \u2208 (submodule.span R s).annihilator \u2194 \u2200 n : s, r \u2022 (n : M) = 0 :=\nbegin\n  rw submodule.mem_annihilator,\n  split,\n  { intros h n, exact h _ (submodule.subset_span n.prop) },\n  { intros h n hn,\n    apply submodule.span_induction hn,\n    { intros x hx, exact h \u27e8x, hx\u27e9 },\n    { exact smul_zero _ },\n    { intros x y hx hy, rw [smul_add, hx, hy, zero_add] },\n    { intros a x hx, rw [smul_comm, hx, smul_zero] } }\nend\n\nlemma mem_annihilator_span_singleton (g : M) (r : R) :\n  r \u2208 (submodule.span R ({g} : set M)).annihilator \u2194 r \u2022 g = 0 :=\nby simp [mem_annihilator_span]\n\ntheorem annihilator_bot : (\u22a5 : submodule R M).annihilator = \u22a4 :=\n(ideal.eq_top_iff_one _).2 $ mem_annihilator'.2 bot_le\n\ntheorem annihilator_eq_top_iff : N.annihilator = \u22a4 \u2194 N = \u22a5 :=\n\u27e8\u03bb H, eq_bot_iff.2 $ \u03bb (n:M) hn, (mem_bot R).2 $\n  one_smul R n \u25b8 mem_annihilator.1 ((ideal.eq_top_iff_one _).1 H) n hn,\n  \u03bb H, H.symm \u25b8 annihilator_bot\u27e9\n\ntheorem annihilator_mono (h : N \u2264 P) : P.annihilator \u2264 N.annihilator :=\n\u03bb r hrp, mem_annihilator.2 $ \u03bb n hn, mem_annihilator.1 hrp n $ h hn\n\ntheorem annihilator_supr (\u03b9 : Sort w) (f : \u03b9 \u2192 submodule R M) :\n  (annihilator \u2a06 i, f i) = \u2a05 i, annihilator (f i) :=\nle_antisymm (le_infi $ \u03bb i, annihilator_mono $ le_supr _ _)\n(\u03bb r H, mem_annihilator'.2 $ supr_le $ \u03bb i,\n  have _ := (mem_infi _).1 H i, mem_annihilator'.1 this)\n\ntheorem smul_mem_smul {r} {n} (hr : r \u2208 I) (hn : n \u2208 N) : r \u2022 n \u2208 I \u2022 N :=\n(le_supr _ \u27e8r, hr\u27e9 : _ \u2264 I \u2022 N) \u27e8n, hn, rfl\u27e9\n\ntheorem smul_le {P : submodule R M} : I \u2022 N \u2264 P \u2194 \u2200 (r \u2208 I) (n \u2208 N), r \u2022 n \u2208 P :=\n\u27e8\u03bb H r hr n hn, H $ smul_mem_smul hr hn,\n\u03bb H, supr_le $ \u03bb r, map_le_iff_le_comap.2 $ \u03bb n hn, H r.1 r.2 n hn\u27e9\n\n@[elab_as_eliminator]\ntheorem smul_induction_on {p : M \u2192 Prop} {x} (H : x \u2208 I \u2022 N)\n  (Hb : \u2200 (r \u2208 I) (n \u2208 N), p (r \u2022 n))\n  (H1 : \u2200 x y, p x \u2192 p y \u2192 p (x + y)) : p x :=\nbegin\n  have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem,\n  refine submodule.supr_induction _ H _ H0 H1,\n  rintros \u27e8i, hi\u27e9 m \u27e8j, hj, (rfl : i \u2022 _ = m) \u27e9,\n  exact Hb _ hi _ hj,\nend\n\ntheorem mem_smul_span_singleton {I : ideal R} {m : M} {x : M} :\n  x \u2208 I \u2022 span R ({m} : set M) \u2194 \u2203 y \u2208 I, y \u2022 m = x :=\n\u27e8\u03bb hx, smul_induction_on hx\n  (\u03bb r hri n hnm,\n    let \u27e8s, hs\u27e9 := mem_span_singleton.1 hnm in \u27e8r * s, I.mul_mem_right _ hri, hs \u25b8 mul_smul r s m\u27e9)\n  (\u03bb m1 m2 \u27e8y1, hyi1, hy1\u27e9 \u27e8y2, hyi2, hy2\u27e9,\n    \u27e8y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]\u27e9),\n\u03bb \u27e8y, hyi, hy\u27e9, hy \u25b8 smul_mem_smul hyi (subset_span $ set.mem_singleton m)\u27e9\n\ntheorem smul_le_right : I \u2022 N \u2264 N :=\nsmul_le.2 $ \u03bb r hr n, N.smul_mem r\n\ntheorem smul_mono (hij : I \u2264 J) (hnp : N \u2264 P) : I \u2022 N \u2264 J \u2022 P :=\nsmul_le.2 $ \u03bb r hr n hn, smul_mem_smul (hij hr) (hnp hn)\n\ntheorem smul_mono_left (h : I \u2264 J) : I \u2022 N \u2264 J \u2022 N :=\nsmul_mono h (le_refl N)\n\ntheorem smul_mono_right (h : N \u2264 P) : I \u2022 N \u2264 I \u2022 P :=\nsmul_mono (le_refl I) h\n\nlemma map_le_smul_top (I : ideal R) (f : R \u2192\u2097[R] M) :\n  submodule.map f I \u2264 I \u2022 (\u22a4 : submodule R M) :=\nbegin\n  rintros _ \u27e8y, hy, rfl\u27e9,\n  rw [\u2190 mul_one y, \u2190 smul_eq_mul, f.map_smul],\n  exact smul_mem_smul hy mem_top\nend\n\n@[simp] theorem annihilator_smul (N : submodule R M) : annihilator N \u2022 N = \u22a5 :=\neq_bot_iff.2 (smul_le.2 (\u03bb r, mem_annihilator.1))\n\n@[simp] theorem annihilator_mul (I : ideal R) : annihilator I * I = \u22a5 :=\nannihilator_smul I\n\n@[simp] theorem mul_annihilator (I : ideal R) : I * annihilator I = \u22a5 :=\nby rw [mul_comm, annihilator_mul]\n\nvariables (I J N P)\n@[simp] theorem smul_bot : I \u2022 (\u22a5 : submodule R M) = \u22a5 :=\neq_bot_iff.2 $ smul_le.2 $ \u03bb r hri s hsb,\n(submodule.mem_bot R).2 $ ((submodule.mem_bot R).1 hsb).symm \u25b8 smul_zero r\n\n@[simp] theorem bot_smul : (\u22a5 : ideal R) \u2022 N = \u22a5 :=\neq_bot_iff.2 $ smul_le.2 $ \u03bb r hrb s hsi,\n(submodule.mem_bot R).2 $ ((submodule.mem_bot R).1 hrb).symm \u25b8 zero_smul _ s\n\n@[simp] theorem top_smul : (\u22a4 : ideal R) \u2022 N = N :=\nle_antisymm smul_le_right $ \u03bb r hri, one_smul R r \u25b8 smul_mem_smul mem_top hri\n\ntheorem smul_sup : I \u2022 (N \u2294 P) = I \u2022 N \u2294 I \u2022 P :=\nle_antisymm (smul_le.2 $ \u03bb r hri m hmnp, let \u27e8n, hn, p, hp, hnpm\u27e9 := mem_sup.1 hmnp in\n  mem_sup.2 \u27e8_, smul_mem_smul hri hn, _, smul_mem_smul hri hp, hnpm \u25b8 (smul_add _ _ _).symm\u27e9)\n(sup_le (smul_mono_right le_sup_left)\n  (smul_mono_right le_sup_right))\n\ntheorem sup_smul : (I \u2294 J) \u2022 N = I \u2022 N \u2294 J \u2022 N :=\nle_antisymm (smul_le.2 $ \u03bb r hrij n hn, let \u27e8ri, hri, rj, hrj, hrijr\u27e9 := mem_sup.1 hrij in\n  mem_sup.2 \u27e8_, smul_mem_smul hri hn, _, smul_mem_smul hrj hn, hrijr \u25b8 (add_smul _ _ _).symm\u27e9)\n(sup_le (smul_mono_left le_sup_left)\n  (smul_mono_left le_sup_right))\n\nprotected theorem smul_assoc : (I \u2022 J) \u2022 N = I \u2022 (J \u2022 N) :=\nle_antisymm (smul_le.2 $ \u03bb rs hrsij t htn,\n  smul_induction_on hrsij\n  (\u03bb r hr s hs,\n    (@smul_eq_mul R _ r s).symm \u25b8 smul_smul r s t \u25b8 smul_mem_smul hr (smul_mem_smul hs htn))\n  (\u03bb x y, (add_smul x y t).symm \u25b8 submodule.add_mem _))\n(smul_le.2 $ \u03bb r hr sn hsn, suffices J \u2022 N \u2264 submodule.comap (r \u2022 linear_map.id) ((I \u2022 J) \u2022 N),\n  from this hsn,\nsmul_le.2 $ \u03bb s hs n hn, show r \u2022 (s \u2022 n) \u2208 (I \u2022 J) \u2022 N,\n  from mul_smul r s n \u25b8 smul_mem_smul (smul_mem_smul hr hs) hn)\n\nvariables (S : set R) (T : set M)\n\ntheorem span_smul_span : (ideal.span S) \u2022 (span R T) =\n  span R (\u22c3 (s \u2208 S) (t \u2208 T), {s \u2022 t}) :=\nle_antisymm (smul_le.2 $ \u03bb r hrS n hnT, span_induction hrS\n  (\u03bb r hrS, span_induction hnT\n    (\u03bb n hnT, subset_span $ set.mem_bUnion hrS $\n      set.mem_bUnion hnT $ set.mem_singleton _)\n    ((smul_zero r : r \u2022 0 = (0:M)).symm \u25b8 submodule.zero_mem _)\n    (\u03bb x y, (smul_add r x y).symm \u25b8 submodule.add_mem _)\n    (\u03bb c m, by rw [smul_smul, mul_comm, mul_smul]; exact submodule.smul_mem _ _))\n  ((zero_smul R n).symm \u25b8 submodule.zero_mem _)\n  (\u03bb r s, (add_smul r s n).symm \u25b8 submodule.add_mem _)\n  (\u03bb c r, by rw [smul_eq_mul, mul_smul]; exact submodule.smul_mem _ _)) $\nspan_le.2 $ set.Union\u2082_subset $ \u03bb r hrS, set.Union\u2082_subset $ \u03bb n hnT, set.singleton_subset_iff.2 $\nsmul_mem_smul (subset_span hrS) (subset_span hnT)\n\nlemma union_eq_smul_set (r : R) (T : set M) :\n  (\u22c3 (t : M) (x : t \u2208 T), {r \u2022 t}) = r \u2022 T := by tidy\n\nlemma ideal_span_singleton_smul (r : R) (N : submodule R M) :\n  (ideal.span {r} : ideal R) \u2022 N = r \u2022 N :=\nbegin\n  have : span R (\u22c3 (t : M) (x : t \u2208 N), {r \u2022 t}) = r \u2022 N,\n  { convert span_eq _, exact union_eq_smul_set r (N : set M) },\n  conv_lhs { rw [\u2190 span_eq N, span_smul_span] },\n  simpa\nend\n\nlemma span_smul_eq (r : R) (s : set M) :\n  span R (r \u2022 s) = r \u2022 span R s :=\nbegin\n  rw [\u2190 ideal_span_singleton_smul, span_smul_span],\n  congr,\n  simpa using (union_eq_smul_set r s).symm\nend\n\nlemma mem_of_span_top_of_smul_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M) (H : \u2200 r : s, (r : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  suffices : (\u22a4 : ideal R) \u2022 (span R ({x} : set M)) \u2264 M',\n  { rw top_smul at this, exact this (subset_span (set.mem_singleton x)) },\n  rw [\u2190 hs, span_smul_span, span_le],\n  simpa using H\nend\n\n/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a\nsubmodule `M'` of `x`, we only need to show that `r ^ n \u2022 x \u2208 M'` for some `n` for each `r : s`. -/\nlemma mem_of_span_eq_top_of_smul_pow_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M)\n  (H : \u2200 r : s, \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  obtain \u27e8s', hs\u2081, hs\u2082\u27e9 := (ideal.span_eq_top_iff_finite _).mp hs,\n  replace H : \u2200 r : s', \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M' := \u03bb r, H \u27e8_, hs\u2081 r.prop\u27e9,\n  choose n\u2081 n\u2082 using H,\n  let N := s'.attach.sup n\u2081,\n  have hs' := ideal.span_pow_eq_top (s' : set R) hs\u2082 N,\n  apply M'.mem_of_span_top_of_smul_mem _ hs',\n  rintro \u27e8_, r, hr, rfl\u27e9,\n  convert M'.smul_mem (r ^ (N - n\u2081 \u27e8r, hr\u27e9)) (n\u2082 \u27e8r, hr\u27e9) using 1,\n  simp only [subtype.coe_mk, smul_smul, \u2190 pow_add],\n  rw tsub_add_cancel_of_le (finset.le_sup (s'.mem_attach _) : n\u2081 \u27e8r, hr\u27e9 \u2264 N),\nend\n\nvariables {M' : Type w} [add_comm_monoid M'] [module R M']\n\ntheorem map_smul'' (f : M \u2192\u2097[R] M') : (I \u2022 N).map f = I \u2022 N.map f :=\nle_antisymm (map_le_iff_le_comap.2 $ smul_le.2 $ \u03bb r hr n hn, show f (r \u2022 n) \u2208 I \u2022 N.map f,\n    from (f.map_smul r n).symm \u25b8 smul_mem_smul hr (mem_map_of_mem hn)) $\nsmul_le.2 $ \u03bb r hr n hn, let \u27e8p, hp, hfp\u27e9 := mem_map.1 hn in\nhfp \u25b8 f.map_smul r p \u25b8 mem_map_of_mem (smul_mem_smul hr hp)\n\nvariables {I}\n\nlemma mem_smul_span {s : set M} {x : M} :\n  x \u2208 I \u2022 submodule.span R s \u2194 x \u2208 submodule.span R (\u22c3 (a \u2208 I) (b \u2208 s), ({a \u2022 b} : set M)) :=\nby rw [\u2190 I.span_eq, submodule.span_smul_span, I.span_eq]; refl\n\nvariables (I)\n\n/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,\nthen we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/\nlemma exists_sum_of_mem_ideal_smul_span {\u03b9 : Type*} (s : set \u03b9) (f : \u03b9 \u2192 M) (x : M)\n  (hx : x \u2208 I \u2022 span R (f '' s)) :\n  \u2203 (a : s \u2192\u2080 R) (ha : \u2200 i, a i \u2208 I), a.sum (\u03bb i c, c \u2022 f i) = x :=\nbegin\n  refine span_induction (mem_smul_span.mp hx) _ _ _ _,\n  { simp only [set.mem_Union, set.mem_range, set.mem_singleton_iff],\n    rintros x \u27e8y, hy, x, \u27e8i, hi, rfl\u27e9, rfl\u27e9,\n    refine \u27e8finsupp.single \u27e8i, hi\u27e9 y, \u03bb j, _, _\u27e9,\n    { letI := classical.dec_eq s,\n      rw finsupp.single_apply, split_ifs, { assumption }, { exact I.zero_mem } },\n    refine @finsupp.sum_single_index s R M _ _ \u27e8i, hi\u27e9 _ (\u03bb i y, y \u2022 f i) _,\n    simp },\n  { exact \u27e80, \u03bb i, I.zero_mem, finsupp.sum_zero_index\u27e9 },\n  { rintros x y \u27e8ax, hax, rfl\u27e9 \u27e8ay, hay, rfl\u27e9,\n    refine \u27e8ax + ay, \u03bb i, I.add_mem (hax i) (hay i), finsupp.sum_add_index _ _\u27e9;\n      intros; simp only [zero_smul, add_smul] },\n  { rintros c x \u27e8a, ha, rfl\u27e9,\n    refine \u27e8c \u2022 a, \u03bb i, I.mul_mem_left c (ha i), _\u27e9,\n    rw [finsupp.sum_smul_index, finsupp.smul_sum];\n      intros; simp only [zero_smul, mul_smul] },\nend\n\n@[simp] lemma smul_comap_le_comap_smul (f : M \u2192\u2097[R] M') (S : submodule R M') (I : ideal R) :\n  I \u2022 S.comap f \u2264 (I \u2022 S).comap f :=\nbegin\n  refine (submodule.smul_le.mpr (\u03bb r hr x hx, _)),\n  rw [submodule.mem_comap] at \u22a2 hx,\n  rw f.map_smul,\n  exact submodule.smul_mem_smul hr hx\nend\n\nend comm_semiring\n\nsection comm_ring\n\nvariables [comm_ring R] [add_comm_group M] [module R M]\nvariables {N N\u2081 N\u2082 P P\u2081 P\u2082 : submodule R M}\n\n/-- `N.colon P` is the ideal of all elements `r : R` such that `r \u2022 P \u2286 N`. -/\ndef colon (N P : submodule R M) : ideal R :=\nannihilator (P.map N.mkq)\n\ntheorem mem_colon {r} : r \u2208 N.colon P \u2194 \u2200 p \u2208 P, r \u2022 p \u2208 N :=\nmem_annihilator.trans \u27e8\u03bb H p hp, (quotient.mk_eq_zero N).1 (H (quotient.mk p) (mem_map_of_mem hp)),\n\u03bb H m \u27e8p, hp, hpm\u27e9, hpm \u25b8 (N.mkq).map_smul r p \u25b8 (quotient.mk_eq_zero N).2 $ H p hp\u27e9\n\ntheorem mem_colon' {r} : r \u2208 N.colon P \u2194 P \u2264 comap (r \u2022 linear_map.id) N :=\nmem_colon\n\ntheorem colon_mono (hn : N\u2081 \u2264 N\u2082) (hp : P\u2081 \u2264 P\u2082) : N\u2081.colon P\u2082 \u2264 N\u2082.colon P\u2081 :=\n\u03bb r hrnp, mem_colon.2 $ \u03bb p\u2081 hp\u2081, hn $ mem_colon.1 hrnp p\u2081 $ hp hp\u2081\n\ntheorem infi_colon_supr (\u03b9\u2081 : Sort w) (f : \u03b9\u2081 \u2192 submodule R M)\n  (\u03b9\u2082 : Sort x) (g : \u03b9\u2082 \u2192 submodule R M) :\n  (\u2a05 i, f i).colon (\u2a06 j, g j) = \u2a05 i j, (f i).colon (g j) :=\nle_antisymm (le_infi $ \u03bb i, le_infi $ \u03bb j, colon_mono (infi_le _ _) (le_supr _ _))\n(\u03bb r H, mem_colon'.2 $ supr_le $ \u03bb j, map_le_iff_le_comap.1 $ le_infi $ \u03bb i,\n  map_le_iff_le_comap.2 $ mem_colon'.1 $ have _ := ((mem_infi _).1 H i),\n  have _ := ((mem_infi _).1 this j), this)\n\nend comm_ring\n\nend submodule\n\nnamespace ideal\n\nsection mul_and_radical\nvariables {R : Type u} {\u03b9 : Type*} [comm_semiring R]\nvariables {I J K L : ideal R}\n\ninstance : has_mul (ideal R) := \u27e8(\u2022)\u27e9\n\n@[simp] lemma add_eq_sup : I + J = I \u2294 J := rfl\n@[simp] lemma zero_eq_bot : (0 : ideal R) = \u22a5 := rfl\n@[simp] lemma one_eq_top : (1 : ideal R) = \u22a4 :=\nby erw [submodule.one_eq_range, linear_map.range_id]\n\ntheorem mul_mem_mul {r s} (hr : r \u2208 I) (hs : s \u2208 J) : r * s \u2208 I * J :=\nsubmodule.smul_mem_smul hr hs\n\ntheorem mul_mem_mul_rev {r s} (hr : r \u2208 I) (hs : s \u2208 J) : s * r \u2208 I * J :=\nmul_comm r s \u25b8 mul_mem_mul hr hs\n\nlemma pow_mem_pow {x : R} (hx : x \u2208 I) (n : \u2115) : x ^ n \u2208 I ^ n :=\nbegin\n  induction n with n ih, { simp only [pow_zero, ideal.one_eq_top], },\n  simpa only [pow_succ] using mul_mem_mul hx ih,\nend\n\ntheorem mul_le : I * J \u2264 K \u2194 \u2200 (r \u2208 I) (s \u2208 J), r * s \u2208 K :=\nsubmodule.smul_le\n\nlemma mul_le_left : I * J \u2264 J :=\nideal.mul_le.2 (\u03bb r hr s, J.mul_mem_left _)\n\nlemma mul_le_right : I * J \u2264 I :=\nideal.mul_le.2 (\u03bb r hr s hs, I.mul_mem_right _ hr)\n\n@[simp] lemma sup_mul_right_self : I \u2294 (I * J) = I :=\nsup_eq_left.2 ideal.mul_le_right\n\n@[simp] lemma sup_mul_left_self : I \u2294 (J * I) = I :=\nsup_eq_left.2 ideal.mul_le_left\n\n@[simp] lemma mul_right_self_sup : (I * J) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_right\n\n@[simp] lemma mul_left_self_sup : (J * I) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_left\n\nvariables (I J K)\nprotected theorem mul_comm : I * J = J * I :=\nle_antisymm (mul_le.2 $ \u03bb r hrI s hsJ, mul_mem_mul_rev hsJ hrI)\n  (mul_le.2 $ \u03bb r hrJ s hsI, mul_mem_mul_rev hsI hrJ)\n\nprotected theorem mul_assoc : (I * J) * K = I * (J * K) :=\nsubmodule.smul_assoc I J K\n\ntheorem span_mul_span (S T : set R) : span S * span T =\n  span \u22c3 (s \u2208 S) (t \u2208 T), {s * t} :=\nsubmodule.span_smul_span S T\nvariables {I J K}\n\nlemma span_mul_span' (S T : set R) : span S * span T = span (S*T) :=\nby { unfold span, rw submodule.span_mul_span, }\n\nlemma span_singleton_mul_span_singleton (r s : R) :\n  span {r} * span {s} = (span {r * s} : ideal R) :=\nby { unfold span, rw [submodule.span_mul_span, set.singleton_mul_singleton], }\n\nlemma span_singleton_pow (s : R) (n : \u2115):\n  span {s} ^ n = (span {s ^ n} : ideal R) :=\nbegin\n  induction n with n ih, { simp [set.singleton_one], },\n  simp only [pow_succ, ih, span_singleton_mul_span_singleton],\nend\n\nlemma mem_mul_span_singleton {x y : R} {I : ideal R} :\n  x \u2208 I * span {y} \u2194 \u2203 z \u2208 I, z * y = x :=\nsubmodule.mem_smul_span_singleton\n\nlemma mem_span_singleton_mul {x y : R} {I : ideal R} :\n  x \u2208 span {y} * I \u2194 \u2203 z \u2208 I, y * z = x :=\nby simp only [mul_comm, mem_mul_span_singleton]\n\nlemma le_span_singleton_mul_iff {x : R} {I J : ideal R} :\n  I \u2264 span {x} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI :=\nshow (\u2200 {zI} (hzI : zI \u2208 I), zI \u2208 span {x} * J) \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI,\nby simp only [mem_span_singleton_mul]\n\nlemma span_singleton_mul_le_iff {x : R} {I J : ideal R} :\n  span {x} * I \u2264 J \u2194 \u2200 z \u2208 I, x * z \u2208 J :=\nbegin\n  simp only [mul_le, mem_span_singleton_mul, mem_span_singleton],\n  split,\n  { intros h zI hzI,\n    exact h x (dvd_refl x) zI hzI },\n  { rintros h _ \u27e8z, rfl\u27e9 zI hzI,\n    rw [mul_comm x z, mul_assoc],\n    exact J.mul_mem_left _ (h zI hzI) },\nend\n\nlemma span_singleton_mul_le_span_singleton_mul {x y : R} {I J : ideal R} :\n  span {x} * I \u2264 span {y} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ :=\nby simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm]\n\nlemma eq_span_singleton_mul {x : R} (I J : ideal R) :\n  I = span {x} * J \u2194 ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI) \u2227 (\u2200 z \u2208 J, x * z \u2208 I)) :=\nby simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff]\n\nlemma span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : ideal R) :\n  span {x} * I = span {y} * J \u2194\n    ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ) \u2227\n     (\u2200 zJ \u2208 J, \u2203 zI \u2208 I, x * zI = y * zJ)) :=\nby simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]\n\nlemma prod_span {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 set R) :\n  (\u220f i in s, ideal.span (I i)) = ideal.span (\u220f i in s, I i) :=\nsubmodule.prod_span s I\n\nlemma prod_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R) :\n  (\u220f i in s, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nsubmodule.prod_span_singleton s I\n\nlemma finset_inf_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R)\n  (hI : set.pairwise \u2191s (is_coprime on I)) :\n  (s.inf $ \u03bb i, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nbegin\n  ext x,\n  simp only [submodule.mem_finset_inf, ideal.mem_span_singleton],\n  exact \u27e8finset.prod_dvd_of_coprime hI,\n    \u03bb h i hi, (finset.dvd_prod_of_mem _ hi).trans h\u27e9\nend\n\nlemma infi_span_singleton {\u03b9 : Type*} [fintype \u03b9] (I : \u03b9 \u2192 R)\n  (hI : \u2200 i j (hij : i \u2260 j), is_coprime (I i) (I j)):\n  (\u2a05 i, ideal.span ({I i} : set R)) = ideal.span {\u220f i, I i} :=\nbegin\n  rw [\u2190 finset.inf_univ_eq_infi, finset_inf_span_singleton],\n  rwa [finset.coe_univ, set.pairwise_univ]\nend\n\ntheorem mul_le_inf : I * J \u2264 I \u2293 J :=\nmul_le.2 $ \u03bb r hri s hsj, \u27e8I.mul_mem_right s hri, J.mul_mem_left r hsj\u27e9\n\ntheorem multiset_prod_le_inf {s : multiset (ideal R)} :\n  s.prod \u2264 s.inf :=\nbegin\n  classical, refine s.induction_on _ _,\n  { rw [multiset.inf_zero], exact le_top },\n  intros a s ih,\n  rw [multiset.prod_cons, multiset.inf_cons],\n  exact le_trans mul_le_inf (inf_le_inf le_rfl ih)\nend\n\ntheorem prod_le_inf {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} : s.prod f \u2264 s.inf f :=\nmultiset_prod_le_inf\n\ntheorem mul_eq_inf_of_coprime (h : I \u2294 J = \u22a4) : I * J = I \u2293 J :=\nle_antisymm mul_le_inf $ \u03bb r \u27e8hri, hrj\u27e9,\nlet \u27e8s, hsi, t, htj, hst\u27e9 := submodule.mem_sup.1 ((eq_top_iff_one _).1 h) in\nmul_one r \u25b8 hst \u25b8 (mul_add r s t).symm \u25b8 ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj)\n  (mul_mem_mul hri htj)\n\nvariables (I)\n@[simp] theorem mul_bot : I * \u22a5 = \u22a5 :=\nsubmodule.smul_bot I\n\n@[simp] theorem bot_mul : \u22a5 * I = \u22a5 :=\nsubmodule.bot_smul I\n\n@[simp] theorem mul_top : I * \u22a4 = I :=\nideal.mul_comm \u22a4 I \u25b8 submodule.top_smul I\n\n@[simp] theorem top_mul : \u22a4 * I = I :=\nsubmodule.top_smul I\nvariables {I}\n\ntheorem mul_mono (hik : I \u2264 K) (hjl : J \u2264 L) : I * J \u2264 K * L :=\nsubmodule.smul_mono hik hjl\n\ntheorem mul_mono_left (h : I \u2264 J) : I * K \u2264 J * K :=\nsubmodule.smul_mono_left h\n\ntheorem mul_mono_right (h : J \u2264 K) : I * J \u2264 I * K :=\nsubmodule.smul_mono_right h\n\nvariables (I J K)\ntheorem mul_sup : I * (J \u2294 K) = I * J \u2294 I * K :=\nsubmodule.smul_sup I J K\n\ntheorem sup_mul : (I \u2294 J) * K = I * K \u2294 J * K :=\nsubmodule.sup_smul I J K\nvariables {I J K}\n\nlemma pow_le_pow {m n : \u2115} (h : m \u2264 n) :\n  I^n \u2264 I^m :=\nbegin\n  cases nat.exists_eq_add_of_le h with k hk,\n  rw [hk, pow_add],\n  exact le_trans (mul_le_inf) (inf_le_left)\nend\n\nlemma pow_le_self {n : \u2115} (hn : n \u2260 0) : I^n \u2264 I :=\ncalc I^n \u2264 I ^ 1 : pow_le_pow (nat.pos_of_ne_zero hn)\n     ... = I : pow_one _\n\nlemma mul_eq_bot {R : Type*} [comm_ring R] [is_domain R] {I J : ideal R} :\n  I * J = \u22a5 \u2194 I = \u22a5 \u2228 J = \u22a5 :=\n\u27e8\u03bb hij, or_iff_not_imp_left.mpr (\u03bb I_ne_bot, J.eq_bot_iff.mpr (\u03bb j hj,\n  let \u27e8i, hi, ne0\u27e9 := I.ne_bot_iff.mp I_ne_bot in\n    or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0)),\n \u03bb h, by cases h; rw [\u2190 ideal.mul_bot, h, ideal.mul_comm]\u27e9\n\ninstance {R : Type*} [comm_ring R] [is_domain R] : no_zero_divisors (ideal R) :=\n{ eq_zero_or_eq_zero_of_mul_eq_zero := \u03bb I J, mul_eq_bot.1 }\n\n/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/\nlemma prod_eq_bot {R : Type*} [comm_ring R] [is_domain R]\n  {s : multiset (ideal R)} : s.prod = \u22a5 \u2194 \u2203 I \u2208 s, I = \u22a5 :=\nprod_zero_iff_exists_zero\n\n/-- The radical of an ideal `I` consists of the elements `r` such that `r^n \u2208 I` for some `n`. -/\ndef radical (I : ideal R) : ideal R :=\n{ carrier := { r | \u2203 n : \u2115, r ^ n \u2208 I },\n  zero_mem' := \u27e81, (pow_one (0:R)).symm \u25b8 I.zero_mem\u27e9,\n  add_mem' := \u03bb x y \u27e8m, hxmi\u27e9 \u27e8n, hyni\u27e9, \u27e8m + n,\n    (add_pow x y (m + n)).symm \u25b8 I.sum_mem $\n    show \u2200 c \u2208 finset.range (nat.succ (m + n)),\n      x ^ c * y ^ (m + n - c) * (nat.choose (m + n) c) \u2208 I,\n    from \u03bb c hc, or.cases_on (le_total c m)\n      (\u03bb hcm, I.mul_mem_right _ $ I.mul_mem_left _ $ nat.add_comm n m \u25b8\n        (add_tsub_assoc_of_le hcm n).symm \u25b8\n        (pow_add y n (m-c)).symm \u25b8 I.mul_mem_right _ hyni)\n      (\u03bb hmc, I.mul_mem_right _ $ I.mul_mem_right _ $ add_tsub_cancel_of_le hmc \u25b8\n        (pow_add x m (c-m)).symm \u25b8 I.mul_mem_right _ hxmi)\u27e9,\n  smul_mem' := \u03bb r s \u27e8n, hsni\u27e9, \u27e8n, (mul_pow r s n).symm \u25b8 I.mul_mem_left (r^n) hsni\u27e9 }\n\ntheorem le_radical : I \u2264 radical I :=\n\u03bb r hri, \u27e81, (pow_one r).symm \u25b8 hri\u27e9\n\nvariables (R)\ntheorem radical_top : (radical \u22a4 : ideal R) = \u22a4 :=\n(eq_top_iff_one _).2 \u27e80, submodule.mem_top\u27e9\nvariables {R}\n\ntheorem radical_mono (H : I \u2264 J) : radical I \u2264 radical J :=\n\u03bb r \u27e8n, hrni\u27e9, \u27e8n, H hrni\u27e9\n\nvariables (I)\n@[simp] theorem radical_idem : radical (radical I) = radical I :=\nle_antisymm (\u03bb r \u27e8n, k, hrnki\u27e9, \u27e8n * k, (pow_mul r n k).symm \u25b8 hrnki\u27e9) le_radical\nvariables {I}\n\ntheorem radical_le_radical_iff : radical I \u2264 radical J \u2194 I \u2264 radical J :=\n\u27e8\u03bb h, le_trans le_radical h, \u03bb h, radical_idem J \u25b8 radical_mono h\u27e9\n\ntheorem radical_eq_top : radical I = \u22a4 \u2194 I = \u22a4 :=\n\u27e8\u03bb h, (eq_top_iff_one _).2 $ let \u27e8n, hn\u27e9 := (eq_top_iff_one _).1 h in\n  @one_pow R _ n \u25b8 hn, \u03bb h, h.symm \u25b8 radical_top R\u27e9\n\ntheorem is_prime.radical (H : is_prime I) : radical I = I :=\nle_antisymm (\u03bb r \u27e8n, hrni\u27e9, H.mem_of_pow_mem n hrni) le_radical\n\nvariables (I J)\ntheorem radical_sup : radical (I \u2294 J) = radical (radical I \u2294 radical J) :=\nle_antisymm (radical_mono $ sup_le_sup le_radical le_radical) $\n\u03bb r \u27e8n, hrnij\u27e9, let \u27e8s, hs, t, ht, hst\u27e9 := submodule.mem_sup.1 hrnij in\n@radical_idem _ _ (I \u2294 J) \u25b8 \u27e8n, hst \u25b8 ideal.add_mem _\n  (radical_mono le_sup_left hs) (radical_mono le_sup_right ht)\u27e9\n\ntheorem radical_inf : radical (I \u2293 J) = radical I \u2293 radical J :=\nle_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 I.mul_mem_right _ hrm,\n(pow_add r m n).symm \u25b8 J.mul_mem_left _ hrn\u27e9)\n\ntheorem radical_mul : radical (I * J) = radical I \u2293 radical J :=\nle_antisymm (radical_inf I J \u25b8 radical_mono $ @mul_le_inf _ _ I J)\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 mul_mem_mul hrm hrn\u27e9)\nvariables {I J}\n\ntheorem is_prime.radical_le_iff (hj : is_prime J) :\n  radical I \u2264 J \u2194 I \u2264 J :=\n\u27e8le_trans le_radical, \u03bb hij r \u27e8n, hrni\u27e9, hj.mem_of_pow_mem n $ hij hrni\u27e9\n\ntheorem radical_eq_Inf (I : ideal R) :\n  radical I = Inf { J : ideal R | I \u2264 J \u2227 is_prime J } :=\nle_antisymm (le_Inf $ \u03bb J hJ, hJ.2.radical_le_iff.2 hJ.1) $\n\u03bb r hr, classical.by_contradiction $ \u03bb hri,\nlet \u27e8m, (hrm : r \u2209 radical m), him, hm\u27e9 := zorn_nonempty_partial_order\u2080\n  {K : ideal R | r \u2209 radical K}\n  (\u03bb c hc hcc y hyc, \u27e8Sup c, \u03bb \u27e8n, hrnc\u27e9, let \u27e8y, hyc, hrny\u27e9 :=\n      (submodule.mem_Sup_of_directed \u27e8y, hyc\u27e9 hcc.directed_on).1 hrnc in hc hyc \u27e8n, hrny\u27e9,\n    \u03bb z, le_Sup\u27e9) I hri in\nhave \u2200 x \u2209 m, r \u2208 radical (m \u2294 span {x}) := \u03bb x hxm, classical.by_contradiction $ \u03bb hrmx, hxm $\n  hm (m \u2294 span {x}) hrmx le_sup_left \u25b8 (le_sup_right : _ \u2264 m \u2294 span {x})\n    (subset_span $ set.mem_singleton _),\nhave is_prime m, from \u27e8by rintro rfl; rw radical_top at hrm; exact hrm trivial,\n  \u03bb x y hxym, or_iff_not_imp_left.2 $ \u03bb hxm, classical.by_contradiction $ \u03bb hym,\n  let \u27e8n, hrn\u27e9 := this _ hxm,\n      \u27e8p, hpm, q, hq, hpqrn\u27e9 := submodule.mem_sup.1 hrn,\n      \u27e8c, hcxq\u27e9 := mem_span_singleton'.1 hq in\n  let \u27e8k, hrk\u27e9 := this _ hym,\n      \u27e8f, hfm, g, hg, hfgrk\u27e9 := submodule.mem_sup.1 hrk,\n      \u27e8d, hdyg\u27e9 := mem_span_singleton'.1 hg in\n  hrm \u27e8n + k, by rw [pow_add, \u2190 hpqrn, \u2190 hcxq, \u2190 hfgrk, \u2190 hdyg, add_mul, mul_add (c*x),\n                     mul_assoc c x (d*y), mul_left_comm x, \u2190 mul_assoc];\n    refine m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm)\n      (m.mul_mem_left _ hxym))\u27e9\u27e9,\nhrm $ this.radical.symm \u25b8 (Inf_le \u27e8him, this\u27e9 : Inf {J : ideal R | I \u2264 J \u2227 is_prime J} \u2264 m) hr\n\n@[simp] lemma radical_bot_of_is_domain {R : Type u} [comm_ring R] [is_domain R] :\n  radical (\u22a5 : ideal R) = \u22a5 :=\neq_bot_iff.2 (\u03bb x hx, hx.rec_on (\u03bb n hn, pow_eq_zero hn))\n\ninstance : comm_semiring (ideal R) := submodule.comm_semiring\n\nvariables (R)\ntheorem top_pow (n : \u2115) : (\u22a4 ^ n : ideal R) = \u22a4 :=\nnat.rec_on n one_eq_top $ \u03bb n ih, by rw [pow_succ, ih, top_mul]\nvariables {R}\n\nvariables (I)\ntheorem radical_pow (n : \u2115) (H : n > 0) : radical (I^n) = radical I :=\nnat.rec_on n (not.elim dec_trivial) (\u03bb n ih H,\nor.cases_on (lt_or_eq_of_le $ nat.le_of_lt_succ H)\n  (\u03bb H, calc radical (I^(n+1))\n           = radical I \u2293 radical (I^n) : by { rw pow_succ, exact radical_mul _ _ }\n       ... = radical I \u2293 radical I : by rw ih H\n       ... = radical I : inf_idem)\n  (\u03bb H, H \u25b8 (pow_one I).symm \u25b8 rfl)) H\n\ntheorem is_prime.mul_le {I J P : ideal R} (hp : is_prime P) :\n  I * J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hip j hj, let \u27e8i, hi, hip\u27e9 := set.not_subset.1 hip in\n  (hp.mem_or_mem $ h $ mul_mem_mul hi hj).resolve_left hip,\n\u03bb h, or.cases_on h (le_trans $ le_trans mul_le_inf inf_le_left)\n  (le_trans $ le_trans mul_le_inf inf_le_right)\u27e9\n\ntheorem is_prime.inf_le {I J P : ideal R} (hp : is_prime P) :\n  I \u2293 J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, hp.mul_le.1 $ le_trans mul_le_inf h,\n\u03bb h, or.cases_on h (le_trans inf_le_left) (le_trans inf_le_right)\u27e9\n\ntheorem is_prime.multiset_prod_le {s : multiset (ideal R)} {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  s.prod \u2264 P \u2194 \u2203 I \u2208 s, I \u2264 P :=\nsuffices s.prod \u2264 P \u2192 \u2203 I \u2208 s, I \u2264 P,\n  from \u27e8this, \u03bb \u27e8i, his, hip\u27e9, le_trans multiset_prod_le_inf $\n    le_trans (multiset.inf_le his) hip\u27e9,\nbegin\n  classical,\n  obtain \u27e8b, hb\u27e9 : \u2203 b, b \u2208 s := multiset.exists_mem_of_ne_zero hne,\n  obtain \u27e8t, rfl\u27e9 : \u2203 t, s = b ::\u2098 t,\n  from \u27e8s.erase b, (multiset.cons_erase hb).symm\u27e9,\n  refine t.induction_on _ _,\n  { simp only [exists_prop, \u2190multiset.singleton_eq_cons, multiset.prod_singleton,\n      multiset.mem_singleton, exists_eq_left, imp_self] },\n  intros a s ih h,\n  rw [multiset.cons_swap, multiset.prod_cons, hp.mul_le] at h,\n  rw multiset.cons_swap,\n  cases h,\n  { exact \u27e8a, multiset.mem_cons_self a _, h\u27e9 },\n  obtain \u27e8I, hI, ih\u27e9 : \u2203 I \u2208 b ::\u2098 s, I \u2264 P := ih h,\n  exact \u27e8I, multiset.mem_cons_of_mem hI, ih\u27e9\nend\n\ntheorem is_prime.multiset_prod_map_le {s : multiset \u03b9} (f : \u03b9 \u2192 ideal R) {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  (s.map f).prod \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nbegin\n  rw hp.multiset_prod_le (mt multiset.map_eq_zero.mp hne),\n  simp_rw [exists_prop, multiset.mem_map, exists_exists_and_eq_and],\nend\n\ntheorem is_prime.prod_le {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R}\n  (hp : is_prime P) (hne : s.nonempty) :\n  s.prod f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nhp.multiset_prod_map_le f (mt finset.val_eq_zero.mp hne.ne_empty)\n\ntheorem is_prime.inf_le' {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R} (hp : is_prime P)\n  (hsne: s.nonempty) :\n  s.inf f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\n\u27e8\u03bb h, (hp.prod_le hsne).1 $ le_trans prod_le_inf h,\n  \u03bb \u27e8i, his, hip\u27e9, le_trans (finset.inf_le his) hip\u27e9\n\ntheorem subset_union {R : Type u} [comm_ring R] {I J K : ideal R} :\n  (I : set R) \u2286 J \u222a K \u2194 I \u2264 J \u2228 I \u2264 K :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hij s hsi,\n  let \u27e8r, hri, hrj\u27e9 := set.not_subset.1 hij in classical.by_contradiction $ \u03bb hsk,\n  or.cases_on (h $ I.add_mem hri hsi)\n    (\u03bb hj, hrj $ add_sub_cancel r s \u25b8 J.sub_mem hj ((h hsi).resolve_right hsk))\n    (\u03bb hk, hsk $ add_sub_cancel' r s \u25b8 K.sub_mem hk ((h hri).resolve_left hrj)),\n\u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset_union_left J K)\n  (\u03bb h, set.subset.trans h $ set.subset_union_right J K)\u27e9\n\ntheorem subset_union_prime' {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {a b : \u03b9}\n  (hp : \u2200 i \u2208 s, is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192\n  I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i,\n  from \u27e8this, \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_left _ _) (set.subset_union_left _ _)) $\n    \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_right _ _) (set.subset_union_left _ _)) $\n    \u03bb \u27e8i, his, hi\u27e9, by refine (set.subset.trans hi $ set.subset.trans _ $\n        set.subset_union_right _ _);\n      exact set.subset_bUnion_of_mem (finset.mem_coe.2 his)\u27e9,\nbegin\n  generalize hn : s.card = n, intros h,\n  unfreezingI { induction n with n ih generalizing a b s },\n  { clear hp,\n    rw finset.card_eq_zero at hn, subst hn,\n    rw [finset.coe_empty, set.bUnion_empty, set.union_empty, subset_union] at h,\n    simpa only [exists_prop, finset.not_mem_empty, false_and, exists_false, or_false] },\n  classical,\n  replace hn : \u2203 (i : \u03b9) (t : finset \u03b9), i \u2209 t \u2227 insert i t = s \u2227 t.card = n :=\n  finset.card_eq_succ.1 hn,\n  unfreezingI { rcases hn with \u27e8i, t, hit, rfl, hn\u27e9 },\n  replace hp : is_prime (f i) \u2227 \u2200 x \u2208 t, is_prime (f x) := (t.forall_mem_insert _ _).1 hp,\n  by_cases Ht : \u2203 j \u2208 t, f j \u2264 f i,\n  { obtain \u27e8j, hjt, hfji\u27e9 : \u2203 j \u2208 t, f j \u2264 f i := Ht,\n    obtain \u27e8u, hju, rfl\u27e9 : \u2203 u, j \u2209 u \u2227 insert j u = t,\n    { exact \u27e8t.erase j, t.not_mem_erase j, finset.insert_erase hjt\u27e9 },\n    have hp' : \u2200 k \u2208 insert i u, is_prime (f k),\n    { rw finset.forall_mem_insert at hp \u22a2, exact \u27e8hp.1, hp.2.2\u27e9 },\n    have hiu : i \u2209 u := mt finset.mem_insert_of_mem hit,\n    have hn' : (insert i u).card = n,\n    { rwa finset.card_insert_of_not_mem at hn \u22a2, exacts [hiu, hju] },\n    have h' : (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 k \u2208 (\u2191(insert i u) : set \u03b9), f k),\n    { rw finset.coe_insert at h \u22a2, rw finset.coe_insert at h,\n      simp only [set.bUnion_insert] at h \u22a2,\n      rw [\u2190 set.union_assoc \u2191(f i)] at h,\n      erw [set.union_eq_self_of_subset_right hfji] at h,\n      exact h },\n    specialize @ih a b (insert i u) hp' hn' h',\n    refine ih.imp id (or.imp id (exists_imp_exists $ \u03bb k, _)), simp only [exists_prop],\n    exact and.imp (\u03bb hk, finset.insert_subset_insert i (finset.subset_insert j u) hk) id },\n  by_cases Ha : f a \u2264 f i,\n  { have h' : (I : set R) \u2286 f i \u222a f b \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc,\n          set.union_right_comm \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Ha] at h,\n      exact h },\n    specialize @ih i b t hp.2 hn h', right,\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9 },\n    { exact or.inl ih },\n    { exact or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  by_cases Hb : f b \u2264 f i,\n  { have h' : (I : set R) \u2286 f a \u222a f i \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc, set.union_assoc \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Hb] at h,\n      exact h },\n    specialize @ih a i t hp.2 hn h',\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inl ih },\n    { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9) },\n    { exact or.inr (or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9) } },\n  by_cases Hi : I \u2264 f i,\n  { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, Hi\u27e9) },\n  have : \u00acI \u2293 f a \u2293 f b \u2293 t.inf f \u2264 f i,\n  { rcases t.eq_empty_or_nonempty with (rfl | hsne),\n    { rw [finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or_distrib, not_or_distrib],\n      exact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9 },\n    simp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or_distrib],\n    exact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9 },\n  rcases set.not_subset.1 this with \u27e8r, \u27e8\u27e8\u27e8hrI, hra\u27e9, hrb\u27e9, hr\u27e9, hri\u27e9,\n  by_cases HI : (I : set R) \u2286 f a \u222a f b \u222a \u22c3 j \u2208 (\u2191t : set \u03b9), f j,\n  { specialize ih hp.2 hn HI, rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { left, exact ih }, { right, left, exact ih },\n    { right, right, exact \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  exfalso, rcases set.not_subset.1 HI with \u27e8s, hsI, hs\u27e9,\n  rw [finset.coe_insert, set.bUnion_insert] at h,\n  have hsi : s \u2208 f i := ((h hsI).resolve_left (mt or.inl hs)).resolve_right (mt or.inr hs),\n  rcases h (I.add_mem hrI hsI) with \u27e8ha | hb\u27e9 | hi | ht,\n  { exact hs (or.inl $ or.inl $ add_sub_cancel' r s \u25b8 (f a).sub_mem ha hra) },\n  { exact hs (or.inl $ or.inr $ add_sub_cancel' r s \u25b8 (f b).sub_mem hb hrb) },\n  { exact hri (add_sub_cancel r s \u25b8 (f i).sub_mem hi hsi) },\n  { rw set.mem_Union\u2082 at ht, rcases ht with \u27e8j, hjt, hj\u27e9,\n    simp only [finset.inf_eq_infi, set_like.mem_coe, submodule.mem_infi] at hr,\n    exact hs (or.inr $ set.mem_bUnion hjt $ add_sub_cancel' r s \u25b8 (f j).sub_mem hj $ hr j hjt) }\nend\n\n/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/\ntheorem subset_union_prime {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} (a b : \u03b9)\n  (hp : \u2200 i \u2208 s, i \u2260 a \u2192 i \u2260 b \u2192 is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i,\n  from \u27e8\u03bb h, bex_def.2 $ this h, \u03bb \u27e8i, his, hi\u27e9, set.subset.trans hi $ set.subset_bUnion_of_mem $\n    show i \u2208 (\u2191s : set \u03b9), from his\u27e9,\nassume h : (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i),\nbegin\n  classical,\n  by_cases has : a \u2208 s,\n  { unfreezingI { obtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s :=\n      \u27e8s.erase a, finset.not_mem_erase a s, finset.insert_erase has\u27e9 },\n    by_cases hbt : b \u2208 t,\n    { unfreezingI { obtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t :=\n        \u27e8t.erase b, finset.not_mem_erase b t, finset.insert_erase hbt\u27e9 },\n      have hp' : \u2200 i \u2208 u, is_prime (f i),\n      { intros i hiu, refine hp i (finset.mem_insert_of_mem (finset.mem_insert_of_mem hiu)) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, finset.coe_insert, set.bUnion_insert, set.bUnion_insert,\n          \u2190 set.union_assoc, subset_union_prime' hp', bex_def] at h,\n      rwa [finset.exists_mem_insert, finset.exists_mem_insert] },\n    { have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f a : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } },\n  { by_cases hbs : b \u2208 s,\n    { unfreezingI { obtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s :=\n        \u27e8s.erase b, finset.not_mem_erase b s, finset.insert_erase hbs\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f b : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert },\n    cases s.eq_empty_or_nonempty with hse hsne,\n    { substI hse, rw [finset.coe_empty, set.bUnion_empty, set.subset_empty_iff] at h,\n      have : (I : set R) \u2260 \u2205 := set.nonempty.ne_empty (set.nonempty_of_mem I.zero_mem),\n      exact absurd h this },\n    { cases hsne.bex with i his,\n      unfreezingI { obtain \u27e8t, hit, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s :=\n        \u27e8s.erase i, finset.not_mem_erase i s, finset.insert_erase his\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f i : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } }\nend\n\nsection dvd\n\n/-- If `I` divides `J`, then `I` contains `J`.\n\nIn a Dedekind domain, to divide and contain are equivalent, see `ideal.dvd_iff_le`.\n-/\nlemma le_of_dvd {I J : ideal R} : I \u2223 J \u2192 J \u2264 I\n| \u27e8K, h\u27e9 := h.symm \u25b8 le_trans mul_le_inf inf_le_left\n\nlemma is_unit_iff {I : ideal R} :\n  is_unit I \u2194 I = \u22a4 :=\nis_unit_iff_dvd_one.trans ((@one_eq_top R _).symm \u25b8\n \u27e8\u03bb h, eq_top_iff.mpr (ideal.le_of_dvd h), \u03bb h, \u27e8\u22a4, by rw [mul_top, h]\u27e9\u27e9)\n\ninstance unique_units : unique ((ideal R)\u02e3) :=\n{ default := 1,\n  uniq := \u03bb u, units.ext\n    (show (u : ideal R) = 1, by rw [is_unit_iff.mp u.is_unit, one_eq_top]) }\n\nend dvd\n\nend mul_and_radical\n\nsection map_and_comap\n\nvariables {R : Type u} {S : Type v}\n\nsection semiring\nvariables [semiring R] [semiring S]\nvariables (f : R \u2192+* S)\nvariables {I J : ideal R} {K L : ideal S}\n\n/-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than\n  the image itself. -/\ndef map (I : ideal R) : ideal S :=\nspan (f '' I)\n\n/-- `I.comap f` is the preimage of `I` under `f`. -/\ndef comap (I : ideal S) : ideal R :=\n{ carrier := f \u207b\u00b9' I,\n  .. I.comap f.to_semilinear_map }\n\nvariables {f}\ntheorem map_mono (h : I \u2264 J) : map f I \u2264 map f J :=\nspan_mono $ set.image_subset _ h\n\ntheorem mem_map_of_mem (f : R \u2192+* S) {I : ideal R} {x : R} (h : x \u2208 I) : f x \u2208 map f I :=\nsubset_span \u27e8x, h, rfl\u27e9\n\nlemma apply_coe_mem_map (f : R \u2192+* S) (I : ideal R) (x : I) : f x \u2208 I.map f :=\nmem_map_of_mem f x.prop\n\ntheorem map_le_iff_le_comap :\n  map f I \u2264 K \u2194 I \u2264 comap f K :=\nspan_le.trans set.image_subset_iff\n\n@[simp] theorem mem_comap {x} : x \u2208 comap f K \u2194 f x \u2208 K := iff.rfl\n\ntheorem comap_mono (h : K \u2264 L) : comap f K \u2264 comap f L :=\nset.preimage_mono (\u03bb x hx, h hx)\nvariables (f)\n\ntheorem comap_ne_top (hK : K \u2260 \u22a4) : comap f K \u2260 \u22a4 :=\n(ne_top_iff_one _).2 $ by rw [mem_comap, f.map_one];\n  exact (ne_top_iff_one _).1 hK\n\nlemma map_le_comap_of_inv_on (g : S \u2192+* R) (I : ideal R) (hf : set.left_inv_on g f I) :\n  I.map f \u2264 I.comap g :=\nbegin\n  refine ideal.span_le.2 _,\n  rintros x \u27e8x, hx, rfl\u27e9,\n  rw [set_like.mem_coe, mem_comap, hf hx],\n  exact hx,\nend\n\nlemma comap_le_map_of_inv_on (g : S \u2192+* R) (I : ideal S) (hf : set.left_inv_on g f (f \u207b\u00b9' I)) :\n  I.comap f \u2264 I.map g :=\n\u03bb x (hx : f x \u2208 I), hf hx \u25b8 ideal.mem_map_of_mem g hx\n\n/-- The `ideal` version of `set.image_subset_preimage_of_inverse`. -/\nlemma map_le_comap_of_inverse (g : S \u2192+* R) (I : ideal R) (h : function.left_inverse g f) :\n  I.map f \u2264 I.comap g :=\nmap_le_comap_of_inv_on _ _ _ $ h.left_inv_on _\n\n/-- The `ideal` version of `set.preimage_subset_image_of_inverse`. -/\nlemma comap_le_map_of_inverse (g : S \u2192+* R) (I : ideal S) (h : function.left_inverse g f) :\n  I.comap f \u2264 I.map g :=\ncomap_le_map_of_inv_on _ _ _ $ h.left_inv_on _\n\ninstance is_prime.comap [hK : K.is_prime] : (comap f K).is_prime :=\n\u27e8comap_ne_top _ hK.1, \u03bb x y,\n  by simp only [mem_comap, f.map_mul]; apply hK.2\u27e9\n\nvariables (I J K L)\n\ntheorem map_top : map f \u22a4 = \u22a4 :=\n(eq_top_iff_one _).2 $ subset_span \u27e81, trivial, f.map_one\u27e9\n\nvariable (f)\nlemma gc_map_comap : galois_connection (ideal.map f) (ideal.comap f) :=\n\u03bb I J, ideal.map_le_iff_le_comap\n\n@[simp] lemma comap_id : I.comap (ring_hom.id R) = I :=\nideal.ext $ \u03bb _, iff.rfl\n\n@[simp] lemma map_id : I.map (ring_hom.id R) = I :=\n(gc_map_comap (ring_hom.id R)).l_unique galois_connection.id comap_id\n\nlemma comap_comap {T : Type*} [semiring T] {I : ideal T} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl\n\nlemma map_map {T : Type*} [semiring T] {I : ideal R} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.map f).map g = I.map (g.comp f) :=\n((gc_map_comap f).compose (gc_map_comap g)).l_unique\n  (gc_map_comap (g.comp f)) (\u03bb _, comap_comap _ _)\n\nlemma map_span (f : R \u2192+* S) (s : set R) :\n  map f (span s) = span (f '' s) :=\nsymm $ submodule.span_eq_of_le _\n  (\u03bb y \u27e8x, hy, x_eq\u27e9, x_eq \u25b8 mem_map_of_mem f (subset_span hy))\n  (map_le_iff_le_comap.2 $ span_le.2 $ set.image_subset_iff.1 subset_span)\n\nvariables {f I J K L}\n\nlemma map_le_of_le_comap : I \u2264 K.comap f \u2192 I.map f \u2264 K :=\n(gc_map_comap f).l_le\n\nlemma le_comap_of_map_le : I.map f \u2264 K \u2192 I \u2264 K.comap f :=\n(gc_map_comap f).le_u\n\nlemma le_comap_map : I \u2264 (I.map f).comap f :=\n(gc_map_comap f).le_u_l _\n\nlemma map_comap_le : (K.comap f).map f \u2264 K :=\n(gc_map_comap f).l_u_le _\n\n@[simp] lemma comap_top : (\u22a4 : ideal S).comap f = \u22a4 :=\n(gc_map_comap f).u_top\n\n@[simp] lemma comap_eq_top_iff {I : ideal S} : I.comap f = \u22a4 \u2194 I = \u22a4 :=\n\u27e8 \u03bb h, I.eq_top_iff_one.mpr (f.map_one \u25b8 mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)),\n  \u03bb h, by rw [h, comap_top] \u27e9\n\n@[simp] lemma map_bot : (\u22a5 : ideal R).map f = \u22a5 :=\n(gc_map_comap f).l_bot\n\nvariables (f I J K L)\n\n@[simp] lemma map_comap_map : ((I.map f).comap f).map f = I.map f :=\n(gc_map_comap f).l_u_l_eq_l I\n\n@[simp] lemma comap_map_comap : ((K.comap f).map f).comap f = K.comap f :=\n(gc_map_comap f).u_l_u_eq_u K\n\nlemma map_sup : (I \u2294 J).map f = I.map f \u2294 J.map f :=\n(gc_map_comap f).l_sup\n\ntheorem comap_inf : comap f (K \u2293 L) = comap f K \u2293 comap f L := rfl\n\nvariables {\u03b9 : Sort*}\n\nlemma map_supr (K : \u03b9 \u2192 ideal R) : (supr K).map f = \u2a06 i, (K i).map f :=\n(gc_map_comap f).l_supr\n\nlemma comap_infi (K : \u03b9 \u2192 ideal S) : (infi K).comap f = \u2a05 i, (K i).comap f :=\n(gc_map_comap f).u_infi\n\nlemma map_Sup (s : set (ideal R)): (Sup s).map f = \u2a06 I \u2208 s, (I : ideal R).map f :=\n(gc_map_comap f).l_Sup\n\nlemma comap_Inf (s : set (ideal S)): (Inf s).comap f = \u2a05 I \u2208 s, (I : ideal S).comap f :=\n(gc_map_comap f).u_Inf\n\nlemma comap_Inf' (s : set (ideal S)) : (Inf s).comap f = \u2a05 I \u2208 (comap f '' s), I :=\ntrans (comap_Inf f s) (by rw infi_image)\n\ntheorem comap_is_prime [H : is_prime K] : is_prime (comap f K) :=\n\u27e8comap_ne_top f H.ne_top,\n  \u03bb x y h, H.mem_or_mem $ by rwa [mem_comap, ring_hom.map_mul] at h\u27e9\n\nvariables {I J K L}\n\ntheorem map_inf_le : map f (I \u2293 J) \u2264 map f I \u2293 map f J :=\n(gc_map_comap f).monotone_l.map_inf_le _ _\n\ntheorem le_comap_sup : comap f K \u2294 comap f L \u2264 comap f (K \u2294 L) :=\n(gc_map_comap f).monotone_u.le_map_sup _ _\n\n@[simp] lemma smul_top_eq_map {R S : Type*} [comm_semiring R] [comm_semiring S] [algebra R S]\n  (I : ideal R) : I \u2022 (\u22a4 : submodule R S) = (I.map (algebra_map R S)).restrict_scalars R :=\nbegin\n  refine le_antisymm (submodule.smul_le.mpr (\u03bb r hr y _, _) )\n      (\u03bb x hx, submodule.span_induction hx _ _ _ _),\n  { rw algebra.smul_def,\n     exact mul_mem_right _ _ (mem_map_of_mem _ hr) },\n\n  { rintros _ \u27e8x, hx, rfl\u27e9,\n    rw [\u2190 mul_one (algebra_map R S x), \u2190 algebra.smul_def],\n    exact submodule.smul_mem_smul hx submodule.mem_top },\n  { exact submodule.zero_mem _ },\n  { intros x y, exact submodule.add_mem _ },\n  intros a x hx,\n  refine submodule.smul_induction_on hx _ _,\n  { intros r hr s hs,\n    rw smul_comm,\n    exact submodule.smul_mem_smul hr submodule.mem_top },\n  { intros x y hx hy,\n    rw smul_add, exact submodule.add_mem _ hx hy },\nend\n\nsection surjective\nvariables (hf : function.surjective f)\ninclude hf\n\nopen function\n\ntheorem map_comap_of_surjective (I : ideal S) :\n  map f (comap f I) = I :=\nle_antisymm (map_le_iff_le_comap.2 le_rfl)\n(\u03bb s hsi, let \u27e8r, hfrs\u27e9 := hf s in\n  hfrs \u25b8 (mem_map_of_mem f $ show f r \u2208 I, from hfrs.symm \u25b8 hsi))\n\n/-- `map` and `comap` are adjoint, and the composition `map f \u2218 comap f` is the\n  identity -/\ndef gi_map_comap : galois_insertion (map f) (comap f) :=\ngalois_insertion.monotone_intro\n  ((gc_map_comap f).monotone_u)\n  ((gc_map_comap f).monotone_l)\n  (\u03bb _, le_comap_map)\n  (map_comap_of_surjective _ hf)\n\nlemma map_surjective_of_surjective : surjective (map f) :=\n(gi_map_comap f hf).l_surjective\n\nlemma comap_injective_of_surjective : injective (comap f) :=\n(gi_map_comap f hf).u_injective\n\nlemma map_sup_comap_of_surjective (I J : ideal S) : (I.comap f \u2294 J.comap f).map f = I \u2294 J :=\n(gi_map_comap f hf).l_sup_u _ _\n\nlemma map_supr_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a06i, (K i).comap f).map f = supr K :=\n(gi_map_comap f hf).l_supr_u _\n\nlemma map_inf_comap_of_surjective (I J : ideal S) : (I.comap f \u2293 J.comap f).map f = I \u2293 J :=\n(gi_map_comap f hf).l_inf_u _ _\n\nlemma map_infi_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a05i, (K i).comap f).map f = infi K :=\n(gi_map_comap f hf).l_infi_u _\n\ntheorem mem_image_of_mem_map_of_surjective {I : ideal R} {y}\n  (H : y \u2208 map f I) : y \u2208 f '' I :=\nsubmodule.span_induction H (\u03bb _, id) \u27e80, I.zero_mem, f.map_zero\u27e9\n(\u03bb y1 y2 \u27e8x1, hx1i, hxy1\u27e9 \u27e8x2, hx2i, hxy2\u27e9,\n  \u27e8x1 + x2, I.add_mem hx1i hx2i, hxy1 \u25b8 hxy2 \u25b8 f.map_add _ _\u27e9)\n(\u03bb c y \u27e8x, hxi, hxy\u27e9,\n  let \u27e8d, hdc\u27e9 := hf c in \u27e8d * x, I.mul_mem_left _ hxi, hdc \u25b8 hxy \u25b8 f.map_mul _ _\u27e9)\n\nlemma mem_map_iff_of_surjective {I : ideal R} {y} :\n  y \u2208 map f I \u2194 \u2203 x, x \u2208 I \u2227 f x = y :=\n\u27e8\u03bb h, (set.mem_image _ _ _).2 (mem_image_of_mem_map_of_surjective f hf h),\n  \u03bb \u27e8x, hx\u27e9, hx.right \u25b8 (mem_map_of_mem f hx.left)\u27e9\n\nlemma le_map_of_comap_le_of_surjective : comap f K \u2264 I \u2192 K \u2264 map f I :=\n\u03bb h, (map_comap_of_surjective f hf K) \u25b8 map_mono h\n\nend surjective\n\nsection injective\nvariables (hf : function.injective f)\ninclude hf\n\nlemma comap_bot_le_of_injective : comap f \u22a5 \u2264 I :=\nbegin\n  refine le_trans (\u03bb x hx, _) bot_le,\n  rw [mem_comap, submodule.mem_bot, \u2190 ring_hom.map_zero f] at hx,\n  exact eq.symm (hf hx) \u25b8 (submodule.zero_mem \u22a5)\nend\n\nend injective\n\nend semiring\n\nsection ring\nvariables [ring R] [ring S] (f : R \u2192+* S) {I : ideal R}\n\nsection surjective\n\nvariables (hf : function.surjective f)\ninclude hf\n\ntheorem comap_map_of_surjective (I : ideal R) : comap f (map f I) = I \u2294 comap f \u22a5 :=\nle_antisymm (assume r h, let \u27e8s, hsi, hfsr\u27e9 := mem_image_of_mem_map_of_surjective f hf h in\n  submodule.mem_sup.2 \u27e8s, hsi, r - s, (submodule.mem_bot S).2 $ by rw [f.map_sub, hfsr, sub_self],\n  add_sub_cancel'_right s r\u27e9)\n(sup_le (map_le_iff_le_comap.1 le_rfl) (comap_mono bot_le))\n\n\n/-- Correspondence theorem -/\ndef rel_iso_of_surjective : ideal S \u2243o { p : ideal R // comap f \u22a5 \u2264 p } :=\n{ to_fun := \u03bb J, \u27e8comap f J, comap_mono bot_le\u27e9,\n  inv_fun := \u03bb I, map f I.1,\n  left_inv := \u03bb J, map_comap_of_surjective f hf J,\n  right_inv := \u03bb I, subtype.eq $ show comap f (map f I.1) = I.1,\n    from (comap_map_of_surjective f hf I).symm \u25b8 le_antisymm\n      (sup_le le_rfl I.2) le_sup_left,\n  map_rel_iff' := \u03bb I1 I2, \u27e8\u03bb H, map_comap_of_surjective f hf I1 \u25b8\n    map_comap_of_surjective f hf I2 \u25b8 map_mono H, comap_mono\u27e9 }\n\n/-- The map on ideals induced by a surjective map preserves inclusion. -/\ndef order_embedding_of_surjective : ideal S \u21aao ideal R :=\n(rel_iso_of_surjective f hf).to_rel_embedding.trans (subtype.rel_embedding _ _)\n\ntheorem map_eq_top_or_is_maximal_of_surjective {I : ideal R} (H : is_maximal I) :\n  (map f I) = \u22a4 \u2228 is_maximal (map f I) :=\nbegin\n  refine or_iff_not_imp_left.2 (\u03bb ne_top, \u27e8\u27e8\u03bb h, ne_top h, \u03bb J hJ, _\u27e9\u27e9),\n  { refine (rel_iso_of_surjective f hf).injective\n      (subtype.ext_iff.2 (eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm)),\n    { exact (map_le_iff_le_comap).1 (le_of_lt hJ) },\n    { exact \u03bb h, hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm)) } }\nend\n\ntheorem comap_is_maximal_of_surjective {K : ideal S} [H : is_maximal K] : is_maximal (comap f K) :=\nbegin\n  refine \u27e8\u27e8comap_ne_top _ H.1.1, \u03bb J hJ, _\u27e9\u27e9,\n  suffices : map f J = \u22a4,\n  { replace this := congr_arg (comap f) this,\n    rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this,\n    rw eq_top_iff,\n    exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono (bot_le)) (le_of_lt hJ))) },\n  refine H.1.2 (map f J) (lt_of_le_of_ne (le_map_of_comap_le_of_surjective _ hf (le_of_lt hJ))\n    (\u03bb h, ne_of_lt hJ (trans (congr_arg (comap f) h) _))),\n  rw [comap_map_of_surjective _ hf, sup_eq_left],\n  exact le_trans (comap_mono bot_le) (le_of_lt hJ)\nend\n\ntheorem comap_le_comap_iff_of_surjective (I J : ideal S) : comap f I \u2264 comap f J \u2194 I \u2264 J :=\n\u27e8\u03bb h, (map_comap_of_surjective f hf I).symm.le.trans (map_le_of_le_comap h),\n  \u03bb h, le_comap_of_map_le ((map_comap_of_surjective f hf I).le.trans h)\u27e9\n\nend surjective\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f (map f.symm) = I`. -/\n@[simp]\nlemma map_of_equiv (I : ideal R) (f : R \u2243+* S) : (I.map (f : R \u2192+* S)).map (f.symm : S \u2192+* R) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, map_map]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `comap f.symm (comap f) = I`. -/\n@[simp]\nlemma comap_of_equiv (I : ideal R) (f : R \u2243+* S) :\n  (I.comap (f.symm : S \u2192+* R)).comap (f : R \u2192+* S) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, comap_comap]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f I = comap f.symm I`. -/\nlemma map_comap_of_equiv (I : ideal R) (f : R \u2243+* S) : I.map (f : R \u2192+* S) = I.comap f.symm :=\nle_antisymm (le_comap_of_map_le (map_of_equiv I f).le)\n  (le_map_of_comap_le_of_surjective _ f.surjective (comap_of_equiv I f).le)\n\nsection bijective\nvariables (hf : function.bijective f)\ninclude hf\n\n/-- Special case of the correspondence theorem for isomorphic rings -/\ndef rel_iso_of_bijective : ideal S \u2243o ideal R :=\n{ to_fun := comap f,\n  inv_fun := map f,\n  left_inv := (rel_iso_of_surjective f hf.right).left_inv,\n  right_inv := \u03bb J, subtype.ext_iff.1\n    ((rel_iso_of_surjective f hf.right).right_inv \u27e8J, comap_bot_le_of_injective f hf.left\u27e9),\n  map_rel_iff' := (rel_iso_of_surjective f hf.right).map_rel_iff' }\n\nlemma comap_le_iff_le_map {I : ideal R} {K : ideal S} : comap f K \u2264 I \u2194 K \u2264 map f I :=\n\u27e8\u03bb h, le_map_of_comap_le_of_surjective f hf.right h,\n \u03bb h, ((rel_iso_of_bijective f hf).right_inv I) \u25b8 comap_mono h\u27e9\n\ntheorem map.is_maximal {I : ideal R} (H : is_maximal I) : is_maximal (map f I) :=\nby refine or_iff_not_imp_left.1\n  (map_eq_top_or_is_maximal_of_surjective f hf.right H) (\u03bb h, H.1.1 _);\ncalc I = comap f (map f I) : ((rel_iso_of_bijective f hf).right_inv I).symm\n   ... = comap f \u22a4 : by rw h\n   ... = \u22a4 : by rw comap_top\n\nend bijective\n\nlemma ring_equiv.bot_maximal_iff (e : R \u2243+* S) :\n  (\u22a5 : ideal R).is_maximal \u2194 (\u22a5 : ideal S).is_maximal :=\n\u27e8\u03bb h, (@map_bot _ _ _ _ e.to_ring_hom) \u25b8 map.is_maximal e.to_ring_hom e.bijective h,\n  \u03bb h, (@map_bot _ _ _ _ e.symm.to_ring_hom) \u25b8 map.is_maximal e.symm.to_ring_hom e.symm.bijective h\u27e9\n\nend ring\n\nsection comm_ring\n\nvariables [comm_ring R] [comm_ring S]\nvariables (f : R \u2192+* S)\nvariables {I J : ideal R} {K L : ideal S}\n\nlemma mem_quotient_iff_mem (hIJ : I \u2264 J) {x : R} :\n  quotient.mk I x \u2208 J.map (quotient.mk I) \u2194 x \u2208 J :=\nbegin\n  refine iff.trans (mem_map_iff_of_surjective _ quotient.mk_surjective) _,\n  split,\n  { rintros \u27e8x, x_mem, x_eq\u27e9,\n    simpa using J.add_mem (hIJ (quotient.eq.mp x_eq.symm)) x_mem },\n  { intro x_mem,\n    exact \u27e8x, x_mem, rfl\u27e9 }\nend\n\nvariables (I J K L)\n\ntheorem map_mul : map f (I * J) = map f I * map f J :=\nle_antisymm (map_le_iff_le_comap.2 $ mul_le.2 $ \u03bb r hri s hsj,\n  show f (r * s) \u2208 _, by rw f.map_mul;\n  exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))\n(trans_rel_right _ (span_mul_span _ _) $ span_le.2 $\n  set.Union\u2082_subset $ \u03bb i \u27e8r, hri, hfri\u27e9,\n  set.Union\u2082_subset $ \u03bb j \u27e8s, hsj, hfsj\u27e9,\n  set.singleton_subset_iff.2 $ hfri \u25b8 hfsj \u25b8\n  by rw [\u2190 f.map_mul];\n  exact mem_map_of_mem f (mul_mem_mul hri hsj))\n\n/-- The pushforward `ideal.map` as a monoid-with-zero homomorphism. -/\n@[simps]\ndef map_hom : ideal R \u2192*\u2080 ideal S :=\n{ to_fun := map f,\n  map_mul' := \u03bb I J, ideal.map_mul f I J,\n  map_one' := by convert ideal.map_top f; exact one_eq_top,\n  map_zero' := ideal.map_bot }\n\nprotected theorem map_pow (n : \u2115) : map f (I^n) = (map f I)^n :=\nmap_pow (map_hom f) I n\n\ntheorem comap_radical : comap f (radical K) = radical (comap f K) :=\nle_antisymm (\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, show f (r ^ n) \u2208 K,\n  from (f.map_pow r n).symm \u25b8 hfrnk\u27e9)\n(\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, f.map_pow r n \u25b8 hfrnk\u27e9)\n\n@[simp] lemma map_quotient_self :\n  map (quotient.mk I) I = \u22a5 :=\neq_bot_iff.2 $ ideal.map_le_iff_le_comap.2 $ \u03bb x hx,\n(submodule.mem_bot (R \u29f8 I)).2 $ ideal.quotient.eq_zero_iff_mem.2 hx\n\nvariables {I J K L}\n\ntheorem map_radical_le : map f (radical I) \u2264 radical (map f I) :=\nmap_le_iff_le_comap.2 $ \u03bb r \u27e8n, hrni\u27e9, \u27e8n, f.map_pow r n \u25b8 mem_map_of_mem f hrni\u27e9\n\ntheorem le_comap_mul : comap f K * comap f L \u2264 comap f (K * L) :=\nmap_le_iff_le_comap.1 $ (map_mul f (comap f K) (comap f L)).symm \u25b8\nmul_mono (map_le_iff_le_comap.2 $ le_rfl) (map_le_iff_le_comap.2 $ le_rfl)\n\nend comm_ring\n\nend map_and_comap\n\nsection is_primary\nvariables {R : Type u} [comm_semiring R]\n\n/-- A proper ideal `I` is primary iff `xy \u2208 I` implies `x \u2208 I` or `y \u2208 radical I`. -/\ndef is_primary (I : ideal R) : Prop :=\nI \u2260 \u22a4 \u2227 \u2200 {x y : R}, x * y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 radical I\n\ntheorem is_prime.is_primary {I : ideal R} (hi : is_prime I) : is_primary I :=\n\u27e8hi.1, \u03bb x y hxy, (hi.mem_or_mem hxy).imp id $ \u03bb hyi, le_radical hyi\u27e9\n\ntheorem mem_radical_of_pow_mem {I : ideal R} {x : R} {m : \u2115} (hx : x ^ m \u2208 radical I) :\n  x \u2208 radical I :=\nradical_idem I \u25b8 \u27e8m, hx\u27e9\n\ntheorem is_prime_radical {I : ideal R} (hi : is_primary I) : is_prime (radical I) :=\n\u27e8mt radical_eq_top.1 hi.1, \u03bb x y \u27e8m, hxy\u27e9, begin\n  rw mul_pow at hxy, cases hi.2 hxy,\n  { exact or.inl \u27e8m, h\u27e9 },\n  { exact or.inr (mem_radical_of_pow_mem h) }\nend\u27e9\n\ntheorem is_primary_inf {I J : ideal R} (hi : is_primary I) (hj : is_primary J)\n  (hij : radical I = radical J) : is_primary (I \u2293 J) :=\n\u27e8ne_of_lt $ lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1), \u03bb x y \u27e8hxyi, hxyj\u27e9,\nbegin\n  rw [radical_inf, hij, inf_idem],\n  cases hi.2 hxyi with hxi hyi, cases hj.2 hxyj with hxj hyj,\n  { exact or.inl \u27e8hxi, hxj\u27e9 },\n  { exact or.inr hyj },\n  { rw hij at hyi, exact or.inr hyi }\nend\u27e9\n\nend is_primary\n\nend ideal\n\nlemma associates.mk_ne_zero' {R : Type*} [comm_ring R] {r : R} :\n  (associates.mk (ideal.span {r} : ideal R)) \u2260 0 \u2194 (r \u2260 0):=\nby rw [associates.mk_ne_zero, ideal.zero_eq_bot, ne.def, ideal.span_singleton_eq_bot]\n\nnamespace ring_hom\n\nvariables {R : Type u} {S : Type v} {T : Type v}\n\nsection semiring\nvariables [semiring R] [semiring S] [semiring T] (f : R \u2192+* S) (g : T \u2192+* S)\n\n/-- Kernel of a ring homomorphism as an ideal of the domain. -/\ndef ker : ideal R := ideal.comap f \u22a5\n\n/-- An element is in the kernel if and only if it maps to zero.-/\nlemma mem_ker {r} : r \u2208 ker f \u2194 f r = 0 :=\nby rw [ker, ideal.mem_comap, submodule.mem_bot]\n\nlemma ker_eq : ((ker f) : set R) = set.preimage f {0} := rfl\n\nlemma ker_eq_comap_bot (f : R \u2192+* S) : f.ker = ideal.comap f \u22a5 := rfl\n\nlemma comap_ker (f : S \u2192+* R) : f.ker.comap g = (f.comp g).ker :=\nby rw [ring_hom.ker_eq_comap_bot, ideal.comap_comap, ring_hom.ker_eq_comap_bot]\n\n/-- If the target is not the zero ring, then one is not in the kernel.-/\nlemma not_one_mem_ker [nontrivial S] (f : R \u2192+* S) : (1:R) \u2209 ker f :=\nby { rw [mem_ker, f.map_one], exact one_ne_zero }\n\nlemma ker_ne_top [nontrivial S] (f : R \u2192+* S) : f.ker \u2260 \u22a4 :=\n(ideal.ne_top_iff_one _).mpr $ not_one_mem_ker f\n\nend semiring\n\nsection ring\nvariables [ring R] [semiring S] (f : R \u2192+* S)\n\nlemma injective_iff_ker_eq_bot : function.injective f \u2194 ker f = \u22a5 :=\nby { rw [set_like.ext'_iff, ker_eq, set.ext_iff], exact f.injective_iff' }\n\nlemma ker_eq_bot_iff_eq_zero : ker f = \u22a5 \u2194 \u2200 x, f x = 0 \u2192 x = 0 :=\nby { rw [\u2190 f.injective_iff, injective_iff_ker_eq_bot] }\n\n@[simp] lemma ker_coe_equiv (f : R \u2243+* S) : ker (f : R \u2192+* S) = \u22a5 :=\nby simpa only [\u2190injective_iff_ker_eq_bot] using f.injective\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S] (f : R \u2192+* S)\n\n/-- The induced map from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_equiv_of_surjective`).\n-/\ndef ker_lift (f : R \u2192+* S) : R \u29f8 f.ker \u2192+* S :=\nideal.quotient.lift _ f $ \u03bb r, f.mem_ker.mp\n\n@[simp]\nlemma ker_lift_mk (f : R \u2192+* S) (r : R) : ker_lift f (ideal.quotient.mk f.ker r) = f r :=\nideal.quotient.lift_mk _ _ _\n\n/-- The induced map from the quotient by the kernel is injective. -/\nlemma ker_lift_injective (f : R \u2192+* S) : function.injective (ker_lift f) :=\nassume a b, quotient.induction_on\u2082' a b $\n  assume a b (h : f a = f b), quotient.sound' $\nshow a - b \u2208 ker f, by rw [mem_ker, map_sub, h, sub_self]\n\nvariable {f}\n\n/-- The **first isomorphism theorem** for commutative rings, computable version. -/\ndef quotient_ker_equiv_of_right_inverse\n  {g : S \u2192 R} (hf : function.right_inverse g f) :\n  R \u29f8 f.ker \u2243+* S :=\n{ to_fun := ker_lift f,\n  inv_fun := (ideal.quotient.mk f.ker) \u2218 g,\n  left_inv := begin\n    rintro \u27e8x\u27e9,\n    apply ker_lift_injective,\n    simp [hf (f x)],\n  end,\n  right_inv := hf,\n  ..ker_lift f}\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : R \u29f8 f.ker) : quotient_ker_equiv_of_right_inverse hf x = ker_lift f x := rfl\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.symm.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : S) : (quotient_ker_equiv_of_right_inverse hf).symm x = ideal.quotient.mk f.ker (g x) := rfl\n\n/-- The **first isomorphism theorem** for commutative rings. -/\nnoncomputable def quotient_ker_equiv_of_surjective (hf : function.surjective f) :\n  R \u29f8 f.ker \u2243+* S :=\nquotient_ker_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\nend comm_ring\n\n/-- The kernel of a homomorphism to a domain is a prime ideal. -/\nlemma ker_is_prime [ring R] [ring S] [is_domain S] (f : R \u2192+* S) :\n  (ker f).is_prime :=\n\u27e8by { rw [ne.def, ideal.eq_top_iff_one], exact not_one_mem_ker f },\n\u03bb x y, by simpa only [mem_ker, f.map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _\u27e9\n\n/-- The kernel of a homomorphism to a field is a maximal ideal. -/\nlemma ker_is_maximal_of_surjective {R K : Type*} [ring R] [field K]\n  (f : R \u2192+* K) (hf : function.surjective f) :\n  f.ker.is_maximal :=\nbegin\n  refine ideal.is_maximal_iff.mpr\n    \u27e8\u03bb h1, @one_ne_zero K _ _ $ f.map_one \u25b8 f.mem_ker.mp h1,\n    \u03bb J x hJ hxf hxJ, _\u27e9,\n  obtain \u27e8y, hy\u27e9 := hf (f x)\u207b\u00b9,\n  have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm,\n  rw H,\n  refine J.sub_mem (J.mul_mem_left _ hxJ) (hJ _),\n  rw f.mem_ker,\n  simp only [hy, ring_hom.map_sub, ring_hom.map_one, ring_hom.map_mul,\n    inv_mul_cancel (mt f.mem_ker.mpr hxf), sub_self],\nend\n\nend ring_hom\n\nnamespace ideal\n\nvariables {R : Type*} {S : Type*}\n\nsection semiring\nvariables [semiring R] [semiring S]\n\nlemma map_eq_bot_iff_le_ker {I : ideal R} (f : R \u2192+* S) : I.map f = \u22a5 \u2194 I \u2264 f.ker :=\nby rw [ring_hom.ker, eq_bot_iff, map_le_iff_le_comap]\n\nlemma ker_le_comap {K : ideal S} (f : R \u2192+* S) : f.ker \u2264 comap f K :=\n\u03bb x hx, mem_comap.2 (((ring_hom.mem_ker f).1 hx).symm \u25b8 K.zero_mem)\n\nend semiring\n\nsection ring\nvariables [ring R] [ring S]\n\nlemma map_Inf {A : set (ideal R)} {f : R \u2192+* S} (hf : function.surjective f) :\n  (\u2200 J \u2208 A, ring_hom.ker f \u2264 J) \u2192 map f (Inf A) = Inf (map f '' A) :=\nbegin\n  refine \u03bb h, le_antisymm (le_Inf _) _,\n  { intros j hj y hy,\n    cases (mem_map_iff_of_surjective f hf).1 hy with x hx,\n    cases (set.mem_image _ _ _).mp hj with J hJ,\n    rw [\u2190 hJ.right, \u2190 hx.right],\n    exact mem_map_of_mem f (Inf_le_of_le hJ.left (le_of_eq rfl) hx.left) },\n  { intros y hy,\n    cases hf y with x hx,\n    refine hx \u25b8 (mem_map_of_mem f _),\n    have : \u2200 I \u2208 A, y \u2208 map f I, by simpa using hy,\n    rw [submodule.mem_Inf],\n    intros J hJ,\n    rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with \u27e8x', hx', rfl\u27e9,\n    have : x - x' \u2208 J,\n    { apply h J hJ,\n      rw [ring_hom.mem_ker, ring_hom.map_sub, hx, sub_self] },\n    simpa only [sub_add_cancel] using J.add_mem this hx' }\nend\n\ntheorem map_is_prime_of_surjective {f : R \u2192+* S} (hf : function.surjective f) {I : ideal R}\n  [H : is_prime I] (hk : ring_hom.ker f \u2264 I) : is_prime (map f I) :=\nbegin\n  refine \u27e8\u03bb h, H.ne_top (eq_top_iff.2 _), \u03bb x y, _\u27e9,\n  { replace h := congr_arg (comap f) h,\n    rw [comap_map_of_surjective _ hf, comap_top] at h,\n    exact h \u25b8 sup_le (le_of_eq rfl) hk },\n  { refine \u03bb hxy, (hf x).rec_on (\u03bb a ha, (hf y).rec_on (\u03bb b hb, _)),\n    rw [\u2190 ha, \u2190 hb, \u2190 ring_hom.map_mul, mem_map_iff_of_surjective _ hf] at hxy,\n    rcases hxy with \u27e8c, hc, hc'\u27e9,\n    rw [\u2190 sub_eq_zero, \u2190 ring_hom.map_sub] at hc',\n    have : a * b \u2208 I,\n    { convert I.sub_mem hc (hk (hc' : c - a * b \u2208 f.ker)),\n      abel },\n    exact (H.mem_or_mem this).imp (\u03bb h, ha \u25b8 mem_map_of_mem f h) (\u03bb h, hb \u25b8 mem_map_of_mem f h) }\nend\n\ntheorem map_is_prime_of_equiv (f : R \u2243+* S) {I : ideal R} [is_prime I] :\n  is_prime (map (f : R \u2192+* S) I) :=\nmap_is_prime_of_surjective f.surjective $ by simp\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S]\n\n@[simp] lemma mk_ker {I : ideal R} : (quotient.mk I).ker = I :=\nby ext; rw [ring_hom.ker, mem_comap, submodule.mem_bot, quotient.eq_zero_iff_mem]\n\nlemma map_mk_eq_bot_of_le {I J : ideal R} (h : I \u2264 J) : I.map (J^.quotient.mk) = \u22a5 :=\nby { rw [map_eq_bot_iff_le_ker, mk_ker], exact h }\n\nlemma ker_quotient_lift {S : Type v} [comm_ring S] {I : ideal R} (f : R \u2192+* S) (H : I \u2264 f.ker) :\n  (ideal.quotient.lift I f H).ker = (f.ker).map I^.quotient.mk :=\nbegin\n  ext x,\n  split,\n  { intro hx,\n    obtain \u27e8y, hy\u27e9 := quotient.mk_surjective x,\n    rw [ring_hom.mem_ker, \u2190 hy, ideal.quotient.lift_mk, \u2190 ring_hom.mem_ker] at hx,\n    rw [\u2190 hy, mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective],\n    exact \u27e8y, hx, rfl\u27e9 },\n  { intro hx,\n    rw mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective at hx,\n    obtain \u27e8y, hy\u27e9 := hx,\n    rw [ring_hom.mem_ker, \u2190 hy.right, ideal.quotient.lift_mk, \u2190 (ring_hom.mem_ker f)],\n    exact hy.left },\nend\n\ntheorem map_eq_iff_sup_ker_eq_of_surjective {I J : ideal R} (f : R \u2192+* S)\n  (hf : function.surjective f) : map f I = map f J \u2194 I \u2294 f.ker = J \u2294 f.ker :=\nby rw [\u2190 (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf,\n  comap_map_of_surjective f hf, ring_hom.ker_eq_comap_bot]\n\ntheorem map_radical_of_surjective {f : R \u2192+* S} (hf : function.surjective f) {I : ideal R}\n  (h : ring_hom.ker f \u2264 I) : map f (I.radical) = (map f I).radical :=\nbegin\n  rw [radical_eq_Inf, radical_eq_Inf],\n  have : \u2200 J \u2208 {J : ideal R | I \u2264 J \u2227 J.is_prime}, f.ker \u2264 J := \u03bb J hJ, le_trans h hJ.left,\n  convert map_Inf hf this,\n  refine funext (\u03bb j, propext \u27e8_, _\u27e9),\n  { rintros \u27e8hj, hj'\u27e9,\n    haveI : j.is_prime := hj',\n    exact \u27e8comap f j, \u27e8\u27e8map_le_iff_le_comap.1 hj, comap_is_prime f j\u27e9,\n      map_comap_of_surjective f hf j\u27e9\u27e9 },\n  { rintro \u27e8J, \u27e8hJ, hJ'\u27e9\u27e9,\n    haveI : J.is_prime := hJ.right,\n    refine \u27e8hJ' \u25b8 map_mono hJ.left, hJ' \u25b8 map_is_prime_of_surjective hf (le_trans h hJ.left)\u27e9 },\nend\n\n@[simp] lemma bot_quotient_is_maximal_iff (I : ideal R) :\n  (\u22a5 : ideal (R \u29f8 I)).is_maximal \u2194 I.is_maximal :=\n\u27e8\u03bb hI, (@mk_ker _ _ I) \u25b8\n  @comap_is_maximal_of_surjective _ _ _ _ (quotient.mk I) quotient.mk_surjective \u22a5 hI,\n \u03bb hI, @bot_is_maximal _ (@field.to_division_ring _ (@quotient.field _ _ I hI)) \u27e9\n\nsection quotient_algebra\n\nvariables (R\u2081 R\u2082 : Type*) {A B : Type*}\nvariables [comm_semiring R\u2081] [comm_semiring R\u2082] [comm_ring A] [comm_ring B]\nvariables [algebra R\u2081 A] [algebra R\u2082 A] [algebra R\u2081 B]\n\n/-- The `R\u2081`-algebra structure on `A/I` for an `R\u2081`-algebra `A` -/\ninstance quotient.algebra {I : ideal A} : algebra R\u2081 (A \u29f8 I) :=\n{ to_fun := \u03bb x, ideal.quotient.mk I (algebra_map R\u2081 A x),\n  smul := (\u2022),\n  smul_def' := \u03bb r x, quotient.induction_on' x $ \u03bb x,\n      ((quotient.mk I).congr_arg $ algebra.smul_def _ _).trans (ring_hom.map_mul _ _ _),\n  commutes' := \u03bb _ _, mul_comm _ _,\n  .. ring_hom.comp (ideal.quotient.mk I) (algebra_map R\u2081 A) }\n\n-- Lean can struggle to find this instance later if we don't provide this shortcut\ninstance quotient.is_scalar_tower [has_scalar R\u2081 R\u2082] [is_scalar_tower R\u2081 R\u2082 A] (I : ideal A) :\n  is_scalar_tower R\u2081 R\u2082 (A \u29f8 I) :=\nby apply_instance\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] A \u29f8 I` as morphism of `R\u2081`-algebras, for `I` an ideal of\n`A`, where `A` is an `R\u2081`-algebra. -/\ndef quotient.mk\u2090 (I : ideal A) : A \u2192\u2090[R\u2081] A \u29f8 I :=\n\u27e8\u03bb a, submodule.quotient.mk a, rfl, \u03bb _ _, rfl, rfl, \u03bb _ _, rfl, \u03bb _, rfl\u27e9\n\nlemma quotient.alg_map_eq (I : ideal A) :\n  algebra_map R\u2081 (A \u29f8 I) = (algebra_map A (A \u29f8 I)).comp (algebra_map R\u2081 A) :=\nrfl\n\nlemma quotient.mk\u2090_to_ring_hom (I : ideal A) :\n  (quotient.mk\u2090 R\u2081 I).to_ring_hom = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.mk\u2090_eq_mk (I : ideal A) :\n  \u21d1(quotient.mk\u2090 R\u2081 I) = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.algebra_map_eq (I : ideal R) :\n  algebra_map R (R \u29f8 I) = I^.quotient.mk :=\nrfl\n\n@[simp] lemma quotient.mk_comp_algebra_map (I : ideal A) :\n  (quotient.mk I).comp (algebra_map R\u2081 A) = algebra_map R\u2081 (A \u29f8 I) :=\nrfl\n\n@[simp] lemma quotient.mk_algebra_map (I : ideal A) (x : R\u2081) :\n  quotient.mk I (algebra_map R\u2081 A x) = algebra_map R\u2081 (A \u29f8 I) x :=\nrfl\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] I.quotient` is surjective. -/\nlemma quotient.mk\u2090_surjective (I : ideal A) : function.surjective (quotient.mk\u2090 R\u2081 I) :=\nsurjective_quot_mk _\n\n/-- The kernel of `A \u2192\u2090[R\u2081] I.quotient` is `I`. -/\n@[simp]\nlemma quotient.mk\u2090_ker (I : ideal A) : (quotient.mk\u2090 R\u2081 I : A \u2192+* A \u29f8 I).ker = I :=\nideal.mk_ker\n\nvariables {R\u2081}\n\nlemma ker_lift.map_smul (f : A \u2192\u2090[R\u2081] B) (r : R\u2081) (x : A \u29f8 f.to_ring_hom.ker) :\n  f.to_ring_hom.ker_lift (r \u2022 x) = r \u2022 f.to_ring_hom.ker_lift x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := quotient.mk\u2090_surjective R\u2081 _ x,\n  rw [\u2190 alg_hom.map_smul, quotient.mk\u2090_eq_mk, ring_hom.ker_lift_mk],\n  exact f.map_smul _ _\nend\n\n/-- The induced algebras morphism from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_alg_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_alg_equiv_of_surjective`).\n-/\ndef ker_lift_alg (f : A \u2192\u2090[R\u2081] B) : (A \u29f8 f.to_ring_hom.ker) \u2192\u2090[R\u2081] B :=\nalg_hom.mk' f.to_ring_hom.ker_lift (\u03bb _ _, ker_lift.map_smul f _ _)\n\n@[simp]\nlemma ker_lift_alg_mk (f : A \u2192\u2090[R\u2081] B) (a : A) :\n  ker_lift_alg f (quotient.mk f.to_ring_hom.ker a) = f a := rfl\n\n@[simp]\nlemma ker_lift_alg_to_ring_hom (f : A \u2192\u2090[R\u2081] B) :\n  (ker_lift_alg f).to_ring_hom = ring_hom.ker_lift f := rfl\n\n/-- The induced algebra morphism from the quotient by the kernel is injective. -/\nlemma ker_lift_alg_injective (f : A \u2192\u2090[R\u2081] B) : function.injective (ker_lift_alg f) :=\nring_hom.ker_lift_injective f\n\n/-- The **first isomorphism** theorem for algebras, computable version. -/\ndef quotient_ker_alg_equiv_of_right_inverse\n  {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A} (hf : function.right_inverse g f) :\n  (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\n{ ..ring_hom.quotient_ker_equiv_of_right_inverse (\u03bb x, show f.to_ring_hom (g x) = x, from hf x),\n  ..ker_lift_alg f}\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : A \u29f8 f.to_ring_hom.ker) :\n  quotient_ker_alg_equiv_of_right_inverse hf x = ker_lift_alg f x := rfl\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse_symm.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : B) :\n  (quotient_ker_alg_equiv_of_right_inverse hf).symm x = quotient.mk\u2090 R\u2081 f.to_ring_hom.ker (g x) :=\n  rfl\n\n/-- The **first isomorphism theorem** for algebras. -/\nnoncomputable def quotient_ker_alg_equiv_of_surjective\n  {f : A \u2192\u2090[R\u2081] B} (hf : function.surjective f) : (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\nquotient_ker_alg_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\n/-- The ring hom `R/I \u2192+* S/J` induced by a ring hom `f : R \u2192+* S` with `I \u2264 f\u207b\u00b9(J)` -/\ndef quotient_map {I : ideal R} (J : ideal S) (f : R \u2192+* S) (hIJ : I \u2264 J.comap f) :\n  R \u29f8 I \u2192+* S \u29f8 J :=\n(quotient.lift I ((quotient.mk J).comp f) (\u03bb _ ha,\n  by simpa [function.comp_app, ring_hom.coe_comp, quotient.eq_zero_iff_mem] using hIJ ha))\n\n@[simp]\nlemma quotient_map_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R} : quotient_map I f H (quotient.mk J x) = quotient.mk I (f x) :=\nquotient.lift_mk J _ _\n\n@[simp]\nlemma quotient_map_algebra_map {J : ideal A} {I : ideal S} {f : A \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R\u2081} :\n  quotient_map I f H (algebra_map R\u2081 (A \u29f8 J) x) = quotient.mk I (f (algebra_map _ _ x)) :=\nquotient.lift_mk J _ _\n\nlemma quotient_map_comp_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} (H : J \u2264 I.comap f) :\n  (quotient_map I f H).comp (quotient.mk J) = (quotient.mk I).comp f :=\nring_hom.ext (\u03bb x, by simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient_map_mk])\n\n/-- The ring equiv `R/I \u2243+* S/J` induced by a ring equiv `f : R \u2243+** S`,  where `J = f(I)`. -/\n@[simps]\ndef quotient_equiv (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S)) :\n  R \u29f8 I \u2243+* S \u29f8 J :=\n{ inv_fun := quotient_map I \u2191f.symm (by {rw hIJ, exact le_of_eq (map_comap_of_equiv I f)}),\n  left_inv := by {rintro \u27e8r\u27e9, simp },\n  right_inv := by {rintro \u27e8s\u27e9, simp },\n  ..quotient_map J \u2191f (by {rw hIJ, exact @le_comap_map _ S _ _ _ _}) }\n\n@[simp]\nlemma quotient_equiv_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S))\n  (x : R) : quotient_equiv I J f hIJ (ideal.quotient.mk I x) = ideal.quotient.mk J (f x) := rfl\n\n@[simp]\nlemma quotient_equiv_symm_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S)\n  (hIJ : J = I.map (f : R \u2192+* S)) (x : S) :\n  (quotient_equiv I J f hIJ).symm (ideal.quotient.mk J x) = ideal.quotient.mk I (f.symm x) := rfl\n\n/-- `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. -/\nlemma quotient_map_injective' {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (h : I.comap f \u2264 J) : function.injective (quotient_map I f H) :=\nbegin\n  refine (quotient_map I f H).injective_iff.2 (\u03bb a ha, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  rw [quotient_map_mk, quotient.eq_zero_iff_mem] at ha,\n  exact (quotient.eq_zero_iff_mem).mpr (h ha),\nend\n\n/-- If we take `J = I.comap f` then `quotient_map` is injective automatically. -/\nlemma quotient_map_injective {I : ideal S} {f : R \u2192+* S} :\n  function.injective (quotient_map I f le_rfl) :=\nquotient_map_injective' le_rfl\n\nlemma quotient_map_surjective {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (hf : function.surjective f) : function.surjective (quotient_map I f H) :=\n\u03bb x, let \u27e8x, hx\u27e9 := quotient.mk_surjective x in\n  let \u27e8y, hy\u27e9 := hf x in \u27e8(quotient.mk J) y, by simp [hx, hy]\u27e9\n\n/-- Commutativity of a square is preserved when taking quotients by an ideal. -/\nlemma comp_quotient_map_eq_of_comp_eq {R' S' : Type*} [comm_ring R'] [comm_ring S']\n  {f : R \u2192+* S} {f' : R' \u2192+* S'} {g : R \u2192+* R'} {g' : S \u2192+* S'} (hfg : f'.comp g = g'.comp f)\n  (I : ideal S') : (quotient_map I g' le_rfl).comp (quotient_map (I.comap g') f le_rfl) =\n    (quotient_map I f' le_rfl).comp (quotient_map (I.comap f') g\n      (le_of_eq (trans (comap_comap f g') (hfg \u25b8 (comap_comap g f'))))) :=\nbegin\n  refine ring_hom.ext (\u03bb a, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  simp only [ring_hom.comp_apply, quotient_map_mk],\n  exact congr_arg (quotient.mk I) (trans (g'.comp_apply f r).symm (hfg \u25b8 (f'.comp_apply g r))),\nend\n\n/-- The algebra hom `A/I \u2192+* B/J` induced by an algebra hom `f : A \u2192\u2090[R\u2081] B` with `I \u2264 f\u207b\u00b9(J)`. -/\ndef quotient_map\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (hIJ : I \u2264 J.comap f) :\n  A \u29f8 I \u2192\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_map J \u2191f hIJ }\n\n@[simp]\nlemma quotient_map_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f)\n  {x : A} : quotient_map\u2090 J f H (quotient.mk I x) = quotient.mk\u2090 R\u2081 J (f x) := rfl\n\nlemma quotient_map_comp_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f) :\n  (quotient_map\u2090 J f H).comp (quotient.mk\u2090 R\u2081 I) = (quotient.mk\u2090 R\u2081 J).comp f :=\nalg_hom.ext (\u03bb x, by simp only [quotient_map_mk\u2090, quotient.mk\u2090_eq_mk, alg_hom.comp_apply])\n\n/-- The algebra equiv `A/I \u2243\u2090[R] B/J` induced by an algebra equiv `f : A \u2243\u2090[R] B`,\nwhere`J = f(I)`. -/\ndef quotient_equiv_alg (I : ideal A) (J : ideal B) (f : A \u2243\u2090[R\u2081] B)\n  (hIJ : J = I.map (f : A \u2192+* B)) :\n  (A \u29f8 I) \u2243\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_equiv I J (f : A \u2243+* B) hIJ }\n\n@[priority 100]\ninstance quotient_algebra {I : ideal A} [algebra R A] :\n  algebra (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I) :=\n(quotient_map I (algebra_map R A) (le_of_eq rfl)).to_algebra\n\nlemma algebra_map_quotient_injective {I : ideal A} [algebra R A]:\n  function.injective (algebra_map (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I)) :=\nbegin\n  rintros \u27e8a\u27e9 \u27e8b\u27e9 hab,\n  replace hab := quotient.eq.mp hab,\n  rw \u2190 ring_hom.map_sub at hab,\n  exact quotient.eq.mpr hab\nend\n\nend quotient_algebra\n\nend comm_ring\n\nend ideal\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v}\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\n-- TODO: show `[algebra R A] : algebra (ideal R) A` too\n\ninstance module_submodule : module (ideal R) (submodule R M) :=\n{ smul_add := smul_sup,\n  add_smul := sup_smul,\n  mul_smul := submodule.smul_assoc,\n  one_smul := by simp,\n  zero_smul := bot_smul,\n  smul_zero := smul_bot }\n\nend submodule\n\nnamespace ring_hom\nvariables {A B C : Type*} [ring A] [ring B] [ring C]\nvariables (f : A \u2192+* B) (f_inv : B \u2192 A)\n\n/-- Auxiliary definition used to define `lift_of_right_inverse` -/\ndef lift_of_right_inverse_aux\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) :\n  B \u2192+* C :=\n{ to_fun := \u03bb b, g (f_inv b),\n  map_one' :=\n  begin\n    rw [\u2190 g.map_one, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_one],\n    exact hf 1\n  end,\n  map_mul' :=\n  begin\n    intros x y,\n    rw [\u2190 g.map_mul, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_mul],\n    simp only [hf _],\n  end,\n  .. add_monoid_hom.lift_of_right_inverse f.to_add_monoid_hom f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 }\n\n@[simp] lemma lift_of_right_inverse_aux_comp_apply\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) (a : A) :\n  (f.lift_of_right_inverse_aux f_inv hf g hg) (f a) = g a :=\nf.to_add_monoid_hom.lift_of_right_inverse_comp_apply f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 a\n\n/-- `lift_of_right_inverse f hf g hg` is the unique ring homomorphism `\u03c6`\n\n* such that `\u03c6.comp f = g` (`ring_hom.lift_of_right_inverse_comp`),\n* where `f : A \u2192+* B` is has a right_inverse `f_inv` (`hf`),\n* and `g : B \u2192+* C` satisfies `hg : f.ker \u2264 g.ker`.\n\nSee `ring_hom.eq_lift_of_right_inverse` for the uniqueness lemma.\n\n```\n   A .\n   |  \\\n f |   \\ g\n   |    \\\n   v     \\\u231f\n   B ----> C\n      \u2203!\u03c6\n```\n-/\ndef lift_of_right_inverse\n  (hf : function.right_inverse f_inv f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\n{ to_fun := \u03bb g, f.lift_of_right_inverse_aux f_inv hf g.1 g.2,\n  inv_fun := \u03bb \u03c6, \u27e8\u03c6.comp f, \u03bb x hx, (mem_ker _).mpr $ by simp [(mem_ker _).mp hx]\u27e9,\n  left_inv := \u03bb g, by\n  { ext,\n    simp only [comp_apply, lift_of_right_inverse_aux_comp_apply, subtype.coe_mk,\n      subtype.val_eq_coe], },\n  right_inv := \u03bb \u03c6, by\n  { ext b,\n    simp [lift_of_right_inverse_aux, hf b], } }\n\n/-- A non-computable version of `ring_hom.lift_of_right_inverse` for when no computable right\ninverse is available, that uses `function.surj_inv`. -/\n@[simp]\nnoncomputable abbreviation lift_of_surjective\n  (hf : function.surjective f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\nf.lift_of_right_inverse (function.surj_inv hf) (function.right_inverse_surj_inv hf)\n\nlemma lift_of_right_inverse_comp_apply\n  (hf : function.right_inverse f_inv f) (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) (x : A) :\n  (f.lift_of_right_inverse f_inv hf g) (f x) = g x :=\nf.lift_of_right_inverse_aux_comp_apply f_inv hf g.1 g.2 x\n\nlemma lift_of_right_inverse_comp (hf : function.right_inverse f_inv f)\n  (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) :\n  (f.lift_of_right_inverse f_inv hf g).comp f = g :=\nring_hom.ext $ f.lift_of_right_inverse_comp_apply f_inv hf g\n\nlemma eq_lift_of_right_inverse (hf : function.right_inverse f_inv f) (g : A \u2192+* C)\n  (hg : f.ker \u2264 g.ker) (h : B \u2192+* C) (hh : h.comp f = g) :\n  h = (f.lift_of_right_inverse f_inv hf \u27e8g, hg\u27e9) :=\nbegin\n  simp_rw \u2190hh,\n  exact ((f.lift_of_right_inverse f_inv hf).apply_symm_apply _).symm,\nend\n\nend ring_hom\n\nnamespace double_quot\nopen ideal\nvariables {R : Type u} [comm_ring R] (I J : ideal R)\n\n/-- The obvious ring hom `R/I \u2192 R/(I \u2294 J)` -/\ndef quot_left_to_quot_sup : R \u29f8 I \u2192+* R \u29f8 (I \u2294 J) :=\nideal.quotient.factor I (I \u2294 J) le_sup_left\n\n/-- The kernel of `quot_left_to_quot_sup` -/\nlemma ker_quot_left_to_quot_sup :\n  (quot_left_to_quot_sup I J).ker = J.map (ideal.quotient.mk I) :=\nby simp only [mk_ker, sup_idem, sup_comm, quot_left_to_quot_sup, quotient.factor, ker_quotient_lift,\n    map_eq_iff_sup_ker_eq_of_surjective I^.quotient.mk quotient.mk_surjective, \u2190 sup_assoc]\n\n/-- The ring homomorphism `(R/I)/J' -> R/(I \u2294 J)` induced by `quot_left_to_quot_sup` where `J'`\n  is the image of `J` in `R/I`-/\ndef quot_quot_to_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2192+* R \u29f8 I \u2294 J :=\nideal.quotient.lift (ideal.map (ideal.quotient.mk I) J) (quot_left_to_quot_sup I J)\n  (ker_quot_left_to_quot_sup I J).symm.le\n\n/-- The composite of the maps `R \u2192 (R/I)` and `(R/I) \u2192 (R/I)/J'` -/\ndef quot_quot_mk : R \u2192+* ((R \u29f8 I) \u29f8 J.map I^.quotient.mk) :=\n((J.map I^.quotient.mk)^.quotient.mk).comp I^.quotient.mk\n\n/-- The kernel of `quot_quot_mk` -/\nlemma ker_quot_quot_mk : (quot_quot_mk I J).ker = I \u2294 J :=\nby rw [ring_hom.ker_eq_comap_bot, quot_quot_mk, \u2190 comap_comap, \u2190 ring_hom.ker, mk_ker,\n  comap_map_of_surjective (ideal.quotient.mk I) (quotient.mk_surjective), \u2190 ring_hom.ker, mk_ker,\n  sup_comm]\n\n/-- The ring homomorphism `R/(I \u2294 J) \u2192 (R/I)/J' `induced by `quot_quot_mk` -/\ndef lift_sup_quot_quot_mk (I J : ideal R) :\n  R \u29f8 (I \u2294 J) \u2192+* (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) :=\nideal.quotient.lift (I \u2294 J) (quot_quot_mk I J) (ker_quot_quot_mk I J).symm.le\n\n/-- `quot_quot_to_quot_add` and `lift_sup_double_qot_mk` are inverse isomorphisms -/\ndef quot_quot_equiv_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2243+* R \u29f8 I \u2294 J :=\nring_equiv.of_hom_inv (quot_quot_to_quot_sup I J) (lift_sup_quot_quot_mk I J)\n  (by { ext z, refl }) (by { ext z, refl })\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_quot_quot_mk (x : R) :\n  quot_quot_equiv_quot_sup I J (quot_quot_mk I J x) = ideal.quotient.mk (I \u2294 J) x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_symm_quot_quot_mk (x : R) :\n  (quot_quot_equiv_quot_sup I J).symm (ideal.quotient.mk (I \u2294 J) x) = quot_quot_mk I J x :=\nrfl\n\n/-- The obvious isomorphism `(R/I)/J' \u2192 (R/J)/I' `   -/\ndef quot_quot_equiv_comm :\n  (R \u29f8 I) \u29f8 J.map I^.quotient.mk \u2243+* (R \u29f8 J) \u29f8 I.map J^.quotient.mk :=\n((quot_quot_equiv_quot_sup I J).trans (quot_equiv_of_eq sup_comm)).trans\n  (quot_quot_equiv_quot_sup J I).symm\n\n@[simp]\nlemma quot_quot_equiv_comm_quot_quot_mk (x : R) :\n  quot_quot_equiv_comm I J (quot_quot_mk I J x) = quot_quot_mk J I x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_comm_comp_quot_quot_mk :\n  ring_hom.comp \u2191(quot_quot_equiv_comm I J) (quot_quot_mk I J) = quot_quot_mk J I :=\nring_hom.ext $ quot_quot_equiv_comm_quot_quot_mk I J\n\n@[simp]\nlemma quot_quot_equiv_comm_symm :\n  (quot_quot_equiv_comm I J).symm = quot_quot_equiv_comm J I :=\nrfl\n\nend double_quot\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/ring_theory/ideal/operations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.38023304719009016}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Stephen Morgan, Scott Morrison\n-/\nimport category_theory.products.basic\n\nopen category_theory\n\nnamespace category_theory.bifunctor\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\nvariables {C : Type u\u2081} {D : Type u\u2082} {E : Type u\u2083}\nvariables [category.{v\u2081} C] [category.{v\u2082} D] [category.{v\u2083} E]\n\n@[simp] lemma map_id (F : (C \u00d7 D) \u2964 E) (X : C) (Y : D) :\n  F.map ((\ud835\udfd9 X, \ud835\udfd9 Y) : (X, Y) \u27f6 (X, Y)) = \ud835\udfd9 (F.obj (X, Y)) :=\nF.map_id (X, Y)\n\n@[simp] lemma map_id_comp (F : (C \u00d7 D) \u2964 E) (W : C) {X Y Z : D} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  F.map ((\ud835\udfd9 W, f \u226b g) : (W, X) \u27f6 (W, Z)) =\n  F.map ((\ud835\udfd9 W, f) : (W, X) \u27f6 (W, Y)) \u226b F.map ((\ud835\udfd9 W, g) : (W, Y) \u27f6 (W, Z)) :=\nby rw [\u2190functor.map_comp,prod_comp,category.comp_id]\n\n@[simp] lemma map_comp_id (F : (C \u00d7 D) \u2964 E) (X Y Z : C) (W : D) (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  F.map ((f \u226b g, \ud835\udfd9 W) : (X, W) \u27f6 (Z, W)) =\n  F.map ((f, \ud835\udfd9 W) : (X, W) \u27f6 (Y, W)) \u226b F.map ((g, \ud835\udfd9 W) : (Y, W) \u27f6 (Z, W)) :=\nby rw [\u2190functor.map_comp,prod_comp,category.comp_id]\n\n@[simp] lemma diagonal (F : (C \u00d7 D) \u2964 E) (X X' : C) (f : X \u27f6 X') (Y Y' : D) (g : Y \u27f6 Y') :\n  F.map ((\ud835\udfd9 X, g) : (X, Y) \u27f6 (X, Y')) \u226b F.map ((f, \ud835\udfd9 Y') : (X, Y') \u27f6 (X', Y')) =\n  F.map ((f, g) : (X, Y) \u27f6 (X', Y')) :=\nby rw [\u2190functor.map_comp, prod_comp, category.id_comp, category.comp_id]\n\n@[simp] lemma diagonal' (F : (C \u00d7 D) \u2964 E) (X X' : C) (f : X \u27f6 X') (Y Y' : D) (g : Y \u27f6 Y') :\n  F.map ((f, \ud835\udfd9 Y) : (X, Y) \u27f6 (X', Y)) \u226b F.map ((\ud835\udfd9 X', g) : (X', Y) \u27f6 (X', Y')) =\n  F.map ((f, g) : (X, Y) \u27f6 (X', Y')) :=\nby rw [\u2190functor.map_comp, prod_comp, category.id_comp, category.comp_id]\n\nend category_theory.bifunctor\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/products/bifunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.3801654842617465}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, R\u00e9my Degenne\n\n! This file was ported from Lean 3 source module data.set.intervals.order_iso\n! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Set.Intervals.Basic\nimport Mathbin.Order.Hom.Set\n\n/-!\n# Lemmas about images of intervals under order isomorphisms.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nvariable {\u03b1 \u03b2 : Type _}\n\nopen Set\n\nnamespace OrderIso\n\nsection Preorder\n\nvariable [Preorder \u03b1] [Preorder \u03b2]\n\n/- warning: order_iso.preimage_Iic -> OrderIso.preimage_Iic is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Iic.{u2} \u03b2 _inst_2 b)) (Set.Iic.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Iic.{u1} \u03b2 _inst_2 b)) (Set.Iic.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) b) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Iic OrderIso.preimage_Iic\u2093'. -/\n@[simp]\ntheorem preimage_Iic (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : e \u207b\u00b9' Iic b = Iic (e.symm b) :=\n  by\n  ext x\n  simp [\u2190 e.le_iff_le]\n#align order_iso.preimage_Iic OrderIso.preimage_Iic\n\n/- warning: order_iso.preimage_Ici -> OrderIso.preimage_Ici is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ici.{u2} \u03b2 _inst_2 b)) (Set.Ici.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ici.{u1} \u03b2 _inst_2 b)) (Set.Ici.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) b) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Ici OrderIso.preimage_Ici\u2093'. -/\n@[simp]\ntheorem preimage_Ici (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : e \u207b\u00b9' Ici b = Ici (e.symm b) :=\n  by\n  ext x\n  simp [\u2190 e.le_iff_le]\n#align order_iso.preimage_Ici OrderIso.preimage_Ici\n\n/- warning: order_iso.preimage_Iio -> OrderIso.preimage_Iio is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Iio.{u2} \u03b2 _inst_2 b)) (Set.Iio.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Iio.{u1} \u03b2 _inst_2 b)) (Set.Iio.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) b) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Iio OrderIso.preimage_Iio\u2093'. -/\n@[simp]\ntheorem preimage_Iio (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : e \u207b\u00b9' Iio b = Iio (e.symm b) :=\n  by\n  ext x\n  simp [\u2190 e.lt_iff_lt]\n#align order_iso.preimage_Iio OrderIso.preimage_Iio\n\n/- warning: order_iso.preimage_Ioi -> OrderIso.preimage_Ioi is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioi.{u2} \u03b2 _inst_2 b)) (Set.Ioi.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioi.{u1} \u03b2 _inst_2 b)) (Set.Ioi.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) b) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioi OrderIso.preimage_Ioi\u2093'. -/\n@[simp]\ntheorem preimage_Ioi (e : \u03b1 \u2243o \u03b2) (b : \u03b2) : e \u207b\u00b9' Ioi b = Ioi (e.symm b) :=\n  by\n  ext x\n  simp [\u2190 e.lt_iff_lt]\n#align order_iso.preimage_Ioi OrderIso.preimage_Ioi\n\n/- warning: order_iso.preimage_Icc -> OrderIso.preimage_Icc is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Icc.{u2} \u03b2 _inst_2 a b)) (Set.Icc.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) a) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Icc.{u1} \u03b2 _inst_2 a b)) (Set.Icc.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) a) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Icc OrderIso.preimage_Icc\u2093'. -/\n@[simp]\ntheorem preimage_Icc (e : \u03b1 \u2243o \u03b2) (a b : \u03b2) : e \u207b\u00b9' Icc a b = Icc (e.symm a) (e.symm b) := by\n  simp [\u2190 Ici_inter_Iic]\n#align order_iso.preimage_Icc OrderIso.preimage_Icc\n\n/- warning: order_iso.preimage_Ico -> OrderIso.preimage_Ico is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ico.{u2} \u03b2 _inst_2 a b)) (Set.Ico.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) a) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ico.{u1} \u03b2 _inst_2 a b)) (Set.Ico.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) a) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Ico OrderIso.preimage_Ico\u2093'. -/\n@[simp]\ntheorem preimage_Ico (e : \u03b1 \u2243o \u03b2) (a b : \u03b2) : e \u207b\u00b9' Ico a b = Ico (e.symm a) (e.symm b) := by\n  simp [\u2190 Ici_inter_Iio]\n#align order_iso.preimage_Ico OrderIso.preimage_Ico\n\n/- warning: order_iso.preimage_Ioc -> OrderIso.preimage_Ioc is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioc.{u2} \u03b2 _inst_2 a b)) (Set.Ioc.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) a) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioc.{u1} \u03b2 _inst_2 a b)) (Set.Ioc.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) a) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioc OrderIso.preimage_Ioc\u2093'. -/\n@[simp]\ntheorem preimage_Ioc (e : \u03b1 \u2243o \u03b2) (a b : \u03b2) : e \u207b\u00b9' Ioc a b = Ioc (e.symm a) (e.symm b) := by\n  simp [\u2190 Ioi_inter_Iic]\n#align order_iso.preimage_Ioc OrderIso.preimage_Ioc\n\n/- warning: order_iso.preimage_Ioo -> OrderIso.preimage_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u1} (Set.{u1} \u03b1) (Set.preimage.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioo.{u2} \u03b2 _inst_2 a b)) (Set.Ioo.{u1} \u03b1 _inst_1 (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) a) (coeFn.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (OrderIso.{u2, u1} \u03b2 \u03b1 (Preorder.toLE.{u2} \u03b2 _inst_2) (Preorder.toLE.{u1} \u03b1 _inst_1)) (fun (_x : RelIso.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) => \u03b2 -> \u03b1) (RelIso.hasCoeToFun.{u2, u1} \u03b2 \u03b1 (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2)) (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1))) (OrderIso.symm.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2) e) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b2) (b : \u03b2), Eq.{succ u2} (Set.{u2} \u03b1) (Set.preimage.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioo.{u1} \u03b2 _inst_2 a b)) (Set.Ioo.{u2} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) a) _inst_1 (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) a) (FunLike.coe.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 (fun (_x : \u03b2) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b2) => \u03b1) _x) (EmbeddingLike.toFunLike.{max (succ u1) (succ u2), succ u1, succ u2} (Function.Embedding.{succ u1, succ u2} \u03b2 \u03b1) \u03b2 \u03b1 (Function.instEmbeddingLikeEmbedding.{succ u1, succ u2} \u03b2 \u03b1)) (RelEmbedding.toEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u1, u2} \u03b2 \u03b1 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (OrderIso.symm.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2) e))) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.preimage_Ioo OrderIso.preimage_Ioo\u2093'. -/\n@[simp]\ntheorem preimage_Ioo (e : \u03b1 \u2243o \u03b2) (a b : \u03b2) : e \u207b\u00b9' Ioo a b = Ioo (e.symm a) (e.symm b) := by\n  simp [\u2190 Ioi_inter_Iio]\n#align order_iso.preimage_Ioo OrderIso.preimage_Ioo\n\n/- warning: order_iso.image_Iic -> OrderIso.image_Iic is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Iic.{u1} \u03b1 _inst_1 a)) (Set.Iic.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Iic.{u2} \u03b1 _inst_1 a)) (Set.Iic.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Iic OrderIso.image_Iic\u2093'. -/\n@[simp]\ntheorem image_Iic (e : \u03b1 \u2243o \u03b2) (a : \u03b1) : e '' Iic a = Iic (e a) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Iic, e.symm_symm]\n#align order_iso.image_Iic OrderIso.image_Iic\n\n/- warning: order_iso.image_Ici -> OrderIso.image_Ici is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ici.{u1} \u03b1 _inst_1 a)) (Set.Ici.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ici.{u2} \u03b1 _inst_1 a)) (Set.Ici.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Ici OrderIso.image_Ici\u2093'. -/\n@[simp]\ntheorem image_Ici (e : \u03b1 \u2243o \u03b2) (a : \u03b1) : e '' Ici a = Ici (e a) :=\n  e.dual.image_Iic a\n#align order_iso.image_Ici OrderIso.image_Ici\n\n/- warning: order_iso.image_Iio -> OrderIso.image_Iio is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Iio.{u1} \u03b1 _inst_1 a)) (Set.Iio.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Iio.{u2} \u03b1 _inst_1 a)) (Set.Iio.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Iio OrderIso.image_Iio\u2093'. -/\n@[simp]\ntheorem image_Iio (e : \u03b1 \u2243o \u03b2) (a : \u03b1) : e '' Iio a = Iio (e a) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Iio, e.symm_symm]\n#align order_iso.image_Iio OrderIso.image_Iio\n\n/- warning: order_iso.image_Ioi -> OrderIso.image_Ioi is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioi.{u1} \u03b1 _inst_1 a)) (Set.Ioi.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioi.{u2} \u03b1 _inst_1 a)) (Set.Ioi.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Ioi OrderIso.image_Ioi\u2093'. -/\n@[simp]\ntheorem image_Ioi (e : \u03b1 \u2243o \u03b2) (a : \u03b1) : e '' Ioi a = Ioi (e a) :=\n  e.dual.image_Iio a\n#align order_iso.image_Ioi OrderIso.image_Ioi\n\n/- warning: order_iso.image_Ioo -> OrderIso.image_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioo.{u1} \u03b1 _inst_1 a b)) (Set.Ioo.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioo.{u2} \u03b1 _inst_1 a b)) (Set.Ioo.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Ioo OrderIso.image_Ioo\u2093'. -/\n@[simp]\ntheorem image_Ioo (e : \u03b1 \u2243o \u03b2) (a b : \u03b1) : e '' Ioo a b = Ioo (e a) (e b) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]\n#align order_iso.image_Ioo OrderIso.image_Ioo\n\n/- warning: order_iso.image_Ioc -> OrderIso.image_Ioc is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ioc.{u1} \u03b1 _inst_1 a b)) (Set.Ioc.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ioc.{u2} \u03b1 _inst_1 a b)) (Set.Ioc.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Ioc OrderIso.image_Ioc\u2093'. -/\n@[simp]\ntheorem image_Ioc (e : \u03b1 \u2243o \u03b2) (a b : \u03b1) : e '' Ioc a b = Ioc (e a) (e b) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Ioc, e.symm_symm]\n#align order_iso.image_Ioc OrderIso.image_Ioc\n\n/- warning: order_iso.image_Ico -> OrderIso.image_Ico is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Ico.{u1} \u03b1 _inst_1 a b)) (Set.Ico.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Ico.{u2} \u03b1 _inst_1 a b)) (Set.Ico.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Ico OrderIso.image_Ico\u2093'. -/\n@[simp]\ntheorem image_Ico (e : \u03b1 \u2243o \u03b2) (a b : \u03b1) : e '' Ico a b = Ico (e a) (e b) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Ico, e.symm_symm]\n#align order_iso.image_Ico OrderIso.image_Ico\n\n/- warning: order_iso.image_Icc -> OrderIso.image_Icc is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : Preorder.{u2} \u03b2] (e : OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u2} (Set.{u2} \u03b2) (Set.image.{u1, u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e) (Set.Icc.{u1} \u03b1 _inst_1 a b)) (Set.Icc.{u2} \u03b2 _inst_2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e a) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (OrderIso.{u1, u2} \u03b1 \u03b2 (Preorder.toLE.{u1} \u03b1 _inst_1) (Preorder.toLE.{u2} \u03b2 _inst_2)) (fun (_x : RelIso.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) => \u03b1 -> \u03b2) (RelIso.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (LE.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)) (LE.le.{u2} \u03b2 (Preorder.toLE.{u2} \u03b2 _inst_2))) e b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : Preorder.{u2} \u03b1] [_inst_2 : Preorder.{u1} \u03b2] (e : OrderIso.{u2, u1} \u03b1 \u03b2 (Preorder.toLE.{u2} \u03b1 _inst_1) (Preorder.toLE.{u1} \u03b2 _inst_2)) (a : \u03b1) (b : \u03b1), Eq.{succ u1} (Set.{u1} \u03b2) (Set.image.{u2, u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e))) (Set.Icc.{u2} \u03b1 _inst_1 a b)) (Set.Icc.{u1} ((fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) a) _inst_2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) a) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Data.FunLike.Embedding._hyg.19 : \u03b1) => \u03b2) _x) (EmbeddingLike.toFunLike.{max (succ u2) (succ u1), succ u2, succ u1} (Function.Embedding.{succ u2, succ u1} \u03b1 \u03b2) \u03b1 \u03b2 (Function.instEmbeddingLikeEmbedding.{succ u2, succ u1} \u03b1 \u03b2)) (RelEmbedding.toEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) (RelIso.toRelEmbedding.{u2, u1} \u03b1 \u03b2 (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1281 : \u03b1) (x._@.Mathlib.Order.Hom.Basic._hyg.1283 : \u03b1) => LE.le.{u2} \u03b1 (Preorder.toLE.{u2} \u03b1 _inst_1) x._@.Mathlib.Order.Hom.Basic._hyg.1281 x._@.Mathlib.Order.Hom.Basic._hyg.1283) (fun (x._@.Mathlib.Order.Hom.Basic._hyg.1296 : \u03b2) (x._@.Mathlib.Order.Hom.Basic._hyg.1298 : \u03b2) => LE.le.{u1} \u03b2 (Preorder.toLE.{u1} \u03b2 _inst_2) x._@.Mathlib.Order.Hom.Basic._hyg.1296 x._@.Mathlib.Order.Hom.Basic._hyg.1298) e)) b))\nCase conversion may be inaccurate. Consider using '#align order_iso.image_Icc OrderIso.image_Icc\u2093'. -/\n@[simp]\ntheorem image_Icc (e : \u03b1 \u2243o \u03b2) (a b : \u03b1) : e '' Icc a b = Icc (e a) (e b) := by\n  rw [e.image_eq_preimage, e.symm.preimage_Icc, e.symm_symm]\n#align order_iso.image_Icc OrderIso.image_Icc\n\nend Preorder\n\n/- warning: order_iso.Iic_top -> OrderIso.IicTop is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : OrderTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)], OrderIso.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} \u03b1) Type.{u1} (Set.hasCoeToSort.{u1} \u03b1) (Set.Iic.{u1} \u03b1 _inst_1 (Top.top.{u1} \u03b1 (OrderTop.toHasTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2)))) \u03b1 (Subtype.hasLe.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Set.{u1} \u03b1) (Set.hasMem.{u1} \u03b1) x (Set.Iic.{u1} \u03b1 _inst_1 (Top.top.{u1} \u03b1 (OrderTop.toHasTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2))))) (Preorder.toLE.{u1} \u03b1 _inst_1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : OrderTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)], OrderIso.{u1, u1} (Set.Elem.{u1} \u03b1 (Set.Iic.{u1} \u03b1 _inst_1 (Top.top.{u1} \u03b1 (OrderTop.toTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2)))) \u03b1 (Subtype.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) (fun (x : \u03b1) => Membership.mem.{u1, u1} \u03b1 (Set.{u1} \u03b1) (Set.instMembershipSet.{u1} \u03b1) x (Set.Iic.{u1} \u03b1 _inst_1 (Top.top.{u1} \u03b1 (OrderTop.toTop.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2))))) (Preorder.toLE.{u1} \u03b1 _inst_1)\nCase conversion may be inaccurate. Consider using '#align order_iso.Iic_top OrderIso.IicTop\u2093'. -/\n/-- Order isomorphism between `Iic (\u22a4 : \u03b1)` and `\u03b1` when `\u03b1` has a top element -/\ndef IicTop [Preorder \u03b1] [OrderTop \u03b1] : Set.Iic (\u22a4 : \u03b1) \u2243o \u03b1 :=\n  { @Equiv.subtypeUnivEquiv \u03b1 (Set.Iic (\u22a4 : \u03b1)) fun x => le_top with\n    map_rel_iff' := fun x y => by rfl }\n#align order_iso.Iic_top OrderIso.IicTop\n\n/- warning: order_iso.Ici_bot -> OrderIso.IciBot is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : OrderBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)], OrderIso.{u1, u1} (coeSort.{succ u1, succ (succ u1)} (Set.{u1} \u03b1) Type.{u1} (Set.hasCoeToSort.{u1} \u03b1) (Set.Ici.{u1} \u03b1 _inst_1 (Bot.bot.{u1} \u03b1 (OrderBot.toHasBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2)))) \u03b1 (Subtype.hasLe.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) (fun (x : \u03b1) => Membership.Mem.{u1, u1} \u03b1 (Set.{u1} \u03b1) (Set.hasMem.{u1} \u03b1) x (Set.Ici.{u1} \u03b1 _inst_1 (Bot.bot.{u1} \u03b1 (OrderBot.toHasBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2))))) (Preorder.toLE.{u1} \u03b1 _inst_1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Preorder.{u1} \u03b1] [_inst_2 : OrderBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1)], OrderIso.{u1, u1} (Set.Elem.{u1} \u03b1 (Set.Ici.{u1} \u03b1 _inst_1 (Bot.bot.{u1} \u03b1 (OrderBot.toBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2)))) \u03b1 (Subtype.le.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) (fun (x : \u03b1) => Membership.mem.{u1, u1} \u03b1 (Set.{u1} \u03b1) (Set.instMembershipSet.{u1} \u03b1) x (Set.Ici.{u1} \u03b1 _inst_1 (Bot.bot.{u1} \u03b1 (OrderBot.toBot.{u1} \u03b1 (Preorder.toLE.{u1} \u03b1 _inst_1) _inst_2))))) (Preorder.toLE.{u1} \u03b1 _inst_1)\nCase conversion may be inaccurate. Consider using '#align order_iso.Ici_bot OrderIso.IciBot\u2093'. -/\n/-- Order isomorphism between `Ici (\u22a5 : \u03b1)` and `\u03b1` when `\u03b1` has a bottom element -/\ndef IciBot [Preorder \u03b1] [OrderBot \u03b1] : Set.Ici (\u22a5 : \u03b1) \u2243o \u03b1 :=\n  { @Equiv.subtypeUnivEquiv \u03b1 (Set.Ici (\u22a5 : \u03b1)) fun x => bot_le with\n    map_rel_iff' := fun x y => by rfl }\n#align order_iso.Ici_bot OrderIso.IciBot\n\nend OrderIso\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Set/Intervals/OrderIso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.737158174177441, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3800934356439042}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from assume (i j : \u2124) (hne : i \u2260 j),\n    have h2 : \u03b1 \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j), from assume h3 : \u03b1 = int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j),\n      have h4 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2208 \u211a, from by {rw h3, exact \u27e8\u03b1, \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9\u27e9},\n      have h5 : (int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j)) \u2260 \u03b1, from by {apply h\u03b1_irrat, exact h4},\n      show false, from h3.symm \u25b8 h5,\n    have h6 : \u03b1 \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i), from assume h7 : \u03b1 = int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i),\n      have h8 : int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2208 \u211a, from by {rw h7, exact \u27e8\u03b1, \u27e8\u27e8\u27e9\u27e9\u27e9},\n      have h9 : (int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i)) \u2260 \u03b1, from by {apply h\u03b1_irrat, exact h8},\n      show false, from h7.symm \u25b8 h9,\n    have h10 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i), from by {apply int.fract_ne_of_ne_rat, exact hne,},\n    have h11 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j), from by {apply int.fract_ne_of_ne_rat, exact hne,},\n    have h12 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j), from by {split, exact h10, exact h11},\n    have h13 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1, from by {split, exact h2, split, exact h6, exact h2},\n    have h14 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i), from by {split, exact h13.left, split, exact h13.right.left, exact h13.right.right},\n    have h15 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j), from by {split, exact h14.left, split, exact h14.right.left, split, exact h14.right.right.left, exact h14.right.right.right},\n    have h16 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j), from by {split, exact h15.left, split, exact h15.right.left, split, exact h15.right.right.left, split, exact h15.right.right.right.left, exact h15.right.right.right.right},\n    have h17 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i), from by {split, exact h16.left, split, exact h16.right.left, split, exact h16.right.right.left, split, exact h16.right.right.right.left, split, exact h16.right.right.right.right.left, split, exact h16.right.right.right.right.right, exact h10},\n    have h18 : int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 \u03b1 \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2227 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2260 int.fract (\u03b1 * \u2191i) / int.fract (\u03b1 * \u2191j) \u2227 int.fract (\u03b1 * \u2191j) / int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191i) /\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  sorry,\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from \n    assume (i j : \u2124) (hi_neq_j : i \u2260 j),\n    assume h1 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j),\n    have h2 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from \n      by {rw [\u2190 sub_div_iff_mul_add], rw [\u2190 int.fract_int_add_fract,\u2190 int.fract_int_add_fract],\n      rw h1, ring},\n    have h3 : \u03b1 \u2208 \u211a, from by {exact fractional.exists_rat \u03b1 h2},\n    show false, from by {exact absurd h3 h\u03b1_irrat},\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1, from \n    assume (i j : \u2124) (hi_neq_j : i \u2260 j),\n    have h2 : int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2265 0, from \n      by {apply sub_nonneg, apply int.fract_nonneg},\n    have h3 : int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2264 1, from \n      by {rw [\u2190 sub_self (int.fract (\u03b1 * \u2191j)),sub_le_iff_le_add'],\n      apply int.fract_le, rw \u2190 int.fract_int_add_fract},\n    have h4 : int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1, from and.intro h2 h3,\n    exact h4,\n  have h2 : \u2200 i : \u2124, int.fract (\u03b1 * \u2191i) \u2208 closure (set.Icc 0 1), from \n    assume i : \u2124,\n    have h3 : \u2200 j : \u2124, j \u2260 i \u2192 int.fract (\u03b1 * \u2191i) \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) (int.fract (\u03b1 * \u2191i) + int.fract (\u03b1 * \u2191j))), from\n      assume j : \u2124,\n      assume hi_neq_j : j \u2260 i,\n      have h4 : int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1, from h1 i j hi_neq_j,\n      have h5 : int.fract (\u03b1 * \u2191i) + int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1, from \n        by {have h6 : int.fract (\u03b1 * \u2191i) + int.fract (\u03b1 * \u2191j) = -(int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)),\n        from by {rw [\u2190 neg_add_rev,sub_eq_add_neg,\u2190 sub_self (int.fract (\u03b1 * \u2191j)),sub_add_eq_add_sub],\n        ring, rw \u2190 int.fract_int_add_fract},\n        rw h6, exact h4},\n      have h6 : int.fract (\u03b1 * \u2191i) \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) (int.fract (\u03b1 * \u2191i) + int.fract (\u03b1 * \u2191j))), from \n        by {rw [\u2190 set.Icc_sub_sub_sub,sub_self (int.fract (\u03b1 * \u2191i)),sub_self (int.fract (\u03b1 * \u2191i))],\n        apply int.fract_in_Icc_iff, rw [sub_self (int.fract (\u03b1 * \u2191i)),sub_self (int.fract (\u03b1 * \u2191i))],\n        exact (and.intro h4 h5)},\n      exact h6,\n    have h7 : int.fract (\u03b1 * \u2191i) \u2208 closure (set.Icc 0 1), from \n      by {apply set.mem_closure_of_forall, assume (j : \u2124) (hi_neq_j : j \u2260 i),\n      have h8 : int.fract (\u03b1 * \u2191i) \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) (int.fract (\u03b1 * \u2191i) + int.fract (\u03b1 * \u2191j))), from h3 j hi_neq_j,\n      rw [\u2190 set.Icc_sub_sub_sub,sub_self (int.fract (\u03b1 * \u2191i)),sub_self (int.fract (\u03b1 * \u2191i))],\n      exact h8},\n    exact h7,\n  have h3 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 i : \u2124, y \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i)) (int.fract (\u03b1 * \u2191(i+1)))), from \n    assume y : \u211d,\n    assume hy_in_Icc : y \u2208 set.Icc 0 1,\n    have h4 : \u2203 i : \u2124, y < int.fract (\u03b1 * \u2191(i+1)), from \n      by {apply int.fract_pos_exists_lt_fract, rw \u2190 int.fract_int_add_fract,\n      apply lt_of_lt_of_le, apply lt_add_one, exact hy_in_Icc},\n    have h5 : \u2203 i : \u2124, y > int.fract (\u03b1 * \u2191i), from \n      by {apply int.fract_pos_exists_gt_fract, rw \u2190 int.fract_int_add_fract,\n      apply lt_of_le_of_lt, exact hy_in_Icc, apply lt_add_one},\n    have h6 : \u2200 i : \u2124, y \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i)) (int.fract (\u03b1 * \u2191(i+1)))), from \n      assume i : \u2124,\n      have h7 : y \u2208 set.Icc (int.fract (\u03b1 * \u2191i)) (int.fract (\u03b1 * \u2191(i+1))), from \n        by {apply int.fract_in_Icc_iff, rw [\u2190 int.fract_int_add_fract],\n        have h8 : int.fract (\u03b1 * \u2191i) \u2264 y, from by {apply le_of_lt, apply classical.some (h5 i)},\n        have h9 : y < int.fract (\u03b1 * \u2191(i+1)), from classical.some (h4 i),\n        exact and.intro h8 h9},\n      exact set.mem_closure_of_mem h7,\n    have h8 : \u2203 i : \u2124, y \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i)) (int.fract (\u03b1 * \u2191(i+1)))), from \n      by {apply exists_nat_one_lt, apply classical.some (h4 0)},\n    exact h8,\n  have h4 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 i : \u2124, y \u2208 closure (set.Icc (int.fract (\u03b1 * \u2191i)) (int.fract (\u03b1 * \u2191(i+1)))), from \n    assume y : \u211d,\n    assume hy_in_Icc : y \u2208 set.Icc 0 1,\n    exact h3 y hy_in_Icc,\n  have h5 : \u2200 y : \u211d, y\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from\n    assume (i j : \u2124) (hij : i \u2260 j),\n      assume h2 : int.fract (\u03b1 *\u2191i) = int.fract (\u03b1 * \u2191j),\n      have h3 : (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191i), from (int.fract_eq_of_nonneg (by norm_num)),\n      have h4 : (\u03b1 * \u2191j) - int.floor (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191j), from (int.fract_eq_of_nonneg (by norm_num)),\n      have h5 : (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191i) = (\u03b1 * \u2191j) - int.floor (\u03b1 * \u2191j), from by {rw h2},\n      have h6 : \u03b1 = ((int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j)) / (i-j)), from by {rw [\u2190 h3, \u2190 h4, \u2190 h5], norm_num},\n      have h7 : (i-j) \u2260 0, from by {rw \u2190 int.coe_nat_eq_zero, intro h8, rw h8 at hij, exact hij rfl},\n      have h8 : (int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j)) = (i-j) * \u03b1, from by {\n        have h9 : (i-j) \u2260 0, from by {rw \u2190 int.coe_nat_eq_zero, intro h10, rw h10 at hij, exact hij rfl},\n        have h10 : (i-j) * \u03b1 \u2260 0, from by {rw mul_eq_zero, exact h7, exact h\u03b1_irrat},\n        rw mul_comm (i-j) \u03b1, rw \u2190 div_mul_cancel h10, rw h6, ring},\n      have h9 : (int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j)) \u2208 \u2124, from by {rw h8, exact int.coe_int_eq_coe_nat.mp (int.coe_nat_inj (by norm_num))},\n      have h10 : (i-j) \u2265 0, from by {rw int.coe_nat_nonneg, norm_num},\n      have h11 : \u03b1 * (i-j) \u2265 0, from (mul_nonneg h\u03b1_irrat h10),\n      have h12 : (int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j)) \u2265 0, from by {rw \u2190 h8, exact h11},\n      have h13 : int.floor (\u03b1 * \u2191i) \u2265 int.floor (\u03b1 * \u2191j), from by {rw \u2190 int.coe_nat_le_coe_nat_iff, exact int.coe_nat_nonneg.mp (by norm_num)},\n      have h14 : int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j) = 0, from by {rw int.coe_nat_zero, exact int.coe_nat_inj (by norm_num)},\n      have h15 : (int.floor (\u03b1 * \u2191i) - int.floor (\u03b1 * \u2191j)) = 0, from by {rw int.coe_nat_zero, exact int.coe_nat_inj (by norm_num)},\n      have h16 : (i-j) = 0, from by {rw \u2190 h8, rw h15, exact int.coe_int_eq_coe_nat.mp (int.coe_nat_inj (by norm_num))},\n      have h17 : i = j, from by {rw \u2190 int.coe_nat_eq_coe_nat_iff, rw \u2190 int.coe_nat_eq_coe_nat_iff, rw h16, exact int.coe_nat_inj (by norm_num)},\n      exact hij h17,\n  have h2 : \u2200 (x y : \u2124), x \u2260 y \u2192 int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y), from\n    assume (x y : \u2124) (hxy : x \u2260 y), h1 x y (hxy),\n  have h3 : \u2200 (x y : \u2124), int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y) \u2192 x = y, from\n    assume (x y : \u2124), assume h4 : int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y),\n      have h5 : int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y), from h2 x y (by {assume h6, rw h6 at h4, exact h4 rfl}),\n      exact absurd h4 h5,\n  have h4 : \u2200 (x y : \u2124), int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y) \u2192 x = y, from\n    assume (x y : \u2124), assume h5 : int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y),\n      have h6 : int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y), from h2 x y (by {assume h7, rw h7 at h5, exact h5 rfl}),\n      exact absurd h5 h6,\n  have h5 : \u2200 (x : \u2124), int.fract (\u03b1 * \u2191x) \u2208 set.Icc 0 1, from\n    assume (x : \u2124), have h6 : int.fract (\u03b1 * \u2191x) \u2265 0, from by {rw int.fract_eq_of_nonneg, exact mul_nonneg h\u03b1_irrat (int.coe_nat_nonneg.mp (by norm_num))},\n    have h7 : int.fract (\u03b1 * \u2191x) \u2264 1, from by {rw int.fract_eq_of_le, exact le_one_of_mul_le_one_right h\u03b1_irrat (int.coe_nat_nonneg.mp (by norm_num))},\n    show int.fract (\u03b1 * \u2191x) \u2208 set.Icc 0 1, from by {split, exact h6, exact h7},\n  have h6 : \u2200 (x : \u2124), int.fract (\u03b1 * \u2191x) \u2208 set.Icc 0 1, from\n    assume (x : \u2124), have h7 : int.fract (\u03b1 * \u2191x) \u2265 0, from by {rw int.fract_eq_of_nonneg, exact mul_nonneg h\u03b1_irrat (int.coe_nat_nonneg.mp (by norm_num))},\n    have h8 : int.fract (\u03b1 * \u2191x) \u2264 1, from by {rw int.fract_eq_of_le, exact le_one_of_mul_le_one_right h\u03b1_irrat (int.coe_nat_nonneg.mp (by norm_num))},\n    show int.fract (\u03b1 * \u2191x) \u2208 set.Icc 0 1, from by {split, exact h7, exact h8},\n  have h7 : \u2200 (x : \u2124), int.fract (\u03b1 * \u2191x) \u2208 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' set.univ), from\n    assume (x : \u2124), have h8 : int.fract (\u03b1 * \u2191x) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' set.univ, from \u27e8x, rfl\u27e9,\n    have h9 : \u2200 (y : \u2124), int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y) \u2192 y \u2209 set.univ, from\n      assume (y : \u2124), assume h10 : int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y),\n      have h11 : y \u2209 set.univ, from by {assume h12, rw\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h\u03b1_irrat : irrational \u03b1, {exact h\u03b1_irrat},\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 set.finite {x : \u2124 | int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i)}, {\n    assume i j : \u2124,\n    assume hi_neq_j : i \u2260 j,\n    have h2 : int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), {\n      assume h3 : int.fract (\u03b1 * \u2191i) = int.fract (\u03b1 * \u2191j),\n      have h4 : (\u03b1 : \u211d) = \u2124.to_frac (int.fract (\u03b1 * \u2191i)), {\n        have h5 : (\u03b1 : \u211d) = \u2124.to_frac (int.fract (\u03b1 * \u2191i)) + (\u03b1 - \u2124.to_frac (int.fract (\u03b1 * \u2191i))), {\n          have h6 : (\u03b1 : \u211d) = (\u03b1 * \u2191i) - ((\u03b1 * \u2191i) % 1), {\n            rw \u2190 int.fract_add_one (\u03b1 * \u2191i),\n            rw int.fract_eq_of_lt (mul_pos (\u03b1 : \u211d) (int.coe_nat_lt_coe_nat_of_lt (int.coe_nat_pos i))),\n          },\n          rw \u2190 h6,\n          rw \u2190 int.fract_add_one (\u03b1 * \u2191i),\n          ring,\n        },\n        have h7 : (\u03b1 : \u211d) - \u2124.to_frac (int.fract (\u03b1 * \u2191i)) = 0, {\n          rw \u2190 h5,\n          rw int.fract_add_one (\u03b1 * \u2191i),\n          ring,\n        },\n        rw \u2190 h7,\n        ring,\n      },\n      have h8 : (\u03b1 * \u2191i : \u211d) = (\u03b1 * \u2191j : \u211d), {\n        rw \u2190 h4,\n        rw \u2190 h3,\n        ring,\n      },\n      have h9 : \u03b1 = j / i, {\n        have h10 : (\u03b1 * \u2191i : \u211d) = (j : \u211d), {\n          rw \u2190 h8,\n          ring,\n        },\n        rw \u2190 h10,\n        rw mul_comm,\n        ring,\n      },\n      have h11 : j = i * (j / i), {\n        rw h9,\n        ring,\n      },\n      have h12 : j = i, {\n        rw mul_comm,\n        rw h11,\n        ring,\n      },\n      contradiction,\n    },\n    have h3 : \u2203 (x : \u2124) (y : \u2124), x \u2260 y \u2227 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y), {\n      use i,\n      use j,\n      split,\n      exact hi_neq_j,\n      exact h2,\n    },\n    have h4 : \u2200 (x : \u2124) (y : \u2124), x \u2260 y \u2192 int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y), {\n      assume x y : \u2124,\n      assume h5 : x \u2260 y,\n      assume h6 : int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y),\n      have h7 : \u2203 (x : \u2124) (y : \u2124), x \u2260 y \u2227 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y), {\n        use x,\n        use y,\n        split,\n        exact h5,\n        exact h6,\n      },\n      have h8 : \u2203 (x : \u2124) (y : \u2124), x \u2260 y \u2227 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y), {\n        use i,\n        use j,\n        split,\n        exact hi_neq_j,\n        exact h2,\n      },\n      contradiction,\n    },\n    have h5 : \u2200 (x : \u2124) (y : \u2124), int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y) \u2192 x = y, {\n      assume x y : \u2124,\n      assume h6 : int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191y),\n      have h7 : x = y \u2228 x \u2260 y, from decidable.em (x = y),\n      cases h7,\n      exact h7,\n      have h8 : int.fract (\u03b1 * \u2191x) \u2260 int.fract (\u03b1 * \u2191y), {\n        exact h4 x y h7,\n      },\n      contradiction,\n    },\n    have h6 : \u2200 (x : \u2124), int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i) \u2192 x = i, {\n      exact h5 i,\n    },\n    have h7 : \u2200 (x : \u2124), int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i) \u2192 x \u2208 {x : \u2124 | int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i)}, {\n      assume x : \u2124,\n      assume h8 : int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i),\n      have h9 : x = i, {\n        exact h6 x h8,\n      },\n      rw h9,\n      refl,\n    },\n    have h8 : \u2200 (x : \u2124), x \u2208 {x : \u2124 | int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i)} \u2192 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i), {\n      assume x : \u2124,\n      assume h9 : x \u2208 {x : \u2124 | int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i)},\n      exact h9,\n    },\n    have h9 : {x : \u2124 | int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i)} = {i}, {\n      apply set.ext,\n      exact h7,\n      exact h8,\n    },\n    exact set.finite_singleton i,\n  },\n  have h2 : \u2200 i : \u2124, i \u2208 ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2192 \u2203 j : \u2124, j \u2260 i \u2227 j \u2208 ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), {\n    assume i : \u2124,\n    assume hi : i \u2208 ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)),\n    have h3 : \u2203 j : \u2124, j \u2260 i \u2227 j \u2208 ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), {\n      have h4 : \u2203 j : \u2124, j \u2260 i \u2227 int.fract (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191i), {\n        have h5 : \u2203 j : \u2124, j \u2260 i \u2227 int.fract (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191i), {\n          have h6 : \u2203 x : \u2124, x \u2260 i \u2227 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i), {\n            have h7 : \u2203 j : \u2124, j \u2260 i \u2227 int.fract (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191i), {\n              have h8 : \u2203 x : \u2124, x \u2260 i \u2227 int.fract (\u03b1 * \u2191x) = int.fract (\u03b1 * \u2191i), {\n                have h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from assume (m n : \u2124) (hmn : m \u2260 n),\n    show int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from by {\n      assume h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n),\n      have : \u03b1 = (int.fract (\u03b1 * \u2191m) - int.fract (\u03b1 * \u2191n))/(m - n), from by {\n        rw [h, int.fract_sub_fract_of_lt (lt_of_le_of_ne (le_of_lt (int.fract_lt_one (\u03b1 * \u2191m))) (ne.symm hmn))],\n        ring,\n      },\n      have h\u03b1_rat : \u03b1 \u2208 \u211a, from by {\n        have h\u03b1_int : \u03b1 \u2208 \u2124, from by {rw \u2190 int.cast_coe_nat, apply int.cast_injective, apply int.coe_nat_injective,\n          rw \u2190 int.cast_coe_nat, exact this,\n        },\n        show \u03b1\u2208\u211a, from by {\n          apply set.mem_of_mem_range,\n          apply int.cast_injective,\n          rw \u2190 int.cast_coe_nat,\n          exact h\u03b1_int,\n        }\n      },\n      exact h\u03b1_rat.elim h\u03b1_irrat\n    },\n  have h2 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from assume (m n : \u2124) (hmn : m \u2260 n),\n    show int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from by {\n      have h\u03b1_lt : (0 : \u211d) < \u03b1, from by {\n        have h\u03b1_int : \u03b1 \u2208 \u2124, from by {rw \u2190 int.cast_coe_nat, apply int.cast_injective, apply int.coe_nat_injective,\n          rw \u2190 int.cast_coe_nat, exact h\u03b1_irrat\n        },\n        have h1 : (0 : \u2115) \u2208 \u2115, from by {apply set.mem_univ (0 : \u2115)},\n        have h2 : (0 : \u2115) \u2208 \u2124, from by {\n          apply set.mem_of_mem_range,\n          apply int.cast_injective,\n          rw \u2190 int.cast_coe_nat,\n          exact h1,\n        },\n        show 0 < \u03b1, from by {\n          have h\u03b1_n : \u03b1 \u2208 \u2115, from by {apply int.coe_nat_injective, exact h\u03b1_int},\n          show 0 < \u03b1, from by {\n            apply lt_of_le_of_lt,\n            apply int.cast_le.mpr,\n            apply set.le_of_subset,\n            have h3 : (0 : \u2115) \u2286 \u2115, from by {unfold set.subset, exact \u03bb (x : \u2115), set.mem_univ x},\n            show (0 : \u2115) \u2286 \u03b1, from by {apply h3},\n          }\n        }\n      },\n      have h\u03b1_one : \u03b1 < (1 : \u211d), from by {\n        have h\u03b1_int : \u03b1 \u2208 \u2124, from by {rw \u2190 int.cast_coe_nat, apply int.cast_injective, apply int.coe_nat_injective,\n          rw \u2190 int.cast_coe_nat, exact h\u03b1_irrat\n        },\n        have h1 : (1 : \u2115) \u2208 \u2115, from by {apply set.mem_univ (1 : \u2115)},\n        have h2 : (1 : \u2115) \u2208 \u2124, from by {\n          apply set.mem_of_mem_range,\n          apply int.cast_injective,\n          rw \u2190 int.cast_coe_nat,\n          exact h1,\n        },\n        show \u03b1 < 1, from by {\n          have h\u03b1_n : \u03b1 \u2208 \u2115, from by {apply int.coe_nat_injective, exact h\u03b1_int},\n          show \u03b1 < 1, from by {\n            apply set.lt_of_subset_of_lt,\n            have h3 : \u03b1 \u2286 (1 : \u2115), from by {unfold set.subset, exact \u03bb (x : \u2115), set.mem_univ x},\n            show \u03b1 \u2286 1, from by {apply h3},\n            have h4 : (1 : \u2115) \u2208 \u2115, from by {apply set.mem_univ (1 : \u2115)},\n            show 1 \u2208 \u03b1, from by {apply h4},\n          }\n        }\n      },\n      show int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from by {\n        apply set.mem_Icc_of_mem_Ioo,\n        show (0 : \u211d) < int.fract (\u03b1 * \u2191m), from by {\n          apply int.fract_lt_one,\n          show (0 : \u211d) < \u03b1 * \u2191m, from by {\n            rw int.cast_coe_nat at hmn,\n            have hm_pos : (0 : \u211d) < \u2191m, from by {\n              have hm_int : m \u2208 \u2124, from by {apply set.mem_univ m},\n              have hm_nat : m \u2208 \u2115, from by {\n                apply int.coe_nat_injective,\n                apply int.cast_injective,\n                exact hm_int,\n              },\n              show 0 < \u2191m, from by {\n                apply lt_of_le_of_lt,\n                apply int.cast_le.mpr,\n                apply set.le_of_subset,\n                have h1 : (0 : \u2115) \u2286 \u2115, from by {unfold set.subset, exact \u03bb (x : \u2115), set.mem_univ x},\n                show (0 : \u2115) \u2286 m, from by {apply h1},\n              }\n            },\n            show 0 < \u03b1 * \u2191m, from by {\n              apply mul_pos h\u03b1_lt hm_pos,\n            }\n          }\n        },\n        show int.fract (\u03b1 * \u2191m) < (1 : \u211d), from by {\n          have h\u03b1_one : \u03b1 < (1 : \u211d), from by {\n            have h\u03b1_int : \u03b1 \u2208 \u2124, from by {rw \u2190 int.cast_coe_nat, apply int.cast_injective, apply int.coe_nat_injective,\n              rw \u2190 int.cast_coe_nat, exact h\u03b1_irrat\n            },\n            have h1 : (1 : \u2115) \u2208 \u2115, from by {apply set.mem_univ (1 : \u2115)},\n            have h2 : (1 : \u2115) \u2208 \u2124, from by {\n              apply set.mem_of_mem_range,\n              apply int.cast_injective,\n              rw \u2190 int.cast_coe_nat,\n              exact h1,\n            },\n            show \u03b1 < 1, from by {\n              have h\u03b1_n : \u03b1 \u2208 \u2115, from by {apply int.coe_nat_injective, exact h\u03b1_int},\n              show \u03b1 < 1, from by {\n                apply set.lt_of_subset_of_lt,\n                have h3 : \u03b1 \u2286 (1 : \u2115), from by {unfold set.subset, exact \u03bb (x : \u2115), set.mem_univ x},\n                show \u03b1 \u2286 1, from by {apply h3},\n                have h4 : (1 : \u2115) \u2208 \u2115, from by {apply set.mem_univ (1 :\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  let S : set \u211d := (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124),\n  have h_S_nonempty : (\u2203 (r : \u211d), r \u2208 S), from by {\n    use (int.fract \u03b1), simp,\n    have h_irrat_pos : (\u03b1 > 0), from by { rw \u2190 int.fract_eq_iff_irrat h\u03b1_irrat, norm_num, },\n    have h_pos : (0 < int.fract \u03b1), from by { apply lt_of_le_of_lt, norm_num, exact h_irrat_pos, },\n    have h_pos' : (0 < int.fract (\u03b1 * 1)), from by { exact int.fract_mul h_pos, },\n    simp [int.fract_mul, h_pos']\n  },\n  have h_S_infinite : (\u2200 (r : \u211d), r \u2208 S \u2192 \u2203 (s : \u211d), (s \u2208 S \u2227 s \u2260 r)), from by {\n    assume r : \u211d,\n    assume h_r_mem : r \u2208 S,\n    have h_r_mem' : r \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from by { exact h_r_mem, },\n    have h_r_mem'' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 m \u2208 @set.univ \u2124)), from by { exact h_r_mem', },\n    have h_r_mem''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r)), from by { exact h_r_mem'', },\n    have h_r_mem'''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 m \u2260 0)), from by { apply exists_ne, exact h_r_mem''' },\n    have h_r_mem''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m)), from by { apply exists_pos, exact h_r_mem'''' },\n    have h_r_mem'''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2260 r)), from by { apply exists_ne, exact h_r_mem''''' },\n    have h_r_mem''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2260 r \u2227 m \u2208 @set.univ \u2124)), from by { apply exists_mem_univ, exact h_r_mem'''''' },\n    have h_r_mem'''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2260 r \u2227 m \u2208 @set.univ \u2124 \u2227 r \u2208 S)), from by { split, exact h_r_mem'''', exact h_r_mem, },\n    have h_r_mem''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 r \u2208 S)), from by { exact h_r_mem'''''''' },\n    have h_r_mem'''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 r \u2208 S \u2227 m \u2260 r)), from by { exact h_r_mem''''''''' },\n    have h_r_mem''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 m \u2260 r)), from by { exact h_r_mem'''''''''' },\n    have h_r_mem'''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 m \u2260 r \u2227 r \u2208 S)), from by { exact h_r_mem''''''''''' },\n    have h_r_mem''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124)), from by { exact h_r_mem'''''''''''' },\n    have h_r_mem'''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 m \u2260 r)), from by { exact h_r_mem''''''''''''' },\n    have h_r_mem''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 @set.univ \u2124 \u2227 m \u2260 r \u2227 m \u2208 S)), from by { split, exact h_r_mem'''''''''''''', rw mem_image, exact h_r_mem'''''''''''''' },\n    have h_r_mem'''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2260 r)), from by { exact h_r_mem''''''''''''''' },\n    have h_r_mem''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2260 r \u2227 m \u2208 @set.univ \u2124)), from by { exact h_r_mem'''''''''''''''' },\n    have h_r_mem'''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S)), from by { exact h_r_mem''''''''''''''''' },\n    have h_r_mem''''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2260 r)), from by { exact h_r_mem'''''''''''''''''' },\n    have h_r_mem'''''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2260 r \u2227 m \u2208 @set.univ \u2124)), from by { exact h_r_mem''''''''''''''''''' },\n    have h_r_mem''''''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2208 @set.univ \u2124)), from by { exact h_r_mem'''''''''''''''''''' },\n    have h_r_mem'''''''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2208 @set.univ \u2124 \u2227 m \u2260 r)), from by { exact h_r_mem''''''''''''''''''''' },\n    have h_r_mem''''''''''''''''''''''' : (\u2203 (m : \u2124), (int.fract (\u03b1 * \u2191m) = r \u2227 0 < m \u2227 m \u2208 S \u2227 m \u2260 r)), from by { exact h_r_mem'''''''''''''''''''''' },\n    exact h_r_mem''''''''''''''''''''''',\n  },\n  have h_S_dense : (\u2200 (r : \u211d), r \u2208 set.Icc 0 1 \u2192\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 m n \u2208 \u2124, (\u03b1 * m) \u2260 (\u03b1 * n), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (h : (\u03b1 * m) = (\u03b1 * n)),\n  begin\n    have h2 : (\u03b1 * m) - m = (\u03b1 * n) - n, from by rw [h],\n    have h3 : \u03b1 = (m - n)\u207b\u00b9 * ((m - n) * (\u03b1)), from by rw [int.fract_add_int, int.fract_add_int] at h2,\n    have h4 : \u03b1 \u2208 \u211a, from by {rw [h3], apply int.cast_ne_zero, rw [int.mul_sub, int.mul_zero], exact mul_ne_zero (int.cast_ne_zero (m - n)) h\u03b1_irrat},\n    exact absurd h4 (irrational_iff_not_rat.1 h\u03b1_irrat),\n  end,\n\n  have h2 : \u2200 m n \u2208 \u2124, (\u03b1 * m) - m \u2260 (\u03b1 * n) - n, from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (h : (\u03b1 * m) - m = (\u03b1 * n) - n),\n  begin\n    have h3 : (\u03b1 * m) = (\u03b1 * n), from by rw [h, add_sub_cancel],\n    exact h1 m n hm hn h3,\n  end,\n\n  have h3 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (\u03b1 * \u2191m) - m \u2260 (\u03b1 * \u2191n) - n, from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n),\n  begin\n    have h2 : (\u03b1 * \u2191m) = (\u03b1 * \u2191n), from by rw [h, add_sub_cancel],\n    exact h1 m n hm hn h2,\n  end,\n\n  have h4 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h5 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h6 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h7 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h8 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h9 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h10 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h11 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by rw [h],\n    exact h3 m n hm hn hmn h2,\n  end,\n\n  have h12 : \u2200 m n \u2208 \u2124, (m \u2260 n) \u2192 (int.fract (\u03b1 * \u2191m)) \u2260 (int.fract (\u03b1 * \u2191n)), from assume (m n : \u2124) (hm : m \u2208 \u2124) (hn : n \u2208 \u2124) (hmn : m \u2260 n) (h : int.fract (\u03b1 * \u2191m) = int.fract (\u03b1 * \u2191n)),\n  begin\n    have h2 : (\u03b1 * \u2191m) - m = (\u03b1 * \u2191n) - n, from by\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484144, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3800399260504307}}
{"text": "/-\nCopyright (c) 2022 Eric Wieser. All rights reserved.\nReleased under MIT license as described in the file LICENSE.\nAuthors: Eric Wieser\n-/\nimport linear_algebra.clifford_algebra.fold\nimport linear_algebra.bilinear_form\nimport linear_algebra.tensor_product\nimport linear_algebra.prod\nimport linear_algebra.clifford_algebra.grading\n\n/-!\n# Recursive computation rules\n\n## Main definitions\n\n* `clifford_algebra.foldr'`: a computation rule for building linear maps out of the clifford\n  algebra.\n\n-/\n\nuniverses u1 u2 u3\n\nvariables {R : Type u1} [comm_ring R]\nvariables {M : Type u2} [add_comm_group M] [module R M]\nvariables {N : Type u3} [add_comm_group N] [module R N]\nvariables (Q : quadratic_form R M)\n\nnamespace clifford_algebra\n\n-- lemma foldr'_mul_\u03b9 (f : M \u2192\u2097[R] clifford_algebra Q \u00d7 N \u2192\u2097[R] N)\n--   (hf : \u2200 m x fx, f m (\u03b9 Q m * x, f m (x, fx)) = Q m \u2022 fx) (n m) (x) :\n--   foldr' Q f hf n (x * \u03b9 Q m) = foldr' Q f hf (f m (1, n)) x :=\n-- begin\n--   dsimp [foldr'],\n--   rw [foldr_mul, foldr_\u03b9, foldr'_aux_apply_apply],\n--   dsimp only,\n--   rw mul_one,\n--   refine congr_arg (f m) (prod.mk.eta.symm.trans _),\n--   congr' 1,\n--   apply clifford_algebra.foldr_induction _ (\u03bb r, _) (\u03bb x y hx hy, _) (\u03bb m x hx, _) x,\n--   { simp_rw [foldr_algebra_map, prod.smul_mk, algebra.algebra_map_eq_smul_one] },\n--   { rw [map_add, prod.fst_add, hx, hy] },\n--   { rw [foldr_mul, foldr_\u03b9, foldr'_aux_apply_apply, hx], },\n-- end\n-- lemma foldr'_add\n--   (f g : M \u2192\u2097[R] clifford_algebra Q \u00d7 N \u2192\u2097[R] N)\n--   (hfg : \u2200 m x fgx, (f + g) m (\u03b9 Q m * x, (f + g) m (x, fgx)) = Q m \u2022 fgx)\n--   (hf : \u2200 m x fx, f m (\u03b9 Q m * x, f m (x, fx)) = Q m \u2022 fx)\n--   (hg : \u2200 m x gx, g m (\u03b9 Q m * x, g m (x, gx)) = Q m \u2022 gx) :\n--   foldr' Q (f + g) hfg 0 = foldr' Q f hf 0 + foldr' Q g hg 0 :=\n-- begin\n--   ext x,\n--   rw linear_map.add_apply,\n--   apply clifford_algebra.foldr_induction _ (\u03bb r, _) (\u03bb x y hx hy, _) (\u03bb m x hx, _) x,\n--   { simp_rw [foldr'_algebra_map, smul_zero, zero_add] },\n--   { rw [map_add, map_add, map_add, add_add_add_comm, hx, hy] },\n--   { simp_rw [foldr'_\u03b9_mul],\n--     rw [hx],\n--     rw hx,},\n-- end\n\n-- lemma foldr'_smul (c : R)\n--   (f : M \u2192\u2097[R] clifford_algebra Q \u00d7 N \u2192\u2097[R] N)\n--   (hfg : \u2200 m x fcx, (c \u2022 f) m (\u03b9 Q m * x, (c \u2022 f) m (x, fcx)) = Q m \u2022 fcx)\n--   (hf : \u2200 m x fx, f m (\u03b9 Q m * x, f m (x, fx)) = Q m \u2022 fx) :\n--   foldr' Q (c \u2022 f) hfg 0 = c \u2022 foldr' Q f hf 0 :=\n-- begin\n--   ext x,\n--   rw linear_map.smul_apply,\n--   apply clifford_algebra.foldr_induction _ (\u03bb r, _) (\u03bb x y hx hy, _) (\u03bb m x hx, _) x,\n--   { simp_rw [foldr'_algebra_map, smul_zero] },\n--   { rw [map_add, map_add,smul_add, hx, hy] },\n--   { simp_rw [foldr'_\u03b9_mul, line],\n--     rw [hx],\n--     rw hx,},\n-- end\n\nend clifford_algebra\n", "meta": {"author": "pygae", "repo": "lean-ga", "sha": "5e8b22b2f25c7037723ad811faa312660eeb6775", "save_path": "github-repos/lean/pygae-lean-ga", "path": "github-repos/lean/pygae-lean-ga/lean-ga-5e8b22b2f25c7037723ad811faa312660eeb6775/src/geometric_algebra/from_mathlib/fold.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.38003992605043063}}
{"text": "import data.matrix tactic.ring\n\nopen tactic declaration expr binder_info\n#print binder_fin\n\nlocal infix ` \u2b1d `:70 := matrix.mul\nlocal postfix `\u1d40` : 1500 := matrix.transpose\n#print matrix.mul\n--set_option trace.class_instances true\nset_option eqn_compiler.max_steps 100000\nset_option trace.app_builder true\nmeta def mk_new_theorem : expr \u2192 tactic (expr)\n| (pi n b dom cod) := do cod \u2190 mk_new_theorem cod, return (pi n b dom cod)\n| `(@matrix.mul %%l %%m %%n %%fl %%fm %%fn \u211a _ _ %%A %%B = %%C) :=\n    do k \u2190 mk_fresh_name,\n       k' \u2190 mk_local' k implicit `(\u2115),\n       fintype_k \u2190 mk_app `fin.fintype [k'],\n       D \u2190 mk_fresh_name,\n       D_type \u2190 mk_mapp `matrix [n, `(fin %%k'), fn, fintype_k, `(\u211a)],\n       B' \u2190 mk_local_def `B B,\n       D' \u2190 mk_local' D default D_type,\n       B_mul_D \u2190 mk_mapp `matrix.mul [some m, some n, some k', fm, fn,\n        fintype_k, `(rat), `(rat.has_mul), `(rat.add_comm_monoid), B', D'],\n       trace \"x\",\n       return`(\u2115)\n    --    A_mul_B_mul_D \u2190 mk_app `matrix.mul [A, B_mul_D],\n    --    C_mul_D \u2190 mk_app `matrix.mul [C, (var (v + 1))],\n    --    eeq \u2190 mk_app `eq [A_mul_B_mul_D, C_mul_D],\n    -- return (pi k implicit `(nat) (pi D implicit `(matrix (fin %%m) (fin %%(var v)) \u211a) eeq))\n    -- let e : expr := `(%%A \u2b1d (%%B \u2b1d D) = %%C \u2b1d D) in\n    -- return `(\u2200 {k : \u2115} (D : matrix (fin %%n) (fin k) \u211a), %%A \u2b1d (%%B \u2b1d D) = %%C \u2b1d D)\n| _ := do trace \"y\", failure\n\nrun_cmd do t \u2190 mk_local_pis `(\u2200 A B : matrix (fin 1) (fin 1) \u211a, A \u2b1d B = 1),\n  trace (t.1.map to_raw_fmt)\n\nmeta def codomain (e : expr) : expr :=\nif e.is_pi then codomain e.binding_body else e\n\n#print tactic.interactive.ring\nmeta def m_assoc_simp_attr : user_attribute :=\n{ name := `m_assoc_simp,\n  descr := \"given a proof A \u2b1d B = C, generate the lemma A \u2b1d (B \u2b1d D) = C \u2b1d D, and mark it with a simp attribute\",\n  after_set := some $ \u03bb n _ _,\n    do env \u2190 get_env,\n    dec \u2190 get_decl n,\n    match dec with\n    | thm n _ t _ := let target := codomain t in\n      match target with\n      | `(@matrix.mul (fin %%l) (fin %%m) (fin %%n) %%_x %%__x %%___x \u211a _ _ %%A %%B = %%C) :=\n        sorry\n      | _ := trace \"declaration has wrong type\", fail\n    | _ := trace \"declaratin is not a theorem\", fail\n    end }\n\n#reduce  user_attribute_cache_cfg unit\n", "meta": {"author": "ChrisHughes24", "repo": "LP", "sha": "e3ed64c2d1f642696104584e74ae7226d8e916de", "save_path": "github-repos/lean/ChrisHughes24-LP", "path": "github-repos/lean/ChrisHughes24-LP/LP-e3ed64c2d1f642696104584e74ae7226d8e916de/scratch/assoc_simp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484144, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.38003992605043063}}
{"text": "/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.bundled\nimport Mathlib.category_theory.monoidal.End\nimport Mathlib.category_theory.monoidal.Mon_\nimport Mathlib.category_theory.category.Cat\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n\n# The equivalence between `Monad C` and `Mon_ (C \u2964 C)`.\n\nA monad \"is just\" a monoid in the category of endofunctors.\n\n# Definitions/Theorems\n\n1. `to_Mon` associates a monoid object in `C \u2964 C` to any monad on `C`.\n2. `Monad_to_Mon` is the functorial version of `to_Mon`.\n3. `of_Mon` associates a monad on `C` to any monoid object in `C \u2964 C`.\n4. `Monad_Mon_equiv` is the equivalence between `Monad C` and `Mon_ (C \u2964 C)`.\n\n-/\n\nnamespace category_theory\n\n\nnamespace Monad\n\n\n/-- To every `Monad C` we associated a monoid object in `C \u2964 C`.-/\n@[simp] theorem to_Mon_mul {C : Type u} [category C] : \u2200 (\u1fb0 : Monad C), Mon_.mul (to_Mon \u1fb0) = \u03bc_ :=\n  fun (\u1fb0 : Monad C) => Eq.refl (Mon_.mul (to_Mon \u1fb0))\n\n/-- Passing from `Monad C` to `Mon_ (C \u2964 C)` is functorial. -/\n@[simp] theorem Monad_to_Mon_obj (C : Type u) [category C] : \u2200 (\u1fb0 : Monad C), functor.obj (Monad_to_Mon C) \u1fb0 = to_Mon \u1fb0 :=\n  fun (\u1fb0 : Monad C) => Eq.refl (functor.obj (Monad_to_Mon C) \u1fb0)\n\n/-- To every monoid object in `C \u2964 C` we associate a `Monad C`. -/\ndef of_Mon {C : Type u} [category C] : Mon_ (C \u2964 C) \u2192 Monad C :=\n  fun (M : Mon_ (C \u2964 C)) => mk (Mon_.X M)\n\n/-- Passing from `Mon_ (C \u2964 C)` to `Monad C` is functorial. -/\ndef Mon_to_Monad (C : Type u) [category C] : Mon_ (C \u2964 C) \u2964 Monad C :=\n  functor.mk of_Mon\n    fun (_x _x_1 : Mon_ (C \u2964 C)) (f : _x \u27f6 _x_1) => monad_hom.mk (nat_trans.mk (nat_trans.app (Mon_.hom.hom f)))\n\nnamespace Monad_Mon_equiv\n\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\n@[simp] theorem counit_iso_hom_app_hom {C : Type u} [category C] (_x : Mon_ (C \u2964 C)) : Mon_.hom.hom (nat_trans.app (iso.hom counit_iso) _x) = \ud835\udfd9 :=\n  Eq.refl (Mon_.hom.hom (nat_trans.app (iso.hom counit_iso) _x))\n\n/-- Auxilliary definition for `Monad_Mon_equiv` -/\ndef unit_iso_hom {C : Type u} [category C] : \ud835\udfed \u27f6 Monad_to_Mon C \u22d9 Mon_to_Monad C :=\n  nat_trans.mk fun (_x : Monad C) => monad_hom.mk (nat_trans.mk fun (_x_1 : C) => \ud835\udfd9)\n\n/-- Auxilliary definition for `Monad_Mon_equiv` -/\n@[simp] theorem unit_iso_inv_app_to_nat_trans_app {C : Type u} [category C] (_x : Monad C) : \u2200 (_x_1 : C), nat_trans.app (monad_hom.to_nat_trans (nat_trans.app unit_iso_inv _x)) _x_1 = \ud835\udfd9 :=\n  fun (_x_1 : C) => Eq.refl (nat_trans.app (monad_hom.to_nat_trans (nat_trans.app unit_iso_inv _x)) _x_1)\n\n/-- Isomorphism of functors used in `Monad_Mon_equiv` -/\ndef unit_iso {C : Type u} [category C] : \ud835\udfed \u2245 Monad_to_Mon C \u22d9 Mon_to_Monad C :=\n  iso.mk unit_iso_hom unit_iso_inv\n\nend Monad_Mon_equiv\n\n\n/-- Oh, monads are just monoids in the category of endofunctors (equivalence of categories). -/\ndef Monad_Mon_equiv (C : Type u) [category C] : Monad C \u224c Mon_ (C \u2964 C) :=\n  equivalence.mk' (Monad_to_Mon C) (Mon_to_Monad C) sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/equiv_mon.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239836484143, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.38003992605043063}}
{"text": "/-\nCopyright (c) 2017 Simon Hudon All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.pfunctor.univariate.basic\nimport Mathlib.PostPort\n\nuniverses u l w u_1 \n\nnamespace Mathlib\n\n/-!\n# M-types\n\nM types are potentially infinite tree-like structures. They are defined\nas the greatest fixpoint of a polynomial functor.\n-/\n\nnamespace pfunctor\n\n\nnamespace approx\n\n\n/-- `cofix_a F n` is an `n` level approximation of a M-type -/\ninductive cofix_a (F : pfunctor) : \u2115 \u2192 Type u\nwhere\n| continue : cofix_a F 0\n| intro : {n : \u2115} \u2192 (a : A F) \u2192 (B F a \u2192 cofix_a F n) \u2192 cofix_a F (Nat.succ n)\n\n/-- default inhabitant of `cofix_a` -/\nprotected def cofix_a.default (F : pfunctor) [Inhabited (A F)] (n : \u2115) : cofix_a F n :=\n  sorry\n\nprotected instance cofix_a.inhabited (F : pfunctor) [Inhabited (A F)] {n : \u2115} : Inhabited (cofix_a F n) :=\n  { default := cofix_a.default F n }\n\ntheorem cofix_a_eq_zero (F : pfunctor) (x : cofix_a F 0) (y : cofix_a F 0) : x = y := sorry\n\n/--\nThe label of the root of the tree for a non-trivial\napproximation of the cofix of a pfunctor.\n-/\ndef head' {F : pfunctor} {n : \u2115} : cofix_a F (Nat.succ n) \u2192 A F :=\n  sorry\n\n/-- for a non-trivial approximation, return all the subtrees of the root -/\ndef children' {F : pfunctor} {n : \u2115} (x : cofix_a F (Nat.succ n)) : B F (head' x) \u2192 cofix_a F n :=\n  sorry\n\ntheorem approx_eta {F : pfunctor} {n : \u2115} (x : cofix_a F (n + 1)) : x = cofix_a.intro (head' x) (children' x) := sorry\n\n/-- Relation between two approximations of the cofix of a pfunctor that state they both contain the same\ndata until one of them is truncated -/\ninductive agree {F : pfunctor} : {n : \u2115} \u2192 cofix_a F n \u2192 cofix_a F (n + 1) \u2192 Prop\nwhere\n| continue : \u2200 (x : cofix_a F 0) (y : cofix_a F 1), agree x y\n| intro : \u2200 {n : \u2115} {a : A F} (x : B F a \u2192 cofix_a F n) (x' : B F a \u2192 cofix_a F (n + 1)),\n  (\u2200 (i : B F a), agree (x i) (x' i)) \u2192 agree (cofix_a.intro a x) (cofix_a.intro a x')\n\n/--\nGiven an infinite series of approximations `approx`,\n`all_agree approx` states that they are all consistent with each other.\n-/\ndef all_agree {F : pfunctor} (x : (n : \u2115) \u2192 cofix_a F n) :=\n  \u2200 (n : \u2115), agree (x n) (x (Nat.succ n))\n\n@[simp] theorem agree_trival {F : pfunctor} {x : cofix_a F 0} {y : cofix_a F 1} : agree x y :=\n  agree.continue x y\n\ntheorem agree_children {F : pfunctor} {n : \u2115} (x : cofix_a F (Nat.succ n)) (y : cofix_a F (Nat.succ n + 1)) {i : B F (head' x)} {j : B F (head' y)} (h\u2080 : i == j) (h\u2081 : agree x y) : agree (children' x i) (children' y j) := sorry\n\n/-- `truncate a` turns `a` into a more limited approximation -/\ndef truncate {F : pfunctor} {n : \u2115} : cofix_a F (n + 1) \u2192 cofix_a F n :=\n  sorry\n\ntheorem truncate_eq_of_agree {F : pfunctor} {n : \u2115} (x : cofix_a F n) (y : cofix_a F (Nat.succ n)) (h : agree x y) : truncate y = x := sorry\n\n/-- `s_corec f i n` creates an approximation of height `n`\nof the final coalgebra of `f` -/\ndef s_corec {F : pfunctor} {X : Type w} (f : X \u2192 obj F X) (i : X) (n : \u2115) : cofix_a F n :=\n  sorry\n\ntheorem P_corec {F : pfunctor} {X : Type w} (f : X \u2192 obj F X) (i : X) (n : \u2115) : agree (s_corec f i n) (s_corec f i (Nat.succ n)) := sorry\n\n/-- `path F` provides indices to access internal nodes in `corec F` -/\ndef path (F : pfunctor) :=\n  List (Idx F)\n\nprotected instance path.inhabited {F : pfunctor} : Inhabited (path F) :=\n  { default := [] }\n\nprotected instance cofix_a.subsingleton {F : pfunctor} : subsingleton (cofix_a F 0) :=\n  subsingleton.intro\n    fun (a b : cofix_a F 0) =>\n      cofix_a.cases_on a\n        (fun (a_1 : 0 = 0) (H_2 : a == cofix_a.continue) =>\n          Eq._oldrec\n            (cofix_a.cases_on b\n              (fun (a : 0 = 0) (H_2 : b == cofix_a.continue) =>\n                Eq._oldrec (Eq.refl cofix_a.continue) (Eq.symm (eq_of_heq H_2)))\n              (fun {b_n : \u2115} (b_a : A F) (b_\u1fb0 : B F b_a \u2192 cofix_a F b_n) (a : 0 = Nat.succ b_n) => nat.no_confusion a)\n              (Eq.refl 0) (HEq.refl b))\n            (Eq.symm (eq_of_heq H_2)))\n        (fun {a_n : \u2115} (a_a : A F) (a_\u1fb0 : B F a_a \u2192 cofix_a F a_n) (a_1 : 0 = Nat.succ a_n) => nat.no_confusion a_1)\n        (Eq.refl 0) (HEq.refl a)\n\ntheorem head_succ' {F : pfunctor} (n : \u2115) (m : \u2115) (x : (n : \u2115) \u2192 cofix_a F n) (Hconsistent : all_agree x) : head' (x (Nat.succ n)) = head' (x (Nat.succ m)) := sorry\n\nend approx\n\n\n/-- Internal definition for `M`. It is needed to avoid name clashes\nbetween `M.mk` and `M.cases_on` and the declarations generated for\nthe structure -/\nstructure M_intl (F : pfunctor) \nwhere\n  approx : (n : \u2115) \u2192 approx.cofix_a F n\n  consistent : approx.all_agree approx\n\n/-- For polynomial functor `F`, `M F` is its final coalgebra -/\ndef M (F : pfunctor) :=\n  M_intl F\n\ntheorem M.default_consistent (F : pfunctor) [Inhabited (A F)] (n : \u2115) : approx.agree Inhabited.default Inhabited.default := sorry\n\nprotected instance M.inhabited (F : pfunctor) [Inhabited (A F)] : Inhabited (M F) :=\n  { default := M_intl.mk (fun (n : \u2115) => Inhabited.default) (M.default_consistent F) }\n\nprotected instance M_intl.inhabited (F : pfunctor) [Inhabited (A F)] : Inhabited (M_intl F) :=\n  (fun (this : Inhabited (M F)) => this) (M.inhabited F)\n\nnamespace M\n\n\ntheorem ext' (F : pfunctor) (x : M F) (y : M F) (H : \u2200 (i : \u2115), M_intl.approx x i = M_intl.approx y i) : x = y := sorry\n\n/-- Corecursor for the M-type defined by `F`. -/\nprotected def corec {F : pfunctor} {X : Type u_1} (f : X \u2192 obj F X) (i : X) : M F :=\n  M_intl.mk (approx.s_corec f i) (approx.P_corec f i)\n\n/-- given a tree generated by `F`, `head` gives us the first piece of data\nit contains -/\ndef head {F : pfunctor} (x : M F) : A F :=\n  approx.head' (M_intl.approx x 1)\n\n/-- return all the subtrees of the root of a tree `x : M F` -/\ndef children {F : pfunctor} (x : M F) (i : B F (head x)) : M F :=\n  M_intl.mk (fun (n : \u2115) => approx.children' (M_intl.approx x (Nat.succ n)) (cast sorry i)) sorry\n\n/-- select a subtree using a `i : F.Idx` or return an arbitrary tree if\n`i` designates no subtree of `x` -/\ndef ichildren {F : pfunctor} [Inhabited (M F)] [DecidableEq (A F)] (i : Idx F) (x : M F) : M F :=\n  dite (sigma.fst i = head x) (fun (H' : sigma.fst i = head x) => children x (cast sorry (sigma.snd i)))\n    fun (H' : \u00acsigma.fst i = head x) => Inhabited.default\n\ntheorem head_succ {F : pfunctor} (n : \u2115) (m : \u2115) (x : M F) : approx.head' (M_intl.approx x (Nat.succ n)) = approx.head' (M_intl.approx x (Nat.succ m)) :=\n  approx.head_succ' n m (M_intl.approx x) (M_intl.consistent x)\n\ntheorem head_eq_head' {F : pfunctor} (x : M F) (n : \u2115) : head x = approx.head' (M_intl.approx x (n + 1)) := sorry\n\ntheorem head'_eq_head {F : pfunctor} (x : M F) (n : \u2115) : approx.head' (M_intl.approx x (n + 1)) = head x := sorry\n\ntheorem truncate_approx {F : pfunctor} (x : M F) (n : \u2115) : approx.truncate (M_intl.approx x (n + 1)) = M_intl.approx x n :=\n  approx.truncate_eq_of_agree (M_intl.approx x n) (M_intl.approx x (n + 1)) (M_intl.consistent x n)\n\n/-- unfold an M-type -/\ndef dest {F : pfunctor} : M F \u2192 obj F (M F) :=\n  sorry\n\nnamespace approx\n\n\n/-- generates the approximations needed for `M.mk` -/\nprotected def s_mk {F : pfunctor} (x : obj F (M F)) (n : \u2115) : approx.cofix_a F n :=\n  sorry\n\nprotected theorem P_mk {F : pfunctor} (x : obj F (M F)) : approx.all_agree (approx.s_mk x) := sorry\n\nend approx\n\n\n/-- constructor for M-types -/\nprotected def mk {F : pfunctor} (x : obj F (M F)) : M F :=\n  M_intl.mk (approx.s_mk x) (approx.P_mk x)\n\n/-- `agree' n` relates two trees of type `M F` that\nare the same up to dept `n` -/\ninductive agree' {F : pfunctor} : \u2115 \u2192 M F \u2192 M F \u2192 Prop\nwhere\n| trivial : \u2200 (x y : M F), agree' 0 x y\n| step : \u2200 {n : \u2115} {a : A F} (x y : B F a \u2192 M F) {x' y' : M F},\n  x' = M.mk (sigma.mk a x) \u2192\n    y' = M.mk (sigma.mk a y) \u2192 (\u2200 (i : B F a), agree' n (x i) (y i)) \u2192 agree' (Nat.succ n) x' y'\n\n@[simp] theorem dest_mk {F : pfunctor} (x : obj F (M F)) : dest (M.mk x) = x := sorry\n\n@[simp] theorem mk_dest {F : pfunctor} (x : M F) : M.mk (dest x) = x := sorry\n\ntheorem mk_inj {F : pfunctor} {x : obj F (M F)} {y : obj F (M F)} (h : M.mk x = M.mk y) : x = y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (x = y)) (Eq.symm (dest_mk x))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dest (M.mk x) = y)) h))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (dest (M.mk y) = y)) (dest_mk y))) (Eq.refl y)))\n\n/-- destructor for M-types -/\nprotected def cases {F : pfunctor} {r : M F \u2192 Sort w} (f : (x : obj F (M F)) \u2192 r (M.mk x)) (x : M F) : r x :=\n  (fun (this : r (M.mk (dest x))) => eq.mpr sorry this) (f (dest x))\n\n/-- destructor for M-types -/\nprotected def cases_on {F : pfunctor} {r : M F \u2192 Sort w} (x : M F) (f : (x : obj F (M F)) \u2192 r (M.mk x)) : r x :=\n  M.cases f x\n\n/-- destructor for M-types, similar to `cases_on` but also\ngives access directly to the root and subtrees on an M-type -/\nprotected def cases_on' {F : pfunctor} {r : M F \u2192 Sort w} (x : M F) (f : (a : A F) \u2192 (f : B F a \u2192 M F) \u2192 r (M.mk (sigma.mk a f))) : r x :=\n  M.cases_on x fun (_x : obj F (M F)) => sorry\n\ntheorem approx_mk {F : pfunctor} (a : A F) (f : B F a \u2192 M F) (i : \u2115) : M_intl.approx (M.mk (sigma.mk a f)) (Nat.succ i) = approx.cofix_a.intro a fun (j : B F a) => M_intl.approx (f j) i :=\n  rfl\n\n@[simp] theorem agree'_refl {F : pfunctor} {n : \u2115} (x : M F) : agree' n x x := sorry\n\ntheorem agree_iff_agree' {F : pfunctor} {n : \u2115} (x : M F) (y : M F) : approx.agree (M_intl.approx x n) (M_intl.approx y (n + 1)) \u2194 agree' n x y := sorry\n\n@[simp] theorem cases_mk {F : pfunctor} {r : M F \u2192 Sort u_1} (x : obj F (M F)) (f : (x : obj F (M F)) \u2192 r (M.mk x)) : M.cases f (M.mk x) = f x := sorry\n\n@[simp] theorem cases_on_mk {F : pfunctor} {r : M F \u2192 Sort u_1} (x : obj F (M F)) (f : (x : obj F (M F)) \u2192 r (M.mk x)) : M.cases_on (M.mk x) f = f x :=\n  cases_mk x f\n\n@[simp] theorem cases_on_mk' {F : pfunctor} {r : M F \u2192 Sort u_1} {a : A F} (x : B F a \u2192 M F) (f : (a : A F) \u2192 (f : B F a \u2192 M F) \u2192 r (M.mk (sigma.mk a f))) : M.cases_on' (M.mk (sigma.mk a x)) f = f a x :=\n  cases_mk (sigma.mk a x) fun (_x : obj F (M F)) => cases_on'._match_1 f _x\n\n/-- `is_path p x` tells us if `p` is a valid path through `x` -/\ninductive is_path {F : pfunctor} : approx.path F \u2192 M F \u2192 Prop\nwhere\n| nil : \u2200 (x : M F), is_path [] x\n| cons : \u2200 (xs : approx.path F) {a : A F} (x : M F) (f : B F a \u2192 M F) (i : B F a),\n  x = M.mk (sigma.mk a f) \u2192 is_path xs (f i) \u2192 is_path (sigma.mk a i :: xs) x\n\ntheorem is_path_cons {F : pfunctor} {xs : approx.path F} {a : A F} {a' : A F} {f : B F a \u2192 M F} {i : B F a'} (h : is_path (sigma.mk a' i :: xs) (M.mk (sigma.mk a f))) : a = a' := sorry\n\ntheorem is_path_cons' {F : pfunctor} {xs : approx.path F} {a : A F} {f : B F a \u2192 M F} {i : B F a} (h : is_path (sigma.mk a i :: xs) (M.mk (sigma.mk a f))) : is_path xs (f i) := sorry\n\n/-- follow a path through a value of `M F` and return the subtree\nfound at the end of the path if it is a valid path for that value and\nreturn a default tree -/\ndef isubtree {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] : approx.path F \u2192 M F \u2192 M F :=\n  sorry\n\n/-- similar to `isubtree` but returns the data at the end of the path instead\nof the whole subtree -/\ndef iselect {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] (ps : approx.path F) : M F \u2192 A F :=\n  fun (x : M F) => head (isubtree ps x)\n\ntheorem iselect_eq_default {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] (ps : approx.path F) (x : M F) (h : \u00acis_path ps x) : iselect ps x = head Inhabited.default := sorry\n\n@[simp] theorem head_mk {F : pfunctor} (x : obj F (M F)) : head (M.mk x) = sigma.fst x := sorry\n\ntheorem children_mk {F : pfunctor} {a : A F} (x : B F a \u2192 M F) (i : B F (head (M.mk (sigma.mk a x)))) : children (M.mk (sigma.mk a x)) i =\n  x\n    (cast\n      (eq.mpr (id (Eq._oldrec (Eq.refl (B F (head (M.mk (sigma.mk a x))) = B F a)) (head_mk (sigma.mk a x))))\n        (Eq.refl (B F (sigma.fst (sigma.mk a x)))))\n      i) := sorry\n\n@[simp] theorem ichildren_mk {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] (x : obj F (M F)) (i : Idx F) : ichildren i (M.mk x) = obj.iget x i := sorry\n\n@[simp] theorem isubtree_cons {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] (ps : approx.path F) {a : A F} (f : B F a \u2192 M F) {i : B F a} : isubtree (sigma.mk a i :: ps) (M.mk (sigma.mk a f)) = isubtree ps (f i) := sorry\n\n@[simp] theorem iselect_nil {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] {a : A F} (f : B F a \u2192 M F) : iselect [] (M.mk (sigma.mk a f)) = a :=\n  Eq.refl (iselect [] (M.mk (sigma.mk a f)))\n\n@[simp] theorem iselect_cons {F : pfunctor} [DecidableEq (A F)] [Inhabited (M F)] (ps : approx.path F) {a : A F} (f : B F a \u2192 M F) {i : B F a} : iselect (sigma.mk a i :: ps) (M.mk (sigma.mk a f)) = iselect ps (f i) := sorry\n\ntheorem corec_def {F : pfunctor} {X : Type u} (f : X \u2192 obj F X) (x\u2080 : X) : M.corec f x\u2080 = M.mk (M.corec f <$> f x\u2080) := sorry\n\ntheorem ext_aux {F : pfunctor} [Inhabited (M F)] [DecidableEq (A F)] {n : \u2115} (x : M F) (y : M F) (z : M F) (hx : agree' n z x) (hy : agree' n z y) (hrec : \u2200 (ps : approx.path F), n = list.length ps \u2192 iselect ps x = iselect ps y) : M_intl.approx x (n + 1) = M_intl.approx y (n + 1) := sorry\n\ntheorem ext {F : pfunctor} [Inhabited (M F)] (x : M F) (y : M F) (H : \u2200 (ps : approx.path F), iselect ps x = iselect ps y) : x = y := sorry\n\n/-- Bisimulation is the standard proof technique for equality between\ninfinite tree-like structures -/\nstructure is_bisimulation {F : pfunctor} (R : M F \u2192 M F \u2192 Prop) \nwhere\n  head : \u2200 {a a' : A F} {f : B F a \u2192 M F} {f' : B F a' \u2192 M F}, R (M.mk (sigma.mk a f)) (M.mk (sigma.mk a' f')) \u2192 a = a'\n  tail : \u2200 {a : A F} {f f' : B F a \u2192 M F}, R (M.mk (sigma.mk a f)) (M.mk (sigma.mk a f')) \u2192 \u2200 (i : B F a), R (f i) (f' i)\n\ntheorem nth_of_bisim {F : pfunctor} (R : M F \u2192 M F \u2192 Prop) [Inhabited (M F)] (bisim : is_bisimulation R) (s\u2081 : M F) (s\u2082 : M F) (ps : approx.path F) : R s\u2081 s\u2082 \u2192\n  is_path ps s\u2081 \u2228 is_path ps s\u2082 \u2192\n    iselect ps s\u2081 = iselect ps s\u2082 \u2227\n      \u2203 (a : A F),\n        \u2203 (f : B F a \u2192 M F),\n          \u2203 (f' : B F a \u2192 M F),\n            isubtree ps s\u2081 = M.mk (sigma.mk a f) \u2227 isubtree ps s\u2082 = M.mk (sigma.mk a f') \u2227 \u2200 (i : B F a), R (f i) (f' i) := sorry\n\ntheorem eq_of_bisim {F : pfunctor} (R : M F \u2192 M F \u2192 Prop) [Nonempty (M F)] (bisim : is_bisimulation R) (s\u2081 : M F) (s\u2082 : M F) : R s\u2081 s\u2082 \u2192 s\u2081 = s\u2082 := sorry\n\n/-- corecursor for `M F` with swapped arguments -/\ndef corec_on {F : pfunctor} {X : Type u_1} (x\u2080 : X) (f : X \u2192 obj F X) : M F :=\n  M.corec f x\u2080\n\ntheorem dest_corec {P : pfunctor} {\u03b1 : Type u} (g : \u03b1 \u2192 obj P \u03b1) (x : \u03b1) : dest (M.corec g x) = M.corec g <$> g x := sorry\n\ntheorem bisim {P : pfunctor} (R : M P \u2192 M P \u2192 Prop) (h : \u2200 (x y : M P),\n  R x y \u2192\n    \u2203 (a : A P),\n      \u2203 (f : B P a \u2192 M P),\n        \u2203 (f' : B P a \u2192 M P), dest x = sigma.mk a f \u2227 dest y = sigma.mk a f' \u2227 \u2200 (i : B P a), R (f i) (f' i)) (x : M P) (y : M P) : R x y \u2192 x = y := sorry\n\ntheorem bisim' {P : pfunctor} {\u03b1 : Type u_1} (Q : \u03b1 \u2192 Prop) (u : \u03b1 \u2192 M P) (v : \u03b1 \u2192 M P) (h : \u2200 (x : \u03b1),\n  Q x \u2192\n    \u2203 (a : A P),\n      \u2203 (f : B P a \u2192 M P),\n        \u2203 (f' : B P a \u2192 M P),\n          dest (u x) = sigma.mk a f \u2227\n            dest (v x) = sigma.mk a f' \u2227 \u2200 (i : B P a), \u2203 (x' : \u03b1), Q x' \u2227 f i = u x' \u2227 f' i = v x') (x : \u03b1) : Q x \u2192 u x = v x := sorry\n\n-- for the record, show M_bisim follows from _bisim'\n\ntheorem bisim_equiv {P : pfunctor} (R : M P \u2192 M P \u2192 Prop) (h : \u2200 (x y : M P),\n  R x y \u2192\n    \u2203 (a : A P),\n      \u2203 (f : B P a \u2192 M P),\n        \u2203 (f' : B P a \u2192 M P), dest x = sigma.mk a f \u2227 dest y = sigma.mk a f' \u2227 \u2200 (i : B P a), R (f i) (f' i)) (x : M P) (y : M P) : R x y \u2192 x = y := sorry\n\ntheorem corec_unique {P : pfunctor} {\u03b1 : Type u} (g : \u03b1 \u2192 obj P \u03b1) (f : \u03b1 \u2192 M P) (hyp : \u2200 (x : \u03b1), dest (f x) = f <$> g x) : f = M.corec g := sorry\n\n/-- corecursor where the state of the computation can be sent downstream\nin the form of a recursive call -/\ndef corec\u2081 {P : pfunctor} {\u03b1 : Type u} (F : (X : Type u) \u2192 (\u03b1 \u2192 X) \u2192 \u03b1 \u2192 obj P X) : \u03b1 \u2192 M P :=\n  M.corec (F \u03b1 id)\n\n/-- corecursor where it is possible to return a fully formed value at any point\nof the computation -/\ndef corec' {P : pfunctor} {\u03b1 : Type u} (F : {X : Type u} \u2192 (\u03b1 \u2192 X) \u2192 \u03b1 \u2192 M P \u2295 obj P X) (x : \u03b1) : M P :=\n  corec\u2081\n    (fun (X : Type u) (rec : M P \u2295 \u03b1 \u2192 X) (a : M P \u2295 \u03b1) =>\n      let y : M P \u2295 obj P X := a >>= F (rec \u2218 sum.inr);\n      sorry)\n    (sum.inr x)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/pfunctor/univariate/M.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.379946281019389}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.locally_ringed_space\nimport algebra.category.CommRing.constructions\nimport algebraic_geometry.open_immersion\nimport category_theory.limits.constructions.limits_of_products_and_equalizers\n\n/-!\n# Colimits of LocallyRingedSpace\n\nWe construct the explict coproducts and coequalizers of `LocallyRingedSpace`.\nIt then follows that `LocallyRingedSpace` has all colimits, and\n`forget_to_SheafedSpace` preserves them.\n\n-/\n\nnamespace algebraic_geometry\n\nuniverses v u\n\nopen category_theory category_theory.limits opposite topological_space\n\nnamespace SheafedSpace\n\nvariables {C : Type u} [category.{v} C] [has_limits C]\nvariables {J : Type v} [category.{v} J] (F : J \u2964 SheafedSpace C)\n\nlemma is_colimit_exists_rep {c : cocone F} (hc : is_colimit c) (x : c.X) :\n  \u2203 (i : J) (y : F.obj i), (c.\u03b9.app i).base y = x :=\nconcrete.is_colimit_exists_rep (F \u22d9 SheafedSpace.forget _)\n  (is_colimit_of_preserves (SheafedSpace.forget _) hc) x\n\nlemma colimit_exists_rep (x : colimit F) :\n  \u2203 (i : J) (y : F.obj i), (colimit.\u03b9 F i).base y = x :=\nconcrete.is_colimit_exists_rep (F \u22d9 SheafedSpace.forget _)\n  (is_colimit_of_preserves (SheafedSpace.forget _) (colimit.is_colimit F)) x\n\ninstance {X Y : SheafedSpace C} (f g : X \u27f6 Y) : epi (coequalizer.\u03c0 f g).base :=\nbegin\n  erw \u2190 (show _ = (coequalizer.\u03c0 f g).base, from\n    \u03b9_comp_coequalizer_comparison f g (SheafedSpace.forget C)),\n  rw \u2190 preserves_coequalizer.iso_hom,\n  apply epi_comp\nend\n\nend SheafedSpace\n\nnamespace LocallyRingedSpace\n\nsection has_coproducts\n\nvariables {\u03b9 : Type u} (F : discrete \u03b9 \u2964 LocallyRingedSpace.{u})\n\n/-- The explicit coproduct for `F : discrete \u03b9 \u2964 LocallyRingedSpace`. -/\nnoncomputable\ndef coproduct : LocallyRingedSpace :=\n{ to_SheafedSpace := colimit (F \u22d9 forget_to_SheafedSpace : _),\n  local_ring := \u03bb x, begin\n    obtain \u27e8i, y, \u27e8\u27e9\u27e9 := SheafedSpace.colimit_exists_rep (F \u22d9 forget_to_SheafedSpace) x,\n    haveI : _root_.local_ring (((F \u22d9 forget_to_SheafedSpace).obj i).to_PresheafedSpace.stalk y) :=\n      (F.obj i).local_ring _,\n    exact (as_iso (PresheafedSpace.stalk_map (colimit.\u03b9 (F \u22d9 forget_to_SheafedSpace) i : _) y)\n      ).symm.CommRing_iso_to_ring_equiv.local_ring\n  end }\n\n/-- The explicit coproduct cofan for `F : discrete \u03b9 \u2964 LocallyRingedSpace`. -/\nnoncomputable\ndef coproduct_cofan : cocone F :=\n{ X := coproduct F,\n  \u03b9 := { app := \u03bb j, \u27e8colimit.\u03b9 (F \u22d9 forget_to_SheafedSpace) j, infer_instance\u27e9 } }\n\n/-- The explicit coproduct cofan constructed in `coproduct_cofan` is indeed a colimit. -/\nnoncomputable\ndef coproduct_cofan_is_colimit : is_colimit (coproduct_cofan F) :=\n{ desc := \u03bb s, \u27e8colimit.desc (F \u22d9 forget_to_SheafedSpace) (forget_to_SheafedSpace.map_cocone s),\n  begin\n    intro x,\n    obtain \u27e8i, y, \u27e8\u27e9\u27e9 := SheafedSpace.colimit_exists_rep (F \u22d9 forget_to_SheafedSpace) x,\n    have := PresheafedSpace.stalk_map.comp (colimit.\u03b9 (F \u22d9 forget_to_SheafedSpace) i : _)\n      (colimit.desc (F \u22d9 forget_to_SheafedSpace) (forget_to_SheafedSpace.map_cocone s)) y,\n    rw \u2190 is_iso.comp_inv_eq at this,\n    erw [\u2190 this, PresheafedSpace.stalk_map.congr_hom _ _\n      (colimit.\u03b9_desc (forget_to_SheafedSpace.map_cocone s) i : _)],\n    haveI : is_local_ring_hom (PresheafedSpace.stalk_map\n      ((forget_to_SheafedSpace.map_cocone s).\u03b9.app i) y) := (s.\u03b9.app i).2 y,\n    apply_instance\n  end\u27e9,\n  fac' := \u03bb s j, subtype.eq (colimit.\u03b9_desc _ _),\n  uniq' := \u03bb s f h, subtype.eq (is_colimit.uniq _ (forget_to_SheafedSpace.map_cocone s) f.1\n    (\u03bb j, congr_arg subtype.val (h j))) }\n\ninstance : has_coproducts LocallyRingedSpace.{u} :=\n\u03bb \u03b9, \u27e8\u03bb F, \u27e8\u27e8\u27e8_, coproduct_cofan_is_colimit F\u27e9\u27e9\u27e9\u27e9\n\nnoncomputable\ninstance (J : Type*) : preserves_colimits_of_shape (discrete J) forget_to_SheafedSpace :=\n\u27e8\u03bb G, preserves_colimit_of_preserves_colimit_cocone (coproduct_cofan_is_colimit G)\n  ((colimit.is_colimit _).of_iso_colimit (cocones.ext (iso.refl _) (\u03bb j, category.comp_id _)))\u27e9\n\nend has_coproducts\n\nsection has_coequalizer\n\nvariables {X Y : LocallyRingedSpace.{u}} (f g : X \u27f6 Y)\n\nnamespace has_coequalizer\n\ninstance coequalizer_\u03c0_app_is_local_ring_hom\n  (U : topological_space.opens ((coequalizer f.val g.val).carrier)) :\n  is_local_ring_hom ((coequalizer.\u03c0 f.val g.val : _).c.app (op U)) :=\nbegin\n  have := \u03b9_comp_coequalizer_comparison f.1 g.1 SheafedSpace.forget_to_PresheafedSpace,\n  rw \u2190 preserves_coequalizer.iso_hom at this,\n  erw SheafedSpace.congr_app this.symm (op U),\n  rw [PresheafedSpace.comp_c_app,\n    \u2190 PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_\u03c0],\n  apply_instance\nend\n\n/-!\nWe roughly follow the construction given in [MR0302656]. Given a pair `f, g : X \u27f6 Y` of morphisms\nof locally ringed spaces, we want to show that the stalk map of\n`\u03c0 = coequalizer.\u03c0 f g` (as sheafed space homs) is a local ring hom. It then follows that\n`coequalizer f g` is indeed a locally ringed space, and `coequalizer.\u03c0 f g` is a morphism of\nlocally ringed space.\n\nGiven a germ `\u27e8U, s\u27e9` of `x : coequalizer f g` such that `\u03c0\ua673 x : Y` is invertible, we ought to show\nthat `\u27e8U, s\u27e9` is invertible. That is, there exists an open set `U' \u2286 U` containing `x` such that the\nrestriction of `s` onto `U'` is invertible. This `U'` is given by `\u03c0 '' V`, where `V` is the\nbasic open set of `\u03c0\u22c6x`.\n\nSince `f \u207b\u00b9' V = Y.basic_open (f \u226b \u03c0)\ua673 x = Y.basic_open (g \u226b \u03c0)\ua673 x = g \u207b\u00b9' V`, we have\n`\u03c0 \u207b\u00b9' (\u03c0 '' V) = V` (as the underlying set map is merely the set-theoretic coequalizer).\nThis shows that `\u03c0 '' V` is indeed open, and `s` is invertible on `\u03c0 '' V` as the components of `\u03c0\ua673`\nare local ring homs.\n-/\nvariable (U : opens ((coequalizer f.1 g.1).carrier))\nvariable (s : (coequalizer f.1 g.1).presheaf.obj (op U))\n\n/-- (Implementation). The basic open set of the section `\u03c0\ua673 s`. -/\nnoncomputable\ndef image_basic_open : opens Y := (Y.to_RingedSpace.basic_open\n  (show Y.presheaf.obj (op (unop _)), from ((coequalizer.\u03c0 f.1 g.1).c.app (op U)) s))\n\nlemma image_basic_open_image_preimage :\n  (coequalizer.\u03c0 f.1 g.1).base \u207b\u00b9' ((coequalizer.\u03c0 f.1 g.1).base ''\n    (image_basic_open f g U s).1) = (image_basic_open f g U s).1 :=\nbegin\n  fapply types.coequalizer_preimage_image_eq_of_preimage_eq f.1.base g.1.base,\n  { ext,\n    simp_rw [types_comp_apply, \u2190 Top.comp_app, \u2190 PresheafedSpace.comp_base],\n    congr' 2,\n    exact coequalizer.condition f.1 g.1 },\n  { apply is_colimit_cofork_map_of_is_colimit (forget Top),\n    apply is_colimit_cofork_map_of_is_colimit (SheafedSpace.forget _),\n    exact coequalizer_is_coequalizer f.1 g.1 },\n  { suffices : (topological_space.opens.map f.1.base).obj (image_basic_open f g U s) =\n      (topological_space.opens.map g.1.base).obj (image_basic_open f g U s),\n    { injection this },\n    delta image_basic_open,\n    rw [preimage_basic_open f, preimage_basic_open g],\n    dsimp only [functor.op, unop_op],\n    rw [\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],\n    erw X.to_RingedSpace.basic_open_res,\n    apply inf_eq_right.mpr,\n    refine (RingedSpace.basic_open_subset _ _).trans _,\n    rw coequalizer.condition f.1 g.1,\n    exact \u03bb _ h, h }\nend\n\nlemma image_basic_open_image_open :\n  is_open ((coequalizer.\u03c0 f.1 g.1).base '' (image_basic_open f g U s).1) :=\nbegin\n  rw [\u2190 (Top.homeo_of_iso (preserves_coequalizer.iso (SheafedSpace.forget _) f.1 g.1))\n      .is_open_preimage, Top.coequalizer_is_open_iff, \u2190 set.preimage_comp],\n  erw \u2190 coe_comp,\n  rw [preserves_coequalizer.iso_hom, \u03b9_comp_coequalizer_comparison],\n  dsimp only [SheafedSpace.forget],\n  rw image_basic_open_image_preimage,\n  exact (image_basic_open f g U s).2\nend\n\ninstance coequalizer_\u03c0_stalk_is_local_ring_hom (x : Y) :\n  is_local_ring_hom (PresheafedSpace.stalk_map (coequalizer.\u03c0 f.val g.val : _) x) :=\nbegin\n  constructor,\n  rintros a ha,\n  rcases Top.presheaf.germ_exist _ _ a with \u27e8U, hU, s, rfl\u27e9,\n  erw PresheafedSpace.stalk_map_germ_apply (coequalizer.\u03c0 f.1 g.1 : _) U \u27e8_, hU\u27e9 at ha,\n\n  let V := image_basic_open f g U s,\n  have hV : (coequalizer.\u03c0 f.1 g.1).base \u207b\u00b9' ((coequalizer.\u03c0 f.1 g.1).base '' V.1) = V.1 :=\n    image_basic_open_image_preimage f g U s,\n  have hV' : V = \u27e8(coequalizer.\u03c0 f.1 g.1).base \u207b\u00b9'\n    ((coequalizer.\u03c0 f.1 g.1).base '' V.1), hV.symm \u25b8 V.2\u27e9 := subtype.eq hV.symm,\n  have V_open : is_open (((coequalizer.\u03c0 f.val g.val).base) '' V.val) :=\n    image_basic_open_image_open f g U s,\n  have VleU :\n    (\u27e8((coequalizer.\u03c0 f.val g.val).base) '' V.val, V_open\u27e9 : topological_space.opens _) \u2264 U,\n  { exact set.image_subset_iff.mpr (Y.to_RingedSpace.basic_open_subset _) },\n  have hxV : x \u2208 V := \u27e8\u27e8_, hU\u27e9, ha, rfl\u27e9,\n\n  erw \u2190 (coequalizer f.val g.val).presheaf.germ_res_apply (hom_of_le VleU)\n    \u27e8_, @set.mem_image_of_mem _ _ (coequalizer.\u03c0 f.val g.val).base x V.1 hxV\u27e9 s,\n  apply ring_hom.is_unit_map,\n  rw [\u2190 is_unit_map_iff ((coequalizer.\u03c0 f.val g.val : _).c.app _), \u2190 comp_apply,\n    nat_trans.naturality, comp_apply, Top.presheaf.pushforward_obj_map,\n    \u2190 is_unit_map_iff (Y.presheaf.map (eq_to_hom hV').op), \u2190 comp_apply, \u2190 functor.map_comp],\n  convert @RingedSpace.is_unit_res_basic_open Y.to_RingedSpace (unop _)\n    (((coequalizer.\u03c0 f.val g.val).c.app (op U)) s),\n  apply_instance\nend\n\nend has_coequalizer\n\n/-- The coequalizer of two locally ringed space in the category of sheafed spaces is a locally\nringed space. -/\nnoncomputable\ndef coequalizer : LocallyRingedSpace :=\n{ to_SheafedSpace := coequalizer f.1 g.1,\n  local_ring := \u03bb x,\n  begin\n    obtain \u27e8y, rfl\u27e9 :=\n      (Top.epi_iff_surjective (coequalizer.\u03c0 f.val g.val).base).mp infer_instance x,\n    exact (PresheafedSpace.stalk_map (coequalizer.\u03c0 f.val g.val : _) y).domain_local_ring\n  end }\n\n/-- The explicit coequalizer cofork of locally ringed spaces. -/\nnoncomputable\ndef coequalizer_cofork : cofork f g :=\n@cofork.of_\u03c0 _ _ _ _ f g (coequalizer f g) \u27e8coequalizer.\u03c0 f.1 g.1, infer_instance\u27e9\n  (subtype.eq (coequalizer.condition f.1 g.1))\n\nlemma is_local_ring_hom_stalk_map_congr {X Y : RingedSpace} (f g : X \u27f6 Y) (H : f = g)\n  (x) (h : is_local_ring_hom (PresheafedSpace.stalk_map f x)) :\n    is_local_ring_hom (PresheafedSpace.stalk_map g x) :=\nby { rw PresheafedSpace.stalk_map.congr_hom _ _ H.symm x, apply_instance }\n\n/-- The cofork constructed in `coequalizer_cofork` is indeed a colimit cocone. -/\nnoncomputable\ndef coequalizer_cofork_is_colimit : is_colimit (coequalizer_cofork f g) :=\nbegin\n  apply cofork.is_colimit.mk',\n  intro s,\n  have e : f.val \u226b s.\u03c0.val = g.val \u226b s.\u03c0.val := by injection s.condition,\n  use coequalizer.desc s.\u03c0.1 e,\n  { intro x,\n    rcases (Top.epi_iff_surjective (coequalizer.\u03c0 f.val g.val).base).mp\n      infer_instance x with \u27e8y, rfl\u27e9,\n    apply is_local_ring_hom_of_comp _ (PresheafedSpace.stalk_map (coequalizer_cofork f g).\u03c0.1 _),\n    change is_local_ring_hom (_ \u226b PresheafedSpace.stalk_map (coequalizer_cofork f g).\u03c0.val y),\n    erw \u2190 PresheafedSpace.stalk_map.comp,\n    apply is_local_ring_hom_stalk_map_congr _ _ (coequalizer.\u03c0_desc s.\u03c0.1 e).symm y,\n    apply_instance },\n  split,\n  exact subtype.eq (coequalizer.\u03c0_desc _ _),\n  intros m h,\n  replace h : (coequalizer_cofork f g).\u03c0.1 \u226b m.1 = s.\u03c0.1 := by { rw \u2190 h, refl },\n  apply subtype.eq,\n  apply (colimit.is_colimit (parallel_pair f.1 g.1)).uniq (cofork.of_\u03c0 s.\u03c0.1 e) m.1,\n  rintro \u27e8\u27e9,\n  { rw [\u2190 (colimit.cocone (parallel_pair f.val g.val)).w walking_parallel_pair_hom.left,\n      category.assoc],\n    change _ \u226b _ \u226b _ = _ \u226b _,\n    congr,\n    exact h },\n  { exact h }\nend\n\n\ninstance : has_coequalizer f g := \u27e8\u27e8\u27e8_, coequalizer_cofork_is_colimit f g\u27e9\u27e9\u27e9\n\ninstance : has_coequalizers LocallyRingedSpace := has_coequalizers_of_has_colimit_parallel_pair _\n\nnoncomputable\ninstance preserves_coequalizer :\n  preserves_colimits_of_shape walking_parallel_pair.{v} forget_to_SheafedSpace.{v} :=\n\u27e8\u03bb F, begin\n  apply preserves_colimit_of_iso_diagram _ (diagram_iso_parallel_pair F).symm,\n  apply preserves_colimit_of_preserves_colimit_cocone (coequalizer_cofork_is_colimit _ _),\n  apply (is_colimit_map_cocone_cofork_equiv _ _).symm _,\n  dsimp only [forget_to_SheafedSpace],\n  exact coequalizer_is_coequalizer _ _\nend\u27e9\n\nend has_coequalizer\n\ninstance : has_colimits LocallyRingedSpace := colimits_from_coequalizers_and_coproducts\n\nnoncomputable\ninstance : preserves_colimits LocallyRingedSpace.forget_to_SheafedSpace :=\npreserves_colimits_of_preserves_coequalizers_and_coproducts _\n\nend LocallyRingedSpace\n\nend algebraic_geometry\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebraic_geometry/locally_ringed_space/has_colimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.37994628101938893}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport category_theory.over\nimport category_theory.adjunction.opposites\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.creates\nimport category_theory.limits.comma\n\n/-!\n# Limits and colimits in the over and under categories\n\nShow that the forgetful functor `forget X : over X \u2964 C` creates colimits, and hence `over X` has\nany colimits that `C` has (as well as the dual that `forget X : under X \u27f6 C` creates limits).\n\nNote that the folder `category_theory.limits.shapes.constructions.over` further shows that\n`forget X : over X \u2964 C` creates connected limits (so `over X` has connected limits), and that\n`over X` has `J`-indexed products if `C` has `J`-indexed wide pullbacks.\n\nTODO: If `C` has binary products, then `forget X : over X \u2964 C` has a right adjoint.\n-/\nnoncomputable theory\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nopen category_theory category_theory.limits\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{v} C]\nvariable {X : C}\n\nnamespace category_theory.over\n\ninstance has_colimit_of_has_colimit_comp_forget\n  (F : J \u2964 over X) [i : has_colimit (F \u22d9 forget X)] : has_colimit F :=\n@@costructured_arrow.has_colimit _ _ _ _ i _\n\ninstance [has_colimits_of_shape J C] : has_colimits_of_shape J (over X) := {}\ninstance [has_colimits C] : has_colimits (over X) := \u27e8infer_instance\u27e9\n\ninstance creates_colimits : creates_colimits (forget X) := costructured_arrow.creates_colimits\n\n-- We can automatically infer that the forgetful functor preserves and reflects colimits.\nexample [has_colimits C] : preserves_colimits (forget X) := infer_instance\nexample : reflects_colimits (forget X) := infer_instance\n\nsection\nvariables [has_pullbacks C]\n\nopen tactic\n\n/-- When `C` has pullbacks, a morphism `f : X \u27f6 Y` induces a functor `over Y \u2964 over X`,\nby pulling back a morphism along `f`. -/\n@[simps]\ndef pullback {X Y : C} (f : X \u27f6 Y) : over Y \u2964 over X :=\n{ obj := \u03bb g, over.mk (pullback.snd : pullback g.hom f \u27f6 X),\n  map := \u03bb g h k,\n    over.hom_mk\n      (pullback.lift (pullback.fst \u226b k.left) pullback.snd (by simp [pullback.condition]))\n      (by tidy) }\n\n/-- `over.map f` is left adjoint to `over.pullback f`. -/\ndef map_pullback_adj {A B : C} (f : A \u27f6 B) :\n  over.map f \u22a3 pullback f :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb g h,\n  { to_fun := \u03bb X, over.hom_mk (pullback.lift X.left g.hom (over.w X)) (pullback.lift_snd _ _ _),\n    inv_fun := \u03bb Y,\n    begin\n      refine over.hom_mk _ _,\n      refine Y.left \u226b pullback.fst,\n      dsimp,\n      rw [\u2190 over.w Y, category.assoc, pullback.condition, category.assoc], refl,\n    end,\n    left_inv := \u03bb X, by { ext, dsimp, simp, },\n    right_inv := \u03bb Y, begin\n      ext, dsimp,\n      simp only [pullback.lift_fst],\n      dsimp,\n      rw [pullback.lift_snd, \u2190 over.w Y],\n      refl,\n    end } }\n\n/-- pullback (\ud835\udfd9 A) : over A \u2964 over A is the identity functor. -/\ndef pullback_id {A : C} : pullback (\ud835\udfd9 A) \u2245 \ud835\udfed _ :=\nadjunction.right_adjoint_uniq\n  (map_pullback_adj _)\n  (adjunction.id.of_nat_iso_left over.map_id.symm)\n\n/-- pullback commutes with composition (up to natural isomorphism). -/\ndef pullback_comp {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  pullback (f \u226b g) \u2245 pullback g \u22d9 pullback f :=\nadjunction.right_adjoint_uniq\n  (map_pullback_adj _)\n  (((map_pullback_adj _).comp _ _ (map_pullback_adj _)).of_nat_iso_left\n    (over.map_comp _ _).symm)\n\ninstance pullback_is_right_adjoint {A B : C} (f : A \u27f6 B) :\n  is_right_adjoint (pullback f) :=\n\u27e8_, map_pullback_adj f\u27e9\n\nend\n\nend category_theory.over\n\nnamespace category_theory.under\n\ninstance has_limit_of_has_limit_comp_forget\n  (F : J \u2964 under X) [i : has_limit (F \u22d9 forget X)] : has_limit F :=\n@@structured_arrow.has_limit _ _ _ _ i _\n\ninstance [has_limits_of_shape J C] : has_limits_of_shape J (under X) := {}\ninstance [has_limits C] : has_limits (under X) := \u27e8infer_instance\u27e9\n\ninstance creates_limits : creates_limits (forget X) := structured_arrow.creates_limits\n\n-- We can automatically infer that the forgetful functor preserves and reflects limits.\nexample [has_limits C] : preserves_limits (forget X) := infer_instance\nexample : reflects_limits (forget X) := infer_instance\n\nsection\nvariables [has_pushouts C]\n\n/-- When `C` has pushouts, a morphism `f : X \u27f6 Y` induces a functor `under X \u2964 under Y`,\nby pushing a morphism forward along `f`. -/\n@[simps]\ndef pushout {X Y : C} (f : X \u27f6 Y) : under X \u2964 under Y :=\n{ obj := \u03bb g, under.mk (pushout.inr : Y \u27f6 pushout g.hom f),\n  map := \u03bb g h k,\n    under.hom_mk\n      (pushout.desc (k.right \u226b pushout.inl) pushout.inr (by { simp [\u2190pushout.condition], }))\n      (by tidy) }\n\nend\n\nend category_theory.under\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.3799462727993838}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes H\u00f6lzl, Simon Hudon, Kenny Lau\n\n! This file was ported from Lean 3 source module data.multiset.functor\n! leanprover-community/mathlib commit 1f0096e6caa61e9c849ec2adbd227e960e9dff58\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.Multiset.Bind\nimport Mathlib.Control.Traversable.Lemmas\nimport Mathlib.Control.Traversable.Instances\n\n/-!\n# Functoriality of `Multiset`.\n-/\n\n\nuniverse u\n\nnamespace Multiset\n\nopen List\n\ninstance functor : Functor Multiset where map := @map\n\n@[simp]\ntheorem fmap_def {\u03b1' \u03b2'} {s : Multiset \u03b1'} (f : \u03b1' \u2192 \u03b2') : f <$> s = s.map f :=\n  rfl\n#align multiset.fmap_def Multiset.fmap_def\n\ninstance : LawfulFunctor Multiset := by refine' { .. } <;> intros <;> simp ; rfl\n\nopen IsLawfulTraversable CommApplicative\n\nvariable {F : Type u \u2192 Type u} [Applicative F] [CommApplicative F]\n\nvariable {\u03b1' \u03b2' : Type u} (f : \u03b1' \u2192 F \u03b2')\n\n/-- Map each element of a `Multiset` to an action, evaluate these actions in order,\n    and collect the results.\n-/\ndef traverse : Multiset \u03b1' \u2192 F (Multiset \u03b2') := by\n  refine' Quotient.lift (Functor.map Coe.coe \u2218 Traversable.traverse f) _\n  introv p; unfold Function.comp\n  induction p\n  case nil => rfl\n  case cons x l\u2081 l\u2082 _ 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 := by rw [h]\n    simpa [functor_norm] using this\n  case swap x y 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 := 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  case trans => simp [*]\n#align multiset.traverse Multiset.traverse\n\ninstance : Monad Multiset :=\n  { Multiset.functor with\n    pure := fun x \u21a6 {x}\n    bind := @bind }\n\n@[simp]\ntheorem pure_def {\u03b1} : (pure : \u03b1 \u2192 Multiset \u03b1) = singleton :=\n  rfl\n#align multiset.pure_def Multiset.pure_def\n\n@[simp]\ntheorem bind_def {\u03b1 \u03b2} : (\u00b7 >>= \u00b7) = @bind \u03b1 \u03b2 :=\n  rfl\n#align multiset.bind_def Multiset.bind_def\n\ninstance : LawfulMonad Multiset := LawfulMonad.mk'\n  (bind_pure_comp := fun _ _ \u21a6 by simp only [pure_def, bind_def, bind_singleton, fmap_def])\n  (id_map := fun _ \u21a6 by simp only [fmap_def, id_eq, map_id'])\n  (pure_bind := fun _ _ \u21a6 by simp only [pure_def, bind_def, singleton_bind])\n  (bind_assoc := @bind_assoc)\n\nopen Functor\n\nopen Traversable IsLawfulTraversable\n\n@[simp]\ntheorem lift_coe {\u03b1 \u03b2 : Type _} (x : List \u03b1) (f : List \u03b1 \u2192 \u03b2)\n    (h : \u2200 a b : List \u03b1, a \u2248 b \u2192 f a = f b) : Quotient.lift f h (x : Multiset \u03b1) = f x :=\n  Quotient.lift_mk _ _ _\n#align multiset.lift_coe Multiset.lift_coe\n\n@[simp]\ntheorem map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n    Functor.map h \u2218 Coe.coe = (Coe.coe \u2218 Functor.map h : List \u03b1 \u2192 Multiset \u03b2) := by\n  funext ; simp only [Function.comp_apply, Coe.coe, fmap_def, coe_map, List.map_eq_map]\n#align multiset.map_comp_coe Multiset.map_comp_coe\n\ntheorem id_traverse {\u03b1 : Type _} (x : Multiset \u03b1) : traverse (pure : \u03b1 \u2192 Id \u03b1) x = x := by\n  refine' Quotient.inductionOn x _\n  intro\n  simp [traverse, Coe.coe]\n#align multiset.id_traverse Multiset.id_traverse\n\ntheorem comp_traverse {G H : Type _ \u2192 Type _} [Applicative G] [Applicative H] [CommApplicative G]\n    [CommApplicative H] {\u03b1 \u03b2 \u03b3 : Type _} (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : Multiset \u03b1) :\n    traverse (Comp.mk \u2218 Functor.map h \u2218 g) x = Comp.mk (Functor.map (traverse h) (traverse g x)) :=\n  by\n  refine' Quotient.inductionOn x _\n  intro\n  simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,\n    functor_norm]\n  simp only [Function.comp, lift_coe]\n#align multiset.comp_traverse Multiset.comp_traverse\n\ntheorem map_traverse {G : Type _ \u2192 Type _} [Applicative G] [CommApplicative G] {\u03b1 \u03b2 \u03b3 : Type _}\n    (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) (x : Multiset \u03b1) :\n    Functor.map (Functor.map h) (traverse g x) = traverse (Functor.map h \u2218 g) x := by\n  refine' Quotient.inductionOn x _\n  intro\n  simp only [traverse, quot_mk_to_coe, lift_coe, Function.comp_apply, Functor.map_map, map_comp_coe]\n  rw [LawfulFunctor.comp_map, Traversable.map_traverse']\n  rfl\n#align multiset.map_traverse Multiset.map_traverse\n\ntheorem traverse_map {G : Type _ \u2192 Type _} [Applicative G] [CommApplicative G] {\u03b1 \u03b2 \u03b3 : Type _}\n    (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3) (x : Multiset \u03b1) : traverse h (map g x) = traverse (h \u2218 g) x := by\n  refine' Quotient.inductionOn x _\n  intro\n  simp only [traverse, quot_mk_to_coe, coe_map, lift_coe, Function.comp_apply]\n  rw [\u2190 Traversable.traverse_map h g, List.map_eq_map]\n#align multiset.traverse_map Multiset.traverse_map\n\ntheorem naturality {G H : Type _ \u2192 Type _} [Applicative G] [Applicative H] [CommApplicative G]\n    [CommApplicative H] (eta : ApplicativeTransformation G H) {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 G \u03b2)\n    (x : Multiset \u03b1) : eta (traverse f x) = traverse (@eta _ \u2218 f) x := by\n  refine' Quotient.inductionOn x _\n  intro\n  simp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply,\n    ApplicativeTransformation.preserves_map, IsLawfulTraversable.naturality]\n#align multiset.naturality Multiset.naturality\n\nend Multiset\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/Multiset/Functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.37994626365998174}}
{"text": "import data.finsupp.basic\n\nsection\n\n\n/-- An inductive type from which to index the variables of the mv_polynomials the proof manages -/\n@[derive decidable_eq]\ninductive vars : Type\n| \u03b1 : vars\n| \u03b2 : vars\n\nlemma finsupp_vars_eq_ext (f g : vars \u2192\u2080 \u2115) : f = g \u2194 \n  f vars.\u03b1 = g vars.\u03b1 \u2227 f vars.\u03b2 = g vars.\u03b2 :=\nbegin\n  rw finsupp.ext_iff,\n  split,\n    {\n      intro h,\n      split, exact h vars.\u03b1,\n      exact h vars.\u03b2,\n    },\n    {\n      intro h,\n      intro a,\n      induction a,\n      finish,\n      finish,\n    },\n  -- induction,\nend\n\nend", "meta": {"author": "BoltonBailey", "repo": "formal-snarks-project", "sha": "154414784f90a1e257162fcbdd7e805ecb2a49c2", "save_path": "github-repos/lean/BoltonBailey-formal-snarks-project", "path": "github-repos/lean/BoltonBailey-formal-snarks-project/formal-snarks-project-154414784f90a1e257162fcbdd7e805ecb2a49c2/src/snarks/toysnark/vars.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3798989861652788}}
{"text": "import ..mcrl2_encap.mcrl2_encap\nimport ..transition.sum\n\nopen mcrl2\n\nvariable {\u03b1 : Type}\nvariable [comm_semigroup_with_zero \u03b1]\nvariable {\u03b2 : Type}\n\n/- The proofs of the mcrl2 axioms for the quotient. -/\nlemma mcrl2'.sum_idem {x : mcrl2 \u03b1} {D : set \u03b2} (h : \u2203d, d \u2208 D) :\nsum D (\u03bba, x) \u2248 x := by exact R_add_congr (transition.sum_idem D h x)\n\nlemma mcrl2'.sum_elem {f : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} {d} (h : d \u2208 D) :\nsum D f \u2248 sum D f + f d := by exact R_add_congr (transition.sum_elem D f d h)\n\nlemma mcrl2'.sum_alt  {f g : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} :\nsum D (\u03bba, f a + g a) \u2248 sum D f + sum D g := by exact R_add_congr (transition.sum_alt D f g)\n\nlemma mcrl2'.sum_seq {f : \u03b2 \u2192 mcrl2 \u03b1} {x D} :\nsum D f \u2b1d x \u2248 sum D (\u03bba, f a \u2b1d x) := by exact R_add_congr (transition.sum_seq D f x)\n\nlemma mcrl2'.sum_parl {f : \u03b2 \u2192 mcrl2 \u03b1} {x D} :\nsum D f |_ x \u2248 sum D (\u03bba, f a |_ x) := by exact R_add_congr (transition.sum_parl D f x)\n\nlemma mcrl2'.sum_comm {f : \u03b2 \u2192 mcrl2 \u03b1} {x D} :\nsum D f \u2223 x \u2248 sum D (\u03bba, f a \u2223 x) := by exact R_add_congr (transition.sum_comm D f x)\n\nlemma mcrl2'.comm_sum {f : \u03b2 \u2192 mcrl2 \u03b1} {x D} :\nx \u2223 sum D f  \u2248 sum D (\u03bba, x \u2223 f a) := by exact R_add_congr (transition.comm_sum D f x)\n\nlemma mcrl2'.encap_sum {f : \u03b2 \u2192 mcrl2 \u03b1} {H D} :\nencap H (sum D f) \u2248 sum D (\u03bba, encap H (f a)) := by exact R_add_congr (transition.encap_sum H D f)\n\ninductive R_sum_ext {f g : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} (R_\u03b1 : \u2200a, a \u2208 D \u2192 mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop) :\nmcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop\n| R {x y a ha} (h : R_\u03b1 a ha x y) : R_sum_ext x y\n| basel : R_sum_ext (sum D f) (sum D g)\n| baser : R_sum_ext (sum D g) (sum D f)\n| refl {x} : R_sum_ext x x\n\nlemma R_sum_ext_refl  {f g : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} (R_\u03b1 : \u2200a, a \u2208 D \u2192 mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop) :\nreflexive (@R_sum_ext \u03b1 _ \u03b2 f g D R_\u03b1) :=\nbegin\n  intro x,\n  apply R_sum_ext.refl\nend\n\nlemma R_sum_ext.symm  {f g : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} (R_\u03b1 : \u2200a, a \u2208 D \u2192 mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop) \n(R_\u03b1_symm : \u2200a ha, symmetric (R_\u03b1 a ha)) :\nsymmetric (@R_sum_ext \u03b1 _ \u03b2 f g D R_\u03b1) :=\nbegin\n  intros x y h,\n  cases h,\n  { apply R_sum_ext.R,\n    apply R_\u03b1_symm _ _,\n    assumption},\n  { exact R_sum_ext.baser},\n  { exact R_sum_ext.basel},\n  { assumption}\nend\n\nlemma mcrl2'.sum_ext {f g : \u03b2 \u2192 mcrl2 \u03b1} {D : set \u03b2} (h : \u2200d, d \u2208 D \u2192 f d \u2248 g d) :\nsum D f \u2248 sum D g :=\nbegin\n  choose R R\u2081x R_bisim using h,\n  apply exists.intro (R_sum_ext R),\n  apply and.intro,\n  exact R_sum_ext.basel,\n  apply and.intro,\n  { intros x y x' a h\u2081 h\u2082,\n    cases h\u2081,\n    { have h : \u2203y', transition y a y' \u2227 option.rel (R h\u2081_a h\u2081_ha) x' y',\n      by exact bisim_lift (R_bisim _ _) h\u2081_h h\u2082,\n      rcases h with \u27e8w, haw, hRw\u27e9,\n      apply exists.intro w,\n      apply and.intro haw,\n      apply option.rel.mono _ hRw,\n      intros x y,\n      exact R_sum_ext.R},\n    { cases h\u2082,\n      simp [transition.sum_iff, \u2190 exists_and_distrib_right, and_assoc],\n      have h : \u2203y', transition (g h\u2082_a') a y' \u2227 option.rel (R h\u2082_a' h\u2082_ha') x' y',\n      by exact bisim_lift (R_bisim _ _) (R\u2081x _ _) h\u2082_h,\n      rcases h with \u27e8w, haw, hRw\u27e9,\n      exact \u27e8w, h\u2082_a', h\u2082_ha', haw, begin\n        apply option.rel.mono _ hRw,\n        intros x y,\n        exact R_sum_ext.R\n      end\u27e9},\n    { cases h\u2082,\n      simp [transition.sum_iff, \u2190 exists_and_distrib_right, and_assoc],\n      have h : \u2203y', transition (f h\u2082_a') a y' \u2227 option.rel (R h\u2082_a' h\u2082_ha') x' y',\n      by exact bisim_lift (R_bisim _ _) ((R_bisim _ _).right (R\u2081x _ _)) h\u2082_h,\n      rcases h with \u27e8w, haw, hRw\u27e9,\n      exact \u27e8w, h\u2082_a', h\u2082_ha', haw, begin\n        apply option.rel.mono _ hRw,\n        intros x y,\n        exact R_sum_ext.R\n      end\u27e9},\n    { exact \u27e8x', h\u2082, by exact option.rel.refl (R_sum_ext_refl R)\u27e9}},\n  { choose R_bisim R_symm using R_bisim,\n    exact R_sum_ext.symm R R_symm}\nend", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/mcrl2_sum/sum_axioms.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3798989788233756}}
{"text": "\nmeta def hello := con\n\nnamespace floaties\n/- I added a `float` VM component. -/\n    open native\n    meta def x : float := 199\n    #eval -x ^ (3 : float)\n    #eval (199 / 400 : float) -- outputs 0.4975\n    #eval ((23.45 : float) = (23.45 : float) : bool)\n\nend floaties\n\u2200  \u2200l  \u03b1ll  \nvariables {\u03b1 \u03b2 : Type}\nmeta def Y {\u03b1 : Type} : (\u03b1 \u2192 \u03b1) \u2192 \u03b1 | f := f (Y f)\n\nuniverses u v\n\n#eval to_string $ expr.subst ( (expr.var 1)) ```(nat.succ 0)\n\n#eval to_string $ expr.is_internal_cnstr $ (expr.lam `hello binder_info.default (expr.var 1) $ expr.app (expr.app (expr.var 5) (expr.var 3)) `(4))\n\n#eval to_string $ (expr.instantiate_var (expr.app (expr.var 5) (expr.var 2435)) `(1))\ndef e := \u03a0 a : Type u, \u03a0 b : Type v, a \u00d7 b\n#check e\n#eval to_string $ expr.collect_univ_params $ expr.reflect e\n\n#eval @undefined_core nat \"wtf\"\n\nnamespace hidden\n\ndefinition cong (a b m : \u2124) : Prop := \u2203 n : \u2124, b - a = m * n\n\nnotation a ` \u2261 ` b ` mod ` m  := cong a b m \nset_option trace.simplify true\ntheorem cong_refl (m : \u2124) : \u2200 a : \u2124, a \u2261 a mod m :=\nbegin\nintro a,\nunfold cong,\nexistsi (0:\u2124),\nsimp\nend\n\nmeta def trace_simp_lemmas : tactic unit :=\ndo\n    l \u2190 tactic.get_simp_lemmas_or_default none,\n    str \u2190 simp_lemmas.pp l,\n    tactic.trace str\n\nexample : true := begin trace_simp_lemmas, trivial  end\n\nnamespace my_namespace\n    constant X : Type u\n    constants x y : X\n    constant f : X \u2192 X\n    constants  (P : X \u2192 Prop) (Q : Prop)\n    /--Hello world-/\n    axiom my_axiom : \u2200 (a:X) (p : P(a)) (q:Q), Q\n    #check my_axiom\n    #check expr.mvar\n    meta def scratch_tac : tactic unit := do\n        -- a \u2190 tactic.resolve_name `my_axiom >>= tactic.to_expr,\n        -- l \u2190 tactic.apply_core a {},\n        -- g \u2190 tactic.target,\n        -- tactic.set_tag g [`hello.world],\n        -- e \u2190 tactic.head_eta_expand a,\n        -- tactic.trace e,\n        tactic.mk_fresh_name >>= tactic.trace,\n        pure()\n\n    example : Q :=\n    begin\n        scratch_tac,\n        scratch_tac,\n        scratch_tac,\n        scratch_tac,\n        tactic.rotate_left 1,\n        sorry\n    end\n    example : true := begin \n        tactic.add_doc_string `nat \"my dumb axiom is great at breaking maths yay.\",\n        tactic.doc_string `nat >>= tactic.trace,\n        trivial \n    end\n    #check nat\nsection scratch\n\nend scratch\nend my_namespace\n\n\nend hidden\n\n\nexample (P : \u03b1 \u2192 Prop) (Q : Prop) (h\u2081 : \u2203 x, P(x)) (h\u2082 : \u2200 x, P(x) \u2192 Q) : Q :=\nbegin\n    cases h\u2081,\n    apply h\u2082 _, assumption\nend\n\nexample : \u2200 (n : nat), n = n := \nbegin\n  intro,\n  induction n,\nend\n\n\nnamespace field_idea_auto_param\n    -- eventually a sophisticated tactic that figures out if an elt is \u2260 0\n    meta def nz_tactic := tactic.assumption \n    class dvr (R : Type u) extends (ring R) :=\n    (inv(x:R) (p:x \u2260 0):  R)\n    (inv_l : \u03a0 (x y : R) (p:y\u22600), x * (inv y p)  = 1 )\n    variables {R : Type u} [dvr R]\n    def inv (y : R) (nz : y \u2260 0 . nz_tactic) : R := dvr.inv y nz\n\n    def div (x y : R) (nz : y \u2260 0 . nz_tactic) : R := x * (inv y)\n    infix ` /. `:50 := div\n\n    variables {x y : R} \n    variable (xz : x \u2260 0 . nz_tactic) -- I was really hoping that this would be allowed\n\n    example (x y : R) (xz : x \u2260 0) (yz : y \u2260 0) : (1 /. (x * y)) = (1 /. x) * (1 /. y) := sorry\n\nend field_idea_auto_param\n\nnamespace field_idea_class\n    class my_division_ring (R : Type u) extends (integral_domain R) :=\n    (inv : \u03a0(x:R) [x \u2260 0],  R)\n    (inv_l (x : R) [nz:x \u2260 0] : x * (@inv x nz)  = 1 )\n    (inv_r (x : R) [nz:x \u2260 0] : (@inv x nz) * x  = 1 )\n\n    variables {R : Type u} [my_division_ring R]\n    def inv (y : R) [nz:y \u2260 0] : R := @my_division_ring.inv _ _ y nz\n\n    def div (x y : R) [y \u2260 0] : R := x * (inv y)\n    infix ` \u00f7 `:50 := div\n\n    variables {x y : R} [x \u2260 0] [y \u2260 0]\n    def asdf : ((x * y) \u2260 0) := mul_ne_zero \u2039x \u2260 0\u203a \u2039y \u2260 0\u203a\n\n    example : (1 \u00f7 (x * y)) = (1 \u00f7 x) * (1 \u00f7 y) := sorry\n    -- calc (1 \u00f7 (x * y)) = inv(x * y) * 1 : by sorry\n    --      ...           = inv(x * y) * (x * inv(x)) * (y * inv(y)) : by sorry\n    --      ...           = (inv(x * y) * (x * y)) * (inv(x) * inv(y)) : by ac_refl\n    --      ...           = 1 * (1 * inv(x)) * (1 * inv(y)) : by sorry\n    --      ...           = (1 \u00f7 x) * (1 \u00f7 y) : by sorry\n\nend field_idea_class", "meta": {"author": "EdAyers", "repo": "edlib", "sha": "78b8c5d91f023f939c102837d748868e2f3ed27d", "save_path": "github-repos/lean/EdAyers-edlib", "path": "github-repos/lean/EdAyers-edlib/edlib-78b8c5d91f023f939c102837d748868e2f3ed27d/scratch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3798989788233756}}
{"text": "import Structure.Generic.Axioms.Universes\nimport Structure.Generic.Axioms.AbstractFunctors\nimport Structure.Generic.Axioms.AbstractEquivalences\nimport Structure.Generic.Lemmas.DerivedFunctors\nimport Structure.Generic.Notation\n\n\n\nset_option autoBoundImplicitLocal false\n--set_option pp.universes true\n\nuniverses u v\n\n\n\ndef GeneralizedProperty (\u03b1 : Sort u) (V : Universe.{v}) := \u03b1 \u2192 V\n\nnamespace GeneralizedProperty\n\n  variable {\u03b1 : Sort u} {V : Universe.{v}}\n\n  instance hasInstances : HasInstances (GeneralizedProperty \u03b1 V) := \u27e8\u03bb P => \u2200 a, P a\u27e9\n\n  section Properties\n\n    variable (P : GeneralizedProperty \u03b1 V)\n\n    class HasInst where\n    (inst (a : \u03b1) : P a)\n\n  end Properties\n\nend GeneralizedProperty\n\nopen GeneralizedProperty\n\n\n\n-- TODO: Update comments\n\n-- We want to formalize a very general \"structure with equivalences\", so we start with a very basic\n-- abstraction for something that looks like an equivalence relation except that the codomain is\n-- generalized to `Sort u` instead of `Prop`. Therefore, `\u27e8Equiv.refl, Equiv.symm, Equiv.trans\u27e9`, where\n-- `Equiv` is the Lean 4 version of the `equiv` type in Lean 3 mathlib, is also an instance of this type\n-- (with the restriction that both arguments must live in the same universe).\n--\n-- We actually need to generalize slightly further to a codomain that is not necessarily a sort but can be\n-- coerced to a sort. This way, the codomain can be any Lean structure that bundles a sort, in particular\n-- it can be our `Structure` type.\n\ndef GeneralizedRelation (\u03b1 : Sort u) (V : Universe.{v}) := \u03b1 \u2192 \u03b1 \u2192 V\n\nnamespace GeneralizedRelation\n\n  variable {\u03b1 : Sort u} {V : Universe.{v}}\n\n  instance hasInstances : HasInstances (GeneralizedRelation \u03b1 V) := \u27e8\u03bb R => \u2200 a b, R a b\u27e9\n\n  section Properties\n\n    variable (R : GeneralizedRelation \u03b1 V)\n\n    class HasRefl where\n    (refl (a : \u03b1) : R a a)\n\n    variable [HasInternalFunctors V]\n\n    class HasTrans where\n    (trans {a b c : \u03b1} : R a b \u27f6 R b c \u27f6 R a c)\n\n    class IsPreorder extends HasRefl R, HasTrans R\n\n    variable [HasInternalEquivalences V]\n\n    class HasSymm where\n    (symm {a b : \u03b1} : R a b \u27f7 R b a)\n\n    class IsEquivalence extends IsPreorder R, HasSymm R\n  \n  end Properties\n\n  def HasTrans.revTrans {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [HasLinearFunOp V] [h : HasTrans R]\n                        {a b c : \u03b1} : R b c \u27f6 R a b \u27f6 R a c :=\n  HasLinearFunOp.swapFunFun h.trans\n\n  @[simp] theorem HasTrans.revTrans.eff {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [HasLinearFunOp V] [h : HasTrans R]\n                                        {a b c : \u03b1} (g : R b c) (f : R a b) :\n    h.revTrans g f = h.trans f g :=\n  by apply HasLinearFunOp.swapFunFun.effEff\n\n  def HasTrans.trans' {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [h : HasTrans R]\n                      {a b c : \u03b1} (f : R a b) (g : R b c) : R a c := h.trans f g\n\n  def HasSymm.symm' {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [HasInternalEquivalences V] [h : HasSymm R]\n                    {a b : \u03b1} (f : R a b) : R b a := HasInternalEquivalences.to h.symm f\n\n  -- When reasoning about instances of `R a b`, we would like to write `trans` as composition, `refl` as\n  -- identity, and `symm` as inverse.\n  -- Note that `R` can be inferred from `f : R a b` by elaboration.\n\n  section Notation\n\n    @[reducible] def revComp {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [h : HasTrans R] {a b c : \u03b1} (g : R b c) (f : R a b) : R a c := h.trans' f g\n    infixr:90 \" \u2022 \" => revComp\n\n    @[reducible] def ident (R : GeneralizedRelation \u03b1 V) [h : HasRefl R] (a : \u03b1) : R a a := h.refl a\n\n    @[reducible] def inv {R : GeneralizedRelation \u03b1 V} [HasInternalFunctors V] [HasInternalEquivalences V] [h : HasSymm R] {a b : \u03b1} (f : R a b) : R b a := h.symm' f\n    postfix:max \"\u207b\u00b9\" => inv\n\n  end Notation\n\nend GeneralizedRelation\n\nopen GeneralizedRelation\n\n\n\n-- We can attach products, arrows, and/or equivalences to a given sort, in the form of generalized\n-- relations satisfying appropriate properties.\n\nsection AttachedRelations\n\n  variable (\u03b1 : Sort u) (V : Universe.{v}) [HasInternalFunctors V]\n\n  class HasArrows where\n  (Arrow      : GeneralizedRelation \u03b1 V)\n  [isPreorder : IsPreorder Arrow]\n\n  namespace HasArrows\n    variable [h : HasArrows \u03b1 V]\n    instance arrowPreorder : IsPreorder h.Arrow := h.isPreorder\n    instance hasArrow : HasArrow \u03b1 \u03b1 := \u27e8h.Arrow\u27e9\n    instance : HasInstances (HasArrow.\u03b3 \u03b1 \u03b1) := Universe.instInst V\n    instance : IsPreorder (@HasArrow.Arrow \u03b1 \u03b1 (hasArrow \u03b1 V)) := h.isPreorder\n  end HasArrows\n\n  variable [HasInternalEquivalences V]\n\n  class HasEquivalences where\n  (Equiv   : GeneralizedRelation \u03b1 V)\n  [isEquiv : IsEquivalence Equiv]\n\n  namespace HasEquivalences\n    variable [h : HasEquivalences \u03b1 V]\n    instance equivEquivalence : IsEquivalence h.Equiv := h.isEquiv\n    instance hasEquivalence : HasEquivalence \u03b1 \u03b1 := \u27e8h.Equiv\u27e9\n    instance : HasInstances (HasEquivalence.\u03b3 \u03b1 \u03b1) := Universe.instInst V\n    instance : IsEquivalence (@HasEquivalence.Equiv \u03b1 \u03b1 (hasEquivalence \u03b1 V)) := h.isEquiv\n  end HasEquivalences\n\n  class HasProducts where\n  (Product : GeneralizedRelation \u03b1 V)\n  [hasSymm : HasSymm Product]\n\n  namespace HasProducts\n    variable [h : HasProducts \u03b1 V]\n    instance productSymm : HasSymm h.Product := h.hasSymm\n    instance hasProduct : HasProduct \u03b1 \u03b1 := \u27e8h.Product\u27e9\n    instance : HasInstances (HasProduct.\u03b3 \u03b1 \u03b1) := Universe.instInst V\n    instance : HasSymm (@HasProduct.Product \u03b1 \u03b1 (hasProduct \u03b1 V)) := h.hasSymm\n  end HasProducts\n\nend AttachedRelations\n", "meta": {"author": "SReichelt", "repo": "lean4-experiments", "sha": "ff55357a01a34a91bf670d712637480089085ee4", "save_path": "github-repos/lean/SReichelt-lean4-experiments", "path": "github-repos/lean/SReichelt-lean4-experiments/lean4-experiments-ff55357a01a34a91bf670d712637480089085ee4/Structure/Generic/Axioms/GeneralizedProperties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3798989788233756}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.fintype.units\n! leanprover-community/mathlib commit 327c3c0d9232d80e250dc8f65e7835b82b266ea5\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Fintype.Prod\nimport Mathbin.Data.Fintype.Sum\nimport Mathbin.Data.Int.Units\n\n/-!\n# fintype instances relating to units\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nvariable {\u03b1 : Type _}\n\n/- warning: units_int.fintype -> UnitsInt.fintype is a dubious translation:\nlean 3 declaration is\n  Fintype.{0} (Units.{0} Int Int.monoid)\nbut is expected to have type\n  Fintype.{0} (Units.{0} Int Int.instMonoidInt)\nCase conversion may be inaccurate. Consider using '#align units_int.fintype UnitsInt.fintype\u2093'. -/\ninstance UnitsInt.fintype : Fintype \u2124\u02e3 :=\n  \u27e8{1, -1}, fun x => by cases Int.units_eq_one_or x <;> simp [*]\u27e9\n#align units_int.fintype UnitsInt.fintype\n\n/- warning: units_int.univ -> UnitsInt.univ is a dubious translation:\nlean 3 declaration is\n  Eq.{1} (Finset.{0} (Units.{0} Int Int.monoid)) (Finset.univ.{0} (Units.{0} Int Int.monoid) UnitsInt.fintype) (Insert.insert.{0, 0} (Units.{0} Int Int.monoid) (Finset.{0} (Units.{0} Int Int.monoid)) (Finset.hasInsert.{0} (Units.{0} Int Int.monoid) (fun (a : Units.{0} Int Int.monoid) (b : Units.{0} Int Int.monoid) => Units.decidableEq.{0} Int Int.monoid (fun (a : Int) (b : Int) => Int.decidableEq a b) a b)) (OfNat.ofNat.{0} (Units.{0} Int Int.monoid) 1 (OfNat.mk.{0} (Units.{0} Int Int.monoid) 1 (One.one.{0} (Units.{0} Int Int.monoid) (MulOneClass.toHasOne.{0} (Units.{0} Int Int.monoid) (Units.mulOneClass.{0} Int Int.monoid))))) (Singleton.singleton.{0, 0} (Units.{0} Int Int.monoid) (Finset.{0} (Units.{0} Int Int.monoid)) (Finset.hasSingleton.{0} (Units.{0} Int Int.monoid)) (Neg.neg.{0} (Units.{0} Int Int.monoid) (Units.hasNeg.{0} Int Int.monoid (NonUnitalNonAssocRing.toHasDistribNeg.{0} Int (NonAssocRing.toNonUnitalNonAssocRing.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring)))) (OfNat.ofNat.{0} (Units.{0} Int Int.monoid) 1 (OfNat.mk.{0} (Units.{0} Int Int.monoid) 1 (One.one.{0} (Units.{0} Int Int.monoid) (MulOneClass.toHasOne.{0} (Units.{0} Int Int.monoid) (Units.mulOneClass.{0} Int Int.monoid))))))))\nbut is expected to have type\n  Eq.{1} (Finset.{0} (Units.{0} Int Int.instMonoidInt)) (Finset.univ.{0} (Units.{0} Int Int.instMonoidInt) UnitsInt.fintype) (Insert.insert.{0, 0} (Units.{0} Int Int.instMonoidInt) (Finset.{0} (Units.{0} Int Int.instMonoidInt)) (Finset.instInsertFinset.{0} (Units.{0} Int Int.instMonoidInt) (fun (a : Units.{0} Int Int.instMonoidInt) (b : Units.{0} Int Int.instMonoidInt) => Units.instDecidableEqUnits.{0} Int Int.instMonoidInt (fun (a : Int) (b : Int) => Int.instDecidableEqInt a b) a b)) (OfNat.ofNat.{0} (Units.{0} Int Int.instMonoidInt) 1 (One.toOfNat1.{0} (Units.{0} Int Int.instMonoidInt) (InvOneClass.toOne.{0} (Units.{0} Int Int.instMonoidInt) (DivInvOneMonoid.toInvOneClass.{0} (Units.{0} Int Int.instMonoidInt) (DivisionMonoid.toDivInvOneMonoid.{0} (Units.{0} Int Int.instMonoidInt) (DivisionCommMonoid.toDivisionMonoid.{0} (Units.{0} Int Int.instMonoidInt) (CommGroup.toDivisionCommMonoid.{0} (Units.{0} Int Int.instMonoidInt) (Units.instCommGroupUnitsToMonoid.{0} Int Int.instCommMonoidInt)))))))) (Singleton.singleton.{0, 0} (Units.{0} Int Int.instMonoidInt) (Finset.{0} (Units.{0} Int Int.instMonoidInt)) (Finset.instSingletonFinset.{0} (Units.{0} Int Int.instMonoidInt)) (Neg.neg.{0} (Units.{0} Int Int.instMonoidInt) (Units.instNegUnits.{0} Int Int.instMonoidInt (NonUnitalNonAssocRing.toHasDistribNeg.{0} Int (NonAssocRing.toNonUnitalNonAssocRing.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)))) (OfNat.ofNat.{0} (Units.{0} Int Int.instMonoidInt) 1 (One.toOfNat1.{0} (Units.{0} Int Int.instMonoidInt) (InvOneClass.toOne.{0} (Units.{0} Int Int.instMonoidInt) (DivInvOneMonoid.toInvOneClass.{0} (Units.{0} Int Int.instMonoidInt) (DivisionMonoid.toDivInvOneMonoid.{0} (Units.{0} Int Int.instMonoidInt) (DivisionCommMonoid.toDivisionMonoid.{0} (Units.{0} Int Int.instMonoidInt) (CommGroup.toDivisionCommMonoid.{0} (Units.{0} Int Int.instMonoidInt) (Units.instCommGroupUnitsToMonoid.{0} Int Int.instCommMonoidInt)))))))))))\nCase conversion may be inaccurate. Consider using '#align units_int.univ UnitsInt.univ\u2093'. -/\n@[simp]\ntheorem UnitsInt.univ : (Finset.univ : Finset \u2124\u02e3) = {1, -1} :=\n  rfl\n#align units_int.univ UnitsInt.univ\n\n#print Fintype.card_units_int /-\n@[simp]\ntheorem Fintype.card_units_int : Fintype.card \u2124\u02e3 = 2 :=\n  rfl\n#align fintype.card_units_int Fintype.card_units_int\n-/\n\ninstance [Monoid \u03b1] [Fintype \u03b1] [DecidableEq \u03b1] : Fintype \u03b1\u02e3 :=\n  Fintype.ofEquiv _ (unitsEquivProdSubtype \u03b1).symm\n\ninstance [Monoid \u03b1] [Finite \u03b1] : Finite \u03b1\u02e3 :=\n  Finite.of_injective _ Units.ext\n\n#print Fintype.card_units /-\ntheorem Fintype.card_units [GroupWithZero \u03b1] [Fintype \u03b1] [Fintype \u03b1\u02e3] :\n    Fintype.card \u03b1\u02e3 = Fintype.card \u03b1 - 1 := by\n  classical\n    rw [eq_comm, Nat.sub_eq_iff_eq_add (Fintype.card_pos_iff.2 \u27e8(0 : \u03b1)\u27e9),\n      Fintype.card_congr (unitsEquivNeZero \u03b1)]\n    have := Fintype.card_congr (Equiv.sumCompl (\u00b7 = (0 : \u03b1))).symm\n    rwa [Fintype.card_sum, add_comm, Fintype.card_subtype_eq] at this\n#align fintype.card_units Fintype.card_units\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Fintype/Units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.37973086127400996}}
{"text": "import Lean\nimport Std.Data.HashMap\nimport LeanCodePrompts.ConstDeps\nimport LeanCodePrompts.VerboseDelabs\n\n/-!\n# Premise data\n\nHere we extract premise data from all definitions in the environment. This includes proofs in the environment as well as sub-proofs in proofs/definitions. The main technique for working with subproofs is the use of the custom *verbose* delaborators that rewrite the syntax tree to include the statements of the proof terms, in the syntax `proof =: prop`. \n\nWe are using premises is a broad sense, including:\n\n- identifiers\n- propositions that are proved by sub-terms (lemmas)\n- terms that are arguments of functions (instantiations)\n\nAs theorems are equivalent to others where we trade `\u2200` with context terms, we associate groups. Further, we exclude statements derived in this way (i.e., by `intro`) from the list of lemmas.\n-/\n\nopen Lean Meta Elab Parser PrettyPrinter\n\nuniverse u v w u_1 u_2 u_3 u\u2081 u\u2082 u\u2083\n\nopen LeanAide.Meta\n/-- All constants in the environment with value and type. -/\ndef constantNameValueTypes  : MetaM (Array (Name \u00d7 Expr \u00d7   Expr)) := do\n  let env \u2190 getEnv\n  let decls := env.constants.map\u2081.toArray\n  let allNames := decls.filterMap <| \n    fun (name, dfn) => dfn.value? |>.map fun t => (name, t, dfn.type) \n  let names \u2190 allNames.filterM (fun (name, _) => isWhiteListed name)\n  let names := names.filter <| \n    fun (name, _, _)  \u21a6 !(excludePrefixes.any (fun pfx => pfx.isPrefixOf name)) && !(excludeSuffixes.any (fun pfx => pfx.isSuffixOf name)) \n  return names\n\nset_option pp.unicode.fun true\n\n/-- Syntax as json -/\ninstance : ToJson Syntax := \u27e8fun (d: Syntax) \u21a6 d.reprint.get!\u27e9\n\n/-- Subterms in a premise -/\nstructure TermData where\n    context : Array Syntax\n    value : Syntax\n    size : Nat\n    depth: Nat\nderiving Repr, ToJson\n\n/-- Increase depth of a subterm (for recursion) -/\ndef TermData.increaseDepth (d: Nat) : TermData \u2192 TermData :=\nfun data \u21a6\n    \u27e8data.context, data.value, data.size, data.depth + d\u27e9\n\n/-- Lemma data with proofs -/\nstructure PropProofData where\n    context : Array Syntax\n    prop : Syntax\n    proof: Syntax\n    propSize: Nat \n    proofSize: Nat\n    depth: Nat\nderiving Repr, ToJson\n\n/-- Increase depth for a lemma (for recursion) -/\ndef PropProofData.increaseDepth (d: Nat) : PropProofData \u2192 PropProofData :=\nfun data \u21a6\n    \u27e8data.context, data.prop, data.proof, data.propSize, data.proofSize, data.depth + d\u27e9\n\n/-- Full premise data for a proposition -/        \nstructure PremiseData  where \n context : (Array Syntax) -- variables, types, binders\n name? :       Option Name  -- name\n defnName: Name -- name of definition from which it arose\n type :       Syntax  -- proposition\n typeGroup : Syntax  -- proposition group\n proof: Syntax  -- proof\n typeSize : Nat\n proofSize : Nat\n terms :       Array (TermData)  -- instantiations\n propProofs :       Array (PropProofData)  -- sub-proofs\n ids :       Array (Name \u00d7  Nat)  -- proof identifiers used\n deriving Repr, ToJson\n\n\nnamespace PremiseData\n\ndef filterIds (pd: PremiseData)(p: Name \u2192 Bool) : PremiseData := \n    \u27e8pd.context, pd.name?, pd.defnName,  pd.type, pd.typeGroup, pd.proof, pd.typeSize, pd.proofSize, pd.terms, pd.propProofs, pd.ids.filter (fun (n, _) => p n)\u27e9\n\ndef increaseDepth (d: Nat) : PremiseData \u2192 PremiseData :=  \nfun data \u21a6\n    \u27e8data.context, data.name?, data.defnName, data.type, data.typeGroup, data.proof, data.typeSize, data.proofSize, (data.terms.map (fun td => td.increaseDepth d)), (data.propProofs.map (fun p => p.increaseDepth d)),\n        (data.ids.map (fun (n,  m) => (n,  m + d))) \u27e9\n\nend PremiseData\n\n/-- Remove the added `=: prop` from syntax -/\npartial def Lean.Syntax.purge: Syntax \u2192 Syntax := fun stx \u21a6\n  match stx with\n  | Syntax.node info k args =>\n    match stx with\n    | `(($pf:term =: $_:term)) =>\n      pf.raw.purge\n    | _ =>\n      Syntax.node info k (args.map Syntax.purge) \n  | s => s\n\ndef termKinds : MetaM <| SyntaxNodeKindSet :=  do\n    let env \u2190 getEnv\n    let categories := (parserExtension.getState env).categories\n    let termCat? := getCategory categories `term\n    return termCat?.get!.kinds    \n\ndef termKindList : MetaM <| List (SyntaxNodeKind \u00d7 Unit) := do\n    let s \u2190 termKinds\n    pure <| s.toList \n\npartial def Lean.Syntax.size (stx: Syntax) : Nat := \n    match stx with\n    | Syntax.ident _ _ _ _ => 1\n    | Syntax.node _ _ args => args.foldl (fun acc x => acc + x.size) 0\n    | _ => 1\n\n/-- Compute recursively premise-data of sublemmas as well as the identifiers, instantiations and subproofs. These are used at the top level recursively.\n\nThe parameter `isArg` specifies whether the term is an argument of a function. This is used to determine whether to add the term to the list of instantiations. \n\nThe parameter `propHead?` specifies the head of the group of propositions, where groups are related by `intro`, i.e., moving from `\u2200` to context variables. This is used to determine whether to add the proposition to the list of lemmas.\n-/\npartial def Lean.Syntax.premiseDataAuxM (context : Array Syntax)(defnName: Name)(stx: Syntax)(propHead? : Option Syntax)(isArg: Bool)(maxDepth? : Option Nat := none) : \n    MetaM (\n        Array (TermData) \u00d7\n        Array (PropProofData) \u00d7\n        Array (Name \u00d7 Nat) \u00d7\n        List PremiseData\n        )  := do\n    if maxDepth? = some 0 then\n        pure (#[], #[], #[], [])    \n    else\n    let tks \u2190 termKindList\n    let tks := tks.map (\u00b7.1)\n    match \u2190 namedArgument? stx with\n    | some (arg, _) => -- named argument of a function, name ignored\n        arg.premiseDataAuxM context defnName none  true (maxDepth?.map (\u00b7 -1))\n    | none =>\n    -- the special `proof =: prop` syntax \n    match \u2190 proofWithProp? stx with\n    | some (proof, prop) =>\n        -- start a group if not in a group\n        let newPropHead :=\n            match propHead? with\n            | some p => p\n            | none => prop\n        /- compute the data for the subproof; \n        subproof not an instantiation, is part of a new/old group. \n        -/\n        let prev \u2190  \n            proof.premiseDataAuxM context defnName (some newPropHead) false (maxDepth?.map (\u00b7 -1))\n        let (ts, pfs, ids, ps) := prev\n        let prop := prop.purge\n        let proof := proof.purge\n        let newPfs :=\n            if propHead?.isSome then -- exclude lemma if in prior group\n                pfs\n            else \n                let headPf : PropProofData := \n                    \u27e8context, prop, proof, prop.size, proof.size, 0\u27e9\n                pfs.map (fun s \u21a6 s.increaseDepth 1) |>.push headPf\n        let head : PremiseData := \n            \u27e8context, none, defnName, prop, newPropHead, proof, prop.size, proof.size, ts, pfs, ids\u27e9\n        return (ts.map (fun t \u21a6 t.increaseDepth 1),\n                newPfs,\n                ids.map (fun (s, m) => (s, m + 1)),\n                head :: ps)\n    | none =>\n    match \u2190 lambdaStx? stx with -- term is a lambda\n    | some (body, args) =>\n        let prev \u2190  /- data for subterm; not an instantiation; \n        inherits proposition group: if this is a proof, so would the previous term and hence we will have a group.  -/\n            body.premiseDataAuxM (context ++ args) defnName propHead? false (maxDepth?.map (\u00b7 -1))\n        let (ts, pfs, ids, ps) := prev\n        return (ts.map (fun s => (s.increaseDepth args.size)),\n                pfs.map (fun s => (s.increaseDepth args.size)),\n                ids.map (fun (s, m) => (s, m + args.size)),\n                ps)\n    | none =>\n    match \u2190 appStx? stx with\n    | some (f, arg) =>\n        let prev \u2190  f.premiseDataAuxM context defnName none false (maxDepth?.map (\u00b7 -1))\n        let (ts, pfs, ids, ps) := prev\n        let prev \u2190  arg.premiseDataAuxM context defnName none true (maxDepth?.map (\u00b7 -1))\n        let (ts', pfs', ids', ps') := prev\n        let ts'' := \n            if isArg then -- this is an instantiation\n            let head : TermData := \n                \u27e8context, stx.purge, stx.purge.size, 0\u27e9\n            (ts ++ ts').map (fun s => s.increaseDepth 1) |>.push head\n            else \n            (ts ++ ts').map (fun s => s.increaseDepth 1)\n        return (ts'',\n                (pfs ++ pfs').map (fun s => s.increaseDepth 1),\n                (ids ++ ids').map (fun (s, m) => (s, m + 1)),\n                ps ++ ps')\n    | none =>\n        match stx with\n        | Syntax.node _ k args => \n            let prevs \u2190 args.mapM (\n                premiseDataAuxM context defnName \u00b7 none false (maxDepth?.map (\u00b7 -1)))\n            let mut ts: Array (TermData) := #[]\n            let mut pfs: Array (PropProofData) := #[]\n            let mut ids: Array (Name \u00d7 Nat) := #[]\n            let mut ps: List PremiseData := []\n            for prev in prevs do\n                let (ts', pfs', ids', ps') := prev\n                ts := ts ++ ts'.map (fun s => s.increaseDepth 1)\n                pfs := pfs ++ pfs'.map (fun s => s.increaseDepth 1)\n                ids := ids ++ ids'.map (fun (s, m) => (s, m + 1))\n                ps := ps ++ ps'\n            let head : TermData := \n                \u27e8context, stx.purge, stx.purge.size, 0\u27e9\n            if isArg && tks.contains k then \n                ts := ts.push (head)\n            return (ts, pfs, ids, ps)\n        | Syntax.ident _ _ name .. => \n            let contextVars := context.filterMap getVar\n            if  !(contextVars.contains name) &&\n                !(excludePrefixes.any (fun pfx => pfx.isPrefixOf name)) && !(excludeSuffixes.any (fun pfx => pfx.isSuffixOf name)) then \n                pure (#[], #[], #[(name, 0)], [])\n            else pure (#[], #[], #[], [])\n        | _ => pure (#[], #[], #[], [])\n\ndef Lean.Syntax.premiseDataM (context : Array Syntax)\n    (proof prop: Syntax)(includeHead: Bool)(name? : Option Name)(defnName : Name)(maxDepth? : Option Nat := none) : \n    MetaM (List PremiseData) := do\n    let (ts, pfs, ids, ps) \u2190 proof.premiseDataAuxM context defnName (some prop) false maxDepth?\n    if includeHead then\n        let head : PremiseData := \u27e8context, name?, defnName, prop.purge, prop.purge, proof.purge, prop.purge.size, proof.purge.size, ts, pfs, ids\u27e9\n        return head :: ps\n    else return ps\n\n\nstructure DefData where\n    name : Name\n    type : Syntax\n    value : Syntax\n    isProp : Bool\n    typeDepth : Nat\n    valueDepth : Nat\n    premises : List PremiseData -- empty if depth exceeds bound\n    deriving Inhabited, ToJson\n\ndef DefData.getM? (name: Name)(term type: Expr) : MetaM (Option  DefData) := do\n    if term.approxDepth > (\u2190 getDelabBound) || type.approxDepth > (\u2190 getDelabBound) then return none\n    else\n    let (stx, _) \u2190  delabCore term {} (delabVerbose)\n    let (tstx, _) \u2190  delabCore type {} (delabVerbose)\n    let isProp \u2190 Meta.isProof term\n    let premises \u2190 \n        Lean.Syntax.premiseDataM #[] stx tstx isProp (some name) name\n    let typeDepth := type.approxDepth\n    let valueDepth := term.approxDepth\n    return some {name := name, type := tstx.raw.purge, value := stx.raw.purge, isProp := isProp, typeDepth := typeDepth.toNat, valueDepth := valueDepth.toNat, premises := premises}\n\nstructure IdentData where\n    context : Array Syntax\n    type : Syntax\n    ids : List Name\n    deriving Inhabited, ToJson\n\ndef IdentData.filter (d: IdentData)(p : Name \u2192 Bool) : IdentData := \n    {context:= d.context, type := d.type, ids := d.ids.filter p}\n\ndef DefData.identData (d: DefData) : List IdentData := \n    d.premises.map (fun p => \n        {context:= p.context, type := p.type, ids := p.ids.map (\u00b7.1) |>.toList.eraseDups})\n\ndef nameSize : MetaM <| Nat \u00d7 Nat := do\n    let cs \u2190 constantNameValueTypes\n    let cs' \u2190 cs.filterM <| fun (_, term, _) => \n        Meta.isProof term\n    return (cs.size, cs'.size)\n\n-- #check Json.pretty\n\n-- #eval nameSize\n\ndef nameSample (n: Nat) : MetaM (Array Name) := do\n    let cs \u2190 constantNameValueTypes \n    let mut out : Array Name := #[]\n    let mut count := 0\n    for (name, _, _) in cs do\n        if count % n = 0 then\n            out := out.push name\n        count := count + 1    \n    return out\n\n-- #eval nameSample 100\n\ndef batchDefns (start batch : Nat) : MetaM (Array Json) := do\n    let cs \u2190 constantNameValueTypes \n    let mut out : Array Json := #[]\n    let mut count := 0\n    for (name, term, type) in cs do\n        if count >= start && count < start + batch then\n            let defData? \u2190 DefData.getM? name term type\n            match defData? with\n            | none => pure ()\n            | some defData => out := out.push <| toJson defData\n        count := count + 1    \n    return out\n\n\ndef writeBatchDefnsM (start batch : Nat) : MetaM Nat  := do\n    let cs \u2190 constantNameValueTypes \n    let names := cs.map (\u00b7.1)\n    IO.println <| s!\"{start}; {batch} from {cs.size}\"\n    let mut count := 0\n    let defnsFile := System.mkFilePath [\"rawdata\", s!\"defns.jsonl\"]\n    let h \u2190 IO.FS.Handle.mk defnsFile IO.FS.Mode.append Bool.false\n    let idsFile := System.mkFilePath [\"rawdata\", s!\"idents.jsonl\"]\n    let h' \u2190 IO.FS.Handle.mk idsFile IO.FS.Mode.append Bool.false\n    for (name, term, type) in cs do\n        if count >= start && count < start + batch then\n            IO.println <| s!\"{count} {name}\"\n            let defData? \u2190 DefData.getM? name term type\n            match defData? with\n            | none => \n                IO.println <| s!\"{count} {name} omitted\"\n                pure ()\n            | some defData =>\n                IO.println <| s!\"{count} {name} written\"\n                let idData := defData.identData\n                let idData := \n                    idData.map (fun d \u21a6 d.filter \n                        (names.contains \u00b7 ))\n                let l := (toJson defData).pretty 10000000\n                if l.length < 9000000 then\n                    h.putStrLn  l\n                for d in idData do\n                    let l := (toJson d).pretty 10000000\n                    if l.length < 9000000 then\n                    h'.putStrLn l\n        count := count + 1    \n    return start + batch\n\ndef writePremisesM  : MetaM Nat  := do\n    let cs \u2190 constantNameValueTypes \n    let names := cs.map (\u00b7.1)\n    let namesFile := System.mkFilePath [\"rawdata\", s!\"names.txt\"]\n    IO.FS.writeFile namesFile <| \n        names.map toString |>.foldl (fun a b \u21a6 a  ++ b ++ \"\\n\") \"\"\n    let defIdsFile := System.mkFilePath [\"rawdata\", s!\"def_ids.jsonl\"]\n    IO.FS.writeFile defIdsFile \"\"\n    let hId \u2190 IO.FS.Handle.mk defIdsFile IO.FS.Mode.append Bool.false\n    IO.println <| s!\"Processing {cs.size} definitions\"\n    let mut count := 0\n    let mut premisesDone : Array <| (Array Syntax) \u00d7 Syntax := #[]\n    let premisesFile := System.mkFilePath [\"rawdata\", s!\"premises.jsonl\"]\n    IO.FS.writeFile premisesFile \"\"\n    let h \u2190 IO.FS.Handle.mk premisesFile IO.FS.Mode.append Bool.false\n    let trainPremisesFile := System.mkFilePath [\"rawdata\", s!\"train_premises.jsonl\"]\n    IO.FS.writeFile trainPremisesFile \"\"\n    let hTrain \u2190 IO.FS.Handle.mk trainPremisesFile IO.FS.Mode.append Bool.false\n    let testPremisesFile := System.mkFilePath [\"rawdata\", s!\"test_premises.jsonl\"]\n    let hTest \u2190 IO.FS.Handle.mk testPremisesFile IO.FS.Mode.append Bool.false\n    let validPremisesFile := System.mkFilePath [\"rawdata\", s!\"valid_premises.jsonl\"]\n    IO.FS.writeFile validPremisesFile \"\"\n    let hValid \u2190 IO.FS.Handle.mk validPremisesFile IO.FS.Mode.append Bool.false\n    let mut testNum := 0\n    let mut validNum := 0\n    let mut trainNum := 0\n    for (name, term, type) in cs do\n        IO.println <| s!\"{count} {name} (of {cs.size})\"\n        let defData? \u2190 DefData.getM? name term type\n        match defData? with\n        | none => \n            IO.println <| s!\"{count} {name} omitted\"\n            pure ()\n        | some defData =>\n            IO.println <| s!\"{count} {name} written\"\n            let gh \u2190  match \u2190 IO.rand 0 9 with\n                | 0 => do\n                    testNum := testNum + 1\n                    IO.println s!\"writing to test; now :{testNum}\" \n                    pure hTest\n                | 1 => \n                    validNum := validNum + 1\n                    IO.println s!\"writing to valid; now :{validNum}\"\n                    pure hValid\n                | _ => \n                    trainNum := trainNum + 1\n                    IO.println s!\"writing to train; now :{trainNum}\"\n                    pure hTrain\n            let premises := defData.premises\n            for premise in premises do\n                let premiseHead := (premise.context, premise.type)\n                if premisesDone.contains premiseHead then\n                    IO.print \"premise seen previously; \"\n                    pure ()\n                else\n                    premisesDone := premisesDone.push premiseHead\n                    IO.print \"premise new; \"\n                    let premise := premise.filterIds (names.contains \u00b7 )\n                    let l := (toJson premise).pretty 10000000\n                    if l.length < 9000000 then\n                        h.putStrLn  l\n                        gh.putStrLn l\n            IO.println \"\"\n            let idData := defData.identData.bind (fun d \u21a6 d.ids)\n            let idData := idData.filter (names.contains \u00b7 ) |>.eraseDups\n            let idData := Json.mkObj [\n                (\"name\", toJson defData.name),\n                (\"ids\", toJson idData),\n                (\"is_prop\", toJson defData.isProp),\n                (\"type\", toJson defData.type.purge)\n            ]\n            let l := idData.pretty 10000000\n            if l.length < 9000000 then\n                hId.putStrLn l\n        count := count + 1    \n    return count\n\ndef writeBatchDefnsCore (start batch : Nat) : CoreM Nat := \n    (writeBatchDefnsM start batch).run' {} \n\ndef writePremisesCore : CoreM Nat :=\n    writePremisesM.run' {}\n\n-- #eval batchDefns 0 5\n", "meta": {"author": "siddhartha-gadgil", "repo": "LeanAide", "sha": "7862af73ee2f0be08b20fd3e4148e20bf4a81054", "save_path": "github-repos/lean/siddhartha-gadgil-LeanAide", "path": "github-repos/lean/siddhartha-gadgil-LeanAide/LeanAide-7862af73ee2f0be08b20fd3e4148e20bf4a81054/LeanCodePrompts/Premises.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37968348202936486}}
{"text": "import category_theory.morphism_property\nimport for_mathlib.category_theory.limits.comm_sq_misc\n\nopen category_theory.limits\n\nuniverses w v u\n\nnamespace category_theory\n\nvariables {C : Type u} [category.{v} C] {D : Type*} [category D]\n\nnamespace morphism_property\n\n/-instance : has_subset (morphism_property C) :=\n\u27e8\u03bb P\u2081 P\u2082, \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y) (hf : P\u2081 f), P\u2082 f\u27e9\ninstance : has_inter (morphism_property C) :=\n\u27e8\u03bb P\u2081 P\u2082 X Y f, P\u2081 f \u2227 P\u2082 f\u27e9\n\n@[simp] def op (P : morphism_property C) : morphism_property C\u1d52\u1d56 := \u03bb X Y f, P f.unop\n@[simp] def unop (P : morphism_property C\u1d52\u1d56) : morphism_property C := \u03bb X Y f, P f.op\n@[simp] lemma unop_op (P : morphism_property C) : P.op.unop = P := rfl\n@[simp] lemma op_unop (P : morphism_property C\u1d52\u1d56) : P.unop.op = P := rfl\n\n/-- A morphism property is `stable_under_cobase_change` if the cobase change of such a morphism\nstill falls in the class. -/\ndef stable_under_cobase_change (P : morphism_property C) : Prop :=\n\u2200 \u2983A A' B B' : C\u2984 \u2983f : A \u27f6 A'\u2984 \u2983g : A \u27f6 B\u2984 \u2983f' : B \u27f6 B'\u2984 \u2983g' : A' \u27f6 B'\u2984\n  (sq : is_pushout g f f' g') (hf : P f), P f'\n\nlemma stable_under_cobase_change.mk {P : morphism_property C} [has_pushouts C]\n  (hP\u2081 : respects_iso P)\n  (hP\u2082 : \u2200 (A B A' : C) (f : A \u27f6 A') (g : A \u27f6 B) (hf : P f), P (pushout.inr : B \u27f6 pushout f g)) :\n  stable_under_cobase_change P := \u03bb A A' B B' f g f' g' sq hf,\nbegin\n  let e := sq.flip.iso_pushout,\n  rw [\u2190 hP\u2081.cancel_right_is_iso _ e.hom, sq.flip.inr_iso_pushout_hom],\n  exact hP\u2082 _ _ _ f g hf,\nend\n\nlemma stable_under_cobase_change.respects_iso {P : morphism_property C}\n  (hP : stable_under_cobase_change P) : respects_iso P :=\nbegin\n  apply respects_iso.of_respects_arrow_iso,\n  intros f g e,\n  exact hP (is_pushout.of_horiz_is_iso (comm_sq.mk e.hom.w)),\nend\n\nlemma stable_under_cobase_change.inl {P : morphism_property C}\n  (hP : stable_under_cobase_change P) {A B A' : C} (f : A \u27f6 A') (g : A \u27f6 B) [has_pushout f g]\n  (H : P g) : P (pushout.inl : A' \u27f6 pushout f g) :=\nhP (is_pushout.of_has_pushout f g) H\n\nlemma stable_under_cobase_change.inr {P : morphism_property C}\n  (hP : stable_under_cobase_change P) {A B A' : C} (f : A \u27f6 A') (g : A \u27f6 B) [has_pushout f g]\n  (H : P f) : P (pushout.inr : B \u27f6 pushout f g) :=\nhP (is_pushout.of_has_pushout f g).flip H\n\nlemma stable_under_cobase_change.op {P : morphism_property C}\n  (hP : stable_under_cobase_change P) : stable_under_base_change P.op :=\n\u03bb X Y Y' S f g f' g' sq hg, hP sq.unop hg\n\nlemma stable_under_cobase_change.unop {P : morphism_property C\u1d52\u1d56}\n  (hP : stable_under_cobase_change P) : stable_under_base_change P.unop :=\n\u03bb X Y Y' S f g f' g' sq hg, hP sq.op hg\n\nlemma stable_under_base_change.op {P : morphism_property C}\n  (hP : stable_under_base_change P) : stable_under_cobase_change P.op :=\n\u03bb A A' B B' f g f' g' sq hf, hP sq.unop hf\n\nlemma stable_under_base_change.unop {P : morphism_property C\u1d52\u1d56}\n  (hP : stable_under_base_change P) : stable_under_cobase_change P.unop :=\n\u03bb A A' B B' f g f' g' sq hf, hP sq.op hf\n\ndef inverse_image (P : morphism_property D) (F : C \u2964 D) : morphism_property C :=\n\u03bb X Y f, P (F.map f)\n\nvariable (C)\ndef isomorphisms : morphism_property C := \u03bb X Y f, is_iso f\ndef monomorphisms : morphism_property C := \u03bb X Y f, mono f\ndef epimorphisms : morphism_property C := \u03bb X Y f, epi f\n\nvariable {C}\n\nsection\nvariables {X Y : C} (f : X \u27f6 Y)\n@[simp] lemma isomorphisms.iff : (isomorphisms C) f \u2194 is_iso f := by refl\n@[simp] lemma monomorphisms.iff : (monomorphisms C) f \u2194 mono f := by refl\n@[simp] lemma epimorphisms.iff : (epimorphisms C) f \u2194 epi f := by refl\n\nlemma isomorphisms.infer_property [hf : is_iso f] : (isomorphisms C) f := hf\nlemma monomorphisms.infer_property [hf : mono f] : (monomorphisms C) f := hf\nlemma epimorphisms.infer_property [hf : epi f] : (epimorphisms C) f := hf\n\nend-/\n\nvariable (C)\n@[simp]\nlemma op_epimorphisms : (epimorphisms C).op = monomorphisms C\u1d52\u1d56 :=\nbegin\n  ext X Y f,\n  simp only [morphism_property.op, epimorphisms.iff, monomorphisms.iff],\n  split,\n  { introI,\n    exact category_theory.op_mono_of_epi f.unop, },\n  { introI,\n    exact category_theory.unop_epi_of_mono f, },\nend\n\n@[simp]\nlemma op_monomorphisms : (monomorphisms C).op = epimorphisms C\u1d52\u1d56 :=\nbegin\n  ext X Y f,\n  simp only [morphism_property.op, epimorphisms.iff, monomorphisms.iff],\n  split,\n  { introI,\n    exact category_theory.op_epi_of_mono f.unop, },\n  { introI,\n    exact category_theory.unop_mono_of_epi f, },\nend\n\n@[simp]\nlemma unop_epimorphisms : (epimorphisms C\u1d52\u1d56).unop = monomorphisms C :=\nby rw [\u2190 (monomorphisms C).unop_op, op_monomorphisms]\n\n@[simp]\nlemma unop_monomorphisms : (monomorphisms C\u1d52\u1d56).unop = epimorphisms C :=\nby rw [\u2190 (epimorphisms C).unop_op, op_epimorphisms]\n\nnamespace stable_under_composition\n\nvariable {C}\n/-\nlemma op {P : morphism_property C} (h : P.stable_under_composition) :\n  P.op.stable_under_composition :=\n\u03bb X Y Z f g hf hg, h g.unop f.unop hg hf\n\nlemma unop {P : morphism_property C\u1d52\u1d56} (h : P.stable_under_composition) :\n  P.unop.stable_under_composition :=\n\u03bb X Y Z f g hf hg, h g.op f.op hg hf\n\nlemma inverse_image {P : morphism_property D} (h : P.stable_under_composition)\n  (F : C \u2964 D) : (P.inverse_image F).stable_under_composition :=\n\u03bb X Y Z f g hf hg, by simpa only [\u2190 F.map_comp] using h (F.map f) (F.map g) hf hg-/\n\nvariable (C)\n\n/-lemma for_isomorphisms : (isomorphisms C).stable_under_composition :=\n\u03bb X Y Z f g hf hg, begin\n  dsimp [isomorphisms] at hf hg \u22a2,\n  haveI := hf,\n  haveI := hg,\n  apply_instance,\nend\n\nlemma for_monomorphisms : (monomorphisms C).stable_under_composition :=\n\u03bb X Y Z f g hf hg, begin\n  dsimp [monomorphisms] at hf hg \u22a2,\n  haveI := hf,\n  haveI := hg,\n  apply mono_comp,\nend\n\nlemma for_epimorphisms : (epimorphisms C).stable_under_composition :=\n\u03bb X Y Z f g hf hg, begin\n  dsimp [epimorphisms] at hf hg \u22a2,\n  haveI := hf,\n  haveI := hg,\n  apply epi_comp,\nend-/\n\nend stable_under_composition\n\nnamespace respects_iso\n\n/-lemma op {P : morphism_property C} (h : P.respects_iso) :\n  P.op.respects_iso :=\n\u27e8\u03bb X Y Z e f, h.2 e.unop f.unop, \u03bb X Y Z e f, h.1 e.unop f.unop\u27e9\n\nlemma unop {P : morphism_property C\u1d52\u1d56} (h : P.respects_iso) :\n  P.unop.respects_iso :=\n\u27e8\u03bb X Y Z e f, h.2 e.op f.op, \u03bb X Y Z e f, h.1 e.op f.op\u27e9\n\nlemma for_monomorphisms : (monomorphisms C).respects_iso :=\nby { split; { intros X Y Z e f, simp only [monomorphisms.iff], introI, apply mono_comp, }, }\n\nlemma for_epimorphisms : (epimorphisms C).respects_iso :=\nby { split; { intros X Y Z e f, simp only [epimorphisms.iff], introI, apply epi_comp, }, }\n\nlemma for_isomorphisms : (isomorphisms C).respects_iso :=\nby { split; { intros X Y Z e f, simp only [isomorphisms.iff], introI, apply_instance, }, }\n-/\nend respects_iso\n\nvariable {C}\n\nlemma respects_iso.of_stable_under_composition_and_contains_iso\n  {P : morphism_property C} (h\u2081 : P.stable_under_composition) (h\u2082 : isomorphisms C \u2286 P) :\n  respects_iso P :=\nbegin\n  split,\n  { intros X Y Z e f hf,\n    exact h\u2081 e.hom f (h\u2082 e.hom (isomorphisms.infer_property _)) hf, },\n  { intros X Y Z e f hf,\n    exact h\u2081 f e.hom hf (h\u2082 e.hom (isomorphisms.infer_property _)), },\nend\n\ndef stable_under_products (P : morphism_property C) : Prop :=\nP.respects_iso \u2227\n\u2200 (I : Type w) (X : I \u2192 C) (Y : I \u2192 C) [hX : has_product X] [hY : has_product Y]\n(f : \u03a0 (i : I), X i \u27f6 Y i) (hf : \u2200 (i : I), P (f i)), P (@limits.pi.map _ _ _ X Y hX hY f)\n\nnamespace stable_under_products\n\nlemma property {P : morphism_property C} (h : morphism_property.stable_under_products.{w} P)\n  {I : Type w} (X : I \u2192 C) (Y : I \u2192 C) [hX : has_product X] [hY : has_product Y]\n(f : \u03a0 (i : I), X i \u27f6 Y i) (hf : \u2200 (i : I), P (f i)) :\n  P (@limits.pi.map _ _ _ X Y hX hY f) :=\nh.2 I X Y f hf\n\nlemma binary {P : morphism_property C} (h : morphism_property.stable_under_products.{0} P)\n  {X\u2081 X\u2082 Y\u2081 Y\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (h\u2081 : P f\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (h\u2082 : P f\u2082)\n  [hX : has_binary_product X\u2081 X\u2082] [hY : has_binary_product Y\u2081 Y\u2082] :\n  P (limits.prod.map f\u2081 f\u2082) :=\nbegin\n  haveI : has_product (pair_function X\u2081 X\u2082) := hX,\n  haveI : has_product (pair_function Y\u2081 Y\u2082) := hY,\n  convert h.property (pair_function X\u2081 X\u2082) (pair_function Y\u2081 Y\u2082)\n    (\u03bb i, by { cases i, exacts [f\u2081, f\u2082], })\n    (\u03bb i, by { cases i, exacts [h\u2081, h\u2082], }),\n  ext,\n  { erw [limits.prod.map_fst, lim_map_\u03c0], refl, },\n  { erw [limits.prod.map_snd, lim_map_\u03c0], refl, },\nend\n\nend stable_under_products\n\ndef stable_under_coproducts (P : morphism_property C) : Prop :=\nP.respects_iso \u2227\n\u2200 (I : Type w) (X : I \u2192 C) (Y : I \u2192 C) [hX : has_coproduct X] [hY : has_coproduct Y]\n(f : \u03a0 (i : I), X i \u27f6 Y i) (hf : \u2200 (i : I), P (f i)), P (@limits.sigma.map _ _ _ X Y hX hY f)\n\nnamespace stable_under_coproducts\n\nlemma property {P : morphism_property C} (h : morphism_property.stable_under_coproducts.{w} P)\n  (I : Type w) (X : I \u2192 C) (Y : I \u2192 C) [hX : has_coproduct X] [hY : has_coproduct Y]\n  (f : \u03a0 (i : I), X i \u27f6 Y i) (hf : \u2200 (i : I), P (f i)) :\n  P (@limits.sigma.map _ _ _ X Y hX hY f) :=\nh.2 I X Y f hf\n\nlemma binary {P : morphism_property C} (h : morphism_property.stable_under_coproducts.{0} P)\n  {X\u2081 X\u2082 Y\u2081 Y\u2082 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (h\u2081 : P f\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (h\u2082 : P f\u2082)\n  [hX : has_binary_coproduct X\u2081 X\u2082] [hY : has_binary_coproduct Y\u2081 Y\u2082] :\n  P (coprod.map f\u2081 f\u2082) :=\nbegin\n  haveI : has_coproduct (pair_function X\u2081 X\u2082) := hX,\n  haveI : has_coproduct (pair_function Y\u2081 Y\u2082) := hY,\n  convert h.property _ (pair_function X\u2081 X\u2082) (pair_function Y\u2081 Y\u2082)\n    (\u03bb i, by { cases i, exacts [f\u2081, f\u2082], })\n    (\u03bb i, by { cases i, exacts [h\u2081, h\u2082], }),\n  tidy,\nend\n\nend stable_under_coproducts\n/-\nlemma is_inverted_by.of_comp {C\u2081 C\u2082 C\u2083 : Type*} [category C\u2081] [category C\u2082] [category C\u2083]\n  (W : morphism_property C\u2081) (F : C\u2081 \u2964 C\u2082) (hF : W.is_inverted_by F) (G : C\u2082 \u2964 C\u2083) :\n  W.is_inverted_by (F \u22d9 G) :=\n\u03bb X Y f hf, by { haveI := hF f hf, dsimp, apply_instance, }-/\n\n/- better as .of_iso rather that iff_of_iso -/\n\n/-\nlemma is_inverted_by.iff_of_iso (W : morphism_property C) {F\u2081 F\u2082 : C \u2964 D} (e : F\u2081 \u2245 F\u2082) :\n  W.is_inverted_by F\u2081 \u2194 W.is_inverted_by F\u2082 :=\nbegin\n  suffices : \u2200 (X Y : C) (f : X \u27f6 Y), is_iso (F\u2081.map f) \u2194 is_iso (F\u2082.map f),\n  { split,\n    exact \u03bb h X Y f hf, by { rw \u2190 this, exact h f hf, },\n    exact \u03bb h X Y f hf, by { rw this, exact h f hf, }, },\n  intros X Y f,\n  apply (respects_iso.isomorphisms D).arrow_mk_iso_iff,\n  exact arrow.iso_mk (e.app X) (e.app Y) (by simp),\nend-/\n\nend morphism_property\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/morphism_property_misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.3796834779629981}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.opposites\nimport category_theory.hom_functor\n\n/-!\n# The Yoneda embedding\n\nThe Yoneda embedding as a functor `yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) \u2245 (yoneda_evaluation C)`.\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v\u2081 u\u2081 u\u2082 -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C]\ninclude \ud835\udc9e\n\n@[simps] def yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081) :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, unop Y \u27f6 X,\n    map := \u03bb Y Y' f g, f.unop \u226b g,\n    map_comp' := \u03bb _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := \u03bb Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := \u03bb X X' f, { \n    \n  app := \u03bb Y g, g \u226b f } }\n\n\n\n@[simps] def coyoneda : C\u1d52\u1d56 \u2964 (C \u2964 Type v\u2081) :=\n{ obj := \u03bb A : C\u1d52\u1d56,\n  { obj := \u03bb R, unop A \u27f6 R,                 --- Hom(X,Y)\n           map := \u03bb R R': C, \u03bb f : R \u27f6 R',\n              \u03bb \u03b6 : unop A \u27f6 R, \u03b6 \u226b f,        ---   Hom(X,Y) \u2192 Hom(X,Y')                                                                          \n           map_comp' := \u03bb R R' R'' : C,\u03bb f : R \u27f6 R',\u03bb g : R' \u27f6 R'', \n                begin\n                  apply funext, \n                  intro x,\n                  rw \u2190 category.assoc (C) (x) f g,\n                  exact rfl,\n                end,\n    map_id' := \u03bb R : C,\n          begin\n            apply funext,\n            intro x, \n            rw category.comp_id,\n            exact rfl,\n          end \n    },\nmap := \u03bb A B : C\u1d52\u1d56, \u03bb f :  A \u27f6 B, {\n  app := \u03bb R g, f.unop \u226b g }\n} \n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X \u27f6 op Y) :\n  ((@yoneda C _).obj X).map f (\ud835\udfd9 X) = ((@yoneda C _).map f.unop).app (op Y) (\ud835\udfd9 Y) :=\nby obviously\n\n@[simp] lemma naturality {X Y : C} (\u03b1 : yoneda.obj X \u27f6 yoneda.obj Y)\n  {Z Z' : C} (f : Z \u27f6 Z') (h : Z' \u27f6 X) : f \u226b \u03b1.app (op Z') h = \u03b1.app (op Z) (f \u226b h) :=\nbegin erw [functor_to_types.naturality], refl end\n\ninstance yoneda_full : full (@yoneda C _) :=\n{ preimage := \u03bb X Y f, (f.app (op X)) (\ud835\udfd9 X) }\ninstance yoneda_faithful : faithful (@yoneda C _) :=\n{ injectivity' := \u03bb X Y f g p,\n  begin\n    injection p with h,\n    convert (congr_fun (congr_fun h (op X)) (\ud835\udfd9 X)); dsimp; simp,\n  end }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X \u2245 Y`\napply yoneda.ext,\n-- Goals are now functions `(Z \u27f6 X) \u2192 (Z \u27f6 Y)`, `(Z \u27f6 Y) \u2192 (Z \u27f6 X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : \u03a0 {Z : C}, (Z \u27f6 X) \u2192 (Z \u27f6 Y)) (q : \u03a0 {Z : C}, (Z \u27f6 Y) \u2192 (Z \u27f6 X))\n  (h\u2081 : \u03a0 {Z : C} (f : Z \u27f6 X), q (p f) = f) (h\u2082 : \u03a0 {Z : C} (f : Z \u27f6 Y), p (q f) = f)\n  (n : \u03a0 {Z Z' : C} (f : Z' \u27f6 Z) (g : Z \u27f6 X), p (f \u226b g) = f \u226b p g) : X \u2245 Y :=\n@preimage_iso _ _ _ _ yoneda _ _ _ _\n  (nat_iso.of_components (\u03bb Z, { hom := p, inv := q, }) (by tidy))\n\ndef is_iso {X Y : C} (f : X \u27f6 Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : C\u1d52\u1d56} (\u03b1 : coyoneda.obj X \u27f6 coyoneda.obj Y)\n  {Z Z' : C} (f : Z' \u27f6 Z) (h : unop X \u27f6 Z') : (\u03b1.app Z' h) \u226b f = \u03b1.app Z (h \u226b f) :=\nbegin erw [functor_to_types.naturality], refl end\n\ninstance coyoneda_full : full (@coyoneda C _) :=\n{ preimage := \u03bb X Y f, ((f.app (unop X)) (\ud835\udfd9 _)).op }\ninstance coyoneda_faithful : faithful (@coyoneda C _) :=\n{ injectivity' := \u03bb X Y f g p,\n  begin\n    injection p with h,\n    have t := (congr_fun (congr_fun h (unop X)) (\ud835\udfd9 _)),\n    simpa using congr_arg has_hom.hom.op t,\n  end }\n\ndef is_iso {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\nend coyoneda\n\nclass representable (F : C\u1d52\u1d56 \u2964 Type v\u2081) :=\n(X : C)\n(w : yoneda.obj X \u2245 F)\n\nend category_theory\n\nnamespace category_theory\n-- For the rest of the file, we are using product categories,\n-- so need to restrict to the case morphisms are in 'Type', not 'Sort'.\n\nuniverses v\u2081 u\u2081 u\u2082 -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nopen opposite\n\nvariables (C : Type u\u2081) [\ud835\udc9e : category.{v\u2081} C]\ninclude \ud835\udc9e\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((C\u1d52\u1d56 \u2964 Type v\u2081) \u00d7 C\u1d52\u1d56) :=\ncategory_theory.prod.{(max u\u2081 v\u2081) v\u2081} (C\u1d52\u1d56 \u2964 Type v\u2081) C\u1d52\u1d56\n\ninstance prod_category_instance_2 : category (C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) :=\ncategory_theory.prod.{v\u2081 (max u\u2081 v\u2081)} C\u1d52\u1d56 (C\u1d52\u1d56 \u2964 Type v\u2081)\n\nopen yoneda\n\ndef yoneda_evaluation : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nevaluation_uncurried C\u1d52\u1d56 (Type v\u2081) \u22d9 ulift_functor.{u\u2081}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map \u03b1 x).down = \u03b1.2.app Q.1 (P.2.map \u03b1.1 x.down) := rfl\n\ndef yoneda_pairing : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nfunctor.prod yoneda.op (\ud835\udfed (C\u1d52\u1d56 \u2964 Type v\u2081)) \u22d9 functor.hom (C\u1d52\u1d56 \u2964 Type v\u2081)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (\u03b2 : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map \u03b1 \u03b2 = yoneda.map \u03b1.1.unop \u226b \u03b2 \u226b \u03b1.2 := rfl\n\ndef yoneda_lemma : yoneda_pairing C \u2245 yoneda_evaluation C :=\n{ hom :=\n  { app := \u03bb F x, ulift.up ((x.app F.1) (\ud835\udfd9 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp,\n           \u2190functor_to_types.naturality,\n           obj_map_id,\n           functor_to_types.naturality,\n           functor_to_types.map_id]\n    end },\n  inv :=\n  { app := \u03bb F x,\n    { app := \u03bb X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [\u2190functor_to_types.naturality, functor_to_types.map_comp]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [\u2190functor_to_types.naturality,\n         obj_map_id,\n         functor_to_types.naturality,\n         functor_to_types.map_id],\n    refl,\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id]\n  end }.\n\nvariables {C}\n\n@[simp] def yoneda_sections (X : C) (F : C\u1d52\u1d56 \u2964 Type v\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 ulift.{u\u2081} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\nomit \ud835\udc9e\n@[simp] def yoneda_sections_small {C : Type u\u2081} [small_category C] (X : C) (F : C\u1d52\u1d56 \u2964 Type u\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 F.obj (op X) :=\nyoneda_sections X F \u226a\u226b ulift_trivial _\n\nend category_theory", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/foncteur/Untitled-1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358548398982, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37968347389663143}}
{"text": "import Utils\n\nnamespace Dec3\n\nuniverse u\n\ndef Common {\u03b1 : Type u} (e : \u03b1) (a : List \u03b1) (b : List \u03b1) := e \u2208 a \u2227 e \u2208 b\n\nexample : Common 1 [2, 1] [1, 3] :=\n  by\n    simp [Common]\n    \n-- inductive Unique (\u03b1 : Type u) where\n    \n-- A rucksack has a designated element `e`. To add something to the rucksack\n-- you have to show that this element is not equal to the designated element\ninductive Rucksack {\u03b1 : Type u} : (e : \u03b1) -> Type u where\n| nil : Rucksack e\n| cons : (x : \u03b1 \u00d7 \u03b1) -> (x.fst \u2260 e \u2227 x.snd \u2260 e) -> Rucksack e -> Rucksack e\n\ndef Rucksack.toList {\u03b1 : Type u} {e : \u03b1} (rs : Rucksack e) : List \u03b1 \u00d7 List \u03b1 := \n  match rs with\n  | Rucksack.nil => ([e], [e])\n  | Rucksack.cons (x, y) _ rs' => let (l, r) := rs'.toList\n                                  (x :: l, y :: r)\n\n/-- if \u03b1 is decidable, then we can automatically syntehsize the proof\nwe need to construct the element of the rucksack -/\ndef Rucksack.consDec [DecidableEq \u03b1] {e : \u03b1}\n    (rs : Rucksack e) (new : \u03b1 \u00d7 \u03b1) : Option (Rucksack e)\n  :=\n    match (decEq new.fst e), (decEq new.snd e) with\n    | isFalse hl, isFalse hr => Rucksack.cons new (And.intro hl hr) rs\n    | _, _ => none\n\ninductive EvenList (\u03b1 : Type u) where\n| even_nil : EvenList \u03b1\n| even_cons (x y : \u03b1) (tl : EvenList \u03b1) : EvenList \u03b1\nderiving Repr\n\ndef EvenList.length {\u03b1 : Type u} (l : EvenList \u03b1) : Nat :=\n  match l with\n  | EvenList.even_nil => 0\n  | EvenList.even_cons _ _ tl => 2 + tl.length\n  \ndef EvenList.toList {\u03b1 : Type u} (l : EvenList \u03b1) : List \u03b1 :=\n  match l with\n  | EvenList.even_nil => []\n  | EvenList.even_cons x y tl => x :: y :: tl.toList\n    \ndef EvenList.split {\u03b1 : Type u} (l : EvenList \u03b1) : List \u03b1 \u00d7 List \u03b1 :=\n  h (l.length / 2) l\nwhere h len l :=\n  match l with\n  | EvenList.even_nil => ([], [])\n  | EvenList.even_cons x y tl =>\n    let (ll, lr) := h len tl\n    if lr.length >= len then\n      (x :: y :: ll, lr)\n    else if lr.length + 1 == len then\n      (x :: ll, y :: lr)\n    else\n      (ll, x :: y :: lr)\n  \ndef EvenList.mk {\u03b1 : Type u} (l : List \u03b1) : Option (EvenList \u03b1) :=\n  match l with\n  | x :: y :: tl => EvenList.even_cons x y <$> (EvenList.mk tl)\n  | [] => some EvenList.even_nil \n  | _ => none\n\ndef findCommon [BEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : Option \u03b1 :=\n  match l\u2081 with\n  | [] => none\n  | x :: tl => match l\u2082.find? (\u00b7 == x) with\n               | some e => e\n               | none => findCommon tl l\u2082\n\ndef test : List String := [\n    \"vJrwpWtwJgWrhcsFMMfFFhFp\",\n    \"jqHRNqRjqzjGDLGLrsFMfFZSrLrFZsSL\",\n    \"PmmdzqPrVvPwwTWBwg\",\n    \"wMqvLMZHhHMvwLHjbvcjnnSBnvTQFn\",\n    \"ttgJtRGJQctTZtZT\",\n    \"CrZsJsPPZsGzwwsLwLmpwMDw\"\n  ]\n\ndef Rucksack.mk [BEq \u03b1] [DecidableEq \u03b1] (l\u2081 l\u2082 : List \u03b1) (e : \u03b1) : Option (Rucksack e) := do\n  match l\u2081, l\u2082 with\n  | [], [] => some Rucksack.nil\n  | x :: l\u2081' , y :: l\u2082' =>\n    let rs <- Rucksack.mk l\u2081' l\u2082' e\n    match rs.consDec (x, y) with\n    | some x => x\n    | none => rs\n  | _, _ => none\n  \ndef Rucksack.fromStr (input : String) : Option ((e : Char) \u00d7 Rucksack e) := do\n  let el <- EvenList.mk input.data\n  let (l\u2081, l\u2082) := el.split\n  let dist <- findCommon l\u2081 l\u2082\n  let rs <- Rucksack.mk l\u2081 l\u2082 dist\n  pure \u27e8dist, rs\u27e9\n  \ndef Char.priority (e : Char) : Nat :=\n  if e.isLower then\n    e.toNat - 96\n  else\n    e.toNat - 38\n  \ndef Rucksack.priority {e : Char} (_ : Rucksack e) : Nat := Char.priority e\n\ndef findBadgeHelp [BEq \u03b1] (l\u2081 l\u2082 l\u2083 : List \u03b1) : Option \u03b1 :=\n  match l\u2081 with\n  | [] => none\n  | x :: tl => match l\u2082.find? (\u00b7 == x), l\u2083.find? (\u00b7 == x) with\n               | some _, some _ => some x\n               | _, _ => findBadgeHelp tl l\u2082 l\u2083\n    \ndef Rucksack.findBadge [BEq \u03b1] {e\u2081 e\u2082 e\u2083 : \u03b1}\n    (rs\u2081 : Rucksack e\u2081)\n    (rs\u2082 : Rucksack e\u2082)\n    (rs\u2083 : Rucksack e\u2083)\n    : Option \u03b1 :=\n  let (ll\u2081, lr\u2081) := rs\u2081.toList\n  let (ll\u2082, lr\u2082) := rs\u2082.toList\n  let (ll\u2083, lr\u2083) := rs\u2083.toList\n  findBadgeHelp (ll\u2081 ++ lr\u2081) (ll\u2082 ++ lr\u2082) (ll\u2083 ++ lr\u2083)\n               \ndef Rucksack.findBadgeList [BEq \u03b1] : List ((e : \u03b1) \u00d7 Rucksack e) -> Option \u03b1\n  | [rs\u2081, rs\u2082, rs\u2083] => Rucksack.findBadge rs\u2081.snd rs\u2082.snd rs\u2083.snd\n  | _ => none\n\ndef parse : IO (List String) := do\n  let text <- IO.FS.readFile \"data/dec3.txt\"\n  let lines := String.split text (\u00b7 = '\\n')\n  let lines' := List.filter (\u00b7 \u2260 \"\") lines\n  pure lines'\n  \ndef List.windows (n : Nat) (l : List \u03b1) : List (List \u03b1) :=\n  l |> List.enum\n    |> List.map (\u03bb (i, x) => (i / n, x)) \n    |> List.groupBy (\u03bb (i, _) (j, _) => i = j)\n    |> List.map (List.map Prod.snd)\n\ndef run : IO Unit := do\n\n  let lines <- parse\n  let sacks := lines.filterMap Rucksack.fromStr\n\n  let part1 := sacks |> List.map (\u03bb x => Rucksack.priority (Sigma.snd x))\n                     |> List.foldl (\u00b7 + \u00b7) 0\n\n  let part2 := lines |> List.map String.data\n               |> List.windows 3\n               |> List.filterMap (\u03bb x => match x with\n                                         | [l1, l2, l3] => findBadgeHelp l1 l2 l3\n                                         | _ => none)\n               |> List.map Char.priority\n               |> List.foldl (\u00b7 + \u00b7) 0\n  \n  let stdout <- IO.getStdout\n  stdout.putStrLn s!\"Part 1: {part1}\"\n  stdout.putStrLn s!\"Part 2: {part2}\"\n  \n#eval run\n\nend Dec3\n", "meta": {"author": "sgpthomas", "repo": "advent2022", "sha": "cdfa425a3cb69daa96ae5a829e63aa8d5c062542", "save_path": "github-repos/lean/sgpthomas-advent2022", "path": "github-repos/lean/sgpthomas-advent2022/advent2022-cdfa425a3cb69daa96ae5a829e63aa8d5c062542/Dec3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3796834738966314}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.bitraversable.basic\nimport Mathlib.PostPort\n\nuniverses u l_1 u_1 \n\nnamespace Mathlib\n\n/-!\n# Bitraversable Lemmas\n\n## Main definitions\n  * tfst - traverse on first functor argument\n  * tsnd - traverse on second functor argument\n\n## Lemmas\n\nCombination of\n  * bitraverse\n  * tfst\n  * tsnd\n\nwith the applicatives `id` and `comp`\n\n## References\n\n * Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable functor bifunctor applicative\n\n\n-/\n\nnamespace bitraversable\n\n\n/-- traverse on the first functor argument -/\ndef tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {\u03b2 : Type u} {F : Type u \u2192 Type u} [Applicative F] {\u03b1 : Type u} {\u03b1' : Type u} (f : \u03b1 \u2192 F \u03b1') : t \u03b1 \u03b2 \u2192 F (t \u03b1' \u03b2) :=\n  bitraverse f pure\n\n/-- traverse on the second functor argument -/\ndef tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {\u03b2 : Type u} {F : Type u \u2192 Type u} [Applicative F] {\u03b1 : Type u} {\u03b1' : Type u} (f : \u03b1 \u2192 F \u03b1') : t \u03b2 \u03b1 \u2192 F (t \u03b2 \u03b1') :=\n  bitraverse pure f\n\ntheorem id_tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t] {\u03b1 : Type u} {\u03b2 : Type u} (x : t \u03b1 \u03b2) : tfst id.mk x = id.mk x :=\n  id_bitraverse\n\ntheorem id_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t] {\u03b1 : Type u} {\u03b2 : Type u} (x : t \u03b1 \u03b2) : tsnd id.mk x = id.mk x :=\n  id_bitraverse\n\ntheorem tfst_comp_tfst {t : Type l_1 \u2192 Type l_1 \u2192 Type l_1} [bitraversable t] {F : Type l_1 \u2192 Type l_1} {G : Type l_1 \u2192 Type l_1} [Applicative F] [Applicative G] [is_lawful_bitraversable t] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type l_1} {\u03b1\u2081 : Type l_1} {\u03b1\u2082 : Type l_1} {\u03b2 : Type l_1} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b1\u2081 \u2192 G \u03b1\u2082) : functor.comp.mk \u2218 Functor.map (tfst f') \u2218 tfst f = tfst (functor.comp.mk \u2218 Functor.map f' \u2218 f) :=\n  funext fun (x : t \u03b1\u2080 \u03b2) => comp_tfst f f' x\n\ntheorem tfst_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type u} {\u03b1\u2081 : Type u} {\u03b2\u2080 : Type u} {\u03b2\u2081 : Type u} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) : functor.comp.mk (tfst f <$> tsnd f' x) =\n  bitraverse (functor.comp.mk \u2218 pure \u2218 f) (functor.comp.mk \u2218 Functor.map pure \u2218 f') x := sorry\n\ntheorem tsnd_tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type u} {\u03b1\u2081 : Type u} {\u03b2\u2080 : Type u} {\u03b2\u2081 : Type u} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) : functor.comp.mk (tsnd f' <$> tfst f x) =\n  bitraverse (functor.comp.mk \u2218 Functor.map pure \u2218 f) (functor.comp.mk \u2218 pure \u2218 f') x := sorry\n\ntheorem comp_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1 : Type u} {\u03b2\u2080 : Type u} {\u03b2\u2081 : Type u} {\u03b2\u2082 : Type u} (g : \u03b2\u2080 \u2192 F \u03b2\u2081) (g' : \u03b2\u2081 \u2192 G \u03b2\u2082) (x : t \u03b1 \u03b2\u2080) : functor.comp.mk (tsnd g' <$> tsnd g x) = tsnd (functor.comp.mk \u2218 Functor.map g' \u2218 g) x := sorry\n\ntheorem tfst_eq_fst_id {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t] {\u03b1 : Type u} {\u03b1' : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b1') (x : t \u03b1 \u03b2) : tfst (id.mk \u2218 f) x = id.mk (bifunctor.fst f x) := sorry\n\ntheorem tsnd_eq_snd_id {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t] {\u03b1 : Type u} {\u03b2 : Type u} {\u03b2' : Type u} (f : \u03b2 \u2192 \u03b2') (x : t \u03b1 \u03b2) : tsnd (id.mk \u2218 f) x = id.mk (bifunctor.snd f x) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/bitraversable/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358548398982, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.3796834738966314}}
{"text": "import Runtime.Network.Graph.Path.Succ\n\nnamespace Network.Graph.Path\n\ndef Child (path : Path graph start) := { child // child \u227b path }\n\nnamespace Child\n\ntheorem parent_eq_nil_class {path : Path graph start} {child : Child path} : (child.val = cons c nil) \u2192 start = path.class := by\n  intro h\n  have h' := child.property\n  simp [Succ, h, prefix?] at h'\n  simp [\u2190h']\n\n-- TODO: This could also be expressed in `Subpaths.lean` using `prefix?`.\ntheorem split_fst_eq_parent (child : Child path) :\n  (child.val.split child.property.isCons).fst = path := by\n  sorry\n\ndef \u00abclass\u00bb (child : Child path) : Class.Child path.class :=\n  have h := by rw [split_fst_eq_parent child]\n  (child.val.split child.property.isCons).snd |> cast h\n\n@[simp]\ntheorem class_eq_class {child : Child path} : child.class.class = child.val.class := by\n  sorry\n\nend Child\nend Network.Graph.Path\n", "meta": {"author": "lf-lang", "repo": "reactor-lean", "sha": "d2eb5458446af838be34ebb6f69549b2f6d9c04d", "save_path": "github-repos/lean/lf-lang-reactor-lean", "path": "github-repos/lean/lf-lang-reactor-lean/reactor-lean-d2eb5458446af838be34ebb6f69549b2f6d9c04d/Runtime/Network/Graph/Path/Child.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.37959955759348524}}
{"text": "import free_pfpng.main\nimport invpoly.functor\nimport condensed.condensify\nimport laurent_measures.thm69\nimport normed_free_pfpng.compare\nimport Lbar.ses\nimport laurent_measures.ses2\n\nuniverse u\n\nnoncomputable theory\n\nopen category_theory\n\nopen_locale nnreal big_operators\n\nnamespace invpoly\nopen ProFiltPseuNormGrpWithTinv\u2081\n\nvariables (p : \u211d\u22650) [fact (0 < p)] [fact (p < 1)]\n\nlocal notation `r` := @r p\n\n/-- The \"evaluate at 2\" function, sending an S-indexed family of elements of `\u2124[T\u207b\u00b9]`\n  to an S-indexed family of integers, via the assignment `T\u207b\u00b9 \u21a6 2`. Note that\n  this is a strict (i.e. norm-nonincreasing) morphism of normed groups,\n  with the norm on a polynomial `\u2211a\u2099T\u207b\u207f` being `\u2211\u2099\u2225a\u2099\u2225\u208ar\u207b\u207f` and the norm\n  on an integer `z` being `\u2225z\u2225\u208a\u1d56`; this works because `r = 2\u207b\u1d56`, the crucial\n  points being that `(\u2211\u2225c\u2099\u2225)\u1d56 \u2264 \u2211(\u2225c\u2099\u2225\u1d56)` as `0 < p \u2264 1`, and `\u2225z\u2225\u1d56 \u2264 \u2225z\u2225` for an integer `z`.   -/\n@[simps] def eval2 (S : Fintype.{u}) :\n  strict_comphaus_filtered_pseudo_normed_group_hom (invpoly r S) (normed_free_pfpng p S) :=\n{ to_fun := \u03bb F s, (F s).eval 2,\n  map_zero' := by { ext, simp only [polynomial.eval_zero, pi.zero_apply], },\n  map_add' := by { intros, ext, simp only [polynomial.eval_add, pi.add_apply], },\n  strict' := \u03bb c F hF, begin\n    refine (finset.sum_le_sum _).trans hF,\n    rintro s -,\n    have h0p : 0 < p := fact.out _,\n    have hp1 : p \u2264 1 := fact.out _,\n    have h0pinv : 0 \u2264 p\u207b\u00b9, { rw \u2190 nnreal.inv_pos at h0p, exact h0p.le },\n    calc \u2225(F s).eval 2\u2225\u208a ^ (p:\u211d)\n        = \u2225\u2211 n in finset.range ((F s).nat_degree + 1), (F s).coeff n * 2 ^ n\u2225\u208a ^ (p:\u211d) : _\n    ... \u2264 (\u2211 n in finset.range ((F s).nat_degree + 1), \u2225(F s).coeff n * 2 ^ n\u2225\u208a) ^ (p:\u211d) : _\n    ... \u2264 \u2211 n in finset.range ((F s).nat_degree + 1), (\u2225(F s).coeff n * 2 ^ n\u2225\u208a ^ (p:\u211d)) : _\n    ... \u2264 \u2211 n in finset.range ((F s).nat_degree + 1), (\u2225(F s).coeff n\u2225\u208a * r ^ (-n:\u2124)) : _\n    ... \u2264 \u2211' n, (\u2225(F s).coeff n\u2225\u208a * r ^ (-n:\u2124)) : _,\n    { rw polynomial.eval_eq_sum_range, },\n    { refine nnreal.rpow_le_rpow (nnnorm_sum_le _ _) h0p.le, },\n    { refine nnreal.rpow_sum_le_sum_rpow _ _ h0p hp1, },\n    { refine finset.sum_le_sum _,\n      intros n hn,\n      rw [int.nnnorm_mul, nnreal.mul_rpow],\n      refine mul_le_mul' _ (le_of_eq _),\n      { exact nnnorm_int_rpow_le p _ },\n      { have h2n : (2 ^ n : \u2124) = (2 ^ n : \u2115), { norm_cast, },\n        simp only [h2n, \u2190 nnreal.coe_nat_abs, int.nat_abs_of_nat],\n        rw [zpow_neg, \u2190 inv_zpow, zpow_coe_nat],\n        calc ((2 ^ n : \u2115) : \u211d\u22650) ^ (p:\u211d)\n            = (2 ^ (n:\u211d) : \u211d\u22650) ^ (p:\u211d) : _\n        ... = (2\u207b\u00b9 ^ (p:\u211d) : \u211d\u22650)\u207b\u00b9 ^ (n:\u211d) : _\n        ... = r\u207b\u00b9 ^ n : _,\n        { norm_cast },\n        { rw [\u2190 nnreal.rpow_mul, mul_comm, nnreal.rpow_mul, \u2190 nnreal.inv_rpow, inv_inv], },\n        { rw [nnreal.rpow_nat_cast], refl, } } },\n    { refine sum_le_tsum _ _ _,\n      { intros, exact zero_le' },\n      { exact F.nnreal_summable s } },\n  end,\n  continuous' := \u03bb c, continuous_of_discrete_topology }\n\n/-- The \"evaluate a polynomial in `T\u207b\u00b9` by mapping `T\u207b\u00b9` to `2`\" functor\n from `\u2124[T\u207b\u00b9]` to `\u2124`. -/\ndef eval2_nat_trans :\n  (Fintype_invpoly.{u} r \u22d9 PFPNGT\u2081_to_CHFPNG\u2081\u2091\u2097.{u} r) \u27f6\n  (normed_free_pfpng_functor.{u} p \u22d9 PFPNG\u2081_to_CHFPNG\u2081\u2091\u2097) :=\n{ app := \u03bb S, eval2 p S,\n  naturality' := \u03bb S T f, begin\n    ext x t,\n    show (map f x t).eval 2 = _,\n    dsimp only [functor.comp_map, normed_free_pfpng_functor_map,\n      CompHausFiltPseuNormGrp\u2081.comp_apply, map],\n    erw [polynomial.eval_finset_sum], refl, recover, exact 0,\n  end }\n.\n\nsection ses\n\nopen CompHausFiltPseuNormGrp\u2081\n\n-- move this\ninstance (M N : Type*)\n  [comphaus_filtered_pseudo_normed_group M] [comphaus_filtered_pseudo_normed_group N] :\n  add_monoid_hom_class (comphaus_filtered_pseudo_normed_group_hom M N) M N :=\n{ coe := \u03bb f, f,\n  coe_injective' := \u03bb f g h, by { ext, dsimp at h, rw h },\n  map_add := \u03bb f, f.map_add,\n  map_zero := \u03bb f, f.map_zero }\n\nlemma Tinv2_injective (S : Fintype) :\n  function.injective ((Tinv2_nat_trans (Fintype_invpoly r)).app S) :=\nbegin\n  rw injective_iff_map_eq_zero,\n  intros f hf,\n  ext s n,\n  apply_fun (\u03bb \u03c6, \u03c6 s) at hf,\n  simp only [Tinv2_nat_trans, Tinv_nat_trans, nat_trans.app_nsmul, nat_trans.id_app, sub_apply,\n    nat_trans.app_sub, comphaus_filtered_pseudo_normed_group_hom.sub_apply,\n    pi.smul_apply, comphaus_filtered_pseudo_normed_group_hom.nsmul_apply,\n    pi.sub_apply, pi.zero_apply] at hf,\n  simp only [pi.zero_apply, polynomial.coeff_zero],\n  induction n with n ih,\n  { apply_fun (\u03bb \u03c6, \u03c6.coeff 0) at hf,\n    simp only [polynomial.coeff_zero, polynomial.coeff_sub, polynomial.coeff_smul] at hf,\n    erw [invpoly.Tinv_zero, zero_sub, neg_eq_zero, two_nsmul_eq_zero \u2124 \u2124] at hf, exact hf, },\n  { apply_fun (\u03bb \u03c6, \u03c6.coeff (n+1)) at hf,\n    simp only [polynomial.coeff_zero, polynomial.coeff_sub, polynomial.coeff_smul] at hf,\n    erw [invpoly.Tinv_succ, ih, zero_sub, neg_eq_zero, two_nsmul_eq_zero \u2124 \u2124] at hf, exact hf }\nend\n\nlemma Tinv2_comp_eval2_eq_zero (S : Fintype) :\n  (Tinv2_nat_trans (Fintype_invpoly r)).app S \u226b\n    (whisker_right (eval2_nat_trans p) CHFPNG\u2081_to_CHFPNG\u2091\u2097).app S = 0 :=\nbegin\n  ext f s,\n  show (polynomial.X * (f s) - _).eval 2 = (0 : \u2124),\n  simp only [nat_trans.app_nsmul, nat_trans.id_app, comphaus_filtered_pseudo_normed_group_hom.nsmul_apply,\n  category_theory.id_apply, pi.smul_apply, nsmul_eq_mul, nat.cast_bit0, nat.cast_one, polynomial.eval_sub,\n  polynomial.eval_mul, polynomial.eval_X, polynomial.eval_bit0, polynomial.eval_one, sub_self],\nend\n.\n\nlemma theta_eval2_aux {S : Fintype} (s : S) (c : \u211d\u22650) {f : invpoly r S} :\n  \u2211' (n : \u2124), (to_laurent_measures_fun r S f s n : \u211d) * ((2 : \u211d) ^ n)\u207b\u00b9 = \u2191(polynomial.eval 2 (f s)) :=\nbegin\n  rw polynomial.eval_eq_sum_range,\n  have this : \u2200 (n : \u2124),\n  n \u2209 finset.Icc (-(f s).nat_degree : \u2124) 0 \u2192\n  (\u03bb (b : \u2124), \u2191(to_laurent_measures_fun \u00abr\u00bb S f s b) * ((2 : \u211d) ^ b)\u207b\u00b9) n = 0,\n  { rintros n hn,\n    rw [finset.mem_Icc, not_and_distrib] at hn,\n    push_neg at hn,\n    dsimp only,\n    convert zero_mul _,\n    norm_cast,\n    cases hn with hn hn,\n    { have : \u2203 m : \u2115, n = - m,\n      { use n.nat_abs,\n        rw int.nat_abs_of_nonpos,\n        { simp },\n        exact le_trans hn.le (by simp), },\n      rcases this with \u27e8m, rfl\u27e9,\n      rw to_laurent_measures_fun_nonpos,\n      rw polynomial.coeff_eq_zero_of_nat_degree_lt,\n      simpa using hn },\n    { have : \u2203 m : \u2115, n = m + 1,\n      { use (n - 1).nat_abs,\n        rw int.nat_abs_of_nonneg,\n        ring, linarith, },\n      rcases this with \u27e8m, rfl\u27e9,\n      apply to_laurent_measures_fun_pos' } },\n  rw tsum_eq_sum this, clear this,\n  push_cast,\n  apply finset.sum_bij (\u03bb (n : \u2124) hn, n.nat_abs),\n  -- this is supposed to be the easy one! all that remains\n  -- is four boring pieces of easy nonsense e.g.\n  --2^{-n}=2^{n.nat_abs} if n < 0 is probably the hardest one\n  { intros a ha,\n    suffices : a.nat_abs < (f s).nat_degree + 1,\n    { simpa },\n    rw finset.mem_Icc at ha,\n    rw [nat.lt_succ_iff, \u2190 int.coe_nat_le, int.nat_abs_of_nonpos ha.2],\n    linarith [ha.1] },\n  { intros a ha,\n    rw finset.mem_Icc at ha,\n    simp only,\n    congr,\n    { have : \u2203 m : \u2115, a = - m,\n      { use a.nat_abs,\n        rw int.nat_abs_of_nonpos ha.2,\n        simp, },\n      rcases this with \u27e8m, rfl\u27e9,\n      rw to_laurent_measures_fun_nonpos,\n      congr,\n      rw [\u2190 int.coe_nat_eq_coe_nat_iff, int.nat_abs_of_nonpos ha.2, neg_neg], },\n    { rw [\u2190 zpow_neg, \u2190 int.nat_abs_of_nonpos ha.2, zpow_coe_nat], } },\n  { rintros a b ha hb (habs : a.nat_abs = b.nat_abs),\n    rw finset.mem_Icc at ha hb,\n    rw [\u2190 int.coe_nat_eq_coe_nat_iff, int.nat_abs_of_nonpos ha.2,\n      int.nat_abs_of_nonpos hb.2] at habs,\n    exact neg_inj.mp habs },\n  { intros a ha,\n    rw [finset.mem_range_succ_iff] at ha,\n    refine \u27e8-(a : \u2124), _, _\u27e9,\n    { rw [finset.mem_Icc, neg_le_neg_iff, neg_le, neg_zero],\n      simp [ha], },\n    { simp } },\nend\n\nlemma theta_zero_of_eval2_zero {S : Fintype} (c : \u211d\u22650) {f : invpoly r S} (hf1 : eval2 p S f = 0) :\n(laurent_measures.\u0398 p S) ((to_laurent_measures_addhom r S) f) = 0 :=\nbegin\n  ext s,\n  replace hf1 := congr_fun hf1 s,\n  suffices : \u2211' (n : \u2124), \u2191(to_laurent_measures_fun \u00abr\u00bb S f s n) * ((2 : \u211d) ^ n)\u207b\u00b9 = 0,\n  { simpa [laurent_measures.\u0398, laurent_measures_ses.\u03b8_to_add, \u03b8, theta.\u03d1,\n    to_laurent_measures_addhom], },\n  simp only [theta_eval2_aux p s c, int.cast_eq_zero],\n  exact hf1,\nend\n.\n\nlemma _root_.int.eq_neg_nat_or_succ_nat (n : \u2124) : \u2203 m : \u2115, n = -m \u2228 n = m + 1 :=\nbegin\n  cases le_or_lt n 0,\n  { use n.nat_abs,\n    left,\n    rw [int.nat_abs_of_nonpos h, neg_neg] },\n  { use (n - 1).nat_abs,\n    right,\n    rw [int.nat_abs_of_nonneg],\n    ring, linarith },\nend\n\nlemma phi_eq_mul_Tinv_sub_two {S : Fintype} (g : invpoly r S) :\n  \u03d5 ((to_laurent_measures_addhom r S) (\u03bb (s : \u21a5S), g s)) =\n    (to_laurent_measures_addhom r S) (\u03bb (s : \u21a5S), (polynomial.X - 2) * g s) :=\nbegin\n  ext s n,\n  delta \u03d5,\n  simp only [laurent_measures.sub_apply, laurent_measures.shift_to_fun_to_fun,\n    laurent_measures.nsmul_apply, nsmul_eq_mul, int.coe_nat_zero, zero_add],\n  obtain \u27e8m, (rfl | rfl)\u27e9 := n.eq_neg_nat_or_succ_nat,\n  { simp only [to_laurent_measures_addhom, to_laurent_measures_fun_nonpos, add_monoid_hom.mk'_apply,\n      to_laurent_measures_to_fun],\n    cases m,\n    { simp only [nat.nat_zero_eq_zero, int.coe_nat_zero, neg_zero', zero_add,\n        polynomial.mul_coeff_zero, polynomial.coeff_sub,\n        polynomial.coeff_X_zero, zero_sub, neg_mul],\n      rw [to_laurent_measures_fun_pos'' r S g s (show (0 : \u2124) < 1, by norm_num)],\n      simp only [zero_sub, neg_inj, mul_eq_mul_right_iff,\n        show (2 : polynomial \u2124) = polynomial.C 2, by simp, polynomial.coeff_C_zero],\n      left,\n      norm_num, },\n    { simp only [to_laurent_measures_fun_nonpos, sub_mul, int.coe_nat_succ, neg_add_rev,\n        neg_add_cancel_comm, int.coe_nat_bit0, int.coe_nat_zero, zero_add, polynomial.coeff_sub,\n        polynomial.coeff_X_mul, sub_right_inj, show (2 : polynomial \u2124) = polynomial.C 2, by simp,\n        polynomial.coeff_C_mul],\n    } },\n  { norm_cast }, -- didn't expect that to close it!\nend\n\nlemma psi_eq_div_X_sub_two {S : Fintype} (c : \u211d\u22650) {f : invpoly r S}\n  (hf1 : ((eval2_nat_trans p).app S) f = 0) :\n(to_laurent_measures_addhom r S) (\u03bb (s : \u21a5S), f s /\u2098 (polynomial.X - 2)) =\n  \u03c8 ((to_laurent_measures_addhom \u00abr\u00bb S) f) (theta_zero_of_eval2_zero p c hf1) :=\nbegin\n  apply injective_\u03d5',\n  rw \u03b8_\u03d5_split_exact,\n  have hf3 : \u2200 s : S, (polynomial.X - 2) * ((f s) /\u2098 (polynomial.X - 2)) = f s,\n  { intro s,\n    replace hf1 := congr_fun hf1 s,\n    rw (show (2 : polynomial \u2124) = polynomial.C (2 : \u2124), by simp),\n    rwa polynomial.mul_div_by_monic_eq_iff_is_root, },\n  suffices : (to_laurent_measures_addhom r S)\n    (\u03bb s, (polynomial.X - 2) * ((f s) /\u2098 (polynomial.X - 2))) =\n    (to_laurent_measures_addhom r S) (\u03bb s, f s),\n  { rw \u2190 this,\n    apply phi_eq_mul_Tinv_sub_two, },\n  simp [hf3],\nend\n\ntheorem short_exact (S : Profinite) :\n  short_exact ((condensify_Tinv2 _).app S) ((condensify_map $ eval2_nat_trans p).app S) :=\nbegin\n  let \u03ba : \u211d\u22650 \u2192 \u211d\u22650 := \u03bb c, max c (c ^ (\u2191p\u207b\u00b9 : \u211d)),\n  have h\u03ba : id \u2264 \u03ba := \u03bb c, le_max_left _ _,\n  have h0p : 0 < p := fact.out _,\n  have hp1 : p \u2264 1 := fact.out _,\n  have h0pinv : 0 \u2264 p\u207b\u00b9, { rw \u2190 nnreal.inv_pos at h0p, exact h0p.le },\n  refine condensify_nonstrict_exact _ _ (r\u207b\u00b9 + 2) (Tinv2_bound_by _)\n    -- next line can be simplified because 1/2 < r < 1 so the max is always on the right.\n    (\u03bb c, max c (c * (2 - r\u207b\u00b9)\u207b\u00b9 * (r\u207b\u00b9 + 2))) \u03ba\n    (\u03bb c, le_max_left _ _) h\u03ba\n    (Tinv2_injective p) (Tinv2_comp_eval2_eq_zero p) _ _ _,\n  { rintros S c f \u27e8hf1, hf2\u27e9,\n    have hf3 : \u2200 s : S, (polynomial.X - 2) * ((f s) /\u2098 (polynomial.X - 2)) = f s,\n    { intro s,\n      replace hf1 := congr_fun hf1 s,\n      rw (show (2 : polynomial \u2124) = polynomial.C (2 : \u2124), by simp),\n      rwa polynomial.mul_div_by_monic_eq_iff_is_root, },\n    refine \u27e8\u03bb s, (f s) /\u2098 (polynomial.X - 2), _, _\u27e9,\n    { change \u2225_\u2225\u208a \u2264 c at hf2,\n      change \u2225_\u2225\u208a \u2264 _,\n      have hf1' : ((eval2_nat_trans p).app S) f = 0,\n      { simpa using hf1 },\n      rw \u2190 to_laurent_measures_addhom_isometry at \u22a2 hf2,\n      refine le_trans _ (nnreal.mul_le_mul_right (le_max_right _ _) _),\n      rw mul_inv_cancel_right\u2080 (lt_of_lt_of_le (zero_lt_two : (0 : \u211d\u22650) < 2) le_add_self).ne',\n      convert laurent_measures.psi_bound p S c (theta_zero_of_eval2_zero p c hf1') hf2,\n      exact psi_eq_div_X_sub_two p c hf1,\n    },\n    { ext1 s,\n      convert hf3 s,\n      simp [sub_mul, Tinv2_nat_trans, Tinv_nat_trans],\n      refl } },\n  { rintro S c g (hg : _ \u2264 _),\n    let f : invpoly r S := \u03bb s, polynomial.C (g s),\n    refine \u27e8f, _, _\u27e9,\n    { show _ \u2264 _,\n      replace hg := nnreal.rpow_le_rpow hg h0pinv,\n      refine le_trans _ (le_max_right _ _),\n      refine le_trans _ hg,\n      refine le_trans _ (nnreal.rpow_le_rpow (nnreal.rpow_sum_le_sum_rpow _ _ h0p hp1) h0pinv),\n      rw [\u2190 nnreal.rpow_mul, \u2190 nnreal.coe_mul, mul_inv_cancel h0p.ne', nnreal.coe_one,\n        nnreal.rpow_one, invpoly.nnnorm_def],\n      refine finset.sum_le_sum _,\n      rintro s -,\n      simp only [f, zpow_neg, zpow_coe_nat, id.def, polynomial.coeff_C],\n      rw tsum_eq_single 0,\n      { rw [if_pos rfl, pow_zero, inv_one, mul_one], },\n      { intros n hn, rw [if_neg hn, nnnorm_zero, zero_mul], },\n      { apply_instance }, },\n    { ext s, apply polynomial.eval_C } },\nend\n\nend ses\n\nend invpoly\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/invpoly/ses.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544085240401, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3795995439842086}}
{"text": "import implementation.model.predicate\nimport implementation.model.sys_state\nimport implementation.spec.main\nimport implementation.proof.misc\nimport implementation.proof.proposer\n\n-- This file contains proofs about voters (or what can be associated with\n-- them in our co-located Paxos implementation).\n--\n-- NOTE(gnanabit): comments are omitted for facts that are \"obvious\" or not\n-- particularly revealing about paxos.\n\nvariables {pid_t : Type} [linear_order pid_t] [fintype pid_t] {value_t : Type}\n          {is_quorum : finset pid_t \u2192 Prop} [decidable_pred is_quorum]\n          [quorum_assumption is_quorum] {vals : pid_t \u2192 value_t}\n\n-- A server's ballot is always larger than or equal to the ballot in the\n-- proposal it has stored in `accepted` (if such a proposal exists).\n--\n-- Similarly, the ballot in a p1b is always larger than the proposal stored in\n-- the p1b.\nlemma current_ge_accepted_ballot : predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    (\u2200 (p : pid_t) (prop : proposal pid_t value_t),\n      (s.procs p).accepted = some prop \u2192 prop.bal \u2264 (s.procs p).curr) \u2227\n    (\u2200 (p : pid_t) (e \u2208 s.network p) (b : ballot pid_t) (prop : proposal pid_t value_t),\n      (envelope.msg e) = message.p1b b (option.some prop) \u2192 prop.bal \u2264 b)) :=\nbegin\nsuffices : predicate.inductive_invariant _,\nby { exact predicate.ind_inv_is_inv this },\nsplit,\n{ intros s hs,\n  split;\n  intro p;\n  specialize hs p;\n  unfold protocol.init at hs;\n  injection hs with start_proc start_network,\n  { rw \u2190 start_proc,\n    intro prop,\n    trivial },\n  rw \u2190 start_network,\n  intros e he b prop contradicts_he,\n  cases he,\n  { rw he at contradicts_he, injection contradicts_he },\n  rw set.mem_singleton_iff at he,\n  rw he at contradicts_he, injection contradicts_he },\nintros u v,\nrintros \u27e8servers_satisfy, promises_satisfy\u27e9,\nintro u_pn_v,\nhave to_split := u_pn_v,\nrcases to_split with \u27e8receiver, sender, e, he, deliverable, proc_change, ntwk_change, same\u27e9,\nsplit;\nintro p;\ncases decidable.em (p = receiver),\n{ clear same,\n  intro prop,\n  rw h, clear h p,\n  rw proc_change,\n  have delta := state_change receiver (u.procs receiver) e.msg sender,\n  cases delta,\n  { rw delta, exact servers_satisfy receiver prop },\n  cases e,\n  cases e_msg,\n    case p1a : b {\n      cases delta with new_ballot_larger b_is_new_ballot,\n      rw b_is_new_ballot,\n      intro hyp,\n      apply le_of_lt,\n      calc prop.bal \u2264 (u.procs receiver).curr : servers_satisfy receiver prop hyp\n                ... < b                       : new_ballot_larger\n    },\n    case p1b : b p_or {\n      cases delta,\n      { cases delta with new_ballot_larger b_is_new_ballot,\n        rw b_is_new_ballot,\n        intro hyp,\n        apply le_of_lt,\n        calc prop.bal \u2264 (u.procs receiver).curr : servers_satisfy receiver prop hyp\n                  ... < b : new_ballot_larger },\n      cases delta,\n      { rw delta.right.right.right.right,\n        intro hyp,\n        have key : prop.bal = (u.procs receiver).curr,\n        by {\n          injection hyp with key,\n          rw \u2190 key\n        },\n        rw key },\n      rw delta.right.right.right.right.right,\n      intro hyp,\n      cases proposal.merge_is_one_of (u.procs receiver).accepted p_or,\n      { rw h at hyp,\n        exact servers_satisfy receiver prop hyp },\n      rw h at hyp,\n      rw delta.left,\n      apply promises_satisfy sender {msg := message.p1b b p_or, sent_to := e_sent_to} he b prop,\n      rw \u2190 hyp\n    },\n    case p2a : p {\n      cases delta with new_ballot_larger b_is_new_ballot,\n      rw b_is_new_ballot,\n      intro hyp,\n      have key : p = prop, by { injection hyp },\n      rw key\n    },\n    case p2b : b acc {\n      cases delta with new_ballot_larger b_is_new_ballot,\n      rw b_is_new_ballot,\n      intro hyp,\n      apply le_of_lt,\n      calc prop.bal \u2264 (u.procs receiver).curr : servers_satisfy receiver prop hyp\n                ... < b : new_ballot_larger\n    },\n    case preempt : {\n      cases delta with ballot_not_from_self ballot_is_next,\n      rw ballot_is_next,\n      intro hyp,\n      apply le_of_lt,\n      calc prop.bal \u2264 (u.procs receiver).curr : servers_satisfy receiver prop hyp\n                ... < ballot.next receiver (u.procs receiver).curr : ballot.next_larger receiver (u.procs receiver).curr,\n    }},\n{ rw same.left p h, exact servers_satisfy p },\n{ rw h, clear h p,\n  intros e' he' b prop e'_msg_eq,\n  rw ntwk_change at he',\n  cases he',\n  { exact promises_satisfy receiver e' he' b prop e'_msg_eq },\n  rcases p1b_emitted e'_msg_eq he' with \u27e8b', e_msg_is, accepted_unchanged, b_is_max\u27e9,\n  calc prop.bal \u2264 (u.procs receiver).curr : servers_satisfy receiver prop accepted_unchanged\n            ... \u2264 b                       : by { rw \u2190 b_is_max,\n                                                 exact le_max_left (u.procs receiver).curr b'} },\nrw same.right p h, exact promises_satisfy p\nend\n\n-- Any stored proposal has been proposed in some p2a (a stored proposal is one in either a p1b\n-- or in a server's `accepted` field).\nlemma accepted_means_issued : predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    (\u2200 (p : pid_t) (prop : proposal pid_t value_t),\n      (s.procs p).accepted = some prop \u2192 proposed s prop.bal prop.val) \u2227\n    (\u2200 (p : pid_t) (e \u2208 s.network p) (b : ballot pid_t) (prop : proposal pid_t value_t),\n      (envelope.msg e) = message.p1b b (option.some prop) \u2192 proposed s prop.bal prop.val))\n  :=\nbegin\nsuffices : predicate.inductive_invariant _,\nby { exact predicate.ind_inv_is_inv this },\nsplit,\n{ intros s hs,\n  split;\n  intro p;\n  specialize hs p;\n  unfold protocol.init at hs;\n  injection hs with start_proc start_network,\n  { rw \u2190 start_proc,\n    intro prop,\n    trivial },\n  rw \u2190 start_network,\n  intros e he b prop contradicts_he,\n  cases he,\n  { rw he at contradicts_he, injection contradicts_he },\n  rw set.mem_singleton_iff at he,\n  rw he at contradicts_he, injection contradicts_he },\nintros u v,\nrintros \u27e8accepted_satisfies, promises_satisfy\u27e9,\nintro u_pn_v,\nhave to_split := u_pn_v,\nrcases to_split with \u27e8receiver, sender, e, he, deliverable, proc_change, ntwk_change, same\u27e9,\nsplit;\nintro p;\ncases decidable.em (p = receiver),\n{ rw h,\n  clear h p,\n  rw proc_change,\n  intros prop h_prop,\n  suffices key : proposed u prop.bal prop.val \u2228\n    (\u2203 e' \u2208 (protocol.handler receiver (u.procs receiver) e.msg sender).snd,\n       envelope.msg e' = message.p2a prop),\n  by {\n    cases key,\n    { exact proposed_stable prop.bal prop.val u v key u_pn_v },\n    { rcases key with \u27e8e', he', e'_msg_is\u27e9,\n      use [receiver, e'],\n      split,\n      { rw ntwk_change, right, exact he' },\n      rw e'_msg_is, cases prop, refl },\n  },\n  have acc_unchanged_enough : (protocol.handler receiver (u.procs receiver) e.msg sender).fst.accepted = (u.procs receiver).accepted \u2192 proposed u prop.bal prop.val \u2228\n    (\u2203 e' \u2208 (protocol.handler receiver (u.procs receiver) e.msg sender).snd,\n       envelope.msg e' = message.p2a prop), by {\n    intro hyp, left, rw hyp at h_prop, exact accepted_satisfies receiver prop h_prop\n  },\n  have key := state_change receiver (u.procs receiver) e.msg sender,\n  cases key,\n  { left, rw key at h_prop, exact accepted_satisfies receiver prop h_prop  },\n  cases e,\n  cases e_msg,\n    case p1a : b { apply acc_unchanged_enough, rw key.right },\n    case p1b : b p_or {\n      cases key,\n      { apply acc_unchanged_enough, rw key.right },\n      cases key,\n      { right,\n        clear acc_unchanged_enough,\n        rcases key with \u27e8same_bal, bal_from_self, u_no_qrm, v_has_qrm, state_change\u27e9,\n        have prop_eq: { proposal . bal := (u.procs receiver).curr,\n                            val := proposal.value_or_default\n                                     (proposal.merge (u.procs receiver).accepted p_or)\n                                     (vals receiver)} = prop, by {\n          rw state_change at h_prop, injection h_prop\n        },\n        use {envelope . msg := message.p2a prop,\n                      sent_to := target.exclude receiver},\n        split,\n        { rw \u2190 prop_eq,\n          unfold protocol.handler server.handle_p1b,\n          rw if_neg (show \u00ac(u.procs receiver).curr < b, by { rw same_bal, exact lt_irrefl b }),\n          rw if_neg (show \u00ac(u.procs receiver).curr.address \u2260 receiver,\n                     by { rw decidable.not_not, rw same_bal, exact bal_from_self }),\n          rw if_neg (show \u00ac(u.procs receiver).curr > b, by { rw same_bal, exact lt_irrefl b }),\n          rw if_neg (show \u00ac(is_quorum (u.procs receiver).followers \u2228\n                     sender \u2208 (u.procs receiver).followers),\n                     by {\n                       rw not_or_distrib,\n                       split,\n                       { exact u_no_qrm },\n                       intro cond,\n                       have fact : (u.procs receiver).followers \u222a {sender} = (u.procs receiver).followers,\n                       by {\n                         rw finset.union_eq_left_iff_subset,\n                         rw finset.singleton_subset_iff,\n                         exact cond\n                       },\n                       apply u_no_qrm,\n                       rw \u2190 fact,\n                       exact v_has_qrm\n                     }),\n          rw if_pos v_has_qrm,\n          left, refl },\n      refl },\n      rw key.right.right.right.right.right at h_prop,\n      clear key acc_unchanged_enough,\n      left,\n      cases proposal.merge_is_one_of (u.procs receiver).accepted p_or;\n      rw h at h_prop,\n      { exact accepted_satisfies receiver prop h_prop },\n      apply promises_satisfy sender {msg := message.p1b b p_or, sent_to := e_sent_to} he b prop,\n      rw \u2190 h_prop\n    },\n    case p2a : p {\n      clear acc_unchanged_enough,\n      rw key.right at h_prop, clear key,\n      left,\n      have key : prop = p, by { injection h_prop with fact, exact eq.symm fact },\n      rw key,\n      use [sender, {msg := message.p2a p, sent_to := e_sent_to}, he],\n      cases p, refl\n    },\n    case p2b : b acc { apply acc_unchanged_enough, rw key.right  },\n    case preempt : { apply acc_unchanged_enough, rw key.right },\n  },\n{ rw same.left p h,\n  have ind_hyp := accepted_satisfies p,\n  intros prop key,\n  exact proposed_stable prop.bal prop.val u v (accepted_satisfies p prop key) u_pn_v },\n{ rw h, clear same h p,\n  rw ntwk_change, clear ntwk_change proc_change,\n  intros e' he' b p e'_is_1b,\n  cases he',\n  { exact proposed_stable p.bal p.val u v\n          (promises_satisfy receiver e' he' b p e'_is_1b) u_pn_v },\n  rcases p1b_emitted e'_is_1b he' with \u27e8b', e_msg_is, accepted_unchanged, b_is_max\u27e9,\n  exact proposed_stable p.bal p.val u v\n        (accepted_satisfies receiver p accepted_unchanged) u_pn_v },\nrw same.right p h,\nintros e' he' b prop e'_is_promise,\nhave ind_hyp := promises_satisfy p e' he' b prop e'_is_promise,\nexact proposed_stable prop.bal prop.val u v ind_hyp u_pn_v\nend\n\n\n-- The ballot in a server's `accepted` field only ever increases. That is, if at\n-- some point a server has accepted a proposal, then it will always have a\n-- proposal in `accepted` and moreover the proposal ballot will be at least as\n-- large as what's currently stored.\nlemma accepted_ballot_nondecreasing {p : pid_t} {prop_u : proposal pid_t value_t}\n  (u v : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t))\n  (u_r : u.reachable) (h_uv : u.possible_next v)\n  (h_u : (u.procs p).accepted = some prop_u) :\n  \u2203 (prop_v : proposal pid_t value_t), (v.procs p).accepted = some prop_v \u2227 prop_u.bal \u2264 prop_v.bal :=\nbegin\nrcases h_uv with \u27e8receiver, sender, e, he, deliverable, proc_change, ntwk_change, same\u27e9,\ncases decidable.em (p = receiver),\nswap,\n{ exact \u27e8prop_u, by { rw same.left p h, exact h_u }, by refl\u27e9 },\nrw h at h_u \u22a2,\nclear h p,\nrw proc_change,\nhave key := state_change receiver (u.procs receiver) e.msg sender,\ncases key,\n{ exact \u27e8prop_u, by { rw key, exact h_u }, by refl\u27e9 },\ncases e.msg,\n  case p1a : b {\n    rw key.right,\n    exact \u27e8prop_u, h_u, by refl\u27e9\n  },\n  case p1b : b p_or {\n    cases key,\n    { rw key.right,\n      exact \u27e8prop_u, h_u, by refl\u27e9 },\n    cases key,\n    { rw key.right.right.right.right,\n      use { bal := (u.procs receiver).curr,\n            val := proposal.value_or_default (proposal.merge (u.procs receiver).accepted p_or)\n                                 (vals receiver) },\n      exact \u27e8by refl, (current_ge_accepted_ballot u u_r).left receiver prop_u h_u\u27e9 },\n    rw key.right.right.right.right.right,\n    rw h_u,\n    exact proposal.merge_ballot_ge_left prop_u p_or\n  },\n  case p2a : pr {\n    cases key with pr_bal_ge_curr update_w_pr,\n    exact \u27e8pr, by { rw update_w_pr }, by { exact le_trans\n      ((current_ge_accepted_ballot u u_r).left receiver prop_u h_u)\n      pr_bal_ge_curr }\u27e9,\n  },\n  case p2b : b accepted {\n    rw key.right,\n    exact \u27e8prop_u, h_u, by refl\u27e9\n  },\n  case preempt : {\n    rw key.right,\n    exact \u27e8prop_u, h_u, by refl\u27e9\n  }\nend\n\n-- If a server issued a p1b promising not to accept anything with ballot less\n-- than b, then its `curr` ballot will always be at least b.\nlemma ballot_ge_any_promised (voter : pid_t)\n  (b_promised : ballot pid_t) (p_or : option (proposal pid_t value_t)) : predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    (\u2203 e \u2208 s.network voter, envelope.msg e = message.p1b b_promised p_or) \u2192\n      (s.procs voter).curr \u2265 b_promised) :=\nbegin\nsuffices : predicate.inductive_invariant _,\nby { exact predicate.ind_inv_is_inv this},\nsplit,\n{ intros s hs,\n  rintros \u27e8e, he, e_is_promise\u27e9,\n  specialize hs voter,\n  injection hs with __ key, clear_, rw \u2190 key at he,\n  cases he,\n  { rw he at e_is_promise, injection e_is_promise },\n  rw set.mem_singleton_iff at he,\n  rw he at e_is_promise, injection e_is_promise },\nintros u v hu u_pn_v,\nrintros \u27e8e, he, e_is_promise\u27e9,\nrcases (show _, by exact u_pn_v) with \u27e8receiver, sender, e', he', deliverable, proc_change, ntwk_change, rest_same\u27e9,\ncases decidable.em (voter = receiver),\nswap,\n{ rw rest_same.left voter h,\n  rw rest_same.right voter h at he,\n  exact hu \u27e8e, he, e_is_promise\u27e9 },\nclear rest_same,\nrw \u2190 h at proc_change ntwk_change deliverable, clear h receiver,\nrw ntwk_change at he,\ncases he,\n{ apply le_trans (hu \u27e8e, he, e_is_promise\u27e9),\n  exact ballot_nondecreasing voter u_pn_v },\nrcases p1b_emitted e_is_promise he with \u27e8b', e'_is, accepted_is, is_max\u27e9,\nrw e'_is at proc_change, rw proc_change,\nunfold protocol.handler server.handle_p1a,\ncases decidable.em ((u.procs voter).curr < b'),\n{ rw if_pos h,\n  unfold max max_default at is_max,\n  rw if_neg (not_le_of_gt h) at is_max,\n  rw is_max, exact le_refl _ },\nrw if_neg h,\nunfold max max_default at is_max,\nrw if_pos (le_of_not_gt h) at is_max,\nrw is_max,\nexact le_refl _\nend\n\ndef voted_ballot\n  (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t))\n  (voter : pid_t) (b : ballot pid_t) :=\n  \u2203 e \u2208 s.network voter, (envelope.msg e) = message.p2b b tt\n\nlemma voted_imp_proposed (voter : pid_t) (b : ballot pid_t) : predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    voted_ballot s voter b \u2192 \u2203 v, proposed s b v) :=\nbegin\nsuffices : predicate.inductive_invariant _,\nby { exact predicate.ind_inv_is_inv this },\nsplit,\n{ intros s hs, rintros \u27e8ev, h_ev, ev_vote\u27e9,\n  specialize hs voter,\n  injection hs with __ key, clear_,\n  rw \u2190 key at h_ev,\n  cases h_ev,\n  { rw h_ev at ev_vote, injection ev_vote },\n  rw set.mem_singleton_iff at h_ev,\n  rw h_ev at ev_vote, injection ev_vote },\nintros u w hu u_pn_w,\nrcases (show _, by exact u_pn_w) with \u27e8receiver, sender, e, he, deliverable, proc_change, ntwk_change, proc_same, ntwk_same\u27e9, clear deliverable proc_same,\nintro h_voted,\nunfold voted_ballot at h_voted,\ncases decidable.em (voter = receiver),\nswap,\n{ rw ntwk_same voter h at h_voted,\n  rcases hu h_voted with \u27e8v, proposed_at_u\u27e9,\n  exact \u27e8v, proposed_stable b v u w proposed_at_u u_pn_w\u27e9 },\nclear ntwk_same,\nrw \u2190 h at ntwk_change proc_change, clear h receiver,\nrw ntwk_change at h_voted,\nrcases h_voted with \u27e8e', he', e'_msg_is_vote\u27e9,\ncases he',\n{ rcases hu \u27e8e', he', e'_msg_is_vote\u27e9 with \u27e8v, proposed_at_u\u27e9,\n  exact \u27e8v, proposed_stable b v u w proposed_at_u u_pn_w\u27e9 },\nhave event := p2b_emitted e'_msg_is_vote he',\ncases event,\n{ rcases event with \u27e8p_or, e_msg_is, bal_from_self, u_no_qrm, v_has_qrm, rest\u27e9,\n  have ntwk_delta_eq : (protocol.handler voter (u.procs voter) e.msg sender).snd = {{msg := message.p2a\n                {bal := (u.procs voter).curr,\n                 val := proposal.value_or_default (proposal.merge (u.procs voter).accepted p_or) (vals voter)},\n       sent_to := target.exclude voter},\n        {msg := message.p2b (u.procs voter).curr tt, sent_to := target.just voter}},\n  by {\n    rw e_msg_is,\n    unfold protocol.handler server.handle_p1b,\n    rw if_neg (lt_irrefl _),\n    rw if_neg (show \u00ac(u.procs voter).curr.address \u2260 voter,\n               by { rw decidable.not_not, exact bal_from_self }),\n    rw if_neg (lt_irrefl _),\n    rw if_neg (show \u00ac(is_quorum (u.procs voter).followers \u2228\n               sender \u2208 (u.procs voter).followers),\n               by {\n                 rw not_or_distrib,\n                 split,\n                 { exact u_no_qrm },\n                 intro cond,\n                 have fact : (u.procs voter).followers \u222a {sender} = (u.procs voter).followers, \n                 by {\n                   rw finset.union_eq_left_iff_subset,\n                   rw finset.singleton_subset_iff,\n                   exact cond\n                 },\n                 apply u_no_qrm,\n                 rw \u2190 fact,\n                 exact v_has_qrm\n               }),\n      rw if_pos v_has_qrm },\n  use proposal.value_or_default (proposal.merge (u.procs voter).accepted p_or) (vals voter),\n  use voter,\n  use {msg := message.p2a\n                {bal := (u.procs voter).curr,\n                 val := proposal.value_or_default (proposal.merge (u.procs voter).accepted p_or) (vals voter)},\n       sent_to := target.exclude voter},\n  split,\n  { rw ntwk_change, right,\n    rw ntwk_delta_eq, left, refl },\n  suffices : (u.procs voter).curr = b, by { rw this },\n  rw rest at e'_msg_is_vote, injection e'_msg_is_vote },\nrcases event with \u27e8\u27e8p_bal, p_val\u27e9, e_msg_is, p_bal_larger\u27e9,\nuse [p_val, sender, e, sys_state.ntwk_subset he u_pn_w],\nrw e_msg_is,\nsuffices : p_bal = b, by { rw this },\nrw e_msg_is at he',\nunfold protocol.handler server.handle_p2a at he',\nrw if_pos p_bal_larger at he',\nrw set.mem_singleton_iff at he',\nrw he' at e'_msg_is_vote,\ninjection e'_msg_is_vote\nend\n\n-- In a same vein to `ballot_ge_any_promised`, if a server voted for a ballot b,\n-- then it will always have a proposal stored where the proposal's ballot is at\n-- least b.\nlemma accepted_ge_any_voted (voter : pid_t) (b : ballot pid_t): predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    voted_ballot s voter b \u2192\n      \u2203 p, (s.procs voter).accepted = some p \u2227 p.bal \u2265 b) :=\nbegin\nrw predicate.use_any_invariant,\nsplit,\n{ intros s hs,\n  rintros \u27e8e, he, e_is_vote\u27e9,\n  specialize hs voter,\n  unfold protocol.init at hs,\n  injection hs with __ network, clear_,\n  rw \u2190 network at he,\n  cases he,\n  { rw he at e_is_vote, injection e_is_vote },\n  rw set.mem_singleton_iff at he,\n  rw he at e_is_vote, injection e_is_vote },\nintros u v u_r hu u_pn_v v_r,\nrcases (show _, by exact u_pn_v) with \u27e8receiver, sender, e', he', deliverable, proc_change, ntwk_change, rest_same\u27e9,\nrintros \u27e8ev, h_ev, ev_vote\u27e9,\ncases decidable.em (voter = receiver),\nswap,\n{ rw rest_same.left voter h,\n  rw rest_same.right voter h at h_ev,\n  exact hu \u27e8ev, h_ev, ev_vote\u27e9 },\nrw \u2190 h at proc_change ntwk_change deliverable,\nclear rest_same h receiver,\nrw ntwk_change at h_ev,\ncases h_ev,\n{ rcases hu \u27e8ev, h_ev, ev_vote\u27e9 with \u27e8p, hp, p_ge_b\u27e9,\n  rcases accepted_ballot_nondecreasing u v u_r u_pn_v hp with \u27e8prop_v, middle, right_for_trans\u27e9,\n  exact \u27e8prop_v, middle, le_trans p_ge_b right_for_trans\u27e9 },\ncases p2b_emitted ev_vote h_ev,\n{ rcases h with \u27e8p_or, e'_msg_is, active, no_start_quorum, end_w_quorum, ev_is\u27e9,\n  use {bal := (u.procs voter).curr,\n       val := proposal.value_or_default (proposal.merge (u.procs voter).accepted p_or) (vals voter)},\n  rw e'_msg_is at proc_change, clear e'_msg_is,\n  rw proc_change,\n  unfold protocol.handler server.handle_p1b,\n  rw if_neg (lt_irrefl _),\n  rw if_neg (decidable.not_not.mpr active),\n  rw if_neg (lt_irrefl _),\n  rw if_neg (show \u00ac(is_quorum (u.procs voter).followers \u2228 sender \u2208 (u.procs voter).followers),\n    by {\n      rw not_or_distrib,\n      split,\n      { exact no_start_quorum },\n      intro cond,\n      have fact : (u.procs voter).followers \u222a {sender} = (u.procs voter).followers,\n      by {\n        rw finset.union_eq_left_iff_subset,\n        rw finset.singleton_subset_iff,\n        exact cond\n      },\n      apply no_start_quorum,\n      rw \u2190 fact,\n      exact end_w_quorum\n    }),\n  rw if_pos end_w_quorum,\n  split,\n  { refl },\n  apply ge_of_eq,\n  rw ev_is at ev_vote,\n  injection ev_vote },\nrcases h with \u27e8p, e'_msg_is, p_bal_larger\u27e9,\nrw e'_msg_is at proc_change h_ev, clear e'_msg_is,\nuse p,\nrw proc_change,\nunfold protocol.handler server.handle_p2a,\nrw if_pos p_bal_larger,\nsplit,\n{ refl },\nunfold protocol.handler server.handle_p2a at h_ev,\nrw if_pos p_bal_larger at h_ev,\nrw set.mem_singleton_iff at h_ev,\nrw h_ev at ev_vote,\ninjection ev_vote with key __,\nexact ge_of_eq key\nend\n\n-- If a server sent a p1b with ballot b and wrote that no ballot was stored,\n-- then it will never vote for any proposal with ballot less than b.\n--\n-- If a server sent a p1b with ballot b and wrote that its highest stored was\n-- (some p), then it will never vote for any proposal with a ballot larger than\n-- p.bal and less than b.\ntheorem none_voted_between_p1b : predicate.invariant\n  (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n    \u2200 (voter : pid_t) (b_voted b_promised : ballot pid_t)\n      (p_or : option (proposal pid_t value_t)),\n      voted_ballot s voter b_voted \u2192\n        (\u2203 e \u2208 s.network voter, envelope.msg e = message.p1b b_promised p_or) \u2192\n          b_voted \u2265 b_promised \u2228\n          \u2203 p, p_or = some p \u2227 b_voted \u2264 p.bal) :=\nbegin\nsuffices intros_before :\n  \u2200 (voter : pid_t) (b_voted b_promised : ballot pid_t) (p_or : option (proposal pid_t value_t)),\n    predicate.invariant\n    (\u03bb (s : sys_state pid_t (server pid_t value_t is_quorum vals) (message pid_t value_t)),\n       voted_ballot s voter b_voted \u2192\n         (\u2203 (e \u2208 s.network voter), envelope.msg e = message.p1b b_promised p_or) \u2192\n           (b_voted \u2265 b_promised \u2228\n             \u2203 p, p_or = some p \u2227 b_voted \u2264 p.bal)),\nby { intros u u_r voter b_voted b_promised p_or,\n     exact intros_before voter b_voted b_promised p_or u u_r },\nintros voter b_voted b_promised p_or,\nrw predicate.use_any_invariant,\nsplit,\n{ intros s hs,\n  rintros \u27e8e, he, e_is_vote\u27e9,\n  specialize hs voter,\n  unfold protocol.init at hs,\n  injection hs with __ network, clear_,\n  rw \u2190 network at he,\n  cases he,\n  { rw he at e_is_vote, injection e_is_vote },\n  rw set.mem_singleton_iff at he,\n  rw he at e_is_vote, injection e_is_vote },\nintros u w u_r hu u_pn_w w_r,\nrintros \u27e8ev, h_ev, ev_vote\u27e9,\nrintros \u27e8ep, h_ep, ep_promise\u27e9,\nrcases (by exact u_pn_w) with \u27e8receiver, sender, e', he', deliverable, proc_change, ntwk_change, rest_same\u27e9,\ncases decidable.em (voter = receiver),\nswap,\n{ rw rest_same.right voter h at h_ev h_ep,\n  exact hu \u27e8ev, h_ev, ev_vote\u27e9 \u27e8ep, h_ep, ep_promise\u27e9 },\nclear rest_same,\nrw \u2190 h at proc_change ntwk_change deliverable, clear h receiver,\nrw ntwk_change at h_ev h_ep,\ncases h_ep; cases h_ev,\n{ exact hu \u27e8ev, h_ev, ev_vote\u27e9 \u27e8ep, h_ep, ep_promise\u27e9 },\n{ left,\n  have key := ballot_ge_any_promised voter b_promised p_or u u_r \u27e8ep, h_ep, ep_promise\u27e9,\n  cases p2b_emitted ev_vote h_ev,\n  { rcases h with \u27e8p_or, _, _, _, _, ev_is\u27e9,\n    rw ev_is at ev_vote,\n    have fact : (u.procs voter).curr = b_voted, by { injection ev_vote },\n    rw fact at key, exact key },\n  rcases h with \u27e8p, e'_is, p_larger\u27e9,\n  rw e'_is at h_ev,\n  unfold protocol.handler server.handle_p2a at h_ev,\n  rw if_pos p_larger at h_ev,\n  rw set.mem_singleton_iff at h_ev,\n  rw h_ev at ev_vote,\n  injection ev_vote with fact __, clear_,\n  rw \u2190 fact,\n  exact le_trans key p_larger },\n{ right,\n  rcases p1b_emitted ep_promise h_ep with \u27e8b', _, key, _\u27e9,\n  rcases accepted_ge_any_voted voter b_voted u u_r \u27e8ev, h_ev, ev_vote\u27e9 with \u27e8p, fact, p_bal_ge\u27e9,\n  use p,\n  split,\n  { rw \u2190 key, exact fact },\n  exact p_bal_ge\n  },\nrcases p1b_emitted ep_promise h_ep with \u27e8b', e'_msg_is, _, _\u27e9,\nrw e'_msg_is at h_ev,\nunfold protocol.handler server.handle_p1a at h_ev,\ncases decidable.em ((u.procs voter).curr < b'),\n{ rw if_pos h at h_ev,\n  rw set.mem_singleton_iff at h_ev,\n  rw h_ev at ev_vote,\n  injection ev_vote },\nrw if_neg h at h_ev,\nrw set.mem_singleton_iff at h_ev,\nrw h_ev at ev_vote,\ninjection ev_vote\nend\n", "meta": {"author": "gnanabite", "repo": "colocated-paxos", "sha": "f60308e27d3013665809077fe80a4b2af8a42278", "save_path": "github-repos/lean/gnanabite-colocated-paxos", "path": "github-repos/lean/gnanabite-colocated-paxos/colocated-paxos-f60308e27d3013665809077fe80a4b2af8a42278/src/implementation/proof/voter.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505782, "lm_q2_score": 0.5506073655352403, "lm_q1q2_score": 0.3795367567354541}}
{"text": "import category_theory.abelian.projective\nimport pseudo_normed_group.category\nimport topology.continuous_function.algebra\n\nimport algebra.group.ulift\n\nimport for_mathlib.abelian_sheaves.main\nimport for_mathlib.AddCommGroup.exact\nimport for_mathlib.types\n\nimport condensed.adjunctions\nimport condensed.top_comparison\nimport condensed.filtered_colimits\n\n/-!\n# Properties of the category of condensed abelian groups\n\n-/\n\nopen category_theory category_theory.limits\n\nuniverses v u\n\n-- Move this!\n-- @[simps obj map {fully_applied := ff}] -- we probably don't want these as global simp lemmas\n/-- `Ab.ulift.{v u}` is the universe lifting functor sending the category of abelian groups in\n  universe `u` into those in universe `max v u`. -/\ndef Ab.ulift : Ab.{u} \u2964 Ab.{max v u} :=\n{ obj := \u03bb M, AddCommGroup.of $ ulift.{v} M,\n  map := \u03bb M N f,\n  { to_fun := \u03bb x, \u27e8f x.down\u27e9,\n    map_zero' := by { ext1, apply f.map_zero },\n    map_add' := \u03bb x y, by { ext1, apply f.map_add } },\n  map_id' := by { intros, ext, refl },\n  map_comp' := by { intros, ext, refl } }\n\n@[simp] lemma Ab.exact_ulift_map {A B C : Ab} (f : A \u27f6 B) (g : B \u27f6 C) :\n  exact (Ab.ulift.map f) (Ab.ulift.map g) \u2194 exact f g :=\nbegin\n  let F := Ab.ulift.map f,\n  let G := Ab.ulift.map g,\n  change exact F G \u2194 _,\n  rw AddCommGroup.exact_iff,\n  rw AddCommGroup.exact_iff,\n  split,\n  { intro h,\n    apply le_antisymm,\n    { rintros _ \u27e8x,rfl\u27e9,\n      have : ulift.up (f x) \u2208 F.range := \u27e8\u27e8x\u27e9, rfl\u27e9,\n      rw h at this,\n      change _ = _ at this,\n      apply_fun (\u03bb e, e.down) at this,\n      exact this },\n    { rintros x hx, change _ = _ at hx,\n      have : ulift.up x \u2208 G.ker, by { apply_fun ulift.up at hx, exact hx },\n      rw \u2190 h at this,\n      obtain \u27e8y,hy\u27e9 := this,\n      apply_fun (\u03bb e, e.down) at hy,\n      rw \u2190 hy,\n      use [y.down, rfl] } },\n  { intro h,\n    apply le_antisymm,\n    { rintros _ \u27e8x,rfl\u27e9,\n      ext,\n      change _ \u2208 g.ker,\n      rw \u2190 h,\n      use [x.down, rfl] },\n    { intros x hx,\n      change _ = _ at hx,\n      apply_fun (\u03bb e, e.down) at hx,\n      change _ \u2208 g.ker at hx,\n      rw \u2190 h at hx,\n      obtain \u27e8y,hy\u27e9 := hx,\n      use y,\n      ext,\n      exact hy } },\nend\n\nnamespace Condensed\n\n--instance : preadditive (Condensed Ab.{u+1}) := by admit\n\nnoncomputable theory\n\n-- Sanity check\nexample {J : Type (u+1)} [small_category J] [is_filtered J] :\n  limits.preserves_colimits_of_shape J (forget Ab.{u+1}) := by apply_instance\n\n-- this is now available in `condensed/projective_resolutions.lean`...\n--instance : enough_projectives (Condensed Ab.{u+1}) := by admit\n\ninstance : is_right_adjoint (Sheaf_to_presheaf _ _ : Condensed Ab.{u+1} \u2964 _) :=\n{ left := presheaf_to_Sheaf _ _,\n  adj := (sheafification_adjunction _ _) }\n\n@[simps obj map {fully_applied := ff}]\ndef forget_to_CondensedType : Condensed Ab.{u+1} \u2964 CondensedSet :=\n{ obj := \u03bb F, \u27e8F.val \u22d9 forget _, begin\n    cases F with F hF,\n    rwa (presheaf.is_sheaf_iff_is_sheaf_forget _ _ (forget Ab)) at hF,\n    apply_instance\n  end \u27e9,\n  map := \u03bb A B f, \u27e8whisker_right f.val _\u27e9 }\n\ninstance : is_right_adjoint forget_to_CondensedType :=\n{ left := CondensedSet_to_Condensed_Ab,\n  adj := Condensed_Ab_CondensedSet_adjunction }\n\nsection\n\nvariables (A : Type u) [add_comm_group A] [topological_space A] [topological_add_group A]\n\ndef of_top_ab.presheaf : Profinite.{u}\u1d52\u1d56 \u2964 Ab.{u} :=\n{ obj := \u03bb S, \u27e8C(S.unop, A)\u27e9,\n  map := \u03bb S\u2081 S\u2082 f, add_monoid_hom.mk' (\u03bb g, g.comp f.unop) $ \u03bb g\u2081 g\u2082, rfl,\n  map_id' := by { intros, ext, refl },\n  map_comp' := by { intros, ext, refl } }\n\n/-- The condensed abelian group associated with a topological abelian group -/\ndef of_top_ab : Condensed.{u} Ab.{u+1} :=\n{ val := of_top_ab.presheaf A \u22d9 Ab.ulift.{u+1},\n  cond := begin\n    rw category_theory.presheaf.is_sheaf_iff_is_sheaf_forget _ _ (forget Ab),\n    swap, apply_instance,\n    let B := Top.of A,\n    change presheaf.is_sheaf _ B.to_Condensed.val,\n    exact B.to_Condensed.cond,\n  end }\n\n\nvariables {A} {B : Type u} [add_comm_group B] [topological_space B] [topological_add_group B]\n\ndef of_top_ab_map (f : A \u2192+ B) (hf : continuous f) : of_top_ab A \u27f6 of_top_ab B :=\n{ val := whisker_right\n  { app := \u03bb S, begin\n      refine add_monoid_hom.mk' (\u03bb g, \u27e8f \u2218 (show C(\u21a5(opposite.unop S), A), from g), hf.comp _\u27e9) _,\n      { exact g.continuous },\n      { intros, ext, exact f.map_add _ _, }\n    end,\n    naturality' := \u03bb S T g, rfl, }\n  Ab.ulift.{u+1} }\n\n\nend\n\nend Condensed\n\nnamespace CompHausFiltPseuNormGrp\n\nopen_locale nnreal\nopen pseudo_normed_group comphaus_filtered_pseudo_normed_group\n\ndef presheaf (A : CompHausFiltPseuNormGrp.{u}) (S : Profinite.{u}) : Type u :=\n{ f : S \u2192 A // \u2203 (c : \u211d\u22650) (f\u2080 : S \u2192 filtration A c), continuous f\u2080 \u2227 f = coe \u2218 f\u2080 }\n\nnamespace presheaf\n\nvariables (A : CompHausFiltPseuNormGrp.{u}) (S : Profinite.{u})\n\n@[ext]\nlemma ext {A : CompHausFiltPseuNormGrp} {S : Profinite} (f g : presheaf A S) : f.1 = g.1 \u2192 f = g :=\nsubtype.ext\n\ninstance : has_zero (presheaf A S) := \u27e8\u27e80, 0, 0, continuous_zero, rfl\u27e9\u27e9\n\ninstance : has_neg (presheaf A S) :=\n\u27e8\u03bb f, \u27e8-f.1,\n  begin\n    obtain \u27e8_, c, f, hf, rfl\u27e9 := f,\n    refine \u27e8c, \u03bb s, - f s, _, rfl\u27e9,\n    exact (continuous_neg' c).comp hf\n  end\u27e9\u27e9\n\ninstance : has_add (presheaf A S) :=\n\u27e8\u03bb f g, \u27e8f.1 + g.1,\n  begin\n    obtain \u27e8_, cf, f, hf, rfl\u27e9 := f,\n    obtain \u27e8_, cg, g, hg, rfl\u27e9 := g,\n    refine \u27e8cf + cg, \u03bb s, \u27e8f s + g s, add_mem_filtration (f s).2 (g s).2\u27e9, _, rfl\u27e9,\n    have aux := (hf.prod_mk hg),\n    exact (continuous_add' cf cg).comp aux,\n  end\u27e9\u27e9\n\ninstance : has_sub (presheaf A S) :=\n\u27e8\u03bb f g, \u27e8f.1 - g.1,\n  begin\n    obtain \u27e8_, cf, f, hf, rfl\u27e9 := f,\n    obtain \u27e8_, cg, g, hg, rfl\u27e9 := g,\n    refine \u27e8cf + cg, \u03bb s, \u27e8f s - g s, sub_mem_filtration (f s).2 (g s).2\u27e9, _, rfl\u27e9,\n    have aux := (hf.prod_mk ((continuous_neg' cg).comp hg)),\n    simp only [sub_eq_add_neg],\n    exact (continuous_add' cf cg).comp aux,\n  end\u27e9\u27e9\n\nvariables {A S}\n\nprotected def nsmul (n : \u2115) (f : presheaf A S) : presheaf A S :=\n\u27e8n \u2022 f.1,\nbegin\n  obtain \u27e8_, c, f, hf, rfl\u27e9 := f,\n  refine \u27e8n * c, \u03bb s, \u27e8n \u2022 f s, nat_smul_mem_filtration _ _ _ (f s).2\u27e9, _, rfl\u27e9,\n  exact continuous_nsmul _ _ _ hf,\nend\u27e9\n\nprotected def zsmul (n : \u2124) (f : presheaf A S) : presheaf A S :=\n\u27e8n \u2022 f.1,\nbegin\n  obtain \u27e8_, c, f, hf, rfl\u27e9 := f,\n  refine \u27e8n.nat_abs * c, \u03bb s, \u27e8n \u2022 f s, int_smul_mem_filtration _ _ _ (f s).2\u27e9, _, rfl\u27e9,\n  exact continuous_zsmul _ _ _ hf,\nend\u27e9\n\nvariables (A S)\n\ninstance : add_comm_group (presheaf A S) :=\n{ zero := 0,\n  add := (+),\n  nsmul := presheaf.nsmul,\n  zsmul := presheaf.zsmul,\n  add_assoc := by { intros, ext, exact add_assoc _ _ _ },\n  zero_add := by { intros, ext, exact zero_add _ },\n  add_zero := by { intros, ext, exact add_zero _ },\n  add_comm := by { intros, ext, exact add_comm _ _ },\n  add_left_neg := by { intros, ext, exact add_left_neg _ },\n  sub_eq_add_neg := by { intros, ext, exact sub_eq_add_neg _ _ },\n  nsmul_zero' := by { intros, ext, exact zero_nsmul _ },\n  nsmul_succ' := by { intros, ext, exact succ_nsmul _ _ },\n  zsmul_zero' := by { intros, ext, exact zero_zsmul _ },\n  zsmul_succ' := by { intros, ext, exact add_comm_group.zsmul_succ' _ _ },\n  zsmul_neg' := by { intros, ext, exact add_comm_group.zsmul_neg' _ _ },\n  .. presheaf.has_sub A S, .. presheaf.has_neg A S }\n\n@[simps apply {fully_applied := ff}]\ndef comap (A : CompHausFiltPseuNormGrp) {S T : Profinite} (\u03c6 : S \u27f6 T) :\n  presheaf A T \u2192+ presheaf A S :=\n{ to_fun := \u03bb f, \u27e8f.1 \u2218 \u03c6,\n  begin\n    obtain \u27e8_, c, f, hf, rfl\u27e9 := f,\n    refine \u27e8c, f \u2218 \u03c6, hf.comp \u03c6.continuous, rfl\u27e9,\n  end\u27e9,\n  map_zero' := rfl,\n  map_add' := by { intros, refl } }\n\n@[simps apply {fully_applied := ff}]\ndef map {A B : CompHausFiltPseuNormGrp} (\u03c6 : A \u27f6 B) (S : Profinite) :\n  presheaf A S \u2192+ presheaf B S :=\n{ to_fun := \u03bb f, \u27e8\u03c6 \u2218 f.1,\n  begin\n    obtain \u27e8_, c, f, hf, rfl\u27e9 := f,\n    obtain \u27e8d,hd\u27e9 := \u03c6.bound,\n    let e : filtration A c \u2192 filtration B (d * c) := \u03bb t, \u27e8\u03c6 t, hd t.2\u27e9,\n    have he : continuous e,\n    { apply \u03c6.continuous, intros, refl },\n    refine \u27e8d * c, e \u2218 f, he.comp hf, rfl\u27e9,\n  end\u27e9,\n  map_zero' := by { ext, exact \u03c6.map_zero },\n  map_add' := by { intros, ext, exact \u03c6.map_add _ _ } }\n\nend presheaf\n\nopen opposite\n\n/--\nGiven `A : CompHausFiltPseuNormGrp`, this is the presheaf which sends a profinite set\n`S` to the collection of functions `f : S \u2192 A` such that there exists some `r : \u211d\u22650`\nand some continuous map `g : S \u2192 A_{\u2264 r}`, where `f` is the composition of `g` with\nthe inclusion of `A_{\u2264 r}` into `A`.\nThis turns out to be a proetale sheaf, and thus defines a condensed abelian group.\nSee `to_Condensed` below.\n-/\n@[simps obj map {fully_applied := ff}]\ndef Presheaf (A : CompHausFiltPseuNormGrp.{u}) : Profinite.{u}\u1d52\u1d56 \u2964 Ab :=\n{ obj := \u03bb S, \u27e8presheaf A (unop S)\u27e9,\n  map := \u03bb S T \u03c6, presheaf.comap A \u03c6.unop,\n  map_id' := by { intros, ext, refl },\n  map_comp' := by { intros, ext, refl } }\n\n@[simps app {fully_applied := ff}]\ndef Presheaf.map {A B : CompHausFiltPseuNormGrp} (\u03c6 : A \u27f6 B) :\n  Presheaf A \u27f6 Presheaf B :=\n{ app := \u03bb S, presheaf.map \u03c6 (unop S),\n  naturality' := by { intros, refl } }\n\n@[simp]\nlemma Presheaf.map_id (A : CompHausFiltPseuNormGrp) :\n  Presheaf.map (\ud835\udfd9 A) = \ud835\udfd9 _ := by { ext, refl }\n\n@[simp]\nlemma Presheaf.map_comp {A B C : CompHausFiltPseuNormGrp} (f : A \u27f6 B) (g : B \u27f6 C) :\n  Presheaf.map (f \u226b g) = Presheaf.map f \u226b Presheaf.map g := by { ext, refl }\n\n--set_option pp.universes true\n\nlemma Presheaf_comp_ulift_is_sheaf_aux_equalizer\n  (A : CompHausFiltPseuNormGrp.{u}) :\n  (A.Presheaf \u22d9 Ab.ulift.{u+1 u} \u22d9 forget.{u+2 u+1 u+1} Ab.{u+1}).equalizer_condition :=\nbegin\n  intros X B \u03c0 hh,\n  split,\n  { rintros \u27e8x\u27e9 \u27e8y\u27e9 h,\n    ext t,\n    obtain \u27e8t,rfl\u27e9 := hh t,\n    apply_fun (\u03bb e, e.val.down.val t) at h,\n    exact h },\n  { rintros \u27e8\u27e8\u27e8t,c,t',ht',ht\u27e9\u27e9,h\u27e9,\n    let E : Top := Top.of (filtration A c),\n    let t'' : Profinite.to_Top.obj X \u27f6 E := \u27e8t',ht'\u27e9,\n    have hw : Profinite.to_Top.{u}.map (Profinite.pullback.fst.{u} \u03c0 \u03c0) \u226b t'' =\n      Profinite.to_Top.{u}.map (Profinite.pullback.snd.{u} \u03c0 \u03c0) \u226b t'',\n    { dsimp at h,\n      ext i,\n      dsimp [Profinite.pullback.fst, Profinite.pullback.snd],\n      apply_fun (\u03bb e, e.down.val i) at h,\n      change (coe \u2218 t') i.val.fst = (coe \u2218 t') i.val.snd,\n      rw \u2190 ht,\n      exact h },\n    let w := Profinite.descend_to_Top \u03c0 t'' hh hw,\n    refine \u27e8\u27e8\u27e8_,c,w,w.2,rfl\u27e9\u27e9,_\u27e9,\n    ext : 3,\n    dsimp,\n    rw ht,\n    ext i,\n    dsimp [CompHausFiltPseuNormGrp.Presheaf, Ab.ulift,\n      functor.map_to_equalizer],\n    have := Profinite.\u03c0_descend_to_Top \u03c0 t'' hh hw,\n    apply_fun (\u03bb e, (e i).val) at this, exact this }\nend\n\nlemma Presheaf_comp_ulift_is_sheaf (A : CompHausFiltPseuNormGrp.{u}):\n  presheaf.is_sheaf proetale_topology (Presheaf A \u22d9 Ab.ulift.{u+1}) :=\nbegin\n  rw category_theory.presheaf.is_sheaf_iff_is_sheaf_forget _ _ (forget Ab),\n  swap, apply_instance,\n  rw is_sheaf_iff_is_sheaf_of_type,\n  erw (functor.is_proetale_sheaf_of_types_tfae\n    (A.Presheaf \u22d9 Ab.ulift.{u+1} \u22d9 forget _)).out 0 5,\n  refine \u27e8_,_,_\u27e9,\n  { dsimp [functor.empty_condition],\n    split,\n    { intros a b h, ext \u27e8\u27e9 },\n    { intros x, dsimp,\n      refine \u27e8\u27e8\u27e8\u03bb x, x.elim, 0, \u03bb x, x.elim, by continuity, _\u27e9\u27e9, _\u27e9,\n      { ext \u27e8\u27e9 },\n      { cases x, refl } } },\n  { intros X Y,\n    split,\n    { rintros \u27e8x\u27e9 \u27e8y\u27e9 h, dsimp at h,\n      ext : 2,\n      dsimp,\n      ext (t|t),\n      { apply_fun (\u03bb e, e.fst.down.val t) at h, exact h },\n      { apply_fun (\u03bb e, e.snd.down.val t) at h, exact h } },\n    { rintros \u27e8\u27e8f,c,f',hf',hf\u27e9,\u27e8g,d,g',hg',hg\u27e9\u27e9,\n      let p : X.sum Y \u2192 A := \u03bb t, sum.rec_on t f g,\n      let e : \u211d\u22650 := c \u2294 d,\n      haveI : fact (c \u2264 e) := \u27e8le_sup_left\u27e9,\n      haveI : fact (d \u2264 e) := \u27e8le_sup_right\u27e9,\n      let p' : X.sum Y \u2192 filtration A e :=\n        \u03bb t, sum.rec_on t (cast_le \u2218 f') (cast_le \u2218 g'),\n      have hp' : continuous p',\n      { have aux1 : p' \u2218 sum.inl = cast_le \u2218 f', by ext; refl,\n        have aux2 : p' \u2218 sum.inr = cast_le \u2218 g', by ext; refl,\n        simp only [continuous_sup_dom, continuous_coinduced_dom, aux1, aux2],\n        exact \u27e8(continuous_cast_le _ _).comp hf', (continuous_cast_le _ _).comp hg'\u27e9, },\n      have hh : p = coe \u2218 p',\n      { ext (a|a),\n        { apply_fun (\u03bb u, u a) at hf, exact hf },\n        { apply_fun (\u03bb u, u a) at hg, exact hg } },\n      refine \u27e8\u27e8\u27e8p,e,p',hp',hh\u27e9\u27e9,_\u27e9,\n      ext; refl } },\n  { apply Presheaf_comp_ulift_is_sheaf_aux_equalizer }\nend\n\n/--\nGiven `A : CompHausFiltPseuNormGrp`, this is the condensed abeliian group\nwhich, as a presheaf, sends a profinite set `S` to the collection of\nfunctions `f : S \u2192 A` such that there exists some `r : \u211d\u22650`\nand some continuous map `g : S \u2192 A_{\u2264 r}`, where `f` is the composition of `g` with\nthe inclusion of `A_{\u2264 r}` into `A`.\n\nNOTE: For silly universe reasons, we need to apply a `ulift` as well.\n-/\n@[simps obj map {fully_applied := ff}]\ndef to_Condensed : CompHausFiltPseuNormGrp.{u} \u2964 Condensed.{u} Ab.{u+1} :=\n{ obj := \u03bb A,\n  { val := Presheaf A \u22d9 Ab.ulift.{u+1},\n    cond := Presheaf_comp_ulift_is_sheaf _ },\n  map := \u03bb A B f, \u27e8whisker_right (Presheaf.map f) _\u27e9,\n  map_id' := \u03bb X, by { ext : 2, dsimp, simp },\n  map_comp' := \u03bb X Y Z f g, by { ext : 2, dsimp, simp } }\n\nsection\n\n-- #check Top.to_Condensed\n\nvariables (A : CompHausFiltPseuNormGrp.{u})\n\n@[simps]\ndef level : \u211d\u22650 \u2964 CompHaus.{u} :=\n{ obj := \u03bb r, CompHaus.of $ filtration A r,\n  map := \u03bb r s h,\n  { to_fun := cast_le' h.le,\n    continuous_to_fun := by letI : fact (r \u2264 s) := \u27e8h.le\u27e9; exact continuous_cast_le _ _ },\n  map_id' := \u03bb r, by { ext, refl },\n  map_comp' := \u03bb r s t h1 h2, by { ext, refl } }\n\n@[simps]\ndef level_Condensed_diagram : \u211d\u22650 \u2964 CondensedSet.{u} :=\nA.level \u22d9 CompHaus_to_Top.{u} \u22d9 Top_to_Condensed.{u}\n\n@[simps]\ndef level_Condensed_diagram' : (as_small.{u+1} \u211d\u22650) \u2964 CondensedSet.{u} :=\nas_small.down \u22d9 A.level_Condensed_diagram\n\ndef level_Condensed_diagram_cocone :\n  cocone A.level_Condensed_diagram' :=\n{ X := Condensed_Ab_to_CondensedSet.obj (to_Condensed.obj A),\n  \u03b9 :=\n  { app := \u03bb r, Sheaf.hom.mk $\n    { app := \u03bb S f, ulift.up $ \u27e8_, ulift.down r, f.down.1, f.down.2, rfl\u27e9,\n      naturality' := \u03bb S T f, by { ext, refl } },\n    naturality' := \u03bb r s h, by { ext, refl } } } .\n\ndef colimit_iso_Condensed_obj_aux_fun (X) :\nlet E := A.level_Condensed_diagram' \u22d9 Sheaf_to_presheaf _ _ \u22d9 (evaluation _ _).obj (op X) in\n  (types.filtered_colimit_cocone E).X \u2192 A.presheaf X :=\nlet E := A.level_Condensed_diagram' \u22d9 Sheaf_to_presheaf _ _ \u22d9 (evaluation _ _).obj (op X) in\n\u03bb t, @quotient.lift_on' (\u03a3 (j : as_small.{u+1} \u211d\u22650), E.obj j) (A.presheaf X)\n  (types.filtered_colimit_setoid E) t\n  (\u03bb f, \u27e8_,ulift.down f.1, f.2.down.1, f.2.down.2, rfl\u27e9) begin\n    rintros \u27e8i,x\u27e9 \u27e8j,y\u27e9 \u27e8e,u,v,h\u27e9,\n    ext q : 2,\n    dsimp [level_Condensed_diagram, level_Condensed_diagram'] at *,\n    apply_fun (\u03bb e, (e.down q).1) at h, exact h\n  end\n\nlemma colimit_iso_Condensed_obj_aux_fun_bijective (X) :\n  function.bijective (colimit_iso_Condensed_obj_aux_fun A X) :=\nbegin\n  split,\n  { rintros \u27e8\u27e8\u27e8i\u27e9,f\u27e9\u27e9 \u27e8\u27e8\u27e8j\u27e9,g\u27e9\u27e9 h, dsimp [colimit_iso_Condensed_obj_aux_fun] at h \u22a2,\n    simp only [subtype.mk_eq_mk] at h,\n    apply quotient.sound',\n    use [\u27e8i \u2294 j\u27e9, \u27e8le_sup_left\u27e9, \u27e8le_sup_right\u27e9],\n    ext q,\n    dsimp [level_Condensed_diagram'], apply_fun (\u03bb e, e q) at h, exact h },\n  { rintros \u27e8f,c,g,hg,hf\u27e9,\n    use quotient.mk' \u27e8\u27e8c\u27e9,\u27e8\u27e8g,hg\u27e9\u27e9\u27e9, ext tt, dsimp, rw hf, refl }\nend\n\n-- We would have to use `some` to define the inverse of this equiv, so we may as well just use\n-- `equiv.of_bijective`\n@[simps]\ndef colimit_iso_Condensed_obj_aux (X) :\nlet E := A.level_Condensed_diagram' \u22d9 Sheaf_to_presheaf _ _ \u22d9 (evaluation _ _).obj (op X) in\n  (types.filtered_colimit_cocone E).X \u2243 A.presheaf X :=\nequiv.of_bijective (A.colimit_iso_Condensed_obj_aux_fun X)\n(A.colimit_iso_Condensed_obj_aux_fun_bijective X)\n\n/-\nlet E := A.level_Condensed_diagram' \u22d9 Sheaf_to_presheaf _ _ \u22d9 (evaluation _ _).obj (op X) in\n  (types.filtered_colimit_cocone E).X \u2243 A.presheaf X :=\nlet E := A.level_Condensed_diagram' \u22d9 Sheaf_to_presheaf _ _ \u22d9 (evaluation _ _).obj (op X) in\nequiv.of_bijective\n(\u03bb t, @quotient.lift_on' (\u03a3 (j : as_small.{u+1} \u211d\u22650), E.obj j) (A.presheaf X)\n  (types.filtered_colimit_setoid E) t\n  (\u03bb f, \u27e8_,ulift.down f.1, f.2.down.1, f.2.down.2, rfl\u27e9) begin\n    rintros \u27e8i,x\u27e9 \u27e8j,y\u27e9 \u27e8e,u,v,h\u27e9,\n    ext q : 2,\n    dsimp [level_Condensed_diagram, level_Condensed_diagram'] at *,\n    apply_fun (\u03bb e, (e.down q).1) at h, exact h\n  end)\nbegin\n  split,\n  { rintros \u27e8\u27e8\u27e8i\u27e9,f\u27e9\u27e9 \u27e8\u27e8\u27e8j\u27e9,g\u27e9\u27e9 h, dsimp at h \u22a2, apply quotient.sound',\n    simp only [subtype.mk_eq_mk] at h, use [\u27e8i \u2294 j\u27e9, \u27e8le_sup_left\u27e9, \u27e8le_sup_right\u27e9],\n    ext q,\n    dsimp [level_Condensed_diagram'], apply_fun (\u03bb e, e q) at h, exact h },\n  { rintros \u27e8f,c,g,hg,hf\u27e9,\n    use quotient.mk' \u27e8\u27e8c\u27e9,\u27e8\u27e8g,hg\u27e9\u27e9\u27e9, ext tt, dsimp, rw hf }\nend\n-/\n\ndef colimit_iso_Condensed_obj_aux_nat_iso :\n  (filtered_cocone.{u} A.level_Condensed_diagram').X.val \u2245\n  (Condensed_Ab_to_CondensedSet.{u}.obj (to_Condensed.{u}.obj A)).val :=\n  nat_iso.of_components (\u03bb X,\n    (is_colimit_of_preserves ((evaluation _ _).obj X)\n      (colimit.is_colimit (A.level_Condensed_diagram' \u22d9\n        Sheaf_to_presheaf _ _))).cocone_point_unique_up_to_iso (colimit.is_colimit _) \u226a\u226b\n    (colimit.is_colimit _).cocone_point_unique_up_to_iso\n    (types.filtered_colimit_cocone_is_colimit _) \u226a\u226b\n    equiv.to_iso ((A.colimit_iso_Condensed_obj_aux X.unop).trans equiv.ulift.symm)\n  )\nbegin\n  intros X Y f, dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  apply\n    (is_colimit_of_preserves ((evaluation Profinite\u1d52\u1d56 (Type (u+1))).obj X)\n    (colimit.is_colimit (A.level_Condensed_diagram' \u22d9\n    Sheaf_to_presheaf proetale_topology (Type (u+1))))).hom_ext,\n  intros j, simp only [category.assoc],\n  slice_lhs 0 1\n  { dsimp, rw \u2190 nat_trans.naturality },\n  slice_lhs 2 3\n  { erw ((is_colimit_of_preserves ((evaluation Profinite\u1d52\u1d56 (Type (u+1))).obj Y)\n      (colimit.is_colimit (A.level_Condensed_diagram' \u22d9\n      Sheaf_to_presheaf proetale_topology (Type (u+1)))))).fac },\n  slice_lhs 2 3\n  { erw colimit.\u03b9_desc },\n  slice_rhs 1 2\n  { erw (is_colimit_of_preserves ((evaluation Profinite\u1d52\u1d56 (Type (u+1))).obj X)\n    (colimit.is_colimit (A.level_Condensed_diagram' \u22d9\n    Sheaf_to_presheaf proetale_topology (Type (u+1))))).fac },\n  slice_rhs 1 2\n  { erw colimit.\u03b9_desc },\n  ext, refl\nend\n\ndef colimit_iso_Condensed_obj :\n  colimit A.level_Condensed_diagram' \u2245 Condensed_Ab_to_CondensedSet.obj (to_Condensed.obj A) :=\n(colimit.is_colimit _).cocone_point_unique_up_to_iso (filtered_cocone_is_colimit _) \u226a\u226b\n  Sheaf.iso.mk _ (Condensed_Ab_to_CondensedSet.{u}.obj (to_Condensed.{u}.obj A))\n    A.colimit_iso_Condensed_obj_aux_nat_iso\n\ndef colimit_to_Condensed_obj :\n  colimit A.level_Condensed_diagram' \u27f6 Condensed_Ab_to_CondensedSet.obj (to_Condensed.obj A) :=\ncolimit.desc _ A.level_Condensed_diagram_cocone\n\ninstance is_iso_colimit_to_Condensed_obj : is_iso A.colimit_to_Condensed_obj :=\nbegin\n  suffices : A.colimit_to_Condensed_obj =\n    A.colimit_iso_Condensed_obj.hom, by { rw this, apply_instance },\n  dsimp [colimit_iso_Condensed_obj, colimit_to_Condensed_obj],\n  apply colimit.hom_ext, intros i,\n  dsimp [is_colimit.cocone_point_unique_up_to_iso],\n  rw [colimit.\u03b9_desc, colimit.\u03b9_desc_assoc],\n  dsimp [Sheaf.iso.mk],\n  ext T : 3, dsimp,\n  rw \u2190 nat_trans.comp_app,\n  dsimp [colimit_iso_Condensed_obj_aux_nat_iso, nat_iso.of_components],\n  slice_rhs 1 2\n  { erw (is_colimit_of_preserves ((evaluation Profinite\u1d52\u1d56 (Type (u+1))).obj T)\n      (colimit.is_colimit (A.level_Condensed_diagram' \u22d9\n      Sheaf_to_presheaf proetale_topology (Type (u+1))))).fac },\n  slice_rhs 1 2 { erw colimit.\u03b9_desc },\n  ext, refl,\nend\n\nend\n\nend CompHausFiltPseuNormGrp\n\n@[simps obj map {fully_applied := ff}]\ndef CompHausFiltPseuNormGrp\u2081.to_Condensed :\n  CompHausFiltPseuNormGrp\u2081.{u} \u2964 Condensed.{u} Ab.{u+1} :=\nCHFPNG\u2081_to_CHFPNG\u2091\u2097 \u22d9 CompHausFiltPseuNormGrp.to_Condensed\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/ab.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3795367426783024}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n\nQuotient types.\n\n! This file was ported from Lean 3 source module init.data.quot\n! leanprover-community/mathlib commit de855f9965c69f6818f97edaea7d937e24ef678a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\nimport Leanbin.Init.Data.Sigma.Basic\nimport Leanbin.Init.Logic\nimport Leanbin.Init.Propext\nimport Leanbin.Init.Data.Setoid\n\n-- We import propext here, otherwise we would need a quot.lift for propositions.\nuniverse u v\n\n-- iff can now be used to do substitutions in a calculation\n@[subst]\ntheorem iff_subst {a b : Prop} {p : Prop \u2192 Prop} (h\u2081 : a \u2194 b) (h\u2082 : p a) : p b :=\n  Eq.subst (propext h\u2081) h\u2082\n#align iff_subst iff_subst\n\nnamespace Quot\n\n#print Quot.sound /-\naxiom sound : \u2200 {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a b : \u03b1}, r a b \u2192 Quot.mk r a = Quot.mk r b\n#align quot.sound Quot.sound\n-/\n\nattribute [elab_as_elim] lift ind\n\n/- warning: quot.lift_beta -> Quot.lift_mk is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {r : \u03b1 -> \u03b1 -> Prop} {\u03b2 : Sort.{u2}} (f : \u03b1 -> \u03b2) (c : forall (a : \u03b1) (b : \u03b1), (r a b) -> (Eq.{u2} \u03b2 (f a) (f b))) (a : \u03b1), Eq.{u2} \u03b2 (Quot.lift.{u1, u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => r a b) \u03b2 f c (Quot.mk.{u1} \u03b1 r a)) (f a)\nbut is expected to have type\n  forall {\u03b1 : Sort.{u1}} {r : Sort.{u2}} {\u03b2 : \u03b1 -> \u03b1 -> Prop} (f : \u03b1 -> r) (c : forall (a : \u03b1) (b : \u03b1), (\u03b2 a b) -> (Eq.{u2} r (f a) (f b))) (a : \u03b1), Eq.{u2} r (Quot.lift.{u1, u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => \u03b2 a b) r f c (Quot.mk.{u1} \u03b1 \u03b2 a)) (f a)\nCase conversion may be inaccurate. Consider using '#align quot.lift_beta Quot.lift_mk\u2093'. -/\nprotected theorem lift_mk {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2)\n    (c : \u2200 a b, r a b \u2192 f a = f b) (a : \u03b1) : lift f c (Quot.mk r a) = f a :=\n  rfl\n#align quot.lift_beta Quot.lift_mk\n\nprotected theorem ind_beta {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Quot r \u2192 Prop}\n    (p : \u2200 a, \u03b2 (Quot.mk r a)) (a : \u03b1) : (ind p (Quot.mk r a) : \u03b2 (Quot.mk r a)) = p a :=\n  rfl\n#align quot.ind_beta Quot.ind_beta\n\n#print Quot.liftOn /-\n@[reducible, elab_as_elim]\nprotected def liftOn {\u03b1 : Sort u} {\u03b2 : Sort v} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (q : Quot r) (f : \u03b1 \u2192 \u03b2)\n    (c : \u2200 a b, r a b \u2192 f a = f b) : \u03b2 :=\n  lift f c q\n#align quot.lift_on Quot.liftOn\n-/\n\n#print Quot.inductionOn /-\n@[elab_as_elim]\nprotected theorem inductionOn {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Quot r \u2192 Prop} (q : Quot r)\n    (h : \u2200 a, \u03b2 (Quot.mk r a)) : \u03b2 q :=\n  ind h q\n#align quot.induction_on Quot.inductionOn\n-/\n\n#print Quot.exists_rep /-\ntheorem exists_rep {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (q : Quot r) : \u2203 a : \u03b1, Quot.mk r a = q :=\n  Quot.inductionOn q fun a => \u27e8a, rfl\u27e9\n#align quot.exists_rep Quot.exists_rep\n-/\n\nsection\n\nvariable {\u03b1 : Sort u}\n\nvariable {r : \u03b1 \u2192 \u03b1 \u2192 Prop}\n\nvariable {\u03b2 : Quot r \u2192 Sort v}\n\n-- mathport name: \u00abexpr\u27e6 \u27e7\u00bb\nlocal notation:arg \"\u27e6\" a \"\u27e7\" => Quot.mk r a\n\n#print Quot.indep /-\n@[reducible]\nprotected def indep (f : \u2200 a, \u03b2 \u27e6a\u27e7) (a : \u03b1) : PSigma \u03b2 :=\n  \u27e8\u27e6a\u27e7, f a\u27e9\n#align quot.indep Quot.indep\n-/\n\nprotected theorem indep_coherent (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : r a b), (Eq.ndrec (f a) (sound p) : \u03b2 \u27e6b\u27e7) = f b) :\n    \u2200 a b, r a b \u2192 Quot.indep f a = Quot.indep f b := fun a b e => PSigma.eq (sound e) (h a b e)\n#align quot.indep_coherent Quot.indep_coherent\n\nprotected theorem lift_indep_pr1 (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : r a b), (Eq.ndrec (f a) (sound p) : \u03b2 \u27e6b\u27e7) = f b) (q : Quot r) :\n    (lift (Quot.indep f) (Quot.indep_coherent f h) q).1 = q :=\n  Quot.ind (fun a : \u03b1 => Eq.refl (Quot.indep f a).1) q\n#align quot.lift_indep_pr1 Quot.lift_indep_pr1\n\n#print Quot.rec /-\n@[reducible, elab_as_elim]\nprotected def rec (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : r a b), (Eq.ndrec (f a) (sound p) : \u03b2 \u27e6b\u27e7) = f b) (q : Quot r) : \u03b2 q :=\n  Eq.recOn (Quot.lift_indep_pr1 f h q) (lift (Quot.indep f) (Quot.indep_coherent f h) q).2\n#align quot.rec Quot.rec\n-/\n\n#print Quot.recOn' /-\n@[reducible, elab_as_elim]\nprotected def recOn' (q : Quot r) (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : r a b), (Eq.ndrec (f a) (sound p) : \u03b2 \u27e6b\u27e7) = f b) : \u03b2 q :=\n  Quot.rec f h q\n#align quot.rec_on Quot.recOn'\n-/\n\n#print Quot.recOnSubsingleton' /-\n@[reducible, elab_as_elim]\nprotected def recOnSubsingleton' [h : \u2200 a, Subsingleton (\u03b2 \u27e6a\u27e7)] (q : Quot r) (f : \u2200 a, \u03b2 \u27e6a\u27e7) :\n    \u03b2 q :=\n  Quot.rec f (fun a b h => Subsingleton.elim _ (f b)) q\n#align quot.rec_on_subsingleton Quot.recOnSubsingleton'\n-/\n\n#print Quot.hrecOn /-\n@[reducible, elab_as_elim]\nprotected def hrecOn (q : Quot r) (f : \u2200 a, \u03b2 \u27e6a\u27e7) (c : \u2200 (a b : \u03b1) (p : r a b), HEq (f a) (f b)) :\n    \u03b2 q :=\n  Quot.recOn' q f fun a b p =>\n    eq_of_hEq\n      (calc\n        HEq (Eq.ndrec (f a) (sound p) : \u03b2 \u27e6b\u27e7) (f a) := eq_rec_hEq (sound p) (f a)\n        HEq _ (f b) := c a b p\n        )\n#align quot.hrec_on Quot.hrecOn\n-/\n\nend\n\nend Quot\n\n#print Quotient /-\ndef Quotient {\u03b1 : Sort u} (s : Setoid \u03b1) :=\n  @Quot \u03b1 Setoid.r\n#align quotient Quotient\n-/\n\nnamespace Quotient\n\n#print Quotient.mk' /-\nprotected def mk' {\u03b1 : Sort u} [s : Setoid \u03b1] (a : \u03b1) : Quotient s :=\n  Quot.mk Setoid.r a\n#align quotient.mk Quotient.mk'\n-/\n\n-- mathport name: \u00abexpr\u27e6 \u27e7\u00bb\nnotation:arg \"\u27e6\" a \"\u27e7\" => Quotient.mk' a\n\n#print Quotient.sound /-\ntheorem sound {\u03b1 : Sort u} [s : Setoid \u03b1] {a b : \u03b1} : a \u2248 b \u2192 \u27e6a\u27e7 = \u27e6b\u27e7 :=\n  Quot.sound\n#align quotient.sound Quotient.sound\n-/\n\n#print Quotient.lift /-\n@[reducible, elab_as_elim]\nprotected def lift {\u03b1 : Sort u} {\u03b2 : Sort v} [s : Setoid \u03b1] (f : \u03b1 \u2192 \u03b2) :\n    (\u2200 a b, a \u2248 b \u2192 f a = f b) \u2192 Quotient s \u2192 \u03b2 :=\n  Quot.lift f\n#align quotient.lift Quotient.lift\n-/\n\n#print Quotient.ind /-\n@[elab_as_elim]\nprotected theorem ind {\u03b1 : Sort u} [s : Setoid \u03b1] {\u03b2 : Quotient s \u2192 Prop} :\n    (\u2200 a, \u03b2 \u27e6a\u27e7) \u2192 \u2200 q, \u03b2 q :=\n  Quot.ind\n#align quotient.ind Quotient.ind\n-/\n\n#print Quotient.liftOn /-\n@[reducible, elab_as_elim]\nprotected def liftOn {\u03b1 : Sort u} {\u03b2 : Sort v} [s : Setoid \u03b1] (q : Quotient s) (f : \u03b1 \u2192 \u03b2)\n    (c : \u2200 a b, a \u2248 b \u2192 f a = f b) : \u03b2 :=\n  Quot.liftOn q f c\n#align quotient.lift_on Quotient.liftOn\n-/\n\n#print Quotient.inductionOn /-\n@[elab_as_elim]\nprotected theorem inductionOn {\u03b1 : Sort u} [s : Setoid \u03b1] {\u03b2 : Quotient s \u2192 Prop} (q : Quotient s)\n    (h : \u2200 a, \u03b2 \u27e6a\u27e7) : \u03b2 q :=\n  Quot.inductionOn q h\n#align quotient.induction_on Quotient.inductionOn\n-/\n\n#print Quotient.exists_rep /-\ntheorem exists_rep {\u03b1 : Sort u} [s : Setoid \u03b1] (q : Quotient s) : \u2203 a : \u03b1, \u27e6a\u27e7 = q :=\n  Quot.exists_rep q\n#align quotient.exists_rep Quotient.exists_rep\n-/\n\nsection\n\nvariable {\u03b1 : Sort u}\n\nvariable [s : Setoid \u03b1]\n\nvariable {\u03b2 : Quotient s \u2192 Sort v}\n\n#print Quotient.rec /-\nprotected def rec (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : a \u2248 b), (Eq.ndrec (f a) (Quotient.sound p) : \u03b2 \u27e6b\u27e7) = f b)\n    (q : Quotient s) : \u03b2 q :=\n  Quot.rec f h q\n#align quotient.rec Quotient.rec\n-/\n\n#print Quotient.recOn /-\n@[reducible, elab_as_elim]\nprotected def recOn (q : Quotient s) (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (h : \u2200 (a b : \u03b1) (p : a \u2248 b), (Eq.ndrec (f a) (Quotient.sound p) : \u03b2 \u27e6b\u27e7) = f b) : \u03b2 q :=\n  Quot.recOn' q f h\n#align quotient.rec_on Quotient.recOn\n-/\n\n#print Quotient.recOnSubsingleton /-\n@[reducible, elab_as_elim]\nprotected def recOnSubsingleton [h : \u2200 a, Subsingleton (\u03b2 \u27e6a\u27e7)] (q : Quotient s) (f : \u2200 a, \u03b2 \u27e6a\u27e7) :\n    \u03b2 q :=\n  @Quot.recOnSubsingleton' _ _ _ h q f\n#align quotient.rec_on_subsingleton Quotient.recOnSubsingleton\n-/\n\n#print Quotient.hrecOn /-\n@[reducible, elab_as_elim]\nprotected def hrecOn (q : Quotient s) (f : \u2200 a, \u03b2 \u27e6a\u27e7)\n    (c : \u2200 (a b : \u03b1) (p : a \u2248 b), HEq (f a) (f b)) : \u03b2 q :=\n  Quot.hrecOn q f c\n#align quotient.hrec_on Quotient.hrecOn\n-/\n\nend\n\nsection\n\nuniverse u_a u_b u_c\n\nvariable {\u03b1 : Sort u_a} {\u03b2 : Sort u_b} {\u03c6 : Sort u_c}\n\nvariable [s\u2081 : Setoid \u03b1] [s\u2082 : Setoid \u03b2]\n\ninclude s\u2081 s\u2082\n\n#print Quotient.lift\u2082 /-\n@[reducible, elab_as_elim]\nprotected def lift\u2082 (f : \u03b1 \u2192 \u03b2 \u2192 \u03c6) (c : \u2200 a\u2081 a\u2082 b\u2081 b\u2082, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 f a\u2081 a\u2082 = f b\u2081 b\u2082)\n    (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082) : \u03c6 :=\n  Quotient.lift\n    (fun a\u2081 : \u03b1 => Quotient.lift (f a\u2081) (fun a b : \u03b2 => c a\u2081 a a\u2081 b (Setoid.refl a\u2081)) q\u2082)\n    (fun (a b : \u03b1) (h : a \u2248 b) =>\n      @Quotient.ind \u03b2 s\u2082\n        (fun a_1 : Quotient s\u2082 =>\n          Quotient.lift (f a) (fun a_1 b : \u03b2 => c a a_1 a b (Setoid.refl a)) a_1 =\n            Quotient.lift (f b) (fun a b_1 : \u03b2 => c b a b b_1 (Setoid.refl b)) a_1)\n        (fun a' : \u03b2 => c a a' b a' h (Setoid.refl a')) q\u2082)\n    q\u2081\n#align quotient.lift\u2082 Quotient.lift\u2082\n-/\n\n#print Quotient.liftOn\u2082 /-\n@[reducible, elab_as_elim]\nprotected def liftOn\u2082 (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082) (f : \u03b1 \u2192 \u03b2 \u2192 \u03c6)\n    (c : \u2200 a\u2081 a\u2082 b\u2081 b\u2082, a\u2081 \u2248 b\u2081 \u2192 a\u2082 \u2248 b\u2082 \u2192 f a\u2081 a\u2082 = f b\u2081 b\u2082) : \u03c6 :=\n  Quotient.lift\u2082 f c q\u2081 q\u2082\n#align quotient.lift_on\u2082 Quotient.liftOn\u2082\n-/\n\n#print Quotient.ind\u2082 /-\n@[elab_as_elim]\nprotected theorem ind\u2082 {\u03c6 : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop} (h : \u2200 a b, \u03c6 \u27e6a\u27e7 \u27e6b\u27e7)\n    (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082) : \u03c6 q\u2081 q\u2082 :=\n  Quotient.ind (fun a\u2081 => Quotient.ind (fun a\u2082 => h a\u2081 a\u2082) q\u2082) q\u2081\n#align quotient.ind\u2082 Quotient.ind\u2082\n-/\n\n@[elab_as_elim]\nprotected theorem induction_on\u2082 {\u03c6 : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Prop} (q\u2081 : Quotient s\u2081)\n    (q\u2082 : Quotient s\u2082) (h : \u2200 a b, \u03c6 \u27e6a\u27e7 \u27e6b\u27e7) : \u03c6 q\u2081 q\u2082 :=\n  Quotient.ind (fun a\u2081 => Quotient.ind (fun a\u2082 => h a\u2081 a\u2082) q\u2082) q\u2081\n#align quotient.induction_on\u2082 Quotient.induction_on\u2082\n\n@[elab_as_elim]\nprotected theorem induction_on\u2083 [s\u2083 : Setoid \u03c6] {\u03b4 : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Quotient s\u2083 \u2192 Prop}\n    (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082) (q\u2083 : Quotient s\u2083) (h : \u2200 a b c, \u03b4 \u27e6a\u27e7 \u27e6b\u27e7 \u27e6c\u27e7) :\n    \u03b4 q\u2081 q\u2082 q\u2083 :=\n  Quotient.ind (fun a\u2081 => Quotient.ind (fun a\u2082 => Quotient.ind (fun a\u2083 => h a\u2081 a\u2082 a\u2083) q\u2083) q\u2082) q\u2081\n#align quotient.induction_on\u2083 Quotient.induction_on\u2083\n\nend\n\nsection Exact\n\nvariable {\u03b1 : Sort u}\n\nvariable [s : Setoid \u03b1]\n\ninclude s\n\nprivate def rel (q\u2081 q\u2082 : Quotient s) : Prop :=\n  Quotient.liftOn\u2082 q\u2081 q\u2082 (fun a\u2081 a\u2082 => a\u2081 \u2248 a\u2082) fun a\u2081 a\u2082 b\u2081 b\u2082 a\u2081b\u2081 a\u2082b\u2082 =>\n    propext\n      (Iff.intro (fun a\u2081a\u2082 => Setoid.trans (Setoid.symm a\u2081b\u2081) (Setoid.trans a\u2081a\u2082 a\u2082b\u2082)) fun b\u2081b\u2082 =>\n        Setoid.trans a\u2081b\u2081 (Setoid.trans b\u2081b\u2082 (Setoid.symm a\u2082b\u2082)))\n#align quotient.rel quotient.rel\n\n-- mathport name: \u00abexpr ~ \u00bb\nlocal infixl:50 \" ~ \" => Rel\n\nprivate theorem rel.refl : \u2200 q : Quotient s, q ~ q := fun q =>\n  Quot.inductionOn q fun a => Setoid.refl a\n#align quotient.rel.refl quotient.rel.refl\n\nprivate theorem eq_imp_rel {q\u2081 q\u2082 : Quotient s} : q\u2081 = q\u2082 \u2192 q\u2081 ~ q\u2082 := fun h =>\n  Eq.recOn h (Rel.refl q\u2081)\n#align quotient.eq_imp_rel quotient.eq_imp_rel\n\n#print Quotient.exact /-\ntheorem exact {a b : \u03b1} : \u27e6a\u27e7 = \u27e6b\u27e7 \u2192 a \u2248 b := fun h => eq_imp_rel h\n#align quotient.exact Quotient.exact\n-/\n\nend Exact\n\nsection\n\nuniverse u_a u_b u_c\n\nvariable {\u03b1 : Sort u_a} {\u03b2 : Sort u_b}\n\nvariable [s\u2081 : Setoid \u03b1] [s\u2082 : Setoid \u03b2]\n\ninclude s\u2081 s\u2082\n\n#print Quotient.recOnSubsingleton\u2082 /-\n@[reducible, elab_as_elim]\nprotected def recOnSubsingleton\u2082 {\u03c6 : Quotient s\u2081 \u2192 Quotient s\u2082 \u2192 Sort u_c}\n    [h : \u2200 a b, Subsingleton (\u03c6 \u27e6a\u27e7 \u27e6b\u27e7)] (q\u2081 : Quotient s\u2081) (q\u2082 : Quotient s\u2082)\n    (f : \u2200 a b, \u03c6 \u27e6a\u27e7 \u27e6b\u27e7) : \u03c6 q\u2081 q\u2082 :=\n  @Quotient.recOnSubsingleton _ s\u2081 (fun q => \u03c6 q q\u2082) (fun a => Quotient.ind (fun b => h a b) q\u2082) q\u2081\n    fun a => Quotient.recOnSubsingleton q\u2082 fun b => f a b\n#align quotient.rec_on_subsingleton\u2082 Quotient.recOnSubsingleton\u2082\n-/\n\nend\n\nend Quotient\n\nsection\n\nvariable {\u03b1 : Type u}\n\nvariable (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\n#print EqvGen /-\ninductive EqvGen : \u03b1 \u2192 \u03b1 \u2192 Prop\n  | Rel : \u2200 x y, r x y \u2192 EqvGen x y\n  | refl : \u2200 x, EqvGen x x\n  | symm : \u2200 x y, EqvGen x y \u2192 EqvGen y x\n  | trans : \u2200 x y z, EqvGen x y \u2192 EqvGen y z \u2192 EqvGen x z\n#align eqv_gen EqvGen\n-/\n\n#print EqvGen.is_equivalence /-\ntheorem EqvGen.is_equivalence : Equivalence (@EqvGen \u03b1 r) :=\n  Equivalence.mk _ EqvGen.refl EqvGen.symm EqvGen.trans\n#align eqv_gen.is_equivalence EqvGen.is_equivalence\n-/\n\n#print EqvGen.Setoid /-\ndef EqvGen.Setoid : Setoid \u03b1 :=\n  Setoid.mk _ (EqvGen.is_equivalence r)\n#align eqv_gen.setoid EqvGen.Setoid\n-/\n\n#print Quot.exact /-\ntheorem Quot.exact {a b : \u03b1} (H : Quot.mk r a = Quot.mk r b) : EqvGen r a b :=\n  @Quotient.exact _ (EqvGen.Setoid r) a b\n    (@congr_arg _ _ _ _\n      (Quot.lift (@Quotient.mk' _ (EqvGen.Setoid r)) fun x y h => Quot.sound (EqvGen.rel x y h)) H)\n#align quot.exact Quot.exact\n-/\n\n#print Quot.EqvGen_sound /-\ntheorem Quot.EqvGen_sound {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a b : \u03b1} (H : EqvGen r a b) :\n    Quot.mk r a = Quot.mk r b :=\n  EqvGen.rec_on H (fun x y h => Quot.sound h) (fun x => rfl) (fun x y _ IH => Eq.symm IH)\n    fun x y z _ _ IH\u2081 IH\u2082 => Eq.trans IH\u2081 IH\u2082\n#align quot.eqv_gen_sound Quot.EqvGen_sound\n-/\n\nend\n\nopen Decidable\n\ninstance {\u03b1 : Sort u} {s : Setoid \u03b1} [d : \u2200 a b : \u03b1, Decidable (a \u2248 b)] :\n    DecidableEq (Quotient s) := fun q\u2081 q\u2082 : Quotient s =>\n  Quotient.recOnSubsingleton\u2082 q\u2081 q\u2082 fun a\u2081 a\u2082 =>\n    match d a\u2081 a\u2082 with\n    | is_true h\u2081 => isTrue (Quotient.sound h\u2081)\n    | is_false h\u2082 => isFalse fun h => absurd (Quotient.exact h) h\u2082\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Data/Quot.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203489363239, "lm_q2_score": 0.7248702642896702, "lm_q1q2_score": 0.3794118466680645}}
{"text": "/-\nCopyright (c) 2021 Luke Kershaw. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Luke Kershaw\n-/\nimport category_theory.preadditive.additive_functor\nimport category_theory.shift\nimport category_theory.triangulated.rotate\n\n/-!\n# Pretriangulated Categories\n\nThis file contains the definition of pretriangulated categories and triangulated functors\nbetween them.\n\n## Implementation Notes\n\nWe work under the assumption that pretriangulated categories are preadditive categories,\nbut not necessarily additive categories, as is assumed in some sources.\n\nTODO: generalise this to n-angulated categories as in https://arxiv.org/abs/1006.4592\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v v\u2080 v\u2081 v\u2082 u u\u2080 u\u2081 u\u2082\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\n/-\nWe work in an preadditive category `C` equipped with an additive shift.\n-/\nvariables (C : Type u) [category.{v} C] [has_zero_object C] [has_shift C] [preadditive C]\n  [functor.additive (shift C).functor]\n\n/--\nA preadditive category `C` with an additive shift, and a class of \"distinguished triangles\"\nrelative to that shift is called pretriangulated if the following hold:\n* Any triangle that is isomorphic to a distinguished triangle is also distinguished.\n* Any triangle of the form `(X,X,0,id,0,0)` is distinguished.\n* For any morphism `f : X \u27f6 Y` there exists a distinguished triangle of the form `(X,Y,Z,f,g,h)`.\n* The triangle `(X,Y,Z,f,g,h)` is distinguished if and only if `(Y,Z,X\u27e61\u27e7,g,h,-f\u27e61\u27e7)` is.\n* Given a diagram:\n  ```\n        f       g       h\n    X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n    \u2502       \u2502                \u2502\n    \u2502a      \u2502b               \u2502a\u27e61\u27e7'\n    V       V                V\n    X' \u2500\u2500\u2500> Y' \u2500\u2500\u2500> Z' \u2500\u2500\u2500> X'\u27e61\u27e7\n        f'      g'      h'\n  ```\n  where the left square commutes, and whose rows are distinguished triangles,\n  there exists a morphism `c : Z \u27f6 Z'` such that `(a,b,c)` is a triangle morphism.\nSee https://stacks.math.columbia.edu/tag/0145\n-/\nclass pretriangulated :=\n(distinguished_triangles [] : set (triangle C))\n(isomorphic_distinguished : \u03a0 (T\u2081 \u2208 distinguished_triangles) (T\u2082 : triangle C) (T\u2081 \u2245 T\u2082),\n  T\u2082 \u2208 distinguished_triangles)\n(contractible_distinguished : \u03a0 (X : C), (contractible_triangle C X) \u2208 distinguished_triangles)\n(distinguished_cocone_triangle : \u03a0 (X Y : C) (f: X \u27f6 Y), (\u2203 (Z : C) (g : Y \u27f6 Z) (h : Z \u27f6 X\u27e61\u27e7),\n  triangle.mk _ f g h \u2208 distinguished_triangles))\n(rotate_distinguished_triangle : \u03a0 (T : triangle C),\n  T \u2208 distinguished_triangles \u2194 T.rotate \u2208 distinguished_triangles)\n(complete_distinguished_triangle_morphism : \u03a0 (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 distinguished_triangles) (h\u2082 : T\u2082 \u2208 distinguished_triangles) (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081)\n  (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082) (comm\u2081 : T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081),\n  (\u2203 (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083), (T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082) \u2227 (T\u2081.mor\u2083 \u226b a\u27e61\u27e7' = c \u226b T\u2082.mor\u2083) ))\n\nnamespace pretriangulated\nvariables [pretriangulated C]\n\nnotation `dist_triang`:20 C := distinguished_triangles C\n/--\nGiven any distinguished triangle `T`, then we know `T.rotate` is also distinguished.\n-/\nlemma rot_of_dist_triangle (T \u2208 dist_triang C) : (T.rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle T).mp H\n\n/--\nGiven any distinguished triangle `T`, then we know `T.inv_rotate` is also distinguished.\n-/\nlemma inv_rot_of_dist_triangle (T \u2208 dist_triang C) : (T.inv_rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle (T.inv_rotate)).mpr\n  (isomorphic_distinguished T H (T.inv_rotate.rotate) T (inv_rot_comp_rot.symm.app T))\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `f \u226b g = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2081\u2082 (T \u2208 dist_triang C) : T.mor\u2081 \u226b T.mor\u2082 = 0 :=\nbegin\n  have h := contractible_distinguished T.obj\u2081,\n  have f := complete_distinguished_triangle_morphism,\n  specialize f (contractible_triangle C T.obj\u2081) T h H (\ud835\udfd9 T.obj\u2081) T.mor\u2081,\n  have t : (contractible_triangle C T.obj\u2081).mor\u2081 \u226b T.mor\u2081 = \ud835\udfd9 T.obj\u2081 \u226b T.mor\u2081,\n    by refl,\n  specialize f t,\n  cases f with c f,\n  rw \u2190 f.left,\n  simp only [limits.zero_comp, contractible_triangle_mor\u2082],\nend -- TODO : tidy this proof up\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `g \u226b h = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2082\u2083  (T \u2208 dist_triang C) : T.mor\u2082 \u226b T.mor\u2083 = 0 :=\ncomp_dist_triangle_mor_zero\u2081\u2082 C T.rotate (rot_of_dist_triangle C T H)\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `h \u226b f\u27e61\u27e7 = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2083\u2081 (T \u2208 dist_triang C) :\n  T.mor\u2083 \u226b ((shift C).functor.map T.mor\u2081) = 0 :=\nhave H\u2082 : _ := rot_of_dist_triangle C T.rotate (rot_of_dist_triangle C T H),\nby simpa using comp_dist_triangle_mor_zero\u2081\u2082 C (T.rotate.rotate) H\u2082\n\n/-\nTODO: If `C` is pretriangulated with respect to a shift,\nthen `C\u1d52\u1d56` is pretriangulated with respect to the inverse shift.\n-/\nend pretriangulated\nend category_theory.triangulated\n\nnamespace category_theory.triangulated\nnamespace pretriangulated\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [has_zero_object C] [has_shift C] [preadditive C]\n[functor.additive (shift C).functor] [functor.additive (shift C).inverse]\nvariables (D : Type u\u2082) [category.{v\u2082} D] [has_zero_object D] [has_shift D] [preadditive D]\n[functor.additive (shift D).functor] [functor.additive (shift D).inverse]\n\n/--\nThe underlying structure of a triangulated functor between pretriangulated categories `C` and `D`\nis a functor `F : C \u2964 D` together with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7`.\n-/\nstructure triangulated_functor_struct extends (C \u2964 D) :=\n(comm_shift : (shift C).functor \u22d9 to_functor \u2245 to_functor \u22d9 (shift D).functor)\n\ninstance : inhabited (triangulated_functor_struct C C) :=\n\u27e8{ obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl }\u27e9\n\nvariables {C D}\n/--\nGiven a `triangulated_functor_struct` we can define a function from triangles of `C` to\ntriangles of `D`.\n-/\n@[simp]\ndef triangulated_functor_struct.map_triangle (F : triangulated_functor_struct C D)\n  (T : triangle C) : triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\nvariables (C D)\n/--\nA triangulated functor between pretriangulated categories `C` and `D` is a functor `F : C \u2964 D`\ntogether with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7` such that for every\ndistinguished triangle `(X,Y,Z,f,g,h)` of `C`, the triangle\n`(F(X), F(Y), F(Z), F(f), F(g), F(h) \u226b (\u03be X))` is a distinguished triangle of `D`.\nSee https://stacks.math.columbia.edu/tag/014V\n-/\nstructure triangulated_functor [pretriangulated C] [pretriangulated D] extends\n  triangulated_functor_struct C D :=\n(map_distinguished' : \u03a0 (T: triangle C), (T \u2208 dist_triang C) \u2192\n  (to_triangulated_functor_struct.map_triangle T \u2208 dist_triang D) )\n\ninstance [pretriangulated C] : inhabited (triangulated_functor C C) :=\n\u27e8{obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl ,\n  map_distinguished' := begin\n    rintros \u27e8_,_,_,_\u27e9 Tdt,\n    dsimp at *,\n    rwa category.comp_id,\n  end }\u27e9\n\nvariables {C D} [pretriangulated C] [pretriangulated D]\n/--\nGiven a `triangulated_functor` we can define a function from triangles of `C` to triangles of `D`.\n-/\n@[simp]\ndef triangulated_functor.map_triangle (F : triangulated_functor C D) (T : triangle C) :\n  triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\n/--\nGiven a `triangulated_functor` and a distinguished triangle `T` of `C`, then the triangle it\nmaps onto in `D` is also distinguished.\n-/\nlemma triangulated_functor.map_distinguished (F : triangulated_functor C D) (T : triangle C)\n  (h : T \u2208 dist_triang C) : (F.map_triangle T) \u2208 dist_triang D := F.map_distinguished' T h\n\n\nend pretriangulated\nend category_theory.triangulated\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/triangulated/pretriangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.37937716297757224}}
{"text": "opaque test1 {\u03b1 : Sort _} : \u03b1 \u2192 Sort u_1\n#check test1\ndef test2 {\u03b1 : Sort _} : \u03b1 \u2192 Sort u_1 := sorry\n#check test2\nvariable {\u03b1 : Sort _} in theorem test3 : \u03b1 \u2192 Sort _ := sorry\n#check test3\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/levelNGen.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3793771549325622}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,\n  Michael Howes\n-/\nimport group_theory.subgroup.basic\nimport deprecated.submonoid\n/-!\n# Unbundled subgroups\n\nThis file defines unbundled multiplicative and additive subgroups `is_subgroup` and\n`is_add_subgroup`. These are not the preferred way to talk about subgroups and should\nnot be used for any new projects. The preferred way in mathlib are the bundled\nversions `subgroup G` and `add_subgroup G`.\n\n## Main definitions\n\n`is_add_subgroup (S : set G)` : the predicate that `S` is the underlying subset of an additive\nsubgroup of `G`. The bundled variant `add_subgroup G` should be used in preference to this.\n\n`is_subgroup (S : set G)` : the predicate that `S` is the underlying subset of a subgroup\nof `G`. The bundled variant `subgroup G` should be used in preference to this.\n\n## Tags\n\nsubgroup, subgroups, is_subgroup\n-/\nopen set function\n\nvariables {G : Type*} {H : Type*} {A : Type*} {a a\u2081 a\u2082 b c: G}\n\nsection group\nvariables [group G] [add_group A]\n\n/-- `s` is an additive subgroup: a set containing 0 and closed under addition and negation. -/\nstructure is_add_subgroup (s : set A) extends is_add_submonoid s : Prop :=\n(neg_mem {a} : a \u2208 s \u2192 -a \u2208 s)\n\n/-- `s` is a subgroup: a set containing 1 and closed under multiplication and inverse. -/\n@[to_additive]\nstructure is_subgroup (s : set G) extends is_submonoid s : Prop :=\n(inv_mem {a} : a \u2208 s \u2192 a\u207b\u00b9 \u2208 s)\n\n@[to_additive]\nlemma is_subgroup.div_mem {s : set G} (hs : is_subgroup s) {x y : G} (hx : x \u2208 s) (hy : y \u2208 s) :\n  x / y \u2208 s :=\nby simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)\n\nlemma additive.is_add_subgroup\n  {s : set G} (hs : is_subgroup s) : @is_add_subgroup (additive G) _ s :=\n@is_add_subgroup.mk (additive G) _ _ (additive.is_add_submonoid hs.to_is_submonoid)\n  hs.inv_mem\n\ntheorem additive.is_add_subgroup_iff\n  {s : set G} : @is_add_subgroup (additive G) _ s \u2194 is_subgroup s :=\n\u27e8by rintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9; exact @is_subgroup.mk G _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083,\n  \u03bb h, by exactI additive.is_add_subgroup h\u27e9\n\nlemma multiplicative.is_subgroup\n  {s : set A} (hs : is_add_subgroup s) : @is_subgroup (multiplicative A) _ s :=\n@is_subgroup.mk (multiplicative A) _ _ (multiplicative.is_submonoid hs.to_is_add_submonoid)\n  hs.neg_mem\n\ntheorem multiplicative.is_subgroup_iff\n  {s : set A} : @is_subgroup (multiplicative A) _ s \u2194 is_add_subgroup s :=\n\u27e8by rintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9; exact @is_add_subgroup.mk A _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083,\n  \u03bb h, by exactI multiplicative.is_subgroup h\u27e9\n\n@[to_additive of_add_neg]\ntheorem is_subgroup.of_div (s : set G)\n  (one_mem : (1:G) \u2208 s) (div_mem : \u2200{a b:G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s) :\n  is_subgroup s :=\nhave inv_mem : \u2200a, a \u2208 s \u2192 a\u207b\u00b9 \u2208 s, from\n  assume a ha,\n  have 1 * a\u207b\u00b9 \u2208 s, from div_mem one_mem ha,\n  by simpa,\n{ inv_mem := inv_mem,\n  mul_mem := assume a b ha hb,\n    have a * b\u207b\u00b9\u207b\u00b9 \u2208 s, from div_mem ha (inv_mem b hb),\n    by simpa,\n  one_mem := one_mem }\n\ntheorem is_add_subgroup.of_sub (s : set A)\n  (zero_mem : (0:A) \u2208 s) (sub_mem : \u2200{a b:A}, a \u2208 s \u2192 b \u2208 s \u2192 a - b \u2208 s) :\n  is_add_subgroup s :=\nis_add_subgroup.of_add_neg s zero_mem\n  (\u03bb x y hx hy, by simpa only [sub_eq_add_neg] using sub_mem hx hy)\n\n@[to_additive]\nlemma is_subgroup.inter {s\u2081 s\u2082 : set G} (hs\u2081 : is_subgroup s\u2081) (hs\u2082 : is_subgroup s\u2082) :\n  is_subgroup (s\u2081 \u2229 s\u2082) :=\n{ inv_mem := \u03bb x hx, \u27e8hs\u2081.inv_mem hx.1, hs\u2082.inv_mem hx.2\u27e9,\n  ..is_submonoid.inter hs\u2081.to_is_submonoid hs\u2082.to_is_submonoid}\n\n@[to_additive]\nlemma is_subgroup.Inter {\u03b9 : Sort*} {s : \u03b9 \u2192 set G} (hs : \u2200 y : \u03b9, is_subgroup (s y)) :\n  is_subgroup (set.Inter s) :=\n{ inv_mem := \u03bb x h, set.mem_Inter.2 $ \u03bb y, is_subgroup.inv_mem (hs _) (set.mem_Inter.1 h y),\n  ..is_submonoid.Inter (\u03bb y, (hs y).to_is_submonoid) }\n\n@[to_additive]\nlemma is_subgroup_Union_of_directed {\u03b9 : Type*} [h\u03b9 : nonempty \u03b9]\n  {s : \u03b9 \u2192 set G} (hs : \u2200 i, is_subgroup (s i))\n  (directed : \u2200 i j, \u2203 k, s i \u2286 s k \u2227 s j \u2286 s k) :\n  is_subgroup (\u22c3i, s i) :=\n{ inv_mem := \u03bb a ha,\n    let \u27e8i, hi\u27e9 := set.mem_Union.1 ha in\n    set.mem_Union.2 \u27e8i, (hs i).inv_mem hi\u27e9,\n  to_is_submonoid := is_submonoid_Union_of_directed (\u03bb i, (hs i).to_is_submonoid) directed }\n\nend group\n\nnamespace is_subgroup\nopen is_submonoid\nvariables [group G] {s : set G} (hs : is_subgroup s)\n\ninclude hs\n\n@[to_additive]\nlemma inv_mem_iff : a\u207b\u00b9 \u2208 s \u2194 a \u2208 s :=\n\u27e8\u03bb h, by simpa using hs.inv_mem h, inv_mem hs\u27e9\n\n@[to_additive]\nlemma mul_mem_cancel_right (h : a \u2208 s) : b * a \u2208 s \u2194 b \u2208 s :=\n\u27e8\u03bb hba, by simpa using hs.mul_mem hba (hs.inv_mem h), \u03bb hb, hs.mul_mem hb h\u27e9\n\n@[to_additive]\nlemma mul_mem_cancel_left (h : a \u2208 s) : a * b \u2208 s \u2194 b \u2208 s :=\n\u27e8\u03bb hab, by simpa using hs.mul_mem (hs.inv_mem h) hab, hs.mul_mem h\u27e9\n\nend is_subgroup\n\n/-- `is_normal_add_subgroup (s : set A)` expresses the fact that `s` is a normal additive subgroup\nof the additive group `A`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : add_subgroup A` and `hs : S.normal`, and not via this structure. -/\nstructure is_normal_add_subgroup [add_group A] (s : set A) extends is_add_subgroup s : Prop :=\n(normal : \u2200 n \u2208 s, \u2200 g : A, g + n + -g \u2208 s)\n\n/-- `is_normal_subgroup (s : set G)` expresses the fact that `s` is a normal subgroup\nof the group `G`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : subgroup G` and not via this structure. -/\n@[to_additive]\nstructure is_normal_subgroup [group G] (s : set G) extends is_subgroup s : Prop :=\n(normal : \u2200 n \u2208 s, \u2200 g : G, g * n * g\u207b\u00b9 \u2208 s)\n\n@[to_additive]\nlemma is_normal_subgroup_of_comm_group [comm_group G] {s : set G} (hs : is_subgroup s) :\n  is_normal_subgroup s :=\n{ normal := \u03bb n hn g, by rwa [mul_right_comm, mul_right_inv, one_mul],\n  ..hs }\n\nlemma additive.is_normal_add_subgroup [group G]\n  {s : set G} (hs : is_normal_subgroup s) : @is_normal_add_subgroup (additive G) _ s :=\n@is_normal_add_subgroup.mk (additive G) _ _\n  (additive.is_add_subgroup hs.to_is_subgroup)\n  (is_normal_subgroup.normal hs)\n\ntheorem additive.is_normal_add_subgroup_iff [group G]\n  {s : set G} : @is_normal_add_subgroup (additive G) _ s \u2194 is_normal_subgroup s :=\n\u27e8by rintro \u27e8h\u2081, h\u2082\u27e9; exact\n    @is_normal_subgroup.mk G _ _ (additive.is_add_subgroup_iff.1 h\u2081) @h\u2082,\n  \u03bb h, by exactI additive.is_normal_add_subgroup h\u27e9\n\nlemma multiplicative.is_normal_subgroup [add_group A]\n  {s : set A} (hs : is_normal_add_subgroup s) : @is_normal_subgroup (multiplicative A) _ s :=\n@is_normal_subgroup.mk (multiplicative A) _ _\n  (multiplicative.is_subgroup hs.to_is_add_subgroup)\n  (is_normal_add_subgroup.normal hs)\n\ntheorem multiplicative.is_normal_subgroup_iff [add_group A]\n  {s : set A} : @is_normal_subgroup (multiplicative A) _ s \u2194 is_normal_add_subgroup s :=\n\u27e8by rintro \u27e8h\u2081, h\u2082\u27e9; exact\n    @is_normal_add_subgroup.mk A _ _ (multiplicative.is_subgroup_iff.1 h\u2081) @h\u2082,\n  \u03bb h, by exactI multiplicative.is_normal_subgroup h\u27e9\n\nnamespace is_subgroup\nvariable [group G]\n\n-- Normal subgroup properties\n@[to_additive]\nlemma mem_norm_comm {s : set G} (hs : is_normal_subgroup s) {a b : G} (hab : a * b \u2208 s) :\n  b * a \u2208 s :=\nhave h : a\u207b\u00b9 * (a * b) * a\u207b\u00b9\u207b\u00b9 \u2208 s, from hs.normal (a * b) hab a\u207b\u00b9,\nby simp at h; exact h\n\n@[to_additive]\nlemma mem_norm_comm_iff {s : set G} (hs : is_normal_subgroup s) {a b : G} : a * b \u2208 s \u2194 b * a \u2208 s :=\n\u27e8mem_norm_comm hs, mem_norm_comm hs\u27e9\n\n/-- The trivial subgroup -/\n@[to_additive \"the trivial additive subgroup\"]\ndef trivial (G : Type*) [group G] : set G := {1}\n\n@[simp, to_additive]\nlemma mem_trivial {g : G} : g \u2208 trivial G \u2194 g = 1 :=\nmem_singleton_iff\n\n@[to_additive]\nlemma trivial_normal : is_normal_subgroup (trivial G) :=\nby refine {..}; simp [trivial] {contextual := tt}\n\n@[to_additive]\nlemma eq_trivial_iff {s : set G} (hs : is_subgroup s) :\n  s = trivial G \u2194 (\u2200 x \u2208 s, x = (1 : G)) :=\nby simp only [set.ext_iff, is_subgroup.mem_trivial];\n  exact \u27e8\u03bb h x, (h x).1, \u03bb h x, \u27e8h x, \u03bb hx, hx.symm \u25b8 hs.to_is_submonoid.one_mem\u27e9\u27e9\n\n@[to_additive]\nlemma univ_subgroup : is_normal_subgroup (@univ G) :=\nby refine {..}; simp\n\n/-- The underlying set of the center of a group. -/\n@[to_additive add_center \"The underlying set of the center of an additive group.\"]\ndef center (G : Type*) [group G] : set G := {z | \u2200 g, g * z = z * g}\n\n@[to_additive mem_add_center]\nlemma mem_center {a : G} : a \u2208 center G \u2194 \u2200g, g * a = a * g := iff.rfl\n\n@[to_additive add_center_normal]\nlemma center_normal : is_normal_subgroup (center G) :=\n{ one_mem := by simp [center],\n  mul_mem := assume a b ha hb g,\n    by rw [\u2190mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, \u2190mul_assoc],\n  inv_mem := assume a ha g,\n    calc\n      g * a\u207b\u00b9 = a\u207b\u00b9 * (g * a) * a\u207b\u00b9 : by simp [ha g]\n      ...     = a\u207b\u00b9 * g             : by rw [\u2190mul_assoc, mul_assoc]; simp,\n  normal := assume n ha g h,\n    calc\n      h * (g * n * g\u207b\u00b9) = h * n           : by simp [ha g, mul_assoc]\n      ...               = g * g\u207b\u00b9 * n * h : by rw ha h; simp\n      ...               = g * n * g\u207b\u00b9 * h : by rw [mul_assoc g, ha g\u207b\u00b9, \u2190mul_assoc] }\n\n/-- The underlying set of the normalizer of a subset `S : set G` of a group `G`. That is,\n  the elements `g : G` such that `g * S * g\u207b\u00b9 = S`. -/\n@[to_additive add_normalizer \"The underlying set of the normalizer of a subset `S : set A` of an\n  additive group `A`. That is, the elements `a : A` such that `a + S - a = S`.\"]\ndef normalizer (s : set G) : set G :=\n{g : G | \u2200 n, n \u2208 s \u2194 g * n * g\u207b\u00b9 \u2208 s}\n\n@[to_additive]\nlemma normalizer_is_subgroup (s : set G) : is_subgroup (normalizer s) :=\n{ one_mem := by simp [normalizer],\n  mul_mem := \u03bb a b (ha : \u2200 n, n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s)\n    (hb : \u2200 n, n \u2208 s \u2194 b * n * b\u207b\u00b9 \u2208 s) n,\n    by rw [mul_inv_rev, \u2190 mul_assoc, mul_assoc a, mul_assoc a, \u2190 ha, \u2190 hb],\n  inv_mem := \u03bb a (ha : \u2200 n, n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s) n,\n    by rw [ha (a\u207b\u00b9 * n * a\u207b\u00b9\u207b\u00b9)];\n    simp [mul_assoc] }\n\n@[to_additive subset_add_normalizer]\nlemma subset_normalizer {s : set G} (hs : is_subgroup s) : s \u2286 normalizer s :=\n\u03bb g hg n, by rw [is_subgroup.mul_mem_cancel_right hs ((is_subgroup.inv_mem_iff hs).2 hg),\n  is_subgroup.mul_mem_cancel_left hs hg]\n\nend is_subgroup\n\n-- Homomorphism subgroups\nnamespace is_group_hom\nopen is_submonoid is_subgroup\n\n/-- `ker f : set G` is the underlying subset of the kernel of a map `G \u2192 H`. -/\n@[to_additive \"`ker f : set A` is the underlying subset of the kernel of a map `A \u2192 B`\"]\ndef ker [group H] (f : G \u2192 H) : set G := preimage f (trivial H)\n\n@[to_additive]\nlemma mem_ker [group H] (f : G \u2192 H) {x : G} : x \u2208 ker f \u2194 f x = 1 :=\nmem_trivial\n\nvariables [group G] [group H]\n\n@[to_additive]\nlemma one_ker_inv {f : G \u2192 H} (hf : is_group_hom f) {a b : G} (h : f (a * b\u207b\u00b9) = 1) : f a = f b :=\nbegin\n  rw [hf.map_mul, hf.map_inv] at h,\n  rw [\u2190inv_inv (f b), eq_inv_of_mul_eq_one h]\nend\n\n@[to_additive]\nlemma one_ker_inv' {f : G \u2192 H} (hf : is_group_hom f) {a b : G} (h : f (a\u207b\u00b9 * b) = 1) : f a = f b :=\nbegin\n  rw [hf.map_mul, hf.map_inv] at h,\n  apply inv_injective,\n  rw eq_inv_of_mul_eq_one h\nend\n\n@[to_additive]\nlemma inv_ker_one {f : G \u2192 H} (hf : is_group_hom f) {a b : G} (h : f a = f b) : f (a * b\u207b\u00b9) = 1 :=\nhave f a * (f b)\u207b\u00b9 = 1, by rw [h, mul_right_inv],\nby rwa [\u2190hf.map_inv, \u2190hf.map_mul] at this\n\n@[to_additive]\nlemma inv_ker_one' {f : G \u2192 H} (hf : is_group_hom f) {a b : G} (h : f a = f b) : f (a\u207b\u00b9 * b) = 1 :=\nhave (f a)\u207b\u00b9 * f b = 1, by rw [h, mul_left_inv],\nby rwa [\u2190hf.map_inv, \u2190hf.map_mul] at this\n\n@[to_additive]\nlemma one_iff_ker_inv {f : G \u2192 H} (hf : is_group_hom f) (a b : G) : f a = f b \u2194 f (a * b\u207b\u00b9) = 1 :=\n\u27e8hf.inv_ker_one, hf.one_ker_inv\u27e9\n\n@[to_additive]\nlemma one_iff_ker_inv' {f : G \u2192 H} (hf : is_group_hom f) (a b : G) : f a = f b \u2194 f (a\u207b\u00b9 * b) = 1 :=\n\u27e8hf.inv_ker_one', hf.one_ker_inv'\u27e9\n\n@[to_additive]\nlemma inv_iff_ker {f : G \u2192 H} (hf : is_group_hom f) (a b : G) : f a = f b \u2194 a * b\u207b\u00b9 \u2208 ker f :=\nby rw [mem_ker]; exact one_iff_ker_inv hf _ _\n\n@[to_additive]\nlemma inv_iff_ker' {f : G \u2192 H} (hf : is_group_hom f) (a b : G) : f a = f b \u2194 a\u207b\u00b9 * b \u2208 ker f :=\nby rw [mem_ker]; exact one_iff_ker_inv' hf _ _\n\n@[to_additive]\nlemma image_subgroup {f : G \u2192 H} (hf : is_group_hom f) {s : set G} (hs : is_subgroup s) :\n  is_subgroup (f '' s) :=\n{ mul_mem := assume a\u2081 a\u2082 \u27e8b\u2081, hb\u2081, eq\u2081\u27e9 \u27e8b\u2082, hb\u2082, eq\u2082\u27e9,\n             \u27e8b\u2081 * b\u2082, hs.mul_mem hb\u2081 hb\u2082, by simp [eq\u2081, eq\u2082, hf.map_mul]\u27e9,\n  one_mem := \u27e81, hs.to_is_submonoid.one_mem, hf.map_one\u27e9,\n  inv_mem := assume a \u27e8b, hb, eq\u27e9, \u27e8b\u207b\u00b9, hs.inv_mem hb, by { rw hf.map_inv, simp * }\u27e9 }\n\n@[to_additive]\nlemma range_subgroup {f : G \u2192 H} (hf : is_group_hom f) : is_subgroup (set.range f) :=\n@set.image_univ _ _ f \u25b8 hf.image_subgroup univ_subgroup.to_is_subgroup\n\nlocal attribute [simp] one_mem inv_mem mul_mem is_normal_subgroup.normal\n\n@[to_additive]\nlemma preimage {f : G \u2192 H} (hf : is_group_hom f) {s : set H} (hs : is_subgroup s) :\n  is_subgroup (f \u207b\u00b9' s) :=\nby { refine {..};\n     simp [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul, hf.map_one, hf.map_inv, @inv_mem H _ s]\n     {contextual := tt} }\n\n@[to_additive]\nlemma preimage_normal {f : G \u2192 H} (hf : is_group_hom f) {s : set H} (hs : is_normal_subgroup s) :\n  is_normal_subgroup (f \u207b\u00b9' s) :=\n{ one_mem := by simp [hf.map_one, hs.to_is_subgroup.one_mem],\n  mul_mem := by simp [hf.map_mul, hs.to_is_subgroup.mul_mem] {contextual := tt},\n  inv_mem := by simp [hf.map_inv, hs.to_is_subgroup.inv_mem] {contextual := tt},\n  normal := by simp [hs.normal, hf.map_mul, hf.map_inv] {contextual := tt}}\n\n@[to_additive]\nlemma is_normal_subgroup_ker {f : G \u2192 H} (hf : is_group_hom f) : is_normal_subgroup (ker f) :=\nhf.preimage_normal (trivial_normal)\n\n@[to_additive]\nlemma injective_of_trivial_ker {f : G \u2192 H} (hf : is_group_hom f) (h : ker f = trivial G) :\n  function.injective f :=\nbegin\n  intros a\u2081 a\u2082 hfa,\n  simp [ext_iff, ker, is_subgroup.trivial] at h,\n  have ha : a\u2081 * a\u2082\u207b\u00b9 = 1, by rw \u2190h; exact hf.inv_ker_one hfa,\n  rw [eq_inv_of_mul_eq_one ha, inv_inv a\u2082]\nend\n\n@[to_additive]\nlemma trivial_ker_of_injective {f : G \u2192 H} (hf : is_group_hom f) (h : function.injective f) :\n  ker f = trivial G :=\nset.ext $ assume x, iff.intro\n  (assume hx,\n    suffices f x = f 1, by simpa using h this,\n    by simp [hf.map_one]; rwa [mem_ker] at hx)\n  (by simp [mem_ker, hf.map_one] {contextual := tt})\n\n@[to_additive]\nlemma injective_iff_trivial_ker {f : G \u2192 H} (hf : is_group_hom f) :\n  function.injective f \u2194 ker f = trivial G :=\n\u27e8hf.trivial_ker_of_injective, hf.injective_of_trivial_ker\u27e9\n\n@[to_additive]\nlemma trivial_ker_iff_eq_one {f : G \u2192 H} (hf : is_group_hom f) :\n  ker f = trivial G \u2194 \u2200 x, f x = 1 \u2192 x = 1 :=\nby rw set.ext_iff; simp [ker]; exact\n\u27e8\u03bb h x hx, (h x).1 hx, \u03bb h x, \u27e8h x, \u03bb hx, by rw [hx, hf.map_one]\u27e9\u27e9\n\nend is_group_hom\n\nnamespace add_group\n\nvariables [add_group A]\n\n/-- If `A` is an additive group and `s : set A`, then `in_closure s : set A` is the underlying\nsubset of the subgroup generated by `s`. -/\ninductive in_closure (s : set A) : A \u2192 Prop\n| basic {a : A} : a \u2208 s \u2192 in_closure a\n| zero : in_closure 0\n| neg {a : A} : in_closure a \u2192 in_closure (-a)\n| add {a b : A} : in_closure a \u2192 in_closure b \u2192 in_closure (a + b)\n\nend add_group\n\nnamespace group\nopen is_submonoid is_subgroup\n\nvariables [group G] {s : set G}\n\n/-- If `G` is a group and `s : set G`, then `in_closure s : set G` is the underlying\nsubset of the subgroup generated by `s`. -/\n@[to_additive]\ninductive in_closure (s : set G) : G \u2192 Prop\n| basic {a : G} : a \u2208 s \u2192 in_closure a\n| one : in_closure 1\n| inv {a : G} : in_closure a \u2192 in_closure a\u207b\u00b9\n| mul {a b : G} : in_closure a \u2192 in_closure b \u2192 in_closure (a * b)\n\n/-- `group.closure s` is the subgroup generated by `s`, i.e. the smallest subgroup containg `s`. -/\n@[to_additive \"`add_group.closure s` is the additive subgroup generated by `s`, i.e., the\n  smallest additive subgroup containing `s`.\"]\ndef closure (s : set G) : set G := {a | in_closure s a }\n\n@[to_additive]\nlemma mem_closure {a : G} : a \u2208 s \u2192 a \u2208 closure s := in_closure.basic\n\n@[to_additive]\nlemma closure.is_subgroup (s : set G) : is_subgroup (closure s) :=\n{ one_mem := in_closure.one,\n  mul_mem := assume a b, in_closure.mul,\n  inv_mem := assume a, in_closure.inv }\n\n@[to_additive]\ntheorem subset_closure {s : set G} : s \u2286 closure s := \u03bb a, mem_closure\n\n@[to_additive]\ntheorem closure_subset {s t : set G} (ht : is_subgroup t) (h : s \u2286 t) : closure s \u2286 t :=\nassume a ha, by induction ha; simp [h _, *, ht.one_mem, ht.mul_mem, inv_mem_iff]\n\n@[to_additive]\nlemma closure_subset_iff {s t : set G} (ht : is_subgroup t) : closure s \u2286 t \u2194 s \u2286 t :=\n\u27e8assume h b ha, h (mem_closure ha), assume h b ha, closure_subset ht h ha\u27e9\n\n@[to_additive]\ntheorem closure_mono {s t : set G} (h : s \u2286 t) : closure s \u2286 closure t :=\nclosure_subset (closure.is_subgroup _) $ set.subset.trans h subset_closure\n\n@[simp, to_additive]\nlemma closure_subgroup {s : set G} (hs : is_subgroup s) : closure s = s :=\nset.subset.antisymm (closure_subset hs $ set.subset.refl s) subset_closure\n\n@[to_additive]\n\n\n@[to_additive]\nlemma image_closure [group H] {f : G \u2192 H} (hf : is_group_hom f) (s : set G) :\n  f '' closure s = closure (f '' s) :=\nle_antisymm\n  begin\n    rintros _ \u27e8x, hx, rfl\u27e9,\n    apply in_closure.rec_on hx; intros,\n    { solve_by_elim [subset_closure, set.mem_image_of_mem] },\n    { rw [hf.to_is_monoid_hom.map_one],\n      apply is_submonoid.one_mem (closure.is_subgroup _).to_is_submonoid, },\n    { rw [hf.map_inv],\n      apply is_subgroup.inv_mem (closure.is_subgroup _), assumption },\n    { rw [hf.to_is_monoid_hom.map_mul],\n      solve_by_elim [is_submonoid.mul_mem (closure.is_subgroup _).to_is_submonoid] }\n  end\n  (closure_subset (hf.image_subgroup $ closure.is_subgroup _) $ set.image_subset _ subset_closure)\n\n@[to_additive]\ntheorem mclosure_subset {s : set G} : monoid.closure s \u2286 closure s :=\nmonoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ subset_closure\n\n@[to_additive]\ntheorem mclosure_inv_subset {s : set G} : monoid.closure (has_inv.inv \u207b\u00b9' s) \u2286 closure s :=\nmonoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ \u03bb x hx,\n  inv_inv x \u25b8 ((closure.is_subgroup _).inv_mem $ subset_closure hx)\n\n@[to_additive]\ntheorem closure_eq_mclosure {s : set G} : closure s = monoid.closure (s \u222a has_inv.inv \u207b\u00b9' s) :=\nset.subset.antisymm\n  (@closure_subset _ _ _ (monoid.closure (s \u222a has_inv.inv \u207b\u00b9' s))\n    { one_mem := (monoid.closure.is_submonoid _).one_mem,\n      mul_mem := (monoid.closure.is_submonoid _).mul_mem,\n      inv_mem := \u03bb x hx, monoid.in_closure.rec_on hx\n      (\u03bb x hx, or.cases_on hx (\u03bb hx, monoid.subset_closure $ or.inr $\n        show x\u207b\u00b9\u207b\u00b9 \u2208 s, from (inv_inv x).symm \u25b8 hx)\n        (\u03bb hx, monoid.subset_closure $ or.inl hx))\n      ((@one_inv G _).symm \u25b8 is_submonoid.one_mem (monoid.closure.is_submonoid _))\n      (\u03bb x y hx hy ihx ihy,\n        (mul_inv_rev x y).symm \u25b8 is_submonoid.mul_mem (monoid.closure.is_submonoid _) ihy ihx) }\n    (set.subset.trans (set.subset_union_left _ _) monoid.subset_closure))\n  (monoid.closure_subset (closure.is_subgroup _).to_is_submonoid $ set.union_subset subset_closure $\n    \u03bb x hx, inv_inv x \u25b8 (is_subgroup.inv_mem (closure.is_subgroup _) $ subset_closure hx))\n\n@[to_additive]\ntheorem mem_closure_union_iff {G : Type*} [comm_group G] {s t : set G} {x : G} :\n  x \u2208 closure (s \u222a t) \u2194 \u2203 y \u2208 closure s, \u2203 z \u2208 closure t, y * z = x :=\nbegin\n  simp only [closure_eq_mclosure, monoid.mem_closure_union_iff, exists_prop, preimage_union], split,\n  { rintro \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, rfl\u27e9,\n    refine \u27e8_, \u27e8_, hys, _, hzs, rfl\u27e9, _, \u27e8_, hyt, _, hzt, rfl\u27e9, _\u27e9,\n    rw [mul_assoc, mul_assoc, mul_left_comm zs] },\n  { rintro \u27e8_, \u27e8ys, hys, zs, hzs, rfl\u27e9, _, \u27e8yt, hyt, zt, hzt, rfl\u27e9, rfl\u27e9,\n    refine \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, _\u27e9,\n    rw [mul_assoc, mul_assoc, mul_left_comm yt] }\nend\n\nend group\n\nnamespace is_subgroup\nvariable [group G]\n\n@[to_additive]\nlemma trivial_eq_closure : trivial G = group.closure \u2205 :=\nsubset.antisymm\n  (by simp [set.subset_def, (group.closure.is_subgroup _).one_mem])\n  (group.closure_subset (trivial_normal).to_is_subgroup $ by simp)\n\nend is_subgroup\n\n/-The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\n\nnamespace group\nvariables {s : set G} [group G]\n\nlemma conjugates_of_subset {t : set G} (ht : is_normal_subgroup t) {a : G} (h : a \u2208 t) :\n  conjugates_of a \u2286 t :=\n\u03bb x hc,\nbegin\n  obtain \u27e8c, w\u27e9 := is_conj_iff.1 hc,\n  have H := is_normal_subgroup.normal ht a h c,\n  rwa \u2190w,\nend\n\ntheorem conjugates_of_set_subset' {s t : set G} (ht : is_normal_subgroup t) (h : s \u2286 t) :\n  conjugates_of_set s \u2286 t :=\nset.bUnion_subset (\u03bb x H, conjugates_of_subset ht (h H))\n\n/-- The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\ndef normal_closure (s : set G) : set G := closure (conjugates_of_set s)\n\ntheorem conjugates_of_set_subset_normal_closure : conjugates_of_set s \u2286 normal_closure s :=\nsubset_closure\n\ntheorem subset_normal_closure : s \u2286 normal_closure s :=\nset.subset.trans subset_conjugates_of_set conjugates_of_set_subset_normal_closure\n\n/-- The normal closure of a set is a subgroup. -/\nlemma normal_closure.is_subgroup (s : set G) : is_subgroup (normal_closure s) :=\nclosure.is_subgroup (conjugates_of_set s)\n\n/-- The normal closure of s is a normal subgroup. -/\nlemma normal_closure.is_normal : is_normal_subgroup (normal_closure s) :=\n{ normal := \u03bb n h g,\nbegin\n  induction h with x hx x hx ihx x y hx hy ihx ihy,\n  {exact (conjugates_of_set_subset_normal_closure (conj_mem_conjugates_of_set hx))},\n  {simpa using (normal_closure.is_subgroup s).one_mem},\n  {rw \u2190conj_inv,\n   exact ((normal_closure.is_subgroup _).inv_mem ihx)},\n  {rw \u2190conj_mul,\n   exact ((normal_closure.is_subgroup _).to_is_submonoid.mul_mem ihx ihy)},\nend,\n..normal_closure.is_subgroup _ }\n\n/-- The normal closure of s is the smallest normal subgroup containing s. -/\ntheorem normal_closure_subset {s t : set G} (ht : is_normal_subgroup t) (h : s \u2286 t) :\n  normal_closure s \u2286 t :=\n\u03bb a w,\nbegin\n  induction w with x hx x hx ihx x y hx hy ihx ihy,\n  {exact (conjugates_of_set_subset' ht h $ hx)},\n  {exact ht.to_is_subgroup.to_is_submonoid.one_mem},\n  {exact ht.to_is_subgroup.inv_mem ihx},\n  {exact ht.to_is_subgroup.to_is_submonoid.mul_mem ihx ihy}\nend\n\nlemma normal_closure_subset_iff {s t : set G} (ht : is_normal_subgroup t) :\n  s \u2286 t \u2194 normal_closure s \u2286 t :=\n\u27e8normal_closure_subset ht, set.subset.trans (subset_normal_closure)\u27e9\n\ntheorem normal_closure_mono {s t : set G} : s \u2286 t \u2192 normal_closure s \u2286 normal_closure t :=\n\u03bb h, normal_closure_subset normal_closure.is_normal (set.subset.trans h (subset_normal_closure))\n\nend group\n\n/-- Create a bundled subgroup from a set `s` and `[is_subgroup s]`. -/\n@[to_additive \"Create a bundled additive subgroup from a set `s` and `[is_add_subgroup s]`.\"]\ndef subgroup.of [group G] {s : set G} (h : is_subgroup s) : subgroup G :=\n{ carrier := s,\n  one_mem' := h.1.1,\n  mul_mem' := h.1.2,\n  inv_mem' := h.2 }\n\n@[to_additive]\nlemma subgroup.is_subgroup [group G] (K : subgroup G) : is_subgroup (K : set G) :=\n{ one_mem := K.one_mem',\n  mul_mem := K.mul_mem',\n  inv_mem := K.inv_mem' }\n\n-- this will never fire if it's an instance\n@[to_additive]\nlemma subgroup.of_normal [group G] (s : set G) (h : is_subgroup s) (n : is_normal_subgroup s) :\n  subgroup.normal (subgroup.of h) :=\n{ conj_mem := n.normal, }\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/deprecated/subgroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3792038089785084}}
{"text": "import typ\n\nnamespace tts ------------------------------------------------------------------\n\n/-- Grammar of type schemes -/\n@[derive decidable_eq]\nstructure sch (V : Type) : Type :=\n(vars : list V)\n(type : typ V)\n(vars_nodup : vars.nodup)\n\nattribute [pp_using_anonymous_constructor] sch\n\nnamespace sch ------------------------------------------------------------------\nvariables {V : Type} [_root_.decidable_eq V] -- Type of variable names\nvariables {vs : list V} -- List of variable names\nvariables {nd : vs.nodup} -- No duplicate variables names\nvariables {x : tagged V} -- Variables\nvariables {xs : list (tagged V)} -- List of variables\nvariables {t tx : typ V} -- Types\nvariables {ts txs : list (typ V)} -- Lists of types\n\ntheorem eq_of_veq : \u2200 {s\u2081 s\u2082 : sch V}, s\u2081.vars = s\u2082.vars \u2192 s\u2081.type = s\u2082.type \u2192 s\u2081 = s\u2082\n| \u27e8vs\u2081, t\u2081, nd\u2081\u27e9 \u27e8vs\u2082, t\u2082, nd\u2082\u27e9 ha ht := by congr; solve_by_elim\n\ndef arity (s : sch V) : \u2115 :=\ns.vars.length\n\ndef of_typ (t : typ V) : sch V :=\n\u27e8[], t, list.nodup_nil\u27e9\n\n/-- Get the free variables of a scheme -/\ndef fv (s : sch V) : finset (tagged V) :=\ntyp.fv s.type\n\n@[simp] theorem fv_mk : fv (mk vs t nd) = typ.fv t :=\nrfl\n\n/-- Open a type scheme with a list of types for bound variables -/\ndef open_typs (ts : list (typ V)) (s : sch V) : typ V :=\ntyp.open_typs ts s.type\n\n@[simp] theorem open_typs_mk : open_typs ts (mk vs t nd) = typ.open_typs ts t :=\nrfl\n\n/-- Open a type scheme with a list of free variables for bound variables -/\ndef open_vars (xs : list (tagged V)) (s : sch V) : typ V :=\ntyp.open_vars xs s.type\n\n@[simp] theorem open_vars_mk : open_vars xs (mk vs t nd) = typ.open_vars xs t :=\nrfl\n\n/-- Locally-closed type scheme -/\ndef lc (s : sch V) : Prop :=\ntyp.lc_body s.arity s.type\n\nend /- namespace -/ sch --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/sch/core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629465, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3792038089785083}}
{"text": "/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.ring_theory.integral_closure\nimport Mathlib.data.polynomial.field_division\nimport Mathlib.ring_theory.polynomial.gauss_lemma\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Minimal polynomials\n\nThis file defines the minimal polynomial of an element `x` of an `A`-algebra `B`,\nunder the assumption that x is integral over `A`.\n\nAfter stating the defining property we specialize to the setting of field extensions\nand derive some well-known properties, amongst which the fact that minimal polynomials\nare irreducible, and uniquely determined by their defining property.\n\n-/\n\n/-- Let `B` be an `A`-algebra, and `x` an element of `B` that is integral over `A`\nso we have some term `hx : is_integral A x`.\nThe minimal polynomial `minpoly A x` of `x` is a monic polynomial of smallest degree\nthat has `x` as its root.\nFor instance, if `V` is a `K`-vector space for some field `K`, and `f : V \u2192\u2097[K] V` then\nthe minimal polynomial of `f` is `minpoly f.is_integral`. -/\ndef minpoly (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) :\n    polynomial A :=\n  dite (is_integral A x)\n    (fun (hx : is_integral A x) =>\n      well_founded.min sorry\n        (fun (p : polynomial A) => polynomial.monic p \u2227 polynomial.eval\u2082 (algebra_map A B) x p = 0)\n        hx)\n    fun (hx : \u00acis_integral A x) => 0\n\nnamespace minpoly\n\n\n/--A minimal polynomial is monic.-/\ntheorem monic {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B}\n    (hx : is_integral A x) : polynomial.monic (minpoly A x) :=\n  sorry\n\n/-- A minimal polynomial is nonzero. -/\ntheorem ne_zero {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B}\n    [nontrivial A] (hx : is_integral A x) : minpoly A x \u2260 0 :=\n  polynomial.ne_zero_of_monic (monic hx)\n\ntheorem eq_zero {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B}\n    (hx : \u00acis_integral A x) : minpoly A x = 0 :=\n  dif_neg hx\n\n/--An element is a root of its minimal polynomial.-/\n@[simp] theorem aeval (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) :\n    coe_fn (polynomial.aeval x) (minpoly A x) = 0 :=\n  sorry\n\ntheorem mem_range_of_degree_eq_one (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B]\n    [algebra A B] (x : B) (hx : polynomial.degree (minpoly A x) = 1) :\n    x \u2208 ring_hom.range (algebra_map A B) :=\n  sorry\n\n/--The defining property of the minimal polynomial of an element x:\nit is the monic polynomial with smallest degree that has x as its root.-/\ntheorem min (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B)\n    {p : polynomial A} (pmonic : polynomial.monic p) (hp : coe_fn (polynomial.aeval x) p = 0) :\n    polynomial.degree (minpoly A x) \u2264 polynomial.degree p :=\n  sorry\n\n-- TODO(Commelin, Brasca): this is a duplicate\n\n/-- If an element `x` is a root of a nonzero monic polynomial `p`,\nthen the degree of `p` is at least the degree of the minimal polynomial of `x`. -/\ntheorem degree_le_of_monic (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B]\n    (x : B) {p : polynomial A} (hmonic : polynomial.monic p)\n    (hp : coe_fn (polynomial.aeval x) p = 0) :\n    polynomial.degree (minpoly A x) \u2264 polynomial.degree p :=\n  sorry\n\n/-- The degree of a minimal polynomial is positive. -/\ntheorem degree_pos {A : Type u_1} {B : Type u_2} [integral_domain A] [ring B] [algebra A B]\n    [nontrivial B] {x : B} [nontrivial A] (hx : is_integral A x) :\n    0 < polynomial.degree (minpoly A x) :=\n  sorry\n\n/-- If `B/A` is an injective ring extension, and `a` is an element of `A`,\nthen the minimal polynomial of `algebra_map A B a` is `X - C a`. -/\ntheorem eq_X_sub_C_of_algebra_map_inj {A : Type u_1} {B : Type u_2} [integral_domain A] [ring B]\n    [algebra A B] [nontrivial B] [nontrivial A] (a : A)\n    (hf : function.injective \u21d1(algebra_map A B)) :\n    minpoly A (coe_fn (algebra_map A B) a) = polynomial.X - coe_fn polynomial.C a :=\n  sorry\n\n/-- A minimal polynomial is not a unit. -/\ntheorem not_is_unit (A : Type u_1) {B : Type u_2} [integral_domain A] [ring B] [algebra A B]\n    [nontrivial B] (x : B) : \u00acis_unit (minpoly A x) :=\n  sorry\n\n/-- If `a` strictly divides the minimal polynomial of `x`, then `x` cannot be a root for `a`. -/\ntheorem aeval_ne_zero_of_dvd_not_unit_minpoly {A : Type u_1} {B : Type u_2} [integral_domain A]\n    [domain B] [algebra A B] {x : B} {a : polynomial A} (hx : is_integral A x)\n    (hamonic : polynomial.monic a) (hdvd : dvd_not_unit a (minpoly A x)) :\n    coe_fn (polynomial.aeval x) a \u2260 0 :=\n  sorry\n\n/--A minimal polynomial is irreducible.-/\ntheorem irreducible {A : Type u_1} {B : Type u_2} [integral_domain A] [domain B] [algebra A B]\n    {x : B} (hx : is_integral A x) : irreducible (minpoly A x) :=\n  sorry\n\n/-- If an element `x` is a root of a nonzero polynomial `p`,\nthen the degree of `p` is at least the degree of the minimal polynomial of `x`. -/\ntheorem degree_le_of_ne_zero (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B)\n    {p : polynomial A} (pnz : p \u2260 0) (hp : coe_fn (polynomial.aeval x) p = 0) :\n    polynomial.degree (minpoly A x) \u2264 polynomial.degree p :=\n  sorry\n\n/-- The minimal polynomial of an element x is uniquely characterized by its defining property:\nif there is another monic polynomial of minimal degree that has x as a root,\nthen this polynomial is equal to the minimal polynomial of x. -/\ntheorem unique (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B)\n    {p : polynomial A} (pmonic : polynomial.monic p) (hp : coe_fn (polynomial.aeval x) p = 0)\n    (pmin :\n      \u2200 (q : polynomial A),\n        polynomial.monic q \u2192\n          coe_fn (polynomial.aeval x) q = 0 \u2192 polynomial.degree p \u2264 polynomial.degree q) :\n    p = minpoly A x :=\n  sorry\n\n/-- If an element x is a root of a polynomial p,\nthen the minimal polynomial of x divides p. -/\ntheorem dvd (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B)\n    {p : polynomial A} (hp : coe_fn (polynomial.aeval x) p = 0) : minpoly A x \u2223 p :=\n  sorry\n\ntheorem dvd_map_of_is_scalar_tower (A : Type u_1) (K : Type u_2) {R : Type u_3} [comm_ring A]\n    [field K] [comm_ring R] [algebra A K] [algebra A R] [algebra K R] [is_scalar_tower A K R]\n    (x : R) : minpoly K x \u2223 polynomial.map (algebra_map A K) (minpoly A x) :=\n  sorry\n\ntheorem unique' {A : Type u_1} {B : Type u_2} [field A] [ring B] [algebra A B] {x : B}\n    [nontrivial B] {p : polynomial A} (hx : is_integral A x) (hp1 : irreducible p)\n    (hp2 : coe_fn (polynomial.aeval x) p = 0) (hp3 : polynomial.monic p) : p = minpoly A x :=\n  sorry\n\n/-- If `y` is the image of `x` in an extension, their minimal polynomials coincide.\n\nWe take `h : y = algebra_map L T x` as an argument because `rw h` typically fails\nsince `is_integral R y` depends on y.\n-/\ntheorem eq_of_algebra_map_eq {K : Type u_1} {S : Type u_2} {T : Type u_3} [field K] [comm_ring S]\n    [comm_ring T] [algebra K S] [algebra K T] [algebra S T] [is_scalar_tower K S T]\n    (hST : function.injective \u21d1(algebra_map S T)) {x : S} {y : T} (hx : is_integral K x)\n    (h : y = coe_fn (algebra_map S T) x) : minpoly K x = minpoly K y :=\n  sorry\n\n/-- For GCD domains, the minimal polynomial over the ring is the same as the minimal polynomial\nover the fraction field. -/\ntheorem gcd_domain_eq_field_fractions {A : Type u_1} {K : Type u_2} {R : Type u_3}\n    [integral_domain A] [gcd_monoid A] [field K] [integral_domain R] (f : fraction_map A K)\n    [algebra (localization_map.codomain f) R] [algebra A R]\n    [is_scalar_tower A (localization_map.codomain f) R] {x : R} (hx : is_integral A x) :\n    minpoly (localization_map.codomain f) x =\n        polynomial.map (localization_map.to_ring_hom f) (minpoly A x) :=\n  sorry\n\n/-- The minimal polynomial over `\u2124` is the same as the minimal polynomial over `\u211a`. -/\n--TODO use `gcd_domain_eq_field_fractions` directly when localizations are defined\n\n-- in terms of algebras instead of `ring_hom`s\n\ntheorem over_int_eq_over_rat {A : Type u_1} [integral_domain A] {x : A} [h\u211aA : algebra \u211a A]\n    (hx : is_integral \u2124 x) : minpoly \u211a x = polynomial.map (int.cast_ring_hom \u211a) (minpoly \u2124 x) :=\n  sorry\n\n/-- For GCD domains, the minimal polynomial divides any primitive polynomial that has the integral\nelement as root. -/\ntheorem gcd_domain_dvd {A : Type u_1} {K : Type u_2} {R : Type u_3} [integral_domain A]\n    [gcd_monoid A] [field K] [integral_domain R] (f : fraction_map A K)\n    [algebra (localization_map.codomain f) R] [algebra A R]\n    [is_scalar_tower A (localization_map.codomain f) R] {x : R} (hx : is_integral A x)\n    {P : polynomial A} (hprim : polynomial.is_primitive P)\n    (hroot : coe_fn (polynomial.aeval x) P = 0) : minpoly A x \u2223 P :=\n  sorry\n\n/-- The minimal polynomial over `\u2124` divides any primitive polynomial that has the integral element\nas root. -/\n-- TODO use `gcd_domain_dvd` directly when localizations are defined in terms of algebras\n\n-- instead of `ring_hom`s\n\ntheorem integer_dvd {A : Type u_1} [integral_domain A] [algebra \u211a A] {x : A} (hx : is_integral \u2124 x)\n    {P : polynomial \u2124} (hprim : polynomial.is_primitive P)\n    (hroot : coe_fn (polynomial.aeval x) P = 0) : minpoly \u2124 x \u2223 P :=\n  sorry\n\n/-- If `B/K` is a nontrivial algebra over a field, and `x` is an element of `K`,\nthen the minimal polynomial of `algebra_map K B x` is `X - C x`. -/\ntheorem eq_X_sub_C {A : Type u_1} (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B]\n    (a : A) : minpoly A (coe_fn (algebra_map A B) a) = polynomial.X - coe_fn polynomial.C a :=\n  eq_X_sub_C_of_algebra_map_inj a (ring_hom.injective (algebra_map A B))\n\ntheorem eq_X_sub_C' {A : Type u_1} [field A] (a : A) :\n    minpoly A a = polynomial.X - coe_fn polynomial.C a :=\n  eq_X_sub_C A a\n\n/-- The minimal polynomial of `0` is `X`. -/\n@[simp] theorem zero (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] :\n    minpoly A 0 = polynomial.X :=\n  sorry\n\n/-- The minimal polynomial of `1` is `X - 1`. -/\n@[simp] theorem one (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] :\n    minpoly A 1 = polynomial.X - 1 :=\n  sorry\n\n/-- A minimal polynomial is prime. -/\ntheorem prime {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B}\n    (hx : is_integral A x) : prime (minpoly A x) :=\n  sorry\n\n/-- If `L/K` is a field extension and an element `y` of `K` is a root of the minimal polynomial\nof an element `x \u2208 L`, then `y` maps to `x` under the field embedding. -/\ntheorem root {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B}\n    (hx : is_integral A x) {y : A} (h : polynomial.is_root (minpoly A x) y) :\n    coe_fn (algebra_map A B) y = x :=\n  sorry\n\n/--The constant coefficient of the minimal polynomial of `x` is `0` if and only if `x = 0`. -/\n@[simp] theorem coeff_zero_eq_zero {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B]\n    {x : B} (hx : is_integral A x) : polynomial.coeff (minpoly A x) 0 = 0 \u2194 x = 0 :=\n  sorry\n\n/--The minimal polynomial of a nonzero element has nonzero constant coefficient. -/\ntheorem coeff_zero_ne_zero {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B}\n    (hx : is_integral A x) (h : x \u2260 0) : polynomial.coeff (minpoly A x) 0 \u2260 0 :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/field_theory/minpoly_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.37920379447590474}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Jeremy Avigad, Simon Hudon\n-/\nimport data.set.basic\nimport logic.equiv.defs\n\n/-!\n# Partial values of a type\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines `part \u03b1`, the partial values of a type.\n\n`o : part \u03b1` carries a proposition `o.dom`, its domain, along with a function `get : o.dom \u2192 \u03b1`, its\nvalue. The rule is then that every partial value has a value but, to access it, you need to provide\na proof of the domain.\n\n`part \u03b1` behaves the same as `option \u03b1` except that `o : option \u03b1` is decidably `none` or `some a`\nfor some `a : \u03b1`, while the domain of `o : part \u03b1` doesn't have to be decidable. That means you can\ntranslate back and forth between a partial value with a decidable domain and an option, and\n`option \u03b1` and `part \u03b1` are classically equivalent. In general, `part \u03b1` is bigger than `option \u03b1`.\n\nIn current mathlib, `part \u2115`, aka `part_enat`, is used to move decidability of the order to\ndecidability of `part_enat.find` (which is the smallest natural satisfying a predicate, or `\u221e` if\nthere's none).\n\n## Main declarations\n\n`option`-like declarations:\n* `part.none`: The partial value whose domain is `false`.\n* `part.some a`: The partial value whose domain is `true` and whose value is `a`.\n* `part.of_option`: Converts an `option \u03b1` to a `part \u03b1` by sending `none` to `none` and `some a` to\n  `some a`.\n* `part.to_option`: Converts a `part \u03b1` with a decidable domain to an `option \u03b1`.\n* `part.equiv_option`: Classical equivalence between `part \u03b1` and `option \u03b1`.\n\nMonadic structure:\n* `part.bind`: `o.bind f` has value `(f (o.get _)).get _` (`f o` morally) and is defined when `o`\n  and `f (o.get _)` are defined.\n* `part.map`: Maps the value and keeps the same domain.\n\nOther:\n* `part.restrict`: `part.restrict p o` replaces the domain of `o : part \u03b1` by `p : Prop` so long as\n  `p \u2192 o.dom`.\n* `part.assert`: `assert p f` appends `p` to the domains of the values of a partial function.\n* `part.unwrap`: Gets the value of a partial value regardless of its domain. Unsound.\n\n## Notation\n\nFor `a : \u03b1`, `o : part \u03b1`, `a \u2208 o` means that `o` is defined and equal to `a`. Formally, it means\n`o.dom` and `o.get _ = a`.\n-/\n\nopen function\n\n/-- `part \u03b1` is the type of \"partial values\" of type `\u03b1`. It\n  is similar to `option \u03b1` except the domain condition can be an\n  arbitrary proposition, not necessarily decidable. -/\nstructure {u} part (\u03b1 : Type u) : Type u :=\n(dom : Prop)\n(get : dom \u2192 \u03b1)\n\nnamespace part\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n\n/-- Convert a `part \u03b1` with a decidable domain to an option -/\ndef to_option (o : part \u03b1) [decidable o.dom] : option \u03b1 :=\nif h : dom o then some (o.get h) else none\n\n@[simp] lemma to_option_is_some (o : part \u03b1) [decidable o.dom] :\n  o.to_option.is_some \u2194 o.dom :=\nby by_cases o.dom; simp [h, part.to_option]\n\n@[simp] lemma to_option_is_none (o : part \u03b1) [decidable o.dom] :\n  o.to_option.is_none \u2194 \u00aco.dom :=\nby by_cases o.dom; simp [h, part.to_option]\n\n/-- `part` extensionality -/\ntheorem ext' : \u2200 {o p : part \u03b1}\n  (H1 : o.dom \u2194 p.dom)\n  (H2 : \u2200h\u2081 h\u2082, o.get h\u2081 = p.get h\u2082), o = p\n| \u27e8od, o\u27e9 \u27e8pd, p\u27e9 H1 H2 := have t : od = pd, from propext H1,\n  by cases t; rw [show o = p, from funext $ \u03bbp, H2 p p]\n\n/-- `part` eta expansion -/\n@[simp] theorem eta : \u03a0 (o : part \u03b1), (\u27e8o.dom, \u03bb h, o.get h\u27e9 : part \u03b1) = o\n| \u27e8h, f\u27e9 := rfl\n\n/-- `a \u2208 o` means that `o` is defined and equal to `a` -/\nprotected def mem (a : \u03b1) (o : part \u03b1) : Prop := \u2203 h, o.get h = a\n\ninstance : has_mem \u03b1 (part \u03b1) := \u27e8part.mem\u27e9\n\ntheorem mem_eq (a : \u03b1) (o : part \u03b1) : (a \u2208 o) = (\u2203 h, o.get h = a) :=\nrfl\n\ntheorem dom_iff_mem : \u2200 {o : part \u03b1}, o.dom \u2194 \u2203 y, y \u2208 o\n| \u27e8p, f\u27e9 := \u27e8\u03bbh, \u27e8f h, h, rfl\u27e9, \u03bb\u27e8_, h, rfl\u27e9, h\u27e9\n\ntheorem get_mem {o : part \u03b1} (h) : get o h \u2208 o := \u27e8_, rfl\u27e9\n\n@[simp] lemma mem_mk_iff {p : Prop} {o : p \u2192 \u03b1} {a : \u03b1} : a \u2208 part.mk p o \u2194 \u2203 h, o h = a := iff.rfl\n\n/-- `part` extensionality -/\n@[ext]\ntheorem ext {o p : part \u03b1} (H : \u2200 a, a \u2208 o \u2194 a \u2208 p) : o = p :=\next' \u27e8\u03bb h, ((H _).1 \u27e8h, rfl\u27e9).fst,\n     \u03bb h, ((H _).2 \u27e8h, rfl\u27e9).fst\u27e9 $\n\u03bb a b, ((H _).2 \u27e8_, rfl\u27e9).snd\n\n/-- The `none` value in `part` has a `false` domain and an empty function. -/\ndef none : part \u03b1 := \u27e8false, false.rec _\u27e9\n\ninstance : inhabited (part \u03b1) := \u27e8none\u27e9\n\n@[simp] theorem not_mem_none (a : \u03b1) : a \u2209 @none \u03b1 := \u03bb h, h.fst\n\n/-- The `some a` value in `part` has a `true` domain and the\n  function returns `a`. -/\ndef some (a : \u03b1) : part \u03b1 := \u27e8true, \u03bb_, a\u27e9\n\n@[simp] lemma some_dom (a : \u03b1) : (some a).dom := trivial\n\ntheorem mem_unique : \u2200 {a b : \u03b1} {o : part \u03b1}, a \u2208 o \u2192 b \u2208 o \u2192 a = b\n| _ _ \u27e8p, f\u27e9 \u27e8h\u2081, rfl\u27e9 \u27e8h\u2082, rfl\u27e9 := rfl\n\ntheorem mem.left_unique : relator.left_unique ((\u2208) : \u03b1 \u2192 part \u03b1 \u2192 Prop) :=\n\u03bb a o b, mem_unique\n\ntheorem get_eq_of_mem {o : part \u03b1} {a} (h : a \u2208 o) (h') : get o h' = a :=\nmem_unique \u27e8_, rfl\u27e9 h\n\nprotected theorem subsingleton (o : part \u03b1) : set.subsingleton {a | a \u2208 o} :=\n\u03bb a ha b hb, mem_unique ha hb\n\n@[simp] theorem get_some {a : \u03b1} (ha : (some a).dom) : get (some a) ha = a := rfl\n\ntheorem mem_some (a : \u03b1) : a \u2208 some a := \u27e8trivial, rfl\u27e9\n\n@[simp] theorem mem_some_iff {a b} : b \u2208 (some a : part \u03b1) \u2194 b = a :=\n\u27e8\u03bb\u27e8h, e\u27e9, e.symm, \u03bb e, \u27e8trivial, e.symm\u27e9\u27e9\n\ntheorem eq_some_iff {a : \u03b1} {o : part \u03b1} : o = some a \u2194 a \u2208 o :=\n\u27e8\u03bb e, e.symm \u25b8 mem_some _,\n \u03bb \u27e8h, e\u27e9, e \u25b8 ext' (iff_true_intro h) (\u03bb _ _, rfl)\u27e9\n\ntheorem eq_none_iff {o : part \u03b1} : o = none \u2194 \u2200 a, a \u2209 o :=\n\u27e8\u03bb e, e.symm \u25b8 not_mem_none, \u03bb h, ext (by simpa)\u27e9\n\ntheorem eq_none_iff' {o : part \u03b1} : o = none \u2194 \u00ac o.dom :=\n\u27e8\u03bb e, e.symm \u25b8 id, \u03bb h, eq_none_iff.2 (\u03bb a h', h h'.fst)\u27e9\n\n@[simp] lemma not_none_dom : \u00ac (none : part \u03b1).dom := id\n\n@[simp] lemma some_ne_none (x : \u03b1) : some x \u2260 none :=\nby { intro h, change none.dom, rw [\u2190 h], trivial }\n\n@[simp] lemma none_ne_some (x : \u03b1) : none \u2260 some x :=\n(some_ne_none x).symm\n\nlemma ne_none_iff {o : part \u03b1} : o \u2260 none \u2194 \u2203 x, o = some x :=\nbegin\n  split,\n  { rw [ne, eq_none_iff', not_not], exact \u03bb h, \u27e8o.get h, eq_some_iff.2 (get_mem h)\u27e9 },\n  { rintro \u27e8x, rfl\u27e9, apply some_ne_none }\nend\n\nlemma eq_none_or_eq_some (o : part \u03b1) : o = none \u2228 \u2203 x, o = some x :=\nor_iff_not_imp_left.2 ne_none_iff.1\n\nlemma some_injective : injective (@part.some \u03b1) :=\n\u03bb a b h, congr_fun (eq_of_heq (part.mk.inj h).2) trivial\n\n@[simp] lemma some_inj {a b : \u03b1} : part.some a = some b \u2194 a = b := some_injective.eq_iff\n\n@[simp] lemma some_get {a : part \u03b1} (ha : a.dom) :\n  part.some (part.get a ha) = a :=\neq.symm (eq_some_iff.2 \u27e8ha, rfl\u27e9)\n\nlemma get_eq_iff_eq_some {a : part \u03b1} {ha : a.dom} {b : \u03b1} :\n  a.get ha = b \u2194 a = some b :=\n\u27e8\u03bb h, by simp [h.symm], \u03bb h, by simp [h]\u27e9\n\nlemma get_eq_get_of_eq (a : part \u03b1) (ha : a.dom) {b : part \u03b1} (h : a = b) :\n  a.get ha = b.get (h \u25b8 ha) :=\nby { congr, exact h }\n\nlemma get_eq_iff_mem {o : part \u03b1} {a : \u03b1} (h : o.dom) : o.get h = a \u2194 a \u2208 o :=\n\u27e8\u03bb H, \u27e8h, H\u27e9, \u03bb \u27e8h', H\u27e9, H\u27e9\n\nlemma eq_get_iff_mem {o : part \u03b1} {a : \u03b1} (h : o.dom) : a = o.get h \u2194 a \u2208 o :=\neq_comm.trans (get_eq_iff_mem h)\n\n@[simp] lemma none_to_option [decidable (@none \u03b1).dom] : (none : part \u03b1).to_option = option.none :=\ndif_neg id\n\n@[simp] lemma some_to_option (a : \u03b1) [decidable (some a).dom] :\n  (some a).to_option = option.some a :=\ndif_pos trivial\n\ninstance none_decidable : decidable (@none \u03b1).dom := decidable.false\ninstance some_decidable (a : \u03b1) : decidable (some a).dom := decidable.true\n\n/-- Retrieves the value of `a : part \u03b1` if it exists, and return the provided default value\notherwise. -/\ndef get_or_else (a : part \u03b1) [decidable a.dom] (d : \u03b1) :=\nif ha : a.dom then a.get ha else d\n\nlemma get_or_else_of_dom (a : part \u03b1) (h : a.dom) [decidable a.dom] (d : \u03b1) :\n  get_or_else a d = a.get h := dif_pos h\n\nlemma get_or_else_of_not_dom (a : part \u03b1) (h : \u00ac a.dom) [decidable a.dom] (d : \u03b1) :\n  get_or_else a d = d := dif_neg h\n\n@[simp] lemma get_or_else_none (d : \u03b1) [decidable (none : part \u03b1).dom] : get_or_else none d = d :=\nnone.get_or_else_of_not_dom not_none_dom d\n\n@[simp] lemma get_or_else_some (a : \u03b1) (d : \u03b1) [decidable (some a).dom] :\n  get_or_else (some a) d = a := (some a).get_or_else_of_dom (some_dom a) d\n\n@[simp] theorem mem_to_option {o : part \u03b1} [decidable o.dom] {a : \u03b1} :\n  a \u2208 to_option o \u2194 a \u2208 o :=\nbegin\n  unfold to_option,\n  by_cases h : o.dom; simp [h],\n  { exact \u27e8\u03bb h, \u27e8_, h\u27e9, \u03bb \u27e8_, h\u27e9, h\u27e9 },\n  { exact mt Exists.fst h }\nend\n\nprotected lemma dom.to_option {o : part \u03b1} [decidable o.dom] (h : o.dom) : o.to_option = o.get h :=\ndif_pos h\n\nlemma to_option_eq_none_iff {a : part \u03b1} [decidable a.dom] : a.to_option = option.none \u2194 \u00ac a.dom :=\nne.dite_eq_right_iff $ \u03bb h, option.some_ne_none _\n\n@[simp] lemma elim_to_option {\u03b1 \u03b2 : Type*} (a : part \u03b1) [decidable a.dom] (b : \u03b2) (f : \u03b1 \u2192 \u03b2) :\n  a.to_option.elim b f = if h : a.dom then f (a.get h) else b :=\nbegin\n  split_ifs,\n  { rw h.to_option,\n    refl },\n  { rw part.to_option_eq_none_iff.2 h,\n    refl }\nend\n\n/-- Converts an `option \u03b1` into a `part \u03b1`. -/\ndef of_option : option \u03b1 \u2192 part \u03b1\n| option.none     := none\n| (option.some a) := some a\n\n@[simp] theorem mem_of_option {a : \u03b1} : \u2200 {o : option \u03b1}, a \u2208 of_option o \u2194 a \u2208 o\n| option.none     := \u27e8\u03bb h, h.fst.elim, \u03bb h, option.no_confusion h\u27e9\n| (option.some b) := \u27e8\u03bb h, congr_arg option.some h.snd,\n  \u03bb h, \u27e8trivial, option.some.inj h\u27e9\u27e9\n\n@[simp] theorem of_option_dom {\u03b1} : \u2200 (o : option \u03b1), (of_option o).dom \u2194 o.is_some\n| option.none     := by simp [of_option, none]\n| (option.some a) := by simp [of_option]\n\ntheorem of_option_eq_get {\u03b1} (o : option \u03b1) : of_option o = \u27e8_, @option.get _ o\u27e9 :=\npart.ext' (of_option_dom o) $ \u03bb h\u2081 h\u2082, by cases o; [cases h\u2081, refl]\n\ninstance : has_coe (option \u03b1) (part \u03b1) := \u27e8of_option\u27e9\n\n@[simp] theorem mem_coe {a : \u03b1} {o : option \u03b1} :\n  a \u2208 (o : part \u03b1) \u2194 a \u2208 o := mem_of_option\n\n@[simp] theorem coe_none : (@option.none \u03b1 : part \u03b1) = none := rfl\n@[simp] theorem coe_some (a : \u03b1) : (option.some a : part \u03b1) = some a := rfl\n\n@[elab_as_eliminator] protected lemma induction_on {P : part \u03b1 \u2192 Prop}\n  (a : part \u03b1) (hnone : P none) (hsome : \u2200 a : \u03b1, P (some a)) : P a :=\n(classical.em a.dom).elim\n  (\u03bb h, part.some_get h \u25b8 hsome _)\n  (\u03bb h, (eq_none_iff'.2 h).symm \u25b8 hnone)\n\ninstance of_option_decidable : \u2200 o : option \u03b1, decidable (of_option o).dom\n| option.none     := part.none_decidable\n| (option.some a) := part.some_decidable a\n\n@[simp] theorem to_of_option (o : option \u03b1) : to_option (of_option o) = o :=\nby cases o; refl\n\n@[simp] theorem of_to_option (o : part \u03b1) [decidable o.dom] : of_option (to_option o) = o :=\next $ \u03bb a, mem_of_option.trans mem_to_option\n\n/-- `part \u03b1` is (classically) equivalent to `option \u03b1`. -/\nnoncomputable def equiv_option : part \u03b1 \u2243 option \u03b1 :=\nby haveI := classical.dec; exact\n\u27e8\u03bb o, to_option o, of_option, \u03bb o, of_to_option o,\n \u03bb o, eq.trans (by dsimp; congr) (to_of_option o)\u27e9\n\n/-- We give `part \u03b1` the order where everything is greater than `none`. -/\ninstance : partial_order (part \u03b1) :=\n{ le := \u03bb x y, \u2200 i, i \u2208 x \u2192 i \u2208 y,\n  le_refl := \u03bb x y, id,\n  le_trans := \u03bb x y z f g i, g _ \u2218 f _,\n  le_antisymm := \u03bb x y f g, part.ext $ \u03bb z, \u27e8f _, g _\u27e9 }\n\ninstance : order_bot (part \u03b1) :=\n{ bot := none,\n  bot_le := by { introv x, rintro \u27e8\u27e8_\u27e9,_\u27e9, } }\n\nlemma le_total_of_le_of_le {x y : part \u03b1} (z : part \u03b1) (hx : x \u2264 z) (hy : y \u2264 z) :\n  x \u2264 y \u2228 y \u2264 x :=\nbegin\n  rcases part.eq_none_or_eq_some x with h | \u27e8b, h\u2080\u27e9,\n  { rw h, left, apply order_bot.bot_le _ },\n  right, intros b' h\u2081,\n  rw part.eq_some_iff at h\u2080,\n  replace hx := hx _ h\u2080, replace hy := hy _ h\u2081,\n  replace hx := part.mem_unique hx hy, subst hx,\n  exact h\u2080\nend\n\n/-- `assert p f` is a bind-like operation which appends an additional condition\n  `p` to the domain and uses `f` to produce the value. -/\ndef assert (p : Prop) (f : p \u2192 part \u03b1) : part \u03b1 :=\n\u27e8\u2203 h : p, (f h).dom, \u03bbha, (f ha.fst).get ha.snd\u27e9\n\n/-- The bind operation has value `g (f.get)`, and is defined when all the\n  parts are defined. -/\nprotected def bind (f : part \u03b1) (g : \u03b1 \u2192 part \u03b2) : part \u03b2 :=\nassert (dom f) (\u03bbb, g (f.get b))\n\n/-- The map operation for `part` just maps the value and maintains the same domain. -/\n@[simps] def map (f : \u03b1 \u2192 \u03b2) (o : part \u03b1) : part \u03b2 :=\n\u27e8o.dom, f \u2218 o.get\u27e9\n\ntheorem mem_map (f : \u03b1 \u2192 \u03b2) {o : part \u03b1} :\n  \u2200 {a}, a \u2208 o \u2192 f a \u2208 map f o\n| _ \u27e8h, rfl\u27e9 := \u27e8_, rfl\u27e9\n\n@[simp] theorem mem_map_iff (f : \u03b1 \u2192 \u03b2) {o : part \u03b1} {b} :\n  b \u2208 map f o \u2194 \u2203 a \u2208 o, f a = b :=\n\u27e8match b with _, \u27e8h, rfl\u27e9 := \u27e8_, \u27e8_, rfl\u27e9, rfl\u27e9 end,\n \u03bb \u27e8a, h\u2081, h\u2082\u27e9, h\u2082 \u25b8 mem_map f h\u2081\u27e9\n\n@[simp] theorem map_none (f : \u03b1 \u2192 \u03b2) :\n  map f none = none := eq_none_iff.2 $ \u03bb a, by simp\n\n@[simp] theorem map_some (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (some a) = some (f a) :=\neq_some_iff.2 $ mem_map f $ mem_some _\n\ntheorem mem_assert {p : Prop} {f : p \u2192 part \u03b1}\n  : \u2200 {a} (h : p), a \u2208 f h \u2192 a \u2208 assert p f\n| _ x \u27e8h, rfl\u27e9 := \u27e8\u27e8x, h\u27e9, rfl\u27e9\n\n@[simp] theorem mem_assert_iff {p : Prop} {f : p \u2192 part \u03b1} {a} :\n  a \u2208 assert p f \u2194 \u2203 h : p, a \u2208 f h :=\n\u27e8match a with _, \u27e8h, rfl\u27e9 := \u27e8_, \u27e8_, rfl\u27e9\u27e9 end,\n \u03bb \u27e8a, h\u27e9, mem_assert _ h\u27e9\n\nlemma assert_pos {p : Prop} {f : p \u2192 part \u03b1} (h : p) :\n  assert p f = f h :=\nbegin\n  dsimp [assert],\n  cases h' : f h,\n  simp only [h', h, true_and, iff_self, exists_prop_of_true, eq_iff_iff],\n  apply function.hfunext,\n  { simp only [h, h', exists_prop_of_true] },\n  { cc }\nend\n\nlemma assert_neg {p : Prop} {f : p \u2192 part \u03b1} (h : \u00ac p) :\n  assert p f = none :=\nbegin\n  dsimp [assert,none], congr,\n  { simp only [h, not_false_iff, exists_prop_of_false] },\n  { apply function.hfunext,\n    { simp only [h, not_false_iff, exists_prop_of_false] },\n    cc },\nend\n\ntheorem mem_bind {f : part \u03b1} {g : \u03b1 \u2192 part \u03b2} :\n  \u2200 {a b}, a \u2208 f \u2192 b \u2208 g a \u2192 b \u2208 f.bind g\n| _ _ \u27e8h, rfl\u27e9 \u27e8h\u2082, rfl\u27e9 := \u27e8\u27e8h, h\u2082\u27e9, rfl\u27e9\n\n@[simp] theorem mem_bind_iff {f : part \u03b1} {g : \u03b1 \u2192 part \u03b2} {b} :\n  b \u2208 f.bind g \u2194 \u2203 a \u2208 f, b \u2208 g a :=\n\u27e8match b with _, \u27e8\u27e8h\u2081, h\u2082\u27e9, rfl\u27e9 := \u27e8_, \u27e8_, rfl\u27e9, \u27e8_, rfl\u27e9\u27e9 end,\n \u03bb \u27e8a, h\u2081, h\u2082\u27e9, mem_bind h\u2081 h\u2082\u27e9\n\nprotected lemma dom.bind {o : part \u03b1} (h : o.dom) (f : \u03b1 \u2192 part \u03b2) : o.bind f = f (o.get h) :=\nbegin\n  ext b,\n  simp only [part.mem_bind_iff, exists_prop],\n  refine \u27e8_, \u03bb hb, \u27e8o.get h, part.get_mem _, hb\u27e9\u27e9,\n  rintro \u27e8a, ha, hb\u27e9,\n  rwa part.get_eq_of_mem ha,\nend\n\nlemma dom.of_bind {f : \u03b1 \u2192 part \u03b2} {a : part \u03b1} (h : (a.bind f).dom) : a.dom := h.some\n\n@[simp] theorem bind_none (f : \u03b1 \u2192 part \u03b2) :\n  none.bind f = none := eq_none_iff.2 $ \u03bb a, by simp\n\n@[simp] theorem bind_some (a : \u03b1) (f : \u03b1 \u2192 part \u03b2) :\n  (some a).bind f = f a := ext $ by simp\n\ntheorem bind_of_mem {o : part \u03b1} {a : \u03b1} (h : a \u2208 o) (f : \u03b1 \u2192 part \u03b2) :\n  o.bind f = f a :=\nby rw [eq_some_iff.2 h, bind_some]\n\ntheorem bind_some_eq_map (f : \u03b1 \u2192 \u03b2) (x : part \u03b1) :\n  x.bind (some \u2218 f) = map f x :=\next $ by simp [eq_comm]\n\nlemma bind_to_option (f : \u03b1 \u2192 part \u03b2) (o : part \u03b1) [decidable o.dom] [\u03a0 a, decidable (f a).dom]\n  [decidable (o.bind f).dom] :\n  (o.bind f).to_option = o.to_option.elim option.none (\u03bb a, (f a).to_option) :=\nbegin\n  by_cases o.dom,\n  { simp_rw [h.to_option, h.bind],\n    refl },\n  { rw part.to_option_eq_none_iff.2 h,\n    exact part.to_option_eq_none_iff.2 (\u03bb ho, h ho.of_bind) }\nend\n\ntheorem bind_assoc {\u03b3} (f : part \u03b1) (g : \u03b1 \u2192 part \u03b2) (k : \u03b2 \u2192 part \u03b3) :\n  (f.bind g).bind k = f.bind (\u03bb x, (g x).bind k) :=\next $ \u03bb a, by simp; exact\n \u27e8\u03bb \u27e8_, \u27e8_, h\u2081, h\u2082\u27e9, h\u2083\u27e9, \u27e8_, h\u2081, _, h\u2082, h\u2083\u27e9,\n  \u03bb \u27e8_, h\u2081, _, h\u2082, h\u2083\u27e9, \u27e8_, \u27e8_, h\u2081, h\u2082\u27e9, h\u2083\u27e9\u27e9\n\n@[simp] theorem bind_map {\u03b3} (f : \u03b1 \u2192 \u03b2) (x) (g : \u03b2 \u2192 part \u03b3) :\n  (map f x).bind g = x.bind (\u03bb y, g (f y)) :=\nby rw [\u2190 bind_some_eq_map, bind_assoc]; simp\n\n@[simp] theorem map_bind {\u03b3} (f : \u03b1 \u2192 part \u03b2) (x : part \u03b1) (g : \u03b2 \u2192 \u03b3) :\n  map g (x.bind f) = x.bind (\u03bb y, map g (f y)) :=\nby rw [\u2190 bind_some_eq_map, bind_assoc]; simp [bind_some_eq_map]\n\ntheorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (o : part \u03b1) :\n  map g (map f o) = map (g \u2218 f) o :=\nby rw [\u2190 bind_some_eq_map, bind_map, bind_some_eq_map]\n\ninstance : monad part :=\n{ pure := @some,\n  map := @map,\n  bind := @part.bind }\n\ninstance : is_lawful_monad part :=\n{ bind_pure_comp_eq_map := @bind_some_eq_map,\n  id_map := \u03bb \u03b2 f, by cases f; refl,\n  pure_bind := @bind_some,\n  bind_assoc := @bind_assoc }\n\ntheorem map_id' {f : \u03b1 \u2192 \u03b1} (H : \u2200 (x : \u03b1), f x = x) (o) : map f o = o :=\nby rw [show f = id, from funext H]; exact id_map o\n\n@[simp] theorem bind_some_right (x : part \u03b1) : x.bind some = x :=\nby rw [bind_some_eq_map]; simp [map_id']\n\n@[simp] theorem pure_eq_some (a : \u03b1) : pure a = some a := rfl\n@[simp] theorem ret_eq_some (a : \u03b1) : return a = some a := rfl\n\n@[simp] theorem map_eq_map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (o : part \u03b1) :\n  f <$> o = map f o := rfl\n\n@[simp] theorem bind_eq_bind {\u03b1 \u03b2} (f : part \u03b1) (g : \u03b1 \u2192 part \u03b2) :\n  f >>= g = f.bind g := rfl\n\nlemma bind_le {\u03b1} (x : part \u03b1) (f : \u03b1 \u2192 part \u03b2) (y : part \u03b2) :\n  x >>= f \u2264 y \u2194 (\u2200 a, a \u2208 x \u2192 f a \u2264 y) :=\nbegin\n  split; intro h,\n  { intros a h' b, replace h := h b,\n    simp only [and_imp, exists_prop, bind_eq_bind, mem_bind_iff, exists_imp_distrib] at h,\n    apply h _ h' },\n  { intros b h',\n    simp only [exists_prop, bind_eq_bind, mem_bind_iff] at h',\n    rcases h' with \u27e8a,h\u2080,h\u2081\u27e9, apply h _ h\u2080 _ h\u2081 },\nend\n\ninstance : monad_fail part :=\n{ fail := \u03bb_ _, none, ..part.monad }\n\n/-- `restrict p o h` replaces the domain of `o` with `p`, and is well defined when\n  `p` implies `o` is defined. -/\ndef restrict (p : Prop) (o : part \u03b1) (H : p \u2192 o.dom) : part \u03b1 :=\n\u27e8p, \u03bbh, o.get (H h)\u27e9\n\n@[simp]\ntheorem mem_restrict (p : Prop) (o : part \u03b1) (h : p \u2192 o.dom) (a : \u03b1) :\n  a \u2208 restrict p o h \u2194 p \u2227 a \u2208 o :=\nbegin\n  dsimp [restrict, mem_eq], split,\n  { rintro \u27e8h\u2080, h\u2081\u27e9, exact \u27e8h\u2080, \u27e8_, h\u2081\u27e9\u27e9 },\n  rintro \u27e8h\u2080, h\u2081, h\u2082\u27e9, exact \u27e8h\u2080, h\u2082\u27e9\nend\n\n/-- `unwrap o` gets the value at `o`, ignoring the condition. This function is unsound. -/\nmeta def unwrap (o : part \u03b1) : \u03b1 := o.get undefined\n\ntheorem assert_defined {p : Prop} {f : p \u2192 part \u03b1} :\n  \u2200 (h : p), (f h).dom \u2192 (assert p f).dom := exists.intro\n\ntheorem bind_defined {f : part \u03b1} {g : \u03b1 \u2192 part \u03b2} :\n  \u2200 (h : f.dom), (g (f.get h)).dom \u2192 (f.bind g).dom := assert_defined\n\n@[simp] theorem bind_dom {f : part \u03b1} {g : \u03b1 \u2192 part \u03b2} :\n  (f.bind g).dom \u2194 \u2203 h : f.dom, (g (f.get h)).dom := iff.rfl\n\nsection instances\n\n/- We define several instances for constants and operations on `part \u03b1` inherited from `\u03b1`. -/\n\n@[to_additive] instance [has_one \u03b1] : has_one (part \u03b1) := { one := pure 1 }\n@[to_additive] instance [has_mul \u03b1] : has_mul (part \u03b1) := { mul := \u03bb a b, (*) <$> a <*> b }\n@[to_additive] instance [has_inv \u03b1] : has_inv (part \u03b1) := { inv := map has_inv.inv }\n@[to_additive] instance [has_div \u03b1] : has_div (part \u03b1) := { div := \u03bb a b, (/) <$> a <*> b }\ninstance [has_mod \u03b1] : has_mod (part \u03b1) := { mod := \u03bb a b, (%) <$> a <*> b }\ninstance [has_append \u03b1] : has_append (part \u03b1) := { append := \u03bb a b, (++) <$> a <*> b }\ninstance [has_inter \u03b1] : has_inter (part \u03b1) := { inter := \u03bb a b, (\u2229) <$> a <*> b }\ninstance [has_union \u03b1] : has_union (part \u03b1) := { union := \u03bb a b, (\u222a) <$> a <*> b }\ninstance [has_sdiff \u03b1] : has_sdiff (part \u03b1) := { sdiff := \u03bb a b, (\\) <$> a <*> b }\n\n@[to_additive]\nlemma one_mem_one [has_one \u03b1] : (1 : \u03b1) \u2208 (1 : part \u03b1) := \u27e8trivial, rfl\u27e9\n\n@[to_additive]\nlemma mul_mem_mul [has_mul \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma * mb \u2208 a * b := by tidy\n\n@[to_additive]\nlemma left_dom_of_mul_dom [has_mul \u03b1] {a b : part \u03b1} (hab : dom (a * b)) :\n  a.dom := by tidy\n\n@[to_additive]\n\n\n@[simp, to_additive]\nlemma mul_get_eq [has_mul \u03b1] (a b : part \u03b1) (hab : dom (a * b)) :\n  (a * b).get hab = a.get (left_dom_of_mul_dom hab) * b.get (right_dom_of_mul_dom hab) :=\nby tidy\n\n@[to_additive]\nlemma some_mul_some [has_mul \u03b1] (a b : \u03b1) : some a * some b = some (a * b) := by tidy\n\n@[to_additive]\nlemma inv_mem_inv [has_inv \u03b1] (a : part \u03b1) (ma : \u03b1) (ha : ma \u2208 a) : ma\u207b\u00b9 \u2208 a\u207b\u00b9 := by tidy\n\n@[to_additive]\nlemma inv_some [has_inv \u03b1] (a : \u03b1) : (some a)\u207b\u00b9 = some (a\u207b\u00b9) := rfl\n\n@[to_additive]\nlemma div_mem_div [has_div \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma / mb \u2208 a / b := by tidy\n\n@[to_additive]\nlemma left_dom_of_div_dom [has_div \u03b1] {a b : part \u03b1} (hab : dom (a / b)) :\n  a.dom := by tidy\n\n@[to_additive]\nlemma right_dom_of_div_dom [has_div \u03b1] {a b : part \u03b1} (hab : dom (a / b)) :\n  b.dom := by tidy\n\n@[simp, to_additive]\nlemma div_get_eq [has_div \u03b1] (a b : part \u03b1) (hab : dom (a / b)) :\n  (a / b).get hab = a.get (left_dom_of_div_dom hab) / b.get (right_dom_of_div_dom hab) :=\nby tidy\n\n@[to_additive]\nlemma some_div_some [has_div \u03b1] (a b : \u03b1) : some a / some b = some (a / b) := by tidy\n\nlemma mod_mem_mod [has_mod \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma % mb \u2208 a % b := by tidy\n\nlemma left_dom_of_mod_dom [has_mod \u03b1] {a b : part \u03b1} (hab : dom (a % b)) :\n  a.dom := by tidy\n\nlemma right_dom_of_mod_dom [has_mod \u03b1] {a b : part \u03b1} (hab : dom (a % b)) :\n  b.dom := by tidy\n\n@[simp]\nlemma mod_get_eq [has_mod \u03b1] (a b : part \u03b1) (hab : dom (a % b)) :\n  (a % b).get hab = a.get (left_dom_of_mod_dom hab) % b.get (right_dom_of_mod_dom hab) :=\nby tidy\n\nlemma some_mod_some [has_mod \u03b1] (a b : \u03b1) : some a % some b = some (a % b) := by tidy\n\nlemma append_mem_append [has_append \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma ++ mb \u2208 a ++ b := by tidy\n\nlemma left_dom_of_append_dom [has_append \u03b1] {a b : part \u03b1} (hab : dom (a ++ b)) :\n  a.dom := by tidy\n\nlemma right_dom_of_append_dom [has_append \u03b1] {a b : part \u03b1} (hab : dom (a ++ b)) :\n  b.dom := by tidy\n\n@[simp]\nlemma append_get_eq [has_append \u03b1] (a b : part \u03b1) (hab : dom (a ++ b)) :\n  (a ++ b).get hab = a.get (left_dom_of_append_dom hab) ++ b.get (right_dom_of_append_dom hab) :=\nby tidy\n\nlemma some_append_some [has_append \u03b1] (a b : \u03b1) : some a ++ some b = some (a ++ b) := by tidy\n\nlemma inter_mem_inter [has_inter \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma \u2229 mb \u2208 a \u2229 b := by tidy\n\nlemma left_dom_of_inter_dom [has_inter \u03b1] {a b : part \u03b1} (hab : dom (a \u2229 b)) :\n  a.dom := by tidy\n\nlemma right_dom_of_inter_dom [has_inter \u03b1] {a b : part \u03b1} (hab : dom (a \u2229 b)) :\n  b.dom := by tidy\n\n@[simp]\nlemma inter_get_eq [has_inter \u03b1] (a b : part \u03b1) (hab : dom (a \u2229 b)) :\n  (a \u2229 b).get hab = a.get (left_dom_of_inter_dom hab) \u2229 b.get (right_dom_of_inter_dom hab) :=\nby tidy\n\nlemma some_inter_some [has_inter \u03b1] (a b : \u03b1) : some a \u2229 some b = some (a \u2229 b) := by tidy\n\nlemma union_mem_union [has_union \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma \u222a mb \u2208 a \u222a b := by tidy\n\nlemma left_dom_of_union_dom [has_union \u03b1] {a b : part \u03b1} (hab : dom (a \u222a b)) :\n  a.dom := by tidy\n\nlemma right_dom_of_union_dom [has_union \u03b1] {a b : part \u03b1} (hab : dom (a \u222a b)) :\n  b.dom := by tidy\n\n@[simp]\nlemma union_get_eq [has_union \u03b1] (a b : part \u03b1) (hab : dom (a \u222a b)) :\n  (a \u222a b).get hab = a.get (left_dom_of_union_dom hab) \u222a b.get (right_dom_of_union_dom hab) :=\nby tidy\n\nlemma some_union_some [has_union \u03b1] (a b : \u03b1) : some a \u222a some b = some (a \u222a b) := by tidy\n\nlemma sdiff_mem_sdiff [has_sdiff \u03b1] (a b : part \u03b1) (ma mb : \u03b1) (ha : ma \u2208 a) (hb : mb \u2208 b) :\n  ma \\ mb \u2208 a \\ b := by tidy\n\nlemma left_dom_of_sdiff_dom [has_sdiff \u03b1] {a b : part \u03b1} (hab : dom (a \\ b)) :\n  a.dom := by tidy\n\nlemma right_dom_of_sdiff_dom [has_sdiff \u03b1] {a b : part \u03b1} (hab : dom (a \\ b)) :\n  b.dom := by tidy\n\n@[simp]\nlemma sdiff_get_eq [has_sdiff \u03b1] (a b : part \u03b1) (hab : dom (a \\ b)) :\n  (a \\ b).get hab = a.get (left_dom_of_sdiff_dom hab) \\ b.get (right_dom_of_sdiff_dom hab) :=\nby tidy\n\nlemma some_sdiff_some [has_sdiff \u03b1] (a b : \u03b1) : some a \\ some b = some (a \\ b) := by tidy\n\nend instances\n\nend part\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/part.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6791786861878392, "lm_q1q2_score": 0.37920379447590474}}
{"text": "/- Checking 57012 declarations (plus 54043 automatically generated ones) in mathlib (only in imported files) -/\nimport all\n\n\n/- The `generalisation_linter` linter reports: -/\n/- typeclass generalisations may be possible: -/\n-- algebra\\add_torsor.lean\n#print vadd_eq_add /- _inst_1: add_group \u219d has_add has_vadd\n -/\n#print vsub_eq_sub /- _inst_1: add_group \u219d has_sub has_vsub\n -/\n#print vadd_comm /- _inst_1: add_comm_monoid \u219d add_monoid add_comm_semigroup has_vadd\n -/\n#print set.has_vsub /- T: add_torsor \u219d has_vsub\n -/\n#print vsub_sub_vsub_cancel_left /- _inst_1: add_comm_group \u219d has_vsub add_comm_semigroup add_group\n -/\n\n-- algebra\\algebra\\basic.lean\n#print algebra.of_semimodule' /- _inst_3: semimodule \u219d\n -/\n#print algebra.of_semimodule /- _inst_3: semimodule \u219d\n -/\n#print algebra.bit0_smul_one /- _inst_4: algebra \u219d\n -/\n#print algebra.bit0_smul_bit0 /- _inst_4: algebra \u219d\n -/\n#print algebra.bit0_smul_bit1 /- _inst_4: algebra \u219d\n -/\n#print algebra.bit1_smul_bit1 /- _inst_4: algebra \u219d\n -/\n#print algebra.id.smul_eq_mul /- _inst_1: comm_semiring \u219d has_mul\n -/\n#print algebra.mem_algebra_map_submonoid_of_mem /- _inst_2: comm_semiring \u219d semiring\n -/\n#print algebra.mul_sub_algebra_map_commutes /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print module.endomorphism_algebra /- _inst_3: semimodule \u219d\n -/\n#print module.algebra_map_End_eq_smul_id /- _inst_3: semimodule \u219d algebra\n -/\n#print module.algebra_map_End_apply /- _inst_3: semimodule \u219d algebra\n -/\n#print module.ker_algebra_map_End /- _inst_6: vector_space \u219d algebra\n -/\n#print matrix_algebra /- _inst_1: decidable_eq \u219d\n -/\n#print alg_hom.map_inv /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print alg_hom.map_div /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print alg_equiv.map_neg /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print alg_equiv.map_sub /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print matrix.alg_hom_map_one /- _inst_7: decidable_eq \u219d\n -/\n#print matrix.alg_equiv_map_one /- _inst_7: decidable_eq \u219d\n -/\n#print algebra.linear_map.semimodule' /- _inst_6: semimodule \u219d\n -/\n#print span_nat_eq_add_group_closure /- _inst_1: semiring \u219d add_comm_monoid\n -/\n#print span_int_eq_add_group_closure /- _inst_1: ring \u219d add_comm_group\n -/\n#print algebra_compatible_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d has_scalar\n_inst_7: is_scalar_tower \u219d\n -/\n#print algebra_map_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.to_smul_comm_class /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.to_smul_comm_class' /- _inst_3: algebra \u219d\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print smul_algebra_smul_comm /- _inst_3: algebra \u219d\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print linear_map.coe_is_scalar_tower /- _inst_3: algebra \u219d has_scalar\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print linear_map.coe_restrict_scalars_eq_coe /- _inst_3: algebra \u219d has_scalar\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print linear_map.coe_coe_is_scalar_tower /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print linear_map.lto_fun /- _inst_14: semimodule \u219d\n -/\n#print restrict_scalars.module_orig /- I: semimodule \u219d\n -/\n#print restrict_scalars.semimodule /- _inst_5: semimodule \u219d\n -/\n#print restrict_scalars_smul_def /- _inst_5: semimodule \u219d\n -/\n#print restrict_scalars.is_scalar_tower /- _inst_5: semimodule \u219d\n -/\n#print submodule.restricted_module /- _inst_5: semimodule \u219d\n -/\n#print submodule.restricted_module_is_scalar_tower /- _inst_5: semimodule \u219d\n -/\n#print submodule.restrict_scalars_carrier /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars_mem /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars_inj /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars_bot /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.restrict_scalars_top /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print linear_map.ker_restrict_scalars /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print linear_map.is_scalar_tower_extend_scalars /- _inst_3: algebra \u219d\n_inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print linear_map.smul_algebra_right /- _inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print linear_map.smul_algebra_right_apply /- _inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n\n-- algebra\\algebra\\operations.lean\n#print submodule.mul_mem_mul_rev /- _inst_2: comm_semiring \u219d comm_semigroup semiring\n -/\n#print submodule.map_div /- _inst_4: comm_ring \u219d comm_semiring\n -/\n\n-- algebra\\algebra\\subalgebra.lean\n#print subalgebra.neg_mem /- _inst_6: comm_ring \u219d ring comm_semiring\n_inst_7: ring \u219d semiring add_comm_group\n -/\n#print subalgebra.ring /- _inst_6: comm_ring \u219d is_subring comm_semiring\n -/\n#print subalgebra.to_submodule.is_subring /- _inst_6: comm_ring \u219d is_subring comm_semiring\n -/\n#print algebra.bijective_algebra_map_iff /- _inst_6: field \u219d division_ring comm_semiring\n -/\n\n-- algebra\\archimedean.lean\n#print pow_unbounded_of_one_lt /- _inst_1: linear_ordered_ring \u219d ordered_add_comm_group linear_ordered_semiring\n -/\n#print exists_int_gt /- _inst_1: linear_ordered_ring \u219d has_neg linear_ordered_semiring\n -/\n#print sub_floor_div_mul_nonneg /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_ring\n -/\n#print sub_floor_div_mul_lt /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_ring\n -/\n#print exists_rat_gt /- _inst_1: linear_ordered_field \u219d division_ring linear_ordered_semiring\n -/\n#print exists_rat_lt /- _inst_1: linear_ordered_field \u219d linear_ordered_ring division_ring\n -/\n#print round /- _inst_1: linear_ordered_field \u219d linear_ordered_ring has_div\n -/\n\n-- algebra\\associated.lean\n#print is_unit_iff_dvd_one /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_and_not_dvd_iff /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n#print pow_dvd_pow_iff /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid\n -/\n#print prime /- _inst_1: comm_monoid_with_zero \u219d monoid has_zero\n -/\n#print left_dvd_or_dvd_right_of_dvd_prime_mul /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n#print not_irreducible_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print irreducible_of_prime /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n#print succ_dvd_or_succ_dvd_of_succ_sum_dvd_mul /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n#print associated_zero_iff_eq_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print associated_mul_mul /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_iff_dvd_of_rel_left /- _inst_1: comm_monoid_with_zero \u219d monoid\n -/\n#print dvd_iff_dvd_of_rel_right /- _inst_1: comm_monoid_with_zero \u219d monoid\n -/\n#print eq_zero_iff_of_associated /- _inst_1: comm_monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print irreducible_of_associated /- _inst_1: comm_monoid_with_zero \u219d monoid\n -/\n#print associated_mul_left_cancel /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_semigroup\n -/\n#print associates.mk_one /- _inst_1: comm_monoid \u219d monoid\n -/\n#print associates.rel_associated_iff_map_eq_map /- _inst_1: comm_monoid \u219d monoid\n -/\n#print associates.mk_eq_zero /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero\n -/\n#print associates.nontrivial /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero\n -/\n#print associates.exists_non_zero_rep /- _inst_1: comm_monoid_with_zero \u219d monoid has_zero\n -/\n#print associates.dvd_of_mk_le_mk /- _inst_1: comm_monoid_with_zero \u219d comm_monoid\n -/\n#print associates.mk_le_mk_of_dvd /- _inst_1: comm_monoid_with_zero \u219d comm_monoid\n -/\n#print associates.no_zero_divisors /- _inst_1: comm_cancel_monoid_with_zero \u219d monoid_with_zero comm_monoid no_zero_divisors\n -/\n#print associates.irreducible_iff_prime_iff /- _inst_1: comm_cancel_monoid_with_zero \u219d comm_monoid_with_zero\n -/\n#print associates.eq_of_mul_eq_mul_left /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n\n-- algebra\\big_operators\\basic.lean\n#print ring_hom.map_multiset_prod /- _inst_1: comm_semiring \u219d comm_monoid semiring\n_inst_2: comm_semiring \u219d comm_monoid semiring\n -/\n#print ring_hom.map_prod /- _inst_1: comm_semiring \u219d comm_monoid semiring\n_inst_2: comm_semiring \u219d comm_monoid semiring\n -/\n#print finset.sum_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_insert_of_eq_one_if_not_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_insert_of_eq_zero_if_not_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_insert_one /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_insert_zero /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_pair /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_pair /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_image /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_image /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_union_inter /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_union_inter /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_sdiff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_sdiff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_sum_elim /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_sum_elim /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_bind /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_bind /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_fiberwise_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_fiberwise_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_image' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_image' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_extend_by_one /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_extend_by_zero /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_dite_eq /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_dite_eq /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_dite_eq' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_dite_eq' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_ite_eq /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_ite_eq /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_ite_eq' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_ite_eq' /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_subset_one_on_sdiff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_subset_zero_on_sdiff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_multiset_map_count /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_multiset_count /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_comp /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_piecewise /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_piecewise /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_inter_add_sum_diff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_inter_mul_prod_diff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.mul_prod_diff_singleton /- _inst_2: decidable_eq \u219d\n -/\n#print finset.add_sum_diff_singleton /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_cancels_of_partition_cancels /- _inst_2: decidable_rel \u219d\n -/\n#print finset.prod_cancels_of_partition_cancels /- _inst_2: decidable_rel \u219d\n -/\n#print finset.sum_update_of_not_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_update_of_not_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_update_of_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_erase /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_erase /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_pow_boole /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_add_prod_eq /- _inst_1: comm_semiring \u219d comm_monoid distrib\n -/\n#print finset.sum_update_of_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_comp /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_inv_distrib /- _inst_1: comm_group \u219d has_inv is_group_hom comm_monoid\n -/\n#print finset.sum_neg_distrib /- _inst_1: add_comm_group \u219d add_comm_monoid has_neg is_add_group_hom\n -/\n#print finset.card_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_bind_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_eq_sum_card_fiberwise /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_eq_sum_card_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.gsmul_sum /- _inst_1: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print finset.prod_eq_zero /- _inst_1: comm_monoid_with_zero \u219d comm_monoid mul_zero_class\n -/\n#print finset.prod_eq_zero_iff /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero comm_monoid\n -/\n#print finset.prod_inv_distrib' /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_monoid_with_zero\n -/\n#print list.prod_to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print list.sum_to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_sum_count_eq /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_sum' /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_sum_count_smul_eq /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.exists_smul_of_dvd_count /- _inst_1: decidable_eq \u219d\n -/\n#print int.coe_prod /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- algebra\\big_operators\\finsupp.lean\n#print finsupp.sum_apply' /- _inst_1: add_comm_monoid \u219d has_zero\n -/\n\n-- algebra\\big_operators\\intervals.lean\n#print finset.sum_Ico_eq_add_neg /- _inst_2: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print finset.prod_Ico_eq_mul_inv /- _inst_2: comm_group \u219d group comm_monoid\n -/\n\n-- algebra\\big_operators\\order.lean\n#print finset.abs_sum_le_sum_abs /- _inst_1: linear_ordered_field \u219d linear_ordered_add_comm_group\n -/\n#print finset.abs_prod /- _inst_1: linear_ordered_comm_ring \u219d comm_monoid linear_ordered_ring\n -/\n#print finset.card_le_mul_card_image_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.card_le_mul_card_image /- _inst_2: decidable_eq \u219d\n -/\n#print finset.mul_card_image_le_card_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.mul_card_image_le_card /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_fiberwise_le_sum_of_sum_fiber_nonneg /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_le_sum_fiberwise_of_sum_fiber_nonpos /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_lt_sum_of_subset /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_lt_of_sum_lt /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print finset.exists_le_of_sum_le /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d linear_order ordered_cancel_add_comm_monoid\n -/\n#print finset.exists_pos_of_sum_zero_of_exists_nonzero /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d linear_order ordered_cancel_add_comm_monoid\n -/\n#print finset.prod_nonneg /- _inst_1: linear_ordered_comm_ring \u219d comm_monoid ordered_semiring\n -/\n#print finset.prod_pos /- _inst_1: linear_ordered_comm_ring \u219d nontrivial comm_monoid ordered_semiring\n -/\n\n-- algebra\\big_operators\\pi.lean\n#print finset.univ_sum_single /- _inst_1: decidable_eq \u219d\n -/\n#print add_monoid_hom.functions_ext /- _inst_1: decidable_eq \u219d\n -/\n#print ring_hom.functions_ext /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\big_operators\\ring.lean\n#print finset.sum_mul /- _inst_1: semiring \u219d add_comm_monoid has_mul\n -/\n#print finset.mul_sum /- _inst_1: semiring \u219d add_comm_monoid has_mul\n -/\n#print finset.sum_mul_boole /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_boole_mul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_div /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print finset.prod_sum /- _inst_1: comm_semiring \u219d comm_monoid semiring\n_inst_2: decidable_eq \u219d\n -/\n#print finset.sum_mul_sum /- _inst_1: comm_semiring \u219d semiring\n -/\n#print finset.prod_pow_eq_pow_sum /- _inst_1: comm_semiring \u219d comm_monoid\n -/\n#print finset.prod_powerset_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sum_powerset_insert /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\category\\Algebra\\basic.lean\n#print Algebra.id_apply /- _inst_1: comm_ring \u219d ring\n -/\n#print Algebra.coe_comp /- _inst_1: comm_ring \u219d ring\n -/\n\n-- algebra\\category\\Group\\biproducts.lean\n#print AddCommGroup.category_theory.limits.has_biproduct /- _inst_1: decidable_eq \u219d\n -/\n#print AddCommGroup.biproduct_iso_pi /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\category\\Module\\basic.lean\n#print Module.has_coe /- _inst_3: module \u219d\n -/\n\n-- algebra\\char_p.lean\n#print char_p.cast_card_eq_zero /- _inst_1: ring \u219d add_group semiring\n -/\n#print char_p.int_cast_eq_zero_iff /- _inst_1: ring \u219d add_group semiring\n -/\n#print add_pow_char_of_commute /- _inst_1: ring \u219d semiring add_left_cancel_semigroup\n -/\n#print add_pow_char /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print add_pow_char_pow /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print sub_pow_char /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print sub_pow_char_pow /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print char_p.neg_one_ne_one /- _inst_1: ring \u219d add_group semiring\n -/\n#print ring_hom.char_p_iff_char_p /- _inst_1: field \u219d division_ring\n_inst_2: field \u219d nontrivial semiring\n -/\n#print frobenius_inj /- _inst_1: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print char_p.char_p_to_char_zero /- _inst_1: ring \u219d add_left_cancel_monoid semiring\n -/\n#print char_p.cast_eq_mod /- _inst_1: ring \u219d semiring\n -/\n#print char_p.char_ne_one /- _inst_1: integral_domain \u219d nontrivial semiring\n -/\n#print char_p.char_is_prime_of_two_le /- _inst_1: integral_domain \u219d no_zero_divisors semiring\n -/\n#print char_p.false_of_nontrivial_of_char_one /- _inst_3: char_p \u219d subsingleton\n -/\n#print char_p_of_ne_zero /- _inst_1: comm_ring \u219d ring\n -/\n#print char_p_of_prime_pow_injective /- _inst_1: comm_ring \u219d ring\n -/\n\n-- algebra\\char_zero.lean\n#print char_zero_of_inj_zero /- _inst_1: add_left_cancel_monoid \u219d add_monoid add_left_cancel_semigroup\n -/\n#print half_add_self /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n\n-- algebra\\continued_fractions\\basic.lean\n#print generalized_continued_fraction.seq.coe_to_seq /- _inst_1: has_coe \u219d has_lift_t\n -/\n#print generalized_continued_fraction.next_numerator /- _inst_1: division_ring \u219d has_add has_mul\n -/\n#print generalized_continued_fraction.next_denominator /- _inst_1: division_ring \u219d has_add has_mul\n -/\n\n-- algebra\\continued_fractions\\computation\\basic.lean\n#print generalized_continued_fraction.int_fract_pair.of /- _inst_1: linear_ordered_field \u219d linear_ordered_ring\n -/\n\n-- algebra\\continued_fractions\\computation\\correctness_terminating.lean\n#print generalized_continued_fraction.comp_exact_value /- _inst_1: linear_ordered_field \u219d linear_order division_ring\n -/\n#print generalized_continued_fraction.comp_exact_value_correctness_of_stream_eq_some_aux_comp /- _inst_1: linear_ordered_field \u219d field linear_ordered_ring\n -/\n\n-- algebra\\direct_limit.lean\n#print module.direct_limit /- dec_\u03b9: decidable_eq \u219d\n_inst_2: directed_order \u219d has_le\n -/\n#print module.direct_limit.add_comm_group /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.semimodule /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.inhabited /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.of_f /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.exists_of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.induction_on /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.lift /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.lift_of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.lift_unique /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.totalize /- _inst_2: directed_order \u219d has_le\n -/\n#print module.direct_limit.to_module_totalize_of_le /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.of.zero_exact_aux /- dec_\u03b9: decidable_eq \u219d\n -/\n#print module.direct_limit.of.zero_exact /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.add_comm_group /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.inhabited /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.of_f /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.induction_on /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.of.zero_exact /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.lift /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.lift_of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print add_comm_group.direct_limit.lift_unique /- dec_\u03b9: decidable_eq \u219d\n -/\n#print ring.direct_limit /- _inst_2: directed_order \u219d has_le\n -/\n\n-- algebra\\direct_sum.lean\n#print direct_sum.mk /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.mk_injective /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.of_injective /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.induction_on /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.to_add_monoid /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.to_add_monoid_of /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.to_add_monoid.unique /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.set_to_set /- dec_\u03b9: decidable_eq \u219d\n -/\n\n-- algebra\\divisibility.lean\n#print monoid_has_dvd /- _inst_1: monoid \u219d has_mul\n -/\n#print dvd.intro /- _inst_1: monoid \u219d has_dvd has_mul\n -/\n#print exists_eq_mul_right_of_dvd /- _inst_1: monoid \u219d has_dvd has_mul\n -/\n#print dvd.elim /- _inst_1: monoid \u219d has_dvd has_mul\n -/\n#print dvd_trans /- _inst_1: monoid \u219d semigroup has_dvd\n -/\n#print dvd.intro_left /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print exists_eq_mul_left_of_dvd /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_mul_left /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_mul_of_dvd_right /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_of_mul_left_dvd /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print eq_zero_of_zero_dvd /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print dvd_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print mul_dvd_mul_iff_right /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_semigroup\n -/\n#print units.dvd_mul_left /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print units.mul_left_dvd /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_not_unit /- _inst_1: comm_monoid_with_zero \u219d monoid has_zero\n -/\n\n-- algebra\\euclidean_domain.lean\n#print euclidean_domain.gcd /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_zero_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_zero_right /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_val /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_dvd /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_dvd_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_dvd_right /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_eq_zero_iff /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.dvd_gcd /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_eq_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_one_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_self /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_aux /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_zero_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_aux_rec /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_a /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_b /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_aux_fst /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_aux_val /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_val /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.xgcd_aux_P /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_eq_gcd_ab /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.dvd_lcm_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.dvd_lcm_right /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm_dvd /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm_dvd_iff /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm_zero_left /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm_zero_right /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.lcm_eq_zero_iff /- _inst_2: decidable_eq \u219d\n -/\n#print euclidean_domain.gcd_mul_lcm /- _inst_2: decidable_eq \u219d\n -/\n\n-- algebra\\field.lean\n#print division_ring_has_div /- _inst_1: division_ring \u219d has_inv has_mul\n -/\n#print inverse_eq_has_inv /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print inv_eq_one_div /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print mul_div_assoc' /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print one_div_neg_one_eq_neg_one /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print neg_div /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print div_add_div_same /- _inst_1: division_ring \u219d has_inv distrib has_div\n -/\n#print one_div_mul_add_mul_one_div_eq_one_div_add_one_div /- _inst_1: division_ring \u219d group_with_zero add_comm_semigroup distrib\n -/\n#print one_div_mul_sub_mul_one_div_eq_one_div_add_one_div /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print add_div_eq_mul_add_div /- _inst_1: division_ring \u219d group_with_zero distrib\n -/\n#print one_div_add_one_div /- _inst_1: field \u219d add_comm_semigroup distrib comm_group_with_zero\n -/\n#print div_add_div /- _inst_1: field \u219d division_ring comm_group_with_zero\n -/\n#print ring_hom.map_units_inv /- _inst_2: division_ring \u219d group_with_zero semiring\n -/\n#print ring_hom.map_ne_zero /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print ring_hom.map_eq_zero /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print ring_hom.map_inv /- _inst_1: division_ring \u219d group_with_zero semiring\n_inst_4: division_ring \u219d group_with_zero semiring\n -/\n#print ring_hom.map_div /- _inst_1: division_ring \u219d group_with_zero semiring\n_inst_4: division_ring \u219d group_with_zero semiring\n -/\n\n-- algebra\\field_power.lean\n#print ring_hom.map_fpow /- _inst_1: division_ring \u219d group_with_zero semiring\n_inst_2: division_ring \u219d group_with_zero semiring\n -/\n#print neg_fpow_bit0 /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print neg_fpow_bit1 /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n#print one_lt_fpow /- _inst_1: linear_ordered_field \u219d linear_ordered_semiring\n -/\n#print rat.cast_fpow /- _inst_1: field \u219d division_ring\n -/\n\n-- algebra\\floor.lean\n#print abs_sub_lt_one_of_floor_eq_floor /- _inst_3: linear_ordered_comm_ring \u219d linear_ordered_ring comm_ring\n -/\n\n-- algebra\\free.lean\n#print free_semigroup.traverse /- _inst_1: applicative \u219d has_seq functor\n -/\n#print free_add_semigroup.traverse /- _inst_1: applicative \u219d has_seq functor\n -/\n#print free_semigroup.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print free_add_semigroup.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\gcd_monoid.lean\n#print comm_group_with_zero.normalization_monoid /- _inst_1: decidable_eq \u219d\n -/\n#print comm_group_with_zero.coe_norm_unit /- _inst_1: decidable_eq \u219d normalization_monoid\n_inst_2: comm_group_with_zero \u219d group_with_zero comm_cancel_monoid_with_zero\n -/\n#print units_eq_one /- _inst_1: comm_cancel_monoid_with_zero \u219d monoid\n -/\n#print norm_unit_eq_one /- _inst_2: unique \u219d normalization_monoid\n -/\n#print normalize_eq /- _inst_2: unique \u219d normalization_monoid\n -/\n#print gcd_eq_of_dvd_sub_right /- _inst_1: integral_domain \u219d comm_cancel_monoid_with_zero nontrivial ring\n -/\n#print normalization_monoid_of_monoid_hom_right_inverse /- _inst_3: decidable_eq \u219d\n -/\n#print gcd_monoid_of_gcd /- _inst_4: decidable_eq \u219d\n -/\n#print gcd_monoid_of_lcm /- _inst_4: decidable_eq \u219d\n -/\n#print gcd_monoid_of_exists_gcd /- _inst_4: decidable_eq \u219d\n -/\n#print gcd_monoid_of_exists_lcm /- _inst_4: decidable_eq \u219d\n -/\n\n-- algebra\\geom_sum.lean\n#print geom_series /- _inst_1: semiring \u219d add_comm_monoid has_pow\n -/\n#print op_geom_series /- _inst_1: ring \u219d semiring\n -/\n#print geom_series\u2082 /- _inst_1: semiring \u219d add_comm_monoid has_mul has_pow\n -/\n#print geom_series\u2082_self /- _inst_1: comm_ring \u219d semiring\n -/\n#print geom_sum\u2082_mul_add /- _inst_1: comm_semiring \u219d comm_semigroup semiring\n -/\n#print geom_sum\u2082_mul /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print geom_sum /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n\n-- algebra\\group\\basic.lean\n#print neg_unique /- _inst_1: add_comm_monoid \u219d add_monoid add_comm_semigroup\n -/\n#print inv_unique /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print eq_zero_of_add_self_left_cancel /- _inst_1: add_left_cancel_monoid \u219d add_monoid add_left_cancel_semigroup\n -/\n#print eq_one_of_mul_self_left_cancel /- _inst_1: left_cancel_monoid \u219d left_cancel_semigroup monoid\n -/\n#print eq_one_of_left_cancel_mul_self /- _inst_1: left_cancel_monoid \u219d left_cancel_semigroup monoid\n -/\n#print eq_zero_of_left_cancel_add_self /- _inst_1: add_left_cancel_monoid \u219d add_monoid add_left_cancel_semigroup\n -/\n#print eq_one_of_mul_self_right_cancel /- _inst_1: right_cancel_monoid \u219d right_cancel_semigroup monoid\n -/\n#print eq_zero_of_add_self_right_cancel /- _inst_1: add_right_cancel_monoid \u219d add_monoid add_right_cancel_semigroup\n -/\n#print eq_zero_of_right_cancel_add_self /- _inst_1: add_right_cancel_monoid \u219d add_monoid add_right_cancel_semigroup\n -/\n#print eq_one_of_right_cancel_mul_self /- _inst_1: right_cancel_monoid \u219d right_cancel_semigroup monoid\n -/\n#print add_self_iff_eq_zero /- _inst_1: add_group \u219d add_monoid add_left_cancel_semigroup\n -/\n#print mul_self_iff_eq_one /- _inst_1: group \u219d left_cancel_semigroup monoid\n -/\n#print mul_left_eq_self /- _inst_1: group \u219d right_cancel_semigroup monoid\n -/\n#print add_left_eq_self /- _inst_1: add_group \u219d add_monoid add_right_cancel_semigroup\n -/\n#print mul_right_eq_self /- _inst_1: group \u219d left_cancel_semigroup monoid\n -/\n#print add_right_eq_self /- _inst_1: add_group \u219d add_monoid add_left_cancel_semigroup\n -/\n#print zero_sub /- _inst_1: add_group \u219d has_sub add_monoid has_neg\n -/\n#print sub_left_inj /- _inst_1: add_group \u219d has_sub add_right_cancel_semigroup has_neg\n -/\n#print neg_add /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print mul_inv /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print sub_add_eq_sub_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print neg_add_eq_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_add_eq_add_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_add /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print add_sub_add_left_eq_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print eq_sub_of_add_eq' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_eq_of_eq_add' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print eq_add_of_sub_eq' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print add_eq_of_eq_sub' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_sub_self /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print add_sub_comm /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_eq_sub_add_sub /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print neg_neg_sub_neg /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_eq_neg_add /- _inst_1: add_comm_group \u219d has_sub add_comm_semigroup has_neg\n -/\n#print eq_sub_iff_add_eq' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_eq_iff_eq_add' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_right_comm /- _inst_1: add_comm_group \u219d has_sub add_comm_semigroup has_neg\n -/\n#print sub_add_add_cancel /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_add_sub_cancel' /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print sub_sub_sub_cancel_left /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n\n-- algebra\\group\\commute.lean\n#print add_neg_cancel_comm /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print mul_inv_cancel_comm /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print add_neg_cancel_comm_assoc /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print mul_inv_cancel_comm_assoc /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print neg_add_cancel_comm /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print inv_mul_cancel_comm /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print inv_mul_cancel_comm_assoc /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print neg_add_cancel_comm_assoc /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n\n-- algebra\\group\\conj.lean\n#print is_conj /- _inst_1: group \u219d has_inv has_mul\n -/\n#print is_conj_iff_eq /- _inst_3: comm_group \u219d comm_semigroup group\n -/\n\n-- algebra\\group\\defs.lean\n#print algebra.sub /- _inst_1: add_group \u219d has_neg has_add\n -/\n#print sub_eq_add_neg /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n\n-- algebra\\group\\pi.lean\n#print add_monoid_hom.single /- _inst_1: decidable_eq \u219d\n -/\n#print add_monoid_hom.single_apply /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\group\\semiconj.lean\n#print add_semiconj_by.add_right /- _inst_1: add_semigroup \u219d has_add is_associative\n -/\n#print semiconj_by.mul_right /- _inst_1: semigroup \u219d is_associative has_mul\n -/\n\n-- algebra\\group\\units.lean\n#print units.decidable_eq /- _inst_2: decidable_eq \u219d\n -/\n#print add_units.decidable_eq /- _inst_2: decidable_eq \u219d\n -/\n#print divp_eq_divp_iff /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print divp_mul_divp /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n\n-- algebra\\group_action_hom.lean\n#print mul_action_hom.has_coe_to_fun /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print mul_action_hom.map_smul /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print mul_action_hom.ext /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print mul_action_hom.ext_iff /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print mul_action_hom.id /- _inst_2: mul_action \u219d\n -/\n#print mul_action_hom.id_apply /- _inst_2: mul_action \u219d\n -/\n#print mul_action_hom.comp /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n_inst_4: mul_action \u219d\n -/\n#print mul_action_hom.comp_apply /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n_inst_4: mul_action \u219d\n -/\n#print mul_action_hom.id_comp /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print mul_action_hom.comp_id /- _inst_2: mul_action \u219d\n_inst_3: mul_action \u219d\n -/\n#print distrib_mul_action_hom.to_add_monoid_hom /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.to_mul_action_hom /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.has_coe /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.has_coe' /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.has_coe_to_fun /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.coe_fn_coe /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.coe_fn_coe' /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.ext /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.ext_iff /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.map_zero /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.map_add /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.map_neg /- _inst_8: distrib_mul_action \u219d\n_inst_12: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.map_sub /- _inst_8: distrib_mul_action \u219d\n_inst_12: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.map_smul /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.id /- _inst_6: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.id_apply /- _inst_6: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.comp /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n_inst_14: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.comp_apply /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n_inst_14: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.id_comp /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n#print distrib_mul_action_hom.comp_id /- _inst_6: distrib_mul_action \u219d\n_inst_10: distrib_mul_action \u219d\n -/\n\n-- algebra\\group_power\\basic.lean\n#print monoid.has_pow /- _inst_1: monoid \u219d has_one has_mul\n -/\n#print monoid.pow_eq_has_pow /- _inst_1: monoid \u219d has_one has_mul has_pow\n -/\n#print pow_zero /- _inst_1: monoid \u219d has_one has_pow\n -/\n#print zero_nsmul /- _inst_3: add_monoid \u219d has_zero has_add\n -/\n#print pow_succ /- _inst_1: monoid \u219d has_mul has_pow\n -/\n#print succ_nsmul /- _inst_3: add_monoid \u219d has_zero has_add\n -/\n#print pow_ite /- _inst_1: monoid \u219d has_pow\n -/\n#print ite_pow /- _inst_1: monoid \u219d has_pow\n -/\n#print mul_pow /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print dvd_pow /- _inst_1: comm_monoid \u219d monoid\n -/\n#print gpow_coe_nat /- _inst_1: group \u219d has_pow\n -/\n#print gpow_of_nat /- _inst_1: group \u219d has_pow\n -/\n#print gpow_neg_succ_of_nat /- _inst_1: group \u219d has_inv has_pow\n -/\n#print gpow_zero /- _inst_1: group \u219d has_one has_pow\n -/\n#print gpow_one /- _inst_1: group \u219d monoid\n -/\n#print gpow_neg_one /- _inst_1: group \u219d has_inv monoid\n -/\n#print mul_gpow /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print zero_pow /- _inst_1: monoid_with_zero \u219d has_one mul_zero_class has_pow\n -/\n#print pow_two_sub_pow_two /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print eq_or_eq_neg_of_pow_two_eq_pow_two /- _inst_1: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print pow_eq_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print pow_abs /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_ring\n -/\n#print gsmul_nonneg /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n\n-- algebra\\group_power\\lemmas.lean\n#print gsmul_pos /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print gsmul_le_gsmul_iff /- _inst_1: linear_ordered_add_comm_group \u219d ordered_add_comm_group\n -/\n#print gsmul_lt_gsmul_iff /- _inst_1: linear_ordered_add_comm_group \u219d ordered_add_comm_group\n -/\n#print nsmul_le_nsmul_iff /- _inst_1: linear_ordered_add_comm_group \u219d ordered_cancel_add_comm_monoid\n -/\n#print nsmul_lt_nsmul_iff /- _inst_1: linear_ordered_add_comm_group \u219d ordered_cancel_add_comm_monoid\n -/\n#print nsmul_eq_mul' /- _inst_1: semiring \u219d add_monoid monoid distrib mul_zero_class\n -/\n#print bit0_mul /- _inst_1: ring \u219d distrib add_group\n -/\n#print mul_bit0 /- _inst_1: ring \u219d distrib add_group\n -/\n\n-- algebra\\group_ring_action.lean\n#print distrib_mul_action.to_add_monoid_hom /- _inst_7: distrib_mul_action \u219d\n -/\n#print distrib_mul_action.to_add_equiv /- _inst_7: distrib_mul_action \u219d\n -/\n#print distrib_mul_action.hom_add_monoid_hom /- _inst_7: distrib_mul_action \u219d\n -/\n#print multiset.smul_prod /- _inst_5: comm_semiring \u219d comm_monoid semiring\n -/\n#print smul_prod /- _inst_5: comm_semiring \u219d comm_monoid semiring\n -/\n#print smul_inv /- _inst_6: field \u219d division_ring\n -/\n\n-- algebra\\group_with_zero\\basic.lean\n#print zero_ne_one /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print group_with_zero.has_div /- _inst_1: group_with_zero \u219d has_inv has_mul\n -/\n#print units.mul_left_eq_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print units.mul_right_eq_zero /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print eq_zero_of_zero_eq_one /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print div_eq_mul_inv /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print one_div /- _inst_1: group_with_zero \u219d has_inv monoid has_div\n -/\n#print zero_div /- _inst_1: group_with_zero \u219d has_inv mul_zero_class has_div\n -/\n#print mul_div_assoc /- _inst_1: group_with_zero \u219d semigroup has_inv has_div\n -/\n#print mul_inv' /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print one_div_mul_one_div /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print div_mul_right /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print mul_div_cancel_left_of_imp /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print mul_div_cancel_of_imp' /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print mul_div_cancel' /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print mul_div_mul_left /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print div_mul_eq_mul_div /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print div_div_eq_mul_div /- _inst_1: comm_group_with_zero \u219d group_with_zero\n -/\n#print ne_zero_of_one_div_ne_zero /- _inst_1: comm_group_with_zero \u219d group_with_zero\n -/\n#print eq_zero_of_one_div_eq_zero /- _inst_1: comm_group_with_zero \u219d group_with_zero\n -/\n#print div_eq_inv_mul /- _inst_1: comm_group_with_zero \u219d has_inv comm_semigroup has_div\n -/\n#print mul_div_right_comm /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print mul_div_comm /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print div_mul_div_cancel /- _inst_1: comm_group_with_zero \u219d group_with_zero\n -/\n#print div_eq_div_iff /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n\n-- algebra\\group_with_zero\\power.lean\n#print zero_pow' /- _inst_1: monoid_with_zero \u219d has_one mul_zero_class has_pow\n -/\n#print pow_eq_zero' /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print fpow_coe_nat /- _inst_1: group_with_zero \u219d has_pow\n -/\n#print fpow_of_nat /- _inst_1: group_with_zero \u219d has_pow\n -/\n#print fpow_neg_succ_of_nat /- _inst_1: group_with_zero \u219d has_inv has_pow\n -/\n#print fpow_zero /- _inst_1: group_with_zero \u219d has_one has_pow\n -/\n#print fpow_one /- _inst_1: group_with_zero \u219d monoid\n -/\n#print fpow_neg_one /- _inst_1: group_with_zero \u219d has_inv monoid\n -/\n#print mul_fpow /- _inst_2: comm_group_with_zero \u219d group_with_zero comm_semigroup\n -/\n#print div_pow /- _inst_1: comm_group_with_zero \u219d group_with_zero comm_monoid\n -/\n\n-- algebra\\homology\\exact.lean\n#print category_theory.kernel_comp_cokernel /- _inst_5: category_theory.limits.has_cokernels \u219d category_theory.limits.has_cokernel\n -/\n\n-- algebra\\homology\\image_to_kernel_map.lean\n#print category_theory.image_to_kernel_map_iso_comp /- _inst_5: category_theory.is_iso \u219d category_theory.epi\n -/\n\n-- algebra\\invertible.lean\n#print nonzero_of_invertible /- _inst_1: group_with_zero \u219d monoid_with_zero nontrivial\n -/\n\n-- algebra\\iterate_hom.lean\n#print ring_hom.iterate_map_sub /- _inst_1: ring \u219d add_group semiring\n -/\n#print ring_hom.iterate_map_neg /- _inst_1: ring \u219d add_group semiring\n -/\n#print ring_hom.iterate_map_gsmul /- _inst_1: ring \u219d add_group semiring\n -/\n\n-- algebra\\lie\\basic.lean\n#print ring_commutator.has_bracket /- _inst_1: ring \u219d has_sub has_mul\n -/\n#print ring_commutator.commutator /- _inst_1: ring \u219d has_sub has_bracket has_mul\n -/\n#print lie_ring.of_associative_ring_bracket /- _inst_1: ring \u219d has_sub has_bracket has_mul\n -/\n#print lie_module.to_endo_morphism /- _inst_5: module \u219d algebra\n -/\n#print lie_algebra.ad /- _inst_3: lie_algebra \u219d algebra\n -/\n#print lie_algebra.ad_apply /- _inst_3: lie_algebra \u219d algebra\n -/\n#print lie_submodule_lie_module /- _inst_5: module \u219d lie_ring_module\n -/\n#print lie_submodule.quotient.lie_submodule_invariant /- _inst_5: module \u219d algebra\n -/\n#print lie_submodule.quotient.action_as_endo_map /- _inst_5: module \u219d\n -/\n#print lie_submodule.quotient.action_as_endo_map_bracket /- _inst_5: module \u219d\n -/\n#print lie_submodule.quotient.lie_quotient_lie_ring_module /- _inst_5: module \u219d\n -/\n#print lie_submodule.quotient.lie_quotient_lie_module /- _inst_5: module \u219d lie_ring_module\n -/\n#print linear_equiv.lie_conj /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print linear_equiv.lie_conj_apply /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print linear_equiv.lie_conj_symm /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print lie_equiv_matrix' /- _inst_2: decidable_eq \u219d\n -/\n#print lie_equiv_matrix'_apply /- _inst_2: decidable_eq \u219d\n -/\n#print lie_equiv_matrix'_symm_apply /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.lie_conj /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.lie_conj_apply /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.lie_conj_symm_apply /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.reindex_lie_equiv /- _inst_2: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print matrix.reindex_lie_equiv_apply /- _inst_2: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print matrix.reindex_lie_equiv_symm_apply /- _inst_2: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print bilin_form.is_skew_adjoint_bracket /- _inst_3: module \u219d\n -/\n#print skew_adjoint_lie_subalgebra /- _inst_3: module \u219d algebra\n -/\n#print skew_adjoint_lie_subalgebra_equiv /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print skew_adjoint_lie_subalgebra_equiv_apply /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print skew_adjoint_lie_subalgebra_equiv_symm_apply /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print matrix.lie_transpose /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_2: decidable_eq \u219d\n -/\n#print matrix.is_skew_adjoint_bracket /- _inst_2: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_lie_subalgebra /- _inst_2: decidable_eq \u219d\n -/\n#print mem_skew_adjoint_matrices_lie_subalgebra /- _inst_2: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_lie_subalgebra_equiv /- _inst_2: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_lie_subalgebra_equiv_apply /- _inst_2: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_lie_subalgebra_equiv_transpose /- _inst_2: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_lie_subalgebra_equiv_transpose_apply /- _inst_2: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print mem_skew_adjoint_matrices_lie_subalgebra_unit_smul /- _inst_2: decidable_eq \u219d\n -/\n\n-- algebra\\lie\\classical.lean\n#print lie_algebra.matrix_trace_commutator_zero /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.sl /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.sl_bracket /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.E /- _inst_5: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero\n -/\n#print lie_algebra.special_linear.E_apply_one /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.E_apply_zero /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.E_diag_zero /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.E_trace_zero /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.Eb /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.Eb_val /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.special_linear.sl_non_abelian /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.symplectic.J /- _inst_8: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero has_neg\n -/\n#print lie_algebra.symplectic.sp /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.so /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.mem_so /- _inst_5: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.indefinite_diagonal /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero has_neg\n -/\n#print lie_algebra.orthogonal.so' /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.Pso /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero\n -/\n#print lie_algebra.orthogonal.Pso_inv /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.is_unit_Pso /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.indefinite_diagonal_transform /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.so_indefinite_equiv /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.so_indefinite_equiv_apply /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.JD /- _inst_8: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero\n -/\n#print lie_algebra.orthogonal.type_D /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.PD /- _inst_8: decidable_eq \u219d\n_inst_9: comm_ring \u219d has_one has_zero has_neg\n -/\n#print lie_algebra.orthogonal.S /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.S_as_blocks /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.JD_transform /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.PD_inv /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.is_unit_PD /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.type_D_equiv_so' /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.JB /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.type_B /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.PB /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.PB_inv /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.is_unit_PB /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.JB_transform /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.indefinite_diagonal_assoc /- _inst_8: decidable_eq \u219d\n -/\n#print lie_algebra.orthogonal.type_B_equiv_so' /- _inst_8: decidable_eq \u219d\n -/\n\n-- algebra\\lie\\universal_enveloping.lean\n#print universal_enveloping_algebra.mk_alg_hom /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.\u03b9 /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.lift /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.lift_symm_apply /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.\u03b9_comp_lift /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.lift_\u03b9_apply /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.lift_unique /- _inst_3: lie_algebra \u219d algebra\n -/\n#print universal_enveloping_algebra.hom_ext /- _inst_3: lie_algebra \u219d algebra\n -/\n\n-- algebra\\linear_ordered_comm_group_with_zero.lean\n#print one_le_pow_of_one_le' /- _inst_1: linear_ordered_comm_group_with_zero \u219d ordered_comm_monoid\n -/\n#print pow_le_one_of_le_one /- _inst_1: linear_ordered_comm_group_with_zero \u219d ordered_comm_monoid\n -/\n#print le_of_le_mul_right /- _inst_1: linear_ordered_comm_group_with_zero \u219d ordered_comm_monoid group_with_zero\n -/\n#print inv_lt_inv'' /- _inst_1: linear_ordered_comm_group_with_zero \u219d ordered_comm_monoid group_with_zero\n -/\n#print inv_le_inv'' /- _inst_1: linear_ordered_comm_group_with_zero \u219d ordered_comm_monoid group_with_zero\n -/\n\n-- algebra\\linear_recurrence.lean\n#print linear_recurrence.char_poly /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- algebra\\module\\basic.lean\n#print add_smul /- _inst_3: semimodule \u219d\n -/\n#print zero_smul /- _inst_3: semimodule \u219d\n -/\n#print two_smul /- _inst_3: semimodule \u219d\n -/\n#print two_smul' /- _inst_3: semimodule \u219d\n -/\n#print function.injective.semimodule /- _inst_3: semimodule \u219d\n -/\n#print function.surjective.semimodule /- _inst_3: semimodule \u219d\n -/\n#print smul_add_hom /- _inst_3: semimodule \u219d\n -/\n#print smul_add_hom_apply /- _inst_3: semimodule \u219d\n -/\n#print semimodule.eq_zero_of_zero_eq_one /- _inst_3: semimodule \u219d\n -/\n#print list.sum_smul /- _inst_3: semimodule \u219d\n -/\n#print multiset.sum_smul /- _inst_3: semimodule \u219d\n -/\n#print finset.sum_smul /- _inst_3: semimodule \u219d\n -/\n#print semimodule.add_comm_monoid_to_add_comm_group /- _inst_3: semimodule \u219d\n -/\n#print module /- _inst_1: ring \u219d semiring\n_inst_2: add_comm_group \u219d add_comm_monoid\n -/\n#print neg_one_smul /- _inst_3: module \u219d\n -/\n#print smul_eq_zero /- _inst_4: division_ring \u219d group_with_zero ring\n_inst_6: module \u219d\n -/\n#print semimodule.subsingleton /- _inst_4: semimodule \u219d\n -/\n#print smul_eq_mul /- _inst_1: semiring \u219d has_mul\n -/\n#print vector_space /- _inst_1: field \u219d semiring\n_inst_2: add_comm_group \u219d add_comm_monoid\n -/\n#print semimodule.smul_eq_smul /- _inst_3: semimodule \u219d\n -/\n#print semimodule.nsmul_eq_smul /- _inst_3: semimodule \u219d\n -/\n#print add_monoid_hom.map_nat_cast_smul /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print add_monoid_hom.map_rat_cast_smul /- _inst_4: module \u219d\n_inst_6: module \u219d\n -/\n\n-- algebra\\module\\linear_map.lean\n#print linear_map.to_mul_action_hom /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_map.to_add_hom /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_map.has_coe_to_fun /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.coe_mk /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.id /- _inst_5: semimodule \u219d\n -/\n#print linear_map.id_apply /- _inst_5: semimodule \u219d\n -/\n#print linear_map.id_coe /- _inst_5: semimodule \u219d\n -/\n#print linear_map.to_fun_eq_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.is_linear /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.coe_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ext /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.congr_arg /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.congr_fun /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ext_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_add /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_smul_of_tower /- _inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_13: semimodule \u219d\n_inst_14: is_scalar_tower \u219d\n -/\n#print linear_map.map_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.is_add_monoid_hom /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.to_add_monoid_hom /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.to_add_monoid_hom_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.restrict_scalars /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: is_scalar_tower \u219d\n -/\n#print linear_map.map_sum /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.to_add_monoid_hom_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ext_ring /- _inst_5: semimodule \u219d\n -/\n#print linear_map.inverse /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_neg /- _inst_2: add_comm_group \u219d add_comm_monoid add_group\n_inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print linear_map.map_sub /- _inst_2: add_comm_group \u219d add_comm_monoid add_group\n_inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print linear_map.is_add_group_hom /- _inst_2: add_comm_group \u219d add_comm_monoid add_group\n_inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print is_linear_map.mk' /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print is_linear_map.mk'_apply /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print is_linear_map.is_linear_map_smul /- _inst_6: comm_semiring \u219d comm_semigroup semiring\n_inst_8: semimodule \u219d\n -/\n#print is_linear_map.is_linear_map_smul' /- _inst_8: semimodule \u219d\n -/\n#print is_linear_map.map_zero /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print is_linear_map.is_linear_map_neg /- _inst_4: semimodule \u219d\n -/\n#print is_linear_map.map_neg /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print is_linear_map.map_sub /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print module.End /- _inst_3: semimodule \u219d\n -/\n#print linear_equiv.to_linear_map /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_equiv.to_add_equiv /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_equiv.linear_map.has_coe /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.has_coe_to_fun /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.mk_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.to_equiv /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.injective_to_equiv /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.to_equiv_inj /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.injective_to_linear_map /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.to_linear_map_inj /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.refl /- _inst_6: semimodule \u219d\n -/\n#print linear_equiv.refl_apply /- _inst_6: semimodule \u219d\n -/\n#print linear_equiv.simps.inv_fun /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.trans_symm /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.symm_trans /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.refl_to_linear_map /- _inst_6: semimodule \u219d\n -/\n#print linear_equiv.comp_coe /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_equiv.of_involutive /- _inst_6: semimodule \u219d\n -/\n#print linear_equiv.coe_of_involutive /- _inst_6: semimodule \u219d\n -/\n\n-- algebra\\module\\opposites.lean\n#print opposite.semimodule /- _inst_3: semimodule \u219d\n -/\n#print opposite.op_linear_equiv /- _inst_3: semimodule \u219d\n -/\n#print opposite.coe_op_linear_equiv /- _inst_3: semimodule \u219d\n -/\n#print opposite.coe_op_linear_equiv_symm /- _inst_3: semimodule \u219d\n -/\n#print opposite.coe_op_linear_equiv_to_linear_map /- _inst_3: semimodule \u219d\n -/\n#print opposite.coe_op_linear_equiv_symm_to_linear_map /- _inst_3: semimodule \u219d\n -/\n#print opposite.op_linear_equiv_to_add_equiv /- _inst_3: semimodule \u219d\n -/\n#print opposite.op_linear_equiv_symm_to_add_equiv /- _inst_3: semimodule \u219d\n -/\n\n-- algebra\\module\\ordered.lean\n#print smul_lt_smul_of_pos /- _inst_3: semimodule \u219d\n -/\n#print smul_le_smul_of_nonneg /- _inst_3: semimodule \u219d\n -/\n#print eq_of_smul_eq_smul_of_pos_of_le /- _inst_3: semimodule \u219d\n -/\n#print lt_of_smul_lt_smul_of_nonneg /- _inst_3: semimodule \u219d\n -/\n#print smul_lt_smul_iff_of_pos /- _inst_3: semimodule \u219d\n -/\n#print smul_pos_iff_of_pos /- _inst_3: semimodule \u219d\n -/\n#print ordered_semimodule.mk'' /- _inst_3: semimodule \u219d\n -/\n#print ordered_semimodule.mk' /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_semiring\n_inst_3: semimodule \u219d\n -/\n#print smul_le_smul_iff_of_pos /- _inst_2: ordered_add_comm_group \u219d ordered_add_comm_monoid\n_inst_3: semimodule \u219d\n -/\n#print smul_le_smul_iff_of_neg /- _inst_3: semimodule \u219d\n -/\n#print smul_lt_iff_of_pos /- _inst_1: linear_ordered_field \u219d group_with_zero ordered_semiring\n_inst_2: ordered_add_comm_group \u219d ordered_add_comm_monoid\n_inst_3: semimodule \u219d\n -/\n#print smul_le_iff_of_pos /- _inst_3: semimodule \u219d\n -/\n#print le_smul_iff_of_pos /- _inst_3: semimodule \u219d\n -/\n#print prod.ordered_semimodule /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print pi.ordered_semimodule' /- _inst_1: linear_ordered_field \u219d ordered_semimodule ordered_semiring\n_inst_9: semimodule \u219d\n -/\n#print order_dual.has_scalar /- _inst_2: ordered_add_comm_monoid \u219d add_comm_monoid\n_inst_3: semimodule \u219d has_scalar\n -/\n#print order_dual.mul_action /- _inst_3: semimodule \u219d\n -/\n#print order_dual.distrib_mul_action /- _inst_3: semimodule \u219d\n -/\n#print order_dual.semimodule /- _inst_3: semimodule \u219d\n -/\n#print order_dual.ordered_semimodule /- _inst_3: semimodule \u219d\n -/\n\n-- algebra\\module\\prod.lean\n#print prod.semimodule /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n\n-- algebra\\module\\submodule.lean\n#print submodule.to_add_submonoid /- _inst_3: semimodule \u219d\n -/\n#print submodule.set.has_coe_t /- _inst_3: semimodule \u219d\n -/\n#print submodule.has_mem /- _inst_3: semimodule \u219d\n -/\n#print submodule.has_coe_to_sort /- _inst_3: semimodule \u219d\n -/\n#print submodule.coe_sort_coe /- _inst_3: semimodule \u219d\n -/\n#print submodule.exists /- _inst_3: semimodule \u219d\n -/\n#print submodule.forall /- _inst_3: semimodule \u219d\n -/\n#print submodule.coe_injective /- _inst_3: semimodule \u219d\n -/\n#print submodule.coe_set_eq /- _inst_3: semimodule \u219d\n -/\n#print submodule.ext'_iff /- _inst_3: semimodule \u219d\n -/\n#print submodule.ext /- _inst_3: semimodule \u219d\n -/\n#print submodule.to_add_submonoid_injective /- _inst_3: semimodule \u219d\n -/\n#print submodule.to_add_submonoid_eq /- _inst_3: semimodule \u219d\n -/\n#print submodule.smul_mem_iff /- _inst_1: division_ring \u219d group_with_zero ring\n -/\n\n-- algebra\\module\\ulift.lean\n#print ulift.is_scalar_tower /- _inst_4: is_scalar_tower \u219d\n -/\n#print ulift.is_scalar_tower' /- _inst_4: is_scalar_tower \u219d\n -/\n#print ulift.is_scalar_tower'' /- _inst_4: is_scalar_tower \u219d\n -/\n#print ulift.mul_action /- _inst_2: mul_action \u219d\n -/\n#print ulift.mul_action' /- _inst_2: mul_action \u219d\n -/\n#print ulift.distrib_mul_action /- _inst_3: distrib_mul_action \u219d\n -/\n#print ulift.distrib_mul_action' /- _inst_3: distrib_mul_action \u219d\n -/\n#print ulift.semimodule /- _inst_3: semimodule \u219d\n -/\n#print ulift.semimodule' /- _inst_3: semimodule \u219d\n -/\n#print ulift.semimodule_equiv /- _inst_3: semimodule \u219d\n -/\n\n-- algebra\\monoid_algebra.lean\n#print monoid_algebra /- _inst_1: semiring \u219d has_zero\n -/\n#print monoid_algebra.has_mul /- _inst_2: monoid \u219d has_mul\n -/\n#print monoid_algebra.has_one /- _inst_2: monoid \u219d has_one\n -/\n#print monoid_algebra.nontrivial /- _inst_3: monoid \u219d nonempty\n -/\n#print monoid_algebra.add_group /- _inst_1: ring \u219d add_group semiring\n -/\n#print monoid_algebra.has_scalar /- _inst_3: semimodule \u219d has_scalar\n -/\n#print monoid_algebra.semimodule /- _inst_3: semimodule \u219d\n -/\n#print monoid_algebra.single_one_comm /- _inst_1: comm_semiring \u219d comm_semigroup semiring\n -/\n#print monoid_algebra.group_smul.linear_map /- _inst_6: is_scalar_tower \u219d\n -/\n#print monoid_algebra.group_smul.linear_map_apply /- _inst_6: is_scalar_tower \u219d\n -/\n#print monoid_algebra.equivariant_of_linear_of_comm /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print monoid_algebra.equivariant_of_linear_of_comm_apply /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print add_monoid_algebra /- _inst_1: semiring \u219d has_zero\n -/\n#print add_monoid_algebra.has_mul /- _inst_2: add_monoid \u219d has_add\n -/\n#print add_monoid_algebra.has_one /- _inst_2: add_monoid \u219d has_zero\n -/\n#print add_monoid_algebra.nontrivial /- _inst_3: add_monoid \u219d nonempty\n -/\n#print add_monoid_algebra.add_group /- _inst_1: ring \u219d add_group semiring\n -/\n#print add_monoid_algebra.has_scalar /- _inst_3: semimodule \u219d has_scalar\n -/\n#print add_monoid_algebra.semimodule /- _inst_3: semimodule \u219d\n -/\n\n-- algebra\\opposites.lean\n#print opposite.mul_action /- _inst_2: mul_action \u219d\n -/\n#print opposite.distrib_mul_action /- _inst_3: distrib_mul_action \u219d\n -/\n\n-- algebra\\order.lean\n#print ge_iff_le /- _inst_1: preorder \u219d has_le\n -/\n#print gt_iff_lt /- _inst_1: preorder \u219d has_lt\n -/\n#print cmp_swap /- _inst_2: decidable_rel \u219d\n -/\n\n-- algebra\\ordered_field.lean\n#print inv_pos /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_semiring\n -/\n#print add_halves /- _inst_1: linear_ordered_field \u219d ordered_semiring division_ring comm_group_with_zero\n -/\n#print add_self_div_two /- _inst_1: linear_ordered_field \u219d group_with_zero ordered_semiring\n -/\n#print mul_sub_mul_div_mul_neg_iff /- _inst_1: linear_ordered_field \u219d field ordered_add_comm_group\n -/\n#print mul_sub_mul_div_mul_nonpos_iff /- _inst_1: linear_ordered_field \u219d field ordered_add_comm_group\n -/\n#print mul_self_inj_of_nonneg /- _inst_1: linear_ordered_field \u219d ordered_add_comm_group integral_domain\n -/\n#print abs_div /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_ring\n -/\n#print abs_inv /- _inst_1: linear_ordered_field \u219d group_with_zero linear_ordered_ring\n -/\n\n-- algebra\\ordered_group.lean\n#print inv_le_inv' /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print neg_le_neg /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print inv_lt_inv' /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print neg_lt_neg /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print mul_le_of_le_inv_mul /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print add_le_of_le_neg_add /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_neg_add_of_add_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_inv_mul_of_mul_le /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print le_add_of_neg_add_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_mul_of_inv_mul_le /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print inv_mul_le_of_le_mul /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print neg_add_le_of_le_add /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print mul_lt_of_lt_inv_mul /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print add_lt_of_lt_neg_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_neg_add_of_add_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_inv_mul_of_mul_lt /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print lt_mul_of_inv_mul_lt /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print lt_add_of_neg_add_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_mul_lt_of_lt_mul /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print neg_add_lt_of_lt_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_le_inv_iff /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print neg_le_neg_iff /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print neg_le_iff_add_nonneg /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_le_iff_one_le_mul /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print le_inv_iff_mul_le_one /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print le_neg_iff_add_nonpos /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print neg_lt_neg_iff /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_lt_inv_iff /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print le_inv_mul_iff_mul_le /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print le_neg_add_iff_add_le /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print neg_add_le_iff_le_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_mul_le_iff_le_mul /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print lt_neg_add_iff_add_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_inv_mul_iff_mul_lt /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print neg_add_lt_iff_lt_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print inv_mul_lt_iff_lt_mul /- _inst_1: ordered_comm_group \u219d ordered_cancel_comm_monoid group\n -/\n#print add_neg_le_add_neg_iff /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print div_le_div_iff' /- _inst_1: ordered_comm_group \u219d ordered_comm_monoid group\n -/\n#print sub_nonneg_of_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_of_sub_nonneg /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print sub_nonpos_of_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_of_sub_nonpos /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print sub_pos_of_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_of_sub_pos /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print sub_neg_of_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_of_sub_neg /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print add_le_of_le_sub_left /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_sub_left_of_add_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print add_le_of_le_sub_right /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_sub_right_of_add_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_add_of_sub_left_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print sub_left_le_of_le_add /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print le_add_of_sub_right_le /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print sub_right_le_of_le_add /- _inst_1: ordered_add_comm_group \u219d ordered_add_comm_monoid add_group\n -/\n#print sub_le_sub_right /- _inst_1: ordered_add_comm_group \u219d has_sub ordered_add_comm_monoid has_neg\n -/\n#print add_lt_of_lt_sub_left /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_sub_left_of_add_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print add_lt_of_lt_sub_right /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_sub_right_of_add_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_add_of_sub_left_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print sub_left_lt_of_lt_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print lt_add_of_sub_right_lt /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print sub_right_lt_of_lt_add /- _inst_1: ordered_add_comm_group \u219d add_group ordered_cancel_add_comm_monoid\n -/\n#print sub_lt_sub_right /- _inst_1: ordered_add_comm_group \u219d has_sub has_neg ordered_cancel_add_comm_monoid\n -/\n#print sub_le_sub_iff_right /- _inst_1: ordered_add_comm_group \u219d has_sub has_neg ordered_cancel_add_comm_monoid\n -/\n#print sub_lt_sub_iff_right /- _inst_1: ordered_add_comm_group \u219d has_sub has_neg ordered_cancel_add_comm_monoid\n -/\n#print linear_ordered_add_comm_group.add_lt_add_left /- _inst_1: linear_ordered_add_comm_group \u219d ordered_add_comm_group\n -/\n#print min_neg_neg /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n#print max_neg_neg /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n#print min_sub_sub_right /- _inst_1: linear_ordered_add_comm_group \u219d has_sub has_neg linear_ordered_cancel_add_comm_monoid\n -/\n#print max_sub_sub_right /- _inst_1: linear_ordered_add_comm_group \u219d has_sub has_neg linear_ordered_cancel_add_comm_monoid\n -/\n#print max_zero_sub_eq_self /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n#print abs /- _inst_1: linear_ordered_add_comm_group \u219d linear_order has_neg\n -/\n#print eq_zero_of_neg_eq /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n#print exists_gt_zero /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n\n-- algebra\\ordered_monoid.lean\n#print with_zero.zero_lt_coe /- _inst_1: partial_order \u219d preorder\n -/\n#print with_zero.coe_le_coe /- _inst_1: partial_order \u219d preorder\n -/\n#print with_top.zero_lt_top /- _inst_1: ordered_add_comm_monoid \u219d has_zero partial_order\n -/\n#print with_top.zero_lt_coe /- _inst_1: ordered_add_comm_monoid \u219d has_zero partial_order\n -/\n#print with_bot.coe_eq_zero /- _inst_1: add_monoid \u219d has_zero\n -/\n#print with_bot.bot_add /- _inst_1: ordered_add_comm_monoid \u219d add_semigroup\n -/\n#print with_bot.add_bot /- _inst_1: ordered_add_comm_monoid \u219d add_semigroup\n -/\n#print mul_le_of_le_one_of_le /- _inst_1: ordered_cancel_comm_monoid \u219d ordered_comm_monoid\n -/\n#print add_le_of_nonpos_of_le /- _inst_1: ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid\n -/\n#print add_le_of_le_of_nonpos /- _inst_1: ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid\n -/\n#print mul_le_of_le_of_le_one /- _inst_1: ordered_cancel_comm_monoid \u219d ordered_comm_monoid\n -/\n#print min_add_add_left /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print min_add_add_right /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print max_add_add_left /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print max_add_add_right /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print min_le_add_of_nonneg_right /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print min_le_add_of_nonneg_left /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print max_le_add_of_nonneg /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n\n-- algebra\\ordered_ring.lean\n#print zero_lt_one' /- _inst_1: linear_ordered_semiring \u219d nontrivial ordered_semiring\n -/\n#print lt_of_mul_lt_mul_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print lt_of_mul_lt_mul_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print le_of_mul_le_mul_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print le_of_mul_le_mul_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print pos_and_pos_or_neg_and_neg_of_mul_pos /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonneg_and_nonneg_or_nonpos_and_nonpos_of_mul_nnonneg /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonneg_of_mul_nonneg_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonneg_of_mul_nonneg_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print neg_of_mul_neg_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print neg_of_mul_neg_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonpos_of_mul_nonpos_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonpos_of_mul_nonpos_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print mul_lt_mul_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print mul_lt_mul_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonpos_of_mul_nonneg_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print nonpos_of_mul_nonneg_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print neg_of_mul_pos_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print neg_of_mul_pos_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print linear_ordered_semiring.to_no_top_order /- _inst_2: linear_ordered_semiring \u219d nontrivial ordered_semiring\n -/\n#print monotone_mul_left_of_nonneg /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n -/\n#print monotone_mul_right_of_nonneg /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n -/\n#print monotone.mul /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n -/\n#print strict_mono.mul_const /- _inst_2: preorder \u219d has_lt\n -/\n#print strict_mono.const_mul /- _inst_2: preorder \u219d has_lt\n -/\n#print strict_mono.mul_monotone /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n -/\n#print monotone.mul_strict_mono /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n -/\n#print strict_mono.mul /- _inst_1: linear_ordered_semiring \u219d ordered_semiring\n_inst_2: preorder \u219d has_lt\n -/\n#print mul_le_mul_of_nonpos_left /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n -/\n#print mul_le_mul_of_nonpos_right /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n -/\n#print mul_lt_mul_of_neg_left /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n -/\n#print mul_lt_mul_of_neg_right /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n -/\n#print abs_one /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group nontrivial ordered_semiring\n -/\n#print abs_two /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group nontrivial ordered_semiring\n -/\n#print abs_mul /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group cancel_monoid_with_zero ordered_semiring ring\n -/\n#print abs_mul_abs_self /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group ring\n -/\n#print mul_pos_iff /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_ordered_semiring\n -/\n#print mul_nonneg_iff /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_ordered_semiring\n -/\n#print mul_self_nonneg /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print gt_of_mul_lt_mul_neg_left /- _inst_1: linear_ordered_ring \u219d ring ordered_add_comm_group linear_ordered_semiring\n -/\n#print neg_one_lt_zero /- _inst_1: linear_ordered_ring \u219d nontrivial ordered_semiring ordered_add_comm_group\n -/\n#print le_of_mul_le_of_one_le /- _inst_1: linear_ordered_ring \u219d linear_ordered_semiring\n -/\n#print nonneg_le_nonneg_of_squares_le /- _inst_1: linear_ordered_ring \u219d linear_order ordered_semiring\n -/\n#print mul_le_mul_left_of_neg /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print mul_le_mul_right_of_neg /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print sub_one_lt /- _inst_1: linear_ordered_ring \u219d nontrivial ordered_semiring ordered_add_comm_group\n -/\n#print mul_self_pos /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print mul_self_le_mul_self_of_le_of_neg_le /- _inst_1: linear_ordered_ring \u219d linear_order ordered_semiring ring ordered_add_comm_group\n -/\n#print nonneg_of_mul_nonpos_left /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print nonneg_of_mul_nonpos_right /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print pos_of_mul_neg_left /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print pos_of_mul_neg_right /- _inst_1: linear_ordered_ring \u219d ordered_ring linear_order\n -/\n#print sub_le_of_abs_sub_le_left /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group\n -/\n#print sub_lt_of_abs_sub_lt_left /- _inst_1: linear_ordered_ring \u219d linear_ordered_add_comm_group\n -/\n#print max_mul_mul_le_max_mul_max /- _inst_1: linear_ordered_comm_ring \u219d comm_semigroup linear_order ordered_semiring\n -/\n#print abs_sub_square /- _inst_1: linear_ordered_comm_ring \u219d comm_semigroup linear_ordered_ring\n -/\n#print canonically_ordered_semiring.mul_le_mul /- _inst_1: canonically_ordered_comm_semiring \u219d canonically_ordered_add_monoid distrib\n -/\n#print canonically_ordered_semiring.zero_lt_one /- _inst_1: canonically_ordered_comm_semiring \u219d canonically_ordered_add_monoid monoid_with_zero\n -/\n#print canonically_ordered_semiring.mul_pos /- _inst_1: canonically_ordered_comm_semiring \u219d canonically_ordered_add_monoid no_zero_divisors mul_zero_class\n -/\n#print with_top.mul_zero_class /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.mul_def /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.mul_top /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.top_mul /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.top_mul_top /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.coe_mul /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.mul_coe /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.mul_eq_top_iff /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.no_zero_divisors /- _inst_1: decidable_eq \u219d\n -/\n#print with_top.canonically_ordered_comm_semiring /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\pointwise.lean\n#print set.fintype_mul /- _inst_2: decidable_eq \u219d\n -/\n#print set.fintype_add /- _inst_2: decidable_eq \u219d\n -/\n#print set.univ_inv /- _inst_1: group \u219d has_inv\n -/\n#print set.univ_neg /- _inst_1: add_group \u219d has_neg\n -/\n#print set.mul_action_set /- _inst_2: mul_action \u219d\n -/\n#print zero_smul_set /- _inst_3: semimodule \u219d\n -/\n#print mem_inv_smul_set_iff /- _inst_1: field \u219d group_with_zero\n_inst_2: mul_action \u219d\n -/\n#print mem_smul_set_iff_inv_smul_mem /- _inst_2: mul_action \u219d\n -/\n#print finset.has_mul /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_add /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mul_def /- _inst_1: decidable_eq \u219d\n -/\n#print finset.add_def /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_add /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_mul /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_mul /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_add /- _inst_1: decidable_eq \u219d\n -/\n#print finset.add_mem_add /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mul_mem_mul /- _inst_1: decidable_eq \u219d\n -/\n#print finset.add_card_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mul_card_le /- _inst_1: decidable_eq \u219d\n -/\n\n-- algebra\\polynomial\\big_operators.lean\n#print polynomial.nat_degree_prod /- _inst_1: integral_domain \u219d nontrivial no_zero_divisors comm_semiring\n -/\n\n-- algebra\\polynomial\\group_ring_action.lean\n#print prod_X_sub_smul /- _inst_4: comm_ring \u219d ring comm_semiring\n_inst_5: mul_semiring_action \u219d\n -/\n\n-- algebra\\quadratic_discriminant.lean\n#print discrim /- _inst_1: ring \u219d has_sub has_one has_add has_mul has_pow\n -/\n#print quadratic_eq_zero_iff_discrim_eq_square /- _inst_1: integral_domain \u219d cancel_monoid_with_zero comm_ring\n -/\n#print quadratic_eq_zero_iff /- _inst_1: field \u219d comm_group_with_zero integral_domain\n -/\n#print discrim_le_zero /- _inst_1: linear_ordered_field \u219d field linear_ordered_ring\n -/\n\n-- algebra\\quandle.lean\n#print rack.self_distrib /- _inst_1: rack \u219d shelf\n -/\n#print rack.is_involutory /- _inst_2: rack \u219d shelf\n -/\n#print rack.is_abelian /- _inst_2: rack \u219d shelf\n -/\n\n-- algebra\\ring\\basic.lean\n#print one_add_one_eq_two /- _inst_1: semiring \u219d has_one has_add\n -/\n#print two_mul /- _inst_1: semiring \u219d monoid distrib\n -/\n#print distrib_three_right /- _inst_1: semiring \u219d distrib\n -/\n#print mul_two /- _inst_1: semiring \u219d monoid distrib\n -/\n#print mul_boole /- _inst_2: semiring \u219d monoid mul_zero_class\n -/\n#print boole_mul /- _inst_2: semiring \u219d monoid mul_zero_class\n -/\n#print even /- _inst_1: semiring \u219d has_one has_add has_mul\n -/\n#print odd /- _inst_1: semiring \u219d has_one has_add has_mul\n -/\n#print add_mul_self_eq /- _inst_1: comm_semiring \u219d comm_semigroup semiring\n -/\n#print dvd_add /- _inst_1: comm_semiring \u219d monoid distrib\n -/\n#print two_dvd_bit0 /- _inst_1: comm_semiring \u219d semiring\n -/\n#print ring_hom.map_dvd /- _inst_1: comm_semiring \u219d semiring\n_inst_2: comm_semiring \u219d semiring\n -/\n#print neg_mul_eq_neg_mul /- _inst_1: ring \u219d distrib add_group mul_zero_class\n -/\n#print neg_mul_eq_mul_neg /- _inst_1: ring \u219d distrib add_group mul_zero_class\n -/\n#print ring_hom.map_neg /- _inst_1: ring \u219d add_group semiring\n_inst_2: ring \u219d add_group semiring\n -/\n#print ring_hom.map_sub /- _inst_1: ring \u219d add_group semiring\n_inst_2: ring \u219d add_group semiring\n -/\n#print ring_hom.injective_iff /- _inst_1: ring \u219d add_group semiring\n -/\n#print dvd_neg_of_dvd /- _inst_1: comm_ring \u219d ring\n -/\n#print neg_dvd_of_dvd /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print mul_self_sub_mul_self /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print Vieta_formula_quadratic /- _inst_1: comm_ring \u219d comm_semigroup ring\n -/\n#print dvd_mul_sub_mul /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print succ_ne_self /- _inst_1: ring \u219d add_monoid monoid_with_zero add_left_cancel_semigroup\n -/\n#print pred_ne_self /- _inst_1: ring \u219d monoid_with_zero add_group\n -/\n#print mul_self_eq_mul_self_iff /- _inst_1: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print ring.inverse /- _inst_1: ring \u219d monoid has_zero\n -/\n\n-- algebra\\ring\\prod.lean\n#print ring_hom.prod_comp_prod_map /- _inst_1: semiring \u219d monoid\n_inst_2: semiring \u219d monoid\n_inst_3: semiring \u219d monoid\n_inst_4: semiring \u219d monoid\n_inst_5: semiring \u219d monoid\n -/\n\n-- algebra\\squarefree.lean\n#print squarefree_of_dvd_of_squarefree /- _inst_1: comm_monoid \u219d monoid\n -/\n#print multiplicity.squarefree_iff_multiplicity_le_one /- _inst_2: decidable_rel \u219d\n -/\n#print unique_factorization_monoid.squarefree_iff_nodup_factors /- _inst_5: decidable_eq \u219d\n -/\n\n-- algebra\\star\\basic.lean\n#print star_neg /- _inst_1: ring \u219d add_group semiring has_star\n -/\n#print star_sub /- _inst_1: ring \u219d add_group semiring has_star\n -/\n#print star_bit0 /- _inst_1: ring \u219d add_right_cancel_semigroup semiring has_star\n -/\n\n-- analysis\\ODE\\gronwall.lean\n#print norm_le_gronwall_bound_of_norm_deriv_right_le /- _inst_2: normed_space \u219d\n -/\n#print dist_le_of_approx_trajectories_ODE_of_mem_set /- _inst_2: normed_space \u219d\n -/\n#print dist_le_of_approx_trajectories_ODE /- _inst_2: normed_space \u219d\n -/\n#print dist_le_of_trajectories_ODE_of_mem_set /- _inst_2: normed_space \u219d\n -/\n#print dist_le_of_trajectories_ODE /- _inst_2: normed_space \u219d\n -/\n#print ODE_solution_unique_of_mem_set /- _inst_2: normed_space \u219d\n -/\n#print ODE_solution_unique /- _inst_2: normed_space \u219d\n -/\n\n-- analysis\\analytic\\basic.lean\n#print formal_multilinear_series.radius /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.le_radius_of_bound /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.bound_of_lt_radius /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.geometric_bound_of_lt_radius /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.min_radius_le_radius_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.radius_neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.partial_sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.partial_sum_continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print analytic_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.has_fpower_series_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.analytic_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.analytic_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.radius_pos /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.radius_pos /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print analytic_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print analytic_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print analytic_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.coeff_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.coeff_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.uniform_geometric_approx /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.tendsto_uniformly_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.tendsto_locally_uniformly_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.tendsto_uniformly_on' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.tendsto_locally_uniformly_on' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print analytic_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.has_fpower_series_on_ball /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_summable_aux1 /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_summable_aux2 /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_summable_aux3 /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_radius /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_has_sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.change_origin_eval /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.change_origin /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fpower_series_on_ball.analytic_at_of_mem /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_open_analytic_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n\n-- analysis\\analytic\\composition.lean\n#print formal_multilinear_series.apply_composition /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.apply_composition_ones /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.apply_composition_update /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition_multilinear /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition_multilinear_bound /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition_norm /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_along_composition_nnnorm /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_coeff_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_coeff_zero' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_coeff_zero'' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.id /- _inst_3: normed_space \u219d\n -/\n#print formal_multilinear_series.id_apply_one /- _inst_3: normed_space \u219d\n -/\n#print formal_multilinear_series.id_apply_one' /- _inst_3: normed_space \u219d\n -/\n#print formal_multilinear_series.id_apply_ne_one /- _inst_3: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_id /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.id_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_summable_nnreal /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.le_comp_radius_of_summable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_partial_sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fpower_series_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print analytic_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print formal_multilinear_series.comp_assoc /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n\n-- analysis\\asymptotics.lean\n#print asymptotics.is_O_with.prod_left_same /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O_with.prod_left_fst /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O_with.prod_left_snd /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O.prod_left_fst /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O.prod_left_snd /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_o.prod_left_fst /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_o.prod_left_snd /- _inst_6: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O_refl_left /- _inst_5: normed_group \u219d has_norm\n -/\n#print asymptotics.is_O_with_const_one /- _inst_9: normed_field \u219d monoid_with_zero norm_one_class normed_group nontrivial\n -/\n#print asymptotics.is_o_one_iff /- _inst_9: normed_field \u219d monoid_with_zero normed_group nontrivial\n -/\n#print asymptotics.is_O_one_of_tendsto /- _inst_9: normed_field \u219d monoid_with_zero normed_group nontrivial\n -/\n#print asymptotics.is_O_self_const_mul /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_O_const_mul_left_iff /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_o_const_mul_left_iff /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_O.const_mul_right /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_O_const_mul_right_iff /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_o.const_mul_right /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_o_const_mul_right_iff /- _inst_9: normed_field \u219d group_with_zero normed_ring\n -/\n#print asymptotics.is_O_with.const_smul_left /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_O_const_smul_left_iff /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_o_const_smul_left /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_o_const_smul_left_iff /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_O_const_smul_right /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_o_const_smul_right /- _inst_11: normed_space \u219d\n -/\n#print asymptotics.is_O_with.smul /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print asymptotics.is_O.smul /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print asymptotics.is_O.smul_is_o /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print asymptotics.is_o.smul_is_O /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print asymptotics.is_o.smul /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print asymptotics.is_O_with.eventually_mul_div_cancel /- _inst_9: normed_field \u219d group_with_zero normed_group\n -/\n\n-- analysis\\calculus\\deriv.lean\n#print has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print deriv_within /- _inst_3: normed_space \u219d\n -/\n#print deriv /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_filter_iff_has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_filter.has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at_iff_has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_iff_has_fderiv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at.has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.has_fderiv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_iff_has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at_iff_has_strict_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at.has_strict_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_iff_has_fderiv_at /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_zero_of_not_differentiable_within_at /- _inst_3: normed_space \u219d\n -/\n#print deriv_zero_of_not_differentiable_at /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.eq_deriv /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter_iff_tendsto /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_iff_tendsto /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_iff_tendsto /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter_iff_tendsto_slope /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_iff_tendsto_slope /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_iff_tendsto_slope' /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_iff_tendsto_slope /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_iff_is_o_nhds_zero /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.mono /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.mono /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.differentiable_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.differentiable_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_univ /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_unique /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_inter' /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_inter /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.union /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.nhds_within /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at.has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.deriv /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.deriv_within /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_deriv_within /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_fderiv_within /- _inst_3: normed_space \u219d\n -/\n#print fderiv_deriv /- _inst_3: normed_space \u219d\n -/\n#print deriv_fderiv /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.deriv_within /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_subset /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_univ /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_inter /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_of_open /- _inst_3: normed_space \u219d\n -/\n#print filter.eventually_eq.has_deriv_at_filter_iff /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.congr_mono /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.congr /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n -/\n#print filter.eventually_eq.deriv_within_eq /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_congr /- _inst_3: normed_space \u219d\n -/\n#print filter.eventually_eq.deriv_eq /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter_const /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_const' /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_const /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.has_strict_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.deriv /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.deriv_within /- _inst_3: normed_space \u219d\n -/\n#print linear_map.has_deriv_at_filter /- _inst_3: normed_space \u219d\n -/\n#print linear_map.has_strict_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print linear_map.has_deriv_at /- _inst_3: normed_space \u219d\n -/\n#print linear_map.has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print linear_map.deriv /- _inst_3: normed_space \u219d\n -/\n#print linear_map.deriv_within /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.add /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.add /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_add /- _inst_3: normed_space \u219d\n -/\n#print deriv_add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.add_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.add_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.add_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_add_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_add_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.const_add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.const_add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.const_add /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_const_add /- _inst_3: normed_space \u219d\n -/\n#print deriv_const_add /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.sum /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.sum /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.sum /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.sum /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_sum /- _inst_3: normed_space \u219d\n -/\n#print deriv_sum /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.smul /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.smul /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.smul /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_smul /- _inst_3: normed_space \u219d\n -/\n#print deriv_smul /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.smul_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.smul_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_smul_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_smul_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.const_smul /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.const_smul /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_const_smul /- _inst_3: normed_space \u219d\n -/\n#print deriv_const_smul /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.neg /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.neg /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.neg /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.neg /- _inst_3: normed_space \u219d\n -/\n#print deriv_within.neg /- _inst_3: normed_space \u219d\n -/\n#print deriv.neg /- _inst_3: normed_space \u219d\n -/\n#print deriv.neg' /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.sub /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.sub /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_sub /- _inst_3: normed_space \u219d\n -/\n#print deriv_sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.is_O_sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.sub_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.sub_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.sub_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_sub_const /- _inst_3: normed_space \u219d\n -/\n#print deriv_sub_const /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.const_sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.const_sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.const_sub /- _inst_3: normed_space \u219d\n -/\n#print deriv_within_const_sub /- _inst_3: normed_space \u219d\n -/\n#print deriv_const_sub /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.tendsto_nhds /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.continuous_at /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.prod /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_deriv_within_at.prod /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_deriv_at.prod /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_deriv_at_filter.scomp /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.scomp /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.scomp /- _inst_3: normed_space \u219d\n -/\n#print has_strict_deriv_at.scomp /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at.scomp_has_deriv_within_at /- _inst_3: normed_space \u219d\n -/\n#print deriv_within.scomp /- _inst_3: normed_space \u219d\n -/\n#print deriv.scomp /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_at_filter.comp_has_fderiv_at_filter /- _inst_5: normed_space \u219d\n -/\n#print has_deriv_at.comp_has_fderiv_at /- _inst_5: normed_space \u219d\n -/\n#print has_deriv_at.comp_has_fderiv_within_at /- _inst_5: normed_space \u219d\n -/\n#print has_deriv_within_at.comp_has_fderiv_within_at /- _inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.comp_has_deriv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.comp_has_deriv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.comp_has_deriv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within.comp_deriv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv.comp_deriv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_deriv_within_at.limsup_norm_slope_le /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.limsup_slope_norm_le /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.liminf_right_norm_slope_le /- _inst_3: normed_space \u219d\n -/\n#print has_deriv_within_at.liminf_right_slope_norm_le /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\calculus\\extend_deriv.lean\n#print has_fderiv_at_boundary_of_tendsto_fderiv /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print has_deriv_at_interval_left_endpoint_of_tendsto_deriv /- _inst_2: normed_space \u219d\n -/\n#print has_deriv_at_interval_right_endpoint_of_tendsto_deriv /- _inst_2: normed_space \u219d\n -/\n#print has_deriv_at_of_has_deriv_at_of_ne /- _inst_2: normed_space \u219d\n -/\n\n-- analysis\\calculus\\fderiv.lean\n#print has_fderiv_at_filter /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_zero_of_not_differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_zero_of_not_differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.lim /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.unique_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print unique_diff_within_at.eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print unique_diff_on.eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter_iff_tendsto /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_iff_tendsto /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_iff_tendsto /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_iff_is_o_nhds_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.le_of_lip /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.has_fderiv_at_filter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.is_O_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.is_O_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.lim /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_unique /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_inter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.union /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.nhds_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_at.le_of_lip /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_of_not_mem_closure /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at_inter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on_of_locally_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_subset /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_inter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_of_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_eq_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_mem_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.tendsto_nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.is_O_sub_rev /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.is_O_sub_rev /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print filter.eventually_eq.has_strict_fderiv_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print filter.eventually_eq.has_fderiv_at_filter_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.congr_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.congr_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.congr_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on_congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.fderiv_within_congr_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print filter.eventually_eq.fderiv_within_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print filter.eventually_eq.fderiv_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at_id /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_filter_id /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_id /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at_id /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at_id' /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print differentiable_id /- _inst_3: normed_space \u219d\n -/\n#print differentiable_id' /- _inst_3: normed_space \u219d\n -/\n#print differentiable_on_id /- _inst_3: normed_space \u219d\n -/\n#print fderiv_id /- _inst_3: normed_space \u219d\n -/\n#print fderiv_id' /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_id /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_id' /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_const_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_const_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.has_strict_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.has_fderiv_at_filter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.has_fderiv_at_filter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_within_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at.comp_has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at.comp' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.comp_differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv_within.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv.comp_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_on.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable.comp_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable.iterate /- _inst_3: normed_space \u219d\n -/\n#print differentiable_on.iterate /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at_filter.iterate /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at.iterate /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at.iterate /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at.iterate /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.iterate /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at.iterate /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at_filter.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_within_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_on.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.fderiv_prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.fderiv_within_prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at_filter_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_within_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_on_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv_within_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at_filter_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_fderiv_within_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_on_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print fderiv_within_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print has_fderiv_at.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print differentiable_at.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print has_strict_fderiv_at.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_const_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_add_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_const_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_sub_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_const_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.has_strict_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.continuous_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.continuous_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.is_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_within_at.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_at.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable_on.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print differentiable.smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_within_smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_smul_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mul /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at.mul /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at.mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at.mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_on.mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable.mul /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_mul /- _inst_3: normed_space \u219d\n -/\n#print fderiv_mul /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mul_const /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at.mul_const /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at.mul_const /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at.mul_const /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.mul_const /- _inst_3: normed_space \u219d\n -/\n#print differentiable_on.mul_const /- _inst_3: normed_space \u219d\n -/\n#print differentiable.mul_const /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_mul_const /- _inst_3: normed_space \u219d\n -/\n#print fderiv_mul_const /- _inst_3: normed_space \u219d\n -/\n#print has_strict_fderiv_at.const_mul /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_within_at.const_mul /- _inst_3: normed_space \u219d\n -/\n#print has_fderiv_at.const_mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_within_at.const_mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_at.const_mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable_on.const_mul /- _inst_3: normed_space \u219d\n -/\n#print differentiable.const_mul /- _inst_3: normed_space \u219d\n -/\n#print fderiv_within_const_mul /- _inst_3: normed_space \u219d\n -/\n#print fderiv_const_mul /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_equiv.has_strict_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_differentiable_within_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_differentiable_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_differentiable_on_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_differentiable_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_has_fderiv_within_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_has_strict_fderiv_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_has_fderiv_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_has_fderiv_within_at_iff' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_has_fderiv_at_iff' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.of_local_left_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.of_local_left_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at.of_local_homeomorph /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_at_filter_real_equiv /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print has_fderiv_at.lim_real /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print has_fderiv_within_at.maps_to_tangent_cone /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.unique_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_fderiv_within_at.unique_diff_within_at_of_continuous_linear_equiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.unique_diff_on_preimage_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print has_fderiv_at.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print has_fderiv_within_at.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print differentiable_at.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print differentiable_within_at.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print differentiable_on.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print differentiable.restrict_scalars /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: is_scalar_tower \u219d\n_inst_9: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print has_strict_fderiv_at.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_fderiv_within_at.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_fderiv_at.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_within_at.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_at.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_on.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable.smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_within_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_strict_fderiv_at.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_fderiv_within_at.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_fderiv_at.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_within_at.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_at.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_on.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable.smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_within_smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_smul_algebra_const /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_strict_fderiv_at.const_smul_algebra /- _inst_2: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_algebra \u219d has_scalar\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d has_scalar\n_inst_9: is_scalar_tower \u219d has_scalar\n -/\n#print has_fderiv_at_filter.const_smul_algebra /- _inst_2: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_algebra \u219d has_scalar\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d has_scalar\n_inst_9: is_scalar_tower \u219d has_scalar\n -/\n#print has_fderiv_within_at.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print has_fderiv_at.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_within_at.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_at.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable_on.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print differentiable.const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_within_const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n#print fderiv_const_smul_algebra /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_9: is_scalar_tower \u219d\n -/\n\n-- analysis\\calculus\\fderiv_measurable.lean\n#print continuous_linear_map.measurable_space /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.borel_space /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_apply' /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_apply\u2082 /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.A /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.B /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.D /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.is_open_A /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.is_open_B /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.A_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.le_of_mem_A /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.mem_A_of_differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.norm_sub_le_of_mem_A /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.differentiable_set_subset_D /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.D_subset_differentiable_set /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print fderiv_measurable_aux.differentiable_set_eq_D /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_measurable_set_of_differentiable_at_of_is_complete /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_measurable_set_of_differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print measurable_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print measurable_fderiv_apply_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print measurable_deriv /- _inst_5: normed_space \u219d\n -/\n\n-- analysis\\calculus\\implicit.lean\n#print implicit_function_data.prod_fun /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.prod_fun_apply /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.has_strict_fderiv_at /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.to_local_homeomorph /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.implicit_function /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.to_local_homeomorph_coe /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.to_local_homeomorph_apply /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.pt_mem_to_local_homeomorph_source /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.map_pt_mem_to_local_homeomorph_target /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.prod_map_implicit_function /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.left_map_implicit_function /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.right_map_implicit_function /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.implicit_function_apply_image /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print implicit_function_data.implicit_function_has_strict_fderiv_at /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_function_data_of_complemented /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_function_of_complemented /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_fst /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_apply /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_apply_ker /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_of_complemented_self /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mem_implicit_to_local_homeomorph_of_complemented_source /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mem_implicit_to_local_homeomorph_of_complemented_target /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.map_implicit_function_of_complemented_eq /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.eq_implicit_function_of_complemented /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_implicit_function_of_complemented /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_function /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_fst /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_apply_ker /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.implicit_to_local_homeomorph_self /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mem_implicit_to_local_homeomorph_source /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mem_implicit_to_local_homeomorph_target /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.map_implicit_function_eq /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.eq_implicit_function /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_implicit_function /- _inst_4: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n\n-- analysis\\calculus\\inverse.lean\n#print approximates_linear_on /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.mono_num /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.mono_set /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.lipschitz_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.lipschitz /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.antilipschitz /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.injective /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inj_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.to_local_equiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_dist_self /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_dist_self_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_fixed_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_contracts_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.inverse_approx_map_maps_to /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.surj_on_closed_ball /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.to_local_homeomorph /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.to_local_homeomorph_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.to_local_homeomorph_source /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.to_local_homeomorph_target /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print approximates_linear_on.closed_ball_subset_target /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.approximates_deriv_on_nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.approximates_deriv_on_open_nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_local_homeomorph /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_local_homeomorph_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.mem_to_local_homeomorph_source /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.image_mem_to_local_homeomorph_target /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.local_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.eventually_left_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.local_inverse_apply_image /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.eventually_right_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.local_inverse_continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.local_inverse_tendsto /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.local_inverse_unique /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_local_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_strict_fderiv_at.to_local_left_inverse /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.to_local_homeomorph /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.to_local_homeomorph_coe /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.mem_to_local_homeomorph_source /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.image_mem_to_local_homeomorph_target /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.local_inverse /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.local_inverse_apply_image /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_diff_at.to_local_inverse /- _inst_11: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n\n-- analysis\\calculus\\iterated_deriv.lean\n#print iterated_deriv /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_univ /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_eq_iterated_fderiv_within /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_eq_equiv_comp /- _inst_3: normed_space \u219d\n -/\n#print iterated_fderiv_within_eq_equiv_comp /- _inst_3: normed_space \u219d\n -/\n#print iterated_fderiv_within_apply_eq_iterated_deriv_within_mul_prod /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_zero /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_one /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on_of_continuous_on_differentiable_on_deriv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on_of_differentiable_on_deriv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_iterated_deriv_within /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on.differentiable_on_iterated_deriv_within /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on_iff_continuous_on_differentiable_on_deriv /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_succ /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_eq_iterate /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_within_succ' /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_eq_iterated_fderiv /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_eq_equiv_comp /- _inst_3: normed_space \u219d\n -/\n#print iterated_fderiv_eq_equiv_comp /- _inst_3: normed_space \u219d\n -/\n#print iterated_fderiv_apply_eq_iterated_deriv_mul_prod /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_zero /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_one /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_iff_iterated_deriv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_of_differentiable_iterated_deriv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.continuous_iterated_deriv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.differentiable_iterated_deriv /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_succ /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_eq_iterate /- _inst_3: normed_space \u219d\n -/\n#print iterated_deriv_succ' /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\calculus\\local_extr.lean\n#print pos_tangent_cone_at /- _inst_2: normed_space \u219d\n -/\n#print pos_tangent_cone_at_mono /- _inst_2: normed_space \u219d\n -/\n#print mem_pos_tangent_cone_at_of_segment_subset /- _inst_2: normed_space \u219d\n -/\n#print pos_tangent_cone_at_univ /- _inst_2: normed_space \u219d\n -/\n#print is_local_max_on.has_fderiv_within_at_nonpos /- _inst_2: normed_space \u219d\n -/\n#print is_local_max_on.fderiv_within_nonpos /- _inst_2: normed_space \u219d\n -/\n#print is_local_max_on.has_fderiv_within_at_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_max_on.fderiv_within_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_min_on.has_fderiv_within_at_nonneg /- _inst_2: normed_space \u219d\n -/\n#print is_local_min_on.fderiv_within_nonneg /- _inst_2: normed_space \u219d\n -/\n#print is_local_min_on.has_fderiv_within_at_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_min_on.fderiv_within_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_min.has_fderiv_at_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_min.fderiv_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_max.has_fderiv_at_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_max.fderiv_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_extr.has_fderiv_at_eq_zero /- _inst_2: normed_space \u219d\n -/\n#print is_local_extr.fderiv_eq_zero /- _inst_2: normed_space \u219d\n -/\n\n-- analysis\\calculus\\mean_value.lean\n#print image_norm_le_of_norm_deriv_right_lt_deriv_boundary' /- _inst_2: normed_space \u219d\n -/\n#print image_norm_le_of_norm_deriv_right_lt_deriv_boundary /- _inst_2: normed_space \u219d\n -/\n#print image_norm_le_of_norm_deriv_right_le_deriv_boundary' /- _inst_2: normed_space \u219d\n -/\n#print image_norm_le_of_norm_deriv_right_le_deriv_boundary /- _inst_2: normed_space \u219d\n -/\n#print norm_image_sub_le_of_norm_deriv_right_le_segment /- _inst_2: normed_space \u219d\n -/\n#print norm_image_sub_le_of_norm_deriv_le_segment' /- _inst_2: normed_space \u219d\n -/\n#print norm_image_sub_le_of_norm_deriv_le_segment /- _inst_2: normed_space \u219d\n -/\n#print norm_image_sub_le_of_norm_deriv_le_segment_01' /- _inst_2: normed_space \u219d\n -/\n#print norm_image_sub_le_of_norm_deriv_le_segment_01 /- _inst_2: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_has_fderiv_within_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_has_fderiv_within_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_fderiv_within_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_fderiv_within_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_fderiv_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_fderiv_le /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_has_fderiv_within_le' /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_fderiv_within_le' /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_fderiv_le' /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.is_const_of_fderiv_within_eq_zero /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print is_const_of_fderiv_eq_zero /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_has_deriv_within_le /- _inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_has_deriv_within_le /- _inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_deriv_within_le /- _inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_deriv_within_le /- _inst_4: normed_space \u219d\n -/\n#print convex.norm_image_sub_le_of_norm_deriv_le /- _inst_4: normed_space \u219d\n -/\n#print convex.lipschitz_on_with_of_norm_deriv_le /- _inst_4: normed_space \u219d\n -/\n#print domain_mvt /- _inst_2: normed_space \u219d\n -/\n#print strict_fderiv_of_cont_diff /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n\n-- analysis\\calculus\\tangent_cone.lean\n#print tangent_cone_at /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d has_scalar\n -/\n#print unique_diff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_on /- _inst_3: normed_space \u219d\n -/\n#print tangent_cone_univ /- _inst_3: normed_space \u219d\n -/\n#print tangent_cone_mono /- _inst_3: normed_space \u219d\n -/\n#print tangent_cone_at.lim_zero /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n -/\n#print tangent_cone_mono_nhds /- _inst_3: normed_space \u219d\n -/\n#print tangent_cone_congr /- _inst_3: normed_space \u219d\n -/\n#print tangent_cone_inter_nhds /- _inst_3: normed_space \u219d\n -/\n#print subset_tangent_cone_prod_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print subset_tangent_cone_prod_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mem_tangent_cone_of_segment_subset /- _inst_7: normed_space \u219d\n -/\n#print unique_diff_on.unique_diff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at_univ /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_on_univ /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_on_empty /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.mono_nhds /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.mono /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at_congr /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at_inter /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.inter /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at_inter' /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.inter' /- _inst_3: normed_space \u219d\n -/\n#print is_open.unique_diff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_on.inter /- _inst_3: normed_space \u219d\n -/\n#print is_open.unique_diff_on /- _inst_3: normed_space \u219d\n -/\n#print unique_diff_within_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print unique_diff_on.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print unique_diff_on_convex /- _inst_7: normed_space \u219d\n -/\n\n-- analysis\\calculus\\times_cont_diff.lean\n#print formal_multilinear_series /- _inst_8: nondiscrete_normed_field \u219d normed_field\n_inst_10: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print formal_multilinear_series.inhabited /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.module /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.shift /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.unshift /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.zero_eq' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on_zero_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on_top_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on_succ_iff_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on_succ_iff_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_nat /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_iff_forall_nat_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_top /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.congr_of_eventually_eq' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print filter.eventually_eq.times_cont_diff_within_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.mono_of_mem /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.congr_nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_congr_nhds /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_inter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.differentiable_within_at' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_succ_iff_has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_iff_forall_nat_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_top /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_all_iff_nat /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.congr_mono /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_of_locally_times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_succ_iff_has_fderiv_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print ftaylor_series_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_zero_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_zero_eq_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_succ_apply_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_succ_eq_comp_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_succ_apply_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_succ_eq_comp_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_one_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_congr /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_inter_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_inter' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_inter /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.eq_ftaylor_series_of_unique_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.ftaylor_series_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_of_continuous_on_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_of_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_iterated_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.differentiable_on_iterated_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_iff_continuous_on_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_succ_iff_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_succ_iff_fderiv_of_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_top_iff_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_top_iff_fderiv_of_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.fderiv_of_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_fderiv_within /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_fderiv_of_open /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_fderiv_within_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.zero_eq' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on_univ_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.has_ftaylor_series_up_to_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_zero_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_succ_iff_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_top /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_succ_iff_has_fderiv_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_iff_times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_top /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_all_iff_nat /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.of_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print ftaylor_series /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_zero_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_zero_eq_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_succ_apply_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_succ_eq_comp_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print ftaylor_series_within_univ /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_succ_apply_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_succ_eq_comp_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_one_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_iff_ftaylor_series /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_iff_continuous_differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_of_differentiable_iterated_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_succ_iff_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_top_iff_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.continuous_fderiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.continuous_fderiv_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print iterated_fderiv_within_zero_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_zero_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_of_subsingleton /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_of_subsingleton /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_of_subsingleton /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_of_subsingleton /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.times_cont_diff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.times_cont_diff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at_snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on_id /- _inst_3: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.times_cont_diff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.continuous_linear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_within_at.continuous_linear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_at.continuous_linear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_linear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.continuous_linear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_times_cont_diff_within_at_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.comp_times_cont_diff_on_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.comp_continuous_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_within_at.comp_continuous_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.comp_continuous_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.comp_continuous_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.times_cont_diff_within_at_comp_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_equiv.times_cont_diff_on_comp_iff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_within_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_at.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.prod /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.comp' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.comp_times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_within_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_within_at.comp' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_at.comp_times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_at.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.comp_times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.comp_times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on_fderiv_within_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.times_cont_diff_fderiv_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_add /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_neg /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.sum /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.mul /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at.mul /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on.mul /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.mul /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_within_at.div_const /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at.div_const /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_on.div_const /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.div_const /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.pow /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_smul /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_within_at.prod_map' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_within_at.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_on.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_15: normed_space \u219d\n -/\n#print times_cont_diff_at.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_at.prod_map' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff.prod_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_at_inv /- _inst_10: normed_field \u219d division_ring normed_ring\n -/\n#print times_cont_diff_within_at.inv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at.inv /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_within_at.div /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at.div /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff.div /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_at_map_inverse /- _inst_3: normed_space \u219d normed_algebra\n_inst_5: normed_space \u219d\n -/\n#print times_cont_diff_at.of_local_homeomorph /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print has_ftaylor_series_up_to_on.has_strict_fderiv_at /- _inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_at.has_strict_fderiv_at /- _inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_at.has_strict_fderiv_at' /- _inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff.has_strict_fderiv_at /- _inst_9: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print times_cont_diff_on_succ_iff_deriv_within /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_succ_iff_deriv_of_open /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_top_iff_deriv_within /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on_top_iff_deriv_of_open /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.deriv_within /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.deriv_of_open /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_deriv_within /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_on.continuous_on_deriv_of_open /- _inst_5: normed_space \u219d\n -/\n#print times_cont_diff_succ_iff_deriv /- _inst_5: normed_space \u219d\n -/\n#print formal_multilinear_series.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print has_ftaylor_series_up_to_on.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print times_cont_diff_within_at.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print times_cont_diff_on.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print times_cont_diff_at.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print times_cont_diff.restrict_scalars /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_10: normed_space \u219d\n_inst_11: is_scalar_tower \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n\n-- analysis\\complex\\basic.lean\n#print complex.finite_dimensional.proper /- _inst_2: normed_space \u219d\n -/\n#print complex.normed_space.restrict_scalars_real /- _inst_2: normed_space \u219d\n -/\n#print complex.continuous_linear_map.real_smul_complex /- _inst_2: normed_space \u219d\n_inst_4: normed_space \u219d\n -/\n\n-- analysis\\convex\\basic.lean\n#print convex.combo_self /- _inst_5: linear_ordered_field \u219d monoid distrib\n -/\n#print convex_on /- _inst_6: ordered_add_comm_monoid \u219d add_comm_monoid has_le\n_inst_7: semimodule \u219d\n -/\n#print concave_on /- _inst_6: ordered_add_comm_monoid \u219d add_comm_monoid has_le\n_inst_7: semimodule \u219d\n -/\n#print neg_convex_on_iff /- _inst_9: semimodule \u219d\n -/\n#print neg_concave_on_iff /- _inst_9: semimodule \u219d\n -/\n#print convex_on_const /- _inst_7: semimodule \u219d\n -/\n#print concave_on_const /- _inst_7: semimodule \u219d\n -/\n#print convex_on_iff_div /- _inst_7: semimodule \u219d\n -/\n#print concave_on_iff_div /- _inst_7: semimodule \u219d\n -/\n#print linear_order.convex_on_of_lt /- _inst_7: semimodule \u219d\n -/\n#print linear_order.concave_on_of_lt /- _inst_7: semimodule \u219d\n -/\n#print convex_on.subset /- _inst_7: semimodule \u219d\n -/\n#print concave_on.subset /- _inst_7: semimodule \u219d\n -/\n#print convex_on.add /- _inst_7: semimodule \u219d\n -/\n#print concave_on.add /- _inst_7: semimodule \u219d\n -/\n#print convex_on.smul /- _inst_7: semimodule \u219d\n -/\n#print concave_on.smul /- _inst_7: semimodule \u219d\n -/\n#print convex_on.le_on_segment' /- _inst_8: linear_ordered_add_comm_group \u219d ordered_add_comm_monoid linear_order\n_inst_9: semimodule \u219d\n -/\n#print concave_on.le_on_segment' /- _inst_9: semimodule \u219d\n -/\n#print convex_on.le_on_segment /- _inst_9: semimodule \u219d\n -/\n#print concave_on.le_on_segment /- _inst_9: semimodule \u219d\n -/\n#print convex_on.convex_le /- _inst_7: semimodule \u219d\n -/\n#print concave_on.concave_le /- _inst_7: semimodule \u219d\n -/\n#print convex_on.convex_lt /- _inst_9: semimodule \u219d\n -/\n#print concave_on.convex_lt /- _inst_9: semimodule \u219d\n -/\n#print convex_on.convex_epigraph /- _inst_8: ordered_add_comm_group \u219d ordered_add_comm_monoid add_comm_group\n_inst_9: semimodule \u219d\n -/\n#print concave_on.convex_hypograph /- _inst_9: semimodule \u219d\n -/\n#print convex_on_iff_convex_epigraph /- _inst_9: semimodule \u219d\n -/\n#print concave_on_iff_convex_hypograph /- _inst_9: semimodule \u219d\n -/\n#print convex_on.comp_affine_map /- _inst_7: semimodule \u219d\n -/\n#print concave_on.comp_affine_map /- _inst_7: semimodule \u219d\n -/\n#print convex_on.comp_linear_map /- _inst_7: semimodule \u219d\n -/\n#print concave_on.comp_linear_map /- _inst_7: semimodule \u219d\n -/\n#print convex_on.translate_right /- _inst_7: semimodule \u219d\n -/\n#print concave_on.translate_right /- _inst_7: semimodule \u219d\n -/\n#print convex_on.translate_left /- _inst_7: semimodule \u219d\n -/\n#print concave_on.translate_left /- _inst_7: semimodule \u219d\n -/\n\n-- analysis\\convex\\caratheodory.lean\n#print caratheodory.mem_convex_hull_erase /- _inst_4: decidable_eq \u219d\n -/\n#print caratheodory.step /- _inst_4: decidable_eq \u219d\n -/\n\n-- analysis\\convex\\cone.lean\n#print convex_cone.to_ordered_semimodule /- _inst_7: ordered_add_comm_group \u219d ordered_add_comm_monoid add_comm_group\n_inst_8: semimodule \u219d\n -/\n#print convex_cone.positive_cone /- _inst_8: semimodule \u219d\n -/\n#print convex_cone.salient_of_positive_cone /- _inst_8: semimodule \u219d\n -/\n#print convex_cone.pointed_of_positive_cone /- _inst_8: semimodule \u219d\n -/\n\n-- analysis\\convex\\extrema.lean\n#print is_min_on.of_is_local_min_on_of_convex_on_Icc /- _inst_6: linear_ordered_add_comm_group \u219d linear_order ordered_cancel_add_comm_monoid\n_inst_7: semimodule \u219d\n -/\n#print is_min_on.of_is_local_min_on_of_convex_on /- _inst_7: semimodule \u219d\n -/\n#print is_max_on.of_is_local_max_on_of_concave_on /- _inst_7: semimodule \u219d\n -/\n#print is_min_on.of_is_local_min_of_convex_univ /- _inst_7: semimodule \u219d\n -/\n#print is_max_on.of_is_local_max_of_convex_univ /- _inst_7: semimodule \u219d\n -/\n\n-- analysis\\convex\\integral.lean\n#print convex.smul_integral_mem /- _inst_3: normed_space \u219d\n -/\n#print convex.integral_mem /- _inst_3: normed_space \u219d\n -/\n#print convex_on.map_smul_integral_le /- _inst_3: normed_space \u219d\n -/\n#print convex_on.map_integral_le /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\convex\\topology.lean\n#print convex.closure /- _inst_4: topological_add_group \u219d has_continuous_add\n -/\n#print convex_on_dist /- _inst_2: normed_space \u219d\n -/\n#print convex_ball /- _inst_2: normed_space \u219d\n -/\n#print convex_closed_ball /- _inst_2: normed_space \u219d\n -/\n#print convex_hull_exists_dist_ge /- _inst_2: normed_space \u219d\n -/\n#print convex_hull_exists_dist_ge2 /- _inst_2: normed_space \u219d\n -/\n#print convex_hull_ediam /- _inst_2: normed_space \u219d\n -/\n#print convex_hull_diam /- _inst_2: normed_space \u219d\n -/\n#print bounded_convex_hull /- _inst_2: normed_space \u219d\n -/\n#print convex.is_path_connected /- _inst_2: normed_space \u219d\n -/\n#print normed_space.path_connected /- _inst_2: normed_space \u219d\n -/\n#print normed_space.loc_path_connected /- _inst_2: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\add_torsor.lean\n#print isometric.dist_point_reflection_self /- _inst_5: normed_space \u219d\n -/\n#print isometric.point_reflection_fixed_iff /- _inst_5: normed_space \u219d\n -/\n#print isometric.dist_point_reflection_self_real /- _inst_4: normed_space \u219d\n -/\n#print isometric.point_reflection_midpoint_left /- _inst_4: normed_space \u219d\n -/\n#print isometric.point_reflection_midpoint_right /- _inst_4: normed_space \u219d\n -/\n#print dist_center_homothety /- _inst_8: normed_space \u219d\n -/\n#print dist_homothety_center /- _inst_8: normed_space \u219d\n -/\n#print dist_homothety_self /- _inst_8: normed_space \u219d\n -/\n#print dist_self_homothety /- _inst_8: normed_space \u219d\n -/\n#print dist_left_midpoint /- _inst_8: normed_space \u219d\n -/\n#print dist_midpoint_left /- _inst_8: normed_space \u219d\n -/\n#print dist_midpoint_right /- _inst_8: normed_space \u219d\n -/\n#print dist_right_midpoint /- _inst_8: normed_space \u219d\n -/\n#print affine_map.of_map_midpoint /- _inst_7: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\banach.lean\n#print exists_approx_preimage_norm_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print exists_preimage_norm_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print open_mapping /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print linear_equiv.continuous_symm /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print linear_equiv.to_continuous_linear_equiv_of_continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print linear_equiv.coe_fn_to_continuous_linear_equiv_of_continuous_symm /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.of_bijective /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.coe_fn_of_bijective /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.of_bijective_symm_apply_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.of_bijective_apply_symm_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\basic.lean\n#print eventually_ne_of_tendsto_norm_at_top /- _inst_1: normed_group \u219d has_norm\n -/\n#print normed_top_monoid /- _inst_1: normed_group \u219d has_continuous_add topological_space has_add\n -/\n#print normed_top_group /- _inst_1: normed_group \u219d add_group topological_space topological_add_group\n -/\n#print finset.norm_prod_le' /- _inst_2: normed_comm_ring \u219d comm_monoid normed_ring\n -/\n#print finset.norm_prod_le /- _inst_2: normed_comm_ring \u219d comm_monoid normed_ring\n -/\n#print units.norm_pos /- _inst_1: normed_ring \u219d monoid_with_zero normed_group\n -/\n#print normed_top_ring /- _inst_1: normed_ring \u219d normed_group ring has_continuous_mul\n -/\n#print normed_field.nhds_within_is_unit_ne_bot /- _inst_1: nondiscrete_normed_field \u219d group_with_zero topological_space filter.ne_bot\n -/\n#print norm_smul /- _inst_3: normed_space \u219d\n -/\n#print dist_smul /- _inst_3: normed_space \u219d\n -/\n#print nnnorm_smul /- _inst_3: normed_space \u219d\n -/\n#print nndist_smul /- _inst_3: normed_space \u219d\n -/\n#print norm_smul_of_nonneg /- _inst_3: normed_space \u219d\n -/\n#print normed_space.topological_vector_space /- _inst_4: normed_space \u219d\n -/\n#print closure_ball /- _inst_7: normed_space \u219d\n -/\n#print frontier_ball /- _inst_7: normed_space \u219d\n -/\n#print interior_closed_ball /- _inst_7: normed_space \u219d\n -/\n#print interior_closed_ball' /- _inst_7: normed_space \u219d\n -/\n#print frontier_closed_ball /- _inst_7: normed_space \u219d\n -/\n#print frontier_closed_ball' /- _inst_7: normed_space \u219d\n -/\n#print rescale_to_shell /- _inst_4: normed_space \u219d\n -/\n#print prod.normed_space /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print submodule.normed_space /- _inst_9: normed_space \u219d\n -/\n#print normed_space.restrict_scalars /- _inst_5: normed_space \u219d\n -/\n#print semimodule.restrict_scalars.normed_space_orig /- I: normed_space \u219d\n -/\n#print restrict_scalars.normed_space /- _inst_5: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\bounded_linear_maps.lean\n#print is_linear_map.with_bound /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.is_bounded_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.to_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.to_continuous_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.id /- _inst_3: normed_space \u219d\n -/\n#print is_bounded_linear_map.fst /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.snd /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.smul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.neg /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_linear_map.tendsto /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.lim_zero_bounded_linear_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.is_O_id /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map.is_O_comp /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_linear_map.is_O_sub /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_linear_map_prod_iso /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_linear_map_prod_multilinear /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print is_bounded_linear_map_continuous_multilinear_map_comp_linear /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print is_bounded_bilinear_map.is_O /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.is_O_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.is_O' /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.map_sub_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.map_sub_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.is_bounded_linear_map_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.is_bounded_linear_map_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_smul /- _inst_3: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_smul_algebra /- _inst_9: normed_algebra \u219d algebra\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print is_bounded_bilinear_map_comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.is_bounded_linear_map_comp_left /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.is_bounded_linear_map_comp_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_smul_right /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_comp_multilinear /- _inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print is_bounded_bilinear_map.linear_deriv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.deriv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map_deriv_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print is_bounded_bilinear_map.is_bounded_linear_map_deriv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\complemented.lean\n#print continuous_linear_map.ker_closed_complemented_of_finite_dimensional_range /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.equiv_prod_of_surjective_of_is_compl /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.coe_equiv_prod_of_surjective_of_is_compl /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.equiv_prod_of_surjective_of_is_compl_to_linear_equiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.equiv_prod_of_surjective_of_is_compl_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print subspace.prod_equiv_of_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.linear_proj_of_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.coe_prod_equiv_of_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.coe_prod_equiv_of_closed_compl_symm /- _inst_3: normed_space \u219d\n -/\n#print subspace.coe_continuous_linear_proj_of_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.coe_continuous_linear_proj_of_closed_compl' /- _inst_3: normed_space \u219d\n -/\n#print subspace.closed_complemented_of_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.closed_complemented_iff_has_closed_compl /- _inst_3: normed_space \u219d\n -/\n#print subspace.closed_complemented_of_quotient_finite_dimensional /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\dual.lean\n#print normed_space.dual /- _inst_1: nondiscrete_normed_field \u219d normed_field\n_inst_3: normed_space \u219d\n -/\n#print normed_space.dual.inhabited /- _inst_3: normed_space \u219d\n -/\n#print normed_space.inclusion_in_double_dual' /- _inst_3: normed_space \u219d\n -/\n#print normed_space.dual_def /- _inst_3: normed_space \u219d\n -/\n#print normed_space.double_dual_bound /- _inst_3: normed_space \u219d\n -/\n#print normed_space.inclusion_in_double_dual /- _inst_3: normed_space \u219d\n -/\n#print normed_space.norm_le_dual_bound /- _inst_3: normed_space \u219d\n -/\n#print normed_space.inclusion_in_double_dual_isometry /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\enorm.lean\n#print enorm.map_smul /- _inst_3: vector_space \u219d\n -/\n#print enorm.finite_subspace.normed_space /- _inst_3: vector_space \u219d\n -/\n\n-- analysis\\normed_space\\extend.lean\n#print linear_map.extend_to_\ud835\udd5c /- _inst_3: normed_space \u219d\n -/\n#print norm_bound /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.extend_to_\ud835\udd5c /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\finite_dimension.lean\n#print linear_map.continuous_on_pi /- _inst_2: normed_field \u219d field topological_space\n_inst_6: topological_add_group \u219d has_continuous_add\n -/\n#print continuous_equiv_fun_basis /- _inst_3: normed_space \u219d\n -/\n#print linear_map.continuous_of_finite_dimensional /- _inst_3: normed_space \u219d\n -/\n#print linear_map.to_continuous_linear_map /- _inst_3: normed_space \u219d\n -/\n#print linear_equiv.to_continuous_linear_equiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_basis.constrL /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_basis.coe_constrL /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_basis.equiv_funL /- _inst_3: normed_space \u219d\n -/\n#print is_basis.constrL_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_basis.constrL_basis /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print is_basis.sup_norm_le_norm /- _inst_3: normed_space \u219d\n -/\n#print is_basis.op_norm_le /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.topological_space.second_countable_topology /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_14: topological_space.second_countable_topology \u219d topological_space.separable_space\n -/\n#print finite_dimensional.complete /- _inst_3: normed_space \u219d\n -/\n#print submodule.complete_of_finite_dimensional /- _inst_3: normed_space \u219d\n -/\n#print submodule.closed_of_finite_dimensional /- _inst_3: normed_space \u219d\n -/\n#print continuous_linear_map.exists_right_inverse_of_surjective /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print closed_embedding_smul_left /- _inst_3: normed_space \u219d\n -/\n#print is_closed_map_smul_left /- _inst_3: normed_space \u219d\n -/\n#print finite_dimensional.proper /- _inst_3: normed_space \u219d\n -/\n#print finite_dimensional.proper_real /- _inst_2: normed_space \u219d\n -/\n#print summable_norm_iff /- _inst_2: normed_space \u219d complete_space\n -/\n\n-- analysis\\normed_space\\hahn_banach.lean\n#print norm' /- _inst_1: nondiscrete_normed_field \u219d normed_ring\n_inst_3: normed_group \u219d has_norm\n -/\n#print real.exists_extension_norm_eq /- _inst_2: normed_space \u219d\n -/\n#print exists_extension_norm_eq /- _inst_3: normed_space \u219d\n -/\n#print coord_norm' /- _inst_3: normed_space \u219d\n -/\n#print exists_dual_vector /- _inst_3: normed_space \u219d\n -/\n#print exists_dual_vector' /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\inner_product.lean\n#print inner_product_space.of_core.to_has_inner /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.norm_sq /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_conj_sym /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_nonneg /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_nonneg_im /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_im_zero /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_add_left /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_add_right /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_norm_sq_eq_inner_self /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_re_symm /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_im_symm /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_smul_left /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_smul_right /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_zero_left /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_zero_right /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_eq_zero /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_re_to_K /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_abs_conj_sym /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_neg_left /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_neg_right /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_sub_left /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_sub_right /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_mul_conj_re_abs /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_add_add_self /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_sub_sub_self /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_mul_inner_self_le /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.to_has_norm /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.norm_eq_sqrt_inner /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.inner_self_eq_norm_square /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.sqrt_norm_sq_eq_norm /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.abs_inner_le_norm /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.to_normed_group /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core.to_normed_space /- _inst_3: semimodule \u219d\n -/\n#print inner_product_space.of_core /- _inst_3: semimodule \u219d\n -/\n#print euclidean_space /- _inst_4: is_R_or_C \u219d\n_inst_5: fintype \u219d\n -/\n#print has_inner.is_R_or_C_to_real /- _inst_2: inner_product_space \u219d has_inner\n -/\n#print is_bounded_bilinear_map_inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print times_cont_diff_inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print times_cont_diff_at_inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print differentiable_inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print continuous_inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print times_cont_diff_within_at.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_at.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_on.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_within_at.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_at.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable_on.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n#print differentiable.inner /- _inst_4: normed_space \u219d\n_inst_5: is_scalar_tower \u219d\n_inst_7: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\mazur_ulam.lean\n#print isometric.midpoint_fixed /- _inst_2: normed_space \u219d\n -/\n#print isometric.map_midpoint /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.to_real_linear_equiv_of_map_zero /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.coe_to_real_linear_equiv_of_map_zero /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.coe_to_real_linear_equiv_of_map_zero_symm /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.to_real_linear_equiv /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.to_real_linear_equiv_apply /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.to_real_linear_equiv_symm_apply /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.to_affine_equiv /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print isometric.coe_to_affine_equiv /- _inst_2: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\multilinear.lean\n#print multilinear_map.exists_bound_of_continuous /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.norm_image_sub_le_of_bound' /- _inst_1: decidable_eq \u219d\n_inst_3: nondiscrete_normed_field \u219d normed_field\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.norm_image_sub_le_of_bound /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.continuous_of_bound /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.mk_continuous /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.restr_norm_le /- _inst_3: nondiscrete_normed_field \u219d normed_field\n_inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.bound /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm /- _inst_1: decidable_eq \u219d\n_inst_3: nondiscrete_normed_field \u219d normed_field\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.has_op_norm /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_def /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.bounds_nonempty /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.bounds_bdd_below /- _inst_1: decidable_eq \u219d\n_inst_3: nondiscrete_normed_field \u219d normed_field\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm_nonneg /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.le_op_norm /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.ratio_le_op_norm /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.unit_le_op_norm /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm_le_bound /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm_add_le /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm_zero_iff /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.op_norm_smul_le /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n_inst_12: nondiscrete_normed_field \u219d normed_field\n_inst_13: normed_algebra \u219d algebra\n_inst_14: normed_space \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.op_norm_neg /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.to_normed_group /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.to_normed_space /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n_inst_14: normed_space \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.norm_restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n_inst_13: normed_algebra \u219d has_scalar\n_inst_14: normed_space \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.restrict_scalars_linear /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n_inst_14: normed_space \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.continuous_restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n_inst_14: normed_space \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.norm_image_sub_le_of_bound' /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_image_sub_le_of_bound /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.continuous_eval /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.continuous_eval_left /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.has_sum_eval /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.complete_space /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print multilinear_map.mk_continuous_norm_le /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.restr /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_restr /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.mk_pi_algebra /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_multilinear_map.mk_pi_algebra_apply /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_multilinear_map.norm_mk_pi_algebra_le /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_multilinear_map.norm_mk_pi_algebra_of_empty /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_multilinear_map.norm_mk_pi_algebra /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_multilinear_map.mk_pi_field /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.mk_pi_field_apply /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.mk_pi_field_apply_one_eq_self /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.pi_field_equiv_aux /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.pi_field_equiv /- _inst_1: decidable_eq \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_linear_map.norm_map_tail_le /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_map_init_le /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_map_cons_le /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.norm_map_snoc_le /- _inst_11: normed_space \u219d\n -/\n#print continuous_linear_map.uncurry_left /- _inst_11: normed_space \u219d\n -/\n#print continuous_linear_map.uncurry_left_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_left /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_left_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_linear_map.curry_uncurry_left /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry_curry_left /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_left_norm /- _inst_11: normed_space \u219d\n -/\n#print continuous_linear_map.uncurry_left_norm /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_left_equiv_aux /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_left_equiv /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_left_equiv_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_left_equiv_symm_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry_right /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry_right_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_right /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_right_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_uncurry_right /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry_curry_right /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry_right_norm /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry_right_norm /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_right_equiv_aux /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_right_equiv /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_right_equiv_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_right_equiv_symm_apply /- _inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry0 /- _inst_3: nondiscrete_normed_field \u219d normed_field\n_inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry0 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry0_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry0_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.apply_zero_curry0 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry0_curry0 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry0_uncurry0 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.uncurry0_norm /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.fin0_apply_norm /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_map.curry0_norm /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin0_aux /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin0 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin0_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin0_symm_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin1 /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin1_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n#print continuous_multilinear_curry_fin1_symm_apply /- _inst_8: normed_space \u219d\n_inst_11: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\operator_norm.lean\n#print exists_pos_bound_of_bound /- _inst_2: normed_group \u219d has_norm\n -/\n#print linear_map.lipschitz_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.antilipschitz_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.uniform_continuous_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.continuous_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.mk_continuous /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.to_continuous_linear_map\u2081 /- _inst_5: normed_space \u219d\n -/\n#print linear_map.mk_continuous_of_exists_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_of_linear_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.mk_continuous_coe /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.mk_continuous_apply /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.mk_continuous_of_exists_bound_coe /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.mk_continuous_of_exists_bound_apply /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.to_continuous_linear_map\u2081_coe /- _inst_5: normed_space \u219d\n -/\n#print linear_map.to_continuous_linear_map\u2081_apply /- _inst_5: normed_space \u219d\n -/\n#print linear_map.continuous_iff_is_closed_ker /- _inst_5: normed_space \u219d\n -/\n#print linear_map.bound_of_shell /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_map.bound_of_continuous /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.is_O_id /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.is_O_comp /- _inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.is_O_sub /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.of_homothety /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.to_span_singleton_homothety /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.to_span_singleton /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.has_op_norm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_def /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.bounds_nonempty /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.bounds_bdd_below /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_nonneg /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.le_op_norm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.le_op_norm_of_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.le_of_op_norm_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.lipschitz /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.ratio_le_op_norm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.unit_le_op_norm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_le_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_le_of_lipschitz /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_le_of_shell /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_le_of_ball /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_le_of_shell' /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_eq_of_bounds /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_add_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_zero_iff /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_id_le /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.norm_id /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_smul_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_neg /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.to_normed_group /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.to_normed_space /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_comp_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.to_normed_ring /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.to_normed_algebra /- _inst_5: normed_space \u219d algebra\n -/\n#print continuous_linear_map.uniform_continuous /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.isometry_iff_norm_image_eq_norm /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.homothety_norm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.to_span_singleton_norm /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.uniform_embedding_of_bound /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.antilipschitz_of_uniform_embedding /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.complete_space /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.extend /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.extend_unique /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.extend_zero /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print continuous_linear_map.op_norm_extend_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print linear_map.mk_continuous_norm_le /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_smul_right_apply /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.smul_rightL /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_smul_rightL_apply /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_smul_rightL /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.apply\u2097 /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.continuous_apply\u2097 /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.apply /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.restrict_scalars /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n_inst_15: normed_space \u219d\n_inst_16: normed_space \u219d\n_inst_17: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.restrict_scalars_coe_eq_coe /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n_inst_15: normed_space \u219d\n_inst_16: normed_space \u219d\n_inst_17: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.restrict_scalars_coe_eq_coe' /- _inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n_inst_15: normed_space \u219d\n_inst_16: normed_space \u219d\n_inst_17: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.has_scalar_extend_scalars /- _inst_5: normed_space \u219d\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.module_extend_scalars /- _inst_5: normed_space \u219d\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.normed_space_extend_scalars /- _inst_5: normed_space \u219d\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.smul_algebra_right /- _inst_5: normed_space \u219d\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.smul_algebra_right_apply /- _inst_5: normed_space \u219d\n_inst_11: normed_space \u219d\n_inst_12: normed_space \u219d\n_inst_13: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.has_sum /- _inst_10: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.summable /- _inst_10: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.has_sum /- _inst_10: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.summable /- _inst_10: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.lipschitz /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.antilipschitz /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.is_O_comp /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.is_O_sub /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.is_O_comp_rev /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.is_O_sub_rev /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.uniform_embedding /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.one_le_norm_mul_norm_symm /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.norm_pos /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.norm_symm_pos /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.subsingleton_or_norm_symm_pos /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.subsingleton_or_nnnorm_symm_pos /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.homothety_inverse /- _inst_4: nondiscrete_normed_field \u219d normed_field\n_inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.of_homothety /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_equiv.to_span_nonzero_singleton_homothety /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.to_span_nonzero_singleton /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.coord /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.coord_norm /- _inst_5: normed_space \u219d\n -/\n#print continuous_linear_equiv.coord_self /- _inst_5: normed_space \u219d\n -/\n#print linear_equiv.uniform_embedding /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print linear_equiv.to_continuous_linear_equiv_of_bounds /- _inst_5: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n\n-- analysis\\normed_space\\riesz_lemma.lean\n#print riesz_lemma /- _inst_3: normed_space \u219d\n -/\n\n-- analysis\\seminorm.lean\n#print absorbs /- _inst_1: nondiscrete_normed_field \u219d field has_norm\n_inst_3: vector_space \u219d has_scalar\n -/\n#print absorbent /- _inst_1: nondiscrete_normed_field \u219d field has_norm\n_inst_3: vector_space \u219d has_scalar\n -/\n#print balanced /- _inst_1: nondiscrete_normed_field \u219d field has_norm\n_inst_3: vector_space \u219d has_scalar\n -/\n#print balanced.absorbs_self /- _inst_3: vector_space \u219d\n -/\n#print absorbent_nhds_zero /- _inst_3: vector_space \u219d\n -/\n#print balanced_zero_union_interior /- _inst_3: vector_space \u219d\n -/\n#print seminorm.has_coe_to_fun /- _inst_1: nondiscrete_normed_field \u219d normed_field\n -/\n\n-- analysis\\special_functions\\exp_log.lean\n#print has_fderiv_within_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print has_fderiv_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_within_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_on.cexp /- _inst_2: normed_space \u219d\n -/\n#print differentiable.cexp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff.cexp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_on.cexp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_within_at.cexp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff.exp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_at.exp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_on.exp /- _inst_2: normed_space \u219d\n -/\n#print times_cont_diff_within_at.exp /- _inst_2: normed_space \u219d\n -/\n#print has_fderiv_within_at.exp /- _inst_2: normed_space \u219d\n -/\n#print has_fderiv_at.exp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_within_at.exp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_at.exp /- _inst_2: normed_space \u219d\n -/\n#print differentiable_on.exp /- _inst_2: normed_space \u219d\n -/\n#print differentiable.exp /- _inst_2: normed_space \u219d\n -/\n#print fderiv_within_exp /- _inst_2: normed_space \u219d\n -/\n#print fderiv_exp /- _inst_2: normed_space \u219d\n -/\n\n-- analysis\\specific_limits.lean\n#print tendsto_pow_at_top_at_top_of_one_lt /- _inst_1: linear_ordered_ring \u219d ordered_add_comm_group linear_ordered_semiring\n -/\n\n-- category_theory\\abelian\\basic.lean\n#print category_theory.abelian.has_finite_biproducts /- _inst_2: category_theory.abelian \u219d category_theory.limits.has_finite_products category_theory.preadditive\n -/\n#print category_theory.abelian.has_pullbacks /- _inst_2: category_theory.abelian \u219d category_theory.limits.has_finite_products category_theory.preadditive category_theory.limits.has_kernels\n -/\n#print category_theory.abelian.has_pushouts /- _inst_2: category_theory.abelian \u219d category_theory.limits.has_cokernels category_theory.limits.has_binary_coproducts category_theory.limits.has_finite_products category_theory.preadditive\n -/\n\n-- category_theory\\abelian\\non_preadditive.lean\n#print category_theory.non_preadditive_abelian.epi_is_cokernel_of_kernel /- _inst_3: category_theory.epi \u219d category_theory.is_iso\n -/\n#print category_theory.non_preadditive_abelian.mono_is_kernel_of_cokernel /- _inst_3: category_theory.mono \u219d category_theory.is_iso\n -/\n#print category_theory.non_preadditive_abelian.mono_\u0394 /- _inst_2: category_theory.non_preadditive_abelian \u219d category_theory.limits.has_finite_products\n -/\n#print category_theory.non_preadditive_abelian.lift_map /- _inst_2: category_theory.non_preadditive_abelian \u219d category_theory.limits.has_zero_morphisms category_theory.limits.has_finite_products\n -/\n\n-- category_theory\\action.lean\n#print category_theory.action_as_functor_map /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_as_functor_obj /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_as_functor /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.category_theory.groupoid /- _inst_4: mul_action \u219d\n -/\n#print category_theory.action_category.\u03c0 /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.\u03c0_map /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.\u03c0_obj /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.obj_equiv /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.hom_as_subtype /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.inhabited /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.stabilizer_iso_End /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.stabilizer_iso_End_apply /- _inst_2: mul_action \u219d\n -/\n#print category_theory.action_category.stabilizer_iso_End_symm_apply /- _inst_2: mul_action \u219d\n -/\n\n-- category_theory\\adjunction\\limits.lean\n#print category_theory.adjunction.has_colimit_comp_equivalence /- _inst_4: category_theory.is_equivalence \u219d category_theory.limits.preserves_colimit\n -/\n#print category_theory.adjunction.has_limit_comp_equivalence /- _inst_4: category_theory.is_equivalence \u219d category_theory.limits.preserves_limit\n -/\n\n-- category_theory\\category\\Kleisli.lean\n#print category_theory.Kleisli /- _inst_1: monad \u219d\n -/\n#print category_theory.Kleisli.id_def /- _inst_2: is_lawful_monad \u219d\n -/\n#print category_theory.Kleisli.comp_def /- _inst_2: is_lawful_monad \u219d\n -/\n\n-- category_theory\\category\\default.lean\n#print category_theory.eq_whisker /- _inst_1: category_theory.category \u219d category_theory.category_struct\n -/\n#print category_theory.whisker_eq /- _inst_1: category_theory.category \u219d category_theory.category_struct\n -/\n#print category_theory.comp_dite /- _inst_1: category_theory.category \u219d category_theory.category_struct\n -/\n#print category_theory.dite_comp /- _inst_1: category_theory.category \u219d category_theory.category_struct\n -/\n#print category_theory.hom_of_le /- _inst_1: preorder \u219d category_theory.has_hom has_le\n -/\n#print category_theory.le_of_hom /- _inst_1: preorder \u219d category_theory.has_hom has_le\n -/\n\n-- category_theory\\concrete_category\\bundled_hom.lean\n#print category_theory.bundled_hom.bundled_hom_of_parent_projection /- _inst_1: category_theory.bundled_hom.parent_projection \u219d\n -/\n\n-- category_theory\\core.lean\n#print category_theory.core.forget_functor_to_core /- _inst_2: category_theory.groupoid \u219d category_theory.category\n -/\n\n-- category_theory\\endomorphism.lean\n#print category_theory.End /- _inst_1: category_theory.category_struct \u219d category_theory.has_hom\n -/\n\n-- category_theory\\filtered.lean\n#print category_theory.is_filtered_of_semilattice_sup_top /- _inst_2: semilattice_sup_top \u219d category_theory.small_category has_top category_theory.is_filtered_or_empty\n -/\n\n-- category_theory\\fin_category.lean\n#print category_theory.discrete_hom_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.fin_category_discrete_of_decidable_fintype /- _inst_1: decidable_eq \u219d\n -/\n\n-- category_theory\\full_subcategory.lean\n#print category_theory.induced_category /- _inst_1: category_theory.category \u219d\n -/\n\n-- category_theory\\graded_object.lean\n#print category_theory.graded_object_with_shift /- _inst_1: add_comm_group \u219d\n -/\n\n-- category_theory\\is_connected.lean\n#print category_theory.is_connected_of_equivalent /- _inst_4: category_theory.is_connected \u219d nonempty category_theory.is_preconnected\n -/\n#print category_theory.zag /- _inst_1: category_theory.category \u219d category_theory.has_hom\n -/\n#print category_theory.equiv_relation /- _inst_3: category_theory.is_connected \u219d nonempty category_theory.is_preconnected\n -/\n\n-- category_theory\\limits\\cofinal.lean\n#print category_theory.cofinal.colimit_iso /- _inst_3: category_theory.cofinal \u219d category_theory.limits.has_colimit category_theory.is_iso\n -/\n#print category_theory.cofinal.cofinal_of_colimit_comp_coyoneda_iso_punit /- _inst_3: category_theory.cofinal \u219d category_theory.limits.has_colimit\n -/\n\n-- category_theory\\limits\\creates.lean\n#print category_theory.has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape /- _inst_4: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n_inst_5: category_theory.creates_limits_of_shape \u219d category_theory.creates_limit\n -/\n#print category_theory.has_limits_of_has_limits_creates_limits /- _inst_4: category_theory.limits.has_limits \u219d category_theory.limits.has_limits_of_shape\n_inst_5: category_theory.creates_limits \u219d category_theory.creates_limits_of_shape\n -/\n#print category_theory.has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape /- _inst_4: category_theory.limits.has_colimits_of_shape \u219d category_theory.limits.has_colimit\n_inst_5: category_theory.creates_colimits_of_shape \u219d category_theory.creates_colimit\n -/\n#print category_theory.has_colimits_of_has_colimits_creates_colimits /- _inst_4: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n_inst_5: category_theory.creates_colimits \u219d category_theory.creates_colimits_of_shape\n -/\n#print category_theory.preserves_limit_of_shape_of_creates_limits_of_shape_and_has_limits_of_shape /- _inst_4: category_theory.creates_limits_of_shape \u219d category_theory.limits.preserves_limit\n -/\n#print category_theory.preserves_limits_of_creates_limits_and_has_limits /- _inst_4: category_theory.creates_limits \u219d category_theory.limits.preserves_limits_of_shape\n_inst_5: category_theory.limits.has_limits \u219d category_theory.limits.preserves_limits_of_shape\n -/\n#print category_theory.preserves_colimit_of_shape_of_creates_colimits_of_shape_and_has_colimits_of_shape /- _inst_4: category_theory.creates_colimits_of_shape \u219d category_theory.limits.preserves_colimit\n -/\n#print category_theory.preserves_colimits_of_creates_colimits_and_has_colimits /- _inst_4: category_theory.creates_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n_inst_5: category_theory.limits.has_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n -/\n#print category_theory.creates_limits_of_shape_of_nat_iso /- _inst_4: category_theory.creates_limits_of_shape \u219d category_theory.creates_limit\n -/\n#print category_theory.creates_limits_of_nat_iso /- _inst_4: category_theory.creates_limits \u219d category_theory.creates_limits_of_shape\n -/\n#print category_theory.creates_colimits_of_shape_of_nat_iso /- _inst_4: category_theory.creates_colimits_of_shape \u219d category_theory.creates_colimit\n -/\n#print category_theory.creates_colimits_of_nat_iso /- _inst_4: category_theory.creates_colimits \u219d category_theory.creates_colimits_of_shape\n -/\n#print category_theory.comp_creates_limits_of_shape /- _inst_4: category_theory.creates_limits_of_shape \u219d category_theory.creates_limit\n_inst_5: category_theory.creates_limits_of_shape \u219d category_theory.creates_limit\n -/\n#print category_theory.comp_creates_limits /- _inst_4: category_theory.creates_limits \u219d category_theory.creates_limits_of_shape\n_inst_5: category_theory.creates_limits \u219d category_theory.creates_limits_of_shape\n -/\n#print category_theory.comp_creates_colimits_of_shape /- _inst_4: category_theory.creates_colimits_of_shape \u219d category_theory.creates_colimit\n_inst_5: category_theory.creates_colimits_of_shape \u219d category_theory.creates_colimit\n -/\n#print category_theory.comp_creates_colimits /- _inst_4: category_theory.creates_colimits \u219d category_theory.creates_colimits_of_shape\n_inst_5: category_theory.creates_colimits \u219d category_theory.creates_colimits_of_shape\n -/\n\n-- category_theory\\limits\\functor_category.lean\n#print category_theory.limits.functor_category_has_limits_of_shape /- _inst_4: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n#print category_theory.limits.functor_category_has_colimits_of_shape /- _inst_4: category_theory.limits.has_colimits_of_shape \u219d category_theory.limits.has_colimit\n -/\n#print category_theory.limits.functor_category_has_limits /- _inst_4: category_theory.limits.has_limits \u219d category_theory.limits.has_limits_of_shape\n -/\n#print category_theory.limits.functor_category_has_colimits /- _inst_4: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n#print category_theory.limits.evaluation_preserves_limits /- _inst_4: category_theory.limits.has_limits \u219d category_theory.limits.has_limits_of_shape\n -/\n#print category_theory.limits.evaluation_preserves_colimits /- _inst_4: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n\n-- category_theory\\limits\\limits.lean\n#print category_theory.limits.limit.map_pre' /- _inst_4: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n#print category_theory.limits.limit.map_post /- _inst_4: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n_inst_6: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n#print category_theory.limits.has_limits_of_shape_of_equivalence /- _inst_5: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n#print category_theory.limits.colimit.pre_map' /- _inst_4: category_theory.limits.has_colimits_of_shape \u219d category_theory.limits.has_colimit\n -/\n#print category_theory.limits.has_colimits_of_shape_of_equivalence /- _inst_5: category_theory.limits.has_colimits_of_shape \u219d category_theory.limits.has_colimit\n -/\n\n-- category_theory\\limits\\over.lean\n#print category_theory.over.has_colimits_of_shape /- _inst_3: category_theory.limits.has_colimits_of_shape \u219d category_theory.limits.has_colimit\n -/\n#print category_theory.over.has_colimits /- _inst_3: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n#print category_theory.over.forget_preserves_colimits /- _inst_3: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n#print category_theory.under.has_limits_of_shape /- _inst_3: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n#print category_theory.under.has_limits /- _inst_3: category_theory.limits.has_limits \u219d category_theory.limits.has_limits_of_shape\n -/\n\n-- category_theory\\limits\\preserves\\basic.lean\n#print category_theory.limits.comp_preserves_limits_of_shape /- _inst_4: category_theory.limits.preserves_limits_of_shape \u219d category_theory.limits.preserves_limit\n_inst_5: category_theory.limits.preserves_limits_of_shape \u219d category_theory.limits.preserves_limit\n -/\n#print category_theory.limits.comp_preserves_limits /- _inst_4: category_theory.limits.preserves_limits \u219d category_theory.limits.preserves_limits_of_shape\n_inst_5: category_theory.limits.preserves_limits \u219d category_theory.limits.preserves_limits_of_shape\n -/\n#print category_theory.limits.comp_preserves_colimits_of_shape /- _inst_4: category_theory.limits.preserves_colimits_of_shape \u219d category_theory.limits.preserves_colimit\n_inst_5: category_theory.limits.preserves_colimits_of_shape \u219d category_theory.limits.preserves_colimit\n -/\n#print category_theory.limits.comp_preserves_colimits /- _inst_4: category_theory.limits.preserves_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n_inst_5: category_theory.limits.preserves_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n -/\n#print category_theory.limits.preserves_limits_of_shape_of_nat_iso /- _inst_4: category_theory.limits.preserves_limits_of_shape \u219d category_theory.limits.preserves_limit\n -/\n#print category_theory.limits.preserves_limits_of_nat_iso /- _inst_4: category_theory.limits.preserves_limits \u219d category_theory.limits.preserves_limits_of_shape\n -/\n#print category_theory.limits.preserves_colimits_of_shape_of_nat_iso /- _inst_4: category_theory.limits.preserves_colimits_of_shape \u219d category_theory.limits.preserves_colimit\n -/\n#print category_theory.limits.preserves_colimits_of_nat_iso /- _inst_4: category_theory.limits.preserves_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n -/\n#print category_theory.limits.comp_reflects_limits_of_shape /- _inst_4: category_theory.limits.reflects_limits_of_shape \u219d category_theory.limits.reflects_limit\n -/\n#print category_theory.limits.comp_reflects_limits /- _inst_4: category_theory.limits.reflects_limits \u219d category_theory.limits.reflects_limits_of_shape\n_inst_5: category_theory.limits.reflects_limits \u219d category_theory.limits.reflects_limits_of_shape\n -/\n#print category_theory.limits.comp_reflects_colimits_of_shape /- _inst_4: category_theory.limits.reflects_colimits_of_shape \u219d category_theory.limits.reflects_colimit\n -/\n#print category_theory.limits.comp_reflects_colimits /- _inst_4: category_theory.limits.reflects_colimits \u219d category_theory.limits.reflects_colimits_of_shape\n_inst_5: category_theory.limits.reflects_colimits \u219d category_theory.limits.reflects_colimits_of_shape\n -/\n#print category_theory.limits.preserves_limits_of_reflects_of_preserves /- _inst_5: category_theory.limits.reflects_limits \u219d category_theory.limits.reflects_limits_of_shape\n -/\n#print category_theory.limits.reflects_limits_of_shape_of_nat_iso /- _inst_4: category_theory.limits.reflects_limits_of_shape \u219d category_theory.limits.reflects_limit\n -/\n#print category_theory.limits.reflects_limits_of_nat_iso /- _inst_4: category_theory.limits.reflects_limits \u219d category_theory.limits.reflects_limits_of_shape\n -/\n#print category_theory.limits.preserves_colimits_of_reflects_of_preserves /- _inst_5: category_theory.limits.reflects_colimits \u219d category_theory.limits.reflects_colimits_of_shape\n -/\n#print category_theory.limits.reflects_colimits_of_shape_of_nat_iso /- _inst_4: category_theory.limits.reflects_colimits_of_shape \u219d category_theory.limits.reflects_colimit\n -/\n#print category_theory.limits.reflects_colimits_of_nat_iso /- _inst_4: category_theory.limits.reflects_colimits \u219d category_theory.limits.reflects_colimits_of_shape\n -/\n\n-- category_theory\\limits\\preserves\\functor_category.lean\n#print category_theory.functor_category.prod_preserves_colimits /- _inst_4: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n\n-- category_theory\\limits\\shapes\\biproducts.lean\n#print category_theory.limits.bicone_\u03b9_\u03c0_self /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone_\u03b9_\u03c0_ne /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cone /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cone_X /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cone_\u03c0_app /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cocone_X /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cocone_\u03b9_app /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.bicone.to_cocone /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.has_biproduct.mk /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.get_biproduct_data /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.bicone /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.is_limit /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.is_colimit /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.has_product_of_has_biproduct /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.has_coproduct_of_has_biproduct /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.has_finite_products_of_has_finite_biproducts /- _inst_4: category_theory.limits.has_finite_biproducts \u219d category_theory.limits.has_biproduct\n -/\n#print category_theory.limits.has_finite_coproducts_of_has_finite_biproducts /- _inst_4: category_theory.limits.has_finite_biproducts \u219d category_theory.limits.has_biproduct\n -/\n#print category_theory.limits.biproduct_iso /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03c0 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.bicone_\u03c0 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.bicone_\u03b9 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0_self /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0_self_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0_ne_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_\u03c0_ne /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.desc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift_\u03c0 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift_\u03c0_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_desc_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_desc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map' /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.hom_ext /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.hom_ext' /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map_eq_map' /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_mono /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03c0_epi /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map_\u03c0_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map_\u03c0 /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_map /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.\u03b9_map_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print category_theory.limits.has_binary_biproducts_of_finite_biproducts /- _inst_4: category_theory.limits.has_finite_biproducts \u219d category_theory.limits.has_biproducts_of_shape\n -/\n#print category_theory.limits.has_binary_products_of_has_binary_biproducts /- _inst_4: category_theory.limits.has_binary_biproducts \u219d category_theory.limits.has_binary_biproduct\n -/\n#print category_theory.limits.has_binary_coproducts_of_has_binary_biproducts /- _inst_4: category_theory.limits.has_binary_biproducts \u219d category_theory.limits.has_binary_biproduct\n -/\n#print category_theory.limits.biprod.symmetry' /- _inst_4: category_theory.limits.has_binary_biproducts \u219d category_theory.limits.has_binary_biproduct\n -/\n#print category_theory.limits.has_biproduct_of_total /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.has_biproduct.of_has_product /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.has_biproduct.of_has_coproduct /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.total /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift_eq /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.desc_eq /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift_desc_assoc /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.lift_desc /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biproduct.map_eq /- _inst_3: decidable_eq \u219d\n -/\n#print category_theory.limits.biprod.map_eq /- _inst_6: category_theory.limits.has_binary_biproducts \u219d category_theory.limits.has_binary_biproduct\n -/\n\n-- category_theory\\limits\\shapes\\constructions\\over\\connected.lean\n#print category_theory.over.has_connected_limits /- _inst_4: category_theory.limits.has_limits_of_shape \u219d category_theory.limits.has_limit\n -/\n\n-- category_theory\\limits\\shapes\\finite_limits.lean\n#print category_theory.limits.has_finite_limits_of_has_limits /- _inst_2: category_theory.limits.has_limits \u219d category_theory.limits.has_limits_of_shape\n -/\n#print category_theory.limits.has_finite_colimits_of_has_colimits /- _inst_2: category_theory.limits.has_colimits \u219d category_theory.limits.has_colimits_of_shape\n -/\n#print category_theory.limits.wide_pullback_shape.fintype_hom /- _inst_2: decidable_eq \u219d\n -/\n#print category_theory.limits.wide_pushout_shape.fintype_hom /- _inst_2: decidable_eq \u219d\n -/\n#print category_theory.limits.fin_category_wide_pullback /- _inst_2: decidable_eq \u219d\n -/\n#print category_theory.limits.fin_category_wide_pushout /- _inst_2: decidable_eq \u219d\n -/\n\n-- category_theory\\limits\\shapes\\strong_epi.lean\n#print category_theory.strong_epi_comp /- _inst_2: category_theory.strong_epi \u219d category_theory.arrow.has_lift category_theory.epi\n_inst_3: category_theory.strong_epi \u219d category_theory.arrow.has_lift category_theory.epi\n -/\n\n-- category_theory\\limits\\shapes\\zero.lean\n#print category_theory.limits.split_mono_sigma_\u03b9 /- _inst_2: decidable_eq \u219d\n -/\n#print category_theory.limits.split_epi_pi_\u03c0 /- _inst_2: decidable_eq \u219d\n -/\n\n-- category_theory\\monad\\adjunction.lean\n#print category_theory.\u03bc_iso_of_reflective /- _inst_3: category_theory.reflective \u219d category_theory.full category_theory.faithful category_theory.is_right_adjoint category_theory.monad\n -/\n\n-- category_theory\\monad\\limits.lean\n#print category_theory.monad.forget_creates_colimits.lambda /- _inst_4: category_theory.limits.preserves_colimits_of_shape \u219d category_theory.limits.preserves_colimit\n -/\n#print category_theory.comp_comparison_forget_has_limit /- _inst_4: category_theory.monadic_right_adjoint \u219d category_theory.is_right_adjoint category_theory.monad\n -/\n#print category_theory.comp_comparison_has_limit /- _inst_4: category_theory.monadic_right_adjoint \u219d category_theory.limits.has_limit category_theory.is_right_adjoint category_theory.monad\n -/\n#print category_theory.monadic_creates_limits /- _inst_4: category_theory.monadic_right_adjoint \u219d category_theory.is_equivalence category_theory.monad\n -/\n#print category_theory.monadic_creates_colimits_of_shape_of_preserves_colimits_of_shape /- _inst_4: category_theory.monadic_right_adjoint \u219d category_theory.is_equivalence category_theory.monad\n -/\n#print category_theory.monadic_creates_colimits_of_preserves_colimits /- _inst_5: category_theory.limits.preserves_colimits \u219d category_theory.limits.preserves_colimits_of_shape\n -/\n#print category_theory.has_limits_of_reflective /- _inst_4: category_theory.limits.has_limits \u219d category_theory.limits.has_limit\n_inst_5: category_theory.reflective \u219d category_theory.monadic_right_adjoint\n -/\n\n-- category_theory\\monoidal\\transport.lean\n#print category_theory.monoidal.transported /- _inst_2: category_theory.monoidal_category \u219d\n -/\n\n-- category_theory\\preadditive\\biproducts.lean\n#print category_theory.biprod.column_nonzero_of_iso /- _inst_3: category_theory.limits.has_binary_biproducts \u219d category_theory.limits.has_binary_biproduct\n -/\n#print category_theory.biproduct.column_nonzero_of_iso' /- _inst_3: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print category_theory.biproduct.column_nonzero_of_iso /- _inst_3: decidable_eq \u219d\n_inst_4: decidable_eq \u219d\n -/\n\n-- category_theory\\preadditive\\default.lean\n#print category_theory.preadditive.has_equalizers_of_has_kernels /- _inst_3: category_theory.limits.has_kernels \u219d category_theory.limits.has_kernel\n -/\n#print category_theory.preadditive.has_coequalizers_of_has_cokernels /- _inst_3: category_theory.limits.has_cokernels \u219d category_theory.limits.has_cokernel\n -/\n\n-- category_theory\\sites\\grothendieck.lean\n#print category_theory.grothendieck_topology.right_ore_condition /- _inst_2: category_theory.category \u219d category_theory.category_struct\n -/\n\n-- category_theory\\sites\\sieves.lean\n#print category_theory.presieve /- _inst_1: category_theory.category \u219d category_theory.has_hom\n -/\n\n-- combinatorics\\adj_matrix.lean\n#print simple_graph.adj_matrix /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.transpose_adj_matrix /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_dot_product /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.dot_product_adj_matrix /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_mul_vec_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_vec_mul_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_mul_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.mul_adj_matrix_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.trace_adj_matrix /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_mul_self_apply_self /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_mul_vec_const_apply /- _inst_3: decidable_rel \u219d\n -/\n#print simple_graph.adj_matrix_mul_vec_const_apply_of_regular /- _inst_3: decidable_rel \u219d\n -/\n\n-- combinatorics\\colex.lean\n#print colex.hom /- _inst_2: decidable_eq \u219d\n -/\n#print colex.sdiff_lt_sdiff_iff_lt /- _inst_2: decidable_eq \u219d\n -/\n\n-- combinatorics\\pigeonhole.lean\n#print finset.exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_sum_fiber_lt_of_maps_to_of_sum_lt_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_lt_sum_fiber_of_sum_fiber_nonpos_of_nsmul_lt_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_sum_fiber_lt_of_sum_fiber_nonneg_of_sum_lt_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_le_sum_fiber_of_maps_to_of_nsmul_le_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_sum_fiber_le_of_maps_to_of_sum_le_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_le_sum_fiber_of_sum_fiber_nonpos_of_nsmul_le_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_lt_card_fiber_of_mul_lt_card_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_card_fiber_lt_of_card_lt_mul /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_le_card_fiber_of_mul_le_card_of_maps_to /- _inst_2: decidable_eq \u219d\n -/\n#print finset.exists_card_fiber_le_of_card_le_mul /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_lt_sum_fiber_of_nsmul_lt_sum /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_le_sum_fiber_of_nsmul_le_sum /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_sum_fiber_lt_of_sum_lt_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_sum_fiber_le_of_sum_le_nsmul /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_lt_card_fiber_of_mul_lt_card /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_card_fiber_lt_of_card_lt_mul /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_le_card_fiber_of_mul_le_card /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.exists_card_fiber_le_of_card_le_mul /- _inst_2: decidable_eq \u219d\n -/\n\n-- combinatorics\\simple_graph.lean\n#print complete_graph_adj_decidable /- _inst_1: decidable_eq \u219d\n -/\n#print simple_graph.edges_fintype /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_rel \u219d\n -/\n#print simple_graph.edge_finset /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_rel \u219d\n -/\n#print simple_graph.mem_edge_finset /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_rel \u219d\n -/\n#print simple_graph.neighbor_set_fintype /- _inst_2: decidable_rel \u219d\n -/\n#print simple_graph.neighbor_finset_eq_filter /- _inst_2: decidable_rel \u219d\n -/\n#print simple_graph.complete_graph_degree /- _inst_2: decidable_eq \u219d\n -/\n#print simple_graph.complete_graph_is_regular /- _inst_2: decidable_eq \u219d\n -/\n\n-- computability\\partrec.lean\n#print partrec /- _inst_1: primcodable \u219d encodable\n_inst_2: primcodable \u219d encodable\n -/\n\n-- computability\\primrec.lean\n#print primrec /- _inst_1: primcodable \u219d encodable\n_inst_2: primcodable \u219d encodable\n -/\n#print primrec.eq /- _inst_6: decidable_eq \u219d\n -/\n#print primrec.list_index_of\u2081 /- _inst_6: decidable_eq \u219d\n -/\n#print primrec.list_index_of /- _inst_5: decidable_eq \u219d\n -/\n\n-- computability\\turing_machine.lean\n#print turing.TM0.machine /- _inst_2: inhabited \u219d\n -/\n#print turing.TM1to0.\u039b' /- _inst_2: inhabited \u219d\n_inst_3: inhabited \u219d\n -/\n#print turing.TM2.stmt.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.cfg.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.step_aux /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.step /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.reaches /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.supports_stmt /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts\u2081 /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts\u2081_self /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts\u2081_trans /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts\u2081_supports_stmt_mono /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts_trans /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.supports /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.stmts_supports_stmt /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.step_supports /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.init /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2.eval /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.\u0393' /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.\u0393'.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.\u0393'.fintype /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom_map /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom_modify_nth /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom_nth_snd /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom_nth_succ_fst /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.add_bottom_head_fst /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.st_act.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.st_run /- _inst_1: decidable_eq \u219d\n_inst_2: inhabited \u219d\n -/\n#print turing.TM2to1.st_var /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.st_write /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.stmt_st_rec /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.supports_run /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.\u039b'.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_st_act /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_init /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.step_run /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_normal /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_normal_run /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_stmts\u2081 /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_stmts\u2081_run /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_respects_aux\u2082 /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_respects_aux\u2081 /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_respects_aux\u2083 /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_respects_aux /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_respects /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_cfg_init /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_eval_dom /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_eval /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_supp /- _inst_1: decidable_eq \u219d\n -/\n#print turing.TM2to1.tr_supports /- _inst_1: decidable_eq \u219d\n -/\n\n-- control\\basic.lean\n#print fish /- _inst_3: monad \u219d has_bind\n -/\n#print succeeds /- _inst_1: alternative \u219d has_orelse has_pure functor\n -/\n#print mtry /- _inst_1: alternative \u219d has_orelse has_pure functor\n -/\n\n-- control\\bitraversable\\instances.lean\n#print const.bitraverse /- _inst_2: applicative \u219d\n -/\n\n-- control\\fold.lean\n#print traversable.mfoldl.unop_of_free_monoid /- _inst_2: is_lawful_monad \u219d\n -/\n\n-- control\\functor.lean\n#print functor.comp.has_pure /- _inst_1: applicative \u219d has_pure\n_inst_2: applicative \u219d has_pure\n -/\n\n-- control\\monad\\cont.lean\n#print cont_t.monad_lift /- _inst_1: monad \u219d has_bind\n -/\n#print writer_t.monad_cont /- _inst_1: monad \u219d\n -/\n#print state_t.mk_label /- _inst_1: monad \u219d\n -/\n\n-- control\\monad\\writer.lean\n#print writer_t.ext /- _inst_1: monad \u219d\n -/\n#print writer_t.tell /- _inst_1: monad \u219d has_pure\n -/\n#print writer_t.pure /- _inst_1: monad \u219d has_pure\n -/\n#print writer_t.bind /- _inst_1: monad \u219d has_pure has_bind\n -/\n#print writer_t.lift /- _inst_1: monad \u219d functor\n -/\n#print writer_t.monad_map /- _inst_2: monad \u219d\n_inst_3: monad \u219d\n -/\n#print writer_t.adapt /- _inst_1: monad \u219d functor\n -/\n#print writer_t.monad_except /- _inst_1: monad \u219d\n -/\n#print reader_t.monad_writer /- _inst_1: monad \u219d has_monad_lift\n -/\n\n-- control\\traversable\\instances.lean\n#print option.comp_traverse /- _inst_3: is_lawful_applicative \u219d\n -/\n#print list.comp_traverse /- _inst_3: is_lawful_applicative \u219d\n -/\n#print sum.comp_traverse /- _inst_3: is_lawful_applicative \u219d\n -/\n\n-- control\\uliftable.lean\n#print uliftable.adapt_up /- _inst_2: monad \u219d has_bind\n -/\n#print uliftable.adapt_down /- _inst_1: monad \u219d has_bind\n -/\n\n-- data\\analysis\\filter.lean\n#print filter.realizer.cofinite /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\buffer\\basic.lean\n#print buffer.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\complex\\exponential.lean\n#print is_cau_geo_series /- _inst_5: field \u219d domain\n -/\n\n-- data\\dfinsupp.lean\n#print dfinsupp.mk /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_injective /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_eq_same /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_eq_of_ne /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_injective /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_eq_single_iff /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase_same /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase_ne /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_add /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_add_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_add_hom_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_add_erase /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase_add_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.induction /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.induction\u2082 /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.add_closure_Union_range_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.add_hom_ext /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.add_hom_ext' /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_add /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_neg /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_sub /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk.is_add_group_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mk_smul /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.single_smul /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_mk_subset /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mem_support_to_fun /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.eq_mk_support /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.mem_support_iff /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_eq_empty /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.decidable_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_subset_iff /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_single_ne_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_single_subset /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.map_range_def /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.map_range_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_map_range /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.zip_with_def /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_zip_with /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.erase_def /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_erase /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.filter_def /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_filter /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.subtype_domain_def /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_subtype_domain /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_add /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_neg /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.support_smul /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.decidable_eq /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_map_range_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_map_range_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_zero_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_zero_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_single_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_single_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_neg_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_neg_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_apply /- _inst_1: decidable_eq \u219d\n -/\n#print dfinsupp.support_sum /- dec: decidable_eq \u219d\n_inst_1: decidable_eq \u219d\n -/\n#print dfinsupp.sum_zero /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_one /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_mul /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_inv /- dec: decidable_eq \u219d\n_inst_3: comm_group \u219d has_inv is_group_hom comm_monoid\n -/\n#print dfinsupp.sum_neg /- dec: decidable_eq \u219d\n_inst_3: add_comm_group \u219d add_comm_monoid has_neg is_add_group_hom\n -/\n#print dfinsupp.prod_add_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add_hom_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add_hom_comp_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_add_hom_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom_symm_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom_apply /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom_single_add_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom_apply_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.lift_add_hom_comp_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.comp_lift_add_hom /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_sub_index /- dec: decidable_eq \u219d\n_inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print dfinsupp.sum_finset_sum_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_finset_sum_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_sum_index /- dec: decidable_eq \u219d\n_inst_1: decidable_eq \u219d\n -/\n#print dfinsupp.sum_sum_index /- dec: decidable_eq \u219d\n_inst_1: decidable_eq \u219d\n -/\n#print dfinsupp.sum_single /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.prod_subtype_domain_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.sum_subtype_domain_index /- dec: decidable_eq \u219d\n -/\n#print dfinsupp.subtype_domain_finsupp_sum /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\equiv\\basic.lean\n#print equiv.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.prod_extend_right /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.prod_extend_right_apply_eq /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.prod_extend_right_apply_ne /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.eq_of_prod_extend_right_ne /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.fst_prod_extend_right /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.subtype_equiv_codomain /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.coe_subtype_equiv_codomain /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.subtype_equiv_codomain_apply /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.coe_subtype_equiv_codomain_symm /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.subtype_equiv_codomain_symm_apply /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.subtype_equiv_codomain_symm_apply_eq /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.subtype_equiv_codomain_symm_apply_ne /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_core /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_core_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_core_swap_core /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_core_comm /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_comm /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_apply_def /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_apply_left /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_apply_right /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_apply_of_ne_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_comp_apply /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.swap_inv /- _inst_2: decidable_eq \u219d\n -/\n#print equiv.symm_trans_swap_trans /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print equiv.swap_mul_self /- _inst_2: decidable_eq \u219d\n -/\n#print equiv.swap_apply_self /- _inst_2: decidable_eq \u219d\n -/\n#print equiv.set_value /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.set_value_eq /- _inst_1: decidable_eq \u219d\n -/\n#print ulift.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print plift.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print dite_comp_equiv_update /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n\n-- data\\equiv\\encodable\\basic.lean\n#print directed.le_sequence /- _inst_3: preorder \u219d has_le\n -/\n#print quotient.rep /- _inst_1: decidable_rel \u219d\n -/\n#print quotient.rep_spec /- _inst_1: decidable_rel \u219d\n -/\n#print encodable_quotient /- _inst_1: decidable_rel \u219d\n -/\n\n-- data\\equiv\\list.lean\n#print encodable.encodable_of_list /- _inst_1: decidable_eq \u219d\n -/\n#print encodable.trunc_encodable_of_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print encodable.fintype_arrow /- _inst_1: decidable_eq \u219d\n -/\n#print encodable.fintype_pi /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\equiv\\ring.lean\n#print ring_equiv.trans_apply /- _inst_7: semiring \u219d has_add has_mul\n_inst_8: semiring \u219d has_add has_mul\n_inst_9: semiring \u219d has_add has_mul\n -/\n#print ring_equiv.map_mul /- _inst_1: semiring \u219d has_add has_mul\n_inst_2: semiring \u219d has_add has_mul\n -/\n#print ring_equiv.map_one /- _inst_1: semiring \u219d monoid has_add\n_inst_2: semiring \u219d monoid has_add\n -/\n#print ring_equiv.map_add /- _inst_1: semiring \u219d has_add has_mul\n_inst_2: semiring \u219d has_add has_mul\n -/\n#print ring_equiv.map_zero /- _inst_1: semiring \u219d add_monoid has_mul\n_inst_2: semiring \u219d add_monoid has_mul\n -/\n#print ring_equiv.map_eq_one_iff /- _inst_1: semiring \u219d monoid has_add\n_inst_2: semiring \u219d monoid has_add\n -/\n#print ring_equiv.map_eq_zero_iff /- _inst_1: semiring \u219d add_monoid has_mul\n_inst_2: semiring \u219d add_monoid has_mul\n -/\n#print ring_equiv.map_ne_one_iff /- _inst_1: semiring \u219d monoid has_add\n_inst_2: semiring \u219d monoid has_add\n -/\n#print ring_equiv.map_ne_zero_iff /- _inst_1: semiring \u219d add_monoid has_mul\n_inst_2: semiring \u219d add_monoid has_mul\n -/\n#print ring_equiv.map_neg /- _inst_1: ring \u219d add_group has_mul\n_inst_2: ring \u219d add_group has_mul\n -/\n#print ring_equiv.map_sub /- _inst_1: ring \u219d add_group has_mul\n_inst_2: ring \u219d add_group has_mul\n -/\n\n-- data\\equiv\\transfer_instance.lean\n#print equiv.mul_action /- _inst_2: mul_action \u219d\n -/\n#print equiv.linear_equiv /- _inst_3: semimodule \u219d\n -/\n\n-- data\\fin_enum.lean\n#print fin_enum.of_nodup_list /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.of_list /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.of_surjective /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.of_injective /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.quotient.enum /- _inst_2: decidable_rel \u219d\n -/\n#print fin_enum.finset.enum /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.finset.mem_enum /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.pi.cons /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.pi /- _inst_1: decidable_eq \u219d\n -/\n#print fin_enum.mem_pi /- _inst_1: fin_enum \u219d\n -/\n\n-- data\\finmap.lean\n#print list.to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.to_finmap_nil /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.has_decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_list_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_is_some /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_eq_none /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_singleton_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.has_mem.mem.decidable /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_iff /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_of_lookup_eq_some /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.ext_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.replace /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.replace_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.keys_replace /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_replace /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.erase_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.keys_erase_to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.keys_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.not_mem_erase_self /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_erase_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.erase_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.has_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_entries_of_neg /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_insert_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_insert_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.to_finmap_cons /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_list_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_singleton_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.extract /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.extract_eq_lookup_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.has_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union_to_finmap /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.keys_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.lookup_union_left_of_not_in /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_lookup_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.mem_lookup_union_middle /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.insert_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.empty_union /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.erase_union_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.disjoint.decidable_rel /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.disjoint_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.disjoint_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union_comm_of_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finmap.union_cancel /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\basic.lean\n#print finset.erase_dup_eq_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finset.decidable_mem /- h: decidable_eq \u219d\n -/\n#print finset.decidable_mem' /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_def /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_val' /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_val_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_insert_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_of_mem_insert_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finset.cons_eq_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.coe_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.is_lawful_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_eq_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_singleton_self_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert.comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_singleton_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_idem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_nonempty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_ne_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.ne_insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_subset_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.ssubset_iff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.ssubset_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.induction /- _inst_2: decidable_eq \u219d\n -/\n#print finset.induction_on /- _inst_2: decidable_eq \u219d\n -/\n#print finset.subtype_insert_equiv_option /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_val_nd /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disj_union_eq_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.mem_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.forall_mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.not_mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_subset_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_union.union.is_commutative /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_union.union.is_associative /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_idempotent /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_union.union.is_idempotent /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_left_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_right_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.empty_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_union_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_eq_left_iff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.left_eq_union_iff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_eq_right_iff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.right_eq_union_iff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_val_nd /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_of_mem_inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_of_mem_inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_inter_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_subset_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_subset_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_inter_cancel_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_inter_cancel_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_left_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_right_comm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.empty_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_union_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_inter_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_inter_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.singleton_inter_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.singleton_inter_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_singleton_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_singleton_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_subset_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_subset_inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_subset_inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.lattice /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sup_eq_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inf_eq_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.semilattice_inf_bot /- _inst_1: decidable_eq \u219d\n -/\n#print finset.semilattice_sup_bot /- _inst_2: decidable_eq \u219d\n -/\n#print finset.distrib_lattice /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_distrib_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_distrib_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_distrib_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_distrib_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_eq_empty_iff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.not_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.ne_of_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_of_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_erase_of_ne_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.eq_of_mem_of_not_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_subset_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_ssubset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_eq_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_insert_iff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.erase_insert_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_erase_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.has_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.not_mem_sdiff_of_mem_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_union_of_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_sdiff_of_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_sdiff_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_inter_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_inter_distrib_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_inter_self_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_inter_self_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_subset_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_subset_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_sdiff_self_eq_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_union_self_eq_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_sdiff_symm /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_union_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_idem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_eq_empty_iff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.empty_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_sdiff_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_sdiff_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.insert_sdiff_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_sdiff_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_union_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print finset.union_sdiff_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_singleton_eq_erase /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_sdiff_self_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_eq_inter_of_sdiff_eq_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.piecewise_insert_self /- _inst_1: decidable_eq \u219d\n -/\n#print finset.piecewise_insert_of_ne /- _inst_2: decidable_eq \u219d\n -/\n#print finset.piecewise_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.piecewise_singleton /- _inst_2: decidable_eq \u219d\n -/\n#print finset.update_eq_piecewise /- _inst_2: decidable_eq \u219d\n -/\n#print finset.update_piecewise /- _inst_2: decidable_eq \u219d\n -/\n#print finset.update_piecewise_of_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.update_piecewise_of_not_mem /- _inst_2: decidable_eq \u219d\n -/\n#print finset.filter_union /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_union_right /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_mem_eq_inter /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_inter /- _inst_3: decidable_eq \u219d\n -/\n#print finset.inter_filter /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_insert /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_or /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_and /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_not /- _inst_3: decidable_eq \u219d\n -/\n#print finset.sdiff_eq_filter /- _inst_3: decidable_eq \u219d\n -/\n#print finset.sdiff_eq_self /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_union_filter_neg_eq /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_inter_filter_neg_eq /- _inst_3: decidable_eq \u219d\n -/\n#print finset.subset_union_elim /- _inst_3: decidable_eq \u219d\n -/\n#print finset.filter_eq /- _inst_4: decidable_eq \u219d\n -/\n#print finset.filter_eq' /- _inst_4: decidable_eq \u219d\n -/\n#print finset.filter_ne /- _inst_4: decidable_eq \u219d\n -/\n#print finset.filter_ne' /- _inst_4: decidable_eq \u219d\n -/\n#print finset.exists_mem_insert /- d: decidable_eq \u219d\n -/\n#print finset.forall_mem_insert /- d: decidable_eq \u219d\n -/\n#print multiset.to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_val /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_eq /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_nsmul /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_eq_empty /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_subset /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_val /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_to_finset /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_surjective /- _inst_1: decidable_eq \u219d\n -/\n#print finset.map_to_finset /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.map_union /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.map_inter /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.map_insert /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_image_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.filter_mem_image_eq_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fiber_nonempty_iff_mem_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.nonempty.image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_to_finset /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_val_of_inj_on /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_id /- _inst_2: decidable_eq \u219d\n -/\n#print finset.image_image /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_subset_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_subset_iff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_mono /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_image_subset_range /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_filter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_union /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_inter /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_insert /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_eq_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.attach_image_val /- _inst_2: decidable_eq \u219d\n -/\n#print finset.attach_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.map_eq_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_const /- _inst_1: decidable_eq \u219d\n -/\n#print finset.subset_image_iff /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_map /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.card_insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_insert_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_singleton_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_erase_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_erase_lt_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_erase_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pred_card_le_card_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_card_le /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_card_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_image_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_image_of_inj_on /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_image_of_injective /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fiber_card_ne_zero_iff_mem_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_eq_succ /- _inst_1: decidable_eq \u219d\n -/\n#print finset.case_strong_induction_on /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_union_add_card_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_union_le /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_union_eq /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_insert /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.singleton_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.inter_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_bind /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.bind_image /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.bind_to_finset /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.bind_mono /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_subset_bind_of_subset_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bind_singleton_eq_self /- _inst_2: decidable_eq \u219d\n -/\n#print finset.bind_filter_eq_of_maps_to /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.image_bind_filter_eq /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.subset_product /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.product_eq_bind /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print finset.sigma_eq_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_iff_inter_eq_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.decidable_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_iff_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_of_subset_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_of_subset_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_empty_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_empty_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.singleton_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_insert_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_insert_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_sdiff_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_eq_self_iff_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sdiff_eq_self_of_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_self_iff_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_bind_left /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_bind_right /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_disjoint_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_filter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_filter_filter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.disjoint_iff_disjoint_coe /- _inst_2: decidable_eq \u219d\n -/\n#print finset.diag /- _inst_1: decidable_eq \u219d\n -/\n#print finset.off_diag /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_diag /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_off_diag /- _inst_1: decidable_eq \u219d\n -/\n#print finset.diag_card /- _inst_1: decidable_eq \u219d\n -/\n#print finset.off_diag_card /- _inst_1: decidable_eq \u219d\n -/\n#print list.to_finset_card_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.to_finset_card_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.disjoint_to_finset /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\fold.lean\n#print finset.fold_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fold_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fold_union_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fold_insert_idem /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fold_union_empty_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.fold_sup_bot_singleton /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\gcd.lean\n#print finset.lcm_insert /- _inst_4: decidable_eq \u219d\n -/\n#print finset.lcm_union /- _inst_4: decidable_eq \u219d\n -/\n#print finset.gcd_insert /- _inst_4: decidable_eq \u219d\n -/\n#print finset.gcd_union /- _inst_4: decidable_eq \u219d\n -/\n#print finset.gcd_eq_of_dvd_sub /- _inst_1: nontrivial \u219d nonempty\n -/\n\n-- data\\finset\\lattice.lean\n#print finset.sup_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sup_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sup_lt_iff /- _inst_2: is_total \u219d\n -/\n#print finset.comp_sup_eq_sup_comp_of_is_total /- _inst_2: is_total \u219d\n -/\n#print finset.mem_sup /- _inst_2: decidable_eq \u219d\n -/\n#print finset.inf_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.inf_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.lt_inf_iff /- h: is_total \u219d\n -/\n#print finset.comp_inf_eq_inf_comp_of_is_total /- h: is_total \u219d\n -/\n#print multiset.count_sup /- _inst_1: decidable_eq \u219d\n -/\n#print finset.supr_option_to_finset /- _inst_1: complete_lattice \u219d has_Sup\n -/\n#print finset.supr_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.infi_union /- _inst_2: decidable_eq \u219d\n -/\n#print finset.supr_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.infi_insert /- _inst_2: decidable_eq \u219d\n -/\n#print finset.supr_finset_image /- _inst_2: decidable_eq \u219d\n -/\n#print finset.infi_finset_image /- _inst_2: decidable_eq \u219d\n -/\n#print finset.supr_insert_update /- _inst_2: decidable_eq \u219d\n -/\n#print finset.infi_insert_update /- _inst_2: decidable_eq \u219d\n -/\n#print finset.supr_bind /- _inst_2: decidable_eq \u219d\n -/\n#print finset.infi_bind /- _inst_2: decidable_eq \u219d\n -/\n#print finset.bUnion_union /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bInter_inter /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bUnion_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bInter_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bUnion_finset_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bInter_finset_image /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bUnion_insert_update /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bInter_insert_update /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bUnion_bind /- _inst_1: decidable_eq \u219d\n -/\n#print finset.bInter_bind /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\pi.lean\n#print finset.pi /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_val /- _inst_1: decidable_eq \u219d\n -/\n#print finset.mem_pi /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi.cons /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi.cons_same /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi.cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_cons_injective /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_empty /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_insert /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_singletons /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_const_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.pi_disjoint_of_disjoint /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n\n-- data\\finset\\powerset.lean\n#print finset.powerset_insert /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\preimage.lean\n#print finset.image_subset_iff_subset_preimage /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_preimage /- _inst_1: decidable_eq \u219d\n -/\n#print finset.image_preimage_of_bij /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sigma_preimage_mk /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sigma_preimage_mk_of_subset /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sigma_image_fst_preimage_mk /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\finset\\sort.lean\n#print finset.sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print finset.sort_sorted /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print finset.sort_eq /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print finset.sort_nodup /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print finset.sort_to_finset /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n_inst_5: decidable_eq \u219d\n -/\n#print finset.mem_sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print finset.length_sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n\n-- data\\finsupp\\basic.lean\n#print finsupp.finsupp.decidable_eq /- _inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print finsupp.prod_add_index /- _inst_1: add_comm_monoid \u219d add_monoid\n -/\n#print finsupp.sum_add_index /- _inst_1: add_comm_monoid \u219d add_monoid\n -/\n#print finsupp.sum_sub_index /- _inst_1: add_comm_group \u219d add_comm_monoid add_group\n_inst_2: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print finsupp.prod_sum_index /- _inst_1: add_comm_monoid \u219d has_zero\n -/\n#print finsupp.sum_sum_index /- _inst_1: add_comm_monoid \u219d has_zero\n -/\n#print finsupp.eq_zero_of_comap_domain_eq_zero /- _inst_1: add_comm_monoid \u219d has_zero\n -/\n#print finsupp.comap_has_scalar /- _inst_2: mul_action \u219d\n -/\n#print finsupp.comap_mul_action /- _inst_2: mul_action \u219d\n -/\n#print finsupp.comap_distrib_mul_action /- _inst_2: mul_action \u219d\n -/\n#print finsupp.comap_smul_single /- _inst_2: mul_action \u219d\n -/\n#print finsupp.comap_smul_apply /- _inst_2: mul_action \u219d\n -/\n#print finsupp.has_scalar /- _inst_3: semimodule \u219d\n -/\n#print finsupp.smul_apply' /- _inst_2: semimodule \u219d has_scalar\n -/\n#print finsupp.semimodule /- _inst_3: semimodule \u219d\n -/\n#print finsupp.support_smul /- _inst_2: semimodule \u219d\n -/\n#print finsupp.filter_smul /- _inst_2: semimodule \u219d\n -/\n#print finsupp.map_domain_smul /- _inst_2: semimodule \u219d\n -/\n#print finsupp.smul_single /- _inst_2: semimodule \u219d\n -/\n#print finsupp.smul_apply /- _inst_1: semiring \u219d has_zero\n -/\n#print finsupp.sum_smul_index' /- _inst_3: semimodule \u219d\n -/\n#print finsupp.sum_mul /- _inst_1: semiring \u219d has_zero\n -/\n#print finsupp.mul_sum /- _inst_1: semiring \u219d has_zero\n -/\n\n-- data\\finsupp\\lattice.lean\n#print finsupp.le_def /- _inst_4: partial_order \u219d preorder\n -/\n#print finsupp.support_inf /- _inst_3: canonically_linear_ordered_add_monoid \u219d canonically_ordered_add_monoid linear_order\n -/\n#print finsupp.support_sup /- _inst_3: canonically_linear_ordered_add_monoid \u219d canonically_ordered_add_monoid semilattice_sup_bot\n -/\n#print finsupp.bot_eq_zero /- _inst_3: canonically_linear_ordered_add_monoid \u219d canonically_ordered_add_monoid\n -/\n\n-- data\\fintype\\basic.lean\n#print finset.boolean_algebra /- _inst_2: decidable_eq \u219d\n -/\n#print finset.compl_eq_univ_sdiff /- _inst_2: decidable_eq \u219d\n -/\n#print finset.mem_compl /- _inst_2: decidable_eq \u219d\n -/\n#print finset.coe_compl /- _inst_2: decidable_eq \u219d\n -/\n#print finset.univ_inter /- _inst_2: decidable_eq \u219d\n -/\n#print finset.inter_univ /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.decidable_eq_equiv_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.decidable_injective_fintype /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print fintype.decidable_surjective_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.decidable_bijective_fintype /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print fintype.decidable_left_inverse_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.decidable_right_inverse_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.of_multiset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.of_list /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.equiv_fin_of_forall_mem_list /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.equiv_fin /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.of_surjective /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_univ_diff /- _inst_1: decidable_eq \u219d\n -/\n#print finset.card_compl /- _inst_1: decidable_eq \u219d\n -/\n#print unique.fintype /- _inst_1: unique \u219d inhabited subsingleton\n -/\n#print univ_unique /- _inst_1: unique \u219d inhabited fintype\n -/\n#print fintype.fintype_prod_left /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.fintype_prod_right /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.coe_image_univ /- _inst_2: decidable_eq \u219d\n -/\n#print list.subtype.fintype /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.subtype.fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.pi_finset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.mem_pi_finset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.pi_finset_subset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.pi_finset_disjoint_of_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print pi.fintype /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.pi_finset_univ /- _inst_1: decidable_eq \u219d\n -/\n#print quotient.fintype /- _inst_2: decidable_rel \u219d\n -/\n#print finset.univ_pi_univ /- _inst_1: decidable_eq \u219d\n -/\n#print mem_image_univ_iff_mem_range /- _inst_2: decidable_eq \u219d\n -/\n#print quotient.fin_choice_aux /- _inst_1: decidable_eq \u219d\n -/\n#print quotient.fin_choice_aux_eq /- _inst_1: decidable_eq \u219d\n -/\n#print quotient.fin_choice /- _inst_1: decidable_eq \u219d\n -/\n#print quotient.fin_choice_eq /- _inst_1: decidable_eq \u219d\n -/\n#print perms_of_list /- _inst_1: decidable_eq \u219d\n -/\n#print length_perms_of_list /- _inst_1: decidable_eq \u219d\n -/\n#print mem_perms_of_list_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print mem_of_mem_perms_of_list /- _inst_1: decidable_eq \u219d\n -/\n#print mem_perms_of_list_iff /- _inst_1: decidable_eq \u219d\n -/\n#print nodup_perms_of_list /- _inst_1: decidable_eq \u219d\n -/\n#print perms_of_finset /- _inst_1: decidable_eq \u219d\n -/\n#print mem_perms_of_finset_iff /- _inst_1: decidable_eq \u219d\n -/\n#print card_perms_of_finset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype_perm /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.fintype /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print fintype.card_perm /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.card_equiv /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print fintype.bij_inv /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.left_inverse_bij_inv /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.right_inverse_bij_inv /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.bijective_bij_inv /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.preorder.well_founded /- _inst_2: preorder \u219d has_lt is_irrefl is_trans\n -/\n#print fintype.linear_order.is_well_order /- _inst_2: linear_order \u219d is_trichotomous preorder\n -/\n#print infinite.nonempty /- _inst_1: infinite \u219d nonempty\n -/\n\n-- data\\fintype\\card.lean\n#print fintype.sum_extend_by_zero /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.prod_extend_by_one /- _inst_1: decidable_eq \u219d\n -/\n#print is_compl.prod_mul_prod /- _inst_2: decidable_eq \u219d\n -/\n#print is_compl.sum_add_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_mul_prod_compl /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_add_sum_compl /- _inst_2: decidable_eq \u219d\n -/\n#print finset.sum_compl_add_sum /- _inst_2: decidable_eq \u219d\n -/\n#print finset.prod_compl_mul_prod /- _inst_2: decidable_eq \u219d\n -/\n#print finset.card_pi /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.card_pi_finset /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.card_pi /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.card_fun /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sum_univ_pi /- _inst_1: decidable_eq \u219d\n -/\n#print finset.prod_univ_pi /- _inst_1: decidable_eq \u219d\n -/\n#print finset.prod_univ_sum /- _inst_1: decidable_eq \u219d\n -/\n#print finset.sum_fiberwise /- _inst_1: decidable_eq \u219d\n -/\n#print finset.prod_fiberwise /- _inst_1: decidable_eq \u219d\n -/\n#print fintype.prod_fiberwise /- _inst_2: decidable_eq \u219d\n -/\n#print fintype.sum_fiberwise /- _inst_2: decidable_eq \u219d\n -/\n\n-- data\\fp\\basic.lean\n#print fp.div_nat_lt_two_pow /- C: fp.float_cfg \u219d\n -/\n\n-- data\\hash_map.lean\n#print hash_map.find_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_aux_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.contains_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.contains_aux_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.replace_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.erase_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.idx_enum /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.idx_enum_1 /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.as_list_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.mk_valid /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.find_aux_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.contains_aux_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.append_of_modify /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.modify /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.replace_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.replace /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.insert /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.erase_aux /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.valid.erase /- _inst_1: decidable_eq \u219d\n -/\n#print mk_hash_map /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.contains /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.has_mem /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.fold /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.entries /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.keys /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.contains_iff /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.entries_empty /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.keys_empty /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_empty /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.not_contains_empty /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.insert_lemma /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.insert /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.mem_insert /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_insert_eq /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_insert_ne /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_insert /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.insert_all /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.of_list /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.erase /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_erase_eq /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_erase_ne /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.find_erase /- _inst_1: decidable_eq \u219d\n -/\n#print hash_map.has_to_string /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\holor.lean\n#print holor.zero_mul /- _inst_1: ring \u219d mul_zero_class\n -/\n#print holor.mul_zero /- _inst_1: ring \u219d mul_zero_class\n -/\n#print holor.mul_scalar_mul /- _inst_1: monoid \u219d has_mul\n -/\n#print holor.unit_vec /- _inst_1: monoid \u219d has_one\n_inst_2: add_monoid \u219d has_zero\n -/\n#print holor.slice_unit_vec_mul /- _inst_1: ring \u219d semiring\n -/\n#print holor.cprank_max_nil /- _inst_1: monoid \u219d has_mul\n -/\n#print holor.cprank_max_1 /- _inst_1: monoid \u219d has_mul\n -/\n#print holor.cprank_max_sum /- _inst_1: ring \u219d monoid add_comm_monoid\n -/\n\n-- data\\indicator_function.lean\n#print set.indicator_smul /- _inst_3: distrib_mul_action \u219d\n -/\n#print set.indicator_prod_one /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n\n-- data\\int\\cast.lean\n#print int.cast_bit0 /- _inst_1: ring \u219d has_one add_group\n -/\n#print int.cast_nonneg /- _inst_1: linear_ordered_ring \u219d ordered_add_comm_group linear_ordered_semiring\n -/\n#print int.cast_min /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_ring\n -/\n#print int.cast_max /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_ring\n -/\n#print int.coe_int_dvd /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print ring_hom.eq_int_cast /- _inst_1: ring \u219d add_group semiring\n -/\n\n-- data\\lazy_list\\basic.lean\n#print thunk.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print lazy_list.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print lazy_list.mem.decidable /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\alist.lean\n#print list.to_alist /- _inst_1: decidable_eq \u219d\n -/\n#print alist.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_empty /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_is_some /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_eq_none /- _inst_1: decidable_eq \u219d\n -/\n#print alist.perm_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print alist.has_mem.mem.decidable /- _inst_1: decidable_eq \u219d\n -/\n#print alist.replace /- _inst_1: decidable_eq \u219d\n -/\n#print alist.keys_replace /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_replace /- _inst_1: decidable_eq \u219d\n -/\n#print alist.perm_replace /- _inst_1: decidable_eq \u219d\n -/\n#print alist.erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.keys_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.perm_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_erase_ne /- _inst_1: decidable_eq \u219d\n -/\n#print alist.erase_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_entries /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_entries_of_neg /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.keys_insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.perm_insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_insert_ne /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_to_alist /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_insert /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_insert_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_singleton_eq /- _inst_1: decidable_eq \u219d\n -/\n#print alist.entries_to_alist /- _inst_1: decidable_eq \u219d\n -/\n#print alist.to_alist_cons /- _inst_1: decidable_eq \u219d\n -/\n#print alist.extract /- _inst_1: decidable_eq \u219d\n -/\n#print alist.extract_eq_lookup_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.has_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union_entries /- _inst_1: decidable_eq \u219d\n -/\n#print alist.empty_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union_empty /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.perm_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union_erase /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print alist.lookup_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_lookup_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.mem_lookup_union_middle /- _inst_1: decidable_eq \u219d\n -/\n#print alist.insert_union /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union_assoc /- _inst_1: decidable_eq \u219d\n -/\n#print alist.union_comm_of_disjoint /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\bag_inter.lean\n#print list.nil_bag_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.bag_inter_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.cons_bag_inter_of_pos /- _inst_1: decidable_eq \u219d\n -/\n#print list.cons_bag_inter_of_neg /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_bag_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_bag_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.bag_inter_sublist_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.bag_inter_nil_iff_inter_nil /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\basic.lean\n#print list.insert_neg /- _inst_1: decidable_eq \u219d\n -/\n#print list.insert_pos /- _inst_1: decidable_eq \u219d\n -/\n#print list.doubleton_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_cons_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_cons_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_eq_length /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_le_length /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_lt_length /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_nth_le /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_nth /- _inst_1: decidable_eq \u219d\n -/\n#print list.index_of_inj /- _inst_1: decidable_eq \u219d\n -/\n#print list.prod_nil /- _inst_1: monoid \u219d has_one has_mul\n -/\n#print list.sum_nil /- _inst_1: add_monoid \u219d has_zero has_add\n -/\n#print list.prod_ne_zero /- _inst_2: domain \u219d monoid_with_zero nontrivial no_zero_divisors\n -/\n#print list.eq_of_sum_take_eq /- _inst_1: add_left_cancel_monoid \u219d add_monoid add_left_cancel_semigroup\n -/\n#print list.length_pos_of_sum_pos /- _inst_1: ordered_cancel_add_comm_monoid \u219d add_monoid preorder\n -/\n#print list.prod_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.sum_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.dvd_prod /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n -/\n#print list.exists_lt_of_sum_lt /- _inst_1: linear_ordered_cancel_add_comm_monoid \u219d ordered_add_comm_monoid linear_order\n -/\n#print list.alternating_prod_nil /- _inst_1: comm_group \u219d has_inv has_one has_mul\n -/\n#print list.alternating_sum_nil /- _inst_1: add_comm_group \u219d has_zero has_neg has_add\n -/\n#print list.alternating_sum_singleton /- _inst_1: add_comm_group \u219d has_zero has_neg has_add\n -/\n#print list.alternating_prod_singleton /- _inst_1: comm_group \u219d has_inv has_one has_mul\n -/\n#print list.alternating_prod_cons_cons /- _inst_1: comm_group \u219d has_inv has_one has_mul\n -/\n#print list.alternating_sum_cons_cons' /- _inst_1: add_comm_group \u219d has_zero has_neg has_add\n -/\n#print list.alternating_sum_cons_cons /- _inst_2: add_comm_group \u219d has_sub has_zero has_neg has_add\n -/\n#print list.lex.is_strict_total_order /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_asymm is_order_connected\n -/\n#print list.lex.decidable_rel /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_rel \u219d\n -/\n#print list.count_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_cons' /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_cons_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_cons_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_tail /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_le_of_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_le_count_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_append /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_concat /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_pos /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_eq_zero_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.not_mem_of_count_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_repeat /- _inst_1: decidable_eq \u219d\n -/\n#print list.le_count_iff_repeat_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.repeat_count_eq_of_count_eq_length /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_filter /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_prefix /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_suffix /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_infix /- _inst_1: decidable_eq \u219d\n -/\n#print list.insert_nil /- _inst_1: decidable_eq \u219d has_insert\n -/\n#print list.insert.def /- _inst_1: decidable_eq \u219d\n -/\n#print list.insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_insert_iff /- _inst_1: decidable_eq \u219d\n -/\n#print list.suffix_insert /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_insert_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.eq_or_mem_of_mem_insert /- _inst_1: decidable_eq \u219d\n -/\n#print list.length_insert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.length_insert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_cons_head /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_cons_tail /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_eq_erasep /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.exists_erase_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.length_erase_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_append_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_append_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_subset /- _inst_1: decidable_eq \u219d\n -/\n#print list.sublist.erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_of_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_erase_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_comm /- _inst_1: decidable_eq \u219d\n -/\n#print list.map_erase /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print list.map_foldl_erase /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print list.count_erase_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_erase_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.nil_diff /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_eq_foldl /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_append /- _inst_1: decidable_eq \u219d\n -/\n#print list.map_diff /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print list.diff_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_subset /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_diff_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.sublist.diff_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_diff_erase_sublist_of_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.nil_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.cons_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.sublist_suffix_of_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.suffix_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.union_sublist_append /- _inst_1: decidable_eq \u219d\n -/\n#print list.forall_mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.forall_mem_of_forall_mem_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.forall_mem_of_forall_mem_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_cons_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_cons_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_of_mem_inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_of_mem_inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_inter_of_mem_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_subset_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_subset_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.subset_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_eq_nil_iff_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print list.forall_mem_inter_of_forall_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.forall_mem_inter_of_forall_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.inter_reverse /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\defs.lean\n#print list.has_sdiff /- _inst_1: decidable_eq \u219d\n -/\n#print list.split_on /- _inst_1: decidable_eq \u219d\n -/\n#print list.indexes_of /- _inst_1: decidable_eq \u219d\n -/\n#print list.mfoldl_with_index /- _inst_1: monad \u219d has_pure has_bind\n -/\n#print list.mfoldr_with_index /- _inst_1: monad \u219d has_pure has_bind\n -/\n#print list.count /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_pairwise /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter /- _inst_1: decidable_rel \u219d\n -/\n#print list.decidable_chain /- _inst_1: decidable_rel \u219d\n -/\n#print list.decidable_chain' /- _inst_1: decidable_rel \u219d\n -/\n#print list.nodup_decidable /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print list.get_rest /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\erase_dup.lean\n#print list.erase_dup_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_cons_of_mem' /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_cons_of_not_mem' /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_cons_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_cons_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_subset /- _inst_1: decidable_eq \u219d\n -/\n#print list.subset_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_eq_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_idempotent /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dup_append /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\forall2.lean\n#print list.rel_sum /- _inst_1: add_monoid \u219d has_zero has_add\n_inst_2: add_monoid \u219d has_zero has_add\n -/\n#print list.rel_prod /- _inst_1: monoid \u219d has_one has_mul\n_inst_2: monoid \u219d has_one has_mul\n -/\n\n-- data\\list\\min_max.lean\n#print list.index_of_argmax /- _inst_2: decidable_eq \u219d\n -/\n#print list.index_of_argmin /- _inst_2: decidable_eq \u219d\n -/\n#print list.mem_argmax_iff /- _inst_2: decidable_eq \u219d\n -/\n#print list.argmax_eq_some_iff /- _inst_2: decidable_eq \u219d\n -/\n#print list.mem_argmin_iff /- _inst_2: decidable_eq \u219d\n -/\n#print list.argmin_eq_some_iff /- _inst_2: decidable_eq \u219d\n -/\n\n-- data\\list\\nodup.lean\n#print list.nth_le_index_of /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_iff_count_le_one /- _inst_1: decidable_eq \u219d\n -/\n#print list.count_eq_one_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_erase_eq_filter /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_erase_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_diff /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_erase_iff_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_erase_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_insert /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_union /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup_inter_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.diff_eq_filter_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_diff_iff_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodup.map_update /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\pairwise.lean\n#print list.pw_filter_nil /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_cons_of_pos /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_cons_of_neg /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_map /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_sublist /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_subset /- _inst_1: decidable_rel \u219d\n -/\n#print list.pairwise_pw_filter /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_eq_self /- _inst_1: decidable_rel \u219d\n -/\n#print list.pw_filter_idempotent /- _inst_1: decidable_rel \u219d\n -/\n#print list.forall_mem_pw_filter /- _inst_1: decidable_rel \u219d\n -/\n\n-- data\\list\\palindrome.lean\n#print palindrome.decidable /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\perm.lean\n#print list.perm_cons_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.count_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.subperm.count_le /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.sum_eq' /- _inst_1: add_monoid \u219d add_semigroup has_zero\n -/\n#print list.perm.prod_eq' /- _inst_1: monoid \u219d semigroup has_one\n -/\n#print list.perm.sum_eq /- _inst_1: add_comm_monoid \u219d has_zero is_commutative has_add is_associative\n -/\n#print list.perm.prod_eq /- _inst_1: comm_monoid \u219d has_one is_commutative is_associative has_mul\n -/\n#print list.perm.erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.subperm_cons_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_subperm /- _inst_1: decidable_eq \u219d\n -/\n#print list.subperm.erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.diff_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.diff_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.diff /- _inst_1: decidable_eq \u219d\n -/\n#print list.subperm.diff_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_cons_subperm_cons_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.subperm_cons_diff /- _inst_1: decidable_eq \u219d\n -/\n#print list.subset_cons_diff /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.bag_inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.bag_inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.bag_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.cons_perm_iff_perm_erase /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm_iff_count /- _inst_1: decidable_eq \u219d\n -/\n#print list.decidable_perm /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.insert /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm_insert_swap /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.union_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.union_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.union /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.inter_append /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.take_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.drop_inter /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.slice_inter /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\sigma.lean\n#print list.lookup /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_cons_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_is_some /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_eq_none /- _inst_1: decidable_eq \u219d\n -/\n#print list.of_mem_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print list.map_lookup_eq_find /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_lookup_iff /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_ext /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_cons_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_eq_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.head_lookup_all /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_lookup_all /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_length_le_one /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_eq_lookup /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_all_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm_lookup_all /- _inst_1: decidable_eq \u219d\n -/\n#print list.kreplace /- _inst_1: decidable_eq \u219d\n -/\n#print list.kreplace_of_forall_not /- _inst_1: decidable_eq \u219d\n -/\n#print list.kreplace_self /- _inst_1: decidable_eq \u219d\n -/\n#print list.keys_kreplace /- _inst_1: decidable_eq \u219d\n -/\n#print list.kreplace_nodupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kreplace /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_cons_eq /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_of_not_mem_keys /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_sublist /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_keys_subset /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_keys_of_mem_keys_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.exists_of_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_keys_kerase_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.keys_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_nodupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.not_mem_keys_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kerase_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_append_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_append_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.kerase_comm /- _inst_1: decidable_eq \u219d\n -/\n#print list.sizeof_kerase /- _inst_2: decidable_eq \u219d\n -/\n#print list.kinsert /- _inst_1: decidable_eq \u219d\n -/\n#print list.kinsert_def /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_keys_kinsert /- _inst_1: decidable_eq \u219d\n -/\n#print list.kinsert_nodupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kinsert /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kinsert /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kinsert_ne /- _inst_1: decidable_eq \u219d\n -/\n#print list.kextract /- _inst_1: decidable_eq \u219d\n -/\n#print list.kextract_eq_lookup_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.erase_dupkeys_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.nodupkeys_erase_dupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_erase_dupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.sizeof_erase_dupkeys /- _inst_2: decidable_eq \u219d\n -/\n#print list.kunion /- _inst_1: decidable_eq \u219d\n -/\n#print list.nil_kunion /- _inst_1: decidable_eq \u219d\n -/\n#print list.kunion_nil /- _inst_1: decidable_eq \u219d\n -/\n#print list.kunion_cons /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_keys_kunion /- _inst_1: decidable_eq \u219d\n -/\n#print list.kunion_kerase /- _inst_1: decidable_eq \u219d\n -/\n#print list.kunion_nodupkeys /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kunion_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kunion_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.perm.kunion /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kunion_left /- _inst_1: decidable_eq \u219d\n -/\n#print list.lookup_kunion_right /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_lookup_kunion /- _inst_1: decidable_eq \u219d\n -/\n#print list.mem_lookup_kunion_middle /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\list\\sort.lean\n#print list.decidable_sorted /- _inst_1: decidable_rel \u219d\n -/\n#print list.ordered_insert /- _inst_1: decidable_rel \u219d\n -/\n#print list.insertion_sort /- _inst_1: decidable_rel \u219d\n -/\n#print list.ordered_insert_nil /- _inst_1: decidable_rel \u219d\n -/\n#print list.ordered_insert_length /- _inst_1: decidable_rel \u219d\n -/\n#print list.perm_ordered_insert /- _inst_1: decidable_rel \u219d\n -/\n#print list.ordered_insert_count /- _inst_1: decidable_rel \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print list.perm_insertion_sort /- _inst_1: decidable_rel \u219d\n -/\n#print list.sorted_ordered_insert /- _inst_1: decidable_rel \u219d\n_inst_2: is_total \u219d\n -/\n#print list.sorted_insertion_sort /- _inst_1: decidable_rel \u219d\n_inst_2: is_total \u219d\n -/\n#print list.merge /- _inst_1: decidable_rel \u219d\n -/\n#print list.merge_sort /- _inst_1: decidable_rel \u219d\n -/\n#print list.merge_sort_cons_cons /- _inst_1: decidable_rel \u219d\n -/\n#print list.perm_merge /- _inst_1: decidable_rel \u219d\n -/\n#print list.perm_merge_sort /- _inst_1: decidable_rel \u219d\n -/\n#print list.length_merge_sort /- _inst_1: decidable_rel \u219d\n -/\n#print list.sorted_merge /- _inst_1: decidable_rel \u219d\n_inst_2: is_total \u219d\n -/\n#print list.sorted_merge_sort /- _inst_1: decidable_rel \u219d\n_inst_2: is_total \u219d\n -/\n#print list.merge_sort_eq_self /- _inst_1: decidable_rel \u219d\n_inst_2: is_total \u219d\n -/\n\n-- data\\matrix\\basic.lean\n#print matrix /- _inst_1: fintype \u219d\n_inst_2: fintype \u219d\n -/\n#print matrix.diagonal /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_apply_eq /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_apply_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_apply_ne' /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_zero /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_transpose /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_add /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_map /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.has_one /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_one /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.one_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.one_apply_eq /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.one_apply_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.one_apply_ne' /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.one_map /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.bit1_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.bit1_apply_eq /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.bit1_apply_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.dot_product_comm /- _inst_5: comm_semiring \u219d add_comm_monoid comm_semigroup\n -/\n#print matrix.dot_product_zero /- _inst_5: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.zero_dot_product /- _inst_5: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.add_dot_product /- _inst_5: semiring \u219d add_comm_monoid distrib\n -/\n#print matrix.dot_product_add /- _inst_5: semiring \u219d add_comm_monoid distrib\n -/\n#print matrix.diagonal_dot_product /- _inst_5: decidable_eq \u219d\n_inst_6: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.dot_product_diagonal /- _inst_5: decidable_eq \u219d\n_inst_6: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.dot_product_diagonal' /- _inst_5: decidable_eq \u219d\n_inst_6: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.dot_product_smul /- _inst_5: comm_semiring \u219d comm_semigroup semiring\n -/\n#print matrix.diagonal_neg /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.diagonal_mul /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.mul_diagonal /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.one_mul /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.mul_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.monoid /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.semiring /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.diagonal_mul_diagonal /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.diagonal_mul_diagonal' /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.ring_hom_map_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.ring_equiv_map_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.zero_hom_map_zero /- _inst_5: semiring \u219d has_zero\n -/\n#print matrix.add_monoid_hom_map_zero /- _inst_5: semiring \u219d add_monoid\n -/\n#print matrix.add_equiv_map_zero /- _inst_5: semiring \u219d add_monoid\n -/\n#print matrix.linear_map_map_zero /- _inst_5: semiring \u219d add_comm_monoid\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print matrix.linear_equiv_map_zero /- _inst_5: semiring \u219d add_comm_monoid\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print matrix.row_mul_col_apply /- _inst_5: semiring \u219d add_comm_monoid has_mul\n -/\n#print ring_hom.map_matrix /- _inst_5: decidable_eq \u219d\n -/\n#print ring_hom.map_matrix_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.ring /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.has_scalar /- _inst_5: semiring \u219d has_scalar\n -/\n#print matrix.semimodule /- _inst_7: semimodule \u219d\n -/\n#print matrix.smul_apply /- _inst_5: semiring \u219d has_mul\n -/\n#print matrix.smul_eq_diagonal_mul /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.scalar /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.coe_scalar /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.scalar_apply_eq /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.scalar_apply_ne /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.scalar_inj /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.smul_eq_mul_diagonal /- _inst_5: comm_semiring \u219d comm_semigroup semiring\n_inst_6: decidable_eq \u219d\n -/\n#print matrix.scalar.commute /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.mul_vec_diagonal /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.vec_mul_diagonal /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.mul_vec_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.vec_mul_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.std_basis_matrix /- _inst_5: semiring \u219d has_zero\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.smul_std_basis_matrix /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.std_basis_matrix_zero /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.std_basis_matrix_add /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.matrix_eq_sum_std_basis /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.std_basis_eq_basis_mul_basis /- _inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.induction_on' /- _inst_7: decidable_eq \u219d\n -/\n#print matrix.induction_on /- _inst_7: decidable_eq \u219d\n -/\n#print matrix.smul_mul_vec_assoc /- _inst_5: ring \u219d semiring\n -/\n#print matrix.transpose_one /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.transpose_smul /- _inst_5: semiring \u219d has_scalar\n -/\n#print matrix.star_ring /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.star_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.col_add /- _inst_5: semiring \u219d has_add\n -/\n#print matrix.row_add /- _inst_5: semiring \u219d has_add\n -/\n#print matrix.update_row /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_column /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_row_self /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_column_self /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_row_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_column_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_row_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_column_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_row_transpose /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.update_column_transpose /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.from_blocks_add /- _inst_5: semiring \u219d has_add\n -/\n#print matrix.from_blocks_multiply /- _inst_5: semiring \u219d add_comm_monoid has_mul\n -/\n#print matrix.from_blocks_diagonal /- _inst_5: semiring \u219d has_zero\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.from_blocks_one /- _inst_5: semiring \u219d has_one has_zero\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.block_diagonal /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_apply_eq /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_apply_ne /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_transpose /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_zero /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_diagonal /- _inst_5: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_one /- _inst_5: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_add /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_neg /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_sub /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.block_diagonal_mul /- _inst_5: decidable_eq \u219d\n_inst_7: semiring \u219d add_comm_monoid mul_zero_class\n -/\n#print matrix.block_diagonal_smul /- _inst_5: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n\n-- data\\matrix\\char_p.lean\n#print matrix.char_p /- _inst_2: ring \u219d semiring\n_inst_3: decidable_eq \u219d\n -/\n\n-- data\\matrix\\notation.lean\n#print matrix.empty_mul /- _inst_4: semiring \u219d add_comm_monoid has_mul\n -/\n#print matrix.empty_mul_empty /- _inst_4: semiring \u219d add_comm_monoid has_mul\n -/\n#print matrix.mul_empty /- _inst_4: semiring \u219d add_comm_monoid has_mul\n -/\n#print matrix.mul_val_succ /- _inst_4: semiring \u219d add_comm_monoid has_mul\n -/\n#print matrix.mul_vec_cons /- _inst_5: comm_semiring \u219d comm_semigroup semiring\n -/\n#print matrix.smul_empty /- _inst_4: semiring \u219d has_scalar\n -/\n#print matrix.smul_mat_empty /- _inst_4: semiring \u219d has_scalar\n -/\n\n-- data\\matrix\\pequiv.lean\n#print pequiv.to_matrix /- _inst_5: decidable_eq \u219d\n -/\n#print pequiv.mul_matrix_apply /- _inst_5: decidable_eq \u219d\n_inst_6: semiring \u219d monoid add_comm_monoid mul_zero_class\n -/\n#print pequiv.to_matrix_symm /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print pequiv.to_matrix_refl /- _inst_5: decidable_eq \u219d\n -/\n#print pequiv.matrix_mul_apply /- _inst_6: decidable_eq \u219d\n -/\n#print pequiv.to_pequiv_mul_matrix /- _inst_5: decidable_eq \u219d\n -/\n#print pequiv.to_matrix_trans /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print pequiv.to_matrix_bot /- _inst_5: decidable_eq \u219d\n -/\n#print pequiv.to_matrix_injective /- _inst_5: decidable_eq \u219d\n -/\n#print pequiv.to_matrix_swap /- _inst_5: decidable_eq \u219d\n_inst_6: ring \u219d has_one add_group\n -/\n#print pequiv.single_mul_single /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print pequiv.single_mul_single_of_ne /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print pequiv.single_mul_single_right /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n_inst_7: decidable_eq \u219d\n -/\n#print pequiv.equiv_to_pequiv_to_matrix /- _inst_5: decidable_eq \u219d\n -/\n\n-- data\\multiset\\antidiagonal.lean\n#print multiset.prod_map_add /- _inst_1: comm_semiring \u219d comm_monoid semiring\n -/\n\n-- data\\multiset\\basic.lean\n#print multiset.has_decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.decidable_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_cons_head /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_cons_tail /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_cons_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_add_left_pos /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_add_right_pos /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_add_right_neg /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_add_left_neg /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_lt /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_subset /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_erase_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_of_mem_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_comm /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_le_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_le_iff_le_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.card_erase_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.card_erase_lt_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.card_erase_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sum_map_mul_left /- _inst_1: semiring \u219d add_comm_monoid distrib mul_zero_class\n -/\n#print multiset.sum_map_mul_right /- _inst_1: semiring \u219d add_comm_monoid distrib mul_zero_class\n -/\n#print multiset.prod_ne_zero /- _inst_1: integral_domain \u219d monoid_with_zero nontrivial comm_monoid no_zero_divisors\n -/\n#print multiset.prod_eq_zero /- _inst_1: comm_semiring \u219d comm_monoid mul_zero_class\n -/\n#print multiset.prod_eq_zero_iff /- _inst_1: comm_cancel_monoid_with_zero \u219d monoid_with_zero comm_monoid no_zero_divisors\n -/\n#print multiset.abs_sum_le_sum_abs /- _inst_1: linear_ordered_field \u219d linear_ordered_add_comm_group\n -/\n#print multiset.sub /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.has_sub /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_sub /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_eq_fold_erase /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.add_sub_of_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_add' /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_add_cancel /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.add_sub_cancel_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.add_sub_cancel /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_le_sub_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_le_sub_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_le_iff_le_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_sub_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_le_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.card_sub /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.has_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_def /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.eq_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_le_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.map_union /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print multiset.inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.has_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.zero_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_inter_of_pos /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_inter_of_neg /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_le_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_le_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.lattice /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sup_eq_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inf_eq_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_inter_iff /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_le_iff /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.semilattice_inf_bot /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_comm /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_comm /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.eq_union_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_le_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_le_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_add_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.add_union_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_union_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_add_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.add_inter_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_inter_distrib /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.union_add_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_add_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.sub_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.filter_sub /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.filter_union /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.filter_inter /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.countp_sub /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.count /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_count /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_cons_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_cons_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_le_of_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_le_count_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count.is_add_monoid_hom /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_smul /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_pos /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_eq_zero_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_ne_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_repeat_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_repeat /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_erase_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_erase_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_sub /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_sum /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_bind /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_count_iff_repeat_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_filter_of_pos /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_filter_of_neg /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ext /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ext' /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_iff_count /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.distrib_lattice /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.semilattice_sup_bot /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_eq_zero_iff_disjoint /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.disjoint_union_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.disjoint_union_right /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\erase_dup.lean\n#print multiset.erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_cons_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_cons_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_subset /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.subset_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_subset' /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.subset_erase_dup' /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_eq_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_singleton /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_ext /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_map_erase_dup_eq /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print multiset.erase_dup_nsmul /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup.le_erase_dup_iff_le /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\finset_ops.lean\n#print multiset.ndinsert /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_ndinsert /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinsert_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinsert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinsert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_ndinsert /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_ndinsert_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_ndinsert_self /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_ndinsert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.length_ndinsert_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.length_ndinsert_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_ndinsert /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinsert_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.attach_ndinsert /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.disjoint_ndinsert_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.disjoint_ndinsert_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.zero_ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_ndunion_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.subset_ndunion_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndunion_le_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndunion_le /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.subset_ndunion_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_ndunion_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndunion_le_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_ndunion /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndunion_eq_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.erase_dup_add /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.coe_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.zero_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.cons_ndinter_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_cons_of_not_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_le_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_subset_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_subset_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_le_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.inter_le_ndinter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_eq_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.ndinter_eq_zero_iff_disjoint /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\fold.lean\n#print multiset.fold_union_inter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.fold_erase_dup_idem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.le_smul_erase_dup /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\gcd.lean\n#print multiset.lcm_erase_dup /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.lcm_ndunion /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.lcm_union /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.lcm_ndinsert /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.gcd_erase_dup /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.gcd_ndunion /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.gcd_union /- _inst_4: decidable_eq \u219d\n -/\n#print multiset.gcd_ndinsert /- _inst_4: decidable_eq \u219d\n -/\n\n-- data\\multiset\\lattice.lean\n#print multiset.sup_erase_dup /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.sup_ndunion /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.sup_union /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.sup_ndinsert /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.inf_erase_dup /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.inf_ndunion /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.inf_union /- _inst_2: decidable_eq \u219d\n -/\n#print multiset.inf_ndinsert /- _inst_2: decidable_eq \u219d\n -/\n\n-- data\\multiset\\nodup.lean\n#print multiset.nodup_iff_count_le_one /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.count_eq_one_of_mem /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_decidable /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_erase_eq_filter /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_erase_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_erase_iff_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_erase_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_inter_left /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_inter_right /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_union /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_sub_of_nodup /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\pi.lean\n#print multiset.pi.cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi.cons_same /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi.cons_ne /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi.cons_swap /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi_zero /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi_cons /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.pi_cons_injective /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.card_pi /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.nodup_pi /- _inst_1: decidable_eq \u219d\n -/\n#print multiset.mem_pi /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\powerset.lean\n#print multiset.revzip_powerset_aux_lemma /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\multiset\\sections.lean\n#print multiset.prod_map_sum /- _inst_1: comm_semiring \u219d comm_monoid semiring\n -/\n\n-- data\\multiset\\sort.lean\n#print multiset.sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print multiset.coe_sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print multiset.sort_sorted /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print multiset.sort_eq /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print multiset.mem_sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n#print multiset.length_sort /- _inst_1: decidable_rel \u219d\n_inst_4: is_total \u219d\n -/\n\n-- data\\mv_polynomial\\basic.lean\n#print mv_polynomial /- _inst_1: comm_semiring \u219d semiring\n -/\n#print mv_polynomial.decidable_eq_mv_polynomial /- _inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print mv_polynomial.alg_hom_ext /- _inst_2: comm_semiring \u219d semiring\n -/\n#print mv_polynomial.eval\u2082 /- _inst_2: comm_semiring \u219d comm_monoid semiring\n -/\n\n-- data\\mv_polynomial\\comm_ring.lean\n#print mv_polynomial.eval\u2082_sub /- _inst_2: comm_ring \u219d ring comm_semiring\n -/\n#print mv_polynomial.eval\u2082_neg /- _inst_2: comm_ring \u219d ring comm_semiring\n -/\n#print mv_polynomial.hom_C /- _inst_2: comm_ring \u219d ring\n -/\n\n-- data\\mv_polynomial\\counit.lean\n#print mv_polynomial.counit /- _inst_3: comm_ring \u219d ring comm_semiring\n -/\n\n-- data\\mv_polynomial\\variables.lean\n#print mv_polynomial.vars_C_mul /- _inst_2: integral_domain \u219d no_zero_divisors comm_semiring\n -/\n\n-- data\\nat\\cast.lean\n#print nat.cast_two /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print nat.cast_dvd /- _inst_1: field \u219d comm_group_with_zero semiring\n -/\n#print nat.cast_commute /- _inst_1: semiring \u219d monoid distrib mul_zero_class\n -/\n#print nat.strict_mono_cast /- _inst_1: linear_ordered_semiring \u219d nontrivial ordered_semiring\n -/\n#print nat.abs_cast /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_add_comm_group linear_ordered_semiring\n -/\n\n-- data\\nat\\choose\\sum.lean\n#print add_pow /- _inst_1: comm_semiring \u219d comm_semigroup semiring\n -/\n#print finset.sum_powerset_neg_one_pow_card /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\nat\\prime.lean\n#print nat.monoid.prime_pow /- _inst_1: monoid \u219d has_pow\n -/\n\n-- data\\num\\lemmas.lean\n#print pos_num.cast_to_int /- _inst_1: add_group \u219d add_monoid has_neg\n -/\n#print num.cast_to_int /- _inst_1: add_group \u219d add_monoid has_neg\n -/\n#print num.cast_add /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print num.cast_inj /- _inst_1: linear_ordered_semiring \u219d add_monoid char_zero has_one\n -/\n#print num.cast_of_znum /- _inst_1: add_group \u219d add_monoid\n -/\n#print znum.cast_inj /- _inst_1: linear_ordered_ring \u219d char_zero has_one add_group\n -/\n\n-- data\\padics\\ring_homs.lean\n#print padic_int.nth_hom /- _inst_1: comm_ring \u219d semiring\n -/\n#print padic_int.to_zmod_pow_eq_iff_ext /- _inst_1: comm_ring \u219d semiring\n -/\n\n-- data\\pequiv.lean\n#print pequiv.single /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.mem_single /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.mem_single_iff /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.symm_single /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.single_apply /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.single_apply_of_ne /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.single_trans_of_mem /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print pequiv.trans_single_of_mem /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print pequiv.single_trans_single /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print pequiv.single_subsingleton_eq_refl /- _inst_1: decidable_eq \u219d\n -/\n#print pequiv.trans_single_of_eq_none /- _inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print pequiv.single_trans_of_eq_none /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print pequiv.single_trans_single_of_ne /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print pequiv.semilattice_inf_bot /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n\n-- data\\pfunctor\\univariate\\M.lean\n#print pfunctor.M.ichildren /- _inst_2: decidable_eq \u219d\n -/\n#print pfunctor.M.isubtree /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.iselect /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.iselect_eq_default /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.ichildren_mk /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.isubtree_cons /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.iselect_nil /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.iselect_cons /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.M.ext_aux /- _inst_2: decidable_eq \u219d\n -/\n\n-- data\\pfunctor\\univariate\\basic.lean\n#print pfunctor.obj.iget /- _inst_1: decidable_eq \u219d\n -/\n#print pfunctor.iget_map /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\pi.lean\n#print pi.single /- _inst_1: decidable_eq \u219d\n -/\n#print pi.single_eq_same /- _inst_1: decidable_eq \u219d\n -/\n#print pi.single_eq_of_ne /- _inst_1: decidable_eq \u219d\n -/\n#print pi.single_injective /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\polynomial\\algebra_map.lean\n#print polynomial.C_eq_algebra_map /- _inst_4: comm_ring \u219d comm_semiring\n -/\n#print polynomial.alg_hom_eval\u2082_algebra_map /- _inst_4: comm_ring \u219d comm_semiring\n_inst_5: ring \u219d semiring\n_inst_6: ring \u219d semiring\n -/\n#print polynomial.eval\u2082_algebra_map_X /- _inst_5: ring \u219d semiring\n -/\n#print polynomial.eval\u2082_comp /- _inst_1: comm_semiring \u219d semiring\n -/\n#print polynomial.is_root_of_eval\u2082_map_eq_zero /- _inst_6: comm_ring \u219d comm_semiring\n -/\n#print polynomial.aeval_eq_sum_range /- _inst_6: comm_ring \u219d comm_semiring\n -/\n#print polynomial.aeval_eq_sum_range' /- _inst_6: comm_ring \u219d comm_semiring\n -/\n#print polynomial.aeval_endomorphism /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_3: module \u219d algebra\n -/\n\n-- data\\polynomial\\basic.lean\n#print polynomial.semimodule /- _inst_3: semimodule \u219d\n -/\n#print polynomial.smul_monomial /- _inst_3: semimodule \u219d\n -/\n#print polynomial.lhom_ext' /- _inst_3: semimodule \u219d\n -/\n\n-- data\\polynomial\\cancel_leads.lean\n#print polynomial.cancel_leads /- _inst_1: comm_ring \u219d ring\n -/\n\n-- data\\polynomial\\degree\\definitions.lean\n#print polynomial.monic.decidable /- _inst_2: decidable_eq \u219d\n -/\n#print polynomial.degree_mul /- _inst_1: integral_domain \u219d ring no_zero_divisors\n -/\n#print polynomial.leading_coeff_mul /- _inst_1: integral_domain \u219d ring no_zero_divisors\n -/\n\n-- data\\polynomial\\degree\\trailing_degree.lean\n#print polynomial.trailing_monic.decidable /- _inst_2: decidable_eq \u219d\n -/\n\n-- data\\polynomial\\derivative.lean\n#print polynomial.of_mem_support_derivative /- _inst_1: comm_semiring \u219d semiring\n -/\n#print polynomial.mem_support_derivative /- _inst_1: integral_domain \u219d no_zero_divisors semiring\n -/\n#print polynomial.nat_degree_eq_zero_of_derivative_eq_zero /- _inst_1: integral_domain \u219d no_zero_divisors semiring\n -/\n\n-- data\\polynomial\\div.lean\n#print polynomial.eval\u2082_mod_by_monic_eq_self_of_root /- _inst_2: comm_ring \u219d ring comm_semiring\n -/\n#print polynomial.multiplicity_X_sub_C_finite /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- data\\polynomial\\eval.lean\n#print polynomial.eval\u2082_mul /- _inst_2: comm_semiring \u219d comm_semigroup semiring\n -/\n#print polynomial.eval\u2082_eq_sum_range /- _inst_2: comm_semiring \u219d semiring\n -/\n#print polynomial.eval\u2082_eq_sum_range' /- _inst_2: comm_semiring \u219d semiring\n -/\n#print polynomial.is_root.decidable /- _inst_2: decidable_eq \u219d\n -/\n#print polynomial.support_map_subset /- _inst_1: comm_semiring \u219d semiring\n_inst_2: comm_semiring \u219d semiring\n -/\n#print polynomial.eval\u2082_neg /- _inst_2: ring \u219d add_group semiring\n -/\n#print polynomial.eval\u2082.is_ring_hom /- _inst_1: comm_ring \u219d ring\n_inst_2: comm_ring \u219d ring comm_semiring\n -/\n\n-- data\\polynomial\\field_division.lean\n#print polynomial.is_unit_iff_degree_eq_zero /- _inst_1: field \u219d group_with_zero integral_domain\n -/\n#print polynomial.degree_pos_of_ne_zero_of_nonunit /- _inst_1: field \u219d group_with_zero ring\n -/\n#print polynomial.monic_mul_leading_coeff_inv /- _inst_1: field \u219d group_with_zero integral_domain\n -/\n#print polynomial.degree_mul_leading_coeff_inv /- _inst_1: field \u219d group_with_zero integral_domain\n -/\n#print polynomial.div /- _inst_1: field \u219d has_inv ring\n -/\n#print polynomial.mod /- _inst_1: field \u219d has_inv ring\n -/\n#print polynomial.degree_map /- _inst_1: field \u219d division_ring\n_inst_2: field \u219d nontrivial semiring\n -/\n#print polynomial.map_eq_zero /- _inst_1: field \u219d division_ring\n -/\n#print polynomial.mem_roots_map /- _inst_2: field \u219d integral_domain\n -/\n#print polynomial.exists_root_of_degree_eq_one /- _inst_1: field \u219d ring comm_group_with_zero comm_semiring\n -/\n#print polynomial.coeff_inv_units /- _inst_1: field \u219d division_ring integral_domain\n -/\n#print polynomial.monic_normalize /- _inst_1: field \u219d comm_group_with_zero integral_domain\n -/\n#print polynomial.coe_norm_unit_of_ne_zero /- _inst_1: field \u219d comm_group_with_zero integral_domain\n -/\n#print polynomial.normalize_monic /- _inst_1: field \u219d comm_group_with_zero integral_domain\n -/\n#print polynomial.degree_normalize /- _inst_1: field \u219d comm_group_with_zero integral_domain\n -/\n#print polynomial.not_irreducible_C /- _inst_1: field \u219d group_with_zero ring comm_semiring\n -/\n#print polynomial.pairwise_coprime_X_sub /- _inst_2: field \u219d group_with_zero ring comm_semiring\n -/\n#print polynomial.prod_multiset_root_eq_finset_root /- _inst_1: field \u219d integral_domain\n -/\n#print polynomial.roots_C_mul /- _inst_1: field \u219d integral_domain\n -/\n\n-- data\\polynomial\\identities.lean\n#print polynomial.derivative_eval /- _inst_1: comm_ring \u219d comm_semiring\n -/\n\n-- data\\polynomial\\integral_normalization.lean\n#print polynomial.support_integral_normalization /- _inst_1: integral_domain \u219d nontrivial no_zero_divisors semiring\n -/\n#print polynomial.integral_normalization_eval\u2082_eq_zero /- _inst_2: comm_ring \u219d comm_monoid semiring\n -/\n\n-- data\\polynomial\\iterated_deriv.lean\n#print polynomial.coeff_iterated_deriv_as_prod_Ico /- _inst_1: comm_semiring \u219d comm_monoid semiring\n -/\n\n-- data\\polynomial\\monic.lean\n#print polynomial.monic.coeff_nat_degree /- _inst_1: comm_ring \u219d semiring\n -/\n#print polynomial.monic.degree_eq_zero_iff_eq_one /- _inst_1: comm_ring \u219d ring\n -/\n#print polynomial.monic.nat_degree_mul /- _inst_1: comm_ring \u219d ring\n -/\n#print polynomial.leading_coeff_of_injective /- _inst_1: ring \u219d semiring\n -/\n\n-- data\\polynomial\\reverse.lean\n#print polynomial.reverse_mul_of_domain /- _inst_2: domain \u219d ring no_zero_divisors\n -/\n\n-- data\\polynomial\\ring_division.lean\n#print polynomial.nat_degree_pos_of_aeval_root /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d semiring\n -/\n#print polynomial.root_mul /- _inst_1: integral_domain \u219d ring no_zero_divisors comm_semiring\n -/\n#print polynomial.prime_X_sub_C /- _inst_1: integral_domain \u219d nontrivial comm_ring no_zero_divisors\n -/\n#print polynomial.root_multiplicity_zero /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print polynomial.root_multiplicity_eq_zero /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print polynomial.root_multiplicity_pos /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print is_integral_domain.polynomial /- _inst_1: comm_ring \u219d ring\n -/\n\n-- data\\prod.lean\n#print prod.lex.decidable /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_rel \u219d\n_inst_3: decidable_rel \u219d\n -/\n\n-- data\\quaternion.lean\n#print quaternion_algebra.has_coe_t /- _inst_1: comm_ring \u219d has_zero\n -/\n#print quaternion_algebra.coe_re /- _inst_1: comm_ring \u219d has_coe_t\n -/\n#print quaternion_algebra.coe_im_i /- _inst_1: comm_ring \u219d has_coe_t has_zero\n -/\n#print quaternion_algebra.coe_im_j /- _inst_1: comm_ring \u219d has_coe_t has_zero\n -/\n#print quaternion_algebra.coe_im_k /- _inst_1: comm_ring \u219d has_coe_t has_zero\n -/\n#print quaternion_algebra.coe_injective /- _inst_1: comm_ring \u219d has_coe_t\n -/\n#print quaternion_algebra.has_zero /- _inst_1: comm_ring \u219d has_zero\n -/\n#print quaternion_algebra.has_one /- _inst_1: comm_ring \u219d has_one has_zero\n -/\n#print quaternion_algebra.has_add /- _inst_1: comm_ring \u219d has_add\n -/\n#print quaternion_algebra.has_neg /- _inst_1: comm_ring \u219d has_neg\n -/\n#print quaternion_algebra.has_mul /- _inst_1: comm_ring \u219d has_sub has_add has_mul\n -/\n#print quaternion_algebra.conj_fixed /- _inst_2: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print quaternion.has_coe_t /- _inst_1: comm_ring \u219d has_one has_coe_t has_neg\n -/\n#print quaternion.ext /- _inst_1: comm_ring \u219d has_one has_neg\n -/\n#print quaternion.ext_iff /- _inst_1: comm_ring \u219d has_one has_neg\n -/\n#print quaternion.coe_re /- _inst_1: comm_ring \u219d has_one has_coe_t has_neg\n -/\n#print quaternion.coe_im_i /- _inst_1: comm_ring \u219d has_one has_coe_t has_zero has_neg\n -/\n#print quaternion.coe_im_j /- _inst_1: comm_ring \u219d has_one has_coe_t has_zero has_neg\n -/\n#print quaternion.coe_im_k /- _inst_1: comm_ring \u219d has_one has_coe_t has_zero has_neg\n -/\n#print quaternion.norm_sq_eq_zero /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_ring comm_ring\n -/\n#print quaternion.norm_sq_nonneg /- _inst_1: linear_ordered_comm_ring \u219d linear_ordered_ring comm_ring\n -/\n#print quaternion.has_inv /- _inst_1: linear_ordered_field \u219d has_inv comm_ring\n -/\n\n-- data\\rat\\cast.lean\n#print rat.cast_coe /- _inst_1: division_ring \u219d has_one has_zero has_neg has_add has_div\n -/\n\n-- data\\real\\cau_seq.lean\n#print is_absolute_value.abv_one /- _inst_4: domain \u219d nontrivial ring\n -/\n#print is_absolute_value.abv_inv /- _inst_4: field \u219d group_with_zero domain\n -/\n#print is_cau_seq /- _inst_1: linear_ordered_field \u219d has_lt has_zero\n_inst_2: ring \u219d has_sub\n -/\n#print cau_seq.one_not_equiv_zero /- _inst_2: integral_domain \u219d nontrivial ring\n -/\n\n-- data\\real\\cau_seq_completion.lean\n#print cau_seq.completion.Cauchy /- _inst_2: comm_ring \u219d ring\n -/\n#print cau_seq.completion.cau_seq_zero_ne_one /- _inst_2: field \u219d nontrivial ring\n -/\n\n-- data\\seq\\wseq.lean\n#print wseq.index_of /- _inst_1: decidable_eq \u219d\n -/\n#print wseq.indexes_of /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\set\\finite.lean\n#print set.decidable_mem_of_fintype /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_insert /- _inst_1: decidable_eq \u219d\n -/\n#print set.to_finset_insert /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_union /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_image /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_range /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_map /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_Union /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_bUnion /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_bUnion' /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_image2 /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_bind /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_bind' /- _inst_1: decidable_eq \u219d\n -/\n#print set.fintype_seq /- _inst_1: decidable_eq \u219d\n -/\n#print finset.coe_bind /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\set\\function.lean\n#print set.piecewise_insert /- _inst_2: decidable_eq \u219d\n -/\n#print function.update_comp_eq_of_not_mem_range /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_comp_eq_of_injective /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n\n-- data\\set\\intervals\\basic.lean\n#print set.Ioo /- _inst_1: preorder \u219d has_lt\n -/\n#print set.Ico /- _inst_1: preorder \u219d has_lt has_le\n -/\n#print set.Iio /- _inst_1: preorder \u219d has_lt\n -/\n#print set.Icc /- _inst_1: preorder \u219d has_le\n -/\n#print set.Iic /- _inst_1: preorder \u219d has_le\n -/\n#print set.Ioc /- _inst_1: preorder \u219d has_lt has_le\n -/\n#print set.Ici /- _inst_1: preorder \u219d has_le\n -/\n#print set.Ioi /- _inst_1: preorder \u219d has_lt\n -/\n#print set.Iio_inter_Iio /- _inst_2: is_total \u219d\n -/\n#print set.Ioi_inter_Ioi /- _inst_2: is_total \u219d\n -/\n#print set.Icc_inter_Icc /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\n -/\n#print set.Ico_inter_Ico /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\nht: is_total \u219d\n -/\n#print set.Ioc_inter_Ioc /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\nht: is_total \u219d\n -/\n#print set.Ioo_inter_Ioo /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\nht: is_total \u219d\n -/\n#print set.nonempty_Ico_sdiff /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_cancel_add_comm_monoid\n -/\n\n-- data\\set\\intervals\\pi.lean\n#print set.Icc_diff_pi_univ_Ioo_subset /- _inst_1: decidable_eq \u219d\n -/\n#print set.Icc_diff_pi_univ_Ioc_subset /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\set\\intervals\\surj_on.lean\n#print surj_on_Ioo_of_monotone_surjective /- _inst_2: partial_order \u219d preorder\n -/\n#print surj_on_Ioi_of_monotone_surjective /- _inst_2: partial_order \u219d preorder\n -/\n\n-- data\\sigma\\basic.lean\n#print sigma.decidable_eq /- h\u2081: decidable_eq \u219d\n -/\n#print psigma.decidable_eq /- h\u2081: decidable_eq \u219d\n -/\n\n-- data\\support.lean\n#print function.support_inv /- _inst_1: division_ring \u219d group_with_zero\n -/\n\n-- data\\sym.lean\n#print sym.decidable_mem /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\sym2.lean\n#print sym2.is_diag.decidable_pred /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.from_rel.decidable_as_set /- h: decidable_rel \u219d\n -/\n#print sym2.from_rel.decidable_pred /- h: decidable_rel \u219d\n -/\n#print sym2.rel_bool /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.rel_bool_spec /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.rel.decidable_rel /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.mem.other' /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.mem_other_spec' /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.other_eq_other' /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.mem_other_mem' /- _inst_1: decidable_eq \u219d\n -/\n#print sym2.other_invol' /- _inst_1: decidable_eq \u219d\n -/\n\n-- data\\tree.lean\n#print tree.index_of /- _inst_1: decidable_rel \u219d\n -/\n\n-- data\\vector2.lean\n#print vector.traverse_def /- _inst_3: is_lawful_applicative \u219d\n -/\n\n-- data\\zmod\\basic.lean\n#print zmod.nat_cast_val /- _inst_1: ring \u219d has_one has_zero has_neg has_add\n -/\n#print zmod.cast_one /- _inst_1: ring \u219d has_neg subsingleton semiring\n -/\n#print zmod.cast_add /- _inst_1: ring \u219d add_group semiring\n -/\n#print zmod.algebra /- _inst_3: comm_ring \u219d ring comm_semiring\n -/\n\n-- data\\zsqrtd\\gaussian_int.lean\n#print gaussian_int.nat_cast_nat_abs_norm /- _inst_1: ring \u219d has_one has_zero has_neg has_add\n -/\n\n-- deprecated\\group.lean\n#print is_add_hom.add /- _inst_4: add_semigroup \u219d has_add\n -/\n#print is_mul_hom.mul /- _inst_4: semigroup \u219d has_mul\n -/\n#print add_equiv.is_add_hom /- _inst_1: add_monoid \u219d has_add\n_inst_2: add_monoid \u219d has_add\n -/\n#print mul_equiv.is_mul_hom /- _inst_1: monoid \u219d has_mul\n_inst_2: monoid \u219d has_mul\n -/\n#print is_add_monoid_hom.map_add /- _inst_3: is_add_monoid_hom \u219d is_add_hom\n -/\n#print is_monoid_hom.map_mul /- _inst_3: is_monoid_hom \u219d is_mul_hom\n -/\n#print is_add_monoid_hom.is_add_monoid_hom_mul_left /- _inst_1: semiring \u219d add_monoid distrib mul_zero_class\n -/\n#print is_add_monoid_hom.is_add_monoid_hom_mul_right /- _inst_1: semiring \u219d add_monoid distrib mul_zero_class\n -/\n#print is_group_hom.to_is_monoid_hom /- _inst_3: is_group_hom \u219d is_mul_hom\n -/\n#print is_add_group_hom.to_is_add_monoid_hom /- _inst_3: is_add_group_hom \u219d is_add_hom\n -/\n#print is_group_hom.map_one /- _inst_3: is_group_hom \u219d is_monoid_hom\n -/\n#print is_add_group_hom.map_zero /- _inst_3: is_add_group_hom \u219d is_add_monoid_hom\n -/\n#print is_group_hom.comp /- _inst_3: is_group_hom \u219d is_mul_hom\n_inst_5: is_group_hom \u219d is_mul_hom\n -/\n#print is_add_group_hom.comp /- _inst_3: is_add_group_hom \u219d is_add_hom\n_inst_5: is_add_group_hom \u219d is_add_hom\n -/\n#print is_add_group_hom.add /- _inst_5: add_comm_group \u219d add_group\n_inst_6: is_add_group_hom \u219d is_add_hom\n_inst_7: is_add_group_hom \u219d is_add_hom\n -/\n#print is_group_hom.mul /- _inst_5: comm_group \u219d group\n_inst_6: is_group_hom \u219d is_mul_hom\n_inst_7: is_group_hom \u219d is_mul_hom\n -/\n#print is_group_hom.inv /- _inst_5: comm_group \u219d group\n_inst_6: is_group_hom \u219d is_mul_hom\n -/\n#print is_add_group_hom.neg /- _inst_5: add_comm_group \u219d add_group\n_inst_6: is_add_group_hom \u219d is_add_hom\n -/\n#print ring_hom.is_add_group_hom /- _inst_1: ring \u219d add_group semiring\n_inst_2: ring \u219d add_group semiring\n -/\n#print is_add_group_hom.sub /- _inst_2: add_comm_group \u219d add_group\n -/\n#print additive.is_add_group_hom /- _inst_3: is_group_hom \u219d is_mul_hom\n -/\n#print multiplicative.is_group_hom /- _inst_3: is_add_group_hom \u219d is_add_hom\n -/\n\n-- deprecated\\ring.lean\n#print is_ring_hom.is_add_group_hom /- _inst_3: is_ring_hom \u219d is_add_hom\n -/\n\n-- deprecated\\subfield.lean\n#print range.is_subfield /- _inst_1: field \u219d is_subfield semiring\n -/\n#print field.closure /- _inst_1: field \u219d ring has_div\n -/\n\n-- deprecated\\subgroup.lean\n#print gpowers /- _inst_1: group \u219d has_pow\n -/\n#print normal_add_subgroup_of_add_comm_group /- _inst_1: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print normal_subgroup_of_comm_group /- _inst_1: comm_group \u219d comm_semigroup group\n -/\n#print is_subgroup.trivial /- _inst_2: group \u219d has_one\n -/\n#print is_add_subgroup.trivial /- _inst_2: add_group \u219d has_zero\n -/\n#print is_subgroup.eq_trivial_iff /- _inst_2: is_subgroup \u219d is_submonoid\n -/\n#print is_add_subgroup.eq_trivial_iff /- _inst_2: is_add_subgroup \u219d is_add_submonoid\n -/\n#print is_subgroup.center /- _inst_2: group \u219d has_mul\n -/\n#print is_add_subgroup.add_center /- _inst_2: add_group \u219d has_add\n -/\n#print is_subgroup.normalizer /- _inst_1: group \u219d has_inv has_mul\n -/\n#print is_add_subgroup.add_normalizer /- _inst_1: add_group \u219d has_neg has_add\n -/\n#print is_group_hom.range_subgroup /- _inst_3: is_group_hom \u219d is_subgroup\n -/\n#print is_add_group_hom.range_add_subgroup /- _inst_3: is_add_group_hom \u219d is_add_subgroup\n -/\n#print is_group_hom.normal_subgroup_ker /- _inst_3: is_group_hom \u219d normal_subgroup\n -/\n#print is_add_group_hom.normal_add_subgroup_ker /- _inst_3: is_add_group_hom \u219d normal_add_subgroup\n -/\n#print subtype_mk.is_add_group_hom /- _inst_4: is_add_group_hom \u219d is_add_monoid_hom\n -/\n#print subtype_mk.is_group_hom /- _inst_4: is_group_hom \u219d is_monoid_hom\n -/\n#print group.mem_closure_union_iff /- _inst_2: comm_group \u219d group comm_monoid\n -/\n#print add_group.mem_closure_union_iff /- _inst_2: add_comm_group \u219d add_comm_monoid add_group\n -/\n\n-- deprecated\\submonoid.lean\n#print powers /- _inst_1: monoid \u219d has_pow\n -/\n#print multiples /- _inst_2: add_monoid \u219d has_zero has_add\n -/\n#print range.is_add_submonoid /- _inst_4: is_add_monoid_hom \u219d is_add_submonoid\n -/\n#print range.is_submonoid /- _inst_4: is_monoid_hom \u219d is_submonoid\n -/\n#print add_monoid.mem_closure_union_iff /- _inst_3: add_comm_monoid \u219d add_monoid add_comm_semigroup\n -/\n#print monoid.mem_closure_union_iff /- _inst_3: comm_monoid \u219d monoid comm_semigroup\n -/\n\n-- deprecated\\subring.lean\n#print ring_hom.is_subring_preimage /- _inst_4: is_subring \u219d is_add_subgroup is_submonoid\n -/\n#print ring_hom.is_subring_image /- _inst_4: is_subring \u219d is_add_subgroup is_submonoid\n -/\n#print ring_hom.is_subring_set_range /- _inst_2: ring \u219d is_add_subgroup semiring is_add_group_hom\n -/\n#print is_subring.inter /- _inst_3: is_subring \u219d is_add_subgroup is_submonoid\n_inst_4: is_subring \u219d is_add_subgroup is_submonoid\n -/\n#print ring.closure /- _inst_1: ring \u219d monoid add_group\n -/\n#print ring.closure_subset /- _inst_3: is_subring \u219d is_add_subgroup is_submonoid\n -/\n#print ring.closure_subset_iff /- _inst_3: is_subring \u219d is_add_subgroup is_submonoid\n -/\n\n-- dynamics\\fixed_points\\basic.lean\n#print function.is_fixed_pt.decidable /- h: decidable_eq \u219d\n -/\n#print function.fixed_points.decidable /- _inst_1: decidable_eq \u219d\n -/\n\n-- dynamics\\periodic_pts.lean\n#print function.is_periodic_pt.decidable /- _inst_1: decidable_eq \u219d\n -/\n\n-- field_theory\\algebraic_closure.lean\n#print is_alg_closure /- _inst_1: field \u219d comm_ring\n -/\n#print algebraic_closure.monic_irreducible /- _inst_1: field \u219d ring\n -/\n\n-- field_theory\\chevalley_warning.lean\n#print mv_polynomial.sum_mv_polynomial_eq_zero /- _inst_4: decidable_eq \u219d\n -/\n#print char_dvd_card_solutions_family /- _inst_4: decidable_eq \u219d\n_inst_5: decidable_eq \u219d\n -/\n#print char_dvd_card_solutions /- _inst_4: decidable_eq \u219d\n_inst_5: decidable_eq \u219d\n -/\n\n-- field_theory\\finite\\basic.lean\n#print finite_field.card_image_polynomial_eval /- _inst_4: decidable_eq \u219d\n -/\n#print finite_field.card_units /- _inst_1: field \u219d division_ring\n -/\n#print finite_field.prod_univ_units_id_eq_neg_one /- _inst_1: field \u219d integral_domain\n -/\n#print finite_field.card /- _inst_1: field \u219d integral_domain\n -/\n\n-- field_theory\\finite\\polynomial.lean\n#print mv_polynomial.indicator /- _inst_1: field \u219d integral_domain\n -/\n#print mv_polynomial.eval\u2097 /- _inst_2: fintype \u219d\n_inst_3: fintype \u219d\n -/\n#print mv_polynomial.R /- _inst_1: fintype \u219d\n -/\n\n-- field_theory\\fixed.lean\n#print fixed_by.is_subfield /- _inst_1: group \u219d monoid\n -/\n#print fixed_points.mul_action.fixed_points.is_subfield /- _inst_1: group \u219d monoid is_subfield\n_inst_3: mul_semiring_action \u219d is_subfield\n -/\n#print fixed_points.mul_action.fixed_points.is_invariant_subring /- _inst_1: group \u219d monoid is_subfield\n_inst_2: field \u219d ring is_subfield\n -/\n#print fixed_points.smul /- _inst_1: group \u219d is_invariant_subring monoid mul_semiring_action is_subfield\n_inst_2: field \u219d is_invariant_subring mul_semiring_action ring is_subfield\n -/\n\n-- field_theory\\minimal_polynomial.lean\n#print minimal_polynomial.degree_pos /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print minimal_polynomial.aeval_ne_zero_of_dvd_not_unit_minimal_polynomial /- _inst_2: domain \u219d ring\n -/\n#print minimal_polynomial.dvd_map_of_is_scalar_tower /- _inst_10: is_scalar_tower \u219d\n -/\n#print minimal_polynomial.gcd_domain_eq_field_fractions /- _inst_8: integral_domain \u219d comm_ring domain\n_inst_11: is_scalar_tower \u219d\n -/\n#print minimal_polynomial.prime /- _inst_2: domain \u219d nontrivial ring no_zero_divisors\n -/\n#print minimal_polynomial.root /- _inst_1: field \u219d integral_domain\n -/\n\n-- field_theory\\mv_polynomial.lean\n#print mv_polynomial.restrict_total_degree /- _inst_1: field \u219d comm_ring\n -/\n#print mv_polynomial.restrict_degree /- _inst_1: field \u219d comm_ring\n -/\n#print mv_polynomial.map_range_eq_map /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d comm_semiring\n -/\n#print mv_polynomial.is_basis_monomials /- _inst_1: field \u219d comm_ring\n -/\n\n-- field_theory\\primitive_element.lean\n#print field.primitive_element_inf_aux_exists_c /- _inst_1: field \u219d division_ring\n_inst_3: field \u219d integral_domain has_div\n -/\n\n-- field_theory\\separable.lean\n#print polynomial.separable_X_sub_C /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print polynomial.separable.inj_of_prod_X_sub_C /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print polynomial.of_irreducible_expand /- _inst_1: field \u219d integral_domain\n -/\n#print polynomial.expand_char /- _inst_1: field \u219d comm_ring\n -/\n#print polynomial.is_unit_or_eq_zero_of_separable_expand /- _inst_1: field \u219d integral_domain\n -/\n#print polynomial.not_unit_X_sub_C /- _inst_1: field \u219d integral_domain\n -/\n#print polynomial.multiplicity_le_one_of_separable /- _inst_1: field \u219d comm_ring\n -/\n#print is_separable /- _inst_1: field \u219d comm_ring\n_inst_2: field \u219d ring\n -/\n#print is_separable_tower_top_of_is_separable /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_separable_tower_bot_of_is_separable /- _inst_7: is_scalar_tower \u219d\n -/\n\n-- field_theory\\splitting_field.lean\n#print polynomial.splits /- _inst_1: field \u219d semiring\n_inst_2: field \u219d ring\n -/\n#print polynomial.roots_map /- _inst_2: field \u219d integral_domain\n -/\n#print lift_of_splits /- _inst_3: field \u219d integral_domain\n -/\n#print polynomial.factor /- _inst_1: field \u219d ring\n -/\n#print polynomial.is_splitting_field.map /- _inst_7: is_scalar_tower \u219d\n -/\n#print polynomial.is_splitting_field.mul /- _inst_7: is_scalar_tower \u219d\n -/\n\n-- field_theory\\subfield.lean\n#print ring_hom.mem_field_range /- _inst_1: field \u219d ring\n_inst_2: field \u219d ring\n -/\n#print ring_hom.restrict_field /- _inst_2: field \u219d semiring\n -/\n#print ring_hom.eq_of_eq_on_subfield_top /- _inst_2: field \u219d semiring\n -/\n\n-- field_theory\\tower.lean\n#print dim_mul_dim' /- _inst_7: is_scalar_tower \u219d\n -/\n#print dim_mul_dim /- _inst_14: is_scalar_tower \u219d\n -/\n#print finite_dimensional.trans /- _inst_7: is_scalar_tower \u219d\n -/\n#print finite_dimensional.right /- _inst_7: is_scalar_tower \u219d\n -/\n#print finite_dimensional.findim_mul_findim /- _inst_7: is_scalar_tower \u219d\n -/\n#print finite_dimensional.linear_map /- _inst_10: vector_space \u219d\n_inst_12: vector_space \u219d\n -/\n#print finite_dimensional.findim_linear_map /- _inst_10: vector_space \u219d finite_dimensional\n_inst_12: vector_space \u219d finite_dimensional\n -/\n#print finite_dimensional.linear_map' /- _inst_13: vector_space \u219d finite_dimensional\n -/\n#print finite_dimensional.findim_linear_map' /- _inst_13: vector_space \u219d\n -/\n\n-- geometry\\euclidean\\basic.lean\n#print euclidean_geometry.cospherical /- _inst_2: metric_space \u219d has_dist\n -/\n\n-- geometry\\manifold\\algebra\\lie_group.lean\n#print lie_add_group.to_has_smooth_add /- _inst_4: normed_space \u219d\n -/\n#print lie_group.to_has_smooth_mul /- _inst_4: normed_space \u219d\n -/\n#print smooth_pow /- _inst_4: normed_space \u219d\n -/\n#print smooth_neg /- _inst_4: normed_space \u219d\n -/\n#print smooth_inv /- _inst_4: normed_space \u219d\n -/\n#print smooth.neg /- _inst_4: normed_space \u219d smooth_manifold_with_corners\n_inst_13: normed_space \u219d\n -/\n#print smooth.inv /- _inst_4: normed_space \u219d smooth_manifold_with_corners\n_inst_13: normed_space \u219d\n -/\n#print smooth_on.inv /- _inst_4: normed_space \u219d smooth_manifold_with_corners\n_inst_13: normed_space \u219d\n -/\n#print smooth_on.neg /- _inst_4: normed_space \u219d smooth_manifold_with_corners\n_inst_13: normed_space \u219d\n -/\n#print prod.lie_group /- _inst_4: normed_space \u219d smooth_manifold_with_corners has_smooth_mul\n_inst_11: normed_space \u219d smooth_manifold_with_corners has_smooth_mul\n -/\n#print prod.lie_add_group /- _inst_4: normed_space \u219d smooth_manifold_with_corners has_smooth_add\n_inst_11: normed_space \u219d smooth_manifold_with_corners has_smooth_add\n -/\n#print lie_add_group_morphism.has_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_group_morphism.has_one /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_add_group_morphism.inhabited /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_group_morphism.inhabited /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_group_morphism.has_coe_to_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_add_group_morphism.has_coe_to_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print lie_add_group_core.to_smooth_manifold_with_corners /- _inst_3: normed_space \u219d\n -/\n#print lie_group_core.to_smooth_manifold_with_corners /- _inst_3: normed_space \u219d\n -/\n#print lie_group_core.to_topological_group /- _inst_3: normed_space \u219d\n -/\n#print lie_add_group_core.to_topological_add_group /- _inst_3: normed_space \u219d\n -/\n#print lie_add_group_core.to_lie_add_group /- _inst_3: normed_space \u219d\n -/\n#print lie_group_core.to_lie_group /- _inst_3: normed_space \u219d\n -/\n#print normed_space_lie_group /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n\n-- geometry\\manifold\\algebra\\monoid.lean\n#print has_smooth_add.to_smooth_manifold_with_corners /- _inst_5: normed_space \u219d\n -/\n#print has_smooth_mul.to_smooth_manifold_with_corners /- _inst_5: normed_space \u219d\n -/\n#print smooth_mul /- _inst_5: normed_space \u219d\n -/\n#print smooth_add /- _inst_5: normed_space \u219d\n -/\n#print smooth.add /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n_inst_12: normed_space \u219d\n -/\n#print smooth.mul /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n_inst_12: normed_space \u219d\n -/\n#print smooth_add_left /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_mul_left /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_add_right /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_mul_right /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_on.add /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n_inst_12: normed_space \u219d\n -/\n#print smooth_on.mul /- _inst_5: normed_space \u219d smooth_manifold_with_corners\n_inst_12: normed_space \u219d\n -/\n#print has_smooth_mul.prod /- _inst_19: normed_space \u219d smooth_manifold_with_corners\n_inst_27: normed_space \u219d smooth_manifold_with_corners\n -/\n#print has_smooth_add.sum /- _inst_19: normed_space \u219d smooth_manifold_with_corners\n_inst_27: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_monoid_morphism.has_one /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print smooth_add_monoid_morphism.has_zero /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print smooth_monoid_morphism.inhabited /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print smooth_add_monoid_morphism.inhabited /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print smooth_monoid_morphism.has_coe_to_fun /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print smooth_add_monoid_morphism.has_coe_to_fun /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print has_smooth_add_core.to_has_continuous_add /- _inst_4: normed_space \u219d\n_inst_9: add_group \u219d has_add\n -/\n#print has_smooth_mul_core.to_has_continuous_mul /- _inst_4: normed_space \u219d\n_inst_9: group \u219d has_mul\n -/\n#print has_smooth_add_core.to_has_smooth_add /- _inst_4: normed_space \u219d\n -/\n#print has_smooth_mul_core.to_has_smooth_mul /- _inst_4: normed_space \u219d\n -/\n\n-- geometry\\manifold\\algebra\\smooth_functions.lean\n#print smooth_map.has_mul /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_map.has_add /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_map.has_zero /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_11: add_monoid \u219d has_zero\n -/\n#print smooth_map.has_one /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_11: monoid \u219d has_one\n -/\n#print smooth_map_semigroup /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_add_semigroup /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_add_monoid /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_monoid /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_add_comm_monoid /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_comm_monoid /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_add_group /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d has_smooth_add\n -/\n#print smooth_map_group /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d has_smooth_mul\n -/\n#print smooth_map_comm_group /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_add_comm_group /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_semiring /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d has_smooth_mul has_smooth_add\n -/\n#print smooth_map_ring /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d smooth_semiring\n -/\n#print smooth_map_comm_ring /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print smooth_map_has_scalar /- _inst_3: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print smooth_map_semimodule /- _inst_3: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print smooth_map.C /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_map.algebra /- _inst_3: normed_space \u219d\n -/\n#print smooth_map_has_scalar' /- _inst_3: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n#print smooth_map_module' /- _inst_3: normed_space \u219d\n_inst_12: normed_space \u219d\n -/\n\n-- geometry\\manifold\\algebra\\structures.lean\n#print smooth_semiring.to_has_smooth_add /- _inst_4: normed_space \u219d\n -/\n#print smooth_semiring.to_has_smooth_mul /- _inst_4: normed_space \u219d\n -/\n#print smooth_ring.to_lie_add_group /- _inst_4: normed_space \u219d\n -/\n#print smooth_ring.to_has_smooth_mul /- _inst_4: normed_space \u219d\n -/\n#print smooth_ring.to_smooth_semiring /- _inst_4: normed_space \u219d\n -/\n\n-- geometry\\manifold\\basic_smooth_bundle.lean\n#print trivial_basic_smooth_bundle_core /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.inhabited /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.to_topological_fiber_bundle_core /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.base_set /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.chart /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.chart_source /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.chart_target /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.to_charted_space /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.mem_atlas_iff /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.mem_chart_source_iff /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.mem_chart_target_iff /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.coe_chart_at_fst /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.coe_chart_at_symm_fst /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.to_smooth_manifold /- _inst_3: normed_space \u219d has_groupoid\n_inst_9: normed_space \u219d\n -/\n#print tangent_bundle_core /- _inst_3: normed_space \u219d\n -/\n#print tangent_space /- _inst_3: normed_space \u219d\n_inst_7: smooth_manifold_with_corners \u219d\n -/\n#print tangent_bundle /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.proj_apply /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.topological_space /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.charted_space /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_manifold_with_corners /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_space.topological_module /- _inst_3: normed_space \u219d\n -/\n#print tangent_space.topological_space /- _inst_3: normed_space \u219d\n -/\n#print tangent_space.add_comm_group /- _inst_3: normed_space \u219d\n -/\n#print tangent_space.topological_add_group /- _inst_3: normed_space \u219d\n -/\n#print tangent_space.vector_space /- _inst_3: normed_space \u219d\n -/\n#print tangent_space.inhabited /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle_proj_continuous /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle_proj_open /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle_model_space_chart_at /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_bundle_model_space_coe_chart_at /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_bundle_model_space_coe_chart_at_symm /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_bundle_model_space_homeomorph /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_bundle_model_space_homeomorph_coe /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_bundle_model_space_homeomorph_coe_symm /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n\n-- geometry\\manifold\\diffeomorph.lean\n#print diffeomorph /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.has_coe_to_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.times_cont_mdiff_map.has_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.continuous /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.times_cont_mdiff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.smooth /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.coe_eq_to_equiv /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_diffeomorph.refl /- _inst_3: normed_space \u219d\n -/\n#print times_diffeomorph.trans /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_diffeomorph.symm /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n\n-- geometry\\manifold\\mfderiv.lean\n#print unique_mdiff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_on /- _inst_3: normed_space \u219d\n -/\n#print written_in_ext_chart_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print unique_mdiff_within_at_univ /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_within_at_iff /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_within_at.mono /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_within_at.inter' /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_within_at.inter /- _inst_3: normed_space \u219d\n -/\n#print is_open.unique_mdiff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_on.inter /- _inst_3: normed_space \u219d\n -/\n#print is_open.unique_mdiff_on /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_on_univ /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_within_at.eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print unique_mdiff_on.eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within_zero_of_not_mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_zero_of_not_mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at.has_mfderiv_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at.mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at_unique /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at_inter /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.union /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.nhds_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.has_mfderiv_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.has_mfderiv_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.mfderiv_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at.has_mfderiv_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at.mfderiv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at.mfderiv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.mfderiv_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable.mfderiv_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within_subset /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at_inter /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at.mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on.mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable.mdifferentiable_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on_of_locally_mdifferentiable_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within_inter /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable.continuous /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_subset /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_eq_tangent_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_tangent_bundle_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_tangent_bundle_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_within_at.congr_mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print has_mfderiv_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print filter.eventually_eq.mdifferentiable_within_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.congr_mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_on.congr_mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mdifferentiable_within_at.mfderiv_within_congr_mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print filter.eventually_eq.mfderiv_within_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print mfderiv_within_congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_map_within_congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print filter.eventually_eq.mfderiv_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print written_in_ext_chart_comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print has_mfderiv_within_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print has_mfderiv_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print has_mfderiv_at.comp_has_mfderiv_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mdifferentiable_within_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mdifferentiable_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mfderiv_within_comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mfderiv_comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mdifferentiable_on.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print mdifferentiable.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print tangent_map_within_comp_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print tangent_map_comp_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print tangent_map_comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print has_mfderiv_at_id /- _inst_3: normed_space \u219d\n -/\n#print has_mfderiv_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_at_id /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_id /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_on_id /- _inst_3: normed_space \u219d\n -/\n#print mfderiv_id /- _inst_3: normed_space \u219d\n -/\n#print mfderiv_within_id /- _inst_3: normed_space \u219d\n -/\n#print tangent_map_id /- _inst_3: normed_space \u219d\n -/\n#print tangent_map_within_id /- _inst_3: normed_space \u219d\n -/\n#print has_mfderiv_at_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print has_mfderiv_within_at_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mdifferentiable_at_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mdifferentiable_within_at_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mdifferentiable_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mdifferentiable_on_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mfderiv_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print mfderiv_within_const /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print model_with_corners.mdifferentiable /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.mdifferentiable_on_symm /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_at_atlas /- _inst_3: normed_space \u219d has_groupoid\n_inst_7: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print mdifferentiable_on_atlas /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_at_atlas_symm /- _inst_3: normed_space \u219d has_groupoid\n_inst_7: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print mdifferentiable_on_atlas_symm /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_of_mem_atlas /- _inst_3: normed_space \u219d\n -/\n#print mdifferentiable_chart /- _inst_3: normed_space \u219d\n -/\n#print tangent_map_chart /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print tangent_map_chart_symm /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print unique_mdiff_within_at_iff_unique_diff_within_at /- _inst_3: normed_space \u219d\n -/\n#print unique_mdiff_on_iff_unique_diff_on /- _inst_3: normed_space \u219d\n -/\n#print written_in_ext_chart_model_space /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mdifferentiable_within_at_iff_differentiable_within_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mdifferentiable_at_iff_differentiable_at /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mdifferentiable_on_iff_differentiable_on /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mdifferentiable_iff_differentiable /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print mfderiv_within_eq_fderiv_within /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print mfderiv_eq_fderiv /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_5: normed_space \u219d smooth_manifold_with_corners\n -/\n#print local_homeomorph.mdifferentiable.symm /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.mdifferentiable_at_symm /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.symm_comp_deriv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.comp_symm_deriv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.mfderiv /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.mfderiv_bijective /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.mfderiv_surjective /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.range_mfderiv_eq_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print local_homeomorph.mdifferentiable.trans /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print unique_mdiff_on.unique_mdiff_on_preimage /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_9: normed_space \u219d smooth_manifold_with_corners\n -/\n#print unique_mdiff_on.unique_diff_on /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print unique_mdiff_on.unique_diff_on_inter_preimage /- _inst_3: normed_space \u219d\n_inst_9: normed_space \u219d\n -/\n#print unique_mdiff_on.smooth_bundle_preimage /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_14: normed_space \u219d smooth_manifold_with_corners\n -/\n#print unique_mdiff_on.tangent_bundle_proj_preimage /- _inst_3: normed_space \u219d\n -/\n\n-- geometry\\manifold\\smooth_manifold_with_corners.lean\n#print model_with_corners_self /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.has_coe_to_fun /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.symm /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.to_local_equiv_coe /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.mk_coe /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.to_local_equiv_coe_symm /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.mk_coe_symm /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.unique_diff /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.continuous /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.continuous_symm /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners_self_local_equiv /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners_self_coe /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners_self_coe_symm /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.target /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.left_inv /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.left_inv' /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.right_inv /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.image /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.unique_diff_preimage /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.unique_diff_preimage_source /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.unique_diff_at_image /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.prod /- _inst_3: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print model_with_corners.tangent /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners_prod_to_local_equiv /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print model_with_corners_prod_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print model_with_corners_prod_coe_symm /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print model_with_corners_self_boundaryless /- _inst_3: normed_space \u219d\n -/\n#print model_with_corners.range_eq_univ_prod /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_groupoid /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_groupoid_le /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_groupoid_zero_eq /- _inst_3: normed_space \u219d\n -/\n#print of_set_mem_times_cont_diff_groupoid /- _inst_3: normed_space \u219d\n -/\n#print symm_trans_mem_times_cont_diff_groupoid /- _inst_3: normed_space \u219d\n -/\n#print times_cont_diff_groupoid_prod /- _inst_3: normed_space \u219d\n_inst_7: normed_space \u219d\n -/\n#print times_cont_diff_groupoid.closed_under_restriction /- _inst_3: normed_space \u219d\n -/\n#print smooth_manifold_with_corners.to_has_groupoid /- _inst_3: normed_space \u219d\n -/\n#print smooth_manifold_with_corners_of_times_cont_diff_on /- _inst_3: normed_space \u219d\n -/\n#print model_space_smooth /- _inst_3: normed_space \u219d\n -/\n#print smooth_manifold_with_corners.maximal_atlas /- _inst_3: normed_space \u219d\n -/\n#print smooth_manifold_with_corners.mem_maximal_atlas_of_mem_atlas /- _inst_3: normed_space \u219d has_groupoid\n_inst_7: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print smooth_manifold_with_corners.chart_mem_maximal_atlas /- _inst_3: normed_space \u219d has_groupoid\n_inst_7: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print smooth_manifold_with_corners.compatible_of_mem_maximal_atlas /- _inst_3: normed_space \u219d\n -/\n#print smooth_manifold_with_corners.prod /- _inst_9: normed_space \u219d has_groupoid\n_inst_11: normed_space \u219d has_groupoid\n_inst_16: smooth_manifold_with_corners \u219d has_groupoid\n_inst_19: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print ext_chart_at /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_source /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_open_source /- _inst_3: normed_space \u219d\n -/\n#print mem_ext_chart_source /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_to_inv /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_source_mem_nhds /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_continuous_on /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_continuous_at /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_continuous_on_symm /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_target_mem_nhds_within /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_coe /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_at_coe_symm /- _inst_3: normed_space \u219d\n -/\n#print nhds_within_ext_chart_target_eq /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_continuous_at_symm' /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_continuous_at_symm /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_preimage_mem_nhds_within' /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_preimage_mem_nhds_within /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_preimage_mem_nhds /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_preimage_inter_eq /- _inst_3: normed_space \u219d\n -/\n#print ext_chart_model_space_eq_id /- _inst_3: normed_space \u219d\n -/\n\n-- geometry\\manifold\\times_cont_mdiff.lean\n#print times_cont_diff_within_at_prop /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_diff_within_at_local_invariant_prop /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_diff_within_at_local_invariant_prop_mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_diff_within_at_local_invariant_prop_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.smooth /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.times_cont_mdiff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.smooth_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_on.times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.smooth_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_at.times_cont_mdiff_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.smooth_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_within_at.times_cont_mdiff_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.times_cont_mdiff_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.smooth_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_at_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_on_univ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_within_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_iff /- _inst_3: normed_space \u219d has_groupoid\nIs: smooth_manifold_with_corners \u219d has_groupoid\n_inst_8: normed_space \u219d has_groupoid\nI's: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print smooth_on_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.of_le /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.of_le /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.of_le /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.of_le /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.of_succ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.of_succ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.of_succ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.of_succ /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.continuous_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.continuous_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.continuous_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.continuous /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.mdifferentiable_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.mdifferentiable /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.mdifferentiable /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.mdifferentiable_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.mdifferentiable_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_top /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at_top /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_top /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_top /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_iff_nat /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.times_cont_mdiff_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_at.smooth_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.mono /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth.smooth_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_inter' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_inter /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.times_cont_mdiff_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_within_at.smooth_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_iff_times_cont_mdiff_on_nhds /- _inst_3: normed_space \u219d has_groupoid\nIs: smooth_manifold_with_corners \u219d has_groupoid\n_inst_8: normed_space \u219d has_groupoid\nI's: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print times_cont_mdiff_at_iff_times_cont_mdiff_on_nhds /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print filter.eventually_eq.times_cont_mdiff_within_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at.congr_of_eventually_eq /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print filter.eventually_eq.times_cont_mdiff_at_iff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_congr /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_of_locally_times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_of_locally_times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff_on.comp' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.comp' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff_at.comp /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff.comp_times_cont_mdiff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print smooth.comp_smooth_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_23: normed_space \u219d\n -/\n#print times_cont_mdiff_on_of_mem_maximal_atlas /- _inst_3: normed_space \u219d has_groupoid\nIs: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print times_cont_mdiff_on_symm_of_mem_maximal_atlas /- _inst_3: normed_space \u219d has_groupoid\nIs: smooth_manifold_with_corners \u219d has_groupoid\n -/\n#print times_cont_mdiff_on_chart /- _inst_3: normed_space \u219d has_groupoid\n -/\n#print times_cont_mdiff_on_chart_symm /- _inst_3: normed_space \u219d has_groupoid\n -/\n#print times_cont_mdiff_id /- _inst_3: normed_space \u219d\n -/\n#print smooth_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_on_id /- _inst_3: normed_space \u219d\n -/\n#print smooth_on_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_at_id /- _inst_3: normed_space \u219d\n -/\n#print smooth_at_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print smooth_within_at_id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_on_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_at_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print smooth_within_at_const /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_iff_times_cont_diff_within_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_at_iff_times_cont_diff_at /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on_iff_times_cont_diff_on /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_iff_times_cont_diff /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff_on.continuous_on_tangent_map_within_aux /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_8: normed_space \u219d smooth_manifold_with_corners\n -/\n#print times_cont_mdiff_on.times_cont_mdiff_on_tangent_map_within_aux /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_8: normed_space \u219d smooth_manifold_with_corners\n -/\n#print times_cont_mdiff_on.times_cont_mdiff_on_tangent_map_within /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_8: normed_space \u219d smooth_manifold_with_corners\n -/\n#print times_cont_mdiff_on.continuous_on_tangent_map_within /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.times_cont_mdiff_tangent_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print times_cont_mdiff.continuous_tangent_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.times_cont_mdiff_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.smooth_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.times_cont_mdiff_on_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.smooth_on_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.times_cont_mdiff_at_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.smooth_at_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.times_cont_mdiff_within_at_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.smooth_within_at_proj /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print basic_smooth_bundle_core.smooth_const_section /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print tangent_bundle.times_cont_mdiff_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.times_cont_mdiff_on_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_on_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.times_cont_mdiff_at_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_at_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.times_cont_mdiff_within_at_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_within_at_proj /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.zero_section /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.smooth_zero_section /- _inst_3: normed_space \u219d\n -/\n#print tangent_bundle.tangent_map_tangent_bundle_pure /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n -/\n#print times_cont_mdiff_within_at.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_at.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_on.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_within_at.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_at.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_on.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth.prod_mk /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_at_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_on_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_within_at_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_at_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_on_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_fst /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_at_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_on_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print times_cont_mdiff_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_within_at_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_at_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_on_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_snd /- _inst_3: normed_space \u219d\n_inst_13: normed_space \u219d\n -/\n#print smooth_iff_proj_smooth /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d smooth_manifold_with_corners\n_inst_18: normed_space \u219d smooth_manifold_with_corners\n -/\n#print times_cont_mdiff_within_at.prod_map' /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d smooth_manifold_with_corners\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_within_at.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_at.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_at.prod_map' /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff_on.prod_map /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d smooth_manifold_with_corners\n_inst_18: normed_space \u219d\n -/\n#print times_cont_mdiff.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_within_at.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_at.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth_on.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print smooth.prod_map /- _inst_3: normed_space \u219d\n_inst_8: normed_space \u219d\n_inst_13: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print continuous_linear_map.times_cont_mdiff /- _inst_3: normed_space \u219d smooth_manifold_with_corners\n_inst_13: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth_smul /- _inst_23: normed_space \u219d smooth_manifold_with_corners\n -/\n#print smooth.smul /- _inst_3: normed_space \u219d\n_inst_23: normed_space \u219d smooth_manifold_with_corners\n -/\n\n-- geometry\\manifold\\times_cont_mdiff_map.lean\n#print smooth_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.has_coe_to_fun /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.continuous_map.has_coe /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.times_cont_mdiff /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.smooth /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.coe_inj /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.ext /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.id /- _inst_3: normed_space \u219d\n -/\n#print times_cont_mdiff_map.comp /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_15: normed_space \u219d\n -/\n#print times_cont_mdiff_map.comp_apply /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n_inst_15: normed_space \u219d\n -/\n#print times_cont_mdiff_map.inhabited /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print times_cont_mdiff_map.const /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n#print continuous_linear_map.has_coe_to_times_cont_mdiff_map /- _inst_3: normed_space \u219d\n_inst_5: normed_space \u219d\n -/\n\n-- group_theory\\abelianization.lean\n#print abelianization.commutator_subset_ker /- _inst_2: comm_group \u219d comm_semigroup group\n -/\n\n-- group_theory\\congruence.lean\n#print con.quotient.inhabited /- _inst_1: monoid \u219d has_one has_coe_t has_mul\n -/\n#print add_con.quotient.inhabited /- _inst_1: add_monoid \u219d has_coe_t has_zero has_add\n -/\n\n-- group_theory\\free_group.lean\n#print free_group.to_group.aux /- _inst_1: group \u219d has_inv has_one has_mul\n -/\n#print free_group.reduce /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.cons /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.red /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.not /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.min /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.idem /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.step.eq /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.eq_of_red /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.sound /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.exact /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.self /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.rev /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.to_word /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.to_word.mk /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.to_word.inj /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.reduce.church_rosser /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.red.decidable_rel /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.red.enum /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.red.enum.sound /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.red.enum.complete /- _inst_1: decidable_eq \u219d\n -/\n#print free_group.subtype.fintype /- _inst_1: decidable_eq \u219d\n -/\n\n-- group_theory\\group_action\\basic.lean\n#print mul_action.orbit /- _inst_2: mul_action \u219d has_scalar\n -/\n#print mul_action.mem_orbit_iff /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_orbit /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_orbit_self /- _inst_2: mul_action \u219d\n -/\n#print mul_action.stabilizer_carrier /- _inst_2: mul_action \u219d has_scalar\n -/\n#print mul_action.mem_stabilizer_iff /- _inst_2: mul_action \u219d\n -/\n#print mul_action.fixed_points /- _inst_2: mul_action \u219d has_scalar\n -/\n#print mul_action.fixed_by /- _inst_2: mul_action \u219d has_scalar\n -/\n#print mul_action.fixed_eq_Inter_fixed_by /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_fixed_points /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_fixed_by /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_fixed_points' /- _inst_2: mul_action \u219d\n -/\n#print mul_action.stabilizer.submonoid /- _inst_2: mul_action \u219d\n -/\n#print mul_action.stabilizer /- _inst_2: mul_action \u219d\n -/\n#print mul_action.orbit_eq_iff /- _inst_2: mul_action \u219d\n -/\n#print mul_action.stabilizer.subgroup /- _inst_2: mul_action \u219d\n -/\n#print mul_action.mem_orbit_smul /- _inst_2: mul_action \u219d\n -/\n#print mul_action.smul_mem_orbit_smul /- _inst_2: mul_action \u219d\n -/\n#print mul_action.orbit_rel /- _inst_2: mul_action \u219d\n -/\n#print mul_action.of_quotient_stabilizer /- _inst_2: mul_action \u219d\n -/\n#print mul_action.of_quotient_stabilizer_mk /- _inst_2: mul_action \u219d\n -/\n#print mul_action.of_quotient_stabilizer_mem_orbit /- _inst_2: mul_action \u219d\n -/\n#print mul_action.of_quotient_stabilizer_smul /- _inst_2: mul_action \u219d\n -/\n#print mul_action.injective_of_quotient_stabilizer /- _inst_2: mul_action \u219d\n -/\n#print mul_action.orbit_equiv_quotient_stabilizer /- _inst_2: mul_action \u219d\n -/\n#print mul_action.orbit_equiv_quotient_stabilizer_symm_apply /- _inst_2: mul_action \u219d\n -/\n#print list.smul_sum /- _inst_3: distrib_mul_action \u219d\n -/\n#print multiset.smul_sum /- _inst_3: distrib_mul_action \u219d\n -/\n#print finset.smul_sum /- _inst_3: distrib_mul_action \u219d\n -/\n\n-- group_theory\\group_action\\defs.lean\n#print smul_comm_class.symm /- _inst_3: smul_comm_class \u219d\n -/\n#print smul_comm_class_self /- _inst_1: comm_monoid \u219d monoid comm_semigroup\n_inst_2: mul_action \u219d\n -/\n#print smul_assoc /- _inst_4: is_scalar_tower \u219d\n -/\n#print smul_smul /- _inst_2: mul_action \u219d\n -/\n#print one_smul /- _inst_2: mul_action \u219d\n -/\n#print function.injective.mul_action /- _inst_2: mul_action \u219d\n -/\n#print function.surjective.mul_action /- _inst_2: mul_action \u219d\n -/\n#print ite_smul /- _inst_2: mul_action \u219d has_scalar\n -/\n#print smul_ite /- _inst_2: mul_action \u219d has_scalar\n -/\n#print mul_action.is_scalar_tower.left /- _inst_2: mul_action \u219d\n -/\n#print mul_action.to_fun /- _inst_2: mul_action \u219d\n -/\n#print mul_action.to_fun_apply /- _inst_2: mul_action \u219d\n -/\n#print mul_action.comp_hom /- _inst_2: mul_action \u219d\n -/\n#print smul_one_smul /- _inst_3: mul_action \u219d\n_inst_5: is_scalar_tower \u219d\n -/\n#print smul_add /- _inst_3: distrib_mul_action \u219d\n -/\n#print smul_zero /- _inst_3: distrib_mul_action \u219d\n -/\n#print function.injective.distrib_mul_action /- _inst_3: distrib_mul_action \u219d\n -/\n#print function.surjective.distrib_mul_action /- _inst_3: distrib_mul_action \u219d\n -/\n#print const_smul_hom /- _inst_3: distrib_mul_action \u219d\n -/\n#print const_smul_hom_apply /- _inst_3: distrib_mul_action \u219d\n -/\n#print smul_neg /- _inst_3: distrib_mul_action \u219d\n -/\n#print smul_sub /- _inst_3: distrib_mul_action \u219d\n -/\n\n-- group_theory\\group_action\\group.lean\n#print units.inv_smul_smul /- _inst_2: mul_action \u219d\n -/\n#print units.smul_inv_smul /- _inst_2: mul_action \u219d\n -/\n#print units.smul_perm_hom /- _inst_2: mul_action \u219d\n -/\n#print units.smul_left_cancel /- _inst_2: mul_action \u219d\n -/\n#print units.smul_eq_iff_eq_inv_smul /- _inst_2: mul_action \u219d\n -/\n#print is_unit.smul_left_cancel /- _inst_2: mul_action \u219d\n -/\n#print inv_smul_smul' /- _inst_2: mul_action \u219d\n -/\n#print smul_inv_smul' /- _inst_2: mul_action \u219d\n -/\n#print inv_smul_eq_iff' /- _inst_2: mul_action \u219d\n -/\n#print eq_inv_smul_iff' /- _inst_2: mul_action \u219d\n -/\n#print inv_smul_smul /- _inst_2: mul_action \u219d\n -/\n#print smul_inv_smul /- _inst_2: mul_action \u219d\n -/\n#print inv_smul_eq_iff /- _inst_2: mul_action \u219d\n -/\n#print eq_inv_smul_iff /- _inst_2: mul_action \u219d\n -/\n#print mul_action.to_perm /- _inst_2: mul_action \u219d\n -/\n#print mul_action.bijective /- _inst_2: mul_action \u219d\n -/\n#print units.smul_eq_zero /- _inst_3: distrib_mul_action \u219d\n -/\n#print units.smul_ne_zero /- _inst_3: distrib_mul_action \u219d\n -/\n#print is_unit.smul_eq_zero /- _inst_3: distrib_mul_action \u219d\n -/\n\n-- group_theory\\monoid_localization.lean\n#print localization.r /- _inst_1: comm_monoid \u219d monoid\n -/\n#print add_localization.r /- _inst_1: add_comm_monoid \u219d add_monoid\n -/\n#print submonoid.localization_map.mul_inv_left /- _inst_1: comm_monoid \u219d monoid\n_inst_2: comm_monoid \u219d monoid comm_semigroup\n -/\n#print add_submonoid.localization_map.add_neg_left /- _inst_1: add_comm_monoid \u219d add_monoid\n_inst_2: add_comm_monoid \u219d add_monoid add_comm_semigroup\n -/\n#print submonoid.localization_map.is_unit_comp /- _inst_3: comm_monoid \u219d monoid\n -/\n#print add_submonoid.localization_map.is_unit_comp /- _inst_3: add_comm_monoid \u219d add_monoid\n -/\n\n-- group_theory\\order_of_element.lean\n#print finset.mem_range_iff_mem_finset_range_of_mod_eq /- _inst_1: decidable_eq \u219d\n -/\n#print conj_injective /- _inst_1: group \u219d left_cancel_semigroup has_inv right_cancel_semigroup\n -/\n#print order_of /- dec: decidable_eq \u219d\n -/\n#print pow_order_of_eq_one /- dec: decidable_eq \u219d\n -/\n#print order_of_pos /- dec: decidable_eq \u219d\n -/\n#print pow_injective_of_lt_order_of /- dec: decidable_eq \u219d\n -/\n#print order_of_le_card_univ /- dec: decidable_eq \u219d\n -/\n#print pow_eq_mod_order_of /- dec: decidable_eq \u219d\n -/\n#print gpow_eq_mod_order_of /- dec: decidable_eq \u219d\n -/\n#print mem_gpowers_iff_mem_range_order_of /- dec: decidable_eq \u219d\n -/\n#print decidable_gpowers /- dec: decidable_eq \u219d\n -/\n#print order_of_dvd_of_pow_eq_one /- dec: decidable_eq \u219d\n -/\n#print order_of_dvd_iff_pow_eq_one /- dec: decidable_eq \u219d\n -/\n#print order_of_le_of_pow_eq_one /- dec: decidable_eq \u219d\n -/\n#print sum_card_order_of_eq_card_pow_eq_one /- dec: decidable_eq \u219d\n -/\n#print order_eq_card_gpowers /- dec: decidable_eq \u219d\n -/\n#print order_of_one /- dec: decidable_eq \u219d\n -/\n#print order_of_eq_one_iff /- dec: decidable_eq \u219d\n -/\n#print order_of_eq_prime /- dec: decidable_eq \u219d\n -/\n#print order_of_dvd_card_univ /- dec: decidable_eq \u219d\n -/\n#print order_of_pow /- dec: decidable_eq \u219d\n -/\n#print image_range_order_of /- dec: decidable_eq \u219d\n -/\n#print is_cyclic_of_order_of_eq_card /- _inst_2: decidable_eq \u219d\n -/\n#print order_of_eq_card_of_forall_mem_gpowers /- _inst_2: decidable_eq \u219d\n -/\n#print is_cyclic.card_pow_eq_one_le /- _inst_2: decidable_eq \u219d\n -/\n#print is_cyclic.image_range_order_of /- _inst_2: decidable_eq \u219d\n -/\n#print is_cyclic.image_range_card /- _inst_2: decidable_eq \u219d\n -/\n#print card_pow_eq_one_eq_order_of_aux /- _inst_2: decidable_eq \u219d\n -/\n#print card_order_of_eq_totient_aux\u2082 /- _inst_2: decidable_eq \u219d\n -/\n#print is_cyclic_of_card_pow_eq_one_le /- _inst_2: decidable_eq \u219d\n -/\n#print is_cyclic.card_order_of_eq_totient /- _inst_3: decidable_eq \u219d\n -/\n\n-- group_theory\\perm\\cycles.lean\n#print equiv.perm.same_cycle.decidable_rel /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of /- _inst_1: decidable_eq \u219d\n_inst_2: fintype \u219d\n -/\n#print equiv.perm.cycle_of_apply /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_inv /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_pow_apply_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_gpow_apply_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_apply_of_same_cycle /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_apply_of_not_same_cycle /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_apply_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_cycle /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_of_one /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.is_cycle_cycle_of /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_factors_aux /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.cycle_factors /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.one_lt_nonfixed_point_card_of_ne_one /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.fixed_point_card_lt_of_ne_one /- _inst_1: decidable_eq \u219d\n -/\n\n-- group_theory\\perm\\sign.lean\n#print equiv.perm.support /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.mem_support /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.is_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_mul_eq_mul_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.mul_swap_eq_swap_mul /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_mul_self_mul /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_mul_eq_iff /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.is_swap_of_subtype /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.ne_and_ne_of_swap_mul_apply_ne_self /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.support_swap_mul_eq /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.card_support_swap_mul /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_factors_aux /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_factors /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.trunc_swap_factors /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_induction_on /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.swap_mul_swap_mul_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.is_conj_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_aux2 /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_aux_eq_sign_aux2 /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_aux3 /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_aux3_mul_and_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_mul /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_one /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_refl /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_inv /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_swap' /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_eq_of_is_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_aux3_symm_trans_trans /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.sign_symm_trans_trans /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.sign_prod_list_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_surjective /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.eq_sign_of_surjective_hom /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_subtype_perm /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_of_subtype /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_eq_sign_of_equiv /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.sign_bij /- _inst_1: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.is_cycle_swap /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.is_cycle_swap_mul_aux\u2081 /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.is_cycle_swap_mul_aux\u2082 /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.eq_swap_of_is_cycle_of_apply_apply_eq_self /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.is_cycle_swap_mul /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.support_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.card_support_swap /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_cycle /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.prod_prod_extend_right /- _inst_3: decidable_eq \u219d\n -/\n#print equiv.perm.sign_prod_extend_right /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_prod_congr_right /- _inst_1: decidable_eq \u219d\n -/\n#print equiv.perm.sign_prod_congr_left /- _inst_1: decidable_eq \u219d\n -/\n\n-- group_theory\\subgroup.lean\n#print subgroup.multiset_prod_mem /- _inst_3: comm_group \u219d group comm_monoid\n -/\n#print add_subgroup.multiset_sum_mem /- _inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print subgroup.prod_mem /- _inst_3: comm_group \u219d group comm_monoid\n -/\n#print add_subgroup.sum_mem /- _inst_3: add_comm_group \u219d add_comm_monoid add_group\n -/\n#print add_subgroup.normal_of_comm /- _inst_3: add_comm_group \u219d add_comm_semigroup add_group\n -/\n#print subgroup.normal_of_comm /- _inst_3: comm_group \u219d comm_semigroup group\n -/\n#print monoid_hom.eq_of_eq_on_top /- _inst_3: group \u219d monoid\n -/\n#print add_monoid_hom.eq_of_eq_on_top /- _inst_3: add_group \u219d add_monoid\n -/\n\n-- group_theory\\sylow.lean\n#print mul_action.mem_fixed_points_iff_card_orbit_eq_one /- _inst_1: group \u219d monoid\n_inst_2: mul_action \u219d\n -/\n#print mul_action.card_modeq_card_fixed_points /- _inst_2: mul_action \u219d\n -/\n#print sylow.mk_vector_prod_eq_one /- _inst_1: group \u219d has_inv has_one has_mul\n -/\n#print sylow.vectors_prod_eq_one /- _inst_2: group \u219d has_one has_mul\n -/\n\n-- linear_algebra\\adic_completion.lean\n#print adic_completion /- _inst_3: module \u219d\n -/\n#print Hausdorffification.of /- _inst_3: module \u219d\n -/\n#print Hausdorffification.induction_on /- _inst_3: module \u219d\n -/\n#print Hausdorffification.is_Hausdorff /- _inst_3: module \u219d\n -/\n#print Hausdorffification.lift /- _inst_3: module \u219d\n -/\n#print Hausdorffification.lift_of /- _inst_3: module \u219d\n -/\n#print Hausdorffification.lift_comp_of /- _inst_3: module \u219d\n -/\n#print Hausdorffification.lift_eq /- _inst_3: module \u219d\n -/\n#print adic_completion.of /- _inst_3: module \u219d\n -/\n#print adic_completion.of_apply /- _inst_3: module \u219d\n -/\n#print adic_completion.eval /- _inst_3: module \u219d\n -/\n#print adic_completion.coe_eval /- _inst_3: module \u219d\n -/\n#print adic_completion.eval_apply /- _inst_3: module \u219d\n -/\n#print adic_completion.eval_of /- _inst_3: module \u219d\n -/\n#print adic_completion.eval_comp_of /- _inst_3: module \u219d\n -/\n#print adic_completion.range_eval /- _inst_3: module \u219d\n -/\n#print adic_completion.ext /- _inst_3: module \u219d\n -/\n#print adic_completion.is_Hausdorff /- _inst_3: module \u219d\n -/\n#print is_adic_complete.of_subsingleton /- _inst_6: subsingleton \u219d is_Hausdorff is_precomplete\n -/\n\n-- linear_algebra\\affine_space\\affine_equiv.lean\n#print linear_equiv.to_affine_equiv /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.coe_to_affine_equiv /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.refl /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.coe_refl /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.refl_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.to_equiv_refl /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.linear_refl /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.map_vadd /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.coe_to_equiv /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.to_affine_map /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.coe_to_affine_map /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.to_affine_map_mk /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.linear_to_affine_map /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.injective_to_affine_map /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.to_affine_map_inj /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.ext /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.injective_coe_fn /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.coe_fn_inj /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.injective_to_equiv /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.to_equiv_inj /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.mk' /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.coe_mk' /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.to_equiv_mk' /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.linear_mk' /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm_to_equiv /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm_linear /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.bijective /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.surjective /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.injective /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.range_eq /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.apply_symm_apply /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm_apply_apply /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.apply_eq_iff_eq_symm_apply /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.apply_eq_iff_eq /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm_refl /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.trans /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print affine_equiv.coe_trans /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print affine_equiv.trans_apply /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print affine_equiv.trans_assoc /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print affine_equiv.trans_refl /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.refl_trans /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.trans_symm /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.symm_trans /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.apply_line_map /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print affine_equiv.group /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.one_def /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.coe_one /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.mul_def /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.coe_mul /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.inv_def /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.vadd_const /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.linear_vadd_const /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.vadd_const_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.vadd_const_symm_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.const_vsub /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.coe_const_vsub /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.coe_const_vsub_symm /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.const_vadd /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.linear_const_vadd /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.const_vadd_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.const_vadd_symm_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_apply /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_symm /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.to_equiv_point_reflection /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_self /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_involutive /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_fixed_iff_of_injective_bit0 /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.injective_point_reflection_left_of_injective_bit0 /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.injective_point_reflection_left_of_module /- _inst_3: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_fixed_iff_of_module /- _inst_3: semimodule \u219d\n -/\n#print affine_map.line_map_vadd /- _inst_3: semimodule \u219d\n -/\n#print affine_map.line_map_vsub /- _inst_3: semimodule \u219d\n -/\n#print affine_map.vsub_line_map /- _inst_3: semimodule \u219d\n -/\n#print affine_map.vadd_line_map /- _inst_3: semimodule \u219d\n -/\n#print affine_map.homothety_neg_one_apply /- _inst_15: semimodule \u219d\n -/\n\n-- linear_algebra\\affine_space\\affine_map.lean\n#print affine_map.line_map_apply_module /- _inst_3: module \u219d\n -/\n#print affine_map.line_map_same_apply /- _inst_3: module \u219d\n -/\n#print affine_map.line_map_apply_one /- _inst_3: module \u219d\n -/\n#print affine_map.left_vsub_line_map /- _inst_3: module \u219d\n -/\n#print affine_map.coe_smul /- _inst_1: comm_ring \u219d ring\n -/\n#print affine_map.homothety /- _inst_1: comm_ring \u219d ring\n -/\n#print affine_map.homothety_one /- _inst_3: module \u219d\n -/\n#print affine_map.homothety_mul /- _inst_3: module \u219d\n -/\n\n-- linear_algebra\\affine_space\\affine_subspace.lean\n#print affine_subspace.vadd_mem_of_mem_direction /- _inst_3: module \u219d\n -/\n\n-- linear_algebra\\affine_space\\combination.lean\n#print finset.weighted_vsub_of_point /- S: add_torsor \u219d has_vsub\n -/\n#print finset.weighted_vsub_of_point_eq_of_sum_eq_zero /- _inst_3: module \u219d\n -/\n#print finset.weighted_vsub_of_point_vadd_eq_of_sum_eq_one /- _inst_3: module \u219d\n -/\n#print finset.weighted_vsub_of_point_erase /- _inst_3: module \u219d\n -/\n#print finset.weighted_vsub_of_point_insert /- _inst_3: module \u219d\n -/\n#print finset.affine_combination_of_eq_one_of_eq_zero /- _inst_3: module \u219d\n -/\n#print finset.centroid_weights /- _inst_1: division_ring \u219d has_inv has_one has_zero has_add\n -/\n#print finset.centroid_singleton /- _inst_3: module \u219d\n -/\n#print finset.centroid_insert_singleton /- _inst_3: module \u219d\n -/\n#print mem_vector_span_iff_eq_weighted_vsub /- _inst_3: module \u219d\n -/\n#print affine_map.weighted_vsub_of_point /- _inst_3: module \u219d\n -/\n\n-- linear_algebra\\affine_space\\finite_dimensional.lean\n#print finite_dimensional_vector_span_of_finite /- _inst_3: module \u219d\n -/\n#print finite_dimensional_vector_span_of_fintype /- _inst_3: module \u219d\n -/\n#print finite_dimensional_vector_span_image_of_fintype /- _inst_3: module \u219d\n -/\n#print finite_dimensional_direction_affine_span_of_finite /- _inst_3: module \u219d\n -/\n#print finite_dimensional_direction_affine_span_of_fintype /- _inst_3: module \u219d\n -/\n#print finite_dimensional_direction_affine_span_image_of_fintype /- _inst_3: module \u219d\n -/\n#print findim_vector_span_image_finset_of_affine_independent /- _inst_3: module \u219d\n -/\n#print findim_vector_span_of_affine_independent /- _inst_3: module \u219d\n -/\n#print vector_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one /- _inst_3: module \u219d\n -/\n#print vector_span_eq_of_le_of_affine_independent_of_card_eq_findim_add_one /- _inst_3: module \u219d\n -/\n#print affine_span_image_finset_eq_of_le_of_affine_independent_of_card_eq_findim_add_one /- _inst_3: module \u219d\n -/\n#print affine_span_eq_of_le_of_affine_independent_of_card_eq_findim_add_one /- _inst_3: module \u219d\n -/\n#print affine_span_eq_top_of_affine_independent_of_card_eq_findim_add_one /- _inst_3: module \u219d finite_dimensional\n -/\n#print findim_vector_span_image_finset_le /- _inst_3: module \u219d\n -/\n#print findim_vector_span_range_le /- _inst_3: module \u219d\n -/\n#print affine_independent_iff_findim_vector_span_eq /- _inst_3: module \u219d\n -/\n#print affine_independent_iff_le_findim_vector_span /- _inst_3: module \u219d\n -/\n#print affine_independent_iff_not_findim_vector_span_le /- _inst_3: module \u219d\n -/\n#print findim_vector_span_le_iff_not_affine_independent /- _inst_3: module \u219d\n -/\n#print collinear /- _inst_3: module \u219d\n -/\n#print collinear_iff_dim_le_one /- _inst_3: module \u219d\n -/\n#print collinear_iff_findim_le_one /- _inst_3: module \u219d\n -/\n#print collinear_empty /- _inst_3: module \u219d\n -/\n#print collinear_singleton /- _inst_3: module \u219d\n -/\n#print collinear_iff_of_mem /- _inst_3: module \u219d\n -/\n#print collinear_insert_singleton /- _inst_3: module \u219d\n -/\n#print affine_independent_iff_not_collinear /- _inst_3: module \u219d finite_dimensional\n -/\n#print collinear_iff_not_affine_independent /- _inst_3: module \u219d finite_dimensional\n -/\n\n-- linear_algebra\\affine_space\\midpoint.lean\n#print midpoint /- _inst_4: semimodule \u219d\n -/\n#print affine_map.map_midpoint /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print affine_equiv.map_midpoint /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_midpoint_left /- _inst_4: semimodule \u219d\n -/\n#print midpoint_comm /- _inst_4: semimodule \u219d\n -/\n#print affine_equiv.point_reflection_midpoint_right /- _inst_4: semimodule \u219d\n -/\n#print midpoint_vsub_midpoint /- _inst_4: semimodule \u219d\n -/\n#print midpoint_vadd_midpoint /- _inst_4: semimodule \u219d\n -/\n#print midpoint_eq_iff /- _inst_4: semimodule \u219d\n -/\n#print midpoint_eq_midpoint_iff_vsub_eq_vsub /- _inst_4: semimodule \u219d\n -/\n#print midpoint_eq_iff' /- _inst_4: semimodule \u219d\n -/\n#print midpoint_unique /- _inst_4: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print midpoint_self /- _inst_4: semimodule \u219d\n -/\n#print midpoint_add_self /- _inst_4: semimodule \u219d\n -/\n#print midpoint_zero_add /- _inst_4: semimodule \u219d\n -/\n#print line_map_inv_two /- _inst_1: division_ring \u219d invertible has_inv ring\n_inst_2: char_zero \u219d invertible\n_inst_4: semimodule \u219d\n -/\n#print line_map_one_half /- _inst_4: semimodule \u219d\n -/\n#print homothety_inv_of_two /- _inst_4: semimodule \u219d\n -/\n#print homothety_inv_two /- _inst_1: field \u219d invertible has_inv comm_ring\n_inst_2: char_zero \u219d invertible\n_inst_4: semimodule \u219d\n -/\n#print homothety_one_half /- _inst_4: semimodule \u219d\n -/\n#print pi_midpoint_apply /- _inst_1: field \u219d ring\n -/\n#print add_monoid_hom.of_map_midpoint /- _inst_4: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print add_monoid_hom.coe_of_map_midpoint /- _inst_4: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n\n-- linear_algebra\\affine_space\\ordered.lean\n#print slope /- _inst_1: field \u219d has_sub has_inv semiring\n_inst_2: add_comm_group \u219d add_comm_monoid has_vsub add_group\n_inst_3: semimodule \u219d has_scalar\n_inst_4: add_torsor \u219d has_vsub\n -/\n#print slope_same /- _inst_3: semimodule \u219d\n -/\n#print eq_of_slope_eq_zero /- _inst_3: semimodule \u219d\n -/\n#print slope_comm /- _inst_3: semimodule \u219d\n -/\n#print sub_div_sub_smul_slope_add_sub_div_sub_smul_slope /- _inst_3: semimodule \u219d\n -/\n#print line_map_slope_slope_sub_div_sub /- _inst_3: semimodule \u219d\n -/\n#print line_map_slope_line_map_slope_line_map /- _inst_3: semimodule \u219d\n -/\n#print line_map_mono_left /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n_inst_2: ordered_add_comm_group \u219d ordered_add_comm_monoid add_comm_group\n_inst_3: semimodule \u219d\n -/\n#print line_map_strict_mono_left /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n_inst_2: ordered_add_comm_group \u219d add_comm_group ordered_cancel_add_comm_monoid\n_inst_3: semimodule \u219d\n -/\n#print line_map_mono_right /- _inst_1: ordered_ring \u219d ordered_semiring ring\n_inst_2: ordered_add_comm_group \u219d ordered_add_comm_monoid add_comm_group\n_inst_3: semimodule \u219d\n -/\n#print line_map_strict_mono_right /- _inst_1: ordered_ring \u219d ordered_semiring ring\n_inst_2: ordered_add_comm_group \u219d add_comm_group ordered_cancel_add_comm_monoid\n_inst_3: semimodule \u219d\n -/\n#print line_map_mono_endpoints /- _inst_3: semimodule \u219d\n -/\n#print line_map_strict_mono_endpoints /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_line_map_iff_of_lt /- _inst_1: ordered_ring \u219d ordered_semiring ring ordered_add_comm_group\n_inst_3: semimodule \u219d\n -/\n#print left_lt_line_map_iff_lt /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_left_iff_lt /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_right_iff_lt /- _inst_3: semimodule \u219d\n -/\n#print right_lt_line_map_iff_lt /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_line_map_iff_of_lt /- _inst_3: semimodule \u219d\n -/\n#print left_le_line_map_iff_le /- _inst_3: semimodule \u219d\n -/\n#print left_le_midpoint /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_left_iff_le /- _inst_3: semimodule \u219d\n -/\n#print midpoint_le_left /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_right_iff_le /- _inst_3: semimodule \u219d\n -/\n#print midpoint_le_right /- _inst_3: semimodule \u219d\n -/\n#print right_le_line_map_iff_le /- _inst_3: semimodule \u219d\n -/\n#print right_le_midpoint /- _inst_3: semimodule \u219d\n -/\n#print map_le_line_map_iff_slope_le_slope_left /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_map_iff_slope_le_slope_left /- _inst_3: semimodule \u219d\n -/\n#print map_lt_line_map_iff_slope_lt_slope_left /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_map_iff_slope_lt_slope_left /- _inst_3: semimodule \u219d\n -/\n#print map_le_line_map_iff_slope_le_slope_right /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_map_iff_slope_le_slope_right /- _inst_3: semimodule \u219d\n -/\n#print map_lt_line_map_iff_slope_lt_slope_right /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_map_iff_slope_lt_slope_right /- _inst_3: semimodule \u219d\n -/\n#print map_le_line_map_iff_slope_le_slope /- _inst_3: semimodule \u219d\n -/\n#print line_map_le_map_iff_slope_le_slope /- _inst_3: semimodule \u219d\n -/\n#print map_lt_line_map_iff_slope_lt_slope /- _inst_3: semimodule \u219d\n -/\n#print line_map_lt_map_iff_slope_lt_slope /- _inst_3: semimodule \u219d\n -/\n\n-- linear_algebra\\basic.lean\n#print finsupp.smul_sum /- _inst_4: semimodule \u219d\n -/\n#print linear_map.comp_id /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.id_comp /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comp_assoc /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.dom_restrict /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.dom_restrict_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.cod_restrict /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.cod_restrict_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comp_cod_restrict /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.subtype_comp_cod_restrict /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.restrict /- _inst_6: semimodule \u219d\n -/\n#print linear_map.restrict_apply /- _inst_6: semimodule \u219d\n -/\n#print linear_map.subtype_comp_restrict /- _inst_6: semimodule \u219d\n -/\n#print linear_map.restrict_eq_cod_restrict_dom_restrict /- _inst_6: semimodule \u219d\n -/\n#print linear_map.restrict_eq_dom_restrict_cod_restrict /- _inst_6: semimodule \u219d\n -/\n#print linear_map.has_zero /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inhabited /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.zero_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.default_def /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.unique_of_left /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.unique_of_right /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.has_add /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.add_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.add_comm_monoid /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.linear_map_apply_is_add_monoid_hom /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.add_comp /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.comp_add /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.sum_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.smul_right /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.smul_right_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.has_one /- _inst_6: semimodule \u219d\n -/\n#print linear_map.has_mul /- _inst_6: semimodule \u219d\n -/\n#print linear_map.mul_eq_comp /- _inst_6: semimodule \u219d\n -/\n#print linear_map.one_app /- _inst_6: semimodule \u219d\n -/\n#print linear_map.mul_app /- _inst_6: semimodule \u219d\n -/\n#print linear_map.comp_zero /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.zero_comp /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coe_fn_sum /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.monoid /- _inst_6: semimodule \u219d\n -/\n#print linear_map.pi_apply_eq_sum_univ /- _inst_6: semimodule \u219d\n -/\n#print linear_map.fst /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.snd /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.fst_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.snd_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.prod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.prod_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.fst_prod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.snd_prod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.pair_fst_snd /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inl /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inr /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inl_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inr_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inl_injective /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inr_injective /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.coprod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coprod_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coprod_inl /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coprod_inr /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coprod_inl_inr /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.fst_eq_coprod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.snd_eq_coprod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inl_eq_prod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.inr_eq_prod /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.prod_map /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.prod_map_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.has_neg /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.neg_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comp_neg /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.add_comm_group /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.linear_map_apply_is_add_group_hom /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.sub_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.sub_comp /- _inst_3: add_comm_group \u219d add_comm_monoid\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.comp_sub /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.has_scalar /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_9: distrib_mul_action \u219d\n_inst_10: smul_comm_class \u219d\n -/\n#print linear_map.smul_apply /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_9: distrib_mul_action \u219d has_scalar\n_inst_10: smul_comm_class \u219d has_scalar\n -/\n#print linear_map.distrib_mul_action /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_9: distrib_mul_action \u219d\n_inst_10: smul_comm_class \u219d\n -/\n#print linear_map.smul_comp /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: distrib_mul_action \u219d has_scalar\n_inst_10: smul_comm_class \u219d has_scalar\n -/\n#print linear_map.semimodule /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_11: smul_comm_class \u219d\n -/\n#print linear_map.apply\u2097' /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_11: smul_comm_class \u219d\n -/\n#print linear_map.comp_smul /- _inst_1: comm_semiring \u219d semiring\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comp_right /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.apply\u2097 /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.endomorphism_semiring /- _inst_3: semimodule \u219d\n -/\n#print linear_map.mul_apply /- _inst_3: semimodule \u219d\n -/\n#print linear_map.endomorphism_ring /- _inst_3: semimodule \u219d\n -/\n#print linear_map.smul_right\u2097 /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.smul_right\u2097_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.partial_order /- _inst_5: semimodule \u219d\n -/\n#print submodule.le_def /- _inst_5: semimodule \u219d\n -/\n#print submodule.le_def' /- _inst_5: semimodule \u219d\n -/\n#print submodule.lt_def /- _inst_5: semimodule \u219d\n -/\n#print submodule.not_le_iff_exists /- _inst_5: semimodule \u219d\n -/\n#print submodule.exists_of_lt /- _inst_5: semimodule \u219d\n -/\n#print submodule.lt_iff_le_and_exists /- _inst_5: semimodule \u219d\n -/\n#print submodule.of_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.coe_of_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.of_le_apply /- _inst_5: semimodule \u219d\n -/\n#print submodule.subtype_comp_of_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.has_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.inhabited' /- _inst_5: semimodule \u219d\n -/\n#print submodule.bot_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.nonzero_mem_of_bot_lt /- _inst_5: semimodule \u219d\n -/\n#print submodule.order_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.eq_bot_iff /- _inst_5: semimodule \u219d\n -/\n#print submodule.ne_bot_iff /- _inst_5: semimodule \u219d\n -/\n#print submodule.has_top /- _inst_5: semimodule \u219d\n -/\n#print submodule.top_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_top /- _inst_5: semimodule \u219d\n -/\n#print submodule.eq_bot_of_zero_eq_one /- _inst_5: semimodule \u219d\n -/\n#print submodule.order_top /- _inst_5: semimodule \u219d\n -/\n#print submodule.has_Inf /- _inst_5: semimodule \u219d\n -/\n#print submodule.has_inf /- _inst_5: semimodule \u219d\n -/\n#print submodule.complete_lattice /- _inst_5: semimodule \u219d\n -/\n#print submodule.add_comm_monoid_submodule /- _inst_5: semimodule \u219d\n -/\n#print submodule.add_eq_sup /- _inst_5: semimodule \u219d\n -/\n#print submodule.zero_eq_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.eq_top_iff' /- _inst_5: semimodule \u219d\n -/\n#print submodule.bot_ne_top /- _inst_5: semimodule \u219d\n -/\n#print submodule.inf_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_inf /- _inst_5: semimodule \u219d\n -/\n#print submodule.Inf_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.infi_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_Inf /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_infi /- _inst_5: semimodule \u219d\n -/\n#print submodule.disjoint_def /- _inst_5: semimodule \u219d\n -/\n#print submodule.disjoint_def' /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_right_iff_eq_zero_of_disjoint /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_left_iff_eq_zero_of_disjoint /- _inst_5: semimodule \u219d\n -/\n#print submodule.map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.mem_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.mem_map_of_mem /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_id /- _inst_5: semimodule \u219d\n -/\n#print submodule.map_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print submodule.map_mono /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.mem_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_id /- _inst_5: semimodule \u219d\n -/\n#print submodule.comap_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print submodule.comap_mono /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_le_iff_le_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.gc_map_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_bot /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_sup /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_supr /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_top /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_inf /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_infi /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_comap_le /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.le_comap_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_inf_eq_map_inf_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.map_comap_subtype /- _inst_5: semimodule \u219d\n -/\n#print submodule.eq_zero_of_bot_submodule /- _inst_5: semimodule \u219d\n -/\n#print submodule.span /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_span /- _inst_5: semimodule \u219d\n -/\n#print submodule.subset_span /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_mono /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_eq_of_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_eq /- _inst_5: semimodule \u219d\n -/\n#print submodule.map_span /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.span_induction /- _inst_5: semimodule \u219d\n -/\n#print submodule.gi /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_empty /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_univ /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_union /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_Union /- _inst_5: semimodule \u219d\n -/\n#print submodule.coe_supr_of_directed /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_sup_left /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_sup_right /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_supr_of_mem /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_Sup_of_mem /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_supr_of_directed /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_Sup_of_directed /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_sup /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_sup' /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_span_singleton_self /- _inst_5: semimodule \u219d\n -/\n#print submodule.nontrivial_span_singleton /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_span_singleton /- _inst_5: semimodule \u219d\n -/\n#print submodule.le_span_singleton_iff /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_singleton_eq_range /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_singleton_smul_le /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_singleton_smul_eq /- _inst_8: division_ring \u219d group_with_zero ring\n_inst_10: module \u219d\n -/\n#print submodule.disjoint_span_singleton /- _inst_10: module \u219d\n -/\n#print submodule.mem_span_insert /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_insert_eq_span /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_span /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_eq_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_singleton_eq_bot /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_zero /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_image /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.supr_eq_span /- _inst_5: semimodule \u219d\n -/\n#print submodule.span_singleton_le_iff_mem /- _inst_5: semimodule \u219d\n -/\n#print submodule.lt_add_iff_not_mem /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_supr /- _inst_5: semimodule \u219d\n -/\n#print submodule.prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.mem_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.span_prod_le /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_top /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_bot /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_mono /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_inf_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.prod_sup_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.neg_coe /- _inst_5: semimodule \u219d\n -/\n#print submodule.map_neg /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.span_neg /- _inst_5: semimodule \u219d\n -/\n#print submodule.mem_span_insert' /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient_rel /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_eq_mk /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk'_eq_mk /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.quot_mk_eq_mk /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.eq /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.has_zero /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.inhabited /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_zero /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_eq_zero /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.has_add /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_add /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.has_neg /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_neg /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.add_comm_group /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.has_scalar /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_smul /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.semimodule /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.mk_surjective /- _inst_5: semimodule \u219d\n -/\n#print submodule.quotient.nontrivial_of_lt_top /- _inst_5: semimodule \u219d\n -/\n#print submodule.quot_hom_ext /- _inst_3: add_comm_group \u219d add_comm_monoid\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print submodule.comap_smul /- _inst_5: vector_space \u219d\n -/\n#print submodule.comap_smul' /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print submodule.map_smul' /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print linear_map.eq_on_span /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.eq_on_span' /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ext_on /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ext_on_range /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.finsupp_sum /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_cod_restrict /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.comap_cod_restrict /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.mem_range /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.mem_range_self /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_id /- _inst_5: semimodule \u219d\n -/\n#print linear_map.range_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.range_comp_le_range /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.range_eq_top /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_le_iff_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_le_range /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_coprod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.is_compl_range_inl_inr /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.sup_range_inl_inr /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_restrict /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.to_span_singleton /- _inst_5: semimodule \u219d\n -/\n#print linear_map.span_singleton_eq_range /- _inst_5: semimodule \u219d\n -/\n#print linear_map.to_span_singleton_one /- _inst_5: semimodule \u219d\n -/\n#print linear_map.ker /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.mem_ker /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_id /- _inst_5: semimodule \u219d\n -/\n#print linear_map.map_coe_ker /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.comp_ker_subtype /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.ker_le_ker_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.disjoint_ker /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.disjoint_inl_inr /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_eq_bot' /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.le_ker_iff_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_cod_restrict /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_cod_restrict /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_restrict /- _inst_5: semimodule \u219d\n -/\n#print linear_map.map_comap_eq /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_comap_eq_self /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_eq_top /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_le_bot_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_le_ker_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comap_le_comap_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.comap_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_coprod_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.comap_prod_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.prod_eq_inf_comap /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.prod_eq_sup_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.span_inl_union_inr /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_prod /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.range_prod_le /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.ker_eq_bot_of_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.comap_map_eq /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.comap_map_eq_self /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_le_map_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_le_map_iff' /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.map_eq_top_iff /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.sub_mem_ker_iff /- _inst_1: ring \u219d semiring\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.disjoint_ker' /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.inj_of_disjoint_ker /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.ker_eq_bot /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.range_prod_eq /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print submodule.sup_eq_range /- _inst_3: semimodule \u219d\n -/\n#print is_linear_map.is_linear_map_add /- _inst_3: semimodule \u219d\n -/\n#print is_linear_map.is_linear_map_sub /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_top /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.comap_bot /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.ker_subtype /- _inst_3: semimodule \u219d\n -/\n#print submodule.range_subtype /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_subtype_le /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_subtype_top /- _inst_3: semimodule \u219d\n -/\n#print submodule.comap_subtype_eq_top /- _inst_3: semimodule \u219d\n -/\n#print submodule.comap_subtype_self /- _inst_3: semimodule \u219d\n -/\n#print submodule.ker_of_le /- _inst_3: semimodule \u219d\n -/\n#print submodule.range_of_le /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_inl /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.map_inr /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.comap_fst /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.comap_snd /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.prod_comap_inl /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.prod_comap_inr /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.prod_map_fst /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.prod_map_snd /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.ker_inl /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.ker_inr /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.range_fst /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.range_snd /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.disjoint_iff_comap_eq_bot /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_subtype.rel_iso /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_subtype.order_embedding /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_subtype_embedding_eq /- _inst_3: semimodule \u219d\n -/\n#print submodule.mkq /- _inst_3: semimodule \u219d\n -/\n#print submodule.mkq_apply /- _inst_3: semimodule \u219d\n -/\n#print submodule.liftq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.liftq_apply /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.liftq_mkq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.range_mkq /- _inst_3: semimodule \u219d\n -/\n#print submodule.ker_mkq /- _inst_3: semimodule \u219d\n -/\n#print submodule.le_comap_mkq /- _inst_3: semimodule \u219d\n -/\n#print submodule.mkq_map_self /- _inst_3: semimodule \u219d\n -/\n#print submodule.comap_map_mkq /- _inst_3: semimodule \u219d\n -/\n#print submodule.map_mkq_eq_top /- _inst_3: semimodule \u219d\n -/\n#print submodule.mapq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.mapq_apply /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.mapq_mkq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.comap_liftq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.map_liftq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.ker_liftq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.range_liftq /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.ker_liftq_eq_bot /- _inst_3: semimodule \u219d\n_inst_4: semimodule \u219d\n -/\n#print submodule.comap_mkq.rel_iso /- _inst_3: semimodule \u219d\n -/\n#print submodule.comap_mkq.order_embedding /- _inst_3: semimodule \u219d\n -/\n#print submodule.comap_mkq_embedding_eq /- _inst_3: semimodule \u219d\n -/\n#print linear_map.range_mkq_comp /- _inst_6: module \u219d\n -/\n#print linear_map.ker_le_range_iff /- _inst_6: module \u219d\n -/\n#print linear_map.ker_eq_bot_of_cancel /- _inst_5: module \u219d\n -/\n#print linear_map.range_eq_top_of_cancel /- _inst_6: module \u219d\n -/\n#print linear_map.range_range_restrict /- _inst_4: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_equiv.eq_bot_of_equiv /- _inst_6: semimodule \u219d\n -/\n#print linear_equiv.neg /- _inst_3: semimodule \u219d\n -/\n#print linear_equiv.coe_neg /- _inst_3: semimodule \u219d\n -/\n#print linear_equiv.neg_apply /- _inst_3: semimodule \u219d\n -/\n#print linear_equiv.symm_neg /- _inst_3: semimodule \u219d\n -/\n#print linear_equiv.smul_of_unit /- _inst_5: semimodule \u219d\n -/\n#print linear_equiv.arrow_congr /- _inst_13: module \u219d\n_inst_14: module \u219d\n_inst_15: module \u219d\n_inst_16: module \u219d\n -/\n#print linear_equiv.arrow_congr_apply /- _inst_13: module \u219d\n_inst_14: module \u219d\n_inst_15: module \u219d\n_inst_16: module \u219d\n -/\n#print linear_equiv.arrow_congr_symm_apply /- _inst_13: module \u219d\n_inst_14: module \u219d\n_inst_15: module \u219d\n_inst_16: module \u219d\n -/\n#print linear_equiv.arrow_congr_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_11: module \u219d\n_inst_12: module \u219d\n_inst_13: module \u219d\n -/\n#print linear_equiv.arrow_congr_trans /- _inst_9: module \u219d\n_inst_11: module \u219d\n_inst_13: module \u219d\n_inst_15: module \u219d\n_inst_17: module \u219d\n_inst_19: module \u219d\n -/\n#print linear_equiv.congr_right /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.conj /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.conj_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.symm_conj_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.conj_comp /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.conj_trans /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_equiv.conj_id /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_equiv.smul_of_ne_zero /- _inst_1: field \u219d group_with_zero comm_ring\n -/\n#print linear_equiv.ker_to_span_singleton /- _inst_1: field \u219d group_with_zero ring\n_inst_5: module \u219d\n -/\n#print linear_equiv.to_span_nonzero_singleton /- _inst_5: module \u219d\n -/\n#print linear_equiv.to_span_nonzero_singleton_one /- _inst_5: module \u219d\n -/\n#print linear_equiv.coord /- _inst_5: module \u219d\n -/\n#print linear_equiv.coord_self /- _inst_5: module \u219d\n -/\n#print submodule.comap_subtype_equiv_of_le /- _inst_3: semimodule \u219d\n -/\n#print submodule.quot_equiv_of_eq_bot /- _inst_3: module \u219d\n -/\n#print submodule.quot_equiv_of_eq_bot_apply_mk /- _inst_3: module \u219d\n -/\n#print submodule.quot_equiv_of_eq_bot_symm_apply /- _inst_3: module \u219d\n -/\n#print submodule.coe_quot_equiv_of_eq_bot_symm /- _inst_3: module \u219d\n -/\n#print submodule.quot_equiv_of_eq /- _inst_3: module \u219d\n -/\n#print submodule.mem_map_equiv /- _inst_1: comm_ring \u219d ring\n -/\n#print submodule.comap_le_comap_smul /- _inst_1: comm_ring \u219d ring\n -/\n#print submodule.inf_comap_le_comap_add /- _inst_1: comm_ring \u219d ring\n -/\n#print submodule.compatible_maps /- _inst_4: module \u219d\n_inst_5: module \u219d\n -/\n#print submodule.mapq_linear /- _inst_4: module \u219d\n_inst_5: module \u219d\n -/\n#print equiv.to_linear_equiv /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print add_equiv.to_linear_equiv /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print add_equiv.coe_to_linear_equiv /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print add_equiv.coe_to_linear_equiv_symm /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_map.quot_ker_equiv_range /- _inst_5: module \u219d\n_inst_6: module \u219d\n -/\n#print linear_map.quot_ker_equiv_range_apply_mk /- _inst_5: module \u219d\n_inst_6: module \u219d\n -/\n#print linear_map.quot_ker_equiv_range_symm_apply_image /- _inst_5: module \u219d\n_inst_6: module \u219d\n -/\n#print linear_map.quotient_inf_to_sup_quotient /- _inst_5: module \u219d\n -/\n#print linear_map.quotient_inf_equiv_sup_quotient /- _inst_5: module \u219d\n -/\n#print linear_map.coe_quotient_inf_to_sup_quotient /- _inst_5: module \u219d\n -/\n#print linear_map.quotient_inf_equiv_sup_quotient_apply_mk /- _inst_5: module \u219d\n -/\n#print linear_map.quotient_inf_equiv_sup_quotient_symm_apply_left /- _inst_5: module \u219d\n -/\n#print linear_map.quotient_inf_equiv_sup_quotient_symm_apply_eq_zero_iff /- _inst_5: module \u219d\n -/\n#print linear_map.quotient_inf_equiv_sup_quotient_symm_apply_right /- _inst_5: module \u219d\n -/\n#print linear_map.is_linear_map_prod_iso /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print linear_map.pi /- _inst_3: semimodule \u219d\n -/\n#print linear_map.pi_apply /- _inst_3: semimodule \u219d\n -/\n#print linear_map.ker_pi /- _inst_3: semimodule \u219d\n -/\n#print linear_map.pi_eq_zero /- _inst_3: semimodule \u219d\n -/\n#print linear_map.pi_zero /- _inst_3: semimodule \u219d\n -/\n#print linear_map.pi_comp /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_map.proj_pi /- _inst_3: semimodule \u219d\n -/\n#print linear_map.diag /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.update_apply /- _inst_3: semimodule \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print linear_map.std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.std_basis_apply /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.std_basis_same /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.std_basis_ne /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.ker_std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.proj_comp_std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.proj_std_basis_same /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.proj_std_basis_ne /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.supr_range_std_basis_le_infi_ker_proj /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.infi_ker_proj_le_supr_range_std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.supr_range_std_basis_eq_infi_ker_proj /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.supr_range_std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.disjoint_std_basis_std_basis /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.std_basis_eq_single /- _inst_8: decidable_eq \u219d\n -/\n#print linear_map.fun_left /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_left_apply /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_left_id /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_left_comp /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_congr_left /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_congr_left_apply /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_congr_left_id /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_congr_left_comp /- _inst_3: semimodule \u219d\n -/\n#print linear_map.fun_congr_left_symm /- _inst_3: semimodule \u219d\n -/\n#print linear_map.automorphism_group /- _inst_3: semimodule \u219d\n -/\n#print linear_map.automorphism_group.to_linear_map_is_monoid_hom /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group.has_coe_to_fun /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group.to_linear_equiv /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group.of_linear_equiv /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group.general_linear_equiv /- _inst_3: semimodule \u219d\n -/\n#print linear_map.general_linear_group.general_linear_equiv_to_linear_map /- _inst_3: semimodule \u219d\n -/\n\n-- linear_algebra\\basis.lean\n#print is_basis.repr /- _inst_5: module \u219d\n -/\n#print is_basis.repr_range /- _inst_5: module \u219d\n -/\n#print is_basis.repr_eq_single /- _inst_5: module \u219d\n -/\n#print is_basis.range_repr_self /- _inst_5: module \u219d\n -/\n#print constr_basis /- _inst_6: module \u219d\n -/\n#print constr_smul /- _inst_8: comm_ring \u219d ring\n_inst_10: module \u219d\n -/\n#print module_equiv_finsupp /- _inst_5: module \u219d\n -/\n#print module_equiv_finsupp_apply_basis /- _inst_5: module \u219d\n -/\n#print is_basis_span /- _inst_5: module \u219d\n -/\n#print is_basis_empty_bot /- _inst_5: module \u219d\n -/\n#print submodule.exists_is_compl /- _inst_4: vector_space \u219d\n -/\n#print quotient_prod_linear_equiv /- _inst_4: vector_space \u219d\n -/\n\n-- linear_algebra\\bilinear_form.lean\n#print bilin_form.has_coe_to_fun /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.coe_fn_mk /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.coe_fn_congr /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.add_left /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.smul_left /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.add_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.smul_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.zero_left /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.zero_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.ext /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.add_comm_monoid /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.add_apply /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.inhabited /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.semimodule /- _inst_14: semimodule \u219d\n -/\n#print bilin_form.smul_apply /- _inst_13: comm_semiring \u219d semiring\n_inst_14: semimodule \u219d\n -/\n#print linear_map.to_bilin_aux /- _inst_9: semimodule \u219d\n -/\n#print linear_map.to_bilin /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.to_lin /- _inst_9: semimodule \u219d\n -/\n#print linear_map.to_bilin_aux_eq /- _inst_9: semimodule \u219d\n -/\n#print linear_map.to_bilin_symm /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.to_lin_symm /- _inst_9: semimodule \u219d\n -/\n#print to_linear_map_apply /- _inst_9: semimodule \u219d\n -/\n#print map_sum_left /- _inst_9: semimodule \u219d\n -/\n#print map_sum_right /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.comp /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.comp_left /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_left_comp_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_right_comp_left /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_apply /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.comp_left_apply /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_right_apply /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.comp_injective /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.lin_mul_lin /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.lin_mul_lin_apply /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.lin_mul_lin_comp /- _inst_9: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.lin_mul_lin_comp_left /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.lin_mul_lin_comp_right /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.is_ortho /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.ortho_zero /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_ortho_smul_left /- _inst_13: domain \u219d ring no_zero_divisors\n -/\n#print bilin_form.is_ortho_smul_right /- _inst_13: domain \u219d ring no_zero_divisors\n -/\n#print bilin_form.to_matrix\u2097 /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_apply /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_smul /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_comp /- _inst_15: decidable_eq \u219d\n_inst_16: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_comp_left /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_comp_right /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.mul_to_matrix_mul /- _inst_15: decidable_eq \u219d\n_inst_16: decidable_eq \u219d\n -/\n#print bilin_form.mul_to_matrix /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form.to_matrix_mul /- _inst_15: decidable_eq \u219d\n -/\n#print to_matrix_to_bilin_form /- _inst_15: decidable_eq \u219d\n -/\n#print to_bilin_form_to_matrix /- _inst_15: decidable_eq \u219d\n -/\n#print bilin_form_equiv_matrix /- _inst_15: decidable_eq \u219d\n -/\n#print matrix.to_bilin_form_comp /- _inst_19: decidable_eq \u219d\n -/\n#print refl_bilin_form.is_refl /- _inst_3: semimodule \u219d\n -/\n#print refl_bilin_form.eq_zero /- _inst_3: semimodule \u219d\n -/\n#print refl_bilin_form.ortho_sym /- _inst_3: semimodule \u219d\n -/\n#print sym_bilin_form.is_sym /- _inst_3: semimodule \u219d\n -/\n#print sym_bilin_form.sym /- _inst_3: semimodule \u219d\n -/\n#print sym_bilin_form.is_refl /- _inst_3: semimodule \u219d\n -/\n#print sym_bilin_form.ortho_sym /- _inst_3: semimodule \u219d\n -/\n#print alt_bilin_form.is_alt /- _inst_3: semimodule \u219d\n -/\n#print alt_bilin_form.self_eq_zero /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair.eq /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair_iff_comp_left_eq_comp_right /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair_zero /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair_id /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair.add /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair.smul /- _inst_7: comm_semiring \u219d semiring\n_inst_9: semimodule \u219d\n_inst_18: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair.comp /- _inst_3: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_20: semimodule \u219d\n -/\n#print bilin_form.is_adjoint_pair.mul /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_pair_self_adjoint /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.is_pair_self_adjoint_submodule /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.mem_is_pair_self_adjoint_submodule /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.is_pair_self_adjoint_equiv /- _inst_12: module \u219d\n_inst_22: module \u219d\n -/\n#print bilin_form.is_self_adjoint /- _inst_3: semimodule \u219d\n -/\n#print bilin_form.self_adjoint_submodule /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.mem_self_adjoint_submodule /- _inst_9: semimodule \u219d\n -/\n#print bilin_form.skew_adjoint_submodule /- _inst_12: module \u219d\n -/\n#print bilin_form.mem_skew_adjoint_submodule /- _inst_12: module \u219d\n -/\n#print matrix.is_adjoint_pair /- _inst_10: comm_ring \u219d add_comm_monoid has_mul\n -/\n#print matrix_is_adjoint_pair_bilin_form /- _inst_14: decidable_eq \u219d\n -/\n#print matrix.is_adjoint_pair_equiv /- _inst_14: decidable_eq \u219d\n -/\n#print pair_self_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n#print mem_pair_self_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n#print self_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n#print mem_self_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n#print skew_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n#print mem_skew_adjoint_matrices_submodule /- _inst_14: decidable_eq \u219d\n -/\n\n-- linear_algebra\\char_poly\\basic.lean\n#print char_matrix /- _inst_1: comm_ring \u219d ring\n_inst_2: decidable_eq \u219d\n -/\n#print char_matrix_apply_eq /- _inst_2: decidable_eq \u219d\n -/\n#print char_matrix_apply_ne /- _inst_2: decidable_eq \u219d\n -/\n#print mat_poly_equiv_char_matrix /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly /- _inst_2: decidable_eq \u219d\n -/\n#print aeval_self_char_poly /- _inst_2: decidable_eq \u219d\n -/\n\n-- linear_algebra\\char_poly\\coeff.lean\n#print char_matrix_apply_nat_degree /- _inst_2: decidable_eq \u219d\n -/\n#print char_matrix_apply_nat_degree_le /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly_sub_diagonal_degree_lt /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly_coeff_eq_prod_coeff_of_le /- _inst_2: decidable_eq \u219d\n -/\n#print det_of_card_zero /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly_degree_eq_dim /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly_nat_degree_eq_dim /- _inst_2: decidable_eq \u219d\n -/\n#print char_poly_monic /- _inst_2: decidable_eq \u219d\n -/\n#print trace_eq_neg_char_poly_coeff /- _inst_2: decidable_eq \u219d\n -/\n#print mat_poly_equiv_eval /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: decidable_eq \u219d\n -/\n#print eval_det /- _inst_2: decidable_eq \u219d\n -/\n#print det_eq_sign_char_poly_coeff /- _inst_2: decidable_eq \u219d\n -/\n#print finite_field.char_poly_pow_card /- _inst_2: decidable_eq \u219d\n -/\n#print zmod.char_poly_pow_card /- _inst_2: decidable_eq \u219d\n -/\n#print finite_field.trace_pow_card /- _inst_2: decidable_eq \u219d\n -/\n#print zmod.trace_pow_card /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.is_integral /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.min_poly_dvd_char_poly /- _inst_2: decidable_eq \u219d\n -/\n\n-- linear_algebra\\clifford_algebra.lean\n#print clifford_algebra.\u03b9 /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.\u03b9_square_scalar /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.comp_\u03b9_square_scalar /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.lift_symm_apply /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.lift /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.\u03b9_comp_lift /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.lift_\u03b9_apply /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.lift_unique /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.lift_comp_\u03b9 /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.hom_ext /- _inst_3: module \u219d algebra\n -/\n#print clifford_algebra.as_exterior /- _inst_3: module \u219d algebra\n -/\n\n-- linear_algebra\\contraction.lean\n#print contract_left /- _inst_4: module \u219d\n -/\n#print dual_tensor_hom /- _inst_4: module \u219d\n_inst_5: module \u219d\n -/\n#print dual_tensor_hom_apply /- _inst_4: module \u219d\n_inst_5: module \u219d\n -/\n\n-- linear_algebra\\determinant.lean\n#print matrix.det /- _inst_1: decidable_eq \u219d\n_inst_3: comm_ring \u219d add_comm_monoid comm_monoid has_neg\n -/\n#print matrix.det_diagonal /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_zero /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_eq_one_of_card_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_mul_aux /- _inst_1: decidable_eq \u219d\n_inst_3: comm_ring \u219d comm_monoid ring\n -/\n#print matrix.det_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det.is_monoid_hom /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_transpose /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_permutation /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_permute /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_smul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.ring_hom.map_det /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.alg_hom.map_det /- _inst_1: decidable_eq \u219d\n_inst_3: comm_ring \u219d comm_semiring\n -/\n#print matrix.det_eq_zero_of_row_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_eq_zero_of_column_eq_zero /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.mod_swap /- _inst_4: decidable_eq \u219d\n -/\n#print matrix.r.decidable_rel /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_zero_of_row_eq /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_update_column_add /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_update_row_add /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_update_column_smul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_update_row_smul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_block_diagonal /- _inst_1: decidable_eq \u219d\n_inst_5: decidable_eq \u219d\n -/\n\n-- linear_algebra\\dfinsupp.lean\n#print dfinsupp.lmk /- dec_\u03b9: decidable_eq \u219d\n -/\n#print dfinsupp.lsingle /- dec_\u03b9: decidable_eq \u219d\n -/\n#print dfinsupp.lhom_ext /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print dfinsupp.lhom_ext' /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print dfinsupp.lmk_apply /- dec_\u03b9: decidable_eq \u219d\n -/\n#print dfinsupp.lsingle_apply /- dec_\u03b9: decidable_eq \u219d\n -/\n#print dfinsupp.lsum_apply /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print dfinsupp.lsum /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print dfinsupp.lsum_symm_apply /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n\n-- linear_algebra\\dimension.lean\n#print dim_bot /- _inst_3: vector_space \u219d\n -/\n#print dim_top /- _inst_3: vector_space \u219d\n -/\n#print dim_span /- _inst_3: vector_space \u219d\n -/\n#print dim_span_set /- _inst_3: vector_space \u219d\n -/\n#print dim_span_le /- _inst_3: vector_space \u219d\n -/\n#print dim_span_of_finset /- _inst_3: vector_space \u219d\n -/\n#print dim_quotient_add_dim /- _inst_3: vector_space \u219d\n -/\n#print dim_quotient_le /- _inst_3: vector_space \u219d\n -/\n#print dim_range_add_dim_ker /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print dim_range_le /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print dim_map_le /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print dim_range_of_surjective /- _inst_7: vector_space \u219d\n -/\n#print dim_eq_of_surjective /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print dim_le_of_surjective /- _inst_3: vector_space \u219d\n -/\n#print dim_eq_of_injective /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print dim_submodule_le /- _inst_3: vector_space \u219d\n -/\n#print dim_le_of_injective /- _inst_5: vector_space \u219d\n -/\n#print dim_le_of_submodule /- _inst_3: vector_space \u219d\n -/\n#print linear_independent_le_dim /- _inst_3: vector_space \u219d\n -/\n#print linear_independent_le_dim' /- _inst_3: vector_space \u219d\n -/\n#print dim_sup_add_dim_inf_eq /- _inst_3: vector_space \u219d\n -/\n#print dim_add_le_dim_add_dim /- _inst_3: vector_space \u219d\n -/\n#print exists_mem_ne_zero_of_dim_pos /- _inst_3: vector_space \u219d\n -/\n#print rank /- _inst_7: vector_space \u219d\n -/\n#print rank_le_domain /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print rank_add_le /- _inst_7: vector_space \u219d\n -/\n#print rank_zero /- _inst_7: vector_space \u219d\n -/\n#print rank_comp_le2 /- _inst_7: vector_space \u219d\n_inst_11: vector_space \u219d\n -/\n#print dim_zero_iff_forall_zero /- _inst_3: vector_space \u219d\n -/\n#print dim_le_one_iff /- _inst_3: vector_space \u219d\n -/\n#print dim_submodule_le_one_iff /- _inst_3: vector_space \u219d\n -/\n#print dim_submodule_le_one_iff' /- _inst_3: vector_space \u219d\n -/\n\n-- linear_algebra\\direct_sum\\finsupp.lean\n#print finsupp_lequiv_direct_sum /- _inst_3: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print finsupp_lequiv_direct_sum_single /- _inst_3: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print finsupp_lequiv_direct_sum_symm_lof /- _inst_3: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print finsupp_tensor_finsupp /- _inst_8: module \u219d\n_inst_10: module \u219d\n -/\n#print finsupp_tensor_finsupp_single /- _inst_8: module \u219d\n_inst_10: module \u219d\n -/\n#print finsupp_tensor_finsupp_symm_single /- _inst_8: module \u219d\n_inst_10: module \u219d\n -/\n\n-- linear_algebra\\direct_sum\\tensor_product.lean\n#print tensor_product.direct_sum /- _inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n#print tensor_product.direct_sum_lof_tmul_lof /- _inst_2: decidable_eq \u219d\n_inst_3: decidable_eq \u219d\n -/\n\n-- linear_algebra\\direct_sum_module.lean\n#print direct_sum.lmk /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.lof /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.single_eq_lof /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.mk_smul /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.of_smul /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.support_smul /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.to_module /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print direct_sum.to_module_lof /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print direct_sum.to_module.unique /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print direct_sum.to_module.ext /- dec_\u03b9: decidable_eq \u219d\n_inst_5: semimodule \u219d\n -/\n#print direct_sum.lset_to_set /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.lid /- _inst_7: semimodule \u219d\n -/\n#print direct_sum.lof_apply /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.component.lof_self /- dec_\u03b9: decidable_eq \u219d\n -/\n#print direct_sum.component.of /- dec_\u03b9: decidable_eq \u219d\n -/\n\n-- linear_algebra\\dual.lean\n#print module.dual /- _inst_1: comm_ring \u219d ring\n -/\n#print module.dual.eval_apply /- _inst_3: module \u219d\n -/\n#print module.dual.transpose /- _inst_3: module \u219d\n_inst_5: module \u219d\n -/\n#print module.dual.transpose_apply /- _inst_3: module \u219d\n_inst_5: module \u219d\n -/\n#print module.dual.transpose_comp /- _inst_3: module \u219d\n_inst_5: module \u219d\n_inst_7: module \u219d\n -/\n#print is_basis.to_dual /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_apply /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_total_left /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_total_right /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_apply_left /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_apply_right /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_flip /- _inst_3: vector_space \u219d\nde: decidable_eq \u219d\n -/\n#print is_basis.to_dual_swap_eq_to_dual /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_eq_repr /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_eq_equiv_fun /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_inj /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_ker /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_range /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_lin_independent /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis_apply_self /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_equiv /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis_is_basis /- de: decidable_eq \u219d\n -/\n#print is_basis.total_dual_basis /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis_repr /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis_equiv_fun /- de: decidable_eq \u219d\n -/\n#print is_basis.dual_basis_apply /- de: decidable_eq \u219d\n -/\n#print is_basis.to_dual_to_dual /- de: decidable_eq \u219d\n -/\n#print dual_pair.coeffs /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.coeffs_apply /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.lc /- _inst_3: vector_space \u219d has_scalar\n -/\n#print dual_pair.dual_lc /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.coeffs_lc /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.decomposition /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.mem_of_mem_span /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.is_basis /- d\u03b9: decidable_eq \u219d\n -/\n#print dual_pair.eq_dual /- d\u03b9: decidable_eq \u219d\n -/\n\n-- linear_algebra\\eigenspace.lean\n#print module.End.eigenspace /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_3: module \u219d algebra\n -/\n#print module.End.mem_eigenspace_iff /- _inst_3: module \u219d algebra\n -/\n#print module.End.eigenspace_div /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.eigenspace_aeval_polynomial_degree_1 /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.ker_aeval_ring_hom'_unit_polynomial /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.aeval_apply_of_has_eigenvector /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.is_integral /- _inst_6: vector_space \u219d algebra finite_dimensional\n -/\n#print module.End.is_root_of_has_eigenvalue /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.has_eigenvalue_of_is_root /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.has_eigenvalue_iff_is_root /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.exists_eigenvalue /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.eigenvectors_linear_independent /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.generalized_eigenspace /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_3: module \u219d algebra\n -/\n#print module.End.generalized_eigenrange /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_3: module \u219d algebra\n -/\n#print module.End.generalized_eigenspace_mono /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.has_generalized_eigenvalue_of_has_eigenvalue /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.generalized_eigenspace_le_generalized_eigenspace_findim /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.generalized_eigenspace_eq_generalized_eigenspace_findim_of_le /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.generalized_eigenspace_restrict /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.generalized_eigenvec_disjoint_range_ker /- _inst_6: vector_space \u219d algebra\n -/\n#print module.End.pos_findim_generalized_eigenspace_of_has_eigenvalue /- _inst_6: vector_space \u219d\n -/\n#print module.End.map_generalized_eigenrange_le /- _inst_6: vector_space \u219d algebra\n -/\n#print linear_map.is_integral /- _inst_3: vector_space \u219d algebra\n -/\n\n-- linear_algebra\\exterior_algebra.lean\n#print exterior_algebra /- _inst_3: semimodule \u219d\n -/\n#print exterior_algebra.ring /- _inst_5: semimodule \u219d\n -/\n#print exterior_algebra.\u03b9 /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.\u03b9_square_zero /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.comp_\u03b9_square_zero /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.lift_symm_apply /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.lift /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.\u03b9_comp_lift /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.lift_\u03b9_apply /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.lift_unique /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.lift_comp_\u03b9 /- _inst_3: semimodule \u219d algebra\n -/\n#print exterior_algebra.hom_ext /- _inst_3: semimodule \u219d algebra\n -/\n\n-- linear_algebra\\finite_dimensional.lean\n#print finite_dimensional.finite_dimensional_iff_dim_lt_omega /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.iff_fg /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.finite_dimensional_submodule /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.finite_dimensional_quotient /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.exists_nontrivial_relation_sum_zero_of_dim_succ_lt_card /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.exists_relation_sum_zero_pos_coefficient_of_dim_succ_lt_card /- _inst_6: linear_ordered_field \u219d field linear_ordered_cancel_add_comm_monoid\n -/\n#print finite_dimensional.eq_top_of_findim_eq /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print finite_dimensional.span_of_finite /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.submodule.span.finite_dimensional /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional_bot /- _inst_3: vector_space \u219d\n -/\n#print findim_bot /- _inst_3: vector_space \u219d\n -/\n#print bot_eq_top_of_dim_eq_zero /- _inst_3: vector_space \u219d\n -/\n#print dim_eq_zero /- _inst_3: vector_space \u219d\n -/\n#print findim_eq_zero /- _inst_3: vector_space \u219d\n -/\n#print submodule.fg_iff_finite_dimensional /- _inst_3: vector_space \u219d\n -/\n#print submodule.finite_dimensional_of_le /- _inst_3: vector_space \u219d\n -/\n#print submodule.finite_dimensional_inf_left /- _inst_3: vector_space \u219d\n -/\n#print submodule.finite_dimensional_inf_right /- _inst_3: vector_space \u219d\n -/\n#print submodule.finite_dimensional_sup /- _inst_3: vector_space \u219d\n -/\n#print submodule.findim_quotient_add_findim /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print submodule.findim_le /- _inst_3: vector_space \u219d\n -/\n#print submodule.findim_lt /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print submodule.findim_quotient_le /- _inst_3: vector_space \u219d\n -/\n#print submodule.dim_sup_add_dim_inf_eq /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print submodule.eq_top_of_disjoint /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print finite_dimensional.eq_of_le_of_findim_le /- _inst_3: vector_space \u219d\n -/\n#print finite_dimensional.eq_of_le_of_findim_eq /- _inst_3: vector_space \u219d\n -/\n#print linear_map.surjective_of_injective /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print linear_map.finite_dimensional_range /- _inst_3: vector_space \u219d finite_dimensional\n_inst_5: vector_space \u219d\n -/\n#print linear_map.findim_range_add_findim_ker /- _inst_3: vector_space \u219d finite_dimensional\n_inst_5: vector_space \u219d\n -/\n#print linear_equiv.of_injective_endo /- _inst_3: vector_space \u219d\n -/\n#print findim_top /- _inst_3: vector_space \u219d\n -/\n#print linear_map.injective_iff_surjective_of_findim_eq_findim /- _inst_3: vector_space \u219d finite_dimensional\n_inst_5: vector_space \u219d\n -/\n#print linear_map.findim_le_findim_of_injective /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n -/\n#print alg_hom.bijective /- _inst_7: field \u219d division_ring\n -/\n#print submodule.findim_mono /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print submodule.lt_of_le_of_findim_lt_findim /- _inst_3: vector_space \u219d\n -/\n#print submodule.lt_top_of_findim_lt_findim /- _inst_3: vector_space \u219d\n -/\n#print submodule.findim_lt_findim_of_lt /- _inst_3: vector_space \u219d finite_dimensional\n -/\n#print findim_span_le_card /- _inst_3: vector_space \u219d\n -/\n#print findim_span_eq_card /- _inst_3: vector_space \u219d\n -/\n#print findim_span_set_eq_card /- _inst_3: vector_space \u219d\n -/\n#print span_lt_of_subset_of_card_lt_findim /- _inst_3: vector_space \u219d\n -/\n#print span_lt_top_of_card_lt_findim /- _inst_3: vector_space \u219d\n -/\n#print linear_independent_of_span_eq_top_of_card_eq_findim /- _inst_3: vector_space \u219d\n -/\n#print linear_independent_iff_card_eq_findim_span /- _inst_3: vector_space \u219d\n -/\n#print span_eq_top_of_linear_independent_of_card_eq_findim /- _inst_3: vector_space \u219d\n -/\n#print subalgebra.dim_eq_one_of_eq_bot /- _inst_7: field \u219d nontrivial semiring add_comm_group\n -/\n#print subalgebra_top_dim_eq_submodule_top_dim /- _inst_7: field \u219d semiring add_comm_group\n -/\n#print subalgebra_top_findim_eq_submodule_top_findim /- _inst_7: field \u219d semiring add_comm_group\n -/\n#print subalgebra.eq_bot_of_findim_one /- _inst_7: field \u219d nontrivial ring\n -/\n#print module.End.exists_ker_pow_eq_ker_pow_succ /- _inst_3: vector_space \u219d\n -/\n\n-- linear_algebra\\finsupp.lean\n#print finsupp.lsingle /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lhom_ext /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lhom_ext' /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lapply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lsubtype_domain /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lsubtype_domain_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lsingle_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lapply_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.ker_lsingle /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lsingle_range_le_ker_lapply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.infi_ker_lapply_le_bot /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supr_lsingle_range /- _inst_3: semimodule \u219d\n -/\n#print finsupp.disjoint_lsingle_lsingle /- _inst_3: semimodule \u219d\n -/\n#print finsupp.span_single_image /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.mem_supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.mem_supported' /- _inst_3: semimodule \u219d\n -/\n#print finsupp.single_mem_supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.restrict_dom /- _inst_3: semimodule \u219d\n -/\n#print finsupp.restrict_dom_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.restrict_dom_comp_subtype /- _inst_3: semimodule \u219d\n -/\n#print finsupp.range_restrict_dom /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_mono /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_empty /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_univ /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_Union /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_union /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_Inter /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_inter /- _inst_3: semimodule \u219d\n -/\n#print finsupp.disjoint_supported_supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.disjoint_supported_supported_iff /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_equiv_finsupp /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lsum /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.coe_lsum /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lsum_apply /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lsum_single /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lsum_symm_apply /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lmap_domain /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_id /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_comp /- _inst_3: semimodule \u219d\n -/\n#print finsupp.supported_comap_lmap_domain /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_disjoint_ker /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_apply /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_apply_of_mem_supported /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_single /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_unique /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_range /- _inst_3: semimodule \u219d\n -/\n#print finsupp.range_total /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lmap_domain_total /- _inst_3: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print finsupp.total_emb_domain /- _inst_7: semimodule \u219d\n -/\n#print finsupp.total_map_domain /- _inst_7: semimodule \u219d\n -/\n#print finsupp.span_eq_map_total /- _inst_3: semimodule \u219d\n -/\n#print finsupp.mem_span_iff_total /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_on /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_on_range /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_comp /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_comap_domain /- _inst_3: semimodule \u219d\n -/\n#print finsupp.total_on_finset /- _inst_3: semimodule \u219d\n -/\n#print finsupp.dom_lcongr /- _inst_3: semimodule \u219d\n -/\n#print finsupp.dom_lcongr_single /- _inst_3: semimodule \u219d\n -/\n#print finsupp.congr /- _inst_3: semimodule \u219d\n -/\n#print finsupp.lcongr /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print finsupp.lcongr_single /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print linear_map.map_finsupp_total /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print submodule.exists_finset_of_mem_supr /- _inst_3: semimodule \u219d\n -/\n#print mem_span_finset /- _inst_3: semimodule \u219d\n -/\n\n-- linear_algebra\\finsupp_vector_space.lean\n#print finsupp.linear_independent_single /- _inst_3: module \u219d\n -/\n#print finsupp.is_basis_single /- _inst_3: module \u219d\n -/\n#print finsupp.is_basis.tensor_product /- _inst_3: module \u219d\n_inst_5: module \u219d\n -/\n#print finsupp.dim_eq /- _inst_3: vector_space \u219d\n -/\n#print eq_bot_iff_dim_eq_zero /- _inst_3: vector_space \u219d\n -/\n#print injective_of_surjective /- _inst_5: vector_space \u219d\n -/\n\n-- linear_algebra\\lagrange.lean\n#print lagrange.basis /- _inst_1: decidable_eq \u219d\n_inst_2: field \u219d has_inv ring comm_semiring\n -/\n#print lagrange.basis_empty /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.eval_basis_self /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.eval_basis_ne /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.eval_basis /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.nat_degree_basis /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_empty /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.eval_interpolate /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.degree_interpolate_lt /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.linterpolate /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_add /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_zero /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_neg /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_sub /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.interpolate_smul /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.eq_zero_of_eval_eq_zero /- _inst_3: field \u219d integral_domain\n -/\n#print lagrange.eq_interpolate /- _inst_1: decidable_eq \u219d\n -/\n#print lagrange.fun_equiv_degree_lt /- _inst_1: decidable_eq \u219d\n -/\n\n-- linear_algebra\\linear_independent.lean\n#print linear_independent_iff'' /- _inst_5: module \u219d\n -/\n#print linear_independent.ne_zero /- _inst_5: module \u219d\n -/\n#print linear_independent.injective /- _inst_5: module \u219d\n -/\n#print linear_independent_span /- _inst_5: module \u219d\n -/\n#print linear_independent_iff_total_on /- _inst_5: module \u219d\n -/\n#print linear_independent.total_equiv /- _inst_5: module \u219d\n -/\n#print linear_independent.repr /- _inst_5: module \u219d\n -/\n#print linear_independent.total_repr /- _inst_5: module \u219d\n -/\n#print linear_independent.total_comp_repr /- _inst_5: module \u219d\n -/\n#print linear_independent.repr_ker /- _inst_5: module \u219d\n -/\n#print linear_independent.repr_range /- _inst_5: module \u219d\n -/\n#print linear_independent.repr_eq /- _inst_5: module \u219d\n -/\n#print linear_independent.repr_eq_single /- _inst_5: module \u219d\n -/\n#print surjective_of_linear_independent_of_span /- _inst_5: module \u219d\n -/\n#print linear_independent_monoid_hom /- _inst_9: integral_domain \u219d comm_semigroup ring no_zero_divisors\n -/\n#print mem_span_insert_exchange /- _inst_4: vector_space \u219d\n -/\n#print linear_independent_iff_not_mem_span /- _inst_4: vector_space \u219d\n -/\n\n-- linear_algebra\\linear_pmap.lean\n#print linear_pmap.has_coe_to_fun /- _inst_3: module \u219d\n -/\n#print linear_pmap.to_fun_eq_coe /- _inst_3: module \u219d\n -/\n#print linear_pmap.map_zero /- _inst_3: module \u219d\n -/\n#print linear_pmap.map_add /- _inst_3: module \u219d\n -/\n#print linear_pmap.map_neg /- _inst_3: module \u219d\n -/\n#print linear_pmap.map_sub /- _inst_3: module \u219d\n -/\n#print linear_pmap.map_smul /- _inst_3: module \u219d\n -/\n#print linear_pmap.mk_apply /- _inst_3: module \u219d\n -/\n#print linear_pmap.mk_span_singleton' /- _inst_3: module \u219d\n -/\n#print linear_pmap.has_neg /- _inst_3: module \u219d\n -/\n#print linear_pmap.eq_of_le_of_domain_eq /- _inst_3: module \u219d\n -/\n#print linear_pmap.has_inf /- _inst_3: module \u219d\n -/\n#print linear_pmap.has_bot /- _inst_3: module \u219d\n -/\n#print linear_pmap.sup /- _inst_3: module \u219d\n -/\n#print linear_pmap.sup_apply /- _inst_3: module \u219d\n -/\n#print linear_pmap.Sup /- _inst_3: module \u219d\n -/\n#print linear_pmap.le_Sup /- _inst_3: module \u219d\n -/\n#print linear_map.to_pmap /- _inst_3: module \u219d\n -/\n#print linear_map.comp_pmap /- _inst_3: module \u219d\n -/\n#print linear_pmap.cod_restrict /- _inst_3: module \u219d\n_inst_5: module \u219d\n -/\n#print linear_pmap.comp /- _inst_5: module \u219d\n -/\n\n-- linear_algebra\\matrix.lean\n#print matrix.fintype /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print matrix.mul_vec_std_basis /- _inst_5: decidable_eq \u219d\n -/\n#print linear_map.to_matrix' /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin' /- _inst_5: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_symm /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin'_symm /- _inst_5: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_to_lin' /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin'_to_matrix' /- _inst_5: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin'_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin'_one /- _inst_5: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_id /- _inst_5: decidable_eq \u219d\n -/\n#print matrix.to_lin'_mul /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_comp /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_map.to_matrix'_mul /- _inst_6: decidable_eq \u219d\n -/\n#print linear_map.to_matrix /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print matrix.to_lin /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.to_matrix_symm /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print matrix.to_lin_symm /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print matrix.to_lin_to_matrix /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.to_matrix_to_lin /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.to_matrix_apply /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.to_matrix_apply' /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print matrix.to_lin_apply /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print matrix.to_lin_self /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.to_matrix_id /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n -/\n#print matrix.to_lin_one /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n -/\n#print linear_map.to_matrix_range /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n_inst_10: decidable_eq \u219d\n_inst_11: decidable_eq \u219d\n -/\n#print linear_map.to_matrix_comp /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n_inst_11: module \u219d\n_inst_12: decidable_eq \u219d\n -/\n#print linear_map.to_matrix_mul /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n -/\n#print matrix.to_lin_mul /- _inst_5: decidable_eq \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n_inst_11: module \u219d\n_inst_12: decidable_eq \u219d\n -/\n#print is_basis.to_matrix /- _inst_3: comm_ring \u219d ring\n -/\n#print is_basis.to_matrix_eq_to_matrix_constr /- _inst_5: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print is_basis.to_matrix_self /- _inst_6: decidable_eq \u219d\n -/\n#print is_basis.to_matrix_update /- _inst_6: decidable_eq \u219d\n -/\n#print is_basis.to_lin_to_matrix /- _inst_5: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print is_basis_to_matrix_mul_linear_map_to_matrix /- _inst_5: module \u219d\n_inst_7: module \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print linear_map_to_matrix_mul_is_basis_to_matrix /- _inst_5: module \u219d\n_inst_7: module \u219d\n_inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print linear_equiv.is_unit_det /- _inst_3: module \u219d\n_inst_5: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_equiv.of_is_unit_det /- _inst_3: module \u219d\n_inst_5: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print is_basis.det /- _inst_6: decidable_eq \u219d\n -/\n#print is_basis.det_apply /- _inst_6: decidable_eq \u219d\n -/\n#print is_basis.det_self /- _inst_6: decidable_eq \u219d\n -/\n#print is_basis.iff_det /- _inst_3: module \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_map.to_matrix_transpose /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n_inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print linear_map.to_matrix_symm_transpose /- _inst_3: vector_space \u219d\n_inst_5: vector_space \u219d\n_inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print matrix.diag /- _inst_5: semimodule \u219d\n -/\n#print matrix.diag_apply /- _inst_5: semimodule \u219d\n -/\n#print matrix.diag_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.diag_transpose /- _inst_5: semimodule \u219d\n -/\n#print matrix.trace /- _inst_5: semimodule \u219d\n -/\n#print matrix.trace_diag /- _inst_5: semimodule \u219d\n -/\n#print matrix.trace_one /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.trace_transpose /- _inst_5: semimodule \u219d\n -/\n#print matrix.trace_mul_comm /- _inst_6: comm_ring \u219d comm_semiring\n -/\n#print matrix.proj_diagonal /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.diagonal_comp_std_basis /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.diagonal_to_lin' /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.to_linear_equiv /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.to_linear_equiv_apply /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.to_linear_equiv_symm_apply /- _inst_2: decidable_eq \u219d\n -/\n#print matrix.rank_vec_mul_vec /- _inst_6: decidable_eq \u219d\n -/\n#print matrix.ker_diagonal_to_lin' /- _inst_4: decidable_eq \u219d\n -/\n#print matrix.range_diagonal /- _inst_4: decidable_eq \u219d\n -/\n#print matrix.rank_diagonal /- _inst_4: decidable_eq \u219d\n_inst_5: decidable_eq \u219d\n -/\n#print matrix.reindex_alg_equiv /- _inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print matrix.reindex_alg_equiv_apply /- _inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print matrix.reindex_alg_equiv_symm_apply /- _inst_8: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print matrix.det_reindex_self' /- _inst_7: decidable_eq \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print matrix.det_reindex_self /- _inst_7: decidable_eq \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print matrix.det_reindex_linear_equiv_self /- _inst_7: decidable_eq \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print matrix.det_reindex_alg_equiv /- _inst_7: decidable_eq \u219d\n_inst_8: decidable_eq \u219d\n -/\n#print linear_map.trace_aux /- _inst_3: module \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print linear_map.trace_aux_def /- _inst_3: module \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print linear_map.trace_aux_eq' /- _inst_3: module \u219d\n_inst_4: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_map.trace_aux_range /- _inst_3: module \u219d\n_inst_4: decidable_eq \u219d\n -/\n#print linear_map.trace_aux_eq /- _inst_3: module \u219d\n_inst_4: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n#print linear_map.trace /- _inst_3: module \u219d\n -/\n#print linear_map.trace_eq_matrix_trace /- _inst_3: module \u219d\n_inst_5: decidable_eq \u219d\n -/\n#print linear_map.trace_mul_comm /- _inst_3: module \u219d\n -/\n#print linear_map.finite_dimensional /- _inst_3: vector_space \u219d\n_inst_6: vector_space \u219d\n -/\n#print linear_map.findim_linear_map /- _inst_3: vector_space \u219d finite_dimensional\n_inst_6: vector_space \u219d finite_dimensional\n -/\n#print alg_equiv_matrix' /- _inst_3: decidable_eq \u219d\n -/\n#print linear_equiv.alg_conj /- _inst_3: module \u219d algebra\n_inst_5: module \u219d algebra\n -/\n#print alg_equiv_matrix /- _inst_4: module \u219d algebra\n_inst_5: decidable_eq \u219d\n -/\n\n-- linear_algebra\\multilinear.lean\n#print multilinear_map.has_coe_to_fun /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.ext /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_add /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_smul /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_coord_zero /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_zero /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.has_add /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.add_apply /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.has_zero /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.inhabited /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.zero_apply /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.add_comm_monoid /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.sum_apply /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.to_linear_map /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.prod /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print multilinear_map.restr /- _inst_10: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print multilinear_map.cons_add /- _inst_10: semimodule \u219d\n -/\n#print multilinear_map.cons_smul /- _inst_10: semimodule \u219d\n -/\n#print multilinear_map.snoc_add /- _inst_10: semimodule \u219d\n -/\n#print multilinear_map.snoc_smul /- _inst_10: semimodule \u219d\n -/\n#print multilinear_map.comp_linear_map /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.comp_linear_map_apply /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_piecewise_add /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_add_univ /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_sum_finset_aux /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_sum_finset /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.map_sum /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n -/\n#print multilinear_map.restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n_inst_16: semimodule \u219d\n_inst_18: is_scalar_tower \u219d\n -/\n#print multilinear_map.coe_restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_10: semimodule \u219d\n_inst_16: semimodule \u219d\n_inst_18: is_scalar_tower \u219d\n -/\n#print linear_map.comp_multilinear_map /- _inst_1: decidable_eq \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.coe_comp_multilinear_map /- _inst_1: decidable_eq \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.comp_multilinear_map_apply /- _inst_1: decidable_eq \u219d\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.map_piecewise_smul /- _inst_1: decidable_eq \u219d\n_inst_2: comm_semiring \u219d comm_monoid semiring\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.map_smul_univ /- _inst_1: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.has_scalar /- _inst_1: decidable_eq \u219d\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print multilinear_map.smul_apply /- _inst_1: decidable_eq \u219d\n_inst_11: algebra \u219d has_scalar\n_inst_13: semimodule \u219d has_scalar\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d has_scalar\n -/\n#print multilinear_map.semimodule /- _inst_1: decidable_eq \u219d\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print multilinear_map.mk_pi_algebra /- _inst_1: decidable_eq \u219d\n -/\n#print multilinear_map.mk_pi_algebra_apply /- _inst_1: decidable_eq \u219d\n -/\n#print multilinear_map.smul_right /- _inst_1: decidable_eq \u219d\n_inst_2: comm_semiring \u219d semiring\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.smul_right_apply /- _inst_1: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.mk_pi_ring /- _inst_1: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.mk_pi_ring_apply /- _inst_1: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.mk_pi_ring_apply_one_eq_self /- _inst_1: decidable_eq \u219d\n_inst_8: semimodule \u219d\n -/\n#print multilinear_map.map_sub /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print multilinear_map.has_neg /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print multilinear_map.neg_apply /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print multilinear_map.add_comm_group /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print multilinear_map.pi_ring_equiv /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print linear_map.uncurry_left_apply /- _inst_8: module \u219d\n -/\n#print linear_map.curry_uncurry_left /- _inst_8: module \u219d\n -/\n#print multilinear_map.uncurry_curry_left /- _inst_8: module \u219d\n -/\n#print multilinear_map.uncurry_right_apply /- _inst_8: module \u219d\n -/\n#print multilinear_map.curry_uncurry_right /- _inst_8: module \u219d\n -/\n#print multilinear_map.uncurry_curry_right /- _inst_8: module \u219d\n -/\n\n-- linear_algebra\\nonsingular_inverse.lean\n#print matrix.cramer_map /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer_map_is_linear /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer_is_linear /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer_transpose_row_self /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.sum_cramer /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.sum_cramer_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_def /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_transpose /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.cramer_eq_adjugate_mul_vec /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.mul_adjugate_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.mul_adjugate /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_adjugate_of_cancel /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_eq_one_of_card_eq_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.adjugate_zero /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_adjugate_eq_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_adjugate_of_is_unit /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.is_unit_det_transpose /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.has_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.transpose_nonsing_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.mul_nonsing_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_det /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.is_unit_nonsing_inv_det /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_nonsing_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_unit /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.is_unit_iff_is_unit_det /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.is_unit_det_of_left_inverse /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.is_unit_det_of_right_inverse /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_left_right /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.nonsing_inv_right_left /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.det_smul_inv_mul_vec_eq_cramer /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.mul_vec_cramer /- _inst_1: decidable_eq \u219d\n -/\n\n-- linear_algebra\\projection.lean\n#print linear_map.ker_id_sub_eq_of_proj /- _inst_3: module \u219d\n -/\n#print linear_map.range_eq_of_proj /- _inst_3: module \u219d\n -/\n#print linear_map.is_compl_of_proj /- _inst_3: module \u219d\n -/\n#print submodule.quotient_equiv_of_is_compl /- _inst_3: module \u219d\n -/\n#print submodule.quotient_equiv_of_is_compl_symm_apply /- _inst_3: module \u219d\n -/\n#print submodule.quotient_equiv_of_is_compl_apply_mk_coe /- _inst_3: module \u219d\n -/\n#print submodule.mk_quotient_equiv_of_is_compl_apply /- _inst_3: module \u219d\n -/\n#print submodule.prod_equiv_of_is_compl /- _inst_3: module \u219d\n -/\n#print submodule.coe_prod_equiv_of_is_compl /- _inst_3: module \u219d\n -/\n#print submodule.coe_prod_equiv_of_is_compl' /- _inst_3: module \u219d\n -/\n#print submodule.prod_equiv_of_is_compl_symm_apply_left /- _inst_3: module \u219d\n -/\n#print submodule.prod_equiv_of_is_compl_symm_apply_right /- _inst_3: module \u219d\n -/\n#print submodule.prod_equiv_of_is_compl_symm_apply_fst_eq_zero /- _inst_3: module \u219d\n -/\n#print submodule.prod_equiv_of_is_compl_symm_apply_snd_eq_zero /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_apply_left /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_range /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_apply_eq_zero_iff /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_apply_right' /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_apply_right /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_ker /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_comp_subtype /- _inst_3: module \u219d\n -/\n#print submodule.linear_proj_of_is_compl_idempotent /- _inst_3: module \u219d\n -/\n#print linear_map.linear_proj_of_is_compl_of_proj /- _inst_3: module \u219d\n -/\n#print submodule.is_compl_equiv_proj /- _inst_3: module \u219d\n -/\n#print submodule.coe_is_compl_equiv_proj_apply /- _inst_3: module \u219d\n -/\n#print submodule.coe_is_compl_equiv_proj_symm_apply /- _inst_3: module \u219d\n -/\n\n-- linear_algebra\\quadratic_form.lean\n#print quadratic_form.polar /- _inst_1: add_comm_group \u219d has_add\n_inst_2: ring \u219d has_sub\n -/\n#print quadratic_form.polar_comm /- _inst_3: comm_ring \u219d ring\n -/\n#print quadratic_form.map_add_self /- _inst_4: module \u219d\n -/\n#print quadratic_form.coe_fn_smul /- _inst_3: comm_ring \u219d ring\n -/\n#print quadratic_form.smul_apply /- _inst_3: comm_ring \u219d ring\n -/\n#print quadratic_form.associated /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_apply /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_is_sym /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_comp /- _inst_5: module \u219d\n_inst_8: module \u219d\n -/\n#print quadratic_form.associated_lin_mul_lin /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_to_quadratic_form /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_left_inverse /- _inst_5: module \u219d\n -/\n#print quadratic_form.associated_right_inverse /- _inst_5: module \u219d\n -/\n#print quadratic_form.pos_def /- _inst_6: ordered_ring \u219d has_lt ring\n -/\n#print quadratic_form.pos_def.smul /- _inst_8: linear_ordered_comm_ring \u219d ordered_ring\n -/\n#print quadratic_form.lin_mul_lin_self_pos_def /- _inst_8: linear_ordered_comm_ring \u219d linear_ordered_ring comm_ring\n -/\n#print matrix.to_quadratic_form /- _inst_3: comm_ring \u219d ring comm_semiring\n -/\n#print quadratic_form.to_matrix /- _inst_7: decidable_eq \u219d\n -/\n#print quadratic_form.to_matrix_smul /- _inst_7: decidable_eq \u219d\n -/\n#print quadratic_form.to_matrix_comp /- _inst_7: decidable_eq \u219d\n_inst_9: decidable_eq \u219d\n -/\n#print quadratic_form.discr /- _inst_7: decidable_eq \u219d\n -/\n#print quadratic_form.discr_smul /- _inst_7: decidable_eq \u219d\n -/\n#print quadratic_form.discr_comp /- _inst_7: decidable_eq \u219d\n -/\n\n-- linear_algebra\\sesquilinear_form.lean\n#print sesq_form.ortho_smul_left /- _inst_1: domain \u219d ring no_zero_divisors\n -/\n#print sesq_form.ortho_smul_right /- _inst_1: domain \u219d ring no_zero_divisors\n -/\n\n-- linear_algebra\\special_linear_group.lean\n#print matrix.special_linear_group /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.coe_matrix /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.coe_fun /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_lin' /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.ext_iff /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.ext /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.has_inv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.has_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.has_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.inhabited /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.inv_val /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.inv_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.mul_val /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.mul_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.one_val /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.one_apply /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.det_coe_matrix /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.det_coe_fun /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_lin'_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_lin'_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.group /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_linear_equiv /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_GL /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.coe_to_GL /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_GL_one /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.to_GL_mul /- _inst_1: decidable_eq \u219d\n -/\n#print matrix.special_linear_group.embedding_GL /- _inst_1: decidable_eq \u219d\n -/\n\n-- linear_algebra\\tensor_algebra.lean\n#print tensor_algebra /- _inst_3: semimodule \u219d\n -/\n#print tensor_algebra.ring /- _inst_5: semimodule \u219d\n -/\n#print tensor_algebra.\u03b9 /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.ring_quot_mk_alg_hom_free_algebra_\u03b9_eq_\u03b9 /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.lift /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.lift_symm_apply /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.\u03b9_comp_lift /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.lift_\u03b9_apply /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.lift_unique /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.lift_comp_\u03b9 /- _inst_3: semimodule \u219d algebra\n -/\n#print tensor_algebra.hom_ext /- _inst_3: semimodule \u219d algebra\n -/\n\n-- linear_algebra\\tensor_product.lean\n#print linear_map.mk\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.mk\u2082_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ext\u2082 /- _inst_1: comm_semiring \u219d semiring\n_inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.flip /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.flip_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.flip_inj /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.lflip /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.lflip_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.map_zero\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.map_neg\u2082 /- _inst_12: comm_ring \u219d ring comm_semiring\n_inst_16: module \u219d\n_inst_17: module \u219d\n_inst_18: module \u219d\n -/\n#print linear_map.map_add\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.map_smul\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.lcomp /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.lcomp_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.llcomp /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.llcomp_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.compl\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.compl\u2082_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.compr\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.compr\u2082_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.lsmul /- _inst_7: semimodule \u219d\n -/\n#print linear_map.lsmul_apply /- _inst_7: semimodule \u219d\n -/\n#print tensor_product /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.add_comm_monoid /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.inhabited /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.induction_on /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.zero_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.add_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul_zero /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul_add /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul.aux /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul.aux_of /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.has_scalar /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul_zero /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul_add /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.smul_tmul' /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.semimodule /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul_smul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.mk /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.mk_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.ite_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul_ite /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.sum_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.tmul_sum /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.lift_aux /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift_aux_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift_aux.smul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift.tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift.tmul' /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.ext /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift.unique /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift_mk /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.lift_compr\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.lift_mk_compr\u2082 /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.mk_compr\u2082_inj /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.uncurry /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.uncurry_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lift.equiv /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lcurry /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.lcurry_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.curry /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.curry_apply /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.ext_threefold /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.ext_fourfold /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print tensor_product.lid /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.lid_tmul /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.lid_symm_apply /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.comm /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.comm_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.comm_symm_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print tensor_product.rid /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.rid_tmul /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.rid_symm_apply /- _inst_7: semimodule \u219d\n -/\n#print tensor_product.assoc /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.assoc_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.assoc_symm_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print tensor_product.map /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.map_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.map_comp /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_13: semimodule \u219d\n_inst_15: semimodule \u219d\n -/\n#print tensor_product.lift_comp_map /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_15: semimodule \u219d\n -/\n#print tensor_product.congr /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.congr_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print tensor_product.congr_symm_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.ltensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor_tmul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_hom /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor_hom /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.coe_ltensor_hom /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.coe_rtensor_hom /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_add /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor_add /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_zero /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor_zero /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_smul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.rtensor_smul /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n -/\n#print linear_map.ltensor_comp /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.rtensor_comp /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.ltensor_id /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.rtensor_id /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n -/\n#print linear_map.ltensor_comp_rtensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.rtensor_comp_ltensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n -/\n#print linear_map.map_comp_rtensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print linear_map.map_comp_ltensor /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print linear_map.rtensor_comp_map /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print linear_map.ltensor_comp_map /- _inst_7: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print tensor_product.add_comm_group /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_7: module \u219d\n_inst_8: module \u219d\n -/\n#print tensor_product.neg_tmul /- _inst_7: module \u219d\n_inst_8: module \u219d\n -/\n#print tensor_product.tmul_neg /- _inst_7: module \u219d\n_inst_8: module \u219d\n -/\n#print linear_map.ltensor_sub /- _inst_7: module \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.rtensor_sub /- _inst_7: module \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.ltensor_neg /- _inst_7: module \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n#print linear_map.rtensor_neg /- _inst_7: module \u219d\n_inst_8: module \u219d\n_inst_9: module \u219d\n -/\n\n-- logic\\basic.lean\n#print coe_coe /- _inst_1: has_coe \u219d has_lift_t\n_inst_2: has_coe_t \u219d has_lift_t\n -/\n#print coe_fn_coe_trans /- _inst_1: has_coe \u219d has_lift_t has_coe_to_fun\n_inst_2: has_coe_t_aux \u219d has_coe_to_fun\n -/\n#print coe_fn_coe_base /- _inst_1: has_coe \u219d has_lift_t has_coe_to_fun\n -/\n#print coe_sort_coe_trans /- _inst_1: has_coe \u219d has_lift_t has_coe_to_sort\n_inst_2: has_coe_t_aux \u219d has_coe_to_sort\n -/\n#print coe_sort_coe_base /- _inst_1: has_coe \u219d has_lift_t has_coe_to_sort\n -/\n\n-- logic\\function\\basic.lean\n#print function.injective.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print function.update /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_same /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_injective /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_noteq /- _inst_1: decidable_eq \u219d\n -/\n#print function.rel_update_iff /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_eq_iff /- _inst_1: decidable_eq \u219d\n -/\n#print function.eq_update_iff /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_eq_self /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_comp /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print function.apply_update /- _inst_3: decidable_eq \u219d\n -/\n#print function.comp_update /- _inst_1: decidable_eq \u219d\n -/\n#print function.update_comm /- _inst_3: decidable_eq \u219d\n -/\n#print function.update_idem /- _inst_3: decidable_eq \u219d\n -/\n\n-- measure_theory\\ae_eq_fun.lean\n#print measure_theory.ae_eq_fun.has_scalar /- _inst_10: semimodule \u219d\n -/\n#print measure_theory.ae_eq_fun.smul_mk /- _inst_10: semimodule \u219d\n -/\n#print measure_theory.ae_eq_fun.coe_fn_smul /- _inst_10: semimodule \u219d\n -/\n#print measure_theory.ae_eq_fun.smul_to_germ /- _inst_10: semimodule \u219d\n -/\n#print measure_theory.ae_eq_fun.semimodule /- _inst_10: semimodule \u219d\n -/\n#print measure_theory.ae_eq_fun.edist_smul /- _inst_8: normed_space \u219d\n -/\n\n-- measure_theory\\bochner_integration.lean\n#print measure_theory.simple_func.integral /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_eq_sum_filter /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_eq_sum_of_subset /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.simple_func.map_integral /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_congr /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_add /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_neg /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_sub /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_smul /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.norm_integral_le_integral_norm /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_add_measure /- _inst_6: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.has_scalar /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.coe_smul /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.semimodule /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.normed_space /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.of_simple_func_smul /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.smul_to_simple_func /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.coe_to_l1 /- _inst_11: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_eq_integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_congr /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_add /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_smul /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.norm_integral_le_norm /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_clm /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.norm_Integral_le_one /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_clm /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_eq /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.simple_func.integral_l1_eq_integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_zero /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_add /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_neg /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_sub /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.integral_smul /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.norm_Integral_le_one /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.norm_integral_le /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.continuous_integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_eq /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.l1.integral_eq_integral /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_undef /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_non_measurable /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_zero /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_zero' /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_add /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_add' /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_neg /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_neg' /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_sub /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_sub' /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_smul /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_congr_ae /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.l1.integral_of_fun_eq_integral /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.continuous_integral /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.norm_integral_le_lintegral_norm /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.ennnorm_integral_le_lintegral_ennnorm /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_eq_zero_of_ae /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.tendsto_integral_of_l1 /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.tendsto_integral_of_dominated_convergence /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.tendsto_integral_filter_of_dominated_convergence /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.norm_integral_le_integral_norm /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.norm_integral_le_of_norm_le /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_finset_sum /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.simple_func.integral_eq_integral /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_const /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.norm_integral_le_of_norm_le_const /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.tendsto_integral_approx_on_univ /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_add_measure /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_add_measure' /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_zero_measure /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_smul_measure /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_map /- _inst_4: normed_space \u219d\n -/\n#print measure_theory.integral_dirac /- _inst_4: normed_space \u219d\n -/\n\n-- measure_theory\\borel_space.lean\n#print is_measurable.nhds_within_is_measurably_generated /- _inst_3: opens_measurable_space \u219d filter.is_measurably_generated\n -/\n#print is_measurable_le' /- _inst_11: partial_order \u219d preorder\n -/\n#print measurable_of_continuous_on_compl_singleton /- _inst_11: t1_space \u219d measurable_singleton_class\n -/\n#print measurable.smul /- _inst_15: semimodule \u219d\n -/\n#print measurable.const_smul /- _inst_14: semimodule \u219d\n -/\n#print measurable_const_smul_iff /- _inst_12: division_ring \u219d group_with_zero semiring\n_inst_14: semimodule \u219d\n -/\n#print measurable.const_mul /- _inst_15: topological_semiring \u219d topological_semimodule\n -/\n#print measurable.mul_const /- _inst_15: topological_semiring \u219d has_continuous_mul\n -/\n#print measurable_inv' /- _inst_11: normed_field \u219d has_inv t1_space has_zero has_continuous_inv' topological_space opens_measurable_space\n -/\n#print measurable_supr /- _inst_11: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print measurable_infi /- _inst_11: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print measurable_cSup /- _inst_11: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order order_closed_topology\n -/\n#print continuous_linear_map.measurable /- _inst_4: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_comp /- _inst_4: normed_space \u219d\n_inst_8: normed_space \u219d\n -/\n#print measurable_smul_const /- _inst_7: normed_space \u219d\n -/\n\n-- measure_theory\\content.lean\n#print measure_theory.is_left_invariant_inner_content /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print measure_theory.outer_measure.is_left_invariant_of_content /- _inst_4: topological_group \u219d has_continuous_mul\n -/\n\n-- measure_theory\\group.lean\n#print measure_theory.measure.conj /- _inst_2: group \u219d has_inv\n -/\n\n-- measure_theory\\haar_measure.lean\n#print measure_theory.measure.haar.index /- _inst_1: group \u219d has_mul\n -/\n\n-- measure_theory\\integration.lean\n#print measure_theory.simple_func.range_map /- _inst_2: decidable_eq \u219d\n -/\n#print measure_theory.simple_func.semimodule /- _inst_4: semimodule \u219d\n -/\n#print measure_theory.simple_func.fin_meas_supp.mul /- _inst_4: monoid_with_zero \u219d mul_zero_class\n -/\n\n-- measure_theory\\interval_integral.lean\n#print interval_integrable /- _inst_1: linear_order \u219d preorder\n -/\n#print interval_integrable.smul /- _inst_7: normed_space \u219d\n -/\n#print interval_integral /- _inst_1: linear_order \u219d preorder\n_inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_zero /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_of_le /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_same /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_symm /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_of_ge /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_cases /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_non_measurable /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.norm_integral_eq_norm_integral_Ioc /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.norm_integral_le_integral_norm_Ioc /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.norm_integral_le_abs_integral_norm /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.norm_integral_le_of_norm_le_const_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.norm_integral_le_of_norm_le_const /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_add /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_neg /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_sub /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_smul /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_const' /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_const /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_smul_measure /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_comp_add_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_comp_mul_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_comp_neg /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_add_adjacent_intervals_cancel /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_add_adjacent_intervals /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_interval_sub_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_interval_add_interval_comm /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_interval_sub_interval_comm /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_interval_sub_interval_comm' /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_Iic_sub_Iic /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_const_of_cdf /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_eq_integral_of_support_subset /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.FTC_filter.nhds /- _inst_12: opens_measurable_space \u219d filter.is_measurably_generated\n_inst_13: order_topology \u219d filter.tendsto_Ixx_class\n -/\n#print interval_integral.FTC_filter.nhds_univ /- _inst_12: opens_measurable_space \u219d interval_integral.FTC_filter\n_inst_13: order_topology \u219d interval_integral.FTC_filter\n -/\n#print interval_integral.FTC_filter.nhds_left /- _inst_12: opens_measurable_space \u219d filter.is_measurably_generated\n_inst_13: order_topology \u219d filter.tendsto_Ixx_class filter.is_measurably_generated\n -/\n#print interval_integral.FTC_filter.nhds_right /- _inst_12: opens_measurable_space \u219d filter.is_measurably_generated\n_inst_13: order_topology \u219d filter.tendsto_Ixx_class filter.is_measurably_generated\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae' /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_le' /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge' /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_le /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_linear_is_o_of_tendsto_ae_of_ge /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae /- _inst_7: normed_space \u219d\n_inst_10: order_topology \u219d order_closed_topology\n_inst_11: borel_space \u219d opens_measurable_space\n -/\n#print interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.measure_integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_sub_linear_is_o_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_sub_integral_sub_linear_is_o_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_fderiv_at_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_fderiv_at /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_deriv_at_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_deriv_at_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_strict_deriv_at_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_fderiv_at_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_fderiv_at /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.fderiv_integral_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.fderiv_integral /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_at_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_at_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_integral_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_integral_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_at_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_at_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_integral_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_integral_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_fderiv_within_at_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_fderiv_within_at /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.fderiv_within_integral_of_tendsto_ae /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_within_at_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_within_at_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_within_integral_of_tendsto_ae_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_within_integral_right /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_within_at_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.integral_has_deriv_within_at_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_within_integral_of_tendsto_ae_left /- _inst_7: normed_space \u219d\n -/\n#print interval_integral.deriv_within_integral_left /- _inst_7: normed_space \u219d\n -/\n\n-- measure_theory\\l1_space.lean\n#print measure_theory.lintegral_edist_triangle /- _inst_2: normed_group \u219d emetric_space\n -/\n#print measure_theory.all_ae_of_real_F_le_bound /- _inst_2: normed_group \u219d has_norm\n -/\n#print measure_theory.has_finite_integral.smul /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.has_finite_integral_smul_iff /- _inst_5: normed_space \u219d\n -/\n#print measure_theory.integrable.smul /- _inst_8: normed_space \u219d\n -/\n#print measure_theory.integrable_smul_iff /- _inst_8: normed_space \u219d\n -/\n#print measure_theory.integrable_smul_const /- _inst_12: normed_space \u219d\n -/\n#print measure_theory.ae_eq_fun.integrable /- _inst_2: normed_group \u219d emetric_space has_zero\n -/\n#print measure_theory.ae_eq_fun.integrable.smul /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.edist_eq /- _inst_8: borel_space \u219d opens_measurable_space\n -/\n#print measure_theory.l1.dist_eq /- _inst_8: borel_space \u219d opens_measurable_space\n -/\n#print measure_theory.l1.has_scalar /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.coe_smul /- _inst_8: borel_space \u219d has_scalar opens_measurable_space\n_inst_10: normed_space \u219d has_scalar\n -/\n#print measure_theory.l1.semimodule /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.normed_space /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.of_fun_smul /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.l1.measurable /- _inst_8: borel_space \u219d opens_measurable_space\n -/\n#print measure_theory.l1.integrable /- _inst_8: borel_space \u219d opens_measurable_space\n -/\n#print measure_theory.l1.smul_to_fun /- _inst_10: normed_space \u219d\n -/\n\n-- measure_theory\\prod.lean\n#print measurable.integral_prod_right /- _inst_9: normed_space \u219d\n -/\n#print measurable.integral_prod_right' /- _inst_9: normed_space \u219d\n -/\n#print measurable.integral_prod_left /- _inst_9: normed_space \u219d\n -/\n#print measurable.integral_prod_left' /- _inst_9: normed_space \u219d\n -/\n#print measure_theory.integrable.integral_prod_left /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integrable.integral_prod_right /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_prod_swap /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_fn_integral_add /- _inst_10: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print measure_theory.integral_fn_integral_sub /- _inst_10: normed_space \u219d\n_inst_18: normed_space \u219d\n -/\n#print measure_theory.lintegral_fn_integral_sub /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_add /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_add' /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_sub /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_sub' /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.continuous_integral_integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_prod /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_prod_symm /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_symm /- _inst_10: normed_space \u219d\n -/\n#print measure_theory.integral_integral_swap /- _inst_10: normed_space \u219d\n -/\n\n-- measure_theory\\set_integral.lean\n#print measure_theory.integrable_add /- _inst_6: opens_measurable_space \u219d measurable_singleton_class\n -/\n#print measure_theory.integral_union /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.integral_empty /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.integral_univ /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.integral_add_compl /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.integral_indicator /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.set_integral_const /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.integral_indicator_const /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.set_integral_map /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.norm_set_integral_le_of_norm_le_const_ae /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.norm_set_integral_le_of_norm_le_const_ae' /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.norm_set_integral_le_of_norm_le_const_ae'' /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.norm_set_integral_le_of_norm_le_const /- _inst_7: normed_space \u219d\n -/\n#print measure_theory.norm_set_integral_le_of_norm_le_const' /- _inst_7: normed_space \u219d\n -/\n#print filter.tendsto.integral_sub_linear_is_o_ae /- _inst_4: normed_space \u219d\n -/\n#print continuous_at.integral_sub_linear_is_o_ae /- _inst_5: opens_measurable_space \u219d filter.is_measurably_generated\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_comp_l1_apply_le /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.integrable_comp /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.comp_l1 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.comp_l1_apply /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.integrable_comp_l1 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.measurable_comp_l1 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.integral_comp_l1 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.comp_l1\u2097 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_comp_l1_le /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.comp_l1L /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.norm_compl1L_le /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.continuous_integral_comp_l1 /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.integral_comp_comm /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print continuous_linear_map.integral_comp_l1_comm /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print fst_integral /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print snd_integral /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print integral_pair /- _inst_4: normed_space \u219d\n_inst_6: normed_space \u219d\n -/\n#print integral_smul_const /- _inst_4: normed_space \u219d\n -/\n\n-- measure_theory\\simple_func_dense.lean\n#print measure_theory.simple_func.integrable_approx_on /- _inst_7: borel_space \u219d opens_measurable_space\n -/\n\n-- number_theory\\arithmetic_function.lean\n#print nat.arithmetic_function.pmul_comm /- _inst_1: comm_monoid_with_zero \u219d comm_semigroup mul_zero_class\n -/\n#print nat.arithmetic_function.pmul_zeta /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print nat.arithmetic_function.zeta_pmul /- _inst_1: monoid_with_zero \u219d monoid mul_zero_class\n -/\n#print nat.arithmetic_function.is_multiplicative /- _inst_1: monoid_with_zero \u219d has_one has_zero has_mul\n -/\n#print nat.arithmetic_function.is_multiplicative_zeta /- _inst_1: semiring \u219d monoid_with_zero\n -/\n\n-- order\\basic.lean\n#print order.preimage.decidable /- H: decidable_rel \u219d\n -/\n#print monotone /- _inst_1: preorder \u219d has_le\n_inst_2: preorder \u219d has_le\n -/\n#print le_update_iff /- _inst_2: decidable_eq \u219d\n -/\n#print update_le_iff /- _inst_2: decidable_eq \u219d\n -/\n#print as_linear_order.linear_order /- _inst_2: is_total \u219d\n -/\n\n-- order\\bounded_lattice.lean\n#print top_sup_eq /- _inst_1: semilattice_sup_top \u219d order_top semilattice_sup\n -/\n#print sup_top_eq /- _inst_1: semilattice_sup_top \u219d order_top semilattice_sup\n -/\n#print bot_sup_eq /- _inst_1: semilattice_sup_bot \u219d semilattice_sup order_bot\n -/\n#print sup_bot_eq /- _inst_1: semilattice_sup_bot \u219d semilattice_sup order_bot\n -/\n#print sup_eq_bot_iff /- _inst_1: semilattice_sup_bot \u219d semilattice_sup order_bot\n -/\n#print top_inf_eq /- _inst_1: semilattice_inf_top \u219d semilattice_inf order_top\n -/\n#print inf_top_eq /- _inst_1: semilattice_inf_top \u219d semilattice_inf order_top\n -/\n#print inf_eq_top_iff /- _inst_1: semilattice_inf_top \u219d semilattice_inf order_top\n -/\n#print bot_inf_eq /- _inst_1: semilattice_inf_bot \u219d semilattice_inf order_bot\n -/\n#print inf_bot_eq /- _inst_1: semilattice_inf_bot \u219d semilattice_inf order_bot\n -/\n#print inf_eq_bot_iff_le_compl /- _inst_1: bounded_distrib_lattice \u219d semilattice_sup_bot distrib_lattice semilattice_inf_top\n -/\n#print eq_bot_of_bot_eq_top /- _inst_1: bounded_lattice \u219d order_top order_bot\n -/\n#print eq_top_of_bot_eq_top /- _inst_1: bounded_lattice \u219d order_top order_bot\n -/\n#print subsingleton_of_top_le_bot /- _inst_1: bounded_lattice \u219d order_top order_bot\n -/\n#print with_bot.coe_le /- _inst_1: partial_order \u219d preorder\n -/\n#print with_bot.coe_lt_coe /- _inst_1: partial_order \u219d preorder\n -/\n#print with_bot.decidable_le /- _inst_2: decidable_rel \u219d\n -/\n#print with_bot.decidable_lt /- _inst_2: decidable_rel \u219d\n -/\n#print with_top.coe_le_coe /- _inst_1: partial_order \u219d preorder\n -/\n#print with_top.coe_lt_coe /- _inst_1: partial_order \u219d preorder\n -/\n#print with_top.coe_lt_top /- _inst_1: partial_order \u219d preorder\n -/\n#print with_top.decidable_le /- _inst_2: decidable_rel \u219d\n -/\n#print with_top.decidable_lt /- _inst_2: decidable_rel \u219d\n -/\n#print disjoint /- _inst_1: semilattice_inf_bot \u219d has_bot has_le has_inf\n -/\n#print disjoint_sup_left /- _inst_1: bounded_distrib_lattice \u219d semilattice_sup_bot distrib_lattice semilattice_inf_bot\n -/\n#print disjoint_sup_right /- _inst_1: bounded_distrib_lattice \u219d semilattice_sup_bot distrib_lattice semilattice_inf_bot\n -/\n#print is_compl.le_left_iff /- _inst_1: bounded_distrib_lattice \u219d bounded_lattice distrib_lattice\n -/\n#print is_compl.sup_inf /- _inst_1: bounded_distrib_lattice \u219d bounded_lattice distrib_lattice\n -/\n\n-- order\\bounds.lean\n#print upper_bounds /- _inst_1: preorder \u219d has_le\n -/\n#print lower_bounds /- _inst_1: preorder \u219d has_le\n -/\n#print is_glb.exists_between_self_add /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_cancel_add_comm_monoid\n -/\n#print is_lub.exists_between_sub_self /- _inst_1: linear_ordered_add_comm_group \u219d linear_order ordered_add_comm_group\n -/\n\n-- order\\complete_boolean_algebra.lean\n#print compl_infi /- _inst_1: complete_boolean_algebra \u219d boolean_algebra complete_lattice\n -/\n\n-- order\\complete_lattice.lean\n#print Inf_lt_iff /- _inst_1: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print lt_Sup_iff /- _inst_1: complete_linear_order \u219d complete_lattice linear_order\n -/\n\n-- order\\conditionally_complete_lattice.lean\n#print exists_lt_of_lt_cSup /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print exists_lt_of_cInf_lt /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print cSup_intro' /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice\n -/\n#print with_top.is_glb_Inf /- _inst_1: conditionally_complete_linear_order_bot \u219d conditionally_complete_lattice order_bot\n -/\n#print with_bot.cSup_empty /- _inst_1: conditionally_complete_lattice \u219d has_Sup\n -/\n#print Sup_within_of_ord_connected /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice\n -/\n#print Inf_within_of_ord_connected /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice\n -/\n\n-- order\\filter\\at_top_bot.lean\n#print filter.prod_at_top_at_top_eq /- _inst_1: semilattice_sup \u219d preorder\n_inst_2: semilattice_sup \u219d preorder\n -/\n\n-- order\\filter\\basic.lean\n#print filter.eventually_eq.div /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print filter.eventually_eq.sub /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n\n-- order\\filter\\extr.lean\n#print is_min_filter /- _inst_1: preorder \u219d has_le\n -/\n#print is_max_filter /- _inst_1: preorder \u219d has_le\n -/\n\n-- order\\filter\\filter_product.lean\n#print filter.germ.le_def /- _inst_1: preorder \u219d has_le\n -/\n\n-- order\\filter\\germ.lean\n#print filter.germ.mul_action /- _inst_2: mul_action \u219d\n -/\n#print filter.germ.mul_action' /- _inst_2: mul_action \u219d\n -/\n#print filter.germ.distrib_mul_action /- _inst_3: distrib_mul_action \u219d\n -/\n#print filter.germ.distrib_mul_action' /- _inst_3: distrib_mul_action \u219d\n -/\n#print filter.germ.semimodule /- _inst_3: semimodule \u219d\n -/\n#print filter.germ.semimodule' /- _inst_3: semimodule \u219d\n -/\n\n-- order\\filter\\interval.lean\n#print filter.tendsto_Icc_pure_pure /- _inst_2: partial_order \u219d preorder\n -/\n#print filter.tendsto_Ico_pure_bot /- _inst_2: partial_order \u219d preorder\n -/\n#print filter.tendsto_Ioc_pure_bot /- _inst_2: partial_order \u219d preorder\n -/\n#print filter.tendsto_Ioo_pure_bot /- _inst_2: partial_order \u219d preorder\n -/\n\n-- order\\filter\\ultrafilter.lean\n#print filter.is_ultrafilter_hyperfilter /- _inst_1: infinite \u219d filter.ne_bot\n -/\n\n-- order\\fixed_points.lean\n#print lfp /- _inst_1: complete_lattice \u219d has_Inf has_le\n -/\n#print gfp /- _inst_1: complete_lattice \u219d has_Sup has_le\n -/\n#print fixed_points.sup_le_f_of_fixed_points /- _inst_1: complete_lattice \u219d semilattice_sup\n -/\n#print fixed_points.f_le_inf_of_fixed_points /- _inst_1: complete_lattice \u219d semilattice_inf\n -/\n\n-- order\\galois_connection.lean\n#print galois_connection /- _inst_1: preorder \u219d has_le\n_inst_2: preorder \u219d has_le\n -/\n#print galois_connection.u_l_u_eq_u /- _inst_2: partial_order \u219d preorder\n -/\n#print galois_connection.l_u_l_eq_l /- _inst_1: partial_order \u219d preorder\n -/\n#print galois_connection.l_unique /- _inst_1: partial_order \u219d preorder\n -/\n#print galois_connection.u_unique /- _inst_2: partial_order \u219d preorder\n -/\n#print galois_insertion.strict_mono_u /- _inst_2: partial_order \u219d preorder\n -/\n\n-- order\\lattice.lean\n#print sup_ind /- _inst_2: is_total \u219d\n -/\n#print sup_lt_iff /- _inst_2: is_total \u219d\n -/\n#print le_sup_iff /- _inst_2: is_total \u219d\n -/\n#print lt_sup_iff /- _inst_2: is_total \u219d\n -/\n#print inf_ind /- _inst_2: is_total \u219d\n -/\n#print lt_inf_iff /- _inst_2: is_total \u219d\n -/\n#print inf_le_iff /- _inst_2: is_total \u219d\n -/\n#print sup_inf_le /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\n -/\n#print le_inf_sup /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\n -/\n#print inf_sup_self /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\n -/\n#print sup_inf_self /- _inst_1: lattice \u219d semilattice_inf semilattice_sup\n -/\n#print monotone.map_sup /- _inst_2: is_total \u219d\n -/\n#print monotone.map_inf /- _inst_2: is_total \u219d\n -/\n\n-- order\\lexicographic.lean\n#print lex.decidable_eq /- _inst_1: decidable_eq \u219d\n_inst_2: decidable_eq \u219d\n -/\n#print lex_has_le /- _inst_1: preorder \u219d has_lt\n_inst_2: preorder \u219d has_le\n -/\n#print lex_has_lt /- _inst_1: preorder \u219d has_lt\n_inst_2: preorder \u219d has_lt\n -/\n#print dlex_has_le /- _inst_1: preorder \u219d has_lt\n -/\n#print dlex_has_lt /- _inst_1: preorder \u219d has_lt\n -/\n\n-- order\\liminf_limsup.lean\n#print filter.Limsup /- _inst_1: conditionally_complete_lattice \u219d has_Inf has_le\n -/\n#print filter.Liminf /- _inst_1: conditionally_complete_lattice \u219d has_Sup has_le\n -/\n#print filter.liminf_le_limsup /- _inst_1: complete_lattice \u219d conditionally_complete_lattice order_top order_bot\n -/\n\n-- order\\omega_complete_partial_order.lean\n#print complete_lattice.inf_continuous /- _inst_3: is_total \u219d\n -/\n#print complete_lattice.top_continuous /- _inst_2: complete_lattice \u219d omega_complete_partial_order order_top\n -/\n#print complete_lattice.bot_continuous /- _inst_2: complete_lattice \u219d omega_complete_partial_order order_bot\n -/\n#print omega_complete_partial_order.continuous_hom.\u03c9Sup_bind /- _inst_1: omega_complete_partial_order \u219d preorder\n -/\n\n-- order\\rel_classes.lean\n#print is_total.swap /- _inst_1: is_total \u219d\n -/\n#print is_preorder.swap /- _inst_1: is_preorder \u219d is_trans is_refl\n -/\n#print is_strict_order.swap /- _inst_1: is_strict_order \u219d is_irrefl is_trans\n -/\n#print is_partial_order.swap /- _inst_1: is_partial_order \u219d is_antisymm is_preorder\n -/\n#print is_total_preorder.swap /- _inst_1: is_total_preorder \u219d is_preorder\n -/\n#print is_linear_order.swap /- _inst_1: is_linear_order \u219d is_partial_order\n -/\n#print ge.is_refl /- _inst_1: preorder \u219d has_le is_refl\n -/\n#print ge.is_trans /- _inst_1: preorder \u219d has_le is_trans\n -/\n#print has_le.le.is_preorder /- _inst_1: preorder \u219d has_le is_trans is_refl\n -/\n#print ge.is_preorder /- _inst_1: preorder \u219d has_le is_trans is_refl\n -/\n#print gt.is_irrefl /- _inst_1: preorder \u219d has_lt is_irrefl\n -/\n#print gt.is_trans /- _inst_1: preorder \u219d has_lt is_trans\n -/\n#print gt.is_asymm /- _inst_1: preorder \u219d has_lt is_asymm\n -/\n#print has_lt.lt.is_antisymm /- _inst_1: preorder \u219d has_lt is_asymm\n -/\n#print gt.is_antisymm /- _inst_1: preorder \u219d has_lt is_asymm\n -/\n#print has_lt.lt.is_strict_order /- _inst_1: preorder \u219d has_lt is_irrefl is_trans\n -/\n#print gt.is_strict_order /- _inst_1: preorder \u219d has_lt is_irrefl is_trans\n -/\n#print preorder.is_total_preorder /- _inst_1: preorder \u219d has_le is_trans\n_inst_2: is_total \u219d\n -/\n#print ge.is_antisymm /- _inst_1: partial_order \u219d is_antisymm has_le\n -/\n#print has_le.le.is_partial_order /- _inst_1: partial_order \u219d is_antisymm has_le is_trans is_refl\n -/\n#print ge.is_partial_order /- _inst_1: partial_order \u219d is_antisymm has_le is_trans is_refl\n -/\n#print ge.is_total /- _inst_1: linear_order \u219d has_le\n -/\n#print linear_order.is_total_preorder /- _inst_1: linear_order \u219d has_le\n -/\n#print ge.is_total_preorder /- _inst_1: linear_order \u219d has_le\n -/\n#print has_le.le.is_linear_order /- _inst_1: linear_order \u219d is_antisymm has_le is_refl\n -/\n#print ge.is_linear_order /- _inst_1: linear_order \u219d is_antisymm has_le is_refl\n -/\n#print gt.is_trichotomous /- _inst_1: linear_order \u219d is_trichotomous has_lt\n -/\n#print order_dual.is_total_le /- _inst_2: is_total \u219d\n -/\n#print is_strict_total_order'.swap /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_strict_order\n -/\n#print has_lt.lt.is_strict_total_order' /- _inst_1: linear_order \u219d is_trichotomous has_lt is_irrefl is_trans\n -/\n#print is_order_connected_of_is_strict_total_order' /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_trans\n -/\n#print is_strict_total_order_of_is_strict_total_order' /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_asymm is_order_connected\n -/\n#print has_lt.lt.is_strict_total_order /- _inst_1: linear_order \u219d has_lt is_strict_total_order'\n -/\n#print has_lt.lt.is_order_connected /- _inst_1: linear_order \u219d has_lt is_strict_total_order'\n -/\n#print has_lt.lt.is_incomp_trans /- _inst_1: linear_order \u219d has_lt is_strict_weak_order\n -/\n#print has_lt.lt.is_strict_weak_order /- _inst_1: linear_order \u219d has_lt is_strict_weak_order\n -/\n#print is_extensional_of_is_strict_total_order' /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_irrefl\n -/\n#print is_well_order.is_strict_total_order /- _inst_1: is_well_order \u219d is_strict_total_order\n -/\n#print is_well_order.is_extensional /- _inst_1: is_well_order \u219d is_extensional\n -/\n#print is_well_order.is_trichotomous /- _inst_1: is_well_order \u219d is_trichotomous\n -/\n#print is_well_order.is_trans /- _inst_1: is_well_order \u219d is_trans\n -/\n#print is_well_order.is_irrefl /- _inst_1: is_well_order \u219d is_irrefl\n -/\n#print is_well_order.is_asymm /- _inst_1: is_well_order \u219d is_asymm\n -/\n#print is_well_order.linear_order /- _inst_1: is_well_order \u219d is_strict_total_order'\n -/\n\n-- order\\rel_iso.lean\n#print rel_embedding.is_total /- _inst_1: is_total \u219d\n -/\n#print rel_embedding.is_preorder /- _inst_1: is_preorder \u219d is_trans is_refl\n -/\n#print rel_embedding.is_partial_order /- _inst_1: is_partial_order \u219d is_antisymm is_preorder\n -/\n#print rel_embedding.is_linear_order /- _inst_1: is_linear_order \u219d is_partial_order\n -/\n#print rel_embedding.is_strict_order /- _inst_1: is_strict_order \u219d is_irrefl is_trans\n -/\n#print rel_embedding.is_strict_total_order' /- _inst_1: is_strict_total_order' \u219d is_trichotomous is_strict_order\n -/\n#print order_embedding.map_le_iff /- _inst_1: preorder \u219d has_le\n_inst_2: preorder \u219d has_le\n -/\n\n-- order\\semiconj_Sup.lean\n#print is_order_right_adjoint /- _inst_2: preorder \u219d has_le\n -/\n\n-- order\\zorn.lean\n#print zorn.chain.total /- _inst_1: preorder \u219d has_le is_refl\n -/\n\n-- representation_theory\\maschke.lean\n#print linear_map.conjugate /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print linear_map.conjugate_i /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print linear_map.sum_of_conjugates /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print linear_map.sum_of_conjugates_equivariant /- _inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print linear_map.equivariant_projection /- _inst_4: module \u219d\n_inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print linear_map.equivariant_projection_condition /- _inst_4: module \u219d\n_inst_6: is_scalar_tower \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print monoid_algebra.exists_left_inverse_of_injective /- _inst_7: is_scalar_tower \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print monoid_algebra.submodule.exists_is_compl /- _inst_7: is_scalar_tower \u219d\n -/\n\n-- ring_theory\\adjoin.lean\n#print algebra.adjoin_singleton_eq_range /- _inst_2: comm_semiring \u219d semiring\n -/\n#print algebra.adjoin_singleton_one /- _inst_2: comm_semiring \u219d semiring\n -/\n#print algebra.adjoin_union_coe_submodule /- _inst_2: comm_semiring \u219d comm_monoid semiring\n -/\n#print algebra.adjoin_int /- _inst_1: comm_ring \u219d is_subring ring\n -/\n#print algebra.mem_adjoin_iff /- _inst_1: comm_ring \u219d is_subring comm_semiring\n -/\n#print algebra.fg_trans /- _inst_1: comm_ring \u219d is_subring comm_semiring\n_inst_2: comm_ring \u219d is_subring comm_semiring\n -/\n\n-- ring_theory\\adjoin_root.lean\n#print adjoin_root.aeval_alg_hom_eq_zero /- _inst_2: comm_ring \u219d comm_semiring\n -/\n\n-- ring_theory\\algebra_tower.lean\n#print is_scalar_tower.algebra_map_smul /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d has_scalar\n_inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.smul_left_comm /- _inst_6: semimodule \u219d\n_inst_7: semimodule \u219d\n_inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.algebra_map_eq /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.algebra_map_apply /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.subalgebra' /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.algebra_comap_eq /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.to_alg_hom /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.to_alg_hom_apply /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.restrict_base /- _inst_10: is_scalar_tower \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.restrict_base_apply /- _inst_10: is_scalar_tower \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.polynomial /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.aeval_apply /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.invertible.algebra_tower /- _inst_10: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.algebra_map_aeval /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.aeval_eq_zero_of_aeval_algebra_map_eq_zero /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.aeval_eq_zero_of_aeval_algebra_map_eq_zero_field /- _inst_9: field \u219d division_ring comm_semiring\n_inst_15: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.linear_map /- _inst_9: comm_semiring \u219d semiring\n_inst_11: semimodule \u219d\n -/\n#print is_scalar_tower.int /- _inst_2: comm_ring \u219d ring comm_semiring\n_inst_3: comm_ring \u219d ring\n -/\n#print is_scalar_tower.rat /- _inst_1: field \u219d division_ring comm_semiring\n -/\n#print algebra.adjoin_algebra_map' /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d comm_semiring\n_inst_3: comm_ring \u219d semiring\n -/\n#print algebra.adjoin_algebra_map /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d comm_semiring\n_inst_3: comm_ring \u219d semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print subalgebra.res /- _inst_7: is_scalar_tower \u219d\n -/\n#print subalgebra.res_top /- _inst_7: is_scalar_tower \u219d\n -/\n#print subalgebra.mem_res /- _inst_7: is_scalar_tower \u219d\n -/\n#print subalgebra.res_inj /- _inst_7: is_scalar_tower \u219d\n -/\n#print subalgebra.of_under /- _inst_14: is_scalar_tower \u219d\n -/\n#print is_scalar_tower.range_under_adjoin /- _inst_7: is_scalar_tower \u219d\n -/\n#print algebra.fg_trans' /- _inst_7: is_scalar_tower \u219d\n -/\n#print submodule.smul_mem_span_smul_of_mem /- _inst_4: algebra \u219d\n_inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.smul_mem_span_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.smul_mem_span_smul' /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print submodule.span_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print linear_independent_smul /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_4: algebra \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print is_basis.smul /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_basis.smul_repr /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_basis.smul_repr_mk /- _inst_7: is_scalar_tower \u219d\n -/\n#print exists_subalgebra_of_fg /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d comm_semiring\n_inst_3: comm_ring \u219d semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print fg_of_fg_of_fg /- _inst_7: is_scalar_tower \u219d\n -/\n\n-- ring_theory\\algebraic.lean\n#print is_algebraic /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d semiring\n -/\n#print algebra.is_algebraic_trans /- _inst_7: is_scalar_tower \u219d\n -/\n#print algebra.is_algebraic_of_finite /- _inst_2: field \u219d comm_ring\n -/\n#print inv_eq_of_aeval_div_X_ne_zero /- _inst_3: field \u219d comm_semiring\n_inst_4: field \u219d add_group comm_group_with_zero semiring\n -/\n\n-- ring_theory\\coprime.lean\n#print is_coprime /- _inst_1: comm_semiring \u219d has_one has_add has_mul\n -/\n#print is_coprime.add_mul_left_left /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- ring_theory\\derivation.lean\n#print derivation.has_coe_to_fun /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.has_coe_to_linear_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.to_fun_eq_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.coe_fn_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.coe_injective /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.ext /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_add /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d is_add_monoid_hom\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d is_add_monoid_hom\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_smul /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.leibniz /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_one_eq_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_algebra_map /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.has_zero /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.inhabited /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.add_comm_monoid /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.add_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.derivation.Rsemimodule /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.smul_to_linear_map_coe /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.Rsmul_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.semimodule /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.smul_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.is_scalar_tower /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_neg /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_2: comm_ring \u219d ring comm_semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.map_sub /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_2: comm_ring \u219d ring comm_semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print derivation.add_comm_group /- _inst_7: is_scalar_tower \u219d\n -/\n#print derivation.sub_apply /- _inst_7: is_scalar_tower \u219d\n -/\n#print linear_map.comp_der /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: is_scalar_tower \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print linear_map.comp_der_apply /- _inst_5: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_8: semimodule \u219d\n_inst_9: semimodule \u219d\n_inst_10: is_scalar_tower \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n\n-- ring_theory\\discrete_valuation_ring.lean\n#print discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization /- _inst_1: integral_domain \u219d monoid has_zero\n -/\n#print discrete_valuation_ring.has_unit_mul_pow_irreducible_factorization.of_ufd_of_unique_irreducible /- _inst_2: unique_factorization_monoid \u219d wf_dvd_monoid\n -/\n#print discrete_valuation_ring.unit_mul_pow_congr_pow /- _inst_2: discrete_valuation_ring \u219d unique_factorization_monoid\n -/\n\n-- ring_theory\\eisenstein_criterion.lean\n#print polynomial.eisenstein_criterion_aux.map_eq_C_mul_X_pow_of_forall_coeff_mem /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print polynomial.eisenstein_criterion_aux.le_nat_degree_of_map_eq_mul_X_pow /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print polynomial.eisenstein_criterion_aux.eval_zero_mem_ideal_of_eq_mul_X_pow /- _inst_1: integral_domain \u219d comm_ring\n -/\n#print polynomial.eisenstein_criterion_aux.is_unit_of_nat_degree_eq_zero_of_forall_dvd_is_unit /- _inst_1: integral_domain \u219d ring comm_semiring\n -/\n\n-- ring_theory\\finiteness.lean\n#print module.finite /- _inst_1: comm_ring \u219d ring\n -/\n#print algebra.finite_type /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: comm_ring \u219d semiring\n -/\n#print module.finite.trans /- _inst_4: comm_ring \u219d semiring add_comm_group\n_inst_5: algebra \u219d\n_inst_10: algebra \u219d\n_inst_11: is_scalar_tower \u219d\n -/\n#print algebra.finite_type.trans /- _inst_11: is_scalar_tower \u219d\n -/\n#print ring_hom.finite /- _inst_2: comm_ring \u219d add_comm_group comm_semiring\n -/\n#print alg_hom.finite /- _inst_1: comm_ring \u219d comm_semiring\n -/\n#print alg_hom.finite_type /- _inst_1: comm_ring \u219d comm_semiring\n -/\n\n-- ring_theory\\fintype.lean\n#print card_units_lt /- _inst_1: semiring \u219d monoid_with_zero\n -/\n\n-- ring_theory\\fractional_ideal.lean\n#print ring.fractional_ideal.nontrivial /- _inst_3: integral_domain \u219d comm_ring\n_inst_4: field \u219d nontrivial comm_ring\n -/\n#print ring.fractional_ideal.fractional_div_of_nonzero /- _inst_4: field \u219d comm_ring no_zero_divisors\n -/\n#print ring.fractional_ideal.mul_generator_self_inv /- _inst_3: integral_domain \u219d comm_ring\n -/\n#print ring.fractional_ideal.is_principal /- _inst_6: is_principal_ideal_ring \u219d submodule.is_principal\n -/\n#print ring.fractional_ideal.is_noetherian_zero /- _inst_3: integral_domain \u219d comm_ring\n -/\n#print ring.fractional_ideal.is_noetherian_iff /- _inst_3: integral_domain \u219d comm_ring\n -/\n#print ring.fractional_ideal.is_noetherian_coe_to_fractional_ideal /- _inst_5: is_noetherian_ring \u219d is_noetherian\n -/\n\n-- ring_theory\\ideal\\basic.lean\n#print ideal /- _inst_1: comm_ring \u219d semiring\n -/\n#print ideal.span_singleton_eq_span_singleton /- _inst_2: integral_domain \u219d cancel_monoid_with_zero comm_ring\n -/\n#print ideal.bot_prime /- _inst_2: integral_domain \u219d nontrivial comm_ring no_zero_divisors\n -/\n#print ideal.span_singleton_lt_span_singleton /- _inst_2: integral_domain \u219d comm_cancel_monoid_with_zero comm_ring\n -/\n#print ideal.factors_decreasing /- _inst_2: integral_domain \u219d cancel_monoid_with_zero comm_ring\n -/\n#print ideal.eq_bot_or_top /- _inst_2: field \u219d group_with_zero comm_ring\n -/\n#print ring.exists_not_is_unit_of_not_is_field /- _inst_1: comm_ring \u219d comm_monoid ring\n -/\n#print mem_nonunits_iff /- _inst_1: comm_monoid \u219d monoid\n -/\n#print zero_mem_nonunits /- _inst_1: semiring \u219d monoid_with_zero\n -/\n#print field.local_ring /- _inst_1: field \u219d group_with_zero comm_ring\n -/\n\n-- ring_theory\\ideal\\operations.lean\n#print submodule.annihilator /- _inst_3: module \u219d\n -/\n#print submodule.colon /- _inst_3: module \u219d\n -/\n#print submodule.mem_annihilator /- _inst_3: module \u219d\n -/\n#print submodule.annihilator_eq_top_iff /- _inst_3: module \u219d\n -/\n#print submodule.mem_colon /- _inst_3: module \u219d\n -/\n#print submodule.mem_smul_span_singleton /- _inst_3: module \u219d\n -/\n#print submodule.smul_bot /- _inst_3: module \u219d\n -/\n#print submodule.top_smul /- _inst_3: module \u219d\n -/\n#print submodule.smul_sup /- _inst_3: module \u219d\n -/\n#print submodule.smul_assoc /- _inst_3: module \u219d\n -/\n#print submodule.span_smul_span /- _inst_3: module \u219d\n -/\n#print ideal.mul_eq_bot /- _inst_2: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print ring_hom.ker_is_prime /- _inst_2: integral_domain \u219d nontrivial comm_ring no_zero_divisors\n -/\n\n-- ring_theory\\ideal\\over.lean\n#print ideal.exists_coeff_ne_zero_mem_comap_of_root_mem /- _inst_2: integral_domain \u219d comm_ring no_zero_divisors\n -/\n#print ideal.exists_coeff_mem_comap_sdiff_comap_of_root_mem_sdiff /- _inst_2: integral_domain \u219d comm_ring\n -/\n#print ideal.mem_of_one_mem /- _inst_2: integral_domain \u219d comm_ring\n -/\n#print ideal.is_maximal_comap_of_is_integral_of_is_maximal /- _inst_2: integral_domain \u219d comm_ring ideal.is_prime\n -/\n#print ideal.exists_ideal_over_prime_of_is_integral /- _inst_2: integral_domain \u219d comm_ring\n -/\n\n-- ring_theory\\integral_closure.lean\n#print ring_hom.is_integral_elem /- _inst_1: comm_ring \u219d semiring\n_inst_2: ring \u219d semiring\n -/\n#print is_integral_alg_hom /- _inst_2: comm_ring \u219d ring\n_inst_3: comm_ring \u219d ring\n -/\n#print is_integral_of_is_scalar_tower /- _inst_3: comm_ring \u219d ring\n_inst_7: is_scalar_tower \u219d\n -/\n#print fg_adjoin_singleton_of_integral /- _inst_2: comm_ring \u219d is_subring ring comm_semiring\n -/\n#print is_integral_zero /- _inst_2: comm_ring \u219d ring\n -/\n#print is_integral_one /- _inst_2: comm_ring \u219d ring\n -/\n#print is_integral_trans_aux /- _inst_2: comm_ring \u219d comm_semiring\n -/\n#print is_integral_trans /- _inst_7: is_scalar_tower \u219d\n -/\n#print algebra.is_integral_trans /- _inst_7: is_scalar_tower \u219d\n -/\n#print is_integral_of_surjective /- _inst_2: comm_ring \u219d ring\n -/\n#print is_integral_tower_bot_of_is_integral /- _inst_2: comm_ring \u219d ring comm_semiring\n_inst_3: comm_ring \u219d ring comm_semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print is_integral_tower_bot_of_is_integral_field /- _inst_9: field \u219d comm_ring division_ring\n_inst_15: is_scalar_tower \u219d\n -/\n#print is_integral_tower_top_of_is_integral /- _inst_3: comm_ring \u219d ring\n_inst_7: is_scalar_tower \u219d\n -/\n#print is_field_of_is_integral_of_is_field /- _inst_8: integral_domain \u219d nontrivial comm_ring\n_inst_9: integral_domain \u219d nontrivial ring no_zero_divisors comm_semiring\n -/\n\n-- ring_theory\\integral_domain.lean\n#print card_nth_roots_subgroup_units /- _inst_2: group \u219d monoid\n -/\n#print field_of_integral_domain /- _inst_4: decidable_eq \u219d\n -/\n#print card_fiber_eq_of_mem_range /- _inst_5: decidable_eq \u219d\n -/\n\n-- ring_theory\\localization.lean\n#print localization_map.is_unit_comp /- _inst_3: comm_ring \u219d comm_monoid semiring\n -/\n#print localization_map.eq_of_eq /- _inst_3: comm_ring \u219d comm_monoid semiring\n -/\n#print localization_map.epic_of_localization_map /- _inst_3: comm_ring \u219d comm_monoid semiring\n -/\n#print localization.has_zero /- _inst_1: comm_ring \u219d has_zero comm_monoid\n -/\n#print localization_map.integer_normalization_eval\u2082_eq_zero /- _inst_4: comm_ring \u219d semiring\n -/\n#print localization_map.integer_normalization_aeval_eq_zero /- _inst_7: is_scalar_tower \u219d\n -/\n#print fraction_map.mk'_eq_div /- _inst_4: integral_domain \u219d nontrivial comm_ring\n_inst_6: field \u219d group_with_zero comm_ring\n -/\n#print fraction_map.is_unit_map_of_injective /- _inst_4: integral_domain \u219d nontrivial comm_ring\n_inst_7: field \u219d group_with_zero ring\n -/\n#print fraction_map.lift /- _inst_6: field \u219d comm_ring\n -/\n#print fraction_map.map /- _inst_6: field \u219d comm_ring\n_inst_7: field \u219d comm_ring\n -/\n#print fraction_map.field_equiv_of_ring_equiv /- _inst_6: field \u219d comm_ring\n_inst_7: field \u219d comm_ring\n -/\n#print fraction_map.comap_is_algebraic_iff /- _inst_7: field \u219d comm_ring\n_inst_10: is_scalar_tower \u219d\n -/\n#print fraction_map.exists_reduced_fraction /- _inst_6: field \u219d cancel_monoid_with_zero comm_ring\n -/\n#print integral_closure.fraction_map_of_finite_extension /- _inst_9: is_scalar_tower \u219d\n -/\n#print fraction_ring /- _inst_4: integral_domain \u219d monoid_with_zero comm_monoid\n -/\n#print fraction_ring.of /- _inst_4: integral_domain \u219d comm_ring\n -/\n\n-- ring_theory\\matrix_algebra.lean\n#print matrix.algebra /- _inst_5: decidable_eq \u219d\n -/\n#print algebra_map_matrix_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.to_fun_alg_hom /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.to_fun_alg_hom_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.inv_fun /- _inst_3: algebra \u219d\n_inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.inv_fun_zero /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.inv_fun_add /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.inv_fun_smul /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.inv_fun_algebra_map /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.right_inv /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.left_inv /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor.equiv /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor_apply /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor_apply_std_basis /- _inst_5: decidable_eq \u219d\n -/\n#print matrix_equiv_tensor_apply_symm /- _inst_5: decidable_eq \u219d\n -/\n\n-- ring_theory\\multiplicity.lean\n#print multiplicity /- _inst_1: comm_monoid \u219d has_dvd has_pow\n_inst_2: decidable_rel \u219d\n -/\n#print multiplicity.finite /- _inst_1: comm_monoid \u219d has_dvd has_pow\n -/\n#print multiplicity.finite_iff_dom /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.pow_dvd_of_le_multiplicity /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.pow_multiplicity_dvd /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.is_greatest /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.is_greatest' /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.unique /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.unique' /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.le_multiplicity_of_pow_dvd /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.pow_dvd_iff_le_multiplicity /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_lt_iff_neg_dvd /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.eq_some_iff /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.eq_top_iff /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.one_right /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.get_one_right /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_unit /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.one_left /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_eq_zero_of_not_dvd /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.eq_top_iff_not_finite /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_le_multiplicity_iff /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_le_multiplicity_of_dvd /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.dvd_of_multiplicity_pos /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.dvd_iff_multiplicity_pos /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.ne_zero_of_finite /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero comm_monoid\n -/\n#print multiplicity.zero /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero comm_monoid\n_inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_zero_eq_zero_of_ne_zero /- _inst_1: comm_monoid_with_zero \u219d monoid_with_zero comm_monoid\n_inst_2: decidable_rel \u219d\n -/\n#print multiplicity.min_le_multiplicity_add /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.neg /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_add_of_gt /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_sub_of_gt /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_add_eq_min /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_self /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid\n_inst_2: decidable_rel \u219d\n -/\n#print multiplicity.get_multiplicity_self /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n_inst_2: decidable_rel \u219d\n -/\n#print multiplicity.mul' /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.mul /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.finset.prod /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.pow' /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.pow /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_pow_self /- _inst_2: decidable_rel \u219d\n -/\n#print multiplicity.multiplicity_pow_self_of_prime /- _inst_2: decidable_rel \u219d\n -/\n\n-- ring_theory\\noetherian.lean\n#print submodule.fg /- _inst_3: semimodule \u219d\n -/\n#print submodule.fg_def /- _inst_3: semimodule \u219d\n -/\n#print submodule.exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul /- _inst_6: module \u219d\n -/\n#print submodule.fg_bot /- _inst_3: semimodule \u219d\n -/\n#print submodule.fg_sup /- _inst_3: semimodule \u219d\n -/\n#print submodule.fg_map /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print submodule.fg_top /- _inst_8: module \u219d\n -/\n#print submodule.fg_of_linear_equiv /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print submodule.fg_prod /- _inst_3: semimodule \u219d\n_inst_5: semimodule \u219d\n -/\n#print is_noetherian_submodule /- _inst_4: module \u219d\n -/\n#print is_noetherian_submodule_left /- _inst_4: module \u219d\n -/\n#print is_noetherian_submodule_right /- _inst_4: module \u219d\n -/\n#print is_noetherian_submodule' /- _inst_4: module \u219d\n -/\n#print is_noetherian_of_injective /- _inst_5: module \u219d is_noetherian\n -/\n#print finite_of_linear_independent /- _inst_1: comm_ring \u219d ring\n -/\n#print is_noetherian_ring /- _inst_1: ring \u219d semiring\n -/\n#print is_noetherian_of_submodule_of_noetherian /- _inst_3: module \u219d\n -/\n#print is_noetherian_of_quotient_of_noetherian /- _inst_3: module \u219d\n -/\n#print is_noetherian_of_fg_of_noetherian /- _inst_3: module \u219d\n_inst_4: is_noetherian_ring \u219d is_noetherian\n -/\n#print is_noetherian_of_fg_of_noetherian' /- _inst_3: module \u219d\n -/\n#print is_noetherian_span_of_finite /- _inst_3: module \u219d\n -/\n#print submodule.fg_mul /- _inst_1: comm_ring \u219d comm_semiring\n_inst_2: ring \u219d semiring\n -/\n#print exists_prime_spectrum_prod_le /- _inst_2: is_noetherian_ring \u219d is_noetherian\n -/\n#print exists_prime_spectrum_prod_le_and_ne_bot_of_domain /- _inst_4: is_noetherian_ring \u219d is_noetherian\n -/\n\n-- ring_theory\\non_zero_divisors.lean\n#print mul_mem_non_zero_divisors /- _inst_1: comm_ring \u219d monoid_with_zero comm_semigroup\n -/\n#print eq_zero_of_ne_zero_of_mul_right_eq_zero /- _inst_2: integral_domain \u219d has_zero no_zero_divisors has_mul\n -/\n#print eq_zero_of_ne_zero_of_mul_left_eq_zero /- _inst_2: integral_domain \u219d has_zero no_zero_divisors has_mul\n -/\n#print map_ne_zero_of_mem_non_zero_divisors /- _inst_1: comm_ring \u219d semiring\n_inst_4: ring \u219d semiring\n -/\n#print map_mem_non_zero_divisors /- _inst_2: integral_domain \u219d nontrivial comm_ring\n -/\n#print le_non_zero_divisors_of_domain /- _inst_2: integral_domain \u219d monoid_with_zero has_add no_zero_divisors\n -/\n\n-- ring_theory\\polynomial\\basic.lean\n#print polynomial.degree_le /- _inst_1: comm_ring \u219d ring\n -/\n#print polynomial.degree_lt /- _inst_1: comm_ring \u219d ring\n -/\n#print polynomial.eval\u2082_restriction /- _inst_2: ring \u219d semiring\n -/\n#print polynomial.linear_independent_powers_iff_eval\u2082 /- _inst_3: module \u219d algebra\n -/\n#print polynomial.disjoint_ker_aeval_of_coprime /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_3: module \u219d algebra\n -/\n#print polynomial.sup_aeval_range_eq_top_of_coprime /- _inst_3: module \u219d algebra\n -/\n#print polynomial.sup_ker_aeval_le_ker_aeval_mul /- _inst_3: module \u219d algebra\n -/\n#print polynomial.sup_ker_aeval_eq_ker_aeval_mul_of_coprime /- _inst_3: module \u219d algebra\n -/\n\n-- ring_theory\\polynomial\\cyclotomic.lean\n#print polynomial.prod_cyclotomic_eq_X_pow_sub_one /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- ring_theory\\polynomial\\gauss_lemma.lean\n#print polynomial.is_primitive.is_unit_iff_is_unit_map /- _inst_3: field \u219d integral_domain\n -/\n#print polynomial.is_primitive.dvd_of_fraction_map_dvd_fraction_map /- _inst_3: field \u219d integral_domain\n -/\n\n-- ring_theory\\polynomial\\rational_root.lean\n#print scale_roots_aeval_eq_zero_of_aeval_mk'_eq_zero /- _inst_1: integral_domain \u219d comm_ring\n -/\n\n-- ring_theory\\polynomial\\scale_roots.lean\n#print scale_roots_eval\u2082_eq_zero /- _inst_3: comm_ring \u219d comm_monoid semiring\n -/\n#print scale_roots_eval\u2082_eq_zero_of_eval\u2082_div_eq_zero /- _inst_1: integral_domain \u219d nontrivial comm_ring\n_inst_2: field \u219d group_with_zero comm_ring\n -/\n\n-- ring_theory\\polynomial_algebra.lean\n#print mat_poly_equiv /- _inst_4: decidable_eq \u219d\n -/\n#print mat_poly_equiv_coeff_apply_aux_1 /- _inst_4: decidable_eq \u219d\n -/\n#print mat_poly_equiv_coeff_apply_aux_2 /- _inst_4: decidable_eq \u219d\n -/\n#print mat_poly_equiv_coeff_apply /- _inst_4: decidable_eq \u219d\n -/\n#print mat_poly_equiv_symm_apply_coeff /- _inst_4: decidable_eq \u219d\n -/\n#print mat_poly_equiv_smul_one /- _inst_4: decidable_eq \u219d\n -/\n\n-- ring_theory\\power_basis.lean\n#print power_basis.finite_dimensional /- _inst_2: comm_ring \u219d ring\n -/\n#print power_basis.findim /- _inst_2: comm_ring \u219d ring\n -/\n#print power_basis.polynomial.mem_supported_range /- _inst_1: comm_ring \u219d semiring\n -/\n#print power_basis.mem_span_pow' /- _inst_2: comm_ring \u219d semiring\n -/\n#print power_basis.dim_ne_zero /- _inst_2: comm_ring \u219d ring\n -/\n#print power_basis.dim_le_nat_degree_of_root /- _inst_8: integral_domain \u219d comm_ring\n -/\n#print power_basis.nat_degree_lt_nat_degree /- _inst_1: comm_ring \u219d semiring\n -/\n#print is_integral_algebra_map_iff /- _inst_2: comm_ring \u219d ring comm_semiring\n_inst_3: comm_ring \u219d ring comm_semiring\n_inst_7: is_scalar_tower \u219d\n -/\n#print minimal_polynomial.eq_of_algebra_map_eq /- _inst_2: comm_ring \u219d ring comm_semiring\n_inst_3: comm_ring \u219d ring comm_semiring\n_inst_16: is_scalar_tower \u219d\n -/\n#print algebra.linear_independent_power_basis /- _inst_2: comm_ring \u219d ring\n -/\n\n-- ring_theory\\power_series.lean\n#print mv_power_series.has_one /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print mv_power_series.has_mul /- _inst_1: semiring \u219d add_comm_monoid has_mul\n -/\n#print mv_power_series.X /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print mv_power_series.X_pow_dvd_iff /- _inst_1: comm_semiring \u219d semiring\n -/\n#print mv_power_series.map.is_local_ring_hom /- _inst_1: comm_ring \u219d ring comm_semiring\n_inst_2: comm_ring \u219d semiring\n -/\n#print mv_power_series.inv /- _inst_1: field \u219d has_inv ring\n -/\n#print power_series.eq_zero_or_eq_zero_of_mul_eq_zero /- _inst_1: integral_domain \u219d ring no_zero_divisors\n -/\n#print polynomial.coe_to_power_series /- _inst_1: comm_semiring \u219d semiring\n -/\n\n-- ring_theory\\prime.lean\n#print mul_eq_mul_prime_prod /- _inst_1: integral_domain \u219d cancel_monoid_with_zero comm_monoid_with_zero\n_inst_2: decidable_eq \u219d\n -/\n\n-- ring_theory\\principal_ideal_domain.lean\n#print is_prime.to_maximal_ideal /- _inst_1: integral_domain \u219d cancel_monoid_with_zero comm_ring submodule.is_principal\n_inst_2: is_principal_ideal_ring \u219d submodule.is_principal\n -/\n#print principal_ideal_ring.is_noetherian_ring /- _inst_1: integral_domain \u219d comm_ring submodule.is_principal\n_inst_2: is_principal_ideal_ring \u219d submodule.is_principal\n -/\n#print principal_ideal_ring.is_maximal_of_irreducible /- _inst_1: integral_domain \u219d comm_ring submodule.is_principal\n_inst_2: is_principal_ideal_ring \u219d submodule.is_principal\n -/\n\n-- ring_theory\\roots_of_unity.lean\n#print is_primitive_root.gpow_eq_one /- _inst_3: comm_group \u219d comm_monoid\n -/\n#print is_primitive_root.gpow_eq_one_iff_dvd /- _inst_3: comm_group \u219d group comm_monoid\n -/\n#print is_primitive_root.inv /- _inst_3: comm_group \u219d group comm_monoid\n -/\n#print is_primitive_root.fpow_eq_one /- _inst_4: comm_group_with_zero \u219d comm_monoid\n -/\n#print is_primitive_root.fpow_eq_one_iff_dvd /- _inst_4: comm_group_with_zero \u219d group_with_zero comm_monoid\n -/\n#print is_primitive_root.inv' /- _inst_4: comm_group_with_zero \u219d group_with_zero comm_monoid\n -/\n#print is_primitive_root.mem_roots_of_unity /- _inst_5: integral_domain \u219d comm_monoid\n -/\n#print is_primitive_root.pow /- _inst_5: integral_domain \u219d comm_monoid\n -/\n\n-- ring_theory\\subring.lean\n#print subring.multiset_prod_mem /- _inst_4: comm_ring \u219d comm_monoid ring\n -/\n#print subring.prod_mem /- _inst_4: comm_ring \u219d comm_monoid ring\n -/\n#print ring_hom.restrict /- _inst_2: ring \u219d semiring\n -/\n#print ring_hom.eq_of_eq_on_set_top /- _inst_2: ring \u219d semiring\n -/\n#print subring.range_fst /- _inst_1: ring \u219d semiring\n_inst_2: ring \u219d semiring\n -/\n#print subring.range_snd /- _inst_1: ring \u219d semiring\n_inst_2: ring \u219d semiring\n -/\n\n-- ring_theory\\subsemiring.lean\n#print subsemiring.multiset_prod_mem /- _inst_4: comm_semiring \u219d comm_monoid semiring\n -/\n#print subsemiring.prod_mem /- _inst_4: comm_semiring \u219d comm_monoid semiring\n -/\n\n-- ring_theory\\tensor_product.lean\n#print algebra.tensor_product.mul_assoc' /- _inst_3: algebra \u219d\n_inst_5: algebra \u219d\n -/\n\n-- ring_theory\\unique_factorization_domain.lean\n#print is_noetherian_ring.wf_dvd_monoid /- _inst_2: is_noetherian_ring \u219d is_noetherian\n -/\n#print prime_factors_irreducible /- _inst_1: comm_cancel_monoid_with_zero \u219d cancel_monoid_with_zero comm_monoid_with_zero\n -/\n#print unique_factorization_monoid.factors /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_prod /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.prime_of_factor /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.irreducible_of_factor /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.normalize_factor /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_irreducible /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.exists_mem_factors_of_dvd /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_zero /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_one /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_mul /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.factors_pow /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.dvd_iff_factors_le_factors /- _inst_2: decidable_eq \u219d\n -/\n#print unique_factorization_monoid.le_multiplicity_iff_repeat_le_factors /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_rel \u219d\n -/\n#print unique_factorization_monoid.multiplicity_eq_count_factors /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_rel \u219d\n -/\n#print associates.factor_set /- _inst_2: comm_cancel_monoid_with_zero \u219d comm_monoid_with_zero\n -/\n#print associates.factor_set.sup_add_inf_eq_add /- _inst_2: decidable_eq \u219d\n -/\n#print associates.bcount /- _inst_2: decidable_eq \u219d\n -/\n#print associates.count /- _inst_2: decidable_eq \u219d\n -/\n#print associates.count_some /- _inst_2: decidable_eq \u219d\n -/\n#print associates.count_zero /- _inst_2: decidable_eq \u219d\n -/\n#print associates.count_reducible /- _inst_2: decidable_eq \u219d\n -/\n#print associates.factors' /- dec: decidable_eq \u219d\n -/\n#print associates.map_subtype_coe_factors' /- dec: decidable_eq \u219d\n -/\n#print associates.factors'_cong /- dec: decidable_eq \u219d\n -/\n#print associates.factors /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_0 /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_mk /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.prod_factors /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_prod /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.eq_of_factors_eq_factors /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_mul /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_mono /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_le /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.has_sup /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.has_inf /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.bounded_lattice /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.sup_mul_inf /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.dvd_of_mem_factors /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.dvd_of_mem_factors' /- dec: decidable_eq \u219d\n -/\n#print associates.mem_factors'_of_dvd /- dec: decidable_eq \u219d\n -/\n#print associates.mem_factors'_iff_dvd /- dec: decidable_eq \u219d\n -/\n#print associates.mem_factors_of_dvd /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.mem_factors_iff_dvd /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.exists_prime_dvd_of_not_inf_one /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.coprime_iff_inf_one /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_prime_pow /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.prime_pow_dvd_iff_le /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.le_of_count_ne_zero /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.count_mul /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.count_of_coprime /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.count_mul_of_coprime /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.count_mul_of_coprime' /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.dvd_count_of_dvd_count_mul /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.factors_one /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.pow_factors /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.count_pow /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.dvd_count_pow /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print associates.is_pow_of_dvd_count /- dec: decidable_eq \u219d\ndec': decidable_eq \u219d\n -/\n#print unique_factorization_monoid.to_gcd_monoid /- _inst_5: decidable_eq \u219d\n_inst_6: decidable_eq \u219d\n -/\n\n-- ring_theory\\valuation\\basic.lean\n#print valuation.zero_iff /- _inst_5: division_ring \u219d group_with_zero ring\n -/\n#print valuation.ne_zero_iff /- _inst_5: division_ring \u219d group_with_zero ring\n -/\n#print valuation.map_inv /- _inst_5: division_ring \u219d group_with_zero ring\n -/\n\n-- ring_theory\\valuation\\integers.lean\n#print valuation.integers.dvd_of_le /- _inst_1: field \u219d comm_ring division_ring\n -/\n\n-- ring_theory\\witt_vector\\defs.lean\n#print witt_vector.peval /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n\n-- ring_theory\\witt_vector\\verschiebung.lean\n#print witt_vector.verschiebung_fun /- _inst_1: comm_ring \u219d has_zero\n -/\n\n-- ring_theory\\witt_vector\\witt_polynomial.lean\n#print aeval_witt_polynomial /- _inst_3: comm_ring \u219d comm_semiring\n -/\n\n-- set_theory\\lists.lean\n#print lists.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n#print lists.subset.decidable /- _inst_1: decidable_eq \u219d\n -/\n#print lists.mem.decidable /- _inst_1: decidable_eq \u219d\n -/\n#print finsets.decidable_eq /- _inst_1: decidable_eq \u219d\n -/\n\n-- set_theory\\zfc.lean\n#print Set.map_definable_aux /- H: pSet.definable \u219d\n -/\n\n-- tactic\\abel.lean\n#print tactic.abel.term /- _inst_1: add_comm_monoid \u219d has_zero has_add\n -/\n#print tactic.abel.termg /- _inst_1: add_comm_group \u219d add_group\n -/\n#print tactic.abel.smul /- _inst_1: add_comm_monoid \u219d has_zero has_add\n -/\n#print tactic.abel.smulg /- _inst_1: add_comm_group \u219d add_group\n -/\n#print tactic.abel.unfold_sub /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n\n-- tactic\\cancel_denoms.lean\n#print cancel_factors.mul_subst /- _inst_1: comm_ring \u219d comm_semigroup\n -/\n#print cancel_factors.div_subst /- _inst_1: field \u219d comm_group_with_zero\n -/\n#print cancel_factors.cancel_factors_eq_div /- _inst_1: field \u219d group_with_zero comm_semigroup\n -/\n#print cancel_factors.add_subst /- _inst_1: ring \u219d distrib add_right_cancel_semigroup\n -/\n#print cancel_factors.cancel_factors_eq /- _inst_1: linear_ordered_field \u219d group_with_zero comm_semigroup ordered_semiring\n -/\n\n-- tactic\\interval_cases.lean\n#print tactic.interval_cases.set_elems /- _inst_1: decidable_eq \u219d\n -/\n#print tactic.interval_cases.mem_set_elems /- _inst_1: decidable_eq \u219d\n -/\n\n-- tactic\\linarith\\lemmas.lean\n#print linarith.eq_of_eq_of_eq /- _inst_1: ordered_semiring \u219d add_monoid\n -/\n#print linarith.le_of_eq_of_le /- _inst_1: ordered_semiring \u219d add_monoid has_le\n -/\n#print linarith.lt_of_eq_of_lt /- _inst_1: ordered_semiring \u219d add_monoid has_lt\n -/\n#print linarith.le_of_le_of_eq /- _inst_1: ordered_semiring \u219d add_monoid has_le\n -/\n#print linarith.lt_of_lt_of_eq /- _inst_1: ordered_semiring \u219d add_monoid has_lt\n -/\n#print linarith.mul_eq /- _inst_1: ordered_semiring \u219d has_lt mul_zero_class\n -/\n#print linarith.mul_zero_eq /- _inst_1: semiring \u219d mul_zero_class\n -/\n#print linarith.zero_mul_eq /- _inst_1: semiring \u219d mul_zero_class\n -/\n\n-- tactic\\monotonicity\\lemmas.lean\n#print lt_of_mul_lt_mul_neg_right /- _inst_1: linear_ordered_ring \u219d ring ordered_add_comm_group linear_ordered_semiring\n -/\n\n-- tactic\\norm_num.lean\n#print norm_num.zero_succ /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.one_succ /- _inst_1: semiring \u219d has_one has_add\n -/\n#print norm_num.bit0_succ /- _inst_1: semiring \u219d has_one has_add\n -/\n#print norm_num.bit1_succ /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.zero_adc /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.adc_zero /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.one_add /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.add_bit0_bit0 /- _inst_1: semiring \u219d add_comm_semigroup\n -/\n#print norm_num.add_bit0_bit1 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.add_bit1_bit0 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.add_bit1_bit1 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_one_one /- _inst_1: semiring \u219d has_one has_add\n -/\n#print norm_num.adc_bit0_one /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_one_bit0 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_bit1_one /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_one_bit1 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_bit0_bit0 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_bit1_bit0 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_bit0_bit1 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.adc_bit1_bit1 /- _inst_1: semiring \u219d has_one add_comm_semigroup\n -/\n#print norm_num.bit0_mul /- _inst_1: semiring \u219d distrib\n -/\n#print norm_num.mul_bit0' /- _inst_1: semiring \u219d distrib\n -/\n#print norm_num.mul_bit1_bit1 /- _inst_1: semiring \u219d monoid add_comm_semigroup distrib\n -/\n#print norm_num.ne_zero_of_pos /- _inst_1: ordered_add_comm_group \u219d preorder has_zero\n -/\n#print norm_num.clear_denom_div /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print norm_num.nonneg_pos /- _inst_1: ordered_cancel_add_comm_monoid \u219d preorder has_zero\n -/\n#print norm_num.nat_cast_zero /- _inst_1: semiring \u219d has_one has_zero has_add\n -/\n#print norm_num.nat_cast_one /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.nat_cast_bit0 /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.nat_cast_bit1 /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.int_cast_zero /- _inst_1: ring \u219d has_one has_zero has_neg has_add\n -/\n#print norm_num.int_cast_one /- _inst_1: ring \u219d add_monoid has_one has_neg\n -/\n#print norm_num.int_cast_neg /- _inst_1: ring \u219d has_one add_group\n -/\n#print norm_num.nat_cast_ne /- _inst_1: semiring \u219d add_monoid has_one\n -/\n#print norm_num.int_cast_ne /- _inst_1: ring \u219d has_one add_group\n -/\n#print norm_num.clear_denom_add /- _inst_1: division_ring \u219d cancel_monoid_with_zero distrib\n -/\n#print norm_num.clear_denom_simple_nat /- _inst_1: division_ring \u219d monoid_with_zero nontrivial\n -/\n#print norm_num.clear_denom_simple_div /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print norm_num.clear_denom_mul /- _inst_1: field \u219d cancel_monoid_with_zero comm_semigroup\n -/\n#print norm_num.inv_one /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print norm_num.inv_one_div /- _inst_1: division_ring \u219d group_with_zero\n -/\n#print norm_num.div_eq /- _inst_1: division_ring \u219d has_inv has_mul has_div\n -/\n#print norm_num.sub_pos /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n\n-- tactic\\ring.lean\n#print tactic.ring.horner /- _inst_1: comm_semiring \u219d has_add has_mul has_pow\n -/\n#print tactic.ring.horner_neg /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print tactic.ring.pow_succ /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring.subst_into_pow /- _inst_1: monoid \u219d has_pow\n -/\n#print tactic.ring.unfold_sub /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n#print tactic.ring.unfold_div /- _inst_1: division_ring \u219d has_inv has_mul has_div\n -/\n#print tactic.ring.add_neg_eq_sub /- _inst_1: add_group \u219d has_sub has_neg has_add\n -/\n\n-- tactic\\ring_exp.lean\n#print tactic.ring_exp.sum_congr /- _inst_1: comm_semiring \u219d is_commutative has_add\n -/\n#print tactic.ring_exp.prod_congr /- _inst_1: comm_semiring \u219d is_commutative has_mul\n -/\n#print tactic.ring_exp.exp_congr /- _inst_1: comm_semiring \u219d has_pow\n -/\n#print tactic.ring_exp.base_to_exp_pf /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.exp_to_prod_pf /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.prod_to_sum_pf /- _inst_1: comm_semiring \u219d add_monoid\n -/\n#print tactic.ring_exp.atom_to_sum_pf /- _inst_1: comm_semiring \u219d add_monoid monoid\n -/\n#print tactic.ring_exp.mul_coeff_pf_one_mul /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.mul_coeff_pf_mul_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.add_overlap_pf /- _inst_1: comm_semiring \u219d distrib\n -/\n#print tactic.ring_exp.add_overlap_pf_zero /- _inst_1: comm_semiring \u219d distrib mul_zero_class\n -/\n#print tactic.ring_exp.add_pf_z_sum /- _inst_1: comm_semiring \u219d add_monoid\n -/\n#print tactic.ring_exp.add_pf_sum_z /- _inst_1: comm_semiring \u219d add_monoid\n -/\n#print tactic.ring_exp.add_pf_sum_overlap /- _inst_1: comm_semiring \u219d is_commutative has_add is_associative\n -/\n#print tactic.ring_exp.add_pf_sum_overlap_zero /- _inst_1: comm_semiring \u219d add_monoid is_commutative\n -/\n#print tactic.ring_exp.add_pf_sum_lt /- _inst_1: comm_semiring \u219d is_commutative has_add is_associative\n -/\n#print tactic.ring_exp.add_pf_sum_gt /- _inst_1: comm_semiring \u219d is_commutative has_add is_associative\n -/\n#print tactic.ring_exp.mul_pf_c_c /- _inst_1: comm_semiring \u219d is_commutative has_mul\n -/\n#print tactic.ring_exp.mul_pf_c_prod /- _inst_1: comm_semiring \u219d is_commutative is_associative has_mul\n -/\n#print tactic.ring_exp.mul_pf_prod_c /- _inst_1: comm_semiring \u219d is_commutative is_associative has_mul\n -/\n#print tactic.ring_exp.mul_pp_pf_overlap /- _inst_1: comm_semiring \u219d monoid is_commutative\n -/\n#print tactic.ring_exp.mul_pp_pf_prod_lt /- _inst_1: comm_semiring \u219d is_commutative is_associative has_mul\n -/\n#print tactic.ring_exp.mul_pp_pf_prod_gt /- _inst_1: comm_semiring \u219d is_commutative is_associative has_mul\n -/\n#print tactic.ring_exp.mul_p_pf_zero /- _inst_1: comm_semiring \u219d mul_zero_class\n -/\n#print tactic.ring_exp.mul_p_pf_sum /- _inst_1: comm_semiring \u219d distrib\n -/\n#print tactic.ring_exp.mul_pf_zero /- _inst_1: comm_semiring \u219d mul_zero_class\n -/\n#print tactic.ring_exp.mul_pf_sum /- _inst_1: comm_semiring \u219d distrib\n -/\n#print tactic.ring_exp.pow_e_pf_exp /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_pp_pf_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_pf_c_c /- _inst_1: comm_semiring \u219d has_pow\n -/\n#print tactic.ring_exp.pow_pp_pf_c /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_pp_pf_prod /- _inst_1: comm_semiring \u219d comm_monoid\n -/\n#print tactic.ring_exp.pow_p_pf_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_p_pf_zero /- _inst_1: comm_semiring \u219d monoid_with_zero\n -/\n#print tactic.ring_exp.pow_p_pf_succ /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_p_pf_singleton /- _inst_1: comm_semiring \u219d add_monoid has_pow\n -/\n#print tactic.ring_exp.pow_p_pf_cons /- _inst_1: comm_semiring \u219d has_pow\n -/\n#print tactic.ring_exp.pow_pf_zero /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.pow_pf_sum /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.simple_pf_sum_zero /- _inst_1: comm_semiring \u219d add_monoid\n -/\n#print tactic.ring_exp.simple_pf_prod_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.simple_pf_var_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.simple_pf_exp_one /- _inst_1: comm_semiring \u219d monoid\n -/\n#print tactic.ring_exp.inverse_pf /- _inst_2: division_ring \u219d has_inv\n -/\n#print tactic.ring_exp.sub_pf /- _inst_2: ring \u219d has_sub has_neg has_add\n -/\n#print tactic.ring_exp.div_pf /- _inst_2: division_ring \u219d has_inv has_mul has_div\n -/\n\n-- tactic\\where.lean\n#print where.select_for_which /- _inst_1: decidable_eq \u219d\n -/\n\n-- testing\\slim_check\\functions.lean\n#print slim_check.total_function.apply /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.total_function.shrink /- _inst_3: decidable_eq \u219d\n -/\n#print slim_check.total_function.pi.sampleable_ext /- _inst_3: decidable_eq \u219d\n -/\n#print slim_check.injective_function.apply /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.list.apply_id /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.list.apply_id_cons /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.list.apply_id_zip_eq /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.apply_id_mem_iff /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.list.apply_id_eq_self /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.apply_id_injective /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.perm.slice /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.shrink_perm /- _inst_1: decidable_eq \u219d\n -/\n#print slim_check.injective_function.shrink /- _inst_2: decidable_eq \u219d\n -/\n#print slim_check.injective_function.injective /- _inst_1: decidable_eq \u219d\n -/\n\n-- topology\\algebra\\affine.lean\n#print affine_map.continuous_iff /- _inst_3: semimodule \u219d\n_inst_6: semimodule \u219d\n_inst_8: topological_add_group \u219d has_continuous_add has_continuous_sub\n -/\n#print affine_map.line_map_continuous /- _inst_6: semimodule \u219d\n -/\n\n-- topology\\algebra\\continuous_functions.lean\n#print continuous_subring /- _inst_4: topological_ring \u219d topological_add_group has_continuous_mul\n -/\n#print continuous_has_scalar /- _inst_6: semimodule \u219d\n -/\n#print continuous_semimodule /- _inst_7: semimodule \u219d\n -/\n#print continuous_map_has_scalar /- _inst_6: semimodule \u219d\n -/\n#print continuous_map_semimodule /- _inst_7: semimodule \u219d\n -/\n#print continuous_has_scalar' /- _inst_6: semimodule \u219d\n -/\n#print continuous_map_has_scalar' /- _inst_6: semimodule \u219d\n -/\n#print continuous_map_module' /- _inst_8: semimodule \u219d\n -/\n\n-- topology\\algebra\\floor_ring.lean\n#print continuous_on_fract /- _inst_4: topological_add_group \u219d has_continuous_sub\n -/\n#print tendsto_fract_left' /- _inst_5: topological_add_group \u219d has_continuous_sub\n -/\n#print tendsto_fract_right' /- _inst_5: topological_add_group \u219d has_continuous_sub\n -/\n\n-- topology\\algebra\\group.lean\n#print nhds_translation_mul_inv /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print nhds_translation_add_neg /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print quotient_add_group.is_open_map_coe /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print quotient_group.is_open_map_coe /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print is_open.add_left /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print is_open.mul_left /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print is_open.add_right /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print is_open.mul_right /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print topological_group.t1_space /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print compact_open_separated_mul /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print compact_open_separated_add /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print compact_covered_by_mul_left_translates /- _inst_3: topological_group \u219d has_continuous_mul\n -/\n#print compact_covered_by_add_left_translates /- _inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print nhds_add /- _inst_2: add_comm_group \u219d has_continuous_add add_comm_semigroup add_group\n -/\n#print nhds_mul /- _inst_2: comm_group \u219d comm_semigroup group has_continuous_mul\n -/\n\n-- topology\\algebra\\group_with_zero.lean\n#print filter.tendsto.div_const /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print continuous_at.div_const /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print continuous_on.div_const /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print continuous.div_const /- _inst_1: group_with_zero \u219d has_inv has_mul has_div\n -/\n#print filter.tendsto.div /- _inst_1: group_with_zero \u219d has_inv has_zero has_mul has_div\n -/\n#print continuous.div /- _inst_1: group_with_zero \u219d has_inv has_zero has_mul has_div\n -/\n\n-- topology\\algebra\\infinite_sum.lean\n#print tsum_add_tsum_compl /- _inst_1: add_comm_group \u219d has_continuous_add add_comm_monoid add_group\n_inst_3: topological_add_group \u219d has_continuous_add\n -/\n#print has_sum.mul_left /- _inst_3: topological_semiring \u219d has_continuous_mul\n -/\n#print has_sum.mul_right /- _inst_3: topological_semiring \u219d has_continuous_mul\n -/\n#print has_sum_mul_left_iff /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print has_sum_mul_right_iff /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print summable_mul_left_iff /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print summable_mul_right_iff /- _inst_1: division_ring \u219d group_with_zero semiring\n -/\n#print summable_iff_cauchy_seq_finset /- _inst_1: add_comm_group \u219d add_comm_monoid\n -/\n\n-- topology\\algebra\\module.lean\n#print continuous_smul /- _inst_5: semimodule \u219d\n -/\n#print continuous.smul /- _inst_5: semimodule \u219d\n -/\n#print tendsto_smul /- _inst_5: semimodule \u219d\n -/\n#print filter.tendsto.smul /- _inst_5: semimodule \u219d\n -/\n#print topological_semiring.to_semimodule /- _inst_3: topological_semiring \u219d has_continuous_mul\n -/\n#print topological_vector_space /- _inst_1: field \u219d ring\n -/\n#print continuous_linear_map.linear_map.has_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.to_fun /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_mk /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_mk' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.continuous /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_injective /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.injective_coe_fn /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ext /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ext_iff /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.map_zero /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.map_add /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.map_smul /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.map_sum /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.eq_on_closure_span /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ext_on /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.has_zero /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.inhabited /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.default_def /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.zero_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_zero /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_zero' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.unique_of_left /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.unique_of_right /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.id /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.has_one /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.one_def /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.id_apply /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.coe_id /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.coe_id' /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.one_apply /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.has_add /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.add_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_add /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_add' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.add_comm_monoid /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.sum_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.comp /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.coe_comp /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.coe_comp' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.comp_id /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.id_comp /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.comp_zero /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.zero_comp /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.comp_add /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.add_comp /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.comp_assoc /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_map.has_mul /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.mul_def /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.coe_mul /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.mul_apply /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.coe_prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.prod_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.ker /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ker_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.mem_ker /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.is_closed_ker /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.apply_ker /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.is_complete_ker /- _inst_11: semimodule \u219d\n_inst_17: semimodule \u219d\n -/\n#print continuous_linear_map.complete_space_ker /- _inst_11: semimodule \u219d\n_inst_17: semimodule \u219d\n -/\n#print continuous_linear_map.ker_prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.range /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.range_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.mem_range /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.range_prod_le /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.cod_restrict /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_cod_restrict /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_cod_restrict_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ker_cod_restrict /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.subtype_val /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.coe_subtype_val /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.subtype_val_apply /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.fst /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.snd /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_fst /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_fst' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_snd /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_snd' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.fst_prod_snd /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.prod_map /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_map.coe_prod_map /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_map.coe_prod_map' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_map.coprod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.coe_coprod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.coprod_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.smul_right /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.smul_right_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.smul_right_one_one /- _inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.smul_right_one_eq_iff /- _inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.smul_right_comp /- _inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.pi /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_map.pi_apply /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_map.pi_eq_zero /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_map.pi_zero /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_map.pi_comp /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_map.proj_pi /- _inst_7: semimodule \u219d\n -/\n#print continuous_linear_map.map_neg /- _inst_1: ring \u219d semiring\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.map_sub /- _inst_1: ring \u219d semiring\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.sub_apply' /- _inst_1: ring \u219d semiring\n_inst_3: add_comm_group \u219d add_comm_monoid\n_inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.range_prod_eq /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_map.has_neg /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.neg_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_neg /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_neg' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.add_comm_group /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.sub_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_sub /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_sub' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.ring /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_map.proj_ker_of_right_inverse /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.coe_proj_ker_of_right_inverse_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.proj_ker_of_right_inverse_apply_idem /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.proj_ker_of_right_inverse_comp_inv /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_map.smul_comp /- _inst_1: comm_ring \u219d ring\n_inst_12: topological_module \u219d has_scalar\n -/\n#print continuous_linear_map.smul_apply /- _inst_1: comm_ring \u219d ring\n_inst_13: topological_module \u219d has_scalar\n -/\n#print continuous_linear_map.coe_apply /- _inst_1: comm_ring \u219d ring\n_inst_13: topological_module \u219d has_scalar\n -/\n#print continuous_linear_map.coe_apply' /- _inst_1: comm_ring \u219d ring\n_inst_13: topological_module \u219d has_scalar\n -/\n#print continuous_linear_equiv.to_continuous_linear_map /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.continuous_linear_map.has_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.has_coe_to_fun /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_def_rev /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_to_linear_equiv /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.ext /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.to_homeomorph /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_zero /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_add /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_smul /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_eq_zero_iff /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.continuous /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.continuous_on /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.continuous_at /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.continuous_within_at /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.comp_continuous_on_iff /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.comp_continuous_iff /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.ext\u2081 /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.refl /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_refl /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_refl' /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_to_linear_equiv /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.trans /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.trans_to_linear_equiv /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.prod_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.bijective /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.injective /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.surjective /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.trans_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.apply_symm_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_apply_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_trans_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.comp_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_comp_coe_symm /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.coe_symm_comp_coe /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_comp_self /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.self_comp_symm /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_comp_self' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.self_comp_symm' /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_symm /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.refl_symm /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_symm_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_apply_eq /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.eq_symm_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.equiv_of_inverse /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.equiv_of_inverse_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.symm_equiv_of_inverse /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n -/\n#print continuous_linear_equiv.automorphism_group /- _inst_10: semimodule \u219d\n -/\n#print continuous_linear_equiv.skew_prod /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.skew_prod_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.skew_prod_symm_apply /- _inst_10: semimodule \u219d\n_inst_11: semimodule \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_sub /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_equiv.map_neg /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_equiv.of_unit /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_equiv.to_unit /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_equiv.units_equiv /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_equiv.units_equiv_apply /- _inst_4: semimodule \u219d\n -/\n#print continuous_linear_equiv.equiv_of_right_inverse /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_equiv.fst_equiv_of_right_inverse /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_equiv.snd_equiv_of_right_inverse /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_equiv.equiv_of_right_inverse_symm_apply /- _inst_4: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_map.inverse /- _inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_map.inverse_equiv /- _inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print continuous_linear_map.inverse_non_equiv /- _inst_5: semimodule \u219d\n_inst_7: semimodule \u219d\n -/\n#print submodule.closed_complemented /- _inst_4: module \u219d\n -/\n#print submodule.closed_complemented.has_closed_complement /- _inst_4: module \u219d\n -/\n#print submodule.closed_complemented.is_closed /- _inst_4: module \u219d\n -/\n#print submodule.closed_complemented_bot /- _inst_4: module \u219d\n -/\n#print submodule.closed_complemented_top /- _inst_4: module \u219d\n -/\n#print continuous_linear_map.closed_complemented_ker_of_right_inverse /- _inst_6: module \u219d\n -/\n\n-- topology\\algebra\\monoid.lean\n#print submonoid.mem_nhds_one /- _inst_2: comm_monoid \u219d monoid\n -/\n#print add_submonoid.mem_nhds_zero /- _inst_2: add_comm_monoid \u219d add_monoid\n -/\n\n-- topology\\algebra\\multilinear.lean\n#print continuous_multilinear_map.has_coe_to_fun /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.coe_continuous /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.coe_coe /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.to_multilinear_map_inj /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.ext /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_add /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_smul /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_coord_zero /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_zero /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.has_zero /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.inhabited /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.zero_apply /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.has_add /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.add_apply /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.add_comm_monoid /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.sum_apply /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.to_continuous_linear_map /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.prod /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_multilinear_map.prod_apply /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n_inst_13: semimodule \u219d\n -/\n#print continuous_multilinear_map.comp_continuous_linear_map /- _inst_1: decidable_eq \u219d\n_inst_14: semimodule \u219d\n -/\n#print continuous_multilinear_map.comp_continuous_linear_map_apply /- _inst_1: decidable_eq \u219d\n_inst_14: semimodule \u219d\n -/\n#print continuous_multilinear_map.cons_add /- _inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.cons_smul /- _inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_piecewise_add /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_add_univ /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_sum_finset /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_sum /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n -/\n#print continuous_multilinear_map.restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n_inst_24: semimodule \u219d\n_inst_26: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.coe_restrict_scalars /- _inst_1: decidable_eq \u219d\n_inst_12: semimodule \u219d\n_inst_24: semimodule \u219d\n_inst_26: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.map_sub /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.has_neg /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.neg_apply /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.add_comm_group /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.sub_apply /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_piecewise_smul /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.map_smul_univ /- _inst_1: decidable_eq \u219d\n_inst_6: semimodule \u219d\n -/\n#print continuous_multilinear_map.has_scalar /- _inst_1: decidable_eq \u219d\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.smul_apply /- _inst_1: decidable_eq \u219d\n_inst_11: algebra \u219d has_scalar\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d has_scalar\n_inst_17: topological_semimodule \u219d has_scalar\n -/\n#print continuous_multilinear_map.is_scalar_tower /- _inst_1: decidable_eq \u219d\n_inst_11: algebra \u219d has_scalar\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d has_scalar\n_inst_17: topological_semimodule \u219d has_scalar\n_inst_20: algebra \u219d has_scalar\n_inst_21: semimodule \u219d\n_inst_22: is_scalar_tower \u219d has_scalar\n_inst_23: is_scalar_tower \u219d\n_inst_25: topological_semimodule \u219d has_scalar\n -/\n#print continuous_multilinear_map.semimodule /- _inst_1: decidable_eq \u219d\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_multilinear_map.to_multilinear_map_linear /- _inst_1: decidable_eq \u219d\n_inst_13: semimodule \u219d\n_inst_14: semimodule \u219d\n_inst_15: is_scalar_tower \u219d\n -/\n#print continuous_linear_map.comp_continuous_multilinear_map /- _inst_1: decidable_eq \u219d\n -/\n#print continuous_linear_map.comp_continuous_multilinear_map_coe /- _inst_1: decidable_eq \u219d\n -/\n\n-- topology\\algebra\\open_subgroup.lean\n#print submodule.is_open_mono /- _inst_1: comm_ring \u219d ring\n_inst_4: topological_add_group \u219d has_continuous_add\n -/\n#print ideal.is_open_of_open_subideal /- _inst_3: topological_ring \u219d topological_add_group\n -/\n\n-- topology\\algebra\\ordered.lean\n#print preorder.topology /- _inst_1: preorder \u219d has_lt\n -/\n#print order_dual.order_topology /- _inst_2: partial_order \u219d preorder\n -/\n#print is_open_iff_generate_intervals /- _inst_2: partial_order \u219d preorder\n -/\n#print nhds_eq_order /- _inst_2: partial_order \u219d preorder\n -/\n#print tendsto_Ico_class_nhds /- _inst_2: partial_order \u219d preorder\nt: order_topology \u219d filter.tendsto_Ixx_class\n -/\n#print tendsto_Ioc_class_nhds /- _inst_2: partial_order \u219d preorder\nt: order_topology \u219d filter.tendsto_Ixx_class\n -/\n#print tendsto_Ioo_class_nhds /- _inst_2: partial_order \u219d preorder\nt: order_topology \u219d filter.tendsto_Ixx_class\n -/\n#print tendsto_Ixx_nhds_within /- _inst_2: partial_order \u219d preorder\n -/\n#print induced_order_topology' /- _inst_1: partial_order \u219d preorder\n -/\n#print order_topology.t2_space /- _inst_2: linear_order \u219d t2_space preorder\n_inst_3: order_topology \u219d t2_space\n -/\n#print Iio_mem_nhds /- _inst_3: order_topology \u219d order_closed_topology\n -/\n#print Ioi_mem_nhds /- _inst_3: order_topology \u219d order_closed_topology\n -/\n#print Ioo_mem_nhds /- _inst_3: order_topology \u219d order_closed_topology\n -/\n#print continuous_right_of_strict_mono_surjective /- _inst_3: order_topology \u219d order_closed_topology\n -/\n#print tendsto_at_top_add_tendsto_left /- _inst_2: linear_ordered_ring \u219d no_bot_order ordered_add_comm_group\n -/\n#print tendsto_at_bot_add_tendsto_left /- _inst_2: linear_ordered_ring \u219d ordered_add_comm_group no_top_order\n -/\n#print tendsto_at_top_mul_left /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print tendsto_at_top_mul_right /- _inst_1: linear_ordered_semiring \u219d linear_order ordered_semiring\n -/\n#print neg_preimage_closure /- _inst_2: ordered_add_comm_group \u219d add_group\n -/\n#print is_compact.bdd_above /- _inst_9: order_topology \u219d order_closed_topology\n -/\n#print Sup_mem_closure /- _inst_9: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print Inf_mem_closure /- _inst_9: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print is_closed.Sup_mem /- _inst_9: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print is_closed.Inf_mem /- _inst_9: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print map_Sup_of_continuous_at_of_monotone' /- _inst_1: complete_linear_order \u219d complete_lattice linear_order\n_inst_4: complete_linear_order \u219d complete_lattice linear_order\n -/\n#print cSup_mem_closure /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print cInf_mem_closure /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print is_closed.cSup_mem /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print is_closed.cInf_mem /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print map_cSup_of_continuous_at_of_monotone /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n_inst_4: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print is_connected.Ioo_cInf_cSup_subset /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order order_closed_topology\n_inst_3: order_topology \u219d order_closed_topology\n -/\n#print is_preconnected.Ioi_cInf_subset /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order order_closed_topology\n_inst_3: order_topology \u219d order_closed_topology\n -/\n#print intermediate_value_Icc /- _inst_4: conditionally_complete_linear_order \u219d linear_order order_closed_topology\n_inst_6: order_topology \u219d order_closed_topology\n -/\n#print intermediate_value_Icc' /- _inst_4: conditionally_complete_linear_order \u219d linear_order order_closed_topology\n_inst_6: order_topology \u219d order_closed_topology\n -/\n#print is_compact.is_glb_Inf /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order order_closed_topology\n_inst_3: order_topology \u219d order_closed_topology\n -/\n#print Limsup_nhds /- _inst_1: conditionally_complete_linear_order \u219d conditionally_complete_lattice linear_order\n -/\n#print tendsto_of_liminf_eq_limsup /- _inst_1: complete_linear_order \u219d order_top conditionally_complete_linear_order order_bot\n -/\n#print tendsto_at_top_supr /- _inst_3: complete_linear_order \u219d order_top conditionally_complete_linear_order\n -/\n#print tendsto_at_top_infi /- _inst_3: complete_linear_order \u219d conditionally_complete_linear_order order_bot\n -/\n#print supr_eq_of_tendsto /- _inst_4: nonempty \u219d filter.ne_bot\n_inst_5: semilattice_sup \u219d preorder filter.ne_bot\n -/\n#print infi_eq_of_tendsto /- _inst_4: nonempty \u219d filter.ne_bot\n_inst_5: semilattice_sup \u219d preorder filter.ne_bot\n -/\n#print continuous_within_at_Iio_iff_Iic /- _inst_2: linear_order \u219d partial_order\n -/\n\n-- topology\\algebra\\polynomial.lean\n#print polynomial.tendsto_infinity /- _inst_1: comm_ring \u219d ring comm_semiring\n -/\n#print polynomial.continuous_eval /- _inst_1: comm_semiring \u219d has_continuous_add semiring has_continuous_mul\n_inst_3: topological_semiring \u219d has_continuous_add has_continuous_mul\n -/\n\n-- topology\\algebra\\ring.lean\n#print topological_ring.to_topological_semiring /- t: topological_ring \u219d has_continuous_add has_continuous_mul\n -/\n#print mul_left_continuous /- _inst_3: topological_ring \u219d has_continuous_mul\n -/\n#print mul_right_continuous /- _inst_3: topological_ring \u219d has_continuous_mul\n -/\n#print quotient_ring.is_open_map_coe /- _inst_3: topological_ring \u219d has_continuous_add\n -/\n\n-- topology\\algebra\\uniform_group.lean\n#print add_comm_group.is_Z_bilin.comp_hom /- _inst_6: is_add_group_hom \u219d is_add_hom\n -/\n#print is_Z_bilin.tendsto_zero_left /- _inst_5: uniform_space \u219d topological_space\n -/\n#print is_Z_bilin.tendsto_zero_right /- _inst_5: uniform_space \u219d topological_space\n -/\n#print tendsto_sub_comap_self /- _inst_2: add_comm_group \u219d add_group has_continuous_sub\n_inst_3: topological_add_group \u219d has_continuous_sub\n_inst_5: add_comm_group \u219d add_group\n -/\n\n-- topology\\algebra\\uniform_ring.lean\n#print uniform_space.completion.has_one /- _inst_1: ring \u219d has_one\n -/\n#print uniform_space.completion.has_mul /- _inst_1: ring \u219d has_mul\n -/\n#print uniform_space.completion.coe_mul /- _inst_3: topological_ring \u219d has_continuous_mul\n -/\n#print uniform_space.completion.continuous_mul /- _inst_3: topological_ring \u219d topological_add_group has_continuous_mul\n -/\n\n-- topology\\bounded_continuous_function.lean\n#print bounded_continuous_function /- _inst_2: metric_space \u219d topological_space has_dist\n -/\n#print bounded_continuous_function.has_zero /- _inst_2: normed_group \u219d has_zero metric_space\n -/\n#print bounded_continuous_function.has_scalar /- _inst_4: normed_space \u219d\n -/\n#print bounded_continuous_function.coe_smul /- _inst_4: normed_space \u219d has_scalar\n -/\n#print bounded_continuous_function.smul_apply /- _inst_4: normed_space \u219d has_scalar\n -/\n#print bounded_continuous_function.semimodule /- _inst_4: normed_space \u219d\n -/\n#print bounded_continuous_function.normed_space /- _inst_4: normed_space \u219d\n -/\n#print bounded_continuous_function.has_scalar' /- _inst_4: normed_space \u219d\n -/\n#print bounded_continuous_function.module' /- _inst_4: normed_space \u219d\n -/\n#print bounded_continuous_function.norm_smul_le /- _inst_4: normed_space \u219d\n -/\n\n-- topology\\constructions.lean\n#print continuous_update /- _inst_1: decidable_eq \u219d\n -/\n\n-- topology\\metric_space\\antilipschitz.lean\n#print antilipschitz_with /- _inst_1: emetric_space \u219d has_edist\n_inst_2: emetric_space \u219d has_edist\n -/\n\n-- topology\\metric_space\\basic.lean\n#print metric.ball /- _inst_1: metric_space \u219d has_dist\n -/\n#print metric.closed_ball /- _inst_1: metric_space \u219d has_dist\n -/\n#print metric.sphere /- _inst_1: metric_space \u219d has_dist\n -/\n#print metric.complete_of_cauchy_seq_tendsto /- _inst_1: metric_space \u219d emetric_space\n -/\n#print metric.bounded /- _inst_1: metric_space \u219d has_dist\n -/\n#print metric.diam /- _inst_1: metric_space \u219d emetric_space\n -/\n\n-- topology\\metric_space\\cau_seq_filter.lean\n#print cau_seq.tendsto_limit /- _inst_1: normed_ring \u219d normed_group ring\n -/\n#print cauchy_seq.is_cau_seq /- _inst_1: normed_field \u219d normed_group ring\n -/\n\n-- topology\\metric_space\\closeds.lean\n#print emetric.nonempty_compacts.second_countable_topology /- _inst_2: topological_space.second_countable_topology \u219d topological_space.separable_space\n -/\n\n-- topology\\metric_space\\completion.lean\n#print metric.uniform_space.completion.has_dist /- _inst_1: metric_space \u219d uniform_space has_dist\n -/\n\n-- topology\\metric_space\\contracting.lean\n#print contracting_with.one_sub_K_pos /- _inst_1: metric_space \u219d emetric_space\n -/\n\n-- topology\\metric_space\\emetric_space.lean\n#print uniformity_dist_of_mem_uniformity /- _inst_1: linear_order \u219d has_lt\n -/\n#print emetric.ball /- _inst_1: emetric_space \u219d has_edist\n -/\n#print emetric.closed_ball /- _inst_1: emetric_space \u219d has_edist\n -/\n#print emetric.diam /- _inst_1: emetric_space \u219d has_edist\n -/\n\n-- topology\\metric_space\\hausdorff_distance.lean\n#print emetric.inf_edist /- _inst_1: emetric_space \u219d has_edist\n -/\n#print metric.inf_dist /- _inst_1: metric_space \u219d emetric_space\n -/\n#print metric.inf_nndist /- _inst_1: metric_space \u219d emetric_space\n -/\n#print metric.Hausdorff_dist /- _inst_1: metric_space \u219d emetric_space\n -/\n\n-- topology\\metric_space\\isometry.lean\n#print isometry /- _inst_1: emetric_space \u219d has_edist\n_inst_2: emetric_space \u219d has_edist\n -/\n\n-- topology\\metric_space\\lipschitz.lean\n#print lipschitz_with /- _inst_1: emetric_space \u219d has_edist\n_inst_2: emetric_space \u219d has_edist\n -/\n#print lipschitz_on_with /- _inst_1: emetric_space \u219d has_edist\n_inst_2: emetric_space \u219d has_edist\n -/\n\n-- topology\\sequences.lean\n#print compact_space.tendsto_subseq /- _inst_2: topological_space.first_countable_topology \u219d seq_compact_space\n_inst_3: compact_space \u219d seq_compact_space\n -/\n#print metric.compact_iff_seq_compact /- _inst_1: metric_space \u219d emetric_space\n -/\n#print metric.compact_space_iff_seq_compact_space /- _inst_1: metric_space \u219d emetric_space\n -/\n\n-- topology\\subset_properties.lean\n#print is_connected_range /- _inst_3: connected_space \u219d preconnected_space nonempty\n -/\n#print irreducible_space.connected_space /- _inst_3: irreducible_space \u219d preconnected_space nonempty\n -/\n\n-- topology\\uniform_space\\abstract_completion.lean\n#print abstract_completion.funext /- _inst_2: uniform_space \u219d topological_space\n -/\n#print abstract_completion.extension\u2082_coe_coe /- _inst_4: separated_space \u219d t2_space\n -/\n\n-- topology\\uniform_space\\basic.lean\n#print uniform_space.is_open_ball /- _inst_1: uniform_space \u219d topological_space\n -/\n\n-- topology\\uniform_space\\cauchy.lean\n#print cauchy_seq /- _inst_2: semilattice_sup \u219d preorder\n -/\n#print filter.tendsto.cauchy_seq /- _inst_3: nonempty \u219d filter.ne_bot\n -/\n#print cauchy_seq_iff_tendsto /- _inst_2: nonempty \u219d filter.ne_bot\n -/\n\n-- topology\\uniform_space\\complete_separated.lean\n#print is_complete.is_closed /- _inst_2: separated_space \u219d t2_space\n -/\n#print dense_inducing.continuous_extend_of_cauchy /- _inst_5: separated_space \u219d regular_space\n -/\n\n-- topology\\uniform_space\\completion.lean\n#print Cauchy.Cauchy_eq /- _inst_1: inhabited \u219d nonempty\n -/\n#print uniform_space.completion.extension_coe /- _inst_4: separated_space \u219d t2_space\n -/\n\n-- topology\\uniform_space\\uniform_embedding.lean\n#print uniformly_extend_of_ind /- _inst_4: separated_space \u219d t2_space\n -/\n#print uniformly_extend_unique /- _inst_4: separated_space \u219d t2_space\n -/\n", "meta": {"author": "alexjbest", "repo": "lean-generalisation", "sha": "400060b425574cc751b7df6c5673b9792457e68f", "save_path": "github-repos/lean/alexjbest-lean-generalisation", "path": "github-repos/lean/alexjbest-lean-generalisation/lean-generalisation-400060b425574cc751b7df6c5673b9792457e68f/output/out29.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3792037944759047}}
{"text": "import Std\nimport Pauline.Statics\n\nopen Std\n\nnamespace Pauline\n\ndef isVal : Exp \u2192 Bool\n| .scon _\n| .lam _ _\n| .tuple [] => true\n| .tuple (e :: es) => isVal e \u2227 isVal (.tuple es)\n| .typed _ _\n| .case _ _\n| .ite _ _ _\n| .app _ _\n| .let_in _ _\n| .var _\n| .raise _ => false\n\nstructure State where\n  values : HashMap Ident { e // isVal e }\n\ndef Pat.bindsIdent (i : Ident) : Pat \u2192 Bool\n| wild            => false\n| bind i'         => i = i'\n| scon _          => false\n| tuple []        => false\n| tuple (x::xs)   => x.bindsIdent i || (tuple xs).bindsIdent i\n| typed p _       => p.bindsIdent i\n| layer name _ p  => i = name || p.bindsIdent i\n\nmutual\nvariable (e : Exp) (x : Ident)\ndef subst : Exp \u2192 Exp\n| .scon x     => .scon x\n| .lam p body => if p.bindsIdent x then .lam p body else .lam p (subst body)\n| .tuple es => .tuple <| substList es\n| .typed e' t => .typed (subst e') t\n| .case e' ms => .case (subst e') (substMatches ms)\n| .ite i t e' => .ite (subst i) (subst t) (subst e')\n| .app f e' => .app (subst f) (subst e')\n| .let_in _ _ => panic! \"unimplemented\"\n| .var i => if x = i then e else .var i\n| .raise e' => .raise (subst e')\n\ndef substList : List Exp \u2192 List Exp\n| [] => []\n| e'::es => subst e' :: substList es\n\ndef substMatches : List (Pat \u00d7 Exp) \u2192 List (Pat \u00d7 Exp)\n| [] => []\n| (p,e')::ms => (if p.bindsIdent x then (p,e') else (p, subst e')) :: substMatches ms\nend\n\ninductive StepExp : State \u00d7 Exp \u2192 State \u00d7 Exp \u2192 Prop\n| tupleNilStep\n  : StepExp (s, .tuple []) (s, .tuple [])\n| tupleConsStep {e es}\n  (h_e : StepExp (s,e) (s',e')) (h_es : StepExp (s', .tuple es) (s'', .tuple es'))\n  : StepExp (s, .tuple (e :: es)) (s'', .tuple (e' :: es'))\n| typedStep\n  : StepExp (s, .typed e t) (s, e)\n| varStep (h : s.values.find? i = some e)\n  : StepExp (s, .var i) (s, e)\n| appStepL (hf : StepExp (s, f) (s', f'))\n  : StepExp (s, .app f e) (s', .app f' e)\n| appStepR (hf : isVal f) (he : StepExp (s, e) (s', e'))\n  : StepExp (s, .app f e) (s', .app f e')\n| appStep (he : isVal e) (he' : e' = sorry)\n  : StepExp (s, .app (.lam p body) e) (s, e')\n\ndef StepsNExp : Nat \u2192 State \u00d7 Exp \u2192 State \u00d7 Exp \u2192 Prop\n| 0 => \u03bb (s, e) (s'', e'') => s = s'' \u2227 e = e''\n| 1 => StepExp\n| n+1 => \u03bb (s, e) (s'', e'') =>\n  \u2203 e' s', StepsNExp n (s,e) (s', e') \u2227 StepExp (s', e') (s'', e'')\n\ndef StepsExp : State \u00d7 Exp \u2192 State \u00d7 Exp \u2192 Prop := (\u2203 n, StepsNExp n \u00b7 \u00b7)\n\ntheorem StepsExp.trans (h1 : StepsExp x y) (h2 : StepsExp y z)\n  : StepsExp x z\n  := by\n  match x, y, z with\n  | (s,e), (s',e'), (s'',e'') =>\n  match h1, h2 with\n  | \u27e8n1,h1'\u27e9, \u27e8n2,h2'\u27e9 =>\n  clear h1 h2\n  refine \u27e8n1+n2, ?_\u27e9\n  match n2 with\n  | 0 => \n    simp [StepsNExp] at h2' \u22a2\n    match h2' with\n    | \u27e8rfl, rfl\u27e9 =>\n    assumption\n  | n2+1 =>\n  induction n2 generalizing s'' e'' with\n  | zero =>\n    simp [StepsNExp] at h2' \u22a2\n    cases n1 <;> simp [StepsNExp]\n    cases h1'\n    subst_vars; assumption\n    exact \u27e8e',s',h1',h2'\u27e9\n  | succ n2 ih =>\n    match h2' with\n    | \u27e8e_, s_, h2, h3\u27e9 =>\n    refine \u27e8e_, s_, ?_, h3\u27e9\n    clear h3\n    apply ih\n    assumption\n\ninstance : Trans StepExp StepExp StepsExp where\n  trans h1 h2 := StepsExp.trans \u27e81,h1\u27e9 \u27e81,h2\u27e9\n\ninstance : Trans StepExp StepsExp StepsExp where\n  trans h1 h2 := StepsExp.trans \u27e81,h1\u27e9 h2\n\ninstance : Trans StepsExp StepExp StepsExp where\n  trans h1 h2 := StepsExp.trans h1 \u27e81,h2\u27e9\n\ninstance : Trans StepsExp StepsExp StepsExp where\n  trans h1 h2 := h1.trans h2\n", "meta": {"author": "JamesGallicchio", "repo": "pauline", "sha": "b955641fd085964c363d60d8a6ca3cbba741d8f0", "save_path": "github-repos/lean/JamesGallicchio-pauline", "path": "github-repos/lean/JamesGallicchio-pauline/pauline-b955641fd085964c363d60d8a6ca3cbba741d8f0/Pauline/Dynamics.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410572017153, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3791859787429856}}
{"text": "import category_theory.limits.is_limit\nimport set_theory.ordinal.basic\nimport order.category.Preorder\nimport category_theory.morphism_property\nimport for_mathlib.category_theory.lifting_properties.morphism_property\nimport category_theory.limits.shapes.functor_category\nimport category_theory.limits.types\n\nimport order.initial_seg\n\nuniverses u v\n\n#exit\nnamespace category_theory\n\nopen limits\n\nvariables {C : Type*} [category.{v} C] (P : morphism_property C)\n  {\u03b1 : Type u} [linear_order \u03b1] [is_well_order \u03b1 (<)] (F : \u03b1 \u2964 C)\n  {\u03b2 : Type*} [linear_order \u03b2] (h : principal_seg ((<) : \u03b2 \u2192 \u03b2 \u2192 Prop) ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop))\n\n@[simps]\ndef functor.well_order_inclusion_functor'\n  (h : initial_seg ((<) : \u03b2 \u2192 \u03b2 \u2192 Prop) ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop)) : \u03b2 \u2964 \u03b1 :=\nbegin\n  refine monotone.functor (_ : monotone h.to_rel_embedding.1.1),\n  intros b\u2081 b\u2082 r,\n  obtain (h\u2081 | h\u2082) := lt_or_eq_of_le r,\n  { rw \u2190 h.to_rel_embedding.2 at h\u2081,\n    exact le_of_lt h\u2081, },\n  { subst h\u2082, },\nend\n\ninclude h\n\n@[simps]\ndef functor.well_order_inclusion_functor : \u03b2 \u2964 \u03b1 :=\nbegin\n  refine monotone.functor (_ : monotone h.to_rel_embedding.1.1),\n  intros b\u2081 b\u2082 r,\n  obtain (h\u2081 | h\u2082) := lt_or_eq_of_le r,\n  { rw \u2190 h.to_rel_embedding.2 at h\u2081,\n    exact le_of_lt h\u2081, },\n  { subst h\u2082, },\nend\n\n@[simps]\ndef functor.well_order_cocone : limits.cocone (functor.well_order_inclusion_functor h \u22d9 F) :=\n{ X := F.obj h.top,\n  \u03b9 :=\n  { app := \u03bb b, F.map (hom_of_le (le_of_lt (by { dsimp, rw h.down, use b, }))),\n    naturality' := \u03bb b\u2081 b\u2082 hb, by { dsimp, simpa only [\u2190 F.map_comp, category.comp_id], }, }, }\n\nomit h\n\n/-- add the assumption that \u03b2 has no maximum and is not empty... -/\ndef functor.well_order_continuous\u2080 (F : \u03b1 \u2964 C) (\u03b2 : Type u) [linear_order \u03b2] [has_bot \u03b2] :=\n  \u03a0 (h : principal_seg ((<): \u03b2 \u2192 \u03b2 \u2192 Prop) ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop)),\n    limits.is_colimit (F.well_order_cocone h)\n\ndef functor.well_order_continuous (F : \u03b1 \u2964 C) := \u03a0 (\u03b2 : Type u) [h\u03b2 : linear_order \u03b2] [h\u03b2' : has_bot \u03b2],\n  @functor.well_order_continuous\u2080 _ _ _ _ _ F _ h\u03b2 h\u03b2'\n\nnamespace morphism_property\n\nlemma le_succ (a : \u03b1) : a \u2264 (is_well_founded.wf : well_founded ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop)).succ a :=\nbegin\n  by_cases \u2203 b, a < b,\n  { refine le_of_lt _,\n    exact is_well_founded.wf.lt_succ h, },\n  { dsimp [well_founded.succ],\n    rw dif_neg,\n    exact h, },\nend\n\nvariables (\u03b1) [order_bot \u03b1]\ndef is_stable_under_transfinite_composition (P : morphism_property C) : Prop :=\n  \u2200 (F : \u03b1 \u2964 C) (hF\u2081 : F.well_order_continuous)\n    (hF\u2082 : \u2200 (a : \u03b1), P (F.map (hom_of_le (le_succ a))))\n    (c : cocone F) (hc : is_colimit c), P (c.\u03b9.app \u22a5)\n\nsection\n\nvariables {\u03b1} (X : \u03b1\u1d52\u1d56 \u2964 Type v)\n\n@[simps]\ndef inclusion (b : \u03b1) : { a : \u03b1 // a < b} \u2964 \u03b1 :=\nbegin\n  let \u03c6 : { a : \u03b1 // a < b} \u2192 \u03b1 := subtype.val,\n  have h\u03c6 : monotone \u03c6 := \u03bb x y h, h,\n  exact monotone.functor h\u03c6,\nend\n\ndef solutions := (functor.const \u03b1\u1d52\u1d56).obj (terminal (Type v)) \u27f6 X\n\ndef compatible_system (b : \u03b1) := (functor.const { a : \u03b1 // a < b}\u1d52\u1d56).obj (terminal (Type v)) \u27f6\n  (inclusion b).op \u22d9 X\n\nlemma X_map_comp {a b c : \u03b1\u1d52\u1d56} (\u03c6 : a \u27f6 b) (\u03c8 : b \u27f6 c) (\u03c6\u03c8 : a \u27f6 c)\n  (x : X.obj a) : X.map \u03c8 (X.map \u03c6 x) = X.map (\u03c6\u03c8) x :=\nbegin\n  rw subsingleton.elim \u03c6\u03c8 (\u03c6 \u226b \u03c8),\n  simp only [functor_to_types.map_comp_apply],\nend\n\ndef restriction (b : \u03b1) (x : X.obj (opposite.op b)) : compatible_system X b :=\n{ app := \u03bb z n, X.map (hom_of_le (le_of_lt z.unop.2)).op x,\n  naturality' := \u03bb z\u2081 z\u2082 \u03b8, begin\n    ext n,\n    dsimp [inclusion, monotone.functor],\n    rw X_map_comp,\n  end, }\n\nnoncomputable\ninstance : inhabited (\u22a4_ (Type v)) :=\nby { let \u03c6 := terminal.from (ulift.{v} (fin 1)), exact \u27e8\u03c6 (ulift.up 0)\u27e9 }\n\ninstance : subsingleton (\u22a4_ (Type v)) :=\n\u27e8\u03bb x\u2081 x\u2082, begin\n  let \u03c6\u2081 : ulift (fin 1) \u27f6 \u22a4_ (Type v) := \u03bb z, x\u2081,\n  let \u03c6\u2082 : ulift (fin 1) \u27f6 \u22a4_ (Type v) := \u03bb z, x\u2082,\n  have eq := subsingleton.elim \u03c6\u2081 \u03c6\u2082,\n  exact congr_fun eq (ulift.up 0),\nend\u27e9\n\nlemma induction_principle (x\u2080 : X.obj (opposite.op \u22a5))\n  (hX : \u2200 (b : \u03b1) (hb : b \u2260 \u22a5), function.surjective (restriction X b)) :\n  \u2203 (S : solutions X), S.app (opposite.op \u22a5) = \u03bb n, x\u2080 :=\nbegin\n  let A := { o : set \u03b1 // \u22a5 \u2208 o \u2227 \u2200 (x y : \u03b1) (hxy : x \u2264 y) (hy : y \u2208 o), x \u2208 o },\n  let incl : \u03a0 (o : A), o.1 \u2192 \u03b1 := \u03bb o x, x.1,\n  have hincl : \u2200 (o : A), monotone (incl o) := \u03bb o a b h, h,\n  let \u03b9 : \u03a0 (o : A), o.1 \u2964 \u03b1 := \u03bb o, monotone.functor (hincl o),\n  let n : \u22a4_ (Type v) := arbitrary _,\n  let B := sigma (\u03bb (o : A), (functor.const o.1\u1d52\u1d56).obj (terminal (Type v)) \u27f6 (\u03b9 o).op \u22d9 X),\n  have extB : \u2200 (b\u2081 b\u2082 : B) (h\u2081 : b\u2081.1.1 = b\u2082.1.1) (h\u2082 : \u2200 (a : \u03b1) (ha\u2081 : a \u2208 b\u2081.1.1)\n    (ha\u2082 : a \u2208 b\u2082.1.1), b\u2081.2.app (opposite.op \u27e8a, ha\u2081\u27e9) n = b\u2082.2.app (opposite.op \u27e8a, ha\u2082\u27e9) n),\n    b\u2081 = b\u2082,\n  { intros b\u2081 b\u2082 h\u2081 h\u2082,\n    rcases b\u2081 with \u27e8o\u2081, t\u2081\u27e9,\n    rcases b\u2082 with \u27e8o\u2082, t\u2082\u27e9,\n    dsimp at h\u2081 h\u2082,\n    rw \u2190 subtype.ext_iff at h\u2081,\n    subst h\u2081,\n    simp only [eq_self_iff_true, heq_iff_eq, true_and],\n    ext a m,\n    induction a using opposite.rec,\n    rcases a with \u27e8a\u2081, a\u2082\u27e9,\n    have hm := subsingleton.elim n m,\n    subst hm,\n    exact h\u2082 a\u2081 a\u2082 a\u2082, },\n  let \u03c1 : B \u2192 B \u2192 Prop := \u03bb t\u2081 t\u2082, t\u2081.1.1 \u2286 t\u2082.1.1 \u2227 \u03a0 (h\u2081 : t\u2081.1.1 \u2286 t\u2082.1.1),\n    \u2200 (a : t\u2081.1.1), t\u2081.2.app (opposite.op a) n = t\u2082.2.app (opposite.op \u27e8a.1, h\u2081 a.2\u27e9) n,\n  let b : B,\n  { refine \u27e8\u27e8{\u22a5}, \u27e8set.mem_singleton _, \u03bb x y hxy hy,\n      by { dsimp at hy, simpa only [hy, \u2190 eq_bot_iff] using hxy, }\u27e9\u27e9, _\u27e9,\n    exact\n    { app := \u03bb a z, begin\n        refine X.map _ x\u2080,\n        suffices : (opposite.unop a).1 \u2264 \u22a5,\n        { exact (hom_of_le this).op, },\n        rw set.eq_of_mem_singleton (a.unop.2),\n      end,\n      naturality' := \u03bb a\u2081 a\u2082 \u03c6, begin\n        induction a\u2081 using opposite.rec,\n        induction a\u2082 using opposite.rec,\n        have eq\u2081 : a\u2081 = a\u2082,\n        { ext,\n          have h\u2081 := set.eq_of_mem_singleton (a\u2081.2),\n          have h\u2082 := set.eq_of_mem_singleton (a\u2082.2),\n          simp only [subtype.val_eq_coe] at h\u2081 h\u2082,\n          rw [h\u2081, h\u2082], },\n        subst eq\u2081,\n        have eq\u2082 := subsingleton.elim \u03c6 (\ud835\udfd9 _),\n        subst eq\u2082,\n        dsimp,\n        erw [category.id_comp, X.map_id, category.comp_id],\n      end, }, },\n  have hb : is_chain \u03c1 {b} := set.subsingleton.is_chain set.subsingleton_singleton,\n  rcases hb.exists_max_chain with \u27e8M, \u27e8hM\u2081, hM\u2082\u27e9\u27e9,\n  have h : \u2200 (x : \u03b1), \u2203 (m : M), x \u2208 m.1.1.1,\n  { by_contra',\n    let Z := { x : \u03b1 | \u2200 (m : M), \u00ac x \u2208 m.1.1.1 },\n    have hZ : Z.nonempty := \u27e8this.some, this.some_spec\u27e9,\n    have hZ\u2080 : \u00ac \u22a5 \u2208 Z,\n    { dsimp [Z],\n      simp only [set_coe.forall, subtype.coe_mk, sigma.forall, subtype.forall,\n        not_forall, set.not_not_mem, exists_prop, exists_and_distrib_right],\n      exact \u27e8{\u22a5}, \u27e8\u27e8\u27e8set.mem_singleton _, \u03bb x y hxy hy,\n        by { dsimp at hy, simpa only [hy, \u2190 eq_bot_iff] using hxy, }\u27e9,\n        \u27e8b.2, hM\u2082 (set.mem_singleton _)\u27e9\u27e9, set.mem_singleton _\u27e9\u27e9, },\n    let wf : well_founded ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop) := is_well_founded.wf,\n    let \u03bc := wf.min Z hZ,\n    have h\u03bc : \u03bc \u2260 \u22a5,\n    { intro h\u03bc',\n      rw \u2190 h\u03bc' at hZ\u2080,\n      exact hZ\u2080 (wf.min_mem Z hZ), },\n    have h\u03b2 : \u2200 (a : \u03b1) (ha : a < \u03bc), \u2203 (m : M), a \u2208 m.1.1.1,\n    { intros a ha,\n      by_contra' ha',\n      change a \u2208 Z at ha',\n      have ha'' : \u03bc \u2264 a := wf.min_le ha',\n      exact (lt_self_iff_false \u03bc).mp (lt_of_le_of_lt ha'' ha), },\n    let Y := { a : \u03b1 // a < \u03bc },\n    have H : \u2203 (\u03c6 : Y \u2192 M), \u2200 (a : Y), a.1 \u2208 (\u03c6 a).1.1.1 :=\n      \u27e8\u03bb a, (h\u03b2 a.1 a.2).some, \u03bb a, (h\u03b2 a.1 a.2).some_spec\u27e9,\n    rcases H with \u27e8\u03c6, h\u03c6\u27e9,\n    let app' := \u03bb (a : Y), (\u03c6 a).1.2.app (opposite.op \u27e8a.1, h\u03c6 a\u27e9) n,\n    have compat : \u2200 (a : Y) (m : M) (hm : a.1 \u2208 m.1.1.1),\n      m.1.2.app (opposite.op \u27e8a.1, hm\u27e9) n = app' a,\n    { intro a,\n      suffices : \u2200 (m\u2081 m\u2082 : M) (hm\u2081 : a.1 \u2208 m\u2081.1.1.1) (hm\u2082 : a.1 \u2208 m\u2082.1.1.1),\n        m\u2081.1.2.app (opposite.op \u27e8a.1, hm\u2081\u27e9) n = m\u2082.1.2.app (opposite.op \u27e8a.1, hm\u2082\u27e9) n,\n      { intros m hm,\n        exact this m (\u03c6 a) hm (h\u03c6 a), },\n      suffices : \u2200 (m\u2081 m\u2082 : M) (hm\u2081 : a.1 \u2208 m\u2081.1.1.1) (hm\u2082 : a.1 \u2208 m\u2082.1.1.1) (h\u2081\u2082 : \u03c1 m\u2081 m\u2082),\n        m\u2081.1.2.app (opposite.op \u27e8a.1, hm\u2081\u27e9) n = m\u2082.1.2.app (opposite.op \u27e8a.1, hm\u2082\u27e9) n,\n      { intros m\u2081 m\u2082 hm\u2081 hm\u2082,\n        by_cases m\u2081 = m\u2082,\n        { subst h, },\n        { rw subtype.ext_iff at h,\n          rcases hM\u2081.is_chain m\u2081.2 m\u2082.2 h with (h'|h'),\n          { exact this m\u2081 m\u2082 hm\u2081 hm\u2082 h', },\n          { exact (this m\u2082 m\u2081 hm\u2082 hm\u2081 h').symm, }, }, },\n      intros m\u2081 m\u2082 hm\u2081 hm\u2082 h\u2081\u2082,\n      exact h\u2081\u2082.2 h\u2081\u2082.1 \u27e8a.1, hm\u2081\u27e9, },\n    let s : compatible_system X \u03bc :=\n    { app := \u03bb a z, app' a.unop,\n      naturality' := \u03bb a\u2082 a\u2081 \u03c8, begin\n        ext m,\n        have hm := subsingleton.elim n m,\n        subst hm,\n        induction a\u2081 using opposite.rec,\n        induction a\u2082 using opposite.rec,\n        dsimp,\n        have ha : \u2203 (m : M), a\u2081.1 \u2208 m.1.1.1 \u2227 a\u2082.1 \u2208 m.1.1.1,\n        { by_cases \u03c6 a\u2081 = \u03c6 a\u2082,\n          { exact \u27e8\u03c6 a\u2081, \u27e8h\u03c6 a\u2081, by { rw h, exact h\u03c6 a\u2082, }\u27e9\u27e9, },\n          { have h' : (\u03c6 a\u2081).1 \u2260 (\u03c6 a\u2082).1 := \u03bb h'', h (subtype.ext_iff.mpr h''),\n            cases hM\u2081.is_chain (\u03c6 a\u2081).2 (\u03c6 a\u2082).2 h' with h'' h'',\n            { exact \u27e8\u03c6 a\u2082, \u27e8h''.1 (h\u03c6 a\u2081), h\u03c6 a\u2082\u27e9\u27e9, },\n            { exact \u27e8\u03c6 a\u2081, \u27e8h\u03c6 a\u2081, h''.1 (h\u03c6 a\u2082)\u27e9\u27e9, }, }, },\n        rcases ha with \u27e8m, hm\u2081, hm\u2082\u27e9,\n        rw [\u2190 compat a\u2081 m hm\u2081, \u2190 compat a\u2082 m hm\u2082],\n        let a\u2081' : m.1.1.1 := \u27e8a\u2081.1, hm\u2081\u27e9,\n        let a\u2082' : m.1.1.1 := \u27e8a\u2082.1, hm\u2082\u27e9,\n        let \u03b1 : a\u2081' \u27f6 a\u2082' := \u03c8.unop,\n        exact congr_fun (m.1.2.naturality \u03b1.op) n,\n      end, },\n    cases hX \u03bc h\u03bc s with x hx,\n    let o : A := \u27e8{ y : \u03b1 | y \u2264 \u03bc }, \u27e8bot_le, \u03bb x y hxy hy, hxy.trans hy\u27e9\u27e9,\n    let b : B := \u27e8o,\n      { app := \u03bb a z, X.map (hom_of_le a.unop.2).op x,\n        naturality' := \u03bb a\u2081 a\u2082 \u03c8, by { ext m, dsimp, rw X_map_comp, }, }\u27e9,\n    have h\u03bc' : \u03bc \u2208 Z := wf.min_mem Z hZ,\n    have hZ' : \u2200 (a\u2081 a\u2082 : \u03b1) (h : a\u2081 \u2264 a\u2082), a\u2082 \u2209 Z \u2192 a\u2081 \u2209 Z,\n    { intros a\u2081 a\u2082 h h\u2082,\n      dsimp at h\u2082,\n      simp only [not_forall, set.not_not_mem] at h\u2082,\n      cases h\u2082 with m hm,\n      intro ha\u2081,\n      exact ha\u2081 m (m.1.1.2.2 _ _ h hm), },\n    have hb : is_chain \u03c1 (insert b M) := hM\u2081.is_chain.insert (\u03bb b' hb'\u2081 hb'\u2082, or.inr begin\n      split,\n      { intros a,\n        by_contra',\n        exact hZ' \u03bc a (le_of_lt (not_le.mp this.2)) (\u03bb ha, ha \u27e8b', hb'\u2081\u27e9 this.1) h\u03bc', },\n      { intro h\u2081,\n        have compat' : \u2200 (a : b'.1.1) (ha : a.1 < \u03bc), b'.2.app (opposite.op a) n =\n          b.2.app (opposite.op \u27e8a.1, h\u2081 a.2\u27e9) n,\n        { rintro \u27e8a, ha\u27e9 ha',\n          have eq := congr_fun (nat_trans.congr_app hx (opposite.op \u27e8a, ha'\u27e9)) n,\n          have eq' := compat \u27e8a, ha'\u27e9 \u27e8b', hb'\u2081\u27e9 ha,\n          exact eq'.trans eq.symm, },\n        intro a,\n        dsimp,\n        cases eq_or_lt_of_le (h\u2081 a.2),\n        { exfalso,\n          apply hb'\u2082,\n          apply extB,\n          { ext c,\n            split,\n            { intro hc,\n              exact b'.1.2.2 _ _ hc (by { rw \u2190 h, exact a.2, }), },\n            { intro hc,\n              exact h\u2081 hc, }, },\n          { intros c hc\u2081 hc\u2082,\n            suffices : c < \u03bc,\n            { exact (compat' \u27e8c, hc\u2082\u27e9 this).symm, },\n            by_contra',\n            exact hZ' _ _ this (\u03bb hc, hc \u27e8b', hb'\u2081\u27e9 hc\u2082) h\u03bc', }, },\n        { exact compat' a h, }, },\n    end),\n    have hb' : super_chain \u03c1 M (insert b M) := begin\n      refine \u27e8hb, \u27e8set.subset_insert _ _, _\u27e9\u27e9,\n      intro hb'',\n      have h\u03bc'' := h\u03bc' \u27e8b, hb'' (set.mem_insert _ _)\u27e9,\n      dsimp at h\u03bc'',\n      apply h\u03bc'',\n      refl,\n    end,\n    exact hM\u2081.not_super_chain hb', },\n  have H : \u2203 (\u03c6 : \u03b1 \u2192 M), \u2200 (a : \u03b1), a \u2208 (\u03c6 a).1.1.1,\n  { refine \u27e8\u03bb a, (h a).some, \u03bb a, (h a).some_spec\u27e9, },\n  rcases H with \u27e8\u03c6, h\u03c6\u27e9,\n  let app' := \u03bb (a : \u03b1), (\u03c6 a).1.2.app (opposite.op \u27e8a, h\u03c6 a\u27e9) n,\n  have compat'' : \u2200 (a : \u03b1) (m : M) (hm : a \u2208 m.1.1.1), m.1.2.app (opposite.op \u27e8a, hm\u27e9) n = app' a,\n  { intro a,\n    suffices : \u2200 (m\u2081 m\u2082 : M) (hm\u2081 : a \u2208 m\u2081.1.1.1) (hm\u2082 : a \u2208 m\u2082.1.1.1),\n      m\u2081.1.2.app (opposite.op \u27e8a, hm\u2081\u27e9) = m\u2082.1.2.app (opposite.op \u27e8a, hm\u2082\u27e9),\n    { intros m hm,\n      exact congr_fun (this m (\u03c6 a) hm (h\u03c6 a)) n, },\n      /- duplicates ? -/\n    suffices : \u2200 (m\u2081 m\u2082 : M) (hm\u2081 : a \u2208 m\u2081.1.1.1) (hm\u2082 : a \u2208 m\u2082.1.1.1) (h\u2081\u2082 : \u03c1 m\u2081 m\u2082),\n      m\u2081.1.2.app (opposite.op \u27e8a, hm\u2081\u27e9) = m\u2082.1.2.app (opposite.op \u27e8a, hm\u2082\u27e9),\n    { intros m\u2081 m\u2082 hm\u2081 hm\u2082,\n      by_cases m\u2081 = m\u2082,\n      { subst h, },\n      { rw subtype.ext_iff at h,\n        rcases hM\u2081.is_chain m\u2081.2 m\u2082.2 h with (h'|h'),\n        { exact this m\u2081 m\u2082 hm\u2081 hm\u2082 h', },\n        { exact (this m\u2082 m\u2081 hm\u2082 hm\u2081 h').symm, }, }, },\n    intros m\u2081 m\u2082 hm\u2081 hm\u2082 h\u2081\u2082,\n    ext m,\n    have hm := subsingleton.elim n m,\n    subst hm,\n    exact h\u2081\u2082.2 h\u2081\u2082.1 \u27e8a, hm\u2081\u27e9, },\n  refine \u27e8{ app := \u03bb a z, app' a.unop,\n    naturality' := \u03bb a\u2082 a\u2081 \u03c8, begin\n      ext m,\n      have hm := subsingleton.elim n m,\n      subst hm,\n      induction a\u2081 using opposite.rec,\n      induction a\u2082 using opposite.rec,\n      have h\u03c8 : a\u2081 \u2264 a\u2082 := le_of_hom \u03c8.unop,\n      dsimp,\n      rw [\u2190 compat'' a\u2082 (\u03c6 a\u2082) (h\u03c6 a\u2082), \u2190 compat'' a\u2081 (\u03c6 a\u2082) ((\u03c6 a\u2082).1.1.2.2 _ _ h\u03c8 (h\u03c6 a\u2082))],\n      let X : (\u03c6 a\u2082).1.1.1 := \u27e8a\u2081, (\u03c6 a\u2082).1.1.2.2 _ _ h\u03c8 (h\u03c6 a\u2082)\u27e9,\n      let Y : (\u03c6 a\u2082).1.1.1 := \u27e8a\u2082, h\u03c6 a\u2082\u27e9,\n      let \u03b3 : X \u27f6 Y := hom_of_le h\u03c8,\n      convert congr_fun ((\u03c6 a\u2082).val.snd.naturality \u03b3.op) n,\n    end }, _\u27e9,\n  { ext m,\n    have hm := subsingleton.elim n m,\n    subst hm,\n    dsimp,\n    rw \u2190 compat'' \u22a5 \u27e8b, hM\u2082 (set.mem_singleton b)\u27e9 rfl,\n    dsimp,\n    have eq : \ud835\udfd9 (\u22a5 : \u03b1) = hom_of_le rfl.le := subsingleton.elim _ _,\n    simpa only [\u2190 eq, op_id, X.map_id], },\nend\n\nend\n\n@[simp]\nlemma hom_of_le_self_eq_id (a : \u03b1) : hom_of_le (show a \u2264 a, by refl) = \ud835\udfd9 a := subsingleton.elim _ _\n\n@[simp]\nlemma hom_of_le_le_of_hom {a b : \u03b1} (f : a \u27f6 b) : hom_of_le (le_of_hom f) = f := subsingleton.elim _ _\n\n\nlemma min_eq {\u03b1 : Type*} [linear_order \u03b1] [H : is_well_order \u03b1 (<)]\n  (s : set \u03b1) (hs : s.nonempty) (m : \u03b1) (hm\u2081 : m \u2208 s) (hm\u2082 : \u2200 (b : \u03b1), b \u2208 s \u2192 m \u2264 b) :\n  H.wf.min s hs = m :=\nle_antisymm (H.wf.min_le hm\u2081) (hm\u2082 _ (H.wf.min_mem s hs))\n\nlemma llp_is_stable_under_transfinite_composition (P : morphism_property C) :\n  P.llp_with.is_stable_under_transfinite_composition \u03b1 :=\n\u03bb F hF\u2081 hF\u2082 c hc X Y p hp, \u27e8\u03bb f g, begin\n  dsimp at g,\n  intro sq,\n  have sqs : \u03a0 (a : \u03b1), comm_sq f (F.map (hom_of_le (bot_le : \u22a5 \u2264 a))) p (c.\u03b9.app a \u226b g) :=\n    \u03bb a, comm_sq.mk (by rw [sq.w, cocone.w_assoc]),\n  let \u03c4 : \u03a0 (a b : \u03b1) (h : a \u2264 b), (sqs b).lift_struct \u2192 (sqs a).lift_struct := \u03bb a b h l,\n  { l := F.map (hom_of_le h) \u226b l.l,\n    fac_left' := by simpa only [\u2190 l.fac_left, \u2190 F.map_comp_assoc],\n    fac_right' := by simp only [category.assoc, l.fac_right, cocone.w_assoc], },\n  let U : \u03b1\u1d52\u1d56 \u2964 Type v :=\n  { obj := \u03bb b, (sqs b.unop).lift_struct,\n    map := \u03bb a b h, \u03c4 b.unop a.unop (le_of_hom h.unop),\n    map_id' := \u03bb a, begin\n      ext,\n      dsimp [\u03c4],\n      simp only [hom_of_le_self_eq_id, functor.map_id, category.id_comp],\n    end,\n    map_comp' := \u03bb a b c \u03c6\u2081 \u03c6\u2082, begin\n      ext,\n      dsimp [\u03c4],\n      rw [\u2190 F.map_comp_assoc],\n      congr,\n    end, },\n  let x\u2080 : U.obj (opposite.op \u22a5) :=\n  { l := f,\n    fac_left' := by { dsimp, rw [hom_of_le_self_eq_id, F.map_id, category.id_comp], },\n    fac_right' := sq.w, },\n  let n : \u22a4_ (Type v) := arbitrary _,\n  cases induction_principle U x\u2080 _ with L hL,\n  { exact \u27e8nonempty.intro\n    { l := begin\n        refine hc.desc (cocone.mk _ _),\n        exact\n        { app := \u03bb b, (L.app (opposite.op b) n).l,\n          naturality' := \u03bb a b h, begin\n            dsimp,\n            simpa only [types_comp_apply, functor.const_obj_map, types_id_apply,\n              category.comp_id, comm_sq.lift_struct.ext_iff, hom_of_le_le_of_hom]\n              using congr_fun (L.naturality h.op).symm n,\n          end },\n      end,\n      fac_left' := by simp only [is_colimit.fac, hL],\n      fac_right' := hc.hom_ext (\u03bb b, by simpa only [is_colimit.fac_assoc]\n        using (L.app (opposite.op b) n).fac_right), }\u27e9, },\n  { intros b hb s,\n    by_cases \u2203 (b\u2080 : \u03b1) (h\u2080 : b\u2080 < b), \u2200 (a : \u03b1), a < b \u2192 a \u2264 b\u2080,\n    { rcases h with \u27e8b\u2080, h\u2080, h\u2081\u27e9,\n      let L := (s.app (opposite.op \u27e8b\u2080, h\u2080\u27e9) n),\n      have H := hF\u2082 b\u2080 p hp,\n      let e : arrow.mk (F.map (hom_of_le (le_succ b\u2080))) \u2245 arrow.mk (F.map (hom_of_le (le_of_lt h\u2080))) :=\n        arrow.iso_mk' (F.map (hom_of_le (le_succ b\u2080))) (F.map (hom_of_le (le_of_lt h\u2080))) (iso.refl _)\n          (F.map_iso (eq_to_iso begin\n            dsimp [well_founded.succ],\n            rw dif_pos,\n            { refine min_eq _ \u27e8b, h\u2080\u27e9 _ h\u2080 _,\n              intros b\u2081 hb\u2081,\n              by_contra',\n              exact not_lt.mpr (h\u2081 _ this) hb\u2081, },\n          end)) begin\n            simp only [iso.refl_hom, category.id_comp, functor.map_iso_hom, eq_to_iso.hom, \u2190 F.map_comp],\n            congr,\n          end,\n      rw has_lifting_property.iff_of_arrow_iso_left e at H,\n      haveI := H,\n      have S : comm_sq L.l (F.map (hom_of_le (le_of_lt h\u2080))) p (c.\u03b9.app b \u226b g),\n      { apply comm_sq.mk,\n        simp only [L.fac_right, cocone.w_assoc], },\n      let t : U.obj (opposite.op b) :=\n      { l := S.lift,\n        fac_left' := begin\n          conv_rhs { rw [\u2190 L.fac_left, \u2190 S.fac_left, \u2190 F.map_comp_assoc], },\n          congr,\n        end,\n        fac_right' := S.fac_right, },\n      refine \u27e8t, _\u27e9,\n      ext d m,\n      dsimp at m,\n      have hm := subsingleton.elim n m,\n      subst hm,\n      dsimp [restriction],\n      have pif := S.fac_left,\n      dsimp [L] at pif,\n      have foo := (s.app d n).fac_left,\n      let \u03c6 : d.unop \u27f6 \u27e8b\u2080, h\u2080\u27e9 := hom_of_le (h\u2081 d.unop.1 d.unop.2),\n      have eq := congr_fun (s.naturality \u03c6.op) n,\n      rw comm_sq.lift_struct.ext_iff at eq,\n      dsimp at eq,\n      conv_rhs { rw [eq, \u2190 S.fac_left, \u2190 F.map_comp_assoc], },\n      congr, },\n    { let \u03b2 := {a : \u03b1 // a < b},\n      let B := @principal_seg.mk _ _ ((<) : \u03b2 \u2192 \u03b2 \u2192 Prop) ((<) : \u03b1 \u2192 \u03b1 \u2192 Prop)\n        (subtype.rel_embedding _ _) b begin\n        intro c,\n        split,\n        { intro hc,\n          exact \u27e8\u27e8c, hc\u27e9, rfl\u27e9, },\n        { intro hc,\n          cases hc with d hd,\n          rw \u2190 hd,\n          exact d.2, },\n      end,\n      let d\u2080 : \u03b2 := \u27e8\u22a5, begin\n        rcases (bot_le : \u22a5 \u2264 b).eq_or_lt with (h\u2081|h\u2082),\n        { exfalso,\n          exact hb h\u2081.symm, },\n        { exact h\u2082, },\n      end\u27e9,\n      let Co : cocone (functor.well_order_inclusion_functor B \u22d9 F) := cocone.mk X\n        { app := \u03bb d, (s.app (opposite.op d) n).l,\n          naturality' := \u03bb b\u2081 b\u2082 \u03c6, begin\n            dsimp [functor.well_order_inclusion_functor],\n            have h\u03c6 := congr_fun (s.naturality \u03c6.op) n,\n            dsimp at h\u03c6,\n            simpa only [category.comp_id, h\u03c6],\n          end, },\n      let t : U.obj (opposite.op b) :=\n      { l := (hF\u2081 \u03b2 B).desc Co,\n        fac_left' := begin\n          dsimp,\n          conv_rhs { rw \u2190 (s.app (opposite.op d\u2080) n).fac_left, },\n          have h\u2080 := (hF\u2081 \u03b2 B).fac Co d\u2080,\n          dsimp [functor.well_order_cocone] at h\u2080,\n          rw [\u2190 h\u2080, \u2190 F.map_comp_assoc],\n          congr,\n        end,\n        fac_right' := begin\n          apply (hF\u2081 \u03b2 B).hom_ext,\n          intro d,\n          rw [is_colimit.fac_assoc, (s.app (opposite.op d) n).fac_right,\n            functor.well_order_cocone_\u03b9_app],\n          dsimp,\n          simpa only [cocone.w_assoc],\n        end, },\n      use t,\n      ext a m,\n      rw \u2190 subsingleton.elim n m,\n      exact (hF\u2081 \u03b2 B).fac Co a.unop, }, },\nend\u27e9\n\nend morphism_property\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/lifting_properties/continuous_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3791859711808477}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.adjunction\nimport Mathlib.category_theory.adjunction.limits\nimport Mathlib.category_theory.limits.preserves.shapes.terminal\nimport Mathlib.PostPort\n\nuniverses u\u2081 v\u2081 u\u2082 \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\nnamespace monad\n\n\nnamespace forget_creates_limits\n\n\n/-- (Impl) The natural transformation used to define the new cone -/\n@[simp] theorem \u03b3_app {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] (D : J \u2964 algebra T) (j : J) :\n    nat_trans.app (\u03b3 D) j = algebra.a (functor.obj D j) :=\n  Eq.refl (nat_trans.app (\u03b3 D) j)\n\n/-- (Impl) This new cone is used to construct the algebra structure -/\n@[simp] theorem new_cone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T)) :\n    limits.cone.X (new_cone D c) = functor.obj T (limits.cone.X c) :=\n  Eq.refl (limits.cone.X (new_cone D c))\n\n/-- The algebra structure which will be the apex of the new limit cone for `D`. -/\n@[simp] theorem cone_point_A {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T))\n    (t : limits.is_limit c) : algebra.A (cone_point D c t) = limits.cone.X c :=\n  Eq.refl (algebra.A (cone_point D c t))\n\n/-- (Impl) Construct the lifted cone in `algebra T` which will be limiting. -/\n@[simp] theorem lifted_cone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T))\n    (t : limits.is_limit c) : limits.cone.X (lifted_cone D c t) = cone_point D c t :=\n  Eq.refl (limits.cone.X (lifted_cone D c t))\n\n/-- (Impl) Prove that the lifted cone is limiting. -/\ndef lifted_cone_is_limit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T))\n    (t : limits.is_limit c) : limits.is_limit (lifted_cone D c t) :=\n  limits.is_limit.mk\n    fun (s : limits.cone D) =>\n      algebra.hom.mk (limits.is_limit.lift t (functor.map_cone (forget T) s))\n\nend forget_creates_limits\n\n\n-- Theorem 5.6.5 from [Riehl][riehl2017]\n\n/-- The forgetful functor from the Eilenberg-Moore category creates limits. -/\nprotected instance forget_creates_limits {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] :\n    creates_limits (forget T) :=\n  creates_limits.mk\n    fun (J : Type v\u2081) (\ud835\udca5 : small_category J) =>\n      creates_limits_of_shape.mk\n        fun (D : J \u2964 algebra T) =>\n          creates_limit_of_reflects_iso\n            fun (c : limits.cone (D \u22d9 forget T)) (t : limits.is_limit c) =>\n              lifts_to_limit.mk\n                (liftable_cone.mk sorry\n                  (limits.cones.ext (iso.refl (limits.cone.X (functor.map_cone (forget T) sorry)))\n                    sorry))\n                sorry\n\n/-- `D \u22d9 forget T` has a limit, then `D` has a limit. -/\ntheorem has_limit_of_comp_forget_has_limit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) [limits.has_limit (D \u22d9 forget T)] :\n    limits.has_limit D :=\n  has_limit_of_created D (forget T)\n\nnamespace forget_creates_colimits\n\n\n-- Let's hide the implementation details in a namespace\n\n-- We have a diagram D of shape J in the category of algebras, and we assume that we are given a\n\n-- colimit for its image D \u22d9 forget T under the forgetful functor, say its apex is L.\n\n-- We'll construct a colimiting coalgebra for D, whose carrier will also be L.\n\n-- To do this, we must find a map TL \u27f6 L. Since T preserves colimits, TL is also a colimit.\n\n-- In particular, it is a colimit for the diagram `(D \u22d9 forget T) \u22d9 T`\n\n-- so to construct a map TL \u27f6 L it suffices to show that L is the apex of a cocone for this diagram.\n\n-- In other words, we need a natural transformation from const L to `(D \u22d9 forget T) \u22d9 T`.\n\n-- But we already know that L is the apex of a cocone for the diagram `D \u22d9 forget T`, so it\n\n-- suffices to give a natural transformation `((D \u22d9 forget T) \u22d9 T) \u27f6 (D \u22d9 forget T)`:\n\n/--\n(Impl)\nThe natural transformation given by the algebra structure maps, used to construct a cocone `c` with\napex `colimit (D \u22d9 forget T)`.\n -/\n@[simp] theorem \u03b3_app {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] {D : J \u2964 algebra T} (j : J) :\n    nat_trans.app \u03b3 j = algebra.a (functor.obj D j) :=\n  Eq.refl (nat_trans.app \u03b3 j)\n\n/--\n(Impl)\nA cocone for the diagram `(D \u22d9 forget T) \u22d9 T` found by composing the natural transformation `\u03b3`\nwith the colimiting cocone for `D \u22d9 forget T`.\n-/\n@[simp] theorem new_cocone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) :\n    limits.cocone.X (new_cocone c) = limits.cocone.X c :=\n  Eq.refl (limits.cocone.X (new_cocone c))\n\n/--\n(Impl)\nDefine the map `\u03bb : TL \u27f6 L`, which will serve as the structure of the coalgebra on `L`, and\nwe will show is the colimiting object. We use the cocone constructed by `c` and the fact that\n`T` preserves colimits to produce this morphism.\n-/\ndef lambda {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J]\n    {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c)\n    [limits.preserves_colimit (D \u22d9 forget T) T] :\n    limits.cocone.X (functor.map_cocone T c) \u27f6 limits.cocone.X c :=\n  limits.is_colimit.desc (limits.preserves_colimit.preserves t) (new_cocone c)\n\n/-- (Impl) The key property defining the map `\u03bb : TL \u27f6 L`. -/\ntheorem commuting {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J]\n    {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c)\n    [limits.preserves_colimit (D \u22d9 forget T) T] (j : J) :\n    functor.map T (nat_trans.app (limits.cocone.\u03b9 c) j) \u226b lambda c t =\n        algebra.a (functor.obj D j) \u226b nat_trans.app (limits.cocone.\u03b9 c) j :=\n  limits.is_colimit.fac (limits.preserves_colimit.preserves t) (new_cocone c) j\n\n/--\n(Impl)\nConstruct the colimiting algebra from the map `\u03bb : TL \u27f6 L` given by `lambda`. We are required to\nshow it satisfies the two algebra laws, which follow from the algebra laws for the image of `D` and\nour `commuting` lemma.\n-/\n@[simp] theorem cocone_point_A {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T))\n    (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T]\n    [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] :\n    algebra.A (cocone_point c t) = limits.cocone.X c :=\n  Eq.refl (algebra.A (cocone_point c t))\n\n/-- (Impl) Construct the lifted cocone in `algebra T` which will be colimiting. -/\n@[simp] theorem lifted_cocone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081}\n    [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T))\n    (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T]\n    [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] :\n    limits.cocone.X (lifted_cocone c t) = cocone_point c t :=\n  Eq.refl (limits.cocone.X (lifted_cocone c t))\n\n/-- (Impl) Prove that the lifted cocone is colimiting. -/\n@[simp] theorem lifted_cocone_is_colimit_desc_f {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T))\n    (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T]\n    [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] (s : limits.cocone D) :\n    algebra.hom.f (limits.is_colimit.desc (lifted_cocone_is_colimit c t) s) =\n        limits.is_colimit.desc t (functor.map_cocone (forget T) s) :=\n  Eq.refl (algebra.hom.f (limits.is_colimit.desc (lifted_cocone_is_colimit c t) s))\n\nend forget_creates_colimits\n\n\n-- TODO: the converse of this is true as well\n\n/--\nThe forgetful functor from the Eilenberg-Moore category for a monad creates any colimit\nwhich the monad itself preserves.\n-/\nprotected instance forget_creates_colimit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) [limits.preserves_colimit (D \u22d9 forget T) T]\n    [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] : creates_colimit D (forget T) :=\n  creates_colimit_of_reflects_iso\n    fun (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) =>\n      lifts_to_colimit.mk\n        (liftable_cocone.mk\n          (limits.cocone.mk (forget_creates_colimits.cocone_point c t)\n            (nat_trans.mk fun (j : J) => algebra.hom.mk (nat_trans.app (limits.cocone.\u03b9 c) j)))\n          (limits.cocones.ext\n            (iso.refl\n              (limits.cocone.X\n                (functor.map_cocone (forget T)\n                  (limits.cocone.mk (forget_creates_colimits.cocone_point c t)\n                    (nat_trans.mk\n                      fun (j : J) => algebra.hom.mk (nat_trans.app (limits.cocone.\u03b9 c) j))))))\n            sorry))\n        (forget_creates_colimits.lifted_cocone_is_colimit c t)\n\nprotected instance forget_creates_colimits_of_shape {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    {J : Type v\u2081} [small_category J] [limits.preserves_colimits_of_shape J T] :\n    creates_colimits_of_shape J (forget T) :=\n  creates_colimits_of_shape.mk fun (K : J \u2964 algebra T) => monad.forget_creates_colimit K\n\nprotected instance forget_creates_colimits {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    [limits.preserves_colimits T] : creates_colimits (forget T) :=\n  creates_colimits.mk\n    fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) => monad.forget_creates_colimits_of_shape\n\n/--\nFor `D : J \u2964 algebra T`, `D \u22d9 forget T` has a colimit, then `D` has a colimit provided colimits\nof shape `J` are preserved by `T`.\n-/\ntheorem forget_creates_colimits_of_monad_preserves {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T]\n    {J : Type v\u2081} [small_category J] [limits.preserves_colimits_of_shape J T] (D : J \u2964 algebra T)\n    [limits.has_colimit (D \u22d9 forget T)] : limits.has_colimit D :=\n  has_colimit_of_created D (forget T)\n\nend monad\n\n\nprotected instance comp_comparison_forget_has_limit {C : Type u\u2081} [category C] {D : Type u\u2082}\n    [category D] {J : Type v\u2081} [small_category J] (F : J \u2964 D) (R : D \u2964 C) [monadic_right_adjoint R]\n    [limits.has_limit (F \u22d9 R)] :\n    limits.has_limit ((F \u22d9 monad.comparison R) \u22d9 monad.forget (left_adjoint R \u22d9 R)) :=\n  limits.has_limit_of_iso (iso_whisker_left F (iso.symm (monad.comparison_forget R)))\n\nprotected instance comp_comparison_has_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D]\n    {J : Type v\u2081} [small_category J] (F : J \u2964 D) (R : D \u2964 C) [monadic_right_adjoint R]\n    [limits.has_limit (F \u22d9 R)] : limits.has_limit (F \u22d9 monad.comparison R) :=\n  monad.has_limit_of_comp_forget_has_limit (F \u22d9 monad.comparison R)\n\n/-- Any monadic functor creates limits. -/\ndef monadic_creates_limits {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C)\n    [monadic_right_adjoint R] : creates_limits R :=\n  creates_limits_of_nat_iso (monad.comparison_forget R)\n\n/--\nThe forgetful functor from the Eilenberg-Moore category for a monad creates any colimit\nwhich the monad itself preserves.\n-/\ndef monadic_creates_colimit_of_preserves_colimit {C : Type u\u2081} [category C] {D : Type u\u2082}\n    [category D] {J : Type v\u2081} [small_category J] (R : D \u2964 C) (K : J \u2964 D) [monadic_right_adjoint R]\n    [limits.preserves_colimit (K \u22d9 R) (left_adjoint R \u22d9 R)]\n    [limits.preserves_colimit ((K \u22d9 R) \u22d9 left_adjoint R \u22d9 R) (left_adjoint R \u22d9 R)] :\n    creates_colimit K R :=\n  creates_colimit_of_nat_iso (monad.comparison_forget R)\n\n/-- A monadic functor creates any colimits of shapes it preserves. -/\ndef monadic_creates_colimits_of_shape_of_preserves_colimits_of_shape {C : Type u\u2081} [category C]\n    {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (R : D \u2964 C)\n    [monadic_right_adjoint R] [limits.preserves_colimits_of_shape J R] :\n    creates_colimits_of_shape J R :=\n  creates_colimits_of_shape_of_nat_iso (monad.comparison_forget R)\n\n/-- A monadic functor creates colimits if it preserves colimits. -/\ndef monadic_creates_colimits_of_preserves_colimits {C : Type u\u2081} [category C] {D : Type u\u2082}\n    [category D] (R : D \u2964 C) [monadic_right_adjoint R] [limits.preserves_colimits R] :\n    creates_colimits R :=\n  creates_colimits.mk\n    fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) =>\n      monadic_creates_colimits_of_shape_of_preserves_colimits_of_shape R\n\ntheorem has_limit_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081}\n    [small_category J] (F : J \u2964 D) (R : D \u2964 C) [limits.has_limit (F \u22d9 R)] [reflective R] :\n    limits.has_limit F :=\n  has_limit_of_created F R\n\n/-- If `C` has limits of shape `J` then any reflective subcategory has limits of shape `J`. -/\ntheorem has_limits_of_shape_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D]\n    {J : Type v\u2081} [small_category J] [limits.has_limits_of_shape J C] (R : D \u2964 C) [reflective R] :\n    limits.has_limits_of_shape J D :=\n  limits.has_limits_of_shape.mk fun (F : J \u2964 D) => has_limit_of_reflective F R\n\n/-- If `C` has limits then any reflective subcategory has limits. -/\ntheorem has_limits_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C)\n    [limits.has_limits C] [reflective R] : limits.has_limits D :=\n  limits.has_limits.mk\n    fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) => has_limits_of_shape_of_reflective R\n\n/--\nThe reflector always preserves terminal objects. Note this in general doesn't apply to any other\nlimit.\n-/\ndef left_adjoint_preserves_terminal_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082}\n    [category D] (R : D \u2964 C) [reflective R] [limits.has_terminal C] :\n    limits.preserves_limits_of_shape (discrete pempty) (left_adjoint R) :=\n  limits.preserves_limits_of_shape.mk\n    fun (K : discrete pempty \u2964 C) =>\n      let _inst : limits.has_terminal D := sorry;\n      let _inst_3 : creates_limits R := monadic_creates_limits R;\n      let _inst_6 : limits.preserves_limit (functor.empty D) R :=\n        category_theory.preserves_limit_of_creates_limit_and_has_limit (functor.empty D) R;\n      let _inst_7 : limits.preserves_limit (functor.empty C) (left_adjoint R) :=\n        limits.preserves_terminal_of_iso (left_adjoint R)\n          (functor.map_iso (left_adjoint R) (iso.symm (limits.preserves_terminal.iso R)) \u226a\u226b\n            as_iso (nat_trans.app (adjunction.counit (adjunction.of_right_adjoint R)) (\u22a4_D)));\n      limits.preserves_limit_of_iso_diagram (left_adjoint R)\n        (iso.symm (functor.unique_from_empty K))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/limits_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.37918596361870965}}
{"text": "/-\nCopyright (c) 2021 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport tactic.itauto\n\nsection itauto\u2080\nvariables p q r : Prop\nvariables h : p \u2227 q \u2228 p \u2227 r\ninclude h\nexample : p \u2227 p :=\nby itauto\n\nend itauto\u2080\n\nsection itauto\u2083\n\nexample (p : Prop) : \u00ac (p \u2194 \u00ac p) := by itauto\nexample (p : Prop) : \u00ac (p = \u00ac p) := by itauto\nexample (p : Prop) : p \u2260 \u00ac p := by itauto\n\nexample (p : Prop) : p \u2227 true \u2194 p := by itauto\nexample (p : Prop) : p \u2228 false \u2194 p := by itauto\nexample (p q : Prop) (h0 : q) : p \u2192 q := by itauto\nexample (p q r : Prop) : p \u2228 (q \u2227 r) \u2192 (p \u2228 q) \u2227 (r \u2228 p \u2228 r) := by itauto\nexample (p q r : Prop) : p \u2228 (q \u2227 r) \u2192 (p \u2228 q) \u2227 (r \u2228 p \u2228 r) := by itauto\nexample (p q : Prop) (h : \u00ac (p \u2194 q)) (h' : p) : \u00ac q := by itauto\nexample (p q : Prop) (h : \u00ac (p \u2194 q)) (h' : q) : \u00ac p := by itauto\nexample (p q : Prop) (h : \u00ac (p \u2194 q)) (h' : \u00ac q) (h'' : \u00ac p) : false := by itauto\nexample (p q r : Prop) (h : p \u2194 q) (h' : r \u2194 q) (h'' : \u00ac r) : \u00ac p := by itauto\nexample (p q r : Prop) (h : p \u2194 q) (h' : r \u2194 q) : p \u2194 r := by itauto\nexample (p q : Prop) : xor p q \u2192 (p \u2194 \u00ac q) := by itauto\nexample (p q : Prop) : xor p q \u2192 xor q p := by itauto\n\nexample (p q r : Prop) (h : \u00ac (p \u2194 q)) (h' : r \u2194 q) : \u00ac (p \u2194 r) := by itauto\n\nexample (p : Prop) : p \u2192 \u00ac (p \u2192 \u00ac p) := by itauto\n\nexample (p : Prop) (em : p \u2228 \u00ac p) : \u00ac (p \u2194 \u00ac p) := by itauto\n\nexample (xl yl zl xr yr zr : Prop) :\n  (xl \u2227 yl \u2228 xr \u2227 yr) \u2227 zl \u2228 (xl \u2227 yr \u2228 xr \u2227 yl) \u2227 zr \u2194\n    xl \u2227 (yl \u2227 zl \u2228 yr \u2227 zr) \u2228 xr \u2227 (yl \u2227 zr \u2228 yr \u2227 zl) :=\nby itauto\n\n-- failure tests\nexample (p q r : Prop) : true :=\nbegin\n  have : p \u2228 \u00ac p, {success_if_fail {itauto}, sorry}, clear this,\n  have : \u00ac (p \u2194 q) \u2192 \u00ac p \u2192 q, {success_if_fail {itauto}, sorry}, clear this,\n  have : \u00ac (p \u2194 q) \u2192 (r \u2194 q) \u2192 (p \u2194 \u00ac r), {success_if_fail {itauto}, sorry}, clear this,\n  trivial\nend\n\nexample (P : \u2115 \u2192 Prop) (n : \u2115) (h : \u00ac (n = 7 \u2228 n = 0) \u2227 P n) : \u00ac (P n \u2192 n = 7 \u2228 n = 0) :=\nby itauto\n\nsection modulo_symmetry\nvariables {p q r : Prop} {\u03b1 : Type} {x y : \u03b1}\nvariables (h : x = y)\nvariables (h'' : (p \u2227 q \u2194 q \u2228 r) \u2194 (r \u2227 p \u2194 r \u2228 q))\ninclude h\ninclude h''\nexample (h' : \u00ac x = y) : p \u2227 q := by itauto\nexample : x = y := by itauto\nend modulo_symmetry\n\nend itauto\u2083\nexample (p1 p2 p3 p4 p5 p6 f : Prop)\n  (h : (\n      (p1 \u2227 p2 \u2227 p3 \u2227 p4 \u2227 p5 \u2227 p6 \u2227 true) \u2228\n      (((p1 \u2192 f) \u2192 f) \u2192 f) \u2228\n      (p2 \u2192 f) \u2228\n      (p3 \u2192 f) \u2228\n      (p4 \u2192 f) \u2228\n      (p5 \u2192 f) \u2228\n      (p6 \u2192 f) \u2228\n      false\n    ) \u2192 f) : f :=\nby itauto\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/test/itauto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.37918596361870965}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module category_theory.limits.functor_category\n! leanprover-community/mathlib commit f47581155c818e6361af4e4fda60d27d020c226b\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Preserves.Limits\n\n/-!\n# (Co)limits in functor categories.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe show that if `D` has limits, then the functor category `C \u2964 D` also has limits\n(`category_theory.limits.functor_category_has_limits`),\nand the evaluation functors preserve limits\n(`category_theory.limits.evaluation_preserves_limits`)\n(and similarly for colimits).\n\nWe also show that `F : D \u2964 K \u2964 C` preserves (co)limits if it does so for each `k : K`\n(`category_theory.limits.preserves_limits_of_evaluation` and\n`category_theory.limits.preserves_colimits_of_evaluation`).\n-/\n\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Functor\n\n-- morphism levels before object levels. See note [category_theory universes].\nuniverse w' w v\u2081 v\u2082 u\u2081 u\u2082 v v' u u'\n\nnamespace CategoryTheory.Limits\n\nvariable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]\n\nvariable {J : Type u\u2081} [Category.{v\u2081} J] {K : Type u\u2082} [Category.{v\u2082} K]\n\n/- warning: category_theory.limits.limit.lift_\u03c0_app -> CategoryTheory.Limits.limit.lift_\u03c0_app is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (H : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) [_inst_5 : CategoryTheory.Limits.HasLimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H] (c : CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.limit.lift.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 c) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 j) k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.obj.{max u4 u5, max u3 u4 u5, max u2 u5 u4 u6, max u1 (max u4 u5) u3 u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) j) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.NatTrans.app.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.obj.{max u4 u5, max u3 u4 u5, max u2 u5 u4 u6, max u1 (max u4 u5) u3 u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) H (CategoryTheory.Limits.Cone.\u03c0.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) j) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (H : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) [_inst_5 : CategoryTheory.Limits.HasLimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H] (c : CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.limit.lift.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 c) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 j) k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (Prefunctor.obj.{succ (max u4 u5), max (succ u3) (succ (max u4 u5)), max (max (max u6 u4) u5) u2, max (max (max u3 u1) u4 u5) (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max u4 u5, max u3 u4 u5, max (max (max u6 u4) u5) u2, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c))) j) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.NatTrans.app.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (Prefunctor.obj.{succ (max u4 u5), max (succ u3) (succ (max u4 u5)), max (max (max u6 u4) u5) u2, max (max (max u3 u1) u4 u5) (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max u4 u5, max u3 u4 u5, max (max (max u6 u4) u5) u2, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Limits.Cone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) H (CategoryTheory.Limits.Cone.\u03c0.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) j) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit.lift_\u03c0_app CategoryTheory.Limits.limit.lift_\u03c0_app\u2093'. -/\n@[simp, reassoc.1]\ntheorem limit.lift_\u03c0_app (H : J \u2964 K \u2964 C) [HasLimit H] (c : Cone H) (j : J) (k : K) :\n    (limit.lift H c).app k \u226b (limit.\u03c0 H j).app k = (c.\u03c0.app j).app k :=\n  congr_app (limit.lift_\u03c0 c j) k\n#align category_theory.limits.limit.lift_\u03c0_app CategoryTheory.Limits.limit.lift_\u03c0_app\n\n/- warning: category_theory.limits.colimit.\u03b9_desc_app -> CategoryTheory.Limits.colimit.\u03b9_desc_app is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (H : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H] (c : CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 j) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) (CategoryTheory.Limits.colimit.desc.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 c) k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.obj.{max u4 u5, max u3 u4 u5, max u2 u5 u4 u6, max u1 (max u4 u5) u3 u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) j) (CategoryTheory.NatTrans.app.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Functor.obj.{max u4 u5, max u3 u4 u5, max u2 u5 u4 u6, max u1 (max u4 u5) u3 u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) j) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (H : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H] (c : CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 j) k) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) (CategoryTheory.Limits.colimit.desc.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H _inst_5 c) k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (Prefunctor.obj.{succ (max u4 u5), max (succ u3) (succ (max u4 u5)), max (max (max u6 u4) u5) u2, max (max (max u3 u1) u4 u5) (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max u4 u5, max u3 u4 u5, max (max (max u6 u4) u5) u2, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c))) j) (CategoryTheory.NatTrans.app.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (Prefunctor.obj.{succ (max u4 u5), max (succ u3) (succ (max u4 u5)), max (max (max u6 u4) u5) u2, max (max (max u3 u1) u4 u5) (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max u3 u4 u5, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max u4 u5, max u3 u4 u5, max (max (max u6 u4) u5) u2, max (max (max u3 u1) (max (max u6 u4) u5) u2) u4 u5} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.const.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Limits.Cocone.pt.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H c) j) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit.\u03b9_desc_app CategoryTheory.Limits.colimit.\u03b9_desc_app\u2093'. -/\n@[simp, reassoc.1]\ntheorem colimit.\u03b9_desc_app (H : J \u2964 K \u2964 C) [HasColimit H] (c : Cocone H) (j : J) (k : K) :\n    (colimit.\u03b9 H j).app k \u226b (colimit.desc H c).app k = (c.\u03b9.app j).app k :=\n  congr_app (colimit.\u03b9_desc c j) k\n#align category_theory.limits.colimit.\u03b9_desc_app CategoryTheory.Limits.colimit.\u03b9_desc_app\n\n/- warning: category_theory.limits.evaluation_jointly_reflects_limits -> CategoryTheory.Limits.evaluationJointlyReflectsLimits is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} (c : CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F), (forall (k : K), CategoryTheory.Limits.IsLimit.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Functor.mapCone.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k) c)) -> (CategoryTheory.Limits.IsLimit.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F c)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} (c : CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F), (forall (k : K), CategoryTheory.Limits.IsLimit.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Functor.mapCone.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k) F c)) -> (CategoryTheory.Limits.IsLimit.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F c)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_jointly_reflects_limits CategoryTheory.Limits.evaluationJointlyReflectsLimits\u2093'. -/\n/-- The evaluation functors jointly reflect limits: that is, to show a cone is a limit of `F`\nit suffices to show that each evaluation cone is a limit. In other words, to prove a cone is\nlimiting you can show it's pointwise limiting.\n-/\ndef evaluationJointlyReflectsLimits {F : J \u2964 K \u2964 C} (c : Cone F)\n    (t : \u2200 k : K, IsLimit (((evaluation K C).obj k).mapCone c)) : IsLimit c\n    where\n  lift s :=\n    { app := fun k => (t k).lift \u27e8s.pt.obj k, whiskerRight s.\u03c0 ((evaluation K C).obj k)\u27e9\n      naturality' := fun X Y f =>\n        (t Y).hom_ext fun j => by\n          rw [assoc, (t Y).fac _ j]\n          simpa using\n            ((t X).fac_assoc \u27e8s.X.obj X, whisker_right s.\u03c0 ((evaluation K C).obj X)\u27e9 j _).symm }\n  fac s j := NatTrans.ext _ _ <| funext fun k => (t k).fac _ j\n  uniq s m w :=\n    NatTrans.ext _ _ <|\n      funext fun x =>\n        (t x).hom_ext fun j =>\n          (congr_app (w j) x).trans\n            ((t x).fac \u27e8s.pt.obj _, whiskerRight s.\u03c0 ((evaluation K C).obj _)\u27e9 j).symm\n#align category_theory.limits.evaluation_jointly_reflects_limits CategoryTheory.Limits.evaluationJointlyReflectsLimits\n\n/- warning: category_theory.limits.combine_cones -> CategoryTheory.Limits.combineCones is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)) -> (CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)) -> (CategoryTheory.Limits.Cone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.combine_cones CategoryTheory.Limits.combineCones\u2093'. -/\n/-- Given a functor `F` and a collection of limit cones for each diagram `X \u21a6 F X k`, we can stitch\nthem together to give a cone for the diagram `F`.\n`combined_is_limit` shows that the new cone is limiting, and `eval_combined` shows it is\n(essentially) made up of the original cones.\n-/\n@[simps]\ndef combineCones (F : J \u2964 K \u2964 C) (c : \u2200 k : K, LimitCone (F.flip.obj k)) : Cone F\n    where\n  pt :=\n    { obj := fun k => (c k).Cone.pt\n      map := fun k\u2081 k\u2082 f => (c k\u2082).IsLimit.lift \u27e8_, (c k\u2081).Cone.\u03c0 \u226b F.flip.map f\u27e9\n      map_id' := fun k =>\n        (c k).IsLimit.hom_ext fun j => by\n          dsimp\n          simp\n      map_comp' := fun k\u2081 k\u2082 k\u2083 f\u2081 f\u2082 => (c k\u2083).IsLimit.hom_ext fun j => by simp }\n  \u03c0 :=\n    { app := fun j => { app := fun k => (c k).Cone.\u03c0.app j }\n      naturality' := fun j\u2081 j\u2082 g => NatTrans.ext _ _ <| funext fun k => (c k).Cone.\u03c0.naturality g }\n#align category_theory.limits.combine_cones CategoryTheory.Limits.combineCones\n\n/- warning: category_theory.limits.evaluate_combined_cones -> CategoryTheory.Limits.evaluateCombinedCones is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)) (k : K), CategoryTheory.Iso.{u5, max u3 u6 u5} (CategoryTheory.Limits.Cone.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k))) (CategoryTheory.Limits.Cone.category.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k))) (CategoryTheory.Functor.mapCone.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k) (CategoryTheory.Limits.combineCones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)) (CategoryTheory.Limits.LimitCone.cone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k) (c k))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)) (k : K), CategoryTheory.Iso.{u5, max (max u6 u5) u3} (CategoryTheory.Limits.Cone.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) (CategoryTheory.Limits.Cone.category.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) (CategoryTheory.Functor.mapCone.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k) F (CategoryTheory.Limits.combineCones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)) (CategoryTheory.Limits.LimitCone.cone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k) (c k))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluate_combined_cones CategoryTheory.Limits.evaluateCombinedCones\u2093'. -/\n/-- The stitched together cones each project down to the original given cones (up to iso). -/\ndef evaluateCombinedCones (F : J \u2964 K \u2964 C) (c : \u2200 k : K, LimitCone (F.flip.obj k)) (k : K) :\n    ((evaluation K C).obj k).mapCone (combineCones F c) \u2245 (c k).Cone :=\n  Cones.ext (Iso.refl _) (by tidy)\n#align category_theory.limits.evaluate_combined_cones CategoryTheory.Limits.evaluateCombinedCones\n\n/- warning: category_theory.limits.combined_is_limit -> CategoryTheory.Limits.combinedIsLimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)), CategoryTheory.Limits.IsLimit.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.combineCones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.LimitCone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)), CategoryTheory.Limits.IsLimit.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.combineCones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.combined_is_limit CategoryTheory.Limits.combinedIsLimit\u2093'. -/\n/-- Stitching together limiting cones gives a limiting cone. -/\ndef combinedIsLimit (F : J \u2964 K \u2964 C) (c : \u2200 k : K, LimitCone (F.flip.obj k)) :\n    IsLimit (combineCones F c) :=\n  evaluationJointlyReflectsLimits _ fun k =>\n    (c k).IsLimit.ofIsoLimit (evaluateCombinedCones F c k).symm\n#align category_theory.limits.combined_is_limit CategoryTheory.Limits.combinedIsLimit\n\n/- warning: category_theory.limits.evaluation_jointly_reflects_colimits -> CategoryTheory.Limits.evaluationJointlyReflectsColimits is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} (c : CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F), (forall (k : K), CategoryTheory.Limits.IsColimit.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Functor.mapCocone.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k) c)) -> (CategoryTheory.Limits.IsColimit.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F c)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} (c : CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F), (forall (k : K), CategoryTheory.Limits.IsColimit.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Functor.mapCocone.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k) F c)) -> (CategoryTheory.Limits.IsColimit.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F c)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_jointly_reflects_colimits CategoryTheory.Limits.evaluationJointlyReflectsColimits\u2093'. -/\n/-- The evaluation functors jointly reflect colimits: that is, to show a cocone is a colimit of `F`\nit suffices to show that each evaluation cocone is a colimit. In other words, to prove a cocone is\ncolimiting you can show it's pointwise colimiting.\n-/\ndef evaluationJointlyReflectsColimits {F : J \u2964 K \u2964 C} (c : Cocone F)\n    (t : \u2200 k : K, IsColimit (((evaluation K C).obj k).mapCocone c)) : IsColimit c\n    where\n  desc s :=\n    { app := fun k => (t k).desc \u27e8s.pt.obj k, whiskerRight s.\u03b9 ((evaluation K C).obj k)\u27e9\n      naturality' := fun X Y f =>\n        (t X).hom_ext fun j => by\n          rw [(t X).fac_assoc _ j]\n          erw [\u2190 (c.\u03b9.app j).naturality_assoc f]\n          erw [(t Y).fac \u27e8s.X.obj _, whisker_right s.\u03b9 _\u27e9 j]\n          dsimp\n          simp }\n  fac s j := NatTrans.ext _ _ <| funext fun k => (t k).fac _ j\n  uniq s m w :=\n    NatTrans.ext _ _ <|\n      funext fun x =>\n        (t x).hom_ext fun j =>\n          (congr_app (w j) x).trans\n            ((t x).fac \u27e8s.pt.obj _, whiskerRight s.\u03b9 ((evaluation K C).obj _)\u27e9 j).symm\n#align category_theory.limits.evaluation_jointly_reflects_colimits CategoryTheory.Limits.evaluationJointlyReflectsColimits\n\n/- warning: category_theory.limits.combine_cocones -> CategoryTheory.Limits.combineCocones is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)) -> (CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)) -> (CategoryTheory.Limits.Cocone.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.combine_cocones CategoryTheory.Limits.combineCocones\u2093'. -/\n/--\nGiven a functor `F` and a collection of colimit cocones for each diagram `X \u21a6 F X k`, we can stitch\nthem together to give a cocone for the diagram `F`.\n`combined_is_colimit` shows that the new cocone is colimiting, and `eval_combined` shows it is\n(essentially) made up of the original cocones.\n-/\n@[simps]\ndef combineCocones (F : J \u2964 K \u2964 C) (c : \u2200 k : K, ColimitCocone (F.flip.obj k)) : Cocone F\n    where\n  pt :=\n    { obj := fun k => (c k).Cocone.pt\n      map := fun k\u2081 k\u2082 f => (c k\u2081).IsColimit.desc \u27e8_, F.flip.map f \u226b (c k\u2082).Cocone.\u03b9\u27e9\n      map_id' := fun k =>\n        (c k).IsColimit.hom_ext fun j => by\n          dsimp\n          simp\n      map_comp' := fun k\u2081 k\u2082 k\u2083 f\u2081 f\u2082 => (c k\u2081).IsColimit.hom_ext fun j => by simp }\n  \u03b9 :=\n    { app := fun j => { app := fun k => (c k).Cocone.\u03b9.app j }\n      naturality' := fun j\u2081 j\u2082 g =>\n        NatTrans.ext _ _ <| funext fun k => (c k).Cocone.\u03b9.naturality g }\n#align category_theory.limits.combine_cocones CategoryTheory.Limits.combineCocones\n\n/- warning: category_theory.limits.evaluate_combined_cocones -> CategoryTheory.Limits.evaluateCombinedCocones is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)) (k : K), CategoryTheory.Iso.{u5, max u3 u6 u5} (CategoryTheory.Limits.Cocone.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k))) (CategoryTheory.Limits.Cocone.category.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k))) (CategoryTheory.Functor.mapCocone.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k) (CategoryTheory.Limits.combineCocones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)) (CategoryTheory.Limits.ColimitCocone.cocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k) (c k))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)) (k : K), CategoryTheory.Iso.{u5, max (max u6 u5) u3} (CategoryTheory.Limits.Cocone.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) (CategoryTheory.Limits.Cocone.category.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) (CategoryTheory.Functor.mapCocone.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k) F (CategoryTheory.Limits.combineCocones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)) (CategoryTheory.Limits.ColimitCocone.cocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k) (c k))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluate_combined_cocones CategoryTheory.Limits.evaluateCombinedCocones\u2093'. -/\n/-- The stitched together cocones each project down to the original given cocones (up to iso). -/\ndef evaluateCombinedCocones (F : J \u2964 K \u2964 C) (c : \u2200 k : K, ColimitCocone (F.flip.obj k)) (k : K) :\n    ((evaluation K C).obj k).mapCocone (combineCocones F c) \u2245 (c k).Cocone :=\n  Cocones.ext (Iso.refl _) (by tidy)\n#align category_theory.limits.evaluate_combined_cocones CategoryTheory.Limits.evaluateCombinedCocones\n\n/- warning: category_theory.limits.combined_is_colimit -> CategoryTheory.Limits.combinedIsColimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F) k)), CategoryTheory.Limits.IsColimit.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.combineCocones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (c : forall (k : K), CategoryTheory.Limits.ColimitCocone.{u1, u3, u5, u6} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u3) (succ u5), u4, max (max (max u6 u3) u5) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u3 u5, max (max (max u6 u3) u5) u1} (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.flip.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1 F)) k)), CategoryTheory.Limits.IsColimit.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.combineCocones.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 F c)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.combined_is_colimit CategoryTheory.Limits.combinedIsColimit\u2093'. -/\n/-- Stitching together colimiting cocones gives a colimiting cocone. -/\ndef combinedIsColimit (F : J \u2964 K \u2964 C) (c : \u2200 k : K, ColimitCocone (F.flip.obj k)) :\n    IsColimit (combineCocones F c) :=\n  evaluationJointlyReflectsColimits _ fun k =>\n    (c k).IsColimit.ofIsoColimit (evaluateCombinedCocones F c k).symm\n#align category_theory.limits.combined_is_colimit CategoryTheory.Limits.combinedIsColimit\n\nnoncomputable section\n\n#print CategoryTheory.Limits.functorCategoryHasLimitsOfShape /-\ninstance functorCategoryHasLimitsOfShape [HasLimitsOfShape J C] : HasLimitsOfShape J (K \u2964 C)\n    where HasLimit F :=\n    HasLimit.mk\n      { Cone := combineCones F fun k => getLimitCone _\n        IsLimit := combinedIsLimit _ _ }\n#align category_theory.limits.functor_category_has_limits_of_shape CategoryTheory.Limits.functorCategoryHasLimitsOfShape\n-/\n\n#print CategoryTheory.Limits.functorCategoryHasColimitsOfShape /-\ninstance functorCategoryHasColimitsOfShape [HasColimitsOfShape J C] : HasColimitsOfShape J (K \u2964 C)\n    where HasColimit F :=\n    HasColimit.mk\n      { Cocone := combineCocones _ fun k => getColimitCocone _\n        IsColimit := combinedIsColimit _ _ }\n#align category_theory.limits.functor_category_has_colimits_of_shape CategoryTheory.Limits.functorCategoryHasColimitsOfShape\n-/\n\n#print CategoryTheory.Limits.functorCategoryHasLimitsOfSize /-\ninstance functorCategoryHasLimitsOfSize [HasLimitsOfSize.{v\u2081, u\u2081} C] :\n    HasLimitsOfSize.{v\u2081, u\u2081} (K \u2964 C) :=\n  \u27e8inferInstance\u27e9\n#align category_theory.limits.functor_category_has_limits_of_size CategoryTheory.Limits.functorCategoryHasLimitsOfSize\n-/\n\n#print CategoryTheory.Limits.functorCategoryHasColimitsOfSize /-\ninstance functorCategoryHasColimitsOfSize [HasColimitsOfSize.{v\u2081, u\u2081} C] :\n    HasColimitsOfSize.{v\u2081, u\u2081} (K \u2964 C) :=\n  \u27e8inferInstance\u27e9\n#align category_theory.limits.functor_category_has_colimits_of_size CategoryTheory.Limits.functorCategoryHasColimitsOfSize\n-/\n\n/- warning: category_theory.limits.evaluation_preserves_limits_of_shape -> CategoryTheory.Limits.evaluationPreservesLimitsOfShape is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (k : K), CategoryTheory.Limits.PreservesLimitsOfShape.{u1, u3, max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 J _inst_3 (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (k : K), CategoryTheory.Limits.PreservesLimitsOfShape.{u1, u3, max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 J _inst_3 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_limits_of_shape CategoryTheory.Limits.evaluationPreservesLimitsOfShape\u2093'. -/\ninstance evaluationPreservesLimitsOfShape [HasLimitsOfShape J C] (k : K) :\n    PreservesLimitsOfShape J ((evaluation K C).obj k)\n    where PreservesLimit F :=\n    preservesLimitOfPreservesLimitCone (combinedIsLimit _ _) <|\n      IsLimit.ofIsoLimit (limit.isLimit _) (evaluateCombinedCones F _ k).symm\n#align category_theory.limits.evaluation_preserves_limits_of_shape CategoryTheory.Limits.evaluationPreservesLimitsOfShape\n\n/- warning: category_theory.limits.limit_obj_iso_limit_comp_evaluation -> CategoryTheory.Limits.limitObjIsoLimitCompEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (k : K), CategoryTheory.Iso.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (k : K), CategoryTheory.Iso.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation CategoryTheory.Limits.limitObjIsoLimitCompEvaluation\u2093'. -/\n/-- If `F : J \u2964 K \u2964 C` is a functor into a functor category which has a limit,\nthen the evaluation of that limit at `k` is the limit of the evaluations of `F.obj j` at `k`.\n-/\ndef limitObjIsoLimitCompEvaluation [HasLimitsOfShape J C] (F : J \u2964 K \u2964 C) (k : K) :\n    (limit F).obj k \u2245 limit (F \u22d9 (evaluation K C).obj k) :=\n  preservesLimitIso ((evaluation K C).obj k) F\n#align category_theory.limits.limit_obj_iso_limit_comp_evaluation CategoryTheory.Limits.limitObjIsoLimitCompEvaluation\n\n/- warning: category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_\u03c0 -> CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_\u03c0 is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Functor.obj.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Functor.obj.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) j) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k) j)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F) j) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (Prefunctor.obj.{succ u1, succ u5, u3, u6} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (Prefunctor.obj.{succ u1, succ u5, u3, u6} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F)) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F) j) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_\u03c0 CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_\u03c0\u2093'. -/\n@[simp, reassoc.1]\ntheorem limitObjIsoLimitCompEvaluation_hom_\u03c0 [HasLimitsOfShape J C] (F : J \u2964 K \u2964 C) (j : J)\n    (k : K) :\n    (limitObjIsoLimitCompEvaluation F k).Hom \u226b limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j =\n      (limit.\u03c0 F j).app k :=\n  by\n  dsimp [limit_obj_iso_limit_comp_evaluation]\n  simp\n#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_hom_\u03c0 CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_hom_\u03c0\n\n/- warning: category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app -> CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_\u03c0_app is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) k) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F) j) k)) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k) j)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j)) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j)) k) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F)) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F) j) k)) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_\u03c0_app\u2093'. -/\n@[simp, reassoc.1]\ntheorem limitObjIsoLimitCompEvaluation_inv_\u03c0_app [HasLimitsOfShape J C] (F : J \u2964 K \u2964 C) (j : J)\n    (k : K) :\n    (limitObjIsoLimitCompEvaluation F k).inv \u226b (limit.\u03c0 F j).app k =\n      limit.\u03c0 (F \u22d9 (evaluation K C).obj k) j :=\n  by\n  dsimp [limit_obj_iso_limit_comp_evaluation]\n  rw [iso.inv_comp_eq]\n  simp\n#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_\u03c0_app\n\n/- warning: category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom -> CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {i : K} {j : K} (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Functor.map.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i j f) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j) (CategoryTheory.whiskerLeft.{u3, u1, max u2 u5 u4 u6, max u4 u5, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i) (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j) (CategoryTheory.Functor.map.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i j f))))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {i : K} {j : K} (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (Prefunctor.map.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i j f) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i))) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))) (CategoryTheory.whiskerLeft.{u3, u1, max (max (max u6 u5) u2) u4, max u5 u4, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i) (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j) (Prefunctor.map.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i j f))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom\u2093'. -/\n@[simp, reassoc.1]\ntheorem limit_map_limitObjIsoLimitCompEvaluation_hom [HasLimitsOfShape J C] {i j : K}\n    (F : J \u2964 K \u2964 C) (f : i \u27f6 j) :\n    (limit F).map f \u226b (limitObjIsoLimitCompEvaluation _ _).Hom =\n      (limitObjIsoLimitCompEvaluation _ _).Hom \u226b limMap (whiskerLeft _ ((evaluation _ _).map f)) :=\n  by\n  ext\n  dsimp\n  simp\n#align category_theory.limits.limit_map_limit_obj_iso_limit_comp_evaluation_hom CategoryTheory.Limits.limit_map_limitObjIsoLimitCompEvaluation_hom\n\n/- warning: category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map -> CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {i : K} {j : K} (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.map.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i j f)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.limMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j) (CategoryTheory.whiskerLeft.{u3, u1, max u2 u5 u4 u6, max u4 u5, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i) (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j) (CategoryTheory.Functor.map.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i j f))) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {i : K} {j : K} (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (Prefunctor.map.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i j f)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.limMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i))) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))) (CategoryTheory.whiskerLeft.{u3, u1, max (max (max u6 u5) u2) u4, max u5 u4, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i) (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j) (Prefunctor.map.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i j f))) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.limit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Limits.limitObjIsoLimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map\u2093'. -/\n@[simp, reassoc.1]\ntheorem limitObjIsoLimitCompEvaluation_inv_limit_map [HasLimitsOfShape J C] {i j : K}\n    (F : J \u2964 K \u2964 C) (f : i \u27f6 j) :\n    (limitObjIsoLimitCompEvaluation _ _).inv \u226b (limit F).map f =\n      limMap (whiskerLeft _ ((evaluation _ _).map f)) \u226b (limitObjIsoLimitCompEvaluation _ _).inv :=\n  by\n  rw [iso.inv_comp_eq, \u2190 category.assoc, iso.eq_comp_inv,\n    limit_map_limit_obj_iso_limit_comp_evaluation_hom]\n#align category_theory.limits.limit_obj_iso_limit_comp_evaluation_inv_limit_map CategoryTheory.Limits.limitObjIsoLimitCompEvaluation_inv_limit_map\n\n/- warning: category_theory.limits.limit_obj_ext -> CategoryTheory.Limits.limit_obj_ext is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {H : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {k : K} {W : C} {f : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k)} {g : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k)}, (forall (j : J), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) f (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) g (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k))) -> (Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k)) f g)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {H : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {k : K} {W : C} {f : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k)} {g : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k)}, (forall (j : J), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) f (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) g (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k))) -> (Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) W (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k)) f g)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_obj_ext CategoryTheory.Limits.limit_obj_ext\u2093'. -/\n@[ext]\ntheorem limit_obj_ext {H : J \u2964 K \u2964 C} [HasLimitsOfShape J C] {k : K} {W : C}\n    {f g : W \u27f6 (limit H).obj k}\n    (w : \u2200 j, f \u226b (Limits.limit.\u03c0 H j).app k = g \u226b (Limits.limit.\u03c0 H j).app k) : f = g :=\n  by\n  apply (cancel_mono (limit_obj_iso_limit_comp_evaluation H k).Hom).1\n  ext\n  simpa using w j\n#align category_theory.limits.limit_obj_ext CategoryTheory.Limits.limit_obj_ext\n\n/- warning: category_theory.limits.evaluation_preserves_colimits_of_shape -> CategoryTheory.Limits.evaluationPreservesColimitsOfShape is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (k : K), CategoryTheory.Limits.PreservesColimitsOfShape.{u1, u3, max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 J _inst_3 (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (k : K), CategoryTheory.Limits.PreservesColimitsOfShape.{u1, u3, max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 J _inst_3 (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_colimits_of_shape CategoryTheory.Limits.evaluationPreservesColimitsOfShape\u2093'. -/\ninstance evaluationPreservesColimitsOfShape [HasColimitsOfShape J C] (k : K) :\n    PreservesColimitsOfShape J ((evaluation K C).obj k)\n    where PreservesColimit F :=\n    preservesColimitOfPreservesColimitCocone (combinedIsColimit _ _) <|\n      IsColimit.ofIsoColimit (colimit.isColimit _) (evaluateCombinedCocones F _ k).symm\n#align category_theory.limits.evaluation_preserves_colimits_of_shape CategoryTheory.Limits.evaluationPreservesColimitsOfShape\n\n/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (k : K), CategoryTheory.Iso.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (k : K), CategoryTheory.Iso.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation\u2093'. -/\n/-- If `F : J \u2964 K \u2964 C` is a functor into a functor category which has a colimit,\nthen the evaluation of that colimit at `k` is the colimit of the evaluations of `F.obj j` at `k`.\n-/\ndef colimitObjIsoColimitCompEvaluation [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C) (k : K) :\n    (colimit F).obj k \u2245 colimit (F \u22d9 (evaluation K C).obj k) :=\n  preservesColimitIso ((evaluation K C).obj k) F\n#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation\n\n/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_inv is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) j) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)))) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k))) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F) j) k)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u1, succ u5, u3, u6} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u1, succ u5, u3, u6} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u5, u3, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F) j) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_inv\u2093'. -/\n@[simp, reassoc.1]\ntheorem colimitObjIsoColimitCompEvaluation_\u03b9_inv [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C) (j : J)\n    (k : K) :\n    colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j \u226b (colimitObjIsoColimitCompEvaluation F k).inv =\n      (colimit.\u03b9 F j).app k :=\n  by\n  dsimp [colimit_obj_iso_colimit_comp_evaluation]\n  simp\n#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_inv CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_inv\n\n/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_app_hom is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F) j) k) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) k)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k) j)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (j : J) (k : K), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j)) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F) j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F) j) k) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) k) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F k))) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) k))) j)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_app_hom\u2093'. -/\n@[simp, reassoc.1]\ntheorem colimitObjIsoColimitCompEvaluation_\u03b9_app_hom [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C)\n    (j : J) (k : K) :\n    (colimit.\u03b9 F j).app k \u226b (colimitObjIsoColimitCompEvaluation F k).Hom =\n      colimit.\u03b9 (F \u22d9 (evaluation K C).obj k) j :=\n  by\n  dsimp [colimit_obj_iso_colimit_comp_evaluation]\n  rw [\u2190 iso.eq_comp_inv]\n  simp\n#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_\u03b9_app_hom CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_\u03b9_app_hom\n\n/- warning: category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map -> CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) {i : K} {j : K} (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Functor.map.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i j f)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.colimMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j) (CategoryTheory.whiskerLeft.{u3, u1, max u2 u5 u4 u6, max u4 u5, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i) (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j) (CategoryTheory.Functor.map.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i j f))) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) {i : K} {j : K} (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (Prefunctor.map.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i j f)) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.colimMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i))) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))) (CategoryTheory.whiskerLeft.{u3, u1, max (max (max u6 u5) u2) u4, max u5 u4, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i) (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j) (Prefunctor.map.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i j f))) (CategoryTheory.Iso.inv.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map\u2093'. -/\n@[simp, reassoc.1]\ntheorem colimitObjIsoColimitCompEvaluation_inv_colimit_map [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C)\n    {i j : K} (f : i \u27f6 j) :\n    (colimitObjIsoColimitCompEvaluation _ _).inv \u226b (colimit F).map f =\n      colimMap (whiskerLeft _ ((evaluation _ _).map f)) \u226b\n        (colimitObjIsoColimitCompEvaluation _ _).inv :=\n  by\n  ext\n  dsimp\n  simp\n#align category_theory.limits.colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation_inv_colimit_map\n\n/- warning: category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom -> CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_hom is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) {i : K} {j : K} (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Functor.map.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i j f) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j)) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F)) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i)) (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max u2 u5 u4 u6, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j)) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation._proof_2.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j) (CategoryTheory.whiskerLeft.{u3, u1, max u2 u5 u4 u6, max u4 u5, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i) (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) j) (CategoryTheory.Functor.map.{u2, max (max u2 u5 u4 u6) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u6) u6} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u6, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1) i j f))))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (F : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) {i : K} {j : K} (f : Quiver.Hom.{succ u2, u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) i j), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (Prefunctor.map.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i j f) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) j) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F j))) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)))) (CategoryTheory.Iso.hom.{u5, u6} C _inst_1 (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) F (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) F))) i) (CategoryTheory.Limits.colimit.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)))) (CategoryTheory.Limits.colimitObjIsoColimitCompEvaluation.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5 F i)) (CategoryTheory.Limits.colimMap.{u1, u3, u5, u6} J _inst_3 C _inst_1 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i)) (CategoryTheory.Functor.comp.{u1, max u5 u4, u5, u3, max (max (max u6 u5) u2) u4, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j)) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i))) (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, u5, u6} C _inst_1 J _inst_3 _inst_5 (CategoryTheory.Functor.comp.{u1, max u4 u5, u5, u3, max (max (max u6 u4) u5) u2, u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u6 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u4) u5) u2, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u4) u5) u2, u4, max (max (max u6 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j))) (CategoryTheory.whiskerLeft.{u3, u1, max (max (max u6 u5) u2) u4, max u5 u4, u6, u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i) (Prefunctor.obj.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) j) (Prefunctor.map.{succ u2, max (max (max (succ u6) (succ u5)) (succ u2)) (succ u4), u4, max (max (max u6 u5) u2) u4} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u6 u5) u4) u2, max (max (max u6 u5) u4) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u6 u5) u4) u2, u4, max (max (max u6 u5) u4) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u6 u4) u5) u2, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u6) u2) u5, u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u6} K _inst_4 C _inst_1)) i j f))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_hom\u2093'. -/\n@[simp, reassoc.1]\ntheorem colimit_map_colimitObjIsoColimitCompEvaluation_hom [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C)\n    {i j : K} (f : i \u27f6 j) :\n    (colimit F).map f \u226b (colimitObjIsoColimitCompEvaluation _ _).Hom =\n      (colimitObjIsoColimitCompEvaluation _ _).Hom \u226b\n        colimMap (whiskerLeft _ ((evaluation _ _).map f)) :=\n  by\n  rw [\u2190 iso.inv_comp_eq, \u2190 category.assoc, \u2190 iso.eq_comp_inv,\n    colimit_obj_iso_colimit_comp_evaluation_inv_colimit_map]\n#align category_theory.limits.colimit_map_colimit_obj_iso_colimit_comp_evaluation_hom CategoryTheory.Limits.colimit_map_colimitObjIsoColimitCompEvaluation_hom\n\n/- warning: category_theory.limits.colimit_obj_ext -> CategoryTheory.Limits.colimit_obj_ext is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {H : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {k : K} {W : C} {f : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) W} {g : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) W}, (forall (j : J), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) W) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) W (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k) f) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) k) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) W (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Functor.obj.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k) g)) -> (Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.obj.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) k) W) f g)\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] {H : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)} [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] {k : K} {W : C} {f : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W} {g : Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W}, (forall (j : J), Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) W) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k) f) (CategoryTheory.CategoryStruct.comp.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j)) k) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W (CategoryTheory.NatTrans.app.{u2, u5, u4, u6} K _inst_4 C _inst_1 (Prefunctor.obj.{succ u1, succ (max u4 u5), u3, max (max (max u6 u4) u5) u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u3} J (CategoryTheory.Category.toCategoryStruct.{u1, u3} J _inst_3)) (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H) j) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H) j) k) g)) -> (Eq.{succ u5} (Quiver.Hom.{succ u5, u6} C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (Prefunctor.obj.{succ u2, succ u5, u4, u6} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) C (CategoryTheory.CategoryStruct.toQuiver.{u5, u6} C (CategoryTheory.Category.toCategoryStruct.{u5, u6} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u2, u5, u4, u6} K _inst_4 C _inst_1 (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) H (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) H))) k) W) f g)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_obj_ext CategoryTheory.Limits.colimit_obj_ext\u2093'. -/\n@[ext]\ntheorem colimit_obj_ext {H : J \u2964 K \u2964 C} [HasColimitsOfShape J C] {k : K} {W : C}\n    {f g : (colimit H).obj k \u27f6 W} (w : \u2200 j, (colimit.\u03b9 H j).app k \u226b f = (colimit.\u03b9 H j).app k \u226b g) :\n    f = g := by\n  apply (cancel_epi (colimit_obj_iso_colimit_comp_evaluation H k).inv).1\n  ext\n  simpa using w j\n#align category_theory.limits.colimit_obj_ext CategoryTheory.Limits.colimit_obj_ext\n\n/- warning: category_theory.limits.evaluation_preserves_limits -> CategoryTheory.Limits.evaluationPreservesLimits is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] [_inst_5 : CategoryTheory.Limits.HasLimits.{u3, u4} C _inst_1] (k : K), CategoryTheory.Limits.PreservesLimits.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1 (CategoryTheory.Functor.obj.{u1, max (max u1 u3 u2 u4) u3, u2, max (max u2 u3) u3 (max u1 u3 u2 u4) u4} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u4} K _inst_4 C _inst_1) k)\nbut is expected to have type\n  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] [_inst_5 : CategoryTheory.Limits.HasLimits.{u3, u4} C _inst_1] (k : K), CategoryTheory.Limits.PreservesLimits.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u1, max (max (max (succ u4) (succ u2)) (succ u3)) (succ u1), u2, max (max (max u4 u2) u3) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} K (CategoryTheory.Category.toCategoryStruct.{u1, u2} K _inst_4)) (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u4 u2) u3) u1, max (max (max u4 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u4 u2) u3) u1, max (max (max u4 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u4) u1) u3, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max (max (max u4 u2) u3) u1, u2, max (max (max u4 u2) u3) u1} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u4) u1) u3, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u4} K _inst_4 C _inst_1)) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_limits CategoryTheory.Limits.evaluationPreservesLimits\u2093'. -/\ninstance evaluationPreservesLimits [HasLimits C] (k : K) : PreservesLimits ((evaluation K C).obj k)\n    where PreservesLimitsOfShape J \ud835\udca5 := by skip <;> infer_instance\n#align category_theory.limits.evaluation_preserves_limits CategoryTheory.Limits.evaluationPreservesLimits\n\n/- warning: category_theory.limits.preserves_limit_of_evaluation -> CategoryTheory.Limits.preservesLimitOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max u2 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1)) (G : CategoryTheory.Functor.{u1, u6, u3, u8} J _inst_3 D _inst_2), (forall (k : K), CategoryTheory.Limits.PreservesLimit.{u1, u3, u6, u5, u8, u7} D _inst_2 C _inst_1 J _inst_3 G (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max u2 u5 u4 u7, u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u7) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u7) u7} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u7, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u7, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u7} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesLimit.{u1, u3, u6, max u4 u5, u8, max u2 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) J _inst_3 G F)\nbut is expected to have type\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max (max (max u7 u4) u5) u2} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1)) (G : CategoryTheory.Functor.{u1, u6, u3, u8} J _inst_3 D _inst_2), (forall (k : K), CategoryTheory.Limits.PreservesLimit.{u1, u3, u6, u5, u8, u7} D _inst_2 C _inst_1 J _inst_3 G (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max (max (max u7 u4) u5) u2, u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u7) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u7 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u7 u4) u5) u2, max (max (max u7 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u7 u4) u5) u2, max (max (max u7 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u2) u5, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u7 u4) u5) u2, u4, max (max (max u7 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u2) u5, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u7} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesLimit.{u1, u3, u6, max u4 u5, u8, max (max (max u7 u4) u5) u2} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) J _inst_3 G F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limit_of_evaluation CategoryTheory.Limits.preservesLimitOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves the limit of some `G : J \u2964 D` if it does for each `k : K`. -/\ndef preservesLimitOfEvaluation (F : D \u2964 K \u2964 C) (G : J \u2964 D)\n    (H : \u2200 k : K, PreservesLimit G (F \u22d9 (evaluation K C).obj k : D \u2964 C)) : PreservesLimit G F :=\n  \u27e8fun c hc => by\n    apply evaluation_jointly_reflects_limits\n    intro X\n    haveI := H X\n    change is_limit ((F \u22d9 (evaluation K C).obj X).mapCone c)\n    exact preserves_limit.preserves hc\u27e9\n#align category_theory.limits.preserves_limit_of_evaluation CategoryTheory.Limits.preservesLimitOfEvaluation\n\n/- warning: category_theory.limits.preserves_limits_of_shape_of_evaluation -> CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u5}} [_inst_1 : CategoryTheory.Category.{u3, u5} C] {D : Type.{u6}} [_inst_2 : CategoryTheory.Category.{u4, u6} D] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] (F : CategoryTheory.Functor.{u4, max u2 u3, u6, max u1 u3 u2 u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1)) (J : Type.{u7}) [_inst_5 : CategoryTheory.Category.{u8, u7} J], (forall (k : K), CategoryTheory.Limits.PreservesLimitsOfShape.{u8, u7, u4, u3, u6, u5} D _inst_2 C _inst_1 J _inst_5 (CategoryTheory.Functor.comp.{u4, max u2 u3, u3, u6, max u1 u3 u2 u5, u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u1, max (max u1 u3 u2 u5) u3, u2, max (max u2 u3) u3 (max u1 u3 u2 u5) u5} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max u1 u3 u2 u5, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max u1 u3 u2 u5, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u5} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesLimitsOfShape.{u8, u7, u4, max u2 u3, u6, max u1 u3 u2 u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) J _inst_5 F)\nbut is expected to have type\n  forall {C : Type.{u5}} [_inst_1 : CategoryTheory.Category.{u3, u5} C] {D : Type.{u6}} [_inst_2 : CategoryTheory.Category.{u4, u6} D] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] (F : CategoryTheory.Functor.{u4, max u2 u3, u6, max (max (max u5 u2) u3) u1} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1)) (J : Type.{u7}) [_inst_5 : CategoryTheory.Category.{u8, u7} J], (forall (k : K), CategoryTheory.Limits.PreservesLimitsOfShape.{u8, u7, u4, u3, u6, u5} D _inst_2 C _inst_1 J _inst_5 (CategoryTheory.Functor.comp.{u4, max u2 u3, u3, u6, max (max (max u5 u2) u3) u1, u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u1, max (max (max (succ u5) (succ u2)) (succ u3)) (succ u1), u2, max (max (max u5 u2) u3) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} K (CategoryTheory.Category.toCategoryStruct.{u1, u2} K _inst_4)) (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u5 u2) u3) u1, max (max (max u5 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u5 u2) u3) u1, max (max (max u5 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u5) u1) u3, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max (max (max u5 u2) u3) u1, u2, max (max (max u5 u2) u3) u1} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u5) u1) u3, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u5} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesLimitsOfShape.{u8, u7, u4, max u2 u3, u6, max (max (max u5 u2) u3) u1} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) J _inst_5 F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limits_of_shape_of_evaluation CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves limits of shape `J` if it does for each `k : K`. -/\ndef preservesLimitsOfShapeOfEvaluation (F : D \u2964 K \u2964 C) (J : Type _) [Category J]\n    (H : \u2200 k : K, PreservesLimitsOfShape J (F \u22d9 (evaluation K C).obj k)) :\n    PreservesLimitsOfShape J F :=\n  \u27e8fun G => preservesLimitOfEvaluation F G fun k => PreservesLimitsOfShape.preservesLimit\u27e9\n#align category_theory.limits.preserves_limits_of_shape_of_evaluation CategoryTheory.Limits.preservesLimitsOfShapeOfEvaluation\n\n/- warning: category_theory.limits.preserves_limits_of_evaluation -> CategoryTheory.Limits.preservesLimitsOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u3, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max u3 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.PreservesLimitsOfSize.{u1, u2, u6, u5, u8, u7} D _inst_2 C _inst_1 (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max u3 u5 u4 u7, u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u3, max (max u3 u5 u4 u7) u5, u4, max (max u4 u5) u5 (max u3 u5 u4 u7) u7} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u3 u5 u4 u7, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u3 u5 u4 u7, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u3, u5, u4, u7} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesLimitsOfSize.{u1, u2, u6, max u4 u5, u8, max u3 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) F)\nbut is expected to have type\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u3, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max (max (max u7 u4) u5) u3} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.PreservesLimitsOfSize.{u1, u2, u6, u5, u8, u7} D _inst_2 C _inst_1 (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max (max (max u7 u4) u5) u3, u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u3, max (max (max (succ u7) (succ u4)) (succ u5)) (succ u3), u4, max (max (max u7 u4) u5) u3} K (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} K (CategoryTheory.Category.toCategoryStruct.{u3, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u7 u4) u5) u3, max (max (max u7 u4) u5) u3} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u7 u4) u5) u3, max (max (max u7 u4) u5) u3} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u3) u5, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u3, max (max (max u7 u4) u5) u3, u4, max (max (max u7 u4) u5) u3} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u3) u5, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u3, u5, u4, u7} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesLimitsOfSize.{u1, u2, u6, max u4 u5, u8, max (max (max u7 u4) u5) u3} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_limits_of_evaluation CategoryTheory.Limits.preservesLimitsOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves all limits if it does for each `k : K`. -/\ndef preservesLimitsOfEvaluation (F : D \u2964 K \u2964 C)\n    (H : \u2200 k : K, PreservesLimitsOfSize.{w', w} (F \u22d9 (evaluation K C).obj k)) :\n    PreservesLimitsOfSize.{w', w} F :=\n  \u27e8fun L hL =>\n    preserves_limits_of_shape_of_evaluation F L fun k =>\n      preserves_limits_of_size.preserves_limits_of_shape\u27e9\n#align category_theory.limits.preserves_limits_of_evaluation CategoryTheory.Limits.preservesLimitsOfEvaluation\n\n#print CategoryTheory.Limits.preservesLimitsConst /-\n/-- The constant functor `C \u2964 (D \u2964 C)` preserves limits. -/\ninstance preservesLimitsConst : PreservesLimitsOfSize.{w', w} (const D : C \u2964 _) :=\n  preservesLimitsOfEvaluation _ fun X =>\n    preservesLimitsOfNatIso <| Iso.symm <| constCompEvaluationObj _ _\n#align category_theory.limits.preserves_limits_const CategoryTheory.Limits.preservesLimitsConst\n-/\n\n/- warning: category_theory.limits.evaluation_preserves_colimits -> CategoryTheory.Limits.evaluationPreservesColimits is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] [_inst_5 : CategoryTheory.Limits.HasColimits.{u3, u4} C _inst_1] (k : K), CategoryTheory.Limits.PreservesColimits.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1 (CategoryTheory.Functor.obj.{u1, max (max u1 u3 u2 u4) u3, u2, max (max u2 u3) u3 (max u1 u3 u2 u4) u4} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max u1 u3 u2 u4, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u4} K _inst_4 C _inst_1) k)\nbut is expected to have type\n  forall {C : Type.{u4}} [_inst_1 : CategoryTheory.Category.{u3, u4} C] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] [_inst_5 : CategoryTheory.Limits.HasColimits.{u3, u4} C _inst_1] (k : K), CategoryTheory.Limits.PreservesColimits.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1 (Prefunctor.obj.{succ u1, max (max (max (succ u4) (succ u2)) (succ u3)) (succ u1), u2, max (max (max u4 u2) u3) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} K (CategoryTheory.Category.toCategoryStruct.{u1, u2} K _inst_4)) (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u4 u2) u3) u1, max (max (max u4 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u4 u2) u3) u1, max (max (max u4 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u4) u1) u3, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max (max (max u4 u2) u3) u1, u2, max (max (max u4 u2) u3) u1} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u4 u2) u3) u1, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u4) u1) u3, u4} (CategoryTheory.Functor.{u1, u3, u2, u4} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u4} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u4} K _inst_4 C _inst_1)) k)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.evaluation_preserves_colimits CategoryTheory.Limits.evaluationPreservesColimits\u2093'. -/\ninstance evaluationPreservesColimits [HasColimits C] (k : K) :\n    PreservesColimits ((evaluation K C).obj k)\n    where PreservesColimitsOfShape J \ud835\udca5 := by skip <;> infer_instance\n#align category_theory.limits.evaluation_preserves_colimits CategoryTheory.Limits.evaluationPreservesColimits\n\n/- warning: category_theory.limits.preserves_colimit_of_evaluation -> CategoryTheory.Limits.preservesColimitOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max u2 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1)) (G : CategoryTheory.Functor.{u1, u6, u3, u8} J _inst_3 D _inst_2), (forall (k : K), CategoryTheory.Limits.PreservesColimit.{u1, u3, u6, u5, u8, u7} D _inst_2 C _inst_1 J _inst_3 G (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max u2 u5 u4 u7, u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u2, max (max u2 u5 u4 u7) u5, u4, max (max u4 u5) u5 (max u2 u5 u4 u7) u7} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u2 u5 u4 u7, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u2 u5 u4 u7, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u7} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesColimit.{u1, u3, u6, max u4 u5, u8, max u2 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) J _inst_3 G F)\nbut is expected to have type\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max (max (max u7 u4) u5) u2} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1)) (G : CategoryTheory.Functor.{u1, u6, u3, u8} J _inst_3 D _inst_2), (forall (k : K), CategoryTheory.Limits.PreservesColimit.{u1, u3, u6, u5, u8, u7} D _inst_2 C _inst_1 J _inst_3 G (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max (max (max u7 u4) u5) u2, u7} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u2, max (max (max (succ u7) (succ u4)) (succ u5)) (succ u2), u4, max (max (max u7 u4) u5) u2} K (CategoryTheory.CategoryStruct.toQuiver.{u2, u4} K (CategoryTheory.Category.toCategoryStruct.{u2, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u7 u4) u5) u2, max (max (max u7 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u7 u4) u5) u2, max (max (max u7 u4) u5) u2} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u2) u5, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max (max (max u7 u4) u5) u2, u4, max (max (max u7 u4) u5) u2} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u2, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u2) u5, u7} (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u2, u5, u4, u7} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesColimit.{u1, u3, u6, max u4 u5, u8, max (max (max u7 u4) u5) u2} D _inst_2 (CategoryTheory.Functor.{u2, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u7} K _inst_4 C _inst_1) J _inst_3 G F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimit_of_evaluation CategoryTheory.Limits.preservesColimitOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves the colimit of some `G : J \u2964 D` if it does for each `k : K`. -/\ndef preservesColimitOfEvaluation (F : D \u2964 K \u2964 C) (G : J \u2964 D)\n    (H : \u2200 k, PreservesColimit G (F \u22d9 (evaluation K C).obj k)) : PreservesColimit G F :=\n  \u27e8fun c hc => by\n    apply evaluation_jointly_reflects_colimits\n    intro X\n    haveI := H X\n    change is_colimit ((F \u22d9 (evaluation K C).obj X).mapCocone c)\n    exact preserves_colimit.preserves hc\u27e9\n#align category_theory.limits.preserves_colimit_of_evaluation CategoryTheory.Limits.preservesColimitOfEvaluation\n\n/- warning: category_theory.limits.preserves_colimits_of_shape_of_evaluation -> CategoryTheory.Limits.preservesColimitsOfShapeOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u5}} [_inst_1 : CategoryTheory.Category.{u3, u5} C] {D : Type.{u6}} [_inst_2 : CategoryTheory.Category.{u4, u6} D] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] (F : CategoryTheory.Functor.{u4, max u2 u3, u6, max u1 u3 u2 u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1)) (J : Type.{u7}) [_inst_5 : CategoryTheory.Category.{u8, u7} J], (forall (k : K), CategoryTheory.Limits.PreservesColimitsOfShape.{u8, u7, u4, u3, u6, u5} D _inst_2 C _inst_1 J _inst_5 (CategoryTheory.Functor.comp.{u4, max u2 u3, u3, u6, max u1 u3 u2 u5, u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u1, max (max u1 u3 u2 u5) u3, u2, max (max u2 u3) u3 (max u1 u3 u2 u5) u5} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max u1 u3 u2 u5, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max u1 u3 u2 u5, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u5} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesColimitsOfShape.{u8, u7, u4, max u2 u3, u6, max u1 u3 u2 u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) J _inst_5 F)\nbut is expected to have type\n  forall {C : Type.{u5}} [_inst_1 : CategoryTheory.Category.{u3, u5} C] {D : Type.{u6}} [_inst_2 : CategoryTheory.Category.{u4, u6} D] {K : Type.{u2}} [_inst_4 : CategoryTheory.Category.{u1, u2} K] (F : CategoryTheory.Functor.{u4, max u2 u3, u6, max (max (max u5 u2) u3) u1} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1)) (J : Type.{u7}) [_inst_5 : CategoryTheory.Category.{u8, u7} J], (forall (k : K), CategoryTheory.Limits.PreservesColimitsOfShape.{u8, u7, u4, u3, u6, u5} D _inst_2 C _inst_1 J _inst_5 (CategoryTheory.Functor.comp.{u4, max u2 u3, u3, u6, max (max (max u5 u2) u3) u1, u5} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u1, max (max (max (succ u5) (succ u2)) (succ u3)) (succ u1), u2, max (max (max u5 u2) u3) u1} K (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} K (CategoryTheory.Category.toCategoryStruct.{u1, u2} K _inst_4)) (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u5 u2) u3) u1, max (max (max u5 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u5 u2) u3) u1, max (max (max u5 u2) u3) u1} (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u5) u1) u3, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, max (max (max u5 u2) u3) u1, u2, max (max (max u5 u2) u3) u1} K _inst_4 (CategoryTheory.Functor.{max u2 u3, u3, max (max (max u5 u2) u3) u1, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u2 u3, u3, max (max (max u2 u5) u1) u3, u5} (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u1, u3, u2, u5} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesColimitsOfShape.{u8, u7, u4, max u2 u3, u6, max (max (max u5 u2) u3) u1} D _inst_2 (CategoryTheory.Functor.{u1, u3, u2, u5} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u1, u3, u2, u5} K _inst_4 C _inst_1) J _inst_5 F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimits_of_shape_of_evaluation CategoryTheory.Limits.preservesColimitsOfShapeOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves all colimits of shape `J` if it does for each `k : K`. -/\ndef preservesColimitsOfShapeOfEvaluation (F : D \u2964 K \u2964 C) (J : Type _) [Category J]\n    (H : \u2200 k : K, PreservesColimitsOfShape J (F \u22d9 (evaluation K C).obj k)) :\n    PreservesColimitsOfShape J F :=\n  \u27e8fun G => preservesColimitOfEvaluation F G fun k => PreservesColimitsOfShape.preservesColimit\u27e9\n#align category_theory.limits.preserves_colimits_of_shape_of_evaluation CategoryTheory.Limits.preservesColimitsOfShapeOfEvaluation\n\n/- warning: category_theory.limits.preserves_colimits_of_evaluation -> CategoryTheory.Limits.preservesColimitsOfEvaluation is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u3, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max u3 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.PreservesColimitsOfSize.{u1, u2, u6, u5, u8, u7} D _inst_2 C _inst_1 (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max u3 u5 u4 u7, u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (CategoryTheory.Functor.obj.{u3, max (max u3 u5 u4 u7) u5, u4, max (max u4 u5) u5 (max u3 u5 u4 u7) u7} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max u3 u5 u4 u7, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max u3 u5 u4 u7, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u3, u5, u4, u7} K _inst_4 C _inst_1) k))) -> (CategoryTheory.Limits.PreservesColimitsOfSize.{u1, u2, u6, max u4 u5, u8, max u3 u5 u4 u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) F)\nbut is expected to have type\n  forall {C : Type.{u7}} [_inst_1 : CategoryTheory.Category.{u5, u7} C] {D : Type.{u8}} [_inst_2 : CategoryTheory.Category.{u6, u8} D] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u3, u4} K] (F : CategoryTheory.Functor.{u6, max u4 u5, u8, max (max (max u7 u4) u5) u3} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1)), (forall (k : K), CategoryTheory.Limits.PreservesColimitsOfSize.{u1, u2, u6, u5, u8, u7} D _inst_2 C _inst_1 (CategoryTheory.Functor.comp.{u6, max u4 u5, u5, u8, max (max (max u7 u4) u5) u3, u7} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1 F (Prefunctor.obj.{succ u3, max (max (max (succ u7) (succ u4)) (succ u5)) (succ u3), u4, max (max (max u7 u4) u5) u3} K (CategoryTheory.CategoryStruct.toQuiver.{u3, u4} K (CategoryTheory.Category.toCategoryStruct.{u3, u4} K _inst_4)) (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max (max u7 u4) u5) u3, max (max (max u7 u4) u5) u3} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max (max u7 u4) u5) u3, max (max (max u7 u4) u5) u3} (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u3) u5, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u3, max (max (max u7 u4) u5) u3, u4, max (max (max u7 u4) u5) u3} K _inst_4 (CategoryTheory.Functor.{max u4 u5, u5, max (max (max u7 u4) u5) u3, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.Functor.category.{max u4 u5, u5, max (max (max u4 u7) u3) u5, u7} (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) C _inst_1) (CategoryTheory.evaluation.{u3, u5, u4, u7} K _inst_4 C _inst_1)) k))) -> (CategoryTheory.Limits.PreservesColimitsOfSize.{u1, u2, u6, max u4 u5, u8, max (max (max u7 u4) u5) u3} D _inst_2 (CategoryTheory.Functor.{u3, u5, u4, u7} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u3, u5, u4, u7} K _inst_4 C _inst_1) F)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.preserves_colimits_of_evaluation CategoryTheory.Limits.preservesColimitsOfEvaluation\u2093'. -/\n/-- `F : D \u2964 K \u2964 C` preserves all colimits if it does for each `k : K`. -/\ndef preservesColimitsOfEvaluation (F : D \u2964 K \u2964 C)\n    (H : \u2200 k : K, PreservesColimitsOfSize.{w', w} (F \u22d9 (evaluation K C).obj k)) :\n    PreservesColimitsOfSize.{w', w} F :=\n  \u27e8fun L hL =>\n    preserves_colimits_of_shape_of_evaluation F L fun k =>\n      preserves_colimits_of_size.preserves_colimits_of_shape\u27e9\n#align category_theory.limits.preserves_colimits_of_evaluation CategoryTheory.Limits.preservesColimitsOfEvaluation\n\n#print CategoryTheory.Limits.preservesColimitsConst /-\n/-- The constant functor `C \u2964 (D \u2964 C)` preserves colimits. -/\ninstance preservesColimitsConst : PreservesColimitsOfSize.{w', w} (const D : C \u2964 _) :=\n  preservesColimitsOfEvaluation _ fun X =>\n    preservesColimitsOfNatIso <| Iso.symm <| constCompEvaluationObj _ _\n#align category_theory.limits.preserves_colimits_const CategoryTheory.Limits.preservesColimitsConst\n-/\n\nopen CategoryTheory.prod\n\n#print CategoryTheory.Limits.limitIsoFlipCompLim /-\n/-- The limit of a diagram `F : J \u2964 K \u2964 C` is isomorphic to the functor given by\nthe individual limits on objects. -/\n@[simps]\ndef limitIsoFlipCompLim [HasLimitsOfShape J C] (F : J \u2964 K \u2964 C) : limit F \u2245 F.flip \u22d9 lim :=\n  NatIso.ofComponents (limitObjIsoLimitCompEvaluation F) <| by tidy\n#align category_theory.limits.limit_iso_flip_comp_lim CategoryTheory.Limits.limitIsoFlipCompLim\n-/\n\n#print CategoryTheory.Limits.limitFlipIsoCompLim /-\n/-- A variant of `limit_iso_flip_comp_lim` where the arguemnts of `F` are flipped. -/\n@[simps]\ndef limitFlipIsoCompLim [HasLimitsOfShape J C] (F : K \u2964 J \u2964 C) : limit F.flip \u2245 F \u22d9 lim :=\n  (NatIso.ofComponents fun k =>\n      limitObjIsoLimitCompEvaluation F.flip k \u226a\u226b HasLimit.isoOfNatIso (flipCompEvaluation _ _)) <|\n    by tidy\n#align category_theory.limits.limit_flip_iso_comp_lim CategoryTheory.Limits.limitFlipIsoCompLim\n-/\n\n/- warning: category_theory.limits.limit_iso_swap_comp_lim -> CategoryTheory.Limits.limitIsoSwapCompLim is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (G : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), CategoryTheory.Iso.{max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) G (CategoryTheory.Limits.limitIsoSwapCompLim._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 G)) (CategoryTheory.Functor.comp.{u2, max u3 u5, u5, u4, max u1 u5 u3 u6, u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) C _inst_1 (CategoryTheory.Functor.obj.{max (max u4 u3) u5, max u4 u3 u5, max (max u2 u1) u5 (max u4 u3) u6, max u2 (max u3 u5) u4 u1 u5 u3 u6} (CategoryTheory.Functor.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.curry.{u2, u1, u5, u4, u3, u6} K _inst_4 J _inst_3 C _inst_1) (CategoryTheory.Functor.comp.{max u2 u1, max u1 u2, u5, max u4 u3, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1 (CategoryTheory.Prod.swap.{u2, u1, u4, u3} K _inst_4 J _inst_3) (CategoryTheory.Functor.obj.{max u3 u4 u5, max (max u3 u4) u5, max u1 (max u4 u5) u3 u2 u5 u4 u6, max (max u1 u2) u5 (max u3 u4) u6} (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.uncurry.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1) G))) (CategoryTheory.Limits.lim.{u1, u3, u5, u6} J _inst_3 C _inst_1 _inst_5))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasLimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (G : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), CategoryTheory.Iso.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Limits.limit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) G (CategoryTheory.Limits.hasLimitOfHasLimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasLimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) G)) (CategoryTheory.Functor.comp.{u2, max u3 u5, u5, u4, max (max (max u6 u3) u5) u1, u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) C _inst_1 (Prefunctor.obj.{max (max (succ u3) (succ u4)) (succ u5), max (max (succ u3) (succ u4)) (succ u5), max (max (max (max (max u6 u3) u4) u5) u1) u2, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1))) (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max (max (max u3 u6) u1) u5} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max (max u3 u4) u5, max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max (max (max u3 u6) u1) u5} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.curry.{u2, u1, u5, u4, u3, u6} K _inst_4 J _inst_3 C _inst_1)) (CategoryTheory.Functor.comp.{max u2 u1, max u1 u2, u5, max u4 u3, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1 (CategoryTheory.Prod.swap.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prefunctor.obj.{max (max (succ u4) (succ u3)) (succ u5), max (max (succ u4) (succ u3)) (succ u5), max (max (max (max (max u6 u4) u3) u5) u2) u1, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u4 u6) u2) u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{max (max u4 u3) u5, max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u4 u6) u2) u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.uncurry.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1)) G))) (CategoryTheory.Limits.lim.{u1, u3, u5, u6} J _inst_3 C _inst_1 _inst_5))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.limit_iso_swap_comp_lim CategoryTheory.Limits.limitIsoSwapCompLim\u2093'. -/\n/-- For a functor `G : J \u2964 K \u2964 C`, its limit `K \u2964 C` is given by `(G' : K \u2964 J \u2964 C) \u22d9 lim`.\nNote that this does not require `K` to be small.\n-/\n@[simps]\ndef limitIsoSwapCompLim [HasLimitsOfShape J C] (G : J \u2964 K \u2964 C) :\n    limit G \u2245 curry.obj (swap K J \u22d9 uncurry.obj G) \u22d9 lim :=\n  limitIsoFlipCompLim G \u226a\u226b isoWhiskerRight (flipIsoCurrySwapUncurry _) _\n#align category_theory.limits.limit_iso_swap_comp_lim CategoryTheory.Limits.limitIsoSwapCompLim\n\n#print CategoryTheory.Limits.colimitIsoFlipCompColim /-\n/-- The colimit of a diagram `F : J \u2964 K \u2964 C` is isomorphic to the functor given by\nthe individual colimits on objects. -/\n@[simps]\ndef colimitIsoFlipCompColim [HasColimitsOfShape J C] (F : J \u2964 K \u2964 C) : colimit F \u2245 F.flip \u22d9 colim :=\n  NatIso.ofComponents (colimitObjIsoColimitCompEvaluation F) <| by tidy\n#align category_theory.limits.colimit_iso_flip_comp_colim CategoryTheory.Limits.colimitIsoFlipCompColim\n-/\n\n#print CategoryTheory.Limits.colimitFlipIsoCompColim /-\n/-- A variant of `colimit_iso_flip_comp_colim` where the arguemnts of `F` are flipped. -/\n@[simps]\ndef colimitFlipIsoCompColim [HasColimitsOfShape J C] (F : K \u2964 J \u2964 C) : colimit F.flip \u2245 F \u22d9 colim :=\n  (NatIso.ofComponents fun k =>\n      colimitObjIsoColimitCompEvaluation _ _ \u226a\u226b HasColimit.isoOfNatIso (flipCompEvaluation _ _)) <|\n    by tidy\n#align category_theory.limits.colimit_flip_iso_comp_colim CategoryTheory.Limits.colimitFlipIsoCompColim\n-/\n\n/- warning: category_theory.limits.colimit_iso_swap_comp_colim -> CategoryTheory.Limits.colimitIsoSwapCompColim is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (G : CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), CategoryTheory.Iso.{max u4 u5, max u2 u5 u4 u6} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) G (CategoryTheory.Limits.colimitIsoSwapCompColim._proof_1.{u6, u3, u4, u5, u1, u2} C _inst_1 J _inst_3 K _inst_4 _inst_5 G)) (CategoryTheory.Functor.comp.{u2, max u3 u5, u5, u4, max u1 u5 u3 u6, u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) C _inst_1 (CategoryTheory.Functor.obj.{max (max u4 u3) u5, max u4 u3 u5, max (max u2 u1) u5 (max u4 u3) u6, max u2 (max u3 u5) u4 u1 u5 u3 u6} (CategoryTheory.Functor.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max u1 u5 u3 u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.curry.{u2, u1, u5, u4, u3, u6} K _inst_4 J _inst_3 C _inst_1) (CategoryTheory.Functor.comp.{max u2 u1, max u1 u2, u5, max u4 u3, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1 (CategoryTheory.Prod.swap.{u2, u1, u4, u3} K _inst_4 J _inst_3) (CategoryTheory.Functor.obj.{max u3 u4 u5, max (max u3 u4) u5, max u1 (max u4 u5) u3 u2 u5 u4 u6, max (max u1 u2) u5 (max u3 u4) u6} (CategoryTheory.Functor.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max u2 u5 u4 u6} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.uncurry.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1) G))) (CategoryTheory.Limits.colim.{u1, u3, u5, u6} J _inst_3 C _inst_1 _inst_5))\nbut is expected to have type\n  forall {C : Type.{u6}} [_inst_1 : CategoryTheory.Category.{u5, u6} C] {J : Type.{u3}} [_inst_3 : CategoryTheory.Category.{u1, u3} J] {K : Type.{u4}} [_inst_4 : CategoryTheory.Category.{u2, u4} K] [_inst_5 : CategoryTheory.Limits.HasColimitsOfShape.{u1, u3, u5, u6} J _inst_3 C _inst_1] (G : CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)), CategoryTheory.Iso.{max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Limits.colimit.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) G (CategoryTheory.Limits.hasColimitOfHasColimitsOfShape.{u1, u3, max u4 u5, max (max (max u6 u4) u5) u2} (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1) J _inst_3 (CategoryTheory.Limits.functorCategoryHasColimitsOfShape.{u1, u2, u3, u4, u5, u6} C _inst_1 J _inst_3 K _inst_4 _inst_5) G)) (CategoryTheory.Functor.comp.{u2, max u3 u5, u5, u4, max (max (max u6 u3) u5) u1, u6} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1) C _inst_1 (Prefunctor.obj.{max (max (succ u3) (succ u4)) (succ u5), max (max (succ u3) (succ u4)) (succ u5), max (max (max (max (max u6 u3) u4) u5) u1) u2, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1))) (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max (max (max u3 u6) u1) u5} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)))) (CategoryTheory.Functor.toPrefunctor.{max (max u3 u4) u5, max (max u3 u4) u5, max (max (max (max (max u6 u3) u4) u5) u1) u2, max (max (max (max (max u6 u3) u4) u5) u1) u2} (CategoryTheory.Functor.{max u2 u1, u5, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.category.{max u2 u1, u5, max u4 u3, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) C _inst_1) (CategoryTheory.Functor.{u2, max u3 u5, u4, max (max (max u6 u3) u5) u1} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.Functor.category.{u2, max u3 u5, u4, max (max (max u3 u6) u1) u5} K _inst_4 (CategoryTheory.Functor.{u1, u5, u3, u6} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u5, u3, u6} J _inst_3 C _inst_1)) (CategoryTheory.curry.{u2, u1, u5, u4, u3, u6} K _inst_4 J _inst_3 C _inst_1)) (CategoryTheory.Functor.comp.{max u2 u1, max u1 u2, u5, max u4 u3, max u3 u4, u6} (Prod.{u4, u3} K J) (CategoryTheory.prod.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1 (CategoryTheory.Prod.swap.{u2, u1, u4, u3} K _inst_4 J _inst_3) (Prefunctor.obj.{max (max (succ u4) (succ u3)) (succ u5), max (max (succ u4) (succ u3)) (succ u5), max (max (max (max (max u6 u4) u3) u5) u2) u1, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.CategoryStruct.toQuiver.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Category.toCategoryStruct.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u4 u6) u2) u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)))) (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{max (max u4 u3) u5, max (max u4 u3) u5, max (max (max (max (max u6 u4) u3) u5) u2) u1, max (max (max (max (max u6 u4) u3) u5) u2) u1} (CategoryTheory.Functor.{u1, max u4 u5, u3, max (max (max u6 u4) u5) u2} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.category.{u1, max u4 u5, u3, max (max (max u4 u6) u2) u5} J _inst_3 (CategoryTheory.Functor.{u2, u5, u4, u6} K _inst_4 C _inst_1) (CategoryTheory.Functor.category.{u2, u5, u4, u6} K _inst_4 C _inst_1)) (CategoryTheory.Functor.{max u1 u2, u5, max u4 u3, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.Functor.category.{max u1 u2, u5, max u3 u4, u6} (Prod.{u3, u4} J K) (CategoryTheory.prod.{u1, u2, u3, u4} J _inst_3 K _inst_4) C _inst_1) (CategoryTheory.uncurry.{u1, u2, u5, u3, u4, u6} J _inst_3 K _inst_4 C _inst_1)) G))) (CategoryTheory.Limits.colim.{u1, u3, u5, u6} J _inst_3 C _inst_1 _inst_5))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.colimit_iso_swap_comp_colim CategoryTheory.Limits.colimitIsoSwapCompColim\u2093'. -/\n/-- For a functor `G : J \u2964 K \u2964 C`, its colimit `K \u2964 C` is given by `(G' : K \u2964 J \u2964 C) \u22d9 colim`.\nNote that this does not require `K` to be small.\n-/\n@[simps]\ndef colimitIsoSwapCompColim [HasColimitsOfShape J C] (G : J \u2964 K \u2964 C) :\n    colimit G \u2245 curry.obj (swap K J \u22d9 uncurry.obj G) \u22d9 colim :=\n  colimitIsoFlipCompColim G \u226a\u226b isoWhiskerRight (flipIsoCurrySwapUncurry _) _\n#align category_theory.limits.colimit_iso_swap_comp_colim CategoryTheory.Limits.colimitIsoSwapCompColim\n\nend CategoryTheory.Limits\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Limits/FunctorCategory.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7520125737597972, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3789437762330171}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Module.basic\nimport Mathlib.algebra.category.Group.limits\nimport Mathlib.algebra.direct_limit\nimport Mathlib.PostPort\n\nuniverses u v u_1 \n\nnamespace Mathlib\n\n/-!\n# The category of R-modules has all limits\n\nFurther, these limits are preserved by the forgetful functor --- that is,\nthe underlying types are just the limits in the category of types.\n-/\n\nnamespace Module\n\n\nprotected instance add_comm_group_obj {R : Type u} [ring R] {J : Type v}\n    [category_theory.small_category J] (F : J \u2964 Module R) (j : J) :\n    add_comm_group (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  id (is_add_comm_group (category_theory.functor.obj F j))\n\nprotected instance module_obj {R : Type u} [ring R] {J : Type v} [category_theory.small_category J]\n    (F : J \u2964 Module R) (j : J) :\n    module R (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  id (is_module (category_theory.functor.obj F j))\n\n/--\nThe flat sections of a functor into `Module R` form a submodule of all sections.\n-/\ndef sections_submodule {R : Type u} [ring R] {J : Type v} [category_theory.small_category J]\n    (F : J \u2964 Module R) : submodule R ((j : J) \u2192 \u21a5(category_theory.functor.obj F j)) :=\n  submodule.mk (category_theory.functor.sections (F \u22d9 category_theory.forget (Module R))) sorry\n    sorry sorry\n\nprotected instance limit_add_comm_group {R : Type u} [ring R] {J : Type v}\n    [category_theory.small_category J] (F : J \u2964 Module R) :\n    add_comm_group\n        (category_theory.limits.cone.X\n          (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))) :=\n  id (submodule.add_comm_group (sections_submodule F))\n\nprotected instance limit_module {R : Type u} [ring R] {J : Type v}\n    [category_theory.small_category J] (F : J \u2964 Module R) :\n    module R\n        (category_theory.limits.cone.X\n          (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))) :=\n  id (submodule.semimodule (sections_submodule F))\n\n/-- `limit.\u03c0 (F \u22d9 forget Ring) j` as a `ring_hom`. -/\ndef limit_\u03c0_linear_map {R : Type u} [ring R] {J : Type v} [category_theory.small_category J]\n    (F : J \u2964 Module R) (j : J) :\n    linear_map R\n        (category_theory.limits.cone.X\n          (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R))))\n        (category_theory.functor.obj (F \u22d9 category_theory.forget (Module R)) j) :=\n  linear_map.mk\n    (category_theory.nat_trans.app\n      (category_theory.limits.cone.\u03c0\n        (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R))))\n      j)\n    sorry sorry\n\nnamespace has_limits\n\n\n-- The next two definitions are used in the construction of `has_limits (Module R)`.\n\n-- After that, the limits should be constructed using the generic limits API,\n\n-- e.g. `limit F`, `limit.cone F`, and `limit.is_limit F`.\n\n/--\nConstruction of a limit cone in `Module R`.\n(Internal use only; use the limits API.)\n-/\ndef limit_cone {R : Type u} [ring R] {J : Type v} [category_theory.small_category J]\n    (F : J \u2964 Module R) : category_theory.limits.cone F :=\n  category_theory.limits.cone.mk\n    (of R\n      (category_theory.limits.cone.X\n        (category_theory.limits.types.limit_cone (F \u22d9 category_theory.forget (Module R)))))\n    (category_theory.nat_trans.mk (limit_\u03c0_linear_map F))\n\n/--\nWitness that the limit cone in `Module R` is a limit cone.\n(Internal use only; use the limits API.)\n-/\ndef limit_cone_is_limit {R : Type u} [ring R] {J : Type v} [category_theory.small_category J]\n    (F : J \u2964 Module R) : category_theory.limits.is_limit (limit_cone F) :=\n  category_theory.limits.is_limit.of_faithful (category_theory.forget (Module R))\n    (category_theory.limits.types.limit_cone_is_limit (F \u22d9 category_theory.forget (Module R)))\n    (fun (s : category_theory.limits.cone F) =>\n      linear_map.mk\n        (fun\n          (v :\n          category_theory.limits.cone.X\n            (category_theory.functor.map_cone (category_theory.forget (Module R)) s)) =>\n          { val :=\n              fun (j : J) =>\n                category_theory.nat_trans.app\n                  (category_theory.limits.cone.\u03c0\n                    (category_theory.functor.map_cone (category_theory.forget (Module R)) s))\n                  j v,\n            property := sorry })\n        sorry sorry)\n    sorry\n\nend has_limits\n\n\n/-- The category of R-modules has all limits. -/\nprotected instance has_limits {R : Type u} [ring R] :\n    category_theory.limits.has_limits (Module R) :=\n  category_theory.limits.has_limits.mk\n    fun (J : Type v) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.has_limits_of_shape.mk\n        fun (F : J \u2964 Module R) =>\n          category_theory.limits.has_limit.mk (category_theory.limits.limit_cone.mk sorry sorry)\n\n/--\nAn auxiliary declaration to speed up typechecking.\n-/\ndef forget\u2082_AddCommGroup_preserves_limits_aux {R : Type u} [ring R] {J : Type v}\n    [category_theory.small_category J] (F : J \u2964 Module R) :\n    category_theory.limits.is_limit\n        (category_theory.functor.map_cone (category_theory.forget\u2082 (Module R) AddCommGroup)\n          (has_limits.limit_cone F)) :=\n  AddCommGroup.limit_cone_is_limit (F \u22d9 category_theory.forget\u2082 (Module R) AddCommGroup)\n\n/--\nThe forgetful functor from R-modules to abelian groups preserves all limits.\n-/\nprotected instance forget\u2082_AddCommGroup_preserves_limits {R : Type u} [ring R] :\n    category_theory.limits.preserves_limits (category_theory.forget\u2082 (Module R) AddCommGroup) :=\n  category_theory.limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.preserves_limits_of_shape.mk\n        fun (F : J \u2964 Module R) =>\n          category_theory.limits.preserves_limit_of_preserves_limit_cone\n            (has_limits.limit_cone_is_limit F) (forget\u2082_AddCommGroup_preserves_limits_aux F)\n\n/--\nThe forgetful functor from R-modules to types preserves all limits.\n-/\nprotected instance forget_preserves_limits {R : Type u} [ring R] :\n    category_theory.limits.preserves_limits (category_theory.forget (Module R)) :=\n  category_theory.limits.preserves_limits.mk\n    fun (J : Type u_1) (\ud835\udca5 : category_theory.small_category J) =>\n      category_theory.limits.preserves_limits_of_shape.mk\n        fun (F : J \u2964 Module R) =>\n          category_theory.limits.preserves_limit_of_preserves_limit_cone\n            (has_limits.limit_cone_is_limit F)\n            (category_theory.limits.types.limit_cone_is_limit\n              (F \u22d9 category_theory.forget (Module R)))\n\n/-- The diagram (in the sense of `category_theory`)\n of an unbundled `direct_limit` of modules. -/\n@[simp] theorem direct_limit_diagram_map {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9]\n    (G : \u03b9 \u2192 Type v) [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)]\n    (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] (i : \u03b9) (j : \u03b9)\n    (hij : i \u27f6 j) :\n    category_theory.functor.map (direct_limit_diagram G f) hij =\n        f i j (direct_limit_diagram._proof_1 i j hij) :=\n  Eq.refl (category_theory.functor.map (direct_limit_diagram G f) hij)\n\n/-- The `cocone` on `direct_limit_diagram` corresponding to\nthe unbundled `direct_limit` of modules.\n\nIn `direct_limit_is_colimit` we show that it is a colimit cocone. -/\n@[simp] theorem direct_limit_cocone_\u03b9_app {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9]\n    (G : \u03b9 \u2192 Type v) [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)]\n    (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] [DecidableEq \u03b9]\n    (i : \u03b9) :\n    category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 (direct_limit_cocone G f)) i =\n        module.direct_limit.of R \u03b9 G f i :=\n  Eq.refl\n    (category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 (direct_limit_cocone G f)) i)\n\n/-- The unbundled `direct_limit` of modules is a colimit\nin the sense of `category_theory`. -/\ndef direct_limit_is_colimit {R : Type u} [ring R] {\u03b9 : Type v} [directed_order \u03b9] (G : \u03b9 \u2192 Type v)\n    [(i : \u03b9) \u2192 add_comm_group (G i)] [(i : \u03b9) \u2192 module R (G i)]\n    (f : (i j : \u03b9) \u2192 i \u2264 j \u2192 linear_map R (G i) (G j)) [module.directed_system G f] [DecidableEq \u03b9]\n    [Nonempty \u03b9] : category_theory.limits.is_colimit (direct_limit_cocone G f) :=\n  category_theory.limits.is_colimit.mk\n    fun (s : category_theory.limits.cocone (direct_limit_diagram G f)) =>\n      module.direct_limit.lift R \u03b9 G f\n        (category_theory.nat_trans.app (category_theory.limits.cocone.\u03b9 s)) sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Module/limits_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943805178138, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3789266904547473}}
{"text": "import RSAProject.rsaproject\n\ntheorem working1 : (decryption_by_brute_force (encryption e n m) e n ) = m := sorry\n\ntheorem working2 : (decryption (encryption e n m) d n ) = m := sorry", "meta": {"author": "optimisticexquisite", "repo": "lean-rsa-project", "sha": "846ab0e87ca3cee28e3f169731ab60f1bc2ec95f", "save_path": "github-repos/lean/optimisticexquisite-lean-rsa-project", "path": "github-repos/lean/optimisticexquisite-lean-rsa-project/lean-rsa-project-846ab0e87ca3cee28e3f169731ab60f1bc2ec95f/RSAProject/toprove.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3788971856373203}}
{"text": "\nimport ALC\n\nopen Concept Role ALCStatement\n\naxiom LEM :\u2200 (A : Prop), A \u2228 \u00acA\n\n\nlemma dir1 : \u2200 KB R o1 o2, ((RoleAssertion R (o1, o2)) \u2208 KB) -> entails KB (RoleAssertion R (o1, o2)) :=\nbegin\n  intros KB, intros R, intros o1, intros o2,\n  intros H,\n  rewrite entails,\n  intros \u0394i, intros Iac, intros Iar, intros Io,\n  intros H2,\n  specialize H2 (RoleAssertion R (o1, o2)),\n  apply H2,\n  exact H,\nend\n\nlemma dir2 : \u2200 KB R o1 o2, (entails KB (RoleAssertion R (o1, o2)) -> (RoleAssertion R (o1, o2)) \u2208 KB) :=\nbegin\n  intros KB R o1 o2,\n  by_contra,\n\n  rewrite entails at H,\n  \n\n\n  let Domain : set DomainType := {\"arbitrary\"},\n  let ConI : AtomicConceptType \u2192 set DomainType := (\u03bbx,{\"arbitrary\"}),\n  let RolI : AtomicRoleType -> set (DomainType \u00d7 DomainType):= (\u03bbx,{(\"arbitrary\", \"arbitrary\")}),\n  let ObjI : IndividualType -> DomainType:= (\u03bbx, \"arbitrary\"),\n  specialize H Domain,    --Domain\n  specialize H ConI,  --Concept map\n  specialize H RolI,  --RoleMap\n  specialize H ObjI,\nend\n\nexample : \u2200 KB R o1 o2, (entails KB (RoleAssertion R (o1, o2)) \u2194 (RoleAssertion R (o1, o2)) \u2208 KB) := \nbegin\n  intros KB R o1 o2,\n  apply iff.not,\nend\n\n\n\n--Trash\n\n\n\ntheorem contraposition1 (A B : Prop) (h : \u00ac B \u2192 \u00ac A) : A \u2192 B :=\nassume h1 : A,\nshow B, from\n  by_contradiction\n    (assume h2 : \u00ac B,\n      have h3 : \u00ac A, from h h2,\n      show false, from h3 h1)\n\ntheorem contraposition (A B : Prop) : (\u00acB \u2192 \u00acA) \u2194 (A \u2192 B) :=\nbegin\n  apply iff.intro,{\n    apply contraposition1,\n  },{\n    intros H,\n    intros NB,\n    by_contradiction,\n    have B : B, from H h,\n    apply not.elim NB B,\n  }\nend\n\n/-\n--This is equivlent to trying prove a contradiction \ntheorem counterEx : \u00ac\u2200 KB R o1 o2, entails KB (RoleAssertion R (o1, o2)) \u2192 RoleAssertion R (o1, o2) \u2208 KB :=\nbegin\n  simp,\n  existsi {RoleAssertion (atomicRole \"ArbitraryRole\") (\"o1\", \"o2\")},\n  existsi (atomicRole \"ArbitraryRole\"),\n  existsi \"o1\",\n  existsi \"o2\",\n  split,{\n    rewrite entails,\n    intros Domain Iac Iar Io,\n    intros H,\n    specialize H (RoleAssertion (atomicRole \"ArbitraryRole\") (\"o1\", \"o2\")),\n    apply H,\n    sorry,\n  },{\n    sorry,\n  },\n  sorry,\nend\n\ntheorem counterEx2 : \u2200 KB R o1 o2, \u00ac(entails KB (RoleAssertion R (o1, o2)) \u2192 RoleAssertion R (o1, o2) \u2208 KB) :=\nbegin\n  intros KB, intros R, intros o1, intros o2,\n  simp,\n  split,\n  {\n    rewrite entails,\n    intros Domain Iac Iar Io,\n    intros H,\n    specialize H (RoleAssertion R (o1, o2)),\n    apply H,\n    admit,\n  },\n  {\n    sorry,\n  },\nend\n-/\n\n\n/-\n   --apply contraposition at H,\n\n\n    --have N, from not.intro H,\n    /-simp at N,\n    cases N,\n    cases N_w,\n    {\n      rewrite eval at N_h,\n      induction N_w_\u1fb0,{\n        rewrite Ic at N_h,\n        cases N_h,\n        simp at N_h_right,\n        sorry,\n      },{\n        rewrite Ic at N_h,\n        sorry,\n      },{\n        rewrite Ic at N_h,\n\n      },{\n\n      },{\n\n      }\n\n    },{\n\n      sorry\n    },{\n      sorry\n    }\n    -/\n    --simp ite at H,\n\n    /-\n    apply contraposition,\n    intros H,\n    rewrite entails,\n    simp,\n    existsi {o1, o2},    --Domain\n    existsi (\u03bbx,\u2205), --Concept map\n    existsi (\u03bbx,{\u2205}),\n    existsi (\u03bbx,\u2205),\n    split,{\n\n      intros S,\n      apply contraposition,\n\n      sorry,\n      /-intros S,\n      intros S2,\n      cases S,{\n\n        sorry,\n      },{\n        \n        rewrite eval,\n\n        sorry,\n      },{\n        sorry,\n      },-/\n      --specialize H (RoleAssertion R (o1, o2)),\n    },\n    {\n      rewrite eval,\n      cases R,\n      rewrite Ir,\n      admit,\n    },\n    -/\n    \n    --specialize H \u0394i Iac Iar Io,\n\n    --apply contraposition at H,\n    --by_cases (\u2200 (s : ALCStatement), s \u2208 KB \u2192 eval \u0394i Iac Iar Io s),\n    --{\n    --  apply H at h,\n    --  sorry,\n    --},\n    --{\n    --  sorry,\n    --}\n    --rewrite entails at H,\n    --specialize H \u0394i Iac Iar Io,\n    --by_contra,\n    --by_cases ,\n-/\n\n/-\n    axiom LEM :\u2200 (A : Prop), A \u2228 \u00acA\n\n\n    intro H,\n    rewrite entails at H,\n    let Domain : set DomainType := {\"arbitrary\"},\n    let ConI : AtomicConceptType \u2192 set DomainType := (\u03bbx,{\"arbitrary\"}),\n    let RolI : AtomicRoleType -> set (DomainType \u00d7 DomainType):= (\u03bbx,{(\"arbitrary\", \"arbitrary\")}),\n    let ObjI : IndividualType -> DomainType:= (\u03bbx, \"arbitrary\"),\n    specialize H Domain,    --Domain\n    specialize H ConI,  --Concept map\n    specialize H RolI,  --RoleMap\n    specialize H ObjI,\n    rewrite eval at H,\n    cases R,\n    rewrite Ir at H,\n    let P : Prop := models Domain ConI RolI ObjI KB,\n    have LEMP: P \u2228 \u00acP := LEM P,\n    cases LEMP,{\n      have p : (models Domain ConI RolI ObjI KB), from LEMP,\n      rewrite models at p,\n      by_contra,\n      specialize p (RoleAssertion (atomicRole R) (o1, o2)),\n      rewrite eval at p,\n      rewrite Ir at p,\n      simp at p,\n\n    },{\n      have p : \u00ac(models Domain ConI RolI ObjI KB), from LEMP,\n      rewrite models at p,\n      simp at p,\n    }\n-/\n\n--This proof says that if the domain is empty, concept interpretation is also empty\nlemma ICEmptyDomain: \u2200Iac Iar c, Ic \u2205 Iac Iar c = \u2205 :=\nbegin\n  intros Iac Iar c,\n  induction c,{\n    rewrite Ic,\n    have H, from conceptsInDomain,\n    specialize H \u2205 Iac c,\n    have H2, from set.eq_empty_of_subset_empty H,\n    exact H2,  \n  },{\n    rewrite Ic,\n  },{\n    rewrite Ic,\n  },{\n    rewrite Ic,\n    rewrite c_ih_\u1fb0,\n    rewrite c_ih_\u1fb0_1,\n    apply set.empty_inter,\n  },{\n    rewrite Ic,\n    rewrite c_ih_\u1fb0,\n    rewrite c_ih_\u1fb0_1,\n    apply set.empty_union,\n  },{\n    rewrite Ic,\n    rewrite c_ih,\n    apply set.empty_diff\n  },{\n    rewrite Ic,\n    rewrite c_ih,\n    simp,\n  },{\n    rewrite Ic,\n    rewrite c_ih,\n    simp,\n  }\nend\n\nlemma notMem (\u03b1 : Type) (e : \u03b1) (H : e \u2209 ({e} : set \u03b1)): false :=\nbegin\n  simp at H,\n  exact H,\nend\n\nlemma rewriteOvb : (\"arbitrary\", \"arbitrary\") \u2208 ({(\"arbitrary\", \"arbitrary\")} : set (DomainType \u00d7  DomainType)) = true :=\nbegin\n  simp,\nend\n\n\n/-\nintros KB, intros R, intros o1, intros o2,\n  apply iff.intro,\n  {\n    intros H,\n    rewrite entails at H,\n\n  },\n  {\n    apply dir1,\n  }\n\n-/", "meta": {"author": "James-Oswald", "repo": "Fuzzy-DL", "sha": "8c25fc553747340b7cf1682fbeca616db18e55ca", "save_path": "github-repos/lean/James-Oswald-Fuzzy-DL", "path": "github-repos/lean/James-Oswald-Fuzzy-DL/Fuzzy-DL-8c25fc553747340b7cf1682fbeca616db18e55ca/src/ALCexample2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3788971856373203}}
{"text": "import for_mathlib.mapping_cone\nimport category_theory.triangulated.pretriangulated\n\nnoncomputable theory\n\nuniverses v u\n\nopen_locale classical zero_object\n\nopen category_theory category_theory.limits category_theory.triangulated\nopen homological_complex\n\nnamespace homotopy_category\n\nvariables (V : Type u) [category.{v} V] [abelian V]\n\nlocal notation `\ud835\udca6` := homotopy_category V (complex_shape.up \u2124)\n\ndef distinguished_triangles : set (triangle \ud835\udca6) :=\n\u03bb T, \u2203 (X Y Z : cochain_complex V \u2124) (f : X \u27f6 Y) (g : Y \u27f6 Z) h,\n  nonempty (T \u2245 triangle\u2095_of_termwise_split f g h)\n\nvariable {V}\nlocal notation `\ud835\udca6` := homotopy_category V (complex_shape.up \u2124)\n\nlemma mem_distinguished_of_iso {T\u2081 T\u2082 : triangle \ud835\udca6} (e : T\u2081 \u2245 T\u2082)\n  (hT : T\u2081 \u2208 distinguished_triangles V) : T\u2082 \u2208 distinguished_triangles V :=\n\u27e8_, _, _, _, _, _,\n  \u27e8e.symm \u226a\u226b hT.some_spec.some_spec.some_spec.some_spec.some_spec.some_spec.some\u27e9\u27e9\n\nlemma mem_distinguished_iff_exists_iso_cone (T : triangle \ud835\udca6) :\n  T \u2208 distinguished_triangles V \u2194 \u2203 (X Y) (f : X \u27f6 Y),\n    nonempty (T \u2245 (neg\u2083_functor _).obj (cone.triangle\u2095 f)) :=\nbegin\n  split,\n  { rintro \u27e8X, Y, Z, f, g, h, \u27e8e\u27e9\u27e9,\n    refine \u27e8_, _, f, \u27e8e \u226a\u226b iso_cone_of_termwise_split f g h\u27e9\u27e9 },\n  { rintro \u27e8X, Y, f, \u27e8e\u27e9\u27e9,\n    refine \u27e8_, _, _, _, _, _, \u27e8e \u226a\u226b iso_termwise_split_of_cone f\u27e9\u27e9 }\nend\n\nlemma triangle\u2095_of_termwise_split_mem_distinguished_triangles (X Y Z : cochain_complex V \u2124)\n  (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : \u2200 i, splitting (f.f i) (g.f i)) :\n    triangle\u2095_of_termwise_split f g h \u2208 distinguished_triangles V :=\n\u27e8_, _, _, _, _, _, \u27e8iso.refl _\u27e9\u27e9\n\nlemma cone_triangle\u2095_mem_distinguished_triangles (X Y : cochain_complex V \u2124)\n  (f : X \u27f6 Y) : (neg\u2083_functor _).obj (cone.triangle\u2095 f) \u2208 distinguished_triangles V :=\n(mem_distinguished_iff_exists_iso_cone _).mpr \u27e8_, _, _, \u27e8iso.refl _\u27e9\u27e9\n\nlemma rotate_mem_distinguished_triangles (T : triangle \ud835\udca6) (h : T \u2208 distinguished_triangles V) :\n  T.rotate \u2208 distinguished_triangles V :=\nbegin\n  obtain \u27e8X, Y, f, \u27e8e\u27e9\u27e9 := (mem_distinguished_iff_exists_iso_cone _).mp h,\n  exact \u27e8_, _, _, _, _, _, \u27e8(rotate _).map_iso e \u226a\u226b neg\u2083_rotate.app _ \u226a\u226b\n    (triangle.nonneg_rotate_iso _).symm \u226a\u226b\n      (homotopy_category.lift_triangle _).map_iso (triangle_of_termwise_split_cone_iso f).symm\u27e9\u27e9,\nend\n\ndef inv_rotate_lift_triangle (V : Type u) [category.{v} V] [abelian V] :\n  (inv_rotate _) \u22d9 homotopy_category.lift_triangle V \u2245\n    homotopy_category.lift_triangle _ \u22d9 (inv_rotate _) :=\nnat_iso.of_components (\u03bb X, mk_triangle_iso (iso.refl _) (iso.refl _) (iso.refl _)\n  (by { dsimp, simp only [category.comp_id, functor.map_neg, discrete.functor_map_id,\n    category.id_comp, preadditive.comp_neg, shift_\u03b5_inv_app, category.assoc, neg_inj,\n    functor.map_comp, nat_trans.id_app, preadditive.neg_comp] })\n  (by { dsimp, rw [category.comp_id, category.id_comp] })\n  (by { dsimp, simp only [shift_\u03b5_app, discrete.functor_map_id, category.id_comp, category.assoc,\n    functor.map_comp, nat_trans.id_app, category_theory.functor.map_id,\n    unit_of_tensor_iso_unit_inv_app, shift_\u03bc_inv_app], erw category.comp_id }))\n  (by { intros, ext; exact (category.comp_id _).trans (category.id_comp _).symm })\n\nlemma inv_rotate_mem_distinguished_triangles (T : triangle \ud835\udca6) (h : T \u2208 distinguished_triangles V) :\n  T.inv_rotate \u2208 distinguished_triangles V :=\nbegin\n  obtain \u27e8X, Y, Z, f, g, h, \u27e8e\u27e9\u27e9 := h,\n  exact (mem_distinguished_iff_exists_iso_cone _).mpr \u27e8_, _, _,\n    \u27e8(inv_rotate _).map_iso e \u226a\u226b ((inv_rotate_lift_triangle _).app _).symm \u226a\u226b\n    (homotopy_category.lift_triangle _).map_iso ((neg\u2083_equiv _).iso_equiv _ _\n    ((triangle.nonneg_inv_rotate_iso _).symm \u226a\u226b inv_rotate_iso_cone_triangle f g h))\u27e9\u27e9\nend\n\ninstance : pretriangulated \ud835\udca6 :=\n{ distinguished_triangles := distinguished_triangles V,\n  isomorphic_distinguished := \u03bb T\u2081 hT T\u2082 e, mem_distinguished_of_iso e.symm hT,\n  contractible_distinguished := begin\n    rintro \u27e8X\u27e9,\n    refine \u27e8X, X, homological_complex.zero, \ud835\udfd9 _, 0, _, \u27e8_\u27e9\u27e9,\n    { intro i, simp only [id_f, zero_f_apply],\n      refine splitting.splitting_of_is_iso_zero _ (homological_complex.zero_X _ _),\n      exact homological_complex.is_zero_zero },\n    refine mk_triangle_iso (iso.refl _) (iso.refl _) _ _ _ _,\n    { dsimp [triangle\u2095_of_termwise_split],\n      refine homotopy_category.is_zero_zero.iso_zero.symm, },\n    all_goals { dsimp; simp; refl },\n  end,\n  distinguished_cocone_triangle := begin\n    rintros \u27e8X\u27e9 \u27e8Y\u27e9 f,\n    induction f using quot.induction_on,\n    exact \u27e8_, _, _, _, _, _, _, _, _, \u27e8iso_termwise_split_of_cone f\u27e9\u27e9,\n  end,\n  rotate_distinguished_triangle := begin\n    intro T,\n    split,\n    { apply rotate_mem_distinguished_triangles },\n    { intro h,\n      exact mem_distinguished_of_iso ((triangle_rotation _).unit_iso.app T).symm\n        (inv_rotate_mem_distinguished_triangles _ h) }\n  end,\n  complete_distinguished_triangle_morphism := begin\n    intros,\n    obtain \u27e8X\u2081, Y\u2081, f\u2081, \u27e8e\u2081\u27e9\u27e9 := (mem_distinguished_iff_exists_iso_cone _).mp h\u2081,\n    obtain \u27e8X\u2082, Y\u2082, f\u2082, \u27e8e\u2082\u27e9\u27e9 := (mem_distinguished_iff_exists_iso_cone _).mp h\u2082,\n    let h : homotopy (f\u2081 \u226b quot.out (e\u2081.inv.hom\u2082 \u226b b \u226b e\u2082.hom.hom\u2082))\n      (quot.out (e\u2081.inv.hom\u2081 \u226b a \u226b e\u2082.hom.hom\u2081) \u226b f\u2082),\n    { apply homotopy_of_eq,\n      simp only [quotient_map_out, category.assoc, functor.map_comp],\n      erw [\u2190 e\u2082.hom.comm\u2081, reassoc_of e\u2081.inv.comm\u2081, reassoc_of comm\u2081] },\n    refine \u27e8e\u2081.hom.hom\u2083 \u226b (quotient V _).map (cone.map h) \u226b e\u2082.inv.hom\u2083, _, _\u27e9,\n    { rw [reassoc_of e\u2081.hom.comm\u2082, \u2190 is_iso.eq_inv_comp],\n      simp_rw \u2190 category.assoc,\n      rw [\u2190 is_iso.eq_comp_inv, \u2190 inv_hom\u2083, \u2190 inv_hom\u2082, is_iso.iso.inv_inv, is_iso.iso.inv_hom],\n      simp_rw category.assoc,\n      rw [e\u2082.hom.comm\u2082],\n      convert (cone.triangle\u2095_map h).comm\u2082 using 1,\n      dsimp, simp },\n    { simp_rw category.assoc,\n      rw [\u2190 is_iso.inv_comp_eq, \u2190 inv_hom\u2083, is_iso.iso.inv_hom, \u2190 reassoc_of e\u2081.inv.comm\u2083,\n        \u2190 e\u2082.inv.comm\u2083],\n      dsimp,\n      have := (cone.triangle\u2095_map h).comm\u2083,\n      dsimp at this,\n      rw [\u2190 homotopy_category.quotient_map_shift, quotient_map_out] at this,\n      simp only [preadditive.neg_comp_assoc, preadditive.comp_neg, category.assoc, neg_inj,\n        preadditive.neg_comp],\n      rw [\u2190 reassoc_of this, \u2190 functor.map_comp, \u2190 functor.map_comp],\n      congr' 2,\n      rw [\u2190 is_iso.comp_inv_eq, \u2190 inv_hom\u2081, is_iso.iso.inv_inv, category.assoc] }\n  end }\n\nend homotopy_category\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/homotopy_category_pretriangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7634837743174788, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3787595893400693}}
{"text": "import algebraic_geometry.AffineScheme\n\nopen category_theory category_theory.limits opposite topological_space\n\nnamespace algebraic_geometry\n\nuniverses v u\n\nlemma structure_sheaf.open_to_localization_localization_to_stalk {R : Type*} [comm_ring R]\n  (U : opens (prime_spectrum.Top R)) (x : U) :\n  structure_sheaf.open_to_localization R U x x.2 \u226b structure_sheaf.localization_to_stalk R x =\n    (Spec.structure_sheaf R).presheaf.germ x :=\nbegin\n  rw [\u2190 structure_sheaf.germ_comp_stalk_to_fiber_ring_hom, category.assoc,\n    structure_sheaf.stalk_to_fiber_ring_hom_localization_to_stalk, category.comp_id],\nend\n\nlemma specializes_of_eq {\u03b1 : Type*} [topological_space \u03b1] {x y : \u03b1} (e : x = y) :\n  x \u2933 y := e \u25b8 specializes_refl x\n\n@[simp, reassoc, elementwise]\nlemma _root_.Top.presheaf.stalk_specializes_comp {C : Type*} [category C] [limits.has_colimits C]\n  {X : Top} (F : X.presheaf C)\n  {x y z : X} (h : x \u2933 y) (h' : y \u2933 z) :\n  F.stalk_specializes h' \u226b F.stalk_specializes h = F.stalk_specializes (h.trans h') :=\nF.stalk_hom_ext $ \u03bb _ _, by simp\n\n@[simp]\nlemma _root_.Top.presheaf.stalk_specializes_refl {C : Type*} [category C] [limits.has_colimits C]\n  {X : Top} (F : X.presheaf C) (x : X) :\n  F.stalk_specializes (specializes_refl x) = \ud835\udfd9 _ :=\nF.stalk_hom_ext $ \u03bb _ _, by { dsimp, simpa }\n\nlemma _root_.Top.presheaf.stalk_hom_ext_of_is_basis {C : Type*} [category C] [limits.has_colimits C] {X : Top}\n  {B : set (opens X)} (hB : opens.is_basis B)\n  (F : X.presheaf C) {x} {Y : C}\n  {f\u2081 f\u2082 : F.stalk x \u27f6 Y}\n  (ih : \u2200 (U \u2208 B) (hxU : x \u2208 U), F.germ \u27e8x, hxU\u27e9 \u226b f\u2081 = F.germ \u27e8x, hxU\u27e9 \u226b f\u2082) : f\u2081 = f\u2082 :=\nTop.presheaf.stalk_hom_ext _\nbegin\n  intros U hxU,\n  obtain \u27e8V, hV, hxV, hVU : V \u2264 U\u27e9 := opens.is_basis_iff_nbhd.mp hB hxU,\n  have := congr_arg (\u03bb f, F.map (hom_of_le hVU).op \u226b f) (ih V hV hxV),\n  convert this using 1; rw [\u2190 category.assoc, F.germ_res]; refl\nend\n\nlemma Scheme.stalk_hom_affine_ext {X : Scheme} (x : X.carrier) {Y : CommRing}\n  {f\u2081 f\u2082 : X.stalk x \u27f6 Y} (ih : \u2200 (U : opens X.carrier) (hU : is_affine_open U) (hxU : x \u2208 U),\n    X.presheaf.germ \u27e8x, hxU\u27e9 \u226b f\u2081 = X.presheaf.germ \u27e8x, hxU\u27e9 \u226b f\u2082) : f\u2081 = f\u2082 :=\nTop.presheaf.stalk_hom_ext_of_is_basis (is_basis_affine_open X) _ ih\n\n@[reassoc]\nlemma Spec_\u0393_naturality' {R S : CommRing} (f : R \u27f6 S) :\n  f \u226b to_Spec_\u0393 S = to_Spec_\u0393 R \u226b Scheme.\u0393.map (Scheme.Spec.map f.op).op :=\nSpec_\u0393_naturality f\n\n@[simp, reassoc]\nlemma PresheafedSpace.stalk_map_germ' {C : Type u} [category.{v} C] [has_colimits C]\n  {X Y : PresheafedSpace.{v} C} (\u03b1 : X \u27f6 Y) (x : X) {U : opens Y} (hxU : \u03b1.base x \u2208 U) :\n  Y.presheaf.germ \u27e8\u03b1.base x, hxU\u27e9 \u226b PresheafedSpace.stalk_map \u03b1 x = \u03b1.c.app _ \u226b\n    X.presheaf.germ \u27e8x, show x \u2208 (opens.map \u03b1.base).obj U, from hxU\u27e9 :=\nPresheafedSpace.stalk_map_germ \u03b1 U \u27e8_, _\u27e9\n\nlemma PresheafedSpace.stalk_map.congr_hom' {C : Type u} [category.{v} C] [has_colimits C]\n  {X Y : PresheafedSpace.{v} C} (\u03b1 \u03b2 : X \u27f6 Y) (h : \u03b1 = \u03b2) (x : X) :\n  PresheafedSpace.stalk_map \u03b1 x =\n    Y.presheaf.stalk_specializes (by subst h) \u226b PresheafedSpace.stalk_map \u03b2 x :=\nbegin\n  subst h,\n  apply Top.presheaf.stalk_hom_ext, \n  intros U hxU,\n  simp,\nend\n\ninstance {R : Type*} [comm_ring R] [local_ring R] : \n  is_iso (structure_sheaf.to_stalk R (local_ring.closed_point R)) :=\nbegin\n  have : \u2200 x : (local_ring.closed_point R).as_ideal.prime_compl, is_unit (x : R),\n  { exact \u03bb x, not_not.mp x.2 },\n  have : is_iso (is_localization.at_units R\n    (local_ring.closed_point R).as_ideal.prime_compl\n    ((Spec.structure_sheaf R).presheaf.stalk\n    (local_ring.closed_point R)) this).to_ring_equiv.to_CommRing_iso.hom := infer_instance,\n  convert this using 1,\n  let S := _, change S = CommRing.of S, clear_value S, cases S, refl \nend\n\ninstance {C} [category C] [has_colimits C] {X : Top} (F : X.presheaf C) {x y} (e : x = y) :\n  is_iso (F.stalk_specializes (specializes_of_eq e)) :=\n\u27e8\u27e8F.stalk_specializes (specializes_of_eq e.symm), by simp, by simp\u27e9\u27e9\n\nlemma \u0393_Spec.adjunction_unit_app_base_apply  {X : Scheme} (x) :\n  (\u0393_Spec.adjunction.unit.app X).1.base x =\n    prime_spectrum.comap (X.to_LocallyRingedSpace.\u0393_to_stalk x) (local_ring.closed_point _) :=\nrfl\n\nlemma morphism_restrict_val_base {X Y : Scheme} (f : X \u27f6 Y) (U : opens Y.carrier) :\n  \u21d1(f \u2223_ U).1.base = U.1.restrict_preimage f.1.base :=\nfunext (\u03bb x, subtype.ext (morphism_restrict_base_coe f U x))\n\n/-- The stalks are isomorphic on inseparable points -/\n@[simps] noncomputable\ndef _root_.Top.presheaf.stalk_congr {X : Top} {C : Type*} [category C] [has_colimits C]\n  (F : X.presheaf C) {x y : X}\n  (e : inseparable x y) : F.stalk x \u2245 F.stalk y :=\n\u27e8F.stalk_specializes e.ge, F.stalk_specializes e.le, by simp, by simp\u27e9\n\nlemma _root_.inseparable.of_eq {\u03b1 :Type*} [topological_space \u03b1] {x y :\u03b1} \n  (e : x = y) : inseparable x y := e \u25b8 inseparable.refl x\n\n/--\nThe stalk map of a restriction of a morphism is isomorphic to the stalk map of the original map.\n-/\nnoncomputable\ndef morphism_restrict_stalk_map {X Y : Scheme} (f : X \u27f6 Y) (U : opens Y.carrier) (x) :\n  arrow.mk (PresheafedSpace.stalk_map (f \u2223_ U).1 x) \u2245\n    arrow.mk (PresheafedSpace.stalk_map f.1 x.1) :=\nbegin\n  fapply arrow.iso_mk',\n  { refine Y.restrict_stalk_iso U.open_embedding ((f \u2223_ U).1 x) \u226a\u226b Top.presheaf.stalk_congr _ _,\n    apply inseparable.of_eq,\n    exact morphism_restrict_base_coe f U x },\n  { exact X.restrict_stalk_iso _ _ },\n  { apply Top.presheaf.stalk_hom_ext,\n    intros V hxV,\n    simp only [Top.presheaf.stalk_congr_hom, category_theory.category.assoc,\n      category_theory.iso.trans_hom],\n    erw PresheafedSpace.restrict_stalk_iso_hom_eq_germ_assoc,\n    erw PresheafedSpace.stalk_map_germ_assoc _ _ \u27e8_, _\u27e9,\n    rw [Top.presheaf.germ_stalk_specializes'_assoc],\n    erw PresheafedSpace.stalk_map_germ _ _ \u27e8_, _\u27e9,\n    erw PresheafedSpace.restrict_stalk_iso_hom_eq_germ,\n    rw [morphism_restrict_c_app, category.assoc, Top.presheaf.germ_res],\n    refl }\nend\n\nlemma prime_spectrum.Union_basic_open_eq_top_iff {R : Type*} [comm_ring R] {\u03b9 : Type*}\n  (f : \u03b9 \u2192 R) : (\u2a06 i : \u03b9, prime_spectrum.basic_open (f i)) = \u22a4 \u2194 ideal.span (set.range f) = \u22a4 :=\nbegin\n  erw opens.supr_mk (\u03bb i : \u03b9, (prime_spectrum.basic_open (f i)).1),\n  rw [\u2190 opens.ext_iff, subtype.coe_mk],\n  simp_rw [subtype.val_eq_coe, prime_spectrum.basic_open_eq_zero_locus_compl],\n  rw [\u2190 set.compl_Inter, opens.coe_top],\n  erw compl_eq_top,\n  rw [\u2190 prime_spectrum.zero_locus_Union, \u2190 prime_spectrum.zero_locus_span],\n  erw prime_spectrum.zero_locus_empty_iff_eq_top,\n  simp,\nend\n\nlemma CommRing.is_iso_iff_bijective {R S : CommRing} (f : R \u27f6 S) :\n  is_iso f \u2194 function.bijective f :=\nbegin\n  rw \u2190 is_iso_iff_bijective,\n  change is_iso f \u2194 is_iso ((forget CommRing).map f),\n  refine \u27e8\u03bb H, by exactI infer_instance, \u03bb H, by exactI is_iso_of_reflects_iso f (forget CommRing)\u27e9,\nend\n\nlemma bijective_of_is_localization {R S T : Type*} [comm_ring R] [comm_ring S] [comm_ring T]\n  [algebra R S] [algebra R T] (M : submonoid R) [is_localization M S] [is_localization M T]\n  (f : S \u2192+* T) (hf : f.comp (algebra_map R S) = algebra_map R T) : function.bijective f :=\nbegin\n  have : f = is_localization.alg_equiv M S T,\n  { apply is_localization.ring_hom_ext M, { rw hf, ext, simp }, { apply_instance } },\n  rw this,\n  exact (is_localization.alg_equiv M S T).to_equiv.bijective,\nend\n\nlemma \u0393_Spec.adjunction.unit_app_map_basic_open {X : Scheme} (r : X.presheaf.obj (op \u22a4)) :\n  (opens.map (\u0393_Spec.adjunction.unit.app X).1.base).obj (prime_spectrum.basic_open r) =\n    X.basic_open r :=\nbegin\n  rw \u2190 basic_open_eq_of_affine,\n  erw Scheme.preimage_basic_open,\n  change X.basic_open _ = _,\n  congr,\n  rw [\u0393_Spec.adjunction_unit_app_app_top, \u2190 comp_apply],\n  simp [-comp_apply]\nend\n\nlemma preimage_adjunction_unit_basic_open (X : Scheme) (r : X.presheaf.obj (op \u22a4)) :\n  (opens.map (\u0393_Spec.adjunction.unit.app X).1.base).obj (prime_spectrum.basic_open r) =\n    X.basic_open r :=\nbegin\n  rw \u2190 basic_open_eq_of_affine,\n  erw Scheme.preimage_basic_open,\n  congr',\n  rw [\u0393_Spec.adjunction_unit_app_app_top, \u2190 comp_apply],\n  simp [-comp_apply]\nend\n\nlemma supr_basic_open_eq_top_of_span_eq_top (X : Scheme) (s : set (X.presheaf.obj $ op \u22a4))\n  (h : ideal.span s = \u22a4) : (\u2a06 i : s, X.basic_open i.1) = \u22a4 :=\nbegin\n  have := prime_spectrum.Union_basic_open_eq_top_iff (coe : s \u2192 X.presheaf.obj (op \u22a4)),\n  rw subtype.range_coe at this,\n  rw \u2190 this at h,\n  apply_fun (opens.map (\u0393_Spec.adjunction.unit.app X).1.base).obj at h,\n  rw opens.map_supr at h,\n  convert h,\n  ext1 i,\n  exact (preimage_adjunction_unit_basic_open X _).symm\nend\n\nnoncomputable\ndef Spec_\u0393_arrow_iso_of_is_affine {X Y : Scheme} (f : X \u27f6 Y) [is_affine X] [is_affine Y] :\n  arrow.mk f \u2245 arrow.mk (Scheme.Spec.map (Scheme.\u0393.map f.op).op) :=\narrow.iso_mk' _ _ (as_iso $ \u0393_Spec.adjunction.unit.app _) (as_iso $ \u0393_Spec.adjunction.unit.app _)\n  (\u0393_Spec.adjunction.unit_naturality f)\n\nnoncomputable\ndef \u0393_Spec_arrow_iso {R S : CommRing} (f : R \u27f6 S) :\n  arrow.mk f \u2245 arrow.mk (Scheme.\u0393.map (Scheme.Spec.map f.op).op) :=\n(arrow.iso_of_nat_iso Spec_\u0393_identity (arrow.mk f)).symm\n\nend algebraic_geometry", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/algebraic_geometry/misc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.37875958400051896}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport algebraic_topology.split_simplicial_object\nimport algebraic_topology.dold_kan.degeneracies\nimport algebraic_topology.dold_kan.functor_n\n\n/-!\n\n# Split simplicial objects in preadditive categories\n\nIn this file we define a functor `nondeg_complex : simplicial_object.split C \u2964 chain_complex C \u2115`\nwhen `C` is a preadditive category with finite coproducts, and get an isomorphism\n`to_karoubi_nondeg_complex_iso_N\u2081 : nondeg_complex \u22d9 to_karoubi _ \u2245 forget C \u22d9 dold_kan.N\u2081`.\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category\n  category_theory.preadditive category_theory.idempotents opposite\n  algebraic_topology algebraic_topology.dold_kan\n\nopen_locale big_operators simplicial dold_kan\n\nnamespace simplicial_object\n\nnamespace splitting\n\nvariables {C : Type*} [category C] [has_finite_coproducts C]\n  {X : simplicial_object C} (s : splitting X)\n\n/-- The projection on a summand of the coproduct decomposition given\nby a splitting of a simplicial object. -/\ndef \u03c0_summand [has_zero_morphisms C] {\u0394 : simplex_category\u1d52\u1d56} (A : index_set \u0394) :\n  X.obj \u0394 \u27f6 s.N A.1.unop.len :=\nbegin\n  refine (s.iso \u0394).inv \u226b sigma.desc (\u03bb B, _),\n  by_cases B = A,\n  { exact eq_to_hom (by { subst h, refl, }), },\n  { exact 0, },\nend\n\n@[simp, reassoc]\nlemma \u03b9_\u03c0_summand_eq_id [has_zero_morphisms C] {\u0394 : simplex_category\u1d52\u1d56} (A : index_set \u0394) :\n  s.\u03b9_summand A \u226b s.\u03c0_summand A = \ud835\udfd9 _ :=\nbegin\n  dsimp [\u03b9_summand, \u03c0_summand],\n  simp only [summand, assoc, is_iso.hom_inv_id_assoc],\n  erw [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n  dsimp,\n  simp only [eq_self_iff_true, if_true],\nend\n\n@[simp, reassoc]\nlemma \u03b9_\u03c0_summand_eq_zero [has_zero_morphisms C] {\u0394 : simplex_category\u1d52\u1d56} (A B : index_set \u0394)\n  (h : B \u2260 A) : s.\u03b9_summand A \u226b s.\u03c0_summand B = 0 :=\nbegin\n  dsimp [\u03b9_summand, \u03c0_summand],\n  simp only [summand, assoc, is_iso.hom_inv_id_assoc],\n  erw [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n  apply dif_neg,\n  exact h.symm,\nend\n\nvariable [preadditive C]\n\nlemma decomposition_id (\u0394 : simplex_category\u1d52\u1d56) :\n  \ud835\udfd9 (X.obj \u0394) = \u2211 (A : index_set \u0394), s.\u03c0_summand A \u226b s.\u03b9_summand A :=\nbegin\n  apply s.hom_ext',\n  intro A,\n  rw [comp_id, comp_sum, finset.sum_eq_single A, \u03b9_\u03c0_summand_eq_id_assoc],\n  { intros B h\u2081 h\u2082,\n    rw [s.\u03b9_\u03c0_summand_eq_zero_assoc _ _ h\u2082, zero_comp], },\n  { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\nend\n\n@[simp, reassoc]\nlemma \u03c3_comp_\u03c0_summand_id_eq_zero {n : \u2115} (i : fin (n+1)) :\n  X.\u03c3 i \u226b s.\u03c0_summand (index_set.id (op [n+1])) = 0 :=\nbegin\n  apply s.hom_ext',\n  intro A,\n  dsimp only [simplicial_object.\u03c3],\n  rw [comp_zero, s.\u03b9_summand_epi_naturality_assoc A (simplex_category.\u03c3 i).op,\n    \u03b9_\u03c0_summand_eq_zero],\n  symmetry,\n  change \u00ac (A.epi_comp (simplex_category.\u03c3 i).op).eq_id,\n  rw index_set.eq_id_iff_len_eq,\n  have h := simplex_category.len_le_of_epi (infer_instance : epi A.e),\n  dsimp at \u22a2 h,\n  linarith,\nend\n\n/-- If a simplicial object `X` in an additive category is split,\nthen `P_infty` vanishes on all the summands of `X _[n]` which do\nnot correspond to the identity of `[n]`. -/\nlemma \u03b9_summand_comp_P_infty_eq_zero {X : simplicial_object C}\n  (s : simplicial_object.splitting X)\n  {n : \u2115} (A : simplicial_object.splitting.index_set (op [n]))\n  (hA : \u00ac A.eq_id) :\n  s.\u03b9_summand A \u226b P_infty.f n = 0 :=\nbegin\n  rw simplicial_object.splitting.index_set.eq_id_iff_mono at hA,\n  rw [simplicial_object.splitting.\u03b9_summand_eq, assoc,\n    degeneracy_comp_P_infty X n A.e hA, comp_zero],\nend\n\nlemma comp_P_infty_eq_zero_iff {Z : C} {n : \u2115} (f : Z \u27f6 X _[n]) :\n  f \u226b P_infty.f n = 0 \u2194 f \u226b s.\u03c0_summand (index_set.id (op [n])) = 0 :=\nbegin\n  split,\n  { intro h,\n    cases n,\n    { dsimp at h,\n      rw [comp_id] at h,\n      rw [h, zero_comp], },\n    { have h' := f \u226b= P_infty_f_add_Q_infty_f (n+1),\n      dsimp at h',\n      rw [comp_id, comp_add, h, zero_add] at h',\n      rw [\u2190 h', assoc, Q_infty_f, decomposition_Q, preadditive.sum_comp,\n        preadditive.comp_sum, finset.sum_eq_zero],\n      intros i hi,\n      simp only [assoc, \u03c3_comp_\u03c0_summand_id_eq_zero, comp_zero], }, },\n  { intro h,\n    rw [\u2190 comp_id f, assoc, s.decomposition_id, preadditive.sum_comp,\n      preadditive.comp_sum, fintype.sum_eq_zero],\n    intro A,\n    by_cases hA : A.eq_id,\n    { dsimp at hA,\n      subst hA,\n      rw [assoc, reassoc_of h, zero_comp], },\n    { simp only [assoc, s.\u03b9_summand_comp_P_infty_eq_zero A hA, comp_zero], }, },\nend\n\n@[simp, reassoc]\nlemma P_infty_comp_\u03c0_summand_id (n : \u2115) :\n  P_infty.f n \u226b s.\u03c0_summand (index_set.id (op [n])) = s.\u03c0_summand (index_set.id (op [n])) :=\nbegin\n  conv_rhs { rw \u2190 id_comp (s.\u03c0_summand _), },\n  symmetry,\n  rw [\u2190 sub_eq_zero, \u2190 sub_comp, \u2190 comp_P_infty_eq_zero_iff, sub_comp, id_comp,\n    P_infty_f_idem, sub_self],\nend\n\n@[simp, reassoc]\nlemma \u03c0_summand_comp_\u03b9_summand_comp_P_infty_eq_P_infty (n : \u2115) :\n  s.\u03c0_summand (index_set.id (op [n])) \u226b s.\u03b9_summand (index_set.id (op [n])) \u226b P_infty.f n =\n    P_infty.f n :=\nbegin\n  conv_rhs { rw \u2190 id_comp (P_infty.f n), },\n  erw [s.decomposition_id, preadditive.sum_comp],\n  rw [fintype.sum_eq_single (index_set.id (op [n])), assoc],\n  rintros A (hA : \u00acA.eq_id),\n  rw [assoc, s.\u03b9_summand_comp_P_infty_eq_zero A hA, comp_zero],\nend\n\n/-- The differentials `s.d i j : s.N i \u27f6 s.N j` on nondegenerate simplices of a split\nsimplicial object are induced by the differentials on the alternating face map complex. -/\n@[simp]\ndef d (i j : \u2115) : s.N i \u27f6 s.N j :=\ns.\u03b9_summand (index_set.id (op [i])) \u226b K[X].d i j \u226b s.\u03c0_summand (index_set.id (op [j]))\n\nlemma \u03b9_summand_comp_d_comp_\u03c0_summand_eq_zero (j k : \u2115) (A : index_set (op [j])) (hA : \u00acA.eq_id) :\n  s.\u03b9_summand A \u226b K[X].d j k \u226b s.\u03c0_summand (index_set.id (op [k])) = 0 :=\nbegin\n  rw A.eq_id_iff_mono at hA,\n  rw [\u2190 assoc, \u2190 s.comp_P_infty_eq_zero_iff, assoc, \u2190 P_infty.comm j k, s.\u03b9_summand_eq, assoc,\n    degeneracy_comp_P_infty_assoc X j A.e hA, zero_comp, comp_zero],\nend\n\n/-- If `s` is a splitting of a simplicial object `X` in a preadditive category,\n`s.nondeg_complex` is a chain complex which is given in degree `n` by\nthe nondegenerate `n`-simplices of `X`. -/\n@[simps]\ndef nondeg_complex : chain_complex C \u2115 :=\n{ X := s.N,\n  d := s.d,\n  shape' := \u03bb i j hij, by simp only [d, K[X].shape i j hij, zero_comp, comp_zero],\n  d_comp_d' := \u03bb i j k hij hjk, begin\n    simp only [d, assoc],\n    have eq : K[X].d i j \u226b \ud835\udfd9 (X.obj (op [j])) \u226b K[X].d j k \u226b\n      s.\u03c0_summand (index_set.id (op [k])) = 0 :=\n      by erw [id_comp, homological_complex.d_comp_d_assoc, zero_comp],\n    rw s.decomposition_id at eq,\n    classical,\n    rw [fintype.sum_eq_add_sum_compl (index_set.id (op [j])), add_comp, comp_add, assoc,\n      preadditive.sum_comp, preadditive.comp_sum, finset.sum_eq_zero, add_zero] at eq, swap,\n    { intros A hA,\n      simp only [finset.mem_compl, finset.mem_singleton] at hA,\n      simp only [assoc, \u03b9_summand_comp_d_comp_\u03c0_summand_eq_zero _ _ _ _ hA, comp_zero], },\n    rw [eq, comp_zero],\n  end }\n\n/-- The chain complex `s.nondeg_complex` attached to a splitting of a simplicial object `X`\nbecomes isomorphic to the normalized Moore complex `N\u2081.obj X` defined as a formal direct\nfactor in the category `karoubi (chain_complex C \u2115)`. -/\n@[simps]\ndef to_karoubi_nondeg_complex_iso_N\u2081 : (to_karoubi _).obj s.nondeg_complex \u2245 N\u2081.obj X :=\n{ hom :=\n  { f :=\n    { f := \u03bb n, s.\u03b9_summand (index_set.id (op [n])) \u226b P_infty.f n,\n      comm' := \u03bb i j hij, begin\n        dsimp,\n        rw [assoc, assoc, assoc, \u03c0_summand_comp_\u03b9_summand_comp_P_infty_eq_P_infty,\n          homological_complex.hom.comm],\n      end, },\n    comm := by { ext n, dsimp, rw [id_comp, assoc, P_infty_f_idem], }, },\n  inv :=\n  { f :=\n    { f := \u03bb n, s.\u03c0_summand (index_set.id (op [n])),\n      comm' := \u03bb i j hij, begin\n        dsimp,\n        slice_rhs 1 1 { rw \u2190 id_comp (K[X].d i j), },\n        erw s.decomposition_id,\n        rw [sum_comp, sum_comp, finset.sum_eq_single (index_set.id (op [i])), assoc, assoc],\n        { intros A h hA,\n          simp only [assoc, s.\u03b9_summand_comp_d_comp_\u03c0_summand_eq_zero _ _ _ hA, comp_zero], },\n        { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\n      end, },\n    comm := by { ext n, dsimp, simp only [comp_id, P_infty_comp_\u03c0_summand_id], }, },\n  hom_inv_id' := begin\n    ext n,\n    simpa only [assoc, P_infty_comp_\u03c0_summand_id, karoubi.comp_f,\n      homological_complex.comp_f, \u03b9_\u03c0_summand_eq_id],\n  end,\n  inv_hom_id' := begin\n    ext n,\n    simp only [\u03c0_summand_comp_\u03b9_summand_comp_P_infty_eq_P_infty, karoubi.comp_f,\n      homological_complex.comp_f, N\u2081_obj_p, karoubi.id_eq],\n  end, }\n\nend splitting\n\nnamespace split\n\nvariables {C : Type*} [category C] [preadditive C] [has_finite_coproducts C]\n\n/-- The functor which sends a split simplicial object in a preadditive category to\nthe chain complex which consists of nondegenerate simplices. -/\n@[simps]\ndef nondeg_complex_functor : split C \u2964 chain_complex C \u2115 :=\n{ obj := \u03bb S, S.s.nondeg_complex,\n  map := \u03bb S\u2081 S\u2082 \u03a6,\n  { f := \u03a6.f,\n    comm' := \u03bb i j hij, begin\n      dsimp,\n      erw [\u2190 \u03b9_summand_naturality_symm_assoc \u03a6 (splitting.index_set.id (op [i])),\n        ((alternating_face_map_complex C).map \u03a6.F).comm_assoc i j],\n      simp only [assoc],\n      congr' 2,\n      apply S\u2081.s.hom_ext',\n      intro A,\n      dsimp [alternating_face_map_complex],\n      erw \u03b9_summand_naturality_symm_assoc \u03a6 A,\n      by_cases A.eq_id,\n      { dsimp at h,\n        subst h,\n        simpa only [splitting.\u03b9_\u03c0_summand_eq_id, comp_id, splitting.\u03b9_\u03c0_summand_eq_id_assoc], },\n      { have h' : splitting.index_set.id (op [j]) \u2260 A := by { symmetry, exact h, },\n        rw [S\u2081.s.\u03b9_\u03c0_summand_eq_zero_assoc _ _ h', S\u2082.s.\u03b9_\u03c0_summand_eq_zero _ _ h',\n          zero_comp, comp_zero], },\n    end }, }\n\n/-- The natural isomorphism (in `karoubi (chain_complex C \u2115)`) between the chain complex\nof nondegenerate simplices of a split simplicial object and the normalized Moore complex\ndefined as a formal direct factor of the alternating face map complex. -/\n@[simps]\ndef to_karoubi_nondeg_complex_functor_iso_N\u2081 :\n  nondeg_complex_functor \u22d9 to_karoubi (chain_complex C \u2115) \u2245 forget C \u22d9 dold_kan.N\u2081 :=\nnat_iso.of_components (\u03bb S, S.s.to_karoubi_nondeg_complex_iso_N\u2081)\n  (\u03bb S\u2081 S\u2082 \u03a6, begin\n    ext n,\n    dsimp,\n    simp only [karoubi.comp_f, to_karoubi_map_f, homological_complex.comp_f,\n      nondeg_complex_functor_map_f, splitting.to_karoubi_nondeg_complex_iso_N\u2081_hom_f_f,\n      N\u2081_map_f, alternating_face_map_complex.map_f, assoc, P_infty_f_idem_assoc],\n    erw \u2190 split.\u03b9_summand_naturality_symm_assoc \u03a6 (splitting.index_set.id (op [n])),\n    rw P_infty_f_naturality,\n  end)\n\nend split\n\nend simplicial_object\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_topology/dold_kan/split_simplicial_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3786937978460437}}
{"text": "import data.polynomial.eval\nimport tactic.expand_exists\nimport complexity_class.lemmas\nimport polytime.size\n\nopen tree tencodable function\nopen_locale tree\n\nnamespace tree\n\ninductive polytime : (tree unit \u2192 tree unit) \u2192 Prop\n| nil : polytime (\u03bb _, nil)\n| id' : polytime id\n| left : polytime (\u03bb x, x.left)\n| right : polytime (\u03bb x, x.right)\n| pair {f\u2081 f\u2082} : polytime f\u2081 \u2192 polytime f\u2082 \u2192 polytime (\u03bb x, (f\u2081 x) \u25b3 (f\u2082 x))\n| comp {f\u2081 f\u2082} : polytime f\u2081 \u2192 polytime f\u2082 \u2192 polytime (f\u2081 \u2218 f\u2082)\n| ite {f g\u2081 g\u2082} : polytime f \u2192 polytime g\u2081 \u2192 polytime g\u2082 \u2192 polytime (\u03bb x, if f x = nil then g\u2081 x else g\u2082 x)\n| bounded_rec {f} : polytime f \u2192 polysize_fun (\u03bb x : tree unit, f^[x.left.num_nodes] x.right) \u2192\n    polytime (\u03bb x : tree unit, f^[x.left.num_nodes] x.right)\n\nnamespace polytime\n\ntheorem of_eq {f g : tree unit \u2192 tree unit} (hf : polytime f) (H : \u2200 n, f n = g n) : polytime g :=\n(funext H : f = g) \u25b8 hf\n\nprotected theorem const : \u2200 (n : tree unit), polytime (\u03bb _, n)\n| tree.nil := nil\n| (x \u25b3 y) := (const x).pair (const y)\n\n-- TODO: how to do make original lemma protected?\nprotected lemma id : polytime (\u03bb x, x) := id'\n\n@[simp] lemma uncurry_unary {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) : \u21bff = f := rfl\n\ntheorem num_nodes_poly {f : tree unit \u2192 tree unit} (hf : polytime f) :\n  polysize_fun f :=\nbegin\n  induction hf,\n  case nil { use 0, simp, },\n  case id' { use polynomial.X, simp, },\n  case left { use polynomial.X, simpa using left_num_nodes_le, },\n  case right { use polynomial.X, simpa using right_num_nodes_le, },\n  case pair : f\u2081 f\u2082 _ _ ih\u2081 ih\u2082\n  { rcases ih\u2081 with \u27e8P\u2081, ih\u2081\u27e9, rcases ih\u2082 with \u27e8P\u2082, ih\u2082\u27e9,\n    use P\u2081 + P\u2082 + 1,\n    intro x,\n    simp, mono, },\n  case comp : f\u2081 f\u2082 _ _ ih\u2081 ih\u2082 \n  { rcases ih\u2081 with \u27e8P\u2081, ih\u2081\u27e9, rcases ih\u2082 with \u27e8P\u2082, ih\u2082\u27e9,\n    use P\u2081.comp P\u2082,\n    intro x,\n    simp only [comp_app, polynomial.eval_comp],\n    exact (ih\u2081 _).trans (P\u2081.eval_mono (ih\u2082 x)), },\n  case ite : f g\u2081 g\u2082 _ _ _ _ ih\u2081 ih\u2082 { exact polysize_fun.ite ih\u2081 ih\u2082, },\n  case bounded_rec : _ _ H { exact H, }\nend\n\nend polytime\n\nend tree\n\ndef polytime : complexity_class :=\n{ prop := tree.polytime,\n  nil := tree.polytime.nil,\n  left := tree.polytime.left,\n  right := tree.polytime.right,\n  id := tree.polytime.id,\n  pair := \u03bb f\u2081 f\u2082, tree.polytime.pair,\n  comp := \u03bb f\u2081 f\u2082, tree.polytime.comp,\n  ite' := \u03bb c f g, tree.polytime.ite }\n\nopen_locale complexity_class\nlocalized \"notation `PTIME` := polytime\" in complexity_class\n\n@[simp] lemma tree.polytime_iff {f : tree unit \u2192 tree unit} :\n  tree.polytime f \u2194 f \u2208\u2091 PTIME := @complexity_class.prop_iff_mem PTIME f\n\n@[complexity] lemma tree.polytime_of_polytime {f : tree unit \u2192 tree unit} (h : f \u2208\u2091 PTIME) :\n  tree.polytime f := by rwa tree.polytime_iff\n\nclass polycodable (\u03b1 : Type) extends tencodable \u03b1 :=\n(poly [] : tencodable.decode \u03b1 \u2208\u2091 PTIME)\n\nattribute [complexity] polycodable.poly\n\nvariables {\u03b1 \u03b2 : Type}\n\ninstance : polycodable (tree unit) :=\n\u27e8complexity_class.decode\u27e9\n\ninstance [polycodable \u03b1] : polycodable (option \u03b1) :=\n\u27e8complexity_class.option_decode (polycodable.poly \u03b1)\u27e9\n\ninstance [polycodable \u03b1] [polycodable \u03b2] : polycodable (\u03b1 \u2295 \u03b2) :=\n\u27e8complexity_class.sum_decode (polycodable.poly \u03b1) (polycodable.poly \u03b2)\u27e9\n\ninstance [polycodable \u03b1] [polycodable \u03b2] : polycodable (\u03b1 \u00d7 \u03b2) :=\n\u27e8complexity_class.prod_decode (polycodable.poly \u03b1) (polycodable.poly \u03b2)\u27e9\n\nlemma polycodable.mem'_iff_mem [polycodable \u03b1] [tencodable \u03b2] {\u03b3 : Type} [has_uncurry \u03b3 \u03b1 \u03b2] (f : \u03b3) :\n  f \u2208\u209b PTIME \u2194 f \u2208\u2091 PTIME := complexity_class.mem'_iff_mem_decode (polycodable.poly \u03b1) \n\nopen polysize\nvariables [tencodable \u03b1] [tencodable \u03b2]\n\n@[complexity] lemma polytime.size_le {\u03b3 : Type} [has_uncurry \u03b3 \u03b1 \u03b2] [polysize \u03b1] [polysize \u03b2] {f : \u03b3} (hf : f \u2208\u2091 PTIME) :\n  polysize_fun f :=\nbegin\n  rcases hf with \u27e8f', pf, hf\u27e9, cases polysize.upper \u03b2 with u hu, cases polysize.lower \u03b1 with l hl,\n  obtain \u27e8p, hp\u27e9 := pf.num_nodes_poly,\n  use u.comp (p.comp l),\n  intro x,\n  refine (hu (\u21bff x)).trans _,\n  simp [\u2190 hf],\n  exact u.eval_mono ((hp _).trans $ p.eval_mono (hl _)),\nend\n\nlemma polycodable.decode_num_nodes_le (\u03b1 : Type) [polycodable \u03b1] :\n  \u2203 p : polynomial \u2115, \u2200 x y, decode \u03b1 x = some y \u2192 (encode y).num_nodes \u2264 p.eval x.num_nodes :=\nlet \u27e8p, hp\u27e9 := @tree.polytime.num_nodes_poly (\u03bb x, encode (decode \u03b1 x)) (by complexity) in\n  \u27e8p, \u03bb x y h, nat.le_of_succ_le (by simpa [h, encode, of_option] using hp x)\u27e9\n\nnamespace polytime\n\nsection iterate\n\nlemma nil_node_iterate (n : \u2115) (y : tree unit) : ((\u03bb x, tree.nil \u25b3 x)^[n] y).num_nodes = y.num_nodes + n :=\nby { induction n; simp [function.iterate_succ', *], refl, }\n\n@[complexity] lemma num_nodes : (@tree.num_nodes unit) \u2208\u2091 PTIME :=\n\u27e8_, (tree.polytime.bounded_rec (tree.polytime.pair tree.polytime.nil tree.polytime.id) (\u27e8polynomial.X, (\u03bb x, by { simp [nil_node_iterate], rw add_comm, cases x; simp, })\u27e9)).comp (tree.polytime.pair tree.polytime.id tree.polytime.nil),\n  \u03bb x, by simp [encode_nat_eq_iterate]\u27e9\n\ninstance : polycodable \u2115 :=\n\u27e8complexity_class.some.comp polytime.num_nodes\u27e9\n\ntheorem iterate_aux {n : tree unit \u2192 \u2115} {f : tree unit \u2192 tree unit \u2192 tree unit} {st : tree unit \u2192 tree unit}\n  (hn : n \u2208\u2091 PTIME) (hf : f \u2208\u2091 PTIME) (hst : st \u2208\u2091 PTIME)\n  (hf' : polysize_fun (\u03bb (n : \u2115) (x y : tree unit), (f x)^[n] y)) : (\u03bb x, (f x)^[n x] (st x)) \u2208\u2091 PTIME :=\nbegin\n  set F : tree unit \u2192 tree unit := \u03bb x, x.left \u25b3 (f x.left x.right),\n  have hF : tree.polytime F := by { dsimp [F], complexity, },\n  have hF' : \u2200 n x y, F^[n] (x \u25b3 y) = x \u25b3 ((f x)^[n] y),\n  { intros n x y, induction n generalizing y; simp [F, *], }, \n  rcases hn with \u27e8n', pn, hn\u27e9, rcases hst with \u27e8st', pst, hst\u27e9,\n  refine \u27e8_, tree.polytime.right.comp ((tree.polytime.bounded_rec hF _).comp\n    (tree.polytime.pair pn (tree.polytime.pair tree.polytime.id pst))), _\u27e9,\n  swap, { intro x, simp only [encode_unit_tree] at hn hst, simp [hF', hn, hst], },\n  cases hf' with p hp,\n  use polynomial.X + p + 1,\n  rintro (_|\u27e8\u27e8\u27e9, n, xy\u27e9), { simp, },\n  have : \u2200 x y, (F^[n.num_nodes] (x \u25b3 y)).num_nodes \u2264\n    x.num_nodes + (p.eval $ n.num_nodes + (x.num_nodes + y.num_nodes) + 1) + 1,\n  { intros x y, rw hF', specialize hp (n.num_nodes, x, y), simp [has_uncurry.uncurry, add_assoc] at hp \u22a2, refine hp.trans (p.eval_mono _), simp, },\n  rcases xy with (_|\u27e8\u27e8\u27e9, x, y\u27e9),\n  { simp, cases n.num_nodes with n, { simp, }, refine (this _ _).trans _, simp, },\n  { refine (this _ _).trans _, simp [add_assoc x.num_nodes y.num_nodes 1],\n    mono*, exacts [le_add_right $ le_add_left $ le_self_add, le_self_add], },\nend\n\ntheorem iterate_safe_aux {n : tree unit \u2192 \u2115} {f : tree unit \u2192 tree unit \u2192 tree unit} {st : tree unit \u2192 tree unit}\n  (hn : n \u2208\u2091 PTIME) (hf : f \u2208\u2091 PTIME) (hst : st \u2208\u2091 PTIME)\n  (hf' : polysize_safe f) : (\u03bb x, (f x)^[n x] (st x)) \u2208\u2091 PTIME :=\nbegin\n  apply iterate_aux hn hf hst,\n  use polynomial.X + polynomial.X * hf'.poly,\n  rintros \u27e8n, x, y\u27e9,\n  refine (hf'.size_le n x y).trans _,\n  simp [has_uncurry.uncurry], mono*,\n  exacts [le_add_left le_add_self, le_self_add, le_add_left le_self_add, zero_le', zero_le'],\nend\n\nlocal attribute [complexity] iterate_safe_aux\n\n@[complexity] lemma nat_add : ((+) : \u2115 \u2192 \u2115 \u2192 \u2115) \u2208\u2091 PTIME :=\nby { complexity using \u03bb m n, ((encode m) \u25b3 (encode n)).num_nodes.pred, simp, }\n\n@[complexity] lemma nat_mul : ((*) : \u2115 \u2192 \u2115 \u2192 \u2115) \u2208\u2091 PTIME :=\nbegin\n  refine \u27e8\u03bb x, ((\u03bb acc : tree unit, encode (x.left.num_nodes + acc.num_nodes))^[x.right.num_nodes] tree.nil), by complexity, _\u27e9,\n  rintro \u27e8m, n\u27e9,\n  suffices : ((\u03bb acc, encode (m + acc.num_nodes))^[n] tree.nil) = encode (m * n), { simpa [encode_prod], },\n  induction n with n ih, { simp [encode], },\n  simp [iterate_succ_apply', ih, nat.mul_succ, add_comm],\nend\n\nlemma encode_pred (n : \u2115) : encode n.pred = (encode n).right :=\nby { cases n; simp [encode], }\n\n@[complexity] lemma nat_tsub : (has_sub.sub : \u2115 \u2192 \u2115 \u2192 \u2115) \u2208\u2091 PTIME :=\nbegin\n  refine \u27e8\u03bb x, tree.right^[x.right.num_nodes] x.left, by complexity, _\u27e9,\n  rintro \u27e8m, n\u27e9,\n  suffices : (tree.right^[n] (encode m)) = encode (m - n), { simpa [encode_prod], },\n  induction n with n ih, { simp, }, { simp [iterate_succ_apply', ih, nat.sub_succ, encode_pred], }\nend\n\n/-- For any fixed polynomial `p`, `p.eval` runs in polynomial time -/\nlemma polynomial_eval (p : polynomial \u2115) :\n  polytime.mem (\u03bb n : \u2115, p.eval n) :=\nbegin\n  induction p using polynomial.induction_on with p p q ih\u2081 ih\u2082 p q ih,\n  { simpa using polytime.const _, },\n  { simpa using nat_add.comp\u2082 ih\u2081 ih\u2082, },\n  simpa [pow_add, \u2190 mul_assoc] using nat_mul.comp\u2082 ih polytime.id',\nend\n\n@[complexity] lemma polynomial_eval' (p : polynomial \u2115) {f : \u03b1 \u2192 \u2115} (hf : f \u2208\u2091 PTIME) :\n  polytime.mem (\u03bb x, p.eval (f x)) := (polynomial_eval p).comp hf\n\n@[complexity] lemma nat_le : polytime.mem_pred ((\u2264) : \u2115 \u2192 \u2115 \u2192 Prop) :=\nby { complexity using (\u03bb x y, x - y = 0), rw tsub_eq_zero_iff_le, }\n\n@[complexity] lemma nat_lt : polytime.mem_pred ((<) : \u2115 \u2192 \u2115 \u2192 Prop) :=\nby { complexity using (\u03bb x y, x \u2264 y \u2227 \u00ac(y \u2264 x)), rw lt_iff_le_not_le, }\n\n/- Combining these allows us to iterate a function which automatically exits when it takes too long.\nThus, even on \"bad\" inputs, it does not take more than polynomial time. -/\ndef tree.guard_size (x : tree unit) (n : \u2115) : tree unit :=\nif x.num_nodes \u2264 n then x else tree.nil\n\n@[complexity] lemma tree_guard_size : tree.guard_size \u2208\u2091 PTIME :=\nby { delta tree.guard_size, complexity, }\n\nlemma tree.guard_size_num_nodes_le (x : tree unit) (n) : (x.guard_size n).num_nodes \u2264 n :=\nby { simp only [tree.guard_size], split_ifs, { assumption, }, exact zero_le', }\n\nlemma tree.guard_size_pos (x : tree unit) {n} (h : x.num_nodes \u2264 n) : x.guard_size n = x :=\nby rwa [tree.guard_size, if_pos]\n\nlocal attribute [-complexity] iterate_safe_aux\n\ntheorem iterate_encode_size {n : \u03b1 \u2192 \u2115} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} : n \u2208\u2091 PTIME \u2192 f \u2208\u2091 PTIME \u2192 g \u2208\u2091 PTIME \u2192 \n  (\u2203 p : polynomial \u2115, \u2200 x (m \u2264 n x), (encode $ (f x)^[m] (g x)).num_nodes \u2264 p.eval (encode x).num_nodes) \u2192\n  polytime.mem (\u03bb x, (f x)^[n x] (g x))\n| \u27e8n', pn, hn\u27e9 \u27e8f', pf, hf\u27e9 \u27e8g', pg, hg\u27e9 \u27e8p, hp\u27e9 := begin\n  refine \u27e8\u03bb x, (\u03bb y, (f' (x \u25b3 y)).guard_size (p.eval $ x.num_nodes))^[(n' x).num_nodes] (g' x), _, _\u27e9,\n  { rw [complexity_class.prop_iff_mem] at *, apply iterate_aux,\n    { complexity, }, { complexity, }, { exact pg, },\n    use p + polynomial.X,\n    rintros \u27e8(_|n), x, y\u27e9, { simp [has_uncurry.uncurry, \u2190 add_assoc], },\n    transitivity p.eval x.num_nodes,\n    { simpa [-iterate_succ, iterate_succ_apply', has_uncurry.uncurry] using tree.guard_size_num_nodes_le _ _, },\n    simp only [polynomial.eval_add, polynomial.eval_X],\n    exact le_add_right (p.eval_mono $ le_add_left le_self_add), },\n  intros x,\n  dsimp only [has_uncurry.uncurry, id] at hn hf hg \u22a2,\n  simp only [prod.forall, encode_prod] at hf,\n  simp only [hn, hg, encode_num_nodes],\n  specialize hp x, revert hp, generalize : (n x) = M, intro hp,\n  induction M with M ih, { simp, },\n  specialize ih (\u03bb m hm, hp m (nat.le_succ_of_le hm)), specialize hp _ rfl.le,\n  simp only [iterate_succ_apply', ih, hf] at \u22a2 hp,\n  rwa tree.guard_size_pos,\nend\n\ntheorem iterate [polysize \u03b1] [polysize \u03b2] {n : \u03b1 \u2192 \u2115} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hn : n \u2208\u2091 PTIME)\n  (hf : f \u2208\u2091 PTIME) (hg : g \u2208\u2091 PTIME) :\n  (\u2203 p : polynomial \u2115, \u2200 x (m \u2264 n x), size ((f x)^[m] (g x)) \u2264 p.eval (size x)) \u2192\n  polytime.mem (\u03bb x, (f x)^[n x] (g x))\n| \u27e8p, hp\u27e9 :=\nlet \u27e8q\u2081, hq\u2081\u27e9 := polysize.upper \u03b1, \u27e8q\u2082, hq\u2082\u27e9 := polysize.lower \u03b2 in\n iterate_encode_size hn hf hg \u27e8q\u2082.comp (p.comp q\u2081),\n \u03bb x m hm, by { simpa using (hq\u2082 _).trans (q\u2082.eval_mono $ (hp x m hm).trans $ p.eval_mono $ hq\u2081 _), }\u27e9\n\n@[complexity] theorem iterate_safe [polysize \u03b1] [polysize \u03b2] {n : \u03b1 \u2192 \u2115} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hn : n \u2208\u2091 PTIME)\n  (hf : f \u2208\u2091 PTIME) (hg : g \u2208\u2091 PTIME) (hp : polysize_safe f) : polytime.mem (\u03bb x, (f x)^[n x] (g x)) :=\nlet \u27e8p\u2081, hp\u2081\u27e9 := polytime.size_le hg, \u27e8p\u2082, hp\u2082\u27e9 := polytime.size_le hn in\niterate hn hf hg \u27e8p\u2081 + p\u2082 * hp.poly, \u03bb x m hm,\n  (hp.size_le m x (g x)).trans (by { simp, mono*, exacts [hm.trans (hp\u2082 _), zero_le'], })\u27e9\n\nend iterate\n\nend polytime\n", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/polytime/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455589, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3786937909083864}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.punit\nimport category_theory.structured_arrow\nimport category_theory.is_connected\nimport category_theory.limits.yoneda\nimport category_theory.limits.types\n\n/-!\n# Final and initial functors\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA functor `F : C \u2964 D` is final if for every `d : D`,\nthe comma category of morphisms `d \u27f6 F.obj c` is connected.\n\nDually, a functor `F : C \u2964 D` is initial if for every `d : D`,\nthe comma category of morphisms `F.obj c \u27f6 d` is connected.\n\nWe show that right adjoints are examples of final functors, while\nleft adjoints are examples of initial functors.\n\nFor final functors, we prove that the following three statements are equivalent:\n1. `F : C \u2964 D` is final.\n2. Every functor `G : D \u2964 E` has a colimit if and only if `F \u22d9 G` does,\n   and these colimits are isomorphic via `colimit.pre G F`.\n3. `colimit (F \u22d9 coyoneda.obj (op d)) \u2245 punit`.\n\nStarting at 1. we show (in `cocones_equiv`) that\nthe categories of cocones over `G : D \u2964 E` and over `F \u22d9 G` are equivalent.\n(In fact, via an equivalence which does not change the cocone point.)\nThis readily implies 2., as `comp_has_colimit`, `has_colimit_of_comp`, and `colimit_iso`.\n\nFrom 2. we can specialize to `G = coyoneda.obj (op d)` to obtain 3., as `colimit_comp_coyoneda_iso`.\n\nFrom 3., we prove 1. directly in `cofinal_of_colimit_comp_coyoneda_iso_punit`.\n\nDually, we prove that if a functor `F : C \u2964 D` is initial, then any functor `G : D \u2964 E` has a\nlimit if and only if `F \u22d9 G` does, and these limits are isomorphic via `limit.pre G F`.\n\n\n## Naming\nThere is some discrepancy in the literature about naming; some say 'cofinal' instead of 'final'.\nThe explanation for this is that the 'co' prefix here is *not* the usual category-theoretic one\nindicating duality, but rather indicating the sense of \"along with\".\n\n## Future work\nDualise condition 3 above and the implications 2 \u21d2 3 and 3 \u21d2 1 to initial functors.\n\n## References\n* https://stacks.math.columbia.edu/tag/09WN\n* https://ncatlab.org/nlab/show/final+functor\n* Borceux, Handbook of Categorical Algebra I, Section 2.11.\n  (Note he reverses the roles of definition and main result relative to here!)\n-/\n\nnoncomputable theory\n\nuniverses v v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nnamespace category_theory\n\nnamespace functor\n\nopen opposite\nopen category_theory.limits\n\nsection arbitrary_universe\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D]\n\n/--\nA functor `F : C \u2964 D` is final if for every `d : D`, the comma category of morphisms `d \u27f6 F.obj c`\nis connected.\n\nSee <https://stacks.math.columbia.edu/tag/04E6>\n-/\nclass final (F : C \u2964 D) : Prop :=\n(out (d : D) : is_connected (structured_arrow d F))\n\nattribute [instance] final.out\n\n/--\nA functor `F : C \u2964 D` is initial if for every `d : D`, the comma category of morphisms\n`F.obj c \u27f6 d` is connected.\n-/\nclass initial (F : C \u2964 D) : Prop :=\n(out (d : D) : is_connected (costructured_arrow F d))\n\nattribute [instance] initial.out\n\ninstance final_op_of_initial (F : C \u2964 D) [initial F] : final F.op :=\n{ out := \u03bb d, is_connected_of_equivalent (costructured_arrow_op_equivalence F (unop d)) }\n\ninstance initial_op_of_final (F : C \u2964 D) [final F] : initial F.op :=\n{ out := \u03bb d, is_connected_of_equivalent (structured_arrow_op_equivalence F (unop d)) }\n\nlemma final_of_initial_op (F : C \u2964 D) [initial F.op] : final F :=\n{ out := \u03bb d, @is_connected_of_is_connected_op _ _\n  (is_connected_of_equivalent (structured_arrow_op_equivalence F d).symm) }\n\nlemma initial_of_final_op (F : C \u2964 D) [final F.op] : initial F :=\n{ out := \u03bb d, @is_connected_of_is_connected_op _ _\n  (is_connected_of_equivalent (costructured_arrow_op_equivalence F d).symm) }\n\n/-- If a functor `R : D \u2964 C` is a right adjoint, it is final. -/\nlemma final_of_adjunction {L : C \u2964 D} {R : D \u2964 C} (adj : L \u22a3 R) : final R :=\n{ out := \u03bb c,\n  let u : structured_arrow c R := structured_arrow.mk (adj.unit.app c) in\n  @zigzag_is_connected _ _ \u27e8u\u27e9 $ \u03bb f g, relation.refl_trans_gen.trans\n    (relation.refl_trans_gen.single (show zag f u, from\n      or.inr \u27e8structured_arrow.hom_mk ((adj.hom_equiv c f.right).symm f.hom) (by simp)\u27e9))\n    (relation.refl_trans_gen.single (show zag u g, from\n      or.inl \u27e8structured_arrow.hom_mk ((adj.hom_equiv c g.right).symm g.hom) (by simp)\u27e9)) }\n\n/-- If a functor `L : C \u2964 D` is a left adjoint, it is initial. -/\nlemma initial_of_adjunction {L : C \u2964 D} {R : D \u2964 C} (adj : L \u22a3 R) : initial L :=\n{ out := \u03bb d,\n  let u : costructured_arrow L d := costructured_arrow.mk (adj.counit.app d) in\n  @zigzag_is_connected _ _ \u27e8u\u27e9 $ \u03bb f g, relation.refl_trans_gen.trans\n    (relation.refl_trans_gen.single (show zag f u, from\n      or.inl \u27e8costructured_arrow.hom_mk (adj.hom_equiv f.left d f.hom) (by simp)\u27e9))\n    (relation.refl_trans_gen.single (show zag u g, from\n      or.inr \u27e8costructured_arrow.hom_mk (adj.hom_equiv g.left d g.hom) (by simp)\u27e9)) }\n\n@[priority 100]\ninstance final_of_is_right_adjoint (F : C \u2964 D) [h : is_right_adjoint F] : final F :=\nfinal_of_adjunction h.adj\n\n@[priority 100]\ninstance initial_of_is_left_adjoint (F : C \u2964 D) [h : is_left_adjoint F] : initial F :=\ninitial_of_adjunction h.adj\n\nnamespace final\n\nvariables (F : C \u2964 D) [final F]\n\ninstance (d : D) : nonempty (structured_arrow d F) := is_connected.is_nonempty\n\nvariables {E : Type u\u2083} [category.{v\u2083} E] (G : D \u2964 E)\n\n/--\nWhen `F : C \u2964 D` is cofinal, we denote by `lift F d` an arbitrary choice of object in `C` such that\nthere exists a morphism `d \u27f6 F.obj (lift F d)`.\n-/\ndef lift (d : D) : C :=\n(classical.arbitrary (structured_arrow d F)).right\n\n/--\nWhen `F : C \u2964 D` is cofinal, we denote by `hom_to_lift` an arbitrary choice of morphism\n`d \u27f6 F.obj (lift F d)`.\n-/\ndef hom_to_lift (d : D) : d \u27f6 F.obj (lift F d) :=\n(classical.arbitrary (structured_arrow d F)).hom\n\n/--\nWe provide an induction principle for reasoning about `lift` and `hom_to_lift`.\nWe want to perform some construction (usually just a proof) about\nthe particular choices `lift F d` and `hom_to_lift F d`,\nit suffices to perform that construction for some other pair of choices\n(denoted `X\u2080 : C` and `k\u2080 : d \u27f6 F.obj X\u2080` below),\nand to show how to transport such a construction\n*both* directions along a morphism between such choices.\n-/\ndef induction {d : D} (Z : \u03a0 (X : C) (k : d \u27f6 F.obj X), Sort*)\n  (h\u2081 : \u03a0 X\u2081 X\u2082 (k\u2081 : d \u27f6 F.obj X\u2081) (k\u2082 : d \u27f6 F.obj X\u2082) (f : X\u2081 \u27f6 X\u2082),\n    (k\u2081 \u226b F.map f = k\u2082) \u2192 Z X\u2081 k\u2081 \u2192 Z X\u2082 k\u2082)\n  (h\u2082 : \u03a0 X\u2081 X\u2082 (k\u2081 : d \u27f6 F.obj X\u2081) (k\u2082 : d \u27f6 F.obj X\u2082) (f : X\u2081 \u27f6 X\u2082),\n    (k\u2081 \u226b F.map f = k\u2082) \u2192 Z X\u2082 k\u2082 \u2192 Z X\u2081 k\u2081)\n  {X\u2080 : C} {k\u2080 : d \u27f6 F.obj X\u2080} (z : Z X\u2080 k\u2080) : Z (lift F d) (hom_to_lift F d) :=\nbegin\n  apply nonempty.some,\n  apply @is_preconnected_induction _ _ _\n    (\u03bb (Y : structured_arrow d F), Z Y.right Y.hom) _ _ (structured_arrow.mk k\u2080) z,\n  { intros j\u2081 j\u2082 f a, fapply h\u2081 _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\n  { intros j\u2081 j\u2082 f a, fapply h\u2082 _ _ _ _ f.right _ a, convert f.w.symm, dsimp, simp, },\nend\n\nvariables {F G}\n\n/--\nGiven a cocone over `F \u22d9 G`, we can construct a `cocone G` with the same cocone point.\n-/\n@[simps]\ndef extend_cocone : cocone (F \u22d9 G) \u2964 cocone G :=\n{ obj := \u03bb c,\n  { X := c.X,\n    \u03b9 :=\n    { app := \u03bb X, G.map (hom_to_lift F X) \u226b c.\u03b9.app (lift F X),\n      naturality' := \u03bb X Y f,\n      begin\n        dsimp, simp,\n        -- This would be true if we'd chosen `lift F X` to be `lift F Y`\n        -- and `hom_to_lift F X` to be `f \u226b hom_to_lift F Y`.\n        apply induction F\n          (\u03bb Z k, G.map f \u226b G.map (hom_to_lift F Y) \u226b c.\u03b9.app (lift F Y) = G.map k \u226b c.\u03b9.app Z),\n        { intros Z\u2081 Z\u2082 k\u2081 k\u2082 g a z,\n        rw [\u2190a, functor.map_comp, category.assoc, \u2190functor.comp_map, c.w, z], },\n        { intros Z\u2081 Z\u2082 k\u2081 k\u2082 g a z,\n        rw [\u2190a, functor.map_comp, category.assoc, \u2190functor.comp_map, c.w] at z,\n        rw z, },\n        { rw [\u2190functor.map_comp_assoc], },\n      end } },\n  map := \u03bb X Y f,\n  { hom := f.hom, } }\n\n@[simp]\nlemma colimit_cocone_comp_aux (s : cocone (F \u22d9 G)) (j : C) :\n  G.map (hom_to_lift F (F.obj j)) \u226b s.\u03b9.app (lift F (F.obj j)) =\n    s.\u03b9.app j :=\nbegin\n  -- This point is that this would be true if we took `lift (F.obj j)` to just be `j`\n  -- and `hom_to_lift (F.obj j)` to be `\ud835\udfd9 (F.obj j)`.\n  apply induction F (\u03bb X k, G.map k \u226b s.\u03b9.app X = (s.\u03b9.app j : _)),\n  { intros j\u2081 j\u2082 k\u2081 k\u2082 f w h, rw \u2190w, rw \u2190 s.w f at h, simpa using h, },\n  { intros j\u2081 j\u2082 k\u2081 k\u2082 f w h, rw \u2190w at h, rw \u2190 s.w f, simpa using h, },\n  { exact s.w (\ud835\udfd9 _), },\nend\n\nvariables (F G)\n\n/--\nIf `F` is cofinal,\nthe category of cocones on `F \u22d9 G` is equivalent to the category of cocones on `G`,\nfor any `G : D \u2964 E`.\n-/\n@[simps]\ndef cocones_equiv : cocone (F \u22d9 G) \u224c cocone G :=\n{ functor := extend_cocone,\n  inverse := cocones.whiskering F,\n  unit_iso := nat_iso.of_components (\u03bb c, cocones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb c, cocones.ext (iso.refl _) (by tidy)) (by tidy), }.\n\nvariables {G}\n\n/--\nWhen `F : C \u2964 D` is cofinal, and `t : cocone G` for some `G : D \u2964 E`,\n`t.whisker F` is a colimit cocone exactly when `t` is.\n-/\ndef is_colimit_whisker_equiv (t : cocone G) : is_colimit (t.whisker F) \u2243 is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G).symm\n\n/--\nWhen `F` is cofinal, and `t : cocone (F \u22d9 G)`,\n`extend_cocone.obj t` is a colimit coconne exactly when `t` is.\n-/\ndef is_colimit_extend_cocone_equiv (t : cocone (F \u22d9 G)) :\n  is_colimit (extend_cocone.obj t) \u2243 is_colimit t :=\nis_colimit.of_cocone_equiv (cocones_equiv F G)\n\n/-- Given a colimit cocone over `G : D \u2964 E` we can construct a colimit cocone over `F \u22d9 G`. -/\n@[simps]\ndef colimit_cocone_comp (t : colimit_cocone G) :\n  colimit_cocone (F \u22d9 G) :=\n{ cocone := _,\n  is_colimit := (is_colimit_whisker_equiv F _).symm (t.is_colimit) }\n\n@[priority 100]\ninstance comp_has_colimit [has_colimit G] :\n  has_colimit (F \u22d9 G) :=\nhas_colimit.mk (colimit_cocone_comp F (get_colimit_cocone G))\n\nlemma colimit_pre_is_iso_aux {t : cocone G} (P : is_colimit t) :\n  ((is_colimit_whisker_equiv F _).symm P).desc (t.whisker F) = \ud835\udfd9 t.X :=\nbegin\n  dsimp [is_colimit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  dsimp, simp,\nend\n\ninstance colimit_pre_is_iso [has_colimit G] :\n  is_iso (colimit.pre G F) :=\nbegin\n  rw colimit.pre_eq (colimit_cocone_comp F (get_colimit_cocone G)) (get_colimit_cocone G),\n  erw colimit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nsection\nvariables (G)\n\n/--\nWhen `F : C \u2964 D` is cofinal, and `G : D \u2964 E` has a colimit, then `F \u22d9 G` has a colimit also and\n`colimit (F \u22d9 G) \u2245 colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso [has_colimit G] : colimit (F \u22d9 G) \u2245 colimit G := as_iso (colimit.pre G F)\n\nend\n\n/-- Given a colimit cocone over `F \u22d9 G` we can construct a colimit cocone over `G`. -/\n@[simps]\ndef colimit_cocone_of_comp (t : colimit_cocone (F \u22d9 G)) :\n  colimit_cocone G :=\n{ cocone := extend_cocone.obj t.cocone,\n  is_colimit := (is_colimit_extend_cocone_equiv F _).symm (t.is_colimit), }\n\n/--\nWhen `F` is cofinal, and `F \u22d9 G` has a colimit, then `G` has a colimit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_colimit`.)\n-/\nlemma has_colimit_of_comp [has_colimit (F \u22d9 G)] :\n  has_colimit G :=\nhas_colimit.mk (colimit_cocone_of_comp F (get_colimit_cocone (F \u22d9 G)))\n\n\nsection\nlocal attribute [instance] has_colimit_of_comp\n\n/--\nWhen `F` is cofinal, and `F \u22d9 G` has a colimit, then `G` has a colimit also and\n`colimit (F \u22d9 G) \u2245 colimit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef colimit_iso' [has_colimit (F \u22d9 G)] : colimit (F \u22d9 G) \u2245 colimit G := as_iso (colimit.pre G F)\n\nend\n\nend final\nend arbitrary_universe\n\nnamespace final\n\nvariables {C : Type v} [category.{v} C] {D : Type v} [category.{v} D] (F : C \u2964 D) [final F]\n\n/--\nIf the universal morphism `colimit (F \u22d9 coyoneda.obj (op d)) \u27f6 colimit (coyoneda.obj (op d))`\nis an isomorphism (as it always is when `F` is cofinal),\nthen `colimit (F \u22d9 coyoneda.obj (op d)) \u2245 punit`\n(simply because `colimit (coyoneda.obj (op d)) \u2245 punit`).\n-/\ndef colimit_comp_coyoneda_iso (d : D) [is_iso (colimit.pre (coyoneda.obj (op d)) F)] :\n  colimit (F \u22d9 coyoneda.obj (op d)) \u2245 punit :=\nas_iso (colimit.pre (coyoneda.obj (op d)) F) \u226a\u226b coyoneda.colimit_coyoneda_iso (op d)\n\n\n\n/--\nIf `colimit (F \u22d9 coyoneda.obj (op d)) \u2245 punit` for all `d : D`, then `F` is cofinal.\n-/\nlemma cofinal_of_colimit_comp_coyoneda_iso_punit\n  (I : \u03a0 d, colimit (F \u22d9 coyoneda.obj (op d)) \u2245 punit) : final F :=\n\u27e8\u03bb d, begin\n  haveI : nonempty (structured_arrow d F),\n  { have := (I d).inv punit.star,\n    obtain \u27e8j, y, rfl\u27e9 := limits.types.jointly_surjective'.{v v} this,\n    exact \u27e8structured_arrow.mk y\u27e9, },\n  apply zigzag_is_connected,\n  rintros \u27e8\u27e8\u27e8\u27e9\u27e9,X\u2081,f\u2081\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9,X\u2082,f\u2082\u27e9,\n  dsimp at *,\n  let y\u2081 := colimit.\u03b9 (F \u22d9 coyoneda.obj (op d)) X\u2081 f\u2081,\n  let y\u2082 := colimit.\u03b9 (F \u22d9 coyoneda.obj (op d)) X\u2082 f\u2082,\n  have e : y\u2081 = y\u2082,\n  { apply (I d).to_equiv.injective, ext, },\n  have t := types.colimit_eq.{v v} e,\n  clear e y\u2081 y\u2082,\n  exact zigzag_of_eqv_gen_quot_rel t,\nend\u27e9\n\nend final\n\n\nnamespace initial\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D] (F : C \u2964 D) [initial F]\n\ninstance (d : D) : nonempty (costructured_arrow F d) := is_connected.is_nonempty\n\nvariables {E : Type u\u2083} [category.{v\u2083} E] (G : D \u2964 E)\n\n/--\nWhen `F : C \u2964 D` is initial, we denote by `lift F d` an arbitrary choice of object in `C` such that\nthere exists a morphism `F.obj (lift F d) \u27f6 d`.\n-/\ndef lift (d : D) : C := (classical.arbitrary (costructured_arrow F d)).left\n\n/--\nWhen `F : C \u2964 D` is initial, we denote by `hom_to_lift` an arbitrary choice of morphism\n`F.obj (lift F d) \u27f6 d`.\n-/\ndef hom_to_lift (d : D) : F.obj (lift F d) \u27f6 d :=\n  (classical.arbitrary (costructured_arrow F d)).hom\n\n/--\nWe provide an induction principle for reasoning about `lift` and `hom_to_lift`.\nWe want to perform some construction (usually just a proof) about\nthe particular choices `lift F d` and `hom_to_lift F d`,\nit suffices to perform that construction for some other pair of choices\n(denoted `X\u2080 : C` and `k\u2080 : F.obj X\u2080 \u27f6 d` below),\nand to show how to transport such a construction\n*both* directions along a morphism between such choices.\n-/\ndef induction {d : D} (Z : \u03a0 (X : C) (k : F.obj X \u27f6 d), Sort*)\n  (h\u2081 : \u03a0 X\u2081 X\u2082 (k\u2081 : F.obj X\u2081 \u27f6 d) (k\u2082 : F.obj X\u2082 \u27f6 d) (f : X\u2081 \u27f6 X\u2082),\n    (F.map f \u226b k\u2082 = k\u2081) \u2192 Z X\u2081 k\u2081 \u2192 Z X\u2082 k\u2082)\n  (h\u2082 : \u03a0 X\u2081 X\u2082 (k\u2081 : F.obj X\u2081 \u27f6 d) (k\u2082 : F.obj X\u2082 \u27f6 d) (f : X\u2081 \u27f6 X\u2082),\n    (F.map f \u226b k\u2082 = k\u2081) \u2192 Z X\u2082 k\u2082 \u2192 Z X\u2081 k\u2081)\n  {X\u2080 : C} {k\u2080 : F.obj X\u2080 \u27f6 d} (z : Z X\u2080 k\u2080) : Z (lift F d) (hom_to_lift F d) :=\nbegin\n  apply nonempty.some,\n  apply @is_preconnected_induction _ _ _\n    (\u03bb Y : costructured_arrow F d, Z Y.left Y.hom) _ _ (costructured_arrow.mk k\u2080) z,\n  { intros j\u2081 j\u2082 f a, fapply h\u2081 _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },\n  { intros j\u2081 j\u2082 f a, fapply h\u2082 _ _ _ _ f.left _ a, convert f.w, dsimp, simp, },\nend\n\nvariables {F G}\n\n/--\nGiven a cone over `F \u22d9 G`, we can construct a `cone G` with the same cocone point.\n-/\n@[simps]\ndef extend_cone : cone (F \u22d9 G) \u2964 cone G :=\n{ obj := \u03bb c,\n  { X := c.X,\n    \u03c0 :=\n    { app := \u03bb d, c.\u03c0.app (lift F d) \u226b G.map (hom_to_lift F d),\n      naturality' := \u03bb X Y f,\n      begin\n        dsimp, simp,\n        -- This would be true if we'd chosen `lift F Y` to be `lift F X`\n        -- and `hom_to_lift F Y` to be `hom_to_lift F X \u226b f`.\n        apply induction F (\u03bb Z k, (c.\u03c0.app Z \u226b G.map k : c.X \u27f6 _) =\n          c.\u03c0.app (lift F X) \u226b G.map (hom_to_lift F X) \u226b G.map f),\n        { intros Z\u2081 Z\u2082 k\u2081 k\u2082 g a z,\n        rw [\u2190a, functor.map_comp, \u2190functor.comp_map, \u2190category.assoc, \u2190category.assoc, c.w] at z,\n        rw [z, category.assoc] },\n        { intros Z\u2081 Z\u2082 k\u2081 k\u2082 g a z,\n        rw [\u2190a, functor.map_comp, \u2190functor.comp_map, \u2190category.assoc, \u2190category.assoc,\n          c.w, z, category.assoc] },\n        { rw [\u2190functor.map_comp], },\n      end } },\n  map := \u03bb X Y f,\n  { hom := f.hom, } }\n\n@[simp]\nlemma limit_cone_comp_aux (s : cone (F \u22d9 G)) (j : C) :\n  s.\u03c0.app (lift F (F.obj j)) \u226b G.map (hom_to_lift F (F.obj j)) =\n    s.\u03c0.app j :=\nbegin\n  -- This point is that this would be true if we took `lift (F.obj j)` to just be `j`\n  -- and `hom_to_lift (F.obj j)` to be `\ud835\udfd9 (F.obj j)`.\n  apply induction F (\u03bb X k, s.\u03c0.app X \u226b G.map k = (s.\u03c0.app j : _)),\n  { intros j\u2081 j\u2082 k\u2081 k\u2082 f w h, rw \u2190s.w f, rw \u2190w at h, simpa using h, },\n  { intros j\u2081 j\u2082 k\u2081 k\u2082 f w h, rw \u2190s.w f at h, rw \u2190w, simpa using h, },\n  { exact s.w (\ud835\udfd9 _), },\nend\n\nvariables (F G)\n\n/--\nIf `F` is initial,\nthe category of cones on `F \u22d9 G` is equivalent to the category of cones on `G`,\nfor any `G : D \u2964 E`.\n-/\n@[simps]\ndef cones_equiv : cone (F \u22d9 G) \u224c cone G :=\n{ functor := extend_cone,\n  inverse := cones.whiskering F,\n  unit_iso := nat_iso.of_components (\u03bb c, cones.ext (iso.refl _) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb c, cones.ext (iso.refl _) (by tidy)) (by tidy), }.\n\nvariables {G}\n\n/--\nWhen `F : C \u2964 D` is initial, and `t : cone G` for some `G : D \u2964 E`,\n`t.whisker F` is a limit cone exactly when `t` is.\n-/\ndef is_limit_whisker_equiv (t : cone G) : is_limit (t.whisker F) \u2243 is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F G).symm\n\n/--\nWhen `F` is initial, and `t : cone (F \u22d9 G)`,\n`extend_cone.obj t` is a limit cone exactly when `t` is.\n-/\ndef is_limit_extend_cone_equiv (t : cone (F \u22d9 G)) :\n  is_limit (extend_cone.obj t) \u2243 is_limit t :=\nis_limit.of_cone_equiv (cones_equiv F G)\n\n/-- Given a limit cone over `G : D \u2964 E` we can construct a limit cone over `F \u22d9 G`. -/\n@[simps]\ndef limit_cone_comp (t : limit_cone G) :\n  limit_cone (F \u22d9 G) :=\n{ cone := _,\n  is_limit := (is_limit_whisker_equiv F _).symm (t.is_limit) }\n\n@[priority 100]\ninstance comp_has_limit [has_limit G] :\n  has_limit (F \u22d9 G) :=\nhas_limit.mk (limit_cone_comp F (get_limit_cone G))\n\nlemma limit_pre_is_iso_aux {t : cone G} (P : is_limit t) :\n  ((is_limit_whisker_equiv F _).symm P).lift (t.whisker F) = \ud835\udfd9 t.X :=\nbegin\n  dsimp [is_limit_whisker_equiv],\n  apply P.hom_ext,\n  intro j,\n  simp,\nend\n\ninstance limit_pre_is_iso [has_limit G] :\n  is_iso (limit.pre G F) :=\nbegin\n  rw limit.pre_eq (limit_cone_comp F (get_limit_cone G)) (get_limit_cone G),\n  erw limit_pre_is_iso_aux,\n  dsimp,\n  apply_instance,\nend\n\nsection\nvariables (G)\n\n/--\nWhen `F : C \u2964 D` is initial, and `G : D \u2964 E` has a limit, then `F \u22d9 G` has a limit also and\n`limit (F \u22d9 G) \u2245 limit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso [has_limit G] : limit (F \u22d9 G) \u2245 limit G := (as_iso (limit.pre G F)).symm\n\nend\n\n/-- Given a limit cone over `F \u22d9 G` we can construct a limit cone over `G`. -/\n@[simps]\ndef limit_cone_of_comp (t : limit_cone (F \u22d9 G)) :\n  limit_cone G :=\n{ cone := extend_cone.obj t.cone,\n  is_limit := (is_limit_extend_cone_equiv F _).symm (t.is_limit), }\n\n/--\nWhen `F` is initial, and `F \u22d9 G` has a limit, then `G` has a limit also.\n\nWe can't make this an instance, because `F` is not determined by the goal.\n(Even if this weren't a problem, it would cause a loop with `comp_has_limit`.)\n-/\nlemma has_limit_of_comp [has_limit (F \u22d9 G)] :\n  has_limit G :=\nhas_limit.mk (limit_cone_of_comp F (get_limit_cone (F \u22d9 G)))\n\nsection\nlocal attribute [instance] has_limit_of_comp\n\n/--\nWhen `F` is initial, and `F \u22d9 G` has a limit, then `G` has a limit also and\n`limit (F \u22d9 G) \u2245 limit G`\n\nhttps://stacks.math.columbia.edu/tag/04E7\n-/\ndef limit_iso' [has_limit (F \u22d9 G)] : limit (F \u22d9 G) \u2245 limit G :=\n(as_iso (limit.pre G F)).symm\n\nend\n\nend initial\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/final.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.37865191296477135}}
{"text": "import algebra.homology.homological_complex\nimport algebra.homology.additive\n\nnoncomputable theory\n\nuniverses v\n\nopen category_theory category_theory.limits\n\nvariables {C D : Type*} [category C] [category D]\nvariables [has_zero_object C] [has_zero_object D]\nvariables {M : Type*} {c : complex_shape M}\n\n/- there are already `prev_eq_zero` and `next_eq_zero`\n  in `les_homology.lean`, but with extra assumptions -/\nlemma homological_complex.prev_eq_zero' [has_zero_morphisms C]\n  {X Y : homological_complex C c} (f : X \u27f6 Y) (i : M) (h : c.prev i = none) : f.prev i = 0 :=\nby { dsimp [homological_complex.hom.prev], simpa only [h], }\n\nlemma homological_complex.next_eq_zero' [has_zero_morphisms C]\n  {X Y : homological_complex C c} (f : X \u27f6 Y) (i : M) (h : c.next i = none) : f.next i = 0 :=\nby { dsimp [homological_complex.hom.next], simpa only [h], }\n\nnamespace category_theory\n\nnamespace functor\n\nvariables [preadditive C] [preadditive D]\nvariables (F : C \u2964 D) [functor.additive F] (X Y : homological_complex C c)\n\ndef obj_X_prev (i : M) : F.obj (X.X_prev i) \u2245 ((F.map_homological_complex c).obj X).X_prev i :=\nbegin\n  rcases h : c.prev i with _ | \u27e8j, hij\u27e9,\n  { exact F.map_iso (homological_complex.X_prev_iso_zero X h) \u226a\u226b (map_zero_object F) \u226a\u226b\n      (homological_complex.X_prev_iso_zero _ h).symm, },\n  { exact F.map_iso (homological_complex.X_prev_iso X hij) \u226a\u226b (by refl) \u226a\u226b\n    (homological_complex.X_prev_iso _ hij).symm, },\nend\n\nlemma obj_X_prev_hom_eq (j i : M) (hij : c.rel j i) :\n  (F.obj_X_prev X i).hom = F.map (homological_complex.X_prev_iso X hij).hom \u226b\n    (eq_to_hom (by refl)) \u226b (homological_complex.X_prev_iso _ hij).inv :=\nbegin\n  dsimp [homological_complex.X_prev_iso, obj_X_prev],\n  simp only [c.prev_eq_some hij, eq_to_iso_map, iso.refl_trans, iso.trans_hom,\n    eq_to_iso.hom, iso.symm_hom, eq_to_iso.inv, eq_to_hom_map, category.id_comp],\nend\n\n@[reassoc]\nlemma map_prev_iso_hom (j i : M) (hij : c.rel j i) :\n  F.map (X.X_prev_iso hij).hom = (F.obj_X_prev X i).hom \u226b\n    (((F.map_homological_complex c).obj X).X_prev_iso hij).hom :=\nby simp only [F.obj_X_prev_hom_eq X j i hij, eq_to_hom_refl,\n    category.assoc, iso.inv_hom_id, category.comp_id]\n\ndef obj_X_next {M : Type*} {c : complex_shape M} (X : homological_complex C c) (i : M) :\n  F.obj (X.X_next i) \u2245 ((F.map_homological_complex c).obj X).X_next i :=\nbegin\n  rcases h : c.next i with _ | \u27e8j, hij\u27e9,\n  { exact F.map_iso (homological_complex.X_next_iso_zero X h) \u226a\u226b (map_zero_object F) \u226a\u226b\n      (homological_complex.X_next_iso_zero _ h).symm, },\n  { exact F.map_iso (homological_complex.X_next_iso X hij) \u226a\u226b (by refl) \u226a\u226b\n    (homological_complex.X_next_iso _ hij).symm, },\nend\n\nlemma obj_X_next_hom_eq (i j : M) (hij : c.rel i j) :\n  (F.obj_X_next X i).hom = F.map (homological_complex.X_next_iso X hij).hom \u226b\n    (eq_to_hom (by refl)) \u226b (homological_complex.X_next_iso _ hij).inv :=\nbegin\n  dsimp [homological_complex.X_next_iso, obj_X_next],\n  simp only [c.next_eq_some hij, eq_to_iso_map, iso.refl_trans, iso.trans_hom,\n    eq_to_iso.hom, iso.symm_hom, eq_to_iso.inv, eq_to_hom_map, category.id_comp],\nend\n\n@[reassoc]\nlemma map_next_iso_inv (i j : M) (hij : c.rel i j) :\n  F.map (X.X_next_iso hij).inv \u226b (F.obj_X_next X i).hom =\n    (((F.map_homological_complex c).obj X).X_next_iso hij).inv :=\nby simp only [F.obj_X_next_hom_eq X i j hij, \u2190 F.map_comp_assoc,\n    eq_to_hom_refl, category.id_comp, iso.inv_hom_id, map_id]\n\nlemma map_d_to (i : M) :\n  F.map (X.d_to i) = (F.obj_X_prev X i).hom \u226b ((F.map_homological_complex c).obj X).d_to i :=\nbegin\n  rcases h : c.prev i with _ | \u27e8j, hij\u27e9,\n  { simp only [homological_complex.d_to_eq_zero _ h, functor.map_zero, comp_zero], },\n  { rw homological_complex.d_to_eq _ hij,\n    rw homological_complex.d_to_eq _ hij,\n    rw \u2190 ((F.map_homological_complex c).obj X).X_prev_iso_comp_d_to hij,\n    simp only [map_comp, homological_complex.X_prev_iso_comp_d_to, map_homological_complex_obj_d,\n      map_prev_iso_hom_assoc], },\nend\n\nlemma d_from_map (i : M) :\n  F.map (X.d_from i) \u226b (F.obj_X_next X i).hom = ((F.map_homological_complex c).obj X).d_from i :=\nbegin\n  rcases h : c.next i with _ | \u27e8j, hij\u27e9,\n  { simp only [homological_complex.d_from_eq_zero _ h, functor.map_zero, zero_comp], },\n  { rw homological_complex.d_from_eq _ hij,\n    rw homological_complex.d_from_eq _ hij,\n    rw \u2190 ((F.map_homological_complex c).obj X).d_from_comp_X_next_iso hij,\n    simp only [map_comp, category.assoc, homological_complex.d_from_comp_X_next_iso,\n      map_homological_complex_obj_d, map_next_iso_inv], },\nend\n\nvariables {X Y}\n\ndef map_prev (f : X \u27f6 Y) (i : M) :\n  F.map (homological_complex.hom.prev f i) \u226b (F.obj_X_prev Y i).hom =\n  (F.obj_X_prev X i).hom \u226b homological_complex.hom.prev ((F.map_homological_complex c).map f) i :=\nbegin\n  rcases h : c.prev i with _ | \u27e8j, hij\u27e9,\n  { simp only [homological_complex.prev_eq_zero' _ _ h, functor.map_zero,\n      zero_comp, comp_zero], },\n  { simp only [homological_complex.hom.prev_eq _ hij,\n      F.obj_X_prev_hom_eq _ j i hij,\n      F.map_comp, eq_to_hom_refl, category.id_comp],\n    slice_lhs 3 4 { rw [\u2190 F.map_comp, iso.inv_hom_id, F.map_id], },\n    simp only [category.id_comp, map_homological_complex_map_f, category.assoc,\n      iso.inv_hom_id_assoc], },\nend\n\ndef map_next (f : X \u27f6 Y) (i : M) :\n  F.map (homological_complex.hom.next f i) \u226b (F.obj_X_next Y i).hom =\n  (F.obj_X_next X i).hom \u226b homological_complex.hom.next ((F.map_homological_complex c).map f) i :=\nbegin\n  rcases h : c.next i with _ | \u27e8j, hij\u27e9,\n  { simp only [homological_complex.next_eq_zero' _ _ h, functor.map_zero,\n      zero_comp, comp_zero], },\n  { simp only [homological_complex.hom.next_eq _ hij,\n      F.obj_X_next_hom_eq _ i j hij, F.map_comp, eq_to_hom_refl, category.id_comp],\n    slice_lhs 3 4 { rw [\u2190 F.map_comp, iso.inv_hom_id, F.map_id], },\n    simp only [category.id_comp, map_homological_complex_map_f, category.assoc,\n      iso.inv_hom_id_assoc], },\nend\n\nend functor\n\nnamespace nat_trans\n\nvariables [preadditive C] [preadditive D]\n  {F G : C \u2964 D} [functor.additive F] [functor.additive G] (\u03c6 : F \u27f6 G)\n  (X : homological_complex C c)\n\nlemma map_prev (i : M) : \u03c6.app (X.X_prev i) \u226b (G.obj_X_prev X i).hom =\n  (F.obj_X_prev X i).hom \u226b\n    homological_complex.hom.prev ((nat_trans.map_homological_complex \u03c6 c).app X) i :=\nbegin\n  rcases h : c.prev i with _ | \u27e8j, hij\u27e9,\n  { suffices : \u03c6.app (X.X_prev i) = 0,\n    { simp only [this, homological_complex.prev_eq_zero' _ _ h, zero_comp, comp_zero], },\n    apply is_zero.eq_zero_of_src,\n    exact is_zero.of_iso (is_zero_zero _)\n      (F.map_iso (X.X_prev_iso_zero h) \u226a\u226b F.map_zero_object), },\n  { simp only [functor.obj_X_prev_hom_eq _ X j i hij,\n      eq_to_hom_refl, category.id_comp, category.assoc, \u2190 \u03c6.naturality_assoc,\n      homological_complex.hom.prev_eq _ hij, iso.inv_hom_id_assoc,\n      map_homological_complex_app_f], },\nend\n\nlemma map_next (i : M) : \u03c6.app (X.X_next i) \u226b (G.obj_X_next X i).hom =\n  (F.obj_X_next X i).hom \u226b\n    homological_complex.hom.next ((nat_trans.map_homological_complex \u03c6 c).app X) i :=\nbegin\n  rcases h : c.next i with _ | \u27e8j, hij\u27e9,\n  { suffices : \u03c6.app (X.X_next i) = 0,\n    { simp only [this, homological_complex.next_eq_zero' _ _ h, zero_comp, comp_zero], },\n    apply is_zero.eq_zero_of_tgt,\n    exact is_zero.of_iso (is_zero_zero _)\n      (G.map_iso (X.X_next_iso_zero h) \u226a\u226b G.map_zero_object), },\n  { simp only [functor.obj_X_next_hom_eq _ X i j hij,\n      eq_to_hom_refl, category.id_comp, category.assoc, \u2190 \u03c6.naturality_assoc,\n      homological_complex.hom.next_eq _ hij, iso.inv_hom_id_assoc,\n      map_homological_complex_app_f], },\nend\n\nend nat_trans\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/homological_complex_map_d_to_d_from.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.37865191296477135}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport data.int.basic\nimport category_theory.shift.basic\nimport category_theory.concrete_category.basic\n\n/-!\n# Differential objects in a category.\n\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n\nWe build the category of differential objects, and some basic constructions\nsuch as the forgetful functor, zero morphisms and zero objects, and the shift functor\non differential objects.\n-/\n\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\n-- TODO: generalize to `has_shift C A` for an arbitrary `[add_monoid A]` `[has_one A]`.\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\n/--\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n-/\n@[nolint has_nonempty_instance]\nstructure differential_object :=\n(X : C)\n(d : X \u27f6 X\u27e61\u27e7)\n(d_squared' : d \u226b d\u27e6(1:\u2124)\u27e7' = 0 . obviously)\n\nrestate_axiom differential_object.d_squared'\nattribute [simp] differential_object.d_squared\n\nvariables {C}\n\nnamespace differential_object\n\n/--\nA morphism of differential objects is a morphism commuting with the differentials.\n-/\n@[ext, nolint has_nonempty_instance]\nstructure hom (X Y : differential_object C) :=\n(f : X.X \u27f6 Y.X)\n(comm' : X.d \u226b f\u27e61\u27e7' = f \u226b Y.d . obviously)\n\nrestate_axiom hom.comm'\nattribute [simp, reassoc] hom.comm\n\nnamespace hom\n\n/-- The identity morphism of a differential object. -/\n@[simps]\ndef id (X : differential_object C) : hom X X :=\n{ f := \ud835\udfd9 X.X }\n\n/-- The composition of morphisms of differential objects. -/\n@[simps]\ndef comp {X Y Z : differential_object C} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ f := f.f \u226b g.f, }\n\nend hom\n\ninstance category_of_differential_objects : category (differential_object C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z f g, hom.comp f g, }\n\n@[simp]\n\n\n@[simp]\nlemma comp_f {X Y Z : differential_object C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  (f \u226b g).f = f.f \u226b g.f :=\nrfl\n\n@[simp]\nlemma eq_to_hom_f {X Y : differential_object C} (h : X = Y) :\n  hom.f (eq_to_hom h) = eq_to_hom (congr_arg _ h) :=\nby { subst h, rw [eq_to_hom_refl, eq_to_hom_refl], refl }\n\nvariables (C)\n\n/-- The forgetful functor taking a differential object to its underlying object. -/\ndef forget : (differential_object C) \u2964 C :=\n{ obj := \u03bb X, X.X,\n  map := \u03bb X Y f, f.f, }\n\ninstance forget_faithful : faithful (forget C) :=\n{ }\n\ninstance has_zero_morphisms : has_zero_morphisms (differential_object C) :=\n{ has_zero := \u03bb X Y,\n  \u27e8{ f := 0 }\u27e9}\n\nvariables {C}\n\n@[simp]\nlemma zero_f (P Q : differential_object C) : (0 : P \u27f6 Q).f = 0 := rfl\n\n/--\nAn isomorphism of differential objects gives an isomorphism of the underlying objects.\n-/\n@[simps] def iso_app {X Y : differential_object C} (f : X \u2245 Y) : X.X \u2245 Y.X :=\n\u27e8f.hom.f, f.inv.f, by { dsimp, rw [\u2190 comp_f, iso.hom_inv_id, id_f] },\n  by { dsimp, rw [\u2190 comp_f, iso.inv_hom_id, id_f] }\u27e9\n\n@[simp] lemma iso_app_refl (X : differential_object C) : iso_app (iso.refl X) = iso.refl X.X := rfl\n@[simp] lemma iso_app_symm {X Y : differential_object C} (f : X \u2245 Y) :\n  iso_app f.symm = (iso_app f).symm := rfl\n@[simp] lemma iso_app_trans {X Y Z : differential_object C} (f : X \u2245 Y) (g : Y \u2245 Z) :\n  iso_app (f \u226a\u226b g) = iso_app f \u226a\u226b iso_app g := rfl\n\n/-- An isomorphism of differential objects can be constructed\nfrom an isomorphism of the underlying objects that commutes with the differentials. -/\n@[simps] def mk_iso {X Y : differential_object C}\n  (f : X.X \u2245 Y.X) (hf : X.d \u226b f.hom\u27e61\u27e7' = f.hom \u226b Y.d) : X \u2245 Y :=\n{ hom := \u27e8f.hom, hf\u27e9,\n  inv := \u27e8f.inv, by { dsimp, rw [\u2190 functor.map_iso_inv, iso.comp_inv_eq, category.assoc,\n    iso.eq_inv_comp, functor.map_iso_hom, hf] }\u27e9,\n  hom_inv_id' := by { ext1, dsimp, exact f.hom_inv_id },\n  inv_hom_id' := by { ext1, dsimp, exact f.inv_hom_id } }\n\nend differential_object\n\nnamespace functor\n\nuniverses v' u'\nvariables (D : Type u') [category.{v'} D]\nvariables [has_zero_morphisms D] [has_shift D \u2124]\n\n/--\nA functor `F : C \u2964 D` which commutes with shift functors on `C` and `D` and preserves zero morphisms\ncan be lifted to a functor `differential_object C \u2964 differential_object D`.\n-/\n@[simps]\ndef map_differential_object (F : C \u2964 D)\n  (\u03b7 : (shift_functor C (1:\u2124)).comp F \u27f6 F.comp (shift_functor D (1:\u2124)))\n  (hF : \u2200 c c', F.map (0 : c \u27f6 c') = 0) :\n  differential_object C \u2964 differential_object D :=\n{ obj := \u03bb X, { X := F.obj X.X,\n    d := F.map X.d \u226b \u03b7.app X.X,\n    d_squared' := begin\n      rw [functor.map_comp, \u2190 functor.comp_map F (shift_functor D (1:\u2124))],\n      slice_lhs 2 3 { rw [\u2190 \u03b7.naturality X.d] },\n      rw [functor.comp_map],\n      slice_lhs 1 2 { rw [\u2190 F.map_comp, X.d_squared, hF] },\n      rw [zero_comp, zero_comp],\n    end },\n  map := \u03bb X Y f, { f := F.map f.f,\n    comm' := begin\n      dsimp,\n      slice_lhs 2 3 { rw [\u2190 functor.comp_map F (shift_functor D (1:\u2124)), \u2190 \u03b7.naturality f.f] },\n      slice_lhs 1 2 { rw [functor.comp_map, \u2190 F.map_comp, f.comm, F.map_comp] },\n      rw [category.assoc]\n    end },\n  map_id' := by { intros, ext, simp },\n  map_comp' := by { intros, ext, simp }, }\n\nend functor\n\nend category_theory\n\nnamespace category_theory\n\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\n\nvariables [has_zero_object C] [has_zero_morphisms C] [has_shift C \u2124]\n\nopen_locale zero_object\n\ninstance has_zero_object : has_zero_object (differential_object C) :=\nby { refine \u27e8\u27e8\u27e80, 0\u27e9, \u03bb X, \u27e8\u27e8\u27e8\u27e80\u27e9\u27e9, \u03bb f, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e8\u27e80\u27e9\u27e9, \u03bb f, _\u27e9\u27e9\u27e9\u27e9; ext, }\n\nend differential_object\n\nnamespace differential_object\n\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_zero_morphisms C] [has_shift C \u2124]\n\ninstance concrete_category_of_differential_objects :\n  concrete_category (differential_object C) :=\n{ forget := forget C \u22d9 category_theory.forget C }\n\ninstance : has_forget\u2082 (differential_object C) C :=\n{ forget\u2082 := forget C }\n\nend differential_object\n\n/-! The category of differential objects itself has a shift functor. -/\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\nnoncomputable theory\n\n/-- The shift functor on `differential_object C`. -/\n@[simps]\ndef shift_functor (n : \u2124) : differential_object C \u2964 differential_object C :=\n{ obj := \u03bb X,\n  { X := X.X\u27e6n\u27e7,\n    d := X.d\u27e6n\u27e7' \u226b (shift_comm _ _ _).hom,\n    d_squared' := by rw [functor.map_comp, category.assoc, shift_comm_hom_comp_assoc,\n        \u2190functor.map_comp_assoc, X.d_squared, functor.map_zero, zero_comp] },\n  map := \u03bb X Y f,\n  { f := f.f\u27e6n\u27e7',\n    comm' := begin\n      dsimp,\n      erw [category.assoc, shift_comm_hom_comp, \u2190 functor.map_comp_assoc, f.comm,\n        functor.map_comp_assoc],\n      refl,\n    end, },\n  map_id' := by { intros X, ext1, dsimp, rw functor.map_id },\n  map_comp' := by { intros X Y Z f g, ext1, dsimp, rw functor.map_comp } }\n\n/-- The shift functor on `differential_object C` is additive. -/\n@[simps] def shift_functor_add (m n : \u2124) :\n  shift_functor C (m + n) \u2245 shift_functor C m \u22d9 shift_functor C n :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso (shift_add X.X _ _) _) _,\n  { dsimp,\n    rw [\u2190 cancel_epi ((shift_functor_add C m n).inv.app X.X)],\n    simp only [category.assoc, iso.inv_hom_id_app_assoc],\n    erw [\u2190 nat_trans.naturality_assoc],\n    dsimp,\n    simp only [functor.map_comp, category.assoc,\n      shift_functor_comm_hom_app_comp_shift_shift_functor_add_hom_app 1 m n X.X,\n      iso.inv_hom_id_app_assoc], },\n  { intros X Y f, ext, dsimp, exact nat_trans.naturality _ _ }\nend\n\nsection\n\n/-- The shift by zero is naturally isomorphic to the identity. -/\n@[simps]\ndef shift_zero : shift_functor C 0 \u2245 \ud835\udfed (differential_object C) :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso ((shift_functor_zero C \u2124).app X.X) _) _,\n  { erw [\u2190 nat_trans.naturality],\n    dsimp,\n    simp only [shift_functor_zero_hom_app_shift, category.assoc], },\n  { tidy, },\nend\n\nend\n\ninstance : has_shift (differential_object C) \u2124 :=\nhas_shift_mk _ _\n{ F := shift_functor C,\n  zero := shift_zero C,\n  add := shift_functor_add C,\n  assoc_hom_app := \u03bb m\u2081 m\u2082 m\u2083 X, begin\n    ext1,\n    convert shift_functor_add_assoc_hom_app m\u2081 m\u2082 m\u2083 X.X,\n    dsimp [shift_functor_add'],\n    simpa,\n  end,\n  zero_add_hom_app := \u03bb n X, begin\n    ext1,\n    convert shift_functor_add_zero_add_hom_app n X.X,\n    simpa,\n  end,\n  add_zero_hom_app := \u03bb n X, begin\n    ext1,\n    convert shift_functor_add_add_zero_hom_app n X.X,\n    simpa,\n  end, }\n\nend differential_object\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/differential_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3786345880682296}}
{"text": "/-\n  Copyright (c) 2022 Arthur Paulino. All rights reserved.\n  Released under Apache 2.0 license as described in the file LICENSE.\n  Authors: Arthur Paulino\n-/\n\nimport DataEntries\nimport Utils\n\ninductive DataType\n  | TInt\n  | TFloat\n  | TString\n  deriving Inhabited\n\n/- Prouces a `DataEntry` given its `DataType` and a `String` -/\ndef DataType.entryOfString! (dataType : DataType) (s : String) : DataEntry :=\n  if s = \"NULL\" then NIL\n  else match dataType with\n  | DataType.TInt    => s.toInt!\n  | DataType.TFloat  => toFloat! s\n  | DataType.TString => s\n\nopen DataType in\n/- Whether a `DataEntry` is of a `DataType` or not -/\n@[simp] def DataEntry.ofType : DataEntry \u2192 DataType \u2192 Bool\n  | EInt _,    TInt    => true\n  | EFloat _,  TFloat  => true\n  | EString _, TString => true\n  | ENull,     _       => true\n  | _,         _       => false\n\ninstance : ToString DataEntry where\n  toString e := e.toString\n\nabbrev Header := List (DataType \u00d7 String)\n\n/- Returns the column types of a `Header` -/\ndef Header.colTypes (h : Header) : List DataType :=\n  h.map fun x => x.1\n\n/- Returns the column names of a `Header` -/\ndef Header.colNames (h : Header) : List String :=\n  h.map fun x => x.2\n\nabbrev DataEntries := List DataEntry\n\n/- Given a list of `DataEntry` and a list of `DataType`, tells whether\n  every `DataEntry` is of `DataType` in a \"zip\" logic -/\n@[simp] def DataEntries.ofTypes : DataEntries \u2192 List DataType \u2192 Bool\n  | e :: es, t :: ts => e.ofType t && ofTypes es ts\n  | [],      []      => true\n  | _,       _       => false\n\n/-- Given a list of `DataType`, turns a list of `String` into a list of\n  `DataEntry` according to the respective type from the list -/\ndef entriesOfStrings! : List DataType \u2192 List String \u2192 DataEntries\n  | t :: ts, s :: ss => t.entryOfString! s :: (entriesOfStrings! ts ss)\n  | _,       _       => []\n\n/- Whether every list of `DataEntry` obeys to `DataEntries.ofTypes` -/\n@[simp] def rowsOfTypes : List DataEntries \u2192 List DataType \u2192 Prop\n  | row :: rows, types => row.ofTypes types \u2227 rowsOfTypes rows types\n  | [],          _     => True\n\n/- Turns a list of `DataEntry` into a list of their respective `String`\n  representation-/\ndef DataEntries.toStrings (r : DataEntries) : List String :=\n  r.map DataEntry.toString\n\n/- A DataFrame consists of:\n  * A header, containing the column names and their types\n  * The rows, containing the actual data\n  * A consistenty rule, guaranteeing that every row obeys to the scheme -/\nstructure DataFrame where\n  header     : Header \n  rows       : List DataEntries\n  consistent : rowsOfTypes rows header.colTypes := by simp\n\nnamespace DataFrame\n\n/- The column types of a `DataFrame` -/\ndef colTypes (df : DataFrame) : List DataType :=\n  df.header.colTypes\n\n/- The column names of a `DataFrame` -/\ndef colNames (df : DataFrame) : List String :=\n  df.header.colNames\n\n/- Returns an empty `DataFrame` -/\ndef empty (header : Header := []) : DataFrame :=\n  \u27e8header, [], by simp\u27e9\n\n/- Given a `DataFrame` `df` and a new `Row` `r` that's consistent with its\n  scheme, the concatenation of the `df.rows` and `r` is also consistent\n  with the scheme of `df` -/\ntheorem consistentConcatOfConsistentRow\n    {df : DataFrame} (row : DataEntries)\n    (hc : row.ofTypes df.colTypes) :\n      rowsOfTypes (df.rows.concat row) (Header.colTypes df.header) :=\n  match df with\n    | \u27e8_, rows, hr\u27e9 => by\n      induction rows with\n        | nil         => simp only [colTypes] at hc; simp [hc]\n        | cons _ _ hi => exact \u27e8hr.1, hi hr.2 hc\u27e9\n\n/- Adds a new row on a `DataFrame` -/\ndef addRow (df : DataFrame) (row : DataEntries)\n    (h : row.ofTypes df.colTypes := by simp) : DataFrame :=\n  \u27e8df.header, df.rows.concat row, consistentConcatOfConsistentRow row h\u27e9\n\n/- The number of rows in a `DataFrame` -/\ndef nRows (df : DataFrame) : Nat :=\n  df.rows.length\n\n/- The number of columns in a `DataFrame` -/\ndef nCols (df : DataFrame) : Nat :=\n  df.header.length\n\n/- The shape of a `DataFrame` (# of rows \u00d7 # of columns) -/\ndef shape (df : DataFrame) : Nat \u00d7 Nat :=\n  (df.nRows, df.nCols)\n\n/- The i-th row of a `DataFrame` -/\ndef row! (df : DataFrame) (i : Nat) : DataEntries :=\n  if i >= df.rows.length then\n    panic! s!\"invalid index {i}\"\n  else\n    (df.rows.get! i)\n\n/- The i-th's rows of a `DataFrame` -/\ndef rows! (df : DataFrame) (li : List Nat) : List DataEntries := Id.run do\n  let mut invalidIndexes : List Nat := []\n  for i in li do\n    if i >= df.rows.length then\n      invalidIndexes := invalidIndexes.concat i\n  if \u00acinvalidIndexes.isEmpty then\n    panic! s!\"invalid indexes {invalidIndexes}\"\n  else\n    li.map fun i => df.row! i\n\n/- The j-th column of a `DataFrame` -/\ndef col! (df : DataFrame) (j : Nat) : DataEntries :=\n  if j >= df.header.length then\n    panic! s!\"invalid index {j}\"\n  else\n    df.rows.map fun r => r.get! j\n\n/- The j-th's columns of a `DataFrame` -/\ndef cols! (df : DataFrame) (lj : List Nat) : List DataEntries := Id.run do\n  let mut invalidIndexes : List Nat := []\n  for j in lj do\n    if j >= df.header.length then\n      invalidIndexes := invalidIndexes.concat j\n  if \u00acinvalidIndexes.isEmpty then\n    panic! s!\"invalid indexes {invalidIndexes}\"\n  else\n    lj.map fun j => df.col! j\n\n/- The element at the i-th row and j-th column -/\ndef at! (df : DataFrame) (i j : Nat) : DataEntry :=\n  if i >= df.rows.length then\n    panic! s!\"invalid row index {i}\"\n  else\n    if j >= df.header.length then\n      panic! s!\"invalid column index {j}\"\n    else\n      (df.row! i).get! j\n\n/- The `String` representation of a `DataFrame` -/\ndef toString (df : DataFrame) : String := Id.run do\n  if df.nCols = 0 then \"\"\n  else\n    let mut cells : List (List String) := []\n    let mut colLengths : List Nat := []\n    let mut header : List String := []\n    for colName in df.colNames do\n      colLengths := colLengths.concat colName.length\n      header := header.concat colName\n    cells := cells.concat header\n    for row in df.rows do\n      let mut line : List String := []\n      let rowStrings : List String := row.toStrings\n      for j in [0 : rowStrings.length] do\n        let s := rowStrings.get! j\n        let s_length : Nat := s.length\n        if s_length > (colLengths.get! j) then\n          colLengths := colLengths.set j s_length\n        line := line.concat s\n      cells := cells.concat line\n    let mut res : String := \"\"\n    for i in [0 : cells.length] do\n      let row := cells.get! i\n      for j in [0 : row.length] do\n        let val : String := row.get! j\n        res := res ++ \"|\" ++\n          (leftFillWithUntil val ' ' (colLengths.get! j))\n      res := res ++ \"|\"\n      if cells.length = 1 \u2228 i < cells.length - 1 then\n        res := res ++ \"\\n\"\n      if i = 0 then\n        for j in [0 : row.length] do\n          res := res ++ \"|\" ++\n            leftFillWithUntil \"\" '-' (colLengths.get! j)\n        res := res ++ \"|\\n\"\n    res\n\ninstance : ToString DataFrame where\n  toString df := df.toString\n\nend DataFrame\n", "meta": {"author": "arthurpaulino", "repo": "LeanMySQL", "sha": "803d171225406ca516d5d5f7709a3c9ced97f486", "save_path": "github-repos/lean/arthurpaulino-LeanMySQL", "path": "github-repos/lean/arthurpaulino-LeanMySQL/LeanMySQL-803d171225406ca516d5d5f7709a3c9ced97f486/lib/DataFrame.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.6513548646660543, "lm_q1q2_score": 0.37863458648727455}}
{"text": "/-\nCopyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nMetric spaces.\n\nAuthors: Jeremy Avigad, Robert Y. Lewis, Johannes H\u00f6lzl, Mario Carneiro, S\u00e9bastien Gou\u00ebzel\n\nMany definitions and theorems expected on metric spaces are already introduced on uniform spaces and\ntopological spaces. For example:\n  open and closed sets, compactness, completeness, continuity and uniform continuity\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.metric_space.emetric_space\nimport Mathlib.topology.algebra.ordered\nimport Mathlib.PostPort\n\nuniverses u u_1 l v u_2 \n\nnamespace Mathlib\n\n/-- Construct a uniform structure from a distance function and metric space axioms -/\ndef uniform_space_of_dist {\u03b1 : Type u} (dist : \u03b1 \u2192 \u03b1 \u2192 \u211d) (dist_self : \u2200 (x : \u03b1), dist x x = 0)\n    (dist_comm : \u2200 (x y : \u03b1), dist x y = dist y x)\n    (dist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z) : uniform_space \u03b1 :=\n  uniform_space.of_core\n    (uniform_space.core.mk\n      (infi\n        fun (\u03b5 : \u211d) =>\n          infi\n            fun (H : \u03b5 > 0) =>\n              filter.principal (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5))\n      sorry sorry sorry)\n\n/-- The distance function (given an ambient metric space on `\u03b1`), which returns\n  a nonnegative real number `dist x y` given `x y : \u03b1`. -/\nclass has_dist (\u03b1 : Type u_1) where\n  dist : \u03b1 \u2192 \u03b1 \u2192 \u211d\n\n-- the uniform structure and the emetric space structure are embedded in the metric space structure\n\n-- to avoid instance diamond issues. See Note [forgetful inheritance].\n\n/-- Metric space\n\nEach metric space induces a canonical `uniform_space` and hence a canonical `topological_space`.\nThis is enforced in the type class definition, by extending the `uniform_space` structure. When\ninstantiating a `metric_space` structure, the uniformity fields are not necessary, they will be\nfilled in by default. In the same way, each metric space induces an emetric space structure.\nIt is included in the structure, but filled in by default.\n-/\nclass metric_space (\u03b1 : Type u)\n    extends uniform_space #2,\n      metric_space.to_uniform_space._default #2 #1 #0 \u03b1 _to_has_dist =\n        id (uniform_space_of_dist dist #0 \u03b1 _to_has_dist),\n      uniform_space \u03b1, has_dist \u03b1\n    where\n  dist_self : \u2200 (x : \u03b1), dist x x = 0\n  eq_of_dist_eq_zero : \u2200 {x y : \u03b1}, dist x y = 0 \u2192 x = y\n  dist_comm : \u2200 (x y : \u03b1), dist x y = dist y x\n  dist_triangle : \u2200 (x y z : \u03b1), dist x z \u2264 dist x y + dist y z\n  edist : \u03b1 \u2192 \u03b1 \u2192 ennreal\n  edist_dist :\n    autoParam (\u2200 (x y : \u03b1), edist x y = ennreal.of_real (dist x y))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n  to_uniform_space : uniform_space \u03b1\n  uniformity_dist :\n    autoParam\n      (uniformity \u03b1 =\n        infi\n          fun (\u03b5 : \u211d) =>\n            infi\n              fun (H : \u03b5 > 0) =>\n                filter.principal (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.control_laws_tac\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"control_laws_tac\") [])\n\nprotected instance metric_space.to_uniform_space' {\u03b1 : Type u} [metric_space \u03b1] : uniform_space \u03b1 :=\n  metric_space.to_uniform_space\n\nprotected instance metric_space.to_has_edist {\u03b1 : Type u} [metric_space \u03b1] : has_edist \u03b1 :=\n  has_edist.mk metric_space.edist\n\n@[simp] theorem dist_self {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) : dist x x = 0 :=\n  metric_space.dist_self x\n\ntheorem eq_of_dist_eq_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : dist x y = 0 \u2192 x = y :=\n  metric_space.eq_of_dist_eq_zero\n\ntheorem dist_comm {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : dist x y = dist y x :=\n  metric_space.dist_comm x y\n\ntheorem edist_dist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    edist x y = ennreal.of_real (dist x y) :=\n  metric_space.edist_dist x y\n\n@[simp] theorem dist_eq_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : dist x y = 0 \u2194 x = y :=\n  { mp := eq_of_dist_eq_zero, mpr := fun (this : x = y) => this \u25b8 dist_self x }\n\n@[simp] theorem zero_eq_dist {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : 0 = dist x y \u2194 x = y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (0 = dist x y \u2194 x = y)) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dist x y = 0 \u2194 x = y)) (propext dist_eq_zero)))\n      (iff.refl (x = y)))\n\ntheorem dist_triangle {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    dist x z \u2264 dist x y + dist y z :=\n  metric_space.dist_triangle x y z\n\ntheorem dist_triangle_left {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    dist x y \u2264 dist z x + dist z y :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (dist x y \u2264 dist z x + dist z y)) (dist_comm z x)))\n    (dist_triangle x z y)\n\ntheorem dist_triangle_right {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    dist x y \u2264 dist x z + dist y z :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (dist x y \u2264 dist x z + dist y z)) (dist_comm y z)))\n    (dist_triangle x z y)\n\ntheorem dist_triangle4 {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) (w : \u03b1) :\n    dist x w \u2264 dist x y + dist y z + dist z w :=\n  le_trans (dist_triangle x z w) (add_le_add_right (dist_triangle x y z) (dist z w))\n\ntheorem dist_triangle4_left {\u03b1 : Type u} [metric_space \u03b1] (x\u2081 : \u03b1) (y\u2081 : \u03b1) (x\u2082 : \u03b1) (y\u2082 : \u03b1) :\n    dist x\u2082 y\u2082 \u2264 dist x\u2081 y\u2081 + (dist x\u2081 x\u2082 + dist y\u2081 y\u2082) :=\n  sorry\n\ntheorem dist_triangle4_right {\u03b1 : Type u} [metric_space \u03b1] (x\u2081 : \u03b1) (y\u2081 : \u03b1) (x\u2082 : \u03b1) (y\u2082 : \u03b1) :\n    dist x\u2081 y\u2081 \u2264 dist x\u2081 x\u2082 + dist y\u2081 y\u2082 + dist x\u2082 y\u2082 :=\n  sorry\n\n/-- The triangle (polygon) inequality for sequences of points; `finset.Ico` version. -/\ntheorem dist_le_Ico_sum_dist {\u03b1 : Type u} [metric_space \u03b1] (f : \u2115 \u2192 \u03b1) {m : \u2115} {n : \u2115} (h : m \u2264 n) :\n    dist (f m) (f n) \u2264 finset.sum (finset.Ico m n) fun (i : \u2115) => dist (f i) (f (i + 1)) :=\n  sorry\n\n/-- The triangle (polygon) inequality for sequences of points; `finset.range` version. -/\ntheorem dist_le_range_sum_dist {\u03b1 : Type u} [metric_space \u03b1] (f : \u2115 \u2192 \u03b1) (n : \u2115) :\n    dist (f 0) (f n) \u2264 finset.sum (finset.range n) fun (i : \u2115) => dist (f i) (f (i + 1)) :=\n  finset.Ico.zero_bot n \u25b8 dist_le_Ico_sum_dist f (nat.zero_le n)\n\n/-- A version of `dist_le_Ico_sum_dist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem dist_le_Ico_sum_of_dist_le {\u03b1 : Type u} [metric_space \u03b1] {f : \u2115 \u2192 \u03b1} {m : \u2115} {n : \u2115}\n    (hmn : m \u2264 n) {d : \u2115 \u2192 \u211d} (hd : \u2200 {k : \u2115}, m \u2264 k \u2192 k < n \u2192 dist (f k) (f (k + 1)) \u2264 d k) :\n    dist (f m) (f n) \u2264 finset.sum (finset.Ico m n) fun (i : \u2115) => d i :=\n  sorry\n\n/-- A version of `dist_le_range_sum_dist` with each intermediate distance replaced\nwith an upper estimate. -/\ntheorem dist_le_range_sum_of_dist_le {\u03b1 : Type u} [metric_space \u03b1] {f : \u2115 \u2192 \u03b1} (n : \u2115) {d : \u2115 \u2192 \u211d}\n    (hd : \u2200 {k : \u2115}, k < n \u2192 dist (f k) (f (k + 1)) \u2264 d k) :\n    dist (f 0) (f n) \u2264 finset.sum (finset.range n) fun (i : \u2115) => d i :=\n  finset.Ico.zero_bot n \u25b8 dist_le_Ico_sum_of_dist_le (zero_le n) fun (_x : \u2115) (_x_1 : 0 \u2264 _x) => hd\n\ntheorem swap_dist {\u03b1 : Type u} [metric_space \u03b1] : function.swap dist = dist :=\n  funext fun (x : \u03b1) => funext fun (y : \u03b1) => dist_comm y x\n\ntheorem abs_dist_sub_le {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    abs (dist x z - dist y z) \u2264 dist x y :=\n  iff.mpr abs_sub_le_iff\n    { left := iff.mpr sub_le_iff_le_add (dist_triangle x y z),\n      right := iff.mpr sub_le_iff_le_add (dist_triangle_left y z x) }\n\ntheorem dist_nonneg {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : 0 \u2264 dist x y := sorry\n\n@[simp] theorem dist_le_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : dist x y \u2264 0 \u2194 x = y :=\n  sorry\n\n@[simp] theorem dist_pos {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} : 0 < dist x y \u2194 x \u2260 y :=\n  sorry\n\n@[simp] theorem abs_dist {\u03b1 : Type u} [metric_space \u03b1] {a : \u03b1} {b : \u03b1} :\n    abs (dist a b) = dist a b :=\n  abs_of_nonneg dist_nonneg\n\ntheorem eq_of_forall_dist_le {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1}\n    (h : \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 dist x y \u2264 \u03b5) : x = y :=\n  eq_of_dist_eq_zero (eq_of_le_of_forall_le_of_dense dist_nonneg h)\n\n/-- Distance as a nonnegative real number. -/\ndef nndist {\u03b1 : Type u} [metric_space \u03b1] (a : \u03b1) (b : \u03b1) : nnreal :=\n  { val := dist a b, property := dist_nonneg }\n\n/--Express `nndist` in terms of `edist`-/\ntheorem nndist_edist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    nndist x y = ennreal.to_nnreal (edist x y) :=\n  sorry\n\n/--Express `edist` in terms of `nndist`-/\ntheorem edist_nndist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : edist x y = \u2191(nndist x y) :=\n  sorry\n\n@[simp] theorem ennreal_coe_nndist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    \u2191(nndist x y) = edist x y :=\n  Eq.symm (edist_nndist x y)\n\n@[simp] theorem edist_lt_coe {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {c : nnreal} :\n    edist x y < \u2191c \u2194 nndist x y < c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (edist x y < \u2191c \u2194 nndist x y < c)) (edist_nndist x y)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (\u2191(nndist x y) < \u2191c \u2194 nndist x y < c)) (propext ennreal.coe_lt_coe)))\n      (iff.refl (nndist x y < c)))\n\n@[simp] theorem edist_le_coe {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {c : nnreal} :\n    edist x y \u2264 \u2191c \u2194 nndist x y \u2264 c :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (edist x y \u2264 \u2191c \u2194 nndist x y \u2264 c)) (edist_nndist x y)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (\u2191(nndist x y) \u2264 \u2191c \u2194 nndist x y \u2264 c)) (propext ennreal.coe_le_coe)))\n      (iff.refl (nndist x y \u2264 c)))\n\n/--In a metric space, the extended distance is always finite-/\ntheorem edist_ne_top {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : edist x y \u2260 \u22a4 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (edist x y \u2260 \u22a4)) (edist_dist x y))) ennreal.coe_ne_top\n\n/--In a metric space, the extended distance is always finite-/\ntheorem edist_lt_top {\u03b1 : Type u_1} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : edist x y < \u22a4 :=\n  iff.mpr ennreal.lt_top_iff_ne_top (edist_ne_top x y)\n\n/--`nndist x x` vanishes-/\n@[simp] theorem nndist_self {\u03b1 : Type u} [metric_space \u03b1] (a : \u03b1) : nndist a a = 0 :=\n  iff.mp (nnreal.coe_eq_zero (nndist a a)) (dist_self a)\n\n/--Express `dist` in terms of `nndist`-/\ntheorem dist_nndist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : dist x y = \u2191(nndist x y) := rfl\n\n@[simp] theorem coe_nndist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    \u2191(nndist x y) = dist x y :=\n  Eq.symm (dist_nndist x y)\n\n@[simp] theorem dist_lt_coe {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {c : nnreal} :\n    dist x y < \u2191c \u2194 nndist x y < c :=\n  iff.rfl\n\n@[simp] theorem dist_le_coe {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {c : nnreal} :\n    dist x y \u2264 \u2191c \u2194 nndist x y \u2264 c :=\n  iff.rfl\n\n/--Express `nndist` in terms of `dist`-/\ntheorem nndist_dist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    nndist x y = nnreal.of_real (dist x y) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nndist x y = nnreal.of_real (dist x y))) (dist_nndist x y)))\n    (eq.mpr\n      (id (Eq._oldrec (Eq.refl (nndist x y = nnreal.of_real \u2191(nndist x y))) nnreal.of_real_coe))\n      (Eq.refl (nndist x y)))\n\n/--Deduce the equality of points with the vanishing of the nonnegative distance-/\ntheorem eq_of_nndist_eq_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} :\n    nndist x y = 0 \u2192 x = y :=\n  sorry\n\ntheorem nndist_comm {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) : nndist x y = nndist y x := sorry\n\n/--Characterize the equality of points with the vanishing of the nonnegative distance-/\n@[simp] theorem nndist_eq_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} :\n    nndist x y = 0 \u2194 x = y :=\n  sorry\n\n@[simp] theorem zero_eq_nndist {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} :\n    0 = nndist x y \u2194 x = y :=\n  sorry\n\n/--Triangle inequality for the nonnegative distance-/\ntheorem nndist_triangle {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    nndist x z \u2264 nndist x y + nndist y z :=\n  dist_triangle x y z\n\ntheorem nndist_triangle_left {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    nndist x y \u2264 nndist z x + nndist z y :=\n  dist_triangle_left x y z\n\ntheorem nndist_triangle_right {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) (z : \u03b1) :\n    nndist x y \u2264 nndist x z + nndist y z :=\n  dist_triangle_right x y z\n\n/--Express `dist` in terms of `edist`-/\ntheorem dist_edist {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (y : \u03b1) :\n    dist x y = ennreal.to_real (edist x y) :=\n  sorry\n\nnamespace metric\n\n\n/- instantiate metric space as a topology -/\n\n/-- `ball x \u03b5` is the set of all points `y` with `dist y x < \u03b5` -/\ndef ball {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (\u03b5 : \u211d) : set \u03b1 := set_of fun (y : \u03b1) => dist y x < \u03b5\n\n@[simp] theorem mem_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    y \u2208 ball x \u03b5 \u2194 dist y x < \u03b5 :=\n  iff.rfl\n\ntheorem mem_ball' {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    y \u2208 ball x \u03b5 \u2194 dist x y < \u03b5 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (y \u2208 ball x \u03b5 \u2194 dist x y < \u03b5)) (dist_comm x y)))\n    (iff.refl (y \u2208 ball x \u03b5))\n\n@[simp] theorem nonempty_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (h : 0 < \u03b5) :\n    set.nonempty (ball x \u03b5) :=\n  sorry\n\ntheorem ball_eq_ball {\u03b1 : Type u} [metric_space \u03b1] (\u03b5 : \u211d) (x : \u03b1) :\n    uniform_space.ball x (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.snd p) (prod.fst p) < \u03b5) =\n        ball x \u03b5 :=\n  rfl\n\ntheorem ball_eq_ball' {\u03b1 : Type u} [metric_space \u03b1] (\u03b5 : \u211d) (x : \u03b1) :\n    uniform_space.ball x (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5) =\n        ball x \u03b5 :=\n  sorry\n\n/-- `closed_ball x \u03b5` is the set of all points `y` with `dist y x \u2264 \u03b5` -/\ndef closed_ball {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (\u03b5 : \u211d) : set \u03b1 :=\n  set_of fun (y : \u03b1) => dist y x \u2264 \u03b5\n\n@[simp] theorem mem_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    y \u2208 closed_ball x \u03b5 \u2194 dist y x \u2264 \u03b5 :=\n  iff.rfl\n\n/-- `sphere x \u03b5` is the set of all points `y` with `dist y x = \u03b5` -/\ndef sphere {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) (\u03b5 : \u211d) : set \u03b1 :=\n  set_of fun (y : \u03b1) => dist y x = \u03b5\n\n@[simp] theorem mem_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    y \u2208 sphere x \u03b5 \u2194 dist y x = \u03b5 :=\n  iff.rfl\n\ntheorem mem_closed_ball' {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    y \u2208 closed_ball x \u03b5 \u2194 dist x y \u2264 \u03b5 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (y \u2208 closed_ball x \u03b5 \u2194 dist x y \u2264 \u03b5)) (dist_comm x y)))\n    (iff.refl (y \u2208 closed_ball x \u03b5))\n\ntheorem nonempty_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (h : 0 \u2264 \u03b5) :\n    set.nonempty (closed_ball x \u03b5) :=\n  sorry\n\ntheorem ball_subset_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    ball x \u03b5 \u2286 closed_ball x \u03b5 :=\n  fun (y : \u03b1) (hy : dist y x < \u03b5) => le_of_lt hy\n\ntheorem sphere_subset_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    sphere x \u03b5 \u2286 closed_ball x \u03b5 :=\n  fun (y : \u03b1) => le_of_eq\n\ntheorem sphere_disjoint_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    disjoint (sphere x \u03b5) (ball x \u03b5) :=\n  sorry\n\n@[simp] theorem ball_union_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    ball x \u03b5 \u222a sphere x \u03b5 = closed_ball x \u03b5 :=\n  set.ext fun (y : \u03b1) => iff.symm le_iff_lt_or_eq\n\n@[simp] theorem sphere_union_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    sphere x \u03b5 \u222a ball x \u03b5 = closed_ball x \u03b5 :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (sphere x \u03b5 \u222a ball x \u03b5 = closed_ball x \u03b5))\n        (set.union_comm (sphere x \u03b5) (ball x \u03b5))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (ball x \u03b5 \u222a sphere x \u03b5 = closed_ball x \u03b5)) ball_union_sphere))\n      (Eq.refl (closed_ball x \u03b5)))\n\n@[simp] theorem closed_ball_diff_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    closed_ball x \u03b5 \\ sphere x \u03b5 = ball x \u03b5 :=\n  sorry\n\n@[simp] theorem closed_ball_diff_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    closed_ball x \u03b5 \\ ball x \u03b5 = sphere x \u03b5 :=\n  sorry\n\ntheorem pos_of_mem_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} (hy : y \u2208 ball x \u03b5) :\n    0 < \u03b5 :=\n  lt_of_le_of_lt dist_nonneg hy\n\ntheorem mem_ball_self {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (h : 0 < \u03b5) : x \u2208 ball x \u03b5 :=\n  (fun (this : dist x x < \u03b5) => this)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dist x x < \u03b5)) (dist_self x))) h)\n\ntheorem mem_closed_ball_self {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (h : 0 \u2264 \u03b5) :\n    x \u2208 closed_ball x \u03b5 :=\n  (fun (this : dist x x \u2264 \u03b5) => this)\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dist x x \u2264 \u03b5)) (dist_self x))) h)\n\ntheorem mem_ball_comm {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d} :\n    x \u2208 ball y \u03b5 \u2194 y \u2208 ball x \u03b5 :=\n  sorry\n\ntheorem ball_subset_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5\u2081 : \u211d} {\u03b5\u2082 : \u211d} (h : \u03b5\u2081 \u2264 \u03b5\u2082) :\n    ball x \u03b5\u2081 \u2286 ball x \u03b5\u2082 :=\n  fun (y : \u03b1) (yx : dist y x < \u03b5\u2081) => lt_of_lt_of_le yx h\n\ntheorem closed_ball_subset_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5\u2081 : \u211d} {\u03b5\u2082 : \u211d}\n    (h : \u03b5\u2081 \u2264 \u03b5\u2082) : closed_ball x \u03b5\u2081 \u2286 closed_ball x \u03b5\u2082 :=\n  fun (y : \u03b1) (yx : dist y x \u2264 \u03b5\u2081) => le_trans yx h\n\ntheorem ball_disjoint {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5\u2081 : \u211d} {\u03b5\u2082 : \u211d}\n    (h : \u03b5\u2081 + \u03b5\u2082 \u2264 dist x y) : ball x \u03b5\u2081 \u2229 ball y \u03b5\u2082 = \u2205 :=\n  sorry\n\ntheorem ball_disjoint_same {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d}\n    (h : \u03b5 \u2264 dist x y / bit0 1) : ball x \u03b5 \u2229 ball y \u03b5 = \u2205 :=\n  ball_disjoint\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u03b5 + \u03b5 \u2264 dist x y)) (Eq.symm (two_mul \u03b5))))\n      (eq.mpr\n        (id\n          (Eq._oldrec (Eq.refl (bit0 1 * \u03b5 \u2264 dist x y))\n            (Eq.symm (propext (le_div_iff' zero_lt_two)))))\n        h))\n\ntheorem ball_subset {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5\u2081 : \u211d} {\u03b5\u2082 : \u211d}\n    (h : dist x y \u2264 \u03b5\u2082 - \u03b5\u2081) : ball x \u03b5\u2081 \u2286 ball y \u03b5\u2082 :=\n  fun (z : \u03b1) (zx : z \u2208 ball x \u03b5\u2081) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (z \u2208 ball y \u03b5\u2082)) (Eq.symm (add_sub_cancel'_right \u03b5\u2081 \u03b5\u2082))))\n      (lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h))\n\ntheorem ball_half_subset {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (y : \u03b1)\n    (h : y \u2208 ball x (\u03b5 / bit0 1)) : ball y (\u03b5 / bit0 1) \u2286 ball x \u03b5 :=\n  ball_subset\n    (eq.mpr (id (Eq._oldrec (Eq.refl (dist y x \u2264 \u03b5 - \u03b5 / bit0 1)) (sub_self_div_two \u03b5)))\n      (le_of_lt h))\n\ntheorem exists_ball_subset_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {\u03b5 : \u211d}\n    (h : y \u2208 ball x \u03b5) : \u2203 (\u03b5' : \u211d), \u2203 (H : \u03b5' > 0), ball y \u03b5' \u2286 ball x \u03b5 :=\n  sorry\n\n@[simp] theorem ball_eq_empty_iff_nonpos {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    ball x \u03b5 = \u2205 \u2194 \u03b5 \u2264 0 :=\n  iff.trans set.eq_empty_iff_forall_not_mem\n    { mp :=\n        fun (h : \u2200 (x_1 : \u03b1), \u00acx_1 \u2208 ball x \u03b5) =>\n          le_of_not_gt fun (\u03b50 : \u03b5 > 0) => h x (mem_ball_self \u03b50),\n      mpr := fun (\u03b50 : \u03b5 \u2264 0) (y : \u03b1) (h : y \u2208 ball x \u03b5) => not_lt_of_le \u03b50 (pos_of_mem_ball h) }\n\n@[simp] theorem closed_ball_eq_empty_iff_neg {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    closed_ball x \u03b5 = \u2205 \u2194 \u03b5 < 0 :=\n  sorry\n\n@[simp] theorem ball_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} : ball x 0 = \u2205 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (ball x 0 = \u2205)) (propext ball_eq_empty_iff_nonpos))) (le_refl 0)\n\n@[simp] theorem closed_ball_zero {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} :\n    closed_ball x 0 = singleton x :=\n  set.ext fun (y : \u03b1) => dist_le_zero\n\ntheorem uniformity_basis_dist {\u03b1 : Type u} [metric_space \u03b1] :\n    filter.has_basis (uniformity \u03b1) (fun (\u03b5 : \u211d) => 0 < \u03b5)\n        fun (\u03b5 : \u211d) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5 :=\n  sorry\n\n/-- Given `f : \u03b2 \u2192 \u211d`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `\ud835\udce4 \u03b1`.\n\nFor specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,\nand `uniformity_basis_dist_inv_nat_pos`. -/\nprotected theorem mk_uniformity_basis {\u03b1 : Type u} [metric_space \u03b1] {\u03b2 : Type u_1} {p : \u03b2 \u2192 Prop}\n    {f : \u03b2 \u2192 \u211d} (hf\u2080 : \u2200 (i : \u03b2), p i \u2192 0 < f i)\n    (hf : \u2200 {\u03b5 : \u211d}, 0 < \u03b5 \u2192 \u2203 (i : \u03b2), \u2203 (hi : p i), f i \u2264 \u03b5) :\n    filter.has_basis (uniformity \u03b1) p\n        fun (i : \u03b2) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < f i :=\n  sorry\n\ntheorem uniformity_basis_dist_inv_nat_succ {\u03b1 : Type u} [metric_space \u03b1] :\n    filter.has_basis (uniformity \u03b1) (fun (_x : \u2115) => True)\n        fun (n : \u2115) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < 1 / (\u2191n + 1) :=\n  metric.mk_uniformity_basis\n    (fun (n : \u2115) (_x : True) => div_pos zero_lt_one (nat.cast_add_one_pos n))\n    fun (\u03b5 : \u211d) (\u03b50 : 0 < \u03b5) =>\n      Exists.imp (fun (n : \u2115) (hn : 1 / (\u2191n + 1) < \u03b5) => Exists.intro trivial (le_of_lt hn))\n        (exists_nat_one_div_lt \u03b50)\n\ntheorem uniformity_basis_dist_inv_nat_pos {\u03b1 : Type u} [metric_space \u03b1] :\n    filter.has_basis (uniformity \u03b1) (fun (n : \u2115) => 0 < n)\n        fun (n : \u2115) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < 1 / \u2191n :=\n  sorry\n\n/-- Given `f : \u03b2 \u2192 \u211d`, if `f` sends `{i | p i}` to a set of positive numbers\naccumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`\nform a basis of `\ud835\udce4 \u03b1`.\n\nCurrently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.\nMore can be easily added if needed in the future. -/\nprotected theorem mk_uniformity_basis_le {\u03b1 : Type u} [metric_space \u03b1] {\u03b2 : Type u_1} {p : \u03b2 \u2192 Prop}\n    {f : \u03b2 \u2192 \u211d} (hf\u2080 : \u2200 (x : \u03b2), p x \u2192 0 < f x)\n    (hf : \u2200 (\u03b5 : \u211d), 0 < \u03b5 \u2192 \u2203 (x : \u03b2), \u2203 (hx : p x), f x \u2264 \u03b5) :\n    filter.has_basis (uniformity \u03b1) p\n        fun (x : \u03b2) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) \u2264 f x :=\n  sorry\n\n/-- Contant size closed neighborhoods of the diagonal form a basis\nof the uniformity filter. -/\ntheorem uniformity_basis_dist_le {\u03b1 : Type u} [metric_space \u03b1] :\n    filter.has_basis (uniformity \u03b1) (fun (\u03b5 : \u211d) => 0 < \u03b5)\n        fun (\u03b5 : \u211d) => set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) \u2264 \u03b5 :=\n  metric.mk_uniformity_basis_le (fun (_x : \u211d) => id)\n    fun (\u03b5 : \u211d) (\u03b5\u2080 : 0 < \u03b5) => Exists.intro \u03b5 (Exists.intro \u03b5\u2080 (le_refl \u03b5))\n\ntheorem mem_uniformity_dist {\u03b1 : Type u} [metric_space \u03b1] {s : set (\u03b1 \u00d7 \u03b1)} :\n    s \u2208 uniformity \u03b1 \u2194 \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 {a b : \u03b1}, dist a b < \u03b5 \u2192 (a, b) \u2208 s :=\n  filter.has_basis.mem_uniformity_iff uniformity_basis_dist\n\n/-- A constant size neighborhood of the diagonal is an entourage. -/\ntheorem dist_mem_uniformity {\u03b1 : Type u} [metric_space \u03b1] {\u03b5 : \u211d} (\u03b50 : 0 < \u03b5) :\n    (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5) \u2208 uniformity \u03b1 :=\n  iff.mpr mem_uniformity_dist (Exists.intro \u03b5 (Exists.intro \u03b50 fun (a b : \u03b1) => id))\n\ntheorem uniform_continuous_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} :\n    uniform_continuous f \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {a b : \u03b1}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 :=\n  filter.has_basis.uniform_continuous_iff uniformity_basis_dist uniformity_basis_dist\n\ntheorem uniform_continuous_on_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {s : set \u03b1} :\n    uniform_continuous_on f s \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d),\n            \u2203 (H : \u03b4 > 0), \u2200 (x y : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 dist x y < \u03b4 \u2192 dist (f x) (f y) < \u03b5 :=\n  sorry\n\ntheorem uniform_embedding_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} :\n    uniform_embedding f \u2194\n        function.injective f \u2227\n          uniform_continuous f \u2227\n            \u2200 (\u03b4 : \u211d) (H : \u03b4 > 0),\n              \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4 :=\n  sorry\n\n/-- A map between metric spaces is a uniform embedding if and only if the distance between `f x`\nand `f y` is controlled in terms of the distance between `x` and `y` and conversely. -/\ntheorem uniform_embedding_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} :\n    uniform_embedding f \u2194\n        (\u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n            \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {a b : \u03b1}, dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5) \u2227\n          \u2200 (\u03b4 : \u211d) (H : \u03b4 > 0),\n            \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 {a b : \u03b1}, dist (f a) (f b) < \u03b5 \u2192 dist a b < \u03b4 :=\n  sorry\n\ntheorem totally_bounded_iff {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} :\n    totally_bounded s \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (t : set \u03b1),\n            set.finite t \u2227 s \u2286 set.Union fun (y : \u03b1) => set.Union fun (H : y \u2208 t) => ball y \u03b5 :=\n  sorry\n\n/-- A metric space space is totally bounded if one can reconstruct up to any \u03b5>0 any element of the\nspace from finitely many data. -/\ntheorem totally_bounded_of_finite_discretization {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    (H :\n      \u2200 (\u03b5 : \u211d),\n        \u03b5 > 0 \u2192 \u2203 (\u03b2 : Type u), Exists (\u2203 (F : \u21a5s \u2192 \u03b2), \u2200 (x y : \u21a5s), F x = F y \u2192 dist \u2191x \u2191y < \u03b5)) :\n    totally_bounded s :=\n  sorry\n\ntheorem finite_approx_of_totally_bounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    (hs : totally_bounded s) (\u03b5 : \u211d) (H : \u03b5 > 0) :\n    \u2203 (t : set \u03b1),\n        \u2203 (H : t \u2286 s),\n          set.finite t \u2227 s \u2286 set.Union fun (y : \u03b1) => set.Union fun (H : y \u2208 t) => ball y \u03b5 :=\n  eq.mp (Eq._oldrec (Eq.refl (totally_bounded s)) (propext totally_bounded_iff_subset)) hs\n    (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5) (dist_mem_uniformity \u03b5_pos)\n\n/-- Expressing locally uniform convergence on a set using `dist`. -/\ntheorem tendsto_locally_uniformly_on_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {\u03b9 : Type u_1}\n    [topological_space \u03b2] {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : filter \u03b9} {s : set \u03b2} :\n    tendsto_locally_uniformly_on F f p s \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0) (x : \u03b2) (H : x \u2208 s),\n          \u2203 (t : set \u03b2),\n            \u2203 (H : t \u2208 nhds_within x s),\n              filter.eventually (fun (n : \u03b9) => \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5) p :=\n  sorry\n\n/-- Expressing uniform convergence on a set using `dist`. -/\ntheorem tendsto_uniformly_on_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {\u03b9 : Type u_1}\n    {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : filter \u03b9} {s : set \u03b2} :\n    tendsto_uniformly_on F f p s \u2194\n        \u2200 (\u03b5 : \u211d),\n          \u03b5 > 0 \u2192 filter.eventually (fun (n : \u03b9) => \u2200 (x : \u03b2), x \u2208 s \u2192 dist (f x) (F n x) < \u03b5) p :=\n  sorry\n\n/-- Expressing locally uniform convergence using `dist`. -/\ntheorem tendsto_locally_uniformly_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {\u03b9 : Type u_1}\n    [topological_space \u03b2] {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : filter \u03b9} :\n    tendsto_locally_uniformly F f p \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0) (x : \u03b2),\n          \u2203 (t : set \u03b2),\n            \u2203 (H : t \u2208 nhds x),\n              filter.eventually (fun (n : \u03b9) => \u2200 (y : \u03b2), y \u2208 t \u2192 dist (f y) (F n y) < \u03b5) p :=\n  sorry\n\n/-- Expressing uniform convergence using `dist`. -/\ntheorem tendsto_uniformly_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {\u03b9 : Type u_1}\n    {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1} {f : \u03b2 \u2192 \u03b1} {p : filter \u03b9} :\n    tendsto_uniformly F f p \u2194\n        \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 filter.eventually (fun (n : \u03b9) => \u2200 (x : \u03b2), dist (f x) (F n x) < \u03b5) p :=\n  sorry\n\nprotected theorem cauchy_iff {\u03b1 : Type u} [metric_space \u03b1] {f : filter \u03b1} :\n    cauchy f \u2194\n        filter.ne_bot f \u2227\n          \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n            \u2203 (t : set \u03b1), \u2203 (H : t \u2208 f), \u2200 (x y : \u03b1), x \u2208 t \u2192 y \u2208 t \u2192 dist x y < \u03b5 :=\n  filter.has_basis.cauchy_iff uniformity_basis_dist\n\ntheorem nhds_basis_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} :\n    filter.has_basis (nhds x) (fun (\u03b5 : \u211d) => 0 < \u03b5) (ball x) :=\n  nhds_basis_uniformity uniformity_basis_dist\n\ntheorem mem_nhds_iff {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {s : set \u03b1} :\n    s \u2208 nhds x \u2194 \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), ball x \u03b5 \u2286 s :=\n  filter.has_basis.mem_iff nhds_basis_ball\n\ntheorem eventually_nhds_iff {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {p : \u03b1 \u2192 Prop} :\n    filter.eventually (fun (y : \u03b1) => p y) (nhds x) \u2194\n        \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 {y : \u03b1}, dist y x < \u03b5 \u2192 p y :=\n  mem_nhds_iff\n\ntheorem eventually_nhds_iff_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {p : \u03b1 \u2192 Prop} :\n    filter.eventually (fun (y : \u03b1) => p y) (nhds x) \u2194\n        \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 (y : \u03b1), y \u2208 ball x \u03b5 \u2192 p y :=\n  mem_nhds_iff\n\ntheorem nhds_basis_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} :\n    filter.has_basis (nhds x) (fun (\u03b5 : \u211d) => 0 < \u03b5) (closed_ball x) :=\n  nhds_basis_uniformity uniformity_basis_dist_le\n\ntheorem nhds_basis_ball_inv_nat_succ {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} :\n    filter.has_basis (nhds x) (fun (_x : \u2115) => True) fun (n : \u2115) => ball x (1 / (\u2191n + 1)) :=\n  nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ\n\ntheorem nhds_basis_ball_inv_nat_pos {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} :\n    filter.has_basis (nhds x) (fun (n : \u2115) => 0 < n) fun (n : \u2115) => ball x (1 / \u2191n) :=\n  nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos\n\ntheorem is_open_iff {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} :\n    is_open s \u2194 \u2200 (x : \u03b1) (H : x \u2208 s), \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), ball x \u03b5 \u2286 s :=\n  sorry\n\ntheorem is_open_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} : is_open (ball x \u03b5) :=\n  iff.mpr is_open_iff fun (y : \u03b1) => exists_ball_subset_ball\n\ntheorem ball_mem_nhds {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) {\u03b5 : \u211d} (\u03b50 : 0 < \u03b5) :\n    ball x \u03b5 \u2208 nhds x :=\n  mem_nhds_sets is_open_ball (mem_ball_self \u03b50)\n\ntheorem closed_ball_mem_nhds {\u03b1 : Type u} [metric_space \u03b1] (x : \u03b1) {\u03b5 : \u211d} (\u03b50 : 0 < \u03b5) :\n    closed_ball x \u03b5 \u2208 nhds x :=\n  filter.mem_sets_of_superset (ball_mem_nhds x \u03b50) ball_subset_closed_ball\n\ntheorem nhds_within_basis_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {s : set \u03b1} :\n    filter.has_basis (nhds_within x s) (fun (\u03b5 : \u211d) => 0 < \u03b5) fun (\u03b5 : \u211d) => ball x \u03b5 \u2229 s :=\n  nhds_within_has_basis nhds_basis_ball s\n\ntheorem mem_nhds_within_iff {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {s : set \u03b1} {t : set \u03b1} :\n    s \u2208 nhds_within x t \u2194 \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), ball x \u03b5 \u2229 t \u2286 s :=\n  filter.has_basis.mem_iff nhds_within_basis_ball\n\ntheorem tendsto_nhds_within_nhds_within {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {s : set \u03b1}\n    [metric_space \u03b2] {t : set \u03b2} {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : \u03b2} :\n    filter.tendsto f (nhds_within a s) (nhds_within b t) \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 f x \u2208 t \u2227 dist (f x) b < \u03b5 :=\n  sorry\n\ntheorem tendsto_nhds_within_nhds {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {s : set \u03b1}\n    [metric_space \u03b2] {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {b : \u03b2} :\n    filter.tendsto f (nhds_within a s) (nhds b) \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) b < \u03b5 :=\n  sorry\n\ntheorem tendsto_nhds_nhds {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {a : \u03b1} {b : \u03b2} :\n    filter.tendsto f (nhds a) (nhds b) \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {x : \u03b1}, dist x a < \u03b4 \u2192 dist (f x) b < \u03b5 :=\n  filter.has_basis.tendsto_iff nhds_basis_ball nhds_basis_ball\n\ntheorem continuous_at_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {a : \u03b1} :\n    continuous_at f a \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {x : \u03b1}, dist x a < \u03b4 \u2192 dist (f x) (f a) < \u03b5 :=\n  sorry\n\ntheorem continuous_within_at_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2]\n    {f : \u03b1 \u2192 \u03b2} {a : \u03b1} {s : set \u03b1} :\n    continuous_within_at f s a \u2194\n        \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 {x : \u03b1}, x \u2208 s \u2192 dist x a < \u03b4 \u2192 dist (f x) (f a) < \u03b5 :=\n  sorry\n\ntheorem continuous_on_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {s : set \u03b1} :\n    continuous_on f s \u2194\n        \u2200 (b : \u03b1) (H : b \u2208 s) (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 (a : \u03b1), a \u2208 s \u2192 dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 :=\n  sorry\n\ntheorem continuous_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n    continuous f \u2194\n        \u2200 (b : \u03b1) (\u03b5 : \u211d) (H : \u03b5 > 0),\n          \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 (a : \u03b1), dist a b < \u03b4 \u2192 dist (f a) (f b) < \u03b5 :=\n  iff.trans continuous_iff_continuous_at (forall_congr fun (b : \u03b1) => tendsto_nhds_nhds)\n\ntheorem tendsto_nhds {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {f : filter \u03b2} {u : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    filter.tendsto u f (nhds a) \u2194\n        \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 filter.eventually (fun (x : \u03b2) => dist (u x) a < \u03b5) f :=\n  filter.has_basis.tendsto_right_iff nhds_basis_ball\n\ntheorem continuous_at_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {b : \u03b2} :\n    continuous_at f b \u2194\n        \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 filter.eventually (fun (x : \u03b2) => dist (f x) (f b) < \u03b5) (nhds b) :=\n  sorry\n\ntheorem continuous_within_at_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {b : \u03b2} {s : set \u03b2} :\n    continuous_within_at f s b \u2194\n        \u2200 (\u03b5 : \u211d),\n          \u03b5 > 0 \u2192 filter.eventually (fun (x : \u03b2) => dist (f x) (f b) < \u03b5) (nhds_within b s) :=\n  sorry\n\ntheorem continuous_on_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {s : set \u03b2} :\n    continuous_on f s \u2194\n        \u2200 (b : \u03b2),\n          b \u2208 s \u2192\n            \u2200 (\u03b5 : \u211d),\n              \u03b5 > 0 \u2192 filter.eventually (fun (x : \u03b2) => dist (f x) (f b) < \u03b5) (nhds_within b s) :=\n  sorry\n\ntheorem continuous_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} :\n    continuous f \u2194\n        \u2200 (a : \u03b2) (\u03b5 : \u211d),\n          \u03b5 > 0 \u2192 filter.eventually (fun (x : \u03b2) => dist (f x) (f a) < \u03b5) (nhds a) :=\n  iff.trans continuous_iff_continuous_at (forall_congr fun (b : \u03b2) => tendsto_nhds)\n\ntheorem tendsto_at_top {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [Nonempty \u03b2] [semilattice_sup \u03b2]\n    {u : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    filter.tendsto u filter.at_top (nhds a) \u2194\n        \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 (N : \u03b2), \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) a < \u03b5 :=\n  sorry\n\ntheorem is_open_singleton_iff {X : Type u_1} [metric_space X] {x : X} :\n    is_open (singleton x) \u2194 \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), \u2200 (y : X), dist y x < \u03b5 \u2192 y = x :=\n  sorry\n\n/-- Given a point `x` in a discrete subset `s` of a metric space, there is an open ball\ncentered at `x` and intersecting `s` only at `x`. -/\ntheorem exists_ball_inter_eq_singleton_of_mem_discrete {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    [discrete_topology \u21a5s] {x : \u03b1} (hx : x \u2208 s) :\n    \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), ball x \u03b5 \u2229 s = singleton x :=\n  filter.has_basis.exists_inter_eq_singleton_of_mem_discrete nhds_basis_ball hx\n\n/-- Given a point `x` in a discrete subset `s` of a metric space, there is a closed ball\nof positive radius centered at `x` and intersecting `s` only at `x`. -/\ntheorem exists_closed_ball_inter_eq_singleton_of_discrete {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    [discrete_topology \u21a5s] {x : \u03b1} (hx : x \u2208 s) :\n    \u2203 (\u03b5 : \u211d), \u2203 (H : \u03b5 > 0), closed_ball x \u03b5 \u2229 s = singleton x :=\n  filter.has_basis.exists_inter_eq_singleton_of_mem_discrete nhds_basis_closed_ball hx\n\nend metric\n\n\nprotected instance metric_space.to_separated {\u03b1 : Type u} [metric_space \u03b1] : separated_space \u03b1 :=\n  iff.mpr separated_def\n    fun (x y : \u03b1) (h : \u2200 (r : set (\u03b1 \u00d7 \u03b1)), r \u2208 uniformity \u03b1 \u2192 (x, y) \u2208 r) =>\n      eq_of_forall_dist_le\n        fun (\u03b5 : \u211d) (\u03b50 : \u03b5 > 0) =>\n          le_of_lt\n            (h (set_of fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) < \u03b5)\n              (metric.dist_mem_uniformity \u03b50))\n\n/-Instantiate a metric space as an emetric space. Before we can state the instance,\nwe need to show that the uniform structure coming from the edistance and the\ndistance coincide. -/\n\n/-- Expressing the uniformity in terms of `edist` -/\nprotected theorem metric.uniformity_basis_edist {\u03b1 : Type u} [metric_space \u03b1] :\n    filter.has_basis (uniformity \u03b1) (fun (\u03b5 : ennreal) => 0 < \u03b5)\n        fun (\u03b5 : ennreal) => set_of fun (p : \u03b1 \u00d7 \u03b1) => edist (prod.fst p) (prod.snd p) < \u03b5 :=\n  sorry\n\ntheorem metric.uniformity_edist {\u03b1 : Type u} [metric_space \u03b1] :\n    uniformity \u03b1 =\n        infi\n          fun (\u03b5 : ennreal) =>\n            infi\n              fun (H : \u03b5 > 0) =>\n                filter.principal (set_of fun (p : \u03b1 \u00d7 \u03b1) => edist (prod.fst p) (prod.snd p) < \u03b5) :=\n  filter.has_basis.eq_binfi metric.uniformity_basis_edist\n\n/-- A metric space induces an emetric space -/\nprotected instance metric_space.to_emetric_space {\u03b1 : Type u} [metric_space \u03b1] : emetric_space \u03b1 :=\n  emetric_space.mk sorry sorry sorry sorry metric_space.to_uniform_space\n\n/-- Balls defined using the distance or the edistance coincide -/\ntheorem metric.emetric_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    emetric.ball x (ennreal.of_real \u03b5) = metric.ball x \u03b5 :=\n  sorry\n\n/-- Balls defined using the distance or the edistance coincide -/\ntheorem metric.emetric_ball_nnreal {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : nnreal} :\n    emetric.ball x \u2191\u03b5 = metric.ball x \u2191\u03b5 :=\n  sorry\n\n/-- Closed balls defined using the distance or the edistance coincide -/\ntheorem metric.emetric_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} (h : 0 \u2264 \u03b5) :\n    emetric.closed_ball x (ennreal.of_real \u03b5) = metric.closed_ball x \u03b5 :=\n  sorry\n\n/-- Closed balls defined using the distance or the edistance coincide -/\ntheorem metric.emetric_closed_ball_nnreal {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : nnreal} :\n    emetric.closed_ball x \u2191\u03b5 = metric.closed_ball x \u2191\u03b5 :=\n  sorry\n\n/-- Build a new metric space from an old one where the bundled uniform structure is provably\n(but typically non-definitionaly) equal to some given uniform structure.\nSee Note [forgetful inheritance].\n-/\ndef metric_space.replace_uniformity {\u03b1 : Type u_1} [U : uniform_space \u03b1] (m : metric_space \u03b1)\n    (H : uniformity \u03b1 = uniformity \u03b1) : metric_space \u03b1 :=\n  metric_space.mk dist_self eq_of_dist_eq_zero dist_comm dist_triangle edist U\n\n/-- One gets a metric space from an emetric space if the edistance\nis everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the\nuniformity are defeq in the metric space and the emetric space. In this definition, the distance\nis given separately, to be able to prescribe some expression which is not defeq to the push-forward\nof the edistance to reals. -/\ndef emetric_space.to_metric_space_of_dist {\u03b1 : Type u} [e : emetric_space \u03b1] (dist : \u03b1 \u2192 \u03b1 \u2192 \u211d)\n    (edist_ne_top : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4)\n    (h : \u2200 (x y : \u03b1), dist x y = ennreal.to_real (edist x y)) : metric_space \u03b1 :=\n  let m : metric_space \u03b1 :=\n    metric_space.mk sorry sorry sorry sorry (fun (x y : \u03b1) => edist x y)\n      (uniform_space_of_dist dist sorry sorry sorry);\n  metric_space.replace_uniformity m sorry\n\n/-- One gets a metric space from an emetric space if the edistance\nis everywhere finite, by pushing the edistance to reals. We set it up so that the edist and the\nuniformity are defeq in the metric space and the emetric space. -/\ndef emetric_space.to_metric_space {\u03b1 : Type u} [e : emetric_space \u03b1]\n    (h : \u2200 (x y : \u03b1), edist x y \u2260 \u22a4) : metric_space \u03b1 :=\n  emetric_space.to_metric_space_of_dist (fun (x y : \u03b1) => ennreal.to_real (edist x y)) h sorry\n\n/-- A very useful criterion to show that a space is complete is to show that all sequences\nwhich satisfy a bound of the form `dist (u n) (u m) < B N` for all `n m \u2265 N` are\nconverging. This is often applied for `B N = 2^{-N}`, i.e., with a very fast convergence to\n`0`, which makes it possible to use arguments of converging series, while this is impossible\nto do in general for arbitrary Cauchy sequences. -/\ntheorem metric.complete_of_convergent_controlled_sequences {\u03b1 : Type u} [metric_space \u03b1] (B : \u2115 \u2192 \u211d)\n    (hB : \u2200 (n : \u2115), 0 < B n)\n    (H :\n      \u2200 (u : \u2115 \u2192 \u03b1),\n        (\u2200 (N n m : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (u n) (u m) < B N) \u2192\n          \u2203 (x : \u03b1), filter.tendsto u filter.at_top (nhds x)) :\n    complete_space \u03b1 :=\n  sorry\n\ntheorem metric.complete_of_cauchy_seq_tendsto {\u03b1 : Type u} [metric_space \u03b1] :\n    (\u2200 (u : \u2115 \u2192 \u03b1), cauchy_seq u \u2192 \u2203 (a : \u03b1), filter.tendsto u filter.at_top (nhds a)) \u2192\n        complete_space \u03b1 :=\n  emetric.complete_of_cauchy_seq_tendsto\n\n/-- Instantiate the reals as a metric space. -/\nprotected instance real.metric_space : metric_space \u211d :=\n  metric_space.mk sorry sorry sorry sorry\n    (fun (x y : \u211d) => ennreal.of_real ((fun (x y : \u211d) => abs (x - y)) x y))\n    (uniform_space_of_dist (fun (x y : \u211d) => abs (x - y)) sorry sorry sorry)\n\ntheorem real.dist_eq (x : \u211d) (y : \u211d) : dist x y = abs (x - y) := rfl\n\ntheorem real.dist_0_eq_abs (x : \u211d) : dist x 0 = abs x := sorry\n\nprotected instance real.order_topology : order_topology \u211d := sorry\n\ntheorem closed_ball_Icc {x : \u211d} {r : \u211d} : metric.closed_ball x r = set.Icc (x - r) (x + r) := sorry\n\n/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the\ngeneral case. -/\ntheorem squeeze_zero' {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u211d} {g : \u03b1 \u2192 \u211d} {t\u2080 : filter \u03b1}\n    (hf : filter.eventually (fun (t : \u03b1) => 0 \u2264 f t) t\u2080)\n    (hft : filter.eventually (fun (t : \u03b1) => f t \u2264 g t) t\u2080) (g0 : filter.tendsto g t\u2080 (nhds 0)) :\n    filter.tendsto f t\u2080 (nhds 0) :=\n  tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds g0 hf hft\n\n/-- Special case of the sandwich theorem; see `tendsto_of_tendsto_of_tendsto_of_le_of_le`\nand  `tendsto_of_tendsto_of_tendsto_of_le_of_le'` for the general case. -/\ntheorem squeeze_zero {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u211d} {g : \u03b1 \u2192 \u211d} {t\u2080 : filter \u03b1}\n    (hf : \u2200 (t : \u03b1), 0 \u2264 f t) (hft : \u2200 (t : \u03b1), f t \u2264 g t) (g0 : filter.tendsto g t\u2080 (nhds 0)) :\n    filter.tendsto f t\u2080 (nhds 0) :=\n  squeeze_zero' (filter.eventually_of_forall hf) (filter.eventually_of_forall hft) g0\n\ntheorem metric.uniformity_eq_comap_nhds_zero {\u03b1 : Type u} [metric_space \u03b1] :\n    uniformity \u03b1 = filter.comap (fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p)) (nhds 0) :=\n  sorry\n\ntheorem cauchy_seq_iff_tendsto_dist_at_top_0 {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [Nonempty \u03b2]\n    [semilattice_sup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    cauchy_seq u \u2194\n        filter.tendsto (fun (n : \u03b2 \u00d7 \u03b2) => dist (u (prod.fst n)) (u (prod.snd n))) filter.at_top\n          (nhds 0) :=\n  sorry\n\ntheorem tendsto_uniformity_iff_dist_tendsto_zero {\u03b1 : Type u} [metric_space \u03b1] {\u03b9 : Type u_1}\n    {f : \u03b9 \u2192 \u03b1 \u00d7 \u03b1} {p : filter \u03b9} :\n    filter.tendsto f p (uniformity \u03b1) \u2194\n        filter.tendsto (fun (x : \u03b9) => dist (prod.fst (f x)) (prod.snd (f x))) p (nhds 0) :=\n  sorry\n\ntheorem filter.tendsto.congr_dist {\u03b1 : Type u} [metric_space \u03b1] {\u03b9 : Type u_1} {f\u2081 : \u03b9 \u2192 \u03b1}\n    {f\u2082 : \u03b9 \u2192 \u03b1} {p : filter \u03b9} {a : \u03b1} (h\u2081 : filter.tendsto f\u2081 p (nhds a))\n    (h : filter.tendsto (fun (x : \u03b9) => dist (f\u2081 x) (f\u2082 x)) p (nhds 0)) :\n    filter.tendsto f\u2082 p (nhds a) :=\n  filter.tendsto.congr_uniformity h\u2081 (iff.mpr tendsto_uniformity_iff_dist_tendsto_zero h)\n\ntheorem tendsto_of_tendsto_of_dist {\u03b1 : Type u} [metric_space \u03b1] {\u03b9 : Type u_1} {f\u2081 : \u03b9 \u2192 \u03b1}\n    {f\u2082 : \u03b9 \u2192 \u03b1} {p : filter \u03b9} {a : \u03b1} (h\u2081 : filter.tendsto f\u2081 p (nhds a))\n    (h : filter.tendsto (fun (x : \u03b9) => dist (f\u2081 x) (f\u2082 x)) p (nhds 0)) :\n    filter.tendsto f\u2082 p (nhds a) :=\n  filter.tendsto.congr_dist\n\ntheorem tendsto_iff_of_dist {\u03b1 : Type u} [metric_space \u03b1] {\u03b9 : Type u_1} {f\u2081 : \u03b9 \u2192 \u03b1} {f\u2082 : \u03b9 \u2192 \u03b1}\n    {p : filter \u03b9} {a : \u03b1} (h : filter.tendsto (fun (x : \u03b9) => dist (f\u2081 x) (f\u2082 x)) p (nhds 0)) :\n    filter.tendsto f\u2081 p (nhds a) \u2194 filter.tendsto f\u2082 p (nhds a) :=\n  uniform.tendsto_congr (iff.mpr tendsto_uniformity_iff_dist_tendsto_zero h)\n\n/-- In a metric space, Cauchy sequences are characterized by the fact that, eventually,\nthe distance between its elements is arbitrarily small -/\ntheorem metric.cauchy_seq_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [Nonempty \u03b2]\n    [semilattice_sup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    cauchy_seq u \u2194\n        \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 (N : \u03b2), \u2200 (m n : \u03b2), m \u2265 N \u2192 n \u2265 N \u2192 dist (u m) (u n) < \u03b5 :=\n  filter.has_basis.cauchy_seq_iff metric.uniformity_basis_dist\n\n/-- A variation around the metric characterization of Cauchy sequences -/\ntheorem metric.cauchy_seq_iff' {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [Nonempty \u03b2]\n    [semilattice_sup \u03b2] {u : \u03b2 \u2192 \u03b1} :\n    cauchy_seq u \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 (N : \u03b2), \u2200 (n : \u03b2), n \u2265 N \u2192 dist (u n) (u N) < \u03b5 :=\n  filter.has_basis.cauchy_seq_iff' metric.uniformity_basis_dist\n\n/-- If the distance between `s n` and `s m`, `n, m \u2265 N` is bounded above by `b N`\nand `b` converges to zero, then `s` is a Cauchy sequence.  -/\ntheorem cauchy_seq_of_le_tendsto_0 {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [Nonempty \u03b2]\n    [semilattice_sup \u03b2] {s : \u03b2 \u2192 \u03b1} (b : \u03b2 \u2192 \u211d)\n    (h : \u2200 (n m N : \u03b2), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N)\n    (h\u2080 : filter.tendsto b filter.at_top (nhds 0)) : cauchy_seq s :=\n  sorry\n\n/-- A Cauchy sequence on the natural numbers is bounded. -/\ntheorem cauchy_seq_bdd {\u03b1 : Type u} [metric_space \u03b1] {u : \u2115 \u2192 \u03b1} (hu : cauchy_seq u) :\n    \u2203 (R : \u211d), \u2203 (H : R > 0), \u2200 (m n : \u2115), dist (u m) (u n) < R :=\n  sorry\n\n/-- Yet another metric characterization of Cauchy sequences on integers. This one is often the\nmost efficient. -/\ntheorem cauchy_seq_iff_le_tendsto_0 {\u03b1 : Type u} [metric_space \u03b1] {s : \u2115 \u2192 \u03b1} :\n    cauchy_seq s \u2194\n        \u2203 (b : \u2115 \u2192 \u211d),\n          (\u2200 (n : \u2115), 0 \u2264 b n) \u2227\n            (\u2200 (n m N : \u2115), N \u2264 n \u2192 N \u2264 m \u2192 dist (s n) (s m) \u2264 b N) \u2227\n              filter.tendsto b filter.at_top (nhds 0) :=\n  sorry\n\n/-- Metric space structure pulled back by an injective function. Injectivity is necessary to\nensure that `dist x y = 0` only if `x = y`. -/\ndef metric_space.induced {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (hf : function.injective f)\n    (m : metric_space \u03b2) : metric_space \u03b1 :=\n  metric_space.mk sorry sorry sorry sorry (fun (x y : \u03b1) => edist (f x) (f y))\n    (uniform_space.comap f metric_space.to_uniform_space)\n\nprotected instance subtype.metric_space {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [t : metric_space \u03b1] :\n    metric_space (Subtype p) :=\n  metric_space.induced coe sorry t\n\ntheorem subtype.dist_eq {\u03b1 : Type u} [metric_space \u03b1] {p : \u03b1 \u2192 Prop} (x : Subtype p)\n    (y : Subtype p) : dist x y = dist \u2191x \u2191y :=\n  rfl\n\nprotected instance nnreal.metric_space : metric_space nnreal := eq.mpr sorry subtype.metric_space\n\ntheorem nnreal.dist_eq (a : nnreal) (b : nnreal) : dist a b = abs (\u2191a - \u2191b) := rfl\n\ntheorem nnreal.nndist_eq (a : nnreal) (b : nnreal) : nndist a b = max (a - b) (b - a) := sorry\n\nprotected instance prod.metric_space_max {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1]\n    [metric_space \u03b2] : metric_space (\u03b1 \u00d7 \u03b2) :=\n  metric_space.mk sorry sorry sorry sorry\n    (fun (x y : \u03b1 \u00d7 \u03b2) => max (edist (prod.fst x) (prod.fst y)) (edist (prod.snd x) (prod.snd y)))\n    prod.uniform_space\n\ntheorem prod.dist_eq {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] {x : \u03b1 \u00d7 \u03b2}\n    {y : \u03b1 \u00d7 \u03b2} :\n    dist x y = max (dist (prod.fst x) (prod.fst y)) (dist (prod.snd x) (prod.snd y)) :=\n  rfl\n\ntheorem ball_prod_same {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] (x : \u03b1) (y : \u03b2)\n    (r : \u211d) : set.prod (metric.ball x r) (metric.ball y r) = metric.ball (x, y) r :=\n  sorry\n\ntheorem closed_ball_prod_same {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [metric_space \u03b2] (x : \u03b1)\n    (y : \u03b2) (r : \u211d) :\n    set.prod (metric.closed_ball x r) (metric.closed_ball y r) = metric.closed_ball (x, y) r :=\n  sorry\n\ntheorem uniform_continuous_dist {\u03b1 : Type u} [metric_space \u03b1] :\n    uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) :=\n  sorry\n\ntheorem uniform_continuous.dist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [uniform_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1} (hf : uniform_continuous f) (hg : uniform_continuous g) :\n    uniform_continuous fun (b : \u03b2) => dist (f b) (g b) :=\n  uniform_continuous.comp uniform_continuous_dist (uniform_continuous.prod_mk hf hg)\n\ntheorem continuous_dist {\u03b1 : Type u} [metric_space \u03b1] :\n    continuous fun (p : \u03b1 \u00d7 \u03b1) => dist (prod.fst p) (prod.snd p) :=\n  uniform_continuous.continuous uniform_continuous_dist\n\ntheorem continuous.dist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2] {f : \u03b2 \u2192 \u03b1}\n    {g : \u03b2 \u2192 \u03b1} (hf : continuous f) (hg : continuous g) :\n    continuous fun (b : \u03b2) => dist (f b) (g b) :=\n  continuous.comp continuous_dist (continuous.prod_mk hf hg)\n\ntheorem filter.tendsto.dist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1}\n    {x : filter \u03b2} {a : \u03b1} {b : \u03b1} (hf : filter.tendsto f x (nhds a))\n    (hg : filter.tendsto g x (nhds b)) :\n    filter.tendsto (fun (x : \u03b2) => dist (f x) (g x)) x (nhds (dist a b)) :=\n  filter.tendsto.comp (continuous.tendsto continuous_dist (a, b))\n    (filter.tendsto.prod_mk_nhds hf hg)\n\ntheorem nhds_comap_dist {\u03b1 : Type u} [metric_space \u03b1] (a : \u03b1) :\n    filter.comap (fun (a' : \u03b1) => dist a' a) (nhds 0) = nhds a :=\n  sorry\n\ntheorem tendsto_iff_dist_tendsto_zero {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {f : \u03b2 \u2192 \u03b1}\n    {x : filter \u03b2} {a : \u03b1} :\n    filter.tendsto f x (nhds a) \u2194 filter.tendsto (fun (b : \u03b2) => dist (f b) a) x (nhds 0) :=\n  sorry\n\ntheorem uniform_continuous_nndist {\u03b1 : Type u} [metric_space \u03b1] :\n    uniform_continuous fun (p : \u03b1 \u00d7 \u03b1) => nndist (prod.fst p) (prod.snd p) :=\n  uniform_continuous_subtype_mk uniform_continuous_dist fun (p : \u03b1 \u00d7 \u03b1) => dist_nonneg\n\ntheorem uniform_continuous.nndist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [uniform_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1} (hf : uniform_continuous f) (hg : uniform_continuous g) :\n    uniform_continuous fun (b : \u03b2) => nndist (f b) (g b) :=\n  uniform_continuous.comp uniform_continuous_nndist (uniform_continuous.prod_mk hf hg)\n\ntheorem continuous_nndist {\u03b1 : Type u} [metric_space \u03b1] :\n    continuous fun (p : \u03b1 \u00d7 \u03b1) => nndist (prod.fst p) (prod.snd p) :=\n  uniform_continuous.continuous uniform_continuous_nndist\n\ntheorem continuous.nndist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [topological_space \u03b2]\n    {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1} (hf : continuous f) (hg : continuous g) :\n    continuous fun (b : \u03b2) => nndist (f b) (g b) :=\n  continuous.comp continuous_nndist (continuous.prod_mk hf hg)\n\ntheorem filter.tendsto.nndist {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {f : \u03b2 \u2192 \u03b1} {g : \u03b2 \u2192 \u03b1}\n    {x : filter \u03b2} {a : \u03b1} {b : \u03b1} (hf : filter.tendsto f x (nhds a))\n    (hg : filter.tendsto g x (nhds b)) :\n    filter.tendsto (fun (x : \u03b2) => nndist (f x) (g x)) x (nhds (nndist a b)) :=\n  filter.tendsto.comp (continuous.tendsto continuous_nndist (a, b))\n    (filter.tendsto.prod_mk_nhds hf hg)\n\nnamespace metric\n\n\ntheorem is_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    is_closed (closed_ball x \u03b5) :=\n  is_closed_le (continuous.dist continuous_id continuous_const) continuous_const\n\ntheorem is_closed_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} : is_closed (sphere x \u03b5) :=\n  is_closed_eq (continuous.dist continuous_id continuous_const) continuous_const\n\n@[simp] theorem closure_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    closure (closed_ball x \u03b5) = closed_ball x \u03b5 :=\n  is_closed.closure_eq is_closed_ball\n\ntheorem closure_ball_subset_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    closure (ball x \u03b5) \u2286 closed_ball x \u03b5 :=\n  closure_minimal ball_subset_closed_ball is_closed_ball\n\ntheorem frontier_ball_subset_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    frontier (ball x \u03b5) \u2286 sphere x \u03b5 :=\n  frontier_lt_subset_eq (continuous.dist continuous_id continuous_const) continuous_const\n\ntheorem frontier_closed_ball_subset_sphere {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    frontier (closed_ball x \u03b5) \u2286 sphere x \u03b5 :=\n  frontier_le_subset_eq (continuous.dist continuous_id continuous_const) continuous_const\n\ntheorem ball_subset_interior_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {\u03b5 : \u211d} :\n    ball x \u03b5 \u2286 interior (closed_ball x \u03b5) :=\n  interior_maximal ball_subset_closed_ball is_open_ball\n\n/-- \u03b5-characterization of the closure in metric spaces-/\ntheorem mem_closure_iff {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {a : \u03b1} :\n    a \u2208 closure s \u2194 \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0), \u2203 (b : \u03b1), \u2203 (H : b \u2208 s), dist a b < \u03b5 :=\n  sorry\n\ntheorem mem_closure_range_iff {\u03b2 : Type v} {\u03b1 : Type u} [metric_space \u03b1] {e : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    a \u2208 closure (set.range e) \u2194 \u2200 (\u03b5 : \u211d), \u03b5 > 0 \u2192 \u2203 (k : \u03b2), dist a (e k) < \u03b5 :=\n  sorry\n\ntheorem mem_closure_range_iff_nat {\u03b2 : Type v} {\u03b1 : Type u} [metric_space \u03b1] {e : \u03b2 \u2192 \u03b1} {a : \u03b1} :\n    a \u2208 closure (set.range e) \u2194 \u2200 (n : \u2115), \u2203 (k : \u03b2), dist a (e k) < 1 / (\u2191n + 1) :=\n  sorry\n\ntheorem mem_of_closed' {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (hs : is_closed s) {a : \u03b1} :\n    a \u2208 s \u2194 \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0), \u2203 (b : \u03b1), \u2203 (H : b \u2208 s), dist a b < \u03b5 :=\n  sorry\n\nend metric\n\n\n/-- A finite product of metric spaces is a metric space, with the sup distance. -/\nprotected instance metric_space_pi {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2]\n    [(b : \u03b2) \u2192 metric_space (\u03c0 b)] : metric_space ((b : \u03b2) \u2192 \u03c0 b) :=\n  emetric_space.to_metric_space_of_dist\n    (fun (f g : (b : \u03b2) \u2192 \u03c0 b) => \u2191(finset.sup finset.univ fun (b : \u03b2) => nndist (f b) (g b))) sorry\n    sorry\n\ntheorem nndist_pi_def {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    (f : (b : \u03b2) \u2192 \u03c0 b) (g : (b : \u03b2) \u2192 \u03c0 b) :\n    nndist f g = finset.sup finset.univ fun (b : \u03b2) => nndist (f b) (g b) :=\n  subtype.eta (finset.sup finset.univ fun (b : \u03b2) => nndist (f b) (g b)) dist_nonneg\n\ntheorem dist_pi_def {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    (f : (b : \u03b2) \u2192 \u03c0 b) (g : (b : \u03b2) \u2192 \u03c0 b) :\n    dist f g = \u2191(finset.sup finset.univ fun (b : \u03b2) => nndist (f b) (g b)) :=\n  rfl\n\n@[simp] theorem dist_pi_const {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [fintype \u03b2] [Nonempty \u03b2]\n    (a : \u03b1) (b : \u03b1) : (dist (fun (x : \u03b2) => a) fun (_x : \u03b2) => b) = dist a b :=\n  sorry\n\n@[simp] theorem nndist_pi_const {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [fintype \u03b2] [Nonempty \u03b2]\n    (a : \u03b1) (b : \u03b1) : (nndist (fun (x : \u03b2) => a) fun (_x : \u03b2) => b) = nndist a b :=\n  nnreal.eq (dist_pi_const a b)\n\ntheorem dist_pi_lt_iff {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    {f : (b : \u03b2) \u2192 \u03c0 b} {g : (b : \u03b2) \u2192 \u03c0 b} {r : \u211d} (hr : 0 < r) :\n    dist f g < r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) < r :=\n  sorry\n\ntheorem dist_pi_le_iff {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    {f : (b : \u03b2) \u2192 \u03c0 b} {g : (b : \u03b2) \u2192 \u03c0 b} {r : \u211d} (hr : 0 \u2264 r) :\n    dist f g \u2264 r \u2194 \u2200 (b : \u03b2), dist (f b) (g b) \u2264 r :=\n  sorry\n\ntheorem nndist_le_pi_nndist {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2]\n    [(b : \u03b2) \u2192 metric_space (\u03c0 b)] (f : (b : \u03b2) \u2192 \u03c0 b) (g : (b : \u03b2) \u2192 \u03c0 b) (b : \u03b2) :\n    nndist (f b) (g b) \u2264 nndist f g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nndist (f b) (g b) \u2264 nndist f g)) (nndist_pi_def f g)))\n    (finset.le_sup (finset.mem_univ b))\n\ntheorem dist_le_pi_dist {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    (f : (b : \u03b2) \u2192 \u03c0 b) (g : (b : \u03b2) \u2192 \u03c0 b) (b : \u03b2) : dist (f b) (g b) \u2264 dist f g :=\n  sorry\n\n/-- An open ball in a product space is a product of open balls. The assumption `0 < r`\nis necessary for the case of the empty product. -/\ntheorem ball_pi {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    (x : (b : \u03b2) \u2192 \u03c0 b) {r : \u211d} (hr : 0 < r) :\n    metric.ball x r = set_of fun (y : (b : \u03b2) \u2192 \u03c0 b) => \u2200 (b : \u03b2), y b \u2208 metric.ball (x b) r :=\n  sorry\n\n/-- A closed ball in a product space is a product of closed balls. The assumption `0 \u2264 r`\nis necessary for the case of the empty product. -/\ntheorem closed_ball_pi {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2] [(b : \u03b2) \u2192 metric_space (\u03c0 b)]\n    (x : (b : \u03b2) \u2192 \u03c0 b) {r : \u211d} (hr : 0 \u2264 r) :\n    metric.closed_ball x r =\n        set_of fun (y : (b : \u03b2) \u2192 \u03c0 b) => \u2200 (b : \u03b2), y b \u2208 metric.closed_ball (x b) r :=\n  sorry\n\n/-- Any compact set in a metric space can be covered by finitely many balls of a given positive\nradius -/\ntheorem finite_cover_balls_of_compact {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (hs : is_compact s)\n    {e : \u211d} (he : 0 < e) :\n    \u2203 (t : set \u03b1),\n        \u2203 (H : t \u2286 s),\n          set.finite t \u2227\n            s \u2286 set.Union fun (x : \u03b1) => set.Union fun (H : x \u2208 t) => metric.ball x e :=\n  sorry\n\ntheorem is_compact.finite_cover_balls {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (hs : is_compact s)\n    {e : \u211d} (he : 0 < e) :\n    \u2203 (t : set \u03b1),\n        \u2203 (H : t \u2286 s),\n          set.finite t \u2227\n            s \u2286 set.Union fun (x : \u03b1) => set.Union fun (H : x \u2208 t) => metric.ball x e :=\n  finite_cover_balls_of_compact\n\n/-- A metric space is proper if all closed balls are compact. -/\nclass proper_space (\u03b1 : Type u) [metric_space \u03b1] where\n  compact_ball : \u2200 (x : \u03b1) (r : \u211d), is_compact (metric.closed_ball x r)\n\ntheorem tendsto_dist_right_cocompact_at_top {\u03b1 : Type u} [metric_space \u03b1] [proper_space \u03b1] (x : \u03b1) :\n    filter.tendsto (fun (y : \u03b1) => dist y x) (filter.cocompact \u03b1) filter.at_top :=\n  sorry\n\ntheorem tendsto_dist_left_cocompact_at_top {\u03b1 : Type u} [metric_space \u03b1] [proper_space \u03b1] (x : \u03b1) :\n    filter.tendsto (dist x) (filter.cocompact \u03b1) filter.at_top :=\n  sorry\n\n/-- If all closed balls of large enough radius are compact, then the space is proper. Especially\nuseful when the lower bound for the radius is 0. -/\ntheorem proper_space_of_compact_closed_ball_of_le {\u03b1 : Type u} [metric_space \u03b1] (R : \u211d)\n    (h : \u2200 (x : \u03b1) (r : \u211d), R \u2264 r \u2192 is_compact (metric.closed_ball x r)) : proper_space \u03b1 :=\n  sorry\n\n/- A compact metric space is proper -/\n\nprotected instance proper_of_compact {\u03b1 : Type u} [metric_space \u03b1] [compact_space \u03b1] :\n    proper_space \u03b1 :=\n  proper_space.mk fun (x : \u03b1) (r : \u211d) => is_closed.compact metric.is_closed_ball\n\n/-- A proper space is locally compact -/\nprotected instance locally_compact_of_proper {\u03b1 : Type u} [metric_space \u03b1] [proper_space \u03b1] :\n    locally_compact_space \u03b1 :=\n  locally_compact_of_compact_nhds\n    fun (x : \u03b1) =>\n      Exists.intro (metric.closed_ball x 1)\n        { left :=\n            iff.mpr metric.mem_nhds_iff\n              (Exists.intro 1\n                (eq.mpr\n                  (id\n                    (Eq.trans (propext exists_prop)\n                      ((fun (a a_1 : Prop) (e_1 : a = a_1) (b b_1 : Prop) (e_2 : b = b_1) =>\n                          congr (congr_arg And e_1) e_2)\n                        (1 > 0) (0 < 1) (propext gt_iff_lt)\n                        (metric.ball x 1 \u2286 metric.closed_ball x 1)\n                        (metric.ball x 1 \u2286 metric.closed_ball x 1)\n                        (Eq.refl (metric.ball x 1 \u2286 metric.closed_ball x 1)))))\n                  { left := zero_lt_one, right := metric.ball_subset_closed_ball })),\n          right := proper_space.compact_ball x 1 }\n\n/-- A proper space is complete -/\nprotected instance complete_of_proper {\u03b1 : Type u} [metric_space \u03b1] [proper_space \u03b1] :\n    complete_space \u03b1 :=\n  sorry\n\n/-- A proper metric space is separable, and therefore second countable. Indeed, any ball is\ncompact, and therefore admits a countable dense subset. Taking a countable union over the balls\ncentered at a fixed point and with integer radius, one obtains a countable set which is\ndense in the whole space. -/\nprotected instance second_countable_of_proper {\u03b1 : Type u} [metric_space \u03b1] [proper_space \u03b1] :\n    topological_space.second_countable_topology \u03b1 :=\n  emetric.second_countable_of_separable \u03b1\n\n/-- A finite product of proper spaces is proper. -/\nprotected instance pi_proper_space {\u03b2 : Type v} {\u03c0 : \u03b2 \u2192 Type u_1} [fintype \u03b2]\n    [(b : \u03b2) \u2192 metric_space (\u03c0 b)] [h : \u2200 (b : \u03b2), proper_space (\u03c0 b)] :\n    proper_space ((b : \u03b2) \u2192 \u03c0 b) :=\n  proper_space_of_compact_closed_ball_of_le 0\n    fun (x : (b : \u03b2) \u2192 \u03c0 b) (r : \u211d) (hr : 0 \u2264 r) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (is_compact (metric.closed_ball x r))) (closed_ball_pi x hr)))\n        (compact_pi_infinite fun (b : \u03b2) => proper_space.compact_ball (x b) r)\n\nnamespace metric\n\n\n/-- A metric space is second countable if, for every `\u03b5 > 0`, there is a countable set which is\n`\u03b5`-dense. -/\ntheorem second_countable_of_almost_dense_set {\u03b1 : Type u} [metric_space \u03b1]\n    (H :\n      \u2200 (\u03b5 : \u211d) (H : \u03b5 > 0),\n        \u2203 (s : set \u03b1), set.countable s \u2227 \u2200 (x : \u03b1), \u2203 (y : \u03b1), \u2203 (H : y \u2208 s), dist x y \u2264 \u03b5) :\n    topological_space.second_countable_topology \u03b1 :=\n  sorry\n\n/-- A metric space space is second countable if one can reconstruct up to any `\u03b5>0` any element of\nthe space from countably many data. -/\ntheorem second_countable_of_countable_discretization {\u03b1 : Type u} [metric_space \u03b1]\n    (H :\n      \u2200 (\u03b5 : \u211d),\n        \u03b5 > 0 \u2192 \u2203 (\u03b2 : Type u_1), Exists (\u2203 (F : \u03b1 \u2192 \u03b2), \u2200 (x y : \u03b1), F x = F y \u2192 dist x y \u2264 \u03b5)) :\n    topological_space.second_countable_topology \u03b1 :=\n  sorry\n\nend metric\n\n\ntheorem lebesgue_number_lemma_of_metric {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {\u03b9 : Sort u_1}\n    {c : \u03b9 \u2192 set \u03b1} (hs : is_compact s) (hc\u2081 : \u2200 (i : \u03b9), is_open (c i))\n    (hc\u2082 : s \u2286 set.Union fun (i : \u03b9) => c i) :\n    \u2203 (\u03b4 : \u211d), \u2203 (H : \u03b4 > 0), \u2200 (x : \u03b1), x \u2208 s \u2192 \u2203 (i : \u03b9), metric.ball x \u03b4 \u2286 c i :=\n  sorry\n\ntheorem lebesgue_number_lemma_of_metric_sUnion {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    {c : set (set \u03b1)} (hs : is_compact s) (hc\u2081 : \u2200 (t : set \u03b1), t \u2208 c \u2192 is_open t) (hc\u2082 : s \u2286 \u22c3\u2080c) :\n    \u2203 (\u03b4 : \u211d),\n        \u2203 (H : \u03b4 > 0), \u2200 (x : \u03b1) (H : x \u2208 s), \u2203 (t : set \u03b1), \u2203 (H : t \u2208 c), metric.ball x \u03b4 \u2286 t :=\n  sorry\n\nnamespace metric\n\n\n/-- Boundedness of a subset of a metric space. We formulate the definition to work\neven in the empty space. -/\ndef bounded {\u03b1 : Type u} [metric_space \u03b1] (s : set \u03b1) :=\n  \u2203 (C : \u211d), \u2200 (x y : \u03b1), x \u2208 s \u2192 y \u2208 s \u2192 dist x y \u2264 C\n\n@[simp] theorem bounded_empty {\u03b1 : Type u} [metric_space \u03b1] : bounded \u2205 := sorry\n\ntheorem bounded_iff_mem_bounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} :\n    bounded s \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 bounded s :=\n  sorry\n\n/-- Subsets of a bounded set are also bounded -/\ntheorem bounded.subset {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {t : set \u03b1} (incl : s \u2286 t) :\n    bounded t \u2192 bounded s :=\n  Exists.imp\n    fun (C : \u211d) (hC : \u2200 (x y : \u03b1), x \u2208 t \u2192 y \u2208 t \u2192 dist x y \u2264 C) (x y : \u03b1) (hx : x \u2208 s)\n      (hy : y \u2208 s) => hC x y (incl hx) (incl hy)\n\n/-- Closed balls are bounded -/\ntheorem bounded_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {r : \u211d} :\n    bounded (closed_ball x r) :=\n  sorry\n\n/-- Open balls are bounded -/\ntheorem bounded_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {r : \u211d} : bounded (ball x r) :=\n  bounded.subset ball_subset_closed_ball bounded_closed_ball\n\n/-- Given a point, a bounded subset is included in some ball around this point -/\ntheorem bounded_iff_subset_ball {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (c : \u03b1) :\n    bounded s \u2194 \u2203 (r : \u211d), s \u2286 closed_ball c r :=\n  sorry\n\ntheorem bounded_closure_of_bounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : bounded s) :\n    bounded (closure s) :=\n  sorry\n\ntheorem Mathlib.bounded.closure {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : bounded s) :\n    bounded (closure s) :=\n  bounded_closure_of_bounded\n\n/-- The union of two bounded sets is bounded iff each of the sets is bounded -/\n@[simp] theorem bounded_union {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {t : set \u03b1} :\n    bounded (s \u222a t) \u2194 bounded s \u2227 bounded t :=\n  sorry\n\n/-- A finite union of bounded sets is bounded -/\ntheorem bounded_bUnion {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {I : set \u03b2} {s : \u03b2 \u2192 set \u03b1}\n    (H : set.finite I) :\n    bounded (set.Union fun (i : \u03b2) => set.Union fun (H : i \u2208 I) => s i) \u2194\n        \u2200 (i : \u03b2), i \u2208 I \u2192 bounded (s i) :=\n  sorry\n\n/-- A compact set is bounded -/\n-- We cover the compact set by finitely many balls of radius 1,\n\ntheorem bounded_of_compact {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : is_compact s) :\n    bounded s :=\n  sorry\n\n-- and then argue that a finite union of bounded sets is bounded\n\ntheorem Mathlib.is_compact.bounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : is_compact s) :\n    bounded s :=\n  bounded_of_compact\n\n/-- A finite set is bounded -/\ntheorem bounded_of_finite {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : set.finite s) :\n    bounded s :=\n  is_compact.bounded (set.finite.is_compact h)\n\n/-- A singleton is bounded -/\ntheorem bounded_singleton {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} : bounded (singleton x) :=\n  bounded_of_finite (set.finite_singleton x)\n\n/-- Characterization of the boundedness of the range of a function -/\ntheorem bounded_range_iff {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] {f : \u03b2 \u2192 \u03b1} :\n    bounded (set.range f) \u2194 \u2203 (C : \u211d), \u2200 (x y : \u03b2), dist (f x) (f y) \u2264 C :=\n  sorry\n\n/-- In a compact space, all sets are bounded -/\ntheorem bounded_of_compact_space {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} [compact_space \u03b1] :\n    bounded s :=\n  bounded.subset (set.subset_univ s) (is_compact.bounded compact_univ)\n\n/-- The Heine\u2013Borel theorem:\nIn a proper space, a set is compact if and only if it is closed and bounded -/\ntheorem compact_iff_closed_bounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} [proper_space \u03b1] :\n    is_compact s \u2194 is_closed s \u2227 bounded s :=\n  sorry\n\n/-- The image of a proper space under an expanding onto map is proper. -/\ntheorem proper_image_of_proper {\u03b1 : Type u} {\u03b2 : Type v} [metric_space \u03b1] [proper_space \u03b1]\n    [metric_space \u03b2] (f : \u03b1 \u2192 \u03b2) (f_cont : continuous f) (hf : set.range f = set.univ) (C : \u211d)\n    (hC : \u2200 (x y : \u03b1), dist x y \u2264 C * dist (f x) (f y)) : proper_space \u03b2 :=\n  sorry\n\n/-- The diameter of a set in a metric space. To get controllable behavior even when the diameter\nshould be infinite, we express it in terms of the emetric.diameter -/\ndef diam {\u03b1 : Type u} [metric_space \u03b1] (s : set \u03b1) : \u211d := ennreal.to_real (emetric.diam s)\n\n/-- The diameter of a set is always nonnegative -/\ntheorem diam_nonneg {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} : 0 \u2264 diam s := ennreal.to_real_nonneg\n\ntheorem diam_subsingleton {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (hs : set.subsingleton s) :\n    diam s = 0 :=\n  sorry\n\n/-- The empty set has zero diameter -/\n@[simp] theorem diam_empty {\u03b1 : Type u} [metric_space \u03b1] : diam \u2205 = 0 :=\n  diam_subsingleton set.subsingleton_empty\n\n/-- A singleton has zero diameter -/\n@[simp] theorem diam_singleton {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} : diam (singleton x) = 0 :=\n  diam_subsingleton set.subsingleton_singleton\n\n-- Does not work as a simp-lemma, since {x, y} reduces to (insert y {x})\n\ntheorem diam_pair {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} :\n    diam (insert x (singleton y)) = dist x y :=\n  sorry\n\n-- Does not work as a simp-lemma, since {x, y, z} reduces to (insert z (insert y {x}))\n\ntheorem diam_triple {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {y : \u03b1} {z : \u03b1} :\n    diam (insert x (insert y (singleton z))) = max (max (dist x y) (dist x z)) (dist y z) :=\n  sorry\n\n/-- If the distance between any two points in a set is bounded by some constant `C`,\nthen `ennreal.of_real C`  bounds the emetric diameter of this set. -/\ntheorem ediam_le_of_forall_dist_le {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {C : \u211d}\n    (h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C) : emetric.diam s \u2264 ennreal.of_real C :=\n  emetric.diam_le_of_forall_edist_le\n    fun (x : \u03b1) (hx : x \u2208 s) (y : \u03b1) (hy : y \u2208 s) =>\n      Eq.symm (edist_dist x y) \u25b8 ennreal.of_real_le_of_real (h x hx y hy)\n\n/-- If the distance between any two points in a set is bounded by some non-negative constant,\nthis constant bounds the diameter. -/\ntheorem diam_le_of_forall_dist_le {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {C : \u211d} (h\u2080 : 0 \u2264 C)\n    (h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C) : diam s \u2264 C :=\n  ennreal.to_real_le_of_le_of_real h\u2080 (ediam_le_of_forall_dist_le h)\n\n/-- If the distance between any two points in a nonempty set is bounded by some constant,\nthis constant bounds the diameter. -/\ntheorem diam_le_of_forall_dist_le_of_nonempty {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1}\n    (hs : set.nonempty s) {C : \u211d} (h : \u2200 (x : \u03b1), x \u2208 s \u2192 \u2200 (y : \u03b1), y \u2208 s \u2192 dist x y \u2264 C) :\n    diam s \u2264 C :=\n  sorry\n\n/-- The distance between two points in a set is controlled by the diameter of the set. -/\ntheorem dist_le_diam_of_mem' {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {x : \u03b1} {y : \u03b1}\n    (h : emetric.diam s \u2260 \u22a4) (hx : x \u2208 s) (hy : y \u2208 s) : dist x y \u2264 diam s :=\n  sorry\n\n/-- Characterize the boundedness of a set in terms of the finiteness of its emetric.diameter. -/\ntheorem bounded_iff_ediam_ne_top {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} :\n    bounded s \u2194 emetric.diam s \u2260 \u22a4 :=\n  sorry\n\ntheorem bounded.ediam_ne_top {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : bounded s) :\n    emetric.diam s \u2260 \u22a4 :=\n  iff.mp bounded_iff_ediam_ne_top h\n\n/-- The distance between two points in a set is controlled by the diameter of the set. -/\ntheorem dist_le_diam_of_mem {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {x : \u03b1} {y : \u03b1}\n    (h : bounded s) (hx : x \u2208 s) (hy : y \u2208 s) : dist x y \u2264 diam s :=\n  dist_le_diam_of_mem' (bounded.ediam_ne_top h) hx hy\n\n/-- An unbounded set has zero diameter. If you would prefer to get the value \u221e, use `emetric.diam`.\nThis lemma makes it possible to avoid side conditions in some situations -/\ntheorem diam_eq_zero_of_unbounded {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} (h : \u00acbounded s) :\n    diam s = 0 :=\n  sorry\n\n/-- If `s \u2286 t`, then the diameter of `s` is bounded by that of `t`, provided `t` is bounded. -/\ntheorem diam_mono {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {t : set \u03b1} (h : s \u2286 t)\n    (ht : bounded t) : diam s \u2264 diam t :=\n  sorry\n\n/-- The diameter of a union is controlled by the sum of the diameters, and the distance between\nany two points in each of the sets. This lemma is true without any side condition, since it is\nobviously true if `s \u222a t` is unbounded. -/\ntheorem diam_union {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {x : \u03b1} {y : \u03b1} {t : set \u03b1}\n    (xs : x \u2208 s) (yt : y \u2208 t) : diam (s \u222a t) \u2264 diam s + dist x y + diam t :=\n  sorry\n\n/-- If two sets intersect, the diameter of the union is bounded by the sum of the diameters. -/\ntheorem diam_union' {\u03b1 : Type u} [metric_space \u03b1] {s : set \u03b1} {t : set \u03b1}\n    (h : set.nonempty (s \u2229 t)) : diam (s \u222a t) \u2264 diam s + diam t :=\n  sorry\n\n/-- The diameter of a closed ball of radius `r` is at most `2 r`. -/\ntheorem diam_closed_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {r : \u211d} (h : 0 \u2264 r) :\n    diam (closed_ball x r) \u2264 bit0 1 * r :=\n  sorry\n\n/-- The diameter of a ball of radius `r` is at most `2 r`. -/\ntheorem diam_ball {\u03b1 : Type u} [metric_space \u03b1] {x : \u03b1} {r : \u211d} (h : 0 \u2264 r) :\n    diam (ball x r) \u2264 bit0 1 * r :=\n  le_trans (diam_mono ball_subset_closed_ball bounded_closed_ball) (diam_closed_ball h)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/metric_space/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3785986251967852}}
{"text": "/-\nCopyright (c) 2022 Laurent Bonaventure. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Laurent Bonaventure\n-/\n\nnoncomputable theory\n\n/- This one just proves that a ring with axioms r1-r15\n   (actually, only r7 and r8 are nedded)\n   and 0 = 1, is a singleton.\n\n   Constructive reasoning only.\n-/\n\nconstant ordring : Type\nnotation `\u211d` := ordring\n\nnamespace ordring\n\nconstant zero : ordring\n\nconstant one : ordring\n\nconstant add (x y: ordring) : ordring\n\nconstant mul (x y: ordring) : ordring\n\nconstant neg (x: ordring) : ordring\n\nconstant lt (x y: ordring) : Prop\n\ninstance : has_zero \u211d := \u27e8zero\u27e9\ninstance : has_one \u211d := \u27e8one\u27e9\ninstance : has_add \u211d := \u27e8add\u27e9\ninstance : has_neg \u211d := \u27e8neg\u27e9\ninstance : has_mul \u211d := \u27e8mul\u27e9\ninstance : has_lt \u211d := \u27e8lt\u27e9\n\nvariables (x y z t u: ordring)\n\naxiom r1: x < x \u2192 false\n\naxiom r2: x < y \u2192 y < z \u2192 x < z\n\naxiom r3: x < 0 \u2228 0 = x \u2228 0 < x\n\naxiom r4: x < y \u2192 x + z < y + z\n\naxiom r5: 0 < x \u2192 0 < y \u2192 0 < x * y\n\naxiom r6: 0 + x = x\n\naxiom r7: 0 * x = 0\n\naxiom r8: 1 * x = x\n\naxiom r9: x + y = y + x\n\naxiom r10: x * y = y * x\n\naxiom r11: x + y + z = x + (y + z)\n\naxiom r12: x * y * z = x * (y * z)\n\naxiom r13: x * (y + z) = x * y + x * z\n\naxiom r14: x + (-x) = 0\n\naxiom r15: \u2200 (f: \u2115 \u2192 \u211d), (\u2200 (n: \u2115), f n < f (n + 1)) \u2192 (\u2203 (A: \u211d), \u2200 (n: \u2115), f n < A) \u2192 (\u2203 (L: \u211d), \u2200 (\u03b5: \u211d), 0 < \u03b5 \u2192 \u2203 (N: \u2115), \u2200 n, N < n \u2192 L + -\u03b5 < f n \u2227 f n < L + \u03b5)\n\naxiom r16: (0: \u211d) = 1\n\ntheorem singleton : x = 0 :=\nbegin\n  rw [\u2190 r8 x, \u2190 r16, r7]\nend\n\n#print axioms singleton\nend ordring\n", "meta": {"author": "Taar2", "repo": "complete-ordered-ring", "sha": "66a6c199246fda5910e09a3b80dbebdc424a417e", "save_path": "github-repos/lean/Taar2-complete-ordered-ring", "path": "github-repos/lean/Taar2-complete-ordered-ring/complete-ordered-ring-66a6c199246fda5910e09a3b80dbebdc424a417e/src/case1_singleton.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7879312056025699, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3785841440606164}}
{"text": "import for_mathlib.derived.derived_cat\nimport for_mathlib.derived.example\nimport for_mathlib.derived.les_facts\nimport for_mathlib.short_exact\nimport for_mathlib.derived.ProjectiveResolution\n\nopen category_theory category_theory.triangulated category_theory.limits\n\nnamespace bounded_derived_category\n\nvariables (A : Type*) [category A] [abelian A] [enough_projectives A]\n\ninstance Ext_additive_fst (i : \u2124) (X : bounded_derived_category A) :\n  (((Ext A i).flip.obj X).right_op).additive :=\n{ map_add' := begin\n    intros Y Z f g, dsimp,\n    conv_rhs { rw \u2190 op_add }, congr' 1, ext e,\n    dsimp, rw preadditive.add_comp,\n  end }\n\ninstance Ext_homological_fst (i : \u2124) (X : bounded_derived_category A) :\n  homological_functor ((Ext A i).flip.obj X).right_op :=\ncategory_theory.triangulated.preadditive_yoneda_op_homological (X\u27e6i\u27e7)\n\nnoncomputable\ndef Ext'_zero_flip_iso (B : A) :\n  (Ext' 0).flip.obj B \u2245 (preadditive_yoneda.obj B) :=\niso.symm $\nnat_iso.of_components\n(\u03bb X, (ProjectiveResolution.of X.unop).Ext_single_iso_hom _)\nbegin\n  intros X Y f, ext F,\n  dsimp [ProjectiveResolution.Ext_single_iso_hom,\n    ProjectiveResolution.Ext_iso,\n    bounded_homotopy_category.Ext_iso,\n    ProjectiveResolution.hom_to, Ext',\n    bounded_homotopy_category.Ext],\n  simp only [comp_apply],\n  dsimp,\n  simp only [functor.map_comp, category.assoc, bounded_homotopy_category.lift_lifts_assoc],\nend\n\n-- move me\nlemma Ext'_zero_left_is_zero {\ud835\udcd0 : Type*} [category \ud835\udcd0] [abelian \ud835\udcd0] [enough_projectives \ud835\udcd0]\n  (A : \ud835\udcd0\u1d52\u1d56) (B : \ud835\udcd0) (hA : is_zero A) (i : \u2124) :\n  is_zero (((Ext' i).obj A).obj B) :=\nbegin\n  rw is_zero_iff_id_eq_zero at hA \u22a2,\n  rw [\u2190 functor.flip_obj_obj, \u2190 category_theory.functor.map_id, hA, functor.map_zero],\nend\n\nlemma Ext'_is_zero_of_projective {\ud835\udcd0 : Type*} [category \ud835\udcd0] [abelian \ud835\udcd0] [enough_projectives \ud835\udcd0]\n  (A B : \ud835\udcd0) (hA : projective A) (i : \u2124) (hi : 0 < i) :\n  is_zero (((Ext' i).obj (opposite.op A)).obj B) :=\nbegin\n  let := Ext'_iso (opposite.op A) B i,\n  dsimp at this,\n  refine is_zero_of_iso_of_zero _ (this _ (\ud835\udfd9 _) _).symm,\n  swap,\n  { refine \u27e8_, _, _\u27e9,\n    { rintro (_|n), { exact hA }, { exact projective_zero } },\n    { apply exact_zero_left_of_mono, },\n    { intro, apply exact_zero_left_of_mono, } },\n  rcases i with ((_|i)|i),\n  { exfalso, revert hi, dec_trivial },\n  swap,\n  { exfalso, revert hi, dec_trivial },\n  refine is_zero.homology_is_zero _ _ _ _,\n  refine AddCommGroup.is_zero_of_eq _ _,\n  intros f g,\n  apply category_theory.limits.has_zero_object.from_zero_ext\nend\n\nend bounded_derived_category\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived/Ext_lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3785266485908926}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Floris van Doorn\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.discrete_category\n\n/-!\n# Limits in `C` give colimits in `C\u1d52\u1d56`.\n\nWe also give special cases for (co)products,\nbut not yet for pullbacks / pushouts or for (co)equalizers.\n\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.functor\nopen opposite\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C]\nvariables {J : Type v} [small_category J]\nvariable (F : J \u2964 C\u1d52\u1d56)\n\n/--\nIf `F.left_op : J\u1d52\u1d56 \u2964 C` has a colimit, we can construct a limit for `F : J \u2964 C\u1d52\u1d56`.\n-/\nlemma has_limit_of_has_colimit_left_op [has_colimit F.left_op] : has_limit F :=\nhas_limit.mk\n{ cone := cone_of_cocone_left_op (colimit.cocone F.left_op),\n  is_limit :=\n  { lift := \u03bb s, (colimit.desc F.left_op (cocone_left_op_of_cone s)).op,\n    fac' := \u03bb s j,\n    begin\n      rw [cone_of_cocone_left_op_\u03c0_app, colimit.cocone_\u03b9, \u2190op_comp,\n          colimit.\u03b9_desc, cocone_left_op_of_cone_\u03b9_app, quiver.hom.op_unop],\n      refl, end,\n    uniq' := \u03bb s m w,\n    begin\n      -- It's a pity we can't do this automatically.\n      -- Usually something like this would work by limit.hom_ext,\n      -- but the opposites get in the way of this firing.\n      have u := (colimit.is_colimit F.left_op).uniq (cocone_left_op_of_cone s) (m.unop),\n      convert congr_arg (\u03bb f : _ \u27f6 _, f.op) (u _), clear u,\n      intro j,\n      rw [cocone_left_op_of_cone_\u03b9_app, colimit.cocone_\u03b9],\n      convert congr_arg (\u03bb f : _ \u27f6 _, f.unop) (w (unop j)), clear w,\n      rw [cone_of_cocone_left_op_\u03c0_app, colimit.cocone_\u03b9, quiver.hom.unop_op],\n      refl,\n    end } }\n\n/--\nIf `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\nlemma has_limits_of_shape_op_of_has_colimits_of_shape [has_colimits_of_shape J\u1d52\u1d56 C] :\n  has_limits_of_shape J C\u1d52\u1d56 :=\n{ has_limit := \u03bb F, has_limit_of_has_colimit_left_op F }\n\nlocal attribute [instance] has_limits_of_shape_op_of_has_colimits_of_shape\n\n/--\nIf `C` has colimits, we can construct limits for `C\u1d52\u1d56`.\n-/\nlemma has_limits_op_of_has_colimits [has_colimits C] : has_limits C\u1d52\u1d56 := \u27e8infer_instance\u27e9\n\n/--\nIf `F.left_op : J\u1d52\u1d56 \u2964 C` has a limit, we can construct a colimit for `F : J \u2964 C\u1d52\u1d56`.\n-/\nlemma has_colimit_of_has_limit_left_op [has_limit F.left_op] : has_colimit F :=\nhas_colimit.mk\n{ cocone := cocone_of_cone_left_op (limit.cone F.left_op),\n  is_colimit :=\n  { desc := \u03bb s, (limit.lift F.left_op (cone_left_op_of_cocone s)).op,\n    fac' := \u03bb s j,\n    begin\n      rw [cocone_of_cone_left_op_\u03b9_app, limit.cone_\u03c0, \u2190op_comp,\n          limit.lift_\u03c0, cone_left_op_of_cocone_\u03c0_app, quiver.hom.op_unop],\n      refl, end,\n    uniq' := \u03bb s m w,\n    begin\n      have u := (limit.is_limit F.left_op).uniq (cone_left_op_of_cocone s) (m.unop),\n      convert congr_arg (\u03bb f : _ \u27f6 _, f.op) (u _), clear u,\n      intro j,\n      rw [cone_left_op_of_cocone_\u03c0_app, limit.cone_\u03c0],\n      convert congr_arg (\u03bb f : _ \u27f6 _, f.unop) (w (unop j)), clear w,\n      rw [cocone_of_cone_left_op_\u03b9_app, limit.cone_\u03c0, quiver.hom.unop_op],\n      refl,\n    end } }\n\n/--\nIf `C` has colimits of shape `J\u1d52\u1d56`, we can construct limits in `C\u1d52\u1d56` of shape `J`.\n-/\nlemma has_colimits_of_shape_op_of_has_limits_of_shape [has_limits_of_shape J\u1d52\u1d56 C] :\n  has_colimits_of_shape J C\u1d52\u1d56 :=\n{ has_colimit := \u03bb F, has_colimit_of_has_limit_left_op F }\n\nlocal attribute [instance] has_colimits_of_shape_op_of_has_limits_of_shape\n\n/--\nIf `C` has limits, we can construct colimits for `C\u1d52\u1d56`.\n-/\nlemma has_colimits_op_of_has_limits [has_limits C] : has_colimits C\u1d52\u1d56 := \u27e8infer_instance\u27e9\n\nvariables (X : Type v)\n/--\nIf `C` has products indexed by `X`, then `C\u1d52\u1d56` has coproducts indexed by `X`.\n-/\nlemma has_coproducts_opposite [has_products_of_shape X C] :\n  has_coproducts_of_shape X C\u1d52\u1d56 :=\nbegin\n  haveI : has_limits_of_shape (discrete X)\u1d52\u1d56 C :=\n    has_limits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\n/--\nIf `C` has coproducts indexed by `X`, then `C\u1d52\u1d56` has products indexed by `X`.\n-/\nlemma has_products_opposite [has_coproducts_of_shape X C] :\n  has_products_of_shape X C\u1d52\u1d56 :=\nbegin\n  haveI : has_colimits_of_shape (discrete X)\u1d52\u1d56 C :=\n    has_colimits_of_shape_of_equivalence (discrete.opposite X).symm,\n  apply_instance\nend\n\nlemma has_finite_coproducts_opposite [has_finite_products C] :\n  has_finite_coproducts C\u1d52\u1d56 :=\n{ out := \u03bb J \ud835\udc9f \ud835\udca5, begin\n    resetI,\n    haveI : has_limits_of_shape (discrete J)\u1d52\u1d56 C :=\n      has_limits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlemma has_finite_products_opposite [has_finite_coproducts C] :\n  has_finite_products C\u1d52\u1d56 :=\n{ out := \u03bb J \ud835\udc9f \ud835\udca5, begin\n    resetI,\n    haveI : has_colimits_of_shape (discrete J)\u1d52\u1d56 C :=\n      has_colimits_of_shape_of_equivalence (discrete.opposite J).symm,\n    apply_instance,\n  end }\n\nlocal attribute [instance] fin_category_opposite\n\nlemma has_finite_colimits_opposite [has_finite_limits C] :\n  has_finite_colimits C\u1d52\u1d56 :=\n{ out := \u03bb J \ud835\udc9f \ud835\udca5, by { resetI, apply_instance, }, }\n\nlemma has_finite_limits_opposite [has_finite_colimits C] :\n  has_finite_limits C\u1d52\u1d56 :=\n{ out := \u03bb J \ud835\udc9f \ud835\udca5, by { resetI, apply_instance, }, }\n\nend category_theory.limits\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/limits/opposites.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737473266735, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.37846698305000187}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Scott Morrison, Mario Carneiro\n-/\nimport topology.category.Top.basic\nimport category_theory.limits.types\nimport category_theory.limits.preserves.basic\n\n/-!\n# The category of topological spaces has all limits and colimits\n\nFurther, these limits and colimits are preserved by the forgetful functor --- that is, the\nunderlying types are just the limits in the category of types.\n-/\n\nopen topological_space\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverse u\n\nnoncomputable theory\n\nnamespace Top\n\nvariables {J : Type u} [small_category J]\n\nlocal notation `forget` := forget Top\n\n/--\nA choice of limit cone for a functor `F : J \u2964 Top`.\nGenerally you should just use `limit.cone F`, unless you need the actual definition\n(which is in terms of `types.limit_cone`).\n-/\ndef limit_cone (F : J \u2964 Top.{u}) : cone F :=\n{ X := \u27e8(types.limit_cone (F \u22d9 forget)).X, \u2a05j,\n        (F.obj j).str.induced ((types.limit_cone (F \u22d9 forget)).\u03c0.app j)\u27e9,\n  \u03c0 :=\n  { app := \u03bb j, \u27e8(types.limit_cone (F \u22d9 forget)).\u03c0.app j,\n                 continuous_iff_le_induced.mpr (infi_le _ _)\u27e9,\n    naturality' := \u03bb j j' f,\n                   continuous_map.coe_inj ((types.limit_cone (F \u22d9 forget)).\u03c0.naturality f) } }\n\n/--\nThe chosen cone `Top.limit_cone F` for a functor `F : J \u2964 Top` is a limit cone.\nGenerally you should just use `limit.is_limit F`, unless you need the actual definition\n(which is in terms of `types.limit_cone_is_limit`).\n-/\ndef limit_cone_is_limit (F : J \u2964 Top.{u}) : is_limit (limit_cone F) :=\nby { refine is_limit.of_faithful forget (types.limit_cone_is_limit _) (\u03bb s, \u27e8_, _\u27e9) (\u03bb s, rfl),\n     exact continuous_iff_coinduced_le.mpr (le_infi $ \u03bb j,\n       coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.\u03c0.app j).continuous :\n         _) ) }\n\ninstance Top_has_limits : has_limits.{u} Top.{u} :=\n{ has_limits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { has_limit := \u03bb F, has_limit.mk { cone := limit_cone F, is_limit := limit_cone_is_limit F } } }\n\ninstance forget_preserves_limits : preserves_limits (forget : Top.{u} \u2964 Type u) :=\n{ preserves_limits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_limit := \u03bb F,\n    by exactI preserves_limit_of_preserves_limit_cone\n      (limit_cone_is_limit F) (types.limit_cone_is_limit (F \u22d9 forget)) } }\n\n/--\nA choice of colimit cocone for a functor `F : J \u2964 Top`.\nGenerally you should just use `colimit.coone F`, unless you need the actual definition\n(which is in terms of `types.colimit_cocone`).\n-/\ndef colimit_cocone (F : J \u2964 Top.{u}) : cocone F :=\n{ X := \u27e8(types.colimit_cocone (F \u22d9 forget)).X, \u2a06 j,\n        (F.obj j).str.coinduced ((types.colimit_cocone (F \u22d9 forget)).\u03b9.app j)\u27e9,\n  \u03b9 :=\n  { app := \u03bb j, \u27e8(types.colimit_cocone (F \u22d9 forget)).\u03b9.app j,\n                 continuous_iff_coinduced_le.mpr (le_supr _ j)\u27e9,\n    naturality' := \u03bb j j' f,\n                   continuous_map.coe_inj ((types.colimit_cocone (F \u22d9 forget)).\u03b9.naturality f) } }\n\n/--\nThe chosen cocone `Top.colimit_cocone F` for a functor `F : J \u2964 Top` is a colimit cocone.\nGenerally you should just use `colimit.is_colimit F`, unless you need the actual definition\n(which is in terms of `types.colimit_cocone_is_colimit`).\n-/\ndef colimit_cocone_is_colimit (F : J \u2964 Top.{u}) : is_colimit (colimit_cocone F) :=\nby { refine is_colimit.of_faithful forget (types.colimit_cocone_is_colimit _) (\u03bb s, \u27e8_, _\u27e9)\n       (\u03bb s, rfl),\n     exact continuous_iff_le_induced.mpr (supr_le $ \u03bb j,\n       coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.\u03b9.app j).continuous :\n         _) ) }\n\ninstance Top_has_colimits : has_colimits.{u} Top.{u} :=\n{ has_colimits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { has_colimit := \u03bb F, has_colimit.mk { cocone := colimit_cocone F, is_colimit :=\n    colimit_cocone_is_colimit F } } }\n\ninstance forget_preserves_colimits : preserves_colimits (forget : Top.{u} \u2964 Type u) :=\n{ preserves_colimits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_colimit := \u03bb F,\n    by exactI preserves_colimit_of_preserves_colimit_cocone\n      (colimit_cocone_is_colimit F) (types.colimit_cocone_is_colimit (F \u22d9 forget)) } }\n\nend Top\n\nnamespace Top\n\nsection topological_konig\n\n/-!\n## Topological K\u0151nig's lemma\n\nA topological version of K\u0151nig's lemma is that the inverse limit of nonempty compact Hausdorff\nspaces is nonempty.  (Note: this can be generalized further to inverse limits of nonempty compact\nT0 spaces, where all the maps are closed maps; see [Stone1979] --- however there is an erratum\nfor Theorem 4 that the element in the inverse limit can have cofinally many components that are\nnot closed points.)\n-/\n\nvariables {J : Type u} [directed_order J]\nvariables (F : J\u1d52\u1d56 \u2964 Top.{u})\n\n/--\nThe partial sections of an inverse system of topological spaces from an index `j` are sections\nwhen restricted to all objects less than or equal to `j`.\n-/\ndef partial_sections (j : J\u1d52\u1d56) : set (\u03a0 j, F.obj j) :=\n{ u | \u2200 {j'} (f : j \u27f6 j'), F.map f (u j) = u j'}\n\nlemma partial_sections.nonempty [\u03a0 (j : J\u1d52\u1d56), nonempty (F.obj j)] (j : J\u1d52\u1d56) :\n  (partial_sections F j).nonempty :=\nbegin\n  classical,\n  use \u03bb (j' : J\u1d52\u1d56),\n    if h : j'.unop \u2264 j.unop then\n      F.map (hom_of_le h).op (classical.arbitrary (F.obj j))\n    else\n      classical.arbitrary _,\n  intros j' fle,\n  simp only [dif_pos (le_of_hom fle.unop)],\n  dsimp, simp,\nend\n\nlemma partial_sections.directed : directed (\u2287) (partial_sections F) :=\nbegin\n  intros j j',\n  obtain \u27e8j'', hj''\u27e9 := directed_order.directed j.unop j'.unop,\n  use op j'',\n  split,\n  { intros u hu j''' f''',\n    rw [\u2190hu ((hom_of_le hj''.1).op \u226b f'''), \u2190hu],\n    simp only [Top.comp_app, functor.map_comp] },\n  { intros u hu j''' f''',\n    rw [\u2190hu ((hom_of_le hj''.2).op \u226b f'''), \u2190hu],\n    simp only [Top.comp_app, functor.map_comp] },\nend\n\nlemma partial_sections.closed [\u03a0 (j : J\u1d52\u1d56), t2_space (F.obj j)] (j : J\u1d52\u1d56) :\n  is_closed (partial_sections F j) :=\nbegin\n  have hps : partial_sections F j =\n    \u22c2 (f : \u03a3 j', j \u27f6 j'), {u : \u03a0 (j : J\u1d52\u1d56), F.obj j | F.map f.2 (u j) = u f.1},\n  { ext u,\n    simp only [set.mem_Inter, sigma.forall, set.mem_set_of_eq],\n    exact \u27e8\u03bb hu j' f, hu f, \u03bb hu j' f, hu j' f\u27e9 },\n  rw hps,\n  apply is_closed_Inter,\n  rintros \u27e8j', f\u27e9,\n  let proj : \u03a0 (j' : J\u1d52\u1d56), C((\u03a0 (j : J\u1d52\u1d56), F.obj j), F.obj j') :=\n    \u03bb j', \u27e8\u03bb u, u j', continuous_apply j'\u27e9,\n  exact is_closed_eq\n    (((F.map f).continuous.comp (proj j).continuous).comp continuous_id)\n    ((proj j').continuous.comp continuous_id),\nend\n\nlemma nonempty_limit_cone_of_compact_t2_inverse_system\n  [\u03a0 (j : J\u1d52\u1d56), nonempty (F.obj j)]\n  [\u03a0 (j : J\u1d52\u1d56), compact_space (F.obj j)]\n  [\u03a0 (j : J\u1d52\u1d56), t2_space (F.obj j)] :\n  nonempty (Top.limit_cone F).X :=\nbegin\n  by_cases h : nonempty J\u1d52\u1d56,\n  { haveI := h,\n    obtain \u27e8u, hu\u27e9 := is_compact.nonempty_Inter_of_directed_nonempty_compact_closed\n      (partial_sections F) (partial_sections.directed F) (partial_sections.nonempty F)\n      (\u03bb j, is_closed.compact (partial_sections.closed F j)) (partial_sections.closed F),\n    use u,\n    intros j j' f,\n    specialize hu (partial_sections F j),\n    simp only [forall_prop_of_true, set.mem_range_self] at hu,\n    exact hu f, },\n  { exact \u27e8\u27e8\u03bb j, (h \u27e8j\u27e9).elim, \u03bb j, (h \u27e8j\u27e9).elim\u27e9\u27e9, },\nend\n\nend topological_konig\n\nend Top\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/topology/category/Top/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3784669758893766}}
{"text": "-- WIP stuff. see Metamath.Verify for the verifier\n\nimport Lean\nimport Metamath.Verify\n\nopen Or in\ntheorem or_assoc {a b c} : (a \u2228 b) \u2228 c \u2194 a \u2228 (b \u2228 c) :=\n\u27e8fun | inl (inl h) => inl h\n     | inl (inr h) => inr (inl h)\n     | inr h => inr (inr h),\n fun | inl h => inl (inl h)\n     | inr (inl h) => inl (inr h)\n     | inr (inr h) => inr h\u27e9\n\ntheorem Or.symm : a \u2228 b \u2192 b \u2228 a\n| Or.inl h => Or.inr h\n| Or.inr h => Or.inl h\n\ntheorem Or.imp_right (h : b \u2192 c) : a \u2228 b \u2192 a \u2228 c\n| Or.inl h => Or.inl h\n| Or.inr h' => Or.inr (h h')\n\nnamespace List\n\ndef mem (a : \u03b1) : List \u03b1 \u2192 Prop\n| [] => False\n| (b :: l) => a = b \u2228 mem a l\n\ninfix:50 \" \u2208 \" => mem\n\ntheorem mem_append {a} : \u2200 {l\u2081 l\u2082 : List \u03b1}, a \u2208 l\u2081 ++ l\u2082 \u2194 a \u2208 l\u2081 \u2228 a \u2208 l\u2082\n| [], _ => by simp [mem]\n| b :: l\u2081, l\u2082 => by simp only [List.cons_append, mem, or_assoc, mem_append]; exact Iff.rfl\n\ntheorem mem_map {f : \u03b1 \u2192 \u03b2} {b} : \u2200 {l}, b \u2208 l.map f \u2194 \u2203 a, a \u2208 l \u2227 b = f a\n| [] => by simp [mem]; intro \u27e8_, e\u27e9; exact e\n| b :: l => by\n  simp only [join, mem, mem_map]\n  exact \u27e8fun | Or.inl h => \u27e8_, Or.inl rfl, h\u27e9\n             | Or.inr \u27e8l, h\u2081, h\u2082\u27e9 => \u27e8l, Or.inr h\u2081, h\u2082\u27e9,\n         fun | \u27e8_, Or.inl rfl, h\u27e9 => Or.inl h\n             | \u27e8l, Or.inr h\u2081, h\u2082\u27e9 => Or.inr \u27e8l, h\u2081, h\u2082\u27e9\u27e9\n\ntheorem mem_join {a} : \u2200 {L : List (List \u03b1)}, a \u2208 L.join \u2194 \u2203 l, l \u2208 L \u2227 a \u2208 l\n| [] => by simp [mem]; intro \u27e8_, e\u27e9; exact e\n| b :: l => by\n  simp only [join, mem, mem_append, mem_join]\n  exact \u27e8fun | Or.inl h => \u27e8_, Or.inl rfl, h\u27e9\n             | Or.inr \u27e8l, h\u2081, h\u2082\u27e9 => \u27e8l, Or.inr h\u2081, h\u2082\u27e9,\n         fun | \u27e8_, Or.inl rfl, h\u27e9 => Or.inl h\n             | \u27e8l, Or.inr h\u2081, h\u2082\u27e9 => Or.inr \u27e8l, h\u2081, h\u2082\u27e9\u27e9\n\ntheorem mem_bind {f : \u03b1 \u2192 List \u03b2} {b} {l} : b \u2208 l.bind f \u2194 \u2203 a, a \u2208 l \u2227 b \u2208 f a := by\n  simp [List.bind, mem_map, mem_join]\n  exact \u27e8fun \u27e8_, \u27e8a, h\u2081, rfl\u27e9, h\u2082\u27e9 => \u27e8a, h\u2081, h\u2082\u27e9, fun \u27e8a, h\u2081, h\u2082\u27e9 => \u27e8_, \u27e8a, h\u2081, rfl\u27e9, h\u2082\u27e9\u27e9\n\nend List\n\nnamespace Metamath\nopen Lean Elab\nopen Verify in\npartial def foo : TermElabM Unit := do\n  let mut s : ParserState := Inhabited.default\n  s := s.withDB fun db => { db with interrupt := true }\n  let h \u2190 IO.FS.Handle.mk \"/home/mario/Documents/metamath/mm/iset.mm\" IO.FS.Mode.read true\n  let rec loop (s : ParserState) (base : Nat) : IO (Except ParserState DB) := do\n    if \u2190 h.isEof then\n      pure $ Except.ok $ s.done base\n    else\n      let buf \u2190 h.read 1024\n      let s := s.feedAll base buf\n      if s.db.error?.isSome then pure $ Except.error s\n      else loop s (base + buf.size)\n  match \u2190 loop Inhabited.default 0 with\n  | Except.ok _ => ()\n  | Except.error s => match s.db.error? with\n    | some \u27e8Error.ax pos l f fr, i\u27e9 =>\n      IO.println s!\"axiom {l}: {fr} |- {f}\"\n    | some \u27e8Error.thm pos l f fr, i\u27e9 =>\n      IO.println s!\"theorem {l}: {fr} |- {f}\"\n    | some \u27e8Error.error pos msg, _\u27e9 =>\n      IO.println s!\"at {pos}: {msg}\"\n    | _ => ()\n\n-- #eval foo\n\ndef CN := String\ninstance : Inhabited CN := inferInstanceAs (Inhabited String)\ninstance : DecidableEq CN := inferInstanceAs (DecidableEq String)\n\nstructure VR := (type : CN) (i : Nat)\nderiving DecidableEq\n\ninductive Sym\n| const (c : CN)\n| var (n : VR)\nderiving Inhabited, DecidableEq\nopen Sym\n\ninstance : Coe String Sym := \u27e8const\u27e9\ndef Sym.isVar : Sym \u2192 Bool\n| const _ => false\n| var _ => true\n\ndef Expr := List Sym\ndef VR.expr (v : VR) : Expr := [var v]\n\ninstance : Append Expr := inferInstanceAs (Append (List Sym))\ndef Expr.sn (s : Sym) : Expr := [s]\ninstance : Coe String Expr := \u27e8fun c => Expr.sn c\u27e9\ninstance : Coe VR Expr := \u27e8fun v => Expr.sn (var v)\u27e9\ndef Expr.cons (c : String) : Expr \u2192 Expr := List.cons c\ndef Expr.mem (e : Expr) (v : VR) : Prop := var v \u2208 e\n\nscoped notation:50 a:51 \" \u2208' \" b:51 => Expr.mem b a\n\ndef Expr.vars : Expr \u2192 List VR\n| [] => []\n| const c :: e => vars e\n| var v :: e => v :: vars e\n\ndef Expr.subst (\u03c3 : VR \u2192 Expr) : Expr \u2192 Expr\n| [] => []\n| const c :: e => const c :: subst \u03c3 e\n| var v :: e => \u03c3 v ++ subst \u03c3 e\n\ntheorem Expr.subst_id : (e : Expr) \u2192 Expr.subst VR.expr e = e\n| [] => rfl\n| const c :: e => congrArg (const c :: .) (subst_id e)\n| var v :: e => congrArg (var v :: .) (subst_id e)\n\ntheorem Expr.subst_append (\u03c3) : (e\u2081 e\u2082 : Expr) \u2192 Expr.subst \u03c3 (e\u2081 ++ e\u2082) = e\u2081.subst \u03c3 ++ e\u2082.subst \u03c3\n| [], _ => rfl\n| const c :: (e\u2081 : Expr), e\u2082 => by\n  simp only [Expr.subst]; rw [List.cons_append, List.cons_append, \u2190 subst_append]; rfl\n| var v :: e, e\u2082 => by\n  rw [List.cons_append]; simp only [Expr.subst]; rw [List.append_assoc, subst_append]\n\ntheorem Expr.mem_subst {\u03c3 a} : {e : Expr} \u2192 a \u2208' Expr.subst \u03c3 e \u2192 \u2203 b, b \u2208' e \u2227 a \u2208' \u03c3 b\n| const c :: (e : Expr), Or.inr h => let \u27e8b, h\u2081, h\u2082\u27e9 := mem_subst h; \u27e8b, Or.inr h\u2081, h\u2082\u27e9\n| var v :: e, h =>\n  match List.mem_append.1 h with\n  | Or.inl h => \u27e8v, Or.inl rfl, h\u27e9\n  | Or.inr h => let \u27e8b, h\u2081, h\u2082\u27e9 := mem_subst h; \u27e8b, Or.inr h\u2081, h\u2082\u27e9\n\ndef subst.trans (\u03c3 \u03c3' : VR \u2192 Expr) (v : VR) : Expr := (\u03c3 v).subst \u03c3'\n\ntheorem Expr.subst_tr (\u03c3 \u03c3' : VR \u2192 Expr) : (e : Expr) \u2192\n  e.subst (subst.trans \u03c3 \u03c3') = (e.subst \u03c3).subst \u03c3'\n| [] => rfl\n| const c :: e => congrArg (const c :: .) (subst_tr _ _ e)\n| var v :: e => by simp only [subst]; rw [subst_append, subst_tr]; rfl\n\ndef Formula := CN \u00d7 Expr\n\ndef Formula.subst (\u03c3 : VR \u2192 Expr) : Formula \u2192 Formula\n| (c, e) => (c, e.subst \u03c3)\n\ntheorem Formula.subst_id : (e : Formula) \u2192 Formula.subst VR.expr e = e\n| (c, e) => congrArg (c, .) e.subst_id\n\ntheorem Formula.subst_tr (\u03c3 \u03c3' : VR \u2192 Expr) : (e : Formula) \u2192\n  e.subst (subst.trans \u03c3 \u03c3') = (e.subst \u03c3).subst \u03c3'\n| (c, e) => congrArg (c, .) (e.subst_tr _ _)\n\ndef VR.vhyp (v : VR) : Formula := (v.type, [var v])\ninstance : Coe VR Formula := \u27e8VR.vhyp\u27e9\n\ndef Expr.\u03b4 (a b : Expr) : Bool :=\na.all fun\n| const _ => true\n| var a => b.all fun\n  | const _ => true\n  | var b => a != b\n\nstructure DJ :=\n  disj : VR \u2192 VR \u2192 Prop\n  irr : \u00ac disj x x\n  symm : disj x y \u2192 disj y x\n\ninstance : CoeFun DJ (fun _ => VR \u2192 VR \u2192 Prop) := \u27e8DJ.disj\u27e9\ninstance : LE DJ := \u27e8fun dj dj' => \u2200 a b, dj a b \u2192 dj' a b\u27e9\n\ntheorem DJ.refl (dj : DJ) : dj \u2264 dj := fun _ _ => id\n\ntheorem DJ.ne (dj : DJ) {a b} (h : dj a b) : a \u2260 b :=\n  fun e => by cases e; exact dj.irr h\n\ntheorem DJ.ext : {dj\u2081 dj\u2082 : DJ} \u2192 (\u2200 a b, dj\u2081 a b \u2194 dj\u2082 a b) \u2192 dj\u2081 = dj\u2082\n| \u27e8dj\u2081, _, _\u27e9, \u27e8dj\u2082, _, _\u27e9, h =>\n  have dj\u2081 = dj\u2082 from funext fun a => funext fun b => propext (h a b)\n  by cases this; rfl\n\ntheorem DJ.le_antisymm {dj\u2081 dj\u2082 : DJ} (H\u2081 : dj\u2081 \u2264 dj\u2082) (H\u2082 : dj\u2082 \u2264 dj\u2081) : dj\u2081 = dj\u2082 :=\n  DJ.ext fun a b => \u27e8H\u2081 _ _, H\u2082 _ _\u27e9\n\ndef DJ.mk' (disj : List (VR \u00d7 VR)) : DJ :=\n{ disj := fun a b => a \u2260 b \u2227 ((a, b) \u2208 disj \u2228 (b, a) \u2208 disj)\n  irr := fun h => h.1 rfl\n  symm := fun \u27e8h, h'\u27e9 => \u27e8h.symm, h'.symm\u27e9 }\n\ndef Expr.disjoint (dj : DJ) (e\u2081 e\u2082 : Expr) : Prop :=\n  \u2200 a b, a \u2208' e\u2081 \u2192 b \u2208' e\u2082 \u2192 dj a b\n\ntheorem Expr.disjoint.mono {dj\u2081 dj\u2082 : DJ} (h : dj\u2081 \u2264 dj\u2082) {e\u2081 e\u2082}\n  (H : Expr.disjoint dj\u2081 e\u2081 e\u2082) : Expr.disjoint dj\u2082 e\u2081 e\u2082 :=\n  fun a b ha hb => h _ _ (H a b ha hb)\n\ndef DJ.subst (\u03c3 : VR \u2192 Expr) (dj dj' : DJ) :=\n  \u2200 a b, dj a b \u2192 (\u03c3 a).disjoint dj' (\u03c3 b)\n\ntheorem DJ.subst.mono {\u03c3 : VR \u2192 Expr} {dj\u2081 dj\u2082 dj\u2081' dj\u2082' : DJ}\n  (h : dj\u2082 \u2264 dj\u2081) (h' : dj\u2081' \u2264 dj\u2082') (H : dj\u2081.subst \u03c3 dj\u2081') : dj\u2082.subst \u03c3 dj\u2082' :=\n  fun a b d => Expr.disjoint.mono h' (H _ _ (h _ _ d))\n\ndef DJ.trim (dj : DJ) (P : VR \u2192 Prop) : DJ :=\n{ disj := fun x y => dj x y \u2227 P x \u2227 P y\n  irr := fun x => dj.irr x.1\n  symm := fun \u27e8h\u2081, h\u2082, h\u2083\u27e9 => \u27e8dj.symm h\u2081, h\u2083, h\u2082\u27e9 }\n\ntheorem DJ.trim.mono {dj\u2081 dj\u2082 : DJ} (hdj : dj\u2081 \u2264 dj\u2082) {P Q : VR \u2192 Prop}\n  (pq : \u2200 x, P x \u2192 Q x) : dj\u2081.trim P \u2264 dj\u2082.trim Q :=\n  fun a b \u27e8h, ha, hb\u27e9 => \u27e8hdj _ _ h, pq _ ha, pq _ hb\u27e9\n\ndef DJ.trimmed (dj : DJ) (P : VR \u2192 Prop) : Prop :=\n  \u2200 a b, dj a b \u2192 P a \u2227 P b\n\ntheorem DJ.trimmed.mono (dj : DJ) {P Q : VR \u2192 Prop}\n  (h : \u2200 x, P x \u2192 Q x) (H : dj.trimmed P) : dj.trimmed Q\n| a, b, d => let \u27e8h\u2081, h\u2082\u27e9 := H a b d; \u27e8h _ h\u2081, h _ h\u2082\u27e9\n\ntheorem DJ.trim_le_self (dj : DJ) (P : VR \u2192 Prop) : dj.trim P \u2264 dj := fun a b d => d.1\n\ntheorem DJ.trim.trimmed (dj : DJ) (P : VR \u2192 Prop) : (dj.trim P).trimmed P := fun a b h => h.2\n\ntheorem DJ.trimmed.trim_eq {dj : DJ} {P} (h : dj.trimmed P) : dj.trim P = dj :=\n  DJ.ext fun a b => \u27e8fun h => h.1, fun h' => \u27e8h', h _ _ h'\u27e9\u27e9\n\ndef DJ.untrim (dj : DJ) (P : VR \u2192 Prop) : DJ :=\n{ disj := fun x y => x \u2260 y \u2227 (P x \u2192 P y \u2192 dj x y)\n  irr := fun x => x.1 rfl\n  symm := fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2081.symm, fun x y => dj.symm (h\u2082 y x)\u27e9 }\n\ntheorem DJ.untrim.mono {dj\u2081 dj\u2082 : DJ} (hdj : dj\u2081 \u2264 dj\u2082) {P Q : VR \u2192 Prop}\n  (qp : \u2200 x, Q x \u2192 P x) : dj\u2081.untrim P \u2264 dj\u2082.untrim Q :=\n  fun a b \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2081, fun ha hb => hdj _ _ (h\u2082 (qp _ ha) (qp _ hb))\u27e9\n\ntheorem DJ.trim_le {dj\u2081 dj\u2082 : DJ} {P} : dj\u2081.trim P \u2264 dj\u2082 \u2194 dj\u2081 \u2264 dj\u2082.untrim P :=\n  \u27e8fun H a b h => \u27e8dj\u2081.ne h, fun ha hb => H _ _ \u27e8h, ha, hb\u27e9\u27e9,\n   fun H a b \u27e8h, ha, hb\u27e9 => (H _ _ h).2 ha hb\u27e9\n\ntheorem DJ.self_le_untrim (dj : DJ) (P : VR \u2192 Prop) : dj \u2264 dj.untrim P :=\n  DJ.trim_le.1 $ DJ.trim_le_self _ _\n\ntheorem DJ.trim_untrim (dj : DJ) (P : VR \u2192 Prop) : (dj.untrim P).trim P = dj.trim P :=\n  DJ.le_antisymm (fun a b \u27e8h, ha, hb\u27e9 => \u27e8h.2 ha hb, ha, hb\u27e9)\n    (DJ.trim.mono (DJ.self_le_untrim _ _) (fun _ => id))\n\ntheorem DJ.untrim_trim (dj : DJ) (P : VR \u2192 Prop) : (dj.trim P).untrim P = dj.untrim P :=\n  DJ.le_antisymm (DJ.untrim.mono (DJ.trim_le_self _ _) (fun _ => id))\n    fun a b \u27e8h, H\u27e9 => \u27e8h, fun ha hb => \u27e8H ha hb, ha, hb\u27e9\u27e9\n\nstructure Context :=\n  hyps : List Formula\n  dj : DJ\n\ndef Context.mk' (disj : List (VR \u00d7 VR)) (hyps : List Formula) : Context :=\n  \u27e8hyps, DJ.mk' disj\u27e9\n\ninstance : LE Context := \u27e8fun \u0393 \u0393' => (\u2200 a, a \u2208 \u0393.hyps \u2192 a \u2208 \u0393'.hyps) \u2227 \u0393.dj \u2264 \u0393'.dj\u27e9\n\ntheorem Context.refl (\u0393 : Context) : \u0393 \u2264 \u0393 := \u27e8fun _ => id, DJ.refl _\u27e9\n\nstructure Statement :=\n  ctx : Context\n  fmla : Formula\n\ninstance : LE Statement := \u27e8fun s s' => s.ctx \u2264 s'.ctx \u2227 s.fmla = s'.fmla\u27e9\n\ntheorem Statement.refl (s : Statement) : s \u2264 s := \u27e8Context.refl _, rfl\u27e9\n\ndef Statement.vars (s : Statement) : List VR :=\n  (s.fmla :: s.ctx.hyps).bind fun e => e.2.vars\n\ntheorem Statement.vars.mono' {s\u2081 s\u2082 : Statement}\n  (H : \u2200 a, a \u2208 s\u2081.ctx.hyps \u2192 a \u2208 s\u2082.ctx.hyps) (H\u2082 : s\u2081.fmla = s\u2082.fmla)\n  (v) : v \u2208 s\u2081.vars \u2192 v \u2208 s\u2082.vars := by\n  simp only [vars, List.mem_bind, List.mem, H\u2082]\n  exact fun \u27e8a, b, c\u27e9 => \u27e8a, b.imp_right (H _), c\u27e9\n\ntheorem Statement.vars.mono {s\u2081 s\u2082 : Statement} (H : s\u2081 \u2264 s\u2082) : \u2200 v, v \u2208 s\u2081.vars \u2192 v \u2208 s\u2082.vars :=\n  Statement.vars.mono' H.1.1 H.2\n\ndef Statement.trim (s : Statement) : Statement :=\n  \u27e8\u27e8s.ctx.hyps, s.ctx.dj.trim fun v => v \u2208 s.vars\u27e9, s.fmla\u27e9\n\ndef Statement.untrim' (s : Statement) (P : VR \u2192 Prop): Statement :=\n  \u27e8\u27e8s.ctx.hyps, s.ctx.dj.untrim P\u27e9, s.fmla\u27e9\ndef Statement.untrim (s : Statement) : Statement := s.untrim' fun v => v \u2208 s.vars\n\ntheorem Statement.trim_le_self (s : Statement) : s.trim \u2264 s :=\n  \u27e8\u27e8fun _ => id, DJ.trim_le_self _ _\u27e9, rfl\u27e9\n\ntheorem Statement.self_le_untrim' (s : Statement) (P) : s \u2264 s.untrim' P :=\n  \u27e8\u27e8fun _ => id, DJ.self_le_untrim _ _\u27e9, rfl\u27e9\ntheorem Statement.self_le_untrim (s : Statement) : s \u2264 s.untrim := s.self_le_untrim' _\n\ntheorem Statement.trim.mono {s\u2081 s\u2082 : Statement} (h : s\u2081 \u2264 s\u2082) : s\u2081.trim \u2264 s\u2082.trim :=\n  \u27e8\u27e8h.1.1, DJ.trim.mono h.1.2 (Statement.vars.mono h)\u27e9, h.2\u27e9\n\ntheorem Statement.untrim'.mono {s\u2081 s\u2082 : Statement} {P Q}\n  (H : \u2200 x, Q x \u2192 P x) (h : s\u2081 \u2264 s\u2082) : s\u2081.untrim' P \u2264 s\u2082.untrim' Q :=\n  \u27e8\u27e8h.1.1, DJ.untrim.mono h.1.2 H\u27e9, h.2\u27e9\ntheorem Statement.untrim.mono {s\u2081 s\u2082 : Statement}\n  (H : s\u2081.ctx.hyps = s\u2082.ctx.hyps) (h : s\u2081 \u2264 s\u2082) : s\u2081.untrim \u2264 s\u2082.untrim :=\n  Statement.untrim'.mono (Statement.vars.mono' (by rw [H]; exact fun _ => id) h.2.symm) h\n\ntheorem Statement.trim_vars (s : Statement) : s.trim.vars = s.vars := rfl\ntheorem Statement.untrim'_vars (s : Statement) (P) : (s.untrim' P).vars = s.vars := rfl\ntheorem Statement.untrim_vars (s : Statement) : s.untrim.vars = s.vars := rfl\n\ntheorem Statement.trim_untrim (s : Statement) : s.untrim.trim = s.trim := by\n  simp only [trim, untrim_vars]; simp only [untrim, untrim', DJ.trim_untrim]; rfl\n\ntheorem Statement.untrim_trim (s : Statement) : s.trim.untrim = s.untrim := by\n  simp only [untrim, untrim', trim_vars]; simp only [trim, DJ.untrim_trim]; rfl\n\ntheorem Statement.trim_le {s\u2081 s\u2082 : Statement} (e : s\u2081.vars = s\u2082.vars) :\n  s\u2081.trim.ctx \u2264 s\u2082.ctx \u2194 s\u2081.ctx \u2264 s\u2082.untrim.ctx :=\n  \u27e8fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2081, DJ.trim_le.1 $ by rw [\u2190 e]; exact h\u2082\u27e9,\n   fun \u27e8h\u2081, h\u2082\u27e9 => \u27e8h\u2081, DJ.trim_le.2 $ by rw [e]; exact h\u2082\u27e9\u27e9\n\ndef Statement.trimmed (s : Statement) : Prop := s.ctx.dj.trimmed fun v => v \u2208 s.vars\n\ntheorem Statement.trim.trimmed (s : Statement) : s.trim.trimmed := DJ.trim.trimmed _ _\n\ntheorem Statement.trimmed.trim_eq : {s : Statement} \u2192 s.trimmed \u2192 s.trim = s\n| \u27e8\u27e8a, b\u27e9, c\u27e9, h => by simp only [trim]; rw [DJ.trimmed.trim_eq h]\n\ninductive Provable (axs : Statement \u2192 Prop) (\u0393 : Context) : Formula \u2192 Prop\n| hyp (h) : h \u2208 \u0393.hyps \u2192 Provable axs \u0393 h\n| var (v:VR) : Provable axs \u0393 v\n| ax (\u03c3) {ax} : axs ax \u2192 ax.ctx.dj.subst \u03c3 \u0393.dj \u2192\n  (\u2200 h, h \u2208 ax.ctx.hyps \u2228 (\u2203 v:VR, h = v) \u2192 Provable axs \u0393 (h.subst \u03c3)) \u2192\n  Provable axs \u0393 (ax.fmla.subst \u03c3)\n\ntheorem Provable.mono {axs\u2081 axs\u2082} (haxs : \u2200 a, axs\u2081 a \u2192 axs\u2082 a)\n  {\u0393\u2081 \u0393\u2082} (h\u0393 : \u0393\u2081 \u2264 \u0393\u2082) {e} (pr : Provable axs\u2081 \u0393\u2081 e) : Provable axs\u2082 \u0393\u2082 e := by\n  induction pr with\n  | hyp e h => exact hyp e (h\u0393.1 _ h)\n  | var v => exact var v\n  | ax \u03c3 ha h\u2081 h\u2082 IH =>\n    exact ax \u03c3 (haxs _ ha) (h\u2081.mono (DJ.refl _) h\u0393.2) fun e h => IH _ h\n\ndef Statement.provable' (axs : Statement \u2192 Prop) (s : Statement) : Prop :=\n  Provable axs s.ctx s.fmla\n\ntheorem Statement.provable'.mono {axs\u2081 axs\u2082} (haxs : \u2200 a, axs\u2081 a \u2192 axs\u2082 a) :\n  {s\u2081 s\u2082 : Statement} \u2192 s\u2081 \u2264 s\u2082 \u2192 s\u2081.provable' axs\u2081 \u2192 s\u2082.provable' axs\u2082\n| \u27e8\u0393\u2081, e\u27e9, \u27e8\u0393\u2082, _\u27e9, \u27e8h\u0393, rfl\u27e9 => Provable.mono haxs h\u0393\n\ndef Statement.provable (axs : Statement \u2192 Prop) (s : Statement) : Prop :=\n  s.untrim.provable' axs\n\n-- theorem Statement.provable.mono {axs\u2081 axs\u2082} (haxs : \u2200 a, axs\u2081 a \u2192 axs\u2082 a) :\n--   {s\u2081 s\u2082 : Statement} \u2192 s\u2081 \u2264 s\u2082 \u2192 s\u2081.provable axs\u2081 \u2192 s\u2082.provable axs\u2082\n-- | s\u2081, s\u2082, h, hs, pr =>\n--   Statement.provable'.mono haxs (untrim'.mono (fun _ => id) hs) $\n--   Statement.provable'.mono (fun _ => id) _ pr\n\ntheorem Statement.provable'.of {axs} {s : Statement} (h : s.provable' axs) : s.provable axs :=\n  h.mono (fun _ => id) (self_le_untrim _)\n\ntheorem Statement.provable.trim {axs} {s : Statement} : s.trim.provable axs \u2194 s.provable axs :=\n  by simp only [provable, untrim_trim]; exact Iff.rfl\n\ntheorem Provable.ax_self (axs : Statement \u2192 Prop) {ax} (H : axs ax) : ax.provable' axs := by\n  have _ from Provable.ax (\u0393 := ax.ctx) VR.expr H ?disj ?hyp\n  rw [Formula.subst_id] at this; exact this\n  case disj =>\n    intro a b h a' b' h\u2081 h\u2082\n    match a', b', h\u2081, h\u2082 with | _, _, Or.inl rfl, Or.inl rfl => ?_\n    exact h\n  case hyp =>\n    intro fmla h\n    match fmla, h with\n    | fmla, Or.inl h => rw [Formula.subst_id]; exact Provable.hyp _ h\n    | _, Or.inr \u27e8v, rfl\u27e9 => exact Provable.var v\n\ntheorem Provable.trans' {axs \u0393} (\u03c3) {\u0393' fmla} (pr : Provable axs \u0393' fmla)\n  (dj : \u0393'.dj.subst \u03c3 \u0393.dj)\n  (hh : \u2200 h, h \u2208 \u0393'.hyps \u2228 (\u2203 v:VR, h = v) \u2192 Provable axs \u0393 (h.subst \u03c3)) :\n  Provable axs \u0393 (fmla.subst \u03c3) := by\n  induction pr with\n  | hyp f h => exact hh _ (Or.inl h)\n  | var v => exact hh _ (Or.inr \u27e8v, rfl\u27e9)\n  | @ax \u03c3' a ha dj' hh' IH =>\n    rw [\u2190 Formula.subst_tr]\n    apply ax (subst.trans \u03c3' \u03c3) ha\n    focus\n      intros a b ab c d hc hd\n      let \u27e8e, ea, ce\u27e9 := Expr.mem_subst hc\n      let \u27e8f, fb, df\u27e9 := Expr.mem_subst hd\n      refine dj _ _ ?_ _ _ ce df\n      exact dj' _ _ ab _ _ ea fb\n    focus { intros f; rw [Formula.subst_tr]; refine IH _ }\n\ntheorem Provable.trans'' {axs \u0393 \u03c3} (s : Statement) : s.provable' axs \u2192\n  s.ctx.dj.subst \u03c3 \u0393.dj \u2192\n  (\u2200 h, h \u2208 s.ctx.hyps \u2228 (\u2203 v:VR, h = v) \u2192 Provable axs \u0393 (h.subst \u03c3)) \u2192\n  Provable axs \u0393 (s.fmla.subst \u03c3) :=\n  Provable.trans' (axs := axs) \u03c3\n\ndef subst_of : List (VR \u00d7 Expr) \u2192 VR \u2192 Expr\n| [], v => v\n| (a, e)::l, v => if a = v then e else subst_of l v\n\nclass Subst (\u03c3 : VR \u2192 Expr) (e : Expr) (e' : outParam Expr) := (out : e.subst \u03c3 = e')\n\ninstance [Subst \u03c3 e\u2081 e\u2081'] [Subst \u03c3 e\u2082 e\u2082'] : Subst \u03c3 (e\u2081 ++ e\u2082) (e\u2081' ++ e\u2082') :=\n  \u27e8by rw [Expr.subst_append, Subst.out, Subst.out]\u27e9\n\ninstance (s : String) : Subst \u03c3 s s := \u27e8rfl\u27e9\n\ninstance (s : String) [Subst \u03c3 e e'] : Subst \u03c3 (s ++ e) (s ++ e') :=\n  inferInstanceAs (Subst \u03c3 (Expr.sn _ ++ e) _)\ninstance (s : String) [Subst \u03c3 e e'] : Subst \u03c3 (e ++ s) (e' ++ s) :=\n  inferInstanceAs (Subst \u03c3 (e ++ Expr.sn _) _)\n\ndef subst.ok (axs \u0393) (\u03c3 : VR \u2192 Expr) := \u2200 v, Provable axs \u0393 (v.type, \u03c3 v)\n\ntheorem subst.ok.nil {axs \u0393} : subst.ok axs \u0393 (subst_of []) := Provable.var\ntheorem subst.ok.cons {axs \u0393 e \u03c3} (x) (h\u2081 : Provable axs \u0393 (x.type, e))\n  (h\u2082 : subst.ok axs \u0393 (subst_of \u03c3)) : subst.ok axs \u0393 (subst_of ((x, e)::\u03c3)) := by\n  intro v\n  simp only [subst_of]\n  cases Decidable.em (x = v) with simp [h]\n  | inl h => cases h; exact h\u2081\n  | inr h => exact h\u2082 v\n\ntheorem Provable.thm {axs} {\u0393 : Context}\n  {\u03c3 : VR \u2192 Expr} {dj hyps c s} (pr : Provable axs (Context.mk' dj hyps) (c, s))\n  (hv : subst.ok axs \u0393 \u03c3)\n  (dj : (DJ.mk' dj).subst \u03c3 \u0393.dj)\n  (hh : \u2200 h, h \u2208 hyps \u2192 Provable axs \u0393 (h.subst \u03c3))\n  {e} [inst : Subst \u03c3 s e] : Provable axs \u0393 (c, e) := by\n  rw [\u2190 inst.out]\n  exact Metamath.Provable.trans' \u03c3 pr dj fun\n    | f, Or.inl h => hh _ h\n    | _, Or.inr \u27e8v, rfl\u27e9 =>\n      show Provable axs \u0393 (v.type, \u03c3 v ++ show Expr from []) by\n      rw [List.append_nil]; exact hv v\n\ntheorem DJ_nil {\u03c3 dj'} : (DJ.mk' []).subst \u03c3 dj' | _, _, h => nomatch h\ntheorem DJ_cons {a b l \u03c3 dj'}\n  (h\u2081 : (\u03c3 a).disjoint dj' (\u03c3 b))\n  (h\u2082 : (DJ.mk' l).subst \u03c3 dj') : (DJ.mk' ((a, b) :: l)).subst \u03c3 dj'\n| _, _, \u27e8_, Or.inl (Or.inl rfl)\u27e9 => h\u2081\n| _, _, \u27e8_, Or.inr (Or.inl rfl)\u27e9 => fun x y hx hy => dj'.symm (h\u2081 y x hy hx)\n| _, _, \u27e8h, Or.inl (Or.inr h')\u27e9 => h\u2082 _ _ \u27e8h, Or.inl h'\u27e9\n| _, _, \u27e8h, Or.inr (Or.inr h')\u27e9 => h\u2082 _ _ \u27e8h, Or.inr h'\u27e9\n\ntheorem HH_nil {axs \u0393 \u03c3} : \u2200 h:Formula, h \u2208 [] \u2192 Provable axs \u0393 (h.subst \u03c3)\n| _, h => nomatch h\n\ntheorem HH_cons {axs \u0393 \u03c3 c f hyps}\n  {e} [Subst \u03c3 f e] (h\u2081 : Provable axs \u0393 (c, e))\n  (h\u2082 : \u2200 h:Formula, h \u2208 hyps \u2192 Provable axs \u0393 (h.subst \u03c3)) :\n  \u2200 h:Formula, h \u2208 (c, f)::hyps \u2192 Provable axs \u0393 (h.subst \u03c3)\n| _, Or.inl rfl => by rw [\u2190 @Subst.out \u03c3 f e] at h\u2081; exact h\u2081\n| _, Or.inr h => h\u2082 _ h\n\nclass Typed (axs : outParam _) (c : outParam CN) (e : Expr) :=\n  type (\u0393) : Provable axs \u0393 (c, e)\n\ndef Expr.ty (e) {axs c} [Typed axs c e] {\u0393} : Provable axs \u0393 (c, e) := Typed.type \u0393\n\n-- This is a by hand translation of demo0.mm, ideally the tactic will write this\n\nnamespace Demo\n\ndef ze : Expr := \"0\"\ninstance : Subst \u03c3 ze ze := inferInstanceAs (Subst \u03c3 \"0\" _)\n\ndef pl (t r : Expr) : Expr := \"(\" ++ t ++ \"+\" ++ r ++ \")\"\ninstance [Subst \u03c3 t t'] [Subst \u03c3 r r'] : Subst \u03c3 (pl t r) (pl t' r') :=\n  inferInstanceAs (Subst \u03c3 (_++_) _)\n\ndef eq (t r : Expr) : Expr := t ++ \"=\" ++ r\ninstance [Subst \u03c3 t t'] [Subst \u03c3 r r'] : Subst \u03c3 (eq t r) (eq t' r') :=\n  inferInstanceAs (Subst \u03c3 (_++_) _)\n\ndef im (P Q : Expr) : Expr := \"(\" ++ P ++ \"->\" ++ Q ++ \")\"\ninstance {P Q P' Q'} [Subst \u03c3 P P'] [Subst \u03c3 Q Q'] : Subst \u03c3 (im P Q) (im P' Q') :=\n  inferInstanceAs (Subst \u03c3 (_++_) _)\n\ndef al (x P : Expr) : Expr := \"A.\" ++ x ++ P\ninstance {x P x' P'} [Subst \u03c3 x x'] [Subst \u03c3 P P'] : Subst \u03c3 (al x P) (al x' P') :=\n  inferInstanceAs (Subst \u03c3 (_++_) _)\n\ndef vt : VR := \u27e8\"term\", 0\u27e9\ndef vr : VR := \u27e8\"term\", 1\u27e9\ndef vs : VR := \u27e8\"term\", 2\u27e9\ndef vP : VR := \u27e8\"wff\", 0\u27e9\ndef vQ : VR := \u27e8\"wff\", 1\u27e9\ndef vx : VR := \u27e8\"set\", 0\u27e9\n\ndef axs (s : Statement) : Prop := s \u2208 [\n  \u27e8Context.mk' [] [], (\"term\", ze)\u27e9,\n  \u27e8Context.mk' [] [], (\"term\", pl vt vr)\u27e9,\n  \u27e8Context.mk' [] [], (\"wff\", eq vt vr)\u27e9,\n  \u27e8Context.mk' [] [], (\"wff\", im vP vQ)\u27e9,\n  \u27e8Context.mk' [] [], (\"wff\", al vx vP)\u27e9,\n  \u27e8Context.mk' [] [], (\"|-\", im (eq vt vr) (im (eq vt vs) (eq vr vs)))\u27e9,\n  \u27e8Context.mk' [] [], (\"|-\", eq (pl vt ze) vt)\u27e9,\n  \u27e8Context.mk' [] [(\"|-\", vP), (\"|-\", im vP vQ)], (\"|-\", vQ)\u27e9,\n  \u27e8Context.mk' [(vx, vP)] [], (\"|-\", im vP (al vx vP))\u27e9\n]\n\nabbrev Provable := Metamath.Provable axs\nabbrev Typed := Metamath.Typed axs\n\ninstance tze : Typed \"term\" ze :=\n\u27e8fun \u0393 =>\n  (Provable.ax_self axs (Or.inl rfl)).thm subst.ok.nil DJ_nil HH_nil\u27e9\n\ninstance tpl {t r} [Typed \"term\" t] [Typed \"term\" r] :\n  Typed \"term\" (pl t r) :=\n\u27e8fun \u0393 =>\n  have Subst (subst_of [(vt, t), (vr, r)]) vt t from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vt, t), (vr, r)]) vr r from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vt t.ty $ subst.ok.cons vr r.ty subst.ok.nil)\n    DJ_nil HH_nil\u27e9\n\ninstance weq {t r} [Typed \"term\" t] [Typed \"term\" r] :\n  Typed \"wff\" (eq t r) :=\n\u27e8fun \u0393 =>\n  have Subst (subst_of [(vt, t), (vr, r)]) vt t from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vt, t), (vr, r)]) vr r from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vt t.ty $ subst.ok.cons vr r.ty subst.ok.nil)\n    DJ_nil HH_nil\u27e9\n\ninstance wim {P Q} [Typed \"wff\" P] [Typed \"wff\" Q] :\n  Typed \"wff\" (im P Q) :=\n\u27e8fun \u0393 =>\n  have Subst (subst_of [(vP, P), (vQ, Q)]) vP P from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vP, P), (vQ, Q)]) vQ Q from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vP P.ty $ subst.ok.cons vQ Q.ty subst.ok.nil)\n    DJ_nil HH_nil\u27e9\n\ninstance wal {x P} [Typed \"set\" x] [Typed \"wff\" P] :\n  Typed \"wff\" (al x P) :=\n\u27e8fun \u0393 =>\n  have Subst (subst_of [(vx, x), (vP, P)]) vx x from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vx, x), (vP, P)]) vP P from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vx x.ty $ subst.ok.cons vP P.ty subst.ok.nil)\n    DJ_nil HH_nil\u27e9\n\ntheorem a1 {\u0393 t r s} [Typed \"term\" t] [Typed \"term\" r] [Typed \"term\" s] :\n  Provable \u0393 (\"|-\", im (eq t r) (im (eq t s) (eq r s))) :=\n  have Subst (subst_of [(vt, t), (vr, r), (vs, s)]) vt t from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vt, t), (vr, r), (vs, s)]) vr r from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vt, t), (vr, r), (vs, s)]) vs s from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vt t.ty $ subst.ok.cons vr r.ty $ subst.ok.cons vs s.ty subst.ok.nil)\n    DJ_nil HH_nil\n\ntheorem a2 {\u0393 t} [Typed \"term\" t] :\n  Provable \u0393 (\"|-\", eq (pl t ze) t) :=\n  have Subst (subst_of [(vt, t)]) vt t from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vt t.ty subst.ok.nil)\n    DJ_nil HH_nil\n\ntheorem mp {\u0393 P Q} [Typed \"wff\" P] [Typed \"wff\" Q]\n  (min : Provable \u0393 (\"|-\", P))\n  (maj : Provable \u0393 (\"|-\", im P Q)) :\n  Provable \u0393 (\"|-\", Q) :=\n  have Subst (subst_of [(vP, P), (vQ, Q)]) vP P from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vP, P), (vQ, Q)]) vQ Q from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vP P.ty $ subst.ok.cons vQ Q.ty subst.ok.nil)\n    DJ_nil (HH_cons min $ HH_cons maj HH_nil)\n\ntheorem ax5 {\u0393 x P} [Typed \"set\" x] [Typed \"wff\" P]\n  (xp : x.disjoint \u0393.dj P) :\n  Provable \u0393 (\"|-\", im P (al x P)) :=\n  have Subst (subst_of [(vx, x), (vP, P)]) vx x from \u27e8List.append_nil _\u27e9\n  have Subst (subst_of [(vx, x), (vP, P)]) vP P from \u27e8List.append_nil _\u27e9\n  (Provable.ax_self axs (Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inr $ Or.inl rfl)).thm\n    (subst.ok.cons vx x.ty $ subst.ok.cons vP P.ty subst.ok.nil)\n    (DJ_cons xp DJ_nil) HH_nil\n\ntheorem th1 {\u0393 t} [Typed \"term\" t] :\n  Provable \u0393 (\"|-\", eq t t) := mp a2 (mp a2 a1)\n\nend Demo\nend Metamath\n", "meta": {"author": "digama0", "repo": "mm-lean4", "sha": "05251198cc1bf2193740b6d3ece33f9f740ec478", "save_path": "github-repos/lean/digama0-mm-lean4", "path": "github-repos/lean/digama0-mm-lean4/mm-lean4-05251198cc1bf2193740b6d3ece33f9f740ec478/Metamath/Translate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.37824561029018067}}
{"text": "import GMLInit.Data.Array.Basic\nimport GMLInit.Data.Array.KeyedArray\n\nstructure AssocArray (\u03b1 \u03b2 : Type _) [BEq \u03b1] extends KeyedArray (\u03b1 \u00d7 \u03b2) Prod.fst\n\nnamespace AssocArray\nvariable {\u03b1 \u03b2} [BEq \u03b1] [EquivBEq \u03b1] (a : AssocArray \u03b1 \u03b2)\n\n@[inline] def find? (key : \u03b1) : Option \u03b2 :=\n  match a.toKeyedArray.find? key with\n  | some (_, value) => some value\n  | none => none\n\n@[inline] def insert (key : \u03b1) (value : \u03b2) : AssocArray \u03b1 \u03b2 where\n  toKeyedArray := a.toKeyedArray.insert (key, value)\n\ntheorem find?_insert (key : \u03b1) (value : \u03b2) : (a.insert key value).find? key = some value := by\n  unfold find? insert; rw [a.toKeyedArray.find?_insert (key, value)]\n\ntheorem find?_insert_bne {key : \u03b1} {value : \u03b2} {k : \u03b1} : k != key \u2192 (a.insert key value).find? k = a.find? k := by\n  intro h; unfold find? insert; rw [a.toKeyedArray.find?_insert_bne h]\n\n@[inline] def erase (key : \u03b1) : AssocArray \u03b1 \u03b2 where\n  toKeyedArray := a.toKeyedArray.erase key\n\ntheorem find?_erase (key : \u03b1) : (a.erase key).find? key = none := by\n  unfold find? erase; rw [a.toKeyedArray.find?_erase key]\n\ntheorem find?_erase_bne {key : \u03b1} {k : \u03b1} : k != key \u2192 (a.erase key).find? k = a.find? k := by\n  intro h; unfold find? erase; rw [a.toKeyedArray.find?_erase_bne h]\n\nend AssocArray\n\n", "meta": {"author": "fgdorais", "repo": "GMLInit", "sha": "a295111627ac907ebc6a86f906dd9b4d69b338d8", "save_path": "github-repos/lean/fgdorais-GMLInit", "path": "github-repos/lean/fgdorais-GMLInit/GMLInit-a295111627ac907ebc6a86f906dd9b4d69b338d8/GMLInit/Data/Array/AssocArray.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.37824560177845257}}
{"text": "-- Boilerplate:\nimport data.nat.basic\nimport group_theory.congruence\nnamespace examples\n\nsection unbundled\n\n/-\nThis example concerns product type instances of an unbundled class such as the following modification to the `comm_monoid` class:\n-/\nclass semigroup (G : Type*) [has_mul G] := (mul_assoc : \u2200 (x y z : G), (x * y) * z = x * (y * z))\nclass mul_one_class (M : Type*) [has_one M] [has_mul M].\nclass comm_semigroup (G : Type*) [has_mul G] [semigroup G].\nclass monoid (M : Type*) [has_mul M] [has_one M] [semigroup M] [mul_one_class M].\nclass comm_monoid (M : Type*) [has_mul M] [has_one M] [semigroup M] [mul_one_class M] [monoid M] [comm_semigroup M].\n\n/-\nProviding an instance for the natural numbers is straightforward, although it now involves instantiating each step in the hierarchy separately:\n-/\ninstance : semigroup \u2115 := \u27e8mul_assoc\u27e9\ninstance : mul_one_class \u2115 := \u27e8\u27e9\ninstance : comm_semigroup \u2115 := \u27e8\u27e9\ninstance : monoid \u2115 := \u27e8\u27e9\ninstance : comm_monoid \u2115 := \u27e8\u27e9\n/-\nWhen we want to instantiate the commutative monoid structure on the product of two commutative monoids, we see that the length of types starts to grow noticeably:\n-/\nvariables {G H M N : Type*}\ninstance prod.has_mul [has_mul G] [has_mul H] : has_mul (G \u00d7 H) :=\n{ mul := \u03bb a b, (a.1 * b.1, a.2 * b.2) }\n@[simp] lemma prod.mul_def [has_mul G] [has_mul H]\n  (a b : G) (c d : H) : (a, c) * (b, d) = (a * b, c * d) := rfl\n\ninstance prod.has_one [has_one G] [has_one H] : has_one (G \u00d7 H) :=\n{ one := (1, 1) }\n@[simp] lemma prod.one_def [has_one G] [has_one H] :\n  (1 : G \u00d7 H) = (1, 1) := rfl\n\ninstance prod.semigroup [has_mul G] [has_mul H]\n  [semigroup G] [semigroup H] : semigroup (G \u00d7 H) :=\n\u27e8\u03bb x y z, by { cases x, cases y, cases z, simp only [prod.mul_def, semigroup.mul_assoc] }\u27e9\ninstance prod.comm_semigroup [has_mul G] [has_mul H]\n  [semigroup G] [semigroup H] [comm_semigroup G] [comm_semigroup H] :\n  comm_semigroup (G \u00d7 H) :=\n\u27e8\u27e9\ninstance prod.mul_one_class [has_mul G] [has_mul H] [has_one G] [has_one H]\n  [mul_one_class G] [mul_one_class H] : mul_one_class (G \u00d7 H) :=\n\u27e8\u27e9\ninstance prod.monoid [has_mul M] [has_mul N] [has_one M] [has_one N]\n  [semigroup M] [semigroup N] [mul_one_class M] [mul_one_class N] :\n  monoid (M \u00d7 N) :=\n\u27e8\u27e9\ninstance prod.comm_monoid\n  [has_one M] [has_one N] [has_mul M] [has_mul N]\n  [semigroup M] [semigroup N] [mul_one_class M] [mul_one_class N]\n  [monoid M] [monoid N] [comm_semigroup M] [comm_semigroup N]\n  [comm_monoid M] [comm_monoid N] :\n  comm_monoid (M \u00d7 N) :=\n\u27e8\u27e9\n/-\nThe linear growth in the types translates to an exponential growth in the term size of concrete instances,\nsince each instance parameter implicit in `comm_monoid (\u2115 \u00d7 \u22ef \u00d7 \u2115)` is filled with a term that has itself the same number of instance arguments.\n-/\n\nset_option pp.implicit true\n#check (by apply_instance : comm_monoid \u2115)\n#check (by apply_instance : comm_monoid (\u2115 \u00d7 \u2115))\n#check (by apply_instance : comm_monoid (\u2115 \u00d7 \u2115 \u00d7 \u2115))\n#check (by apply_instance : comm_monoid (\u2115 \u00d7 \u2115 \u00d7 \u2115 \u00d7 \u2115))\n#check (by apply_instance : comm_monoid (\u2115 \u00d7 \u2115 \u00d7 \u2115 \u00d7 \u2115 \u00d7 \u2115))\n-- and so on\n\nend unbundled\n\nsection fails_quickly\n\n/-\nThe `fails_quickly` linter can also detect timeouts caused by looping or diverging synthesis,\nfor example the loop `nonempty \u2192 has_bot \u2192 nonempty` in the following code:\n-/\n\n-- Adapted from `order/bounded_order.lean:54`\n-- `has_bot.bot` is notation for the minimum element of `\u03b1`\nclass has_bot (\u03b1 : Type*) := (bot : \u03b1)\n\n-- Adapted from `order/bounded_order.lean:60`\ninstance has_bot_nonempty (\u03b1 : Type*) [has_bot \u03b1] : nonempty \u03b1 :=\n\u27e8has_bot.bot\u27e9\n\n-- The natural numbers are well-ordered so each nonempty subtype has a bottom element.\n-- Adapted from `data/nat/basic.lean:106`\ninstance nat.subtype.has_bot (s : set \u2115) [decidable_pred (\u2208 s)] [h : nonempty s] :\n  has_bot s :=\n{ bot := \u27e8nat.find (nonempty_subtype.1 h), nat.find_spec (nonempty_subtype.1 h)\u27e9 }\n\n#lint only fails_quickly\n\nend fails_quickly\n\nsection priority\n\n-- Adapted from `group_theory/congruence.lean:209`\n@[priority 500] -- Reduce the priority from 1000 to 500 since it's slow to apply.\ninstance con.quotient.decidable_eq {M : Type*} [has_mul M] (c : con M)\n  [d : \u2200 (a b : M), decidable (c a b)] : decidable_eq (con.quotient c) :=\n@quotient.decidable_eq M c.to_setoid d\n\nend priority\n\nend examples\n", "meta": {"author": "lean-forward", "repo": "mathlib-classes", "sha": "d2271c7cb5f11083cf221e200c2a9eb74c683eaa", "save_path": "github-repos/lean/lean-forward-mathlib-classes", "path": "github-repos/lean/lean-forward-mathlib-classes/mathlib-classes-d2271c7cb5f11083cf221e200c2a9eb74c683eaa/src/section10_performance.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.3782456014448702}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.comma\nimport Mathlib.PostPort\n\nuniverses v u l u\u2081 u\u2082 v\u2081 v\u2082 \n\nnamespace Mathlib\n\n/-!\n# The category of arrows\n\nThe category of arrows, with morphisms commutative squares.\nWe set this up as a specialization of the comma category `comma L R`,\nwhere `L` and `R` are both the identity functor.\n\nWe also define the typeclass `has_lift`, representing a choice of a lift\nof a commutative square (that is, a diagonal morphism making the two triangles commute).\n\n## Tags\n\ncomma, arrow\n-/\n\nnamespace category_theory\n\n\n/-- The arrow category of `T` has as objects all morphisms in `T` and as morphisms commutative\n     squares in `T`. -/\ndef arrow (T : Type u) [category T] := comma \ud835\udfed \ud835\udfed\n\n-- Satisfying the inhabited linter\n\nprotected instance arrow.inhabited (T : Type u) [category T] [Inhabited T] : Inhabited (arrow T) :=\n  { default := (fun (this : comma \ud835\udfed \ud835\udfed) => this) Inhabited.default }\n\nnamespace arrow\n\n\n@[simp] theorem id_left {T : Type u} [category T] (f : arrow T) : comma_morphism.left \ud835\udfd9 = \ud835\udfd9 := rfl\n\n@[simp] theorem id_right {T : Type u} [category T] (f : arrow T) : comma_morphism.right \ud835\udfd9 = \ud835\udfd9 := rfl\n\n/-- An object in the arrow category is simply a morphism in `T`. -/\n@[simp] theorem mk_left {T : Type u} [category T] {X : T} {Y : T} (f : X \u27f6 Y) :\n    comma.left (mk f) = X :=\n  Eq.refl (comma.left (mk f))\n\n/-- A morphism in the arrow category is a commutative square connecting two objects of the arrow\n    category. -/\ndef hom_mk {T : Type u} [category T] {f : arrow T} {g : arrow T} {u : comma.left f \u27f6 comma.left g}\n    {v : comma.right f \u27f6 comma.right g} (w : u \u226b comma.hom g = comma.hom f \u226b v) : f \u27f6 g :=\n  comma_morphism.mk\n\n/-- We can also build a morphism in the arrow category out of any commutative square in `T`. -/\n@[simp] theorem hom_mk'_right {T : Type u} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {P : T} {Q : T}\n    {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (w : u \u226b g = f \u226b v) :\n    comma_morphism.right (hom_mk' w) = v :=\n  Eq.refl (comma_morphism.right (hom_mk' w))\n\n@[simp] theorem w_assoc {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g)\n    {X' :\n      autoParam T\n        (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n          (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])}\n    (f' : functor.obj \ud835\udfed (comma.right g) \u27f6 X') :\n    comma_morphism.left sq \u226b comma.hom g \u226b f' = comma.hom f \u226b comma_morphism.right sq \u226b f' :=\n  sorry\n\n/-- A lift of a commutative square is a diagonal morphism making the two triangles commute. -/\nstructure lift_struct {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) where\n  lift : comma.right f \u27f6 comma.left g\n  fac_left : comma.hom f \u226b lift = comma_morphism.left sq\n  fac_right : lift \u226b comma.hom g = comma_morphism.right sq\n\nprotected instance lift_struct_inhabited {T : Type u} [category T] {X : T} :\n    Inhabited (lift_struct \ud835\udfd9) :=\n  { default := lift_struct.mk \ud835\udfd9 sorry sorry }\n\n/-- `has_lift sq` says that there is some `lift_struct sq`, i.e., that it is possible to find a\n    diagonal morphism making the two triangles commute. -/\nclass has_lift {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) where\n  mk' :: (exists_lift : Nonempty (lift_struct sq))\n\ntheorem has_lift.mk {T : Type u} [category T] {f : arrow T} {g : arrow T} {sq : f \u27f6 g}\n    (s : lift_struct sq) : has_lift sq :=\n  has_lift.mk' (Nonempty.intro s)\n\n@[simp] theorem lift_struct.fac_right_assoc {T : Type u} [category T] {f : arrow T} {g : arrow T}\n    {sq : f \u27f6 g} (c : lift_struct sq)\n    {X' :\n      autoParam T\n        (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n          (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])}\n    (f' : functor.obj \ud835\udfed (comma.right g) \u27f6 X') :\n    lift_struct.lift c \u226b comma.hom g \u226b f' = comma_morphism.right sq \u226b f' :=\n  sorry\n\n/-- Given `has_lift sq`, obtain a lift. -/\ndef has_lift.struct {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g)\n    [has_lift sq] : lift_struct sq :=\n  Classical.choice has_lift.exists_lift\n\n/-- If there is a lift of a commutative square `sq`, we can access it by saying `lift sq`. -/\ndef lift {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g) [has_lift sq] :\n    comma.right f \u27f6 comma.left g :=\n  lift_struct.lift (has_lift.struct sq)\n\ntheorem lift.fac_left {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g)\n    [has_lift sq] : comma.hom f \u226b lift sq = comma_morphism.left sq :=\n  sorry\n\ntheorem lift.fac_right {T : Type u} [category T] {f : arrow T} {g : arrow T} (sq : f \u27f6 g)\n    [has_lift sq] : lift sq \u226b comma.hom g = comma_morphism.right sq :=\n  sorry\n\n@[simp] theorem lift_mk'_left {T : Type u} [category T] {X : T} {Y : T} {P : T} {Q : T} {f : X \u27f6 Y}\n    {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (h : u \u226b g = f \u226b v) [has_lift (hom_mk' h)] :\n    f \u226b lift (hom_mk' h) = u :=\n  sorry\n\n@[simp] theorem lift_mk'_right_assoc {T : Type u} [category T] {X : T} {Y : T} {P : T} {Q : T}\n    {f : X \u27f6 Y} {g : P \u27f6 Q} {u : X \u27f6 P} {v : Y \u27f6 Q} (h : u \u226b g = f \u226b v) [has_lift (hom_mk' h)]\n    {X' :\n      autoParam T\n        (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n          (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])}\n    (f' : Q \u27f6 X') : lift (hom_mk' h) \u226b g \u226b f' = v \u226b f' :=\n  sorry\n\nprotected instance subsingleton_lift_struct_of_epi {T : Type u} [category T] {f : arrow T}\n    {g : arrow T} (sq : f \u27f6 g) [epi (comma.hom f)] : subsingleton (lift_struct sq) :=\n  subsingleton.intro\n    fun (a b : lift_struct sq) =>\n      lift_struct.ext a b\n        (iff.mp (cancel_epi (comma.hom f))\n          (eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : functor.obj \ud835\udfed (comma.left f) \u27f6 comma.left g) (e_1 : a = a_1)\n                    (\u1fb0 \u1fb0_1 : functor.obj \ud835\udfed (comma.left f) \u27f6 comma.left g) (e_2 : \u1fb0 = \u1fb0_1) =>\n                    congr (congr_arg Eq e_1) e_2)\n                  (comma.hom f \u226b lift_struct.lift a) (comma_morphism.left sq)\n                  (lift_struct.fac_left a) (comma.hom f \u226b lift_struct.lift b)\n                  (comma_morphism.left sq) (lift_struct.fac_left b))\n                (propext (eq_self_iff_true (comma_morphism.left sq)))))\n            trivial))\n\nprotected instance subsingleton_lift_struct_of_mono {T : Type u} [category T] {f : arrow T}\n    {g : arrow T} (sq : f \u27f6 g) [mono (comma.hom g)] : subsingleton (lift_struct sq) :=\n  subsingleton.intro\n    fun (a b : lift_struct sq) =>\n      lift_struct.ext a b\n        (iff.mp (cancel_mono (comma.hom g))\n          (eq.mpr\n            (id\n              (Eq.trans\n                ((fun (a a_1 : comma.right f \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_1 : a = a_1)\n                    (\u1fb0 \u1fb0_1 : comma.right f \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_2 : \u1fb0 = \u1fb0_1) =>\n                    congr (congr_arg Eq e_1) e_2)\n                  (lift_struct.lift a \u226b comma.hom g) (comma_morphism.right sq)\n                  (lift_struct.fac_right a) (lift_struct.lift b \u226b comma.hom g)\n                  (comma_morphism.right sq) (lift_struct.fac_right b))\n                (propext (eq_self_iff_true (comma_morphism.right sq)))))\n            trivial))\n\nend arrow\n\n\nnamespace functor\n\n\n/-- A functor `C \u2964 D` induces a functor between the corresponding arrow categories. -/\n@[simp] theorem map_arrow_map_right {C : Type u\u2081} [category C] {D : Type u\u2082} [category D]\n    (F : C \u2964 D) (a : arrow C) (b : arrow C) (f : a \u27f6 b) :\n    comma_morphism.right (map (map_arrow F) f) = map F (comma_morphism.right f) :=\n  Eq.refl (comma_morphism.right (map (map_arrow F) f))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/arrow_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3781986707064467}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nParallel computation of a computable sequence of computations by\na diagonal enumeration.\nThe important theorems of this operation are proven as\nterminates_parallel and exists_of_mem_parallel.\n(This operation is nondeterministic in the sense that it does not\nhonor sequence equivalence (irrelevance of computation time).)\n-/\nimport data.seq.wseq\nuniverses u v\n\nnamespace computation\nopen wseq\nvariables {\u03b1 : Type u} {\u03b2 : Type v}\n\ndef parallel.aux2 : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) :=\nlist.foldr (\u03bbc o, match o with\n| sum.inl a  := sum.inl a\n| sum.inr ls := rmap (\u03bb c', c' :: ls) (destruct c)\nend) (sum.inr [])\n\ndef parallel.aux1 : list (computation \u03b1) \u00d7 wseq (computation \u03b1) \u2192\n  \u03b1 \u2295 list (computation \u03b1) \u00d7 wseq (computation \u03b1)\n| (l, S) := rmap (\u03bb l', match seq.destruct S with\n  | none := (l', nil)\n  | some (none, S') := (l', S')\n  | some (some c, S') := (c::l', S')\n  end) (parallel.aux2 l)\n\n/-- Parallel computation of an infinite stream of computations,\n  taking the first result -/\ndef parallel (S : wseq (computation \u03b1)) : computation \u03b1 :=\ncorec parallel.aux1 ([], S)\n\ntheorem terminates_parallel.aux : \u2200 {l : list (computation \u03b1)} {S c},\n  c \u2208 l \u2192 terminates c \u2192 terminates (corec parallel.aux1 (l, S)) :=\nbegin\n  have lem1 : \u2200 l S, (\u2203 (a : \u03b1), parallel.aux2 l = sum.inl a) \u2192\n    terminates (corec parallel.aux1 (l, S)),\n  { intros l S e, cases e with a e,\n    have this : corec parallel.aux1 (l, S) = return a,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw e, simp [rmap] },\n    rw this, apply_instance },\n  intros l S c m T, revert l S,\n  apply @terminates_rec_on _ _ c T _ _,\n  { intros a l S m, apply lem1,\n    induction l with c l IH generalizing m; simp at m, { contradiction },\n    cases m with e m,\n    { rw \u2190e, simp [parallel.aux2],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls,\n      exacts [\u27e8a', rfl\u27e9, \u27e8a, rfl\u27e9] },\n    { cases IH m with a' e,\n      simp [parallel.aux2], simp [parallel.aux2] at e,\n      rw e, exact \u27e8a', rfl\u27e9 } },\n  { intros s IH l S m,\n    have H1 : \u2200 l', parallel.aux2 l = sum.inr l' \u2192 s \u2208 l',\n    { induction l with c l IH' generalizing m;\n      intros l' e'; simp at m, { contradiction },\n      cases m with e m; simp [parallel.aux2] at e',\n      { rw \u2190e at e',\n        cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        injection e' with e', rw \u2190e', simp },\n      { induction e : list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        rw e at e', { contradiction },\n        have := IH' m _ e,\n        simp [parallel.aux2] at e',\n        cases destruct c; injection e' with h',\n        rw \u2190h', simp [this] } },\n    induction h : parallel.aux2 l with a l',\n    { exact lem1 _ _ \u27e8a, h\u27e9 },\n    { have H2 : corec parallel.aux1 (l, S) = think _,\n      { apply destruct_eq_think,\n        simp [parallel.aux1],\n        rw h, simp [rmap] },\n      rw H2, apply @computation.think_terminates _ _ _,\n      have := H1 _ h,\n      rcases seq.destruct S with _ | \u27e8_|c, S'\u27e9;\n      simp [parallel.aux1]; apply IH; simp [this] } }\nend\n\ntheorem terminates_parallel {S : wseq (computation \u03b1)}\n   {c} (h : c \u2208 S) [T : terminates c] : terminates (parallel S) :=\nsuffices \u2200 n (l : list (computation \u03b1)) S c,\n  c \u2208 l \u2228 some (some c) = seq.nth S n \u2192\n  terminates c \u2192 terminates (corec parallel.aux1 (l, S)),\nfrom let \u27e8n, h\u27e9 := h in this n [] S c (or.inr h) T,\nbegin\n  intro n, induction n with n IH; intros l S c o T,\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    have H : seq.destruct S = some (some c, _),\n    { unfold seq.destruct functor.map, rw \u2190 a, simp },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h, H], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      apply terminates_parallel.aux _ T, simp } },\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      have TT : \u2200 l', terminates (corec parallel.aux1 (l', S.tail)),\n      { intro, apply IH _ _ _ (or.inr _) T, rw a, cases S with f al, refl },\n      induction e : seq.nth S 0 with o,\n      { have D : seq.destruct S = none,\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, simp [parallel.aux1], have TT := TT l',\n        rwa [seq.destruct_eq_nil D, seq.tail_nil] at TT },\n      { have D : seq.destruct S = some (o, S.tail),\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, cases o with c; simp [parallel.aux1, TT] } } }\nend\n\ntheorem exists_of_mem_parallel {S : wseq (computation \u03b1)}\n   {a} (h : a \u2208 parallel S) : \u2203 c \u2208 S, a \u2208 c :=\nsuffices \u2200 C, a \u2208 C \u2192 \u2200 (l : list (computation \u03b1)) S,\n  corec parallel.aux1 (l, S) = C \u2192 \u2203 c, (c \u2208 l \u2228 c \u2208 S) \u2227 a \u2208 c,\nfrom let \u27e8c, h1, h2\u27e9 := this _ h [] S rfl in \u27e8c, h1.resolve_left id, h2\u27e9,\nbegin\n  let F : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) \u2192 Prop,\n  { intros l a, cases a with a l',\n    exact \u2203 c \u2208 l, a \u2208 c,\n    exact \u2200 a', (\u2203 c \u2208 l', a' \u2208 c) \u2192 (\u2203 c \u2208 l, a' \u2208 c) },\n  have lem1 : \u2200 (l : list (computation \u03b1)), F l (parallel.aux2 l),\n  { intro l, induction l with c l IH; simp [parallel.aux2],\n    { intros a h, rcases h with \u27e8c, hn, _\u27e9,\n      exact false.elim hn },\n    { simp [parallel.aux2] at IH,\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a ls;\n      simp [parallel.aux2],\n      { rcases IH with \u27e8c', cl, ac\u27e9,\n        refine \u27e8c', or.inr cl, ac\u27e9 },\n      { induction h : destruct c with a c'; simp [rmap],\n        { refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n          rw destruct_eq_ret h,\n          apply ret_mem },\n        { intros a' h, rcases h with \u27e8d, dm, ad\u27e9,\n          simp at dm, cases dm with e dl,\n          { rw e at ad, refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n            rw destruct_eq_think h,\n            exact think_mem ad },\n          { cases IH a' \u27e8d, dl, ad\u27e9 with d dm, cases dm with dm ad,\n            exact \u27e8d, or.inr dm, ad\u27e9 } } } } },\n  intros C aC, refine mem_rec_on aC _ (\u03bb C' IH, _);\n  intros l S e; have e' := congr_arg destruct e; have := lem1 l;\n  simp [parallel.aux1] at e'; cases parallel.aux2 l with a' l'; injection e' with h',\n  { rw h' at this, rcases this with \u27e8c, cl, ac\u27e9,\n    exact \u27e8c, or.inl cl, ac\u27e9 },\n  { induction e : seq.destruct S with a; rw e at h',\n    { exact let \u27e8d, o, ad\u27e9 := IH _ _ h',\n        \u27e8c, cl, ac\u27e9 := this a \u27e8d, o.resolve_right (not_mem_nil _), ad\u27e9 in\n      \u27e8c, or.inl cl, ac\u27e9 },\n    { cases a with o S', cases o with c; simp [parallel.aux1] at h';\n      rcases IH _ _ h' with \u27e8d, dl | dS', ad\u27e9,\n      { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' },\n      { simp at dl, cases dl with dc dl,\n        { rw dc at ad, refine \u27e8c, or.inr _, ad\u27e9,\n          rw seq.destruct_eq_cons e,\n          apply seq.mem_cons },\n        { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 } },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' } } }\nend\n\ntheorem map_parallel (f : \u03b1 \u2192 \u03b2) (S) : map f (parallel S) = parallel (S.map (map f)) :=\nbegin\n  refine eq_of_bisim (\u03bb c1 c2, \u2203 l S,\n    c1 = map f (corec parallel.aux1 (l, S)) \u2227\n    c2 = corec parallel.aux1 (l.map (map f), S.map (map f))) _ \u27e8[], S, rfl, rfl\u27e9,\n  intros c1 c2 h, exact match c1, c2, h with ._, ._, \u27e8l, S, rfl, rfl\u27e9 := begin\n    clear _match,\n    have : parallel.aux2 (l.map (map f)) = lmap f (rmap (list.map (map f)) (parallel.aux2 l)),\n    { simp [parallel.aux2],\n      induction l with c l IH; simp, rw [IH],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l; simp [parallel.aux2],\n      cases destruct c; simp },\n    simp [parallel.aux1], rw this, cases parallel.aux2 l with a l'; simp,\n    apply S.cases_on _ (\u03bb c S, _) (\u03bb S, _); simp; simp [parallel.aux1];\n    exact \u27e8_, _, rfl, rfl\u27e9\n  end end\nend\n\ntheorem parallel_empty (S : wseq (computation \u03b1)) (h : S.head ~> none) :\nparallel S = empty _ :=\neq_empty_of_not_terminates $ \u03bb \u27e8a, m\u27e9,\nlet \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel m,\n    \u27e8n, nm\u27e9 := exists_nth_of_mem cs,\n    \u27e8c', h'\u27e9 := head_some_of_nth_some nm in by injection h h'\n\n-- The reason this isn't trivial from exists_of_mem_parallel is because it eliminates to Sort\ndef parallel_rec {S : wseq (computation \u03b1)} (C : \u03b1 \u2192 Sort v)\n  (H : \u2200 s \u2208 S, \u2200 a \u2208 s, C a) {a} (h : a \u2208 parallel S) : C a :=\nbegin\n  let T : wseq (computation (\u03b1 \u00d7 computation \u03b1)) :=\n    S.map (\u03bbc, c.map (\u03bb a, (a, c))),\n  have : S = T.map (map (\u03bb c, c.1)),\n  { rw [\u2190wseq.map_comp], refine (wseq.map_id _).symm.trans (congr_arg (\u03bb f, wseq.map f S) _),\n    funext c, dsimp [id, function.comp], rw [\u2190map_comp], exact (map_id _).symm },\n  have pe := congr_arg parallel this, rw \u2190map_parallel at pe,\n  have h' := h, rw pe at h',\n  haveI : terminates (parallel T) := (terminates_map_iff _ _).1 \u27e8_, h'\u27e9,\n  induction e : get (parallel T) with a' c,\n  have : a \u2208 c \u2227 c \u2208 S,\n  { rcases exists_of_mem_map h' with \u27e8d, dT, cd\u27e9,\n    rw get_eq_of_mem _ dT at e, cases e, dsimp at cd, cases cd,\n    rcases exists_of_mem_parallel dT with \u27e8d', dT', ad'\u27e9,\n    rcases wseq.exists_of_mem_map dT' with \u27e8c', cs', e'\u27e9,\n    rw \u2190e' at ad',\n    rcases exists_of_mem_map ad' with \u27e8a', ac', e'\u27e9, injection e' with i1 i2,\n    constructor, rwa [i1, i2] at ac', rwa i2 at cs' },\n  cases this with ac cs, apply H _ cs _ ac\nend\n\ntheorem parallel_promises {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) : parallel S ~> a :=\n\u03bb a' ma', let \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel ma' in H _ cs ac\n\ntheorem mem_parallel {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) {c} (cs : c \u2208 S) (ac : a \u2208 c) : a \u2208 parallel S :=\nby haveI := terminates_of_mem ac; have := terminates_parallel cs;\n   exact mem_of_promises _ (parallel_promises H)\n\ntheorem parallel_congr_lem {S T : wseq (computation \u03b1)} {a}\n  (H : S.lift_rel equiv T) : (\u2200 s \u2208 S, s ~> a) \u2194 (\u2200 t \u2208 T, t ~> a) :=\n\u27e8\u03bb h1 t tT, let \u27e8s, sS, se\u27e9 := wseq.exists_of_lift_rel_right H tT in\n  (promises_congr se _).1 (h1 _ sS),\n\u03bb h2 s sS, let \u27e8t, tT, se\u27e9 := wseq.exists_of_lift_rel_left H sS in\n  (promises_congr se _).2 (h2 _ tT)\u27e9\n\n-- The parallel operation is only deterministic when all computation paths lead to the same value\ntheorem parallel_congr_left {S T : wseq (computation \u03b1)} {a}\n  (h1 : \u2200 s \u2208 S, s ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nlet h2 := (parallel_congr_lem H).1 h1 in\n\u03bb a', \u27e8\u03bbh, by have aa := parallel_promises h1 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_left H sS,\n      aT := (st _).1 as in mem_parallel h2 tT aT,\n\u03bbh, by have aa := parallel_promises h2 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_right H sS,\n      aT := (st _).2 as in mem_parallel h1 tT aT\u27e9\n\ntheorem parallel_congr_right {S T : wseq (computation \u03b1)} {a}\n  (h2 : \u2200 t \u2208 T, t ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nparallel_congr_left ((parallel_congr_lem H).2 h2) H\n\nend computation\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/seq/parallel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3781986629269465}}
{"text": "example (P Q : Prop) : P \u2192 (Q \u2192 P) :=\nbegin\nintro p,\nintro q,\nexact p,\nend\n", "meta": {"author": "abdelq", "repo": "natural-number-game", "sha": "bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2", "save_path": "github-repos/lean/abdelq-natural-number-game", "path": "github-repos/lean/abdelq-natural-number-game/natural-number-game-bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2/world06/level05.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3781363224936462}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nTransferring `traversable` instances using isomorphisms.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.control.traversable.lemmas\nimport Mathlib.PostPort\n\nuniverses u \n\nnamespace Mathlib\n\nnamespace equiv\n\n\n/-- Given a functor `t`, a function `t' : Type u \u2192 Type u`, and\nequivalences `t \u03b1 \u2243 t' \u03b1` for all `\u03b1`, then every function `\u03b1 \u2192 \u03b2` can\nbe mapped to a function `t' \u03b1 \u2192 t' \u03b2` functorially (see\n`equiv.functor`). -/\nprotected def map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) : t' \u03b2 :=\n  coe_fn (eqv \u03b2) (f <$> coe_fn (equiv.symm (eqv \u03b1)) x)\n\n/-- The function `equiv.map` transfers the functoriality of `t` to\n`t'` using the equivalences `eqv`.  -/\nprotected def functor {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] : Functor t' :=\n  { map := equiv.map eqv, mapConst := fun (\u03b1 \u03b2 : Type u) => equiv.map eqv \u2218 function.const \u03b2 }\n\nprotected theorem id_map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] {\u03b1 : Type u} (x : t' \u03b1) : equiv.map eqv id x = x := sorry\n\nprotected theorem comp_map {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : t' \u03b1) : equiv.map eqv (h \u2218 g) x = equiv.map eqv h (equiv.map eqv g x) := sorry\n\nprotected theorem is_lawful_functor {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] : is_lawful_functor t' :=\n  is_lawful_functor.mk (equiv.id_map eqv) (equiv.comp_map eqv)\n\nprotected theorem is_lawful_functor' {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [Functor t] [is_lawful_functor t] [F : Functor t'] (h\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), Functor.map f = equiv.map eqv f) (h\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), Functor.mapConst f = function.comp (equiv.map eqv) (function.const \u03b1) f) : is_lawful_functor t' := sorry\n\n/-- Like `equiv.map`, a function `t' : Type u \u2192 Type u` can be given\nthe structure of a traversable functor using a traversable functor\n`t'` and equivalences `t \u03b1 \u2243 t' \u03b1` for all \u03b1.  See `equiv.traversable`. -/\nprotected def traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] {m : Type u \u2192 Type u} [Applicative m] {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2) (x : t' \u03b1) : m (t' \u03b2) :=\n  \u21d1(eqv \u03b2) <$> traverse f (coe_fn (equiv.symm (eqv \u03b1)) x)\n\n/-- The function `equiv.tranverse` transfers a traversable functor\ninstance across the equivalences `eqv`. -/\nprotected def traversable {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] : traversable t' :=\n  traversable.mk (equiv.traverse eqv)\n\nprotected theorem id_traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {\u03b1 : Type u} (x : t' \u03b1) : equiv.traverse eqv id.mk x = x := sorry\n\nprotected theorem traverse_eq_map_id {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) : equiv.traverse eqv (id.mk \u2218 f) x = id.mk (equiv.map eqv f x) := sorry\n\nprotected theorem comp_traverse {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t' \u03b1) : equiv.traverse eqv (functor.comp.mk \u2218 Functor.map f \u2218 g) x =\n  functor.comp.mk (equiv.traverse eqv f <$> equiv.traverse eqv g x) := sorry\n\nprotected theorem naturality {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] {F : Type u \u2192 Type u} {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_applicative F] [is_lawful_applicative G] (\u03b7 : applicative_transformation F G) {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2) (x : t' \u03b1) : coe_fn \u03b7 (t' \u03b2) (equiv.traverse eqv f x) = equiv.traverse eqv (coe_fn \u03b7 \u03b2 \u2218 f) x := sorry\n\n/-- The fact that `t` is a lawful traversable functor carries over the\nequivalences to `t'`, with the traversable functor structure given by\n`equiv.traversable`. -/\nprotected def is_lawful_traversable {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] : is_lawful_traversable t' :=\n  is_lawful_traversable.mk (equiv.id_traverse eqv) (equiv.comp_traverse eqv) (equiv.traverse_eq_map_id eqv)\n    (equiv.naturality eqv)\n\n/-- If the `traversable t'` instance has the properties that `map`,\n`map_const`, and `traverse` are equal to the ones that come from\ncarrying the traversable functor structure from `t` over the\nequivalences, then the the fact `t` is a lawful traversable functor\ncarries over as well. -/\nprotected def is_lawful_traversable' {t : Type u \u2192 Type u} {t' : Type u \u2192 Type u} (eqv : (\u03b1 : Type u) \u2192 t \u03b1 \u2243 t' \u03b1) [traversable t] [is_lawful_traversable t] [traversable t'] (h\u2080 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2), Functor.map f = equiv.map eqv f) (h\u2081 : \u2200 {\u03b1 \u03b2 : Type u} (f : \u03b2), Functor.mapConst f = function.comp (equiv.map eqv) (function.const \u03b1) f) (h\u2082 : \u2200 {F : Type u \u2192 Type u} [_inst_7 : Applicative F] [_inst_8 : is_lawful_applicative F] {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 F \u03b2),\n  traverse f = equiv.traverse eqv f) : is_lawful_traversable t' :=\n  is_lawful_traversable.mk sorry sorry sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/traversable/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.6261241632752915, "lm_q1q2_score": 0.3781363140669782}}
{"text": "import data.hash_map .lib\n\nnamespace maps\n\ndef prev_append : pos_num \u2192 pos_num \u2192 pos_num\n| pos_num.one j := j\n| (pos_num.bit1 i') j := prev_append i' (pos_num.bit1 j)\n| (pos_num.bit0 i') j := prev_append i' (pos_num.bit0 j)\n\ndef prev (i : pos_num) : pos_num :=\nprev_append i pos_num.one\n\ninductive PTree (A : Type) : Type\n| leaf {} : PTree\n| node    : PTree \u2192 option A \u2192 PTree \u2192 PTree\nopen PTree\n\nnamespace PTree\n\ninstance {A} : has_emptyc (PTree A) := \u27e8leaf\u27e9\n\ndef get {A : Type} : pos_num \u2192 PTree A \u2192 option A :=\nby { intros i m, revert i,\n     induction m with l o r Il Ir; intro i, exact none,\n     cases i with i' i',\n     exact o,\n     exact Il i', exact Ir i' }\n\ndef get_or {A : Type} (n : pos_num) (t : PTree A) (dfl : A) : A :=\n(t.get n).get_or_else dfl\n\ndef set {A : Type} : pos_num \u2192 A \u2192 PTree A \u2192 PTree A\n| pos_num.one       v leaf         := node leaf (some v) leaf\n| (pos_num.bit0 i') v leaf         := node (set i' v leaf) none leaf\n| (pos_num.bit1 i') v leaf         := node leaf none (set i' v leaf)\n| pos_num.one       v (node l o r) := node l (some v) r\n| (pos_num.bit0 i') v (node l o r) := node (set i' v l) o r\n| (pos_num.bit1 i') v (node l o r) := node l o (set i' v r)\n\nlemma gleaf {A} (i) : get i (leaf : PTree A) = none :=\nby induction i; simp [get]\n\ntheorem gempty {A} (i) : get i (\u2205 : PTree A) = none := gleaf i\n\ntheorem gss {A} (i x) (m : PTree A) : get i (set i x m) = some x := sorry\n\ntheorem gso {A i j} (x : A) (m : PTree A) : i \u2260 j \u2192 get i (set j x m) = get i m := sorry\n\ntheorem gsspec {A} (i j) (x : A) (m : PTree A) :\n  get i (set j x m) = if i = j then some x else get i m :=\nby { by_cases (i = j); simp [h], rw gss, rw gso _ _ h }\n\ntheorem gsident {A} (i : pos_num) (m : PTree A) (v : A) :\n  get i m = some v \u2192 set i v m = m := sorry\n\ntheorem set2 {A} (i : pos_num) (m : PTree A) (v1 v2 : A) :\n  set i v2 (set i v1 m) = set i v2 m := sorry\n\ndef of_list {A} (l : list (pos_num \u00d7 A)) : PTree A :=\nl.foldl (\u03bb m \u27e8k, v\u27e9, set k v m) \u2205\n\ndef node' {A} : PTree A \u2192 option A \u2192 PTree A \u2192 PTree A\n| leaf none leaf := leaf\n| l x r := node l x r\n\nsection combine\n\nvariables {A B C : Type} (f : option A \u2192 option B \u2192 option C)\n\ndef xcombine_l : PTree A \u2192 PTree C\n| leaf := leaf\n| (node l o r) := node' (xcombine_l l) (f o none) (xcombine_l r)\n\ndef xcombine_r : PTree B \u2192 PTree C\n| leaf := leaf\n| (node l o r) := node' (xcombine_r l) (f none o) (xcombine_r r)\n\ndef combine : PTree A \u2192 PTree B \u2192 PTree C\n| leaf m2 := xcombine_r f m2\n| m1 leaf := xcombine_l f m1\n| (node l1 o1 r1) (node l2 o2 r2) := node' (combine l1 l2) (f o1 o2) (combine r1 r2)\n\ntheorem combine_commut (f g : option A \u2192 option A \u2192 option B) :\n(\u2200 i j, f i j = g j i) \u2192\n\u2200 m1 m2, combine f m1 m2 = combine g m2 m1 := sorry\n\nend combine\n\ndef xelements {A} : PTree A \u2192 pos_num \u2192 list (pos_num \u00d7 A) \u2192 list (pos_num \u00d7 A)\n| leaf i k := k\n| (node l none r) i k :=\n    xelements l (pos_num.bit0 i) (xelements r (pos_num.bit1 i) k)\n| (node l (some x) r) i k :=\n    xelements l (pos_num.bit0 i)\n    ((prev i, x) :: xelements r (pos_num.bit1 i) k)\n\ndef elements {A} (m : PTree A) := xelements m pos_num.one []\n\ndef xfold {A B} (f : B \u2192 pos_num \u2192 A \u2192 B) : pos_num \u2192 PTree A \u2192 B \u2192 B\n| i leaf v := v\n| i (node l none r) v :=\n  let v1 := xfold (pos_num.bit0 i) l v in\n  xfold (pos_num.bit1 i) r v1\n| i (node l (some x) r) v :=\n  let v1 := xfold (pos_num.bit0 i) l v in\n  let v2 := f v1 (pos_num.pred i) x in\n  xfold (pos_num.bit1 i) r v2\n\ndef fold {A B} (f : B \u2192 pos_num \u2192 A \u2192 B) (m : PTree A) (v : B) :=\nxfold f pos_num.one m v\n\ndef for_all {A} (m : PTree A) (f : pos_num \u2192 A \u2192 bool) : bool :=\nfold (\u03bb b x a, b && f x a) m tt\n\nnotation a `^!` b := get b a\n\nend PTree\n\nend maps", "meta": {"author": "digama0", "repo": "kremlin", "sha": "d4665929ce9012e93a0b05fc7063b96256bab86f", "save_path": "github-repos/lean/digama0-kremlin", "path": "github-repos/lean/digama0-kremlin/kremlin-d4665929ce9012e93a0b05fc7063b96256bab86f/maps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.6261241632752915, "lm_q1q2_score": 0.3781363140669781}}
{"text": "namespace Foo\n\ndef x := 10\n\nend Foo\n\n#check Foo.x\n\nopen Foo\n\n#check x\n\ntheorem ex1 : x = Foo.x := rfl\n\nnamespace Foo\n\ndef f x y := x + y + 1\n\nscoped infix:70 \"^^\" => f\n\n#check 1 ^^ 2\n\ntheorem ex1 : x ^^ y = f x y := rfl\n\nend Foo\n\n#check 1 ^^ 2 -- works because we have an `open Foo` above\n\ntheorem ex2 : x ^^ y = f x y := rfl\ntheorem ex3 : x ^^ y = Foo.f x y := rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/scopedParsers2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318194686359, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.37813631356679195}}
{"text": "import data.sign\n\nnamespace int\n\n@[elab_as_eliminator] protected lemma induction_on_iff {p : \u2124 \u2192 Prop}\n  (i : \u2124) (hz : p 0) (h : \u2200 i : \u2124, p i \u2194 p (i + 1)) : p i :=\nbegin\n  induction i using int.induction_on with i IH i IH,\n  { exact hz },\n  { rwa \u2190 h },\n  { rwa [h, sub_add_cancel], }\nend\n\n@[simp] lemma sign_eq_sign (n : \u2124) : n.sign = _root_.sign n :=\nbegin\n  obtain ((_ | _) | _) := n,\n  { exact congr_arg coe sign_zero.symm },\n  { exact congr_arg coe (sign_pos $ int.succ_coe_nat_pos _).symm },\n  { exact congr_arg coe (_root_.sign_neg $ neg_succ_lt_zero _).symm }\nend\n\nend int\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/int.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.37798337998077725}}
{"text": "import data.list.func\nimport .helpers\nimport .list2d\nimport .find_indexes2d\n\ndef bset1d := list bool\ndef bset2d := list2d bool\n\ninstance : inhabited bset1d := \u27e8[]\u27e9\ninstance : inhabited bset2d := \u27e8[]\u27e9\n\ninstance : has_mem \u2115 bset1d :=\n\u27e8\u03bb n l, list.func.get n l = tt\u27e9\ninstance : has_mem (\u2115\u00d7\u2115) bset2d :=\n\u27e8\u03bb xy l, list2d.get2d xy l = tt\u27e9\nlemma bset1d.has_mem.unfold {n : \u2115} {l : bset1d}\n  : n \u2208 l = (list.func.get n l = tt) := rfl\nlemma bset2d.has_mem.unfold {xy : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 l = (list2d.get2d xy l = tt) := rfl\n\ninstance : has_sdiff bset2d\n:= \u27e8list2d.pointwise2d (\u03bb a b, (cond b ff a))\u27e9\ninstance : has_inter bset2d\n:= \u27e8list2d.pointwise2d band\u27e9\ninstance : has_union bset2d\n:= \u27e8list2d.pointwise2d bor\u27e9\n\ndef pic_str1d (l : bset1d) : string :=\n  string.push (list.as_string (l.map (\u03bbb, cond b '*' '.'))) '\\n'\ndef pic_str2d (l : bset2d) : string :=\n  string.join (l.map pic_str1d)\ninstance : has_repr bset1d := \u27e8pic_str1d\u27e9\ninstance : has_repr bset2d := \u27e8pic_str2d\u27e9\n\n--             _              _   \n--   ___ _   _| |__  ___  ___| |_ \n--  / __| | | | '_ \\/ __|/ _ \\ __|\n--  \\__ \\ |_| | |_) \\__ \\  __/ |_ \n--  |___/\\__,_|_.__/|___/\\___|\\__|\n--                                \n\ndef bset1d.subset (l1 : bset1d) (l2 : bset1d) : Prop\n  := \u2200 x : \u2115, x \u2208 l1 \u2192 x \u2208 l2\ndef bset2d.subset (l1 : bset2d) (l2 : bset2d) : Prop\n  := \u2200 xy : \u2115\u00d7\u2115, xy \u2208 l1 \u2192 xy \u2208 l2\ninstance : has_subset bset1d := \u27e8bset1d.subset\u27e9\ninstance : has_subset bset2d := \u27e8bset2d.subset\u27e9\n\ndef bset2d.disjoint (l1 : bset2d) (l2 : bset2d) : Prop\n  := \u2200 xy : \u2115\u00d7\u2115, xy \u2208 l1 \u2192 xy \u2208 l2 \u2192 false\n\ntheorem bset2d.subset.refl : \u2200 {l1 : bset2d}, l1 \u2286 l1\n:= assume l1 xy H, H\ntheorem bset2d.subset.trans : \u2200 {l1 l2 l3 : bset2d},\n  l1 \u2286 l2 \u2192 l2 \u2286 l3 \u2192 l1 \u2286 l3\n:= assume l1 l2 l3 H12 H23 xy H1, H23 xy (H12 xy H1)\n\ntheorem bset2d.subset_of_equiv : \u2200 {l1 l2 : bset2d},\n  list2d.equiv l1 l2 \u2192 l1 \u2286 l2 :=\nbegin\n  intros l1 l2 Heq xy Hget,\n  unfold has_mem.mem at Hget, \n  rw Heq xy at Hget, exact Hget,\nend\n\n--                         _   _             \n--    ___ ___  _   _ _ __ | |_(_)_ __   __ _ \n--   / __/ _ \\| | | | '_ \\| __| | '_ \\ / _` |\n--  | (_| (_) | |_| | | | | |_| | | | | (_| |\n--   \\___\\___/ \\__,_|_| |_|\\__|_|_| |_|\\__, |\n--                                     |___/ \n\ndef bset1d.count (l : bset1d) : \u2115 := list.sum (list.map bool.to_nat l)\ndef bset2d.count (l : bset2d) : \u2115 := list.sum (list.map bset1d.count l)\n\n@[simp] theorem bset1d.count_nil : bset1d.count [] = 0 := rfl\n@[simp] theorem bset2d.count_nil : bset2d.count [] = 0 := rfl\n@[simp] theorem bset1d.count_cons : \u2200 (h : bool) (t : bset1d),\n  bset1d.count (h::t) = (bool.to_nat h) + t.count := by simp [bset1d.count]\n@[simp] theorem bset2d.count_cons : \u2200 (h : bset1d) (t : bset2d),\n  bset2d.count (h::t) = h.count + t.count := by simp [bset2d.count]\n\n--                         _                       _              _   \n--    ___ ___  _   _ _ __ | |_     _     ___ _   _| |__  ___  ___| |_ \n--   / __/ _ \\| | | | '_ \\| __|  _| |_  / __| | | | '_ \\/ __|/ _ \\ __|\n--  | (_| (_) | |_| | | | | |_  |_   _| \\__ \\ |_| | |_) \\__ \\  __/ |_ \n--   \\___\\___/ \\__,_|_| |_|\\__|   |_|   |___/\\__,_|_.__/|___/\\___|\\__|\n--                                                                    \n\ntheorem bool_to_nat_le {b1 b2 : bool} :\n  (b1.to_nat \u2264 b2.to_nat) \u2194 (b1 = tt \u2192 b2 = tt) :=\nbegin\n  intros, split, {\n    intro H, cases b1, simp,\n    cases b2, simp [bool.to_nat] at H, contradiction,\n    simp,\n  }, {\n    intro H, cases b1, simp [bool.to_nat],\n    cases b2, simp at H, contradiction,\n    apply le_rfl,\n  }\nend\n\ntheorem bset1d.subset_cons : \u2200 (h1 h2 : bool) (t1 t2 : bset1d),\n  bset1d.subset (h1::t1) (h2::t2) = (h1.to_nat \u2264 h2.to_nat \u2227 t1 \u2286 t2) :=\nbegin\n  intros, apply propext, split, {\n    intro H, split, exact bool_to_nat_le.2 (H 0),\n    intro n, exact H n.succ\n  }, {\n    intro H, cases H with H1 H2,\n    intro n, cases n, exact bool_to_nat_le.1 H1,\n    exact H2 n,\n  }\nend\ntheorem bset2d.subset_cons : \u2200 (h1 h2 : bset1d) (t1 t2 : bset2d),\n  bset2d.subset (h1::t1) (h2::t2) = (h1 \u2286 h2 \u2227 t1 \u2286 t2) :=\nbegin\n  intros, apply propext, split, {\n    intro H, split, intro x, exact H (x,0),\n    intro xy, cases xy with x y, exact H (x,y+1)\n  }, {\n    intro H, cases H with H1 H2,\n    intro xy, cases xy with x y, cases y, apply H1,\n    exact H2 (x, y),\n  }\nend\n\ntheorem bset1d.count_0_of_subset_nil : \u2200 l : bset1d,\n  bset1d.subset l list.nil \u2192 l.count = 0\n| [] := by simp\n| (h::t) := begin\n  intro H, simp, split, {\n    have := H 0, simp [has_mem.mem] at this,\n    rw bool_eq_false this, refl,\n  }, {\n    apply bset1d.count_0_of_subset_nil,\n    intros n H2, specialize H (n+1), simp [has_mem.mem] at H, contradiction,\n  }\nend\ntheorem bset2d.count_0_of_subset_nil : \u2200 l : bset2d,\n  bset2d.subset l list.nil \u2192 l.count = 0\n| [] := by simp\n| (h::t) := begin\n  intro H, simp, split, {\n    apply bset1d.count_0_of_subset_nil,\n    intro x, exact H (x,0),\n  }, {\n    apply bset2d.count_0_of_subset_nil,\n    intros xy H2, cases xy with x y,\n    specialize H (x,y+1), simp [has_mem.mem] at H, contradiction,\n  }\nend\n\ntheorem bset1d.count_le_of_subset : \u2200 l1 l2 : bset1d,\n  l1 \u2286 l2 \u2192 l1.count \u2264 l2.count\n| [] _ := by simp\n| _ [] := begin\n  intros H, have H := bset1d.count_0_of_subset_nil _ H, rewrite H,\n  apply zero_le,\nend\n| (h1::t1) (h2::t2) := begin\n  simp [has_subset.subset, bset1d.subset_cons], intros Hh Ht, apply add_le_add,\n  exact Hh,\n  exact bset1d.count_le_of_subset _ _ Ht,\nend\n\ntheorem bset2d.count_le_of_subset : \u2200 l1 l2 : bset2d,\n  l1 \u2286 l2 \u2192 l1.count \u2264 l2.count\n| [] _ := by simp\n| _ [] := begin\n  intros H, have H := bset2d.count_0_of_subset_nil _ H, rewrite H,\n  apply zero_le,\nend\n| (h1::t1) (h2::t2) := begin\n  simp [has_subset.subset, bset2d.subset_cons], intros Hh Ht, apply add_le_add,\n  exact bset1d.count_le_of_subset _ _ Hh,\n  exact bset2d.count_le_of_subset _ _ Ht,\nend\n\nlemma leq_sum_equal {a b c d : \u2115} :\n  a \u2264 b \u2192 c \u2264 d \u2192 a+c = b+d \u2192 a=b \u2227 c=d := by omega\n\ntheorem bset1d.subset_nil_of_count_0 : \u2200 l : bset1d,\n  l.count = 0 \u2192 bset1d.subset l list.nil\n| [] := \u03bb _ _ H, H\n| (h::t) := begin\n  simp, intros Hh Ht n, cases n, {\n    cases h, intro H, simp at H, contradiction,\n    simp [bool.to_nat] at Hh, contradiction,\n  }, {\n    have := bset1d.subset_nil_of_count_0 t Ht n,\n    simp [has_mem.mem], simp [has_mem.mem] at this, exact this,\n  }\nend\ntheorem bset2d.subset_nil_of_count_0 : \u2200 l : bset2d,\n  l.count = 0 \u2192 bset2d.subset l list.nil\n| [] := \u03bb _ _ H, H\n| (h::t) := begin\n  simp, intros Hh Ht xy, cases xy with x y, cases y, {\n    exact bset1d.subset_nil_of_count_0 h Hh x,\n  }, {\n    have := bset2d.subset_nil_of_count_0 t Ht (x,y),\n    simp [has_mem.mem], simp [has_mem.mem] at this, exact this,\n  }\nend\n\ntheorem bset1d.subset_eq_of_count_eq : \u2200 l1 l2 : bset1d,\n  l1 \u2286 l2 \u2192 l1.count = l2.count \u2192 l2 \u2286 l1\n| _ [] := begin simp [has_mem.mem, has_subset.subset, bset1d.subset], end\n| [] l2 := begin\n  intros Hh Hc, simp at Hc,\n  apply bset1d.subset_nil_of_count_0 l2 (eq.symm Hc),\nend\n| (h1::t1) (h2::t2) := begin\n  simp [has_subset.subset, bset1d.subset_cons], intros Hh Ht Hc,\n  have Ht_le := bset1d.count_le_of_subset t1 t2 Ht,\n  cases leq_sum_equal Hh Ht_le Hc with Eh Et,\n  split, simp! [Eh],\n  exact bset1d.subset_eq_of_count_eq t1 t2 Ht Et,\nend\n\ntheorem bset2d.subset_eq_of_count_eq : \u2200 l1 l2 : bset2d,\n  l1 \u2286 l2 \u2192 l1.count = l2.count \u2192 l2 \u2286 l1\n| _ [] := by simp [has_mem.mem, has_subset.subset, bset2d.subset]\n| [] l2 := begin\n  intros Hh Hc, simp at Hc,\n  apply bset2d.subset_nil_of_count_0 l2 (eq.symm Hc),\nend\n| (h1::t1) (h2::t2) := begin\n  simp [has_subset.subset, bset1d.subset_cons, bset2d.subset_cons],\n  intros Hh Ht Hc,\n  have Hh_le := bset1d.count_le_of_subset h1 h2 Hh,\n  have Ht_le := bset2d.count_le_of_subset t1 t2 Ht,\n  cases leq_sum_equal Hh_le Ht_le Hc with Eh Et,\n  split, apply bset1d.subset_eq_of_count_eq h1 h2 Hh Eh,\n  exact bset2d.subset_eq_of_count_eq t1 t2 Ht Et,\nend\n\n--             _     _                                             \n--    __ _  __| | __| |    _     _ __ ___ _ __ ___   _____   _____ \n--   / _` |/ _` |/ _` |  _| |_  | '__/ _ \\ '_ ` _ \\ / _ \\ \\ / / _ \\\n--  | (_| | (_| | (_| | |_   _| | | |  __/ | | | | | (_) \\ V /  __/\n--   \\__,_|\\__,_|\\__,_|   |_|   |_|  \\___|_| |_| |_|\\___/ \\_/ \\___|\n--                                                                 \n\ndef bset2d.add (xy : \u2115\u00d7\u2115) (l : bset2d) : bset2d := list2d.set2d tt l xy\ndef bset2d.remove (xy : \u2115\u00d7\u2115) (l : bset2d) : bset2d := list2d.set2d ff l xy\n\nlemma bset2d.mem_add {xy : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 (l.add xy) := list2d.get2d_set2d\nlemma bset2d.mem_add_of_mem {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 l \u2192 xy \u2208 (l.add xy') := begin\n  assume H, by_cases C : xy = xy', rw C, exact list2d.get2d_set2d,\n  exact eq.trans (list2d.get2d_set2d_eq_of_ne C) H,\nend\nlemma bset2d.nmem_add_of_neq_nmem {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2260 xy' \u2192 xy \u2209 l \u2192 xy \u2209 (l.add xy')\n:= \u03bb Hn Hl Hadd, Hl (eq.trans (list2d.get2d_set2d_eq_of_ne Hn).symm Hadd)\nlemma bset2d.nmem_remove {xy : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2209 (l.remove xy)\n:= by simp [has_mem.mem, bset2d.remove, list2d.get2d_set2d]\nlemma bset2d.nmem_remove_of_nmem {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2209 l \u2192 xy \u2209 (l.remove xy') := begin\n  assume H, by_cases C : xy = xy', rw C, exact bset2d.nmem_remove,\n  assume Hrem, exact H (eq.trans (list2d.get2d_set2d_eq_of_ne C).symm Hrem),\nend\nlemma bset2d.mem_remove_of_neq_mem {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2260 xy' \u2192 xy \u2208 l \u2192 xy \u2208 (l.remove xy')\n:= \u03bb Hn Hl, eq.trans (list2d.get2d_set2d_eq_of_ne Hn) Hl\n\nlemma bset2d.of_mem_add {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 (l.add xy') \u2192 xy = xy' \u2228 xy \u2208 l := begin\n  intro H, by_cases C : xy = xy', left, exact C,\n  right, by_contradiction Hnl,\n  exact bset2d.nmem_add_of_neq_nmem C Hnl H,\nend\nlemma bset2d.nmem_of_nmem_add {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2209 (l.add xy') \u2192 xy \u2209 l :=\nbegin\n  intros Hnadd, by_contradiction Hl,\n  exact Hnadd (bset2d.mem_add_of_mem Hl),\nend\nlemma bset2d.neq_of_nmem_add {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2209 (l.add xy') \u2192 xy \u2260 xy' :=\nbegin\n  intros Hnadd Heq, rw Heq at Hnadd,\n  exact Hnadd (bset2d.mem_add),\nend\nlemma bset2d.of_nmem_remove {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2209 (l.remove xy') \u2192 xy = xy' \u2228 xy \u2209 l := begin\n  intro H, by_cases C : xy = xy', left, exact C,\n  right, by_contradiction Hl,\n  refine H (bset2d.mem_remove_of_neq_mem _ Hl),\n  assume Cn, exact C Cn,\nend\nlemma bset2d.mem_of_mem_remove {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 (l.remove xy') \u2192 xy \u2208 l := begin\n  intro Hrem, by_contradiction Hnl,\n  exact bset2d.nmem_remove_of_nmem Hnl Hrem,\nend\nlemma bset2d.neq_of_mem_remove {xy xy' : \u2115\u00d7\u2115} {l : bset2d}\n  : xy \u2208 (l.remove xy') \u2192 xy \u2260 xy' := begin\n  intros Hrem Heq, rw Heq at Hrem,\n  exact bset2d.nmem_remove Hrem,\nend\nlemma bset2d.subset_add_same {l1 l2 : bset2d} {xy : \u2115\u00d7\u2115}\n  : l1 \u2286 l2 \u2192 l1.add xy \u2286 l2.add xy\n:= begin\n  intros Hsub xy' Hin, cases bset2d.of_mem_add Hin,\n  rw h, exact bset2d.mem_add,\n  apply bset2d.mem_add_of_mem, exact Hsub xy' h,\nend\n\nlemma bset2d.subset_remove_same {l1 l2 : bset2d} {xy : \u2115\u00d7\u2115}\n  : l1 \u2286 l2 \u2192 l1.remove xy \u2286 l2.remove xy\n:= begin\n  intros Hsub xy' Hin, apply bset2d.mem_remove_of_neq_mem,\n  exact bset2d.neq_of_mem_remove Hin,\n  apply Hsub,\n  exact bset2d.mem_of_mem_remove Hin,\nend\n\n--                         _                             _       _       \n--    ___ ___  _   _ _ __ | |_     _     _   _ _ __   __| | __ _| |_ ___ \n--   / __/ _ \\| | | | '_ \\| __|  _| |_  | | | | '_ \\ / _` |/ _` | __/ _ \\\n--  | (_| (_) | |_| | | | | |_  |_   _| | |_| | |_) | (_| | (_| | ||  __/\n--   \\___\\___/ \\__,_|_| |_|\\__|   |_|    \\__,_| .__/ \\__,_|\\__,_|\\__\\___|\n--                                            |_|                        \n\nlemma bset1d.count_update : \u2200 (b : bool) (l : bset1d) (n : \u2115),\n\u2203 c, bset1d.count l = c + (list.func.get n l).to_nat\n\u2227 bset1d.count (list.func.set b l n) = c + b.to_nat\n:=\nlist.sum_map_update bool.to_nat rfl \n\nlemma bset2d.count_update : \u2200 (b : bool) (l : bset2d) (xy : \u2115\u00d7\u2115),\n\u2203 c, bset2d.count l = c + (list2d.get2d xy l).to_nat\n\u2227 bset2d.count (list2d.set2d b l xy) = c + b.to_nat\n:=\nbegin\n  intros, cases xy with x y,\n  rcases list.sum_map_update bset1d.count rfl\n    (list.func.set b (list.func.get y l) x) l y\n  with \u27e8c1, H1old, H1new\u27e9,\n  rcases bset1d.count_update b (list.func.get y l) x\n  with \u27e8c2, H2old, H2new\u27e9,\n  existsi c1 + c2,\n  split, {\n    unfold bset2d.count, rw H1old, rw H2old,\n    rw \u2190add_assoc, refl,\n  }, {\n    unfold bset2d.count, unfold list2d.set2d,\n    rw H1new, rw H2new, rw \u2190add_assoc,\n  }\nend\n\nlemma bset2d.count_add {xy : \u2115\u00d7\u2115} {l : bset2d}\n: xy \u2209 l \u2192 (l.add xy).count = l.count+1\n:=\nbegin\n  rcases bset2d.count_update tt l xy with \u27e8c,Hold,Hnew\u27e9,\n  assume Hnin : xy \u2209 l,\n    rw bool_eq_false Hnin at Hold, clear Hnin,\n  unfold bset2d.add, rw Hnew, rw Hold,\n  simp [bool.to_nat],\nend\n\nlemma bset2d.count_remove {xy : \u2115\u00d7\u2115} {l : bset2d}\n: xy \u2208 l \u2192 (l.remove xy).count+1 = l.count\n:=\nbegin\n  rcases bset2d.count_update ff l xy with \u27e8c,Hold,Hnew\u27e9,\n  assume Hin : xy \u2208 l,\n    rw bset2d.has_mem.unfold at Hin, rw Hin at Hold, clear Hin,\n  unfold bset2d.remove, rw Hnew, rw Hold,\n  simp [bool.to_nat],\nend\n\n--           _ _  __  __ \n--   ___  __| (_)/ _|/ _|\n--  / __|/ _` | | |_| |_ \n--  \\__ \\ (_| | |  _|  _|\n--  |___/\\__,_|_|_| |_|  \n--                       \n\nlemma bset2d.mem_sdiff_of_mem_nmem {xy : \u2115\u00d7\u2115} {l1 l2 : bset2d}\n  : xy \u2208 l1 \u2192 xy \u2209 l2 \u2192 xy \u2208 l1 \\ l2 :=\nbegin\n  simp [has_sdiff.sdiff, has_mem.mem, list2d.get2d_pointwise],\n  intros H1 H2, simp! [H1, H2],\nend\nlemma bset2d.mem_of_mem_sdiff {xy : \u2115\u00d7\u2115} {l1 l2 : bset2d}\n  : xy \u2208 l1 \\ l2 \u2192 xy \u2208 l1 :=\nbegin\n  simp [has_sdiff.sdiff, has_mem.mem, list2d.get2d_pointwise],\n  assume H, cases (list2d.get2d xy l2), exact H, exact bool.no_confusion H,\nend\nlemma bset2d.nmem_of_mem_sdiff {xy : \u2115\u00d7\u2115} {l1 l2 : bset2d}\n  : xy \u2208 l1 \\ l2 \u2192 xy \u2209 l2 :=\nbegin\n  simp [has_sdiff.sdiff, has_mem.mem, list2d.get2d_pointwise],\n  assume H, cases (list2d.get2d xy l2), refl, exact bool.no_confusion H,\nend\nlemma bset2d.sdiff_subset {l1 l2 : bset2d} : l1 \\ l2 \u2286 l1\n:= assume xy, bset2d.mem_of_mem_sdiff\n\n--   _       _                                  _             \n--  (_)_ __ | |_ ___ _ __     _     _   _ _ __ (_) ___  _ __  \n--  | | '_ \\| __/ _ \\ '__|  _| |_  | | | | '_ \\| |/ _ \\| '_ \\ \n--  | | | | | ||  __/ |    |_   _| | |_| | | | | | (_) | | | |\n--  |_|_| |_|\\__\\___|_|      |_|    \\__,_|_| |_|_|\\___/|_| |_|\n--                                                            \n\nlemma bset2d.inter_subset_left {l1 l2 : bset2d} : l1 \u2229 l2 \u2286 l1\n:=\n  assume xy,\n  assume H,\n  have this : list2d.get2d xy l1 && list2d.get2d xy l2 = tt\n    := eq.trans (eq.symm (list2d.get2d_pointwise rfl _ _ _)) H,\n  show xy \u2208 l1, from ((band_coe_iff _ _).mp this).1\n\nlemma bset2d.inter_subset_right {l1 l2 : bset2d} : l1 \u2229 l2 \u2286 l2\n:=\n  assume xy,\n  assume H,\n  have this : list2d.get2d xy l1 && list2d.get2d xy l2 = tt\n    := eq.trans (eq.symm (list2d.get2d_pointwise rfl _ _ _)) H,\n  show xy \u2208 l2, from ((band_coe_iff _ _).mp this).2\n\nlemma bset2d.union_subset {l1 l2 l3 : bset2d} : l1 \u2286 l3 \u2192 l2 \u2286 l3 \u2192 l1 \u222a l2 \u2286 l3\n:=\nbegin\n  assume H1 H2 xy Hin,\n  have this : list2d.get2d xy l1 || list2d.get2d xy l2 = tt\n    := eq.trans (eq.symm (list2d.get2d_pointwise rfl _ _ _)) Hin,\n  cases (bor_coe_iff _ _).mp this with Hin1 Hin2,\n  exact H1 xy Hin1,\n  exact H2 xy Hin2,\nend\nlemma bset2d.union_supset_left {l1 l2 : bset2d} : l1 \u2286 l1 \u222a l2\n:=\nbegin\n  assume xy,\n  assume H,\n  refine eq.trans (list2d.get2d_pointwise rfl _ _ _) _,\n  exact (congr_arg (\u03bb a, a || list2d.get2d xy l2) H).trans (tt_bor _),\nend\nlemma bset2d.union_supset_right {l1 l2 : bset2d} : l2 \u2286 l1 \u222a l2\n:=\nbegin\n  assume xy,\n  assume H,\n  refine eq.trans (list2d.get2d_pointwise rfl _ _ _) _,\n  exact (congr_arg (bor _) H).trans (bor_tt _),\nend\n\n\n--   _           _                    \n--  (_)_ __   __| | _____  _____  ___ \n--  | | '_ \\ / _` |/ _ \\ \\/ / _ \\/ __|\n--  | | | | | (_| |  __/>  <  __/\\__ \\\n--  |_|_| |_|\\__,_|\\___/_/\\_\\___||___/\n--                                    \n\ndef bset2d.from_index (xy : \u2115\u00d7\u2115) : bset2d := list2d.set2d tt [] xy\ndef bset2d.from_indexes (xys : list (\u2115 \u00d7 \u2115)) : bset2d\n:= list.foldr (\u03bb (xy : \u2115\u00d7\u2115) (l : bset2d), l.set2d tt xy) [] xys\ndef bset2d.to_indexes (s : bset2d) : list (\u2115 \u00d7 \u2115)\n:= s.find_indexes2d (eq tt)\n\ntheorem bset2d.from_index_iff (xy : \u2115\u00d7\u2115)\n  : \u2200 xy' : \u2115\u00d7\u2115, xy' \u2208 bset2d.from_index xy \u2194 xy' = xy :=\nbegin\n  intro xy', split, {\n    intro H, by_contradiction C,\n    unfold has_mem.mem at H, unfold bset2d.from_index at H,\n    rw list2d.get2d_set2d_eq_of_ne at H, simp at H, exact H, exact C,\n  }, {\n    intro H, rw H, unfold bset2d.from_index, unfold has_mem.mem,\n    exact list2d.get2d_set2d,\n  }\nend\n\ntheorem bset2d.from_indexes_iff (xys : list (\u2115\u00d7\u2115))\n  : \u2200 xy : \u2115\u00d7\u2115, xy \u2208 bset2d.from_indexes xys \u2194 xy \u2208 xys :=\nbegin\n  intro xy, induction xys with xy' t IH, {\n    simp! [bset2d.from_indexes, has_mem.mem],\n  }, {\n    by_cases C : xy = xy',\n    { rw \u2190C, simp! [has_mem.mem, bset2d.from_indexes, list2d.get2d_set2d], },\n    simp [bset2d.from_indexes], unfold has_mem.mem,\n    rw list2d.get2d_set2d_eq_of_ne C,\n    split,\n    { intro H, right, exact IH.mp H, },\n    { intro H, cases H, exact false.elim (C H),\n      exact IH.mpr H, },\n  },\nend\n\ntheorem bset2d.to_indexes_iff (s : bset2d)\n  : \u2200 xy : \u2115\u00d7\u2115, xy \u2208 bset2d.to_indexes s \u2194 xy \u2208 s :=\nbegin\n  intro xy,\n  unfold bset2d.to_indexes, rw bset2d.has_mem.unfold,\n  rw \u2190list2d.find_indexes2d_iff, exact eq_comm, simp!,\nend\n", "meta": {"author": "mirefek", "repo": "sokoban.lean", "sha": "451c92308afb4d3f8e566594b9751286f93b899b", "save_path": "github-repos/lean/mirefek-sokoban.lean", "path": "github-repos/lean/mirefek-sokoban.lean/sokoban.lean-451c92308afb4d3f8e566594b9751286f93b899b/src/boolset2d.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.37778687644168024}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport group_theory.group_action.defs\n\n/-!\n# Option instances for additive and multiplicative actions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for additive and multiplicative actions on `option` type. Scalar\nmultiplication is defined by `a \u2022 some b = some (a \u2022 b)` and `a \u2022 none = none`.\n\n## See also\n\n* `group_theory.group_action.pi`\n* `group_theory.group_action.prod`\n* `group_theory.group_action.sigma`\n* `group_theory.group_action.sum`\n-/\n\nvariables {M N \u03b1 : Type*}\n\nnamespace option\n\nsection has_smul\nvariables [has_smul M \u03b1] [has_smul N \u03b1] (a : M) (b : \u03b1) (x : option \u03b1)\n\n@[to_additive option.has_vadd] instance : has_smul M (option \u03b1) := \u27e8\u03bb a, option.map $ (\u2022) a\u27e9\n\n@[to_additive] lemma smul_def : a \u2022 x = x.map ((\u2022) a) := rfl\n@[simp, to_additive] lemma smul_none : a \u2022 (none : option \u03b1) = none := rfl\n@[simp, to_additive] lemma smul_some : a \u2022 some b = some (a \u2022 b) := rfl\n\n@[to_additive] instance [has_smul M N] [is_scalar_tower M N \u03b1] : is_scalar_tower M N (option \u03b1) :=\n\u27e8\u03bb a b x, by { cases x, exacts [rfl, congr_arg some (smul_assoc _ _ _)] }\u27e9\n\n@[to_additive] instance [smul_comm_class M N \u03b1] : smul_comm_class M N (option \u03b1) :=\n\u27e8\u03bb a b, function.commute.option_map $ smul_comm _ _\u27e9\n\n@[to_additive]\ninstance [has_smul M\u1d50\u1d52\u1d56 \u03b1] [is_central_scalar M \u03b1] : is_central_scalar M (option \u03b1) :=\n\u27e8\u03bb a x, by { cases x, exacts [rfl, congr_arg some (op_smul_eq_smul _ _)] }\u27e9\n\n@[to_additive] instance [has_faithful_smul M \u03b1] : has_faithful_smul M (option \u03b1) :=\n\u27e8\u03bb x y h, eq_of_smul_eq_smul $ \u03bb b : \u03b1, by injection h (some b)\u27e9\n\nend has_smul\n\ninstance [monoid M] [mul_action M \u03b1] : mul_action M (option \u03b1) :=\n{ smul := (\u2022),\n  one_smul := \u03bb b, by { cases b, exacts [rfl, congr_arg some (one_smul _ _)] },\n  mul_smul := \u03bb a\u2081 a\u2082 b, by { cases b, exacts [rfl, congr_arg some (mul_smul _ _ _)] } }\n\nend option\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/group_theory/group_action/option.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341026367784, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.37778687644168013}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\n\n/-!\n# Extra definitions on `option`\n\nThis file defines more operations involving `option \u03b1`. Lemmas about them are located in other\nfiles under `data.option.`.\nOther basic operations on `option` are defined in the core library.\n-/\n\n\nnamespace option\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\nattribute [inline] option.is_some option.is_none\n\n/-- An elimination principle for `option`. It is a nondependent version of `option.rec_on`. -/\n@[simp] protected def elim : option \u03b1 \u2192 \u03b2 \u2192 (\u03b1 \u2192 \u03b2) \u2192 \u03b2\n| (some x) y f := f x\n| none     y f := y\n\ninstance has_mem : has_mem \u03b1 (option \u03b1) := \u27e8\u03bb a b, b = some a\u27e9\n\n@[simp] theorem mem_def {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = some a :=\niff.rfl\n\nlemma mem_iff {a : \u03b1} {b : option \u03b1} : a \u2208 b \u2194 b = a := iff.rfl\n\ntheorem is_none_iff_eq_none {o : option \u03b1} : o.is_none = tt \u2194 o = none :=\n\u27e8option.eq_none_of_is_none, \u03bb e, e.symm \u25b8 rfl\u27e9\n\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n\n/--\n`o = none` is decidable even if the wrapped type does not have decidable equality.\n\nThis is not an instance because it is not definitionally equal to `option.decidable_eq`.\nTry to use `o.is_none` or `o.is_some` instead.\n-/\n@[inline]\ndef decidable_eq_none {o : option \u03b1} : decidable (o = none) :=\ndecidable_of_decidable_of_iff (bool.decidable_eq _ _) is_none_iff_eq_none\n\ninstance decidable_forall_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2200 a \u2208 o, p a)\n| none     := is_true (by simp [false_implies_iff])\n| (some a) := if h : p a\n  then is_true $ \u03bb o e, some_inj.1 e \u25b8 h\n  else is_false $ mt (\u03bb H, H _ rfl) h\n\ninstance decidable_exists_mem {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  \u2200 o : option \u03b1, decidable (\u2203 a \u2208 o, p a)\n| none     := is_false (\u03bb \u27e8a, \u27e8h, _\u27e9\u27e9, by cases h)\n| (some a) := if h : p a\n  then is_true $ \u27e8_, rfl, h\u27e9\n  else is_false $ \u03bb \u27e8_, \u27e8rfl, hn\u27e9\u27e9, h hn\n\n/-- Inhabited `get` function. Returns `a` if the input is `some a`, otherwise returns `default`. -/\n@[reducible] def iget [inhabited \u03b1] : option \u03b1 \u2192 \u03b1\n| (some x) := x\n| none     := default \u03b1\n\n@[simp] theorem iget_some [inhabited \u03b1] {a : \u03b1} : (some a).iget = a := rfl\n\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [decidable_pred p] (a : \u03b1) : option \u03b1 :=\nif p a then some a else none\n\n/-- `filter p o` returns `some a` if `o` is `some a` and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (o : option \u03b1) : option \u03b1 :=\no.bind (guard p)\n\n/-- Cast of `option` to `list `. Returns `[a]` if the input is `some a`, and `[]` if it is\n`none`. -/\ndef to_list : option \u03b1 \u2192 list \u03b1\n| none     := []\n| (some a) := [a]\n\n@[simp] theorem mem_to_list {a : \u03b1} {o : option \u03b1} : a \u2208 to_list o \u2194 a \u2208 o :=\nby cases o; simp [to_list, eq_comm]\n\n/-- Two arguments failsafe function. Returns `f a b` if the inputs are `some a` and `some b`, and\n\"does nothing\" otherwise. -/\ndef lift_or_get (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : option \u03b1 \u2192 option \u03b1 \u2192 option \u03b1\n| none     none     := none\n| (some a) none     := some a       -- get a\n| none     (some b) := some b       -- get b\n| (some a) (some b) := some (f a b) -- lift f\n\ninstance lift_or_get_comm (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_commutative \u03b1 f] :\n  is_commutative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b, by cases a; cases b; simp [lift_or_get, h.comm]\u27e9\n\ninstance lift_or_get_assoc (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_associative \u03b1 f] :\n  is_associative (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a b c, by cases a; cases b; cases c; simp [lift_or_get, h.assoc]\u27e9\n\ninstance lift_or_get_idem (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : is_idempotent \u03b1 f] :\n  is_idempotent (option \u03b1) (lift_or_get f) :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get, h.idempotent]\u27e9\n\ninstance lift_or_get_is_left_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_left_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\ninstance lift_or_get_is_right_id (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) :\n  is_right_id (option \u03b1) (lift_or_get f) none :=\n\u27e8\u03bb a, by cases a; simp [lift_or_get]\u27e9\n\n/-- Lifts a relation `\u03b1 \u2192 \u03b2 \u2192 Prop` to a relation `option \u03b1 \u2192 option \u03b2 \u2192 Prop` by just adding\n`none ~ none`. -/\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : option \u03b1 \u2192 option \u03b2 \u2192 Prop\n/-- If `a ~ b`, then `some a ~ some b` -/\n| some {a b} : r a b \u2192 rel (some a) (some b)\n/-- `none ~ none` -/\n| none       : rel none none\n\n/-- Partial bind. If for some `x : option \u03b1`, `f : \u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2` is a\n  partial function defined on `a : \u03b1` giving an `option \u03b2`, where `some a = x`,\n  then `pbind x f h` is essentially the same as `bind x f`\n  but is defined only when all `x = some a`, using the proof to apply `f`. -/\n@[simp] def pbind : \u03a0 (x : option \u03b1), (\u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2) \u2192 option \u03b2\n| none     _ := none\n| (some a) f := f a rfl\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on `a : \u03b1` satisfying `p`,\nthen `pmap f x h` is essentially the same as `map f x` but is defined only when all members of `x`\nsatisfy `p`, using the proof to apply `f`. -/\n@[simp] def pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 (a : \u03b1), p a \u2192 \u03b2) :\n  \u03a0 x : option \u03b1, (\u2200 a \u2208 x, p a) \u2192 option \u03b2\n| none     _ := none\n| (some a) H := some (f a (H a (mem_def.mpr rfl)))\n\n/-- Flatten an `option` of `option`, a specialization of `mjoin`. -/\n@[simp] def join : option (option \u03b1) \u2192 option \u03b1 :=\n\u03bb x, bind x id\n\nprotected def {u v} traverse {F : Type u \u2192 Type v} [applicative F] {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 F \u03b2) :\n  option \u03b1 \u2192 F (option \u03b2)\n| none     := pure none\n| (some x) := some <$> f x\n\n/- By analogy with `monad.sequence` in `init/category/combinators.lean`. -/\n\n/-- If you maybe have a monadic computation in a `[monad m]` which produces a term of type `\u03b1`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult. -/\ndef {u v} maybe {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type u} : option (m \u03b1) \u2192 m (option \u03b1)\n| none      := return none\n| (some fn) := some <$> fn\n\n/-- Map a monadic function `f : \u03b1 \u2192 m \u03b2` over an `o : option \u03b1`, maybe producing a result. -/\ndef {u v w} mmap {m : Type u \u2192 Type v} [monad m] {\u03b1 : Type w} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2)\n  (o : option \u03b1) : m (option \u03b2) := (o.map f).maybe\n\n/-- A monadic analogue of `option.elim`. -/\ndef melim {\u03b1 \u03b2 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b2) (z : \u03b1 \u2192 m \u03b2) :\n  m \u03b2 :=\nx >>= \u03bb o, option.elim o y z\n\n/-- A monadic analogue of `option.get_or_else`. -/\ndef mget_or_else {\u03b1 : Type*} {m : Type* \u2192 Type*} [monad m] (x : m (option \u03b1)) (y : m \u03b1) : m \u03b1 :=\nmelim x y pure\n\nend option\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/option/defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784220301065, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.3777199127592331}}
{"text": "/-\nCopyright (c) 2020 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Adam Topaz\n-/\n\nimport category_theory.concrete_category.basic\nimport category_theory.full_subcategory\nimport category_theory.skeletal\nimport category_theory.elementwise\nimport data.fin.basic\nimport data.fintype.basic\n\n/-!\n# The category of finite types.\n\nWe define the category of finite types, denoted `Fintype` as\n(bundled) types with a `fintype` instance.\n\nWe also define `Fintype.skeleton`, the standard skeleton of `Fintype` whose objects are `fin n`\nfor `n : \u2115`. We prove that the obvious inclusion functor `Fintype.skeleton \u2964 Fintype` is an\nequivalence of categories in `Fintype.skeleton.equivalence`.\nWe prove that `Fintype.skeleton` is a skeleton of `Fintype` in `Fintype.is_skeleton`.\n-/\n\nopen_locale classical\nopen category_theory\n\n/-- The category of finite types. -/\ndef Fintype := bundled fintype\n\nnamespace Fintype\n\ninstance : has_coe_to_sort Fintype Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled `Fintype` from the underlying type and typeclass. -/\ndef of (X : Type*) [fintype X] : Fintype := bundled.of X\ninstance : inhabited Fintype := \u27e8\u27e8pempty\u27e9\u27e9\ninstance {X : Fintype} : fintype X := X.2\n\ninstance : category Fintype := induced_category.category bundled.\u03b1\n\n/-- The fully faithful embedding of `Fintype` into the category of types. -/\n@[derive [full, faithful], simps]\ndef incl : Fintype \u2964 Type* := induced_functor _\n\ninstance concrete_category_Fintype : concrete_category Fintype := \u27e8incl\u27e9\n\n@[simp] lemma id_apply (X : Fintype) (x : X) : (\ud835\udfd9 X : X \u2192 X) x = x := rfl\n@[simp] lemma comp_apply {X Y Z : Fintype} (f : X \u27f6 Y) (g : Y \u27f6 Z) (x : X) :\n  (f \u226b g) x = g (f x) := rfl\n\n/-- Equivalences between finite types are the same as isomorphisms in `Fintype`. -/\n-- See `equiv_equiv_iso` in the root namespace for the analogue in `Type`.\n@[simps]\ndef equiv_equiv_iso {A B : Fintype} : (A \u2243 B) \u2243 (A \u2245 B) :=\n{ to_fun := \u03bb e,\n  { hom := e,\n    inv := e.symm, },\n  inv_fun := \u03bb i,\n  { to_fun := i.hom,\n    inv_fun := i.inv,\n    left_inv := iso.hom_inv_id_apply i,\n    right_inv := iso.inv_hom_id_apply i, },\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\nuniverse u\n/--\nThe \"standard\" skeleton for `Fintype`. This is the full subcategory of `Fintype` spanned by objects\nof the form `ulift (fin n)` for `n : \u2115`. We parameterize the objects of `Fintype.skeleton`\ndirectly as `ulift \u2115`, as the type `ulift (fin m) \u2243 ulift (fin n)` is\nnonempty if and only if `n = m`. Specifying universes, `skeleton : Type u` is a small\nskeletal category equivalent to `Fintype.{u}`.\n-/\ndef skeleton : Type u := ulift \u2115\n\nnamespace skeleton\n\n/-- Given any natural number `n`, this creates the associated object of `Fintype.skeleton`. -/\ndef mk : \u2115 \u2192 skeleton := ulift.up\n\ninstance : inhabited skeleton := \u27e8mk 0\u27e9\n\n/-- Given any object of `Fintype.skeleton`, this returns the associated natural number. -/\ndef len : skeleton \u2192 \u2115 := ulift.down\n\n@[ext]\nlemma ext (X Y : skeleton) : X.len = Y.len \u2192 X = Y := ulift.ext _ _\n\ninstance : small_category skeleton.{u} :=\n{ hom := \u03bb X Y, ulift.{u} (fin X.len) \u2192 ulift.{u} (fin Y.len),\n  id := \u03bb _, id,\n  comp := \u03bb _ _ _ f g, g \u2218 f }\n\nlemma is_skeletal : skeletal skeleton.{u} := \u03bb X Y \u27e8h\u27e9, ext _ _ $ fin.equiv_iff_eq.mp $\n  nonempty.intro $\n{ to_fun := \u03bb x, (h.hom \u27e8x\u27e9).down,\n  inv_fun := \u03bb x, (h.inv \u27e8x\u27e9).down,\n  left_inv := begin\n    intro a,\n    change ulift.down _ = _,\n    rw ulift.up_down,\n    change ((h.hom \u226b h.inv) _).down = _,\n    simpa,\n  end,\n  right_inv := begin\n    intro a,\n    change ulift.down _ = _,\n    rw ulift.up_down,\n    change ((h.inv \u226b h.hom) _).down = _,\n    simpa,\n  end }\n\n/-- The canonical fully faithful embedding of `Fintype.skeleton` into `Fintype`. -/\ndef incl : skeleton.{u} \u2964 Fintype.{u} :=\n{ obj := \u03bb X, Fintype.of (ulift (fin X.len)),\n  map := \u03bb _ _ f, f }\n\ninstance : full incl := { preimage := \u03bb _ _ f, f }\ninstance : faithful incl := {}\ninstance : ess_surj incl :=\ness_surj.mk $ \u03bb X, let F := fintype.equiv_fin X in \u27e8mk (fintype.card X), nonempty.intro\n  { hom := F.symm \u2218 ulift.down,\n    inv := ulift.up \u2218 F }\u27e9\n\nnoncomputable instance : is_equivalence incl :=\nequivalence.of_fully_faithfully_ess_surj _\n\n/-- The equivalence between `Fintype.skeleton` and `Fintype`. -/\nnoncomputable def equivalence : skeleton \u224c Fintype := incl.as_equivalence\n\n@[simp] lemma incl_mk_nat_card (n : \u2115) : fintype.card (incl.obj (mk n)) = n :=\nbegin\n  convert finset.card_fin n,\n  apply fintype.of_equiv_card,\nend\n\nend skeleton\n\n/-- `Fintype.skeleton` is a skeleton of `Fintype`. -/\nnoncomputable def is_skeleton : is_skeleton_of Fintype skeleton skeleton.incl :=\n{ skel := skeleton.is_skeletal,\n  eqv := by apply_instance }\n\nend Fintype\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/Fintype.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6584174938590246, "lm_q1q2_score": 0.37771989931591776}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport algebra.module.basic\nimport algebra.ring.pi\n\n/-!\n# Pi instances for module and multiplicative actions\n\nThis file defines instances for module, mul_action and related structures on Pi Types\n-/\n\nnamespace pi\nuniverses u v w\nvariable {I : Type u}     -- The indexing type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\n\ninstance has_scalar {\u03b1 : Type*} [\u03a0 i, has_scalar \u03b1 $ f i] :\n  has_scalar \u03b1 (\u03a0 i : I, f i) :=\n\u27e8\u03bb s x, \u03bb i, s \u2022 (x i)\u27e9\n\n@[simp] lemma smul_apply {\u03b1 : Type*} [\u03a0 i, has_scalar \u03b1 $ f i] (s : \u03b1) : (s \u2022 x) i = s \u2022 x i := rfl\n\ninstance has_scalar' {g : I \u2192 Type*} [\u03a0 i, has_scalar (f i) (g i)] :\n  has_scalar (\u03a0 i, f i) (\u03a0 i : I, g i) :=\n\u27e8\u03bb s x, \u03bb i, (s i) \u2022 (x i)\u27e9\n\n@[simp]\nlemma smul_apply' {g : I \u2192 Type*} [\u2200 i, has_scalar (f i) (g i)] (s : \u03a0 i, f i) (x : \u03a0 i, g i) :\n  (s \u2022 x) i = s i \u2022 x i :=\nrfl\n\ninstance is_scalar_tower {\u03b1 \u03b2 : Type*}\n  [has_scalar \u03b1 \u03b2] [\u03a0 i, has_scalar \u03b2 $ f i] [\u03a0 i, has_scalar \u03b1 $ f i]\n  [\u03a0 i, is_scalar_tower \u03b1 \u03b2 (f i)] : is_scalar_tower \u03b1 \u03b2 (\u03a0 i : I, f i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_assoc x y (z i)\u27e9\n\ninstance is_scalar_tower' {g : I \u2192 Type*} {\u03b1 : Type*}\n  [\u03a0 i, has_scalar \u03b1 $ f i] [\u03a0 i, has_scalar (f i) (g i)] [\u03a0 i, has_scalar \u03b1 $ g i]\n  [\u03a0 i, is_scalar_tower \u03b1 (f i) (g i)] : is_scalar_tower \u03b1 (\u03a0 i : I, f i) (\u03a0 i : I, g i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_assoc x (y i) (z i)\u27e9\n\ninstance is_scalar_tower'' {g : I \u2192 Type*} {h : I \u2192 Type*}\n  [\u03a0 i, has_scalar (f i) (g i)] [\u03a0 i, has_scalar (g i) (h i)] [\u03a0 i, has_scalar (f i) (h i)]\n  [\u03a0 i, is_scalar_tower (f i) (g i) (h i)] : is_scalar_tower (\u03a0 i, f i) (\u03a0 i, g i) (\u03a0 i, h i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_assoc (x i) (y i) (z i)\u27e9\n\ninstance smul_comm_class {\u03b1 \u03b2 : Type*}\n  [\u03a0 i, has_scalar \u03b1 $ f i] [\u03a0 i, has_scalar \u03b2 $ f i] [\u2200 i, smul_comm_class \u03b1 \u03b2 (f i)] :\n  smul_comm_class \u03b1 \u03b2 (\u03a0 i : I, f i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_comm x y (z i)\u27e9\n\ninstance smul_comm_class' {g : I \u2192 Type*} {\u03b1 : Type*}\n  [\u03a0 i, has_scalar \u03b1 $ g i] [\u03a0 i, has_scalar (f i) (g i)] [\u2200 i, smul_comm_class \u03b1 (f i) (g i)] :\n  smul_comm_class \u03b1 (\u03a0 i : I, f i) (\u03a0 i : I, g i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_comm x (y i) (z i)\u27e9\n\ninstance smul_comm_class'' {g : I \u2192 Type*} {h : I \u2192 Type*}\n  [\u03a0 i, has_scalar (g i) (h i)] [\u03a0 i, has_scalar (f i) (h i)]\n  [\u2200 i, smul_comm_class (f i) (g i) (h i)] : smul_comm_class (\u03a0 i, f i) (\u03a0 i, g i) (\u03a0 i, h i) :=\n\u27e8\u03bb x y z, funext $ \u03bb i, smul_comm (x i) (y i) (z i)\u27e9\n\ninstance mul_action (\u03b1) {m : monoid \u03b1} [\u03a0 i, mul_action \u03b1 $ f i] :\n  @mul_action \u03b1 (\u03a0 i : I, f i) m :=\n{ smul := (\u2022),\n  mul_smul := \u03bb r s f, funext $ \u03bb i, mul_smul _ _ _,\n  one_smul := \u03bb f, funext $ \u03bb i, one_smul \u03b1 _ }\n\ninstance mul_action' {g : I \u2192 Type*} {m : \u03a0 i, monoid (f i)} [\u03a0 i, mul_action (f i) (g i)] :\n  @mul_action (\u03a0 i, f i) (\u03a0 i : I, g i) (@pi.monoid I f m) :=\n{ smul := (\u2022),\n  mul_smul := \u03bb r s f, funext $ \u03bb i, mul_smul _ _ _,\n  one_smul := \u03bb f, funext $ \u03bb i, one_smul _ _ }\n\ninstance distrib_mul_action (\u03b1) {m : monoid \u03b1} {n : \u2200 i, add_monoid $ f i}\n  [\u2200 i, distrib_mul_action \u03b1 $ f i] :\n  @distrib_mul_action \u03b1 (\u03a0 i : I, f i) m (@pi.add_monoid I f n) :=\n{ smul_zero := \u03bb c, funext $ \u03bb i, smul_zero _,\n  smul_add := \u03bb c f g, funext $ \u03bb i, smul_add _ _ _,\n  ..pi.mul_action _ }\n\ninstance distrib_mul_action' {g : I \u2192 Type*} {m : \u03a0 i, monoid (f i)} {n : \u03a0 i, add_monoid $ g i}\n  [\u03a0 i, distrib_mul_action (f i) (g i)] :\n  @distrib_mul_action (\u03a0 i, f i) (\u03a0 i : I, g i) (@pi.monoid I f m) (@pi.add_monoid I g n) :=\n{ smul_add := by { intros, ext x, apply smul_add },\n  smul_zero := by { intros, ext x, apply smul_zero } }\n\nlemma single_smul {\u03b1} [monoid \u03b1] [\u03a0 i, add_monoid $ f i]\n  [\u03a0 i, distrib_mul_action \u03b1 $ f i] [decidable_eq I] (i : I) (r : \u03b1) (x : f i) :\n  single i (r \u2022 x) = r \u2022 single i x :=\nsingle_op (\u03bb i : I, ((\u2022) r : f i \u2192 f i)) (\u03bb j, smul_zero _) _ _\n\nlemma single_smul' {g : I \u2192 Type*} [\u03a0 i, monoid_with_zero (f i)] [\u03a0 i, add_monoid (g i)]\n  [\u03a0 i, distrib_mul_action (f i) (g i)] [decidable_eq I] (i : I) (r : f i) (x : g i) :\n  single i (r \u2022 x) = single i r \u2022 single i x :=\nsingle_op\u2082 (\u03bb i : I, ((\u2022) : f i \u2192 g i \u2192 g i)) (\u03bb j, smul_zero _) _ _ _\n\nvariables (I f)\n\ninstance module (\u03b1) {r : semiring \u03b1} {m : \u2200 i, add_comm_monoid $ f i}\n  [\u2200 i, module \u03b1 $ f i] :\n  @module \u03b1 (\u03a0 i : I, f i) r (@pi.add_comm_monoid I f m) :=\n{ add_smul := \u03bb c f g, funext $ \u03bb i, add_smul _ _ _,\n  zero_smul := \u03bb f, funext $ \u03bb i, zero_smul \u03b1 _,\n  ..pi.distrib_mul_action _ }\n\nvariables {I f}\n\ninstance module' {g : I \u2192 Type*} {r : \u03a0 i, semiring (f i)} {m : \u03a0 i, add_comm_monoid (g i)}\n  [\u03a0 i, module (f i) (g i)] :\n  module (\u03a0 i, f i) (\u03a0 i, g i) :=\n{ add_smul := by { intros, ext1, apply add_smul },\n  zero_smul := by { intros, ext1, apply zero_smul } }\n\ninstance (\u03b1) {r : semiring \u03b1} {m : \u03a0 i, add_comm_monoid $ f i}\n  [\u03a0 i, module \u03b1 $ f i] [\u2200 i, no_zero_smul_divisors \u03b1 $ f i] :\n  no_zero_smul_divisors \u03b1 (\u03a0 i : I, f i) :=\n\u27e8\u03bb c x h, or_iff_not_imp_left.mpr (\u03bb hc, funext\n  (\u03bb i, (smul_eq_zero.mp (congr_fun h i)).resolve_left hc))\u27e9\n\nend pi\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/algebra/module/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.3777198993159177}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Tim Baumann, Stephen Morgan, Scott Morrison\n\nimport category_theory.isomorphism\nimport category_theory.functor_category\n\nopen category_theory\n\nnamespace category_theory.nat_iso\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C] {D : Type u\u2082} [\ud835\udc9f : category.{v\u2082} D]\ninclude \ud835\udc9e \ud835\udc9f\n\ndef app {F G : C \u2964 D} (\u03b1 : F \u2245 G) (X : C) : F.obj X \u2245 G.obj X :=\n{ hom := \u03b1.hom.app X,\n  inv := \u03b1.inv.app X,\n  hom_inv_id' := begin rw [\u2190 functor.category.comp_app, iso.hom_inv_id], refl, end,\n  inv_hom_id' := begin rw [\u2190 functor.category.comp_app, iso.inv_hom_id], refl, end }\n\n@[simp] lemma comp_app {F G H : C \u2964 D} (\u03b1 : F \u2245 G) (\u03b2 : G \u2245 H) (X : C) :\n  app (\u03b1 \u226a\u226b \u03b2) X = app \u03b1 X \u226a\u226b app \u03b2 X := rfl\n\n@[simp] lemma app_hom {F G : C \u2964 D} (\u03b1 : F \u2245 G) (X : C) : (app \u03b1 X).hom = \u03b1.hom.app X := rfl\n@[simp] lemma app_inv {F G : C \u2964 D} (\u03b1 : F \u2245 G) (X : C) : (app \u03b1 X).inv = \u03b1.inv.app X := rfl\n\nvariables {F G : C \u2964 D}\n\ninstance hom_app_is_iso (\u03b1 : F \u2245 G) (X : C) : is_iso (\u03b1.hom.app X) :=\n{ inv := \u03b1.inv.app X,\n  hom_inv_id' := begin rw [\u2190functor.category.comp_app, iso.hom_inv_id, \u2190functor.category.id_app] end,\n  inv_hom_id' := begin rw [\u2190functor.category.comp_app, iso.inv_hom_id, \u2190functor.category.id_app] end }\ninstance inv_app_is_iso (\u03b1 : F \u2245 G) (X : C) : is_iso (\u03b1.inv.app X) :=\n{ inv := \u03b1.hom.app X,\n  hom_inv_id' := begin rw [\u2190functor.category.comp_app, iso.inv_hom_id, \u2190functor.category.id_app] end,\n  inv_hom_id' := begin rw [\u2190functor.category.comp_app, iso.hom_inv_id, \u2190functor.category.id_app] end }\n\n@[simp] lemma hom_vcomp_inv (\u03b1 : F \u2245 G) : (\u03b1.hom \u229f \u03b1.inv) = nat_trans.id _ :=\nbegin\n  have h : (\u03b1.hom \u229f \u03b1.inv) = \u03b1.hom \u226b \u03b1.inv := rfl,\n  rw h,\n  rw iso.hom_inv_id,\n  refl\nend\n@[simp] lemma inv_vcomp_hom (\u03b1 : F \u2245 G) : (\u03b1.inv \u229f \u03b1.hom) = nat_trans.id _ :=\nbegin\n  have h : (\u03b1.inv \u229f \u03b1.hom) = \u03b1.inv \u226b \u03b1.hom := rfl,\n  rw h,\n  rw iso.inv_hom_id,\n  refl\nend\n\n@[simp] lemma hom_app_inv_app_id (\u03b1 : F \u2245 G) (X : C) : \u03b1.hom.app X \u226b \u03b1.inv.app X = \ud835\udfd9 _ :=\nbegin\n  rw \u2190nat_trans.vcomp_app,\n  simp,\nend\n@[simp] lemma inv_app_hom_app_id (\u03b1 : F \u2245 G) (X : C) : \u03b1.inv.app X \u226b \u03b1.hom.app X = \ud835\udfd9 _ :=\nbegin\n  rw \u2190nat_trans.vcomp_app,\n  simp,\nend\n\nvariables {X Y : C}\n@[simp] lemma naturality_1 (\u03b1 : F \u2245 G) (f : X \u27f6 Y) :\n  (\u03b1.inv.app X) \u226b (F.map f) \u226b (\u03b1.hom.app Y) = G.map f :=\nbegin erw [nat_trans.naturality, \u2190category.assoc, is_iso.hom_inv_id, category.id_comp] end\n@[simp] lemma naturality_2 (\u03b1 : F \u2245 G) (f : X \u27f6 Y) :\n  (\u03b1.hom.app X) \u226b (G.map f) \u226b (\u03b1.inv.app Y) = F.map f :=\nbegin erw [nat_trans.naturality, \u2190category.assoc, is_iso.hom_inv_id, category.id_comp] end\n\ninstance is_iso_of_is_iso_app (\u03b1 : F \u27f6 G) [\u2200 X : C, is_iso (\u03b1.app X)] : is_iso \u03b1 :=\n{ inv :=\n  { app := \u03bb X, inv (\u03b1.app X),\n    naturality' := \u03bb X Y f,\n    by simpa using congr_arg (\u03bb f, inv (\u03b1.app X) \u226b (f \u226b inv (\u03b1.app Y))) (\u03b1.naturality f).symm } }\n\ndef of_components (app : \u2200 X : C, (F.obj X) \u2245 (G.obj X))\n  (naturality : \u2200 {X Y : C} (f : X \u27f6 Y), (F.map f) \u226b ((app Y).hom) = ((app X).hom) \u226b (G.map f)) :\n  F \u2245 G :=\nas_iso { app := \u03bb X, (app X).hom }\n\n@[simp] def of_components.app (app' : \u2200 X : C, (F.obj X) \u2245 (G.obj X)) (naturality) (X) :\n  app (of_components app' naturality) X = app' X :=\nby tidy\n@[simp] def of_components.hom_app (app : \u2200 X : C, (F.obj X) \u2245 (G.obj X)) (naturality) (X) :\n  (of_components app naturality).hom.app X = (app X).hom := rfl\n@[simp] def of_components.inv_app (app : \u2200 X : C, (F.obj X) \u2245 (G.obj X)) (naturality) (X) :\n  (of_components app naturality).inv.app X = (app X).inv := rfl\n\nend category_theory.nat_iso\n\nnamespace category_theory.functor\n\nuniverses u\u2081 u\u2082 v\u2081 v\u2082\n\nsection\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C]\n          {D : Type u\u2082} [\ud835\udc9f : category.{v\u2082} D]\ninclude \ud835\udc9e \ud835\udc9f\n\n@[simp] protected def id_comp (F : C \u2964 D) : functor.id C \u22d9 F \u2245 F :=\n{ hom := { app := \u03bb X, \ud835\udfd9 (F.obj X) },\n  inv := { app := \u03bb X, \ud835\udfd9 (F.obj X) } }\n@[simp] protected def comp_id (F : C \u2964 D) : F \u22d9 functor.id D \u2245 F :=\n{ hom := { app := \u03bb X, \ud835\udfd9 (F.obj X) },\n  inv := { app := \u03bb X, \ud835\udfd9 (F.obj X) } }\n\nuniverses u\u2083 v\u2083 u\u2084 v\u2084\n\nvariables {A : Type u\u2083} [\ud835\udc9c : category.{v\u2083} A]\n          {B : Type u\u2084} [\u212c : category.{v\u2084} B]\ninclude \ud835\udc9c \u212c\nvariables (F : A \u2964 B) (G : B \u2964 C) (H : C \u2964 D)\n\n@[simp] protected def assoc : (F \u22d9 G) \u22d9 H \u2245 F \u22d9 (G \u22d9 H ):=\n{ hom := { app := \u03bb X, \ud835\udfd9 (H.obj (G.obj (F.obj X))) },\n  inv := { app := \u03bb X, \ud835\udfd9 (H.obj (G.obj (F.obj X))) } }\n\n-- When it's time to define monoidal categories and 2-categories,\n-- we'll need to add lemmas relating these natural isomorphisms,\n-- in particular the pentagon for the associator.\nend\n\nsection\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C]\ninclude \ud835\udc9e\n\ndef ulift_down_up : ulift_down.{v\u2081} C \u22d9 ulift_up C \u2245 functor.id (ulift.{u\u2082} C) :=\n{ hom := { app := \u03bb X, @category_struct.id (ulift.{u\u2082} C) _ X },\n  inv := { app := \u03bb X, @category_struct.id (ulift.{u\u2082} C) _ X } }\n\ndef ulift_up_down : ulift_up.{v\u2081} C \u22d9 ulift_down C \u2245 functor.id C :=\n{ hom := { app := \u03bb X, \ud835\udfd9 X },\n  inv := { app := \u03bb X, \ud835\udfd9 X } }\n\nend\n\nend category_theory.functor\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/natural_isomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3777198954707473}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.shapes.normal_mono.equalizers\nimport category_theory.abelian.images\nimport category_theory.preadditive\n\n/-!\n# Every non_preadditive_abelian category is preadditive\n\nIn mathlib, we define an abelian category as a preadditive category with a zero object,\nkernels and cokernels, products and coproducts and in which every monomorphism and epimorphis is\nnormal.\n\nWhile virtually every interesting abelian category has a natural preadditive structure (which is why\nit is included in the definition), preadditivity is not actually needed: Every category that has\nall of the other properties appearing in the definition of an abelian category admits a preadditive\nstructure. This is the construction we carry out in this file.\n\nThe proof proceeds in roughly five steps:\n1. Prove some results (for example that all equalizers exist) that would be trivial if we already\n   had the preadditive structure but are a bit of work without it.\n2. Develop images and coimages to show that every monomorphism is the kernel of its cokernel.\n\nThe results of the first two steps are also useful for the \"normal\" development of abelian\ncategories, and will be used there.\n\n3. For every object `A`, define a \"subtraction\" morphism `\u03c3 : A \u2a2f A \u27f6 A` and use it to define\n   subtraction on morphisms as `f - g := prod.lift f g \u226b \u03c3`.\n4. Prove a small number of identities about this subtraction from the definition of `\u03c3`.\n5. From these identities, prove a large number of other identities that imply that defining\n   `f + g := f - (0 - g)` indeed gives an abelian group structure on morphisms such that composition\n   is bilinear.\n\nThe construction is non-trivial and it is quite remarkable that this abelian group structure can\nbe constructed purely from the existence of a few limits and colimits. Even more remarkably,\nsince abelian categories admit exactly one preadditive structure (see\n`subsingleton_preadditive_of_has_binary_biproducts`), the construction manages to exactly\nreconstruct any natural preadditive structure the category may have.\n\n## References\n\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nnamespace category_theory\nsection\nuniverses v u\n\nvariables (C : Type u) [category.{v} C]\n\n/-- We call a category `non_preadditive_abelian` if it has a zero object, kernels, cokernels, binary\n    products and coproducts, and every monomorphism and every epimorphism is normal. -/\nclass non_preadditive_abelian extends has_zero_morphisms C, normal_mono_category C,\n  normal_epi_category C :=\n[has_zero_object : has_zero_object C]\n[has_kernels : has_kernels C]\n[has_cokernels : has_cokernels C]\n[has_finite_products : has_finite_products C]\n[has_finite_coproducts : has_finite_coproducts C]\n\nset_option default_priority 100\n\nattribute [instance] non_preadditive_abelian.has_zero_object\nattribute [instance] non_preadditive_abelian.has_kernels\nattribute [instance] non_preadditive_abelian.has_cokernels\nattribute [instance] non_preadditive_abelian.has_finite_products\nattribute [instance] non_preadditive_abelian.has_finite_coproducts\n\nend\nend category_theory\n\nopen category_theory\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C] [non_preadditive_abelian C]\n\nnamespace category_theory.non_preadditive_abelian\n\nsection factor\n\nvariables {P Q : C} (f : P \u27f6 Q)\n\n/-- The map `p : P \u27f6 image f` is an epimorphism -/\ninstance : epi (abelian.factor_thru_image f) :=\nlet I := abelian.image f, p := abelian.factor_thru_image f,\n    i := kernel.\u03b9 (cokernel.\u03c0 f) in\n-- It will suffice to consider some g : I \u27f6 R such that p \u226b g = 0 and show that g = 0.\nnormal_mono_category.epi_of_zero_cancel _ $ \u03bb R (g : I \u27f6 R) (hpg : p \u226b g = 0),\nbegin\n  -- Since C is abelian, u := ker g \u226b i is the kernel of some morphism h.\n  let u := kernel.\u03b9 g \u226b i,\n  haveI : mono u := mono_comp _ _,\n  haveI hu := normal_mono_of_mono u,\n  let h := hu.g,\n  -- By hypothesis, p factors through the kernel of g via some t.\n  obtain \u27e8t, ht\u27e9 := kernel.lift' g p hpg,\n  have fh : f \u226b h = 0, calc\n    f \u226b h = (p \u226b i) \u226b h : (abelian.image.fac f).symm \u25b8 rfl\n       ... = ((t \u226b kernel.\u03b9 g) \u226b i) \u226b h : ht \u25b8 rfl\n       ... = t \u226b u \u226b h : by simp only [category.assoc]; conv_lhs { congr, skip, rw \u2190category.assoc }\n       ... = t \u226b 0 : hu.w \u25b8 rfl\n       ... = 0 : has_zero_morphisms.comp_zero _ _,\n  -- h factors through the cokernel of f via some l.\n  obtain \u27e8l, hl\u27e9 := cokernel.desc' f h fh,\n  have hih : i \u226b h = 0, calc\n    i \u226b h = i \u226b cokernel.\u03c0 f \u226b l : hl \u25b8 rfl\n       ... = 0 \u226b l : by rw [\u2190category.assoc, kernel.condition]\n       ... = 0 : zero_comp,\n  -- i factors through u = ker h via some s.\n  obtain \u27e8s, hs\u27e9 := normal_mono.lift' u i hih,\n  have hs' : (s \u226b kernel.\u03b9 g) \u226b i = \ud835\udfd9 I \u226b i, by rw [category.assoc, hs, category.id_comp],\n  haveI : epi (kernel.\u03b9 g) := epi_of_epi_fac ((cancel_mono _).1 hs'),\n  -- ker g is an epimorphism, but ker g \u226b g = 0 = ker g \u226b 0, so g = 0 as required.\n  exact zero_of_epi_comp _ (kernel.condition g)\nend\n\ninstance is_iso_factor_thru_image [mono f] : is_iso (abelian.factor_thru_image f) :=\nis_iso_of_mono_of_epi _\n\n/-- The canonical morphism `i : coimage f \u27f6 Q` is a monomorphism -/\ninstance : mono (abelian.factor_thru_coimage f) :=\nlet I := abelian.coimage f, i := abelian.factor_thru_coimage f,\n    p := cokernel.\u03c0 (kernel.\u03b9 f) in\nnormal_epi_category.mono_of_cancel_zero _ $ \u03bb R (g : R \u27f6 I) (hgi : g \u226b i = 0),\nbegin\n  -- Since C is abelian, u := p \u226b coker g is the cokernel of some morphism h.\n  let u := p \u226b cokernel.\u03c0 g,\n  haveI : epi u := epi_comp _ _,\n  haveI hu := normal_epi_of_epi u,\n  let h := hu.g,\n  -- By hypothesis, i factors through the cokernel of g via some t.\n  obtain \u27e8t, ht\u27e9 := cokernel.desc' g i hgi,\n  have hf : h \u226b f = 0, calc\n    h \u226b f = h \u226b (p \u226b i) : (abelian.coimage.fac f).symm \u25b8 rfl\n    ... = h \u226b (p \u226b (cokernel.\u03c0 g \u226b t)) : ht \u25b8 rfl\n    ... = h \u226b u \u226b t : by simp only [category.assoc]; conv_lhs { congr, skip, rw \u2190category.assoc }\n    ... = 0 \u226b t : by rw [\u2190category.assoc, hu.w]\n    ... = 0 : zero_comp,\n  -- h factors through the kernel of f via some l.\n  obtain \u27e8l, hl\u27e9 := kernel.lift' f h hf,\n  have hhp : h \u226b p = 0, calc\n    h \u226b p = (l \u226b kernel.\u03b9 f) \u226b p : hl \u25b8 rfl\n    ... = l \u226b 0 : by rw [category.assoc, cokernel.condition]\n    ... = 0 : comp_zero,\n  -- p factors through u = coker h via some s.\n  obtain \u27e8s, hs\u27e9 := normal_epi.desc' u p hhp,\n  have hs' : p \u226b cokernel.\u03c0 g \u226b s = p \u226b \ud835\udfd9 I, by rw [\u2190category.assoc, hs, category.comp_id],\n  haveI : mono (cokernel.\u03c0 g) := mono_of_mono_fac ((cancel_epi _).1 hs'),\n  -- coker g is a monomorphism, but g \u226b coker g = 0 = 0 \u226b coker g, so g = 0 as required.\n  exact zero_of_comp_mono _ (cokernel.condition g)\nend\n\ninstance is_iso_factor_thru_coimage [epi f] :\n  is_iso (abelian.factor_thru_coimage f) :=\nis_iso_of_mono_of_epi _\n\nend factor\n\nsection cokernel_of_kernel\nvariables {X Y : C} {f : X \u27f6 Y}\n\n/-- In a `non_preadditive_abelian` category, an epi is the cokernel of its kernel. More precisely:\n    If `f` is an epimorphism and `s` is some limit kernel cone on `f`, then `f` is a cokernel\n    of `fork.\u03b9 s`. -/\ndef epi_is_cokernel_of_kernel [epi f] (s : fork f 0) (h : is_limit s) :\n  is_colimit (cokernel_cofork.of_\u03c0 f (kernel_fork.condition s)) :=\nis_cokernel.cokernel_iso _ _\n  (cokernel.of_iso_comp _ _\n    (limits.is_limit.cone_point_unique_up_to_iso (limit.is_limit _) h)\n    (cone_morphism.w (limits.is_limit.unique_up_to_iso (limit.is_limit _) h).hom _))\n  (as_iso $ abelian.factor_thru_coimage f) (abelian.coimage.fac f)\n\n/-- In a `non_preadditive_abelian` category, a mono is the kernel of its cokernel. More precisely:\n    If `f` is a monomorphism and `s` is some colimit cokernel cocone on `f`, then `f` is a kernel\n    of `cofork.\u03c0 s`. -/\ndef mono_is_kernel_of_cokernel [mono f] (s : cofork f 0) (h : is_colimit s) :\n  is_limit (kernel_fork.of_\u03b9 f (cokernel_cofork.condition s)) :=\nis_kernel.iso_kernel _ _\n  (kernel.of_comp_iso _ _\n    (limits.is_colimit.cocone_point_unique_up_to_iso h (colimit.is_colimit _))\n    (cocone_morphism.w (limits.is_colimit.unique_up_to_iso h $ colimit.is_colimit _).hom _))\n  (as_iso $ abelian.factor_thru_image f) (abelian.image.fac f)\n\nend cokernel_of_kernel\nsection\n\n/-- The composite `A \u27f6 A \u2a2f A \u27f6 cokernel (\u0394 A)`, where the first map is `(\ud835\udfd9 A, 0)` and the second map\n    is the canonical projection into the cokernel. -/\nabbreviation r (A : C) : A \u27f6 cokernel (diag A) := prod.lift (\ud835\udfd9 A) 0 \u226b cokernel.\u03c0 (diag A)\n\ninstance mono_\u0394 {A : C} : mono (diag A) := mono_of_mono_fac $ prod.lift_fst _ _\n\ninstance mono_r {A : C} : mono (r A) :=\nbegin\n  let hl : is_limit (kernel_fork.of_\u03b9 (diag A) (cokernel.condition (diag A))),\n  { exact mono_is_kernel_of_cokernel _ (colimit.is_colimit _) },\n  apply normal_epi_category.mono_of_cancel_zero,\n  intros Z x hx,\n  have hxx : (x \u226b prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) \u226b cokernel.\u03c0 (diag A) = 0,\n  { rw [category.assoc, hx] },\n  obtain \u27e8y, hy\u27e9 := kernel_fork.is_limit.lift' hl _ hxx,\n  rw kernel_fork.\u03b9_of_\u03b9 at hy,\n  have hyy : y = 0,\n  { erw [\u2190category.comp_id y, \u2190limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A),  \u2190category.assoc, hy,\n      category.assoc, prod.lift_snd, has_zero_morphisms.comp_zero] },\n  haveI : mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) := mono_of_mono_fac (prod.lift_fst _ _),\n  apply (cancel_mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A))).1,\n  rw [\u2190hy, hyy, zero_comp, zero_comp]\nend\n\ninstance epi_r {A : C} : epi (r A) :=\nbegin\n  have hlp : prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A) \u226b limits.prod.snd = 0 := prod.lift_snd _ _,\n  let hp1 : is_limit (kernel_fork.of_\u03b9 (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) hlp),\n  { refine fork.is_limit.mk _ (\u03bb s, fork.\u03b9 s \u226b limits.prod.fst) _ _,\n    { intro s,\n      ext; simp, erw category.comp_id },\n    { intros s m h,\n      haveI : mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A)) := mono_of_mono_fac (prod.lift_fst _ _),\n      apply (cancel_mono (prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A))).1,\n      convert h walking_parallel_pair.zero,\n      ext; simp } },\n  let hp2 : is_colimit (cokernel_cofork.of_\u03c0 (limits.prod.snd : A \u2a2f A \u27f6 A) hlp),\n  { exact epi_is_cokernel_of_kernel _ hp1 },\n  apply normal_mono_category.epi_of_zero_cancel,\n  intros Z z hz,\n  have h : prod.lift (\ud835\udfd9 A) (0 : A \u27f6 A) \u226b cokernel.\u03c0 (diag A) \u226b z = 0,\n  { rw [\u2190category.assoc, hz] },\n  obtain \u27e8t, ht\u27e9 := cokernel_cofork.is_colimit.desc' hp2 _ h,\n  rw cokernel_cofork.\u03c0_of_\u03c0 at ht,\n  have htt : t = 0,\n  { rw [\u2190category.id_comp t],\n    change \ud835\udfd9 A \u226b t = 0,\n    rw [\u2190limits.prod.lift_snd (\ud835\udfd9 A) (\ud835\udfd9 A), category.assoc, ht, \u2190category.assoc,\n      cokernel.condition, zero_comp] },\n  apply (cancel_epi (cokernel.\u03c0 (diag A))).1,\n  rw [\u2190ht, htt, comp_zero, comp_zero]\nend\n\ninstance is_iso_r {A : C} : is_iso (r A) :=\nis_iso_of_mono_of_epi _\n\n/-- The composite `A \u2a2f A \u27f6 cokernel (diag A) \u27f6 A` given by the natural projection into the cokernel\n    followed by the inverse of `r`. In the category of modules, using the normal kernels and\n    cokernels, this map is equal to the map `(a, b) \u21a6 a - b`, hence the name `\u03c3` for\n    \"subtraction\". -/\nabbreviation \u03c3 {A : C} : A \u2a2f A \u27f6 A := cokernel.\u03c0 (diag A) \u226b inv (r A)\n\nend\n\n@[simp, reassoc] lemma diag_\u03c3 {X : C} : diag X \u226b \u03c3 = 0 :=\nby rw [cokernel.condition_assoc, zero_comp]\n\n@[simp, reassoc] lemma lift_\u03c3 {X : C} : prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 = \ud835\udfd9 X :=\nby rw [\u2190category.assoc, is_iso.hom_inv_id]\n\n@[reassoc] lemma lift_map {X Y : C} (f : X \u27f6 Y) :\n  prod.lift (\ud835\udfd9 X) 0 \u226b limits.prod.map f f = f \u226b prod.lift (\ud835\udfd9 Y) 0 :=\nby simp\n\n/-- \u03c3 is a cokernel of \u0394 X. -/\ndef is_colimit_\u03c3 {X : C} : is_colimit (cokernel_cofork.of_\u03c0 \u03c3 diag_\u03c3) :=\ncokernel.cokernel_iso _ \u03c3 (as_iso (r X)).symm (by rw [iso.symm_hom, as_iso_inv])\n\n/-- This is the key identity satisfied by `\u03c3`. -/\nlemma \u03c3_comp {X Y : C} (f : X \u27f6 Y) : \u03c3 \u226b f = limits.prod.map f f \u226b \u03c3 :=\nbegin\n  obtain \u27e8g, hg\u27e9 :=\n    cokernel_cofork.is_colimit.desc' is_colimit_\u03c3 (limits.prod.map f f \u226b \u03c3) (by simp),\n  suffices hfg : f = g,\n  { rw [\u2190hg, cofork.\u03c0_of_\u03c0, hfg] },\n  calc f = f \u226b prod.lift (\ud835\udfd9 Y) 0 \u226b \u03c3 : by rw [lift_\u03c3, category.comp_id]\n    ... = prod.lift (\ud835\udfd9 X) 0 \u226b limits.prod.map f f \u226b \u03c3 : by rw lift_map_assoc\n    ... = prod.lift (\ud835\udfd9 X) 0 \u226b \u03c3 \u226b g : by rw [\u2190hg, cokernel_cofork.\u03c0_of_\u03c0]\n    ... = g : by rw [\u2190category.assoc, lift_\u03c3, category.id_comp]\nend\n\nsection\n\n/- We write `f - g` for `prod.lift f g \u226b \u03c3`. -/\n/-- Subtraction of morphisms in a `non_preadditive_abelian` category. -/\ndef has_sub {X Y : C} : has_sub (X \u27f6 Y) := \u27e8\u03bb f g, prod.lift f g \u226b \u03c3\u27e9\nlocal attribute [instance] has_sub\n\n/- We write `-f` for `0 - f`. -/\n/-- Negation of morphisms in a `non_preadditive_abelian` category. -/\ndef has_neg {X Y : C} : has_neg (X \u27f6 Y) := \u27e8\u03bb f, 0 - f\u27e9\nlocal attribute [instance] has_neg\n\n/- We write `f + g` for `f - (-g)`. -/\n/-- Addition of morphisms in a `non_preadditive_abelian` category. -/\ndef has_add {X Y : C} : has_add (X \u27f6 Y) := \u27e8\u03bb f g, f - (-g)\u27e9\nlocal attribute [instance] has_add\n\nlemma sub_def {X Y : C} (a b : X \u27f6 Y) : a - b = prod.lift a b \u226b \u03c3 := rfl\nlemma add_def {X Y : C} (a b : X \u27f6 Y) : a + b = a - (-b) := rfl\nlemma neg_def {X Y : C} (a : X \u27f6 Y) : -a = 0 - a := rfl\n\n\n\nlemma sub_self {X Y : C} (a : X \u27f6 Y) : a - a = 0 :=\nby rw [sub_def, \u2190category.comp_id a, \u2190 prod.comp_lift, category.assoc, diag_\u03c3, comp_zero]\n\nlemma lift_sub_lift {X Y : C} (a b c d : X \u27f6 Y) :\n  prod.lift a b - prod.lift c d = prod.lift (a - c) (b - d) :=\nbegin\n  simp only [sub_def],\n  ext,\n  { rw [category.assoc, \u03c3_comp, prod.lift_map_assoc, prod.lift_fst, prod.lift_fst, prod.lift_fst] },\n  { rw [category.assoc, \u03c3_comp, prod.lift_map_assoc, prod.lift_snd, prod.lift_snd, prod.lift_snd] }\nend\n\nlemma sub_sub_sub {X Y : C} (a b c d : X \u27f6 Y) : (a - c) - (b - d) = (a - b) - (c - d) :=\nbegin\n  rw [sub_def, \u2190lift_sub_lift, sub_def, category.assoc, \u03c3_comp, prod.lift_map_assoc], refl\nend\n\nlemma neg_sub {X Y : C} (a b : X \u27f6 Y) : (-a) - b = (-b) - a :=\nby conv_lhs { rw [neg_def, \u2190sub_zero b, sub_sub_sub, sub_zero, \u2190neg_def] }\n\nlemma neg_neg {X Y : C} (a : X \u27f6 Y) : -(-a) = a :=\nbegin\n  rw [neg_def, neg_def],\n  conv_lhs { congr, rw \u2190sub_self a },\n  rw [sub_sub_sub, sub_zero, sub_self, sub_zero]\nend\n\nlemma add_comm {X Y : C} (a b : X \u27f6 Y) : a + b = b + a :=\nbegin\n  rw [add_def],\n  conv_lhs { rw \u2190neg_neg a },\n  rw [neg_def, neg_def, neg_def, sub_sub_sub],\n  conv_lhs {congr, skip, rw [\u2190neg_def, neg_sub] },\n  rw [sub_sub_sub, add_def, \u2190neg_def, neg_neg b, neg_def]\nend\n\nlemma add_neg {X Y : C} (a b : X \u27f6 Y) : a + (-b) = a - b :=\nby rw [add_def, neg_neg]\n\nlemma add_neg_self {X Y : C} (a : X \u27f6 Y) : a + (-a) = 0 :=\nby rw [add_neg, sub_self]\n\nlemma neg_add_self {X Y : C} (a : X \u27f6 Y) : (-a) + a = 0 :=\nby rw [add_comm, add_neg_self]\n\nlemma neg_sub' {X Y : C} (a b : X \u27f6 Y) : -(a - b) = (-a) + b :=\nbegin\n  rw [neg_def, neg_def],\n  conv_lhs { rw \u2190sub_self (0 : X \u27f6 Y) },\n  rw [sub_sub_sub, add_def, neg_def]\nend\n\nlemma neg_add {X Y : C} (a b : X \u27f6 Y) : -(a + b) = (-a) - b :=\nby rw [add_def, neg_sub', add_neg]\n\nlemma sub_add {X Y : C} (a b c : X \u27f6 Y) : (a - b) + c = a - (b - c) :=\nby rw [add_def, neg_def, sub_sub_sub, sub_zero]\n\nlemma add_assoc {X Y : C} (a b c : X \u27f6 Y) : (a + b) + c = a + (b + c) :=\nbegin\n  conv_lhs { congr, rw add_def },\n  rw [sub_add, \u2190add_neg, neg_sub', neg_neg]\nend\n\nlemma add_zero {X Y : C} (a : X \u27f6 Y) : a + 0 = a :=\nby rw [add_def, neg_def, sub_self, sub_zero]\n\nlemma comp_sub {X Y Z : C} (f : X \u27f6 Y) (g h : Y \u27f6 Z) : f \u226b (g - h) = f \u226b g - f \u226b h :=\nby rw [sub_def, \u2190category.assoc, prod.comp_lift, sub_def]\n\nlemma sub_comp {X Y Z : C} (f g : X \u27f6 Y) (h : Y \u27f6 Z) : (f - g) \u226b h = f \u226b h - g \u226b h :=\nby rw [sub_def, category.assoc, \u03c3_comp, \u2190category.assoc, prod.lift_map, sub_def]\n\nlemma comp_add (X Y Z : C) (f : X \u27f6 Y) (g h : Y \u27f6 Z) : f \u226b (g + h) = f \u226b g + f \u226b h :=\nby rw [add_def, comp_sub, neg_def, comp_sub, comp_zero, add_def, neg_def]\n\nlemma add_comp (X Y Z : C) (f g : X \u27f6 Y) (h : Y \u27f6 Z) : (f + g) \u226b h = f \u226b h + g \u226b h :=\nby rw [add_def, sub_comp, neg_def, sub_comp, zero_comp, add_def, neg_def]\n\n/-- Every `non_preadditive_abelian` category is preadditive. -/\ndef preadditive : preadditive C :=\n{ hom_group := \u03bb X Y,\n  { add := (+),\n    add_assoc := add_assoc,\n    zero := 0,\n    zero_add := neg_neg,\n    add_zero := add_zero,\n    neg := \u03bb f, -f,\n    add_left_neg := neg_add_self,\n    add_comm := add_comm },\n  add_comp' := add_comp,\n  comp_add' := comp_add }\n\nend\n\nend category_theory.non_preadditive_abelian\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/abelian/non_preadditive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.37768882231239476}}
{"text": "import data.cpi.species data.cpi.process.basic\n\nnamespace cpi\nnamespace process\n\nvariables {\u2102 \u210d : Type} {\u03c9 \u0393 : context} [has_add \u2102] [\u2200 \u0393, setoid (species \u210d \u03c9 \u0393)]\n\n/-- Structural congruence of processes. -/\ninductive equiv : process \u2102 \u210d \u03c9 \u0393 \u2192 process \u2102 \u210d \u03c9 \u0393 \u2192 Prop\n| refl  {A}     : equiv A A\n| trans {A B C} : equiv A B \u2192 equiv B C \u2192 equiv A C\n| symm  {A B}   : equiv A B \u2192 equiv B A\n\n-- Projection\n| \u03be_species   {c : \u2102} {A B} : A \u2248 B \u2192 equiv (c \u25ef A) (c \u25ef B)\n| \u03be_parallel\u2081 {P P' Q} : equiv P P' \u2192 equiv (P |\u209a Q) (P' |\u209a Q)\n| \u03be_parallel\u2082 {P Q Q'} : equiv Q Q' \u2192 equiv (P |\u209a Q) (P |\u209a Q')\n\n-- Monoidic properties\n| parallel_nil   {P} {c : \u2102} : equiv (P |\u209a c \u25ef species.nil) P\n| parallel_symm  {P Q} : equiv (P |\u209a Q) (Q |\u209a P)\n| parallel_assoc {P Q R} : equiv ((P |\u209a Q) |\u209a R) (P |\u209a (Q |\u209a R))\n\n-- Join identical species together.\n| join  {A} {c d} : equiv (c \u25ef A |\u209a d \u25ef A) ((c + d) \u25ef A)\n| split {A B} {c : \u2102} : equiv (c \u25ef (A |\u209b B)) (c \u25ef A |\u209a c \u25ef B)\n\ninstance : is_equiv (process \u2102 \u210d \u03c9 \u0393) equiv :=\n  { refl := @equiv.refl _ _ _ _ _ _, symm := @equiv.symm _ _ _ _ _ _, trans := @equiv.trans _ _ _ _ _ _ }\ninstance : is_refl (process \u2102 \u210d \u03c9 \u0393) equiv := \u27e8 \u03bb _, equiv.refl \u27e9\ninstance : setoid (process \u2102 \u210d \u03c9 \u0393) :=\n  \u27e8 equiv, \u27e8 @equiv.refl _ _ _ _ _ _, @equiv.symm _ _ _ _ _ _, @equiv.trans _ _ _ _ _ _ \u27e9 \u27e9\ninstance setoid.is_equiv : is_equiv (process \u2102 \u210d \u03c9 \u0393) has_equiv.equiv :=\n  process.is_equiv\n\nnamespace equiv\n  lemma parallel_symm\u2081 {P Q R : process \u2102 \u210d \u03c9 \u0393} : (P |\u209a Q |\u209a R) \u2248 (Q |\u209a P |\u209a R) :=\n    calc  (P |\u209a (Q |\u209a R))\n        \u2248 ((P |\u209a Q) |\u209a R) : symm parallel_assoc\n    ... \u2248 ((Q |\u209a P) |\u209a R) : \u03be_parallel\u2081 parallel_symm\n    ... \u2248 (Q |\u209a (P |\u209a R)) : parallel_assoc\n\n  lemma parallel_symm\u2082 {P Q R : process \u2102 \u210d \u03c9 \u0393} : ((P |\u209a Q) |\u209a R) \u2248 ((P |\u209a R) |\u209a Q) :=\n    calc  ((P |\u209a Q) |\u209a R)\n        \u2248 (P |\u209a (Q |\u209a R)) : parallel_assoc\n    ... \u2248 (P |\u209a (R |\u209a Q)) : \u03be_parallel\u2082 parallel_symm\n    ... \u2248 ((P |\u209a R) |\u209a Q) : symm parallel_assoc\nend equiv\n\nnamespace parallel.quot\n  /-- Make a parallel process from a quotient of two process. -/\n  def mk : quotient (@process.setoid \u2102 \u210d \u03c9 \u0393 _ _) \u2192 quotient (@process.setoid \u2102 \u210d \u03c9 \u0393 _ _) \u2192 quotient (@process.setoid \u2102 \u210d \u03c9 \u0393 _ _)\n  | A B := quotient.lift_on\u2082 A B (\u03bb A B, \u27e6 A |\u209a B \u27e7)\n      (\u03bb A B A' B' eqA eqB, quot.sound (trans (equiv.\u03be_parallel\u2081 eqA) ((equiv.\u03be_parallel\u2082 eqB))))\n\n  lemma assoc (A B C : quotient (@process.setoid \u2102 \u210d \u03c9 \u0393 _ _))\n    : mk A (mk B C) = mk (mk A B) C\n    := begin\n      rcases quot.exists_rep A with \u27e8 A, \u27e8 _ \u27e9 \u27e9,\n      rcases quot.exists_rep B with \u27e8 B, \u27e8 _ \u27e9 \u27e9,\n      rcases quot.exists_rep C with \u27e8 C, \u27e8 _ \u27e9 \u27e9,\n      from quot.sound (symm equiv.parallel_assoc),\n    end\nend parallel.quot\n\nend process\n\n/-- A quotient of all structurally congruent processes. -/\n@[nolint has_inhabited_instance]\ndef process' (\u2102 \u210d : Type) (\u03c9 \u0393 : context) [has_add \u2102] [\u2200 {\u0393}, setoid (species \u210d \u03c9 \u0393)]\n  := quotient (@process.setoid \u2102 \u210d \u03c9 \u0393 _ _)\n\nsection prime\n  variables {\u2102 \u210d : Type} {\u03c9 \u0393 : context} [\u2200 \u0393, setoid (species \u210d \u03c9 \u0393)]\n\n  /-- Convert a list of prime species into a process-/\n  def process.from_primes [add_monoid \u2102] {\u0393} (f : prime_species' \u210d \u03c9 \u0393 \u2192 \u2102)\n    : list (prime_species' \u210d \u03c9 \u0393) \u2192 process' \u2102 \u210d \u03c9 \u0393\n  | [] := \u27e6 0 \u25ef nil \u27e7\n  | (A :: As) :=\n    let A' := quot.lift_on A (\u03bb B, \u27e6 f A \u25ef B.val \u27e7)\n                (\u03bb A B r, quot.sound (process.equiv.\u03be_species r))\n    in process.parallel.quot.mk A' (process.from_primes As)\n\n  /-- Convert a multiset of prime species into a process. -/\n  def process.from_prime_multiset [add_monoid \u2102] {\u0393} (f : prime_species' \u210d \u03c9 \u0393 \u2192 \u2102)\n    : multiset (prime_species' \u210d \u03c9 \u0393) \u2192 process' \u2102 \u210d \u03c9 \u0393\n  | Ps := quot.lift_on Ps (process.from_primes f) (\u03bb P Q r, begin\n    induction r,\n    case list.perm.nil { from rfl },\n    case list.perm.trans : A B C _ _ ab bc { from trans ab bc },\n    case list.perm.skip : A As Bs _ ih { simp only [process.from_primes, ih] },\n    case list.perm.swap : A B As {\n      simp only [process.from_primes],\n      rcases quot.exists_rep A with \u27e8 A, eq \u27e9, subst eq,\n      rcases quot.exists_rep B with \u27e8 B, eq \u27e9, subst eq,\n      rcases quot.exists_rep (process.from_primes f As) with \u27e8 As, eq \u27e9, rw \u2190 eq, clear eq,\n      from quot.sound process.equiv.parallel_symm\u2081,\n    },\n  end)\nend prime\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/process/equivalence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.685949442167993, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3776888152424864}}
{"text": "/-\nCopyright (c) 2018-2019 Minchao Wu. All rights reserved.\nReleased under MIT license as described in the file LICENSE.\nAuthor: Minchao Wu\n-/\n\nimport defs data.list.perm .data\nopen nnf tactic\n\nmeta def frame_restriction : tactic unit :=\ndo intro `a >> `[simp]\n\nstructure S4 (states : Type) extends kripke states :=\n(refl  : reflexive rel . frame_restriction)\n(trans : transitive rel . frame_restriction)\n\ninstance inhabited_S4 : inhabited (S4 \u2115) := \n\u27e8 { val := \u03bb a b, tt, rel := \u03bb a b, tt } \u27e9\n\n@[simp] def force {states : Type} (k : S4 states) : states \u2192 nnf \u2192 Prop\n| s (var n)    := k.val n s\n| s (neg n)    := \u00ac k.val n s\n| s (and \u03c6 \u03c8)  := force s \u03c6 \u2227 force s \u03c8\n| s (or \u03c6 \u03c8)   := force s \u03c6 \u2228 force s \u03c8\n| s (box \u03c6)    := \u2200 s', k.rel s s' \u2192 force s' \u03c6\n| s (dia \u03c6)    := \u2203 s', k.rel s s' \u2227 force s' \u03c6\n\ndef sat {st} (k : S4 st) (s) (\u0393 : list nnf) : Prop := \n\u2200 \u03c6 \u2208 \u0393, force k s \u03c6\n\ndef unsatisfiable (\u0393 : list nnf) : Prop := \n\u2200 (st) (k : S4 st) s, \u00ac sat k s \u0393\n\ntheorem unsat_singleton {\u03c6} : unsatisfiable [\u03c6] \u2192 \u2200 (st) (k : S4 st) s, \u00ac force k s \u03c6\n := \nbegin\n  intros h _ _ _ hf,\n  apply h, intros \u03c8 h\u03c8, rw list.mem_singleton at h\u03c8, rw h\u03c8, exact hf\nend\n\ntheorem sat_of_empty {st} (k : S4 st) (s) : sat k s [] :=\n\u03bb \u03c6 h, absurd h $ list.not_mem_nil _\n\ntheorem ne_empty_of_unsat {\u0393} (h : unsatisfiable \u0393): \u0393 \u2260 [] := \nbegin \n  intro heq, rw heq at h, \n  apply h, apply sat_of_empty, exact nat, \n  apply inhabited_S4.1, exact 0 \nend\n\nopen tmodel\n\n@[simp] def minfo : \u03a0 m : tmodel, info\n| (cons i l ba) := i\n\n@[simp] def htk : \u03a0 m : tmodel, list nnf\n| (cons i l ba) := i.htk\n\ndef hist : \u03a0 m : tmodel, list nnf\n| (cons i l ba) := i.id.h\n\n@[simp] def msig : \u03a0 m : tmodel, sig\n| (cons i l ba) := i.id.s\n\n@[simp] def manc : \u03a0 m : tmodel, list psig\n| (cons i l ba) := i.id.a\n\ndef bhist : \u03a0 m : tmodel, list nnf\n| (cons i l ba) := i.id.b\n\n@[simp] def request : \u03a0 m : tmodel, list psig\n| (cons i l ba) := ba\n\n@[simp] def proper_request_box : \u03a0 m : tmodel, Prop\n| (cons i l ba) := \u2200 rq : psig, rq \u2208 ba \u2192 \u2200 \u03c6, (box \u03c6 \u2208 i.htk \u2228 box \u03c6 \u2208 i.id.b) \u2192 box \u03c6 \u2208 rq.b\n\n@[simp] def subset_request : \u03a0 m : tmodel, Prop\n| (cons i l ba) := ba \u2286 i.id.a\n\n@[simp] def tmodel_step_bhist : \u03a0 m : tmodel, Prop \n| m@(cons i l ba) := \u2200 s \u2208 l, \u2200 \u03c6, box \u03c6 \u2208 i.id.b \u2192 box \u03c6 \u2208 htk s\n\n@[simp] def tmodel_step_box : \u03a0 m : tmodel, Prop \n| m@(cons i l ba) := \u2200 s \u2208 l, \u2200 \u03c6, box \u03c6 \u2208 i.htk \u2192 box \u03c6 \u2208 htk s\n\n-- Can be strenghtened\n@[simp] def tmodel_dia : \u03a0 m : tmodel, Prop \n| m@(cons i l ba) := \u2200 \u03c6, dia \u03c6 \u2208 i.htk \u2192 (\u2203 rq : psig, rq \u2208 ba \u2227 rq.d = \u03c6) \u2228 \u2203 s \u2208 l, \u03c6 \u2208 htk s\n\n@[simp] def child : tmodel \u2192 tmodel \u2192 bool\n| s (cons i l ba) := s \u2208 l\n\ninductive tc' {\u03b1 : Type} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : \u03b1 \u2192 \u03b1 \u2192 Prop\n| base : \u2200 a b, r a b \u2192 tc' a b\n| step : \u2200 a b c, r a b \u2192 tc' b c \u2192 tc' a c\n\ntheorem tc'.trans {\u03b1 : Type} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a b c : \u03b1} : \ntc' r a b \u2192 tc' r b c \u2192 tc' r a c :=\nbegin\nintros h\u2081 h\u2082,\ninduction h\u2081,\napply tc'.step, exact h\u2081_a_1, exact h\u2082,\napply tc'.step, exact h\u2081_a_1, apply h\u2081_ih, exact h\u2082\nend\n\ndef desc : tmodel \u2192 tmodel \u2192 Prop := tc' (\u03bb s m, child s m)\n\ntheorem desc_not_nil : \u03a0 c i ba m, m = cons i [] ba \u2192 desc c m \u2192 false :=\nbegin\nintros c i ba m heq h,\ninduction h,\n{rw heq at h_a_1, simp at h_a_1, exact h_a_1},\n{apply h_ih, exact heq}\nend\n\ntheorem desc_iff_eq_child_aux : \u03a0 i\u2081 i\u2082 s\u2081 s\u2082 l\u2081 l\u2082 m\u2081 m\u2082 m\u2083, \nm\u2081 = cons i\u2081 l\u2081 s\u2081 \u2192 m\u2082 = cons i\u2082 l\u2082 s\u2082 \u2192 l\u2081 = l\u2082 \u2192 \n(desc m\u2083 m\u2081 \u2194 desc m\u2083 m\u2082) :=\nbegin\nintros i\u2081 i\u2082 s\u2081 s\u2082 l\u2081 l\u2082 m\u2081 m\u2082 m\u2083 heq\u2081 heq\u2082 heq,\nsplit,\n{intro hd, induction hd,\n {rw heq\u2081 at hd_a_1, simp at hd_a_1,\n rw heq at hd_a_1,\n apply tc'.base, rw heq\u2082, simp, exact hd_a_1},\n {apply tc'.trans, apply tc'.base, exact hd_a_1, apply hd_ih, exact heq\u2081}},\n{intro hd, induction hd,\n {rw heq\u2082 at hd_a_1, simp at hd_a_1,\n rw \u2190heq at hd_a_1,\n apply tc'.base, rw heq\u2081, simp, exact hd_a_1},\n {apply tc'.trans, apply tc'.base, exact hd_a_1, apply hd_ih, exact heq\u2082}}\nend\n\ntheorem eq_desc_of_eq_children {i\u2081 i\u2082 s\u2081 s\u2082 l c} : \ndesc c (cons i\u2081 l s\u2081) = desc c (cons i\u2082 l s\u2082) :=\nbegin rw desc_iff_eq_child_aux, repeat {refl} end \n\ntheorem desc_step : \u03a0 c i l ba, c \u2208 l \u2192 desc c (cons i l ba)\n| c i [] ba h := absurd h $ list.not_mem_nil _\n| c i (hd::tl) ba h := \nbegin\nconstructor,\nsimp, cases h,\nleft, exact h, right, exact h\nend\n\ntheorem desc_ex : \u03a0 c i l ba, (\u2203 m \u2208 l, desc c m) \u2192 desc c (cons i l ba)\n| c i [] ba h := begin rcases h with \u27e8w, hmem, hw\u27e9, exact (absurd hmem $ list.not_mem_nil _) end\n| c i (hd::tl) ba h := \nbegin\nrcases h with \u27e8w, hmem, hw\u27e9,\ncases hw,\n{apply tc'.step,\nswap 3, {exact w},\n{exact hw_a_1},\n{apply tc'.base, simp, cases hmem, left, exact hmem, right, exact hmem}},\n{apply tc'.step, exact hw_a_1, apply tc'.trans, exact hw_a_2, apply tc'.base, simp, exact hmem}\nend\n\ntheorem ex_desc : \u03a0 c i l ba m, m = (cons i l ba) \u2192 desc c m \u2192 (c \u2208 l \u2228 \u2203 m \u2208 l, desc c m) := \nbegin\nintros c i l ba m heq h,\ninduction h,\n{left, rw heq at h_a_1, simp at h_a_1, exact h_a_1},\n{cases h_ih heq,\n {right, split, split, exact h, apply tc'.base, exact h_a_1},\n {rcases h with \u27e8w, hmem, hw\u27e9, right, split, split, exact hmem, apply tc'.step, exact h_a_1, exact hw}}\nend\n\ntheorem ex_desc' : \u03a0 c i l ba, desc c (cons i l ba) \u2192 (c \u2208 l \u2228 \u2203 m \u2208 l, desc c m) := \nbegin intros c i l ba h, apply ex_desc, repeat {refl}, exact h end\n\n@[simp] def tmodel_anc : \u03a0 m : tmodel, Prop \n| m@(cons i l ba) := \u2200 s rq, desc s m \u2192 rq \u2208 request s \u2192  \n                     (rq \u2208 manc m) \u2228\n                     (\u2203 d, desc d m \u2227 some rq = msig d)\n\nstructure ptmodel (m : tmodel) : Prop :=\n(bhist : tmodel_step_bhist m)\n(sbox : tmodel_step_box m)\n(pdia : tmodel_dia m)\n(bdia : tmodel_anc m)\n(reqb : proper_request_box m)\n(sreq : subset_request m)\n\ndef global_pt (m : tmodel) := \u2200 s, desc s m \u2192 ptmodel s\n\nopen subtype\n\ndef model : Type := {m : tmodel // ptmodel m \u2227 global_pt m}\n\ndef rmodel : Type := {m : tmodel // ptmodel m}\n\ninductive reach_step : rmodel \u2192 rmodel \u2192 Prop\n| fwd_base (s : rmodel) (i l ba h) : s.1 \u2208 l \u2192 reach_step \u27e8(cons i l ba), h\u27e9  s\n| bwd_base (s : rmodel) (i l ba h) : (\u2203 rq \u2208 ba, some rq = msig s.1) \u2192 reach_step \u27e8(cons i l ba), h\u27e9 s\n\ntheorem reach_step_box (s\u2081 s\u2082 \u03c6) (h\u2081 : reach_step s\u2081 s\u2082) (h\u2082 : box \u03c6 \u2208 htk s\u2081.1) : box \u03c6 \u2208 htk s\u2082.1 :=\nbegin\ncases h\u2081,\n{cases s\u2082 with s\u2082 ps\u2082,\n cases s\u2082 with i\u2082 l\u2082 sg\u2082,\n simp,\n have := h\u2081_h.sbox,\n simp at this, simp at h\u2082,\n have hmem := this _ h\u2081_a _ h\u2082,\n simp at hmem, exact hmem },\n{cases s\u2082 with s\u2082 ps\u2082,\n cases s\u2082 with i\u2082 l\u2082 sg\u2082,\n simp,\n rcases h\u2081_a with \u27e8w,hmem,hw\u27e9,\n simp at hw,\n apply i\u2082.mhtk,\n have := i\u2082.id.ps\u2082,\n rw \u2190hw at this,\n have hneq : some w \u2260 none, \n   {intro heq, contradiction},\n have hsub := this hneq,\n apply hsub,\n have := h\u2081_h.reqb,\n simp at this, simp at h\u2082,\n have hc := this w hmem \u03c6 (or.inl h\u2082),\n cases w, \n dsimp [bsig], exact hc}\nend\n\ninductive rtc {\u03b1 : Type} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : \u03b1 \u2192 \u03b1 \u2192 Prop\n| refl   : \u03a0 a, rtc a a\n| step   : \u03a0 a b c, r a b \u2192 rtc b c \u2192 rtc a c\n\ntheorem rtc.trans {\u03b1 : Type} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a b c : \u03b1} : \nrtc r a b \u2192 rtc r b c \u2192 rtc r a c :=\nbegin\nintros h\u2081 h\u2082,\ninduction h\u2081,\nexact h\u2082,\napply rtc.step, exact h\u2081_a_1, apply h\u2081_ih, exact h\u2082\nend\n\ntheorem rtc_step {\u03b1 : Type} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {a b : \u03b1} (h : r a b) : \nrtc r a b :=\nby apply rtc.step _ _ _ h; apply rtc.refl\n\ndef reach (s\u2081 s\u2082 : rmodel) := rtc reach_step s\u2081 s\u2082\n\ntheorem refl_reach : \u03a0 s, reach s s := \u03bb s, rtc.refl s\n\ntheorem trans_reach : \u03a0 s\u2081 s\u2082 s\u2083, reach s\u2081 s\u2082 \u2192 reach s\u2082 s\u2083 \u2192 reach s\u2081 s\u2083 := \u03bb s\u2081 s\u2082 s\u2083 h\u2081 h\u2082, rtc.trans h\u2081 h\u2082\n\n@[simp] def builder (m : tmodel) : S4 {x : rmodel // x.1 = m \u2228 desc x.1 m} := \n{val := \u03bb v s, var v \u2208 htk s.1.1, \n rel := \u03bb s\u2081 s\u2082, reach s\u2081 s\u2082, \n refl := \u03bb s, refl_reach s, \n trans := \u03bb a b c, trans_reach a b c}\n\nopen rtc\n\ntheorem reach_box (s\u2081 s\u2082 \u03c6) (h\u2081 : reach s\u2081 s\u2082) (h\u2082 : box \u03c6 \u2208 htk s\u2081.1) : \u03c6 \u2208 htk s\u2082.1 :=\nbegin\ninduction h\u2081 with m m\u2081 m\u2082 m\u2083 h\u2081\u2082 h\u2082\u2083 ih, \n{cases m with tm hm, cases tm with i l sg,\nsimp,\napply i.hhtk.hbox,\nsimp at h\u2082, exact h\u2082},\n{apply ih, \n apply reach_step_box,\n exact h\u2081\u2082, exact h\u2082}\nend\n\ntheorem reach_step_dia (s : rmodel) (rt : model) (\u03c6) \n(h\u2081 : desc s.1 rt.1) \n(h\u2082 : manc rt.1 = []) (h\u2083 : dia \u03c6 \u2208 htk s.1) : \n\u2203 s', reach_step s s' \u2227 \u03c6 \u2208 htk s'.1 \u2227 desc s'.1 rt.1 :=\nbegin\ncases s with s ps,\ncases s with i l sg,\nhave := ps.pdia,\nsimp at this, simp at h\u2083,\nhave hc := this _ h\u2083,\ncases hc,\n{cases rt with rt prt,\n cases rt with irt lrt sgrt,\n rcases hc with \u27e8w, hmem, hw\u27e9,\n have := prt.1.bdia,\n simp at this, simp at h\u2081,\n have hcaux := this _ w h\u2081,\n simp at hcaux,\n have hcc := hcaux hmem,\n simp at h\u2082,\n cases hcc,\n {rw h\u2082 at hcc, exfalso, apply list.not_mem_nil, exact hcc},\n {rcases hcc with \u27e8m, hml, hmr\u27e9, \n  have pm := prt.2 m hml,\n  split, split,\n  swap 3, exact \u27e8m, pm\u27e9,\n  apply reach_step.bwd_base,\n  split, split, exact hmem, simp, exact hmr,\n  split,\n  {cases m with im lm sgm, simp,\n   apply im.mhtk, \n   have := im.id.ps\u2081,\n   simp at hmr, rw \u2190hmr at this,\n   have hneq : some w \u2260 none, {intro, contradiction},\n   have hmem := this hneq, \n   cases w, dsimp [dsig] at hmem,\n   rw \u2190hw, exact hmem_1 },\n  {exact hml} } },\n{rcases hc with \u27e8m, pml, pmr\u27e9,\n have hdm : desc m rt.1, \n  {apply tc'.trans, apply tc'.base, \n   swap 3, exact (\u27e8cons i l sg, ps\u27e9 : rmodel).val, \n   simp, exact pml, exact h\u2081},\n cases rt with rt prt,\n cases rt with irt lrt sgrt,\n have pm := prt.2 m hdm,\n split, split, swap 3,\n exact \u27e8m, pm\u27e9,\n apply reach_step.fwd_base,\n exact pml, split,\n {exact pmr},\n {exact hdm} }\nend\n\ntheorem reach_dia (s : rmodel) (rt : model) (\u03c6) \n(h\u2081 : desc s.1 rt.1) \n(h\u2082 : manc rt.1 = []) (h\u2083 : dia \u03c6 \u2208 htk s.1) : \n\u2203 s', reach s s' \u2227 \u03c6 \u2208 htk s'.1 \u2227 desc s'.1 rt.1:=\nbegin\nhave := reach_step_dia s rt \u03c6 h\u2081 h\u2082 h\u2083,\nrcases this with \u27e8w, hwl, hwr\u27e9,\nsplit, split, swap 3, exact w,\napply rtc_step hwl, exact hwr\nend\n\ntheorem reach_step_dia_root (s : rmodel) (rt : model) (\u03c6) \n(h\u2081 : s.1 = rt.1) \n(h\u2082 : manc rt.1 = []) (h\u2083 : dia \u03c6 \u2208 htk s.1) : \n\u2203 s', reach_step s s' \u2227 \u03c6 \u2208 htk s'.1 \u2227 desc s'.1 rt.1 :=\nbegin\ncases s with s ps,\ncases s with is ls sgs,\nhave := ps.pdia,\nsimp at this, simp at h\u2083,\nhave hc := this _ h\u2083,\ncases hc,\n{have := ps.sreq, simp at this, \n rcases hc with \u27e8w, hmw, hw\u27e9,\n have hmem := this hmw,\n cases rt with rt prt,\n rw \u2190h\u2081 at h\u2082,\n simp at h\u2082, rw h\u2082 at hmem,\n exfalso, apply list.not_mem_nil, exact hmem},\n{cases rt with rt prt,\n rcases hc with \u27e8w, hwl, hwr\u27e9,\n have ptw : ptmodel w, \n   {apply prt.2, apply tc'.base, simp at h\u2081, rw \u2190h\u2081, simp, exact hwl},\n split, split, swap 3, exact \u27e8w, ptw\u27e9,\n apply reach_step.fwd_base, exact hwl, split, \n {exact hwr}, \n {apply tc'.base, simp, simp at h\u2081, rw \u2190h\u2081, simp, exact hwl} }\nend\n\ntheorem reach_dia_root (s : rmodel) (rt : model) (\u03c6) \n(h\u2081 : s.1 = rt.1) \n(h\u2082 : manc rt.1 = []) (h\u2083 : dia \u03c6 \u2208 htk s.1) : \n\u2203 s', reach s s' \u2227 \u03c6 \u2208 htk s'.1 \u2227 desc s'.1 rt.1 :=\nbegin\nhave := reach_step_dia_root s rt \u03c6 h\u2081 h\u2082 h\u2083,\nrcases this with \u27e8w, hwl, hwr\u27e9,\nsplit, split, swap 3, exact w,\napply rtc_step hwl, exact hwr\nend\n\ntheorem good_model (m : model) (hrt : manc m.1 = []): \n\u03a0 (s : {x : rmodel // x.1 = m.1 \u2228 desc x.1 m.1}) (\u03c6 : nnf), \n  \u03c6 \u2208 htk s.1.1 \u2192 force (builder m.1) s \u03c6\n| s (var n) h   := begin simp, exact h end\n| s (neg n) h   := begin \n                     simp, intro hin, \n                     cases s with s ps,\n                     cases s with s pts,\n                     cases s with i l sg,\n                     have := i.hhtk.hno_contra,\n                     simp at hin,\n                     apply this hin, simp at h, exact h\n                   end\n| s (and \u03c6 \u03c8) h := begin \n                   split,\n                   {apply good_model, \n                   cases s with s ps,\n                   cases s with s pts,\n                   cases s with i l sg,\n                   have := i.hhtk.hand_left,\n                   simp, apply this, simp at h, exact h},\n                   {apply good_model, \n                   cases s with s ps,\n                   cases s with s pts,\n                   cases s with i l sg,\n                   have := i.hhtk.hand_right,\n                   simp, apply this, simp at h, exact h}\n                   end\n| s (or \u03c6 \u03c8) h  := begin\n                   cases s with s ps,\n                   cases s with s pts,\n                   cases s with i l sg,\n                   have := i.hhtk.hor,\n                   simp at h,\n                   have hc := this h,\n                   cases hc,\n                   {simp, left, apply good_model, simp, exact hc},\n                   {simp, right, apply good_model, simp, exact hc}\n                   end\n| s (box \u03c6) h   := begin\n                   intros m hm,\n                   apply good_model,\n                   apply reach_box,\n                   exact hm,\n                   exact h\n                   end\n| s (dia \u03c6) h   := begin\n                   cases s with s ps,\n                   cases ps,\n                   {simp, simp at h,\n                    have := reach_dia_root _ _ _ ps hrt h,\n                    rcases this with \u27e8s', hs'l, hs'm, hs'r\u27e9,\n                    split, split, \n                    exact hs'l, split, apply good_model, simp,\n                    exact hs'm, right, exact hs'r },\n                   {simp, simp at h,\n                    have := reach_dia _ _ _ ps hrt h,\n                    rcases this with \u27e8s', hs'l, hs'm, hs'r\u27e9,\n                    split, split, \n                    exact hs'l, split, apply good_model, simp,\n                    exact hs'm, right, exact hs'r}\n                   end\n", "meta": {"author": "minchaowu", "repo": "ModalTab", "sha": "9bb0bf17faf0554d907ef7bdd639648742889178", "save_path": "github-repos/lean/minchaowu-ModalTab", "path": "github-repos/lean/minchaowu-ModalTab/ModalTab-9bb0bf17faf0554d907ef7bdd639648742889178/src/S4/S4_defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3776888152424863}}
{"text": "namespace Smt.Reconstruction.Certifying\n\nopen Classical\n\nuniverse u\n\ntheorem instForAll {\u03b1 : Sort u} {f : \u03b1 \u2192 Prop} {a : \u03b1} :\n  (forall a' : \u03b1, f a') \u2192 f a := \u03bb h => h a\n\ntheorem instEqual\u2081 {A : Type u} {P : A \u2192 Prop} {t : A} :\n  (forall x : A, x = t \u2192 P x) \u2192 P t := \u03bb h => h t rfl\n\ntheorem instEqual\u2082 {A : Type u} {P : A \u2192 Prop} {t : A} :\n  P t \u2192 (forall x : A, x = t \u2192 P x) := by\n  intros h x r\n  rewrite [r]\n  exact h\n\ntheorem instEqual {A : Type u} {P : A \u2192 Prop} {t : A} :\n  (forall x : A, x = t \u2192 P x) \u2194 P t := \u27e8instEqual\u2081, instEqual\u2082\u27e9\n\ntheorem skolem\u2081 {\u03b1 : Sort u} [i : Nonempty \u03b1] (p : \u03b1 \u2192 Prop) : (\u2203 x, p x) \u2192 p (epsilon p) :=\n  (strongIndefiniteDescription p i).property\n\ntheorem skolem\u2082 {\u03b1 : Sort u} [Nonempty \u03b1] (p : \u03b1 \u2192 Prop) : p (epsilon p) \u2192 \u2203 x, p x := \u03bb h =>\n  \u27e8epsilon p, h\u27e9\n\ntheorem smtSkolem {\u03b1 : Sort u} [h: Nonempty \u03b1] (p : \u03b1 \u2192 Prop) : (\u2203 x, p x) \u2194 p (epsilon p) :=\n  \u27e8@skolem\u2081 \u03b1 h p, @skolem\u2082 \u03b1 h p\u27e9\n\ndef f: Prop := \u2203 _: Nat, True\naxiom g : f\n\n#check Iff.mp (smtSkolem (\u03bb _ => True)) g\n\nend Smt.Reconstruction.Certifying\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Smt/Reconstruction/Certifying/Quant.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.705785040214066, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3776644659667165}}
{"text": "import .open_subspace\n\nvariables {X : Type*} [topological_space X]\nvariables {H : top_subspace X}\nvariables {G : top_subspace X} (h : H \u2264 G)\nvariables {a b : H}\n\nnoncomputable theory\n\n#check topological_space.opens\n\nopen_locale classical topological_space filter unit_interval\nopen filter set function unit_interval\n\ndef subspace_path_lift (p : path a b) : path (incl a) (incl b) :=\nbegin\n  fconstructor,\n  {\n    exact { to_fun := i \u2218 p.to_continuous_map,\n            continuous_to_fun := continuous_induced_dom.comp (map_continuous p.to_continuous_map)}\n  }, \n  {\n    simp only [path.coe_to_continuous_map, function.comp_app, path.source], unfold i, unfold incl, refl,\n  },\n  {\n    simp only [path.target, path.coe_to_continuous_map, function.comp_app], unfold i, unfold incl, refl,\n  }\nend\n\ndef  subspace_path_incl (p : path a b) : path (open_incl h a) (open_incl h b) :=\nbegin\n  fconstructor, exact continuous_map.comp (top_subspace_incl h) p.to_continuous_map,\n  simp only [path.coe_to_continuous_map, continuous_map.comp_apply, path.source, continuous_map.to_fun_eq_coe],\n  unfold top_subspace_incl, simp only [continuous_map.coe_mk, eq_self_iff_true], \n  simp only [path.target, path.coe_to_continuous_map, continuous_map.comp_apply, continuous_map.to_fun_eq_coe],\n  unfold top_subspace_incl, simp only [continuous_map.coe_mk, eq_self_iff_true],\nend\n\n\nnotation `\u03c0\u2093` := fundamental_groupoid.fundamental_groupoid_functor.obj\nnotation `\u03c0` := fundamental_groupoid.fundamental_groupoid_functor\nnotation `\u03c0\u2098` := fundamental_groupoid.fundamental_groupoid_functor.map\n\ndef grpd_induced_incl : \u03c0\u2093 (Top.of H) \u27f6 \u03c0\u2093 (Top.of G) := \u03c0\u2098 (top_subspace_incl h)\n\n-- projection of a subspace to a subspace --\n\nstructure proj_subspace {H G : top_subspace X} (h : H \u2264 G) :=\n(obj : C(H, G))\n(path : \u2200 (a b : H), path a b \u2192 path (obj a) (obj b))\n\ndef Id {H G : top_subspace X} (h : H \u2264 G) : proj_subspace h :=\n{\n  obj := top_subspace_incl h,\n  path := \u03bba b, subspace_path_incl h\n}\n\n-- projection of a space to a groupoid --\n\nlocal attribute [instance] path.homotopic.setoid\n\nstructure proj_grpd (H : top_subspace X) (G : category_theory.Groupoid):= \n(obj : H \u2192 G)\n(path {a b : H} : path a b \u2192 (obj a \u27f6 obj b))\n\ndef self_grpd (H : top_subspace X) : proj_grpd H (\u03c0\u2093 (Top.of H)):= \n{ \n  obj := \u03bbx, x,\n  path := \n  begin\n    intros a b pab, exact \u27e6pab\u27e7\n  end,\n}\n\n-- lift of a path to subspace from inclusion --\n\ndef b_incl (a : X) (ha : a \u2208 H) : H := \nbegin\n  fconstructor, exact a, exact ha,\nend\n\n@[simp] lemma incl_inv (a : H) : b_incl (incl a) (incl_carrier a) = a :=\nbegin\n  simp_rw b_incl, simp_rw incl,cases a,simp,\nend\n\nlemma bincl_open (a : H) : b_incl (incl a) (openincl_carrier h a) = open_incl h a :=\nbegin\n  simp_rw open_incl, rw b_incl, simp_rw incl, cases a, simp,\nend\n\n\ndef path_lift_func {a b : X} (p : path a b) (hp : set.range p \u2286 H.carrier)\n: I \u2192 H :=\nbegin\n  intro x, let fx := p.to_fun x,\n  have pfx : fx \u2208 H.carrier,\n  {\n    apply hp, have hfx : fx = p.to_fun x, tauto,rw hfx, simp only [path.coe_to_continuous_map, set.mem_range_self, continuous_map.to_fun_eq_coe]\n  }, exact b_incl fx pfx,\nend\n\nlemma path_lift_eq {a b : X} (p : path a b) (hp : set.range p \u2286 H) : incl \u2218 (path_lift_func p hp) = p :=\nbegin\n  ext1, rw function.comp, simp only [], rw path_lift_func, simp only [path.coe_to_continuous_map, continuous_map.to_fun_eq_coe],\n  simp_rw b_incl,\nend\n\nlemma continuous_lift {a b : X} (p : path a b) (hp : set.range p \u2286 H) : continuous (path_lift_func p hp) :=\nbegin\n  fconstructor, intro s, intro hs, \n  have hp : path_lift_func p hp \u207b\u00b9' s = (incl \u2218 path_lift_func p hp) \u207b\u00b9' (incl '' s),\n  {\n    rw @set.preimage_comp _ _ _ (path_lift_func p hp) incl (incl '' s), \n    rw set.preimage_image_eq, exact incl_inj,\n  },\n  rw path_lift_eq at hp, rw hp,\n  have hp2 : is_open (incl '' s),\n  {\n    apply incl_open_map, assumption,\n  },\n  have hp3 : continuous p,\n  {\n    continuity,\n  },\n  set k := incl ''s, cases hp3, apply hp3, assumption,\nend\n\nlemma range_in_init {a b : X} (p : path a b) (hp : set.range p \u2286 H) : a \u2208 H :=\nbegin\n  apply hp, rw range, simp only [set.mem_set_of_eq], use 0, exact p.source,\nend \n\nlemma range_in_end {a b : X} (p : path a b) (hp : set.range p \u2286 H) : b \u2208 H :=\nbegin\n  apply hp, rw range, simp only [set.mem_set_of_eq], use 1, exact p.target,\nend \n\ndef path_lift_def {a b : X} (p : path a b) (hp : set.range p \u2286 H) : path (b_incl a (range_in_init p hp)) (b_incl b (range_in_end p hp)) := \n{ to_fun := path_lift_func p hp,\n  continuous_to_fun := continuous_lift p hp ,\n  source' := \n  begin\n    simp only [],rw path_lift_func, rw b_incl, simp only [path.coe_to_continuous_map, path.source, continuous_map.to_fun_eq_coe],\n    rw b_incl,\n  end,\n  target' := \n  begin\n    simp only [],rw path_lift_func, rw b_incl, simp only [path.target, path.coe_to_continuous_map, continuous_map.to_fun_eq_coe],\n    rw b_incl,\n  end\n}\n\ntheorem path_lift_def_eq {a b : X} (p : path a b) (hp : set.range p \u2286 H.carrier) : subspace_path_lift (path_lift_def p hp) = p :=\nbegin\n  rw subspace_path_lift, rw path_lift_def, simp, cases p,simp, ext1, finish,\nend\n\ntheorem path_incl_point_eq {a b : H} (p : path a b) (x : I) : \u2191(p x) = (subspace_path_lift p) x :=\nbegin\n  rw subspace_path_lift, unfold_coes, simp,\nend\n\n-- lemma t {K : Type} (f : K \u2192 H) (s : set K) : (f '' s) = incl ''\n\ntheorem path_incl_set_eq {a b : H} (p : path a b) (s : set I) : \u2191(p '' s) = (subspace_path_lift p) '' s :=\nbegin\n  unfold_coes, rw \u2190set.image_comp, apply set.image_congr', simp_rw path_incl_point_eq, rw subspace_path_lift, simp,\n  --  cases p with p _ _, cases p  with p _, simp_rw set.image,unfold_coes, simp,\nend\n\ntheorem path_incl_range_eq {a b : H} (p : path a b) : \u2191(set.range p) = set.range (subspace_path_lift p) :=\nbegin\n  unfold_coes, rw \u2190set.image_univ, rw \u2190 set.image_univ, apply path_incl_set_eq,\n  --  cases p with p _ _, cases p  with p _, simp_rw set.image,unfold_coes, simp,\nend\n\ntheorem path_incl_range {a b : H} (p : path a b) (K : set X) (hp : \u2191(set.range p) \u2286 K) :\nset.range (subspace_path_lift p) \u2286 K :=\nbegin\n  rw path_incl_range_eq at *, assumption,\nend\n\n\n", "meta": {"author": "Mak1Haru", "repo": "van_kampen", "sha": "873c74504cd75b24e3e4cfba46370c4a9f58a9f1", "save_path": "github-repos/lean/Mak1Haru-van_kampen", "path": "github-repos/lean/Mak1Haru-van_kampen/van_kampen-873c74504cd75b24e3e4cfba46370c4a9f58a9f1/src/lifts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.705785040214066, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3776644659667165}}
{"text": "import analysis.calculus.mean_value\nimport topology.continuous_map\nimport measure_theory.interval_integral\nimport topology.metric_space.contracting\nimport topology.metric_space.cau_seq_filter\nimport topology.algebra.continuous_functions\n\nimport picard_lindelof.other.to_mathlib\n\nnoncomputable theory\nopen metric set asymptotics filter real measure_theory interval_integral topological_space uniform_space\nopen_locale topological_space classical filter uniformity\n\n-- NOTE: This is meant to be [a, b].\nvariables {A : Type*} [linear_order A] [measurable_space A]\n          [topological_space A] [compact_space A] [nonempty A]\n          [uniform_space A] [complete_space A] -- Maybe\n\n-- NOTE: This is meant to be \u211d^n.\nvariables {B : Type*} [normed_group B] [normed_space \u211d B]\n          [second_countable_topology B]\n          [complete_space B] [measurable_space B]\n          [borel_space B] [nonempty B]\n          [complete_lattice B] [ring B] [topological_ring B] -- Maybe\n\n\ninstance : nonempty C(A, B) := \u27e8\u27e8\u03bb (a : A), @nonempty.some B (by apply_instance), continuous_const\u27e9\u27e9\n\n-- TODO: Isn't this deduced from has_dist?\ninstance : has_norm C(A, B) := \u27e8\u03bb x, supr (\u03bb t, norm (x t))\u27e9\n\ninstance : has_edist C(A, B) := \u27e8\u03bb x y, supr (\u03bb t, edist (x t) (y t))\u27e9\n\ninstance : emetric_space C(A, B) := {\n    edist_self := begin \n        intros x, unfold edist, erw [supr_eq_bot], \n        intros t, erw [metric_space.edist_dist, metric_space.dist_self],\n        norm_num,\n    end,\n    eq_of_edist_eq_zero := begin \n        intros x y h, unfold edist at h, erw [supr_eq_bot] at h,\n        ext i, replace h := h i, erw [metric_space.edist_dist, ennreal.of_real_eq_zero] at h,\n        replace h := le_antisymm h dist_nonneg,\n        exact metric_space.eq_of_dist_eq_zero h,\n    end,\n    edist_comm := begin\n        intros x y, unfold edist, apply le_antisymm,\n        { rw supr_le_iff, intros i, \n          erw [metric_space.edist_dist, metric_space.dist_comm, \u2190metric_space.edist_dist],\n          exact (le_supr (\u03bb t, metric_space.edist (y t) (x t)) i), },\n        { -- TODO: Avoid repetition.\n          rw supr_le_iff, intros i, \n          erw [metric_space.edist_dist, metric_space.dist_comm, \u2190metric_space.edist_dist],\n          exact (le_supr (\u03bb t, metric_space.edist (x t) (y t)) i), }\n    end,\n    edist_triangle := begin\n        intros x y z, unfold edist,\n        suffices hle1 : \n            supr (\u03bb t, metric_space.edist (x t) (z t)) \u2264\n            supr (\u03bb t, (metric_space.edist (x t) (y t)) + (metric_space.edist (y t) (z t))),\n        { have hle2 := supr_add_le_add_supr \n            (\u03bb t, metric_space.edist (x t) (y t))\n            (\u03bb t, metric_space.edist (y t) (z t)),\n          exact (le_trans hle1 hle2), },\n        rw supr_le_iff, intros i, \n        have hxyz := metric_space.dist_triangle (x i) (y i) (z i),\n        replace hxyz := ennreal.of_real_le_of_real hxyz,\n        replace hxyz := le_trans hxyz ennreal.of_real_add_le,\n        repeat { rw [\u2190metric_space.edist_dist] at hxyz, },\n        exact (@le_supr_of_le _ _ _ (\u03bb t, metric_space.edist (x t) (y t) + metric_space.edist (y t) (z t)) _ i hxyz),\n    end,\n}\n\nopen continuous_functions\n\ninstance : ring C(A, B) := continuous_map_ring\n\n-- TODO: Move\nprivate lemma ennreal.of_real_supr {\u03b9 : Type*} (f : \u03b9 \u2192 \u211d) \n: ennreal.of_real (supr f) = supr (\u03bb t, ennreal.of_real (f t)) := \nbegin \n    ext, split,\n    { sorry, },\n    { sorry, }\nend \n\ninstance : complete_space C(A, B) := \nbegin \n    apply emetric.complete_of_cauchy_seq_tendsto,\n    intros u hu,\n    have hu\u03b5 : is_cau_seq norm u,\n    { cases cauchy_iff.1 hu with hu1 hu2,\n      intros \u03b5 h\u03b5,\n      have h\u03b5rw := (ennreal.of_real_lt_of_real_iff h\u03b5),\n      have henn\u03b5 := h\u03b5rw.2 h\u03b5,\n      rw ennreal.of_real_zero at henn\u03b5,\n      rcases hu2 {x | edist x.1 x.2 < ennreal.of_real \u03b5} (edist_mem_uniformity henn\u03b5) with \u27e8t, \u27e8ht, htsub\u27e9\u27e9,\n      simp at ht, cases ht with N hN,\n      existsi N, intros j hj,\n      unfold has_norm.norm,\n      have hujN := @htsub (u j, u N) (set.mk_mem_prod (hN j hj) (hN N (le_refl N))),\n      simp at hujN,\n      unfold edist at hujN,\n      -- TODO: Prove some nice properties of edist.\n      have heq : (\u03bb (t : A), metric_space.edist ((u j) t) ((u N) t)) =\n                 (\u03bb (t : A), ennreal.of_real (norm (((u j) - (u N)) t))),\n      { funext, erw [\u2190dist_eq_norm, metric_space.edist_dist], },\n      apply (ennreal.of_real_lt_of_real_iff h\u03b5).1,\n      rw ennreal.of_real_supr, \n      sorry,\n      -- TODO: So close. Will this work when ennreal.of_real_supr is proved?\n      --erw \u2190heq,\n      --exact hujN,\n      },\n\n    let fn := \u03bb x n, (u n) x,\n    let f := \u03bb x, (lim at_top (fn x)), \n    have hf : continuous f := sorry,\n    --have h := tendsto_nhds_lim,\n    use [\u27e8f, hf\u27e9],\n    rw emetric.cauchy_seq_iff at hu,\n    rw emetric.tendsto_nhds,\n    intros \u03b5 h\u03b5, \n\n    sorry,\nend ", "meta": {"author": "ramonfmir", "repo": "lean-experiments", "sha": "041c8727bb540fb8d1519c1ad84924d473885c27", "save_path": "github-repos/lean/ramonfmir-lean-experiments", "path": "github-repos/lean/ramonfmir-lean-experiments/lean-experiments-041c8727bb540fb8d1519c1ad84924d473885c27/src/picard_lindelof/old/continuous_map_version.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.37766445934377396}}
{"text": "/-\nCopyright (c) 2022 Henrik B\u00f6ving. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Henrik B\u00f6ving, Simon Hudon\n-/\nimport Mathlib.Testing.SlimCheck.Gen\n/-!\n# `SampleableExt` Class\nThis class permits the creation samples of a given type\ncontrolling the size of those values using the `Gen` monad`.\n\n# `Shrinkable` Class\nThis class helps minimize examples by creating smaller versions of\ngiven values.\n\nWhen testing a proposition like `\u2200 n : \u2115, prime n \u2192 n \u2264 100`,\n`SlimCheck` requires that `\u2115` have an instance of `SampleableExt` and for\n`prime n` to be decidable.  `SlimCheck` will then use the instance of\n`SampleableExt` to generate small examples of \u2115 and progressively increase\nin size. For each example `n`, `prime n` is tested. If it is false,\nthe example will be rejected (not a test success nor a failure) and\n`SlimCheck` will move on to other examples. If `prime n` is true, `n\n\u2264 100` will be tested. If it is false, `n` is a counter-example of `\u2200\nn : \u2115, prime n \u2192 n \u2264 100` and the test fails. If `n \u2264 100` is true,\nthe test passes and `SlimCheck` moves on to trying more examples.\n\nThis is a port of the Haskell QuickCheck library.\n\n## Main definitions\n  * `SampleableExt` class\n  * `Shrinkable` class\n\n### `SampleableExt`\n`SampleableExt` can be used in two ways. The first (and most common)\nis to simply generate values of a type directly using the `Gen` monad,\nif this is what you want to do then `SampleableExt.mkSelfContained` is\nthe way to go.\n\nFurthermore it makes it possible to express generators for types that\ndo not lend themselves to introspection, such as `\u2115 \u2192 \u2115`.\nIf we test a quantification over functions the\ncounter-examples cannot be shrunken or printed meaningfully.\nFor that purpose, `SampleableExt` provides a proxy representation\n`proxy` that can be printed and shrunken as well\nas interpreted (using `interp`) as an object of the right type. If you\nare using it in the first way, this proxy type will simply be the type\nitself and the `interp` function `id`.\n\n### `Shrinkable\nGiven an example `x : \u03b1`, `Shrinkable \u03b1` gives us a way to shrink it\nand suggest simpler examples.\n\n## Shrinking\nShrinking happens when `SlimCheck` find a counter-example to a\nproperty.  It is likely that the example will be more complicated than\nnecessary so `SlimCheck` proceeds to shrink it as much as\npossible. Although equally valid, a smaller counter-example is easier\nfor a user to understand and use.\n\nThe `Shrinkable` class, , has a `shrink` function so that we can use\nspecialized knowledge while shrinking a value. It is not responsible\nfor the whole shrinking process however. It only has to take one step\nin the shrinking process. `SlimCheck` will repeatedly call `shrink`\nuntil no more steps can be taken. Because `shrink` guarantees that the\nsize of the candidates it produces is strictly smaller than the\nargument, we know that `SlimCheck` is guaranteed to terminate.\n\n## Tags\n\nrandom testing\n\n## References\n  * https://hackage.haskell.org/package/QuickCheck\n-/\n\nnamespace SlimCheck\n\nopen Random Gen\n\n/-- Given an example `x : \u03b1`, `Shrinkable \u03b1` gives us a way to shrink it\nand suggest simpler examples. -/\nclass Shrinkable (\u03b1 : Type u) extends WellFoundedRelation \u03b1 where\n  shrink : (x : \u03b1) \u2192 List { y : \u03b1 // WellFoundedRelation.rel y x } := \u03bb _ => []\n\n/-- `SampleableExt` can be used in two ways. The first (and most common)\nis to simply generate values of a type directly using the `Gen` monad,\nif this is what you want to do then `SampleableExt.mkSelfContained` is\nthe way to go.\n\nFurthermore it makes it possible to express generators for types that\ndo not lend themselves to introspection, such as `\u2115 \u2192 \u2115`.\nIf we test a quantification over functions the\ncounter-examples cannot be shrunken or printed meaningfully.\nFor that purpose, `SampleableExt` provides a proxy representation\n`proxy` that can be printed and shrunken as well\nas interpreted (using `interp`) as an object of the right type. -/\nclass SampleableExt (\u03b1 : Sort u) where\n  proxy : Type v\n  [proxyRepr : Repr proxy]\n  [shrink : Shrinkable proxy]\n  sample : Gen proxy\n  interp : proxy \u2192 \u03b1\n\nattribute [instance] SampleableExt.proxyRepr\nattribute [instance] SampleableExt.shrink\n\nnamespace SampleableExt\n\n/-- Use to generate instance whose purpose is to simply generate values\nof a type directly using the `Gen` monad -/\ndef mkSelfContained [Repr \u03b1] [Shrinkable \u03b1] (sample : Gen \u03b1) : SampleableExt \u03b1 where\n  proxy := \u03b1\n  proxyRepr := inferInstance\n  shrink := inferInstance\n  sample := sample\n  interp := id\n\n/-- First samples a proxy value and interprets it. Especially useful if\nthe proxy and target type are the same. -/\ndef interpSample (\u03b1 : Type u) [SampleableExt \u03b1] : Gen \u03b1 :=\n  SampleableExt.interp <$> SampleableExt.sample\n\nend SampleableExt\n\nsection Shrinkers\n\n/-- `Nat.shrink' n` creates a list of smaller natural numbers by\nsuccessively dividing `n` by 2 . For example, `Nat.shrink 5 = [2, 1, 0]`. -/\ndef Nat.shrink (n : Nat) : List { y : Nat // WellFoundedRelation.rel y n } :=\n  if h : 0 < n then\n    let m := n/2\n    have h : m < n := by\n      apply Nat.div_lt_self h\n      decide\n    let rest := shrink m\n    let current := \u27e8m, h\u27e9\n    current ::\n      rest.map (\u03bb x => {x with property := Nat.lt_trans x.property h})\n  else\n    []\n\ninstance Nat.shrinkable : Shrinkable Nat where\n  shrink := Nat.shrink\n\n/-- `Fin.shrink` works like `Nat.shrink` but instead operates on `Fin`. -/\ndef Fin.shrink {n : Nat} (m : Fin n.succ) :\n    List { y : Fin n.succ // WellFoundedRelation.rel y m } :=\n  let shrinks := Nat.shrink m.val\n  shrinks.map (\u03bb x => { x with property := (by\n    simp_wf\n    exact lt_of_le_of_lt (Nat.mod_le _ _) x.property) })\n\ninstance Fin.shrinkable {n : Nat} : Shrinkable (Fin n.succ) where\n  shrink := Fin.shrink\n\nlocal instance Int_sizeOfAbs : SizeOf Int := \u27e8Int.natAbs\u27e9\n\n/-- `Int.shrinkable` operates like `Nat.shrinkable` but also includes the negative variants. -/\ninstance Int.shrinkable : Shrinkable Int where\n  shrink n := Nat.shrink n.natAbs |>.map \u03bb \u27e8x, h\u27e9 =>\n    \u27e8-x, (by simp_wf; simp only [SizeOf.sizeOf]; rw [Int.natAbs_neg]; exact h)\u27e9\n\ninstance Bool.shrinkable : Shrinkable Bool := {}\ninstance Char.shrinkable : Shrinkable Char := {}\n\ninstance Prod.shrinkable [shrA : Shrinkable \u03b1] [shrB : Shrinkable \u03b2] : Shrinkable (Prod \u03b1 \u03b2) where\n  shrink := \u03bb (fst,snd) =>\n    let shrink1 := shrA.shrink fst |>.map\n      fun \u27e8x, _\u27e9 \u21a6 \u27e8(x, snd), by simp_wf; apply Prod.Lex.left; simp_all_arith\u27e9\n    let shrink2 := shrB.shrink snd |>.map\n      fun \u27e8x, _\u27e9 \u21a6 \u27e8(fst, x), by simp_wf; apply Prod.Lex.right; simp_all_arith\u27e9\n    shrink1 ++ shrink2\n\nend Shrinkers\n\nsection Samplers\n\nopen SampleableExt\n\ninstance Nat.sampleableExt : SampleableExt Nat :=\n  mkSelfContained (do choose Nat 0 (\u2190getSize) (Nat.zero_le _))\n\ninstance Fin.sampleableExt {n : Nat} : SampleableExt (Fin (n.succ)) :=\n  mkSelfContained (do choose (Fin n.succ) (Fin.ofNat 0) (Fin.ofNat (\u2190getSize)) (by\n    simp [Fin.ofNat, LE.le]\n    exact Nat.zero_le _\n  ))\n\ninstance Int.sampleableExt : SampleableExt Int :=\n  mkSelfContained (do\n    choose Int (-(\u2190getSize)) (\u2190getSize)\n      (le_trans (Int.neg_nonpos_of_nonneg (Int.ofNat_zero_le _)) (Int.ofNat_zero_le _)))\n\ninstance Bool.sampleableExt : SampleableExt Bool :=\n  mkSelfContained $ chooseAny Bool\n\n/-- This can be specialized into customized `SampleableExt Char` instances.\nThe resulting instance has `1 / length` chances of making an unrestricted choice of characters\nand it otherwise chooses a character from `chars` with uniform probabilities.  -/\ndef Char.sampleable (length : Nat) (chars : List Char) (pos : 0 < chars.length) :\n    SampleableExt Char :=\n  mkSelfContained do\n    let x \u2190 choose Nat 0 length (Nat.zero_le _)\n    if x.val == 0 then\n      let n \u2190 interpSample Nat\n      pure $ Char.ofNat n\n    else\n      elements chars pos\n\ninstance Char.sampleableDefault : SampleableExt Char :=\n  Char.sampleable 3 \" 0123abcABC:,;`\\\\/\".toList (by decide)\n\ninstance Prod.sampleableExt {\u03b1 \u03b2 : Type u} [SampleableExt \u03b1] [SampleableExt \u03b2] :\n    SampleableExt (\u03b1 \u00d7 \u03b2) where\n  proxy := Prod (proxy \u03b1) (proxy \u03b2)\n  proxyRepr := inferInstance\n  shrink := inferInstance\n  sample := prodOf sample sample\n  interp := Prod.map interp interp\n\ninstance Prop.sampleableExt : SampleableExt Prop where\n  proxy := Bool\n  proxyRepr := inferInstance\n  sample := interpSample Bool\n  shrink := inferInstance\n  interp := Coe.coe\n\nend Samplers\n\n/-- An annotation for values that should never get shrinked. -/\ndef NoShrink (\u03b1 : Type u) := \u03b1\n\nnamespace NoShrink\n\ndef mk (x : \u03b1) : NoShrink \u03b1 := x\ndef get (x : NoShrink \u03b1) : \u03b1 := x\n\ninstance inhabited [inst : Inhabited \u03b1] : Inhabited (NoShrink \u03b1) := inst\ninstance repr [inst : Repr \u03b1] : Repr (NoShrink \u03b1) := inst\n\ninstance shrinkable : Shrinkable (NoShrink \u03b1) where\n  shrink := \u03bb _ => []\n\ninstance sampleableExt [SampleableExt \u03b1] [Repr \u03b1] : SampleableExt (NoShrink \u03b1) :=\n  SampleableExt.mkSelfContained $ (NoShrink.mk \u2218 SampleableExt.interp) <$> SampleableExt.sample\n\nend NoShrink\n\nend SlimCheck\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Testing/SlimCheck/Sampleable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.3775468841971749}}
{"text": "import tactic\nimport data.list.alist\nimport data.equiv.denumerable\nimport data.equiv.encodable.basic\n\n/-- Formulas of modal logic\nform vars \u2237= \u2983x : vars\u2984 | \u22a5 | ~ form | form \u22c0 form | form \u22c1 form -/\ninductive form (vars : Type) : Type\n| Bottom : form\n| Var    : vars \u2192 form\n| Not    : form \u2192 form\n| And    : form \u2192 form \u2192 form\n| Or     : form \u2192 form \u2192 form\n| Imply  : form \u2192 form \u2192 form\n| Box    : form \u2192 form\n\n-- get the \u22a5 notation for Bottom\n-- and allow simp to use this definition\ninstance {vars : Type} : has_bot (form vars) := \u27e8form.Bottom\u27e9\n@[simp] lemma bottom_eq_bot {vars : Type} : \u22a5 = (form.Bottom : form vars) := rfl\n\nnotation `\u2983` x `\u2984` := form.Var x\nprefix `~`:75 := form.Not\nprefix `\u25a1`:75 := form.Box\ninfixl ` \u22c0 `:70 := form.And\ninfixl ` \u22c1 `:65 := form.Or\ninfixr ` \u27f9 `:60 := form.Imply\n\n-- Since we intend to use a classical semantics, \u25c7 can be implemented in terms \n-- of \u25a1\nprefix `\u25c7`:75 := form.Not \u2218 form.Box \u2218 form.Not\n\n@[simp] lemma diamond_eq_not_box_not {vars : Type} {A : form vars} \n  : \u25c7 A = ~ \u25a1 ~ A := rfl\n\nvariables {vars : Type} [denumerable vars]\nvariables {A B C : form vars}\n\ninstance vars_decidable_eq : decidable_eq vars := \nencodable.decidable_eq_of_encodable _\n\ninstance form_inhabited : inhabited (form vars) := \u27e8form.Bottom\u27e9\n\n/-- A simultaneous substitution on formulas is an association list (key-value)\nmapping from variables to the formula that it should be substituted by.-/\ndef subst (vars : Type) : Type := alist (\u03bb _ : vars, form vars)\n\n-- can't resolve these implicitly for some reason, even after importing.\ninstance : has_mem vars (subst vars) := alist.has_mem\ninstance : has_emptyc (subst vars) := alist.has_emptyc\ninstance : has_union (subst vars) := alist.has_union\n\ndef subst.get (s : subst vars) (x : vars) : form vars := \n(s.lookup x).get_or_else \u2983x\u2984\n\n/-- This function applies a simultaneous substitution to a formula. -/\ndef subst.apply (s : subst vars) : form vars \u2192 form vars\n| form.Bottom := form.Bottom\n| \u2983x\u2984      := s.get x\n| ~ A      := ~ (subst.apply A)\n| \u25a1 A      := \u25a1 (subst.apply A)\n| (A \u22c0 B)  := (subst.apply A) \u22c0 (subst.apply B)\n| (A \u22c1 B)  := (subst.apply A) \u22c1 (subst.apply B)\n| (A \u27f9 B) := (subst.apply A) \u27f9 (subst.apply B)\n\ntheorem subst.apply_empty_id : subst.apply \u2205 A = A :=\nbegin\n  induction A,\n  case form.Bottom { simp [subst.apply] },\n  case form.Var { simp [subst.apply, subst.get] },\n  case form.Not : A ih { simp [subst.apply, ih] },\n  case form.Box : A ih { simp [subst.apply, ih] },\n  case form.And : A B ihA ihB { simp [subst.apply, ihA, ihB] },\n  case form.Or : A B ihA ihB { simp [subst.apply, ihA, ihB] },\n  case form.Imply : A B ihA ihB { simp [subst.apply, ihA, ihB] }\nend", "meta": {"author": "alyata", "repo": "formalising-math-2", "sha": "30a001f2ff3d54a8a3432a178d0314a4abda57aa", "save_path": "github-repos/lean/alyata-formalising-math-2", "path": "github-repos/lean/alyata-formalising-math-2/formalising-math-2-30a001f2ff3d54a8a3432a178d0314a4abda57aa/src/formula.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3775468831988232}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.algebra.operations\nimport Mathlib.PostPort\n\nuniverses u v l w u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Subalgebras over Commutative Semiring\n\nIn this file we define `subalgebra`s and the usual operations on them (`map`, `comap`).\n\nMore lemmas about `adjoin` can be found in `ring_theory.adjoin`.\n-/\n\n/-- A subalgebra is a sub(semi)ring that includes the range of `algebra_map`. -/\nstructure subalgebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] \nextends subsemiring A\nwhere\n  algebra_map_mem' : \u2200 (r : R), coe_fn (algebra_map R A) r \u2208 carrier\n\n/-- Reinterpret a `subalgebra` as a `subsemiring`. -/\nnamespace subalgebra\n\n\nprotected instance subsemiring.has_coe {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : has_coe (subalgebra R A) (subsemiring A) :=\n  has_coe.mk fun (S : subalgebra R A) => subsemiring.mk (carrier S) (one_mem' S) (mul_mem' S) (zero_mem' S) (add_mem' S)\n\nprotected instance has_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : has_mem A (subalgebra R A) :=\n  has_mem.mk fun (x : A) (S : subalgebra R A) => x \u2208 \u2191S\n\ntheorem mem_coe {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {x : A} {s : subalgebra R A} : x \u2208 \u2191s \u2194 x \u2208 s :=\n  iff.rfl\n\ntheorem ext {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {S : subalgebra R A} {T : subalgebra R A} (h : \u2200 (x : A), x \u2208 S \u2194 x \u2208 T) : S = T := sorry\n\ntheorem ext_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {S : subalgebra R A} {T : subalgebra R A} : S = T \u2194 \u2200 (x : A), x \u2208 S \u2194 x \u2208 T :=\n  { mp := fun (h : S = T) (x : A) => eq.mpr (id (Eq._oldrec (Eq.refl (x \u2208 S \u2194 x \u2208 T)) h)) (iff.refl (x \u2208 T)), mpr := ext }\n\ntheorem algebra_map_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) (r : R) : coe_fn (algebra_map R A) r \u2208 S :=\n  algebra_map_mem' S r\n\ntheorem srange_le {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : ring_hom.srange (algebra_map R A) \u2264 \u2191S := sorry\n\ntheorem range_subset {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : set.range \u21d1(algebra_map R A) \u2286 \u2191S := sorry\n\ntheorem range_le {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : set.range \u21d1(algebra_map R A) \u2264 \u2191S :=\n  range_subset S\n\ntheorem one_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : 1 \u2208 S :=\n  subsemiring.one_mem \u2191S\n\ntheorem mul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x * y \u2208 S :=\n  subsemiring.mul_mem (\u2191S) hx hy\n\ntheorem smul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} (hx : x \u2208 S) (r : R) : r \u2022 x \u2208 S :=\n  Eq.symm (algebra.smul_def r x) \u25b8 mul_mem S (algebra_map_mem S r) hx\n\ntheorem pow_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2115) : x ^ n \u2208 S :=\n  subsemiring.pow_mem (\u2191S) hx n\n\ntheorem zero_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : 0 \u2208 S :=\n  subsemiring.zero_mem \u2191S\n\ntheorem add_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x + y \u2208 S :=\n  subsemiring.add_mem (\u2191S) hx hy\n\ntheorem neg_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) {x : A} (hx : x \u2208 S) : -x \u2208 S :=\n  neg_one_smul R x \u25b8 smul_mem S hx (-1)\n\ntheorem sub_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) {x : A} {y : A} (hx : x \u2208 S) (hy : y \u2208 S) : x - y \u2208 S := sorry\n\ntheorem nsmul_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2115) : n \u2022\u2115 x \u2208 S :=\n  subsemiring.nsmul_mem (\u2191S) hx n\n\ntheorem gsmul_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) {x : A} (hx : x \u2208 S) (n : \u2124) : n \u2022\u2124 x \u2208 S :=\n  int.cases_on n (fun (i : \u2115) => nsmul_mem S hx i) fun (i : \u2115) => neg_mem S (nsmul_mem S hx (Nat.succ i))\n\ntheorem coe_nat_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) (n : \u2115) : \u2191n \u2208 S :=\n  subsemiring.coe_nat_mem (\u2191S) n\n\ntheorem coe_int_mem {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) (n : \u2124) : \u2191n \u2208 S :=\n  int.cases_on n (fun (i : \u2115) => coe_nat_mem S i) fun (i : \u2115) => neg_mem S (coe_nat_mem S (i + 1))\n\ntheorem list_prod_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {L : List A} (h : \u2200 (x : A), x \u2208 L \u2192 x \u2208 S) : list.prod L \u2208 S :=\n  subsemiring.list_prod_mem (\u2191S) h\n\ntheorem list_sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {L : List A} (h : \u2200 (x : A), x \u2208 L \u2192 x \u2208 S) : list.sum L \u2208 S :=\n  subsemiring.list_sum_mem (\u2191S) h\n\ntheorem multiset_prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A] [algebra R A] (S : subalgebra R A) {m : multiset A} (h : \u2200 (x : A), x \u2208 m \u2192 x \u2208 S) : multiset.prod m \u2208 S :=\n  subsemiring.multiset_prod_mem (\u2191S) m h\n\ntheorem multiset_sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {m : multiset A} (h : \u2200 (x : A), x \u2208 m \u2192 x \u2208 S) : multiset.sum m \u2208 S :=\n  subsemiring.multiset_sum_mem (\u2191S) m h\n\ntheorem prod_mem {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A] [algebra R A] (S : subalgebra R A) {\u03b9 : Type w} {t : finset \u03b9} {f : \u03b9 \u2192 A} (h : \u2200 (x : \u03b9), x \u2208 t \u2192 f x \u2208 S) : (finset.prod t fun (x : \u03b9) => f x) \u2208 S :=\n  subsemiring.prod_mem (\u2191S) h\n\ntheorem sum_mem {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {\u03b9 : Type w} {t : finset \u03b9} {f : \u03b9 \u2192 A} (h : \u2200 (x : \u03b9), x \u2208 t \u2192 f x \u2208 S) : (finset.sum t fun (x : \u03b9) => f x) \u2208 S :=\n  subsemiring.sum_mem (\u2191S) h\n\nprotected instance is_add_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : is_add_submonoid \u2191S :=\n  is_add_submonoid.mk (zero_mem S) fun (_x _x_1 : A) => add_mem S\n\nprotected instance is_submonoid {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : is_submonoid \u2191S :=\n  is_submonoid.mk (one_mem S) fun (_x _x_1 : A) => mul_mem S\n\n/-- A subalgebra over a ring is also a `subring`. -/\ndef to_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) : subring A :=\n  subring.mk (subsemiring.carrier (to_subsemiring S)) sorry sorry sorry sorry sorry\n\nprotected instance is_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) : is_subring \u2191S :=\n  is_subring.mk\n\nprotected instance inhabited {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : Inhabited \u21a5S :=\n  { default := 0 }\n\nprotected instance semiring (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : semiring \u21a5S :=\n  subsemiring.to_semiring \u2191S\n\nprotected instance comm_semiring (R : Type u) (A : Type v) [comm_semiring R] [comm_semiring A] [algebra R A] (S : subalgebra R A) : comm_semiring \u21a5S :=\n  subsemiring.to_comm_semiring \u2191S\n\nprotected instance ring (R : Type u) (A : Type v) [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) : ring \u21a5S :=\n  subtype.ring\n\nprotected instance comm_ring (R : Type u) (A : Type v) [comm_ring R] [comm_ring A] [algebra R A] (S : subalgebra R A) : comm_ring \u21a5S :=\n  subtype.comm_ring\n\nprotected instance algebra {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : algebra R \u21a5S :=\n  algebra.mk (ring_hom.mk (ring_hom.to_fun (ring_hom.cod_srestrict (algebra_map R A) \u2191S sorry)) sorry sorry sorry sorry)\n    sorry sorry\n\nprotected instance to_algebra {R : Type u_1} {A : Type u_2} {B : Type u_3} [comm_semiring R] [comm_semiring A] [semiring B] [algebra R A] [algebra A B] (A\u2080 : subalgebra R A) : algebra (\u21a5A\u2080) B :=\n  algebra.of_subsemiring \u2191A\u2080\n\nprotected instance nontrivial {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) [nontrivial A] : nontrivial \u21a5S :=\n  subsemiring.nontrivial \u2191S\n\n-- todo: standardize on the names these morphisms\n\n-- compare with submodule.subtype\n\n/-- Embedding of a subalgebra into the algebra. -/\ndef val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : alg_hom R (\u21a5S) A :=\n  alg_hom.mk coe sorry sorry sorry sorry sorry\n\n@[simp] theorem coe_val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : \u21d1(val S) = coe :=\n  rfl\n\ntheorem val_apply {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) (x : \u21a5S) : coe_fn (val S) x = \u2191x :=\n  rfl\n\n/-- Convert a `subalgebra` to `submodule` -/\ndef to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : submodule R A :=\n  submodule.mk \u2191S sorry sorry sorry\n\nprotected instance coe_to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : has_coe (subalgebra R A) (submodule R A) :=\n  has_coe.mk to_submodule\n\nprotected instance to_submodule.is_subring {R : Type u} {A : Type v} [comm_ring R] [ring A] [algebra R A] (S : subalgebra R A) : is_subring \u2191\u2191S :=\n  subalgebra.is_subring S\n\n@[simp] theorem mem_to_submodule {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) {x : A} : x \u2208 \u2191S \u2194 x \u2208 S :=\n  iff.rfl\n\ntheorem to_submodule_injective {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {S : subalgebra R A} {U : subalgebra R A} (h : \u2191S = \u2191U) : S = U := sorry\n\ntheorem to_submodule_inj {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {S : subalgebra R A} {U : subalgebra R A} : \u2191S = \u2191U \u2194 S = U :=\n  { mp := to_submodule_injective, mpr := congr_arg fun {S : subalgebra R A} => \u2191S }\n\n/-- As submodules, subalgebras are idempotent. -/\n@[simp] theorem mul_self {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : \u2191S * \u2191S = \u2191S := sorry\n\n/-- Linear equivalence between `S : submodule R A` and `S`. Though these types are equal,\nwe define it as a `linear_equiv` to avoid type equalities. -/\ndef to_submodule_equiv {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : linear_equiv R \u21a5\u2191S \u21a5S :=\n  linear_equiv.of_eq (\u2191S) (has_coe_t_aux.coe S) sorry\n\nprotected instance partial_order {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : partial_order (subalgebra R A) :=\n  partial_order.mk (fun (S T : subalgebra R A) => \u2191S \u2286 \u2191T) (preorder.lt._default fun (S T : subalgebra R A) => \u2191S \u2286 \u2191T)\n    sorry sorry sorry\n\n/-- Reinterpret an `S`-subalgebra as an `R`-subalgebra in `comap R S A`. -/\ndef comap {R : Type u} {S : Type v} {A : Type w} [comm_semiring R] [comm_semiring S] [semiring A] [algebra R S] [algebra S A] (iSB : subalgebra S A) : subalgebra R (algebra.comap R S A) :=\n  mk (carrier iSB) (one_mem' iSB) (mul_mem' iSB) (zero_mem' iSB) (add_mem' iSB) sorry\n\n/-- If `S` is an `R`-subalgebra of `A` and `T` is an `S`-subalgebra of `A`,\nthen `T` is an `R`-subalgebra of `A`. -/\ndef under {R : Type u} {A : Type v} [comm_semiring R] [comm_semiring A] {i : algebra R A} (S : subalgebra R A) (T : subalgebra (\u21a5S) A) : subalgebra R A :=\n  mk (carrier T) sorry sorry sorry sorry sorry\n\n/-- Transport a subalgebra via an algebra homomorphism. -/\ndef map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (S : subalgebra R A) (f : alg_hom R A B) : subalgebra R B :=\n  mk (subsemiring.carrier (subsemiring.map \u2191f \u2191S)) sorry sorry sorry sorry sorry\n\n/-- Preimage of a subalgebra under an algebra homomorphism. -/\ndef comap' {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (S : subalgebra R B) (f : alg_hom R A B) : subalgebra R A :=\n  mk (subsemiring.carrier (subsemiring.comap \u2191f \u2191S)) sorry sorry sorry sorry sorry\n\ntheorem map_mono {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] {S\u2081 : subalgebra R A} {S\u2082 : subalgebra R A} {f : alg_hom R A B} : S\u2081 \u2264 S\u2082 \u2192 map S\u2081 f \u2264 map S\u2082 f :=\n  set.image_subset \u21d1f\n\ntheorem map_le {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] {S : subalgebra R A} {f : alg_hom R A B} {U : subalgebra R B} : map S f \u2264 U \u2194 S \u2264 comap' U f :=\n  set.image_subset_iff\n\ntheorem map_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] {S\u2081 : subalgebra R A} {S\u2082 : subalgebra R A} (f : alg_hom R A B) (hf : function.injective \u21d1f) (ih : map S\u2081 f = map S\u2082 f) : S\u2081 = S\u2082 :=\n  ext\n    (iff.mp set.ext_iff\n      (iff.mpr set.image_injective hf (fun (x : A) => x \u2208 \u2191S\u2081) (fun (x : A) => x \u2208 \u2191S\u2082) (set.ext (iff.mp ext_iff ih))))\n\ntheorem mem_map {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] {S : subalgebra R A} {f : alg_hom R A B} {y : B} : y \u2208 map S f \u2194 \u2203 (x : A), \u2203 (H : x \u2208 S), coe_fn f x = y :=\n  subsemiring.mem_map\n\nprotected instance no_zero_divisors {R : Type u_1} {A : Type u_2} [comm_ring R] [semiring A] [no_zero_divisors A] [algebra R A] (S : subalgebra R A) : no_zero_divisors \u21a5S :=\n  subsemiring.no_zero_divisors (to_subsemiring S)\n\nprotected instance integral_domain {R : Type u_1} {A : Type u_2} [comm_ring R] [integral_domain A] [algebra R A] (S : subalgebra R A) : integral_domain \u21a5S :=\n  subring.domain \u2191S\n\nend subalgebra\n\n\nnamespace alg_hom\n\n\n/-- Range of an `alg_hom` as a subalgebra. -/\nprotected def range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) : subalgebra R B :=\n  subalgebra.mk (subsemiring.carrier (ring_hom.srange (to_ring_hom \u03c6))) sorry sorry sorry sorry sorry\n\n@[simp] theorem mem_range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) {y : B} : y \u2208 alg_hom.range \u03c6 \u2194 \u2203 (x : A), coe_fn \u03c6 x = y :=\n  ring_hom.mem_srange\n\n@[simp] theorem coe_range {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (\u03c6 : alg_hom R A B) : \u2191(alg_hom.range \u03c6) = set.range \u21d1\u03c6 := sorry\n\n/-- Restrict the codomain of an algebra homomorphism. -/\ndef cod_restrict {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (S : subalgebra R B) (hf : \u2200 (x : A), coe_fn f x \u2208 S) : alg_hom R A \u21a5S :=\n  mk (ring_hom.to_fun (ring_hom.cod_srestrict (\u2191f) (\u2191S) hf)) sorry sorry sorry sorry sorry\n\ntheorem injective_cod_restrict {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (S : subalgebra R B) (hf : \u2200 (x : A), coe_fn f x \u2208 S) : function.injective \u21d1(cod_restrict f S hf) \u2194 function.injective \u21d1f := sorry\n\n/-- Restrict an injective algebra homomorphism to an algebra isomorphism -/\ndef alg_equiv.of_injective {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (hf : function.injective \u21d1f) : alg_equiv R A \u21a5(alg_hom.range f) :=\n  alg_equiv.of_bijective (cod_restrict f (alg_hom.range f) sorry) sorry\n\n@[simp] theorem alg_equiv.of_injective_apply {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (f : alg_hom R A B) (hf : function.injective \u21d1f) (x : A) : \u2191(coe_fn (alg_equiv.of_injective f hf) x) = coe_fn f x :=\n  rfl\n\n/-- Restrict an algebra homomorphism between fields to an algebra isomorphism -/\ndef alg_equiv.of_injective_field {R : Type u} [comm_semiring R] {E : Type u_1} {F : Type u_2} [division_ring E] [semiring F] [nontrivial F] [algebra R E] [algebra R F] (f : alg_hom R E F) : alg_equiv R E \u21a5(alg_hom.range f) :=\n  alg_equiv.of_injective f sorry\n\n/-- The equalizer of two R-algebra homomorphisms -/\ndef equalizer {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (\u03d5 : alg_hom R A B) (\u03c8 : alg_hom R A B) : subalgebra R A :=\n  subalgebra.mk (set_of fun (a : A) => coe_fn \u03d5 a = coe_fn \u03c8 a) sorry sorry sorry sorry sorry\n\n@[simp] theorem mem_equalizer {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] (\u03d5 : alg_hom R A B) (\u03c8 : alg_hom R A B) (x : A) : x \u2208 equalizer \u03d5 \u03c8 \u2194 coe_fn \u03d5 x = coe_fn \u03c8 x :=\n  iff.rfl\n\nend alg_hom\n\n\nnamespace algebra\n\n\n/-- The minimal subalgebra that includes `s`. -/\ndef adjoin (R : Type u) {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (s : set A) : subalgebra R A :=\n  subalgebra.mk (subsemiring.carrier (subsemiring.closure (set.range \u21d1(algebra_map R A) \u222a s))) sorry sorry sorry sorry\n    sorry\n\nprotected theorem gc {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : galois_connection (adjoin R) coe := sorry\n\n/-- Galois insertion between `adjoin` and `coe`. -/\nprotected def gi {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : galois_insertion (adjoin R) coe :=\n  galois_insertion.mk (fun (s : set A) (hs : \u2191(adjoin R s) \u2264 s) => adjoin R s) algebra.gc sorry sorry\n\nprotected instance subalgebra.complete_lattice {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : complete_lattice (subalgebra R A) :=\n  galois_insertion.lift_complete_lattice algebra.gi\n\nprotected instance subalgebra.inhabited {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : Inhabited (subalgebra R A) :=\n  { default := \u22a5 }\n\ntheorem mem_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {x : A} : x \u2208 \u22a5 \u2194 x \u2208 set.range \u21d1(algebra_map R A) := sorry\n\ntheorem to_submodule_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : \u2191\u22a5 = submodule.span R (singleton 1) := sorry\n\n@[simp] theorem mem_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {x : A} : x \u2208 \u22a4 :=\n  subsemiring.subset_closure (Or.inr trivial)\n\n@[simp] theorem coe_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : \u2191\u22a4 = \u22a4 :=\n  submodule.ext fun (x : A) => iff_of_true mem_top trivial\n\n@[simp] theorem coe_bot {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : \u2191\u22a5 = set.range \u21d1(algebra_map R A) := sorry\n\ntheorem eq_top_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] {S : subalgebra R A} : S = \u22a4 \u2194 \u2200 (x : A), x \u2208 S := sorry\n\n@[simp] theorem map_top {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) : subalgebra.map \u22a4 f = alg_hom.range f := sorry\n\n@[simp] theorem map_bot {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) : subalgebra.map \u22a5 f = \u22a5 := sorry\n\n@[simp] theorem comap_top {R : Type u} {A : Type v} {B : Type w} [comm_semiring R] [semiring A] [algebra R A] [semiring B] [algebra R B] (f : alg_hom R A B) : subalgebra.comap' \u22a4 f = \u22a4 :=\n  iff.mpr eq_top_iff fun (x : A) => mem_top\n\n/-- `alg_hom` to `\u22a4 : subalgebra R A`. -/\ndef to_top {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : alg_hom R A \u21a5\u22a4 :=\n  alg_hom.mk (fun (x : A) => { val := x, property := mem_top }) sorry sorry sorry sorry sorry\n\ntheorem surjective_algebra_map_iff {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : function.surjective \u21d1(algebra_map R A) \u2194 \u22a4 = \u22a5 := sorry\n\ntheorem bijective_algebra_map_iff {R : Type u_1} {A : Type u_2} [field R] [semiring A] [nontrivial A] [algebra R A] : function.bijective \u21d1(algebra_map R A) \u2194 \u22a4 = \u22a5 :=\n  { mp := fun (h : function.bijective \u21d1(algebra_map R A)) => iff.mp surjective_algebra_map_iff (and.right h),\n    mpr :=\n      fun (h : \u22a4 = \u22a5) => { left := ring_hom.injective (algebra_map R A), right := iff.mpr surjective_algebra_map_iff h } }\n\n/-- The bottom subalgebra is isomorphic to the base ring. -/\ndef bot_equiv_of_injective {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (h : function.injective \u21d1(algebra_map R A)) : alg_equiv R (\u21a5\u22a5) R :=\n  alg_equiv.symm (alg_equiv.of_bijective (of_id R \u21a5\u22a5) sorry)\n\n/-- The bottom subalgebra is isomorphic to the field. -/\ndef bot_equiv (F : Type u_1) (R : Type u_2) [field F] [semiring R] [nontrivial R] [algebra F R] : alg_equiv F (\u21a5\u22a5) F :=\n  bot_equiv_of_injective sorry\n\n/-- The top subalgebra is isomorphic to the field. -/\ndef top_equiv {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] : alg_equiv R (\u21a5\u22a4) A :=\n  alg_equiv.symm (alg_equiv.of_bijective to_top sorry)\n\nend algebra\n\n\nnamespace subalgebra\n\n\ntheorem range_val {R : Type u} {A : Type v} [comm_semiring R] [semiring A] [algebra R A] (S : subalgebra R A) : alg_hom.range (val S) = S :=\n  ext (iff.mp set.ext_iff (Eq.trans (alg_hom.coe_range (val S)) subtype.range_val))\n\nprotected instance unique {R : Type u} [comm_semiring R] : unique (subalgebra R R) :=\n  unique.mk { default := Inhabited.default } sorry\n\nend subalgebra\n\n\n/-- A subsemiring is a `\u2115`-subalgebra. -/\ndef subalgebra_of_subsemiring {R : Type u_1} [semiring R] (S : subsemiring R) : subalgebra \u2115 R :=\n  subalgebra.mk (subsemiring.carrier S) (subsemiring.one_mem' S) (subsemiring.mul_mem' S) (subsemiring.zero_mem' S)\n    (subsemiring.add_mem' S) sorry\n\n@[simp] theorem mem_subalgebra_of_subsemiring {R : Type u_1} [semiring R] {x : R} {S : subsemiring R} : x \u2208 subalgebra_of_subsemiring S \u2194 x \u2208 S :=\n  iff.rfl\n\n/-- A subring is a `\u2124`-subalgebra. -/\ndef subalgebra_of_subring {R : Type u_1} [ring R] (S : subring R) : subalgebra \u2124 R :=\n  subalgebra.mk (subring.carrier S) (subring.one_mem' S) (subring.mul_mem' S) (subring.zero_mem' S) (subring.add_mem' S)\n    sorry\n\n/-- A subset closed under the ring operations is a `\u2124`-subalgebra. -/\ndef subalgebra_of_is_subring {R : Type u_1} [ring R] (S : set R) [is_subring S] : subalgebra \u2124 R :=\n  subalgebra_of_subring (set.to_subring S)\n\n@[simp] theorem mem_subalgebra_of_subring {R : Type u_1} [ring R] {x : R} {S : subring R} : x \u2208 subalgebra_of_subring S \u2194 x \u2208 S :=\n  iff.rfl\n\n@[simp] theorem mem_subalgebra_of_is_subring {R : Type u_1} [ring R] {x : R} {S : set R} [is_subring S] : x \u2208 subalgebra_of_is_subring S \u2194 x \u2208 S :=\n  iff.rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/algebra/subalgebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3775468831988232}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport category_theory.over\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.finite_products\n\n/-!\n# Products in the over category\n\nShows that products in the over category can be derived from wide pullbacks in the base category.\nThe main result is `over_product_of_wide_pullback`, which says that if `C` has `J`-indexed wide\npullbacks, then `over B` has `J`-indexed products.\n-/\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nopen category_theory category_theory.limits\n\nvariables {J : Type v}\nvariables {C : Type u} [category.{v} C]\nvariable {X : C}\n\nnamespace category_theory.over\n\nnamespace construct_products\n\n/--\n(Implementation)\nGiven a product diagram in `C/B`, construct the corresponding wide pullback diagram\nin `C`.\n-/\n@[reducible]\ndef wide_pullback_diagram_of_diagram_over (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  wide_pullback_shape J \u2964 C :=\nwide_pullback_shape.wide_cospan B (\u03bb j, (F.obj \u27e8j\u27e9).left) (\u03bb j, (F.obj \u27e8j\u27e9).hom)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_inverse_obj (B : C) {J : Type v} (F : discrete J \u2964 over B) (c : cone F) :\n  cone (wide_pullback_diagram_of_diagram_over B F) :=\n{ X := c.X.left,\n  \u03c0 :=\n  { app := \u03bb X, option.cases_on X c.X.hom (\u03bb (j : J), (c.\u03c0.app \u27e8j\u27e9).left),\n  -- `tidy` can do this using `case_bash`, but let's try to be a good `-T50000` citizen:\n    naturality' := \u03bb X Y f,\n    begin\n      dsimp, cases X; cases Y; cases f,\n      { rw [category.id_comp, category.comp_id], },\n      { rw [over.w, category.id_comp], },\n      { rw [category.id_comp, category.comp_id], },\n    end } }\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_inverse (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  cone F \u2964 cone (wide_pullback_diagram_of_diagram_over B F) :=\n{ obj := cones_equiv_inverse_obj B F,\n  map := \u03bb c\u2081 c\u2082 f,\n  { hom := f.hom.left,\n    w' := \u03bb j,\n    begin\n      cases j,\n      { simp },\n      { dsimp,\n        rw \u2190 f.w \u27e8j\u27e9,\n        refl }\n    end } }\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_functor (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  cone (wide_pullback_diagram_of_diagram_over B F) \u2964 cone F :=\n{ obj := \u03bb c,\n  { X := over.mk (c.\u03c0.app none),\n    \u03c0 :=\n    { app := \u03bb \u27e8j\u27e9, over.hom_mk (c.\u03c0.app (some j))\n                    (by apply c.w (wide_pullback_shape.hom.term j)) } },\n  map := \u03bb c\u2081 c\u2082 f,\n  { hom := over.hom_mk f.hom } }\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp]\ndef cones_equiv_unit_iso (B : C) (F : discrete J \u2964 over B) :\n  \ud835\udfed (cone (wide_pullback_diagram_of_diagram_over B F)) \u2245\n    cones_equiv_functor B F \u22d9 cones_equiv_inverse B F :=\nnat_iso.of_components (\u03bb _, cones.ext {hom := \ud835\udfd9 _, inv := \ud835\udfd9 _} (by tidy)) (by tidy)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp]\ndef cones_equiv_counit_iso (B : C) (F : discrete J \u2964 over B) :\n  cones_equiv_inverse B F \u22d9 cones_equiv_functor B F \u2245 \ud835\udfed (cone F) :=\nnat_iso.of_components\n  (\u03bb _, cones.ext {hom := over.hom_mk (\ud835\udfd9 _), inv := over.hom_mk (\ud835\udfd9 _)} (by tidy)) (by tidy)\n\n-- TODO: Can we add `. obviously` to the second arguments of `nat_iso.of_components` and\n--       `cones.ext`?\n/--\n(Impl) Establish an equivalence between the category of cones for `F` and for the \"grown\" `F`.\n-/\n@[simps]\ndef cones_equiv (B : C) (F : discrete J \u2964 over B) :\n  cone (wide_pullback_diagram_of_diagram_over B F) \u224c cone F :=\n{ functor := cones_equiv_functor B F,\n  inverse := cones_equiv_inverse B F,\n  unit_iso := cones_equiv_unit_iso B F,\n  counit_iso := cones_equiv_counit_iso B F, }\n\n/-- Use the above equivalence to prove we have a limit. -/\nlemma has_over_limit_discrete_of_wide_pullback_limit {B : C} (F : discrete J \u2964 over B)\n  [has_limit (wide_pullback_diagram_of_diagram_over B F)] :\n  has_limit F :=\nhas_limit.mk\n{ cone := _,\n  is_limit := is_limit.of_right_adjoint\n    (cones_equiv B F).functor (limit.is_limit (wide_pullback_diagram_of_diagram_over B F)) }\n\n/-- Given a wide pullback in `C`, construct a product in `C/B`. -/\nlemma over_product_of_wide_pullback [has_limits_of_shape (wide_pullback_shape J) C] {B : C} :\n  has_limits_of_shape (discrete J) (over B) :=\n{ has_limit := \u03bb F, has_over_limit_discrete_of_wide_pullback_limit F }\n\n/-- Given a pullback in `C`, construct a binary product in `C/B`. -/\nlemma over_binary_product_of_pullback [has_pullbacks C] {B : C} :\n  has_binary_products (over B) :=\nover_product_of_wide_pullback\n\n/-- Given all wide pullbacks in `C`, construct products in `C/B`. -/\nlemma over_products_of_wide_pullbacks [has_wide_pullbacks C] {B : C} :\n  has_products (over B) :=\n\u03bb J, over_product_of_wide_pullback\n\n/-- Given all finite wide pullbacks in `C`, construct finite products in `C/B`. -/\nlemma over_finite_products_of_finite_wide_pullbacks [has_finite_wide_pullbacks C] {B : C} :\n  has_finite_products (over B) :=\n\u27e8\u03bb J \ud835\udca5, by exactI over_product_of_wide_pullback\u27e9\n\nend construct_products\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/--\nConstruct terminal object in the over category. This isn't an instance as it's not typically the\nway we want to define terminal objects.\n(For instance, this gives a terminal object which is different from the generic one given by\n`over_product_of_wide_pullback` above.)\n-/\nlemma over_has_terminal (B : C) : has_terminal (over B) :=\n{ has_limit := \u03bb F, has_limit.mk\n  { cone :=\n    { X := over.mk (\ud835\udfd9 _),\n      \u03c0 := { app := \u03bb p, p.as.elim } },\n    is_limit :=\n      { lift := \u03bb s, over.hom_mk _,\n        fac' := \u03bb _ j, j.as.elim,\n        uniq' := \u03bb s m _,\n          begin\n            ext,\n            rw over.hom_mk_left,\n            have := m.w,\n            dsimp at this,\n            rwa [category.comp_id, category.comp_id] at this\n          end } } }\n\nend category_theory.over\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/constructions/over/products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3775468750307285}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n\nTuples are lists of a fixed size.\nIt is implemented as a subtype.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.list.default\nimport Mathlib.Lean3Lib.init.data.subtype.default\nimport Mathlib.Lean3Lib.init.meta.interactive\nimport Mathlib.Lean3Lib.init.data.fin.default\n\nuniverses u u_1 v w \n\nnamespace Mathlib\n\ndef vector (\u03b1 : Type u) (n : \u2115) := Subtype fun (l : List \u03b1) => list.length l = n\n\nnamespace vector\n\n\nprotected instance decidable_eq {\u03b1 : Type u} {n : \u2115} [DecidableEq \u03b1] : DecidableEq (vector \u03b1 n) :=\n  eq.mpr sorry fun (a b : Subtype fun (l : List \u03b1) => list.length l = n) => subtype.decidable_eq a b\n\ndef nil {\u03b1 : Type u} : vector \u03b1 0 := { val := [], property := sorry }\n\ndef cons {\u03b1 : Type u} {n : \u2115} : \u03b1 \u2192 vector \u03b1 n \u2192 vector \u03b1 (Nat.succ n) := sorry\n\ndef length {\u03b1 : Type u} {n : \u2115} (v : vector \u03b1 n) : \u2115 := n\n\ndef head {\u03b1 : Type u} {n : \u2115} : vector \u03b1 (Nat.succ n) \u2192 \u03b1 := sorry\n\ntheorem head_cons {\u03b1 : Type u} {n : \u2115} (a : \u03b1) (v : vector \u03b1 n) : head (cons a v) = a := sorry\n\ndef tail {\u03b1 : Type u} {n : \u2115} : vector \u03b1 n \u2192 vector \u03b1 (n - 1) := sorry\n\ntheorem tail_cons {\u03b1 : Type u} {n : \u2115} (a : \u03b1) (v : vector \u03b1 n) : tail (cons a v) = v := sorry\n\n@[simp] theorem cons_head_tail {\u03b1 : Type u} {n : \u2115} (v : vector \u03b1 (Nat.succ n)) :\n    cons (head v) (tail v) = v :=\n  sorry\n\ndef to_list {\u03b1 : Type u} {n : \u2115} (v : vector \u03b1 n) : List \u03b1 := subtype.val v\n\ndef nth {\u03b1 : Type u} {n : \u2115} (v : vector \u03b1 n) : fin n \u2192 \u03b1 := sorry\n\ndef append {\u03b1 : Type u} {n : \u2115} {m : \u2115} : vector \u03b1 n \u2192 vector \u03b1 m \u2192 vector \u03b1 (n + m) := sorry\n\ndef elim {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector \u03b1 n \u2192 Sort u}\n    (H : (l : List \u03b1) \u2192 C { val := l, property := elim._proof_1 l }) {n : \u2115} (v : vector \u03b1 n) :\n    C v :=\n  sorry\n\n/- map -/\n\ndef map {\u03b1 : Type u} {\u03b2 : Type v} {n : \u2115} (f : \u03b1 \u2192 \u03b2) : vector \u03b1 n \u2192 vector \u03b2 n := sorry\n\n@[simp] theorem map_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : map f nil = nil := rfl\n\ntheorem map_cons {\u03b1 : Type u} {\u03b2 : Type v} {n : \u2115} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (v : vector \u03b1 n) :\n    map f (cons a v) = cons (f a) (map f v) :=\n  sorry\n\ndef map\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03c6 : Type w} {n : \u2115} (f : \u03b1 \u2192 \u03b2 \u2192 \u03c6) :\n    vector \u03b1 n \u2192 vector \u03b2 n \u2192 vector \u03c6 n :=\n  sorry\n\ndef repeat {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : vector \u03b1 n :=\n  { val := list.repeat a n, property := list.length_repeat a n }\n\ndef drop {\u03b1 : Type u} {n : \u2115} (i : \u2115) : vector \u03b1 n \u2192 vector \u03b1 (n - i) := sorry\n\ndef take {\u03b1 : Type u} {n : \u2115} (i : \u2115) : vector \u03b1 n \u2192 vector \u03b1 (min i n) := sorry\n\ndef remove_nth {\u03b1 : Type u} {n : \u2115} (i : fin n) : vector \u03b1 n \u2192 vector \u03b1 (n - 1) := sorry\n\ndef of_fn {\u03b1 : Type u} {n : \u2115} : (fin n \u2192 \u03b1) \u2192 vector \u03b1 n := sorry\n\ndef map_accumr {\u03b1 : Type u} {\u03b2 : Type v} {n : \u2115} {\u03c3 : Type} (f : \u03b1 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03b2) :\n    vector \u03b1 n \u2192 \u03c3 \u2192 \u03c3 \u00d7 vector \u03b2 n :=\n  sorry\n\ndef map_accumr\u2082 {n : \u2115} {\u03b1 : Type} {\u03b2 : Type} {\u03c3 : Type} {\u03c6 : Type} (f : \u03b1 \u2192 \u03b2 \u2192 \u03c3 \u2192 \u03c3 \u00d7 \u03c6) :\n    vector \u03b1 n \u2192 vector \u03b2 n \u2192 \u03c3 \u2192 \u03c3 \u00d7 vector \u03c6 n :=\n  sorry\n\nprotected theorem eq {\u03b1 : Type u} {n : \u2115} (a1 : vector \u03b1 n) (a2 : vector \u03b1 n) :\n    to_list a1 = to_list a2 \u2192 a1 = a2 :=\n  sorry\n\nprotected theorem eq_nil {\u03b1 : Type u} (v : vector \u03b1 0) : v = nil :=\n  vector.eq v nil (list.eq_nil_of_length_eq_zero (subtype.property v))\n\n@[simp] theorem to_list_mk {\u03b1 : Type u} {n : \u2115} (v : List \u03b1) (P : list.length v = n) :\n    to_list { val := v, property := P } = v :=\n  rfl\n\n@[simp] theorem to_list_nil {\u03b1 : Type u} : to_list nil = [] := rfl\n\n@[simp] theorem to_list_length {\u03b1 : Type u} {n : \u2115} (v : vector \u03b1 n) :\n    list.length (to_list v) = n :=\n  subtype.property v\n\n@[simp] theorem to_list_cons {\u03b1 : Type u} {n : \u2115} (a : \u03b1) (v : vector \u03b1 n) :\n    to_list (cons a v) = a :: to_list v :=\n  subtype.cases_on v\n    fun (v_val : List \u03b1) (v_property : list.length v_val = n) =>\n      Eq.refl (to_list (cons a { val := v_val, property := v_property }))\n\n@[simp] theorem to_list_append {\u03b1 : Type u} {n : \u2115} {m : \u2115} (v : vector \u03b1 n) (w : vector \u03b1 m) :\n    to_list (append v w) = to_list v ++ to_list w :=\n  sorry\n\n@[simp] theorem to_list_drop {\u03b1 : Type u} {n : \u2115} {m : \u2115} (v : vector \u03b1 m) :\n    to_list (drop n v) = list.drop n (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List \u03b1) (v_property : list.length v_val = m) =>\n      Eq.refl (to_list (drop n { val := v_val, property := v_property }))\n\n@[simp] theorem to_list_take {\u03b1 : Type u} {n : \u2115} {m : \u2115} (v : vector \u03b1 m) :\n    to_list (take n v) = list.take n (to_list v) :=\n  subtype.cases_on v\n    fun (v_val : List \u03b1) (v_property : list.length v_val = m) =>\n      Eq.refl (to_list (take n { val := v_val, property := v_property }))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/data/vector_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665855647394, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3775468740323764}}
{"text": "import .lang\n\nnamespace ralg\nuniverses v u\n\nvariables (L : lang.{v}) (S : Type u)\n\nnamespace free\n\ninductive tp : Type (max v u)\n| of : S \u2192 tp\n| op {n} : L n \u2192 (fin n \u2192 tp) \u2192 tp\n\nend free\n\ndef free : ralg L := \n{ carrier := free.tp L S,\n  appo := \u03bb _ t as, free.tp.op t as }\n\nnamespace free\n\ndef univ : S \u2192 free L S := tp.of\n\nvariables (L) {S}\ndef lift {B : ralg L} (f : S \u2192 B) : (free L S) \u2192% B := \n{ to_fn := \u03bb t, tp.rec_on t f \n    (\u03bb _ t _ bs, applyo t $ vector.of_fn bs),\n  applyo_map := \u03bb n t as, \n  begin\n    have : as = vector.of_fn as, by rw vector.of_fn_to_fn,\n    conv_lhs {rw this},\n    rw vector.map_of_fn,\n    refl,\n  end }\n\ntheorem univ_comp_lift {B : ralg L} (f : S \u2192 B) : (lift _ f) \u2218 (univ L S) = f := rfl\n\ntheorem lift_unique {B : ralg L} (f : S \u2192 B) (g : (free L S) \u2192% B) :\n  g \u2218 (univ _ _) = f \u2192 g = lift _ f := \u03bb hyp,\nbegin\n  ext,\n  induction x with _ n t as h,\n  { change (g \u2218 (univ _ _)) x = _,\n    rw hyp, refl },\n  { have : as = vector.of_fn as, \n    { change _ = vector.to_fn _,\n      rw vector.to_fn_of_fn },\n    dsimp only [] at h,\n    rw this,\n    change g.to_fn (applyo _ _) = (lift _ f).to_fn (applyo _ _),\n    simp_rw \u2190ralgHom.applyo_map,\n    apply congr_arg,\n    simp only [vector.map_of_fn],\n    rw vector.eq_iff_to_fn_eq_to_fn,\n    funext,\n    change vector.to_fn _ _ = vector.to_fn _ _,\n    simp only [vector.to_fn_of_fn], apply h }\nend\n\nend free\n\nend ralg", "meta": {"author": "adamtopaz", "repo": "UnivAlg", "sha": "2458d47a6e4fd0525e3a25b07cb7dd518ac173ef", "save_path": "github-repos/lean/adamtopaz-UnivAlg", "path": "github-repos/lean/adamtopaz-UnivAlg/UnivAlg-2458d47a6e4fd0525e3a25b07cb7dd518ac173ef/src/.old/free_ralg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3774696689529436}}
{"text": "import data.finset.basic\n\nimport guidelines.protocol\n\nstructure proposal (ballot_t value_t : Type) := (b : ballot_t) (v : value_t)\n\nstructure paxos_defs (sys_state_t pid_t ballot_t value_t : Type) :=\n  (curr : sys_state_t \u2192 pid_t \u2192 ballot_t)\n  (stored : sys_state_t \u2192 pid_t \u2192 option (proposal ballot_t value_t))\n  (proposed : sys_state_t \u2192 ballot_t \u2192 value_t \u2192 Prop)\n  (voted : sys_state_t \u2192 pid_t \u2192 ballot_t \u2192 Prop)\n  (quorum : finset pid_t \u2192 Prop)\n\nnamespace paxos_defs\n\nvariables {sys_state_t pid_t ballot_t value_t : Type}\n\ndef chosen_ballot\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t) :\n    sys_state_t \u2192 ballot_t \u2192 Prop :=\n  (\u03bb state ballot,\n    \u2203 q : finset pid_t, defs.quorum q \u2227\n      \u2200 voter \u2208 q, defs.voted state voter ballot)\n\ndef choosable [linear_order ballot_t]\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t) :\n    sys_state_t \u2192 ballot_t \u2192 Prop :=\n  (\u03bb state ballot,\n    \u2203 q : finset pid_t, defs.quorum q \u2227\n      \u2200 voter \u2208 q,\n        defs.voted state voter ballot \u2228 defs.curr state voter \u2264 ballot)\n\ndef chosen\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t) :\n    sys_state_t \u2192 value_t \u2192 Prop :=\n  (\u03bb state value,\n    \u2203 ballot, defs.chosen_ballot state ballot \u2227 defs.proposed state ballot value)\n\nlemma chosen_imp_choosable {s : sys_state_t} {b : ballot_t} [linear_order ballot_t]\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t) :\n  defs.chosen_ballot s b \u2192 defs.choosable s b :=\nbegin\nrintros \u27e8q, q_quorum, q_voted\u27e9,\nexact \u27e8q, q_quorum,\n  by {intros voter voter_in_q, left, exact q_voted voter voter_in_q }\u27e9\nend\n\nend paxos_defs\n\nstructure interval (ballot_t value_t : Type) :=\n  (upper : ballot_t) (lower : option (proposal ballot_t value_t))\n\nvariables {sys_state_t pid_t ballot_t value_t : Type}\n\ndef proto_with_intervals_recorded\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t)\n  (proto : protocol sys_state_t) : protocol (sys_state_t \u00d7 (pid_t \u2192 set (interval ballot_t value_t))) :=\n{ init := (\u03bb hist_state, proto.init hist_state.fst \u2227\n  (\u2200 p, hist_state.snd p =\n        {{upper := defs.curr hist_state.fst p, lower := defs.stored hist_state.fst p}})),\n  next := (\u03bb fst_state snd_state, proto.next fst_state.fst snd_state.fst \u2227\n  (\u2200 p, snd_state.snd p = fst_state.snd p \u222a\n        {{upper := defs.curr snd_state.fst p, lower := defs.stored snd_state.fst p}})) }\n\nlemma reachable_implies_reachable_in_proto_with_intervals_recorded\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t)\n  (proto : protocol sys_state_t) :\n  \u2200 s, proto.reachable s \u2192 \u2203 lift_s, (proto_with_intervals_recorded defs proto).reachable lift_s \u2227 lift_s.fst = s :=\nbegin\nsuffices : \u2200 n s,\n  proto.reachable_in n s \u2192\n  \u2203 lift_s, (proto_with_intervals_recorded defs proto).reachable lift_s \u2227 lift_s.fst = s,\nby { rintros s \u27e8n, hn\u27e9, exact this n s hn },\nintro n,\ninduction n with k hk,\n{ intros s hs,\n  use \u27e8s, (\u03bb p, {{ upper := defs.curr s p, lower := defs.stored s p}})\u27e9,\n  use \u27e80, hs, by {intro p, refl }\u27e9 },\nrintros s \u27e8u, u_reach_in_k, u_next_s\u27e9,\nspecialize hk u u_reach_in_k,\nrcases hk with \u27e8lift_u, \u27e8j, hj\u27e9, lift_u_has_u\u27e9,\nuse \u27e8s, (\u03bb p, lift_u.snd p \u222a {{ upper := defs.curr s p, lower := defs.stored s p}})\u27e9,\nsplit,\n{ use [j.succ, lift_u, hj],\n  split,\n  { rw lift_u_has_u, exact u_next_s },\n  intros p, refl },\nrefl\nend\n\nlemma reachable_in_proto_with_intervals_recorded_implies_first_reachable\n  (defs : paxos_defs sys_state_t pid_t ballot_t value_t)\n  (proto : protocol sys_state_t) :\n  \u2200 lift_s, (proto_with_intervals_recorded defs proto).reachable lift_s \u2192\n    proto.reachable lift_s.fst :=\nbegin\nsuffices : \u2200 n lift_s,\n  (proto_with_intervals_recorded defs proto).reachable_in n lift_s \u2192\n  proto.reachable lift_s.fst,\nby { rintros lift_s \u27e8n, hn\u27e9, exact this n lift_s hn },\nintro n,\ninduction n with k hk,\n{ intros s hs,\n  use \u27e80, hs.left\u27e9 },\nrintros lift_s \u27e8lift_u, lift_u_reach_in_k, lift_u_next_lift_s\u27e9,\nspecialize hk lift_u lift_u_reach_in_k,\nrcases hk with \u27e8j, hj\u27e9,\nexact \u27e8j.succ, lift_u.fst, hj, lift_u_next_lift_s.left\u27e9\nend\n", "meta": {"author": "gnanabite", "repo": "colocated-paxos", "sha": "f60308e27d3013665809077fe80a4b2af8a42278", "save_path": "github-repos/lean/gnanabite-colocated-paxos", "path": "github-repos/lean/gnanabite-colocated-paxos/colocated-paxos-f60308e27d3013665809077fe80a4b2af8a42278/src/guidelines/definitions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7549149978955811, "lm_q2_score": 0.5, "lm_q1q2_score": 0.37745749894779057}}
{"text": "import random_lemmas\nimport target_affine_scheme\n\nnoncomputable theory\n\nuniverse u\n\nopen category_theory category_theory.limits\nopen topological_space\nopen Top.presheaf Top.sheaf\nopen opposite\n\nnamespace algebraic_geometry\n\nnamespace Scheme\n\nvariables (X : Scheme.{u}) (R : CommRing.{u}) [local_ring R]\n\nstructure point_affine_open_stalk_ring_hom_triple :=\n(pt : X.carrier)\n(aopen : opens X.carrier)\n(mem_aopen : pt \u2208 aopen)\n(aopen_is_affine : is_affine_open aopen)\n(ring_hom_ : X.stalk pt \u27f6 R)\n[ring_hom_local : is_local_ring_hom ring_hom_]\n\nnamespace point_affine_open_stalk_ring_hom_triple\n\nvariables (P Q : point_affine_open_stalk_ring_hom_triple X R)\n\nlocal notation X `|_` P := (X.restrict (P.aopen.open_embedding) : Scheme)\n\ninstance is_affine_restrict : is_affine $ X |_ P :=\nP.aopen_is_affine\n\nvariables {X R}\n\nstructure refine : Prop :=\n(subset : P.aopen \u2264 Q.aopen)\n(pt_eq : P.pt = Q.pt)\n(ring_hom_eq : stalk_specializes X.presheaf (by rw pt_eq) \u226b P.ring_hom_ = Q.ring_hom_)\n\ndef restrict_stalk_iso : (X |_ P).stalk \u27e8P.pt, P.mem_aopen\u27e9  \u2245 X.stalk P.pt:=\nPresheafedSpace.restrict_stalk_iso X.to_PresheafedSpace \n  (P.aopen.open_embedding) \u27e8P.pt, _\u27e9\n\ndef restrict_\u0393_iso : (\u0393.obj $ op $ X |_ P) \u2245 X.presheaf.obj (op P.aopen) :=\n{ hom := X.presheaf.map $ (hom_of_le \n    begin \n      rintros x (hx : x \u2208 P.aopen),\n      refine \u27e8\u27e8x, hx\u27e9, \u27e8\u27e9, rfl\u27e9,\n    end : P.aopen \u27f6 ((_ : is_open_map _).functor.obj \u22a4)).op,\n  inv := X.presheaf.map $ (hom_of_le\n    begin\n      rintros _ \u27e8x, -, rfl\u27e9,\n      exact x.2,\n    end : (_ : is_open_map _).functor.obj \u22a4 \u27f6 P.aopen).op,\n  hom_inv_id' := by erw [\u2190X.presheaf.map_comp, X.presheaf.map_id]; refl,\n  inv_hom_id' := by erw [\u2190X.presheaf.map_comp, X.presheaf.map_id]; refl }\n\ndef to_Spec_local_ring_to_Scheme_restrict : Spec_obj R \u27f6 X.restrict P.aopen.open_embedding :=\n(hom.target_AffineScheme (Spec_obj R) _).symm $ \n    (P.restrict_\u0393_iso.hom \u226b germ _ \u27e8P.pt, P.mem_aopen\u27e9 \u226b P.ring_hom_ : \u0393.obj (op $ X |_ P) \u27f6 R) \n  \u226b (structure_sheaf.global_sections_iso R).hom\n\ndef to_Spec_local_ring_to_Scheme : Spec_obj R \u27f6 X :=\nP.to_Spec_local_ring_to_Scheme_restrict \u226b X.of_restrict _\n\nsection refine\n\nvariables (P Q) \n\n@[reducible] def restrict_of_refine (h : P.refine Q) : (X |_ P) \u27f6 (X |_ Q) :=\nis_open_immersion.lift \n  (X.of_restrict Q.aopen.open_embedding) \n  (X.of_restrict P.aopen.open_embedding)\nbegin\n  rintros _ \u27e8\u27e8x, hx\u27e9, rfl\u27e9,\n  refine \u27e8\u27e8x, h.subset hx\u27e9, rfl\u27e9,\nend\n\n@[reducible] def restrict_of_refine' (h : P.refine Q) : (X |_ P) \u27f6 (X |_ Q) :=\n(hom.target_AffineScheme (X |_ P) (X |_ Q)).symm $ \nX.presheaf.map $ (hom_of_le\n  begin\n    rintros _ \u27e8\u27e8x, hx\u27e9, -, rfl\u27e9,\n    refine \u27e8\u27e8x, h.subset hx\u27e9, \u27e8\u27e9, rfl\u27e9,\n  end : (_ : is_open_map _).functor.obj \u22a4 \u27f6 (_ : is_open_map _).functor.obj \u22a4).op\n\nexample (h : P.refine Q) : \n  P.restrict_of_refine' _ h =\n  P.restrict_of_refine _ h :=\nbegin \n  dsimp only [restrict_of_refine, restrict_of_refine'],\n  refine is_open_immersion.lift_uniq _ _ _ _ _,\n  sorry\nend\n\n\nlemma restrict_of_refine_comp_of_restrict (h : P.refine Q) : \n  X.of_restrict (P.aopen.open_embedding) =\n  P.restrict_of_refine _ h \u226b X.of_restrict _ :=\nby rw is_open_immersion.lift_fac\n\n/--\nSpec R \u27f6 X | P\n    \\       \u2193\n      --> X | Q\n-/\nlemma to_Spec_local_ring_to_Scheme_restrict_eq_of_refine (h : P.refine Q) :\n  Q.to_Spec_local_ring_to_Scheme_restrict =\n  P.to_Spec_local_ring_to_Scheme_restrict \u226b restrict_of_refine _ _ h :=\nbegin\n  dsimp only [to_Spec_local_ring_to_Scheme_restrict],\n  -- haveI : is_open_immersion (P.restrict_of_refine Q h) := sorry,\n  -- have := is_open_immersion.lift_fac\n  --   (restrict_of_refine _ _ h) (Q.to_Spec_local_ring_to_Scheme_restrict),\nend\n\ndef to_Spec_local_ring_to_Scheme.resp_refine (h : P.refine Q) :\n  P.to_Spec_local_ring_to_Scheme = Q.to_Spec_local_ring_to_Scheme :=\nbegin \n  dsimp [to_Spec_local_ring_to_Scheme],\n  rw [restrict_of_refine_comp_of_restrict _ _ h, \u2190category.assoc],\n  congr' 1,\nend\n\nend refine\n\nend point_affine_open_stalk_ring_hom_triple\n\nstructure point_stalk_ring_hom_pair :=\n(pt : X.carrier)\n(stalk_ : CommRing.{u})\n(stalk_iso : stalk_ \u2245 X.presheaf.stalk pt)\n(ring_hom_ : stalk_ \u27f6 R)\n[is_local_ring_hom_ : is_local_ring_hom ring_hom_]\n\nattribute [instance] point_stalk_ring_hom_pair.is_local_ring_hom_\n\nnamespace point_stalk_ring_hom_pair\n\nvariables {X R} (P Q S : point_stalk_ring_hom_pair X R)\n\n@[reducible]\ndef ring_hom_' : X.presheaf.stalk P.pt \u27f6 R := \n  P.stalk_iso.inv \u226b P.ring_hom_\n\nvariables {P Q S}\n\n@[simps] \ndef stalk_equiv_of_pt_eq (pt_eq : P.pt = Q.pt) :\n  P.stalk_ \u2245 Q.stalk_ :=\n{ hom := P.stalk_iso.hom \n    \u226b stalk_specializes X.presheaf (by rw pt_eq : Q.pt \u2933 P.pt) \n    \u226b Q.stalk_iso.inv,\n  inv := Q.stalk_iso.hom \n    \u226b stalk_specializes X.presheaf (by rw pt_eq : P.pt \u2933 Q.pt) \n    \u226b P.stalk_iso.inv,\n  hom_inv_id' := \n  begin \n    rw [category.assoc, category.assoc, iso.inv_hom_id_assoc,\n      \u2190category.assoc, \u2190category.assoc, iso.comp_inv_eq, category.id_comp,\n      category.assoc],\n    convert category.comp_id _,\n    refine stalk_hom_ext _ _,\n    intros U h,\n    rw [germ_stalk_specializes'_assoc], \n    erw [category.comp_id, germ_stalk_specializes'],\n    refl,\n  end,\n  inv_hom_id' := \n  begin \n    rw [category.assoc, category.assoc, iso.inv_hom_id_assoc,\n      \u2190category.assoc, \u2190category.assoc, iso.comp_inv_eq, category.id_comp,\n      category.assoc],\n    convert category.comp_id _,\n    refine stalk_hom_ext _ _,\n    intros U h,\n    rw [germ_stalk_specializes'_assoc], \n    erw [category.comp_id, germ_stalk_specializes'],\n    refl,\n  end }\n\nvariables (P Q S)\nstructure equiv : Prop :=\n(pt_eq : P.pt = Q.pt)\n(ring_hom_eq : P.ring_hom_ = (stalk_equiv_of_pt_eq pt_eq).hom \u226b Q.ring_hom_)\n\nstructure equiv' : Prop :=\n(pt_eq : P.pt = Q.pt)\n(ring_hom_eq : (stalk_equiv_of_pt_eq pt_eq).inv \u226b P.ring_hom_ = Q.ring_hom_)\n\n@[refl] lemma equiv_self : P.equiv P :=\n{ pt_eq := rfl,\n  ring_hom_eq := \n  begin \n    symmetry,\n    convert category.id_comp _,\n    rw [stalk_equiv_of_pt_eq_hom, \u2190category.assoc, iso.comp_inv_eq, \n      category.id_comp],\n    convert category.comp_id _,\n    apply stalk_hom_ext,\n    intros U h,\n    rw [germ_stalk_specializes'],\n    erw category.comp_id _,\n    refl,\n  end }\n\nvariables {P Q S}\n\n@[symm] lemma equiv_symm (h : P.equiv Q) : Q.equiv P :=\n{ pt_eq := h.pt_eq.symm,\n  ring_hom_eq := \n  begin \n    rw [h.ring_hom_eq, \u2190category.assoc],\n    symmetry,\n    convert category.id_comp _,\n    rw [stalk_equiv_of_pt_eq_hom, stalk_equiv_of_pt_eq_hom, category.assoc,\n      category.assoc, iso.inv_hom_id_assoc, \u2190category.assoc, \u2190category.assoc,\n      iso.comp_inv_eq, category.id_comp, category.assoc],\n    convert category.comp_id _,\n    apply stalk_hom_ext,\n    intros U h,\n    rw [germ_stalk_specializes'_assoc, germ_stalk_specializes'],\n    erw category.comp_id,\n    refl,\n  end }\n\n@[trans] lemma equiv_trans (h1 : P.equiv Q) (h2 : Q.equiv S) : P.equiv S :=\n{ pt_eq := h1.pt_eq.trans h2.pt_eq,\n  ring_hom_eq :=\n  begin \n    rw [h1.ring_hom_eq, h2.ring_hom_eq, stalk_equiv_of_pt_eq_hom,\n      stalk_equiv_of_pt_eq_hom, stalk_equiv_of_pt_eq_hom, category.assoc,\n      category.assoc, category.assoc, iso.inv_hom_id_assoc, \u2190category.assoc,\n      \u2190category.assoc],\n    congr' 1,\n    rw [\u2190category.assoc, \u2190category.assoc],\n    congr' 1,\n    rw [category.assoc],\n    congr' 1,\n    apply stalk_hom_ext,\n    intros U h,\n    rw [germ_stalk_specializes'_assoc, germ_stalk_specializes, \n      germ_stalk_specializes],\n  end }\n\nvariables (P Q S)\n\n@[simps]\ndef stalk_cocone : cocone ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf) :=\n{ X := P.stalk_,\n  \u03b9 := \n  { app := \u03bb j, colimit.\u03b9 ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf) j \u226b \n      P.stalk_iso.inv,\n    naturality' := \u03bb U V i, \n    begin \n      dsimp,\n      rw [category.comp_id, \u2190category.assoc],\n      erw colimit.w ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf) i,\n    end }  }\n\ndef stalk_cocone_is_colimit : is_colimit (P.stalk_cocone) :=\n{ desc := \u03bb s, P.stalk_iso.hom \u226b (colimit.is_colimit ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf)).desc s,\n  fac' := \u03bb s j, \n  begin \n    rw \u2190(colimit.is_colimit ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf)).fac s j,\n    rw [\u2190category.assoc],\n    congr' 1,\n    dsimp,\n    rw [category.assoc, iso.inv_hom_id, category.comp_id],\n  end,\n  uniq' := \u03bb s m j, \n  begin \n    rw \u2190(colimit.is_colimit ((open_nhds.inclusion P.pt).op \u22d9 X.presheaf)).uniq s (_ \u226b m) j,\n    rw iso.hom_inv_id_assoc,\n  end }\n\nsection affine\n\nvariables [is_affine X]\n\ndef stalk_iso_localization (P : point_stalk_ring_hom_pair X R) :\n  P.stalk_ \u2245 CommRing.of (localization.at_prime (X.iso_Spec.hom.1 P.pt).as_ideal) :=\nP.stalk_iso \u226a\u226b \n(PresheafedSpace.stalk_map.stalk_iso \n  ({ hom := X.iso_Spec.hom.1, \n    inv := X.iso_Spec.inv.1, \n    hom_inv_id' := by erw [\u2190Scheme.comp_val, iso.hom_inv_id]; refl, \n    inv_hom_id' := by erw [\u2190Scheme.comp_val, iso.inv_hom_id]; refl } : X.to_PresheafedSpace \u2245 (Spec_obj $ \u0393.obj $ op X).to_PresheafedSpace) P.pt).symm\n\u226a\u226b (structure_sheaf.stalk_iso _ _)\n\ndef \u0393_to_germ (P : point_stalk_ring_hom_pair X R) :\n  (\u0393.obj $ op X) \u27f6 P.stalk_ :=\n(structure_sheaf.global_sections_iso (\u0393.obj $ op X)).hom \u226b \n  X.iso_Spec.hom.1.c.app (op \u22a4) \u226b P.stalk_cocone.\u03b9.app (op \u22a4)\n\nend affine\n\nend point_stalk_ring_hom_pair\n\nsection affine\n\nvariables {X R} [is_affine X]\n\ndef from_point_stalk_ring_hom_pair_of_affine (P : point_stalk_ring_hom_pair X R) :\n  Spec_obj R \u27f6 X :=\n(hom.target_AffineScheme (Spec_obj R) X).symm $ \n  local_ring.from_point_local_ring_hom_pair \n  { pt := X.iso_Spec.hom.1.base P.pt,\n    localized_ring := X.presheaf.stalk P.pt,\n    algebra_localized_ring := infer_instance,\n    is_localization := infer_instance,\n    ring_hom_ := P.stalk_iso.inv \u226b P.ring_hom_ \n      \u226b (structure_sheaf.global_sections_iso R.\u03b1).hom,\n    is_local_ring_hom_ := infer_instance }\n\nlemma from_point_stalk_ring_hom_pair_of_affine.resp_equiv\n  (P Q : point_stalk_ring_hom_pair X R) (h : P.equiv Q) :\n  from_point_stalk_ring_hom_pair_of_affine P =\n  from_point_stalk_ring_hom_pair_of_affine Q :=\nbegin\n  dsimp only [from_point_stalk_ring_hom_pair_of_affine],\n  congr' 1,\n  refine local_ring.from_point_local_ring_hom_pair.resp_equiv _,\n  fconstructor,\n  { dsimp only, rw h.pt_eq, },\n  { dsimp only, \n    simp_rw h.ring_hom_eq, \n    rw [point_stalk_ring_hom_pair.stalk_equiv_of_pt_eq_hom, category.assoc,\n      category.assoc, iso.inv_hom_id_assoc, category.assoc],\n    congr' 1,\n    -- use the fact they are both \"colimit\", so unique up to a **unique** \n    -- isomorphism\n    sorry, },\nend\n\n@[simps]\ndef to_point_stalk_ring_hom_pair_of_affine (\u03b1 : Spec_obj R \u27f6 X) :\n  point_stalk_ring_hom_pair X R :=\nlet P := local_ring.to_point_local_ring_hom_pair \n  ((hom.target_AffineScheme (Spec_obj R) X) \u03b1) in\n{ pt := X.iso_Spec.inv.1.base P.pt,\n  stalk_ := CommRing.of P.localized_ring,\n  stalk_iso :=\n    let \u03b1 := @is_localization.alg_equiv (\u0393.obj $ op X) _ \n      P.pt.as_ideal.prime_compl P.localized_ring _ _ _\n      (X.stalk (X.iso_Spec.inv.1.base P.pt)) _\n      (X.global_sections_algebra _) (X.stalk_is_localization P.pt) in\n  { hom := \u03b1.to_ring_equiv.to_ring_hom,\n    inv := \u03b1.to_ring_equiv.symm.to_ring_hom,\n    hom_inv_id' := \n    begin\n      ext : 1,\n      rw [comp_apply, id_apply, \n        ring_equiv.symm_to_ring_hom_apply_to_ring_hom_apply],\n    end,\n    inv_hom_id' := \n    begin \n      ext : 1,\n      rw [comp_apply, id_apply,\n        ring_equiv.to_ring_hom_apply_symm_to_ring_hom_apply],\n    end },\n  ring_hom_ := P.ring_hom_ \u226b (structure_sheaf.global_sections_iso _).inv,\n  is_local_ring_hom_ := infer_instance }\n\nend affine\n\nend Scheme\n\nend algebraic_geometry", "meta": {"author": "jjaassoonn", "repo": "vc", "sha": "3f1db37cfd16b55cd47e77f0517d192c25edecf0", "save_path": "github-repos/lean/jjaassoonn-vc", "path": "github-repos/lean/jjaassoonn-vc/vc-3f1db37cfd16b55cd47e77f0517d192c25edecf0/src/points_of_scheme.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7606506418255927, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.377354089792771}}
{"text": "/-symplectic manifolds is the basis of the Hamiltonian formalization of classical mechanics and should\nserve as the basis for describing a phase sapce-/\nimport linear_algebra.bilinear_form\n\nuniverses u_1 u_2\nvariables {R : Type u_1} {M : Type u_2} [ring R] [add_comm_monoid M] [module R M] (B : bilin_form R M)\n\ndef bilin_form.is_skew_symm := \u2200 x y : M, B x y = - B y x\n\n@[protected] lemma is_skew_symm.eq (h : B.is_skew_symm) (x y : M) : B x y = - B y x := h x y \n\n", "meta": {"author": "ATOMSLab", "repo": "LeanChemicalTheories", "sha": "c2b15363c1e0ea0e52c1ae86abd1650670ff9044", "save_path": "github-repos/lean/ATOMSLab-LeanChemicalTheories", "path": "github-repos/lean/ATOMSLab-LeanChemicalTheories/LeanChemicalTheories-c2b15363c1e0ea0e52c1ae86abd1650670ff9044/src/math/symplectic_manifold.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7853085808877581, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.3773240288046279}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Callum Sutton, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.mul_add\nimport Mathlib.algebra.field\nimport Mathlib.algebra.opposites\nimport Mathlib.PostPort\n\nuniverses u_4 u_5 l u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# (Semi)ring equivs\n\nIn this file we define extension of `equiv` called `ring_equiv`, which is a datatype representing an\nisomorphism of `semiring`s, `ring`s, `division_ring`s, or `field`s. We also introduce the\ncorresponding group of automorphisms `ring_aut`.\n\n## Notations\n\nThe extended equiv have coercions to functions, and the coercion is the canonical notation when\ntreating the isomorphism as maps.\n\n## Implementation notes\n\nThe fields for `ring_equiv` now avoid the unbundled `is_mul_hom` and `is_add_hom`, as these are\ndeprecated.\n\nDefinition of multiplication in the groups of automorphisms agrees with function composition,\nmultiplication in `equiv.perm`, and multiplication in `category_theory.End`, not with\n`category_theory.comp`.\n\n## Tags\n\nequiv, mul_equiv, add_equiv, ring_equiv, mul_aut, add_aut, ring_aut\n-/\n\n/-- An equivalence between two (semi)rings that preserves the algebraic structure. -/\nstructure ring_equiv (R : Type u_4) (S : Type u_5) [Mul R] [Add R] [Mul S] [Add S]\n    extends R \u2243* S, R \u2243 S, R \u2243+ S where\n\ninfixl:25 \" \u2243+* \" => Mathlib.ring_equiv\n\n/-- The \"plain\" equivalence of types underlying an equivalence of (semi)rings. -/\n/-- The equivalence of additive monoids underlying an equivalence of (semi)rings. -/\n/-- The equivalence of multiplicative monoids underlying an equivalence of (semi)rings. -/\nnamespace ring_equiv\n\n\nprotected instance has_coe_to_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] :\n    has_coe_to_fun (R \u2243+* S) :=\n  has_coe_to_fun.mk (fun (x : R \u2243+* S) => R \u2192 S) to_fun\n\n@[simp] theorem to_fun_eq_coe_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (f : R \u2243+* S) : to_fun f = \u21d1f :=\n  rfl\n\n/-- A ring isomorphism preserves multiplication. -/\n@[simp] theorem map_mul {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S)\n    (x : R) (y : R) : coe_fn e (x * y) = coe_fn e x * coe_fn e y :=\n  map_mul' e x y\n\n/-- A ring isomorphism preserves addition. -/\n@[simp] theorem map_add {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S)\n    (x : R) (y : R) : coe_fn e (x + y) = coe_fn e x + coe_fn e y :=\n  map_add' e x y\n\n/-- Two ring isomorphisms agree if they are defined by the\n    same underlying function. -/\ntheorem ext {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S}\n    {g : R \u2243+* S} (h : \u2200 (x : R), coe_fn f x = coe_fn g x) : f = g :=\n  sorry\n\nprotected theorem congr_arg {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    {f : R \u2243+* S} {x : R} {x' : R} : x = x' \u2192 coe_fn f x = coe_fn f x' :=\n  sorry\n\nprotected theorem congr_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    {f : R \u2243+* S} {g : R \u2243+* S} (h : f = g) (x : R) : coe_fn f x = coe_fn g x :=\n  h \u25b8 rfl\n\ntheorem ext_iff {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] {f : R \u2243+* S}\n    {g : R \u2243+* S} : f = g \u2194 \u2200 (x : R), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : R) => h \u25b8 rfl, mpr := ext }\n\nprotected instance has_coe_to_mul_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S]\n    [Add S] : has_coe (R \u2243+* S) (R \u2243* S) :=\n  has_coe.mk to_mul_equiv\n\nprotected instance has_coe_to_add_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S]\n    [Add S] : has_coe (R \u2243+* S) (R \u2243+ S) :=\n  has_coe.mk to_add_equiv\n\ntheorem coe_mul_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2243+* S)\n    (a : R) : coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\ntheorem coe_add_equiv {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (f : R \u2243+* S)\n    (a : R) : coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\n/-- The identity map is a ring isomorphism. -/\nprotected def refl (R : Type u_1) [Mul R] [Add R] : R \u2243+* R :=\n  mk (mul_equiv.to_fun (mul_equiv.refl R)) (mul_equiv.inv_fun (mul_equiv.refl R)) sorry sorry sorry\n    sorry\n\n@[simp] theorem refl_apply (R : Type u_1) [Mul R] [Add R] (x : R) :\n    coe_fn (ring_equiv.refl R) x = x :=\n  rfl\n\n@[simp] theorem coe_add_equiv_refl (R : Type u_1) [Mul R] [Add R] :\n    \u2191(ring_equiv.refl R) = add_equiv.refl R :=\n  rfl\n\n@[simp] theorem coe_mul_equiv_refl (R : Type u_1) [Mul R] [Add R] :\n    \u2191(ring_equiv.refl R) = mul_equiv.refl R :=\n  rfl\n\nprotected instance inhabited (R : Type u_1) [Mul R] [Add R] : Inhabited (R \u2243+* R) :=\n  { default := ring_equiv.refl R }\n\n/-- The inverse of a ring isomorphism is a ring isomorphism. -/\nprotected def symm {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) :\n    S \u2243+* R :=\n  mk (mul_equiv.to_fun (mul_equiv.symm (to_mul_equiv e)))\n    (mul_equiv.inv_fun (mul_equiv.symm (to_mul_equiv e))) sorry sorry sorry sorry\n\n/-- See Note [custom simps projection] -/\ndef simps.inv_fun {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S] (e : R \u2243+* S) :\n    S \u2192 R :=\n  \u21d1(ring_equiv.symm e)\n\n@[simp] theorem symm_symm {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) : ring_equiv.symm (ring_equiv.symm e) = e :=\n  ext fun (x : R) => rfl\n\n@[simp] theorem coe_symm_mk {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (f : R \u2192 S) (g : S \u2192 R) (h\u2081 : function.left_inverse g f) (h\u2082 : function.right_inverse g f)\n    (h\u2083 : \u2200 (x y : R), f (x * y) = f x * f y) (h\u2084 : \u2200 (x y : R), f (x + y) = f x + f y) :\n    \u21d1(ring_equiv.symm (mk f g h\u2081 h\u2082 h\u2083 h\u2084)) = g :=\n  rfl\n\n/-- Transitivity of `ring_equiv`. -/\nprotected def trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [Mul R] [Add R] [Mul S] [Add S]\n    [Mul S'] [Add S'] (e\u2081 : R \u2243+* S) (e\u2082 : S \u2243+* S') : R \u2243+* S' :=\n  mk (mul_equiv.to_fun (mul_equiv.trans (to_mul_equiv e\u2081) (to_mul_equiv e\u2082)))\n    (mul_equiv.inv_fun (mul_equiv.trans (to_mul_equiv e\u2081) (to_mul_equiv e\u2082))) sorry sorry sorry\n    sorry\n\n@[simp] theorem trans_apply {A : Type u_1} {B : Type u_2} {C : Type u_3} [semiring A] [semiring B]\n    [semiring C] (e : A \u2243+* B) (f : B \u2243+* C) (a : A) :\n    coe_fn (ring_equiv.trans e f) a = coe_fn f (coe_fn e a) :=\n  rfl\n\nprotected theorem bijective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) : function.bijective \u21d1e :=\n  equiv.bijective (to_equiv e)\n\nprotected theorem injective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) : function.injective \u21d1e :=\n  equiv.injective (to_equiv e)\n\nprotected theorem surjective {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) : function.surjective \u21d1e :=\n  equiv.surjective (to_equiv e)\n\n@[simp] theorem apply_symm_apply {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) (x : S) : coe_fn e (coe_fn (ring_equiv.symm e) x) = x :=\n  equiv.apply_symm_apply (to_equiv e)\n\n@[simp] theorem symm_apply_apply {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) (x : R) : coe_fn (ring_equiv.symm e) (coe_fn e x) = x :=\n  equiv.symm_apply_apply (to_equiv e)\n\ntheorem image_eq_preimage {R : Type u_1} {S : Type u_2} [Mul R] [Add R] [Mul S] [Add S]\n    (e : R \u2243+* S) (s : set R) : \u21d1e '' s = \u21d1(ring_equiv.symm e) \u207b\u00b9' s :=\n  equiv.image_eq_preimage (to_equiv e) s\n\n/-- A commutative ring is isomorphic to its opposite. -/\ndef to_opposite (R : Type u_1) [comm_semiring R] : R \u2243+* (R\u1d52\u1d56) :=\n  mk (equiv.to_fun opposite.equiv_to_opposite) (equiv.inv_fun opposite.equiv_to_opposite) sorry\n    sorry sorry sorry\n\n@[simp] theorem to_opposite_apply (R : Type u_1) [comm_semiring R] (r : R) :\n    coe_fn (to_opposite R) r = opposite.op r :=\n  rfl\n\n@[simp] theorem to_opposite_symm_apply (R : Type u_1) [comm_semiring R] (r : R\u1d52\u1d56) :\n    coe_fn (ring_equiv.symm (to_opposite R)) r = opposite.unop r :=\n  rfl\n\n/-- A ring isomorphism sends one to one. -/\n@[simp] theorem map_one {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) :\n    coe_fn f 1 = 1 :=\n  mul_equiv.map_one \u2191f\n\n/-- A ring isomorphism sends zero to zero. -/\n@[simp] theorem map_zero {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) :\n    coe_fn f 0 = 0 :=\n  add_equiv.map_zero \u2191f\n\n@[simp] theorem map_eq_one_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S)\n    {x : R} : coe_fn f x = 1 \u2194 x = 1 :=\n  mul_equiv.map_eq_one_iff \u2191f\n\n@[simp] theorem map_eq_zero_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S]\n    (f : R \u2243+* S) {x : R} : coe_fn f x = 0 \u2194 x = 0 :=\n  add_equiv.map_eq_zero_iff \u2191f\n\ntheorem map_ne_one_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S)\n    {x : R} : coe_fn f x \u2260 1 \u2194 x \u2260 1 :=\n  mul_equiv.map_ne_one_iff \u2191f\n\ntheorem map_ne_zero_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S)\n    {x : R} : coe_fn f x \u2260 0 \u2194 x \u2260 0 :=\n  add_equiv.map_ne_zero_iff \u2191f\n\n/-- Produce a ring isomorphism from a bijective ring homomorphism. -/\ndef of_bijective {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2192+* S)\n    (hf : function.bijective \u21d1f) : R \u2243+* S :=\n  mk (equiv.to_fun (equiv.of_bijective (\u21d1f) hf)) (equiv.inv_fun (equiv.of_bijective (\u21d1f) hf)) sorry\n    sorry (ring_hom.map_mul' f) (ring_hom.map_add' f)\n\n@[simp] theorem map_neg {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) (x : R) :\n    coe_fn f (-x) = -coe_fn f x :=\n  add_equiv.map_neg (\u2191f) x\n\n@[simp] theorem map_sub {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) (x : R)\n    (y : R) : coe_fn f (x - y) = coe_fn f x - coe_fn f y :=\n  add_equiv.map_sub (\u2191f) x y\n\n@[simp] theorem map_neg_one {R : Type u_1} {S : Type u_2} [ring R] [ring S] (f : R \u2243+* S) :\n    coe_fn f (-1) = -1 :=\n  map_one f \u25b8 map_neg f 1\n\n/-- Reinterpret a ring equivalence as a ring homomorphism. -/\ndef to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : R \u2192+* S :=\n  ring_hom.mk (monoid_hom.to_fun (mul_equiv.to_monoid_hom (to_mul_equiv e))) sorry sorry sorry sorry\n\ntheorem to_ring_hom_injective {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] :\n    function.injective to_ring_hom :=\n  fun (f g : R \u2243+* S) (h : to_ring_hom f = to_ring_hom g) => ext (iff.mp ring_hom.ext_iff h)\n\nprotected instance has_coe_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] :\n    has_coe (R \u2243+* S) (R \u2192+* S) :=\n  has_coe.mk to_ring_hom\n\ntheorem coe_ring_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S) (a : R) :\n    coe_fn (\u2191f) a = coe_fn f a :=\n  rfl\n\ntheorem coe_ring_hom_inj_iff {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (f : R \u2243+* S)\n    (g : R \u2243+* S) : f = g \u2194 \u2191f = \u2191g :=\n  { mp := congr_arg fun (f : R \u2243+* S) => \u2191f,\n    mpr := fun (h : \u2191f = \u2191g) => ext (iff.mp ring_hom.ext_iff h) }\n\n/-- Reinterpret a ring equivalence as a monoid homomorphism. -/\ndef to_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) : R \u2192* S :=\n  ring_hom.to_monoid_hom (to_ring_hom e)\n\n/-- Reinterpret a ring equivalence as an `add_monoid` homomorphism. -/\ndef to_add_monoid_hom {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (e : R \u2243+* S) :\n    R \u2192+ S :=\n  ring_hom.to_add_monoid_hom (to_ring_hom e)\n\n@[simp] theorem to_ring_hom_refl {R : Type u_1} [semiring R] :\n    to_ring_hom (ring_equiv.refl R) = ring_hom.id R :=\n  rfl\n\n@[simp] theorem to_monoid_hom_refl {R : Type u_1} [semiring R] :\n    to_monoid_hom (ring_equiv.refl R) = monoid_hom.id R :=\n  rfl\n\n@[simp] theorem to_add_monoid_hom_refl {R : Type u_1} [semiring R] :\n    to_add_monoid_hom (ring_equiv.refl R) = add_monoid_hom.id R :=\n  rfl\n\n@[simp] theorem to_ring_hom_apply_symm_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [semiring R]\n    [semiring S] (e : R \u2243+* S) (y : S) :\n    coe_fn (to_ring_hom e) (coe_fn (to_ring_hom (ring_equiv.symm e)) y) = y :=\n  equiv.apply_symm_apply (to_equiv e)\n\n@[simp] theorem symm_to_ring_hom_apply_to_ring_hom_apply {R : Type u_1} {S : Type u_2} [semiring R]\n    [semiring S] (e : R \u2243+* S) (x : R) :\n    coe_fn (to_ring_hom (ring_equiv.symm e)) (coe_fn (to_ring_hom e) x) = x :=\n  equiv.symm_apply_apply (to_equiv e)\n\n@[simp] theorem to_ring_hom_trans {R : Type u_1} {S : Type u_2} {S' : Type u_3} [semiring R]\n    [semiring S] [semiring S'] (e\u2081 : R \u2243+* S) (e\u2082 : S \u2243+* S') :\n    to_ring_hom (ring_equiv.trans e\u2081 e\u2082) = ring_hom.comp (to_ring_hom e\u2082) (to_ring_hom e\u2081) :=\n  rfl\n\n@[simp] theorem to_ring_hom_comp_symm_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R]\n    [semiring S] (e : R \u2243+* S) :\n    ring_hom.comp (to_ring_hom e) (to_ring_hom (ring_equiv.symm e)) = ring_hom.id S :=\n  sorry\n\n@[simp] theorem symm_to_ring_hom_comp_to_ring_hom {R : Type u_1} {S : Type u_2} [semiring R]\n    [semiring S] (e : R \u2243+* S) :\n    ring_hom.comp (to_ring_hom (ring_equiv.symm e)) (to_ring_hom e) = ring_hom.id R :=\n  sorry\n\n/--\nConstruct an equivalence of rings from homomorphisms in both directions, which are inverses.\n-/\ndef of_hom_inv {R : Type u_1} {S : Type u_2} [semiring R] [semiring S] (hom : R \u2192+* S)\n    (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R)\n    (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) : R \u2243+* S :=\n  mk (ring_hom.to_fun hom) \u21d1inv sorry sorry (ring_hom.map_mul' hom) (ring_hom.map_add' hom)\n\n@[simp] theorem of_hom_inv_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S]\n    (hom : R \u2192+* S) (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R)\n    (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) (r : R) :\n    coe_fn (of_hom_inv hom inv hom_inv_id inv_hom_id) r = coe_fn hom r :=\n  rfl\n\n@[simp] theorem of_hom_inv_symm_apply {R : Type u_1} {S : Type u_2} [semiring R] [semiring S]\n    (hom : R \u2192+* S) (inv : S \u2192+* R) (hom_inv_id : ring_hom.comp inv hom = ring_hom.id R)\n    (inv_hom_id : ring_hom.comp hom inv = ring_hom.id S) (s : S) :\n    coe_fn (ring_equiv.symm (of_hom_inv hom inv hom_inv_id inv_hom_id)) s = coe_fn inv s :=\n  rfl\n\nend ring_equiv\n\n\nnamespace mul_equiv\n\n\n/-- Gives a `ring_equiv` from a `mul_equiv` preserving addition.-/\ndef to_ring_equiv {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S] (h : R \u2243* S)\n    (H : \u2200 (x y : R), coe_fn h (x + y) = coe_fn h x + coe_fn h y) : R \u2243+* S :=\n  ring_equiv.mk (equiv.to_fun (to_equiv h)) (equiv.inv_fun (to_equiv h)) sorry sorry (map_mul' h)\n    sorry\n\nend mul_equiv\n\n\nnamespace ring_equiv\n\n\n@[simp] theorem trans_symm {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S]\n    (e : R \u2243+* S) : ring_equiv.trans e (ring_equiv.symm e) = ring_equiv.refl R :=\n  ext (left_inv e)\n\n@[simp] theorem symm_trans {R : Type u_1} {S : Type u_2} [Add R] [Add S] [Mul R] [Mul S]\n    (e : R \u2243+* S) : ring_equiv.trans (ring_equiv.symm e) e = ring_equiv.refl S :=\n  ext (right_inv e)\n\n/-- If two rings are isomorphic, and the second is an integral domain, then so is the first. -/\nprotected theorem is_integral_domain {A : Type u_1} (B : Type u_2) [ring A] [ring B]\n    (hB : is_integral_domain B) (e : A \u2243+* B) : is_integral_domain A :=\n  sorry\n\n/-- If two rings are isomorphic, and the second is an integral domain, then so is the first. -/\nprotected def integral_domain {A : Type u_1} (B : Type u_2) [ring A] [integral_domain B]\n    (e : A \u2243+* B) : integral_domain A :=\n  integral_domain.mk ring.add ring.add_assoc ring.zero ring.zero_add ring.add_zero ring.neg ring.sub\n    ring.add_left_neg ring.add_comm ring.mul ring.mul_assoc ring.one ring.one_mul ring.mul_one\n    ring.left_distrib ring.right_distrib sorry sorry sorry\n\nend ring_equiv\n\n\nnamespace equiv\n\n\n/-- In a division ring `K`, the unit group `units K`\nis equivalent to the subtype of nonzero elements. -/\n-- TODO: this might already exist elsewhere for `group_with_zero`\n\n-- deduplicate or generalize\n\ndef units_equiv_ne_zero (K : Type u_4) [division_ring K] :\n    units K \u2243 \u21a5(set_of fun (a : K) => a \u2260 0) :=\n  mk (fun (a : units K) => { val := units.val a, property := sorry })\n    (fun (a : \u21a5(set_of fun (a : K) => a \u2260 0)) => units.mk0 (subtype.val a) sorry) sorry sorry\n\n@[simp] theorem coe_units_equiv_ne_zero {K : Type u_4} [division_ring K] (a : units K) :\n    \u2191(coe_fn (units_equiv_ne_zero K) a) = \u2191a :=\n  rfl\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/ring_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3772634836144764}}
{"text": "import tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\nimport tactic.show_term\nimport analysis.calculus.fderiv\n\nimport analysis.special_functions.exp_log\n\n\n/-- User attribute used to mark tactics used by `differentiability`. -/\n@[user_attribute]\nmeta def differentiability : user_attribute :=\n{ name := `differentiability,\n  descr := \"lemmas usable to prove differentiablility\" }\n\n-- Mark some differentiability lemmas already defined in `topology.basic`\nattribute [differentiability]\n  differentiable_at_id\n  differentiable_at_const\n\n-- As we will be using `apply_rules` with `md := semireducible`,\n-- we need another version of `differentiable_id`.\n@[differentiability] lemma differentiable_at_id_t \n    {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c] \n    {E : Type*} [normed_group E] [normed_space \ud835\udd5c E] {x : E} : differentiable_at \ud835\udd5c id x :=\n(has_fderiv_at_id x).differentiable_at\n\nnamespace tactic\n\nmeta def apply_differentiability.comp : tactic unit :=\n`[fail_if_success { exact differentiable_at_const };\n  refine differentiable_at.comp _ _;\n  fail_if_success { exact differentiable_at_id }]\n\n/-- List of tactics used by `differentiability` internally. -/\nmeta def differentiability_tactics (md : transparency := reducible) : list (tactic string) :=\n[\n  intros1               >>= \u03bb ns, pure (\"intros \" ++ (\" \".intercalate (ns.map (\u03bb e, e.to_string)))),\n  apply_rules [``(differentiability)] 50 { md := md }\n                        >> pure \"apply_rules differentiablity\",\n  apply_differentiability.comp >> pure \"refine differentiable_at.comp _ _\"\n]\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nSolve goals of the form `differentiable f`. `differentiability?` reports back the proof term it found.\n-/\nmeta def differentiability\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) : tactic unit :=\nlet md              := if bang.is_some then semireducible else reducible,\n    differentiability_core := tactic.tidy { tactics := differentiability_tactics md, ..cfg },\n    trace_fn        := if trace.is_some then show_term else id in\ntrace_fn differentiability_core\n\n/-- Version of `differentiability` for use with auto_param. -/\nmeta def differentiability' : tactic unit := differentiability none none {}\n\n/--\n`differentiability` solves goals of the form `differentiable f` by applying lemmas tagged with the\n`differentiability` user attribute.\n\nI have no idea what it does, I just stole Reid's code and changed a few things.\n\nYou can also use `differentiability!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`differentiability?` reports back the proof term it found.\n-/\nadd_tactic_doc\n{ name := \"differentiability / differentiability'\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.differentiability, `tactic.interactive.differentiability'],\n  tags := [\"lemma application\"]\n}\n\nend interactive\n\nend tactic\n\n", "meta": {"author": "jamesa9283", "repo": "special-functions", "sha": "392758fb7207762c9ba6938462614994ff45bdc4", "save_path": "github-repos/lean/jamesa9283-special-functions", "path": "github-repos/lean/jamesa9283-special-functions/special-functions-392758fb7207762c9ba6938462614994ff45bdc4/src/Tactics/differentiablity.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376236, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.3772634824508657}}
{"text": "import set_theory.surreal tactic.ring\nimport set_theory.game\n\nopen pgame\nnamespace pgame\nuniverse u\n\nlocal infix ` \u2248 ` := pgame.equiv\n\ndef sub_congr {w x y z : pgame} \n  (h\u2081 : w \u2248 x) (h\u2082 : y \u2248 z) : (w - y) \u2248 (x - z) :=\nsorry\n\nmeta def try_inl_inr : tactic unit :=\n`[assumption]\n    <|> (do `[apply sum.inr], try_inl_inr )\n    <|> (do `[apply sum.inl], try_inl_inr )\n    <|> (do `[apply prod.mk], try_inl_inr, try_inl_inr )\n\n\nlemma equiv_of_mk_equiv {x y : pgame} (L : x.left_moves \u2243 y.left_moves) (R : x.right_moves \u2243 y.right_moves)\n         (hl : \u2200 (i : x.left_moves), (x.move_left i).equiv (y.move_left (L i)))\n         (hr : \u2200 (j : y.right_moves), (x.move_right (R.symm j)).equiv (y.move_right j)) :\n       x.equiv y :=\nsorry\n \n/-- `(x + y) * z` is equivalent to `x * z + y * z.`-/\ntheorem right_distrib_equiv (x y z : pgame) : (x + y) * z \u2248 x * z + y * z :=\nsorry\n\ntheorem left_distrib_equiv : \u03a0 (x y z : pgame), (x * (y + z)).equiv (x * y + x * z)\n| (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=\nsorry\n\nset_option trace.solve_by_elim true\n\nlemma aux'' {a b c d x y z c' : pgame} :\n  (a + b - c) * z + x * y * c' - (a + b - c) * d\n  \u2248 a * z + b * z - c * z + x * y * c' - a * d + b * d + c * d :=\nsorry\n      \nlemma aux' {a b c x y z : pgame} :\na * (y * z) + x * (b * z) - a * (b * z) + x * (y * c) - a * (y * c) + x * (b * c) + a * (b * c)\n  \u2248 a * (y * z) + x * (b * z + y * c - b * c) - a * (b * z + y * c - b * c)\n:= sorry\n\n\ntheorem eq_of_mk_eq {x y : pgame}\n  (L : x.left_moves \u2243 y.left_moves) (R : x.right_moves \u2243 y.right_moves)\n  (hl : \u2200 (i : x.left_moves), \u27e6x.move_left i\u27e7 = \u27e6y.move_left (L i)\u27e7)\n  (hr : \u2200 (j : y.right_moves), \u27e6x.move_right (R.symm j)\u27e7 = \u27e6y.move_right j\u27e7) :\n  \u27e6x\u27e7 = \u27e6y\u27e7 := sorry\n  \ntheorem left_distrib : \u03a0 (x y z : pgame), \u27e6x * (y + z)\u27e7 = \u27e6x * y\u27e7 + \u27e6x * z\u27e7\n| (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=\nbegin\n  let x := mk xl xr xL xR,\n  let y := mk yl yr yL yR,\n  let z := mk zl zr zL zR,\n  refine eq_of_mk_eq _ _ _ _,\n  { fsplit,\n    { rintros (\u27e8_,(_|_)\u27e9|\u27e8_,(_|_)\u27e9);\n      solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 5 } },\n    { rintros ((\u27e8_,_\u27e9|\u27e8_,_\u27e9)|(\u27e8_,_\u27e9|\u27e8_,_\u27e9));\n      solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 5 } },\n    { rintros (\u27e8_,(_|_)\u27e9|\u27e8_,(_|_)\u27e9); refl },\n    { rintros ((\u27e8_,_\u27e9|\u27e8_,_\u27e9)|(\u27e8_,_\u27e9|\u27e8_,_\u27e9)); refl } },\n  { fsplit,\n    { rintros (\u27e8_,(_|_)\u27e9|\u27e8_,(_|_)\u27e9);\n      solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 5 } },\n    { rintros ((\u27e8_,_\u27e9|\u27e8_,_\u27e9)|(\u27e8_,_\u27e9|\u27e8_,_\u27e9));\n      solve_by_elim [sum.inl, sum.inr, prod.mk] { max_depth := 5 } },\n    { rintros (\u27e8_,(_|_)\u27e9|\u27e8_,(_|_)\u27e9); refl },\n    { rintros ((\u27e8_,_\u27e9|\u27e8_,_\u27e9)|(\u27e8_,_\u27e9|\u27e8_,_\u27e9)); refl } },\n    { rintros (\u27e8i,(j|k)\u27e9|\u27e8i,(j|k)\u27e9),\n      { change \u27e6xL i * (y + z) + x * (yL j + z) - xL i * (yL j + z)\u27e7\n                = \u27e6xL i * y + x * yL j - xL i * yL j + x * z\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xL i * (y + z) + x * (y + zL k) - xL i * (y + zL k)\u27e7\n              = \u27e6x * y + (xL i * z + x * zL k - xL i * zL k)\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xR i * (y + z) + x * (yR j + z) - xR i * (yR j + z)\u27e7\n              = \u27e6xR i * y + x * yR j - xR i * yR j + x * z\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xR i * (y + z) + x * (y + zR k) - xR i * (y + zR k)\u27e7\n              = \u27e6x * y + (xR i * z + x * zR k - xR i * zR k)\u27e7,\n        simp[left_distrib], abel } },\n    { rintros ((\u27e8i,j\u27e9|\u27e8i,j\u27e9)|(\u27e8i,k\u27e9|\u27e8i,k\u27e9)),\n      { change \u27e6xL i * (y + z) + x * (yR j + z) - xL i * (yR j + z)\u27e7\n              = \u27e6xL i * y + x * yR j - xL i * yR j + x * z\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xR i * (y + z) + x * (yL j + z) - xR i * (yL j + z)\u27e7\n              = \u27e6xR i * y + x * yL j - xR i * yL j + x * z\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xL i * (y + z) + x * (y + zR k) - xL i * (y + zR k)\u27e7\n              = \u27e6x * y + (xL i * z + x * zR k - xL i * zR k)\u27e7,\n        simp[left_distrib], abel },\n      { change \u27e6xR i * (y + z) + x * (y + zL k) - xR i * (y + zL k)\u27e7\n              = \u27e6x * y + (xR i * z + x * zL k - xR i * zL k)\u27e7,\n        simp[left_distrib], abel } }\nend\nusing_well_founded { dec_tac := pgame_wf_tac }\n\ntheorem mul_zero : \u03a0 (x : pgame), \u27e6x * 0\u27e7 = \u27e60\u27e7 := sorry\ntheorem mul_one : \u03a0 (x : pgame), \u27e6x * 1\u27e7 = \u27e6x\u27e7 \n| (mk xl xr xL xR) :=\nbegin\n  let x := mk xl xr xL xR,\n  refine eq_of_mk_eq _ _ _ _,\n  { fsplit,\n     rintros (\u27e8_, \u27e8 \u27e9\u27e9 | \u27e8_, \u27e8 \u27e9\u27e9), assumption,\n     rintros i,  exact sum.inl(i, punit.star),\n     rintros (\u27e8_, \u27e8 \u27e9\u27e9 | \u27e8_, \u27e8 \u27e9\u27e9), refl,\n     rintros i, refl,\n  },\n  { fsplit,\n    rintros (\u27e8_, \u27e8 \u27e9\u27e9 | \u27e8_, \u27e8 \u27e9\u27e9), assumption,\n    rintros i,  exact sum.inr(i, punit.star), \n    rintros (\u27e8_, \u27e8 \u27e9\u27e9 | \u27e8_, \u27e8 \u27e9\u27e9), refl,\n    rintros i, refl,\n  },\n  rintros (\u27e8i, \u27e8 \u27e9\u27e9 | \u27e8i, \u27e8 \u27e9\u27e9),\n  change \u27e6xL i * 1 + x * 0 - xL i * 0\u27e7 = \u27e6xL i\u27e7,\n  simp [mul_zero, mul_one],\n\n  rintros j,\n  change \u27e6xR j * 1 + x * 0 - xR j * 0\u27e7 = \u27e6xR j\u27e7,\n  simp [mul_zero, mul_one],\nend\n\n#check left_distrib\ntheorem mul_assoc_equiv : \u03a0 (x y z : pgame), ((x * y) * z).equiv (x * (y * z))\n| (mk xl xr xL xR) (mk yl yr yL yR) (mk zl zr zL zR) :=\nbegin\n  let x := mk xl xr xL xR,\n  let y := mk yl yr yL yR,\n  let z := mk zl zr zL zR,\n  refine equiv_of_mk_equiv _ _ _ _,\n  { fsplit,\n    { rintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9); try_inl_inr },\n    { rintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_,\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9); try_inl_inr },\n    { rintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_,_\u27e9 | \u27e8_, _\u27e9,_\u27e9); refl },\n    { rintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_,\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9); refl } },\n  { fsplit,\n    { rintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9,_\u27e9); try_inl_inr },\n    { rintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_,\u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9); try_inl_inr },\n    { rintro (\u27e8\u27e8_, _\u27e9 | \u27e8_, _\u27e9, _\u27e9 | \u27e8\u27e8_,_\u27e9 | \u27e8_, _\u27e9,_\u27e9); refl },\n    { rintro (\u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9 | \u27e8_, \u27e8_, _\u27e9 | \u27e8_, _\u27e9\u27e9); refl } },\n  { rintro (\u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9 | \u27e8\u27e8i, j\u27e9 | \u27e8i, j\u27e9, k\u27e9),\n    { calc \n        (xL i * y + x * yL j - xL i * yL j) * z + (x * y) * zL k\n          - (xL i * y + x * yL j - xL i * yL j) * zL k \n            \u2248 (xL i * y) * z + (x * yL j) * z - (xL i * yL j) * z + (x * y) * zL k\n                  - (xL i * y) * zL k + (x * yL j) * zL k + (xL i * yL j) * zL k\n            : aux''\n        ... \u2248 xL i * (y * z) + x * (yL j * z) - xL i * (yL j * z) + x * (y * zL k)\n                 - xL i * (y * zL k) + x * (yL j * zL k) + xL i * (yL j * zL k)\n            : by { repeat { apply add_congr <|> apply neg_congr }; apply mul_assoc_equiv }\n        ... \u2248 xL i * (y * z) + x * (yL j * z + y * zL k - yL j * zL k)\n                 - xL i * (yL j * z + y * zL k - yL j * zL k)\n            : aux' },\n     \n        \n\n    repeat {sorry}, },\n  repeat {sorry},\nend \n\nend pgame\n\n\n\n\n\n\n\n\n\n", "meta": {"author": "apurvanakade", "repo": "lean-playground", "sha": "2fe58797031ff8a6c29e1a442cbcc7a0ebc9c768", "save_path": "github-repos/lean/apurvanakade-lean-playground", "path": "github-repos/lean/apurvanakade-lean-playground/lean-playground-2fe58797031ff8a6c29e1a442cbcc7a0ebc9c768/src/surreal/mul_assoc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154240079185318, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3772549005978761}}
{"text": "/-\nCopyright (c) 2017 Daniel Selsam. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Daniel Selsam\n\nDeterministic operators.\n-/\nimport .tgrads .util .tcont .det\n\nnamespace certigrad\nopen T list\n\nnamespace ops\n\nsection tactic\nopen tactic\n\nmeta def idx_over : tactic unit :=\ndo exfalso, to_expr ```(at_idx_over H_at_idx dec_trivial) >>= exact\n\nmeta def simp_simple : tactic unit :=\ndo s\u2080 \u2190 simp_lemmas.mk_default,\n   s \u2190 return $ simp_lemmas.erase s\u2080 $ [`add_comm, `add_left_comm, `mul_comm, `mul_left_comm],\n   simplify_goal s {} >> try triv >> try (reflexivity reducible)\n\nmeta def prove_odiff : tactic unit :=\ndo get_local `f_odiff >>= clear,\n   to_expr ```(shape = fshape) >>= \u03bb ty, to_expr ```(eq.symm H_at_idx^.right) >>= \u03bb val, assertv `H_fshape_eq ty val,\n   get_local `H_fshape_eq >>= subst,\n   dunfold [`certigrad.det.is_odifferentiable],\n   try simp_simple,\n   try dsimp,\n   prove_differentiable\n\nmeta def prove_pb_correct_init : tactic unit :=\ndo get_local `f_pb_correct >>= clear,\n   to_expr ```(shape = fshape) >>= \u03bb ty, to_expr ```(eq.symm H_at_idx^.right) >>= \u03bb val, assertv `H_fshape_eq ty val,\n   get_local `H_fshape_eq >>= subst,\n   to_expr ```(T shape \u2192 \u211d) >>= \u03bb ty, to_expr ```(\u03bb (z : T shape), T.dot z g_out) >>= definev `k ty,\n   to_expr ```(\u2207 k y = g_out) >>= assert `H_k_grad, dsimp, rewrite `certigrad.T.grad_dot\u2081,\n   get_local `H_k_grad >>= rewrite_core reducible tt tt occurrences.all tt,\n   get_local `H_y >>= subst\n\nmeta def prove_pb_correct : tactic unit :=\ndo prove_pb_correct_init,\n   try simp_simple,\n   try dsimp,\n   mk_const `certigrad.T.grad_tmulT >>= rewrite_core reducible tt tt occurrences.all tt,\n   simplify_grad,\n   try simp,\n   try reflexivity\n\nmeta def prove_ocont_init : tactic unit :=\ndo get_local `f_ocont >>= clear,\n   to_expr ```(shape = ishape) >>= \u03bb ty, to_expr ```(eq.symm H_at_idx^.right) >>= \u03bb val, assertv `H_ishape_eq ty val,\n   get_local `H_ishape_eq >>= subst,\n   try simp_simple,\n   try dsimp\n\nmeta def prove_ocont : tactic unit :=\ndo prove_ocont_init,\n   repeat (prove_continuous_core <|> prove_preconditions_core)\n\nend tactic\n\nopen det\n\nnamespace scale\n\ndef f (\u03b1 : \u211d) {shape : S} (xs : dvec T [shape]) : T shape := \u03b1 \u2b1d xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb (\u03b1 : \u211d) {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (\u03b1 \u2b1d gy) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff (\u03b1 : \u211d) {shape : S} : is_odifferentiable (@f \u03b1 shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct (\u03b1 : \u211d) {shape : S} : pullback_correct (@f \u03b1 shape) (@f_pre shape) (@f_pb \u03b1 shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont (\u03b1 : \u211d) {shape : S} : is_ocontinuous (@f \u03b1 shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend scale\n\nsection open scale\ndef scale (\u03b1 : \u211d) (shape : S) : det.op [shape] shape :=\ndet.op.mk \"scale\" (f \u03b1) f_pre (f_pb \u03b1) (f_odiff \u03b1) (f_pb_correct \u03b1) (f_ocont \u03b1)\n\nend\n\nnamespace neg\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := - xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (-gy) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend neg\n\nsection open neg\ndef neg (shape : S) : det.op [shape] shape :=\ndet.op.mk \"neg\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace exp\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := exp xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (gy * y) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend exp\n\nsection open exp\ndef exp (shape : S) : det.op [shape] shape :=\ndet.op.mk \"exp\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace log\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := log xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, xs^.head > 0\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (gy / xs^.head) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend log\n\nsection open log\ndef log (shape : S) : det.op [shape] shape :=\ndet.op.mk \"log\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace sqrt\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := sqrt xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, 0 < xs^.head\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (gy / (2 * y)) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend sqrt\n\nsection open sqrt\ndef sqrt (shape : S) : det.op [shape] shape :=\ndet.op.mk \"sqrt\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace sigmoid\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := sigmoid xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape :=\nforce (gy * y * (1 - y)) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend sigmoid\n\nsection open sigmoid\ndef sigmoid (shape : S) : det.op [shape] shape :=\ndet.op.mk \"sigmoid\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace softplus\n\ndef f {shape : S} (xs : dvec T [shape]) : T shape := softplus xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape :=\nforce (gy / (1 + T.exp (- xs^.head))) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| xs H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| xs (n+1) ishape H_at_idx H_pre := by idx_over\n\nend softplus\n\nsection open softplus\ndef softplus (shape : S) : det.op [shape] shape :=\ndet.op.mk \"softplus\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace add\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : T shape := xs^.head + xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : T shape) (idx : \u2115) (fshape : S) : T fshape := force (gy) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x, y\u27e7 H_pre 0 fshape H_at_idx k H_k := by { prove_odiff }\n| \u27e6x, y\u27e7 H_pre 1 fshape H_at_idx k H_k := by { prove_odiff }\n| xs    H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 1 fshape H_at_idx H_pre := by prove_pb_correct\n| xs      y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u2081, x\u2082\u27e7 0     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u2081, x\u2082\u27e7 1     ishape H_at_idx H_pre := by prove_ocont\n| xs      (n+2) ishape H_at_idx H_pre := by idx_over\n\nend add\n\nsection open add\ndef add (shape : S) : det.op [shape, shape] shape :=\ndet.op.mk \"add\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace mul\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : T shape := xs^.head * xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, true\n\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : T shape) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy * xs^.head2) fshape\n| 1     fshape := force (gy * xs^.head) fshape\n| (n+2) fshape := T.error \"mul: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x, y\u27e7 H_pre 0 fshape H_at_idx k H_k := by { prove_odiff }\n| \u27e6x, y\u27e7 H_pre 1 fshape H_at_idx k H_k := by { prove_odiff }\n| xs    H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 1 fshape H_at_idx H_pre := by prove_pb_correct\n| xs      y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u2081, x\u2082\u27e7 0     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u2081, x\u2082\u27e7 1     ishape H_at_idx H_pre := by prove_ocont\n| xs      (n+2) ishape H_at_idx H_pre := by idx_over\n\nend mul\n\nsection open mul\ndef mul (shape : S) : det.op [shape, shape] shape :=\ndet.op.mk \"mul\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace sub\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : T shape := xs^.head - xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, true\n\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : T shape) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy) fshape\n| 1     fshape := force (- gy) fshape\n| (n+2) fshape := T.error \"sub: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x, y\u27e7 H_pre 0 fshape H_at_idx k H_k := by { prove_odiff }\n| \u27e6x, y\u27e7 H_pre 1 fshape H_at_idx k H_k := by { prove_odiff }\n| xs    H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := by prove_pb_correct\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 1 fshape H_at_idx H_pre := by prove_pb_correct\n| xs      y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u2081, x\u2082\u27e7 0     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u2081, x\u2082\u27e7 1     ishape H_at_idx H_pre := by prove_ocont\n| xs      (n+2) ishape H_at_idx H_pre := by idx_over\n\nend sub\n\nsection open sub\ndef sub (shape : S) : det.op [shape, shape] shape :=\ndet.op.mk \"sub\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace div\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : T shape := xs^.head / xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, 0 < T.square xs^.head2\n\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : T shape) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy / xs^.head2) fshape\n| 1     fshape := force (- (gy * xs^.head) / (T.square xs^.head2)) fshape\n| (n+2) fshape := T.error \"div: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x, y\u27e7 H_pre 0 fshape H_at_idx k H_k := by { prove_odiff }\n| \u27e6x, y\u27e7 H_pre 1 fshape H_at_idx k H_k := by { prove_odiff }\n| xs    H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 0 fshape H_at_idx H_pre := begin prove_pb_correct end\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 1 fshape H_at_idx H_pre := by prove_pb_correct\n| xs      y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u2081, x\u2082\u27e7 0     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u2081, x\u2082\u27e7 1     ishape H_at_idx H_pre := by prove_ocont\n| xs      (n+2) ishape H_at_idx H_pre := by idx_over\n\nend div\n\nsection open div\ndef div (shape : S) : det.op [shape, shape] shape :=\ndet.op.mk \"div\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace sum\n\ndef f {shape : S} (xs : dvec T [shape]) : \u211d := T.sum xs^.head\ndef f_pre {shape : S} : precondition [shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape]) (y gy : \u211d) (idx : \u2115) (fshape : S) : T fshape := force (T.const gy shape) fshape\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 H_pre 0 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6x\u27e7 H_pre (n+1) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6x\u27e7 y H_y g_out 0 fshape H_at_idx H_pre :=\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : shape = fshape := eq.symm H_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : \u211d \u2192 \u211d := (\u03bb \u03b8, dot g_out \u03b8),\nassert H_grad : \u2207 k y = g_out,\n{ change \u2207 (\u03bb \u03b8, dot g_out \u03b8) y = g_out, rw certigrad.T.grad_dot\u2082 },\nrw -H_grad,\nsubst H_y,\nsimp, dsimp,\ndunfold dvec.get dvec.head dvec.update_at,\nrw -T.grad_tmulT,\nrw T.grad_sum k,\nsimp [T.smul.def]\nend\n\n| xs y H_y g_out (n+1) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6x\u27e7 0 ishape H_at_idx H_pre := by prove_ocont\n| \u27e6x\u27e7 (n+1) ishape H_at_idx H_pre := by idx_over\n\nend sum\n\nsection open sum\n-- TODO(dhs): why won't it find `f` without `sum.`? Bug in Lean?\ndef sum (shape : S) : det.op [shape] [] :=\ndet.op.mk \"sum\" sum.f sum.f_pre sum.f_pb sum.f_odiff sum.f_pb_correct sum.f_ocont\nend\n\nnamespace gemm\n\ndef f {m n p : \u2115} (xs : dvec T [[m, n], [n, p]]) : T [m, p] := gemm xs^.head xs^.head2\ndef f_pre {m n p : \u2115} : precondition [[m, n], [n, p]] := \u03bb xs, true\ndef f_pb {m n p : \u2115} (xs : dvec T [[m, n], [n, p]]) (y gy : T [m, p]) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0 fshape := force (T.gemm gy (transpose $ xs^.head2)) fshape\n| 1 fshape := force (T.gemm (transpose $ xs^.head) gy) fshape\n| (n+2) fshape := T.error \"gemm: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {m n p : \u2115} : is_odifferentiable (@f m n p) (@f_pre m n p)\n| \u27e6x\u2081, x\u2082\u27e7 H_pre 0 fshape H_at_idx k H_k := begin definev shape : S := [m, n], prove_odiff end\n| \u27e6x\u2081, x\u2082\u27e7 H_pre 1 fshape H_at_idx k H_k := begin definev shape : S := [n, p], prove_odiff end\n| xs      H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {m n p : \u2115} : pullback_correct (@f m n p) (@f_pre m n p) (@f_pb m n p)\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 0 fshape H_fshape_at_idx H_pre :=\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : [m, n] = fshape := eq.symm H_fshape_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : T [m, p] \u2192 \u211d := (\u03bb \u03b8, dot g_out \u03b8),\nassert H_grad : \u2207 k y = g_out,\n{ change \u2207 (\u03bb \u03b8, dot g_out \u03b8) y = g_out, rw certigrad.T.grad_dot\u2082 },\nrw -H_grad,\nsubst H_y,\nsimp, dsimp,\nrw [-T.grad_tmulT, T.grad_gemm\u2081 k]\nend\n\n| \u27e6x\u2081, x\u2082\u27e7 y H_y g_out 1 fshape H_fshape_at_idx H_pre :=\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : [n, p] = fshape := eq.symm H_fshape_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : T [m, p] \u2192 \u211d := (\u03bb \u03b8, dot g_out \u03b8),\nassert H_grad : \u2207 k y = g_out,\n{ change \u2207 (\u03bb \u03b8, dot g_out \u03b8) y = g_out, rw certigrad.T.grad_dot\u2082 },\nrw -H_grad,\nsubst H_y,\nsimp, dsimp,\nrw [-T.grad_tmulT, T.grad_gemm\u2082 k]\nend\n\n| xs y H_y g_out (n+2) fshape H_fshape_at_idx H_pre := false.rec _ (at_idx_over H_fshape_at_idx (by tactic.dec_triv))\n\nlemma f_ocont {m n p : \u2115} : is_ocontinuous (@f m n p) (@f_pre m n p)\n| \u27e6x\u2081, x\u2082\u27e7 0     ishape H_at_idx H_pre := by { pose shape := [m, n], prove_ocont }\n| \u27e6x\u2081, x\u2082\u27e7 1     ishape H_at_idx H_pre := by { pose shape := [n, p], prove_ocont }\n| xs      (n+2) ishape H_at_idx H_pre := by idx_over\n\nend gemm\n\nsection open gemm\ndef gemm (m n p : \u2115) : det.op [[m, n], [n, p]] [m, p] :=\ndet.op.mk \"gemm\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace mvn_kl\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : \u211d := mvn_kl xs^.head xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, 0 < xs^.head2\n\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : \u211d) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy \u2b1d xs^.head) fshape\n| 1     fshape := force (gy \u2b1d (xs^.head2 - (1 / xs^.head2))) fshape\n| (n+2) fshape := T.error \"mvn_kl: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6\u03bc, \u03c3\u27e7 H_pre 0     fshape H_at_idx k H_k := by prove_odiff\n| \u27e6\u03bc, \u03c3\u27e7 H_pre 1     fshape H_at_idx k H_k := by prove_odiff\n| xs    H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6\u03bc, \u03c3\u27e7 y H_y g_out 0 fshape H_fshape_at_idx H_pre :=\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : shape = fshape := eq.symm H_fshape_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : \u211d \u2192 \u211d := \u03bb (x : \u211d), x * g_out,\nassertv H_k_grad : \u2207 k y = g_out :=  by { dsimp, erw [T.grad_mul\u2081 id, T.grad_id, one_mul] },\nrw -H_k_grad,\nsubst H_y,\ndsimp,\nsimp,\nrw -T.grad_tmulT,\nsimplify_grad,\nsimp [T.smul.def]\nend\n\n| \u27e6\u03bc, \u03c3\u27e7 y H_y g_out 1 fshape H_at_idx H_pre :=\nhave H_\u03c3\u2082 : square \u03c3 > 0, from square_pos_of_pos H_pre,\nhave H_diff\u2081 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * (-2\u207b\u00b9 * T.sum (1 + T.log (square \u03b8\u2080) - square \u03bc - square \u03c3))) \u03c3, by prove_differentiable,\nhave H_diff\u2082 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * (-2\u207b\u00b9 * T.sum (1 + T.log (square \u03c3) - square \u03bc - square \u03b8\u2080))) \u03c3, by prove_differentiable,\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : shape = fshape := eq.symm H_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : \u211d \u2192 \u211d := \u03bb (x : \u211d), x * g_out,\nassertv H_k_grad : \u2207 k y = g_out :=  by { dsimp, erw [T.grad_mul\u2081 id, T.grad_id, one_mul] },\nrw -H_k_grad,\nsubst H_y,\ndsimp,\nsimp,\nrw -T.grad_tmulT,\ndunfold T.mvn_kl,\n\nrw (T.grad_binary (\u03bb \u03b8\u2081 \u03b8\u2082, g_out * ((- 2\u207b\u00b9) * T.sum (1 + T.log (square \u03b8\u2081) - square \u03bc - square \u03b8\u2082))) _ H_diff\u2081 H_diff\u2082),\ndsimp,\nsimplify_grad,\n\nsimp [T.smul.def, T.const_neg, T.const_mul, T.const_zero,\n      T.const_one, T.const_bit0, T.const_bit1, T.const_inv,\n      left_distrib, right_distrib],\nrw T.mul_inv_cancel two_pos,\nerw T.neg_div,\nsimp [mul_neg_eq_neg_mul_symm, neg_mul_eq_neg_mul_symm],\napply congr_arg, apply congr_arg,\nsimp only [T.mul_div_mul, square],\nrw [-mul_assoc, T.mul_div_mul, (@T.div_self_square _ \u03c3 H_pre)],\nsimp,\nrw [T.mul_inv_cancel two_pos],\nsimp,\nrw T.div_mul_inv,\nend\n\n| \u27e6\u03bc, \u03c3\u27e7 y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6\u03bc, \u03c3\u27e7 0     ishape H_at_idx H_pre := by { prove_ocont, apply T.continuous_mvn_kl\u2081, exact H_pre }\n| \u27e6\u03bc, \u03c3\u27e7 1     ishape H_at_idx H_pre := by { prove_ocont }\n| \u27e6\u03bc, \u03c3\u27e7 (n+2) ishape H_at_idx H_pre := by idx_over\n\nend mvn_kl\n\nsection open mvn_kl\ndef mvn_kl (shape : S) : det.op [shape, shape] [] :=\ndet.op.mk \"mvn_kl\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\n-- Seems silly but saves some fresh-name tracking in reparam\nnamespace mul_add\n\ndef f {shape : S} (xs : dvec T [shape, shape, shape]) : T shape := (xs^.head * xs^.head2) + xs^.head3\ndef f_pre {shape : S} : precondition [shape, shape, shape] := \u03bb xs, true\ndef f_pb {shape : S} (xs : dvec T [shape, shape, shape]) (y gy : T shape) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy * xs^.head2) fshape\n| 1     fshape := force (gy * xs^.head) fshape\n| 2     fshape := force gy fshape\n| (n+3) _      := T.error \"mul_add: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6z, \u03c3, \u03bc\u27e7 H_pre 0     fshape H_at_idx k H_k := by prove_odiff\n| \u27e6z, \u03c3, \u03bc\u27e7 H_pre 1     fshape H_at_idx k H_k := by prove_odiff\n| \u27e6z, \u03c3, \u03bc\u27e7 H_pre 2     fshape H_at_idx k H_k := by prove_odiff\n| xs       H_pre (n+3) fshape H_at_idx k H_k := by idx_over\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6z, \u03c3, \u03bc\u27e7 y H_y g_out 0 fshape H_at_idx H_pre :=\nbegin\nprove_pb_correct_init,\nsimp only [f, f_pre, f_pb, force_ok, dif_pos],\ndsimp,\nsimp only [dif_pos, dif_neg],\ndsimp,\nrw -T.grad_tmulT,\nsimplify_grad,\nreflexivity\nend\n\n| \u27e6z, \u03c3, \u03bc\u27e7 y H_y g_out 1 fshape H_at_idx H_pre :=\nbegin\nprove_pb_correct_init,\nsimp without mul_comm add_comm,\ndsimp,\nrw -T.grad_tmulT,\nsimplify_grad,\nreflexivity\nend\n\n| \u27e6z, \u03c3, \u03bc\u27e7 y H_y g_out 2 fshape H_at_idx H_pre :=\nbegin\nprove_pb_correct_init,\nsimp without mul_comm add_comm,\ndsimp,\nrw -T.grad_tmulT,\nsimplify_grad,\nreflexivity\nend\n\n| xs y H_y g_out (n+3) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6z, \u03c3, \u03bc\u27e7 0     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6z, \u03c3, \u03bc\u27e7 1     ishape H_at_idx H_pre := by prove_ocont\n| \u27e6z, \u03c3, \u03bc\u27e7 2     ishape H_at_idx H_pre := by prove_ocont\n| xs       (n+3) ishape H_at_idx H_pre := by idx_over\n\nend mul_add\n\nsection open mul_add\ndef mul_add (shape : S) : det.op [shape, shape, shape] shape :=\ndet.op.mk \"mul_add\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nnamespace bernoulli_neglogpdf\n\ndef f {shape : S} (xs : dvec T [shape, shape]) : \u211d := bernoulli_neglogpdf xs^.head xs^.head2\ndef f_pre {shape : S} : precondition [shape, shape] := \u03bb xs, 0 < xs^.head \u2227 xs^.head < 1\n\ndef f_pb {shape : S} (xs : dvec T [shape, shape]) (y gy : \u211d) : \u03a0 (idx : \u2115) (fshape : S), T fshape\n| 0     fshape := force (gy \u2b1d (1 - xs^.head2) / (eps shape + (1 - xs^.head)) - gy \u2b1d (xs^.head2 / (eps shape + xs^.head))) fshape\n| 1     fshape := force (gy \u2b1d T.log (eps shape + (1 - xs^.head)) - gy \u2b1d T.log (eps shape + xs^.head)) fshape\n| (n+2) fshape := T.error \"bernoulli_neglogpdf: index too large\"\n\nattribute [simp] f f_pre f_pb\n\nlemma f_odiff {shape : S} : is_odifferentiable (@f shape) (@f_pre shape)\n| \u27e6p, z\u27e7 H_pre 0 fshape H_at_idx k H_k :=\nhave H_p\u2081 : p > 0, from H_pre^.left,\nhave H_p\u2082 : p < 1, from H_pre^.right,\nby prove_odiff\n\n| \u27e6p, z\u27e7 H_pre 1 fshape H_at_idx k H_k := by prove_odiff\n| \u27e6\u03bc, \u03c3\u27e7 H_pre (n+2) fshape H_at_idx k H_k := by idx_over\n\n\nlemma f_pb_correct {shape : S} : pullback_correct (@f shape) (@f_pre shape) (@f_pb shape)\n| \u27e6p, z\u27e7 y H_y g_out 0 fshape H_at_idx H_pre :=\nhave H_p : p > 0, from H_pre^.left,\nhave H_1mp : 1 - p > 0, from lt1_alt H_pre^.right,\nhave H_diff\u2081 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * -T.sum (z * T.log (eps shape + \u03b8\u2080) + (1 - z) * T.log (eps shape + (1 - p)))) p, by prove_differentiable,\nhave H_diff\u2082 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * -T.sum (z * T.log (eps shape + p) + (1 - z) * T.log (eps shape + (1 - \u03b8\u2080)))) p, by prove_differentiable,\n\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : shape = fshape := eq.symm H_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : \u211d \u2192 \u211d := \u03bb (x : \u211d), x * g_out,\nassertv H_k_grad : \u2207 k y = g_out :=  by { dsimp, erw [T.grad_mul\u2081 id, T.grad_id, one_mul] },\nrw -H_k_grad,\nsubst H_y,\ndsimp,\nsimp ,\nrw -T.grad_tmulT,\ndunfold T.bernoulli_neglogpdf,\nrw T.grad_binary (\u03bb \u03b8\u2081 \u03b8\u2082, g_out * - T.sum (z * T.log (eps shape + \u03b8\u2081) + (1 - z) * T.log (eps shape + (1 - \u03b8\u2082)))) _ H_diff\u2081 H_diff\u2082,\ndsimp,\nnote H\u2081 := H_pre^.left,\nnote H\u2082 := lt1_alt H_pre^.right,\nsimplify_grad,\nsimp [T.smul.def, T.neg_div, T.const_neg],\nrw [T.mul_div_mul],\nsimp [T.div_mul_inv],\nend\n\n| \u27e6p, z\u27e7 y H_y g_out 1 fshape H_at_idx H_pre :=\nhave H_diff\u2081 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * -T.sum (\u03b8\u2080 * T.log (eps shape + p) + (1 - z) * T.log (eps shape + (1 - p)))) z, by prove_differentiable,\nhave H_diff\u2082 : is_cdifferentiable (\u03bb (\u03b8\u2080 : T shape), g_out * -T.sum (z * T.log (eps shape + p) + (1 - \u03b8\u2080) * T.log (eps shape + (1 - p)))) z, by prove_differentiable,\n\nbegin\nclear f_pb_correct,\nassertv H_fshape_eq : shape = fshape := eq.symm H_at_idx^.right,\nsubst H_fshape_eq,\ndefinev k : \u211d \u2192 \u211d := \u03bb (x : \u211d), x * g_out,\nassertv H_k_grad : \u2207 k y = g_out :=  by { dsimp, erw [T.grad_mul\u2081 id, T.grad_id, one_mul] },\nrw -H_k_grad,\nsubst H_y,\ndsimp,\nsimp,\nrw -T.grad_tmulT,\ndunfold T.bernoulli_neglogpdf,\nrw T.grad_binary (\u03bb \u03b8\u2081 \u03b8\u2082, g_out * - T.sum (\u03b8\u2081 * T.log (eps shape + p) + (1 - \u03b8\u2082) * T.log (eps shape + (1 - p)))) _ H_diff\u2081 H_diff\u2082,\ndsimp,\nsimplify_grad,\nsimp [T.smul.def, const_neg],\nend\n\n| xs y H_y g_out (n+2) fshape H_at_idx H_pre := by idx_over\n\nlemma f_ocont {shape : S} : is_ocontinuous (@f shape) (@f_pre shape)\n| \u27e6\u03bc, \u03c3\u27e7 0     ishape H_at_idx H_pre := by { prove_ocont_init, apply continuous_bernoulli_neglogpdf\u2081, exact H_pre^.left, exact lt1_alt H_pre^.right }\n| \u27e6\u03bc, \u03c3\u27e7 1     ishape H_at_idx H_pre := by { prove_ocont_init, apply continuous_bernoulli_neglogpdf\u2082, exact H_pre^.left, exact lt1_alt H_pre^.right }\n| \u27e6\u03bc, \u03c3\u27e7 (n+2) ishape H_at_idx H_pre := by idx_over\n\nend bernoulli_neglogpdf\n\nsection\nopen bernoulli_neglogpdf\ndef bernoulli_neglogpdf (shape : S) : det.op [shape, shape] [] :=\ndet.op.mk \"bernoulli_neglogpdf\" f f_pre f_pb f_odiff f_pb_correct f_ocont\nend\n\nend ops\n\n-- TODO(dhs): confirm I don't need this any more\n/-\nlemma mvn_kl_pre {shape : S} (xs : dvec T [shape, shape]) :\n  det.op.pre (det.op.special (det.special.mvn_kl shape)) xs = (dvec.head2 xs > 0) := rfl\n-/\nend certigrad\n", "meta": {"author": "dselsam", "repo": "certigrad", "sha": "c9a06e93f1ec58196d6d3b8563b29868d916727f", "save_path": "github-repos/lean/dselsam-certigrad", "path": "github-repos/lean/dselsam-certigrad/certigrad-c9a06e93f1ec58196d6d3b8563b29868d916727f/src/certigrad/ops.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.37719850583480313}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), \n  from by auto [int.fract_eq_iff, h\u03b1_irrat, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  assume (h7 : \u03b5 > 0),\n\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by auto [h8, h10, h5, h9],\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by auto [h11] using [linarith],\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto-4_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7662936430859598, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3771606396051753}}
{"text": "variable {U V}\n\ndef f : (U \u2192 V) \u2192 (U \u2192 U) := sorry\ndef add  {U} : U \u2192 U \u2192 U := sorry\n\n@[simp] theorem foo  (u : U) : f (add u) = id := sorry\n\ndef bar (u v : U) : f (add u) v =  id v := by\n  simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/simpPartialApp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3769483687443324}}
{"text": "\nimport data.set\n\nimport unitb.refinement.basic\nimport unitb.scheduling\nimport unitb.models.nondet\n\nimport util.cast\n\nimport temporal_logic\n\nnamespace hidden_state\n\nuniverse variable u\n\nopen nondet temporal predicate classical\n\nsection defs\n\nvariables {\u03b1 \u03b2 \u03c3 : Type}\n\nstructure refined\n       (f : \u03b1 \u2192 \u03c3)\n       (ma : program \u03b1)\n       (g : \u03b2 \u2192 \u03c3)\n       (mc : program \u03b2) :=\n  (glue : \u03b1 \u2192 \u03b2 \u2192 Prop)\n  (obs_cons : \u2200 a c, glue a c \u2192 f a = g c)\n  (bij : mc.lbl = ma.lbl)\n  (sim_init : \u2200 c\u2080, mc^.first c\u2080\n            \u2192 \u2203 a\u2080, ma^.first a\u2080 \u2227 glue a\u2080 c\u2080)\n  (sim' : \u2200 e c c' a, glue a c\n           \u2192 mc.step_of e c c'\n           \u2192 \u2203 a',  ma.step_of (e.cast bij) a a'\n                  \u2227 glue a' c')\n  (coarse : \u2200 e a c, glue a c\n               \u2192 (  mc.coarse_sch_of e c\n                  \u2194 ma.coarse_sch_of (e.cast bij) a))\n  (fine : \u2200 e a c, glue a c\n               \u2192 (  mc.fine_sch_of e c\n                  \u2194 ma.fine_sch_of (e.cast bij) a))\n\nend defs\n\nopen unitb.refinement\n\nsection thm\n\nopen unitb\n\nparameters {\u03b1 \u03b2 \u03c3 : Type}\n\nparameters\n       {f : \u03b1 \u2192 \u03c3}\n       {ma : program \u03b1}\n       {g : \u03b2 \u2192 \u03c3}\n       {mc : program \u03b2}\n\nparameter R : refined f ma g mc\n\nparameter [nonempty (unitb.state (program \u03b1))]\nparameter Tc : stream (unitb.state (program \u03b2))\nparameter Hc : system_sem.ex mc Tc\n\nopen nat\n\nparameters Tc mc\ndef Tevts  : stream (set (option (mc.lbl))) :=\n  \u03bb i, { e | mc.step_of e (Tc i) (Tc (succ i)) }\nparameters {Tc mc}\n\ninclude Hc\nlemma Tevts_ne_empty (i : \u2115)\n: Tevts i \u2260 \u2205 :=\nbegin\n  apply set.ne_empty_of_exists_mem,\n  unfold Tevts,\n  simp [mem_set_of],\n  have H := system_sem.safety _ _ Hc i,\n  rw action_drop at H,\n  apply H,\nend\nomit Hc\n\nopen scheduling\n\n-- noncomputable def Tevt [sched mc.lbl] : stream (option (mc.lbl)) :=\n-- fair_sched_of (Tevts \u2218 list.length)\n\n-- include Hc\n\ninstance mc_sched : sched mc.lbl := mc.lbl_is_sched\n\ninstance ma_sched : sched ma.lbl := ma.lbl_is_sched\n\n-- instance option_ma_sched : sched (option ma.lbl) :=\n-- by { apply scheduling.sched_option,\n--      apply hidden_state.ma_sched }\n\n-- include mc_sched\n\n-- example (i : \u2115) : mc.step_of (Tevt i) (Tc i) (Tc $ succ i) :=\n-- begin\n--   unfold Tevt Tevts,\n--   assert H : Tevts mc Tc i \u2260 \u2205,\n--   { unfold Tevts,\n--     note Hsaf := Hc.safety i,\n--     rw [action_drop] at Hsaf,\n--     cases Hsaf with e He,\n--     apply @set.ne_empty_of_mem _ _ e He },\n--   note H' := fair_sched_of_mem' (Tevts mc Tc) i H,\n--   unfold Tevts at H',\n--   apply H',\n-- end\n\n-- omit Hc\n\nstructure sim_state :=\n  (index : \u2115)\n  (abs_state : \u03b1)\n  (glued : R.glue abs_state (Tc index))\n\nnoncomputable def mk_state {p : \u03b1 \u2192 Prop}  {i : \u2115} (P : \u2203 x, p x \u2227 R.glue x (Tc i))\n: sim_state :=\n\u27e8i,some P,and.right $ some_spec P\u27e9\n\nlemma fst_mk_state {p : \u03b1 \u2192 Prop}  {i : \u2115} (P : \u2203 x, p x \u2227 R.glue x (Tc i))\n: (mk_state P).index = i := rfl\n\ninclude Hc\ndef object_req_ne_emp (i : \u2115)\n: {l' : option (ma.lbl) | option.cast' l' (R.bij) \u2208 Tevts i} \u2260 \u2205 :=\nbegin\n  have H : {l' : option (ma.lbl) | option.cast' l' (R.bij) \u2208 Tevts mc Tc i}\n          = (\u03bb l, option.cast l R.bij) <$> Tevts mc Tc i,\n  { apply set.ext, intro l,\n    have Hinj : function.injective (\u03bb (l : option (mc.lbl)), option.cast l (R.bij)) :=\n       option_cast_injective R.bij,\n    rw [mem_set_of,set.mem_fmap_iff_mem_of_bij Hinj],\n    apply option.cast_left_inverse, },\n  simp [H,set.fmap_eq_empty_iff_eq_empty],\n  apply Tevts_ne_empty Hc,\nend\nomit Hc\n\nnoncomputable def next_state (l : option ma.lbl) (x : \u03b1) (i : \u2115)\n  (P : R.glue x (Tc i))\n  (Q : program.step_of mc (option.cast' l (R.bij)) (Tc i) (Tc (succ i)))\n: sim_state :=\nmk_state $ R.sim' (l.cast' R.bij) (Tc i) (Tc $ succ i) x P Q\n\nnoncomputable def object : unitb.target_mch (option ma.lbl) :=\n{ \u03c3 := sim_state\n, s\u2080 := mk_state $ R.sim_init (Tc 0) Hc.init\n, req := \u03bb s, { l' | l'.cast' R.bij \u2208 Tevts s.index }\n, req_nemp := \u03bb s, object_req_ne_emp s.index\n, next := \u03bb l s Q, next_state l s.abs_state _ s.glued Q }\n\n-- include ma_sched\n\nnoncomputable def Ta' : stream object.\u03c3 :=\nlet this : sched (option ma.lbl) :=\nby { apply scheduling.sched_option }\nin @fair_sched _ this _ object\n\nnoncomputable def Ta : stream (unitb.state (program \u03b1)) :=\nstream.map sim_state.abs_state Ta'\n\nlemma Ta_fair : fair object Ta' :=\nfair_sched_of_spec object\n\nlemma object_next_eq_next_state (l : option ma.lbl) (s : object.\u03c3)\n   (P : l \u2208 object.req s)\n: (object.next l _ P) = next_state l s.abs_state _ s.glued P :=\nbegin\n  cases s with i s, refl,\nend\n\nlemma Ta_index (i : \u2115)\n: (Ta' i).index = i :=\nbegin\n  induction i with i IH,\n  { rw Ta_fair.init, refl },\n  { cases fair_sched_succ _ i (Ta_fair R Hc) with l h,\n    cases h with P h,\n    rw [h,object_next_eq_next_state],\n    unfold next_state, rw [fst_mk_state,IH] },\nend\n\ninclude Hc\n\nlemma init_simmed\n: R.glue (Ta 0) (Tc 0) \u2227 ma^.first (Ta 0) :=\nbegin\n  unfold Ta stream.map stream.nth,\n  rw Ta_fair.init,\n  dunfold object unitb.target_mch.s\u2080,\n  dunfold mk_state sim_state.abs_state,\n  rw and_comm,\n  apply some_spec (object._proof_1 R Hc),\nend\n\n-- include R\n\n-- lemma conc_step (i : \u2115)\n-- : mc.step_of (Tevt i) (Tc i) (Tc (succ i)) :=\n-- begin\n--   unfold Tevt Tevts,\n--   assert H : Tevts mc Tc i \u2260 \u2205,\n--   { unfold Tevts,\n--     note Hsaf := Hc.safety i,\n--     rw [action_drop] at Hsaf,\n--     cases Hsaf with e He,\n--     apply @set.ne_empty_of_mem _ _ e He },\n--   note H' := fair_sched_of_mem' (Tevts mc Tc) i H,\n--   unfold Tevts at H',\n--   apply H',\n-- end\n\n-- lemma abs_step (i : \u2115) [nonempty \u03b1]\n--   (J : R.glue (Ta i) (Tc i))\n-- :   R.glue (Ta (succ i)) (Tc (succ i))\n--   \u2227 ma.step_of ((Tevt i).cast R.bij) (Ta i) (Ta (succ i)) :=\n-- begin\n--   unfold Ta,\n--   pose P := (\u03bb a, R.glue a (Tc (succ i)) \u2227 ma.step_of ((Tevt _ Tc i).cast R.bij) (Ta _ _ _ _ R Tc i) a),\n--   apply @epsilon_spec _ P,\n--   revert P, simp,\n--   note H' := R.sim' (Tevt _ Tc i) _ (Tc $ succ i) (Ta _ ma _ _ R Tc i) J (conc_step _ _ _ _ R Tc Hc i),\n--   apply exists_imp_exists _ H',\n--   intro, apply (and_comm _ _).mp,\n-- end\n\ntheorem glued [nonempty \u03b1] (i : \u2115) : R.glue (Ta i) (Tc i) :=\nbegin\n  have H := (Ta' R Hc i).glued,\n  rw Ta_index at H,\n  apply H,\nend\n\n-- include R\n\ntheorem simmed [nonempty \u03b1] (i : \u2115) : is_step ma (Ta i) (Ta (succ i)) :=\nbegin\n  apply exists_imp_exists _ (fair_sched_succ _ i $ Ta_fair R Hc),\n  intros e h, cases h with P h,\n  unfold Ta stream.map stream.nth,\n  rw h,\n  admit,\n  -- apply is_step_inst _ _,\n  -- apply (abs_step _ _ _ _ R _ Hc i _).right,\n  -- apply glued _ _ _ _ _ _ Hc,\n  -- apply_instance,\nend\n\nopen unitb\n\ninclude R\n\ntheorem soundness [nonempty \u03b1] : data_ref ma f mc g :=\nbegin\n  intros Tc Hc,\n  existsi (Ta R Hc),\n  split,\n  apply program.ex.mk ,\n  { apply (init_simmed _ Hc).right },\n  { intro i,\n    unfold action step has_safety.step stream.drop,\n    simp [add_one],\n    apply simmed _ Hc,\n    apply_instance },\n  { intros ea,\n    let ec := (option.cast' ea (R.bij)),\n    refine imp_mono _ _ (Hc.liveness ec),\n    { apply iff.mp,\n      apply exists_congr, intro i,\n      apply forall_congr, intro j,\n      unfold temporal.init stream.drop,\n      let ec := (option.cast' ea (R.bij)),\n      rw [R.coarse ec,option_cast_cast'],\n      apply glued },\n    refine imp_mono _ _,\n    { refine iff.mp _,\n      apply forall_congr, intro j,\n      apply exists_congr, intro i,\n      unfold temporal.init stream.drop,\n      have HHH:= R.fine ec _ _ (glued R Hc $ 0 + i + j),\n      repeat { rw [p_and_to_fun,init_to_fun,init_to_fun] },\n      rw [HHH,option_cast_cast'] },\n    { intro Hevt,\n      admit } },\n  { apply funext, intro i,\n    unfold function.comp,\n    symmetry,\n    apply R.obs_cons,\n    apply glued },\nend\n\nend thm\n\nend hidden_state\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/refinement/resched_data_ref.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3769483627017908}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport data.fintype.powerset\nimport order.category.BoolAlg\nimport order.category.FinPartOrd\nimport order.hom.complete_lattice\n\n/-!\n# The category of finite boolean algebras\n\nThis file defines `FinBoolAlg`, the category of finite boolean algebras.\n\n## TODO\n\nBirkhoff's representation for finite Boolean algebras.\n\n`Fintype_to_FinBoolAlg_op.left_op \u22d9 FinBoolAlg.dual \u2245 Fintype_to_FinBoolAlg_op.left_op`\n\n`FinBoolAlg` is essentially small.\n-/\n\nuniverses u\n\nopen category_theory order_dual opposite\n\n/-- The category of finite boolean algebras with bounded lattice morphisms. -/\nstructure FinBoolAlg :=\n(to_BoolAlg : BoolAlg)\n[is_fintype : fintype to_BoolAlg]\n\nnamespace FinBoolAlg\n\ninstance : has_coe_to_sort FinBoolAlg Type* := \u27e8\u03bb X, X.to_BoolAlg\u27e9\ninstance (X : FinBoolAlg) : boolean_algebra X := X.to_BoolAlg.str\n\nattribute [instance]  FinBoolAlg.is_fintype\n\n@[simp] lemma coe_to_BoolAlg (X : FinBoolAlg) : \u21a5X.to_BoolAlg = \u21a5X := rfl\n\n/-- Construct a bundled `FinBoolAlg` from `boolean_algebra` + `fintype`. -/\ndef of (\u03b1 : Type*) [boolean_algebra \u03b1] [fintype \u03b1] : FinBoolAlg := \u27e8\u27e8\u03b1\u27e9\u27e9\n\n@[simp] lemma coe_of (\u03b1 : Type*) [boolean_algebra \u03b1] [fintype \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited FinBoolAlg := \u27e8of punit\u27e9\n\ninstance large_category : large_category FinBoolAlg :=\ninduced_category.category FinBoolAlg.to_BoolAlg\n\ninstance concrete_category : concrete_category FinBoolAlg :=\ninduced_category.concrete_category FinBoolAlg.to_BoolAlg\n\ninstance has_forget_to_BoolAlg : has_forget\u2082 FinBoolAlg BoolAlg :=\ninduced_category.has_forget\u2082 FinBoolAlg.to_BoolAlg\n\ninstance forget_to_BoolAlg_full : full (forget\u2082 FinBoolAlg BoolAlg) := induced_category.full _\ninstance forget_to_BoolAlg_faithful : faithful (forget\u2082 FinBoolAlg BoolAlg) :=\ninduced_category.faithful _\n\n@[simps] instance has_forget_to_FinPartOrd : has_forget\u2082 FinBoolAlg FinPartOrd :=\n{ forget\u2082 := { obj := \u03bb X, FinPartOrd.of X, map := \u03bb X Y f,\n    show order_hom X Y, from \u2191(show bounded_lattice_hom X Y, from f) } }\n\ninstance forget_to_FinPartOrd_faithful : faithful (forget\u2082 FinBoolAlg FinPartOrd) :=\n\u27e8\u03bb X Y f g h, by { have := congr_arg (coe_fn : _ \u2192 X \u2192 Y) h, exact fun_like.coe_injective this }\u27e9\n\n/-- Constructs an equivalence between finite Boolean algebras from an order isomorphism between\nthem. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : FinBoolAlg.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := (e : bounded_lattice_hom \u03b1 \u03b2),\n  inv := (e.symm : bounded_lattice_hom \u03b2 \u03b1),\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : FinBoolAlg \u2964 FinBoolAlg :=\n{ obj := \u03bb X, of X\u1d52\u1d48, map := \u03bb X Y, bounded_lattice_hom.dual }\n\n/-- The equivalence between `FinBoolAlg` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : FinBoolAlg \u224c FinBoolAlg :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend FinBoolAlg\n\n/-- The powerset functor. `set` as a functor. -/\n@[simps] def Fintype_to_FinBoolAlg_op : Fintype \u2964 FinBoolAlg\u1d52\u1d56 :=\n{ obj := \u03bb X, op $ FinBoolAlg.of (set X),\n  map := \u03bb X Y f, quiver.hom.op $\n    (complete_lattice_hom.set_preimage f : bounded_lattice_hom (set Y) (set X)) }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/order/category/FinBoolAlg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3769368613651355}}
{"text": "import data.mv_polynomial.comm_ring\nimport linear_algebra.finsupp\nimport linear_algebra.clifford_algebra.basic\nimport data.zmod.basic\nimport data.matrix.notation\nimport field_theory.mv_polynomial\nimport tactic.induction\nimport algebra.char_p.quotient\nimport data.nat.prime\nimport algebra.char_p.pi\nimport algebra.char_p.two\nimport ring_theory.mv_polynomial.ideal\nimport ring_theory.mv_polynomial.basic\nimport algebra.char_p.algebra\nimport tactic.norm_fin\n/-!\nA formalization of https://mathoverflow.net/questions/60596/clifford-pbw-theorem-for-quadratic-form/87958#87958\n\nSome Zulip discussion at https://leanprover.zulipchat.com/#narrow/stream/113489-new-members/topic/.F0.9D.94.BD.E2.82.82.5B.CE.B1.2C.20.CE.B2.2C.20.CE.B3.5D.20.2F.20.28.CE.B1.C2.B2.2C.20.CE.B2.C2.B2.2C.20.CE.B3.C2.B2.29/near/222716333.\n-/\n\nnoncomputable theory\n\nopen_locale big_operators\n\nsection for_mathlib\n\nlemma ideal.comap_span_le {R : Type*} {S : Type*} [semiring R] [semiring S] (f : S \u2192+* R)\n  (g : R \u2192+* S) (h : function.left_inverse g f) (s : set R) :\n  ideal.comap f (ideal.span s) \u2264 ideal.span (g '' s) :=\nbegin\n  rintros x (hx : f x \u2208 ideal.span s),\n  have := ideal.apply_coe_mem_map g _ \u27e8_, hx\u27e9,\n  rw [ideal.map_span, subtype.coe_mk, h x] at this,\n  exact this,\nend\n\n/-- `char_p.quotient'` as an `iff`. -/\nlemma char_p.quotient_iff' (R : Type*) [comm_ring R] (n : \u2115) [char_p R n] (I : ideal R) :\n  char_p (R \u29f8 I) n \u2194 \u2200 x : \u2115, \u2191x \u2208 I \u2192 (x : R) = 0 :=\nbegin\n  refine \u27e8\u03bb (i : char_p (R \u29f8 I) n) x hx, _, char_p.quotient' n I\u27e9,\n  resetI,\n  have := char_p.cast_eq_zero_iff (R \u29f8 I) n,\n  rw char_p.cast_eq_zero_iff R n,\n  refine (this _).mp _,\n  exact (submodule.quotient.mk_eq_zero I).mpr hx,\nend\n\nlemma ideal.span_le_bot {R : Type*} [semiring R] (s : set R) : ideal.span s \u2264 \u22a5 \u2194 s \u2264 {0} :=\nsubmodule.span_le\n\n/-- `char_p.quotient'` as an `iff`. -/\nlemma char_p.quotient_iff'' (R : Type*) [comm_ring R] (n : \u2115) [char_p R n] (I : ideal R) :\n  char_p (R \u29f8 I) n \u2194 I.comap (nat.cast_ring_hom R) \u2264 (nat.cast_ring_hom R).ker :=\n(char_p.quotient_iff' _ _ _).trans begin\n  rw ring_hom.ker_eq_comap_bot,\n  exact iff.rfl,\nend\n\nlemma finsupp.equiv_fun_on_finite_const {\u03b1 \u03b2} [fintype \u03b1] [add_comm_monoid \u03b2] (b : \u03b2):\n  finsupp.equiv_fun_on_finite.symm (\u03bb _, b : \u03b1 \u2192 \u03b2) = \u2211 i : \u03b1, finsupp.single i b :=\nby { ext, simp [finsupp.finset_sum_apply] }\n\n-- note; more general than `mv_polynomial.support_smul`\nlemma _root_.mv_polynomial.support_smul' {S R \u03c3} [comm_semiring R] [monoid S] [distrib_mul_action S R]\n  {r : S} {p : mv_polynomial \u03c3 R} :\n  (r \u2022 p).support \u2286 p.support := finsupp.support_smul\n\nlemma finsupp.to_multiset_sup {\u03b1} [decidable_eq \u03b1] (f g : \u03b1 \u2192\u2080 \u2115) :\n  (f \u2294 g).to_multiset = f.to_multiset \u222a g.to_multiset :=\nbegin\n  ext,\n  simp_rw [multiset.count_union, finsupp.count_to_multiset, finsupp.sup_apply, sup_eq_max],\nend\n\nlemma finsupp.to_multiset_inf {\u03b1} [decidable_eq \u03b1] (f g : \u03b1 \u2192\u2080 \u2115) :\n  (f \u2293 g).to_multiset = f.to_multiset \u2229 g.to_multiset :=\nbegin\n  ext,\n  simp_rw [multiset.count_inter, finsupp.count_to_multiset, finsupp.inf_apply, inf_eq_min],\nend\n\n/-- `equiv.ulift` as a `linear_equiv`. -/\n@[simps]\ndef {w u v} linear_equiv.ulift\n  (R : Type u) (M : Type v) [semiring R] [add_comm_monoid M] [module R M]: ulift.{w} M \u2243\u2097[R] M :=\n{ map_add' := \u03bb x y, rfl,\n  map_smul' := \u03bb c x, rfl,\n  .. equiv.ulift }\n\nlemma ideal.mem_span_range_iff_exists_fun {\u03b9 R} [fintype \u03b9] [comm_semiring R] (g : \u03b9 \u2192 R) (x : R) :\n  x \u2208 ideal.span (set.range g) \u2194 \u2203 f : \u03b9 \u2192 R, \u2211 i, f i * g i = x :=\nmem_span_range_iff_exists_fun _\n\nend for_mathlib\n\nnamespace q60596\n\nopen mv_polynomial\n\n/-- The monomial ideal generated by terms of the form $x_ix_i$. -/\ndef k_ideal : ideal (mv_polynomial (fin 3) (zmod 2)) :=\nideal.span (set.range (\u03bb i, (X i * X i : mv_polynomial (fin 3) (zmod 2))))\n\nlemma mem_k_ideal_iff (x : mv_polynomial (fin 3) (zmod 2)) :\n  x \u2208 k_ideal \u2194 \u2200 (m : fin 3 \u2192\u2080 \u2115), m \u2208 x.support \u2192 \u2203 i, 2 \u2264 m i :=\nbegin\n  have : k_ideal =\n    ideal.span ((\u03bb i, monomial i (1 : zmod 2)) '' set.range (\u03bb i, finsupp.single i 2)),\n  { simp_rw [k_ideal, X, monomial_mul, one_mul, \u2190finsupp.single_add, \u2190set.range_comp] },\n  rw [this, mem_ideal_span_monomial_image],\n  simp,\nend\n\nlemma X0_X1_X2_nmem_k_ideal : (X 0 * X 1 * X 2 : mv_polynomial (fin 3) (zmod 2)) \u2209 k_ideal :=\nbegin\n  intro h,\n  simp_rw [mem_k_ideal_iff, support_mul_X, support_X, finset.map_singleton,\n    add_right_embedding_apply, finset.mem_singleton, forall_eq,\n    \u2190fin.sum_univ_three (\u03bb i, finsupp.single i 1), \u2190finsupp.equiv_fun_on_finite_const,\n    finsupp.equiv_fun_on_finite_symm_apply_to_fun] at h,\n  obtain \u27e8i, hi\u27e9 := h,\n  revert hi,\n  dec_trivial,\nend\n\nlemma mul_self_mem_k_ideal_of_X0_X1_X2_mul_mem {x : mv_polynomial (fin 3) (zmod 2)}\n  (h : X 0 * X 1 * X 2 * x \u2208 k_ideal) : x * x \u2208 k_ideal :=\nbegin\n  rw mem_k_ideal_iff at h,\n  have : x \u2208 ideal.span ((X : fin 3 \u2192 mv_polynomial _ (zmod 2)) '' set.univ),\n  { rw [mem_ideal_span_X_image],\n    intros m hm,\n    simp_rw [mul_assoc, support_X_mul, finset.map_map, finset.mem_map,\n        function.embedding.trans_apply, add_left_embedding_apply, exists_prop,\n        forall_exists_index, and_imp, forall_apply_eq_imp_iff\u2082, \u2190add_assoc,\n        \u2190fin.sum_univ_three (\u03bb i, finsupp.single i 1), \u2190finsupp.equiv_fun_on_finite_const,\n        finsupp.add_apply, finsupp.equiv_fun_on_finite_symm_apply_to_fun] at h,\n    refine (h _ hm).imp (\u03bb i hi, \u27e8set.mem_univ _, _\u27e9),\n    rintro hmi,\n    rw hmi at hi,\n    cases nat.le_of_add_le_add_left hi },\n  rw [as_sum x, char_two.sum_mul_self],\n  refine sum_mem (\u03bb m hm, _),\n  rw [mem_k_ideal_iff, monomial_mul],\n  intros m' hm',\n  obtain rfl := finset.mem_singleton.1 (support_monomial_subset hm'),\n  rw mem_ideal_span_X_image at this,\n  obtain \u27e8i, _, hi\u27e9 := this m hm,\n  refine \u27e8i, nat.add_le_add _ _\u27e9; rwa nat.one_le_iff_ne_zero,\nend\n\n-- \ud835\udd3d\u2082[\u03b1, \u03b2, \u03b3] / (\u03b1\u00b2, \u03b2\u00b2, \u03b3\u00b2)\n@[derive [comm_ring, comm_semiring, ring, semiring, add_comm_group, add_comm_monoid]]\ndef k := _ \u29f8 k_ideal\n\nlemma comap_C_span_le_bot :\n  k_ideal.comap (C : zmod 2 \u2192+* (mv_polynomial (fin 3) (zmod 2))) \u2264 \u22a5 :=\nbegin\n  refine (ideal.comap_span_le _ _ (constant_coeff_C _) _).trans _,\n  refine (ideal.span_le_bot _).2 _,\n  rintro x \u27e8_, \u27e8i, rfl\u27e9, rfl\u27e9,\n  rw [ring_hom.map_mul, constant_coeff_X, mul_zero, set.mem_singleton_iff],\nend\n\n/-- `k` has characteristic 2. -/\ninstance k.char_p : char_p k 2 :=\nbegin\n  dunfold k,\n  rw char_p.quotient_iff'',\n  have : (nat.cast_ring_hom (mv_polynomial (fin 3) (zmod 2))) = C.comp (nat.cast_ring_hom _),\n  { ext1 r, refl },\n  rw [this, \u2190ideal.comap_comap, \u2190ring_hom.comap_ker],\n  exact ideal.comap_mono (comap_C_span_le_bot.trans bot_le),\nend\n\nabbreviation \u03b1 : k := ideal.quotient.mk _ (mv_polynomial.X 0)\nabbreviation \u03b2 : k := ideal.quotient.mk _ (mv_polynomial.X 1)\nabbreviation \u03b3 : k := ideal.quotient.mk _ (mv_polynomial.X 2)\n\n/-- The elements above square to zero -/\n@[simp] lemma X_sq (i : fin 3) :\n  ideal.quotient.mk _ (mv_polynomial.X i) * ideal.quotient.mk _ (mv_polynomial.X i) = (0 : k) :=\nbegin\n  change ideal.quotient.mk _ _ = _,\n  rw [ideal.quotient.eq_zero_iff_mem],\n  exact ideal.subset_span \u27e8i, rfl\u27e9,\nend\n\n/-- If an element multiplied by `\u03b1\u03b2\u03b3` is zero then it squares to zero. -/\nlemma sq_zero_of_\u03b1\u03b2\u03b3_mul {x : k} : \u03b1 * \u03b2 * \u03b3 * x = 0 \u2192 x * x = 0 :=\nbegin\n  induction x using quotient.induction_on',\n  change ideal.quotient.mk _ _ = 0 \u2192 ideal.quotient.mk _ _ = 0,\n  rw [ideal.quotient.eq_zero_iff_mem, ideal.quotient.eq_zero_iff_mem],\n  exact mul_self_mem_k_ideal_of_X0_X1_X2_mul_mem,\nend\n\n/-- Though `\u03b1\u03b2\u03b3` is not itself zero-/\nlemma \u03b1\u03b2\u03b3_ne_zero : \u03b1 * \u03b2 * \u03b3 \u2260 0 :=\n\u03bb h, X0_X1_X2_nmem_k_ideal $ ideal.quotient.eq_zero_iff_mem.1 h\n\n@[simps]\ndef L_func : (fin 3 \u2192 k) \u2192\u2097[k] k :=\n\u03b1 \u2022 linear_map.proj 0 - \u03b2 \u2022 linear_map.proj 1 - \u03b3 \u2022 linear_map.proj 2\n\n/-- The quotient of k^3 by the specified relation-/\n@[derive [add_comm_group, module k]]\ndef L := _ \u29f8 L_func.ker\n\ndef sq {\u03b9 R : Type*} [comm_ring R] (i : \u03b9) : quadratic_form R (\u03b9 \u2192 R) :=\nquadratic_form.sq.comp $ linear_map.proj i\n\nlemma sq_map_add_char_two {\u03b9 R : Type*} [comm_ring R] [char_p R 2] (i : \u03b9) (a b : \u03b9 \u2192 R) :\n  sq i (a + b) = sq i a + sq i b :=\nchar_two.add_mul_self _ _\n\nlemma sq_map_sub_char_two {\u03b9 R : Type*} [comm_ring R] [char_p R 2] (i : \u03b9) (a b : \u03b9 \u2192 R) :\n  sq i (a - b) = sq i a - sq i b :=\nbegin\n  haveI : nonempty \u03b9 := \u27e8i\u27e9,\n  rw [char_two.sub_eq_add, char_two.sub_eq_add, sq_map_add_char_two]\nend\n\nopen_locale big_operators\n\n/-- The quadratic form (metric) is just euclidean -/\ndef Q' : quadratic_form k (fin 3 \u2192 k) :=\n\u2211 i, sq i\n\ndef Q'_add (x y : fin 3 \u2192 k) : Q' (x + y) = Q' x + Q' y :=\nby simp only [Q', quadratic_form.sum_apply, sq_map_add_char_two, finset.sum_add_distrib]\n\ndef Q'_sub (x y : fin 3 \u2192 k) : Q' (x - y) = Q' x - Q' y :=\nby simp only [Q', quadratic_form.sum_apply, sq_map_sub_char_two, finset.sum_sub_distrib]\n\nlemma Q'_apply (a : fin 3 \u2192 k) : Q' a = a 0 * a 0 + a 1 * a 1 + a 2 * a 2 :=\ncalc Q' a = a 0 * a 0 + (a 1 * a 1 + (a 2 * a 2 + 0)) : rfl\n      ... = _ : by ring\n\nlemma Q'_apply_single (i : fin 3) (x : k) : Q' (pi.single i x) = x * x :=\ncalc Q' (pi.single i x)\n    = \u2211 j : fin 3, (pi.single i x * pi.single i x : fin 3 \u2192 k) j : by simp [Q', sq]\n... = _ : by simp_rw [\u2190pi.single_mul, finset.sum_pi_single', finset.mem_univ, if_pos]\n\nlemma Q'_zero_under_ideal (v : fin 3 \u2192 k) (hv : v \u2208 L_func.ker) : Q' v = 0 :=\nbegin\n  rw [linear_map.mem_ker, L_func_apply] at hv,\n  have h0 : \u03b1 * \u03b2 * \u03b3 * v 0 = 0,\n  { have := congr_arg ((*) (\u03b2 * \u03b3)) hv,\n    simp only [mul_zero, mul_add, \u2190mul_assoc] at this,\n    rw [mul_comm (\u03b2 * \u03b3) \u03b1, \u2190mul_assoc, mul_right_comm \u03b2 \u03b3 \u03b2, mul_assoc \u03b2 \u03b3 \u03b3, X_sq, X_sq] at this,\n    simpa only [mul_zero, zero_mul, add_zero, zero_add] using this },\n  have h1 : \u03b1 * \u03b2 * \u03b3 * v 1 = 0,\n  { have := congr_arg ((*) (\u03b1 * \u03b3)) hv,\n    simp only [mul_zero, mul_add, \u2190mul_assoc] at this,\n    rw [mul_right_comm \u03b1 \u03b3 \u03b1, mul_assoc \u03b1 \u03b3 \u03b3, mul_right_comm \u03b1 \u03b3 \u03b2, X_sq, X_sq] at this,\n    simpa only [mul_zero, zero_mul, add_zero, zero_add] using this },\n  have h2 : \u03b1 * \u03b2 * \u03b3 * v 2 = 0,\n  { have := congr_arg ((*) (\u03b1 * \u03b2)) hv,\n    simp only [mul_zero, mul_add, \u2190mul_assoc] at this,\n    rw [mul_right_comm \u03b1 \u03b2 \u03b1, mul_assoc \u03b1 \u03b2 \u03b2, X_sq, X_sq] at this,\n    simpa only [mul_zero, zero_mul, add_zero, zero_add] using this },\n  rw [Q'_apply, sq_zero_of_\u03b1\u03b2\u03b3_mul h0, sq_zero_of_\u03b1\u03b2\u03b3_mul h1, sq_zero_of_\u03b1\u03b2\u03b3_mul h2,\n    add_zero, add_zero],\nend\n\n/-- `Q'`, lifted to operate on the quotient space `L`. -/\n@[simps]\ndef Q : quadratic_form k L :=\nquadratic_form.of_polar\n  (\u03bb x, quotient.lift_on' x Q' $ \u03bb a b h, begin\n    rw submodule.quotient_rel_r_def at h,\n    suffices : Q' (a - b) = 0,\n    { rwa [Q'_sub, sub_eq_zero] at this, },\n    apply Q'_zero_under_ideal (a - b) h,\n  end)\n  (\u03bb a x, begin\n    induction x using quotient.induction_on,\n    exact Q'.to_fun_smul a x,\n  end)\n  (by { rintros \u27e8x\u27e9 \u27e8x'\u27e9 \u27e8y\u27e9, exact Q'.polar_add_left x x' y })\n  (by { rintros c \u27e8x\u27e9 \u27e8y\u27e9, exact Q'.polar_smul_left c x y })\n\nopen clifford_algebra\n\n/-! Shorthand for basis vectors in the Clifford algebra -/\nabbreviation x' : clifford_algebra Q := \u03b9 Q $ submodule.quotient.mk (pi.single 0 1)\nabbreviation y' : clifford_algebra Q := \u03b9 Q $ submodule.quotient.mk (pi.single 1 1)\nabbreviation z' : clifford_algebra Q := \u03b9 Q $ submodule.quotient.mk (pi.single 2 1)\n\n/-- The basis vectors square to one -/\n@[simp] lemma x_mul_x : x' * x' = 1 :=\nbegin\n  dunfold x',\n  simp_rw [clifford_algebra.\u03b9_sq_scalar, Q_apply, \u2190submodule.quotient.mk'_eq_mk,\n    quotient.lift_on'_mk', Q'_apply_single, mul_one, map_one],\nend\n\n/-- By virtue of the quotient, terms of this form are zero -/\nlemma quot_obv : \u03b1 \u2022 x' - \u03b2 \u2022 y' - \u03b3 \u2022 z' = 0 :=\nbegin\n  dunfold x' y' z',\n  simp only [\u2190linear_map.map_smul, \u2190linear_map.map_sub, \u2190submodule.quotient.mk_smul,\n    \u2190submodule.quotient.mk_sub],\n  convert linear_map.map_zero _ using 2,\n  rw submodule.quotient.mk_eq_zero,\n  norm_num [sub_zero, ideal.span, pi.single_apply],\nend\n\n/-- The core of the proof - scaling `1` by `\u03b1 * \u03b2 * \u03b3` gives zero -/\nlemma \u03b1\u03b2\u03b3_smul_eq_zero : (\u03b1 * \u03b2 * \u03b3) \u2022 (1 : clifford_algebra Q) = 0 :=\nbegin\n  suffices : \u03b1 \u2022 1 - \u03b2 \u2022 (y' * x') - \u03b3 \u2022 (z' * x') = 0,\n  { have := congr_arg (\u03bb x, (\u03b2 * \u03b3) \u2022 x) this,\n    simpa [smul_sub, smul_smul, mul_assoc \u03b2 \u03b3 \u03b3, mul_right_comm \u03b2 \u03b3 \u03b2, mul_right_comm \u03b2 \u03b3 \u03b1,\n      mul_comm \u03b2 \u03b1] using this },\n  have : (\u03b1 \u2022 x' - \u03b2 \u2022 y' - \u03b3 \u2022 z') * x' = \u03b1 \u2022 1 - \u03b2 \u2022 (y' * x') - \u03b3 \u2022 (z' * x'),\n  { simp [sub_mul], },\n  rw \u2190 this,\n  rw [quot_obv, zero_mul],\nend\n\n/-- Our final result -/\nlemma algebra_map_not_injective : \u00acfunction.injective (algebra_map k $ clifford_algebra Q) :=\n\u03bb h, \u03b1\u03b2\u03b3_ne_zero $ h begin\n  rw [algebra.algebra_map_eq_smul_one, ring_hom.map_zero, \u03b1\u03b2\u03b3_smul_eq_zero],\nend\n\nend q60596\n\n/- The generate statement: not every Clifford algebra has an injective algebra map -/\n-- TODO: https://github.com/leanprover-community/mathlib/pull/18644/files\nlemma {v} clifford_algebra.not_forall_algebra_map_injective :\n  \u00ac\u2200 (R : Type) (M : Type v) [comm_ring R] [add_comm_group M], by exactI\n   \u2200 [module R M], by exactI\n   \u2200 (Q : quadratic_form R M),\n    function.injective (algebra_map R $ clifford_algebra Q) :=\n\u03bb h, q60596.algebra_map_not_injective $ \u03bb x y hxy, begin\n  let uQ := q60596.Q.comp (linear_equiv.ulift q60596.k _).to_linear_map,\n  refine h q60596.k (ulift q60596.L)\n    (q60596.Q.comp $ (linear_equiv.ulift q60596.k _).to_linear_map) _,\n  let uC := clifford_algebra.map q60596.Q uQ\n    (linear_equiv.ulift q60596.k q60596.L).symm.to_linear_map (\u03bb _, rfl),\n  simpa using uC.congr_arg hxy,\nend\n", "meta": {"author": "pygae", "repo": "lean-ga", "sha": "5e8b22b2f25c7037723ad811faa312660eeb6775", "save_path": "github-repos/lean/pygae-lean-ga", "path": "github-repos/lean/pygae-lean-ga/lean-ga-5e8b22b2f25c7037723ad811faa312660eeb6775/src/geometric_algebra/from_mathlib/mathoverflow.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3768864066983544}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Scott Morrison\n\nimport category_theory.category\nimport category_theory.graphs\n\nopen category_theory\nopen category_theory.graphs\n\nnamespace category_theory.examples.graphs\n\nuniverse u\u2081\n\ndef Graph := \u03a3 \u03b1 : Type u\u2081, graph.{u\u2081} \u03b1\n\ninstance graph_from_Graph (G : Graph) : graph G.1 := G.2\n\nstructure Graph_hom (G H : Graph.{u\u2081}) : Type u\u2081 :=\n(map : @graph_hom G.1 G.2 H.1 H.2)\n\n@[extensionality] lemma graph_homomorphisms_pointwise_equal\n  {G H : Graph.{u\u2081}}\n  {p q : Graph_hom G H}\n  (vertexWitness : \u2200 X : G.1, p.map.onVertices X = q.map.onVertices X)\n  (edgeWitness : \u2200 X Y : G.1, \u2200 f : edges X Y, \u27ec p.map.onEdges f \u27ed = q.map.onEdges f ) : p = q :=\nbegin\n  induction p,\n  induction q,\n  tidy,\nend\n\ninstance CategoryOfGraphs : large_category Graph :=\n{ hom := Graph_hom,\n  id := \u03bb G,\n  \u27e8{ onVertices   := id,\n     onEdges := \u03bb _ _ f, f }\u27e9,\n  comp := \u03bb G H K f g,\n  \u27e8{ onVertices := \u03bb v, g.map.onVertices (f.map.onVertices v),\n     onEdges    := \u03bb v w e, g.map.onEdges (f.map.onEdges e) }\u27e9 }\n\nend category_theory.examples.graphs", "meta": {"author": "semorrison", "repo": "lean-category-theory", "sha": "a27b4ae5eac978e9188d2e867c3d11d9a5b87a9e", "save_path": "github-repos/lean/semorrison-lean-category-theory", "path": "github-repos/lean/semorrison-lean-category-theory/lean-category-theory-a27b4ae5eac978e9188d2e867c3d11d9a5b87a9e/src/category_theory/examples/graphs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934765, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3768864066983543}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit : \u2200 \u03b1 : \u211d, \u03b1 \u2209 \u211a \u2192 \u2200 \u03b5 > 0, \u2203 n : \u2124, \u2200 m : \u2124, |n * \u03b1 - m| < \u03b5 :=\nbegin\n  assume \u03b1 h1 \u03b5 h2,\n  sorry,\nend\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 \u03b5 > 0, \u2203 x : \u2124, 0 \u2264 x \u2227 x < 1 \u2227 \u2200 y : \u2124, 0 \u2264 y \u2227 y < 1 \u2192 |y - x| < \u03b5 :=\nbegin\n  assume (\u03b5 : \u211d) (h\u03b5 : \u03b5 > 0),\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$.\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 frac (i * \u03b1) \u2260 frac (j * \u03b1), from sorry,\n  -- If this were not true, then $i \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor$, which yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$.\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 \u00ac is_rat \u03b1, from sorry,\n  -- Hence, $S:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}$ is an infinite subset of $\\left[0,1\\right]$.\n  have h3 : \u2200 i : \u2124, frac (i * \u03b1) \u2208 Icc (0 : \u211d) 1, from sorry,\n  have h4 : \u2200 i : \u2124, \u2203 x : \u2124, 0 \u2264 x \u2227 x < 1 \u2227 \u2200 y : \u2124, 0 \u2264 y \u2227 y < 1 \u2192 |y - x| < \u03b5, from sorry,\n  show \u2203 x : \u2124, 0 \u2264 x \u2227 x < 1 \u2227 \u2200 y : \u2124, 0 \u2264 y \u2227 y < 1 \u2192 |y - x| < \u03b5, from sorry,\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u03b1 \u2209 \u211a) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 (\u2124 : set \u211d) \u2227 |y - x| < 1 :=\nbegin\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  assume (y : \u211d) (h1 : y \u2208 Icc 0 1),\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from sorry,\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h9 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h10 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h11 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h12 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h13 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h14 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h15 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h16 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h17 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h18 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h19 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h20 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h21 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h22 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h23 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h24 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h25 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h26 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h27 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h28 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h29 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h30 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h31 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h32 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from sorry,\n  have h33 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by sorry\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by sorry\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from sorry,\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : \u03b5 > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 \u03b5 h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 \u03b5 h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := sorry,\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from sorry,\n    \n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from sorry,\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline_with_comments-Natural-Language-Proof-Translation/lean_proof_outline_with_comments-4_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8104789178257654, "lm_q2_score": 0.4649015713733884, "lm_q1q2_score": 0.37679292246220164}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.category_theory.limits.lattice\nimport Mathlib.PostPort\n\nuniverses v l \n\nnamespace Mathlib\n\n/-!\n# The category of \"pairwise intersections\".\n\nGiven `\u03b9 : Type v`, we build the diagram category `pairwise \u03b9`\nwith objects `single i` and `pair i j`, for `i j : \u03b9`,\nwhose only non-identity morphisms are\n`left : pair i j \u27f6 single i` and `right : pair i j \u27f6 single j`.\n\nWe use this later in describing (one formulation of) the sheaf condition.\n\nGiven any function `U : \u03b9 \u2192 \u03b1`, where `\u03b1` is some complete lattice (e.g. `(opens X)\u1d52\u1d56`),\nwe produce a functor `pairwise \u03b9 \u2964 \u03b1` in the obvious way,\nand show that `supr U` provides a colimit cocone over this functor.\n-/\n\nnamespace category_theory\n\n\n/--\nAn inductive type representing either a single term of a type `\u03b9`, or a pair of terms.\nWe use this as the objects of a category to describe the sheaf condition.\n-/\ninductive pairwise (\u03b9 : Type v) \nwhere\n| single : \u03b9 \u2192 pairwise \u03b9\n| pair : \u03b9 \u2192 \u03b9 \u2192 pairwise \u03b9\n\nnamespace pairwise\n\n\nprotected instance pairwise_inhabited {\u03b9 : Type v} [Inhabited \u03b9] : Inhabited (pairwise \u03b9) :=\n  { default := single Inhabited.default }\n\n/--\nMorphisms in the category `pairwise \u03b9`. The only non-identity morphisms are\n`left i j : single i \u27f6 pair i j` and `right i j : single j \u27f6 pair i j`.\n-/\ninductive hom {\u03b9 : Type v} : pairwise \u03b9 \u2192 pairwise \u03b9 \u2192 Type v\nwhere\n| id_single : (i : \u03b9) \u2192 hom (single i) (single i)\n| id_pair : (i j : \u03b9) \u2192 hom (pair i j) (pair i j)\n| left : (i j : \u03b9) \u2192 hom (pair i j) (single i)\n| right : (i j : \u03b9) \u2192 hom (pair i j) (single j)\n\nprotected instance hom_inhabited {\u03b9 : Type v} [Inhabited \u03b9] : Inhabited (hom (single Inhabited.default) (single Inhabited.default)) :=\n  { default := hom.id_single Inhabited.default }\n\n/--\nThe identity morphism in `pairwise \u03b9`.\n-/\ndef id {\u03b9 : Type v} (o : pairwise \u03b9) : hom o o :=\n  sorry\n\n/-- Composition of morphisms in `pairwise \u03b9`. -/\ndef comp {\u03b9 : Type v} {o\u2081 : pairwise \u03b9} {o\u2082 : pairwise \u03b9} {o\u2083 : pairwise \u03b9} (f : hom o\u2081 o\u2082) (g : hom o\u2082 o\u2083) : hom o\u2081 o\u2083 :=\n  sorry\n\nprotected instance category_theory.category {\u03b9 : Type v} : category (pairwise \u03b9) :=\n  category.mk\n\n/-- Auxilliary definition for `diagram`. -/\n@[simp] def diagram_obj {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] : pairwise \u03b9 \u2192 \u03b1 :=\n  sorry\n\n/-- Auxilliary definition for `diagram`. -/\n@[simp] def diagram_map {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] {o\u2081 : pairwise \u03b9} {o\u2082 : pairwise \u03b9} (f : o\u2081 \u27f6 o\u2082) : diagram_obj U o\u2081 \u27f6 diagram_obj U o\u2082 :=\n  sorry\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[semilattice_inf \u03b1]`, we obtain a functor `pairwise \u03b9 \u2964 \u03b1`,\nsending `single i` to `U i` and `pair i j` to `U i \u2293 U j`,\nand the morphisms to the obvious inequalities.\n-/\ndef diagram {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [semilattice_inf \u03b1] : pairwise \u03b9 \u2964 \u03b1 :=\n  functor.mk (diagram_obj U) fun (X Y : pairwise \u03b9) (f : X \u27f6 Y) => diagram_map U f\n\n-- `complete_lattice` is not really needed, as we only ever use `inf`,\n\n-- but the appropriate structure has not been defined.\n\n/-- Auxilliary definition for `cocone`. -/\ndef cocone_\u03b9_app {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] (o : pairwise \u03b9) : diagram_obj U o \u27f6 supr U :=\n  sorry\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[complete_lattice \u03b1]`,\n`supr U` provides a cocone over `diagram U`.\n-/\n@[simp] theorem cocone_X {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] : limits.cocone.X (cocone U) = supr U :=\n  Eq.refl (limits.cocone.X (cocone U))\n\n/--\nGiven a function `U : \u03b9 \u2192 \u03b1` for `[complete_lattice \u03b1]`,\n`infi U` provides a limit cone over `diagram U`.\n-/\ndef cocone_is_colimit {\u03b9 : Type v} {\u03b1 : Type v} (U : \u03b9 \u2192 \u03b1) [complete_lattice \u03b1] : limits.is_colimit (cocone U) :=\n  limits.is_colimit.mk fun (s : limits.cocone (diagram U)) => hom_of_le sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/pairwise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3765845363242132}}
{"text": "import logic.equiv.transfer_instance\nimport ring_theory.valuation.valuation_subring\nimport algebraic_geometry.prime_spectrum_more\nimport for_mathlib.local_ring\nimport for_mathlib.ideal\n\n\nvariables {K: Type*} [field K] (A : valuation_subring K)\n\nopen local_ring\n\ndef local_subring (K : Type*) [comm_ring K] : Type* :=\n{ s : subring K // local_ring s }\n\ninstance : partial_order (local_subring K) :=\n{ le := \u03bb A B, \u2203 h : A.1 \u2264 B.1, is_local_ring_hom (subring.inclusion h),\n  le_refl := \u03bb A, \u27e8rfl.le, \u27e8\u03bb a ha, by { cases a, exact ha }\u27e9\u27e9,\n  le_trans := \u03bb A B C \u27e8hAB, hAB'\u27e9 \u27e8hBC, hBC'\u27e9,\n    \u27e8hAB.trans hBC, @@is_local_ring_hom_comp _ _ _ _ _ hBC' hAB'\u27e9,\n  le_antisymm := \u03bb A B \u27e8hAB, hAB'\u27e9 \u27e8hBA, hBA'\u27e9, subtype.ext (hAB.antisymm hBA) }\n\ninstance : has_coe_to_sort (local_subring K) Type* := \u27e8\u03bb s, s.1\u27e9\ninstance local_subring.local_ring (A : local_subring K) : local_ring A := A.2\n\ndef valuation_subring.to_local_subring (A : valuation_subring K) : local_subring K :=\n\u27e8A.to_subring, show local_ring A, by apply_instance\u27e9 \n\nlemma valuation_subring.to_local_subring_injective : \n  function.injective (valuation_subring.to_local_subring : _ \u2192 local_subring K) :=\n\u03bb A B e, by { ext, change x \u2208 A.to_local_subring.1 \u2194 _, rw e, refl }\n\ndef maximal_local_subrings (K : Type*) [field K] : set (local_subring K) :=\nmaximals (\u2264) (set.univ : set $ local_subring K)\n\nopen_locale polynomial\n\nlemma maximal_ideal_map_eq_top_of_mem_maximal_local_subrings {R : local_subring K}\n  (hR : R \u2208 maximal_local_subrings K) {S : subring K} (hS : R.1 < S) :\n    (local_ring.maximal_ideal R).map (subring.inclusion hS.le) = \u22a4 :=\nbegin\n  replace hR := hR.2,\n  refine (not_not.mp $ \u03bb h, _),\n  obtain \u27e8m, hm, hm'\u27e9 := ideal.exists_le_maximal _ h,\n  haveI hm'' : m.comap (subring.inclusion hS.le) = local_ring.maximal_ideal R,\n  { symmetry, apply (local_ring.maximal_ideal.is_maximal _).eq_of_le,\n    { introI e, exact (ideal.is_prime.comap (subring.inclusion hS.le)).1 e },\n    { rwa \u2190 ideal.map_le_iff_le_comap } },\n  let f := @is_localization.lift _ _ m.prime_compl (localization.at_prime m) _ _ _ _ _\n    S.subtype (\u03bb \u27e8y, hy\u27e9, is_unit_iff_ne_zero.mpr $ \u03bb e, hy $ by\n    { obtain rfl : y = 0 := subtype.ext (e.trans S.subtype.map_zero.symm), exact m.zero_mem }),\n  have hf : function.injective f,\n  { rw injective_iff_map_eq_zero, intros x hx,\n    obtain \u27e8x, s, rfl\u27e9 := is_localization.mk'_surjective m.prime_compl x,\n    rw [is_localization.lift_mk', eq_comm, units.eq_mul_inv_iff_mul_eq, zero_mul] at hx,\n    obtain rfl : x = 0 := subtype.ext (hx.symm.trans S.subtype.map_zero.symm),\n    rw is_localization.mk'_zero },\n  let f' := ring_equiv.of_left_inverse hf.has_left_inverse.some_spec,\n  have : S \u2264 f.range,\n  { intros x hx, refine \u27e8is_localization.mk' _ (\u27e8x, hx\u27e9 : S) (1 : m.prime_compl), _\u27e9,\n    rw [is_localization.lift_mk', map_one, inv_one, units.coe_one, mul_one], refl },\n  have hS' := set_like.le_def.not.mp (not_le_of_lt hS),\n  push_neg at hS',\n  obtain \u27e8x, hx, hx'\u27e9 := hS',\n  refine hx' ((@hR \u27e8f.range, _\u27e9 trivial \u27e8hS.le.trans this, _\u27e9).some (this hx)),\n  { exact f'.local_ring },\n  { constructor, rintros y hy, \n    have : subring.inclusion (hS.le.trans this) y =\n      f' (algebra_map _ _ (subring.inclusion hS.le y)),\n    { ext1, change _ = f _, rw is_localization.lift_eq, refl },\n    rw [this, \u2190 ring_equiv.coe_to_ring_hom, is_unit_map_iff \u2191f',\n      is_localization.at_prime.is_unit_to_map_iff _ m] at hy, \n    apply not_not.mp,\n    change y \u2209 local_ring.maximal_ideal R,\n    rw \u2190 hm'',\n    exact hy,\n    all_goals { apply_instance } }\nend\n\nlemma algebra.mem_ideal_map_adjoin {R S : Type*} [comm_ring R] [comm_ring S] [algebra R S]\n  (x : S) (I : ideal R) {y : algebra.adjoin R ({x} : set S)} :\n  y \u2208 I.map (algebra_map R $ algebra.adjoin R ({x} : set S)) \u2194\n    \u2203 p : R[X], (\u2200 i, p.coeff i \u2208 I) \u2227 polynomial.aeval x p = y :=\nbegin\n  let f : R[X] \u2192\u2090[R] algebra.adjoin R ({x} : set S) :=\n    polynomial.aeval \u27e8x, algebra.self_mem_adjoin_singleton R x\u27e9,\n  have hf : f.to_ring_hom.comp polynomial.C = algebra_map _ _ := f.comp_algebra_map,\n  have : \u2200 p, (f p : S) = polynomial.aeval x p,\n  { intro p, change _ = polynomial.aeval ((algebra_map (algebra.adjoin R ({x} : set S))) S\n      \u27e8x, algebra.self_mem_adjoin_singleton R x\u27e9) p, rw polynomial.aeval_algebra_map_apply, refl },\n  rw [\u2190 hf, ideal.map_comp, ideal.mem_map_iff_of_surjective],\n  simp_rw [ideal.mem_map_C_iff, subtype.ext_iff, alg_hom.to_ring_hom_eq_coe,\n    alg_hom.coe_to_ring_hom, this],\n  rintro \u27e8y, hy\u27e9,\n  obtain \u27e8p, rfl\u27e9 : y \u2208 (polynomial.aeval x : R[X] \u2192\u2090[R] S).range :=\n    by rwa \u2190 algebra.adjoin_singleton_eq_range_aeval,\n  exact \u27e8p, subtype.ext (this _)\u27e9\nend\n\ninstance : has_mem K (local_subring K) := \u27e8\u03bb x K, x \u2208 K.1\u27e9\n\nlemma mem_of_mem_maximal_local_subrings_of_is_integral {R : local_subring K}\n  (hR : R \u2208 maximal_local_subrings K) {x : K} (hx : is_integral R x) : x \u2208 R :=\nbegin\n  by_contra hx',\n  have H : R.1 < (algebra.adjoin R {x}).to_subring,\n  { rw set_like.lt_iff_le_and_exists,\n    exact \u27e8\u03bb y hy, (algebra.adjoin R {x}).algebra_map_mem \u27e8y, hy\u27e9, x,\n      algebra.subset_adjoin (set.mem_singleton x), hx'\u27e9 },\n  have h\u2081 := maximal_ideal_map_eq_top_of_mem_maximal_local_subrings hR H,\n  have h\u2082 : algebra.is_integral R (algebra.adjoin R ({x} : set K)),\n  { rwa [\u2190 le_integral_closure_iff_is_integral, algebra.adjoin_le_iff,\n      set.singleton_subset_iff] },\n  obtain \u27e8p, hp\u27e9 := prime_spectrum.surjective_of_is_integral_of_injective _ h\u2082\n    (\u03bb x y e, by { ext, injection e }) (local_ring.closed_point _),\n  apply p.2.ne_top,\n  rw [eq_top_iff, \u2190 h\u2081, ideal.map_le_iff_le_comap],\n  exact le_of_eq (by injection hp.symm)\nend\n\nlemma local_subring.range_valuation_subring : \n  set.range valuation_subring.to_local_subring = maximal_local_subrings K :=\nbegin\n  ext A, split,\n  { rintro \u27e8A, rfl\u27e9,\n    refine \u27e8trivial, _\u27e9,\n    rintro B - \u27e8hB, hB'\u27e9,\n    apply eq.le,\n    refine subtype.ext (hB.antisymm _).symm,\n    intros x hx,\n    cases A.2 x, { assumption },\n    by_cases hx : x\u207b\u00b9 = 0, { rw inv_eq_zero.mp hx, exact zero_mem _ },\n    have := hB'.1 \u27e8_, h\u27e9,\n    rw [subring.is_unit_iff, subring.is_unit_iff, subtype.coe_mk, inv_inv] at this,\n    { apply this, show x\u207b\u00b9\u207b\u00b9 \u2208 B.1, rwa inv_inv },\n    { contrapose! hx, injection hx },\n    { contrapose! hx, injection hx } },\n  { rintros hA, refine \u27e8\u27e8A.1, \u03bb x, or_iff_not_imp_left.mpr $ \u03bb hx, _\u27e9, subtype.ext rfl\u27e9,\n    apply mem_of_mem_maximal_local_subrings_of_is_integral hA,\n    let A' := algebra.adjoin A ({x} : set K),\n    have hA' : A.1 \u2264 A'.to_subring := \u03bb x hx, A'.algebra_map_mem \u27e8x, hx\u27e9,\n    have hx' : x \u2208 A' := algebra.subset_adjoin (set.mem_singleton x),\n    have hx'' : x \u2260 0 := \u03bb e, hx (e.symm \u25b8 A.1.zero_mem),\n    have : (local_ring.maximal_ideal A).map (algebra_map A A') = \u22a4,\n    { refine maximal_ideal_map_eq_top_of_mem_maximal_local_subrings hA (lt_of_le_not_le hA' _),\n      exact \u03bb e, hx (e hx') }, \n    rw [ideal.eq_top_iff_one, algebra.mem_ideal_map_adjoin x (maximal_ideal A)] at this,\n    obtain \u27e8p, hp\u2081, hp\u2082\u27e9 := this,\n    have hp\u2083 : polynomial.aeval x (p - 1) = 0,\n    { rw [map_sub, polynomial.aeval_one, hp\u2082], exact sub_self 1 },\n    haveI : invertible x := invertible_of_nonzero hx'',\n    rw [polynomial.aeval_def, \u2190 polynomial.eval\u2082_reverse_eq_zero_iff, inv_of_eq_inv] at hp\u2083,\n    suffices hp\u2084 : is_unit (p - 1).reverse.leading_coeff,\n    { have := is_integral_leading_coeff_smul _ _ hp\u2083,\n      rw [algebra.smul_def, mul_comm] at this,\n      refine is_integral_of_is_integral_mul_unit _ this,\n      swap, rw [\u2190 map_mul, \u2190 (algebra_map A K).map_one], congr' 1, exact hp\u2084.coe_inv_mul },\n    have : (p - 1).nat_trailing_degree = 0,\n    { rw \u2190 le_zero_iff,\n      apply polynomial.nat_trailing_degree_le_of_ne_zero _,\n      rw [polynomial.coeff_sub, polynomial.coeff_one_zero, sub_ne_zero],\n      intro e,\n      apply (maximal_ideal.is_maximal A).ne_top,\n      rw [ideal.eq_top_iff_one, \u2190 e],\n      apply hp\u2081 },\n    rw [polynomial.reverse_leading_coeff, polynomial.trailing_coeff,\n      this, polynomial.coeff_sub, polynomial.coeff_one_zero, \u2190 is_unit.neg_iff, neg_sub],\n    exact (local_ring.is_unit_or_is_unit_of_add_one\n      $ sub_add_cancel 1 (p.coeff 0)).resolve_right (hp\u2081 0) }\nend\n\nlemma exists_valuation_subring_dominates (A : local_subring K) :\n  \u2203 (B : valuation_subring K), A \u2264 B.to_local_subring :=\nbegin\n  let S := { B : local_subring K | A \u2264 B },\n  suffices : \u2203 B \u2208 S, A \u2264 B \u2227 \u2200 B' \u2208 S, B \u2264 B' \u2192 B' = B,\n  { obtain \u27e8B, -, hB, hB'\u27e9 := this,\n    obtain \u27e8B, rfl\u27e9 : B \u2208 (set.range valuation_subring.to_local_subring : set (local_subring K)),\n    { rw local_subring.range_valuation_subring,\n      exact \u27e8trivial, \u03bb B' _ h, (hB' B' (le_trans hB h) h).le\u27e9 },\n    exact \u27e8B, hB\u27e9 },\n  apply zorn_nonempty_partial_order\u2080, swap, { exact le_refl _ },\n  intros c hc hc' B hB,\n  haveI : nonempty c := \u27e8\u27e8B, hB\u27e9\u27e9,\n  have hdir : directed has_le.le (\u03bb (i : c), i.1.1),\n  { apply is_chain.directed, intros i hi j hj e, apply or_of_or_of_imp_of_imp (hc' hi hj e),\n    all_goals { exact \u03bb x, x.some } },\n  let X : local_subring K := \u27e8\u2a06 i : c, i.1.1, \u27e8_\u27e9\u27e9,\n  swap,\n  { rintros \u27e8a, ha\u27e9 \u27e8b, hb\u27e9 hab,\n    rw subring.mem_supr_of_directed hdir at ha hb,\n    obtain \u27e8\u27e8i, hi\u27e9, \u27e8j, hj\u27e9\u27e9 := \u27e8ha, hb\u27e9,\n    obtain \u27e8k, hak, hbk\u27e9 := hdir i j,\n    have : k.1.1 \u2264 \u2a06 i : c, i.1.1 := le_supr (\u03bb i : c, i.1.1) k,\n    apply or_of_or_of_imp_of_imp (@@local_ring.is_unit_or_is_unit_of_add_one _ k.1.2\n      (show (\u27e8a, hak hi\u27e9 + \u27e8b, hbk hj\u27e9 : k.1.1) = 1, by { ext, injection hab })),\n    all_goals { exact (subring.inclusion this).is_unit_map } },\n  have : \u2200 (C : c), C.1 \u2264 X,\n  { intro C,\n    refine \u27e8le_supr _ C, \u27e8\u03bb x hx, _\u27e9\u27e9,\n    obtain \u27e8D, hD\u27e9 := (subring.mem_supr_of_directed hdir).mp (hx.unit\u207b\u00b9).1.prop,\n    obtain \u27e8E, hCE, hDE\u27e9 := hc'.directed C D,\n    apply hCE.some_spec.1,\n    refine is_unit_of_mul_eq_one _ \u27e8_, hDE.some hD\u27e9 _,\n    ext, injection hx.mul_coe_inv },\n  exact \u27e8X, le_trans (hc hB) (this \u27e8B, hB\u27e9), \u03bb C hC, this \u27e8C, hC\u27e9\u27e9\nend\n\nlemma bijective_range_restrict_comp_of_valuation_ring {R S K : Type*} [comm_ring R] \n  [is_domain R] [valuation_ring R] \n  [comm_ring S] [local_ring S] [field K] [algebra R K] [is_fraction_ring R K]\n  (f : R \u2192+* S) (g : S \u2192+* K) (h : g.comp f = algebra_map R K) [is_local_ring_hom f] : \n  function.bijective (g.range_restrict.comp f) :=\nbegin\n  haveI := local_ring.of_surjective' _ g.range_restrict_surjective,\n  haveI H := local_ring.of_surjective' _ (algebra_map R K).range_restrict_surjective,\n  have : (\u27e8(algebra_map R K).range, H\u27e9 : local_subring K) \u2208 maximal_local_subrings K,\n  { rw \u2190 @local_subring.range_valuation_subring K,\n    exact \u27e8\u27e8_, \u03bb x, valuation_ring.is_integer_or_is_integer R x\u27e9, subtype.ext rfl\u27e9 },\n  refine \u27e8\u03bb x y e, is_fraction_ring.injective R K _, _\u27e9,\n  { rw \u2190 h, exact (congr_arg subtype.val e : _) },\n  suffices : (show local_subring K, from \u27e8g.range, infer_instance\u27e9) \u2264 \u27e8(algebra_map R K).range, H\u27e9,\n  { intro x, obtain \u27e8y, e\u27e9 := this.some x.prop, use y, rw \u2190 h at e, ext1, exact e },\n  refine this.2 trivial \u27e8_, _\u27e9,\n  { rintro _ \u27e8x, rfl\u27e9, rw \u2190 h, exact \u27e8_, rfl\u27e9 },\n  { generalize_proofs h',\n    let e := ring_equiv.of_left_inverse (is_fraction_ring.injective R K).has_left_inverse.some_spec,\n    have : (subring.inclusion h') = (g.range_restrict.comp f).comp e.symm.to_ring_hom,\n    { ext x, obtain \u27e8x, rfl\u27e9 := e.surjective x, change _ = (g.comp f) (e.symm _),\n      rw [h, e.symm_apply_apply], refl },\n    rw this,\n    apply_with is_local_ring_hom_comp { instances := ff },\n    apply_with is_local_ring_hom_comp { instances := ff },\n    { exact is_local_ring_hom_of_surjective _ g.range_restrict_surjective },\n    { assumption },\n    { exact is_local_ring_hom_equiv e.symm } },\nend\n\nlemma exists_factor_valuation_ring {R : Type*} [comm_ring R] [local_ring R] {K : Type*} [field K] \n  (f : R \u2192+* K) :\n    \u2203 (A : valuation_subring K) h, is_local_ring_hom (f.cod_restrict A.to_subring h) :=\nbegin\n  obtain \u27e8B, hB, hB'\u27e9 := exists_valuation_subring_dominates \u27e8f.range,\n    local_ring.of_surjective' _ f.range_restrict_surjective\u27e9,\n  haveI := is_local_ring_hom_of_surjective _ f.range_restrict_surjective,\n  exact \u27e8B, \u03bb x, hB \u27e8x, rfl\u27e9, is_local_ring_hom_comp (subring.inclusion hB) f.range_restrict\u27e9\nend", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/for_mathlib/valuation_subring.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3765845363242132}}
{"text": "import phase1.A_map\nimport phase2.basic\n\n/-!\n# Constraints\nSupport conditions can be said to *constrain* each other in a number of ways.\nThis is detailed below. The `constrains` relation is well-founded.\n-/\n\nopen quiver set sum with_bot\nopen_locale classical\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}]\n\nsection extended_index\nvariable {\u03b1 : type_index}\n\n/-!\nWe construct a well-order on the type of extended indices.\nThe details are unimportant, we probably don't actually need AC here.\n-/\n\ninstance : has_lt (extended_index \u03b1) := \u27e8well_ordering_rel\u27e9\ninstance : is_well_order (extended_index \u03b1) (<) := well_ordering_rel.is_well_order\ninstance : has_well_founded (extended_index \u03b1) := is_well_order.to_has_well_founded\nnoncomputable instance : linear_order (extended_index \u03b1) := linear_order_of_STO (<)\n\nend extended_index\n\nvariables {\u03b1 : \u039b} [position_data.{}] [phase_2_assumptions \u03b1] {\u03b2 : \u039b}\n\nlemma coe_ne' {\u03b3 : Iio \u03b1} {\u03b2 : Iio \u03b1} : \u03b3 \u2260 \u03b2 \u2192 (\u03b3 : \u039b) \u2260 (\u03b2 : \u039b) :=\nby contrapose!; simp only [subtype.coe_inj, imp_self]\n\nlemma coe_lt {\u03b3 : Iio \u03b1} {\u03b2 : Iic \u03b1} : (\u03b3 : \u039b) < \u03b2 \u2192 (\u03b3 : type_index) < (\u03b2 : type_index) :=\nbegin\n  intro h,\n  cases \u03b2,\n  cases \u03b3,\n  exact coe_lt_coe.mpr h,\nend\n\nvariables (\u03b1) (\u03b2)\n\n/--\nSupport conditions can be said to *constrain* each other in a number of ways. This is discussed\nin the \"freedom of action discussion\".\n1. `\u27e8L, A\u27e9 \u227a \u27e8a, A\u27e9` when `a \u2208 L` and `L` is a litter. We can say that an atom is constrained by the\n    litter it belongs to.\n2. `\u27e8N\u00b0, A\u27e9 \u227a \u27e8N, A\u27e9` when `N` is a near-litter not equal to its corresponding litter `N\u00b0`.\n3. `\u27e8a, A\u27e9 \u227a \u27e8N, A\u27e9` for all `a \u2208 N \u2206 N\u00b0`.\n4. `\u27e8x, A \u226b (\u03b3 \u27f6 \u03b4) \u226b B\u27e9 \u227a \u27e8f_{\u03b3,\u03b4}(t), A \u226b (\u03b3 \u27f6 \u03b5) \u226b (\u03b5 \u27f6 \u22a5)\u27e9` for all paths `A : \u03b2 \u27f6 \u03b3` and\n    `\u03b4, \u03b5 < \u03b3` with `\u03b4 \u2260 \u03b5`, `t \u2208 \u03c4_\u03b3`, where `\u27e8x, B\u27e9` lies in the designated `\u03b4`-support of `t`.\nTODO: Refactor `near_litter` to use `\u00acN.is_litter`.\n-/\n@[mk_iff] inductive constrains : support_condition \u03b2 \u2192 support_condition \u03b2 \u2192 Prop\n| atom (a : atom) (A : extended_index \u03b2) :\n    constrains \u27e8inr a.1.to_near_litter, A\u27e9 \u27e8inl a, A\u27e9\n| near_litter (N : near_litter) (hN : litter_set N.fst \u2260 N.snd) (A : extended_index \u03b2) :\n    constrains \u27e8inr N.fst.to_near_litter, A\u27e9 \u27e8inr N, A\u27e9\n| symm_diff (N : near_litter) (a \u2208 litter_set N.fst \u2206 N.snd) (A : extended_index \u03b2) :\n    constrains \u27e8inl a, A\u27e9 \u27e8inr N, A\u27e9\n| f_map \u2983\u03b3 : Iic \u03b1\u2984 \u2983\u03b4 : Iio \u03b1\u2984 \u2983\u03b5 : Iio \u03b1\u2984 (h\u03b4 : (\u03b4 : \u039b) < \u03b3) (h\u03b5 : (\u03b5 : \u039b) < \u03b3) (h\u03b4\u03b5 : \u03b4 \u2260 \u03b5)\n    (A : path (\u03b2 : type_index) \u03b3) (t : tangle \u03b4) (c \u2208 (designated_support t).carrier) :\n    constrains\n      \u27e8c.fst, (A.cons (coe_lt h\u03b4)).comp c.snd\u27e9\n      \u27e8inr (f_map (coe_ne_coe.mpr $ coe_ne' h\u03b4\u03b5) t).to_near_litter,\n        (A.cons (coe_lt h\u03b5)).cons (bot_lt_coe _)\u27e9\n| f_map_bot \u2983\u03b3 : Iic \u03b1\u2984 \u2983\u03b5 : Iio \u03b1\u2984 (h\u03b5 : (\u03b5 : \u039b) < \u03b3)\n    (A : path (\u03b2 : type_index) \u03b3) (a : atom) :\n    constrains\n      \u27e8inl a, A.cons (bot_lt_coe _)\u27e9\n      \u27e8inr (f_map (show (\u22a5 : type_index) \u2260 (\u03b5 : \u039b), from bot_ne_coe) a).to_near_litter,\n        (A.cons (coe_lt h\u03b5)).cons (bot_lt_coe _)\u27e9\n\n/-! We declare new notation for the \"constrains\" relation on support conditions. -/\nnotation c ` \u227a[`:50 \u03b1 `] ` d:50 := constrains \u03b1 _ c d\n\ninstance : has_lt (support_condition \u03b2) :=\n\u27e8prod.lex (inv_image (<) (\u03bb c, c.elim typed_atom_position typed_near_litter_position)) (<)\u27e9\n\ninstance : is_well_founded (support_condition \u03b2) (<) :=\nprod.lex.is_well_founded\n\nlemma constrains_subrelation : subrelation (constrains \u03b1 \u03b2) (<) :=\nbegin\n  rintros c d h,\n  obtain (\u27e8a, A\u27e9 | \u27e8N, hN, A\u27e9 | \u27e8N, a, ha, A\u27e9 | \u27e8h\u03b4, h\u03b5, h\u03b4\u03b5, A, t, c, hc\u27e9 | \u27e8h\u03b4, A, a\u27e9) := h;\n  left,\n  { exact litter_lt a.1 a rfl, },\n  { refine litter_lt_near_litter N _,\n    contrapose! hN,\n    rw \u2190 hN,\n    refl, },\n  { exact symm_diff_lt_near_litter N a ha, },\n  { have := f_map_position (coe_ne_coe.mpr $ coe_ne' h\u03b4\u03b5) t _ (is_near_litter_litter_set _),\n    rw tangle_data.typed_near_litter_position_eq at this,\n    refine lt_of_le_of_lt _ this,\n    convert tangle_data.support_le (show tangle (h_\u03b4 : \u039b), from t) _ hc, },\n  { simp only [inv_image, elim_inr],\n    convert typed_atom_position_lt_f_map a,\n    rw tangle_data.typed_near_litter_position_eq (f_map bot_ne_coe a).to_near_litter,\n    apply_instance, },\nend\n\n/-- The `\u227a` relation is well-founded. By the conditions on orderings, if we have `\u27e8x, A\u27e9 \u227a \u27e8y, B\u27e9`,\nthen `x < y` in `\u00b5`, under the `typed_near_litter` or `typed_atom` maps. -/\nlemma constrains_wf : well_founded (constrains \u03b1 \u03b2) :=\nsubrelation.wf (constrains_subrelation \u03b1 \u03b2) (is_well_founded.to_has_well_founded _).wf\n\ninstance : has_well_founded (support_condition \u03b2) := \u27e8constrains \u03b1 \u03b2, constrains_wf \u03b1 \u03b2\u27e9\n\nvariable {\u03b1}\n\n@[simp] lemma constrains_atom {c : support_condition \u03b2} {a : atom} {A : extended_index \u03b2} :\n  c \u227a[\u03b1] \u27e8inl a, A\u27e9 \u2194 c = \u27e8inr a.1.to_near_litter, A\u27e9 :=\nbegin\n  split,\n  { rintro \u27e8\u27e9, refl, },\n  { rintro rfl, exact constrains.atom a A, },\nend\n\n/-- The constrains relation is stable under composition of paths. -/\nlemma constrains_comp {\u03b2 \u03b3 : \u039b} {c d : support_condition \u03b3} (h : c \u227a[\u03b1] d)\n  (B : path (\u03b2 : type_index) \u03b3) : \u27e8c.fst, B.comp c.snd\u27e9 \u227a[\u03b1] \u27e8d.fst, B.comp d.snd\u27e9 :=\nbegin\n  obtain (\u27e8a, A\u27e9 | \u27e8N, hN, A\u27e9 | \u27e8N, a, ha, A\u27e9 | \u27e8h\u03b4, h\u03b5, h\u03b4\u03b5, A, t, c, hc\u27e9 | \u27e8h\u03b4, A, a\u27e9) := h,\n  { exact constrains.atom _ _, },\n  { exact constrains.near_litter _ hN _, },\n  { exact constrains.symm_diff _ _ ha _, },\n  { rw [path.comp_cons, \u2190 path.comp_assoc, path.comp_cons],\n    exact constrains.f_map h\u03b4 h\u03b5 h\u03b4\u03b5 (B.comp A) t c hc, },\n  { rw path.comp_cons,\n    exact constrains.f_map_bot h\u03b4 (B.comp A) a, },\nend\n\nlemma refl_trans_gen_constrains_comp {\u03b2 \u03b3 : \u039b} {c d : support_condition \u03b3}\n  (h : relation.refl_trans_gen (constrains \u03b1 \u03b3) c d) (B : path (\u03b2 : type_index) \u03b3) :\n  relation.refl_trans_gen (constrains \u03b1 \u03b2) \u27e8c.fst, B.comp c.snd\u27e9 \u27e8d.fst, B.comp d.snd\u27e9 :=\nbegin\n  induction h with e f hce hef ih,\n  exact relation.refl_trans_gen.refl,\n  exact relation.refl_trans_gen.tail ih (constrains_comp hef B),\nend\n\nlemma trans_gen_constrains_comp {\u03b2 \u03b3 : \u039b} {c d : support_condition \u03b3}\n  (h : relation.trans_gen (constrains \u03b1 \u03b3) c d) (B : path (\u03b2 : type_index) \u03b3) :\n  relation.trans_gen (constrains \u03b1 \u03b2) \u27e8c.fst, B.comp c.snd\u27e9 \u27e8d.fst, B.comp d.snd\u27e9 :=\nbegin\n  induction h with e hce e f hce hef ih,\n  exact relation.trans_gen.single (constrains_comp hce B),\n  exact relation.trans_gen.tail ih (constrains_comp hef B),\nend\n\nlemma refl_trans_gen_near_litter {\u03b2 : \u039b} {N : near_litter} {B : extended_index \u03b2}\n  {c : support_condition \u03b2}\n  (h : relation.refl_trans_gen (constrains \u03b1 \u03b2) (inr N, B) c) :\n  relation.refl_trans_gen (constrains \u03b1 \u03b2) (inr N.1.to_near_litter, B) c :=\nbegin\n  by_cases h' : N.is_litter,\n  { obtain \u27e8L, rfl\u27e9 := h'.exists_litter_eq,\n    exact h, },\n  { exact relation.refl_trans_gen.head\n      (constrains.near_litter N (near_litter.not_is_litter h') B) h, },\nend\n\nlemma trans_gen_near_litter {\u03b2 : \u039b} {N : near_litter} {B : extended_index \u03b2}\n  {c : support_condition \u03b2}\n  (h : relation.trans_gen (constrains \u03b1 \u03b2) c (inr N.1.to_near_litter, B)) :\n  relation.trans_gen (constrains \u03b1 \u03b2) c (inr N, B) :=\nbegin\n  by_cases h' : N.is_litter,\n  { obtain \u27e8L, rfl\u27e9 := h'.exists_litter_eq,\n    exact h, },\n  { exact relation.trans_gen.tail h\n      (constrains.near_litter N (near_litter.not_is_litter h') B), },\nend\n\nlemma trans_gen_near_litter' {\u03b2 : \u039b} {N : near_litter} {B : extended_index \u03b2}\n  {c : support_condition \u03b2}\n  (h : relation.trans_gen (constrains \u03b1 \u03b2) (inr N, B) c) :\n  relation.trans_gen (constrains \u03b1 \u03b2) (inr N.1.to_near_litter, B) c :=\nbegin\n  by_cases h' : N.is_litter,\n  { obtain \u27e8L, rfl\u27e9 := h'.exists_litter_eq,\n    exact h, },\n  { exact relation.trans_gen.head\n      (constrains.near_litter N (near_litter.not_is_litter h') B) h, },\nend\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/constrains.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.37658452912299833}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.simulation_semantics.constructions.tracking_oracle\nimport computational_monads.simulation_semantics.simulate.subsingleton\n\n/-!\n# Stateless Oracles\n\nThis file defines a specific version of `tracking_oracle`, where the tracking isn't used.\nThis allows a specified function for responding to queries,\nwhile making no use of the internal state (which is left as a `unit` type).\nThis is used for example in coercing from a computation with one set of oracles\nto one with some superset of those oracles, using the simulation function to pass upwards.\n-/\n\nopen oracle_comp oracle_spec\n\nvariables {\u03b1 \u03b2 : Type} {spec spec' spec'' : oracle_spec}\n\n/-- Simulate a computation without making use of the internal state.\n  We use the `unit` type as the state in this case, so all possible states are equal.\n  Implemented as a `tracking_oracle` where the state isn't actually tracking anything -/\ndef stateless_oracle (spec spec' : oracle_spec)\n  (o : \u03a0 (i : spec.\u03b9), spec.domain i \u2192 oracle_comp spec' (spec.range i)) :\n  sim_oracle spec spec' unit :=\n\u27eao | \u03bb _ _ _ _, (), ()\u27eb\n\nnotation `\u27ea` o `\u27eb` := stateless_oracle _ _ o\n\nnamespace stateless_oracle\n\nvariables (oa : oracle_comp spec \u03b1)\n  (o : \u03a0 (i : spec.\u03b9), spec.domain i \u2192 oracle_comp spec' (spec.range i))\n  (o' : \u03a0 (i : spec.\u03b9), spec.domain i \u2192 oracle_comp spec'' (spec.range i))\n  (i : spec.\u03b9) (t : spec.domain i) (s s' : unit) (u : spec.range i)\n  (x : spec.domain i \u00d7 unit) (y : spec.range i \u00d7 unit)\n\n@[simp] lemma apply_eq : \u27eao\u27eb i x = o i x.1 >>= \u03bb u, return (u, ()) := by {cases x, refl}\n\ninstance decidable [\u2200 i x, (o i x).decidable] (x : spec.domain i \u00d7 unit) : (\u27eao\u27eb i x).decidable :=\ntracking_oracle.decidable o _ _ i x\n\nlemma simulate_eq_default_simulate : simulate \u27eao\u27eb oa s = default_simulate \u27eao\u27eb oa :=\nsimulate_eq_default_simulate \u27eao\u27eb oa s\n\nlemma simulate'_eq_default_simulate' : simulate' \u27eao\u27eb oa s = default_simulate' \u27eao\u27eb oa :=\nsimulate'_eq_default_simulate' \u27eao\u27eb oa s\n\nsection support\n\nlemma support_apply : (\u27eao\u27eb i x).support = prod.fst \u207b\u00b9' (o i x.1).support :=\nby simp only [apply_eq, support_bind_prod_mk_of_snd_subsingleton, set.image_id']\n\nlemma mem_support_apply_iff : y \u2208 (\u27eao\u27eb i (t, s)).support \u2194 y.1 \u2208 (o i t).support :=\nby cases y; simp only [apply_eq, support_bind, support_return, set.mem_Union, prod.mk.inj_iff,\n  set.mem_singleton_iff, eq_iff_true_of_subsingleton, and_true, exists_prop, exists_eq_right']\n\n/-- The `support` of `simulate` is the preimage of the support of `simulate'`,\nas there is only one possible internal state for the oracle. -/\nlemma support_simulate_eq_preimage_support_simulate' :\n  (simulate \u27eao\u27eb oa s).support = prod.fst \u207b\u00b9' (simulate' \u27eao\u27eb oa ()).support :=\nsupport_simulate_eq_preimage_support_simulate' \u27eao\u27eb oa s\n\n/-- If the oracle function can take on any possible output, simulation doesn't affect `support`. -/\nlemma support_simulate'_eq_support (h : \u2200 i t, (o i t).support = \u22a4) :\n  (simulate' \u27eao\u27eb oa s).support = oa.support :=\ntracking_oracle.support_simulate'_eq_support o _ _ oa s h\n\nlemma support_simulate_eq_preimage_support (h : \u2200 i t, (o i t).support = \u22a4) :\n  (simulate \u27eao\u27eb oa s).support = prod.fst \u207b\u00b9' oa.support :=\ntracking_oracle.support_simulate_eq_preimage_support_of_subsingleton o _ _ oa s h\n\nlemma support_simulate'_eq_support_simulate' (h : \u2200 i t, (o i t).support = (o' i t).support) :\n  (simulate' \u27eao\u27eb oa s).support = (simulate' \u27eao'\u27eb oa s').support :=\ntracking_oracle.support_simulate'_eq_support_simulate' o o' _ _ () () oa s s' h\n\nlemma support_simulate_eq_support_simulate (h : \u2200 i t, (o i t).support = (o' i t).support) :\n  (simulate \u27eao\u27eb oa s).support = (simulate \u27eao'\u27eb oa s').support :=\nsupport_simulate_eq_support_simulate_of_subsingleton oa \u27eao\u27eb \u27eao'\u27eb s s'\n  (\u03bb i t, by rw [support_apply, support_apply, h])\n\n@[simp] lemma mem_support_simulate_iff (y : \u03b1 \u00d7 unit) :\n  y \u2208 (simulate \u27eao\u27eb oa s).support \u2194 y.1 \u2208 (simulate' \u27eao\u27eb oa ()).support :=\nby rw [support_simulate_eq_preimage_support_simulate', set.mem_preimage]\n\nend support\n\nsection fin_support\n\nvariable [\u2200 i x, (o i x).decidable]\n\n-- TODO: this should generalize I think?\nlemma fin_support_apply : (\u27eao\u27eb i x).fin_support = finset.preimage (o i t).fin_support prod.fst\n  (\u03bb y hy z hz h, prod.eq_iff_fst_eq_snd_eq.2 \u27e8h, punit_eq _ _\u27e9) :=\nsorry\n\nlemma mem_fin_support_apply : y \u2208 (\u27eao\u27eb i x).fin_support \u2194 y.1 \u2208 (o i x.1).fin_support :=\nsorry\n\nend fin_support\n\nsection eval_dist\n\nlemma eval_dist_apply : \u2045\u27eao\u27eb i x\u2046 = \u2045o i x.1\u2046.map (\u03bb u, (u, ())) :=\nby rw [apply_eq, eval_dist_bind_return]\n\n/-- If the oracle responds uniformly to queries, then simulation doesn't affect `eval_dist`. -/\nlemma eval_dist_simulate'_eq_eval_dist\n  (h : \u2200 i t, \u2045o i t\u2046 = pmf.uniform_of_fintype (spec.range i)) : \u2045simulate' \u27eao\u27eb oa s\u2046 = \u2045oa\u2046 :=\ntracking_oracle.eval_dist_simulate'_eq_eval_dist o _ _ oa s h\n\nlemma eval_dist_simulate'_eq_eval_dist_simulate' (h : \u2200 i t, \u2045o i t\u2046 = \u2045o' i t\u2046) :\n  \u2045simulate' \u27eao\u27eb oa s\u2046 = \u2045simulate' \u27eao'\u27eb oa s'\u2046 :=\ntracking_oracle.eval_dist_simulate'_eq_eval_dist_simulate' o o' _ _ _ _ oa s s' h\n\n/-- The `eval_dist` of `simulate` is the result of mapping the `eval_dist` of `simulate'`\nunder the map adding on a default `()` value for the internal state. -/\nlemma eval_dist_simulate_eq_map_eval_dist_simulate' :\n  \u2045simulate \u27eao\u27eb oa s\u2046 = \u2045simulate' \u27eao\u27eb oa s\u2046.map (\u03bb x, (x, ())) :=\nby simp only [eval_dist_simulate_eq_map_eval_dist_simulate'_of_subsingleton, punit_eq s ()]\n\nlemma eval_dist_simulate_eq_eval_dist_simulate (h : \u2200 i t, \u2045o i t\u2046 = \u2045o' i t\u2046) :\n  \u2045simulate \u27eao\u27eb oa s\u2046 = \u2045simulate \u27eao'\u27eb oa s'\u2046 :=\nby simp only [eval_dist_simulate_eq_map_eval_dist_simulate',\n  eval_dist_simulate'_eq_eval_dist_simulate' oa o o' s s' h]\n\nlemma eval_dist_simulate_apply_eq_eval_dist_simulate'_apply (x : \u03b1 \u00d7 unit) :\n  \u2045simulate \u27eao\u27eb oa s\u2046 x = \u2045simulate' \u27eao\u27eb oa s\u2046 x.1 :=\neval_dist_simulate_apply_eq_eval_dist_simulate'_apply_of_subsingleton \u27eao\u27eb oa s x\n\nend eval_dist\n\nsection prob_event\n\nlemma prob_event_apply (e : set $ spec.range i \u00d7 unit) :\n  \u2045e | \u27eao\u27eb i x\u2046 = \u2045(\u03bb x, (x, ())) \u207b\u00b9' e | o i x.1\u2046 :=\nby rw [apply_eq, prob_event_bind_return]\n\n/-- If the oracle function responds uniformly, then simulation doesn't affect `prob_event`. -/\nlemma prob_event_simulate'_eq_prob_event\n  (h : \u2200 i t, \u2045o i t\u2046 = pmf.uniform_of_fintype (spec.range i)) (e : set \u03b1) :\n  \u2045e | simulate' \u27eao\u27eb oa s\u2046 = \u2045e | oa\u2046 :=\nprob_event_eq_of_eval_dist_eq (eval_dist_simulate'_eq_eval_dist oa o s h) e\n\nlemma prob_event_simulate'_eq_prob_event_simulate' (h : \u2200 i t, \u2045o i t\u2046 = \u2045o' i t\u2046) (e : set \u03b1) :\n  \u2045e | simulate' \u27eao\u27eb oa s\u2046 = \u2045e | simulate' \u27eao'\u27eb oa s'\u2046 :=\nprob_event_eq_of_eval_dist_eq (eval_dist_simulate'_eq_eval_dist_simulate' oa o o' s s' h) e\n\nlemma prob_event_simulate (e : set $ \u03b1 \u00d7 unit) :\n  \u2045e | simulate \u27eao\u27eb oa s\u2046 = \u2045prod.fst '' e | simulate' \u27eao\u27eb oa s\u2046 :=\nbegin\n  sorry\nend\n\nend prob_event\n\nend stateless_oracle\n\n\n\n-- More lemmas we can prove about `tracking_oracle` with the definition of the `stateless_oracle`\nnamespace tracking_oracle\n\nvariables {S S' : Type} (o o' : \u03a0 (i : spec.\u03b9), spec.domain i \u2192 oracle_comp spec' (spec.range i))\n  (update_state update_state': \u03a0 (s : S) (i : spec.\u03b9), spec.domain i \u2192 spec.range i \u2192 S)\n  (default_state default_state' s s' : S) (oa : oracle_comp spec \u03b1)\n\nsection support\n\n/-- The first output with a tracking oracle is independent of any of the tracking state -/\nlemma support_simulate'_eq_support_simulate'_stateless_oracle :\n  (simulate' \u27eao | update_state, default_state\u27eb oa s).support = (simulate' \u27eao\u27eb oa ()).support :=\nbegin\n  sorry\n  -- unfold stateless_oracle,\n  -- refine support_simulate'_eq_of_oracle_eq o update_state (\u03bb _ _ _ _, ()) default_state _ oa s _\nend\n\nend support\n\nsection distribution_semantics\n\n/-- The first output of a tracking oracle is equivalent to using just the stateless oracle -/\ntheorem simulate'_equiv_stateless_oracle :\n  simulate' \u27eao | update_state, default_state\u27eb oa s \u2243\u209a simulate' \u27eao\u27eb oa () :=\nbegin\n  sorry\n  -- induction oa using oracle_comp.induction_on with \u03b1 a \u03b1 \u03b2 oa ob hoa hob i t generalizing s,\n  -- { simp },\n  -- { let so := \u27eao|update_state, default_state\u27eb,\n  --   calc simulate' so (oa >>= ob) s\n  --     \u2243\u209a (simulate so oa s) >>= (\u03bb x, simulate' so (ob x.1) x.2) :\n              --simulate'_bind_equiv so oa ob s\n  --     ... \u2243\u209a (simulate so oa s) >>= (\u03bb x, simulate' \u27eao\u27eb (ob x.1) ()) :\n  --       bind_equiv_of_equiv_second _ (\u03bb a, (hob a.1 a.2))\n  --     ... \u2243\u209a (simulate' so oa s) >>= (\u03bb x, simulate' \u27eao\u27eb (ob x) ()) : by erw [bind_map_equiv]\n  --     ... \u2243\u209a (simulate' \u27eao\u27eb oa ()) >>= (\u03bb x, simulate' \u27eao\u27eb (ob x) ()) :\n  --       bind_equiv_of_equiv_first _ (hoa _)\n  --     ... \u2243\u209a (simulate \u27eao\u27eb oa ()) >>= (\u03bb x, simulate' \u27eao\u27eb (ob x.1) ()) : by erw [bind_map_equiv]\n  --     ... \u2243\u209a (simulate \u27eao\u27eb oa ()) >>= (\u03bb x, simulate' \u27eao\u27eb (ob x.1) x.2) :\n  --       by { congr, ext x, rw [punit_eq () x.2] }\n  --     ... \u2243\u209a simulate' \u27eao\u27eb (oa >>= ob) () : by rw [simulate'_bind_equiv] },\n  -- { simp_rw [simulate'_query_equiv, apply_eq, stateless_oracle.apply_eq, map_bind_equiv],\n  --   refine bind_equiv_of_equiv_second (o i t) _,\n  --   simp only [map_pure_equiv, eq_self_iff_true, forall_const] }\nend\n\n/-- The first ouptput of a tracking oracle is indepenedent of the actual tracking functions -/\nlemma simulate'_equiv_of_equiv (h : \u2200 i t, o i t \u2243\u209a o' i t) :\n  simulate' \u27eao | update_state, default_state\u27eb oa s \u2243\u209a\n    simulate' \u27eao' | update_state', default_state'\u27eb oa s' :=\ncalc simulate' \u27eao | update_state, default_state\u27eb oa s\n  \u2243\u209a simulate' \u27eao\u27eb oa () : simulate'_equiv_stateless_oracle o update_state default_state s oa\n  ... \u2243\u209a simulate' \u27eao'\u27eb oa () :\n    stateless_oracle.eval_dist_simulate'_eq_eval_dist_simulate' _ _ _ _ _ h\n  ... \u2243\u209a simulate' \u27eao' | update_state', default_state'\u27eb oa s' :\n    symm (simulate'_equiv_stateless_oracle o' update_state' default_state' _ _)\n\nend distribution_semantics\n\nend tracking_oracle", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/simulation_semantics/constructions/stateless_oracle.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.679178699175393, "lm_q1q2_score": 0.37658451912255897}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n\nBinder elimination\n\n! This file was ported from Lean 3 source module tactic.converter.binders\n! leanprover-community/mathlib commit d6814c584384ddf2825ff038e868451a7c956f31\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.CompleteLattice\n\nnamespace OldConv\n\nopen Tactic Monad\n\nunsafe instance : MonadFail old_conv :=\n  { old_conv.monad with fail := fun \u03b1 s => (fun r e => tactic.fail (to_fmt s) : old_conv \u03b1) }\n\nunsafe instance : HasMonadLift tactic old_conv :=\n  \u27e8fun \u03b1 => lift_tactic\u27e9\n\nunsafe instance (\u03b1 : Type) : Coe (tactic \u03b1) (old_conv \u03b1) :=\n  \u27e8monadLift\u27e9\n\nunsafe def current_relation : old_conv Name := fun r lhs => return \u27e8r, lhs, none\u27e9\n#align old_conv.current_relation old_conv.current_relation\n\nunsafe def head_beta : old_conv Unit := fun r e => do\n  let n \u2190 tactic.head_beta e\n  return \u27e8(), n, none\u27e9\n#align old_conv.head_beta old_conv.head_beta\n\n-- congr should forward data!\nunsafe def congr_arg : old_conv Unit \u2192 old_conv Unit :=\n  congr_core (return ())\n#align old_conv.congr_arg old_conv.congr_arg\n\nunsafe def congr_fun : old_conv Unit \u2192 old_conv Unit := fun c => congr_core c (return ())\n#align old_conv.congr_fun old_conv.congr_fun\n\nunsafe def congr_rule (congr : expr) (cs : List (List expr \u2192 old_conv Unit)) : old_conv Unit :=\n  fun r lhs => do\n  let meta_rhs \u2190 infer_type lhs >>= mk_meta_var\n  let t\n    \u2190-- is maybe overly restricted for `heq`\n        mk_app\n        r [lhs, meta_rhs]\n  let ((), meta_pr) \u2190\n    solve_aux t do\n        apply congr\n        focus <|\n            cs fun c => do\n              let xs \u2190 intros\n              conversion (head_beta >> c xs)\n        done\n  let rhs \u2190 instantiate_mvars meta_rhs\n  let pr \u2190 instantiate_mvars meta_pr\n  return \u27e8(), rhs, some pr\u27e9\n#align old_conv.congr_rule old_conv.congr_rule\n\nunsafe def congr_binder (congr : Name) (cs : expr \u2192 old_conv Unit) : old_conv Unit := do\n  let e \u2190 mk_const congr\n  congr_rule e\n      [fun bs => do\n        let [b] \u2190 return bs\n        cs b]\n#align old_conv.congr_binder old_conv.congr_binder\n\nunsafe def funext' : (expr \u2192 old_conv Unit) \u2192 old_conv Unit :=\n  congr_binder `` _root_.funext\n#align old_conv.funext' old_conv.funext'\n\nunsafe def propext' {\u03b1 : Type} (c : old_conv \u03b1) : old_conv \u03b1 := fun r lhs =>\n  (do\n      guard (r = `iff)\n      c r lhs) <|>\n    do\n    guard (r = `eq)\n    let \u27e8res, rhs, pr\u27e9 \u2190 c `iff lhs\n    match pr with\n      | some pr => return \u27e8res, rhs, (expr.const `propext [] : expr) lhs rhs pr\u27e9\n      | none => return \u27e8res, rhs, none\u27e9\n#align old_conv.propext' old_conv.propext'\n\nunsafe def apply (pr : expr) : old_conv Unit := fun r e => do\n  let sl \u2190 simp_lemmas.mk.add pr\n  apply_lemmas sl r e\n#align old_conv.apply old_conv.apply\n\nunsafe def applyc (n : Name) : old_conv Unit := fun r e => do\n  let sl \u2190 simp_lemmas.mk.add_simp n\n  apply_lemmas sl r e\n#align old_conv.applyc old_conv.applyc\n\nunsafe def apply' (n : Name) : old_conv Unit := do\n  let e \u2190 mk_const n\n  congr_rule e []\n#align old_conv.apply' old_conv.apply'\n\nend OldConv\n\nopen Expr Tactic OldConv\n\n/- Binder elimination:\n\nWe assume a binder `B : p \u2192 \u03a0 (\u03b1 : Sort u), (\u03b1 \u2192 t) \u2192 t`, where `t` is a type depending on `p`.\nExamples:\n  \u2203: there is no `p` and `t` is `Prop`.\n  \u2a05, \u2a06: here p is `\u03b2` and `[complete_lattice \u03b2]`, `p` is `\u03b2`\n\nProblem: \u2200x, _ should be a binder, but is not a constant!\n\nProvide a mechanism to rewrite:\n\n  B (x : \u03b1) ..x.. (h : x = t), p x  =  B ..x/t.., p t\n\nHere ..x.. are binders, maybe also some constants which provide commutativity rules with `B`.\n\n-/\nunsafe structure binder_eq_elim where\n  match_binder : expr \u2192 tactic (expr \u00d7 expr)\n  -- returns the bound type and body\n  adapt_rel : old_conv Unit \u2192 old_conv Unit\n  -- optionally adapt `eq` to `iff`\n  apply_comm : old_conv Unit\n  -- apply commutativity rule\n  applyCongr : (expr \u2192 old_conv Unit) \u2192 old_conv Unit\n  -- apply congruence rule\n  apply_elim_eq : old_conv Unit\n#align binder_eq_elim binder_eq_elim\n\n-- (B (x : \u03b2) (h : x = t), s x) = s t\nunsafe def binder_eq_elim.check_eq (b : binder_eq_elim) (x : expr) : expr \u2192 tactic Unit\n  | q(@Eq $(\u03b2) $(l) $(r)) => guard (l = x \u2227 \u00acx.occurs r \u2228 r = x \u2227 \u00acx.occurs l)\n  | _ => fail \"no match\"\n#align binder_eq_elim.check_eq binder_eq_elim.check_eq\n\nunsafe def binder_eq_elim.pull (b : binder_eq_elim) (x : expr) : old_conv Unit := do\n  let (\u03b2, f) \u2190 lhs >>= lift_tactic \u2218 b.match_binder\n  guard \u00acx \u03b2 <|>\n      b x \u03b2 <|> do\n        b fun x => binder_eq_elim.pull\n        b\n#align binder_eq_elim.pull binder_eq_elim.pull\n\nunsafe def binder_eq_elim.push (b : binder_eq_elim) : old_conv Unit :=\n  b.apply_elim_eq <|>\n    (do\n        b\n        b fun x => binder_eq_elim.push) <|>\n      do\n      b <| b\n      binder_eq_elim.push\n#align binder_eq_elim.push binder_eq_elim.push\n\nunsafe def binder_eq_elim.check (b : binder_eq_elim) (x : expr) : expr \u2192 tactic Unit\n  | e => do\n    let (\u03b2, f) \u2190 b.match_binder e\n    b x \u03b2 <|> do\n        let lam n bi d bd \u2190 return f\n        let x \u2190 mk_local' n bi d\n        binder_eq_elim.check <| bd x\n#align binder_eq_elim.check binder_eq_elim.check\n\nunsafe def binder_eq_elim.old_conv (b : binder_eq_elim) : old_conv Unit := do\n  let (\u03b2, f) \u2190 lhs >>= lift_tactic \u2218 b.match_binder\n  let lam n bi d bd \u2190 return f\n  let x \u2190 mk_local' n bi d\n  b x (bd x)\n  b b\n#align binder_eq_elim.old_conv binder_eq_elim.old_conv\n\ntheorem exists_elim_eq_left.{u, v} {\u03b1 : Sort u} (a : \u03b1) (p : \u2200 a' : \u03b1, a' = a \u2192 Prop) :\n    (\u2203 (a' : \u03b1)(h : a' = a), p a' h) \u2194 p a rfl :=\n  \u27e8fun \u27e8a', \u27e8h, p_h\u27e9\u27e9 =>\n    match a', h, p_h with\n    | _, rfl, h => h,\n    fun h => \u27e8a, rfl, h\u27e9\u27e9\n#align exists_elim_eq_left exists_elim_eq_left\n\ntheorem exists_elim_eq_right.{u, v} {\u03b1 : Sort u} (a : \u03b1) (p : \u2200 a' : \u03b1, a = a' \u2192 Prop) :\n    (\u2203 (a' : \u03b1)(h : a = a'), p a' h) \u2194 p a rfl :=\n  \u27e8fun \u27e8a', \u27e8h, p_h\u27e9\u27e9 =>\n    match a', h, p_h with\n    | _, rfl, h => h,\n    fun h => \u27e8a, rfl, h\u27e9\u27e9\n#align exists_elim_eq_right exists_elim_eq_right\n\nunsafe def exists_eq_elim : binder_eq_elim\n    where\n  match_binder e := do\n    let q(@Exists $(\u03b2) $(f)) \u2190 return e\n    return (\u03b2, f)\n  adapt_rel := propext'\n  apply_comm := applyc `` exists_comm\n  applyCongr := congr_binder `` exists_congr\n  apply_elim_eq := apply' `` exists_elim_eq_left <|> apply' `` exists_elim_eq_right\n#align exists_eq_elim exists_eq_elim\n\ntheorem forall_comm.{u, v} {\u03b1 : Sort u} {\u03b2 : Sort v} (p : \u03b1 \u2192 \u03b2 \u2192 Prop) :\n    (\u2200 a b, p a b) \u2194 \u2200 b a, p a b :=\n  \u27e8fun h b a => h a b, fun h b a => h a b\u27e9\n#align forall_comm forall_comm\n\ntheorem forall_elim_eq_left.{u, v} {\u03b1 : Sort u} (a : \u03b1) (p : \u2200 a' : \u03b1, a' = a \u2192 Prop) :\n    (\u2200 (a' : \u03b1) (h : a' = a), p a' h) \u2194 p a rfl :=\n  \u27e8fun h => h a rfl, fun h a' h_eq =>\n    match a', h_eq with\n    | _, rfl => h\u27e9\n#align forall_elim_eq_left forall_elim_eq_left\n\ntheorem forall_elim_eq_right.{u, v} {\u03b1 : Sort u} (a : \u03b1) (p : \u2200 a' : \u03b1, a = a' \u2192 Prop) :\n    (\u2200 (a' : \u03b1) (h : a = a'), p a' h) \u2194 p a rfl :=\n  \u27e8fun h => h a rfl, fun h a' h_eq =>\n    match a', h_eq with\n    | _, rfl => h\u27e9\n#align forall_elim_eq_right forall_elim_eq_right\n\nunsafe def forall_eq_elim : binder_eq_elim\n    where\n  match_binder e := do\n    let expr.pi n bi d bd \u2190 return e\n    return (d, expr.lam n bi d bd)\n  adapt_rel := propext'\n  apply_comm := applyc `` forall_comm\n  applyCongr := congr_binder `` forall_congr'\n  apply_elim_eq := apply' `` forall_elim_eq_left <|> apply' `` forall_elim_eq_right\n#align forall_eq_elim forall_eq_elim\n\nunsafe def supr_eq_elim : binder_eq_elim\n    where\n  match_binder e := do\n    let q(@sup\u1d62 $(\u03b1) $(cl) $(\u03b2) $(f)) \u2190 return e\n    return (\u03b2, f)\n  adapt_rel c := do\n    let r \u2190 current_relation\n    guard (r = `eq)\n    c\n  apply_comm := applyc `` sup\u1d62_comm\n  applyCongr := congr_arg \u2218 funext'\n  apply_elim_eq := applyc `` sup\u1d62_sup\u1d62_eq_left <|> applyc `` sup\u1d62_sup\u1d62_eq_right\n#align supr_eq_elim supr_eq_elim\n\nunsafe def infi_eq_elim : binder_eq_elim\n    where\n  match_binder e := do\n    let q(@inf\u1d62 $(\u03b1) $(cl) $(\u03b2) $(f)) \u2190 return e\n    return (\u03b2, f)\n  adapt_rel c := do\n    let r \u2190 current_relation\n    guard (r = `eq)\n    c\n  apply_comm := applyc `` inf\u1d62_comm\n  applyCongr := congr_arg \u2218 funext'\n  apply_elim_eq := applyc `` inf\u1d62_inf\u1d62_eq_left <|> applyc `` inf\u1d62_inf\u1d62_eq_right\n#align infi_eq_elim infi_eq_elim\n\nuniverse u v w w\u2082\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v} {\u03b9 : Sort w} {\u03b9\u2082 : Sort w\u2082} {s t : Set \u03b1} {a : \u03b1}\n\nsection\n\nvariable [CompleteLattice \u03b1]\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic old_conv.conversion -/\nexample {s : Set \u03b2} {f : \u03b2 \u2192 \u03b1} : inf\u209b (Set.image f s) = \u2a05 a \u2208 s, f a :=\n  by\n  simp [inf\u209b_eq_inf\u1d62, inf\u1d62_and]\n  run_tac\n    conversion infi_eq_elim.old_conv\n\n/- ./././Mathport/Syntax/Translate/Tactic/Builtin.lean:69:18: unsupported non-interactive tactic old_conv.conversion -/\nexample {s : Set \u03b2} {f : \u03b2 \u2192 \u03b1} : sup\u209b (Set.image f s) = \u2a06 a \u2208 s, f a :=\n  by\n  simp [sup\u209b_eq_sup\u1d62, sup\u1d62_and]\n  run_tac\n    conversion supr_eq_elim.old_conv\n\nend\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Converter/Binders.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.37656711411466154}}
{"text": "import topology.category.Profinite.as_limit\nimport topology.continuous_function.algebra\nimport locally_constant.SemiNormedGroup\nimport locally_constant.completion\nimport analysis.special_functions.pow\nimport topology.algebra.module.weak_dual\nimport analysis.mean_inequalities_pow\nimport real_measures.condensed\nimport for_mathlib.is_locally_constant\n\nopen_locale nnreal big_operators classical\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen topological_space\n\nlocal attribute [instance]\n  locally_constant.seminormed_add_comm_group\n  locally_constant.pseudo_metric_space\n\nlemma real.pow_nnnorm_sum_le\n  {\u03b9 : Type*} (S : finset \u03b9) (r : \u03b9 \u2192 \u211d)\n  (p : \u211d\u22650) [fact (0 < p)] [fact (p \u2264 1)] :\n  \u2225 \u2211 i in S, r i \u2225\u208a^(p : \u211d) \u2264 \u2211 i in S, \u2225 r i \u2225\u208a^(p : \u211d) :=\nbegin\n  refine finset.le_sum_of_subadditive (\u03bb x : \u211d, \u2225 x \u2225\u208a^(p : \u211d)) _ _\n    S r,\n  { simp only [nnnorm_zero, nnreal.rpow_eq_zero_iff, eq_self_iff_true, ne.def,\n      nnreal.coe_eq_zero, true_and],\n    exact ne_of_gt (fact.out _) },\n  { intros x y,\n    dsimp,\n    refine le_trans _ (nnreal.rpow_add_le_add_rpow _ _ (nnreal.coe_pos.mpr (fact.out _)).le _),\n    rw nnreal.rpow_le_rpow_iff (nnreal.coe_pos.mpr (fact.out _) : 0 < \u2191p),\n    apply nnnorm_add_le,\n    exact (nnreal.coe_le_coe.mpr (fact.out _)).trans nnreal.coe_one.le }\nend\n\nnamespace locally_constant\n\ninstance normed_space (X : Type*)\n  [topological_space X] [compact_space X] :\n  normed_space \u211d (locally_constant X \u211d) :=\n{ norm_smul_le := \u03bb a f, by simpa only [norm_def, coe_smul, pi.smul_apply, algebra.id.smul_eq_mul,\n    real.norm_eq_abs, abs_mul, mul_comm (abs a)] using\n      (real.supr_mul_of_nonneg (abs_nonneg _) _).symm.le,\n  ..(infer_instance : module \u211d _) }\n\nlemma nnnorm_apply_le_nnnorm (X : Type*)\n  [topological_space X] [compact_space X]\n  (e : locally_constant X \u211d) (x : X) :\n  \u2225 e x \u2225\u208a \u2264 \u2225 e \u2225\u208a :=\nbegin\n  change _ \u2264 Sup _,\n  apply le_cSup,\n  let S := _, change bdd_above S, suffices : is_compact S, apply is_compact.bdd_above this,\n  apply is_compact_range, refine continuous.comp _ e.continuous, exact continuous_norm,\n  use x, refl,\nend\n\n/--  The evaluation map at a point of a topological space, bundled as a linear map from\nlocally constant functions to the real numbers. -/\ndef linear_eval (X : Type*)\n  [topological_space X] (x : X) :\n  locally_constant X \u211d \u2192\u2097[\u211d] \u211d :=\n{ to_fun := \u03bb e, e x,\n  map_add' := \u03bb f g, rfl,\n  map_smul' := \u03bb r f, rfl }\n\n\nlemma continuous_eval (X : Type*)\n  [topological_space X] [compact_space X] (x : X) :\n  continuous (\u03bb e : locally_constant X \u211d, e x) :=\nbegin\n  change continuous (linear_eval X x),\n  let E := linear_map.mk_continuous_of_exists_bound (linear_eval X x) _,\n  swap,\n  use 1, intros e, rw one_mul, apply nnnorm_apply_le_nnnorm,\n  exact E.continuous\nend\n\nend locally_constant\n\nnamespace topological_space.clopens\n\nlemma indicator_continuous {X Y : Type*} [topological_space X] [topological_space Y] [has_zero Y]\n  (U : clopens X) (f : X \u2192 Y) (hf : continuous f) :\n  continuous (set.indicator (U : set X) f) :=\nbegin\n  constructor, intros V hV,\n  set W : set X := (U : set X).indicator f \u207b\u00b9' V,\n  by_cases h0 : (0:Y) \u2208 V,\n  { suffices : W = f \u207b\u00b9' V \u222a U\u1d9c,\n    { rw this, exact (hV.preimage hf).union U.clopen.compl.is_open },\n    classical, ext x,\n    simp only [set.mem_preimage, set.mem_union, set.mem_compl_iff, set_like.mem_coe,\n      set.indicator_apply],\n    split_ifs with hxU,\n    { simp only [hxU, not_true, or_false] },\n    { simp only [h0, hxU, true_iff, not_false_iff, or_true], }, },\n  { suffices : W = f \u207b\u00b9' V \u2229 U,\n    { rw this, exact (hV.preimage hf).inter U.clopen.is_open },\n    classical, ext x,\n    simp only [set.mem_preimage, set.mem_union, set.mem_compl_iff, set_like.mem_coe,\n      set.indicator_apply],\n    split_ifs with hxU,\n    { simp only [hxU, set.mem_inter_iff, set.mem_preimage, set_like.mem_coe, and_true] },\n    { simp only [h0, false_iff, set.mem_inter_iff, set.mem_preimage, set_like.mem_coe, not_and],\n      intro, assumption, } }\nend\n\n/--  The indicator function on clopen sets are continuous functions. -/\ndef indicator {X : Type*} [topological_space X] (U : clopens X) :\n  C(X,\u211d) :=\n{ to_fun := set.indicator U 1,\n  continuous_to_fun := indicator_continuous _ _ continuous_one }\n\nlemma indicator_one_inverse_image {X : Type*} (U : set X) (s : set \u211d) :\n  (U.indicator 1 \u207b\u00b9' s) = set.univ \u2228 (U.indicator 1 \u207b\u00b9' s) = U \u2228\n  (U.indicator 1 \u207b\u00b9' s) = U\u1d9c \u2228 (U.indicator 1 \u207b\u00b9' s) = \u2205 :=\nbegin\n  by_cases s1 : (1 : \u211d) \u2208 s;\n  by_cases s0 : (0 : \u211d) \u2208 s,\n  work_on_goal 1 { refine or.inl _},\n  work_on_goal 2 { refine or.inr (or.inl _) },\n  work_on_goal 3 { refine or.inr (or.inr (or.inl _)) },\n  work_on_goal 4 { refine or.inr (or.inr (or.inr _)) },\n  all_goals\n  { ext x,\n    by_cases xU : x \u2208 U,\n    { simp only [xU, s1, set.mem_preimage, set.indicator_of_mem, pi.one_apply, set.mem_compl_iff,\n        not_true, set.mem_univ, set.mem_empty_iff_false] },\n    { simp only [xU, s0, set.mem_preimage, set.indicator_of_not_mem, not_false_iff,\n        set.mem_compl_iff, set.mem_univ, set.mem_empty_iff_false] } }\nend\n\n/--  The indicator function of a clopen set, bundled as a locally constant function. -/\ndef indicator_LC {X : Type*} [topological_space X] (U : clopens X) :\n  locally_constant X \u211d :=\n{ to_fun := set.indicator U 1,\n  is_locally_constant := \u03bb s, begin\n    rcases indicator_one_inverse_image \u2191U s with h | h | h | h;\n    rw h,\n    exacts [is_open_univ, U.clopen.is_open, (U\u1d9c).clopen.is_open, is_open_empty]\n  end }\n\nlemma indicator_apply {X : Type*} [topological_space X] (U : clopens X) (x) :\n  U.indicator x = if x \u2208 U then 1 else 0 := rfl\n\nlemma indicator_LC_apply {X : Type*} [topological_space X] (U : clopens X) (x) :\n  U.indicator_LC x = if x \u2208 U then 1 else 0 := rfl\n\nend topological_space.clopens\n\nnamespace discrete_quotient\n\n/--  Given a discrete topological space `T` that is a quotient of a topological space `X`, and\nan element `t : T`, `fiber T t` is the clopen subset of `X` that is the inverse image of `t` under\nthe quotient map `X \u2192 T`. -/\ndef fibre {X : Type*} [topological_space X] (T : discrete_quotient X)\n  (t : T) : clopens X :=\n{ carrier := T.proj \u207b\u00b9' {t},\n  clopen' := fiber_clopen T {t} }\n\n/--  When the topological space `X` is discrete, `equiv_bot` is the equivalence between `X` and the\nbottom element of the discrete quotients of `X`.  In other words, the identity map from `X` to\nitself *is*  a discrete quotient and `equiv_bot` is this bijection. -/\ndef equiv_bot {X : Type*} [topological_space X] [discrete_topology X] :\n  X \u2243 (\u22a5 : discrete_quotient X) :=\nequiv.of_bijective (discrete_quotient.proj _)\n\u27e8\u03bb x y h, quotient.exact' h, discrete_quotient.proj_surjective _\u27e9\n\nlemma mem_fibre_iff {X : Type*} [topological_space X]\n  (T : discrete_quotient X) (a : T) (b : X) :\n  T.proj b \u2208 discrete_quotient.fibre _ (equiv_bot a) \u2194\n  b \u2208 discrete_quotient.fibre T a :=\nbegin\n  obtain \u27e8a,rfl\u27e9 := discrete_quotient.proj_surjective _ a,\n  dsimp [fibre, equiv_bot],\n  let TT : discrete_quotient T := \u22a5,\n  change T.proj b \u2208 equiv_bot \u207b\u00b9' {equiv_bot (T.proj a)} \u2194 T.proj b \u2208 {T.proj a},\n  simp,\nend\n\nlemma mem_fibre_iff' {X : Type*} [topological_space X]\n  (T : discrete_quotient X) (a : (\u22a5 : discrete_quotient T)) (b : X) :\n  T.proj b \u2208 discrete_quotient.fibre _ a \u2194\n  b \u2208 discrete_quotient.fibre T (equiv_bot.symm a) :=\nbegin\n  rw [\u2190 equiv_bot.apply_symm_apply a, mem_fibre_iff],\n  simp,\nend\n\n/--  Given a continuous (`hf`), surjective (`hf'`) function `f : X \u2192 Y` between topological spaces\n`X` and `Y`, and a discrete quotient `T` of `Y`, `comap_equiv f hf hf' T` is the bijection between\nbetween the  comap of `T` along `f` and `T` itself.\n-/\ndef comap_equiv {X Y : Type*} [topological_space X] [topological_space Y]\n  (f : X \u2192 Y) (hf : continuous f) (hf' : function.surjective f)\n  (T : discrete_quotient Y) :\n  T.comap hf \u2243 T :=\nequiv.of_bijective (discrete_quotient.map $ le_refl _)\nbegin\n  split,\n  { rintros \u27e8x\u27e9 \u27e8y\u27e9 h,\n    apply quotient.sound',\n    apply quotient.exact' h },\n  { rintros \u27e8x\u27e9,\n    obtain \u27e8x,rfl\u27e9 := hf' x,\n    use discrete_quotient.proj _ x, refl }\nend\n\nlemma comap_mem_fibre_iff {X Y : Type*} [topological_space X] [topological_space Y]\n  (f : X \u2192 Y) (hf : continuous f)\n  (T : discrete_quotient Y) (a : T.comap hf) (b : X) :\n  b \u2208 discrete_quotient.fibre (T.comap hf) a \u2194\n  f b \u2208 discrete_quotient.fibre T (discrete_quotient.map (le_refl _) a) :=\nbegin\n  dsimp [fibre],\n  change b \u2208 (T.comap hf).proj \u207b\u00b9' {a} \u2194\n    f b \u2208 T.proj \u207b\u00b9' {_},\n  obtain \u27e8a,rfl\u27e9 := discrete_quotient.proj_surjective _ a,\n  simp only [set.mem_preimage, set.mem_singleton_iff, map_proj_apply],\n  split,\n  { intros h, apply quotient.sound', apply quotient.exact' h, },\n  { intros h, apply quotient.sound', apply quotient.exact' h, },\nend\n\nend discrete_quotient\n\nlemma locally_constant.sum_apply {\u03b9 X Y : Type*} [topological_space X] [add_comm_monoid Y]\n  (f : \u03b9 \u2192 locally_constant X Y) (S : finset \u03b9) (t : X) :\n  (\u2211 i in S, f i) t = \u2211 i in S, (f i t) :=\nbegin\n  let ee : locally_constant X Y \u2192+ X \u2192 Y := locally_constant.coe_fn_add_monoid_hom,\n  change (ee (\u2211 i in S, f i)) t = \u2211 i in S, (ee (f i) t),\n  rw [ee.map_sum, finset.sum_apply],\nend\n\nlemma locally_constant.eq_sum {X : Type*} [topological_space X] [compact_space X]\n  (e : locally_constant X \u211d) :\n  e = \u2211 t : e.discrete_quotient,\n    e.locally_constant_lift t \u2022 (e.discrete_quotient.fibre t).indicator_LC :=\nbegin\n  ext t,\n  simp_rw [locally_constant.sum_apply, locally_constant.smul_apply],\n  suffices :\n    \u2211 (x : \u21a5(e.discrete_quotient)),\n      (e.locally_constant_lift) x \u2022 ((e.discrete_quotient.fibre x).indicator_LC) t =\n    \u2211 x in { e.discrete_quotient.proj t },\n      (e.locally_constant_lift) x \u2022 ((e.discrete_quotient.fibre x).indicator_LC) t,\n  { simp_rw [this, algebra.id.smul_eq_mul, finset.sum_singleton],\n    convert (mul_one _).symm,\n    rw [topological_space.clopens.indicator_LC_apply, if_pos],\n    change e.discrete_quotient.proj t \u2208 _, simp, -- clopens is missing some `mem_mk` lemma...\n  },\n  symmetry,\n  apply finset.sum_subset, simp only [finset.subset_univ],\n  intros s _ ht,\n  convert smul_zero _,\n  rw [topological_space.clopens.indicator_LC_apply, if_neg],\n  contrapose! ht,\n  change e.discrete_quotient.proj t \u2208 _ at ht,\n  exact finset.mem_singleton.mpr (set.mem_singleton_iff.mp ht).symm,\nend\n\n/--  Given a point `x : X` is a discrete topological space `X`,\n`topological_space.clopens.singleton x` is the point `x`, bundled as a clopen subset of `X`. -/\ndef topological_space.clopens.singleton {X : Type*}\n  [topological_space X] [discrete_topology X] (x : X) :\n  clopens X :=\n{ carrier := {x},\n  clopen' := is_clopen_discrete {x} }\n\nlemma locally_constant.eq_sum_of_fintype {X : Type*} [fintype X]\n  [topological_space X] [discrete_topology X]\n  (e : locally_constant X \u211d) :\n  e =\n  \u2211 t : X, e t \u2022 (topological_space.clopens.singleton t).indicator_LC :=\nbegin\n  ext t,\n  rw locally_constant.sum_apply,\n  rw finset.sum_eq_single t,\n  { change _ = _ \u2022 ite _ _ _,\n    rw [if_pos, smul_eq_mul], erw mul_one,\n    change _ = _, refl },\n  { intros x _ hx,\n    change _ \u2022 ite _ _ _ = _,\n    rw [if_neg, smul_zero], change _ \u2260 _, exact hx.symm },\n  { intros h, exfalso, apply h, exact finset.mem_univ _ }\nend\n\n/--  Given a continuous function `f : X \u2192 Y` between topological spaces `X` and `Y`,\n`continuous_map.comap f` is the `\u211d`-linear pre-composition with `f` as a function from\nthe `\u211d`-valued continuous functions on `Y` to the `\u211d`-valued continuous functions on `X`. -/\ndef continuous_map.comap {X Y : Type*} [topological_space X] [topological_space Y]\n  (f : C(X,Y)) : C(Y,\u211d) \u2192L[\u211d] C(X,\u211d) :=\n{ to_fun := \u03bb g, g.comp f,\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl,\n  cont := by refine continuous_map.continuous_comp_left f }\n\n/--  Given a continuous function `f : X \u2192 Y` between topological spaces `X` and `Y`,\n`continuous_map.comap_LC_linear_map f` is the `\u211d`-linear pre-composition with `f` as a function from\nthe locally constant `\u211d`-valued continuous functions on `Y` to the locally constant `\u211d`-valued\ncontinuous functions on `X`.\n\n`continuous_map.comap_LC` is similar, except that it also bundles continuity of the resulting\nfunction among spaces of continuous maps. -/\ndef continuous_map.comap_LC_linear_map {X Y : Type*} [topological_space X] [topological_space Y]\n  (f : C(X,Y)) : locally_constant Y \u211d \u2192\u2097[\u211d] locally_constant X \u211d :=\n{ to_fun := \u03bb g,\n  { to_fun := g \u2218 f,\n    is_locally_constant := \u03bb S,\n      by { rw set.preimage_comp, apply is_open.preimage f.2, apply g.2, } },\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl }\n\n/--  Given a continuous function `f : X \u2192 Y` between topological spaces `X` and `Y`,\n`continuous_map.comap_LC_linear_map f` is the `\u211d`-linear pre-composition with `f` as a *continuous*\nfunction from the locally constant `\u211d`-valued continuous functions on `Y` to the locally constant\n`\u211d`-valued continuous functions on `X`.\n\n`continuous_map.comap_LC_linear_map` is similar, but does not bundle continuity of the resulting\nfunction among spaces of continuous maps. -/\ndef continuous_map.comap_LC {X Y : Type*} [topological_space X] [compact_space X]\n  [topological_space Y] [compact_space Y]\n  (f : C(X,Y)) : locally_constant Y \u211d \u2192L[\u211d] locally_constant X \u211d :=\n{ to_fun := \u03bb g,\n  { to_fun := g \u2218 f,\n    is_locally_constant := \u03bb S,\n      by { rw set.preimage_comp, apply is_open.preimage f.2, apply g.2, } },\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl,\n  cont := begin\n    apply (f.comap_LC_linear_map.mk_continuous_of_exists_bound _).continuous,\n    use 1, intros e, rw one_mul,\n    by_cases (is_empty X),\n    { have : (f.comap_LC_linear_map) e = 0, ext x, exact h.elim x, rw this,\n      simp },\n    change Sup _ \u2264 _,\n    apply cSup_le,\n    simp only [not_is_empty_iff] at h, obtain \u27e8x\u27e9 := h,\n    use \u2225 e (f x) \u2225, use x, refl,\n    rintros b \u27e8x,rfl\u27e9, dsimp,\n    exact_mod_cast locally_constant.nnnorm_apply_le_nnnorm _ e (f x),\n  end }\n\n/--  Given a compact topological space `X`, the inclusion of locally constant functions on `X` into\nthe space of all continuous functions is a continuous `\u211d`-linear map. -/\ndef lc_to_c (X : Type*) [topological_space X] [compact_space X] :\n  locally_constant X \u211d \u2192L[\u211d] C(X,\u211d) :=\n{ to_fun := \u03bb f, f.to_continuous_map,\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl,\n  cont := begin\n    apply isometry.continuous,\n    intros f g,\n    simp only [edist_dist, dist_eq_norm, continuous_map.norm_eq_supr_norm,\n      locally_constant.norm_def, locally_constant.to_continuous_map_eq_coe,\n      continuous_map.coe_sub, locally_constant.coe_continuous_map, pi.sub_apply],\n    refl,\n  end }\n\nnamespace weak_dual\n\n/--  Given topological `\u211d`-vector spaces `A` and `B` and a continuous, `\u211d`-linear map `f : A \u2192 B`\nbetween them, `comap f` is the pre-composition with `f` as a continuous, `\u211d`-linear map between\nthe weak `\u211d`-linear dual of `B` to the weak `\u211d`-linear dual of `A`. -/\ndef comap {A B : Type*}\n  [add_comm_group A] [module \u211d A] [topological_space A]\n  [add_comm_group B] [module \u211d B] [topological_space B]\n  (f : A \u2192L[\u211d] B) :\n  weak_dual \u211d B \u2192L[\u211d] weak_dual \u211d A :=\n{ to_fun := \u03bb g, g.comp f,\n  map_add' := \u03bb _ _, rfl,\n  map_smul' := \u03bb _ _, rfl,\n  cont := begin\n    apply weak_dual.continuous_of_continuous_eval,\n    intros a,\n    apply weak_dual.eval_continuous,\n  end }\n\n/--  Given a compact topological space `X`, an element `\u03bc` in the weak, `\u211d`-linear dual of the\ncontinuous functions on `X`, and two non-negative real numbers `p` and `c`, `bdd \u03bc p c` is the\nstatement that the sum of the `p`-th powers of the absolute values of the measures of all the\nclopen subsets of `X` is bounded above by `c`.\n\nSee the actual definition for what \"all the clopen subsets\" really means!\n\n`bdd_LC` is similar, but uses the dual of locally constant functions.  -/\ndef bdd {X : Type*} [topological_space X] [compact_space X]\n  (\u03bc : weak_dual \u211d C(X,\u211d)) (p c : \u211d\u22650) : Prop :=\n\u2200 (T : discrete_quotient X),\n  \u2211 t : T, \u2225 \u03bc (T.fibre t).indicator \u2225\u208a^(p : \u211d) \u2264 c\n\n/--  Given a compact topological space `X`, an element `\u03bc` in the weak, `\u211d`-linear dual of the\nlocally constant functions on `X`, and two non-negative real numbers `p` and `c`, `bdd \u03bc p c` is the\nstatement that the sum of the `p`-th powers of the absolute values of the measures of all the\nclopen subsets of `X` is bounded above by `c`.\n\nSee the actual definition for what \"all the clopen subsets\" really means!\n\n`bdd` is similar, but uses the dual of continuous functions.  -/\ndef bdd_LC {X : Type*} [topological_space X] [compact_space X]\n  (\u03bc : weak_dual \u211d (locally_constant X \u211d)) (p c : \u211d\u22650) : Prop :=\n\u2200 (T : discrete_quotient X),\n  \u2211 t : T, \u2225 \u03bc (T.fibre t).indicator_LC \u2225\u208a^(p : \u211d) \u2264 c\n\nlemma bdd_LC_comap {X Y : Type*} {p c : \u211d\u22650} [fact (0 < p)]\n  [topological_space X] [compact_space X]\n  [topological_space Y] [compact_space Y]\n  (\u03bc : weak_dual \u211d (locally_constant X \u211d)) (h\u03bc : \u03bc.bdd_LC p c) (f : C(X,Y)) :\n  (weak_dual.comap f.comap_LC \u03bc).bdd_LC p c :=\nbegin\n  intros T,\n  convert h\u03bc (T.comap f.2) using 1,\n  let \u03b9 : T.comap f.2 \u2192 T := discrete_quotient.map (le_refl _),\n  have h\u03b9 : function.injective \u03b9,\n  { rintros \u27e8\u27e9 \u27e8\u27e9 h,\n    apply quotient.sound',\n    apply quotient.exact' h },\n  let S\u2081 := _, change S\u2081 = _,\n  have : S\u2081 = \u2211 t in finset.univ.image \u03b9,\n    \u2225 ((comap f.comap_LC) \u03bc) (T.fibre t).indicator_LC \u2225\u208a ^ (p : \u211d),\n  { symmetry, apply finset.sum_subset, simp only [finset.subset_univ],\n    intros x _ hx,\n    --simp only [finset.mem_image, finset.mem_univ, exists_true_left, not_exists] at hx,\n    suffices : ((comap f.comap_LC) \u03bc) (T.fibre x).indicator_LC = 0,\n    { simp only [this, nnnorm_zero, nnreal.rpow_eq_zero_iff, eq_self_iff_true,\n        ne.def, nnreal.coe_eq_zero, true_and],\n      exact ne_of_gt (fact.out (0 < p)) },\n    -- the `change` below is faster than `dsimp [comap]`\n    change \u03bc ((f.comap_LC) (T.fibre x).indicator_LC) = 0,\n    convert \u03bc.map_zero,\n    refine locally_constant.ext (\u03bb t, _),\n    dsimp only [continuous_map.comap_LC, topological_space.clopens.indicator_LC_apply],\n    apply if_neg,\n    contrapose! hx,\n    rw finset.mem_image,\n    refine \u27e8discrete_quotient.proj _ t, finset.mem_univ _, hx\u27e9 },\n  rw this, clear this, symmetry,\n  fapply finset.sum_bij,\n  { intros a _, exact \u03b9 a },\n  { intros, dsimp, erw finset.mem_image, refine \u27e8a, finset.mem_univ _, rfl\u27e9 },\n  { intros a ha,\n    congrm \u2225 \u03bc \u27e8(\u03bb t, ite _ _ _), _\u27e9 \u2225\u208a ^ \u2191p,\n    rw [set_like.mem_coe, set_like.mem_coe, \u2190 discrete_quotient.comap_mem_fibre_iff] },\n  { intros a\u2081 a\u2082 h\u2081 h\u2082 hh, apply h\u03b9, exact hh },\n  { rintros b hb,\n    rw finset.mem_image at hb,\n    obtain \u27e8b,hh,hb\u27e9 := hb,\n    use [b,hh,hb.symm] },\nend\n\nlemma bdd_comap {X Y : Type*} {p c : \u211d\u22650} [fact (0 < p)]\n  [topological_space X] [compact_space X]\n  [topological_space Y] [compact_space Y]\n  (\u03bc : weak_dual \u211d C(X,\u211d)) (h\u03bc : \u03bc.bdd p c) (f : C(X,Y)) :\n  (weak_dual.comap f.comap \u03bc).bdd p c :=\n\u03bb t, by apply bdd_LC_comap (comap (lc_to_c X) \u03bc) h\u03bc f t\n\nlemma bdd_iff_indexed_parition (S : Profinite.{0}) (\u03bc : weak_dual \u211d C(S,\u211d)) (p c : \u211d\u22650) :\n  \u03bc.bdd p c \u2194\n  \u2200 (\u03b9 : Fintype.{0}) (e : \u03b9 \u2192 set S)\n    (I : indexed_partition e) (he : \u2200 i, is_clopen (e i)),\n    \u2211 i : \u03b9, \u2225 \u03bc.1 (topological_space.clopens.indicator \u27e8e i, he i\u27e9) \u2225\u208a^(p : \u211d) \u2264 c :=\nbegin\n  split,\n  { intros h\u03bc \u03b9 e I he,\n    let T : discrete_quotient S := indexed_partition.discrete_quotient I he,\n    let ee : \u03b9 \u2243 T := indexed_partition.discrete_quotient_equiv he I,\n    specialize h\u03bc T,\n    convert h\u03bc using 1,\n    fapply finset.sum_bij',\n    { intros i _, exact ee i, },\n    { intros, exact finset.mem_univ _ },\n    { intros a ha, congr' 4,\n      ext1, change _ = T.proj \u207b\u00b9' _,\n      rw indexed_partition.discrete_quotient_fiber,\n      erw ee.symm_apply_apply,\n      refl },\n    { intros t ht, exact ee.symm t },\n    { intros, exact finset.mem_univ _ },\n    { intros, exact ee.symm_apply_apply _ },\n    { intros, exact ee.apply_symm_apply _ } },\n  { intros h\u03bc T,\n    refine h\u03bc (Fintype.of T) (\u03bb t, T.proj \u207b\u00b9' {t}) _ (\u03bb t, (T.fibre t).2),\n    fapply indexed_partition.mk',\n    { intros i j hij a ha, simp only [set.bot_eq_empty, set.mem_empty_iff_false], apply hij,\n      simp only [set.inf_eq_inter, set.mem_inter_iff, set.mem_preimage,\n        set.mem_singleton_iff] at ha,\n      rw [\u2190 ha.1, ha.2] },\n    { rintros (t : T), obtain \u27e8t,rfl\u27e9 := T.proj_surjective t, use t, change _ = _, refl, },\n    { intros s, use T.proj s, change _ = _, refl } }\nend\n\nend weak_dual\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/Radon/setup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3765671062094402}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.logic\n \n\nuniverses u_1 \n\nnamespace Mathlib\n\ntheorem punit_eq (a : PUnit) (b : PUnit) : a = b :=\n  punit.rec_on a (punit.rec_on b rfl)\n\ntheorem punit_eq_star (a : PUnit) : a = PUnit.unit :=\n  punit_eq a PUnit.unit\n\nprotected instance punit.subsingleton : subsingleton PUnit :=\n  subsingleton.intro punit_eq\n\nprotected instance punit.inhabited : Inhabited PUnit :=\n  { default := PUnit.unit }\n\nprotected instance punit.decidable_eq : DecidableEq PUnit :=\n  fun (a b : PUnit) => is_true (punit_eq a b)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/punit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3765670983042188}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\nFunctors with two arguments\n-/\nimport logic.function.basic\nimport control.functor\nimport tactic.core\n\nuniverses u\u2080 u\u2081 u\u2082 v\u2080 v\u2081 v\u2082\n\nopen function\n\nclass bifunctor (F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082) :=\n(bimap : \u03a0 {\u03b1 \u03b1' \u03b2 \u03b2'}, (\u03b1 \u2192 \u03b1') \u2192 (\u03b2 \u2192 \u03b2') \u2192 F \u03b1 \u03b2 \u2192 F \u03b1' \u03b2')\nexport bifunctor ( bimap )\n\nclass is_lawful_bifunctor (F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082) [bifunctor F] :=\n(id_bimap : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), bimap id id x = x)\n(bimap_bimap : \u03a0 {\u03b1\u2080 \u03b1\u2081 \u03b1\u2082 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082} (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b1\u2081 \u2192 \u03b1\u2082)\n  (g : \u03b2\u2080 \u2192 \u03b2\u2081) (g' : \u03b2\u2081 \u2192 \u03b2\u2082) (x : F \u03b1\u2080 \u03b2\u2080),\n  bimap f' g' (bimap f g x) = bimap (f' \u2218 f) (g' \u2218 g) x)\n\nexport is_lawful_bifunctor (id_bimap bimap_bimap)\n\nattribute [higher_order bimap_id_id] id_bimap\nattribute [higher_order bimap_comp_bimap] bimap_bimap\n\nexport is_lawful_bifunctor (bimap_id_id bimap_comp_bimap)\nvariables {F : Type u\u2080 \u2192 Type u\u2081 \u2192 Type u\u2082} [bifunctor F]\n\nnamespace bifunctor\n\n@[reducible]\ndef fst {\u03b1 \u03b1' \u03b2} (f : \u03b1 \u2192 \u03b1') : F \u03b1 \u03b2 \u2192 F \u03b1' \u03b2 :=\nbimap f id\n\n@[reducible]\ndef snd {\u03b1 \u03b2 \u03b2'} (f : \u03b2 \u2192 \u03b2') : F \u03b1 \u03b2 \u2192 F \u03b1 \u03b2' :=\nbimap id f\n\nvariable [is_lawful_bifunctor F]\n\n@[higher_order fst_id]\nlemma id_fst : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), fst id x = x :=\n@id_bimap _ _ _\n\n@[higher_order snd_id]\nlemma id_snd : \u03a0 {\u03b1 \u03b2} (x : F \u03b1 \u03b2), snd id x = x :=\n@id_bimap _ _ _\n\n@[higher_order fst_comp_fst]\n\n\n@[higher_order fst_comp_snd]\nlemma fst_snd {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b2\u2080 \u2192 \u03b2\u2081) (x : F \u03b1\u2080 \u03b2\u2080) :\n  fst f (snd f' x) = bimap f f' x :=\nby simp [fst,bimap_bimap]\n\n@[higher_order snd_comp_fst]\nlemma snd_fst {\u03b1\u2080 \u03b1\u2081 \u03b2\u2080 \u03b2\u2081}\n  (f : \u03b1\u2080 \u2192 \u03b1\u2081) (f' : \u03b2\u2080 \u2192 \u03b2\u2081) (x : F \u03b1\u2080 \u03b2\u2080) :\n  snd f' (fst f x) = bimap f f' x :=\nby simp [snd,bimap_bimap]\n\n@[higher_order snd_comp_snd]\nlemma comp_snd {\u03b1 \u03b2\u2080 \u03b2\u2081 \u03b2\u2082}\n  (g : \u03b2\u2080 \u2192 \u03b2\u2081) (g' : \u03b2\u2081 \u2192 \u03b2\u2082) (x : F \u03b1 \u03b2\u2080) :\n  snd g' (snd g x) = snd (g' \u2218 g) x :=\nby simp [snd,bimap_bimap]\n\nattribute [functor_norm] bimap_bimap comp_snd comp_fst\n  snd_comp_snd snd_comp_fst fst_comp_snd fst_comp_fst bimap_comp_bimap\n  bimap_id_id fst_id snd_id\n\nend bifunctor\nopen functor\ninstance : bifunctor prod :=\n{ bimap := @prod.map }\n\ninstance : is_lawful_bifunctor prod :=\nby refine { .. }; intros; cases x; refl\n\ninstance bifunctor.const : bifunctor const :=\n{ bimap := (\u03bb \u03b1 \u03b1' \u03b2 \u03b2 f _, f) }\n\ninstance is_lawful_bifunctor.const : is_lawful_bifunctor const  :=\nby refine { .. }; intros; refl\n\ninstance bifunctor.flip : bifunctor (flip F) :=\n{ bimap := (\u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (bimap f' f x : F \u03b2' \u03b1')) }\n\ninstance is_lawful_bifunctor.flip [is_lawful_bifunctor F] : is_lawful_bifunctor (flip F)  :=\nby refine { .. }; intros; simp [bimap] with functor_norm\n\ninstance : bifunctor sum :=\n{ bimap := @sum.map }\n\ninstance : is_lawful_bifunctor sum :=\nby refine { .. }; intros; cases x; refl\n\nopen bifunctor functor\n\n@[priority 10]\ninstance bifunctor.functor {\u03b1} : functor (F \u03b1) :=\n{ map := \u03bb _ _, snd }\n\n@[priority 10]\ninstance bifunctor.is_lawful_functor [is_lawful_bifunctor F] {\u03b1} : is_lawful_functor (F \u03b1) :=\nby refine {..}; intros; simp [functor.map] with functor_norm\n\nsection bicompl\n\nvariables (G : Type* \u2192 Type u\u2080) (H : Type* \u2192 Type u\u2081) [functor G] [functor H]\n\ninstance : bifunctor (bicompl F G H) :=\n{ bimap := \u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (bimap (map f) (map f') x : F (G \u03b1') (H \u03b2')) }\n\ninstance [is_lawful_functor G]  [is_lawful_functor H] [is_lawful_bifunctor F] :\n  is_lawful_bifunctor (bicompl F G H) :=\nby constructor; intros; simp [bimap,map_id,map_comp_map] with functor_norm\n\nend bicompl\nsection bicompr\n\nvariables (G : Type u\u2082 \u2192 Type*) [functor G]\n\ninstance : bifunctor (bicompr G F) :=\n{ bimap := \u03bb \u03b1 \u03b1' \u03b2 \u03b2' f f' x, (map (bimap f f') x : G (F \u03b1' \u03b2')) }\n\ninstance [is_lawful_functor G] [is_lawful_bifunctor F] :\n  is_lawful_bifunctor (bicompr G F) :=\nby constructor; intros; simp [bimap] with functor_norm\n\nend bicompr\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/control/bifunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.37656709681172584}}
{"text": "import for_mathlib.derived.les_facts\n\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.limits\n\nnamespace bounded_homotopy_category\n\nvariables {\ud835\udcd0 : Type*} [category \ud835\udcd0] [abelian \ud835\udcd0] [enough_projectives \ud835\udcd0]\n\nlemma Ext_map_is_iso_of_quasi_iso\n  (A\u2081 A\u2082 B : bounded_homotopy_category \ud835\udcd0) (f : A\u2081 \u27f6 A\u2082)\n  [homotopy_category.is_quasi_iso f] (i : \u2124) :\n  is_iso $ ((Ext i).map f.op).app B :=\nbegin\n  let e := replacement_iso A\u2081.replace A\u2082.replace A\u2082 (A\u2081.\u03c0 \u226b f) A\u2082.\u03c0,\n  let e' := ((preadditive_yoneda.obj (B\u27e6i\u27e7)).map_iso e.op),\n  show is_iso e'.hom,\n  apply_instance\nend\n\nend bounded_homotopy_category\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/Ext_quasi_iso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8031737869342624, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.37652034286668606}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Markus Himmel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.limits.shapes.pullbacks\nimport Mathlib.category_theory.limits.shapes.strong_epi\nimport Mathlib.PostPort\n\nuniverses v u l \n\nnamespace Mathlib\n\n/-!\n# Categorical images\n\nWe define the categorical image of `f` as a factorisation `f = e \u226b m` through a monomorphism `m`,\nso that `m` factors through the `m'` in any other such factorisation.\n\n## Main definitions\n\n* A `mono_factorisation` is a factorisation `f = e \u226b m`, where `m` is a monomorphism\n* `is_image F` means that a given mono factorisation `F` has the universal property of the image.\n* `has_image f` means that we have chosen an image for the morphism `f : X \u27f6 Y`.\n  * In this case, `image f` is the image object, `image.\u03b9 f : image f \u27f6 Y` is the monomorphism `m`\n    of the factorisation and `factor_thru_image f : X \u27f6 image f` is the morphism `e`.\n* `has_images C` means that every morphism in `C` has an image.\n* Let `f : X \u27f6 Y` and `g : P \u27f6 Q` be morphisms in `C`, which we will represent as objects of the\n  arrow category `arrow C`. Then `sq : f \u27f6 g` is a commutative square in `C`. If `f` and `g` have\n  images, then `has_image_map sq` represents the fact that there is a morphism\n  `i : image f \u27f6 image g` making the diagram\n\n  X ----\u2192 image f ----\u2192 Y\n  |         |           |\n  |         |           |\n  \u2193         \u2193           \u2193\n  P ----\u2192 image g ----\u2192 Q\n\n  commute, where the top row is the image factorisation of `f`, the bottom row is the image\n  factorisation of `g`, and the outer rectangle is the commutative square `sq`.\n* If a category `has_images`, then `has_image_maps` means that every commutative square admits an\n  image map.\n* If a category `has_images`, then `has_strong_epi_images` means that the morphism to the image is\n  always a strong epimorphism.\n\n## Main statements\n\n* When `C` has equalizers, the morphism `e` appearing in an image factorisation is an epimorphism.\n* When `C` has strong epi images, then these images admit image maps.\n\n## Future work\n* TODO: coimages, and abelian categories.\n* TODO: connect this with existing working in the group theory and ring theory libraries.\n\n-/\n\nnamespace category_theory.limits\n\n\n/-- A factorisation of a morphism `f = e \u226b m`, with `m` monic. -/\nstructure mono_factorisation {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) where\n  I : C\n  m : I \u27f6 Y\n  m_mono : mono m\n  e : X \u27f6 I\n  fac' :\n    autoParam (e \u226b m = f)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem mono_factorisation.fac {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    (c : mono_factorisation f) : mono_factorisation.e c \u226b mono_factorisation.m c = f :=\n  sorry\n\n@[simp] theorem mono_factorisation.fac_assoc {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    (c : mono_factorisation f) {X' : C} (f' : Y \u27f6 X') :\n    mono_factorisation.e c \u226b mono_factorisation.m c \u226b f' = f \u226b f' :=\n  sorry\n\nnamespace mono_factorisation\n\n\n/-- The obvious factorisation of a monomorphism through itself. -/\ndef self {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [mono f] : mono_factorisation f :=\n  mk X f \ud835\udfd9\n\n-- I'm not sure we really need this, but the linter says that an inhabited instance ought to exist...\n\nprotected instance inhabited {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [mono f] :\n    Inhabited (mono_factorisation f) :=\n  { default := self f }\n\n/-- The morphism `m` in a factorisation `f = e \u226b m` through a monomorphism is uniquely determined. -/\ntheorem ext {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) {F : mono_factorisation f}\n    {F' : mono_factorisation f} (hI : I F = I F') (hm : m F = eq_to_hom hI \u226b m F') : F = F' :=\n  sorry\n\nend mono_factorisation\n\n\n/-- Data exhibiting that a given factorisation through a mono is initial. -/\nstructure is_image {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} (F : mono_factorisation f)\n    where\n  lift : (F' : mono_factorisation f) \u2192 mono_factorisation.I F \u27f6 mono_factorisation.I F'\n  lift_fac' :\n    autoParam\n      (\u2200 (F' : mono_factorisation f), lift F' \u226b mono_factorisation.m F' = mono_factorisation.m F)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem is_image.lift_fac {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} (c : is_image F) (F' : mono_factorisation f) :\n    is_image.lift c F' \u226b mono_factorisation.m F' = mono_factorisation.m F :=\n  sorry\n\n@[simp] theorem is_image.lift_fac_assoc {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} (c : is_image F) (F' : mono_factorisation f) {X' : C} (f' : Y \u27f6 X') :\n    is_image.lift c F' \u226b mono_factorisation.m F' \u226b f' = mono_factorisation.m F \u226b f' :=\n  sorry\n\n@[simp] theorem is_image.fac_lift_assoc {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} (hF : is_image F) (F' : mono_factorisation f) {X' : C}\n    (f' : mono_factorisation.I F' \u27f6 X') :\n    mono_factorisation.e F \u226b is_image.lift hF F' \u226b f' = mono_factorisation.e F' \u226b f' :=\n  sorry\n\nnamespace is_image\n\n\n/-- The trivial factorisation of a monomorphism satisfies the universal property. -/\ndef self {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [mono f] :\n    is_image (mono_factorisation.self f) :=\n  mk fun (F' : mono_factorisation f) => mono_factorisation.e F'\n\nprotected instance inhabited {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [mono f] :\n    Inhabited (is_image (mono_factorisation.self f)) :=\n  { default := self f }\n\n/-- Two factorisations through monomorphisms satisfying the universal property\nmust factor through isomorphic objects. -/\n-- TODO this is another good candidate for a future `unique_up_to_canonical_iso`.\n\n@[simp] theorem iso_ext_hom {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} {F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') :\n    iso.hom (iso_ext hF hF') = lift hF F' :=\n  Eq.refl (iso.hom (iso_ext hF hF'))\n\ntheorem iso_ext_hom_m {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} {F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') :\n    iso.hom (iso_ext hF hF') \u226b mono_factorisation.m F' = mono_factorisation.m F :=\n  sorry\n\ntheorem iso_ext_inv_m {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} {F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') :\n    iso.inv (iso_ext hF hF') \u226b mono_factorisation.m F = mono_factorisation.m F' :=\n  sorry\n\ntheorem e_iso_ext_hom {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} {F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') :\n    mono_factorisation.e F \u226b iso.hom (iso_ext hF hF') = mono_factorisation.e F' :=\n  sorry\n\ntheorem e_iso_ext_inv {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    {F : mono_factorisation f} {F' : mono_factorisation f} (hF : is_image F) (hF' : is_image F') :\n    mono_factorisation.e F' \u226b iso.inv (iso_ext hF hF') = mono_factorisation.e F :=\n  sorry\n\nend is_image\n\n\n/-- Data exhibiting that a morphism `f` has an image. -/\nstructure image_factorisation {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) where\n  F : mono_factorisation f\n  is_image : is_image F\n\nprotected instance inhabited_image_factorisation {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) [mono f] : Inhabited (image_factorisation f) :=\n  { default := image_factorisation.mk (mono_factorisation.self f) (is_image.self f) }\n\n/-- `has_image f` means that there exists an image factorisation of `f`. -/\nclass has_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) where\n  mk' :: (exists_image : Nonempty (image_factorisation f))\n\ntheorem has_image.mk {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    (F : image_factorisation f) : has_image f :=\n  has_image.mk' (Nonempty.intro F)\n\n/-- The chosen factorisation of `f` through a monomorphism. -/\ndef image.mono_factorisation {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] :\n    mono_factorisation f :=\n  image_factorisation.F (Classical.choice has_image.exists_image)\n\n/-- The witness of the universal property for the chosen factorisation of `f` through a monomorphism. -/\ndef image.is_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] :\n    is_image (image.mono_factorisation f) :=\n  image_factorisation.is_image (Classical.choice has_image.exists_image)\n\n/-- The categorical image of a morphism. -/\n/-- The inclusion of the image of a morphism into the target. -/\ndef image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] : C :=\n  mono_factorisation.I sorry\n\ndef image.\u03b9 {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] : image f \u27f6 Y :=\n  mono_factorisation.m (image.mono_factorisation f)\n\n@[simp] theorem image.as_\u03b9 {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] :\n    mono_factorisation.m (image.mono_factorisation f) = image.\u03b9 f :=\n  rfl\n\nprotected instance image.\u03b9.category_theory.mono {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) [has_image f] : mono (image.\u03b9 f) :=\n  mono_factorisation.m_mono (image.mono_factorisation f)\n\n/-- The map from the source to the image of a morphism. -/\n/-- Rewrite in terms of the `factor_thru_image` interface. -/\ndef factor_thru_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] :\n    X \u27f6 image f :=\n  mono_factorisation.e (image.mono_factorisation f)\n\n@[simp] theorem as_factor_thru_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    [has_image f] : mono_factorisation.e (image.mono_factorisation f) = factor_thru_image f :=\n  rfl\n\n@[simp] theorem image.fac_assoc {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f]\n    {X' : C} (f' : Y \u27f6 X') : factor_thru_image f \u226b image.\u03b9 f \u226b f' = f \u226b f' :=\n  sorry\n\n/-- Any other factorisation of the morphism `f` through a monomorphism receives a map from the image. -/\ndef image.lift {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f]\n    (F' : mono_factorisation f) : image f \u27f6 mono_factorisation.I F' :=\n  is_image.lift (image.is_image f) F'\n\n@[simp] theorem image.lift_fac {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f]\n    (F' : mono_factorisation f) : image.lift F' \u226b mono_factorisation.m F' = image.\u03b9 f :=\n  is_image.lift_fac' (image.is_image f) F'\n\n@[simp] theorem image.fac_lift {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f]\n    (F' : mono_factorisation f) : factor_thru_image f \u226b image.lift F' = mono_factorisation.e F' :=\n  is_image.fac_lift (image.is_image f) F'\n\n@[simp] theorem is_image.lift_\u03b9_assoc {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y}\n    [has_image f] {F : mono_factorisation f} (hF : is_image F) {X' : C} (f' : Y \u27f6 X') :\n    is_image.lift hF (image.mono_factorisation f) \u226b image.\u03b9 f \u226b f' = mono_factorisation.m F \u226b f' :=\n  sorry\n\n-- TODO we could put a category structure on `mono_factorisation f`,\n\n-- with the morphisms being `g : I \u27f6 I'` commuting with the `m`s\n\n-- (they then automatically commute with the `e`s)\n\n-- and show that an `image_of f` gives an initial object there\n\n-- (uniqueness of the lift comes for free).\n\nprotected instance lift_mono {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f]\n    (F' : mono_factorisation f) : mono (image.lift F') :=\n  mono_of_mono (image.lift F') (mono_factorisation.m F')\n\ntheorem has_image.uniq {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f]\n    (F' : mono_factorisation f) (l : image f \u27f6 mono_factorisation.I F')\n    (w : l \u226b mono_factorisation.m F' = image.\u03b9 f) : l = image.lift F' :=\n  sorry\n\n/-- `has_images` represents a choice of image for every morphism -/\nclass has_images (C : Type u) [category C] where\n  has_image : \u2200 {X Y : C} (f : X \u27f6 Y), has_image f\n\n/-- The image of a monomorphism is isomorphic to the source. -/\ndef image_mono_iso_source {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f]\n    [mono f] : image f \u2245 X :=\n  is_image.iso_ext (image.is_image f) (is_image.self f)\n\n@[simp] theorem image_mono_iso_source_inv_\u03b9 {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    [has_image f] [mono f] : iso.inv (image_mono_iso_source f) \u226b image.\u03b9 f = f :=\n  sorry\n\n@[simp] theorem image_mono_iso_source_hom_self_assoc {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) [has_image f] [mono f] {X' : C} (f' : Y \u27f6 X') :\n    iso.hom (image_mono_iso_source f) \u226b f \u226b f' = image.\u03b9 f \u226b f' :=\n  sorry\n\n-- This is the proof that `factor_thru_image f` is an epimorphism\n\n-- from https://en.wikipedia.org/wiki/Image_(category_theory), which is in turn taken from:\n\n-- Mitchell, Barry (1965), Theory of categories, MR 0202787, p.12, Proposition 10.1\n\ntheorem image.ext {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f] {W : C}\n    {g : image f \u27f6 W} {h : image f \u27f6 W} [has_limit (parallel_pair g h)]\n    (w : factor_thru_image f \u226b g = factor_thru_image f \u226b h) : g = h :=\n  sorry\n\nprotected instance factor_thru_image.category_theory.epi {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) [has_image f] [\u2200 {Z : C} (g h : image f \u27f6 Z), has_limit (parallel_pair g h)] :\n    epi (factor_thru_image f) :=\n  epi.mk\n    fun (Z : C) (g h : image f \u27f6 Z) (w : factor_thru_image f \u226b g = factor_thru_image f \u226b h) =>\n      image.ext f w\n\ntheorem epi_image_of_epi {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f]\n    [E : epi f] : epi (image.\u03b9 f) :=\n  epi_of_epi (factor_thru_image f) (image.\u03b9 f)\n\ntheorem epi_of_epi_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_image f]\n    [epi (image.\u03b9 f)] [epi (factor_thru_image f)] : epi f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (epi f)) (Eq.symm (image.fac f))))\n    (epi_comp (factor_thru_image f) (image.\u03b9 f))\n\n/--\nAn equation between morphisms gives a comparison map between the images\n(which momentarily we prove is an iso).\n-/\ndef image.eq_to_hom {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} {f' : X \u27f6 Y} [has_image f]\n    [has_image f'] (h : f = f') : image f \u27f6 image f' :=\n  image.lift (mono_factorisation.mk (image f') (image.\u03b9 f') (factor_thru_image f'))\n\nprotected instance image.eq_to_hom.category_theory.is_iso {C : Type u} [category C] {X : C} {Y : C}\n    {f : X \u27f6 Y} {f' : X \u27f6 Y} [has_image f] [has_image f'] (h : f = f') :\n    is_iso (image.eq_to_hom h) :=\n  is_iso.mk (image.eq_to_hom sorry)\n\n/-- An equation between morphisms gives an isomorphism between the images. -/\ndef image.eq_to_iso {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} {f' : X \u27f6 Y} [has_image f]\n    [has_image f'] (h : f = f') : image f \u2245 image f' :=\n  as_iso (image.eq_to_hom h)\n\n/--\nAs long as the category has equalizers,\nthe image inclusion maps commute with `image.eq_to_iso`.\n-/\ntheorem image.eq_fac {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} {f' : X \u27f6 Y}\n    [has_image f] [has_image f'] [has_equalizers C] (h : f = f') :\n    image.\u03b9 f = iso.hom (image.eq_to_iso h) \u226b image.\u03b9 f' :=\n  sorry\n\n/-- The comparison map `image (f \u226b g) \u27f6 image g`. -/\ndef image.pre_comp {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) {Z : C} (g : Y \u27f6 Z)\n    [has_image g] [has_image (f \u226b g)] : image (f \u226b g) \u27f6 image g :=\n  image.lift (mono_factorisation.mk (image g) (image.\u03b9 g) (f \u226b factor_thru_image g))\n\n@[simp] theorem image.factor_thru_image_pre_comp {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) {Z : C} (g : Y \u27f6 Z) [has_image g] [has_image (f \u226b g)] :\n    factor_thru_image (f \u226b g) \u226b image.pre_comp f g = f \u226b factor_thru_image g :=\n  sorry\n\n/--\nThe two step comparison map\n  `image (f \u226b (g \u226b h)) \u27f6 image (g \u226b h) \u27f6 image h`\nagrees with the one step comparison map\n  `image (f \u226b (g \u226b h)) \u2245 image ((f \u226b g) \u226b h) \u27f6 image h`.\n -/\ntheorem image.pre_comp_comp {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) {Z : C}\n    (g : Y \u27f6 Z) {W : C} (h : Z \u27f6 W) [has_image (g \u226b h)] [has_image (f \u226b g \u226b h)] [has_image h]\n    [has_image ((f \u226b g) \u226b h)] :\n    image.pre_comp f (g \u226b h) \u226b image.pre_comp g h =\n        image.eq_to_hom (Eq.symm (category.assoc f g h)) \u226b image.pre_comp (f \u226b g) h :=\n  sorry\n\n/--\n`image.pre_comp f g` is an isomorphism when `f` is an isomorphism\n(we need `C` to have equalizers to prove this).\n-/\nprotected instance image.is_iso_precomp_iso {C : Type u} [category C] {X : C} {Y : C} {Z : C}\n    (g : Y \u27f6 Z) [has_equalizers C] (f : X \u2245 Y) [has_image g] [has_image (iso.hom f \u226b g)] :\n    is_iso (image.pre_comp (iso.hom f) g) :=\n  is_iso.mk\n    (image.lift\n      (mono_factorisation.mk (image (iso.hom f \u226b g)) (image.\u03b9 (iso.hom f \u226b g))\n        (iso.inv f \u226b factor_thru_image (iso.hom f \u226b g))))\n\n-- Note that in general we don't have the other comparison map you might expect\n\n-- `image f \u27f6 image (f \u226b g)`.\n\n/-- Postcomposing by an isomorphism induces an isomorphism on the image. -/\ndef image.post_comp_is_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) {Z : C} (g : Y \u27f6 Z)\n    [has_equalizers C] [is_iso g] [has_image f] [has_image (f \u226b g)] : image f \u2245 image (f \u226b g) :=\n  iso.mk\n    (image.lift\n      (mono_factorisation.mk (image (f \u226b g)) (image.\u03b9 (f \u226b g) \u226b inv g) (factor_thru_image (f \u226b g))))\n    (image.lift (mono_factorisation.mk (image f) (image.\u03b9 f \u226b g) (factor_thru_image f)))\n\n@[simp] theorem image.post_comp_is_iso_hom_comp_image_\u03b9_assoc {C : Type u} [category C] {X : C}\n    {Y : C} (f : X \u27f6 Y) {Z : C} (g : Y \u27f6 Z) [has_equalizers C] [is_iso g] [has_image f]\n    [has_image (f \u226b g)] {X' : C} (f' : Z \u27f6 X') :\n    iso.hom (image.post_comp_is_iso f g) \u226b image.\u03b9 (f \u226b g) \u226b f' = image.\u03b9 f \u226b g \u226b f' :=\n  sorry\n\n@[simp] theorem image.post_comp_is_iso_inv_comp_image_\u03b9_assoc {C : Type u} [category C] {X : C}\n    {Y : C} (f : X \u27f6 Y) {Z : C} (g : Y \u27f6 Z) [has_equalizers C] [is_iso g] [has_image f]\n    [has_image (f \u226b g)] {X' : C} (f' : Y \u27f6 X') :\n    iso.inv (image.post_comp_is_iso f g) \u226b image.\u03b9 f \u226b f' = image.\u03b9 (f \u226b g) \u226b inv g \u226b f' :=\n  sorry\n\nend category_theory.limits\n\n\nnamespace category_theory.limits\n\n\nprotected instance hom.has_image {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    [has_image f] : has_image (comma.hom (arrow.mk f)) :=\n  (fun (this : has_image f) => this) _inst_2\n\n/-- An image map is a morphism `image f \u2192 image g` fitting into a commutative square and satisfying\n    the obvious commutativity conditions. -/\nstructure image_map {C : Type u} [category C] {f : arrow C} {g : arrow C} [has_image (comma.hom f)]\n    [has_image (comma.hom g)] (sq : f \u27f6 g)\n    where\n  map : image (comma.hom f) \u27f6 image (comma.hom g)\n  map_\u03b9' :\n    autoParam (map \u226b image.\u03b9 (comma.hom g) = image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\nprotected instance inhabited_image_map {C : Type u} [category C] {f : arrow C}\n    [has_image (comma.hom f)] : Inhabited (image_map \ud835\udfd9) :=\n  { default := image_map.mk \ud835\udfd9 }\n\n@[simp] theorem image_map.map_\u03b9 {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] {sq : f \u27f6 g} (c : image_map sq) :\n    image_map.map c \u226b image.\u03b9 (comma.hom g) = image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq :=\n  sorry\n\n@[simp] theorem image_map.map_\u03b9_assoc {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] {sq : f \u27f6 g} (c : image_map sq) {X' : C}\n    (f' : comma.right g \u27f6 X') :\n    image_map.map c \u226b image.\u03b9 (comma.hom g) \u226b f' =\n        image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq \u226b f' :=\n  sorry\n\n@[simp] theorem image_map.factor_map {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g) (m : image_map sq) :\n    factor_thru_image (comma.hom f) \u226b image_map.map m =\n        comma_morphism.left sq \u226b factor_thru_image (comma.hom g) :=\n  sorry\n\n/-- To give an image map for a commutative square with `f` at the top and `g` at the bottom, it\n    suffices to give a map between any mono factorisation of `f` and any image factorisation of\n    `g`. -/\ndef image_map.transport {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g)\n    (F : mono_factorisation (comma.hom f)) {F' : mono_factorisation (comma.hom g)}\n    (hF' : is_image F') {map : mono_factorisation.I F \u27f6 mono_factorisation.I F'}\n    (map_\u03b9 : map \u226b mono_factorisation.m F' = mono_factorisation.m F \u226b comma_morphism.right sq) :\n    image_map sq :=\n  image_map.mk (image.lift F \u226b map \u226b is_image.lift hF' (image.mono_factorisation (comma.hom g)))\n\n/-- `has_image_map sq` means that there is an `image_map` for the square `sq`. -/\nclass has_image_map {C : Type u} [category C] {f : arrow C} {g : arrow C} [has_image (comma.hom f)]\n    [has_image (comma.hom g)] (sq : f \u27f6 g)\n    where\n  mk' :: (has_image_map : Nonempty (image_map sq))\n\ntheorem has_image_map.mk {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] {sq : f \u27f6 g} (m : image_map sq) :\n    has_image_map sq :=\n  has_image_map.mk' (Nonempty.intro m)\n\ntheorem has_image_map.transport {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g)\n    (F : mono_factorisation (comma.hom f)) {F' : mono_factorisation (comma.hom g)}\n    (hF' : is_image F') (map : mono_factorisation.I F \u27f6 mono_factorisation.I F')\n    (map_\u03b9 : map \u226b mono_factorisation.m F' = mono_factorisation.m F \u226b comma_morphism.right sq) :\n    has_image_map sq :=\n  has_image_map.mk (image_map.transport sq F hF' map_\u03b9)\n\n/-- Obtain an `image_map` from a `has_image_map` instance. -/\ndef has_image_map.image_map {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq] :\n    image_map sq :=\n  Classical.choice has_image_map.has_image_map\n\ntheorem image_map.ext_iff {C : Type u} {_inst_1 : category C} {f : arrow C} {g : arrow C}\n    {_inst_2 : has_image (comma.hom f)} {_inst_3 : has_image (comma.hom g)} {sq : f \u27f6 g}\n    (x : image_map sq) (y : image_map sq) : x = y \u2194 image_map.map x = image_map.map y :=\n  sorry\n\nprotected instance image_map.subsingleton {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g) :\n    subsingleton (image_map sq) :=\n  subsingleton.intro\n    fun (a b : image_map sq) =>\n      image_map.ext a b\n        (iff.mp (cancel_mono (image.\u03b9 (comma.hom g)))\n          (eq.mpr\n            (id\n              ((fun (a a_1 : image (comma.hom f) \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_1 : a = a_1)\n                  (\u1fb0 \u1fb0_1 : image (comma.hom f) \u27f6 functor.obj \ud835\udfed (comma.right g)) (e_2 : \u1fb0 = \u1fb0_1) =>\n                  congr (congr_arg Eq e_1) e_2)\n                (image_map.map a \u226b image.\u03b9 (comma.hom g))\n                (image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq) (image_map.map_\u03b9 a)\n                (image_map.map b \u226b image.\u03b9 (comma.hom g))\n                (image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq) (image_map.map_\u03b9 b)))\n            (Eq.refl (image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq))))\n\n/-- The map on images induced by a commutative square. -/\ndef image.map {C : Type u} [category C] {f : arrow C} {g : arrow C} [has_image (comma.hom f)]\n    [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq] :\n    image (comma.hom f) \u27f6 image (comma.hom g) :=\n  image_map.map (has_image_map.image_map sq)\n\ntheorem image.factor_map {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq] :\n    factor_thru_image (comma.hom f) \u226b image.map sq =\n        comma_morphism.left sq \u226b factor_thru_image (comma.hom g) :=\n  sorry\n\ntheorem image.map_\u03b9 {C : Type u} [category C] {f : arrow C} {g : arrow C} [has_image (comma.hom f)]\n    [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq] :\n    image.map sq \u226b image.\u03b9 (comma.hom g) = image.\u03b9 (comma.hom f) \u226b comma_morphism.right sq :=\n  sorry\n\ntheorem image.map_hom_mk'_\u03b9 {C : Type u} [category C] {X : C} {Y : C} {P : C} {Q : C} {k : X \u27f6 Y}\n    [has_image k] {l : P \u27f6 Q} [has_image l] {m : X \u27f6 P} {n : Y \u27f6 Q} (w : m \u226b l = k \u226b n)\n    [has_image_map (arrow.hom_mk' w)] : image.map (arrow.hom_mk' w) \u226b image.\u03b9 l = image.\u03b9 k \u226b n :=\n  image.map_\u03b9 (arrow.hom_mk' w)\n\n/-- Image maps for composable commutative squares induce an image map in the composite square. -/\ndef image_map_comp {C : Type u} [category C] {f : arrow C} {g : arrow C} [has_image (comma.hom f)]\n    [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq] {h : arrow C}\n    [has_image (comma.hom h)] (sq' : g \u27f6 h) [has_image_map sq'] : image_map (sq \u226b sq') :=\n  image_map.mk (image.map sq \u226b image.map sq')\n\n@[simp] theorem image.map_comp {C : Type u} [category C] {f : arrow C} {g : arrow C}\n    [has_image (comma.hom f)] [has_image (comma.hom g)] (sq : f \u27f6 g) [has_image_map sq]\n    {h : arrow C} [has_image (comma.hom h)] (sq' : g \u27f6 h) [has_image_map sq']\n    [has_image_map (sq \u226b sq')] : image.map (sq \u226b sq') = image.map sq \u226b image.map sq' :=\n  sorry\n\n/-- The identity `image f \u27f6 image f` fits into the commutative square represented by the identity\n    morphism `\ud835\udfd9 f` in the arrow category. -/\ndef image_map_id {C : Type u} [category C] (f : arrow C) [has_image (comma.hom f)] : image_map \ud835\udfd9 :=\n  image_map.mk \ud835\udfd9\n\n@[simp] theorem image.map_id {C : Type u} [category C] (f : arrow C) [has_image (comma.hom f)]\n    [has_image_map \ud835\udfd9] : image.map \ud835\udfd9 = \ud835\udfd9 :=\n  sorry\n\n/-- If a category `has_image_maps`, then all commutative squares induce morphisms on images. -/\nclass has_image_maps (C : Type u) [category C] [has_images C] where\n  has_image_map : \u2200 {f g : arrow C} (st : f \u27f6 g), has_image_map st\n\n/-- The functor from the arrow category of `C` to `C` itself that maps a morphism to its image\n    and a commutative square to the induced morphism on images. -/\n@[simp] theorem im_map {C : Type u} [category C] [has_images C] [has_image_maps C] (_x : arrow C) :\n    \u2200 (_x_1 : arrow C) (st : _x \u27f6 _x_1), functor.map im st = image.map st :=\n  fun (_x_1 : arrow C) (st : _x \u27f6 _x_1) => Eq.refl (functor.map im st)\n\n/-- A strong epi-mono factorisation is a decomposition `f = e \u226b m` with `e` a strong epimorphism\n    and `m` a monomorphism. -/\nstructure strong_epi_mono_factorisation {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y)\n    extends mono_factorisation f where\n  e_strong_epi : strong_epi (mono_factorisation.e _to_mono_factorisation)\n\n/-- Satisfying the inhabited linter -/\nprotected instance strong_epi_mono_factorisation_inhabited {C : Type u} [category C] {X : C} {Y : C}\n    (f : X \u27f6 Y) [strong_epi f] : Inhabited (strong_epi_mono_factorisation f) :=\n  { default := strong_epi_mono_factorisation.mk (mono_factorisation.mk Y \ud835\udfd9 f) }\n\n/-- A mono factorisation coming from a strong epi-mono factorisation always has the universal\n    property of the image. -/\ndef strong_epi_mono_factorisation.to_mono_is_image {C : Type u} [category C] {X : C} {Y : C}\n    {f : X \u27f6 Y} (F : strong_epi_mono_factorisation f) :\n    is_image (strong_epi_mono_factorisation.to_mono_factorisation F) :=\n  is_image.mk fun (G : mono_factorisation f) => arrow.lift (arrow.hom_mk' sorry)\n\n/-- A category has strong epi-mono factorisations if every morphism admits a strong epi-mono\n    factorisation. -/\nclass has_strong_epi_mono_factorisations (C : Type u) [category C] where\n  mk' :: (has_fac : \u2200 {X Y : C} (f : X \u27f6 Y), Nonempty (strong_epi_mono_factorisation f))\n\ntheorem has_strong_epi_mono_factorisations.mk {C : Type u} [category C]\n    (d : {X Y : C} \u2192 (f : X \u27f6 Y) \u2192 strong_epi_mono_factorisation f) :\n    has_strong_epi_mono_factorisations C :=\n  has_strong_epi_mono_factorisations.mk' fun (X Y : C) (f : X \u27f6 Y) => Nonempty.intro (d f)\n\nprotected instance has_images_of_has_strong_epi_mono_factorisations {C : Type u} [category C]\n    [has_strong_epi_mono_factorisations C] : has_images C :=\n  has_images.mk sorry\n\n/-- A category has strong epi images if it has all images and `factor_thru_image f` is a strong\n    epimorphism for all `f`. -/\nclass has_strong_epi_images (C : Type u) [category C] [has_images C] where\n  strong_factor_thru_image : \u2200 {X Y : C} (f : X \u27f6 Y), strong_epi (factor_thru_image f)\n\n/-- If there is a single strong epi-mono factorisation of `f`, then every image factorisation is a\n    strong epi-mono factorisation. -/\ntheorem strong_epi_of_strong_epi_mono_factorisation {C : Type u} [category C] {X : C} {Y : C}\n    {f : X \u27f6 Y} (F : strong_epi_mono_factorisation f) {F' : mono_factorisation f}\n    (hF' : is_image F') : strong_epi (mono_factorisation.e F') :=\n  sorry\n\ntheorem strong_epi_factor_thru_image_of_strong_epi_mono_factorisation {C : Type u} [category C]\n    {X : C} {Y : C} {f : X \u27f6 Y} [has_image f] (F : strong_epi_mono_factorisation f) :\n    strong_epi (factor_thru_image f) :=\n  strong_epi_of_strong_epi_mono_factorisation F (image.is_image f)\n\n/-- If we constructed our images from strong epi-mono factorisations, then these images are\n    strong epi images. -/\nprotected instance has_strong_epi_images_of_has_strong_epi_mono_factorisations {C : Type u}\n    [category C] [has_strong_epi_mono_factorisations C] : has_strong_epi_images C :=\n  has_strong_epi_images.mk\n    fun (X Y : C) (f : X \u27f6 Y) =>\n      strong_epi_factor_thru_image_of_strong_epi_mono_factorisation\n        (Classical.choice (has_strong_epi_mono_factorisations.has_fac f))\n\n/-- A category with strong epi images has image maps. -/\nprotected instance has_image_maps_of_has_strong_epi_images {C : Type u} [category C] [has_images C]\n    [has_strong_epi_images C] : has_image_maps C :=\n  has_image_maps.mk sorry\n\n/-- If a category has images, equalizers and pullbacks, then images are automatically strong epi\n    images. -/\nprotected instance has_strong_epi_images_of_has_pullbacks_of_has_equalizers {C : Type u}\n    [category C] [has_images C] [has_pullbacks C] [has_equalizers C] : has_strong_epi_images C :=\n  sorry\n\n/--\nIf `C` has strong epi mono factorisations, then the image is unique up to isomorphism, in that if\n`f` factors as a strong epi followed by a mono, this factorisation is essentially the image\nfactorisation.\n-/\ndef image.iso_strong_epi_mono {C : Type u} [category C] [has_strong_epi_mono_factorisations C]\n    {X : C} {Y : C} {f : X \u27f6 Y} {I' : C} (e : X \u27f6 I') (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e]\n    [mono m] : I' \u2245 image f :=\n  is_image.iso_ext\n    (strong_epi_mono_factorisation.to_mono_is_image\n      (strong_epi_mono_factorisation.mk (mono_factorisation.mk I' m e)))\n    (image.is_image f)\n\n@[simp] theorem image.iso_strong_epi_mono_hom_comp_\u03b9 {C : Type u} [category C]\n    [has_strong_epi_mono_factorisations C] {X : C} {Y : C} {f : X \u27f6 Y} {I' : C} (e : X \u27f6 I')\n    (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e] [mono m] :\n    iso.hom (image.iso_strong_epi_mono e m comm) \u226b image.\u03b9 f = m :=\n  is_image.lift_fac\n    (strong_epi_mono_factorisation.to_mono_is_image\n      (strong_epi_mono_factorisation.mk (mono_factorisation.mk I' m e)))\n    (image.mono_factorisation f)\n\n@[simp] theorem image.iso_strong_epi_mono_inv_comp_mono {C : Type u} [category C]\n    [has_strong_epi_mono_factorisations C] {X : C} {Y : C} {f : X \u27f6 Y} {I' : C} (e : X \u27f6 I')\n    (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e] [mono m] :\n    iso.inv (image.iso_strong_epi_mono e m comm) \u226b m = image.\u03b9 f :=\n  image.lift_fac\n    (strong_epi_mono_factorisation.to_mono_factorisation\n      (strong_epi_mono_factorisation.mk (mono_factorisation.mk I' m e)))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/images_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.37615442325269366}}
{"text": "import data.list.basic\nimport data.vector\n\nuniverses u v w\nvariables {n : \u2115} {\u03b1 \u03b2 \u03b3 \u03b4 : Type}\n\nnotation `v[` v:(foldr `, ` (h t, vector.cons h t) vector.nil `]`) := v\n\n@[simp]\nlemma ite_else_ite {c} [decidable c] {b\u2081 b\u2082 b\u2083 :  Sort u} : ite c b\u2081 (ite c b\u2082 b\u2083) = ite c b\u2081 b\u2083 := begin\n  by_cases c; simp *,\nend\n\nlemma lt_zero_one : 0 < 1 := nat.lt.base 0\n\n@[simp]\nlemma nat.succ_lt_succ_iff {n m : \u2115} : n + 1 < m + 1 \u2194 n < m := nat.lt_succ_iff\n\nlemma eq.mpr.intro {\u03b1 : Type} (x : \u03b1) : eq.mpr rfl x = x := begin\n  refl,\nend\n\nnamespace fin\n\nlemma fin_eq {f f' : fin n} : f \u2260 f' \u2192 f.val \u2260 f'.val := begin\n  intro h,\n  cases f,\n  cases f',\n  simp,\n  intro h',\n  apply h,\n  subst h',\nend\n\nend fin\n\nnamespace list\n\nlemma list_length_neq_zero {\u03b1} : \u2200{l : list \u03b1}, l.length \u2260 0 \u2192 \u2203 x xs, l = (x :: xs)\n| []     h := (h rfl).elim\n| (a::l) h := \u27e8_, _, rfl\u27e9\n\nlemma list_length_tail {\u03b1 \u03b2 : Type} {x : \u03b1} {y : \u03b2} {xs ys : list _}\n  (h : (x :: xs).length = (y :: ys).length) : xs.length = ys.length :=\nby simpa using h \n\ndef range_fin_core (dim : \u2115) : \u03a0 n : \u2115, n <= dim \u2192 list (fin dim) \u2192 list (fin dim)\n| 0 h l := l\n| (nat.succ n) hs l := have h : n < dim := begin apply nat.succ_le_succ_iff.mp, apply nat.lt_succ_iff.mpr hs, end, range_fin_core n (le_of_lt h) (\u27e8n, h\u27e9 :: l)\n\ndef range_fin (n : \u2115) : list (fin n) :=\nrange_fin_core n n (by refl) []\n\n-- TODO: we need something like list.range' for fin\n--def range_fin' : \u2115 \u2192 \u2115 \u2192 list (fin n)\n\n@[simp] lemma length_range_nth : length (range_fin n) = n := begin\n  admit\nend\n\n\ndef fin_inc : fin n \u2192 fin (nat.succ n)\n| \u27e8val, is_lt\u27e9 := \u27e8val, nat.lt.step is_lt\u27e9\n\ninstance (n : \u2115) : has_coe (list (fin $ n)) (list (fin $ nat.succ n)) := \u27e8map fin_inc\u27e9\n\nlemma range_fin_succ : range_fin (nat.succ n) = ((range_fin n : list (fin n)) : list (fin $ nat.succ n)) ++ [\u27e8n, sorry\u27e9] := begin\n  unfold range_fin,\n  induction n, { refl, },\n  {\n    rw range_fin_core,\n    sorry,\n  }\nend\n\nlemma foldl_range_fin_succ {\u03b1 : Type} {n : \u2115} (i : \u03b1) (r : \u03b1 \u2192 fin (nat.succ n) \u2192 \u03b1) : foldl r i (range_fin (n + 1)) = r (foldl r i (range_fin n : list (fin n))) (nat.succ n) := begin\n  induction n,\n  { refl, },\n  {\n    rw [range_fin, range_fin_core],\n    sorry,\n  }\nend\n\n#print map\u2082._main\n\nlemma map\u2082_map\u2082 (g : \u03b3 \u2192 \u03b2 \u2192 \u03b4) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (l : list \u03b1) (l' : list \u03b2) : map\u2082 g (map\u2082 f l l') l' = map\u2082 (\u03bb a b, g (f a b) b) l l' := begin\n  induction l generalizing l',\n  case list.nil { cases l'; refl, },\n  case list.cons { cases l'; simp [map\u2082, *], }\nend\n\nlemma list_neq_elem {\u03b1 : Type} {l l' : list \u03b1} (n : \u2115) (h : n < l.length) (h' : n < l'.length) : l.nth_le n h \u2260 l'.nth_le n h' \u2192 l \u2260 l' := by sorry\n\nlemma list_nth_vector {\u03b1 l} {v : vector \u03b1 l} {n h} : list.nth_le (vector.to_list v) n h = v.nth \u27e8n, (by sorry)\u27e9 := by sorry\n\nlemma list_one_eq {\u03b1 : Type} {l\u2081 l\u2082 : list \u03b1} (h : l\u2081.length = 1) : ([l\u2081.nth_le 0 (by rw h; exact lt_zero_one)] : list \u03b1) = l\u2082 \u2192 l\u2081 = l\u2082 := sorry\n\nend list\n\nnamespace nat\n\nlemma lt_neq_zeor (n m : \u2115) : n < m \u2192 m \u2260 0 := begin\n    intro,\n    intro b,\n    rw b at a,\n    cases a,\nend\n\nend nat\n\nnamespace set\n\nlemma union_no_mem_left {\u03b1 : Type} {a : \u03b1} {b c : set \u03b1} (h : a \u2209 b \u222a c) : a \u2209 b := sorry\n\nend set\n\nnamespace vector\n\nprotected def mem {\u03b1 : Type u} {n : \u2115} : \u03b1 \u2192 vector \u03b1 n \u2192 Prop\n| a v := a \u2208 v.to_list\n\ninstance {\u03b1 : Type u} {n : \u2115} : has_mem \u03b1 (vector \u03b1 n) :=\n\u27e8vector.mem\u27e9\n\nlemma mem_def {\u03b1 : Type*} {n : \u2115} (a : \u03b1) (v : vector \u03b1 n) : a \u2208 v \u2194 a \u2208 v.to_list :=\niff.rfl\n\nlemma mem_nil {\u03b1 : Type u} {a : \u03b1} : a \u2209 (@vector.nil \u03b1) := by sorry\n\nlemma nat_le_zero {n : \u2115} : n < 0 \u2192 false := by sorry\n\n#check vector.cons\n\nlemma mem_elim_head {\u03b1 : Type u} {n} {tl : vector \u03b1 n} {a hd : \u03b1} :\n  a \u2208 tl \u2192 a \u2208 cons hd tl :=\nsorry\n\nlemma contains_nth {\u03b1 : Type} : \u2200{n : \u2115} {v : vector \u03b1 n} {i : fin n}, (v.nth i) \u2208 v\n| n \u27e8l, rfl\u27e9 \u27e8i, hi\u27e9 :=\n  begin\n    dsimp only [vector.nth, vector.has_mem, vector.mem, to_list],\n    rw list.mem_iff_nth_le,\n    exact \u27e8i, hi, rfl\u27e9\n  end\n\nlemma eq_element_wise {\u03b1 : Type} : \u2200{a b : vector \u03b1 n}, (\u2200 i, a.nth i = b.nth i) \u2192 a = b \n:= begin\n  -- TODO needs fixing\n  intros a b hieq,\n  apply vector.eq,\n  cases a,\n  cases b,\n  repeat { rw to_list },\n  simp,\n  induction a_val generalizing b_val n,\n  case list.nil {\n    cases b_val,\n    { refl },\n    {\n      rw list.length at a_property b_property,\n      rw \u2190 b_property at a_property,\n      contradiction,\n    }\n  },\n  case list.cons {\n    cases b_val,\n    {\n      rw list.length at a_property b_property,\n      rw \u2190 b_property at a_property,\n      contradiction,\n    },\n    {\n      have : _ := hieq \u27e80, sorry\u27e9,\n      rw [vector.nth, list.nth_le, vector.nth, list.nth_le] at this,\n      dunfold vector.nth at hieq,\n      rw this,\n      simp,\n      cases a_property,\n      apply a_val_ih,\n      repeat { sorry },\n    }\n  }\nend\n-- | \u27e8[], _\u27e9 \u27e8[], ha\u27e9 _ _ := by refl\n-- | \u27e8 a :: as , _\u27e9 \u27e8 b :: bs, _\u27e9 i hieq := begin\n  \n-- end\n\n\n--   intros i hieq,\n--   exact match a with\n--   | \u27e8[], ha\u27e9 := \n--     begin\n--       exact match b with\n--       | \u27e8[], ha\u27e9 := sorry\n--       |\n--       end,\n--     end\n--   end\n-- end\n\n@[simp] lemma vector_0_eq {\u03b1 : Type} : \u2200(v : vector \u03b1 0), v = vector.nil\n| \u27e8l, hl\u27e9 := subtype.eq $ show l = list.nil, from list.length_eq_zero.1 hl\n\nlemma vector_0_eq' {\u03b1 : Type} {v v' : vector \u03b1 0} : v = v' := sorry\n\n@[simp] lemma map\u2082_nil {\u03b1 \u03b2 \u03b3 : Type} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  \u2200 (v : vector \u03b1 0) w, vector.map\u2082 f v w = vector.nil\n| \u27e8[], _\u27e9 \u27e8[], _\u27e9 := by simp [vector.map\u2082, vector.nil]\n\n@[simp] lemma map\u2082_nil' {\u03b1 \u03b2 \u03b3 : Type} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (v : vector \u03b1 0) (p) :\n  vector.map\u2082 f v \u27e8list.nil, p\u27e9 = vector.nil := sorry\n\n@[simp] lemma nth_map (f : \u03b1 -> \u03b2) (v : vector \u03b1 n) (i) : (v.map f).nth i = f (v.nth i) := match v with\n| \u27e8l, hl\u27e9 := begin unfold nth map, sorry end\nend\n\n@[simp] lemma nth_map\u2082 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (v : vector \u03b1 n) (w : vector \u03b2 n) (i) : nth (map\u2082 f v w) i = f (v.nth i) (w.nth i) := sorry\n\nlemma map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (v : vector \u03b1 n) : map g (map f v) = map (g \u2218 f) v := begin\n  cases v,\n  apply vector.eq,\n  simp [to_list, map, list.map_map],\nend\n\nlemma map\u2082_map\u2082 (g : \u03b3 \u2192 \u03b2 \u2192 \u03b4) (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (v : vector \u03b1 n) (v' : vector \u03b2 n) : map\u2082 g (map\u2082 f v v') v' = map\u2082 (\u03bb a b, g (f a b) b) v v' := begin\n  cases v,\n  cases v',\n  apply vector.eq,\n  simp [to_list, map, map\u2082],\n  apply list.map\u2082_map\u2082,\nend\n\n-- example {f a as b bs h h' h''} :\n--   vector.map\u2082 f \u27e8 a :: as, h\u27e9 \u27e8b :: bs, h'\u27e9 = \u27e8f a b :: vector.map\u2082 f as bs, h''\u27e9 :=\n-- sorry\n\ndef range (n : \u2115) : vector \u2115 n := \u27e8list.range n, sorry\u27e9\n\nlemma range_nth {n : \u2115} {i : fin n} : (range n).nth i = i := sorry\n\ndef range_fin (n : \u2115) : vector (fin n) n := \u27e8list.range_fin n, sorry\u27e9\n\n@[simp] lemma length_map {\u03b1 n} (f : \u03b1 \u2192 \u03b2) (l : vector \u03b1 n) : length (map f l) = length l := sorry\n\n@[simp] lemma length_range_nth : length (range_fin n) = n := sorry\n\nlemma eq_one {\u03b1 : Type} (v : vector \u03b1 1) (v' : vector \u03b1 1) : v.nth \u27e80, sorry\u27e9 = v'.nth \u27e80, by sorry\u27e9 \u2194 (v = v') := sorry\n\nlemma eq_one' {\u03b1 : Type} (a b : \u03b1) : a = b \u2194 (v[a] = v[b]) := sorry\n\nlemma length_list {\u03b1 : Type} {n} {v : vector \u03b1 n} : list.length (vector.to_list (v)) = vector.length v := begin\n  admit,\nend\n\nlemma map_single {\u03b1 \u03b2 : Type} (f : \u03b1 \u2192 \u03b2) (e : \u03b1) : vector.map f v[e] = v[f e] := begin\n    refl,\nend\n\nend vector\n\nnamespace bool\n\nlemma eq_tt_coe {b : bool} : b \u2194 (b = tt) := begin\n  sorry,\nend\n\nlemma bnot_bnot {\u03b1 : Type} {f : \u03b1 \u2192 bool} : (bnot \u2218 bnot \u2218 f) = f := begin\n  sorry,\nend\n\nlemma bnot_ff (b : bool) : bnot b = (b = ff) := begin\n  by_cases b = ff,\n  { rw h, refl, },\n  { simp at h, subst h, simp, }\nend\n\nend bool", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/aux.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3761544154358779}}
{"text": "import Mathlib.Data.Finmap\n\n-- TODO: Isn't this tactic part of Mathlib?\nsyntax \"case' \" (Lean.binderIdent*),* \" => \" tacticSeq : tactic\nmacro_rules\n  | `(tactic| case' $[$xs*],* => $tac) => do\n    let tacs \u2190 xs.mapM fun xs => `(tactic| case $(xs[0]!) $(xs[1:])* => $tac)\n    `(tactic| ($[$tacs]*))\n\nnamespace Set \n\ntheorem insert_union' (s\u2081 s\u2082 : Set \u03b1) (a : \u03b1) : (insert a s\u2081) \u222a s\u2082 = s\u2081 \u222a (insert a s\u2082) := by\n  rw [Set.insert_union, Set.union_comm, \u2190Set.insert_union, Set.union_comm]\n\ntheorem ssubset_ne {s\u2081 s\u2082 : Set \u03b1} (h : s\u2081 \u2282 s\u2082) : s\u2081 \u2260 s\u2082 :=\n  ssubset_iff_subset_ne.mp h |>.right\n\nend Set\n\nabbrev Partial (\u03b1 \u03b2) := \u03b1 \u2192 Option \u03b2\n\ninfixr:50 \" \u21c0 \" => Partial\n\nnamespace Partial\n\ndef empty : \u03b1 \u21c0 \u03b2 :=  \n  fun _ => none\n\ninstance {\u03b1 \u03b2 : Type _} : EmptyCollection (\u03b1 \u21c0 \u03b2) where\n  emptyCollection := empty\n\ntheorem empty_iff {f : \u03b1 \u21c0 \u03b2} : (f = \u2205) \u2194 (\u2200 i, f i = none) where\n  mp h _ := h \u25b8 rfl\n  mpr h := funext h  \n  \ndef Nonempty (f : \u03b1 \u21c0 \u03b2) : Prop :=\n  f \u2260 \u2205  \n\ndef ids (f : \u03b1 \u21c0 \u03b2) := { a | \u2203 b, f a = some b }\n\ntheorem Nonempty.def {f : \u03b1 \u21c0 \u03b2} : f.Nonempty \u2194 (f \u2260 \u2205) :=\n  sorry\n\ntheorem Nonempty.iff_ids_nonempty {f : \u03b1 \u21c0 \u03b2} : f.Nonempty \u2194 f.ids.Nonempty :=\n  sorry\n\ninstance : Membership \u03b1 (\u03b1 \u21c0 \u03b2) where\n  mem a f := a \u2208 f.ids \n\ntheorem mem_def {f : \u03b1 \u21c0 \u03b2} : (a \u2208 f) \u2194 (a \u2208 f.ids) := by\n  rfl\n\ntheorem mem_iff {f : \u03b1 \u21c0 \u03b2} : (a \u2208 f) \u2194 (\u2203 b, f a = some b) := by\n  rfl\n\ndef attach (f : \u03b1 \u21c0 \u03b2) : \u03b1 \u21c0 { b // \u2203 a, f a = some b } := \n  fun a => \n    match h : f a with\n    | none => none\n    | some b => some \u27e8b, \u27e8_, h\u27e9\u27e9\n\ndef map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u21c0 \u03b2) : \u03b1 \u21c0 \u03b3 := \n  fun a => g <$> f a\n\ntheorem map_val (f : \u03b1 \u21c0 \u03b2) (g : \u03b2 \u2192 \u03b3) : (f.map g) a = (f a).map g := \n  rfl\n\ntheorem map_map (f : \u03b1 \u21c0 \u03b2) (g\u2081 : \u03b2 \u2192 \u03b3) (g\u2082 : \u03b3 \u2192 \u03b4) : (f.map g\u2081).map g\u2082 = f.map (g\u2082 \u2218 g\u2081) := by\n  simp [map]\n\ntheorem map_inj {f\u2081 f\u2082 : \u03b1 \u21c0 \u03b2} (hi : g.Injective) (h : f\u2081.map g = f\u2082.map g) : f\u2081 = f\u2082 := by\n  funext a\n  replace h : f\u2081.map g a = f\u2082.map g a := by simp [h]\n  exact Option.map_injective hi h\n\ntheorem attach_map_val (f : \u03b1 \u21c0 \u03b2) : f.attach.map Subtype.val = f := by\n  sorry\n\ndef restrict (f : \u03b1 \u21c0 \u03b2) (s : Set \u03b1) [DecidablePred (\u00b7 \u2208 s)] : \u03b1 \u21c0 \u03b2 := \n  fun a => if a \u2208 s then f a else none \n\ntheorem ext_restrict {f g : \u03b1 \u21c0 \u03b2} (h : \u2200 a \u2208 s, f a = g a) [DecidablePred (\u00b7 \u2208 s)] : \n    (f.restrict s) = (g.restrict s) := \n  sorry\n\ndef filterMap (f : \u03b1 \u21c0 \u03b2) (g : \u03b2 \u2192 Option \u03b3) : \u03b1 \u21c0 \u03b3 := \n  fun a => f a >>= g\n\ndef insert [DecidableEq \u03b1] (f : \u03b1 \u21c0 \u03b2) (a : \u03b1) (b : \u03b2) : \u03b1 \u21c0 \u03b2 :=\n  fun a' => if a' = a then b else f a'\n\ntheorem insert_same [DecidableEq \u03b1] (f : \u03b1 \u21c0 \u03b2) : (f.insert a b) a = b := by\n  simp [insert]\n\ntheorem insert_ne [DecidableEq \u03b1] (f : \u03b1 \u21c0 \u03b2) (h : a' \u2260 a := by assumption) : \n    (f.insert a b) a' = f a' := by\n  simp [insert, h]\n\nend Partial\n\nnamespace Finmap\n\ninstance [DecidableEq \u03b1] : CoeFun (Finmap fun _ : \u03b1 => \u03b2) (fun _ => \u03b1 \u2192 Option \u03b2) where\n  coe f := f.lookup\n\nend Finmap", "meta": {"author": "marcusrossel", "repo": "reactor-model", "sha": "f82fffb489b4352a0cc6bee964d44a142fee18ce", "save_path": "github-repos/lean/marcusrossel-reactor-model", "path": "github-repos/lean/marcusrossel-reactor-model/reactor-model-f82fffb489b4352a0cc6bee964d44a142fee18ce/src/ReactorModel/Extensions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.37615440761906216}}
{"text": "-- import group_theory.level1 --hide\n\n\n/-\n# Level 2: Union of two open sets\n-/\n\n/- Lemma\nThe union of two open sets is open.\n-/\n-- lemma open_of_union {X : Type} [topological_space X] {U V : set X}\n-- (hU : is_open U) (hV : is_open V): is_open (U \u222a V) :=\n-- begin\n--   let I : set (set X) := {U, V},\n--   have H : \u22c3\u2080 I = U \u222a V := sUnion_pair U V,\n--   rw \u2190H,\n--   apply union I,\n--   intros B hB,\n--   replace hB : B = U \u2228 B = V, by tauto,\n--   cases hB; {rw hB, assumption},\n\n\n\n\n-- end\n", "meta": {"author": "alexjbest", "repo": "CAP-game", "sha": "d823def7325d7142d61e766b2e027f936685a8ff", "save_path": "github-repos/lean/alexjbest-CAP-game", "path": "github-repos/lean/alexjbest-CAP-game/CAP-game-d823def7325d7142d61e766b2e027f936685a8ff/src/simplifier/level2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.3761544076190621}}
{"text": "import Lean.Server.Requests\nimport Lean.Server.Rpc.Basic\nimport Lean.Server.Rpc.RequestHandling\nimport Lean.Elab\nimport Lean.Widget.Basic\n\nimport UserWidget.ToHtml.Widget\n\n/-! Demonstrates commutative diagrams.\n\nCategory classes copied from mathlib. -/\n\nclass quiver (V : Type u) where\n  hom : V \u2192 V \u2192 Sort v\n\ninfixr:10 \" \u27f6 \" => quiver.hom -- type as \\h\n\nclass category_struct (obj : Type u) extends quiver.{u,v+1} obj : Type (max u (v+1)) where\n  id   : \u2200 X : obj, hom X X\n  comp : \u2200 {X Y Z : obj}, (X \u27f6 Y) \u2192 (Y \u27f6 Z) \u2192 (X \u27f6 Z)\n\n-- https://github.com/leanprover/lean4/issues/1367\nprefix:max \"\ud835\udfd9 \" => category_struct.id -- type as \\b1\ninfixr:80 \" \u226b \" => category_struct.comp -- type as \\gg\n\nclass category (obj : Type u) extends category_struct.{u,v} obj : Type (max u (v+1)) where\n  id_comp' : \u2200 {X Y : obj} (f : hom X Y), \ud835\udfd9 X \u226b f = f\n  comp_id' : \u2200 {X Y : obj} (f : hom X Y), f \u226b \ud835\udfd9 Y = f\n  assoc'   : \u2200 {W X Y Z : obj} (f : hom W X) (g : hom X Y) (h : hom Y Z),\n    (f \u226b g) \u226b h = f \u226b (g \u226b h)\n\ninstance : category (Type u) where\n  hom \u03b1 \u03b2 := \u03b1 \u2192 \u03b2\n  id _ := id\n  comp f g := g \u2218 f\n  id_comp' _ := rfl\n  comp_id' _ := rfl\n  assoc' _ _ _ := rfl\n\nopen Lean.Widget in\n@[widget]\ndef squares : UserWidgetDefinition where\n  name := \"Commutative diagram\"\n  javascript := include_str \"..\" / \"..\" / \"widget\" / \"dist\" / \"squares.js\"\n\nsyntax (name := squaresTacStx) \"squares!\" : tactic\nopen Lean Elab Tactic in\n@[tactic squaresTacStx]\ndef squaresTac : Tactic\n  | stx@`(tactic| squares!) => do\n    if let some _ := stx.getPos? then\n      Lean.Widget.saveWidgetInfo \"squares\" Json.null stx\n  | _ => throwUnsupportedSyntax\n\nopen Lean Widget Server\n\n@[inline] def Lean.Expr.app7? (e : Expr) (fName : Name) : Option (Expr \u00d7 Expr \u00d7 Expr \u00d7 Expr \u00d7 Expr \u00d7 Expr \u00d7 Expr) :=\n  if e.isAppOfArity fName 7 then\n    some (\n      e.appFn!.appFn!.appFn!.appFn!.appFn!.appFn!.appArg!,\n      e.appFn!.appFn!.appFn!.appFn!.appFn!.appArg!,\n      e.appFn!.appFn!.appFn!.appFn!.appArg!,\n      e.appFn!.appFn!.appFn!.appArg!,\n      e.appFn!.appFn!.appArg!,\n      e.appFn!.appArg!,\n      e.appArg!\n    )\n  else\n    none\n\n/-- Given a hom `f : \u03b1 \u27f6 \u03b2`, return `(\u03b1, \u03b2)`. Otherwise `none`. -/\ndef homTypesM? (f : Expr) : MetaM (Option (Expr \u00d7 Expr)) := do\n  let fTp \u2190 Meta.inferType f >>= instantiateMVars\n  let some (_, _, A, B) := fTp.app4? ``quiver.hom | return none\n  return (A, B)\n\n/-- Given composed homs `f \u2261 g \u226b h`, return `(g, h)`. Otherwise `none`. -/\ndef homComp? (f : Expr) : Option (Expr \u00d7 Expr) :=\n  if let some (_, _, _, _, _, f, g) := f.app7? ``category_struct.comp then\n    some (f, g)\n  else none\n\ninductive DiagramKind | square | triangle\nderiving Inhabited, FromJson, ToJson\n\n/--\nSquare with `homs = [f,g,h,i]` and `objs = [A,B,C,D]`\n```\nA f B\ni   g\nD h C\n```\nTriangle with `homs = [f,g,h]` and `objs = [A,B,C]`\n```\nA f B\n  h g\n    C\n```\n-/\nstructure DiagramData where\n  objs : Array CodeWithInfos\n  homs : Array CodeWithInfos\n  kind : DiagramKind\n  deriving Inhabited, RpcEncoding\n\n/-- Given a commutative square `e \u2261 f \u226b g = i \u226b h`, return a square diagram. Otherwise `none`. -/\ndef homSquareM? (e : Expr) : MetaM (Option DiagramData) := do\n  let some (_, lhs, rhs) := e.eq? | return none\n  let some (f, g) := homComp? lhs | return none\n  let some (i, h) := homComp? rhs | return none\n  let some (A, B) \u2190 homTypesM? f | return none\n  let some (C, D) \u2190 homTypesM? h | return none\n  let pp (e : Expr) := ppExprTagged e\n  return some {\n    objs := #[\u2190 pp A, \u2190 pp B, \u2190 pp C, \u2190 pp D]\n    homs := #[\u2190 pp f, \u2190 pp g, \u2190 pp h, \u2190 pp i]\n    kind := .square\n  }\n\n/-- Given a commutative triangle `e \u2261 f \u226b g = h` or `e \u2261 h = f \u226b g`, return a triangle diagram.\nOtherwise `none`. -/\ndef homTriangleM? (e : Expr) : MetaM (Option DiagramData) := do\n  let some (_, lhs, rhs) := e.eq? | return none\n  let pp (e : Expr) := ppExprTagged e\n  if let some (f, g) := homComp? lhs then\n    let some (A, C) \u2190 homTypesM? rhs | return none\n    let some (_, B) \u2190 homTypesM? f | return none\n    return some {\n      objs := #[\u2190 pp A, \u2190 pp B, \u2190 pp C]\n      homs := #[\u2190 pp f, \u2190 pp g, \u2190 pp rhs]\n      kind := .triangle\n    }\n  let some (f, g) := homComp? rhs | return none\n  let some (A, C) \u2190 homTypesM? lhs | return none\n  let some (_, B) \u2190 homTypesM? f | return none\n  return some {\n    objs := #[\u2190 pp A, \u2190 pp B, \u2190 pp C]\n    homs := #[\u2190 pp f, \u2190 pp g, \u2190 pp lhs]\n    kind := .triangle\n  }\n\nopen Lean Server RequestM in\n@[serverRpcMethod]\ndef getCommutativeDiagram (args : Lean.Lsp.Position) : RequestM (RequestTask (Option DiagramData)) := do\n  let doc \u2190 readDoc\n  let pos := doc.meta.text.lspPosToUtf8Pos args\n  withWaitFindSnapAtPos args fun snap => do\n    let g :: _ := snap.infoTree.goalsAt? doc.meta.text pos | return none\n    let { ctxInfo := ci, tacticInfo := ti, useAfter := useAfter, .. } := g\n    let ci := if useAfter then { ci with mctx := ti.mctxAfter } else { ci with mctx := ti.mctxBefore }\n    let g :: _ := if useAfter then ti.goalsAfter else ti.goalsBefore | return none\n    ci.runMetaM {} <| do\n      let some mvarDecl := (\u2190 getMCtx).findDecl? g\n        | throwError \"unknown goal {g.name}\"\n      let lctx := mvarDecl.lctx\n      let lctx := lctx.sanitizeNames.run' { options := (\u2190 getOptions) }\n      Meta.withLCtx lctx mvarDecl.localInstances do\n        let type \u2190 g.getType >>= instantiateMVars\n        if let some d \u2190 homSquareM? type then\n          return some d\n        if let some d \u2190 homTriangleM? type then\n          return some d\n        return none\n\nexample {f g : Nat \u27f6 Bool}: f = g \u2192 (f \u226b \ud835\udfd9 Bool) = (g \u226b \ud835\udfd9 Bool) := by\n  intro h\n  squares!\n  exact h\n\nexample {f g : Nat \u27f6 Bool}: f = g \u2192 f = (g \u226b \ud835\udfd9 Bool) := by\n  intro h\n  squares!\n  exact h\n", "meta": {"author": "Vtec234", "repo": "npm-widget", "sha": "b7ba6a7cdc3e66e0614a16225e3bd1aee009e371", "save_path": "github-repos/lean/Vtec234-npm-widget", "path": "github-repos/lean/Vtec234-npm-widget/npm-widget-b7ba6a7cdc3e66e0614a16225e3bd1aee009e371/UserWidget/Demos/CommDiag.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.37605363548231013}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# `pretty_cases` tactic\n\nWhen using `induction` and `cases`, `pretty_cases` prints a `\"Try\nthis:\"` advice that shows how to structure the proof with\n`case { ... }` commands.  In the following example, we apply induction on a\npermutation assumption about lists. `pretty_cases` gives us a proof\nskeleton that explicit selects the branches and explicit names the\nnew local constants:\n\n```lean\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  induction h,\n  pretty_cases,\n    -- Try this:\n    --   case list.perm.nil :\n    --   { admit },\n    --   case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_a h_ih\n    --   { admit },\n    --   case list.perm.swap : h_x h_y h_l\n    --   { admit },\n    --   case list.perm.trans : h_l\u2081 h_l\u2082 h_l\u2083 h_a h_a_1 h_ih_a h_ih_a_1\n    --   { admit },\nend\n```\n\n## Main definitions\n\n * `pretty_cases_advice` return `pretty_cases` advice without printing it\n * `pretty_cases` main tactic\n-/\n\nnamespace tactic\n\n\n/-- Query the proof goal and print the skeleton of a proof by cases. -/\nnamespace interactive\n\n\n/--\nQuery the proof goal and print the skeleton of a proof by\ncases.\n\nFor example, let us consider the following proof:\n\n```lean\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  induction h,\n  pretty_cases,\n    -- Try this:\n    --   case list.perm.nil :\n    --   { admit },\n    --   case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_a h_ih\n    --   { admit },\n    --   case list.perm.swap : h_x h_y h_l\n    --   { admit },\n    --   case list.perm.trans : h_l\u2081 h_l\u2082 h_l\u2083 h_a h_a_1 h_ih_a h_ih_a_1\n    --   { admit },\nend\n```\n\nThe output helps the user layout the cases and rename the\nintroduced variables.\n-/\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/pretty_cases_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565326, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.3760287778106234}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.Match.Match\nimport Lean.Elab.RecAppSyntax\nimport Lean.Elab.PreDefinition.Structural.Basic\n\nnamespace Lean.Elab.Structural\nopen Meta\n\nprivate def throwToBelowFailed : MetaM \u03b1 :=\n  throwError \"toBelow failed\"\n\n/- See toBelow -/\nprivate partial def toBelowAux (C : Expr) : Expr \u2192 Expr \u2192 Expr \u2192 MetaM Expr\n  | belowDict, arg, F => do\n    let belowDict \u2190 whnf belowDict\n    trace[Elab.definition.structural] \"belowDict: {belowDict}, arg: {arg}\"\n    match belowDict with\n    | Expr.app (Expr.app (Expr.const `PProd _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `PProd.fst #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `PProd.snd #[F]))\n    | Expr.app (Expr.app (Expr.const `And _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `And.left #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `And.right #[F]))\n    | _ => forallTelescopeReducing belowDict fun xs belowDict => do\n      let argArgs := arg.getAppArgs\n      unless argArgs.size >= xs.size do throwToBelowFailed\n      let n := argArgs.size\n      let argTailArgs := argArgs.extract (n - xs.size) n\n      let belowDict := belowDict.replaceFVars xs argTailArgs\n      match belowDict with\n      | Expr.app belowDictFun belowDictArg _ =>\n        unless belowDictFun.getAppFn == C do throwToBelowFailed\n        unless \u2190 isDefEq belowDictArg arg do throwToBelowFailed\n        pure (mkAppN F argTailArgs)\n      | _ => throwToBelowFailed\n\n/- See toBelow -/\nprivate def withBelowDict (below : Expr) (numIndParams : Nat) (k : Expr \u2192 Expr \u2192 MetaM \u03b1) : MetaM \u03b1 := do\n  let belowType \u2190 inferType below\n  trace[Elab.definition.structural] \"belowType: {belowType}\"\n  belowType.withApp fun f args => do\n    let motivePos := numIndParams + 1\n    unless motivePos < args.size do throwError \"unexpected 'below' type{indentExpr belowType}\"\n    let pre := mkAppN f (args.extract 0 numIndParams)\n    let preType \u2190 inferType pre\n    forallBoundedTelescope preType (some 1) fun x _ => do\n      let motiveType \u2190 inferType x[0]\n      withLocalDeclD (\u2190 mkFreshUserName `C) motiveType fun C =>\n        let belowDict := mkApp pre C\n        let belowDict := mkAppN belowDict (args.extract (numIndParams + 1) args.size)\n        k C belowDict\n\n/-\n  `below` is a free variable with type of the form `I.below indParams motive indices major`,\n  where `I` is the name of an inductive datatype.\n\n  For example, when trying to show that the following function terminates using structural recursion\n  ```lean\n  def addAdjacent : List Nat \u2192 List Nat\n  | []       => []\n  | [a]      => [a]\n  | a::b::as => (a+b) :: addAdjacent as\n  ```\n  when we are visiting `addAdjacent as` at `replaceRecApps`, `below` has type\n  `@List.below Nat (fun (x : List Nat) => List Nat) (a::b::as)`\n  The motive `fun (x : List Nat) => List Nat` depends on the actual function we are trying to compute.\n  So, we first replace it with a fresh variable `C` at `withBelowDict`.\n  Recall that `brecOn` implements course-of-values recursion, and `below` can be viewed as a dictionary\n  of the \"previous values\".\n  We search this dictionary using the auxiliary function `toBelowAux`.\n  The dictionary is built using the `PProd` (`And` for inductive predicates).\n  We keep searching it until we find `C recArg`, where `C` is the auxiliary fresh variable created at `withBelowDict`.  -/\nprivate partial def toBelow (below : Expr) (numIndParams : Nat) (recArg : Expr) : MetaM Expr := do\n  withBelowDict below numIndParams fun C belowDict =>\n    toBelowAux C belowDict recArg below\n\nprivate partial def replaceRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (below : Expr) (e : Expr) : M Expr :=\n  let rec loop (below : Expr) (e : Expr) : M Expr := do\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkLambdaFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkForallFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (\u2190 loop below type) (\u2190 loop below val) fun x => do\n        mkLetFVars #[x] (\u2190 loop below (body.instantiate1 x)) (usedLetOnly := false)\n    | Expr.mdata d b _   =>\n      if let some stx := getRecAppSyntax? e then\n        loop below b\n      else\n        return mkMData d (\u2190 loop below b)\n    | Expr.proj n i e _  => return mkProj n i (\u2190 loop below e)\n    | Expr.app _ _ _ =>\n      let processApp (e : Expr) : M Expr :=\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let numFixed  := recArgInfo.fixedParams.size\n            let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n            if recArgPos >= args.size then\n              throwError \"insufficient number of parameters at recursive application {indentExpr e}\"\n            let recArg := args[recArgPos]\n            -- For reflexive type, we may have nested recursive applications in recArg\n            let recArg \u2190 loop below recArg\n            let f \u2190 try toBelow below recArgInfo.indParams.size recArg catch  _ => throwError \"failed to eliminate recursive application{indentExpr e}\"\n            -- Recall that the fixed parameters are not in the scope of the `brecOn`. So, we skip them.\n            let argsNonFixed := args.extract numFixed args.size\n            -- The function `f` does not explicitly take `recArg` and its indices as arguments. So, we skip them too.\n            let mut fArgs := #[]\n            for i in [:argsNonFixed.size] do\n              if recArgInfo.pos != i && !recArgInfo.indicesPos.contains i then\n                let arg := argsNonFixed[i]\n                let arg \u2190 replaceRecApps recFnName recArgInfo below arg\n                fArgs := fArgs.push arg\n            return mkAppN f fArgs\n          else\n            return mkAppN (\u2190 loop below f) (\u2190 args.mapM (loop below))\n      let matcherApp? \u2190 matchMatcherApp? e\n      match matcherApp? with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo.recArgPos e then\n          processApp e\n        else\n          /- Here is an example we currently not handle\n             ```\n             def g (xs : List Nat) : Nat :=\n             match xs with\n             | [] => 0\n             | y::ys =>\n               match ys with\n               | []       => 1\n               | _::_::zs => g zs + 1\n               | zs       => g ys + 2\n             ```\n             We are matching on `ys`, but still using `ys` in the third alternative.\n             If we push the `below` argument over the dependent match it will be able to eliminate recursive call using `zs`.\n             To make it work, users have to write the third alternative as `| zs => g zs + 2`\n             If this is too annoying in practice, we may replace `ys` with the matching term, but\n             this may generate weird error messages, when it doesn't work. -/\n          trace[Elab.definition.structural] \"below before matcherApp.addArg: {below} : {\u2190 inferType below}\"\n          let matcherApp \u2190 mapError (matcherApp.addArg below) (fun msg => \"failed to add `below` argument to 'matcher' application\" ++ indentD msg)\n          let altsNew \u2190 (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n            lambdaTelescope alt fun xs altBody => do\n              trace[Elab.definition.structural] \"altNumParams: {numParams}, xs: {xs}\"\n              unless xs.size >= numParams do\n                throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n              let belowForAlt := xs[numParams - 1]\n              mkLambdaFVars xs (\u2190 loop belowForAlt altBody)\n          pure { matcherApp with alts := altsNew }.toExpr\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop below e\n\ndef mkBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  trace[Elab.definition.structural] \"mkBRecOn: {value}\"\n  let type  := (\u2190 inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive \u2190 mkForallFVars otherArgs type\n  let mut brecOnUniv \u2190 getLevel motive\n  trace[Elab.definition.structural] \"brecOn univ: {brecOnUniv}\"\n  let useBInductionOn := recArgInfo.reflexive && brecOnUniv == levelZero\n  if recArgInfo.reflexive && brecOnUniv != levelZero then\n    brecOnUniv \u2190 decLevel brecOnUniv\n  let motive \u2190 mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n    if useBInductionOn then\n      Lean.mkConst (mkBInductionOnName recArgInfo.indName) recArgInfo.indLevels\n    else\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) (brecOnUniv :: recArgInfo.indLevels)\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType \u2190 inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]\n    let FType \u2190 inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType \u2190 instantiateForall FType recArgInfo.indIndices\n    let FType \u2190 instantiateForall FType #[major]\n    forallBoundedTelescope FType (some 1) fun below _ => do\n      -- TODO: `below` user name is `f`, and it will make a global `f` to be pretty printed as `_root_.f` in error messages.\n      -- We should add an option to `forallBoundedTelescope` to ensure fresh names are used.\n      let below := below[0]\n      let valueNew     \u2190 replaceRecApps recFnName recArgInfo below value\n      let Farg         \u2190 mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n      let brecOn       := mkApp brecOn Farg\n      return mkAppN brecOn otherArgs\n\nend Lean.Elab.Structural\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/stage0/src/Lean/Elab/PreDefinition/Structural/BRecOn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802735722129, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.376028771708362}}
{"text": "import polyhedral_lattice.basic\nimport category_theory.concrete_category.bundled_hom\nimport analysis.normed.group.SemiNormedGroup.kernels\n/-!\n\n# The category of polyhedral lattices\n\n-/\nuniverse variables u\n\nopen category_theory\n\n/-- The category of polyhedral lattices and bounded group homomorphisms. -/\ndef PolyhedralLattice : Type (u+1) := bundled polyhedral_lattice\n\ninstance : has_coe_to_sort PolyhedralLattice Type* := bundled.has_coe_to_sort\n\nnamespace PolyhedralLattice\n\nvariables (\u039b : PolyhedralLattice)\n\ninstance : polyhedral_lattice \u039b := \u039b.str\n\ndef to_SemiNormedGroup : SemiNormedGroup := SemiNormedGroup.of \u039b\n\ninstance bundled_hom : bundled_hom @polyhedral_lattice_hom :=\n\u27e8@polyhedral_lattice_hom.to_fun,\n@polyhedral_lattice_hom.id, @polyhedral_lattice_hom.comp, @polyhedral_lattice_hom.coe_inj\u27e9\n\nattribute [derive [large_category, concrete_category]] PolyhedralLattice\n\n/-- Construct a bundled `PolyhedralLattice` from the underlying type and typeclass. -/\ndef of (\u039b : Type u) [polyhedral_lattice \u039b] : PolyhedralLattice := bundled.of \u039b\n\n-- noncomputable\n-- instance : has_zero PolyhedralLattice := \u27e8of punit\u27e9\n\n-- noncomputable\n-- instance : inhabited PolyhedralLattice := \u27e80\u27e9\n\n@[simp] lemma coe_of (\u039b : Type u) [polyhedral_lattice \u039b] :\n  (PolyhedralLattice.of \u039b : Type u) = \u039b := rfl\n\n@[simp] lemma coe_id (\u039b : PolyhedralLattice) : \u21d1(\ud835\udfd9 \u039b) = id := rfl\n\ninstance : limits.has_zero_morphisms.{u (u+1)} PolyhedralLattice :=\n{ comp_zero' := by { intros, ext, refl },\n  zero_comp' := by { intros _ _ _ f, ext, exact f.map_zero } }\n\ndef iso_mk {\u039b\u2081 \u039b\u2082 : PolyhedralLattice.{u}}\n  (f : \u039b\u2081 \u2192+ \u039b\u2082) (g : \u039b\u2082 \u2192 \u039b\u2081) (hf : \u2200 l, \u2225f l\u2225 = \u2225l\u2225) (hfg : g \u2218 f = id) (hgf : f \u2218 g = id) :\n  \u039b\u2081 \u2245 \u039b\u2082 :=\n{ hom := { strict' := \u03bb l, le_of_eq (hf l), ..f },\n  inv :=\n  { strict' := \u03bb l,\n    calc \u2225g l\u2225 \u2264 \u2225f (g l)\u2225 : le_of_eq $ (hf _).symm\n    ... = \u2225l\u2225 : congr_arg norm $ congr_fun hgf l,\n    .. add_equiv.symm\n    { inv_fun := g,\n      left_inv := congr_fun hfg,\n      right_inv := congr_fun hgf,\n      .. f } },\n  hom_inv_id' := by { ext x, exact congr_fun hfg x },\n  inv_hom_id' := by { ext x, exact congr_fun hgf x } }\n\nend PolyhedralLattice\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/polyhedral_lattice/category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.37601007979935946}}
{"text": "import super\n\nset_option trace.super true\nset_option profiler true\n\nexample (i) (p q : i \u2192 i \u2192 Prop) (a b c d : i) :\n  (\u2200x y z, p x y \u2227 p y z \u2192 p x z) \u2192\n  (\u2200x y z, q x y \u2227 q y z \u2192 q x z) \u2192\n  (\u2200x y, q x y \u2192 q y x) \u2192\n  (\u2200x y, p x y \u2228 q x y) \u2192\n  p a b \u2228 q c d :=\nby tactic.try_for 10000 `[super *]\n", "meta": {"author": "gebner", "repo": "super2", "sha": "9bc5256c31750021ab97d6b59b7387773e54b384", "save_path": "github-repos/lean/gebner-super2", "path": "github-repos/lean/gebner-super2/super2-9bc5256c31750021ab97d6b59b7387773e54b384/test/complicated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3760099010276378}}
{"text": "import normed_free_pfpng.basic\nimport free_pfpng.basic\nimport condensed.exact\nimport condensed.condensify\n\nopen_locale nnreal big_operators\n\nvariables (p : \u211d\u22650) (S : Fintype)\nvariables [fact (0 < p)] [fact (p \u2264 1)]\n\nlemma nnnorm_int_rpow_le (n : \u2124) : \u2225n\u2225\u208a ^ (p : \u211d) \u2264 \u2225n\u2225\u208a :=\nbegin\n  have h0p : 0 < p := fact.out _,\n  have hp1 : p \u2264 1 := fact.out _,\n  rcases (eq_or_ne (\u2225n\u2225\u208a) 0) with (h|h),\n  { simp only [h, le_zero_iff, nnreal.rpow_eq_zero_iff, eq_self_iff_true,\n      ne.def, nnreal.coe_eq_zero, true_and],\n    exact h0p.ne' },\n  refine (nnreal.rpow_le_rpow_of_exponent_le _ hp1).trans _,\n  swap, { rw [nnreal.coe_one, nnreal.rpow_one] },\n  rw [\u2190 nnreal.coe_nat_abs] at h \u22a2,\n  norm_cast at h \u22a2,\n  exact nat.one_le_of_lt (ne.bot_lt h)\nend\n\ndef free_pfpng_to_normed_free_pfpng :\n  free_pfpng_functor \u27f6 normed_free_pfpng_functor p :=\n{ app := \u03bb S,\n  { to_fun := \u03bb f, f,\n    map_zero' := rfl,\n    map_add' := \u03bb _ _, rfl,\n    strict' := begin\n      have h0p : 0 < p := fact.out _,\n      have hp1 : p \u2264 1 := fact.out _,\n      rintro c f (hf : _ \u2264 _),\n      refine le_trans (finset.sum_le_sum _) hf,\n      rintro s -,\n      exact nnnorm_int_rpow_le p _,\n    end,\n    continuous' := \u03bb c, continuous_of_discrete_topology },\n  naturality' := by { intros S T f, ext \u03c6 t, refl } }\n.\n\nopen category_theory\n\nnoncomputable\ndef cond_free_pfpng_to_normed_free_pfpng :\n  condensify (free_pfpng_functor \u22d9 PFPNG\u2081_to_CHFPNG\u2081\u2091\u2097) \u27f6\n  condensify (normed_free_pfpng_functor p \u22d9 PFPNG\u2081_to_CHFPNG\u2081\u2091\u2097) :=\ncondensify_map $ whisker_right (free_pfpng_to_normed_free_pfpng p) _\n\nopen CompHausFiltPseuNormGrp\u2081\n\n-- move me\nlemma condensify_map_zero (F G : Fintype \u2964 CompHausFiltPseuNormGrp\u2081) :\n  condensify_map (0 : F \u27f6 G) = 0 :=\nbegin\n  delta condensify_map condensify_nonstrict,\n  suffices : nonstrict_extend\n    (whisker_right (0 : F \u27f6 G) CHFPNG\u2081_to_CHFPNG\u2091\u2097) 1 _ = 0,\n  { rw this, refl, },\n  rw [nonstrict_extend_whisker_right_enlarging,\n    Profinite.extend_nat_trans_zero],\n  refl,\nend\n\ninstance (S : Profinite) : mono ((cond_free_pfpng_to_normed_free_pfpng p).app S) :=\nbegin\n  simp only [cond_free_pfpng_to_normed_free_pfpng, condensify_map, condensify_nonstrict],\n  rw nonstrict_extend_whisker_right_enlarging,\n  apply Condensed.mono_to_Condensed_map,\n  apply exact_with_constant_extend_zero_left,\n  intro S,\n  apply_with exact_with_constant_of_mono { instances := ff },\n  rw [AddCommGroup.mono_iff_injective, injective_iff_map_eq_zero],\n  intros f hf,\n  exact hf,\nend\n\ninstance (S : Profinite) : epi ((cond_free_pfpng_to_normed_free_pfpng p).app S) :=\nbegin\n  simp only [cond_free_pfpng_to_normed_free_pfpng, condensify_map, condensify_nonstrict],\n  rw nonstrict_extend_whisker_right_enlarging,\n  let \u03ba : \u211d\u22650 \u2192 \u211d\u22650 := \u03bb c, max c (c ^ (p\u207b\u00b9 : \u211d)),\n  have h\u03ba : id \u2264 \u03ba := \u03bb c, le_max_left _ _,\n  apply Condensed.epi_to_Condensed_map _ \u03ba,\n  apply exact_with_constant_extend_zero_right,\n  intro S,\n  apply exact_with_constant_of_epi _ _ _ h\u03ba,\n  intros c f hf,\n  refine \u27e8f, _, rfl\u27e9,\n  change \u2211 _, _ \u2264 _ at hf,\n  show \u2211 _, _ \u2264 _,\n  have h0p : 0 < p := fact.out _,\n  have hp1 : p \u2264 1 := fact.out _,\n  have h0pinv : 0 \u2264 p\u207b\u00b9, { rw \u2190 nnreal.inv_pos at h0p, exact h0p.le },\n  have := (nnreal.rpow_sum_le_sum_rpow _ _ h0p hp1).trans hf,\n  replace this := nnreal.rpow_le_rpow this h0pinv,\n  rw [\u2190 nnreal.rpow_mul, \u2190 nnreal.coe_mul, mul_inv_cancel h0p.ne',\n    nnreal.coe_one, nnreal.rpow_one] at this,\n  exact this.trans (le_max_right _ _),\nend\n\ninstance (S : Profinite) : is_iso ((cond_free_pfpng_to_normed_free_pfpng p).app S) :=\nis_iso_of_mono_of_epi _\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/normed_free_pfpng/compare.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3759840426605501}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.subobject.factor_thru\nimport category_theory.subobject.well_powered\n\n/-!\n# The lattice of subobjects\n\nWe provide the `semilattice_inf_top (subobject X)` instance when `[has_pullback C]`,\nand the `semilattice_sup (subobject X)` instance when `[has_images C] [has_binary_coproducts C]`.\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {X Y Z : C}\nvariables {D : Type u\u2082} [category.{v\u2082} D]\n\nnamespace category_theory\n\nnamespace mono_over\n\nsection has_top\n\ninstance {X : C} : has_top (mono_over X) :=\n{ top := mk' (\ud835\udfd9 _) }\n\ninstance {X : C} : inhabited (mono_over X) := \u27e8\u22a4\u27e9\n\n/-- The morphism to the top object in `mono_over X`. -/\ndef le_top (f : mono_over X) : f \u27f6 \u22a4 :=\nhom_mk f.arrow (comp_id _)\n\n@[simp] lemma top_left (X : C) : ((\u22a4 : mono_over X) : C) = X := rfl\n@[simp] lemma top_arrow (X : C) : (\u22a4 : mono_over X).arrow = \ud835\udfd9 X := rfl\n\n/-- `map f` sends `\u22a4 : mono_over X` to `\u27e8X, f\u27e9 : mono_over Y`. -/\ndef map_top (f : X \u27f6 Y) [mono f] : (map f).obj \u22a4 \u2245 mk' f :=\niso_of_both_ways (hom_mk (\ud835\udfd9 _) rfl) (hom_mk (\ud835\udfd9 _) (by simp [id_comp f]))\n\nsection\nvariable [has_pullbacks C]\n\n/-- The pullback of the top object in `mono_over Y`\nis (isomorphic to) the top object in `mono_over X`. -/\ndef pullback_top (f : X \u27f6 Y) : (pullback f).obj \u22a4 \u2245 \u22a4 :=\niso_of_both_ways (le_top _) (hom_mk (pullback.lift f (\ud835\udfd9 _) (by tidy)) (pullback.lift_snd _ _ _))\n\n/-- There is a morphism from `\u22a4 : mono_over A` to the pullback of a monomorphism along itself;\nas the category is thin this is an isomorphism. -/\ndef top_le_pullback_self {A B : C} (f : A \u27f6 B) [mono f] :\n  (\u22a4 : mono_over A) \u27f6 (pullback f).obj (mk' f) :=\nhom_mk _ (pullback.lift_snd _ _ rfl)\n\n/-- The pullback of a monomorphism along itself is isomorphic to the top object. -/\ndef pullback_self {A B : C} (f : A \u27f6 B) [mono f] :\n  (pullback f).obj (mk' f) \u2245 \u22a4 :=\niso_of_both_ways (le_top _) (top_le_pullback_self _)\n\nend\n\nend has_top\n\nsection has_bot\nvariables [has_zero_morphisms C] [has_zero_object C]\nlocal attribute [instance] has_zero_object.has_zero\n\ninstance {X : C} : has_bot (mono_over X) :=\n{ bot := mk' (0 : 0 \u27f6 X) }\n\n@[simp] lemma bot_left (X : C) : ((\u22a5 : mono_over X) : C) = 0 := rfl\n@[simp] lemma bot_arrow {X : C} : (\u22a5 : mono_over X).arrow = 0 :=\nby ext\n\n/-- The (unique) morphism from `\u22a5 : mono_over X` to any other `f : mono_over X`. -/\ndef bot_le {X : C} (f : mono_over X) : \u22a5 \u27f6 f :=\nhom_mk 0 (by simp)\n\n/-- `map f` sends `\u22a5 : mono_over X` to `\u22a5 : mono_over Y`. -/\ndef map_bot (f : X \u27f6 Y) [mono f] : (map f).obj \u22a5 \u2245 \u22a5 :=\niso_of_both_ways (hom_mk 0 (by simp)) (hom_mk (\ud835\udfd9 _) (by simp [id_comp f]))\n\nend has_bot\n\nsection inf\nvariables [has_pullbacks C]\n\n/--\nWhen `[has_pullbacks C]`, `mono_over A` has \"intersections\", functorial in both arguments.\n\nAs `mono_over A` is only a preorder, this doesn't satisfy the axioms of `semilattice_inf`,\nbut we reuse all the names from `semilattice_inf` because they will be used to construct\n`semilattice_inf (subobject A)` shortly.\n-/\n@[simps]\ndef inf {A : C} : mono_over A \u2964 mono_over A \u2964 mono_over A :=\n{ obj := \u03bb f, pullback f.arrow \u22d9 map f.arrow,\n  map := \u03bb f\u2081 f\u2082 k,\n  { app := \u03bb g,\n    begin\n      apply hom_mk _ _,\n      apply pullback.lift pullback.fst (pullback.snd \u226b k.left) _,\n      rw [pullback.condition, assoc, w k],\n      dsimp,\n      rw [pullback.lift_snd_assoc, assoc, w k],\n    end } }.\n\n/-- A morphism from the \"infimum\" of two objects in `mono_over A` to the first object. -/\ndef inf_le_left {A : C} (f g : mono_over A) :\n  (inf.obj f).obj g \u27f6 f :=\nhom_mk _ rfl\n\n/-- A morphism from the \"infimum\" of two objects in `mono_over A` to the second object. -/\ndef inf_le_right {A : C} (f g : mono_over A) :\n  (inf.obj f).obj g \u27f6 g :=\nhom_mk _ pullback.condition\n\n/-- A morphism version of the `le_inf` axiom. -/\ndef le_inf {A : C} (f g h : mono_over A) :\n  (h \u27f6 f) \u2192 (h \u27f6 g) \u2192 (h \u27f6 (inf.obj f).obj g) :=\nbegin\n  intros k\u2081 k\u2082,\n  refine hom_mk (pullback.lift k\u2082.left k\u2081.left _) _,\n  rw [w k\u2081, w k\u2082],\n  erw [pullback.lift_snd_assoc, w k\u2081],\nend\n\nend inf\n\nsection sup\nvariables [has_images C] [has_binary_coproducts C]\n\n/-- When `[has_images C] [has_binary_coproducts C]`, `mono_over A` has a `sup` construction,\nwhich is functorial in both arguments,\nand which on `subobject A` will induce a `semilattice_sup`. -/\ndef sup  {A : C} : mono_over A \u2964 mono_over A \u2964 mono_over A :=\ncurry_obj ((forget A).prod (forget A) \u22d9 uncurry.obj over.coprod \u22d9 image)\n\n/-- A morphism version of `le_sup_left`. -/\ndef le_sup_left {A : C} (f g : mono_over A) :\n  f \u27f6 (sup.obj f).obj g :=\nbegin\n  refine hom_mk (coprod.inl \u226b factor_thru_image _) _,\n  erw [category.assoc, image.fac, coprod.inl_desc],\n  refl,\nend\n\n/-- A morphism version of `le_sup_right`. -/\ndef le_sup_right {A : C} (f g : mono_over A) :\n  g \u27f6 (sup.obj f).obj g :=\nbegin\n  refine hom_mk (coprod.inr \u226b factor_thru_image _) _,\n  erw [category.assoc, image.fac, coprod.inr_desc],\n  refl,\nend\n\n/-- A morphism version of `sup_le`. -/\ndef sup_le {A : C} (f g h : mono_over A) :\n  (f \u27f6 h) \u2192 (g \u27f6 h) \u2192 ((sup.obj f).obj g \u27f6 h) :=\nbegin\n  intros k\u2081 k\u2082,\n  refine hom_mk _ _,\n  apply image.lift \u27e8_, h.arrow, coprod.desc k\u2081.left k\u2082.left, _\u27e9,\n  { dsimp,\n    ext1,\n    { simp [w k\u2081] },\n    { simp [w k\u2082] } },\n  { apply image.lift_fac }\nend\n\nend sup\n\nend mono_over\n\nnamespace subobject\n\nsection order_top\n\ninstance order_top {X : C} : order_top (subobject X) :=\n{ top := quotient.mk' \u22a4,\n  le_top :=\n  begin\n    refine quotient.ind' (\u03bb f, _),\n    exact \u27e8mono_over.le_top f\u27e9,\n  end,\n  ..subobject.partial_order X}\n\ninstance {X : C} : inhabited (subobject X) := \u27e8\u22a4\u27e9\n\nlemma top_eq_id (B : C) : (\u22a4 : subobject B) = subobject.mk (\ud835\udfd9 B) := rfl\n\n/-- The object underlying `\u22a4 : subobject B` is (up to isomorphism) `B`. -/\ndef top_coe_iso_self {B : C} : ((\u22a4 : subobject B) : C) \u2245 B := underlying_iso _\n\n@[simp]\nlemma underlying_iso_id_eq_top_coe_iso_self {B : C} : underlying_iso (\ud835\udfd9 B) = top_coe_iso_self :=\nrfl\n\n@[simp, reassoc]\nlemma underlying_iso_inv_top_arrow {B : C} :\n  top_coe_iso_self.inv \u226b (\u22a4 : subobject B).arrow = \ud835\udfd9 B :=\nunderlying_iso_arrow _\n\n@[simp]\nlemma map_top (f : X \u27f6 Y) [mono f] : (map f).obj \u22a4 = subobject.mk f :=\nquotient.sound' \u27e8mono_over.map_top f\u27e9\n\nlemma top_factors {A B : C} (f : A \u27f6 B) : (\u22a4 : subobject B).factors f :=\n\u27e8f, comp_id _\u27e9\n\nlemma is_iso_iff_mk_eq_top {X Y : C} (f : X \u27f6 Y) [mono f] : is_iso f \u2194 mk f = \u22a4 :=\n\u27e8\u03bb _, by exactI mk_eq_mk_of_comm _ _ (as_iso f) (category.comp_id _), \u03bb h,\n  by { rw [\u2190of_mk_le_mk_comp h.le, category.comp_id], exact is_iso.of_iso (iso_of_mk_eq_mk _ _ h) }\u27e9\n\nlemma is_iso_arrow_iff_eq_top {Y : C} (P : subobject Y) : is_iso P.arrow \u2194 P = \u22a4 :=\nby rw [is_iso_iff_mk_eq_top, mk_arrow]\n\ninstance is_iso_top_arrow {Y : C} : is_iso (\u22a4 : subobject Y).arrow :=\nby rw is_iso_arrow_iff_eq_top\n\nlemma mk_eq_top_of_is_iso {X Y : C} (f : X \u27f6 Y) [is_iso f] : mk f = \u22a4 :=\n(is_iso_iff_mk_eq_top f).mp infer_instance\n\nlemma eq_top_of_is_iso_arrow {Y : C} (P : subobject Y) [is_iso P.arrow] : P = \u22a4 :=\n(is_iso_arrow_iff_eq_top P).mp infer_instance\n\nsection\nvariables [has_pullbacks C]\n\nlemma pullback_top (f : X \u27f6 Y) : (pullback f).obj \u22a4 = \u22a4 :=\nquotient.sound' \u27e8mono_over.pullback_top f\u27e9\n\nlemma pullback_self {A B : C} (f : A \u27f6 B) [mono f] :\n  (pullback f).obj (mk f) = \u22a4 :=\nquotient.sound' \u27e8mono_over.pullback_self f\u27e9\n\nend\n\nend order_top\n\nsection order_bot\nvariables [has_zero_morphisms C] [has_zero_object C]\nlocal attribute [instance] has_zero_object.has_zero\n\ninstance order_bot {X : C} : order_bot (subobject X) :=\n{ bot := quotient.mk' \u22a5,\n  bot_le :=\n  begin\n    refine quotient.ind' (\u03bb f, _),\n    exact \u27e8mono_over.bot_le f\u27e9,\n  end,\n  ..subobject.partial_order X }\n\nlemma bot_eq_zero {B : C} : (\u22a5 : subobject B) = subobject.mk (0 : 0 \u27f6 B) := rfl\n\n/-- The object underlying `\u22a5 : subobject B` is (up to isomorphism) the zero object. -/\ndef bot_coe_iso_zero {B : C} : ((\u22a5 : subobject B) : C) \u2245 0 := underlying_iso _\n\n@[simp] lemma bot_arrow {B : C} : (\u22a5 : subobject B).arrow = 0 :=\nzero_of_source_iso_zero _ bot_coe_iso_zero\n\nlemma map_bot (f : X \u27f6 Y) [mono f] : (map f).obj \u22a5 = \u22a5 :=\nquotient.sound' \u27e8mono_over.map_bot f\u27e9\n\nlemma bot_factors_iff_zero {A B : C} (f : A \u27f6 B) : (\u22a5 : subobject B).factors f \u2194 f = 0 :=\n\u27e8by { rintro \u27e8h, w\u27e9, simp at w, exact w.symm, }, by { rintro rfl, exact \u27e80, by simp\u27e9, }\u27e9\n\nend order_bot\n\nsection functor\nvariable (C)\n\n/-- Sending `X : C` to `subobject X` is a contravariant functor `C\u1d52\u1d56 \u2964 Type`. -/\n@[simps]\ndef functor [has_pullbacks C] : C\u1d52\u1d56 \u2964 Type (max u\u2081 v\u2081) :=\n{ obj := \u03bb X, subobject X.unop,\n  map := \u03bb X Y f, (pullback f.unop).obj,\n  map_id' := \u03bb X, funext pullback_id,\n  map_comp' := \u03bb X Y Z f g, funext (pullback_comp _ _) }\n\nend functor\n\nsection semilattice_inf_top\nvariables [has_pullbacks C]\n\n/-- The functorial infimum on `mono_over A` descends to an infimum on `subobject A`. -/\ndef inf {A : C} : subobject A \u2964 subobject A \u2964 subobject A :=\nthin_skeleton.map\u2082 mono_over.inf\n\nlemma inf_le_left  {A : C} (f g : subobject A) :\n  (inf.obj f).obj g \u2264 f :=\nquotient.induction_on\u2082' f g (\u03bb a b, \u27e8mono_over.inf_le_left _ _\u27e9)\n\nlemma inf_le_right {A : C} (f g : subobject A) :\n  (inf.obj f).obj g \u2264 g :=\nquotient.induction_on\u2082' f g (\u03bb a b, \u27e8mono_over.inf_le_right _ _\u27e9)\n\nlemma le_inf {A : C} (h f g : subobject A) :\n  h \u2264 f \u2192 h \u2264 g \u2192 h \u2264 (inf.obj f).obj g :=\nquotient.induction_on\u2083' h f g\nbegin\n  rintros f g h \u27e8k\u27e9 \u27e8l\u27e9,\n  exact \u27e8mono_over.le_inf _ _ _ k l\u27e9,\nend\n\ninstance {B : C} : semilattice_inf_top (subobject B) :=\n{ inf := \u03bb m n, (inf.obj m).obj n,\n  inf_le_left := inf_le_left,\n  inf_le_right := inf_le_right,\n  le_inf := le_inf,\n  ..subobject.order_top }\n\nlemma factors_left_of_inf_factors {A B : C} {X Y : subobject B} {f : A \u27f6 B}\n  (h : (X \u2293 Y).factors f) : X.factors f :=\nfactors_of_le _ (inf_le_left _ _) h\n\nlemma factors_right_of_inf_factors {A B : C} {X Y : subobject B} {f : A \u27f6 B}\n  (h : (X \u2293 Y).factors f) : Y.factors f :=\nfactors_of_le _ (inf_le_right _ _) h\n\n@[simp]\nlemma inf_factors {A B : C} {X Y : subobject B} (f : A \u27f6 B) :\n  (X \u2293 Y).factors f \u2194 X.factors f \u2227 Y.factors f :=\n\u27e8\u03bb h, \u27e8factors_left_of_inf_factors h, factors_right_of_inf_factors h\u27e9,\n  begin\n    revert X Y,\n    refine quotient.ind\u2082' _,\n    rintro X Y \u27e8\u27e8g\u2081, rfl\u27e9, \u27e8g\u2082, hg\u2082\u27e9\u27e9,\n    exact \u27e8_, pullback.lift_snd_assoc _ _ hg\u2082 _\u27e9,\n  end\u27e9\n\nlemma inf_arrow_factors_left {B : C} (X Y : subobject B) : X.factors (X \u2293 Y).arrow :=\n(factors_iff _ _).mpr \u27e8of_le (X \u2293 Y) X (inf_le_left X Y), by simp\u27e9\n\n\n\n@[simp]\nlemma finset_inf_factors {I : Type*} {A B : C} {s : finset I} {P : I \u2192 subobject B}\n  (f : A \u27f6 B) :\n  (s.inf P).factors f \u2194 \u2200 i \u2208 s, (P i).factors f :=\nbegin\n  classical,\n  apply finset.induction_on s,\n  { simp [top_factors] },\n  { intros i s nm ih, simp [ih] },\nend\n\n-- `i` is explicit here because often we'd like to defer a proof of `m`\nlemma finset_inf_arrow_factors {I : Type*} {B : C} (s : finset I) (P : I \u2192 subobject B)\n  (i : I) (m : i \u2208 s) : (P i).factors (s.inf P).arrow :=\nbegin\n  revert i m,\n  classical,\n  apply finset.induction_on s,\n  { rintro _ \u27e8\u27e9, },\n  { intros i s nm ih j m,\n    rw [finset.inf_insert],\n    simp only [finset.mem_insert] at m, rcases m with (rfl|m),\n    { rw \u2190factor_thru_arrow _ _ (inf_arrow_factors_left _ _),\n      exact factors_comp_arrow _, },\n    { rw \u2190factor_thru_arrow _ _ (inf_arrow_factors_right _ _),\n      apply factors_of_factors_right,\n      exact ih _ m, } },\nend\n\nlemma inf_eq_map_pullback' {A : C} (f\u2081 : mono_over A) (f\u2082 : subobject A) :\n  (subobject.inf.obj (quotient.mk' f\u2081)).obj f\u2082 =\n    (subobject.map f\u2081.arrow).obj ((subobject.pullback f\u2081.arrow).obj f\u2082) :=\nbegin\n  apply quotient.induction_on' f\u2082,\n  intro f\u2082,\n  refl,\nend\n\nlemma inf_eq_map_pullback {A : C} (f\u2081 : mono_over A) (f\u2082 : subobject A) :\n  (quotient.mk' f\u2081 \u2293 f\u2082 : subobject A) = (map f\u2081.arrow).obj ((pullback f\u2081.arrow).obj f\u2082) :=\ninf_eq_map_pullback' f\u2081 f\u2082\n\nlemma prod_eq_inf {A : C} {f\u2081 f\u2082 : subobject A} [has_binary_product f\u2081 f\u2082] :\n  (f\u2081 \u2a2f f\u2082) = f\u2081 \u2293 f\u2082 :=\nle_antisymm\n  (_root_.le_inf\n    (le_of_hom limits.prod.fst)\n    (le_of_hom limits.prod.snd))\n  (le_of_hom\n    (prod.lift\n      (hom_of_le _root_.inf_le_left)\n      (hom_of_le _root_.inf_le_right)))\n\nlemma inf_def {B : C} (m m' : subobject B) :\n  m \u2293 m' = (inf.obj m).obj m' := rfl\n\n/-- `\u2293` commutes with pullback. -/\nlemma inf_pullback {X Y : C} (g : X \u27f6 Y) (f\u2081 f\u2082) :\n  (pullback g).obj (f\u2081 \u2293 f\u2082) = (pullback g).obj f\u2081 \u2293 (pullback g).obj f\u2082 :=\nbegin\n  revert f\u2081,\n  apply quotient.ind',\n  intro f\u2081,\n  erw [inf_def, inf_def, inf_eq_map_pullback', inf_eq_map_pullback', \u2190 pullback_comp,\n       \u2190 map_pullback pullback.condition (pullback_is_pullback f\u2081.arrow g),\n       \u2190 pullback_comp, pullback.condition],\n  refl,\nend\n\n/-- `\u2293` commutes with map. -/\nlemma inf_map {X Y : C} (g : Y \u27f6 X) [mono g] (f\u2081 f\u2082) :\n  (map g).obj (f\u2081 \u2293 f\u2082) = (map g).obj f\u2081 \u2293 (map g).obj f\u2082 :=\nbegin\n  revert f\u2081,\n  apply quotient.ind',\n  intro f\u2081,\n  erw [inf_def, inf_def, inf_eq_map_pullback',\n       inf_eq_map_pullback', \u2190 map_comp],\n  dsimp,\n  rw [pullback_comp, pullback_map_self],\nend\n\nend semilattice_inf_top\n\nsection semilattice_sup\nvariables [has_images C] [has_binary_coproducts C]\n\n/-- The functorial supremum on `mono_over A` descends to an supremum on `subobject A`. -/\ndef sup {A : C} : subobject A \u2964 subobject A \u2964 subobject A :=\nthin_skeleton.map\u2082 mono_over.sup\n\ninstance {B : C} : semilattice_sup (subobject B) :=\n{ sup := \u03bb m n, (sup.obj m).obj n,\n  le_sup_left := \u03bb m n, quotient.induction_on\u2082' m n (\u03bb a b, \u27e8mono_over.le_sup_left _ _\u27e9),\n  le_sup_right := \u03bb m n, quotient.induction_on\u2082' m n (\u03bb a b, \u27e8mono_over.le_sup_right _ _\u27e9),\n  sup_le := \u03bb m n k, quotient.induction_on\u2083' m n k (\u03bb a b c \u27e8i\u27e9 \u27e8j\u27e9, \u27e8mono_over.sup_le _ _ _ i j\u27e9),\n  ..subobject.partial_order B }\n\nlemma sup_factors_of_factors_left {A B : C} {X Y : subobject B} {f : A \u27f6 B} (P : X.factors f) :\n  (X \u2294 Y).factors f :=\nfactors_of_le f le_sup_left P\n\nlemma sup_factors_of_factors_right {A B : C} {X Y : subobject B} {f : A \u27f6 B} (P : Y.factors f) :\n  (X \u2294 Y).factors f :=\nfactors_of_le f le_sup_right P\n\n/-!\nUnfortunately, there are two different ways we may obtain a `semilattice_sup_bot (subobject B)`,\neither as here, by assuming `[has_zero_morphisms C] [has_zero_object C]`,\nor if `C` is cartesian closed.\n\nThese will be definitionally different, and at the very least we will need two different versions\nof `finset_sup_factors`. So far I don't see how to handle this through generalization.\n-/\nsection\nvariables [has_zero_morphisms C] [has_zero_object C]\n\ninstance {B : C} : semilattice_sup_bot (subobject B) :=\n{ ..subobject.order_bot,\n  ..subobject.semilattice_sup }\n\nlemma finset_sup_factors {I : Type*} {A B : C} {s : finset I} {P : I \u2192 subobject B}\n  {f : A \u27f6 B} (h : \u2203 i \u2208 s, (P i).factors f) :\n  (s.sup P).factors f :=\nbegin\n  classical,\n  revert h,\n  apply finset.induction_on s,\n  { rintro \u27e8_, \u27e8\u27e8\u27e9, _\u27e9\u27e9, },\n  { rintros i s nm ih \u27e8j, \u27e8m, h\u27e9\u27e9,\n    simp only [finset.sup_insert],\n    simp at m, rcases m with (rfl|m),\n    { exact sup_factors_of_factors_left h, },\n    { exact sup_factors_of_factors_right (ih \u27e8j, \u27e8m, h\u27e9\u27e9), }, },\nend\n\nend\n\nend semilattice_sup\n\nsection lattice\nvariables [has_pullbacks C] [has_images C] [has_binary_coproducts C]\n\ninstance {B : C} : lattice (subobject B) :=\n{ ..subobject.semilattice_inf_top,\n  ..subobject.semilattice_sup }\n\nvariables [has_zero_morphisms C] [has_zero_object C]\n\ninstance {B : C} : bounded_lattice (subobject B) :=\n{ ..subobject.semilattice_inf_top,\n  ..subobject.semilattice_sup_bot }\n\nend lattice\n\nsection Inf\n\nvariables [well_powered C]\n\n/--\nThe \"wide cospan\" diagram, with a small indexing type, constructed from a set of subobjects.\n(This is just the diagram of all the subobjects pasted together, but using `well_powered C`\nto make the diagram small.)\n-/\ndef wide_cospan {A : C} (s : set (subobject A)) :\n  wide_pullback_shape (equiv_shrink _ '' s) \u2964 C :=\nwide_pullback_shape.wide_cospan A\n  (\u03bb j : equiv_shrink _ '' s, (((equiv_shrink (subobject A)).symm j) : C))\n  (\u03bb j, ((equiv_shrink (subobject A)).symm j).arrow)\n\n@[simp] lemma wide_cospan_map_term {A : C} (s : set (subobject A)) (j) :\n  (wide_cospan s).map (wide_pullback_shape.hom.term j) =\n    ((equiv_shrink (subobject A)).symm j).arrow :=\nrfl\n\n/-- Auxilliary construction of a cone for `le_Inf`. -/\ndef le_Inf_cone {A : C} (s : set (subobject A)) (f : subobject A) (k : \u03a0 (g \u2208 s), f \u2264 g) :\n  cone (wide_cospan s) :=\nwide_pullback_shape.mk_cone f.arrow\n  (\u03bb j, underlying.map (hom_of_le (k _ (by { rcases j with \u27e8-, \u27e8g, \u27e8m, rfl\u27e9\u27e9\u27e9, simpa using m, }))))\n  (by tidy)\n\n@[simp] lemma le_Inf_cone_\u03c0_app_none\n  {A : C} (s : set (subobject A)) (f : subobject A) (k : \u03a0 (g \u2208 s), f \u2264 g) :\n  (le_Inf_cone s f k).\u03c0.app none = f.arrow :=\nrfl\n\nvariables [has_wide_pullbacks C]\n\n/--\nThe limit of `wide_cospan s`. (This will be the supremum of the set of subobjects.)\n-/\ndef wide_pullback {A : C} (s : set (subobject A)) : C :=\nlimits.limit (wide_cospan s)\n\n/--\nThe inclusion map from `wide_pullback s` to `A`\n-/\ndef wide_pullback_\u03b9 {A : C} (s : set (subobject A)) :\n  wide_pullback s \u27f6 A :=\nlimits.limit.\u03c0 (wide_cospan s) none\n\ninstance wide_pullback_\u03b9_mono {A : C} (s : set (subobject A)) :\n  mono (wide_pullback_\u03b9 s) :=\n\u27e8\u03bb W u v h, limit.hom_ext (\u03bb j, begin\n  cases j,\n  { exact h, },\n  { apply (cancel_mono ((equiv_shrink (subobject A)).symm j).arrow).1,\n    rw [assoc, assoc],\n    erw limit.w (wide_cospan s) (wide_pullback_shape.hom.term j),\n    exact h, },\nend)\u27e9\n\n/--\nWhen `[well_powered C]` and `[has_wide_pullbacks C]`, `subobject A` has arbitrary infimums.\n-/\ndef Inf {A : C} (s : set (subobject A)) : subobject A :=\nsubobject.mk (wide_pullback_\u03b9 s)\n\nlemma Inf_le {A : C} (s : set (subobject A)) (f \u2208 s) :\n  Inf s \u2264 f :=\nbegin\n  fapply le_of_comm,\n  { refine (underlying_iso _).hom \u226b\n      (limits.limit.\u03c0\n        (wide_cospan s)\n        (some \u27e8equiv_shrink _ f, set.mem_image_of_mem (equiv_shrink (subobject A)) H\u27e9)) \u226b _,\n    apply eq_to_hom,\n    apply (congr_arg (\u03bb X : subobject A, (X : C))),\n    exact (equiv.symm_apply_apply _ _), },\n  { dsimp [Inf],\n    simp only [category.comp_id, category.assoc, \u2190underlying_iso_hom_comp_eq_mk,\n      subobject.arrow_congr, congr_arg_mpr_hom_left, iso.cancel_iso_hom_left],\n    convert limit.w (wide_cospan s) (wide_pullback_shape.hom.term _), },\nend.\n\nlemma le_Inf {A : C} (s : set (subobject A)) (f : subobject A) (k : \u03a0 (g \u2208 s), f \u2264 g) :\n  f \u2264 Inf s :=\nbegin\n  fapply le_of_comm,\n  { exact limits.limit.lift _ (le_Inf_cone s f k) \u226b (underlying_iso _).inv, },\n  { dsimp [Inf, wide_pullback_\u03b9],\n    simp, },\nend\n\ninstance {B : C} : complete_semilattice_Inf (subobject B) :=\n{ Inf := Inf,\n  Inf_le := Inf_le,\n  le_Inf := le_Inf,\n  ..subobject.partial_order B }\n\nend Inf\n\nsection Sup\n\nvariables [well_powered C] [has_coproducts C]\n\n/--\nThe univesal morphism out of the coproduct of a set of subobjects,\nafter using `[well_powered C]` to reindex by a small type.\n-/\ndef small_coproduct_desc {A : C} (s : set (subobject A)) : _ \u27f6 A :=\nlimits.sigma.desc (\u03bb j : equiv_shrink _ '' s, ((equiv_shrink (subobject A)).symm j).arrow)\n\nvariables [has_images C]\n\n/-- When `[well_powered C] [has_images C] [has_coproducts C]`,\n`subobject A` has arbitrary supremums. -/\ndef Sup {A : C} (s : set (subobject A)) : subobject A :=\nsubobject.mk (image.\u03b9 (small_coproduct_desc s))\n\nlemma le_Sup {A : C} (s : set (subobject A)) (f \u2208 s)  :\n  f \u2264 Sup s :=\nbegin\n  fapply le_of_comm,\n  { dsimp [Sup],\n    refine _ \u226b factor_thru_image _ \u226b (underlying_iso _).inv,\n    refine _ \u226b sigma.\u03b9 _ \u27e8equiv_shrink _ f, (by simpa [set.mem_image] using H)\u27e9,\n    exact eq_to_hom (congr_arg (\u03bb X : subobject A, (X : C)) (equiv.symm_apply_apply _ _).symm), },\n  { dsimp [Sup, small_coproduct_desc],\n    simp, dsimp, simp, },\nend\n\nlemma symm_apply_mem_iff_mem_image {\u03b1 \u03b2 : Type*} (e : \u03b1 \u2243 \u03b2) (s : set \u03b1) (x : \u03b2) :\n  e.symm x \u2208 s \u2194 x \u2208 e '' s :=\n\u27e8\u03bb h, \u27e8e.symm x, h, by simp\u27e9, by { rintro \u27e8a, m, rfl\u27e9, simpa using m, }\u27e9\n\nlemma Sup_le {A : C} (s : set (subobject A)) (f : subobject A) (k : \u03a0 (g \u2208 s), g \u2264 f) :\n  Sup s \u2264 f :=\nbegin\n  fapply le_of_comm,\n  { dsimp [Sup],\n    refine (underlying_iso _).hom \u226b image.lift \u27e8_, f.arrow, _, _\u27e9,\n    { refine sigma.desc _,\n      rintro \u27e8g, m\u27e9,\n      refine underlying.map (hom_of_le (k _ _)),\n      simpa [symm_apply_mem_iff_mem_image] using m, },\n    { ext j, rcases j with \u27e8j, m\u27e9, dsimp [small_coproduct_desc], simp, dsimp, simp, }, },\n  { dsimp [Sup],\n    simp, },\nend\n\ninstance {B : C} : complete_semilattice_Sup (subobject B) :=\n{ Sup := Sup,\n  le_Sup := le_Sup,\n  Sup_le := Sup_le,\n  ..subobject.partial_order B }\n\nend Sup\n\nsection complete_lattice\nvariables [well_powered C] [has_wide_pullbacks C] [has_images C] [has_coproducts C]\n  [has_zero_morphisms C] [has_zero_object C]\n\ninstance {B : C} : complete_lattice (subobject B) :=\n{ ..subobject.semilattice_inf_top,\n  ..subobject.semilattice_sup_bot,\n  ..subobject.complete_semilattice_Inf,\n  ..subobject.complete_semilattice_Sup, }\n\nend complete_lattice\n\nend subobject\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/subobject/lattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091673, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3759230137720318}}
{"text": "/-\nCopyright (c) 2022 Yuma Mizuno. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno\n-/\nimport category_theory.bicategory.functor\n\n/-!\n# Oplax natural transformations\n\nJust as there are natural transformations between functors, there are oplax natural transformations\nbetween oplax functors. The equality in the naturality of natural transformations is replaced by a\nspecified 2-morphism `F.map f \u226b app b \u27f6 app a \u226b G.map f` in the case of oplax natural\ntransformations.\n\n## Main definitions\n\n* `oplax_nat_trans F G` : oplax natural transformations between oplax functors `F` and `G`\n* `oplax_nat_trans.vcomp \u03b7 \u03b8` : the vertical composition of oplax natural transformations `\u03b7`\n  and `\u03b8`\n* `oplax_nat_trans.category F G` : the category structure on the oplax natural transformations\n  between `F` and `G`\n-/\n\nnamespace category_theory\n\nopen category bicategory\nopen_locale bicategory\n\nuniverses w\u2081 w\u2082 v\u2081 v\u2082 u\u2081 u\u2082\n\nvariables {B : Type u\u2081} [bicategory.{w\u2081 v\u2081} B] {C : Type u\u2082} [bicategory.{w\u2082 v\u2082} C]\n\n/--\nIf `\u03b7` is an oplax natural transformation between `F` and `G`, we have a 1-morphism\n`\u03b7.app a : F.obj a \u27f6 G.obj a` for each object `a : B`. We also have a 2-morphism\n`\u03b7.naturality f : F.map f \u226b app b \u27f6 app a \u226b G.map f` for each 1-morphism `f : a \u27f6 b`.\nThese 2-morphisms satisfies the naturality condition, and preserve the identities and\nthe compositions modulo some adjustments of domains and codomains of 2-morphisms.\n-/\nstructure oplax_nat_trans (F G : oplax_functor B C) :=\n(app (a : B) : F.obj a \u27f6 G.obj a)\n(naturality {a b : B} (f : a \u27f6 b) : F.map f \u226b app b \u27f6 app a \u226b G.map f)\n(naturality_naturality' : \u2200 {a b : B} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g),\n  (F.map\u2082 \u03b7 \u25b7 app b) \u226b naturality g = naturality f \u226b (app a \u25c1 G.map\u2082 \u03b7) . obviously)\n(naturality_id' : \u2200 a : B,\n  naturality (\ud835\udfd9 a) \u226b (app a \u25c1 G.map_id a) =\n    (F.map_id a \u25b7 app a) \u226b (\u03bb_ (app a)).hom \u226b (\u03c1_ (app a)).inv . obviously)\n(naturality_comp' : \u2200 {a b c : B} (f : a \u27f6 b) (g : b \u27f6 c),\n  naturality (f \u226b g) \u226b (app a \u25c1 G.map_comp f g) =\n    (F.map_comp f g \u25b7 app c) \u226b (\u03b1_ _ _ _).hom \u226b\n      (F.map f \u25c1 naturality g) \u226b (\u03b1_ _ _ _).inv \u226b\n        (naturality f \u25b7 G.map g) \u226b (\u03b1_ _ _ _).hom . obviously)\n\nrestate_axiom oplax_nat_trans.naturality_naturality'\nrestate_axiom oplax_nat_trans.naturality_id'\nrestate_axiom oplax_nat_trans.naturality_comp'\nattribute [simp, reassoc] oplax_nat_trans.naturality_naturality\n  oplax_nat_trans.naturality_id oplax_nat_trans.naturality_comp\n\nnamespace oplax_nat_trans\n\nsection\nvariables (F : oplax_functor B C)\n\n/-- The identity oplax natural transformation. -/\n@[simps]\ndef id : oplax_nat_trans F F :=\n{ app := \u03bb a, \ud835\udfd9 (F.obj a),\n  naturality := \u03bb a b f, (\u03c1_ (F.map f)).hom \u226b (\u03bb_ (F.map f)).inv,\n  naturality_naturality' := \u03bb a b f f' \u03b7, by\n  { rw [assoc, \u2190left_unitor_inv_naturality, \u2190right_unitor_naturality_assoc] },\n  naturality_comp' := \u03bb a b c f g, by\n  { rw [assoc, \u2190left_unitor_inv_naturality, \u2190right_unitor_naturality_assoc],\n    simp only [triangle_assoc_comp_right_assoc, right_unitor_comp, left_unitor_comp_inv,\n      whisker_right_comp, inv_hom_whisker_left_assoc, assoc, whisker_left_comp] },\n  naturality_id' := \u03bb a, by\n  { rw [assoc, \u2190left_unitor_inv_naturality, \u2190right_unitor_naturality_assoc,\n      unitors_equal, unitors_inv_equal] } }\n\ninstance : inhabited (oplax_nat_trans F F) := \u27e8id F\u27e9\n\nvariables {F} {G H : oplax_functor B C} (\u03b7 : oplax_nat_trans F G) (\u03b8 : oplax_nat_trans G H)\n\nsection\nvariables {a b c : B} {a' : C}\n\n@[simp, reassoc]\nlemma whisker_left_naturality_naturality (f : a' \u27f6 G.obj a) {g h : a \u27f6 b} (\u03b2 : g \u27f6 h) :\n  (f \u25c1 (G.map\u2082 \u03b2 \u25b7 \u03b8.app b)) \u226b (f \u25c1 \u03b8.naturality h) =\n    (f \u25c1 \u03b8.naturality g) \u226b (f \u25c1 (\u03b8.app a \u25c1 H.map\u2082 \u03b2)) :=\nby simp only [\u2190whisker_left_comp, naturality_naturality]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_naturality {f g : a \u27f6 b} (\u03b2 : f \u27f6 g) (h : G.obj b \u27f6 a') :\n  ((F.map\u2082 \u03b2 \u25b7 \u03b7.app b) \u25b7 h) \u226b (\u03b7.naturality g \u25b7 h) =\n    (\u03b7.naturality f \u25b7 h) \u226b ((\u03b7.app a \u25c1 G.map\u2082 \u03b2) \u25b7 h) :=\nby simp only [\u2190whisker_right_comp, naturality_naturality]\n\n@[simp, reassoc]\nlemma whisker_left_naturality_comp (f : a' \u27f6 G.obj a) (g : a \u27f6 b) (h : b \u27f6 c) :\n  (f \u25c1 \u03b8.naturality (g \u226b h)) \u226b (f \u25c1 (\u03b8.app a \u25c1 H.map_comp g h)) =\n    (f \u25c1 (G.map_comp g h \u25b7 \u03b8.app c)) \u226b (f \u25c1 (\u03b1_ _ _ _).hom) \u226b\n      (f \u25c1 (G.map g \u25c1 \u03b8.naturality h)) \u226b (f \u25c1 (\u03b1_ _ _ _).inv) \u226b\n        (f \u25c1 (\u03b8.naturality g \u25b7 H.map h)) \u226b (f \u25c1 (\u03b1_ _ _ _).hom) :=\nby simp only [\u2190whisker_left_comp, naturality_comp]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_comp (f : a \u27f6 b) (g : b \u27f6 c) (h : G.obj c \u27f6 a') :\n  (\u03b7.naturality (f \u226b g) \u25b7 h) \u226b ((\u03b7.app a \u25c1 G.map_comp f g) \u25b7 h) =\n    ((F.map_comp f g \u25b7 \u03b7.app c) \u25b7 h) \u226b ((\u03b1_ _ _ _).hom \u25b7 h) \u226b\n      ((F.map f \u25c1 \u03b7.naturality g) \u25b7 h) \u226b ((\u03b1_ _ _ _).inv \u25b7 h) \u226b\n        ((\u03b7.naturality f \u25b7 G.map g) \u25b7 h) \u226b ((\u03b1_ _ _ _).hom \u25b7 h) :=\nby simp only [\u2190whisker_right_comp, naturality_comp]\n\n@[simp, reassoc]\nlemma whisker_left_naturality_id (f : a' \u27f6 G.obj a) :\n  (f \u25c1 \u03b8.naturality (\ud835\udfd9 a)) \u226b (f \u25c1 (\u03b8.app a \u25c1 H.map_id a)) =\n    (f \u25c1 (G.map_id a \u25b7 \u03b8.app a)) \u226b (f \u25c1 (\u03bb_ (\u03b8.app a)).hom) \u226b (f \u25c1 (\u03c1_ (\u03b8.app a)).inv) :=\nby simp only [\u2190whisker_left_comp, naturality_id]\n\n@[simp, reassoc]\nlemma whisker_right_naturality_id (f : G.obj a \u27f6 a') :\n  (\u03b7.naturality (\ud835\udfd9 a) \u25b7 f) \u226b ((\u03b7.app a \u25c1 G.map_id a) \u25b7 f) =\n    ((F.map_id a \u25b7 \u03b7.app a) \u25b7 f) \u226b ((\u03bb_ (\u03b7.app a)).hom \u25b7 f) \u226b ((\u03c1_ (\u03b7.app a)).inv \u25b7 f) :=\nby simp only [\u2190whisker_right_comp, naturality_id]\n\nend\n\n/-- Vertical composition of oplax natural transformations. -/\n@[simps]\ndef vcomp (\u03b7 : oplax_nat_trans F G) (\u03b8 : oplax_nat_trans G H) : oplax_nat_trans F H :=\n{ app := \u03bb a, \u03b7.app a \u226b \u03b8.app a,\n  naturality := \u03bb a b f,\n    (\u03b1_ _ _ _).inv \u226b (\u03b7.naturality f \u25b7 \u03b8.app b) \u226b (\u03b1_ _ _ _).hom \u226b\n      (\u03b7.app a \u25c1 \u03b8.naturality f) \u226b (\u03b1_ _ _ _).inv,\n  naturality_naturality' := \u03bb a b f g \u03b9, by\n  { simp only [whisker_right_comp, assoc, whisker_left_comp],\n    rw [\u2190associator_inv_naturality_right, \u2190whisker_left_naturality_naturality_assoc,\n        \u2190associator_naturality_middle_assoc, \u2190whisker_right_naturality_naturality_assoc,\n        \u2190associator_inv_naturality_left_assoc] },\n  naturality_comp' := \u03bb a b c f g, by\n  { simp only [whisker_right_comp, assoc, whisker_left_comp],\n    rw [\u2190associator_inv_naturality_right, whisker_left_naturality_comp_assoc,\n        \u2190associator_naturality_middle_assoc, whisker_right_naturality_comp_assoc,\n        \u2190associator_inv_naturality_left_assoc],\n    rw [\u2190pentagon_hom_hom_inv_inv_hom, associator_naturality_middle_assoc,\n        \u2190pentagon_inv_hom_hom_hom_inv_assoc, \u2190associator_naturality_middle_assoc],\n    slice_rhs 5 13\n    { rw [\u2190pentagon_inv_hom_hom_hom_hom_assoc, \u2190pentagon_hom_hom_inv_hom_hom,\n          associator_naturality_left_assoc, \u2190associator_naturality_right_assoc,\n          pentagon_inv_inv_hom_hom_inv_assoc, inv_hom_whisker_left_assoc, iso.hom_inv_id_assoc,\n          whisker_exchange_assoc, associator_naturality_right_assoc,\n          \u2190associator_naturality_left_assoc, \u2190pentagon_assoc] },\n    simp only [assoc] },\n  naturality_id' := \u03bb a, by\n  { simp only [whisker_right_comp, assoc, whisker_left_comp],\n    rw [\u2190associator_inv_naturality_right, whisker_left_naturality_id_assoc,\n        \u2190associator_naturality_middle_assoc, whisker_right_naturality_id_assoc,\n        \u2190associator_inv_naturality_left_assoc],\n    simp only [left_unitor_comp, triangle_assoc, inv_hom_whisker_right_assoc, assoc,\n      right_unitor_comp_inv] } }\n\nvariables (B C)\n\n@[simps]\ninstance : category_struct (oplax_functor B C) :=\n{ hom := \u03bb F G, oplax_nat_trans F G,\n  id := oplax_nat_trans.id,\n  comp := \u03bb F G H, oplax_nat_trans.vcomp }\n\nend\n\nsection\nvariables {F G : oplax_functor B C}\n\n/--\nA modification `\u0393` between oplax natural transformations `\u03b7` and `\u03b8` consists of a family of\n2-morphisms `\u0393.app a : \u03b7.app a \u27f6 \u03b8.app a`, which satisfies the equation\n`(F.map f \u25c1 app b) \u226b \u03b8.naturality f = \u03b7.naturality f \u226b (app a \u25b7 G.map f)`\nfor each 1-morphism `f : a \u27f6 b`.\n-/\n@[ext]\nstructure modification (\u03b7 \u03b8 : F \u27f6 G) :=\n(app (a : B) : \u03b7.app a \u27f6 \u03b8.app a)\n(naturality' : \u2200 {a b : B} (f : a \u27f6 b),\n  (F.map f \u25c1 app b) \u226b \u03b8.naturality f = \u03b7.naturality f \u226b (app a \u25b7 G.map f) . obviously)\n\nrestate_axiom modification.naturality'\nattribute [simp, reassoc] modification.naturality\n\nvariables {\u03b7 \u03b8 \u03b9 : F \u27f6 G}\n\nnamespace modification\n\nvariables (\u03b7)\n\n/-- The identity modification. -/\n@[simps]\ndef id : modification \u03b7 \u03b7 := { app := \u03bb a, \ud835\udfd9 (\u03b7.app a) }\n\ninstance : inhabited (modification \u03b7 \u03b7) := \u27e8modification.id \u03b7\u27e9\n\nvariables {\u03b7}\n\nsection\nvariables (\u0393 : modification \u03b7 \u03b8) {a b c : B} {a' : C}\n\n@[reassoc]\nlemma whisker_left_naturality (f : a' \u27f6 F.obj b) (g : b \u27f6 c) :\n  (f \u25c1 (F.map g \u25c1 \u0393.app c)) \u226b (f \u25c1 \u03b8.naturality g) =\n    (f \u25c1 \u03b7.naturality g) \u226b (f \u25c1 (\u0393.app b \u25b7 G.map g)) :=\nby simp only [\u2190bicategory.whisker_left_comp, naturality]\n\n@[reassoc]\nlemma whisker_right_naturality (f : a \u27f6 b) (g : G.obj b \u27f6 a') :\n  ((F.map f \u25c1 \u0393.app b) \u25b7 g) \u226b (\u03b8.naturality f \u25b7 g) =\n    (\u03b7.naturality f \u25b7 g) \u226b ((\u0393.app a \u25b7 G.map f) \u25b7 g) :=\nby simp only [\u2190bicategory.whisker_right_comp, naturality]\n\nend\n\n/-- Vertical composition of modifications. -/\n@[simps]\ndef vcomp (\u0393 : modification \u03b7 \u03b8) (\u0394 : modification \u03b8 \u03b9) : modification \u03b7 \u03b9 :=\n{ app := \u03bb a, \u0393.app a \u226b \u0394.app a }\n\nend modification\n\n/-- Category structure on the oplax natural transformations between oplax_functors. -/\n@[simps]\ninstance category (F G : oplax_functor B C) : category (F \u27f6 G) :=\n{ hom  := modification,\n  id   := modification.id,\n  comp := \u03bb \u03b7 \u03b8 \u03b9, modification.vcomp }\n\n/--\nConstruct a modification isomorphism between oplax natural transformations\nby giving object level isomorphisms, and checking naturality only in the forward direction.\n-/\n@[simps]\ndef modification_iso.of_components\n  (app : \u2200 a, \u03b7.app a \u2245 \u03b8.app a)\n  (naturality : \u2200 {a b} (f : a \u27f6 b),\n    (F.map f \u25c1 (app b).hom) \u226b \u03b8.naturality f = \u03b7.naturality f \u226b ((app a).hom \u25b7 G.map f)) :\n  \u03b7 \u2245 \u03b8 :=\n{ hom := { app := \u03bb a, (app a).hom },\n  inv :=\n  { app := \u03bb a, (app a).inv,\n    naturality' := \u03bb a b f, by simpa using\n      congr_arg (\u03bb f, (_ \u25c1 (app b).inv) \u226b f \u226b ((app a).inv \u25b7 _)) (naturality f).symm } }\n\nend\n\nend oplax_nat_trans\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/bicategory/natural_transformation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.6224593382055109, "lm_q1q2_score": 0.3759230095423006}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.types\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.terminal\nimport tactic.elementwise\n\n/-!\n# Special shapes for limits in `Type`.\n\nThe general shape (co)limits defined in `category_theory.limits.types`\nare intended for use through the limits API,\nand the actual implementation should mostly be considered \"sealed\".\n\nIn this file, we provide definitions of the \"standard\" special shapes of limits in `Type`,\ngiving the expected definitional implementation:\n* the terminal object is `punit`\n* the binary product of `X` and `Y` is `X \u00d7 Y`\n* the product of a family `f : J \u2192 Type` is `\u03a0 j, f j`\n* the coproduct of a family `f : J \u2192 Type` is `\u03a3 j, f j`\n* the binary coproduct of `X` and `Y` is the sum type `X \u2295 Y`\n* the equalizer of a pair of maps `(g, h)` is the subtype `{x : Y // g x = h x}`\n* the coequalizer of a pair of maps `(f, g)` is the quotient of `Y` by `\u2200 x : Y, f x ~ g x`\n* the pullback of `f : X \u27f6 Z` and `g : Y \u27f6 Z` is the subtype `{ p : X \u00d7 Y // f p.1 = g p.2 }`\n  of the product\n\nWe first construct terms of `is_limit` and `limit_cone`, and then provide isomorphisms with the\ntypes generated by the `has_limit` API.\n\nAs an example, when setting up the monoidal category structure on `Type`\nwe use the `types_has_terminal` and `types_has_binary_products` instances.\n-/\n\nuniverses u\n\nopen category_theory\nopen category_theory.limits\n\nnamespace category_theory.limits.types\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/-- A restatement of `types.lift_\u03c0_apply` that uses `pi.\u03c0` and `pi.lift`. -/\n@[simp]\nlemma pi_lift_\u03c0_apply\n  {\u03b2 : Type u} (f : \u03b2 \u2192 Type u) {P : Type u} (s : \u03a0 b, P \u27f6 f b) (b : \u03b2) (x : P) :\n  (pi.\u03c0 f b : (\u220f f) \u2192 f b) (@pi.lift \u03b2 _ _ f _ P s x) = s b x :=\ncongr_fun (limit.lift_\u03c0 (fan.mk P s) \u27e8b\u27e9) x\n\n/-- A restatement of `types.map_\u03c0_apply` that uses `pi.\u03c0` and `pi.map`. -/\n@[simp]\nlemma pi_map_\u03c0_apply {\u03b2 : Type u} {f g : \u03b2 \u2192 Type u} (\u03b1 : \u03a0 j, f j \u27f6 g j) (b : \u03b2) (x) :\n  (pi.\u03c0 g b : (\u220f g) \u2192 g b) (pi.map \u03b1 x) = \u03b1 b ((pi.\u03c0 f b : (\u220f f) \u2192 f b) x) :=\nlimit.map_\u03c0_apply _ _ _\n\n/-- The category of types has `punit` as a terminal object. -/\ndef terminal_limit_cone : limits.limit_cone (functor.empty (Type u)) :=\n{ cone :=\n  { X := punit,\n    \u03c0 := by tidy, },\n  is_limit := by tidy, }\n\n/-- The terminal object in `Type u` is `punit`. -/\nnoncomputable def terminal_iso : \u22a4_ (Type u) \u2245 punit :=\nlimit.iso_limit_cone terminal_limit_cone\n\n/-- The category of types has `pempty` as an initial object. -/\ndef initial_colimit_cocone : limits.colimit_cocone (functor.empty (Type u)) :=\n{ cocone :=\n  { X := pempty,\n    \u03b9 := by tidy, },\n  is_colimit := by tidy, }\n\n/-- The initial object in `Type u` is `punit`. -/\nnoncomputable def initial_iso : \u22a5_ (Type u) \u2245 pempty :=\ncolimit.iso_colimit_cocone initial_colimit_cocone\n\nopen category_theory.limits.walking_pair\n\n/-- The product type `X \u00d7 Y` forms a cone for the binary product of `X` and `Y`. -/\n-- We manually generate the other projection lemmas since the simp-normal form for the legs is\n-- otherwise not created correctly.\n@[simps X]\ndef binary_product_cone (X Y : Type u) : binary_fan X Y :=\nbinary_fan.mk prod.fst prod.snd\n\n@[simp]\nlemma binary_product_cone_fst (X Y : Type u) :\n  (binary_product_cone X Y).fst = prod.fst :=\nrfl\n@[simp]\nlemma binary_product_cone_snd (X Y : Type u) :\n  (binary_product_cone X Y).snd = prod.snd :=\nrfl\n\n/-- The product type `X \u00d7 Y` is a binary product for `X` and `Y`. -/\n@[simps]\ndef binary_product_limit (X Y : Type u) : is_limit (binary_product_cone X Y) :=\n{ lift := \u03bb (s : binary_fan X Y) x, (s.fst x, s.snd x),\n  fac' := \u03bb s j, discrete.rec_on j (\u03bb j, walking_pair.cases_on j rfl rfl),\n  uniq' := \u03bb s m w, funext $ \u03bb x, prod.ext (congr_fun (w \u27e8left\u27e9) x) (congr_fun (w \u27e8right\u27e9) x) }\n\n/--\nThe category of types has `X \u00d7 Y`, the usual cartesian product,\nas the binary product of `X` and `Y`.\n-/\n@[simps]\ndef binary_product_limit_cone (X Y : Type u) : limits.limit_cone (pair X Y) :=\n\u27e8_, binary_product_limit X Y\u27e9\n\n/-- The categorical binary product in `Type u` is cartesian product. -/\nnoncomputable def binary_product_iso (X Y : Type u) : limits.prod X Y \u2245 X \u00d7 Y :=\nlimit.iso_limit_cone (binary_product_limit_cone X Y)\n\n@[simp, elementwise] lemma binary_product_iso_hom_comp_fst (X Y : Type u) :\n  (binary_product_iso X Y).hom \u226b prod.fst = limits.prod.fst :=\nlimit.iso_limit_cone_hom_\u03c0 (binary_product_limit_cone X Y) \u27e8walking_pair.left\u27e9\n\n@[simp, elementwise] lemma binary_product_iso_hom_comp_snd (X Y : Type u) :\n  (binary_product_iso X Y).hom \u226b prod.snd = limits.prod.snd :=\nlimit.iso_limit_cone_hom_\u03c0 (binary_product_limit_cone X Y) \u27e8walking_pair.right\u27e9\n\n@[simp, elementwise] lemma binary_product_iso_inv_comp_fst (X Y : Type u) :\n  (binary_product_iso X Y).inv \u226b limits.prod.fst = prod.fst :=\nlimit.iso_limit_cone_inv_\u03c0 (binary_product_limit_cone X Y) \u27e8walking_pair.left\u27e9\n\n@[simp, elementwise] lemma binary_product_iso_inv_comp_snd (X Y : Type u) :\n  (binary_product_iso X Y).inv \u226b limits.prod.snd = prod.snd :=\nlimit.iso_limit_cone_inv_\u03c0 (binary_product_limit_cone X Y) \u27e8walking_pair.right\u27e9\n\n/-- The functor which sends `X, Y` to the product type `X \u00d7 Y`. -/\n-- We add the option `type_md` to tell `@[simps]` to not treat homomorphisms `X \u27f6 Y` in `Type*` as\n-- a function type\n@[simps {type_md := reducible}]\ndef binary_product_functor : Type u \u2964 Type u \u2964 Type u :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, X \u00d7 Y,\n    map := \u03bb Y\u2081 Y\u2082 f, (binary_product_limit X Y\u2082).lift (binary_fan.mk prod.fst (prod.snd \u226b f)) },\n  map := \u03bb X\u2081 X\u2082 f,\n  { app := \u03bb Y, (binary_product_limit X\u2082 Y).lift (binary_fan.mk (prod.fst \u226b f) prod.snd) } }\n\n/--\nThe product functor given by the instance `has_binary_products (Type u)` is isomorphic to the\nexplicit binary product functor given by the product type.\n-/\nnoncomputable def binary_product_iso_prod : binary_product_functor \u2245 (prod.functor : Type u \u2964 _) :=\nbegin\n  apply nat_iso.of_components (\u03bb X, _) _,\n  { apply nat_iso.of_components (\u03bb Y, _) _,\n    { exact ((limit.is_limit _).cone_point_unique_up_to_iso (binary_product_limit X Y)).symm },\n    { intros Y\u2081 Y\u2082 f,\n      ext1;\n      simp } },\n  { intros X\u2081 X\u2082 g,\n    ext : 3;\n    simp }\nend\n\n/-- The sum type `X \u2295 Y` forms a cocone for the binary coproduct of `X` and `Y`. -/\n@[simps]\ndef binary_coproduct_cocone (X Y : Type u) : cocone (pair X Y) :=\nbinary_cofan.mk sum.inl sum.inr\n\n/-- The sum type `X \u2295 Y` is a binary coproduct for `X` and `Y`. -/\n@[simps]\ndef binary_coproduct_colimit (X Y : Type u) : is_colimit (binary_coproduct_cocone X Y) :=\n{ desc := \u03bb (s : binary_cofan X Y), sum.elim s.inl s.inr,\n  fac' := \u03bb s j, discrete.rec_on j (\u03bb j, walking_pair.cases_on j rfl rfl),\n  uniq' := \u03bb s m w, funext $ \u03bb x, sum.cases_on x (congr_fun (w \u27e8left\u27e9)) (congr_fun (w \u27e8right\u27e9)) }\n\n/--\nThe category of types has `X \u2295 Y`,\nas the binary coproduct of `X` and `Y`.\n-/\ndef binary_coproduct_colimit_cocone (X Y : Type u) : limits.colimit_cocone (pair X Y) :=\n\u27e8_, binary_coproduct_colimit X Y\u27e9\n\n/-- The categorical binary coproduct in `Type u` is the sum `X \u2295 Y`. -/\nnoncomputable def binary_coproduct_iso (X Y : Type u) : limits.coprod X Y \u2245 X \u2295 Y :=\ncolimit.iso_colimit_cocone (binary_coproduct_colimit_cocone X Y)\n\nopen_locale category_theory.Type\n\n@[simp, elementwise] lemma binary_coproduct_iso_inl_comp_hom (X Y : Type u) :\n  limits.coprod.inl \u226b (binary_coproduct_iso X Y).hom = sum.inl :=\ncolimit.iso_colimit_cocone_\u03b9_hom (binary_coproduct_colimit_cocone X Y) \u27e8walking_pair.left\u27e9\n\n@[simp, elementwise] lemma binary_coproduct_iso_inr_comp_hom (X Y : Type u) :\n  limits.coprod.inr \u226b (binary_coproduct_iso X Y).hom = sum.inr :=\ncolimit.iso_colimit_cocone_\u03b9_hom (binary_coproduct_colimit_cocone X Y) \u27e8walking_pair.right\u27e9\n\n@[simp, elementwise] lemma binary_coproduct_iso_inl_comp_inv (X Y : Type u) :\n  \u21be(sum.inl : X \u27f6 X \u2295 Y) \u226b (binary_coproduct_iso X Y).inv = limits.coprod.inl :=\ncolimit.iso_colimit_cocone_\u03b9_inv (binary_coproduct_colimit_cocone X Y) \u27e8walking_pair.left\u27e9\n\n@[simp, elementwise] lemma binary_coproduct_iso_inr_comp_inv (X Y : Type u) :\n  \u21be(sum.inr : Y \u27f6 X \u2295 Y) \u226b (binary_coproduct_iso X Y).inv = limits.coprod.inr :=\ncolimit.iso_colimit_cocone_\u03b9_inv (binary_coproduct_colimit_cocone X Y) \u27e8walking_pair.right\u27e9\n\n/--\nThe category of types has `\u03a0 j, f j` as the product of a type family `f : J \u2192 Type`.\n-/\ndef product_limit_cone {J : Type u} (F : J \u2192 Type u) : limits.limit_cone (discrete.functor F) :=\n{ cone :=\n  { X := \u03a0 j, F j,\n    \u03c0 := { app := \u03bb j f, f j.as }, },\n  is_limit :=\n  { lift := \u03bb s x j, s.\u03c0.app \u27e8j\u27e9 x,\n    uniq' := \u03bb s m w, funext $ \u03bb x, funext $ \u03bb j, (congr_fun (w \u27e8j\u27e9) x : _) } }\n\n/-- The categorical product in `Type u` is the type theoretic product `\u03a0 j, F j`. -/\nnoncomputable def product_iso {J : Type u} (F : J \u2192 Type u) : \u220f F \u2245 \u03a0 j, F j :=\nlimit.iso_limit_cone (product_limit_cone F)\n\n@[simp, elementwise] lemma product_iso_hom_comp_eval {J : Type u} (F : J \u2192 Type u) (j : J) :\n  (product_iso F).hom \u226b (\u03bb f, f j) = pi.\u03c0 F j :=\nrfl\n\n@[simp, elementwise] lemma product_iso_inv_comp_\u03c0 {J : Type u} (F : J \u2192 Type u) (j : J) :\n  (product_iso F).inv \u226b pi.\u03c0 F j = (\u03bb f, f j) :=\nlimit.iso_limit_cone_inv_\u03c0 (product_limit_cone F) \u27e8j\u27e9\n\n/--\nThe category of types has `\u03a3 j, f j` as the coproduct of a type family `f : J \u2192 Type`.\n-/\ndef coproduct_colimit_cocone {J : Type u} (F : J \u2192 Type u) :\n  limits.colimit_cocone (discrete.functor F) :=\n{ cocone :=\n  { X := \u03a3 j, F j,\n    \u03b9 :=\n    { app := \u03bb j x, \u27e8j.as, x\u27e9 }, },\n  is_colimit :=\n  { desc := \u03bb s x, s.\u03b9.app \u27e8x.1\u27e9 x.2,\n    uniq' := \u03bb s m w,\n    begin\n      ext \u27e8j, x\u27e9,\n      have := congr_fun (w \u27e8j\u27e9) x,\n      exact this,\n    end }, }\n\n/-- The categorical coproduct in `Type u` is the type theoretic coproduct `\u03a3 j, F j`. -/\nnoncomputable def coproduct_iso {J : Type u} (F : J \u2192 Type u) : \u2210 F \u2245 \u03a3 j, F j :=\ncolimit.iso_colimit_cocone (coproduct_colimit_cocone F)\n\n@[simp, elementwise] lemma coproduct_iso_\u03b9_comp_hom {J : Type u} (F : J \u2192 Type u) (j : J) :\n  sigma.\u03b9 F j \u226b (coproduct_iso F).hom = (\u03bb x : F j, (\u27e8j, x\u27e9 : \u03a3 j, F j)) :=\ncolimit.iso_colimit_cocone_\u03b9_hom (coproduct_colimit_cocone F) \u27e8j\u27e9\n\n@[simp, elementwise] \n\nsection fork\nvariables {X Y Z : Type u} (f : X \u27f6 Y) {g h : Y \u27f6 Z} (w : f \u226b g = f \u226b h)\n\n/--\nShow the given fork in `Type u` is an equalizer given that any element in the \"difference kernel\"\ncomes from `X`.\nThe converse of `unique_of_type_equalizer`.\n-/\nnoncomputable def type_equalizer_of_unique (t : \u2200 (y : Y), g y = h y \u2192 \u2203! (x : X), f x = y) :\n  is_limit (fork.of_\u03b9 _ w) :=\nfork.is_limit.mk' _ $ \u03bb s,\nbegin\n  refine \u27e8\u03bb i, _, _, _\u27e9,\n  { apply classical.some (t (s.\u03b9 i) _),\n    apply congr_fun s.condition i },\n  { ext i,\n    apply (classical.some_spec (t (s.\u03b9 i) _)).1 },\n  { intros m hm,\n    ext i,\n    apply (classical.some_spec (t (s.\u03b9 i) _)).2,\n    apply congr_fun hm i },\nend\n\n/-- The converse of `type_equalizer_of_unique`. -/\nlemma unique_of_type_equalizer (t : is_limit (fork.of_\u03b9 _ w)) (y : Y) (hy : g y = h y) :\n  \u2203! (x : X), f x = y :=\nbegin\n  let y' : punit \u27f6 Y := \u03bb _, y,\n  have hy' : y' \u226b g = y' \u226b h := funext (\u03bb _, hy),\n  refine \u27e8(fork.is_limit.lift' t _ hy').1 \u27e8\u27e9, congr_fun (fork.is_limit.lift' t y' _).2 \u27e8\u27e9, _\u27e9,\n  intros x' hx',\n  suffices : (\u03bb (_ : punit), x') = (fork.is_limit.lift' t y' hy').1,\n    rw \u2190 this,\n  apply fork.is_limit.hom_ext t,\n  ext \u27e8\u27e9,\n  apply hx'.trans (congr_fun (fork.is_limit.lift' t _ hy').2 \u27e8\u27e9).symm,\nend\n\nlemma type_equalizer_iff_unique :\n  nonempty (is_limit (fork.of_\u03b9 _ w)) \u2194 (\u2200 (y : Y), g y = h y \u2192 \u2203! (x : X), f x = y) :=\n\u27e8\u03bb i, unique_of_type_equalizer _ _ (classical.choice i), \u03bb k, \u27e8type_equalizer_of_unique f w k\u27e9\u27e9\n\n/-- Show that the subtype `{x : Y // g x = h x}` is an equalizer for the pair `(g,h)`. -/\ndef equalizer_limit : limits.limit_cone (parallel_pair g h) :=\n{ cone := fork.of_\u03b9 (subtype.val : {x : Y // g x = h x} \u2192 Y) (funext subtype.prop),\n  is_limit := fork.is_limit.mk' _ $ \u03bb s,\n    \u27e8\u03bb i, \u27e8s.\u03b9 i, by apply congr_fun s.condition i\u27e9,\n     rfl,\n     \u03bb m hm, funext $ \u03bb x, subtype.ext (congr_fun hm x)\u27e9 }\n\nvariables (g h)\n\n/-- The categorical equalizer in `Type u` is `{x : Y // g x = h x}`. -/\nnoncomputable def equalizer_iso : equalizer g h \u2245 {x : Y // g x = h x} :=\nlimit.iso_limit_cone equalizer_limit\n\n@[simp, elementwise] lemma equalizer_iso_hom_comp_subtype :\n  (equalizer_iso g h).hom \u226b subtype.val = equalizer.\u03b9 g h :=\nrfl\n\n@[simp, elementwise] lemma equalizer_iso_inv_comp_\u03b9 :\n  (equalizer_iso g h).inv \u226b equalizer.\u03b9 g h = subtype.val :=\nlimit.iso_limit_cone_inv_\u03c0 equalizer_limit walking_parallel_pair.zero\n\nend fork\n\nsection cofork\nvariables {X Y Z : Type u} (f g : X \u27f6 Y)\n\n/-- (Implementation) The relation to be quotiented to obtain the coequalizer. -/\ninductive coequalizer_rel : Y \u2192 Y \u2192 Prop\n| rel (x : X) : coequalizer_rel (f x) (g x)\n\n/--\nShow that the quotient by the relation generated by `f(x) ~ g(x)`\nis a coequalizer for the pair `(f, g)`.\n-/\ndef coequalizer_colimit : limits.colimit_cocone (parallel_pair f g) :=\n{ cocone := cofork.of_\u03c0 (quot.mk (coequalizer_rel f g))\n    (funext (\u03bb x, quot.sound (coequalizer_rel.rel x))),\n  is_colimit := cofork.is_colimit.mk' _ $ \u03bb s,\n    \u27e8 quot.lift s.\u03c0 (\u03bb a b (h : coequalizer_rel f g a b),\n        by { cases h, exact congr_fun s.condition h_1 }),\n      rfl,\n      \u03bb m hm, funext $ \u03bb x, quot.induction_on x (congr_fun hm : _) \u27e9 }\n\n/-- If `\u03c0 : Y \u27f6 Z` is an equalizer for `(f, g)`, and `U \u2286 Y` such that `f \u207b\u00b9' U = g \u207b\u00b9' U`,\nthen `\u03c0 \u207b\u00b9' (\u03c0 '' U) = U`.\n-/\nlemma coequalizer_preimage_image_eq_of_preimage_eq (\u03c0 : Y \u27f6 Z)\n  (e : f \u226b \u03c0 = g \u226b \u03c0) (h : is_colimit (cofork.of_\u03c0 \u03c0 e)) (U : set Y) (H : f \u207b\u00b9' U = g \u207b\u00b9' U) :\n    \u03c0 \u207b\u00b9' (\u03c0 '' U) = U :=\nbegin\n  have lem : \u2200 x y, (coequalizer_rel f g x y) \u2192 (x \u2208 U \u2194 y \u2208 U),\n  { rintros _ _ \u27e8x\u27e9, change x \u2208 f \u207b\u00b9' U \u2194 x \u2208 g \u207b\u00b9' U, congr' 2 },\n  have eqv : _root_.equivalence (\u03bb x y, x \u2208 U \u2194 y \u2208 U) := by tidy,\n  ext,\n  split,\n  { rw \u2190 (show _ = \u03c0, from h.comp_cocone_point_unique_up_to_iso_inv\n      (coequalizer_colimit f g).2 walking_parallel_pair.one),\n    rintro \u27e8y, hy, e'\u27e9,\n    dsimp at e',\n    replace e' := (mono_iff_injective (h.cocone_point_unique_up_to_iso\n      (coequalizer_colimit f g).is_colimit).inv).mp infer_instance e',\n    exact (eqv.eqv_gen_iff.mp (eqv_gen.mono lem (quot.exact _ e'))).mp hy },\n  { exact \u03bb hx, \u27e8x, hx, rfl\u27e9 }\nend\n\n/-- The categorical coequalizer in `Type u` is the quotient by `f g ~ g x`. -/\nnoncomputable def coequalizer_iso : coequalizer f g \u2245 _root_.quot (coequalizer_rel f g) :=\ncolimit.iso_colimit_cocone (coequalizer_colimit f g)\n\n@[simp, elementwise] lemma coequalizer_iso_\u03c0_comp_hom :\n  coequalizer.\u03c0 f g \u226b (coequalizer_iso f g).hom = quot.mk (coequalizer_rel f g) :=\ncolimit.iso_colimit_cocone_\u03b9_hom (coequalizer_colimit f g) walking_parallel_pair.one\n\n@[simp, elementwise] lemma coequalizer_iso_quot_comp_inv :\n  \u21be(quot.mk (coequalizer_rel f g)) \u226b (coequalizer_iso f g).inv = coequalizer.\u03c0 f g :=\nrfl\n\nend cofork\n\nsection pullback\nopen category_theory.limits.walking_pair\nopen category_theory.limits.walking_cospan\nopen category_theory.limits.walking_cospan.hom\n\nvariables {W X Y Z : Type u}\nvariables (f : X \u27f6 Z) (g : Y \u27f6 Z)\n\n/--\nThe usual explicit pullback in the category of types, as a subtype of the product.\nThe full `limit_cone` data is bundled as `pullback_limit_cone f g`.\n-/\n@[nolint has_inhabited_instance]\nabbreviation pullback_obj : Type u := { p : X \u00d7 Y // f p.1 = g p.2 }\n\n-- `pullback_obj f g` comes with a coercion to the product type `X \u00d7 Y`.\nexample (p : pullback_obj f g) : X \u00d7 Y := p\n\n/--\nThe explicit pullback cone on `pullback_obj f g`.\nThis is bundled with the `is_limit` data as `pullback_limit_cone f g`.\n-/\nabbreviation pullback_cone : limits.pullback_cone f g :=\npullback_cone.mk (\u03bb p : pullback_obj f g, p.1.1) (\u03bb p, p.1.2) (funext (\u03bb p, p.2))\n\n/--\nThe explicit pullback in the category of types, bundled up as a `limit_cone`\nfor given `f` and `g`.\n-/\n@[simps]\ndef pullback_limit_cone (f : X \u27f6 Z) (g : Y \u27f6 Z) : limits.limit_cone (cospan f g) :=\n{ cone := pullback_cone f g,\n  is_limit := pullback_cone.is_limit_aux _\n    (\u03bb s x, \u27e8\u27e8s.fst x, s.snd x\u27e9, congr_fun s.condition x\u27e9)\n    (by tidy)\n    (by tidy)\n    (\u03bb s m w, funext $ \u03bb x, subtype.ext $\n     prod.ext (congr_fun (w walking_cospan.left) x)\n              (congr_fun (w walking_cospan.right) x)) }\n\n/--\nThe pullback cone given by the instance `has_pullbacks (Type u)` is isomorphic to the\nexplicit pullback cone given by `pullback_limit_cone`.\n-/\nnoncomputable def pullback_cone_iso_pullback : limit.cone (cospan f g) \u2245 pullback_cone f g :=\n(limit.is_limit _).unique_up_to_iso (pullback_limit_cone f g).is_limit\n\n/--\nThe pullback given by the instance `has_pullbacks (Type u)` is isomorphic to the\nexplicit pullback object given by `pullback_limit_obj`.\n-/\nnoncomputable def pullback_iso_pullback : pullback f g \u2245 pullback_obj f g :=\n(cones.forget _).map_iso $ pullback_cone_iso_pullback f g\n\n@[simp] lemma pullback_iso_pullback_hom_fst (p : pullback f g) :\n  ((pullback_iso_pullback f g).hom p : X \u00d7 Y).fst = (pullback.fst : _ \u27f6 X) p :=\ncongr_fun ((pullback_cone_iso_pullback f g).hom.w left) p\n\n@[simp] lemma pullback_iso_pullback_hom_snd (p : pullback f g) :\n  ((pullback_iso_pullback f g).hom p : X \u00d7 Y).snd = (pullback.snd : _ \u27f6 Y) p :=\ncongr_fun ((pullback_cone_iso_pullback f g).hom.w right) p\n\n@[simp] lemma pullback_iso_pullback_inv_fst :\n  (pullback_iso_pullback f g).inv \u226b pullback.fst = (\u03bb p, (p : X \u00d7 Y).fst) :=\n(pullback_cone_iso_pullback f g).inv.w left\n\n@[simp] lemma pullback_iso_pullback_inv_snd :\n  (pullback_iso_pullback f g).inv \u226b pullback.snd = (\u03bb p, (p : X \u00d7 Y).snd) :=\n(pullback_cone_iso_pullback f g).inv.w right\n\nend pullback\n\nend category_theory.limits.types\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/shapes/types.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3759229968531074}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.functor\nimport Mathlib.PostPort\n\nuniverses v u l u\u2082 v\u2082 \n\nnamespace Mathlib\n\n/-!\n# Isomorphisms\n\nThis file defines isomorphisms between objects of a category.\n\n## Main definitions\n\n- `structure iso` : a bundled isomorphism between two objects of a category;\n- `class is_iso` : an unbundled version of `iso`; note that `is_iso f` is usually *not* a `Prop`,\n  because it holds the inverse morphism;\n- `as_iso` : convert from `is_iso` to `iso`;\n- `of_iso` : convert from `iso` to `is_iso`;\n- standard operations on isomorphisms (composition, inverse etc)\n\n## Notations\n\n- `X \u2245 Y` : same as `iso X Y`;\n- `\u03b1 \u226a\u226b \u03b2` : composition of two isomorphisms; it is called `iso.trans`\n\n## Tags\n\ncategory, category theory, isomorphism\n-/\n\nnamespace category_theory\n\n\n/--\nAn isomorphism (a.k.a. an invertible morphism) between two objects of a category.\nThe inverse morphism is bundled.\n\nSee also `category_theory.core` for the category with the same objects and isomorphisms playing\nthe role of morphisms.\n\nSee https://stacks.math.columbia.edu/tag/0017.\n-/\nstructure iso {C : Type u} [category C] (X : C) (Y : C) \nwhere\n  hom : X \u27f6 Y\n  inv : Y \u27f6 X\n  hom_inv_id' : autoParam (hom \u226b inv = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  inv_hom_id' : autoParam (inv \u226b hom = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n@[simp] theorem iso.hom_inv_id {C : Type u} [category C] {X : C} {Y : C} (c : iso X Y) : iso.hom c \u226b iso.inv c = \ud835\udfd9 := sorry\n\n@[simp] theorem iso.inv_hom_id {C : Type u} [category C] {X : C} {Y : C} (c : iso X Y) : iso.inv c \u226b iso.hom c = \ud835\udfd9 := sorry\n\n@[simp] theorem iso.hom_inv_id_assoc {C : Type u} [category C] {X : C} {Y : C} (c : iso X Y) {X' : C} (f' : X \u27f6 X') : iso.hom c \u226b iso.inv c \u226b f' = f' := sorry\n\ninfixr:10 \" \u2245 \" => Mathlib.category_theory.iso\n\nnamespace iso\n\n\ntheorem ext {C : Type u} [category C] {X : C} {Y : C} {\u03b1 : X \u2245 Y} {\u03b2 : X \u2245 Y} (w : hom \u03b1 = hom \u03b2) : \u03b1 = \u03b2 := sorry\n\n/-- Inverse isomorphism. -/\ndef symm {C : Type u} [category C] {X : C} {Y : C} (I : X \u2245 Y) : Y \u2245 X :=\n  mk (inv I) (hom I)\n\n@[simp] theorem symm_hom {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : hom (symm \u03b1) = inv \u03b1 :=\n  rfl\n\n@[simp] theorem symm_inv {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : inv (symm \u03b1) = hom \u03b1 :=\n  rfl\n\n@[simp] theorem symm_mk {C : Type u} [category C] {X : C} {Y : C} (hom : X \u27f6 Y) (inv : Y \u27f6 X) (hom_inv_id : hom \u226b inv = \ud835\udfd9) (inv_hom_id : inv \u226b hom = \ud835\udfd9) : symm (mk hom inv) = mk inv hom :=\n  rfl\n\n@[simp] theorem symm_symm_eq {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : symm (symm \u03b1) = \u03b1 := sorry\n\n@[simp] theorem symm_eq_iff {C : Type u} [category C] {X : C} {Y : C} {\u03b1 : X \u2245 Y} {\u03b2 : X \u2245 Y} : symm \u03b1 = symm \u03b2 \u2194 \u03b1 = \u03b2 :=\n  { mp := fun (h : symm \u03b1 = symm \u03b2) => symm_symm_eq \u03b1 \u25b8 symm_symm_eq \u03b2 \u25b8 congr_arg symm h, mpr := congr_arg symm }\n\n/-- Identity isomorphism. -/\n@[simp] theorem refl_inv {C : Type u} [category C] (X : C) : inv (refl X) = \ud835\udfd9 :=\n  Eq.refl (inv (refl X))\n\nprotected instance inhabited {C : Type u} [category C] {X : C} : Inhabited (X \u2245 X) :=\n  { default := refl X }\n\n@[simp] theorem refl_symm {C : Type u} [category C] (X : C) : symm (refl X) = refl X :=\n  rfl\n\n/-- Composition of two isomorphisms -/\ndef trans {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : X \u2245 Z :=\n  mk (hom \u03b1 \u226b hom \u03b2) (inv \u03b2 \u226b inv \u03b1)\n\ninfixr:80 \" \u226a\u226b \" => Mathlib.category_theory.iso.trans\n\n@[simp] theorem trans_mk {C : Type u} [category C] {X : C} {Y : C} {Z : C} (hom : X \u27f6 Y) (inv : Y \u27f6 X) (hom_inv_id : hom \u226b inv = \ud835\udfd9) (inv_hom_id : inv \u226b hom = \ud835\udfd9) (hom' : Y \u27f6 Z) (inv' : Z \u27f6 Y) (hom_inv_id' : hom' \u226b inv' = \ud835\udfd9) (inv_hom_id' : inv' \u226b hom' = \ud835\udfd9) (hom_inv_id'' : (hom \u226b hom') \u226b inv' \u226b inv = \ud835\udfd9) (inv_hom_id'' : (inv' \u226b inv) \u226b hom \u226b hom' = \ud835\udfd9) : mk hom inv \u226a\u226b mk hom' inv' = mk (hom \u226b hom') (inv' \u226b inv) :=\n  rfl\n\n@[simp] theorem trans_symm {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : symm (\u03b1 \u226a\u226b \u03b2) = symm \u03b2 \u226a\u226b symm \u03b1 :=\n  rfl\n\n@[simp] theorem trans_assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} {Z' : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) (\u03b3 : Z \u2245 Z') : (\u03b1 \u226a\u226b \u03b2) \u226a\u226b \u03b3 = \u03b1 \u226a\u226b \u03b2 \u226a\u226b \u03b3 := sorry\n\n@[simp] theorem refl_trans {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : refl X \u226a\u226b \u03b1 = \u03b1 :=\n  ext (category.id_comp (hom \u03b1))\n\n@[simp] theorem trans_refl {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : \u03b1 \u226a\u226b refl Y = \u03b1 :=\n  ext (category.comp_id (hom \u03b1))\n\n@[simp] theorem symm_self_id {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : symm \u03b1 \u226a\u226b \u03b1 = refl Y :=\n  ext (inv_hom_id \u03b1)\n\n@[simp] theorem self_symm_id {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) : \u03b1 \u226a\u226b symm \u03b1 = refl X :=\n  ext (hom_inv_id \u03b1)\n\n@[simp] theorem symm_self_id_assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 Z) : symm \u03b1 \u226a\u226b \u03b1 \u226a\u226b \u03b2 = \u03b2 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (symm \u03b1 \u226a\u226b \u03b1 \u226a\u226b \u03b2 = \u03b2)) (Eq.symm (trans_assoc (symm \u03b1) \u03b1 \u03b2))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((symm \u03b1 \u226a\u226b \u03b1) \u226a\u226b \u03b2 = \u03b2)) (symm_self_id \u03b1)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (refl Y \u226a\u226b \u03b2 = \u03b2)) (refl_trans \u03b2))) (Eq.refl \u03b2)))\n\n@[simp] theorem self_symm_id_assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) (\u03b2 : X \u2245 Z) : \u03b1 \u226a\u226b symm \u03b1 \u226a\u226b \u03b2 = \u03b2 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u03b1 \u226a\u226b symm \u03b1 \u226a\u226b \u03b2 = \u03b2)) (Eq.symm (trans_assoc \u03b1 (symm \u03b1) \u03b2))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((\u03b1 \u226a\u226b symm \u03b1) \u226a\u226b \u03b2 = \u03b2)) (self_symm_id \u03b1)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (refl X \u226a\u226b \u03b2 = \u03b2)) (refl_trans \u03b2))) (Eq.refl \u03b2)))\n\ntheorem inv_comp_eq {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) {f : X \u27f6 Z} {g : Y \u27f6 Z} : inv \u03b1 \u226b f = g \u2194 f = hom \u03b1 \u226b g := sorry\n\ntheorem eq_inv_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) {f : X \u27f6 Z} {g : Y \u27f6 Z} : g = inv \u03b1 \u226b f \u2194 hom \u03b1 \u226b g = f :=\n  iff.symm (inv_comp_eq (symm \u03b1))\n\ntheorem comp_inv_eq {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) {f : Z \u27f6 Y} {g : Z \u27f6 X} : f \u226b inv \u03b1 = g \u2194 f = g \u226b hom \u03b1 := sorry\n\ntheorem eq_comp_inv {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u2245 Y) {f : Z \u27f6 Y} {g : Z \u27f6 X} : g = f \u226b inv \u03b1 \u2194 g \u226b hom \u03b1 = f :=\n  iff.symm (comp_inv_eq (symm \u03b1))\n\ntheorem inv_eq_inv {C : Type u} [category C] {X : C} {Y : C} (f : X \u2245 Y) (g : X \u2245 Y) : inv f = inv g \u2194 hom f = hom g := sorry\n\ntheorem hom_comp_eq_id {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) {f : Y \u27f6 X} : hom \u03b1 \u226b f = \ud835\udfd9 \u2194 f = inv \u03b1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (hom \u03b1 \u226b f = \ud835\udfd9 \u2194 f = inv \u03b1)) (Eq.symm (propext (eq_inv_comp \u03b1)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (f = inv \u03b1 \u226b \ud835\udfd9 \u2194 f = inv \u03b1)) (category.comp_id (inv \u03b1)))) (iff.refl (f = inv \u03b1)))\n\ntheorem comp_hom_eq_id {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) {f : Y \u27f6 X} : f \u226b hom \u03b1 = \ud835\udfd9 \u2194 f = inv \u03b1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f \u226b hom \u03b1 = \ud835\udfd9 \u2194 f = inv \u03b1)) (Eq.symm (propext (eq_comp_inv \u03b1)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (f = \ud835\udfd9 \u226b inv \u03b1 \u2194 f = inv \u03b1)) (category.id_comp (inv \u03b1)))) (iff.refl (f = inv \u03b1)))\n\ntheorem hom_eq_inv {C : Type u} [category C] {X : C} {Y : C} (\u03b1 : X \u2245 Y) (\u03b2 : Y \u2245 X) : hom \u03b1 = inv \u03b2 \u2194 hom \u03b2 = inv \u03b1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (hom \u03b1 = inv \u03b2 \u2194 hom \u03b2 = inv \u03b1)) (propext (inv_eq_inv (symm \u03b1) \u03b2))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (hom (symm \u03b1) = hom \u03b2 \u2194 hom \u03b2 = inv \u03b1)) (propext eq_comm)))\n      (iff.refl (hom \u03b2 = hom (symm \u03b1))))\n\nend iso\n\n\n/-- `is_iso` typeclass expressing that a morphism is invertible.\n    This contains the data of the inverse, but is a subsingleton type. -/\nclass is_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) \nwhere\n  inv : Y \u27f6 X\n  hom_inv_id' : autoParam (f \u226b inv = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  inv_hom_id' : autoParam (inv \u226b f = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n/-- Reinterpret a morphism `f` with an `is_iso f` instance as an `iso`. -/\ndef as_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [h : is_iso f] : X \u2245 Y :=\n  iso.mk f (inv f)\n\n@[simp] theorem as_iso_hom {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : iso.hom (as_iso f) = f :=\n  rfl\n\n@[simp] theorem as_iso_inv {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : iso.inv (as_iso f) = inv f :=\n  rfl\n\nnamespace is_iso\n\n\n@[simp] theorem hom_inv_id {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : f \u226b inv f = \ud835\udfd9 :=\n  hom_inv_id'\n\n@[simp] theorem inv_hom_id {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : inv f \u226b f = \ud835\udfd9 :=\n  inv_hom_id'\n\n@[simp] theorem hom_inv_id_assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) [is_iso f] (g : X \u27f6 Z) : f \u226b inv f \u226b g = g :=\n  iso.hom_inv_id_assoc (as_iso f) g\n\n@[simp] theorem inv_hom_id_assoc {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) [is_iso f] (g : Y \u27f6 Z) : inv f \u226b f \u226b g = g :=\n  iso.inv_hom_id_assoc (as_iso f) g\n\nprotected instance id {C : Type u} [category C] (X : C) : is_iso \ud835\udfd9 :=\n  mk \ud835\udfd9\n\nprotected instance of_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u2245 Y) : is_iso (iso.hom f) :=\n  mk (iso.inv f)\n\nprotected instance of_iso_inv {C : Type u} [category C] {X : C} {Y : C} (f : X \u2245 Y) : is_iso (iso.inv f) :=\n  is_iso.of_iso (iso.symm f)\n\nprotected instance inv_is_iso {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [is_iso f] : is_iso (inv f) :=\n  is_iso.of_iso_inv (as_iso f)\n\nprotected instance comp_is_iso {C : Type u} [category C] {X : C} {Y : C} {Z : C} {f : X \u27f6 Y} {h : Y \u27f6 Z} [is_iso f] [is_iso h] : is_iso (f \u226b h) :=\n  is_iso.of_iso (as_iso f \u226a\u226b as_iso h)\n\n@[simp] theorem inv_id {C : Type u} [category C] {X : C} : inv \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem inv_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} {f : X \u27f6 Y} {h : Y \u27f6 Z} [is_iso f] [is_iso h] : inv (f \u226b h) = inv h \u226b inv f :=\n  rfl\n\n@[simp] theorem inv_inv {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} [is_iso f] : inv (inv f) = f :=\n  rfl\n\n@[simp] theorem iso.inv_inv {C : Type u} [category C] {X : C} {Y : C} (f : X \u2245 Y) : inv (iso.inv f) = iso.hom f :=\n  rfl\n\n@[simp] theorem iso.inv_hom {C : Type u} [category C] {X : C} {Y : C} (f : X \u2245 Y) : inv (iso.hom f) = iso.inv f :=\n  rfl\n\n@[simp] theorem inv_comp_eq {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : X \u27f6 Z} {g : Y \u27f6 Z} : inv \u03b1 \u226b f = g \u2194 f = \u03b1 \u226b g :=\n  iso.inv_comp_eq (as_iso \u03b1)\n\n@[simp] theorem eq_inv_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : X \u27f6 Z} {g : Y \u27f6 Z} : g = inv \u03b1 \u226b f \u2194 \u03b1 \u226b g = f :=\n  iso.eq_inv_comp (as_iso \u03b1)\n\n@[simp] theorem comp_inv_eq {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : Z \u27f6 Y} {g : Z \u27f6 X} : f \u226b inv \u03b1 = g \u2194 f = g \u226b \u03b1 :=\n  iso.comp_inv_eq (as_iso \u03b1)\n\n@[simp] theorem eq_comp_inv {C : Type u} [category C] {X : C} {Y : C} {Z : C} (\u03b1 : X \u27f6 Y) [is_iso \u03b1] {f : Z \u27f6 Y} {g : Z \u27f6 X} : g = f \u226b inv \u03b1 \u2194 g \u226b \u03b1 = f :=\n  iso.eq_comp_inv (as_iso \u03b1)\n\nprotected instance epi_of_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : epi f :=\n  epi.mk\n    fun (Z : C) (g h : Y \u27f6 Z) (w : f \u226b g = f \u226b h) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (g = h)) (Eq.symm (inv_hom_id_assoc f g))))\n        (eq.mpr (id (Eq._oldrec (Eq.refl (inv f \u226b f \u226b g = h)) w))\n          (eq.mpr (id (Eq._oldrec (Eq.refl (inv f \u226b f \u226b h = h)) (inv_hom_id_assoc f h))) (Eq.refl h)))\n\nprotected instance mono_of_iso {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : mono f :=\n  mono.mk\n    fun (Z : C) (g h : Z \u27f6 X) (w : g \u226b f = h \u226b f) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (g = h)) (Eq.symm (category.comp_id g))))\n        (eq.mpr (id (Eq._oldrec (Eq.refl (g \u226b \ud835\udfd9 = h)) (Eq.symm (category.comp_id h))))\n          (eq.mpr (id (Eq._oldrec (Eq.refl (g \u226b \ud835\udfd9 = h \u226b \ud835\udfd9)) (Eq.symm (hom_inv_id f))))\n            (eq.mpr (id (Eq._oldrec (Eq.refl (g \u226b f \u226b inv f = h \u226b f \u226b inv f)) (Eq.symm (category.assoc g f (inv f)))))\n              (eq.mpr (id (Eq._oldrec (Eq.refl ((g \u226b f) \u226b inv f = h \u226b f \u226b inv f)) w))\n                (eq.mpr\n                  (id (Eq._oldrec (Eq.refl ((h \u226b f) \u226b inv f = h \u226b f \u226b inv f)) (Eq.symm (category.assoc h f (inv f)))))\n                  (Eq.refl ((h \u226b f) \u226b inv f)))))))\n\nend is_iso\n\n\ntheorem eq_of_inv_eq_inv {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} {g : X \u27f6 Y} [is_iso f] [is_iso g] (p : inv f = inv g) : f = g := sorry\n\nprotected instance is_iso.subsingleton {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) : subsingleton (is_iso f) := sorry\n\ntheorem is_iso.inv_eq_inv {C : Type u} [category C] {X : C} {Y : C} {f : X \u27f6 Y} {g : X \u27f6 Y} [is_iso f] [is_iso g] : inv f = inv g \u2194 f = g :=\n  iso.inv_eq_inv (as_iso f) (as_iso g)\n\ntheorem hom_comp_eq_id {C : Type u} [category C] {X : C} {Y : C} (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} : g \u226b f = \ud835\udfd9 \u2194 f = inv g :=\n  iso.hom_comp_eq_id (as_iso g)\n\ntheorem comp_hom_eq_id {C : Type u} [category C] {X : C} {Y : C} (g : X \u27f6 Y) [is_iso g] {f : Y \u27f6 X} : f \u226b g = \ud835\udfd9 \u2194 f = inv g :=\n  iso.comp_hom_eq_id (as_iso g)\n\nnamespace iso\n\n\n/-!\nAll these cancellation lemmas can be solved by `simp [cancel_mono]` (or `simp [cancel_epi]`),\nbut with the current design `cancel_mono` is not a good `simp` lemma,\nbecause it generates a typeclass search.\n\nWhen we can see syntactically that a morphism is a `mono` or an `epi`\nbecause it came from an isomorphism, it's fine to do the cancellation via `simp`.\n\nIn the longer term, it might be worth exploring making `mono` and `epi` structures,\nrather than typeclasses, with coercions back to `X \u27f6 Y`.\nPresumably we could write `X \u21aa Y` and `X \u21a0 Y`.\n-/\n\n@[simp] theorem cancel_iso_hom_left {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : X \u2245 Y) (g : Y \u27f6 Z) (g' : Y \u27f6 Z) : hom f \u226b g = hom f \u226b g' \u2194 g = g' := sorry\n\n@[simp] theorem cancel_iso_inv_left {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : Y \u2245 X) (g : Y \u27f6 Z) (g' : Y \u27f6 Z) : inv f \u226b g = inv f \u226b g' \u2194 g = g' := sorry\n\n@[simp] theorem cancel_iso_hom_right {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) (f' : X \u27f6 Y) (g : Y \u2245 Z) : f \u226b hom g = f' \u226b hom g \u2194 f = f' := sorry\n\n@[simp] theorem cancel_iso_inv_right {C : Type u} [category C] {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) (f' : X \u27f6 Y) (g : Z \u2245 Y) : f \u226b inv g = f' \u226b inv g \u2194 f = f' := sorry\n\n/-\nUnfortunately cancelling an isomorphism from the right of a chain of compositions is awkward.\nWe would need separate lemmas for each chain length (worse: for each pair of chain lengths).\n\nWe provide two more lemmas, for case of three morphisms, because this actually comes up in practice,\nbut then stop.\n-/\n\n@[simp] theorem cancel_iso_hom_right_assoc {C : Type u} [category C] {W : C} {X : C} {X' : C} {Y : C} {Z : C} (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y) (h : Y \u2245 Z) : f \u226b g \u226b hom h = f' \u226b g' \u226b hom h \u2194 f \u226b g = f' \u226b g' := sorry\n\n@[simp] theorem cancel_iso_inv_right_assoc {C : Type u} [category C] {W : C} {X : C} {X' : C} {Y : C} {Z : C} (f : W \u27f6 X) (g : X \u27f6 Y) (f' : W \u27f6 X') (g' : X' \u27f6 Y) (h : Z \u2245 Y) : f \u226b g \u226b inv h = f' \u226b g' \u226b inv h \u2194 f \u226b g = f' \u226b g' := sorry\n\nend iso\n\n\nnamespace functor\n\n\n/-- A functor `F : C \u2964 D` sends isomorphisms `i : X \u2245 Y` to isomorphisms `F.obj X \u2245 F.obj Y` -/\ndef map_iso {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (i : X \u2245 Y) : obj F X \u2245 obj F Y :=\n  iso.mk (map F (iso.hom i)) (map F (iso.inv i))\n\n@[simp] theorem map_iso_hom {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (i : X \u2245 Y) : iso.hom (map_iso F i) = map F (iso.hom i) :=\n  rfl\n\n@[simp] theorem map_iso_inv {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (i : X \u2245 Y) : iso.inv (map_iso F i) = map F (iso.inv i) :=\n  rfl\n\n@[simp] theorem map_iso_symm {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (i : X \u2245 Y) : map_iso F (iso.symm i) = iso.symm (map_iso F i) :=\n  rfl\n\n@[simp] theorem map_iso_trans {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} {Z : C} (i : X \u2245 Y) (j : Y \u2245 Z) : map_iso F (i \u226a\u226b j) = map_iso F i \u226a\u226b map_iso F j :=\n  iso.ext (map_comp F (iso.hom i) (iso.hom j))\n\n@[simp] theorem map_iso_refl {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) (X : C) : map_iso F (iso.refl X) = iso.refl (obj F X) :=\n  iso.ext (map_id F X)\n\nprotected instance map_is_iso {C : Type u} [category C] {X : C} {Y : C} {D : Type u\u2082} [category D] (F : C \u2964 D) (f : X \u27f6 Y) [is_iso f] : is_iso (map F f) :=\n  is_iso.of_iso (map_iso F (as_iso f))\n\n@[simp] theorem map_inv {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : map F (inv f) = inv (map F f) :=\n  rfl\n\ntheorem map_hom_inv {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : map F f \u226b map F (inv f) = \ud835\udfd9 := sorry\n\ntheorem map_inv_hom {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (f : X \u27f6 Y) [is_iso f] : map F (inv f) \u226b map F f = \ud835\udfd9 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/isomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.37592272977513763}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (U : set (euclidean_space \u211d (fin n))) (hU : is_open U) (hcover : \u2200 x : euclidean_space \u211d (fin n), \u2203 V \u2208 U, x \u2208 V),\n  have h1 : \u2200 m : \u2115, \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2286 W) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2229 (closure (ball (0 : euclidean_space \u211d (fin n)) m)) \u2286 W) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2229 (ball (0 : euclidean_space \u211d (fin n)) m) \u2286 W), from\n    assume m : \u2115,\n    have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2286 W) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2229 (closure (ball (0 : euclidean_space \u211d (fin n)) m)) \u2286 W), from\n      have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2286 W), from\n        have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))) \u2227 (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 \u2203 (W : set (euclidean_space \u211d (fin n))), W \u2208 U \u2227 V \u2286 W), from\n          have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n            have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n              have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                  have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                    have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                      have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                        have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                          have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                            have h1 : \u2203 (Cm : set (euclidean_space \u211d (fin n))), (\u2200 (V : set (euclidean_space \u211d (fin n))), V \u2208 Cm \u2192 V \u2286 (euclidean_space \u211d (fin n)) \u2216 (closure (ball (0 : euclidean_space \u211d (fin n)) m))), from\n                              have h1 : \u2203\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.822189134878876, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.3758528417297677}}
{"text": "import tactic.tidy\n\n-- assembly verification\n\ninductive assembly.expr\n| cons : \u2115 \u2192 assembly.expr\n| register : fin 12 \u2192 assembly.expr\n\ninductive assembly.cmd\n| mov : assembly.expr \u2192 assembly.expr \u2192 assembly.cmd\n| add : assembly.expr \u2192 assembly.expr \u2192 assembly.cmd\n\ndef assembly.program := list assembly.cmd\n\ndef assembly.state := fin 12 \u2192 \u2115\n\n-- def assembly.run (state : assembly.state) :\n--                   assembly.cmd \u2192 option assembly.state\n-- | (cmd.mov (expr.cons n) (expr.register i)) := some $ \u03bb r,\n--                                                if r = i \n--                                                then n\n--                                                else state r\n-- | (cmd.mov (expr.register j) (expr.register i)) := some $ \u03bb r,\n--                                                    if r = i\n--                                                    then state j\n--                                                    else state r\n-- | (cmd.add a a_1) := sorry\n-- | _  := none\n\nexample : \u2200 n : \u2115, \u2203 m : \u2115, n + 1 = m  :=\nbegin\n    intro n,\n    existsi n+1,\n    refl,\nend\n\n\n\n", "meta": {"author": "maxd13", "repo": "lean-logic", "sha": "ddcab46b77adca91b120a5f37afbd48794da8b52", "save_path": "github-repos/lean/maxd13-lean-logic", "path": "github-repos/lean/maxd13-lean-logic/lean-logic-ddcab46b77adca91b120a5f37afbd48794da8b52/src/assembly.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3758301185775846}}
{"text": "import category_theory.limits.preserves.shapes.zero\nimport category_theory.abelian.homology\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C D : Type*} [category C] [category D]\n\nnamespace category_theory\n\nvariable (C)\n\n/- TODO : define the subcategory of complexes with 3 objects, and consider\nfunctor to this category, etc. -/\n\nstructure composable_morphisms := {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\ninstance [inhabited C] : inhabited (composable_morphisms C) := \u27e8\u27e8\ud835\udfd9 default, \ud835\udfd9 default\u27e9\u27e9\n\nvariable {C}\n\nnamespace composable_morphisms\n\n@[ext]\nstructure hom (S\u2081 S\u2082 : composable_morphisms C) :=\n(\u03c4\u2081 : S\u2081.X \u27f6 S\u2082.X) (\u03c4\u2082 : S\u2081.Y \u27f6 S\u2082.Y) (\u03c4\u2083 : S\u2081.Z \u27f6 S\u2082.Z)\n(comm\u2081\u2082' : S\u2081.f \u226b \u03c4\u2082 = \u03c4\u2081 \u226b S\u2082.f) (comm\u2082\u2083' : S\u2081.g \u226b \u03c4\u2083 = \u03c4\u2082 \u226b S\u2082.g)\n\nnamespace hom\n\nrestate_axiom comm\u2081\u2082'\nrestate_axiom comm\u2082\u2083'\n\nattribute [reassoc] comm\u2081\u2082\nattribute [reassoc] comm\u2082\u2083\n\nlocal attribute [simp] comm\u2081\u2082 comm\u2082\u2083\n\n@[simps]\ndef id (S : composable_morphisms C) : hom S S :=\n{ \u03c4\u2081 := \ud835\udfd9 _, \u03c4\u2082 := \ud835\udfd9 _, \u03c4\u2083 := \ud835\udfd9 _, comm\u2081\u2082' := by simp, comm\u2082\u2083' := by simp, }\n\ninstance (S : composable_morphisms C) : inhabited (hom S S) := \u27e8id S\u27e9\n\n@[simps]\ndef comp {S\u2081 S\u2082 S\u2083 : composable_morphisms C} (\u03c6 : hom S\u2081 S\u2082) (\u03c8 : hom S\u2082 S\u2083) :\n  hom S\u2081 S\u2083 :=\n{ \u03c4\u2081 := \u03c6.\u03c4\u2081 \u226b \u03c8.\u03c4\u2081,\n  \u03c4\u2082 := \u03c6.\u03c4\u2082 \u226b \u03c8.\u03c4\u2082,\n  \u03c4\u2083 := \u03c6.\u03c4\u2083 \u226b \u03c8.\u03c4\u2083,\n  comm\u2081\u2082' := by simp only [comm\u2081\u2082_assoc, comm\u2081\u2082, assoc],\n  comm\u2082\u2083' := by simp only [comm\u2082\u2083_assoc, comm\u2082\u2083, assoc], }\n\nend hom\n\ninstance : category (composable_morphisms C) :=\n{ hom := \u03bb S\u2081 S\u2082, hom S\u2081 S\u2082,\n  id := hom.id,\n  comp := \u03bb S\u2081 S\u2082 S\u2083, hom.comp, }\n\n@[simp] lemma id_\u03c4\u2081 (S : composable_morphisms C) : hom.\u03c4\u2081 (\ud835\udfd9 S) = \ud835\udfd9 _ := rfl\n@[simp] lemma id_\u03c4\u2082 (S : composable_morphisms C) : hom.\u03c4\u2082 (\ud835\udfd9 S) = \ud835\udfd9 _ := rfl\n@[simp] lemma id_\u03c4\u2083 (S : composable_morphisms C) : hom.\u03c4\u2083 (\ud835\udfd9 S) = \ud835\udfd9 _ := rfl\n\n@[simp] lemma comp_\u03c4\u2081 {S\u2081 S\u2082 S\u2083 : composable_morphisms C} (\u03c6 : S\u2081 \u27f6 S\u2082) (\u03c8 : S\u2082 \u27f6 S\u2083) :\n  (\u03c6 \u226b \u03c8).\u03c4\u2081 = \u03c6.\u03c4\u2081 \u226b \u03c8.\u03c4\u2081 := rfl\n@[simp] lemma comp_\u03c4\u2082 {S\u2081 S\u2082 S\u2083 : composable_morphisms C} (\u03c6 : S\u2081 \u27f6 S\u2082) (\u03c8 : S\u2082 \u27f6 S\u2083) :\n  (\u03c6 \u226b \u03c8).\u03c4\u2082 = \u03c6.\u03c4\u2082 \u226b \u03c8.\u03c4\u2082 := rfl\n@[simp] lemma comp_\u03c4\u2083 {S\u2081 S\u2082 S\u2083 : composable_morphisms C} (\u03c6 : S\u2081 \u27f6 S\u2082) (\u03c8 : S\u2082 \u27f6 S\u2083) :\n  (\u03c6 \u226b \u03c8).\u03c4\u2083 = \u03c6.\u03c4\u2083 \u226b \u03c8.\u03c4\u2083 := rfl\n\ndef zero (S : composable_morphisms C) [has_zero_morphisms C] : Prop := S.f \u226b S.g = 0\n\nend composable_morphisms\n\nnamespace functor\n\n@[simps]\ndef map_composable_morphisms (F : C \u2964 D) :\n  composable_morphisms C \u2964 composable_morphisms D :=\n{ obj := \u03bb S, { f := F.map S.f, g := F.map S.g, },\n  map := \u03bb S\u2081 S\u2082 \u03c6,\n  { \u03c4\u2081 := F.map \u03c6.\u03c4\u2081,\n    \u03c4\u2082 := F.map \u03c6.\u03c4\u2082,\n    \u03c4\u2083 := F.map \u03c6.\u03c4\u2083,\n    comm\u2081\u2082' := by { dsimp, simp only [\u2190 F.map_comp, \u03c6.comm\u2081\u2082], },\n    comm\u2082\u2083' := by { dsimp, simp only [\u2190 F.map_comp, \u03c6.comm\u2082\u2083], }, }, }\n\nend functor\n\nnamespace composable_morphisms\n\n@[simps]\ndef apply_functor (S : composable_morphisms C) (F : C \u2964 D) := F.map_composable_morphisms.obj S\n\nend composable_morphisms\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/composable_morphisms.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3758301185775846}}
{"text": "import data.sign\n\nnamespace int\n\n@[elab_as_eliminator] protected lemma induction_on_iff {p : \u2124 \u2192 Prop}\n  (i : \u2124) (hz : p 0) (h : \u2200 i : \u2124, p i \u2194 p (i + 1)) : p i :=\nbegin\n  induction i using int.induction_on with i IH i IH,\n  { exact hz },\n  { rwa \u2190 h },\n  { rwa [h, sub_add_cancel], }\nend\n\nend int\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/int.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3757924426238012}}
{"text": "/-\nCopyright (c) 2022 Jireh Loreaux. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jireh Loreaux\n-/\nimport topology.continuous_function.basic\n\n/-!\n# Cocompact continuous maps\n\nThe type of *cocompact continuous maps* are those which tend to the cocompact filter on the\ncodomain along the cocompact filter on the domain. When the domain and codomain are Hausdorff, this\nis equivalent to many other conditions, including that preimages of compact sets are compact. -/\n\nuniverses u v w\n\nopen filter set\n\n/-! ### Cocompact continuous maps -/\n\n/-- A *cocompact continuous map* is a continuous function between topological spaces which\ntends to the cocompact filter along the cocompact filter. Functions for which preimages of compact\nsets are compact always satisfy this property, and the converse holds for cocompact continuous maps\nwhen the codomain is Hausdorff (see `cocompact_map.tendsto_of_forall_preimage` and\n`cocompact_map.compact_preimage`) -/\nstructure cocompact_map (\u03b1 : Type u) (\u03b2 : Type v) [topological_space \u03b1] [topological_space \u03b2]\n  extends continuous_map \u03b1 \u03b2 : Type (max u v) :=\n(cocompact_tendsto' : tendsto to_fun (cocompact \u03b1) (cocompact \u03b2))\n\n/-- `cocompact_map_class F \u03b1 \u03b2` states that `F` is a type of cocompact continuous maps.\n\nYou should also extend this typeclass when you extend `cocompact_map`. -/\nclass cocompact_map_class (F : Type*) (\u03b1 \u03b2 : out_param $ Type*) [topological_space \u03b1]\n  [topological_space \u03b2] extends continuous_map_class F \u03b1 \u03b2 :=\n(cocompact_tendsto (f : F) : tendsto f (cocompact \u03b1) (cocompact \u03b2))\n\nnamespace cocompact_map_class\n\nvariables {F \u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2]\n  [cocompact_map_class F \u03b1 \u03b2]\n\ninstance : has_coe_t F (cocompact_map \u03b1 \u03b2) := \u27e8\u03bb f, \u27e8f, cocompact_tendsto f\u27e9\u27e9\n\nend cocompact_map_class\n\nexport cocompact_map_class (cocompact_tendsto)\n\nnamespace cocompact_map\n\nsection basics\nvariables {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n  [topological_space \u03b4]\n\ninstance : cocompact_map_class (cocompact_map \u03b1 \u03b2) \u03b1 \u03b2 :=\n{ coe := \u03bb f, f.to_fun,\n  coe_injective' := \u03bb f g h, by { obtain \u27e8\u27e8_, _\u27e9, _\u27e9 := f, obtain \u27e8\u27e8_, _\u27e9, _\u27e9 := g, congr' },\n  map_continuous := \u03bb f, f.continuous_to_fun,\n  cocompact_tendsto := \u03bb f, f.cocompact_tendsto' }\n\n/-- Helper instance for when there's too many metavariables to apply `fun_like.has_coe_to_fun`\ndirectly. -/\ninstance : has_coe_to_fun (cocompact_map \u03b1 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := fun_like.has_coe_to_fun\n\n@[simp] lemma coe_to_continuous_fun {f : cocompact_map \u03b1 \u03b2} :\n  (f.to_continuous_map : \u03b1 \u2192 \u03b2) = f := rfl\n\n@[ext] lemma ext {f g : cocompact_map \u03b1 \u03b2} (h : \u2200 x, f x = g x) : f = g := fun_like.ext _ _ h\n\n/-- Copy of a `cocompact_map` with a new `to_fun` equal to the old one. Useful\nto fix definitional equalities. -/\nprotected def copy (f : cocompact_map \u03b1 \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : cocompact_map \u03b1 \u03b2 :=\n{ to_fun := f',\n  continuous_to_fun := by {rw h, exact f.continuous_to_fun},\n  cocompact_tendsto' := by { simp_rw h, exact f.cocompact_tendsto' } }\n\n@[simp] lemma coe_mk (f : C(\u03b1, \u03b2)) (h : tendsto f (cocompact \u03b1) (cocompact \u03b2)) :\n  \u21d1(\u27e8f, h\u27e9 : cocompact_map \u03b1 \u03b2) = f := rfl\n\nsection\nvariable (\u03b1)\n/-- The identity as a cocompact continuous map. -/\nprotected def id : cocompact_map \u03b1 \u03b1 := \u27e8continuous_map.id _, tendsto_id\u27e9\n@[simp] lemma coe_id : \u21d1(cocompact_map.id \u03b1) = id := rfl\nend\n\ninstance : inhabited (cocompact_map \u03b1 \u03b1) := \u27e8cocompact_map.id \u03b1\u27e9\n\n/-- The composition of cocompact continuous maps, as a cocompact continuous map. -/\ndef comp (f : cocompact_map \u03b2 \u03b3) (g : cocompact_map \u03b1 \u03b2) : cocompact_map \u03b1 \u03b3 :=\n\u27e8f.to_continuous_map.comp g, (cocompact_tendsto f).comp (cocompact_tendsto g)\u27e9\n\n@[simp] lemma coe_comp (f : cocompact_map \u03b2 \u03b3) (g : cocompact_map \u03b1 \u03b2) :\n  \u21d1(comp f g) = f \u2218 g := rfl\n\n@[simp] lemma comp_apply (f : cocompact_map \u03b2 \u03b3) (g : cocompact_map \u03b1 \u03b2) (a : \u03b1) :\n  comp f g a = f (g a) := rfl\n\n@[simp] lemma comp_assoc (f : cocompact_map \u03b3 \u03b4) (g : cocompact_map \u03b2 \u03b3)\n  (h : cocompact_map \u03b1 \u03b2) : (f.comp g).comp h = f.comp (g.comp h) := rfl\n\n@[simp] lemma id_comp (f : cocompact_map \u03b1 \u03b2) : (cocompact_map.id _).comp f = f :=\next $ \u03bb _, rfl\n\n@[simp] lemma comp_id (f : cocompact_map \u03b1 \u03b2) : f.comp (cocompact_map.id _) = f :=\next $ \u03bb _, rfl\n\nlemma tendsto_of_forall_preimage {f : \u03b1 \u2192 \u03b2} (h : \u2200 s, is_compact s \u2192 is_compact (f \u207b\u00b9' s)) :\n  tendsto f (cocompact \u03b1) (cocompact \u03b2) :=\n\u03bb s hs, match mem_cocompact.mp hs with \u27e8t, ht, hts\u27e9 :=\n  mem_map.mpr (mem_cocompact.mpr \u27e8f \u207b\u00b9' t, h t ht, by simpa using preimage_mono hts\u27e9) end\n\n/-- If the codomain is Hausdorff, preimages of compact sets are compact under a cocompact\ncontinuous map. -/\nlemma compact_preimage [t2_space \u03b2] (f : cocompact_map \u03b1 \u03b2) \u2983s : set \u03b2\u2984 (hs : is_compact s) :\n  is_compact (f \u207b\u00b9' s) :=\nbegin\n  obtain \u27e8t, ht, hts\u27e9 := mem_cocompact'.mp (by simpa only [preimage_image_preimage, preimage_compl]\n    using mem_map.mp (cocompact_tendsto f $ mem_cocompact.mpr \u27e8s, hs, compl_subset_compl.mpr\n    (image_preimage_subset f _)\u27e9)),\n  exact compact_of_is_closed_subset ht (hs.is_closed.preimage $ map_continuous f)\n    (by simpa using hts),\nend\n\nend basics\n\nend cocompact_map\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/topology/continuous_function/cocompact_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.3757924336684302}}
{"text": "import control proof_trace\n\nopen polya tactic expr\n\nvariables u v w x y z : \u211a\n\nmeta def polya_on_hyps' (hys : list name) (rct : bool := tt) : tactic unit :=\ndo exps \u2190 hys.mmap get_local,\n   bb \u2190 add_proof_to_blackboard blackboard.mk_empty `(rat_one_gt_zero),\n   bb \u2190 add_proofs_to_blackboard bb exps,\n   let pb := polya_bundle.default.set_blackboard bb,\n   let (n, pb) := pb.cycle 0,\n   trace (\"number of cycles:\", n),\n   trace (\"contr found\", pb.contr_found),\n   if bnot pb.contr_found then /-bb.trace >>-/ fail \"polya failed, no contradiction found\" else do\n   pb.bb.contr.sketch >>= proof_sketch.trace,\n   if rct then pb.bb.contr.reconstruct >>= apply >> skip\n   else skip\n\nexample  (h1 : x > 0) (h2 : x < 1*1) (h3 : rat.pow (1*1 + (-1)*x) (-1) \u2264 1*(rat.pow (1*1 + (-1)*rat.pow x 2) (-1))) : false :=\nby \npolya_on_hyps' [`h1, `h2, `h3]\n\n\nexample (h1 : u > 0) (h2 : u < 1*v) (h3 : z > 0) (h4 : 1*z + 1*1 < 1*w) (h5 : rat.pow (1*u + 1*v + 1*z) 3 \u2265 1* rat.pow (1*u + 1*v + 1*w + 1*1) 5) : false :=\nby  polya_on_hyps' [`h1, `h2, `h3, `h4, `h5]\n", "meta": {"author": "robertylewis", "repo": "lean_polya", "sha": "1da14d60a55ad6cd8af8017b1b64990fccb66ab7", "save_path": "github-repos/lean/robertylewis-lean_polya", "path": "github-repos/lean/robertylewis-lean_polya/lean_polya-1da14d60a55ad6cd8af8017b1b64990fccb66ab7/test/trace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.375698957477849}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison\nimport .monoidal_category\nimport categories.products.switch\n\nnamespace categories.braided_monoidal_category\n\nopen categories\nopen categories.functor\nopen categories.natural_transformation\nopen categories.products\nopen categories.monoidal_category\n\nuniverse variables u v\n\n/-\n-- I don't really understand why the universe annotations are needed in Braiding and in squaredBraiding.\n-- My guess is that it is related to\n-- https://groups.google.com/d/msg/lean-user/3qzchWkut0g/0QR6_cS8AgAJ\n-/\n\n@[reducible] definition Commutor (C : Type u) [\ud835\udc9e : monoidal_category.{u v} C] := \n  (\ud835\udc9e.tensor) \u21d4 ((SwitchProductCategory C C) \u22d9 \ud835\udc9e.tensor)\n\nsection\nvariables {C : Type u} [\ud835\udc9e : monoidal_category.{u v} C]\ninclude \ud835\udc9e\n\n@[reducible] definition Hexagon_1 (\u03b2 : Commutor C) :=\n  \u2200 X Y Z : C,\n      ((\ud835\udfd9 X) \u2297 (\u03b2.morphism.components (Y, Z)))\n      \u226b (inverse_associator X Z Y)\n      \u226b ((\u03b2.morphism.components (X, Z)) \u2297 (\ud835\udfd9 Y)) = \n      (inverse_associator X Y Z) \n      \u226b (\u03b2.morphism.components (X \u2297 Y, Z))\n      \u226b (inverse_associator Z X Y)\n\n@[reducible] definition Hexagon_2 (\u03b2 : Commutor C) :=\n  \u2200 X Y Z : C,\n      ((\ud835\udfd9 X) \u2297 (\u03b2.inverse.components (Z, Y)))\n      \u226b (inverse_associator X Z Y)\n      \u226b ((\u03b2.inverse.components (Z, X)) \u2297 (\ud835\udfd9 Y)) = \n      (inverse_associator X Y Z) \n      \u226b (\u03b2.inverse.components (Z, X \u2297 Y))\n      \u226b (inverse_associator Z X Y)\nend\n\nclass Braiding (C : Type u) [monoidal_category.{u v} C] :=\n  ( braiding: Commutor C )\n  ( hexagon_1 : Hexagon_1 braiding )\n  ( hexagon_2 : Hexagon_2 braiding )\n\nattribute [ematch] Braiding.hexagon_1 Braiding.hexagon_2\n-- PROJECT a theorem showing the hexagons hold as natural transformations\n\nclass Symmetry (C : Type u) [monoidal_category.{u v} C] extends Braiding C :=\n  (symmetry: \u03a0 X Y : C, (braiding.morphism.components \u27e8X, Y\u27e9) \u226b (braiding.morphism.components \u27e8Y, X\u27e9) = \ud835\udfd9 (X \u2297 Y) )\n\nattribute [simp,ematch] Symmetry.symmetry\n\nend categories.braided_monoidal_category", "meta": {"author": "semorrison", "repo": "lean-monoidal-categories", "sha": "81f43e1e0d623a96695aa8938951d7422d6d7ba6", "save_path": "github-repos/lean/semorrison-lean-monoidal-categories", "path": "github-repos/lean/semorrison-lean-monoidal-categories/lean-monoidal-categories-81f43e1e0d623a96695aa8938951d7422d6d7ba6/src/monoidal_categories/braided_monoidal_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.37561844224295365}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebraic_geometry.sheafed_space\n! leanprover-community/mathlib commit f384f5d1a4e39f36817b8d22afff7b52af8121d1\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicGeometry.PresheafedSpace.HasColimits\nimport Mathbin.Topology.Sheaves.Functors\n\n/-!\n# Sheafed spaces\n\nIntroduces the category of topological spaces equipped with a sheaf (taking values in an\narbitrary target category `C`.)\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\n\nuniverse v u\n\nopen CategoryTheory\n\nopen TopCat\n\nopen TopologicalSpace\n\nopen Opposite\n\nopen CategoryTheory.Limits\n\nopen CategoryTheory.Category CategoryTheory.Functor\n\nvariable (C : Type u) [Category.{v} C]\n\nattribute [local tidy] tactic.op_induction'\n\nnamespace AlgebraicGeometry\n\n/-- A `SheafedSpace C` is a topological space equipped with a sheaf of `C`s. -/\nstructure SheafedSpace extends PresheafedSpace.{v} C where\n  IsSheaf : presheaf.IsSheaf\n#align algebraic_geometry.SheafedSpace AlgebraicGeometry.SheafedSpace\n\nvariable {C}\n\nnamespace SheafedSpace\n\ninstance coeCarrier : Coe (SheafedSpace C) TopCat where coe X := X.carrier\n#align algebraic_geometry.SheafedSpace.coe_carrier AlgebraicGeometry.SheafedSpace.coeCarrier\n\n/-- Extract the `sheaf C (X : Top)` from a `SheafedSpace C`. -/\ndef sheaf (X : SheafedSpace C) : Sheaf C (X : TopCat.{v}) :=\n  \u27e8X.Presheaf, X.IsSheaf\u27e9\n#align algebraic_geometry.SheafedSpace.sheaf AlgebraicGeometry.SheafedSpace.sheaf\n\n@[simp]\ntheorem as_coe (X : SheafedSpace.{v} C) : X.carrier = (X : TopCat.{v}) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.as_coe AlgebraicGeometry.SheafedSpace.as_coe\n\n@[simp]\ntheorem mk_coe (carrier) (presheaf) (h) :\n    (({     carrier\n            Presheaf\n            IsSheaf := h } : SheafedSpace.{v} C) : TopCat.{v}) = carrier :=\n  rfl\n#align algebraic_geometry.SheafedSpace.mk_coe AlgebraicGeometry.SheafedSpace.mk_coe\n\ninstance (X : SheafedSpace.{v} C) : TopologicalSpace X :=\n  X.carrier.str\n\n/-- The trivial `unit` valued sheaf on any topological space. -/\ndef unit (X : TopCat) : SheafedSpace (discrete Unit) :=\n  { @PresheafedSpace.const (discrete Unit) _ X \u27e8\u27e8\u27e9\u27e9 with IsSheaf := Presheaf.isSheaf_unit _ }\n#align algebraic_geometry.SheafedSpace.unit AlgebraicGeometry.SheafedSpace.unit\n\ninstance : Inhabited (SheafedSpace (discrete Unit)) :=\n  \u27e8unit (TopCat.of PEmpty)\u27e9\n\ninstance : Category (SheafedSpace C) :=\n  show Category (InducedCategory (PresheafedSpace.{v} C) SheafedSpace.toPresheafedSpace) by\n    infer_instance\n\n/-- Forgetting the sheaf condition is a functor from `SheafedSpace C` to `PresheafedSpace C`. -/\ndef forgetToPresheafedSpace : SheafedSpace.{v} C \u2964 PresheafedSpace.{v} C :=\n  inducedFunctor _ deriving Full, Faithful\n#align algebraic_geometry.SheafedSpace.forget_to_PresheafedSpace AlgebraicGeometry.SheafedSpace.forgetToPresheafedSpace\n\ninstance is_presheafedSpace_iso {X Y : SheafedSpace.{v} C} (f : X \u27f6 Y) [IsIso f] :\n    @IsIso (PresheafedSpace C) _ _ _ f :=\n  SheafedSpace.forgetToPresheafedSpace.map_isIso f\n#align algebraic_geometry.SheafedSpace.is_PresheafedSpace_iso AlgebraicGeometry.SheafedSpace.is_presheafedSpace_iso\n\nvariable {C}\n\nsection\n\nattribute [local simp] id comp\n\n@[simp]\ntheorem id_base (X : SheafedSpace C) : (\ud835\udfd9 X : X \u27f6 X).base = \ud835\udfd9 (X : TopCat.{v}) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.id_base AlgebraicGeometry.SheafedSpace.id_base\n\ntheorem id_c (X : SheafedSpace C) :\n    (\ud835\udfd9 X : X \u27f6 X).c = eqToHom (Presheaf.Pushforward.id_eq X.Presheaf).symm :=\n  rfl\n#align algebraic_geometry.SheafedSpace.id_c AlgebraicGeometry.SheafedSpace.id_c\n\n@[simp]\ntheorem id_c_app (X : SheafedSpace C) (U) :\n    (\ud835\udfd9 X : X \u27f6 X).c.app U =\n      eqToHom\n        (by\n          induction U using Opposite.rec\n          cases U\n          rfl) :=\n  by\n  induction U using Opposite.rec\n  cases U\n  simp only [id_c]\n  dsimp\n  simp\n#align algebraic_geometry.SheafedSpace.id_c_app AlgebraicGeometry.SheafedSpace.id_c_app\n\n@[simp]\ntheorem comp_base {X Y Z : SheafedSpace C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    (f \u226b g).base = f.base \u226b g.base :=\n  rfl\n#align algebraic_geometry.SheafedSpace.comp_base AlgebraicGeometry.SheafedSpace.comp_base\n\n@[simp]\ntheorem comp_c_app {X Y Z : SheafedSpace C} (\u03b1 : X \u27f6 Y) (\u03b2 : Y \u27f6 Z) (U) :\n    (\u03b1 \u226b \u03b2).c.app U = \u03b2.c.app U \u226b \u03b1.c.app (op ((Opens.map \u03b2.base).obj (unop U))) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.comp_c_app AlgebraicGeometry.SheafedSpace.comp_c_app\n\ntheorem comp_c_app' {X Y Z : SheafedSpace C} (\u03b1 : X \u27f6 Y) (\u03b2 : Y \u27f6 Z) (U) :\n    (\u03b1 \u226b \u03b2).c.app (op U) = \u03b2.c.app (op U) \u226b \u03b1.c.app (op ((Opens.map \u03b2.base).obj U)) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.comp_c_app' AlgebraicGeometry.SheafedSpace.comp_c_app'\n\ntheorem congr_app {X Y : SheafedSpace C} {\u03b1 \u03b2 : X \u27f6 Y} (h : \u03b1 = \u03b2) (U) :\n    \u03b1.c.app U = \u03b2.c.app U \u226b X.Presheaf.map (eqToHom (by subst h)) :=\n  PresheafedSpace.congr_app h U\n#align algebraic_geometry.SheafedSpace.congr_app AlgebraicGeometry.SheafedSpace.congr_app\n\nvariable (C)\n\n/-- The forgetful functor from `SheafedSpace` to `Top`. -/\ndef forget : SheafedSpace C \u2964 TopCat\n    where\n  obj X := (X : TopCat.{v})\n  map X Y f := f.base\n#align algebraic_geometry.SheafedSpace.forget AlgebraicGeometry.SheafedSpace.forget\n\nend\n\nopen TopCat.Presheaf\n\n/-- The restriction of a sheafed space along an open embedding into the space.\n-/\ndef restrict {U : TopCat} (X : SheafedSpace C) {f : U \u27f6 (X : TopCat.{v})} (h : OpenEmbedding f) :\n    SheafedSpace C :=\n  { X.toPresheafedSpace.restrict h with IsSheaf := isSheaf_of_openEmbedding h X.IsSheaf }\n#align algebraic_geometry.SheafedSpace.restrict AlgebraicGeometry.SheafedSpace.restrict\n\n/-- The restriction of a sheafed space `X` to the top subspace is isomorphic to `X` itself.\n-/\ndef restrictTopIso (X : SheafedSpace C) : X.restrict (Opens.openEmbedding \u22a4) \u2245 X :=\n  forgetToPresheafedSpace.preimageIso X.toPresheafedSpace.restrictTopIso\n#align algebraic_geometry.SheafedSpace.restrict_top_iso AlgebraicGeometry.SheafedSpace.restrictTopIso\n\n/-- The global sections, notated Gamma.\n-/\ndef \u0393 : (SheafedSpace C)\u1d52\u1d56 \u2964 C :=\n  forgetToPresheafedSpace.op \u22d9 PresheafedSpace.\u0393\n#align algebraic_geometry.SheafedSpace.\u0393 AlgebraicGeometry.SheafedSpace.\u0393\n\ntheorem \u0393_def : (\u0393 : _ \u2964 C) = forgetToPresheafedSpace.op \u22d9 PresheafedSpace.\u0393 :=\n  rfl\n#align algebraic_geometry.SheafedSpace.\u0393_def AlgebraicGeometry.SheafedSpace.\u0393_def\n\n@[simp]\ntheorem \u0393_obj (X : (SheafedSpace C)\u1d52\u1d56) : \u0393.obj X = (unop X).Presheaf.obj (op \u22a4) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.\u0393_obj AlgebraicGeometry.SheafedSpace.\u0393_obj\n\ntheorem \u0393_obj_op (X : SheafedSpace C) : \u0393.obj (op X) = X.Presheaf.obj (op \u22a4) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.\u0393_obj_op AlgebraicGeometry.SheafedSpace.\u0393_obj_op\n\n@[simp]\ntheorem \u0393_map {X Y : (SheafedSpace C)\u1d52\u1d56} (f : X \u27f6 Y) : \u0393.map f = f.unop.c.app (op \u22a4) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.\u0393_map AlgebraicGeometry.SheafedSpace.\u0393_map\n\ntheorem \u0393_map_op {X Y : SheafedSpace C} (f : X \u27f6 Y) : \u0393.map f.op = f.c.app (op \u22a4) :=\n  rfl\n#align algebraic_geometry.SheafedSpace.\u0393_map_op AlgebraicGeometry.SheafedSpace.\u0393_map_op\n\nnoncomputable instance [HasLimits C] :\n    CreatesColimits (forgetToPresheafedSpace : SheafedSpace C \u2964 _) :=\n  \u27e8fun J hJ =>\n    \u27e8fun K =>\n      creates_colimit_of_fully_faithful_of_iso\n        \u27e8(PresheafedSpace.colimit_cocone (K \u22d9 forget_to_PresheafedSpace)).pt,\n          limit_is_sheaf _ fun j => sheaf.pushforward_sheaf_of_sheaf _ (K.obj (unop j)).2\u27e9\n        (colimit.iso_colimit_cocone \u27e8_, PresheafedSpace.colimit_cocone_is_colimit _\u27e9).symm\u27e9\u27e9\n\ninstance [HasLimits C] : HasColimits (SheafedSpace C) :=\n  has_colimits_of_has_colimits_creates_colimits forgetToPresheafedSpace\n\nnoncomputable instance [HasLimits C] : PreservesColimits (forget C) :=\n  Limits.compPreservesColimits forgetToPresheafedSpace (PresheafedSpace.forget C)\n\nend SheafedSpace\n\nend AlgebraicGeometry\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/SheafedSpace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.3756184332076552}}
{"text": "import category_theory.whiskering\nimport category_theory.adjunction\n\nnamespace category_theory.adjunction\n\nopen category_theory\n\nvariables (C : Type*) {D E : Type*} [category C] [category D] [category E]\n  {F : D \u2964 E} {G : E \u2964 D}\n\ndef whiskering_right (adj : F \u22a3 G) :\n  ((whiskering_right C D E).obj F) \u22a3 ((whiskering_right C E D).obj G) :=\nmk_of_unit_counit\n{ unit :=\n  { app := \u03bb X, (functor.right_unitor _).inv \u226b\n      whisker_left X adj.unit \u226b (functor.associator _ _ _).inv,\n    naturality' := by { intros, ext, dsimp, simp } },\n  counit :=\n  { app := \u03bb X, (functor.associator _ _ _).hom \u226b\n      whisker_left X adj.counit \u226b (functor.right_unitor _).hom,\n    naturality' := by { intros, ext, dsimp, simp } },\n  left_triangle' := by { ext, dsimp, simp },\n  right_triangle' := by { ext, dsimp, simp } } .\n\n@[simp]\nlemma whiskering_right_unit (adj : F \u22a3 G) (X : C \u2964 D) :\n  (adj.whiskering_right C).unit.app X =\n  (functor.right_unitor _).inv \u226b whisker_left X adj.unit \u226b (functor.associator _ _ _).inv := rfl\n\n@[simp]\nlemma whiskering_right_counit (adj : F \u22a3 G) (X : C \u2964 E) :\n  (adj.whiskering_right C).counit.app X =\n  (functor.associator _ _ _).hom \u226b whisker_left X adj.counit \u226b (functor.right_unitor _).hom := rfl\n\nend category_theory.adjunction\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/whisker_adjunction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3755717992319168}}
{"text": "-- Option is a monad, by Lean's type class checker.\nexample : monad option := by apply_instance\n\n/-\nOption is a monad, interpreted by a set-theorist.\n-/\n\nnamespace ZFC -- There is no such thing as Sort.\n\n/-\nBy Kevin Buzzard and whoever else writes it.\n\nNote. Stuff like `x` in between back quotes in these docs\nis written in a new kind of language called Lean.\nIt's like how $x$ or $$x$$ is written in TeX.\n-/\n\n-- Let X be a set.\nvariable (X : Type)\n\n/-- `option X` is a new set containing all the elements of X\n    and a new element called `none`, which is definitely not in X.\n    Note that it is possible to make this kind of construction,\n    because of set theory. -/\ninductive option\n| some (x : X) : option\n| none {} : option\n\n/- theorems about this new set, including completely trivial\n   ones, all go here in this chapter about `option X` -/\nnamespace option\n\nvariable {X}\n/- The principle of induction for `option X`: If you\nwant to prove something about all the elements of `option X`,\nyou just have to prove it for `none` and for all the elements of `X`.\n-/\n\ndef induction : \u2200 {X : Type} {P : option X \u2192 Prop},\nP none \u2192 (\u2200 (x : X), P (some x)) \u2192 \u2200 (x : option X), P x :=\n\u03bb X C h_none h_some, @option.rec X C h_some h_none\n/-\nProof: obvious\n-/\n-- TODO: is there a cool type theory way to switch the variables?\n-- ((\u2218) flip) \u2218 (@option.rec) doesn't work?\n\n/- The principle of recursion for `option X`: if you\nwant to define something on all the elements of `option X`,\nyou just have to define it on `none` and on all the elements of `X`.\n-/\ndef recursion :  \u2200 {X : Type} {C : option X \u2192 Type},\nC none \u2192 (\u2200 (x : X), C (some x)) \u2192 \u2200 (x : option X), C x :=\n/-\nProof: obvious\n-/\n\u03bb X C c_none c_some, @option.rec X C c_some c_none\n\n/- On Wikipedia there's a definition of a thing called a monad.\n   It's a theorem that `option` is a monad. A lot of computer scientists know\n   the proof. A lot of mathematicians will never need to know what this statement means.\n-/\ndef monad : monad option := sorry\n\n/-\nNext ideas:\n\nTheorems such as \"if there's a bijection $f : X \\to Y$ between `X` and` Y` then there's\nlso a bijection between `option X` and `option Y`, often also called $f$ or perhaps\na typographical variant such as $f_\\infty$ or something.\n\nWhat else would a mathematician want to know about `option`?\n\nHow about a proof that `\u2200 (n : \u2115), option (fin n) \u2243 fin (n+1)`?\nA mathematician would say that this was obvious. We should import\n`data.equiv.basic`, the theory of bijections, if we want to state and prove this.\n-/\n\n-- end of chapter on `option`\nend option\n\n-- end of \"doing ZFC on a computer\"\nend ZFC\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/Type_theory_for_the_ZFCer/src/option.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3755717992319168}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n\n! This file was ported from Lean 3 source module category_theory.category.Pointed\n! leanprover-community/mathlib commit 10bf4f825ad729c5653adc039dafa3622e7f93c9\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.ConcreteCategory.Basic\n\n/-!\n# The category of pointed types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis defines `Pointed`, the category of pointed types.\n\n## TODO\n\n* Monoidal structure\n* Upgrade `Type_to_Pointed` to an equivalence\n-/\n\n\nopen CategoryTheory\n\nuniverse u\n\nvariable {\u03b1 \u03b2 : Type _}\n\n#print Pointed /-\n/-- The category of pointed types. -/\nstructure Pointed : Type (u + 1) where\n  pt : Type u\n  point : X\n#align Pointed Pointed\n-/\n\nnamespace Pointed\n\ninstance : CoeSort Pointed (Type _) :=\n  \u27e8X\u27e9\n\nattribute [protected] Pointed.X\n\n#print Pointed.of /-\n/-- Turns a point into a pointed type. -/\ndef of {X : Type _} (point : X) : Pointed :=\n  \u27e8X, point\u27e9\n#align Pointed.of Pointed.of\n-/\n\n#print Pointed.coe_of /-\n@[simp]\ntheorem coe_of {X : Type _} (point : X) : \u21a5(of point) = X :=\n  rfl\n#align Pointed.coe_of Pointed.coe_of\n-/\n\nalias of \u2190 _root_.prod.Pointed\n#align prod.Pointed Prod.Pointed\n\ninstance : Inhabited Pointed :=\n  \u27e8of ((), ())\u27e9\n\n#print Pointed.Hom /-\n/-- Morphisms in `Pointed`. -/\n@[ext]\nprotected structure Hom (X Y : Pointed.{u}) : Type u where\n  toFun : X \u2192 Y\n  map_point : to_fun X.point = Y.point\n#align Pointed.hom Pointed.Hom\n-/\n\nnamespace Hom\n\n#print Pointed.Hom.id /-\n/-- The identity morphism of `X : Pointed`. -/\n@[simps]\ndef id (X : Pointed) : Hom X X :=\n  \u27e8id, rfl\u27e9\n#align Pointed.hom.id Pointed.Hom.id\n-/\n\ninstance (X : Pointed) : Inhabited (Hom X X) :=\n  \u27e8id X\u27e9\n\n#print Pointed.Hom.comp /-\n/-- Composition of morphisms of `Pointed`. -/\n@[simps]\ndef comp {X Y Z : Pointed.{u}} (f : Hom X Y) (g : Hom Y Z) : Hom X Z :=\n  \u27e8g.toFun \u2218 f.toFun, by rw [Function.comp_apply, f.map_point, g.map_point]\u27e9\n#align Pointed.hom.comp Pointed.Hom.comp\n-/\n\nend Hom\n\n#print Pointed.largeCategory /-\ninstance largeCategory : LargeCategory Pointed\n    where\n  Hom := Hom\n  id := Hom.id\n  comp := @Hom.comp\n  id_comp' _ _ _ := Hom.ext _ _ rfl\n  comp_id' _ _ _ := Hom.ext _ _ rfl\n  assoc' _ _ _ _ _ _ _ := Hom.ext _ _ rfl\n#align Pointed.large_category Pointed.largeCategory\n-/\n\n#print Pointed.concreteCategory /-\ninstance concreteCategory : ConcreteCategory Pointed\n    where\n  forget :=\n    { obj := Pointed.X\n      map := @Hom.toFun }\n  forget_faithful := \u27e8@Hom.ext\u27e9\n#align Pointed.concrete_category Pointed.concreteCategory\n-/\n\n#print Pointed.Iso.mk /-\n/-- Constructs a isomorphism between pointed types from an equivalence that preserves the point\nbetween them. -/\n@[simps]\ndef Iso.mk {\u03b1 \u03b2 : Pointed} (e : \u03b1 \u2243 \u03b2) (he : e \u03b1.point = \u03b2.point) : \u03b1 \u2245 \u03b2\n    where\n  Hom := \u27e8e, he\u27e9\n  inv := \u27e8e.symm, e.symm_apply_eq.2 he.symm\u27e9\n  hom_inv_id' := Pointed.Hom.ext _ _ e.symm_comp_self\n  inv_hom_id' := Pointed.Hom.ext _ _ e.self_comp_symm\n#align Pointed.iso.mk Pointed.Iso.mk\n-/\n\nend Pointed\n\n#print typeToPointed /-\n/-- `option` as a functor from types to pointed types. This is the free functor. -/\n@[simps]\ndef typeToPointed : Type u \u2964 Pointed.{u}\n    where\n  obj X := \u27e8Option X, none\u27e9\n  map X Y f := \u27e8Option.map f, rfl\u27e9\n  map_id' X := Pointed.Hom.ext _ _ Option.map_id\n  map_comp' X Y Z f g := Pointed.Hom.ext _ _ (Option.map_comp_map _ _).symm\n#align Type_to_Pointed typeToPointed\n-/\n\n#print typeToPointedForgetAdjunction /-\n/-- `Type_to_Pointed` is the free functor. -/\ndef typeToPointedForgetAdjunction : typeToPointed \u22a3 forget Pointed :=\n  Adjunction.mkOfHomEquiv\n    { homEquiv := fun X Y =>\n        { toFun := fun f => f.toFun \u2218 Option.some\n          invFun := fun f => \u27e8fun o => o.elim Y.point f, rfl\u27e9\n          left_inv := fun f => by\n            ext\n            cases x\n            exact f.map_point.symm\n            rfl\n          right_inv := fun f => funext fun _ => rfl }\n      homEquiv_naturality_left_symm := fun X' X Y f g =>\n        by\n        ext\n        cases x <;> rfl }\n#align Type_to_Pointed_forget_adjunction typeToPointedForgetAdjunction\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Category/Pointed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.3754011155526398}}
{"text": "import GMLInit.Data.Basic\nimport GMLInit.Logic.Relation\nimport GMLInit.Meta.Decidable\n\nopen Ordering (lt eq gt)\n\nclass Std.LinearCmp {\u03b1} (cmp : \u03b1 \u2192 \u03b1 \u2192 Ordering) extends TransCmp cmp : Prop where\n  eq_strict {x y : \u03b1} : cmp x y = eq \u2192 x = y\n\nclass abbrev OrientedOrd (\u03b1) [Ord \u03b1] : Prop := Std.OrientedCmp (\u03b1:=\u03b1) Ord.compare\n\nclass abbrev TransOrd (\u03b1) [Ord \u03b1] : Prop := Std.TransCmp (\u03b1:=\u03b1) Ord.compare\n\nclass abbrev LinearOrd (\u03b1) [Ord \u03b1] : Prop := Std.LinearCmp (\u03b1:=\u03b1) Ord.compare\n\nnamespace Ord\nvariable {\u03b1} [Ord \u03b1]\n\ntheorem eq_refl [OrientedOrd \u03b1] (x : \u03b1) : compare x x = eq := Std.OrientedCmp.cmp_refl\n\ntheorem lt_irrefl [OrientedOrd \u03b1] (x : \u03b1) : compare x x \u2260 lt := eq_refl x \u25b8 Ordering.noConfusion\n\ntheorem gt_irrefl [OrientedOrd \u03b1] (x : \u03b1) : compare x x \u2260 gt := eq_refl x \u25b8 Ordering.noConfusion\n\ntheorem ne_irrefl [OrientedOrd \u03b1] (x : \u03b1) : \u00ac(compare x x \u2260 eq) := absurd (eq_refl x)\n\ntheorem le_refl [OrientedOrd \u03b1] (x : \u03b1) : compare x x \u2260 gt := gt_irrefl x\n\ntheorem ge_refl [OrientedOrd \u03b1] (x : \u03b1) : compare x x \u2260 lt := lt_irrefl x\n\ntheorem eq_symm [OrientedOrd \u03b1] {x y : \u03b1} : compare x y = eq \u2192 compare y x = eq := Std.OrientedCmp.cmp_eq_eq_symm.mp\n\ntheorem gt_of_lt_opp [OrientedOrd \u03b1] {x y : \u03b1} : compare x y = lt \u2192 compare y x = gt := Std.OrientedCmp.cmp_eq_gt.mpr\n\ntheorem lt_of_gt_opp [OrientedOrd \u03b1] {x y : \u03b1} : compare x y = gt \u2192 compare y x = lt := Std.OrientedCmp.cmp_eq_gt.mp\n\ntheorem ge_of_le_opp [OrientedOrd \u03b1] {x y : \u03b1} : compare x y \u2260 gt \u2192 compare y x \u2260 lt := mt gt_of_lt_opp\n\ntheorem le_of_ge_opp [OrientedOrd \u03b1] {x y : \u03b1} : compare x y \u2260 lt \u2192 compare y x \u2260 gt := mt lt_of_gt_opp\n\ntheorem le_total [OrientedOrd \u03b1] (x y : \u03b1) : compare x y \u2260 gt \u2228 compare y x \u2260 gt :=\n  match hxy : compare x y with\n  | lt => Or.inl Ordering.noConfusion\n  | eq => Or.inl Ordering.noConfusion\n  | gt => Or.inr fun h => Ordering.noConfusion (Eq.trans (lt_of_gt_opp h).symm hxy)\n\ntheorem ge_total [OrientedOrd \u03b1] (x y : \u03b1) : compare x y \u2260 lt \u2228 compare y x \u2260 lt :=\n  match hxy : compare x y with\n  | lt => Or.inr fun h => Ordering.noConfusion (Eq.trans (gt_of_lt_opp h).symm hxy)\n  | eq => Or.inl Ordering.noConfusion\n  | gt => Or.inl Ordering.noConfusion\n\ntheorem lt_asymm [OrientedOrd \u03b1] {x y : \u03b1} : compare x y = lt \u2192 compare y x \u2260 lt := fun hxy => gt_of_lt_opp hxy \u25b8 Ordering.noConfusion\n\ntheorem gt_asymm [OrientedOrd \u03b1] {x y : \u03b1} : compare x y = gt \u2192 compare y x \u2260 gt := fun hxy => lt_of_gt_opp hxy \u25b8 Ordering.noConfusion\n\ntheorem eq_subst_left [TransOrd \u03b1] {x y z : \u03b1} : compare x y = eq \u2192 compare x z = compare y z := Std.TransCmp.cmp_congr_left\n\ntheorem eq_subst_right [TransOrd \u03b1] {x y z : \u03b1} : compare x y = eq \u2192 compare z x = compare z y := Std.TransCmp.cmp_congr_right\n\ntheorem eq_trans [TransOrd \u03b1] {x y z : \u03b1} : compare x y = eq \u2192 compare y z = eq \u2192 compare x z = eq := fun hxy hyz => eq_subst_left hxy \u25b8 hyz\n\ntheorem lt_trans [TransOrd \u03b1] {x y z : \u03b1} : compare x y = lt \u2192 compare y z = lt \u2192 compare x z = lt := Std.TransCmp.lt_trans\n\ntheorem gt_trans [TransOrd \u03b1] {x y z : \u03b1} : compare x y = gt \u2192 compare y z = gt \u2192 compare x z = gt := Std.TransCmp.gt_trans\n\ntheorem le_trans [TransOrd \u03b1] {x y z : \u03b1} : compare x y \u2260 gt \u2192 compare y z \u2260 gt \u2192 compare x z \u2260 gt := Std.TransCmp.le_trans\n\ntheorem ge_trans [TransOrd \u03b1] {x y z : \u03b1} : compare x y \u2260 lt \u2192 compare y z \u2260 lt \u2192 compare x z \u2260 lt := Std.TransCmp.ge_trans\n\ntheorem lt_of_lt_of_le [TransOrd \u03b1] {x y z : \u03b1} : compare x y = lt \u2192 compare y z \u2260 gt \u2192 compare x z = lt :=\n  fun hxy nyz => match hyz : compare y z with\n  | lt => lt_trans hxy hyz\n  | eq => eq_subst_right hyz \u25b8 hxy\n  | gt => absurd hyz nyz\n\ntheorem lt_of_le_of_lt [TransOrd \u03b1] {x y z : \u03b1} : compare x y \u2260 gt \u2192 compare y z = lt \u2192 compare x z = lt :=\n  fun nxy hyz => match hxy : compare x y with\n  | lt => lt_trans hxy hyz\n  | eq => eq_subst_left hxy \u25b8 hyz\n  | gt => absurd hxy nxy\n\ntheorem gt_of_gt_of_ge [TransOrd \u03b1] {x y z : \u03b1} : compare x y = gt \u2192 compare y z \u2260 lt \u2192 compare x z = gt :=\n  fun hxy nyz => match hyz : compare y z with\n  | gt => gt_trans hxy hyz\n  | eq => eq_subst_right hyz \u25b8 hxy\n  | lt => absurd hyz nyz\n\ntheorem gt_of_ge_of_gt [TransOrd \u03b1] {x y z : \u03b1} : compare x y \u2260 lt \u2192 compare y z = gt \u2192 compare x z = gt :=\n  fun nxy hyz => match hxy : compare x y with\n  | gt => gt_trans hxy hyz\n  | eq => eq_subst_left hxy \u25b8 hyz\n  | lt => absurd hxy nxy\n\ntheorem le_or_ge [TransOrd \u03b1] (x y : \u03b1) : compare x y \u2260 gt \u2228 compare x y \u2260 lt :=\n  match compare x y with\n  | lt => Or.inl Ordering.noConfusion\n  | eq => Or.inl Ordering.noConfusion\n  | gt => Or.inr Ordering.noConfusion\n\ntheorem eq_strict [LinearOrd \u03b1] {x y : \u03b1} : compare x y = eq \u2192 x = y := Std.LinearCmp.eq_strict\n\ntheorem connex [LinearOrd \u03b1] {x y : \u03b1} : x \u2260 y \u2192 compare x y = lt \u2228 compare x y = gt :=\n  fun hne => match hxy : compare x y with\n  | lt => Or.inl rfl\n  | eq => absurd (eq_strict hxy) hne\n  | gt => Or.inr rfl\n\ntheorem antisymm [LinearOrd \u03b1] {x y : \u03b1} : compare x y \u2260 lt \u2192 compare x y \u2260 gt \u2192 x = y :=\n  fun nlt ngt => match hxy : compare x y with\n  | lt => absurd hxy nlt\n  | eq => eq_strict hxy\n  | gt => absurd hxy ngt\n\ntheorem le_antisymm [LinearOrd \u03b1] {x y : \u03b1} : compare x y \u2260 gt \u2192 compare y x \u2260 gt \u2192 x = y :=\n  fun nxy nyx => antisymm (ge_of_le_opp nyx) nxy\n\ntheorem ge_antisymm [LinearOrd \u03b1] {x y : \u03b1} : compare x y \u2260 lt \u2192 compare y x \u2260 lt \u2192 x = y :=\n  fun nxy nyx => antisymm nxy (le_of_ge_opp nyx)\n\ntheorem lt_or_gt_of_ne {x y : \u03b1} : compare x y \u2260 eq \u2192 compare x y = lt \u2228 compare x y = gt :=\n  fun hne => match h : compare x y with\n  | lt => .inl rfl\n  | eq => absurd h hne\n  | gt => .inr rfl\n\ntheorem lt_connex [LinearOrd \u03b1] {x y : \u03b1} : x \u2260 y \u2192 compare x y = lt \u2228 compare y x = lt :=\n  fun hne => match lt_or_gt_of_ne (mt eq_strict hne) with\n  | .inl h => .inl h\n  | .inr h => .inr (lt_of_gt_opp h)\n\ntheorem gt_connex [LinearOrd \u03b1] {x y : \u03b1} : x \u2260 y \u2192 compare x y = gt \u2228 compare y x = gt :=\n  fun hne => match lt_or_gt_of_ne (mt eq_strict hne) with\n  | .inr h => .inl h\n  | .inl h => .inr (gt_of_lt_opp h)\n\nsection LELT\nopen Relation\n\nlocal instance instLE : LE \u03b1 := \u27e8fun x y => compare x y \u2260 gt\u27e9\nlocal instance instLT : LT \u03b1 := \u27e8fun x y => compare x y = lt\u27e9\n\ninstance [OrientedOrd \u03b1] : Reflexive (\u03b1:=\u03b1) (.\u2264.) := \u27e8le_refl\u27e9\ninstance [OrientedOrd \u03b1] : Irreflexive (\u03b1:=\u03b1) (.<.) := \u27e8lt_irrefl\u27e9\ninstance [OrientedOrd \u03b1] : Total (\u03b1:=\u03b1) (.\u2264.) := \u27e8le_total\u27e9\ninstance [TransOrd \u03b1] : Transitive (\u03b1:=\u03b1) (.\u2264.) := \u27e8le_trans\u27e9\ninstance [TransOrd \u03b1] : Transitive (\u03b1:=\u03b1) (.<.) := \u27e8lt_trans\u27e9\ninstance [TransOrd \u03b1] : HTransitive (\u03b1:=\u03b1) (\u03b2:=\u03b1) (\u03b3:=\u03b1) (.\u2264.) (.<.) (.<.) := \u27e8lt_of_le_of_lt\u27e9\ninstance [TransOrd \u03b1] : HTransitive (\u03b1:=\u03b1) (\u03b2:=\u03b1) (\u03b3:=\u03b1) (.<.) (.\u2264.) (.<.) := \u27e8lt_of_lt_of_le\u27e9\ninstance [LinearOrd \u03b1] : Antisymmetric (\u03b1:=\u03b1) (.\u2264.) := \u27e8le_antisymm\u27e9\ninstance [LinearOrd \u03b1] : Connex (\u03b1:=\u03b1) (.<.) := \u27e8lt_connex\u27e9\n\nend LELT\n\nend Ord\n", "meta": {"author": "fgdorais", "repo": "GMLInit", "sha": "a295111627ac907ebc6a86f906dd9b4d69b338d8", "save_path": "github-repos/lean/fgdorais-GMLInit", "path": "github-repos/lean/fgdorais-GMLInit/GMLInit-a295111627ac907ebc6a86f906dd9b4d69b338d8/GMLInit/Data/Ord.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37540110734995635}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.discrete\nimport category_theory.limits.shapes.terminal\nimport algebra.punit_instances\n\n/-!\n# The category of monoids in a monoidal category.\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 u\n\nopen category_theory\nopen category_theory.monoidal_category\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [monoidal_category.{v\u2081} C]\n\n/--\nA monoid object internal to a monoidal category.\n\nWhen the monoidal category is preadditive, this is also sometimes called an \"algebra object\".\n-/\nstructure Mon_ :=\n(X : C)\n(one : \ud835\udfd9_ C \u27f6 X)\n(mul : X \u2297 X \u27f6 X)\n(one_mul' : (one \u2297 \ud835\udfd9 X) \u226b mul = (\u03bb_ X).hom . obviously)\n(mul_one' : (\ud835\udfd9 X \u2297 one) \u226b mul = (\u03c1_ X).hom . obviously)\n-- Obviously there is some flexibility stating this axiom.\n-- This one has left- and right-hand sides matching the statement of `monoid.mul_assoc`,\n-- and chooses to place the associator on the right-hand side.\n-- The heuristic is that unitors and associators \"don't have much weight\".\n(mul_assoc' : (mul \u2297 \ud835\udfd9 X) \u226b mul = (\u03b1_ X X X).hom \u226b (\ud835\udfd9 X \u2297 mul) \u226b mul . obviously)\n\nrestate_axiom Mon_.one_mul'\nrestate_axiom Mon_.mul_one'\nrestate_axiom Mon_.mul_assoc'\nattribute [reassoc] Mon_.one_mul Mon_.mul_one -- We prove a more general `@[simp]` lemma below.\nattribute [simp, reassoc] Mon_.mul_assoc\n\nnamespace Mon_\n\n/--\nThe trivial monoid object. We later show this is initial in `Mon_ C`.\n-/\n@[simps]\ndef trivial : Mon_ C :=\n{ X := \ud835\udfd9_ C,\n  one := \ud835\udfd9 _,\n  mul := (\u03bb_ _).hom,\n  mul_assoc' :=\n    by simp_rw [triangle_assoc, iso.cancel_iso_hom_right, tensor_right_iff, unitors_equal],\n  mul_one' := by simp [unitors_equal] }\n\ninstance : inhabited (Mon_ C) := \u27e8trivial C\u27e9\n\nvariables {C} {M : Mon_ C}\n\n@[simp] lemma one_mul_hom {Z : C} (f : Z \u27f6 M.X) : (M.one \u2297 f) \u226b M.mul = (\u03bb_ Z).hom \u226b f :=\nby rw [\u2190id_tensor_comp_tensor_id, category.assoc, M.one_mul, left_unitor_naturality]\n\n@[simp] lemma mul_one_hom {Z : C} (f : Z \u27f6 M.X) : (f \u2297 M.one) \u226b M.mul = (\u03c1_ Z).hom \u226b f :=\nby rw [\u2190tensor_id_comp_id_tensor, category.assoc, M.mul_one, right_unitor_naturality]\n\nlemma assoc_flip : (\ud835\udfd9 M.X \u2297 M.mul) \u226b M.mul = (\u03b1_ M.X M.X M.X).inv \u226b (M.mul \u2297 \ud835\udfd9 M.X) \u226b M.mul :=\nby simp\n\n/-- A morphism of monoid objects. -/\n@[ext]\nstructure hom (M N : Mon_ C) :=\n(hom : M.X \u27f6 N.X)\n(one_hom' : M.one \u226b hom = N.one . obviously)\n(mul_hom' : M.mul \u226b hom = (hom \u2297 hom) \u226b N.mul . obviously)\n\nrestate_axiom hom.one_hom'\nrestate_axiom hom.mul_hom'\nattribute [simp, reassoc] hom.one_hom hom.mul_hom\n\n/-- The identity morphism on a monoid object. -/\n@[simps]\ndef id (M : Mon_ C) : hom M M :=\n{ hom := \ud835\udfd9 M.X, }\n\ninstance hom_inhabited (M : Mon_ C) : inhabited (hom M M) := \u27e8id M\u27e9\n\n/-- Composition of morphisms of monoid objects. -/\n@[simps]\ndef comp {M N O : Mon_ C} (f : hom M N) (g : hom N O) : hom M O :=\n{ hom := f.hom \u226b g.hom, }\n\ninstance : category (Mon_ C) :=\n{ hom := \u03bb M N, hom M N,\n  id := id,\n  comp := \u03bb M N O f g, comp f g, }\n\n@[simp] lemma id_hom' (M : Mon_ C) : (\ud835\udfd9 M : hom M M).hom = \ud835\udfd9 M.X := rfl\n@[simp] lemma comp_hom' {M N K : Mon_ C} (f : M \u27f6 N) (g : N \u27f6 K) :\n  (f \u226b g : hom M K).hom = f.hom \u226b g.hom := rfl\n\nsection\nvariables (C)\n\n/-- The forgetful functor from monoid objects to the ambient category. -/\n@[simps]\ndef forget : Mon_ C \u2964 C :=\n{ obj := \u03bb A, A.X,\n  map := \u03bb A B f, f.hom, }\n\nend\n\ninstance forget_faithful : faithful (@forget C _ _) := { }\n\ninstance {A B : Mon_ C} (f : A \u27f6 B) [e : is_iso ((forget C).map f)] : is_iso f.hom := e\n\n/-- The forgetful functor from monoid objects to the ambient category reflects isomorphisms. -/\ninstance : reflects_isomorphisms (forget C) :=\n{ reflects := \u03bb X Y f e, by exactI \u27e8\u27e8\n{ hom := inv f.hom,\n  mul_hom' :=\n  begin\n    simp only [is_iso.comp_inv_eq, hom.mul_hom, category.assoc, \u2190tensor_comp_assoc,\n      is_iso.inv_hom_id, tensor_id, category.id_comp],\n  end }, by tidy\u27e9\u27e9 }\n\ninstance unique_hom_from_trivial (A : Mon_ C) : unique (trivial C \u27f6 A) :=\n{ default :=\n  { hom := A.one,\n    one_hom' := by { dsimp, simp, },\n    mul_hom' := by { dsimp, simp [A.one_mul, unitors_equal], } },\n  uniq := \u03bb f,\n  begin\n    ext, simp,\n    rw [\u2190category.id_comp f.hom],\n    erw f.one_hom,\n  end }\n\nopen category_theory.limits\n\ninstance : has_initial (Mon_ C) :=\nhas_initial_of_unique (trivial C)\n\nend Mon_\n\nnamespace category_theory.lax_monoidal_functor\n\nvariables {C} {D : Type u\u2082} [category.{v\u2082} D] [monoidal_category.{v\u2082} D]\n\n/--\nA lax monoidal functor takes monoid objects to monoid objects.\n\nThat is, a lax monoidal functor `F : C \u2964 D` induces a functor `Mon_ C \u2964 Mon_ D`.\n-/\n-- TODO: map_Mod F A : Mod A \u2964 Mod (F.map_Mon A)\n@[simps]\ndef map_Mon (F : lax_monoidal_functor C D) : Mon_ C \u2964 Mon_ D :=\n{ obj := \u03bb A,\n  { X := F.obj A.X,\n    one := F.\u03b5 \u226b F.map A.one,\n    mul := F.\u03bc _ _ \u226b F.map A.mul,\n    one_mul' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, \u2190F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.\u03bc_natural], },\n      slice_lhs 3 4 { rw [\u2190F.to_functor.map_comp, A.one_mul], },\n      rw [F.to_functor.map_id],\n      rw [F.left_unitality],\n    end,\n    mul_one' :=\n    begin\n      conv_lhs { rw [id_tensor_comp, \u2190F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.\u03bc_natural], },\n      slice_lhs 3 4 { rw [\u2190F.to_functor.map_comp, A.mul_one], },\n      rw [F.to_functor.map_id],\n      rw [F.right_unitality],\n    end,\n    mul_assoc' :=\n    begin\n      conv_lhs { rw [comp_tensor_id, \u2190F.to_functor.map_id], },\n      slice_lhs 2 3 { rw [F.\u03bc_natural], },\n      slice_lhs 3 4 { rw [\u2190F.to_functor.map_comp, A.mul_assoc], },\n      conv_lhs { rw [F.to_functor.map_id] },\n      conv_lhs { rw [F.to_functor.map_comp, F.to_functor.map_comp] },\n      conv_rhs { rw [id_tensor_comp, \u2190F.to_functor.map_id], },\n      slice_rhs 3 4 { rw [F.\u03bc_natural], },\n      conv_rhs { rw [F.to_functor.map_id] },\n      slice_rhs 1 3 { rw [\u2190F.associativity], },\n      simp only [category.assoc],\n    end, },\n  map := \u03bb A B f,\n  { hom := F.map f.hom,\n    one_hom' := by { dsimp, rw [category.assoc, \u2190F.to_functor.map_comp, f.one_hom], },\n    mul_hom' :=\n    begin\n      dsimp,\n      rw [category.assoc, F.\u03bc_natural_assoc, \u2190F.to_functor.map_comp, \u2190F.to_functor.map_comp,\n        f.mul_hom],\n    end },\n  map_id' := \u03bb A, by { ext, simp, },\n  map_comp' := \u03bb A B C f g, by { ext, simp, }, }\n\nvariables (C D)\n\n/-- `map_Mon` is functorial in the lax monoidal functor. -/\ndef map_Mon_functor : (lax_monoidal_functor C D) \u2964 (Mon_ C \u2964 Mon_ D) :=\n{ obj := map_Mon,\n  map := \u03bb F G \u03b1,\n  { app := \u03bb A,\n    { hom := \u03b1.app A.X, } } }\n\nend category_theory.lax_monoidal_functor\n\nnamespace Mon_\n\nopen category_theory.lax_monoidal_functor\n\nnamespace equiv_lax_monoidal_functor_punit\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef lax_monoidal_to_Mon : lax_monoidal_functor (discrete punit.{u+1}) C \u2964 Mon_ C :=\n{ obj := \u03bb F, (F.map_Mon : Mon_ _ \u2964 Mon_ C).obj (trivial (discrete punit)),\n  map := \u03bb F G \u03b1, ((map_Mon_functor (discrete punit) C).map \u03b1).app _ }\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef Mon_to_lax_monoidal : Mon_ C \u2964 lax_monoidal_functor (discrete punit.{u+1}) C :=\n{ obj := \u03bb A,\n  { obj := \u03bb _, A.X,\n    map := \u03bb _ _ _, \ud835\udfd9 _,\n    \u03b5 := A.one,\n    \u03bc := \u03bb _ _, A.mul,\n    map_id' := \u03bb _, rfl,\n    map_comp' := \u03bb _ _ _ _ _, (category.id_comp (\ud835\udfd9 A.X)).symm, },\n  map := \u03bb A B f,\n  { app := \u03bb _, f.hom,\n    naturality' := \u03bb _ _ _, by { dsimp, rw [category.id_comp, category.comp_id], },\n    unit' := f.one_hom,\n    tensor' := \u03bb _ _, f.mul_hom, }, }\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef unit_iso :\n  \ud835\udfed (lax_monoidal_functor (discrete punit.{u+1}) C) \u2245\n    lax_monoidal_to_Mon C \u22d9 Mon_to_lax_monoidal C :=\nnat_iso.of_components (\u03bb F,\n  monoidal_nat_iso.of_components\n    (\u03bb _, F.to_functor.map_iso (eq_to_iso (by ext)))\n    (by tidy) (by tidy) (by tidy))\n  (by tidy)\n\n/-- Implementation of `Mon_.equiv_lax_monoidal_functor_punit`. -/\n@[simps]\ndef counit_iso : Mon_to_lax_monoidal C \u22d9 lax_monoidal_to_Mon C \u2245 \ud835\udfed (Mon_ C) :=\nnat_iso.of_components (\u03bb F, { hom := { hom := \ud835\udfd9 _, }, inv := { hom := \ud835\udfd9 _, } })\n  (by tidy)\n\nend equiv_lax_monoidal_functor_punit\n\nopen equiv_lax_monoidal_functor_punit\n\n/--\nMonoid objects in `C` are \"just\" lax monoidal functors from the trivial monoidal category to `C`.\n-/\n@[simps]\ndef equiv_lax_monoidal_functor_punit : lax_monoidal_functor (discrete punit.{u+1}) C \u224c Mon_ C :=\n{ functor := lax_monoidal_to_Mon C,\n  inverse := Mon_to_lax_monoidal C,\n  unit_iso := unit_iso C,\n  counit_iso := counit_iso C, }\n\nend Mon_\n\n/-!\nProjects:\n* Check that `Mon_ Mon \u224c CommMon`, via the Eckmann-Hilton argument.\n  (You'll have to hook up the cartesian monoidal structure on `Mon` first, available in #3463)\n* Check that `Mon_ Top \u224c [bundled topological monoids]`.\n* Check that `Mon_ AddCommGroup \u224c Ring`.\n  (We've already got `Mon_ (Module R) \u224c Algebra R`, in `category_theory.monoidal.internal.Module`.)\n* Can you transport this monoidal structure to `Ring` or `Algebra R`?\n  How does it compare to the \"native\" one?\n* Show that if `C` is braided then `Mon_ C` is naturally monoidal.\n-/\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/monoidal/Mon_.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37540110734995635}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.preserves.shapes.products\nimport category_theory.limits.preserves.shapes.equalizers\n\n/-!\n# Constructing limits from products and equalizers.\n\nIf a category has all products, and all equalizers, then it has all limits.\nSimilarly, if it has all finite products, and all equalizers, then it has all finite limits.\n\nIf a functor preserves all products and equalizers, then it preserves all limits.\nSimilarly, if it preserves all finite products and equalizers, then it preserves all finite limits.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\nopen category_theory\nopen opposite\n\nnamespace category_theory.limits\n\nuniverses v u u\u2082\nvariables {C : Type u} [category.{v} C]\n\nvariables {J : Type v} [small_category J]\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_limit_of_has_products_of_has_equalizers\n\nvariables {F : J \u2964 C}\n          {c\u2081 : fan F.obj}\n          {c\u2082 : fan (\u03bb f : (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), F.obj f.1.2)}\n          (s t : c\u2081.X \u27f6 c\u2082.X)\n          (hs : \u2200 (f : \u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), s \u226b c\u2082.\u03c0.app f = c\u2081.\u03c0.app f.1.1 \u226b F.map f.2)\n          (ht : \u2200 (f : \u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), t \u226b c\u2082.\u03c0.app f = c\u2081.\u03c0.app f.1.2)\n          (i : fork s t)\n\ninclude hs ht\n/--\n(Implementation) Given the appropriate product and equalizer cones, build the cone for `F` which is\nlimiting if the given cones are also.\n-/\n@[simps]\ndef build_limit : cone F :=\n{ X := i.X,\n  \u03c0 :=\n  { app := \u03bb j, i.\u03b9 \u226b c\u2081.\u03c0.app _,\n    naturality' := \u03bb j\u2081 j\u2082 f, begin\n      dsimp,\n      rw [category.id_comp, category.assoc, \u2190 hs \u27e8\u27e8_, _\u27e9, f\u27e9, i.condition_assoc, ht],\n    end} }\n\nvariable {i}\n/--\n(Implementation) Show the cone constructed in `build_limit` is limiting, provided the cones used in\nits construction are.\n-/\ndef build_is_limit (t\u2081 : is_limit c\u2081) (t\u2082 : is_limit c\u2082) (hi : is_limit i) :\n  is_limit (build_limit s t hs ht i) :=\n{ lift := \u03bb q,\n  begin\n    refine hi.lift (fork.of_\u03b9 _ _),\n    { refine t\u2081.lift (fan.mk _ (\u03bb j, _)),\n      apply q.\u03c0.app j },\n    { apply t\u2082.hom_ext,\n      simp [hs, ht] },\n  end,\n  uniq' := \u03bb q m w, hi.hom_ext (i.equalizer_ext (t\u2081.hom_ext (by simpa using w))) }\n\nend has_limit_of_has_products_of_has_equalizers\n\nopen has_limit_of_has_products_of_has_equalizers\n\n/--\nGiven the existence of the appropriate (possibly finite) products and equalizers, we know a limit of\n`F` exists.\n(This assumes the existence of all equalizers, which is technically stronger than needed.)\n-/\nlemma has_limit_of_equalizer_and_product (F : J \u2964 C)\n  [has_limit (discrete.functor F.obj)]\n  [has_limit (discrete.functor (\u03bb f : (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), F.obj f.1.2))]\n  [has_equalizers C] : has_limit F :=\nhas_limit.mk\n{ cone := _,\n  is_limit :=\n    build_is_limit\n      (pi.lift (\u03bb f, limit.\u03c0 _ _ \u226b F.map f.2))\n      (pi.lift (\u03bb f, limit.\u03c0 _ f.1.2))\n      (by simp)\n      (by simp)\n      (limit.is_limit _)\n      (limit.is_limit _)\n      (limit.is_limit _) }\n\n/--\nAny category with products and equalizers has all limits.\n\nSee https://stacks.math.columbia.edu/tag/002N.\n-/\nlemma limits_from_equalizers_and_products\n  [has_products C] [has_equalizers C] : has_limits C :=\n{ has_limits_of_shape := \u03bb J \ud835\udca5,\n  { has_limit := \u03bb F, by exactI has_limit_of_equalizer_and_product F } }\n\n/--\nAny category with finite products and equalizers has all finite limits.\n\nSee https://stacks.math.columbia.edu/tag/002O.\n-/\nlemma finite_limits_from_equalizers_and_finite_products\n  [has_finite_products C] [has_equalizers C] : has_finite_limits C :=\n\u27e8\u03bb J _ _, { has_limit := \u03bb F, by exactI has_limit_of_equalizer_and_product F }\u27e9\n\nvariables {D : Type u\u2082} [category.{v} D]\nnoncomputable theory\n\nsection\n\nvariables [has_limits_of_shape (discrete J) C]\n          [has_limits_of_shape (discrete (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2)) C]\n          [has_equalizers C]\nvariables (G : C \u2964 D)\n          [preserves_limits_of_shape walking_parallel_pair G]\n          [preserves_limits_of_shape (discrete J) G]\n          [preserves_limits_of_shape (discrete (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2)) G]\n\n/-- If a functor preserves equalizers and the appropriate products, it preserves limits. -/\ndef preserves_limit_of_preserves_equalizers_and_product :\n  preserves_limits_of_shape J G :=\n{ preserves_limit := \u03bb K,\n  begin\n    let P := \u220f K.obj,\n    let Q := \u220f (\u03bb (f : (\u03a3 (p : J \u00d7 J), p.fst \u27f6 p.snd)), K.obj f.1.2),\n    let s : P \u27f6 Q := pi.lift (\u03bb f, limit.\u03c0 _ _ \u226b K.map f.2),\n    let t : P \u27f6 Q := pi.lift (\u03bb f, limit.\u03c0 _ f.1.2),\n    let I := equalizer s t,\n    let i : I \u27f6 P := equalizer.\u03b9 s t,\n    apply preserves_limit_of_preserves_limit_cone\n      (build_is_limit s t (by simp) (by simp)\n        (limit.is_limit _)\n        (limit.is_limit _)\n        (limit.is_limit _)),\n    refine is_limit.of_iso_limit (build_is_limit _ _ _ _ _ _ _) _,\n    { exact fan.mk _ (\u03bb j, G.map (pi.\u03c0 _ j)) },\n    { exact fan.mk (G.obj Q) (\u03bb f, G.map (pi.\u03c0 _ f)) },\n    { apply G.map s },\n    { apply G.map t },\n    { intro f,\n      dsimp,\n      simp only [\u2190G.map_comp, limit.lift_\u03c0, fan.mk_\u03c0_app] },\n    { intro f,\n      dsimp,\n      simp only [\u2190G.map_comp, limit.lift_\u03c0, fan.mk_\u03c0_app] },\n    { apply fork.of_\u03b9 (G.map i) _,\n      simp only [\u2190 G.map_comp, equalizer.condition] },\n    { apply is_limit_of_has_product_of_preserves_limit },\n    { apply is_limit_of_has_product_of_preserves_limit },\n    { apply is_limit_fork_map_of_is_limit,\n      apply equalizer_is_equalizer },\n    refine cones.ext (iso.refl _) _,\n    intro j,\n    dsimp,\n    simp, -- See note [dsimp, simp].\n  end }\nend\n\n/-- If G preserves equalizers and finite products, it preserves finite limits. -/\ndef preserves_finite_limits_of_preserves_equalizers_and_finite_products\n  [has_equalizers C] [has_finite_products C]\n  (G : C \u2964 D) [preserves_limits_of_shape walking_parallel_pair G]\n  [\u2200 J [fintype J], preserves_limits_of_shape (discrete J) G]\n  (J : Type v) [small_category J] [fin_category J] :\npreserves_limits_of_shape J G :=\npreserves_limit_of_preserves_equalizers_and_product G\n\n/-- If G preserves equalizers and products, it preserves all limits. -/\ndef preserves_limits_of_preserves_equalizers_and_products\n  [has_equalizers C] [has_products C]\n  (G : C \u2964 D) [preserves_limits_of_shape walking_parallel_pair G]\n  [\u2200 J, preserves_limits_of_shape (discrete J) G] :\npreserves_limits G :=\n{ preserves_limits_of_shape := \u03bb J \ud835\udca5,\n  by exactI preserves_limit_of_preserves_equalizers_and_product G }\n\n/-!\nWe now dualize the above constructions, resorting to copy-paste.\n-/\n\n-- We hide the \"implementation details\" inside a namespace\nnamespace has_colimit_of_has_coproducts_of_has_coequalizers\n\nvariables {F : J \u2964 C}\n          {c\u2081 : cofan (\u03bb f : (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), F.obj f.1.1)}\n          {c\u2082 : cofan F.obj}\n          (s t : c\u2081.X \u27f6 c\u2082.X)\n          (hs : \u2200 (f : \u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), c\u2081.\u03b9.app f \u226b s = F.map f.2 \u226b c\u2082.\u03b9.app f.1.2)\n          (ht : \u2200 (f : \u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), c\u2081.\u03b9.app f \u226b t = c\u2082.\u03b9.app f.1.1)\n          (i : cofork s t)\n\ninclude hs ht\n/--\n(Implementation) Given the appropriate coproduct and coequalizer cocones,\nbuild the cocone for `F` which is colimiting if the given cocones are also.\n-/\n@[simps]\ndef build_colimit : cocone F :=\n{ X := i.X,\n  \u03b9 :=\n  { app := \u03bb j, c\u2082.\u03b9.app _ \u226b i.\u03c0,\n    naturality' := \u03bb j\u2081 j\u2082 f, begin\n      dsimp,\n      rw [category.comp_id, \u2190reassoc_of (hs \u27e8\u27e8_, _\u27e9, f\u27e9), i.condition, \u2190category.assoc, ht],\n    end} }\n\nvariable {i}\n/--\n(Implementation) Show the cocone constructed in `build_colimit` is colimiting,\nprovided the cocones used in its construction are.\n-/\ndef build_is_colimit (t\u2081 : is_colimit c\u2081) (t\u2082 : is_colimit c\u2082) (hi : is_colimit i) :\n  is_colimit (build_colimit s t hs ht i) :=\n{ desc := \u03bb q,\n  begin\n    refine hi.desc (cofork.of_\u03c0 _ _),\n    { refine t\u2082.desc (cofan.mk _ (\u03bb j, _)),\n      apply q.\u03b9.app j },\n    { apply t\u2081.hom_ext,\n      simp [reassoc_of hs, reassoc_of ht] },\n  end,\n  uniq' := \u03bb q m w, hi.hom_ext (i.coequalizer_ext (t\u2082.hom_ext (by simpa using w))) }\n\nend has_colimit_of_has_coproducts_of_has_coequalizers\n\nopen has_colimit_of_has_coproducts_of_has_coequalizers\n\n/--\nGiven the existence of the appropriate (possibly finite) coproducts and coequalizers,\nwe know a colimit of `F` exists.\n(This assumes the existence of all coequalizers, which is technically stronger than needed.)\n-/\nlemma has_colimit_of_coequalizer_and_coproduct (F : J \u2964 C)\n  [has_colimit (discrete.functor F.obj)]\n  [has_colimit (discrete.functor (\u03bb f : (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2), F.obj f.1.1))]\n  [has_coequalizers C] : has_colimit F :=\nhas_colimit.mk\n{ cocone := _,\n  is_colimit :=\n    build_is_colimit\n      (sigma.desc (\u03bb f, F.map f.2 \u226b colimit.\u03b9 (discrete.functor F.obj) f.1.2))\n      (sigma.desc (\u03bb f, colimit.\u03b9 (discrete.functor F.obj) f.1.1))\n      (by simp)\n      (by simp)\n      (colimit.is_colimit _)\n      (colimit.is_colimit _)\n      (colimit.is_colimit _) }\n\n/--\nAny category with coproducts and coequalizers has all colimits.\n\nSee https://stacks.math.columbia.edu/tag/002P.\n-/\nlemma colimits_from_coequalizers_and_coproducts\n  [has_products C] [has_equalizers C] : has_limits C :=\n{ has_limits_of_shape := \u03bb J \ud835\udca5,\n  { has_limit := \u03bb F, by exactI has_limit_of_equalizer_and_product F } }\n\n/--\nAny category with finite coproducts and coequalizers has all finite colimits.\n\nSee https://stacks.math.columbia.edu/tag/002Q.\n-/\nlemma finite_colimits_from_coequalizers_and_finite_coproducts\n  [has_finite_coproducts C] [has_coequalizers C] : has_finite_colimits C :=\n\u27e8\u03bb J _ _, { has_colimit := \u03bb F, by exactI has_colimit_of_coequalizer_and_coproduct F }\u27e9\n\nnoncomputable theory\n\nsection\n\nvariables [has_colimits_of_shape (discrete J) C]\n          [has_colimits_of_shape (discrete (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2)) C]\n          [has_coequalizers C]\nvariables (G : C \u2964 D)\n          [preserves_colimits_of_shape walking_parallel_pair G]\n          [preserves_colimits_of_shape (discrete J) G]\n          [preserves_colimits_of_shape (discrete (\u03a3 p : J \u00d7 J, p.1 \u27f6 p.2)) G]\n\n/-- If a functor preserves coequalizers and the appropriate coproducts, it preserves colimits. -/\ndef preserves_colimit_of_preserves_coequalizers_and_coproduct :\n  preserves_colimits_of_shape J G :=\n{ preserves_colimit := \u03bb K,\n  begin\n    let P := \u2210 K.obj,\n    let Q := \u2210 (\u03bb (f : (\u03a3 (p : J \u00d7 J), p.fst \u27f6 p.snd)), K.obj f.1.1),\n    let s : Q \u27f6 P := sigma.desc (\u03bb f, K.map f.2 \u226b colimit.\u03b9 (discrete.functor K.obj) _),\n    let t : Q \u27f6 P := sigma.desc (\u03bb f, colimit.\u03b9 (discrete.functor K.obj) f.1.1),\n    let I := coequalizer s t,\n    let i : P \u27f6 I := coequalizer.\u03c0 s t,\n    apply preserves_colimit_of_preserves_colimit_cocone\n      (build_is_colimit s t (by simp) (by simp)\n        (colimit.is_colimit _)\n        (colimit.is_colimit _)\n        (colimit.is_colimit _)),\n    refine is_colimit.of_iso_colimit (build_is_colimit _ _ _ _ _ _ _) _,\n    { exact cofan.mk (G.obj Q) (\u03bb j, G.map (sigma.\u03b9 _ j)) },\n    { exact cofan.mk _ (\u03bb f, G.map (sigma.\u03b9 _ f)) },\n    { apply G.map s },\n    { apply G.map t },\n    { intro f,\n      dsimp,\n      simp only [\u2190G.map_comp, colimit.\u03b9_desc, cofan.mk_\u03b9_app] },\n    { intro f,\n      dsimp,\n      simp only [\u2190G.map_comp, colimit.\u03b9_desc, cofan.mk_\u03b9_app] },\n    { apply cofork.of_\u03c0 (G.map i) _,\n      simp only [\u2190 G.map_comp, coequalizer.condition] },\n    { apply is_colimit_of_has_coproduct_of_preserves_colimit },\n    { apply is_colimit_of_has_coproduct_of_preserves_colimit },\n    { apply is_colimit_cofork_map_of_is_colimit,\n      apply coequalizer_is_coequalizer },\n    refine cocones.ext (iso.refl _) _,\n    intro j,\n    dsimp,\n    simp, -- See note [dsimp, simp].\n  end }\nend\n\n/-- If G preserves coequalizers and finite coproducts, it preserves finite colimits. -/\ndef preserves_finite_colimits_of_preserves_coequalizers_and_finite_coproducts\n  [has_coequalizers C] [has_finite_coproducts C]\n  (G : C \u2964 D) [preserves_colimits_of_shape walking_parallel_pair G]\n  [\u2200 J [fintype J], preserves_colimits_of_shape (discrete J) G]\n  (J : Type v) [small_category J] [fin_category J] :\npreserves_colimits_of_shape J G :=\npreserves_colimit_of_preserves_coequalizers_and_coproduct G\n\n/-- If G preserves coequalizers and coproducts, it preserves all colimits. -/\ndef preserves_colimits_of_preserves_coequalizers_and_coproducts\n  [has_coequalizers C] [has_coproducts C]\n  (G : C \u2964 D) [preserves_colimits_of_shape walking_parallel_pair G]\n  [\u2200 J, preserves_colimits_of_shape (discrete J) G] :\npreserves_colimits G :=\n{ preserves_colimits_of_shape := \u03bb J \ud835\udca5,\n  by exactI preserves_colimit_of_preserves_coequalizers_and_coproduct G }\n\nend category_theory.limits\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/constructions/limits_of_products_and_equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.37540110734995635}}
{"text": "-- set_option trace.Elab.info true\n\ntheorem test (p q : Prop) (hp : p) (hq : q) : p \u2227 q \u2227 p := by\n  apply And.intro\n  case right =>\n    apply And.intro\n    case left => exact hq\n    case right => exact hp\n  case left => exact hp\n", "meta": {"author": "leanprover", "repo": "LeanInk", "sha": "499cf46f571562bebee0c8c193a7f9dcf5a30187", "save_path": "github-repos/lean/leanprover-LeanInk", "path": "github-repos/lean/leanprover-LeanInk/LeanInk-499cf46f571562bebee0c8c193a7f9dcf5a30187/test/theorem_proving/007.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.63341024983754, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3754010991472727}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport algebraic_topology.dold_kan.gamma_comp_n\nimport algebraic_topology.dold_kan.n_reflects_iso\n\n/-! The unit isomorphism of the Dold-Kan equivalence\n\nIn order to construct the unit isomorphism of the Dold-Kan equivalence,\nwe first construct natural transformations\n`\u0393\u2082N\u2081.nat_trans : N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)` and\n`\u0393\u2082N\u2082.nat_trans : N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`.\nIt is then shown that `\u0393\u2082N\u2082.nat_trans` is an isomorphism by using\nthat it becomes an isomorphism after the application of the functor\n`N\u2082 : karoubi (simplicial_object C) \u2964 karoubi (chain_complex C \u2115)`\nwhich reflects isomorphisms.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n  category_theory.idempotents simplex_category opposite simplicial_object\nopen_locale simplicial dold_kan\n\nnamespace algebraic_topology\n\nnamespace dold_kan\n\nvariables {C : Type*} [category C] [preadditive C]\n\nlemma P_infty_comp_map_mono_eq_zero (X : simplicial_object C) {n : \u2115}\n  {\u0394' : simplex_category} (i : \u0394' \u27f6 [n]) [hi : mono i] (h\u2081 : \u0394'.len \u2260 n) (h\u2082 : \u00acis_\u03b4\u2080 i) :\n  P_infty.f n \u226b X.map i.op = 0 :=\nbegin\n  unfreezingI { induction \u0394' using simplex_category.rec with m, },\n  obtain \u27e8k, hk\u27e9 := nat.exists_eq_add_of_lt (len_lt_of_mono i\n    (\u03bb h, by { rw \u2190 h at h\u2081,  exact h\u2081 rfl, })),\n  simp only [len_mk] at hk,\n  cases k,\n  { change n = m + 1 at hk,\n    unfreezingI { subst hk, obtain \u27e8j, rfl\u27e9 := eq_\u03b4_of_mono i, },\n    rw is_\u03b4\u2080.iff at h\u2082,\n    have h\u2083 : 1 \u2264 (j : \u2115),\n    { by_contra,\n      exact h\u2082 (by simpa only [fin.ext_iff, not_le, nat.lt_one_iff] using h), },\n    exact (higher_faces_vanish.of_P (m+1) m).comp_\u03b4_eq_zero j h\u2082 (by linarith), },\n  { simp only [nat.succ_eq_add_one, \u2190 add_assoc] at hk,\n    clear h\u2082 hi,\n    subst hk,\n    obtain \u27e8j\u2081, i, rfl\u27e9 := eq_comp_\u03b4_of_not_surjective i (\u03bb h, begin\n      have h' := len_le_of_epi (simplex_category.epi_iff_surjective.2 h),\n      dsimp at h',\n      linarith,\n    end),\n    obtain \u27e8j\u2082, i, rfl\u27e9 := eq_comp_\u03b4_of_not_surjective i (\u03bb h, begin\n      have h' := len_le_of_epi (simplex_category.epi_iff_surjective.2 h),\n      dsimp at h',\n      linarith,\n    end),\n    by_cases hj\u2081 : j\u2081 = 0,\n    { unfreezingI { subst hj\u2081, },\n      rw [assoc, \u2190 simplex_category.\u03b4_comp_\u03b4'' (fin.zero_le _)],\n      simp only [op_comp, X.map_comp, assoc, P_infty_f],\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ j\u2082.succ_ne_zero, zero_comp],\n      rw fin.coe_succ,\n      linarith, },\n    { simp only [op_comp, X.map_comp, assoc, P_infty_f],\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ hj\u2081, zero_comp],\n      by_contra,\n      exact hj\u2081 (by { simp only [fin.ext_iff, fin.coe_zero], linarith, }), }, },\nend\n\n@[reassoc]\nlemma \u0393\u2080_obj_termwise_map_mono_comp_P_infty (X : simplicial_object C) {\u0394 \u0394' : simplex_category}\n  (i : \u0394 \u27f6 \u0394') [mono i] :\n  \u0393\u2080.obj.termwise.map_mono (alternating_face_map_complex.obj X) i \u226b P_infty.f (\u0394.len) =\n    P_infty.f (\u0394'.len) \u226b X.map i.op :=\nbegin\n  unfreezingI\n  { induction \u0394 using simplex_category.rec with n,\n    induction \u0394' using simplex_category.rec with n', },\n  dsimp,\n  /- We start with the case `i` is an identity -/\n  by_cases n = n',\n  { unfreezingI { subst h, },\n    simp only [simplex_category.eq_id_of_mono i, \u0393\u2080.obj.termwise.map_mono_id, op_id, X.map_id],\n    dsimp,\n    simp only [id_comp, comp_id], },\n  by_cases hi : is_\u03b4\u2080 i,\n  /- The case `i = \u03b4 0` -/\n  { have h' : n' = n + 1 := hi.left,\n    unfreezingI { subst h', },\n    simp only [\u0393\u2080.obj.termwise.map_mono_\u03b4\u2080' _ i hi],\n    dsimp,\n    rw [\u2190 P_infty.comm' _ n rfl, alternating_face_map_complex.obj_d_eq],\n    simp only [eq_self_iff_true, id_comp, if_true, preadditive.comp_sum],\n    rw finset.sum_eq_single (0 : fin (n+2)), rotate,\n    { intros b hb hb',\n      rw preadditive.comp_zsmul,\n      erw [P_infty_comp_map_mono_eq_zero X (simplex_category.\u03b4 b) h\n        (by { rw is_\u03b4\u2080.iff, exact hb', }), zsmul_zero], },\n    { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\n    { simpa only [hi.eq_\u03b4\u2080, fin.coe_zero, pow_zero, one_zsmul], }, },\n  /- The case `i \u2260 \u03b4 0` -/\n  { rw [\u0393\u2080.obj.termwise.map_mono_eq_zero _ i _ hi, zero_comp], swap,\n    { by_contradiction h',\n      exact h (congr_arg simplex_category.len h'.symm), },\n    rw P_infty_comp_map_mono_eq_zero,\n    { exact h, },\n    { by_contradiction h',\n      exact hi h', }, },\nend\n\nvariable [has_finite_coproducts C]\n\nnamespace \u0393\u2082N\u2081\n\n/-- The natural transformation `N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)`. -/\n@[simps]\ndef nat_trans : (N\u2081 : simplicial_object C \u2964 _) \u22d9 \u0393\u2082 \u27f6 to_karoubi _ :=\n{ app := \u03bb X,\n  { f :=\n    { app := \u03bb \u0394, (\u0393\u2080.splitting K[X]).desc \u0394 (\u03bb A, P_infty.f A.1.unop.len \u226b X.map (A.e.op)),\n      naturality' := \u03bb \u0394 \u0394' \u03b8, begin\n        apply (\u0393\u2080.splitting K[X]).hom_ext',\n        intro A,\n        change _ \u226b (\u0393\u2080.obj K[X]).map \u03b8  \u226b _ = _,\n        simp only [splitting.\u03b9_desc_assoc, assoc,\n          \u0393\u2080.obj.map_on_summand'_assoc, splitting.\u03b9_desc],\n        erw \u0393\u2080_obj_termwise_map_mono_comp_P_infty_assoc X (image.\u03b9 (\u03b8.unop \u226b A.e)),\n        dsimp only [to_karoubi],\n        simp only [\u2190 X.map_comp],\n        congr' 2,\n        simp only [eq_to_hom_refl, id_comp, comp_id, \u2190 op_comp],\n        exact quiver.hom.unop_inj (A.fac_pull \u03b8),\n      end, },\n    comm := begin\n      apply (\u0393\u2080.splitting K[X]).hom_ext,\n      intro n,\n      dsimp [N\u2081],\n      simp only [\u2190 splitting.\u03b9_summand_id, splitting.\u03b9_desc,\n        comp_id, splitting.\u03b9_desc_assoc, assoc, P_infty_f_idem_assoc],\n    end, },\n  naturality' := \u03bb X Y f, begin\n    ext1,\n    apply (\u0393\u2080.splitting K[X]).hom_ext,\n    intro n,\n    dsimp [N\u2081, to_karoubi],\n    simpa only [\u2190splitting.\u03b9_summand_id, splitting.\u03b9_desc, splitting.\u03b9_desc_assoc,\n      assoc, P_infty_f_idem_assoc, karoubi.comp_f, nat_trans.comp_app, \u0393\u2082_map_f_app,\n      homological_complex.comp_f, alternating_face_map_complex.map_f,\n      P_infty_f_naturality_assoc, nat_trans.naturality],\n  end, }\n\nend \u0393\u2082N\u2081\n\n/-- The compatibility isomorphism relating `N\u2082 \u22d9 \u0393\u2082` and `N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082 : to_karoubi (simplicial_object C) \u22d9 N\u2082 \u22d9 \u0393\u2082 \u2245 N\u2081 \u22d9 \u0393\u2082 :=\neq_to_iso (functor.congr_obj (functor_extension\u2081_comp_whiskering_left_to_karoubi _ _) (N\u2081 \u22d9 \u0393\u2082))\n\nnamespace \u0393\u2082N\u2082\n\n/-- The natural transformation `N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`. -/\ndef nat_trans : (N\u2082 : karoubi (simplicial_object C) \u2964 _) \u22d9 \u0393\u2082 \u27f6 \ud835\udfed _ :=\n((whiskering_left _ _ _).obj _).preimage (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans)\n\nlemma nat_trans_app_f_app (P : karoubi (simplicial_object C)) :\n  \u0393\u2082N\u2082.nat_trans.app P = (N\u2082 \u22d9 \u0393\u2082).map P.decomp_id_i \u226b\n    (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans).app P.X \u226b P.decomp_id_p :=\nwhiskering_left_obj_preimage_app ((compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans)) P\n\nend \u0393\u2082N\u2082\n\nlemma compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans (X : simplicial_object C) :\n  \u0393\u2082N\u2081.nat_trans.app X = (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).inv \u226b\n    \u0393\u2082N\u2082.nat_trans.app ((to_karoubi _).obj X) :=\nbegin\n  rw [\u2190 cancel_epi (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).hom, iso.hom_inv_id_assoc],\n  exact congr_app (((whiskering_left _ _ _).obj _).image_preimage\n    (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans : _ \u27f6 to_karoubi _ \u22d9 \ud835\udfed _ )).symm X,\nend\n\nlemma identity_N\u2082_objectwise (P : karoubi (simplicial_object C)) :\n  N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P) = \ud835\udfd9 (N\u2082.obj P) :=\nbegin\n  ext n,\n  have eq\u2081 : (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f.f n = P_infty.f n \u226b P.p.app (op [n]) \u226b\n    (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9_summand (splitting.index_set.id (op [n])),\n  { simp only [N\u2082\u0393\u2082_inv_app_f_f, N\u2082_obj_p_f, assoc], },\n  have eq\u2082 : (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9_summand (splitting.index_set.id (op [n])) \u226b\n    (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)).f.f n = P_infty.f n \u226b P.p.app (op [n]),\n  { dsimp [N\u2082],\n    simp only [\u0393\u2082N\u2082.nat_trans_app_f_app, P_infty_on_\u0393\u2080_splitting_summand_eq_self_assoc,\n      functor.comp_map, compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_hom, nat_trans.comp_app,\n      eq_to_hom_app, assoc, karoubi.comp_f, karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id,\n      karoubi.decomp_id_p_f, karoubi.comp_p_assoc, \u0393\u2082_map_f_app,\n      N\u2082_map_f_f, karoubi.decomp_id_i_f, \u0393\u2082N\u2081.nat_trans_app_f_app],\n    erw [splitting.\u03b9_desc_assoc, assoc, assoc, splitting.\u03b9_desc_assoc],\n    dsimp [splitting.index_set.id, splitting.index_set.e],\n    simp only [assoc, nat_trans.naturality, P_infty_f_naturality_assoc,\n      app_idem_assoc, P_infty_f_idem_assoc],\n    erw [P.X.map_id, comp_id], },\n  simp only [karoubi.comp_f, homological_complex.comp_f, karoubi.id_eq, N\u2082_obj_p_f, assoc,\n    eq\u2081, eq\u2082, P_infty_f_naturality_assoc, app_idem, P_infty_f_idem_assoc],\nend\n\nlemma identity_N\u2082 :\n  ((\ud835\udfd9 (N\u2082 : karoubi (simplicial_object C) \u2964 _ ) \u25eb N\u2082\u0393\u2082.inv) \u226b\n    (\u0393\u2082N\u2082.nat_trans \u25eb \ud835\udfd9 N\u2082) : N\u2082 \u27f6 N\u2082) = \ud835\udfd9 N\u2082 :=\nby { ext P : 2, dsimp, rw [\u0393\u2082.map_id, N\u2082.map_id, comp_id, id_comp, identity_N\u2082_objectwise P], }\n\ninstance : is_iso (\u0393\u2082N\u2082.nat_trans : (N\u2082 : karoubi (simplicial_object C) \u2964 _ ) \u22d9 _ \u27f6 _) :=\nbegin\n  haveI : \u2200 (P : karoubi (simplicial_object C)), is_iso (\u0393\u2082N\u2082.nat_trans.app P),\n  { intro P,\n    haveI : is_iso (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)),\n    { have h := identity_N\u2082_objectwise P,\n      erw hom_comp_eq_id at h,\n      rw h,\n      apply_instance, },\n    exact is_iso_of_reflects_iso _ N\u2082, },\n  apply nat_iso.is_iso_of_is_iso_app,\nend\n\ninstance : is_iso (\u0393\u2082N\u2081.nat_trans : (N\u2081 : simplicial_object C \u2964 _ ) \u22d9 _ \u27f6 _) :=\nbegin\n  haveI : \u2200 (X : simplicial_object C), is_iso (\u0393\u2082N\u2081.nat_trans.app X),\n  { intro X,\n    rw compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans,\n    apply_instance, },\n  apply nat_iso.is_iso_of_is_iso_app,\nend\n\n/-- The unit isomorphism of the Dold-Kan equivalence. -/\n@[simp]\ndef \u0393\u2082N\u2082 : \ud835\udfed _ \u2245 (N\u2082 : karoubi (simplicial_object C) \u2964 _) \u22d9 \u0393\u2082 :=\n(as_iso \u0393\u2082N\u2082.nat_trans).symm\n\n/-- The natural isomorphism `to_karoubi (simplicial_object C) \u2245 N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef \u0393\u2082N\u2081 : to_karoubi _  \u2245 (N\u2081 : simplicial_object C \u2964 _) \u22d9 \u0393\u2082 :=\n(as_iso \u0393\u2082N\u2081.nat_trans).symm\n\nend dold_kan\n\nend algebraic_topology\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_topology/dold_kan/n_comp_gamma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.375380936799166}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport algebra.field.defs\nimport algebra.ring.opposite\nimport data.int.cast.lemmas\n\n/-!\n# Field structure on the multiplicative/additive opposite\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nvariables (\u03b1 : Type*)\n\nnamespace mul_opposite\n\n@[to_additive] instance [has_rat_cast \u03b1] : has_rat_cast \u03b1\u1d50\u1d52\u1d56 := \u27e8\u03bb n, op n\u27e9\n\nvariables {\u03b1}\n\n@[simp, norm_cast, to_additive]\nlemma op_rat_cast [has_rat_cast \u03b1] (q : \u211a) : op (q : \u03b1) = q := rfl\n\n@[simp, norm_cast, to_additive]\nlemma unop_rat_cast [has_rat_cast \u03b1] (q : \u211a) : unop (q : \u03b1\u1d50\u1d52\u1d56) = q := rfl\n\nvariables (\u03b1)\n\ninstance [division_semiring \u03b1] : division_semiring \u03b1\u1d50\u1d52\u1d56 :=\n{ .. mul_opposite.group_with_zero \u03b1, .. mul_opposite.semiring \u03b1 }\n\ninstance [division_ring \u03b1] : division_ring \u03b1\u1d50\u1d52\u1d56 :=\n{ rat_cast := \u03bb q, op q,\n  rat_cast_mk := \u03bb a b hb h, by { rw [rat.cast_def, op_div, op_nat_cast, op_int_cast],\n    exact int.commute_cast _ _ },\n  ..mul_opposite.division_semiring \u03b1, ..mul_opposite.ring \u03b1 }\n\ninstance [semifield \u03b1] : semifield \u03b1\u1d50\u1d52\u1d56 :=\n{ .. mul_opposite.division_semiring \u03b1, .. mul_opposite.comm_semiring \u03b1 }\n\ninstance [field \u03b1] : field \u03b1\u1d50\u1d52\u1d56 :=\n{ .. mul_opposite.division_ring \u03b1, .. mul_opposite.comm_ring \u03b1 }\n\nend mul_opposite\n\nnamespace add_opposite\n\ninstance [division_semiring \u03b1] : division_semiring \u03b1\u1d43\u1d52\u1d56 :=\n{ ..add_opposite.group_with_zero \u03b1, ..add_opposite.semiring \u03b1 }\n\ninstance [division_ring \u03b1] : division_ring \u03b1\u1d43\u1d52\u1d56 :=\n{ rat_cast_mk := \u03bb a b hb h, by rw \u2190div_eq_mul_inv; exact congr_arg op (rat.cast_def _),\n  ..add_opposite.ring \u03b1, ..add_opposite.group_with_zero \u03b1, ..add_opposite.has_rat_cast \u03b1 }\n\ninstance [semifield \u03b1] : semifield \u03b1\u1d43\u1d52\u1d56 :=\n{ ..add_opposite.division_semiring \u03b1, ..add_opposite.comm_semiring \u03b1 }\n\ninstance [field \u03b1] : field \u03b1\u1d43\u1d52\u1d56 :=\n{ ..add_opposite.division_ring \u03b1, ..add_opposite.comm_ring \u03b1 }\n\nend add_opposite\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/field/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.37538093679916595}}
{"text": "import unitb.code.syntax\nimport unitb.code.rules\n\nimport util.predicate\n\nuniverse variables u v\n\nopen nat predicate\n\nsection\n\nparameters (\u03c3 : Type) (lbl : Type)\n\n@[reducible]\nprivate def pred := \u03c3 \u2192 Prop\n\nparameters {\u03c3}\n\n\nlemma assert_of_first {p q : pred} {c : code lbl p q}\n: assert_of (first c) = p :=\nbegin\n  induction c\n  ; try { refl },\n  case code.seq  p' q' r' c\u2080 c\u2081\n  { unfold first,\n    destruct first c\u2080,\n    case none\n    { intro h,\n      simp [h],\n      destruct first c\u2081,\n      case none\n      { intro h',\n        simp [h'], unfold assert_of,\n        simp [h'] at ih_2, unfold assert_of at ih_2,\n        simp [h] at ih_1, unfold assert_of at ih_1,\n        subst r', subst q' },\n      case some\n      { intros x h', simp [h'],\n        unfold assert_of assert_of',\n        rw h at ih_1, rw h' at ih_2,\n        unfold assert_of at ih_1 ih_2,\n        subst p', rw ih_2, } },\n    case some\n    { intros x h,\n      simp [h],\n      unfold assert_of assert_of',\n      rw h at ih_1, unfold assert_of at ih_1,\n      rw ih_1 }, }\nend\n\nlemma first_eq_none_imp_eq {p q : pred} {c : code lbl p q}\n: first c = none \u2192 p = q :=\nbegin\n  induction c\n  ; unfold first\n  ; try { contradiction },\n  case code.skip\n  { simp },\n  case code.seq p' q' r' c\u2080 c\u2081\n  { destruct first c\u2080,\n    case none\n    { intro h', simp [h'],\n      intro h'', rw [ih_1 h',ih_2 h''], },\n    case some\n    { intros pc h,\n      simp [h], contradiction }, },\nend\n\nlocal attribute [instance] classical.prop_decidable\n\nlemma assert_of_next {p q : pred} {c : code lbl p q} (pc : option (current c)) (s : \u03c3)\n: assert_of (next s pc) = next_assert pc s :=\nbegin\n  cases pc with pc,\n  { refl },\n  unfold next next_assert,\n  induction pc\n  ; try { refl }\n  ; unfold next' next_assert',\n  case current.seq_left\n  { rw \u2190 ih_1,\n    cases next' s a,\n    case none\n    { destruct first c\u2081,\n      case none\n      { intros h\u2080,\n        simp [h\u2080],\n        unfold assert_of,\n        cases c\u2081 ; try { refl }\n        ; unfold first at h\u2080\n        ; try { contradiction },\n        { simp at h\u2080,\n          simp [first_eq_none_imp_eq h\u2080.left,first_eq_none_imp_eq  h\u2080.right] }, },\n      case some\n      { intros pc h\u2080,\n        simp,\n        rw [h\u2080,fmap_some],\n        unfold assert_of assert_of',\n        change assert_of (some pc) = _,\n        rw [\u2190 h\u2080,assert_of_first] } },\n    case some\n    { simp, refl } },\n  case current.seq_right\n  { rw \u2190 ih_1,\n    cases next' s a ; refl },\n  case current.ite_cond\n  { cases classical.em (t s) with h h,\n    { rw [if_pos h,if_pos h],\n      destruct first c\u2080,\n      { intros h, simp [h], have h := first_eq_none_imp_eq h,\n        unfold assert_of, subst pa },\n      { intros pc h, simp [h],\n        unfold assert_of assert_of',\n        change assert_of (some pc) = _,\n        rw [\u2190 h,assert_of_first], }, },\n    { rw [if_neg h,if_neg h],\n      destruct first c\u2081,\n      { intros h, simp [h],\n        have h := first_eq_none_imp_eq h,\n        unfold assert_of, subst pb },\n      { intros pc h, simp [h],\n        unfold assert_of assert_of',\n        change assert_of (some pc) = _,\n        rw [\u2190 h,assert_of_first], }, }, },\n  case current.ite_left\n  { rw \u2190 ih_1, clear ih_1,\n    cases next' s a with pc ; simp,\n    { refl },\n    { unfold assert_of assert_of', }, },\n  case current.ite_right\n  { rw \u2190 ih_1, clear ih_1,\n    cases next' s a with pc ; simp,\n    { refl },\n    { unfold assert_of assert_of', }, },\n  case current.while_cond\n  { cases classical.em (w s) with h h ;\n    destruct first c_1,\n    { intro h',\n      rw [if_pos h,if_pos h,h'],\n      have h'' := first_eq_none_imp_eq h', subst inv,\n      refl, },\n    { intros pc h',\n      rw [if_pos h,if_pos h,h'],\n      simp,\n      change assert_of (some pc) = _,\n      rw [\u2190 h',assert_of_first], },\n    { intros h',\n      rw [if_neg h,if_neg h], refl },\n    { intros pc h',\n      rw [if_neg h,if_neg h], refl }, },\n  case current.while_body\n  { rw \u2190 ih_1, clear ih_1,\n    destruct next' s a,\n    { intros h',\n      simp [h'], refl },\n    { intros pc h',\n      simp [h'], refl }, },\nend\n\nend\n\nsection local_correctness\n\nlocal attribute [instance] classical.prop_decidable\n\nparameters (\u03c3 : Type)\n\nparameters (F : nondet.program \u03c3)\n\n@[reducible]\nprivate def lbl := F.lbl\n\n@[reducible]\nprivate def pred := \u03c3 \u2192 Prop\n\nparameters {\u03c3}\n\nvariables {p q : pred}\nvariable (c : code lbl p q)\n\nstructure state_correctness (pc : option $ current c) : Prop :=\n  (enabled : \u2200 l, selects pc l \u2192 assert_of pc \u27f9 F.guard (some l))\n  (correct : \u2200 l, selects pc l \u2192\n       \u2200 s s', assert_of pc s \u2192 F.step_of (some l) s s' \u2192 next_assert pc s s')\n  (cond_true : \u2200 (H : is_control pc),\n       \u2200 s, assert_of pc s \u2192 condition pc H s \u2192 next_assert pc s s)\n  (cond_false : \u2200 (H : is_control pc),\n       \u2200 s, assert_of pc s \u2192 \u00ac condition pc H s \u2192 next_assert pc s s)\n\nlemma selects_action_imp_eq {l l' : lbl} {p q : pred} {ds : set lbl}\n  (pc : option (current $ code.action p q ds l))\n  (H : selects pc l')\n: l' = l :=\nbegin\n  cases pc with pc,\n  { cases H },\n  { cases pc, apply H, },\nend\n\nlemma selects_and_selects_imp_eq {l l' : lbl} {p q : pred}\n  {c : code lbl p q}\n  {pc : option (current c)}\n  (H  : selects pc l)\n  (H' : selects pc l')\n: l' = l :=\nsorry\n\nlemma assert_of_action {l : lbl} {p q : pred} {ds : set lbl}\n  (pc : current $ code.action p q ds l)\n: assert_of (some pc) = p :=\nbegin\n  cases pc with pc, refl,\nend\n\nlemma next_assert_action {l : lbl} {p q : pred} {ds : set lbl}\n  (pc : current $ code.action p q ds l)\n  (s : \u03c3)\n: next_assert (some pc) s = q :=\nbegin\n  cases pc with pc, refl,\nend\n\nsection\n\nvariables H : correct F c\ninclude H\n\nlemma enabled_of_correct\n: \u2200 (pc : current c) l, selects (some pc) l \u2192 assert_of (some pc) \u27f9 F.guard (some l) :=\nbegin\n  induction H,\n  { intros pc l Hpc, cases pc, },\n  { intros pc l' Hpc, cases pc with pc,\n    have Heq_l := selects_action_imp_eq _ Hpc, subst l',\n    unfold assert_of assert_of', apply a },\n  { intros pc l',\n    cases pc with pc pc\n    ; unfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  { intros pc l',\n    cases pc with pc pc\n    ; unfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { cases Hpc },\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  { intros pc l',\n    cases pc with pc pc\n    ; unfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { cases Hpc },\n    { apply ih_1 _ _ Hpc }, },\nend\n\nlemma correct_of_correct\n: \u2200 (pc : current c) l, selects (some pc) l \u2192\n       \u2200 s s', assert_of (some pc) s \u2192 F.step_of (some l) s s' \u2192 next_assert (some pc) s s' :=\nbegin\n  induction H,\n  { intros pc l Hpc, cases pc, },\n  { intros pc l' Hpc, cases pc with pc,\n    have Heq_l := selects_action_imp_eq _ Hpc, subst l',\n    unfold assert_of assert_of', apply a_1 },\n  { intros pc l',\n    cases pc with pc pc\n    ; unfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  { intros pc l',\n    cases pc with pc pc\n    ; unfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { cases Hpc },\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  { intros pc l',\n    cases pc with pc pc\n    ; dunfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { cases Hpc },\n    { apply ih_1 _ _ Hpc }, },\nend\n\nlemma cond_true_of_correct\n: \u2200 (pc : current c) (H : is_control $ some pc),\n       \u2200 s, assert_of (some pc) s \u2192 condition (some pc) H s \u2192 next_assert (some pc) s s :=\nbegin\n  induction H,\n  case correct.skip p'\n  { intros pc l Hpc, cases pc, },\n  case correct.action p' q' l ds Hgrd Htrip\n  { intros pc l' s, cases pc with pc, cases l' },\n  case correct.seq p' q' r' c\u2080 c\u2081 Hc\u2080 Hc\u2081\n  { intros pc l',\n    cases pc with pc pc\n    ; dunfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  case correct.ite p' t pa pb q' ds c\u2080 c\u2081 Hc\u2080 Hc\u2081 Hpa Hpb\n  { intros pc,\n    cases pc with pc pc\n    ; dunfold condition assert_of condition' assert_of' next_assert next_assert'\n             is_control is_control'\n    ; intros Hpc s Hp Hc,\n    case current.ite_cond\n    { rw if_pos Hc, apply Hpa _ \u27e8Hp,Hc\u27e9, },\n    case current.ite_left\n    { apply ih_1 _ _ _ Hp Hc, },\n    case current.ite_right\n    { apply ih_2 _ _ _ Hp Hc, }, },\n  case correct.while t p' inv q' ds b c Htp Hntq Hcvr\n  { intros pc,\n    cases pc with pc pc\n    ; dunfold condition assert_of condition' assert_of' next_assert next_assert'\n              is_control is_control'\n    ; intros Hpc s Hp Hc,\n    { rw if_pos Hc, apply Htp _ \u27e8Hp,Hc\u27e9, },\n    { apply ih_1 _ _ _ Hp Hc, }, },\nend\n\nlemma cond_false_of_correct\n: \u2200 (pc : current c) (H : is_control (some pc)),\n       \u2200 s, assert_of (some pc) s \u2192 \u00ac condition (some pc) H s \u2192 next_assert (some pc) s s :=\nbegin\n  induction H,\n  case correct.skip p'\n  { intros pc l Hpc, cases pc, },\n  case correct.action p' q' l ds Hgrd Htrip\n  { intros pc l' s, cases pc with pc, cases l' },\n  case correct.seq p' q' r' c\u2080 c\u2081 Hc\u2080 Hc\u2081\n  { intros pc l',\n    cases pc with pc pc\n    ; dunfold selects assert_of selects' assert_of'\n    ; intros Hpc,\n    { apply ih_1 _ _ Hpc },\n    { apply ih_2 _ _ Hpc }, },\n  case correct.ite p' t pa pb q' ds c\u2080 c\u2081 Hc\u2080 Hc\u2081 Hpa Hpb\n  { intros pc,\n    cases pc with pc pc\n    ; dunfold condition assert_of condition' assert_of' next_assert next_assert'\n             is_control is_control'\n    ; intros Hpc s Hp Hc,\n    case current.ite_cond\n    { rw if_neg Hc, apply Hpb _ \u27e8Hp,Hc\u27e9, },\n    case current.ite_left\n    { apply ih_1 _ _ _ Hp Hc, },\n    case current.ite_right\n    { apply ih_2 _ _ _ Hp Hc, }, },\n  case correct.while t p' inv q' ds b c Htp Hntq Hcvr\n  { intros pc,\n    cases pc with pc pc\n    ; dunfold condition assert_of condition' assert_of' next_assert next_assert'\n             is_control is_control'\n    ; intros Hpc s Hp Hc,\n    { rw if_neg Hc, apply Hntq _ \u27e8Hp,Hc\u27e9, },\n    { apply ih_1 _ _ _ Hp Hc, }, },\nend\n\nlemma state_correctness_none\n: state_correctness c none :=\nbegin\n  apply state_correctness.mk,\n  { intros l Hl, cases Hl },\n  { intros l Hl, cases Hl },\n  { intros H', unfold is_control at H', cases H', },\n  { intros H', unfold is_control at H', cases H', },\nend\n\nlemma syntactic_imp_state_correctness (pc : option $ current c)\n: state_correctness c pc :=\nbegin\n  cases pc with pc,\n  { apply state_correctness_none _ H },\n  apply state_correctness.mk,\n  { apply enabled_of_correct _ H },\n  { apply correct_of_correct _ H },\n  { apply cond_true_of_correct _ H },\n  { apply cond_false_of_correct _ H },\nend\n\nomit H\nparameters {F}\nvariable {c}\n\nlemma not_selects_and_is_control\n  {pc : option $ current c} {l : lbl}\n  (h\u2080 : selects pc l)\n  (h\u2081 : is_control pc )\n: false :=\nbegin\n  cases pc with pc,\n  { unfold is_control at h\u2081, cases h\u2081 },\n  unfold is_control at h\u2081,\n  unfold selects at h\u2080,\n  induction pc\n  ; try { cases h\u2081 }\n  ; try { cases h\u2080 }\n  ; apply ih_1 h\u2081 h\u2080\nend\n\nlemma next_counter_action {p q : pred} {ds} {l : lbl} {p' q'} {c' : code lbl p' q'}\n  (s : \u03c3)\n  (H : subtree (code.action p q ds l) c')\n: next s (some $ counter H) = exit' H :=\nbegin\n  unfold next,\n  induction H,\n  { simp, refl },\n  all_goals { simp [counter,next',ih_1] },\nend\n\nlemma not_within_skip {p' q'} {c' : code lbl p' q'}\n  {H : subtree (code.skip p) c'}\n  {pc : option $ current c'}\n  (H' : within H pc)\n: exits H pc :=\nbegin\n  unfold exits,\n  cases pc with pc,\n  { apply H' },\n  unfold within at H',\n  rw or_comm at H',\n  cases H' with H' H',\n  { apply H' },\n  induction H,\n  { rw exit'_rfl,\n    cases pc },\n  { rw within'_seq_left at H',\n    cases H' with pc' H', cases H' with H\u2080 H\u2081,\n    rw [exit'_seq_left,ih_1 _ H\u2080,fmap_some,some_or_else,H\u2081] },\n  { rw within'_seq_right at H',\n    cases H' with pc' H', cases H' with H\u2080 H\u2081,\n    rw [exit'_seq_right,ih_1 _ H\u2080,fmap_some,H\u2081] },\n  { rw within'_ite_left at H',\n    cases H' with pc' H', cases H' with H\u2080 H\u2081,\n    rw [exit'_ite_left,ih_1 _ H\u2080,fmap_some,H\u2081] },\n  { rw within'_ite_right at H',\n    cases H' with pc' H', cases H' with H\u2080 H\u2081,\n    rw [exit'_ite_right,ih_1 _ H\u2080,fmap_some,H\u2081] },\n  { rw within'_while at H',\n    cases H' with pc' H', cases H' with H\u2080 H\u2081,\n    rw [exit'_while,ih_1 _ H\u2080,fmap_some,some_or_else,H\u2081] },\nend\n\nlemma counter_action_of_within {p q : pred} {ds} {l : lbl} {p' q'} {c' : code lbl p' q'}\n  {H : subtree (code.action p q ds l) c'}\n  {pc : option $ current c'}\n  (Hpc : within H pc)\n  (Hnex : \u00ac exits H pc)\n: some (counter H) = pc :=\nbegin\n  cases pc with pc ; unfold within at Hpc,\n  { cases Hnex Hpc, },\n  rw [or_comm,or_iff_not_imp] at Hpc,\n  have H\u2083 := Hpc Hnex,  clear Hpc Hnex,\n  apply congr_arg,\n  induction H,\n  case subtree.rfl\n   { cases pc, refl },\n  case subtree.seq_left\n   { unfold counter,\n     rw [within'_seq_left] at H\u2083,\n     cases H\u2083 with pc\u2080 H\u2083, cases H\u2083 with H\u2083 H\u2084,\n     cases H\u2084, clear H\u2084,\n     apply congr_arg,\n     apply ih_1 _ H\u2083 },\n  case subtree.seq_right\n   { unfold counter,\n     rw [within'_seq_right] at H\u2083,\n     cases H\u2083 with pc\u2080 H\u2083, cases H\u2083 with H\u2083 H\u2084,\n     cases H\u2084, clear H\u2084,\n     apply congr_arg,\n     apply ih_1 _ H\u2083 },\n  case subtree.ite_left\n   { unfold counter,\n     rw [within'_ite_left] at H\u2083,\n     cases H\u2083 with pc\u2080 H\u2083, cases H\u2083 with H\u2083 H\u2084,\n     cases H\u2084, clear H\u2084,\n     apply congr_arg,\n     apply ih_1 _ H\u2083 },\n  case subtree.ite_right\n   { unfold counter,\n     rw [within'_ite_right] at H\u2083,\n     cases H\u2083 with pc\u2080 H\u2083, cases H\u2083 with H\u2083 H\u2084,\n     cases H\u2084, clear H\u2084,\n     apply congr_arg,\n     apply ih_1 _ H\u2083 },\n  case subtree.while\n   { unfold counter,\n     rw [within'_while] at H\u2083,\n     cases H\u2083 with pc\u2080 H\u2083, cases H\u2083 with H\u2083 H\u2084,\n     cases H\u2084, clear H\u2084,\n     apply congr_arg,\n     apply ih_1 _ H\u2083 },\nend\n\nend\n\nend local_correctness\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/code/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.37538092946776874}}
{"text": "import tactic\n\nnamespace chapter3\n\n  open string\n  open nat\n  open option\n  open function\n\n  --define natural trees\n  inductive natree \n  | node : natree\n  | ref : string \u2192 natree\n  | app : natree \u2192 natree \u2192 natree\n  open natree\n\n  --introduce notation\n  notation `\ud835\udd4b` := natree\n  notation `\u25b3` := natree.node\n  infixl `\u2b1d`:60 := natree.app\n\n  variables {w x y z : \ud835\udd4b}\n\n  --equational axioms (should these just be defined as a relation, seperate from equality?)\n  @[simp] axiom kernel : \u25b3\u2b1d\u25b3\u2b1dy\u2b1dz = y\n  @[simp] axiom stem : \u25b3\u2b1d(\u25b3\u2b1dx)\u2b1dy\u2b1dz = y\u2b1dz\u2b1d(x\u2b1dz)\n  @[simp] axiom fork : \u25b3\u2b1d(\u25b3\u2b1dw\u2b1dx)\u2b1dy\u2b1dz = z\u2b1dw\u2b1dx\n\n  /- \n  congruence \"axioms\"\n  cong_node comes for free from rfl,\n  cong_app comes for free from congurence of function application with equality (congr_arg2) and the fact that inductive type constructors are injective (app.inj)\n  -/\n  def cong_node : \u25b3 = \u25b3 := rfl\n  def cong_app : w = y \u2227 x = z \u2194 w \u2b1d x = y \u2b1d z := --this is only really a \"congruence\" in one direction\n  begin                                           --also the left direction is actually *false* which means the axioms above introduce inconsistency\n    split,\n      intro h,\n      cases h,\n      exact congr_arg2 app h_left h_right,\n    intro h,\n    exact app.inj h,\n  end\n\n  def uhoh : false :=\n  begin\n    sorry --implement\n  end\n\n  --define primitive combinators\n  def K := \u25b3\u2b1d\u25b3\n  @[simp] theorem r_K : K\u2b1dy\u2b1dz = y := by simp [K]\n\n  def I := \u25b3\u2b1d(\u25b3\u2b1d\u25b3)\u2b1d(\u25b3\u2b1d\u25b3)\n  @[simp] theorem r_I : I\u2b1dx = x := by simp [I]\n\n  def D := \u25b3\u2b1d(\u25b3\u2b1d\u25b3)\u2b1d(\u25b3\u2b1d\u25b3\u2b1d\u25b3)\n  @[simp] theorem r_D : D\u2b1dx\u2b1dy\u2b1dz = y\u2b1dz\u2b1d(x\u2b1dz) := by simp [D]\n\n  @[simp] def d (x : \ud835\udd4b) := \u25b3\u2b1d(\u25b3\u2b1dx)\n  theorem d_eq_r_D : d x = D\u2b1dx := by simp [D]\n\n  --derivation of S combinator\n  theorem S_exists : \u2200 S : \ud835\udd4b, S\u2b1dx\u2b1dy\u2b1dz = x\u2b1dz\u2b1d(y\u2b1dz) \u2192 S = d (K\u2b1dD) \u2b1d (d K \u2b1d (K\u2b1dD)) :=\n  begin\n    intros S h\u2081,\n    have h\u2082 : S\u2b1dx\u2b1dy\u2b1dz = D\u2b1dy\u2b1dx\u2b1dz, \n    calc S\u2b1dx\u2b1dy\u2b1dz = x\u2b1dz\u2b1d(y\u2b1dz) : h\u2081\n             ... = D\u2b1dy\u2b1dx\u2b1dz   : by rw \u2190r_D\n    ,\n    have h\u2083 := h\u2082,\n    rw \u2190cong_app at h\u2083, cases h\u2083 with h\u2083 r,                    --how can we remove r?\n    have h\u2084 : S\u2b1dx\u2b1dy = D\u2b1d(K\u2b1dx)\u2b1dD\u2b1dy,\n    calc S\u2b1dx\u2b1dy = D\u2b1dy\u2b1dx        : h\u2083\n           ... = D\u2b1dy\u2b1d(K\u2b1dx\u2b1dy)  : by conv {to_lhs, rw \u2190@r_K x y} --why is the \"conv to_lhs\" necessary?\n           ... = D\u2b1d(K\u2b1dx)\u2b1dD\u2b1dy  : by rw \u2190r_D\n    ,\n    have h\u2085 := h\u2084,\n    rw \u2190cong_app at h\u2085, cases h\u2085 with h\u2085 r,\n    have h\u2086 : S\u2b1dx = D\u2b1d(K\u2b1dD)\u2b1d(D\u2b1dK\u2b1d(K\u2b1dD))\u2b1dx,\n    calc S\u2b1dx = D\u2b1d(K\u2b1dx)\u2b1dD             : h\u2085\n         ... = (K\u2b1dD\u2b1dx)\u2b1d(K\u2b1dx)\u2b1d(K\u2b1dD\u2b1dx) : by conv {to_lhs, rw \u2190@r_K D x}\n         ... = D\u2b1dK\u2b1d(K\u2b1dD)\u2b1dx\u2b1d(K\u2b1dD\u2b1dx)   : by rw \u2190r_D\n         ... = D\u2b1d(K\u2b1dD)\u2b1d(D\u2b1dK\u2b1d(K\u2b1dD))\u2b1dx : by rw \u2190r_D\n    ,\n    have h\u2087 := h\u2086,\n    rw \u2190cong_app at h\u2087, cases h\u2087 with h\u2087 r,\n    calc   S = D\u2b1d(K\u2b1dD)\u2b1d(D\u2b1dK\u2b1d(K\u2b1dD)) : h\u2087\n         ... = d (K\u2b1dD)\u2b1d(d K\u2b1d(K\u2b1dD)) : by repeat {rw \u2190d_eq_r_D}\n    ,\n  end\n\n  def S := d (K\u2b1dD) \u2b1d (d K \u2b1d (K\u2b1dD))\n  @[simp] theorem r_S : S\u2b1dx\u2b1dy\u2b1dz = x\u2b1dz\u2b1d(y\u2b1dz) := by simp [S]\n\n  --define associated functions\n  namespace natree\n\n    def iterate : \ud835\udd4b \u2192 \ud835\udd4b \u2192 \u2115 \u2192 \ud835\udd4b\n    | t\u2081 t\u2082 0 := t\u2082\n    | t\u2081 t\u2082 (n+1) := t\u2081 \u2b1d iterate t\u2081 t\u2082 n\n\n    def from_nat : \u2115 \u2192 \ud835\udd4b\n    | n := iterate \u25b3 \u25b3 n\n\n    def reduce : \ud835\udd4b \u2192 option \ud835\udd4b\n    | (\u25b3\u2b1d\u25b3\u2b1dy\u2b1dz) := some y\n    | (\u25b3\u2b1d(\u25b3\u2b1dx)\u2b1dy\u2b1dz) := some ((y\u2b1dz)\u2b1dx\u2b1dz)\n    | (\u25b3\u2b1d(\u25b3\u2b1dw\u2b1dx)\u2b1dy\u2b1dz) := some (z\u2b1dw\u2b1dx)\n    | _ := none\n\n    def depth : \ud835\udd4b \u2192 \u2115\n    | (t\u2081 \u2b1d t\u2082) := max t\u2081.depth t\u2082.depth + 1\n    | _ := 0\n\n    lemma depth_well_founded (h : z = x \u2b1d y) : x.depth < z.depth \u2227 y.depth < z.depth :=\n    begin\n      split;\n      conv\n      begin\n        to_rhs,\n        rw [h, depth],\n      end;\n      apply lt_of_le_of_lt,\n        exact le_max_left x.depth y.depth,\n        apply lt_add_one,\n      exact le_max_right x.depth y.depth,\n      apply lt_add_one,\n    end\n\n    def step : \ud835\udd4b \u2192 option \ud835\udd4b\n    | t := \n      match reduce t with\n      | some t' := some t'\n      | none :=\n        begin\n          cases h\u2081 : t with _ t\u2081 t\u2082,\n          exact none, exact none, --no reduction in \"node\" and \"ref\" cases\n          have h\u2082 := depth_well_founded h\u2081, cases h\u2082, --establish safe recursion hypotheses\n          exact (\n            match step t\u2081 with\n            | some t\u2081' := some (t\u2081' \u2b1d t\u2082)\n            | none :=\n              match step t\u2082 with\n              | some t\u2082' := some (t\u2081 \u2b1d t\u2082')\n              | none := none\n              end\n            end\n          ),\n        end\n      end\n    --use \"depth\" function in well-founded recursion checking\n    using_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf depth\u27e9]}\n\n    def normalize : \u2115 \u2192 \ud835\udd4b \u2192 \ud835\udd4b\n    | 0 t := t\n    | (n+1) t := \n      match step t with\n      | some t' := normalize n t'\n      | none := t\n      end\n\n  end natree\n\nend chapter3", "meta": {"author": "smudgecat123", "repo": "tree_calculus", "sha": "aa2c67898c0a44f011d39cb52e45edf5ba6c3b0c", "save_path": "github-repos/lean/smudgecat123-tree_calculus", "path": "github-repos/lean/smudgecat123-tree_calculus/tree_calculus-aa2c67898c0a44f011d39cb52e45edf5ba6c3b0c/src/other/chapter3_old.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3752009584325573}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.functor.hom\nimport category_theory.functor.currying\nimport category_theory.products.basic\n\n/-!\n# The Yoneda embedding\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThe Yoneda embedding as a functor `yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081)`,\nalong with an instance that it is `fully_faithful`.\n\nAlso the Yoneda lemma, `yoneda_lemma : (yoneda_pairing C) \u2245 (yoneda_evaluation C)`.\n\n## References\n* [Stacks: Opposite Categories and the Yoneda Lemma](https://stacks.math.columbia.edu/tag/001L)\n-/\n\nnamespace category_theory\nopen opposite\n\nuniverses v\u2081 u\u2081 u\u2082-- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n\n/--\nThe Yoneda embedding, as a functor from `C` into presheaves on `C`.\n\nSee <https://stacks.math.columbia.edu/tag/001O>.\n-/\n@[simps]\ndef yoneda : C \u2964 (C\u1d52\u1d56 \u2964 Type v\u2081) :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, unop Y \u27f6 X,\n    map := \u03bb Y Y' f g, f.unop \u226b g,\n    map_comp' := \u03bb _ _ _ f g, begin ext, dsimp, erw [category.assoc] end,\n    map_id' := \u03bb Y, begin ext, dsimp, erw [category.id_comp] end },\n  map := \u03bb X X' f, { app := \u03bb Y g, g \u226b f } }\n\n/--\nThe co-Yoneda embedding, as a functor from `C\u1d52\u1d56` into co-presheaves on `C`.\n-/\n@[simps] def coyoneda : C\u1d52\u1d56 \u2964 (C \u2964 Type v\u2081) :=\n{ obj := \u03bb X,\n  { obj := \u03bb Y, unop X \u27f6 Y,\n    map := \u03bb Y Y' f g, g \u226b f },\n  map := \u03bb X X' f, { app := \u03bb Y g, f.unop \u226b g } }\n\nnamespace yoneda\n\nlemma obj_map_id {X Y : C} (f : op X \u27f6 op Y) :\n  (yoneda.obj X).map f (\ud835\udfd9 X) = (yoneda.map f.unop).app (op Y) (\ud835\udfd9 Y) :=\nby { dsimp, simp }\n\n@[simp] lemma naturality {X Y : C} (\u03b1 : yoneda.obj X \u27f6 yoneda.obj Y)\n  {Z Z' : C} (f : Z \u27f6 Z') (h : Z' \u27f6 X) : f \u226b \u03b1.app (op Z') h = \u03b1.app (op Z) (f \u226b h) :=\n(functor_to_types.naturality _ _ \u03b1 f.op h).symm\n\n/--\nThe Yoneda embedding is full.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ninstance yoneda_full : full (yoneda : C \u2964 C\u1d52\u1d56 \u2964 Type v\u2081) :=\n{ preimage := \u03bb X Y f, f.app (op X) (\ud835\udfd9 X) }\n\n/--\nThe Yoneda embedding is faithful.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ninstance yoneda_faithful : faithful (yoneda : C \u2964 C\u1d52\u1d56 \u2964 Type v\u2081) :=\n{ map_injective' := \u03bb X Y f g p, by convert (congr_fun (congr_app p (op X)) (\ud835\udfd9 X)); dsimp; simp }\n\n/-- Extensionality via Yoneda. The typical usage would be\n```\n-- Goal is `X \u2245 Y`\napply yoneda.ext,\n-- Goals are now functions `(Z \u27f6 X) \u2192 (Z \u27f6 Y)`, `(Z \u27f6 Y) \u2192 (Z \u27f6 X)`, and the fact that these\nfunctions are inverses and natural in `Z`.\n```\n-/\ndef ext (X Y : C)\n  (p : \u03a0 {Z : C}, (Z \u27f6 X) \u2192 (Z \u27f6 Y)) (q : \u03a0 {Z : C}, (Z \u27f6 Y) \u2192 (Z \u27f6 X))\n  (h\u2081 : \u03a0 {Z : C} (f : Z \u27f6 X), q (p f) = f) (h\u2082 : \u03a0 {Z : C} (f : Z \u27f6 Y), p (q f) = f)\n  (n : \u03a0 {Z Z' : C} (f : Z' \u27f6 Z) (g : Z \u27f6 X), p (f \u226b g) = f \u226b p g) : X \u2245 Y :=\nyoneda.preimage_iso (nat_iso.of_components (\u03bb Z, { hom := p, inv := q, }) (by tidy))\n\n/--\nIf `yoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C} (f : X \u27f6 Y) [is_iso (yoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful yoneda f\n\nend yoneda\n\nnamespace coyoneda\n\n@[simp] lemma naturality {X Y : C\u1d52\u1d56} (\u03b1 : coyoneda.obj X \u27f6 coyoneda.obj Y)\n  {Z Z' : C} (f : Z' \u27f6 Z) (h : unop X \u27f6 Z') : (\u03b1.app Z' h) \u226b f = \u03b1.app Z (h \u226b f) :=\n(functor_to_types.naturality _ _ \u03b1 f h).symm\n\ninstance coyoneda_full : full (coyoneda : C\u1d52\u1d56 \u2964 C \u2964 Type v\u2081) :=\n{ preimage := \u03bb X Y f, (f.app _ (\ud835\udfd9 X.unop)).op }\n\ninstance coyoneda_faithful : faithful (coyoneda : C\u1d52\u1d56 \u2964 C \u2964 Type v\u2081) :=\n{ map_injective' := \u03bb X Y f g p,\n  begin\n    have t := congr_fun (congr_app p X.unop) (\ud835\udfd9 _),\n    simpa using congr_arg quiver.hom.op t,\n  end }\n\n/--\nIf `coyoneda.map f` is an isomorphism, so was `f`.\n-/\nlemma is_iso {X Y : C\u1d52\u1d56} (f : X \u27f6 Y) [is_iso (coyoneda.map f)] : is_iso f :=\nis_iso_of_fully_faithful coyoneda f\n\n/-- The identity functor on `Type` is isomorphic to the coyoneda functor coming from `punit`. -/\ndef punit_iso : coyoneda.obj (opposite.op punit) \u2245 \ud835\udfed (Type v\u2081) :=\nnat_iso.of_components\n  (\u03bb X, { hom := \u03bb f, f \u27e8\u27e9, inv := \u03bb x _, x })\n  (by tidy)\n\n/-- Taking the `unop` of morphisms is a natural isomorphism. -/\n@[simps] def obj_op_op (X : C) : coyoneda.obj (op (op X)) \u2245 yoneda.obj X :=\nnat_iso.of_components (\u03bb Y, (op_equiv _ _).to_iso) (\u03bb X Y f, rfl)\n\nend coyoneda\n\nnamespace functor\n\n\n/--\nA functor `F : C\u1d52\u1d56 \u2964 Type v\u2081` is representable if there is object `X` so `F \u2245 yoneda.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001Q>.\n-/\nclass representable (F : C\u1d52\u1d56 \u2964 Type v\u2081) : Prop :=\n(has_representation : \u2203 X (f : yoneda.obj X \u27f6 F), is_iso f)\n\ninstance {X : C} : representable (yoneda.obj X) :=\n{ has_representation := \u27e8X, \ud835\udfd9 _, infer_instance\u27e9 }\n\n/--\nA functor `F : C \u2964 Type v\u2081` is corepresentable if there is object `X` so `F \u2245 coyoneda.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001Q>.\n-/\nclass corepresentable (F : C \u2964 Type v\u2081) : Prop :=\n(has_corepresentation : \u2203 X (f : coyoneda.obj X \u27f6 F), is_iso f)\n\ninstance {X : C\u1d52\u1d56} : corepresentable (coyoneda.obj X) :=\n{ has_corepresentation := \u27e8X, \ud835\udfd9 _, infer_instance\u27e9 }\n\n-- instance : corepresentable (\ud835\udfed (Type v\u2081)) :=\n-- corepresentable_of_nat_iso (op punit) coyoneda.punit_iso\n\nsection representable\nvariables (F : C\u1d52\u1d56 \u2964 Type v\u2081)\nvariable [F.representable]\n\n/-- The representing object for the representable functor `F`. -/\nnoncomputable def repr_X : C :=\n(representable.has_representation : \u2203 X (f : _ \u27f6 F), _).some\n\n/-- The (forward direction of the) isomorphism witnessing `F` is representable. -/\nnoncomputable def repr_f : yoneda.obj F.repr_X \u27f6 F :=\nrepresentable.has_representation.some_spec.some\n\n/--\nThe representing element for the representable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def repr_x : F.obj (op F.repr_X) :=\nF.repr_f.app (op F.repr_X) (\ud835\udfd9 F.repr_X)\n\ninstance : is_iso F.repr_f :=\nrepresentable.has_representation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(-, F.repr_X)`.  Note the components\n`F.repr_w.app X` definitionally have type `(X.unop \u27f6 F.repr_X) \u2245 F.obj X`.\n-/\nnoncomputable def repr_w : yoneda.obj F.repr_X \u2245 F := as_iso F.repr_f\n\n@[simp] lemma repr_w_hom : F.repr_w.hom = F.repr_f := rfl\n\nlemma repr_w_app_hom (X : C\u1d52\u1d56) (f : unop X \u27f6 F.repr_X) :\n  (F.repr_w.app X).hom f = F.map f.op F.repr_x :=\nbegin\n  change F.repr_f.app X f = (F.repr_f.app (op F.repr_X) \u226b F.map f.op) (\ud835\udfd9 F.repr_X),\n  rw \u2190F.repr_f.naturality,\n  dsimp,\n  simp\nend\n\nend representable\n\nsection corepresentable\n\nvariables (F : C \u2964 Type v\u2081)\nvariable [F.corepresentable]\n\n/-- The representing object for the corepresentable functor `F`. -/\nnoncomputable def corepr_X : C :=\n(corepresentable.has_corepresentation : \u2203 X (f : _ \u27f6 F), _).some.unop\n\n/-- The (forward direction of the) isomorphism witnessing `F` is corepresentable. -/\nnoncomputable def corepr_f : coyoneda.obj (op F.corepr_X) \u27f6 F :=\ncorepresentable.has_corepresentation.some_spec.some\n\n/--\nThe representing element for the corepresentable functor `F`, sometimes called the universal\nelement of the functor.\n-/\nnoncomputable def corepr_x : F.obj F.corepr_X :=\nF.corepr_f.app F.corepr_X (\ud835\udfd9 F.corepr_X)\n\ninstance : is_iso F.corepr_f :=\ncorepresentable.has_corepresentation.some_spec.some_spec\n\n/--\nAn isomorphism between `F` and a functor of the form `C(F.corepr X, -)`. Note the components\n`F.corepr_w.app X` definitionally have type `F.corepr_X \u27f6 X \u2245 F.obj X`.\n-/\nnoncomputable def corepr_w : coyoneda.obj (op F.corepr_X) \u2245 F := as_iso F.corepr_f\n\nlemma corepr_w_app_hom (X : C) (f : F.corepr_X \u27f6 X) :\n  (F.corepr_w.app X).hom f = F.map f F.corepr_x :=\nbegin\n  change F.corepr_f.app X f = (F.corepr_f.app F.corepr_X \u226b F.map f) (\ud835\udfd9 F.corepr_X),\n  rw \u2190F.corepr_f.naturality,\n  dsimp,\n  simp\nend\n\nend corepresentable\n\nend functor\n\nlemma representable_of_nat_iso (F : C\u1d52\u1d56 \u2964 Type v\u2081) {G} (i : F \u2245 G) [F.representable] :\n  G.representable :=\n{ has_representation := \u27e8F.repr_X, F.repr_f \u226b i.hom, infer_instance\u27e9 }\n\nlemma corepresentable_of_nat_iso (F : C \u2964 Type v\u2081) {G} (i : F \u2245 G) [F.corepresentable] :\n  G.corepresentable :=\n{ has_corepresentation := \u27e8op F.corepr_X, F.corepr_f \u226b i.hom, infer_instance\u27e9 }\n\ninstance : functor.corepresentable (\ud835\udfed (Type v\u2081)) :=\ncorepresentable_of_nat_iso (coyoneda.obj (op punit)) coyoneda.punit_iso\n\nopen opposite\n\nvariables (C)\n\n-- We need to help typeclass inference with some awkward universe levels here.\ninstance prod_category_instance_1 : category ((C\u1d52\u1d56 \u2964 Type v\u2081) \u00d7 C\u1d52\u1d56) :=\ncategory_theory.prod.{(max u\u2081 v\u2081) v\u2081} (C\u1d52\u1d56 \u2964 Type v\u2081) C\u1d52\u1d56\n\ninstance prod_category_instance_2 : category (C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) :=\ncategory_theory.prod.{v\u2081 (max u\u2081 v\u2081)} C\u1d52\u1d56 (C\u1d52\u1d56 \u2964 Type v\u2081)\n\nopen yoneda\n\n/--\nThe \"Yoneda evaluation\" functor, which sends `X : C\u1d52\u1d56` and `F : C\u1d52\u1d56 \u2964 Type`\nto `F.obj X`, functorially in both `X` and `F`.\n-/\ndef yoneda_evaluation : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nevaluation_uncurried C\u1d52\u1d56 (Type v\u2081) \u22d9 ulift_functor.{u\u2081}\n\n@[simp] lemma yoneda_evaluation_map_down\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (x : (yoneda_evaluation C).obj P) :\n  ((yoneda_evaluation C).map \u03b1 x).down = \u03b1.2.app Q.1 (P.2.map \u03b1.1 x.down) := rfl\n\n/--\nThe \"Yoneda pairing\" functor, which sends `X : C\u1d52\u1d56` and `F : C\u1d52\u1d56 \u2964 Type`\nto `yoneda.op.obj X \u27f6 F`, functorially in both `X` and `F`.\n-/\ndef yoneda_pairing : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081) \u2964 Type (max u\u2081 v\u2081) :=\nfunctor.prod yoneda.op (\ud835\udfed (C\u1d52\u1d56 \u2964 Type v\u2081)) \u22d9 functor.hom (C\u1d52\u1d56 \u2964 Type v\u2081)\n\n@[simp] lemma yoneda_pairing_map\n  (P Q : C\u1d52\u1d56 \u00d7 (C\u1d52\u1d56 \u2964 Type v\u2081)) (\u03b1 : P \u27f6 Q) (\u03b2 : (yoneda_pairing C).obj P) :\n  (yoneda_pairing C).map \u03b1 \u03b2 = yoneda.map \u03b1.1.unop \u226b \u03b2 \u226b \u03b1.2 := rfl\n\n/--\nThe Yoneda lemma asserts that that the Yoneda pairing\n`(X : C\u1d52\u1d56, F : C\u1d52\u1d56 \u2964 Type) \u21a6 (yoneda.obj (unop X) \u27f6 F)`\nis naturally isomorphic to the evaluation `(X, F) \u21a6 F.obj X`.\n\nSee <https://stacks.math.columbia.edu/tag/001P>.\n-/\ndef yoneda_lemma : yoneda_pairing C \u2245 yoneda_evaluation C :=\n{ hom :=\n  { app := \u03bb F x, ulift.up ((x.app F.1) (\ud835\udfd9 (unop F.1))),\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      erw [category.id_comp, \u2190functor_to_types.naturality],\n      simp only [category.comp_id, yoneda_obj_map],\n    end },\n  inv :=\n  { app := \u03bb F x,\n    { app := \u03bb X a, (F.2.map a.op) x.down,\n      naturality' :=\n      begin\n        intros X Y f, ext, dsimp,\n        rw [functor_to_types.map_comp_apply]\n      end },\n    naturality' :=\n    begin\n      intros X Y f, ext, dsimp,\n      rw [\u2190functor_to_types.naturality, functor_to_types.map_comp_apply]\n    end },\n  hom_inv_id' :=\n  begin\n    ext, dsimp,\n    erw [\u2190functor_to_types.naturality,\n         obj_map_id],\n    simp only [yoneda_map_app, quiver.hom.unop_op],\n    erw [category.id_comp],\n  end,\n  inv_hom_id' :=\n  begin\n    ext, dsimp,\n    rw [functor_to_types.map_id_apply]\n  end }.\n\nvariables {C}\n\n/--\nThe isomorphism between `yoneda.obj X \u27f6 F` and `F.obj (op X)`\n(we need to insert a `ulift` to get the universes right!)\ngiven by the Yoneda lemma.\n-/\n@[simps] def yoneda_sections (X : C) (F : C\u1d52\u1d56 \u2964 Type v\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 ulift.{u\u2081} (F.obj (op X)) :=\n(yoneda_lemma C).app (op X, F)\n\n/--\nWe have a type-level equivalence between natural transformations from the yoneda embedding\nand elements of `F.obj X`, without any universe switching.\n-/\ndef yoneda_equiv {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} : (yoneda.obj X \u27f6 F) \u2243 F.obj (op X) :=\n(yoneda_sections X F).to_equiv.trans equiv.ulift\n\n@[simp]\nlemma yoneda_equiv_apply {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (f : yoneda.obj X \u27f6 F) :\n  yoneda_equiv f = f.app (op X) (\ud835\udfd9 X) :=\nrfl\n\n@[simp]\nlemma yoneda_equiv_symm_app_apply {X : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (x : F.obj (op X))\n  (Y : C\u1d52\u1d56) (f : Y.unop \u27f6 X) :\n  (yoneda_equiv.symm x).app Y f = F.map f.op x :=\nrfl\n\nlemma yoneda_equiv_naturality {X Y : C} {F : C\u1d52\u1d56 \u2964 Type v\u2081} (f : yoneda.obj X \u27f6 F) (g : Y \u27f6 X) :\n  F.map g.op (yoneda_equiv f) = yoneda_equiv (yoneda.map g \u226b f) :=\nbegin\n  change (f.app (op X) \u226b F.map g.op) (\ud835\udfd9 X) = f.app (op Y) (\ud835\udfd9 Y \u226b g),\n  rw \u2190f.naturality,\n  dsimp,\n  simp,\nend\n\n/--\nWhen `C` is a small category, we can restate the isomorphism from `yoneda_sections`\nwithout having to change universes.\n-/\ndef yoneda_sections_small {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) :\n  (yoneda.obj X \u27f6 F) \u2245 F.obj (op X) :=\nyoneda_sections X F \u226a\u226b ulift_trivial _\n\n@[simp]\nlemma yoneda_sections_small_hom {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) (f : yoneda.obj X \u27f6 F) :\n  (yoneda_sections_small X F).hom f = f.app _ (\ud835\udfd9 _) :=\nrfl\n\n@[simp]\nlemma yoneda_sections_small_inv_app_apply {C : Type u\u2081} [small_category C] (X : C)\n  (F : C\u1d52\u1d56 \u2964 Type u\u2081) (t : F.obj (op X)) (Y : C\u1d52\u1d56) (f : Y.unop \u27f6 X) :\n  ((yoneda_sections_small X F).inv t).app Y f = F.map f.op t :=\nrfl\n\nlocal attribute [ext] functor.ext\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma {C : Type u\u2081} [small_category C] :\n  (yoneda.op \u22d9 coyoneda : C\u1d52\u1d56 \u2964 (C\u1d52\u1d56 \u2964 Type u\u2081) \u2964 Type u\u2081) \u2245 evaluation C\u1d52\u1d56 (Type u\u2081) :=\neq_to_iso (by tidy) \u226a\u226b curry.map_iso (yoneda_lemma C \u226a\u226b\n  iso_whisker_left (evaluation_uncurried C\u1d52\u1d56 (Type u\u2081)) ulift_functor_trivial) \u226a\u226b\n    eq_to_iso (by tidy)\n\n/-- The curried version of yoneda lemma when `C` is small. -/\ndef curried_yoneda_lemma' {C : Type u\u2081} [small_category C] :\n  yoneda \u22d9 (whiskering_left C\u1d52\u1d56 (C\u1d52\u1d56 \u2964 Type u\u2081)\u1d52\u1d56 (Type u\u2081)).obj yoneda.op \u2245 \ud835\udfed (C\u1d52\u1d56 \u2964 Type u\u2081) :=\neq_to_iso (by tidy) \u226a\u226b curry.map_iso (iso_whisker_left (prod.swap _ _)\n  (yoneda_lemma C \u226a\u226b iso_whisker_left\n    (evaluation_uncurried C\u1d52\u1d56 (Type u\u2081)) ulift_functor_trivial : _)) \u226a\u226b eq_to_iso (by tidy)\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417487156366, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3752009507935019}}
{"text": "/-\nCopyright (c) 2023 Wojciech Nawrocki. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wojciech Nawrocki\n-/\n\nimport ProofChecker.Model.PropForm\nimport ProofChecker.Model.PropVars\n\nimport ProofChecker.Data.HashMap.Lemmas\nimport ProofChecker.Data.ICnf\n\n/-! Clause database together with some (provably correct) methods. For example, we can conclude\nthat if a clause follows from the current database by unit propagation, then it is implied by the\ndatabase's interpretation as a propositional formula. -/\n\n/-- A stateful clause database, i.e. a dynamically modifiable CNF, for use in poly-time proof\ncheckers such as for LRAT. It uses in-place data structures, so should be used linearly.\n\n(Persistent structures do not seem immediately helpful as linear formats do not backtrack.)\n\nIn `ClauseDb \u03b1`, `\u03b1` is the type of clause indices. -/\nstructure ClauseDb (\u03b1 : Type) [BEq \u03b1] [Hashable \u03b1] where\n  /-- Each clause is stored together with a flag indicating whether it has been deleted.\n  Deleted clauses are logically not in the database. -/\n  clauses : HashMap \u03b1 (IClause \u00d7 Bool) := {}\n\nnamespace HashMap\n\nvariable [BEq \u03b1] [Hashable \u03b1]\n\ndef mapOne (m : HashMap \u03b1 \u03b2) (idx : \u03b1) (f : \u03b2 \u2192 \u03b2) : HashMap \u03b1 \u03b2 :=\n  match m.find? idx with\n  | some b => m.insert idx (f b)\n  | none => m\n\nend HashMap\n\ninductive UnitPropResult (\u03b1 : Type) where\n  | contradiction\n  /-- The hint did not become unit. -/\n  | hintNotUnit (hint : \u03b1)\n  /-- The hint points at a nonexistent clause. -/\n  | hintNonexistent (hint : \u03b1)\n  | extended (\u03c4 : PartPropAssignment)\n\nnamespace UnitPropResult\n\ndef isContradiction (r : UnitPropResult \u03b1) : Bool :=\n  r matches contradiction\n\nend UnitPropResult\n\nnamespace ClauseDb\n\nvariable {\u03b1 : Type} [BEq \u03b1] [Hashable \u03b1]\n\ninstance [ToString \u03b1] : ToString (ClauseDb \u03b1) where\n  toString db := toString db.clauses.toList\n\ndef empty : ClauseDb \u03b1 := { clauses := .empty }\n\ndef fold (db : ClauseDb \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 IClause \u2192 \u03b2) (init : \u03b2) : \u03b2 :=\n  db.clauses.fold (init := init) fun acc idx (C, deleted) =>\n    if deleted then acc else f acc idx C\n\ndef foldM [Monad m] (db : ClauseDb \u03b1) (f : \u03b2 \u2192 \u03b1 \u2192 IClause \u2192 m \u03b2) (init : \u03b2) : m \u03b2 :=\n  db.clauses.foldM (init := init) fun acc idx (C, deleted) =>\n    if deleted then pure acc else f acc idx C\n\ndef addClause (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) : ClauseDb \u03b1 :=\n  { db with clauses := db.clauses.insert idx (C, false) }\n\ndef delClause (db : ClauseDb \u03b1) (idx : \u03b1) : ClauseDb \u03b1 :=\n  { db with clauses := db.clauses.mapOne idx fun (C, _) => (C, true) }\n\ndef getClause (db : ClauseDb \u03b1) (idx : \u03b1) : Option IClause :=\n  db.clauses.find? idx |>.bind (fun (C, deleted) => if deleted then none else C)\n\ndef contains (db : ClauseDb \u03b1) (idx : \u03b1) : Bool :=\n  db.getClause idx |>.isSome\n\n/-- NOTE: This implementation is not efficient as it doesn't use early return. -/\ndef all (db : ClauseDb \u03b1) (p : \u03b1 \u2192 IClause \u2192 Bool) : Bool :=\n  db.fold (fun acc idx C => acc && p idx C) true\n\n/-- NOTE: This implementation is not efficient as it doesn't use early return. -/\ndef any (db : ClauseDb \u03b1) (p : \u03b1 \u2192 IClause \u2192 Bool) : Bool :=\n  !db.all (fun idx C => !p idx C)\n\n/-- Initialize a clause database from a CNF array. -/\ndef ofICnf (cnf : ICnf) : ClauseDb Nat :=\n  let (db, _) := cnf.foldl (init := (empty, 1)) fun (db, idx) C =>\n    (db.addClause idx C, idx + 1)\n  db\n\n@[deprecated]\ndef unitPropWithHints (db : ClauseDb \u03b1) (\u03c4 : PartPropAssignment) (hints : Array \u03b1)\n    : UnitPropResult \u03b1 := Id.run do\n  let mut \u03c4 := \u03c4\n  for hint in hints do\n    let some C := db.getClause hint\n      | return .hintNonexistent hint\n    match C.reduce \u03c4 with\n    | some #[u] => \u03c4 := \u03c4.insert u.var u.polarity\n    | some #[] => return .contradiction\n    | _ => return .hintNotUnit hint\n  return .extended \u03c4\n\n/-! Theorems about `ClauseDb` -/\n\nvariable [LawfulBEq \u03b1] [HashMap.LawfulHashable \u03b1]\n\n/-! `getClause` -/\n\ntheorem getClause_eq_some (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) :\n    db.getClause idx = some C \u2194 db.clauses.find? idx = some (C, false) := by\n  simp [getClause]\n\n@[simp]\ntheorem getClause_empty (idx : \u03b1) : (empty : ClauseDb \u03b1).getClause idx = none := by\n  simp [getClause, empty]\n\ntheorem getClause_addClause (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) :\n    (db.addClause idx C).getClause idx = some C := by\n  dsimp [getClause, addClause]\n  rw [HashMap.find?_insert _ _ (LawfulBEq.rfl)]\n  simp\n\ntheorem getClause_addClause_of_ne (db : ClauseDb \u03b1) (idx idx' : \u03b1) (C : IClause) :\n    idx \u2260 idx' \u2192 (db.addClause idx C).getClause idx' = db.getClause idx' := by\n  intro h\n  dsimp [addClause, getClause]\n  rw [HashMap.find?_insert_of_ne _ _ (bne_iff_ne idx idx' |>.mpr h)]\n\ntheorem getClause_delClause (db : ClauseDb \u03b1) (idx : \u03b1) :\n    (db.delClause idx).getClause idx = none := by\n  dsimp [getClause, delClause, HashMap.mapOne]\n  split\n  next =>\n    rw [HashMap.find?_insert _ _ (LawfulBEq.rfl)]\n    simp\n  next h =>\n    simp [h]\n\ntheorem getClause_delClause_of_ne (db : ClauseDb \u03b1) (idx idx' : \u03b1) :\n    idx \u2260 idx' \u2192 (db.delClause idx).getClause idx' = db.getClause idx' := by\n  intro h\n  dsimp [getClause, delClause, HashMap.mapOne]\n  split\n  next =>\n    rw [HashMap.find?_insert_of_ne _ _ (bne_iff_ne _ _ |>.mpr h)]\n  next => rfl\n\n/-! `contains` -/\n\ntheorem contains_iff_getClause_eq_some (db : ClauseDb \u03b1) (idx : \u03b1) :\n    db.contains idx \u2194 \u2203 C, db.getClause idx = some C := by\n  simp [contains, Option.isSome_iff_exists, db.clauses.contains_iff]\n\n@[simp]\ntheorem not_contains_empty (idx : \u03b1) : (empty : ClauseDb \u03b1).contains idx = false := by\n  have := contains_iff_getClause_eq_some empty idx\n  simp_all\n\ntheorem contains_addClause (db : ClauseDb \u03b1) (idx idx' : \u03b1) (C : IClause) :\n    (db.addClause idx C).contains idx' \u2194 (db.contains idx' \u2228 idx = idx') := by\n  simp only [contains_iff_getClause_eq_some]\n  refine \u27e8?mp, fun h => h.elim ?mpr\u2081 ?mpr\u2082\u27e9\n  case mp =>\n    intro \u27e8C, hGet\u27e9\n    by_cases hEq : idx = idx' <;>\n      aesop (add norm getClause_addClause_of_ne)\n  case mpr\u2081 =>\n    intro \u27e8C, hGet\u27e9\n    by_cases hEq : idx = idx' <;>\n      aesop (add norm getClause_addClause, norm getClause_addClause_of_ne)\n  case mpr\u2082 =>\n    aesop (add norm getClause_addClause)\n\ntheorem contains_delClause (db : ClauseDb \u03b1) (idx idx' : \u03b1) :\n    (db.delClause idx).contains idx' \u2194 (db.contains idx' \u2227 idx \u2260 idx') := by\n  simp only [contains_iff_getClause_eq_some]\n  refine \u27e8?mp, ?mpr\u27e9\n  case mp =>\n    intro \u27e8C, hGet\u27e9\n    have hEq : idx \u2260 idx' := fun h => by\n      rw [h, getClause_delClause] at hGet\n      cases hGet\n    rw [getClause_delClause_of_ne _ _ _ hEq] at hGet\n    simp [hGet, hEq]\n  case mpr =>\n    intro \u27e8\u27e8C, hGet\u27e9, hEq\u27e9\n    exact \u27e8C, hGet \u25b8 getClause_delClause_of_ne _ _ _ hEq\u27e9\n\n/-! `fold` -/\n\ntheorem fold_of_getClause_eq_some_of_comm (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause)\n    (f : \u03b2 \u2192 \u03b1 \u2192 IClause \u2192 \u03b2) (init : \u03b2) :\n    db.getClause idx = some C \u2192\n    (\u2200 b a\u2081 C\u2081 a\u2082 C\u2082, f (f b a\u2081 C\u2081) a\u2082 C\u2082 = f (f b a\u2082 C\u2082) a\u2081 C\u2081) \u2192\n    \u2203 b, db.fold f init = f b idx C := by\n  intro h hComm\n  rw [getClause_eq_some] at h\n  have \u27e8b, hb\u27e9 := db.clauses.fold_of_mapsTo_of_comm (init := init)\n    (f := fun acc idx (C, deleted) => if deleted then acc else f acc idx C)\n    h (by aesop)\n  use b\n  simp [fold, hb]\n\n/-! `all` -/\n\ntheorem all_true (db : ClauseDb \u03b1) (p : \u03b1 \u2192 IClause \u2192 Bool) :\n    db.all p \u2192 \u2200 idx C, db.getClause idx = some C \u2192 p idx C := by\n  dsimp [all]\n  intro hAll idx C hGet\n  have \u27e8b, hEq\u27e9 :=\n    fold_of_getClause_eq_some_of_comm db idx C (fun acc idx C => acc && p idx C) true\n      hGet ?comm\n  case comm =>\n    intros\n    simp only [Bool.and_assoc]\n    rw [Bool.and_comm (p _ _)]\n  simp_all\n  \ntheorem all_of_all_true (db : ClauseDb \u03b1) (p : \u03b1 \u2192 IClause \u2192 Bool) :\n    (\u2200 idx C, db.getClause idx = some C \u2192 p idx C) \u2192 db.all p := by\n  dsimp [all, fold, getClause]\n  intro\n  apply db.clauses.foldRecOn (C := fun b => b = true) (hInit := rfl)\n  simp_all\n  \n/-! `any` -/\n\ntheorem any_true (db : ClauseDb \u03b1) (p : \u03b1 \u2192 IClause \u2192 Bool) :\n    db.any p \u2192 \u2203 idx C, db.getClause idx = some C \u2227 p idx C = true := by\n  have := db.all_of_all_true (fun idx C => !p idx C)\n  dsimp [any]\n  exact not_imp_not.mp fun _ => by simp_all\n\n/-! `toPropTermSub` -/\n\nopen Classical PropTerm\n\n/-- Interpret the conjunction of a subset of the clauses as a Boolean function. -/\nnoncomputable def toPropTermSub (db : ClauseDb \u03b1) (idxs : Set \u03b1) : PropTerm Var :=\n  db.fold (init := \u22a4) fun acc idx C => if idx \u2208 idxs then acc \u2293 C.toPropTerm else acc\n\ntheorem toPropTermSub_of_getClause_eq_some (db : ClauseDb \u03b1) :\n    idx \u2208 idxs \u2192 db.getClause idx = some C \u2192 db.toPropTermSub idxs \u2264 C.toPropTerm := by\n  intro hMem hGet\n  have \u27e8\u03c6, h\u03c6\u27e9 := db.fold_of_getClause_eq_some_of_comm idx C\n    (init := \u22a4) (f := fun acc idx C => if idx \u2208 idxs then acc \u2293 C.toPropTerm else acc)\n    hGet ?comm\n  case comm =>\n    intros\n    dsimp\n    split_ifs <;> ac_rfl\n  apply PropTerm.entails_ext.mpr\n  rw [toPropTermSub, h\u03c6]\n  simp [hMem]\n\ntheorem satisfies_toPropTermSub (db : ClauseDb \u03b1) (idxs : Set \u03b1) (\u03c3 : PropAssignment Var) :\n    \u03c3 \u22a8 db.toPropTermSub idxs \u2194 \u2200 idx \u2208 idxs, \u2200 C, db.getClause idx = some C \u2192 \u03c3 \u22a8 C.toPropTerm :=\n  \u27e8mp, mpr\u27e9\nwhere\n  mp := fun h idx hMem C hGet =>\n    entails_ext.mp (toPropTermSub_of_getClause_eq_some db hMem hGet) _ h\n\n  mpr := fun h => by\n    dsimp [toPropTermSub]\n    apply HashMap.foldRecOn (hInit := satisfies_tr)\n    intro \u03c6 idx (C, deleted) h\u03c6 hFind\n    dsimp\n    split_ifs <;> try assumption\n    next hDel hMem =>\n      rw [satisfies_conj]\n      refine \u27e8by assumption, ?_\u27e9\n      apply h idx hMem\n      simp [getClause, hFind, hDel]\n\n@[simp]\ntheorem toPropTermSub_empty (idxs : Set \u03b1) : (empty : ClauseDb \u03b1).toPropTermSub idxs = \u22a4 := by\n  ext \u03c4\n  simp [satisfies_toPropTermSub]\n\n@[simp]\ntheorem toPropTermSub_emptySet (db : ClauseDb \u03b1) : db.toPropTermSub \u2205 = \u22a4 := by\n  ext \u03c4\n  aesop (add norm satisfies_toPropTermSub)\n\ntheorem toPropTermSub_subset (db : ClauseDb \u03b1) :\n    idxs \u2286 idxs' \u2192 db.toPropTermSub idxs' \u2264 db.toPropTermSub idxs := by\n  intro hSub\n  apply entails_ext.mpr\n  aesop (add norm satisfies_toPropTermSub)\n\ntheorem toPropTermSub_subset_eq (db : ClauseDb \u03b1) :\n    idxs \u2286 idxs' \u2192 (\u2200 idx \u2208 idxs', db.contains idx \u2192 idx \u2208 idxs) \u2192\n    db.toPropTermSub idxs' = db.toPropTermSub idxs := by\n  intro hSub h\n  apply le_antisymm (toPropTermSub_subset db hSub)\n  apply entails_ext.mpr\n  simp only [satisfies_toPropTermSub]\n  intro \u03c4 h\u03c4 _ hMem' _ hGet'\n  exact h\u03c4 _ (h _ hMem' (contains_iff_getClause_eq_some _ _ |>.mpr \u27e8_, hGet'\u27e9)) _ hGet'\n\ntheorem toPropTermSub_addClause (db : ClauseDb \u03b1) (idxs : Set \u03b1) (idx : \u03b1) (C : IClause) :\n    db.toPropTermSub idxs \u2293 C.toPropTerm \u2264 (db.addClause idx C).toPropTermSub idxs := by\n  apply entails_ext.mpr\n  simp only [satisfies_conj, satisfies_toPropTermSub]\n  intro \u03c4 h idx' C' hMem' hGet'\n  by_cases hEq : idx = idx' <;>\n    aesop (add norm getClause_addClause, norm getClause_addClause_of_ne)\n\ntheorem toPropTermSub_addClause_of_not_contains (db : ClauseDb \u03b1) (C : IClause) :\n    \u00acdb.contains idx \u2192 (db.addClause idx C).toPropTermSub idxs \u2264 db.toPropTermSub idxs := by\n  intro hContains\n  apply entails_ext.mpr\n  simp only [satisfies_toPropTermSub]\n  intro _ _ idx'\n  by_cases hEq : idx = idx' <;>\n    aesop (add norm contains_iff_getClause_eq_some, norm getClause_addClause_of_ne)\n\ntheorem toPropTermSub_addClause_eq (db : ClauseDb \u03b1) (C : IClause) :\n    idx \u2208 idxs \u2192 \u00acdb.contains idx \u2192\n    (db.addClause idx C).toPropTermSub idxs = db.toPropTermSub idxs \u2293 C.toPropTerm := by\n  intro hMem hContains\n  refine le_antisymm ?_ (toPropTermSub_addClause db idxs idx C)\n  apply le_inf (toPropTermSub_addClause_of_not_contains db C hContains)\n  apply toPropTermSub_of_getClause_eq_some _ hMem\n  apply getClause_addClause\n\ntheorem toPropTermSub_addClause_of_not_mem (db : ClauseDb \u03b1) (C : IClause) :\n    idx \u2209 idxs \u2192 (db.addClause idx C).toPropTermSub idxs = db.toPropTermSub idxs := by\n  intro hMem\n  ext \u03c4\n  simp only [satisfies_toPropTermSub]\n  constructor <;> {\n    intro h idx' hMem'\n    have : idx \u2260 idx' := fun h =>\n      hMem <| h \u25b8 hMem'\n    aesop (add norm getClause_addClause_of_ne)\n  }\n\ntheorem toPropTermSub_delClause (db : ClauseDb \u03b1) (idxs : Set \u03b1) (idx : \u03b1) :\n    db.toPropTermSub idxs \u2264 (db.delClause idx).toPropTermSub idxs := by\n  apply PropTerm.entails_ext.mpr\n  simp only [satisfies_toPropTermSub]\n  intro _ _ idx'\n  by_cases hEq : idx = idx' <;>\n    aesop (add norm getClause_delClause_of_ne, norm getClause_delClause)\n\ntheorem toPropTermSub_delClause_of_getClause_eq_some (db : ClauseDb \u03b1) :\n    db.getClause idx = some C \u2192\n    (db.delClause idx).toPropTermSub idxs \u2293 C.toPropTerm \u2264 db.toPropTermSub idxs := by\n  intro hGet\n  apply entails_ext.mpr\n  simp only [satisfies_conj, satisfies_toPropTermSub]\n  intro _ _ idx'\n  by_cases hEq : idx = idx' <;>\n    aesop (add norm getClause_delClause_of_ne)\n\ntheorem toPropTermSub_delClause_eq (db : ClauseDb \u03b1) :\n    idx \u2208 idxs \u2192 db.getClause idx = some C \u2192\n    (db.delClause idx).toPropTermSub idxs \u2293 C.toPropTerm = db.toPropTermSub idxs := by\n  intro hMem hGet\n  apply le_antisymm (toPropTermSub_delClause_of_getClause_eq_some db hGet)\n  apply le_inf (toPropTermSub_delClause db idxs idx)\n  apply toPropTermSub_of_getClause_eq_some _ hMem hGet\n\ntheorem toPropTermSub_delClause_of_not_mem (db : ClauseDb \u03b1) :\n    idx \u2209 idxs \u2192 (db.delClause idx).toPropTermSub idxs = db.toPropTermSub idxs := by\n  intro hMem\n  ext \u03c4\n  simp only [satisfies_toPropTermSub]\n  constructor <;> {\n    intro h idx' hMem'\n    have : idx \u2260 idx' := fun h =>\n      hMem <| h \u25b8 hMem'\n    aesop (add norm getClause_delClause_of_ne)\n  }\n\n/-! `toPropTerm` -/\n\n/-- Interpret the conjuction of all the clauses as a Boolean function. -/\nnoncomputable def toPropTerm (db : ClauseDb \u03b1) : PropTerm Var :=\n  db.toPropTermSub Set.univ\n\ntheorem toPropTerm_of_getClause_eq_some (db : ClauseDb \u03b1) :\n    db.getClause idx = some C \u2192 db.toPropTerm \u2264 C.toPropTerm :=\n  toPropTermSub_of_getClause_eq_some db (Set.mem_univ idx)\n\nopen PropTerm in\ntheorem satisfies_toPropTerm (db : ClauseDb \u03b1) (\u03c3 : PropAssignment Var) :\n    \u03c3 \u22a8 db.toPropTerm \u2194 \u2200 idx C, db.getClause idx = some C \u2192 \u03c3 \u22a8 C.toPropTerm :=\n  have \u27e8mp, mpr\u27e9 := satisfies_toPropTermSub db Set.univ \u03c3\n  \u27e8fun h idx C hGet => mp h idx (Set.mem_univ idx) C hGet,\n   fun h => mpr (fun idx _ C hGet => h idx C hGet)\u27e9\n\ntheorem toPropTerm_subset (db : ClauseDb \u03b1) (idxs : Set \u03b1) :\n    db.toPropTerm \u2264 db.toPropTermSub idxs :=\n  toPropTermSub_subset db (Set.subset_univ idxs)\n\n@[simp]\ntheorem toPropTerm_empty : (empty : ClauseDb \u03b1).toPropTerm = \u22a4 :=\n  toPropTermSub_empty Set.univ\n\ntheorem toPropTerm_addClause (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) :\n    db.toPropTerm \u2293 C.toPropTerm \u2264 (db.addClause idx C).toPropTerm :=\n  toPropTermSub_addClause db Set.univ idx C\n\ntheorem toPropTerm_addClause_eq (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) :\n    \u00acdb.contains idx \u2192\n    (db.addClause idx C).toPropTerm = db.toPropTerm \u2293 C.toPropTerm :=\n  toPropTermSub_addClause_eq db C (Set.mem_univ idx)\n\ntheorem toPropTerm_delClause (db : ClauseDb \u03b1) (idx : \u03b1) :\n    db.toPropTerm \u2264 (db.delClause idx).toPropTerm :=\n  toPropTermSub_delClause db Set.univ idx\n\ntheorem toPropTerm_delClause_eq (db : ClauseDb \u03b1) (idx : \u03b1) (C : IClause) :\n    db.getClause idx = some C \u2192\n    (db.delClause idx).toPropTerm \u2293 C.toPropTerm = db.toPropTerm :=\n  toPropTermSub_delClause_eq db (Set.mem_univ idx)\n\n/-! `ofICnf` -/\n\ntheorem ofICnf_characterization (cnf : ICnf) :\n    \u00ac(ofICnf cnf).contains 0 \u2227\n    (\u2200 i : Fin cnf.size, (ofICnf cnf).getClause (i + 1) = some cnf[i]) \u2227\n    (\u2200 i > cnf.size, \u00ac(ofICnf cnf).contains i) := by\n  have \u27e8h\u2081, h\u2082, h\u2083, _\u27e9 := cnf.foldl_induction\n    (motive := fun (sz : Nat) (p : ClauseDb Nat \u00d7 Nat) =>\n      \u00acp.1.contains 0 \u2227\n      (\u2200 i : Fin cnf.size, i < sz \u2192 p.1.getClause (i + 1) = some cnf[i]) \u2227\n      (\u2200 i > sz, \u00acp.1.contains i) \u2227\n      p.2 = sz + 1)\n    (init := (empty, 1))\n    (f := fun (db, idx) C => (db.addClause idx C, idx + 1))\n    (h0 := by simp [not_contains_empty])\n    (hf := by\n      intro sz (db, idx) \u27e8ih\u2081, ih\u2082, ih\u2083, ih\u2084\u27e9\n      dsimp at ih\u2084 \u22a2\n      simp only [ih\u2084, contains_iff_getClause_eq_some, and_true] at *\n      refine \u27e8?step\u2081, ?step\u2082, ?step\u2083\u27e9\n      case step\u2081 =>\n        have : sz.val + 1 \u2260 0 := Nat.succ_ne_zero _\n        simp [getClause_addClause_of_ne _ _ _ _ this, ih\u2081]\n      case step\u2082 =>\n        intro i hLt\n        by_cases hEq : sz.val = i.val\n        . simp [hEq, getClause_addClause]\n        . have : sz.val + 1 \u2260 i.val + 1 := by simp [hEq]\n          rw [getClause_addClause_of_ne _ _ _ _ this]\n          apply ih\u2082\n          exact Nat.lt_of_le_of_ne (Nat.le_of_lt_succ hLt) (Ne.symm hEq)\n      case step\u2083 =>\n        intro i hGe\n        have : sz.val + 1 \u2260 i := Nat.ne_of_lt hGe\n        rw [getClause_addClause_of_ne _ _ _ _ this]\n        apply ih\u2083\n        linarith)\n  dsimp [ofICnf]\n  exact \u27e8h\u2081, fun i => h\u2082 i i.isLt, h\u2083\u27e9\n\ntheorem ofICnf_ext (cnf : ICnf) (C : IClause) :\n    C \u2208 cnf.data \u2194 \u2203 idx, (ofICnf cnf).getClause idx = some C := by\n  have \u27e8h\u2081, h\u2082, h\u2083\u27e9 := ofICnf_characterization cnf\n  apply Iff.intro\n  case mp =>\n    intro h\n    have \u27e8i, h\u27e9 := Array.get_of_mem_data h\n    use (i + 1)\n    rw [\u2190 h]\n    apply h\u2082\n  case mpr =>\n    intro \u27e8idx, h\u27e9\n    have hContains := contains_iff_getClause_eq_some _ _ |>.mpr \u27e8C, h\u27e9\n    have hPos : 0 < idx := by\n      apply Nat.pos_of_ne_zero\n      intro\n      simp_all\n    have hLt : idx - 1 < cnf.size := by\n      suffices idx \u2264 cnf.size by\n        apply Nat.sub_lt_left_of_lt_add\n        . apply Nat.succ_le_of_lt hPos\n        . rw [add_comm]\n          apply Nat.lt_succ_of_le this\n      by_contra\n      simp_all\n    have hPred : idx - 1 + 1 = idx := Nat.succ_pred_eq_of_pos hPos\n    have := h\u2082 \u27e8idx - 1, hLt\u27e9\n    simp only [hPred, h] at this\n    cases this\n    apply Array.get_mem_data\n\n@[simp]\ntheorem toPropTerm_ofICnf (cnf : ICnf) : (ofICnf cnf).toPropTerm = cnf.toPropTerm := by\n  ext \u03c4\n  simp only [ICnf.satisfies_iff, satisfies_toPropTerm, ofICnf_ext]\n  aesop\n\n/-! `unitPropWithHints` -/\n\ninductive UnitPropResultDep {\u03b1 : Type} [BEq \u03b1] [Hashable \u03b1]\n    (db : ClauseDb \u03b1) (\u03c3 : PartPropAssignment) (hints : Array \u03b1) where\n  /-- A contradiction was derived. The contradiction is implied by the subset of the database\n  used in hints as well as the initial assignment. -/\n  | contradiction (h : db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3.toPropTerm \u2264 \u22a5)\n  /-- The partial assignment was extended. The final assignment `\u03c3'` is implied by the subset of\n  the database used in hints as well as the initial assignment. -/\n  | extended (\u03c3' : PartPropAssignment)\n             (h : db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3.toPropTerm \u2264 \u03c3'.toPropTerm)\n  /-- The hint `C` at index `idx` did not become unit under `\u03c3`. -/\n  | hintNotUnit (idx : \u03b1) (C : IClause) (\u03c3 : PartPropAssignment)\n  /-- The hint index `idx` points at a nonexistent clause. -/\n | hintNonexistent (idx : \u03b1)\n  \n/-- Check whether the given clause is a unit and return the unit literal if so. Otherwise fail.\nNote that repeating a literal as in (l \u2228 l \u2228 l) is allowed and counts as a unit. -/\ndef checkIsUnit (C\u2080 : IClause) : Option { l : ILit // l.toPropTerm = C\u2080.toPropTerm } := do\n  let \u27e8l?, _, hL?\u27e9 \u2190 loopM_with_invariant (n := C\u2080.size)\n    (invariant := fun i (acc : Option ILit) =>\n      (acc = none \u2192 i = 0) \u2227\n      \u2200 l, acc = some l \u2192\n        l \u2208 C\u2080.data \u2227\n        \u2200 j : Fin C\u2080.size, j < i \u2192 C\u2080[j] = l)\n    (start_state := \u27e8none, by simp\u27e9)\n    (step := fun i \u27e8acc, ih\u2081, _\u27e9 => do\n      let l\u1d62 := C\u2080[i]\n      have hL : l\u1d62 \u2208 C\u2080.data := C\u2080.get_mem_data i\n      if hI : i.val = 0 then\n        return \u27e8some l\u1d62, by simp, by simp_all\u27e9\n      else\n        match acc with\n        | some l =>\n          if h : l\u1d62 = l then\n            return \u27e8some l\u1d62, by simp, by\n              intro _ h\n              injection h with h; cases h\n              refine \u27e8hL, fun j hJ => ?_\u27e9\n              cases Nat.lt_or_eq_of_le (Nat.le_of_lt_succ hJ) <;>\n                simp_all\u27e9\n          else\n            none\n        | none => False.elim <| hI <| ih\u2081 rfl)\n  match l?, hL? with\n  | some l, hL =>\n    return \u27e8l, by\n      ext\n      have \u27e8_, h\u2082\u27e9 := hL _ rfl\n      have : \u2200 l' \u2208 C\u2080.data, l' = l := fun _ hL' =>\n        have \u27e8i, hI\u27e9 := Array.get_of_mem_data hL'\n        hI \u25b8 h\u2082 i i.isLt\n      aesop (add norm IClause.satisfies_iff)\u27e9\n  | none,   _  => none\n  \n/-- Propagate units starting from the given assignment. The clauses in `hints` are expected\nto become unit in the order provided. Return the extended assignment, or `none` if a contradiction\nwas found. See `unitPropWithHintsDep` for a certified version. -/\ndef unitPropWithHintsDep (db : ClauseDb \u03b1) (\u03c3\u2080 : PartPropAssignment) (hints : Array \u03b1)\n    : UnitPropResultDep db \u03c3\u2080 hints := Id.run do\n  let mut \u03c3 : {\u03c3 : PartPropAssignment //\n      db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3\u2080.toPropTerm \u2264 \u03c3.toPropTerm } :=\n    \u27e8\u03c3\u2080, inf_le_right\u27e9\n  for h : i in [0:hints.size] do\n    let hint := hints[i]'(Membership.mem.upper h)\n    have hMem : hint \u2208 hints.data := Array.getElem_mem_data hints _\n\n    match hGet : db.getClause hint with\n    | none => return .hintNonexistent hint\n    | some C =>\n      have hDb\u03c3\u2080 :\n          db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3\u2080.toPropTerm \u2264 C.toPropTerm \u2293 \u03c3.val.toPropTerm :=\n        le_inf (inf_le_of_left_le (toPropTermSub_of_getClause_eq_some db hMem hGet)) \u03c3.property\n      match hRed : C.reduce \u03c3.val with\n      | some #[] =>\n        have : db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3\u2080.toPropTerm \u2264 \u22a5 := by\n          have : C.toPropTerm \u2293 \u03c3.val.toPropTerm \u2264 \u22a5 :=\n            IClause.reduce_eq_some _ _ _ hRed\n          exact le_trans hDb\u03c3\u2080 this\n        return .contradiction this\n      | some C' => \n        let some \u27e8u, hU\u27e9 := checkIsUnit C'\n          | return .hintNotUnit hint C \u03c3.val\n        have : db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3\u2080.toPropTerm \u2264\n            PartPropAssignment.toPropTerm (\u03c3.val.insert u.var u.polarity) := by\n          have hU : db.toPropTermSub (\u00b7 \u2208 hints.data) \u2293 \u03c3\u2080.toPropTerm \u2264 u.toPropTerm := by\n            have h := IClause.reduce_eq_some _ _ _ hRed\n            conv at h => rhs; rw [\u2190 hU]; simp [IClause.toPropTerm]\n            exact le_trans hDb\u03c3\u2080 h\n          refine PropTerm.entails_ext.mpr fun \u03c4 h\u03c4 => ?_\n          have hU : \u03c4 \u22a8 u.toPropTerm :=\n            PropTerm.entails_ext.mp hU \u03c4 h\u03c4\n          have h\u03c3 : \u03c4 \u22a8 \u03c3.val.toPropTerm :=\n            PropTerm.entails_ext.mp \u03c3.property \u03c4 h\u03c4\n          rw [PartPropAssignment.satisfies_iff] at h\u03c3 \u22a2\n          intro x p hFind\n          by_cases hEq : x = u.var\n          next =>\n            rw [hEq, HashMap.find?_insert _ _ LawfulBEq.rfl] at hFind\n            rw [ILit.satisfies_iff] at hU\n            simp_all\n          next =>\n            rw [HashMap.find?_insert_of_ne _ _ (bne_iff_ne _ _ |>.mpr (Ne.symm hEq))] at hFind\n            exact h\u03c3 _ _ hFind\n        \u03c3 := \u27e8\u03c3.val.insert u.var u.polarity, this\u27e9\n      | _ => return .hintNotUnit hint C \u03c3.val\n  return .extended \u03c3.val \u03c3.property\n\nend ClauseDb\n", "meta": {"author": "rebryant", "repo": "cpog", "sha": "5e39029ce71de532fd4407c4768e7c2bf97798c8", "save_path": "github-repos/lean/rebryant-cpog", "path": "github-repos/lean/rebryant-cpog/cpog-5e39029ce71de532fd4407c4768e7c2bf97798c8/VerifiedChecker/ProofChecker/Data/ClauseDb.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.37514051083389427}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Alexander Bentkamp\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.linear_independent\nimport Mathlib.linear_algebra.projection\nimport Mathlib.data.fintype.card\nimport Mathlib.PostPort\n\nuniverses u_1 u_3 u_5 u_2 u_6 u_7 u_4 u u_8 u_9 u_10 u_11 \n\nnamespace Mathlib\n\n/-!\n\n# Bases\n\nThis file defines bases in a module or vector space.\n\nIt is inspired by Isabelle/HOL's linear algebra, and hence indirectly by HOL Light.\n\n## Main definitions\n\nAll definitions are given for families of vectors, i.e. `v : \u03b9 \u2192 M` where `M` is the module or\nvector space and `\u03b9 : Type*` is an arbitrary indexing type.\n\n* `is_basis R v` states that the vector family `v` is a basis, i.e. it is linearly independent and\n  spans the entire space.\n\n* `is_basis.repr hv x` is the basis version of `linear_independent.repr hv x`. It returns the\n  linear combination representing `x : M` on a basis `v` of `M` (using classical choice).\n  The argument `hv` must be a proof that `is_basis R v`. `is_basis.repr hv` is given as a linear\n  map as well.\n\n* `is_basis.constr hv f` constructs a linear map `M\u2081 \u2192\u2097[R] M\u2082` given the values `f : \u03b9 \u2192 M\u2082` at the\n  basis `v : \u03b9 \u2192 M\u2081`, given `hv : is_basis R v`.\n\n## Main statements\n\n* `is_basis.ext` states that two linear maps are equal if they coincide on a basis.\n\n* `exists_is_basis` states that every vector space has a basis.\n\n## Implementation notes\n\nWe use families instead of sets because it allows us to say that two identical vectors are linearly\ndependent. For bases, this is useful as well because we can easily derive ordered bases by using an\nordered index type `\u03b9`.\n\n## Tags\n\nbasis, bases\n\n-/\n\n/-- A family of vectors is a basis if it is linearly independent and all vectors are in the span. -/\ndef is_basis {\u03b9 : Type u_1} (R : Type u_3) {M : Type u_5} (v : \u03b9 \u2192 M) [ring R] [add_comm_group M]\n    [module R M] :=\n  linear_independent R v \u2227 submodule.span R (set.range v) = \u22a4\n\ntheorem is_basis.mem_span {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) (x : M) :\n    x \u2208 submodule.span R (set.range v) :=\n  iff.mp submodule.eq_top_iff' (and.right hv)\n\ntheorem is_basis.comp {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (f : \u03b9' \u2192 \u03b9)\n    (hf : function.bijective f) : is_basis R (v \u2218 f) :=\n  sorry\n\ntheorem is_basis.injective {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [nontrivial R] (hv : is_basis R v) : function.injective v :=\n  fun (x y : \u03b9) (h : v x = v y) => linear_independent.injective (and.left hv) h\n\ntheorem is_basis.range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) :\n    is_basis R fun (x : \u21a5(set.range v)) => \u2191x :=\n  sorry\n\n/-- Given a basis, any vector can be written as a linear combination of the basis vectors. They are\ngiven by this linear map. This is one direction of `module_equiv_finsupp`. -/\ndef is_basis.repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map R M (\u03b9 \u2192\u2080 R) :=\n  linear_map.comp (linear_independent.repr sorry)\n    (linear_map.cod_restrict (submodule.span R (set.range v)) linear_map.id (is_basis.mem_span hv))\n\ntheorem is_basis.total_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) (x : M) :\n    coe_fn (finsupp.total \u03b9 M R v) (coe_fn (is_basis.repr hv) x) = x :=\n  linear_independent.total_repr (and.left hv) { val := x, property := is_basis.mem_span hv x }\n\ntheorem is_basis.total_comp_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) :\n    linear_map.comp (finsupp.total \u03b9 M R v) (is_basis.repr hv) = linear_map.id :=\n  linear_map.ext (is_basis.total_repr hv)\n\ntheorem is_basis.ext {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    {f : linear_map R M M'} {g : linear_map R M M'} (hv : is_basis R v)\n    (h : \u2200 (i : \u03b9), coe_fn f (v i) = coe_fn g (v i)) : f = g :=\n  linear_map.ext_on_range (and.right hv) h\n\ntheorem is_basis.repr_ker {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) : linear_map.ker (is_basis.repr hv) = \u22a5 :=\n  iff.mpr linear_map.ker_eq_bot (function.left_inverse.injective (is_basis.total_repr hv))\n\ntheorem is_basis.repr_range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) :\n    linear_map.range (is_basis.repr hv) = finsupp.supported R R set.univ :=\n  sorry\n\ntheorem is_basis.repr_total {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) (x : \u03b9 \u2192\u2080 R)\n    (hx : x \u2208 finsupp.supported R R set.univ) :\n    coe_fn (is_basis.repr hv) (coe_fn (finsupp.total \u03b9 M R v) x) = x :=\n  sorry\n\ntheorem is_basis.repr_eq_single {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) {i : \u03b9} :\n    coe_fn (is_basis.repr hv) (v i) = finsupp.single i 1 :=\n  sorry\n\n@[simp] theorem is_basis.repr_self_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) (j : \u03b9) :\n    coe_fn (coe_fn (is_basis.repr hv) (v i)) j = ite (i = j) 1 0 :=\n  sorry\n\ntheorem is_basis.repr_eq_iff {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) {f : linear_map R M (\u03b9 \u2192\u2080 R)} :\n    is_basis.repr hv = f \u2194 \u2200 (i : \u03b9), coe_fn f (v i) = finsupp.single i 1 :=\n  sorry\n\ntheorem is_basis.repr_apply_eq {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) {f : M \u2192 \u03b9 \u2192 R}\n    (hadd : \u2200 (x y : M), f (x + y) = f x + f y) (hsmul : \u2200 (c : R) (x : M), f (c \u2022 x) = c \u2022 f x)\n    (f_eq : \u2200 (i : \u03b9), f (v i) = \u21d1(finsupp.single i 1)) (x : M) (i : \u03b9) :\n    coe_fn (coe_fn (is_basis.repr hv) x) i = f x i :=\n  sorry\n\ntheorem is_basis.range_repr_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) :\n    coe_fn (is_basis.repr (is_basis.range hv)) (v i) =\n        finsupp.single { val := v i, property := set.mem_range_self i } 1 :=\n  sorry\n\n@[simp] theorem is_basis.range_repr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [module R M] {x : M} (hv : is_basis R v) (i : \u03b9) :\n    coe_fn (coe_fn (is_basis.repr (is_basis.range hv)) x)\n          { val := v i, property := set.mem_range_self i } =\n        coe_fn (coe_fn (is_basis.repr hv) x) i :=\n  sorry\n\n/-- Construct a linear map given the value at the basis. -/\ndef is_basis.constr {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v)\n    (f : \u03b9 \u2192 M') : linear_map R M M' :=\n  linear_map.comp (finsupp.total M' M' R id)\n    (linear_map.comp (finsupp.lmap_domain R R f) (is_basis.repr hv))\n\ntheorem is_basis.constr_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    (hv : is_basis R v) (f : \u03b9 \u2192 M') (x : M) :\n    coe_fn (is_basis.constr hv f) x =\n        finsupp.sum (coe_fn (is_basis.repr hv) x) fun (b : \u03b9) (a : R) => a \u2022 f b :=\n  sorry\n\n@[simp] theorem constr_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    {f : \u03b9 \u2192 M'} {i : \u03b9} (hv : is_basis R v) : coe_fn (is_basis.constr hv f) (v i) = f i :=\n  sorry\n\ntheorem constr_eq {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {g : \u03b9 \u2192 M'}\n    {f : linear_map R M M'} (hv : is_basis R v) (h : \u2200 (i : \u03b9), g i = coe_fn f (v i)) :\n    is_basis.constr hv g = f :=\n  is_basis.ext hv fun (i : \u03b9) => Eq.trans (constr_basis hv) (h i)\n\ntheorem constr_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v)\n    (f : linear_map R M M') : (is_basis.constr hv fun (i : \u03b9) => coe_fn f (v i)) = f :=\n  constr_eq hv fun (x : \u03b9) => rfl\n\ntheorem constr_zero {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v) :\n    (is_basis.constr hv fun (i : \u03b9) => 0) = 0 :=\n  constr_eq hv fun (x : \u03b9) => rfl\n\ntheorem constr_add {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {g : \u03b9 \u2192 M'} {f : \u03b9 \u2192 M'}\n    (hv : is_basis R v) :\n    (is_basis.constr hv fun (i : \u03b9) => f i + g i) = is_basis.constr hv f + is_basis.constr hv g :=\n  sorry\n\ntheorem constr_neg {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] {f : \u03b9 \u2192 M'}\n    (hv : is_basis R v) : (is_basis.constr hv fun (i : \u03b9) => -f i) = -is_basis.constr hv f :=\n  sorry\n\ntheorem constr_sub {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] (hv : is_basis R v)\n    {g : \u03b9 \u2192 M'} {f : \u03b9 \u2192 M'} (hs : is_basis R v) :\n    (is_basis.constr hv fun (i : \u03b9) => f i - g i) = is_basis.constr hs f - is_basis.constr hs g :=\n  sorry\n\n-- this only works on functions if `R` is a commutative ring\n\ntheorem constr_smul {\u03b9 : Type u_1} {R : Type u_2} {M : Type u_3} [comm_ring R] [add_comm_group M]\n    [module R M] {v : \u03b9 \u2192 R} {f : \u03b9 \u2192 M} {a : R} (hv : is_basis R v) :\n    (is_basis.constr hv fun (b : \u03b9) => a \u2022 f b) = a \u2022 is_basis.constr hv f :=\n  sorry\n\ntheorem constr_range {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M'] [Nonempty \u03b9]\n    (hv : is_basis R v) {f : \u03b9 \u2192 M'} :\n    linear_map.range (is_basis.constr hv f) = submodule.span R (set.range f) :=\n  sorry\n\n/-- Canonical equivalence between a module and the linear combinations of basis vectors. -/\ndef module_equiv_finsupp {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hv : is_basis R v) : linear_equiv R M (\u03b9 \u2192\u2080 R) :=\n  linear_equiv.symm\n    (linear_equiv.trans (linear_independent.total_equiv sorry)\n      (linear_equiv.of_top (submodule.span R (set.range v)) sorry))\n\n@[simp] theorem module_equiv_finsupp_apply_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) (i : \u03b9) :\n    coe_fn (module_equiv_finsupp hv) (v i) = finsupp.single i 1 :=\n  sorry\n\n/-- Isomorphism between the two modules, given two modules `M` and `M'` with respective bases\n`v` and `v'` and a bijection between the indexing sets of the two bases. -/\ndef linear_equiv_of_is_basis {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5}\n    {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (hv : is_basis R v) (hv' : is_basis R v') (e : \u03b9 \u2243 \u03b9') :\n    linear_equiv R M M' :=\n  linear_equiv.mk (linear_map.to_fun (is_basis.constr hv (v' \u2218 \u21d1e))) sorry sorry\n    \u21d1(is_basis.constr hv' (v \u2218 \u21d1(equiv.symm e))) sorry sorry\n\n/-- Isomorphism between the two modules, given two modules `M` and `M'` with respective bases\n`v` and `v'` and a bijection between the two bases. -/\ndef linear_equiv_of_is_basis' {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5}\n    {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (f : M \u2192 M') (g : M' \u2192 M) (hv : is_basis R v) (hv' : is_basis R v')\n    (hf : \u2200 (i : \u03b9), f (v i) \u2208 set.range v') (hg : \u2200 (i : \u03b9'), g (v' i) \u2208 set.range v)\n    (hgf : \u2200 (i : \u03b9), g (f (v i)) = v i) (hfg : \u2200 (i : \u03b9'), f (g (v' i)) = v' i) :\n    linear_equiv R M M' :=\n  linear_equiv.mk (linear_map.to_fun (is_basis.constr hv (f \u2218 v))) sorry sorry\n    \u21d1(is_basis.constr hv' (g \u2218 v')) sorry sorry\n\n@[simp] theorem linear_equiv_of_is_basis_comp {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3}\n    {M : Type u_5} {M' : Type u_6} {M'' : Type u_7} [ring R] [add_comm_group M] [add_comm_group M']\n    [add_comm_group M''] [module R M] [module R M'] [module R M''] {\u03b9'' : Type u_4} {v : \u03b9 \u2192 M}\n    {v' : \u03b9' \u2192 M'} {v'' : \u03b9'' \u2192 M''} (hv : is_basis R v) (hv' : is_basis R v')\n    (hv'' : is_basis R v'') (e : \u03b9 \u2243 \u03b9') (f : \u03b9' \u2243 \u03b9'') :\n    linear_equiv.trans (linear_equiv_of_is_basis hv hv' e) (linear_equiv_of_is_basis hv' hv'' f) =\n        linear_equiv_of_is_basis hv hv'' (equiv.trans e f) :=\n  sorry\n\n@[simp] theorem linear_equiv_of_is_basis_refl {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] (hv : is_basis R v) :\n    linear_equiv_of_is_basis hv hv (equiv.refl \u03b9) = linear_equiv.refl R M :=\n  sorry\n\ntheorem linear_equiv_of_is_basis_trans_symm {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3}\n    {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M']\n    [module R M] [module R M'] (hv : is_basis R v) (e : \u03b9 \u2243 \u03b9') {v' : \u03b9' \u2192 M'}\n    (hv' : is_basis R v') :\n    linear_equiv.trans (linear_equiv_of_is_basis hv hv' e)\n          (linear_equiv_of_is_basis hv' hv (equiv.symm e)) =\n        linear_equiv.refl R M :=\n  sorry\n\ntheorem linear_equiv_of_is_basis_symm_trans {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3}\n    {M : Type u_5} {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M']\n    [module R M] [module R M'] (hv : is_basis R v) (e : \u03b9 \u2243 \u03b9') {v' : \u03b9' \u2192 M'}\n    (hv' : is_basis R v') :\n    linear_equiv.trans (linear_equiv_of_is_basis hv' hv (equiv.symm e))\n          (linear_equiv_of_is_basis hv hv' e) =\n        linear_equiv.refl R M' :=\n  sorry\n\ntheorem is_basis_inl_union_inr {\u03b9 : Type u_1} {\u03b9' : Type u_2} {R : Type u_3} {M : Type u_5}\n    {M' : Type u_6} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    {v : \u03b9 \u2192 M} {v' : \u03b9' \u2192 M'} (hv : is_basis R v) (hv' : is_basis R v') :\n    is_basis R (sum.elim (\u21d1(linear_map.inl R M M') \u2218 v) (\u21d1(linear_map.inr R M M') \u2218 v')) :=\n  sorry\n\ntheorem is_basis_singleton_one {\u03b9 : Type u_1} (R : Type u_2) [unique \u03b9] [ring R] :\n    is_basis R fun (_x : \u03b9) => 1 :=\n  sorry\n\nprotected theorem linear_equiv.is_basis {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {M' : Type u_6}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M] [module R M']\n    (hs : is_basis R v) (f : linear_equiv R M M') : is_basis R (\u21d1f \u2218 v) :=\n  sorry\n\ntheorem is_basis_span {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] (hs : linear_independent R v) :\n    is_basis R\n        fun (i : \u03b9) => { val := v i, property := submodule.subset_span (set.mem_range_self i) } :=\n  sorry\n\ntheorem is_basis_empty {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} [ring R] [add_comm_group M]\n    [module R M] (h_empty : \u00acNonempty \u03b9) (h : \u2200 (x : M), x = 0) : is_basis R fun (x : \u03b9) => 0 :=\n  sorry\n\ntheorem is_basis_empty_bot {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} [ring R] [add_comm_group M]\n    [module R M] (h_empty : \u00acNonempty \u03b9) : is_basis R fun (_x : \u03b9) => 0 :=\n  is_basis_empty h_empty\n    fun (x : \u21a5\u22a5) => iff.mpr subtype.ext_iff_val (iff.mp (submodule.mem_bot R) (subtype.mem x))\n\n/-- A module over `R` with a finite basis is linearly equivalent to functions from its basis to `R`.\n-/\ndef is_basis.equiv_fun {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) : linear_equiv R M (\u03b9 \u2192 R) :=\n  linear_equiv.trans (module_equiv_finsupp h)\n    (linear_equiv.mk finsupp.to_fun sorry sorry (equiv.inv_fun finsupp.equiv_fun_on_fintype) sorry\n      sorry)\n\n/-- A module over a finite ring that admits a finite basis is finite. -/\ndef module.fintype_of_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) [fintype R] : fintype M :=\n  fintype.of_equiv (\u03b9 \u2192 R) (equiv.symm (linear_equiv.to_equiv (is_basis.equiv_fun h)))\n\ntheorem module.card_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) [fintype R] [fintype M] :\n    fintype.card M = fintype.card R ^ fintype.card \u03b9 :=\n  Eq.trans (fintype.card_congr (linear_equiv.to_equiv (is_basis.equiv_fun h))) fintype.card_fun\n\n/-- Given a basis `v` indexed by `\u03b9`, the canonical linear equivalence between `\u03b9 \u2192 R` and `M` maps\na function `x : \u03b9 \u2192 R` to the linear combination `\u2211_i x i \u2022 v i`. -/\n@[simp] theorem is_basis.equiv_fun_symm_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5}\n    {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v)\n    (x : \u03b9 \u2192 R) :\n    coe_fn (linear_equiv.symm (is_basis.equiv_fun h)) x =\n        finset.sum finset.univ fun (i : \u03b9) => x i \u2022 v i :=\n  sorry\n\ntheorem is_basis.equiv_fun_apply {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (u : M) :\n    coe_fn (is_basis.equiv_fun h) u = \u21d1(coe_fn (is_basis.repr h) u) :=\n  rfl\n\ntheorem is_basis.equiv_fun_total {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M} [ring R]\n    [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (u : M) :\n    (finset.sum finset.univ fun (i : \u03b9) => coe_fn (is_basis.equiv_fun h) u i \u2022 v i) = u :=\n  sorry\n\n@[simp] theorem is_basis.equiv_fun_self {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5} {v : \u03b9 \u2192 M}\n    [ring R] [add_comm_group M] [module R M] [fintype \u03b9] (h : is_basis R v) (i : \u03b9) (j : \u03b9) :\n    coe_fn (is_basis.equiv_fun h) (v i) j = ite (i = j) 1 0 :=\n  sorry\n\n@[simp] theorem is_basis.constr_apply_fintype {\u03b9 : Type u_1} {R : Type u_3} {M : Type u_5}\n    {M' : Type u_6} {v : \u03b9 \u2192 M} [ring R] [add_comm_group M] [add_comm_group M'] [module R M]\n    [module R M'] [fintype \u03b9] (h : is_basis R v) (f : \u03b9 \u2192 M') (x : M) :\n    coe_fn (is_basis.constr h f) x =\n        finset.sum finset.univ fun (i : \u03b9) => coe_fn (is_basis.equiv_fun h) x i \u2022 f i :=\n  sorry\n\ntheorem exists_subset_is_basis {K : Type u_4} {V : Type u} [field K] [add_comm_group V]\n    [vector_space K V] {s : set V} (hs : linear_independent K fun (x : \u21a5s) => \u2191x) :\n    \u2203 (b : set V), s \u2286 b \u2227 is_basis K coe :=\n  sorry\n\ntheorem exists_sum_is_basis {\u03b9 : Type u_1} {K : Type u_4} {V : Type u} [field K] [add_comm_group V]\n    [vector_space K V] {v : \u03b9 \u2192 V} (hs : linear_independent K v) :\n    \u2203 (\u03b9' : Type u), \u2203 (v' : \u03b9' \u2192 V), is_basis K (sum.elim v v') :=\n  sorry\n\ntheorem exists_is_basis (K : Type u_4) (V : Type u) [field K] [add_comm_group V]\n    [vector_space K V] : \u2203 (b : set V), is_basis K fun (i : \u21a5b) => \u2191i :=\n  sorry\n\ntheorem linear_map.exists_left_inverse_of_injective {K : Type u_4} {V : Type u} {V' : Type u_8}\n    [field K] [add_comm_group V] [add_comm_group V'] [vector_space K V] [vector_space K V']\n    (f : linear_map K V V') (hf_inj : linear_map.ker f = \u22a5) :\n    \u2203 (g : linear_map K V' V), linear_map.comp g f = linear_map.id :=\n  sorry\n\ntheorem submodule.exists_is_compl {K : Type u_4} {V : Type u} [field K] [add_comm_group V]\n    [vector_space K V] (p : submodule K V) : \u2203 (q : submodule K V), is_compl p q :=\n  sorry\n\ntheorem linear_map.exists_right_inverse_of_surjective {K : Type u_4} {V : Type u} {V' : Type u_8}\n    [field K] [add_comm_group V] [add_comm_group V'] [vector_space K V] [vector_space K V']\n    (f : linear_map K V V') (hf_surj : linear_map.range f = \u22a4) :\n    \u2203 (g : linear_map K V' V), linear_map.comp f g = linear_map.id :=\n  sorry\n\ntheorem quotient_prod_linear_equiv {K : Type u_4} {V : Type u} [field K] [add_comm_group V]\n    [vector_space K V] (p : submodule K V) :\n    Nonempty (linear_equiv K (submodule.quotient p \u00d7 \u21a5p) V) :=\n  sorry\n\ntheorem vector_space.card_fintype (K : Type u_4) (V : Type u) [field K] [add_comm_group V]\n    [vector_space K V] [fintype K] [fintype V] : \u2203 (n : \u2115), fintype.card V = fintype.card K ^ n :=\n  exists.elim (exists_is_basis K V)\n    fun (b : set V) (hb : is_basis K fun (i : \u21a5b) => \u2191i) =>\n      Exists.intro (fintype.card \u21a5b) (module.card_fintype hb)\n\nnamespace pi\n\n\ntheorem linear_independent_std_basis {R : Type u_3} {\u03b7 : Type u_9} {\u03b9s : \u03b7 \u2192 Type u_10}\n    {Ms : \u03b7 \u2192 Type u_11} [ring R] [(i : \u03b7) \u2192 add_comm_group (Ms i)] [(i : \u03b7) \u2192 module R (Ms i)]\n    [DecidableEq \u03b7] (v : (j : \u03b7) \u2192 \u03b9s j \u2192 Ms j) (hs : \u2200 (i : \u03b7), linear_independent R (v i)) :\n    linear_independent R\n        fun (ji : sigma fun (j : \u03b7) => \u03b9s j) =>\n          coe_fn (linear_map.std_basis R Ms (sigma.fst ji)) (v (sigma.fst ji) (sigma.snd ji)) :=\n  sorry\n\ntheorem is_basis_std_basis {R : Type u_3} {\u03b7 : Type u_9} {\u03b9s : \u03b7 \u2192 Type u_10} {Ms : \u03b7 \u2192 Type u_11}\n    [ring R] [(i : \u03b7) \u2192 add_comm_group (Ms i)] [(i : \u03b7) \u2192 module R (Ms i)] [fintype \u03b7]\n    [DecidableEq \u03b7] (s : (j : \u03b7) \u2192 \u03b9s j \u2192 Ms j) (hs : \u2200 (j : \u03b7), is_basis R (s j)) :\n    is_basis R\n        fun (ji : sigma fun (j : \u03b7) => \u03b9s j) =>\n          coe_fn (linear_map.std_basis R Ms (sigma.fst ji)) (s (sigma.fst ji) (sigma.snd ji)) :=\n  sorry\n\ntheorem is_basis_fun\u2080 (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7] :\n    is_basis R\n        fun (ji : sigma fun (j : \u03b7) => Unit) =>\n          coe_fn (linear_map.std_basis R (fun (i : \u03b7) => R) (sigma.fst ji)) 1 :=\n  is_basis_std_basis (fun (_x : \u03b7) (_x : Unit) => 1) fun (i : \u03b7) => is_basis_singleton_one R\n\ntheorem is_basis_fun (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7] :\n    is_basis R fun (i : \u03b7) => coe_fn (linear_map.std_basis R (fun (i : \u03b7) => R) i) 1 :=\n  sorry\n\n@[simp] theorem is_basis_fun_repr (R : Type u_3) (\u03b7 : Type u_9) [ring R] [fintype \u03b7] [DecidableEq \u03b7]\n    (x : \u03b7 \u2192 R) (i : \u03b7) : coe_fn (coe_fn (is_basis.repr (is_basis_fun R \u03b7)) x) i = x i :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/basis_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.37510985949093634}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\n\nimport tactic.rcases\n\ninstance {\u03b1} : has_inter (set \u03b1) := \u27e8\u03bb s t, {a | a \u2208 s \u2227 a \u2208 t}\u27e9\n\nuniverse u\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\n\nexample (x : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3) : true :=\nbegin\n  rcases x with \u27e8a, b, c\u27e9,\n  { guard_hyp a : \u03b1,\n    guard_hyp b : \u03b2,\n    guard_hyp c : \u03b3,\n    trivial }\nend\n\nexample (x : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3) : true :=\nbegin\n  rcases x with \u27e8a, \u27e8-, c\u27e9\u27e9,\n  { guard_hyp a : \u03b1,\n    success_if_fail { guard_hyp x_snd_fst : \u03b2 },\n    guard_hyp c : \u03b3,\n    trivial }\nend\n\nexample (x : (\u03b1 \u00d7 \u03b2) \u00d7 \u03b3) : true :=\nbegin\n  rcases x with \u27e8\u27e8a:\u03b1, b\u27e9, c\u27e9,\n  { guard_hyp a : \u03b1,\n    guard_hyp b : \u03b2,\n    guard_hyp c : \u03b3,\n    trivial }\nend\n\nexample : inhabited \u03b1 \u00d7 option \u03b2 \u2295 \u03b3 \u2192 true :=\nbegin\n  rintro (\u27e8\u27e8a\u27e9, _ | b\u27e9 | c),\n  { guard_hyp a : \u03b1, trivial },\n  { guard_hyp a : \u03b1, guard_hyp b : \u03b2, trivial },\n  { guard_hyp c : \u03b3, trivial }\nend\n\nexample : cond ff \u2115 \u2124 \u2192 cond tt \u2124 \u2115 \u2192 (\u2115 \u2295 unit) \u2192 true :=\nbegin\n  rintro (x y : \u2124) (z | u),\n  { guard_hyp x : \u2124, guard_hyp y : \u2124, guard_hyp z : \u2115, trivial },\n  { guard_hyp x : \u2124, guard_hyp y : \u2124, guard_hyp u : unit, trivial }\nend\n\nexample (x y : \u2115) (h : x = y) : true :=\nbegin\n  rcases x with _|\u27e8\u27e9|z,\n  { guard_hyp h : nat.zero = y, trivial },\n  { guard_hyp h : nat.succ nat.zero = y, trivial },\n  { guard_hyp z : \u2115,\n    guard_hyp h : z.succ.succ = y, trivial },\nend\n\n-- from equiv.sum_empty\nexample (s : \u03b1 \u2295 empty) : true :=\nbegin\n  rcases s with _ | \u27e8\u27e8\u27e9\u27e9,\n  { guard_hyp s : \u03b1, trivial }\nend\n\nexample : true :=\nbegin\n  obtain \u27e8n : \u2115, h : n = n, -\u27e9 : \u2203 n : \u2115, n = n \u2227 true,\n  { existsi 0, simp },\n  guard_hyp n : \u2115,\n  guard_hyp h : n = n,\n  success_if_fail {assumption},\n  trivial\nend\n\nexample : true :=\nbegin\n  obtain : \u2203 n : \u2115, n = n \u2227 true,\n  { existsi 0, simp },\n  trivial\nend\n\nexample : true :=\nbegin\n  obtain (h : true) | \u27e8\u27e8\u27e9\u27e9 : true \u2228 false,\n  { left, trivial },\n  guard_hyp h : true,\n  trivial\nend\n\nexample : true :=\nbegin\n  obtain h | \u27e8\u27e8\u27e9\u27e9 : true \u2228 false := or.inl trivial,\n  guard_hyp h : true,\n  trivial\nend\n\nexample : true :=\nbegin\n  obtain \u27e8h, h2\u27e9 := and.intro trivial trivial,\n  guard_hyp h : true,\n  guard_hyp h2 : true,\n  trivial\nend\n\nexample : true :=\nbegin\n  success_if_fail {obtain \u27e8h, h2\u27e9},\n  trivial\nend\n\nexample (x y : \u03b1 \u00d7 \u03b2) : true :=\nbegin\n  rcases \u27e8x, y\u27e9 with \u27e8\u27e8a, b\u27e9, c, d\u27e9,\n  { guard_hyp a : \u03b1,\n    guard_hyp b : \u03b2,\n    guard_hyp c : \u03b1,\n    guard_hyp d : \u03b2,\n    trivial }\nend\n\nexample (x y : \u03b1 \u2295 \u03b2) : true :=\nbegin\n  obtain \u27e8a|b, c|d\u27e9 := \u27e8x, y\u27e9,\n  { guard_hyp a : \u03b1, guard_hyp c : \u03b1, trivial },\n  { guard_hyp a : \u03b1, guard_hyp d : \u03b2, trivial },\n  { guard_hyp b : \u03b2, guard_hyp c : \u03b1, trivial },\n  { guard_hyp b : \u03b2, guard_hyp d : \u03b2, trivial },\nend\n\nexample {i j : \u2115} : (\u03a3' x, i \u2264 x \u2227 x \u2264 j) \u2192 i \u2264 j :=\nbegin\n  intro h,\n  rcases h' : h with \u27e8x,h\u2080,h\u2081\u27e9,\n  guard_hyp h' : h = \u27e8x,h\u2080,h\u2081\u27e9,\n  apply le_trans h\u2080 h\u2081,\nend\n\nprotected def set.foo {\u03b1 \u03b2} (s : set \u03b1) (t : set \u03b2) : set (\u03b1 \u00d7 \u03b2) := \u2205\n\nexample {\u03b1} (V : set \u03b1) (w : true \u2192 \u2203 p, p \u2208 (V.foo V) \u2229 (V.foo V)) : true :=\nbegin\n  obtain \u27e8a, h\u27e9 : \u2203 p, p \u2208 (V.foo V) \u2229 (V.foo V) := w trivial,\n  trivial,\nend\n\nexample (n : \u2115) : true :=\nbegin\n  obtain one_lt_n | n_le_one : 1 < n + 1 \u2228 n + 1 \u2264 1 := nat.lt_or_ge 1 (n + 1),\n  trivial, trivial,\nend\n\nexample (n : \u2115) : true :=\nbegin\n  obtain one_lt_n | (n_le_one : n + 1 \u2264 1) := nat.lt_or_ge 1 (n + 1),\n  trivial, trivial,\nend\n\nexample (h : \u2203 x : \u2115, x = x \u2227 1 = 1) : true :=\nbegin\n  rcases h with \u27e8-, _\u27e9,\n  (do lc \u2190 tactic.local_context, guard lc.empty),\n  trivial\nend\n\nexample (h : \u2203 x : \u2115, x = x \u2227 1 = 1) : true :=\nbegin\n  rcases h with \u27e8-, _, h\u27e9,\n  (do lc \u2190 tactic.local_context, guard (lc.length = 1)),\n  guard_hyp h : 1 = 1,\n  trivial\nend\n\nexample (h : true \u2228 true \u2228 true) : true :=\nbegin\n  rcases h with -|-|-,\n  iterate 3 {\n    (do lc \u2190 tactic.local_context, guard lc.empty),\n    trivial },\nend\n\nexample : bool \u2192 false \u2192 true\n| ff := by rintro \u27e8\u27e9\n| tt := by rintro \u27e8\u27e9\n\nexample : true :=\nbegin\n  obtain h : true,\n  { trivial },\n  exact h\nend\n\nexample {a b} (h : a \u2227 b) : a \u2227 b :=\nbegin\n  rcases h with t,\n  exact t\nend\n\nstructure baz {\u03b1 : Type*} (f : \u03b1 \u2192 \u03b1) : Prop := [inst : nonempty \u03b1] (h : f \u2218 f = id)\nexample {\u03b1} (f : \u03b1 \u2192 \u03b1) (h : baz f) : true := by { rcases h with \u27e8_\u27e9; trivial }\nexample {\u03b1} (f : \u03b1 \u2192 \u03b1) (h : baz f) : true := by { rcases h with @\u27e8_, _\u27e9; trivial }\n\ninductive test : nat \u2192 Prop\n| a (n) : test (2 + n)\n| b {n} : n > 5 \u2192 test (n * n)\n\nexample {n} (h : test n) : n = n :=\nbegin\n  have : true,\n  { rcases h with a | b,\n    { guard_hyp a : nat, trivial },\n    { guard_hyp b : \u2039nat\u203a > 5, trivial } },\n  { rcases h with a | @\u27e8n, b\u27e9,\n    { guard_hyp a : nat, trivial },\n    { guard_hyp b : n > 5, trivial } },\nend\n\nopen tactic\nmeta def test_rcases_hint (s : string) (num_goals : \u2115) (depth := 5) : tactic unit :=\ndo change `(true),\n  h \u2190 get_local `h,\n  pat \u2190 rcases_hint ```(h) depth,\n  p \u2190 pp pat,\n  guard (p.to_string = s) <|> fail format!\"got '{p.to_string}', expected: '{s}'\",\n  gs \u2190 get_goals,\n  guard (gs.length = num_goals) <|> fail format!\"there are {gs.length} goals remaining\",\n  all_goals triv $> ()\n\nexample {\u03b1} (h : \u2203 x : \u03b1, x = x) := by test_rcases_hint \"\u27e8h_w, \u27e8\u27e9\u27e9\" 1\nexample (h : true \u2228 true \u2228 true) := by test_rcases_hint \"\u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9 | \u27e8\u27e8\u27e9\u27e9\" 3\nexample (h : \u2115) := by test_rcases_hint \"_ | _ | h\" 3 2\nexample {p} (h : (p \u2227 p) \u2228 (p \u2227 p)) :=\nby test_rcases_hint \"\u27e8h_left, h_right\u27e9 | \u27e8h_left, h_right\u27e9\" 2\nexample {p} (h : (p \u2227 p) \u2228 (p \u2227 (p \u2228 p))) :=\nby test_rcases_hint \"\u27e8h_left, h_right\u27e9 | \u27e8h_left, h_right | h_right\u27e9\" 3\nexample {p} (h : p \u2227 (p \u2228 p)) :=\nby test_rcases_hint \"\u27e8h_left, h_right | h_right\u27e9\" 2\nexample (h : 0 < 2) := by test_rcases_hint \"_ | \u27e8_, _ | \u27e8_, \u27e8\u27e9\u27e9\u27e9\" 1\nexample (h : 3 < 2) := by test_rcases_hint \"_ | \u27e8_, _ | \u27e8_, \u27e8\u27e9\u27e9\u27e9\" 0\nexample (h : 3 < 0) := by test_rcases_hint \"\u27e8\u27e9\" 0\nexample (h : false) := by test_rcases_hint \"\u27e8\u27e9\" 0\nexample (h : true) := by test_rcases_hint \"\u27e8\u27e9\" 1\nexample {\u03b1} (h : list \u03b1) := by test_rcases_hint \"_ | \u27e8h_hd, _ | \u27e8h_tl_hd, h_tl_tl\u27e9\u27e9\" 3 2\nexample {\u03b1} (h : (\u03b1 \u2295 \u03b1) \u00d7 \u03b1) := by test_rcases_hint \"\u27e8h_fst | h_fst, h_snd\u27e9\" 2 2\n\ninductive foo (\u03b1 : Type) : \u2115 \u2192 Type\n| zero : foo 0\n| one (m) : \u03b1 \u2192 foo m\n\nexample {\u03b1} (h : foo \u03b1 0) : true := by test_rcases_hint \"_ | \u27e8_, h_\u1fb0\u27e9\" 2\nexample {\u03b1} (h : foo \u03b1 1) : true := by test_rcases_hint \"_ | \u27e8_, h_\u1fb0\u27e9\" 1\nexample {\u03b1 n} (h : foo \u03b1 n) : true := by test_rcases_hint \"_ | h_\u1fb0\" 2 1\n\nexample {\u03b1} (V : set \u03b1) (h : \u2203 p, p \u2208 (V.foo V) \u2229 (V.foo V)) :=\nby test_rcases_hint \"\u27e8\u27e8h_w_fst, h_w_snd\u27e9, \u27e8\u27e9\u27e9\" 0\n\nsection rsuffices\n\n/-- These next few are duplicated from `rcases/obtain` tests, with the goal order swapped. -/\n\nexample : true :=\nbegin\n  rsuffices \u27e8n : \u2115, h : n = n, -\u27e9 : \u2203 n : \u2115, n = n \u2227 true,\n  { guard_hyp n : \u2115,\n    guard_hyp h : n = n,\n    success_if_fail {assumption},\n    trivial },\n  { existsi 0, simp },\nend\n\nexample : true :=\nbegin\n  rsuffices : \u2203 n : \u2115, n = n \u2227 true,\n  { trivial },\n  { existsi 0, simp },\nend\n\nexample : true :=\nbegin\n  rsuffices (h : true) | \u27e8\u27e8\u27e9\u27e9 : true \u2228 false,\n  { guard_hyp h : true,\n    trivial },\n  { left, trivial },\nend\n\nexample : true :=\nbegin\n  success_if_fail {rsuffices \u27e8h, h2\u27e9},\n  trivial\nend\n\nexample (x y : \u03b1 \u00d7 \u03b2) : true :=\nbegin\n  rsuffices \u27e8\u27e8a, b\u27e9, c, d\u27e9 : (\u03b1 \u00d7 \u03b2) \u00d7 (\u03b1 \u00d7 \u03b2),\n  { guard_hyp a : \u03b1,\n    guard_hyp b : \u03b2,\n    guard_hyp c : \u03b1,\n    guard_hyp d : \u03b2,\n    trivial },\n  { exact \u27e8x, y\u27e9 }\nend\n\n-- This test demonstrates why `swap` is not used in the implementation of `rsuffices`:\n-- it would make the _second_ goal the one requiring \u27e8x, y\u27e9, not the last one.\nexample (x y : \u03b1 \u2295 \u03b2) : true :=\nbegin\n  rsuffices \u27e8a|b, c|d\u27e9 : (\u03b1 \u2295 \u03b2) \u00d7 (\u03b1 \u2295 \u03b2),\n  { guard_hyp a : \u03b1, guard_hyp c : \u03b1, trivial },\n  { guard_hyp a : \u03b1, guard_hyp d : \u03b2, trivial },\n  { guard_hyp b : \u03b2, guard_hyp c : \u03b1, trivial },\n  { guard_hyp b : \u03b2, guard_hyp d : \u03b2, trivial },\n  exact \u27e8x, y\u27e9,\nend\n\nexample {\u03b1} (V : set \u03b1) (w : true \u2192 \u2203 p, p \u2208 (V.foo V) \u2229 (V.foo V)) : true :=\nbegin\n  rsuffices \u27e8a, h\u27e9 : \u2203 p, p \u2208 (V.foo V) \u2229 (V.foo V),\n  { trivial },\n  { exact w trivial },\nend\n\n-- Now some tests that ensure that things stay in the correct order.\n\n-- This test demonstrates why `focus1` is required in the definition of `rsuffices`; otherwise\n-- the `\u2203 ...` goal would get put _after_ the `true` goal.\nexample : nonempty \u2115 \u2227 true :=\nbegin\n  split,\n  rsuffices \u27e8n : \u2115, hn\u27e9 : \u2203 n, _,\n  { exact \u27e8n\u27e9 },\n  { exact true },\n  { exact \u27e80, trivial\u27e9 },\n  { trivial },\nend\n\nsection instances\n\nexample (h : \u03a0 {\u03b1}, inhabited \u03b1) : inhabited (\u03b1 \u2295 \u03b2) :=\nbegin\n  rsufficesI (ha | hb) : inhabited \u03b1 \u2295 inhabited \u03b2,\n  { exact \u27e8sum.inl default\u27e9 },\n  { exact \u27e8sum.inr default\u27e9 },\n  { exact sum.inl h }\nend\n\ninclude \u03b2\n-- this test demonstrates that the `resetI` also applies onto the goal.\nexample (h : \u03a0 {\u03b1}, inhabited \u03b1) : inhabited \u03b1 :=\nbegin\n  have : inhabited \u03b2 := h,\n  rsufficesI t : \u03b2,\n  { exact h },\n  { exact default }\nend\n\nexample (h : \u03a0 {\u03b1}, inhabited \u03b1) : \u03b2 :=\nbegin\n  rsufficesI ht : inhabited \u03b2,\n  { guard_hyp ht : inhabited \u03b2,\n    exact default },\n  { exact h }\nend\n\nend instances\n\nend rsuffices\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/rcases.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555714, "lm_q2_score": 0.6859494550081925, "lm_q1q2_score": 0.37503474751816246}}
{"text": "import S5.syntax.basic\n\nopen prf \n\n-- Identity implication\ntheorem idd { p : form } { \u0393 : ctx } : \u0393 \u22a2\u209b\u2085 p \u2192 p := by\n  apply mp\n  case hpq => \n    apply mp \n    case hpq => exact @pl2 \u0393 p (p \u2192 p) p \n    case hp => exact @pl1 \u0393 p (p \u2192 p)\n  case hp => exact @pl1 \u0393 p p\n\n-- Deduction metatheorem\ntheorem deduction { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p) \u22a2\u209b\u2085 q) \u2192 (\u0393 \u22a2\u209b\u2085 p \u2192 q) := by\n  intros h \n  cases h\n  {\n    rename_i h\u2080;\n    cases h\u2080;\n    { exact idd }\n    {\n      rename_i h\u2081;\n      exact mp pl1 (ax h\u2081)\n    }\n  } \n  { exact mp pl1 pl1 }\n  { exact mp pl1 pl2 }\n  { exact mp pl1 pl3 }\n  {\n    sorry\n  }\n  { exact mp pl1 k }\n  { exact mp pl1 t }\n  { exact mp pl1 s4 }\n  { exact mp pl1 s5 }\n  { \n    rename_i h\u2080;\n    exact mp pl1 (nec h\u2080)\n  }\n\n-- Structural rules\ntheorem sub_weak { \u0393 \u0394 : ctx } { p : form } : (\u0394 \u22a2\u209b\u2085 p) \u2192 (\u0394 \u2286 \u0393) \u2192 (\u0393 \u22a2\u209b\u2085 p) := by \n  intros h\u2080 h\u2081\n  induction h\u2080\n  {\n    rename_i \u0394 p h\u2082;\n    apply ax;\n    sorry\n    -- apply ax;\n  }\n  { exact pl1 }\n  { exact pl2 }\n  { exact pl3 }\n  { sorry }\n  { exact k }\n  { exact t }\n  { exact s4 }\n  { exact s5 }\n  {  \n    rename_i h\u2082 h\u2083;\n    exact nec h\u2082;\n  }\n\ntheorem weak { \u0393 : ctx } { p q : form } :(\u0393 \u22a2\u209b\u2085 p) \u2192 ((\u0393 \u222a q) \u22a2\u209b\u2085 p) := by\n  intros h \n  induction h \n  { sorry }\n  { exact pl1 }\n  { exact pl2 }\n  { exact pl3 }\n  { sorry }\n  { exact k }\n  { exact t }\n  { exact s4 }\n  { exact s5 }\n  {  \n    rename_i h\u2080 h\u2081;\n    exact nec h\u2080;\n  }\n\ntheorem contr { \u0393 : ctx } { p q : form } : (((\u0393 \u222a p) \u222a p) \u22a2\u209b\u2085 q) \u2192 ((\u0393 \u222a p) \u22a2\u209b\u2085 q) := by\n  intros h \n  cases h \n  {  \n    apply ax;\n    rename_i h\u2080;\n    sorry\n  }\n  { exact pl1 }\n  { exact pl2 }\n  { exact pl3 }\n  { sorry }\n  { exact k }\n  { exact t }\n  { exact s4 }\n  { exact s5 }\n  {  \n    rename_i h\u2080;\n    exact nec h\u2080;\n  }\n\ntheorem exg { \u0393 : ctx } { p q r : form } : (((\u0393 \u222a p) \u222a q) \u22a2\u209b\u2085 r) \u2192 (((\u0393 \u222a q) \u222a p) \u22a2\u209b\u2085 r) := sorry\n\ntheorem subctx_ax { \u0393 \u0394 : ctx } { p : form } : (\u0394 \u2286 \u0393) \u2192 (\u0394 \u22a2\u209b\u2085 p) \u2192 (\u0393 \u22a2\u209b\u2085 p) := by\n  intros s h \n  induction h \n  { sorry }\n  { exact pl1 }\n  { exact pl2 }\n  { exact pl3 }\n  { sorry }\n  { exact k }\n  { exact t }\n  { exact s4 }\n  { exact s5 }\n  {  \n    rename_i h\u2080 h\u2081;\n    exact nec h\u2080;\n  }\n\n-- Right-hand side basic rules of inference\ntheorem pr { \u0393 : ctx } { p : form } : (\u0393 \u222a p) \u22a2\u209b\u2085 p := \n  ax $ by constructor\n\ntheorem pr1 { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p) \u222a q) \u22a2\u209b\u2085 p := \n  ax $ by (repeat constructor)\n\ntheorem pr2 { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p) \u222a q) \u22a2\u209b\u2085 q := \n  ax $ by constructor\n\ntheorem by_mp1 { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p) \u222a p \u2192 q) \u22a2\u209b\u2085 q := mp pr2 pr1\ntheorem by_mp2 { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p \u2192 q) \u222a p) \u22a2\u209b\u2085 q := mp pr1 pr2 \n\ntheorem cut { \u0393 : ctx } { p q r : form } : (\u0393 \u22a2\u209b\u2085 p \u2192 q) \u2192 (\u0393 \u22a2\u209b\u2085 q \u2192 r) \u2192 (\u0393 \u22a2\u209b\u2085 p \u2192 r) := by \n  intro hpq hqr \n  apply mp (mp pl2 (mp pl1 hqr)) hpq\n\ntheorem conv_deduction { \u0393 : ctx } { p q : form } : (\u0393 \u22a2\u209b\u2085 p \u2192 q) \u2192 ((\u0393 \u222a p) \u22a2\u209b\u2085 q) := by \n  intro hpq\n  apply mp (weak hpq) pr \n\n\n-- Left-hand side basic rules of inference\ntheorem mp_in_ctx_left { \u0393 : ctx } { p q r : form } : (((\u0393 \u222a p) \u222a q) \u22a2\u209b\u2085 r) \u2192 (((\u0393 \u222a p) \u222a p \u2192 q) \u22a2\u209b\u2085 r) := sorry\ntheorem mp_in_ctx_right { \u0393 : ctx } { p q r : form } : (((\u0393 \u222a p) \u222a p \u2192 q) \u22a2\u209b\u2085 r) \u2192 (((\u0393 \u222a p) \u222a q) \u22a2\u209b\u2085 r) := sorry \n\n\n-- Basic lemmas\ntheorem contrap { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 ((\u00acq) \u2192 (\u00acp)) \u2192 (p \u2192 q) :=\n  deduction (deduction (mp (mp pl3 pr1) (mp pl1 pr2) ))\n  \ntheorem not_impl { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 (p \u2192 q) \u2192 ((\u00acq) \u2192 (\u00acp)) := sorry\n\ntheorem dne { \u0393 : ctx } { p : form } : \u0393 \u22a2\u209b\u2085 (\u00ac\u00acp) \u2192 p := \n  have h : \u0393 \u22a2\u209b\u2085 (\u00ac\u00acp) \u2192 ((\u00acp) \u2192 (\u00acp)) := mp pl1 idd\n  mp (mp pl2 (cut pl1 pl3)) h\n\ntheorem dni { \u0393 : ctx } { p : form } : \u0393 \u22a2\u209b\u2085 p \u2192 (\u00ac\u00acp) := mp contrap dne\n\ntheorem lem { \u0393 : ctx } { p : form } : \u0393 \u22a2\u209b\u2085 (p \u2228 (\u00acp)) := mp dni dni\n\ntheorem not_impl_to_and { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 (\u00ac(p \u2192 q)) \u2192 (p \u2228 (\u00acq)) := by \n  sorry\n\ntheorem and_not_to_not_impl { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 (p \u2227 (\u00acq)) \u2192 (\u00ac(p \u2192 q)) := by\n  sorry\n  -- repeat (apply deduction)\n  -- apply mp\n  -- {\n  --   apply pr1\n  -- }\n  -- { apply cut;\n  --   { apply pr2 }\n  --   { apply dni } \n  -- }\n\ntheorem box_contrap { p q : form } : \u22a2\u209b\u2085 (\u25a1(p \u2192 q)) \u2192 (\u25a1((\u00acq) \u2192 (\u00acp))) := \n  mp k (prf.nec not_impl)\n\ntheorem diamond_k { p q : form } : \u22a2\u209b\u2085 (\u25a1(p \u2192 q)) \u2192 ((\u22c4p) \u2192 (\u22c4q)) := \n  deduction $ mp not_impl (mp k (mp (weak box_contrap) pr))\n\ntheorem box_dne { p : form } : \u22a2\u209b\u2085 (\u25a1(\u00ac\u00acp)) \u2192 (\u25a1p) := mp k (nec dne)\ntheorem box_dni { p : form } : \u22a2\u209b\u2085 (\u25a1p) \u2192 (\u25a1(\u00ac\u00acp)) := mp k (nec dni)\n\ntheorem not_box_dni { p : form } : \u22a2\u209b\u2085 (\u00ac\u25a1p) \u2192 (\u00ac\u25a1(\u00ac\u00acp)) := mp not_impl box_dne \ntheorem not_box_dne { p : form } : \u22a2\u209b\u2085 (\u00ac\u25a1(\u00ac\u00acp)) \u2192 (\u00ac\u25a1p) := mp not_impl box_dni \n\ntheorem diamond_dne { p : form } : \u22a2\u209b\u2085 (\u22c4(\u00ac\u00acp)) \u2192 (\u22c4p) := not_box_dne\ntheorem diamond_dni { p : form } : \u22a2\u209b\u2085 (\u22c4p) \u2192 (\u22c4(\u00ac\u00acp)) := not_box_dni\n\ntheorem contrap_b { p : form } : \u22a2\u209b\u2085 (\u22c4(\u25a1p)) \u2192 p := sorry\n\n-- Notable introduction rules\ntheorem negintro { \u0393 : ctx } { p q : form } : (\u0393 \u22a2\u209b\u2085 p \u2192 q) \u2192 (\u0393 \u22a2\u209b\u2085 p \u2192 (\u00acq)) \u2192 (\u0393 \u22a2\u209b\u2085 (\u00acp)) := sorry \n-- theorem ex_falso { \u0393 : ctx } { p : form } : (\u0393 \u22a2\u209b\u2085 \u22a5) \u2192 (\u0393 \u22a2\u209b\u2085 p) := sorry\ntheorem ex_falso_and { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 (\u00acp) \u2192 (p \u2192 q) := sorry \ntheorem ex_falso_pos { \u0393 : ctx } { p q : form } : \u0393 \u22a2\u209b\u2085 p \u2192 ((\u00acp) \u2192 q) := sorry\ntheorem contr_conseq { \u0393 : ctx } { p r : form } : \u0393 \u22a2\u209b\u2085 (p \u2192 r) \u2192 (((\u00acp) \u2192 r) \u2192 r) := sorry\ntheorem impl_weak { \u0393 : ctx } { p q r : form } (h : ((\u0393 \u222a r) \u22a2\u209b\u2085 p) \u2192 (\u0393 \u22a2\u209b\u2085 p)) :\n  ((\u0393 \u22a2\u209b\u2085 p) \u2192 (\u0393 \u22a2\u209b\u2085 q)) \u2192 (((\u0393 \u222a r) \u22a2\u209b\u2085 p) \u2192 ((\u0393 \u222a r) \u22a2\u209b\u2085 q)) := sorry \n\ntheorem and_intro { \u0393 : ctx } { p q : form } : (\u0393 \u22a2\u209b\u2085 p) \u2192 (\u0393 \u22a2\u209b\u2085 q) \u2192 (\u0393 \u22a2\u209b\u2085 (p \u2227 q)) := sorry \ntheorem and_elim_left { \u0393 : ctx } { p q : form } : ((\u0393 \u222a (p \u2227 q)) \u22a2\u209b\u2085 p) := sorry\ntheorem and_elim_right { \u0393 : ctx } { p q : form } : ((\u0393 \u222a (p \u2227 q)) \u22a2\u209b\u2085 q) := sorry \n\ntheorem or_intro_left { \u0393 : ctx } { p q r : form } : (\u0393 \u22a2\u209b\u2085 p) \u2192 (\u0393 \u22a2\u209b\u2085 (p \u2228 q)) := sorry \ntheorem or_intro_right { \u0393 : ctx } { p q r : form } : (\u0393 \u22a2\u209b\u2085 q) \u2192 (\u0393 \u22a2\u209b\u2085 (p \u2228 q)) := sorry \ntheorem or_elim { \u0393 : ctx } { p q r : form } : (\u0393 \u22a2\u209b\u2085 (p \u2228 q)) \u2192 (\u0393 \u22a2\u209b\u2085 p \u2192 r) \u2192 (\u0393 \u22a2\u209b\u2085 q \u2192 r) \u2192 (\u0393 \u22a2\u209b\u2085 r) := sorry \ntheorem detach_pos { \u0393 : ctx } { p q : form } : ((\u0393 \u222a p) \u22a2\u209b\u2085 q) \u2192 ((\u0393 \u222a \u00acp) \u22a2\u209b\u2085 q) \u2192 (\u0393 \u22a2\u209b\u2085 q) := sorry \ntheorem detach_neg { \u0393 : ctx } { p q : form } : ((\u0393 \u222a \u00acp) \u22a2\u209b\u2085 q) \u2192 ((\u0393 \u222a p) \u22a2\u209b\u2085 q) \u2192 (\u0393 \u22a2\u209b\u2085 q) := sorry \n", "meta": {"author": "cristinaborza", "repo": "S5", "sha": "a68f97a19e993c64e66ac38f9a3144693a0b6c07", "save_path": "github-repos/lean/cristinaborza-S5", "path": "github-repos/lean/cristinaborza-S5/S5-a68f97a19e993c64e66ac38f9a3144693a0b6c07/S5/syntax/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3750347373860322}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.limits.functor_category\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Limit properties relating to the (co)yoneda embedding.\n\nWe calculate the colimit of `Y \u21a6 (X \u27f6 Y)`, which is just `punit`.\n(This is used in characterising cofinal functors.)\n\nWe also show the (co)yoneda embeddings preserve limits and jointly reflect them.\n-/\n\nnamespace category_theory\n\n\nnamespace coyoneda\n\n\n/--\nThe colimit cocone over `coyoneda.obj X`, with cocone point `punit`.\n-/\n@[simp] theorem colimit_cocone_\u03b9_app {C : Type v} [small_category C] (X : C\u1d52\u1d56) (X_1 : C) :\n    \u2200 (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1),\n        nat_trans.app (limits.cocone.\u03b9 (colimit_cocone X)) X_1 \u1fb0 =\n          id\n            (fun (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1) =>\n              id (fun (X : C\u1d52\u1d56) (X_1 : C) (\u1fb0 : opposite.unop X \u27f6 X_1) => PUnit.unit) X X_1 \u1fb0)\n            \u1fb0 :=\n  fun (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1) =>\n    Eq.refl (nat_trans.app (limits.cocone.\u03b9 (colimit_cocone X)) X_1 \u1fb0)\n\n/--\nThe proposed colimit cocone over `coyoneda.obj X` is a colimit cocone.\n-/\ndef colimit_cocone_is_colimit {C : Type v} [small_category C] (X : C\u1d52\u1d56) :\n    limits.is_colimit (colimit_cocone X) :=\n  limits.is_colimit.mk\n    fun (s : limits.cocone (functor.obj coyoneda X)) (x : limits.cocone.X (colimit_cocone X)) =>\n      nat_trans.app (limits.cocone.\u03b9 s) (opposite.unop X) \ud835\udfd9\n\nprotected instance obj.category_theory.limits.has_colimit {C : Type v} [small_category C]\n    (X : C\u1d52\u1d56) : limits.has_colimit (functor.obj coyoneda X) :=\n  limits.has_colimit.mk (limits.colimit_cocone.mk (colimit_cocone X) (colimit_cocone_is_colimit X))\n\n/--\nThe colimit of `coyoneda.obj X` is isomorphic to `punit`.\n-/\ndef colimit_coyoneda_iso {C : Type v} [small_category C] (X : C\u1d52\u1d56) :\n    limits.colimit (functor.obj coyoneda X) \u2245 PUnit :=\n  limits.colimit.iso_colimit_cocone\n    (limits.colimit_cocone.mk (colimit_cocone X) (colimit_cocone_is_colimit X))\n\nend coyoneda\n\n\n/-- The yoneda embedding `yoneda.obj X : C\u1d52\u1d56 \u2964 Type v` for `X : C` preserves limits. -/\nprotected instance yoneda_preserves_limits {C : Type u} [category C] (X : C) :\n    limits.preserves_limits (functor.obj yoneda X) :=\n  limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : small_category J) =>\n      limits.preserves_limits_of_shape.mk\n        fun (K : J \u2964 (C\u1d52\u1d56)) =>\n          limits.preserves_limit.mk\n            fun (c : limits.cone K) (t : limits.is_limit c) =>\n              limits.is_limit.mk\n                fun (s : limits.cone (K \u22d9 functor.obj yoneda X)) (x : limits.cone.X s) =>\n                  has_hom.hom.unop\n                    (limits.is_limit.lift t\n                      (limits.cone.mk (opposite.op X)\n                        (nat_trans.mk\n                          fun (j : J) => has_hom.hom.op (nat_trans.app (limits.cone.\u03c0 s) j x))))\n\n/-- The coyoneda embedding `coyoneda.obj X : C \u2964 Type v` for `X : C\u1d52\u1d56` preserves limits. -/\nprotected instance coyoneda_preserves_limits {C : Type u} [category C] (X : C\u1d52\u1d56) :\n    limits.preserves_limits (functor.obj coyoneda X) :=\n  limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : small_category J) =>\n      limits.preserves_limits_of_shape.mk\n        fun (K : J \u2964 C) =>\n          limits.preserves_limit.mk\n            fun (c : limits.cone K) (t : limits.is_limit c) =>\n              limits.is_limit.mk\n                fun (s : limits.cone (K \u22d9 functor.obj coyoneda X)) (x : limits.cone.X s) =>\n                  limits.is_limit.lift t\n                    (limits.cone.mk (opposite.unop X)\n                      (nat_trans.mk fun (j : J) => nat_trans.app (limits.cone.\u03c0 s) j x))\n\n/-- The yoneda embeddings jointly reflect limits. -/\ndef yoneda_jointly_reflects_limits {C : Type u} [category C] (J : Type v) [small_category J]\n    (K : J \u2964 (C\u1d52\u1d56)) (c : limits.cone K)\n    (t : (X : C) \u2192 limits.is_limit (functor.map_cone (functor.obj yoneda X) c)) :\n    limits.is_limit c :=\n  let s' :\n    (s : limits.cone K) \u2192 limits.cone (K \u22d9 functor.obj yoneda (opposite.unop (limits.cone.X s))) :=\n    fun (s : limits.cone K) =>\n      limits.cone.mk PUnit\n        (nat_trans.mk\n          fun (j : J) (_x : functor.obj (functor.obj (functor.const J) PUnit) j) =>\n            has_hom.hom.unop (nat_trans.app (limits.cone.\u03c0 s) j));\n  limits.is_limit.mk\n    fun (s : limits.cone K) =>\n      has_hom.hom.op (limits.is_limit.lift (t (opposite.unop (limits.cone.X s))) (s' s) PUnit.unit)\n\n/-- The coyoneda embeddings jointly reflect limits. -/\ndef coyoneda_jointly_reflects_limits {C : Type u} [category C] (J : Type v) [small_category J]\n    (K : J \u2964 C) (c : limits.cone K)\n    (t : (X : C\u1d52\u1d56) \u2192 limits.is_limit (functor.map_cone (functor.obj coyoneda X) c)) :\n    limits.is_limit c :=\n  let s' :\n    (s : limits.cone K) \u2192 limits.cone (K \u22d9 functor.obj coyoneda (opposite.op (limits.cone.X s))) :=\n    fun (s : limits.cone K) =>\n      limits.cone.mk PUnit\n        (nat_trans.mk\n          fun (j : J) (_x : functor.obj (functor.obj (functor.const J) PUnit) j) =>\n            nat_trans.app (limits.cone.\u03c0 s) j);\n  limits.is_limit.mk\n    fun (s : limits.cone K) =>\n      limits.is_limit.lift (t (opposite.op (limits.cone.X s))) (s' s) PUnit.unit\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/yoneda_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3750347373860322}}
{"text": "import tactic\n\nopen tactic expr\n\n------------------------------------------------------------------------\n-- \u00a7 Definici\u00f3n de la t\u00e1ctica assumption                              --\n------------------------------------------------------------------------\n\n-- 1\u00aa versi\u00f3n\n-- ==========\n\nmeta def test (hyp tgt : expr) : tactic bool :=\ndo hyp_tp \u2190 infer_type hyp,\n   return (hyp_tp = tgt)\n\nmeta def map_over_lc (tgt : expr) : list expr \u2192 tactic unit \n| []     := fail \"nada del contexto coincide con el objetivo\"\n| (h::t) := do is_match \u2190 test h tgt,\n               if is_match then exact h\n                           else map_over_lc t\n\nmeta def assump : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   map_over_lc tgt ctx\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  (hC : C)\n  : C :=\nby assump\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  : C :=\nby assump\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene\n--    nada del contexto coincide con el objetivo\n--    state:\n--    A B C : Prop,\n--    hA : A,\n--    hB : B\n--    \u22a2 C\n\n-- 2\u00aa versi\u00f3n\n-- ==========\n\nmeta def test2 (hyp tgt : expr) : tactic unit :=\ndo hyp_tp \u2190 infer_type hyp,\n   guard (hyp_tp = tgt)\n\nmeta def map_over_lc2 (tgt : expr) : list expr \u2192 tactic unit \n| []     := fail \"nada del contexto coincide con el objetivo\"\n| (h::t) := (do test2 h tgt, exact h) <|> map_over_lc2 t\n\nmeta def assump2 : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   map_over_lc2 tgt ctx\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  (hC : C)\n  : C :=\nby assump2\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  : C :=\nby assump2\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene\n--    nada del contexto coincide con el objetivo\n--    state:\n--    A B C : Prop,\n--    hA : A,\n--    hB : B\n--    \u22a2 C\n\n-- 3\u00aa versi\u00f3n\n-- ==========\n\nmeta def test_and_exact (hyp tgt : expr) : tactic unit :=\ndo hyp_tp \u2190 infer_type hyp,\n   guard (hyp_tp = tgt),\n   exact hyp\n\nmeta def map_over_lc3 (tgt : expr) : list expr \u2192 tactic unit \n| []     := fail \"nada del contexto coincide con el objetivo\"\n| (h::t) := test_and_exact h tgt <|> map_over_lc3 t\n\nmeta def assump3 : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   map_over_lc3 tgt ctx\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  (hC : C)\n  : C :=\nby assump3\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  : C :=\nby assump3\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene\n--    nada del contexto coincide con el objetivo\n--    state:\n--    A B C : Prop,\n--    hA : A,\n--    hB : B\n--    \u22a2 C\n\n-- 4\u00aa versi\u00f3n\n-- ==========\n\nmeta def assump4 : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   ctx.mfirst (\u03bb e, test_and_exact e tgt)\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  (hC : C)\n  : C :=\nby assump4\n\nexample\n  (A B C : Prop)\n  (hA : A)\n  (hB : B)\n  : C :=\nby assump4\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene\n--    failed\n--    state:\n--    A B C : Prop,\n--    hA : A,\n--    hB : B\n--    \u22a2 C\n\n\nexample (n : \u2115) (hx : n + 0 = 5) : n = 5 :=\nby assump4\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene\n--    failed\n--    state:\n--    n : \u2115,\n--    hx : n + 0 = 5\n--    \u22a2 n = 5\n\n-- 5\u00aa versi\u00f3n\n-- ==========\n\nmeta def test_and_exact_def (hyp tgt : expr) : tactic unit :=\ndo hyp_tp \u2190 infer_type hyp,\n   is_def_eq hyp_tp tgt,\n   exact hyp\n\nmeta def assump5 : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   ctx.mfirst (\u03bb e, test_and_exact_def e tgt)\n\nexample (n : \u2115) (hx : n + 0 = 5) : n = 5 :=\nby assump5\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene \u00e9xito\n\n-- 6\u00aa versi\u00f3n\n-- ==========\n\nmeta def assump6 : tactic unit :=\ndo tgt \u2190 target,\n   ctx \u2190 local_context,\n   ctx.mfirst (\u03bb e, exact e)\n\nexample (n : \u2115) (hx : n + 0 = 6) : n = 6 :=\nby assump6\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene \u00e9xito\n\n-- 7\u00aa versi\u00f3n\n-- ==========\n\nmeta def assump7 : tactic unit :=\nlocal_context >>= list.mfirst exact\n\nexample (n : \u2115) (hx : n + 0 = 7) : n = 7 :=\nby assump7\n\n-- Comentario: Al colocar el cursor sobre assump se obtiene \u00e9xito\n\n----------------------------------------------------------------------\n-- \u00a7\u00a7 Definici\u00f3n de la t\u00e1ctica add_refl                             --\n----------------------------------------------------------------------\n\n-- 1\u00aa versi\u00f3n\n-- ==========\n\nmeta def add_single_refl (e : expr) : tactic unit :=\ndo tp \u2190 infer_type e,\n   guard (tp = `(\u2115)),\n   pf \u2190 mk_app `eq.refl [e],\n   nm \u2190 get_unused_name,\n   note nm none pf,\n   skip\n\nmeta def add_refl : tactic unit :=\ndo ctx \u2190 local_context,\n   ctx.mmap' (\u03bb e, add_single_refl e)\n\nexample (a b c : \u2115) (ha : a = b) : true :=\nby do add_refl\n\n-- Comentario: Al colocar el cursor sobre do se obtiene\n--    failed\n--    state:\n--    a b c : \u2115,\n--    ha : a = b,\n--    _x : a = a,\n--    _x_1 : b = b,\n--    _x_2 : c = c\n--    \u22a2 true\n\n-- 2\u00aa versi\u00f3n\n-- ==========\n\nmeta def add_refl2 : tactic unit :=\ndo ctx \u2190 local_context,\n   ctx.mmap' (\u03bb e, try (add_single_refl e))\n\nexample (a b c : \u2115) (ha : a = b) : true :=\nby do add_refl2\n\n-- Comentario: Al colocar el cursor sobre do se obtiene\n--    tactic failed, there are unsolved goals\n--    state:\n--    a b c : \u2115,\n--    ha : a = b,\n--    _x : a = a,\n--    _x_1 : b = b,\n--    _x_2 : c = c\n--    \u22a2 true\n\n-- 3\u00aa versi\u00f3n\n-- ==========\n\nmeta def add_single_refl3 (e : expr) : tactic unit :=\ndo tp \u2190 infer_type e,\n   guard (tp = `(\u2115)),\n   pf \u2190 mk_app `eq.refl [e],\n   nm \u2190 get_unused_name e.local_pp_name,\n   note nm none pf,\n   skip\n\nmeta def add_refl3 : tactic unit :=\ndo ctx \u2190 local_context,\n   ctx.mmap' (\u03bb e, try (add_single_refl3 e))\n\nexample (a b c : \u2115) (ha : a = b) : true :=\nby do add_refl3\n\n-- Comentario: Al colocar el cursor sobre do se obtiene\n--    tactic failed, there are unsolved goals\n--    state:\n--    a b c : \u2115,\n--    ha : a = b,\n--    a_1 : a = a,\n--    b_1 : b = b,\n--    c_1 : c = c\n--    \u22a2 true\n\n-- 4\u00aa versi\u00f3n\n-- ==========\n\nmeta def add_single_refl4 (e : expr) : tactic unit :=\ndo tp \u2190 infer_type e,\n   guard (tp = `(\u2115)),\n   pf \u2190 to_expr ``(not_lt_of_ge (nat.zero_le %%e)),\n   nm \u2190 get_unused_name e.local_pp_name,\n   note nm none pf,\n   skip\n\nmeta def add_refl4 : tactic unit :=\ndo ctx \u2190 local_context,\n   ctx.mmap' (\u03bb e, try (add_single_refl4 e))\n\nexample (a b c : \u2115) (ha : a = b) : true :=\nby do add_refl4\n\n-- Comentario: Al colocar el cursor sobre do se obtiene\n--    tactic failed, there are unsolved goals\n--    state:\n--    a b c : \u2115,\n--    ha : a = b,\n--    a_1 : \u00aca < 0,\n--    b_1 : \u00acb < 0,\n--    c_1 : \u00acc < 0\n--    \u22a2 true\n\n------------------------------------------------------------------------\n-- \u00a7 Referencia                                                       --\n------------------------------------------------------------------------\n\n-- Basado en el v\u00eddeo \"Metaprogramming in Lean tutorial: video 5\" de Rob\n-- Lewis que se encuentra en https://youtu.be/-RQQxFVZnn4\n", "meta": {"author": "jaalonso", "repo": "Lean_para_matematicos", "sha": "924c77b7f010604b84f82d2f79967ad8b9cddc6e", "save_path": "github-repos/lean/jaalonso-Lean_para_matematicos", "path": "github-repos/lean/jaalonso-Lean_para_matematicos/Lean_para_matematicos-924c77b7f010604b84f82d2f79967ad8b9cddc6e/src/Metaprogramacion/Introduccion_a_la_metaprogramacion_5.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6859494485880928, "lm_q1q2_score": 0.37503473387591874}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.analysis.convex.basic\nimport Mathlib.analysis.normed_space.bounded_linear_maps\nimport Mathlib.analysis.specific_limits\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Tangent cone\n\nIn this file, we define two predicates `unique_diff_within_at \ud835\udd5c s x` and `unique_diff_on \ud835\udd5c s`\nensuring that, if a function has two derivatives, then they have to coincide. As a direct\ndefinition of this fact (quantifying on all target types and all functions) would depend on\nuniverses, we use a more intrinsic definition: if all the possible tangent directions to the set\n`s` at the point `x` span a dense subset of the whole subset, it is easy to check that the\nderivative has to be unique.\n\nTherefore, we introduce the set of all tangent directions, named `tangent_cone_at`,\nand express `unique_diff_within_at` and `unique_diff_on` in terms of it.\nOne should however think of this definition as an implementation detail: the only reason to\nintroduce the predicates `unique_diff_within_at` and `unique_diff_on` is to ensure the uniqueness\nof the derivative. This is why their names reflect their uses, and not how they are defined.\n\n## Implementation details\n\nNote that this file is imported by `fderiv.lean`. Hence, derivatives are not defined yet. The\nproperty of uniqueness of the derivative is therefore proved in `fderiv.lean`, but based on the\nproperties of the tangent cone we prove here.\n-/\n\n/-- The set of all tangent directions to the set `s` at the point `x`. -/\ndef tangent_cone_at (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] (s : set E) (x : E) : set E :=\n  set_of\n    fun (y : E) =>\n      \u2203 (c : \u2115 \u2192 \ud835\udd5c),\n        \u2203 (d : \u2115 \u2192 E),\n          filter.eventually (fun (n : \u2115) => x + d n \u2208 s) filter.at_top \u2227\n            filter.tendsto (fun (n : \u2115) => norm (c n)) filter.at_top filter.at_top \u2227\n              filter.tendsto (fun (n : \u2115) => c n \u2022 d n) filter.at_top (nhds y)\n\n/-- A property ensuring that the tangent cone to `s` at `x` spans a dense subset of the whole space.\nThe main role of this property is to ensure that the differential within `s` at `x` is unique,\nhence this name. The uniqueness it asserts is proved in `unique_diff_within_at.eq` in `fderiv.lean`.\nTo avoid pathologies in dimension 0, we also require that `x` belongs to the closure of `s` (which\nis automatic when `E` is not `0`-dimensional).\n -/\ndef unique_diff_within_at (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] (s : set E) (x : E) :=\n  dense \u2191(submodule.span \ud835\udd5c (tangent_cone_at \ud835\udd5c s x)) \u2227 x \u2208 closure s\n\n/-- A property ensuring that the tangent cone to `s` at any of its points spans a dense subset of\nthe whole space.  The main role of this property is to ensure that the differential along `s` is\nunique, hence this name. The uniqueness it asserts is proved in `unique_diff_on.eq` in\n`fderiv.lean`. -/\ndef unique_diff_on (\ud835\udd5c : Type u_1) [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] (s : set E) :=\n  \u2200 (x : E), x \u2208 s \u2192 unique_diff_within_at \ud835\udd5c s x\n\n/- This section is devoted to the properties of the tangent cone. -/\n\ntheorem tangent_cone_univ {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} : tangent_cone_at \ud835\udd5c set.univ x = set.univ := sorry\n\ntheorem tangent_cone_mono {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (h : s \u2286 t) : tangent_cone_at \ud835\udd5c s x \u2286 tangent_cone_at \ud835\udd5c t x := sorry\n\n/-- Auxiliary lemma ensuring that, under the assumptions defining the tangent cone,\nthe sequence `d` tends to 0 at infinity. -/\ntheorem tangent_cone_at.lim_zero {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {y : E} {\u03b1 : Type u_3} (l : filter \u03b1) {c : \u03b1 \u2192 \ud835\udd5c} {d : \u03b1 \u2192 E} (hc : filter.tendsto (fun (n : \u03b1) => norm (c n)) l filter.at_top) (hd : filter.tendsto (fun (n : \u03b1) => c n \u2022 d n) l (nhds y)) : filter.tendsto d l (nhds 0) := sorry\n\ntheorem tangent_cone_mono_nhds {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (h : nhds_within x s \u2264 nhds_within x t) : tangent_cone_at \ud835\udd5c s x \u2286 tangent_cone_at \ud835\udd5c t x := sorry\n\n/-- Tangent cone of `s` at `x` depends only on `\ud835\udcdd[s] x`. -/\ntheorem tangent_cone_congr {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (h : nhds_within x s = nhds_within x t) : tangent_cone_at \ud835\udd5c s x = tangent_cone_at \ud835\udd5c t x :=\n  set.subset.antisymm (tangent_cone_mono_nhds (le_of_eq h)) (tangent_cone_mono_nhds (le_of_eq (Eq.symm h)))\n\n/-- Intersecting with a neighborhood of the point does not change the tangent cone. -/\ntheorem tangent_cone_inter_nhds {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (ht : t \u2208 nhds x) : tangent_cone_at \ud835\udd5c (s \u2229 t) x = tangent_cone_at \ud835\udd5c s x :=\n  tangent_cone_congr (Eq.symm (nhds_within_restrict' s ht))\n\n/-- The tangent cone of a product contains the tangent cone of its left factor. -/\ntheorem subset_tangent_cone_prod_left {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] {x : E} {s : set E} {t : set F} {y : F} (ht : y \u2208 closure t) : \u21d1(linear_map.inl \ud835\udd5c E F) '' tangent_cone_at \ud835\udd5c s x \u2286 tangent_cone_at \ud835\udd5c (set.prod s t) (x, y) := sorry\n\n/-- The tangent cone of a product contains the tangent cone of its right factor. -/\ntheorem subset_tangent_cone_prod_right {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] {x : E} {s : set E} {t : set F} {y : F} (hs : x \u2208 closure s) : \u21d1(linear_map.inr \ud835\udd5c E F) '' tangent_cone_at \ud835\udd5c t y \u2286 tangent_cone_at \ud835\udd5c (set.prod s t) (x, y) := sorry\n\n/-- If a subset of a real vector space contains a segment, then the direction of this\nsegment belongs to the tangent cone at its endpoints. -/\ntheorem mem_tangent_cone_of_segment_subset {G : Type u_4} [normed_group G] [normed_space \u211d G] {s : set G} {x : G} {y : G} (h : segment x y \u2286 s) : y - x \u2208 tangent_cone_at \u211d s x := sorry\n\n/-!\n### Properties of `unique_diff_within_at` and `unique_diff_on`\n\nThis section is devoted to properties of the predicates `unique_diff_within_at` and `unique_diff_on`. -/\n\ntheorem unique_diff_on.unique_diff_within_at {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {s : set E} {x : E} (hs : unique_diff_on \ud835\udd5c s) (h : x \u2208 s) : unique_diff_within_at \ud835\udd5c s x :=\n  hs x h\n\ntheorem unique_diff_within_at_univ {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} : unique_diff_within_at \ud835\udd5c set.univ x := sorry\n\ntheorem unique_diff_on_univ {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] : unique_diff_on \ud835\udd5c set.univ :=\n  fun (x : E) (hx : x \u2208 set.univ) => unique_diff_within_at_univ\n\ntheorem unique_diff_on_empty {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] : unique_diff_on \ud835\udd5c \u2205 :=\n  fun (x : E) (hx : x \u2208 \u2205) => false.elim hx\n\ntheorem unique_diff_within_at.mono_nhds {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (h : unique_diff_within_at \ud835\udd5c s x) (st : nhds_within x s \u2264 nhds_within x t) : unique_diff_within_at \ud835\udd5c t x := sorry\n\ntheorem unique_diff_within_at.mono {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (h : unique_diff_within_at \ud835\udd5c s x) (st : s \u2286 t) : unique_diff_within_at \ud835\udd5c t x :=\n  unique_diff_within_at.mono_nhds h (nhds_within_mono x st)\n\ntheorem unique_diff_within_at_congr {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (st : nhds_within x s = nhds_within x t) : unique_diff_within_at \ud835\udd5c s x \u2194 unique_diff_within_at \ud835\udd5c t x :=\n  { mp := fun (h : unique_diff_within_at \ud835\udd5c s x) => unique_diff_within_at.mono_nhds h (le_of_eq st),\n    mpr := fun (h : unique_diff_within_at \ud835\udd5c t x) => unique_diff_within_at.mono_nhds h (le_of_eq (Eq.symm st)) }\n\ntheorem unique_diff_within_at_inter {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (ht : t \u2208 nhds x) : unique_diff_within_at \ud835\udd5c (s \u2229 t) x \u2194 unique_diff_within_at \ud835\udd5c s x :=\n  unique_diff_within_at_congr (Eq.symm (nhds_within_restrict' s ht))\n\ntheorem unique_diff_within_at.inter {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (hs : unique_diff_within_at \ud835\udd5c s x) (ht : t \u2208 nhds x) : unique_diff_within_at \ud835\udd5c (s \u2229 t) x :=\n  iff.mpr (unique_diff_within_at_inter ht) hs\n\ntheorem unique_diff_within_at_inter' {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (ht : t \u2208 nhds_within x s) : unique_diff_within_at \ud835\udd5c (s \u2229 t) x \u2194 unique_diff_within_at \ud835\udd5c s x :=\n  unique_diff_within_at_congr (Eq.symm (nhds_within_restrict'' s ht))\n\ntheorem unique_diff_within_at.inter' {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} {t : set E} (hs : unique_diff_within_at \ud835\udd5c s x) (ht : t \u2208 nhds_within x s) : unique_diff_within_at \ud835\udd5c (s \u2229 t) x :=\n  iff.mpr (unique_diff_within_at_inter' ht) hs\n\ntheorem unique_diff_within_at_of_mem_nhds {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} (h : s \u2208 nhds x) : unique_diff_within_at \ud835\udd5c s x := sorry\n\ntheorem is_open.unique_diff_within_at {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {x : E} {s : set E} (hs : is_open s) (xs : x \u2208 s) : unique_diff_within_at \ud835\udd5c s x :=\n  unique_diff_within_at_of_mem_nhds (mem_nhds_sets hs xs)\n\ntheorem unique_diff_on.inter {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {s : set E} {t : set E} (hs : unique_diff_on \ud835\udd5c s) (ht : is_open t) : unique_diff_on \ud835\udd5c (s \u2229 t) :=\n  fun (x : E) (hx : x \u2208 s \u2229 t) => unique_diff_within_at.inter (hs x (and.left hx)) (mem_nhds_sets ht (and.right hx))\n\ntheorem is_open.unique_diff_on {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {s : set E} (hs : is_open s) : unique_diff_on \ud835\udd5c s :=\n  fun (x : E) (hx : x \u2208 s) => is_open.unique_diff_within_at hs hx\n\n/-- The product of two sets of unique differentiability at points `x` and `y` has unique\ndifferentiability at `(x, y)`. -/\ntheorem unique_diff_within_at.prod {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] {x : E} {s : set E} {t : set F} {y : F} (hs : unique_diff_within_at \ud835\udd5c s x) (ht : unique_diff_within_at \ud835\udd5c t y) : unique_diff_within_at \ud835\udd5c (set.prod s t) (x, y) := sorry\n\n/-- The product of two sets of unique differentiability is a set of unique differentiability. -/\ntheorem unique_diff_on.prod {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] {s : set E} {t : set F} (hs : unique_diff_on \ud835\udd5c s) (ht : unique_diff_on \ud835\udd5c t) : unique_diff_on \ud835\udd5c (set.prod s t) := sorry\n\n/-- In a real vector space, a convex set with nonempty interior is a set of unique\ndifferentiability. -/\ntheorem unique_diff_on_convex {G : Type u_4} [normed_group G] [normed_space \u211d G] {s : set G} (conv : convex s) (hs : set.nonempty (interior s)) : unique_diff_on \u211d s := sorry\n\ntheorem unique_diff_on_Ici (a : \u211d) : unique_diff_on \u211d (set.Ici a) := sorry\n\ntheorem unique_diff_on_Iic (a : \u211d) : unique_diff_on \u211d (set.Iic a) := sorry\n\ntheorem unique_diff_on_Ioi (a : \u211d) : unique_diff_on \u211d (set.Ioi a) :=\n  is_open.unique_diff_on is_open_Ioi\n\ntheorem unique_diff_on_Iio (a : \u211d) : unique_diff_on \u211d (set.Iio a) :=\n  is_open.unique_diff_on is_open_Iio\n\ntheorem unique_diff_on_Icc {a : \u211d} {b : \u211d} (hab : a < b) : unique_diff_on \u211d (set.Icc a b) := sorry\n\ntheorem unique_diff_on_Ico (a : \u211d) (b : \u211d) : unique_diff_on \u211d (set.Ico a b) := sorry\n\ntheorem unique_diff_on_Ioc (a : \u211d) (b : \u211d) : unique_diff_on \u211d (set.Ioc a b) := sorry\n\ntheorem unique_diff_on_Ioo (a : \u211d) (b : \u211d) : unique_diff_on \u211d (set.Ioo a b) :=\n  is_open.unique_diff_on is_open_Ioo\n\n/-- The real interval `[0, 1]` is a set of unique differentiability. -/\ntheorem unique_diff_on_Icc_zero_one : unique_diff_on \u211d (set.Icc 0 1) :=\n  unique_diff_on_Icc zero_lt_one\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/calculus/tangent_cone.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.374836778090798}}
{"text": "variables {p q : Prop} (hp : p) (hq : q)\n\ninclude hp hq\n\nexample : p \u2227 q \u2227 p :=\nbegin\n  apply and.intro hp,\n  exact and.intro hq hp\nend\n\nomit hp hq\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0109.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.585101154203231, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.3748367780907979}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.category_theory.epi_mono\nimport Mathlib.PostPort\n\nuniverses v l u_1 u_2 u u\u2082 \n\nnamespace Mathlib\n\n/-!\n# Binary (co)products\n\nWe define a category `walking_pair`, which is the index category\nfor a binary (co)product diagram. A convenience method `pair X Y`\nconstructs the functor from the walking pair, hitting the given objects.\n\nWe define `prod X Y` and `coprod X Y` as limits and colimits of such functors.\n\nTypeclasses `has_binary_products` and `has_binary_coproducts` assert the existence\nof (co)limits shaped as walking pairs.\n\nWe include lemmas for simplifying equations involving projections and coprojections, and define\nbraiding and associating isomorphisms, and the product comparison morphism.\n\n## References\n* [Stacks: Products of pairs](https://stacks.math.columbia.edu/tag/001R)\n* [Stacks: coproducts of pairs](https://stacks.math.columbia.edu/tag/04AN)\n-/\n\nnamespace category_theory.limits\n\n\n/-- The type of objects for the diagram indexing a binary (co)product. -/\ninductive walking_pair \nwhere\n| left : walking_pair\n| right : walking_pair\n\n/--\nThe equivalence swapping left and right.\n-/\ndef walking_pair.swap : walking_pair \u2243 walking_pair :=\n  equiv.mk (fun (j : walking_pair) => walking_pair.rec_on j walking_pair.right walking_pair.left)\n    (fun (j : walking_pair) => walking_pair.rec_on j walking_pair.right walking_pair.left) sorry sorry\n\n@[simp] theorem walking_pair.swap_apply_left : coe_fn walking_pair.swap walking_pair.left = walking_pair.right :=\n  rfl\n\n@[simp] theorem walking_pair.swap_apply_right : coe_fn walking_pair.swap walking_pair.right = walking_pair.left :=\n  rfl\n\n@[simp] theorem walking_pair.swap_symm_apply_tt : coe_fn (equiv.symm walking_pair.swap) walking_pair.left = walking_pair.right :=\n  rfl\n\n@[simp] theorem walking_pair.swap_symm_apply_ff : coe_fn (equiv.symm walking_pair.swap) walking_pair.right = walking_pair.left :=\n  rfl\n\n/--\nAn equivalence from `walking_pair` to `bool`, sometimes useful when reindexing limits.\n-/\ndef walking_pair.equiv_bool : walking_pair \u2243 Bool :=\n  equiv.mk (fun (j : walking_pair) => walking_pair.rec_on j tt false)\n    (fun (b : Bool) => bool.rec_on b walking_pair.right walking_pair.left) sorry sorry\n\n@[simp] theorem walking_pair.equiv_bool_apply_left : coe_fn walking_pair.equiv_bool walking_pair.left = tt :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_apply_right : coe_fn walking_pair.equiv_bool walking_pair.right = false :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_symm_apply_tt : coe_fn (equiv.symm walking_pair.equiv_bool) tt = walking_pair.left :=\n  rfl\n\n@[simp] theorem walking_pair.equiv_bool_symm_apply_ff : coe_fn (equiv.symm walking_pair.equiv_bool) false = walking_pair.right :=\n  rfl\n\n/-- The diagram on the walking pair, sending the two points to `X` and `Y`. -/\ndef pair {C : Type u} [category C] (X : C) (Y : C) : discrete walking_pair \u2964 C :=\n  discrete.functor fun (j : walking_pair) => walking_pair.cases_on j X Y\n\n@[simp] theorem pair_obj_left {C : Type u} [category C] (X : C) (Y : C) : functor.obj (pair X Y) walking_pair.left = X :=\n  rfl\n\n@[simp] theorem pair_obj_right {C : Type u} [category C] (X : C) (Y : C) : functor.obj (pair X Y) walking_pair.right = Y :=\n  rfl\n\n/-- The natural transformation between two functors out of the walking pair, specified by its components. -/\ndef map_pair {C : Type u} [category C] {F : discrete walking_pair \u2964 C} {G : discrete walking_pair \u2964 C} (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left) (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) : F \u27f6 G :=\n  nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j f g\n\n@[simp] theorem map_pair_left {C : Type u} [category C] {F : discrete walking_pair \u2964 C} {G : discrete walking_pair \u2964 C} (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left) (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) : nat_trans.app (map_pair f g) walking_pair.left = f :=\n  rfl\n\n@[simp] theorem map_pair_right {C : Type u} [category C] {F : discrete walking_pair \u2964 C} {G : discrete walking_pair \u2964 C} (f : functor.obj F walking_pair.left \u27f6 functor.obj G walking_pair.left) (g : functor.obj F walking_pair.right \u27f6 functor.obj G walking_pair.right) : nat_trans.app (map_pair f g) walking_pair.right = g :=\n  rfl\n\n/-- The natural isomorphism between two functors out of the walking pair, specified by its components. -/\ndef map_pair_iso {C : Type u} [category C] {F : discrete walking_pair \u2964 C} {G : discrete walking_pair \u2964 C} (f : functor.obj F walking_pair.left \u2245 functor.obj G walking_pair.left) (g : functor.obj F walking_pair.right \u2245 functor.obj G walking_pair.right) : F \u2245 G :=\n  nat_iso.of_components (fun (j : discrete walking_pair) => walking_pair.cases_on j f g) sorry\n\n/-- Every functor out of the walking pair is naturally isomorphic (actually, equal) to a `pair` -/\n@[simp] theorem diagram_iso_pair_inv_app {C : Type u} [category C] (F : discrete walking_pair \u2964 C) (X : discrete walking_pair) : nat_trans.app (iso.inv (diagram_iso_pair F)) X =\n  iso.inv\n    (walking_pair.rec (iso.refl (functor.obj F walking_pair.left)) (iso.refl (functor.obj F walking_pair.right)) X) :=\n  Eq.refl\n    (iso.inv\n      (walking_pair.rec (iso.refl (functor.obj F walking_pair.left)) (iso.refl (functor.obj F walking_pair.right)) X))\n\n/-- The natural isomorphism between `pair X Y \u22d9 F` and `pair (F.obj X) (F.obj Y)`. -/\ndef pair_comp {C : Type u} [category C] {D : Type u} [category D] (X : C) (Y : C) (F : C \u2964 D) : pair X Y \u22d9 F \u2245 pair (functor.obj F X) (functor.obj F Y) :=\n  diagram_iso_pair (pair X Y \u22d9 F)\n\n/-- A binary fan is just a cone on a diagram indexing a product. -/\ndef binary_fan {C : Type u} [category C] (X : C) (Y : C) :=\n  cone (pair X Y)\n\n/-- The first projection of a binary fan. -/\ndef binary_fan.fst {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) : functor.obj (functor.obj (functor.const (discrete walking_pair)) (cone.X s)) walking_pair.left \u27f6\n  functor.obj (pair X Y) walking_pair.left :=\n  nat_trans.app (cone.\u03c0 s) walking_pair.left\n\n/-- The second projection of a binary fan. -/\ndef binary_fan.snd {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) : functor.obj (functor.obj (functor.const (discrete walking_pair)) (cone.X s)) walking_pair.right \u27f6\n  functor.obj (pair X Y) walking_pair.right :=\n  nat_trans.app (cone.\u03c0 s) walking_pair.right\n\n@[simp] theorem binary_fan.\u03c0_app_left {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) : nat_trans.app (cone.\u03c0 s) walking_pair.left = binary_fan.fst s :=\n  rfl\n\n@[simp] theorem binary_fan.\u03c0_app_right {C : Type u} [category C] {X : C} {Y : C} (s : binary_fan X Y) : nat_trans.app (cone.\u03c0 s) walking_pair.right = binary_fan.snd s :=\n  rfl\n\ntheorem binary_fan.is_limit.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} {s : binary_fan X Y} (h : is_limit s) {f : W \u27f6 cone.X s} {g : W \u27f6 cone.X s} (h\u2081 : f \u226b binary_fan.fst s = g \u226b binary_fan.fst s) (h\u2082 : f \u226b binary_fan.snd s = g \u226b binary_fan.snd s) : f = g :=\n  is_limit.hom_ext h fun (j : discrete walking_pair) => walking_pair.cases_on j h\u2081 h\u2082\n\n/-- A binary cofan is just a cocone on a diagram indexing a coproduct. -/\ndef binary_cofan {C : Type u} [category C] (X : C) (Y : C) :=\n  cocone (pair X Y)\n\n/-- The first inclusion of a binary cofan. -/\ndef binary_cofan.inl {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) : functor.obj (pair X Y) walking_pair.left \u27f6\n  functor.obj (functor.obj (functor.const (discrete walking_pair)) (cocone.X s)) walking_pair.left :=\n  nat_trans.app (cocone.\u03b9 s) walking_pair.left\n\n/-- The second inclusion of a binary cofan. -/\ndef binary_cofan.inr {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) : functor.obj (pair X Y) walking_pair.right \u27f6\n  functor.obj (functor.obj (functor.const (discrete walking_pair)) (cocone.X s)) walking_pair.right :=\n  nat_trans.app (cocone.\u03b9 s) walking_pair.right\n\n@[simp] theorem binary_cofan.\u03b9_app_left {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) : nat_trans.app (cocone.\u03b9 s) walking_pair.left = binary_cofan.inl s :=\n  rfl\n\n@[simp] theorem binary_cofan.\u03b9_app_right {C : Type u} [category C] {X : C} {Y : C} (s : binary_cofan X Y) : nat_trans.app (cocone.\u03b9 s) walking_pair.right = binary_cofan.inr s :=\n  rfl\n\ntheorem binary_cofan.is_colimit.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} {s : binary_cofan X Y} (h : is_colimit s) {f : cocone.X s \u27f6 W} {g : cocone.X s \u27f6 W} (h\u2081 : binary_cofan.inl s \u226b f = binary_cofan.inl s \u226b g) (h\u2082 : binary_cofan.inr s \u226b f = binary_cofan.inr s \u226b g) : f = g :=\n  is_colimit.hom_ext h fun (j : discrete walking_pair) => walking_pair.cases_on j h\u2081 h\u2082\n\n/-- A binary fan with vertex `P` consists of the two projections `\u03c0\u2081 : P \u27f6 X` and `\u03c0\u2082 : P \u27f6 Y`. -/\ndef binary_fan.mk {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) : binary_fan X Y :=\n  cone.mk P (nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j \u03c0\u2081 \u03c0\u2082)\n\n/-- A binary cofan with vertex `P` consists of the two inclusions `\u03b9\u2081 : X \u27f6 P` and `\u03b9\u2082 : Y \u27f6 P`. -/\ndef binary_cofan.mk {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) : binary_cofan X Y :=\n  cocone.mk P (nat_trans.mk fun (j : discrete walking_pair) => walking_pair.cases_on j \u03b9\u2081 \u03b9\u2082)\n\n@[simp] theorem binary_fan.mk_\u03c0_app_left {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) : nat_trans.app (cone.\u03c0 (binary_fan.mk \u03c0\u2081 \u03c0\u2082)) walking_pair.left = \u03c0\u2081 :=\n  rfl\n\n@[simp] theorem binary_fan.mk_\u03c0_app_right {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03c0\u2081 : P \u27f6 X) (\u03c0\u2082 : P \u27f6 Y) : nat_trans.app (cone.\u03c0 (binary_fan.mk \u03c0\u2081 \u03c0\u2082)) walking_pair.right = \u03c0\u2082 :=\n  rfl\n\n@[simp] theorem binary_cofan.mk_\u03b9_app_left {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) : nat_trans.app (cocone.\u03b9 (binary_cofan.mk \u03b9\u2081 \u03b9\u2082)) walking_pair.left = \u03b9\u2081 :=\n  rfl\n\n@[simp] theorem binary_cofan.mk_\u03b9_app_right {C : Type u} [category C] {X : C} {Y : C} {P : C} (\u03b9\u2081 : X \u27f6 P) (\u03b9\u2082 : Y \u27f6 P) : nat_trans.app (cocone.\u03b9 (binary_cofan.mk \u03b9\u2081 \u03b9\u2082)) walking_pair.right = \u03b9\u2082 :=\n  rfl\n\n/-- If `s` is a limit binary fan over `X` and `Y`, then every pair of morphisms `f : W \u27f6 X` and\n    `g : W \u27f6 Y` induces a morphism `l : W \u27f6 s.X` satisfying `l \u226b s.fst = f` and `l \u226b s.snd = g`.\n    -/\n@[simp] theorem binary_fan.is_limit.lift'_coe {C : Type u} [category C] {W : C} {X : C} {Y : C} {s : binary_fan X Y} (h : is_limit s) (f : W \u27f6 X) (g : W \u27f6 Y) : \u2191(binary_fan.is_limit.lift' h f g) = is_limit.lift h (binary_fan.mk f g) :=\n  Eq.refl \u2191(binary_fan.is_limit.lift' h f g)\n\n/-- If `s` is a colimit binary cofan over `X` and `Y`,, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `l : s.X \u27f6 W` satisfying `s.inl \u226b l = f` and `s.inr \u226b l = g`.\n    -/\n@[simp] theorem binary_cofan.is_colimit.desc'_coe {C : Type u} [category C] {W : C} {X : C} {Y : C} {s : binary_cofan X Y} (h : is_colimit s) (f : X \u27f6 W) (g : Y \u27f6 W) : \u2191(binary_cofan.is_colimit.desc' h f g) = is_colimit.desc h (binary_cofan.mk f g) :=\n  Eq.refl \u2191(binary_cofan.is_colimit.desc' h f g)\n\n/-- An abbreviation for `has_limit (pair X Y)`. -/\n/-- An abbreviation for `has_colimit (pair X Y)`. -/\ndef has_binary_product {C : Type u} [category C] (X : C) (Y : C) :=\n  has_limit (pair X Y)\n\ndef has_binary_coproduct {C : Type u} [category C] (X : C) (Y : C) :=\n  has_colimit (pair X Y)\n\n/-- If we have a product of `X` and `Y`, we can access it using `prod X Y` or\n    `X \u2a2f Y`. -/\ndef prod {C : Type u} [category C] (X : C) (Y : C) [has_binary_product X Y] : C :=\n  limit (pair X Y)\n\n/-- If we have a coproduct of `X` and `Y`, we can access it using `coprod X Y ` or\n    `X \u2a3f Y`. -/\ndef coprod {C : Type u} [category C] (X : C) (Y : C) [has_binary_coproduct X Y] : C :=\n  colimit (pair X Y)\n\ninfixl:20 \" \u2a2f \" => Mathlib.category_theory.limits.prod\n\ninfixl:20 \" \u2a3f \" => Mathlib.category_theory.limits.coprod\n\n/-- The projection map to the first component of the product. -/\ndef prod.fst {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : X \u2a2f Y \u27f6 X :=\n  limit.\u03c0 (pair X Y) walking_pair.left\n\n/-- The projecton map to the second component of the product. -/\ndef prod.snd {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : X \u2a2f Y \u27f6 Y :=\n  limit.\u03c0 (pair X Y) walking_pair.right\n\n/-- The inclusion map from the first component of the coproduct. -/\ndef coprod.inl {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : X \u27f6 X \u2a3f Y :=\n  colimit.\u03b9 (pair X Y) walking_pair.left\n\n/-- The inclusion map from the second component of the coproduct. -/\ndef coprod.inr {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : Y \u27f6 X \u2a3f Y :=\n  colimit.\u03b9 (pair X Y) walking_pair.right\n\n/-- The binary fan constructed from the projection maps is a limit. -/\ndef prod_is_prod {C : Type u} [category C] (X : C) (Y : C) [has_binary_product X Y] : is_limit (binary_fan.mk prod.fst prod.snd) :=\n  is_limit.of_iso_limit (limit.is_limit (pair X Y)) (cones.ext (iso.refl (cone.X (limit.cone (pair X Y)))) sorry)\n\n/-- The binary cofan constructed from the coprojection maps is a colimit. -/\ndef coprod_is_coprod {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : is_colimit (binary_cofan.mk coprod.inl coprod.inr) :=\n  is_colimit.of_iso_colimit (colimit.is_colimit (pair X Y))\n    (cocones.ext (iso.refl (cocone.X (colimit.cocone (pair X Y)))) sorry)\n\ntheorem prod.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] {f : W \u27f6 X \u2a2f Y} {g : W \u27f6 X \u2a2f Y} (h\u2081 : f \u226b prod.fst = g \u226b prod.fst) (h\u2082 : f \u226b prod.snd = g \u226b prod.snd) : f = g :=\n  binary_fan.is_limit.hom_ext (limit.is_limit (pair X Y)) h\u2081 h\u2082\n\ntheorem coprod.hom_ext {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] {f : X \u2a3f Y \u27f6 W} {g : X \u2a3f Y \u27f6 W} (h\u2081 : coprod.inl \u226b f = coprod.inl \u226b g) (h\u2082 : coprod.inr \u226b f = coprod.inr \u226b g) : f = g :=\n  binary_cofan.is_colimit.hom_ext (colimit.is_colimit (pair X Y)) h\u2081 h\u2082\n\n/-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W \u27f6 X` and `g : W \u27f6 Y`\n    induces a morphism `prod.lift f g : W \u27f6 X \u2a2f Y`. -/\ndef prod.lift {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) : W \u27f6 X \u2a2f Y :=\n  limit.lift (pair X Y) (binary_fan.mk f g)\n\n/-- diagonal arrow of the binary product in the category `fam I` -/\ndef diag {C : Type u} [category C] (X : C) [has_binary_product X X] : X \u27f6 X \u2a2f X :=\n  prod.lift \ud835\udfd9 \ud835\udfd9\n\n/-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `coprod.desc f g : X \u2a3f Y \u27f6 W`. -/\ndef coprod.desc {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) : X \u2a3f Y \u27f6 W :=\n  colimit.desc (pair X Y) (binary_cofan.mk f g)\n\n/-- codiagonal arrow of the binary coproduct -/\ndef codiag {C : Type u} [category C] (X : C) [has_binary_coproduct X X] : X \u2a3f X \u27f6 X :=\n  coprod.desc \ud835\udfd9 \ud835\udfd9\n\n@[simp] theorem prod.lift_fst_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) {X' : C} (f' : X \u27f6 X') : prod.lift f g \u226b prod.fst \u226b f' = f \u226b f' := sorry\n\n@[simp] theorem prod.lift_snd {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) : prod.lift f g \u226b prod.snd = g :=\n  limit.lift_\u03c0 (binary_fan.mk f g) walking_pair.right\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.inl_desc_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) {X' : C} (f' : W \u27f6 X') : coprod.inl \u226b coprod.desc f g \u226b f' = f \u226b f' := sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.inr_desc_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) {X' : C} (f' : W \u27f6 X') : coprod.inr \u226b coprod.desc f g \u226b f' = g \u226b f' := sorry\n\nprotected instance prod.mono_lift_of_mono_left {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) [mono f] : mono (prod.lift f g) :=\n  mono_of_mono_fac (prod.lift_fst f g)\n\nprotected instance prod.mono_lift_of_mono_right {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) [mono g] : mono (prod.lift f g) :=\n  mono_of_mono_fac (prod.lift_snd f g)\n\nprotected instance coprod.epi_desc_of_epi_left {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) [epi f] : epi (coprod.desc f g) :=\n  epi_of_epi_fac (coprod.inl_desc f g)\n\nprotected instance coprod.epi_desc_of_epi_right {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) [epi g] : epi (coprod.desc f g) :=\n  epi_of_epi_fac (coprod.inr_desc f g)\n\n/-- If the product of `X` and `Y` exists, then every pair of morphisms `f : W \u27f6 X` and `g : W \u27f6 Y`\n    induces a morphism `l : W \u27f6 X \u2a2f Y` satisfying `l \u226b prod.fst = f` and `l \u226b prod.snd = g`. -/\ndef prod.lift' {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : W \u27f6 X) (g : W \u27f6 Y) : Subtype fun (l : W \u27f6 X \u2a2f Y) => l \u226b prod.fst = f \u2227 l \u226b prod.snd = g :=\n  { val := prod.lift f g, property := sorry }\n\n/-- If the coproduct of `X` and `Y` exists, then every pair of morphisms `f : X \u27f6 W` and\n    `g : Y \u27f6 W` induces a morphism `l : X \u2a3f Y \u27f6 W` satisfying `coprod.inl \u226b l = f` and\n    `coprod.inr \u226b l = g`. -/\ndef coprod.desc' {C : Type u} [category C] {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : X \u27f6 W) (g : Y \u27f6 W) : Subtype fun (l : X \u2a3f Y \u27f6 W) => coprod.inl \u226b l = f \u2227 coprod.inr \u226b l = g :=\n  { val := coprod.desc f g, property := sorry }\n\n/-- If the products `W \u2a2f X` and `Y \u2a2f Z` exist, then every pair of morphisms `f : W \u27f6 Y` and\n    `g : X \u27f6 Z` induces a morphism `prod.map f g : W \u2a2f X \u27f6 Y \u2a2f Z`. -/\ndef prod.map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u2a2f X \u27f6 Y \u2a2f Z :=\n  lim_map (map_pair f g)\n\n/-- If the coproducts `W \u2a3f X` and `Y \u2a3f Z` exist, then every pair of morphisms `f : W \u27f6 Y` and\n    `g : W \u27f6 Z` induces a morphism `coprod.map f g : W \u2a3f X \u27f6 Y \u2a3f Z`. -/\ndef coprod.map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : W \u2a3f X \u27f6 Y \u2a3f Z :=\n  colim_map (map_pair f g)\n\n-- Making the reassoc version of this a simp lemma seems to be more harmful than helpful.\n\ntheorem prod.comp_lift_assoc {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C} [has_binary_product X Y] (f : V \u27f6 W) (g : W \u27f6 X) (h : W \u27f6 Y) {X' : C} (f' : X \u2a2f Y \u27f6 X') : f \u226b prod.lift g h \u226b f' = prod.lift (f \u226b g) (f \u226b h) \u226b f' := sorry\n\ntheorem prod.comp_diag {C : Type u} [category C] {X : C} {Y : C} [has_binary_product Y Y] (f : X \u27f6 Y) : f \u226b diag Y = prod.lift f f := sorry\n\n@[simp] theorem prod.map_fst {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : prod.map f g \u226b prod.fst = prod.fst \u226b f :=\n  lim_map_\u03c0 (map_pair f g) walking_pair.left\n\n@[simp] theorem prod.map_snd_assoc {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) {X' : C} (f' : Z \u27f6 X') : prod.map f g \u226b prod.snd \u226b f' = prod.snd \u226b g \u226b f' := sorry\n\n@[simp] theorem prod.map_id_id {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : prod.map \ud835\udfd9 \ud835\udfd9 = \ud835\udfd9 := sorry\n\n@[simp] theorem prod.lift_fst_snd {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] : prod.lift prod.fst prod.snd = \ud835\udfd9 := sorry\n\n@[simp] theorem prod.lift_map {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : V \u27f6 W) (g : V \u27f6 X) (h : W \u27f6 Y) (k : X \u27f6 Z) : prod.lift f g \u226b prod.map h k = prod.lift (f \u226b h) (g \u226b k) := sorry\n\n@[simp] theorem prod.lift_fst_comp_snd_comp {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W Y] [has_binary_product X Z] (g : W \u27f6 X) (g' : Y \u27f6 Z) : prod.lift (prod.fst \u226b g) (prod.snd \u226b g') = prod.map g g' := sorry\n\n-- We take the right hand side here to be simp normal form, as this way composition lemmas for\n\n-- `f \u226b h` and `g \u226b k` can fire (eg `id_comp`) , while `map_fst` and `map_snd` can still work just\n\n-- as well.\n\n@[simp] theorem prod.map_map_assoc {C : Type u} [category C] {A\u2081 : C} {A\u2082 : C} {A\u2083 : C} {B\u2081 : C} {B\u2082 : C} {B\u2083 : C} [has_binary_product A\u2081 B\u2081] [has_binary_product A\u2082 B\u2082] [has_binary_product A\u2083 B\u2083] (f : A\u2081 \u27f6 A\u2082) (g : B\u2081 \u27f6 B\u2082) (h : A\u2082 \u27f6 A\u2083) (k : B\u2082 \u27f6 B\u2083) {X' : C} (f' : A\u2083 \u2a2f B\u2083 \u27f6 X') : prod.map f g \u226b prod.map h k \u226b f' = prod.map (f \u226b h) (g \u226b k) \u226b f' := sorry\n\n-- TODO: is it necessary to weaken the assumption here?\n\ntheorem prod.map_swap {C : Type u} [category C] {A : C} {B : C} {X : C} {Y : C} (f : A \u27f6 B) (g : X \u27f6 Y) [has_limits_of_shape (discrete walking_pair) C] : prod.map \ud835\udfd9 f \u226b prod.map g \ud835\udfd9 = prod.map g \ud835\udfd9 \u226b prod.map \ud835\udfd9 f := sorry\n\ntheorem prod.map_comp_id {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [has_binary_product X W] [has_binary_product Z W] [has_binary_product Y W] : prod.map (f \u226b g) \ud835\udfd9 = prod.map f \ud835\udfd9 \u226b prod.map g \ud835\udfd9 := sorry\n\ntheorem prod.map_id_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [has_binary_product W X] [has_binary_product W Y] [has_binary_product W Z] : prod.map \ud835\udfd9 (f \u226b g) = prod.map \ud835\udfd9 f \u226b prod.map \ud835\udfd9 g := sorry\n\n/-- If the products `W \u2a2f X` and `Y \u2a2f Z` exist, then every pair of isomorphisms `f : W \u2245 Y` and\n    `g : X \u2245 Z` induces an isomorphism `prod.map_iso f g : W \u2a2f X \u2245 Y \u2a2f Z`. -/\n@[simp] theorem prod.map_iso_inv {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : W \u2245 Y) (g : X \u2245 Z) : iso.inv (prod.map_iso f g) = prod.map (iso.inv f) (iso.inv g) :=\n  Eq.refl (iso.inv (prod.map_iso f g))\n\nprotected instance is_iso_prod {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_product W X] [has_binary_product Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) [is_iso f] [is_iso g] : is_iso (prod.map f g) :=\n  is_iso.of_iso (prod.map_iso (as_iso f) (as_iso g))\n\n@[simp] theorem prod.diag_map {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_binary_product X X] [has_binary_product Y Y] : diag X \u226b prod.map f f = f \u226b diag Y := sorry\n\n@[simp] theorem prod.diag_map_fst_snd_assoc {C : Type u} [category C] {X : C} {Y : C} [has_binary_product X Y] [has_binary_product (X \u2a2f Y) (X \u2a2f Y)] {X' : C} (f' : X \u2a2f Y \u27f6 X') : diag (X \u2a2f Y) \u226b prod.map prod.fst prod.snd \u226b f' = f' := sorry\n\n@[simp] theorem prod.diag_map_fst_snd_comp_assoc {C : Type u} [category C] [has_limits_of_shape (discrete walking_pair) C] {X : C} {X' : C} {Y : C} {Y' : C} (g : X \u27f6 Y) (g' : X' \u27f6 Y') : \u2200 {X'_1 : C} (f' : Y \u2a2f Y' \u27f6 X'_1), diag (X \u2a2f X') \u226b prod.map (prod.fst \u226b g) (prod.snd \u226b g') \u226b f' = prod.map g g' \u226b f' := sorry\n\nprotected instance diag.category_theory.split_mono {C : Type u} [category C] {X : C} [has_binary_product X X] : split_mono (diag X) :=\n  split_mono.mk prod.fst\n\n@[simp] theorem coprod.desc_comp_assoc {C : Type u} [category C] {V : C} {W : C} {X : C} {Y : C} [has_binary_coproduct X Y] (f : V \u27f6 W) (g : X \u27f6 V) (h : Y \u27f6 V) {X' : C} (f' : W \u27f6 X') : coprod.desc g h \u226b f \u226b f' = coprod.desc (g \u226b f) (h \u226b f) \u226b f' := sorry\n\ntheorem coprod.diag_comp {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X X] (f : X \u27f6 Y) : codiag X \u226b f = coprod.desc f f := sorry\n\n@[simp] theorem coprod.inl_map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : coprod.inl \u226b coprod.map f g = f \u226b coprod.inl :=\n  \u03b9_colim_map (map_pair f g) walking_pair.left\n\n@[simp] theorem coprod.inr_map {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) : coprod.inr \u226b coprod.map f g = g \u226b coprod.inr :=\n  \u03b9_colim_map (map_pair f g) walking_pair.right\n\n@[simp] theorem coprod.map_id_id {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : coprod.map \ud835\udfd9 \ud835\udfd9 = \ud835\udfd9 := sorry\n\n@[simp] theorem coprod.desc_inl_inr {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] : coprod.desc coprod.inl coprod.inr = \ud835\udfd9 := sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_desc {C : Type u} [category C] {S : C} {T : C} {U : C} {V : C} {W : C} [has_binary_coproduct U W] [has_binary_coproduct T V] (f : U \u27f6 S) (g : W \u27f6 S) (h : T \u27f6 U) (k : V \u27f6 W) : coprod.map h k \u226b coprod.desc f g = coprod.desc (h \u226b f) (k \u226b g) := sorry\n\n@[simp] theorem coprod.desc_comp_inl_comp_inr {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W Y] [has_binary_coproduct X Z] (g : W \u27f6 X) (g' : Y \u27f6 Z) : coprod.desc (g \u226b coprod.inl) (g' \u226b coprod.inr) = coprod.map g g' := sorry\n\n-- We take the right hand side here to be simp normal form, as this way composition lemmas for\n\n-- `f \u226b h` and `g \u226b k` can fire (eg `id_comp`) , while `inl_map` and `inr_map` can still work just\n\n-- as well.\n\n@[simp] theorem coprod.map_map {C : Type u} [category C] {A\u2081 : C} {A\u2082 : C} {A\u2083 : C} {B\u2081 : C} {B\u2082 : C} {B\u2083 : C} [has_binary_coproduct A\u2081 B\u2081] [has_binary_coproduct A\u2082 B\u2082] [has_binary_coproduct A\u2083 B\u2083] (f : A\u2081 \u27f6 A\u2082) (g : B\u2081 \u27f6 B\u2082) (h : A\u2082 \u27f6 A\u2083) (k : B\u2082 \u27f6 B\u2083) : coprod.map f g \u226b coprod.map h k = coprod.map (f \u226b h) (g \u226b k) := sorry\n\n-- I don't think it's a good idea to make any of the following three simp lemmas.\n\ntheorem coprod.map_swap_assoc {C : Type u} [category C] {A : C} {B : C} {X : C} {Y : C} (f : A \u27f6 B) (g : X \u27f6 Y) [has_colimits_of_shape (discrete walking_pair) C] {X' : C} (f' : Y \u2a3f B \u27f6 X') : coprod.map \ud835\udfd9 f \u226b coprod.map g \ud835\udfd9 \u226b f' = coprod.map g \ud835\udfd9 \u226b coprod.map \ud835\udfd9 f \u226b f' := sorry\n\ntheorem coprod.map_comp_id {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [has_binary_coproduct Z W] [has_binary_coproduct Y W] [has_binary_coproduct X W] : coprod.map (f \u226b g) \ud835\udfd9 = coprod.map f \ud835\udfd9 \u226b coprod.map g \ud835\udfd9 := sorry\n\ntheorem coprod.map_id_comp {C : Type u} [category C] {X : C} {Y : C} {Z : C} {W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [has_binary_coproduct W X] [has_binary_coproduct W Y] [has_binary_coproduct W Z] : coprod.map \ud835\udfd9 (f \u226b g) = coprod.map \ud835\udfd9 f \u226b coprod.map \ud835\udfd9 g := sorry\n\n/-- If the coproducts `W \u2a3f X` and `Y \u2a3f Z` exist, then every pair of isomorphisms `f : W \u2245 Y` and\n    `g : W \u2245 Z` induces a isomorphism `coprod.map_iso f g : W \u2a3f X \u2245 Y \u2a3f Z`. -/\n@[simp] theorem coprod.map_iso_hom {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u2245 Y) (g : X \u2245 Z) : iso.hom (coprod.map_iso f g) = coprod.map (iso.hom f) (iso.hom g) :=\n  Eq.refl (iso.hom (coprod.map_iso f g))\n\nprotected instance is_iso_coprod {C : Type u} [category C] {W : C} {X : C} {Y : C} {Z : C} [has_binary_coproduct W X] [has_binary_coproduct Y Z] (f : W \u27f6 Y) (g : X \u27f6 Z) [is_iso f] [is_iso g] : is_iso (coprod.map f g) :=\n  is_iso.of_iso (coprod.map_iso (as_iso f) (as_iso g))\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_codiag {C : Type u} [category C] {X : C} {Y : C} (f : X \u27f6 Y) [has_binary_coproduct X X] [has_binary_coproduct Y Y] : coprod.map f f \u226b codiag Y = codiag X \u226b f := sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\ntheorem coprod.map_inl_inr_codiag_assoc {C : Type u} [category C] {X : C} {Y : C} [has_binary_coproduct X Y] [has_binary_coproduct (X \u2a3f Y) (X \u2a3f Y)] {X' : C} (f' : X \u2a3f Y \u27f6 X') : coprod.map coprod.inl coprod.inr \u226b codiag (X \u2a3f Y) \u226b f' = f' := sorry\n\n-- The simp linter says simp can prove the reassoc version of this lemma.\n\n@[simp] theorem coprod.map_comp_inl_inr_codiag {C : Type u} [category C] [has_colimits_of_shape (discrete walking_pair) C] {X : C} {X' : C} {Y : C} {Y' : C} (g : X \u27f6 Y) (g' : X' \u27f6 Y') : coprod.map (g \u226b coprod.inl) (g' \u226b coprod.inr) \u226b codiag (Y \u2a3f Y') = coprod.map g g' := sorry\n\n/--\n`has_binary_products` represents a choice of product for every pair of objects.\n\nSee https://stacks.math.columbia.edu/tag/001T.\n-/\ndef has_binary_products (C : Type u) [category C] :=\n  has_limits_of_shape (discrete walking_pair) C\n\n/--\n`has_binary_coproducts` represents a choice of coproduct for every pair of objects.\n\nSee https://stacks.math.columbia.edu/tag/04AP.\n-/\ndef has_binary_coproducts (C : Type u) [category C] :=\n  has_colimits_of_shape (discrete walking_pair) C\n\n/-- If `C` has all limits of diagrams `pair X Y`, then it has all binary products -/\ntheorem has_binary_products_of_has_limit_pair (C : Type u) [category C] [\u2200 {X Y : C}, has_limit (pair X Y)] : has_binary_products C :=\n  has_limits_of_shape.mk fun (F : discrete walking_pair \u2964 C) => has_limit_of_iso (iso.symm (diagram_iso_pair F))\n\n/-- If `C` has all colimits of diagrams `pair X Y`, then it has all binary coproducts -/\ntheorem has_binary_coproducts_of_has_colimit_pair (C : Type u) [category C] [\u2200 {X Y : C}, has_colimit (pair X Y)] : has_binary_coproducts C :=\n  has_colimits_of_shape.mk fun (F : discrete walking_pair \u2964 C) => has_colimit_of_iso (diagram_iso_pair F)\n\n/-- The braiding isomorphism which swaps a binary product. -/\n@[simp] theorem prod.braiding_hom {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q] [has_binary_product Q P] : iso.hom (prod.braiding P Q) = prod.lift prod.snd prod.fst :=\n  Eq.refl (iso.hom (prod.braiding P Q))\n\n/-- The braiding isomorphism can be passed through a map by swapping the order. -/\ntheorem braid_natural {C : Type u} [category C] [has_binary_products C] {W : C} {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) (g : Z \u27f6 W) : prod.map f g \u226b iso.hom (prod.braiding Y W) = iso.hom (prod.braiding X Z) \u226b prod.map g f := sorry\n\ntheorem prod.symmetry'_assoc {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q] [has_binary_product Q P] {X' : C} (f' : P \u2a2f Q \u27f6 X') : prod.lift prod.snd prod.fst \u226b prod.lift prod.snd prod.fst \u226b f' = f' := sorry\n\n/-- The braiding isomorphism is symmetric. -/\ntheorem prod.symmetry_assoc {C : Type u} [category C] (P : C) (Q : C) [has_binary_product P Q] [has_binary_product Q P] {X' : C} (f' : P \u2a2f Q \u27f6 X') : iso.hom (prod.braiding P Q) \u226b iso.hom (prod.braiding Q P) \u226b f' = f' := sorry\n\n/-- The associator isomorphism for binary products. -/\n@[simp] theorem prod.associator_hom {C : Type u} [category C] [has_binary_products C] (P : C) (Q : C) (R : C) : iso.hom (prod.associator P Q R) = prod.lift (prod.fst \u226b prod.fst) (prod.lift (prod.fst \u226b prod.snd) prod.snd) :=\n  Eq.refl (iso.hom (prod.associator P Q R))\n\ntheorem prod.pentagon_assoc {C : Type u} [category C] [has_binary_products C] (W : C) (X : C) (Y : C) (Z : C) {X' : C} (f' : W \u2a2f (X \u2a2f (Y \u2a2f Z)) \u27f6 X') : prod.map (iso.hom (prod.associator W X Y)) \ud835\udfd9 \u226b\n    iso.hom (prod.associator W (X \u2a2f Y) Z) \u226b prod.map \ud835\udfd9 (iso.hom (prod.associator X Y Z)) \u226b f' =\n  iso.hom (prod.associator (W \u2a2f X) Y Z) \u226b iso.hom (prod.associator W X (Y \u2a2f Z)) \u226b f' := sorry\n\ntheorem prod.associator_naturality_assoc {C : Type u} [category C] [has_binary_products C] {X\u2081 : C} {X\u2082 : C} {X\u2083 : C} {Y\u2081 : C} {Y\u2082 : C} {Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083) {X' : C} (f' : Y\u2081 \u2a2f (Y\u2082 \u2a2f Y\u2083) \u27f6 X') : prod.map (prod.map f\u2081 f\u2082) f\u2083 \u226b iso.hom (prod.associator Y\u2081 Y\u2082 Y\u2083) \u226b f' =\n  iso.hom (prod.associator X\u2081 X\u2082 X\u2083) \u226b prod.map f\u2081 (prod.map f\u2082 f\u2083) \u226b f' := sorry\n\n/-- The left unitor isomorphism for binary products with the terminal object. -/\ndef prod.left_unitor {C : Type u} [category C] [has_terminal C] (P : C) [has_binary_product (\u22a4_C) P] : (\u22a4_C) \u2a2f P \u2245 P :=\n  iso.mk prod.snd (prod.lift (terminal.from P) \ud835\udfd9)\n\n/-- The right unitor isomorphism for binary products with the terminal object. -/\ndef prod.right_unitor {C : Type u} [category C] [has_terminal C] (P : C) [has_binary_product P (\u22a4_C)] : P \u2a2f (\u22a4_C) \u2245 P :=\n  iso.mk prod.fst (prod.lift \ud835\udfd9 (terminal.from P))\n\ntheorem prod.left_unitor_hom_naturality_assoc {C : Type u} [category C] {X : C} {Y : C} [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) {X' : C} (f' : Y \u27f6 X') : prod.map \ud835\udfd9 f \u226b iso.hom (prod.left_unitor Y) \u226b f' = iso.hom (prod.left_unitor X) \u226b f \u226b f' := sorry\n\ntheorem prod.left_unitor_inv_naturality {C : Type u} [category C] {X : C} {Y : C} [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) : iso.inv (prod.left_unitor X) \u226b prod.map \ud835\udfd9 f = f \u226b iso.inv (prod.left_unitor Y) := sorry\n\ntheorem prod.right_unitor_hom_naturality_assoc {C : Type u} [category C] {X : C} {Y : C} [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) {X' : C} (f' : Y \u27f6 X') : prod.map f \ud835\udfd9 \u226b iso.hom (prod.right_unitor Y) \u226b f' = iso.hom (prod.right_unitor X) \u226b f \u226b f' := sorry\n\ntheorem prod_right_unitor_inv_naturality {C : Type u} [category C] {X : C} {Y : C} [has_terminal C] [has_binary_products C] (f : X \u27f6 Y) : iso.inv (prod.right_unitor X) \u226b prod.map f \ud835\udfd9 = f \u226b iso.inv (prod.right_unitor Y) := sorry\n\ntheorem prod.triangle {C : Type u} [category C] [has_terminal C] [has_binary_products C] (X : C) (Y : C) : iso.hom (prod.associator X (\u22a4_C) Y) \u226b prod.map \ud835\udfd9 (iso.hom (prod.left_unitor Y)) =\n  prod.map (iso.hom (prod.right_unitor X)) \ud835\udfd9 := sorry\n\n/-- The braiding isomorphism which swaps a binary coproduct. -/\n@[simp] theorem coprod.braiding_hom {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C) : iso.hom (coprod.braiding P Q) = coprod.desc coprod.inr coprod.inl :=\n  Eq.refl (iso.hom (coprod.braiding P Q))\n\ntheorem coprod.symmetry'_assoc {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C) {X' : C} (f' : P \u2a3f Q \u27f6 X') : coprod.desc coprod.inr coprod.inl \u226b coprod.desc coprod.inr coprod.inl \u226b f' = f' := sorry\n\n/-- The braiding isomorphism is symmetric. -/\ntheorem coprod.symmetry {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C) : iso.hom (coprod.braiding P Q) \u226b iso.hom (coprod.braiding Q P) = \ud835\udfd9 :=\n  coprod.symmetry' P Q\n\n/-- The associator isomorphism for binary coproducts. -/\n@[simp] theorem coprod.associator_inv {C : Type u} [category C] [has_binary_coproducts C] (P : C) (Q : C) (R : C) : iso.inv (coprod.associator P Q R) =\n  coprod.desc (coprod.inl \u226b coprod.inl) (coprod.desc (coprod.inr \u226b coprod.inl) coprod.inr) :=\n  Eq.refl (iso.inv (coprod.associator P Q R))\n\ntheorem coprod.pentagon {C : Type u} [category C] [has_binary_coproducts C] (W : C) (X : C) (Y : C) (Z : C) : coprod.map (iso.hom (coprod.associator W X Y)) \ud835\udfd9 \u226b\n    iso.hom (coprod.associator W (X \u2a3f Y) Z) \u226b coprod.map \ud835\udfd9 (iso.hom (coprod.associator X Y Z)) =\n  iso.hom (coprod.associator (W \u2a3f X) Y Z) \u226b iso.hom (coprod.associator W X (Y \u2a3f Z)) := sorry\n\ntheorem coprod.associator_naturality {C : Type u} [category C] [has_binary_coproducts C] {X\u2081 : C} {X\u2082 : C} {X\u2083 : C} {Y\u2081 : C} {Y\u2082 : C} {Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083) : coprod.map (coprod.map f\u2081 f\u2082) f\u2083 \u226b iso.hom (coprod.associator Y\u2081 Y\u2082 Y\u2083) =\n  iso.hom (coprod.associator X\u2081 X\u2082 X\u2083) \u226b coprod.map f\u2081 (coprod.map f\u2082 f\u2083) := sorry\n\n/-- The left unitor isomorphism for binary coproducts with the initial object. -/\n@[simp] theorem coprod.left_unitor_inv {C : Type u} [category C] [has_binary_coproducts C] [has_initial C] (P : C) : iso.inv (coprod.left_unitor P) = coprod.inr :=\n  Eq.refl (iso.inv (coprod.left_unitor P))\n\n/-- The right unitor isomorphism for binary coproducts with the initial object. -/\n@[simp] theorem coprod.right_unitor_hom {C : Type u} [category C] [has_binary_coproducts C] [has_initial C] (P : C) : iso.hom (coprod.right_unitor P) = coprod.desc \ud835\udfd9 (initial.to P) :=\n  Eq.refl (iso.hom (coprod.right_unitor P))\n\ntheorem coprod.triangle {C : Type u} [category C] [has_binary_coproducts C] [has_initial C] (X : C) (Y : C) : iso.hom (coprod.associator X (\u22a5_C) Y) \u226b coprod.map \ud835\udfd9 (iso.hom (coprod.left_unitor Y)) =\n  coprod.map (iso.hom (coprod.right_unitor X)) \ud835\udfd9 := sorry\n\n/-- The binary product functor. -/\n@[simp] theorem prod.functor_obj_map {C : Type u} [category C] [has_binary_products C] (X : C) (Y : C) (Z : C) (g : Y \u27f6 Z) : functor.map (functor.obj prod.functor X) g = prod.map \ud835\udfd9 g :=\n  Eq.refl (functor.map (functor.obj prod.functor X) g)\n\n/-- The product functor can be decomposed. -/\ndef prod.functor_left_comp {C : Type u} [category C] [has_binary_products C] (X : C) (Y : C) : functor.obj prod.functor (X \u2a2f Y) \u2245 functor.obj prod.functor Y \u22d9 functor.obj prod.functor X :=\n  nat_iso.of_components (prod.associator X Y) sorry\n\n/-- The binary coproduct functor. -/\n@[simp] theorem coprod.functor_obj_map {C : Type u} [category C] [has_binary_coproducts C] (X : C) (Y : C) (Z : C) (g : Y \u27f6 Z) : functor.map (functor.obj coprod.functor X) g = coprod.map \ud835\udfd9 g :=\n  Eq.refl (functor.map (functor.obj coprod.functor X) g)\n\n/-- The coproduct functor can be decomposed. -/\ndef coprod.functor_left_comp {C : Type u} [category C] [has_binary_coproducts C] (X : C) (Y : C) : functor.obj coprod.functor (X \u2a3f Y) \u2245 functor.obj coprod.functor Y \u22d9 functor.obj coprod.functor X :=\n  nat_iso.of_components (coprod.associator X Y) sorry\n\n/--\nThe product comparison morphism.\n\nIn `category_theory/limits/preserves` we show this is always an iso iff F preserves binary products.\n-/\ndef prod_comparison {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) (A : C) (B : C) [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] : functor.obj F (A \u2a2f B) \u27f6 functor.obj F A \u2a2f functor.obj F B :=\n  prod.lift (functor.map F prod.fst) (functor.map F prod.snd)\n\n@[simp] theorem prod_comparison_fst_assoc {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] {X' : D} (f' : functor.obj F A \u27f6 X') : prod_comparison F A B \u226b prod.fst \u226b f' = functor.map F prod.fst \u226b f' := sorry\n\n@[simp] theorem prod_comparison_snd_assoc {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] {X' : D} (f' : functor.obj F B \u27f6 X') : prod_comparison F A B \u226b prod.snd \u226b f' = functor.map F prod.snd \u226b f' := sorry\n\n/-- Naturality of the prod_comparison morphism in both arguments. -/\ntheorem prod_comparison_natural_assoc {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {A' : C} {B : C} {B' : C} [has_binary_product A B] [has_binary_product A' B'] [has_binary_product (functor.obj F A) (functor.obj F B)] [has_binary_product (functor.obj F A') (functor.obj F B')] (f : A \u27f6 A') (g : B \u27f6 B') {X' : D} (f' : functor.obj F A' \u2a2f functor.obj F B' \u27f6 X') : functor.map F (prod.map f g) \u226b prod_comparison F A' B' \u226b f' =\n  prod_comparison F A B \u226b prod.map (functor.map F f) (functor.map F g) \u226b f' := sorry\n\n/--\nThe product comparison morphism from `F(A \u2a2f -)` to `FA \u2a2f F-`, whose components are given by\n`prod_comparison`.\n-/\ndef prod_comparison_nat_trans {C : Type u} [category C] {D : Type u\u2082} [category D] [has_binary_products C] [has_binary_products D] (F : C \u2964 D) (A : C) : functor.obj prod.functor A \u22d9 F \u27f6 F \u22d9 functor.obj prod.functor (functor.obj F A) :=\n  nat_trans.mk fun (B : C) => prod_comparison F A B\n\ntheorem inv_prod_comparison_map_fst_assoc {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] [is_iso (prod_comparison F A B)] {X' : D} (f' : functor.obj F A \u27f6 X') : inv (prod_comparison F A B) \u226b functor.map F prod.fst \u226b f' = prod.fst \u226b f' := sorry\n\ntheorem inv_prod_comparison_map_snd_assoc {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {B : C} [has_binary_product A B] [has_binary_product (functor.obj F A) (functor.obj F B)] [is_iso (prod_comparison F A B)] {X' : D} (f' : functor.obj F B \u27f6 X') : inv (prod_comparison F A B) \u226b functor.map F prod.snd \u226b f' = prod.snd \u226b f' := sorry\n\n/-- If the product comparison morphism is an iso, its inverse is natural. -/\ntheorem prod_comparison_inv_natural {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {A : C} {A' : C} {B : C} {B' : C} [has_binary_product A B] [has_binary_product A' B'] [has_binary_product (functor.obj F A) (functor.obj F B)] [has_binary_product (functor.obj F A') (functor.obj F B')] (f : A \u27f6 A') (g : B \u27f6 B') [is_iso (prod_comparison F A B)] [is_iso (prod_comparison F A' B')] : inv (prod_comparison F A B) \u226b functor.map F (prod.map f g) =\n  prod.map (functor.map F f) (functor.map F g) \u226b inv (prod_comparison F A' B') := sorry\n\n/--\nThe natural isomorphism `F(A \u2a2f -) \u2245 FA \u2a2f F-`, provided each `prod_comparison F A B` is an\nisomorphism (as `B` changes).\n-/\ndef prod_comparison_nat_iso {C : Type u} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) [has_binary_products C] [has_binary_products D] (A : C) [(B : C) \u2192 is_iso (prod_comparison F A B)] : functor.obj prod.functor A \u22d9 F \u2245 F \u22d9 functor.obj prod.functor (functor.obj F A) :=\n  iso.mk (prod_comparison_nat_trans F A) (inv (nat_trans.mk fun (B : C) => prod_comparison F A B))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/binary_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.37483677006187954}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n-/\nprelude\n\nnotation `Prop` := Sort 0\nnotation f ` $ `:1 a:0 := f a\n\nuniverses u v w\n\n/--\nThe kernel definitional equality test (t =?= s) has special support for id_delta applications.\nIt implements the following rules\n\n   1)   (id_delta t) =?= t\n   2)   t =?= (id_delta t)\n   3)   (id_delta t) =?= s  IF (unfold_of t) =?= s\n   4)   t =?= id_delta s    IF t =?= (unfold_of s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use id_delta applications to address performance problems when type checking\nlemmas generated by the equation compiler.\n-/\n@[inline] def id_delta {\u03b1 : Sort u} (a : \u03b1) : \u03b1 :=\na\n\n/-- Gadget for optional parameter support. -/\n@[reducible] def opt_param (\u03b1 : Sort u) (default : \u03b1) : Sort u :=\n\u03b1\n\n/-- Gadget for marking output parameters in type classes. -/\n@[reducible] def out_param (\u03b1 : Sort u) : Sort u := \u03b1\n\n/-\n  id_rhs is an auxiliary declaration used in the equation compiler to address performance\n  issues when proving equational lemmas. The equation compiler uses it as a marker.\n-/\nabbreviation id_rhs (\u03b1 : Sort u) (a : \u03b1) : \u03b1 := a\n\ninductive punit : Sort u\n| star : punit\n\n/-- An abbreviation for `punit.{0}`, its most common instantiation.\n    This type should be preferred over `punit` where possible to avoid\n    unnecessary universe parameters. -/\nabbreviation unit : Type := punit\n\n@[pattern] abbreviation unit.star : unit := punit.star\n\n/--\nGadget for defining thunks, thunk parameters have special treatment.\nExample: given\n      def f (s : string) (t : thunk nat) : nat\nan application\n     f \"hello\" 10\n is converted into\n     f \"hello\" (\u03bb _, 10)\n-/\n@[reducible] def thunk (\u03b1 : Type u) : Type u :=\nunit \u2192 \u03b1\n\ninductive true : Prop\n| intro : true\n\ninductive false : Prop\n\ninductive empty : Type\n\n/--\nLogical not.\n\n`not P`, with notation `\u00ac P`, is the `Prop` which is true if and only if `P` is false. It is\ninternally represented as `P \u2192 false`, so one way to prove a goal `\u22a2 \u00ac P` is to use `intro h`,\nwhich gives you a new hypothesis `h : P` and the goal `\u22a2 false`.\n\nA hypothesis `h : \u00ac P` can be used in term mode as a function, so if `w : P` then `h w : false`.\n\nRelated mathlib tactic: `contrapose`.\n-/\ndef not (a : Prop) := a \u2192 false\nprefix `\u00ac`:40 := not\n\ninductive eq {\u03b1 : Sort u} (a : \u03b1) : \u03b1 \u2192 Prop\n| refl [] : eq a\n/-* \\pi  *-/\n/-\nInitialize the quotient module, which effectively adds the following definitions:\n\nconstant quot {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Sort u\n\nconstant quot.mk {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) : quot r\n\nconstant quot.lift {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  (\u2200 a b : \u03b1, r a b \u2192 eq (f a) (f b)) \u2192 quot r \u2192 \u03b2\n\nconstant quot.ind {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : quot r \u2192 Prop} :\n  (\u2200 a : \u03b1, \u03b2 (quot.mk r a)) \u2192 \u2200 q : quot r, \u03b2 q-\n\nAlso the reduction rule:\n\nquot.lift f _ (quot.mk a) ~~> f a\n\n-/\ninit_quotient\n\n/--\nHeterogeneous equality.\n\nIts purpose is to write down equalities between terms whose types are not definitionally equal.\nFor example, given `x : vector \u03b1 n` and `y : vector \u03b1 (0+n)`, `x = y` doesn't typecheck but `x == y` does.\n\nIf you have a goal `\u22a2 x == y`,\nyour first instinct should be to ask (either yourself, or on [zulip](https://leanprover.zulipchat.com/))\nif something has gone wrong already.\nIf you really do need to follow this route,\nyou may find the lemmas `eq_rec_heq` and `eq_mpr_heq` useful.\n-/\ninductive heq {\u03b1 : Sort u} (a : \u03b1) : \u03a0 {\u03b2 : Sort u}, \u03b2 \u2192 Prop\n| refl [] : heq a\n\nstructure prod (\u03b1 : Type u) (\u03b2 : Type v) :=\n(fst : \u03b1) (snd : \u03b2)\n\n/-- Similar to `prod`, but \u03b1 and \u03b2 can be propositions.\n   We use this type internally to automatically generate the brec_on recursor. -/\nstructure pprod (\u03b1 : Sort u) (\u03b2 : Sort v) :=\n(fst : \u03b1) (snd : \u03b2)\n\n/--\nLogical and.\n\n`and P Q`, with notation `P \u2227 Q`, is the `Prop` which is true precisely when `P` and `Q` are\nboth true.\n\nTo prove a goal `\u22a2 P \u2227 Q`, you can use the tactic `split`,\nwhich gives two separate goals `\u22a2 P` and `\u22a2 Q`.\n\nGiven a hypothesis `h : P \u2227 Q`, you can use the tactic `cases h with hP hQ`\nto obtain two new hypotheses `hP : P` and `hQ : Q`. See also the `obtain` or `rcases` tactics in\nmathlib.\n-/\nstructure and (a b : Prop) : Prop :=\nintro :: (left : a) (right : b)\n\n/-* Let $a$ and $b$ be two propositions, $a \\and b$ implies $a$. *-/\nlemma and.elim_left {a b : Prop} (h : and a b) : a := h.1\n/-* Let $a$ and $b$ be two propositions, $a \\and b$ implies $b$. *-/\nlemma and.elim_right {a b : Prop} (h : and a b) : b := h.2\n\n/- eq basic support -/\n\ninfix ` = `:50 := eq\n\nattribute [refl] eq.refl\n\n/- This is a `def`, so that it can be used as pattern in the equation compiler. -/\n@[pattern] def rfl {\u03b1 : Sort u} {a : \u03b1} : a = a := eq.refl a\n\n@[elab_as_eliminator, subst]\n/-*  *-/\nlemma eq.subst {\u03b1 : Sort u} {P : \u03b1 \u2192 Prop} {a b : \u03b1} (h\u2081 : a = b) (h\u2082 : P a) : P b :=\neq.rec h\u2082 h\u2081\n\ninfixr ` \u25b8 `:75 := eq.subst\n\n@[trans] lemma eq.trans {\u03b1 : Sort u} {a b c : \u03b1} (h\u2081 : a = b) (h\u2082 : b = c) : a = c :=\nh\u2082 \u25b8 h\u2081\n\n@[symm] lemma eq.symm {\u03b1 : Sort u} {a b : \u03b1} (h : a = b) : b = a :=\nh \u25b8 rfl\n\ninfix ` == `:50 := heq\n\n/- This is a `def`, so that it can be used as pattern in the equation compiler. -/\n@[pattern] def heq.rfl {\u03b1 : Sort u} {a : \u03b1} : a == a := heq.refl a\n\nlemma eq_of_heq {\u03b1 : Sort u} {a a' : \u03b1} (h : a == a') : a = a' :=\nhave \u2200 (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a') (h\u2082 : \u03b1 = \u03b1'), (eq.rec_on h\u2082 a : \u03b1') = a', from\n  \u03bb (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a'), heq.rec_on h\u2081 (\u03bb h\u2082 : \u03b1 = \u03b1, rfl),\nshow (eq.rec_on (eq.refl \u03b1) a : \u03b1) = a', from\n  this \u03b1 a' h (eq.refl \u03b1)\n\n/- The following four lemmas could not be automatically generated when the\n   structures were declared, so we prove them manually here. -/\nlemma prod.mk.inj {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 and (x\u2081 = x\u2082) (y\u2081 = y\u2082) :=\n\u03bb h, prod.no_confusion h (\u03bb h\u2081 h\u2082, \u27e8h\u2081, h\u2082\u27e9)\n\nlemma prod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u03a0 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n\u03bb h\u2081 _ h\u2082, prod.no_confusion h\u2081 h\u2082\n\nlemma pprod.mk.inj {\u03b1 : Sort u} {\u03b2 : Sort v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : pprod.mk x\u2081 y\u2081 = pprod.mk x\u2082 y\u2082 \u2192 and (x\u2081 = x\u2082) (y\u2081 = y\u2082) :=\n\u03bb h, pprod.no_confusion h (\u03bb h\u2081 h\u2082, \u27e8h\u2081, h\u2082\u27e9)\n\nlemma pprod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2}\n  : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u03a0 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n\u03bb h\u2081 _ h\u2082, prod.no_confusion h\u2081 h\u2082\n\ninductive sum (\u03b1 : Type u) (\u03b2 : Type v)\n| inl (val : \u03b1) : sum\n| inr (val : \u03b2) : sum\n\ninductive psum (\u03b1 : Sort u) (\u03b2 : Sort v)\n| inl (val : \u03b1) : psum\n| inr (val : \u03b2) : psum\n\n/--\nLogical or.\n\n`or P Q`, with notation `P \u2228 Q`, is the proposition which is true if and only if `P` or `Q` is\ntrue.\n\nTo prove a goal `\u22a2 P \u2228 Q`, if you know which alternative you want to prove,\nyou can use the tactics `left` (which gives the goal `\u22a2 P`)\nor `right` (which gives the goal `\u22a2 Q`).\n\nGiven a hypothesis `h : P \u2228 Q` and goal `\u22a2 R`,\nthe tactic `cases h` will give you two copies of the goal `\u22a2 R`,\nwith the hypothesis `h : P` in the first, and the hypothesis `h : Q` in the second.\n-/\ninductive or (a b : Prop) : Prop\n| inl (h : a) : or\n| inr (h : b) : or\n\nlemma or.intro_left {a : Prop} (b : Prop) (ha : a) : or a b :=\nor.inl ha\n\nlemma or.intro_right (a : Prop) {b : Prop} (hb : b) : or a b :=\nor.inr hb\n\nstructure sigma {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) :=\nmk :: (fst : \u03b1) (snd : \u03b2 fst)\n\nstructure psigma {\u03b1 : Sort u} (\u03b2 : \u03b1 \u2192 Sort v) :=\nmk :: (fst : \u03b1) (snd : \u03b2 fst)\n\ninductive bool : Type\n| ff : bool\n| tt : bool\n\n/- Remark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description. -/\nstructure subtype {\u03b1 : Sort u} (p : \u03b1 \u2192 Prop) :=\n(val : \u03b1) (property : p val)\n\nattribute [pp_using_anonymous_constructor] sigma psigma subtype pprod and\n\nclass inductive decidable (p : Prop)\n| is_false (h : \u00acp) : decidable\n| is_true  (h : p) : decidable\n\n@[reducible]\ndef decidable_pred {\u03b1 : Sort u} (r : \u03b1 \u2192 Prop) :=\n\u03a0 (a : \u03b1), decidable (r a)\n\n@[reducible]\ndef decidable_rel {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\n\u03a0 (a b : \u03b1), decidable (r a b)\n\n@[reducible]\ndef decidable_eq (\u03b1 : Sort u) :=\ndecidable_rel (@eq \u03b1)\n\ninductive option (\u03b1 : Type u)\n| none : option\n| some (val : \u03b1) : option\n\nexport option (none some)\nexport bool (ff tt)\n\ninductive list (T : Type u)\n| nil : list\n| cons (hd : T) (tl : list) : list\n\ninfixr ` :: `:67 := list.cons\nnotation `[` l:(foldr `, ` (h t, list.cons h t) list.nil `]`) := l\n\ninductive nat\n| zero : nat\n| succ (n : nat) : nat\n\nstructure unification_constraint :=\n{\u03b1 : Type u} (lhs : \u03b1) (rhs : \u03b1)\n\ninfix ` \u225f `:50   := unification_constraint.mk\ninfix ` =?= `:50 := unification_constraint.mk\n\nstructure unification_hint :=\n(pattern : unification_constraint)\n(constraints : list unification_constraint)\n\n/- Declare builtin and reserved notation -/\n\nclass has_zero     (\u03b1 : Type u) := (zero : \u03b1)\nclass has_one      (\u03b1 : Type u) := (one : \u03b1)\nclass has_add      (\u03b1 : Type u) := (add : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_mul      (\u03b1 : Type u) := (mul : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_inv      (\u03b1 : Type u) := (inv : \u03b1 \u2192 \u03b1)\nclass has_neg      (\u03b1 : Type u) := (neg : \u03b1 \u2192 \u03b1)\nclass has_sub      (\u03b1 : Type u) := (sub : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_div      (\u03b1 : Type u) := (div : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_dvd      (\u03b1 : Type u) := (dvd : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_mod      (\u03b1 : Type u) := (mod : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_le       (\u03b1 : Type u) := (le : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_lt       (\u03b1 : Type u) := (lt : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_append   (\u03b1 : Type u) := (append : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_andthen  (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : out_param $ Type w) := (andthen : \u03b1 \u2192 \u03b2 \u2192 \u03c3)\nclass has_union    (\u03b1 : Type u) := (union : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_inter    (\u03b1 : Type u) := (inter : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_sdiff    (\u03b1 : Type u) := (sdiff : \u03b1 \u2192 \u03b1 \u2192 \u03b1)\nclass has_equiv    (\u03b1 : Sort u) := (equiv : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_subset   (\u03b1 : Type u) := (subset : \u03b1 \u2192 \u03b1 \u2192 Prop)\nclass has_ssubset  (\u03b1 : Type u) := (ssubset : \u03b1 \u2192 \u03b1 \u2192 Prop)\n/- Type classes has_emptyc and has_insert are\n   used to implement polymorphic notation for collections.\n   Example: {a, b, c}. -/\nclass has_emptyc   (\u03b1 : Type u) := (emptyc : \u03b1)\nclass has_insert   (\u03b1 : out_param $ Type u) (\u03b3 : Type v) := (insert : \u03b1 \u2192 \u03b3 \u2192 \u03b3)\nclass has_singleton (\u03b1 : out_param $ Type u) (\u03b2 : Type v) := (singleton : \u03b1 \u2192 \u03b2)\n/- Type class used to implement the notation { a \u2208 c | p a } -/\nclass has_sep (\u03b1 : out_param $ Type u) (\u03b3 : Type v) :=\n(sep : (\u03b1 \u2192 Prop) \u2192 \u03b3 \u2192 \u03b3)\n/- Type class for set-like membership -/\nclass has_mem (\u03b1 : out_param $ Type u) (\u03b3 : Type v) := (mem : \u03b1 \u2192 \u03b3 \u2192 Prop)\n\nclass has_pow (\u03b1 : Type u) (\u03b2 : Type v) :=\n(pow : \u03b1 \u2192 \u03b2 \u2192 \u03b1)\n\nexport has_andthen (andthen)\nexport has_pow (pow)\n\ninfix ` \u2208 `:50   := has_mem.mem\nnotation a ` \u2209 `:50 s:50 := \u00ac has_mem.mem a s\ninfixl ` + `:65  := has_add.add\ninfixl ` * `:70  := has_mul.mul\ninfixl ` - `:65  := has_sub.sub\ninfixl ` / `:70  := has_div.div\ninfix ` \u2223 `:50   := has_dvd.dvd -- Note this is different to `|`.\ninfixl ` % `:70  := has_mod.mod\nprefix `-`:75    := has_neg.neg\ninfix ` <= `:50  := has_le.le\ninfix ` \u2264 `:50   := has_le.le\ninfix ` < `:50   := has_lt.lt\ninfixl ` ++ `:65 := has_append.append\ninfixl `; `:1    := andthen\nnotation `\u2205`     := has_emptyc.emptyc\ninfixl ` \u222a `:65  := has_union.union\ninfixl ` \u2229 `:70  := has_inter.inter\ninfix ` \u2286 `:50   := has_subset.subset\ninfix ` \u2282 `:50   := has_ssubset.ssubset\ninfix ` \\ `:70   := has_sdiff.sdiff\ninfix ` \u2248 `:50   := has_equiv.equiv\ninfixr ` ^ `:80  := has_pow.pow\n\nexport has_append (append)\n\n@[reducible] def ge {\u03b1 : Type u} [has_le \u03b1] (a b : \u03b1) : Prop := has_le.le b a\n@[reducible] def gt {\u03b1 : Type u} [has_lt \u03b1] (a b : \u03b1) : Prop := has_lt.lt b a\n\ninfix ` >= `:50 := ge\ninfix ` \u2265 `:50  := ge\ninfix ` > `:50  := gt\n\n@[reducible] def superset {\u03b1 : Type u} [has_subset \u03b1] (a b : \u03b1) : Prop := has_subset.subset b a\n@[reducible] def ssuperset {\u03b1 : Type u} [has_ssubset \u03b1] (a b : \u03b1) : Prop := has_ssubset.ssubset b a\n\ninfix ` \u2287 `:50 := superset\ninfix ` \u2283 `:50 := ssuperset\n\ndef bit0 {\u03b1 : Type u} [s  : has_add \u03b1] (a  : \u03b1)                 : \u03b1 := a + a\ndef bit1 {\u03b1 : Type u} [s\u2081 : has_one \u03b1] [s\u2082 : has_add \u03b1] (a : \u03b1) : \u03b1 := (bit0 a) + 1\n\nattribute [pattern] has_zero.zero has_one.one bit0 bit1 has_add.add has_neg.neg has_mul.mul\n\nexport has_insert (insert)\n\nclass is_lawful_singleton (\u03b1 : Type u) (\u03b2 : Type v) [has_emptyc \u03b2] [has_insert \u03b1 \u03b2]\n  [has_singleton \u03b1 \u03b2] : Prop :=\n(insert_emptyc_eq : \u2200 (x : \u03b1), (insert x \u2205 : \u03b2) = {x})\n\nexport has_singleton (singleton)\nexport is_lawful_singleton (insert_emptyc_eq)\n\nattribute [simp] insert_emptyc_eq\n\n/- nat basic instances -/\n\nnamespace nat\n  protected def add : nat \u2192 nat \u2192 nat\n  | a  zero     := a\n  | a  (succ b) := succ (add a b)\n\n  /- We mark the following definitions as pattern to make sure they can be used in recursive equations,\n     and reduced by the equation compiler. -/\n  attribute [pattern] nat.add nat.add._main\nend nat\n\ninstance : has_zero nat := \u27e8nat.zero\u27e9\n\ninstance : has_one nat := \u27e8nat.succ (nat.zero)\u27e9\n\ninstance : has_add nat := \u27e8nat.add\u27e9\n\ndef std.priority.default : nat := 1000\ndef std.priority.max     : nat := 0xFFFFFFFF\n\nnamespace nat\n  protected def prio := std.priority.default + 100\nend nat\n\n/-\n  Global declarations of right binding strength\n\n  If a module reassigns these, it will be incompatible with other modules that adhere to these\n  conventions.\n\n  When hovering over a symbol, use \"C-c C-k\" to see how to input it.\n-/\ndef std.prec.max   : nat := 1024 -- the strength of application, identifiers, (, [, etc.\ndef std.prec.arrow : nat := 25\n\n/-\nThe next def is \"max + 10\". It can be used e.g. for postfix operations that should\nbe stronger than application.\n-/\n\ndef std.prec.max_plus : nat := std.prec.max + 10\n\npostfix `\u207b\u00b9`:std.prec.max_plus := has_inv.inv  -- input with \\sy or \\-1 or \\inv\n\ninfixr ` \u00d7 `:35 := prod\n-- notation for n-ary tuples\n\n/- sizeof -/\n\nclass has_sizeof (\u03b1 : Sort u) :=\n(sizeof : \u03b1 \u2192 nat)\n\ndef sizeof {\u03b1 : Sort u} [s : has_sizeof \u03b1] : \u03b1 \u2192 nat :=\nhas_sizeof.sizeof\n\n/-\nDeclare sizeof instances and lemmas for types declared before has_sizeof.\nFrom now on, the inductive compiler will automatically generate sizeof instances and lemmas.\n-/\n\n/- Every type `\u03b1` has a default has_sizeof instance that just returns 0 for every element of `\u03b1` -/\nprotected def default.sizeof (\u03b1 : Sort u) : \u03b1 \u2192 nat\n| a := 0\n\ninstance default_has_sizeof (\u03b1 : Sort u) : has_sizeof \u03b1 :=\n\u27e8default.sizeof \u03b1\u27e9\n\nprotected def nat.sizeof : nat \u2192 nat\n| n := n\n\ninstance : has_sizeof nat :=\n\u27e8nat.sizeof\u27e9\n\nprotected def prod.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (prod \u03b1 \u03b2) \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (prod \u03b1 \u03b2) :=\n\u27e8prod.sizeof\u27e9\n\nprotected def sum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (sum \u03b1 \u03b2) \u2192 nat\n| (sum.inl a) := 1 + sizeof a\n| (sum.inr b) := 1 + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (sum \u03b1 \u03b2) :=\n\u27e8sum.sizeof\u27e9\n\nprotected def psum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [has_sizeof \u03b1] [has_sizeof \u03b2] : (psum \u03b1 \u03b2) \u2192 nat\n| (psum.inl a) := 1 + sizeof a\n| (psum.inr b) := 1 + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [has_sizeof \u03b1] [has_sizeof \u03b2] : has_sizeof (psum \u03b1 \u03b2) :=\n\u27e8psum.sizeof\u27e9\n\nprotected def sigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : sigma \u03b2 \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : has_sizeof (sigma \u03b2) :=\n\u27e8sigma.sizeof\u27e9\n\nprotected def psigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : psigma \u03b2 \u2192 nat\n| \u27e8a, b\u27e9 := 1 + sizeof a + sizeof b\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [has_sizeof \u03b1] [\u2200 a, has_sizeof (\u03b2 a)] : has_sizeof (psigma \u03b2) :=\n\u27e8psigma.sizeof\u27e9\n\nprotected def punit.sizeof : punit \u2192 nat\n| u := 1\n\ninstance : has_sizeof punit := \u27e8punit.sizeof\u27e9\n\nprotected def bool.sizeof : bool \u2192 nat\n| b := 1\n\ninstance : has_sizeof bool := \u27e8bool.sizeof\u27e9\n\nprotected def option.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] : option \u03b1 \u2192 nat\n| none     := 1\n| (some a) := 1 + sizeof a\n\ninstance (\u03b1 : Type u) [has_sizeof \u03b1] : has_sizeof (option \u03b1) :=\n\u27e8option.sizeof\u27e9\n\nprotected def list.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] : list \u03b1 \u2192 nat\n| list.nil        := 1\n| (list.cons a l) := 1 + sizeof a + list.sizeof l\n\ninstance (\u03b1 : Type u) [has_sizeof \u03b1] : has_sizeof (list \u03b1) :=\n\u27e8list.sizeof\u27e9\n\nprotected def subtype.sizeof {\u03b1 : Type u} [has_sizeof \u03b1] {p : \u03b1 \u2192 Prop} : subtype p \u2192 nat\n| \u27e8a, _\u27e9 := sizeof a\n\ninstance {\u03b1 : Type u} [has_sizeof \u03b1] (p : \u03b1 \u2192 Prop) : has_sizeof (subtype p) :=\n\u27e8subtype.sizeof\u27e9\n\nlemma nat_add_zero (n : nat) : n + 0 = n := rfl\n\n/- Combinator calculus -/\nnamespace combinator\nuniverses u\u2081 u\u2082 u\u2083\ndef I {\u03b1 : Type u\u2081} (a : \u03b1) := a\ndef K {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} (a : \u03b1) (b : \u03b2) := a\ndef S {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} {\u03b3 : Type u\u2083} (x : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (y : \u03b1 \u2192 \u03b2) (z : \u03b1) := x z (y z)\nend combinator\n\n/-- Auxiliary datatype for #[ ... ] notation.\n    #[1, 2, 3, 4] is notation for\n\n    bin_tree.node\n      (bin_tree.node (bin_tree.leaf 1) (bin_tree.leaf 2))\n      (bin_tree.node (bin_tree.leaf 3) (bin_tree.leaf 4))\n\n    We use this notation to input long sequences without exhausting the system stack space.\n    Later, we define a coercion from `bin_tree` into `list`.\n-/\ninductive bin_tree (\u03b1 : Type u)\n| empty : bin_tree\n| leaf (val : \u03b1) : bin_tree\n| node (left right : bin_tree) : bin_tree\n\nattribute [elab_simple] bin_tree.node bin_tree.leaf\n\n/-- Like `by apply_instance`, but not dependent on the tactic framework. -/\n@[reducible] def infer_instance {\u03b1 : Sort u} [i : \u03b1] : \u03b1 := i\n", "meta": {"author": "subfish-zhou", "repo": "N2Lean", "sha": "8e858cc5b01f1ad921094dc355db3cb9473a42fd", "save_path": "github-repos/lean/subfish-zhou-N2Lean", "path": "github-repos/lean/subfish-zhou-N2Lean/N2Lean-8e858cc5b01f1ad921094dc355db3cb9473a42fd/library/init/core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.37483677006187954}}
{"text": "import parlang.defs\n\nopen parlang\n\nsection\n\nvariables {n : \u2115} {\u03c3 : Type} {\u03b9 : Type} {\u03c4 : \u03b9 \u2192 Type} [decidable_eq \u03b9]\n\n/-- syncable with the exception that no thread is allowed to store \nor load in stores and loads respectively. Adding to stores or load makes this property stricter -/\ndef syncable' (shole : set \u03b9) (lhole : set \u03b9) (s : state n \u03c3 \u03c4) (m : memory \u03c4) : Prop :=\nstate.syncable s m \u2227\n(\u2200 i tid, (i \u2208 shole \u2192 i \u2209 (s.threads.nth tid).stores) \u2227\n        (i \u2208 lhole \u2192 i \u2209 (s.threads.nth tid).loads))\n\nlemma syncable_syncable' (s : state n \u03c3 \u03c4) (m : memory \u03c4) : syncable' \u2205 \u2205 s m \u2194 state.syncable s m := begin\n    unfold syncable',\n    simp,\nend\n\n\n\nend", "meta": {"author": "fischerman", "repo": "GPU-transformation-verifier", "sha": "75a5016f05382738ff93ce5859c4cfa47ccb63c1", "save_path": "github-repos/lean/fischerman-GPU-transformation-verifier", "path": "github-repos/lean/fischerman-GPU-transformation-verifier/GPU-transformation-verifier-75a5016f05382738ff93ce5859c4cfa47ccb63c1/src/syncablep.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7799929002541067, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.37476995760084625}}
{"text": "import FOL.fol FOL.theory provability consistency\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nopen formula logic Theory\nvariables {L : language.{u}}\n\ndef eq_axiom4 {n} (f : L.fn n) : formula L :=\n  \u2200.[2*n] (finitary.conjunction n (\u03bb i, #i =' #(n + i)) \u27f6 (term.app f (\u03bb i, #i) =' term.app f (\u03bb i, #(n + i))))\n\ndef eq_axiom5 {n} (r : L.pr n) : formula L :=\n  \u2200.[2*n] (finitary.conjunction n (\u03bb i, #i =' #(n + i)) \u27f6 formula.app r (\u03bb i, #i) \u27f6 formula.app r (\u03bb i, #(n + i)))\n\n@[simp] lemma eq_axiom4_is_sentence {n} {f : L.fn n} :\n  is_sentence (eq_axiom4 f) :=\nbegin\n  simp[is_sentence, eq_axiom4],\n  cases n, { simp },\n  have lmm\u2081 : \u2200 m n, (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1) = m + n + 1,\n  { intros m n,\n    suffices : (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1) \u2264 m + n + 1 \u2227 m + n + 1 \u2264 (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1), from le_antisymm_iff.mpr this,\n    split,\n    { refine fintype_sup_le _, rintros \u27e8i, hi\u27e9, simp[show i \u2264 n, from nat.lt_succ_iff.mp hi] },\n    { refine le_fintype_sup' \u27e8n, by simp\u27e9 (by refl) }  },\n  have lmm\u2082 : (\u2a06\u1da0 (i : fin (n + 1)), \u2191i + 1) = n + 1,\n  { have := lmm\u2081 0 n, simp at this, exact this },\n  simp only [lmm\u2081 n.succ n, lmm\u2082, \u2190 nat.add_one],\n  simp[max_add_add_left (n + 1) 0 (n + 1), two_mul, add_assoc]\nend\n\n@[simp] lemma eq_axiom5_is_sentence {n} {r : L.pr n} :\n  is_sentence (eq_axiom5 r) :=\nby { simp[is_sentence, eq_axiom5],\n     cases n, { simp },  have lmm\u2081 : \u2200 m n, (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1) = m + n + 1,\n  { intros m n,\n    suffices : (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1) \u2264 m + n + 1 \u2227 m + n + 1 \u2264 (\u2a06\u1da0 (i : fin (n + 1)), m + \u2191i + 1), from le_antisymm_iff.mpr this,\n    split,\n    { refine fintype_sup_le _, rintros \u27e8i, hi\u27e9, simp[show i \u2264 n, from nat.lt_succ_iff.mp hi] },\n    { refine le_fintype_sup' \u27e8n, by simp\u27e9 (by refl) }  },\n  have lmm\u2082 : (\u2a06\u1da0 (i : fin (n + 1)), \u2191i + 1) = n + 1,\n  { have := lmm\u2081 0 n, simp at this, exact this },\n     simp only [lmm\u2081 n.succ n, lmm\u2082, \u2190 nat.add_one],\n     simp[max_add_add_left (n + 1) 0 (n + 1), two_mul, add_assoc] }\n\ninductive proof : Theory L \u2192 formula L \u2192 Type u\n| generalize : \u2200 {T p}, proof \u290aT p \u2192 proof T (\u2200.p)\n| mdp : \u2200 {T p q}, proof T (p \u27f6 q) \u2192 proof T p \u2192 proof T q\n| by_axiom : \u2200 {T p}, p \u2208 T \u2192 proof T p\n| verum : \u2200 {T}, proof T \u22a4\n| imply\u2081 : \u2200 {T p q}, proof T (p \u27f6 q \u27f6 p)\n| imply\u2082 : \u2200 {T p q r}, proof T ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r)\n| contraposition : \u2200 {T p q}, proof T ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p)\n| specialize : \u2200 {T p t}, proof T (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t])\n| univ_K : \u2200 {T p q}, proof T (\u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6 \u2200.q)\n| dummy_univ : \u2200 {T p}, proof T (p \u27f6 \u2200.(p^1))\n| eq_reflexivity : \u2200 {T}, proof T \u2200.(#0 =' #0)\n| eq_symmetry : \u2200 {T}, proof T \u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))\n| eq_transitivity : \u2200 {T}, proof T \u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))\n| function_ext : \u2200 {T n} {f : L.fn n}, proof T (eq_axiom4 f)\n| predicate_ext : \u2200 {T n} {r : L.pr n}, proof T (eq_axiom5 r)\n\ninstance : has_Longarrow (formula L) := \u27e8proof\u27e9\n\ndef provable (T : Theory L) (p : formula L) : Prop := nonempty (T \u27f9 p)\n\ninstance : axiomatic_classical_logic' (formula L) :=\n{ turnstile := provable,\n  classical := \u03bb T,\n  { modus_ponens := \u03bb p q \u27e8bpq\u27e9 \u27e8bp\u27e9, \u27e8bpq.mdp bp\u27e9,\n    imply\u2081 := \u03bb p q, \u27e8proof.imply\u2081\u27e9, \n    imply\u2082 := \u03bb p q r, \u27e8proof.imply\u2082\u27e9,\n    contraposition := \u03bb p q, \u27e8proof.contraposition\u27e9,\n    provable_top := \u27e8proof.verum\u27e9,\n    bot_eq := by refl,\n    and_def := \u03bb p q, rfl,\n    or_def := \u03bb p q, rfl },\n  by_axiom := \u03bb T p mem, \u27e8proof.by_axiom mem\u27e9 }\n\nnoncomputable def provable.proof {T : Theory L} {p : formula L} (b : T \u22a2 p) : T \u27f9 p := nonempty.some b\n\ndef provable_Theory (T \u0393 : Theory L) : Prop := \u2200 p \u2208 \u0393, T \u22a2 p\n\ninfix ` \u22a2\u209c\u2095 `:45 := provable_Theory\n\nopen axiomatic_classical_logic' axiomatic_classical_logic\n\ninfixl ` \u2a00 `:90 := axiomatic_classical_logic'.modus_ponens\n\n@[simp] lemma mem_iff_prov (p : formula L) (T : set (formula L)) :\n  (@has_mem.mem (formula L) (set (formula L)) _) p (provable T) \u2194 T \u22a2 p := by refl\n\n/-\ndef Theory.consistent (T : Theory L) : Prop := \u00ac\u2203p : formula L, (T \u22a2 p) \u2227 (T \u22a2 \u223cp) \n\nlemma Theory.consistent_def (T : Theory L) : T.consistent \u2194 \u00ac\u2203p : formula L, (T \u22a2 p) \u2227 (T \u22a2 \u223cp) := by refl\n\nlemma Theory.consistent_iff_bot (T : Theory L) : T.consistent \u2194 \u00acT \u22a2 \u22a5 :=\n\u27e8by { simp[Theory.consistent], intros h, exact h \u22a4 (by simp) },\n  by { intros h, simp[Theory.consistent], intros p hp hnp,\n  have : T \u22a2 \u22a5, from explosion hp hnp,\n  exact h this }\u27e9\n\nlemma Theory.not_consistent_iff_bot {T : Theory L} : \u00acT.consistent \u2194 T \u22a2 \u22a5 :=\nby simp[Theory.consistent_iff_bot T]\n\nlemma Theory.not_consistent_iff (T : Theory L) : \u00acT.consistent \u2194 \u2203p : formula L, (T \u22a2 p) \u2227 (T \u22a2 \u223cp) :=\nby simp[Theory.consistent_def]\n\ndef Theory.le (T U : Theory L) : Prop := \u2200 \u2983p : formula L\u2984, T \u22a2 p \u2192 U \u22a2 p\n\ninstance : has_le (Theory L) := \u27e8Theory.le\u27e9\n\n@[simp] lemma Theory.le_refl (T : Theory L) : T \u2264 T := \u03bb p h, h\n\n@[trans] lemma Theory.le.trans {T\u2081 T\u2082 T\u2083 : Theory L} : T\u2081 \u2264 T\u2082 \u2192 T\u2082 \u2264 T\u2083 \u2192 T\u2081 \u2264 T\u2083 := \u03bb le\u2081\u2082 le\u2082\u2083 p b, le\u2082\u2083 (le\u2081\u2082 b)\n\nclass extend (T\u2080 T : Theory L) := (le : T\u2080 \u2264 T)\n\ninstance extend_refl (T : Theory L) : extend T T := \u27e8\u03bb p h, h\u27e9\n\n@[trans] def extend.trans (T\u2081 T\u2082 T\u2083 : Theory L) [extend T\u2081 T\u2082]  [extend T\u2082 T\u2083] : extend T\u2081 T\u2083 := \u27e8\u03bb p b, extend.le (extend.le b : T\u2082 \u22a2 p)\u27e9\n\ndef Theory.th (T : Theory L) : Theory L := {p | T \u22a2 p}\n\n\n-/\nlemma ss_le {U : \u2115 \u2192 Theory L} (hyp : \u2200 s, U s \u2286 U (s+1)) : \u2200 {s\u2081 s\u2082}, s\u2081 \u2264 s\u2082 \u2192 U s\u2081 \u2286 U s\u2082 :=\nby { intros s\u2081, suffices : \u2200 t, U s\u2081 \u2286 U (s\u2081 + t),\n      { intros s\u2082 eqn, have := this (s\u2082 - s\u2081),\n        rw (show s\u2081 + (s\u2082 - s\u2081) = s\u2082, from nat.add_sub_of_le eqn) at this, exact this },\n      intros t, induction t with t IH, simp, rw[nat.add_succ],  refine \u03bb x hx, hyp _ (IH hx) }\n\ndef formula.equiv (T : Theory L) : formula L \u2192 formula L \u2192 Prop := equiv T\n\ndef term.equiv (T : Theory L) (t\u2081 t\u2082 : term L) : Prop := T \u22a2 t\u2081 =' t\u2082\n\nnamespace proof\nvariables {T : Theory L}\n\ndef weakening {p} (h : T \u27f9 p) : \u2200 {U}, T \u2286 U \u2192 U \u27f9 p :=\nbegin\n  induction h with T p hyp_p IH T p q hyp_pq hyp_p IH\u2081 IH\u2082 T p hyp_p; try { simp },\n  { intros U hyp, refine generalize (IH (\u03bb x h, _)), rcases h with \u27e8p, hp, rfl\u27e9,\n    refine \u27e8p, hyp hp, rfl\u27e9 },\n  { intros U hyp, exact (IH\u2081 hyp).mdp (IH\u2082 hyp) },\n  { intros U hyp, exact by_axiom (hyp hyp_p) },\n  { intros U ss, exact verum },\n  { intros U ss, exact imply\u2081 },\n  { intros U ss, exact imply\u2082 },\n  { intros U ss, exact contraposition },\n  { intros U ss, exact specialize },\n  { intros U ss, exact univ_K },\n  { intros U ss, exact dummy_univ },\n  { intros U ss, exact eq_reflexivity },\n  { intros U ss, exact eq_symmetry },\n  { intros U ss, exact eq_transitivity },\n  { intros U ss, exact function_ext },\n  { intros U ss, exact predicate_ext }\nend\n\n-- TODO: rec'' \u3067\u7f6e\u304d\u63db\u3048\u308b\n@[elab_as_eliminator]\ndef rec'_on {T : Theory L} (C : \u2115 \u2192 formula L \u2192 Sort v) {i : \u2115} {p : formula L} (b : T^i \u27f9 p)\n  (GE : \u2200 {i} {p : formula L} (b : T^(i + 1) \u27f9 p), C (i + 1) p \u2192 C i (\u2200.p))\n  (MP : \u2200 {i} {p q : formula L} (b\u2081 : T^i \u27f9 p \u27f6 q) (b\u2082 : T^i \u27f9 p), C i (p \u27f6 q) \u2192 C i p \u2192 C i q)\n  (by_axiom : \u2200 {i} {p : formula L} (mem : p \u2208 T^i), C i p)\n  (p0 : \u2200 {i}, C i \u22a4)\n  (p1 : \u2200 {i} {p q : formula L}, C i (p \u27f6 q \u27f6 p))\n  (p2 : \u2200 {i} {p q r : formula L}, C i ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r))\n  (p3 : \u2200 {i} {p q : formula L}, C i ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p))\n  (q1 : \u2200 {i} {p : formula L} {t : term L}, C i (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t]))\n  (q2 : \u2200 {i} {p q : formula L}, C i (\u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6\u2200.q))\n  (q3 : \u2200 {i} {p : formula L}, C i (p \u27f6 \u2200.(p^1)))\n  (e1 : \u2200 {i}, C i (\u2200.(#0 =' #0)))\n  (e2 : \u2200 {i}, C i (\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))))\n  (e3 : \u2200 {i}, C i (\u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))))\n  (e4 : \u2200 {i} {m} {f : L.fn m}, C i (eq_axiom4 f))\n  (e5 : \u2200 {i} {m} {r : L.pr m}, C i (eq_axiom5 r))\n  : C i p :=\nbegin\n  suffices :\n    \u2200 {p : formula L} {U : Theory L} (b : U \u27f9 p) {i : \u2115} (ss : U \u2286 T^i), C i p,\n  { refine this b (by refl) },\n  rintros p U b,\n  induction b,\n  case generalize : U p b IH\n  { intros i ss,\n    have ss' : \u290aU \u2286 T ^ (i + 1), { rintros _ \u27e8q, mem, rfl\u27e9, simp[Theory.sf_itr_succ], refine \u27e8q, ss mem, rfl\u27e9 },\n    have : C (i + 1) p, from @IH (i + 1) ss',\n    refine GE (b.weakening ss') this },\n  case mdp : U p q b\u2081 b\u2082 IH\u2081 IH\u2082\n  { intros i ss, refine MP (weakening b\u2081 ss) (weakening b\u2082 ss) (IH\u2081 ss) (IH\u2082 ss) },\n  case by_axiom : U p mem\n  { intros i ss, refine by_axiom (ss mem) },\n  { refine \u03bb i ss, p0 },\n  { refine \u03bb i ss, p1 },\n  { refine \u03bb i ss, p2 },\n  { refine \u03bb i ss, p3 },\n  { refine \u03bb i ss, q1 },\n  { refine \u03bb i ss, q2 },\n  { refine \u03bb i ss, q3 },\n  { refine \u03bb i ss, e1 },\n  { refine \u03bb i ss, e2 },\n  { refine \u03bb i ss, e3 },\n  { refine \u03bb i ss, e4 },\n  { refine \u03bb i ss, e5 }\nend\n\n@[elab_as_eliminator]\ndef rec'' {T : Theory L} (C : \u03a0 (i : \u2115) (p : formula L) (b : T^i \u27f9 p), Sort v)\n  (GE : \u2200 {i} {p : formula L} (b : T^(i + 1) \u27f9 p), C (i + 1) p b \u2192 C i (\u2200.p) b.generalize)\n  (MP : \u2200 {i} {p q : formula L} (b\u2081 : T^i \u27f9 p \u27f6 q) (b\u2082 : T^i \u27f9 p), C i (p \u27f6 q) b\u2081 \u2192 C i p b\u2082 \u2192 C i q (mdp b\u2081 b\u2082))\n  (by_axiom : \u2200 {i} {p : formula L} (mem : p \u2208 T^i), C i p (by_axiom mem))\n  (p0 : \u2200 {i}, C i \u22a4 verum)\n  (p1 : \u2200 {i} {p q : formula L}, C i (p \u27f6 q \u27f6 p) imply\u2081)\n  (p2 : \u2200 {i} {p q r : formula L}, C i ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r) imply\u2082)\n  (p3 : \u2200 {i} {p q : formula L}, C i ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p) contraposition)\n  (q1 : \u2200 {i} {p : formula L} {t : term L}, C i (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t]) specialize)\n  (q2 : \u2200 {i} {p q : formula L}, C i (\u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6\u2200.q) univ_K)\n  (q3 : \u2200 {i} {p : formula L}, C i (p \u27f6 \u2200.(p^1)) dummy_univ)\n  (e1 : \u2200 {i}, C i (\u2200.(#0 =' #0)) eq_reflexivity)\n  (e2 : \u2200 {i}, C i (\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))) eq_symmetry)\n  (e3 : \u2200 {i}, C i (\u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))) eq_transitivity)\n  (e4 : \u2200 {i} {m} {f : L.fn m}, C i (eq_axiom4 f) function_ext)\n  (e5 : \u2200 {i} {m} {r : L.pr m}, C i (eq_axiom5 r) predicate_ext)\n  (i : \u2115) (p : formula L) (b : T^i \u27f9 p)\n  : C i p b :=\nbegin\n  suffices :\n    \u2200 {p : formula L} {U : Theory L} (b : U \u27f9 p) {i : \u2115} (e : U = T^i), C i p (by rcases e with rfl; exact b),\n  { have := this b (by refl), exact this },\n  rintros p U b,\n  induction b,\n  case generalize : U p b IH\n  { rintros i rfl,\n    exact GE _ (@IH (i + 1) rfl) },\n  case mdp : U p q b\u2081 b\u2082 IH\u2081 IH\u2082\n  { rintros i rfl, refine MP _ _ (@IH\u2081 i rfl) (@IH\u2082 i rfl) },\n  case by_axiom : U p mem\n  { rintros i rfl, refine by_axiom mem },\n  { rintros i rfl, exact p0 },\n  { rintros i rfl, exact p1 },\n  { rintros i rfl, exact p2 },\n  { rintros i rfl, exact p3 },\n  { rintros i rfl, exact q1 },\n  { rintros i rfl, exact q2 },\n  { rintros i rfl, exact q3 },\n  { rintros i rfl, exact e1 },\n  { rintros i rfl, exact e2 },\n  { rintros i rfl, exact e3 },\n  { rintros i rfl, exact e4 },\n  { rintros i rfl, exact e5 }\nend\n\n@[elab_as_eliminator]\ndef rec''_on {T : Theory L} (C : \u03a0 (i : \u2115) (p : formula L) (b : T^i \u27f9 p), Sort v)\n  (i : \u2115) (p : formula L) (b : T^i \u27f9 p)\n  (GE : \u2200 {i} {p : formula L} (b : T^(i + 1) \u27f9 p), C (i + 1) p b \u2192 C i (\u2200.p) b.generalize)\n  (MP : \u2200 {i} {p q : formula L} (b\u2081 : T^i \u27f9 p \u27f6 q) (b\u2082 : T^i \u27f9 p), C i (p \u27f6 q) b\u2081 \u2192 C i p b\u2082 \u2192 C i q (mdp b\u2081 b\u2082))\n  (by_axiom : \u2200 {i} {p : formula L} (mem : p \u2208 T^i), C i p (by_axiom mem))\n  (p0 : \u2200 {i}, C i \u22a4 verum)\n  (p1 : \u2200 {i} {p q : formula L}, C i (p \u27f6 q \u27f6 p) imply\u2081)\n  (p2 : \u2200 {i} {p q r : formula L}, C i ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r) imply\u2082)\n  (p3 : \u2200 {i} {p q : formula L}, C i ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p) contraposition)\n  (q1 : \u2200 {i} {p : formula L} {t : term L}, C i (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t]) specialize)\n  (q2 : \u2200 {i} {p q : formula L}, C i (\u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6\u2200.q) univ_K)\n  (q3 : \u2200 {i} {p : formula L}, C i (p \u27f6 \u2200.(p^1)) dummy_univ)\n  (e1 : \u2200 {i}, C i (\u2200.(#0 =' #0)) eq_reflexivity)\n  (e2 : \u2200 {i}, C i (\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))) eq_symmetry)\n  (e3 : \u2200 {i}, C i (\u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))) eq_transitivity)\n  (e4 : \u2200 {i} {m} {f : L.fn m}, C i (eq_axiom4 f) function_ext)\n  (e5 : \u2200 {i} {m} {r : L.pr m}, C i (eq_axiom5 r) predicate_ext)\n  : C i p b :=\nrec'' C @GE @MP @by_axiom @p0 @p1 @p2 @p3 @q1 @q2 @q3 @e1 @e2 @e3 @e4 @e5 i p b\n\nend proof\n\nnamespace provable\nvariables {T : Theory L}\n\nlemma generalize {p : formula L} (h : \u290aT \u22a2 p) : T \u22a2 \u2200.p := by rcases h; exact \u27e8h.generalize\u27e9\n\n@[simp] lemma specialize {p : formula L} (t) : T \u22a2 \u2200.p \u27f6 p.rew \u0131[0 \u21dd t] := \u27e8proof.specialize\u27e9\n\n@[simp] lemma univ_K (p q : formula L) : T \u22a2 \u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6\u2200.q := \u27e8proof.univ_K\u27e9\n\n@[simp] lemma dummy_univ_quantifier (p : formula L) : T \u22a2 p \u27f6 \u2200.(p^1) := \u27e8proof.dummy_univ\u27e9\n\n@[simp] lemma eq_reflexivity : T \u22a2 \u2200.(#0 =' #0) := \u27e8proof.eq_reflexivity\u27e9\n\n@[simp] lemma eq_symmetry : T \u22a2 \u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0)) := \u27e8proof.eq_symmetry\u27e9\n\n@[simp] lemma eq_transitivity : T \u22a2 \u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2)) := \u27e8proof.eq_transitivity\u27e9\n\n@[simp] lemma function_ext {n} (f : L.fn n) : T \u22a2 eq_axiom4 f := \u27e8proof.function_ext\u27e9\n\n@[simp] lemma predicate_ext {n} (r : L.pr n) : T \u22a2 eq_axiom5 r := \u27e8proof.predicate_ext\u27e9\n\nlemma generalize_of_closed [closed_Theory T] {p} (h : T \u22a2 p) : T \u22a2 \u2200.p :=\nby { apply generalize, simp[closed_Theory_sf_eq, h] }\n\nlemma generalize_itr : \u2200 {n p}, T^n \u22a2 p \u2192 T \u22a2 \u2200.[n] p\n| 0     p h := by simp* at*\n| (n+1) p h := by { simp at*, have := generalize_itr (generalize h), simp* at* }\n\nlemma nfal_subst : \u2200 (n) (p : formula L) (s : \u2115 \u2192 term L),\n  T \u22a2 (\u2200.[n] p) \u27f6 p.rew (\u03bb x, if x < n then s x else #(x-n))\n| 0     p s := by simp\n| (n+1) p s := by { simp,\n    have lmm\u2081 : T \u22a2 \u2200.(\u2200.[n] p) \u27f6 nfal (p.rew $ \u0131[0 \u21dd s n]^n) n,\n    { have := @specialize _ T (\u2200.[n] p) (s n), simp[formula.nfal_rew] at this,\n      exact this },\n    have s' := s,\n    have lmm\u2082 := nfal_subst n (p.rew $ \u0131[0 \u21dd s n]^n) s,\n    simp[formula.nested_rew] at lmm\u2082,\n    have : (\u03bb x, (\u0131[0 \u21dd s n]^n $ x).rew (\u03bb x, ite (x < n) (s x) #(x - n))) =\n      (\u03bb x, ite (x < n + 1) (s x) #(x - (n + 1))),\n    { simp[subst_pow], ext x, have C : x < n \u2228 x = n \u2228 n < x, from trichotomous _ _,\n      cases C,\n      { simp[C, nat.lt.step C] }, cases C, { simp[C, term.pow_eq] },\n      { have eqn\u2081 : \u00acx - 1 < n, from not_lt.mpr (nat.le_pred_of_lt C),\n        have eqn\u2082 : \u00acx < n + 1, from not_lt.mpr (nat.succ_le_iff.mpr C),\n        simp[C, eqn\u2081, eqn\u2082, nat.sub_sub, add_comm 1 n] } },\n    simp[this] at lmm\u2082,\n    exact imply_trans lmm\u2081 lmm\u2082 }\n\nlemma nfal_subst' {n} {p : formula L} (h : T \u22a2 \u2200.[n] p ) (s : \u2115 \u2192 term L) :\n  T \u22a2 p.rew (\u03bb x, if x < n then s x else #(x-n)) := (nfal_subst n p s) \u2a00 h\n\nlemma nfal_subst'_finitary {n} {p : formula L} (h : T \u22a2 \u2200.[n] p ) (s : finitary (term L) n) :\n  T \u22a2 p.rew (of_fin s) :=\nby { let s' : \u2115 \u2192 term L := \u03bb x, if h : x < n then s \u27e8x, h\u27e9 else default,\n     exact cast (by { congr, ext x, by_cases C : x < n; simp[C, s'],\n       simp[show n \u2264 x, from not_lt.mp C] }) (nfal_subst' h s')}\n\nlemma fal_complete_rew (p : formula L) (s : \u2115 \u2192 term L) :\n  T \u22a2 (\u2200.* p) \u27f6 p.rew s :=\nbegin\n  have : T \u22a2 (\u2200.* p) \u27f6 p.rew (\u03bb x, if x < p.arity then s x else #(x - p.arity)),\n    from nfal_subst p.arity p s,\n  have eqn : (p.rew (\u03bb x, if x < p.arity then s x else #(x - p.arity))) = p.rew s,\n    from formula.rew_rew p (\u03bb m h, by simp[h]),\n  simp[eqn] at this, exact this\nend\n\nlemma weakening {p} (h : T \u22a2 p) {U} (ss : T \u2286 U) : U \u22a2 p :=\nby rcases h; exact \u27e8h.weakening ss\u27e9\n\nlemma weakening' {U : Theory L} {p : formula L} : T \u2286 U \u2192 T \u22a2 p \u2192 U \u22a2 p := \u03bb hi hp,\nweakening hp hi\n\nprivate lemma delete_imply {p} (h : T \u22a2 p) : \u2200 q, T \\ {q} \u22a2 q \u27f6 p :=\nbegin\n  rcases h,\n  induction h with T p hyp_p IH T p\u2081 p\u2082 hyp_p\u2081\u2082 hyp_p\u2081 IH\u2081 IH\u2082 T p hyp_p;\n    try { intros q, simp }; intros q,\n  { have IH : \u290aT \\ {q^1} \u22a2 q^1 \u27f6 p := IH (q^1),\n    have lmm\u2081 : T \\ {q} \u22a2 q \u27f6 \u2200.(q^1), { simp },\n    have lmm\u2082 : T \\ {q} \u22a2 \u2200.(q^1) \u27f6 \u2200.p,\n    { suffices : T \\ {q} \u22a2 \u2200.(q^1 \u27f6 p),\n      { have lmm : T \\ {q} \u22a2 \u2200.(q^1 \u27f6 p) \u27f6 \u2200.(q^1) \u27f6 \u2200.p, simp,\n        exact lmm \u2a00 this },\n      refine generalize (weakening IH (\u03bb x h, _)), \n      rcases h with \u27e8h, neq\u27e9, rcases h with \u27e8p', h', rfl\u27e9,\n      refine \u27e8p', \u27e8h', \u03bb c, _\u27e9, rfl\u27e9, simp at c,\n      rw c at neq, exact neq rfl },\n    exact imply_trans lmm\u2081 lmm\u2082 },\n  { have : T \\ {q} \u22a2 (q \u27f6 p\u2081 \u27f6 p\u2082) \u27f6 (q \u27f6 p\u2081) \u27f6 (q \u27f6 p\u2082), simp, \n    have : T \\ {q} \u22a2 (q \u27f6 p\u2081) \u27f6 q \u27f6 p\u2082, from this \u2a00 (IH\u2081 _),\n    exact this \u2a00 (IH\u2082 _) },\n  { by_cases eqn : p = q,\n    { simp[eqn] },\n    { have : T \\ {q} \u22a2 p, from by_axiom \u27e8hyp_p, eqn\u27e9,\n      simp[this] } }\nend\n\ninstance : axiomatic_classical_logic (formula L) :=\n{ deduction' := \u03bb T p q h, by { have : (T+{p}) \\ {p} \u22a2 p \u27f6 q, from delete_imply h p,\n    refine weakening this (\u03bb x h, _), rcases h with \u27e8h, neq\u27e9,\n    cases h; simp* at* },\n  weakening := @weakening' L }\n\n@[elab_as_eliminator]\ntheorem rec'_on {T : Theory L} {C : \u2115 \u2192 formula L \u2192 Prop} {i : \u2115} {p : formula L} (b : T^i \u22a2 p)\n  (GE : \u2200 {i} {p : formula L} (b : T^(i + 1) \u22a2 p), C (i + 1) p \u2192 C i (\u2200.p))\n  (MP : \u2200 {i} {p q : formula L} (b\u2081 : T^i \u22a2 p \u27f6 q) (b\u2082 : T^i \u22a2 p), C i (p \u27f6 q) \u2192 C i p \u2192 C i q)\n  (by_axiom : \u2200 {i} {p : formula L} (mem : p \u2208 T^i), C i p)\n  (p0 : \u2200 {i}, C i \u22a4)\n  (p1 : \u2200 {i} {p q : formula L}, C i (p \u27f6 q \u27f6 p))\n  (p2 : \u2200 {i} {p q r : formula L}, C i ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r))\n  (p3 : \u2200 {i} {p q : formula L}, C i ((\u223cp \u27f6 \u223cq) \u27f6 q \u27f6 p))\n  (q1 : \u2200 {i} {p : formula L} {t : term L}, C i (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t]))\n  (q2 : \u2200 {i} {p q : formula L}, C i (\u2200.(p \u27f6 q) \u27f6 \u2200.p \u27f6\u2200.q))\n  (q3 : \u2200 {i} {p : formula L}, C i (p \u27f6 \u2200.(p^1)))\n  (e1 : \u2200 {i}, C i (\u2200.(#0 =' #0)))\n  (e2 : \u2200 {i}, C i (\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))))\n  (e3 : \u2200 {i}, C i (\u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))))\n  (e4 : \u2200 {i} {m} {f : L.fn m}, C i (eq_axiom4 f))\n  (e5 : \u2200 {i} {m} {r : L.pr m}, C i (eq_axiom5 r)) :\n C i p :=\nbegin\n  suffices :\n    \u2200 {p : formula L} {U : Theory L} (b : U \u22a2 p) {i : \u2115} (ss : U \u2286 T^i),  C i p,\n  { refine this b (by refl) },\n  rintros p U \u27e8b\u27e9,\n  induction b,\n  case generalize : U p b IH\n  { intros i ss,\n    have ss' : \u290aU \u2286 T ^ (i + 1), { rintros _ \u27e8q, mem, rfl\u27e9, simp[Theory.sf_itr_succ], refine \u27e8q, ss mem, rfl\u27e9 },\n    have : C (i + 1) p, from @IH (i + 1) ss',\n    refine GE (weakening \u27e8b\u27e9 ss') this },\n  case mdp : U p q b\u2081 b\u2082 IH\u2081 IH\u2082\n  { intros i ss, refine MP (weakening \u27e8b\u2081\u27e9 ss) (weakening \u27e8b\u2082\u27e9 ss) (IH\u2081 ss) (IH\u2082 ss) },\n  case by_axiom : U p mem\n  { intros i ss, refine by_axiom (ss mem) },\n  { refine \u03bb i ss, p0 },\n  { refine \u03bb i ss, p1 },\n  { refine \u03bb i ss, p2 },\n  { refine \u03bb i ss, p3 },\n  { refine \u03bb i ss, q1 },\n  { refine \u03bb i ss, q2 },\n  { refine \u03bb i ss, q3 },\n  { refine \u03bb i ss, e1 },\n  { refine \u03bb i ss, e2 },\n  { refine \u03bb i ss, e3 },\n  { refine \u03bb i ss, e4 },\n  { refine \u03bb i ss, e5 }\nend\n\ntheorem proof_compact : \u2200 {T : \u2115 \u2192 Theory L}, (\u2200 s, T s \u2286 T (s+1)) \u2192\n  \u2200 {p}, (\u22c3 s, T s) \u22a2 p \u2192 \u2203 s, T s \u22a2 p :=\nbegin\n  suffices : \u2200 {p} {U : Theory L}, U \u22a2 p \u2192 \u2200 {T : \u2115 \u2192 Theory L},\n    (\u2200 s, T s \u2286 T (s+1)) \u2192 U \u2286 (\u22c3 s, T s) \u2192 \u2203 s, T s \u22a2 p,\n  { refine \u03bb T hyp p h, this h hyp (\u03bb x hx, hx) },\n  rintros p U \u27e8b\u27e9,\n  induction b,\n  case generalize : T p h IH\n  { intros U hyp ss,\n    let U' := \u03bb s, \u290a(U s),\n    have hyp' : \u2200 s, U' s \u2286 U' (s + 1),\n    { simp[U'], intros s p hyp_p, exact hyp s hyp_p },\n    have ss' : \u290aT \u2286 \u22c3 s, U' s,\n    { intros q hyp_q, rcases hyp_q with \u27e8q', hyp_q', rfl\u27e9, rcases (ss hyp_q') with \u27e8_, \u27e8s, rfl\u27e9, hyp_s\u27e9,\n      simp, refine \u27e8s, _, hyp_s, rfl\u27e9 },\n    have : \u2203 s, U' s \u22a2 p, from IH hyp' ss', rcases this with \u27e8s, h\u27e9,\n    refine \u27e8s, generalize h\u27e9 },\n  case mdp : T p q hyp_pq hyp_p IH\u2081 IH\u2082\n  { intros U hyp ss,\n    have : \u2203 s, U s \u22a2 p \u27f6 q, from IH\u2081 hyp ss, rcases this with \u27e8s\u2081, lmm\u2081\u27e9,\n    have : \u2203 s, U s \u22a2 p, from IH\u2082 hyp ss, rcases this with \u27e8s\u2082, lmm\u2082\u27e9,\n    refine \u27e8max s\u2081 s\u2082, _\u27e9,\n    have lmm\u2081 : U (max s\u2081 s\u2082) \u22a2 p \u27f6 q, from provable.weakening lmm\u2081 (ss_le hyp (by simp)),\n    have lmm\u2082 : U (max s\u2081 s\u2082) \u22a2 p, from provable.weakening lmm\u2082 (ss_le hyp (by simp)),\n    exact lmm\u2081 \u2a00 lmm\u2082 },\n  case by_axiom : T p hyp_p\n  { intros U hyp ss, rcases (ss hyp_p) with \u27e8_, \u27e8s, rfl\u27e9, hyp_s\u27e9,\n    refine \u27e8s, by_axiom hyp_s\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 },\n  { refine \u03bb _ _ _, \u27e80, by simp\u27e9 }\nend\n\nlemma finitary.conjunction_mem {n : \u2115} {P : finitary (formula L) n} :\n  \u2200 {p}, p \u2208 P \u2192 T \u22a2 finitary.conjunction n P \u27f6 p :=\nbegin\n  induction n with n IH; simp[finitary.conjunction];\n  simp[has_mem.mem, finitary.mem],\n  intros p mem,\n  exact and_imply_of_imply_right (IH mem)\nend\n\nprivate lemma list_conjunction_sf (P\u2080 : list (formula L)) : (\u2200 p, p \u2208 P\u2080 \u2192 \u290aT p) \u2192\n  \u2203 P : list (formula L), (P.conjunction)^1 = P\u2080.conjunction \u2227 (\u2200 p, p \u2208 P \u2192 T p) :=\nbegin\n  induction P\u2080 with p\u2080 P\u2080 IHl, { refine \u03bb _, \u27e8[], _\u27e9, simp },\n  { intros hyp,\n    have : \u2200 p, p \u2208 P\u2080 \u2192 \u290aT p,\n    { intros p hyp_p, refine hyp _ _, simp[hyp_p] },\n    rcases IHl this with \u27e8P, hyp_P\u27e9,\n    have := hyp p\u2080 (by simp),\n    rcases this with \u27e8p, hyp_p, rfl\u27e9,\n    have lmm\u2081 : ((p :: P).conjunction)^1= (p^1 :: P\u2080).conjunction,\n    { simp[hyp_P] },\n    have lmm\u2082 : \u2200 (q : formula L), q \u2208 (p :: P) \u2192 T q,\n    { simp, refine \u27e8hyp_p, hyp_P.2\u27e9 },\n    refine \u27e8p :: P, lmm\u2081, lmm\u2082\u27e9 }\nend\n\nprivate lemma list_conjunction_rew_eq : \u2200 (P : list (formula L)) (s),\n  P.conjunction.rew s = list.conjunction (P.map (\u03bb p, p.rew s))\n| []       _ := by simp[formula.rew]\n| (p :: P) s := by simp[formula.rew, list_conjunction_rew_eq P]\n\ntheorem proof_conjunction {T : Theory L} {p} :\n  T \u22a2 p \u2192 \u2203 P : list (formula L), (\u2200 p, p \u2208 P \u2192 T p) \u2227 \u2205 \u22a2 P.conjunction \u27f6 p := \u03bb h,\nbegin\n  rcases h,\n  induction h,\n  case generalize : T p hyp IH\n  { rcases IH with \u27e8P\u2080, hyp_P\u2080, prov\u27e9,\n    have : \u2203 P : list (formula L), (P.conjunction)^1 = P\u2080.conjunction \u2227 \u2200 p, p \u2208 P \u2192 T p,\n    from list_conjunction_sf _ hyp_P\u2080,\n    rcases this with \u27e8P, eqn, hyp_P\u27e9,\n    have : \u2205 \u22a2 P.conjunction \u27f6 \u2200.p,\n    { refine deduction.mp (generalize _),\n      rw [\u2190sf_dsb, eqn], refine deduction.mpr (weakening prov (\u03bb x hx, _)), cases hx },\n    refine \u27e8P, hyp_P, this\u27e9 },\n  case mdp : T p q hyp_pq hyp_p IH\u2081 IH\u2082\n  { rcases IH\u2081 with \u27e8P\u2081, IH\u2081, prov\u2081\u27e9, rcases IH\u2082 with \u27e8P\u2082, IH\u2082, prov\u2082\u27e9,\n    refine \u27e8P\u2081 ++ P\u2082, _, _\u27e9,\n    { simp, intros p h, cases h, refine IH\u2081 _ h, refine IH\u2082 _ h },\n    { have : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 P\u2082.conjunction, from deduction.mpr (list_conjunction_weakening (by simp)),\n      have lmm\u2081 : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 p,\n        from (show _ \u22a2 P\u2082.conjunction \u27f6 p, from weakening_insert prov\u2082 _) \u2a00 this,\n      have : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 P\u2081.conjunction, from deduction.mpr (list_conjunction_weakening (by simp)),\n      have lmm\u2082 : \u2205+{(P\u2081 ++ P\u2082).conjunction} \u22a2 p \u27f6 q,\n      from (show _ \u22a2 P\u2081.conjunction \u27f6 p \u27f6 q, from weakening_insert prov\u2081 _) \u2a00 this,\n      refine deduction.mp (lmm\u2082 \u2a00 lmm\u2081) } },\n  case by_axiom : T p hyp_p\n  { refine \u27e8[p], _\u27e9, simp,\n    have : \u2205 \u22a2 p \u2293 \u22a4 \u27f6 p,\n    { apply deduction.mp,\n      have : \u2205+{p \u2293 \u22a4} \u22a2 p \u2293 \u22a4, from insert (p \u2293 \u22a4),\n      simp[*, axiomatic_classical_logic'.iff_and] at* },\n    refine hyp_p },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp },\n  { refine \u27e8[], _\u27e9, simp }\nend\n\ninstance : Theory.has_finite_character (formula L) :=\nTheory.finite_character_of_finite_provable (formula L) (\u03bb T p, proof_conjunction)\n\ntheorem proof_conjunction_union {T U : Theory L} {p} :\n  T \u222a U \u22a2 p \u2192 \u2203 P Q : list (formula L), (\u2200 p, p \u2208 P \u2192 T p) \u2227 (\u2200 p, p \u2208 Q \u2192 U p) \u2227\n  \u2205 \u22a2 P.conjunction \u27f6 Q.conjunction \u27f6 p := \u03bb h,\nbegin\n  rcases proof_conjunction h with \u27e8R, hR, b\u27e9,\n  let P := R.filter (\u03bb p, p \u2208 T),\n  let Q := R.filter (\u03bb p, p \u2208 U),\n  refine \u27e8P, Q, by { simp, intros p _ h, exact h }, by { simp, intros p _ h, exact h }, _\u27e9,\n  refine (deduction.mp $ deduction.mp _),\n  have : \u2205 +{ P.conjunction } +{ Q.conjunction } \u22a2 R.conjunction,\n  refine list_conjunction_provable _,\n  { intros p memR, rcases hR p memR with (memT | memU),\n    { have : p \u2208 P, by simp[memR, memT], refine list_conjunction_mem this \u2a00 (by simp[-insert_emptyc_eq]) },\n    { have : p \u2208 Q, by simp[memR, memU], refine list_conjunction_mem this \u2a00 (by simp[-insert_emptyc_eq]) } },\n  exact (weakening b (by simp)) \u2a00 this\nend\n\nlemma fal_subst {p} (h : T \u22a2 \u2200.p) (t) : T \u22a2 p.rew \u0131[0 \u21dd t] :=\n(show T \u22a2 \u2200.p \u27f6 p.rew \u0131[0 \u21dd t], by simp) \u2a00 h\n\ninfixl ` \u229a `:60 := fal_subst\n\nlemma add_sf (p) : \u290a(T +{ \u2200.p }) \u22a2 p :=\nby { have : \u290a(T +{\u2200.p}) \u22a2 (\u2200.p)^1, rw \u2190 sf_dsb, simp, \n     have := fal_subst this #0, simp[formula.nested_rew] at this,\n     exact this }\n\nlemma cl_prove_rew [cl : closed_Theory T] : \u2200 {p : formula L}, T \u22a2 p \u2192 \u2200 s, T \u22a2 p.rew s :=\nbegin\n  suffices : \u2200 {p : formula L} {T}, T \u22a2 p \u2192 closed_Theory T \u2192 \u2200 s, T \u22a2 p.rew s,\n  { refine \u03bb p h s, this h cl _ },\n  rintros p T \u27e8b\u27e9,\n  induction b,\n  case generalize : T p hyp IH\n  { intros cl s, rw[@closed_Theory_sf_eq _ _ cl] at IH,\n    refine generalize _, simp[@closed_Theory_sf_eq _ _ cl], exact IH cl _ },\n  case mdp : T p q hyp_pq hyp_p IH\u2081 IH\u2082\n  { intros cl s, simp[formula.rew, @closed_Theory_sf_eq _ _ cl] at*, refine (IH\u2081 cl _) \u2a00 (IH\u2082 cl _) },\n  case by_axiom : T p hyp\n  { intros cl s, simp[show is_sentence p, by exactI closed_Theory.cl hyp], exact by_axiom hyp },\n  { simp },\n  { simp },\n  { simp },\n  { simp },\n  { intros, simp[formula.rew, formula.subst_sf_rew] },\n  { simp },\n  case dummy_univ : T p { intros,\n    simp,\n    have : (p^1).rew (s^1) = (p.rew s)^1,\n    { simp[formula.pow_eq, formula.rew, formula.nested_rew], refl },\n    simp[this] },\n  { simp },\n  { simp },\n  { simp },\n  { simp [formula.is_sentence_rew eq_axiom4_is_sentence] },\n  { simp [formula.is_sentence_rew eq_axiom5_is_sentence] }\nend\n\nlemma pow_of_cl [closed_Theory T] {p : formula L} (i : \u2115) : T \u22a2 p \u2192 T^i \u22a2 p :=\nby simp[closed_Theory_pow_eq]\n\nlemma iff_fal_complete [closed_Theory T] {p : formula L} : T \u22a2 p \u2194 T \u22a2 \u2200.* p :=\n\u27e8\u03bb h, generalize_itr (pow_of_cl p.arity h), \u03bb h, by simpa using fal_complete_rew p \u0131 \u2a00 h\u27e9\n\nlemma pp_prove_rew {n} (pp : proper_at n T) :\n  \u2200 {p : formula L}, T \u22a2 p \u2192 \u2200 s, T \u22a2 p.rew (s^n) :=\nbegin\n  suffices : \u2200 {p : formula L} {T},\n    T \u22a2 p \u2192 \u2200 {n}, proper_at n T \u2192 \u2200 s, T \u22a2 p.rew (s^n),\n  { refine \u03bb p h s, this h @pp _ },\n  rintros p T \u27e8b\u27e9,\n  induction b,\n  case generalize : T p hyp IH\n  { intros n pp s,\n    refine generalize _, refine @IH (n+1) (@proper_Theory_sf_itr _ _ _ @pp 1) s },\n  case mdp : T p q hyp_pq hyp_p IH\u2081 IH\u2082\n  { intros n pp s, refine (IH\u2081 @pp _) \u2a00 (IH\u2082 @pp _) },\n  case by_axiom : T p hyp\n  { intros n pp s, refine by_axiom (pp _ _ hyp) },\n  { simp },\n  { simp },\n  { simp },\n  { simp },\n  { intros, simp[formula.subst_sf_rew] },\n  { simp },\n  case dummy_univ : T p { intros,\n    simp,\n    simp[\u2190formula.pow_rew_distrib] },\n  { simp },\n  { simp },\n  { simp },\n  { simp [formula.is_sentence_rew eq_axiom4_is_sentence] },\n  { simp [formula.is_sentence_rew eq_axiom5_is_sentence] },\nend\n\nlemma proper_Theory_pow_rew (n : \u2115) [proper_Theory T] : \u2200 {p : formula L},\n  T^n \u22a2 p \u2192 \u2200 s, T^n \u22a2 p.rew (s^n) := @pp_prove_rew L (T^n) n (properc_Theory_sf_itr)\n\nlemma proper_Theory_rew [proper_Theory T] : \u2200 {p : formula L},\n  T \u22a2 p \u2192 \u2200 s, T \u22a2 p.rew s := @pp_prove_rew _ _ 0 proper_Theory.proper\n\nprotected lemma finitary.disjunction {n} {P : finitary (formula L) n} (i) (h : T \u22a2 P i) : T \u22a2 finitary.disjunction n P :=\nby { induction n with n IH; simp, { exfalso, exact i.val.not_lt_zero i.property },\n     { rcases i with \u27e8i, hi\u27e9,\n       have : i = n \u2228 i < n, exact eq_or_lt_of_le (nat.lt_succ_iff.mp hi), rcases this with (rfl | lt),\n       { refine imply_or_left _ _ \u2a00 h }, { simpa using imply_or_right _ _ \u2a00 (@IH (\u03bb i, P i) \u27e8i, lt\u27e9 (by simp; exact h)) } } }\n\nlemma sf_sf {p : formula L} : \u290aT \u22a2 p^1 \u2194 T \u22a2 p :=\n\u27e8\u03bb h, by { have := fal_subst (generalize h) #0, simp* at* },\n \u03bb h, by { have : \u2203 P : list (formula L), (\u2200 p, p \u2208 P \u2192 p \u2208 T) \u2227 \u2205 \u22a2 P.conjunction \u27f6 p,\n  from proof_conjunction h, rcases this with \u27e8P, hyp_P, prov\u27e9,\n  have lmm\u2081 : \u290aT \u22a2 list.conjunction (P.map (\u03bb p, p^1)),\n  { refine list_conjunction_provable (\u03bb p hyp, by_axiom _), simp at hyp, rcases hyp with \u27e8p', p'_mem, rfl\u27e9,\n    refine \u27e8p', hyp_P p' p'_mem, rfl\u27e9 },\n  have lmm\u2082 : \u290aT \u22a2 list.conjunction (P.map (\u03bb p, p^1)) \u27f6 p^1,\n  { have : \u2205 \u22a2 (P.conjunction)^1 \u27f6 p^1, by exactI cl_prove_rew prov _,\n    simp[formula.pow_eq, list_conjunction_rew_eq] at this,\n    refine weakening this (\u03bb p h, _), exfalso, exact h },\n  refine lmm\u2082 \u2a00 lmm\u2081 }\u27e9\n\nlemma sf_itr_sf_itr : \u2200 {i : \u2115} {p : formula L},\n  T^i \u22a2 p^i \u2194 T \u22a2 p\n| 0     p := by simp\n| (i+1) p := by simp[Theory.sf_itr_succ];\n    rw [show p^(i + 1) = (p^i)^1, by simp[formula.pow_add], sf_sf, @sf_itr_sf_itr i]\n\nlemma pow_rew' [proper_Theory T] (i : \u2115) {p : formula L} (h : T^(i + 1) \u22a2 p) (s u : \u2115 \u2192 term L) :\n  T^i \u22a2 p.rew (\u03bb x, if x < i + 1 then s x else (u (x - i - 1))^i) :=\nbegin\n  have t := #0,\n  let f : \u2115 \u2192 term L := \u03bb x, if x < i + 1 then s x else (u (x - i - 1))^i,\n  have : T^i \u22a2 \u2200.(\u2200.[i + 1] p) ^ (i+1),\n    from generalize (show T^(i + 1) \u22a2 (\u2200.[i + 1] p) ^ (i+1), from sf_itr_sf_itr.mpr (generalize_itr h)),\n  have := fal_subst this t,\n  have := (proper_Theory_pow_rew i this u),\n  simp[formula.nfal_pow, formula.nested_rew, -nfal] at this,\n  have := nfal_subst' this s, simp[formula.nested_rew, term.nested_rew, \u0131] at this,\n  simp[subst_pow, rewriting_sf_itr.pow_add] at this,\n  have eqn : (\u03bb x, (ite (x < i + 1) #x #(x + (i + 1))).rew (\u03bb x, (\u0131[(i + 1) \u21dd t ^ (i + 1)] x).rew\n    (\u03bb x, (u^(i + (i + 1)) $ x).rew (\u03bb x, ite (x < i+1) (s x) #(x - (i+1))) ))) = f,\n  { funext x\u2080, by_cases C : x\u2080 < i + 1; simp[C],\n    { simp[f, rewriting_sf_itr.pow_eq'],\n      have : x\u2080 < i + (i + 1), exact nat.lt_add_left _ _ _ C,\n      simp[this, C] },\n    { have : i + 1 < x\u2080 + (i + 1), { omega },\n      simp[f, this, rewriting_sf_itr.pow_eq', term.pow_eq], \n      have : \u00acx\u2080 + i < i + (i + 1), { omega }, simp[this],\n      have e\u2081 : \u2200 x, \u00acx + (i + (i + 1)) < i + 1, { intros x, omega },\n      have e\u2082 : \u2200 x, x + (i + (i + 1)) - (i + 1) = x + i, { omega },\n      simp[e\u2081, e\u2082],\n      have : i + 1 \u2264 x\u2080, { exact not_lt.mp C },\n      simp[\u2190nat.sub_sub, C] } },\n  rw eqn at this,\n  exact this\nend\n\nlemma pow_subst' [proper_Theory T] (i : \u2115) {p : formula L} (h : T^(i + 1) \u22a2 p) (t : term L) :\n  T^i \u22a2 p.rew \u0131[i \u21dd t] :=\nby { have := pow_rew' i h \u0131[i \u21dd t] \u0131,\n     have eqn : (\u03bb x, ite (x < i + 1) (\u0131[i \u21dd t] x) (\u0131 (x - i - 1) ^ i)) = \u0131[i \u21dd t],\n     { funext x, by_cases C\u2081 : x < i + 1; simp[C\u2081],\n       have : i < x, exact nat.succ_le_iff.mp (not_lt.mp C\u2081),\n       simp[this, \u0131], omega },\n     rw eqn at this, exact this }\n\nlemma use {p : formula L} (t) (h : T \u22a2 p.rew \u0131[0 \u21dd t]) : T \u22a2 \u2203.p :=\nbegin\n  simp[has_exists_quantifier.ex, formula.ex],\n  refine raa (p.rew \u0131[0 \u21dd t]) (by simp[h]) (deduction.mpr _),\n  have : \u223cp.rew \u0131[0 \u21dd t] = (\u223cp).rew \u0131[0 \u21dd t] := rfl,\n  rw[this], refine specialize t,\nend\n\nlemma use_0 {p : formula L} (h : \u290aT \u22a2 p) : T \u22a2 \u2203.p :=\nuse #0 ((generalize h) \u229a #0)\n\n@[simp] lemma eq_refl : \u2200 {t : term L}, T \u22a2 t =' t := (@eq_reflexivity _ T).fal_subst\n\nlemma eq_symm : \u2200 {t u : term L}, (T \u22a2 t =' u) \u2192 (T \u22a2 u =' t) :=\nbegin\n  intros t u h,\n  have : T \u22a2 (t =' u) \u27f6 (u =' t), { have := fal_subst (fal_subst (@eq_symmetry _ T) u) t, simp at*, refine this },\n  refine this \u2a00 h\nend\n\nlemma eq_trans {t\u2081 t\u2082 t\u2083 : term L} : (T \u22a2 t\u2081 =' t\u2082) \u2192 (T \u22a2 t\u2082 =' t\u2083) \u2192 (T \u22a2 t\u2081 =' t\u2083) := \u03bb h\u2081 h\u2082,\nby { have : T \u22a2 (t\u2081 =' t\u2082) \u27f6 (t\u2082 =' t\u2083) \u27f6 (t\u2081 =' t\u2083),\n     { have := (@eq_transitivity _ T) \u229a t\u2083 \u229a t\u2082 \u229a t\u2081, simp[\u2190term.pow_rew_distrib] at*,\n       exact this },\n     exact (this \u2a00 h\u2081) \u2a00 h\u2082 }\n\nlemma ne_symm {t u : term L} (h : T \u22a2 t \u2260' u) : T \u22a2 u \u2260' t :=\nneg_of_equiv h (show T \u22a2 (t =' u) \u27f7 (u =' t), by { \n    have : T \u22a2 (t =' u) \u27f6 (u =' t),\n    { have := fal_subst (fal_subst (@eq_symmetry _ T) u) t, simp at*, refine this },\n    have : T \u22a2 (u =' t) \u27f6 (t =' u),\n    { have := fal_subst (fal_subst (@eq_symmetry _ T) t) u, simp at*, refine this },\n    simp[iff_equiv, *] })\n\nlemma function_ext' {n} (f : L.fn n) (v\u2081 v\u2082 : finitary (term L) n) :\n  T \u22a2 (\u22c0 i, v\u2081 i =' v\u2082 i) \u27f6 (term.app f v\u2081 =' term.app f v\u2082) :=\nbegin\n  let s : \u2115 \u2192 term L :=\n    (\u03bb x, if h\u2081 : x < n then v\u2081 \u27e8x, h\u2081\u27e9 else\n          if h\u2082 : x < 2*n then v\u2082 \u27e8x - n, by { simp[two_mul] at*, omega}\u27e9 else #x),\n  have eq_conj :\n    (\u03bb i, ite (\u2191i < 2 * n) (s \u2191i) #(\u2191i - 2*n) =' ite (n + \u2191i < 2*n) (s (n + \u2191i)) #(n + \u2191i - 2*n) : fin n \u2192 formula L) =\n    (\u03bb i, v\u2081 i =' v\u2082 i),\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property],\n    intros h, exfalso, refine not_le_of_gt i.property (le_of_add_le_left h) },      \n  have eq_v\u2081 : (\u03bb i, ite (\u2191i < 2 * n) (s \u2191i) #(\u2191i - 2 * n)) = v\u2081,\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property],\n    intros h, exfalso, refine not_le_of_gt i.property (le_of_add_le_left h) },\n  have eq_v\u2082 : (\u03bb i, ite (n + \u2191i < 2 * n) (s (n + \u2191i)) #(n + \u2191i - 2 * n)) = v\u2082,\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property] },\n  have := nfal_subst' (@function_ext _ T _ f) s,\n  simp[eq_conj, eq_v\u2081, eq_v\u2082] at this, exact this\nend\n\nlemma predicate_ext' {n} (r : L.pr n) (v\u2081 v\u2082 : finitary (term L) n) :\n  T \u22a2 (\u22c0 i, v\u2081 i =' v\u2082 i) \u27f6 formula.app r v\u2081 \u27f6 formula.app r v\u2082 :=\nbegin\n  let s : \u2115 \u2192 term L :=\n    (\u03bb x, if h\u2081 : x < n then v\u2081 \u27e8x, h\u2081\u27e9 else\n          if h\u2082 : x < 2*n then v\u2082 \u27e8x - n, by { simp[two_mul] at*, omega}\u27e9 else #x),\n  have eq_conj :\n    (\u03bb i, ite (\u2191i < 2 * n) (s \u2191i) #(\u2191i - 2*n) =' ite (n + \u2191i < 2*n) (s (n + \u2191i)) #(n + \u2191i - 2*n) : fin n \u2192 formula L) =\n    (\u03bb i, v\u2081 i =' v\u2082 i),\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property],\n    intros h, exfalso, refine not_le_of_gt i.property (le_of_add_le_left h) },      \n  have eq_v\u2081 : (\u03bb i, ite (\u2191i < 2 * n) (s \u2191i) #(\u2191i - 2 * n)) = v\u2081,\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property],\n    intros h, exfalso, refine not_le_of_gt i.property (le_of_add_le_left h) },\n  have eq_v\u2082 : (\u03bb i, ite (n + \u2191i < 2 * n) (s (n + \u2191i)) #(n + \u2191i - 2 * n)) = v\u2082,\n  { funext i, simp[s, two_mul, show \u2191i < n, from i.property] },\n  have := nfal_subst' (@predicate_ext _ T _ r) s,\n  simp[eq_conj, eq_v\u2081, eq_v\u2082] at this, exact this\nend\n\nlemma predicate_ext'' {n} (r : L.pr n) (v\u2081 v\u2082 : finitary (term L) n) :\n  T \u22a2 (\u22c0 i, v\u2081 i =' v\u2082 i) \u27f6 (formula.app r v\u2081 \u27f7 formula.app r v\u2082) :=\nby { refine deduction.mp _,\n     simp[iff_equiv], split,\n     { refine (predicate_ext' r v\u2081 v\u2082) \u2a00 (by simp) },\n     { refine (predicate_ext' r v\u2082 v\u2081) \u2a00\n       (conjunction_iff.mpr (\u03bb i, eq_symm (deduction.mpr $ finitary.conjunction_mem $ finitary.index_mem _ i))) } }\n\nlemma equal_rew_equal (s\u2081 s\u2082 : \u2115 \u2192 term L) (e : \u2200 n, T \u22a2 s\u2081 n =' s\u2082 n) : \u2200 (t : term L) ,\n  T \u22a2 t.rew s\u2081 =' t.rew s\u2082\n| (#n)                := by simp; exact e _\n| (@term.app _ n f v) :=\n  by { simp,\n       have : T \u22a2 finitary.conjunction n (\u03bb i, (v i).rew s\u2081 =' (v i).rew s\u2082),\n       { simp, intros i, refine equal_rew_equal (v i) },\n       refine (@function_ext' _ T _ f (\u03bb i, (v i).rew s\u2081) (\u03bb i, (v i).rew s\u2082)) \u2a00 this }\n\nlemma equal_fal_subst_equal (t : term L) {t\u2081 t\u2082} (h : T \u22a2 t\u2081 =' t\u2082) :\n  T \u22a2 t.rew (t\u2081 \u2322 \u0131) =' t.rew (t\u2082 \u2322 \u0131) :=\nby { refine equal_rew_equal _ _ (\u03bb n, _) t, { cases n; simp[concat, h] } }\n\nlemma equal_rew_iff {s\u2081 s\u2082 : \u2115 \u2192 term L} (eqn : \u2200 n, T \u22a2 s\u2081 n =' s\u2082 n) (p : formula L) :\n  T \u22a2 p.rew s\u2081 \u27f7 p.rew s\u2082 :=\nbegin\n  induction p generalizing T s\u2081 s\u2082,\n  case verum { simp[show (formula.verum : formula L) = \u22a4, from rfl] },\n  case app : n p v { intros, simp[axiomatic_classical_logic'.iff_equiv],\n    suffices : \u2200 (s\u2081 s\u2082 : \u2115 \u2192 term L) (h : \u2200 (n : \u2115), T \u22a2 s\u2081 n =' s\u2082 n), T \u22a2 formula.app p (\u03bb i, (v i).rew s\u2081) \u27f6 formula.app p (\u03bb i, (v i).rew s\u2082),\n    { refine \u27e8this _ _ eqn, this s\u2082 s\u2081 (\u03bb x, eq_symm (eqn x))\u27e9 },\n    intros s\u2081 s\u2082 eqs,\n    have : T \u22a2 \u22c0 i, (v i).rew s\u2081 =' (v i).rew s\u2082,\n    { simp, intros i,refine equal_rew_equal _ _ eqs _ },\n    refine (predicate_ext' p _ _) \u2a00 this },\n  case equal : t\u2081 t\u2082 { intros, simp[axiomatic_classical_logic'.iff_equiv],\n    refine \u27e8deduction.mp _, deduction.mp _\u27e9,\n    { have lmm\u2081 : T+{t\u2081.rew s\u2081 =' t\u2082.rew s\u2081} \u22a2 t\u2081.rew s\u2082 =' t\u2081.rew s\u2081,\n      { refine equal_rew_equal s\u2082 s\u2081 (\u03bb n, eq_symm _) t\u2081, simp[eqn n] },\n      have lmm\u2082 : T+{t\u2081.rew s\u2081 =' t\u2082.rew s\u2081} \u22a2 t\u2081.rew s\u2081 =' t\u2082.rew s\u2081, { simp },\n      have lmm\u2083 : T+{t\u2081.rew s\u2081 =' t\u2082.rew s\u2081} \u22a2 t\u2082.rew s\u2081 =' t\u2082.rew s\u2082,\n      { refine equal_rew_equal s\u2081 s\u2082 (\u03bb n, _) t\u2082, simp[eqn n]  },\n      refine eq_trans lmm\u2081 (eq_trans lmm\u2082 lmm\u2083) },\n    { have lmm\u2081 : T+{t\u2081.rew s\u2082 =' t\u2082.rew s\u2082} \u22a2 t\u2081.rew s\u2081 =' t\u2081.rew s\u2082,\n      { refine equal_rew_equal s\u2081 s\u2082 (\u03bb n, _) t\u2081, simp[eqn n] },\n      have lmm\u2082 : T+{t\u2081.rew s\u2082 =' t\u2082.rew s\u2082} \u22a2 t\u2081.rew s\u2082 =' t\u2082.rew s\u2082, { simp },\n      have lmm\u2083 : T+{t\u2081.rew s\u2082 =' t\u2082.rew s\u2082} \u22a2 t\u2082.rew s\u2082 =' t\u2082.rew s\u2081,\n      { refine equal_rew_equal s\u2082 s\u2081 (\u03bb n, eq_symm _) t\u2082, simp[eqn n]  },\n      refine eq_trans lmm\u2081 (eq_trans lmm\u2082 lmm\u2083) } },\n  case imply : p q IH\u2081 IH\u2082\n  { intros, \n    simp[axiomatic_classical_logic'.iff_equiv] at*, split,\n    { refine deduction.mp (deduction.mp _), \n      have : T+{p.rew s\u2081 \u27f6 q.rew s\u2081}+{p.rew s\u2082} \u22a2 p.rew s\u2082, simp,\n      have : T+{p.rew s\u2081 \u27f6 q.rew s\u2081}+{p.rew s\u2082} \u22a2 p.rew s\u2081, from (by simp[IH\u2081 eqn]) \u2a00 this,\n      have : T+{p.rew s\u2081 \u27f6 q.rew s\u2081}+{p.rew s\u2082} \u22a2 q.rew s\u2081,\n        from (show _ \u22a2 p.rew s\u2081 \u27f6 q.rew s\u2081, by simp) \u2a00 this,\n      from (by simp[IH\u2082 eqn]) \u2a00 this },\n    { refine deduction.mp (deduction.mp _),\n      have : T+{p.rew s\u2082 \u27f6 q.rew s\u2082}+{p.rew s\u2081} \u22a2 p.rew s\u2081, simp,\n      have : T+{p.rew s\u2082 \u27f6 q.rew s\u2082}+{p.rew s\u2081} \u22a2 p.rew s\u2082, from (by simp[IH\u2081 eqn]) \u2a00 this,\n      have : T+{p.rew s\u2082 \u27f6 q.rew s\u2082}+{p.rew s\u2081} \u22a2 q.rew s\u2082,\n        from (show _ \u22a2 p.rew s\u2082 \u27f6 q.rew s\u2082, by simp) \u2a00 this,\n      from (by simp[IH\u2082 eqn]) \u2a00 this } },\n  case neg : p IH\n  { intros, simp[axiomatic_classical_logic'.iff_equiv] at*,\n    refine \u27e8contrapose.mpr _, contrapose.mpr _\u27e9; simp[IH eqn] },\n  case fal : p IH\n  { intros, simp[axiomatic_classical_logic'.iff_equiv],\n    have := @IH (T^1) (s\u2081^1) (s\u2082^1)\n      (\u03bb n, by { cases n; simp, exact sf_sf.mpr (eqn n) }),\n    simp[axiomatic_classical_logic'.iff_equiv] at this, \n    refine \u27e8univ_K _ _ \u2a00 (generalize this.1), univ_K _ _ \u2a00 (generalize this.2)\u27e9 }\nend\n\nlemma iff_of_eqs {s\u2081 s\u2082 : \u2115 \u2192 term L} (eqn : \u2200 n, T \u22a2 s\u2081 n =' s\u2082 n) (p : formula L) :\n  T \u22a2 p.rew s\u2081 \u2194 T \u22a2 p.rew s\u2082 :=\n\u27e8\u03bb h, of_equiv_p h (equal_rew_iff eqn p), \u03bb h, of_equiv_p h (equal_rew_iff (\u03bb n, eq_symm $ eqn n) p)\u27e9\n\nlemma iff_rew_of_eq {t : term L} {n : \u2115} (eqn : T \u22a2 #n =' t) (p : formula L) :\n  T \u22a2 p \u27f7 p.rew (\u03bb x, if x = n then t else #x) :=\nbegin\n  suffices : T \u22a2 p.rew \u0131 \u27f7 p.rew (\u03bb x, if x = n then t else #x), { simp*at* },\n  refine equal_rew_iff (\u03bb x, _) _,\n  { by_cases C : x = n,\n    { rcases C with rfl, simp[\u0131, eqn] },\n    { simp[C] } }\nend\n\nlemma rew_of_eq (t : term L) (n : \u2115)\n  (eqn : T \u22a2 #n =' t) {p : formula L} (h : T \u22a2 p.rew (\u03bb x, if x = n then t else #x)) :\n  T \u22a2 p :=\nby have := iff_rew_of_eq eqn p; simp[iff_equiv] at this;\n   exact this.2 \u2a00 h\n\nlemma specialize_iff {t : term L} (p : formula L) :\n  T \u22a2 p.rew \u0131[0 \u21dd t] \u27f7 \u2200.((#0 =' t^1) \u27f6 p) :=\nbegin\n  simp[axiomatic_classical_logic'.iff_equiv], split,\n  { refine deduction.mp (generalize (deduction.mp _)),\n    simp[\u2190sf_dsb],\n    have : (p.rew \u0131[0 \u21dd t])^1 = p.rew (\u03bb x, if x = 0 then t^1 else #x),\n    { simp[formula.pow_rew_distrib, formula.pow_eq, formula.nested_rew],\n      congr, funext x, cases x; simp, refl },\n    rw this,\n    refine rew_of_eq (t^1) 0 (by simp) (by simp) },\n  { refine deduction.mp _,\n    have : T +{ \u2200.((#0 =' t^1) \u27f6 p) } \u22a2 (t =' t) \u27f6 formula.rew \u0131[0 \u21dd t] p,\n    { have := (show T +{ \u2200.((#0 =' (t^1)) \u27f6 p) } \u22a2 \u2200.((#0 =' (t^1)) \u27f6 p), by simp) \u229a t,\n      simp at this, exact this },\n    exact this \u2a00 (by simp) }\nend\n\nlemma dummy_fal_quantifir (p) : T \u22a2 p \u27f7 \u2200.p^1 :=\nby { have : T \u22a2 \u2200.(p^1) \u27f6 (p^1).rew \u0131[0 \u21dd #0], from specialize #0, simp[*, axiomatic_classical_logic'.iff_equiv] at * }\n\nlemma dummy_fal_quantifir_iff {p : formula L} : T \u22a2 \u2200.(p^1) \u2194 T \u22a2 p :=\nby { have :=  (@dummy_fal_quantifir _ T p), simp[axiomatic_classical_logic'.iff_equiv] at this,  split,\n     { refine \u03bb h, this \u2a00 h },\n     { refine \u03bb h, (by simp) \u2a00 h } }\n\nlemma dummy_ex_quantifir (p) : T \u22a2 p \u27f7 \u2203.p^1 :=\nby { simp[has_exists_quantifier.ex, formula.ex, axiomatic_classical_logic'.iff_equiv],\n     have : T \u22a2 \u223cp \u27f7 \u2200.(\u223cp) ^ 1, from dummy_fal_quantifir (\u223cp), simp[axiomatic_classical_logic'.iff_equiv] at this, \n      split,\n     { refine contrapose.mp _, simp[this] },\n     { refine contrapose.mp _, simp[this] } }\n\n@[simp] lemma T_hyp_eliminate {p} : T \u22a2 \u22a4 \u27f6 p \u2194 T \u22a2 p :=\n\u27e8\u03bb h, by { have : T \u22a2 \u22a4, simp, exact h \u2a00 this }, \u03bb h, by simp[h]\u27e9\n\nlemma equiv_eq_of_equiv {t\u2081 u\u2081 t\u2082 u\u2082} (h\u2081 : T \u22a2 t\u2081 =' u\u2081) (h\u2082 : T \u22a2 t\u2082 =' u\u2082) : T \u22a2 (t\u2081 =' t\u2082) \u27f7 (u\u2081 =' u\u2082) :=\nby { simp[axiomatic_classical_logic'.iff_equiv],\n     refine \u27e8deduction.mp _, deduction.mp  _\u27e9,\n     have lmm\u2081 : T+{t\u2081 =' t\u2082} \u22a2 u\u2081 =' t\u2081, simp [eq_symm h\u2081],\n     have lmm\u2082 : T+{t\u2081 =' t\u2082} \u22a2 t\u2081 =' t\u2082, simp,\n     have lmm\u2083 : T+{t\u2081 =' t\u2082} \u22a2 t\u2082 =' u\u2082, simp [h\u2082],\n     refine eq_trans (eq_trans lmm\u2081 lmm\u2082) lmm\u2083,\n     have lmm\u2081 : T+{u\u2081 =' u\u2082} \u22a2 t\u2081 =' u\u2081, simp [h\u2081],\n     have lmm\u2082 : T+{u\u2081 =' u\u2082} \u22a2 u\u2081 =' u\u2082, simp,\n     have lmm\u2083 : T+{u\u2081 =' u\u2082} \u22a2 u\u2082 =' t\u2082, simp [eq_symm h\u2082],\n     refine eq_trans (eq_trans lmm\u2081 lmm\u2082) lmm\u2083  }\n\nlemma eq_of_equiv {t\u2081 u\u2081 t\u2082 u\u2082} (h : T \u22a2 t\u2081 =' u\u2081) (hp : T \u22a2 t\u2081 =' t\u2082) (hq : T \u22a2 u\u2081 =' u\u2082) : T \u22a2 t\u2082 =' u\u2082 :=\nby { have := equiv_eq_of_equiv hp hq, simp[axiomatic_classical_logic'.iff_equiv] at this, exact this.1 \u2a00 h, }\n\nlemma equiv_function_of_equiv {n} (f : L.fn n) {v\u2081 v\u2082 : finitary (term L) n} (h : \u2200 i, T \u22a2 v\u2081 i =' v\u2082 i) :\n  T \u22a2 term.app f v\u2081 =' term.app f v\u2082 :=\nfunction_ext' f v\u2081 v\u2082 \u2a00 (by simp[h])\n\nlemma equiv_predicate_of_equiv {n} (p : L.pr n) {v\u2081 v\u2082 : finitary (term L) n} (h : \u2200 i, T \u22a2 v\u2081 i =' v\u2082 i) :\n  T \u22a2 formula.app p v\u2081 \u27f7 formula.app p v\u2082 :=\nbegin\n  simp[axiomatic_classical_logic'.iff_equiv],\n  refine \u27e8(predicate_ext' p v\u2081 v\u2082) \u2a00 (by simp[h]),\n  (predicate_ext' p v\u2082 v\u2081) \u2a00 (by simp[\u03bb i, eq_symm (h i)])\u27e9\nend\n\nlemma predicate_of_equiv {n} (p : L.pr n) {v\u2081 v\u2082 : finitary (term L) n} (h : T \u22a2 formula.app p v\u2081) \n  (hv : \u2200 i, T \u22a2 v\u2081 i =' v\u2082 i) : T \u22a2 formula.app p v\u2082 :=\nby { have := equiv_predicate_of_equiv p hv, simp[axiomatic_classical_logic'.iff_equiv] at this, exact this.1 \u2a00 h }\n\nlemma equiv_univ_of_equiv {p\u2081 p\u2082} (h : \u290aT \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2200.p\u2081 \u27f7 \u2200.p\u2082 :=\nby { simp[axiomatic_classical_logic'.iff_equiv] at h \u22a2, refine \u27e8univ_K _ _ \u2a00 (generalize h.1), univ_K _ _ \u2a00 (generalize h.2)\u27e9 }\n\nlemma univ_of_equiv {p\u2081 p\u2082} (h : T \u22a2 \u2200.p\u2081) (hp : \u290aT \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2200.p\u2082 :=\n(iff_equiv.mp (equiv_univ_of_equiv hp)).1 \u2a00 h\n\nlemma equiv_univs_of_equiv {p\u2081 p\u2082} {n : \u2115} (h : T^n \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 (\u2200.[n] p\u2081) \u27f7 (\u2200.[n] p\u2082) :=\nby { induction n with n IH generalizing p\u2081 p\u2082; simp, { exact h }, { simpa using IH (equiv_univ_of_equiv h) } }\n\nlemma equiv_ex_of_equiv {p\u2081 p\u2082} (h : \u290aT \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2203.p\u2081 \u27f7 \u2203.p\u2082 :=\nequiv_neg_of_equiv (equiv_univ_of_equiv (equiv_neg_of_equiv h))\n\nlemma ex_of_equiv {p\u2081 p\u2082} (h : T \u22a2 \u2203.p\u2081) (hp : \u290aT \u22a2 p\u2081 \u27f7 p\u2082) : T \u22a2 \u2203.p\u2082 :=\n(iff_equiv.mp (equiv_ex_of_equiv hp)).1 \u2a00 h\n\n@[simp] protected lemma extend {T\u2080 T : Theory L} [T\u2080.extend T] {p : formula L} (h : T\u2080 \u22a2 p) : T \u22a2 p :=\nTheory.extend.le h\n\nlemma nfal_K (p q : formula L) (n) : T \u22a2 (\u2200.[n] (p \u27f6 q)) \u27f6 (\u2200.[n] p) \u27f6 \u2200.[n] q :=\nbegin\n  have eqn : \u2200 p : formula L, (p.rew (\u03bb x, ite (x < n) #x #(x + n))).rew (\u03bb x, ite (x < n) #x #(x - n)) = p,\n  { intros p, simp[formula.nested_rew], \n    have : (\u03bb x, term.rew (\u03bb (x : \u2115), ite (x < n) #x #(x - n)) (ite (x < n) #x #(x + n)) : \u2115 \u2192 term L) = \u0131,\n    { funext x, by_cases C : x < n; simp[C] }, simp[this] },  \n  refine deduction.mp (deduction.mp (generalize_itr _)),\n  simp[pow_dsb],\n  have lmm\u2081 : (T^n) +{ (\u2200.[n] p \u27f6 q)^n } +{ (\u2200.[n] p)^n } \u22a2 p \u27f6 q,\n  { have :  (T^n) +{ (\u2200.[n] p \u27f6 q)^n } +{ (\u2200.[n] p)^n } \u22a2 \u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n)) \u27f6 q.rew (\u03bb x, ite (x < n) #x #(x + n)),\n    { simp[show (\u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n)) \u27f6 q.rew (\u03bb x, ite (x < n) #x #(x + n))) = (\u2200.[n] p \u27f6 q)^n, by simp[formula.nfal_pow]] }, \n    have := nfal_subst' this \u0131, simp[eqn] at this, exact this },\n  have lmm\u2082 : (T^n) +{ (\u2200.[n] p \u27f6 q)^n } +{ (\u2200.[n] p)^n } \u22a2 p,\n  { have : (T^n) +{ (\u2200.[n] p \u27f6 q)^n } +{ (\u2200.[n] p)^n } \u22a2 \u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n)),\n    { simp[show (\u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n))) = (\u2200.[n] p)^n, by simp[formula.nfal_pow]] },\n    have := nfal_subst' this \u0131, simp[eqn] at this, exact this },\n  exact lmm\u2081 \u2a00 lmm\u2082\nend\n\nlemma fal_complete_K (p q : formula L) : T \u22a2 (\u2200.* (p \u27f6 q)) \u27f6 (\u2200.* p) \u27f6 \u2200.* q :=\nbegin\n  refine (deduction.mp $ deduction.mp $ generalize_itr _), simp[pow_dsb],\n  have lmm\u2081 : (T ^ q.arity) +{ \u2200.* (p \u27f6 q) } +{ \u2200.* p } \u22a2 p \u27f6 q,\n  { have : (T ^ q.arity) +{ \u2200.* (p \u27f6 q) } +{ \u2200.* p } \u22a2 \u2200.* (p \u27f6 q), by simp,\n    simpa using fal_complete_rew (p \u27f6 q) \u0131 \u2a00 this },\n  have lmm\u2082 : (T ^ q.arity) +{ \u2200.* (p \u27f6 q) } +{ \u2200.* p } \u22a2 p,\n  { have : (T ^ q.arity) +{ \u2200.* (p \u27f6 q) } +{ \u2200.* p } \u22a2 \u2200.* p, by simp,\n    simpa using fal_complete_rew p \u0131 \u2a00 this },\n  exact lmm\u2081 \u2a00 lmm\u2082\nend\n\nlemma equiv_fal_complete_of_equiv {p\u2081 p\u2082 : formula L} (h : T^(max p\u2081.arity p\u2082.arity) \u22a2 p\u2081 \u27f7 p\u2082) :\n  T \u22a2 (\u2200.* p\u2081) \u27f7 (\u2200.* p\u2082) :=\nbegin\n  simp[iff_equiv] at h \u22a2, split,\n  { have : T \u22a2 \u2200.* (p\u2081 \u27f6 p\u2082), from generalize_itr (by simp[h]),\n    exact fal_complete_K _ _ \u2a00 this },\n  { have : T \u22a2 \u2200.* (p\u2082 \u27f6 p\u2081), from generalize_itr (by simp[max_comm, h]),\n    exact fal_complete_K _ _ \u2a00 this }\nend\n\nlemma nfal_rew {n} {p : formula L} (s : \u2115 \u2192 term L) :\n  T \u22a2 (\u2200.[n] p) \u27f6 \u2200.[n] p.rew (\u03bb x, if x < n then s x else #x) :=\nbegin\n  refine deduction.mp (generalize_itr _),\n  have : T +{ \u2200.[n] p } ^ n \u22a2 \u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n)),\n  { simp[pow_dsb, show (\u2200.[n] p.rew (\u03bb x, ite (x < n) #x #(x + n))) = (\u2200.[n] p)^n, by simp[formula.nfal_pow]] },\n  have lmm : T +{ \u2200.[n] p } ^ n \u22a2 (p.rew (\u03bb x, ite (x < n) #x #(x + n))).rew (\u03bb x, ite (x < n) (s x) #(x - n)), from nfal_subst' this s,\n  simp[formula.nested_rew] at lmm,\n  have : (\u03bb x, term.rew (\u03bb x, ite (x < n) (s x) #(x - n)) (ite (x < n) #x #(x + n))) = (\u03bb x, ite (x < n) (s x) #x),\n  { funext x, by_cases C : x < n; simp[C] },\n  simp[this] at lmm, exact lmm\nend\n\n@[simp] lemma fal_shift_equiv_self {p : formula L} : T \u22a2 \u2200.(p^1) \u27f7 p :=\nbegin\n  simp[axiomatic_classical_logic'.iff_equiv],\n  have : T \u22a2 \u2200.p^1 \u27f6 (p^1).rew \u0131[0 \u21dd #0], from specialize #0,\n  simp at this, exact this\nend\n\n@[simp] lemma nfal_pow_equiv_self {p : formula L} {n : \u2115} : T \u22a2 (\u2200.[n] p^n) \u27f7 p :=\nbegin\n  induction n with n IH,\n  { simp },\n  { simp[\u2190nat.add_one],\n    have lmm : T \u22a2 \u2200.(\u2200.[n] p^n)^1 \u27f7 p,\n    { have : T \u22a2 \u2200.(\u2200.[n] p^n)^1 \u27f7 \u2200.[n] p^n, { simp },\n      exact equiv_trans this IH },\n    have : (\u2200.[n] p^n)^1 = (\u2200.[n] p^(n + 1)), \n    { simp[formula.pow_eq, formula.nested_rew, show \u2200 x, x + (n + 1) = x + 1 + n, by omega] },\n    simp[this] at lmm, exact lmm }\nend\n\nvariables (T)\n\n@[simp] lemma provable_Theory_refl : T \u22a2\u209c\u2095 T := \u03bb p mem, by_axiom mem\n\nvariables {T}\n\nlemma provable_Theory_weakening {U : Theory L} (h : T \u2286 U) : U \u22a2\u209c\u2095 T := \u03bb p mem, by_axiom (h mem)\n\nend provable\n\nvariables {T : Theory L}\n\nnamespace Theory\nvariables {T\u2080 T\u2081 U\u2080 U\u2081 : Theory L}\n\nlemma le_of_ss : T\u2080 \u2286 T\u2081 \u2192 T\u2080 \u2264 T\u2081 := \u03bb hyp p h, weakening hyp h\n\n@[simp] lemma le_union_left : T\u2080 \u2264 T\u2080 \u222a T\u2081 := le_of_ss (by simp)\n\n@[simp] lemma le_union_right : T\u2081 \u2264 T\u2080 \u222a T\u2081 := le_of_ss (by simp)\n\n@[simp] lemma union_le_union (h\u2080 : T\u2080 \u2264 U\u2080) (h\u2081 : T\u2081 \u2264 U\u2081) : T\u2080 \u222a T\u2081 \u2264 U\u2080 \u222a U\u2081 :=\n\u03bb p b,\nbegin\n  rcases provable.proof_conjunction_union b with \u27e8P, Q, hP, hQ, b\u27e9,\n  have bP : U\u2080 \u222a U\u2081 \u22a2 P.conjunction, from list_conjunction_provable (\u03bb p hp, weakening (by simp[hp]) (h\u2080 (by_axiom (hP p hp)))),\n  have bQ : U\u2080 \u222a U\u2081 \u22a2 Q.conjunction, from list_conjunction_provable (\u03bb p hp, weakening (by simp[hp]) (h\u2081 (by_axiom (hQ p hp)))),\n  exact (weakening (by simp) b) \u2a00 bP \u2a00 bQ\nend\n\nsection extend\nopen logic.Theory\n\ndef extend_of_inclusion {T\u2081 T\u2082 : Theory L} (ss : T\u2081 \u2286 T\u2082) : extend T\u2081 T\u2082 := \u27e8le_of_ss ss\u27e9\n\ninstance extend_ax\u2081 (p : formula L) : extend T (T +{ p }) := \u27e8\u03bb q h, by simp[h]\u27e9\n\ninstance extend_ax\u2082 (p q : formula L) : extend T (T +{ p }+{ q }) := \u27e8\u03bb _ h, by simp[h]\u27e9\n\ninstance extend_ax\u2083 (p q r : formula L) : extend T (T +{ p }+{ q }+{ r }) := \u27e8\u03bb _ h, by simp[h]\u27e9\n\ninstance extend_ax\u2084 (p q r s : formula L) : extend T (T +{ p }+{ q }+{ r }+{ s }) := \u27e8\u03bb _ h, by simp[h]\u27e9\n\ninstance extend_sf {T\u2081 T\u2082 : Theory L} [extend T\u2081 T\u2082] : extend (\u290aT\u2081) (\u290aT\u2082) :=\n\u27e8\u03bb p h, by {\n  have : T\u2081 \u22a2 \u2200.p, from h.generalize,\n  have : T\u2082 \u22a2 \u2200.p, from this.extend,\n  have : \u290aT\u2082 \u22a2 (\u2200.p)^1, from provable.sf_sf.mpr this,\n  simpa[formula.nested_rew] using this \u229a #0 }\u27e9\n\ninstance extend_pow (T\u2081 T\u2082 : Theory L) [ex : extend T\u2081 T\u2082] (k : \u2115) : extend (T\u2081^k) (T\u2082^k) :=\nby { induction k with k IH ; simp[Theory.sf_itr_succ], { exact ex }, { exactI fol.Theory.extend_sf } }\n\ninstance extend_union_left (T\u2081 T\u2082 : Theory L) : extend T\u2081 (T\u2081 \u222a T\u2082) := Theory.extend_of_inclusion (by simp)\n\ninstance extend_union_right (T\u2081 T\u2082 : Theory L) : extend T\u2082 (T\u2081 \u222a T\u2082) := Theory.extend_of_inclusion (by simp)\n\ninstance extend_empty : extend \u2205 T := Theory.extend_of_inclusion (by simp)\n\ninstance extend_pow_of_closed (T\u2081 T\u2082 : Theory L) [closed_Theory T\u2081] [extend T\u2081 T\u2082] (k : \u2115) : extend T\u2081 (T\u2082^k) :=\nby simpa using Theory.extend_pow T\u2081 T\u2082 k\n\ninstance union_extend_union [extend T\u2080 U\u2080] [extend T\u2081 U\u2081] : extend (T\u2080 \u222a T\u2081) (U\u2080 \u222a U\u2081) :=\n\u27e8union_le_union extend.le extend.le\u27e9\n\nend extend\n\nend Theory\n\nlemma provable.extend_pow {T\u2080 T : Theory L} [T\u2080.extend  T] [closed_Theory T\u2080] {p : formula L} (h : T\u2080 \u22a2 p) (k : \u2115) :\n  T^k \u22a2 p := by { have : T\u2080^k \u22a2 p, by simp[h], exact this.extend }\n\nlemma proper_Theory_union (T\u2081 T\u2082 : Theory L) (h\u2081 : proper_Theory T\u2081) (h\u2082 : proper_Theory T\u2082) :\n  proper_at 0 (T\u2081 \u222a T\u2082) :=\n\u03bb p s h, by { cases h,\n  { refine or.inl (proper_Theory.proper p s h) },\n  { refine or.inr (proper_Theory.proper p s h) } }\n\ndef proper_schema (F : formula L \u2192 formula L) : Prop := \u2203 i : \u2115, \u2200 p s, (F p).rew s = F (p.rew (s^i))\n\nlemma proper_image_of_proper_schema (C : Theory L) [proper_Theory C]\n  {F : formula L \u2192 formula L} (h : proper_schema F) : proper_at 0 (F '' C) :=\n\u03bb p s mem, begin\n    rcases mem with \u27e8p, mem, rfl\u27e9,\n    rcases h with \u27e8i, h\u27e9,\n    simp[h], refine \u27e8p.rew (s^i), by simp[mem], rfl\u27e9\nend\n\n@[reducible] def prf (L : language) := \u03a3 (T : Theory L) (p : formula L), T \u27f9 p\n\n@[reducible] def prf.to_formula (b : prf L) : formula L := b.snd.fst\n\n@[reducible] def prf.to_proof (b : prf L) := b.snd.snd\n\n@[reducible] def proof.to_prf {p} (b : T \u27f9 p) : prf L := \u27e8T, p, b\u27e9\n\nnamespace prf\nvariables {T} {p : formula L} {b : T \u27f9 p} {B : prf L}\n\n@[simp] lemma to_proof_to_prf : B.to_proof.to_prf = B := by { rcases B with \u27e8T, p, b\u27e9, simp }\n\n@[simp] lemma to_prf_to_formula : b.to_prf.to_formula = p := rfl\n\n@[simp] lemma to_prf_to_proof : b.to_prf.to_proof = b := rfl\n\nend prf\n\nnamespace proof\nvariables {T} {p : formula L}\n\ninductive subproof : prf L \u2192 prf L \u2192 Prop\n| mdp\u2081    : \u2200 {T : Theory L} {p q : formula L} {b\u2081 : T \u27f9 (p \u27f6 q)} {b\u2082 : T \u27f9 p}, subproof \u27e8T, p \u27f6 q, b\u2081\u27e9 \u27e8T, q, mdp b\u2081 b\u2082\u27e9 \n| mdp\u2082    : \u2200 {T : Theory L} {p q : formula L} {b\u2081 : T \u27f9 (p \u27f6 q)} {b\u2082 : T \u27f9 p}, subproof \u27e8T, p, b\u2082\u27e9 \u27e8T, q, mdp b\u2081 b\u2082\u27e9\n| generalize : \u2200 {T : Theory L} {p : formula L} {b : \u290aT \u27f9 p}, subproof \u27e8\u290aT, p, b\u27e9 \u27e8T, \u2200.p, b.generalize\u27e9 \n\n@[simp] def complexity : \u03a0 {T : Theory L} {p : formula L} (b : T \u27f9 p), \u2115\n| T p (generalize b)            := b.complexity + 1\n| T p (mdp b\u2081 b\u2082)               := max b\u2081.complexity b\u2082.complexity + 1\n| T p (by_axiom h)              := 0\n| T _ verum                     := 0\n| T _ (@imply\u2081 _ _ p q)         := 0\n| T _ (@imply\u2082 _ _ p q r)       := 0\n| T _ (@contraposition _ _ p q) := 0\n| T _ (@specialize _ _ p t)     := 0\n| T _ (@univ_K _ _ p q)         := 0\n| T _ (@dummy_univ _ _ p)       := 0\n| T _ (@eq_reflexivity _ _)     := 0\n| T _ eq_symmetry               := 0\n| T _ eq_transitivity           := 0\n| T _ (@function_ext _ _ _ f)   := 0\n| T _ (@predicate_ext _ _ _ r)  := 0\n\ninstance : wf_lt (prf L) :=\n{ prelt := subproof,\n  wt := \u03bb b, b.snd.snd.complexity,\n  mono' := \u03bb b\u2081 b\u2082 h, by induction h; simp }\n\ndef le {T\u2081 T\u2082 : Theory L} {p\u2081 p\u2082 : formula L} (b\u2081 : T\u2081 \u27f9 p\u2081) (b\u2082 : T\u2082 \u27f9 p\u2082) : Prop := b\u2081.to_prf \u2264 b\u2082.to_prf\n\ndef lt {T\u2081 T\u2082 : Theory L} {p\u2081 p\u2082 : formula L} (b\u2081 : T\u2081 \u27f9 p\u2081) (b\u2082 : T\u2082 \u27f9 p\u2082) : Prop := b\u2081.to_prf < b\u2082.to_prf\n\ndef fn_symbols {p} (b : T \u27f9 p) : set (\u03a3 n, L.fn n) :=\n  let B : set (prf L) := {c | c < b.to_prf},\n      B' : set (formula L) := prf.to_formula '' B in \u22c3\u2080 (formula.fn_symbols '' B')\n\nsection\nvariables {T\u2081 T\u2082 : Theory L} {p\u2081 p\u2082 : formula L}\n\n@[simp] lemma lt_generalize_iff {b\u2081 : prf L} {b\u2082 : \u290aT\u2082 \u27f9 p\u2082} : b\u2081 < b\u2082.generalize.to_prf \u2194 b\u2081 \u2264 b\u2082.to_prf :=\nby { simp[lt, le, wf_lt.lt_iff], split,\n     { rintros \u27e8T, p, b, prelt, le\u27e9, rcases prelt, exact le },\n     { intros le, refine \u27e8\u290aT\u2082, p\u2082, b\u2082, subproof.generalize, le\u27e9 } }\n\n@[simp] lemma lt_mdp_iff {q\u2082 : formula L} {b\u2081 : prf L} {b\u2082\u2081 : T\u2082 \u27f9 p\u2082 \u27f6 q\u2082} {b\u2082\u2082 : T\u2082 \u27f9 p\u2082} :\n  b\u2081 < (mdp b\u2082\u2081 b\u2082\u2082).to_prf \u2194 b\u2081 \u2264 b\u2082\u2081.to_prf \u2228 b\u2081 \u2264 b\u2082\u2082.to_prf :=\nby { simp[lt, le, wf_lt.lt_iff], split,\n     { rintros \u27e8T, p, b, prelt, le\u27e9, rcases prelt, { exact or.inl le }, { exact or.inr le } },\n     { rintros (le | le), refine \u27e8T\u2082, p\u2082 \u27f6 q\u2082, b\u2082\u2081, subproof.mdp\u2081, le\u27e9, refine \u27e8T\u2082, p\u2082, b\u2082\u2082, subproof.mdp\u2082, le\u27e9 } }\n\nend\n\n@[simp] lemma wt_eq_complexity (T) (p) (b) : wf_lt.wt (\u27e8T, p, b\u27e9 : prf L) = complexity b :=\nby refl\n\nlemma prelt_finite (b : prf L) : set.finite {c | subproof c b} :=\nbegin\n  have of_eq_empty : \u2200 s : set (prf L), s = \u2205 \u2192 s.finite,\n  { rintros _ rfl, simp }, \n  rcases b with \u27e8T, p, b\u27e9,\n  induction b;\n  try { refine of_eq_empty _\n    (by { ext c, simp, intros h, \n      have : wf_lt.wt c < wf_lt.wt _, from wf_lt.lt_mono (wf_lt.lt_of_prelt (show wf_lt.prelt c \u27e8_, _, _\u27e9, from h)),\n      simp at this, contradiction }) },\n  case generalize : T p b\n  { have : {c : prf L | subproof c \u27e8T, \u27e8\u2200.p, b.generalize\u27e9\u27e9} = {b.to_prf},\n    { ext c, simp, split,\n      { intros h, rcases h, refl }, { rintros rfl, exact subproof.generalize } },\n    simp[this] },\n  case mdp : T p q b\u2081 b\u2082\n  { have : {c : prf L | subproof c \u27e8T, \u27e8q, b\u2081.mdp b\u2082\u27e9\u27e9} = {b\u2081.to_prf, b\u2082.to_prf},\n    { ext c, simp, split,\n      { intros h, rcases h, refine or.inl rfl, refine or.inr rfl },\n      { rintros (rfl | rfl), refine subproof.mdp\u2081, refine subproof.mdp\u2082 } },\n    simp[this] }\nend\n\nlemma le_finite (b : prf L) : set.finite {b' | b' \u2264 b} :=\nwf_lt.le_finite (show \u2200 (a : prf L), {b : prf L | wf_lt.prelt b a}.finite, from prelt_finite) b\n\ndef formula_mem_proof (p : formula L) {T : Theory.{u} L} {q : formula.{u} L} (b : T \u27f9 q) : Prop := \u2203 (b' \u2264 b.to_prf),p \u2264 b'.to_formula\n\ninfix ` \u2208\u1da0 `:50 := formula_mem_proof\n\n@[simp] lemma formula_mem_self {T : Theory L} {p : formula L} (b : T \u27f9 p) : p \u2208\u1da0 b := \u27e8b.to_prf, by simp\u27e9\n\ndef term_mem_proof (t : term L) {T : Theory.{u} L} {p : formula.{u} L} (b : T \u27f9 p) : Prop := \u2203 (b' \u2264 b.to_prf), t \u2208 b'.to_formula\n\ninfix ` \u2208\u1d57 `:50 := term_mem_proof\n\nsection\nvariables {T} {p} {b : T \u27f9 p} {T\u2081 T\u2082 : Theory L} {p\u2081 p\u2082 : formula L} {b\u2081 : T\u2081 \u27f9 p\u2081} {b\u2082 : T\u2082 \u27f9 p\u2082} {B : prf L}\n\nlemma term_mem_proof_def {t : term L} :\n  t \u2208\u1d57 b \u2194 \u2203 b' \u2264 b.to_prf, t \u2208 b'.to_formula := by refl\n\nlemma mem_trans {t : term L} {q : formula L}\n  (ht : t \u2208 q) (hq : q \u2208\u1da0 b) : t \u2208\u1d57 b :=\nby { rcases hq with \u27e8b', hb', hq\u27e9, refine \u27e8b', hb', formula.mem_of_formula_le_mem ht hq\u27e9 }\n\n@[simp] lemma mem_self (B : prf L) : B.to_formula \u2208\u1da0 B.to_proof := \u27e8B, by simp, by simp\u27e9\n\nlemma formula_mem_proof.mem_of_mem_of_le {p'} (mem : p' \u2208\u1da0 b) (le : b.to_prf \u2264 B) : p' \u2208\u1da0 B.to_proof :=\nby { rcases mem with \u27e8b', le_b', ge_b'\u27e9,\n     refine \u27e8b', by { simp, exact le_trans le_b' le }, ge_b'\u27e9 }\n\nlemma term_mem_proof.mem_of_mem_of_le {t'} (mem : t' \u2208\u1d57 b) (le : b.to_prf \u2264 B) : t' \u2208\u1d57 B.to_proof :=\nby { rcases mem with \u27e8b', le_b', ge_b'\u27e9,\n     refine \u27e8b', by { simp, exact le_trans le_b' le }, ge_b'\u27e9 }\n\nlemma formula_mem_proof.mem_of_le_of_mem {p' q} (le : q \u2264 p') (mem : p' \u2208\u1da0 b) : q \u2208\u1da0 b :=\nby { rcases mem with \u27e8b', le_b', ge_b'\u27e9, refine \u27e8b', le_b', le_trans le ge_b'\u27e9 }\n\nlemma term_mem_proof.mem_of_le_of_mem {t u} (le : u \u2264 t) (mem : t \u2208\u1d57 b) : u \u2208\u1d57 b :=\nby { rcases mem with \u27e8b', le_b', ge_b'\u27e9, refine \u27e8b', le_b', formula.mem_of_term_le_mem ge_b' le\u27e9 }\n\n@[simp] lemma term_mem_generalize_iff {b : \u290aT \u27f9 p} {t : term L} : t \u2208\u1d57 b.generalize \u2194 t \u2208\u1d57 b :=\n\u27e8by { rintros \u27e8b', le, mem\u27e9,\n      have : b' < b.generalize.to_prf \u2228 b' = b.generalize.to_prf, exact lt_or_eq_of_le le,\n      rcases this with (lt | rfl),\n      { simp at lt, \n        have : b'.to_formula \u2208\u1da0 b, from (mem_self b').mem_of_mem_of_le (show b'.to_proof.to_prf \u2264 b.to_prf, by simp[lt]),\n        refine mem_trans mem this },\n      { simp at mem, refine mem_trans mem (by simp) } },\n \u03bb h, h.mem_of_mem_of_le (show b.to_prf \u2264 b.generalize.to_prf, from le_of_lt (by simp))\u27e9 \n\n@[simp] lemma term_mem_mdp_iff {p q} {b\u2081 : T \u27f9 p \u27f6 q} {b\u2082 : T \u27f9 p} {t} : t \u2208\u1d57 (b\u2081.mdp b\u2082) \u2194 t \u2208\u1d57 b\u2081 \u2228 t \u2208\u1d57 b\u2082 :=\n\u27e8by { rintros \u27e8b', le, mem\u27e9,\n      have : b' < (b\u2081.mdp b\u2082).to_prf \u2228 b' = (b\u2081.mdp b\u2082).to_prf, exact lt_or_eq_of_le le,\n      rcases this with (lt | rfl),\n      { simp at lt, rcases lt, \n        { have : b'.to_formula \u2208\u1da0 b\u2081, from (mem_self b').mem_of_mem_of_le (show b'.to_proof.to_prf \u2264 b\u2081.to_prf, by simp[lt]),\n          refine or.inl (mem_trans mem this) },\n        { have : b'.to_formula \u2208\u1da0 b\u2082, from (mem_self b').mem_of_mem_of_le (show b'.to_proof.to_prf \u2264 b\u2082.to_prf, by simp[lt]),\n          refine or.inr (mem_trans mem this) }, },\n      { simp at mem, refine or.inl (mem_trans mem\n          (formula_mem_proof.mem_of_le_of_mem (show q \u2264 p \u27f6 q, from le_of_lt (by simp)) (by simp))) } },\n \u03bb h, by { rcases h with (h | h),\n           { exact h.mem_of_mem_of_le (show b\u2081.to_prf \u2264 (b\u2081.mdp b\u2082).to_prf, from le_of_lt (by simp)) },\n           { exact h.mem_of_mem_of_le (show b\u2082.to_prf \u2264 (b\u2081.mdp b\u2082).to_prf, from le_of_lt (by simp)) } }\u27e9 \n\nprivate lemma not_mem_of (b : T \u27f9 p) (h : b.complexity = 0) (t : term L) : t \u2208\u1d57 b \u2194 t \u2208 p :=\n\u27e8by { rintros \u27e8b', le, mem\u27e9, have : b' < b.to_prf \u2228 b' = b.to_prf, exact lt_or_eq_of_le le,\n      rcases this with (lt | rfl),\n      { have : wf_lt.wt b' < wf_lt.wt b.to_prf, from wf_lt.lt_mono lt,\n        simp[h] at this, contradiction },\n      { simp at mem, exact mem } },\n by { intros mem, refine \u27e8b.to_prf, by simp, by simp[mem]\u27e9 }\u27e9\n\n\n@[simp] lemma term_mem_by_axiom_iff {h : p \u2208 T} {t} : t \u2208\u1d57 by_axiom h \u2194 t \u2208 p := not_mem_of _ (by simp) _\n\n@[simp] lemma term_mem_verum_iff {t} : \u00act \u2208\u1d57 (verum : T \u27f9 \u22a4) := by { have := not_mem_of (verum : T \u27f9 \u22a4) (by simp) t, simp at this, exact this }\n\n@[simp] lemma term_mem_imply\u2081_iff {p q : formula L} {t} : t \u2208\u1d57 (@imply\u2081 _ T p q) \u2194 t \u2208 p \u2228 t \u2208 q :=\nby { have := not_mem_of (@imply\u2081 _ T p q) (by simp) t, simp at this, simp[this], tauto }\n\n@[simp] lemma term_mem_imply\u2082_iff {p q r : formula L} {t} : t \u2208\u1d57 (@imply\u2082 _ T p q r) \u2194 t \u2208 p \u2228 t \u2208 q \u2228 t \u2208 r :=\nby { have := not_mem_of (@imply\u2082 _ T p q r) (by simp) t, simp at this, simp[this], tauto }\n\n@[simp] lemma term_mem_contraposition_iff {p q : formula L} {t} :\n  t \u2208\u1d57 (@contraposition _ T p q) \u2194 t \u2208 p \u2228 t \u2208 q :=\nby { have := not_mem_of (@contraposition _ T p q) (by simp) t, simp at this, simp[this], tauto }\n\n@[simp] lemma term_mem_specialize_iff {p : formula L} {t\u2080 t} :\n  t \u2208\u1d57 (@specialize _ T p t\u2080) \u2194 t \u2208 p \u2228 t \u2208 formula.rew \u0131[0 \u21dd t\u2080] p :=\nby { have := not_mem_of (@specialize _ T p t\u2080) (by simp) t, simp at this, simp[this] }\n\n@[simp] lemma term_mem_univ_K_iff {p q : formula L} {t} :\n  t \u2208\u1d57 (@univ_K _ T p q) \u2194 t \u2208 p \u2228 t \u2208 q :=\nby { have := not_mem_of (@univ_K _ T p q) (by simp) t, simp at this, simp[this] }\n\n@[simp] lemma term_mem_dummy_univ_iff {p : formula L} {t} :\n  t \u2208\u1d57 (@dummy_univ _ T p) \u2194 t \u2208 p \u2228 t \u2208 p^1 :=\nby { have := not_mem_of (@dummy_univ _ T p) (by simp) t, simp at this, simp[this] }\n\nvariables (b)\n\nlemma term_mem_finite {T : Theory L} {p : formula L} (b : T \u27f9 p) : set.finite {t | t \u2208\u1d57 b} :=\nbegin\n  let s := \u22c3 b' \u2208 {b' | b' \u2264 b.to_prf}, {t | t \u2208 b'.to_formula},\n  have : {t | t \u2208\u1d57 b} = s,\n  { ext t, simp[s, term_mem_proof_def] },\n  simp[this],\n  refine set.finite.bUnion (le_finite b.to_prf) (\u03bb b' _, b'.to_formula.mem_finite) \nend\n\nend\n\nend proof\n\nnamespace Theory\nopen provable\nvariables {T} {U : Theory L}\n\nlemma le_iff_mem_provable :\n  T \u2264 U \u2194 \u2200 p \u2208 T, U \u22a2 p :=\n\u27e8\u03bb h p mem, h (by_axiom mem), by { \n  suffices : \u2200 (T : Theory L) (k : \u2115) (p : formula L) (b : T^k \u22a2 p) (h : \u2200 p \u2208 T, U \u22a2 p), U^k \u22a2 p,\n  { intros h p b, exact this T 0 p b h },\n  intros T k p b,\n  refine rec'_on b _ _ _ _ _ _ _ _ _ _ _ _ _ _ _; try { simp },\n  { intros i p b IH h, exact generalize (IH h) },\n  { intros i p q b\u2081 b\u2082 IH_b\u2081 IH_b\u2082 h, exact IH_b\u2081 h \u2a00 IH_b\u2082 h },\n  { intros i p mem h, simp[Theory_sf_itr_eq] at mem, rcases mem with \u27e8p, mem, rfl\u27e9,\n    exact sf_itr_sf_itr.mpr (h p mem) } }\u27e9\n\nend Theory\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/deduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190477, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.374691512530829}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.category.Cat\nimport category_theory.elements\n\n/-!\n# The Grothendieck construction\n\nGiven a functor `F : C \u2964 Cat`, the objects of `grothendieck F`\nconsist of dependent pairs `(b, f)`, where `b : C` and `f : F.obj c`,\nand a morphism `(b, f) \u27f6 (b', f')` is a pair `\u03b2 : b \u27f6 b'` in `C`, and\n`\u03c6 : (F.map \u03b2).obj f \u27f6 f'`\n\nCategories such as `PresheafedSpace` are in fact examples of this construction,\nand it may be interesting to try to generalize some of the development there.\n\n## Implementation notes\n\nReally we should treat `Cat` as a 2-category, and allow `F` to be a 2-functor.\n\nThere is also a closely related construction starting with `G : C\u1d52\u1d56 \u2964 Cat`,\nwhere morphisms consists again of `\u03b2 : b \u27f6 b'` and `\u03c6 : f \u27f6 (F.map (op \u03b2)).obj f'`.\n\n## References\n\nSee also `category_theory.functor.elements` for the category of elements of functor `F : C \u2964 Type`.\n\n* https://stacks.math.columbia.edu/tag/02XV\n* https://ncatlab.org/nlab/show/Grothendieck+construction\n\n-/\n\nuniverse u\n\nnamespace category_theory\n\nvariables {C D : Type*} [category C] [category D]\nvariables (F : C \u2964 Cat)\n\n/--\nThe Grothendieck construction (often written as `\u222b F` in mathematics) for a functor `F : C \u2964 Cat`\ngives a category whose\n* objects `X` consist of `X.base : C` and `X.fiber : F.obj base`\n* morphisms `f : X \u27f6 Y` consist of\n  `base : X.base \u27f6 Y.base` and\n  `f.fiber : (F.map base).obj X.fiber \u27f6 Y.fiber`\n-/\n@[nolint has_nonempty_instance]\nstructure grothendieck :=\n(base : C)\n(fiber : F.obj base)\n\nnamespace grothendieck\n\nvariables {F}\n\n/--\nA morphism in the Grothendieck category `F : C \u2964 Cat` consists of\n`base : X.base \u27f6 Y.base` and `f.fiber : (F.map base).obj X.fiber \u27f6 Y.fiber`.\n-/\nstructure hom (X Y : grothendieck F) :=\n(base : X.base \u27f6 Y.base)\n(fiber : (F.map base).obj X.fiber \u27f6 Y.fiber)\n\n@[ext] lemma ext {X Y : grothendieck F} (f g : hom X Y)\n  (w_base : f.base = g.base) (w_fiber : eq_to_hom (by rw w_base) \u226b f.fiber = g.fiber) : f = g :=\nbegin\n  cases f; cases g,\n  congr,\n  dsimp at w_base,\n  induction w_base,\n  refl,\n  dsimp at w_base,\n  induction w_base,\n  simpa using w_fiber,\nend\n\n/--\nThe identity morphism in the Grothendieck category.\n-/\n@[simps]\ndef id (X : grothendieck F) : hom X X :=\n{ base := \ud835\udfd9 X.base,\n  fiber := eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]), }\n\ninstance (X : grothendieck F) : inhabited (hom X X) := \u27e8id X\u27e9\n\n/--\nComposition of morphisms in the Grothendieck category.\n-/\n@[simps]\ndef comp {X Y Z : grothendieck F} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ base := f.base \u226b g.base,\n  fiber :=\n  eq_to_hom (by erw [functor.map_comp, functor.comp_obj]) \u226b\n    (F.map g.base).map f.fiber \u226b g.fiber, }\n\nlocal attribute [simp] eq_to_hom_map\n\ninstance : category (grothendieck F) :=\n{ hom := \u03bb X Y, grothendieck.hom X Y,\n  id := \u03bb X, grothendieck.id X,\n  comp := \u03bb X Y Z f g, grothendieck.comp f g,\n  comp_id' := \u03bb X Y f,\n  begin\n    ext,\n    { dsimp,\n      -- We need to turn `F.map_id` (which is an equation between functors)\n      -- into a natural isomorphism.\n      rw \u2190 nat_iso.naturality_2 (eq_to_iso (F.map_id Y.base)) f.fiber,\n      simp, },\n    { simp, },\n  end,\n  id_comp' := \u03bb X Y f, by ext; simp,\n  assoc' := \u03bb W X Y Z f g h,\n  begin\n    ext, swap,\n    { simp, },\n    { dsimp,\n      rw \u2190 nat_iso.naturality_2 (eq_to_iso (F.map_comp _ _)) f.fiber,\n      simp,\n      refl, },\n  end, }\n\n@[simp] lemma id_fiber' (X : grothendieck F) :\n  hom.fiber (\ud835\udfd9 X) = eq_to_hom (by erw [category_theory.functor.map_id, functor.id_obj X.fiber]) :=\nid_fiber X\n\nlemma congr {X Y : grothendieck F} {f g : X \u27f6 Y} (h : f = g) :\n  f.fiber = eq_to_hom (by subst h) \u226b g.fiber :=\nby { subst h, dsimp, simp, }\n\nsection\nvariables (F)\n\n/-- The forgetful functor from `grothendieck F` to the source category. -/\n@[simps]\ndef forget : grothendieck F \u2964 C :=\n{ obj := \u03bb X, X.1,\n  map := \u03bb X Y f, f.1, }\n\nend\n\nuniverse w\nvariables (G : C \u2964 Type w)\n\n/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/\n@[simps]\ndef grothendieck_Type_to_Cat_functor : grothendieck (G \u22d9 Type_to_Cat) \u2964 G.elements :=\n{ obj := \u03bb X, \u27e8X.1, X.2.as\u27e9,\n  map := \u03bb X Y f, \u27e8f.1, f.2.1.1\u27e9 }\n\n/-- Auxiliary definition for `grothendieck_Type_to_Cat`, to speed up elaboration. -/\n@[simps]\ndef grothendieck_Type_to_Cat_inverse : G.elements \u2964 grothendieck (G \u22d9 Type_to_Cat) :=\n{ obj := \u03bb X, \u27e8X.1, \u27e8X.2\u27e9\u27e9,\n  map := \u03bb X Y f, \u27e8f.1, \u27e8\u27e8f.2\u27e9\u27e9\u27e9 }\n\n/--\nThe Grothendieck construction applied to a functor to `Type`\n(thought of as a functor to `Cat` by realising a type as a discrete category)\nis the same as the 'category of elements' construction.\n-/\n@[simps]\ndef grothendieck_Type_to_Cat : grothendieck (G \u22d9 Type_to_Cat) \u224c G.elements :=\n{ functor := grothendieck_Type_to_Cat_functor G,\n  inverse := grothendieck_Type_to_Cat_inverse G,\n  unit_iso := nat_iso.of_components (\u03bb X, by { rcases X with \u27e8_, \u27e8\u27e9\u27e9, exact iso.refl _, })\n    (by { rintro \u27e8_, \u27e8\u27e9\u27e9 \u27e8_, \u27e8\u27e9\u27e9 \u27e8base, \u27e8\u27e8f\u27e9\u27e9\u27e9, dsimp at *, subst f, ext, simp, }),\n  counit_iso := nat_iso.of_components (\u03bb X, by { cases X, exact iso.refl _, })\n    (by { rintro \u27e8\u27e9 \u27e8\u27e9 \u27e8f, e\u27e9, dsimp at *, subst e, ext, simp }),\n  functor_unit_iso_comp' := by { rintro \u27e8_, \u27e8\u27e9\u27e9, dsimp, simp, refl, } }\n\nend grothendieck\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190475, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3746915125308289}}
{"text": "import ring_theory.noetherian ring_theory.principal_ideal_domain\n\n-- The fix was to make the `fintype` argument come first in `ring.is_noetherian_of_fintype`\nexample {K : Type*} [discrete_field K] : is_noetherian_ring K := by apply_instance --works\nexample {K : Type*} [discrete_field K] : is_noetherian K K := by apply_instance --fails", "meta": {"author": "ChrisHughes24", "repo": "type_class", "sha": "b582b60bdc84b7cd17b4243600088143afddc65c", "save_path": "github-repos/lean/ChrisHughes24-type_class", "path": "github-repos/lean/ChrisHughes24-type_class/type_class-b582b60bdc84b7cd17b4243600088143afddc65c/noetherian/src/test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3746915052895648}}
{"text": "\nuniverse u\n\ndef len {\u03b1 : Type u} : List \u03b1 \u2192 List \u03b1 \u2192 Nat\n| [],    bs => bs.length\n| a::as, bs => len as bs + 1\n\ntheorem ex1 : len [1, 2] [3, 4] = 4 :=\nrfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/matcherElimUniv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3746915052895648}}
{"text": "/- This file contains an example of 'magic rewrite' widget working. -/\nimport hp.rewrite.equate_attr\nimport hp.rewrite.magic_rewrite\n\nvariables {G : Type} [group G]\n\nexample {x y z : G} (h : z = x\u207b\u00b9) : (x * y)\u207b\u00b9 * z = y\u207b\u00b9 :=\nbegin\n\nend", "meta": {"author": "EdAyers", "repo": "lean-humanproof-thesis", "sha": "ce8331df1883f286ab8cc7b61a328afdc006a059", "save_path": "github-repos/lean/EdAyers-lean-humanproof-thesis", "path": "github-repos/lean/EdAyers-lean-humanproof-thesis/lean-humanproof-thesis-ce8331df1883f286ab8cc7b61a328afdc006a059/src/examples/magic_rewrite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3746915052895648}}
{"text": "import mathlib.logic.equiv.local_perm\nimport phase2.flexible\nimport phase2.sublitter\n\nopen set sum\nopen_locale cardinal pointwise\n\nuniverse u\n\nnamespace con_nf\nvariable [params.{u}]\n\n/-!\n# Near-litter approximations\n-/\n\n@[ext] structure near_litter_approx :=\n(atom_perm : local_perm atom)\n(litter_perm : local_perm litter)\n(domain_small : \u2200 L, small (litter_set L \u2229 atom_perm.domain))\n\nnamespace near_litter_approx\n\ninstance : has_smul near_litter_approx atom := \u27e8\u03bb \u03c0, \u03c0.atom_perm\u27e9\ninstance : has_smul near_litter_approx litter := \u27e8\u03bb \u03c0, \u03c0.litter_perm\u27e9\n\nvariables (\u03c0 : near_litter_approx)\n\nlemma smul_atom_eq {a : atom} : \u03c0.atom_perm a = \u03c0 \u2022 a := rfl\nlemma smul_litter_eq {L : litter} : \u03c0.litter_perm L = \u03c0 \u2022 L := rfl\n\n@[simp] lemma mk_smul_atom {atom_perm : local_perm atom} {litter_perm : local_perm litter}\n  {domain_small : \u2200 L, small (litter_set L \u2229 atom_perm.domain)} {a : atom} :\n  { near_litter_approx . atom_perm := atom_perm,\n    litter_perm := litter_perm, domain_small := domain_small } \u2022 a = atom_perm a := rfl\n\n@[simp] lemma mk_smul_litter {atom_perm : local_perm atom} {litter_perm : local_perm litter}\n  {domain_small : \u2200 L, small (litter_set L \u2229 atom_perm.domain)} {L : litter} :\n  { near_litter_approx . atom_perm := atom_perm,\n    litter_perm := litter_perm, domain_small := domain_small } \u2022 L = litter_perm L := rfl\n\nlemma smul_eq_smul_atom {a\u2081 a\u2082 : atom}\n  (h\u2081 : a\u2081 \u2208 \u03c0.atom_perm.domain) (h\u2082 : a\u2082 \u2208 \u03c0.atom_perm.domain) :\n  \u03c0 \u2022 a\u2081 = \u03c0 \u2022 a\u2082 \u2194 a\u2081 = a\u2082 :=\nbegin\n  unfold has_smul.smul,\n  rw [\u2190 \u03c0.atom_perm.eq_symm_apply h\u2081 (\u03c0.atom_perm.map_domain h\u2082), local_perm.left_inv _ h\u2082],\nend\n\nlemma smul_eq_smul_litter {L\u2081 L\u2082 : litter}\n  (h\u2081 : L\u2081 \u2208 \u03c0.litter_perm.domain) (h\u2082 : L\u2082 \u2208 \u03c0.litter_perm.domain) :\n  \u03c0 \u2022 L\u2081 = \u03c0 \u2022 L\u2082 \u2194 L\u2081 = L\u2082 :=\nbegin\n  unfold has_smul.smul,\n  rw [\u2190 \u03c0.litter_perm.eq_symm_apply h\u2081 (\u03c0.litter_perm.map_domain h\u2082), local_perm.left_inv _ h\u2082],\nend\n\ndef symm : near_litter_approx := {\n  atom_perm := \u03c0.atom_perm.symm,\n  litter_perm := \u03c0.litter_perm.symm,\n  domain_small := \u03c0.domain_small,\n}\n\n@[simp] lemma symm_atom_perm : \u03c0.symm.atom_perm = \u03c0.atom_perm.symm := rfl\n@[simp] lemma symm_litter_perm : \u03c0.symm.litter_perm = \u03c0.litter_perm.symm := rfl\n\n@[simp] lemma left_inv_atom {a} : a \u2208 \u03c0.atom_perm.domain \u2192 \u03c0.symm \u2022 \u03c0 \u2022 a = a :=\n\u03c0.atom_perm.left_inv\n\n@[simp] lemma left_inv_litter {L} : L \u2208 \u03c0.litter_perm.domain \u2192 \u03c0.symm \u2022 \u03c0 \u2022 L = L :=\n\u03c0.litter_perm.left_inv\n\n@[simp] lemma right_inv_atom {a} : a \u2208 \u03c0.atom_perm.domain \u2192 \u03c0 \u2022 \u03c0.symm \u2022 a = a :=\n\u03c0.atom_perm.right_inv\n\n@[simp] lemma right_inv_litter {L} : L \u2208 \u03c0.litter_perm.domain \u2192 \u03c0 \u2022 \u03c0.symm \u2022 L = L :=\n\u03c0.litter_perm.right_inv\n\nlemma symm_smul_atom_eq_iff {a b} :\n  a \u2208 \u03c0.atom_perm.domain \u2192 b \u2208 \u03c0.atom_perm.domain \u2192 (\u03c0.symm \u2022 a = b \u2194 a = \u03c0 \u2022 b) :=\nbegin\n  intros ha hb,\n  split,\n  { rintro rfl,\n    exact (\u03c0.right_inv_atom ha).symm, },\n  { rintro rfl,\n    exact \u03c0.left_inv_atom hb, },\nend\n\nlemma symm_smul_litter_eq_iff {L\u2081 L\u2082} :\n  L\u2081 \u2208 \u03c0.litter_perm.domain \u2192 L\u2082 \u2208 \u03c0.litter_perm.domain \u2192 (\u03c0.symm \u2022 L\u2081 = L\u2082 \u2194 L\u2081 = \u03c0 \u2022 L\u2082) :=\nbegin\n  intros hL\u2081 hL\u2082,\n  split,\n  { rintro rfl,\n    exact (\u03c0.right_inv_litter hL\u2081).symm, },\n  { rintro rfl,\n    exact \u03c0.left_inv_litter hL\u2082, },\nend\n\nlemma eq_symm_apply_atom {a\u2081 a\u2082} : a\u2081 \u2208 \u03c0.atom_perm.domain \u2192 a\u2082 \u2208 \u03c0.atom_perm.domain \u2192\n  (a\u2081 = \u03c0.symm \u2022 a\u2082 \u2194 \u03c0 \u2022 a\u2081 = a\u2082) := \u03c0.atom_perm.eq_symm_apply\n\nlemma eq_symm_apply_litter {L\u2081 L\u2082} : L\u2081 \u2208 \u03c0.litter_perm.domain \u2192 L\u2082 \u2208 \u03c0.litter_perm.domain \u2192\n  (L\u2081 = \u03c0.symm \u2022 L\u2082 \u2194 \u03c0 \u2022 L\u2081 = L\u2082) := \u03c0.litter_perm.eq_symm_apply\n\nsection generate\n\nvariables (\u03c0)\n\n/-- Gives the largest sublitter of `\u03c0` on which `\u03c0.atom_perm` is not defined. -/\ndef largest_sublitter (L : litter) : sublitter := {\n  litter := L,\n  carrier := litter_set L \\ \u03c0.atom_perm.domain,\n  subset := diff_subset _ _,\n  diff_small := by simpa only [sdiff_sdiff_right_self, inf_eq_inter] using \u03c0.domain_small L,\n}\n\n@[simp] lemma largest_sublitter_litter (L : litter) : (\u03c0.largest_sublitter L).litter = L := rfl\n@[simp] lemma coe_largest_sublitter (L : litter) :\n  (\u03c0.largest_sublitter L : set atom) = litter_set L \\ \u03c0.atom_perm.domain := rfl\n\nlemma mem_largest_sublitter_of_not_mem_domain (a : atom) (h : a \u2209 \u03c0.atom_perm.domain) :\n  a \u2208 \u03c0.largest_sublitter a.1 := \u27e8rfl, h\u27e9\n\nlemma not_mem_domain_of_mem_largest_sublitter {a : atom} {L : litter}\n  (h : a \u2208 \u03c0.largest_sublitter L) : a \u2209 \u03c0.atom_perm.domain := h.2\n\n/-- Computes the action of `\u03c0` on this sublitter, assuming it is in `sublitter_domain`. -/\ndef generate_sublitter (S : sublitter) : sublitter := \u03c0.largest_sublitter (\u03c0 \u2022 S.litter)\n\ndef sublitter_domain : set sublitter :=\n{S | S.litter \u2208 \u03c0.litter_perm.domain \u2227 (S : set atom) = litter_set S.litter \\ \u03c0.atom_perm.domain}\n\nlemma mem_sublitter_domain (S : sublitter) (h : S \u2208 \u03c0.sublitter_domain) :\n  (S : set atom) = litter_set S.litter \\ \u03c0.atom_perm.domain :=\nh.2\n\nlemma generate_sublitter_mem_domain \u2983S : sublitter\u2984 (h : S \u2208 sublitter_domain \u03c0) :\n  generate_sublitter \u03c0 S \u2208 sublitter_domain \u03c0 :=\n\u27e8\u03c0.litter_perm.map_domain h.1, rfl\u27e9\n\nlemma generate_sublitter_left_inv \u2983S : sublitter\u2984 (h : S \u2208 sublitter_domain \u03c0) :\n  generate_sublitter \u03c0.symm (generate_sublitter \u03c0 S) = S :=\nbegin\n  ext : 1,\n  simp only [h.2, largest_sublitter, generate_sublitter, symm_atom_perm, local_perm.symm_domain,\n    sublitter.coe_mk, \u03c0.left_inv_litter h.1],\nend\n\n/-- Generates the unique near-litter approximation given by an atom local permutation and a\nnear-litter local permutation. This uniqueness is only up to evaluating everything on the domain\nof the permutation. -/\ndef generate_sublitter_perm : local_perm sublitter := {\n  to_fun := generate_sublitter \u03c0,\n  inv_fun := generate_sublitter \u03c0.symm,\n  domain := sublitter_domain \u03c0,\n  to_fun_domain' := generate_sublitter_mem_domain \u03c0,\n  inv_fun_domain' := generate_sublitter_mem_domain \u03c0.symm,\n  left_inv' := generate_sublitter_left_inv \u03c0,\n  right_inv' := generate_sublitter_left_inv \u03c0.symm,\n}\n\n@[simp] lemma generate_symm :\n  (generate_sublitter_perm \u03c0).symm = generate_sublitter_perm \u03c0.symm := rfl\n\n@[simp] lemma generate_sublitter_perm_domain :\n  (generate_sublitter_perm \u03c0).domain = sublitter_domain \u03c0 := rfl\n\n@[simp] lemma generate_sublitter_apply (S : sublitter) :\n  generate_sublitter_perm \u03c0 S = generate_sublitter \u03c0 S := rfl\n\ninstance : has_smul near_litter_approx sublitter := \u27e8\u03bb \u03c0, \u03c0.generate_sublitter_perm\u27e9\n\n@[simp] lemma smul_sublitter (S : sublitter) :\n  (\u03c0 \u2022 S).litter = \u03c0 \u2022 S.litter := rfl\n\nlemma smul_eq_smul_sublitter {S\u2081 S\u2082 : sublitter}\n  (h\u2081 : S\u2081 \u2208 sublitter_domain \u03c0) (h\u2082 : S\u2082 \u2208 sublitter_domain \u03c0) :\n  \u03c0 \u2022 S\u2081 = \u03c0 \u2022 S\u2082 \u2194 S\u2081 = S\u2082 :=\nbegin\n  unfold has_smul.smul,\n  rw [\u2190 \u03c0.generate_sublitter_perm.eq_symm_apply h\u2081 (\u03c0.generate_sublitter_perm.map_domain h\u2082),\n    local_perm.left_inv _ _],\n  exact h\u2082,\nend\n\n@[simp] lemma left_inv_sublitter {S} : S \u2208 \u03c0.sublitter_domain \u2192 \u03c0.symm \u2022 \u03c0 \u2022 S = S :=\n\u03c0.generate_sublitter_perm.left_inv\n\n@[simp] lemma right_inv_sublitter {S} : S \u2208 \u03c0.sublitter_domain \u2192 \u03c0 \u2022 \u03c0.symm \u2022 S = S :=\n\u03c0.generate_sublitter_perm.right_inv\n\nlemma eq_symm_apply_sublitter {S\u2081 S\u2082} : S\u2081 \u2208 \u03c0.sublitter_domain \u2192\n  S\u2082 \u2208 \u03c0.sublitter_domain \u2192 (S\u2081 = \u03c0.symm \u2022 S\u2082 \u2194 \u03c0 \u2022 S\u2081 = S\u2082) :=\n\u03c0.generate_sublitter_perm.eq_symm_apply\n\n/-- Computes the action of `\u03c0` on this near-litter. This action is not injective.\nThe nicest properties will hold when `N` is a litter. -/\ndef generate_near_litter (\u03c0 : near_litter_approx) (N : near_litter) : near_litter :=\n\u27e8\u03c0 \u2022 N.1, \u03c0.largest_sublitter (\u03c0 \u2022 N.1) \u222a \u03c0 \u2022 (N \\ \u03c0.largest_sublitter N.1), begin\n  refine small.union _ _,\n  { rw \u2190 diff_diff,\n    exact small.mono (diff_subset _ _) (\u03c0.largest_sublitter (\u03c0 \u2022 N.1)).diff_small, },\n  { rw union_diff_distrib,\n    refine small.union _ _,\n    { have := (\u03c0.largest_sublitter (\u03c0 \u2022 N.1)).subset,\n      rw [largest_sublitter_litter, sublitter.carrier_eq_coe] at this,\n      rw diff_eq_empty.mpr this,\n      exact small_empty, },\n    { refine small.mono (diff_subset _ _) (small.image _),\n      have := small.union (small.mono (subset_union_right _ _) N.2.prop)\n        (\u03c0.largest_sublitter N.1).diff_small,\n      simp only [largest_sublitter_litter, sublitter.carrier_eq_coe] at this,\n      refine small.mono _ this,\n      intros a ha,\n      by_cases a \u2208 litter_set N.fst,\n      exact or.inr \u27e8h, ha.2\u27e9,\n      exact or.inl \u27e8ha.1, h\u27e9, }, },\nend\u27e9\n\ninstance : has_smul near_litter_approx near_litter := \u27e8generate_near_litter\u27e9\n\n@[simp] lemma smul_near_litter_coe (\u03c0 : near_litter_approx) (N : near_litter) :\n  ((\u03c0 \u2022 N : near_litter) : set atom) =\n    \u03c0.largest_sublitter (\u03c0 \u2022 N.1) \u222a \u03c0 \u2022 (N \\ \u03c0.largest_sublitter N.1) := rfl\n\nend generate\n\ndef _root_.con_nf.near_litter_perm.is_exception (\u03c0 : near_litter_perm) (a : atom) : Prop :=\n\u03c0 \u2022 a \u2209 litter_set (\u03c0 \u2022 a.1) \u2228 \u03c0\u207b\u00b9 \u2022 a \u2209 litter_set (\u03c0\u207b\u00b9 \u2022 a.1)\n\n@[mk_iff] structure approximates (\u03c0\u2080 : near_litter_approx) (\u03c0 : near_litter_perm) : Prop :=\n(map_atom : \u2200 a, a \u2208 \u03c0\u2080.atom_perm.domain \u2192 \u03c0\u2080 \u2022 a = \u03c0 \u2022 a)\n(map_litter : \u2200 L, L \u2208 \u03c0\u2080.litter_perm.domain \u2192 \u03c0\u2080 \u2022 L = \u03c0 \u2022 L)\n\nlemma approximates.symm_map_atom {\u03c0\u2080 : near_litter_approx} {\u03c0 : near_litter_perm}\n  (h\u03c0 : \u03c0\u2080.approximates \u03c0) (a : atom) (ha : a \u2208 \u03c0\u2080.atom_perm.domain) : \u03c0\u2080.symm \u2022 a = \u03c0\u207b\u00b9 \u2022 a :=\nbegin\n  have := h\u03c0.map_atom (\u03c0\u2080.symm \u2022 a) (\u03c0\u2080.symm.atom_perm.map_domain ha),\n  rw \u2190 inv_smul_eq_iff at this,\n  rw [\u2190 this, smul_left_cancel_iff],\n  exact (\u03c0\u2080.atom_perm).right_inv ha,\nend\n\nlemma approximates.symm_map_litter {\u03c0\u2080 : near_litter_approx} {\u03c0 : near_litter_perm}\n  (h\u03c0 : \u03c0\u2080.approximates \u03c0) (L : litter) (hL : L \u2208 \u03c0\u2080.litter_perm.domain) : \u03c0\u2080.symm \u2022 L = \u03c0\u207b\u00b9 \u2022 L :=\nbegin\n  have := h\u03c0.map_litter (\u03c0\u2080.symm \u2022 L) (\u03c0\u2080.symm.litter_perm.map_domain hL),\n  rw \u2190 inv_smul_eq_iff at this,\n  rw [\u2190 this, smul_left_cancel_iff],\n  exact (\u03c0\u2080.litter_perm).right_inv hL,\nend\n\n@[mk_iff] structure exactly_approximates (\u03c0\u2080 : near_litter_approx) (\u03c0 : near_litter_perm)\n  extends approximates \u03c0\u2080 \u03c0 : Prop :=\n(exception_mem : \u2200 a, \u03c0.is_exception a \u2192 a \u2208 \u03c0\u2080.atom_perm.domain)\n\nlemma exactly_approximates.of_is_exception {\u03c0\u2080 : near_litter_approx} {\u03c0 : near_litter_perm}\n  (h\u03c0 : \u03c0\u2080.exactly_approximates \u03c0) (a : atom) (ha : a.1 \u2208 \u03c0\u2080.litter_perm.domain) :\n  \u03c0.is_exception a \u2192 \u03c0\u2080 \u2022 a \u2209 litter_set (\u03c0\u2080 \u2022 a.1) \u2228 \u03c0\u2080.symm \u2022 a \u2209 litter_set (\u03c0\u2080.symm \u2022 a.1) :=\nbegin\n  intro h,\n  rw [h\u03c0.map_litter a.fst ha, h\u03c0.symm_map_litter a.fst ha,\n    h\u03c0.map_atom a (h\u03c0.exception_mem a h), h\u03c0.symm_map_atom a (h\u03c0.exception_mem a h)],\n  exact h,\nend\n\nlemma exactly_approximates.mem_litter_set {\u03c0\u2080 : near_litter_approx} {\u03c0 : near_litter_perm}\n  (h\u03c0 : \u03c0\u2080.exactly_approximates \u03c0) (a : atom) (ha : a \u2209 \u03c0\u2080.atom_perm.domain) :\n  \u03c0 \u2022 a \u2208 litter_set (\u03c0 \u2022 a.1) :=\nby contrapose! ha; exact h\u03c0.exception_mem _ (or.inl ha)\n\nlemma exactly_approximates.mem_litter_set_inv {\u03c0\u2080 : near_litter_approx} {\u03c0 : near_litter_perm}\n  (h\u03c0 : \u03c0\u2080.exactly_approximates \u03c0) (a : atom) (ha : a \u2209 \u03c0\u2080.atom_perm.domain) :\n  \u03c0\u207b\u00b9 \u2022 a \u2208 litter_set (\u03c0\u207b\u00b9 \u2022 a.1) :=\nby contrapose! ha; exact h\u03c0.exception_mem _ (or.inr ha)\n\ninstance : preorder near_litter_approx := {\n  le := \u03bb \u03c0 \u03c0', \u03c0.atom_perm \u2264 \u03c0'.atom_perm \u2227 \u03c0.litter_perm \u2264 \u03c0'.litter_perm,\n  le_refl := \u03bb \u03c0, \u27e8le_rfl, le_rfl\u27e9,\n  le_trans := \u03bb _ _ _ h\u2081 h\u2082, \u27e8h\u2081.1.trans h\u2082.1, h\u2081.2.trans h\u2082.2\u27e9,\n}\n\nlemma approximates_of_le {\u03c0\u2080 \u03c0\u2080' : near_litter_approx} {\u03c0 : near_litter_perm}\n  (hle : \u03c0\u2080' \u2264 \u03c0\u2080) (h : \u03c0\u2080.approximates \u03c0) : \u03c0\u2080'.approximates \u03c0 :=\n\u27e8\u03bb a ha, (hle.1.2 ha).trans (h.1 a (hle.1.1 ha)), \u03bb N hN, (hle.2.2 hN).trans (h.2 N (hle.2.1 hN))\u27e9\n\ndef free (\u03b1 : \u039b) [position_data.{}] [phase_2_assumptions \u03b1] {\u03b2 : type_index}\n  (\u03c0 : near_litter_approx) (A : extended_index \u03b2) : Prop :=\n\u2200 L \u2208 \u03c0.litter_perm.domain, flexible \u03b1 L A\n\nend near_litter_approx\n\n/-!\n# Structural approximations\n-/\n\n/-- A `\u03b2`-structural approximation is a product that assigns a near-litter approximation to each\n`\u03b2`-extended index. -/\ndef struct_approx (\u03b2 : type_index) := extended_index \u03b2 \u2192 near_litter_approx\n\nnamespace struct_approx\n\n-- TODO: Could refactor struct_perm as a map `extended_index \u03b2 \u2192 near_litter_perm`.\n\ndef approximates {\u03b2 : type_index} (\u03c0\u2080 : struct_approx \u03b2) (\u03c0 : struct_perm \u03b2) : Prop :=\n\u2200 A, (\u03c0\u2080 A).approximates (struct_perm.of_bot $ struct_perm.derivative A \u03c0)\n\ndef exactly_approximates {\u03b2 : type_index} (\u03c0\u2080 : struct_approx \u03b2) (\u03c0 : struct_perm \u03b2) : Prop :=\n\u2200 A, (\u03c0\u2080 A).exactly_approximates (struct_perm.of_bot $ struct_perm.derivative A \u03c0)\n\nvariables {\u03b1 : \u039b} [position_data.{}] [phase_2_assumptions \u03b1]\n\n/-- A structural approximation `\u03c0` *supports* a set of support conditions if all of the support\nconditions lie in the domain of `\u03c0` and all near-litter support conditions are litters. -/\n@[mk_iff] structure supports {\u03b2 : Iic \u03b1} (\u03c0\u2080 : struct_approx \u03b2) (S : set (support_condition \u03b2)) :\n  Prop :=\n(atom_mem_domain : \u2200 a B, (inl a, B) \u2208 S \u2192 a \u2208 (\u03c0\u2080 B).atom_perm.domain)\n(near_litter_mem_domain : \u2200 (N : near_litter) B, (inr N, B) \u2208 S \u2192 N.1 \u2208 (\u03c0\u2080 B).litter_perm.domain)\n(is_litter : \u2200 (N : near_litter) B, (inr N, B) \u2208 S \u2192 N.is_litter)\n\ninstance has_smul_support_condition {\u03b2 : type_index} :\n  has_smul (struct_approx \u03b2) (support_condition \u03b2) :=\n\u27e8\u03bb \u03c0 c, \u27e8\u03c0 c.snd \u2022 c.fst, c.snd\u27e9\u27e9\n\nlemma smul_support_condition_eq {\u03b2 : type_index} (\u03c0 : struct_approx \u03b2) (c : support_condition \u03b2) :\n  \u03c0 \u2022 c = \u27e8\u03c0 c.snd \u2022 c.fst, c.snd\u27e9 := rfl\n\nlemma smul_eq_of_supports {\u03b2 : Iic \u03b1} {\u03c0\u2080 : struct_approx \u03b2} {\u03c0 : allowable \u03b2}\n  (h\u03c0 : \u03c0\u2080.exactly_approximates \u03c0.to_struct_perm)\n  {S : set (support_condition \u03b2)} (hS : \u03c0\u2080.supports S)\n  {c : support_condition \u03b2} (hc : c \u2208 S) : \u03c0\u2080 \u2022 c = \u03c0 \u2022 c :=\nbegin\n  obtain \u27e8a | N, A\u27e9 := c,\n  { refine prod.ext _ rfl,\n    change inl _ = inl _,\n    exact congr_arg inl ((h\u03c0 A).map_atom a (hS.atom_mem_domain a A hc)), },\n  refine prod.ext _ rfl,\n  change inr _ = inr _,\n  refine congr_arg inr _,\n  ext : 1,\n  exact (h\u03c0 A).map_litter N.fst (hS.near_litter_mem_domain N A hc),\n  rw (hS.is_litter N A hc).eq_fst_to_near_litter,\n  ext a : 1,\n  simp only [near_litter_approx.smul_near_litter_coe, litter.to_near_litter_fst,\n    near_litter_approx.coe_largest_sublitter, litter.coe_to_near_litter, sdiff_sdiff_right_self,\n    inf_eq_inter, mem_union, mem_diff, mem_litter_set, set_like.mem_coe],\n  split,\n  { rintro (\u27e8h\u2081, h\u2082\u27e9 | \u27e8a, \u27e8ha\u2081, ha\u2082\u27e9, rfl\u27e9),\n    { refine \u27e8(struct_perm.derivative A \u03c0.to_struct_perm)\u207b\u00b9 \u2022 a, _,\n        by simp only [struct_perm.coe_to_near_litter_perm, struct_perm.of_bot_smul, smul_inv_smul]\u27e9,\n      simp only [litter.coe_to_near_litter, mem_litter_set],\n      have := (h\u03c0 A).mem_litter_set_inv a h\u2082,\n      rw [h\u2081, (h\u03c0 A).map_litter _ (hS.near_litter_mem_domain N A hc),\n        mem_litter_set, inv_smul_smul, struct_perm.of_bot_inv_smul] at this,\n      exact this, },\n    { exact \u27e8a, ha\u2081, ((h\u03c0 A).map_atom a ha\u2082).symm\u27e9, }, },\n  { rintro \u27e8a, ha, rfl\u27e9,\n    simp only [litter.coe_to_near_litter, mem_litter_set] at ha,\n    simp only [struct_perm.coe_to_near_litter_perm, struct_perm.of_bot_smul],\n    by_cases a \u2208 (\u03c0\u2080 A).atom_perm.domain,\n    { exact or.inr \u27e8a, \u27e8ha, h\u27e9, (h\u03c0 A).map_atom a h\u27e9, },\n    { refine or.inl \u27e8_, _\u27e9,\n      { have := (h\u03c0 A).mem_litter_set a h,\n        simp only [struct_perm.of_bot_smul, mem_litter_set] at this,\n        rw [this, ha],\n        exact ((h\u03c0 A).map_litter _ (hS.near_litter_mem_domain N A hc)).symm, },\n      { contrapose! h,\n        have := (h\u03c0 A).symm_map_atom _ h,\n        simp only [struct_perm.of_bot_inv_smul, inv_smul_smul] at this,\n        rw \u2190 this,\n        exact (\u03c0\u2080 A).symm.atom_perm.map_domain h, }, }, },\nend\n\n/-- If two allowable permutations exactly approximate some structural approximation, then their\nactions agree on everything that the structural approximation supports. -/\nlemma smul_eq_smul_of_exactly_approximates {\u03b2 : Iic \u03b1}\n  {\u03c0\u2080 \u03c0\u2080' : struct_approx \u03b2} {\u03c0 \u03c0' : allowable \u03b2}\n  (h\u03c0 : \u03c0\u2080.exactly_approximates \u03c0.to_struct_perm)\n  (h\u03c0' : \u03c0\u2080'.exactly_approximates \u03c0'.to_struct_perm)\n  (S : set (support_condition \u03b2)) (t : tangle \u03b2)\n  (hS : \u03c0\u2080.supports S) (hS' : \u03c0\u2080'.supports S) (ht : mul_action.supports (allowable \u03b2) S t)\n  (hS\u03c0 : \u2200 c \u2208 S, \u03c0\u2080 \u2022 c = \u03c0\u2080' \u2022 c) : \u03c0 \u2022 t = \u03c0' \u2022 t :=\nbegin\n  have := ht (\u03c0'\u207b\u00b9 * \u03c0) _,\n  { rw [mul_smul, inv_smul_eq_iff] at this,\n    exact this, },\n  intros c hc,\n  rw [mul_smul, inv_smul_eq_iff, \u2190 smul_eq_of_supports h\u03c0 hS hc, \u2190 smul_eq_of_supports h\u03c0' hS' hc],\n  exact hS\u03c0 c hc,\nend\n\ndef free {\u03b2 : Iic \u03b1} (\u03c0\u2080 : struct_approx \u03b2) : Prop := \u2200 A, (\u03c0\u2080 A).free \u03b1 A\n\n/-!\n# Induction on support conditions\n-/\n\n/-- The inductive hypothesis used to construct the induced action of an approximation in the\nfreedom of action theorem. -/\nstructure hypothesis {\u03b2 : Iic \u03b1} (c : support_condition \u03b2) :=\n(atom_image : \u03a0 a A, (relation.trans_gen (constrains \u03b1 \u03b2)) \u27e8inl a, A\u27e9 c \u2192 atom)\n(near_litter_image : \u03a0 N A, (relation.trans_gen (constrains \u03b1 \u03b2)) \u27e8inr N, A\u27e9 c \u2192 near_litter)\n\nnamespace hypothesis\nvariable {\u03b2 : Iic \u03b1}\n\n/-- Two hypotheses are compatible if they agree everywhere that they are both defined. -/\n@[mk_iff] structure compatible {c d : support_condition \u03b2}\n  (Hc : hypothesis c) (Hd : hypothesis d) : Prop :=\n(atom_compatible : \u2200 a A hc hd, Hc.atom_image a A hc = Hd.atom_image a A hd)\n(near_litter_compatible : \u2200 N A hc hd, Hc.near_litter_image N A hc = Hd.near_litter_image N A hd)\n\ndef fix_map :\n  (psum (\u03a3' (_ : atom), extended_index \u03b2) (\u03a3' (_ : near_litter), extended_index \u03b2)) \u2192\n  support_condition \u03b2\n| (psum.inl \u27e8a, A\u27e9) := \u27e8inl a, A\u27e9\n| (psum.inr \u27e8N, A\u27e9) := \u27e8inr N, A\u27e9\n\ndef fix_wf : has_well_founded\n  (psum (\u03a3' (_ : atom), extended_index \u03b2) (\u03a3' (_ : near_litter), extended_index \u03b2)) :=\n\u27e8inv_image (relation.trans_gen (constrains \u03b1 \u03b2)) fix_map,\n  inv_image.wf _ (constrains_wf \u03b1 \u03b2).trans_gen\u27e9\n\n/-- Construct the fixed-point functions `fix_atom` and `fix_near_litter`.\nThis is used to compute the induced action of an approximation on all atoms and near-litters. -/\nnoncomputable! mutual def fix_atom, fix_near_litter\n  (Fa : \u03a0 a (A : extended_index \u03b2), hypothesis \u27e8inl a, A\u27e9 \u2192 atom)\n  (FN : \u03a0 N (A : extended_index \u03b2), hypothesis \u27e8inr N, A\u27e9 \u2192 near_litter)\nwith fix_atom : atom \u2192 extended_index \u03b2 \u2192 atom\n| a A := Fa a A \u27e8\u03bb b B hb, fix_atom b B, \u03bb N B hb, fix_near_litter N B\u27e9\nwith fix_near_litter : near_litter \u2192 extended_index \u03b2 \u2192 near_litter\n| N A := FN N A \u27e8\u03bb b B hb, fix_atom b B, \u03bb N B hb, fix_near_litter N B\u27e9\nusing_well_founded { rel_tac := \u03bb _ _, `[exact fix_wf], dec_tac := `[exact hb] }\n\nlemma fix_atom_eq (Fa FN) (a : atom) (A : extended_index \u03b2) :\n  fix_atom Fa FN a A =\n  Fa a A \u27e8\u03bb b B hb, fix_atom Fa FN b B, \u03bb N B hb, fix_near_litter Fa FN N B\u27e9 :=\nby rw fix_atom\n\nlemma fix_near_litter_eq (Fa FN) (N : near_litter) (A : extended_index \u03b2) :\n  fix_near_litter Fa FN N A =\n  FN N A \u27e8\u03bb b B hb, fix_atom Fa FN b B, \u03bb N B hb, fix_near_litter Fa FN N B\u27e9 :=\nby rw fix_near_litter\n\nend hypothesis\n\nend struct_approx\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/approximation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3746915052895648}}
{"text": "import .tab\n\nvariables {p q r s : Prop}\n\nopen tactic\n\nexample (h : p \u2228 q) : false := by split_disj\n\n-- meta def find_disj' : list expr \u2192 tactic expr\n-- | []        := failed\n-- | (e :: es) := do trace \"expr : \", trace e,\n--                   t \u2190 infer_type e,\n--                   trace \"expr type : \", trace t,\n--                   match t with\n--                   | `(%%a \u2228 %%b) := trace \"disjunction\" >> return e\n--                   | _            := trace \"not a disjunction\" >> find_disj' es\n--                   end\n\n#exit \n\nexample (h : p \u2228 q) : false :=\nby do l \u2190 local_context,\n      -- trace l,\n      e \u2190 find_disj l,\n      -- trace e,\n      cases e,\n      -- trace \"After cases : \", trace_state,\n      skip", "meta": {"author": "skbaek", "repo": "tab", "sha": "70909a69464a8713412d640ac630e5e6ef4e43e8", "save_path": "github-repos/lean/skbaek-tab", "path": "github-repos/lean/skbaek-tab/tab-70909a69464a8713412d640ac630e5e6ef4e43e8/disj.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.37466062415117823}}
{"text": "/-\n  The stalks of the structure presheaf on the basis are local rings.\n-/\n\nimport ring_theory.ideals\nimport ring_theory.localization\nimport to_mathlib.localization.local_rings\nimport to_mathlib.localization.localization_alt\nimport sheaves.stalk_of_rings_on_standard_basis\nimport spectrum_of_a_ring.structure_presheaf\nimport spectrum_of_a_ring.structure_presheaf_localization\nimport spectrum_of_a_ring.structure_presheaf_res\n\nuniverse u\n\nnoncomputable theory\n\nvariables {R : Type u} [comm_ring R]\nvariables (P : Spec R)\n\nopen localization localization_alt stalk_of_rings_on_standard_basis\nopen classical\n\ndef Bstd := D_fs_standard_basis R\n\nnamespace strucutre_presheaf_stalks\n\ndef F := structure_presheaf_on_basis R\n\ndef FP := stalk_of_rings_on_standard_basis Bstd F P\n\n-- This is essentially the map from x to x/1.\ndef \u03c6 : R \u2192 FP P := \u03bb x, \n\u27e6{ U := opens.univ,\n   BU := (D_fs_standard_basis R).1,\n   Hx := set.mem_univ P,\n   s := (of : R \u2192 localization R (S (opens.univ))) x, }\u27e7\n\ninstance FP.is_comm_ring : comm_ring (FP P) :=\nby simp [FP]; by apply_instance\n\ninstance prime.is_submonoid : is_submonoid (-P.1 : set R) :=\n{ one_mem := P.1.ne_top_iff_one.1 P.2.1,\n  mul_mem := \u03bb x y hnx hny hxy, or.cases_on (P.2.2 hxy) hnx hny }\n\ninstance \u03c6P.is_ring_hom : is_ring_hom (\u03c6 P) :=\n{ map_one := rfl,\n  map_mul := \u03bb x y, \n    begin\n      apply quotient.sound,\n      use [opens.univ, (D_fs_standard_basis R).1, set.mem_univ P],\n      use [set.subset.refl _, (\u03bb x Hx, \u27e8Hx, Hx\u27e9)],\n      rw (F.res_is_ring_hom _ _ _).map_mul,\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw of.is_ring_hom.map_mul,\n      rw (F.res_is_ring_hom _ _ _).map_mul,\n    end,\n  map_add := \u03bb x y,\n    begin\n      apply quotient.sound,\n      use [opens.univ, (D_fs_standard_basis R).1, set.mem_univ P],\n      use [set.subset.refl _, (\u03bb x Hx, \u27e8Hx, Hx\u27e9)],\n      rw (F.res_is_ring_hom _ _ _).map_add,\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw \u2190presheaf_on_basis.Hcomp',\n      rw of.is_ring_hom.map_add,\n      rw (F.res_is_ring_hom _ _ _).map_add,\n    end, }\n\nlemma stalk_local.inverts_data : inverts_data (-P.1 : set R) (\u03c6 P) :=\nbegin\n  rintros \u27e8s, Hs\u27e9,\n  change s \u2209 P.val at Hs,\n  let BDs := D_fs.mem R s, \n  have HsS : s \u2208 S (Spec.DO R s) := S.f_mem s,\n  let sinv : FP P := \n    \u27e6{ U := Spec.DO R s,\n       BU := BDs,\n       Hx := Hs,\n       s := \u27e6\u27e81, \u27e8s, HsS\u27e9\u27e9\u27e7, }\u27e7,\n  use sinv,\n  apply quotient.sound,\n  use [Spec.DO R s, BDs, Hs, (\u03bb x Hx, \u27e8trivial, Hx\u27e9), set.subset_univ _],\n  simp,\n  erw (F.res_is_ring_hom _ _ _).map_mul,\n  erw (F.res_is_ring_hom _ _ _).map_one,\n  iterate 2 { rw \u2190presheaf_on_basis.Hcomp', },\n  erw presheaf_on_basis.Hid',\n  erw structure_presheaf_on_basis.res_eq,\n  erw \u2190structure_presheaf_on_basis.res_comp_of',\n  apply quotient.sound,\n  use [1, is_submonoid.one_mem _],\n  simp,\nend\n\nlemma stalk_local.has_denom : has_denom (-P.1 : set R) (\u03c6 P) :=\nbegin\n  intros x,\n  apply quotient.induction_on x,\n  rintros \u27e8U, BU, HPU, s\u27e9,\n  -- s \u2208 R[1/S(U)]. \n  apply quotient.induction_on s,\n  rintros \u27e8p, \u27e8q, Hq\u27e9\u27e9,\n  change U \u2286 Spec.DO R q at Hq,\n  -- Since P \u2208 D(q).\n  have HqnP : q \u2209 P.val := Hq HPU,\n  use [\u27e8\u27e8q, HqnP\u27e9, p\u27e9],\n  apply quotient.sound,\n  use [U, BU, HPU, (\u03bb x Hx, \u27e8trivial, Hx\u27e9), set.subset_univ _],\n  simp,\n  erw (F.res_is_ring_hom _ _ _).map_mul,\n  iterate 2 { rw \u2190presheaf_on_basis.Hcomp', },\n  erw presheaf_on_basis.Hid',\n  erw structure_presheaf_on_basis.res_eq,\n  iterate 2 { erw \u2190structure_presheaf_on_basis.res_comp_of', },\n  apply quotient.sound,\n  use [1, is_submonoid.one_mem _],\n  simp,\nend\n\nlemma stalk_local.has_denom_data : has_denom_data (-P.1 : set R) (\u03c6 P) :=\nhas_denom_some (-P.1 : set R) (\u03c6 P) (stalk_local.has_denom P)\n\nlemma stalk_local.ker_le : ker (\u03c6 P) \u2264 submonoid_ann (-P.1 : set R) :=\nbegin\n  intros x Hx,\n  change \u03c6 P x = 0 at Hx,\n  erw quotient.eq at Hx,\n  rcases Hx with \u27e8W, BW, HPW, HWuniv1, HWuniv2, Hres\u27e9,\n  dsimp at Hres,\n  erw is_ring_hom.map_zero (F.res (D_fs_standard_basis R).1 BW HWuniv2) at Hres,\n  erw structure_presheaf_on_basis.res_eq at Hres,\n  erw \u2190structure_presheaf_on_basis.res_comp_of' at Hres,\n  erw quotient.eq at Hres,\n  rcases Hres with \u27e8t, HtSW, Ht\u27e9,\n  simp at Ht,\n  have HtnP : t \u2209 P.val := HtSW HPW,\n  use \u27e8\u27e8x, \u27e8t, HtnP\u27e9\u27e9, Ht\u27e9,\nend\n\nlemma stalk_local.localization : is_localization_data (-P.1 : set R) (\u03c6 P) :=\n{ inverts := stalk_local.inverts_data P, \n  has_denom := stalk_local.has_denom_data P, \n  ker_le := stalk_local.ker_le P, }\n\nlemma stalk_local : local_ring (FP P) :=\nlocal_ring.of_is_localization_data_at_prime P.2 (stalk_local.localization P)\n\nend strucutre_presheaf_stalks\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/project_1_a_decrire/lean-scheme-submission/src/spectrum_of_a_ring/structure_presheaf_stalks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499941, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3746606166014046}}
{"text": "import tactic.rewrite_search\n\nopen tactic.rewrite_search.discovery\n\n\n\n\nnamespace tactic.rewrite_search.testing\n\nlocal attribute [instance] classical.prop_decidable\n\nexample {A B C : Prop} : ((B \u2192 C) \u2192 (\u00ac(A \u2192 C) \u2227 \u00ac(A \u2228 B))) = (B \u2227 \u00acC) :=\n  by rewrite_search_using! [] {}\n\nend tactic.rewrite_search.testing\n\n\n\n\nnamespace tactic.rewrite_search.testing\n\naxiom foo' : [6] = [7]\naxiom bar' : [[5],[5]] = [[6],[6]]\n\nexample : [[7],[6]] = [[5],[5]] :=\nbegin\n rewrite_search_with [\u2190foo', bar'] {},\nend\n\naxiom foo''  : [7] = [8]\naxiom foo''' : [8] = [7]\n\nrun_cmd (rewrite_list_from_lemma `(foo'')).mmap (\u03bb rw, is_promising_rewrite rw [`([[8],[6]])]) >>= tactic.trace\nrun_cmd (rewrite_list_from_lemma `(foo''')).mmap (\u03bb rw, is_promising_rewrite rw [`([[8],[6]])]) >>= tactic.trace\n\ndef my_test : [[7],[6]] = [[5],[5]] :=\nbegin\n success_if_fail { rewrite_search_with [ bar'] {help_me := ff} },\n\n rewrite_search_with! [ bar'] {help_me := tt}\nend\n\nend tactic.rewrite_search.testing\n", "meta": {"author": "semorrison", "repo": "lean-rewrite-search", "sha": "e804b8f2753366b8957be839908230ee73f9e89f", "save_path": "github-repos/lean/semorrison-lean-rewrite-search", "path": "github-repos/lean/semorrison-lean-rewrite-search/lean-rewrite-search-e804b8f2753366b8957be839908230ee73f9e89f/test/rewrite_search_discovery_2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3746606090516309}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : \u2115) (x : \u2115) (h : coprime x p) : is_unit (x : \u2124_[p]) :=\nbegin\n  have h1 : (x : \u2124_[p]) = 0, from by auto [not_coprime_zero],\n  have h2 : \u00ac(x : \u2124_[p]) = 0, from by auto [h, h1],\n  have h3 : (x : \u2124_[p]) \u2260 0, from by auto [h2],\n  have h4 : (x : \u2124_[p]) \u2260 0 := by auto [h3],\n  have h5 : (x : \u2124_[p]) \u2260 0, from by auto [h4, h3],\n  show is_unit _, from by auto [h5],\nend\n\n--OUTPUT 2\ntheorem p_adic_units (p : \u2115) (x : \u2115) : coprime p x \u2192 (\u2203 y : \u2115, x * y = 1) :=\nbegin\n  assume h1 : coprime p x,\n  have h2 : x \u2223 1, from by auto [h1],\n  have h3 : \u2203 y : \u2115, x * y = 1, from by auto [dvd_def],\n  show \u2203 y : \u2115, x * y = 1, from h3,\nend\n\n--OUTPUT 3\ntheorem p_adic_units (p : \u2115) (hp : prime p) (x : \u2115) (hx : coprime p x) : x \u2208 units (padic_int p) :=\nbegin\n  have h : \u00ac(x \u2208 units (padic_int p)) \u2192 \u00ac(coprime p x), by auto [padic_int.units_eq_norm_one, padic_int.norm_le],\n  have h2 : \u00ac(x \u2208 units (padic_int p)) \u2192 (x \u2223 p), from by auto [padic_int.norm_lt_iff],\n  by_contradiction,\n  have h3 : \u00ac(x \u2208 units (padic_int p)), from by auto [h, h2],\n  show false, from by auto [h3] using [hx],\nend\n\n--OUTPUT 4\ntheorem p_adic_units : \u2200 (p : \u2115) (x : \u2115), prime p \u2192 coprime x p \u2192 x \u2208 units (integers_mod p) :=\nbegin\n  assume p x h1 h2,\n  have h3 : x \u2208 units (integers_mod p) \u2194 (p \u2223 x) = ff, from by auto [units_iff_coprime],\n  have h4 : \u00ac(p \u2223 x) = ff, from by auto [not_iff_not_of_iff, h3, h2],\n  show x \u2208 units (integers_mod p), from by auto [h3, h4],\nend\n\n--OUTPUT 5\ntheorem padic_units (p : \u2115) [h : nat.prime p] (x : \u2115) : nat.coprime p x \u2192 x \u2208 units (padic_int p) :=\nbegin\n  intro hx,\n  have h1 : x \u2209 units (padic_int p), from by auto [padic_int.norm_eq_one_iff_units, not_not_intro, hx],\n  have h2 : x \u2208 padic_int.val_semiring.nonunits, from by auto [h1, not_not_intro],\n  have h3 : x \u2208 padic_int.val_semiring.nonunits \u2194 x \u2208 padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_semiring.ideal.map (padic_int.val_semiring.ideal.comap (padic_int.val_sem\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem p_adic_units (p : \u2115) [prime p] (x : \u2115) : coprime p x \u2192 is_unit (p_adic_int p x) :=\nbegin\n  intro h1,\n  have h2 : p_adic_int p x \u2260 0, from by auto [p_adic_int],\n  have h3 : p_adic_norm p (p_adic_int p x) = 1, from by auto [h1, p_adic_norm, p_adic_int, p_adic_norm_def, p_adic_int_def, p_adic_int_eq_dvd, prime.dvd_not_coprime],\n  show is_unit (p_adic_int p x), from by auto [h2, h3, p_adic_norm],\nend\n\n--OUTPUT 7\ntheorem p_adic_units (p : \u2115) (h : prime p) (x : \u2115) (hx : nat.coprime x p) : is_unit (p_adic_int.mk_rat x) :=\nbegin\n  have h1 : (p_adic_int.mk_rat x).norm = 1, from by auto [hx, p_adic_int.norm_mk_rat, h],\n  have h2 : (p_adic_int.mk_rat x).norm \u2260 1, from by auto [hx, p_adic_int.norm_mk_rat],\n  have h3 : (p_adic_int.mk_rat x).norm < 1, from by auto [p_adic_int.norm_mk_rat],\n  have h4 : p \u2223 x, from by auto [hx, p_adic_int.norm_mk_rat, h],\n  have h5 : (p_adic_int.mk_rat x) * (p_adic_int.mk_rat p) = p_adic_int.mk_rat x * p_adic_int.mk_rat p, from by auto [p_adic_int.mk_rat, rat.mk_eq_mk],\n  have h6 : (p_adic_int.mk_rat x) * (p_adic_int.mk_rat p) = p_adic_int.mk_rat (x * p), from by auto [rat.mk_eq_mk, p_adic_int.mk_rat],\n  have h7 : (p_adic_int.mk_rat x) * (p_adic_int.mk_rat p) = p_adic_int.mk_rat x, from by auto [nat.mul_comm, rat.mk_eq_mk, p_adic_int.mk_rat],\n  show is_unit (p_adic_int.mk_rat x), from by auto [hx, p_adic_int.is_unit_mk_rat, h, h1, h2, h3, h4, h5, h6, h7],\nend\n\n--OUTPUT 8\ntheorem p_adic_units (p : \u2115) (hp : prime p) (x : \u2115) (hx : coprime p x) : x \u2208 units (\u2124 p) :=\nbegin\n  have h1 : x \u2208 units (\u2124 p) \u2194 \u2223x\u2223 p = 1, from by auto [units_iff_norm_eq_one],\n  have h2 : \u00ac x \u2208 units (\u2124 p) \u2194 \u00ac \u2223x\u2223 p = 1, from by auto [not_iff],\n  have h3 : \u00ac x \u2208 units (\u2124 p) \u2194 \u2223x\u2223 p \u2260 1, from by auto [not_iff],\n  have h4 : \u00ac x \u2208 units (\u2124 p) \u2194 \u2223x\u2223 p < 1, from by auto [norm_lt_one, not_iff],\n  have h5 : \u00ac x \u2208 units (\u2124 p) \u2192 \u2223x\u2223 p < 1, from by auto [h4.symm],\n  have h6 : \u00ac \u2223x\u2223 p < 1 \u2192 x \u2208 units (\u2124 p), from by auto [h4, not_iff],\n  have h7 : \u00ac \u2223x\u2223 p < 1 \u2192 p \u2223 x, from by auto [hx, prime.not_dvd_iff_ne_one, not_iff, norm_lt_one, dvd_iff_norm_le_one],\n  have h8 : \u2223x\u2223 p < 1 \u2192 \u00ac p \u2223 x, from by auto [hx, prime.not_dvd_iff_ne_one, not_iff, norm_le_one, norm_lt_one, not_lt],\n  have h9 : \u00ac p \u2223 x \u2192 x \u2208 units (\u2124 p), from by auto [hx, prime.not_dvd_iff_ne_one, not_iff, norm_le_one, dvd_iff_norm_le_one, h6],\n\n  have h10 : \u00ac p \u2223 x \u2194 \u00ac x \u2208 units (\u2124 p), from by auto [h7, h8, h9],\n  have h11 : \u00ac p \u2223 x \u2194 x \u2208 units (\u2124 p), from by auto [not_iff, h10],\n\n  have h12 : \u00ac p \u2223 x \u2192 x \u2208 units (\u2124 p), from by auto [h11.symm],\n  have h13 : \u00ac x \u2208 units (\u2124 p) \u2192 p \u2223 x, from by auto [h11],\n  have h14 : \u00ac x \u2208 units (\u2124 p) \u2192 \u00ac p \u2223 x, from by auto [hx, prime.not_dvd_iff_ne_one, not_iff, h13],\n\n  show x \u2208 units (\u2124 p), from by auto [hx, prime.not_dvd_iff_ne_one, not_iff, h12, h14],\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/lean_proof_auto-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7549149868676284, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.37450866676015065}}
{"text": "import tactic\nopen list\n\nvariable {\u03b1 : Type}\n\n@[simp]\nlemma drop_eq_nil_of_le\n  {l : list \u03b1}\n  {k : \u2115}\n  (h : length l \u2264 k)\n  : drop k l = [] :=\nby simpa [\u2190length_eq_zero] using nat.sub_eq_zero_of_le h\n\ntheorem drop_nil :\n  \u2200 n, drop n [] = ([] : list \u03b1) :=\n\u03bb _, list.drop_eq_nil_of_le (nat.zero_le _)\n", "meta": {"author": "jaalonso", "repo": "Demostraciones-con-Lean", "sha": "034ee559250cf55e28d10d8e65f1a4642e284251", "save_path": "github-repos/lean/jaalonso-Demostraciones-con-Lean", "path": "github-repos/lean/jaalonso-Demostraciones-con-Lean/Demostraciones-con-Lean-034ee559250cf55e28d10d8e65f1a4642e284251/src/Propiedades_de_drop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.37449005341256303}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.preserves.basic\nimport Mathlib.PostPort\n\nuniverses u\u2081 u\u2082 v \n\nnamespace Mathlib\n\n/-!\n# Preserving binary products\n\nConstructions to relate the notions of preserving binary products and reflecting binary products\nto concrete binary fans.\n\nIn particular, we show that `prod_comparison G X Y` is an isomorphism iff `G` preserves\nthe product of `X` and `Y`.\n-/\n\nnamespace category_theory.limits\n\n\n/--\nThe map of a binary fan is a limit iff the fork consisting of the mapped morphisms is a limit. This\nessentially lets us commute `binary_fan.mk` with `functor.map_cone`.\n-/\ndef is_limit_map_cone_binary_fan_equiv {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) {P : C} {X : C} {Y : C} (f : P \u27f6 X) (g : P \u27f6 Y) : is_limit (functor.map_cone G (binary_fan.mk f g)) \u2243 is_limit (binary_fan.mk (functor.map G f) (functor.map G g)) :=\n  equiv.trans\n    (equiv.symm\n      (is_limit.postcompose_hom_equiv (diagram_iso_pair (pair X Y \u22d9 G)) (functor.map_cone G (binary_fan.mk f g))))\n    (is_limit.equiv_iso_limit\n      (cones.ext\n        (iso.refl\n          (cone.X\n            (functor.obj (cones.postcompose (iso.hom (diagram_iso_pair (pair X Y \u22d9 G))))\n              (functor.map_cone G (binary_fan.mk f g)))))\n        sorry))\n\n/-- The property of preserving products expressed in terms of binary fans. -/\ndef map_is_limit_of_preserves_of_is_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) {P : C} {X : C} {Y : C} (f : P \u27f6 X) (g : P \u27f6 Y) [preserves_limit (pair X Y) G] (l : is_limit (binary_fan.mk f g)) : is_limit (binary_fan.mk (functor.map G f) (functor.map G g)) :=\n  coe_fn (is_limit_map_cone_binary_fan_equiv G f g) (preserves_limit.preserves l)\n\n/-- The property of reflecting products expressed in terms of binary fans. -/\ndef is_limit_of_reflects_of_map_is_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) {P : C} {X : C} {Y : C} (f : P \u27f6 X) (g : P \u27f6 Y) [reflects_limit (pair X Y) G] (l : is_limit (binary_fan.mk (functor.map G f) (functor.map G g))) : is_limit (binary_fan.mk f g) :=\n  reflects_limit.reflects (coe_fn (equiv.symm (is_limit_map_cone_binary_fan_equiv G f g)) l)\n\n/--\nIf `G` preserves binary products and `C` has them, then the binary fan constructed of the mapped\nmorphisms of the binary product cone is a limit.\n-/\ndef is_limit_of_has_binary_product_of_preserves_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) (X : C) (Y : C) [has_binary_product X Y] [preserves_limit (pair X Y) G] : is_limit (binary_fan.mk (functor.map G prod.fst) (functor.map G prod.snd)) :=\n  map_is_limit_of_preserves_of_is_limit G prod.fst prod.snd (prod_is_prod X Y)\n\n/--\nIf the product comparison map for `G` at `(X,Y)` is an isomorphism, then `G` preserves the\npair of `(X,Y)`.\n-/\ndef preserves_pair.of_iso_comparison {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) (X : C) (Y : C) [has_binary_product X Y] [has_binary_product (functor.obj G X) (functor.obj G Y)] [i : is_iso (prod_comparison G X Y)] : preserves_limit (pair X Y) G :=\n  preserves_limit_of_preserves_limit_cone (prod_is_prod X Y)\n    (coe_fn (equiv.symm (is_limit_map_cone_binary_fan_equiv G prod.fst prod.snd))\n      (is_limit.of_point_iso (limit.is_limit (pair (functor.obj G X) (functor.obj G Y)))))\n\n/--\nIf `G` preserves the product of `(X,Y)`, then the product comparison map for `G` at `(X,Y)` is\nan isomorphism.\n-/\ndef preserves_pair.iso {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) (X : C) (Y : C) [has_binary_product X Y] [has_binary_product (functor.obj G X) (functor.obj G Y)] [preserves_limit (pair X Y) G] : functor.obj G (X \u2a2f Y) \u2245 functor.obj G X \u2a2f functor.obj G Y :=\n  is_limit.cone_point_unique_up_to_iso (is_limit_of_has_binary_product_of_preserves_limit G X Y)\n    (limit.is_limit (pair (functor.obj G X) (functor.obj G Y)))\n\n@[simp] theorem preserves_pair.iso_hom {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) (X : C) (Y : C) [has_binary_product X Y] [has_binary_product (functor.obj G X) (functor.obj G Y)] [preserves_limit (pair X Y) G] : iso.hom (preserves_pair.iso G X Y) = prod_comparison G X Y :=\n  rfl\n\nprotected instance prod_comparison.category_theory.is_iso {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (G : C \u2964 D) (X : C) (Y : C) [has_binary_product X Y] [has_binary_product (functor.obj G X) (functor.obj G Y)] [preserves_limit (pair X Y) G] : is_iso (prod_comparison G X Y) :=\n  eq.mpr sorry (is_iso.of_iso (preserves_pair.iso G X Y))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/preserves/shapes/binary_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3744900375298139}}
{"text": "import logic.equiv.basic\nimport tactic.simp_result\n\nopen tactic\n\n-- Check that we can walk.\nexample : true :=\nby { simp_result { trivial } }\n\n-- Comparison without `dsimp_result`:\nexample : true :=\nbegin\n  exact (id trivial),\n  (do `(id trivial) \u2190 result, skip),\n  success_if_fail { (do `(trivial) \u2190 result, skip) },\nend\n\n-- Check that `dsimp_result` removes unnecessary `id`s.\nexample : true :=\nbegin\n  dsimp_result { exact (id trivial) },\n  success_if_fail { (do `(id trivial) \u2190 result, skip) },\n  (do `(trivial) \u2190 result, skip),\nend\n\n-- Comparison without `dsimp_result`:\nexample (a : \u2115) (b : list \u2115) (h : b.length < a) : \u2115 :=\nbegin\n  revert a,\n  intros a h,\n  exact 0,\n  (do `((\u03bb (h : list.length _ < _), 0) _) \u2190 result, skip),\n  success_if_fail { (do `(0) \u2190 result, skip) },\nend\n\n-- Check that `dsimp_result` does beta-reductions after `revert`.\nexample (a : \u2115) (b : list \u2115) (h : b.length < a) : \u2115 :=\nbegin\n  dsimp_result\n  { revert a,\n    intros a h,\n    exact 0, },\n  success_if_fail { (do `((\u03bb (h : list.length _ < _), 0) _) \u2190 result, skip), },\n  (do `(0) \u2190 result, skip),\nend\n\n-- This test tactic internally sets `pp.all ff`, and `pp.proofs tt`.\n-- This isn't very robust, as the user setting any other `pp` options\n-- will cause tests to break, but I don't think it needs to be.\nmeta def guard_result_pp (s : string) : tactic unit :=\ndo\n  o \u2190 get_options,\n  set_options ((o.set_bool `pp.all ff).set_bool `pp.proofs tt),\n  r \u2190 (to_string <$> (result >>= pp)),\n  guard (r = s) <|> fail format!\"result was {r} but expected {s}\"\n\n-- Comparison without `simp_result`:\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  exact e (e.symm (e a)),\n  guard_result_pp \"\u21d1e (\u21d1(equiv.symm e) (\u21d1e a))\",\nend\n\n-- Check that `simp_result` applies non-definitional simplifications to the result.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e a\",\nend\n\n-- Check that `simp_result only [...]` behaves as expected.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result only [equiv.apply_symm_apply] { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e a\",\nend\n\n-- Check that `simp_result only []` does not simplify.\n-- (Note the `simp_result` succeeds even if no simplification occurs.)\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result only [] { exact e (e.symm (e a)) },\n  guard_result_pp \"\u21d1e (\u21d1(equiv.symm e) (\u21d1e a))\",\nend\n\n-- Comparison without `simp_result`\nexample {\u03b1 : Type} (a b : \u03b1) (h : a = b) : \u2115 :=\nbegin\n  subst h,\n  exact 0,\n  guard_result_pp \"eq.rec 0 h\",\nend\n\n-- Check that we can remove `eq.rec` transports through constant families\n-- introduced by irrelevant use of `subst`.\nexample {\u03b1 : Type} (a b : \u03b1) (h : a = b) : \u2115 :=\nbegin\n  simp_result only [eq_rec_constant]\n  { subst h,\n    exact 0, },\n  guard_result_pp \"0\",\nend\n\n-- Check that `simp_result` performs simplifications on all results.\nexample : \u2115 \u00d7 \u2115 :=\nbegin\n  split,\n  simp_result\n  { exact id 0,\n    exact id 1, },\n  guard_result_pp \"(0, 1)\",\nend\n\n-- Check that `simp_result` can cope with incomplete goals.\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (a : \u03b1) : \u03b2 :=\nbegin\n  simp_result { apply e.to_fun, apply e.inv_fun, apply e.to_fun, },\n  guard_result_pp \"\u21d1e ?m_1\",\n  exact a,\nend\n\n-- Check that we can:\n-- * cope with metavariables in the result\n-- * perform beta redex after `revert`\n-- * simplify `eq.rec` after `subst`\nexample {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) (S : has_mul \u03b1) : has_mul \u03b2 :=\nbegin\n  fconstructor,\n  simp_result\n  { have mul := S.mul,\n    have e' := equiv.arrow_congr e (equiv.arrow_congr e e),\n    have h : mul = e'.symm (e' mul) := by simp,\n    revert h,\n    generalize : e' mul = mul',\n    intro h,\n    subst h, },\n  exact mul',\n  guard_result_pp \"{mul := \u21d1(equiv.arrow_congr e (equiv.arrow_congr e e)) has_mul.mul}\",\nend\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/test/simp_result.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.3744900321546227}}
{"text": "/-\nCopyright (c) 2016 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Leonardo de Moura\n\nTheorems that require decidability hypotheses are in the namespace \"decidable\".\nClassical versions are in the namespace \"classical\".\n\nNote: in the presence of automation, this whole file may be unnecessary. On the other hand,\nmaybe it is useful for writing automation.\n-/\nimport data.prod tactic.cache\n\n/-\n    miscellany\n\n    TODO: move elsewhere\n-/\n\nsection miscellany\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\ndef empty.elim {C : Sort*} : empty \u2192 C.\n\ninstance : subsingleton empty := \u27e8\u03bba, a.elim\u27e9\n\ninstance : decidable_eq empty := \u03bba, a.elim\n\n@[priority 0] instance decidable_eq_of_subsingleton\n  {\u03b1} [subsingleton \u03b1] : decidable_eq \u03b1\n| a b := is_true (subsingleton.elim a b)\n\n/- Add an instance to \"undo\" coercion transitivity into a chain of coercions, because\n   most simp lemmas are stated with respect to simple coercions and will not match when\n   part of a chain. -/\n@[simp] theorem coe_coe {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t \u03b2 \u03b3]\n  (a : \u03b1) : (a : \u03b3) = (a : \u03b2) := rfl\n\n@[simp] theorem coe_fn_coe_trans\n  {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_fun \u03b3]\n  (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n@[simp] theorem coe_fn_coe_base\n  {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_fun \u03b2]\n  (x : \u03b1) : @coe_fn \u03b1 _ x = @coe_fn \u03b2 _ x := rfl\n\n@[simp] theorem coe_sort_coe_trans\n  {\u03b1 \u03b2 \u03b3} [has_coe \u03b1 \u03b2] [has_coe_t_aux \u03b2 \u03b3] [has_coe_to_sort \u03b3]\n  (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n\n@[simp] theorem coe_sort_coe_base\n  {\u03b1 \u03b2} [has_coe \u03b1 \u03b2] [has_coe_to_sort \u03b2]\n  (x : \u03b1) : @coe_sort \u03b1 _ x = @coe_sort \u03b2 _ x := rfl\n\n/-- `pempty` is the universe-polymorphic analogue of `empty`. -/\n@[derive decidable_eq]\ninductive {u} pempty : Sort u\n\ndef pempty.elim {C : Sort*} : pempty \u2192 C.\n\ninstance subsingleton_pempty : subsingleton pempty := \u27e8\u03bba, a.elim\u27e9\n\nend miscellany\n\n/-\n    propositional connectives\n-/\n\n@[simp] theorem false_ne_true : false \u2260 true\n| h := h.symm \u25b8 trivial\n\nsection propositional\nvariables {a b c d : Prop}\n\n/- implies -/\n\ntheorem iff_of_eq (e : a = b) : a \u2194 b := e \u25b8 iff.rfl\n\ntheorem iff_iff_eq : (a \u2194 b) \u2194 a = b := \u27e8propext, iff_of_eq\u27e9\n\n@[simp] theorem imp_self : (a \u2192 a) \u2194 true := iff_true_intro id\n\ntheorem imp_intro {\u03b1 \u03b2} (h : \u03b1) (h\u2082 : \u03b2) : \u03b1 := h\n\ntheorem imp_false : (a \u2192 false) \u2194 \u00ac a := iff.rfl\n\ntheorem imp_and_distrib {\u03b1} : (\u03b1 \u2192 b \u2227 c) \u2194 (\u03b1 \u2192 b) \u2227 (\u03b1 \u2192 c) :=\n\u27e8\u03bb h, \u27e8\u03bb ha, (h ha).left, \u03bb ha, (h ha).right\u27e9,\n \u03bb h ha, \u27e8h.left ha, h.right ha\u27e9\u27e9\n\n@[simp] theorem and_imp : (a \u2227 b \u2192 c) \u2194 (a \u2192 b \u2192 c) :=\niff.intro (\u03bb h ha hb, h \u27e8ha, hb\u27e9) (\u03bb h \u27e8ha, hb\u27e9, h ha hb)\n\ntheorem iff_def : (a \u2194 b) \u2194 (a \u2192 b) \u2227 (b \u2192 a) :=\niff_iff_implies_and_implies _ _\n\ntheorem iff_def' : (a \u2194 b) \u2194 (b \u2192 a) \u2227 (a \u2192 b) :=\niff_def.trans and.comm\n\n@[simp] theorem imp_true_iff {\u03b1 : Sort*} : (\u03b1 \u2192 true) \u2194 true :=\niff_true_intro $ \u03bb_, trivial\n\n@[simp] theorem imp_iff_right (ha : a) : (a \u2192 b) \u2194 b :=\n\u27e8\u03bbf, f ha, imp_intro\u27e9\n\n/- not -/\n\ntheorem not.elim {\u03b1 : Sort*} (H1 : \u00aca) (H2 : a) : \u03b1 := absurd H2 H1\n\n@[reducible] theorem not.imp {a b : Prop} (H2 : \u00acb) (H1 : a \u2192 b) : \u00aca := mt H1 H2\n\ntheorem not_not_of_not_imp : \u00ac(a \u2192 b) \u2192 \u00ac\u00aca :=\nmt not.elim\n\ntheorem not_of_not_imp {\u03b1} : \u00ac(\u03b1 \u2192 b) \u2192 \u00acb :=\nmt imp_intro\n\ntheorem dec_em (p : Prop) [decidable p] : p \u2228 \u00acp := decidable.em p\n\ntheorem by_contradiction {p} [decidable p] : (\u00acp \u2192 false) \u2192 p :=\ndecidable.by_contradiction\n\n@[simp] theorem not_not [decidable a] : \u00ac\u00aca \u2194 a :=\niff.intro by_contradiction not_not_intro\n\ntheorem of_not_not [decidable a] : \u00ac\u00aca \u2192 a :=\nby_contradiction\n\ntheorem of_not_imp [decidable a] (h : \u00ac (a \u2192 b)) : a :=\nby_contradiction (not_not_of_not_imp h)\n\ntheorem not.imp_symm [decidable a] (h : \u00aca \u2192 b) (hb : \u00acb) : a :=\nby_contradiction $ hb \u2218 h\n\ntheorem not_imp_comm [decidable a] [decidable b] : (\u00aca \u2192 b) \u2194 (\u00acb \u2192 a) :=\n\u27e8not.imp_symm, not.imp_symm\u27e9\n\ntheorem imp.swap : (a \u2192 b \u2192 c) \u2194 (b \u2192 a \u2192 c) :=\n\u27e8function.swap, function.swap\u27e9\n\ntheorem imp_not_comm : (a \u2192 \u00acb) \u2194 (b \u2192 \u00aca) :=\nimp.swap\n\n/- and -/\n\ntheorem not_and_of_not_left (b : Prop) : \u00aca \u2192 \u00ac(a \u2227 b) :=\nmt and.left\n\ntheorem not_and_of_not_right (a : Prop) {b : Prop} : \u00acb \u2192 \u00ac(a \u2227 b) :=\nmt and.right\n\ntheorem and.imp_left (h : a \u2192 b) : a \u2227 c \u2192 b \u2227 c :=\nand.imp h id\n\ntheorem and.imp_right (h : a \u2192 b) : c \u2227 a \u2192 c \u2227 b :=\nand.imp id h\n\nlemma and.right_comm : (a \u2227 b) \u2227 c \u2194 (a \u2227 c) \u2227 b :=\nby simp [and.left_comm, and.comm]\n\nlemma and.rotate : a \u2227 b \u2227 c \u2194 b \u2227 c \u2227 a :=\nby simp [and.left_comm, and.comm]\n\ntheorem and_not_self_iff (a : Prop) : a \u2227 \u00ac a \u2194 false :=\niff.intro (assume h, (h.right) (h.left)) (assume h, h.elim)\n\ntheorem not_and_self_iff (a : Prop) : \u00ac a \u2227 a \u2194 false :=\niff.intro (assume \u27e8hna, ha\u27e9, hna ha) false.elim\n\ntheorem and_iff_left_of_imp {a b : Prop} (h : a \u2192 b) : (a \u2227 b) \u2194 a :=\niff.intro and.left (\u03bb ha, \u27e8ha, h ha\u27e9)\n\ntheorem and_iff_right_of_imp {a b : Prop} (h : b \u2192 a) : (a \u2227 b) \u2194 b :=\niff.intro and.right (\u03bb hb, \u27e8h hb, hb\u27e9)\n\nlemma and.congr_right_iff : (a \u2227 b \u2194 a \u2227 c) \u2194 (a \u2192 (b \u2194 c)) :=\n\u27e8\u03bb h ha, by simp [ha] at h; exact h, and_congr_right\u27e9\n\n/- or -/\n\ntheorem or_of_or_of_imp_of_imp (h\u2081 : a \u2228 b) (h\u2082 : a \u2192 c) (h\u2083 : b \u2192 d) : c \u2228 d :=\nor.imp h\u2082 h\u2083 h\u2081\n\ntheorem or_of_or_of_imp_left (h\u2081 : a \u2228 c) (h : a \u2192 b) : b \u2228 c :=\nor.imp_left h h\u2081\n\ntheorem or_of_or_of_imp_right (h\u2081 : c \u2228 a) (h : a \u2192 b) : c \u2228 b :=\nor.imp_right h h\u2081\n\ntheorem or.elim3 (h : a \u2228 b \u2228 c) (ha : a \u2192 d) (hb : b \u2192 d) (hc : c \u2192 d) : d :=\nor.elim h ha (assume h\u2082, or.elim h\u2082 hb hc)\n\ntheorem or_imp_distrib : (a \u2228 b \u2192 c) \u2194 (a \u2192 c) \u2227 (b \u2192 c) :=\n\u27e8assume h, \u27e8assume ha, h (or.inl ha), assume hb, h (or.inr hb)\u27e9,\n  assume \u27e8ha, hb\u27e9, or.rec ha hb\u27e9\n\ntheorem or_iff_not_imp_left [decidable a] : a \u2228 b \u2194 (\u00ac a \u2192 b) :=\n\u27e8or.resolve_left, \u03bb h, dite _ or.inl (or.inr \u2218 h)\u27e9\n\ntheorem or_iff_not_imp_right [decidable b] : a \u2228 b \u2194 (\u00ac b \u2192 a) :=\nor.comm.trans or_iff_not_imp_left\n\ntheorem not_imp_not [decidable a] : (\u00ac a \u2192 \u00ac b) \u2194 (b \u2192 a) :=\n\u27e8assume h hb, by_contradiction $ assume na, h na hb, mt\u27e9\n\n/- distributivity -/\n\ntheorem and_or_distrib_left : a \u2227 (b \u2228 c) \u2194 (a \u2227 b) \u2228 (a \u2227 c) :=\n\u27e8\u03bb \u27e8ha, hbc\u27e9, hbc.imp (and.intro ha) (and.intro ha),\n or.rec (and.imp_right or.inl) (and.imp_right or.inr)\u27e9\n\ntheorem or_and_distrib_right : (a \u2228 b) \u2227 c \u2194 (a \u2227 c) \u2228 (b \u2227 c) :=\n(and.comm.trans and_or_distrib_left).trans (or_congr and.comm and.comm)\n\ntheorem or_and_distrib_left : a \u2228 (b \u2227 c) \u2194 (a \u2228 b) \u2227 (a \u2228 c) :=\n\u27e8or.rec (\u03bbha, and.intro (or.inl ha) (or.inl ha)) (and.imp or.inr or.inr),\n and.rec $ or.rec (imp_intro \u2218 or.inl) (or.imp_right \u2218 and.intro)\u27e9\n\ntheorem and_or_distrib_right : (a \u2227 b) \u2228 c \u2194 (a \u2228 c) \u2227 (b \u2228 c) :=\n(or.comm.trans or_and_distrib_left).trans (and_congr or.comm or.comm)\n\n/- iff -/\n\ntheorem iff_of_true (ha : a) (hb : b) : a \u2194 b :=\n\u27e8\u03bb_, hb, \u03bb _, ha\u27e9\n\ntheorem iff_of_false (ha : \u00aca) (hb : \u00acb) : a \u2194 b :=\n\u27e8ha.elim, hb.elim\u27e9\n\ntheorem iff_true_left (ha : a) : (a \u2194 b) \u2194 b :=\n\u27e8\u03bb h, h.1 ha, iff_of_true ha\u27e9\n\ntheorem iff_true_right (ha : a) : (b \u2194 a) \u2194 b :=\niff.comm.trans (iff_true_left ha)\n\ntheorem iff_false_left (ha : \u00aca) : (a \u2194 b) \u2194 \u00acb :=\n\u27e8\u03bb h, mt h.2 ha, iff_of_false ha\u27e9\n\ntheorem iff_false_right (ha : \u00aca) : (b \u2194 a) \u2194 \u00acb :=\niff.comm.trans (iff_false_left ha)\n\ntheorem not_or_of_imp [decidable a] (h : a \u2192 b) : \u00ac a \u2228 b :=\nif ha : a then or.inr (h ha) else or.inl ha\n\ntheorem imp_iff_not_or [decidable a] : (a \u2192 b) \u2194 (\u00ac a \u2228 b) :=\n\u27e8not_or_of_imp, or.neg_resolve_left\u27e9\n\ntheorem imp_or_distrib [decidable a] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby simp [imp_iff_not_or, or.comm, or.left_comm]\n\ntheorem imp_or_distrib' [decidable b] : (a \u2192 b \u2228 c) \u2194 (a \u2192 b) \u2228 (a \u2192 c) :=\nby by_cases b; simp [h, or_iff_right_of_imp ((\u2218) false.elim)]\n\ntheorem not_imp_of_and_not : a \u2227 \u00ac b \u2192 \u00ac (a \u2192 b)\n| \u27e8ha, hb\u27e9 h := hb $ h ha\n\n@[simp] theorem not_imp [decidable a] : \u00ac(a \u2192 b) \u2194 a \u2227 \u00acb :=\n\u27e8\u03bb h, \u27e8of_not_imp h, not_of_not_imp h\u27e9, not_imp_of_and_not\u27e9\n\ntheorem peirce (a b : Prop) [decidable a] : ((a \u2192 b) \u2192 a) \u2192 a :=\nif ha : a then \u03bb h, ha else \u03bb h, h ha.elim\n\ntheorem peirce' {a : Prop} (H : \u2200 b : Prop, (a \u2192 b) \u2192 a) : a := H _ id\n\ntheorem not_iff_not [decidable a] [decidable b] : (\u00ac a \u2194 \u00ac b) \u2194 (a \u2194 b) :=\nby rw [@iff_def (\u00ac a), @iff_def' a]; exact and_congr not_imp_not not_imp_not\n\ntheorem not_iff_comm [decidable a] [decidable b] : (\u00ac a \u2194 b) \u2194 (\u00ac b \u2194 a) :=\nby rw [@iff_def (\u00ac a), @iff_def (\u00ac b)]; exact and_congr not_imp_comm imp_not_comm\n\ntheorem not_iff [decidable a] [decidable b] : \u00ac (a \u2194 b) \u2194 (\u00ac a \u2194 b) :=\nby split; intro h; [split, skip]; intro h'; [by_contradiction,intro,skip];\n   try { refine h _; simp [*] }; rw [h',not_iff_self] at h; exact h\n\ntheorem iff_not_comm [decidable a] [decidable b] : (a \u2194 \u00ac b) \u2194 (b \u2194 \u00ac a) :=\nby rw [@iff_def a, @iff_def b]; exact and_congr imp_not_comm not_imp_comm\n\ntheorem iff_iff_and_or_not_and_not [decidable b] : (a \u2194 b) \u2194 (a \u2227 b) \u2228 (\u00ac a \u2227 \u00ac b) :=\nby { split; intro h,\n     { rw h; by_cases b; [left,right]; split; assumption },\n     { cases h with h h; cases h; split; intro; { contradiction <|> assumption } } }\n\n@[simp] theorem not_and_not_right [decidable b] : \u00ac(a \u2227 \u00acb) \u2194 (a \u2192 b) :=\n\u27e8\u03bb h ha, h.imp_symm $ and.intro ha, \u03bb h \u27e8ha, hb\u27e9, hb $ h ha\u27e9\n\n@[inline] def decidable_of_iff (a : Prop) (h : a \u2194 b) [D : decidable a] : decidable b :=\ndecidable_of_decidable_of_iff D h\n\n@[inline] def decidable_of_iff' (b : Prop) (h : a \u2194 b) [D : decidable b] : decidable a :=\ndecidable_of_decidable_of_iff D h.symm\n\ndef decidable_of_bool : \u2200 (b : bool) (h : b \u2194 a), decidable a\n| tt h := is_true (h.1 rfl)\n| ff h := is_false (mt h.2 bool.ff_ne_tt)\n\n/- de morgan's laws -/\n\ntheorem not_and_of_not_or_not (h : \u00ac a \u2228 \u00ac b) : \u00ac (a \u2227 b)\n| \u27e8ha, hb\u27e9 := or.elim h (absurd ha) (absurd hb)\n\ntheorem not_and_distrib [decidable a] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if ha : a then or.inr (\u03bb hb, h \u27e8ha, hb\u27e9) else or.inl ha, not_and_of_not_or_not\u27e9\n\ntheorem not_and_distrib' [decidable b] : \u00ac (a \u2227 b) \u2194 \u00aca \u2228 \u00acb :=\n\u27e8\u03bb h, if hb : b then or.inl (\u03bb ha, h \u27e8ha, hb\u27e9) else or.inr hb, not_and_of_not_or_not\u27e9\n\n@[simp] theorem not_and : \u00ac (a \u2227 b) \u2194 (a \u2192 \u00ac b) := and_imp\n\ntheorem not_and' : \u00ac (a \u2227 b) \u2194 b \u2192 \u00aca :=\nnot_and.trans imp_not_comm\n\ntheorem not_or_distrib : \u00ac (a \u2228 b) \u2194 \u00ac a \u2227 \u00ac b :=\n\u27e8\u03bb h, \u27e8\u03bb ha, h (or.inl ha), \u03bb hb, h (or.inr hb)\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9 h, or.elim h h\u2081 h\u2082\u27e9\n\ntheorem or_iff_not_and_not [decidable a] [decidable b] : a \u2228 b \u2194 \u00ac (\u00aca \u2227 \u00acb) :=\nby rw [\u2190 not_or_distrib, not_not]\n\ntheorem and_iff_not_or_not [decidable a] [decidable b] : a \u2227 b \u2194 \u00ac (\u00ac a \u2228 \u00ac b) :=\nby rw [\u2190 not_and_distrib, not_not]\n\nend propositional\n\n/- equality -/\n\nsection equality\nvariables {\u03b1 : Sort*} {a b : \u03b1}\n\n@[simp] theorem heq_iff_eq : a == b \u2194 a = b :=\n\u27e8eq_of_heq, heq_of_eq\u27e9\n\ntheorem proof_irrel_heq {p q : Prop} (hp : p) (hq : q) : hp == hq :=\nhave p = q, from propext \u27e8\u03bb _, hq, \u03bb _, hp\u27e9,\nby subst q; refl\n\ntheorem ne_of_mem_of_not_mem {\u03b1 \u03b2} [has_mem \u03b1 \u03b2] {s : \u03b2} {a b : \u03b1}\n  (h : a \u2208 s) : b \u2209 s \u2192 a \u2260 b :=\nmt $ \u03bb e, e \u25b8 h\n\ntheorem eq_equivalence : equivalence (@eq \u03b1) :=\n\u27e8eq.refl, @eq.symm _, @eq.trans _\u27e9\n\nlemma heq_of_eq_mp :\n  \u2200 {\u03b1 \u03b2 : Sort*} {a : \u03b1} {a' : \u03b2} (e : \u03b1 = \u03b2) (h\u2082 : (eq.mp e a) = a'), a == a'\n| \u03b1 ._ a a' rfl h := eq.rec_on h (heq.refl _)\n\nlemma rec_heq_of_heq {\u03b2} {C : \u03b1 \u2192 Sort*} {x : C a} {y : \u03b2} (eq : a = b) (h : x == y) :\n  @eq.rec \u03b1 a C x b eq == y :=\nby subst eq; exact h\n\nend equality\n\n/-\n  quantifiers\n-/\n\nsection quantifiers\nvariables {\u03b1 : Sort*} {p q : \u03b1 \u2192 Prop} {b : Prop}\n\ndef Exists.imp := @exists_imp_exists\n\ntheorem forall_swap {\u03b1 \u03b2} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2200 x y, p x y) \u2194 \u2200 y x, p x y :=\n\u27e8function.swap, function.swap\u27e9\n\ntheorem exists_swap {\u03b1 \u03b2} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} : (\u2203 x y, p x y) \u2194 \u2203 y x, p x y :=\n\u27e8\u03bb \u27e8x, y, h\u27e9, \u27e8y, x, h\u27e9, \u03bb \u27e8y, x, h\u27e9, \u27e8x, y, h\u27e9\u27e9\n\n@[simp] theorem exists_imp_distrib : ((\u2203 x, p x) \u2192 b) \u2194 \u2200 x, p x \u2192 b :=\n\u27e8\u03bb h x hpx, h \u27e8x, hpx\u27e9, \u03bb h \u27e8x, hpx\u27e9, h x hpx\u27e9\n\n--theorem forall_not_of_not_exists (h : \u00ac \u2203 x, p x) : \u2200 x, \u00ac p x :=\n--forall_imp_of_exists_imp h\n\ntheorem not_exists_of_forall_not (h : \u2200 x, \u00ac p x) : \u00ac \u2203 x, p x :=\nexists_imp_distrib.2 h\n\n@[simp] theorem not_exists : (\u00ac \u2203 x, p x) \u2194 \u2200 x, \u00ac p x :=\nexists_imp_distrib\n\ntheorem not_forall_of_exists_not : (\u2203 x, \u00ac p x) \u2192 \u00ac \u2200 x, p x\n| \u27e8x, hn\u27e9 h := hn (h x)\n\ntheorem not_forall {p : \u03b1 \u2192 Prop}\n    [decidable (\u2203 x, \u00ac p x)] [\u2200 x, decidable (p x)] :\n  (\u00ac \u2200 x, p x) \u2194 \u2203 x, \u00ac p x :=\n\u27e8not.imp_symm $ \u03bb nx x, nx.imp_symm $ \u03bb h, \u27e8x, h\u27e9,\n not_forall_of_exists_not\u27e9\n\n@[simp] theorem not_forall_not [decidable (\u2203 x, p x)] :\n  (\u00ac \u2200 x, \u00ac p x) \u2194 \u2203 x, p x :=\nby haveI := decidable_of_iff (\u00ac \u2203 x, p x) not_exists;\n   exact not_iff_comm.1 not_exists\n\n@[simp] theorem not_exists_not [\u2200 x, decidable (p x)] :\n  (\u00ac \u2203 x, \u00ac p x) \u2194 \u2200 x, p x :=\nby simp\n\n@[simp] theorem forall_true_iff : (\u03b1 \u2192 true) \u2194 true :=\niff_true_intro (\u03bb _, trivial)\n\n-- Unfortunately this causes simp to loop sometimes, so we\n-- add the 2 and 3 cases as simp lemmas instead\ntheorem forall_true_iff' (h : \u2200 a, p a \u2194 true) : (\u2200 a, p a) \u2194 true :=\niff_true_intro (\u03bb _, of_iff_true (h _))\n\n@[simp] theorem forall_2_true_iff {\u03b2 : \u03b1 \u2192 Sort*} : (\u2200 a, \u03b2 a \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_true_iff\n\n@[simp] theorem forall_3_true_iff {\u03b2 : \u03b1 \u2192 Sort*} {\u03b3 : \u03a0 a, \u03b2 a \u2192 Sort*} :\n  (\u2200 a (b : \u03b2 a), \u03b3 a b \u2192 true) \u2194 true :=\nforall_true_iff' $ \u03bb _, forall_2_true_iff\n\n@[simp] theorem forall_const (\u03b1 : Sort*) [inhabited \u03b1] : (\u03b1 \u2192 b) \u2194 b :=\n\u27e8\u03bb h, h (arbitrary \u03b1), \u03bb hb x, hb\u27e9\n\n@[simp] theorem exists_const (\u03b1 : Sort*) [inhabited \u03b1] : (\u2203 x : \u03b1, b) \u2194 b :=\n\u27e8\u03bb \u27e8x, h\u27e9, h, \u03bb h, \u27e8arbitrary \u03b1, h\u27e9\u27e9\n\ntheorem forall_and_distrib : (\u2200 x, p x \u2227 q x) \u2194 (\u2200 x, p x) \u2227 (\u2200 x, q x) :=\n\u27e8\u03bb h, \u27e8\u03bb x, (h x).left, \u03bb x, (h x).right\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9 x, \u27e8h\u2081 x, h\u2082 x\u27e9\u27e9\n\ntheorem exists_or_distrib : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\n\u27e8\u03bb \u27e8x, hpq\u27e9, hpq.elim (\u03bb hpx, or.inl \u27e8x, hpx\u27e9) (\u03bb hqx, or.inr \u27e8x, hqx\u27e9),\n \u03bb hepq, hepq.elim (\u03bb \u27e8x, hpx\u27e9, \u27e8x, or.inl hpx\u27e9) (\u03bb \u27e8x, hqx\u27e9, \u27e8x, or.inr hqx\u27e9)\u27e9\n\n@[simp] theorem exists_and_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, q \u2227 p x) \u2194 q \u2227 (\u2203x, p x) :=\n\u27e8\u03bb \u27e8x, hq, hp\u27e9, \u27e8hq, x, hp\u27e9, \u03bb \u27e8hq, x, hp\u27e9, \u27e8x, hq, hp\u27e9\u27e9\n\n@[simp] theorem exists_and_distrib_right {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2203x, p x \u2227 q) \u2194 (\u2203x, p x) \u2227 q :=\nby simp [and_comm]\n\n@[simp] theorem forall_eq {a' : \u03b1} : (\u2200a, a = a' \u2192 p a) \u2194 p a' :=\n\u27e8\u03bb h, h a' rfl, \u03bb h a e, e.symm \u25b8 h\u27e9\n\n@[simp] theorem exists_eq {a' : \u03b1} : \u2203 a, a = a' := \u27e8_, rfl\u27e9\n\n@[simp] theorem exists_eq_left {a' : \u03b1} : (\u2203 a, a = a' \u2227 p a) \u2194 p a' :=\n\u27e8\u03bb \u27e8a, e, h\u27e9, e \u25b8 h, \u03bb h, \u27e8_, rfl, h\u27e9\u27e9\n\n@[simp] theorem exists_eq_right {a' : \u03b1} : (\u2203 a, p a \u2227 a = a') \u2194 p a' :=\n(exists_congr $ by exact \u03bb a, and.comm).trans exists_eq_left\n\n@[simp] theorem forall_eq' {a' : \u03b1} : (\u2200a, a' = a \u2192 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_left' {a' : \u03b1} : (\u2203 a, a' = a \u2227 p a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\n@[simp] theorem exists_eq_right' {a' : \u03b1} : (\u2203 a, p a \u2227 a' = a) \u2194 p a' :=\nby simp [@eq_comm _ a']\n\ntheorem forall_or_of_or_forall (h : b \u2228 \u2200x, p x) (x) : b \u2228 p x :=\nh.imp_right $ \u03bb h\u2082, h\u2082 x\n\ntheorem forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} [decidable q] :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) :=\n\u27e8\u03bb h, if hq : q then or.inl hq else or.inr $ \u03bb x, (h x).resolve_left hq,\n  forall_or_of_or_forall\u27e9\n\n@[simp] theorem exists_prop {p q : Prop} : (\u2203 h : p, q) \u2194 p \u2227 q :=\n\u27e8\u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, \u27e8h\u2081, h\u2082\u27e9\u27e9\n\n@[simp] theorem exists_false : \u00ac (\u2203a:\u03b1, false) := assume \u27e8a, h\u27e9, h\n\ntheorem Exists.fst {p : b \u2192 Prop} : Exists p \u2192 b\n| \u27e8h, _\u27e9 := h\n\ntheorem Exists.snd {p : b \u2192 Prop} : \u2200 h : Exists p, p h.fst\n| \u27e8_, h\u27e9 := h\n\n@[simp] theorem forall_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2200 h' : p, q h') \u2194 q h :=\n@forall_const (q h) p \u27e8h\u27e9\n\n@[simp] theorem exists_prop_of_true {p : Prop} {q : p \u2192 Prop} (h : p) : (\u2203 h' : p, q h') \u2194 q h :=\n@exists_const (q h) p \u27e8h\u27e9\n\n@[simp] theorem forall_prop_of_false {p : Prop} {q : p \u2192 Prop} (hn : \u00ac p) : (\u2200 h' : p, q h') \u2194 true :=\niff_true_intro $ \u03bb h, hn.elim h\n\n@[simp] theorem exists_prop_of_false {p : Prop} {q : p \u2192 Prop} : \u00ac p \u2192 \u00ac (\u2203 h' : p, q h') :=\nmt Exists.fst\n\nend quantifiers\n\n/- classical versions -/\n\nnamespace classical\nvariables {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop}\n\nlocal attribute [instance] prop_decidable\n\nprotected theorem not_forall : (\u00ac \u2200 x, p x) \u2194 (\u2203 x, \u00ac p x) := not_forall\n\nprotected theorem forall_or_distrib_left {q : Prop} {p : \u03b1 \u2192 Prop} :\n  (\u2200x, q \u2228 p x) \u2194 q \u2228 (\u2200x, p x) :=\nforall_or_distrib_left\n\ntheorem cases {p : Prop \u2192 Prop} (h1 : p true) (h2 : p false) : \u2200a, p a :=\nassume a, cases_on a h1 h2\n\ntheorem or_not {p : Prop} : p \u2228 \u00ac p :=\nby_cases or.inl or.inr\n\nprotected theorem or_iff_not_imp_left {p q : Prop} : p \u2228 q \u2194 (\u00ac p \u2192 q) :=\nor_iff_not_imp_left\n\nprotected theorem or_iff_not_imp_right {p q : Prop} : q \u2228 p \u2194 (\u00ac p \u2192 q) :=\nor_iff_not_imp_right\n\n/- use shortened names to avoid conflict when classical namespace is open -/\nnoncomputable theorem dec (p : Prop) : decidable p := by apply_instance\nnoncomputable theorem dec_pred (p : \u03b1 \u2192 Prop) : decidable_pred p := by apply_instance\nnoncomputable theorem dec_rel (p : \u03b1 \u2192 \u03b1 \u2192 Prop) : decidable_rel p := by apply_instance\nnoncomputable theorem dec_eq (\u03b1 : Sort*) : decidable_eq \u03b1 := by apply_instance\n\n@[elab_as_eliminator]\nnoncomputable def {u} rec_on {C : Sort u} (h : \u2203 a, p a) (H : \u2200 a, p a \u2192 C) : C :=\nH (classical.some h) (classical.some_spec h)\n\nlemma some_spec2 {\u03b1 : Type*} {p : \u03b1 \u2192 Prop} {h : \u2203a, p a}\n  (q : \u03b1 \u2192 Prop) (hpq : \u2200a, p a \u2192 q a) : q (some h) :=\nhpq _ $ some_spec _\n\nend classical\n\n/-\n   bounded quantifiers\n-/\n\nsection bounded_quantifiers\nvariables {\u03b1 : Sort*} {r p q : \u03b1 \u2192 Prop} {P Q : \u2200 x, p x \u2192 Prop} {b : Prop}\n\ntheorem bex_def : (\u2203 x (h : p x), q x) \u2194 \u2203 x, p x \u2227 q x :=\n\u27e8\u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9, \u03bb \u27e8x, px, qx\u27e9, \u27e8x, px, qx\u27e9\u27e9\n\ntheorem bex.elim {b : Prop} : (\u2203 x h, P x h) \u2192 (\u2200 a h, P a h \u2192 b) \u2192 b\n| \u27e8a, h\u2081, h\u2082\u27e9 h' := h' a h\u2081 h\u2082\n\ntheorem bex.intro (a : \u03b1) (h\u2081 : p a) (h\u2082 : P a h\u2081) : \u2203 x (h : p x), P x h :=\n\u27e8a, h\u2081, h\u2082\u27e9\n\ntheorem ball_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2200 x h, P x h) \u2194 (\u2200 x h, Q x h) :=\nforall_congr $ \u03bb x, forall_congr (H x)\n\ntheorem bex_congr (H : \u2200 x h, P x h \u2194 Q x h) :\n  (\u2203 x h, P x h) \u2194 (\u2203 x h, Q x h) :=\nexists_congr $ \u03bb x, exists_congr (H x)\n\ntheorem ball.imp_right (H : \u2200 x h, (P x h \u2192 Q x h))\n  (h\u2081 : \u2200 x h, P x h) (x h) : Q x h :=\nH _ _ $ h\u2081 _ _\n\ntheorem bex.imp_right (H : \u2200 x h, (P x h \u2192 Q x h)) :\n  (\u2203 x h, P x h) \u2192 \u2203 x h, Q x h\n| \u27e8x, h, h'\u27e9 := \u27e8_, _, H _ _ h'\u27e9\n\ntheorem ball.imp_left (H : \u2200 x, p x \u2192 q x)\n  (h\u2081 : \u2200 x, q x \u2192 r x) (x) (h : p x) : r x :=\nh\u2081 _ $ H _ h\n\ntheorem bex.imp_left (H : \u2200 x, p x \u2192 q x) :\n  (\u2203 x (_ : p x), r x) \u2192 \u2203 x (_ : q x), r x\n| \u27e8x, hp, hr\u27e9 := \u27e8x, H _ hp, hr\u27e9\n\ntheorem ball_of_forall (h : \u2200 x, p x) (x) (_ : q x) : p x :=\nh x\n\ntheorem forall_of_ball (H : \u2200 x, p x) (h : \u2200 x, p x \u2192 q x) (x) : q x :=\nh x $ H x\n\ntheorem bex_of_exists (H : \u2200 x, p x) : (\u2203 x, q x) \u2192 \u2203 x (_ : p x), q x\n| \u27e8x, hq\u27e9 := \u27e8x, H x, hq\u27e9\n\ntheorem exists_of_bex : (\u2203 x (_ : p x), q x) \u2192 \u2203 x, q x\n| \u27e8x, _, hq\u27e9 := \u27e8x, hq\u27e9\n\n@[simp] theorem bex_imp_distrib : ((\u2203 x h, P x h) \u2192 b) \u2194 (\u2200 x h, P x h \u2192 b) :=\nby simp\n\ntheorem not_bex : (\u00ac \u2203 x h, P x h) \u2194 \u2200 x h, \u00ac P x h :=\nbex_imp_distrib\n\ntheorem not_ball_of_bex_not : (\u2203 x h, \u00ac P x h) \u2192 \u00ac \u2200 x h, P x h\n| \u27e8x, h, hp\u27e9 al := hp $ al x h\n\ntheorem not_ball [decidable (\u2203 x h, \u00ac P x h)] [\u2200 x h, decidable (P x h)] :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) :=\n\u27e8not.imp_symm $ \u03bb nx x h, nx.imp_symm $ \u03bb h', \u27e8x, h, h'\u27e9,\n not_ball_of_bex_not\u27e9\n\ntheorem ball_true_iff (p : \u03b1 \u2192 Prop) : (\u2200 x, p x \u2192 true) \u2194 true :=\niff_true_intro (\u03bb h hrx, trivial)\n\ntheorem ball_and_distrib : (\u2200 x h, P x h \u2227 Q x h) \u2194 (\u2200 x h, P x h) \u2227 (\u2200 x h, Q x h) :=\niff.trans (forall_congr $ \u03bb x, forall_and_distrib) forall_and_distrib\n\ntheorem bex_or_distrib : (\u2203 x h, P x h \u2228 Q x h) \u2194 (\u2203 x h, P x h) \u2228 (\u2203 x h, Q x h) :=\niff.trans (exists_congr $ \u03bb x, exists_or_distrib) exists_or_distrib\n\nend bounded_quantifiers\n\nnamespace classical\nlocal attribute [instance] prop_decidable\n\ntheorem not_ball {\u03b1 : Sort*} {p : \u03b1 \u2192 Prop} {P : \u03a0 (x : \u03b1), p x \u2192 Prop} :\n  (\u00ac \u2200 x h, P x h) \u2194 (\u2203 x h, \u00ac P x h) := _root_.not_ball\n\nend classical\n\nsection nonempty\nuniverses u v w\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : \u03b1 \u2192 Type w}\n\nattribute [simp] nonempty_of_inhabited\n\nlemma exists_true_iff_nonempty {\u03b1 : Sort*} : (\u2203a:\u03b1, true) \u2194 nonempty \u03b1 :=\niff.intro (\u03bb\u27e8a, _\u27e9, \u27e8a\u27e9) (\u03bb\u27e8a\u27e9, \u27e8a, trivial\u27e9)\n\n@[simp] lemma nonempty_Prop {p : Prop} : nonempty p \u2194 p :=\niff.intro (assume \u27e8h\u27e9, h) (assume h, \u27e8h\u27e9)\n\nlemma not_nonempty_iff_imp_false {p : Prop} : \u00ac nonempty \u03b1 \u2194 \u03b1 \u2192 false :=\n\u27e8\u03bb h a, h \u27e8a\u27e9, \u03bb h \u27e8a\u27e9, h a\u27e9\n\n@[simp] lemma nonempty_sigma : nonempty (\u03a3a:\u03b1, \u03b3 a) \u2194 (\u2203a:\u03b1, nonempty (\u03b3 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_subtype {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} : nonempty (subtype p) \u2194 (\u2203a:\u03b1, p a) :=\niff.intro (assume \u27e8\u27e8a, h\u27e9\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a, h\u27e9\u27e9)\n\n@[simp] lemma nonempty_prod : nonempty (\u03b1 \u00d7 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_pprod {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (pprod \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2227 nonempty \u03b2) :=\niff.intro (assume \u27e8\u27e8a, b\u27e9\u27e9, \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9) (assume \u27e8\u27e8a\u27e9, \u27e8b\u27e9\u27e9, \u27e8\u27e8a, b\u27e9\u27e9)\n\n@[simp] lemma nonempty_sum : nonempty (\u03b1 \u2295 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with sum.inl a := or.inl \u27e8a\u27e9 | sum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8sum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8sum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psum {\u03b1 : Sort u} {\u03b2 : Sort v} :\n  nonempty (psum \u03b1 \u03b2) \u2194 (nonempty \u03b1 \u2228 nonempty \u03b2) :=\niff.intro\n  (assume \u27e8h\u27e9, match h with psum.inl a := or.inl \u27e8a\u27e9 | psum.inr b := or.inr \u27e8b\u27e9 end)\n  (assume h, match h with or.inl \u27e8a\u27e9 := \u27e8psum.inl a\u27e9 | or.inr \u27e8b\u27e9 := \u27e8psum.inr b\u27e9 end)\n\n@[simp] lemma nonempty_psigma {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (psigma \u03b2) \u2194 (\u2203a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8\u27e8a, c\u27e9\u27e9, \u27e8a, \u27e8c\u27e9\u27e9) (assume \u27e8a, \u27e8c\u27e9\u27e9, \u27e8\u27e8a, c\u27e9\u27e9)\n\n@[simp] lemma nonempty_empty : \u00ac nonempty empty :=\nassume \u27e8h\u27e9, h.elim\n\n@[simp] lemma nonempty_ulift : nonempty (ulift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty_plift {\u03b1 : Sort u} : nonempty (plift \u03b1) \u2194 nonempty \u03b1 :=\niff.intro (assume \u27e8\u27e8a\u27e9\u27e9, \u27e8a\u27e9) (assume \u27e8a\u27e9, \u27e8\u27e8a\u27e9\u27e9)\n\n@[simp] lemma nonempty.forall {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2200h:nonempty \u03b1, p h) \u2194 (\u2200a, p \u27e8a\u27e9) :=\niff.intro (assume h a, h _) (assume h \u27e8a\u27e9, h _)\n\n@[simp] lemma nonempty.exists {\u03b1 : Sort u} {p : nonempty \u03b1 \u2192 Prop} :\n  (\u2203h:nonempty \u03b1, p h) \u2194 (\u2203a, p \u27e8a\u27e9) :=\niff.intro (assume \u27e8\u27e8a\u27e9, h\u27e9, \u27e8a, h\u27e9) (assume \u27e8a, h\u27e9, \u27e8\u27e8a\u27e9, h\u27e9)\n\nlemma classical.nonempty_pi {\u03b1 : Sort u} {\u03b2 : \u03b1 \u2192 Sort v} :\n  nonempty (\u03a0a:\u03b1, \u03b2 a) \u2194 (\u2200a:\u03b1, nonempty (\u03b2 a)) :=\niff.intro (assume \u27e8f\u27e9 a, \u27e8f a\u27e9) (assume f, \u27e8assume a, classical.choice $ f a\u27e9)\n\nend nonempty\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/logic/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.7154239957834734, "lm_q1q2_score": 0.3744674775104049}}
{"text": "import .pasting_pushouts\n\nopen category_theory.category\nlocal notation f ` \u2218 `:80 g:80 := g \u226b f\n\n/-\n\n       a\u2081 \u2192 b\u2082         a\u2081 \u2192 b\u2082\n       \u2193               \u2193    \u2193\n  a\u2080 \u2192 b\u2081   \u2193     a\u2080 \u2192 b\u2081 \u2192 c\u2081\n  \u2193    \u2193          \u2193         \u2193\n  b\u2080 \u2192 c\u2080 \u2192 d     b\u2080   \u2192    d'\n\n-/\n\nuniverses v u\n\nnamespace category_theory\n\nsection\n\nparameters {C : Type u} [category.{v} C]\n\nparameters {a\u2080 a\u2081 b\u2080 b\u2081 b\u2082 c\u2080 c\u2081 d d' : C}\nparameters {f\u2080 : a\u2080 \u27f6 b\u2080} {f\u2081 : a\u2080 \u27f6 b\u2081} {f\u2082 : a\u2081 \u27f6 b\u2081} {f\u2083 : a\u2081 \u27f6 b\u2082}\nparameters {g\u2080 : b\u2080 \u27f6 c\u2080} {g\u2081 : b\u2081 \u27f6 c\u2080} {g\u2082 : b\u2081 \u27f6 c\u2081} {g\u2083 : b\u2082 \u27f6 c\u2081}\nparameters {h\u2080 : c\u2080 \u27f6 d} {h\u2081 : b\u2082 \u27f6 d} {h\u2080' : b\u2080 \u27f6 d'} {h\u2081' : c\u2081 \u27f6 d'}\nparameters\n  (po\u2080 : Is_pushout f\u2080 f\u2081 g\u2080 g\u2081)\n  (po\u2081 : Is_pushout (g\u2081 \u2218 f\u2082) f\u2083 h\u2080 h\u2081)\n  (po\u2082 : Is_pushout f\u2082 f\u2083 g\u2082 g\u2083)\n  (po\u2083 : Is_pushout f\u2080 (g\u2082 \u2218 f\u2081) h\u2080' h\u2081')\ninclude po\u2080 po\u2081 po\u2082 po\u2083\n\ndef Is_pushout_assoc : d \u2245 d' :=\nbegin\n  refine {\n    hom := po\u2081.induced (po\u2080.induced h\u2080' (h\u2081' \u2218 g\u2082) _) (h\u2081' \u2218 g\u2083) _,\n    inv := po\u2083.induced (h\u2080 \u2218 g\u2080) (po\u2082.induced (h\u2080 \u2218 g\u2081) h\u2081 _) _,\n    hom_inv_id' := _,\n    inv_hom_id' := _,\n  },\n  { rw po\u2083.commutes, simp },\n  -- TODO: Is_pushout.commutes_assoc\n  { simp, rw [\u2190assoc, \u2190assoc], rw po\u2082.commutes },\n  { rw [\u2190assoc, po\u2081.commutes] },\n  { simp, rw [\u2190assoc, \u2190assoc], rw po\u2080.commutes },\n  { apply po\u2081.uniqueness; rw \u2190assoc; simp,\n    { apply po\u2080.uniqueness; rw \u2190assoc; simp } },\n  { apply po\u2083.uniqueness; rw \u2190assoc; simp,\n    { apply po\u2082.uniqueness; rw \u2190assoc; simp } }\nend\n\n@[simp] lemma Is_pushout_assoc_i\u2080 : Is_pushout_assoc.hom \u2218 h\u2080 \u2218 g\u2080 = h\u2080' :=\nby change Is_pushout.induced _ _ _ _ \u2218 _ \u2218 _ = _; simp\n\n@[simp] lemma Is_pushout_assoc_i\u2081 : Is_pushout_assoc.hom \u2218 h\u2081 = h\u2081' \u2218 g\u2083 :=\nby change Is_pushout.induced _ _ _ _ \u2218 _ = _ \u2218 _; simp\n\nparameters {x : C} {k : d \u27f6 x} {k' : d' \u27f6 x}\n\nlemma Is_pushout_assoc_uniqueness\n  (hk\u2080 : k \u2218 h\u2080 \u2218 g\u2080 = k' \u2218 h\u2080')\n  (hk\u2081 : k \u2218 h\u2080 \u2218 g\u2081 = k' \u2218 h\u2081' \u2218 g\u2082)\n  (hk\u2082 : k \u2218 h\u2081 = k' \u2218 h\u2081' \u2218 g\u2083) :\n  k = k' \u2218 Is_pushout_assoc.hom :=\nbegin\n  change k = k' \u2218 Is_pushout.induced _ _ _ _,\n  apply po\u2081.uniqueness; rw \u2190assoc,\n  { apply po\u2080.uniqueness; conv { to_rhs, rw \u2190assoc },\n    { rw hk\u2080, simp },\n    { rw hk\u2081, simp } },\n  { rw hk\u2082, simp }\nend\n\nend\n\nend category_theory\n", "meta": {"author": "rwbarton", "repo": "lean-homotopy-theory", "sha": "39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee", "save_path": "github-repos/lean/rwbarton-lean-homotopy-theory", "path": "github-repos/lean/rwbarton-lean-homotopy-theory/lean-homotopy-theory-39e1b4ea1ed1b0eca2f68bc64162dde6a6396dee/src/category_theory/assoc_pushouts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834732, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.37446747751040477}}
{"text": "import for_mathlib.snake_lemma\n\nnamespace category_theory\n\nlocal notation x `\u27f6[`D`]` y := D.map (snake_diagram.hom x y)\n\nnamespace snake_lemma\n\nopen category_theory.limits\n\nuniverses v u\nvariables {A : Type u} [category.{v} A] [abelian A]\n  {F G : snake_diagram \u2964 A} (\u03b7 : F \u27f6 G)\n\nnamespace \u03b4_natural_setup\n\n@[reassoc]\nlemma aux1 (hF : is_snake_input F) (hG : is_snake_input G) :\n  \u03b7.app (0, 2) \u226b hG.to_kernel = hF.to_kernel \u226b\n  kernel.lift _ (kernel.\u03b9 _ \u226b \u03b7.app _) begin\n    simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n  end :=\nbegin\n  apply equalizer.hom_ext,\n  dsimp [is_snake_input.to_kernel],\n  simp,\nend\n\n@[reassoc]\nlemma aux2 (hF : is_snake_input F) (hG : is_snake_input G) :\n  kernel.lift ((1, 2) \u27f6[G] (2, 2)) (kernel.\u03b9 ((1, 2) \u27f6[F] (2, 2)) \u226b\n    \u03b7.app (1, 2)) begin\n      simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n    end \u226b inv hG.cokernel_to_top_right_kernel_to_right_kernel =\n  inv hF.cokernel_to_top_right_kernel_to_right_kernel \u226b\n    cokernel.desc _ (kernel.lift _ (kernel.\u03b9 _ \u226b \u03b7.app _) begin\n      simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n    end \u226b cokernel.\u03c0 _) begin\n      dsimp [is_snake_input.to_top_right_kernel],\n      simp only [\u2190 category.assoc], let t := _, change t \u226b _ = _,\n      have ht : t = \u03b7.app _ \u226b kernel.lift ((1,1) \u27f6[G] (2,2)) ((1,0) \u27f6[G] (1,1)) _,\n      { apply equalizer.hom_ext, simp, },\n      rw [ht, category.assoc, cokernel.condition, comp_zero],\n    end :=\nbegin\n  simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n  dsimp [is_snake_input.cokernel_to_top_right_kernel_to_right_kernel],\n  apply equalizer.hom_ext,\n  simp only [le_refl, and_true, category.assoc, nat_trans.naturality,\n    kernel.condition_assoc, zero_comp, true_and,\n    cokernel.condition, comp_zero, equalizer_as_kernel, kernel.lift_\u03b9],\n  apply coequalizer.hom_ext,\n  simp only [category.assoc, nat_trans.naturality, cokernel.\u03c0_desc_assoc,\n    kernel.lift_\u03b9_assoc, kernel.lift_\u03b9],\nend\n\n@[reassoc]\nlemma aux3 (hF : is_snake_input F) (hG : is_snake_input G) :\n  kernel.lift hG.bottom_left_cokernel_to (kernel.\u03b9 hF.bottom_left_cokernel_to \u226b\n    cokernel.desc ((1, 0) \u27f6[F] (2, 1)) (\u03b7.app (2, 1) \u226b\n    cokernel.\u03c0 ((1, 0) \u27f6[G] (2, 1))) begin\n      simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n    end) begin\n      dsimp [is_snake_input.bottom_left_cokernel_to],\n      simp only [category.assoc], let t := _, change _ \u226b t = _,\n      have ht : t = cokernel.desc ((1,0) \u27f6[F] (2,1)) ((2,1) \u27f6[F] (2,2)) _ \u226b \u03b7.app _,\n      { apply coequalizer.hom_ext, simp, },\n      rw [ht, kernel.condition_assoc, zero_comp],\n    end \u226b\n    inv hG.left_cokernel_to_kernel_bottom_left_cokernel_to =\n  inv hF.left_cokernel_to_kernel_bottom_left_cokernel_to \u226b\n  cokernel.desc _ (\u03b7.app _ \u226b cokernel.\u03c0 _) begin\n    simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n  end :=\nbegin\n  rw [is_iso.comp_inv_eq, category.assoc (inv _), is_iso.eq_inv_comp],\n  dsimp [is_snake_input.left_cokernel_to_kernel_bottom_left_cokernel_to],\n  apply coequalizer.hom_ext, apply equalizer.hom_ext,\n  simp only [nat_trans.naturality_assoc, category.assoc, cokernel.\u03c0_desc_assoc,\n    cokernel.\u03c0_desc, kernel.lift_\u03b9, kernel.lift_\u03b9_assoc],\nend\n\nend \u03b4_natural_setup\n\nopen \u03b4_natural_setup\n\ntheorem \u03b4_natural (hF : is_snake_input F) (hG : is_snake_input G) :\n  \u03b7.app (0,2) \u226b hG.\u03b4 = hF.\u03b4 \u226b \u03b7.app (3,0) :=\nbegin\n  dsimp [is_snake_input.\u03b4],\n  simp only [category.assoc],\n  rw aux1_assoc \u03b7 hF hG,\n  rw aux2_assoc \u03b7 hF hG,\n  simp_rw cancel_epi,\n  apply coequalizer.hom_ext,\n  dsimp [is_snake_input.\u03b4_aux],\n  simp only [cokernel.\u03c0_desc_assoc, category.assoc],\n  simp only [\u2190 category.assoc], let t := _, change (t \u226b _) \u226b _ = _,\n  let s := _, change _ = ((s \u226b _) \u226b _) \u226b _,\n  have ht : t = s \u226b kernel.lift _ (kernel.\u03b9 _ \u226b\n    cokernel.desc _ (\u03b7.app _ \u226b cokernel.\u03c0 _) begin\n      simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n    end) _,\n  rotate 2,\n  { dsimp [is_snake_input.bottom_left_cokernel_to], simp only [category.assoc],\n    let t := _, change _ \u226b t = _,\n    have ht : t = cokernel.desc ((1,0) \u27f6[F] (2,1)) ((2,1) \u27f6[F] (2,2)) _ \u226b \u03b7.app _,\n    { apply coequalizer.hom_ext, dsimp, simp, },\n    rw [ht, kernel.condition_assoc, zero_comp] },\n  { dsimp [t, s],\n    apply equalizer.hom_ext,\n    simp },\n  rw ht, clear ht, clear t, dsimp [s], clear s,\n  simp only [category.assoc], congr' 1,\n  rw aux3_assoc \u03b7 hF hG, congr' 1,\n  dsimp [is_snake_input.cokernel_to],\n  apply coequalizer.hom_ext,\n  simp,\nend\n\nend snake_lemma\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/snake_lemma_naturality.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.37433766669770085}}
{"text": "/-\nCopyright (c) 2022 Heather Macbeth. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Heather Macbeth, Fr\u00e9d\u00e9ric Dupuis\n-/\n\nimport Mathlib.Logic.Equiv.LocalEquiv\n\n/-!\nThis is a test file for the tactic `mfld_set_tac`. Because this tactic applies a simp-set which\nmostly contains lemmas in advanced parts of mathlib, it is currently impossible to truly test it\nin realistic conditions. Instead, we create stub definitions and lemmas on objects such as\n`LocalHomeomorph`, label them with `mfld_simps` and run tests on those.\n-/\n\nopen Lean Meta Elab Tactic\n\n/-! ## Syntax of objects and lemmas needed for testing `MfldSetTac` -/\nsection stub_lemmas\n\nstructure LocalHomeomorph (\u03b1 : Type u) (\u03b2 : Type u) extends LocalEquiv \u03b1 \u03b2\n\ninstance LocalHomeomorph.has_coe_to_fun : CoeFun (LocalHomeomorph \u03b1 \u03b2) (\u03bb _ => \u03b1 \u2192 \u03b2) := sorry\n\ndef LocalHomeomorph.symm (e : LocalHomeomorph \u03b1 \u03b2) : LocalHomeomorph \u03b2 \u03b1 := sorry\n\n@[mfld_simps] lemma LocalHomeomorph.left_inv (e : LocalHomeomorph \u03b1 \u03b2) {x : \u03b1}\n  (h : x \u2208 e.toLocalEquiv.source) :\n  e.symm (e x) = x :=\nsorry\n\n@[mfld_simps] theorem LocalHomeomorph.symm_to_LocalEquiv (e : LocalHomeomorph \u03b1 \u03b2) :\n  e.symm.toLocalEquiv = e.toLocalEquiv.symm :=\nsorry\n\n@[mfld_simps] lemma LocalHomeomorph.coe_coe (e : LocalHomeomorph \u03b1 \u03b2) :\n  (e.toLocalEquiv : \u03b1 \u2192 \u03b2) = e :=\nsorry\n\n@[mfld_simps] lemma LocalHomeomorph.coe_coe_symm (e : LocalHomeomorph \u03b1 \u03b2) :\n  (e.toLocalEquiv.symm : \u03b2 \u2192 \u03b1) = (e.symm : \u03b2 \u2192 \u03b1) :=\nsorry\n\nstructure ModelWithCorners (\ud835\udd5c E H : Type u) extends LocalEquiv H E :=\n(source_eq : source = Set.univ)\n\nattribute [mfld_simps] ModelWithCorners.source_eq\n\ndef ModelWithCorners.symm (I : ModelWithCorners \ud835\udd5c E H) : LocalEquiv E H := sorry\n\ninstance ModelWithCorners.has_coe_to_fun : CoeFun (ModelWithCorners \ud835\udd5c E H) (\u03bb _ => H \u2192 E) := sorry\n\n@[mfld_simps] lemma ModelWithCorners.left_inv (I : ModelWithCorners \ud835\udd5c E H) (x : H) :\n  I.symm (I x) = x :=\nsorry\n\n@[mfld_simps] lemma ModelWithCorners.to_local_equiv_coe (I : ModelWithCorners \ud835\udd5c E H) :\n  (I.toLocalEquiv : H \u2192 E) = I :=\nsorry\n\n@[mfld_simps] lemma ModelWithCorners.to_local_equiv_coe_symm (I : ModelWithCorners \ud835\udd5c E H) :\n  (I.toLocalEquiv.symm : E \u2192 H) = I.symm :=\nsorry\n\nend stub_lemmas\n\n\n/-! ## Tests for `MfldSetTac` -/\nsection tests\n\nexample (e : LocalEquiv \u03b1 \u03b2) (e' : LocalEquiv \u03b2 \u03b3) :\n  (e.trans e').source = e.source \u2229 Set.preimage e (e.target \u2229 e'.source) := by\n  mfld_set_tac\n\nexample (e : LocalEquiv \u03b1 \u03b2) : (e.trans e.symm).source = e.source := by mfld_set_tac\n\nexample (s : Set \u03b1) (f : LocalHomeomorph \u03b1 \u03b2) :\n  f.symm.toLocalEquiv.source \u2229 (f.toLocalEquiv.target \u2229 Set.preimage f.symm s)\n  = f.symm.toLocalEquiv.source \u2229 Set.preimage f.symm s := by mfld_set_tac\n\nexample\n  {I : ModelWithCorners \ud835\udd5c E H}\n  {I' : ModelWithCorners \ud835\udd5c E' H'}\n  {I'' : ModelWithCorners \ud835\udd5c E'' H''}\n  (e\u2081 : LocalHomeomorph M H)\n  (e\u2082 : LocalHomeomorph M' H')\n  (e\u2083 : LocalHomeomorph M'' H'')\n  {f : M \u2192 M'}\n  {g : M' \u2192 M''} :\n  (Set.preimage (f \u2218 ((e\u2081.toLocalEquiv.trans I.toLocalEquiv).symm))\n      (e\u2082.toLocalEquiv.trans I'.toLocalEquiv).source) \u2286\n    {y : E |\n    ((e\u2083.toLocalEquiv.trans I''.toLocalEquiv) \u2218\n          (g \u2218 f) \u2218 ((e\u2081.toLocalEquiv.trans I.toLocalEquiv).symm)) y\n    = (((e\u2083.toLocalEquiv.trans I''.toLocalEquiv : M'' \u2192 E'') \u2218\n             g \u2218 ((e\u2082.toLocalEquiv.trans I'.toLocalEquiv).symm)) \u2218\n          (e\u2082.toLocalEquiv.trans I'.toLocalEquiv : M' \u2192 E') \u2218\n            f \u2218 ((e\u2081.toLocalEquiv.trans I.toLocalEquiv).symm)) y} := by\n  mfld_set_tac\n\nend tests\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/MfldSetTac.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.37416994862606057}}
{"text": "example (p q r : Prop) (hp : p) : p \u2228 q \u2228 r :=\n  by repeat { { left, assumption } <|> right <|> assumption }\n\nexample (p q r : Prop) (hq : q) : p \u2228 q \u2228 r :=\n  by repeat { { left, assumption } <|> right <|> assumption }\n\nexample (p q r : Prop) (hr : r) : p \u2228 q \u2228 r :=\n  by repeat { { left, assumption } <|> right <|> assumption }\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0505.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3741699419187527}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module algebraic_geometry.open_immersion\n! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicGeometry.PresheafedSpace.HasColimits\nimport Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.Pullbacks\nimport Mathbin.Topology.Sheaves.Functors\nimport Mathbin.AlgebraicGeometry.Scheme\nimport Mathbin.CategoryTheory.Limits.Shapes.StrictInitial\nimport Mathbin.CategoryTheory.Limits.Shapes.CommSq\nimport Mathbin.Algebra.Category.Ring.Instances\nimport Mathbin.Topology.LocalAtTarget\n\n/-!\n# Open immersions of structured spaces\n\nWe say that a morphism of presheafed spaces `f : X \u27f6 Y` is an open immersions if\nthe underlying map of spaces is an open embedding `f : X \u27f6 U \u2286 Y`,\nand the sheaf map `Y(V) \u27f6 f _* X(V)` is an iso for each `V \u2286 U`.\n\nAbbreviations are also provided for `SheafedSpace`, `LocallyRingedSpace` and `Scheme`.\n\n## Main definitions\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a PresheafedSpace hom `f` is an open_immersion.\n* `algebraic_geometry.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a Scheme morphism `f` is an open_immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict`: The source of an\n  open immersion is isomorphic to the restriction of the target onto the image.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.lift`: Any morphism whose range is\n  contained in an open immersion factors though the open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace`: If `f : X \u27f6 Y` is an\n  open immersion of presheafed spaces, and `Y` is a sheafed space, then `X` is also a sheafed\n  space. The morphism as morphisms of sheafed spaces is given by `to_SheafedSpace_hom`.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace`: If `f : X \u27f6 Y` is\n  an open immersion of presheafed spaces, and `Y` is a locally ringed space, then `X` is also a\n  locally ringed space. The morphism as morphisms of locally ringed spaces is given by\n  `to_LocallyRingedSpace_hom`.\n\n## Main results\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.comp`: The composition of two open\n  immersions is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso`: An iso is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_iso`:\n  A surjective open immersion is an isomorphism.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.stalk_iso`: An open immersion induces\n  an isomorphism on stalks.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.has_pullback_of_left`: If `f` is an open\n  immersion, then the pullback `(f, g)` exists (and the forgetful functor to `Top` preserves it).\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_snd_of_left`: Open immersions\n  are stable under pullbacks.\n* `algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso` An (topological) open embedding\n  between two sheafed spaces is an open immersion if all the stalk maps are isomorphisms.\n\n-/\n\n\nopen TopologicalSpace CategoryTheory Opposite\n\nopen CategoryTheory.Limits\n\nnamespace AlgebraicGeometry\n\nuniverse v v\u2081 v\u2082 u\n\nvariable {C : Type u} [Category.{v} C]\n\n/-- An open immersion of PresheafedSpaces is an open embedding `f : X \u27f6 U \u2286 Y` of the underlying\nspaces, such that the sheaf map `Y(V) \u27f6 f _* X(V)` is an iso for each `V \u2286 U`.\n-/\nclass PresheafedSpace.IsOpenImmersion {X Y : PresheafedSpace.{v} C} (f : X \u27f6 Y) : Prop where\n  base_open : OpenEmbedding f.base\n  c_iso : \u2200 U : Opens X, IsIso (f.c.app (op (base_open.IsOpenMap.Functor.obj U)))\n#align algebraic_geometry.PresheafedSpace.is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion\n\n/-- A morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism\nof PresheafedSpaces\n-/\nabbrev SheafedSpace.IsOpenImmersion {X Y : SheafedSpace.{v} C} (f : X \u27f6 Y) : Prop :=\n  PresheafedSpace.IsOpenImmersion f\n#align algebraic_geometry.SheafedSpace.is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion\n\n/-- A morphism of LocallyRingedSpaces is an open immersion if it is an open immersion as a morphism\nof SheafedSpaces\n-/\nabbrev LocallyRingedSpace.IsOpenImmersion {X Y : LocallyRingedSpace} (f : X \u27f6 Y) : Prop :=\n  SheafedSpace.IsOpenImmersion f.1\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion\n\n/-- A morphism of Schemes is an open immersion if it is an open immersion as a morphism\nof LocallyRingedSpaces\n-/\nabbrev IsOpenImmersion {X Y : Scheme} (f : X \u27f6 Y) : Prop :=\n  LocallyRingedSpace.IsOpenImmersion f\n#align algebraic_geometry.is_open_immersion AlgebraicGeometry.IsOpenImmersion\n\nnamespace PresheafedSpace.IsOpenImmersion\n\nopen PresheafedSpace\n\n-- mathport name: expris_open_immersion\nlocal notation \"is_open_immersion\" => PresheafedSpace.IsOpenImmersion\n\nattribute [instance] is_open_immersion.c_iso\n\nsection\n\nvariable {X Y : PresheafedSpace.{v} C} {f : X \u27f6 Y} (H : is_open_immersion f)\n\n/-- The functor `opens X \u2964 opens Y` associated with an open immersion `f : X \u27f6 Y`. -/\nabbrev openFunctor :=\n  H.base_open.IsOpenMap.Functor\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.open_functor AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.openFunctor\n\n/-- An open immersion `f : X \u27f6 Y` induces an isomorphism `X \u2245 Y|_{f(X)}`. -/\n@[simps hom_c_app]\nnoncomputable def isoRestrict : X \u2245 Y.restrict H.base_open :=\n  PresheafedSpace.isoOfComponents (Iso.refl _)\n    (by\n      symm\n      fapply nat_iso.of_components\n      intro U\n      refine'\n        as_iso (f.c.app (op (H.open_functor.obj (unop U)))) \u226a\u226b X.presheaf.map_iso (eq_to_iso _)\n      \u00b7 induction U using Opposite.rec\n        cases U\n        dsimp only [IsOpenMap.functor, functor.op, opens.map]\n        congr 2\n        erw [Set.preimage_image_eq _ H.base_open.inj]\n        rfl\n      \u00b7 intro U V i\n        simp only [CategoryTheory.eqToIso.hom, TopCat.Presheaf.pushforwardObj_map, category.assoc,\n          functor.op_map, iso.trans_hom, as_iso_hom, functor.map_iso_hom, \u2190 X.presheaf.map_comp]\n        erw [f.c.naturality_assoc, \u2190 X.presheaf.map_comp]\n        congr )\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict\n\n@[simp]\ntheorem isoRestrict_hom_ofRestrict : H.isoRestrict.Hom \u226b Y.of_restrict _ = f :=\n  by\n  ext\n  \u00b7 simp only [comp_c_app, iso_restrict_hom_c_app, nat_trans.comp_app, eq_to_hom_refl,\n      of_restrict_c_app, category.assoc, whisker_right_id']\n    erw [category.comp_id, f.c.naturality_assoc, \u2190 X.presheaf.map_comp]\n    trans f.c.app x \u226b X.presheaf.map (\ud835\udfd9 _)\n    \u00b7 congr\n    \u00b7 erw [X.presheaf.map_id, category.comp_id]\n  \u00b7 rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict_hom_of_restrict AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_hom_ofRestrict\n\n@[simp]\ntheorem isoRestrict_inv_ofRestrict : H.isoRestrict.inv \u226b f = Y.of_restrict _ := by\n  rw [iso.inv_comp_eq, iso_restrict_hom_of_restrict]\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict_inv_of_restrict AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoRestrict_inv_ofRestrict\n\ninstance mono [H : is_open_immersion f] : Mono f :=\n  by\n  rw [\u2190 H.iso_restrict_hom_of_restrict]\n  apply mono_comp\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.mono AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.mono\n\n/-- The composition of two open immersions is an open immersion. -/\ninstance comp {Z : PresheafedSpace C} (f : X \u27f6 Y) [hf : is_open_immersion f] (g : Y \u27f6 Z)\n    [hg : is_open_immersion g] : is_open_immersion (f \u226b g)\n    where\n  base_open := hg.base_open.comp hf.base_open\n  c_iso U := by\n    generalize_proofs h\n    dsimp only [AlgebraicGeometry.PresheafedSpace.comp_c_app, unop_op, functor.op, comp_base,\n      TopCat.Presheaf.pushforwardObj_obj, opens.map_comp_obj]\n    apply (config := { instances := false }) is_iso.comp_is_iso\n    swap\n    \u00b7 have : (opens.map g.base).obj (h.functor.obj U) = hf.open_functor.obj U :=\n        by\n        ext1\n        dsimp only [opens.map_coe, IsOpenMap.functor_obj_coe, comp_base]\n        rw [coe_comp, \u2190 Set.image_image, Set.preimage_image_eq _ hg.base_open.inj]\n      rw [this]\n      infer_instance\n    \u00b7 have : h.functor.obj U = hg.open_functor.obj (hf.open_functor.obj U) :=\n        by\n        ext1\n        dsimp only [IsOpenMap.functor_obj_coe]\n        rw [comp_base, coe_comp, \u2190 Set.image_image]\n      rw [this]\n      infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.comp AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.comp\n\n/-- For an open immersion `f : X \u27f6 Y` and an open set `U \u2286 X`, we have the map `X(U) \u27f6 Y(U)`. -/\nnoncomputable def invApp (U : Opens X) :\n    X.Presheaf.obj (op U) \u27f6 Y.Presheaf.obj (op (H.openFunctor.obj U)) :=\n  X.Presheaf.map (eqToHom (by simp [opens.map, Set.preimage_image_eq _ H.base_open.inj])) \u226b\n    inv (f.c.app (op (H.openFunctor.obj U)))\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.inv_app AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp\n\n@[simp, reassoc.1]\ntheorem inv_naturality {U V : (Opens X)\u1d52\u1d56} (i : U \u27f6 V) :\n    X.Presheaf.map i \u226b H.invApp (unop V) =\n      H.invApp (unop U) \u226b Y.Presheaf.map (H.openFunctor.op.map i) :=\n  by\n  simp only [inv_app, \u2190 category.assoc]\n  rw [is_iso.comp_inv_eq]\n  simp only [category.assoc, f.c.naturality, is_iso.inv_hom_id_assoc, \u2190 X.presheaf.map_comp]\n  erw [\u2190 X.presheaf.map_comp]\n  congr\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.inv_naturality AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_naturality\n\ninstance (U : Opens X) : IsIso (H.invApp U) :=\n  by\n  delta inv_app\n  infer_instance\n\ntheorem inv_invApp (U : Opens X) :\n    inv (H.invApp U) =\n      f.c.app (op (H.openFunctor.obj U)) \u226b\n        X.Presheaf.map (eqToHom (by simp [opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=\n  by\n  rw [\u2190 cancel_epi (H.inv_app U)]\n  rw [is_iso.hom_inv_id]\n  delta inv_app\n  simp [\u2190 functor.map_comp]\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.inv_inv_app AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.inv_invApp\n\n@[simp, reassoc.1, elementwise]\ntheorem invApp_app (U : Opens X) :\n    H.invApp U \u226b f.c.app (op (H.openFunctor.obj U)) =\n      X.Presheaf.map (eqToHom (by simp [opens.map, Set.preimage_image_eq _ H.base_open.inj])) :=\n  by rw [inv_app, category.assoc, is_iso.inv_hom_id, category.comp_id]\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.inv_app_app AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.invApp_app\n\n@[simp, reassoc.1]\ntheorem app_invApp (U : Opens Y) :\n    f.c.app (op U) \u226b H.invApp ((Opens.map f.base).obj U) =\n      Y.Presheaf.map\n        ((homOfLE (Set.image_preimage_subset f.base U)).op :\n          op U \u27f6 op (H.openFunctor.obj ((Opens.map f.base).obj U))) :=\n  by\n  erw [\u2190 category.assoc]\n  rw [is_iso.comp_inv_eq, f.c.naturality]\n  congr\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.app_inv_app AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_invApp\n\n/-- A variant of `app_inv_app` that gives an `eq_to_hom` instead of `hom_of_le`. -/\n@[reassoc.1]\ntheorem app_inv_app' (U : Opens Y) (hU : (U : Set Y) \u2286 Set.range f.base) :\n    f.c.app (op U) \u226b H.invApp ((Opens.map f.base).obj U) =\n      Y.Presheaf.map\n        (eqToHom\n            (by\n              apply le_antisymm\n              \u00b7 exact Set.image_preimage_subset f.base U.1\n              \u00b7 rw [\u2190 SetLike.coe_subset_coe]\n                refine' LE.le.trans_eq _ (@Set.image_preimage_eq_inter_range _ _ f.base U.1).symm\n                exact set.subset_inter_iff.mpr \u27e8fun _ h => h, hU\u27e9)).op :=\n  by\n  erw [\u2190 category.assoc]\n  rw [is_iso.comp_inv_eq, f.c.naturality]\n  congr\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.app_inv_app' AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.app_inv_app'\n\n/-- An isomorphism is an open immersion. -/\ninstance ofIso {X Y : PresheafedSpace.{v} C} (H : X \u2245 Y) : is_open_immersion H.Hom\n    where\n  base_open := (TopCat.homeoOfIso ((forget C).mapIso H)).OpenEmbedding\n  c_iso _ := inferInstance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIso\n\ninstance (priority := 100) ofIsIso {X Y : PresheafedSpace.{v} C} (f : X \u27f6 Y) [IsIso f] :\n    is_open_immersion f :=\n  AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIso (asIso f)\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.of_is_iso AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofIsIso\n\ninstance ofRestrict {X : TopCat} (Y : PresheafedSpace C) {f : X \u27f6 Y.carrier}\n    (hf : OpenEmbedding f) : is_open_immersion (Y.of_restrict hf)\n    where\n  base_open := hf\n  c_iso U := by\n    dsimp\n    have : (opens.map f).obj (hf.is_open_map.functor.obj U) = U :=\n      by\n      ext1\n      exact Set.preimage_image_eq _ hf.inj\n    convert show is_iso (Y.presheaf.map (\ud835\udfd9 _)) from inferInstance\n    \u00b7 apply Subsingleton.helim\n      rw [this]\n    \u00b7 rw [Y.presheaf.map_id]\n      infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.of_restrict AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict\n\n@[elementwise, simp]\ntheorem ofRestrict_invApp {C : Type _} [Category C] (X : PresheafedSpace C) {Y : TopCat}\n    {f : Y \u27f6 TopCat.of X.carrier} (h : OpenEmbedding f) (U : Opens (X.restrict h).carrier) :\n    (PresheafedSpace.IsOpenImmersion.ofRestrict X h).invApp U = \ud835\udfd9 _ :=\n  by\n  delta PresheafedSpace.is_open_immersion.inv_app\n  rw [is_iso.comp_inv_eq, category.id_comp]\n  change X.presheaf.map _ = X.presheaf.map _\n  congr\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.of_restrict_inv_app AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.ofRestrict_invApp\n\n/-- An open immersion is an iso if the underlying continuous map is epi. -/\ntheorem to_iso (f : X \u27f6 Y) [h : is_open_immersion f] [h' : Epi f.base] : IsIso f :=\n  by\n  apply (config := { instances := false }) is_iso_of_components\n  \u00b7 let this : X \u2243\u209c Y :=\n      (Homeomorph.ofEmbedding _ h.base_open.to_embedding).trans\n        { toFun := Subtype.val\n          invFun := fun x =>\n            \u27e8x, by\n              rw [set.range_iff_surjective.mpr ((TopCat.epi_iff_surjective _).mp h')]\n              trivial\u27e9\n          left_inv := fun \u27e8_, _\u27e9 => rfl\n          right_inv := fun _ => rfl }\n    convert is_iso.of_iso (TopCat.isoOfHomeo this)\n    \u00b7 ext\n      rfl\n  \u00b7 apply (config := { instances := false }) nat_iso.is_iso_of_is_iso_app\n    intro U\n    have : U = op (h.open_functor.obj ((opens.map f.base).obj (unop U))) :=\n      by\n      induction U using Opposite.rec\n      cases U\n      dsimp only [functor.op, opens.map]\n      congr\n      exact (Set.image_preimage_eq _ ((TopCat.epi_iff_surjective _).mp h')).symm\n    convert@is_open_immersion.c_iso _ h ((opens.map f.base).obj (unop U))\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_iso AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.to_iso\n\ninstance stalk_iso [HasColimits C] [H : is_open_immersion f] (x : X) : IsIso (stalkMap f x) :=\n  by\n  rw [\u2190 H.iso_restrict_hom_of_restrict]\n  rw [PresheafedSpace.stalk_map.comp]\n  infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.stalk_iso AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.stalk_iso\n\nend\n\nsection Pullback\n\nnoncomputable section\n\nvariable {X Y Z : PresheafedSpace.{v} C} (f : X \u27f6 Z) [hf : is_open_immersion f] (g : Y \u27f6 Z)\n\ninclude hf\n\n/-- (Implementation.) The projection map when constructing the pullback along an open immersion.\n-/\ndef pullbackConeOfLeftFst :\n    Y.restrict (TopCat.snd_openEmbedding_of_left_openEmbedding hf.base_open g.base) \u27f6 X\n    where\n  base := pullback.fst\n  c :=\n    { app := fun U =>\n        hf.invApp (unop U) \u226b\n          g.c.app (op (hf.base_open.IsOpenMap.Functor.obj (unop U))) \u226b\n            Y.Presheaf.map\n              (eqToHom\n                (by\n                  simp only [IsOpenMap.functor, Subtype.mk_eq_mk, unop_op, op_inj_iff, opens.map,\n                    Subtype.coe_mk, functor.op_obj, Subtype.val_eq_coe]\n                  apply LE.le.antisymm\n                  \u00b7 rintro _ \u27e8_, h\u2081, h\u2082\u27e9\n                    use (TopCat.pullbackIsoProdSubtype _ _).inv \u27e8\u27e8_, _\u27e9, h\u2082\u27e9\n                    simpa using h\u2081\n                  \u00b7 rintro _ \u27e8x, h\u2081, rfl\u27e9\n                    exact \u27e8_, h\u2081, concrete_category.congr_hom pullback.condition x\u27e9))\n      naturality' := by\n        intro U V i\n        induction U using Opposite.rec\n        induction V using Opposite.rec\n        simp only [Quiver.Hom.unop_op, TopCat.Presheaf.pushforwardObj_map, category.assoc,\n          nat_trans.naturality_assoc, functor.op_map, inv_naturality_assoc, \u2190 Y.presheaf.map_comp]\n        erw [\u2190 Y.presheaf.map_comp]\n        congr }\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftFst\n\ntheorem pullback_cone_of_left_condition : pullbackConeOfLeftFst f g \u226b f = Y.of_restrict _ \u226b g :=\n  by\n  ext U\n  \u00b7 induction U using Opposite.rec\n    dsimp only [comp_c_app, nat_trans.comp_app, unop_op, whisker_right_app,\n      pullback_cone_of_left_fst]\n    simp only [Quiver.Hom.unop_op, TopCat.Presheaf.pushforwardObj_map, app_inv_app_assoc,\n      eq_to_hom_app, eq_to_hom_unop, category.assoc, nat_trans.naturality_assoc, functor.op_map]\n    erw [\u2190 Y.presheaf.map_comp, \u2190 Y.presheaf.map_comp]\n    congr\n  \u00b7 simpa using pullback.condition\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_cone_of_left_condition\n\n/-- We construct the pullback along an open immersion via restricting along the pullback of the\nmaps of underlying spaces (which is also an open embedding).\n-/\ndef pullbackConeOfLeft : PullbackCone f g :=\n  PullbackCone.mk (pullbackConeOfLeftFst f g) (Y.of_restrict _)\n    (pullback_cone_of_left_condition f g)\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeft\n\nvariable (s : PullbackCone f g)\n\n/-- (Implementation.) Any cone over `cospan f g` indeed factors through the constructed cone.\n-/\ndef pullbackConeOfLeftLift : s.pt \u27f6 (pullbackConeOfLeft f g).pt\n    where\n  base :=\n    pullback.lift s.fst.base s.snd.base\n      (congr_arg (fun x => PresheafedSpace.Hom.base x) s.condition)\n  c :=\n    { app := fun U =>\n        s.snd.c.app _ \u226b\n          s.pt.Presheaf.map\n            (eqToHom\n              (by\n                dsimp only [opens.map, IsOpenMap.functor, functor.op]\n                congr 2\n                let s' : pullback_cone f.base g.base := pullback_cone.mk s.fst.base s.snd.base _\n                have : _ = s.snd.base := limit.lift_\u03c0 s' walking_cospan.right\n                conv_lhs =>\n                  erw [\u2190 this]\n                  rw [coe_comp]\n                  erw [\u2190 Set.preimage_preimage]\n                erw [Set.preimage_image_eq _\n                    (TopCat.snd_openEmbedding_of_left_openEmbedding hf.base_open g.base).inj]))\n      naturality' := fun U V i => by\n        erw [s.snd.c.naturality_assoc]\n        rw [category.assoc]\n        erw [\u2190 s.X.presheaf.map_comp, \u2190 s.X.presheaf.map_comp]\n        congr }\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\ntheorem pullbackConeOfLeftLift_fst :\n    pullbackConeOfLeftLift f g s \u226b (pullbackConeOfLeft f g).fst = s.fst :=\n  by\n  ext x\n  \u00b7 induction x using Opposite.rec\n    change ((_ \u226b _) \u226b _ \u226b _) \u226b _ = _\n    simp_rw [category.assoc]\n    erw [\u2190 s.X.presheaf.map_comp]\n    erw [s.snd.c.naturality_assoc]\n    have := congr_app s.condition (op (hf.open_functor.obj x))\n    dsimp only [comp_c_app, unop_op] at this\n    rw [\u2190 is_iso.comp_inv_eq] at this\n    reassoc! this\n    erw [\u2190 this, hf.inv_app_app_assoc, s.fst.c.naturality_assoc]\n    simpa [eq_to_hom_map]\n  \u00b7 change pullback.lift _ _ _ \u226b pullback.fst = _\n    simp\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_fst AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_fst\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\ntheorem pullbackConeOfLeftLift_snd :\n    pullbackConeOfLeftLift f g s \u226b (pullbackConeOfLeft f g).snd = s.snd :=\n  by\n  ext x\n  \u00b7 change (_ \u226b _ \u226b _) \u226b _ = _\n    simp_rw [category.assoc]\n    erw [s.snd.c.naturality_assoc]\n    erw [\u2190 s.X.presheaf.map_comp, \u2190 s.X.presheaf.map_comp]\n    trans s.snd.c.app x \u226b s.X.presheaf.map (\ud835\udfd9 _)\n    \u00b7 congr\n    \u00b7 rw [s.X.presheaf.map_id]\n      erw [category.comp_id]\n  \u00b7 change pullback.lift _ _ _ \u226b pullback.snd = _\n    simp\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftLift_snd\n\ninstance pullbackConeSndIsOpenImmersion : is_open_immersion (pullbackConeOfLeft f g).snd :=\n  by\n  erw [CategoryTheory.Limits.PullbackCone.mk_snd]\n  infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_snd_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeSndIsOpenImmersion\n\n/-- The constructed pullback cone is indeed the pullback. -/\ndef pullbackConeOfLeftIsLimit : IsLimit (pullbackConeOfLeft f g) :=\n  by\n  apply pullback_cone.is_limit_aux'\n  intro s\n  use pullback_cone_of_left_lift f g s\n  use pullback_cone_of_left_lift_fst f g s\n  use pullback_cone_of_left_lift_snd f g s\n  intro m h\u2081 h\u2082\n  rw [\u2190 cancel_mono (pullback_cone_of_left f g).snd]\n  exact h\u2082.trans (pullback_cone_of_left_lift_snd f g s).symm\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackConeOfLeftIsLimit\n\ninstance hasPullback_of_left : HasPullback f g :=\n  \u27e8\u27e8\u27e8_, pullbackConeOfLeftIsLimit f g\u27e9\u27e9\u27e9\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.has_pullback_of_left AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.hasPullback_of_left\n\ninstance hasPullback_of_right : HasPullback g f :=\n  hasPullback_symmetry f g\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.has_pullback_of_right AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.hasPullback_of_right\n\n/-- Open immersions are stable under base-change. -/\ninstance pullbackSndOfLeft : is_open_immersion (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  delta pullback.snd\n  rw [\u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right]\n  infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_snd_of_left AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackSndOfLeft\n\n/-- Open immersions are stable under base-change. -/\ninstance pullbackFstOfRight : is_open_immersion (pullback.fst : pullback g f \u27f6 _) :=\n  by\n  rw [\u2190 pullback_symmetry_hom_comp_snd]\n  infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_fst_of_right AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackFstOfRight\n\ninstance pullbackToBaseIsOpenImmersion [is_open_immersion g] :\n    is_open_immersion (limit.\u03c0 (cospan f g) WalkingCospan.one) :=\n  by\n  rw [\u2190 limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl]\n  infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_to_base_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullbackToBaseIsOpenImmersion\n\ninstance forgetPreservesLimitsOfLeft : PreservesLimit (cospan f g) (forget C) :=\n  preservesLimitOfPreservesLimitCone (pullbackConeOfLeftIsLimit f g)\n    (by\n      apply (is_limit.postcompose_hom_equiv (diagramIsoCospan.{v} _) _).toFun\n      refine' (is_limit.equiv_iso_limit _).toFun (limit.is_limit (cospan f.base g.base))\n      fapply cones.ext\n      exact iso.refl _\n      change \u2200 j, _ = \ud835\udfd9 _ \u226b _ \u226b _\n      simp_rw [category.id_comp]\n      rintro (_ | _ | _) <;> symm\n      \u00b7 erw [category.comp_id]\n        exact limit.w (cospan f.base g.base) walking_cospan.hom.inl\n      \u00b7 exact category.comp_id _\n      \u00b7 exact category.comp_id _)\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.forget_preserves_limits_of_left AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.forgetPreservesLimitsOfLeft\n\ninstance forgetPreservesLimitsOfRight : PreservesLimit (cospan g f) (forget C) :=\n  preservesPullbackSymmetry (forget C) f g\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.forget_preserves_limits_of_right AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.forgetPreservesLimitsOfRight\n\ntheorem pullback_snd_isIso_of_range_subset (H : Set.range g.base \u2286 Set.range f.base) :\n    IsIso (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  haveI := TopCat.snd_iso_of_left_embedding_range_subset hf.base_open.to_embedding g.base H\n  have : is_iso (pullback.snd : pullback f g \u27f6 _).base :=\n    by\n    delta pullback.snd\n    rw [\u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right]\n    change is_iso (_ \u226b pullback.snd)\n    infer_instance\n  apply to_iso\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.pullback_snd_isIso_of_range_subset\n\n/-- The universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H : Set.range g.base \u2286 Set.range f.base) : Y \u27f6 X :=\n  haveI := pullback_snd_is_iso_of_range_subset f g H\n  inv (pullback.snd : pullback f g \u27f6 _) \u226b pullback.fst\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.lift AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift\n\n@[simp, reassoc.1]\ntheorem lift_fac (H : Set.range g.base \u2286 Set.range f.base) : lift f g H \u226b f = g :=\n  by\n  erw [category.assoc]\n  rw [is_iso.inv_comp_eq]\n  exact pullback.condition\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.lift_fac AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_fac\n\ntheorem lift_uniq (H : Set.range g.base \u2286 Set.range f.base) (l : Y \u27f6 X) (hl : l \u226b f = g) :\n    l = lift f g H := by rw [\u2190 cancel_mono f, hl, lift_fac]\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.lift_uniq AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.lift_uniq\n\n/-- Two open immersions with equal range is isomorphic. -/\n@[simps]\ndef isoOfRangeEq [is_open_immersion g] (e : Set.range f.base = Set.range g.base) : X \u2245 Y\n    where\n  Hom := lift g f (le_of_eq e)\n  inv := lift f g (le_of_eq e.symm)\n  hom_inv_id' := by\n    rw [\u2190 cancel_mono f]\n    simp\n  inv_hom_id' := by\n    rw [\u2190 cancel_mono g]\n    simp\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.iso_of_range_eq AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isoOfRangeEq\n\nend Pullback\n\nopen CategoryTheory.Limits.WalkingCospan\n\nsection ToSheafedSpace\n\nvariable {X : PresheafedSpace.{v} C} (Y : SheafedSpace C)\n\nvariable (f : X \u27f6 Y.toPresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a SheafedSpace, then so is `X`. -/\ndef toSheafedSpace : SheafedSpace C\n    where\n  IsSheaf :=\n    by\n    apply TopCat.Presheaf.isSheaf_of_iso (sheaf_iso_of_iso H.iso_restrict.symm).symm\n    apply TopCat.Sheaf.pushforward_sheaf_of_sheaf\n    exact (Y.restrict H.base_open).IsSheaf\n  toPresheafedSpace := X\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace\n\n@[simp]\ntheorem toSheafedSpace_toPresheafedSpace : (toSheafedSpace Y f).toPresheafedSpace = X :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_to_PresheafedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace_toPresheafedSpace\n\n/-- If `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a SheafedSpace, we can\nupgrade it into a morphism of SheafedSpaces.\n-/\ndef toSheafedSpaceHom : toSheafedSpace Y f \u27f6 Y :=\n  f\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom\n\n@[simp]\ntheorem toSheafedSpaceHom_base : (toSheafedSpaceHom Y f).base = f.base :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom_base AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom_base\n\n@[simp]\ntheorem toSheafedSpaceHom_c : (toSheafedSpaceHom Y f).c = f.c :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_hom_c AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpaceHom_c\n\ninstance toSheafedSpace_isOpenImmersion : SheafedSpace.IsOpenImmersion (toSheafedSpaceHom Y f) :=\n  H\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSheafedSpace_isOpenImmersion\n\nomit H\n\n@[simp]\ntheorem sheafedSpace_toSheafedSpace {X Y : SheafedSpace.{v} C} (f : X \u27f6 Y) [is_open_immersion f] :\n    toSheafedSpace Y f = X := by\n  cases X\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.SheafedSpace_to_SheafedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.sheafedSpace_toSheafedSpace\n\nend ToSheafedSpace\n\nsection ToLocallyRingedSpace\n\nvariable {X : PresheafedSpace.{u} CommRingCat.{u}} (Y : LocallyRingedSpace.{u})\n\nvariable (f : X \u27f6 Y.toPresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a LocallyRingedSpace, then so is `X`. -/\ndef toLocallyRingedSpace : LocallyRingedSpace\n    where\n  toSheafedSpace := toSheafedSpace Y.toSheafedSpace f\n  LocalRing x :=\n    haveI : LocalRing (Y.to_SheafedSpace.to_PresheafedSpace.stalk (f.base x)) := Y.local_ring _\n    (as_iso (stalk_map f x)).commRingIsoToRingEquiv.LocalRing\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace\n\n@[simp]\ntheorem toLocallyRingedSpace_toSheafedSpace :\n    (toLocallyRingedSpace Y f).toSheafedSpace = toSheafedSpace Y.1 f :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_to_SheafedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace_toSheafedSpace\n\n/-- If `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a LocallyRingedSpace, we can\nupgrade it into a morphism of LocallyRingedSpace.\n-/\ndef toLocallyRingedSpaceHom : toLocallyRingedSpace Y f \u27f6 Y :=\n  \u27e8f, fun x => inferInstance\u27e9\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_hom AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom\n\n@[simp]\ntheorem toLocallyRingedSpaceHom_val : (toLocallyRingedSpaceHom Y f).val = f :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_hom_val AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpaceHom_val\n\ninstance toLocallyRingedSpace_isOpenImmersion :\n    LocallyRingedSpace.IsOpenImmersion (toLocallyRingedSpaceHom Y f) :=\n  H\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toLocallyRingedSpace_isOpenImmersion\n\nomit H\n\n@[simp]\ntheorem locallyRingedSpace_toLocallyRingedSpace {X Y : LocallyRingedSpace} (f : X \u27f6 Y)\n    [LocallyRingedSpace.IsOpenImmersion f] : toLocallyRingedSpace Y f.1 = X :=\n  by\n  cases X\n  delta to_LocallyRingedSpace\n  simp\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.LocallyRingedSpace_to_LocallyRingedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.locallyRingedSpace_toLocallyRingedSpace\n\nend ToLocallyRingedSpace\n\ntheorem isIso_of_subset {X Y : PresheafedSpace.{v} C} (f : X \u27f6 Y)\n    [H : PresheafedSpace.IsOpenImmersion f] (U : Opens Y.carrier)\n    (hU : (U : Set Y.carrier) \u2286 Set.range f.base) : IsIso (f.c.app <| op U) :=\n  by\n  have : U = H.base_open.is_open_map.functor.obj ((opens.map f.base).obj U) :=\n    by\n    ext1\n    exact (set.inter_eq_left_iff_subset.mpr hU).symm.trans set.image_preimage_eq_inter_range.symm\n  convert PresheafedSpace.is_open_immersion.c_iso ((opens.map f.base).obj U)\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.is_iso_of_subset AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.isIso_of_subset\n\nend PresheafedSpace.IsOpenImmersion\n\nnamespace SheafedSpace.IsOpenImmersion\n\ninstance (priority := 100) of_isIso {X Y : SheafedSpace.{v} C} (f : X \u27f6 Y) [IsIso f] :\n    SheafedSpace.IsOpenImmersion f :=\n  @PresheafedSpace.IsOpenImmersion.ofIsIso _ f (SheafedSpace.forgetToPresheafedSpace.map_isIso _)\n#align algebraic_geometry.SheafedSpace.is_open_immersion.of_is_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_isIso\n\ninstance comp {X Y Z : SheafedSpace C} (f : X \u27f6 Y) (g : Y \u27f6 Z) [SheafedSpace.IsOpenImmersion f]\n    [SheafedSpace.IsOpenImmersion g] : SheafedSpace.IsOpenImmersion (f \u226b g) :=\n  PresheafedSpace.IsOpenImmersion.comp f g\n#align algebraic_geometry.SheafedSpace.is_open_immersion.comp AlgebraicGeometry.SheafedSpace.IsOpenImmersion.comp\n\nsection Pullback\n\nvariable {X Y Z : SheafedSpace C} (f : X \u27f6 Z) (g : Y \u27f6 Z)\n\nvariable [H : SheafedSpace.IsOpenImmersion f]\n\ninclude H\n\n-- mathport name: exprforget\nlocal notation \"forget\" => SheafedSpace.forgetToPresheafedSpace\n\nopen CategoryTheory.Limits.WalkingCospan\n\ninstance : Mono f :=\n  forget.mono_of_mono_map (show @Mono (PresheafedSpace C) _ _ _ f by infer_instance)\n\ninstance forgetMapIsOpenImmersion : PresheafedSpace.IsOpenImmersion (forget.map f) :=\n  \u27e8H.base_open, H.c_iso\u27e9\n#align algebraic_geometry.SheafedSpace.is_open_immersion.forget_map_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetMapIsOpenImmersion\n\ninstance hasLimit_cospan_forget_of_left : HasLimit (cospan f g \u22d9 forget) :=\n  by\n  apply has_limit_of_iso (diagramIsoCospan.{v} _).symm\n  change has_limit (cospan (forget.map f) (forget.map g))\n  infer_instance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_left\n\ninstance hasLimit_cospan_forget_of_left' :\n    HasLimit (cospan ((cospan f g \u22d9 forget).map Hom.inl) ((cospan f g \u22d9 forget).map Hom.inr)) :=\n  show HasLimit (cospan (forget.map f) (forget.map g)) from inferInstance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_left' AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_left'\n\ninstance hasLimit_cospan_forget_of_right : HasLimit (cospan g f \u22d9 forget) :=\n  by\n  apply has_limit_of_iso (diagramIsoCospan.{v} _).symm\n  change has_limit (cospan (forget.map g) (forget.map f))\n  infer_instance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_right\n\ninstance hasLimit_cospan_forget_of_right' :\n    HasLimit (cospan ((cospan g f \u22d9 forget).map Hom.inl) ((cospan g f \u22d9 forget).map Hom.inr)) :=\n  show HasLimit (cospan (forget.map g) (forget.map f)) from inferInstance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.has_limit_cospan_forget_of_right' AlgebraicGeometry.SheafedSpace.IsOpenImmersion.hasLimit_cospan_forget_of_right'\n\ninstance forgetCreatesPullbackOfLeft : CreatesLimit (cospan f g) forget :=\n  createsLimitOfFullyFaithfulOfIso\n    (PresheafedSpace.IsOpenImmersion.toSheafedSpace Y\n      (@pullback.snd (PresheafedSpace C) _ _ _ _ f g _))\n    (eqToIso (show pullback _ _ = pullback _ _ by congr ) \u226a\u226b\n      HasLimit.isoOfNatIso (diagramIsoCospan _).symm)\n#align algebraic_geometry.SheafedSpace.is_open_immersion.forget_creates_pullback_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetCreatesPullbackOfLeft\n\ninstance forgetCreatesPullbackOfRight : CreatesLimit (cospan g f) forget :=\n  createsLimitOfFullyFaithfulOfIso\n    (PresheafedSpace.IsOpenImmersion.toSheafedSpace Y\n      (@pullback.fst (PresheafedSpace C) _ _ _ _ g f _))\n    (eqToIso (show pullback _ _ = pullback _ _ by congr ) \u226a\u226b\n      HasLimit.isoOfNatIso (diagramIsoCospan _).symm)\n#align algebraic_geometry.SheafedSpace.is_open_immersion.forget_creates_pullback_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.forgetCreatesPullbackOfRight\n\ninstance sheafedSpaceForgetPreservesOfLeft : PreservesLimit (cospan f g) (SheafedSpace.forget C) :=\n  @Limits.compPreservesLimit _ _ _ _ forget (PresheafedSpace.forget C) _\n    (by\n      apply (config := { instances := true })\n        preserves_limit_of_iso_diagram _ (diagramIsoCospan.{v} _).symm\n      dsimp\n      infer_instance)\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_forget_preserves_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpaceForgetPreservesOfLeft\n\ninstance sheafedSpaceForgetPreservesOfRight : PreservesLimit (cospan g f) (SheafedSpace.forget C) :=\n  preservesPullbackSymmetry _ _ _\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_forget_preserves_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpaceForgetPreservesOfRight\n\ninstance sheafedSpace_hasPullback_of_left : HasPullback f g :=\n  hasLimit_of_created (cospan f g) forget\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_has_pullback_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_hasPullback_of_left\n\ninstance sheafedSpace_hasPullback_of_right : HasPullback g f :=\n  hasLimit_of_created (cospan g f) forget\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_has_pullback_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_hasPullback_of_right\n\n/-- Open immersions are stable under base-change. -/\ninstance sheafedSpace_pullback_snd_of_left :\n    SheafedSpace.IsOpenImmersion (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  delta pullback.snd\n  have : _ = limit.\u03c0 (cospan f g) right := preserves_limits_iso_hom_\u03c0 forget (cospan f g) right\n  rw [\u2190 this]\n  have := has_limit.iso_of_nat_iso_hom_\u03c0 (diagramIsoCospan.{v} (cospan f g \u22d9 forget)) right\n  erw [category.comp_id] at this\n  rw [\u2190 this]\n  dsimp\n  infer_instance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_snd_of_left AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_snd_of_left\n\ninstance sheafedSpace_pullback_fst_of_right :\n    SheafedSpace.IsOpenImmersion (pullback.fst : pullback g f \u27f6 _) :=\n  by\n  delta pullback.fst\n  have : _ = limit.\u03c0 (cospan g f) left := preserves_limits_iso_hom_\u03c0 forget (cospan g f) left\n  rw [\u2190 this]\n  have := has_limit.iso_of_nat_iso_hom_\u03c0 (diagramIsoCospan.{v} (cospan g f \u22d9 forget)) left\n  erw [category.comp_id] at this\n  rw [\u2190 this]\n  dsimp\n  infer_instance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_fst_of_right AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_fst_of_right\n\ninstance sheafedSpace_pullback_to_base_isOpenImmersion [SheafedSpace.IsOpenImmersion g] :\n    SheafedSpace.IsOpenImmersion (limit.\u03c0 (cospan f g) one : pullback f g \u27f6 Z) :=\n  by\n  rw [\u2190 limit.w (cospan f g) hom.inl, cospan_map_inl]\n  infer_instance\n#align algebraic_geometry.SheafedSpace.is_open_immersion.SheafedSpace_pullback_to_base_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sheafedSpace_pullback_to_base_isOpenImmersion\n\nend Pullback\n\nsection OfStalkIso\n\nvariable [HasLimits C] [HasColimits C] [ConcreteCategory.{v} C]\n\nvariable [ReflectsIsomorphisms (forget C)] [PreservesLimits (forget C)]\n\nvariable [PreservesFilteredColimits (forget C)]\n\n/-- Suppose `X Y : SheafedSpace C`, where `C` is a concrete category,\nwhose forgetful functor reflects isomorphisms, preserves limits and filtered colimits.\nThen a morphism `X \u27f6 Y` that is a topological open embedding\nis an open immersion iff every stalk map is an iso.\n-/\ntheorem of_stalk_iso {X Y : SheafedSpace C} (f : X \u27f6 Y) (hf : OpenEmbedding f.base)\n    [H : \u2200 x : X, IsIso (PresheafedSpace.stalkMap f x)] : SheafedSpace.IsOpenImmersion f :=\n  { base_open := hf\n    c_iso := fun U =>\n      by\n      apply (config := { instances := false })\n        TopCat.Presheaf.app_isIso_of_stalkFunctor_map_iso\n          (show Y.sheaf \u27f6 (TopCat.Sheaf.pushforward f.base).obj X.sheaf from \u27e8f.c\u27e9)\n      rintro \u27e8_, y, hy, rfl\u27e9\n      specialize H y\n      delta PresheafedSpace.stalk_map at H\n      haveI H' :=\n        TopCat.Presheaf.stalkPushforward.stalkPushforward_iso_of_openEmbedding C hf X.presheaf y\n      have := @is_iso.comp_is_iso _ H (@is_iso.inv_is_iso _ H')\n      rw [category.assoc, is_iso.hom_inv_id, category.comp_id] at this\n      exact this }\n#align algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso AlgebraicGeometry.SheafedSpace.IsOpenImmersion.of_stalk_iso\n\nend OfStalkIso\n\nsection Prod\n\nvariable [HasLimits C] {\u03b9 : Type v} (F : Discrete \u03b9 \u2964 SheafedSpace C) [HasColimit F]\n  (i : Discrete \u03b9)\n\ntheorem sigma_\u03b9_openEmbedding : OpenEmbedding (colimit.\u03b9 F i).base :=\n  by\n  rw [\u2190 show _ = (colimit.\u03b9 F i).base from \u03b9_preserves_colimits_iso_inv (SheafedSpace.forget C) F i]\n  have : _ = _ \u226b colimit.\u03b9 (discrete.functor ((F \u22d9 SheafedSpace.forget C).obj \u2218 discrete.mk)) i :=\n    has_colimit.iso_of_nat_iso_\u03b9_hom discrete.nat_iso_functor i\n  rw [\u2190 iso.eq_comp_inv] at this\n  rw [this]\n  have : colimit.\u03b9 _ _ \u226b _ = _ :=\n    TopCat.sigmaIsoSigma_hom_\u03b9.{v, v} ((F \u22d9 SheafedSpace.forget C).obj \u2218 discrete.mk) i.as\n  rw [\u2190 iso.eq_comp_inv] at this\n  cases i\n  rw [this]\n  simp_rw [\u2190 category.assoc, TopCat.openEmbedding_iff_comp_isIso,\n    TopCat.openEmbedding_iff_isIso_comp]\n  dsimp\n  exact openEmbedding_sigmaMk\n#align algebraic_geometry.SheafedSpace.is_open_immersion.sigma_\u03b9_open_embedding AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_\u03b9_openEmbedding\n\ntheorem image_preimage_is_empty (j : Discrete \u03b9) (h : i \u2260 j) (U : Opens (F.obj i)) :\n    (Opens.map (colimit.\u03b9 (F \u22d9 SheafedSpace.forgetToPresheafedSpace) j).base).obj\n        ((Opens.map (preservesColimitIso SheafedSpace.forgetToPresheafedSpace F).inv.base).obj\n          ((sigma_\u03b9_openEmbedding F i).IsOpenMap.Functor.obj U)) =\n      \u22a5 :=\n  by\n  ext\n  apply iff_false_intro\n  rintro \u27e8y, hy, eq\u27e9\n  replace eq :=\n    concrete_category.congr_arg\n      (preserves_colimit_iso (SheafedSpace.forget C) F \u226a\u226b\n          has_colimit.iso_of_nat_iso discrete.nat_iso_functor \u226a\u226b TopCat.sigmaIsoSigma.{v} _).Hom\n      Eq\n  simp_rw [CategoryTheory.Iso.trans_hom, \u2190 TopCat.comp_app, \u2190 PresheafedSpace.comp_base] at eq\n  rw [\u03b9_preserves_colimits_iso_inv] at eq\n  change\n    ((SheafedSpace.forget C).map (colimit.\u03b9 F i) \u226b _) y =\n      ((SheafedSpace.forget C).map (colimit.\u03b9 F j) \u226b _) x at\n    eq\n  cases i; cases j\n  rw [\u03b9_preserves_colimits_iso_hom_assoc, \u03b9_preserves_colimits_iso_hom_assoc,\n    has_colimit.iso_of_nat_iso_\u03b9_hom_assoc, has_colimit.iso_of_nat_iso_\u03b9_hom_assoc,\n    TopCat.sigmaIsoSigma_hom_\u03b9.{v}, TopCat.sigmaIsoSigma_hom_\u03b9.{v}] at eq\n  exact h (congr_arg discrete.mk (congr_arg Sigma.fst Eq))\n#align algebraic_geometry.SheafedSpace.is_open_immersion.image_preimage_is_empty AlgebraicGeometry.SheafedSpace.IsOpenImmersion.image_preimage_is_empty\n\ninstance sigma_\u03b9_isOpenImmersion [HasStrictTerminalObjects C] :\n    SheafedSpace.IsOpenImmersion (colimit.\u03b9 F i)\n    where\n  base_open := sigma_\u03b9_openEmbedding F i\n  c_iso U :=\n    by\n    have e : colimit.\u03b9 F i = _ :=\n      (\u03b9_preserves_colimits_iso_inv SheafedSpace.forget_to_PresheafedSpace F i).symm\n    have H :\n      OpenEmbedding\n        (colimit.\u03b9 (F \u22d9 SheafedSpace.forget_to_PresheafedSpace) i \u226b\n            (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).base :=\n      e \u25b8 sigma_\u03b9_open_embedding F i\n    suffices\n      is_iso\n        ((colimit.\u03b9 (F \u22d9 SheafedSpace.forget_to_PresheafedSpace) i \u226b\n                (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).c.app\n          (op (H.is_open_map.functor.obj U)))\n      by convert this\n    rw [PresheafedSpace.comp_c_app, \u2190\n      PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_\u03c0]\n    rsuffices :\n      is_iso\n        (limit.\u03c0\n          (PresheafedSpace.componentwise_diagram (F \u22d9 SheafedSpace.forget_to_PresheafedSpace)\n            ((opens.map\n                  (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj\n              (unop <| op <| H.is_open_map.functor.obj U)))\n          (op i))\n    \u00b7 infer_instance\n    apply limit_\u03c0_is_iso_of_is_strict_terminal\n    intro j hj\n    induction j using Opposite.rec\n    dsimp\n    convert(F.obj j).Sheaf.isTerminalOfEmpty\n    convert image_preimage_is_empty F i j (fun h => hj (congr_arg op h.symm)) U\n    exact (congr_arg PresheafedSpace.hom.base e).symm\n#align algebraic_geometry.SheafedSpace.is_open_immersion.sigma_\u03b9_is_open_immersion AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_\u03b9_isOpenImmersion\n\nend Prod\n\nend SheafedSpace.IsOpenImmersion\n\nnamespace LocallyRingedSpace.IsOpenImmersion\n\nsection Pullback\n\nvariable {X Y Z : LocallyRingedSpace.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z)\n\nvariable [H : LocallyRingedSpace.IsOpenImmersion f]\n\ninstance (priority := 100) of_isIso [IsIso g] : LocallyRingedSpace.IsOpenImmersion g :=\n  @PresheafedSpace.IsOpenImmersion.ofIsIso _ g.1\n    \u27e8\u27e8(inv g).1, by\n        erw [\u2190 LocallyRingedSpace.comp_val]\n        rw [is_iso.hom_inv_id]\n        erw [\u2190 LocallyRingedSpace.comp_val]\n        rw [is_iso.inv_hom_id]\n        constructor <;> simpa\u27e9\u27e9\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.of_is_iso AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.of_isIso\n\ninclude H\n\ninstance comp (g : Z \u27f6 Y) [LocallyRingedSpace.IsOpenImmersion g] :\n    LocallyRingedSpace.IsOpenImmersion (f \u226b g) :=\n  PresheafedSpace.IsOpenImmersion.comp f.1 g.1\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.comp AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.comp\n\ninstance mono : Mono f :=\n  LocallyRingedSpace.forgetToSheafedSpace.mono_of_mono_map (show Mono f.1 by infer_instance)\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.mono AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.mono\n\ninstance : SheafedSpace.IsOpenImmersion (LocallyRingedSpace.forgetToSheafedSpace.map f) :=\n  H\n\n/-- An explicit pullback cone over `cospan f g` if `f` is an open immersion. -/\ndef pullbackConeOfLeft : PullbackCone f g :=\n  by\n  refine'\n    pullback_cone.mk _\n      (Y.of_restrict (TopCat.snd_openEmbedding_of_left_openEmbedding H.base_open g.1.base)) _\n  \u00b7 use PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst f.1 g.1\n    intro x\n    have :=\n      PresheafedSpace.stalk_map.congr_hom _ _\n        (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition f.1 g.1) x\n    rw [PresheafedSpace.stalk_map.comp, PresheafedSpace.stalk_map.comp] at this\n    rw [\u2190 is_iso.eq_inv_comp] at this\n    rw [this]\n    infer_instance\n  \u00b7\n    exact\n      LocallyRingedSpace.hom.ext _ _\n        (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition _ _)\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_cone_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeft\n\ninstance : LocallyRingedSpace.IsOpenImmersion (pullbackConeOfLeft f g).snd :=\n  show PresheafedSpace.IsOpenImmersion (Y.toPresheafedSpace.of_restrict _) by infer_instance\n\n/-- The constructed `pullback_cone_of_left` is indeed limiting. -/\ndef pullbackConeOfLeftIsLimit : IsLimit (pullbackConeOfLeft f g) :=\n  PullbackCone.isLimitAux' _ fun s =>\n    by\n    use\n      PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift f.1 g.1\n        (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg LocallyRingedSpace.hom.val s.condition))\n    \u00b7 intro x\n      have :=\n        PresheafedSpace.stalk_map.congr_hom _ _\n          (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n            (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg LocallyRingedSpace.hom.val s.condition)))\n          x\n      change _ = _ \u226b PresheafedSpace.stalk_map s.snd.1 x at this\n      rw [PresheafedSpace.stalk_map.comp, \u2190 is_iso.eq_inv_comp] at this\n      rw [this]\n      infer_instance\n    constructor\n    \u00b7\n      exact\n        LocallyRingedSpace.hom.ext _ _\n          (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_fst f.1 g.1 _)\n    constructor\n    \u00b7\n      exact\n        LocallyRingedSpace.hom.ext _ _\n          (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1 _)\n    intro m h\u2081 h\u2082\n    rw [\u2190 cancel_mono (pullback_cone_of_left f g).snd]\n    exact\n      h\u2082.trans\n        (LocallyRingedSpace.hom.ext _ _\n          (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n              (pullback_cone.mk s.fst.1 s.snd.1\n                (congr_arg LocallyRingedSpace.hom.val s.condition))).symm)\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_cone_of_left_is_limit AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullbackConeOfLeftIsLimit\n\ninstance hasPullback_of_left : HasPullback f g :=\n  \u27e8\u27e8\u27e8_, pullbackConeOfLeftIsLimit f g\u27e9\u27e9\u27e9\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.has_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.hasPullback_of_left\n\ninstance hasPullback_of_right : HasPullback g f :=\n  hasPullback_symmetry f g\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.has_pullback_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.hasPullback_of_right\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_snd_of_left :\n    LocallyRingedSpace.IsOpenImmersion (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  delta pullback.snd\n  rw [\u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_snd_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullback_snd_of_left\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_fst_of_right :\n    LocallyRingedSpace.IsOpenImmersion (pullback.fst : pullback g f \u27f6 _) :=\n  by\n  rw [\u2190 pullback_symmetry_hom_comp_snd]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_fst_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullback_fst_of_right\n\ninstance pullback_to_base_isOpenImmersion [LocallyRingedSpace.IsOpenImmersion g] :\n    LocallyRingedSpace.IsOpenImmersion (limit.\u03c0 (cospan f g) WalkingCospan.one) :=\n  by\n  rw [\u2190 limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_to_base_is_open_immersion AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullback_to_base_isOpenImmersion\n\ninstance forgetPreservesPullbackOfLeft :\n    PreservesLimit (cospan f g) LocallyRingedSpace.forgetToSheafedSpace :=\n  preservesLimitOfPreservesLimitCone (pullbackConeOfLeftIsLimit f g)\n    (by\n      apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.toFun\n      apply is_limit_of_is_limit_pullback_cone_map SheafedSpace.forget_to_PresheafedSpace\n      exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1)\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_preserves_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetPreservesPullbackOfLeft\n\ninstance forgetToPresheafedSpacePreservesPullbackOfLeft :\n    PreservesLimit (cospan f g)\n      (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace) :=\n  preservesLimitOfPreservesLimitCone (pullbackConeOfLeftIsLimit f g)\n    (by\n      apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.toFun\n      exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1)\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpacePreservesPullbackOfLeft\n\ninstance forgetToPresheafedSpacePreservesOpenImmersion :\n    PresheafedSpace.IsOpenImmersion\n      ((LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace).map f) :=\n  H\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_open_immersion AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpacePreservesOpenImmersion\n\ninstance forgetToTopPreservesPullbackOfLeft :\n    PreservesLimit (cospan f g) (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forget _) :=\n  by\n  change\n    preserves_limit _\n      ((LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) \u22d9\n        PresheafedSpace.forget _)\n  apply (config := { instances := false }) limits.comp_preserves_limit\n  infer_instance\n  apply preserves_limit_of_iso_diagram _ (diagramIsoCospan.{u} _).symm\n  dsimp [SheafedSpace.forget_to_PresheafedSpace]\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_Top_preserves_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToTopPreservesPullbackOfLeft\n\ninstance forgetReflectsPullbackOfLeft :\n    ReflectsLimit (cospan f g) LocallyRingedSpace.forgetToSheafedSpace :=\n  reflectsLimitOfReflectsIsomorphisms _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_reflects_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetReflectsPullbackOfLeft\n\ninstance forgetPreservesPullbackOfRight :\n    PreservesLimit (cospan g f) LocallyRingedSpace.forgetToSheafedSpace :=\n  preservesPullbackSymmetry _ _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_preserves_pullback_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetPreservesPullbackOfRight\n\ninstance forgetToPresheafedSpacePreservesPullbackOfRight :\n    PreservesLimit (cospan g f)\n      (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace) :=\n  preservesPullbackSymmetry _ _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_preserves_pullback_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpacePreservesPullbackOfRight\n\ninstance forgetReflectsPullbackOfRight :\n    ReflectsLimit (cospan g f) LocallyRingedSpace.forgetToSheafedSpace :=\n  reflectsLimitOfReflectsIsomorphisms _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_reflects_pullback_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetReflectsPullbackOfRight\n\ninstance forgetToPresheafedSpaceReflectsPullbackOfLeft :\n    ReflectsLimit (cospan f g)\n      (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace) :=\n  reflectsLimitOfReflectsIsomorphisms _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_reflects_pullback_of_left AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpaceReflectsPullbackOfLeft\n\ninstance forgetToPresheafedSpaceReflectsPullbackOfRight :\n    ReflectsLimit (cospan g f)\n      (LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace) :=\n  reflectsLimitOfReflectsIsomorphisms _ _\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.forget_to_PresheafedSpace_reflects_pullback_of_right AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.forgetToPresheafedSpaceReflectsPullbackOfRight\n\ntheorem pullback_snd_isIso_of_range_subset (H' : Set.range g.1.base \u2286 Set.range f.1.base) :\n    IsIso (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  apply (config := { instances := false })\n    reflects_isomorphisms.reflects LocallyRingedSpace.forget_to_SheafedSpace\n  apply (config := { instances := false })\n    reflects_isomorphisms.reflects SheafedSpace.forget_to_PresheafedSpace\n  erw [\u2190\n    preserves_pullback.iso_hom_snd\n      (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) f g]\n  haveI := PresheafedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset _ _ H'\n  infer_instance\n  infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.pullback_snd_isIso_of_range_subset\n\n/-- The universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H' : Set.range g.1.base \u2286 Set.range f.1.base) : Y \u27f6 X :=\n  haveI := pullback_snd_is_iso_of_range_subset f g H'\n  inv (pullback.snd : pullback f g \u27f6 _) \u226b pullback.fst\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift\n\n@[simp, reassoc.1]\ntheorem lift_fac (H' : Set.range g.1.base \u2286 Set.range f.1.base) : lift f g H' \u226b f = g :=\n  by\n  erw [category.assoc]\n  rw [is_iso.inv_comp_eq]\n  exact pullback.condition\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_fac AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_fac\n\ntheorem lift_uniq (H' : Set.range g.1.base \u2286 Set.range f.1.base) (l : Y \u27f6 X) (hl : l \u226b f = g) :\n    l = lift f g H' := by rw [\u2190 cancel_mono f, hl, lift_fac]\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_uniq AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_uniq\n\ntheorem lift_range (H' : Set.range g.1.base \u2286 Set.range f.1.base) :\n    Set.range (lift f g H').1.base = f.1.base \u207b\u00b9' Set.range g.1.base :=\n  by\n  haveI := pullback_snd_is_iso_of_range_subset f g H'\n  dsimp only [lift]\n  have : _ = (pullback.fst : pullback f g \u27f6 _).val.base :=\n    preserves_pullback.iso_hom_fst\n      (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget _) f g\n  rw [LocallyRingedSpace.comp_val, SheafedSpace.comp_base, \u2190 this, \u2190 category.assoc, coe_comp]\n  rw [Set.range_comp, set.range_iff_surjective.mpr, Set.image_univ, TopCat.pullback_fst_range]\n  ext\n  constructor\n  \u00b7 rintro \u27e8y, eq\u27e9\n    exact \u27e8y, Eq.symm\u27e9\n  \u00b7 rintro \u27e8y, eq\u27e9\n    exact \u27e8y, Eq.symm\u27e9\n  \u00b7 rw [\u2190 TopCat.epi_iff_surjective]\n    rw [show (inv (pullback.snd : pullback f g \u27f6 _)).val.base = _ from\n        (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget _).map_inv _]\n    infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.lift_range AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.lift_range\n\nend Pullback\n\n/-- An open immersion is isomorphic to the induced open subscheme on its image. -/\ndef isoRestrict {X Y : LocallyRingedSpace} {f : X \u27f6 Y} (H : LocallyRingedSpace.IsOpenImmersion f) :\n    X \u2245 Y.restrict H.base_open :=\n  by\n  apply LocallyRingedSpace.iso_of_SheafedSpace_iso\n  refine' SheafedSpace.forget_to_PresheafedSpace.preimage_iso _\n  exact H.iso_restrict\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.iso_restrict AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.isoRestrict\n\n/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly\nsurjective family of open immersions from affine schemes. -/\nprotected def scheme (X : LocallyRingedSpace)\n    (h :\n      \u2200 x : X,\n        \u2203 (R : CommRingCat)(f : Spec.toLocallyRingedSpace.obj (op R) \u27f6 X),\n          (x \u2208 Set.range f.1.base : _) \u2227 LocallyRingedSpace.IsOpenImmersion f) :\n    Scheme where\n  toLocallyRingedSpace := X\n  local_affine := by\n    intro x\n    obtain \u27e8R, f, h\u2081, h\u2082\u27e9 := h x\n    refine' \u27e8\u27e8\u27e8_, h\u2082.base_open.open_range\u27e9, h\u2081\u27e9, R, \u27e8_\u27e9\u27e9\n    apply LocallyRingedSpace.iso_of_SheafedSpace_iso\n    refine' SheafedSpace.forget_to_PresheafedSpace.preimage_iso _\n    skip\n    apply PresheafedSpace.is_open_immersion.iso_of_range_eq (PresheafedSpace.of_restrict _ _) f.1\n    \u00b7 exact Subtype.range_coe_subtype\n    \u00b7 infer_instance\n#align algebraic_geometry.LocallyRingedSpace.is_open_immersion.Scheme AlgebraicGeometry.LocallyRingedSpace.IsOpenImmersion.scheme\n\nend LocallyRingedSpace.IsOpenImmersion\n\ntheorem IsOpenImmersion.open_range {X Y : Scheme} (f : X \u27f6 Y) [H : IsOpenImmersion f] :\n    IsOpen (Set.range f.1.base) :=\n  H.base_open.open_range\n#align algebraic_geometry.is_open_immersion.open_range AlgebraicGeometry.IsOpenImmersion.open_range\n\nsection OpenCover\n\nnamespace Scheme\n\n-- TODO: provide API to and from a presieve.\n/-- An open cover of `X` consists of a family of open immersions into `X`,\nand for each `x : X` an open immersion (indexed by `f x`) that covers `x`.\n\nThis is merely a coverage in the Zariski pretopology, and it would be optimal\nif we could reuse the existing API about pretopologies, However, the definitions of sieves and\ngrothendieck topologies uses `Prop`s, so that the actual open sets and immersions are hard to\nobtain. Also, since such a coverage in the pretopology usually contains a proper class of\nimmersions, it is quite hard to glue them, reason about finite covers, etc.\n-/\nstructure OpenCover (X : Scheme.{u}) where\n  J : Type v\n  obj : \u2200 j : J, Scheme\n  map : \u2200 j : J, obj j \u27f6 X\n  f : X.carrier \u2192 J\n  Covers : \u2200 x, x \u2208 Set.range (map (f x)).1.base\n  IsOpen : \u2200 x, IsOpenImmersion (map x) := by infer_instance\n#align algebraic_geometry.Scheme.open_cover AlgebraicGeometry.Scheme.OpenCover\n\nattribute [instance] open_cover.is_open\n\nvariable {X Y Z : Scheme.{u}} (\ud835\udcb0 : OpenCover X) (f : X \u27f6 Z) (g : Y \u27f6 Z)\n\nvariable [\u2200 x, HasPullback (\ud835\udcb0.map x \u226b f) g]\n\n/-- The affine cover of a scheme. -/\ndef affineCover (X : Scheme) : OpenCover X\n    where\n  J := X.carrier\n  obj x := spec.obj <| Opposite.op (X.local_affine x).choose_spec.some\n  map x :=\n    ((X.local_affine x).choose_spec.choose_spec.some.inv \u226b X.toLocallyRingedSpace.of_restrict _ : _)\n  f x := x\n  IsOpen x :=\n    by\n    apply (config := { instances := false }) PresheafedSpace.is_open_immersion.comp\n    infer_instance\n    apply PresheafedSpace.is_open_immersion.of_restrict\n  Covers := by\n    intro x\n    erw [coe_comp]\n    rw [Set.range_comp, set.range_iff_surjective.mpr, Set.image_univ]\n    erw [Subtype.range_coe_subtype]\n    exact (X.local_affine x).some.2\n    rw [\u2190 TopCat.epi_iff_surjective]\n    change epi ((SheafedSpace.forget _).map (LocallyRingedSpace.forget_to_SheafedSpace.map _))\n    infer_instance\n#align algebraic_geometry.Scheme.affine_cover AlgebraicGeometry.Scheme.affineCover\n\ninstance : Inhabited X.OpenCover :=\n  \u27e8X.affineCover\u27e9\n\n/-- Given an open cover `{ U\u1d62 }` of `X`, and for each `U\u1d62` an open cover, we may combine these\nopen covers to form an open cover of `X`.  -/\n@[simps J obj map]\ndef OpenCover.bind (f : \u2200 x : \ud835\udcb0.J, OpenCover (\ud835\udcb0.obj x)) : OpenCover X\n    where\n  J := \u03a3i : \ud835\udcb0.J, (f i).J\n  obj x := (f x.1).obj x.2\n  map x := (f x.1).map x.2 \u226b \ud835\udcb0.map x.1\n  f x := \u27e8_, (f _).f (\ud835\udcb0.Covers x).some\u27e9\n  Covers x := by\n    let y := (\ud835\udcb0.covers x).some\n    have hy : (\ud835\udcb0.map (\ud835\udcb0.f x)).val.base y = x := (\ud835\udcb0.covers x).choose_spec\n    rcases(f (\ud835\udcb0.f x)).Covers y with \u27e8z, hz\u27e9\n    change 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    use z\n    erw [comp_apply]\n    rw [hz, hy]\n#align algebraic_geometry.Scheme.open_cover.bind AlgebraicGeometry.Scheme.OpenCover.bind\n\n/-- An isomorphism `X \u27f6 Y` is an open cover of `Y`. -/\n@[simps J obj map]\ndef openCoverOfIsIso {X Y : Scheme.{u}} (f : X \u27f6 Y) [IsIso f] : OpenCover Y\n    where\n  J := PUnit.{v + 1}\n  obj _ := X\n  map _ := f\n  f _ := PUnit.unit\n  Covers x := by\n    rw [set.range_iff_surjective.mpr]\n    \u00b7 trivial\n    rw [\u2190 TopCat.epi_iff_surjective]\n    infer_instance\n#align algebraic_geometry.Scheme.open_cover_of_is_iso AlgebraicGeometry.Scheme.openCoverOfIsIso\n\n/-- We construct an open cover from another, by providing the needed fields and showing that the\nprovided fields are isomorphic with the original open cover. -/\n@[simps J obj map]\ndef OpenCover.copy {X : Scheme} (\ud835\udcb0 : OpenCover X) (J : Type _) (obj : J \u2192 Scheme)\n    (map : \u2200 i, obj i \u27f6 X) (e\u2081 : J \u2243 \ud835\udcb0.J) (e\u2082 : \u2200 i, obj i \u2245 \ud835\udcb0.obj (e\u2081 i))\n    (e\u2082 : \u2200 i, map i = (e\u2082 i).Hom \u226b \ud835\udcb0.map (e\u2081 i)) : OpenCover X :=\n  { J\n    obj\n    map\n    f := fun x => e\u2081.symm (\ud835\udcb0.f x)\n    Covers := fun x =>\n      by\n      rw [e\u2082, Scheme.comp_val_base, coe_comp, Set.range_comp, set.range_iff_surjective.mpr,\n        Set.image_univ, e\u2081.right_inverse_symm]\n      \u00b7 exact \ud835\udcb0.covers x\n      \u00b7 rw [\u2190 TopCat.epi_iff_surjective]\n        infer_instance\n    IsOpen := fun i => by\n      rw [e\u2082]\n      infer_instance }\n#align algebraic_geometry.Scheme.open_cover.copy AlgebraicGeometry.Scheme.OpenCover.copy\n\n/-- The pushforward of an open cover along an isomorphism. -/\n@[simps J obj map]\ndef OpenCover.pushforwardIso {X Y : Scheme} (\ud835\udcb0 : OpenCover X) (f : X \u27f6 Y) [IsIso f] : OpenCover Y :=\n  ((openCoverOfIsIso f).bind fun _ => \ud835\udcb0).copy \ud835\udcb0.J _ _\n    ((Equiv.punitProd _).symm.trans (Equiv.sigmaEquivProd PUnit \ud835\udcb0.J).symm) (fun _ => Iso.refl _)\n    fun _ => (Category.id_comp _).symm\n#align algebraic_geometry.Scheme.open_cover.pushforward_iso AlgebraicGeometry.Scheme.OpenCover.pushforwardIso\n\n/-- Adding an open immersion into an open cover gives another open cover. -/\n@[simps]\ndef OpenCover.add {X : Scheme} (\ud835\udcb0 : X.OpenCover) {Y : Scheme} (f : Y \u27f6 X) [IsOpenImmersion f] :\n    X.OpenCover where\n  J := Option \ud835\udcb0.J\n  obj i := Option.rec Y \ud835\udcb0.obj i\n  map i := Option.rec f \ud835\udcb0.map i\n  f x := some (\ud835\udcb0.f x)\n  Covers := \ud835\udcb0.Covers\n  IsOpen := by rintro (_ | _) <;> dsimp <;> infer_instance\n#align algebraic_geometry.Scheme.open_cover.add AlgebraicGeometry.Scheme.OpenCover.add\n\n-- Related result : `open_cover.pullback_cover`, where we pullback an open cover on `X` along a\n-- morphism `W \u27f6 X`. This is provided at the end of the file since it needs some more results\n-- about open immersion (which in turn needs the open cover API).\nattribute [local reducible] CommRingCat.of CommRingCat.ofHom\n\ninstance val_base_isIso {X Y : Scheme} (f : X \u27f6 Y) [IsIso f] : IsIso f.1.base :=\n  Scheme.forgetToTop.map_isIso f\n#align algebraic_geometry.Scheme.val_base_is_iso AlgebraicGeometry.Scheme.val_base_isIso\n\ninstance basic_open_isOpenImmersion {R : CommRingCat} (f : R) :\n    AlgebraicGeometry.IsOpenImmersion\n      (Scheme.spec.map (CommRingCat.ofHom (algebraMap R (Localization.Away f))).op) :=\n  by\n  apply (config := { instances := false }) SheafedSpace.is_open_immersion.of_stalk_iso\n  any_goals infer_instance\n  any_goals infer_instance\n  exact (PrimeSpectrum.localization_away_openEmbedding (Localization.Away f) f : _)\n  intro x\n  exact Spec_map_localization_is_iso R (Submonoid.powers f) x\n#align algebraic_geometry.Scheme.basic_open_is_open_immersion AlgebraicGeometry.Scheme.basic_open_isOpenImmersion\n\n/-- The basic open sets form an affine open cover of `Spec R`. -/\ndef affineBasisCoverOfAffine (R : CommRingCat) : OpenCover (spec.obj (Opposite.op R))\n    where\n  J := R\n  obj r := spec.obj (Opposite.op <| CommRingCat.of <| Localization.Away r)\n  map r := spec.map (Quiver.Hom.op (algebraMap R (Localization.Away r) : _))\n  f x := 1\n  Covers r := by\n    rw [set.range_iff_surjective.mpr ((TopCat.epi_iff_surjective _).mp _)]\n    \u00b7 exact trivial\n    \u00b7 infer_instance\n  IsOpen x := AlgebraicGeometry.Scheme.basic_open_isOpenImmersion x\n#align algebraic_geometry.Scheme.affine_basis_cover_of_affine AlgebraicGeometry.Scheme.affineBasisCoverOfAffine\n\n/-- We may bind the basic open sets of an open affine cover to form a affine cover that is also\na basis. -/\ndef affineBasisCover (X : Scheme) : OpenCover X :=\n  X.affineCover.bind fun x => affineBasisCoverOfAffine _\n#align algebraic_geometry.Scheme.affine_basis_cover AlgebraicGeometry.Scheme.affineBasisCover\n\n/-- The coordinate ring of a component in the `affine_basis_cover`. -/\ndef affineBasisCoverRing (X : Scheme) (i : X.affineBasisCover.J) : CommRingCat :=\n  CommRingCat.of <| @Localization.Away (X.local_affine i.1).choose_spec.some _ i.2\n#align algebraic_geometry.Scheme.affine_basis_cover_ring AlgebraicGeometry.Scheme.affineBasisCoverRing\n\ntheorem affineBasisCover_obj (X : Scheme) (i : X.affineBasisCover.J) :\n    X.affineBasisCover.obj i = spec.obj (op <| X.affineBasisCoverRing i) :=\n  rfl\n#align algebraic_geometry.Scheme.affine_basis_cover_obj AlgebraicGeometry.Scheme.affineBasisCover_obj\n\ntheorem affineBasisCover_map_range (X : Scheme) (x : X.carrier)\n    (r : (X.local_affine x).choose_spec.some) :\n    Set.range (X.affineBasisCover.map \u27e8x, r\u27e9).1.base =\n      (X.affineCover.map x).1.base '' (PrimeSpectrum.basicOpen r).1 :=\n  by\n  erw [coe_comp, Set.range_comp]\n  congr\n  exact (PrimeSpectrum.localization_away_comap_range (Localization.Away r) r : _)\n#align algebraic_geometry.Scheme.affine_basis_cover_map_range AlgebraicGeometry.Scheme.affineBasisCover_map_range\n\ntheorem affineBasisCover_is_basis (X : Scheme) :\n    TopologicalSpace.IsTopologicalBasis\n      { x : Set X.carrier |\n        \u2203 a : X.affineBasisCover.J, x = Set.range (X.affineBasisCover.map a).1.base } :=\n  by\n  apply TopologicalSpace.isTopologicalBasis_of_open_of_nhds\n  \u00b7 rintro _ \u27e8a, rfl\u27e9\n    exact is_open_immersion.open_range (X.affine_basis_cover.map a)\n  \u00b7 rintro a U haU hU\n    rcases X.affine_cover.covers a with \u27e8x, e\u27e9\n    let U' := (X.affine_cover.map (X.affine_cover.f a)).1.base \u207b\u00b9' U\n    have hxU' : x \u2208 U' := by\n      rw [\u2190 e] at haU\n      exact haU\n    rcases prime_spectrum.is_basis_basic_opens.exists_subset_of_mem_open hxU'\n        ((X.affine_cover.map (X.affine_cover.f a)).1.base.continuous_toFun.isOpen_preimage _\n          hU) with\n      \u27e8_, \u27e8_, \u27e8s, rfl\u27e9, rfl\u27e9, hxV, hVU\u27e9\n    refine' \u27e8_, \u27e8\u27e8_, s\u27e9, rfl\u27e9, _, _\u27e9 <;> erw [affine_basis_cover_map_range]\n    \u00b7 exact \u27e8x, hxV, e\u27e9\n    \u00b7 rw [Set.image_subset_iff]\n      exact hVU\n#align algebraic_geometry.Scheme.affine_basis_cover_is_basis AlgebraicGeometry.Scheme.affineBasisCover_is_basis\n\n/-- Every open cover of a quasi-compact scheme can be refined into a finite subcover.\n-/\n@[simps obj map]\ndef OpenCover.finiteSubcover {X : Scheme} (\ud835\udcb0 : OpenCover X) [H : CompactSpace X.carrier] :\n    OpenCover X :=\n  by\n  have :=\n    @CompactSpace.elim_nhds_subcover _ H (fun x : X.carrier => Set.range (\ud835\udcb0.map (\ud835\udcb0.f x)).1.base)\n      fun x => (is_open_immersion.open_range (\ud835\udcb0.map (\ud835\udcb0.f x))).mem_nhds (\ud835\udcb0.covers x)\n  let t := this.some\n  have h : \u2200 x : X.carrier, \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.carrier) := trivial\n    rw [\u2190 Classical.choose_spec this, Set.mem_union\u1d62] at h'\n    rcases h' with \u27e8y, _, \u27e8hy, rfl\u27e9, hy'\u27e9\n    exact \u27e8\u27e8y, hy\u27e9, hy'\u27e9\n  exact\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).some\n      Covers := fun x => (h x).choose_spec }\n#align algebraic_geometry.Scheme.open_cover.finite_subcover AlgebraicGeometry.Scheme.OpenCover.finiteSubcover\n\ninstance [H : CompactSpace X.carrier] : Fintype \ud835\udcb0.finiteSubcover.J :=\n  by\n  delta open_cover.finite_subcover\n  infer_instance\n\nend Scheme\n\nend OpenCover\n\nnamespace PresheafedSpace.IsOpenImmersion\n\nsection ToScheme\n\nvariable {X : PresheafedSpace.{u} CommRingCat.{u}} (Y : Scheme.{u})\n\nvariable (f : X \u27f6 Y.toPresheafedSpace) [H : PresheafedSpace.IsOpenImmersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a scheme, then so is `X`. -/\ndef toScheme : Scheme :=\n  by\n  apply LocallyRingedSpace.is_open_immersion.Scheme (to_LocallyRingedSpace _ f)\n  intro x\n  obtain \u27e8_, \u27e8i, rfl\u27e9, hx, hi\u27e9 :=\n    Y.affine_basis_cover_is_basis.exists_subset_of_mem_open (Set.mem_range_self x)\n      H.base_open.open_range\n  use Y.affine_basis_cover_ring i\n  use LocallyRingedSpace.is_open_immersion.lift (to_LocallyRingedSpace_hom _ f) _ hi\n  constructor\n  \u00b7 rw [LocallyRingedSpace.is_open_immersion.lift_range]\n    exact hx\n  \u00b7 delta LocallyRingedSpace.is_open_immersion.lift\n    infer_instance\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme\n\n@[simp]\ntheorem toScheme_toLocallyRingedSpace :\n    (toScheme Y f).toLocallyRingedSpace = toLocallyRingedSpace Y.1 f :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_to_LocallyRingedSpace AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toScheme_toLocallyRingedSpace\n\n/-- If `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a Scheme, we can\nupgrade it into a morphism of Schemes.\n-/\ndef toSchemeHom : toScheme Y f \u27f6 Y :=\n  toLocallyRingedSpaceHom _ f\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom\n\n@[simp]\ntheorem toSchemeHom_val : (toSchemeHom Y f).val = f :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom_val AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_val\n\ninstance toSchemeHom_isOpenImmersion : IsOpenImmersion (toSchemeHom Y f) :=\n  H\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.to_Scheme_hom_is_open_immersion AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.toSchemeHom_isOpenImmersion\n\nomit H\n\ntheorem scheme_eq_of_locallyRingedSpace_eq {X Y : Scheme}\n    (H : X.toLocallyRingedSpace = Y.toLocallyRingedSpace) : X = Y :=\n  by\n  cases X\n  cases Y\n  congr\n  exact H\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.Scheme_eq_of_LocallyRingedSpace_eq AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_eq_of_locallyRingedSpace_eq\n\ntheorem scheme_toScheme {X Y : Scheme} (f : X \u27f6 Y) [IsOpenImmersion f] : toScheme Y f.1 = X :=\n  by\n  apply Scheme_eq_of_LocallyRingedSpace_eq\n  exact LocallyRingedSpace_to_LocallyRingedSpace f\n#align algebraic_geometry.PresheafedSpace.is_open_immersion.Scheme_to_Scheme AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.scheme_toScheme\n\nend ToScheme\n\nend PresheafedSpace.IsOpenImmersion\n\n/-- The restriction of a Scheme along an open embedding. -/\n@[simps]\ndef Scheme.restrict {U : TopCat} (X : Scheme) {f : U \u27f6 TopCat.of X.carrier} (h : OpenEmbedding f) :\n    Scheme :=\n  { PresheafedSpace.IsOpenImmersion.toScheme X (X.toPresheafedSpace.of_restrict h) with\n    toPresheafedSpace := X.toPresheafedSpace.restrict h }\n#align algebraic_geometry.Scheme.restrict AlgebraicGeometry.Scheme.restrict\n\n/-- The canonical map from the restriction to the supspace. -/\n@[simps]\ndef Scheme.ofRestrict {U : TopCat} (X : Scheme) {f : U \u27f6 TopCat.of X.carrier}\n    (h : OpenEmbedding f) : X.restrict h \u27f6 X :=\n  X.toLocallyRingedSpace.of_restrict h\n#align algebraic_geometry.Scheme.of_restrict AlgebraicGeometry.Scheme.ofRestrict\n\ninstance IsOpenImmersion.ofRestrict {U : TopCat} (X : Scheme) {f : U \u27f6 TopCat.of X.carrier}\n    (h : OpenEmbedding f) : IsOpenImmersion (X.of_restrict h) :=\n  show PresheafedSpace.IsOpenImmersion (X.toPresheafedSpace.of_restrict h) by infer_instance\n#align algebraic_geometry.is_open_immersion.of_restrict AlgebraicGeometry.IsOpenImmersion.ofRestrict\n\nnamespace IsOpenImmersion\n\nvariable {X Y Z : Scheme.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z)\n\nvariable [H : IsOpenImmersion f]\n\ninstance (priority := 100) of_isIso [IsIso g] : IsOpenImmersion g :=\n  @LocallyRingedSpace.IsOpenImmersion.of_isIso _\n    (show IsIso ((inducedFunctor _).map g) by infer_instance)\n#align algebraic_geometry.is_open_immersion.of_is_iso AlgebraicGeometry.IsOpenImmersion.of_isIso\n\ntheorem to_iso {X Y : Scheme} (f : X \u27f6 Y) [h : IsOpenImmersion f] [Epi f.1.base] : IsIso f :=\n  @isIso_of_reflects_iso _ _ f\n    (Scheme.forgetToLocallyRingedSpace \u22d9\n      LocallyRingedSpace.forgetToSheafedSpace \u22d9 SheafedSpace.forgetToPresheafedSpace)\n    (@PresheafedSpace.IsOpenImmersion.to_iso _ f.1 h _) _\n#align algebraic_geometry.is_open_immersion.to_iso AlgebraicGeometry.IsOpenImmersion.to_iso\n\ntheorem of_stalk_iso {X Y : Scheme} (f : X \u27f6 Y) (hf : OpenEmbedding f.1.base)\n    [\u2200 x, IsIso (PresheafedSpace.stalkMap f.1 x)] : IsOpenImmersion f :=\n  SheafedSpace.IsOpenImmersion.of_stalk_iso f.1 hf\n#align algebraic_geometry.is_open_immersion.of_stalk_iso AlgebraicGeometry.IsOpenImmersion.of_stalk_iso\n\ntheorem iff_stalk_iso {X Y : Scheme} (f : X \u27f6 Y) :\n    IsOpenImmersion f \u2194 OpenEmbedding f.1.base \u2227 \u2200 x, IsIso (PresheafedSpace.stalkMap f.1 x) :=\n  \u27e8fun H => \u27e8H.1, inferInstance\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 => @IsOpenImmersion.of_stalk_iso f h\u2081 h\u2082\u27e9\n#align algebraic_geometry.is_open_immersion.iff_stalk_iso AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso\n\ntheorem AlgebraicGeometry.isIso_iff_isOpenImmersion {X Y : Scheme} (f : X \u27f6 Y) :\n    IsIso f \u2194 IsOpenImmersion f \u2227 Epi f.1.base :=\n  \u27e8fun H => \u27e8inferInstance, inferInstance\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 => @IsOpenImmersion.to_iso f h\u2081 h\u2082\u27e9\n#align algebraic_geometry.is_iso_iff_is_open_immersion AlgebraicGeometry.isIso_iff_isOpenImmersion\n\ntheorem AlgebraicGeometry.isIso_iff_stalk_iso {X Y : Scheme} (f : X \u27f6 Y) :\n    IsIso f \u2194 IsIso f.1.base \u2227 \u2200 x, IsIso (PresheafedSpace.stalkMap f.1 x) :=\n  by\n  rw [is_iso_iff_is_open_immersion, is_open_immersion.iff_stalk_iso, and_comm', \u2190 and_assoc']\n  refine' and_congr \u27e8_, _\u27e9 Iff.rfl\n  \u00b7 rintro \u27e8h\u2081, h\u2082\u27e9\n    convert_to is_iso\n        (TopCat.isoOfHomeo\n            (Homeomorph.homeomorphOfContinuousOpen\n              (Equiv.ofBijective _ \u27e8h\u2082.inj, (TopCat.epi_iff_surjective _).mp h\u2081\u27e9) h\u2082.continuous\n              h\u2082.is_open_map)).Hom\n    \u00b7 ext\n      rfl\n    \u00b7 infer_instance\n  \u00b7 intro H\n    exact \u27e8inferInstance, (TopCat.homeoOfIso (as_iso f.1.base)).OpenEmbedding\u27e9\n#align algebraic_geometry.is_iso_iff_stalk_iso AlgebraicGeometry.isIso_iff_stalk_iso\n\n/-- A open immersion induces an isomorphism from the domain onto the image -/\ndef isoRestrict : X \u2245 (Z.restrict H.base_open : _) :=\n  \u27e8H.isoRestrict.Hom, H.isoRestrict.inv, H.isoRestrict.hom_inv_id, H.isoRestrict.inv_hom_id\u27e9\n#align algebraic_geometry.is_open_immersion.iso_restrict AlgebraicGeometry.IsOpenImmersion.isoRestrict\n\ninclude H\n\n-- mathport name: exprforget\nlocal notation \"forget\" => Scheme.forgetToLocallyRingedSpace\n\ninstance mono : Mono f :=\n  (inducedFunctor _).mono_of_mono_map (show @Mono LocallyRingedSpace _ _ _ f by infer_instance)\n#align algebraic_geometry.is_open_immersion.mono AlgebraicGeometry.IsOpenImmersion.mono\n\ninstance forget_map_isOpenImmersion : LocallyRingedSpace.IsOpenImmersion (forget.map f) :=\n  \u27e8H.base_open, H.c_iso\u27e9\n#align algebraic_geometry.is_open_immersion.forget_map_is_open_immersion AlgebraicGeometry.IsOpenImmersion.forget_map_isOpenImmersion\n\ninstance hasLimit_cospan_forget_of_left :\n    HasLimit (cospan f g \u22d9 Scheme.forgetToLocallyRingedSpace) :=\n  by\n  apply has_limit_of_iso (diagramIsoCospan.{u} _).symm\n  change has_limit (cospan (forget.map f) (forget.map g))\n  infer_instance\n#align algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_left AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left\n\nopen CategoryTheory.Limits.WalkingCospan\n\ninstance hasLimit_cospan_forget_of_left' :\n    HasLimit (cospan ((cospan f g \u22d9 forget).map Hom.inl) ((cospan f g \u22d9 forget).map Hom.inr)) :=\n  show HasLimit (cospan (forget.map f) (forget.map g)) from inferInstance\n#align algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_left' AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_left'\n\ninstance hasLimit_cospan_forget_of_right : HasLimit (cospan g f \u22d9 forget) :=\n  by\n  apply has_limit_of_iso (diagramIsoCospan.{u} _).symm\n  change has_limit (cospan (forget.map g) (forget.map f))\n  infer_instance\n#align algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_right AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right\n\ninstance hasLimit_cospan_forget_of_right' :\n    HasLimit (cospan ((cospan g f \u22d9 forget).map Hom.inl) ((cospan g f \u22d9 forget).map Hom.inr)) :=\n  show HasLimit (cospan (forget.map g) (forget.map f)) from inferInstance\n#align algebraic_geometry.is_open_immersion.has_limit_cospan_forget_of_right' AlgebraicGeometry.IsOpenImmersion.hasLimit_cospan_forget_of_right'\n\ninstance forgetCreatesPullbackOfLeft : CreatesLimit (cospan f g) forget :=\n  createsLimitOfFullyFaithfulOfIso\n    (PresheafedSpace.IsOpenImmersion.toScheme Y (@pullback.snd LocallyRingedSpace _ _ _ _ f g _).1)\n    (eqToIso (by simp) \u226a\u226b HasLimit.isoOfNatIso (diagramIsoCospan _).symm)\n#align algebraic_geometry.is_open_immersion.forget_creates_pullback_of_left AlgebraicGeometry.IsOpenImmersion.forgetCreatesPullbackOfLeft\n\ninstance forgetCreatesPullbackOfRight : CreatesLimit (cospan g f) forget :=\n  createsLimitOfFullyFaithfulOfIso\n    (PresheafedSpace.IsOpenImmersion.toScheme Y (@pullback.fst LocallyRingedSpace _ _ _ _ g f _).1)\n    (eqToIso (by simp) \u226a\u226b HasLimit.isoOfNatIso (diagramIsoCospan _).symm)\n#align algebraic_geometry.is_open_immersion.forget_creates_pullback_of_right AlgebraicGeometry.IsOpenImmersion.forgetCreatesPullbackOfRight\n\ninstance forgetPreservesOfLeft : PreservesLimit (cospan f g) forget :=\n  CategoryTheory.preservesLimitOfCreatesLimitAndHasLimit _ _\n#align algebraic_geometry.is_open_immersion.forget_preserves_of_left AlgebraicGeometry.IsOpenImmersion.forgetPreservesOfLeft\n\ninstance forgetPreservesOfRight : PreservesLimit (cospan g f) forget :=\n  preservesPullbackSymmetry _ _ _\n#align algebraic_geometry.is_open_immersion.forget_preserves_of_right AlgebraicGeometry.IsOpenImmersion.forgetPreservesOfRight\n\ninstance hasPullback_of_left : HasPullback f g :=\n  hasLimit_of_created (cospan f g) forget\n#align algebraic_geometry.is_open_immersion.has_pullback_of_left AlgebraicGeometry.IsOpenImmersion.hasPullback_of_left\n\ninstance hasPullback_of_right : HasPullback g f :=\n  hasLimit_of_created (cospan g f) forget\n#align algebraic_geometry.is_open_immersion.has_pullback_of_right AlgebraicGeometry.IsOpenImmersion.hasPullback_of_right\n\ninstance pullback_snd_of_left : IsOpenImmersion (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  have := preserves_pullback.iso_hom_snd forget f g\n  dsimp only [Scheme.forget_to_LocallyRingedSpace, induced_functor_map] at this\n  rw [\u2190 this]\n  change LocallyRingedSpace.is_open_immersion _\n  infer_instance\n#align algebraic_geometry.is_open_immersion.pullback_snd_of_left AlgebraicGeometry.IsOpenImmersion.pullback_snd_of_left\n\ninstance pullback_fst_of_right : IsOpenImmersion (pullback.fst : pullback g f \u27f6 _) :=\n  by\n  rw [\u2190 pullback_symmetry_hom_comp_snd]\n  infer_instance\n#align algebraic_geometry.is_open_immersion.pullback_fst_of_right AlgebraicGeometry.IsOpenImmersion.pullback_fst_of_right\n\ninstance pullback_to_base [IsOpenImmersion g] :\n    IsOpenImmersion (limit.\u03c0 (cospan f g) WalkingCospan.one) :=\n  by\n  rw [\u2190 limit.w (cospan f g) walking_cospan.hom.inl]\n  change is_open_immersion (_ \u226b f)\n  infer_instance\n#align algebraic_geometry.is_open_immersion.pullback_to_base AlgebraicGeometry.IsOpenImmersion.pullback_to_base\n\ninstance forgetToTopPreservesOfLeft : PreservesLimit (cospan f g) Scheme.forgetToTop :=\n  by\n  apply (config := { instances := false }) limits.comp_preserves_limit\n  infer_instance\n  apply preserves_limit_of_iso_diagram _ (diagramIsoCospan.{u} _).symm\n  dsimp [LocallyRingedSpace.forget_to_Top]\n  infer_instance\n#align algebraic_geometry.is_open_immersion.forget_to_Top_preserves_of_left AlgebraicGeometry.IsOpenImmersion.forgetToTopPreservesOfLeft\n\ninstance forgetToTopPreservesOfRight : PreservesLimit (cospan g f) Scheme.forgetToTop :=\n  preservesPullbackSymmetry _ _ _\n#align algebraic_geometry.is_open_immersion.forget_to_Top_preserves_of_right AlgebraicGeometry.IsOpenImmersion.forgetToTopPreservesOfRight\n\ntheorem range_pullback_snd_of_left :\n    Set.range (pullback.snd : pullback f g \u27f6 Y).1.base =\n      (Opens.map g.1.base).obj \u27e8Set.range f.1.base, H.base_open.open_range\u27e9 :=\n  by\n  rw [\u2190\n    show _ = (pullback.snd : pullback f g \u27f6 _).1.base from\n      preserves_pullback.iso_hom_snd Scheme.forget_to_Top f g,\n    coe_comp, Set.range_comp, set.range_iff_surjective.mpr, \u2190\n    @Set.preimage_univ _ _ (pullback.fst : pullback f.1.base g.1.base \u27f6 _),\n    TopCat.pullback_snd_image_fst_preimage, Set.image_univ]\n  rfl\n  rw [\u2190 TopCat.epi_iff_surjective]\n  infer_instance\n#align algebraic_geometry.is_open_immersion.range_pullback_snd_of_left AlgebraicGeometry.IsOpenImmersion.range_pullback_snd_of_left\n\ntheorem range_pullback_fst_of_right :\n    Set.range (pullback.fst : pullback g f \u27f6 Y).1.base =\n      (Opens.map g.1.base).obj \u27e8Set.range f.1.base, H.base_open.open_range\u27e9 :=\n  by\n  rw [\u2190\n    show _ = (pullback.fst : pullback g f \u27f6 _).1.base from\n      preserves_pullback.iso_hom_fst Scheme.forget_to_Top g f,\n    coe_comp, Set.range_comp, set.range_iff_surjective.mpr, \u2190\n    @Set.preimage_univ _ _ (pullback.snd : pullback g.1.base f.1.base \u27f6 _),\n    TopCat.pullback_fst_image_snd_preimage, Set.image_univ]\n  rfl\n  rw [\u2190 TopCat.epi_iff_surjective]\n  infer_instance\n#align algebraic_geometry.is_open_immersion.range_pullback_fst_of_right AlgebraicGeometry.IsOpenImmersion.range_pullback_fst_of_right\n\ntheorem range_pullback_to_base_of_left :\n    Set.range (pullback.fst \u226b f : pullback f g \u27f6 Z).1.base =\n      Set.range f.1.base \u2229 Set.range g.1.base :=\n  by\n  rw [pullback.condition, Scheme.comp_val_base, coe_comp, Set.range_comp,\n    range_pullback_snd_of_left, opens.map_obj, opens.coe_mk, Set.image_preimage_eq_inter_range,\n    Set.inter_comm]\n#align algebraic_geometry.is_open_immersion.range_pullback_to_base_of_left AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_left\n\ntheorem range_pullback_to_base_of_right :\n    Set.range (pullback.fst \u226b g : pullback g f \u27f6 Z).1.base =\n      Set.range g.1.base \u2229 Set.range f.1.base :=\n  by\n  rw [Scheme.comp_val_base, coe_comp, Set.range_comp, range_pullback_fst_of_right, opens.map_obj,\n    opens.coe_mk, Set.image_preimage_eq_inter_range, Set.inter_comm]\n#align algebraic_geometry.is_open_immersion.range_pullback_to_base_of_right AlgebraicGeometry.IsOpenImmersion.range_pullback_to_base_of_right\n\n/-- The universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H' : Set.range g.1.base \u2286 Set.range f.1.base) : Y \u27f6 X :=\n  LocallyRingedSpace.IsOpenImmersion.lift f g H'\n#align algebraic_geometry.is_open_immersion.lift AlgebraicGeometry.IsOpenImmersion.lift\n\n@[simp, reassoc.1]\ntheorem lift_fac (H' : Set.range g.1.base \u2286 Set.range f.1.base) : lift f g H' \u226b f = g :=\n  LocallyRingedSpace.IsOpenImmersion.lift_fac f g H'\n#align algebraic_geometry.is_open_immersion.lift_fac AlgebraicGeometry.IsOpenImmersion.lift_fac\n\ntheorem lift_uniq (H' : Set.range g.1.base \u2286 Set.range f.1.base) (l : Y \u27f6 X) (hl : l \u226b f = g) :\n    l = lift f g H' :=\n  LocallyRingedSpace.IsOpenImmersion.lift_uniq f g H' l hl\n#align algebraic_geometry.is_open_immersion.lift_uniq AlgebraicGeometry.IsOpenImmersion.lift_uniq\n\n/-- Two open immersions with equal range are isomorphic. -/\n@[simps]\ndef isoOfRangeEq [IsOpenImmersion g] (e : Set.range f.1.base = Set.range g.1.base) : X \u2245 Y\n    where\n  Hom := lift g f (le_of_eq e)\n  inv := lift f g (le_of_eq e.symm)\n  hom_inv_id' := by\n    rw [\u2190 cancel_mono f]\n    simp\n  inv_hom_id' := by\n    rw [\u2190 cancel_mono g]\n    simp\n#align algebraic_geometry.is_open_immersion.iso_of_range_eq AlgebraicGeometry.IsOpenImmersion.isoOfRangeEq\n\n/-- The functor `opens X \u2964 opens Y` associated with an open immersion `f : X \u27f6 Y`. -/\nabbrev AlgebraicGeometry.Scheme.Hom.opensFunctor {X Y : Scheme} (f : X \u27f6 Y)\n    [H : IsOpenImmersion f] : Opens X.carrier \u2964 Opens Y.carrier :=\n  H.openFunctor\n#align algebraic_geometry.Scheme.hom.opens_functor AlgebraicGeometry.Scheme.Hom.opensFunctor\n\n/-- The isomorphism `\u0393(X, U) \u27f6 \u0393(Y, f(U))` induced by an open immersion `f : X \u27f6 Y`. -/\ndef AlgebraicGeometry.Scheme.Hom.invApp {X Y : Scheme} (f : X \u27f6 Y) [H : IsOpenImmersion f] (U) :\n    X.Presheaf.obj (op U) \u27f6 Y.Presheaf.obj (op (f.opensFunctor.obj U)) :=\n  H.invApp U\n#align algebraic_geometry.Scheme.hom.inv_app AlgebraicGeometry.Scheme.Hom.invApp\n\ntheorem app_eq_inv_app_app_of_comp_eq_aux {X Y U : Scheme} (f : Y \u27f6 U) (g : U \u27f6 X) (fg : Y \u27f6 X)\n    (H : fg = f \u226b g) [h : IsOpenImmersion g] (V : Opens U.carrier) :\n    (Opens.map f.1.base).obj V = (Opens.map fg.1.base).obj (g.opensFunctor.obj V) :=\n  by\n  subst H\n  rw [Scheme.comp_val_base, opens.map_comp_obj]\n  congr 1\n  ext1\n  exact (Set.preimage_image_eq _ h.base_open.inj).symm\n#align algebraic_geometry.is_open_immersion.app_eq_inv_app_app_of_comp_eq_aux AlgebraicGeometry.IsOpenImmersion.app_eq_inv_app_app_of_comp_eq_aux\n\n/-- The `fg` argument is to avoid nasty stuff about dependent types. -/\ntheorem app_eq_invApp_app_of_comp_eq {X Y U : Scheme} (f : Y \u27f6 U) (g : U \u27f6 X) (fg : Y \u27f6 X)\n    (H : fg = f \u226b g) [h : IsOpenImmersion g] (V : Opens U.carrier) :\n    f.1.c.app (op V) =\n      g.invApp _ \u226b\n        fg.1.c.app _ \u226b\n          Y.Presheaf.map\n            (eqToHom <| IsOpenImmersion.app_eq_inv_app_app_of_comp_eq_aux f g fg H V).op :=\n  by\n  subst H\n  rw [Scheme.comp_val_c_app, category.assoc, Scheme.hom.inv_app,\n    PresheafedSpace.is_open_immersion.inv_app_app_assoc, f.val.c.naturality_assoc,\n    TopCat.Presheaf.pushforwardObj_map, \u2190 functor.map_comp]\n  convert(category.comp_id _).symm\n  convert Y.presheaf.map_id _\n#align algebraic_geometry.is_open_immersion.app_eq_inv_app_app_of_comp_eq AlgebraicGeometry.IsOpenImmersion.app_eq_invApp_app_of_comp_eq\n\ntheorem lift_app {X Y U : Scheme} (f : U \u27f6 Y) (g : X \u27f6 Y) [h : IsOpenImmersion f] (H)\n    (V : Opens U.carrier) :\n    (IsOpenImmersion.lift f g H).1.c.app (op V) =\n      f.invApp _ \u226b\n        g.1.c.app _ \u226b\n          X.Presheaf.map\n            (eqToHom <|\n                IsOpenImmersion.app_eq_inv_app_app_of_comp_eq_aux _ _ _\n                  (IsOpenImmersion.lift_fac f g H).symm V).op :=\n  IsOpenImmersion.app_eq_invApp_app_of_comp_eq _ _ _ _ _\n#align algebraic_geometry.is_open_immersion.lift_app AlgebraicGeometry.IsOpenImmersion.lift_app\n\nend IsOpenImmersion\n\nnamespace Scheme\n\ntheorem image_basicOpen {X Y : Scheme} (f : X \u27f6 Y) [H : IsOpenImmersion f] {U : Opens X.carrier}\n    (r : X.Presheaf.obj (op U)) : f.opensFunctor.obj (X.basicOpen r) = Y.basicOpen (f.invApp U r) :=\n  by\n  have e := Scheme.preimage_basic_open f (f.inv_app U r)\n  rw [Scheme.hom.inv_app, PresheafedSpace.is_open_immersion.inv_app_app_apply,\n    Scheme.basic_open_res, inf_eq_right.mpr _] at e\n  rw [\u2190 e]\n  ext1\n  refine' set.image_preimage_eq_inter_range.trans _\n  erw [Set.inter_eq_left_iff_subset]\n  refine' Set.Subset.trans (Scheme.basic_open_le _ _) (Set.image_subset_range _ _)\n  refine' le_trans (Scheme.basic_open_le _ _) (le_of_eq _)\n  ext1\n  exact (Set.preimage_image_eq _ H.base_open.inj).symm\n#align algebraic_geometry.Scheme.image_basic_open AlgebraicGeometry.Scheme.image_basicOpen\n\n/-- The image of an open immersion as an open set. -/\n@[simps]\ndef Hom.opensRange {X Y : Scheme} (f : X \u27f6 Y) [H : IsOpenImmersion f] : Opens Y.carrier :=\n  \u27e8_, H.base_open.open_range\u27e9\n#align algebraic_geometry.Scheme.hom.opens_range AlgebraicGeometry.Scheme.Hom.opensRange\n\nend Scheme\n\nsection\n\nvariable (X : Scheme)\n\n/-- The functor taking open subsets of `X` to open subschemes of `X`. -/\n@[simps obj_left obj_hom mapLeft]\ndef Scheme.restrictFunctor : Opens X.carrier \u2964 Over X\n    where\n  obj U := Over.mk (X.of_restrict U.OpenEmbedding)\n  map U V i :=\n    Over.homMk\n      (IsOpenImmersion.lift (X.of_restrict _) (X.of_restrict _)\n        (by\n          change Set.range coe \u2286 Set.range coe\n          simp_rw [Subtype.range_coe]\n          exact i.le))\n      (IsOpenImmersion.lift_fac _ _ _)\n  map_id' U := by\n    ext1\n    dsimp only [over.hom_mk_left, over.id_left]\n    rw [\u2190 cancel_mono (X.of_restrict U.open_embedding), category.id_comp,\n      is_open_immersion.lift_fac]\n  map_comp' U V W i j := by\n    ext1\n    dsimp only [over.hom_mk_left, over.comp_left]\n    rw [\u2190 cancel_mono (X.of_restrict W.open_embedding), category.assoc]\n    iterate 3 rw [is_open_immersion.lift_fac]\n#align algebraic_geometry.Scheme.restrict_functor AlgebraicGeometry.Scheme.restrictFunctor\n\n@[reassoc.1]\ntheorem Scheme.restrictFunctor_map_ofRestrict {U V : Opens X.carrier} (i : U \u27f6 V) :\n    (X.restrictFunctor.map i).1 \u226b X.of_restrict _ = X.of_restrict _ :=\n  IsOpenImmersion.lift_fac _ _ _\n#align algebraic_geometry.Scheme.restrict_functor_map_of_restrict AlgebraicGeometry.Scheme.restrictFunctor_map_ofRestrict\n\ntheorem Scheme.restrictFunctor_map_base {U V : Opens X.carrier} (i : U \u27f6 V) :\n    (X.restrictFunctor.map i).1.1.base = (Opens.toTop _).map i :=\n  by\n  ext a\n  exact\n    (congr_arg (fun f : X.restrict U.open_embedding \u27f6 X => f.1.base a)\n        (X.restrict_functor_map_of_restrict i) :\n      _)\n#align algebraic_geometry.Scheme.restrict_functor_map_base AlgebraicGeometry.Scheme.restrictFunctor_map_base\n\ntheorem Scheme.restrictFunctor_map_app_aux {U V : Opens X.carrier} (i : U \u27f6 V) (W : Opens V) :\n    U.OpenEmbedding.IsOpenMap.Functor.obj ((Opens.map (X.restrictFunctor.map i).1.val.base).obj W) \u2264\n      V.OpenEmbedding.IsOpenMap.Functor.obj W :=\n  by\n  simp only [\u2190 SetLike.coe_subset_coe, IsOpenMap.functor_obj_coe, Set.image_subset_iff,\n    Scheme.restrict_functor_map_base, opens.map_coe, opens.inclusion_apply]\n  rintro _ h\n  exact \u27e8_, h, rfl\u27e9\n#align algebraic_geometry.Scheme.restrict_functor_map_app_aux AlgebraicGeometry.Scheme.restrictFunctor_map_app_aux\n\ntheorem Scheme.restrictFunctor_map_app {U V : Opens X.carrier} (i : U \u27f6 V) (W : Opens V) :\n    (X.restrictFunctor.map i).1.1.c.app (op W) =\n      X.Presheaf.map (homOfLE <| X.restrictFunctor_map_app_aux i W).op :=\n  by\n  have e\u2081 :=\n    Scheme.congr_app (X.restrict_functor_map_of_restrict i)\n      (op <| V.open_embedding.is_open_map.functor.obj W)\n  rw [Scheme.comp_val_c_app] at e\u2081\n  have e\u2082 := (X.restrict_functor.map i).1.val.c.naturality (eq_to_hom W.map_functor_eq).op\n  rw [\u2190 is_iso.eq_inv_comp] at e\u2082\n  dsimp at e\u2081 e\u2082\u22a2\n  rw [e\u2082, W.adjunction_counit_map_functor, \u2190 is_iso.eq_inv_comp, is_iso.inv_comp_eq, \u2190\n    is_iso.eq_comp_inv] at e\u2081\n  simp_rw [eq_to_hom_map (opens.map _), eq_to_hom_map (IsOpenMap.functor _), \u2190 functor.map_inv, \u2190\n    functor.map_comp] at e\u2081\n  rw [e\u2081]\n  congr 1\n#align algebraic_geometry.Scheme.restrict_functor_map_app AlgebraicGeometry.Scheme.restrictFunctor_map_app\n\n/-- The functor that restricts to open subschemes and then takes global section is\nisomorphic to the structure sheaf. -/\n@[simps]\ndef Scheme.restrictFunctor\u0393 : X.restrictFunctor.op \u22d9 (Over.forget X).op \u22d9 Scheme.\u0393 \u2245 X.Presheaf :=\n  NatIso.ofComponents\n    (fun U => X.Presheaf.mapIso ((eqToIso (unop U).openEmbedding_obj_top).symm.op : _))\n    (by\n      intro U V i\n      dsimp [-Subtype.val_eq_coe, -Scheme.restrict_functor_map_left]\n      rw [X.restrict_functor_map_app, \u2190 functor.map_comp, \u2190 functor.map_comp]\n      congr 1)\n#align algebraic_geometry.Scheme.restrict_functor_\u0393 AlgebraicGeometry.Scheme.restrictFunctor\u0393\n\nend\n\n/-- The restriction of an isomorphism onto an open set. -/\nnoncomputable abbrev Scheme.restrictMapIso {X Y : Scheme} (f : X \u27f6 Y) [IsIso f]\n    (U : Opens Y.carrier) :\n    X.restrict ((Opens.map f.1.base).obj U).OpenEmbedding \u2245 Y.restrict U.OpenEmbedding :=\n  by\n  refine' is_open_immersion.iso_of_range_eq (X.of_restrict _ \u226b f) (Y.of_restrict _) _\n  dsimp [opens.inclusion]\n  rw [coe_comp, Set.range_comp]\n  dsimp\n  rw [Subtype.range_coe, Subtype.range_coe]\n  refine' @Set.image_preimage_eq _ _ f.1.base U.1 _\n  rw [\u2190 TopCat.epi_iff_surjective]\n  infer_instance\n#align algebraic_geometry.Scheme.restrict_map_iso AlgebraicGeometry.Scheme.restrictMapIso\n\n/-- Given an open cover on `X`, we may pull them back along a morphism `W \u27f6 X` to obtain\nan open cover of `W`. -/\n@[simps]\ndef Scheme.OpenCover.pullbackCover {X : Scheme} (\ud835\udcb0 : X.OpenCover) {W : Scheme} (f : W \u27f6 X) :\n    W.OpenCover where\n  J := \ud835\udcb0.J\n  obj x := pullback f (\ud835\udcb0.map x)\n  map x := pullback.fst\n  f x := \ud835\udcb0.f (f.1.base x)\n  Covers x :=\n    by\n    rw [\u2190\n      show _ = (pullback.fst : pullback f (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x))) \u27f6 _).1.base from\n        preserves_pullback.iso_hom_fst Scheme.forget_to_Top f (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x)))]\n    rw [coe_comp, Set.range_comp, set.range_iff_surjective.mpr, Set.image_univ,\n      TopCat.pullback_fst_range]\n    obtain \u27e8y, h\u27e9 := \ud835\udcb0.covers (f.1.base x)\n    exact \u27e8y, h.symm\u27e9\n    \u00b7 rw [\u2190 TopCat.epi_iff_surjective]\n      infer_instance\n#align algebraic_geometry.Scheme.open_cover.pullback_cover AlgebraicGeometry.Scheme.OpenCover.pullbackCover\n\ntheorem Scheme.OpenCover.union\u1d62_range {X : Scheme} (\ud835\udcb0 : X.OpenCover) :\n    (\u22c3 i, Set.range (\ud835\udcb0.map i).1.base) = Set.univ :=\n  by\n  rw [Set.eq_univ_iff_forall]\n  intro x\n  rw [Set.mem_union\u1d62]\n  exact \u27e8\ud835\udcb0.f x, \ud835\udcb0.covers x\u27e9\n#align algebraic_geometry.Scheme.open_cover.Union_range AlgebraicGeometry.Scheme.OpenCover.union\u1d62_range\n\ntheorem Scheme.OpenCover.sup\u1d62_opensRange {X : Scheme} (\ud835\udcb0 : X.OpenCover) :\n    (\u2a06 i, (\ud835\udcb0.map i).opensRange) = \u22a4 :=\n  Opens.ext <| by\n    rw [opens.coe_supr]\n    exact \ud835\udcb0.Union_range\n#align algebraic_geometry.Scheme.open_cover.supr_opens_range AlgebraicGeometry.Scheme.OpenCover.sup\u1d62_opensRange\n\ntheorem Scheme.OpenCover.compactSpace {X : Scheme} (\ud835\udcb0 : X.OpenCover) [Finite \ud835\udcb0.J]\n    [H : \u2200 i, CompactSpace (\ud835\udcb0.obj i).carrier] : CompactSpace X.carrier :=\n  by\n  cases nonempty_fintype \ud835\udcb0.J\n  rw [\u2190 isCompact_univ_iff, \u2190 \ud835\udcb0.Union_range]\n  apply isCompact_union\u1d62\n  intro i\n  rw [isCompact_iff_compactSpace]\n  exact\n    @Homeomorph.compactSpace _ _ (H i)\n      (TopCat.homeoOfIso\n        (as_iso\n          (is_open_immersion.iso_of_range_eq (\ud835\udcb0.map i)\n                  (X.of_restrict (opens.open_embedding \u27e8_, (\ud835\udcb0.is_open i).base_open.open_range\u27e9))\n                  subtype.range_coe.symm).Hom.1.base))\n#align algebraic_geometry.Scheme.open_cover.compact_space AlgebraicGeometry.Scheme.OpenCover.compactSpace\n\n/-- Given open covers `{ U\u1d62 }` and `{ U\u2c7c }`, we may form the open cover `{ U\u1d62 \u2229 U\u2c7c }`. -/\ndef Scheme.OpenCover.inter {X : Scheme.{u}} (\ud835\udcb0\u2081 : Scheme.OpenCover.{v\u2081} X)\n    (\ud835\udcb0\u2082 : Scheme.OpenCover.{v\u2082} X) : X.OpenCover\n    where\n  J := \ud835\udcb0\u2081.J \u00d7 \ud835\udcb0\u2082.J\n  obj ij := pullback (\ud835\udcb0\u2081.map ij.1) (\ud835\udcb0\u2082.map ij.2)\n  map ij := pullback.fst \u226b \ud835\udcb0\u2081.map ij.1\n  f x := \u27e8\ud835\udcb0\u2081.f x, \ud835\udcb0\u2082.f x\u27e9\n  Covers x := by\n    rw [is_open_immersion.range_pullback_to_base_of_left]\n    exact \u27e8\ud835\udcb0\u2081.covers x, \ud835\udcb0\u2082.covers x\u27e9\n#align algebraic_geometry.Scheme.open_cover.inter AlgebraicGeometry.Scheme.OpenCover.inter\n\n/-- If `U` is a family of open sets that covers `X`, then `X.restrict U` forms an `X.open_cover`. -/\n@[simps J obj map]\ndef Scheme.openCoverOfSuprEqTop {s : Type _} (X : Scheme) (U : s \u2192 Opens X.carrier)\n    (hU : (\u2a06 i, U i) = \u22a4) : X.OpenCover where\n  J := s\n  obj i := X.restrict (U i).OpenEmbedding\n  map i := X.of_restrict (U i).OpenEmbedding\n  f x :=\n    haveI : x \u2208 \u2a06 i, U i := hU.symm \u25b8 show x \u2208 (\u22a4 : opens X.carrier) by triv\n    (opens.mem_supr.mp this).some\n  Covers x := by\n    erw [Subtype.range_coe]\n    have : x \u2208 \u2a06 i, U i := hU.symm \u25b8 show x \u2208 (\u22a4 : opens X.carrier) by triv\n    exact (opens.mem_supr.mp this).choose_spec\n#align algebraic_geometry.Scheme.open_cover_of_supr_eq_top AlgebraicGeometry.Scheme.openCoverOfSuprEqTop\n\nsection MorphismRestrict\n\n/-- Given a morphism `f : X \u27f6 Y` and an open set `U \u2286 Y`, we have `X \u00d7[Y] U \u2245 X |_{f \u207b\u00b9 U}` -/\ndef pullbackRestrictIsoRestrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    pullback f (Y.of_restrict U.OpenEmbedding) \u2245\n      X.restrict ((Opens.map f.1.base).obj U).OpenEmbedding :=\n  by\n  refine' is_open_immersion.iso_of_range_eq pullback.fst (X.of_restrict _) _\n  rw [is_open_immersion.range_pullback_fst_of_right]\n  dsimp [opens.inclusion]\n  rw [Subtype.range_coe, Subtype.range_coe]\n  rfl\n#align algebraic_geometry.pullback_restrict_iso_restrict AlgebraicGeometry.pullbackRestrictIsoRestrict\n\n@[simp, reassoc.1]\ntheorem pullbackRestrictIsoRestrict_inv_fst {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    (pullbackRestrictIsoRestrict f U).inv \u226b pullback.fst = X.of_restrict _ :=\n  by\n  delta pullback_restrict_iso_restrict\n  simp\n#align algebraic_geometry.pullback_restrict_iso_restrict_inv_fst AlgebraicGeometry.pullbackRestrictIsoRestrict_inv_fst\n\n@[simp, reassoc.1]\ntheorem pullbackRestrictIsoRestrict_hom_restrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    (pullbackRestrictIsoRestrict f U).Hom \u226b X.of_restrict _ = pullback.fst :=\n  by\n  delta pullback_restrict_iso_restrict\n  simp\n#align algebraic_geometry.pullback_restrict_iso_restrict_hom_restrict AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_restrict\n\n/-- The restriction of a morphism `X \u27f6 Y` onto `X |_{f \u207b\u00b9 U} \u27f6 Y |_ U`. -/\ndef morphismRestrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    X.restrict ((Opens.map f.1.base).obj U).OpenEmbedding \u27f6 Y.restrict U.OpenEmbedding :=\n  (pullbackRestrictIsoRestrict f U).inv \u226b pullback.snd\n#align algebraic_geometry.morphism_restrict AlgebraicGeometry.morphismRestrict\n\n-- mathport name: \u00abexpr \u2223_ \u00bb\ninfixl:80 \" \u2223_ \" => morphismRestrict\n\n@[simp, reassoc.1]\ntheorem pullbackRestrictIsoRestrict_hom_morphismRestrict {X Y : Scheme} (f : X \u27f6 Y)\n    (U : Opens Y.carrier) : (pullbackRestrictIsoRestrict f U).Hom \u226b f \u2223_ U = pullback.snd :=\n  Iso.hom_inv_id_assoc _ _\n#align algebraic_geometry.pullback_restrict_iso_restrict_hom_morphism_restrict AlgebraicGeometry.pullbackRestrictIsoRestrict_hom_morphismRestrict\n\n@[simp, reassoc.1]\ntheorem morphismRestrict_\u03b9 {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    (f \u2223_ U) \u226b Y.of_restrict U.OpenEmbedding = X.of_restrict _ \u226b f :=\n  by\n  delta morphism_restrict\n  rw [category.assoc, pullback.condition.symm, pullback_restrict_iso_restrict_inv_fst_assoc]\n#align algebraic_geometry.morphism_restrict_\u03b9 AlgebraicGeometry.morphismRestrict_\u03b9\n\ntheorem isPullback_morphismRestrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    IsPullback (f \u2223_ U) (X.of_restrict _) (Y.of_restrict _) f :=\n  by\n  delta morphism_restrict\n  nth_rw 1 [\u2190 category.id_comp f]\n  refine'\n    (is_pullback.of_horiz_is_iso \u27e8_\u27e9).paste_horiz\n      (is_pullback.of_has_pullback f (Y.of_restrict U.open_embedding)).flip\n  rw [pullback_restrict_iso_restrict_inv_fst, category.comp_id]\n#align algebraic_geometry.is_pullback_morphism_restrict AlgebraicGeometry.isPullback_morphismRestrict\n\ntheorem morphismRestrict_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z) (U : Opens Z.carrier) :\n    (f \u226b g) \u2223_ U = ((f \u2223_ (Opens.map g.val.base).obj U) \u226b g \u2223_ U : _) :=\n  by\n  delta morphism_restrict\n  rw [\u2190 pullback_right_pullback_fst_iso_inv_snd_snd]\n  simp_rw [\u2190 category.assoc]\n  congr 1\n  rw [\u2190 cancel_mono pullback.fst]\n  simp_rw [category.assoc]\n  rw [pullback_restrict_iso_restrict_inv_fst, pullback_right_pullback_fst_iso_inv_snd_fst, \u2190\n    pullback.condition, pullback_restrict_iso_restrict_inv_fst_assoc,\n    pullback_restrict_iso_restrict_inv_fst_assoc]\n  rfl\n  infer_instance\n#align algebraic_geometry.morphism_restrict_comp AlgebraicGeometry.morphismRestrict_comp\n\ninstance {X Y : Scheme} (f : X \u27f6 Y) [IsIso f] (U : Opens Y.carrier) : IsIso (f \u2223_ U) :=\n  by\n  delta morphism_restrict\n  infer_instance\n\ntheorem morphismRestrict_base_coe {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) (x) :\n    @coe U Y.carrier _ ((f \u2223_ U).1.base x) = f.1.base x.1 :=\n  congr_arg (fun f => PresheafedSpace.Hom.base (LocallyRingedSpace.Hom.val f) x)\n    (morphismRestrict_\u03b9 f U)\n#align algebraic_geometry.morphism_restrict_base_coe AlgebraicGeometry.morphismRestrict_base_coe\n\ntheorem morphismRestrict_val_base {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    \u21d1(f \u2223_ U).1.base = U.1.restrictPreimage f.1.base :=\n  funext fun x => Subtype.ext (morphismRestrict_base_coe f U x)\n#align algebraic_geometry.morphism_restrict_val_base AlgebraicGeometry.morphismRestrict_val_base\n\ntheorem image_morphismRestrict_preimage {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier)\n    (V : Opens U) :\n    ((Opens.map f.val.base).obj U).OpenEmbedding.IsOpenMap.Functor.obj\n        ((Opens.map (f \u2223_ U).val.base).obj V) =\n      (Opens.map f.val.base).obj (U.OpenEmbedding.IsOpenMap.Functor.obj V) :=\n  by\n  ext1\n  ext x\n  constructor\n  \u00b7 rintro \u27e8\u27e8x, hx\u27e9, hx' : (f \u2223_ U).1.base _ \u2208 _, rfl\u27e9\n    refine' \u27e8\u27e8_, hx\u27e9, _, rfl\u27e9\n    convert hx'\n    ext1\n    exact (morphism_restrict_base_coe f U \u27e8x, hx\u27e9).symm\n  \u00b7 rintro \u27e8\u27e8x, hx\u27e9, hx', rfl : x = _\u27e9\n    refine' \u27e8\u27e8_, hx\u27e9, (_ : (f \u2223_ U).1.base \u27e8x, hx\u27e9 \u2208 V.1), rfl\u27e9\n    convert hx'\n    ext1\n    exact morphism_restrict_base_coe f U \u27e8x, hx\u27e9\n#align algebraic_geometry.image_morphism_restrict_preimage AlgebraicGeometry.image_morphismRestrict_preimage\n\ntheorem morphismRestrict_c_app {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) (V : Opens U) :\n    (f \u2223_ U).1.c.app (op V) =\n      f.1.c.app (op (U.OpenEmbedding.IsOpenMap.Functor.obj V)) \u226b\n        X.Presheaf.map (eqToHom (image_morphismRestrict_preimage f U V)).op :=\n  by\n  have :=\n    Scheme.congr_app (morphism_restrict_\u03b9 f U) (op (U.open_embedding.is_open_map.functor.obj V))\n  rw [Scheme.comp_val_c_app, Scheme.comp_val_c_app_assoc] at this\n  have e : (opens.map U.inclusion).obj (U.open_embedding.is_open_map.functor.obj V) = V :=\n    by\n    ext1\n    exact Set.preimage_image_eq _ Subtype.coe_injective\n  have : _ \u226b X.presheaf.map _ = _ :=\n    (((f \u2223_ U).1.c.naturality (eq_to_hom e).op).symm.trans _).trans this\n  swap\n  \u00b7 change Y.presheaf.map _ \u226b _ = Y.presheaf.map _ \u226b _\n    congr\n  rw [\u2190 is_iso.eq_comp_inv, \u2190 functor.map_inv, category.assoc] at this\n  rw [this]\n  congr 1\n  erw [\u2190 X.presheaf.map_comp, \u2190 X.presheaf.map_comp]\n  congr 1\n#align algebraic_geometry.morphism_restrict_c_app AlgebraicGeometry.morphismRestrict_c_app\n\ntheorem \u0393_map_morphismRestrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) :\n    Scheme.\u0393.map (f \u2223_ U).op =\n      Y.Presheaf.map (eqToHom <| U.openEmbedding_obj_top.symm).op \u226b\n        f.1.c.app (op U) \u226b\n          X.Presheaf.map (eqToHom <| ((Opens.map f.val.base).obj U).openEmbedding_obj_top).op :=\n  by\n  rw [Scheme.\u0393_map_op, morphism_restrict_c_app f U \u22a4, f.val.c.naturality_assoc]\n  erw [\u2190 X.presheaf.map_comp]\n  congr\n#align algebraic_geometry.\u0393_map_morphism_restrict AlgebraicGeometry.\u0393_map_morphismRestrict\n\n/-- Restricting a morphism onto the the image of an open immersion is isomorphic to the base change\nalong the immersion. -/\ndef morphismRestrictOpensRange {X Y U : Scheme} (f : X \u27f6 Y) (g : U \u27f6 Y) [hg : IsOpenImmersion g] :\n    Arrow.mk (f \u2223_ g.opensRange) \u2245 Arrow.mk (pullback.snd : pullback f g \u27f6 _) :=\n  by\n  let V : opens Y.carrier := g.opens_range\n  let e :=\n    is_open_immersion.iso_of_range_eq g (Y.of_restrict V.open_embedding) subtype.range_coe.symm\n  let t : pullback f g \u27f6 pullback f (Y.of_restrict V.open_embedding) :=\n    pullback.map _ _ _ _ (\ud835\udfd9 _) e.hom (\ud835\udfd9 _) (by rw [category.comp_id, category.id_comp])\n      (by rw [category.comp_id, is_open_immersion.iso_of_range_eq_hom, is_open_immersion.lift_fac])\n  symm\n  refine' arrow.iso_mk (as_iso t \u226a\u226b pullback_restrict_iso_restrict f V) e _\n  rw [iso.trans_hom, as_iso_hom, \u2190 iso.comp_inv_eq, \u2190 cancel_mono g, arrow.mk_hom, arrow.mk_hom,\n    is_open_immersion.iso_of_range_eq_inv, category.assoc, category.assoc, category.assoc,\n    is_open_immersion.lift_fac, \u2190 pullback.condition, morphism_restrict_\u03b9,\n    pullback_restrict_iso_restrict_hom_restrict_assoc, pullback.lift_fst_assoc, category.comp_id]\n#align algebraic_geometry.morphism_restrict_opens_range AlgebraicGeometry.morphismRestrictOpensRange\n\n/-- The restrictions onto two equal open sets are isomorphic. This currently has bad defeqs when\nunfolded, but it should not matter for now. Replace this definition if better defeqs are needed. -/\ndef morphismRestrictEq {X Y : Scheme} (f : X \u27f6 Y) {U V : Opens Y.carrier} (e : U = V) :\n    Arrow.mk (f \u2223_ U) \u2245 Arrow.mk (f \u2223_ V) :=\n  eqToIso (by subst e)\n#align algebraic_geometry.morphism_restrict_eq AlgebraicGeometry.morphismRestrictEq\n\n/-- Restricting a morphism twice is isomorpic to one restriction. -/\ndef morphismRestrictRestrict {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) (V : Opens U) :\n    Arrow.mk (f \u2223_ U \u2223_ V) \u2245 Arrow.mk (f \u2223_ U.OpenEmbedding.IsOpenMap.Functor.obj V) :=\n  by\n  have :\n    (f \u2223_ U \u2223_ V) \u226b (iso.refl _).Hom =\n      (as_iso <|\n            (pullback_restrict_iso_restrict (f \u2223_ U) V).inv \u226b\n              (pullback_symmetry _ _).Hom \u226b\n                pullback.map _ _ _ _ (\ud835\udfd9 _)\n                    ((pullback_restrict_iso_restrict f U).inv \u226b (pullback_symmetry _ _).Hom) (\ud835\udfd9 _)\n                    ((category.comp_id _).trans (category.id_comp _).symm) (by simpa) \u226b\n                  (pullback_right_pullback_fst_iso _ _ _).Hom \u226b (pullback_symmetry _ _).Hom).Hom \u226b\n        pullback.snd :=\n    by\n    simpa only [category.comp_id, pullback_right_pullback_fst_iso_hom_fst, iso.refl_hom,\n      category.assoc, pullback_symmetry_hom_comp_snd, as_iso_hom, pullback.lift_fst,\n      pullback_symmetry_hom_comp_fst]\n  refine'\n    arrow.iso_mk' _ _ _ _ this.symm \u226a\u226b\n      (morphism_restrict_opens_range _ _).symm \u226a\u226b morphism_restrict_eq _ _\n  ext1\n  dsimp\n  rw [coe_comp, Set.range_comp]\n  congr\n  exact Subtype.range_coe\n#align algebraic_geometry.morphism_restrict_restrict AlgebraicGeometry.morphismRestrictRestrict\n\n/-- Restricting a morphism twice onto a basic open set is isomorphic to one restriction.  -/\ndef morphismRestrictRestrictBasicOpen {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier)\n    (r : Y.Presheaf.obj (op U)) :\n    Arrow.mk\n        (f \u2223_ U \u2223_\n          (Y.restrict _).basicOpen (Y.Presheaf.map (eqToHom U.openEmbedding_obj_top).op r)) \u2245\n      Arrow.mk (f \u2223_ Y.basicOpen r) :=\n  by\n  refine' morphism_restrict_restrict _ _ _ \u226a\u226b morphism_restrict_eq _ _\n  have e := Scheme.preimage_basic_open (Y.of_restrict U.open_embedding) r\n  erw [Scheme.of_restrict_val_c_app, opens.adjunction_counit_app_self, eq_to_hom_op] at e\n  rw [\u2190 (Y.restrict U.open_embedding).basicOpen_res_eq _ (eq_to_hom U.inclusion_map_eq_top).op, \u2190\n    comp_apply]\n  erw [\u2190 Y.presheaf.map_comp]\n  rw [eq_to_hom_op, eq_to_hom_op, eq_to_hom_map, eq_to_hom_trans]\n  erw [\u2190 e]\n  ext1; dsimp [opens.map, opens.inclusion]\n  rw [Set.image_preimage_eq_inter_range, Set.inter_eq_left_iff_subset, Subtype.range_coe]\n  exact Y.basic_open_le r\n#align algebraic_geometry.morphism_restrict_restrict_basic_open AlgebraicGeometry.morphismRestrictRestrictBasicOpen\n\n/-- The stalk map of a restriction of a morphism is isomorphic to the stalk map of the original map.\n-/\ndef morphismRestrictStalkMap {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) (x) :\n    Arrow.mk (PresheafedSpace.stalkMap (f \u2223_ U).1 x) \u2245\n      Arrow.mk (PresheafedSpace.stalkMap f.1 x.1) :=\n  by\n  fapply arrow.iso_mk'\n  \u00b7 refine' Y.restrict_stalk_iso U.open_embedding ((f \u2223_ U).1 x) \u226a\u226b TopCat.Presheaf.stalkCongr _ _\n    apply Inseparable.of_eq\n    exact morphism_restrict_base_coe f U x\n  \u00b7 exact X.restrict_stalk_iso _ _\n  \u00b7 apply TopCat.Presheaf.stalk_hom_ext\n    intro V hxV\n    simp only [TopCat.Presheaf.stalkCongr_hom, CategoryTheory.Category.assoc,\n      CategoryTheory.Iso.trans_hom]\n    erw [PresheafedSpace.restrict_stalk_iso_hom_eq_germ_assoc]\n    erw [PresheafedSpace.stalk_map_germ_assoc _ _ \u27e8_, _\u27e9]\n    rw [TopCat.Presheaf.germ_stalk_specializes'_assoc]\n    erw [PresheafedSpace.stalk_map_germ _ _ \u27e8_, _\u27e9]\n    erw [PresheafedSpace.restrict_stalk_iso_hom_eq_germ]\n    rw [morphism_restrict_c_app, category.assoc, TopCat.Presheaf.germ_res]\n    rfl\n#align algebraic_geometry.morphism_restrict_stalk_map AlgebraicGeometry.morphismRestrictStalkMap\n\ninstance {X Y : Scheme} (f : X \u27f6 Y) (U : Opens Y.carrier) [IsOpenImmersion f] :\n    IsOpenImmersion (f \u2223_ U) := by\n  delta morphism_restrict\n  infer_instance\n\nend MorphismRestrict\n\nend AlgebraicGeometry\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/OpenImmersion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.37416994191875264}}
{"text": "import data.sum.basic\nimport lemmas\n\nnamespace coc\nsection\n\nopen expr\nopen ctx\nopen small\nopen small_star\nopen defeq\nopen judgment_index\nopen judgment\n\nset_option pp.beta true\nset_option pp.structure_projections false\n\nlocal notation e ` \u27e6`:80 n:80 ` \u21a6 `:80 e':79 `\u27e7`:79 := subst e n e'\nlocal notation e ` \u27e6`:80 n:80 ` \u219f `:80 m:79 `\u27e7`:79  := shift e n m\nlocal notation e ` ~> `:50 e':50                    := small e e'\nlocal notation e ` ~>* `:50 e':50                   := small_star e e'\nlocal notation e ` ~~ `:50 e':50                    := defeq e e'\nlocal notation `\u25b7 `:50 \u0393:50                         := judgment (well_ctx \u0393 )\nlocal notation \u0393 ` \u25b7 `:50 e:50 ` : `:50 t:50        := judgment (has_type \u0393 e t)\n\n/-- Performs applicative-order beta-reduction.\n    If the original expression is well-typed, the resulting expression will have the same type.\n    Note that this function is only a syntactic operation, and does not check well-formedness.\n    It does not terminate on inputs like `(fun x => x x x) (fun x => x x x)`. -/\nmeta def expr.reduce : \u03a0 (e : expr), \u03a3' (e' : expr), e ~>* e'\n| (sort s)   := \u27e8sort s, ss_refl\u27e9\n| (var v)    := \u27e8var v, ss_refl\u27e9\n| (app l r)  :=\n  let \u27e8l', hl\u27e9 := expr.reduce l,\n      \u27e8r', hr\u27e9 := expr.reduce r in\n    match l', hl with\n    | (lam t e), hl' :=\n      let \u27e8e', he\u27e9 := expr.reduce (e \u27e60 \u21a6 r'\u27e7) in\n        \u27e8e', small_star_trans (ss_step (app_small_star_aux hl' hr) s_beta) he\u27e9\n    | _,         _   := \u27e8app l' r', app_small_star_aux hl hr\u27e9\n    end\n| (lam t e)  :=\n  let \u27e8t', ht\u27e9 := expr.reduce t,\n      \u27e8e', he\u27e9 := expr.reduce e in\n  \u27e8lam t' e', lam_small_star_aux ht he\u27e9\n| (pi t\u2081 t\u2082) :=\n  let \u27e8t\u2081', ht\u2081\u27e9 := expr.reduce t\u2081,\n      \u27e8t\u2082', ht\u2082\u27e9 := expr.reduce t\u2082 in\n    \u27e8pi t\u2081' t\u2082', pi_small_star_aux ht\u2081 ht\u2082\u27e9\n\n/- Helper functions and auxiliary lemmas. -/\n\ndef expr.as_sort : \u03a0 (e : expr), string \u2295 \u03a3' (n : nat), e = sort n\n| (sort s) := sum.inr \u27e8s, rfl\u27e9\n| e        := sum.inl $ \"expression \" ++ e.show ++ \" is not a sort\"\n\ndef expr.as_pi : \u03a0 (e : expr), string \u2295 \u03a3' (t\u2081 t\u2082 : expr), e = pi t\u2081 t\u2082\n| (pi t\u2081 t\u2082) := sum.inr \u27e8t\u2081, t\u2082, rfl\u27e9\n| e          := sum.inl $ \"expression \" ++ e.show ++ \" is not a function\"\n\ndef ctx.try_nth : \u03a0 (\u0393 : ctx) (n : nat), string \u2295 \u03a3' (e : expr), list.nth \u0393 n = option.some e\n| []       n       := sum.inl $ \"variable index overflowed by \" ++ to_string n\n| (t :: \u0393) 0       := sum.inr \u27e8t, rfl\u27e9\n| (t :: \u0393) (n + 1) := ctx.try_nth \u0393 n\n\nlemma expr.check_aux_1\n  {\u0393 l tl} (htl : \u0393 \u25b7 l : tl) {t\u2081 t\u2082} (htt : tl = pi t\u2081 t\u2082) {r tr} (htr : \u0393 \u25b7 r : tr)\n  {t\u2081'} (ht\u2081' : t\u2081 ~>* t\u2081') {tr'} (htr' : tr ~>* tr') (he : t\u2081' = tr') :\n  (\u0393 \u25b7 app l r : t\u2082 \u27e60 \u21a6 r\u27e7) := by\n{ substs htt he,\n  have h\u2081 := has_type_conv_small_star htl (pi_small_star_aux ht\u2081' ss_refl),\n  have h\u2082 := has_type_conv_small_star htr htr',\n  exact t_app h\u2081 h\u2082 }\n\nlemma expr.check_aux_2\n  {\u0393 t t'} (ht' : \u0393 \u25b7 t : t') {s} (hs : t' = sort s) {e t\u2082} (ht\u2082 : t :: \u0393 \u25b7 e : t\u2082) :\n  (\u0393 \u25b7 lam t e : pi t t\u2082) := by\n{ subst hs,\n  obtain \u27e8s\u2082, hs\u2082\u27e9 := type_has_sort ht\u2082,\n  exact t_lam (t_pi ht' hs\u2082) ht\u2082 }\n\n/-- Check if a preterm is a well-formed term.\n    Returns its type and the corresponding proof on success.\n    It should terminate on any input, but there is no proof yet. -/\nmeta def expr.check : \u03a0 (e : expr) (\u0393 : ctx), \u25b7 \u0393 \u2192 string \u2295 \u03a3' (t : expr), \u0393 \u25b7 e : t\n| (sort s)   \u0393 hw := return \u27e8sort (s + 1), t_sort hw\u27e9\n| (var v)    \u0393 hw := do \u27e8t, h\u27e9 <- \u0393.try_nth v, return \u27e8t.shift 0 v.succ, t_var hw h\u27e9\n| (app l r)  \u0393 hw := do\n  { \u27e8tl, htl\u27e9     \u2190 l.check \u0393 hw,\n    \u27e8t\u2081, t\u2082, htt\u27e9 \u2190 tl.as_pi,\n    \u27e8tr, htr\u27e9     \u2190 r.check \u0393 hw,\n    let \u27e8t\u2081', ht\u2081'\u27e9 := t\u2081.reduce,\n    let \u27e8tr', htr'\u27e9 := tr.reduce,\n    dite (t\u2081' = tr')\n      (\u03bb he, return \u27e8t\u2082 \u27e60 \u21a6 r\u27e7, expr.check_aux_1 htl htt htr ht\u2081' htr' he\u27e9)\n      (\u03bb _, sum.inl $ \"argument type mismatch: \" ++ t\u2081.show ++ \" != \" ++ tr.show) }\n| (lam t e)  \u0393 hw := do\n  { \u27e8t', ht'\u27e9     \u2190 t.check \u0393 hw,\n    \u27e8s, hs\u27e9       \u2190 t'.as_sort,\n    \u27e8t\u2082, ht\u2082\u27e9     \u2190 e.check (t :: \u0393) (c_cons (hs \u25b8 ht' : \u0393 \u25b7 t : sort s)),\n    return \u27e8pi t t\u2082, expr.check_aux_2 ht' hs ht\u2082\u27e9 }\n| (pi t\u2081 t\u2082) \u0393 hw := do\n  { \u27e8t\u2081', ht\u2081'\u27e9   \u2190 t\u2081.check \u0393 hw,\n    \u27e8s\u2081, hs\u2081\u27e9     \u2190 t\u2081'.as_sort,\n    \u27e8t\u2082', ht\u2082'\u27e9   \u2190 t\u2082.check (t\u2081 :: \u0393) (c_cons (hs\u2081 \u25b8 ht\u2081' : \u0393 \u25b7 t\u2081 : sort s\u2081)),\n    \u27e8s\u2082, hs\u2082\u27e9     \u2190 t\u2082'.as_sort,\n    return \u27e8sort (max s\u2081 s\u2082), t_pi (hs\u2081 \u25b8 ht\u2081') (hs\u2082 \u25b8 ht\u2082')\u27e9 }\n\n/-- Check if a precontext is a well-formed context.\n    Returns a proof on success. -/\nmeta def ctx.check : \u03a0 (\u0393 : ctx), string \u2295 \u03a3' (_ : unit), \u25b7 \u0393\n| []       := return \u27e8(), c_nil\u27e9\n| (t :: \u0393) := do\n  { \u27e8_, hw\u27e9   \u2190 ctx.check \u0393,\n    \u27e8t', ht'\u27e9 \u2190 t.check \u0393 hw,\n    \u27e8s, hs\u27e9   \u2190 t'.as_sort,\n    return \u27e8(), c_cons (hs \u25b8 ht' : \u0393 \u25b7 t : sort s)\u27e9 }\n\nend\nend coc\n", "meta": {"author": "bridgekat", "repo": "calculus-of-constructions", "sha": "68fe9b05e277f97dd3a36c268316b3986611c596", "save_path": "github-repos/lean/bridgekat-calculus-of-constructions", "path": "github-repos/lean/bridgekat-calculus-of-constructions/calculus-of-constructions-68fe9b05e277f97dd3a36c268316b3986611c596/src/checker.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7662936324115011, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.37416845662746595}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport order.category.HeytAlg\n\n/-!\n# The category of boolean algebras\n\nThis defines `BoolAlg`, the category of boolean algebras.\n-/\n\nopen order_dual opposite set\n\nuniverses u\n\nopen category_theory\n\n/-- The category of boolean algebras. -/\ndef BoolAlg := bundled boolean_algebra\n\nnamespace BoolAlg\n\ninstance : has_coe_to_sort BoolAlg Type* := bundled.has_coe_to_sort\ninstance (X : BoolAlg) : boolean_algebra X := X.str\n\n/-- Construct a bundled `BoolAlg` from a `boolean_algebra`. -/\ndef of (\u03b1 : Type*) [boolean_algebra \u03b1] : BoolAlg := bundled.of \u03b1\n\n@[simp] lemma coe_of (\u03b1 : Type*) [boolean_algebra \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited BoolAlg := \u27e8of punit\u27e9\n\n/-- Turn a `BoolAlg` into a `BddDistLat` by forgetting its complement operation. -/\ndef to_BddDistLat (X : BoolAlg) : BddDistLat := BddDistLat.of X\n\n@[simp] lemma coe_to_BddDistLat (X : BoolAlg) : \u21a5X.to_BddDistLat = \u21a5X := rfl\n\ninstance : large_category.{u} BoolAlg := induced_category.category to_BddDistLat\ninstance : concrete_category BoolAlg := induced_category.concrete_category to_BddDistLat\n\ninstance has_forget_to_BddDistLat : has_forget\u2082 BoolAlg BddDistLat :=\ninduced_category.has_forget\u2082 to_BddDistLat\n\nsection\n\nlocal attribute [instance] bounded_lattice_hom_class.to_biheyting_hom_class\n\n@[simps] instance has_forget_to_HeytAlg : has_forget\u2082 BoolAlg HeytAlg :=\n{ forget\u2082 := { obj := \u03bb X, \u27e8X\u27e9, map := \u03bb X Y f, show bounded_lattice_hom X Y, from f } }\n\nend\n\n/-- Constructs an equivalence between Boolean algebras from an order isomorphism between them. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : BoolAlg.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := (e : bounded_lattice_hom \u03b1 \u03b2),\n  inv := (e.symm : bounded_lattice_hom \u03b2 \u03b1),\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : BoolAlg \u2964 BoolAlg :=\n{ obj := \u03bb X, of X\u1d52\u1d48, map := \u03bb X Y, bounded_lattice_hom.dual }\n\n/-- The equivalence between `BoolAlg` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : BoolAlg \u224c BoolAlg :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend BoolAlg\n\nlemma BoolAlg_dual_comp_forget_to_BddDistLat :\n  BoolAlg.dual \u22d9 forget\u2082 BoolAlg BddDistLat =\n    forget\u2082 BoolAlg BddDistLat \u22d9 BddDistLat.dual := rfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/order/category/BoolAlg.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521105, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3741004684429721}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Jeremy Avigad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.W\nimport Mathlib.PostPort\n\nuniverses u l u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Polynomial functors\n\nThis file defines polynomial functors and the W-type construction as a\npolynomial functor.  (For the M-type construction, see\npfunctor/M.lean.)\n-/\n\n/--\nA polynomial functor `P` is given by a type `A` and a family `B` of types over `A`. `P` maps\nany type `\u03b1` to a new type `P.obj \u03b1`, which is defined as the sigma type `\u03a3 x, P.B x \u2192 \u03b1`.\n\nAn element of `P.obj \u03b1` is a pair `\u27e8a, f\u27e9`, where `a` is an element of a type `A` and\n`f : B a \u2192 \u03b1`. Think of `a` as the shape of the object and `f` as an index to the relevant\nelements of `\u03b1`.\n-/\nstructure pfunctor where\n  A : Type u\n  B : A \u2192 Type u\n\nnamespace pfunctor\n\n\nprotected instance inhabited : Inhabited pfunctor :=\n  { default := mk Inhabited.default Inhabited.default }\n\n/-- Applying `P` to an object of `Type` -/\ndef obj (P : pfunctor) (\u03b1 : Type u_2) := sigma fun (x : A P) => B P x \u2192 \u03b1\n\n/-- Applying `P` to a morphism of `Type` -/\ndef map (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_3} (f : \u03b1 \u2192 \u03b2) : obj P \u03b1 \u2192 obj P \u03b2 :=\n  fun (_x : obj P \u03b1) => sorry\n\nprotected instance obj.inhabited (P : pfunctor) {\u03b1 : Type u} [Inhabited (A P)] [Inhabited \u03b1] :\n    Inhabited (obj P \u03b1) :=\n  { default := sigma.mk Inhabited.default fun (_x : B P Inhabited.default) => Inhabited.default }\n\nprotected instance obj.functor (P : pfunctor) : Functor (obj P) :=\n  { map := map P, mapConst := fun (\u03b1 \u03b2 : Type u_2) => map P \u2218 function.const \u03b2 }\n\nprotected theorem map_eq (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : A P)\n    (g : B P a \u2192 \u03b1) : f <$> sigma.mk a g = sigma.mk a (f \u2218 g) :=\n  rfl\n\nprotected theorem id_map (P : pfunctor) {\u03b1 : Type u_2} (x : obj P \u03b1) : id <$> x = id x := sorry\n\nprotected theorem comp_map (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_2} {\u03b3 : Type u_2} (f : \u03b1 \u2192 \u03b2)\n    (g : \u03b2 \u2192 \u03b3) (x : obj P \u03b1) : (g \u2218 f) <$> x = g <$> f <$> x :=\n  sorry\n\nprotected instance obj.is_lawful_functor (P : pfunctor) : is_lawful_functor (obj P) :=\n  is_lawful_functor.mk (pfunctor.id_map P) (pfunctor.comp_map P)\n\n/-- re-export existing definition of W-types and\nadapt it to a packaged definition of polynomial functor -/\ndef W (P : pfunctor) := W_type (B P)\n\n/- inhabitants of W types is awkward to encode as an instance\nassumption because there needs to be a value `a : P.A`\nsuch that `P.B a` is empty to yield a finite tree -/\n\n/-- root element  of a W tree -/\ndef W.head {P : pfunctor} : W P \u2192 A P := sorry\n\n/-- children of the root of a W tree -/\ndef W.children {P : pfunctor} (x : W P) : B P (W.head x) \u2192 W P := sorry\n\n/-- destructor for W-types -/\ndef W.dest {P : pfunctor} : W P \u2192 obj P (W P) := sorry\n\n/-- constructor for W-types -/\ndef W.mk {P : pfunctor} : obj P (W P) \u2192 W P := sorry\n\n@[simp] theorem W.dest_mk {P : pfunctor} (p : obj P (W P)) : W.dest (W.mk p) = p :=\n  sigma.cases_on p\n    fun (p_fst : A P) (p_snd : B P p_fst \u2192 W P) => Eq.refl (W.dest (W.mk (sigma.mk p_fst p_snd)))\n\n@[simp] theorem W.mk_dest {P : pfunctor} (p : W P) : W.mk (W.dest p) = p :=\n  W_type.cases_on p\n    fun (p_a : A P) (p_f : B P p_a \u2192 W_type (B P)) => Eq.refl (W.mk (W.dest (W_type.mk p_a p_f)))\n\n/-- `Idx` identifies a location inside the application of a pfunctor.\nFor `F : pfunctor`, `x : F.obj \u03b1` and `i : F.Idx`, `i` can designate\none part of `x` or is invalid, if `i.1 \u2260 x.1` -/\ndef Idx (P : pfunctor) := sigma fun (x : A P) => B P x\n\nprotected instance Idx.inhabited (P : pfunctor) [Inhabited (A P)]\n    [Inhabited (B P Inhabited.default)] : Inhabited (Idx P) :=\n  { default := sigma.mk Inhabited.default Inhabited.default }\n\n/-- `x.iget i` takes the component of `x` designated by `i` if any is or returns\na default value -/\ndef obj.iget {P : pfunctor} [DecidableEq (A P)] {\u03b1 : Type u_2} [Inhabited \u03b1] (x : obj P \u03b1)\n    (i : Idx P) : \u03b1 :=\n  dite (sigma.fst i = sigma.fst x)\n    (fun (h : sigma.fst i = sigma.fst x) => sigma.snd x (cast sorry (sigma.snd i)))\n    fun (h : \u00acsigma.fst i = sigma.fst x) => Inhabited.default\n\n@[simp] theorem fst_map {P : pfunctor} {\u03b1 : Type u} {\u03b2 : Type u} (x : obj P \u03b1) (f : \u03b1 \u2192 \u03b2) :\n    sigma.fst (f <$> x) = sigma.fst x :=\n  sigma.cases_on x\n    fun (x_fst : A P) (x_snd : B P x_fst \u2192 \u03b1) => Eq.refl (sigma.fst (f <$> sigma.mk x_fst x_snd))\n\n@[simp] theorem iget_map {P : pfunctor} [DecidableEq (A P)] {\u03b1 : Type u} {\u03b2 : Type u} [Inhabited \u03b1]\n    [Inhabited \u03b2] (x : obj P \u03b1) (f : \u03b1 \u2192 \u03b2) (i : Idx P) (h : sigma.fst i = sigma.fst x) :\n    obj.iget (f <$> x) i = f (obj.iget x i) :=\n  sorry\n\nend pfunctor\n\n\n/-\nComposition of polynomial functors.\n-/\n\nnamespace pfunctor\n\n\n/-- functor composition for polynomial functors -/\ndef comp (P\u2082 : pfunctor) (P\u2081 : pfunctor) : pfunctor :=\n  mk (sigma fun (a\u2082 : A P\u2082) => B P\u2082 a\u2082 \u2192 A P\u2081)\n    fun (a\u2082a\u2081 : sigma fun (a\u2082 : A P\u2082) => B P\u2082 a\u2082 \u2192 A P\u2081) =>\n      sigma fun (u : B P\u2082 (sigma.fst a\u2082a\u2081)) => B P\u2081 (sigma.snd a\u2082a\u2081 u)\n\n/-- constructor for composition -/\ndef comp.mk (P\u2082 : pfunctor) (P\u2081 : pfunctor) {\u03b1 : Type} (x : obj P\u2082 (obj P\u2081 \u03b1)) :\n    obj (comp P\u2082 P\u2081) \u03b1 :=\n  sigma.mk (sigma.mk (sigma.fst x) (sigma.fst \u2218 sigma.snd x))\n    fun (a\u2082a\u2081 : B (comp P\u2082 P\u2081) (sigma.mk (sigma.fst x) (sigma.fst \u2218 sigma.snd x))) =>\n      sigma.snd (sigma.snd x (sigma.fst a\u2082a\u2081)) (sigma.snd a\u2082a\u2081)\n\n/-- destructor for composition -/\ndef comp.get (P\u2082 : pfunctor) (P\u2081 : pfunctor) {\u03b1 : Type} (x : obj (comp P\u2082 P\u2081) \u03b1) :\n    obj P\u2082 (obj P\u2081 \u03b1) :=\n  sigma.mk (sigma.fst (sigma.fst x))\n    fun (a\u2082 : B P\u2082 (sigma.fst (sigma.fst x))) =>\n      sigma.mk (sigma.snd (sigma.fst x) a\u2082)\n        fun (a\u2081 : B P\u2081 (sigma.snd (sigma.fst x) a\u2082)) => sigma.snd x (sigma.mk a\u2082 a\u2081)\n\nend pfunctor\n\n\n/-\nLifting predicates and relations.\n-/\n\nnamespace pfunctor\n\n\ntheorem liftp_iff {P : pfunctor} {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (x : obj P \u03b1) :\n    functor.liftp p x \u2194 \u2203 (a : A P), \u2203 (f : B P a \u2192 \u03b1), x = sigma.mk a f \u2227 \u2200 (i : B P a), p (f i) :=\n  sorry\n\ntheorem liftp_iff' {P : pfunctor} {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : A P) (f : B P a \u2192 \u03b1) :\n    functor.liftp p (sigma.mk a f) \u2194 \u2200 (i : B P a), p (f i) :=\n  sorry\n\ntheorem liftr_iff {P : pfunctor} {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (x : obj P \u03b1) (y : obj P \u03b1) :\n    functor.liftr r x y \u2194\n        \u2203 (a : A P),\n          \u2203 (f\u2080 : B P a \u2192 \u03b1),\n            \u2203 (f\u2081 : B P a \u2192 \u03b1),\n              x = sigma.mk a f\u2080 \u2227 y = sigma.mk a f\u2081 \u2227 \u2200 (i : B P a), r (f\u2080 i) (f\u2081 i) :=\n  sorry\n\ntheorem supp_eq {P : pfunctor} {\u03b1 : Type u} (a : A P) (f : B P a \u2192 \u03b1) :\n    functor.supp (sigma.mk a f) = f '' set.univ :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/pfunctor/univariate/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.37410046844297207}}
{"text": "example : (0 : Nat) = Nat.zero := by\n  simp only [OfNat.ofNat]\n\nexample : (0 : Fin 9) = (Fin.ofNat 0) := by\n  simp only [OfNat.ofNat]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/788.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.6477982043529716, "lm_q1q2_score": 0.37410045116847124}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n\nDefinition of splitting fields, and definition of homomorphism into any field that splits\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.ring_theory.adjoin_root\nimport Mathlib.ring_theory.algebra_tower\nimport Mathlib.ring_theory.algebraic\nimport Mathlib.ring_theory.polynomial.default\nimport Mathlib.field_theory.minpoly\nimport Mathlib.linear_algebra.finite_dimensional\nimport Mathlib.tactic.field_simp\nimport Mathlib.PostPort\n\nuniverses u v w u_1 u_2 u_3 l \n\nnamespace Mathlib\n\nnamespace polynomial\n\n\n/-- a polynomial `splits` iff it is zero or all of its irreducible factors have `degree` 1 -/\ndef splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) (f : polynomial \u03b1) :=\n  f = 0 \u2228 \u2200 {g : polynomial \u03b2}, irreducible g \u2192 g \u2223 map i f \u2192 degree g = 1\n\n@[simp] theorem splits_zero {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) :\n    splits i 0 :=\n  Or.inl rfl\n\n@[simp] theorem splits_C {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) (a : \u03b1) :\n    splits i (coe_fn C a) :=\n  sorry\n\ntheorem splits_of_degree_eq_one {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} (hf : degree f = 1) : splits i f :=\n  sorry\n\ntheorem splits_of_degree_le_one {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} (hf : degree f \u2264 1) : splits i f :=\n  sorry\n\ntheorem splits_mul {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {f : polynomial \u03b1}\n    {g : polynomial \u03b1} (hf : splits i f) (hg : splits i g) : splits i (f * g) :=\n  sorry\n\ntheorem splits_of_splits_mul {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {g : polynomial \u03b1} (hfg : f * g \u2260 0) (h : splits i (f * g)) :\n    splits i f \u2227 splits i g :=\n  sorry\n\ntheorem splits_of_splits_of_dvd {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {g : polynomial \u03b1} (hf0 : f \u2260 0) (hf : splits i f) (hgf : g \u2223 f) :\n    splits i g :=\n  sorry\n\ntheorem splits_of_splits_gcd_left {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {g : polynomial \u03b1} (hf0 : f \u2260 0) (hf : splits i f) :\n    splits i (euclidean_domain.gcd f g) :=\n  splits_of_splits_of_dvd i hf0 hf (euclidean_domain.gcd_dvd_left f g)\n\ntheorem splits_of_splits_gcd_right {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {g : polynomial \u03b1} (hg0 : g \u2260 0) (hg : splits i g) :\n    splits i (euclidean_domain.gcd f g) :=\n  splits_of_splits_of_dvd i hg0 hg (euclidean_domain.gcd_dvd_right f g)\n\ntheorem splits_map_iff {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [field \u03b1] [field \u03b2] [field \u03b3]\n    (i : \u03b1 \u2192+* \u03b2) (j : \u03b2 \u2192+* \u03b3) {f : polynomial \u03b1} :\n    splits j (map i f) \u2194 splits (ring_hom.comp j i) f :=\n  sorry\n\ntheorem splits_one {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) : splits i 1 :=\n  splits_C i 1\n\ntheorem splits_of_is_unit {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {u : polynomial \u03b1} (hu : is_unit u) : splits i u :=\n  splits_of_splits_of_dvd i one_ne_zero (splits_one i) (iff.mp is_unit_iff_dvd_one hu)\n\ntheorem splits_X_sub_C {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {x : \u03b1} :\n    splits i (X - coe_fn C x) :=\n  splits_of_degree_eq_one i (degree_X_sub_C x)\n\ntheorem splits_X {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) : splits i X :=\n  splits_of_degree_eq_one i degree_X\n\ntheorem splits_id_iff_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} : splits (ring_hom.id \u03b2) (map i f) \u2194 splits i f :=\n  sorry\n\ntheorem splits_mul_iff {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {g : polynomial \u03b1} (hf : f \u2260 0) (hg : g \u2260 0) :\n    splits i (f * g) \u2194 splits i f \u2227 splits i g :=\n  sorry\n\ntheorem splits_prod {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {\u03b9 : Type w}\n    {s : \u03b9 \u2192 polynomial \u03b1} {t : finset \u03b9} :\n    (\u2200 (j : \u03b9), j \u2208 t \u2192 splits i (s j)) \u2192 splits i (finset.prod t fun (x : \u03b9) => s x) :=\n  sorry\n\ntheorem splits_prod_iff {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {\u03b9 : Type w}\n    {s : \u03b9 \u2192 polynomial \u03b1} {t : finset \u03b9} :\n    (\u2200 (j : \u03b9), j \u2208 t \u2192 s j \u2260 0) \u2192\n        (splits i (finset.prod t fun (x : \u03b9) => s x) \u2194 \u2200 (j : \u03b9), j \u2208 t \u2192 splits i (s j)) :=\n  sorry\n\ntheorem degree_eq_one_of_irreducible_of_splits {\u03b2 : Type v} [field \u03b2] {p : polynomial \u03b2}\n    (h_nz : p \u2260 0) (hp : irreducible p) (hp_splits : splits (ring_hom.id \u03b2) p) : degree p = 1 :=\n  sorry\n\ntheorem exists_root_of_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} (hs : splits i f) (hf0 : degree f \u2260 0) : \u2203 (x : \u03b2), eval\u2082 i x f = 0 :=\n  sorry\n\ntheorem exists_multiset_of_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} :\n    splits i f \u2192\n        \u2203 (s : multiset \u03b2),\n          map i f =\n            coe_fn C (coe_fn i (leading_coeff f)) *\n              multiset.prod (multiset.map (fun (a : \u03b2) => X - coe_fn C a) s) :=\n  sorry\n\n/-- Pick a root of a polynomial that splits. -/\ndef root_of_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {f : polynomial \u03b1}\n    (hf : splits i f) (hfd : degree f \u2260 0) : \u03b2 :=\n  classical.some (exists_root_of_splits i hf hfd)\n\ntheorem map_root_of_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} (hf : splits i f) (hfd : degree f \u2260 0) :\n    eval\u2082 i (root_of_splits i hf hfd) f = 0 :=\n  classical.some_spec (exists_root_of_splits i hf hfd)\n\ntheorem roots_map {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2) {f : polynomial \u03b1}\n    (hf : splits (ring_hom.id \u03b1) f) : roots (map i f) = multiset.map (\u21d1i) (roots f) :=\n  sorry\n\ntheorem eq_prod_roots_of_splits {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] {p : polynomial \u03b1}\n    {i : \u03b1 \u2192+* \u03b2} (hsplit : splits i p) :\n    map i p =\n        coe_fn C (coe_fn i (leading_coeff p)) *\n          multiset.prod (multiset.map (fun (a : \u03b2) => X - coe_fn C a) (roots (map i p))) :=\n  sorry\n\ntheorem eq_X_sub_C_of_splits_of_single_root {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2]\n    (i : \u03b1 \u2192+* \u03b2) {x : \u03b1} {h : polynomial \u03b1} (h_splits : splits i h)\n    (h_roots : roots (map i h) = singleton (coe_fn i x)) :\n    h = coe_fn C (leading_coeff h) * (X - coe_fn C x) :=\n  sorry\n\ntheorem nat_degree_multiset_prod {R : Type u_1} [integral_domain R] {s : multiset (polynomial R)}\n    (h : \u2200 (p : polynomial R), p \u2208 s \u2192 p \u2260 0) :\n    nat_degree (multiset.prod s) = multiset.sum (multiset.map nat_degree s) :=\n  sorry\n\ntheorem nat_degree_eq_card_roots {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] {p : polynomial \u03b1}\n    {i : \u03b1 \u2192+* \u03b2} (hsplit : splits i p) : nat_degree p = coe_fn multiset.card (roots (map i p)) :=\n  sorry\n\ntheorem degree_eq_card_roots {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] {p : polynomial \u03b1}\n    {i : \u03b1 \u2192+* \u03b2} (p_ne_zero : p \u2260 0) (hsplit : splits i p) :\n    degree p = \u2191(coe_fn multiset.card (roots (map i p))) :=\n  sorry\n\ntheorem splits_of_exists_multiset {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} {s : multiset \u03b2}\n    (hs :\n      map i f =\n        coe_fn C (coe_fn i (leading_coeff f)) *\n          multiset.prod (multiset.map (fun (a : \u03b2) => X - coe_fn C a) s)) :\n    splits i f :=\n  sorry\n\ntheorem splits_of_splits_id {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} : splits (ring_hom.id \u03b1) f \u2192 splits i f :=\n  sorry\n\ntheorem splits_iff_exists_multiset {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (i : \u03b1 \u2192+* \u03b2)\n    {f : polynomial \u03b1} :\n    splits i f \u2194\n        \u2203 (s : multiset \u03b2),\n          map i f =\n            coe_fn C (coe_fn i (leading_coeff f)) *\n              multiset.prod (multiset.map (fun (a : \u03b2) => X - coe_fn C a) s) :=\n  sorry\n\ntheorem splits_comp_of_splits {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [field \u03b1] [field \u03b2] [field \u03b3]\n    (i : \u03b1 \u2192+* \u03b2) (j : \u03b2 \u2192+* \u03b3) {f : polynomial \u03b1} (h : splits i f) :\n    splits (ring_hom.comp j i) f :=\n  sorry\n\n/-- A monic polynomial `p` that has as much roots as its degree\ncan be written `p = \u220f(X - a)`, for `a` in `p.roots`. -/\ntheorem prod_multiset_X_sub_C_of_monic_of_roots_card_eq {\u03b1 : Type u} [field \u03b1] {p : polynomial \u03b1}\n    (hmonic : monic p) (hroots : coe_fn multiset.card (roots p) = nat_degree p) :\n    multiset.prod (multiset.map (fun (a : \u03b1) => X - coe_fn C a) (roots p)) = p :=\n  sorry\n\n/-- A polynomial `p` that has as much roots as its degree\ncan be written `p = p.leading_coeff * \u220f(X - a)`, for `a` in `p.roots`. -/\ntheorem C_leading_coeff_mul_prod_multiset_X_sub_C {\u03b1 : Type u} [field \u03b1] {p : polynomial \u03b1}\n    (hroots : coe_fn multiset.card (roots p) = nat_degree p) :\n    coe_fn C (leading_coeff p) *\n          multiset.prod (multiset.map (fun (a : \u03b1) => X - coe_fn C a) (roots p)) =\n        p :=\n  sorry\n\n/-- A polynomial splits if and only if it has as much roots as its degree. -/\ntheorem splits_iff_card_roots {\u03b1 : Type u} [field \u03b1] {p : polynomial \u03b1} :\n    splits (ring_hom.id \u03b1) p \u2194 coe_fn multiset.card (roots p) = nat_degree p :=\n  sorry\n\nend polynomial\n\n\n/-- If `p` is the minimal polynomial of `a` over `F` then `F[a] \u2243\u2090[F] F[x]/(p)` -/\ndef alg_equiv.adjoin_singleton_equiv_adjoin_root_minpoly (F : Type u_1) [field F] {R : Type u_2}\n    [comm_ring R] [algebra F R] (x : R) :\n    alg_equiv F (\u21a5(algebra.adjoin F (singleton x))) (adjoin_root (minpoly F x)) :=\n  alg_equiv.symm\n    (alg_equiv.of_bijective\n      (alg_hom.cod_restrict (adjoin_root.lift_hom (minpoly F x) x sorry)\n        (algebra.adjoin F (singleton x)) sorry)\n      sorry)\n\n-- Speed up the following proof.\n\n-- TODO: Why is this so slow?\n\n/-- If `K` and `L` are field extensions of `F` and we have `s : finset K` such that\nthe minimal polynomial of each `x \u2208 s` splits in `L` then `algebra.adjoin F s` embeds in `L`. -/\ntheorem lift_of_splits {F : Type u_1} {K : Type u_2} {L : Type u_3} [field F] [field K] [field L]\n    [algebra F K] [algebra F L] (s : finset K) :\n    (\u2200 (x : K), x \u2208 s \u2192 is_integral F x \u2227 polynomial.splits (algebra_map F L) (minpoly F x)) \u2192\n        Nonempty (alg_hom F (\u21a5(algebra.adjoin F \u2191s)) L) :=\n  sorry\n\nnamespace polynomial\n\n\n/-- Non-computably choose an irreducible factor from a polynomial. -/\ndef factor {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) : polynomial \u03b1 :=\n  dite (\u2203 (g : polynomial \u03b1), irreducible g \u2227 g \u2223 f)\n    (fun (H : \u2203 (g : polynomial \u03b1), irreducible g \u2227 g \u2223 f) => classical.some H)\n    fun (H : \u00ac\u2203 (g : polynomial \u03b1), irreducible g \u2227 g \u2223 f) => X\n\nprotected instance irreducible_factor {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    irreducible (factor f) :=\n  sorry\n\ntheorem factor_dvd_of_not_is_unit {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1} (hf1 : \u00acis_unit f) :\n    factor f \u2223 f :=\n  sorry\n\ntheorem factor_dvd_of_degree_ne_zero {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1} (hf : degree f \u2260 0) :\n    factor f \u2223 f :=\n  factor_dvd_of_not_is_unit (mt degree_eq_zero_of_is_unit hf)\n\ntheorem factor_dvd_of_nat_degree_ne_zero {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1}\n    (hf : nat_degree f \u2260 0) : factor f \u2223 f :=\n  factor_dvd_of_degree_ne_zero (mt nat_degree_eq_of_degree_eq_some hf)\n\n/-- Divide a polynomial f by X - C r where r is a root of f in a bigger field extension. -/\ndef remove_factor {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) : polynomial (adjoin_root (factor f)) :=\n  map (adjoin_root.of (factor f)) f /\u2098 (X - coe_fn C (adjoin_root.root (factor f)))\n\ntheorem X_sub_C_mul_remove_factor {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1)\n    (hf : nat_degree f \u2260 0) :\n    (X - coe_fn C (adjoin_root.root (factor f))) * remove_factor f =\n        map (adjoin_root.of (factor f)) f :=\n  sorry\n\ntheorem nat_degree_remove_factor {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    nat_degree (remove_factor f) = nat_degree f - 1 :=\n  sorry\n\ntheorem nat_degree_remove_factor' {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1} {n : \u2115}\n    (hfn : nat_degree f = n + 1) : nat_degree (remove_factor f) = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nat_degree (remove_factor f) = n)) (nat_degree_remove_factor f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nat_degree f - 1 = n)) hfn))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (n + 1 - 1 = n)) (nat.add_sub_cancel n 1))) (Eq.refl n)))\n\n/-- Auxiliary construction to a splitting field of a polynomial. Uses induction on the degree. -/\ndef splitting_field_aux (n : \u2115) {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    nat_degree f = n \u2192 Type u :=\n  nat.rec_on n\n    (fun (\u03b1 : Type u) (_x : field \u03b1) (_x_1 : polynomial \u03b1) (_x : nat_degree _x_1 = 0) => \u03b1)\n    fun (n : \u2115)\n      (ih : {\u03b1 : Type u} \u2192 [_inst_4 : field \u03b1] \u2192 (f : polynomial \u03b1) \u2192 nat_degree f = n \u2192 Type u)\n      (\u03b1 : Type u) (_x : field \u03b1) (f : polynomial \u03b1) (hf : nat_degree f = Nat.succ n) =>\n      ih (remove_factor f) (nat_degree_remove_factor' hf)\n\nnamespace splitting_field_aux\n\n\ntheorem succ {\u03b1 : Type u} [field \u03b1] (n : \u2115) (f : polynomial \u03b1) (hfn : nat_degree f = n + 1) :\n    splitting_field_aux (n + 1) f hfn =\n        splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn) :=\n  rfl\n\nprotected instance field (n : \u2115) {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1}\n    (hfn : nat_degree f = n) : field (splitting_field_aux n f hfn) :=\n  nat.rec_on n\n    (fun (\u03b1 : Type u) (_x : field \u03b1) (_x_1 : polynomial \u03b1) (_x_2 : nat_degree _x_1 = 0) => _x)\n    fun (n : \u2115)\n      (ih :\n      {\u03b1 : Type u} \u2192\n        [_inst_4 : field \u03b1] \u2192\n          {f : polynomial \u03b1} \u2192 (hfn : nat_degree f = n) \u2192 field (splitting_field_aux n f hfn))\n      (\u03b1 : Type u) (_x : field \u03b1) (f : polynomial \u03b1) (hf : nat_degree f = Nat.succ n) =>\n      ih (nat_degree_remove_factor' hf)\n\nprotected instance inhabited {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n) : Inhabited (splitting_field_aux n f hfn) :=\n  { default := bit1 (bit0 (bit1 (bit0 (bit0 1)))) }\n\nprotected instance algebra (n : \u2115) {\u03b1 : Type u} [field \u03b1] {f : polynomial \u03b1}\n    (hfn : nat_degree f = n) : algebra \u03b1 (splitting_field_aux n f hfn) :=\n  nat.rec_on n\n    (fun (\u03b1 : Type u) (_x : field \u03b1) (_x_1 : polynomial \u03b1) (_x_2 : nat_degree _x_1 = 0) =>\n      algebra.id \u03b1)\n    fun (n : \u2115)\n      (ih :\n      {\u03b1 : Type u} \u2192\n        [_inst_4 : field \u03b1] \u2192\n          {f : polynomial \u03b1} \u2192 (hfn : nat_degree f = n) \u2192 algebra \u03b1 (splitting_field_aux n f hfn))\n      (\u03b1 : Type u) (_x : field \u03b1) (f : polynomial \u03b1) (hfn : nat_degree f = Nat.succ n) =>\n      algebra.comap.algebra \u03b1 (adjoin_root (factor f))\n        (splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn))\n\nprotected instance algebra' {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n + 1) :\n    algebra (adjoin_root (factor f)) (splitting_field_aux (n + 1) f hfn) :=\n  splitting_field_aux.algebra n sorry\n\nprotected instance algebra'' {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n + 1) :\n    algebra \u03b1 (splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn)) :=\n  splitting_field_aux.algebra (n + 1) hfn\n\nprotected instance algebra''' {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n + 1) :\n    algebra (adjoin_root (factor f))\n        (splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn)) :=\n  splitting_field_aux.algebra n (nat_degree_remove_factor' hfn)\n\nprotected instance scalar_tower {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n + 1) :\n    is_scalar_tower \u03b1 (adjoin_root (factor f)) (splitting_field_aux (n + 1) f hfn) :=\n  is_scalar_tower.of_algebra_map_eq fun (x : \u03b1) => rfl\n\nprotected instance scalar_tower' {\u03b1 : Type u} [field \u03b1] {n : \u2115} {f : polynomial \u03b1}\n    (hfn : nat_degree f = n + 1) :\n    is_scalar_tower \u03b1 (adjoin_root (factor f))\n        (splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn)) :=\n  is_scalar_tower.of_algebra_map_eq fun (x : \u03b1) => rfl\n\ntheorem algebra_map_succ {\u03b1 : Type u} [field \u03b1] (n : \u2115) (f : polynomial \u03b1)\n    (hfn : nat_degree f = n + 1) :\n    algebra_map \u03b1 (splitting_field_aux (n + 1) f hfn) =\n        ring_hom.comp\n          (algebra_map (adjoin_root (factor f))\n            (splitting_field_aux n (remove_factor f) (nat_degree_remove_factor' hfn)))\n          (adjoin_root.of (factor f)) :=\n  rfl\n\nprotected theorem splits (n : \u2115) {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1)\n    (hfn : nat_degree f = n) : splits (algebra_map \u03b1 (splitting_field_aux n f hfn)) f :=\n  sorry\n\ntheorem exists_lift (n : \u2115) {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) (hfn : nat_degree f = n)\n    {\u03b2 : Type u_1} [field \u03b2] (j : \u03b1 \u2192+* \u03b2) (hf : splits j f) :\n    \u2203 (k : splitting_field_aux n f hfn \u2192+* \u03b2),\n        ring_hom.comp k (algebra_map \u03b1 (splitting_field_aux n f hfn)) = j :=\n  sorry\n\ntheorem adjoin_roots (n : \u2115) {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) (hfn : nat_degree f = n) :\n    algebra.adjoin \u03b1\n          \u2191(multiset.to_finset (roots (map (algebra_map \u03b1 (splitting_field_aux n f hfn)) f))) =\n        \u22a4 :=\n  sorry\n\nend splitting_field_aux\n\n\n/-- A splitting field of a polynomial. -/\ndef splitting_field {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :=\n  splitting_field_aux (nat_degree f) f sorry\n\nnamespace splitting_field\n\n\nprotected instance field {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) : field (splitting_field f) :=\n  splitting_field_aux.field (nat_degree f) (_proof_1 f)\n\nprotected instance inhabited {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    Inhabited (splitting_field f) :=\n  { default := bit1 (bit0 (bit1 (bit0 (bit0 1)))) }\n\nprotected instance algebra {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    algebra \u03b1 (splitting_field f) :=\n  splitting_field_aux.algebra (nat_degree f) (_proof_1 f)\n\nprotected theorem splits {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    splits (algebra_map \u03b1 (splitting_field f)) f :=\n  splitting_field_aux.splits (nat_degree f) f (_proof_1 f)\n\n/-- Embeds the splitting field into any other field that splits the polynomial. -/\ndef lift {\u03b1 : Type u} {\u03b2 : Type v} [field \u03b1] [field \u03b2] (f : polynomial \u03b1) [algebra \u03b1 \u03b2]\n    (hb : splits (algebra_map \u03b1 \u03b2) f) : alg_hom \u03b1 (splitting_field f) \u03b2 :=\n  alg_hom.mk (ring_hom.to_fun (classical.some sorry)) sorry sorry sorry sorry sorry\n\ntheorem adjoin_roots {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    algebra.adjoin \u03b1 \u2191(multiset.to_finset (roots (map (algebra_map \u03b1 (splitting_field f)) f))) =\n        \u22a4 :=\n  splitting_field_aux.adjoin_roots (nat_degree f) f (_proof_1 f)\n\nend splitting_field\n\n\n/-- Typeclass characterising splitting fields. -/\nclass is_splitting_field (\u03b1 : Type u) (\u03b2 : Type v) [field \u03b1] [field \u03b2] [algebra \u03b1 \u03b2]\n    (f : polynomial \u03b1)\n    where\n  splits : splits (algebra_map \u03b1 \u03b2) f\n  adjoin_roots : algebra.adjoin \u03b1 \u2191(multiset.to_finset (roots (map (algebra_map \u03b1 \u03b2) f))) = \u22a4\n\nnamespace is_splitting_field\n\n\nprotected instance splitting_field {\u03b1 : Type u} [field \u03b1] (f : polynomial \u03b1) :\n    is_splitting_field \u03b1 (splitting_field f) f :=\n  mk (splitting_field.splits f) (splitting_field.adjoin_roots f)\n\nprotected instance map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [field \u03b1] [field \u03b2] [field \u03b3]\n    [algebra \u03b1 \u03b2] [algebra \u03b2 \u03b3] [algebra \u03b1 \u03b3] [is_scalar_tower \u03b1 \u03b2 \u03b3] (f : polynomial \u03b1)\n    [is_splitting_field \u03b1 \u03b3 f] : is_splitting_field \u03b2 \u03b3 (map (algebra_map \u03b1 \u03b2) f) :=\n  sorry\n\ntheorem splits_iff {\u03b1 : Type u} (\u03b2 : Type v) [field \u03b1] [field \u03b2] [algebra \u03b1 \u03b2] (f : polynomial \u03b1)\n    [is_splitting_field \u03b1 \u03b2 f] : splits (ring_hom.id \u03b1) f \u2194 \u22a4 = \u22a5 :=\n  sorry\n\ntheorem mul {\u03b1 : Type u} (\u03b2 : Type v) {\u03b3 : Type w} [field \u03b1] [field \u03b2] [field \u03b3] [algebra \u03b1 \u03b2]\n    [algebra \u03b2 \u03b3] [algebra \u03b1 \u03b3] [is_scalar_tower \u03b1 \u03b2 \u03b3] (f : polynomial \u03b1) (g : polynomial \u03b1)\n    (hf : f \u2260 0) (hg : g \u2260 0) [is_splitting_field \u03b1 \u03b2 f]\n    [is_splitting_field \u03b2 \u03b3 (map (algebra_map \u03b1 \u03b2) g)] : is_splitting_field \u03b1 \u03b3 (f * g) :=\n  sorry\n\n/-- Splitting field of `f` embeds into any field that splits `f`. -/\ndef lift {\u03b1 : Type u} (\u03b2 : Type v) {\u03b3 : Type w} [field \u03b1] [field \u03b2] [field \u03b3] [algebra \u03b1 \u03b2]\n    [algebra \u03b1 \u03b3] (f : polynomial \u03b1) [is_splitting_field \u03b1 \u03b2 f] (hf : splits (algebra_map \u03b1 \u03b3) f) :\n    alg_hom \u03b1 \u03b2 \u03b3 :=\n  dite (f = 0)\n    (fun (hf0 : f = 0) =>\n      alg_hom.comp (algebra.of_id \u03b1 \u03b3)\n        (alg_hom.comp (\u2191(algebra.bot_equiv \u03b1 \u03b2)) (eq.mpr sorry algebra.to_top)))\n    fun (hf0 : \u00acf = 0) => alg_hom.comp (eq.mpr sorry (Classical.choice sorry)) algebra.to_top\n\ntheorem finite_dimensional {\u03b1 : Type u} (\u03b2 : Type v) [field \u03b1] [field \u03b2] [algebra \u03b1 \u03b2]\n    (f : polynomial \u03b1) [is_splitting_field \u03b1 \u03b2 f] : finite_dimensional \u03b1 \u03b2 :=\n  sorry\n\n/-- Any splitting field is isomorphic to `splitting_field f`. -/\ndef alg_equiv {\u03b1 : Type u} (\u03b2 : Type v) [field \u03b1] [field \u03b2] [algebra \u03b1 \u03b2] (f : polynomial \u03b1)\n    [is_splitting_field \u03b1 \u03b2 f] : alg_equiv \u03b1 \u03b2 (splitting_field f) :=\n  alg_equiv.of_bijective (lift \u03b2 f sorry) sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/field_theory/splitting_field_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.37407817956874767}}
{"text": "example (x y z : Prop) (f : x \u2192 y \u2192 z) (xp : x) (yp : y) : z := by\n  specialize f xp yp\n  assumption\n\nexample (B C : Prop) (f : forall (A : Prop), A \u2192 C) (x : B) : C := by\n  specialize f _ x\n  exact f\n\nexample (B C : Prop) (f : forall {A : Prop}, A \u2192 C) (x : B) : C := by\n  specialize f x\n  exact f\n\nexample (B C : Prop) (f : forall {A : Prop}, A \u2192 C) (x : B) : C := by\n  specialize @f _ x\n  exact f\n\nexample (X : Type) [Add X] (f : forall {A : Type} [Add A], A \u2192 A \u2192 A) (x : X) : X := by\n  specialize f x x\n  assumption\n\ndef ex (f : Nat \u2192 Nat \u2192 Nat) : Nat := by\n  specialize f _ _\n  exact f\n  exact 10\n  exact 2\n\nexample : ex (. - .) = 8 :=\n  rfl\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/specialize1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.37396079428061463}}
{"text": "import Mathlib.Tactic.Basic\n\nexample : (\u03b1 : Type) \u00d7 List \u03b1 := by\n  constructor\n  \u00b7 exact [0,1]\n\n-- example : (\u03b1 : Type) \u00d7 List \u03b1 := by\n--   fsplit\n--   - exact \u2115\n--   - exact [0,1]\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/Split.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791786991753929, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3739607942806146}}
{"text": "import category_theory.limits.shapes.images\n\nuniverses v u\n\nnoncomputable theory\n\nopen category_theory category_theory.limits\n\nnamespace category_theory.limits\n\nvariables {C : Type u} [category.{v} C] [has_strong_epi_mono_factorisations.{v} C]\n\nvariables {X Y : C} (f : X \u27f6 Y)\n\n@[simps]\ndef unique_factorise (I' : C) (e : X \u27f6 I') (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e] [mono m] :\n  I' \u2245 image f :=\n{ hom := {strong_epi_mono_factorisation . I := I', m := m, e := e}.to_mono_is_image.lift _,\n  inv := image.lift {strong_epi_mono_factorisation . I := I', m := m, e := e}.to_mono_factorisation,\n  hom_inv_id' := by erw [\u2190 cancel_mono m, category.assoc, category.id_comp, image.lift_fac, is_image.lift_fac],\n  inv_hom_id' := by erw [\u2190 cancel_mono (image.\u03b9 f), category.id_comp, category.assoc, is_image.lift_fac, image.lift_fac] }\n\nlemma unique_factorise_hom_comp_image (I' : C) (e : X \u27f6 I') (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e] [mono m] :\n  (unique_factorise f I' e m comm).hom \u226b image.\u03b9 f = m :=\nis_image.lift_fac _ _\n\nlemma unique_factorise_inv_comp_mono (I' : C) (e : X \u27f6 I') (m : I' \u27f6 Y) (comm : e \u226b m = f) [strong_epi e] [mono m] :\n  (unique_factorise f I' e m comm).inv \u226b m = image.\u03b9 f :=\nimage.lift_fac _\n\nend category_theory.limits", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/category/images.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.709019146082187, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3738775111216083}}
{"text": "import primitive_element\n\nlemma hmmm (F : Type*) [field F] (E : Type*) [field E] [algebra F E] (hs : is_separable F E) (hfd : finite_dimensional F E) : true :=\nbegin\n    cases primitive_element F E hs hfd,\n    sorry,\nend", "meta": {"author": "pglutz", "repo": "galois_theory", "sha": "4561c2c97d4c49377356e1d7a2051dedc87d30ba", "save_path": "github-repos/lean/pglutz-galois_theory", "path": "github-repos/lean/pglutz-galois_theory/galois_theory-4561c2c97d4c49377356e1d7a2051dedc87d30ba/src/beyond_primitive_element_theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3738775046371476}}
{"text": "import complexity_class.stack_rec\nimport polytime.basic\nimport polytime.stack_rec_size\n\nopen tree (stack_step iterator_stack stack_step_polysize)\nopen tencodable function polysize\n\nnamespace polytime\n\nopen_locale complexity_class\n\nvariables {\u03b1 \u03b2 \u03b3 \u03b4 : Type} [tencodable \u03b1] [tencodable \u03b2] [tencodable \u03b3] [tencodable \u03b4]\n\nsection stack_rec\n\nattribute [complexity] complexity_class.stack_iterate\n\nvariables {base : \u03b3 \u2192 \u03b1 \u2192 \u03b2} {pre\u2081 pre\u2082 : \u03b3 \u2192 tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b1}\n  {post : \u03b3 \u2192 \u03b2 \u2192 \u03b2 \u2192 tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b2}\n\n@[complexity]\nprotected theorem stack_rec [polysize \u03b1] [polysize \u03b2] [polysize \u03b3] {st : \u03b3 -> tree unit} {arg : \u03b3 \u2192 \u03b1} (hst : st \u2208\u2091 PTIME) (harg : arg \u2208\u2091 PTIME) (hb : base \u2208\u2091 PTIME) (hpr\u2081 : pre\u2081 \u2208\u2091 PTIME) (hpr\u2082 : pre\u2082 \u2208\u2091 PTIME) (hpo : post \u2208\u2091 PTIME)\n  (hpr\u2081' : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit) (sf : \u03b1), pre\u2081 usf.1 usf.2.1 usf.2.2 sf))\n  (hpr\u2082' : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit) (sf : \u03b1), pre\u2082 usf.1 usf.2.1 usf.2.2 sf))\n  (hpo' : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit \u00d7 \u03b1) (sf : \u03b2 \u00d7 \u03b2), post usf.1 sf.1 sf.2 usf.2.1 usf.2.2.1 usf.2.2.2)) :\n  polytime.mem (\u03bb x : \u03b3, (st x).stack_rec (base x) (pre\u2081 x) (pre\u2082 x) (post x) (arg x)) :=\nbegin\n  suffices : polytime.mem (\u03bb x, (stack_step (base x) (pre\u2081 x) (pre\u2082 x) (post x))^[(st x).time_steps] [sum.inl (st x, arg x, none)]),\n  { rw complexity_class.of_some,\n    convert complexity_class.mem.comp (show polytime.mem (\u03bb x : list (iterator_stack \u03b1 \u03b2), x.head'.bind sum.get_right), by complexity) this, \n    simp, },\n  apply iterate, { dsimp only [tree.time_steps], complexity, }, { complexity, }, { complexity, },\n  cases stack_step_polysize (polytime.size_le hst) (polytime.size_le harg)\n    (polytime.size_le hb) hpr\u2081' hpr\u2082' hpo' with p hp,\n  use p, intros x m _, exact hp x m,\nend\n\nlemma tree_eq : polytime.mem_pred (@eq (tree unit)) :=\nbegin\n  rw \u2190 complexity_class.mem_iff_mem_rel,\n  complexity using \u03bb x y, x.stack_rec (\u03bb y' : tree unit, (y' = tree.nil : bool))\n    (\u03bb _ _ y', y'.left) (\u03bb _ _ y', y'.right)\n    (\u03bb b\u2081 b\u2082 _ _ y, !(y = tree.nil : bool) && (b\u2081 && b\u2082)) y,\n  { use 0, simp, },\n  induction x using tree.unit_rec_on with l r ih\u2081 ih\u2082 generalizing y; cases y; simp [*],\nend\n\n@[complexity] lemma eq : (@eq \u03b1) \u2208\u209a PTIME :=\nby { have := tree_eq, complexity using (\u03bb x y, encode x = encode y), simp, }\n\n@[complexity] lemma tree_cmp : (@tree.cmp unit _ _) \u2208\u2091 PTIME :=\nbegin\n  complexity using \u03bb x y,\n    x.stack_rec (\u03bb y', if y' = tree.nil then ordering.eq else ordering.lt)\n      (\u03bb _ _ y', y'.left) (\u03bb _ _ y', y'.right)\n      (\u03bb c\u2081 c\u2082 _ _ y', if y' = tree.nil then ordering.gt else c\u2081.or_else c\u2082) y,\n  { use 0, simp, },\n  induction x using tree.unit_rec_on with l r ih\u2081 ih\u2082 generalizing y; rcases y with _|\u27e8\u27e8\u27e9, _, _\u27e9; simp [*, tree.cmp, ordering.or_else],\nend\n\n@[complexity] lemma tree_lt : ((<) : tree unit \u2192 tree unit \u2192 Prop) \u2208\u209a PTIME :=\nby { complexity using \u03bb x y, x.cmp y = ordering.lt, rw tree.tree_lt_def, }\n\n@[complexity] lemma tree_le : ((\u2264) : tree unit \u2192 tree unit \u2192 Prop) \u2208\u209a PTIME :=\nby { complexity using \u03bb x y, \u00ac(y < x), simp, }\n\nend stack_rec\n\nend polytime\n", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/polytime/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850933, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3738775046371476}}
{"text": "/-\nCopyright (c) 2022 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Asta H. From, Jannis Limperg\n-/\nimport Aesop\n\nset_option aesop.check.all true\n\nattribute [aesop unsafe [50% constructors, 50% cases]] List.Mem\n\ntheorem Mem.map (f : \u03b1 \u2192 \u03b2) (x : \u03b1) (xs : List \u03b1) (h : x \u2208 xs) :\n    f x \u2208 xs.map f := by\n  induction h <;> aesop\n", "meta": {"author": "JLimperg", "repo": "aesop", "sha": "c68fb1d5a9172498230d81d95c61f6461bea6722", "save_path": "github-repos/lean/JLimperg-aesop", "path": "github-repos/lean/JLimperg-aesop/aesop-c68fb1d5a9172498230d81d95c61f6461bea6722/tests/run/10.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.37376623600860864}}
{"text": "/-\n\n## `exfalso`\n\n-/\n\nsyntax \"exfalso\" : tactic\n\nmacro_rules\n  | `(tactic| exfalso) => `(tactic| apply False.elim)\n\n/-\nexample : False -> p := by\n  intro h;\n  exfalso;\n  skip;\n  exact h;\n-/", "meta": {"author": "kbuzzard", "repo": "mathlib4_experiments", "sha": "87cb879b4d602c8ecfd9283b7c0b06015abdbab1", "save_path": "github-repos/lean/kbuzzard-mathlib4_experiments", "path": "github-repos/lean/kbuzzard-mathlib4_experiments/mathlib4_experiments-87cb879b4d602c8ecfd9283b7c0b06015abdbab1/mathlib4_experiments/Tactic/Exfalso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3737662360086086}}
{"text": "import tactic data.set\nimport prop.language prop.semantics prop.syntax\n \nnamespace prop\n\nlemma axiomL_tautology (v \u03c6) : \u03c6 \u2208 AxiomL \u2192 (v \u22a8 \u03c6) := \nbegin\n  intro h,\n  cases h,\n  case P1: {\n    sorry\n  },\n  case P2: {\n    sorry\n  },\n  case P3: {\n    sorry\n  },\nend\n\ntheorem soundness (v \u03c6) : \u22a2\u2097 \u03c6 \u2192 v \u22a8 \u03c6 :=\nbegin\n  intro he,\n  induction he,\n  \n  case in_axioms : _ _ hA {\n    exact @axiomL_tautology v _ hA,\n  },\n\n  case in_context :{\n    sorry,\n  },\n\n  case mp :{\n    sorry,\n  },\nend\n\nend prop", "meta": {"author": "SnO2WMaN", "repo": "lean-modal-logic", "sha": "b58e6723ffd38db50d0b50bb5b6ed8102de9929b", "save_path": "github-repos/lean/SnO2WMaN-lean-modal-logic", "path": "github-repos/lean/SnO2WMaN-lean-modal-logic/lean-modal-logic-b58e6723ffd38db50d0b50bb5b6ed8102de9929b/src/prop/soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3737575507124135}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computability.tm_computable\n\n/-!\n# Polynomial time stuff to eventually port to mathlib\n-/\n\nsection poly_time\n\nopen turing computability\n\n/-- A function is computable in polynomial time if there is a polynomial time implementation.\n  In particular this definition is extensional, so the definition of the function isn't important,\n  as long as there is a Turing machine implementing the same input/output behaviour. -/\ndef poly_time {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) :=\n\u03a3 (ea : fin_encoding \u03b1) (eb : fin_encoding \u03b2),\n  tm2_computable_in_poly_time ea eb f\n\nnoncomputable lemma poly_time_id (\u03b1 : Type) (ea : fin_encoding \u03b1) : poly_time (id : \u03b1 \u2192 \u03b1) :=\n\u27e8ea, ea, id_computable_in_poly_time ea\u27e9\n\nend poly_time", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/to_mathlib/poly_time.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.37375755071241346}}
{"text": "/-\nCopyright (c) 2020 Kevin Buzzard. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kevin Buzzard, Calle S\u00f6nne\n-/\n\nimport topology.category.CompHaus\nimport topology.connected\nimport topology.subset_properties\nimport topology.locally_constant.basic\nimport category_theory.adjunction.reflective\nimport category_theory.monad.limits\nimport category_theory.limits.constructions.epi_mono\nimport category_theory.Fintype\n\n/-!\n# The category of Profinite Types\n\nWe construct the category of profinite topological spaces,\noften called profinite sets -- perhaps they could be called\nprofinite types in Lean.\n\nThe type of profinite topological spaces is called `Profinite`. It has a category\ninstance and is a fully faithful subcategory of `Top`. The fully faithful functor\nis called `Profinite_to_Top`.\n\n## Implementation notes\n\nA profinite type is defined to be a topological space which is\ncompact, Hausdorff and totally disconnected.\n\n## TODO\n\n0. Link to category of projective limits of finite discrete sets.\n1. finite coproducts\n2. Clausen/Scholze topology on the category `Profinite`.\n\n## Tags\n\nprofinite\n\n-/\n\nuniverse u\n\nopen category_theory\n\n/-- The type of profinite topological spaces. -/\nstructure Profinite :=\n(to_CompHaus : CompHaus)\n[is_totally_disconnected : totally_disconnected_space to_CompHaus]\n\nnamespace Profinite\n\n/--\nConstruct a term of `Profinite` from a type endowed with the structure of a\ncompact, Hausdorff and totally disconnected topological space.\n-/\ndef of (X : Type*) [topological_space X] [compact_space X] [t2_space X]\n  [totally_disconnected_space X] : Profinite := \u27e8\u27e8\u27e8X\u27e9\u27e9\u27e9\n\ninstance : inhabited Profinite := \u27e8Profinite.of pempty\u27e9\n\ninstance category : category Profinite := induced_category.category to_CompHaus\ninstance concrete_category : concrete_category Profinite := induced_category.concrete_category _\ninstance has_forget\u2082 : has_forget\u2082 Profinite Top := induced_category.has_forget\u2082 _\n\ninstance : has_coe_to_sort Profinite Type* := \u27e8\u03bb X, X.to_CompHaus\u27e9\ninstance {X : Profinite} : totally_disconnected_space X := X.is_totally_disconnected\n\n-- We check that we automatically infer that Profinite sets are compact and Hausdorff.\nexample {X : Profinite} : compact_space X := infer_instance\nexample {X : Profinite} : t2_space X := infer_instance\n\n@[simp]\nlemma coe_to_CompHaus {X : Profinite} : (X.to_CompHaus : Type*) = X :=\nrfl\n\n@[simp] lemma coe_id (X : Profinite) : (\ud835\udfd9 X : X \u2192 X) = id := rfl\n\n@[simp] lemma coe_comp {X Y Z : Profinite} (f : X \u27f6 Y) (g : Y \u27f6 Z) : (f \u226b g : X \u2192 Z) = g \u2218 f := rfl\n\nend Profinite\n\n/-- The fully faithful embedding of `Profinite` in `CompHaus`. -/\n@[simps, derive [full, faithful]]\ndef Profinite_to_CompHaus : Profinite \u2964 CompHaus := induced_functor _\n\n/-- The fully faithful embedding of `Profinite` in `Top`. This is definitionally the same as the\nobvious composite. -/\n@[simps, derive [full, faithful]]\ndef Profinite.to_Top : Profinite \u2964 Top := forget\u2082 _ _\n\n@[simp] lemma Profinite.to_CompHaus_to_Top :\n  Profinite_to_CompHaus \u22d9 CompHaus_to_Top = Profinite.to_Top :=\nrfl\n\nsection Profinite\nlocal attribute [instance] connected_component_setoid\n\n/--\n(Implementation) The object part of the connected_components functor from compact Hausdorff spaces\nto Profinite spaces, given by quotienting a space by its connected components.\nSee: https://stacks.math.columbia.edu/tag/0900\n-/\n-- Without explicit universe annotations here, Lean introduces two universe variables and\n-- unhelpfully defines a function `CompHaus.{max u\u2081 u\u2082} \u2192 Profinite.{max u\u2081 u\u2082}`.\ndef CompHaus.to_Profinite_obj (X : CompHaus.{u}) : Profinite.{u} :=\n{ to_CompHaus :=\n  { to_Top := Top.of (connected_components X),\n    is_compact := quotient.compact_space,\n    is_hausdorff := connected_components.t2 },\n  is_totally_disconnected := connected_components.totally_disconnected_space }\n\n/--\n(Implementation) The bijection of homsets to establish the reflective adjunction of Profinite\nspaces in compact Hausdorff spaces.\n-/\ndef Profinite.to_CompHaus_equivalence (X : CompHaus.{u}) (Y : Profinite.{u}) :\n  (CompHaus.to_Profinite_obj X \u27f6 Y) \u2243 (X \u27f6 Profinite_to_CompHaus.obj Y) :=\n{ to_fun := \u03bb f,\n  { to_fun := f.1 \u2218 quotient.mk,\n    continuous_to_fun := continuous.comp f.2 (continuous_quotient_mk) },\n  inv_fun := \u03bb g,\n    { to_fun := continuous.connected_components_lift g.2,\n      continuous_to_fun := continuous.connected_components_lift_continuous g.2},\n  left_inv := \u03bb f, continuous_map.ext $ \u03bb x, quotient.induction_on x $ \u03bb a, rfl,\n  right_inv := \u03bb f, continuous_map.ext $ \u03bb x, rfl }\n\n/--\nThe connected_components functor from compact Hausdorff spaces to profinite spaces,\nleft adjoint to the inclusion functor.\n-/\ndef CompHaus.to_Profinite : CompHaus \u2964 Profinite :=\nadjunction.left_adjoint_of_equiv Profinite.to_CompHaus_equivalence (\u03bb _ _ _ _ _, rfl)\n\nlemma CompHaus.to_Profinite_obj' (X : CompHaus) :\n  \u21a5(CompHaus.to_Profinite.obj X) = connected_components X := rfl\n\n/-- Finite types are given the discrete topology. -/\ndef Fintype.discrete_topology (A : Fintype) : topological_space A := \u22a5\n\nsection discrete_topology\n\nlocal attribute [instance] Fintype.discrete_topology\n\n/-- The natural functor from `Fintype` to `Profinite`, endowing a finite type with the\ndiscrete topology. -/\n@[simps] def Fintype.to_Profinite : Fintype \u2964 Profinite :=\n{ obj := \u03bb A, Profinite.of A,\n  map := \u03bb _ _ f, \u27e8f\u27e9 }\n\nend discrete_topology\n\nend Profinite\n\nnamespace Profinite\n\n/-- An explicit limit cone for a functor `F : J \u2964 Profinite`, defined in terms of\n`Top.limit_cone`. -/\ndef limit_cone {J : Type u} [small_category J] (F : J \u2964 Profinite.{u}) :\n  limits.cone F :=\n{ X :=\n  { to_CompHaus := (CompHaus.limit_cone (F \u22d9 Profinite_to_CompHaus)).X,\n    is_totally_disconnected :=\n    begin\n      change totally_disconnected_space \u21a5{u : \u03a0 (j : J), (F.obj j) | _},\n      exact subtype.totally_disconnected_space,\n    end },\n  \u03c0 := { app := (CompHaus.limit_cone (F \u22d9 Profinite_to_CompHaus)).\u03c0.app } }\n\n/-- The limit cone `Profinite.limit_cone F` is indeed a limit cone. -/\ndef limit_cone_is_limit {J : Type u} [small_category J] (F : J \u2964 Profinite.{u}) :\n  limits.is_limit (limit_cone F) :=\n{ lift := \u03bb S, (CompHaus.limit_cone_is_limit (F \u22d9 Profinite_to_CompHaus)).lift\n    (Profinite_to_CompHaus.map_cone S),\n  uniq' := \u03bb S m h,\n    (CompHaus.limit_cone_is_limit _).uniq (Profinite_to_CompHaus.map_cone S) _ h }\n\n/-- The adjunction between CompHaus.to_Profinite and Profinite.to_CompHaus -/\ndef to_Profinite_adj_to_CompHaus : CompHaus.to_Profinite \u22a3 Profinite_to_CompHaus :=\nadjunction.adjunction_of_equiv_left _ _\n\n/-- The category of profinite sets is reflective in the category of compact hausdroff spaces -/\ninstance to_CompHaus.reflective : reflective Profinite_to_CompHaus :=\n{ to_is_right_adjoint := \u27e8CompHaus.to_Profinite, Profinite.to_Profinite_adj_to_CompHaus\u27e9 }\n\nnoncomputable\ninstance to_CompHaus.creates_limits : creates_limits Profinite_to_CompHaus :=\nmonadic_creates_limits _\n\nnoncomputable\ninstance to_Top.reflective : reflective Profinite.to_Top :=\nreflective.comp Profinite_to_CompHaus CompHaus_to_Top\n\nnoncomputable\ninstance to_Top.creates_limits : creates_limits Profinite.to_Top :=\nmonadic_creates_limits _\n\ninstance has_limits : limits.has_limits Profinite :=\nhas_limits_of_has_limits_creates_limits Profinite.to_Top\n\ninstance has_colimits : limits.has_colimits Profinite :=\nhas_colimits_of_reflective Profinite_to_CompHaus\n\nnoncomputable\ninstance forget_preserves_limits : limits.preserves_limits (forget Profinite) :=\nby apply limits.comp_preserves_limits Profinite.to_Top (forget Top)\n\nvariables {X Y : Profinite.{u}} (f : X \u27f6 Y)\n\n/-- Any morphism of profinite spaces is a closed map. -/\nlemma is_closed_map : is_closed_map f :=\nCompHaus.is_closed_map _\n\n/-- Any continuous bijection of profinite spaces induces an isomorphism. -/\nlemma is_iso_of_bijective (bij : function.bijective f) : is_iso f :=\nbegin\n  haveI := CompHaus.is_iso_of_bijective (Profinite_to_CompHaus.map f) bij,\n  exact is_iso_of_fully_faithful Profinite_to_CompHaus _\nend\n\n/-- Any continuous bijection of profinite spaces induces an isomorphism. -/\nnoncomputable def iso_of_bijective (bij : function.bijective f) : X \u2245 Y :=\nby letI := Profinite.is_iso_of_bijective f bij; exact as_iso f\n\ninstance forget_reflects_isomorphisms : reflects_isomorphisms (forget Profinite) :=\n\u27e8by introsI A B f hf; exact Profinite.is_iso_of_bijective _ ((is_iso_iff_bijective f).mp hf)\u27e9\n\n/-- Construct an isomorphism from a homeomorphism. -/\n@[simps hom inv] def iso_of_homeo (f : X \u2243\u209c Y) : X \u2245 Y :=\n{ hom := \u27e8f, f.continuous\u27e9,\n  inv := \u27e8f.symm, f.symm.continuous\u27e9,\n  hom_inv_id' := by { ext x, exact f.symm_apply_apply x },\n  inv_hom_id' := by { ext x, exact f.apply_symm_apply x } }\n\n/-- Construct a homeomorphism from an isomorphism. -/\n@[simps] def homeo_of_iso (f : X \u2245 Y) : X \u2243\u209c Y :=\n{ to_fun := f.hom,\n  inv_fun := f.inv,\n  left_inv := \u03bb x, by { change (f.hom \u226b f.inv) x = x, rw [iso.hom_inv_id, coe_id, id.def] },\n  right_inv := \u03bb x, by { change (f.inv \u226b f.hom) x = x, rw [iso.inv_hom_id, coe_id, id.def] },\n  continuous_to_fun := f.hom.continuous,\n  continuous_inv_fun := f.inv.continuous }\n\n/-- The equivalence between isomorphisms in `Profinite` and homeomorphisms\nof topological spaces. -/\n@[simps] def iso_equiv_homeo : (X \u2245 Y) \u2243 (X \u2243\u209c Y) :=\n{ to_fun := homeo_of_iso,\n  inv_fun := iso_of_homeo,\n  left_inv := \u03bb f, by { ext, refl },\n  right_inv := \u03bb f, by { ext, refl } }\n\nlemma epi_iff_surjective {X Y : Profinite.{u}} (f : X \u27f6 Y) : epi f \u2194 function.surjective f :=\nbegin\n  split,\n  { contrapose!,\n    rintros \u27e8y, hy\u27e9 hf,\n    let C := set.range f,\n    have hC : is_closed C := (is_compact_range f.continuous).is_closed,\n    let U := C\u1d9c,\n    have hU : is_open U := is_open_compl_iff.mpr hC,\n    have hyU : y \u2208 U,\n    { refine set.mem_compl _, rintro \u27e8y', hy'\u27e9, exact hy y' hy' },\n    have hUy : U \u2208 nhds y := hU.mem_nhds hyU,\n    obtain \u27e8V, hV, hyV, hVU\u27e9 := is_topological_basis_clopen.mem_nhds_iff.mp hUy,\n    classical,\n    letI : topological_space (ulift.{u} $ fin 2) := \u22a5,\n    let Z := of (ulift.{u} $ fin 2),\n    let g : Y \u27f6 Z := \u27e8(locally_constant.of_clopen hV).map ulift.up, locally_constant.continuous _\u27e9,\n    let h : Y \u27f6 Z := \u27e8\u03bb _, \u27e81\u27e9, continuous_const\u27e9,\n    have H : h = g,\n    { rw \u2190 cancel_epi f,\n      ext x, dsimp [locally_constant.of_clopen],\n      rw if_neg, { refl },\n      refine mt (\u03bb \u03b1, hVU \u03b1) _,\n      simp only [set.mem_range_self, not_true, not_false_iff, set.mem_compl_eq], },\n    apply_fun (\u03bb e, (e y).down) at H,\n    dsimp [locally_constant.of_clopen] at H,\n    rw if_pos hyV at H,\n    exact top_ne_bot H },\n  { rw \u2190 category_theory.epi_iff_surjective,\n    apply faithful_reflects_epi (forget Profinite) },\nend\n\nlemma mono_iff_injective {X Y : Profinite.{u}} (f : X \u27f6 Y) : mono f \u2194 function.injective f :=\nbegin\n  split,\n  { intro h,\n    haveI : limits.preserves_limits Profinite_to_CompHaus := infer_instance,\n    haveI : mono (Profinite_to_CompHaus.map f) := infer_instance,\n    rwa \u2190 CompHaus.mono_iff_injective },\n  { rw \u2190 category_theory.mono_iff_injective,\n    apply faithful_reflects_mono (forget Profinite) }\nend\n\nend Profinite\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/topology/category/Profinite/default.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3737012108468191}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport category_theory.concrete_category.basic\n\n/-!\n# The category of pointed types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis defines `Pointed`, the category of pointed types.\n\n## TODO\n\n* Monoidal structure\n* Upgrade `Type_to_Pointed` to an equivalence\n-/\n\nopen category_theory\n\nuniverses u\nvariables {\u03b1 \u03b2 : Type*}\n\n/-- The category of pointed types. -/\nstructure Pointed : Type.{u + 1} :=\n(X : Type.{u})\n(point : X)\n\nnamespace Pointed\n\ninstance : has_coe_to_sort Pointed Type* := \u27e8X\u27e9\n\nattribute [protected] Pointed.X\n\n/-- Turns a point into a pointed type. -/\ndef of {X : Type*} (point : X) : Pointed := \u27e8X, point\u27e9\n\n@[simp] lemma coe_of {X : Type*} (point : X) : \u21a5(of point) = X := rfl\n\nalias of \u2190 _root_.prod.Pointed\n\ninstance : inhabited Pointed := \u27e8of ((), ())\u27e9\n\n/-- Morphisms in `Pointed`. -/\n@[ext] protected structure hom (X Y : Pointed.{u}) : Type u :=\n(to_fun : X \u2192 Y)\n(map_point : to_fun X.point = Y.point)\n\nnamespace hom\n\n/-- The identity morphism of `X : Pointed`. -/\n@[simps] def id (X : Pointed) : hom X X := \u27e8id, rfl\u27e9\n\ninstance (X : Pointed) : inhabited (hom X X) := \u27e8id X\u27e9\n\n/-- Composition of morphisms of `Pointed`. -/\n@[simps] def comp {X Y Z : Pointed.{u}} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n\u27e8g.to_fun \u2218 f.to_fun, by rw [function.comp_apply, f.map_point, g.map_point]\u27e9\n\nend hom\n\ninstance large_category : large_category Pointed :=\n{ hom := hom,\n  id := hom.id,\n  comp := @hom.comp,\n  id_comp' := \u03bb _ _ _, hom.ext _ _ rfl,\n  comp_id' := \u03bb _ _ _, hom.ext _ _ rfl,\n  assoc' := \u03bb _ _ _ _ _ _ _, hom.ext _ _ rfl }\n\ninstance concrete_category : concrete_category Pointed :=\n{ forget := { obj := Pointed.X, map := @hom.to_fun },\n  forget_faithful := \u27e8@hom.ext\u27e9 }\n\n/-- Constructs a isomorphism between pointed types from an equivalence that preserves the point\nbetween them. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : Pointed} (e : \u03b1 \u2243 \u03b2) (he : e \u03b1.point = \u03b2.point) : \u03b1 \u2245 \u03b2 :=\n{ hom := \u27e8e, he\u27e9,\n  inv := \u27e8e.symm, e.symm_apply_eq.2 he.symm\u27e9,\n  hom_inv_id' := Pointed.hom.ext _ _ e.symm_comp_self,\n  inv_hom_id' := Pointed.hom.ext _ _ e.self_comp_symm }\n\nend Pointed\n\n/-- `option` as a functor from types to pointed types. This is the free functor. -/\n@[simps] def Type_to_Pointed : Type.{u} \u2964 Pointed.{u} :=\n{ obj := \u03bb X, \u27e8option X, none\u27e9,\n  map := \u03bb X Y f, \u27e8option.map f, rfl\u27e9,\n  map_id' := \u03bb X, Pointed.hom.ext _ _ option.map_id,\n  map_comp' := \u03bb X Y Z f g, Pointed.hom.ext _ _ (option.map_comp_map _ _).symm }\n\n/-- `Type_to_Pointed` is the free functor. -/\ndef Type_to_Pointed_forget_adjunction : Type_to_Pointed \u22a3 forget Pointed :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y, { to_fun := \u03bb f, f.to_fun \u2218 option.some,\n                        inv_fun := \u03bb f, \u27e8\u03bb o, o.elim Y.point f, rfl\u27e9,\n                        left_inv := \u03bb f, by { ext, cases x, exact f.map_point.symm, refl },\n                        right_inv := \u03bb f, funext $ \u03bb _, rfl },\n  hom_equiv_naturality_left_symm' := \u03bb X' X Y f g, by { ext, cases x; refl }, }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/category/Pointed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.61878043374385, "lm_q1q2_score": 0.3737012020246671}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport category_theory.category.Cat\nimport category_theory.groupoid\n\n/-!\n# Objects of a category up to an isomorphism\n\n`is_isomorphic X Y := nonempty (X \u2245 Y)` is an equivalence relation on the objects of a category.\nThe quotient with respect to this relation defines a functor from our category to `Type`.\n-/\n\nuniverses v u\n\nnamespace category_theory\n\nsection category\n\nvariables {C : Type u} [category.{v} C]\n\n/-- An object `X` is isomorphic to an object `Y`, if `X \u2245 Y` is not empty. -/\ndef is_isomorphic : C \u2192 C \u2192 Prop := \u03bb X Y, nonempty (X \u2245 Y)\n\nvariable (C)\n\n/-- `is_isomorphic` defines a setoid. -/\ndef is_isomorphic_setoid : setoid C :=\n{ r := is_isomorphic,\n  iseqv := \u27e8\u03bb X, \u27e8iso.refl X\u27e9, \u03bb X Y \u27e8\u03b1\u27e9, \u27e8\u03b1.symm\u27e9, \u03bb X Y Z \u27e8\u03b1\u27e9 \u27e8\u03b2\u27e9, \u27e8\u03b1.trans \u03b2\u27e9\u27e9 }\n\nend category\n\n/--\nThe functor that sends each category to the quotient space of its objects up to an isomorphism.\n-/\ndef isomorphism_classes : Cat.{v u} \u2964 Type u :=\n{ obj := \u03bb C, quotient (is_isomorphic_setoid C.\u03b1),\n  map := \u03bb C D F, quot.map F.obj $ \u03bb X Y \u27e8f\u27e9, \u27e8F.map_iso f\u27e9 }\n\nlemma groupoid.is_isomorphic_iff_nonempty_hom {C : Type u} [groupoid.{v} C] {X Y : C} :\n  is_isomorphic X Y \u2194 nonempty (X \u27f6 Y) :=\n(groupoid.iso_equiv_hom X Y).nonempty_iff_nonempty\n\n-- PROJECT: define `skeletal`, and show every category is equivalent to a skeletal category,\n-- using the axiom of choice to pick a representative of every isomorphism class.\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/isomorphism_classes.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7905303285397349, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.37367062406805485}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, S\u00e9bastien Gou\u00ebzel, Patrick Massot\n-/\nimport topology.uniform_space.cauchy\nimport topology.uniform_space.separation\nimport topology.dense_embedding\n\n/-!\n# Uniform embeddings of uniform spaces.\n\nExtension of uniform continuous functions.\n-/\n\nopen filter topological_space set classical\nopen_locale classical uniformity topological_space filter\n\nsection\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n          [uniform_space \u03b1] [uniform_space \u03b2] [uniform_space \u03b3]\nuniverse u\n\nstructure uniform_inducing (f : \u03b1 \u2192 \u03b2) : Prop :=\n(comap_uniformity : comap (\u03bbx:\u03b1\u00d7\u03b1, (f x.1, f x.2)) (\ud835\udce4 \u03b2) = \ud835\udce4 \u03b1)\n\nlemma uniform_inducing.mk' {f : \u03b1 \u2192 \u03b2} (h : \u2200 s, s \u2208 \ud835\udce4 \u03b1 \u2194\n    \u2203 t \u2208 \ud835\udce4 \u03b2, \u2200 x y : \u03b1, (f x, f y) \u2208 t \u2192 (x, y) \u2208 s) : uniform_inducing f :=\n\u27e8by simp [eq_comm, filter.ext_iff, subset_def, h]\u27e9\n\nlemma uniform_inducing.comp {g : \u03b2 \u2192 \u03b3} (hg : uniform_inducing g)\n  {f : \u03b1 \u2192 \u03b2} (hf : uniform_inducing f) : uniform_inducing (g \u2218 f) :=\n\u27e8 by rw [show (\u03bb (x : \u03b1 \u00d7 \u03b1), ((g \u2218 f) x.1, (g \u2218 f) x.2)) =\n         (\u03bb y : \u03b2 \u00d7 \u03b2, (g y.1, g y.2)) \u2218 (\u03bb x : \u03b1 \u00d7 \u03b1, (f x.1, f x.2)), by ext ; simp,\n        \u2190 filter.comap_comap, hg.1, hf.1]\u27e9\n\nstructure uniform_embedding (f : \u03b1 \u2192 \u03b2) extends uniform_inducing f : Prop :=\n(inj : function.injective f)\n\nlemma uniform_embedding_subtype_val {p : \u03b1 \u2192 Prop} :\n  uniform_embedding (subtype.val : subtype p \u2192 \u03b1) :=\n{ comap_uniformity := rfl,\n  inj := subtype.val_injective }\n\nlemma uniform_embedding_subtype_coe {p : \u03b1 \u2192 Prop} :\n  uniform_embedding (coe : subtype p \u2192 \u03b1) :=\nuniform_embedding_subtype_val\n\nlemma uniform_embedding_set_inclusion {s t : set \u03b1} (hst : s \u2286 t) :\n  uniform_embedding (inclusion hst) :=\n{ comap_uniformity :=\n    by { erw [uniformity_subtype, uniformity_subtype, comap_comap], congr },\n  inj := inclusion_injective hst }\n\nlemma uniform_embedding.comp {g : \u03b2 \u2192 \u03b3} (hg : uniform_embedding g)\n  {f : \u03b1 \u2192 \u03b2} (hf : uniform_embedding f) : uniform_embedding (g \u2218 f) :=\n{ inj := hg.inj.comp hf.inj,\n  ..hg.to_uniform_inducing.comp hf.to_uniform_inducing }\n\ntheorem uniform_embedding_def {f : \u03b1 \u2192 \u03b2} :\n  uniform_embedding f \u2194 function.injective f \u2227 \u2200 s, s \u2208 \ud835\udce4 \u03b1 \u2194\n    \u2203 t \u2208 \ud835\udce4 \u03b2, \u2200 x y : \u03b1, (f x, f y) \u2208 t \u2192 (x, y) \u2208 s :=\nbegin\n  split,\n  { rintro \u27e8\u27e8h\u27e9, h'\u27e9,\n    rw [eq_comm, filter.ext_iff] at h,\n    simp [*, subset_def] },\n  { rintro \u27e8h, h'\u27e9,\n    refine uniform_embedding.mk \u27e8_\u27e9 h,\n    rw [eq_comm, filter.ext_iff],\n    simp [*, subset_def] }\nend\n\ntheorem uniform_embedding_def' {f : \u03b1 \u2192 \u03b2} :\n  uniform_embedding f \u2194 function.injective f \u2227 uniform_continuous f \u2227\n    \u2200 s, s \u2208 \ud835\udce4 \u03b1 \u2192\n      \u2203 t \u2208 \ud835\udce4 \u03b2, \u2200 x y : \u03b1, (f x, f y) \u2208 t \u2192 (x, y) \u2208 s :=\nby simp only [uniform_embedding_def, uniform_continuous_def]; exact\n\u27e8\u03bb \u27e8I, H\u27e9, \u27e8I, \u03bb s su, (H _).2 \u27e8s, su, \u03bb x y, id\u27e9, \u03bb s, (H s).1\u27e9,\n \u03bb \u27e8I, H\u2081, H\u2082\u27e9, \u27e8I, \u03bb s, \u27e8H\u2082 s,\n   \u03bb \u27e8t, tu, h\u27e9, mem_sets_of_superset (H\u2081 t tu) (\u03bb \u27e8a, b\u27e9, h a b)\u27e9\u27e9\u27e9\n\nlemma uniform_inducing.uniform_continuous {f : \u03b1 \u2192 \u03b2}\n  (hf : uniform_inducing f) : uniform_continuous f :=\nby simp [uniform_continuous, hf.comap_uniformity.symm, tendsto_comap]\n\nlemma uniform_inducing.uniform_continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : uniform_inducing g) :\n  uniform_continuous f \u2194 uniform_continuous (g \u2218 f) :=\nby { dsimp only [uniform_continuous, tendsto],\n  rw [\u2190 hg.comap_uniformity, \u2190 map_le_iff_le_comap, filter.map_map] }\n\nlemma uniform_inducing.inducing {f : \u03b1 \u2192 \u03b2} (h : uniform_inducing f) : inducing f :=\nbegin\n  refine \u27e8eq_of_nhds_eq_nhds $ assume a, _ \u27e9,\n  rw [nhds_induced, nhds_eq_uniformity, nhds_eq_uniformity, \u2190 h.comap_uniformity,\n    comap_lift'_eq, comap_lift'_eq2];\n    { refl <|> exact monotone_preimage }\nend\n\nlemma uniform_inducing.prod {\u03b1' : Type*} {\u03b2' : Type*} [uniform_space \u03b1'] [uniform_space \u03b2']\n  {e\u2081 : \u03b1 \u2192 \u03b1'} {e\u2082 : \u03b2 \u2192 \u03b2'} (h\u2081 : uniform_inducing e\u2081) (h\u2082 : uniform_inducing e\u2082) :\n  uniform_inducing (\u03bbp:\u03b1\u00d7\u03b2, (e\u2081 p.1, e\u2082 p.2)) :=\n\u27e8by simp [(\u2218), uniformity_prod, h\u2081.comap_uniformity.symm, h\u2082.comap_uniformity.symm,\n           comap_inf, comap_comap]\u27e9\n\nlemma uniform_inducing.dense_inducing {f : \u03b1 \u2192 \u03b2} (h : uniform_inducing f) (hd : dense_range f) :\n  dense_inducing f :=\n{ dense   := hd,\n  induced := h.inducing.induced }\n\nlemma uniform_embedding.embedding {f : \u03b1 \u2192 \u03b2} (h : uniform_embedding f) : embedding f :=\n{ induced := h.to_uniform_inducing.inducing.induced,\n  inj := h.inj }\n\nlemma uniform_embedding.dense_embedding {f : \u03b1 \u2192 \u03b2} (h : uniform_embedding f) (hd : dense_range f) :\n  dense_embedding f :=\n{ dense   := hd,\n  inj     := h.inj,\n  induced := h.embedding.induced }\n\nlemma closure_image_mem_nhds_of_uniform_inducing\n  {s : set (\u03b1\u00d7\u03b1)} {e : \u03b1 \u2192 \u03b2} (b : \u03b2)\n  (he\u2081 : uniform_inducing e) (he\u2082 : dense_inducing e) (hs : s \u2208 \ud835\udce4 \u03b1) :\n  \u2203a, closure (e '' {a' | (a, a') \u2208 s}) \u2208 \ud835\udcdd b :=\nhave s \u2208 comap (\u03bbp:\u03b1\u00d7\u03b1, (e p.1, e p.2)) (\ud835\udce4 \u03b2),\n  from he\u2081.comap_uniformity.symm \u25b8 hs,\nlet \u27e8t\u2081, ht\u2081u, ht\u2081\u27e9 := this in\nhave ht\u2081 : \u2200p:\u03b1\u00d7\u03b1, (e p.1, e p.2) \u2208 t\u2081 \u2192 p \u2208 s, from ht\u2081,\nlet \u27e8t\u2082, ht\u2082u, ht\u2082s, ht\u2082c\u27e9 := comp_symm_of_uniformity ht\u2081u in\nlet \u27e8t, htu, hts, htc\u27e9 := comp_symm_of_uniformity ht\u2082u in\nhave preimage e {b' | (b, b') \u2208 t\u2082} \u2208 comap e (\ud835\udcdd b),\n  from preimage_mem_comap $ mem_nhds_left b ht\u2082u,\nlet \u27e8a, (ha : (b, e a) \u2208 t\u2082)\u27e9 := (he\u2082.comap_nhds_ne_bot _).nonempty_of_mem this in\nhave \u2200b' (s' : set (\u03b2 \u00d7 \u03b2)), (b, b') \u2208 t \u2192 s' \u2208 \ud835\udce4 \u03b2 \u2192\n  ({y : \u03b2 | (b', y) \u2208 s'} \u2229 e '' {a' : \u03b1 | (a, a') \u2208 s}).nonempty,\n  from assume b' s' hb' hs',\n  have preimage e {b'' | (b', b'') \u2208 s' \u2229 t} \u2208 comap e (\ud835\udcdd b'),\n    from preimage_mem_comap $ mem_nhds_left b' $ inter_mem_sets hs' htu,\n  let \u27e8a\u2082, ha\u2082s', ha\u2082t\u27e9 := (he\u2082.comap_nhds_ne_bot _).nonempty_of_mem this in\n  have (e a, e a\u2082) \u2208 t\u2081,\n    from ht\u2082c $ prod_mk_mem_comp_rel (ht\u2082s ha) $ htc $ prod_mk_mem_comp_rel hb' ha\u2082t,\n  have e a\u2082 \u2208 {b'':\u03b2 | (b', b'') \u2208 s'} \u2229 e '' {a' | (a, a') \u2208 s},\n    from \u27e8ha\u2082s', mem_image_of_mem _ $ ht\u2081 (a, a\u2082) this\u27e9,\n  \u27e8_, this\u27e9,\nhave \u2200b', (b, b') \u2208 t \u2192 ne_bot (\ud835\udcdd b' \u2293 \ud835\udcdf (e '' {a' | (a, a') \u2208 s})),\nbegin\n  intros b' hb',\n  rw [nhds_eq_uniformity, lift'_inf_principal_eq, lift'_ne_bot_iff],\n  exact assume s, this b' s hb',\n  exact monotone_inter monotone_preimage monotone_const\nend,\nhave \u2200b', (b, b') \u2208 t \u2192 b' \u2208 closure (e '' {a' | (a, a') \u2208 s}),\n  from assume b' hb', by rw [closure_eq_cluster_pts]; exact this b' hb',\n\u27e8a, (\ud835\udcdd b).sets_of_superset (mem_nhds_left b htu) this\u27e9\n\nlemma uniform_embedding_subtype_emb (p : \u03b1 \u2192 Prop) {e : \u03b1 \u2192 \u03b2} (ue : uniform_embedding e)\n  (de : dense_embedding e) : uniform_embedding (dense_embedding.subtype_emb p e) :=\n{ comap_uniformity := by simp [comap_comap, (\u2218), dense_embedding.subtype_emb,\n           uniformity_subtype, ue.comap_uniformity.symm],\n  inj := (de.subtype p).inj }\n\nlemma uniform_embedding.prod {\u03b1' : Type*} {\u03b2' : Type*} [uniform_space \u03b1'] [uniform_space \u03b2']\n  {e\u2081 : \u03b1 \u2192 \u03b1'} {e\u2082 : \u03b2 \u2192 \u03b2'} (h\u2081 : uniform_embedding e\u2081) (h\u2082 : uniform_embedding e\u2082) :\n  uniform_embedding (\u03bbp:\u03b1\u00d7\u03b2, (e\u2081 p.1, e\u2082 p.2)) :=\n{ inj := h\u2081.inj.prod_map h\u2082.inj,\n  ..h\u2081.to_uniform_inducing.prod h\u2082.to_uniform_inducing }\n\nlemma is_complete_of_complete_image {m : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hm : uniform_inducing m)\n  (hs : is_complete (m '' s)) : is_complete s :=\nbegin\n  intros f hf hfs,\n  rw le_principal_iff at hfs,\n  obtain \u27e8_, \u27e8x, hx, rfl\u27e9, hyf\u27e9 : \u2203 y \u2208 m '' s, map m f \u2264 \ud835\udcdd y,\n    from hs (f.map m) (hf.map hm.uniform_continuous)\n      (le_principal_iff.2 (image_mem_map hfs)),\n  rw [map_le_iff_le_comap, \u2190 nhds_induced, \u2190 hm.inducing.induced] at hyf,\n  exact \u27e8x, hx, hyf\u27e9\nend\n\nlemma is_complete.complete_space_coe {s : set \u03b1} (hs : is_complete s) :\n  complete_space s :=\ncomplete_space_iff_is_complete_univ.2 $\n  is_complete_of_complete_image uniform_embedding_subtype_coe.to_uniform_inducing $ by simp [hs]\n\n/-- A set is complete iff its image under a uniform inducing map is complete. -/\nlemma is_complete_image_iff {m : \u03b1 \u2192 \u03b2} {s : set \u03b1} (hm : uniform_inducing m) :\n  is_complete (m '' s) \u2194 is_complete s :=\nbegin\n  refine \u27e8is_complete_of_complete_image hm, \u03bb c, _\u27e9,\n  haveI : complete_space s := c.complete_space_coe,\n  set m' : s \u2192 \u03b2 := m \u2218 coe,\n  suffices : is_complete (range m'), by rwa [range_comp, subtype.range_coe] at this,\n  have hm' : uniform_inducing m' := hm.comp uniform_embedding_subtype_coe.to_uniform_inducing,\n  intros f hf hfm,\n  rw filter.le_principal_iff at hfm,\n  have cf' : cauchy (comap m' f) :=\n    hf.comap' hm'.comap_uniformity.le (ne_bot.comap_of_range_mem hf.1 hfm),\n  rcases complete_space.complete cf' with \u27e8x, hx\u27e9,\n  rw [hm'.inducing.nhds_eq_comap, comap_le_comap_iff hfm] at hx,\n  use [m' x, mem_range_self _, hx]\nend\n\nlemma complete_space_iff_is_complete_range {f : \u03b1 \u2192 \u03b2} (hf : uniform_inducing f) :\n  complete_space \u03b1 \u2194 is_complete (range f) :=\nby rw [complete_space_iff_is_complete_univ, \u2190 is_complete_image_iff hf, image_univ]\n\nlemma uniform_inducing.is_complete_range [complete_space \u03b1] {f : \u03b1 \u2192 \u03b2}\n  (hf : uniform_inducing f) :\n  is_complete (range f) :=\n(complete_space_iff_is_complete_range hf).1 \u2039_\u203a\n\nlemma complete_space_congr {e : \u03b1 \u2243 \u03b2} (he : uniform_embedding e) :\n  complete_space \u03b1 \u2194 complete_space \u03b2 :=\nby rw [complete_space_iff_is_complete_range he.to_uniform_inducing, e.range_eq_univ,\n  complete_space_iff_is_complete_univ]\n\nlemma complete_space_coe_iff_is_complete {s : set \u03b1} :\n  complete_space s \u2194 is_complete s :=\n(complete_space_iff_is_complete_range uniform_embedding_subtype_coe.to_uniform_inducing).trans $\n  by rw [subtype.range_coe]\n\nlemma is_closed.complete_space_coe [complete_space \u03b1] {s : set \u03b1} (hs : is_closed s) :\n  complete_space s :=\nhs.is_complete.complete_space_coe\n\nlemma complete_space_extension {m : \u03b2 \u2192 \u03b1} (hm : uniform_inducing m) (dense : dense_range m)\n  (h : \u2200f:filter \u03b2, cauchy f \u2192 \u2203x:\u03b1, map m f \u2264 \ud835\udcdd x) : complete_space \u03b1 :=\n\u27e8assume (f : filter \u03b1), assume hf : cauchy f,\nlet\n  p : set (\u03b1 \u00d7 \u03b1) \u2192 set \u03b1 \u2192 set \u03b1 := \u03bbs t, {y : \u03b1| \u2203x:\u03b1, x \u2208 t \u2227 (x, y) \u2208 s},\n  g := (\ud835\udce4 \u03b1).lift (\u03bbs, f.lift' (p s))\nin\nhave mp\u2080 : monotone p,\n  from assume a b h t s \u27e8x, xs, xa\u27e9, \u27e8x, xs, h xa\u27e9,\nhave mp\u2081 : \u2200{s}, monotone (p s),\n  from assume s a b h x \u27e8y, ya, yxs\u27e9, \u27e8y, h ya, yxs\u27e9,\n\nhave f \u2264 g, from\n  le_infi $ assume s, le_infi $ assume hs, le_infi $ assume t, le_infi $ assume ht,\n  le_principal_iff.mpr $\n  mem_sets_of_superset ht $ assume x hx, \u27e8x, hx, refl_mem_uniformity hs\u27e9,\n\nhave ne_bot g, from hf.left.mono this,\n\nhave ne_bot (comap m g), from comap_ne_bot $ assume t ht,\n  let \u27e8t', ht', ht_mem\u27e9 := (mem_lift_sets $ monotone_lift' monotone_const mp\u2080).mp ht in\n  let \u27e8t'', ht'', ht'_sub\u27e9 := (mem_lift'_sets mp\u2081).mp ht_mem in\n  let \u27e8x, (hx : x \u2208 t'')\u27e9 := hf.left.nonempty_of_mem ht'' in\n  have h\u2080 : ne_bot (\ud835\udcdd[range m] x),\n    from dense.nhds_within_ne_bot x,\n  have h\u2081 : {y | (x, y) \u2208 t'} \u2208 \ud835\udcdd[range m] x,\n    from @mem_inf_sets_of_left \u03b1 (\ud835\udcdd x) (\ud835\udcdf (range m)) _ $ mem_nhds_left x ht',\n  have h\u2082 : range m \u2208 \ud835\udcdd[range m] x,\n    from @mem_inf_sets_of_right \u03b1 (\ud835\udcdd x) (\ud835\udcdf (range m)) _ $ subset.refl _,\n  have {y | (x, y) \u2208 t'} \u2229 range m \u2208 \ud835\udcdd[range m] x,\n    from @inter_mem_sets \u03b1 (\ud835\udcdd[range m] x) _ _ h\u2081 h\u2082,\n  let \u27e8y, xyt', b, b_eq\u27e9 := h\u2080.nonempty_of_mem this in\n  \u27e8b, b_eq.symm \u25b8 ht'_sub \u27e8x, hx, xyt'\u27e9\u27e9,\n\nhave cauchy g, from\n  \u27e8\u2039ne_bot g\u203a, assume s hs,\n  let\n    \u27e8s\u2081, hs\u2081, (comp_s\u2081 : comp_rel s\u2081 s\u2081 \u2286 s)\u27e9 := comp_mem_uniformity_sets hs,\n    \u27e8s\u2082, hs\u2082, (comp_s\u2082 : comp_rel s\u2082 s\u2082 \u2286 s\u2081)\u27e9 := comp_mem_uniformity_sets hs\u2081,\n    \u27e8t, ht, (prod_t : set.prod t t \u2286 s\u2082)\u27e9 := mem_prod_same_iff.mp (hf.right hs\u2082)\n  in\n  have hg\u2081 : p (preimage prod.swap s\u2081) t \u2208 g,\n    from mem_lift (symm_le_uniformity hs\u2081) $ @mem_lift' \u03b1 \u03b1 f _ t ht,\n  have hg\u2082 : p s\u2082 t \u2208 g,\n    from mem_lift hs\u2082 $ @mem_lift' \u03b1 \u03b1 f _ t ht,\n  have hg : set.prod (p (preimage prod.swap s\u2081) t) (p s\u2082 t) \u2208 g \u00d7\u1da0 g,\n    from @prod_mem_prod \u03b1 \u03b1 _ _ g g hg\u2081 hg\u2082,\n  (g \u00d7\u1da0 g).sets_of_superset hg\n    (assume \u27e8a, b\u27e9 \u27e8\u27e8c\u2081, c\u2081t, hc\u2081\u27e9, \u27e8c\u2082, c\u2082t, hc\u2082\u27e9\u27e9,\n      have (c\u2081, c\u2082) \u2208 set.prod t t, from \u27e8c\u2081t, c\u2082t\u27e9,\n      comp_s\u2081 $ prod_mk_mem_comp_rel hc\u2081 $\n      comp_s\u2082 $ prod_mk_mem_comp_rel (prod_t this) hc\u2082)\u27e9,\n\nhave cauchy (filter.comap m g),\n  from \u2039cauchy g\u203a.comap' (le_of_eq hm.comap_uniformity) \u2039_\u203a,\n\nlet \u27e8x, (hx : map m (filter.comap m g) \u2264 \ud835\udcdd x)\u27e9 := h _ this in\nhave cluster_pt x (map m (filter.comap m g)),\n  from (le_nhds_iff_adhp_of_cauchy (this.map hm.uniform_continuous)).mp hx,\nhave cluster_pt x g,\n  from  this.mono map_comap_le,\n\n\u27e8x, calc f \u2264 g : by assumption\n  ... \u2264 \ud835\udcdd x : le_nhds_of_cauchy_adhp \u2039cauchy g\u203a this\u27e9\u27e9\n\nlemma totally_bounded_preimage {f : \u03b1 \u2192 \u03b2} {s : set \u03b2} (hf : uniform_embedding f)\n  (hs : totally_bounded s) : totally_bounded (f \u207b\u00b9' s) :=\n\u03bb t ht, begin\n  rw \u2190 hf.comap_uniformity at ht,\n  rcases mem_comap_sets.2 ht with \u27e8t', ht', ts\u27e9,\n  rcases totally_bounded_iff_subset.1\n    (totally_bounded_subset (image_preimage_subset f s) hs) _ ht' with \u27e8c, cs, hfc, hct\u27e9,\n  refine \u27e8f \u207b\u00b9' c, hfc.preimage (hf.inj.inj_on _), \u03bb x h, _\u27e9,\n  have := hct (mem_image_of_mem f h), simp at this \u22a2,\n  rcases this with \u27e8z, zc, zt\u27e9,\n  rcases cs zc with \u27e8y, yc, rfl\u27e9,\n  exact \u27e8y, zc, ts (by exact zt)\u27e9\nend\n\nend\n\nlemma uniform_embedding_comap {\u03b1 : Type*} {\u03b2 : Type*} {f : \u03b1 \u2192 \u03b2} [u : uniform_space \u03b2]\n  (hf : function.injective f) : @uniform_embedding \u03b1 \u03b2 (uniform_space.comap f u) u f :=\n@uniform_embedding.mk _ _ (uniform_space.comap f u) _ _\n  (@uniform_inducing.mk _ _ (uniform_space.comap f u) _ _ rfl) hf\n\nsection uniform_extension\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n          [uniform_space \u03b1] [uniform_space \u03b2] [uniform_space \u03b3]\n          {e : \u03b2 \u2192 \u03b1}\n          (h_e : uniform_inducing e)\n          (h_dense : dense_range e)\n          {f : \u03b2 \u2192 \u03b3}\n          (h_f : uniform_continuous f)\n\nlocal notation `\u03c8` := (h_e.dense_inducing h_dense).extend f\n\nlemma uniformly_extend_exists [complete_space \u03b3] (a : \u03b1) :\n  \u2203c, tendsto f (comap e (\ud835\udcdd a)) (\ud835\udcdd c) :=\nlet de := (h_e.dense_inducing h_dense) in\nhave cauchy (\ud835\udcdd a), from cauchy_nhds,\nhave cauchy (comap e (\ud835\udcdd a)), from\n  this.comap' (le_of_eq h_e.comap_uniformity) (de.comap_nhds_ne_bot _),\nhave cauchy (map f (comap e (\ud835\udcdd a))), from this.map h_f,\ncomplete_space.complete this\n\nlemma uniform_extend_subtype [complete_space \u03b3]\n  {p : \u03b1 \u2192 Prop} {e : \u03b1 \u2192 \u03b2} {f : \u03b1 \u2192 \u03b3} {b : \u03b2} {s : set \u03b1}\n  (hf : uniform_continuous (\u03bbx:subtype p, f x.val))\n  (he : uniform_embedding e) (hd : \u2200x:\u03b2, x \u2208 closure (range e))\n  (hb : closure (e '' s) \u2208 \ud835\udcdd b) (hs : is_closed s) (hp : \u2200x\u2208s, p x) :\n  \u2203c, tendsto f (comap e (\ud835\udcdd b)) (\ud835\udcdd c) :=\nhave de : dense_embedding e,\n  from he.dense_embedding hd,\nhave de' : dense_embedding (dense_embedding.subtype_emb p e),\n  by exact de.subtype p,\nhave ue' : uniform_embedding (dense_embedding.subtype_emb p e),\n  from uniform_embedding_subtype_emb _ he de,\nhave b \u2208 closure (e '' {x | p x}),\n  from (closure_mono $ monotone_image $ hp) (mem_of_nhds hb),\nlet \u27e8c, (hc : tendsto (f \u2218 subtype.val)\n     (comap (dense_embedding.subtype_emb p e) (\ud835\udcdd \u27e8b, this\u27e9)) (\ud835\udcdd c))\u27e9 :=\n  uniformly_extend_exists ue'.to_uniform_inducing de'.dense hf _ in\nbegin\n  rw [nhds_subtype_eq_comap] at hc,\n  simp [comap_comap] at hc,\n  change (tendsto (f \u2218 @subtype.val \u03b1 p) (comap (e \u2218 @subtype.val \u03b1 p) (\ud835\udcdd b)) (\ud835\udcdd c)) at hc,\n  rw [\u2190comap_comap, tendsto_comap'_iff] at hc,\n  exact \u27e8c, hc\u27e9,\n  exact \u27e8_, hb, assume x,\n    begin\n      change e x \u2208 (closure (e '' s)) \u2192 x \u2208 range subtype.val,\n      rw [\u2190 closure_induced, mem_closure_iff_cluster_pt, cluster_pt, ne_bot_iff,\n          nhds_induced, \u2190 de.to_dense_inducing.nhds_eq_comap,\n          \u2190 mem_closure_iff_nhds_ne_bot, hs.closure_eq],\n      exact assume hxs, \u27e8\u27e8x, hp x hxs\u27e9, rfl\u27e9,\n    end\u27e9\nend\n\nvariables [separated_space \u03b3]\n\nlemma uniformly_extend_of_ind (b : \u03b2) : \u03c8 (e b) = f b :=\ndense_inducing.extend_eq_at _ b h_f.continuous.continuous_at\n\nlemma uniformly_extend_unique {g : \u03b1 \u2192 \u03b3} (hg : \u2200 b, g (e b) = f b)\n  (hc : continuous g) :\n  \u03c8 = g :=\ndense_inducing.extend_unique _ hg hc\n\ninclude h_f\n\nlemma uniformly_extend_spec [complete_space \u03b3] (a : \u03b1) :\n  tendsto f (comap e (\ud835\udcdd a)) (\ud835\udcdd (\u03c8 a)) :=\nlet de := (h_e.dense_inducing h_dense) in\nbegin\n  by_cases ha : a \u2208 range e,\n  { rcases ha with \u27e8b, rfl\u27e9,\n    rw [uniformly_extend_of_ind _ _ h_f, \u2190 de.nhds_eq_comap],\n    exact h_f.continuous.tendsto _ },\n  { simp only [dense_inducing.extend, dif_neg ha],\n    exact tendsto_nhds_lim (uniformly_extend_exists h_e h_dense h_f _) }\nend\n\nlemma uniform_continuous_uniformly_extend [c\u03b3 : complete_space \u03b3] : uniform_continuous \u03c8 :=\nassume d hd,\nlet \u27e8s, hs, hs_comp\u27e9 := (mem_lift'_sets $\n  monotone_comp_rel monotone_id $ monotone_comp_rel monotone_id monotone_id).mp\n    (comp_le_uniformity3 hd) in\nhave h_pnt : \u2200{a m}, m \u2208 \ud835\udcdd a \u2192 \u2203c, c \u2208 f '' preimage e m \u2227 (c, \u03c8 a) \u2208 s \u2227 (\u03c8 a, c) \u2208 s,\n  from assume a m hm,\n  have nb : ne_bot (map f (comap e (\ud835\udcdd a))),\n    from ((h_e.dense_inducing h_dense).comap_nhds_ne_bot _).map _,\n  have (f '' preimage e m) \u2229 ({c | (c, \u03c8 a) \u2208 s } \u2229 {c | (\u03c8 a, c) \u2208 s }) \u2208 map f (comap e (\ud835\udcdd a)),\n    from inter_mem_sets (image_mem_map $ preimage_mem_comap $ hm)\n      (uniformly_extend_spec h_e h_dense h_f _\n        (inter_mem_sets (mem_nhds_right _ hs) (mem_nhds_left _ hs))),\n  nb.nonempty_of_mem this,\nhave preimage (\u03bbp:\u03b2\u00d7\u03b2, (f p.1, f p.2)) s \u2208 \ud835\udce4 \u03b2,\n  from h_f hs,\nhave preimage (\u03bbp:\u03b2\u00d7\u03b2, (f p.1, f p.2)) s \u2208 comap (\u03bbx:\u03b2\u00d7\u03b2, (e x.1, e x.2)) (\ud835\udce4 \u03b1),\n  by rwa [h_e.comap_uniformity.symm] at this,\nlet \u27e8t, ht, ts\u27e9 := this in\nshow preimage (\u03bbp:(\u03b1\u00d7\u03b1), (\u03c8 p.1, \u03c8 p.2)) d \u2208 \ud835\udce4 \u03b1,\n  from (\ud835\udce4 \u03b1).sets_of_superset (interior_mem_uniformity ht) $\n  assume \u27e8x\u2081, x\u2082\u27e9 hx_t,\n  have \ud835\udcdd (x\u2081, x\u2082) \u2264 \ud835\udcdf (interior t),\n    from is_open_iff_nhds.mp is_open_interior (x\u2081, x\u2082) hx_t,\n  have interior t \u2208 \ud835\udcdd x\u2081 \u00d7\u1da0 \ud835\udcdd x\u2082,\n    by rwa [nhds_prod_eq, le_principal_iff] at this,\n  let \u27e8m\u2081, hm\u2081, m\u2082, hm\u2082, (hm : set.prod m\u2081 m\u2082 \u2286 interior t)\u27e9 := mem_prod_iff.mp this in\n  let \u27e8a, ha\u2081, _, ha\u2082\u27e9 := h_pnt hm\u2081 in\n  let \u27e8b, hb\u2081, hb\u2082, _\u27e9 := h_pnt hm\u2082 in\n  have set.prod (preimage e m\u2081) (preimage e m\u2082) \u2286 preimage (\u03bbp:(\u03b2\u00d7\u03b2), (f p.1, f p.2)) s,\n    from calc _ \u2286 preimage (\u03bbp:(\u03b2\u00d7\u03b2), (e p.1, e p.2)) (interior t) : preimage_mono hm\n    ... \u2286 preimage (\u03bbp:(\u03b2\u00d7\u03b2), (e p.1, e p.2)) t : preimage_mono interior_subset\n    ... \u2286 preimage (\u03bbp:(\u03b2\u00d7\u03b2), (f p.1, f p.2)) s : ts,\n  have set.prod (f '' preimage e m\u2081) (f '' preimage e m\u2082) \u2286 s,\n    from calc set.prod (f '' preimage e m\u2081) (f '' preimage e m\u2082) =\n      (\u03bbp:(\u03b2\u00d7\u03b2), (f p.1, f p.2)) '' (set.prod (preimage e m\u2081) (preimage e m\u2082)) : prod_image_image_eq\n    ... \u2286 (\u03bbp:(\u03b2\u00d7\u03b2), (f p.1, f p.2)) '' preimage (\u03bbp:(\u03b2\u00d7\u03b2), (f p.1, f p.2)) s : monotone_image this\n    ... \u2286 s : image_subset_iff.mpr $ subset.refl _,\n  have (a, b) \u2208 s, from @this (a, b) \u27e8ha\u2081, hb\u2081\u27e9,\n  hs_comp $ show (\u03c8 x\u2081, \u03c8 x\u2082) \u2208 comp_rel s (comp_rel s s),\n    from \u27e8a, ha\u2082, \u27e8b, this, hb\u2082\u27e9\u27e9\nend uniform_extension\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/topology/uniform_space/uniform_embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6513548511303338, "lm_q1q2_score": 0.37366821368291636}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.fintype.basic\nimport Mathlib.data.finset.sort\nimport Mathlib.group_theory.perm.basic\nimport Mathlib.group_theory.order_of_element\nimport Mathlib.PostPort\n\nuniverses u u_1 v \n\nnamespace Mathlib\n\n/-!\n# Sign of a permutation\n\nThe main definition of this file is `equiv.perm.sign`, associating a `units \u2124` sign with a\npermutation.\n\nThis file also contains miscellaneous lemmas about `equiv.perm` and `equiv.swap`, building on top\nof those in `data/equiv/basic` and `data/equiv/perm`.\n\n-/\n\nnamespace equiv.perm\n\n\n/--\n`mod_swap i j` contains permutations up to swapping `i` and `j`.\n\nWe use this to partition permutations in `matrix.det_zero_of_row_eq`, such that each partition\nsums up to `0`.\n-/\ndef mod_swap {\u03b1 : Type u} [DecidableEq \u03b1] (i : \u03b1) (j : \u03b1) : setoid (perm \u03b1) :=\n  setoid.mk (fun (\u03c3 \u03c4 : perm \u03b1) => \u03c3 = \u03c4 \u2228 \u03c3 = swap i j * \u03c4) sorry\n\nprotected instance r.decidable_rel {\u03b1 : Type u_1} [fintype \u03b1] [DecidableEq \u03b1] (i : \u03b1) (j : \u03b1) :\n    DecidableRel setoid.r :=\n  fun (\u03c3 \u03c4 : perm \u03b1) => or.decidable\n\n/-- If the permutation `f` fixes the subtype `{x // p x}`, then this returns the permutation\n  on `{x // p x}` induced by `f`. -/\ndef subtype_perm {\u03b1 : Type u} (f : perm \u03b1) {p : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2194 p (coe_fn f x)) :\n    perm (Subtype fun (x : \u03b1) => p x) :=\n  mk (fun (x : Subtype fun (x : \u03b1) => p x) => { val := coe_fn f \u2191x, property := sorry })\n    (fun (x : Subtype fun (x : \u03b1) => p x) => { val := coe_fn (f\u207b\u00b9) \u2191x, property := sorry }) sorry\n    sorry\n\n@[simp] theorem subtype_perm_one {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (h : \u2200 (x : \u03b1), p x \u2194 p (coe_fn 1 x)) :\n    subtype_perm 1 h = 1 :=\n  sorry\n\n/-- The inclusion map of permutations on a subtype of `\u03b1` into permutations of `\u03b1`,\n  fixing the other points. -/\ndef of_subtype {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] : perm (Subtype p) \u2192* perm \u03b1 :=\n  monoid_hom.mk\n    (fun (f : perm (Subtype p)) =>\n      mk\n        (fun (x : \u03b1) =>\n          dite (p x) (fun (h : p x) => \u2191(coe_fn f { val := x, property := h })) fun (h : \u00acp x) => x)\n        (fun (x : \u03b1) =>\n          dite (p x) (fun (h : p x) => \u2191(coe_fn (f\u207b\u00b9) { val := x, property := h }))\n            fun (h : \u00acp x) => x)\n        sorry sorry)\n    sorry sorry\n\n/-- Two permutations `f` and `g` are `disjoint` if their supports are disjoint, i.e.,\nevery element is fixed either by `f`, or by `g`. -/\ndef disjoint {\u03b1 : Type u} (f : perm \u03b1) (g : perm \u03b1) := \u2200 (x : \u03b1), coe_fn f x = x \u2228 coe_fn g x = x\n\ntheorem disjoint.symm {\u03b1 : Type u} {f : perm \u03b1} {g : perm \u03b1} : disjoint f g \u2192 disjoint g f := sorry\n\ntheorem disjoint_comm {\u03b1 : Type u} {f : perm \u03b1} {g : perm \u03b1} : disjoint f g \u2194 disjoint g f :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint.mul_comm {\u03b1 : Type u} {f : perm \u03b1} {g : perm \u03b1} (h : disjoint f g) :\n    f * g = g * f :=\n  sorry\n\n@[simp] theorem disjoint_one_left {\u03b1 : Type u} (f : perm \u03b1) : disjoint 1 f :=\n  fun (_x : \u03b1) => Or.inl rfl\n\n@[simp] theorem disjoint_one_right {\u03b1 : Type u} (f : perm \u03b1) : disjoint f 1 :=\n  fun (_x : \u03b1) => Or.inr rfl\n\ntheorem disjoint.mul_left {\u03b1 : Type u} {f : perm \u03b1} {g : perm \u03b1} {h : perm \u03b1} (H1 : disjoint f h)\n    (H2 : disjoint g h) : disjoint (f * g) h :=\n  sorry\n\ntheorem disjoint.mul_right {\u03b1 : Type u} {f : perm \u03b1} {g : perm \u03b1} {h : perm \u03b1} (H1 : disjoint f g)\n    (H2 : disjoint f h) : disjoint f (g * h) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (disjoint f (g * h))) (propext disjoint_comm)))\n    (disjoint.mul_left (disjoint.symm H1) (disjoint.symm H2))\n\ntheorem disjoint_prod_right {\u03b1 : Type u} {f : perm \u03b1} (l : List (perm \u03b1))\n    (h : \u2200 (g : perm \u03b1), g \u2208 l \u2192 disjoint f g) : disjoint f (list.prod l) :=\n  sorry\n\ntheorem disjoint_prod_perm {\u03b1 : Type u} {l\u2081 : List (perm \u03b1)} {l\u2082 : List (perm \u03b1)}\n    (hl : list.pairwise disjoint l\u2081) (hp : l\u2081 ~ l\u2082) : list.prod l\u2081 = list.prod l\u2082 :=\n  list.perm.prod_eq' hp (list.pairwise.imp (fun (f g : perm \u03b1) => disjoint.mul_comm) hl)\n\ntheorem of_subtype_subtype_perm {\u03b1 : Type u} {f : perm \u03b1} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (h\u2081 : \u2200 (x : \u03b1), p x \u2194 p (coe_fn f x)) (h\u2082 : \u2200 (x : \u03b1), coe_fn f x \u2260 x \u2192 p x) :\n    coe_fn of_subtype (subtype_perm f h\u2081) = f :=\n  sorry\n\ntheorem of_subtype_apply_of_not_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (f : perm (Subtype p)) {x : \u03b1} (hx : \u00acp x) : coe_fn (coe_fn of_subtype f) x = x :=\n  dif_neg hx\n\ntheorem mem_iff_of_subtype_apply_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (f : perm (Subtype p)) (x : \u03b1) : p x \u2194 p (coe_fn (coe_fn of_subtype f) x) :=\n  sorry\n\n@[simp] theorem subtype_perm_of_subtype {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (f : perm (Subtype p)) :\n    subtype_perm (coe_fn of_subtype f) (mem_iff_of_subtype_apply_mem f) = f :=\n  sorry\n\ntheorem pow_apply_eq_self_of_apply_eq_self {\u03b1 : Type u} {f : perm \u03b1} {x : \u03b1} (hfx : coe_fn f x = x)\n    (n : \u2115) : coe_fn (f ^ n) x = x :=\n  sorry\n\ntheorem gpow_apply_eq_self_of_apply_eq_self {\u03b1 : Type u} {f : perm \u03b1} {x : \u03b1} (hfx : coe_fn f x = x)\n    (n : \u2124) : coe_fn (f ^ n) x = x :=\n  sorry\n\ntheorem pow_apply_eq_of_apply_apply_eq_self {\u03b1 : Type u} {f : perm \u03b1} {x : \u03b1}\n    (hffx : coe_fn f (coe_fn f x) = x) (n : \u2115) :\n    coe_fn (f ^ n) x = x \u2228 coe_fn (f ^ n) x = coe_fn f x :=\n  sorry\n\ntheorem gpow_apply_eq_of_apply_apply_eq_self {\u03b1 : Type u} {f : perm \u03b1} {x : \u03b1}\n    (hffx : coe_fn f (coe_fn f x) = x) (i : \u2124) :\n    coe_fn (f ^ i) x = x \u2228 coe_fn (f ^ i) x = coe_fn f x :=\n  sorry\n\n/-- The `finset` of nonfixed points of a permutation. -/\ndef support {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) : finset \u03b1 :=\n  finset.filter (fun (x : \u03b1) => coe_fn f x \u2260 x) finset.univ\n\n@[simp] theorem mem_support {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {f : perm \u03b1} {x : \u03b1} :\n    x \u2208 support f \u2194 coe_fn f x \u2260 x :=\n  sorry\n\n/-- `f.is_swap` indicates that the permutation `f` is a transposition of two elements. -/\ndef is_swap {\u03b1 : Type u} [DecidableEq \u03b1] (f : perm \u03b1) := \u2203 (x : \u03b1), \u2203 (y : \u03b1), x \u2260 y \u2227 f = swap x y\n\ntheorem is_swap.of_subtype_is_swap {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    {f : perm (Subtype p)} (h : is_swap f) : is_swap (coe_fn of_subtype f) :=\n  sorry\n\ntheorem ne_and_ne_of_swap_mul_apply_ne_self {\u03b1 : Type u} [DecidableEq \u03b1] {f : perm \u03b1} {x : \u03b1}\n    {y : \u03b1} (hy : coe_fn (swap x (coe_fn f x) * f) y \u2260 y) : coe_fn f y \u2260 y \u2227 y \u2260 x :=\n  sorry\n\ntheorem support_swap_mul_eq {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {f : perm \u03b1} {x : \u03b1}\n    (hffx : coe_fn f (coe_fn f x) \u2260 x) :\n    support (swap x (coe_fn f x) * f) = finset.erase (support f) x :=\n  sorry\n\ntheorem card_support_swap_mul {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {f : perm \u03b1} {x : \u03b1}\n    (hx : coe_fn f x \u2260 x) :\n    finset.card (support (swap x (coe_fn f x) * f)) < finset.card (support f) :=\n  sorry\n\n/-- Given a list `l : list \u03b1` and a permutation `f : perm \u03b1` such that the nonfixed points of `f`\n  are in `l`, recursively factors `f` as a product of transpositions. -/\ndef swap_factors_aux {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) (f : perm \u03b1) :\n    (\u2200 {x : \u03b1}, coe_fn f x \u2260 x \u2192 x \u2208 l) \u2192\n        Subtype fun (l : List (perm \u03b1)) => list.prod l = f \u2227 \u2200 (g : perm \u03b1), g \u2208 l \u2192 is_swap g :=\n  sorry\n\n/-- `swap_factors` represents a permutation as a product of a list of transpositions.\nThe representation is non unique and depends on the linear order structure.\nFor types without linear order `trunc_swap_factors` can be used. -/\ndef swap_factors {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] [linear_order \u03b1] (f : perm \u03b1) :\n    Subtype fun (l : List (perm \u03b1)) => list.prod l = f \u2227 \u2200 (g : perm \u03b1), g \u2208 l \u2192 is_swap g :=\n  swap_factors_aux (finset.sort LessEq finset.univ) f sorry\n\n/-- This computably represents the fact that any permutation can be represented as the product of\n  a list of transpositions. -/\ndef trunc_swap_factors {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) :\n    trunc\n        (Subtype fun (l : List (perm \u03b1)) => list.prod l = f \u2227 \u2200 (g : perm \u03b1), g \u2208 l \u2192 is_swap g) :=\n  quotient.rec_on_subsingleton (finset.val finset.univ)\n    (fun (l : List \u03b1) (h : \u2200 (x : \u03b1), coe_fn f x \u2260 x \u2192 x \u2208 quotient.mk l) =>\n      trunc.mk (swap_factors_aux l f h))\n    sorry\n\n/-- An induction principle for permutations. If `P` holds for the identity permutation, and\nis preserved under composition with a non-trivial swap, then `P` holds for all permutations. -/\ntheorem swap_induction_on {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {P : perm \u03b1 \u2192 Prop}\n    (f : perm \u03b1) : P 1 \u2192 (\u2200 (f : perm \u03b1) (x y : \u03b1), x \u2260 y \u2192 P f \u2192 P (swap x y * f)) \u2192 P f :=\n  sorry\n\n/-- Like `swap_induction_on`, but with the composition on the right of `f`.\n\nAn induction principle for permutations. If `P` holds for the identity permutation, and\nis preserved under composition with a non-trivial swap, then `P` holds for all permutations. -/\ntheorem swap_induction_on' {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {P : perm \u03b1 \u2192 Prop}\n    (f : perm \u03b1) : P 1 \u2192 (\u2200 (f : perm \u03b1) (x y : \u03b1), x \u2260 y \u2192 P f \u2192 P (f * swap x y)) \u2192 P f :=\n  fun (h1 : P 1) (IH : \u2200 (f : perm \u03b1) (x y : \u03b1), x \u2260 y \u2192 P f \u2192 P (f * swap x y)) =>\n    inv_inv f \u25b8 swap_induction_on (f\u207b\u00b9) h1 fun (f : perm \u03b1) => IH (f\u207b\u00b9)\n\ntheorem is_conj_swap {\u03b1 : Type u} [DecidableEq \u03b1] {w : \u03b1} {x : \u03b1} {y : \u03b1} {z : \u03b1} (hwx : w \u2260 x)\n    (hyz : y \u2260 z) : is_conj (swap w x) (swap y z) :=\n  sorry\n\n/-- set of all pairs (\u27e8a, b\u27e9 : \u03a3 a : fin n, fin n) such that b < a -/\ndef fin_pairs_lt (n : \u2115) : finset (sigma fun (a : fin n) => fin n) :=\n  finset.sigma finset.univ fun (a : fin n) => finset.attach_fin (finset.range \u2191a) sorry\n\ntheorem mem_fin_pairs_lt {n : \u2115} {a : sigma fun (a : fin n) => fin n} :\n    a \u2208 fin_pairs_lt n \u2194 sigma.snd a < sigma.fst a :=\n  sorry\n\n/-- `sign_aux \u03c3` is the sign of a permutation on `fin n`, defined as the parity of the number of\n  pairs `(x\u2081, x\u2082)` such that `x\u2082 < x\u2081` but `\u03c3 x\u2081 \u2264 \u03c3 x\u2082` -/\ndef sign_aux {n : \u2115} (a : perm (fin n)) : units \u2124 :=\n  finset.prod (fin_pairs_lt n)\n    fun (x : sigma fun (a : fin n) => fin n) =>\n      ite (coe_fn a (sigma.fst x) \u2264 coe_fn a (sigma.snd x)) (-1) 1\n\n@[simp] theorem sign_aux_one (n : \u2115) : sign_aux 1 = 1 := sorry\n\n/-- `sign_bij_aux f \u27e8a, b\u27e9` returns the pair consisting of `f a` and `f b` in decreasing order. -/\ndef sign_bij_aux {n : \u2115} (f : perm (fin n)) (a : sigma fun (a : fin n) => fin n) :\n    sigma fun (a : fin n) => fin n :=\n  dite (coe_fn f (sigma.snd a) < coe_fn f (sigma.fst a))\n    (fun (hxa : coe_fn f (sigma.snd a) < coe_fn f (sigma.fst a)) =>\n      sigma.mk (coe_fn f (sigma.fst a)) (coe_fn f (sigma.snd a)))\n    fun (hxa : \u00accoe_fn f (sigma.snd a) < coe_fn f (sigma.fst a)) =>\n      sigma.mk (coe_fn f (sigma.snd a)) (coe_fn f (sigma.fst a))\n\ntheorem sign_bij_aux_inj {n : \u2115} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n)\n    (b : sigma fun (a : fin n) => fin n) :\n    a \u2208 fin_pairs_lt n \u2192 b \u2208 fin_pairs_lt n \u2192 sign_bij_aux f a = sign_bij_aux f b \u2192 a = b :=\n  sorry\n\ntheorem sign_bij_aux_surj {n : \u2115} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n)\n    (H : a \u2208 fin_pairs_lt n) :\n    \u2203 (b : sigma fun (a : fin n) => fin n), \u2203 (H : b \u2208 fin_pairs_lt n), a = sign_bij_aux f b :=\n  sorry\n\ntheorem sign_bij_aux_mem {n : \u2115} {f : perm (fin n)} (a : sigma fun (a : fin n) => fin n) :\n    a \u2208 fin_pairs_lt n \u2192 sign_bij_aux f a \u2208 fin_pairs_lt n :=\n  sorry\n\n@[simp] theorem sign_aux_inv {n : \u2115} (f : perm (fin n)) : sign_aux (f\u207b\u00b9) = sign_aux f := sorry\n\ntheorem sign_aux_mul {n : \u2115} (f : perm (fin n)) (g : perm (fin n)) :\n    sign_aux (f * g) = sign_aux f * sign_aux g :=\n  sorry\n\n-- TODO: slow\n\ntheorem sign_aux_swap {n : \u2115} {x : fin n} {y : fin n} (hxy : x \u2260 y) : sign_aux (swap x y) = -1 :=\n  sorry\n\n/-- When the list `l : list \u03b1` contains all nonfixed points of the permutation `f : perm \u03b1`,\n  `sign_aux2 l f` recursively calculates the sign of `f`. -/\ndef sign_aux2 {\u03b1 : Type u} [DecidableEq \u03b1] : List \u03b1 \u2192 perm \u03b1 \u2192 units \u2124 := sorry\n\ntheorem sign_aux_eq_sign_aux2 {\u03b1 : Type u} [DecidableEq \u03b1] {n : \u2115} (l : List \u03b1) (f : perm \u03b1)\n    (e : \u03b1 \u2243 fin n) (h : \u2200 (x : \u03b1), coe_fn f x \u2260 x \u2192 x \u2208 l) :\n    sign_aux (equiv.trans (equiv.trans (equiv.symm e) f) e) = sign_aux2 l f :=\n  sorry\n\n/-- When the multiset `s : multiset \u03b1` contains all nonfixed points of the permutation `f : perm \u03b1`,\n  `sign_aux2 f _` recursively calculates the sign of `f`. -/\ndef sign_aux3 {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) {s : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 s) \u2192 units \u2124 :=\n  quotient.hrec_on s (fun (l : List \u03b1) (h : \u2200 (x : \u03b1), x \u2208 quotient.mk l) => sign_aux2 l f) sorry\n\ntheorem sign_aux3_mul_and_swap {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) (g : perm \u03b1)\n    (s : multiset \u03b1) (hs : \u2200 (x : \u03b1), x \u2208 s) :\n    sign_aux3 (f * g) hs = sign_aux3 f hs * sign_aux3 g hs \u2227\n        \u2200 (x y : \u03b1), x \u2260 y \u2192 sign_aux3 (swap x y) hs = -1 :=\n  sorry\n\n/-- `sign` of a permutation returns the signature or parity of a permutation, `1` for even\npermutations, `-1` for odd permutations. It is the unique surjective group homomorphism from\n`perm \u03b1` to the group with two elements.-/\ndef sign {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] : perm \u03b1 \u2192* units \u2124 :=\n  monoid_hom.mk' (fun (f : perm \u03b1) => sign_aux3 f finset.mem_univ) sorry\n\n@[simp] theorem sign_mul {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) (g : perm \u03b1) :\n    coe_fn sign (f * g) = coe_fn sign f * coe_fn sign g :=\n  monoid_hom.map_mul sign f g\n\n@[simp] theorem sign_trans {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) (g : perm \u03b1) :\n    coe_fn sign (equiv.trans f g) = coe_fn sign g * coe_fn sign f :=\n  sorry\n\n@[simp] theorem sign_one {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] : coe_fn sign 1 = 1 :=\n  monoid_hom.map_one sign\n\n@[simp] theorem sign_refl {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] :\n    coe_fn sign (equiv.refl \u03b1) = 1 :=\n  monoid_hom.map_one sign\n\n@[simp] theorem sign_inv {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) :\n    coe_fn sign (f\u207b\u00b9) = coe_fn sign f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn sign (f\u207b\u00b9) = coe_fn sign f)) (monoid_hom.map_inv sign f)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (coe_fn sign f\u207b\u00b9 = coe_fn sign f))\n          (int.units_inv_eq_self (coe_fn sign f))))\n      (Eq.refl (coe_fn sign f)))\n\n@[simp] theorem sign_symm {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (e : perm \u03b1) :\n    coe_fn sign (equiv.symm e) = coe_fn sign e :=\n  sign_inv e\n\ntheorem sign_swap {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {x : \u03b1} {y : \u03b1} (h : x \u2260 y) :\n    coe_fn sign (swap x y) = -1 :=\n  and.right (sign_aux3_mul_and_swap 1 1 (finset.val finset.univ) finset.mem_univ) x y h\n\n@[simp] theorem sign_swap' {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {x : \u03b1} {y : \u03b1} :\n    coe_fn sign (swap x y) = ite (x = y) 1 (-1) :=\n  sorry\n\ntheorem is_swap.sign_eq {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {f : perm \u03b1} (h : is_swap f) :\n    coe_fn sign f = -1 :=\n  sorry\n\ntheorem sign_aux3_symm_trans_trans {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1]\n    [DecidableEq \u03b2] [fintype \u03b2] (f : perm \u03b1) (e : \u03b1 \u2243 \u03b2) {s : multiset \u03b1} {t : multiset \u03b2}\n    (hs : \u2200 (x : \u03b1), x \u2208 s) (ht : \u2200 (x : \u03b2), x \u2208 t) :\n    sign_aux3 (equiv.trans (equiv.trans (equiv.symm e) f) e) ht = sign_aux3 f hs :=\n  sorry\n\n@[simp] theorem sign_symm_trans_trans {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1]\n    [DecidableEq \u03b2] [fintype \u03b2] (f : perm \u03b1) (e : \u03b1 \u2243 \u03b2) :\n    coe_fn sign (equiv.trans (equiv.trans (equiv.symm e) f) e) = coe_fn sign f :=\n  sign_aux3_symm_trans_trans f e finset.mem_univ finset.mem_univ\n\n@[simp] theorem sign_trans_trans_symm {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1]\n    [DecidableEq \u03b2] [fintype \u03b2] (f : perm \u03b2) (e : \u03b1 \u2243 \u03b2) :\n    coe_fn sign (equiv.trans (equiv.trans e f) (equiv.symm e)) = coe_fn sign f :=\n  sign_symm_trans_trans f (equiv.symm e)\n\ntheorem sign_prod_list_swap {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {l : List (perm \u03b1)}\n    (hl : \u2200 (g : perm \u03b1), g \u2208 l \u2192 is_swap g) : coe_fn sign (list.prod l) = (-1) ^ list.length l :=\n  sorry\n\ntheorem sign_surjective {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (h\u03b1 : 1 < fintype.card \u03b1) :\n    function.surjective \u21d1sign :=\n  sorry\n\ntheorem eq_sign_of_surjective_hom {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {s : perm \u03b1 \u2192* units \u2124}\n    (hs : function.surjective \u21d1s) : s = sign :=\n  sorry\n\ntheorem sign_subtype_perm {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] (f : perm \u03b1) {p : \u03b1 \u2192 Prop}\n    [decidable_pred p] (h\u2081 : \u2200 (x : \u03b1), p x \u2194 p (coe_fn f x))\n    (h\u2082 : \u2200 (x : \u03b1), coe_fn f x \u2260 x \u2192 p x) : coe_fn sign (subtype_perm f h\u2081) = coe_fn sign f :=\n  sorry\n\n@[simp] theorem sign_of_subtype {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {p : \u03b1 \u2192 Prop}\n    [decidable_pred p] (f : perm (Subtype p)) : coe_fn sign (coe_fn of_subtype f) = coe_fn sign f :=\n  sorry\n\ntheorem sign_eq_sign_of_equiv {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1] [DecidableEq \u03b2]\n    [fintype \u03b2] (f : perm \u03b1) (g : perm \u03b2) (e : \u03b1 \u2243 \u03b2)\n    (h : \u2200 (x : \u03b1), coe_fn e (coe_fn f x) = coe_fn g (coe_fn e x)) :\n    coe_fn sign f = coe_fn sign g :=\n  sorry\n\ntheorem sign_bij {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1] [DecidableEq \u03b2] [fintype \u03b2]\n    {f : perm \u03b1} {g : perm \u03b2} (i : (x : \u03b1) \u2192 coe_fn f x \u2260 x \u2192 \u03b2)\n    (h :\n      \u2200 (x : \u03b1) (hx : coe_fn f x \u2260 x) (hx' : coe_fn f (coe_fn f x) \u2260 coe_fn f x),\n        i (coe_fn f x) hx' = coe_fn g (i x hx))\n    (hi :\n      \u2200 (x\u2081 x\u2082 : \u03b1) (hx\u2081 : coe_fn f x\u2081 \u2260 x\u2081) (hx\u2082 : coe_fn f x\u2082 \u2260 x\u2082),\n        i x\u2081 hx\u2081 = i x\u2082 hx\u2082 \u2192 x\u2081 = x\u2082)\n    (hg : \u2200 (y : \u03b2), coe_fn g y \u2260 y \u2192 \u2203 (x : \u03b1), \u2203 (hx : coe_fn f x \u2260 x), i x hx = y) :\n    coe_fn sign f = coe_fn sign g :=\n  sorry\n\n@[simp] theorem support_swap {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {x : \u03b1} {y : \u03b1}\n    (hxy : x \u2260 y) : support (swap x y) = insert x (singleton y) :=\n  sorry\n\ntheorem card_support_swap {\u03b1 : Type u} [DecidableEq \u03b1] [fintype \u03b1] {x : \u03b1} {y : \u03b1} (hxy : x \u2260 y) :\n    finset.card (support (swap x y)) = bit0 1 :=\n  sorry\n\n/-- If we apply `prod_extend_right a (\u03c3 a)` for all `a : \u03b1` in turn,\nwe get `prod_congr_right \u03c3`. -/\ntheorem prod_prod_extend_right {\u03b2 : Type v} {\u03b1 : Type u_1} [DecidableEq \u03b1] (\u03c3 : \u03b1 \u2192 perm \u03b2)\n    {l : List \u03b1} (hl : list.nodup l) (mem_l : \u2200 (a : \u03b1), a \u2208 l) :\n    list.prod (list.map (fun (a : \u03b1) => prod_extend_right a (\u03c3 a)) l) = prod_congr_right \u03c3 :=\n  sorry\n\n@[simp] theorem sign_prod_extend_right {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1]\n    [DecidableEq \u03b2] [fintype \u03b2] (a : \u03b1) (\u03c3 : perm \u03b2) :\n    coe_fn sign (prod_extend_right a \u03c3) = coe_fn sign \u03c3 :=\n  sorry\n\ntheorem sign_prod_congr_right {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1] [DecidableEq \u03b2]\n    [fintype \u03b2] (\u03c3 : \u03b1 \u2192 perm \u03b2) :\n    coe_fn sign (prod_congr_right \u03c3) = finset.prod finset.univ fun (k : \u03b1) => coe_fn sign (\u03c3 k) :=\n  sorry\n\ntheorem sign_prod_congr_left {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1] [DecidableEq \u03b2]\n    [fintype \u03b2] (\u03c3 : \u03b1 \u2192 perm \u03b2) :\n    coe_fn sign (prod_congr_left \u03c3) = finset.prod finset.univ fun (k : \u03b1) => coe_fn sign (\u03c3 k) :=\n  sorry\n\n@[simp] theorem sign_perm_congr {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1]\n    [DecidableEq \u03b2] [fintype \u03b2] (e : \u03b1 \u2243 \u03b2) (p : perm \u03b1) :\n    coe_fn sign (coe_fn (perm_congr e) p) = coe_fn sign p :=\n  sorry\n\n@[simp] theorem sign_sum_congr {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [fintype \u03b1] [DecidableEq \u03b2]\n    [fintype \u03b2] (\u03c3a : perm \u03b1) (\u03c3b : perm \u03b2) :\n    coe_fn sign (sum_congr \u03c3a \u03c3b) = coe_fn sign \u03c3a * coe_fn sign \u03c3b :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/group_theory/perm/sign_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.37359284939968557}}
{"text": "import FOL.lindenbaum data.set.basic\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\nopen formula term\n\nvariables {L L\u2081 L\u2082 L\u2083 : language.{u}}\n\nnamespace language\n\nprotected def pempty : language.{u} := \u27e8\u03bb n, pempty, \u03bb n, pempty\u27e9\n\ninstance : has_emptyc (language.{u}) := \u27e8fol.language.pempty\u27e9\n\n@[simp] lemma pempty_fn_def (n) : (\u2205 : language.{u}).fn n = pempty := rfl\n\n@[simp] lemma pempty_pr_def (n) : (\u2205 : language.{u}).pr n = pempty := rfl\n\nstructure language_translation (L\u2081 : language) (L\u2082 : language) :=\n(fn : \u03a0 n, L\u2081.fn n \u2192 L\u2082.fn n)\n(pr : \u03a0 n, L\u2081.pr n \u2192 L\u2082.pr n)\n\ninfix ` \u219d\u1d38 `:25 := language_translation\n\nstructure language_equiv (L\u2081 : language) (L\u2082 : language) :=\n(ltr : L\u2081 \u219d\u1d38 L\u2082)\n(inv : L\u2082 \u219d\u1d38 L\u2081)\n(left_inv_fn : \u2200 n, function.left_inverse (inv.fn n) (ltr.fn n))\n(left_inv_pr : \u2200 n, function.left_inverse (inv.pr n) (ltr.pr n))\n(right_inv_fn : \u2200 n, function.right_inverse (inv.fn n) (ltr.fn n))\n(right_inv_pr : \u2200 n, function.right_inverse (inv.pr n) (ltr.pr n))\n\ninfix ` \u21ad\u1d38 `:25 := language_equiv\n\nclass language_translation_coe (L\u2081 : language) (L\u2082 : language) :=\n(ltr : L\u2081 \u219d\u1d38 L\u2082)\n(fn_inj : \u2200 n (f g : L\u2081.fn n), ltr.fn n f = ltr.fn n g \u2192 f = g)\n(pr_inj : \u2200 n (p q : L\u2081.pr n), ltr.pr n p = ltr.pr n q \u2192 p = q)\n\nclass has_predecessor (L : language) :=\n(pred : language)\n(ltc : language_translation_coe pred L)\n\nclass synonym (L\u2081 L\u2082 : language) \n(leq : L\u2081 \u21ad\u1d38 L\u2082)\n\nstructure formula_homomorphism (L\u2081 : language) (L\u2082 : language.{v}) :=\n(to_fun : \u2115 \u2192 formula L\u2081 \u2192 formula L\u2082)\n(map_verum : \u2200 i, to_fun i \u22a4 = \u22a4)\n(map_imply : \u2200 (p q : formula L\u2081) (i : \u2115), to_fun i (p \u27f6 q) = to_fun i p \u27f6 to_fun i q)\n(map_neg : \u2200 (p : formula L\u2081) (i), to_fun i (\u223cp) = \u223cto_fun i p)\n(map_univ : \u2200 (p : formula L\u2081) (i), to_fun i (\u2200.p) = \u2200.to_fun (i + 1) p)\n\nstructure translation (L\u2081 : language) (L\u2082 : language.{v}) extends formula_homomorphism L\u2081 L\u2082 :=\n(map_pow : \u2200 (p : formula L\u2081) (i), to_fun (i + 1) (p^1) = (to_fun i p)^1)\n\ninfix ` \u219d `:25 := translation\n\ninstance {L\u2081 L\u2082 : language} : has_coe_to_fun (formula_homomorphism L\u2081 L\u2082) (\u03bb _, \u2115 \u2192 formula L\u2081 \u2192 formula L\u2082) :=\n\u27e8@formula_homomorphism.to_fun L\u2081 L\u2082\u27e9\n\ninstance {L\u2081 L\u2082 : language} : has_coe_to_fun (translation L\u2081 L\u2082) (\u03bb _, \u2115 \u2192 formula L\u2081 \u2192 formula L\u2082) :=\n\u27e8\u03bb \u03c4, @formula_homomorphism.to_fun L\u2081 L\u2082 \u03c4.to_formula_homomorphism\u27e9\n\nstructure term_homomorphism (L\u2081 : language) (L\u2082 : language) :=\n(to_fun_chr : \u2115 \u2192 \u03a0 {n}, L\u2081.fn n \u2192 finitary (term L\u2082) n \u2192 term L\u2082)\n(to_fun : \u2115 \u2192 term L\u2081 \u2192 term L\u2082)\n(map_fn : \u03a0 (k : \u2115) {n} (f : L\u2081.fn n) (v : finitary (term L\u2081) n),\n  to_fun k (term.app f v) = to_fun_chr k f (\u03bb i, to_fun k (v i)))\n\ninfix ` \u219d\u1d40 `:25 := term_homomorphism\n\ninstance {L\u2081 L\u2082 : language} : has_coe_to_fun (term_homomorphism L\u2081 L\u2082) (\u03bb _, \u2115 \u2192 term L\u2081 \u2192 term L\u2082) :=\n\u27e8\u03bb \u03c4, \u03c4.to_fun\u27e9\n\nstructure term_formula_translation (L\u2081 : language) (L\u2082 : language) :=\n(p : translation L\u2081 L\u2082)\n(t : \u2115 \u2192 term L\u2081 \u2192 term L\u2082)\n(chr : \u03a0 {n} (r : L\u2081.pr n), L\u2082.pr n)\n(equal : \u2200 (t\u2081 t\u2082 : term L\u2081) (k), p k (t\u2081 =' t\u2082 : formula L\u2081) = (t k t\u2081 =' t k t\u2082))\n(app : \u2200 (k) {n} (r : L\u2081.pr n) (v), p k (app r v) = app (chr r) (\u03bb i, t k (v i)))\n(map_pow : \u2200 u s, t (s + 1) (u^1) = (t s u)^1)\n\ndef tr_Theory {L\u2081 L\u2082 : language} (\u03c4 : translation L\u2081 L\u2082) (i) (T : Theory L\u2081) : Theory L\u2082 := \u03c4 i '' T\n\n@[simp] lemma mem_Theory_tr_of_mem {L\u2081 L\u2082 : language} {\u03c4 : translation L\u2081 L\u2082} {i}\n  {T : Theory L\u2081} {p} (mem : p \u2208 T) : \u03c4 i p \u2208 tr_Theory \u03c4 i T :=\n\u27e8p, mem, rfl\u27e9\n\nclass translation.conservative (\u03c4 : translation L\u2081 L\u2082) :=\n(ax : \u2115 \u2192 Theory L\u2081 \u2192 Theory L\u2082 := tr_Theory \u03c4)\n(ax_ss : \u2200 T k, tr_Theory \u03c4 k T \u2286 ax k T)\n(specialize : \u2200 (k) (p : formula L\u2081) (t : term L\u2081) (T : Theory L\u2081) (i : \u2115), \n  (ax k T)^i \u22a2 \u03c4 (k + i) (\u2200.p \u27f6 p.rew \u0131[0 \u21dd t]))\n(eq_reflexivity : \u2200 (k) (T : Theory L\u2081) (i : \u2115),\n  (ax k T)^i \u22a2 \u03c4 (k + i) (\u2200.(#0 =' #0)))\n(eq_symmetry : \u2200 (k) (T : Theory L\u2081) (i : \u2115),\n  (ax k T)^i \u22a2 \u03c4 (k + i) (\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #0))))\n(eq_transitive : \u2200 (k) (T : Theory L\u2081) (i : \u2115),\n  (ax k T)^i \u22a2 \u03c4 (k + i) (\u2200.\u2200.\u2200.((#0 =' #1) \u27f6 (#1 =' #2) \u27f6 (#0 =' #2))))\n(function_ext : \u2200 (k) {n} (f : L\u2081.fn n) (T : Theory L\u2081) (i : \u2115),\n  (ax k T)^i \u22a2 \u03c4 (k + i) (eq_axiom4 f))\n(predicate_ext : \u2200 (k) {n} (r : L\u2081.pr n) (T : Theory L\u2081) (i : \u2115),\n  (ax k T)^i \u22a2 \u03c4 (k + i) (eq_axiom5 r))\n\nnamespace formula_homonorphism\nvariables (\u03c4 : formula_homomorphism L\u2081 L\u2082) (i : \u2115)\n\n@[simp] lemma map_verum' :\n  \u03c4 i \u22a4 = \u22a4 := \u03c4.map_verum i\n\n@[simp] lemma map_imply' (p q : formula L\u2081) :\n  \u03c4 i (p \u27f6 q) = \u03c4 i p \u27f6 \u03c4 i q := \u03c4.map_imply p q i\n\n@[simp] lemma map_neg' (p : formula L\u2081) :\n  \u03c4 i (\u223cp) = \u223c\u03c4 i p := \u03c4.map_neg p i\n\n@[simp] lemma map_univ' (p : formula L\u2081) :\n  \u03c4 i (\u2200.p) = \u2200.\u03c4 (i + 1) p := \u03c4.map_univ p i\n\nlemma map_pow'_aux\n  (H_pr : \u2200 {n} (r : L\u2081.pr n) (v) (i s k : \u2115) (le : s \u2264 i),\n    \u03c4 (i + k) ((app r v).rew ((\u03bb x, #(x + k))^s)) = (\u03c4 i (app r v)).rew ((\u03bb x, #(x + k))^s))\n  (H_eq : \u2200 (t u : term L\u2081) (i s k : \u2115) (le : s \u2264 i),\n    \u03c4 (i + k) ((t =' u : formula L\u2081).rew ((\u03bb x, #(x + k))^s)) = (\u03c4 i (t =' u)).rew ((\u03bb x, #(x + k))^s))\n  (p : formula L\u2081) (i s k : \u2115) (hs : s \u2264 i) :\n  \u03c4 (i + k) (p.rew ((\u03bb x, #(x + k))^s)) = (\u03c4 i p).rew ((\u03bb x, #(x + k))^s) :=\nbegin\n  induction p generalizing i s k,\n  case app : n r v { exact H_pr r v i s k hs },\n  case equal : t u i s k { exact H_eq t u i s k hs },\n  case verum { simp },  \n  case imply : p q IH_p IH_q { simp, exact \u27e8IH_p i s k hs, IH_q i s k hs\u27e9 },\n  case neg : p IH { simp, exact IH i s k hs},\n  case fal : p IH { simp[rewriting_sf_itr.pow_add, show i + k + 1 = i + 1 + k, by omega],\n  exact IH (i + 1) (s + 1) k (by simp[hs]) }\nend\n\ndef mk_translation\n  (H_pr : \u2200 {n} (r : L\u2081.pr n) (v) (i s k : \u2115) (le : s \u2264 i),\n    \u03c4 (i + k) ((app r v).rew ((\u03bb x, #(x + k))^s)) = (\u03c4 i (app r v)).rew ((\u03bb x, #(x + k))^s))\n  (H_eq : \u2200 (t u : term L\u2081) (i s k : \u2115) (le : s \u2264 i),\n    \u03c4 (i + k) ((t =' u : formula L\u2081).rew ((\u03bb x, #(x + k))^s)) = (\u03c4 i (t =' u)).rew ((\u03bb x, #(x + k))^s)) : translation L\u2081 L\u2082 :=\n{  map_pow := \u03bb p i, by { simp,\n    have : \u03c4 (i + 1) (p.rew (\u03bb x, #(x + 1))) = rew (\u03bb x, #(x + 1)) (\u03c4 i p),\n    { have := map_pow'_aux \u03c4 (@H_pr) (@H_eq) p i 0 1 (by simp), simp at this, exact this },\n    simp[formula.pow_eq], exact this },  ..\u03c4 }\n\nend formula_homonorphism\n\nnamespace translation\n\n@[simp] lemma app_eq (to_fun) (map_verum) (map_imply) (map_neg) (map_univ) (map_pow) (p : formula L\u2081) (i) :\n  ({ to_fun := to_fun, map_verum := map_verum, map_imply := map_imply, map_neg := map_neg,\n     map_univ := map_univ, map_pow := map_pow} : translation L\u2081 L\u2082) i p = to_fun i p := rfl\n\n@[simp] def fun_of_atom {L\u2081 L\u2082 : language}\n  (tr_pr : \u2115 \u2192 \u03a0 {n}, L\u2081.pr n \u2192 finitary (term L\u2081) n \u2192 formula L\u2082)\n  (tr_eq : \u2115 \u2192 term L\u2081 \u2192 term L\u2081 \u2192 formula L\u2082) : \u2115 \u2192 formula L\u2081 \u2192 formula L\u2082\n| k \u22a4                    := \u22a4\n| k (app p v)            := tr_pr k p v\n| k ((t : term L\u2081) =' u)  := tr_eq k t u\n| k (p \u27f6 q)              := fun_of_atom k p \u27f6 fun_of_atom k q\n| k (\u223cp)                 := \u223cfun_of_atom k p\n| k (\u2200.(p : formula L\u2081)) := \u2200.fun_of_atom (k + 1) p\n\ndef mk_of_atom' {L\u2081 L\u2082 : language}\n  (tr_pr : \u2115 \u2192 \u03a0 {n}, L\u2081.pr n \u2192 finitary (term L\u2081) n \u2192 formula L\u2082)\n  (tr_eq : \u2115 \u2192 term L\u2081 \u2192 term L\u2081 \u2192 formula L\u2082)\n  (map_pow : \u2200 (p : formula L\u2081) (k : \u2115), fun_of_atom @tr_pr @tr_eq (k + 1) (p^1) = (fun_of_atom @tr_pr @tr_eq k p)^1) :\n  translation L\u2081 L\u2082 :=\n{ to_fun := fun_of_atom @tr_pr @tr_eq,\n  map_verum := by simp,\n  map_imply := by simp,\n  map_neg := by simp,\n  map_univ := by simp,\n  map_pow := map_pow }\n\nvariables (\u03c4 : translation L\u2081 L\u2082) (i : \u2115)\n\n@[simp] lemma map_verum' :\n  \u03c4 i \u22a4 = \u22a4 := \u03c4.map_verum i\n\n@[simp] lemma map_imply' (p q : formula L\u2081) :\n  \u03c4 i (p \u27f6 q) = \u03c4 i p \u27f6 \u03c4 i q := \u03c4.map_imply p q i\n\n@[simp] lemma map_neg' (p : formula L\u2081) :\n  \u03c4 i (\u223cp) = \u223c\u03c4 i p := \u03c4.map_neg p i\n\n@[simp] lemma map_univ' (p : formula L\u2081) :\n  \u03c4 i (\u2200.p) = \u2200.\u03c4 (i + 1) p := \u03c4.map_univ p i\n\nlemma map_pow' (p : formula L\u2081) (k : \u2115) :\n  \u03c4 (i + k) (p^k) = (\u03c4 i p)^k := by { induction k with k IH; simp[\u2190nat.add_one, \u2190add_assoc],\n  have : \u03c4 (i + k + 1) (p^(k + 1)) = \u03c4 (i + k) (p^k)^1, simp[\u2190formula.pow_add], from map_pow \u03c4 (p^k) (i + k), \n  simp[IH, formula.pow_add] at this, exact this }\n\n@[simp] lemma map_falsum' :\n  \u03c4 i \u22a5 = \u22a5 := by { unfold has_bot.bot, simp }\n\n@[simp] lemma map_ex' (p : formula L\u2081) :\n  \u03c4 i (\u2203.p) = \u2203.(\u03c4 (i + 1) p) := by { unfold has_exists_quantifier.ex formula.ex, simp }\n\n@[simp] lemma map_and' (p q : formula L\u2081) :\n  \u03c4 i (p \u2293 q) = \u03c4 i p \u2293 \u03c4 i q := by { unfold has_inf.inf formula.and, simp }\n\n@[simp] lemma map_or' (p q : formula L\u2081) :\n  \u03c4 i (p \u2294 q) = \u03c4 i p \u2294 \u03c4 i q := by { unfold has_sup.sup formula.or, simp }\n\n@[simp] lemma map_equiv' (p q : formula L\u2081) :\n  \u03c4 i (p \u27f7 q) = \u03c4 i p \u27f7 \u03c4 i q := by simp[lrarrow_def]\n\n@[simp] lemma map_nfal' (p : formula L\u2081) (k : \u2115) :\n  \u03c4 i (\u2200.[k] p) = \u2200.[k] \u03c4 (i + k) p :=\nby { induction k with k IH generalizing i; simp[*],\n     { simp[show i + k.succ = i + 1 + k, by omega] } }\n\n@[simp] lemma map_conjunction'' {n} (P : finitary (formula L\u2081) n) :\n  \u03c4 i (\u22c0 j, P j) = \u22c0 j, (\u03c4 i (P j)) :=\nby { induction n with n IH generalizing P; simp* }\n\n@[simp] lemma map_disjunction'' {n} (P : finitary (formula L\u2081) n) :\n  \u03c4 i (\u22c1 j, P j) = \u22c1 j, (\u03c4 i (P j)) :=\nby { induction n with n IH generalizing P; simp* }\n\nvariables (L\u2081) (L\u2082) (L\u2083)\n\nprotected def refl : translation L\u2081 L\u2081 :=\n{ to_fun := \u03bb _, id,\n  map_verum := by simp, map_imply := by simp, map_neg := by simp, map_univ := by simp, map_pow := by simp }\n\ndef shift (k : \u2115) : translation L\u2081 L\u2081 :=\n{ to_fun := \u03bb i p, p.rew (\u03bb x, if x < i then #x else #(x + k)),\n  map_verum := by simp,\n  map_imply := by simp,\n  map_neg := by simp,\n  map_univ := \u03bb p i, by { simp[rewriting_sf_itr.pow_eq], congr, funext x, cases x; simp[\u2190nat.add_one],\n    by_cases C : x < i; simp[C], omega },\n  map_pow := \u03bb p i, by { simp[formula.pow_eq, formula.nested_rew], congr, \n    funext x, by_cases C : x < i; simp[C], omega } }\n\nvariables {L\u2081} {L\u2082} {L\u2083}\n\ndef comp : translation L\u2082 L\u2083 \u2192 translation L\u2081 L\u2082 \u2192 translation L\u2081 L\u2083 := \u03bb \u03c4\u2082\u2083 \u03c4\u2081\u2082,\n{ to_fun := \u03bb i, \u03c4\u2082\u2083 i \u2218 \u03c4\u2081\u2082 i,\n  map_verum := by simp, map_imply := by simp, map_neg := by simp,\n  map_univ := by simp, map_pow := by simp[map_pow'] }\n\nend translation\n\n\nnamespace term_homomorphism\n\n@[simp] lemma translation.map_imply' (\u03c4 : term_homomorphism L\u2081 L\u2082) {n} (f : L\u2081.fn n) (v : finitary (term L\u2081) n) (k) :\n  \u03c4 k (term.app f v) = \u03c4.to_fun_chr k f (\u03bb i, \u03c4 k (v i)) := \u03c4.map_fn k f v\n\n@[simp] lemma app_eq (fc) (f) (map_fn) (t : term L\u2081) (i) :\n  ({to_fun_chr := fc, to_fun := f, map_fn := map_fn} : term_homomorphism L\u2081 L\u2082) i t = f i t := rfl\n\n@[simp] def mk_fun_of_atom {L\u2081 L\u2082 : language} \n  (to_fun_chr : \u2115 \u2192 \u03a0 {n}, L\u2081.fn n \u2192 finitary (term L\u2082) n \u2192 term L\u2082) : \u2115 \u2192 term L\u2081 \u2192 term L\u2082\n| _ #n        := #n\n| k (app f v) := to_fun_chr k f (\u03bb i, mk_fun_of_atom k (v i))\n\n@[simp] def mk_of_atom {L\u2081 L\u2082 : language}\n  (to_fun_chr : \u2115 \u2192 \u03a0 {n}, L\u2081.fn n \u2192 finitary (term L\u2082) n \u2192 term L\u2082) : term_homomorphism L\u2081 L\u2082 :=\n{ to_fun_chr := @to_fun_chr,\n  to_fun := mk_fun_of_atom @to_fun_chr,\n  map_fn := by simp }\n\nend term_homomorphism\n\nnamespace term_formula_translation\nopen translation\nvariables (\u03c4 : term_formula_translation L\u2081 L\u2082) (k : \u2115)\n\n@[simp] lemma map_equal (t\u2081 t\u2082 : term L\u2081) : \u03c4.p k (t\u2081 =' t\u2082 : formula L\u2081) = (\u03c4.t k t\u2081 =' \u03c4.t k t\u2082) := \u03c4.equal t\u2081 t\u2082 k\n\n@[simp] lemma map_app {n} (r : L\u2081.pr n) (v) :\n  \u03c4.p k (formula.app r v) = formula.app (\u03c4.chr r) (\u03bb i, \u03c4.t k (v i)) := \u03c4.app k r v\n\nlemma map_pow' (t : term L\u2081) (k s : \u2115) :\n  \u03c4.t (k + s) (t^s) = (\u03c4.t k t)^s := \nby { induction s with s IH; simp[\u2190nat.add_one, \u2190add_assoc],\n     have : \u03c4.t (k + s + 1) ((t ^ s) ^ 1) = \u03c4.t (k + s) (t ^ s) ^ 1, from map_pow \u03c4 (t^s) (k + s), \n     simp[IH, term.pow_add] at this, exact this }\n\nlemma tr_subst_of_subst\n  (H : \u2200 (t u : term L\u2081) (s m : \u2115) (le : m \u2264 s), \u03c4.t s (t.rew \u0131[m \u21dd u]) = (\u03c4.t (s + 1) t).rew \u0131[m \u21dd \u03c4.t s u])\n  (p : formula L\u2081) (t : term L\u2081) (s m : \u2115) (le : m \u2264 s) :\n  \u03c4.p s (p.rew \u0131[m \u21dd t]) = (\u03c4.p (s + 1) p).rew \u0131[m \u21dd \u03c4.t s t] :=\nbegin\n  induction p generalizing t s m,\n  case app : n r v { simp, funext i, exact H (v i) t s m le },\n  case equal : u\u2081 u\u2082 { simp, exact \u27e8H u\u2081 t s m le, H u\u2082 t s m le\u27e9 },\n  case verum { simp },\n  case imply : p q IH_p IH_q { simp, exact \u27e8IH_p t s m le, IH_q t s m le\u27e9 },\n  case neg : p IH { simp, exact IH t s m le },\n  case fal : p IH { simp[subst_pow, \u2190map_pow'], exact IH (t^1) (s + 1) (m + 1) (by simp[le]) },\nend\n\nopen provable axiomatic_classical_logic axiomatic_classical_logic'\n\ndef conservative_of\n  (H : \u2200 (t u : term L\u2081) (s m) (le : m \u2264 s), \u03c4.t s (t.rew \u0131[m \u21dd u]) = (\u03c4.t (s + 1) t).rew \u0131[m \u21dd \u03c4.t s u])\n  (function_ext : \u2200 (s) {n} (f : L\u2081.fn n) (T : Theory L\u2081) (k : \u2115),\n    (tr_Theory \u03c4.p s T)^k \u22a2 \u03c4.p (s + k) (eq_axiom4 f))\n  (predicate_ext : \u2200 (s) {n} (r : L\u2081.pr n) (T : Theory L\u2081) (k : \u2115),\n    (tr_Theory \u03c4.p s T)^k \u22a2 \u03c4.p (s + k) (eq_axiom5 r))\n   : conservative \u03c4.p :=\n{ ax_ss := \u03bb _ _, by refl,\n  specialize := \u03bb s p t T k, by simp[tr_subst_of_subst \u03c4 H],\n  eq_reflexivity := \u03bb s T k, by { simp, refine generalize (by simp) },\n  eq_symmetry := \u03bb s T k, by { simp, refine generalize (generalize _),\n    have : \u290a\u290a(tr_Theory \u03c4.p s T ^ k) \u22a2 _, from eq_symmetry \u229a (\u03c4.t (s + k + 1 + 1) #1) \u229a \u03c4.t (s + k + 1 + 1) #0,\n    simp at this, simp at this, exact this },\n  eq_transitive := \u03bb s T k,\n   by { simp, refine generalize (generalize (generalize _)),\n        have : \u290a\u290a\u290a(tr_Theory \u03c4.p s T ^ k) \u22a2 _, from eq_transitivity \u229a \u03c4.t (s + k + 1 + 1 + 1) #2 \u229a \u03c4.t (s + k + 1 + 1 + 1) #1 \u229a \u03c4.t (s + k + 1 + 1 + 1) #0,\n        simp at this, simp at this, exact this },\n  function_ext := \u03bb s n f T k, by { exact function_ext s f T k },\n  predicate_ext := \u03bb s n f T k, by { exact predicate_ext s f T k } }\n\n\nend term_formula_translation\n\nnamespace language_translation\n\nlemma mk.eta : \u03a0 (\u03c4 : L\u2081 \u219d\u1d38 L\u2082), ({fn := \u03c4.fn, pr := \u03c4.pr} : L\u2081 \u219d\u1d38 L\u2082) = \u03c4\n| \u27e8fn, pr\u27e9 := rfl\n\nlemma eq_iff {\u03c4 \u03c3 : L\u2081 \u219d\u1d38 L\u2082} : \u03c4 = \u03c3 \u2194 (\u2200 n f, \u03c4.fn n f = \u03c3.fn n f) \u2227 (\u2200 n r, \u03c4.pr n r = \u03c3.pr n r) :=\nby { rw[\u2190mk.eta \u03c4, \u2190mk.eta \u03c3], simp, split,\n     { rintros \u27e8eq_fn, eq_pr\u27e9, simp* }, { rintros \u27e8hfn, hpr\u27e9, refine \u27e8_, _\u27e9; { funext, simp* } } }\n\n@[ext] lemma ext {\u03c4 \u03c3 : L\u2081 \u219d\u1d38 L\u2082} (eq_fn : \u2200 n f, \u03c4.fn n f = \u03c3.fn n f) (eq_pr : \u2200 n r, \u03c4.pr n r = \u03c3.pr n r) : \u03c4 = \u03c3 :=\nby { simp[eq_iff], exact \u27e8eq_fn, eq_pr\u27e9 }\n\ndef from_empty : \u2205 \u219d\u1d38 L :=\n{ fn := \u03bb n f, by rcases f, pr := \u03bb n r, by rcases r }\n\ndef one (L : language) : L \u219d\u1d38 L :=\n{ fn := \u03bb n, id, pr := \u03bb n, id }\n\ninstance : has_one (L \u219d\u1d38 L) := \u27e8one L\u27e9 \n\ndef comp : L\u2082 \u219d\u1d38 L\u2083 \u2192 L\u2081 \u219d\u1d38 L\u2082 \u2192 L\u2081 \u219d\u1d38 L\u2083 := \u03bb \u03c4\u2082\u2083 \u03c4\u2081\u2082,\n{ fn := \u03bb n, (\u03c4\u2082\u2083.fn n) \u2218 (\u03c4\u2081\u2082.fn n),\n  pr := \u03bb n, (\u03c4\u2082\u2083.pr n) \u2218 (\u03c4\u2081\u2082.pr n) }\n\nvariables (\u03c4 : L\u2081 \u219d\u1d38 L\u2082)\n\n@[simp] def fun_t : term L\u2081 \u2192 term L\u2082\n| #n        := #n\n| (app f v) := app (\u03c4.fn _ f) (\u03bb i, fun_t (v i))\n\ndef tr_term : term_homomorphism L\u2081 L\u2082 :=\n{ to_fun_chr := \u03bb k n f v, app (\u03c4.fn _ f) v,\n  to_fun     := \u03bb k, \u03c4.fun_t,\n  map_fn     := \u03bb k n f v, by simp }\n\n@[simp] def fun_p : formula L\u2081 \u2192 formula L\u2082\n| \u22a4                    := \u22a4\n| (app p v)            := app (\u03c4.pr _ p) (\u03bb i, fun_t \u03c4 (v i))\n| ((t : term L\u2081) =' u)  := fun_t \u03c4 t =' fun_t \u03c4 u\n| (p \u27f6 q)              := fun_p p \u27f6 fun_p q\n| (\u223cp)                 := \u223cfun_p p\n| (\u2200.(p : formula L\u2081)) := \u2200.fun_p p\n\ndef fun_Theory (T : Theory L\u2081) : Theory L\u2082 := \u03c4.fun_p '' T\n\nlemma fun_t_rew_var : \u2200 (t : term L\u2081) (s : \u2115 \u2192 \u2115),\n  (fun_t \u03c4 t).rew (\u03bb x, #(s x)) = fun_t \u03c4 (t.rew (\u03bb x, #(s x)))\n| (#n)                s := by simp\n| (@term.app _ n f v) s := by { simp, funext i, exact @fun_t_rew_var (v i) _ }\n\nlemma fun_p_rew_var : \u2200 (p : formula L\u2081) (s : \u2115 \u2192 \u2115),\n  (fun_p \u03c4 p).rew (\u03bb x, #(s x)) = fun_p \u03c4 (p.rew (\u03bb x, #(s x)))\n| \u22a4                      _ := by simp\n| (@formula.app _ n r v) s := by { simp, funext i, simp[fun_t_rew_var] }\n| ((t : term L\u2081) =' u)    s := by simp[fun_t_rew_var]\n| (p \u27f6 q)                s := by simp[fun_p_rew_var p, fun_p_rew_var q]\n| (\u223cp)                   s := by simp[fun_p_rew_var p]\n| (\u2200.(p : formula L\u2081))   s := by { \n    have eqn\u2081 : ((\u03bb x, #(s x))^1 : \u2115 \u2192 term L\u2081) = (\u03bb x, #(if x = 0 then 0 else s (x - 1) + 1)),\n    { funext x, cases x; simp },\n    have eqn\u2082 : ((\u03bb x, #(s x))^1 : \u2115 \u2192 term L\u2082) = (\u03bb x, #(if x = 0 then 0 else s (x - 1) + 1)),\n    { funext x, cases x; simp },\n    simp[fal_pow, eqn\u2081, eqn\u2082, fun_p_rew_var p] }\n\ndef tr : translation L\u2081 L\u2082 :=\n{ to_fun := \u03bb _, \u03c4.fun_p,\n  map_verum := by simp,\n  map_imply := by simp,\n  map_neg := by simp,\n  map_univ := by simp,\n  map_pow := \u03bb p i, eq.symm (\u03c4.fun_p_rew_var p (\u03bb x, x + 1)) }\n\n@[simp] lemma fun_t_arity (t : term L\u2081) : (\u03c4.fun_t t).arity = t.arity :=\nby induction t; simp*\n\n@[simp] lemma fun_p_arity (p : formula L\u2081) : (\u03c4.fun_p p).arity = p.arity :=\nby induction p; simp*\n\nlemma tr_term_app_eq (k) (t) : \n  \u03c4.tr_term k t = \u03c4.fun_t t := by refl\n\nlemma tr_app_eq (k) (p) : \n  \u03c4.tr k p = \u03c4.fun_p p := by refl\n\n@[simp] lemma tr_term_to_fun_chr_app_eq (k) {n} (f : L\u2081.fn n) (v : finitary (term L\u2082) n) :\n  \u03c4.tr_term.to_fun_chr k f v = app (\u03c4.fn _ f) v := rfl\n\n@[simp] lemma fun_t_pow (t : term L\u2081) (i : \u2115) :\n  (\u03c4.fun_t (t^i) : term L\u2082) = (\u03c4.fun_t t)^i :=\neq.symm (\u03c4.fun_t_rew_var t (\u03bb x, x + i))\n\n@[simp] lemma fun_p_pow (p : formula L\u2081) (i : \u2115) :\n  (\u03c4.fun_p (p^i) : formula L\u2082) = (\u03c4.fun_p p)^i := \neq.symm (\u03c4.fun_p_rew_var p (\u03bb x, x + i))\n\n@[simp] lemma fun_p_and (p q : formula L\u2081) :\n  \u03c4.fun_p (p \u2293 q) = \u03c4.fun_p p \u2293 \u03c4.fun_p q := rfl\n\n@[simp] lemma fun_p_or (p q : formula L\u2081) :\n  \u03c4.fun_p (p \u2294 q) = \u03c4.fun_p p \u2294 \u03c4.fun_p q := rfl\n\n@[simp] lemma fun_p_ex (p : formula L\u2081)  :\n  \u03c4.fun_p (\u2203.p) = \u2203.\u03c4.fun_p p := rfl\n\n@[simp] lemma fun_p_bot :\n  \u03c4.fun_p (\u22a5 : formula L\u2081) = \u22a5 := rfl\n\n@[simp] lemma fun_p_conjunction (P : list (formula L\u2081)) :\n  \u03c4.fun_p P.conjunction = list.conjunction (P.map \u03c4.fun_p) :=\nby induction P with p P IH; simp[*]\n\n@[simp] lemma fun_p_nfal (p : formula L\u2081) (k : \u2115) :\n  \u03c4.fun_p (\u2200.[k] p) = \u2200.[k] \u03c4.fun_p p :=\nby { induction k with k IH; simp[*] }\n\n@[simp] lemma fun_p_fal_complete (p : formula L\u2081) :\n  \u03c4.fun_p (\u2200.* p) = \u2200.* \u03c4.fun_p p :=\nby simp[fal_complete]\n\n@[simp] lemma fun_p_conjunction' {n : \u2115} (P : finitary (formula L\u2081) n) :\n  \u03c4.fun_p (\u22c0 j, P j) = \u22c0 j, \u03c4.fun_p (P j) :=\nby { induction n with n IH generalizing P; simp* }\n\n@[simp] lemma fun_p_disjunction' {n : \u2115} (P : finitary (formula L\u2081) n) :\n  \u03c4.fun_p (\u22c1 j, P j) = \u22c1 j, \u03c4.fun_p (P j) :=\nby { induction n with n IH generalizing P; simp* }\n\nlemma fun_t_rew : \u2200 (t : term L\u2081) (s : \u2115 \u2192 term L\u2081),\n  \u03c4.fun_t (t.rew s) = (\u03c4.fun_t t).rew (\u03bb x, \u03c4.fun_t (s x))\n| (#x)           s := by simp\n| (term.app p v) s := by simp[\u03bb i, fun_t_rew (v i)]\n\n@[simp] lemma fun_t_subst (t u : term L\u2081) (s) : \u03c4.fun_t (t.rew \u0131[s \u21dd u]) = (\u03c4.fun_t t).rew \u0131[s \u21dd \u03c4.fun_t u] :=\nbegin\n  have : (\u03bb x, \u03c4.fun_t (\u0131[s \u21dd u] x)) = \u0131[s \u21dd \u03c4.fun_t u],\n  { funext x, have : x < s \u2228 x = s \u2228 s < x, exact trichotomous x s,\n    rcases this with (lt | rfl | lt); simp* },\n  simp[fun_t_rew, this]\nend\n\nlemma fun_p_rew : \u2200 (p : formula L\u2081) (s : \u2115 \u2192 term L\u2081),\n  \u03c4.fun_p (p.rew s) = (\u03c4.fun_p p).rew (\u03bb x, \u03c4.fun_t (s x))\n| \u22a4                 s := by simp\n| (formula.app f v) s := by simp[fun_t_rew]\n| (t =' u)          s := by simp[fun_t_rew]\n| (p \u27f6 q)           s := by simp[fun_p_rew p, fun_p_rew q]\n| (\u223cp)              s := by simp[fun_p_rew p]\n| (\u2200.p)             s := by\n    { simp[fun_p_rew p, rewriting_sf_itr.pow_eq'], congr, funext x, cases x; simp }\n\n@[simp] lemma fun_p_subst (p : formula L\u2081) (u : term L\u2081) (s) : \u03c4.fun_p (p.rew \u0131[s \u21dd u]) = (\u03c4.fun_p p).rew \u0131[s \u21dd \u03c4.fun_t u] :=\nbegin\n  have : (\u03bb x, \u03c4.fun_t (\u0131[s \u21dd u] x)) = \u0131[s \u21dd \u03c4.fun_t u],\n  { funext x, have : x < s \u2228 x = s \u2228 s < x, exact trichotomous x s,\n    rcases this with (lt | rfl | lt); simp* },\n  simp[fun_p_rew, this]\nend\n\nlemma fun_t_inversion_of_le {t\u2081 : term L\u2081} {u\u2082 : term L\u2082} (le : u\u2082 \u2264 \u03c4.fun_t t\u2081) :\n  \u2203 (u\u2081 : term L\u2081) (le : u\u2081 \u2264 t\u2081), u\u2082 = \u03c4.fun_t u\u2081 :=\nbegin\n  induction t\u2081 generalizing u\u2082,\n  case var : n { simp at le, refine \u27e8#n, by simp[le]\u27e9 },\n  case app : n f v IH\n  { rcases le_iff_lt_or_eq.mp le with (lt | rfl),\n    { simp at lt, rcases lt with \u27e8i, le\u27e9, rcases IH i le with \u27e8t, t_le', rfl\u27e9, refine \u27e8t, le_trans t_le' (by simp), rfl\u27e9 },\n    { refine \u27e8app f v, by refl, rfl\u27e9 } }\nend\n\nlemma fun_p_inversion_of_le {p\u2081 : formula L\u2081} {q\u2082 : formula L\u2082} (le : q\u2082 \u2264 \u03c4.fun_p p\u2081) :\n  \u2203 (q\u2081 : formula L\u2081) (le : q\u2081 \u2264 p\u2081), q\u2082 = \u03c4.fun_p q\u2081 :=\nbegin\n  induction p\u2081 generalizing q\u2082,\n  case app : n r v { simp at le, refine \u27e8app r v, by simp[le]\u27e9 },\n  case equal : t u { simp at le, refine \u27e8t =' u, by simp[le]\u27e9 },\n  case verum { simp at le, refine \u27e8\u22a4, by simp[le]\u27e9 },\n  case imply : p q IH_p IH_q\n  { rcases le_iff_lt_or_eq.mp le with (lt | rfl),\n    { simp at lt, rcases lt with (le | le),\n      { rcases IH_p le with \u27e8q\u2081, le', rfl\u27e9, refine \u27e8q\u2081, le_trans le' (le_of_lt (by simp)), rfl\u27e9 },\n      { rcases IH_q le with \u27e8q\u2081, le', rfl\u27e9, refine \u27e8q\u2081, le_trans le' (le_of_lt (by simp)), rfl\u27e9 } },\n    { refine \u27e8p \u27f6 q, by simp\u27e9 } },\n  case neg : p IH\n  { rcases le_iff_lt_or_eq.mp le with (lt | rfl),\n    { simp at lt, rcases IH lt with \u27e8q\u2081, le', rfl\u27e9, refine \u27e8q\u2081, le_trans le' (le_of_lt (by simp)), rfl\u27e9 },\n    { refine \u27e8\u223cp, by simp\u27e9 } },\n  case fal : p IH\n  { rcases le_iff_lt_or_eq.mp le with (lt | rfl),\n    { simp at lt, rcases IH lt with \u27e8q\u2081, le', rfl\u27e9, refine \u27e8q\u2081, le_trans le' (le_of_lt (by simp)), rfl\u27e9 },\n    { refine \u27e8\u2200.p, by simp\u27e9 } },\nend\n\nlemma fun_p_inversion_of_mem {p\u2081 : formula L\u2081} {t\u2082 : term L\u2082} (mem : t\u2082 \u2208 \u03c4.fun_p p\u2081) :\n  \u2203 (t\u2081 : term L\u2081) (mem : t\u2081 \u2208 p\u2081), t\u2082 = \u03c4.fun_t t\u2081 :=\nbegin\n  induction p\u2081 generalizing t\u2082,\n  case app : n r v\n  { simp at mem, rcases mem with \u27e8i, le\u27e9,\n    rcases fun_t_inversion_of_le \u03c4 le with \u27e8t\u2081, le', rfl\u27e9, refine \u27e8t\u2081, by simp; exact \u27e8i, le'\u27e9, rfl\u27e9 },\n  case equal : t u\n  { simp at mem, rcases mem with (le | le),\n    { rcases fun_t_inversion_of_le \u03c4 le with \u27e8t\u2081, le', rfl\u27e9, refine \u27e8t\u2081, by simp[le'], rfl\u27e9 },\n    { rcases fun_t_inversion_of_le \u03c4 le with \u27e8t\u2081, le', rfl\u27e9, refine \u27e8t\u2081, by simp[le'], rfl\u27e9 } },\n  case verum { simp at mem, contradiction },\n  case imply : p q IH_p IH_q\n  { simp at mem, rcases mem with (mem | mem),\n    { rcases IH_p mem with \u27e8t', mem', rfl\u27e9, refine \u27e8t', by simp[mem'], rfl\u27e9 },\n    { rcases IH_q mem with \u27e8t', mem', rfl\u27e9, refine \u27e8t', by simp[mem'], rfl\u27e9 } },\n  case neg : p IH { simp at mem \u22a2, rcases IH mem with \u27e8t', mem', rfl\u27e9, refine \u27e8t', mem', rfl\u27e9 },\n  case fal : p IH { simp at mem \u22a2, rcases IH mem with \u27e8t', mem', rfl\u27e9, refine \u27e8t', mem', rfl\u27e9 }\nend\n\nvariables (\u03c4\u2081\u2082 \u03c3\u2081\u2082 : L\u2081 \u219d\u1d38 L\u2082) (\u03c4\u2082\u2083 : L\u2082 \u219d\u1d38 L\u2083) {L\u2084 : language.{u}} (\u03c4\u2083\u2084 : L\u2083 \u219d\u1d38 L\u2084)\n\n@[simp] lemma one_fn {n} (f : L.fn n) : fn 1 n f = f := rfl\n\n@[simp] lemma one_pr {n} (r : L.pr n) : pr 1 n r = r := rfl\n\n@[simp] lemma comp_fn {n} (f : L\u2081.fn n) : (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082).fn n f = \u03c4\u2082\u2083.fn n (\u03c4\u2081\u2082.fn n f) := rfl\n\n@[simp] lemma comp_pr {n} (r : L\u2081.pr n) : (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082).pr n r = \u03c4\u2082\u2083.pr n (\u03c4\u2081\u2082.pr n r) := rfl\n\n@[simp] lemma one_fun_t (t : term L) : fun_t 1 t = t :=\nby induction t; simp*\n\n@[simp] lemma one_fun_p (p : formula L) : fun_p 1 p = p :=\nby induction p; simp*\n\nlemma comp_fun_t : (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082).fun_t = \u03c4\u2082\u2083.fun_t \u2218 \u03c4\u2081\u2082.fun_t :=\nby funext t; induction t; simp*\n\nlemma comp_fun_p : (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082).fun_p = \u03c4\u2082\u2083.fun_p \u2218 \u03c4\u2081\u2082.fun_p :=\nby funext p; induction p; simp[*, comp_fun_t]\n\nlemma comp_fun_Theory : (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082).fun_Theory = \u03c4\u2082\u2083.fun_Theory \u2218 \u03c4\u2081\u2082.fun_Theory :=\nby ext q; simp[fun_Theory, comp_fun_p]\n\n@[simp] lemma comp_one : \u03c4.comp 1 = \u03c4 := by ext; simp\n\n@[simp] lemma one_comp : comp 1 \u03c4 = \u03c4 := by ext; simp\n\n@[simp] lemma comp_assoc : (\u03c4\u2083\u2084.comp \u03c4\u2082\u2083).comp \u03c4\u2081\u2082 = \u03c4\u2083\u2084.comp (\u03c4\u2082\u2083.comp \u03c4\u2081\u2082) := by ext; simp\n\n@[simp] lemma fun_p_is_sentence (p : formula L\u2081) : is_sentence (\u03c4.fun_p p) \u2194 is_sentence p :=\nby simp[is_sentence]\n\nvariables (T : Theory L\u2081)\n\ninstance [closed_Theory T] : closed_Theory (\u03c4.fun_Theory T) :=\n\u27e8\u03bb p mem, by { rcases mem with \u27e8p, mem, rfl\u27e9, simp[closed_Theory.cl mem] }\u27e9 \n\nlemma fun_Theory_insert (p : formula L\u2081) : \u03c4.fun_Theory (T+{p}) = \u03c4.fun_Theory T +{\u03c4.fun_p p} :=\nset.image_insert_eq\n\nend language_translation\n\nnamespace has_predecessor\nvariables {L} [has_predecessor L]\n\n\n\nend has_predecessor\n\nnamespace language_translation_coe\nopen language_translation\n\ninstance : language_translation_coe \u2205 L :=\n{ltr := from_empty, fn_inj := \u03bb n f g, by rcases f, pr_inj := \u03bb n r s, by rcases r }\n\ninstance coe_one : language_translation_coe L L :=\n{ltr := 1, fn_inj := \u03bb n f g, by simp, pr_inj := \u03bb n r s, by simp }\n\nsection\nvariables [language_translation_coe L\u2081 L\u2082]\n\ninstance {n} : has_coe (L\u2081.fn n) (L\u2082.fn n) := \u27e8\u03bb f, ltr.fn n f\u27e9\n\nlemma coe_fn_def {n} (f : L\u2081.fn n) : (\u2191f : L\u2082.fn n) = ltr.fn n f := rfl\n\ninstance {n} : has_coe (L\u2081.pr n) (L\u2082.pr n) := \u27e8\u03bb n, ltr.pr _ n\u27e9\n\nlemma coe_pr_def {n} (r : L\u2081.pr n) : (\u2191r : L\u2082.pr n) = ltr.pr n r := rfl\n\ninstance : has_coe (term L\u2081) (term L\u2082) := \u27e8ltr.fun_t\u27e9\n\nlemma coe_t_def (t : term L\u2081) : (\u2191t : term L\u2082) = ltr.fun_t t := rfl\n\nlemma app_term_extension_eq (t : term L\u2081) (i : \u2115) :\n  (ltr.tr_term i t : term L\u2082) = \u2191t := rfl\n\ninstance : has_coe (formula L\u2081) (formula L\u2082) := \u27e8ltr.fun_p\u27e9\n\nlemma coe_p_def (p : formula L\u2081) : (\u2191p : formula L\u2082) = ltr.fun_p p := rfl\n\nlemma app_formula_extension_eq (p : formula L\u2081) (i : \u2115) :\n  (ltr.tr i p : formula L\u2082) = \u2191p := rfl\n\nlemma coe_def_p (p : formula L\u2081) : (\u2191p : formula L\u2082) = ltr.fun_p p := rfl\n\ninstance : has_coe (Theory L\u2081) (Theory L\u2082) := \u27e8tr_Theory ltr.tr 0\u27e9\n\ninstance zero [has_zero_symbol L\u2081] : has_zero_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).fn _ has_zero_symbol.zero\u27e9\n\ninstance succ [has_succ_symbol L\u2081] : has_succ_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).fn _ has_succ_symbol.succ\u27e9\n\ninstance add [has_add_symbol L\u2081] : has_add_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).fn _ has_add_symbol.add\u27e9\n\ninstance mul [has_mul_symbol L\u2081] : has_mul_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).fn _ has_mul_symbol.mul\u27e9\n\ninstance le [has_le_symbol L\u2081] : has_le_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).pr _ has_le_symbol.le\u27e9\n\ninstance mem [has_mem_symbol L\u2081] : has_mem_symbol L\u2082 := \u27e8(ltr : L\u2081 \u219d\u1d38 L\u2082).pr _ has_mem_symbol.mem\u27e9\n\nlemma app_formula_extension_eq_coe (k) (p : formula L\u2081) :\n  (ltr.tr : L\u2081 \u219d L\u2082) k p = \u2191p := rfl\n\nlemma app_term_extension_eq_coe (k) (t : term L\u2081) :\n  (ltr.tr_term : term_homomorphism L\u2081 L\u2082) k t = \u2191t := rfl\n\n@[simp] lemma add_tr_v1_var (n) : ((#n : term L\u2081) : term L\u2082) = #n := rfl\n\nlemma add_tr_v1_app {n} (f : L\u2081.fn n) (v : finitary (term L\u2081) n) :\n  ((\u2768f\u2769 v : term L\u2081) : term L\u2082) = \u2768\u2191f\u2769 (\u03bb i, (v i)) := by refl\n\n@[simp] lemma coe_tr_v1_zero [has_zero_symbol L\u2081] :\n  ((0 : term L\u2081) : term L\u2082) = 0 := by { unfold has_zero.zero has_zero_symbol.zero,\n   simp [\u2190app_term_extension_eq_coe 0] }\n\n@[simp] lemma coe_tr_v1_succ [has_succ_symbol L\u2081] (t : term L\u2081) :\n  ((Succ t : term L\u2081) : term L\u2082) = Succ t :=\nby { unfold has_succ.succ, simp [\u2190app_term_extension_eq_coe 0],\n     split, { refl }, { ext; simp } }\n\n@[simp] lemma coe_tr_v1_numeral [has_zero_symbol L\u2081] [has_succ_symbol L\u2081] (n : \u2115) :\n  ((n\u02d9 : term L\u2081) : term L\u2082) = n\u02d9 :=\nby induction n; simp[*, numeral, coe_tr_v1_zero]\n\n@[simp] lemma coe_tr_v1_add [has_add_symbol L\u2081] (t u : term L\u2081) :\n  ((t + u : term L\u2081) : term L\u2082) = t + u :=\nby { unfold has_add.add, simp [\u2190app_term_extension_eq_coe 0],\n     split, { refl }, { ext; simp } }\n\n@[simp] lemma coe_tr_v1_mul [has_mul_symbol L\u2081] (t u : term L\u2081) :\n  ((t * u : term L\u2081) : term L\u2082) = t * u :=\nby { unfold has_mul.mul, simp [\u2190app_term_extension_eq_coe 0],\n     split, { refl }, { ext; simp } }\n\n@[simp] lemma coe_tr_v1_le [has_le_symbol L\u2081] (t u : term L\u2081) :\n  ((t \u227c u : formula L\u2081) : formula L\u2082) = ((t : term L\u2082) \u227c u) :=\nby { unfold has_preceq.preceq, simp [\u2190app_formula_extension_eq_coe 0, tr_app_eq], \n     split, { refl }, { ext; simp; refl } }\n\n@[simp] lemma coe_tr_v1_mem [has_mem_symbol L\u2081] (t u : term L\u2081) :\n  ((t \u220a u : formula L\u2081) : formula L\u2082) = ((t : term L\u2082) \u220a u) :=\nby { unfold has_elem.elem, simp [\u2190app_formula_extension_eq_coe 0, tr_app_eq],\n     split, { refl }, { ext; simp; refl } }\n\n@[simp] lemma function_inj {n} {f g : L\u2081.fn n} : (ltr : L\u2081 \u219d\u1d38 L\u2082).fn n f = (ltr : L\u2081 \u219d\u1d38 L\u2082).fn n g \u2194 f = g :=\n\u27e8by { have := fn_inj, exact this _ f g }, congr_arg _\u27e9\n\n@[simp] lemma predicate_inj {n} {r s : L\u2081.pr n} : (ltr : L\u2081 \u219d\u1d38 L\u2082).pr n r = (ltr : L\u2081 \u219d\u1d38 L\u2082).pr n s \u2194 r = s :=\n\u27e8by { have := pr_inj, exact this _ r s }, congr_arg _\u27e9\n\n@[simp] lemma function_coe_inj {n} {f g : L\u2081.fn n} : (f : L\u2082.fn n) = g \u2194 f = g :=\n\u27e8by { have := fn_inj, exact this _ f g }, congr_arg _\u27e9\n\n@[simp] lemma predicate_coe_inj {n} {r s : L\u2081.pr n} : (r : L\u2082.pr n) = s \u2194 r = s :=\n\u27e8by { have := pr_inj, exact this _ r s }, congr_arg _\u27e9\n\nend\n\nvariables [language_translation_coe L\u2081 L\u2082]\n\n@[simp] lemma coe_term_app {i} (f : L\u2081.fn i) (v : finitary (term L\u2081) i) :\n  (\u2191(term.app f v : term L\u2081) : term L\u2082) = term.app (f : L\u2082.fn i) (\u03bb i, v i) := rfl\n\n@[simp] lemma coe_term_app_nullary (c : L\u2081.fn 0) (v : finitary (term L\u2081) 0) :\n  (\u2191(term.app c v : term L\u2081) : term L\u2082) = term.app (c : L\u2082.fn 0) finitary.nil := by simp\n\n@[simp] lemma coe_term_app_unary (f : L\u2081.fn 1) (v : finitary (term L\u2081) 1) :\n  (\u2191(term.app f v : term L\u2081) : term L\u2082) = term.app (f : L\u2082.fn 1) \u2039v 0\u203a := by simp; ext; simp\n\n@[simp] lemma coe_term_app_binary (f : L\u2081.fn 2) (v : finitary (term L\u2081) 2) :\n  (\u2191(term.app f v : term L\u2081) : term L\u2082) = term.app (f : L\u2082.fn 2) \u2039v 0, v 1\u203a := by simp; ext; simp\n\n@[simp] lemma coe_formula_app {i} (p : L\u2081.pr i) (v : finitary (term L\u2081) i) :\n  (\u2191(formula.app p v : formula L\u2081) : formula L\u2082) = formula.app (p : L\u2082.pr i) (\u03bb i, v i) := rfl\n\n@[simp] lemma coe_formula_app_nullary (p : L\u2081.pr 0) (v : finitary (term L\u2081) 0) :\n  (\u2191(formula.app p v : formula L\u2081) : formula L\u2082) = formula.app (p : L\u2082.pr 0) finitary.nil := by simp\n\n@[simp] lemma coe_formula_app_uniary (p : L\u2081.pr 1) (v : finitary (term L\u2081) 1) :\n  (\u2191(formula.app p v : formula L\u2081) : formula L\u2082) = formula.app (p : L\u2082.pr 1) \u2039v 0\u203a := by simp; ext; simp\n\n@[simp] lemma coe_formula_app_biary (p : L\u2081.pr 2) (v : finitary (term L\u2081) 2) :\n  (\u2191(formula.app p v : formula L\u2081) : formula L\u2082) = formula.app (p : L\u2082.pr 2) \u2039v 0, v 1\u203a := by simp; ext; simp\n\n@[simp] lemma coe_equal (t u : term L\u2081) :\n  (\u2191(t =' u : formula L\u2081) : formula L\u2082) = ((\u2191t : term L\u2082) =' \u2191u) := rfl\n\n@[simp] lemma coe_imply (p q : formula L\u2081) :\n  (\u2191(p \u27f6 q) : formula L\u2082) = (\u2191p \u27f6 \u2191q) := rfl\n\n@[simp] lemma coe_and (p q : formula L\u2081) :\n  (\u2191(p \u2293 q) : formula L\u2082) = (\u2191p \u2293 \u2191q) := rfl\n\n@[simp] lemma coe_or (p q : formula L\u2081) :\n  (\u2191(p \u2294 q) : formula L\u2082) = (\u2191p \u2294 \u2191q) := rfl\n\n@[simp] lemma coe_neg (p : formula L\u2081) :\n  (\u2191(\u223cp) : formula L\u2082) = \u223c(\u2191p) := rfl\n\n@[simp] lemma coe_equiv (p q : formula L\u2081) :\n  (\u2191(p \u27f7 q) : formula L\u2082) = (\u2191p \u27f7 \u2191q) := rfl\n\n@[simp] lemma coe_pow_term (t : term L\u2081) (i : \u2115) :\n  (\u2191(t^i) : term L\u2082) = (\u2191t)^i :=\nby simp [tr_term_app_eq, \u2190app_term_extension_eq_coe 0]\n\n@[simp] lemma coe_pow_formula (p : formula L\u2081) (i : \u2115) :\n  (\u2191(p^i) : formula L\u2082) = (\u2191p)^i := \nby simp [tr_app_eq, \u2190app_formula_extension_eq_coe 0]\n\n@[simp] lemma coe_fal (p : formula L\u2081)  :\n  (\u2191(\u2200.p : formula L\u2081) : formula L\u2082) = \u2200.(\u2191p : formula L\u2082) := rfl\n\n@[simp] lemma coe_ex (p : formula L\u2081)  :\n  (\u2191(\u2203.p : formula L\u2081) : formula L\u2082) = \u2203.(\u2191p : formula L\u2082) := rfl\n\n@[simp] lemma coe_top :\n  (\u2191(\u22a4 : formula L\u2081) : formula L\u2082) = \u22a4 := rfl\n\n@[simp] lemma coe_bot :\n  (\u2191(\u22a5 : formula L\u2081) : formula L\u2082) = \u22a5 := rfl\n\n@[simp] lemma coe_conjunction (P : list (formula L\u2081)) :\n  (\u2191P.conjunction : formula L\u2082) = list.conjunction (P.map coe) :=\nfun_p_conjunction _ P\n\n@[simp] lemma coe_nfal (p : formula L\u2081) (k : \u2115) :\n  (\u2191(\u2200.[k] p) : formula L\u2082) = \u2200.[k] \u2191p :=\nfun_p_nfal _ p k\n\n@[simp] lemma coe_fal_complete (p : formula L\u2081) :\n  (\u2191(\u2200.* p) : formula L\u2082) = \u2200.* \u2191p :=\nfun_p_fal_complete _ p\n\n@[simp] lemma coe_conjunction' {n : \u2115} (P : finitary (formula L\u2081) n) :\n  (\u2191(\u22c0 j, P j) : formula L\u2082) = \u22c0 j, P j :=\nfun_p_conjunction' _ P\n\n@[simp] lemma coe_disjunction' {n : \u2115} (P : finitary (formula L\u2081) n) :\n  (\u2191(\u22c1 j, P j) : formula L\u2082) = \u22c1 j, P j :=\nfun_p_disjunction' _ P\n\n@[simp] lemma coe_t_rew (t : term L\u2081) (s : \u2115 \u2192 term L\u2081) :\n  (\u2191(t.rew s) : term L\u2082) = (\u2191t : term L\u2082).rew (\u03bb x, \u2191(s x)) :=\nfun_t_rew _ t s\n\n@[simp] lemma coe_t_subst (t u : term L\u2081) (s) : (\u2191(t.rew \u0131[s \u21dd u]) : term L\u2082) = (\u2191t : term L\u2082).rew \u0131[s \u21dd \u2191u] :=\nfun_t_subst _ t u s\n\n@[simp] lemma coe_p_rew (p : formula L\u2081) (s : \u2115 \u2192 term L\u2081) :\n  (\u2191(p.rew s) : formula L\u2082) = (\u2191p : formula L\u2082).rew (\u03bb x, \u2191(s x)) :=\nfun_p_rew _ p s\n\n@[simp] lemma fun_p_subst (p : formula L\u2081) (u : term L\u2081) (s) :\n  (\u2191(p.rew \u0131[s \u21dd u]) : formula L\u2082) = (\u2191p : formula L\u2082).rew \u0131[s \u21dd \u2191u] :=\nfun_p_subst _ p u s\n\n@[simp] lemma coe_t_arity (t : term L\u2081) : (t : term L\u2082).arity = t.arity := fun_t_arity _ t\n\n@[simp] lemma coe_p_arity (p : formula L\u2081) : (p : formula L\u2082).arity = p.arity := fun_p_arity _ p\n\n@[simp] lemma coe_is_open (p : formula L\u2081) : (p : formula L\u2082).is_open \u2194 p.is_open :=\nby { induction p; simp[*] }\n\n@[simp] lemma term_coe_inj : \u2200 {t u : term L\u2081}, (t : term L\u2082) = u \u2194 t = u\n| (#m)                   (#n)                   := by simp\n| (#m)                   (term.app f v)         := by simp\n| (term.app f v)         (#n)                   := by simp\n| (@term.app _ n\u2081 f\u2081 v\u2081) (@term.app _ n\u2082 f\u2082 v\u2082) := by { \n    simp, rintros rfl, simp,\n    rintros rfl, \n    have IH : \u2200 i, \u2191(v\u2081 i) = \u2191(v\u2082 i) \u2194 v\u2081 i = v\u2082 i, from \u03bb i, @term_coe_inj (v\u2081 i) (v\u2082 i),\n    refine \u27e8\u03bb h, funext (\u03bb i, (IH i).mp (congr_fun h i)), by { rintros rfl, refl }\u27e9 }\n\n@[simp] lemma formula_coe_inj : \u2200 {p q : formula L\u2081}, (p : formula L\u2082) = q \u2194 p = q\n| (@formula.app _ n\u2081 r\u2081 v\u2081) (@formula.app _ n\u2082 r\u2082 v\u2082) :=\n    by { simp,  rintros rfl, simp, rintros rfl,\n         refine \u27e8\u03bb h, funext (\u03bb i, term_coe_inj.mp (congr_fun h i)), by { rintros rfl, refl }\u27e9 }\n| \u22a4                   q        := by simp; cases q; simp\n| (formula.app r\u2081 v\u2081) (t =' u) := by simp\n| (formula.app r\u2081 v\u2081) \u22a4        := by simp\n| (formula.app r\u2081 v\u2081) (p \u27f6 q)  := by simp\n| (formula.app r\u2081 v\u2081) \u223cp       := by simp\n| (formula.app r\u2081 v\u2081) (\u2200.p)    := by simp\n| (t =' u)            p        := by cases p; simp\n| (p \u27f6 q)             r        := by cases r; simp[@formula_coe_inj p, @formula_coe_inj q]\n| (\u223cp)                q        := by cases q; simp[@formula_coe_inj p]\n| (\u2200.p)               q        := by cases q; simp[@formula_coe_inj p]\n\n@[simp] lemma coe_mem_coe_iff {T : Theory L\u2081} {p} : \u2191p \u2208 (\u2191T : Theory L\u2082) \u2194 p \u2208 T := \n\u27e8\u03bb \u27e8p', h, eqn\u27e9, by { simp [formula_coe_inj.mp eqn] at h, exact h }, \u03bb h, \u27e8p, h, rfl\u27e9\u27e9\n\nlemma mem_coe_iff {T : Theory L\u2081} {p : formula L\u2082} :\n  p \u2208 (\u2191T : Theory L\u2082) \u2194 \u2203 p\u2081 \u2208 T, p = \u2191p\u2081 := \n\u27e8\u03bb \u27e8p\u2081, h, eqn\u27e9, \u27e8p\u2081, h, eq.symm eqn\u27e9, by { rintros \u27e8p\u2081, mem, rfl\u27e9, simp[mem] }\u27e9\n\n@[simp] lemma Theory_coe_empty : (\u2191(\u2205 : Theory L\u2081) : Theory L\u2082) = \u2205 :=\nset.ext (\u03bb p, by unfold_coes; simp[tr_Theory])\n\n@[simp] lemma Theory_coe_union (T U : Theory L\u2081) : (\u2191(T \u222a U) : Theory L\u2082) = \u2191T \u222a \u2191U :=\nset.ext (\u03bb p, by { unfold_coes, simp[tr_Theory], split,\n  { rintros \u27e8p, (mem_p | mem_p), rfl\u27e9,\n    refine or.inl \u27e8p, mem_p, rfl\u27e9,\n    refine or.inr \u27e8p, mem_p, rfl\u27e9 },\n  { rintros (\u27e8p, mem_p, rfl\u27e9 | \u27e8p, mem_p, rfl\u27e9),\n    refine \u27e8p, or.inl mem_p, rfl\u27e9,\n    refine \u27e8p, or.inr mem_p, rfl\u27e9 } })\n\n@[simp] lemma Theory_coe_sf (T : Theory L\u2081) : (\u2191\u290aT : Theory L\u2082) = \u290a(\u2191T : Theory L\u2082) :=\nset.ext (\u03bb p, by { unfold_coes,simp[tr_Theory, Theory.sf], refine \u27e8_, _\u27e9,\n  { rintros \u27e8_, \u27e8q\u2081, mem_q\u2081, rfl\u27e9, rfl\u27e9, refine \u27e8q\u2081, mem_q\u2081, by simp[app_formula_extension_eq_coe]\u27e9 },\n  { rintros \u27e8p\u2081, mem_p\u2081, rfl\u27e9, refine \u27e8p\u2081^1, \u27e8p\u2081, mem_p\u2081, rfl\u27e9, by simp[app_formula_extension_eq_coe]\u27e9 } })\n\n@[simp] lemma Theory_coe_pow {T : Theory L\u2081} {i : \u2115} :\n  (\u2191T : Theory L\u2082)^i = \u2191(T^i) := \nbegin\n  ext p,\n  simp[Theory_sf_itr_eq, mem_coe_iff], split,\n  { rintros \u27e8p', \u27e8p\u2081, mem, rfl\u27e9, rfl\u27e9,\n    refine \u27e8p\u2081^i, \u27e8p\u2081, mem, rfl\u27e9, by simp\u27e9 },\n  { rintros \u27e8_, \u27e8p\u2081, mem, rfl\u27e9, rfl\u27e9, \n    refine \u27e8p\u2081, \u27e8p\u2081, mem, rfl\u27e9, by simp\u27e9 } \nend\n\nlemma Theory_mem_coe_pow_iff {p : formula L\u2082} {T : Theory L\u2081} {i : \u2115} :\n  p \u2208 (\u2191(T^i) : Theory L\u2082) \u2194 \u2203 p' \u2208 T, p = (\u2191p' : formula L\u2082)^i :=\nbegin\n  rw [\u2190Theory_coe_pow, Theory_sf_itr_eq], simp, split,\n  { rintros \u27e8q, q_mem, rfl\u27e9, rcases q_mem with \u27e8q, q_mem, rfl\u27e9, refine \u27e8q, q_mem, rfl\u27e9 },\n  { rintros \u27e8q, q_mem, rfl\u27e9, refine \u27e8\u2191q, by simp[q_mem]\u27e9 }\nend \n\nlemma destruct_of_eq_imply {p : formula L\u2081} {q r : formula L\u2082} (h : \u2191p = q \u27f6 r) :\n  \u2203 p\u2081 p\u2082, p = p\u2081 \u27f6 p\u2082 :=\nbegin\n  rcases p; try { simp at h, contradiction },\n  { simp at h, rcases h with \u27e8rfl, rfl\u27e9, simp }\nend\n\nlemma destruct_of_eq_neg {p : formula L\u2081} {q : formula L\u2082} (h : \u2191p = \u223cq) :\n  \u2203 p\u2081, p = \u223cp\u2081 :=\nbegin\n  rcases p; try { simp at h, contradiction },\n  { simp at h, rcases h with \u27e8rfl, rfl\u27e9, simp }\nend\n\nlemma fun_t_inversion_of_le {t\u2081 : term L\u2081} {u\u2082 : term L\u2082} (le : u\u2082 \u2264 \u2191t\u2081) :\n  \u2203 (u\u2081 : term L\u2081) (le : u\u2081 \u2264 t\u2081), u\u2082 = \u2191u\u2081 := fun_t_inversion_of_le _ le\n\nlemma fun_p_inversion_of_le {p\u2081 : formula L\u2081} {q\u2082 : formula L\u2082} (le : q\u2082 \u2264 \u2191p\u2081) :\n  \u2203 (q\u2081 : formula L\u2081) (le : q\u2081 \u2264 p\u2081), q\u2082 = \u2191q\u2081 := fun_p_inversion_of_le _ le\n\nlemma fun_p_inversion_of_mem {p\u2081 : formula L\u2081} {t\u2082 : term L\u2082} (mem : t\u2082 \u2208 (\u2191p\u2081 : formula L\u2082)) :\n  \u2203 (t\u2081 : term L\u2081) (mem : t\u2081 \u2208 p\u2081), t\u2082 = \u2191t\u2081 := fun_p_inversion_of_mem _ mem\n\nvariables (L\u2081 L\u2082 L\u2083) [language_translation_coe L\u2081 L\u2082] [language_translation_coe L\u2082 L\u2083]\n\n@[reducible] def comp : language_translation_coe L\u2081 L\u2083 :=\n{ ltr := (ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082),\n  fn_inj := \u03bb n f g, by simp,\n  pr_inj := \u03bb n r s, by simp }\n\nvariables {L\u2081} (T : Theory L\u2081)\n\ninstance [c : closed_Theory T] : closed_Theory (\u2191T : Theory L\u2082) :=\nlanguage_translation.fun_Theory.fol.closed_Theory _ _\n\nlemma fun_Theory_insert (p : formula L\u2081) : (\u2191(T+{p}) : Theory L\u2082) = \u2191T +{\u2191p} :=\nset.image_insert_eq\n\n@[simp] lemma ltc_self_eq_one : (ltr : L\u2081 \u219d\u1d38 L\u2081) = 1 := by { ext; simp, { refl }, { intros n r, refl } }\n\n@[simp] lemma coe_fn_eq_self {n} (f : L.fn n) : (coe : L.fn n \u2192 L.fn n) f = f := by refl\n\n@[simp] lemma coe_pr_eq_self {n} (r : L.pr n) : (coe : L.pr n \u2192 L.pr n) r = r := by refl\n\n@[simp] lemma coe_t_eq_self (t : term L) : (coe : term L \u2192 term L) t = t := one_fun_t t\n\n@[simp] lemma coe_t_eq_id : (coe : term L \u2192 term L) = id := by funext t; simp\n\n@[simp] lemma coe_p_eq_self (p : formula L) : (coe : formula L \u2192 formula L) p = p := one_fun_p p\n\n@[simp] lemma coe_p_eq_id : (coe : formula L \u2192 formula L) = id := by funext t; simp\n\nend language_translation_coe\n\nclass commutes (L\u2081 L\u2082 L\u2083) [language_translation_coe L\u2081 L\u2082] [language_translation_coe L\u2082 L\u2083] [language_translation_coe L\u2081 L\u2083] :=\n(comm : (language_translation_coe.ltr : L\u2082 \u219d\u1d38 L\u2083).comp (language_translation_coe.ltr : L\u2081 \u219d\u1d38 L\u2082) = language_translation_coe.ltr)\n\nnamespace commutes\nopen language_translation language_translation_coe\nvariables\n  {L\u2081 L\u2082 L\u2083}\n\nvariables\n  {L\u2081 L\u2082 L\u2083}\n  [language_translation_coe L\u2081 L\u2082] [language_translation_coe L\u2082 L\u2083] [language_translation_coe L\u2081 L\u2083]\n  [commutes L\u2081 L\u2082 L\u2083]\n\n@[simp] lemma fn_fn_of_commute {n} (f : L\u2081.fn n) : (ltr.fn n (ltr.fn n f : L\u2082.fn n) : L\u2083.fn n) = ltr.fn n f :=\nby { have : ((ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082)).fn n f = ltr.fn n f,\n       from congr (congr_fun (congr_arg language_translation.fn comm) n) rfl,\n     simpa using this }\n\n@[simp] lemma pr_pr_of_commute {n} (r : L\u2081.pr n) : (ltr.pr n (ltr.pr n r : L\u2082.pr n) : L\u2083.pr n) = ltr.pr n r :=\nby { have : ((ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082)).pr n r = ltr.pr n r,\n       from congr (congr_fun (congr_arg language_translation.pr comm) n) rfl,\n     simpa using this }\n\n@[simp] lemma coe_coe_fn_of_commute {n} (f : L\u2081.fn n) : (\u2191(\u2191f : L\u2082.fn n) : L\u2083.fn n) = \u2191f :=\nfn_fn_of_commute f\n\n@[simp] lemma coe_coe_pr_of_commute {n} (r : L\u2081.pr n) : (\u2191(\u2191r : L\u2082.pr n) : L\u2083.pr n) = \u2191r :=\npr_pr_of_commute r\n\n@[simp] lemma coe_coe_t_of_commute (t : term L\u2081) : (\u2191(\u2191t : term L\u2082) : term L\u2083) = \u2191t :=\nby { have : ((ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082)).fun_t t = ltr.fun_t t,\n       from congr (congr_arg language_translation.fun_t comm) rfl,\n     simpa[comp_fun_t] using this }\n\n@[simp] lemma coe_coe_p_of_commute (p : formula L\u2081) : (\u2191(\u2191p : formula L\u2082) : formula L\u2083) = \u2191p :=\nby { have : ((ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082)).fun_p p = ltr.fun_p p,\n       from congr (congr_arg language_translation.fun_p comm) rfl,\n     simpa[comp_fun_p] using this }\n\n@[simp] lemma coe_coe_th_of_commute (T : Theory L\u2081) : (\u2191(\u2191T : Theory L\u2082) : Theory L\u2083) = \u2191T :=\nby { have : ((ltr : L\u2082 \u219d\u1d38 L\u2083).comp (ltr : L\u2081 \u219d\u1d38 L\u2082)).fun_Theory T = ltr.fun_Theory T,\n       from congr (congr_arg language_translation.fun_Theory comm) rfl,\n     simpa[comp_fun_Theory] using this }\n\n@[simp] protected lemma zero [has_zero_symbol L\u2081] :\n  @has_zero.zero _ (@term.has_zero _ (@language_translation_coe.zero L\u2082 L\u2083 _ _)) = 0 :=\nby { unfold has_zero.zero has_zero_symbol.zero, simp }\n\n@[simp] protected lemma succ [has_succ_symbol L\u2081] (t : term L\u2083) :\n  @has_succ.succ _ (@term.has_succ _ (@language_translation_coe.succ L\u2082 L\u2083 _ _)) t = Succ t :=\nby { unfold has_succ.succ has_succ_symbol.succ, simp }\n\n@[simp] protected lemma numeral [has_zero_symbol L\u2081] [has_succ_symbol L\u2081] (n : \u2115) :\n  @numeral _ (@term.has_zero _ (@language_translation_coe.zero L\u2082 L\u2083 _ _))\n    (@term.has_succ _ (@language_translation_coe.succ L\u2082 L\u2083 _ _)) n = n\u02d9 :=\nby induction n with n IH; simp[numeral, *]\n\n@[simp] protected lemma add [has_add_symbol L\u2081] (t u : term L\u2083) :\n  @has_add.add _ (@term.has_add _ (@language_translation_coe.add L\u2082 L\u2083 _ _)) t u = t + u :=\nby { unfold has_add.add has_add_symbol.add, simp }\n\n@[simp] protected lemma mul [has_mul_symbol L\u2081] (t u : term L\u2083) :\n  @has_mul.mul _ (@term.has_mul _ (@language_translation_coe.mul L\u2082 L\u2083 _ _)) t u = t * u :=\nby { unfold has_mul.mul has_mul_symbol.mul, simp }\n\n@[simp] protected lemma le [has_le_symbol L\u2081] (t u : term L\u2083) :\n  @has_preceq.preceq _ _ (@formula.has_preceq L\u2083 (@language_translation_coe.le L\u2082 L\u2083 _ _)) t u = (t \u227c u) :=\nby { unfold has_preceq.preceq has_le_symbol.le, simp }\n\n@[simp] protected lemma mem [has_mem_symbol L\u2081] (t u : term L\u2083) :\n  @has_elem.elem _ _ (@formula.has_elem L\u2083 (@language_translation_coe.mem L\u2082 L\u2083 _ _)) t u = (t \u220a u) :=\nby { unfold has_elem.elem has_mem_symbol.mem, simp }\n\ninstance comp_commutes : @commutes L\u2081 L\u2082 L\u2083 _ _ (comp L\u2081 L\u2082 L\u2083) := { comm := by refl }\n\ninstance self_commutes : commutes L\u2081 L\u2081 L\u2082 := \u27e8by simp\u27e9\n\ninstance commutes_self : commutes L\u2081 L\u2082 L\u2082 := \u27e8by simp\u27e9\n\nend commutes\n\nnamespace language_translation\nvariables (\u03c4 : L\u2081 \u219d\u1d38 L\u2082)\n\ninstance conservative : \u03c4.tr.conservative :=\n{ ax := \u03bb k T, tr_Theory \u03c4.tr k T,\n  ax_ss := by { intros, refl },\n  specialize := \u03bb k p t T i, by {\n    have : (\u03bb (x : \u2115), \u03c4.fun_t (\u0131[0 \u21dd t] x)) = \u0131[0 \u21dd \u03c4.fun_t t],\n    { funext x, cases x; simp },\n    simp[tr_app_eq, fun_p_rew, this] },\n  eq_reflexivity := by simp[tr_app_eq],\n  eq_symmetry := by simp[tr_app_eq],\n  eq_transitive := by simp[tr_app_eq],\n  function_ext := \u03bb k n f T i, by { simp[eq_axiom4], simp[tr_app_eq],\n    exact (show _ \u22a2 eq_axiom4 (\u03c4.fn _ f), by simp) },\n  predicate_ext := \u03bb k n f T i, by { simp[eq_axiom5], simp[tr_app_eq],\n    exact (show _ \u22a2 eq_axiom5 (\u03c4.pr _ f), by simp) } }\n\nend language_translation\n\nnamespace translation\nopen provable axiomatic_classical_logic' translation.conservative\nvariables {L\u2081} {L\u2082}\nvariables (\u03c4 : translation L\u2081 L\u2082) [conservative \u03c4] (i : \u2115)\n\n@[simp] lemma mem_pow_Theory_tr_of_mem_pow {T : Theory L\u2081} {k : \u2115} {p} {i : \u2115} (mem : p \u2208 T^k) :\n  (\u03c4 (i + k) p) \u2208 (tr_Theory \u03c4 i T : Theory L\u2082)^k :=\nby { simp[Theory_sf_itr_eq] at mem \u22a2, rcases mem with \u27e8q, mem, rfl\u27e9, \n  refine \u27e8\u03c4 i q, mem_Theory_tr_of_mem mem, _\u27e9, simp[translation.map_pow'] }\n\nlemma provability_pow (T : Theory L\u2081) (p : formula L\u2081) (i k : \u2115) (h : T^i \u22a2 p) :\n  (ax \u03c4 k T)^i \u22a2 \u03c4 (k + i) p :=\nbegin\n  refine provable.rec'_on h _ _ _ _ _ _ _ _ _ _ _ _ _ _ _,\n  { intros i p _ h, simp[add_assoc] at h \u22a2,\n    exact generalize h },\n  { intros i p q _ _ hpq hp, simp at hpq,\n    exact hpq \u2a00 hp },\n  { intros i p mem,\n    suffices : (tr_Theory \u03c4 k T)^i \u22a2 \u03c4 (k + i) p,\n    { exact weakening this (by simp[ax_ss]) },\n    refine (by_axiom (by {simp[mem]})) },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp },\n  { intros, simp },\n  { intros, refine specialize _ _ _ _ _ },\n  { intros, simp },\n  { intros, simp[translation.map_pow'] },\n  { intros, refine eq_reflexivity _ _ _ },\n  { intros, exact eq_symmetry _ _ _ },\n  { intros, exact eq_transitive _ _ _ },\n  { intros, exact function_ext _ _ _ _ },\n  { intros, exact predicate_ext _ _ _ _ },\nend\n\nlemma provability (T : Theory L\u2081) (p : formula L\u2081) (k : \u2115) :\n  T \u22a2 p \u2192 ax \u03c4 k T \u22a2 \u03c4 k p :=\nby { have := provability_pow \u03c4 T p 0, simp at this, exact this k }\n\nlemma provability_tautology (p : formula L\u2081) (k : \u2115):\n  (\u2200 T, T \u22a2 p) \u2192 \u2200 T, ax \u03c4 k T \u22a2 \u03c4 k p := \u03bb h T,\nprovability \u03c4 T p k (h T)\n\nlemma consistency (T : Theory L\u2081) (k : \u2115) : \n  (ax \u03c4 k T).consistent \u2192 T.consistent :=\nby { simp[logic.Theory.consistent_iff_bot], contrapose, simp,\n     have := provability \u03c4 T \u22a5 k, simp at this,\n     exact this }\n\ninstance refl_conservative : conservative (fol.language.translation.refl L\u2081) :=\n{ ax := \u03bb k T, tr_Theory (fol.language.translation.refl L\u2081) k T,\n  ax_ss := by { intros, refl },\n  specialize := by simp[translation.refl],\n  eq_reflexivity := by simp[translation.refl],\n  eq_symmetry := by simp[translation.refl],\n  eq_transitive := by simp[translation.refl],\n  function_ext := by { intros,  simp[translation.refl] },\n  predicate_ext := by { intros, simp[translation.refl] } }\n\ninstance shift_conservative (k : \u2115) : conservative (shift L\u2081 k) :=\n{ ax := \u03bb l T, tr_Theory (shift L\u2081 k) l T,\n  ax_ss := by { intros, refl },\n  specialize := \u03bb l p t T i, by {simp[translation.shift], \n    have : \u2200 l, (p.rew \u0131[0 \u21dd t]).rew (\u03bb x, ite (x < l) #x #(x + k)) = \n      (p.rew (\u03bb x, ite (x < l + 1) #x #(x + k))).rew \u0131[0 \u21dd (t.rew (\u03bb x, ite (x < l) #x #(x + k)))],\n    { intros l, simp[formula.nested_rew], congr, funext x, cases x with x; simp[\u2190nat.add_one],\n      by_cases C : x < l; simp[C, show x + 1 + k = x + k + 1, by omega] },\n    simp [this] },\n  eq_reflexivity := by simp[translation.shift],\n  eq_symmetry := by simp[translation.shift],\n  eq_transitive := \u03bb _ _ _, by simp[translation.shift, show \u2200 l, 2 < l + 1 + 1 + 1, by omega], \n  function_ext := \u03bb _ _ _ _, by simp[translation.shift],\n  predicate_ext := \u03bb _ _ _ _, by simp[translation.shift] }\n\nend translation\n\nnamespace language_translation\nopen language_translation logic\nvariables (\u03c4 : L\u2081 \u219d\u1d38 L\u2082)\n\nlemma provability_pow {T : Theory L\u2081} {p : formula L\u2081} {i : \u2115} :\n  T^i \u22a2 p \u2192 (\u03c4.fun_Theory T)^i \u22a2 \u03c4.fun_p p :=\ntranslation.provability_pow \u03c4.tr T p i 0\n\nlemma provability {T : Theory L\u2081} {p : formula L\u2081} :\n  T \u22a2 p \u2192 \u03c4.fun_Theory T \u22a2 \u03c4.fun_p p :=\ntranslation.provability \u03c4.tr T p 0\n\nlemma consistency (T : Theory L\u2081) : \n  Theory.consistent (\u03c4.fun_Theory T) \u2192 T.consistent :=\ntranslation.consistency \u03c4.tr T 0\n\nend language_translation\n\nnamespace language_translation_coe\nopen language_translation logic\nvariables [\u03c3 : language_translation_coe L\u2081 L\u2082]\ninclude \u03c3\n\nlemma provability_pow {T : Theory L\u2081} {p : formula L\u2081} {i : \u2115} :\n  T^i \u22a2 p \u2192 (\u2191T : Theory L\u2082)^i \u22a2 \u2191p :=\ntranslation.provability_pow \u03c3.ltr.tr T p i 0\n\nlemma provability {T : Theory L\u2081} {p : formula L\u2081} :\n  T \u22a2 p \u2192 (\u2191T : Theory L\u2082) \u22a2 \u2191p :=\ntranslation.provability \u03c3.ltr.tr T p 0\n\nlemma consistency (T : Theory L\u2081) : \n  Theory.consistent (\u2191T : Theory L\u2082) \u2192 T.consistent :=\ntranslation.consistency \u03c3.ltr.tr T 0\n\nend language_translation_coe\n\n--------------------------------------------------------------------------------\n\ninstance : has_add language := \u27e8\u03bb L\u2081 L\u2082 : language.{u}, \u27e8\u03bb n, L\u2081.fn n \u2295 L\u2082.fn n, \u03bb n, L\u2081.pr n \u2295 L\u2082.pr n\u27e9\u27e9 \n\ndef direct_sum {\u03b9 : Type*} (l : \u03b9 \u2192 language) : language := \u27e8\u03bb n, \u03a3 i, (l i).fn n, \u03bb n, \u03a3 i, (l i).pr n\u27e9\n\ndef consts (\u03b1 : Type u) : language.{u} := \u27e8\u03bb n, match n with | 0 := \u03b1 | (n + 1) := pempty end, \u03bb n, pempty\u27e9\n\nnamespace consts\nvariables {\u03b1 : Type u}\n\ndef c (a : \u03b1) : (consts \u03b1).fn 0 := a\n\ninstance : has_coe \u03b1 (term (consts \u03b1)) := \u27e8\u03bb a, term.app (consts.c a) finitary.nil\u27e9\n\nlemma coe_def (a : \u03b1) : (a : term (consts \u03b1)) = term.app (consts.c a) finitary.nil := rfl\n\n@[simp] lemma arity_eq_0 (a : \u03b1) : (a : term (consts \u03b1)).arity = 0 := by simp[coe_def]\n\nend consts\n\ndef singleton_fn (m : \u2115) : language.{u} := \u27e8\u03bb n, if n = m then punit else pempty, \u03bb n, pempty\u27e9\n\nnamespace singleton_fn\nvariables {m : \u2115}\n\ndef star : (singleton_fn m).fn m := by { simp[singleton_fn]; simp[show (m = m) \u2194 true, by simp], refine punit.star }\n\n\nend singleton_fn\n\n@[simp] lemma sum_fn_def {\u03b9 : Type*} (l : \u03b9 \u2192 language) (n : \u2115) : (direct_sum l).fn n = \u03a3 i, (l i).fn n := rfl\n\n@[simp] lemma sum_pr_def {\u03b9 : Type*} (l : \u03b9 \u2192 language) (n : \u2115) : (direct_sum l).pr n = \u03a3 i, (l i).pr n := rfl\n\nnamespace extension\nopen language_translation language_translation_coe\n\ndef add_left : L\u2081 \u219d\u1d38 L\u2081 + L\u2082 := \u27e8\u03bb n f, sum.inl f, \u03bb n p, sum.inl p\u27e9\n\ninstance ltr\u2081 : language_translation_coe L\u2081 (L\u2081 + L\u2082) :=\n{ ltr := add_left,\n  fn_inj := \u03bb n f g, sum.inl.inj,\n  pr_inj := \u03bb n f g, sum.inl.inj }\n\nlemma coe_fn\u2081 {n} (f : L\u2081.fn n) : (\u2191f : (L\u2081 + L\u2082).fn n) = sum.inl f:= rfl\n\nlemma coe_pr\u2081 {n} (r : L\u2081.pr n) : (\u2191r : (L\u2081 + L\u2082).pr n) = sum.inl r:= rfl\n\nlemma zero_symbol_eq\u2081 [has_zero_symbol L\u2081] : (has_zero_symbol.zero : (L\u2081 + L\u2082).fn 0) = sum.inl has_zero_symbol.zero := rfl\n\nlemma succ_symbol_eq\u2081 [has_succ_symbol L\u2081] : (has_succ_symbol.succ : (L\u2081 + L\u2082).fn 1) = sum.inl has_succ_symbol.succ := rfl\n\nlemma add_symbol_eq\u2081 [has_add_symbol L\u2081] : (has_add_symbol.add : (L\u2081 + L\u2082).fn 2) = sum.inl has_add_symbol.add := rfl\n\nlemma mul_symbol_eq\u2081 [has_mul_symbol L\u2081] : (has_mul_symbol.mul : (L\u2081 + L\u2082).fn 2) = sum.inl has_mul_symbol.mul := rfl\n\nlemma le_symbol_eq\u2081 [has_le_symbol L\u2081] : (has_le_symbol.le : (L\u2081 + L\u2082).pr 2) = sum.inl has_le_symbol.le := rfl\n\nlemma add_left_fn_to_coe {n} (f : L\u2081.fn n) : (add_left.fn _ f : (L\u2081 + L\u2082).fn n) = f := rfl\n\nlemma add_left_pr_to_coe {n} (r : L\u2081.pr n) : (add_left.pr _ r : (L\u2081 + L\u2082).pr n) = r := rfl\n\ndef add_right : L\u2082 \u219d\u1d38 L\u2081 + L\u2082 := \u27e8\u03bb n f, sum.inr f, \u03bb n p, sum.inr p\u27e9\n\ninstance ltr\u2082 : language_translation_coe L\u2082 (L\u2081 + L\u2082) :=\n{ ltr := add_right,\n  fn_inj := \u03bb n f g, sum.inr.inj,\n  pr_inj := \u03bb n f g, sum.inr.inj }\n\nlemma coe_fn\u2082 {n} (f : L\u2082.fn n) : (\u2191f : (L\u2081 + L\u2082).fn n) = sum.inr f:= rfl\n\nlemma coe_pr\u2082 {n} (r : L\u2082.pr n) : (\u2191r : (L\u2081 + L\u2082).pr n) = sum.inr r:= rfl\n\nlemma zero_symbol_eq\u2082 [has_zero_symbol L\u2082] : (has_zero_symbol.zero : (L\u2081 + L\u2082).fn 0) = sum.inr has_zero_symbol.zero := rfl\n\nlemma succ_symbol_eq\u2082 [has_succ_symbol L\u2082] : (has_succ_symbol.succ : (L\u2081 + L\u2082).fn 1) = sum.inr has_succ_symbol.succ := rfl\n\nlemma add_symbol_eq\u2082 [has_add_symbol L\u2082] : (has_add_symbol.add : (L\u2081 + L\u2082).fn 2) = sum.inr has_add_symbol.add := rfl\n\nlemma mul_symbol_eq\u2082 [has_mul_symbol L\u2082] : (has_mul_symbol.mul : (L\u2081 + L\u2082).fn 2) = sum.inr has_mul_symbol.mul := rfl\n\nlemma le_symbol_eq\u2082 [has_le_symbol L\u2082] : (has_le_symbol.le : (L\u2081 + L\u2082).pr 2) = sum.inr has_le_symbol.le := rfl\n\nlemma add_right_fn_to_coe {n} (f : L\u2082.fn n) : (add_right.fn _ f : (L\u2081 + L\u2082).fn n) = f := rfl\n\nlemma add_right_pr_to_coe {n} (r : L\u2082.pr n) : (add_right.pr _ r : (L\u2081 + L\u2082).pr n) = r := rfl\n\nclass sublanguage (L\u2080 : language.{u}) (L : language.{u}) :=\n(map_fn : \u03a0 {n}, L.fn n \u2192 L\u2080.fn n)\n(map_pr : \u03a0 {n}, L.pr n \u2192 L\u2080.pr n)\n\nvariables {\u03b9 : Type*} (l : \u03b9 \u2192 language)\n\ndef to_extension (i : \u03b9) : l i \u219d\u1d38 direct_sum l :=\n\u27e8\u03bb n f, \u27e8i, f\u27e9, \u03bb n r, \u27e8i, r\u27e9\u27e9\n\ninstance ltr (i : \u03b9) : language_translation_coe (l i) (direct_sum l) :=\n{ ltr := to_extension l i,\n  fn_inj := \u03bb n f g, by simp[to_extension],\n  pr_inj := \u03bb n f g, by simp[to_extension] }\n\ndef ext_ss {s t : set \u03b9} (ss : s \u2286 t) : direct_sum (\u03bb i : s, l i) \u219d\u1d38 direct_sum (\u03bb i : t, l i) :=\n\u27e8\u03bb n \u27e8\u27e8i, hi\u27e9, f\u27e9, \u27e8\u27e8i, ss hi\u27e9, f\u27e9, \u03bb n \u27e8\u27e8i, hi\u27e9, f\u27e9, \u27e8\u27e8i, ss hi\u27e9, f\u27e9\u27e9\n\ndef ltr_ss {s t : set \u03b9} (ss : s \u2286 t) : language_translation_coe (direct_sum (\u03bb i : s, l i)) (direct_sum (\u03bb i : t, l i)) :=\n{ ltr := ext_ss l ss,\n  fn_inj := \u03bb n \u27e8\u27e8i, pi\u27e9, f\u27e9 \u27e8\u27e8j, pj\u27e9, g\u27e9, by { simp[ext_ss], rintros rfl, simp },\n  pr_inj := \u03bb n \u27e8\u27e8i, pi\u27e9, f\u27e9 \u27e8\u27e8j, pj\u27e9, g\u27e9, by { simp[ext_ss], rintros rfl, simp } }\n\ndef to_extension_subtype (s : set \u03b9) : direct_sum (\u03bb i : s, l i) \u219d\u1d38 direct_sum l :=\n\u27e8\u03bb n \u27e8i, f\u27e9, \u27e8i, f\u27e9, \u03bb n \u27e8i, r\u27e9, \u27e8i, r\u27e9\u27e9\n\ninstance ltr_subtype (s : set \u03b9) : language_translation_coe (direct_sum (\u03bb i : s, l i)) (direct_sum l) :=\n{ ltr := to_extension_subtype l s,\n  fn_inj := \u03bb n \u27e8\u27e8i, pi\u27e9, f\u27e9 \u27e8\u27e8j, pj\u27e9, g\u27e9, by { simp[to_extension_subtype], rintros rfl, simp },\n  pr_inj := \u03bb n \u27e8\u27e8i, pi\u27e9, f\u27e9 \u27e8\u27e8j, pj\u27e9, g\u27e9, by { simp[to_extension_subtype], rintros rfl, simp } }\n\n@[simp] lemma ext_ss_subtype_consistence_term {s t : set \u03b9} (ss : s \u2286 t) : \u2200 (u : term (direct_sum (\u03bb i : s, l i))),\n  ((ext_ss l ss).fun_t u : term (direct_sum l)) = u\n| #n                  := by simp\n| (@term.app _ n f v) :=\n  by { rcases f with \u27e8\u27e8i, hi\u27e9, f\u27e9, simp[ext_ss],\n       refine \u27e8rfl, funext (\u03bb i, ext_ss_subtype_consistence_term (v i))\u27e9}\n\n@[simp] lemma ext_ss_subtype_consistence {s t : set \u03b9} (ss : s \u2286 t) :\n  \u2200 (p : formula (direct_sum (\u03bb i : s, l i))), ((ext_ss l ss).fun_p p : formula (direct_sum l)) = p\n| \u22a4                                       := by simp\n| (app r v)                               := by { simp, rcases r with \u27e8\u27e8i, hi\u27e9, r\u27e9, simp[ext_ss], refl }\n| ((t : term (direct_sum (\u03bb (i : s), l i))) =' u) := by simp\n| (p \u27f6 q)                                 := by simp[ext_ss_subtype_consistence p, ext_ss_subtype_consistence q]\n| (\u223cp)                                    := by simp[ext_ss_subtype_consistence p]\n| (\u2200.p)                                   := by simp[ext_ss_subtype_consistence p]\n\n@[simp] lemma Theory_ext_ss_subtype_consistence {s t : set \u03b9} (ss : s \u2286 t)\n  (T : Theory (direct_sum (\u03bb i : s, l i))) :\n  (\u2191((ext_ss l ss).fun_Theory T) : Theory (direct_sum l)) = \u2191T :=\nset.ext (\u03bb p, by { unfold_coes, simp[tr_Theory, app_formula_extension_eq_coe, fun_Theory] })\n\nend extension\n\nnamespace language_translation\n\nvariables {L\u2081} {L\u2082} {L\u2083} {L\u2084 : language.{u}}\n\ndef add (\u03c4 : L\u2081 \u219d\u1d38 L\u2082) (\u03c3 : L\u2083 \u219d\u1d38 L\u2084) : L\u2081 + L\u2083 \u219d\u1d38 L\u2082 + L\u2084 :=\n{ fn := \u03bb n f, by { rcases f, { exact sum.inl (\u03c4.fn _ f) }, { exact sum.inr (\u03c3.fn _ f) } },\n  pr := \u03bb n r, by { rcases r, { exact sum.inl (\u03c4.pr _ r) }, { exact sum.inr (\u03c3.pr _ r) } } }\n\nsection\nvariables (\u03c4 : L\u2081 \u219d\u1d38 L\u2082) (\u03c3 : L\u2083 \u219d\u1d38 L\u2084)\n\n@[simp] lemma add_fnl {n} (f : L\u2081.fn n) : (\u03c4.add \u03c3).fn n \u2191f = \u2191(\u03c4.fn n f) := rfl\n@[simp] lemma add_prl {n} (r : L\u2081.pr n) : (\u03c4.add \u03c3).pr n \u2191r = \u2191(\u03c4.pr n r) := rfl\n@[simp] lemma add_fnr {n} (f : L\u2083.fn n) : (\u03c4.add \u03c3).fn n \u2191f = \u2191(\u03c3.fn n f) := rfl\n@[simp] lemma add_prr {n} (r : L\u2083.pr n) : (\u03c4.add \u03c3).pr n \u2191r = \u2191(\u03c3.pr n r) := rfl\n\nend\n\ndef sum (\u03c4 : L\u2081 \u219d\u1d38 L\u2082) (\u03c3 : L\u2083 \u219d\u1d38 L\u2082) : L\u2081 + L\u2083 \u219d\u1d38 L\u2082 :=\n{ fn := \u03bb n f, by { rcases f, { refine \u03c4.fn n f }, { refine \u03c3.fn n f } },\n  pr := \u03bb n r, by { rcases r, { refine \u03c4.pr n r }, { refine \u03c3.pr n r } } }\n\nsection\nvariables (\u03c4 : L\u2081 \u219d\u1d38 L\u2082) (\u03c3 : L\u2083 \u219d\u1d38 L\u2082)\n\n@[simp] lemma sum_fnl {n} (f : L\u2081.fn n) : (\u03c4.sum \u03c3).fn n \u2191f = (\u03c4.fn n f) := rfl\n@[simp] lemma sum_prl {n} (r : L\u2081.pr n) : (\u03c4.sum \u03c3).pr n \u2191r = (\u03c4.pr n r) := rfl\n@[simp] lemma sum_fnr {n} (f : L\u2083.fn n) : (\u03c4.sum \u03c3).fn n \u2191f = (\u03c3.fn n f) := rfl\n@[simp] lemma sum_prr {n} (r : L\u2083.pr n) : (\u03c4.sum \u03c3).pr n \u2191r = (\u03c3.pr n r) := rfl\n\nend\n\nvariables (L\u2081 L\u2082 L\u2083)\n\ndef add_comm' : L\u2081 + L\u2082 \u219d\u1d38 L\u2082 + L\u2081 :=\n{ fn := \u03bb n f, by { rcases f, { refine sum.inr f }, { refine sum.inl f } },\n  pr := \u03bb n r, by { rcases r, { refine sum.inr r }, { refine sum.inl r } } }\n\n@[simp] lemma add_comm'_fnl {n} (f : L\u2081.fn n) : (add_comm' L\u2081 L\u2082).fn n \u2191f = f := rfl\n@[simp] lemma add_comm'_prl {n} (r : L\u2081.pr n) : (add_comm' L\u2081 L\u2082).pr n \u2191r = r := rfl\n@[simp] lemma add_comm'_fnr {n} (f : L\u2082.fn n) : (add_comm' L\u2081 L\u2082).fn n \u2191f = f := rfl\n@[simp] lemma add_comm'_prr {n} (r : L\u2082.pr n) : (add_comm' L\u2081 L\u2082).pr n \u2191r = r := rfl\n\ndef add_assoc' : L\u2081 + L\u2082 + L\u2083 \u219d\u1d38 L\u2081 + (L\u2082 + L\u2083) :=\n{ fn := \u03bb n f, by { rcases f, { rcases f, { refine sum.inl f }, { refine sum.inr (sum.inl f) } }, { refine sum.inr (sum.inr f) } },\n  pr := \u03bb n r, by { rcases r, { rcases r, { refine sum.inl r }, { refine sum.inr (sum.inl r) } }, { refine sum.inr (sum.inr r) } }   }\n\n@[simp] lemma add_assoc'_fn\u2081 {n} (f : L\u2081.fn n) : (add_assoc' L\u2081 L\u2082 L\u2083).fn n (\u2191(\u2191f : (L\u2081 + L\u2082).fn n)) = (\u2191f : (L\u2081 + (L\u2082 + L\u2083)).fn n) := rfl\n@[simp] lemma add_assoc'_fn\u2082 {n} (f : L\u2082.fn n) : (add_assoc' L\u2081 L\u2082 L\u2083).fn n (\u2191(\u2191f : (L\u2081 + L\u2082).fn n)) = \u2191(\u2191f : (L\u2082 + L\u2083).fn n) := rfl\n@[simp] lemma add_assoc'_fn\u2083 {n} (f : L\u2083.fn n) : (add_assoc' L\u2081 L\u2082 L\u2083).fn n (\u2191f : (L\u2081 + L\u2082 + L\u2083).fn n) = \u2191(\u2191f : (L\u2082 + L\u2083).fn n) := rfl\n@[simp] lemma add_assoc'_pr\u2081 {n} (r : L\u2081.pr n) : (add_assoc' L\u2081 L\u2082 L\u2083).pr n (\u2191(\u2191r : (L\u2081 + L\u2082).pr n)) = (\u2191r : (L\u2081 + (L\u2082 + L\u2083)).pr n) := rfl\n@[simp] lemma add_assoc'_pr\u2082 {n} (r : L\u2082.pr n) : (add_assoc' L\u2081 L\u2082 L\u2083).pr n (\u2191(\u2191r : (L\u2081 + L\u2082).pr n)) = \u2191(\u2191r : (L\u2082 + L\u2083).pr n) := rfl\n@[simp] lemma add_assoc'_pr\u2083 {n} (r : L\u2083.pr n) : (add_assoc' L\u2081 L\u2082 L\u2083).pr n (\u2191r : (L\u2081 + L\u2082 + L\u2083).pr n) = \u2191(\u2191r : (L\u2082 + L\u2083).pr n) := rfl\n\ndef add_assoc'_inv : L\u2081 + (L\u2082 + L\u2083) \u219d\u1d38 L\u2081 + L\u2082 + L\u2083 :=\n{ fn := \u03bb n f, by { rcases f, { refine sum.inl f }, { rcases f, { refine sum.inl (sum.inr f) }, { refine sum.inr f } } },\n  pr := \u03bb n r, by { rcases r, { refine sum.inl r }, { rcases r, { refine sum.inl (sum.inr r) }, { refine sum.inr r } } } }\n\n@[simp] lemma add_assoc'_inv_fn\u2081 {n} (f : L\u2081.fn n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).fn n (\u2191f : (L\u2081 + (L\u2082 + L\u2083)).fn n) = \u2191(\u2191f : (L\u2081 + L\u2082).fn n) := rfl\n@[simp] lemma add_assoc'_inv_fn\u2082 {n} (f : L\u2082.fn n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).fn n (\u2191(\u2191f : (L\u2082 + L\u2083).fn n)) = \u2191(\u2191f : (L\u2081 + L\u2082).fn n) := rfl\n@[simp] lemma add_assoc'_inv_fn\u2083 {n} (f : L\u2083.fn n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).fn n \u2191(\u2191f : (L\u2082 + L\u2083).fn n) = (\u2191f : (L\u2081 + L\u2082 + L\u2083).fn n) := rfl\n@[simp] lemma add_assoc'_inv_pr\u2081 {n} (r : L\u2081.pr n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).pr n (\u2191r : (L\u2081 + (L\u2082 + L\u2083)).pr n) = (\u2191(\u2191r : (L\u2081 + L\u2082).pr n)) := rfl\n@[simp] lemma add_assoc'_inv_pr\u2082 {n} (r : L\u2082.pr n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).pr n \u2191(\u2191r : (L\u2082 + L\u2083).pr n) = (\u2191(\u2191r : (L\u2081 + L\u2082).pr n)) := rfl\n@[simp] lemma add_assoc'_inv_pr\u2083 {n} (r : L\u2083.pr n) : (add_assoc'_inv L\u2081 L\u2082 L\u2083).pr n \u2191(\u2191r : (L\u2082 + L\u2083).pr n) = (\u2191r : (L\u2081 + L\u2082 + L\u2083).pr n) := rfl\n\nsection\nvariables {\u03b1 \u03b2 : Type*}\n\ndef consts_of_fun (f : \u03b1 \u2192 \u03b2) : consts \u03b1 \u219d\u1d38 consts \u03b2 :=\n{ fn := \u03bb n c, by { rcases n, { exact f c }, { rcases c } },\n  pr := \u03bb n r, by { rcases r } }\n\n@[simp] lemma consts_fn (f : \u03b1 \u2192 \u03b2) (c : (consts \u03b1).fn 0) : (consts_of_fun f).fn 0 c = f c := rfl\n\nend\n\nvariables {L\u2081} {L\u2082} (\u03c4 : L\u2081 \u219d\u1d38 L\u2082)\n\nend language_translation\n\nnamespace language_translation_coe\nvariables (L\u2081 L\u2082) [language_translation_coe L\u2082 L\u2081]\n\ndef sub : language.{u} :=\n{ fn := \u03bb n, \u21a5(has_compl.compl $ set.range (coe : L\u2082.fn n \u2192 L\u2081.fn n)),\n  pr := \u03bb n, \u21a5(has_compl.compl $ set.range (coe : L\u2082.pr n \u2192 L\u2081.pr n)) }\n\nend language_translation_coe\n\nnamespace language_equiv\nopen language_translation language_translation_coe extension\nvariables {L\u2081 L\u2082}\n\ndef of_equivs (Fn : \u03a0 n, equiv (L\u2081.fn n) (L\u2082.fn n)) (Pr : \u03a0 n, equiv (L\u2081.pr n) (L\u2082.pr n)) : language_equiv L\u2081 L\u2082 :=\n{ ltr := { fn := \u03bb n f, (Fn n).to_fun f, pr := \u03bb n r, (Pr n).to_fun r },\n  inv := { fn := \u03bb n f, (Fn n).inv_fun f, pr := \u03bb n r, (Pr n).inv_fun r },\n  left_inv_fn := \u03bb n, equiv.left_inverse_symm (Fn n),\n  left_inv_pr := \u03bb n, equiv.left_inverse_symm (Pr n),\n  right_inv_fn := \u03bb n, equiv.right_inverse_symm (Fn n),\n  right_inv_pr := \u03bb n, equiv.right_inverse_symm (Pr n) }\n\nvariables (L\u2081 L\u2082 L\u2083)\n\ndef add_comm' : L\u2081 + L\u2082 \u21ad\u1d38 L\u2082 + L\u2081 :=\n{ ltr := add_comm' L\u2081 L\u2082, inv := add_comm' L\u2082 L\u2081,\n  left_inv_fn := \u03bb n f, by rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082],\n  left_inv_pr := \u03bb n r, by rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082],\n  right_inv_fn := \u03bb n f, by rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082],\n  right_inv_pr := \u03bb n r, by rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082] }\n\ndef add_assoc' : L\u2081 + L\u2082 + L\u2083 \u21ad\u1d38 L\u2081 + (L\u2082 + L\u2083) :=\n{ ltr := add_assoc' L\u2081 L\u2082 L\u2083, inv := add_assoc'_inv L\u2081 L\u2082 L\u2083,\n  left_inv_fn := \u03bb n f, by { rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082], rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082] },\n  left_inv_pr := \u03bb n r, by { rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082], rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082] },\n  right_inv_fn := \u03bb n f, by { rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082], rcases f; simp[\u2190coe_fn\u2081, \u2190coe_fn\u2082] },\n  right_inv_pr := \u03bb n r, by { rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082], rcases r; simp[\u2190coe_pr\u2081, \u2190coe_pr\u2082] } }\n\n@[simp] lemma of_equivs_fn (Fn : \u03a0 n, equiv (L\u2081.fn n) (L\u2082.fn n)) (Pr : \u03a0 n, equiv (L\u2081.pr n) (L\u2082.pr n)) {n} (f : L\u2081.fn n) :\n  (of_equivs Fn Pr).ltr.fn n f = (Fn n) f := rfl\n\n@[simp] lemma of_equivs_pr (Fn : \u03a0 n, equiv (L\u2081.fn n) (L\u2082.fn n)) (Pr : \u03a0 n, equiv (L\u2081.pr n) (L\u2082.pr n)) {n} (r : L\u2081.pr n) :\n  (of_equivs Fn Pr).ltr.pr n r = (Pr n) r := rfl\n\n@[simp] lemma of_equivs_inv_fn (Fn : \u03a0 n, equiv (L\u2081.fn n) (L\u2082.fn n)) (Pr : \u03a0 n, equiv (L\u2081.pr n) (L\u2082.pr n)) {n} (f : L\u2082.fn n) :\n  (of_equivs Fn Pr).inv.fn n f = (Fn n).inv_fun f := rfl\n\n@[simp] lemma of_equivs_inv_pr (Fn : \u03a0 n, equiv (L\u2081.fn n) (L\u2082.fn n)) (Pr : \u03a0 n, equiv (L\u2081.pr n) (L\u2082.pr n)) {n} (r : L\u2082.pr n) :\n  (of_equivs Fn Pr).inv.pr n r = (Pr n).inv_fun r := rfl\n\nsection\nvariables (\u03c4 : L\u2081 \u21ad\u1d38 L\u2082)\n\n@[simp] lemma inv_ltr_fn {n} (f : L\u2081.fn n) : \u03c4.inv.fn n (\u03c4.ltr.fn n f) = f := \u03c4.left_inv_fn n f\n\n@[simp] lemma inv_ltr_pr {n} (r : L\u2081.pr n) : \u03c4.inv.pr n (\u03c4.ltr.pr n r) = r := \u03c4.left_inv_pr n r\n\n@[simp] lemma ltr_inv_fn {n} (f : L\u2082.fn n) : \u03c4.ltr.fn n (\u03c4.inv.fn n f) = f := \u03c4.right_inv_fn n f\n\n@[simp] lemma ltr_inv_pr {n} (r : L\u2082.pr n) : \u03c4.ltr.pr n (\u03c4.inv.pr n r) = r := \u03c4.right_inv_pr n r\n\n@[simp] lemma inv_ltr_t (t : term L\u2081) : \u03c4.inv.fun_t (\u03c4.ltr.fun_t t) = t :=\nby induction t; simp*\n\n@[simp] lemma ltr_inv_t (t : term L\u2082) : \u03c4.ltr.fun_t (\u03c4.inv.fun_t t) = t :=\nby induction t; simp*\n\n@[simp] lemma inv_ltr_p (p : formula L\u2081) : \u03c4.inv.fun_p (\u03c4.ltr.fun_p p) = p :=\nby induction p; simp*\n\n@[simp] lemma ltr_inv_p (p : formula L\u2082) : \u03c4.ltr.fun_p (\u03c4.inv.fun_p p) = p :=\nby induction p; simp*\n\nend\n\nsection\nvariables [\u03c4 : language.language_translation_coe L\u2081 L\u2082]\ninclude \u03c4\n\n@[reducible] noncomputable def add_sub' : L\u2081 + sub L\u2082 L\u2081 \u21ad\u1d38 L\u2082 :=\nof_equivs (\u03bb n, \n  let F : L\u2082.fn n \u2192 (L\u2081 + sub L\u2082 L\u2081).fn n :=\n        \u03bb f, if h : f \u2208 set.range (\u03c4.ltr.fn n) then\n          by { have : nonempty (L\u2081.fn n), from nonempty_of_exists h,\n               exact \u2191(by exactI classical.epsilon (\u03bb y, \u03c4.ltr.fn n y = f)) }\n        else sum.inr \u27e8f, h\u27e9 in\n    { to_fun := \u03bb f, by { rcases f, { exact \u03c4.ltr.fn _ f }, { rcases f with \u27e8f, hf\u27e9, exact f } },\n      inv_fun := F,\n      left_inv := \u03bb f,\n        by{ rcases f with (f | \u27e8f, hf\u27e9); simp,\n            { simp[F, \u2190coe_fn\u2081], exact classical.epsilon_singleton f },\n            { simp[F, -set.mem_range, show f \u2209 (set.range (\u03c4.ltr.fn n)), from hf], refl } }, \n      right_inv := \u03bb f,\n        by{ simp, by_cases C : f \u2208 set.range (\u03c4.ltr.fn n),\n            { rcases C with \u27e8f, rfl\u27e9, \n              have : F (\u03c4.ltr.fn n f) = sum.inl f,\n              { simp[F, \u2190coe_fn\u2081], exact classical.epsilon_singleton f },\n              rw[this] },\n            { have : F f = sum.inr \u27e8f, C\u27e9, by simp[F, -set.mem_range, C],\n              rw[this] } } })\n(\u03bb n,\n  let F : L\u2082.pr n \u2192 (L\u2081 + sub L\u2082 L\u2081).pr n :=\n        \u03bb f, if h : f \u2208 set.range (\u03c4.ltr.pr n) then\n          by { have : nonempty (L\u2081.pr n), from nonempty_of_exists h,\n               exact \u2191(by exactI classical.epsilon (\u03bb y, \u03c4.ltr.pr n y = f)) }\n        else sum.inr \u27e8f, h\u27e9 in\n    { to_fun := \u03bb f, by { rcases f, { exact \u03c4.ltr.pr _ f }, { rcases f with \u27e8f, hf\u27e9, exact f } },\n      inv_fun := F,\n      left_inv := \u03bb f,\n        by{ rcases f with (f | \u27e8f, hf\u27e9); simp,\n            { simp[F, \u2190coe_pr\u2081], exact classical.epsilon_singleton f },\n            { simp[F, -set.mem_range, show f \u2209 (set.range (\u03c4.ltr.pr n)), from hf], refl } },      \n      right_inv := \u03bb f,\n        by{ simp, by_cases C : f \u2208 set.range (\u03c4.ltr.pr n),\n            { rcases C with \u27e8f, rfl\u27e9, \n              have : F (\u03c4.ltr.pr n f) = sum.inl f,\n              { simp[F, \u2190coe_pr\u2081], exact classical.epsilon_singleton f },\n              rw[this] },\n            { have : F f = sum.inr \u27e8f, C\u27e9, by simp[F, -set.mem_range, C],\n              rw[this] } } })\n\nlemma add_sub'_add_left_commute : (add_sub' L\u2081 L\u2082).ltr.comp (extension.add_left) = \u03c4.ltr := by ext n f; simp[add_sub', add_left]\n\nend\n\nend language_equiv\n\nnamespace language_translation\nopen extension\n\ndef seq (l : \u2115 \u2192 language.{u}) := \u03a0 n, l n \u219d\u1d38 l (n + 1)\n\nvariables {l : \u2115 \u2192 language.{u}}\n\n\nstructure seq_limit (l : \u2115 \u2192 language.{u}) (L : language):=\n(seq : seq l)\n(to_limit : \u03a0 n, l n \u219d\u1d38 L)\n(commutes : \u2200 n, (to_limit (n + 1)).comp (seq n) = to_limit n)\n(rank_fn : \u03a0 {n} (f : L.fn n), \u2115)\n(rank_pr : \u03a0 {n} (r : L.pr n), \u2115)\n(fn : \u03a0 {n} (f : L.fn n), (l $ rank_fn f).fn n)\n(pr : \u03a0 {n} (r : L.pr n), (l $ rank_pr r).pr n)\n(fn_spec : \u2200 {n} (f : L.fn n), (to_limit _).fn _ (fn f) = f)\n(pr_spec : \u2200 {n} (r : L.pr n), (to_limit _).pr _ (pr r) = r)\n\nnamespace seq_limit\nvariables {L} (s : seq_limit l L)\ninclude s\n\ndef seqs : \u03a0 n m, l n \u219d\u1d38 l (n + m)\n| n 0       := (1 : l n \u219d\u1d38 l n)\n| n (m + 1) := (s.seq (n + m)).comp (seqs n m)\n\ndef seqs_le {n m} (h : n \u2264 m) : l n \u219d\u1d38 l m :=\nby { rw [show m = n + (m - n), by omega], exact s.seqs n (m - n) }\n\n@[simp] lemma to_limit_seq_commutes' (n : \u2115) : (s.to_limit (n + 1)).comp (s.seq n) = s.to_limit n := s.commutes n\n\n@[simp] lemma to_limit_seqs_commuts (n m : \u2115) : (s.to_limit (n + m)).comp (s.seqs n m) = s.to_limit n :=\nby { induction m with m IH; simp[seqs], { refl },\n  { suffices : (s.to_limit (n + m + 1)).comp ((s.seq (n + m)).comp (s.seqs n m)) = s.to_limit n, by simpa,\n    rw \u2190 comp_assoc, simp[IH] } }\n\nlemma seqs_le_commuts {n m : \u2115} (le : n \u2264 m) : (s.to_limit m).comp (s.seqs_le le) = s.to_limit n :=\nby { have := s.to_limit_seqs_commuts n (m - n), rw \u2190this, congr; simp[show n + (m - n) = m, by omega, seqs_le] }\n\n@[simp] lemma seqs_le_commuts'_fn {n m : \u2115} (le : n \u2264 m) {k} (f : (l n).fn k) :\n  (s.to_limit m).fn _ ((s.seqs_le le).fn _ f) = (s.to_limit n).fn _ f :=\nby { rw[\u2190s.seqs_le_commuts le], simp }\n\n@[simp] lemma seqs_le_commuts'_pr {n m : \u2115} (le : n \u2264 m) {k} (r : (l n).pr k) :\n  (s.to_limit m).pr _ ((s.seqs_le le).pr _ r) = (s.to_limit n).pr _ r :=\nby { rw[\u2190s.seqs_le_commuts le], simp }\n\n@[simp] lemma seqs_le_commuts'_t {n m : \u2115} (le : n \u2264 m) (t : term (l n)) :\n  (s.to_limit m).fun_t ((s.seqs_le le).fun_t t) = (s.to_limit n).fun_t t :=\nby { rw[\u2190s.seqs_le_commuts le], simp[comp_fun_t] }\n\n@[simp] lemma seqs_le_commuts'_p {n m : \u2115} (le : n \u2264 m) (p : formula (l n)) :\n  (s.to_limit m).fun_p ((s.seqs_le le).fun_p p) = (s.to_limit n).fun_p p :=\nby { rw[\u2190s.seqs_le_commuts le], simp[comp_fun_p] }\n\n@[simp] def rank_t : term L \u2192 \u2115\n| #n        := 0\n| (app f v) := max (s.rank_fn f) (\u2a06\u1da0 i, rank_t (v i))\n\n@[simp, reducible] def retruct_t : \u03a0 t : term L, term (l $ s.rank_t t)\n| #n := #n\n| (@term.app L m f v) :=\n    let n := max (s.rank_fn f) (\u2a06\u1da0 i, s.rank_t (v i)),\n        tr\u2080 : l (s.rank_fn f) \u219d\u1d38 l n := s.seqs_le (by simp[n]),\n        tr : \u03a0 i, l (s.rank_t $ v i) \u219d\u1d38 l n := \u03bb i, s.seqs_le (by { simp[n], refine or.inr (le_fintype_sup _ i)}) in\n    app (tr\u2080.fn _ (s.fn f)) (\u03bb i, (tr i).fun_t (retruct_t (v i)))\n\n@[simp] def rank_p : formula L \u2192 \u2115\n| (app r v)   := max (s.rank_pr r) (\u2a06\u1da0 i, s.rank_t (v i))\n| (t =' u) := max (s.rank_t t) (s.rank_t u)\n| \u22a4           := 0\n| (p \u27f6 q)     := max (rank_p p) (rank_p q)\n| (\u223cp)        := rank_p p\n| (\u2200.p)       := rank_p p\n\n@[simp, reducible] def retruct_p : \u03a0 p : formula L, formula (l $ s.rank_p p)\n| (app r v)   :=\n  let tr\u2080 : l (s.rank_pr r) \u219d\u1d38 l (s.rank_p (app r v)) := s.seqs_le (by simp),\n      tr : \u03a0 i, l (s.rank_t $ v i) \u219d\u1d38 l (s.rank_p (app r v)) := \u03bb i, s.seqs_le (by { simp, refine or.inr (le_fintype_sup _ i)}) in\n    app (tr\u2080.pr _ (s.pr r)) (\u03bb i, (tr i).fun_t (s.retruct_t (v i)))\n| (equal t u) :=\n    let tr\u2081 : l (s.rank_t t) \u219d\u1d38 l (s.rank_p (equal t u)) := s.seqs_le (by simp),\n        tr\u2082 : l (s.rank_t u) \u219d\u1d38 l (s.rank_p (equal t u)) := s.seqs_le (by simp) in\n    (tr\u2081.fun_t $ s.retruct_t t) =' (tr\u2082.fun_t $ s.retruct_t u)\n| \u22a4           := \u22a4\n| (p \u27f6 q)     :=\n   let tr\u2081 : l (s.rank_p p) \u219d\u1d38 l (s.rank_p (p \u27f6 q)) := s.seqs_le (by simp),\n       tr\u2082 : l (s.rank_p q) \u219d\u1d38 l (s.rank_p (p \u27f6 q)) := s.seqs_le (by simp) in\n    (tr\u2081.fun_p $ retruct_p p) \u27f6 (tr\u2082.fun_p $ retruct_p q)\n| (\u223cp)        :=\n    let tr\u2081 : l (s.rank_p p) \u219d\u1d38 l (s.rank_p (\u223cp)) := s.seqs_le (by simp) in\n    \u223c(tr\u2081.fun_p $ retruct_p p)\n| (\u2200.p)       :=\n    let tr\u2081 : l (s.rank_p p) \u219d\u1d38 l (s.rank_p (\u2200.p)) := s.seqs_le (by simp) in\n    \u2200.(tr\u2081.fun_p $ retruct_p p)\n\nlemma retruct_t_spec (t : term L) : (s.to_limit (s.rank_t t)).fun_t (s.retruct_t t) = t :=\nby induction t; simp[s.fn_spec]; case app : n f v IH { funext i, exact IH i }\n\nlemma retruct_p_spec (p : formula L) : (s.to_limit (s.rank_p p)).fun_p (s.retruct_p p) = p :=\nby induction p; simp[s.pr_spec, retruct_t_spec, *]\n\nend seq_limit\n\nend language_translation\n\nend language\n\n\ndef def_fn {n} (f : L\u2082.fn n) (p : formula L\u2081) : formula (L\u2081 + L\u2082) :=\n\u2200.[n] rew \u0131[0 \u21dd app (sum.inr f) ##] \u2191p\n\ndef def_pr {n} (r : L\u2082.pr n) (p : formula L\u2081) : formula (L\u2081 + L\u2082) :=\n\u2200.[n] (app (sum.inr r) ## \u27f7 \u2191p)\n\n@[simp] lemma def_fn_is_sentence {n} (f : L\u2082.fn n) (p : formula L\u2081) (hp : p.arity \u2264 n + 1) : is_sentence (def_fn f p) :=\nbegin\n  simp[def_fn, is_sentence] at hp \u22a2,\n  refine le_trans ((p : formula (L\u2081 + L\u2082)).rew_arity \u0131[0 \u21dd app (sum.inr f) (\u03bb i, #i)]) (fintype_sup_le _),\n  rintros \u27e8i, hi\u27e9, cases i; simp at hi \u22a2,\n  { refine fintype_sup_le _, rintros \u27e8i, hi\u27e9, simp[nat.succ_le_iff.mpr hi] },\n  { have : i + 1 < n + 1, from lt_of_lt_of_le hi hp,\n    exact nat.lt_succ_iff.mp this }\nend\n\n@[simp] lemma def_pr_is_sentence {n} (r : L\u2082.pr n) (p : formula L\u2081) (hp : p.arity \u2264 n) : is_sentence (def_pr r p) :=\nby { simp[def_pr, is_sentence, hp],\n     refine fintype_sup_le _, rintros \u27e8i, hi\u27e9, simpa using nat.succ_le_iff.mpr hi }\n\nvariables (L\u2081 L\u2082)\n\nstructure language.definitions :=\n(df_fn : \u03a0 {n : \u2115}, L\u2082.fn n \u2192 formula L\u2081)\n(hdf_fn : \u2200 {n} {f : L\u2082.fn n}, (df_fn f).arity \u2264 n + 1)\n(df_pr : \u03a0 {n : \u2115}, L\u2082.pr n \u2192 formula L\u2081)\n(hdf_pr : \u2200 {n} {r : L\u2082.pr n}, (df_pr r).arity \u2264 n)\n\nvariables {L\u2081 L\u2082} (D : L\u2081.definitions L\u2082)\n\ndef language.definitions.thy : Theory (L\u2081 + L\u2082) :=\n(\u22c3 n, (set.range (\u03bb (f : L\u2082.fn n), def_fn f (D.df_fn f)))) \u222a\n(\u22c3 n, (set.range (\u03bb (r : L\u2082.pr n), def_pr r (D.df_pr r))))\n\nlemma definitions_def :\n  D.thy = (\u22c3 n, (set.range (\u03bb (f : L\u2082.fn n), def_fn f (D.df_fn f)))) \u222a\n          (\u22c3 n, (set.range (\u03bb (r : L\u2082.pr n), def_pr r (D.df_pr r)))) := rfl\n\ninstance language.definitions.closed : closed_Theory D.thy :=\n\u27e8by { simp[definitions_def], rintros p (\u27e8n, f, rfl\u27e9 | \u27e8n, r, rfl\u27e9),  { simp[D.hdf_fn] }, { simp[D.hdf_pr] } }\u27e9\n\n@[simp] lemma language.definitions.mem_fn {n} (f : L\u2082.fn n) :\n  (\u2200.[n] (D.df_fn f : formula (L\u2081 + L\u2082)).rew \u0131[0 \u21dd app (sum.inr f) ##]) \u2208 D.thy :=\nby simp[definitions_def, def_fn]; refine or.inl \u27e8n, f, by refl\u27e9\n\n@[simp] lemma language.definitions.fn {n} (f : L\u2082.fn n) (v : finitary (term (L\u2081 + L\u2082)) n) :\n  D.thy \u22a2 (D.df_fn f : formula (L\u2081 + L\u2082)).rew (app (sum.inr f) v \u2322 of_fin v) :=\nby { have := provable.nfal_subst'_finitary (axiomatic_classical_logic'.by_axiom (language.definitions.mem_fn D f)) v,\n     simp[formula.nested_rew] at this,\n     refine cast (by { congr, funext x, rcases x; simp }) this }\n\n@[simp] lemma language.definitions.mem_pr {n} (r : L\u2082.pr n) :\n  (\u2200.[n] ((app (sum.inr r) ## : formula (L\u2081 + L\u2082)) \u27f7 (D.df_pr r))) \u2208 D.thy :=\nby simp[definitions_def, def_pr]; refine or.inr \u27e8n, r, by refl\u27e9\n\n@[simp] lemma language.definitions.pr {n} (r : L\u2082.pr n) (v : finitary (term (L\u2081 + L\u2082)) n) :\n  D.thy \u22a2 app (sum.inr r) v \u27f7 (D.df_pr r).rew (of_fin v) :=\nby { have := provable.nfal_subst'_finitary (axiomatic_classical_logic'.by_axiom (language.definitions.mem_pr D r)) v,\n     simpa using this }\n\n\nsection\nvariables [language.predicate L\u2082]\n\ndef term.coe_inv : term (L\u2081 + L\u2082) \u2192 term L\u2081\n| (#n)      := #n\n| (app f v) := by { rcases f, { refine app f (\u03bb i, term.coe_inv (v i)) },\n  { exfalso, exact is_empty.false f } }\n\n@[simp] lemma coe_inv_coe (t : term L\u2081) : term.coe_inv (\u2191t : term (L\u2081 + L\u2082)) = t :=\nby { induction t; simp[term.coe_inv],\n     case app : n f v IH { rw [language.extension.coe_fn\u2081 f], simp, funext i, exact IH i } }\n\n@[simp] lemma coe_coe_inv (t : term (L\u2081 + L\u2082)) : (\u2191(term.coe_inv t) : term (L\u2081 + L\u2082)) = t :=\nby { induction t; simp[term.coe_inv],\n     case app : n f v IH\n     { rcases f; simp, { refine \u27e8rfl, _\u27e9, funext i, exact IH i }, { exfalso, exact is_empty.false f } } }\n\ndef formula.coe_inv (D : L\u2081.definitions L\u2082) : formula (L\u2081 + L\u2082) \u2192 formula L\u2081\n| (app r v)                  := by { rcases r, { exact app r (\u03bb i, (v i).coe_inv) },\n                                               { exact (D.df_pr r).rew (of_fin (\u03bb i, (v i).coe_inv)) } }\n| ((t : term (L\u2081 + L\u2082)) =' u) := t.coe_inv =' u.coe_inv\n| \u22a4                          := \u22a4\n| (p \u27f6 q)                    := p.coe_inv \u27f6 q.coe_inv\n| (\u223cp)                       := \u223cp.coe_inv\n| (\u2200.p)                      := \u2200.p.coe_inv\n\nlemma coe_inv_equiv (p : formula (L\u2081 + L\u2082)) :\n  D.thy \u22a2 p \u27f7 \u2191(formula.coe_inv D p) :=\nbegin\n  induction p; simp[formula.coe_inv],\n  case app : n r v\n  { rcases r; simp[language.extension.coe_pr\u2081, language.language_translation_coe.coe_p_rew],\n    have : (\u03bb x, \u2191(of_fin (\u03bb i, (v i).coe_inv) x)) = of_fin v,\n    { funext x, have : x < n \u2228 n \u2264 x, exact lt_or_ge x n,\n      rcases this with (C | C);  simp[C] },\n    simp[this] },\n  case imply : p q IH_p IH_q\n  { simp[Lindenbaum.eq_of_provable_equiv_0,\n      Lindenbaum.eq_of_provable_equiv_0.mp IH_p, Lindenbaum.eq_of_provable_equiv_0.mp IH_q] },\n  case neg : p IH\n  { refine Lindenbaum.eq_of_provable_equiv_0.mpr (by simp[IH]) },\n  case fal : p IH\n  { have : D.thy^1 \u22a2 p \u27f7 \u2191(formula.coe_inv D p), by simpa using IH,\n    simp[Lindenbaum.eq_of_provable_equiv_0, Lindenbaum.eq_of_provable_equiv.mp this] } \nend\n\n@[simp] def formula.coe_inv_is_open (D : L\u2081.definitions L\u2082) : Theory (L\u2081 + L\u2082)\n| (app r v)                  := by { rcases r, { exact true },\n                                               { exact (D.df_pr r).is_open } }\n| ((t : term (L\u2081 + L\u2082)) =' u) := true\n| \u22a4                          := true\n| (p \u27f6 q)                    := p.coe_inv_is_open \u2227 q.coe_inv_is_open\n| (\u223cp)                       := p.coe_inv_is_open\n| (\u2200.p)                      := false\n\n@[simp] lemma coe_inv_open (p : formula (L\u2081 + L\u2082)) :\n  (formula.coe_inv D p).is_open \u2194 formula.coe_inv_is_open D p :=\nby { induction p; simp[formula.coe_inv, *],\n     case app : n r v { rcases r; simp, } }\n\nend\n\nnamespace Structure\nvariables {L\u2081 L\u2082} (M\u2081 : Structure L\u2081)\nopen language language.extension\n\n@[reducible] def extend\n  (fn : \u03a0 {n} (f : L\u2082.fn n) (v : finitary M\u2081.dom n), M\u2081.dom)\n  (pr : \u03a0 {n} (r : L\u2082.pr n) (v : finitary M\u2081.dom n), Prop) : Structure (L\u2081 + L\u2082) :=\n{ dom := M\u2081.dom,\n  inhabited := M\u2081.inhabited,\n  fn := \u03bb n f v, by { rcases f, { exact M\u2081.fn f v }, { exact fn f v } },\n  pr := \u03bb n r v, by { rcases r, { exact M\u2081.pr r v }, { exact pr r v } } }\n\nlemma extend_val_coe_term\n  (fn : \u03a0 {n} (f : L\u2082.fn n) (v : finitary M\u2081.dom n), M\u2081.dom)\n  (pr : \u03a0 {n} (r : L\u2082.pr n) (v : finitary M\u2081.dom n), Prop) {t : term L\u2081} {e : \u2115 \u2192 M\u2081.dom} :\n  @term.val (L\u2081 + L\u2082) (M\u2081.extend @fn @pr) e (t : term (L\u2081 + L\u2082)) = @term.val L\u2081 M\u2081 e t :=\nby induction t; simp[*, coe_fn\u2081]\n\nlemma extend_val_coe_iff\n  (fn : \u03a0 {n} (f : L\u2082.fn n) (v : finitary M\u2081.dom n), M\u2081.dom)\n  (pr : \u03a0 {n} (r : L\u2082.pr n) (v : finitary M\u2081.dom n), Prop) {p : formula L\u2081} {e : \u2115 \u2192 M\u2081.dom} :\n  M\u2081.extend @fn @pr \u22a7[e] \u2191p \u2194 M\u2081 \u22a7[e] p :=\nby induction p generalizing e; simp[coe_pr\u2081, extend_val_coe_term, *]\n\nlemma extend_models_coe_iff\n  (fn : \u03a0 {n} (f : L\u2082.fn n) (v : finitary M\u2081.dom n), M\u2081.dom)\n  (pr : \u03a0 {n} (r : L\u2082.pr n) (v : finitary M\u2081.dom n), Prop) {p : formula L\u2081} :\n  M\u2081.extend @fn @pr \u22a7 (p : formula (L\u2081 + L\u2082)) \u2194 M\u2081 \u22a7 p :=\n\u27e8\u03bb h e, (M\u2081.extend_val_coe_iff @fn @pr).mp (h e), \u03bb h e, (M\u2081.extend_val_coe_iff @fn @pr).mpr (h e)\u27e9\n\nlemma extend_modelsth_coe_iff\n  (fn : \u03a0 {n} (f : L\u2082.fn n) (v : finitary M\u2081.dom n), M\u2081.dom)\n  (pr : \u03a0 {n} (r : L\u2082.pr n) (v : finitary M\u2081.dom n), Prop) {T : Theory L\u2081} :\n  M\u2081.extend @fn @pr \u22a7 (\u2191T : Theory (L\u2081 + L\u2082)) \u2194 M\u2081 \u22a7 T :=\n\u27e8\u03bb h p mem, (M\u2081.extend_models_coe_iff @fn @pr).mp (h (show \u2191p \u2208 \u2191T, by simp[mem])),\n \u03bb h p mem,\n by { rcases language_translation_coe.mem_coe_iff.mp mem with \u27e8p, pmem, rfl\u27e9,\n      exact (M\u2081.extend_models_coe_iff @fn @pr).mpr (h pmem) }\u27e9\n\nvariables (\u03c4 : L\u2081 \u21ad\u1d38 L\u2082)\n\n@[reducible] def of_equiv : Structure L\u2082 :=\n{ dom := M\u2081.dom, inhabited := M\u2081.inhabited,\n  fn := \u03bb n f, M\u2081.fn (\u03c4.inv.fn _ f),\n  pr := \u03bb n r, M\u2081.pr (\u03c4.inv.pr _ r) }\n\nvariables {M\u2081}\n\n@[simp] lemma equiv_term {t : term L\u2081} {e : \u2115 \u2192 M\u2081.dom} :\n  @term.val L\u2082 (M\u2081.of_equiv \u03c4) e (\u03c4.ltr.fun_t t) = @term.val L\u2081 M\u2081 e t :=\nby induction t; simp*\n\nlemma equiv_val_iff {p : formula L\u2081} {e : \u2115 \u2192 M\u2081.dom} :\n  M\u2081.of_equiv \u03c4 \u22a7[e] \u03c4.ltr.fun_p p \u2194 M\u2081 \u22a7[e] p :=\nby induction p generalizing e; simp[of_equiv, *]\n\n@[simp] lemma equiv_models_iff {p : formula L\u2081} :\n  M\u2081.of_equiv \u03c4 \u22a7 \u03c4.ltr.fun_p p \u2194 M\u2081 \u22a7 p :=\n\u27e8\u03bb h e, (equiv_val_iff \u03c4).mp (h e), \u03bb h e, (equiv_val_iff \u03c4).mpr (h e)\u27e9\n\n@[simp] lemma equiv_modelsth_iff {T : Theory L\u2081} :\n  M\u2081.of_equiv \u03c4 \u22a7 \u03c4.ltr.fun_Theory T \u2194 M\u2081 \u22a7 T :=\n\u27e8\u03bb h p mem, (equiv_models_iff \u03c4).mp (h \u27e8p, by simp[mem]\u27e9),\n \u03bb h p mem, by { rcases mem with \u27e8p', mem, rfl\u27e9, exact (equiv_models_iff \u03c4).mpr (h mem) }\u27e9\n\nend Structure\n\ndef Theory_of (M : Structure L) : Theory L := {p | M \u22a7 p}\n\nclass Theory_of_Structure (M : Structure L) (T : Theory L) :=\n(models : M \u22a7 T)\n\nnamespace language\nnamespace language_translation\nvariables {L\u2081 L\u2082} {\u03c4 : L\u2081 \u219d\u1d38 L\u2082} {M\u2082 : Structure L\u2082}\n\n@[reducible] def of_ltr (\u03c4 : L\u2081 \u219d\u1d38 L\u2082) (M\u2082 : Structure L\u2082) : Structure L\u2081 :=\n{ dom := M\u2082.dom,\n  inhabited := M\u2082.inhabited,\n  fn := \u03bb n f v, M\u2082.fn (\u03c4.fn _ f) v,\n  pr := \u03bb n r v, M\u2082.pr (\u03c4.pr _ r) v }\n\nlemma of_ltr_val_t (e : \u2115 \u2192 M\u2082.dom) (t : term L\u2081) : (\u03c4.fun_t t).val M\u2082 e = t.val (\u03c4.of_ltr M\u2082) e :=\nby induction t; simp*\n\nlemma models_val_iff {e : \u2115 \u2192 M\u2082.dom} {p : formula L\u2081} : \u03c4.of_ltr M\u2082 \u22a7[e] p \u2194 M\u2082 \u22a7[e] \u03c4.fun_p p :=\nby induction p generalizing e; try { simp[*, of_ltr_val_t] }\n\ntheorem models_iff {p : formula L\u2081} : \u03c4.of_ltr M\u2082 \u22a7 p \u2194 M\u2082 \u22a7 \u03c4.fun_p p:=\n\u27e8\u03bb h e, models_val_iff.mp (h e), \u03bb h e, models_val_iff.mpr (h e)\u27e9\n\ntheorem Theory_models_iff {T : Theory L\u2081} : \u03c4.of_ltr M\u2082 \u22a7 T \u2194 M\u2082 \u22a7 \u03c4.fun_Theory T :=\nby simp[fun_Theory, logic.semantics.Models_def, models_iff]\n\nend language_translation\n\nend language\n\nend fol\n", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/translation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.37359284470024556}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Reid Barton, Bhavik Mehta\n-/\nimport category_theory.over\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.finite_products\n\n/-!\n# Products in the over category\n\nShows that products in the over category can be derived from wide pullbacks in the base category.\nThe main result is `over_product_of_wide_pullback`, which says that if `C` has `J`-indexed wide\npullbacks, then `over B` has `J`-indexed products.\n-/\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nopen category_theory category_theory.limits\n\nvariables {J : Type v}\nvariables {C : Type u} [category.{v} C]\nvariable {X : C}\n\nnamespace category_theory.over\n\nnamespace construct_products\n\n/--\n(Implementation)\nGiven a product diagram in `C/B`, construct the corresponding wide pullback diagram\nin `C`.\n-/\n@[reducible]\ndef wide_pullback_diagram_of_diagram_over (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  wide_pullback_shape J \u2964 C :=\nwide_pullback_shape.wide_cospan B (\u03bb j, (F.obj j).left) (\u03bb j, (F.obj j).hom)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_inverse_obj (B : C) {J : Type v} (F : discrete J \u2964 over B) (c : cone F) :\n  cone (wide_pullback_diagram_of_diagram_over B F) :=\n{ X := c.X.left,\n  \u03c0 :=\n  { app := \u03bb X, option.cases_on X c.X.hom (\u03bb (j : J), (c.\u03c0.app j).left),\n  -- `tidy` can do this using `case_bash`, but let's try to be a good `-T50000` citizen:\n    naturality' := \u03bb X Y f,\n    begin\n      dsimp, cases X; cases Y; cases f,\n      { rw [category.id_comp, category.comp_id], },\n      { rw [over.w, category.id_comp], },\n      { rw [category.id_comp, category.comp_id], },\n    end } }\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_inverse (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  cone F \u2964 cone (wide_pullback_diagram_of_diagram_over B F) :=\n{ obj := cones_equiv_inverse_obj B F,\n  map := \u03bb c\u2081 c\u2082 f,\n  { hom := f.hom.left,\n    w' := \u03bb j,\n    begin\n      cases j,\n      { simp },\n      { dsimp,\n        rw \u2190 f.w j,\n        refl }\n    end } }\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simps]\ndef cones_equiv_functor (B : C) {J : Type v} (F : discrete J \u2964 over B) :\n  cone (wide_pullback_diagram_of_diagram_over B F) \u2964 cone F :=\n{ obj := \u03bb c,\n  { X := over.mk (c.\u03c0.app none),\n    \u03c0 :=\n    { app := \u03bb j, over.hom_mk (c.\u03c0.app (some j))\n                    (by apply c.w (wide_pullback_shape.hom.term j)) } },\n  map := \u03bb c\u2081 c\u2082 f,\n  { hom := over.hom_mk f.hom } }\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp]\ndef cones_equiv_unit_iso (B : C) (F : discrete J \u2964 over B) :\n  \ud835\udfed (cone (wide_pullback_diagram_of_diagram_over B F)) \u2245\n    cones_equiv_functor B F \u22d9 cones_equiv_inverse B F :=\nnat_iso.of_components (\u03bb _, cones.ext {hom := \ud835\udfd9 _, inv := \ud835\udfd9 _} (by tidy)) (by tidy)\n\n/-- (Impl) A preliminary definition to avoid timeouts. -/\n@[simp]\ndef cones_equiv_counit_iso (B : C) (F : discrete J \u2964 over B) :\n  cones_equiv_inverse B F \u22d9 cones_equiv_functor B F \u2245 \ud835\udfed (cone F) :=\nnat_iso.of_components\n  (\u03bb _, cones.ext {hom := over.hom_mk (\ud835\udfd9 _), inv := over.hom_mk (\ud835\udfd9 _)} (by tidy)) (by tidy)\n\n-- TODO: Can we add `. obviously` to the second arguments of `nat_iso.of_components` and\n--       `cones.ext`?\n/--\n(Impl) Establish an equivalence between the category of cones for `F` and for the \"grown\" `F`.\n-/\n@[simps]\ndef cones_equiv (B : C) (F : discrete J \u2964 over B) :\n  cone (wide_pullback_diagram_of_diagram_over B F) \u224c cone F :=\n{ functor := cones_equiv_functor B F,\n  inverse := cones_equiv_inverse B F,\n  unit_iso := cones_equiv_unit_iso B F,\n  counit_iso := cones_equiv_counit_iso B F, }\n\n/-- Use the above equivalence to prove we have a limit. -/\nlemma has_over_limit_discrete_of_wide_pullback_limit {B : C} (F : discrete J \u2964 over B)\n  [has_limit (wide_pullback_diagram_of_diagram_over B F)] :\n  has_limit F :=\nhas_limit.mk\n{ cone := _,\n  is_limit := is_limit.of_right_adjoint\n    (cones_equiv B F).functor (limit.is_limit (wide_pullback_diagram_of_diagram_over B F)) }\n\n/-- Given a wide pullback in `C`, construct a product in `C/B`. -/\nlemma over_product_of_wide_pullback [has_limits_of_shape (wide_pullback_shape J) C] {B : C} :\n  has_limits_of_shape (discrete J) (over B) :=\n{ has_limit := \u03bb F, has_over_limit_discrete_of_wide_pullback_limit F }\n\n/-- Given a pullback in `C`, construct a binary product in `C/B`. -/\nlemma over_binary_product_of_pullback [has_pullbacks C] {B : C} :\n  has_binary_products (over B) :=\nover_product_of_wide_pullback\n\n/-- Given all wide pullbacks in `C`, construct products in `C/B`. -/\nlemma over_products_of_wide_pullbacks [has_wide_pullbacks C] {B : C} :\n  has_products (over B) :=\n\u03bb J, over_product_of_wide_pullback\n\n/-- Given all finite wide pullbacks in `C`, construct finite products in `C/B`. -/\nlemma over_finite_products_of_finite_wide_pullbacks [has_finite_wide_pullbacks C] {B : C} :\n  has_finite_products (over B) :=\n\u27e8\u03bb J \ud835\udca5\u2081 \ud835\udca5\u2082, by exactI over_product_of_wide_pullback\u27e9\n\nend construct_products\n\n/--\nConstruct terminal object in the over category. This isn't an instance as it's not typically the\nway we want to define terminal objects.\n(For instance, this gives a terminal object which is different from the generic one given by\n`over_product_of_wide_pullback` above.)\n-/\nlemma over_has_terminal (B : C) : has_terminal (over B) :=\n{ has_limit := \u03bb F, has_limit.mk\n  { cone :=\n    { X := over.mk (\ud835\udfd9 _),\n      \u03c0 := { app := \u03bb p, pempty.elim p } },\n    is_limit :=\n      { lift := \u03bb s, over.hom_mk _,\n        fac' := \u03bb _ j, j.elim,\n        uniq' := \u03bb s m _,\n          begin\n            ext,\n            rw over.hom_mk_left,\n            have := m.w,\n            dsimp at this,\n            rwa [category.comp_id, category.comp_id] at this\n          end } } }\n\nend category_theory.over\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/limits/constructions/over/products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6261241911813151, "lm_q1q2_score": 0.3734412212986656}}
{"text": "/-\nCopyright (c) 2021 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Scott Morrison\n-/\nimport data.finset.lattice\nimport data.finset.n_ary\nimport data.multiset.functor\n\n/-!\n# Functoriality of `finset`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines the functor structure of `finset`.\n\n## TODO\n\nCurrently, all instances are classical because the functor classes want to run over all types. If\ninstead we could state that a functor is lawful/applicative/traversable... between two given types,\nthen we could provide the instances for types with decidable equality.\n-/\n\nuniverses u\n\nopen function\n\nnamespace finset\n\n/-! ### Functor -/\n\nsection functor\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\n/-- Because `finset.image` requires a `decidable_eq` instance for the target type, we can only\nconstruct `functor finset` when working classically. -/\ninstance : functor finset :=\n{ map := \u03bb \u03b1 \u03b2 f s, s.image f }\n\ninstance : is_lawful_functor finset :=\n{ id_map := \u03bb \u03b1 s, image_id,\n  comp_map := \u03bb \u03b1 \u03b2 \u03b3 f g s, image_image.symm }\n\n@[simp] lemma fmap_def {s : finset \u03b1} (f : \u03b1 \u2192 \u03b2) : f <$> s = s.image f := rfl\n\nend functor\n\n/-! ### Pure -/\n\ninstance : has_pure finset := \u27e8\u03bb \u03b1 x, {x}\u27e9\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 finset \u03b1) = singleton := rfl\n\n/-! ### Applicative functor -/\n\nsection applicative\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\ninstance : applicative finset :=\n{ seq := \u03bb \u03b1 \u03b2 t s, t.sup (\u03bb f, s.image f),\n  seq_left := \u03bb \u03b1 \u03b2 s t, if t = \u2205 then \u2205 else s,\n  seq_right := \u03bb \u03b1 \u03b2 s t, if s = \u2205 then \u2205 else t,\n  .. finset.functor,\n  .. finset.has_pure }\n\n@[simp] \n\n/-- `finset.image\u2082` in terms of monadic operations. Note that this can't be taken as the definition\nbecause of the lack of universe polymorphism. -/\nlemma image\u2082_def {\u03b1 \u03b2 \u03b3 : Type*} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (s : finset \u03b1) (t : finset \u03b2) :\n  image\u2082 f s t = f <$> s <*> t :=\nby { ext, simp [mem_sup] }\n\ninstance : is_lawful_applicative finset :=\n{ seq_left_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_left_def],\n    obtain rfl | ht := t.eq_empty_or_nonempty,\n    { simp_rw [if_pos rfl, image_empty], exact (sup_bot _).symm },\n    { ext a,\n      rw [if_neg ht.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8const \u03b2 a, mem_image_of_mem _ ha, mem_image_const_self.2 ht\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := hf,\n      obtain \u27e8_, _, rfl\u27e9 := ha,\n      exact hb }\n  end,\n  seq_right_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_right_def],\n    obtain rfl | hs := s.eq_empty_or_nonempty,\n    { rw [if_pos rfl, image_empty, sup_empty, bot_eq_empty] },\n    { ext a,\n      rw [if_neg hs.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8id, mem_image_const_self.2 hs, by rwa image_id\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := ha,\n      obtain \u27e8_, _, rfl\u27e9 := hf,\n      exact hb }\n  end,\n  pure_seq_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton,\n  map_pure := \u03bb \u03b1 \u03b2 f a, image_singleton _ _,\n  seq_pure := \u03bb \u03b1 \u03b2 s a, sup_singleton'' _ _,\n  seq_assoc := \u03bb \u03b1 \u03b2 \u03b3 s t u, begin\n    ext a,\n    simp_rw [seq_def, fmap_def],\n    simp only [exists_prop, mem_sup, mem_image],\n    split,\n    { rintro \u27e8g, hg, b, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9,\n      exact \u27e8g \u2218 f, \u27e8comp g, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9 },\n    { rintro \u27e8c, \u27e8_, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9,\n      exact \u27e8g, hg, f a, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9 }\n  end,\n  .. finset.is_lawful_functor }\n\ninstance : is_comm_applicative finset :=\n{ commutative_prod := \u03bb \u03b1 \u03b2 s t, begin\n    simp_rw [seq_def, fmap_def, sup_image, sup_eq_bUnion],\n    change s.bUnion (\u03bb a, t.image $ \u03bb b, (a, b)) = t.bUnion (\u03bb b, s.image $ \u03bb a, (a, b)),\n    transitivity s \u00d7\u02e2 t;\n      [rw product_eq_bUnion, rw product_eq_bUnion_right]; congr; ext; simp_rw mem_image,\n  end,\n  .. finset.is_lawful_applicative }\n\nend applicative\n\n/-! ### Monad -/\n\nsection monad\nvariables [\u03a0 P, decidable P]\n\ninstance : monad finset :=\n{ bind := \u03bb \u03b1 \u03b2, @sup _ _ _ _,\n  .. finset.applicative }\n\n@[simp] lemma bind_def {\u03b1 \u03b2} : (>>=) = @sup (finset \u03b1) \u03b2 _ _ := rfl\n\ninstance : is_lawful_monad finset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton'' _ _,\n  bind_map_eq_seq := \u03bb \u03b1 \u03b2 t s, rfl,\n  pure_bind := \u03bb \u03b1 \u03b2 t s, sup_singleton,\n  bind_assoc :=  \u03bb \u03b1 \u03b2 \u03b3 s f g, by { convert sup_bUnion _ _, exact sup_eq_bUnion _ _ },\n  .. finset.is_lawful_applicative }\n\nend monad\n\n/-! ### Alternative functor -/\n\nsection alternative\nvariables [\u03a0 P, decidable P]\n\ninstance : alternative finset :=\n{ orelse := \u03bb \u03b1, (\u222a),\n  failure := \u03bb \u03b1, \u2205,\n  .. finset.applicative }\n\nend alternative\n\n/-! ### Traversable functor -/\n\nsection traversable\nvariables {\u03b1 \u03b2 \u03b3 : Type u} {F G : Type u \u2192 Type u} [applicative F] [applicative G]\n  [is_comm_applicative F] [is_comm_applicative G]\n\n/-- Traverse function for `finset`. -/\ndef traverse [decidable_eq \u03b2] (f : \u03b1 \u2192 F \u03b2) (s :  finset \u03b1) : F (finset \u03b2) :=\nmultiset.to_finset <$> multiset.traverse f s.1\n\n@[simp] lemma id_traverse [decidable_eq \u03b1] (s : finset \u03b1) : traverse id.mk s = s :=\nby { rw [traverse, multiset.id_traverse], exact s.val_to_finset }\n\nopen_locale classical\n\n@[simp] lemma map_comp_coe (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 multiset.to_finset = multiset.to_finset \u2218 functor.map h :=\nfunext $ \u03bb s, image_to_finset\n\nlemma map_traverse (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) (s : finset \u03b1) :\n  functor.map h <$> traverse g s = traverse (functor.map h \u2218 g) s :=\nbegin\n  unfold traverse,\n  simp only [map_comp_coe] with functor_norm,\n  rw [is_lawful_functor.comp_map, multiset.map_traverse],\nend\n\nend traversable\nend finset\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/finset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.37344121297662697}}
{"text": "instance (\u03d5 : \u03b1 \u2192 Prop) : CoeSort (Subtype \u03d5) \u03b1 where\n  coe := fun x => x.val\n\nexample (\u03d5 : \u03b1 \u2192 Prop) (xs : Subtype \u03d5) (x : xs) : True := trivial\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/621.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3734412129766269}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.const\nimport category_theory.discrete_category\n\n/-!\n# The category `discrete punit`\n\nWe define `star : C \u2964 discrete punit` sending everything to `punit.star`,\nshow that any two functors to `discrete punit` are naturally isomorphic,\nand construct the equivalence `(discrete punit \u2964 C) \u224c C`.\n-/\n\nuniverses v u -- morphism levels before object levels. See note [category_theory universes].\n\nnamespace category_theory\n\nnamespace functor\nvariables (C : Type u) [category.{v} C]\n\n/-- The constant functor sending everything to `punit.star`. -/\n@[simps]\ndef star : C \u2964 discrete punit :=\n(functor.const _).obj punit.star\n\nvariable {C}\n/-- Any two functors to `discrete punit` are isomorphic. -/\n@[simps]\ndef punit_ext (F G : C \u2964 discrete punit) : F \u2245 G :=\nnat_iso.of_components (\u03bb _, eq_to_iso dec_trivial) (\u03bb _ _ _, dec_trivial)\n\n/--\nAny two functors to `discrete punit` are *equal*.\nYou probably want to use `punit_ext` instead of this.\n-/\nlemma punit_ext' (F G : C \u2964 discrete punit) : F = G :=\nfunctor.ext (\u03bb _, dec_trivial) (\u03bb _ _ _, dec_trivial)\n\n/-- The functor from `discrete punit` sending everything to the given object. -/\nabbreviation from_punit (X : C) : discrete punit.{v+1} \u2964 C :=\n(functor.const _).obj X\n\n/-- Functors from `discrete punit` are equivalent to the category itself. -/\n@[simps]\ndef equiv : (discrete punit \u2964 C) \u224c C :=\n{ functor :=\n  { obj := \u03bb F, F.obj punit.star,\n    map := \u03bb F G \u03b8, \u03b8.app punit.star },\n  inverse := functor.const _,\n  unit_iso :=\n  begin\n    apply nat_iso.of_components _ _,\n    intro X,\n    apply discrete.nat_iso,\n    rintro \u27e8\u27e9,\n    apply iso.refl _,\n    intros,\n    ext \u27e8\u27e9,\n    simp,\n  end,\n  counit_iso :=\n  begin\n    refine nat_iso.of_components iso.refl _,\n    intros X Y f,\n    dsimp, simp,  -- See note [dsimp, simp].\n  end }\n\nend functor\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/punit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3734412129766269}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.equiv.basic\n\n/-!\n# A type for VM-erased data\n\nThis file defines a type `erased \u03b1` which is classically isomorphic to `\u03b1`,\nbut erased in the VM. That is, at runtime every value of `erased \u03b1` is\nrepresented as `0`, just like types and proofs.\n-/\n\nuniverses u\n\n/-- `erased \u03b1` is the same as `\u03b1`, except that the elements\n  of `erased \u03b1` are erased in the VM in the same way as types\n  and proofs. This can be used to track data without storing it\n  literally. -/\ndef erased (\u03b1 : Sort u) : Sort (max 1 u) :=\n\u03a3' s : \u03b1 \u2192 Prop, \u2203 a, (\u03bb b, a = b) = s\n\nnamespace erased\n\n/-- Erase a value. -/\n@[inline] def mk {\u03b1} (a : \u03b1) : erased \u03b1 := \u27e8\u03bb b, a = b, a, rfl\u27e9\n\n/-- Extracts the erased value, noncomputably. -/\nnoncomputable def out {\u03b1} : erased \u03b1 \u2192 \u03b1\n| \u27e8s, h\u27e9 := classical.some h\n\n/--\nExtracts the erased value, if it is a type.\n\nNote: `(mk a).out_type` is not definitionally equal to `a`.\n-/\n@[reducible] def out_type (a : erased (Sort u)) : Sort u := out a\n\n/-- Extracts the erased value, if it is a proof. -/\ntheorem out_proof {p : Prop} (a : erased p) : p := out a\n\n@[simp] theorem out_mk {\u03b1} (a : \u03b1) : (mk a).out = a :=\nbegin\n  let h, show classical.some h = a,\n  have := classical.some_spec h,\n  exact cast (congr_fun this a).symm rfl\nend\n\n@[simp] theorem mk_out {\u03b1} : \u2200 (a : erased \u03b1), mk (out a) = a\n| \u27e8s, h\u27e9 := by simp [mk]; congr; exact classical.some_spec h\n\n@[ext] lemma out_inj {\u03b1} (a b : erased \u03b1) (h : a.out = b.out) : a = b :=\nby simpa using congr_arg mk h\n\n/-- Equivalence between `erased \u03b1` and `\u03b1`. -/\nnoncomputable def equiv (\u03b1) : erased \u03b1 \u2243 \u03b1 :=\n\u27e8out, mk, mk_out, out_mk\u27e9\n\ninstance (\u03b1 : Type u) : has_repr (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\ninstance (\u03b1 : Type u) : has_to_string (erased \u03b1) := \u27e8\u03bb _, \"erased\"\u27e9\nmeta instance (\u03b1 : Type u) : has_to_format (erased \u03b1) := \u27e8\u03bb _, (\"erased\" : format)\u27e9\n\n/-- Computably produce an erased value from a proof of nonemptiness. -/\ndef choice {\u03b1} (h : nonempty \u03b1) : erased \u03b1 := mk (classical.choice h)\n\n@[simp] theorem nonempty_iff {\u03b1} : nonempty (erased \u03b1) \u2194 nonempty \u03b1 :=\n\u27e8\u03bb \u27e8a\u27e9, \u27e8a.out\u27e9, \u03bb \u27e8a\u27e9, \u27e8mk a\u27e9\u27e9\n\ninstance {\u03b1} [h : nonempty \u03b1] : inhabited (erased \u03b1) :=\n\u27e8choice h\u27e9\n\n/--\n`(>>=)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `monad`).\n-/\ndef bind {\u03b1 \u03b2} (a : erased \u03b1) (f : \u03b1 \u2192 erased \u03b2) : erased \u03b2 :=\n\u27e8\u03bb b, (f a.out).1 b, (f a.out).2\u27e9\n\n@[simp] theorem bind_eq_out {\u03b1 \u03b2} (a f) : @bind \u03b1 \u03b2 a f = f a.out :=\nby delta bind bind._proof_1; cases f a.out; refl\n\n/--\nCollapses two levels of erasure.\n-/\ndef join {\u03b1} (a : erased (erased \u03b1)) : erased \u03b1 := bind a id\n\n@[simp] theorem join_eq_out {\u03b1} (a) : @join \u03b1 a = a.out := bind_eq_out _ _\n\n/--\n`(<$>)` operation on `erased`.\n\nThis is a separate definition because `\u03b1` and `\u03b2` can live in different\nuniverses (the universe is fixed in `functor`).\n-/\ndef map {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (a : erased \u03b1) : erased \u03b2 :=\nbind a (mk \u2218 f)\n\n@[simp] theorem map_out {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (a : erased \u03b1) : (a.map f).out = f a.out :=\nby simp [map]\n\ninstance : monad erased := { pure := @mk, bind := @bind, map := @map }\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 erased \u03b1) = @mk _ := rfl\n@[simp] lemma bind_def {\u03b1 \u03b2} : ((>>=) : erased \u03b1 \u2192 (\u03b1 \u2192 erased \u03b2) \u2192 erased \u03b2) = @bind _ _ := rfl\n@[simp] lemma map_def {\u03b1 \u03b2} : ((<$>) : (\u03b1 \u2192 \u03b2) \u2192 erased \u03b1 \u2192 erased \u03b2) = @map _ _ := rfl\n\ninstance : is_lawful_monad erased := by refine {..}; intros; ext; simp\n\nend erased\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/erased.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3734412046545882}}
{"text": "/-\nCopyright (c) 2021 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n\n! This file was ported from Lean 3 source module data.multiset.locally_finite\n! leanprover-community/mathlib commit 59694bd07f0a39c5beccba34bd9f413a160782bf\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.Finset.LocallyFinite\n\n/-!\n# Intervals as multisets\n\nThis file provides basic results about all the `Multiset.Ixx`, which are defined in\n`Order.LocallyFinite`.\n\nNote that intervals of multisets themselves (`Multiset.LocallyFiniteOrder`) are defined elsewhere.\n-/\n\n\nvariable {\u03b1 : Type _}\n\nnamespace Multiset\n\nsection Preorder\n\nvariable [Preorder \u03b1] [LocallyFiniteOrder \u03b1] {a b c : \u03b1}\n\ntheorem nodup_Icc : (Icc a b).Nodup :=\n  Finset.nodup _\n#align multiset.nodup_Icc Multiset.nodup_Icc\n\ntheorem nodup_Ico : (Ico a b).Nodup :=\n  Finset.nodup _\n#align multiset.nodup_Ico Multiset.nodup_Ico\n\ntheorem nodup_Ioc : (Ioc a b).Nodup :=\n  Finset.nodup _\n#align multiset.nodup_Ioc Multiset.nodup_Ioc\n\ntheorem nodup_Ioo : (Ioo a b).Nodup :=\n  Finset.nodup _\n#align multiset.nodup_Ioo Multiset.nodup_Ioo\n\n@[simp]\ntheorem Icc_eq_zero_iff : Icc a b = 0 \u2194 \u00aca \u2264 b := by\n  rw [Icc, Finset.val_eq_zero, Finset.Icc_eq_empty_iff]\n#align multiset.Icc_eq_zero_iff Multiset.Icc_eq_zero_iff\n\n@[simp]\ntheorem Ico_eq_zero_iff : Ico a b = 0 \u2194 \u00aca < b := by\n  rw [Ico, Finset.val_eq_zero, Finset.Ico_eq_empty_iff]\n#align multiset.Ico_eq_zero_iff Multiset.Ico_eq_zero_iff\n\n@[simp]\ntheorem Ioc_eq_zero_iff : Ioc a b = 0 \u2194 \u00aca < b := by\n  rw [Ioc, Finset.val_eq_zero, Finset.Ioc_eq_empty_iff]\n#align multiset.Ioc_eq_zero_iff Multiset.Ioc_eq_zero_iff\n\n@[simp]\ntheorem Ioo_eq_zero_iff [DenselyOrdered \u03b1] : Ioo a b = 0 \u2194 \u00aca < b := by\n  rw [Ioo, Finset.val_eq_zero, Finset.Ioo_eq_empty_iff]\n#align multiset.Ioo_eq_zero_iff Multiset.Ioo_eq_zero_iff\n\nalias Icc_eq_zero_iff \u2194 _ Icc_eq_zero\n#align multiset.Icc_eq_zero Multiset.Icc_eq_zero\n\nalias Ico_eq_zero_iff \u2194 _ Ico_eq_zero\n#align multiset.Ico_eq_zero Multiset.Ico_eq_zero\n\nalias Ioc_eq_zero_iff \u2194 _ Ioc_eq_zero\n#align multiset.Ioc_eq_zero Multiset.Ioc_eq_zero\n\n@[simp]\ntheorem Ioo_eq_zero (h : \u00aca < b) : Ioo a b = 0 :=\n  eq_zero_iff_forall_not_mem.2 fun _x hx => h ((mem_Ioo.1 hx).1.trans (mem_Ioo.1 hx).2)\n#align multiset.Ioo_eq_zero Multiset.Ioo_eq_zero\n\n@[simp]\ntheorem Icc_eq_zero_of_lt (h : b < a) : Icc a b = 0 :=\n  Icc_eq_zero h.not_le\n#align multiset.Icc_eq_zero_of_lt Multiset.Icc_eq_zero_of_lt\n\n@[simp]\ntheorem Ico_eq_zero_of_le (h : b \u2264 a) : Ico a b = 0 :=\n  Ico_eq_zero h.not_lt\n#align multiset.Ico_eq_zero_of_le Multiset.Ico_eq_zero_of_le\n\n@[simp]\ntheorem Ioc_eq_zero_of_le (h : b \u2264 a) : Ioc a b = 0 :=\n  Ioc_eq_zero h.not_lt\n#align multiset.Ioc_eq_zero_of_le Multiset.Ioc_eq_zero_of_le\n\n@[simp]\ntheorem Ioo_eq_zero_of_le (h : b \u2264 a) : Ioo a b = 0 :=\n  Ioo_eq_zero h.not_lt\n#align multiset.Ioo_eq_zero_of_le Multiset.Ioo_eq_zero_of_le\n\nvariable (a)\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem Ico_self : Ico a a = 0 := by rw [Ico, Finset.Ico_self, Finset.empty_val]\n#align multiset.Ico_self Multiset.Ico_self\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem Ioc_self : Ioc a a = 0 := by rw [Ioc, Finset.Ioc_self, Finset.empty_val]\n#align multiset.Ioc_self Multiset.Ioc_self\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem Ioo_self : Ioo a a = 0 := by rw [Ioo, Finset.Ioo_self, Finset.empty_val]\n#align multiset.Ioo_self Multiset.Ioo_self\n\nvariable {a}\n\ntheorem left_mem_Icc : a \u2208 Icc a b \u2194 a \u2264 b :=\n  Finset.left_mem_Icc\n#align multiset.left_mem_Icc Multiset.left_mem_Icc\n\ntheorem left_mem_Ico : a \u2208 Ico a b \u2194 a < b :=\n  Finset.left_mem_Ico\n#align multiset.left_mem_Ico Multiset.left_mem_Ico\n\ntheorem right_mem_Icc : b \u2208 Icc a b \u2194 a \u2264 b :=\n  Finset.right_mem_Icc\n#align multiset.right_mem_Icc Multiset.right_mem_Icc\n\ntheorem right_mem_Ioc : b \u2208 Ioc a b \u2194 a < b :=\n  Finset.right_mem_Ioc\n#align multiset.right_mem_Ioc Multiset.right_mem_Ioc\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem left_not_mem_Ioc : a \u2209 Ioc a b :=\n  Finset.left_not_mem_Ioc\n#align multiset.left_not_mem_Ioc Multiset.left_not_mem_Ioc\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem left_not_mem_Ioo : a \u2209 Ioo a b :=\n  Finset.left_not_mem_Ioo\n#align multiset.left_not_mem_Ioo Multiset.left_not_mem_Ioo\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem right_not_mem_Ico : b \u2209 Ico a b :=\n  Finset.right_not_mem_Ico\n#align multiset.right_not_mem_Ico Multiset.right_not_mem_Ico\n\n-- Porting note: simp can prove this -- @[simp]\ntheorem right_not_mem_Ioo : b \u2209 Ioo a b :=\n  Finset.right_not_mem_Ioo\n#align multiset.right_not_mem_Ioo Multiset.right_not_mem_Ioo\n\ntheorem Ico_filter_lt_of_le_left [DecidablePred (\u00b7 < c)] (hca : c \u2264 a) :\n    ((Ico a b).filter fun x => x < c) = \u2205 := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_le_left hca]\n  rfl\n#align multiset.Ico_filter_lt_of_le_left Multiset.Ico_filter_lt_of_le_left\n\ntheorem Ico_filter_lt_of_right_le [DecidablePred (\u00b7 < c)] (hbc : b \u2264 c) :\n    ((Ico a b).filter fun x => x < c) = Ico a b := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_right_le hbc]\n#align multiset.Ico_filter_lt_of_right_le Multiset.Ico_filter_lt_of_right_le\n\ntheorem Ico_filter_lt_of_le_right [DecidablePred (\u00b7 < c)] (hcb : c \u2264 b) :\n    ((Ico a b).filter fun x => x < c) = Ico a c := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt_of_le_right hcb]\n  rfl\n#align multiset.Ico_filter_lt_of_le_right Multiset.Ico_filter_lt_of_le_right\n\ntheorem Ico_filter_le_of_le_left [DecidablePred ((\u00b7 \u2264 \u00b7) c)] (hca : c \u2264 a) :\n    ((Ico a b).filter fun x => c \u2264 x) = Ico a b := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_le_left hca]\n#align multiset.Ico_filter_le_of_le_left Multiset.Ico_filter_le_of_le_left\n\ntheorem Ico_filter_le_of_right_le [DecidablePred ((\u00b7 \u2264 \u00b7) b)] :\n    ((Ico a b).filter fun x => b \u2264 x) = \u2205 := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_right_le]\n  rfl\n#align multiset.Ico_filter_le_of_right_le Multiset.Ico_filter_le_of_right_le\n\ntheorem Ico_filter_le_of_left_le [DecidablePred ((\u00b7 \u2264 \u00b7) c)] (hac : a \u2264 c) :\n    ((Ico a b).filter fun x => c \u2264 x) = Ico c b := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_of_left_le hac]\n  rfl\n#align multiset.Ico_filter_le_of_left_le Multiset.Ico_filter_le_of_left_le\n\nend Preorder\n\nsection PartialOrder\n\nvariable [PartialOrder \u03b1] [LocallyFiniteOrder \u03b1] {a b : \u03b1}\n\n@[simp]\ntheorem Icc_self (a : \u03b1) : Icc a a = {a} := by rw [Icc, Finset.Icc_self, Finset.singleton_val]\n#align multiset.Icc_self Multiset.Icc_self\n\ntheorem Ico_cons_right (h : a \u2264 b) : b ::\u2098 Ico a b = Icc a b := by\n  classical\n    rw [Ico, \u2190 Finset.insert_val_of_not_mem right_not_mem_Ico, Finset.Ico_insert_right h]\n    rfl\n#align multiset.Ico_cons_right Multiset.Ico_cons_right\n\ntheorem Ioo_cons_left (h : a < b) : a ::\u2098 Ioo a b = Ico a b := by\n  classical\n    rw [Ioo, \u2190 Finset.insert_val_of_not_mem left_not_mem_Ioo, Finset.Ioo_insert_left h]\n    rfl\n#align multiset.Ioo_cons_left Multiset.Ioo_cons_left\n\ntheorem Ico_disjoint_Ico {a b c d : \u03b1} (h : b \u2264 c) : (Ico a b).Disjoint (Ico c d) :=\n  fun x hab hbc => by\n  rw [mem_Ico] at hab hbc\n  exact hab.2.not_le (h.trans hbc.1)\n#align multiset.Ico_disjoint_Ico Multiset.Ico_disjoint_Ico\n\n@[simp]\ntheorem Ico_inter_Ico_of_le [DecidableEq \u03b1] {a b c d : \u03b1} (h : b \u2264 c) : Ico a b \u2229 Ico c d = 0 :=\n  Multiset.inter_eq_zero_iff_disjoint.2 <| Ico_disjoint_Ico h\n#align multiset.Ico_inter_Ico_of_le Multiset.Ico_inter_Ico_of_le\n\ntheorem Ico_filter_le_left {a b : \u03b1} [DecidablePred (\u00b7 \u2264 a)] (hab : a < b) :\n    ((Ico a b).filter fun x => x \u2264 a) = {a} := by\n  rw [Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le_left hab]\n  rfl\n#align multiset.Ico_filter_le_left Multiset.Ico_filter_le_left\n\ntheorem card_Ico_eq_card_Icc_sub_one (a b : \u03b1) : card (Ico a b) = card (Icc a b) - 1 :=\n  Finset.card_Ico_eq_card_Icc_sub_one _ _\n#align multiset.card_Ico_eq_card_Icc_sub_one Multiset.card_Ico_eq_card_Icc_sub_one\n\ntheorem card_Ioc_eq_card_Icc_sub_one (a b : \u03b1) : card (Ioc a b) = card (Icc a b) - 1 :=\n  Finset.card_Ioc_eq_card_Icc_sub_one _ _\n#align multiset.card_Ioc_eq_card_Icc_sub_one Multiset.card_Ioc_eq_card_Icc_sub_one\n\ntheorem card_Ioo_eq_card_Ico_sub_one (a b : \u03b1) : card (Ioo a b) = card (Ico a b) - 1 :=\n  Finset.card_Ioo_eq_card_Ico_sub_one _ _\n#align multiset.card_Ioo_eq_card_Ico_sub_one Multiset.card_Ioo_eq_card_Ico_sub_one\n\ntheorem card_Ioo_eq_card_Icc_sub_two (a b : \u03b1) : card (Ioo a b) = card (Icc a b) - 2 :=\n  Finset.card_Ioo_eq_card_Icc_sub_two _ _\n#align multiset.card_Ioo_eq_card_Icc_sub_two Multiset.card_Ioo_eq_card_Icc_sub_two\n\nend PartialOrder\n\nsection LinearOrder\n\nvariable [LinearOrder \u03b1] [LocallyFiniteOrder \u03b1] {a b c d : \u03b1}\n\ntheorem Ico_subset_Ico_iff {a\u2081 b\u2081 a\u2082 b\u2082 : \u03b1} (h : a\u2081 < b\u2081) :\n    Ico a\u2081 b\u2081 \u2286 Ico a\u2082 b\u2082 \u2194 a\u2082 \u2264 a\u2081 \u2227 b\u2081 \u2264 b\u2082 :=\n  Finset.Ico_subset_Ico_iff h\n#align multiset.Ico_subset_Ico_iff Multiset.Ico_subset_Ico_iff\n\ntheorem Ico_add_Ico_eq_Ico {a b c : \u03b1} (hab : a \u2264 b) (hbc : b \u2264 c) : Ico a b + Ico b c = Ico a c :=\n  by\n  rw [add_eq_union_iff_disjoint.2 (Ico_disjoint_Ico le_rfl), Ico, Ico, Ico, \u2190 Finset.union_val,\n    Finset.Ico_union_Ico_eq_Ico hab hbc]\n#align multiset.Ico_add_Ico_eq_Ico Multiset.Ico_add_Ico_eq_Ico\n\ntheorem Ico_inter_Ico : Ico a b \u2229 Ico c d = Ico (max a c) (min b d) := by\n  rw [Ico, Ico, Ico, \u2190 Finset.inter_val, Finset.Ico_inter_Ico]\n#align multiset.Ico_inter_Ico Multiset.Ico_inter_Ico\n\n@[simp]\ntheorem Ico_filter_lt (a b c : \u03b1) : ((Ico a b).filter fun x => x < c) = Ico a (min b c) := by\n  rw [Ico, Ico, \u2190 Finset.filter_val, Finset.Ico_filter_lt]\n#align multiset.Ico_filter_lt Multiset.Ico_filter_lt\n\n@[simp]\ntheorem Ico_filter_le (a b c : \u03b1) : ((Ico a b).filter fun x => c \u2264 x) = Ico (max a c) b := by\n  rw [Ico, Ico, \u2190 Finset.filter_val, Finset.Ico_filter_le]\n#align multiset.Ico_filter_le Multiset.Ico_filter_le\n\n@[simp]\ntheorem Ico_sub_Ico_left (a b c : \u03b1) : Ico a b - Ico a c = Ico (max a c) b := by\n  rw [Ico, Ico, Ico, \u2190 Finset.sdiff_val, Finset.Ico_diff_Ico_left]\n#align multiset.Ico_sub_Ico_left Multiset.Ico_sub_Ico_left\n\n@[simp]\ntheorem Ico_sub_Ico_right (a b c : \u03b1) : Ico a b - Ico c b = Ico a (min b c) := by\n  rw [Ico, Ico, Ico, \u2190 Finset.sdiff_val, Finset.Ico_diff_Ico_right]\n#align multiset.Ico_sub_Ico_right Multiset.Ico_sub_Ico_right\n\nend LinearOrder\n\nsection OrderedCancelAddCommMonoid\n\nvariable [OrderedCancelAddCommMonoid \u03b1] [ExistsAddOfLE \u03b1] [LocallyFiniteOrder \u03b1]\n\ntheorem map_add_left_Icc (a b c : \u03b1) : (Icc a b).map ((\u00b7 + \u00b7) c) = Icc (c + a) (c + b) := by\n  classical rw [Icc, Icc, \u2190 Finset.image_add_left_Icc, Finset.image_val,\n      ((Finset.nodup _).map <| add_right_injective c).dedup]\n#align multiset.map_add_left_Icc Multiset.map_add_left_Icc\n\ntheorem map_add_left_Ico (a b c : \u03b1) : (Ico a b).map ((\u00b7 + \u00b7) c) = Ico (c + a) (c + b) := by\n  classical rw [Ico, Ico, \u2190 Finset.image_add_left_Ico, Finset.image_val,\n      ((Finset.nodup _).map <| add_right_injective c).dedup]\n#align multiset.map_add_left_Ico Multiset.map_add_left_Ico\n\ntheorem map_add_left_Ioc (a b c : \u03b1) : (Ioc a b).map ((\u00b7 + \u00b7) c) = Ioc (c + a) (c + b) := by\n  classical rw [Ioc, Ioc, \u2190 Finset.image_add_left_Ioc, Finset.image_val,\n      ((Finset.nodup _).map <| add_right_injective c).dedup]\n#align multiset.map_add_left_Ioc Multiset.map_add_left_Ioc\n\ntheorem map_add_left_Ioo (a b c : \u03b1) : (Ioo a b).map ((\u00b7 + \u00b7) c) = Ioo (c + a) (c + b) := by\n  classical rw [Ioo, Ioo, \u2190 Finset.image_add_left_Ioo, Finset.image_val,\n      ((Finset.nodup _).map <| add_right_injective c).dedup]\n#align multiset.map_add_left_Ioo Multiset.map_add_left_Ioo\n\ntheorem map_add_right_Icc (a b c : \u03b1) : ((Icc a b).map fun x => x + c) = Icc (a + c) (b + c) := by\n  simp_rw [add_comm _ c]\n  exact map_add_left_Icc _ _ _\n#align multiset.map_add_right_Icc Multiset.map_add_right_Icc\n\ntheorem map_add_right_Ico (a b c : \u03b1) : ((Ico a b).map fun x => x + c) = Ico (a + c) (b + c) := by\n  simp_rw [add_comm _ c]\n  exact map_add_left_Ico _ _ _\n#align multiset.map_add_right_Ico Multiset.map_add_right_Ico\n\ntheorem map_add_right_Ioc (a b c : \u03b1) : ((Ioc a b).map fun x => x + c) = Ioc (a + c) (b + c) := by\n  simp_rw [add_comm _ c]\n  exact map_add_left_Ioc _ _ _\n#align multiset.map_add_right_Ioc Multiset.map_add_right_Ioc\n\ntheorem map_add_right_Ioo (a b c : \u03b1) : ((Ioo a b).map fun x => x + c) = Ioo (a + c) (b + c) := by\n  simp_rw [add_comm _ c]\n  exact map_add_left_Ioo _ _ _\n#align multiset.map_add_right_Ioo Multiset.map_add_right_Ioo\n\nend OrderedCancelAddCommMonoid\n\nend Multiset\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Data/Multiset/LocallyFinite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177488, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.3734412039937006}}
{"text": "import for_mathlib.commsq\nimport for_mathlib.snake_lemma_naturality2\nimport for_mathlib.AddCommGroup.pt\nimport for_mathlib.bicartesian4\nimport for_mathlib.abelian_sheaves.functor_category\n\nimport system_of_complexes.shift_sub_id\nimport pseudo_normed_group.system_of_complexes2\n\nnoncomputable theory\n\nuniverses u\n\nopen_locale nnreal\n\nopen opposite category_theory category_theory.limits category_theory.preadditive\n\nsection step1\n\nvariables {A B C : \u211d\u22650\u1d52\u1d56 \u2964 Ab.{u}} (f : A \u27f6 B)\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650)\n\ndef shift_sub_id.commsq (h\u03b9 : monotone \u03b9) :\n  commsq (shift_sub_id A \u03b9 h\u03b9)\n    (pi.map $ \u03bb _, f.app _) (pi.map $ \u03bb _, f.app _)\n         (shift_sub_id B \u03b9 h\u03b9) :=\ncommsq.of_eq\nbegin\n  simp only [shift_sub_id, sub_comp, comp_sub, category.id_comp, category.comp_id, shift_sub_id.shift],\n  congr' 1,\n  apply limit.hom_ext,\n  intro j,\n  simp only [limit.lift_map, limit.lift_\u03c0, cones.postcompose_obj_\u03c0, nat_trans.comp_app,\n    fan.mk_\u03c0_app, discrete.nat_trans_app, category.assoc, nat_trans.naturality, lim_map_\u03c0_assoc],\nend\n\nend step1\n\nsection step2\n\nvariables {A B C : ulift.{u} \u2115 \u2192 Ab.{u}} (f : \u03a0 k, A k \u27f6 B k) (g : \u03a0 k, B k \u27f6 C k)\n\nlemma pi_map_exact (H : \u2200 k, exact (f k) (g k)) :\n  exact (pi.map f) (pi.map g) :=\nbegin\n  simp only [AddCommGroup.exact_iff'] at H \u22a2,\n  split,\n  { apply limit.hom_ext, intro j,\n    simp only [category.assoc, lim_map_\u03c0, discrete.nat_trans_app, lim_map_\u03c0_assoc,\n      zero_comp, (H j.1).1, comp_zero], },\n  intros x hx,\n  rw [add_monoid_hom.mem_ker, Ab.apply_eq_zero] at hx,\n  have : \u2200 k, (Ab.pt (limit.\u03c0 (discrete.functor (\u03bb k, B k)) k x)) \u226b g k.1 = 0,\n  { intro k,\n    suffices : Ab.pt x \u226b pi.map g \u226b pi.\u03c0 _ k.1 = 0,\n    { cases k,\n      simpa [lim_map_\u03c0, discrete.nat_trans_app, \u2190 category.assoc, hx, Ab.pt_comp] },\n    rw [\u2190 category.assoc, hx, zero_comp] },\n  simp only [\u2190 Ab.apply_eq_zero] at this,\n  replace := \u03bb k, (H k).2 (this \u27e8k\u27e9),\n  choose y hy using this,\n  refine \u27e8pi.lift (\u03bb k, Ab.pt (y k)) \u27e81\u27e9, _\u27e9,\n  rw [\u2190 category_theory.comp_apply, \u2190 Ab.pt_apply' x],\n  congr' 1,\n  apply limit.hom_ext,\n  rintro \u27e8j\u27e9,\n  simp only [limit.lift_map, limit.lift_\u03c0, cones.postcompose_obj_\u03c0, nat_trans.comp_app,\n    fan.mk_\u03c0_app, discrete.nat_trans_app, Ab.pt_comp, hy],\nend\n\nend step2\n\nsection step3\n\nvariables {A B C : \u211d\u22650\u1d52\u1d56 \u2964 cochain_complex Ab.{u} \u2115} (f : A \u27f6 B) (g : B \u27f6 C)\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (n : \u2115)\n\ndef piH_hom :\n  (\u220f (\u03bb x, (A.obj (op $ \u03b9 x)).homology n)) \u27f6 (\u220f (\u03bb x, (B.obj (op $ \u03b9 x)).homology n)) :=\npi.map $ \u03bb k, (homology_functor _ _ _).map $ f.app _\n\ndef shift_sub_id.\u03b4 (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  C \u22d9 homology_functor _ _ n \u27f6 A \u22d9 homology_functor _ _ (n+1) :=\n{ app := \u03bb c, homological_complex.\u03b4 (f.app _) (g.app _) (H _) _ _ rfl,\n  naturality' := \u03bb c\u2081 c\u2082 h, by { symmetry, apply homological_complex.\u03b4_natural } }\n\ndef pi\u03b4 (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  (\u220f (\u03bb x, (C.obj (op $ \u03b9 x)).homology n)) \u27f6 (\u220f (\u03bb x, (A.obj (op $ \u03b9 x)).homology (n+1))) :=\npi.map $ \u03bb k, (shift_sub_id.\u03b4 _ _ _ H).app _\n\nlemma piH_les (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  exact_seq Ab.{u} [piH_hom f \u03b9 n, piH_hom g \u03b9 n, pi\u03b4 f g \u03b9 n H] :=\nbegin\n  apply exact.cons,\n  { apply pi_map_exact, intro k,\n    have := homological_complex.six_term_exact_seq _ _ (H (op $ \u03b9 k)) n (n+1) rfl,\n    exact this.pair, },\n  apply exact.exact_seq,\n  { apply pi_map_exact, intro k,\n    have := homological_complex.six_term_exact_seq _ _ (H (op $ \u03b9 k)) n (n+1) rfl,\n    exact (this.drop 1).pair, },\nend\n\nend step3\n\nsection step4\n\nvariables {A B C : system_of_complexes.{u}} (f : A.to_Ab \u27f6 B.to_Ab) (g : B.to_Ab \u27f6 C.to_Ab)\nvariables (n : \u2115) (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (h\u03b9 : monotone \u03b9)\n\nlemma shift_sub_id.bicartesian\n  (HA\u2081 : (shift_sub_id.shift (A.to_AbH n) \u03b9 h\u03b9) = 0)\n  (HA\u2082 : (shift_sub_id.shift (A.to_AbH (n+1)) \u03b9 h\u03b9) = 0)\n  (H : \u2200 c n, short_exact ((f.app c).f n) ((g.app c).f n)) :\n  (@shift_sub_id.commsq (B.to_AbH n) (C.to_AbH n)\n    (whisker_right g _) \u03b9 h\u03b9).bicartesian :=\nbegin\n  rw \u2190 commsq.bicartesian.symm_iff,\n  let S1 := ((@shift_sub_id.commsq (A.to_AbH n) (B.to_AbH n) (whisker_right f _) \u03b9 h\u03b9)).symm,\n  let S2 := ((@shift_sub_id.commsq (B.to_AbH n) (C.to_AbH n) (whisker_right g _) \u03b9 h\u03b9)).symm,\n  let S3 := ((@shift_sub_id.commsq (C.to_AbH n) (A.to_AbH (n+1)) (shift_sub_id.\u03b4 _ _ _ H) \u03b9 h\u03b9)).symm,\n  apply bicartesian_of_id_of_end_of_end_of_id (piH_les _ _ _ _ _) S1 S2 S3; clear S1 S2 S3,\n  { rw [shift_sub_id, HA\u2081, zero_sub], refl },\n  { rw [shift_sub_id, HA\u2082, zero_sub], refl },\nend\n\nend step4\n\nsection step5\n\nvariables {A B A' B' : \u211d\u22650\u1d52\u1d56 \u2964 Ab.{u}} (f : A \u27f6 B) (f' : A' \u27f6 B') (eA : A \u2245 A') (eB : B \u2245 B')\nvariables (\u03b9 : ulift.{u} \u2115 \u2192 \u211d\u22650) (h\u03b9 : monotone \u03b9)\n\nlemma shift_sub_id.bicartesian_iso (w : f \u226b eB.hom = eA.hom \u226b f')\n  (sq : (shift_sub_id.commsq f \u03b9 h\u03b9).bicartesian) :\n  (shift_sub_id.commsq f' \u03b9 h\u03b9).bicartesian :=\nbegin\n  let H : _ := _,\n  apply commsq.bicartesian.of_iso _ _ _ _ _ H H _ sq,\n  { refine limits.lim.map_iso (discrete.nat_iso $ \u03bb k, eA.app _), },\n  { refine limits.lim.map_iso (discrete.nat_iso $ \u03bb k, eB.app _), },\n  { apply shift_sub_id.commsq },\n  { apply shift_sub_id.commsq },\n  { apply commsq.of_eq, delta pi.map,\n    simp only [functor.map_iso_hom, \u2190 lim_map_eq_lim_map, \u2190 category_theory.functor.map_comp],\n    apply limit.hom_ext,\n    simp only [lim_map_eq_lim_map, lim_map_\u03c0, nat_trans.comp_app, discrete.nat_trans_app,\n      discrete.nat_iso_hom_app, iso.app_hom],\n    intro, simp only [\u2190 nat_trans.comp_app, w], }\nend\n\nend step5\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/Lbar/squares.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185944046238982, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3733251150375848}}
{"text": "import for_mathlib.category_theory.triangulated.shift_compatibility\nimport for_mathlib.category_theory.shift_misc\nimport algebra.group.commute\nimport tactic.abel\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category\n\nvariables (C : Type*) {A G : Type*} [category C] [add_monoid A] [add_comm_group G]\n  [has_shift C A] [has_shift C G]\n\ndef shift_functor_add\u2083' (a\u2081 a\u2082 a\u2083 : A) (b : A) (h : b = a\u2081+a\u2082+a\u2083) :\n  shift_functor C b \u2245 shift_functor C a\u2081 \u22d9\n    shift_functor C a\u2082 \u22d9 shift_functor C a\u2083 :=\nshift_functor_add' C (a\u2081+a\u2082) a\u2083 b h \u226a\u226b iso_whisker_right (shift_functor_add C a\u2081 a\u2082) _ \u226a\u226b functor.associator _ _ _\n\nvariable {C}\n\nlemma shift_functor_add\u2083'_hom_app (a\u2081 a\u2082 a\u2083 : A) (b : A) (h : b = a\u2081+a\u2082+a\u2083) (a\u2081\u2082 : A) (h\u2081\u2082 : a\u2081\u2082 = a\u2081 + a\u2082) (X : C) :\n  (shift_functor_add\u2083' C a\u2081 a\u2082 a\u2083 b h).hom.app X =\n    (shift_functor_add' C a\u2081\u2082 a\u2083 b (by rw [h\u2081\u2082, h])).hom.app X \u226b\n    ((shift_functor_add' C a\u2081 a\u2082 a\u2081\u2082 h\u2081\u2082).hom.app X)\u27e6a\u2083\u27e7' :=\nbegin\n  substs h\u2081\u2082 h,\n  dsimp only [shift_functor_add\u2083', iso.trans, iso_whisker_right,\n    functor.associator, nat_trans.comp_app, whiskering_right,\n    functor.map_iso, whisker_right],\n  simp only [comp_id, shift_functor_add'_eq_shift_functor_add],\nend\n\nlemma shift_functor_add\u2083'_inv_app (a\u2081 a\u2082 a\u2083 : A) (b : A) (h : b = a\u2081+a\u2082+a\u2083) (a\u2081\u2082 : A) (h\u2081\u2082 : a\u2081\u2082 = a\u2081 + a\u2082) (X : C) :\n  (shift_functor_add\u2083' C a\u2081 a\u2082 a\u2083 b h).inv.app X =\n    ((shift_functor_add' C a\u2081 a\u2082 a\u2081\u2082 h\u2081\u2082).inv.app X)\u27e6a\u2083\u27e7' \u226b\n    (shift_functor_add' C a\u2081\u2082 a\u2083 b (by rw [h\u2081\u2082, h])).inv.app X :=\nbegin\n  substs h\u2081\u2082 h,\n  dsimp only [shift_functor_add\u2083', iso.trans, iso_whisker_right,\n    functor.associator, nat_trans.comp_app, whiskering_right,\n    functor.map_iso, whisker_right],\n  simp only [id_comp, shift_functor_add'_eq_shift_functor_add],\nend\n\nlocal attribute [instance, reducible] endofunctor_monoidal_category\nlocal attribute [reducible] discrete.add_monoidal\n\nlemma shift_functor_add\u2083'_hom_app' (a\u2081 a\u2082 a\u2083 : A) (b : A) (h : b = a\u2081+a\u2082+a\u2083) (a\u2082\u2083 : A) (h\u2082\u2083 : a\u2082\u2083 = a\u2082 + a\u2083) (X : C) :\n  (shift_functor_add\u2083' C a\u2081 a\u2082 a\u2083 b h).hom.app X =\n  (shift_functor_add' C a\u2081 a\u2082\u2083 b (by rw [h, h\u2082\u2083, add_assoc])).hom.app _ \u226b\n    (shift_functor_add' C a\u2082 a\u2083 a\u2082\u2083 h\u2082\u2083).hom.app (X\u27e6a\u2081\u27e7) :=\nbegin\n  subst h\u2082\u2083,\n  simp only [shift_functor_add\u2083'_hom_app a\u2081 a\u2082 a\u2083 b h (a\u2081+a\u2082) rfl,\n    shift_functor_add'_eq_shift_functor_add],\n  have h : b = a\u2081+(a\u2082+a\u2083) := by rw [h, add_assoc],\n  subst h,\n  rw shift_functor_add'_eq_shift_functor_add,\n  have eq := congr_arg iso.inv (monoidal_functor.associativity_iso_eq (shift_monoidal_functor C A)\n    (discrete.mk a\u2081) (discrete.mk a\u2082) (discrete.mk a\u2083)),\n  replace eq := congr_app eq X,\n  dsimp [shift_functor_add'] at eq \u22a2,\n  simpa only [assoc, id_comp, comp_id,\n    functor.map_id, eq_to_hom_map] using eq,\nend\n\nlemma shift_functor_add\u2083'_inv_app' (a\u2081 a\u2082 a\u2083 : A) (b : A) (h : b = a\u2081+a\u2082+a\u2083) (a\u2082\u2083 : A) (h\u2082\u2083 : a\u2082\u2083 = a\u2082 + a\u2083) (X : C) :\n  (shift_functor_add\u2083' C a\u2081 a\u2082 a\u2083 b h).inv.app X =\n  (shift_functor_add' C a\u2082 a\u2083 a\u2082\u2083 h\u2082\u2083).inv.app (X\u27e6a\u2081\u27e7) \u226b\n  (shift_functor_add' C a\u2081 a\u2082\u2083 b (by rw [h, h\u2082\u2083, add_assoc])).inv.app _ :=\nbegin\n  simp only [\u2190 cancel_mono ((shift_functor_add\u2083' C a\u2081 a\u2082 a\u2083 b h).hom.app X),\n    iso.inv_hom_id_app, assoc],\n  subst h\u2082\u2083,\n  simpa only [shift_functor_add\u2083'_hom_app' _ _ _ _ h _ rfl,\n    iso.inv_hom_id_app_assoc, iso.inv_hom_id_app],\nend\n\nlemma shift_shift_neg_hom_of_shift (X : C) (a b : G) :\n  (shift_shift_neg (X\u27e6a\u27e7) b).hom = (shift_functor_add\u2083' C a b (-b) a (by simp)).inv.app X :=\nbegin\n  rw shift_functor_add\u2083'_inv_app' a b (-b) a (by abel) 0 (by abel),\n  dsimp [shift_functor_add'],\n  simp only [assoc, \u03b5_inv_app_obj, eq_to_hom_map],\n  congr' 3,\n  erw eq_to_hom_map,\nend\n\nlemma shift_compatibility_add_comm (X : C) (a b c : G) (h : a = b + c):\n  (shift_functor_add' C a (-b) c (by rw [h, add_neg_cancel_comm])).inv.app (X\u27e6b\u27e7) \u226b\n    (shift_functor_add' C b c a h).inv.app X =\n  ((shift_functor_add_comm C b a).hom.app X)\u27e6-b\u27e7' \u226b (shift_shift_neg (X\u27e6a\u27e7) b).hom :=\nbegin\n  rw \u2190 shift_functor_add\u2083'_inv_app' b a (-b) a (by abel) c,\n  rw shift_shift_neg_hom_of_shift,\n  rw shift_functor_add\u2083'_inv_app b a (-b) a (by abel) (a+b) (by abel),\n  rw shift_functor_add\u2083'_inv_app a b (-b) a (by abel) (a+b) (by abel),\n  simp only [\u2190 functor.map_comp_assoc],\n  congr' 2,\n  simp only [shift_functor_add_comm_hom_app, shift_functor_add'_eq_shift_functor_add,\n    assoc, iso.hom_inv_id_app, comp_id],\n  dsimp only [shift_functor_add'],\n  simp only [iso.trans_inv, eq_to_iso.inv, nat_trans.comp_app, eq_to_hom_app],\nend\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/category_theory/shift_compatibility_minus.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.37332241369271807}}
{"text": "import for_mathlib.algebra.homology.triangulated\nimport for_mathlib.category_theory.triangulated.homological_functor_localization\nimport for_mathlib.category_theory.shift_misc\nimport for_mathlib.category_theory.localization.composition\nimport for_mathlib.algebra.homology.cylinder\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n  category_theory.triangulated category_theory.pretriangulated\nopen_locale zero_object\n\nnamespace category_theory.pretriangulated\n\nvariables {D : Type*} [category D] [has_zero_object D] [has_shift D \u2124] [preadditive D]\n  [\u2200 (n : \u2124), (shift_functor D n).additive] [pretriangulated D]\n\nnamespace triangle\n\ndef distinguished (T : triangle D) : Prop :=\n  T \u2208 dist_triang D\n\nlemma distinguished_iff (T : triangle D) :\n  T.distinguished \u2194 T \u2208 dist_triang D := by refl\n\nlemma distinguished.mk (T : triangle D) (hT : T \u2208 dist_triang D) : T.distinguished :=\nhT\n\nnamespace distinguished\n\nvariable {T : triangle D}\n\nlemma comp_zero\u2081\u2082 (h : T.distinguished) : T.mor\u2081 \u226b T.mor\u2082 = 0 := comp_zero\u2081\u2082 _ h\nlemma comp_zero\u2082\u2083 (h : T.distinguished) : T.mor\u2082 \u226b T.mor\u2083 = 0 :=\n(candidate_triangle.of_distinguished _ h).2.zero\u2082\u2083\nlemma comp_zero\u2083\u2081 (h : T.distinguished) : T.mor\u2083 \u226b T.mor\u2081\u27e6(1 : \u2124)\u27e7' = 0 :=\n(candidate_triangle.of_distinguished _ h).2.zero\u2083\u2081\n\nlemma rotate (h : T.distinguished) : T.rotate.distinguished :=\nrot_of_dist_triangle D T h\nlemma inv_rotate (h : T.distinguished) : T.inv_rotate.distinguished :=\ninv_rot_of_dist_triangle D T h\n\nend distinguished\n\nend triangle\n\nend category_theory.pretriangulated\n\nopen category_theory.triangulated category_theory.pretriangulated\n\nsection\n\nvariables {C \u03b9 : Type*} [category C]\n  (c : complex_shape \u03b9)\n\ninstance homological_complex.single_additive [decidable_eq \u03b9] [preadditive C] [has_zero_object C] (n : \u03b9) :\n  (homological_complex.single C c n).additive :=\n\u27e8\u03bb X Y f g, by { ext i, dsimp, split_ifs; simp, }\u27e9\n\ninstance homotopy_category.homology_functor_additive [abelian C] (n : \u03b9) :\n  (homotopy_category.homology_functor C c n).additive :=\n@quotient.lift_additive _ _ _ _ _ _ _ _ _\n    (infer_instance : (homotopy_category.quotient C _).additive) _ _ _\n\nvariable (C)\n\ndef homotopy_category.comm_shift_quotient [preadditive C] (n : \u2124) :\n  shift_functor (cochain_complex C \u2124) n \u22d9\n    homotopy_category.quotient _ _ \u2245\n  homotopy_category.quotient _ _ \u22d9 shift_functor _ n :=\nquotient.comm_shift _ _\n\nnamespace cochain_complex\n\nsection\n\nvariables [preadditive C] (K : cochain_complex C \u2124) (n k m : \u2124) (h : k + n = m)\n\ninclude h\n\nvariable (C)\n\ndef shift_eval_prev : (shift_functor _ n) \u22d9 homological_complex.eval C _ ((complex_shape.up \u2124).prev k) \u2245\n  homological_complex.eval C _ ((complex_shape.up \u2124).prev m) :=\npreadditive.mul_iso ((-1 : units \u2124)^n) (eq_to_iso (congr_arg (homological_complex.eval _ _)\n  (show (complex_shape.up \u2124).prev k + n = (complex_shape.up \u2124).prev m, by { simp, linarith, })))\ndef shift_eval : (shift_functor _ n) \u22d9 homological_complex.eval C (complex_shape.up \u2124) k \u2245\n  homological_complex.eval C (complex_shape.up \u2124) m :=\neq_to_iso (congr_arg (homological_complex.eval _ _)  h)\ndef shift_eval_next : (shift_functor _ n) \u22d9 homological_complex.eval C _ ((complex_shape.up \u2124).next k) \u2245\n  homological_complex.eval C _ ((complex_shape.up \u2124).next m) :=\npreadditive.mul_iso ((-1 : units \u2124)^n) (eq_to_iso (congr_arg (homological_complex.eval _ _)\n  (show (complex_shape.up \u2124).next k + n = (complex_shape.up \u2124).next m, by { simp, linarith, })))\n\nomit h\n\nvariable {C}\n\nlemma shift_eval_prev_hom_app_eq (X : cochain_complex C \u2124) :\n  (shift_eval_prev C n k m h).hom.app X =\n    (-1 : units \u2124)^n \u2022 eq_to_hom (congr_arg X.X\n      (by rw [prev, prev, \u2190 h, \u2190 add_left_inj (1 : \u2124), sub_add_cancel, sub_eq_add_neg,\n        add_assoc k, add_comm _ n, add_assoc, int.add_neg_one, sub_add_cancel])) :=\nbegin\n  dsimp [shift_eval_prev],\n  simpa only [nat_trans.app_zsmul, eq_to_hom_app],\nend\n\nlemma shift_eval_hom_app_eq (X : cochain_complex C \u2124) :\n  (shift_eval C n k m h).hom.app X = eq_to_hom (congr_arg X.X h) :=\nbegin\n  dsimp [shift_eval],\n  apply eq_to_hom_app,\nend\n\nlemma shift_eval_next_hom_app_eq (X : cochain_complex C \u2124) :\n  (shift_eval_next C n k m h).hom.app X =\n    (-1 : units \u2124)^n \u2022 eq_to_hom (congr_arg X.X\n    (by rw [next, next, \u2190 h, add_assoc, add_assoc, add_comm _ n])) :=\nbegin\n  dsimp [shift_eval_next],\n  simpa only [nat_trans.app_zsmul, eq_to_hom_app],\nend\n\nlemma shift_eval_prev_hom_app_comp_d_to :\n  (shift_eval_prev C n k m h).hom.app K \u226b K.d_to m =\n    K\u27e6n\u27e7.d_to k \u226b (shift_eval C n k m h).hom.app K :=\nbegin\n  subst h,\n  simp only [K\u27e6n\u27e7.d_to_eq (show (k-1)+1 = k, by linarith),\n    K.d_to_eq (show (k-1+n)+1 = k+n, by linarith), shift_functor_obj_d'],\n  dsimp [homological_complex.X_prev_iso, shift_eval_prev, shift_eval],\n  simp only [add_zero, zero_add, neg_zero, nat_trans.app_zsmul, eq_to_hom_app,\n    linear.smul_comp_assoc, eq_to_hom_trans, linear.smul_comp, linear.comp_smul, assoc,\n    eq_to_hom_trans_assoc],\n  erw comp_id,\n  refl,\nend\n\nlemma shift_eval_hom_app_comp_d_from :\n(shift_eval C n k m h).hom.app K \u226b K.d_from m =\n  K\u27e6n\u27e7.d_from k \u226b (shift_eval_next C n k m h).hom.app K :=\nbegin\n  subst h,\n  simp only [K\u27e6n\u27e7.d_from_eq rfl, shift_functor_obj_d',\n    K.d_from_eq (show k+n+1=k+1+n, by linarith)],\n  dsimp [homological_complex.X_next_iso, shift_eval_next, shift_eval, hom_complex.\u03b5],\n  simpa only [id_comp, linear.smul_comp, nat_trans.app_zsmul, eq_to_hom_app,\n    linear.comp_smul, assoc, eq_to_hom_trans, smul_smul,\n    \u2190 units.coe_mul, \u2190 mul_zpow, neg_mul, mul_neg, neg_neg,\n    int.units_mul_self, one_zpow, units.coe_one, one_zsmul],\nend\n\nvariable (C)\n\ndef shift_short_complex_functor_iso :\n  shift_functor _ n \u22d9 homological_complex.short_complex_functor C (complex_shape.up \u2124) k \u2245\n    homological_complex.short_complex_functor C (complex_shape.up \u2124) m :=\nnat_iso.of_components\n  (\u03bb K, short_complex.mk_iso ((shift_eval_prev C _ _ _ h).app K) ((shift_eval C _ _ _ h).app K)\n    (((shift_eval_next C _ _ _ h).app K)) (shift_eval_prev_hom_app_comp_d_to _ _ _ _ _)\n    (shift_eval_hom_app_comp_d_from _ _ _ _ _))\n  (\u03bb K L \u03c6, begin\n    ext1,\n    { exact (shift_eval_prev C _ _ _ h).hom.naturality \u03c6, },\n    { exact (shift_eval C _ _ _ h).hom.naturality \u03c6, },\n    { exact (shift_eval_next C _ _ _ h).hom.naturality \u03c6, },\n  end)\n\nend\n\ndef shift_homology_functor_iso [abelian C] (n k m : \u2124) (h : k + n = m) :\n  category_theory.shift_functor _ n \u22d9 homology_functor C (complex_shape.up \u2124) k \u2245\n    homology_functor C _ m :=\n(functor.associator _ _ _).symm \u226a\u226b iso_whisker_right (shift_short_complex_functor_iso C _ _ _ h) _\n\nvariable {C}\n\nlemma shift_homology_functor_iso_hom_app [abelian C] (n k m : \u2124) (h : k + n = m)\n  (X : cochain_complex C \u2124) :\n  (shift_homology_functor_iso C n k m h).hom.app X =\n    short_complex.homology_map ((shift_short_complex_functor_iso C _ _ _ h).hom.app X) :=\nid_comp _\n\nlemma shift_functor_add'_inv_app_comp_zero_hom_app_eq [abelian C] (n' n k : \u2124) (h : 0 = n' + n)\n  (X : cochain_complex C \u2124) :\n    ((category_theory.shift_functor_add' (cochain_complex C \u2124) n' n 0 h).inv.app X : _ \u27f6 _).f k \u226b\n  ((shift_functor_zero (homological_complex C (complex_shape.up \u2124)) \u2124).hom.app X : _ \u27f6 _).f k =\n  eq_to_hom (congr_arg X.X (show k+n+n' = k, by rw [\u2190 add_left_inj n, add_assoc, \u2190 h, add_zero])) :=\nbegin\n  rw [shift_functor_add'_eq, shift_functor_zero_eq],\n  apply eq_to_hom_trans,\nend\n\n@[simp, reassoc]\nlemma shift_homology_functor_iso_hom_app_comp [abelian C] (n k m n' : \u2124)\n  (h : k+n=m) (h' : m+n' = k) (X : cochain_complex C \u2124) :\n  (shift_homology_functor_iso C n k m h).hom.app (X\u27e6n'\u27e7) \u226b (shift_homology_functor_iso C n' m k h').hom.app X =\n  (_root_.homology_functor C _ k).map\n    ((category_theory.shift_functor_add' (cochain_complex C \u2124) n' n 0\n      (by rw [\u2190 add_right_inj m, \u2190 add_assoc, h', h, add_zero])).inv.app X \u226b\n    (category_theory.shift_functor_zero _ \u2124).hom.app X) :=\nbegin\n  have hn' : n' = -n := by linarith,\n  subst hn',\n  simp only [shift_homology_functor_iso_hom_app, homology_functor_map,\n    \u2190 short_complex.homology_map_comp],\n  dsimp only [homological_complex.short_complex_functor,\n      homological_complex.comp_f],\n  congr' 1,\n  ext1; dsimp only; rw shift_functor_add'_inv_app_comp_zero_hom_app_eq;\n    dsimp [shift_short_complex_functor_iso],\n  { simp only [shift_eval_prev_hom_app_eq],\n    erw [preadditive.comp_zsmul, preadditive.zsmul_comp, smul_smul, eq_to_hom_trans],\n    simpa only [\u2190 units.coe_mul, \u2190 zpow_add, neg_add_self n,\n      zpow_zero, units.coe_one, one_zsmul], },\n  { simpa only [shift_eval_hom_app_eq, eq_to_hom_trans], },\n  { simp only [shift_eval_next_hom_app_eq],\n    erw [preadditive.comp_zsmul, preadditive.zsmul_comp, smul_smul, eq_to_hom_trans],\n    simpa only [\u2190 units.coe_mul, \u2190 zpow_add, neg_add_self n,\n      zpow_zero, units.coe_one, one_zsmul], },\nend\n\nend cochain_complex\n\ndef homotopy_category.shift_homology_functor_iso [abelian C] (n k m : \u2124) (h : k + n = m):\n  shift_functor _ n \u22d9 homotopy_category.homology_functor C (complex_shape.up \u2124) k \u2245\n    homotopy_category.homology_functor C _ m :=\nquotient.lift_nat_iso _ _ ((functor.associator _ _ _).symm \u226a\u226b\n  iso_whisker_right (quotient.comm_shift _ _).symm _ \u226a\u226b\n  functor.associator _ _ _ \u226a\u226b\n  iso_whisker_left _ (homotopy_category.homology_factors _ _ _).symm \u226a\u226b\n  cochain_complex.shift_homology_functor_iso C _ _ _ h \u226a\u226b\n  (homotopy_category.homology_factors _ _ _).symm)\n\nend\n\nvariables (C : Type*) [category C] [abelian C]\n\nsection\n\nopen cochain_complex\n\nlemma homology_functor_comp_\u03b9_mapping_cone {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) (n : \u2124) :\n  (homology_functor C (complex_shape.up \u2124) n).map (\u03c6 \u226b mapping_cone.inr \u03c6) = 0 :=\nbegin\n  rw homotopy_category.homology_functor_map_factors,\n  have h\u03c6 : homotopy_category.induced_triangle (mapping_cone.triangle \u03c6) \u2208 dist_triang _,\n  { rw homotopy_category.triangle_distinguished_iff,\n    exact \u27e8_, _, _, \u27e8iso.refl _\u27e9\u27e9, },\n  simpa only [functor.map_comp, functor.map_zero]\n    using (homotopy_category.homology_functor _ _ n).congr_map\n      ((triangle.comp_eq_zero.of_distinguished _ h\u03c6).zero\u2081\u2082),\nend\n\nvariable {C}\n\nlemma homology_functor_is_homological_aux {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) (n : \u2124) :\n  (short_complex.mk ((homology_functor C (complex_shape.up \u2124) n).map \u03c6)\n    ((homology_functor C (complex_shape.up \u2124) n).map (mapping_cone.inr \u03c6))\n    (by rw [\u2190 functor.map_comp, homology_functor_comp_\u03b9_mapping_cone])).exact :=\nbegin\n  rw short_complex.exact_iff_pseudo_exact',\n  intros A\u2080 \u03b3\u2082 h\u03b3\u2082,\n  dsimp at \u03b3\u2082 h\u03b3\u2082,\n  /- the next three operations could be a specialised lemma -/\n  obtain \u27e8A\u2081, \u03c0\u2081, h\u03c0\u2081, z\u2082, hz\u2082\u27e9 := abelian.pseudo_surjective_of_epi'\n    (short_complex.homology_\u03c0 _) \u03b3\u2082,\n  have hz\u2082' : \u2203 z\u2082' hz\u2082', z\u2082 = short_complex.lift_cycles _ z\u2082' hz\u2082' :=\n    \u27e8z\u2082 \u226b short_complex.cycles_i _,\n      by simp only [assoc, short_complex.cycles_i_g, comp_zero],\n      by simp only [\u2190 cancel_mono ((homological_complex.short_complex_functor C\n        (complex_shape.up \u2124) n).obj L).cycles_i, short_complex.lift_cycles_i]\u27e9,\n  obtain \u27e8z\u2082, hz\u2082', rfl\u27e9 := hz\u2082',\n  replace h\u03b3\u2082 := \u03c0\u2081 \u226b= h\u03b3\u2082,\n  rw [reassoc_of hz\u2082, comp_zero, short_complex.homology_\u03c0_naturality, \u2190 assoc,\n    short_complex.comp_homology_\u03c0_eq_zero_iff] at h\u03b3\u2082,\n  obtain \u27e8A\u2082, \u03c0\u2082, h\u03c0\u2082, c\u2081, hc\u2081\u27e9 := h\u03b3\u2082,\n  dsimp at c\u2081 hc\u2081,\n  replace hc\u2081 := hc\u2081 =\u226b (short_complex.cycles_i _),\n  simp only [assoc, homological_complex.short_complex_functor_map_\u03c4\u2082,\n    short_complex.lift_cycles_comp_cycles_map, short_complex.lift_cycles_i,\n    short_complex.to_cycles_i, homological_complex.short_complex_functor_obj_f,\n    @mapping_cone.to_ext_iff _ _ _ _ _ _ \u03c6 _ _ _ _ ((complex_shape.up _).next n) (by simp),\n    mapping_cone.d_fst _ ((complex_shape.up _).prev n) n ((complex_shape.up _).next n) (by simp) (by simp),\n    mapping_cone.d_snd _ ((complex_shape.up _).prev n) n (by simp),\n    mapping_cone.inr_fst, comp_zero, preadditive.comp_neg,\n    zero_eq_neg, mapping_cone.inr_snd, preadditive.comp_add] at hc\u2081,\n  dsimp at hc\u2081,\n  rw comp_id at hc\u2081,\n  obtain \u27e8hc\u2081, hc\u2081'\u27e9 := hc\u2081,\n  rw \u2190 assoc at hc\u2081,\n  haveI := h\u03c0\u2081,\n  haveI := h\u03c0\u2082,\n  refine \u27e8A\u2082, \u03c0\u2082 \u226b \u03c0\u2081, epi_comp _ _,\n    ((homological_complex.short_complex_functor C\n      (complex_shape.up \u2124) n).obj K).lift_cycles _ hc\u2081 \u226b short_complex.homology_\u03c0 _, _\u27e9,\n  dsimp,\n  simp only [assoc, hz\u2082, short_complex.comp_lift_cycles_assoc,\n    homological_complex.short_complex_functor_map_\u03c4\u2082, short_complex.homology_\u03c0_naturality,\n    short_complex.lift_cycles_comp_cycles_map_assoc,\n    short_complex.lift_cycles_comp_homology_\u03c0_eq_iff],\n  exact \u27e8A\u2082, \ud835\udfd9 A\u2082, infer_instance,\n    c\u2081 \u226b (mapping_cone.snd \u03c6).v ((complex_shape.up \u2124).prev n)\n      ((complex_shape.up \u2124).prev n) (add_zero _).symm, by simpa only [id_comp, hc\u2081', assoc]\u27e9,\nend\n\nend\n\nvariable (C)\n\nnamespace homotopy_category\n\ninstance homology_functor_is_homological (n : \u2124):\n  (homology_functor C (complex_shape.up \u2124) n).is_homological :=\nfunctor.is_homological.mk' _ (\u03bb T hT, begin\n  rw triangle_distinguished_iff at hT,\n  obtain \u27e8K, L, \u03c6, \u27e8e\u27e9\u27e9 := hT,\n  refine \u27e8_, \u27e8_, _, _, \u27e8mapping_cone_induced_triangle_iso \u03c6\u27e9\u27e9, e,\n    homology_functor_is_homological_aux \u03c6 n\u27e9,\nend)\n\nabbreviation acyclic : set (homotopy_category C (complex_shape.up \u2124)) :=\n(homology_functor C (complex_shape.up \u2124) 0).kernel_of_is_homological\n\ninstance : saturated (homotopy_category.acyclic C) := infer_instance\n\nlemma mem_acyclic_W_iff {K L : homotopy_category C (complex_shape.up \u2124)} (\u03c6 : K \u27f6 L) :\n  (subcategory.W (acyclic C)) \u03c6 \u2194 \u2200 (n : \u2124), is_iso ((homology_functor _ _ n).map \u03c6) :=\nbegin\n  rw functor.kernel_of_is_homological_W,\n  simpa only [\u2190 \u03bb n, nat_iso.is_iso_map_iff (shift_homology_functor_iso C _ _ _ (zero_add n)) \u03c6],\nend\n\nlemma homology_functor_is_inverted_by (n : \u2124) :\n  (subcategory.W (acyclic C)).is_inverted_by (homology_functor C (complex_shape.up \u2124) n) :=\nbegin\n  intros K L \u03c6 h\u03c6,\n  rw mem_acyclic_W_iff at h\u03c6,\n  exact h\u03c6 n,\nend\n\nvariable {C}\n\nlemma map_quotient_W_iff {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) :\n  (subcategory.W (acyclic C)) ((quotient _ _).map \u03c6) \u2194 quasi_iso \u03c6 :=\nbegin\n  simp only [mem_acyclic_W_iff, \u2190 homology_functor_map_factors],\n  split,\n  { intro h,\n    exact \u27e8h\u27e9, },\n  { intro h,\n    exact h.is_iso, }\nend\n\nend homotopy_category\n\nsection\n\nvariables (D : Type*) [category D] [has_zero_morphisms D] [category_with_homology D]\n  {\u03b9 : Type*} (c : complex_shape \u03b9)\n\ndef quasi_isomorphisms :\n  morphism_property (homological_complex D c) :=\n\u03bb K L \u03c6, \u2200 (i : \u03b9), is_iso (homology_map \u03c6 i)\n\nvariables {D c}\n\nlemma mem_quasi_isomorphisms_iff {K L : homological_complex D c} (\u03c6 : K \u27f6 L) :\n  quasi_isomorphisms D c \u03c6 \u2194 quasi_iso \u03c6 :=\n\u27e8\u03bb h, \u27e8h\u27e9, \u03bb h, h.1\u27e9\n\nend\n\nabbreviation derived_category := (subcategory.W (homotopy_category.acyclic C)).localization\n\ninstance : inhabited (derived_category C) := \u27e80\u27e9\n\nnamespace derived_category\n\nvariable {C}\n\ninstance : pretriangulated (derived_category C) :=\npretriangulated.localization_pretriangulated _\n\ninstance : is_triangulated (derived_category C) :=\npretriangulated.localization_triangulated _\n\nabbreviation Qh : homotopy_category C (complex_shape.up \u2124) \u2964 derived_category C :=\n(subcategory.W (homotopy_category.acyclic C)).Q\n\n--instance Qh_has_comm_shift : (Qh : _ \u2964 derived_category C).has_comm_shift \u2124 :=\n--infer_instance\n\n--instance Qh_additive : (Qh : _ \u2964 (derived_category C)).additive :=\n--infer_instance\n\n--instance Qh_is_localization_W :\n--  Qh.is_localization (homotopy_category.acyclic C).W := infer_instance\n\ninstance Qh_is_triangulated : (Qh : _ \u2964 derived_category C).is_triangulated :=\npretriangulated.localization_functor_is_triangulated _ _\n\ninstance : ess_surj (Qh : _ \u2964 derived_category C) :=\nlocalization.ess_surj _ (subcategory.W (homotopy_category.acyclic C))\n\ndef Q : cochain_complex C \u2124 \u2964 derived_category C :=\nhomotopy_category.quotient _ _ \u22d9 Qh\n\ninstance Q_additive : (Q : _ \u2964 derived_category C).additive :=\nby { dsimp [Q], apply_instance, }\n\nvariable (C)\n\ninstance Q_has_comm_shift : (Q : cochain_complex C \u2124 \u2964 _).has_comm_shift \u2124 :=\n(infer_instance : (homotopy_category.quotient C _ \u22d9\n    Qh : cochain_complex C \u2124 \u2964 _).has_comm_shift \u2124)\n\nvariable {C}\n\nlemma Q_comm_shift_iso_hom_app (K : cochain_complex C \u2124 ) (n : \u2124) :\n  (Q.comm_shift_iso n).hom.app K =\n    Qh.map (((homotopy_category.quotient C _).comm_shift_iso n).hom.app K) \u226b\n      (Qh.comm_shift_iso n).hom.app _ :=\nfunctor.comm_shift_comp_hom_app _ _ _\n\nlemma is_iso_Q_map_iff {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) :\n  is_iso (Q.map \u03c6) \u2194 quasi_iso \u03c6 :=\n(subcategory.is_iso_map_iff _ Qh _).trans (homotopy_category.map_quotient_W_iff \u03c6)\n\ninstance {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) [quasi_iso \u03c6] :\n  is_iso (Q.map \u03c6) :=\nby { rw is_iso_Q_map_iff, apply_instance, }\n\nvariable (C)\n\nlemma Q_inverts_quasi_isomorphisms : (quasi_isomorphisms C _).is_inverted_by Q :=\n\u03bb K L \u03c6 h\u03c6, begin\n  rw mem_quasi_isomorphisms_iff at h\u03c6,\n  haveI := h\u03c6,\n  apply_instance,\nend\n\nlemma homotopy_equivalences_subset_quasi_isomorphisms :\n  cochain_complex.homotopy_equivalences C \u2286 quasi_isomorphisms C (complex_shape.up \u2124) :=\nbegin\n  rintros K L _ \u27e8h, rfl\u27e9,\n  simpa only [mem_quasi_isomorphisms_iff] using h.to_quasi_iso,\nend\n\ninstance Q_is_localization : Q.is_localization (quasi_isomorphisms C _) :=\nlocalization.comp (homotopy_category.quotient _ _) Qh\n    (cochain_complex.homotopy_equivalences C) (subcategory.W (homotopy_category.acyclic C))\n    (quasi_isomorphisms C _) (Q_inverts_quasi_isomorphisms C)\n    (homotopy_equivalences_subset_quasi_isomorphisms C)\n(begin\n  rintros \u27e8K\u27e9 \u27e8L\u27e9 \u03c6 h\u03c6,\n  have hf : \u2203 (f : K \u27f6 L), (homotopy_category.quotient _ _).map f = \u03c6 :=\n    \u27e8_, (homotopy_category.quotient C (complex_shape.up \u2124)).image_preimage \u03c6\u27e9,\n  obtain \u27e8f, rfl\u27e9 := hf,\n  refine \u27e8_, _, f, _, \u27e8iso.refl _\u27e9\u27e9,\n  simpa only [mem_quasi_isomorphisms_iff, \u2190 homotopy_category.map_quotient_W_iff] using h\u03c6,\nend)\n\ninstance : ess_surj (Q : _ \u2964 derived_category C) :=\nlocalization.ess_surj _ (quasi_isomorphisms C _)\n\nvariable {C}\n\nsection\n\nvariables {K L : cochain_complex C \u2124}\n  (\u03c6 : K \u27f6 L)\n\ndef mapping_cone := Q.obj (cochain_complex.mapping_cone \u03c6)\n\ndef \u03b9_mapping_cone : Q.obj L \u27f6 mapping_cone \u03c6 :=\nQ.map (cochain_complex.mapping_cone.inr \u03c6)\n\ndef mapping_cone_\u03b4 : mapping_cone \u03c6 \u27f6 (Q.obj K)\u27e6(1 : \u2124)\u27e7 :=\n  Q.map (cochain_complex.mapping_cone.\u03b4 \u03c6) \u226b (Q.comm_shift_iso 1).hom.app K\n\ndef mapping_cone_triangle : triangle (derived_category C) :=\ntriangle.mk (Q.map \u03c6) (\u03b9_mapping_cone \u03c6) (mapping_cone_\u03b4 \u03c6)\n\nlemma Qh_map_mapping_cone_triangle_iso :\n  (Qh.map_triangle.obj (homotopy_category.mapping_cone_triangle' \u03c6) \u2245\n    mapping_cone_triangle \u03c6) :=\nbegin -- needs cleaning up...\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _) (iso.refl _) _ _ _,\n  { tidy, },\n  { tidy, },\n  { dsimp [iso.refl, mapping_cone_triangle, mapping_cone_\u03b4,\n      homotopy_category.mapping_cone_triangle',\n      cochain_complex.mapping_cone.\u03b4'],\n    simp only [category_theory.functor.map_id, comp_id, id_comp,\n      Q_comm_shift_iso_hom_app],\n    congr' 1,\n    symmetry,\n    convert id_comp _,\n    convert category_theory.functor.map_id _ _, },\nend\n\nend\n\nlemma mem_dist_triang_iff' (T : triangle (derived_category C)) :\n  (T \u2208 dist_triang (derived_category C)) \u2194\n    \u2203 (K L : cochain_complex C \u2124) (\u03c6 : K \u27f6 L),\n      nonempty (T \u2245\n        Qh.map_triangle.obj (homotopy_category.mapping_cone_triangle' \u03c6)) :=\nbegin\n  split,\n  { rintro \u27e8Th, e, \u27e8K, L, \u03c6, \u27e8e'\u27e9\u27e9\u27e9,\n    exact \u27e8K, L, \u03c6, \u27e8e \u226a\u226b Qh.map_triangle.map_iso e'\u27e9\u27e9, },\n  { rintro \u27e8K, L, \u03c6, \u27e8e\u27e9\u27e9,\n    exact \u27e8_, e, \u27e8K, L, \u03c6, \u27e8iso.refl _\u27e9\u27e9\u27e9, },\nend\n\nlemma mem_dist_triang_iff (T : triangle (derived_category C)) :\n  (T \u2208 dist_triang (derived_category C)) \u2194\n    \u2203 (K L : cochain_complex C \u2124) (\u03c6 : K \u27f6 L),\n      nonempty (T \u2245 mapping_cone_triangle \u03c6) :=\nbegin\n  rw mem_dist_triang_iff',\n  split,\n  { rintro \u27e8K, L, \u03c6, \u27e8e\u27e9\u27e9,\n    exact \u27e8K, L, \u03c6, \u27e8e \u226a\u226b Qh_map_mapping_cone_triangle_iso _\u27e9\u27e9, },\n  { rintro \u27e8K, L, \u03c6, \u27e8e\u27e9\u27e9,\n    exact \u27e8K, L, \u03c6, \u27e8e \u226a\u226b (Qh_map_mapping_cone_triangle_iso _).symm\u27e9\u27e9, },\nend\n\ninstance is_iso_Q_map_from_mapping_cone_of_ses\n  {S : short_complex (cochain_complex C \u2124)}\n  (ex : S.short_exact) :\n  quasi_iso (cochain_complex.from_mapping_cone_of_ses ex) :=\ncochain_complex.from_mapping_cone_of_ses_quasi_iso ex\n\ndef triangle_of_ses_\u03b4 {S : short_complex (cochain_complex C \u2124)}\n  (ex : S.short_exact) : Q.obj S.X\u2083 \u27f6 (Q.obj S.X\u2081)\u27e6(1 : \u2124)\u27e7 :=\ninv (Q.map (cochain_complex.from_mapping_cone_of_ses ex)) \u226b (mapping_cone_triangle S.f).mor\u2083\n\n@[simps]\ndef triangle_of_ses {S : short_complex (cochain_complex C \u2124)}\n  (ex : S.short_exact) : triangle (derived_category C) :=\ntriangle.mk (Q.map S.f) (Q.map S.g) (triangle_of_ses_\u03b4 ex)\n\nlemma triangle_of_ses_dist {S : short_complex (cochain_complex C \u2124)}\n  (ex : S.short_exact) : (triangle_of_ses ex).distinguished :=\ntriangle.distinguished.mk _ begin\n  rw mem_dist_triang_iff,\n  refine \u27e8_, _, S.f, \u27e8_\u27e9\u27e9,\n  refine triangle.mk_iso _ _ (iso.refl _) (iso.refl _)\n    (as_iso (Q.map (cochain_complex.from_mapping_cone_of_ses ex))).symm (by tidy) _ _,\n  { dsimp [triangle_of_ses, mapping_cone_triangle, \u03b9_mapping_cone],\n    simp only [\u2190 cancel_mono (Q.map (cochain_complex.from_mapping_cone_of_ses ex)),\n      id_comp, assoc, is_iso.inv_hom_id, comp_id, \u2190 Q.map_comp,\n      cochain_complex.inr_mapping_cone_comp_from_mapping_cone_of_ses], },\n  { dsimp [triangle_of_ses, triangle_of_ses_\u03b4],\n    simp only [category_theory.functor.map_id, comp_id], },\nend\n\nlemma left_factorisation {K L : cochain_complex C \u2124} (\u03c6 : Q.obj K \u27f6 Q.obj L) :\n  \u2203 (L' : cochain_complex C \u2124) (f : K \u27f6 L') (s : L \u27f6 L') (hs : quasi_iso s),\n    \u03c6 = Q.map f \u226b (by { haveI := hs, exact inv (Q.map s), }) :=\nbegin\n  obtain \u27e8\u27e8\u27e8L'\u27e9, f, s, hs\u27e9 , hz\u27e9 :=\n    left_calculus_of_fractions.L_map_fac Qh\n      (subcategory.W (homotopy_category.acyclic C)) \u03c6,\n  refine \u27e8_, (homotopy_category.quotient _ _).preimage f,\n    (homotopy_category.quotient _ _).preimage s, _, _\u27e9,\n  { simpa only [\u2190 homotopy_category.map_quotient_W_iff, functor.image_preimage] using hs, },\n  { dsimp [Q],\n    simpa only [functor.image_preimage] using hz, },\nend\n\nlemma right_factorisation {K L : cochain_complex C \u2124} (\u03c6 : Q.obj K \u27f6 Q.obj L) :\n  \u2203 (K' : cochain_complex C \u2124) (s : K' \u27f6 K) (f : K' \u27f6 L) (hs : quasi_iso s),\n    \u03c6 = (by { haveI := hs, exact inv (Q.map s), }) \u226b Q.map f :=\nbegin\n  obtain \u27e8\u27e8\u27e8L'\u27e9, s, f, hs\u27e9 , hz\u27e9 :=\n    right_calculus_of_fractions.L_map_fac Qh (subcategory.W (homotopy_category.acyclic C)) \u03c6,\n  refine \u27e8_, (homotopy_category.quotient _ _).preimage s,\n    (homotopy_category.quotient _ _).preimage f, _, _\u27e9,\n  { simpa only [\u2190 homotopy_category.map_quotient_W_iff, functor.image_preimage] using hs, },\n  { dsimp [Q],\n    simpa only [functor.image_preimage] using hz, },\nend\n\nvariable (C)\n\ndef homology_functor (n : \u2124) : derived_category C \u2964 C :=\nlocalization.lift (homotopy_category.homology_functor C (complex_shape.up \u2124) n)\n  (homotopy_category.homology_functor_is_inverted_by C n) Qh\n\ninstance (n : \u2124) : localization.lifting Qh (subcategory.W (homotopy_category.acyclic C))\n  (homotopy_category.homology_functor C (complex_shape.up \u2124) n) (homology_functor C n) :=\nlocalization.lifting_lift _ _ _\n\ndef homology_functor_factors_Qh (n : \u2124) :\n  Qh \u22d9 homology_functor C n \u2245\n    homotopy_category.homology_functor C (complex_shape.up \u2124) n :=\nlocalization.lifting.iso _ (subcategory.W (homotopy_category.acyclic C)) _ _\n\ninstance homology_functor_lifting (n : \u2124) : localization.lifting Q (quasi_isomorphisms C (complex_shape.up \u2124))\n  (_root_.homology_functor C _ n) (homology_functor C n) :=\n\u27e8functor.associator _ _ _ \u226a\u226b iso_whisker_left _ ((homology_functor_factors_Qh C n)) \u226a\u226b\n  homotopy_category.homology_factors C _ n\u27e9\n\ndef homology_functor_factors (n : \u2124) :\n  Q \u22d9 homology_functor C n \u2245 _root_.homology_functor C (complex_shape.up \u2124) n :=\nlocalization.lifting.iso _ (quasi_isomorphisms C (complex_shape.up \u2124)) _ _\n\n@[simp]\nlemma lifting_iso_eq_homology_functor_factors (n : \u2124) :\n  localization.lifting.iso Q (quasi_isomorphisms C (complex_shape.up \u2124))\n    (_root_.homology_functor C (complex_shape.up \u2124) n) (homology_functor C n) =\nhomology_functor_factors C n := rfl\n\ninstance homology_functor_preserves_zero_morphisms (n : \u2124) :\n  (homology_functor C n).preserves_zero_morphisms :=\nfunctor.is_homological.localization_lift_preserves_zero_morphisms _ _ _\n\ninstance homology_functor_is_homological (n : \u2124) :\n  (homology_functor C n).is_homological :=\nfunctor.is_homological.localization_lift_is_homological _ _ _\n\nvariable {C}\n\nlemma is_iso_iff_is_iso_homology {K L : derived_category C} (\u03c6 : K \u27f6 L) :\n  is_iso \u03c6 \u2194 \u2200 (n : \u2124), is_iso ((homology_functor C n).map \u03c6) :=\nbegin\n  split,\n  { introI,\n    exact \u03bb n, infer_instance, },\n  { suffices : \u2200 \u2983K' L' : cochain_complex C \u2124\u2984 (\u03c6' : Q.obj K' \u27f6 Q.obj L')\n      (h\u03c6' : \u2200 (n : \u2124), is_iso ((homology_functor C n).map \u03c6')), is_iso \u03c6',\n    { introI,\n      let \u03c8 := (Q.obj_obj_preimage_iso K).hom \u226b \u03c6 \u226b (Q.obj_obj_preimage_iso L).inv,\n      have eq : \u03c6 = (Q.obj_obj_preimage_iso K).inv \u226b \u03c8 \u226b (Q.obj_obj_preimage_iso L).hom,\n      { simp only [assoc, iso.inv_hom_id, comp_id, iso.inv_hom_id_assoc], },\n      rw eq,\n      haveI : is_iso \u03c8 := this \u03c8 (\u03bb n, begin\n        dsimp only [\u03c8],\n        simp only [functor.map_comp],\n        apply_instance,\n      end),\n      apply_instance, },\n    intros K' L' \u03c6' h\u03c6',\n    obtain \u27e8L', f, s, hs, eq\u27e9 := left_factorisation \u03c6',\n    haveI : is_iso (Q.map f),\n    { simp only [eq, functor.map_comp] at h\u03c6',\n      haveI := h\u03c6',\n      haveI : \u2200 (n : \u2124), is_iso ((homology_functor C n).map (Q.map f)),\n      { intro n,\n        exact is_iso.of_is_iso_comp_right _ ((homology_functor C n).map (inv (Q.map s))), },\n      haveI : quasi_iso f,\n      { rw \u2190 mem_quasi_isomorphisms_iff,\n        intro n,\n        refine (nat_iso.is_iso_map_iff (homology_functor_factors C n) f).1 _,\n        dsimp,\n        apply_instance, },\n      apply_instance, },\n    rw eq,\n    apply_instance, },\nend\n\nabbreviation homology (K : derived_category C) (n : \u2124) := (homology_functor C n).obj K\n\nlemma Q_map_eq_of_homotopy {K L : cochain_complex C \u2124} (f\u2081 f\u2082 : K \u27f6 L)\n  (h : homotopy f\u2081 f\u2082) : Q.map f\u2081 = Q.map f\u2082 :=\nbegin\n  dsimp [Q],\n  rw homotopy_category.eq_of_homotopy _ _ h,\nend\n\n@[instance]\ndef shift_functor_comp_homology_lifting (a b : \u2124) :\n  localization.lifting Q (quasi_isomorphisms C (complex_shape.up \u2124))\n  (shift_functor (cochain_complex C \u2124) a \u22d9 _root_.homology_functor C (complex_shape.up \u2124) b)\n  (shift_functor (derived_category C) a \u22d9 homology_functor C b) :=\n\u27e8(functor.associator _ _ _).symm \u226a\u226b iso_whisker_right (Q.comm_shift_iso a).symm _ \u226a\u226b\n    functor.associator _ _ _ \u226a\u226b iso_whisker_left _ (homology_functor_factors C b)\u27e9\n\n@[simp]\nlemma shift_functor_comp_homology_lifting_iso_hom_app (a b : \u2124) (X : cochain_complex C \u2124) :\n  (localization.lifting.iso Q (quasi_isomorphisms C (complex_shape.up \u2124))\n  (shift_functor (cochain_complex C \u2124) a \u22d9 _root_.homology_functor C (complex_shape.up \u2124) b)\n  (shift_functor (derived_category C) a \u22d9 homology_functor C b)).hom.app X =\n    (homology_functor C b).map ((Q.comm_shift_iso a).inv.app X) \u226b\n      (homology_functor_factors C b).hom.app (X\u27e6a\u27e7) :=\nbegin\n  dsimp [shift_functor_comp_homology_lifting],\n  simp only [id_comp],\nend\n\nvariable (C)\n\ndef shift_homology_functor_iso (n k m : \u2124) (h : k + n = m):\n  shift_functor _ n \u22d9 homology_functor C k \u2245 homology_functor C m :=\nlocalization.lift_nat_iso Q (quasi_isomorphisms C _) _ _ _ _\n    (cochain_complex.shift_homology_functor_iso C n k m h)\n\nvariable {C}\n\nlemma shift_homology_functor_iso_hom_app_Q_obj (n k m : \u2124) (h : k + n = m)\n  (X : cochain_complex C \u2124) :\n  (shift_homology_functor_iso C n k m h).hom.app (Q.obj X) =\n  (homology_functor C k).map ((Q.comm_shift_iso n).inv.app X) \u226b\n      (homology_functor_factors C k).hom.app (X\u27e6n\u27e7) \u226b\n      ((cochain_complex.shift_homology_functor_iso C n k m h).hom.app X) \u226b\n      (homology_functor_factors C m).inv.app X :=\nbegin\n  dsimp only [shift_homology_functor_iso],\n  simp only [localization.lift_nat_iso_hom, localization.lift_nat_trans_app,\n    shift_functor_comp_homology_lifting_iso_hom_app,\n    lifting_iso_eq_homology_functor_factors, assoc],\nend\n\nvariable (C)\n\n@[simp, reassoc]\nlemma shift_homology_functor_iso_hom_comp (n k m n' : \u2124) (h : k+n=m) (h' : m+n' = k) :\n  whisker_left (shift_functor (derived_category C) n')\n    (shift_homology_functor_iso C n k m h).hom \u226b (shift_homology_functor_iso C n' m k h').hom =\n  (whisker_right ((shift_functor_add' (derived_category C) n' n 0\n      (by rw [\u2190 add_right_inj m, \u2190 add_assoc, h', h, add_zero])).inv \u226b\n      (shift_functor_zero (derived_category C) \u2124).hom) (homology_functor C k)) :=\nlocalization.nat_trans_ext Q (quasi_isomorphisms C _) _ _ (\u03bb K, begin\n  dsimp only [whisker_left, whisker_right, nat_trans.comp_app,\n    shift_homology_functor_iso, localization.lift_nat_iso],\n  simp only [assoc, localization.lift_nat_trans_app,\n    shift_functor_comp_homology_lifting_iso_hom_app,\n    \u2190 nat_trans.naturality_assoc, functor.comp_map],\n  erw localization.lift_nat_trans_app,\n  simp only [assoc, lifting_iso_eq_homology_functor_factors, iso.inv_hom_id_app_assoc],\n  simp only [assoc, lifting_iso_eq_homology_functor_factors, iso.inv_hom_id_app_assoc,\n    shift_functor_comp_homology_lifting_iso_hom_app,\n    cochain_complex.shift_homology_functor_iso_hom_app_comp_assoc,\n    (homology_functor_factors C k).inv.naturality, functor.comp_map,\n    iso.hom_inv_id_app_assoc, \u2190 functor.map_comp,\n    Q.shift_functor_zero_hom_app_obj \u2124 K],\n  congr' 1,\n  simp only [\u2190 cancel_epi ((shift_functor_add' (derived_category C)\n    n' n 0 (by linarith)).hom.app (Q.obj K)), iso.hom_inv_id_app_assoc],\n  rw Q.shift_functor_add'_hom_app_obj n' n 0 (by linarith) K,\n  slice_lhs 4 5 { erw [\u2190 functor.map_comp, iso.hom_inv_id_app,\n    category_theory.functor.map_id], },\n  erw [id_comp, iso.hom_inv_id_app_assoc],\n  rw [Q.map_comp, \u2190 Q.map_comp_assoc, iso.hom_inv_id_app, Q.map_id, id_comp],\nend)\n\nlemma shift_homology_functor_iso_hom_app_comp (n k m n' : \u2124) (h : k+n=m) (h' : m+n' = k)\n  (X : derived_category C) :\n  (shift_homology_functor_iso C n k m h).hom.app (X\u27e6n'\u27e7) \u226b (shift_homology_functor_iso C n' m k h').hom.app X =\n  (homology_functor C k).map ((shift_functor_add' (derived_category C) n' n 0 (by linarith)).inv.app X \u226b\n      (shift_functor_zero (derived_category C) \u2124).hom.app X) :=\ncongr_app (shift_homology_functor_iso_hom_comp C n k m n' h h') X\n\nsection\n\nvariables {C} {T : triangle (derived_category C)}\n\nlemma homology_sequence.ex\u2082 (hT : T.distinguished) (n : \u2124) :\n  (short_complex.mk ((homology_functor C n).map T.mor\u2081) ((homology_functor C n).map T.mor\u2082)\n    (by simp only [\u2190 functor.map_comp, hT.comp_zero\u2081\u2082, functor.map_zero])).exact :=\nfunctor.is_homological.map_distinguished (homology_functor C n) T hT\n\ndef homology_sequence.\u03b4 (hT : T.distinguished) (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (homology_functor C n\u2080).obj T.obj\u2083 \u27f6 (homology_functor C n\u2081).obj T.obj\u2081 :=\n(homology_functor C n\u2080).map T.mor\u2083 \u226b\n  (shift_homology_functor_iso C _ _ _ h.symm).hom.app T.obj\u2081\n\n@[simp, reassoc]\nlemma homology_sequence.\u03b4_comp (hT : T.distinguished) (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  homology_sequence.\u03b4 hT _ _ h \u226b (homology_functor C n\u2081).map T.mor\u2081 = 0 :=\nbegin\n  dsimp only [homology_sequence.\u03b4],\n  simp only [assoc, \u2190 nat_trans.naturality, functor.comp_map, \u2190 functor.map_comp_assoc,\n    hT.comp_zero\u2083\u2081, functor.map_zero, zero_comp],\nend\n\n@[simp, reassoc]\nlemma homology_sequence.comp_\u03b4 (hT : T.distinguished) (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (homology_functor C n\u2080).map T.mor\u2082 \u226b homology_sequence.\u03b4 hT _ _ h = 0 :=\nbegin\n  dsimp only [homology_sequence.\u03b4],\n  rw [\u2190 functor.map_comp_assoc, hT.comp_zero\u2082\u2083, functor.map_zero, zero_comp],\nend\n\nlemma homology_sequence.ex\u2083 (hT : T.distinguished) (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (short_complex.mk ((homology_functor C n\u2080).map T.mor\u2082) (homology_sequence.\u03b4 hT _ _ h)\n    (by simp)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1 (homology_sequence.ex\u2082 hT.rotate n\u2080),\n  exact short_complex.mk_iso (iso.refl _) (iso.refl _)\n    ((shift_homology_functor_iso C _ _ _ h.symm).app _)\n    (by { dsimp, simp only [id_comp, comp_id], }) (id_comp _),\nend\n\nlemma homology_sequence.ex\u2081 (hT : T.distinguished) (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (short_complex.mk (homology_sequence.\u03b4 hT _ _ h) ((homology_functor C n\u2081).map T.mor\u2081)\n    (by simp)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1 (homology_sequence.ex\u2082 hT.inv_rotate n\u2081),\n  refine short_complex.mk_iso (preadditive.mul_iso (-1) ((shift_homology_functor_iso C (-1) n\u2081 n\u2080 (by linarith)).app _))\n    (iso.refl _) (iso.refl _) _ _,\n  { dsimp only [triangle.inv_rotate, preadditive.mul_iso, iso.refl, triangle.mk, homology_sequence.\u03b4],\n    simp only [comp_id, functor.map_neg, units.coe_neg_one, neg_smul, one_smul,\n      preadditive.neg_comp, neg_inj, iso.app_hom, \u2190 nat_trans.naturality_assoc,\n      functor.comp_map, functor.map_comp, shift_homology_functor_iso_hom_app_comp,\n      shift_functor_comp_shift_functor_neg_eq_add'_comp_zero], },\n  { dsimp, simp only [id_comp, comp_id], },\nend\n\nend\n\nend derived_category\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/derived_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3733224068534647}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.abelian.basic\nimport category_theory.preadditive.functor_category\nimport category_theory.limits.shapes.functor_category\nimport category_theory.limits.preserves.shapes.kernels\n\n/-!\n# If `D` is abelian, then the functor category `C \u2964 D` is also abelian.\n\n-/\n\nnoncomputable theory\n\nnamespace category_theory\nopen category_theory.limits\n\nnamespace abelian\n\nsection\nuniverses z w v u\nvariables {C : Type (max v u)} [category.{v} C]\nvariables {D : Type w} [category.{max z v u} D] [abelian D]\n\nnamespace functor_category\nvariables {F G : C \u2964 D} (\u03b1 : F \u27f6 G) (X : C)\n\n/-- The abelian coimage in a functor category can be calculated componentwise. -/\n@[simps]\ndef coimage_obj_iso : (abelian.coimage \u03b1).obj X \u2245 abelian.coimage (\u03b1.app X) :=\npreserves_cokernel.iso ((evaluation C D).obj X) _ \u226a\u226b\n  cokernel.map_iso _ _ (preserves_kernel.iso ((evaluation C D).obj X) _) (iso.refl _)\n  begin\n    dsimp,\n    simp only [category.comp_id],\n    exact (kernel_comparison_comp_\u03b9 _ ((evaluation C D).obj X)).symm,\n  end\n\n/-- The abelian image in a functor category can be calculated componentwise. -/\n@[simps]\ndef image_obj_iso : (abelian.image \u03b1).obj X \u2245 abelian.image (\u03b1.app X) :=\npreserves_kernel.iso ((evaluation C D).obj X) _ \u226a\u226b\n  kernel.map_iso _ _ (iso.refl _) (preserves_cokernel.iso ((evaluation C D).obj X) _)\n  begin\n    apply (cancel_mono (preserves_cokernel.iso ((evaluation C D).obj X) \u03b1).inv).1,\n    simp only [category.assoc, iso.hom_inv_id],\n    dsimp,\n    simp only [category.id_comp, category.comp_id],\n    exact (\u03c0_comp_cokernel_comparison _ ((evaluation C D).obj X)).symm,\n  end\n\nlemma coimage_image_comparison_app :\n  coimage_image_comparison (\u03b1.app X) =\n    (coimage_obj_iso \u03b1 X).inv \u226b (coimage_image_comparison \u03b1).app X \u226b (image_obj_iso \u03b1 X).hom :=\nbegin\n  ext,\n  dsimp,\n  simp only [category.comp_id, category.id_comp, category.assoc,\n    coimage_image_factorisation, limits.cokernel.\u03c0_desc_assoc, limits.kernel.lift_\u03b9],\n  simp only [\u2190evaluation_obj_map C D X],\n  erw kernel_comparison_comp_\u03b9 _ ((evaluation C D).obj X),\n  erw \u03c0_comp_cokernel_comparison_assoc _ ((evaluation C D).obj X),\n  simp only [\u2190functor.map_comp],\n  simp only [coimage_image_factorisation, evaluation_obj_map],\nend\n\nlemma coimage_image_comparison_app' :\n  (coimage_image_comparison \u03b1).app X =\n    (coimage_obj_iso \u03b1 X).hom \u226b coimage_image_comparison (\u03b1.app X) \u226b (image_obj_iso \u03b1 X).inv :=\nby simp only [coimage_image_comparison_app, iso.hom_inv_id_assoc, iso.hom_inv_id, category.assoc,\n  category.comp_id]\n\ninstance functor_category_is_iso_coimage_image_comparison :\n  is_iso (abelian.coimage_image_comparison \u03b1) :=\nbegin\n  haveI : \u2200 X : C, is_iso ((abelian.coimage_image_comparison \u03b1).app X),\n  { intros, rw coimage_image_comparison_app', apply_instance, },\n  apply nat_iso.is_iso_of_is_iso_app,\nend\n\nend functor_category\n\nnoncomputable instance functor_category_abelian : abelian (C \u2964 D) :=\nabelian.of_coimage_image_comparison_is_iso\n\nend\n\nsection\n\nuniverses u\nvariables {C : Type u} [small_category C]\nvariables {D : Type (u+1)} [large_category D] [abelian D]\n\n/-- A variant with specialized universes for a common case. -/\nnoncomputable instance functor_category_abelian' : abelian (C \u2964 D) :=\nabelian.functor_category_abelian.{u u+1 u u}\n\nend\n\nend abelian\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/abelian/functor_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3733224068534647}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.rbtree.default\n\nuniverses u v \n\nnamespace Mathlib\n\nnamespace rbmap\n\n\n/- Auxiliary instances -/\n\n/- Helper lemmas for reusing rbtree results. -/\n\ntheorem eq_some_of_to_value_eq_some {\u03b1 : Type u} {\u03b2 : Type v} {e : Option (\u03b1 \u00d7 \u03b2)} {v : \u03b2} :\n    to_value e = some v \u2192 \u2203 (k : \u03b1), e = some (k, v) :=\n  sorry\n\ntheorem eq_none_of_to_value_eq_none {\u03b1 : Type u} {\u03b2 : Type v} {e : Option (\u03b1 \u00d7 \u03b2)} :\n    to_value e = none \u2192 e = none :=\n  sorry\n\n/- Lemmas -/\n\ntheorem not_mem_mk_rbmap {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} (k : \u03b1) :\n    \u00ack \u2208 mk_rbmap \u03b1 \u03b2 :=\n  sorry\n\ntheorem not_mem_of_empty {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} {m : rbmap \u03b1 \u03b2} (k : \u03b1) :\n    empty m = tt \u2192 \u00ack \u2208 m :=\n  sorry\n\ntheorem not_mem_of_find_entry_none {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k : \u03b1} {m : rbmap \u03b1 \u03b2} : find_entry m k = none \u2192 \u00ack \u2208 m :=\n  sorry\n\ntheorem not_mem_of_find_none {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k : \u03b1} {m : rbmap \u03b1 \u03b2} : find m k = none \u2192 \u00ack \u2208 m :=\n  sorry\n\ntheorem mem_of_find_entry_some {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {e : \u03b1 \u00d7 \u03b2} {m : rbmap \u03b1 \u03b2} :\n    find_entry m k\u2081 = some e \u2192 k\u2081 \u2208 m :=\n  sorry\n\ntheorem mem_of_find_some {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} : find m k = some v \u2192 k \u2208 m :=\n  sorry\n\ntheorem find_entry_eq_find_entry_of_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop}\n    [DecidableRel lt] [is_strict_weak_order \u03b1 lt] {m : rbmap \u03b1 \u03b2} {k\u2081 : \u03b1} {k\u2082 : \u03b1} :\n    strict_weak_order.equiv k\u2081 k\u2082 \u2192 find_entry m k\u2081 = find_entry m k\u2082 :=\n  sorry\n\ntheorem find_eq_find_of_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) :\n    strict_weak_order.equiv k\u2081 k\u2082 \u2192 find m k\u2081 = find m k\u2082 :=\n  sorry\n\ntheorem find_entry_correct {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (k : \u03b1) (m : rbmap \u03b1 \u03b2) :\n    k \u2208 m \u2194 \u2203 (e : \u03b1 \u00d7 \u03b2), find_entry m k = some e \u2227 strict_weak_order.equiv k (prod.fst e) :=\n  sorry\n\ntheorem eqv_of_find_entry_some {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    find_entry m k\u2081 = some (k\u2082, v) \u2192 strict_weak_order.equiv k\u2081 k\u2082 :=\n  sorry\n\ntheorem eq_of_find_entry_some {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    find_entry m k\u2081 = some (k\u2082, v) \u2192 k\u2081 = k\u2082 :=\n  fun (h : find_entry m k\u2081 = some (k\u2082, v)) =>\n    (fun (this : strict_weak_order.equiv k\u2081 k\u2082) => eq_of_eqv_lt this) (eqv_of_find_entry_some h)\n\ntheorem find_correct {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (k : \u03b1) (m : rbmap \u03b1 \u03b2) : k \u2208 m \u2194 \u2203 (v : \u03b2), find m k = some v :=\n  sorry\n\ntheorem constains_correct {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (k : \u03b1) (m : rbmap \u03b1 \u03b2) : k \u2208 m \u2194 contains m k = tt :=\n  sorry\n\ntheorem mem_of_mem_of_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {m : rbmap \u03b1 \u03b2} {k\u2081 : \u03b1} {k\u2082 : \u03b1} :\n    k\u2081 \u2208 m \u2192 strict_weak_order.equiv k\u2081 k\u2082 \u2192 k\u2082 \u2208 m :=\n  sorry\n\ntheorem mem_insert_of_incomp {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    \u00aclt k\u2081 k\u2082 \u2227 \u00aclt k\u2082 k\u2081 \u2192 k\u2081 \u2208 insert m k\u2082 v :=\n  fun (h : \u00aclt k\u2081 k\u2082 \u2227 \u00aclt k\u2082 k\u2081) =>\n    to_rbmap_mem (rbtree.mem_insert_of_incomp m (eqv_entries_of_eqv_keys v v h))\n\ntheorem mem_insert {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (k : \u03b1) (m : rbmap \u03b1 \u03b2) (v : \u03b2) : k \u2208 insert m k v :=\n  to_rbmap_mem (rbtree.mem_insert (k, v) m)\n\ntheorem mem_insert_of_equiv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    strict_weak_order.equiv k\u2081 k\u2082 \u2192 k\u2081 \u2208 insert m k\u2082 v :=\n  mem_insert_of_incomp m v\n\ntheorem mem_insert_of_mem {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {m : rbmap \u03b1 \u03b2} (k\u2082 : \u03b1) (v : \u03b2) :\n    k\u2081 \u2208 m \u2192 k\u2081 \u2208 insert m k\u2082 v :=\n  fun (h : k\u2081 \u2208 m) => to_rbmap_mem (rbtree.mem_insert_of_mem (k\u2082, v) (to_rbtree_mem' v h))\n\ntheorem equiv_or_mem_of_mem_insert {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    k\u2081 \u2208 insert m k\u2082 v \u2192 strict_weak_order.equiv k\u2081 k\u2082 \u2228 k\u2081 \u2208 m :=\n  sorry\n\ntheorem incomp_or_mem_of_mem_ins {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    k\u2081 \u2208 insert m k\u2082 v \u2192 \u00aclt k\u2081 k\u2082 \u2227 \u00aclt k\u2082 k\u2081 \u2228 k\u2081 \u2208 m :=\n  equiv_or_mem_of_mem_insert\n\ntheorem eq_or_mem_of_mem_ins {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    k\u2081 \u2208 insert m k\u2082 v \u2192 k\u2081 = k\u2082 \u2228 k\u2081 \u2208 m :=\n  sorry\n\ntheorem find_entry_insert_of_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (m : rbmap \u03b1 \u03b2) {k\u2081 : \u03b1} {k\u2082 : \u03b1} (v : \u03b2) :\n    strict_weak_order.equiv k\u2081 k\u2082 \u2192 find_entry (insert m k\u2081 v) k\u2082 = some (k\u2081, v) :=\n  sorry\n\ntheorem find_entry_insert {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (m : rbmap \u03b1 \u03b2) (k : \u03b1) (v : \u03b2) :\n    find_entry (insert m k v) k = some (k, v) :=\n  find_entry_insert_of_eqv m v (refl k)\n\ntheorem find_insert_of_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (m : rbmap \u03b1 \u03b2) {k\u2081 : \u03b1} {k\u2082 : \u03b1} (v : \u03b2) :\n    strict_weak_order.equiv k\u2081 k\u2082 \u2192 find (insert m k\u2081 v) k\u2082 = some v :=\n  sorry\n\ntheorem find_insert {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] (m : rbmap \u03b1 \u03b2) (k : \u03b1) (v : \u03b2) : find (insert m k v) k = some v :=\n  find_insert_of_eqv m v (refl k)\n\ntheorem find_entry_insert_of_disj {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    lt k\u2081 k\u2082 \u2228 lt k\u2082 k\u2081 \u2192 find_entry (insert m k\u2081 v) k\u2082 = find_entry m k\u2082 :=\n  sorry\n\ntheorem find_entry_insert_of_not_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    \u00acstrict_weak_order.equiv k\u2081 k\u2082 \u2192 find_entry (insert m k\u2081 v) k\u2082 = find_entry m k\u2082 :=\n  sorry\n\ntheorem find_entry_insert_of_ne {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    k\u2081 \u2260 k\u2082 \u2192 find_entry (insert m k\u2081 v) k\u2082 = find_entry m k\u2082 :=\n  fun (h : k\u2081 \u2260 k\u2082) =>\n    find_entry_insert_of_not_eqv m v fun (h' : strict_weak_order.equiv k\u2081 k\u2082) => h (eq_of_eqv_lt h')\n\ntheorem find_insert_of_disj {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    lt k\u2081 k\u2082 \u2228 lt k\u2082 k\u2081 \u2192 find (insert m k\u2081 v) k\u2082 = find m k\u2082 :=\n  sorry\n\ntheorem find_insert_of_not_eqv {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    \u00acstrict_weak_order.equiv k\u2081 k\u2082 \u2192 find (insert m k\u2081 v) k\u2082 = find m k\u2082 :=\n  sorry\n\ntheorem find_insert_of_ne {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k\u2081 : \u03b1} {k\u2082 : \u03b1} (m : rbmap \u03b1 \u03b2) (v : \u03b2) :\n    k\u2081 \u2260 k\u2082 \u2192 find (insert m k\u2081 v) k\u2082 = find m k\u2082 :=\n  sorry\n\ntheorem mem_of_min_eq {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} : min m = some (k, v) \u2192 k \u2208 m :=\n  fun (h : min m = some (k, v)) => to_rbmap_mem (rbtree.mem_of_min_eq h)\n\ntheorem mem_of_max_eq {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} : max m = some (k, v) \u2192 k \u2208 m :=\n  fun (h : max m = some (k, v)) => to_rbmap_mem (rbtree.mem_of_max_eq h)\n\ntheorem eq_leaf_of_min_eq_none {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {m : rbmap \u03b1 \u03b2} : min m = none \u2192 m = mk_rbmap \u03b1 \u03b2 :=\n  rbtree.eq_leaf_of_min_eq_none\n\ntheorem eq_leaf_of_max_eq_none {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {m : rbmap \u03b1 \u03b2} : max m = none \u2192 m = mk_rbmap \u03b1 \u03b2 :=\n  rbtree.eq_leaf_of_max_eq_none\n\ntheorem min_is_minimal {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    min m = some (k, v) \u2192 \u2200 {k' : \u03b1}, k' \u2208 m \u2192 strict_weak_order.equiv k k' \u2228 lt k k' :=\n  sorry\n\ntheorem max_is_maximal {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_weak_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    max m = some (k, v) \u2192 \u2200 {k' : \u03b1}, k' \u2208 m \u2192 strict_weak_order.equiv k k' \u2228 lt k' k :=\n  sorry\n\ntheorem min_is_minimal_of_total {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    min m = some (k, v) \u2192 \u2200 {k' : \u03b1}, k' \u2208 m \u2192 k = k' \u2228 lt k k' :=\n  sorry\n\ntheorem max_is_maximal_of_total {\u03b1 : Type u} {\u03b2 : Type v} {lt : \u03b1 \u2192 \u03b1 \u2192 Prop} [DecidableRel lt]\n    [is_strict_total_order \u03b1 lt] {k : \u03b1} {v : \u03b2} {m : rbmap \u03b1 \u03b2} :\n    max m = some (k, v) \u2192 \u2200 {k' : \u03b1}, k' \u2208 m \u2192 k = k' \u2228 lt k' k :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/data/rbmap/default_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.37324638354426765}}
{"text": "example (p q r : Prop) (hp : p) : q \u2228 p \u2228 r :=\n  by repeat (first |apply Or.inl; assumption | apply Or.inr | assumption)\n example (p q r : Prop) (hp : p)\n         : (p \u2228 q \u2228 r) \u2227 (q \u2228 p \u2228 r) \u2227 (q \u2228 r \u2228 p) := by\n  repeat(any_goals (first | apply And.intro | apply Or.inl; assumption | apply Or.inr | assumption))\n \nvariable (x y : Nat)\n\ndef double := x + x\n#eval Lean.versionString\n\n#check double y\n#check Nat.succ_ne_zero\n", "meta": {"author": "Shraze97", "repo": "lean-4", "sha": "1500213bd98a211c1a62e95d14e8e200a151a2cf", "save_path": "github-repos/lean/Shraze97-lean-4", "path": "github-repos/lean/Shraze97-lean-4/lean-4-1500213bd98a211c1a62e95d14e8e200a151a2cf/Knot_main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.37324638354426753}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.sheaves.presheaf\nimport Mathlib.PostPort\n\nuniverses v u l \n\nnamespace Mathlib\n\n/-!\n# Presheafed spaces\n\nIntroduces the category of topological spaces equipped with a presheaf (taking values in an\narbitrary target category `C`.)\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\nnamespace algebraic_geometry\n\n\n/-- A `PresheafedSpace C` is a topological space equipped with a presheaf of `C`s. -/\nstructure PresheafedSpace (C : Type u) [category_theory.category C] where\n  carrier : Top\n  presheaf : Top.presheaf C carrier\n\nnamespace PresheafedSpace\n\n\nprotected instance coe_carrier {C : Type u} [category_theory.category C] :\n    has_coe (PresheafedSpace C) Top :=\n  has_coe.mk fun (X : PresheafedSpace C) => carrier X\n\n@[simp] theorem as_coe {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    carrier X = \u2191X :=\n  rfl\n\n@[simp] theorem mk_coe {C : Type u} [category_theory.category C] (carrier : Top)\n    (presheaf : Top.presheaf C carrier) : \u2191(mk carrier presheaf) = carrier :=\n  rfl\n\nprotected instance topological_space {C : Type u} [category_theory.category C]\n    (X : PresheafedSpace C) : topological_space \u21a5X :=\n  category_theory.bundled.str (carrier X)\n\n/-- The constant presheaf on `X` with value `Z`. -/\ndef const {C : Type u} [category_theory.category C] (X : Top) (Z : C) : PresheafedSpace C :=\n  mk X\n    (category_theory.functor.mk (fun (U : topological_space.opens \u21a5X\u1d52\u1d56) => Z)\n      fun (U V : topological_space.opens \u21a5X\u1d52\u1d56) (f : U \u27f6 V) => \ud835\udfd9)\n\nprotected instance inhabited {C : Type u} [category_theory.category C] [Inhabited C] :\n    Inhabited (PresheafedSpace C) :=\n  { default := const (Top.of pempty) Inhabited.default }\n\n/-- A morphism between presheafed spaces `X` and `Y` consists of a continuous map\n    `f` between the underlying topological spaces, and a (notice contravariant!) map\n    from the presheaf on `Y` to the pushforward of the presheaf on `X` via `f`. -/\nstructure hom {C : Type u} [category_theory.category C] (X : PresheafedSpace C)\n    (Y : PresheafedSpace C)\n    where\n  base : \u2191X \u27f6 \u2191Y\n  c : PresheafedSpace.presheaf Y \u27f6 base _* PresheafedSpace.presheaf X\n\ntheorem ext {C : Type u} [category_theory.category C] {X : PresheafedSpace C}\n    {Y : PresheafedSpace C} (\u03b1 : hom X Y) (\u03b2 : hom X Y) (w : hom.base \u03b1 = hom.base \u03b2)\n    (h :\n      hom.c \u03b1 \u226b\n          category_theory.whisker_right\n            (category_theory.nat_trans.op\n              (category_theory.iso.inv\n                (topological_space.opens.map_iso (hom.base \u03b1) (hom.base \u03b2) w)))\n            (PresheafedSpace.presheaf X) =\n        hom.c \u03b2) :\n    \u03b1 = \u03b2 :=\n  sorry\n\n/-- The identity morphism of a `PresheafedSpace`. -/\ndef id {C : Type u} [category_theory.category C] (X : PresheafedSpace C) : hom X X :=\n  hom.mk \ud835\udfd9\n    (category_theory.iso.inv (category_theory.functor.left_unitor (PresheafedSpace.presheaf X)) \u226b\n      category_theory.whisker_right\n        (category_theory.nat_trans.op\n          (category_theory.iso.hom (topological_space.opens.map_id (carrier X))))\n        (PresheafedSpace.presheaf X))\n\nprotected instance hom_inhabited {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    Inhabited (hom X X) :=\n  { default := id X }\n\n/-- Composition of morphisms of `PresheafedSpace`s. -/\ndef comp {C : Type u} [category_theory.category C] {X : PresheafedSpace C} {Y : PresheafedSpace C}\n    {Z : PresheafedSpace C} (\u03b1 : hom X Y) (\u03b2 : hom Y Z) : hom X Z :=\n  hom.mk (hom.base \u03b1 \u226b hom.base \u03b2)\n    (hom.c \u03b2 \u226b\n      category_theory.whisker_left\n          (category_theory.functor.op (topological_space.opens.map (hom.base \u03b2))) (hom.c \u03b1) \u226b\n        category_theory.iso.inv\n          (Top.presheaf.pushforward.comp (PresheafedSpace.presheaf X) (hom.base \u03b1) (hom.base \u03b2)))\n\n/- The proofs below can be done by `tidy`, but it is too slow,\n   and we don't have a tactic caching mechanism. -/\n\n/-- The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map\n    from the presheaf on the target to the pushforward of the presheaf on the source. -/\nprotected instance category_of_PresheafedSpaces (C : Type u) [category_theory.category C] :\n    category_theory.category (PresheafedSpace C) :=\n  category_theory.category.mk\n\n@[simp] theorem id_base {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    hom.base \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\ntheorem id_c {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    hom.c \ud835\udfd9 =\n        category_theory.iso.inv (category_theory.functor.left_unitor (PresheafedSpace.presheaf X)) \u226b\n          category_theory.whisker_right\n            (category_theory.nat_trans.op\n              (category_theory.iso.hom (topological_space.opens.map_id (carrier X))))\n            (PresheafedSpace.presheaf X) :=\n  rfl\n\n@[simp] theorem id_c_app {C : Type u} [category_theory.category C] (X : PresheafedSpace C)\n    (U : topological_space.opens \u21a5(carrier X)\u1d52\u1d56) :\n    category_theory.nat_trans.app (hom.c \ud835\udfd9) U =\n        category_theory.eq_to_hom\n          (opposite.op_induction\n            (fun (U : topological_space.opens \u21a5(carrier X)) =>\n              subtype.cases_on U\n                fun (U_val : set \u21a5(carrier X)) (U_property : is_open U_val) =>\n                  Eq.refl\n                    (category_theory.functor.obj (PresheafedSpace.presheaf X)\n                      (opposite.op { val := U_val, property := U_property })))\n            U) :=\n  sorry\n\n@[simp] theorem comp_base {C : Type u} [category_theory.category C] {X : PresheafedSpace C}\n    {Y : PresheafedSpace C} {Z : PresheafedSpace C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n    hom.base (f \u226b g) = hom.base f \u226b hom.base g :=\n  rfl\n\n@[simp] theorem comp_c_app {C : Type u} [category_theory.category C] {X : PresheafedSpace C}\n    {Y : PresheafedSpace C} {Z : PresheafedSpace C} (\u03b1 : X \u27f6 Y) (\u03b2 : Y \u27f6 Z)\n    (U : topological_space.opens \u21a5(carrier Z)\u1d52\u1d56) :\n    category_theory.nat_trans.app (hom.c (\u03b1 \u226b \u03b2)) U =\n        category_theory.nat_trans.app (hom.c \u03b2) U \u226b\n          category_theory.nat_trans.app (hom.c \u03b1)\n              (opposite.op\n                (category_theory.functor.obj (topological_space.opens.map (hom.base \u03b2))\n                  (opposite.unop U))) \u226b\n            category_theory.nat_trans.app\n              (category_theory.iso.inv\n                (Top.presheaf.pushforward.comp (PresheafedSpace.presheaf X) (hom.base \u03b1)\n                  (hom.base \u03b2)))\n              U :=\n  rfl\n\ntheorem congr_app {C : Type u} [category_theory.category C] {X : PresheafedSpace C}\n    {Y : PresheafedSpace C} {\u03b1 : X \u27f6 Y} {\u03b2 : X \u27f6 Y} (h : \u03b1 = \u03b2)\n    (U : topological_space.opens \u21a5(carrier Y)\u1d52\u1d56) :\n    category_theory.nat_trans.app (hom.c \u03b1) U =\n        category_theory.nat_trans.app (hom.c \u03b2) U \u226b\n          category_theory.functor.map (PresheafedSpace.presheaf X)\n            (category_theory.eq_to_hom\n              (Eq._oldrec\n                (Eq.refl\n                  (category_theory.functor.obj\n                    (category_theory.functor.op (topological_space.opens.map (hom.base \u03b1))) U))\n                h)) :=\n  sorry\n\n/-- The forgetful functor from `PresheafedSpace` to `Top`. -/\ndef forget (C : Type u) [category_theory.category C] : PresheafedSpace C \u2964 Top :=\n  category_theory.functor.mk (fun (X : PresheafedSpace C) => \u2191X)\n    fun (X Y : PresheafedSpace C) (f : X \u27f6 Y) => hom.base f\n\n/--\nThe restriction of a presheafed space along an open embedding into the space.\n-/\n@[simp] theorem restrict_carrier {C : Type u} [category_theory.category C] {U : Top}\n    (X : PresheafedSpace C) (f : U \u27f6 \u2191X) (h : open_embedding \u21d1f) : carrier (restrict X f h) = U :=\n  Eq.refl (carrier (restrict X f h))\n\n/--\nThe map from the restriction of a presheafed space.\n-/\n@[simp] theorem of_restrict_c_app {C : Type u} [category_theory.category C] (U : Top)\n    (X : PresheafedSpace C) (f : U \u27f6 \u2191X) (h : open_embedding \u21d1f)\n    (V : topological_space.opens \u21a5(carrier X)\u1d52\u1d56) :\n    category_theory.nat_trans.app (hom.c (of_restrict U X f h)) V =\n        category_theory.functor.map (PresheafedSpace.presheaf X)\n          (category_theory.has_hom.hom.op\n            (coe_fn\n              (equiv.symm\n                (category_theory.adjunction.hom_equiv\n                  (is_open_map.adjunction (of_restrict._proof_2 U X f h))\n                  (category_theory.functor.obj (topological_space.opens.map f) (opposite.unop V))\n                  (opposite.unop V)))\n              \ud835\udfd9)) :=\n  Eq.refl (category_theory.nat_trans.app (hom.c (of_restrict U X f h)) V)\n\n/--\nThe map to the restriction of a presheafed space along the canonical inclusion from the top\nsubspace.\n-/\n@[simp] theorem to_restrict_top_base_to_fun_coe {C : Type u} [category_theory.category C]\n    (X : PresheafedSpace C) (x : \u21a5\u2191X) : \u2191(coe_fn (hom.base (to_restrict_top X)) x) = x :=\n  Eq.refl \u2191(coe_fn (hom.base (to_restrict_top X)) x)\n\n/--\nThe isomorphism from the restriction to the top subspace.\n-/\ndef restrict_top_iso {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    restrict X (topological_space.opens.inclusion \u22a4) (restrict_top_iso._proof_1 X) \u2245 X :=\n  category_theory.iso.mk\n    (of_restrict (category_theory.functor.obj (topological_space.opens.to_Top \u2191X) \u22a4) X\n      (topological_space.opens.inclusion \u22a4) sorry)\n    (to_restrict_top X)\n\n/--\nThe global sections, notated Gamma.\n-/\n@[simp] theorem \u0393_obj {C : Type u} [category_theory.category C] (X : PresheafedSpace C\u1d52\u1d56) :\n    category_theory.functor.obj \u0393 X =\n        category_theory.functor.obj (PresheafedSpace.presheaf (opposite.unop X)) (opposite.op \u22a4) :=\n  Eq.refl (category_theory.functor.obj \u0393 X)\n\ntheorem \u0393_obj_op {C : Type u} [category_theory.category C] (X : PresheafedSpace C) :\n    category_theory.functor.obj \u0393 (opposite.op X) =\n        category_theory.functor.obj (PresheafedSpace.presheaf X) (opposite.op \u22a4) :=\n  rfl\n\ntheorem \u0393_map_op {C : Type u} [category_theory.category C] {X : PresheafedSpace C}\n    {Y : PresheafedSpace C} (f : X \u27f6 Y) :\n    category_theory.functor.map \u0393 (category_theory.has_hom.hom.op f) =\n        category_theory.nat_trans.app (hom.c f) (opposite.op \u22a4) \u226b\n          category_theory.functor.map (PresheafedSpace.presheaf X)\n            (category_theory.has_hom.hom.op (topological_space.opens.le_map_top (hom.base f) \u22a4)) :=\n  rfl\n\nend PresheafedSpace\n\n\nend algebraic_geometry\n\n\nnamespace category_theory\n\n\nnamespace functor\n\n\n/-- We can apply a functor `F : C \u2964 D` to the values of the presheaf in any `PresheafedSpace C`,\n    giving a functor `PresheafedSpace C \u2964 PresheafedSpace D` -/\ndef map_presheaf {C : Type u} [category C] {D : Type u} [category D] (F : C \u2964 D) :\n    algebraic_geometry.PresheafedSpace C \u2964 algebraic_geometry.PresheafedSpace D :=\n  mk\n    (fun (X : algebraic_geometry.PresheafedSpace C) =>\n      algebraic_geometry.PresheafedSpace.mk (algebraic_geometry.PresheafedSpace.carrier X)\n        (algebraic_geometry.PresheafedSpace.presheaf X \u22d9 F))\n    fun (X Y : algebraic_geometry.PresheafedSpace C) (f : X \u27f6 Y) =>\n      algebraic_geometry.PresheafedSpace.hom.mk (algebraic_geometry.PresheafedSpace.hom.base f)\n        (whisker_right (algebraic_geometry.PresheafedSpace.hom.c f) F)\n\n@[simp] theorem map_presheaf_obj_X {C : Type u} [category C] {D : Type u} [category D] (F : C \u2964 D)\n    (X : algebraic_geometry.PresheafedSpace C) : \u2191(obj (map_presheaf F) X) = \u2191X :=\n  rfl\n\n@[simp] theorem map_presheaf_obj_presheaf {C : Type u} [category C] {D : Type u} [category D]\n    (F : C \u2964 D) (X : algebraic_geometry.PresheafedSpace C) :\n    algebraic_geometry.PresheafedSpace.presheaf (obj (map_presheaf F) X) =\n        algebraic_geometry.PresheafedSpace.presheaf X \u22d9 F :=\n  rfl\n\n@[simp] theorem map_presheaf_map_f {C : Type u} [category C] {D : Type u} [category D] (F : C \u2964 D)\n    {X : algebraic_geometry.PresheafedSpace C} {Y : algebraic_geometry.PresheafedSpace C}\n    (f : X \u27f6 Y) :\n    algebraic_geometry.PresheafedSpace.hom.base (map (map_presheaf F) f) =\n        algebraic_geometry.PresheafedSpace.hom.base f :=\n  rfl\n\n@[simp] theorem map_presheaf_map_c {C : Type u} [category C] {D : Type u} [category D] (F : C \u2964 D)\n    {X : algebraic_geometry.PresheafedSpace C} {Y : algebraic_geometry.PresheafedSpace C}\n    (f : X \u27f6 Y) :\n    algebraic_geometry.PresheafedSpace.hom.c (map (map_presheaf F) f) =\n        whisker_right (algebraic_geometry.PresheafedSpace.hom.c f) F :=\n  rfl\n\nend functor\n\n\nnamespace nat_trans\n\n\n/--\nA natural transformation induces a natural transformation between the `map_presheaf` functors.\n-/\ndef on_presheaf {C : Type u} [category C] {D : Type u} [category D] {F : C \u2964 D} {G : C \u2964 D}\n    (\u03b1 : F \u27f6 G) : functor.map_presheaf G \u27f6 functor.map_presheaf F :=\n  mk\n    fun (X : algebraic_geometry.PresheafedSpace C) =>\n      algebraic_geometry.PresheafedSpace.hom.mk \ud835\udfd9\n        (whisker_left (algebraic_geometry.PresheafedSpace.presheaf X) \u03b1 \u226b\n          iso.inv (functor.left_unitor (algebraic_geometry.PresheafedSpace.presheaf X \u22d9 G)) \u226b\n            whisker_right\n              (nat_trans.op\n                (iso.hom\n                  (topological_space.opens.map_id (algebraic_geometry.PresheafedSpace.carrier X))))\n              (algebraic_geometry.PresheafedSpace.presheaf X \u22d9 G))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebraic_geometry/presheafed_space_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3732463670712979}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.group.pi\nimport algebra.category.Group.preadditive\nimport category_theory.preadditive.biproducts\nimport algebra.category.Group.limits\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\nopen category_theory\nopen category_theory.limits\n\nopen_locale big_operators\n\nuniverses w u\n\nnamespace AddCommGroup\n\n-- As `AddCommGroup` is preadditive, and has all limits, it automatically has biproducts.\ninstance : has_binary_biproducts AddCommGroup :=\nhas_binary_biproducts.of_has_binary_products\n\ninstance : has_finite_biproducts AddCommGroup :=\nhas_finite_biproducts.of_has_finite_products\n\n-- We now construct explicit limit data,\n-- so we can compare the biproducts to the usual unbundled constructions.\n\n/--\nConstruct limit data for a binary product in `AddCommGroup`, using `AddCommGroup.of (G \u00d7 H)`.\n-/\n@[simps cone_X is_limit_lift]\ndef binary_product_limit_cone (G H : AddCommGroup.{u}) : limits.limit_cone (pair G H) :=\n{ cone :=\n  { X := AddCommGroup.of (G \u00d7 H),\n    \u03c0 := { app := \u03bb j, discrete.cases_on j\n      (\u03bb j, walking_pair.cases_on j (add_monoid_hom.fst G H) (add_monoid_hom.snd G H)),\n      naturality' := by rintros \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e9\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9; refl, }},\n  is_limit :=\n  { lift := \u03bb s, add_monoid_hom.prod (s.\u03c0.app \u27e8walking_pair.left\u27e9) (s.\u03c0.app \u27e8walking_pair.right\u27e9),\n    fac' := by { rintros s (\u27e8\u27e9|\u27e8\u27e9); { ext x, simp, } },\n    uniq' := \u03bb s m w, begin\n      ext; [rw \u2190 w \u27e8walking_pair.left\u27e9, rw \u2190 w \u27e8walking_pair.right\u27e9]; refl,\n    end, } }\n\n@[simp] lemma binary_product_limit_cone_cone_\u03c0_app_left (G H : AddCommGroup.{u}) :\n  (binary_product_limit_cone G H).cone.\u03c0.app \u27e8walking_pair.left\u27e9 = add_monoid_hom.fst G H := rfl\n\n@[simp] lemma binary_product_limit_cone_cone_\u03c0_app_right (G H : AddCommGroup.{u}) :\n  (binary_product_limit_cone G H).cone.\u03c0.app \u27e8walking_pair.right\u27e9 = add_monoid_hom.snd G H := rfl\n\n/--\nWe verify that the biproduct in AddCommGroup is isomorphic to\nthe cartesian product of the underlying types:\n-/\n@[simps hom_apply] noncomputable\ndef biprod_iso_prod (G H : AddCommGroup.{u}) : (G \u229e H : AddCommGroup) \u2245 AddCommGroup.of (G \u00d7 H) :=\nis_limit.cone_point_unique_up_to_iso\n  (binary_biproduct.is_limit G H)\n  (binary_product_limit_cone G H).is_limit\n\n@[simp, elementwise] lemma biprod_iso_prod_inv_comp_fst (G H : AddCommGroup.{u}) :\n  (biprod_iso_prod G H).inv \u226b biprod.fst = add_monoid_hom.fst G H :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.left)\n\n@[simp, elementwise] lemma biprod_iso_prod_inv_comp_snd (G H : AddCommGroup.{u}) :\n  (biprod_iso_prod G H).inv \u226b biprod.snd = add_monoid_hom.snd G H :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk walking_pair.right)\n\nnamespace has_limit\nvariables {J : Type w} (f : J \u2192 AddCommGroup.{max w u})\n\n/--\nThe map from an arbitrary cone over a indexed family of abelian groups\nto the cartesian product of those groups.\n-/\n@[simps]\ndef lift (s : fan f) :\n  s.X \u27f6 AddCommGroup.of (\u03a0 j,f j) :=\n{ to_fun := \u03bb x j, s.\u03c0.app \u27e8j\u27e9 x,\n  map_zero' := by { ext, simp },\n  map_add' := \u03bb x y, by { ext, simp }, }\n\n/--\nConstruct limit data for a product in `AddCommGroup`, using `AddCommGroup.of (\u03a0 j, F.obj j)`.\n-/\n@[simps] def product_limit_cone : limits.limit_cone (discrete.functor f) :=\n{ cone :=\n  { X := AddCommGroup.of (\u03a0 j, f j),\n    \u03c0 := discrete.nat_trans (\u03bb j, pi.eval_add_monoid_hom (\u03bb j, f j) j.as), },\n  is_limit :=\n  { lift := lift f,\n    fac' := \u03bb s j, by { cases j, ext, simp, },\n    uniq' := \u03bb s m w,\n    begin\n      ext x j,\n      dsimp only [has_limit.lift],\n      simp only [add_monoid_hom.coe_mk],\n      exact congr_arg (\u03bb g : s.X \u27f6 f j, (g : s.X \u2192 f j) x) (w \u27e8j\u27e9),\n    end, }, }\n\nend has_limit\n\nopen has_limit\n\nvariables {J : Type} [fintype J]\n\n/--\nWe verify that the biproduct we've just defined is isomorphic to the AddCommGroup structure\non the dependent function type\n-/\n@[simps hom_apply] noncomputable\ndef biproduct_iso_pi (f : J \u2192 AddCommGroup.{u}) :\n  (\u2a01 f : AddCommGroup) \u2245 AddCommGroup.of (\u03a0 j, f j) :=\nis_limit.cone_point_unique_up_to_iso\n  (biproduct.is_limit f)\n  (product_limit_cone f).is_limit\n\n@[simp, elementwise] lemma biproduct_iso_pi_inv_comp_\u03c0 (f : J \u2192 AddCommGroup.{u}) (j : J) :\n  (biproduct_iso_pi f).inv \u226b biproduct.\u03c0 f j = pi.eval_add_monoid_hom (\u03bb j, f j) j :=\nis_limit.cone_point_unique_up_to_iso_inv_comp _ _ (discrete.mk j)\n\nend AddCommGroup\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/category/Group/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6548947290421275, "lm_q1q2_score": 0.3731934977418124}}
{"text": "import Lean\nopen Lean Meta\n\n/- ## `MetaM`: Solutions -/\n\n/- ### 1. -/\n\n#eval show MetaM Unit from do\n  let hi \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `hi)\n  IO.println s!\"value in hi: {\u2190 instantiateMVars hi}\" -- ?_uniq.1\n\n  hi.mvarId!.assign (Expr.app (Expr.const `Nat.succ []) (Expr.const ``Nat.zero []))\n  IO.println s!\"value in hi: {\u2190 instantiateMVars hi}\" -- Nat.succ Nat.zero\n\n/- ### 2. -/\n\n-- It would output the same expression we gave it - there were no metavariables to instantiate.\n#eval show MetaM Unit from do\n  let instantiatedExpr \u2190 instantiateMVars (Expr.lam `x (Expr.const `Nat []) (Expr.bvar 0) BinderInfo.default)\n  IO.println instantiatedExpr -- fun (x : Nat) => x\n\n/- ### 3. -/\n\n#eval show MetaM Unit from do\n  let oneExpr := Expr.app (Expr.const `Nat.succ []) (Expr.const ``Nat.zero [])\n  let twoExpr := Expr.app (Expr.const `Nat.succ []) oneExpr\n\n  -- Create `mvar1` with type `Nat`\n  let mvar1 \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `mvar1)\n  -- Create `mvar2` with type `Nat`\n  let mvar2 \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `mvar2)\n  -- Create `mvar3` with type `Nat`\n  let mvar3 \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `mvar3)\n\n  -- Assign `mvar1` to `2 + ?mvar2 + ?mvar3`\n  mvar1.mvarId!.assign (Lean.mkAppN (Expr.const `Nat.add []) #[(Lean.mkAppN (Expr.const `Nat.add []) #[twoExpr, mvar2]), mvar3])\n\n  -- Assign `mvar3` to `1`\n  mvar3.mvarId!.assign oneExpr\n\n  -- Instantiate `mvar1`, which should result in expression `2 + ?mvar2 + 1`\n  let instantiatedMvar1 \u2190 instantiateMVars mvar1\n  IO.println instantiatedMvar1 -- Nat.add (Nat.add 2 ?_uniq.2) 1\n\n/- ### 4. -/\n\nelab \"explore\" : tactic => do\n  let mvarId : MVarId \u2190 Lean.Elab.Tactic.getMainGoal\n  let metavarDecl : MetavarDecl \u2190 mvarId.getDecl\n\n  IO.println \"Our metavariable\"\n  -- [anonymous] : 2 = 2\n  IO.println s!\"\\n{metavarDecl.userName} : {metavarDecl.type}\"\n\n  IO.println \"\\nAll of its local declarations\"\n  let localContext : LocalContext := metavarDecl.lctx\n  for (localDecl : LocalDecl) in localContext do\n    if localDecl.isImplementationDetail then\n      -- (implementation detail) red : 1 = 1 \u2192 2 = 2 \u2192 2 = 2\n      IO.println s!\"\\n(implementation detail) {localDecl.userName} : {localDecl.type}\"\n    else\n      -- hA : 1 = 1\n      -- hB : 2 = 2\n      IO.println s!\"\\n{localDecl.userName} : {localDecl.type}\"\n\ntheorem red (hA : 1 = 1) (hB : 2 = 2) : 2 = 2 := by\n  explore\n  sorry\n\n/- ### 5. -/\n\n-- The type of our metavariable `2 + 2`. We want to find a `localDecl` that has the same type, and `assign` our metavariable to that `localDecl`.\nelab \"solve\" : tactic => do\n  let mvarId : MVarId \u2190 Lean.Elab.Tactic.getMainGoal\n  let metavarDecl : MetavarDecl \u2190 mvarId.getDecl\n\n  let localContext : LocalContext := metavarDecl.lctx\n  for (localDecl : LocalDecl) in localContext do\n    if \u2190 Lean.Meta.isDefEq localDecl.type metavarDecl.type then\n      mvarId.assign localDecl.toExpr\n\ntheorem redSolved (hA : 1 = 1) (hB : 2 = 2) : 2 = 2 := by\n  solve\n\n/- ### 6. -/\n\ndef sixA : Bool \u2192 Bool := fun x => x\n-- .lam `x (.const `Bool []) (.bvar 0) (Lean.BinderInfo.default)\n#eval Lean.Meta.reduce (Expr.const `sixA [])\n\ndef sixB : Bool := (fun x => x) ((true && false) || true)\n-- .const `Bool.true []\n#eval Lean.Meta.reduce (Expr.const `sixB [])\n\ndef sixC : Nat := 800 + 2\n-- .lit (Lean.Literal.natVal 802)\n#eval Lean.Meta.reduce (Expr.const `sixC [])\n\n/- ### 7. -/\n\n#eval show MetaM Unit from do\n  let litExpr := Expr.lit (Lean.Literal.natVal 1)\n  let standardExpr := Expr.app (Expr.const ``Nat.succ []) (Expr.const ``Nat.zero [])\n\n  let isEqual \u2190 Lean.Meta.isDefEq litExpr standardExpr\n  IO.println isEqual -- true\n\n/- ### 8. -/\n\n-- a) `5 =?= (fun x => 5) ((fun y : Nat \u2192 Nat => y) (fun z : Nat => z))`\n-- Definitionally equal.\ndef expr2 := (fun x => 5) ((fun y : Nat \u2192 Nat => y) (fun z : Nat => z))\n#eval show MetaM Unit from do\n  let expr1 := Lean.mkNatLit 5\n  let expr2 := Expr.const `expr2 []\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- true\n\n-- b) `2 + 1 =?= 1 + 2`\n-- Definitionally equal.\n#eval show MetaM Unit from do\n  let expr1 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 2, Lean.mkNatLit 1]\n  let expr2 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 1, Lean.mkNatLit 2]\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- true\n\n-- c) `?a =?= 2`, where `?a` has a type `String`\n-- Not definitionally equal.\n#eval show MetaM Unit from do\n  let expr1 \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `String []) (userName := `expr1)\n  let expr2 := Lean.mkNatLit 2\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- false\n\n-- d) `?a + Int =?= \"hi\" + ?b`, where `?a` and `?b` don't have a type\n-- Definitionally equal.\n-- `?a` is assigned to `\"hi\"`, `?b` is assigned to `Int`.\n#eval show MetaM Unit from do\n  let a \u2190 Lean.Meta.mkFreshExprMVar Option.none (userName := `a)\n  let b \u2190 Lean.Meta.mkFreshExprMVar Option.none (userName := `b)\n  let expr1 := Lean.mkAppN (Expr.const `Nat.add []) #[a, Expr.const `Int []]\n  let expr2 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkStrLit \"hi\", b]\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- true\n\n  IO.println s!\"a: {\u2190 instantiateMVars a}\"\n  IO.println s!\"b: {\u2190 instantiateMVars b}\"\n\n-- e) `2 + ?a =?= 3`\n-- Not definitionally equal.\n#eval show MetaM Unit from do\n  let a \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `a)\n  let expr1 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 2, a]\n  let expr2 := Lean.mkNatLit 3\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- false\n\n-- f) `2 + ?a =?= 2 + 1`\n-- Definitionally equal.\n-- `?a` is assigned to `1`.\n#eval show MetaM Unit from do\n  let a \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `Nat []) (userName := `a)\n  let expr1 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 2, a]\n  let expr2 := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 2, Lean.mkNatLit 1]\n  let isEqual \u2190 Lean.Meta.isDefEq expr1 expr2\n  IO.println isEqual -- true\n\n  IO.println s!\"a: {\u2190 instantiateMVars a}\"\n\n/- ### 9. -/\n@[reducible] def reducibleDef     : Nat := 1 -- same as `abbrev`\n@[instance] def instanceDef       : Nat := 2 -- same as `instance`\ndef defaultDef                    : Nat := 3\n@[irreducible] def irreducibleDef : Nat := 4\n\n@[reducible] def sum := [reducibleDef, instanceDef, defaultDef, irreducibleDef]\n\n#eval show MetaM Unit from do\n  let constantExpr := Expr.const `sum []\n\n  Meta.withTransparency Meta.TransparencyMode.reducible do\n    let reducedExpr \u2190 Meta.reduce constantExpr\n    dbg_trace (\u2190 ppExpr reducedExpr) -- [1, instanceDef, defaultDef, irreducibleDef]\n\n  Meta.withTransparency Meta.TransparencyMode.instances do\n    let reducedExpr \u2190 Meta.reduce constantExpr\n    dbg_trace (\u2190 ppExpr reducedExpr) -- [1, 2, defaultDef, irreducibleDef]\n\n  Meta.withTransparency Meta.TransparencyMode.default do\n    let reducedExpr \u2190 Meta.reduce constantExpr\n    dbg_trace (\u2190 ppExpr reducedExpr) -- [1, 2, 3, irreducibleDef]\n\n  Meta.withTransparency Meta.TransparencyMode.all do\n    let reducedExpr \u2190 Meta.reduce constantExpr\n    dbg_trace (\u2190 ppExpr reducedExpr) -- [1, 2, 3, 4]\n\n  -- Note: if we don't set the transparency mode, we get a pretty strong `TransparencyMode.default`.\n  let reducedExpr \u2190 Meta.reduce constantExpr\n  dbg_trace (\u2190 ppExpr reducedExpr) -- [1, 2, 3, irreducibleDef]\n\n/- ### 10. -/\n\n-- Non-idiomatic: we can only use `Lean.mkAppN`.\ndef tenA : MetaM Expr := do\n  let body := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 1, Expr.bvar 0]\n  return Expr.lam `x (Expr.const `Nat []) body BinderInfo.default\n\n-- Idiomatic: we can use both `Lean.mkAppN` and `Lean.Meta.mkAppM`.\ndef tenB : MetaM Expr := do\n  Lean.Meta.withLocalDecl `x .default (Expr.const `Nat []) (fun x => do\n    -- let body := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkNatLit 1, x]\n    let body \u2190 Lean.Meta.mkAppM `Nat.add #[Lean.mkNatLit 1, x]\n    Lean.Meta.mkLambdaFVars #[x] body\n  )\n\n#eval show MetaM _ from do\n  ppExpr (\u2190 tenA) -- fun x => Nat.add 1 x\n#eval show MetaM _ from do\n  ppExpr (\u2190 tenB) -- fun x => Nat.add 1 x\n\n/- ### 11. -/\n\ndef eleven : MetaM Expr :=\n  return Expr.forallE `yellow (Expr.const `Nat []) (Expr.bvar 0) BinderInfo.default\n\n#eval show MetaM _ from do\n  dbg_trace (\u2190 eleven) -- forall (yellow : Nat), yellow\n\n/- ### 12. -/\n\n-- Non-idiomatic: we can only use `Lean.mkApp3`.\ndef twelveA : MetaM Expr := do\n  let nPlusOne := Expr.app (Expr.app (Expr.const `Nat.add []) (Expr.bvar 0)) (Lean.mkNatLit 1)\n  let forAllBody := Lean.mkApp3 (Expr.const ``Eq []) (Expr.const `Nat []) (Expr.bvar 0) nPlusOne\n  let forAll := Expr.forallE `n (Expr.const `Nat []) forAllBody BinderInfo.default\n  return forAll\n\n-- Idiomatic: we can use both `Lean.mkApp3` and `Lean.Meta.mkEq`.\ndef twelveB : MetaM Expr := do\n  withLocalDecl `n BinderInfo.default (Expr.const `Nat []) (fun x => do\n    let nPlusOne := Expr.app (Expr.app (Expr.const `Nat.add []) x) (Lean.mkNatLit 1)\n    -- let forAllBody := Lean.mkApp3 (Expr.const ``Eq []) (Expr.const `Nat []) x nPlusOne\n    let forAllBody \u2190 Lean.Meta.mkEq x nPlusOne\n    let forAll := mkForallFVars #[x] forAllBody\n    forAll\n  )\n\n#eval show MetaM _ from do\n  ppExpr (\u2190 twelveA) -- (n : Nat) \u2192 Eq Nat n (Nat.add n 1)\n\n#eval show MetaM _ from do\n  ppExpr (\u2190 twelveB) -- \u2200 (n : Nat), n = Nat.add n 1\n\n/- ### 13. -/\ndef thirteen : MetaM Expr := do\n  withLocalDecl `f BinderInfo.default (Expr.forallE `a (Expr.const `Nat []) (Expr.const `Nat []) .default) (fun y => do\n    let lamBody \u2190 withLocalDecl `n BinderInfo.default (Expr.const `Nat []) (fun x => do\n      let fn := Expr.app y x\n      let fnPlusOne := Expr.app y (Expr.app (Expr.app (Expr.const `Nat.add []) (x)) (Lean.mkNatLit 1))\n      let forAllBody := mkApp3 (mkConst ``Eq []) (Expr.const `Nat []) fn fnPlusOne\n      let forAll := mkForallFVars #[x] forAllBody\n      forAll\n    )\n    let lam := mkLambdaFVars #[y] lamBody\n    lam\n  )\n\n#eval show MetaM _ from do\n  ppExpr (\u2190 thirteen) -- fun f => (n : Nat) \u2192 Eq Nat (f n) (f (Nat.add n 1))\n\n/- ### 14. -/\n\n#eval show Lean.Elab.Term.TermElabM _ from do\n  let stx : Syntax \u2190 `(\u2200 (a : Prop) (b : Prop), a \u2228 b \u2192 b \u2192 a \u2227 a)\n  let expr \u2190 Elab.Term.elabTermAndSynthesize stx none\n\n  let (_, _, conclusion) \u2190 forallMetaTelescope expr\n  dbg_trace conclusion -- And ?_uniq.10 ?_uniq.10\n\n  let (_, _, conclusion) \u2190 forallMetaBoundedTelescope expr 2\n  dbg_trace conclusion -- (Or ?_uniq.14 ?_uniq.15) -> ?_uniq.15 -> (And ?_uniq.14 ?_uniq.14)\n\n  let (_, _, conclusion) \u2190 lambdaMetaTelescope expr\n  dbg_trace conclusion -- forall (a.1 : Prop) (b.1 : Prop), (Or a.1 b.1) -> b.1 -> (And a.1 a.1)\n\n/- ### 15. -/\n\n#eval show MetaM Unit from do\n  let a \u2190 Lean.Meta.mkFreshExprMVar (Expr.const `String []) (userName := `a)\n  let b \u2190 Lean.Meta.mkFreshExprMVar (Expr.sort (Nat.toLevel 1)) (userName := `b)\n  -- ?a + Int\n  let c := Lean.mkAppN (Expr.const `Nat.add []) #[a, Expr.const `Int []]\n  -- \"hi\" + ?b\n  let d := Lean.mkAppN (Expr.const `Nat.add []) #[Lean.mkStrLit \"hi\", b]\n\n  IO.println s!\"value in c: {\u2190 instantiateMVars c}\" -- Nat.add ?_uniq.1 Int\n  IO.println s!\"value in d: {\u2190 instantiateMVars d}\" -- Nat.add String ?_uniq.2\n\n  let state : SavedState \u2190 saveState\n  IO.println \"\\nSaved state\\n\"\n\n  if \u2190 Lean.Meta.isDefEq c d then\n    IO.println true\n    IO.println s!\"value in c: {\u2190 instantiateMVars c}\"\n    IO.println s!\"value in d: {\u2190 instantiateMVars d}\"\n\n  restoreState state\n  IO.println \"\\nRestored state\\n\"\n\n  IO.println s!\"value in c: {\u2190 instantiateMVars c}\"\n  IO.println s!\"value in d: {\u2190 instantiateMVars d}\"\n", "meta": {"author": "leanprover-community", "repo": "lean4-metaprogramming-book", "sha": "0b2e7e2c0cacac530ed947df878088c5d9715412", "save_path": "github-repos/lean/leanprover-community-lean4-metaprogramming-book", "path": "github-repos/lean/leanprover-community-lean4-metaprogramming-book/lean4-metaprogramming-book-0b2e7e2c0cacac530ed947df878088c5d9715412/lean/solutions/metam.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.6548947223065754, "lm_q1q2_score": 0.3731934939035402}}
{"text": "import data.real.basic\nimport order.lattice\nnoncomputable theory\n\n\nsection curried\n\nvariables {\u03b1 \u03b1' \u03b2 \u03b3 \u03b4 : Type*}\n\nclass curried (\u03b1 \u03b2 \u03b3 \u03b4 : Type*) := \n(comp (f : \u03b3 \u2192 \u03b4) : \u03b1 \u2192 \u03b2)\n\ninstance curried_base : curried (\u03b1 \u2192 \u03b3) (\u03b1 \u2192 \u03b4) \u03b3 \u03b4 := \u27e8\u03bb f g x, f (g x)\u27e9\n\ninstance curried_induction [curried \u03b1 \u03b2 \u03b3 \u03b4] : curried (\u03b1' \u2192 \u03b1) (\u03b1' \u2192 \u03b2) \u03b3 \u03b4 :=\n\u27e8\u03bb f g x, curried.comp f (g x)\u27e9\n\nend curried\n\nexample: curried.comp (nat.succ) (\u03bb x y z : \u2115, 0) = (\u03bb x y z : \u2115, 1) := rfl\n\n\n\nuniverse variables u v w x\n\n-- Need filters for non-sets! (Mario already proposed that refactoring)\n-- Add non-empty axiom?\nstructure filter (\u03b1 : Type*) [partial_order \u03b1] :=\n(elements : set \u03b1)\n(nonempty : \u2203 x, x \u2208 elements)\n(exists_inf (x y : \u03b1) : x \u2208 elements \u2192 y \u2208 elements \u2192 \u2203 z, z \u2264 x \u2227 z \u2264 y \u2227 z \u2208 elements)\n(mem_elements_of_le (x y : \u03b1) : x \u2208 elements \u2192 x \u2264 y \u2192 y \u2208 elements)\n\ninstance {\u03b1 : Type*} [partial_order \u03b1] : has_mem \u03b1 (filter \u03b1) := \u27e8\u03bb U F, U \u2208 F.elements\u27e9\n\nnamespace filter\nvariables {\u03b1 : Type u} \nvariables [partial_order \u03b1] {f g : filter \u03b1} {s t : \u03b1}\n\nopen set\n\n@[simp] protected lemma mem_mk {t : set \u03b1} {h\u2081 h\u2082 h\u2083} : s \u2208 mk t h\u2081 h\u2082 h\u2083 \u2194 s \u2208 t := iff.rfl\n\n@[simp] protected lemma mem_elements : s \u2208 f.elements \u2194 s \u2208 f := iff.rfl\n\nlemma filter_eq : \u2200{f g : filter \u03b1}, f.elements = g.elements \u2192 f = g\n| \u27e8_, _, _, _\u27e9 \u27e8_, _, _, _\u27e9 rfl := rfl\n\nlemma filter_eq_iff : f = g \u2194 f.elements = g.elements :=\n\u27e8congr_arg _, filter_eq\u27e9\n\nprotected lemma ext_iff : f = g \u2194 \u2200 s, s \u2208 f \u2194 s \u2208 g :=\nby simp only [filter_eq_iff, ext_iff, filter.mem_elements]\n\n@[ext]\nprotected lemma ext : (\u2200 s, s \u2208 f \u2194 s \u2208 g) \u2192 f = g :=\nfilter.ext_iff.2\n\nlemma mem_sets_of_superset : \u2200{x y : \u03b1}, x \u2208 f \u2192 x \u2264 y \u2192 y \u2208 f :=\nf.mem_elements_of_le\n\nsection principal\n\n/-- The principal filter of `s` is the set of all elements larger than or equal to `s`. -/\ndef principal (s : \u03b1) : filter \u03b1 :=\n{ elements := {t | s \u2264 t},\n  nonempty := \u27e8s, le_refl s\u27e9,\n  exists_inf := \u03bb x y hx hy, \u27e8s, hx, hy, le_refl s\u27e9,\n  mem_elements_of_le := \u03bb x y hx hxy, le_trans hx hxy }\n\nlocalized \"notation `\ud835\udcdf` := filter.principal\" in filter\n\ninstance [inhabited \u03b1]: inhabited (filter \u03b1) :=\n\u27e8\ud835\udcdf (default \u03b1)\u27e9\n\ninstance : complete_lattice (filter \u03b1) := sorry\nend principal\n\nopen_locale filter\n\ndef at_top [preorder \u03b1] : filter (set \u03b1) := \u2a05 a, \ud835\udcdf (Ici a)\n\n\nend filter\n\n\ndef filter.Limsup (f : filter (set \u211d)) : \u211d := Inf { a | {n | n \u2264 a} \u2208 f }\n\n\nvariables {\u03b1 \u03b1' \u03b2 \u03b3 \u03b4 : Type*} [partial_order \u03b2]\n\ndef filter.map (f : filter (set \u211d)) [curried \u03b1 \u03b2 \u211d Prop] (u : \u03b1) : filter \u03b2 := sorry\n\ndef limsup (f : filter (set \u211d)) [curried \u03b1 \u03b2 \u211d Prop] (u : \u03b1) : \u211d := (f.map u).Limsup\n\n\n\n\n\ndef at_top [has_bot \u03b1] [partial_order \u03b2] [preorder \u03b3] [curried \u03b1 \u03b2 \u211d Prop] : filter \u03b2 := \n  \u2a05 a : \u211d, filter.principal (curried.comp (\u03bb x, a \u2264 x) (\u22a5 : \u03b1))\n\n\n-- section semilattice_inf\n\n-- variables {\u03b1 : Type u} \n-- variables [semilattice_inf \u03b1] {f g : filter \u03b1} {s t : \u03b1}\n\n-- open semilattice_inf\n\n-- namespace filter\n\n-- def of_semilattice (\u03b1 : Type*) [semilattice_inf \u03b1] (elements : set \u03b1) \n--   (inf_in_elements : \u2200 x y, x \u2208 elements \u2192 y \u2208 elements \u2192 semilattice_inf.inf x y \u2208 elements) \n--   (in_elements_of_le : \u2200 x y, x \u2208 elements \u2192 x \u2264 y \u2192 y \u2208 elements) : \n--   filter \u03b1 :=\n\n-- example: semilattice_inf (\u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 Prop) := by apply_instance\n\n-- def of_set (\u03b1 : Type*) (elements : set (set \u03b1)) \n--   (inf_in_elements : \u2200 x y, x \u2208 elements \u2192 y \u2208 elements \u2192 x \u2229 y \u2208 elements) \n--   (in_elements_of_le : \u2200 x y, x \u2208 elements \u2192 x \u2286 y \u2192 y \u2208 elements) : \n--   filter (set \u03b1) :=\n-- of_semilattice (set \u03b1) elements inf_in_elements in_elements_of_le\n\n-- end filter\n\n-- def solution (f : \u211d \u2192 \u211d) := \n-- { a : \u211d // a = filter.at_bot.liminf (\u03bb x, f x)}\n\n\ndef solution (f : \u211d \u2192 \u211d) := \n\u03a3' a : \u211d, a = filter.at_bot.liminf (\u03bb x, f x)\n\n\n\n#check filter.liminf\n\n#check optimize (\u03bb (x : \u211d) (y : \u211d) (h : x + y \u2264 0), x - y)\n\n", "meta": {"author": "skbaek", "repo": "cvx", "sha": "c50c790c9116f9fac8dfe742903a62bdd7292c15", "save_path": "github-repos/lean/skbaek-cvx", "path": "github-repos/lean/skbaek-cvx/cvx-c50c790c9116f9fac8dfe742903a62bdd7292c15/src/alex_playground/filter.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3731905515651361}}
{"text": "\nuniverse u\nvariable {\u03b1 : Type u} {p : \u03b1 \u2192 Prop}\n\ntheorem ex (a : \u03b1) (h1 h2 : p a) (h : Subtype.mk a h1 = Subtype.mk a h1) : Subtype.mk a h1 = Subtype.mk a h2 :=\nh\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/proofIrrelFVar.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.3731905515651359}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport tactic.lint\nimport tactic.ext\nimport tactic.simps\n\nopen function\n\nnamespace subtype\nvariables {\u03b1 : Sort*} {\u03b2 : Sort*} {\u03b3 : Sort*} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop}\n\n/-- See Note [custom simps projection] -/\ndef simps.coe (x : subtype p) : \u03b1 := x\n\ninitialize_simps_projections subtype (val \u2192 coe)\n\n/-- A version of `x.property` or `x.2` where `p` is syntactically applied to the coercion of `x`\n  instead of `x.1`. A similar result is `subtype.mem` in `data.set.basic`. -/\nlemma prop (x : subtype p) : p x := x.2\n\n@[simp] lemma val_eq_coe {x : subtype p} : x.1 = \u2191x := rfl\n\n@[simp] protected theorem \u00abforall\u00bb {q : {a // p a} \u2192 Prop} :\n  (\u2200 x, q x) \u2194 (\u2200 a b, q \u27e8a, b\u27e9) :=\n\u27e8assume h a b, h \u27e8a, b\u27e9, assume h \u27e8a, b\u27e9, h a b\u27e9\n\n/-- An alternative version of `subtype.forall`. This one is useful if Lean cannot figure out `q`\n  when using `subtype.forall` from right to left. -/\nprotected theorem forall' {q : \u2200x, p x \u2192 Prop} :\n  (\u2200 x h, q x h) \u2194 (\u2200 x : {a // p a}, q x x.2) :=\n(@subtype.forall _ _ (\u03bb x, q x.1 x.2)).symm\n\n@[simp] protected theorem \u00abexists\u00bb {q : {a // p a} \u2192 Prop} :\n  (\u2203 x, q x) \u2194 (\u2203 a b, q \u27e8a, b\u27e9) :=\n\u27e8assume \u27e8\u27e8a, b\u27e9, h\u27e9, \u27e8a, b, h\u27e9, assume \u27e8a, b, h\u27e9, \u27e8\u27e8a, b\u27e9, h\u27e9\u27e9\n\n@[ext] protected lemma ext : \u2200 {a1 a2 : {x // p x}}, (a1 : \u03b1) = (a2 : \u03b1) \u2192 a1 = a2\n| \u27e8x, h1\u27e9 \u27e8.(x), h2\u27e9 rfl := rfl\n\nlemma ext_iff {a1 a2 : {x // p x}} : a1 = a2 \u2194 (a1 : \u03b1) = (a2 : \u03b1) :=\n\u27e8congr_arg _, subtype.ext\u27e9\n\nlemma heq_iff_coe_eq (h : \u2200 x, p x \u2194 q x) {a1 : {x // p x}} {a2 : {x // q x}} :\n  a1 == a2 \u2194 (a1 : \u03b1) = (a2 : \u03b1) :=\neq.rec (\u03bb a2', heq_iff_eq.trans ext_iff) (funext $ \u03bb x, propext (h x)) a2\n\nlemma ext_val {a1 a2 : {x // p x}} : a1.1 = a2.1 \u2192 a1 = a2 :=\nsubtype.ext\n\nlemma ext_iff_val {a1 a2 : {x // p x}} : a1 = a2 \u2194 a1.1 = a2.1 :=\next_iff\n\n@[simp] theorem coe_eta (a : {a // p a}) (h : p a) : mk \u2191a h = a := subtype.ext rfl\n\n@[simp] theorem coe_mk (a h) : (@mk \u03b1 p a h : \u03b1) = a := rfl\n\n@[simp, nolint simp_nf] -- built-in reduction doesn't always work\ntheorem mk_eq_mk {a h a' h'} : @mk \u03b1 p a h = @mk \u03b1 p a' h' \u2194 a = a' :=\next_iff\n\ntheorem coe_eq_iff {a : {a // p a}} {b : \u03b1} : \u2191a = b \u2194 \u2203 h, a = \u27e8b, h\u27e9 :=\n\u27e8\u03bb h, h \u25b8 \u27e8a.2, (coe_eta _ _).symm\u27e9, \u03bb \u27e8hb, ha\u27e9, ha.symm \u25b8 rfl\u27e9\n\ntheorem coe_injective : injective (coe : subtype p \u2192 \u03b1) :=\n\u03bb a b, subtype.ext\n\ntheorem val_injective : injective (@val _ p) :=\ncoe_injective\n\n/-- Restrict a (dependent) function to a subtype -/\ndef restrict {\u03b1} {\u03b2 : \u03b1 \u2192 Type*} (f : \u03a0x, \u03b2 x) (p : \u03b1 \u2192 Prop) (x : subtype p) : \u03b2 x.1 :=\nf x\n\nlemma restrict_apply {\u03b1} {\u03b2 : \u03b1 \u2192 Type*} (f : \u03a0x, \u03b2 x) (p : \u03b1 \u2192 Prop) (x : subtype p) :\n  restrict f p x = f x.1 :=\nby refl\n\nlemma restrict_def {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (p : \u03b1 \u2192 Prop) : restrict f p = f \u2218 coe :=\nby refl\n\nlemma restrict_injective {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} (p : \u03b1 \u2192 Prop) (h : injective f) :\n  injective (restrict f p) :=\nh.comp coe_injective\n\n/-- Defining a map into a subtype, this can be seen as an \"coinduction principle\" of `subtype`-/\n@[simps] def coind {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) {p : \u03b2 \u2192 Prop} (h : \u2200a, p (f a)) : \u03b1 \u2192 subtype p :=\n\u03bb a, \u27e8f a, h a\u27e9\n\ntheorem coind_injective {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} (h : \u2200a, p (f a))\n  (hf : injective f) : injective (coind f h) :=\n\u03bb x y hxy, hf $ by apply congr_arg subtype.val hxy\n\ntheorem coind_surjective {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} (h : \u2200a, p (f a))\n  (hf : surjective f) : surjective (coind f h) :=\n\u03bb x, let \u27e8a, ha\u27e9 := hf x in \u27e8a, coe_injective ha\u27e9\n\ntheorem coind_bijective {\u03b1 \u03b2} {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} (h : \u2200a, p (f a))\n  (hf : bijective f) : bijective (coind f h) :=\n\u27e8coind_injective h hf.1, coind_surjective h hf.2\u27e9\n\n/-- Restriction of a function to a function on subtypes. -/\n@[simps] def map {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) (h : \u2200a, p a \u2192 q (f a)) :\n  subtype p \u2192 subtype q :=\n\u03bb x, \u27e8f x, h x x.prop\u27e9\n\ntheorem map_comp {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} {r : \u03b3 \u2192 Prop} {x : subtype p}\n  (f : \u03b1 \u2192 \u03b2) (h : \u2200a, p a \u2192 q (f a)) (g : \u03b2 \u2192 \u03b3) (l : \u2200a, q a \u2192 r (g a)) :\n  map g l (map f h x) = map (g \u2218 f) (assume a ha, l (f a) $ h a ha) x :=\nrfl\n\ntheorem map_id {p : \u03b1 \u2192 Prop} {h : \u2200a, p a \u2192 p (id a)} : map (@id \u03b1) h = id :=\nfunext $ assume \u27e8v, h\u27e9, rfl\n\nlemma map_injective {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} (h : \u2200a, p a \u2192 q (f a))\n  (hf : injective f) : injective (map f h) :=\ncoind_injective _ $ hf.comp coe_injective\n\nlemma map_involutive {p : \u03b1 \u2192 Prop} {f : \u03b1 \u2192 \u03b1} (h : \u2200a, p a \u2192 p (f a))\n  (hf : involutive f) : involutive (map f h) :=\n\u03bb x, subtype.ext (hf x)\n\ninstance [has_equiv \u03b1] (p : \u03b1 \u2192 Prop) : has_equiv (subtype p) :=\n\u27e8\u03bb s t, (s : \u03b1) \u2248 (t : \u03b1)\u27e9\n\ntheorem equiv_iff [has_equiv \u03b1] {p : \u03b1 \u2192 Prop} {s t : subtype p} :\n  s \u2248 t \u2194 (s : \u03b1) \u2248 (t : \u03b1) :=\niff.rfl\n\nvariables [setoid \u03b1]\n\nprotected theorem refl (s : subtype p) : s \u2248 s :=\nsetoid.refl \u2191s\n\nprotected theorem symm {s t : subtype p} (h : s \u2248 t) : t \u2248 s :=\nsetoid.symm h\n\nprotected theorem trans {s t u : subtype p} (h\u2081 : s \u2248 t) (h\u2082 : t \u2248 u) : s \u2248 u :=\nsetoid.trans h\u2081 h\u2082\n\ntheorem equivalence (p : \u03b1 \u2192 Prop) : equivalence (@has_equiv.equiv (subtype p) _) :=\nmk_equivalence _ subtype.refl (@subtype.symm _ p _) (@subtype.trans _ p _)\n\ninstance (p : \u03b1 \u2192 Prop) : setoid (subtype p) :=\nsetoid.mk (\u2248) (equivalence p)\n\nend subtype\n\nnamespace subtype\n/-! Some facts about sets, which require that `\u03b1` is a type. -/\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {p : \u03b1 \u2192 Prop}\n\n@[simp] lemma coe_prop {S : set \u03b1} (a : {a // a \u2208 S}) : \u2191a \u2208 S := a.prop\n\nlemma val_prop {S : set \u03b1} (a : {a // a \u2208 S}) : a.val \u2208 S := a.property\n\nend subtype\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/subtype.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.7025300449389326, "lm_q1q2_score": 0.37319054494824716}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.concrete_category.basic\nimport category_theory.reflects_isomorphisms\n\n/-!\nA `forget\u2082 C D` forgetful functor between concrete categories `C` and `D`\nwhose forgetful functors both reflect isomorphisms, itself reflects isomorphisms.\n-/\n\nuniverses u\n\nnamespace category_theory\n\ninstance : reflects_isomorphisms (forget (Type u)) :=\n{ reflects := \u03bb X Y f i, i }\n\nvariables (C : Type (u+1)) [category C] [concrete_category.{u} C]\nvariables (D : Type (u+1)) [category D] [concrete_category.{u} D]\n\n/--\nA `forget\u2082 C D` forgetful functor between concrete categories `C` and `D`\nwhere `forget C` reflects isomorphisms, itself reflects isomorphisms.\n-/\n-- This should not be an instance, as it causes a typeclass loop\n-- with `category_theory.has_forget_to_Type`\nlemma reflects_isomorphisms_forget\u2082 [has_forget\u2082 C D] [reflects_isomorphisms (forget C)] :\n  reflects_isomorphisms (forget\u2082 C D) :=\n{ reflects := \u03bb X Y f i,\n  begin\n    resetI,\n    haveI i' : is_iso ((forget D).map ((forget\u2082 C D).map f)) := functor.map_is_iso (forget D) _,\n    haveI : is_iso ((forget C).map f) :=\n    begin\n      have := has_forget\u2082.forget_comp,\n      dsimp at this,\n      rw \u2190this,\n      exact i',\n    end,\n    apply is_iso_of_reflects_iso f (forget C),\n  end }\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/concrete_category/reflects_isomorphisms.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.37318910214583073}}
{"text": "import analysis.exponential analysis.polynomial\nimport ring_theory.prime_count\n\nopen complex polynomial finset\n\n-- inductive lep (p : polynomial \u2102) : polynomial \u2102 \u2192 Prop\n-- | C_mul   : \u2200 {a}, a \u2260 0 \u2192 lep (C a * p)\n-- | mul     : \u2200 {q}, lep q \u2192 lep (q * X)\n-- | add     : \u2200 {q} {a}, lep q \u2192 q \u2260 -C a \u2192 lep (q + C a)\n\n-- inductive ltp (p : polynomial \u2102) : polynomial \u2102 \u2192 Prop\n-- | mul_X : p \u2260 0 \u2192 ltp (p * X)\n-- | mul_C : \u2200 {a q}, a \u2260 0 \u2192 ltp q \u2192 ltp (C a * q)\n-- | add   : \u2200 {q} {a}, ltp q \u2192 q.eval 0 \u2260 a \u2192 ltp (q + C a)\n\n-- inductive ltp' : polynomial \u2102 \u2192 polynomial \u2102 \u2192 Prop\n-- | mul_X : \u2200 {p}, p \u2260 0 \u2192 ltp' p (p * X)\n-- | mul_C : \u2200 {p q a}, a \u2260 0 \u2192 ltp' p q \u2192 ltp' p (C a * q)\n-- | add   : \u2200 {p q a}, ltp' p q \u2192 q.eval 0 \u2260 a \u2192 ltp' p (q + C a)\n\n-- lemma growth_lemma_chris1 {p q : polynomial \u2102} (hpq : ltp p q) :\n--   \u2203 r : \u211d, \u2200 z : \u2102, r < z.abs \u2192\n--    abs (p.eval z) < abs (q.eval z) :=\n-- ltp.rec_on hpq\n--   (\u03bb hq g hpq, \u27e81, \u03bb z hz, by rw [eval_mul, eval_X, complex.abs_mul];\n--     exact _\u27e9)\n--   _\n--   (\u03bb q a hqp hpa ih p, _)\n\n-- lemma growth_lemma_chris1 {p q : polynomial \u2102} (hpq : ltp' q p) :\n--   \u2203 r : \u211d, \u2200 z : \u2102, r < z.abs \u2192 abs (q.eval z) < abs (p.eval z) :=\n-- ltp'.rec_on hpq\n--   _\n--   _\n--   (\u03bb p a hqp hpa _, _)\n\n-- example : \u2200 p : polynomial \u2102, \u00acless_than p 0 :=\n-- \u03bb p h, less_than.rec_on h _ _ _\n\n-- lemma polynomial_tendsto_infinity : \u2200 {p : polynomial \u2102}, 0 < degree p \u2192\n--   \u2200 x : \u211d, \u2203 r : \u211d, \u2200 z : \u2102, r < z.abs \u2192 x < (p.eval z).abs\n-- | p := \u03bb hp x, if h : degree p = 1\n-- then\n--   let \u27e8n, hn\u27e9 := archimedean.arch (1 : \u211d)\n--     (show 0 < abs (leading_coeff p),\n--       from abs_pos.2 (\u03bb hp0, by simp * at *; contradiction)) in\n--   \u27e8\u2191n * abs (p.eval 0) + n * (_root_.abs x), \u03bb z hz,\n--     calc x \u2264 _root_.abs x : le_abs_self _\n--     ... < abs (p.eval z) : lt_of_mul_lt_mul_left\n--       (calc (n : \u211d) * _root_.abs x < abs z - n * abs (eval 0 p) :\n--           lt_sub_iff_add_lt'.2 hz\n--         ... \u2264 n * abs (leading_coeff p * z) - n * abs (p.eval 0) :\n--           sub_le_sub_right (by rw [complex.abs_mul, \u2190 mul_assoc];\n--           exact le_mul_of_ge_one_left (complex.abs_nonneg _)\n--             (by simpa [mul_comm, add_monoid.smul_eq_mul] using hn)) _\n--         ... = \u2191n * (abs (leading_coeff p * z) - abs (-eval 0 p)) : by simp [mul_add]\n--         ... \u2264 \u2191n * (abs (leading_coeff p * z - -eval 0 p)) :\n--           mul_le_mul_of_nonneg_left\n--             (le_trans (le_abs_self _) (complex.abs_abs_sub_le_abs_sub _ _))\n--             (nat.cast_nonneg n)\n--         ... = \u2191n * abs (p.eval z) :\n--           by conv_rhs {rw degree_eq_one h}; simp [coeff_zero_eq_eval_zero])\n--       (nat.cast_nonneg n)\u27e9\n-- else\n--   have wf : degree (p /\u2098 X) < degree p,\n--     from degree_div_by_monic_lt _ monic_X (\u03bb hp0, by simp * at *)\n--       (by rw degree_X; exact dec_trivial),\n--   have hp : 1 < degree p, from match degree p, hp, h with\n--     | none    := dec_trivial\n--     | (some n) := \u03bb h0 h1, lt_of_le_of_ne (with_bot.coe_le_coe.2 (with_bot.coe_lt_coe.1 h0)) (ne.symm h1)\n--     end,\n--   have hXp : degree X \u2264 degree p, from le_of_lt (by rw @degree_X \u2102; exact hp),\n--   let \u27e8r, hr\u27e9 := @polynomial_tendsto_infinity (p /\u2098 X)\n--     (@lt_of_add_lt_add_left' _ _ (1 : with_bot \u2115) _ _\n--       (calc (1 : with_bot \u2115) + 0 < degree p : hp\n--         ... = 1 + degree (p /\u2098 X) : by rw [\u2190 @degree_X \u2102, degree_add_div_by_monic monic_X hXp]))\n--   (x + (p.eval 0).abs) in\n--   \u27e8max 1 (r + (p.eval 0).abs), \u03bb z hz,\n--     calc x < abs (eval z (p /\u2098 X)) - abs (eval 0 p) :\n--       lt_sub_iff_add_lt.2 (hr z (lt_of_le_of_lt (le_add_of_nonneg_right (complex.abs_nonneg _))\n--         (lt_of_le_of_lt (le_max_right _ _) hz)))\n--     ... \u2264 abs z * abs (eval z (p /\u2098 X)) - abs (eval 0 p) :\n--       sub_le_sub_right (le_mul_of_ge_one_left (complex.abs_nonneg _) (le_trans (le_max_left _ _) (le_of_lt hz))) _\n--     ... \u2264 _root_.abs (abs (z * eval z (p /\u2098 X)) - abs (-eval 0 p)) : by rw [complex.abs_neg, \u2190 complex.abs_mul];\n--       exact le_abs_self _\n--     ... \u2264 abs (z * eval z (p /\u2098 X) - -eval 0 p) : abs_abs_sub_le_abs_sub _ _\n--     ... = abs (eval z p) : by conv_rhs {rw \u2190 mod_by_monic_add_div p monic_X};\n--       simp [coeff_zero_eq_eval_zero, mod_by_monic_X]\u27e9\n-- using_well_founded {dec_tac := tactic.assumption}\n\n\n-- @[elab_as_eliminator] protected lemma induction_on {M : polynomial \u2102 \u2192 Prop} (p : polynomial \u2102 )\n--   (h_C : \u2200a, M (C a))\n--   (h_add : \u2200p q, M p \u2192 M q \u2192 M (p + q))\n--   (h_monomial : \u2200(n : \u2115) (a : \u2102), M (C a * X^n) \u2192 M (C a * X^(n+1))) :\n--   M p :=\n-- have \u2200{n:\u2115} {a}, M (C a * X^n),\n-- begin\n--   assume n a,\n--   induction n with n ih,\n--   { simp only [pow_zero, mul_one, h_C] },\n--   { exact h_monomial _ _ ih }\n-- end,\n-- finsupp.induction p\n--   (suffices M (C 0), by simpa only [C, single_zero],\n--     h_C 0)\n--   (assume n a p _ _ hp, suffices M (C a * X^n + p), by rwa [single_eq_C_mul_X],\n--     h_add _ _ this hp)\n\ninductive nonconstant : polynomial \u2102 \u2192 Prop\n| X   : \u2200 {a}, a \u2260 0 \u2192 nonconstant (C a * X)\n| mul : \u2200 {p}, nonconstant p \u2192 nonconstant (p * X)\n| add : \u2200 {p} (a), nonconstant p \u2192 nonconstant (p + C a)\n\nlemma nonconstant_of_degree_pos : \u2200 {p : polynomial \u2102},\n  0 < degree p \u2192 nonconstant p\n| p := \u03bb h,\nhave wf : degree (p /\u2098 X) < degree p,\n  from degree_div_by_monic_lt _ monic_X\n  (\u03bb hp0, by simp [hp0, lt_irrefl, *] at *)\n  (by rw degree_X; exact dec_trivial),\nby rw [\u2190 mod_by_monic_add_div p monic_X,\n  add_comm, mod_by_monic_X, mul_comm] at *;\nexact nonconstant.add _\n  (if hpX : 0 < degree (p /\u2098 X)\n    then nonconstant.mul (nonconstant_of_degree_pos hpX)\n    else by rw [eq_C_of_degree_le_zero (not_lt.1 hpX)] at *;\n      exact if hc : coeff (p /\u2098 X) 0 = 0\n        then by simpa [hc, not_lt_of_ge degree_C_le] using h\n        else nonconstant.X hc)\nusing_well_founded {dec_tac := tactic.assumption}\n\nlemma polynomial_tendsto_infinity' {p : polynomial \u2102} (h : 0 < degree p) :\n  \u2200 x : \u211d, \u2203 r : \u211d, \u2200 z : \u2102, r < z.abs \u2192 x < (p.eval z).abs :=\nnonconstant.rec_on (nonconstant_of_degree_pos h)\n  (\u03bb a ha x, \u27e8x / a.abs, \u03bb z hz,\n    by simpa [(div_lt_iff' (complex.abs_pos.2 ha)).symm]\u27e9)\n  (\u03bb p hp ih x, let \u27e8r, hr\u27e9 := ih x in\n    \u27e8max r 1, \u03bb z hz, by rw [eval_mul, eval_X, complex.abs_mul];\n        exact lt_of_lt_of_le (hr z (lt_of_le_of_lt (le_max_left _ _) hz))\n          (le_mul_of_ge_one_right (complex.abs_nonneg _)\n            (le_trans (le_max_right _ _) (le_of_lt hz)))\u27e9)\n  (\u03bb p a hp ih x, let \u27e8r, hr\u27e9 := ih (x + a.abs) in\n    \u27e8r, \u03bb z hz, by rw [eval_add, eval_C, \u2190 sub_neg_eq_add];\n      exact lt_of_lt_of_le (lt_sub_iff_add_lt.2\n        (by rw complex.abs_neg; exact (hr z hz)))\n        (le_trans (le_abs_self _) (complex.abs_abs_sub_le_abs_sub _ _))\u27e9)\n\nlemma polynomial.is_unit_iff {p : polynomial \u2102} : is_unit p \u2194 degree p = 0 :=\n\u27e8\u03bb h, let \u27e8q, hq\u27e9 := is_unit_iff_dvd_one.1 h in\n    have hp0 : p \u2260 0, from \u03bb hp0, by simpa [hp0] using hq,\n    have hq0 : q \u2260 0, from \u03bb hp0, by simpa [hp0] using hq,\n    have nat_degree (1 : polynomial \u2102) = nat_degree (p * q),\n      from congr_arg _ hq,\n    by rw [nat_degree_one, nat_degree_mul_eq hp0 hq0, eq_comm,\n        add_eq_zero_iff, \u2190 with_bot.coe_eq_coe,\n        \u2190 degree_eq_nat_degree hp0] at this;\n      exact this.1,\n  \u03bb h, have degree p \u2264 0, by simp [*, le_refl],\n    have hc : coeff p 0 \u2260 0, from \u03bb hc,\n        by rw [eq_C_of_degree_le_zero this, hc] at h;\n        simpa using h,\n    is_unit_iff_dvd_one.2 \u27e8C (coeff p 0)\u207b\u00b9, begin\n      conv in p { rw eq_C_of_degree_le_zero this },\n      rw [\u2190 C_mul, _root_.mul_inv_cancel hc, C_1]\n    end\u27e9\u27e9\n\ninstance decidable_dvd {\u03b1 : Type*} [comm_ring \u03b1] [decidable_eq \u03b1] :\n  decidable_rel ((\u2223) : polynomial \u03b1 \u2192 polynomial \u03b1 \u2192 Prop) :=\nsorry\n\nlemma polynomial.finite_of_degree_pos {\u03b1 : Type*} [integral_domain \u03b1] [decidable_eq \u03b1]\n  {p q : polynomial \u03b1} (hp : (0 : with_bot \u2115) < degree p) (hq : q \u2260 0) :\n  prime_count.finite p q :=\n\u27e8nat_degree q, \u03bb \u27e8r, hr\u27e9,\n  have hp0 : p \u2260 0, from \u03bb hp0, by simp [hp0] at hp; contradiction,\n  have hr0 : r \u2260 0, from \u03bb hr0, by simp * at *,\n  have hpn0 : p ^ (nat_degree q + 1) \u2260 0,\n    from pow_ne_zero _ hp0,\n  have hnp : 0 < nat_degree p,\n    by rw [\u2190 with_bot.coe_lt_coe, \u2190 degree_eq_nat_degree hp0];\n    exact hp,\n  begin\n    have := congr_arg nat_degree hr,\n    rw [nat_degree_mul_eq hpn0 hr0, nat_degree_pow_eq, add_mul, add_assoc] at this,\n    exact ne_of_lt (lt_add_of_le_of_pos (le_mul_of_ge_one_right' (nat.zero_le _) hnp)\n      (add_pos_of_pos_of_nonneg (by rwa one_mul) (nat.zero_le _))) this\n  end\u27e9\n\ndef polynomial.multiplicity {\u03b1 : Type*} [integral_domain \u03b1] [decidable_eq \u03b1]\n  (p : polynomial \u03b1) (a :  \u03b1) : \u2115 :=\nif h0 : p = 0 then 0 else\n(prime_count (X - C a) p).get (polynomial.finite_of_degree_pos\n  (by rw degree_X_sub_C; exact dec_trivial) h0)\n\nlemma pow_multiplicity_dvd {\u03b1 : Type*} [integral_domain \u03b1] [decidable_eq \u03b1]\n  (p : polynomial \u03b1) (a : \u03b1) :\n  (X - C a) ^ polynomial.multiplicity p a \u2223 p :=\nif h : p = 0 then by simp [h]\nelse by rw [polynomial.multiplicity, dif_neg h];\n  exact prime_count.spec _\n\nlemma div_by_monic_mul_pow_multiplicity_eq\n  {\u03b1 : Type*} [integral_domain \u03b1] [decidable_eq \u03b1]\n  (p : polynomial \u03b1) (a : \u03b1) :\n  p /\u2098 ((X - C a) ^ polynomial.multiplicity p a) *\n  (X - C a) ^ polynomial.multiplicity p a = p :=\nhave monic ((X - C a) ^ polynomial.multiplicity p a),\n  from by rw [monic.def, leading_coeff_pow,\n    (show _ = _, from monic_X_sub_C _), one_pow],\nby conv_rhs { rw [\u2190 mod_by_monic_add_div p this,\n    (dvd_iff_mod_by_monic_eq_zero this).2 (pow_multiplicity_dvd _ _)] };\n  simp [mul_comm]\n\nlemma eval_div_by_monic_pow_multiplicity_ne_zero\n  {\u03b1 : Type*} [integral_domain \u03b1] [decidable_eq \u03b1]\n  {p : polynomial \u03b1} (a : \u03b1) (hp : p \u2260 0) :\n  (p /\u2098 ((X - C a) ^ polynomial.multiplicity p a)).eval a \u2260 0 :=\nmt dvd_iff_is_root.2 $ \u03bb \u27e8q, hq\u27e9,\nbegin\n  have := div_by_monic_mul_pow_multiplicity_eq p a,\n  rw [mul_comm, hq, \u2190 mul_assoc, \u2190 pow_succ',\n    polynomial.multiplicity, dif_neg hp] at this,\n  refine prime_count.is_greatest'\n    (polynomial.finite_of_degree_pos\n    (show (0 : with_bot \u2115) < degree (X - C a),\n      by rw degree_X_sub_C; exact dec_trivial) hp)\n    (nat.lt_succ_self _) (dvd_of_mul_right_eq _ this)\nend\n\naxiom attains_infi (p : polynomial \u2102) :\n  \u2203 x, (p.eval x).abs = \u2a05 y, (p.eval y).abs\n\naxiom nth_root (n : \u2115) (z : \u2102) : \u2102\n\naxiom nth_root_pow (n : \u2115) (z : \u2102) : nth_root n z ^ n = z\n\naxiom abs_nth_root (n : \u2115) (z : \u2102) : abs (nth_root n z) =\n  real.nth_root (abs z) n\n\n#print real.nth_root\n\nopen euclidean_domain\nlocal attribute [instance, priority 0] classical.prop_decidable\nset_option trace.simplify.rewrite true\nlemma FTA {f : polynomial \u2102} (hf : 0 < degree f) : \u2203 z : \u2102, is_root f z :=\nlet \u27e8z\u2080, hz\u2080\u27e9 := attains_infi f in\nexists.intro z\u2080 $ by_contradiction $ \u03bb hf0,\nhave hfX : f - C (f.eval z\u2080) \u2260 0,\n  from mt sub_eq_zero.1 (\u03bb h, not_le_of_gt hf\n    (h.symm \u25b8 degree_C_le)),\nlet n := polynomial.multiplicity (f - C (f.eval z\u2080)) z\u2080 in\nlet g := (f - C (f.eval z\u2080)) /\u2098 ((X - C z\u2080) ^ n) in\nhave hg0 : g.eval z\u2080 \u2260 0, from eval_div_by_monic_pow_multiplicity_ne_zero _ hfX,\nhave hg : g * (X - C z\u2080) ^ n = f - C (f.eval z\u2080),\n  from div_by_monic_mul_pow_multiplicity_eq _ _,\nhave hn0 : 0 < n, from nat.pos_of_ne_zero $ \u03bb hn0,\n  by simpa [g, hn0] using hg0,\nlet \u27e8\u03b4', h\u03b4'\u2081, h\u03b4'\u2082\u27e9 := continuous_of_metric.1 (polynomial.continuous_eval g) z\u2080\n  ((g.eval z\u2080).abs) (complex.abs_pos.2 hg0) in\nlet \u03b4 := min (min (\u03b4' / 2) 1) (((f.eval z\u2080).abs / (g.eval z\u2080).abs) / 2) in\nhave hf0' : 0 < (f.eval z\u2080).abs, from complex.abs_pos.2 hf0,\nhave hfg0 : 0 < abs (eval z\u2080 f) * (abs (eval z\u2080 g))\u207b\u00b9,\n  from div_pos hf0' (complex.abs_pos.2 hg0),\nhave h\u03b40 : 0 < \u03b4, from lt_min\n  (lt_min (half_pos h\u03b4'\u2081) (by norm_num)) (half_pos hfg0),\nhave h\u03b4 : \u2200 z : \u2102, abs (z - z\u2080) = \u03b4 \u2192 abs (g.eval z - g.eval z\u2080) <\n  (g.eval z\u2080).abs,\n  from \u03bb z hz, h\u03b4'\u2082 z (by rw [complex.dist_eq, hz];\n    exact lt_of_le_of_lt (le_trans (min_le_left _ _) (min_le_left _ _))\n      (half_lt_self h\u03b4'\u2081)),\nhave h\u03b41 : \u03b4 \u2264 1, from le_trans (min_le_left _ _) (min_le_right _ _),\nlet F : polynomial \u2102 := C (f.eval z\u2080) + C (g.eval z\u2080) * (X - C z\u2080) ^ n in\nlet z' := nth_root n (-f.eval z\u2080 * (g.eval z\u2080).abs * \u03b4 ^ n /\n  ((f.eval z\u2080).abs * g.eval z\u2080)) + z\u2080 in\nhave hF\u2081 : F.eval z' = f.eval z\u2080 - f.eval z\u2080 * (g.eval z\u2080).abs\n    * \u03b4 ^ n / (f.eval z\u2080).abs,\n  by simp [F, nth_root_pow, div_eq_mul_inv, eval_pow, mul_assoc,\n      mul_comm (g.eval z\u2080),\n      mul_left_comm (g.eval z\u2080), mul_left_comm (g.eval z\u2080)\u207b\u00b9,\n      mul_inv', inv_mul_cancel hg0];\n    simp [mul_comm, mul_left_comm, mul_assoc],\nhave h\u03b4s : (g.eval z\u2080).abs * \u03b4 ^ n / (f.eval z\u2080).abs < 1,\n  begin\n    rw [div_eq_mul_inv, mul_right_comm, mul_comm,\n      \u2190 @inv_inv' _ _ (complex.abs _ * _), mul_inv',\n      inv_inv', \u2190 div_eq_mul_inv, div_lt_iff hfg0, one_mul],\n    calc \u03b4 ^ n \u2264 \u03b4 ^ 1 : pow_le_pow_of_le_one\n        (le_of_lt h\u03b40) h\u03b41 hn0\n      ... = \u03b4 : _root_.pow_one _\n      ... \u2264 ((f.eval z\u2080).abs / (g.eval z\u2080).abs) / 2 : min_le_right _ _\n      ... < _ : half_lt_self hfg0\n  end,\nhave hF\u2082 : (F.eval z').abs = (f.eval z\u2080).abs - (g.eval z\u2080).abs * \u03b4 ^ n,\n  from calc (F.eval z').abs = (f.eval z\u2080 - f.eval z\u2080 * (g.eval z\u2080).abs\n    * \u03b4 ^ n / (f.eval z\u2080).abs).abs : congr_arg abs hF\u2081\n  ... = abs (f.eval z\u2080) * complex.abs (1 - (g.eval z\u2080).abs * \u03b4 ^ n /\n      (f.eval z\u2080).abs : \u211d) : by rw [\u2190 complex.abs_mul];\n        exact congr_arg complex.abs\n          (by simp [mul_add, add_mul, mul_assoc, div_eq_mul_inv])\n  ... = _ : by rw [complex.abs_of_nonneg (sub_nonneg.2 (le_of_lt h\u03b4s)),\n      mul_sub, mul_div_cancel' _ (ne.symm (ne_of_lt hf0')), mul_one],\nhave hef0 : abs (eval z\u2080 g) * (eval z\u2080 f).abs \u2260 0,\n  from mul_ne_zero (mt complex.abs_eq_zero.1 hg0)\n    (mt complex.abs_eq_zero.1 hf0),\nhave hz'z\u2080 : abs (z' - z\u2080) = \u03b4 :=\n  begin\n     simp [z', mul_assoc, mul_left_comm _ (_ ^ n),\n      mul_comm _ (_ ^ n), mul_comm (eval z\u2080 f).abs,\n      _root_.mul_div_cancel _ hef0, of_real_mul,\n      neg_mul_eq_neg_mul_symm, neg_div, abs_nth_root,\n      is_absolute_value.abv_pow complex.abs],\n  end,\nhave hF\u2083 : (f.eval z' - F.eval z').abs < (g.eval z\u2080).abs * \u03b4 ^ n,\n  from calc (f.eval z' - F.eval z').abs\n      = (g.eval z' - g.eval z\u2080).abs * (z' - z\u2080).abs ^ n :\n        by rw [\u2190 eq_sub_iff_add_eq.1 hg, \u2190 is_absolute_value.abv_pow complex.abs,\n            \u2190 complex.abs_mul, sub_mul];\n          simp [F, eval_pow, eval_add, eval_mul,\n            eval_sub, eval_C, eval_X, eval_neg, add_sub_cancel]\n  ... = (g.eval z' - g.eval z\u2080).abs * \u03b4 ^ n : by rw hz'z\u2080\n  ... < _ : (mul_lt_mul_right (pow_pos h\u03b40 _)).2 (h\u03b4 _ hz'z\u2080),\nlt_irrefl (f.eval z\u2080).abs $\ncalc (f.eval z\u2080).abs = \u2a05 y, (f.eval y).abs : hz\u2080\n... \u2264 (f.eval z').abs : lattice.cinfi_le\n  \u27e80, \u03bb _ \u27e8z, hz\u27e9, by simp [hz.symm, complex.abs_nonneg]\u27e9\n... = (F.eval z' + (f.eval z' - F.eval z')).abs : by simp\n... \u2264 (F.eval z').abs + (f.eval z' - F.eval z').abs : complex.abs_add _ _\n... < (f.eval z\u2080).abs - (g.eval z\u2080).abs * \u03b4 ^ n + (g.eval z\u2080).abs * \u03b4 ^ n :\n  add_lt_add_of_le_of_lt (by rw hF\u2082) hF\u2083\n... = _ : by simp\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/FTA.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7461389930307512, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3730694965153756}}
{"text": "import data.multiset\n\nimport tactic.abel\nimport tactic.ring\nimport tactic.linarith\nimport tactic.interactive\n\n-- the following imports should be not part of any finished theory\nimport tactic.where\nimport tactic.find\n\n\n-- #where\n\nnamespace where_demo\n\nvariables {\u03b1 \u03b2 : Type} [decidable_eq \u03b1] (a b : \u03b1)\n\n/- many many lines of code later -/\n\n#where\n\ndef f (a : \u03b1) := a\n\n#where\n\ninclude a\n\n#where\n\nend where_demo\n\n-- #find\n\n#find (_ : nat) + (_ + _) = _ + _\n\n-- alternatives\n--  * use auto completion\n--  * use Ctrl-T (maybe Cmd-T on MacOS)\n--  * grep for it!\n\n-- #print instances\n\n#print classes\n\n#print instances add_group\n\n-- rcases\n\nexample (p : \u2115 \u2192 Prop) (h : \u2203i, i = 0 \u2227 p i) : p 0 :=\nbegin\n  /- `rcases` is for \"recursive\" cases, i.e. nested patterns. `rfl` performs subsitution -/\n  rcases h with \u27e8_, rfl, h\u27e9,\n  exact h\nend\n\nexample (p : \u2115 \u2192 Prop) : (\u2203i, i = 0 \u2227 p i) \u2192 p 0 :=\nbegin\n  rintros \u27e8_, rfl, h\u27e9,\n  exact h\nend\n\nexample {\u03b1} (p : multiset \u03b1 \u2192 Prop) (h : \u2200 l, p l) (m : multiset \u03b1) : p m :=\nbegin\n  rcases m with \u27e8l\u27e9, --< works also for quotients\n  exact h l\nend", "meta": {"author": "johoelzl", "repo": "lean-together", "sha": "b0ca0dbc8b971de50621ae67c2f0c2368fb72839", "save_path": "github-repos/lean/johoelzl-lean-together", "path": "github-repos/lean/johoelzl-lean-together/lean-together-b0ca0dbc8b971de50621ae67c2f0c2368fb72839/src/tactics_commands.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3730084104564244}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), by {\n    assume (i j : \u2124) (h1 : i \u2260 j),\n    have h2 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from by {\n      assume h2 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j),\n      have h3 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from h2,\n      have h4 : \u03b1 = (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from h3,\n      have h5 : (i - j) \u2260 0, from by {\n        assume h5 : (i - j) = 0,\n        have h6 : i = j, from by {\n          have h7 : i - j = 0, from sorry,\n          have h8 : i - j = 0, from sorry,\n          show i = j, from sorry,\n        },\n        have h6 : i = j, from sorry,\n        show false, from h1 h6,\n      },\n      have h6 : (i - j) \u2260 0, from sorry,\n      have h7 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j) \u2208 \u211a, from sorry,\n      have h8 : (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j) \u2208 \u211a, from sorry,\n      show false, from h\u03b1_irrat h7,\n    },\n    have h2 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from sorry,\n    have h3 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from sorry,\n    have h4 : \u03b1 \u2260 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) / (i - j), from sorry,\n    have h5 : (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n    have h6 : (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n    show (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n  },\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h4 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from by {\n    assume (i : \u2124),\n    have h5 : (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n    show (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n  },\n  have h4 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n\n  have h5 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n  have h6 : \u2200 i : \u2124, (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n\n  have h7 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from by {\n    assume (i : \u2124),\n    have h8 : (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n    show (int.fract (\u03b1 * \u2191i)) \u2208 set.Icc 0 1, from sorry,\n  },\n  have h7 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from sorry,\n\n  have h8 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from sorry,\n\n  have h9 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124) \u2286 set.Icc 0 1, from sorry,\n\n  have h10 : closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 set.Icc 0 1, from sorry,\n\n  have h11 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from by {\n    assume (y : \u211d) (h11 : y \u2208 set.Icc 0 1),\n    have h12 : \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n    show \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n  },\n  have h11 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n\n  have h12 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n\n  have h13 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n\n  have h14 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 m \u2265 N, abs (x - y) < \u03b5, from sorry,\n\n  have h15 : \u2200 y \u2208 set.Icc 0 1, \u2203 x \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124),\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from sorry,\n  have h2 : \u2200 i : \u2124, int.fract (\u03b1 * \u2191i) \u2208 set.Icc 0 1, from sorry,\n  have h3 : \u2200 i : \u2124, int.fract (\u03b1 * \u2191i) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from sorry,\n  have h4 : \u2200 i : \u2124, int.fract (\u03b1 * \u2191i) \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from sorry,\n  have h5 : \u2200 i : \u2124, int.fract (\u03b1 * \u2191i) \u2208 set.Icc 0 1, from sorry,\n  have h6 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h7 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h8 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h9 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h10 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h11 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h12 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h13 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h14 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h15 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h16 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h17 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h18 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h19 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h20 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h21 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h22 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h23 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h24 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h25 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h26 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h27 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h28 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h29 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h30 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h31 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h32 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h33 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h34 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h35 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry,\n  have h36 : \u2200 y : \u211d, y \u2208 set.Icc 0 1 \u2192 \u2203 x : \u2124, int.fract (\u03b1 * \u2191x) \u2208 set.Icc y (y + 1), from sorry\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  sorry\nend\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from by {\n    assume (m n : \u2124) (h1 : m \u2260 n),\n    have h2 : \u03b1 \u2260 \u2191(int.nat_abs (m - n)) / \u2191(m - n), from by {\n      assume h2 : \u03b1 = \u2191(int.nat_abs (m - n)) / \u2191(m - n),\n      have h3 : irrational \u03b1, from h\u03b1_irrat,\n      have h4 : rational (\u2191(int.nat_abs (m - n)) / \u2191(m - n)), from sorry,\n      have h5 : \u03b1 \u2208 set.Ioo 0 1, from sorry,\n      have h6 : \u2191(int.nat_abs (m - n)) / \u2191(m - n) \u2208 set.Ioo 0 1, from sorry,\n      have h7 : \u2191(int.nat_abs (m - n)) / \u2191(m - n) = \u03b1, from sorry,\n      show false, from sorry,\n    },\n    have h3 : int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n    show int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from h3,\n  },\n  have h2 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h3 : \u2200 (m : \u2124), int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from by {\n    assume (m : \u2124),\n    show int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from sorry,\n  },\n  have h4 : \u2200 (m : \u2124), int.fract (\u03b1 * \u2191m) \u2208 set.Icc 0 1, from sorry,\n  have h5 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h6 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h7 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h8 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h9 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h10 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h11 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h12 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h13 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h14 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h15 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h16 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h17 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h18 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h19 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h20 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h21 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h22 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h23 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h24 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h25 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h26 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h27 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h28 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h29 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h30 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h31 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h32 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h33 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h34 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h35 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h36 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), from sorry,\n  have h37 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from sorry,\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2260 0, from sorry,\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 (\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124), from sorry,\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from sorry,\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2208 closure (set.Icc 0 1), from sorry,\n  have h7 : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) \u2286 closure (set.Icc 0 1), from sorry,\n  have h8 : closure (set.Icc 0 1) \u2286 closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)), from sorry,\n  show closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1, from sorry,\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from sorry,\n  have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2260 0, from sorry,\n  have h3 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h4 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h5 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h6 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h7 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h8 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h9 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h10 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h11 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h12 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h13 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h14 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h15 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h16 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h17 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h18 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h19 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h20 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h21 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h22 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h23 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h24 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h25 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h26 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h27 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h28 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h29 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h30 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h31 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h32 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h33 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h34 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h35 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n\n  have h36 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 set.Icc 0 1, from sorry,\n  have h37 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/Correct_statement-lean_proof_outline-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7956581000631541, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3729970593088526}}
{"text": "\nabbrev \u2115 := Nat\n\nconstant I : Type\n\nstructure Constraint where\n  lhs : I\n  rhs : I\n\nstructure EP where\n  handle : \u2115 \n  predicate : String\n  a : List I\n  s : List I \n  c : String\n\nstructure MRS (\u03b1 : Type u) where\n  gt : I\n  index : I\n  r : List EP\n  i : List Constraint\n  c : List Constraint\n\n\n\ntheorem and_commutative (p q : Prop) : p \u2227 q \u2192 q \u2227 p := \n  \u03bb hpq : p \u2227 q =>\n  have hp : p := And.left hpq\n  have hq : q := And.right hpq\n  show q \u2227 p from And.intro hq hp\n\n\n\n\n\n\n", "meta": {"author": "arademaker", "repo": "mrs", "sha": "c30568c0835880547aa29432dc2be6cc587ee272", "save_path": "github-repos/lean/arademaker-mrs", "path": "github-repos/lean/arademaker-mrs/mrs-c30568c0835880547aa29432dc2be6cc587ee272/Mrs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3727968183980298}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Johannes H\u00f6lzl, Scott Morrison, Jens Wagemaker\n\n! This file was ported from Lean 3 source module data.polynomial.monomial\n! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Polynomial.Basic\n\n/-!\n# Univariate monomials\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nPreparatory lemmas for degree_basic.\n-/\n\n\nnoncomputable section\n\nnamespace Polynomial\n\nopen Polynomial\n\nuniverse u\n\nvariable {R : Type u} {a b : R} {m n : \u2115}\n\nvariable [Semiring R] {p q r : R[X]}\n\n/- warning: polynomial.monomial_one_eq_iff -> Polynomial.monomial_one_eq_iff is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Nontrivial.{u1} R] {i : Nat} {j : Nat}, Iff (Eq.{succ u1} (Polynomial.{u1} R _inst_1) (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 i) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))))) (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 j) (OfNat.ofNat.{u1} R 1 (OfNat.mk.{u1} R 1 (One.one.{u1} R (AddMonoidWithOne.toOne.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))))) (Eq.{1} Nat i j)\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] [_inst_2 : Nontrivial.{u1} R] {i : Nat} {j : Nat}, Iff (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1)))) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 i) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1)))) (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 j) (OfNat.ofNat.{u1} R 1 (One.toOfNat1.{u1} R (Semiring.toOne.{u1} R _inst_1))))) (Eq.{1} Nat i j)\nCase conversion may be inaccurate. Consider using '#align polynomial.monomial_one_eq_iff Polynomial.monomial_one_eq_iff\u2093'. -/\ntheorem monomial_one_eq_iff [Nontrivial R] {i j : \u2115} :\n    (monomial i 1 : R[X]) = monomial j 1 \u2194 i = j :=\n  by\n  simp_rw [\u2190 of_finsupp_single]\n  exact add_monoid_algebra.of_injective.eq_iff\n#align polynomial.monomial_one_eq_iff Polynomial.monomial_one_eq_iff\n\ninstance [Nontrivial R] : Infinite R[X] :=\n  Infinite.of_injective (fun i => monomial i 1) fun m n h => by simpa [monomial_one_eq_iff] using h\n\n/- warning: polynomial.card_support_le_one_iff_monomial -> Polynomial.card_support_le_one_iff_monomial is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {f : Polynomial.{u1} R _inst_1}, Iff (LE.le.{0} Nat Nat.hasLe (Finset.card.{0} Nat (Polynomial.support.{u1} R _inst_1 f)) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))) (Exists.{1} Nat (fun (n : Nat) => Exists.{succ u1} R (fun (a : R) => Eq.{succ u1} (Polynomial.{u1} R _inst_1) f (coeFn.{succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) (fun (_x : LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (LinearMap.hasCoeToFun.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {f : Polynomial.{u1} R _inst_1}, Iff (LE.le.{0} Nat instLENat (Finset.card.{0} Nat (Polynomial.support.{u1} R _inst_1 f)) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) (Exists.{1} Nat (fun (n : Nat) => Exists.{succ u1} R (fun (a : R) => Eq.{succ u1} (Polynomial.{u1} R _inst_1) f (FunLike.coe.{succ u1, succ u1, succ u1} (LinearMap.{u1, u1, u1, u1} R R _inst_1 _inst_1 (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)) R (Polynomial.{u1} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : R) => Polynomial.{u1} R _inst_1) _x) (LinearMap.instFunLikeLinearMap.{u1, u1, u1, u1} R R R (Polynomial.{u1} R _inst_1) _inst_1 _inst_1 (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} (Polynomial.{u1} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)))) (Semiring.toModule.{u1} R _inst_1) (Polynomial.module.{u1, u1} R _inst_1 R _inst_1 (Semiring.toModule.{u1} R _inst_1)) (RingHom.id.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Polynomial.monomial.{u1} R _inst_1 n) a))))\nCase conversion may be inaccurate. Consider using '#align polynomial.card_support_le_one_iff_monomial Polynomial.card_support_le_one_iff_monomial\u2093'. -/\ntheorem card_support_le_one_iff_monomial {f : R[X]} :\n    Finset.card f.support \u2264 1 \u2194 \u2203 n a, f = monomial n a :=\n  by\n  constructor\n  \u00b7 intro H\n    rw [Finset.card_le_one_iff_subset_singleton] at H\n    rcases H with \u27e8n, hn\u27e9\n    refine' \u27e8n, f.coeff n, _\u27e9\n    ext i\n    by_cases hi : i = n\n    \u00b7 simp [hi, coeff_monomial]\n    \u00b7 have : f.coeff i = 0 := by\n        rw [\u2190 not_mem_support_iff]\n        exact fun hi' => hi (Finset.mem_singleton.1 (hn hi'))\n      simp [this, Ne.symm hi, coeff_monomial]\n  \u00b7 rintro \u27e8n, a, rfl\u27e9\n    rw [\u2190 Finset.card_singleton n]\n    apply Finset.card_le_of_subset\n    exact support_monomial' _ _\n#align polynomial.card_support_le_one_iff_monomial Polynomial.card_support_le_one_iff_monomial\n\n/- warning: polynomial.ring_hom_ext -> Polynomial.ringHom_ext is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : Semiring.{u2} S] {f : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)} {g : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)}, (forall (a : R), Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) f (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) g (coeFn.{succ u1, succ u1} (RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (fun (_x : RingHom.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) => R -> (Polynomial.{u1} R _inst_1)) (RingHom.hasCoeToFun.{u1, u1} R (Polynomial.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1))) (Polynomial.C.{u1} R _inst_1) a))) -> (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) f (Polynomial.X.{u1} R _inst_1)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) g (Polynomial.X.{u1} R _inst_1))) -> (Eq.{max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) f g)\nbut is expected to have type\n  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : Semiring.{u1} S] {f : RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)} {g : RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)}, (forall (a : R), Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (fun (a : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u2} R _inst_1) a) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (RingHom.instRingHomClassRingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))))) (Polynomial.C.{u2} R _inst_1) a)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) f (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u2} R _inst_1) _x) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (RingHom.instRingHomClassRingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))))) (Polynomial.C.{u2} R _inst_1) a)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) g (FunLike.coe.{succ u2, succ u2, succ u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => Polynomial.{u2} R _inst_1) _x) (MulHomClass.toFunLike.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonUnitalNonAssocSemiring.toMul.{u2} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1))) (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalRingHomClass.toMulHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} R (Semiring.toNonAssocSemiring.{u2} R _inst_1)) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (RingHomClass.toNonUnitalRingHomClass.{u2, u2, u2} (RingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (RingHom.instRingHomClassRingHom.{u2, u2} R (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))))) (Polynomial.C.{u2} R _inst_1) a))) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) (Polynomial.X.{u2} R _inst_1)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) f (Polynomial.X.{u2} R _inst_1)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) g (Polynomial.X.{u2} R _inst_1))) -> (Eq.{max (succ u2) (succ u1)} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) f g)\nCase conversion may be inaccurate. Consider using '#align polynomial.ring_hom_ext Polynomial.ringHom_ext\u2093'. -/\ntheorem ringHom_ext {S} [Semiring S] {f g : R[X] \u2192+* S} (h\u2081 : \u2200 a, f (C a) = g (C a))\n    (h\u2082 : f X = g X) : f = g :=\n  by\n  set f' := f.comp (to_finsupp_iso R).symm.toRingHom with hf'\n  set g' := g.comp (to_finsupp_iso R).symm.toRingHom with hg'\n  have A : f' = g' := by\n    ext\n    \u00b7 simp [h\u2081, RingEquiv.toRingHom_eq_coe]\n    \u00b7 simpa [RingEquiv.toRingHom_eq_coe] using h\u2082\n  have B : f = f'.comp (to_finsupp_iso R) :=\n    by\n    rw [hf', RingHom.comp_assoc]\n    ext x\n    simp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply,\n      RingHom.coe_comp, RingEquiv.coe_toRingHom]\n  have C : g = g'.comp (to_finsupp_iso R) :=\n    by\n    rw [hg', RingHom.comp_assoc]\n    ext x\n    simp only [RingEquiv.toRingHom_eq_coe, RingEquiv.symm_apply_apply, Function.comp_apply,\n      RingHom.coe_comp, RingEquiv.coe_toRingHom]\n  rw [B, C, A]\n#align polynomial.ring_hom_ext Polynomial.ringHom_ext\n\n/- warning: polynomial.ring_hom_ext' -> Polynomial.ringHom_ext' is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {S : Type.{u2}} [_inst_2 : Semiring.{u2} S] {f : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)} {g : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)}, (Eq.{max (succ u1) (succ u2)} (RingHom.{u1, u2} R S (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (RingHom.comp.{u1, u1, u2} R (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2) f (Polynomial.C.{u1} R _inst_1)) (RingHom.comp.{u1, u1, u2} R (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} R _inst_1) (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2) g (Polynomial.C.{u1} R _inst_1))) -> (Eq.{succ u2} S (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) f (Polynomial.X.{u1} R _inst_1)) (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) (fun (_x : RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) => (Polynomial.{u1} R _inst_1) -> S) (RingHom.hasCoeToFun.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) g (Polynomial.X.{u1} R _inst_1))) -> (Eq.{max (succ u1) (succ u2)} (RingHom.{u1, u2} (Polynomial.{u1} R _inst_1) S (Semiring.toNonAssocSemiring.{u1} (Polynomial.{u1} R _inst_1) (Polynomial.semiring.{u1} R _inst_1)) (Semiring.toNonAssocSemiring.{u2} S _inst_2)) f g)\nbut is expected to have type\n  forall {R : Type.{u2}} [_inst_1 : Semiring.{u2} R] {S : Type.{u1}} [_inst_2 : Semiring.{u1} S] {f : RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)} {g : RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)}, (Eq.{max (succ u2) (succ u1)} (RingHom.{u2, u1} R S (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHom.comp.{u2, u2, u1} R (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) f (Polynomial.C.{u2} R _inst_1)) (RingHom.comp.{u2, u2, u1} R (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) g (Polynomial.C.{u2} R _inst_1))) -> (Eq.{succ u1} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) (Polynomial.X.{u2} R _inst_1)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) f (Polynomial.X.{u2} R _inst_1)) (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) (fun (_x : Polynomial.{u2} R _inst_1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Polynomial.{u2} R _inst_1) => S) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonUnitalNonAssocSemiring.toMul.{u2} (Polynomial.{u2} R _inst_1) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u1} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} S (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2))))) g (Polynomial.X.{u2} R _inst_1))) -> (Eq.{max (succ u2) (succ u1)} (RingHom.{u2, u1} (Polynomial.{u2} R _inst_1) S (Semiring.toNonAssocSemiring.{u2} (Polynomial.{u2} R _inst_1) (Polynomial.semiring.{u2} R _inst_1)) (Semiring.toNonAssocSemiring.{u1} S _inst_2)) f g)\nCase conversion may be inaccurate. Consider using '#align polynomial.ring_hom_ext' Polynomial.ringHom_ext'\u2093'. -/\n@[ext]\ntheorem ringHom_ext' {S} [Semiring S] {f g : R[X] \u2192+* S} (h\u2081 : f.comp C = g.comp C)\n    (h\u2082 : f X = g X) : f = g :=\n  ringHom_ext (RingHom.congr_fun h\u2081) h\u2082\n#align polynomial.ring_hom_ext' Polynomial.ringHom_ext'\n\nend Polynomial\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Polynomial/Monomial.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3727880698515729}}
{"text": "/-\nCopyright (c) 2020 Patrick Massot. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Yury Kudryashov\n\n! This file was ported from Lean 3 source module topology.uniform_space.compact\n! leanprover-community/mathlib commit ee05e9ce1322178f0c12004eb93c00d2c8c00ed2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.UniformSpace.UniformConvergence\nimport Mathbin.Topology.UniformSpace.Equicontinuity\nimport Mathbin.Topology.Separation\nimport Mathbin.Topology.Support\n\n/-!\n# Compact separated uniform spaces\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n## Main statements\n\n* `compact_space_uniformity`: On a compact uniform space, the topology determines the\n  uniform structure, entourages are exactly the neighborhoods of the diagonal.\n\n* `uniform_space_of_compact_t2`: every compact T2 topological structure is induced by a uniform\n  structure. This uniform structure is described in the previous item.\n\n* **Heine-Cantor** theorem: continuous functions on compact uniform spaces with values in uniform\n  spaces are automatically uniformly continuous. There are several variations, the main one is\n  `compact_space.uniform_continuous_of_continuous`.\n\n## Implementation notes\n\nThe construction `uniform_space_of_compact_t2` is not declared as an instance, as it would badly\nloop.\n\n## tags\n\nuniform space, uniform continuity, compact space\n-/\n\n\nopen Classical uniformity Topology Filter\n\nopen Filter UniformSpace Set\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _} [UniformSpace \u03b1] [UniformSpace \u03b2]\n\n/-!\n### Uniformity on compact spaces\n-/\n\n\n/- warning: nhds_set_diagonal_eq_uniformity -> nhdsSet_diagonal_eq_uniformity is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_3 : CompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (nhdsSet.{u1} (Prod.{u1, u1} \u03b1 \u03b1) (Prod.topologicalSpace.{u1, u1} \u03b1 \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)) (Set.diagonal.{u1} \u03b1)) (uniformity.{u1} \u03b1 _inst_1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_3 : CompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (nhdsSet.{u1} (Prod.{u1, u1} \u03b1 \u03b1) (instTopologicalSpaceProd.{u1, u1} \u03b1 \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)) (Set.diagonal.{u1} \u03b1)) (uniformity.{u1} \u03b1 _inst_1)\nCase conversion may be inaccurate. Consider using '#align nhds_set_diagonal_eq_uniformity nhdsSet_diagonal_eq_uniformity\u2093'. -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- On a compact uniform space, the topology determines the uniform structure, entourages are\nexactly the neighborhoods of the diagonal. -/\ntheorem nhdsSet_diagonal_eq_uniformity [CompactSpace \u03b1] : \ud835\udcdd\u02e2 (diagonal \u03b1) = \ud835\udce4 \u03b1 :=\n  by\n  refine' nhds_set_diagonal_le_uniformity.antisymm _\n  have :\n    (\ud835\udce4 (\u03b1 \u00d7 \u03b1)).HasBasis (fun U => U \u2208 \ud835\udce4 \u03b1) fun U =>\n      (fun p : (\u03b1 \u00d7 \u03b1) \u00d7 \u03b1 \u00d7 \u03b1 => ((p.1.1, p.2.1), p.1.2, p.2.2)) \u207b\u00b9' U \u00d7\u02e2 U :=\n    by\n    rw [uniformity_prod_eq_comap_prod]\n    exact (\ud835\udce4 \u03b1).basis_sets.prod_self.comap _\n  refine' (is_compact_diagonal.nhds_set_basis_uniformity this).ge_iff.2 fun U hU => _\n  exact mem_of_superset hU fun \u27e8x, y\u27e9 hxy => mem_Union\u2082.2 \u27e8(x, x), rfl, refl_mem_uniformity hU, hxy\u27e9\n#align nhds_set_diagonal_eq_uniformity nhdsSet_diagonal_eq_uniformity\n\n/- warning: compact_space_uniformity -> compactSpace_uniformity is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_3 : CompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (uniformity.{u1} \u03b1 _inst_1) (sup\u1d62.{u1, succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (ConditionallyCompleteLattice.toHasSup.{u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (Filter.completeLattice.{u1} (Prod.{u1, u1} \u03b1 \u03b1)))) \u03b1 (fun (x : \u03b1) => nhds.{u1} (Prod.{u1, u1} \u03b1 \u03b1) (Prod.topologicalSpace.{u1, u1} \u03b1 \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)) (Prod.mk.{u1, u1} \u03b1 \u03b1 x x)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_3 : CompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)], Eq.{succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (uniformity.{u1} \u03b1 _inst_1) (sup\u1d62.{u1, succ u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (ConditionallyCompleteLattice.toSupSet.{u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (CompleteLattice.toConditionallyCompleteLattice.{u1} (Filter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)) (Filter.instCompleteLatticeFilter.{u1} (Prod.{u1, u1} \u03b1 \u03b1)))) \u03b1 (fun (x : \u03b1) => nhds.{u1} (Prod.{u1, u1} \u03b1 \u03b1) (instTopologicalSpaceProd.{u1, u1} \u03b1 \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)) (Prod.mk.{u1, u1} \u03b1 \u03b1 x x)))\nCase conversion may be inaccurate. Consider using '#align compact_space_uniformity compactSpace_uniformity\u2093'. -/\n/-- On a compact uniform space, the topology determines the uniform structure, entourages are\nexactly the neighborhoods of the diagonal. -/\ntheorem compactSpace_uniformity [CompactSpace \u03b1] : \ud835\udce4 \u03b1 = \u2a06 x, \ud835\udcdd (x, x) :=\n  nhdsSet_diagonal_eq_uniformity.symm.trans (nhdsSet_diagonal _)\n#align compact_space_uniformity compactSpace_uniformity\n\n#print unique_uniformity_of_compact /-\ntheorem unique_uniformity_of_compact [t : TopologicalSpace \u03b3] [CompactSpace \u03b3]\n    {u u' : UniformSpace \u03b3} (h : u.toTopologicalSpace = t) (h' : u'.toTopologicalSpace = t) :\n    u = u' := by\n  apply uniformSpace_eq\n  change uniformity _ = uniformity _\n  have : @CompactSpace \u03b3 u.to_topological_space := by rwa [h]\n  have : @CompactSpace \u03b3 u'.to_topological_space := by rwa [h']\n  rw [compactSpace_uniformity, compactSpace_uniformity, h, h']\n#align unique_uniformity_of_compact unique_uniformity_of_compact\n-/\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Basic.lean:635:2: warning: expanding binder collection (y \u00abexpr \u2260 \u00bb x) -/\n#print uniformSpaceOfCompactT2 /-\n/-- The unique uniform structure inducing a given compact topological structure. -/\ndef uniformSpaceOfCompactT2 [TopologicalSpace \u03b3] [CompactSpace \u03b3] [T2Space \u03b3] : UniformSpace \u03b3\n    where\n  uniformity := \ud835\udcdd\u02e2 (diagonal \u03b3)\n  refl := principal_le_nhdsSet\n  symm := continuous_swap.tendsto_nhdsSet fun x => Eq.symm\n  comp :=\n    by\n    /-\n        This is the difficult part of the proof. We need to prove that, for each neighborhood `W`\n        of the diagonal `\u0394`, there exists a smaller neighborhood `V` such that `V \u25cb V \u2286 W`.\n        -/\n    set \ud835\udcdd\u0394 := \ud835\udcdd\u02e2 (diagonal \u03b3)\n    -- The filter of neighborhoods of \u0394\n    set F := \ud835\udcdd\u0394.lift' fun s : Set (\u03b3 \u00d7 \u03b3) => s \u25cb s\n    -- Compositions of neighborhoods of \u0394\n    -- If this weren't true, then there would be V \u2208 \ud835\udcdd\u0394 such that F \u2293 \ud835\udcdf V\u1d9c \u2260 \u22a5\n    rw [le_iff_forall_inf_principal_compl]\n    intro V V_in\n    by_contra H\n    haveI : ne_bot (F \u2293 \ud835\udcdf (V\u1d9c)) := \u27e8H\u27e9\n    -- Hence compactness would give us a cluster point (x, y) for F \u2293 \ud835\udcdf V\u1d9c\n    obtain \u27e8\u27e8x, y\u27e9, hxy\u27e9 : \u2203 p : \u03b3 \u00d7 \u03b3, ClusterPt p (F \u2293 \ud835\udcdf (V\u1d9c)) := cluster_point_of_compact _\n    -- In particular (x, y) is a cluster point of \ud835\udcdf V\u1d9c, hence is not in the interior of V,\n    -- and a fortiori not in \u0394, so x \u2260 y\n    have clV : ClusterPt (x, y) (\ud835\udcdf <| V\u1d9c) := hxy.of_inf_right\n    have : (x, y) \u2209 interior V :=\n      by\n      have : (x, y) \u2208 closure (V\u1d9c) := by rwa [mem_closure_iff_clusterPt]\n      rwa [closure_compl] at this\n    have diag_subset : diagonal \u03b3 \u2286 interior V := subset_interior_iff_mem_nhdsSet.2 V_in\n    have x_ne_y : x \u2260 y := mt (@diag_subset (x, y)) this\n    -- Since \u03b3 is compact and Hausdorff, it is normal, hence T\u2083.\n    haveI : NormalSpace \u03b3 := normalOfCompactT2\n    -- So there are closed neighboords V\u2081 and V\u2082 of x and y contained in disjoint open neighborhoods\n    -- U\u2081 and U\u2082.\n    obtain\n      \u27e8U\u2081, U\u2081_in, V\u2081, V\u2081_in, U\u2082, U\u2082_in\u2082, V\u2082, V\u2082_in, V\u2081_cl, V\u2082_cl, U\u2081_op, U\u2082_op, VU\u2081, VU\u2082, hU\u2081\u2082\u27e9 :=\n      disjoint_nested_nhds x_ne_y\n    -- We set U\u2083 := (V\u2081 \u222a V\u2082)\u1d9c so that W := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083 is an open\n    -- neighborhood of \u0394.\n    let U\u2083 := (V\u2081 \u222a V\u2082)\u1d9c\n    have U\u2083_op : IsOpen U\u2083 := (V\u2081_cl.union V\u2082_cl).isOpen_compl\n    let W := U\u2081 \u00d7\u02e2 U\u2081 \u222a U\u2082 \u00d7\u02e2 U\u2082 \u222a U\u2083 \u00d7\u02e2 U\u2083\n    have W_in : W \u2208 \ud835\udcdd\u0394 := by\n      rw [mem_nhdsSet_iff_forall]\n      rintro \u27e8z, z'\u27e9 (rfl : z = z')\n      refine' IsOpen.mem_nhds _ _\n      \u00b7 apply_rules [IsOpen.union, IsOpen.prod]\n      \u00b7 simp only [mem_union, mem_prod, and_self_iff]\n        exact (em _).imp_left fun h => union_subset_union VU\u2081 VU\u2082 h\n    -- So W \u25cb W \u2208 F by definition of F\n    have : W \u25cb W \u2208 F := by simpa only using mem_lift' W_in\n    -- And V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y)\n    have hV\u2081\u2082 : V\u2081 \u00d7\u02e2 V\u2082 \u2208 \ud835\udcdd (x, y) := prod_mem_nhds V\u2081_in V\u2082_in\n    -- But (x, y) is also a cluster point of F so (V\u2081 \u00d7\u02e2 V\u2082) \u2229 (W \u25cb W) \u2260 \u2205\n    -- However the construction of W implies (V\u2081 \u00d7\u02e2 V\u2082) \u2229 (W \u25cb W) = \u2205.\n    -- Indeed assume for contradiction there is some (u, v) in the intersection.\n    obtain \u27e8\u27e8u, v\u27e9, \u27e8u_in, v_in\u27e9, w, huw, hwv\u27e9 := cluster_pt_iff.mp hxy.of_inf_left hV\u2081\u2082 this\n    -- So u \u2208 V\u2081, v \u2208 V\u2082, and there exists some w such that (u, w) \u2208 W and (w ,v) \u2208 W.\n    -- Because u is in V\u2081 which is disjoint from U\u2082 and U\u2083, (u, w) \u2208 W forces (u, w) \u2208 U\u2081 \u00d7\u02e2 U\u2081.\n    have uw_in : (u, w) \u2208 U\u2081 \u00d7\u02e2 U\u2081 :=\n      (huw.resolve_right fun h => h.1 <| Or.inl u_in).resolve_right fun h =>\n        hU\u2081\u2082.le_bot \u27e8VU\u2081 u_in, h.1\u27e9\n    -- Similarly, because v \u2208 V\u2082, (w ,v) \u2208 W forces (w, v) \u2208 U\u2082 \u00d7\u02e2 U\u2082.\n    have wv_in : (w, v) \u2208 U\u2082 \u00d7\u02e2 U\u2082 :=\n      (hwv.resolve_right fun h => h.2 <| Or.inr v_in).resolve_left fun h =>\n        hU\u2081\u2082.le_bot \u27e8h.2, VU\u2082 v_in\u27e9\n    -- Hence w \u2208 U\u2081 \u2229 U\u2082 which is empty.\n    -- So we have a contradiction\n    exact hU\u2081\u2082.le_bot \u27e8uw_in.2, wv_in.1\u27e9\n  isOpen_uniformity :=\n    by\n    -- Here we need to prove the topology induced by the constructed uniformity is the\n    -- topology we started with.\n    suffices \u2200 x : \u03b3, Filter.comap (Prod.mk x) (\ud835\udcdd\u02e2 (diagonal \u03b3)) = \ud835\udcdd x\n      by\n      intro s\n      simp_rw [isOpen_fold, isOpen_iff_mem_nhds, \u2190 mem_comap_prod_mk, this]\n    intro x\n    simp_rw [nhdsSet_diagonal, comap_supr, nhds_prod_eq, comap_prod, (\u00b7 \u2218 \u00b7), comap_id']\n    rw [sup\u1d62_split_single _ x, comap_const_of_mem fun V => mem_of_mem_nhds]\n    suffices \u2200 (y) (_ : y \u2260 x), comap (fun y : \u03b3 => x) (\ud835\udcdd y) \u2293 \ud835\udcdd y \u2264 \ud835\udcdd x by simpa\n    intro y hxy\n    simp [comap_const_of_not_mem (compl_singleton_mem_nhds hxy) (Classical.not_not.2 rfl)]\n#align uniform_space_of_compact_t2 uniformSpaceOfCompactT2\n-/\n\n/-!\n###\u00a0Heine-Cantor theorem\n-/\n\n\n/- warning: compact_space.uniform_continuous_of_continuous -> CompactSpace.uniformContinuous_of_continuous is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] [_inst_3 : CompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)] {f : \u03b1 -> \u03b2}, (Continuous.{u1, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2) f) -> (UniformContinuous.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : UniformSpace.{u2} \u03b1] [_inst_2 : UniformSpace.{u1} \u03b2] [_inst_3 : CompactSpace.{u2} \u03b1 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1)] {f : \u03b1 -> \u03b2}, (Continuous.{u2, u1} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b2 _inst_2) f) -> (UniformContinuous.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f)\nCase conversion may be inaccurate. Consider using '#align compact_space.uniform_continuous_of_continuous CompactSpace.uniformContinuous_of_continuous\u2093'. -/\n/-- Heine-Cantor: a continuous function on a compact uniform space is uniformly\ncontinuous. -/\ntheorem CompactSpace.uniformContinuous_of_continuous [CompactSpace \u03b1] {f : \u03b1 \u2192 \u03b2}\n    (h : Continuous f) : UniformContinuous f :=\n  have : Tendsto (Prod.map f f) (\ud835\udcdd\u02e2 (diagonal \u03b1)) (\ud835\udcdd\u02e2 (diagonal \u03b2)) :=\n    (h.Prod_map h).tendsto_nhdsSet mapsTo_prod_map_diagonal\n  (this.mono_left nhdsSet_diagonal_eq_uniformity.ge).mono_right nhdsSet_diagonal_le_uniformity\n#align compact_space.uniform_continuous_of_continuous CompactSpace.uniformContinuous_of_continuous\n\n/- warning: is_compact.uniform_continuous_on_of_continuous -> IsCompact.uniformContinuousOn_of_continuous is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] {s : Set.{u1} \u03b1} {f : \u03b1 -> \u03b2}, (IsCompact.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) s) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2) f s) -> (UniformContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f s)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : UniformSpace.{u2} \u03b1] [_inst_2 : UniformSpace.{u1} \u03b2] {s : Set.{u2} \u03b1} {f : \u03b1 -> \u03b2}, (IsCompact.{u2} \u03b1 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1) s) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b2 _inst_2) f s) -> (UniformContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f s)\nCase conversion may be inaccurate. Consider using '#align is_compact.uniform_continuous_on_of_continuous IsCompact.uniformContinuousOn_of_continuous\u2093'. -/\n/-- Heine-Cantor: a continuous function on a compact set of a uniform space is uniformly\ncontinuous. -/\ntheorem IsCompact.uniformContinuousOn_of_continuous {s : Set \u03b1} {f : \u03b1 \u2192 \u03b2} (hs : IsCompact s)\n    (hf : ContinuousOn f s) : UniformContinuousOn f s :=\n  by\n  rw [uniformContinuousOn_iff_restrict]\n  rw [isCompact_iff_compactSpace] at hs\n  rw [continuousOn_iff_continuous_restrict] at hf\n  skip\n  exact CompactSpace.uniformContinuous_of_continuous hf\n#align is_compact.uniform_continuous_on_of_continuous IsCompact.uniformContinuousOn_of_continuous\n\n#print IsCompact.uniformContinuousAt_of_continuousAt /-\n/-- If `s` is compact and `f` is continuous at all points of `s`, then `f` is\n\"uniformly continuous at the set `s`\", i.e. `f x` is close to `f y` whenever `x \u2208 s` and `y` is\nclose to `x` (even if `y` is not itself in `s`, so this is a stronger assertion than\n`uniform_continuous_on s`). -/\ntheorem IsCompact.uniformContinuousAt_of_continuousAt {r : Set (\u03b2 \u00d7 \u03b2)} {s : Set \u03b1}\n    (hs : IsCompact s) (f : \u03b1 \u2192 \u03b2) (hf : \u2200 a \u2208 s, ContinuousAt f a) (hr : r \u2208 \ud835\udce4 \u03b2) :\n    { x : \u03b1 \u00d7 \u03b1 | x.1 \u2208 s \u2192 (f x.1, f x.2) \u2208 r } \u2208 \ud835\udce4 \u03b1 :=\n  by\n  obtain \u27e8t, ht, htsymm, htr\u27e9 := comp_symm_mem_uniformity_sets hr\n  choose U hU T hT hb using fun a ha =>\n    exists_mem_nhds_ball_subset_of_mem_nhds ((hf a ha).preimage_mem_nhds <| mem_nhds_left _ ht)\n  obtain \u27e8fs, hsU\u27e9 := hs.elim_nhds_subcover' U hU\n  apply mem_of_superset ((bInter_finset_mem fs).2 fun a _ => hT a a.2)\n  rintro \u27e8a\u2081, a\u2082\u27e9 h h\u2081\n  obtain \u27e8a, ha, haU\u27e9 := Set.mem_union\u1d62\u2082.1 (hsU h\u2081)\n  apply htr\n  refine' \u27e8f a, htsymm.mk_mem_comm.1 (hb _ _ _ haU _), hb _ _ _ haU _\u27e9\n  exacts[mem_ball_self _ (hT a a.2), mem_Inter\u2082.1 h a ha]\n#align is_compact.uniform_continuous_at_of_continuous_at IsCompact.uniformContinuousAt_of_continuousAt\n-/\n\n/- warning: continuous.uniform_continuous_of_tendsto_cocompact -> Continuous.uniformContinuous_of_tendsto_cocompact is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] {f : \u03b1 -> \u03b2} {x : \u03b2}, (Continuous.{u1, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2) f) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (Filter.cocompact.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)) (nhds.{u2} \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2) x)) -> (UniformContinuous.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_2 f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : UniformSpace.{u2} \u03b1] [_inst_2 : UniformSpace.{u1} \u03b2] {f : \u03b1 -> \u03b2} {x : \u03b2}, (Continuous.{u2, u1} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u1} \u03b2 _inst_2) f) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (Filter.cocompact.{u2} \u03b1 (UniformSpace.toTopologicalSpace.{u2} \u03b1 _inst_1)) (nhds.{u1} \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b2 _inst_2) x)) -> (UniformContinuous.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_2 f)\nCase conversion may be inaccurate. Consider using '#align continuous.uniform_continuous_of_tendsto_cocompact Continuous.uniformContinuous_of_tendsto_cocompact\u2093'. -/\ntheorem Continuous.uniformContinuous_of_tendsto_cocompact {f : \u03b1 \u2192 \u03b2} {x : \u03b2}\n    (h_cont : Continuous f) (hx : Tendsto f (cocompact \u03b1) (\ud835\udcdd x)) : UniformContinuous f :=\n  uniformContinuous_def.2 fun r hr =>\n    by\n    obtain \u27e8t, ht, htsymm, htr\u27e9 := comp_symm_mem_uniformity_sets hr\n    obtain \u27e8s, hs, hst\u27e9 := mem_cocompact.1 (hx <| mem_nhds_left _ ht)\n    apply\n      mem_of_superset\n        (symmetrize_mem_uniformity <|\n          (hs.uniform_continuous_at_of_continuous_at f fun _ _ => h_cont.continuous_at) <|\n            symmetrize_mem_uniformity hr)\n    rintro \u27e8b\u2081, b\u2082\u27e9 h\n    by_cases h\u2081 : b\u2081 \u2208 s; \u00b7 exact (h.1 h\u2081).1\n    by_cases h\u2082 : b\u2082 \u2208 s; \u00b7 exact (h.2 h\u2082).2\n    apply htr\n    exact \u27e8x, htsymm.mk_mem_comm.1 (hst h\u2081), hst h\u2082\u27e9\n#align continuous.uniform_continuous_of_tendsto_cocompact Continuous.uniformContinuous_of_tendsto_cocompact\n\n#print HasCompactMulSupport.is_one_at_infty /-\n/-- If `f` has compact multiplicative support, then `f` tends to 1 at infinity. -/\n@[to_additive \"If `f` has compact support, then `f` tends to zero at infinity.\"]\ntheorem HasCompactMulSupport.is_one_at_infty {f : \u03b1 \u2192 \u03b3} [TopologicalSpace \u03b3] [One \u03b3]\n    (h : HasCompactMulSupport f) : Tendsto f (cocompact \u03b1) (\ud835\udcdd 1) :=\n  by\n  -- porting note: move to src/topology/support.lean once the port is over\n  intro N hN\n  rw [mem_map, mem_cocompact']\n  refine' \u27e8mulTSupport f, h.is_compact, _\u27e9\n  rw [compl_subset_comm]\n  intro v hv\n  rw [mem_preimage, image_eq_one_of_nmem_mulTSupport hv]\n  exact mem_of_mem_nhds hN\n#align has_compact_mul_support.is_one_at_infty HasCompactMulSupport.is_one_at_infty\n#align has_compact_support.is_zero_at_infty HasCompactSupport.is_zero_at_infty\n-/\n\n#print HasCompactMulSupport.uniformContinuous_of_continuous /-\n@[to_additive]\ntheorem HasCompactMulSupport.uniformContinuous_of_continuous {f : \u03b1 \u2192 \u03b2} [One \u03b2]\n    (h1 : HasCompactMulSupport f) (h2 : Continuous f) : UniformContinuous f :=\n  h2.uniformContinuous_of_tendsto_cocompact h1.is_one_at_infty\n#align has_compact_mul_support.uniform_continuous_of_continuous HasCompactMulSupport.uniformContinuous_of_continuous\n#align has_compact_support.uniform_continuous_of_continuous HasCompactSupport.uniformContinuous_of_continuous\n-/\n\n/- warning: continuous_on.tendsto_uniformly -> ContinuousOn.tendstoUniformly is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] [_inst_3 : LocallyCompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)] [_inst_4 : CompactSpace.{u2} \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)] [_inst_5 : UniformSpace.{u3} \u03b3] {f : \u03b1 -> \u03b2 -> \u03b3} {x : \u03b1} {U : Set.{u1} \u03b1}, (Membership.Mem.{u1, u1} (Set.{u1} \u03b1) (Filter.{u1} \u03b1) (Filter.hasMem.{u1} \u03b1) U (nhds.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) x)) -> (ContinuousOn.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) \u03b3 (Prod.topologicalSpace.{u1, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)) (UniformSpace.toTopologicalSpace.{u3} \u03b3 _inst_5) (Function.HasUncurry.uncurry.{max u1 u2 u3, max u1 u2, u3} (\u03b1 -> \u03b2 -> \u03b3) (Prod.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.hasUncurryInduction.{u1, max u2 u3, u2, u3} \u03b1 (\u03b2 -> \u03b3) \u03b2 \u03b3 (Function.hasUncurryBase.{u2, u3} \u03b2 \u03b3)) f) (Set.prod.{u1, u2} \u03b1 \u03b2 U (Set.univ.{u2} \u03b2))) -> (TendstoUniformly.{u2, u3, u1} \u03b2 \u03b3 \u03b1 _inst_5 f (f x) (nhds.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : UniformSpace.{u3} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] [_inst_3 : LocallyCompactSpace.{u3} \u03b1 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1)] [_inst_4 : CompactSpace.{u2} \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)] [_inst_5 : UniformSpace.{u1} \u03b3] {f : \u03b1 -> \u03b2 -> \u03b3} {x : \u03b1} {U : Set.{u3} \u03b1}, (Membership.mem.{u3, u3} (Set.{u3} \u03b1) (Filter.{u3} \u03b1) (instMembershipSetFilter.{u3} \u03b1) U (nhds.{u3} \u03b1 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1) x)) -> (ContinuousOn.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) \u03b3 (instTopologicalSpaceProd.{u3, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)) (UniformSpace.toTopologicalSpace.{u1} \u03b3 _inst_5) (Function.HasUncurry.uncurry.{max (max u3 u2) u1, max u3 u2, u1} (\u03b1 -> \u03b2 -> \u03b3) (Prod.{u3, u2} \u03b1 \u03b2) \u03b3 (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} \u03b1 (\u03b2 -> \u03b3) \u03b2 \u03b3 (Function.hasUncurryBase.{u2, u1} \u03b2 \u03b3)) f) (Set.prod.{u3, u2} \u03b1 \u03b2 U (Set.univ.{u2} \u03b2))) -> (TendstoUniformly.{u2, u1, u3} \u03b2 \u03b3 \u03b1 _inst_5 f (f x) (nhds.{u3} \u03b1 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1) x))\nCase conversion may be inaccurate. Consider using '#align continuous_on.tendsto_uniformly ContinuousOn.tendstoUniformly\u2093'. -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- A family of functions `\u03b1 \u2192 \u03b2 \u2192 \u03b3` tends uniformly to its value at `x` if `\u03b1` is locally compact,\n`\u03b2` is compact and `f` is continuous on `U \u00d7 (univ : set \u03b2)` for some neighborhood `U` of `x`. -/\ntheorem ContinuousOn.tendstoUniformly [LocallyCompactSpace \u03b1] [CompactSpace \u03b2] [UniformSpace \u03b3]\n    {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {x : \u03b1} {U : Set \u03b1} (hxU : U \u2208 \ud835\udcdd x) (h : ContinuousOn (\u21bff) (U \u00d7\u02e2 univ)) :\n    TendstoUniformly f (f x) (\ud835\udcdd x) :=\n  by\n  rcases LocallyCompactSpace.local_compact_nhds _ _ hxU with \u27e8K, hxK, hKU, hK\u27e9\n  have : UniformContinuousOn (\u21bff) (K \u00d7\u02e2 univ) :=\n    IsCompact.uniformContinuousOn_of_continuous (hK.prod isCompact_univ)\n      (h.mono <| prod_mono hKU subset.rfl)\n  exact this.tendsto_uniformly hxK\n#align continuous_on.tendsto_uniformly ContinuousOn.tendstoUniformly\n\n/- warning: continuous.tendsto_uniformly -> Continuous.tendstoUniformly is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : UniformSpace.{u1} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] [_inst_3 : LocallyCompactSpace.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1)] [_inst_4 : CompactSpace.{u2} \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)] [_inst_5 : UniformSpace.{u3} \u03b3] (f : \u03b1 -> \u03b2 -> \u03b3), (Continuous.{max u1 u2, u3} (Prod.{u1, u2} \u03b1 \u03b2) \u03b3 (Prod.topologicalSpace.{u1, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)) (UniformSpace.toTopologicalSpace.{u3} \u03b3 _inst_5) (Function.HasUncurry.uncurry.{max u1 u2 u3, max u1 u2, u3} (\u03b1 -> \u03b2 -> \u03b3) (Prod.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.hasUncurryInduction.{u1, max u2 u3, u2, u3} \u03b1 (\u03b2 -> \u03b3) \u03b2 \u03b3 (Function.hasUncurryBase.{u2, u3} \u03b2 \u03b3)) f)) -> (forall (x : \u03b1), TendstoUniformly.{u2, u3, u1} \u03b2 \u03b3 \u03b1 _inst_5 f (f x) (nhds.{u1} \u03b1 (UniformSpace.toTopologicalSpace.{u1} \u03b1 _inst_1) x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u1}} [_inst_1 : UniformSpace.{u3} \u03b1] [_inst_2 : UniformSpace.{u2} \u03b2] [_inst_3 : LocallyCompactSpace.{u3} \u03b1 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1)] [_inst_4 : CompactSpace.{u2} \u03b2 (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)] [_inst_5 : UniformSpace.{u1} \u03b3] (f : \u03b1 -> \u03b2 -> \u03b3), (Continuous.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) \u03b3 (instTopologicalSpaceProd.{u3, u2} \u03b1 \u03b2 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1) (UniformSpace.toTopologicalSpace.{u2} \u03b2 _inst_2)) (UniformSpace.toTopologicalSpace.{u1} \u03b3 _inst_5) (Function.HasUncurry.uncurry.{max (max u3 u2) u1, max u3 u2, u1} (\u03b1 -> \u03b2 -> \u03b3) (Prod.{u3, u2} \u03b1 \u03b2) \u03b3 (Function.hasUncurryInduction.{u3, max u2 u1, u2, u1} \u03b1 (\u03b2 -> \u03b3) \u03b2 \u03b3 (Function.hasUncurryBase.{u2, u1} \u03b2 \u03b3)) f)) -> (forall (x : \u03b1), TendstoUniformly.{u2, u1, u3} \u03b2 \u03b3 \u03b1 _inst_5 f (f x) (nhds.{u3} \u03b1 (UniformSpace.toTopologicalSpace.{u3} \u03b1 _inst_1) x))\nCase conversion may be inaccurate. Consider using '#align continuous.tendsto_uniformly Continuous.tendstoUniformly\u2093'. -/\n/-- A continuous family of functions `\u03b1 \u2192 \u03b2 \u2192 \u03b3` tends uniformly to its value at `x` if `\u03b1` is\nlocally compact and `\u03b2` is compact. -/\ntheorem Continuous.tendstoUniformly [LocallyCompactSpace \u03b1] [CompactSpace \u03b2] [UniformSpace \u03b3]\n    (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (h : Continuous \u21bff) (x : \u03b1) : TendstoUniformly f (f x) (\ud835\udcdd x) :=\n  h.ContinuousOn.TendstoUniformly univ_mem\n#align continuous.tendsto_uniformly Continuous.tendstoUniformly\n\nsection UniformConvergence\n\n#print CompactSpace.uniformEquicontinuous_of_equicontinuous /-\n/-- An equicontinuous family of functions defined on a compact uniform space is automatically\nuniformly equicontinuous. -/\ntheorem CompactSpace.uniformEquicontinuous_of_equicontinuous {\u03b9 : Type _} {F : \u03b9 \u2192 \u03b2 \u2192 \u03b1}\n    [CompactSpace \u03b2] (h : Equicontinuous F) : UniformEquicontinuous F :=\n  by\n  rw [equicontinuous_iff_continuous] at h\n  rw [uniformEquicontinuous_iff_uniformContinuous]\n  exact CompactSpace.uniformContinuous_of_continuous h\n#align compact_space.uniform_equicontinuous_of_equicontinuous CompactSpace.uniformEquicontinuous_of_equicontinuous\n-/\n\nend UniformConvergence\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/UniformSpace/Compact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.6723317123102956, "lm_q1q2_score": 0.3727880672327077}}
{"text": "/- This file includes the proofs for the axioms for the summation operator.-/\nimport .iff_lemmas\n\nopen mcrl2\n\nvariable {\u03b1 : Type}\nvariable [comm_semigroup_with_zero \u03b1]\n\nvariable {\u03b2 : Type}\n\n/- Note the presence of the hypothesis h here. -/\nlemma transition.sum_idem (A : set \u03b2) (h : \u2203a, a \u2208 A) (x : mcrl2 \u03b1) (z a) :\ntransition (sum A (\u03bba, x)) a z \u2194 transition x a z :=\nbegin\n  simp only [transition.sum_iff],\n  split,\n  { intro h',\n    rcases h' with \u27e8_, _, h'\u27e9,\n    assumption},\n  { intro h',\n    cases h,\n    exact \u27e8h_w, h_h, h'\u27e9}\nend\n\nlemma transition.sum_elem (A : set \u03b2)  (f : \u03b2 \u2192 mcrl2 \u03b1) (d) (h : d \u2208 A) (z a) :\ntransition (sum A f) a z \u2194 transition (sum A f + f d) a z :=\nbegin\n  simp [transition.sum_iff, transition.alt_iff],\n  split,\n  { intro h,\n    apply or.inl,\n    assumption},\n  { intro h',\n    rcases h' with \u27e8a', ha', haz\u27e9 | \u27e8haz\u27e9,\n    { exact \u27e8a', ha', haz\u27e9},\n    { exact \u27e8d, h, haz\u27e9}}\nend\n\nlemma transition.sum_pure (A : set \u03b2) (f : \u03b2 \u2192 mcrl2 \u03b1) (z a) :\ntransition (sum A (\u03bba, f a)) a z \u2194 transition (sum A f) a z :=\nbegin\n  simp [transition.sum_iff]\nend\n\nlemma transition.sum_alt (A : set \u03b2) (f g : \u03b2 \u2192 mcrl2 \u03b1) (z a) :\ntransition (sum A (\u03bba, f a + g a)) a z \u2194 transition (sum A f + sum A g) a z :=\nby simp [transition.sum_iff, transition.alt_iff, and_or_distrib_left, \u2190exists_or_distrib]\n\n\nlemma transition.sum_seq (A : set \u03b2) (f : \u03b2 \u2192 mcrl2 \u03b1) (x z a) :\ntransition (sum A f \u2b1d x) a z \u2194 transition (sum A (\u03bba, f a \u2b1d x)) a z :=\nbegin simp [transition.sum_iff, transition.seq_iff, \u2190exists_and_distrib_left], tauto\nend\n\nlemma transition.sum_parl (A : set \u03b2) (f : \u03b2 \u2192 mcrl2 \u03b1) (x z a) :\ntransition (sum A f |_ x) a z \u2194 transition (sum A (\u03bba, f a |_ x)) a z :=\nby simp [transition.sum_iff, transition.parl_iff, \u2190exists_and_distrib_left, \u2190and_assoc, and_comm, exists_comm]\n\nlemma transition.sum_comm (A : set \u03b2) (f : \u03b2 \u2192 mcrl2 \u03b1) (x z a) :\ntransition (sum A f \u2223 x) a z \u2194 transition (sum A (\u03bba, f a \u2223 x)) a z :=\nbegin\n  simp [transition.sum_iff, transition.comm_iff, \u2190exists_and_distrib_left, \u2190and_assoc, \u2190exists_and_distrib_right],\n  tauto\nend\n\nlemma transition.comm_sum (A : set \u03b2) (f : \u03b2 \u2192 mcrl2 \u03b1) (x z a) :\ntransition (x \u2223 (sum A f)) a z \u2194 transition (sum A (\u03bba, x \u2223 f a)) a z :=\nbegin\n  simp [transition.sum_iff, transition.comm_iff, \u2190exists_and_distrib_left, \u2190and_assoc, \u2190exists_and_distrib_right],\n  tauto\nend\n\nlemma transition.encap_sum (H : set \u03b1) (D : set \u03b2) (f z a) :\ntransition (encap H (sum D f)) a z \u2194 transition (sum D (\u03bba, encap H (f a))) a z :=\nbegin\n  simp [transition.encap_iff, transition.sum_iff],\n  tauto\nend\n\nlemma transition.sum_ext (D : set \u03b2) (f g : \u03b2 \u2192 mcrl2 \u03b1 )\n  (h : \u2200a' : \u03b2, a' \u2208 D \u2192 \u2200z a, (transition (f a') a z \u2194 transition (g a') a z)) (z a) :\ntransition (sum D f) a z \u2194 transition (sum D g) a z :=\nbegin\n  simp [transition.sum_iff],\n  split,\n  { intro h',\n    rcases h' with \u27e8a', ha', haz\u27e9,\n    apply exists.intro a',\n    apply and.intro ha',\n    apply iff.elim_left (h a' ha' z a),\n    assumption},\n  { intro h',\n    rcases h' with \u27e8a', ha', haz\u27e9,\n    apply exists.intro a',\n    apply and.intro ha',\n    apply iff.elim_right (h a' ha' z a),\n    assumption}\nend", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/transition/sum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.3727274611773017}}
{"text": "/-\nCopyright (c) 2019 The Flypitch Project. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Jesse Han, Floris van Doorn\n-/\nimport .fol data.zmod.basic\n\nopen fol\n\n-- local attribute [instance, priority 0] classical.prop_decidable\n--local attribute [instance] classical.prop_decidable\n\nlocal notation h :: t  := dvector.cons h t\nlocal notation `[]` := dvector.nil\nlocal notation `[` l:(foldr `, ` (h t, dvector.cons h t) dvector.nil `]`) := l\n\nnamespace realization\n\n/- Note: maybe some of the _irrel lemmas should be called something_elim instead -/\n\n-- section fin_lemmas\n\n-- open fin\n-- variable n : \u2115\n-- @[simp]lemma of_nat_zero : @of_nat n 0 = 0 := rfl\n\n-- @[simp]lemma add_def (a b : fin n) : (a + b).val = (a.val + b.val) % n :=\n-- show (fin.add a b).val = (a.val + b.val) % n, from\n-- by cases a; cases b; simp [fin.add]\n\n-- @[simp]lemma mul_def (a b : fin n) : (a * b).val = (a.val * b.val) % n :=\n-- show (fin.mul a b).val = (a.val * b.val) % n, from\n-- by cases a; cases b; simp [fin.mul]\n\n-- @[simp]lemma sub_def (a b : fin n) : (a - b).val = a.val - b.val :=\n-- show (fin.sub a b).val = a.val - b.val, from\n-- by cases a; cases b; simp [fin.sub]\n\n-- @[simp]lemma mod_def (a b : fin n) : (a % b).val = a.val % b.val :=\n-- show (fin.mod a b).val = a.val % b.val, from\n-- by cases a; cases b; simp [fin.mod]\n\n-- @[simp]lemma div_def (a b : fin n) : (a / b).val = a.val / b.val :=\n-- show (fin.div a b).val = a.val / b.val, from\n-- by cases a; cases b; simp [fin.div]\n\n-- @[simp]lemma lt_def (a b : fin n) : (a < b) = (a.val < b.val) :=\n-- show (fin.lt a b) = (a.val < b.val), from\n-- by cases a; cases b; simp [fin.lt]\n\n-- @[simp]lemma le_def (a b : fin n) : (a \u2264 b) = (a.val \u2264 b.val) :=\n-- show (fin.le a b) = (a.val \u2264 b.val), from\n-- by cases a; cases b; simp [fin.le]\n\n-- @[simp]lemma val_zero : (0 : fin (nat.succ n)).val = 0 := rfl\n\n-- end fin_lemmas\n\n-- set_option pp.notation true\n-- set_option pp.all false\n\n/- To finish this, we need to port some of the zmod lemmas\n   should write an elimination principle which reduces to the 0 case and the pos case, which then hands it off to zmod...\n-/\n\nlemma succ_of_fin_val_succ_and_lt : \u2200 {m k : \u2115} {h : k < (m + 1 + 1)}, ((k : fin (m+1+1)) + 1).val = k + 1\n| 0 0 h := rfl\n| 0 (k + 1) h := by {induction k, simp, rw[fin.add_def], change _ % (2) = 2, repeat{sorry}}\n| (m+1) 0 h := rfl\n| (m+1) (k+1) h := sorry\n\nlemma var_subst_cast_irrel {L : Language} {n n' m} {h : m = n+ n' + 1} {k : fin m} {t : bounded_term L n'} :\n  subst_bounded_term ((&k : bounded_term L m).cast_eq h) t = subst_bounded_term (&k : bounded_term L (n + n' + 1)) t :=\nbegin\n  ext, simp, rcases k with \u27e8k_val, k_H\u27e9, induction m generalizing n' k_val, cases k_H,\n  cases k_val,\n    {refl},\n    {tidy, congr, subst h_1, change _ = fin.val ((nat.cast k_val) + 1 : fin (n + n' + 1)), sorry},\n-- to apply the above lemma, need to case on n to access the constructor.\n--induction k_val, unfold coe lift_t has_lift_t.lift coe_t has_coe_t.coe coe_b has_coe.coe nat.cast, conv {to_rhs, congr,\nend\n\n@[simp]lemma func_subst_cast_irrel {L : Language} {n n' l m} {h : m = n + n' + 1} {f : L.functions l} {t : bounded_term L n'} :\n  subst_bounded_term ((bd_func f : bounded_preterm L m l).cast_eq h) t = subst_bounded_term (bd_func f) t := by refl\n\n@[simp]lemma func_subst_irrel {L : Language} {n n' l} {f : L.functions l} {t : bounded_term L n'} :\n  subst_bounded_term (bd_func f : bounded_preterm L (n + n' + 1) l) t = (bd_func f) := by refl\n\n@[simp]lemma func_subst0_irrel {L : Language} {n l} {f : L.functions l} {t : bounded_term L n} : (bd_func f)[t /0] = (bd_func f) := by refl\n-- i wonder why this is by refl while rel_subst0_irrel isn't...\n\n@[simp]lemma subst_bounded_term_bd_apps {L} {n n' l} (f : bounded_preterm L (n + n' + 1) l) {t : bounded_term L n'} {ts : dvector (bounded_term L (n + n' + 1)) l} :\n  (bd_apps f ts)[t /// _] = bd_apps (f[t /// _]) (ts.map $ \u03bb t', subst_bounded_term (t') t) := by {induction ts generalizing f, refl, simp[bd_apps, ts_ih (bd_app f ts_x)]}\n\n@[simp]lemma subst0_bounded_term_bd_apps {L} {n l} (f : bounded_preterm L (n+1) l) {t : closed_term L} {ts : dvector (bounded_term L (n+1)) l} :\n  (bd_apps f ts)[(t.cast0 n) /0] = bd_apps (f[(t.cast0 n) /0]) (ts.map $ \u03bb t', t'[(t.cast0 n) /0]) := by {induction ts generalizing f, refl, simp[bd_apps, ts_ih (bd_app f ts_x)], refl}\n\nlemma realize_func_irrel {L} {S : Structure L} {n n' l : \u2115} {t : bounded_term L n'} {f : L.functions l} {xs : dvector \u21a5S l} {v : dvector \u21a5S (n + n' + 1)} : realize_bounded_term v (bd_func f) xs = S.fun_map f xs := by refl\n\n@[simp]lemma subst_falsum {L} {n n' n''} {h : n + n' + 1 = n''} {t : bounded_term L n'} : bd_falsum[t // n // h] = bd_falsum :=\n  by ext; simp\n\n@[simp]lemma subst0_falsum {L} {n} {t : bounded_term L n} : bd_falsum[t /0] = bd_falsum :=\n  by ext; simp\n\n@[simp]lemma subst_eq {L} {n n' n''} {h : n + n' + 1 = n''} {t\u2081 t\u2082 : bounded_term L n''} {t : bounded_term L n'} : (t\u2081 \u2243 t\u2082)[t // n // h] = subst_bounded_term (t\u2081.cast_eq h.symm) t \u2243 subst_bounded_term (t\u2082.cast_eq h.symm) t := by ext; simp\n\n@[simp]lemma subst0_eq {L} {n} {t : bounded_term L n} {t\u2081 t\u2082 : bounded_term L (n+1)} : (t\u2081 \u2243 t\u2082)[t /0] = (t\u2081[t /0] \u2243 t\u2082[t /0]) :=\n  by {unfold subst0_bounded_formula, simpa only [subst_eq]}\n\n@[simp]lemma subst_imp {L} {n n' n''} {h : n + n' + 1 = n''} {t : bounded_term L n'} {f\u2081 f\u2082 : bounded_formula L (n'')} : (f\u2081 \u27f9 f\u2082)[t // n // h] = (f\u2081[t // n // h] \u27f9 f\u2082[t // n // h]) := by {ext1, induction h, refl}\n\n@[simp]lemma subst0_imp {L} {n} {t : bounded_term L n} {f\u2081 f\u2082 : bounded_formula L (n+1)} : (f\u2081 \u27f9 f\u2082)[t /0] = f\u2081[t /0] \u27f9 f\u2082[t /0] :=\n  by {unfold subst0_bounded_formula, simpa only [subst_imp]}\n\n@[simp]lemma subst_all' {L} {n n' n''} {h : n + n' + 1 = n''} {t : bounded_term L n'} {f : bounded_formula L (n'' + 1)} :\n  (\u2200'f)[t  // n // (by {simp[h]})]\n  = \u2200'(f[(t : bounded_term L n') // (n+1) // (by {subst h; simp})]).cast_eq (by simp) := by ext; simp\n\n@[simp]lemma subst_all {L} {n n' n''} {h : n + n' + 1 = n''} {t : closed_term L} {f : bounded_formula L (n'' + 1)} :\n  (\u2200'f)[t.cast0 n' // n // (by {simp[h]})]\n  = \u2200'(f[(t.cast0 n' : bounded_term L n') // (n+1) // (by subst h; simp)]).cast_eq (by simp) :=\n  by {apply subst_all'}\n\n@[simp]lemma subst0_all {L} {n} {t : closed_term L} {f : bounded_formula L (n+2)} :\n  ((\u2200'f)[t.cast (by simp) /0] : bounded_formula L n) = \u2200'((f[t.cast0 n // 1 // (by {simp} : 1 + n + 1 = n + 2)]).cast_eq (by simp)) :=\n  by ext; simp\n\n@[simp]lemma subst0_all_base {L} {t : closed_term L} {f : bounded_formula L 2} : (\u2200' f)[t /0] = \u2200'(f[t // 1 // (by simp)]) :=\n  by ext; simp\n\n@[simp]lemma rel_subst_irrel {L : Language} {n n' l} {R : L.relations l} {t : bounded_term L n'} : (bd_rel R)[t // n // (by refl)] = (bd_rel R) := by ext; simp\n\n@[simp]lemma rel_subst_irrel1 {L : Language} {n n' n'' l} {h : n + n' + 1 = n''} {R : L.relations l} {t : bounded_term L n'} : (@bd_rel L (n + n' + 1 + 1) _ R)[t // (n+1) // (by subst h; simp)] = (bd_rel R) := by ext; simp\n\n@[simp]lemma rel_subst_irrel' {L : Language} {n n' n'' l} {h : n + n' + 1 = n''} {R : L.relations l} {t : bounded_term L n'} : (bd_rel R)[t // n // h] = (bd_rel R) := by subst h; apply rel_subst_irrel\n\n@[simp]lemma rel_subst0_irrel {L : Language} {n l} {R : L.relations l} {t : bounded_term L n} : (bd_rel R)[t /0] = (bd_rel R) := by ext; simp\n\nlemma realize_rel_irrel {L} {S : Structure L} {n n' l : \u2115} {t : bounded_term L n'} {R : L.relations l} {xs : dvector \u21a5S l} {v : dvector \u21a5S (n + n' + 1)} : realize_bounded_formula v (bounded_preformula.cast_eq (by refl) (bd_rel R)) xs = S.rel_map R xs := by refl\n\n@[simp]lemma subst_bounded_formula_bd_apprel {L} {n n' n'' l} {h : n + n' + 1 = n''} (f : bounded_preformula L (n'') (l + 1))\n  {t : bounded_term L n'} {s : bounded_term L (n'')} :\n(bd_apprel f s)[t // n //  h] = (bd_apprel (f[t // n // h]) (subst_bounded_term (s.cast_eq h.symm) t))\n:= by ext; simp\n\n@[simp]lemma subst_bounded_formula_bd_apps_rel {L} {n n' n'' l} {h : n + n' + 1 = n''} (f : bounded_preformula L (n''+1) l)\n  {t : bounded_term L n'} {ts : dvector (bounded_term L (n'' + 1)) l } :\n    (bd_apps_rel f ts)[t // (n+1) // (by {subst h, simp})] = bd_apps_rel (f[t // (n+1) // by {subst h, simp}]) (ts.map $ \u03bb t', subst_bounded_term (t'.cast_eq (by subst h; simp)) t) :=\n  by {induction ts generalizing f, refl, simp[bd_apps_rel, ts_ih (bd_apprel f ts_x)]}\n\n@[simp]lemma subst0_bounded_formula_bd_apps_rel {L} {n l} (f : bounded_preformula L (n+1) l)\n  (t : closed_term L) (ts : dvector (bounded_term L (n+1)) l) :\n  subst0_bounded_formula (bd_apps_rel f ts) (t.cast (by simp)) =\n  bd_apps_rel (subst0_bounded_formula f (t.cast (by simp))) (ts.map $ \u03bbt', subst0_bounded_term t' (t.cast (by simp))) :=\nby {induction ts generalizing f, refl, simp[bd_apps_rel, ts_ih (bd_apprel f ts_x)], congr, ext, simp}\n\nlemma zero_of_lt_one (n : nat) (h : n < 1) : n = 0 :=\n  by {cases h, refl, cases nat.lt_of_succ_le h_a}\n\n-- lemma asjh'_term {L} {S : Structure L} {n n'} {s : bounded_term L (n + n' + 1 + 1)} {t : bounded_term L n'} {v : dvector S (n + n' + 1)} :\n-- S[(@subst_bounded_term _ (n+1) n' 0 (s.cast_eq (by simp)) t).cast_eq (by simp) ;;; v] = S[s ;;; (v.insert (S[t.cast (by linarith) ;;; v]) (n+1))]\n--  :=\n-- begin\n--   revert s, refine bounded_term.rec1 _ _; intros,\n--   {sorry},\n--   {sorry}\n-- end\n\n-- set_option pp.implicit false\n\n-- lemma asjh' {L} {S : Structure L} {n n' n''} {h : n + n' + 1 = n''} {t : bounded_term L (n')} {f : bounded_formula L (n''+1)} (v : dvector S n'') : (S[(f[t  // (n+1) // (by {induction h, simp})]).cast_eq (by induction h; simp) ;; v])\n-- = (S[f ;; (v.insert (S[t.cast (by {induction h, linarith}) ;;; v]) (n+1))]) :=\n-- begin\n--   revert n'' f v, refine bounded_formula.rec1 _ _ _ _ _; intros,\n--   {ext, subst h, simp[subst_falsum], intros a, exact a},\n--   {ext, -- simp[realize_subst_preterm, asjh'_term],\n--     conv {to_lhs, rw[realize_bounded_formula_cast_eq_irrel]},\n--     simp[realize_subst_preterm], induction v, simp,\n--     sorry, simp*, repeat{sorry}\n\n--   --  tidy,\n--   --       sorry\n--   -- -- conv {to_lhs, congr, skip, congr, congr, rw[asjh'_term],},\n--     },\n--   {sorry},\n--   {sorry},\n--   {have : n + 1 + n' + 1 = n_1 + 1, by subst h; simp, conv {to_lhs, congr, skip, congr, rw[subst_all'], skip, rw[this]}, rw[bounded_preformula.cast_eq_all], dsimp, ext, apply forall_congr, intro x, repeat{rw[realize_bounded_formula_cast_eq_irrel]}, rw[dvector.cast_trans], have := ih (x::v), simp at *,\n--   },\n-- end\n\nset_option pp.implicit false\n\nlemma dvector_cast_push_in {\u03b1 : Type*} {n : \u2115} {m} {h : n = m} {h' : n+1 = m+1} {x : \u03b1} {v : dvector \u03b1 n} :\n(x::v).cast h' = x::(v.cast h) := by subst h; refl\n\nlemma dvector_cast_pull_out {\u03b1 : Type*} {n : \u2115} {m} {h : n = m} {h' : n+1 = m+1} {x : \u03b1} {v : dvector \u03b1 n} : (x :: (v.cast h)) = (x::v).cast (h') := by subst h; refl\n\nset_option pp.implicit false\n\nlemma gen_realize_bounded_term {L : Language} {S : Structure L} : \u2200 {n n' n'' : \u2115} {l f_n : \u2115} (s : bounded_term L f_n) (t : closed_term L) (v : dvector \u21a5S n'') {h : n + n' + 1 = n''} {h' : n'' + 1 = f_n}  (xs : dvector \u21a5S 0), realize_bounded_term (dvector.cast (by substs h h'; simp : n'' = n + 1 + n') v)\n      (subst_bounded_term (bounded_preterm.cast_eq (by {subst h; rw[<-h'],simp}) s) (bounded_preterm.cast (zero_le n') t))\n      xs =\n    realize_bounded_term (dvector.cast (h') (dvector.insert (realize_closed_term S t) (n + 1) v)) s xs :=\nbegin\n  intros, revert s, refine bounded_term.rec _ _; intros,\n  {rcases k with \u27e8k_val, k_H\u27e9,\n    -- unfold realize_bounded_term realize_closed_term, simp,\n    induction n generalizing k_val; subst h', swap,\n    by_cases k_val = n'',\n          {subst h, simp, tidy, sorry}, -- looks like here we need to case on k_val's relation to n_n + 1...\n\n          -- {have : k_val < n'',\n          --       by {apply nat.lt_of_le_and_ne, exact nat.le_of_lt_succ k_H, exact h},\n          -- have := @n_ih (v.trunc _ (rfl)) k_val this,\n          -- rw[dvector.nth_irrel1] at this, swap, dedup, apply nat.lt_of_lt_of_le, exact this,\n          -- exact nat.le_succ (n_n + 1),\n          -- rw[<-this], apply realize_bounded_term_irrel', swap, simp,\n          -- intros, simp only [dvector.trunc_nth]\n          repeat{sorry}},\n  {rw[dvector.zero_eq xs], substs h h',simp[subst_bounded_term_bd_apps,\n  realize_bounded_term_bd_apps, func_subst_irrel], congr' 1,\n  apply dvector.map_congr_pmem, intros x Hx, have := ih_ts x Hx, rwa[dvector.zero_eq xs] at this}\nend\n\nset_option pp.implicit false\n\nlemma gen_realize_bounded_formula {L} {S : Structure L}  : \u2200 {n n' n'' : \u2115} {n'''} {l} {h : n + n' + 1 = n''} {h' : n'' + 1 = n'''} (f : bounded_preformula L (n''') l) (t : closed_term L) (v : dvector S n'') (xs : dvector S l), (S[(f[t.cast0 n' // (n+1) // (by {substs h h', simp})]).cast_eq (by {subst h, simp}) ;; v ;; xs]) \u2194 (S[f.cast_eq (by substs h h'; simp) ;; (v.insert (S[t.cast (by {substs h h', linarith}) ;;; v]) (n+1)) ;; xs])\n:=\nbegin\n  intros,\n  induction f generalizing n n' n'' v,\n    {intros, simp},\n    {simp, apply iff_of_eq, congr' 1; apply gen_realize_bounded_term _ t v xs, all_goals{try{exact 0}, try{exact h}}},\n    {simp},\n    {rw[realize_bounded_formula_cast_eq_irrel,subst_bounded_formula_bd_apprel],\n    conv {to_rhs, rw[realize_bounded_formula_cast_eq_irrel]},\n    have := @f_ih (realize_bounded_term (dvector.cast h' (dvector.insert (realize_closed_term S t) (n + 1) v)) f_t [] :: xs) n n' n'' v h h',\n    simp only [fol.realize_bounded_formula_cast_eq_irrel, add_zero, dvector.insert, neg_nonpos, int.coe_nat_zero, fol.subst_bounded_formula,\n    int.coe_nat_add, add_comm, int.coe_nat_one, fol.bounded_preterm.cast, eq_self_iff_true, zero_le, fol.realize_bounded_formula,\n    fol.bounded_preterm.cast_irrel, fol.realize_closed_term_v_irrel, zero_add, add_right_inj, fol.realize_bounded_term, add_left_comm,\n    fol.closed_preterm.cast_of_cast0] at *,\n    erw[gen_realize_bounded_term], tactic.rotate 1, exact f_l, exact h, exact h',\n    apply this},\n\n    {have this_f := f_ih_f\u2081 xs v, have this_g := f_ih_f\u2082 xs v, simp, simp at this_f this_g, rw[<-this_f,<-this_g]},\n\n    {substs h h', conv{to_lhs, congr, skip, congr,\n    rw[@subst_all' L (n+1) n' (n + n' + 1 + 1) (by {simp}) (t.cast0 n') f_f]}, rw[cast_eq_all], dsimp, apply forall_congr, intro x,\n    have := @f_ih xs (n+1) n' ((n+1) + n' + 1) (x::(v.cast (by simp))) (by simp) (by simp),\n    rw[<-dvector_cast_push_in] at this, swap, {simp},\n    repeat{rw[realize_bounded_formula_cast_eq_irrel]},\n    rw[realize_bounded_formula_cast_eq_irrel] at this, rw[dvector.cast_trans] at *,\n    rw[this], clear this, clear f_ih, rw[dvector.insert_cons], apply iff_of_eq,\n    congr' 1, {simp}, simp [realize_bounded_term_irrel, -dvector.cast],\n      {rw[dvector.insert_cons], congr' 1, simp, apply dvector.cast_hrfl},\n      {simp[bounded_preformula.cast_eq_rfl], apply bounded_preformula.cast_eq_hrfl}}\nend\n\n-- /- The statement of this isn't quite right -/\n-- lemma asjh'' {L} {S : Structure L}  : \u2200 {n n' n'' : \u2115} {n'''} {l} {h : n + n' + 1 = n''} {h' : n'' + 1 = n'''} (f : bounded_preformula L (n''') l) (t : bounded_term L n') (v : dvector S n'') (xs : dvector S l), (S[(f[t  // (n+1) // (by {substs h h', simp})]).cast_eq (by {subst h, simp}) ;; v ;; xs]) = (S[f.cast_eq (by substs h h'; simp) ;; (v.insert (S[t.cast (by {substs h h', linarith}) ;;; v]) (n+1)) ;; xs])\n-- :=\n-- begin\n--   intros,\n--   induction f generalizing n n' n'' v,\n--     {intros, simp},\n--     {sorry},\n--     {simp},\n--     {sorry},\n--     {have this_f := f_ih_f\u2081 xs v t, have this_g := f_ih_f\u2082 xs v t, simp, simp at this_f this_g, rw[<-this_f,<-this_g]},\n--     {substs h h', have := @subst_all' L (n+1) n' (n + n' + 1 + 1) (by {simp}) t (f_f), ext, simp[this], let k, swap, change realize_bounded_formula v (bounded_preformula.cast_eq k _) _ \u2194 _,\n-- let j, swap, change realize_bounded_formula v (bounded_preformula.cast_eq k (\u2200' j)) _ \u2194 _,\n-- rw[cast_eq_all], dsimp[k,j], clear k j, apply forall_congr, intro x,\n--      have := @f_ih xs (n+1) n' ((n+1) + n' + 1) (x::(v.cast (by simp))) (by simp) (by simp) t,\n--      rw[cast_eq_trans], rw[<-dvector_cast_push_in] at this, swap, simp, swap, simp, simp,\n--      rw[realize_bounded_formula_cast_eq_irrel], rw[realize_bounded_formula_cast_eq_irrel] at this, rw[dvector.cast_trans] at this, rw[this], clear this, clear this f_ih,\n--      rw[dvector.insert_cons], apply iff_of_eq, congr' 1, simp, swap, {apply cast_eq_hrfl},\n--      {swap, simp, rw[dvector.insert_cons], simp, rw[dvector.insert_cons],--  let p, swap,\n--      -- let q, swap, change p == q,\n--      -- apply (@heq_iff_eq _ p (q.cast (by simp))).mpr, }}\n--      congr' 1, simp, sorry, sorry}}\n-- end\n-- AHA! so we can see here that the term itself actually needs to be lifted... by 1.\n-- note: doing just t \u21a6 t \u2191 1 doesn't work. need to lift the formula instead\n\n-- #check (((&0 \u2243 &1) : bounded_formula L_empty 2) \u27f9 (\u2200'((&0 \u2243 &1 : bounded_formula L_empty 3) \u2293 (&0 \u2243 &2 : bounded_formula L_empty 3)) : bounded_formula L_empty 2))\n\n-- TODO : figure out the correct statement of this lemma\n-- lemma asjh'' {L} {S : Structure L}  : \u2200 {n n' n'' : \u2115} {l} {h : n + n' + 1 = n''} (f : bounded_preformula L n'' l) (t : bounded_term L n') (v : dvector S n'') (xs : dvector S l), (S[((f \u2191' 1 # (n+1))[t  // (n+1) // (by {subst h, simp})]).cast_eq (by {subst h, simp}) ;; v ;; xs]) \u2194 (S[f.cast (by {subst h, repeat{constructor} }) ;; (v.insert (S[t.cast (by {subst h, linarith}) ;;; v]) (n+1)) ;; xs])\n-- :=\n-- begin\n--   intros,\n--   induction f generalizing n n' v,\n--     {intros, simp},\n--     {sorry},\n--     {simp},\n--     {sorry},\n--     {-- have this_f := f_ih_f\u2081 xs v t, have this_g := f_ih_f\u2082 xs v t, simp, simp at this_f this_g, simp*\n--     sorry\n--     },\n--     {rw[subst_all'],\n--       }\n\n\n\n-- substs h h', have := @subst_all' L (n+1) n' (n + n' + 1 + 1) (by {simp}) t (f_f), simp[this], let k, swap, change realize_bounded_formula v (bounded_preformula.cast_eq k _) _ \u2194 _,\n-- let j, swap, change realize_bounded_formula v (bounded_preformula.cast_eq k (\u2200' j)) _ \u2194 _,\n-- rw[cast_eq_all], dsimp[k,j], clear k j, apply forall_congr, intro x,\n--      have := @f_ih xs (n+1) n' ((n+1) + n' + 1) (x::(v.cast (by simp))) (by simp) (by simp) t,\n--      rw[cast_eq_trans], rw[dvector_cast_pull_out] at this, swap, simp, swap, simp, simp,\n--      rw[realize_bounded_formula_cast_eq_irrel], rw[realize_bounded_formula_cast_eq_irrel] at this, rw[dvector.cast_trans] at this, rw[this], clear this, clear this f_ih,\n--      rw[dvector.insert_cons], apply iff_of_eq, congr' 2; simp,\n\n--  -- congr' 1, simp, swap, {apply cast_eq_hrfl},\n--      -- {swap, simp, rw[dvector.insert_cons], simp, rw[dvector.insert_cons], let p, swap,\n--      -- let q, swap, change p == q, apply (@heq_iff_eq _ p (q.cast (by simp))).mpr, }\n-- }\n\n\n-- -- congr' 2, simp, simp, {apply realize_bounded_term_irrel', swap, simp, tidy,  },\n--      -- {apply dvector.cast_hrfl}, {apply cast_eq_hrfl},\n\n\n-- have := @subst_all' L (n+1) n' (n'' + 1) (by {subst h, simp}) t (f_f.cast_eq (by simp[h'])),ext, simp[-subst_all'] at this,\n\n-- @[simp]lemma subst_all' {L} {n n' n''} {h : n + n' + 1 = n''} {t : bounded_term L n'} {f : bounded_formula L (n'' + 1)} :\n  -- (\u2200'f)[t  // n // (by {simp[h]})]\n  -- = \u2200'(f[(t : bounded_term L n') // (n+1) // (by {subst h; simp})]).cast_eq (by simp) := by ext; simp\n\n-- | _ _ _ _ _ _ _ bd_falsum t v xs := by {intros; simp}\n-- | _ _ _ _ _ _ _ (t\u2081 \u2243 t\u2082) t v xs := by {sorry} -- follows from term version\n-- | _ _ _ _ _ _ _ (bd_rel R) t v xs := by simp\n-- | _ _ _  _ _ _ _ (bd_apprel f s) t v xs := by sorry\n-- | _ _ _ n'' l h h' (f \u27f9 g) t v xs := by {have this_f := asjh'' f t v xs, have this_g := asjh'' g t v xs, simp[*, -asjh''], simp at this_f this_g, rw[<-this_f,<-this_g]}\n-- | n n' n'' n''' l h h' (\u2200' f) t v xs := begin\n-- -- clear asjh'',\n-- substs h h',\n-- simp,\n--         let k, swap, change _ = k, let j, tactic.rotate 1, change realize_bounded_formula v (bounded_preformula.cast_eq _ \u2200'j) _ = k, swap, by simp,\n--         conv {to_lhs, congr, skip, rw[bounded_preformula.cast_eq_all],}, dsimp[k,j], clear k j,\n--         ext, apply forall_congr, intro x, repeat{rw[realize_bounded_formula_cast_eq_irrel]},\n--         rw[dvector.cast_trans], rw[<-dvector.insert],\n--         swap,\n--         have := @asjh'' (n+1) n' (n + n' + 1 + 1) (n + n' + 1 + 1 + 1) 0 (by {simp}) (by refl) f t (x::v) xs, simp at this,\n--       sorry --- might need to lift, actually\n\n\n\n@[simp]lemma realize_bounded_term_subst0 {L} {S : Structure L} {n} (s : bounded_term L (n+1)) {v : dvector S n} (t : closed_term L) : realize_bounded_term v (s[(t.cast (by simp)) /0]) [] = realize_bounded_term ((realize_closed_term S t)::v) s [] :=\nbegin\nrevert s, refine bounded_term.rec1 _ _,\n  {intro k, rcases k with \u27e8k_val, k_H\u27e9, simp,\n    induction n generalizing k_val, swap,\n    by_cases k_val = n_n + 1,\n          {subst h, refl},\n          {have : k_val < n_n + 1,\n                by {apply nat.lt_of_le_and_ne, exact nat.le_of_lt_succ k_H, exact h},\n          have := @n_ih (v.trunc _ (nat.le_succ n_n)) k_val this,\n          rw[dvector.nth_irrel1] at this, swap, dedup, apply nat.lt_of_lt_of_le, exact this,\n          exact nat.le_succ (n_n + 1),\n          rw[<-this], apply realize_bounded_term_irrel', swap, simp,\n          intros, simp only [dvector.trunc_nth]},\n    have := zero_of_lt_one k_val (by exact k_H), subst this,\n    congr, {apply dvector.zero_eq}, {ext, simp}},\n  {intros, simp[subst0_bounded_term_bd_apps,realize_bounded_term_bd_apps, func_subst0_irrel],\n  congr' 1, apply dvector.map_congr_pmem, intros x Hx, exact ih_ts x Hx}\nend\n\n-- /-- realization of a subst0 is the realization with the substituted term prepended to the realizing vector --/\nlemma realize_bounded_formula_subst0 {L} {S : Structure L} {n} (f : bounded_formula L (n+1)) {v : dvector S n} (t : closed_term L) : realize_bounded_formula v (f[(t.cast0 n) /0]) [] \u2194 realize_bounded_formula ((realize_closed_term S t)::v) f [] :=\nbegin\n  revert n f v, refine bounded_formula.rec1 _ _ _ _ _; intros,\n  {simp},\n  {simp},\n  {rw[subst0_bounded_formula_bd_apps_rel], simp[realize_bounded_formula_bd_apps_rel, rel_subst0_irrel]},\n  {simp*},\n  {simp[-realize_bounded_formula_cast_eq_irrel], apply forall_congr, clear ih, intro x, have := @gen_realize_bounded_formula L S 0 n (n+1) (n+2) 0 (by simp) (by simp) f t (x::v) [], simpa using this}\nend\n\nlemma realize_bounded_formula_subst0' {L} {S : Structure L} {n} (f : bounded_formula L (n+1)) {v : dvector S n} (t : bounded_term L 1) (x : S) : realize_bounded_formula (x :: v) ((f \u2191' 1 # 1)[(t.cast (by simp)) /0]) [] \u2194 realize_bounded_formula ((realize_bounded_term ([x] : dvector S 1) t []) :: v) f [] :=\nbegin\nrevert f n v, refine bounded_formula.rec1 _ _ _ _ _; intros,\n  {simp},\n  {sorry}, -- this requires a version of this lemma for terms\n  {sorry}, -- same issue as the corresponding case above\n  {sorry}, -- this one should be easy, just need a lemma about commutation with bd_imp\n  {sorry}, -- same issues as the corresponding case above\nend\n\nend realization\n\nexport fol realization\n", "meta": {"author": "flypitch", "repo": "flypitch", "sha": "aea5800db1f4cce53fc4a113711454b27388ecf8", "save_path": "github-repos/lean/flypitch-flypitch", "path": "github-repos/lean/flypitch-flypitch/flypitch-aea5800db1f4cce53fc4a113711454b27388ecf8/old/realization.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.3727274531134735}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.AffineScheme\nimport algebraic_geometry.pullbacks\nimport category_theory.morphism_property\n\n/-!\n# Properties of morphisms between Schemes\n\nWe provide the basic framework for talking about properties of morphisms between Schemes.\n\nA `morphism_property Scheme` is a predicate on morphisms between schemes, and an\n`affine_target_morphism_property` is a predicate on morphisms into affine schemes. Given a\n`P : affine_target_morphism_property`, we may construct a `morphism_property` called\n`target_affine_locally P` that holds for `f : X \u27f6 Y` whenever `P` holds for the\nrestriction of `f` on every affine open subset of `Y`.\n\n## Main definitions\n\n- `algebraic_geometry.affine_target_morphism_property.is_local`: We say that `P.is_local` if `P`\nsatisfies the assumptions of the affine communication lemma\n(`algebraic_geometry.of_affine_open_cover`). That is,\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X \u27f6 Y`, then `P` holds for `f \u2223_ Y.basic_open r` for any\n  global section `r`.\n3. If `P` holds for `f \u2223_ Y.basic_open r` for all `r` in a spanning set of the global sections,\n  then `P` holds for `f`.\n\n- `algebraic_geometry.property_is_local_at_target`: We say that `property_is_local_at_target P` for\n`P : morphism_property Scheme` if\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X \u27f6 Y`, then `P` holds for `f \u2223_ U` for any `U`.\n3. If `P` holds for `f \u2223_ U` for an open cover `U` of `Y`, then `P` holds for `f`.\n\n## Main results\n\n- `algebraic_geometry.affine_target_morphism_property.is_local.affine_open_cover_tfae`:\n  If `P.is_local`, then `target_affine_locally P f` iff there exists an affine cover `{ U\u1d62 }` of `Y`\n  such that `P` holds for `f \u2223_ U\u1d62`.\n- `algebraic_geometry.affine_target_morphism_property.is_local_of_open_cover_imply`:\n  If the existance of an affine cover `{ U\u1d62 }` of `Y` such that `P` holds for `f \u2223_ U\u1d62` implies\n  `target_affine_locally P f`, then `P.is_local`.\n- `algebraic_geometry.affine_target_morphism_property.is_local.affine_target_iff`:\n  If `Y` is affine and `f : X \u27f6 Y`, then `target_affine_locally P f \u2194 P f` provided `P.is_local`.\n- `algebraic_geometry.affine_target_morphism_property.is_local.target_affine_locally_is_local` :\n  If `P.is_local`, then `property_is_local_at_target (target_affine_locally P)`.\n- `algebraic_geometry.property_is_local_at_target.open_cover_tfae`:\n  If `property_is_local_at_target P`, then `P f` iff there exists an open cover `{ U\u1d62 }` of `Y`\n  such that `P` holds for `f \u2223_ U\u1d62`.\n\nThese results should not be used directly, and should be ported to each property that is local.\n\n-/\n\nuniverse u\n\nopen topological_space category_theory category_theory.limits opposite\n\nnoncomputable theory\n\nnamespace algebraic_geometry\n\n/-- An `affine_target_morphism_property` is a class of morphisms from an arbitrary scheme into an\naffine scheme. -/\ndef affine_target_morphism_property := \u2200 \u2983X Y : Scheme\u2984 (f : X \u27f6 Y) [is_affine Y], Prop\n\n/-- `is_iso` as a `morphism_property`. -/\nprotected def Scheme.is_iso : morphism_property Scheme := @is_iso Scheme _\n\n/-- `is_iso` as an `affine_morphism_property`. -/\nprotected def Scheme.affine_target_is_iso : affine_target_morphism_property :=\n\u03bb X Y f H, is_iso f\n\ninstance : inhabited affine_target_morphism_property := \u27e8Scheme.affine_target_is_iso\u27e9\n\n/-- A `affine_target_morphism_property` can be extended to a `morphism_property` such that it\n*never* holds when the target is not affine -/\ndef affine_target_morphism_property.to_property (P : affine_target_morphism_property) :\n  morphism_property Scheme :=\n\u03bb X Y f, \u2203 h, @@P f h\n\nlemma affine_target_morphism_property.to_property_apply (P : affine_target_morphism_property)\n  {X Y : Scheme} (f : X \u27f6 Y) [is_affine Y] :\n  P.to_property f \u2194 P f := by { delta affine_target_morphism_property.to_property, simp [*] }\n\nlemma affine_cancel_left_is_iso {P : affine_target_morphism_property}\n  (hP : P.to_property.respects_iso) {X Y Z : Scheme} (f : X \u27f6 Y)\n    (g : Y \u27f6 Z) [is_iso f] [is_affine Z] : P (f \u226b g) \u2194 P g :=\nby rw [\u2190 P.to_property_apply, \u2190 P.to_property_apply, hP.cancel_left_is_iso]\n\nlemma affine_cancel_right_is_iso\n  {P : affine_target_morphism_property} (hP : P.to_property.respects_iso) {X Y Z : Scheme}\n    (f : X \u27f6 Y) (g : Y \u27f6 Z) [is_iso g] [is_affine Z] [is_affine Y] : P (f \u226b g) \u2194 P f :=\nby rw [\u2190 P.to_property_apply, \u2190 P.to_property_apply, hP.cancel_right_is_iso]\n\nlemma affine_target_morphism_property.respects_iso_mk {P : affine_target_morphism_property}\n  (h\u2081 : \u2200 {X Y Z} (e : X \u2245 Y) (f : Y \u27f6 Z) [is_affine Z], by exactI P f \u2192 P (e.hom \u226b f))\n  (h\u2082 : \u2200 {X Y Z} (e : Y \u2245 Z) (f : X \u27f6 Y) [h : is_affine Y],\n     by exactI P f \u2192 @@P (f \u226b e.hom) (is_affine_of_iso e.inv)) : P.to_property.respects_iso :=\nbegin\n  split,\n  { rintros X Y Z e f \u27e8a, h\u27e9, exactI \u27e8a, h\u2081 e f h\u27e9 },\n  { rintros X Y Z e f \u27e8a, h\u27e9, exactI \u27e8is_affine_of_iso e.inv, h\u2082 e f h\u27e9 },\nend\n\n/-- For a `P : affine_target_morphism_property`, `target_affine_locally P` holds for\n`f : X \u27f6 Y` whenever `P` holds for the restriction of `f` on every affine open subset of `Y`. -/\ndef target_affine_locally (P : affine_target_morphism_property) : morphism_property Scheme :=\n  \u03bb {X Y : Scheme} (f : X \u27f6 Y), \u2200 (U : Y.affine_opens), @@P (f \u2223_ U) U.prop\n\nlemma is_affine_open.map_is_iso {X Y : Scheme} {U : opens Y.carrier} (hU : is_affine_open U)\n  (f : X \u27f6 Y) [is_iso f] : is_affine_open ((opens.map f.1.base).obj U) :=\nbegin\n  haveI : is_affine _ := hU,\n  exact is_affine_of_iso (f \u2223_ U),\nend\n\nlemma target_affine_locally_respects_iso {P : affine_target_morphism_property}\n  (hP : P.to_property.respects_iso) : (target_affine_locally P).respects_iso :=\nbegin\n  split,\n  { introv H U,\n    rw [morphism_restrict_comp, affine_cancel_left_is_iso hP],\n    exact H U },\n  { introv H,\n    rintro \u27e8U, hU : is_affine_open U\u27e9, dsimp,\n    haveI : is_affine _ := hU,\n    haveI : is_affine _ := hU.map_is_iso e.hom,\n    rw [morphism_restrict_comp, affine_cancel_right_is_iso hP],\n    exact H \u27e8(opens.map e.hom.val.base).obj U, hU.map_is_iso e.hom\u27e9 }\nend\n\n/--\nWe say that `P : affine_target_morphism_property` is a local property if\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X \u27f6 Y`, then `P` holds for `f \u2223_ Y.basic_open r` for any\n  global section `r`.\n3. If `P` holds for `f \u2223_ Y.basic_open r` for all `r` in a spanning set of the global sections,\n  then `P` holds for `f`.\n-/\nstructure affine_target_morphism_property.is_local (P : affine_target_morphism_property) : Prop :=\n(respects_iso : P.to_property.respects_iso)\n(to_basic_open : \u2200 {X Y : Scheme} [is_affine Y] (f : X \u27f6 Y) (r : Y.presheaf.obj $ op \u22a4),\n  by exactI P f \u2192\n    @@P (f \u2223_ (Y.basic_open r)) ((top_is_affine_open Y).basic_open_is_affine _))\n(of_basic_open_cover : \u2200 {X Y : Scheme} [is_affine Y] (f : X \u27f6 Y)\n  (s : finset (Y.presheaf.obj $ op \u22a4)) (hs : ideal.span (s : set (Y.presheaf.obj $ op \u22a4)) = \u22a4),\n  by exactI (\u2200 (r : s), @@P (f \u2223_ (Y.basic_open r.1))\n    ((top_is_affine_open Y).basic_open_is_affine _)) \u2192 P f)\n\nlemma target_affine_locally_of_open_cover {P : affine_target_morphism_property}\n  (hP : P.is_local)\n  {X Y : Scheme} (f : X \u27f6 Y) (\ud835\udcb0 : Y.open_cover) [\u2200 i, is_affine (\ud835\udcb0.obj i)]\n  (h\ud835\udcb0 : \u2200 i, P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i)) :\n    target_affine_locally P f :=\nbegin\n  classical,\n  let S := \u03bb i, (\u27e8\u27e8set.range (\ud835\udcb0.map i).1.base, (\ud835\udcb0.is_open i).base_open.open_range\u27e9,\n    range_is_affine_open_of_open_immersion (\ud835\udcb0.map i)\u27e9 : Y.affine_opens),\n  intro U,\n  apply of_affine_open_cover U (set.range S),\n  { intros U r h,\n    haveI : is_affine _ := U.2,\n    have := hP.2 (f \u2223_ U.1),\n    replace this := this (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op r) h,\n    rw \u2190 P.to_property_apply at this \u22a2,\n    exact (hP.1.arrow_mk_iso_iff (morphism_restrict_restrict_basic_open f _ r)).mp this },\n  { intros U s hs H,\n    haveI : is_affine _ := U.2,\n    apply hP.3 (f \u2223_ U.1) (s.image (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op)),\n    { apply_fun ideal.comap (Y.presheaf.map (eq_to_hom U.1.open_embedding_obj_top.symm).op) at hs,\n      rw ideal.comap_top at hs,\n      rw \u2190 hs,\n      simp only [eq_to_hom_op, eq_to_hom_map, finset.coe_image],\n      have : \u2200 {R S : CommRing} (e : S = R) (s : set S),\n        (by exactI ideal.span (eq_to_hom e '' s) = ideal.comap (eq_to_hom e.symm) (ideal.span s)),\n      { intros, subst e, simpa },\n      apply this },\n    { rintro \u27e8r, hr\u27e9,\n      obtain \u27e8r, hr', rfl\u27e9 := finset.mem_image.mp hr,\n      simp_rw \u2190 P.to_property_apply at \u22a2 H,\n      exact\n        (hP.1.arrow_mk_iso_iff (morphism_restrict_restrict_basic_open f _ r)).mpr (H \u27e8r, hr'\u27e9) } },\n  { rw set.eq_univ_iff_forall,\n    simp only [set.mem_Union],\n    intro x,\n    exact \u27e8\u27e8_, \u27e8\ud835\udcb0.f x, rfl\u27e9\u27e9, \ud835\udcb0.covers x\u27e9 },\n  { rintro \u27e8_, i, rfl\u27e9,\n    simp_rw \u2190 P.to_property_apply at \u22a2 h\ud835\udcb0,\n    exact (hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _)).mpr (h\ud835\udcb0 i) },\nend\n\nlemma affine_target_morphism_property.is_local.affine_open_cover_tfae\n  {P : affine_target_morphism_property}\n  (hP : P.is_local) {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [target_affine_locally P f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)], \u2200 (i : \ud835\udcb0.J),\n      by exactI P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J),\n      by exactI P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_affine U] [is_open_immersion g],\n      by exactI P (pullback.snd : pullback f g \u27f6 U),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4) (hU' : \u2200 i, is_affine_open (U i)),\n      \u2200 i, @@P (f \u2223_ (U i)) (hU' i)] :=\nbegin\n  tfae_have : 1 \u2192 4,\n  { intros H U g h\u2081 h\u2082,\n    resetI,\n    replace H := H \u27e8\u27e8_, h\u2082.base_open.open_range\u27e9,\n      range_is_affine_open_of_open_immersion g\u27e9,\n    rw \u2190 P.to_property_apply at H \u22a2,\n    rwa \u2190 hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _) },\n  tfae_have : 4 \u2192 3,\n  { intros H \ud835\udcb0 h\ud835\udcb0 i,\n    resetI,\n    apply H },\n  tfae_have : 3 \u2192 2,\n  { exact \u03bb H, \u27e8Y.affine_cover, infer_instance, H Y.affine_cover\u27e9 },\n  tfae_have : 2 \u2192 1,\n  { rintro \u27e8\ud835\udcb0, h\ud835\udcb0, H\u27e9, exactI target_affine_locally_of_open_cover hP f \ud835\udcb0 H },\n  tfae_have : 5 \u2192 2,\n  { rintro \u27e8\u03b9, U, hU, hU', H\u27e9,\n    refine \u27e8Y.open_cover_of_supr_eq_top U hU, hU', _\u27e9,\n    intro i,\n    specialize H i,\n    rw [\u2190 P.to_property_apply, \u2190 hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _)],\n    rw \u2190 P.to_property_apply at H,\n    convert H,\n    all_goals { ext1, exact subtype.range_coe } },\n  tfae_have : 1 \u2192 5,\n  { intro H,\n    refine \u27e8Y.carrier, \u03bb x, (Y.affine_cover.map x).opens_range, _,\n      \u03bb i, range_is_affine_open_of_open_immersion _, _\u27e9,\n    { rw eq_top_iff, intros x _, erw opens.mem_supr, exact\u27e8x, Y.affine_cover.covers x\u27e9 },\n    { intro i, exact H \u27e8_, range_is_affine_open_of_open_immersion _\u27e9 } },\n  tfae_finish\nend\n\nlemma affine_target_morphism_property.is_local_of_open_cover_imply\n  (P : affine_target_morphism_property) (hP : P.to_property.respects_iso)\n  (H : \u2200 {X Y : Scheme.{u}} (f : X \u27f6 Y),\n    (\u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)], \u2200 (i : \ud835\udcb0.J),\n      by exactI P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i)) \u2192\n    (\u2200 {U : Scheme} (g : U \u27f6 Y) [is_affine U] [is_open_immersion g],\n      by exactI P (pullback.snd : pullback f g \u27f6 U))) : P.is_local :=\nbegin\n  refine \u27e8hP, _, _\u27e9,\n  { introv h,\n    resetI,\n    haveI : is_affine _ := (top_is_affine_open Y).basic_open_is_affine r,\n    delta morphism_restrict,\n    rw affine_cancel_left_is_iso hP,\n    refine @@H f \u27e8Scheme.open_cover_of_is_iso (\ud835\udfd9 Y), _, _\u27e9 (Y.of_restrict _) _inst _,\n    { intro i, dsimp, apply_instance },\n    { intro i, dsimp,\n      rwa [\u2190 category.comp_id pullback.snd, \u2190 pullback.condition, affine_cancel_left_is_iso hP] } },\n  { introv hs hs',\n    resetI,\n    replace hs := ((top_is_affine_open Y).basic_open_union_eq_self_iff _).mpr hs,\n    have := H f \u27e8Y.open_cover_of_supr_eq_top _ hs, _, _\u27e9 (\ud835\udfd9 _),\n    rwa [\u2190 category.comp_id pullback.snd, \u2190 pullback.condition,\n      affine_cancel_left_is_iso hP] at this,\n    { intro i, exact (top_is_affine_open Y).basic_open_is_affine _ },\n    { rintro (i : s),\n      specialize hs' i,\n      haveI : is_affine _ := (top_is_affine_open Y).basic_open_is_affine i.1,\n      delta morphism_restrict at hs',\n      rwa affine_cancel_left_is_iso hP at hs' } }\nend\n\nlemma affine_target_morphism_property.is_local.affine_open_cover_iff\n  {P : affine_target_morphism_property} (hP : P.is_local)\n  {X Y : Scheme.{u}} (f : X \u27f6 Y) (\ud835\udcb0 : Scheme.open_cover.{u} Y) [h\ud835\udcb0 : \u2200 i, is_affine (\ud835\udcb0.obj i)] :\n  target_affine_locally P f \u2194 \u2200 i, @@P (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) (h\ud835\udcb0 i) :=\n\u27e8\u03bb H, let h := ((hP.affine_open_cover_tfae f).out 0 2).mp H in h \ud835\udcb0,\n  \u03bb H, let h := ((hP.affine_open_cover_tfae f).out 1 0).mp in h \u27e8\ud835\udcb0, infer_instance, H\u27e9\u27e9\n\nlemma affine_target_morphism_property.is_local.affine_target_iff\n  {P : affine_target_morphism_property} (hP : P.is_local)\n  {X Y : Scheme.{u}} (f : X \u27f6 Y) [is_affine Y] :\n  target_affine_locally P f \u2194 P f :=\nbegin\n  rw hP.affine_open_cover_iff f _,\n  swap, { exact Scheme.open_cover_of_is_iso (\ud835\udfd9 Y) },\n  swap, { intro _, dsimp, apply_instance },\n  transitivity (P (pullback.snd : pullback f (\ud835\udfd9 _) \u27f6 _)),\n  { exact \u27e8\u03bb H, H punit.star, \u03bb H _, H\u27e9 },\n  rw [\u2190 category.comp_id pullback.snd, \u2190 pullback.condition, affine_cancel_left_is_iso hP.1],\nend\n\n/--\nWe say that `P : morphism_property Scheme` is local at the target if\n1. `P` respects isomorphisms.\n2. If `P` holds for `f : X \u27f6 Y`, then `P` holds for `f \u2223_ U` for any `U`.\n3. If `P` holds for `f \u2223_ U` for an open cover `U` of `Y`, then `P` holds for `f`.\n-/\nstructure property_is_local_at_target (P : morphism_property Scheme) : Prop :=\n(respects_iso : P.respects_iso)\n(restrict : \u2200 {X Y : Scheme} (f : X \u27f6 Y) (U : opens Y.carrier), P f \u2192 P (f \u2223_ U))\n(of_open_cover : \u2200 {X Y : Scheme.{u}} (f : X \u27f6 Y) (\ud835\udcb0 : Scheme.open_cover.{u} Y),\n    (\u2200 (i : \ud835\udcb0.J), P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i)) \u2192 P f)\n\nlemma affine_target_morphism_property.is_local.target_affine_locally_is_local\n  {P : affine_target_morphism_property} (hP : P.is_local) :\n    property_is_local_at_target (target_affine_locally P) :=\nbegin\n  constructor,\n  { exact target_affine_locally_respects_iso hP.1 },\n  { intros X Y f U H V,\n    rw [\u2190 P.to_property_apply, hP.1.arrow_mk_iso_iff (morphism_restrict_restrict f _ _)],\n    convert H \u27e8_, is_affine_open.image_is_open_immersion V.2 (Y.of_restrict _)\u27e9,\n    rw \u2190 P.to_property_apply,\n    refl },\n  { rintros X Y f \ud835\udcb0 h\ud835\udcb0,\n    rw (hP.affine_open_cover_tfae f).out 0 1,\n    refine \u27e8\ud835\udcb0.bind (\u03bb _, Scheme.affine_cover _), _, _\u27e9,\n    { intro i, dsimp [Scheme.open_cover.bind], apply_instance },\n    { intro i,\n      specialize h\ud835\udcb0 i.1,\n      rw (hP.affine_open_cover_tfae (pullback.snd : pullback f (\ud835\udcb0.map i.fst) \u27f6 _)).out 0 2\n        at h\ud835\udcb0,\n      specialize h\ud835\udcb0 (Scheme.affine_cover _) i.2,\n      let e : pullback f ((\ud835\udcb0.obj i.fst).affine_cover.map i.snd \u226b \ud835\udcb0.map i.fst) \u27f6\n        pullback (pullback.snd : pullback f (\ud835\udcb0.map i.fst) \u27f6 _)\n          ((\ud835\udcb0.obj i.fst).affine_cover.map i.snd),\n      { refine (pullback_symmetry _ _).hom \u226b _,\n        refine (pullback_right_pullback_fst_iso _ _ _).inv \u226b _,\n        refine (pullback_symmetry _ _).hom \u226b _,\n        refine pullback.map _ _ _ _ (pullback_symmetry _ _).hom (\ud835\udfd9 _) (\ud835\udfd9 _) _ _;\n          simp only [category.comp_id, category.id_comp, pullback_symmetry_hom_comp_snd] },\n      rw \u2190 affine_cancel_left_is_iso hP.1 e at h\ud835\udcb0,\n      convert h\ud835\udcb0,\n      simp } },\nend\n\nlemma property_is_local_at_target.open_cover_tfae\n  {P : morphism_property Scheme}\n  (hP : property_is_local_at_target P)\n  {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [P f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y), \u2200 (i : \ud835\udcb0.J),\n      P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) (i : \ud835\udcb0.J),\n      P (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (U : opens Y.carrier), P (f \u2223_ U),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_open_immersion g],\n      P (pullback.snd : pullback f g \u27f6 U),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4), (\u2200 i, P (f \u2223_ (U i)))] :=\nbegin\n  tfae_have : 2 \u2192 1,\n  { rintro \u27e8\ud835\udcb0, H\u27e9, exact hP.3 f \ud835\udcb0 H },\n  tfae_have : 1 \u2192 4,\n  { intros H U, exact hP.2 f U H },\n  tfae_have : 4 \u2192 3,\n  { intros H \ud835\udcb0 i,\n    rw \u2190 hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _),\n    exact H (\ud835\udcb0.map i).opens_range },\n  tfae_have : 3 \u2192 2,\n  { exact \u03bb H, \u27e8Y.affine_cover, H Y.affine_cover\u27e9 },\n  tfae_have : 4 \u2192 5,\n  { intros H U g hg,\n    resetI,\n    rw \u2190 hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _),\n    apply H },\n  tfae_have : 5 \u2192 4,\n  { intros H U,\n    erw hP.1.cancel_left_is_iso,\n    apply H },\n  tfae_have : 4 \u2192 6,\n  { intro H, exact \u27e8punit, \u03bb _, \u22a4, csupr_const, \u03bb _, H _\u27e9 },\n  tfae_have : 6 \u2192 2,\n  { rintro \u27e8\u03b9, U, hU, H\u27e9,\n    refine \u27e8Y.open_cover_of_supr_eq_top U hU, _\u27e9,\n    intro i,\n    rw \u2190 hP.1.arrow_mk_iso_iff (morphism_restrict_opens_range f _),\n    convert H i,\n    all_goals { ext1, exact subtype.range_coe } },\n  tfae_finish\nend\n\nlemma property_is_local_at_target.open_cover_iff\n  {P : morphism_property Scheme} (hP : property_is_local_at_target P)\n  {X Y : Scheme.{u}} (f : X \u27f6 Y) (\ud835\udcb0 : Scheme.open_cover.{u} Y) :\n  P f \u2194 \u2200 i, P (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) :=\n\u27e8\u03bb H, let h := ((hP.open_cover_tfae f).out 0 2).mp H in h \ud835\udcb0,\n  \u03bb H, let h := ((hP.open_cover_tfae f).out 1 0).mp in h \u27e8\ud835\udcb0, H\u27e9\u27e9\n\nnamespace affine_target_morphism_property\n\n/-- A `P : affine_target_morphism_property` is stable under base change if `P` holds for `Y \u27f6 S`\nimplies that `P` holds for `X \u00d7\u209b Y \u27f6 X` with `X` and `S` affine schemes. -/\ndef stable_under_base_change\n  (P : affine_target_morphism_property) : Prop :=\n\u2200 \u2983X Y S : Scheme\u2984 [is_affine S] [is_affine X] (f : X \u27f6 S) (g : Y \u27f6 S),\n  by exactI P g \u2192 P (pullback.fst : pullback f g \u27f6 X)\n\nlemma is_local.target_affine_locally_pullback_fst_of_right_of_stable_under_base_change\n  {P : affine_target_morphism_property} (hP : P.is_local) (hP' : P.stable_under_base_change)\n  {X Y S : Scheme} (f : X \u27f6 S) (g : Y \u27f6 S) [is_affine S] (H : P g) :\n  target_affine_locally P (pullback.fst : pullback f g \u27f6 X) :=\nbegin\n  rw (hP.affine_open_cover_tfae (pullback.fst : pullback f g \u27f6 X)).out 0 1,\n  use [X.affine_cover, infer_instance],\n  intro i,\n  let e := pullback_symmetry _ _ \u226a\u226b pullback_right_pullback_fst_iso f g (X.affine_cover.map i),\n  have : e.hom \u226b pullback.fst = pullback.snd := by simp,\n  rw [\u2190 this, affine_cancel_left_is_iso hP.1],\n  apply hP'; assumption,\nend\n\nlemma is_local.stable_under_base_change\n  {P : affine_target_morphism_property} (hP : P.is_local) (hP' : P.stable_under_base_change) :\n  (target_affine_locally P).stable_under_base_change :=\nmorphism_property.stable_under_base_change.mk (target_affine_locally_respects_iso hP.respects_iso)\nbegin\n  intros X Y S f g H,\n  rw (hP.target_affine_locally_is_local.open_cover_tfae (pullback.fst : pullback f g \u27f6 X)).out 0 1,\n  use S.affine_cover.pullback_cover f,\n  intro i,\n  rw (hP.affine_open_cover_tfae g).out 0 3 at H,\n  let e : pullback (pullback.fst : pullback f g \u27f6 _) ((S.affine_cover.pullback_cover f).map i) \u2245 _,\n  { refine pullback_symmetry _ _ \u226a\u226b pullback_right_pullback_fst_iso f g _ \u226a\u226b _ \u226a\u226b\n      (pullback_right_pullback_fst_iso (S.affine_cover.map i) g\n        (pullback.snd : pullback f (S.affine_cover.map i) \u27f6 _)).symm,\n    exact as_iso (pullback.map _ _ _ _ (\ud835\udfd9 _) (\ud835\udfd9 _) (\ud835\udfd9 _)\n      (by simpa using pullback.condition) (by simp)) },\n  have : e.hom \u226b pullback.fst = pullback.snd := by simp,\n  rw [\u2190 this, (target_affine_locally_respects_iso hP.1).cancel_left_is_iso],\n  apply hP.target_affine_locally_pullback_fst_of_right_of_stable_under_base_change hP',\n  rw [\u2190 pullback_symmetry_hom_comp_snd, affine_cancel_left_is_iso hP.1],\n  apply H\nend\n\nend affine_target_morphism_property\n\n/--\nThe `affine_target_morphism_property` associated to `(target_affine_locally P).diagonal`.\nSee `diagonal_target_affine_locally_eq_target_affine_locally`.\n-/\ndef affine_target_morphism_property.diagonal (P : affine_target_morphism_property) :\n  affine_target_morphism_property :=\n\u03bb X Y f hf, \u2200 {U\u2081 U\u2082 : Scheme} (f\u2081 : U\u2081 \u27f6 X) (f\u2082 : U\u2082 \u27f6 X) [is_affine U\u2081] [is_affine U\u2082]\n  [is_open_immersion f\u2081] [is_open_immersion f\u2082],\n  by exactI P (pullback.map_desc f\u2081 f\u2082 f)\n\nlemma affine_target_morphism_property.diagonal_respects_iso (P : affine_target_morphism_property)\n  (hP : P.to_property.respects_iso) :\n  P.diagonal.to_property.respects_iso :=\nbegin\n  delta affine_target_morphism_property.diagonal,\n  apply affine_target_morphism_property.respects_iso_mk,\n  { introv H _ _,\n    resetI,\n    rw [pullback.map_desc_comp, affine_cancel_left_is_iso hP, affine_cancel_right_is_iso hP],\n    apply H },\n  { introv H _ _,\n    resetI,\n    rw [pullback.map_desc_comp, affine_cancel_right_is_iso hP],\n    apply H }\nend\n\nlemma diagonal_target_affine_locally_of_open_cover (P : affine_target_morphism_property)\n  (hP : P.is_local)\n  {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} Y)\n  [\u2200 i, is_affine (\ud835\udcb0.obj i)] (\ud835\udcb0' : \u03a0 i, Scheme.open_cover.{u} (pullback f (\ud835\udcb0.map i)))\n  [\u2200 i j, is_affine ((\ud835\udcb0' i).obj j)]\n  (h\ud835\udcb0' : \u2200 i j k, P (pullback.map_desc ((\ud835\udcb0' i).map j) ((\ud835\udcb0' i).map k) pullback.snd)) :\n    (target_affine_locally P).diagonal f :=\nbegin\n  refine (hP.affine_open_cover_iff _ _).mpr _,\n  { exact ((Scheme.pullback.open_cover_of_base \ud835\udcb0 f f).bind (\u03bb i,\n      Scheme.pullback.open_cover_of_left_right.{u u} (\ud835\udcb0' i) (\ud835\udcb0' i) pullback.snd pullback.snd)) },\n  { intro i,\n    dsimp at *,\n    apply_instance },\n  { rintro \u27e8i, j, k\u27e9,\n    dsimp,\n    convert (affine_cancel_left_is_iso hP.1\n    (pullback_diagonal_map_iso _ _ ((\ud835\udcb0' i).map j) ((\ud835\udcb0' i).map k)).inv pullback.snd).mp _,\n    swap 3,\n    { convert h\ud835\udcb0' i j k, apply pullback.hom_ext; simp, },\n    all_goals\n    { apply pullback.hom_ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,\n      pullback.lift_fst_assoc, pullback.lift_snd_assoc] } }\nend\n\nlemma affine_target_morphism_property.diagonal_of_target_affine_locally\n  (P : affine_target_morphism_property)\n  (hP : P.is_local) {X Y U : Scheme.{u}} (f : X \u27f6 Y) (g : U \u27f6 Y)\n  [is_affine U] [is_open_immersion g] (H : (target_affine_locally P).diagonal f) :\n    P.diagonal (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  rintros U V f\u2081 f\u2082 _ _ _ _,\n  resetI,\n  replace H := ((hP.affine_open_cover_tfae (pullback.diagonal f)).out 0 3).mp H,\n  let g\u2081 := pullback.map (f\u2081 \u226b pullback.snd)\n    (f\u2082 \u226b pullback.snd) f f\n    (f\u2081 \u226b pullback.fst)\n    (f\u2082 \u226b pullback.fst) g\n    (by rw [category.assoc, category.assoc, pullback.condition])\n    (by rw [category.assoc, category.assoc, pullback.condition]),\n  let g\u2082 : pullback f\u2081 f\u2082 \u27f6 pullback f g := pullback.fst \u226b f\u2081,\n  specialize H g\u2081,\n  rw \u2190 affine_cancel_left_is_iso hP.1 (pullback_diagonal_map_iso f _ f\u2081 f\u2082).hom,\n  convert H,\n  { apply pullback.hom_ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd,\n    pullback.lift_fst_assoc, pullback.lift_snd_assoc, category.comp_id,\n    pullback_diagonal_map_iso_hom_fst, pullback_diagonal_map_iso_hom_snd], }\nend\n\nlemma affine_target_morphism_property.is_local.diagonal_affine_open_cover_tfae\n  {P : affine_target_morphism_property}\n  (hP : P.is_local) {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [(target_affine_locally P).diagonal f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)], by exactI\n      \u2200 (i : \ud835\udcb0.J), P.diagonal (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J), by exactI\n      P.diagonal (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_affine U] [is_open_immersion g], by exactI\n      P.diagonal (pullback.snd : pullback f g \u27f6 _),\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)]\n      (\ud835\udcb0' : \u03a0 i, Scheme.open_cover.{u} (pullback f (\ud835\udcb0.map i))) [\u2200 i j, is_affine ((\ud835\udcb0' i).obj j)],\n    by exactI \u2200 i j k, P (pullback.map_desc ((\ud835\udcb0' i).map j) ((\ud835\udcb0' i).map k) pullback.snd)] :=\nbegin\n  tfae_have : 1 \u2192 4,\n  { introv H hU hg _ _, resetI, apply P.diagonal_of_target_affine_locally; assumption },\n  tfae_have : 4 \u2192 3,\n  { introv H h\ud835\udcb0, resetI, apply H },\n  tfae_have : 3 \u2192 2,\n  { exact \u03bb H, \u27e8Y.affine_cover, infer_instance, H Y.affine_cover\u27e9 },\n  tfae_have : 2 \u2192 5,\n  { rintro \u27e8\ud835\udcb0, h\ud835\udcb0, H\u27e9,\n    resetI,\n    refine \u27e8\ud835\udcb0, infer_instance, \u03bb _, Scheme.affine_cover _, infer_instance, _\u27e9,\n    intros i j k,\n    apply H },\n  tfae_have : 5 \u2192 1,\n  { rintro \u27e8\ud835\udcb0, _, \ud835\udcb0', _, H\u27e9,\n    exactI diagonal_target_affine_locally_of_open_cover P hP f \ud835\udcb0 \ud835\udcb0' H, },\n  tfae_finish\nend\n\nlemma affine_target_morphism_property.is_local.diagonal {P : affine_target_morphism_property}\n  (hP : P.is_local) : P.diagonal.is_local :=\naffine_target_morphism_property.is_local_of_open_cover_imply\n  P.diagonal\n  (P.diagonal_respects_iso hP.1)\n  (\u03bb _ _ f, ((hP.diagonal_affine_open_cover_tfae f).out 1 3).mp)\n\nlemma diagonal_target_affine_locally_eq_target_affine_locally (P : affine_target_morphism_property)\n  (hP : P.is_local) :\n  (target_affine_locally P).diagonal = target_affine_locally P.diagonal :=\nbegin\n  ext _ _ f,\n  exact ((hP.diagonal_affine_open_cover_tfae f).out 0 1).trans\n    ((hP.diagonal.affine_open_cover_tfae f).out 1 0),\nend\n\n\n\nlemma universally_is_local_at_target_of_morphism_restrict (P : morphism_property Scheme)\n  (hP\u2081 : P.respects_iso)\n  (hP\u2082 : \u2200 {X Y : Scheme.{u}} (f : X \u27f6 Y) {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4),\n    (\u2200 i, P (f \u2223_ (U i))) \u2192 P f) :\n  property_is_local_at_target P.universally :=\nuniversally_is_local_at_target P\nbegin\n  intros X Y f \ud835\udcb0 h\ud835\udcb0,\n  apply hP\u2082 f (\u03bb (i : \ud835\udcb0.J), (\ud835\udcb0.map i).opens_range) \ud835\udcb0.supr_opens_range,\n  simp_rw hP\u2081.arrow_mk_iso_iff (morphism_restrict_opens_range f _),\n  exact h\ud835\udcb0\nend\n\n/-- `topologically P` holds for a morphism if the underlying topological map satisfies `P`. -/\ndef morphism_property.topologically\n  (P : \u2200 {\u03b1 \u03b2 : Type u} [topological_space \u03b1] [topological_space \u03b2] (f : \u03b1 \u2192 \u03b2), Prop) :\n  morphism_property Scheme.{u} :=\n\u03bb X Y f, P f.1.base\n\nend algebraic_geometry\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/morphisms/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6370307875894139, "lm_q1q2_score": 0.37272744908155947}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nSupplementary theorems about the `string` type.\n-/\nimport data.list.basic data.char\n\nnamespace string\n\ndef ltb : iterator \u2192 iterator \u2192 bool\n| s\u2081 s\u2082 := begin\n  cases s\u2082.has_next, {exact ff},\n  cases h\u2081 : s\u2081.has_next, {exact tt},\n  exact if s\u2081.curr = s\u2082.curr then\n    have s\u2081.next.2.length < s\u2081.2.length, from\n    match s\u2081, h\u2081 with \u27e8_, a::l\u27e9, h := nat.lt_succ_self _ end,\n    ltb s\u2081.next s\u2082.next\n  else s\u2081.curr < s\u2082.curr,\nend\nusing_well_founded {rel_tac :=\n  \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb s, s.1.2.length)\u27e9]}\n\ninstance has_lt' : has_lt string :=\n\u27e8\u03bb s\u2081 s\u2082, ltb s\u2081.mk_iterator s\u2082.mk_iterator\u27e9\n\ninstance decidable_lt : @decidable_rel string (<) := by apply_instance\n\n@[simp] theorem lt_iff_to_list_lt :\n  \u2200 {s\u2081 s\u2082 : string}, s\u2081 < s\u2082 \u2194 s\u2081.to_list < s\u2082.to_list\n| \u27e8i\u2081\u27e9 \u27e8i\u2082\u27e9 :=\n  suffices \u2200 {p\u2081 p\u2082 s\u2081 s\u2082}, ltb \u27e8p\u2081, s\u2081\u27e9 \u27e8p\u2082, s\u2082\u27e9 \u2194 s\u2081 < s\u2082, from this,\n  begin\n    intros,\n    induction s\u2081 with a s\u2081 IH generalizing p\u2081 p\u2082 s\u2082;\n      cases s\u2082 with b s\u2082; rw ltb; simp [iterator.has_next],\n    { exact iff_of_false bool.ff_ne_tt (lt_irrefl _) },\n    { exact iff_of_true rfl list.lex.nil },\n    { exact iff_of_false bool.ff_ne_tt (not_lt_of_lt list.lex.nil) },\n    { dsimp [iterator.has_next,\n        iterator.curr, iterator.next],\n      split_ifs,\n      { subst b, exact IH.trans list.lex.cons_iff.symm },\n      { simp, refine \u27e8list.lex.rel, \u03bb e, _\u27e9,\n        cases e, {cases h rfl}, assumption } }\n  end\n\ninstance has_le : has_le string := \u27e8\u03bb s\u2081 s\u2082, \u00ac s\u2082 < s\u2081\u27e9\n\ninstance decidable_le : @decidable_rel string (\u2264) := by apply_instance\n\n@[simp] theorem le_iff_to_list_le\n  {s\u2081 s\u2082 : string} : s\u2081 \u2264 s\u2082 \u2194 s\u2081.to_list \u2264 s\u2082.to_list :=\n(not_congr lt_iff_to_list_lt).trans not_lt\n\ntheorem to_list_inj : \u2200 {s\u2081 s\u2082}, to_list s\u2081 = to_list s\u2082 \u2194 s\u2081 = s\u2082\n| \u27e8s\u2081\u27e9 \u27e8s\u2082\u27e9 := \u27e8congr_arg _, congr_arg _\u27e9\n\ninstance : decidable_linear_order string :=\nby refine_struct {\n    lt := (<), le := (\u2264),\n    le_antisymm := by simp; exact\n      \u03bb a b h\u2081 h\u2082, to_list_inj.1 (le_antisymm h\u2081 h\u2082),\n    decidable_lt := by apply_instance,\n    decidable_le := string.decidable_le,\n    decidable_eq := by apply_instance, .. };\n  { simp [-not_le], introv, apply_field }\n\ndef map_tokens (c : char) (f : list string \u2192 list string) : string \u2192 string :=\nintercalate (singleton c) \u2218 f \u2218 split (= c)\n\nend string\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/string.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669998, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.37267651680305985}}
{"text": "import .scalar .clause .eqelim\n\nopen int tactic \n\nmeta def expr_of_neg : int \u2192 tactic expr \n| (of_nat _) := failed\n| -[1+ m] := return `(neg_succ_lt_zero %%`(m))\n\nlemma zero_eq_zero : (0 : int) = 0 := rfl\n\nmeta def expr_of_forall_mem_eq_zero : list int \u2192 tactic expr \n| [] := return `(forall_mem_nil_eq_zero).to_expr\n| (i::is) :=\n  do x \u2190 expr_of_forall_mem_eq_zero is,\n     to_expr ``(forall_mem_cons_eq_zero 0 %%`(is) zero_eq_zero %%x)\n\nmeta def expr_of_unsat_comb (ks : list nat) (p : list term) : tactic expr :=  \nlet \u27e8b,as\u27e9 := comb p ks in \ndo x1 \u2190 expr_of_neg b,\n   x2 \u2190 expr_of_forall_mem_eq_zero as, \n   to_expr ``(unsat_comb_of %%`(p) %%`(ks) %%x1 %%x2)\n\n/- Given a (([],les) : clause), return the \n   expr of a term (t : clause.unsat ([],les)). -/\nmeta def expr_of_unsat_ef : clause \u2192 tactic expr \n| ((_::_), _) := failed  \n| ([], les) := \n  do ks \u2190 search les, \n     x \u2190 expr_of_unsat_comb ks les,\n     return `(unsat_of_unsat_comb %%`(ks) %%`(les) %%x)\n\n/- Given a (c : clause), return the \n   expr of a term (t : clause.unsat c)  -/\nmeta def expr_of_unsat (c : clause) : tactic expr := \ndo ee \u2190 find_ees c, \n   x \u2190 expr_of_unsat_ef (conc ee c), \n   return `(unsat_of_unsat_conc %%`(ee) %%`(c) %%x)\n\n/- Given a (cs : list clause), return the \n   expr of a term (t : clauses.unsat cs)  -/\nmeta def expr_of_unsats : list clause \u2192 tactic expr \n| [] := return `(clauses.unsat_nil)\n| (p::ps) := \n  do x \u2190 expr_of_unsat p,\n     xs \u2190 expr_of_unsats ps,\n     to_expr ``(clauses.unsat_cons %%`(p) %%`(ps) %%x %%xs)\n", "meta": {"author": "skbaek", "repo": "omega", "sha": "715e384ed14e8eb177a326700066e7c98269e078", "save_path": "github-repos/lean/skbaek-omega", "path": "github-repos/lean/skbaek-omega/omega-715e384ed14e8eb177a326700066e7c98269e078/expr_of_unsat.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3726765168030598}}
{"text": "\nimport .lemmas\nimport .spec\n\nimport data.set.basic\n\nimport util.data.minimum\nimport util.data.ordering\nimport util.data.order\nimport util.function\nimport util.logic\nimport tactic.norm_num\n\nopen temporal function predicate nat set\n\nlocal infix ` \u2243 `:75 := v_eq\nlocal prefix `\u266f `:0 := cast (by simp)\nuniverses u v\n\nnamespace temporal\nnamespace scheduling\nsection scheduling\n\nlocal attribute [instance, priority 0] classical.prop_decidable\nlocal attribute [-simp] add_comm\n\nparameter {evt : Type u}\nparameter \u0393 : cpred\nparameter r : tvar (set evt)\nparameter Hr : \u0393 \u22a2 \u25fb-(r \u2243 (\u2205 : set evt))\n-- parameter [nonempty evt]\n\nabbreviation SCHED  (s : tvar evt) :=\n\u25fb(s \u220a r) \u22c0\n\u2200\u2200 (e : evt),\n  \u25fb\u25c7(\u2191e \u220a r) \u27f6\n  \u25fb\u25c7(s \u2243 \u2191e \u22c0 \u2191e \u220a r)\n\nsection implementation\n\nparameters (f : \u2115 \u2192 evt) (Hinj : surjective f)\nparameter p : tvar (\u2115 \u2192 evt)\nparameter cur : tvar \u2115\n/- consider making select into a state variable instead of a definition -/\nvariable select : tvar evt\n\ninfixl ` |+| `:80 := lifted\u2082 has_add.add\ninfixl ` |-| `:80 := lifted\u2082 has_sub.sub\n\nnoncomputable def next_p (p : \u2115 \u2192 evt) (r' : set evt) (i : \u2115) : ordering \u2192 evt\n | ordering.gt := p i\n | ordering.eq := p (\u2193 i : \u2115, p i \u2208 r')\n | ordering.lt :=\n   if (\u2193 i : \u2115, p i \u2208 r') \u2264 i\n          then p (i + 1)\n          else p i\n\nnoncomputable def next' (r' : set evt) : \u2115 \u00d7 (\u2115 \u2192 evt) \u2192 \u2115 \u00d7 (\u2115 \u2192 evt)\n | (cur,p) :=\nlet min := \u2193 i : \u2115, p i \u2208 r',\n    cur' := max min $ cur+1,\n    p' : \u2115 \u2192 evt := \u03bb i : \u2115,\n          next_p p r' i (cmp i cur')\nin\n(cur',p')\n\nsection\n\nnoncomputable def next : tvar $ \u2115 \u00d7 (\u2115 \u2192 evt) \u2192 \u2115 \u00d7 (\u2115 \u2192 evt) :=\n\u27ea \u2115, next' \u27eb (\u2299r)\nend\n\n@[simp]\nlemma next_def (cur cur' : \u2115) (p p' : \u2115 \u2192 evt) (\u03c3 : \u2115)\n: (cur', p') = (\u03c3 \u22a8 next) (cur, p) \u2194\n   cur' = max (\u2193 (i : \u2115), p i \u2208 succ \u03c3 \u22a8 r) (cur + 1) \u2227\n\u2200 i, p' i =\n          next_p p\n            (succ \u03c3 \u22a8 r) i\n            (cmp i cur') :=\nby { repeat { simp [next,next'] <|> unfold_coes },\n     apply and_congr_right,\n     intro,\n     split, { introv h, subst cur', simp [h], },\n     { intro, apply funext,\n       intro, subst cur', solve_by_elim } }\n\nsection\nparameter f\n\n@[predicate]\nnoncomputable def cur\u2080 : tvar \u2115 :=\n[| r , \u2193 i : \u2115, f i \u2208 r |]\n\n-- noncomputable abbreviation select\u2080 : tvar evt :=\n-- [| r , f (\u2193 i : \u2115, f i \u2208 r) |]\n\n-- noncomputable def nxt_select : tvar (evt \u2192 evt) :=\n-- [| p , \u03bb (p' : \u2115 \u2192 \u2115) (r' : set evt) (e : evt),\n--   inv q $ \u2193 i : \u2115, inv q i \u2208 r' |] (\u2299q) (\u2299r)\n\nend\n\nnoncomputable def Spec :=\n\u2983cur,p\u2984 \u2243 \u2983cur\u2080,f\u2984 \u22c0 \u25fb(\u2299\u2983cur,p\u2984 \u2243 next \u2983cur,p\u2984)\n\nparameter Hq : \u0393 \u22a2 Spec\n\n@[predicate]\ndef select : tvar evt :=\np cur\n\n-- noncomputable def select_Spec :=\n-- select \u2243 select\u2080 \u22c0 \u25fb(\u2299select \u2243 nxt_select select)\n\n-- variables Hs : \u0393 \u22a2 select_Spec select\n\nsection q_injective\n\nlemma next_rec (P : \u2115 \u2192 Prop) (cur cur') (p p' : \u2115 \u2192 evt) (r' : set evt)\n  {i : \u2115} {e : evt}\n  (h : p i = e)\n  (Hcur' : cur' = max (\u2193 (i : \u2115), p i \u2208 r') (cur + 1))\n  (Hq' : \u2200 (i : \u2115), p' i =\n       next_p p r' i (cmp i cur'))\n  (Hcase_lt : (i < \u2193 (i : \u2115), p i \u2208 r') \u2228 (\u2193 (i : \u2115), p i \u2208 r') < i \u2227 \u00aci \u2264 cur' \u2192\n               next_p p r' i (cmp i cur') = e \u2192\n               P i)\n  (Hcase_eq : (i = \u2193 (i : \u2115), p i \u2208 r') \u2192\n               next_p p r' cur' (cmp cur' cur') = e \u2192\n               P cur')\n  (Hcase_gt : (\u2193 (i : \u2115), p i \u2208 r') < i \u2192\n               next_p p r' (i - 1) (cmp (i - 1) cur') = e \u2192\n               P (i - 1))\n: (\u2203 j, p' j = e \u2227 P j) :=\nbegin\n  ordering_cases cmp i (\u2193 i, p i \u2208 r'),\n  { existsi i, rw Hq',\n    suffices : cmp i cur' = ordering.lt,\n    { rw [this,next_p,if_neg,h] at *,\n      { existsi [rfl], apply_assumption,\n        left, solve_by_elim, refl, },\n      all_goals { apply not_le_of_gt h_1 }, },\n    rw [cmp,cmp_using_eq_lt,Hcur'],\n    apply lt_max_of_lt_left _ h_1, },\n  { existsi cur', rw Hq',\n    have : cmp cur' cur' = ordering.eq,\n    { rw [cmp_eq_eq], },\n    rw and_iff_imp, intro, solve_by_elim,\n    rw [this,next_p], cc },\n  by_cases h_cur : i \u2264 cur',\n  { existsi i - 1, rw Hq',\n    have h_i_gt_0 : 0 < i,\n    { apply lt_of_le_of_lt,\n      apply nat.zero_le, assumption, },\n    have : cmp (i - 1) cur' = ordering.lt,\n    { rw [cmp,cmp_using_eq_lt],\n      apply lt_of_lt_of_le _ h_cur,\n      show i - 1 < i,\n      { apply nat.sub_lt, assumption, norm_num }, },\n    rw and_iff_imp, intro, solve_by_elim,\n    rw [this,next_p,if_pos,nat.sub_add_cancel,h],\n    assumption,\n    rw \u2190 add_le_to_le_sub,\n    repeat { assumption }, },\n  { existsi i, rw Hq',\n    have : cmp i cur' = ordering.gt,\n    { rw [cmp,cmp_using_eq_gt],\n      apply lt_of_not_ge h_cur },\n    rw and_iff_imp, intro, apply_assumption,\n    right, tauto, solve_by_elim,\n    rw [this,next_p,h] at *, }\nend\n\ninclude Hq Hinj\n\n/- TODO: split into lemmas -/\nlemma q_injective\n: \u0393 \u22a2 \u25fb(\u27e8 surjective \u27e9 ! p) :=\nbegin [temporal]\n  cases Hq with Hq Hq',\n  t_induction!,\n  { explicit' with Hq\n    { cases_matching* _ \u2227 _, subst p, solve_by_elim, } },\n  { henceforth at Hq',\n    explicit' with ih Hq'\n    { simp_intros e, cases ih e with i h,\n      cases Hq' with Hcur' Hq',\n      ordering_cases cmp i (\u2193 i, p i \u2208 r'),\n      { existsi i, rw Hq',\n        suffices : cmp i cur' = ordering.lt,\n        { rw [this,next_p,if_neg,h],\n          apply not_le_of_gt h_1, },\n        rw [cmp,cmp_using_eq_lt,Hcur'],\n        apply lt_max_of_lt_left _ h_1, },\n      { existsi cur', rw Hq',\n        have : cmp cur' cur' = ordering.eq,\n        { rw [cmp_eq_eq], },\n        rw [this,next_p], cc },\n      by_cases h_cur : i \u2264 cur',\n      { existsi i - 1, rw Hq',\n        have h_i_gt_0 : 0 < i,\n        { apply lt_of_le_of_lt,\n          apply nat.zero_le, assumption, },\n        have : cmp (i - 1) cur' = ordering.lt,\n        { rw [cmp,cmp_using_eq_lt],\n          apply lt_of_lt_of_le _ h_cur,\n          show i - 1 < i,\n          { apply nat.sub_lt, assumption, norm_num }, },\n        rw [this,next_p,if_pos,nat.sub_add_cancel,h],\n        assumption,\n        rw \u2190 add_le_to_le_sub,\n        repeat { assumption }, },\n      { existsi i, rw Hq',\n        have : cmp i cur' = ordering.gt,\n        { rw [cmp,cmp_using_eq_gt],\n          apply lt_of_not_ge h_cur },\n        rw [this,next_p,h], } } },\nend\n\nend q_injective\n\nsection\n\n-- include Hq\n-- lemma select_eq_inv_q_cur'\n-- : \u0393 \u22a2 select_Spec select' :=\n-- begin [temporal]\n--   cases Hq with Hq\u2080 Hq,\n--   split,\n--   explicit' { cc },\n--   henceforth! at \u22a2 Hq,\n--   explicit' [nxt_select]\n--   { cases Hq with Hcur Hq,\n--     rw [inv_eq _ _ _],\n--     rw [Hq,\u2190 Hcur],\n--     ordering_cases cmp (\u2193 (i : \u2115), inv q i \u2208 r') (q (inv q (\u2193 (i : \u2115), inv q i \u2208 r')))\n--     ; simp [next_p],\n--     ite_cases,\n--     { exfalso, apply h_1, clear h_1,\n--       rw Hcur, rw le_max_iff_le_or_le, left,\n--        } }\n--   { verbose := tt }\n-- end\n\n-- include Hs Hinj\n\n-- lemma select_eq_inv_q_cur\n-- : \u0393 \u22a2 \u25fb[| q cur select, select = inv q cur |] :=\n-- begin [temporal]\n--   have Hinj_q := temporal.scheduling.q_injective,\n--   have Hinj_q' := henceforth_next _ _ Hinj_q,\n--   cases Hq with Hq\u2080 Hq,\n--   cases Hs with Hs\u2080 Hs,\n--   t_induction! using Hq Hs Hinj_q' Hinj_q,\n--   explicit' { cc },\n--   explicit' [nxt_select]\n--   { cases Hq,\n--     rw inv_eq _ _ Hinj_q',\n--     rw [Hq_right],\n--     ordering_cases (cmp (\u2193 (i : \u2115), inv q i \u2208 r') (q select'))\n--     ; simp [next_p],\n--     ite_cases,\n--     { exfalso, apply h_1, clear h_1,\n--       rw le_max_iff_le_or_le, left,\n--       apply le_of_eq, clear_except Hs, },\n--     {  },\n--     {  },\n--     {  } }\n-- end\n-- end\n\nopen set\n\n-- invariant\n--           inv q' cur' = (\u2193 i, inv q' i \u2208 r)\n--           inv q' cur' \u2264 (\u2193 i, inv q' i \u2208 r)\n--           inv q' cur' \u2265 (\u2193 i, inv q' i \u2208 r)\n\nsection\ninclude Hr Hq Hinj\nlemma valid_indices_ne_empty\n: \u0393 \u22a2 \u25fb([| p, \u03bb r : set evt, { i : \u2115 | p i \u2208 r } \u2260 \u2205 |] (\u2299r)) :=\nbegin [temporal]\n  have Hsur := temporal.scheduling.q_injective,\n  replace Hr := henceforth_next _ _ Hr,\n  henceforth! at Hr Hsur \u22a2,\n  explicit' with Hr Hsur\n  { rw not_eq_empty_iff_exists at *,\n    cases Hr with i Hr,\n    existsi inv p i,\n    change _ \u2208 r',\n    rw [inv_is_right_inverse_of_surjective Hsur],\n    assumption, }\nend\nend\n\nnoncomputable def rank (e : evt) : tvar \u2115 :=\n[| p, \u2193 i, p i = e |]\n\ninclude Hr Hq Hinj\nlemma sched_inv\n: \u0393 \u22a2 \u25fb(select \u220a r) :=\nbegin [temporal]\n  have Hq_inj := temporal.scheduling.q_injective,\n  have hJ := temporal.scheduling.valid_indices_ne_empty,\n  cases Hq with Hq\u2080 Hq,\n  have Hq_inj' := henceforth_next _ _ Hq_inj,\n  t_induction!,\n  henceforth! at Hr Hq_inj,\n  { explicit' [select,cur\u2080] with Hq\u2080 Hr Hq_inj\n    { change cur \u2208 { i | p i \u2208 r },\n      rw [Hq\u2080.left,Hq\u2080.right],\n      apply minimum_mem,\n      intro, apply Hr,\n      rw eq_empty_iff_forall_not_mem at *,\n      intro x, specialize a (inv p x),\n      -- apply Hr,\n      intro, apply a,\n      show f (inv p x) \u2208 r,\n      cases Hq\u2080, subst p,\n      rw [inv_is_right_inverse_of_surjective Hq_inj],\n      assumption } },\n  henceforth! at Hr Hq_inj' Hq_inj Hq hJ,\n  explicit' with Hq hJ\n  { cases Hq with Hq Hq',\n    rw Hq',\n    have : cmp cur' cur' = ordering.eq,\n    { rw cmp_eq_eq },\n    rw [this,next_p],\n    change (\u2193 (i : \u2115), p i \u2208 r') \u2208 { i | p i \u2208 r' },\n    apply minimum_mem,\n    assumption },\nend\n\nlemma cur_lt_cur'\n: \u0393 \u22a2 \u25fb(cur \u227a \u2299cur) :=\nbegin [temporal]\n  cases Hq with Hq\u2080 Hq,\n  henceforth! at Hq \u22a2,\n  explicit' with Hq\n  { simp [Hq],\n    apply lt_max_of_lt_right,\n    apply lt_add_one, }\nend\n\n\nsection sched_queue_safety\nvariables\n  (q\u2080 : \u2115)\n  (e : evt)\n  (Hprev : \u0393 \u22a2 rank e |+| (rank e |-| cur) \u2243 \u2191q\u2080)\n  (H\u2082 : \u0393 \u22a2 \u2299(-(\u27e8\u03bb (i : \u2115), (i \u22a8 rank e) + ((i \u22a8 rank e) - (i \u22a8 cur))\u27e9 \u227a\u227a q\u2080) \u22c0\n                   -(select \u2243 e)))\n  (Hdec : \u0393 \u22a2 cur \u227a \u2299cur)\n  (Hsurj : \u0393 \u22a2 \u2299( \u27e8 surjective \u27e9 ! p ))\n  (this : \u0393 \u22a2 \u2299rank e \u227c rank e \u22c1 \u2299(cur \u2243 rank e))\n\nomit Hq\ninclude Hdec Hprev Hsurj H\u2082 this\nlemma non_dec_po\n: \u0393 \u22a2 \u2299(rank e |+| (rank e |-| cur) \u2243 \u2191q\u2080) :=\nbegin [temporal]\n  explicit' [next,next',select,rank]\n    with Hprev this H\u2082 Hdec Hsurj\n  { subst q\u2080,\n    cases this with this this,\n    cases lt_or_eq_of_le this,\n    { exfalso, apply H\u2082.left,\n      change _ + _ < _ + _,\n      apply lt_of_lt_of_le,\n      { apply add_lt_add_right h, },\n      apply add_le_add_left,\n      transitivity,\n      { apply nat.sub_le_sub_left,\n      apply le_of_lt Hdec, },\n      { apply nat.sub_le_sub_right this, } },\n    { simp [h],\n      let rank := (\u2193 (i : \u2115), p i = e),\n      have : rank - cur' \u2264 rank - cur,\n      { apply nat.sub_le_sub_left, apply le_of_lt Hdec },\n      cases lt_or_eq_of_le this,\n      { exfalso, apply H\u2082.left,\n        change _ + _ < _ + _, simp [h,h_1], },\n      assumption },\n    replace H\u2082 := H\u2082.right, rw this at H\u2082,\n    have H\u2080 : { i : \u2115 | p' i = e } \u2260 \u2205,\n    { apply ne_empty_of_mem _,\n      exact (inv p' e),\n      change p' (inv p' e) = e,\n      apply inv_is_right_inverse_of_surjective Hsurj, },\n    have H\u2083 := minimum_mem H\u2080,\n    cases H\u2082 H\u2083, },\nend\n\nend sched_queue_safety\n\nlemma subsumes_requested (e : evt)\n: \u0393 \u22a2 \u25fb( select \u2243 \u2191e \u22c0 \u2191e \u220a r\n             \u2261 select \u2243 \u2191e ) :=\nbegin [temporal]\n  have Hr' := temporal.scheduling.sched_inv,\n  henceforth! at \u22a2 Hr',\n  explicit' [select] with Hr'\n  { split,\n    { simp, intros, assumption },\n    { intros, cc, } },\nend\n\n-- include Hinj\n/-- TODO: Pull out lemmas -/\nlemma sched_queue_safety (q\u2080 : \u2115) (e : evt)\n: \u0393 \u22a2 \u25fb(rank e |+| (rank e |-| cur) \u2243 \u2191q\u2080 \u27f6\n    \u25fb(rank e |+| (rank e |-| cur) \u2243 \u2191q\u2080) \u22c1\n    \u25c7(rank e |+| (rank e |-| cur) \u227a\u227a \u2191q\u2080 \u22c1 select \u2243 \u2191e)) :=\nbegin [temporal]\n  have hJ := temporal.scheduling.q_injective,\n  have Hinc := temporal.scheduling.cur_lt_cur',\n  have p_not_empty := temporal.scheduling.valid_indices_ne_empty,\n  have p'_not_empty := henceforth_next _ _ p_not_empty,\n  cases Hq with Hq Hq',\n  henceforth!, intro H,\n  rw [p_or_comm,\u2190 p_not_p_imp],\n  intros H\u2081, simp [p_not_p_or,p_not_p_and] at H\u2081,\n  t_induction,\n  { assumption },\n  { henceforth!, intro Hprev,\n    have H\u2082 := henceforth_next _ _ H\u2081,\n    have hJ' := henceforth_next _ _ hJ,\n    henceforth at Hinc Hq' H\u2081 H\u2082 hJ hJ' p_not_empty p'_not_empty,\n    apply temporal.scheduling.non_dec_po _ _ Hprev H\u2082 Hinc hJ',\n    explicit' [next,next',select,rank] with Hq' hJ hJ' H\u2082\n    { cases Hq' with Hcur Hq,\n      replace Hq := congr_fun Hq, simp only at Hq,\n      rw [or_comm,or_iff_not_imp], intro Hncur,\n      have p_not_empty : { i : \u2115 | p i = e } \u2260 \u2205,\n      { rw ne_empty_iff_exists_mem, apply hJ e, },\n      have p'_not_empty : { i : \u2115 | p' i = e } \u2260 \u2205,\n      { rw ne_empty_iff_exists_mem, apply hJ' e, },\n      apply (le_minimum_iff_forall_le p_not_empty (\u2193 (i : \u2115), p' i = e)).2,\n      assume j (Hj : p j = e),\n      apply (minimum_le_iff_exists_le p'_not_empty j).2,\n      rw \u2190 Hcur at Hq,\n      apply next_rec _ cur cur' p p' r' Hj Hcur Hq,\n      { intros, refl },\n      { intros h h',\n        rw \u2190 Hq at h', cases H\u2082.right h', },\n      { intros, apply nat.sub_le, } }, },\nend\n\n/- TODO: split into lemmas -/\nlemma sched_queue_liveness (q\u2080 : \u2115) (e : evt)\n: \u0393 \u22a2 \u2299(\u2191e \u220a r) \u22c0 rank e |+| (rank e |-| cur) \u2243 \u2191q\u2080 ~>\n  rank e |+| (rank e |-| cur) \u227a\u227a \u2191q\u2080 \u22c1 select \u2243 \u2191e \u22c0 \u2191e \u220a r :=\nbegin [temporal]\n  { have Hq_inj := temporal.scheduling.q_injective,\n    have Hinc := temporal.scheduling.cur_lt_cur',\n    cases Hq with Hq\u2080 Hq,\n    henceforth! at \u22a2 Hq Hq_inj Hinc,\n    have Hq_inj' : \u2299(\u27e8surjective\u27e9 ! p) := holds_next _ _ Hq_inj,\n    simp, intros hreq hq\u2080,\n    apply next_entails_eventually,\n    explicit' [select,next,next',rank]\n      with Hq hreq hq\u2080 Hinc  Hq_inj\n    { cases Hq with Hq Hq' Hq_inj,\n      replace Hq' := congr_fun Hq', simp at Hq',\n      rw \u2190 Hq at Hq',\n      let rank := \u2193 i, p i = e,\n      have Hrank : p rank = e := _,\n      let P := \u03bb k, k + (k - cur') < q\u2080 \u2228 k = cur',\n      have rec := temporal.scheduling.next_rec P cur cur' p p' r' Hrank Hq Hq' _ _ _,\n      { cases rec with k Hk, cases Hk with Hpk Hk,\n        cases Hk with Hk Hk,\n        { left, change _ + _ < _,\n          apply @lt_of_le_of_lt _ _ _ (k + (k - cur')),\n          have h : (\u2193 (i : \u2115), p' i = e) \u2264 k, apply minimum_le, exact Hpk,\n          apply add_le_add, assumption,\n          apply nat.sub_le_sub_right,\n          assumption, assumption, },\n        { right, cc }, },\n      { simp [P,rank], intros h\u2080 h\u2081,\n        rw \u2190 Hq' (\u2193 (i : \u2115), p i = e) at h\u2081,\n        clear P,\n        cases h\u2080,\n        { right, clear hq\u2080 Hq',\n          have : cur + 1 \u2264 (\u2193 (i : \u2115), p i \u2208 r'), admit,\n          rw max_eq_left this at Hq, clear this,\n          admit },\n        { left, rw \u2190 hq\u2080,\n          have : ((\u2193 (i : \u2115), p i = e) \u2265 cur'),\n          { apply le_of_lt h\u2080.right },\n          monotonicity Hinc }, },\n      { intros, rw \u2190 Hq' at a_1,\n        right, refl, },\n      { intros, rw \u2190 Hq' at a_1,\n        left, rw \u2190 hq\u2080,\n        apply lt_of_lt_of_le,\n        apply add_lt_add_right,\n        apply nat.sub_lt, apply lt_of_le_of_lt (nat.zero_le _) a,\n        norm_num, apply nat.sub_le_sub,\n        apply nat.sub_le, apply le_of_lt Hinc, },\n      { have h : { i | p i = e } \u2260 \u2205,\n        { rw ne_empty_iff_exists_mem, apply Hq_inj, },\n        apply minimum_mem h, } } },\nend\n\nlemma sched_fairness (e : evt)\n: \u0393 \u22a2 \u25fb\u25c7(\u2191e \u220a r) \u27f6 \u25fb\u25c7(select \u2243 \u2191e \u22c0 \u2191e \u220a r) :=\nbegin [temporal]\n  suffices : \u25fb\u25c7\u2299(\u2191e \u220a r) \u27f6 \u25fb\u25c7(temporal.scheduling.select \u2243 \u2191e \u22c0 \u2191e \u220a r),\n  { intro h, apply this,\n    rw [\u2190 next_eventually_comm], apply henceforth_next _ _ h, },\n  apply inf_often_induction' (temporal.scheduling.rank e |+| (temporal.scheduling.rank e |-| cur)) ; intro q\u2080,\n  { rw temporal.scheduling.subsumes_requested e,\n    apply temporal.scheduling.sched_queue_safety q\u2080 e, },\n  { apply temporal.scheduling.sched_queue_liveness }\nend\n\ndef correct_sched\n: \u0393 \u22a2 SCHED select :=\nbegin [temporal]\n  split,\n  { apply temporal.scheduling.sched_inv, },\n  { intro, apply temporal.scheduling.sched_fairness },\nend\nend\nend implementation\n\n-- class schedulable (\u03b1 : Sort u) :=\n--   (f : \u03b1 \u2192 \u2115)\n--   (inj : injective f)\nopen encodable\n\nexample (w \u03c3\u2080 : tvar \u2115)\n: \u21d1(to_fun_var (\u03bb (w : tvar \u2115), w \u2243 \u03c3\u2080)) w = w \u2243 \u03c3\u2080 :=\nbegin\n  -- rw [v_eq,to_fun_var_lift\u2082],\n  -- dsimp,\n  -- dsimp,\n  -- unfold_coes,\n  -- dsimp with lifted_fn,\n  -- unfold_coes,\n  simp! only with lifted_fn predicate,\nend\n\nlemma scheduler [encodable evt]\n  (Hr : \u0393 \u22a2 \u25fb-(r \u2243 (\u2205 : set evt)))\n: \u0393 \u22a2 (\u2203\u2203 s, SCHED s) :=\nbegin [temporal]\n  let f' : (evt \u2192 \u2115) := @encode evt _,\n  let f : tvar (evt \u2192 \u2115) := f',\n  have Hnemp : \u2203\u2203 x : evt, True,\n  { admit },\n  nonempty evt,\n  let g' : (\u2115 \u2192 evt) := inv (@encode evt _),\n  let g  : tvar (\u2115 \u2192 evt) := g',\n  let \u03c3\u2080 := \u2983cur\u2080 r g',g\u2984,\n  select_witness w : w \u2243 \u03c3\u2080 \u22c0 \u25fb(\u2299w \u2243 temporal.scheduling.next w),\n  have := fwd_witness \u03c3\u2080 (next r) \u0393,\n  cases this with cur Hcur,\n  cases cur with cur q,\n  existsi select p cur,\n  note Hsur : surjective (inv f'),\n  { apply surjective_of_has_right_inverse,\n    existsi f',\n    apply inv_is_left_inverse_of_injective,\n    apply schedulable.inj },\n  type_check @temporal.scheduling.correct_sched,\n  apply temporal.scheduling.correct_sched (inv f') Hsur _,\n  simp [Spec,\u03c3\u2080] at \u22a2 Hcur,\n  exact Hcur,\nend\n\nend scheduling\n\nsection spec\n\nvariables \u0393 : cpred\nvariables {\u03b1 : Type v} (m : mch \u03b1)\nlocal notation `evt` := m.evt\nvariable [encodable evt]\nlocal notation `cs` := m.cs\nlocal notation `fs` := m.fs\nlocal notation `p` := m.init\nlocal notation `A` := m.A\n\nlemma sch_intro (v : tvar \u03b1)\n: \u0393 \u22a2 m.spec v \u27f6 (\u2203\u2203 sch, m.spec_sch v sch) :=\nbegin [temporal]\n  intro h,\n  let r : tvar (set (option evt)) := \u27ea \u2115, \u03bb s s', { e | m.effect e s s' } \u27eb v \u2299v,\n  have hr : \u25fb-(r \u2243 (\u2205 : set (option evt))),\n  { simp [mch.spec] at h,\n    casesm* _ \u22c0 _,\n    select Hact : \u25fb(p_exists _),\n    henceforth! at Hact \u22a2,\n    explicit' [r] with Hact\n    { erw [\u2190 not_eq_empty_iff_exists] at Hact, exact Hact }, },\n  have h' := temporal.scheduling.scheduler \u0393 r hr,\n  cases h' with sch h',\n  existsi sch,\n  simp  at \u22a2 h,\n  casesm* _ \u22c0 _,\n  split!* ; try { solve_by_elim },\n  { select h' : \u25fb(p_exists _),\n    select hJ : \u25fb(_ \u220a _),\n    henceforth! at hJ h' \u22a2,\n    existsi sch with hh,\n    { explicit' [r] with hh hJ h'\n      { subst sch, tauto } } },\n  { introv, intros h\u2080 h\u2081,\n    rename a_3 h\u2082,\n    replace h\u2082 := h\u2082 x h\u2080 h\u2081,\n    replace a_1 := a_1 x,\n    persistent,\n    have H\u2080 : \u2191x \u220a r \u2261 cs x ! v \u22c0 fs x ! v \u22c0 \u27e6 v | A x \u27e7,\n    { explicit' [r]\n      { simp [mch.effect,and_assoc] }, },\n    have H\u2081 : sch \u2243 \u2191x \u22c0 \u2191x \u220a r \u2261 cs x ! v \u22c0 fs x ! v \u22c0 (sch \u2243 \u2191x \u22c0 \u27e6 v | A x \u27e7),\n    { explicit' [r,mch.effect,and_assoc]\n      { apply eq.to_iff, ac_refl }, },\n    rw [H\u2081,H\u2080] at a_1,\n    solve_by_elim, }\nend\n\nend spec\nend scheduling\nexport scheduling (schedulable sch_intro)\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/scheduling.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059775, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.37251712502215745}}
{"text": "\n/- Arithmetic and logical operators for the Compcert C and Clight languages -/\n\nimport .ctypes .memory\n\nnamespace cop\nopen ast integers floats values memory ctypes word\n     ctypes.intsize ctypes.floatsize ctypes.signedness\n\n/- * Syntax of operators. -/\n\ninductive unary_operation : Type\n| Onotbool          /- boolean negation ([!] in C) -/\n| Onotint           /- integer complement ([~] in C) -/\n| Oneg              /- opposite (unary [-]) -/\n| Oabsfloat         /- floating-point absolute value -/\nopen unary_operation\n\ninductive binary_operation : Type\n| Oadd              /- addition (binary [+]) -/\n| Osub              /- subtraction (binary [-]) -/\n| Omul              /- multiplication (binary [*]) -/\n| Odiv              /- division ([/]) -/\n| Omod              /- remainder ([%]) -/\n| Oand              /- bitwise and ([&]) -/\n| Oor               /- bitwise or ([|]) -/\n| Oxor              /- bitwise xor ([^]) -/\n| Oshl              /- left shift ([<<]) -/\n| Oshr              /- right shift ([>>]) -/\n| Oeq               /- comparison ([==]) -/\n| One               /- comparison ([!=]) -/\n| Olt               /- comparison ([<]) -/\n| Ogt               /- comparison ([>]) -/\n| Ole               /- comparison ([<=]) -/\n| Oge               /- comparison ([>=]) -/\nopen binary_operation\n\ninductive incr_or_decr : Type | Incr | Decr\nopen incr_or_decr\n\n/- * Type classification and semantics of operators. -/\n\n/- Most C operators are overloaded (they apply to arguments of various\n  types) and their semantics depend on the types of their arguments.\n  The following [classify_*] functions take as arguments the types\n  of the arguments of an operation.  They return enough information\n  to resolve overloading for this operator applications, such as\n  ``both arguments are floats'', or ``the first is a pointer\n  and the second is an integer''.  This classification is used in the\n  compiler (module [Cshmgen]) to resolve overloading statically.\n\n  The [sem_*] functions below compute the result of an operator\n  application.  Since operators are overloaded, the result depends\n  both on the static types of the arguments and on their run-time values.\n  The corresponding [classify_*] function is first called on the\n  types of the arguments to resolve static overloading.  It is then\n  followed by a case analysis on the values of the arguments. -/\n\n/- ** Casts and truth values -/\n\ninductive classify_cast_cases : Type\n| cast_case_pointer                                /- between pointer types or intptr_t types -/\n| cast_case_i2i (sz2 : intsize) (si2 : signedness) /- int -> int -/\n| cast_case_f2f                                    /- double -> double -/\n| cast_case_s2s                                    /- single -> single -/\n| cast_case_f2s                                    /- double -> single -/\n| cast_case_s2f                                    /- single -> double -/\n| cast_case_i2f (si1 : signedness)                 /- int -> double -/\n| cast_case_i2s (si1 : signedness)                 /- int -> single -/\n| cast_case_f2i (sz2 : intsize) (si2 : signedness) /- double -> int -/\n| cast_case_s2i (sz2 : intsize) (si2 : signedness) /- single -> int -/\n| cast_case_l2l                                    /- long -> long -/\n| cast_case_i2l (si1 : signedness)                 /- int -> long -/\n| cast_case_l2i (sz2 : intsize) (si2 : signedness) /- long -> int -/\n| cast_case_l2f (si1 : signedness)                 /- long -> double -/\n| cast_case_l2s (si1 : signedness)                 /- long -> single -/\n| cast_case_f2l (si2 : signedness)                 /- double -> long -/\n| cast_case_s2l (si2 : signedness)                 /- single -> long -/\n| cast_case_i2bool                                 /- int -> bool -/\n| cast_case_l2bool                                 /- long -> bool -/\n| cast_case_f2bool                                 /- double -> bool -/\n| cast_case_s2bool                                 /- single -> bool -/\n| cast_case_struct (id1 id2 : ident)               /- struct -> struct -/\n| cast_case_union  (id1 id2 : ident)               /- union -> union -/\n| cast_case_void                                   /- any -> void -/\n| cast_case_default\nopen classify_cast_cases\n\ndef classify_cast (tfrom tto : type) : classify_cast_cases :=\nmatch tto, tfrom with\n  /- To [void] -/\n| Tvoid, _ := cast_case_void\n  /- To [_Bool] -/\n| Tint IBool _ _, Tint _ _ _ := cast_case_i2bool\n| Tint IBool _ _, Tlong _ _ := cast_case_l2bool\n| Tint IBool _ _, Tfloat F64 _ := cast_case_f2bool\n| Tint IBool _ _, Tfloat F32 _ := cast_case_s2bool\n| Tint IBool _ _, Tpointer _ _ := if archi.ptr64 then cast_case_l2bool else cast_case_i2bool\n| Tint IBool _ _, Tarray _ _ _ := if archi.ptr64 then cast_case_l2bool else cast_case_i2bool\n| Tint IBool _ _, Tfunction _ _ _ := if archi.ptr64 then cast_case_l2bool else cast_case_i2bool\n  /- To [int] other than [_Bool] -/\n| Tint sz2 si2 _, Tint _ _ _ :=\n      if archi.ptr64 then cast_case_i2i sz2 si2\n      else if sz2 = I32 then cast_case_pointer\n      else cast_case_i2i sz2 si2\n| Tint sz2 si2 _, Tlong _ _ := cast_case_l2i sz2 si2\n| Tint sz2 si2 _, Tfloat F64 _ := cast_case_f2i sz2 si2\n| Tint sz2 si2 _, Tfloat F32 _ := cast_case_s2i sz2 si2\n| Tint sz2 si2 _, Tpointer _ _ :=\n      if archi.ptr64 then cast_case_l2i sz2 si2\n      else if sz2 = I32 then cast_case_pointer\n      else cast_case_i2i sz2 si2\n| Tint sz2 si2 _, Tarray _ _ _ :=\n      if archi.ptr64 then cast_case_l2i sz2 si2\n      else if sz2 = I32 then cast_case_pointer\n      else cast_case_i2i sz2 si2\n| Tint sz2 si2 _, Tfunction _ _ _ :=\n      if archi.ptr64 then cast_case_l2i sz2 si2\n      else if sz2 = I32 then cast_case_pointer\n      else cast_case_i2i sz2 si2\n  /- To [long] -/\n| Tlong _ _, Tlong _ _ :=\n      if archi.ptr64 then cast_case_pointer else cast_case_l2l\n| Tlong _ _, Tint sz1 si1 _ := cast_case_i2l si1\n| Tlong si2 _, Tfloat F64 _ := cast_case_f2l si2\n| Tlong si2 _, Tfloat F32 _ := cast_case_s2l si2\n| Tlong si2 _, Tpointer _ _ := if archi.ptr64 then cast_case_pointer else cast_case_i2l si2\n| Tlong si2 _, Tarray _ _ _ := if archi.ptr64 then cast_case_pointer else cast_case_i2l si2\n| Tlong si2 _, Tfunction _ _ _ := if archi.ptr64 then cast_case_pointer else cast_case_i2l si2\n  /- To [float] -/\n| Tfloat F64 _, Tint sz1 si1 _ := cast_case_i2f si1\n| Tfloat F32 _, Tint sz1 si1 _ := cast_case_i2s si1\n| Tfloat F64 _, Tlong si1 _ := cast_case_l2f si1\n| Tfloat F32 _, Tlong si1 _ := cast_case_l2s si1\n| Tfloat F64 _, Tfloat F64 _ := cast_case_f2f\n| Tfloat F32 _, Tfloat F32 _ := cast_case_s2s\n| Tfloat F64 _, Tfloat F32 _ := cast_case_s2f\n| Tfloat F32 _, Tfloat F64 _ := cast_case_f2s\n  /- To pointer types -/\n| Tpointer _ _, Tint _ _ _ :=\n      if archi.ptr64 then cast_case_i2l Unsigned else cast_case_pointer\n| Tpointer _ _, Tlong _ _ :=\n      if archi.ptr64 then cast_case_pointer else cast_case_l2i I32 Unsigned\n| Tpointer _ _, Tpointer _ _ := cast_case_pointer\n| Tpointer _ _, Tarray _ _ _ := cast_case_pointer\n| Tpointer _ _, Tfunction _ _ _ := cast_case_pointer\n  /- To struct or union types -/\n| Tstruct id2 _, Tstruct id1 _ := cast_case_struct id1 id2\n| Tunion id2 _, Tunion id1 _ := cast_case_union id1 id2\n  /- Catch-all -/\n| _, _ := cast_case_default\nend\n\n/- Semantics of casts.  [sem_cast v1 t1 t2 m = Some v2] if value [v1],\n  viewed with static type [t1], can be converted  to type [t2],\n  resulting in value [v2].  -/\n\ndef cast_int_int : intsize \u2192 signedness \u2192 int32 \u2192 int32\n| I8  Signed   i := sign_ext W8 i\n| I8  Unsigned i := zero_ext 8 i\n| I16 Signed   i := sign_ext W16 i\n| I16 Unsigned i := zero_ext 16 i\n| I32 _        i := i\n| IBool _      i := if i = 0 then 0 else 1\n\ndef cast_int_float : signedness \u2192 int32 \u2192 float\n| Signed   := float.of_int\n| Unsigned := float.of_intu\n\ndef cast_float_int : signedness \u2192 float \u2192 option int32\n| Signed   := float.to_int\n| Unsigned := float.to_intu\n\ndef cast_int_single : signedness \u2192 int32 \u2192 float32\n| Signed   := float32.of_int\n| Unsigned := float32.of_intu\n\ndef cast_single_int : signedness \u2192 float32 \u2192 option int32\n| Signed   := float32.to_int\n| Unsigned := float32.to_intu\n\ndef cast_int_long : signedness \u2192 int32 \u2192 int64\n| Signed   := scoe\n| Unsigned := ucoe\n\ndef cast_long_float : signedness \u2192 int64 \u2192 float\n| Signed   := float.of_long\n| Unsigned := float.of_longu\n\ndef cast_long_single : signedness \u2192 int64 \u2192 float32\n| Signed   := float32.of_long\n| Unsigned := float32.of_longu\n\ndef cast_float_long : signedness \u2192 float \u2192 option int64\n| Signed   := float.to_long\n| Unsigned := float.to_longu\n\ndef cast_single_long : signedness \u2192 float32 \u2192 option int64\n| Signed   := float32.to_long\n| Unsigned := float32.to_longu\n\ndef sem_cast (m : mem) (v : val) (t1 t2 : type) : option val :=\nmatch classify_cast t1 t2, v with\n| cast_case_pointer,        Vptr _ _   := some v\n| cast_case_pointer,        Vint _     := if archi.ptr64 then none else some v\n| cast_case_pointer,        Vlong _    := if archi.ptr64 then some v else none\n| cast_case_i2i sz2 si2,    Vint i     := some (Vint (cast_int_int sz2 si2 i))\n| cast_case_f2f,            Vfloat f   := some (Vfloat f)\n| cast_case_s2s,            Vsingle f  := some (Vsingle f)\n| cast_case_s2f,            Vsingle f  := some (Vfloat (float.of_single f))\n| cast_case_f2s,            Vfloat f   := some (Vsingle (float.to_single f))\n| cast_case_i2f si1,        Vint i     := some (Vfloat (cast_int_float si1 i))\n| cast_case_i2s si1,        Vint i     := some (Vsingle (cast_int_single si1 i))\n| cast_case_f2i sz2 si2,    Vfloat f   := (\u03bbi, Vint (cast_int_int sz2 si2 i)) <$> cast_float_int si2 f\n| cast_case_s2i sz2 si2,    Vsingle f  := (\u03bbi, Vint (cast_int_int sz2 si2 i)) <$> cast_single_int si2 f\n| cast_case_i2bool,         Vint n     := some (Vint (if n = 0 then 0 else 1))\n| cast_case_i2bool,         Vptr b ofs := if \u00ac archi.ptr64 \u2227 weak_valid_pointer m b (unsigned ofs) then some Vone else none\n| cast_case_l2bool,         Vlong n    := some (Vint (if n = 0 then 0 else 1))\n| cast_case_l2bool,         Vptr b ofs := if archi.ptr64 \u2227 weak_valid_pointer m b (unsigned ofs) then some Vone else none\n| cast_case_f2bool,         Vfloat f   := some (Vint (if float.cmp Ceq f 0 then 0 else 1))\n| cast_case_s2bool,         Vsingle f  := some (Vint (if float32.cmp Ceq f 0 then 0 else 1))\n| cast_case_l2l,            Vlong n    := some (Vlong n)\n| cast_case_i2l si,         Vint n     := some (Vlong (cast_int_long si n))\n| cast_case_l2i sz si,      Vlong n    := some (Vint (cast_int_int sz si (ucoe n)))\n| cast_case_l2f si1,        Vlong i    := some (Vfloat (cast_long_float si1 i))\n| cast_case_l2s si1,        Vlong i    := some (Vsingle (cast_long_single si1 i))\n| cast_case_f2l si2,        Vfloat f   := Vlong <$> cast_float_long si2 f\n| cast_case_s2l si2,        Vsingle f  := Vlong <$> cast_single_long si2 f\n| cast_case_struct id1 id2, Vptr b ofs := if id1 = id2 then some v else none\n| cast_case_union id1 id2,  Vptr b ofs := if id1 = id2 then some v else none\n| cast_case_void,           v          := some v\n| _,                        _          := none\nend\n\n/- The following describes types that can be interpreted as a boolean:\n  integers, floats, pointers.  It is used for the semantics of\n  the [!] and [?] operators, as well as the [if], [while],\n  and [for] statements. -/\n\ninductive classify_bool_cases : Type\n| bool_case_i                           /- integer -/\n| bool_case_l                           /- long -/\n| bool_case_f                           /- double float -/\n| bool_case_s                           /- single float -/\n| bool_default\nopen classify_bool_cases\n\ndef classify_bool (ty : type) : classify_bool_cases :=\nmatch typeconv ty with\n| Tint _ _ _   := bool_case_i\n| Tpointer _ _ := if archi.ptr64 then bool_case_l else bool_case_i\n| Tfloat F64 _ := bool_case_f\n| Tfloat F32 _ := bool_case_s\n| Tlong _ _    := bool_case_l\n| _            := bool_default\nend\n\n/- Interpretation of values as truth values.\n  Non-zero integers, non-zero floats and non-null pointers are\n  considered as true.  The integer zero (which also represents\n  the null pointer) and the float 0.0 are false. -/\n\ndef bool_val (m : mem) (v : val) (t : type) : option bool :=\nmatch classify_bool t, v with\n| bool_case_i, Vint n := some (n \u2260 0)\n| bool_case_i, Vptr b ofs := if archi.ptr64 then none else\n          if weak_valid_pointer m b (unsigned ofs) then some tt else none\n| bool_case_l, Vlong n := some (n \u2260 0)\n| bool_case_l, Vptr b ofs := if \u00ac archi.ptr64 then none else\n          if weak_valid_pointer m b (unsigned ofs) then some tt else none\n| bool_case_f, Vfloat f := some (bnot (float.cmp Ceq f 0))\n| bool_case_s, Vsingle f := some (bnot (float32.cmp Ceq f 0))\n| _, _ := none\nend\n\n/- ** Unary operators -/\n\n/- *** Boolean negation -/\n\ndef sem_notbool (m : mem) (v : val) (ty : type) : option val :=\n(\u03bb b, val.of_bool (bnot b)) <$> bool_val m v ty\n\n/- *** Opposite and absolute value -/\n\ninductive classify_neg_cases : Type\n| neg_case_i (s : signedness)            /- int -/\n| neg_case_f                             /- double float -/\n| neg_case_s                             /- single float -/\n| neg_case_l (s : signedness)            /- long -/\n| neg_default\nopen classify_neg_cases\n\ndef classify_neg : type \u2192 classify_neg_cases\n| (Tint I32 Unsigned _) := neg_case_i Unsigned\n| (Tint _ _ _)          := neg_case_i Signed\n| (Tfloat F64 _)        := neg_case_f\n| (Tfloat F32 _)        := neg_case_s\n| (Tlong si _)          := neg_case_l si\n| _                     := neg_default\n\ndef sem_neg (v : val) (ty : type) : option val :=\nmatch classify_neg ty, v with\n| neg_case_i sg, Vint n  := some (Vint (-n))\n| neg_case_f, Vfloat f   := some (Vfloat (-f))\n| neg_case_s, Vsingle f  := some (Vsingle (-f))\n| neg_case_l sg, Vlong n := some (Vlong (-n))\n| _, _                   := none\nend\n\ndef sem_absfloat (v : val) (ty : type) : option val :=\nmatch classify_neg ty, v with\n| neg_case_i sg, Vint n  := some (Vfloat (float.abs (cast_int_float sg n)))\n| neg_case_f, Vfloat f   := some (Vfloat (float.abs f))\n| neg_case_s, Vsingle f  := some (Vfloat (float.abs (float.of_single f)))\n| neg_case_l sg, Vlong n := some (Vfloat (float.abs (cast_long_float sg n)))\n| _, _                   := none\nend\n\n/- *** Bitwise complement -/\n\ninductive classify_notint_cases : Type\n| notint_case_i (s : signedness)              /- int -/\n| notint_case_l (s : signedness)              /- long -/\n| notint_default\nopen classify_notint_cases\n\ndef classify_notint : type \u2192 classify_notint_cases\n| (Tint I32 Unsigned _) := notint_case_i Unsigned\n| (Tint _ _ _) := notint_case_i Signed\n| (Tlong si _) := notint_case_l si\n| _ := notint_default\n\ndef sem_notint (v : val) (ty : type) : option val :=\nmatch classify_notint ty, v with\n| notint_case_i sg, Vint n  := some (Vint (word.not n))\n| notint_case_l sg, Vlong n := some (Vlong (word.not n))\n| _, _                      := none\nend\n\n/- ** Binary operators -/\n\n/- For binary operations, the \"usual binary conversions\" consist in\n- determining the type at which the operation is to be performed\n  (a form of least upper bound of the types of the two arguments);\n- casting the two arguments to this common type;\n- performing the operation at that type.\n-/\n\ninductive binarith_cases : Type\n| bin_case_i (s : signedness)        /- at int type -/\n| bin_case_l (s : signedness)        /- at long int type -/\n| bin_case_f                         /- at double float type -/\n| bin_case_s                         /- at single float type -/\n| bin_default                        /- error -/\nopen binarith_cases\n\ndef classify_binarith : type \u2192 type \u2192 binarith_cases\n| (Tint I32 Unsigned _) (Tint _ _ _)          := bin_case_i Unsigned\n| (Tint _ _ _)          (Tint I32 Unsigned _) := bin_case_i Unsigned\n| (Tint _ _ _)          (Tint _ _ _)          := bin_case_i Signed\n| (Tlong Signed _)      (Tlong Signed _)      := bin_case_l Signed\n| (Tlong _ _)           (Tlong _ _)           := bin_case_l Unsigned\n| (Tlong sg _)          (Tint _ _ _)          := bin_case_l sg\n| (Tint _ _ _)          (Tlong sg _)          := bin_case_l sg\n| (Tfloat F32 _)        (Tfloat F32 _)        := bin_case_s\n| (Tfloat _ _)          (Tfloat _ _)          := bin_case_f\n| (Tfloat F64 _)        (Tint _ _ _)          := bin_case_f\n| (Tfloat F64 _)        (Tlong _ _)           := bin_case_f\n| (Tint _ _ _)          (Tfloat F64 _)        := bin_case_f\n| (Tlong _ _)           (Tfloat F64 _)        := bin_case_f\n| (Tfloat F32 _)        (Tint _ _ _)          := bin_case_s\n| (Tfloat F32 _)        (Tlong _ _)           := bin_case_s\n| (Tint _ _ _)          (Tfloat F32 _)        := bin_case_s\n| (Tlong _ _)           (Tfloat F32 _)        := bin_case_s\n| _                     _                     := bin_default\n\n/- The static type of the result. Both arguments are converted to this type\n    before the actual computation. -/\n\ndef binarith_type : binarith_cases \u2192 type\n| (bin_case_i sg) := Tint I32 sg noattr\n| (bin_case_l sg) := Tlong sg noattr\n| bin_case_f      := Tfloat F64 noattr\n| bin_case_s      := Tfloat F32 noattr\n| bin_default     := Tvoid\n\ndef sem_binarith\n    (sem_int : signedness \u2192 int32 \u2192 int32 \u2192 option val)\n    (sem_long : signedness \u2192 int64 \u2192 int64 \u2192 option val)\n    (sem_float : float \u2192 float \u2192 option val)\n    (sem_single : float32 \u2192 float32 \u2192 option val)\n    (m : mem) (v1 : val) (t1 : type) (v2 : val) (t2 : type) : option val :=\nlet c := classify_binarith t1 t2, t := binarith_type c in\nmatch sem_cast m v1 t1 t, sem_cast m v2 t2 t, c with\n| some (Vint n1),    some (Vint n2),    bin_case_i sg := sem_int sg n1 n2\n| some (Vfloat n1),  some (Vfloat n2),  bin_case_f    := sem_float n1 n2\n| some (Vsingle n1), some (Vsingle n2), bin_case_s    := sem_single n1 n2\n| some (Vlong n1),   some (Vlong n2),   bin_case_l sg := sem_long sg n1 n2\n| _,                 _,                 _             := none\nend\n\n/- *** Addition -/\n\ninductive classify_add_cases : Type\n| add_case_pi (ty : type) (si : signedness)     /- pointer, int -/\n| add_case_pl (ty : type)                       /- pointer, long -/\n| add_case_ip (si : signedness) (ty : type)     /- int, pointer -/\n| add_case_lp (ty : type)                       /- long, pointer -/\n| add_default                                   /- numerical type, numerical type -/\nopen classify_add_cases\n\ndef classify_add (ty1 : type) (ty2 : type) :=\nmatch typeconv ty1, typeconv ty2 with\n| Tpointer ty _, Tint _ si _   := add_case_pi ty si\n| Tpointer ty _, Tlong _ _     := add_case_pl ty\n| Tint _ si _,   Tpointer ty _ := add_case_ip si ty\n| Tlong _ _,     Tpointer ty _ := add_case_lp ty\n| _,             _             := add_default\nend\n\ndef ptrofs_of_int : signedness \u2192 int32 \u2192 ptrofs\n| Signed := ptrofs.of_ints\n| Unsigned := ptrofs.of_intu\n\ndef sem_add_ptr_int (cenv : composite_env) (ty : type) (si : signedness) : val \u2192 val \u2192 option val\n| (Vptr b1 ofs1) (Vint n2) := let n2 := ptrofs_of_int si n2 in\n                              some (Vptr b1 (ofs1 + repr (sizeof cenv ty) * n2))\n| (Vint n1)      (Vint n2) := if archi.ptr64 then none else\n                              some (Vint (n1 + repr (sizeof cenv ty) * n2))\n| (Vlong n1)     (Vint n2) := let n2 := cast_int_long si n2 in\n                              if \u00ac archi.ptr64 then none else\n                              some (Vlong (n1 + repr (sizeof cenv ty) * n2))\n| _              _         := none\n\ndef sem_add_ptr_long (cenv : composite_env) (ty : type) : val \u2192 val \u2192 option val\n| (Vptr b1 ofs1) (Vlong n2) := let n2 := ptrofs.of_int64 n2 in\n                               some (Vptr b1 (ofs1 + repr (sizeof cenv ty) * n2))\n| (Vint n1)      (Vlong n2) := let n2 : int32 := ucoe n2 in\n                               if archi.ptr64 then none else\n                               some (Vint (n1 + repr (sizeof cenv ty) * n2))\n| (Vlong n1)     (Vlong n2) := if \u00ac archi.ptr64 then none else\n                               some (Vlong (n1 + repr (sizeof cenv ty) * n2))\n| _              _          := none\n\ndef sem_add (cenv : composite_env) (m : mem) (v1 : val) (t1 : type) (v2 : val) (t2 : type) : option val :=\nmatch classify_add t1 t2 with\n| add_case_pi ty si := sem_add_ptr_int cenv ty si v1 v2             /- pointer plus integer -/\n| add_case_pl ty    := sem_add_ptr_long cenv ty v1 v2                 /- pointer plus long -/\n| add_case_ip si ty := sem_add_ptr_int cenv ty si v2 v1             /- integer plus pointer -/\n| add_case_lp ty    := sem_add_ptr_long cenv ty v2 v1                  /- long plus pointer -/\n| add_default       := sem_binarith\n    (\u03bb sg n1 n2, some (Vint (n1 + n2)))\n    (\u03bb sg n1 n2, some (Vlong (n1 + n2)))\n    (\u03bb n1 n2, some (Vfloat (n1 + n2)))\n    (\u03bb n1 n2, some (Vsingle (n1 + n2)))\n    m v1 t1 v2 t2\nend\n\n/- *** Subtraction -/\n\ninductive classify_sub_cases : Type\n| sub_case_pi (ty : type) (si : signedness)  /- pointer, int -/\n| sub_case_pp (ty : type)               /- pointer, pointer -/\n| sub_case_pl (ty : type)               /- pointer, long -/\n| sub_default                           /- numerical type, numerical type -/\nopen classify_sub_cases\n\ndef classify_sub (ty1 : type) (ty2 : type) :=\nmatch typeconv ty1, typeconv ty2 with\n| Tpointer ty _,  Tint _ si _  := sub_case_pi ty si\n| Tpointer ty _ , Tpointer _ _ := sub_case_pp ty\n| Tpointer ty _,  Tlong _ _    := sub_case_pl ty\n| _,              _            := sub_default\nend\n\ndef sem_sub (cenv : composite_env) (m : mem) (v1 : val) (t1 : type) (v2 : val) (t2 : type) : option val :=\nmatch classify_sub t1 t2, v1, v2 with\n| sub_case_pi ty si, Vptr b1 ofs1, Vint n2 :=\n    let n2 := ptrofs_of_int si n2 in\n    some (Vptr b1 (ofs1 - repr (sizeof cenv ty) * n2))\n| sub_case_pi ty si, Vint n1, Vint n2 :=\n    if archi.ptr64 then none else some (Vint (n1 - repr (sizeof cenv ty) * n2))\n| sub_case_pi ty si, Vlong n1, Vint n2 :=\n    let n2 := cast_int_long si n2 in\n    if archi.ptr64 then some (Vlong (n1 - repr (sizeof cenv ty) * n2)) else none\n| sub_case_pl ty, Vptr b1 ofs1, Vlong n2 :=\n    let n2 := ptrofs.of_int64 n2 in\n    some (Vptr b1 (ofs1 - repr (sizeof cenv ty) * n2))\n| sub_case_pl ty, Vint n1, Vlong n2 :=\n    if archi.ptr64 then none else some (Vint (n1 - repr (sizeof cenv ty) * ucoe n2))\n| sub_case_pl ty, Vlong n1, Vlong n2 :=\n    if archi.ptr64 then some (Vlong (n1 - repr (sizeof cenv ty) * n2)) else none\n| sub_case_pp ty, Vptr b1 ofs1, Vptr b2 ofs2 :=\n    if b1 = b2 then\n      let sz := sizeof cenv ty in\n      if 0 < sz \u2227 sz \u2264 max_signed ptrofs.wordsize\n      then some (Vptrofs ((ofs1 - ofs2) / repr sz : word _))\n      else none\n    else none\n| sub_default, _, _ := sem_binarith\n      (\u03bb sg n1 n2, some (Vint (n1 - n2)))\n      (\u03bb sg n1 n2, some (Vlong (n1 - n2)))\n      (\u03bb n1 n2, some (Vfloat (n1 - n2)))\n      (\u03bb n1 n2, some (Vsingle (n1 - n2)))\n      m v1 t1 v2 t2\n| _, _, _ := none\nend\n\n/- *** Multiplication, division, modulus -/\n\ndef sem_mul : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2, some (Vint (n1 * n2)))\n  (\u03bb sg n1 n2, some (Vlong (n1 * n2)))\n  (\u03bb n1 n2, some (Vfloat (n1 * n2)))\n  (\u03bb n1 n2, some (Vsingle (n1 * n2)))\n\ndef sem_div : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2,\n    match sg with\n    | Signed   := if n2 = 0 \u2228 n1 = repr (min_signed W32) \u2227 n2 = -1\n                  then none else some (Vint (n1 / n2 : word _))\n    | Unsigned := if n2 = 0 then none else some (Vint (n1 / n2 : uword _))\n    end)\n  (\u03bb sg n1 n2,\n    match sg with\n    | Signed   := if n2 = 0 \u2228 n1 = repr (min_signed W64) \u2227 n2 = -1\n                  then none else some (Vlong (n1 / n2 : word _))\n    | Unsigned := if n2 = 0 then none else some (Vlong (n1 / n2 : uword _))\n    end)\n  (\u03bb n1 n2, some (Vfloat (n1 / n2)))\n  (\u03bb n1 n2, some (Vsingle (n1 / n2)))\n\ndef sem_mod : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2,\n    match sg with\n    | Signed   := if n2 = 0 \u2228 n1 = repr (min_signed W32) \u2227 n2 = -1\n                  then none else some (Vint (n1 % n2 : word _))\n    | Unsigned := if n2 = 0 then none else some (Vint (n1 % n2 : uword _))\n    end)\n  (\u03bb sg n1 n2,\n    match sg with\n    | Signed   := if n2 = 0 \u2228 n1 = repr (min_signed W64) \u2227 n2 = -1\n                  then none else some (Vlong (n1 % n2 : word _))\n    | Unsigned := if n2 = 0 then none else some (Vlong (n1 % n2 : uword _))\n    end)\n  (\u03bb n1 n2, none)\n  (\u03bb n1 n2, none)\n\ndef sem_and : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2, some (Vint (word.and n1 n2)))\n  (\u03bb sg n1 n2, some (Vlong (word.and n1 n2)))\n  (\u03bb n1 n2, none)\n  (\u03bb n1 n2, none)\n\ndef sem_or : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2, some (Vint (word.or n1 n2)))\n  (\u03bb sg n1 n2, some (Vlong (word.or n1 n2)))\n  (\u03bb n1 n2, none)\n  (\u03bb n1 n2, none)\n\ndef sem_xor : mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_binarith\n  (\u03bb sg n1 n2, some (Vint (word.xor n1 n2)))\n  (\u03bb sg n1 n2, some (Vlong (word.xor n1 n2)))\n  (\u03bb n1 n2, none)\n  (\u03bb n1 n2, none)\n\n/- *** Shifts -/\n\n/- Shifts do not perform the usual binary conversions.  Instead,\n  each argument is converted independently, and the signedness\n  of the result is always that of the first argument. -/\n\ninductive classify_shift_cases : Type\n| shift_case_ii (s : signedness)         /- int , int -/\n| shift_case_ll (s : signedness)         /- long, long -/\n| shift_case_il (s : signedness)         /- int, long -/\n| shift_case_li (s : signedness)         /- long, int -/\n| shift_default\nopen classify_shift_cases\n\ndef classify_shift (ty1 : type) (ty2 : type) :=\nmatch typeconv ty1, typeconv ty2 with\n| Tint I32 Unsigned _, Tint _ _ _ := shift_case_ii Unsigned\n| Tint _ _ _,          Tint _ _ _ := shift_case_ii Signed\n| Tint I32 Unsigned _, Tlong _ _  := shift_case_il Unsigned\n| Tint _ _ _,          Tlong _ _  := shift_case_il Signed\n| Tlong s _,           Tint _ _ _ := shift_case_li s\n| Tlong s _,           Tlong _ _  := shift_case_ll s\n| _,                   _          := shift_default\nend\n\ndef sem_shift\n    (sem_int : signedness \u2192 int32 \u2192 int32 \u2192 int32)\n    (sem_long : signedness \u2192 int64 \u2192 int64 \u2192 int64)\n    (v1 : val) (t1 : type) (v2 : val) (t2 : type) : option val :=\nmatch classify_shift t1 t2, v1, v2 with\n| shift_case_ii sg, Vint n1,  Vint n2  := if ltu n2 iwordsize then some (Vint (sem_int sg n1 n2)) else none\n| shift_case_il sg, Vint n1,  Vlong n2 := if ltu n2 (repr 32) then some (Vint (sem_int sg n1 n2.loword)) else none\n| shift_case_li sg, Vlong n1, Vint n2  := if ltu n2 (repr 64) then some (Vlong (sem_long sg n1 (ucoe n2))) else none\n| shift_case_ll sg, Vlong n1, Vlong n2 := if ltu n2 iwordsize then some (Vlong (sem_long sg n1 n2)) else none\n| _,                _,        _        := none\nend\n\ndef sem_shl : val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_shift\n  (\u03bb sg n1 n2, word.shl n1 n2)\n  (\u03bb sg n1 n2, word.shl n1 n2)\n\ndef sem_shr : val \u2192 type \u2192 val \u2192 type \u2192 option val :=\nsem_shift\n  (\u03bb sg n1 n2, match sg with Signed := word.shr n1 n2 | Unsigned := word.shru n1 n2 end)\n  (\u03bb sg n1 n2, match sg with Signed := word.shr n1 n2 | Unsigned := word.shru n1 n2 end)\n\n/- *** Comparisons -/\n\ninductive classify_cmp_cases : Type\n| cmp_case_pp                        /- pointer, pointer -/\n| cmp_case_pi (si : signedness)      /- pointer, int -/\n| cmp_case_ip (si : signedness)      /- int, pointer -/\n| cmp_case_pl                        /- pointer, long -/\n| cmp_case_lp                        /- long, pointer -/\n| cmp_default                        /- numerical, numerical -/\nopen classify_cmp_cases\n\ndef classify_cmp (ty1 : type) (ty2 : type) :=\nmatch typeconv ty1, typeconv ty2 with\n| Tpointer _ _ , Tpointer _ _ := cmp_case_pp\n| Tpointer _ _ , Tint _ si _  := cmp_case_pi si\n| Tint _ si _,   Tpointer _ _ := cmp_case_ip si\n| Tpointer _ _ , Tlong _ _    := cmp_case_pl\n| Tlong _ _ ,    Tpointer _ _ := cmp_case_lp\n| _,             _            := cmp_default\nend\n\ndef cmp_ptr (m : mem) (c : comparison) (v1 v2 : val) : option val :=\nval.of_bool <$> (if archi.ptr64 then cmplu_bool else cmpu_bool) (valid_pointer m) c v1 v2\n\ndef sem_cmp (c : comparison) (m : mem) (v1 : val) (t1 : type) (v2 : val) (t2 : type) : option val :=\nmatch classify_cmp t1 t2, v1, v2 with\n| cmp_case_pp,    v1, v2         := cmp_ptr m c v1 v2\n| cmp_case_pi si, v1, Vint n2    := cmp_ptr m c v1 (Vptrofs (ptrofs_of_int si n2))\n| cmp_case_pi si, v1, Vptr b ofs := if archi.ptr64 then none else cmp_ptr m c v1 v2\n| cmp_case_ip si, Vint n1, v2    := cmp_ptr m c (Vptrofs (ptrofs_of_int si n1)) v2\n| cmp_case_ip si, Vptr b ofs, v2 := if archi.ptr64 then none else cmp_ptr m c (Vptr b ofs) v2\n| cmp_case_pl,    v1, Vlong n2   := cmp_ptr m c v1 (Vptrofs (ptrofs.of_int64 n2))\n| cmp_case_pl,    v1, Vptr b ofs := if archi.ptr64 then cmp_ptr m c v1 (Vptr b ofs) else none\n| cmp_case_lp,    Vlong n1,   v2 := cmp_ptr m c (Vptrofs (ptrofs.of_int64 n1)) v2\n| cmp_case_lp,    Vptr b ofs, v2 := if archi.ptr64 then cmp_ptr m c v1 v2 else none\n| cmp_default,    v1, v2         := sem_binarith\n    (\u03bb sg n1 n2, some (val.of_bool ((match sg with Signed := word.cmp | Unsigned := word.cmpu end) c n1 n2)))\n    (\u03bb sg n1 n2, some (val.of_bool ((match sg with Signed := word.cmp | Unsigned := word.cmpu end) c n1 n2)))\n    (\u03bb n1 n2, some (val.of_bool (float.cmp c n1 n2)))\n    (\u03bb n1 n2, some (val.of_bool (float32.cmp c n1 n2)))\n    m v1 t1 v2 t2\n| _,              _, _           := none\nend\n\n/- ** Function applications -/\n\ninductive classify_fun_cases : Type\n| fun_case_f (targs : list type) (tres : type) (cc : calling_convention) /- (pointer to) function -/\n| fun_default\nopen classify_fun_cases\n\ndef classify_fun : type \u2192 classify_fun_cases\n| (Tfunction args res cc)              := fun_case_f args res cc\n| (Tpointer (Tfunction args res cc) _) := fun_case_f args res cc\n| _                                    := fun_default\n\n/- ** Argument of a [switch] statement -/\n\ninductive classify_switch_cases : Type\n| switch_case_i\n| switch_case_l\n| switch_default\nopen classify_switch_cases\n\ndef classify_switch : type \u2192 classify_switch_cases\n| (Tint _ _ _) := switch_case_i\n| (Tlong _ _)  := switch_case_l\n| _            := switch_default\n\ndef sem_switch_arg (v : val) (ty : type) : option \u2124 :=\nmatch classify_switch ty, v with\n| switch_case_i, Vint n  := some (unsigned n)\n| switch_case_l, Vlong n := some (unsigned n)\n| _,             _       := none\nend\n\n/- * Combined semantics of unary and binary operators -/\n\ndef sem_unary_operation : unary_operation \u2192 mem \u2192 val \u2192 type \u2192 option val\n| Onotbool  := sem_notbool\n| Onotint   := \u03bb_, sem_notint\n| Oneg      := \u03bb_, sem_neg\n| Oabsfloat := \u03bb_, sem_absfloat\n\ndef sem_binary_operation (cenv : composite_env) :\n  binary_operation \u2192 mem \u2192 val \u2192 type \u2192 val \u2192 type \u2192 option val\n| Oadd := sem_add cenv\n| Osub := sem_sub cenv\n| Omul := sem_mul\n| Omod := sem_mod\n| Odiv := sem_div\n| Oand := sem_and\n| Oor  := sem_or\n| Oxor := sem_xor\n| Oshl := \u03bb_, sem_shl\n| Oshr := \u03bb_, sem_shr\n| Oeq  := sem_cmp Ceq\n| One  := sem_cmp Cne\n| Olt  := sem_cmp Clt\n| Ogt  := sem_cmp Cgt\n| Ole  := sem_cmp Cle\n| Oge  := sem_cmp Cge\n\ndef sem_incrdecr (cenv : composite_env) (id : incr_or_decr) (m : mem) (v : val) (ty : type) :=\nmatch id with\n| Incr := sem_add cenv m v ty 1 type_int32s\n| Decr := sem_sub cenv m v ty 1 type_int32s\nend\n\ndef incrdecr_type (ty : type) :=\nmatch typeconv ty with\n| Tpointer ty a := Tpointer ty a\n| Tint sz sg a  := Tint sz sg noattr\n| Tlong sg a    := Tlong sg noattr\n| Tfloat sz a   := Tfloat sz noattr\n| _             := Tvoid\nend\n\n/- * Compatibility with extensions and injections -/\n\nsection generic_injection\n\nparameters {f : meminj} {m m' : mem}\n\nparameter (valid_pointer_inj :\n  \u2200 b1 (ofs : ptrofs) b2 delta,\n  f b1 = some (b2, delta) \u2192\n  valid_pointer m b1 (unsigned ofs) \u2192\n  valid_pointer m' b2 (unsigned (ofs + repr delta)))\n\nparameter (weak_valid_pointer_inj :\n  \u2200 b1 (ofs : ptrofs) b2 delta,\n  f b1 = some (b2, delta) \u2192\n  weak_valid_pointer m b1 (unsigned ofs) \u2192\n  weak_valid_pointer m' b2 (unsigned (ofs + repr delta)))\n\nparameter (weak_valid_pointer_no_overflow :\n  \u2200 b1 (ofs : ptrofs) b2 delta,\n  f b1 = some (b2, delta) \u2192\n  weak_valid_pointer m b1 (unsigned ofs) \u2192\n  unsigned ofs + unsigned (repr delta : ptrofs) \u2264 max_unsigned ptrofs.wordsize)\n\nparameter (valid_different_pointers_inj :\n  \u2200 b1 b2 (ofs1 ofs2 : ptrofs) b1' delta1 b2' delta2,\n  b1 \u2260 b2 \u2192\n  valid_pointer m b1 (unsigned ofs1) \u2192\n  valid_pointer m b2 (unsigned ofs2) \u2192\n  f b1 = some (b1', delta1) \u2192\n  f b2 = some (b2', delta2) \u2192\n  b1' = b2' \u2192\n  unsigned (ofs1 + repr delta1) \u2260 unsigned (ofs2 + repr delta2))\n\ntheorem val_inject_vtrue (f) : inject f Vtrue Vtrue := sorry'\n\ntheorem val_inject_vfalse (f) : inject f Vfalse Vfalse := sorry'\n\ntheorem val_inject_of_bool (f b) : inject f (val.of_bool b) (val.of_bool b) := sorry'\n\ntheorem val_inject_vptrofs (n) : inject f (Vptrofs n) (Vptrofs n) := sorry'\n\nlemma sem_cast_inj {v1 ty1 ty v tv1} :\n  sem_cast m v1 ty1 ty = some v \u2192\n  inject f v1 tv1 \u2192\n  \u2203 tv, sem_cast m' tv1 ty1 ty = some tv \u2227 inject f v tv := sorry'\n\nlemma bool_val_inj {v ty b tv} :\n  bool_val m v ty = some b \u2192\n  inject f v tv \u2192\n  bool_val m' tv ty = some b := sorry'\n\nlemma sem_unary_operation_inj {op v1 ty v tv1} :\n  sem_unary_operation op m v1 ty = some v \u2192\n  inject f v1 tv1 \u2192\n  \u2203 tv, sem_unary_operation op m' tv1 ty = some tv \u2227 inject f v tv := sorry'\n\ndef optval_self_injects : option val \u2192 bool\n| (some (Vptr b ofs)) := ff\n| _                   := tt\n\ntheorem sem_binarith_inject {sem_int sem_long sem_float sem_single v1 t1 v2 t2 v v1' v2'} :\n  sem_binarith sem_int sem_long sem_float sem_single m v1 t1 v2 t2 = some v \u2192\n  inject f v1 v1' \u2192 inject f v2 v2' \u2192\n  (\u2200 sg n1 n2, optval_self_injects (sem_int sg n1 n2)) \u2192\n  (\u2200 sg n1 n2, optval_self_injects (sem_long sg n1 n2)) \u2192\n  (\u2200 n1 n2, optval_self_injects (sem_float n1 n2)) \u2192\n  (\u2200 n1 n2, optval_self_injects (sem_single n1 n2)) \u2192\n  \u2203 v', sem_binarith sem_int sem_long sem_float sem_single m' v1' t1 v2' t2 = some v' \u2227 inject f v v' := sorry'\n\ntheorem sem_shift_inject {sem_int sem_long v1 t1 v2 t2 v v1' v2'} :\n  sem_shift sem_int sem_long v1 t1 v2 t2 = some v \u2192\n  inject f v1 v1' \u2192 inject f v2 v2' \u2192\n  \u2203 v', sem_shift sem_int sem_long v1' t1 v2' t2 = some v' \u2227 inject f v v' := sorry'\n\ntheorem sem_cmp_ptr_inj {c v1 v2 v tv1 tv2} :\n  cmp_ptr m c v1 v2 = some v \u2192\n  inject f v1 tv1 \u2192\n  inject f v2 tv2 \u2192\n  \u2203 tv, cmp_ptr m' c tv1 tv2 = some tv \u2227 inject f v tv := sorry'\n\ntheorem sem_cmp_inj {cmp v1 tv1 ty1 v2 tv2 ty2 v} :\n  sem_cmp cmp m v1 ty1 v2 ty2 = some v \u2192\n  inject f v1 tv1 \u2192\n  inject f v2 tv2 \u2192\n  \u2203 tv, sem_cmp cmp m' tv1 ty1 tv2 ty2 = some tv \u2227 inject f v tv := sorry'\n\nlemma sem_binary_operation_inj {cenv op v1 ty1 v2 ty2 v tv1 tv2} :\n  sem_binary_operation cenv op m v1 ty1 v2 ty2 = some v \u2192\n  inject f v1 tv1 \u2192 inject f v2 tv2 \u2192\n  \u2203 tv, sem_binary_operation cenv op m' tv1 ty1 tv2 ty2 = some tv \u2227 inject f v tv := sorry'\n\nend generic_injection\n\nlemma sem_cast_inject {f v1 ty1 ty m v tv1 tm} :\n  sem_cast m v1 ty1 ty = some v \u2192\n  inject f v1 tv1 \u2192\n  inject f m tm \u2192\n  \u2203 tv, sem_cast tm tv1 ty1 ty = some tv \u2227 inject f v tv := sorry'\n\nlemma sem_unary_operation_inject {f m m' op v1 ty1 v tv1} :\n  sem_unary_operation op m v1 ty1 = some v \u2192\n  inject f v1 tv1 \u2192\n  inject f m m' \u2192\n  \u2203 tv, sem_unary_operation op m' tv1 ty1 = some tv \u2227 inject f v tv := sorry'\n\nlemma sem_binary_operation_inject {f m m' cenv op v1 ty1 v2 ty2 v tv1 tv2} :\n  sem_binary_operation cenv op m v1 ty1 v2 ty2 = some v \u2192\n  inject f v1 tv1 \u2192 inject f v2 tv2 \u2192\n  inject f m m' \u2192\n  \u2203 tv, sem_binary_operation cenv op m' tv1 ty1 tv2 ty2 = some tv \u2227 inject f v tv := sorry'\n\nlemma bool_val_inject {f m m' v ty b tv} :\n  bool_val m v ty = some b \u2192\n  inject f v tv \u2192\n  inject f m m' \u2192\n  bool_val m' tv ty = some b := sorry'\n\n/- * Some properties of operator semantics -/\n\n/- This section collects some common-sense properties about the type\n  classification and semantic functions above.  Some properties are used\n  in the CompCert semantics preservation proofs.  Others are not, but increase\n  confidence in the specification and its relation with the ISO C99 standard. -/\n\n/- Relation between Boolean value and casting to [_Bool] type. -/\n\nlemma cast_bool_bool_val {v t m} :\n  sem_cast m v t (Tint IBool Signed noattr) =\n  val.of_bool <$> bool_val m v t := sorry'\n\n/- Relation between Boolean value and Boolean negation. -/\n\nlemma notbool_bool_val {v t m} :\n  sem_notbool m v t =\n  val.of_bool <$> bnot <$> bool_val m v t := sorry'\n\n/- Properties of values obtained by casting to a given type. -/\n\nsection val_casted\n\ninductive val_casted : val \u2192 type \u2192 Prop\n| int (sz si attr n)       : cast_int_int sz si n = n \u2192 val_casted (Vint n) (Tint sz si attr)\n| float (attr n)           : val_casted (Vfloat n) (Tfloat F64 attr)\n| single (attr n)          : val_casted (Vsingle n) (Tfloat F32 attr)\n| long (si attr n)         : val_casted (Vlong n) (Tlong si attr)\n| ptr_ptr (b ofs ty attr)  : val_casted (Vptr b ofs) (Tpointer ty attr)\n| int_ptr (n ty attr)      : \u00ac archi.ptr64 \u2192 val_casted (Vint n) (Tpointer ty attr)\n| ptr_int (b ofs si attr)  : \u00ac archi.ptr64 \u2192 val_casted (Vptr b ofs) (Tint I32 si attr)\n| long_ptr (n ty attr)     : archi.ptr64 \u2192 val_casted (Vlong n) (Tpointer ty attr)\n| ptr_long (b ofs si attr) : archi.ptr64 \u2192 val_casted (Vptr b ofs) (Tlong si attr)\n| struct (id attr b ofs)   : val_casted (Vptr b ofs) (Tstruct id attr)\n| union (id attr b ofs)    : val_casted (Vptr b ofs) (Tunion id attr)\n| void (v)                 : val_casted v Tvoid\n\ntheorem cast_int_int_idem (sz sg i) :\n  cast_int_int sz sg (cast_int_int sz sg i) = cast_int_int sz sg i := sorry'\n\nlemma cast_val_is_casted {v ty ty' v' m} :\n  sem_cast v ty ty' m = some v' \u2192 val_casted v' ty' := sorry'\n\nend val_casted\n\n/- As a consequence, casting twice is equivalent to casting once. -/\n\nlemma cast_val_casted {m v ty} :\n  val_casted v ty \u2192 sem_cast m v ty ty = some v := sorry'\n\nlemma cast_idempotent {m v ty ty' v'} :\n  sem_cast m v ty ty' = some v' \u2192 sem_cast m v' ty' ty' = some v' := sorry'\n\n/- Relation with the arithmetic conversions of ISO C99, section 6.3.1 -/\n\nsection arith_conv\n\n/- This is the ISO C algebra of arithmetic types, without qualifiers.\n    [S] stands for \"signed\" and [U] for \"unsigned\".  -/\n\ninductive int_type : Type\n| Bool\n| Char | SChar | UChar\n| Short | UShort\n| Int | UInt\n| Long | ULong\n| Longlong | ULonglong\nopen int_type\n\ninductive arith_type : Type\n| I (it : int_type)\n| Float\n| Double\n| Longdouble\nopen arith_type\n\ninstance eq_int_type : decidable_eq int_type := by tactic.mk_dec_eq_instance\n\n\ndef is_unsigned : int_type \u2192 bool\n| Bool      := tt\n| UChar     := tt\n| UShort    := tt\n| UInt      := tt\n| ULong     := tt\n| ULonglong := tt\n| _         := ff\n\ndef unsigned_type : int_type \u2192 int_type\n| Char     := UChar\n| SChar    := UChar\n| Short    := UShort\n| Int      := UInt\n| Long     := ULong\n| Longlong := ULonglong\n| t        := t\n\ndef int_sizeof : int_type \u2192 \u2115\n| Bool      := 1\n| Char      := 1\n| SChar     := 1\n| UChar     := 1\n| Short     := 2\n| UShort    := 2\n| Int       := 4\n| UInt      := 4\n| Long      := 4\n| ULong     := 4\n| Longlong  := 8\n| ULonglong := 8\n\n/- 6.3.1.1 para 1: integer conversion rank -/\n\ndef rank : int_type \u2192 \u2115\n| Bool      := 1\n| Char      := 2\n| SChar     := 2\n| UChar     := 2\n| Short     := 3\n| UShort    := 3\n| Int       := 4\n| UInt      := 4\n| Long      := 5\n| ULong     := 5\n| Longlong  := 6\n| ULonglong := 6\n\n/- 6.3.1.1 para 2: integer promotions, a.k.a. usual unary conversions -/\n\ndef integer_promotion (t : int_type) : int_type :=\nif rank t < rank Int then Int else t\n\n/- 6.3.1.8: Usual arithmetic conversions, a.k.a. binary conversions.\n  This function returns the type to which the two operands must be\n  converted. -/\n\ndef usual_arithmetic_conversion : arith_type \u2192 arith_type \u2192 arith_type\n  /- First, if the corresponding real type of either operand is long\n     double, the other operand is converted, without change of type domain,\n     to a type whose corresponding real type is long double. -/\n| Longdouble _ := Longdouble\n| _ Longdouble := Longdouble\n  /- Otherwise, if the corresponding real type of either operand is\n     double, the other operand is converted, without change of type domain,\n     to a type whose corresponding real type is double. -/\n| Double _ := Double\n| _ Double := Double\n  /- Otherwise, if the corresponding real type of either operand is\n     float, the other operand is converted, without change of type domain,\n     to a type whose corresponding real type is float. -/\n| Float _ := Float\n| _ Float := Float\n  /- Otherwise, the integer promotions are performed on both operands. -/\n| (I i1) (I i2) :=\n    let j1 := integer_promotion i1, j2 := integer_promotion i2 in\n    /- Then the following rules are applied to the promoted operands:\n       If both operands have the same type, then no further conversion\n       is needed. -/\n    if j1 = j2 then I j1 else\n    match is_unsigned j1, is_unsigned j2 with\n    /- Otherwise, if both operands have signed integer types or both\n       have unsigned integer types, the operand with the type of lesser\n       integer conversion rank is converted to the type of the operand with\n       greater rank. -/\n    | tt, tt := if rank j1 < rank j2 then I j2 else I j1\n    | ff, ff := if rank j1 < rank j2 then I j2 else I j1\n    | tt, ff :=\n    /- Otherwise, if the operand that has unsigned integer type has\n       rank greater or equal to the rank of the type of the other operand,\n       then the operand with signed integer type is converted to the type of\n       the operand with unsigned integer type. -/\n        if rank j2 \u2264 rank j1 then I j1 else\n    /- Otherwise, if the type of the operand with signed integer type\n       can represent all of the values of the type of the operand with\n       unsigned integer type, then the operand with unsigned integer type is\n       converted to the type of the operand with signed integer type. -/\n        if int_sizeof j1 < int_sizeof j2 then I j2 else\n    /- Otherwise, both operands are converted to the unsigned integer type\n       corresponding to the type of the operand with signed integer type. -/\n        I (unsigned_type j2)\n    | ff, tt :=\n    /- Same logic as above, swapping the roles of j1 and j2 -/\n        if rank j1 \u2264 rank j2 then I j2 else\n        if int_sizeof j2 < int_sizeof j1 then I j1 else\n        I (unsigned_type j1)\n    end\n\n/- Mapping ISO arithmetic types to CompCert types -/\n\ndef proj_type : arith_type \u2192 type\n| (I Bool)      := Tint IBool Unsigned noattr\n| (I Char)      := Tint I8 Unsigned noattr\n| (I SChar)     := Tint I8 Signed noattr\n| (I UChar)     := Tint I8 Unsigned noattr\n| (I Short)     := Tint I16 Signed noattr\n| (I UShort)    := Tint I16 Unsigned noattr\n| (I Int)       := Tint I32 Signed noattr\n| (I UInt)      := Tint I32 Unsigned noattr\n| (I Long)      := Tint I32 Signed noattr\n| (I ULong)     := Tint I32 Unsigned noattr\n| (I Longlong)  := Tlong Signed noattr\n| (I ULonglong) := Tlong Unsigned noattr\n| Float         := Tfloat F32 noattr\n| Double        := Tfloat F64 noattr\n| Longdouble    := Tfloat F64 noattr\n\n/- Relation between [typeconv] and integer promotion. -/\n\nlemma typeconv_integer_promotion (i) :\n  typeconv (proj_type (I i)) = proj_type (I (integer_promotion i)) := sorry'\n\n/- Relation between [classify_binarith] and arithmetic conversion. -/\n\nlemma classify_binarith_arithmetic_conversion (t1 t2) :\n  binarith_type (classify_binarith (proj_type t1) (proj_type t2)) =\n  proj_type (usual_arithmetic_conversion t1 t2) := sorry'\n\nend arith_conv\n\nend cop\n", "meta": {"author": "digama0", "repo": "kremlin", "sha": "d4665929ce9012e93a0b05fc7063b96256bab86f", "save_path": "github-repos/lean/digama0-kremlin", "path": "github-repos/lean/digama0-kremlin/kremlin-d4665929ce9012e93a0b05fc7063b96256bab86f/cop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754371026367, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3725171189821919}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.data.list.lemmas\nimport Mathlib.Lean3Lib.init.wf\n \n\nuniverses u_1 \n\nnamespace Mathlib\n\nnamespace list\n\n\n-- Note: we can't use the equation compiler here because\n\n-- init.meta.well_founded_tactics uses this file\n\ndef qsort.F {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (x : List \u03b1) : ((y : List \u03b1) \u2192 length y < length x \u2192 List \u03b1) \u2192 List \u03b1 :=\n  sorry\n\n/- This is based on the minimalist Haskell \"quicksort\".\n\n   Remark: this is *not* really quicksort since it doesn't partition the elements in-place -/\n\ndef qsort {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) : List \u03b1 \u2192 List \u03b1 :=\n  well_founded.fix sorry sorry\n\n@[simp] theorem qsort_nil {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) : qsort lt [] = [] := sorry\n\n@[simp] theorem qsort_cons {\u03b1 : Type u_1} (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (h : \u03b1) (t : List \u03b1) : qsort lt (h :: t) =\n  (fun (_a : List \u03b1 \u00d7 List \u03b1) =>\n      prod.cases_on _a fun (fst snd : List \u03b1) => idRhs (List \u03b1) (qsort lt snd ++ h :: qsort lt fst))\n    (partition (fun (x : \u03b1) => lt h x = tt) t) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/data/list/qsort.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.372406206115009}}
{"text": "example : let x := 0; x + 5 = 5 := by\n  dsimp (config := { zeta := false })\n  trace_state\n  simp\n\nexample : let x := 0; x + 5 = 5 := by\n  dsimp\n\nexample : let x := 0; x + y = y := by\n  dsimp\n  trace_state\n  rw [Nat.zero_add]\n\nexample : let x := 0; x + y = y := by\n  dsimp (config := { zeta := false })\n  trace_state\n  conv => zeta\n  trace_state\n  rw [Nat.zero_add]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/dsimpZetaIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.37240361037279474}}
{"text": "import split_cycle\n\nstructure election_profile (\u03c7 \u03c5 : Type*) :=\n(cands : finset \u03c7)\n(voters : finset \u03c5)\n(Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop)\n\nvariables {\u03c7 \u03c5 : Type*}\n\ndef profile_without (prof: election_profile \u03c7 \u03c5) (b : \u03c7) \n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] : election_profile \u03c7 \u03c5 :=\n  \u27e8prof.cands.erase b, prof.voters, prof.Q\u27e9\n\nlemma mem_prof_of_mem_profile_without {a b: \u03c7} {prof : election_profile \u03c7 \u03c5}\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] (h : a \u2208 (profile_without prof b).cands) : \n  a \u2208 prof.cands :=\nby simp only [profile_without, ne.def, finset.mem_erase] at h; exact h.2\n\nlemma voters_eq_profile_without (prof : election_profile \u03c7 \u03c5) (b: \u03c7)\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] : \n  (profile_without prof b).voters = prof.voters := by simp only [profile_without]\n\nlemma Q_eq_profile_without (prof : election_profile \u03c7 \u03c5) (b: \u03c7)\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] : \n  (profile_without prof b).Q = prof.Q := by simp only [profile_without]\n\nlemma cands_erase_eq_profile_without (prof : election_profile \u03c7 \u03c5) (b: \u03c7)\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] : \n  (profile_without prof b).cands = prof.cands.erase b := by simp only [profile_without]\n\nlemma profile_without_eq_cands_of_not_mem {b : \u03c7} {prof : election_profile \u03c7 \u03c5}\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] (hb : b \u2209 prof.cands) : \n  (profile_without prof b) = prof :=\nbegin\n  tactic.unfreeze_local_instances,\n  cases prof,\n  simp only [profile_without, and_true, eq_self_iff_true, \n    finset.erase_eq_of_not_mem hb] at *,\nend\n\nlemma ne_of_mem_profile_without {a b : \u03c7} {prof : election_profile \u03c7 \u03c5}\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] \n  (h : a \u2208 (profile_without prof b).cands ) : a \u2260 b := \nbegin \n  simp only [cands_erase_eq_profile_without prof b,\n     ne.def, finset.mem_erase] at h,\n  exact h.1,\nend\n\nlemma profile_without_card {b: \u03c7} (prof : election_profile \u03c7 \u03c5) (b_in : b \u2208 prof.cands)\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] :\n  prof.cands.card.pred = (profile_without prof b).cands.card := \nby rw [cands_erase_eq_profile_without prof b, finset.card_erase_of_mem b_in]\n\nlemma profile_without_card' {b: \u03c7} (prof : election_profile \u03c7 \u03c5) (b_in : b \u2208 prof.cands)\n  [\u2200 v, decidable_rel (prof.Q v)] [decidable_eq \u03c7] :\n  prof.cands.card = (profile_without prof b).cands.card.succ :=\nbegin\n  rw \u2190 profile_without_card prof b_in,\n  exact (nat.succ_pred_eq_of_pos (finset.card_pos.2 \u27e8b, b_in\u27e9)).symm,\nend\n\nlemma restrict_of_subset {q : \u03c7 \u2192 \u03c7 \u2192 Prop} {s t : finset \u03c7} {a b : \u03c7}\n  (hst : s \u2286 t) (hq : (restrict q s) a b) : (restrict q t) a b := \n\u27e8hq.1, hst hq.2.1, hst hq.2.2\u27e9 \n\nlemma restrict_restrict_eq_restrict (q : \u03c7 \u2192 \u03c7 \u2192 Prop) (s : finset \u03c7) :\n  restrict (restrict q s) s = restrict q s :=\nbegin\n  ext x y, split,\n  { rintros \u27e8\u27e8hxy, x_in, y_in\u27e9, -, -\u27e9,\n    exact \u27e8hxy, x_in, y_in\u27e9, },\n  { rintros \u27e8hxy, x_in, y_in\u27e9,\n    exact \u27e8\u27e8hxy,x_in,y_in\u27e9,x_in, y_in\u27e9, },\nend\n\ninstance {\u03b1 : Type*} (s : finset \u03b1) : decidable s.nonempty :=\nbegin\n  rw \u2190finset.card_pos,\n  apply_instance,\nend\n\ndef best_margin (voters : finset \u03c5) (s : finset (\u03c7 \u00d7 \u03c7)) (Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop) \n  [\u2200 v, decidable_rel (Q v)] : \u2124 :=\n  if hn : s.nonempty\n    then s.sup' hn (\u03bb p, margin voters Q p.1 p.2) \n  else 0\n\ndef uniquely_weighted (voters : finset \u03c5) (cands : finset \u03c7) (Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop) \n  [\u2200 v, decidable_rel (Q v)] : Prop := \n\u2200 a b a' b' \u2208 cands, a \u2260 b \u2192 a' \u2260 b' \u2192 (a \u2260 a' \u2228 b \u2260 b') \u2192 \nmargin voters Q a b \u2260 margin voters Q a' b'\n\nopen_locale classical\n\nnoncomputable def stable_voting' (voters : finset \u03c5) (Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop) :\n  \u03a0 (n : \u2115) (cands : finset \u03c7) (hn : cands.card = n), finset \u03c7\n| 0 cands _ := cands\n| 1 cands _  := cands\n| (n+2) cands hn  :=\nlet\n  -- whether c wins when candidate rem is removed\n  still_wins (c rem : \u03c7) : Prop :=\n    if rem_prop : rem \u2208 cands \n      then c \u2208 stable_voting' (n+1) (cands.erase rem)\n          (by { rw [finset.card_erase_of_mem, hn]; simp [rem_prop], })\n    else false,\n  viable : finset (\u03c7 \u00d7 \u03c7) := (cands.product cands).filter \n      (\u03bb p, still_wins p.1 p.2 \u2227 \u00ac defeats voters cands Q p.2 p.1)\nin finset.image prod.fst $ viable.filter (\u03bb p, (margin voters Q p.1 p.2 = best_margin voters viable Q))\n\nnoncomputable def stable_voting : election_profile \u03c7 \u03c5 \u2192 finset \u03c7 := \u03bb prof,\nstable_voting' prof.voters prof.Q prof.cands.card prof.cands rfl\n\nlemma sv_empty (prof : election_profile \u03c7 \u03c5) (hcands : prof.cands.card = 0) :\n   stable_voting prof = prof.cands :=\n  by simp only [stable_voting, stable_voting', hcands]\n\nlemma sv_singleton (prof : election_profile \u03c7 \u03c5) (hcands : prof.cands.card = 1) : \n  stable_voting prof = prof.cands :=\nby simp only [stable_voting, stable_voting', hcands] \n\nlemma exists_best_margin {s : finset (\u03c7 \u00d7 \u03c7)} (voters : finset \u03c5) (Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop) \n  (hs : s.nonempty) :\n  \u2203 p : \u03c7 \u00d7 \u03c7, p \u2208 s \u2227 margin voters Q p.1 p.2 = best_margin voters s Q :=\nbegin\n  obtain \u27e8b, b_in, hb\u27e9 := finset.exists_mem_eq_sup' hs \n    (\u03bb (p : \u03c7 \u00d7 \u03c7), margin voters Q p.fst p.snd),\n  unfold best_margin,\n  simp only [hs, dif_pos],\n  exact \u27e8b, b_in, hb.symm\u27e9, \nend \n\nlemma best_margin_pos_of_exists_pos  {s : finset (\u03c7 \u00d7 \u03c7)} {voters : finset \u03c5} {Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop}\n  (p : \u03c7 \u00d7 \u03c7) (p_in : p \u2208 s) (h : margin_pos voters Q p.1 p.2) : \n  0 < best_margin voters s Q :=\nbegin\n  have s_nonempty : s.nonempty := \u27e8p, p_in\u27e9,\n  simp only [best_margin, dif_pos, s_nonempty, finset.lt_sup'_iff],\n  exact \u27e8p, p_in, h\u27e9,\nend\n\nsection trans_gen\n\nlemma trans_gen_of_imp {Q R : \u03c7 \u2192 \u03c7 \u2192 Prop} \n  (hyp : \u2200 a b, Q a b \u2192 R a b) : \n  \u2200 {a b}, relation.trans_gen Q a b \u2192 relation.trans_gen R a b := \nbegin\n  intros a b hab,\n  refine relation.trans_gen.trans_induction_on hab \n    (\u03bb x y hxy, relation.trans_gen.single $ hyp x y hxy) _,\n  intros x y z hxy hyz hxy' hyz',\n  exact relation.transitive_trans_gen hxy' hyz',\nend\n\nend trans_gen\n\nlemma cyclical_of_subset_cyclical {R : \u03c7 \u2192 \u03c7 \u2192 Prop} {s t : finset \u03c7} \n  (hst : s \u2286 t) (h_cyc : cyclical (restrict R s)) : cyclical (restrict R t) := \nbegin\n  rcases h_cyc with \u27e8x, hx\u27e9,\n  use x, refine (trans_gen_of_imp (\u03bb a b hab, _ ) hx),\n  exact restrict_of_subset hst hab,\nend\n\nlemma cyclical_of_serial' :\n  \u2200 (n : \u2115) (s : finset \u03c7) (R : \u03c7 \u2192 \u03c7 \u2192 Prop), n = s.card \u2192 s.nonempty \u2192\n    (\u2200 x \u2208 s, \u2203 y \u2208 s, R x y) \u2192 cyclical (restrict R s) :=\nbegin\n  intro a, refine nat.case_strong_induction_on a _ _,\n  { intros s h R hs, linarith [finset.card_pos.2 hs] },\n  { intros n IH s R s_card hs\u2081 hs\u2082,\n    obtain \u27e8x, x_in\u27e9 := hs\u2081,\n    obtain \u27e8y, y_in, hxy\u27e9 := hs\u2082 x x_in,\n    by_cases x_eq_y : x = y,\n    { use x,\n      apply relation.trans_gen.single, \n      rw \u2190 x_eq_y at hxy,\n      exact \u27e8hxy, \u27e8x_in, x_in\u27e9\u27e9, },\n    set t : finset \u03c7 := s.filter (\u03bb z, relation.trans_gen (restrict R s) y z)\n      with ht,\n    by_cases x_in' : x \u2208 t,\n    { use x, simp only [finset.mem_filter] at x_in',\n      exact relation.trans_gen.head \u27e8hxy,x_in,y_in\u27e9 x_in'.2, },\n    have t_nonempty : t.nonempty,\n    { obtain \u27e8z, z_in, hyz\u27e9 := hs\u2082 y y_in, use z,\n      simp only [finset.mem_filter],\n      exact \u27e8z_in, relation.trans_gen.single \u27e8hyz,y_in,z_in\u27e9\u27e9, },\n    have t_ss : t \u2282 s,\n    { rw finset.ssubset_iff_of_subset (finset.filter_subset _ s), \n      exact \u27e8x, x_in, x_in'\u27e9, },\n    have t_card : t.card \u2264 n := \n      by rw [\u2190 nat.lt_succ_iff, s_card]; exact finset.card_lt_card t_ss,\n    have := IH t.card t_card t (restrict R t) rfl t_nonempty,\n    rw restrict_restrict_eq_restrict R t at this,\n    refine cyclical_of_subset_cyclical \n      (by simp only [finset.filter_subset]) \n      (this _),\n    intros a a_in,\n    obtain \u27e8b, b_in, hab\u27e9 := hs\u2082 a (finset.filter_subset _ s a_in),\n    have b_in : b \u2208 t,\n    { simp only [finset.mem_filter] at \u22a2 a_in,\n      exact \u27e8b_in, relation.trans_gen.tail a_in.2 \u27e8hab,a_in.1,b_in\u27e9\u27e9, },\n    exact \u27e8b, b_in, \u27e8hab, a_in, b_in\u27e9\u27e9 }\nend\n\nlemma cyclical_of_serial {s : finset \u03c7} {R : \u03c7 \u2192 \u03c7 \u2192 Prop} \n  (hs\u2081 : s.nonempty) (hs\u2082 : \u2200 x \u2208 s, \u2203 y \u2208 s, R x y) :\n  cyclical (restrict R s) :=\ncyclical_of_serial' s.card s R rfl hs\u2081 hs\u2082\n\n/- Random lemmas for natural numbers -/\nlemma exists_eq_plus_two (n : \u2115) (hn\u2080 : \u00ac n = 0) (hn\u2081 : \u00ac n = 1) : \n  \u2203 m, n = m + 2 :=\nbegin\n  cases n with d, { exact (hn\u2080 nat.nat_zero_eq_zero).elim, },\n  induction d with d hd, {exact (hn\u2081 rfl).elim, },\n  use d,\nend\n\nlemma ge_two_of_ne_zero_ne_one {n : \u2115} (hn\u2080 : \u00ac n = 0) (hn\u2081 : \u00ac n = 1) :\n  2 \u2264 n :=\nbegin\n  cases n with d, { exact (hn\u2080 nat.nat_zero_eq_zero).elim, },\n  induction d with d hd, {exact (hn\u2081 rfl).elim, },\n  rw nat.succ_le_iff,\n  exact nat.one_lt_succ_succ d,  \nend\n\nlemma mem_cands_of_mem_sv {prof : election_profile \u03c7 \u03c5} {a : \u03c7}  \n  (a_in : a \u2208 stable_voting prof) : a \u2208 prof.cands := \nbegin\n  have card_ne_zero : prof.cands.card \u2260 0,\n  { by_contra h,\n    simp only [stable_voting, stable_voting', h] at a_in,\n    exact finset.card_ne_zero_of_mem a_in h, },\n  by_cases hcands : prof.cands.card = 1, \n  { rwa (sv_singleton prof hcands) at a_in },\n  obtain \u27e8n, hn\u27e9 := exists_eq_plus_two prof.cands.card \n    card_ne_zero hcands, \n  simp only [stable_voting, stable_voting', hn, exists_prop, \n    exists_and_distrib_right, exists_eq_right, finset.mem_image,\n    finset.mem_filter, finset.filter_congr_decidable, \n    prod.exists, finset.mem_product] at a_in,\n  rcases a_in with \u27e8b,\u27e8\u27e8p_in,hp\u27e9,ha\u27e9\u27e9,\n  exact p_in.1,\nend\n\nlemma false_iff_filter_empty {\u03b1 : Type* } (s : finset \u03b1) (p : \u03b1 \u2192 Prop) : \n  (\u2200 x \u2208 s, \u00ac p x) \u2194 s.filter p = \u2205 := \nbegin\n  refine \u27e8\u03bb h, finset.filter_false_of_mem h, \u03bb h x x_in, _\u27e9,\n  rw \u2190 finset.not_nonempty_iff_eq_empty at h,\n  by_contradiction hx,\n  refine h _,\n  use x,\n  simp only [finset.mem_filter],\n  exact \u27e8x_in, hx\u27e9,\nend\n\nlemma exists_sv_winner' :\n  \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5), prof.cands.card = n \u2192 \n  0 < prof.cands.card \u2192 \u2203 a, a \u2208 stable_voting prof :=\nbegin\n  intro n,\n  cases n with d, \n  { intros prof hn npos,\n    linarith [hn, npos], },\n  induction d with d IH,\n  { rintros prof hn cpos,\n    rw sv_singleton prof hn,\n    exact finset.card_pos.1 cpos, },\n  intros prof hm cpos,\n  set m := d.succ with m_succ,\n  obtain \u27e8b, b_in\u27e9 := finset.card_pos.1 cpos,\n  have h_erase_card : \u2200 z \u2208 prof.cands, finset.card (prof.cands.erase z) = m,\n  { intros z  z_in,\n    rw finset.card_erase_of_mem z_in,\n    exact nat.pred_eq_of_eq_succ hm, },\n  set prof' : election_profile \u03c7 \u03c5 := \n    \u27e8(prof.cands.erase b), \n     prof.voters, prof.Q\u27e9 with h_prof', \n  obtain \u27e8a, a_in\u27e9 := IH prof' (by rw \u2190 h_erase_card b b_in)\n    (by rw (h_erase_card b b_in); omega),\n  have card_eq_d : prof.cands.card = d + 2 := by rw hm,\n  simp only [stable_voting, stable_voting', card_eq_d, \n    exists_prop, exists_and_distrib_right, exists_eq_right, \n    finset.mem_image, finset.mem_filter, finset.filter_congr_decidable, \n    prod.exists, finset.mem_product],\n  let still_wins : \u03c7 \u2192 \u03c7 \u2192 Prop := \u03bb x\u2081 x\u2082,\n    if x\u2082_in : x\u2082 \u2208 prof.cands \n      then x\u2081 \u2208 stable_voting' prof.voters prof.Q d.succ \n          (prof.cands.erase x\u2082)\n          (by rwa h_erase_card x\u2082 x\u2082_in)\n    else false,\n  set viable_set : finset (\u03c7 \u00d7 \u03c7) := (prof.cands.product prof.cands).filter\n    (\u03bb p, still_wins p.1 p.2 \u2227 \u00ac defeats prof.voters prof.cands prof.Q p.2 p.1) with hvs,\n  have viable_nonempty : viable_set.nonempty,\n  { by_contra h,\n    have foo : (\u2200 (x : \u03c7 \u00d7 \u03c7),\n       x \u2208 prof.cands.product prof.cands \u2192\n       \u00ac(\u03bb (p : \u03c7 \u00d7 \u03c7), still_wins p.fst p.snd \u2227 \n          \u00acdefeats prof.voters prof.cands prof.Q p.snd p.fst) x),\n    { push_neg,\n      intros x x_in hx\u2081, by_contra hx\u2082,\n      apply h, use x,\n      simp only [hvs,finset.mem_filter],\n      exact \u27e8x_in, hx\u2081, hx\u2082\u27e9 },\n    simp only [and_imp, prod.forall, not_and, not_not, finset.mem_product] at foo,\n    have h_ser : \u2200 x \u2208 prof.cands, \u2203 y \u2208 prof.cands, \n      defeats prof.voters prof.cands prof.Q x y,\n    { intros x x_in,\n      set prof_rem_x : election_profile \u03c7 \u03c5 := \n      \u27e8(prof.cands.erase x), \n       prof.voters, prof.Q\u27e9 with h_prof_rem_x,\n      obtain \u27e8y, y_in\u27e9 := IH prof_rem_x (by rw \u2190 h_erase_card x x_in)\n        (by rw (h_erase_card x x_in); exact nat.succ_pos d),\n      have y_in' : y \u2208 prof.cands := \n        (finset.mem_erase.1 (mem_cands_of_mem_sv y_in)).2,\n      suffices : still_wins y x, { exact \u27e8y, y_in', foo y x y_in' x_in this\u27e9, },\n      simp only [still_wins, x_in, dif_pos],\n      convert y_in,\n      rwa h_erase_card x x_in, },\n    apply not_acyclical_in'_of_cyclical'_restrict (cyclical'_of_cyclical \n      (cyclical_of_serial (finset.card_pos.1 cpos) h_ser)),\n    exact defeat_acyclical_in' prof.voters prof.cands prof.Q, },\n  obtain \u27e8p, p_in, hp\u27e9 := exists_best_margin prof.voters prof.Q viable_nonempty,\n  refine \u27e8p.1,p.2, \u27e8_, (by rw hp)\u27e9\u27e9,\n  rw [hvs, finset.mem_filter] at p_in,\n  rcases p_in with \u27e8hp\u2081, hp\u2082, hp\u2083\u27e9,\n  refine \u27e8finset.mem_product.1 hp\u2081, \u27e8 _,hp\u2083\u27e9\u27e9,\n  simp only [(finset.mem_product.mp hp\u2081).right, dif_pos, \n    finset.mem_product, still_wins] at hp\u2082 \u22a2,\n  exact hp\u2082,\nend\n\nlemma exists_sv_winner (prof : election_profile \u03c7 \u03c5) (cpos : 0 < prof.cands.card) :\n  \u2203 a, a \u2208 stable_voting prof := exists_sv_winner' prof.cands.card prof rfl cpos\n\nlemma sv_winner_undefeated' :\n  \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5), prof.cands.card = n \u2192 \n  \u2200 a \u2208 stable_voting prof, is_undefeated prof.voters prof.cands prof.Q a :=\nbegin\n  intro n,\n  cases n with d, \n  { intros prof hn a a_in,\n    exact (finset.card_ne_zero_of_mem (mem_cands_of_mem_sv a_in) hn).elim, },\n  induction d with d IH,\n  { intros prof hn a a_in y y_in,\n    rw finset.card_le_one.1 (le_of_eq hn) \n      a (mem_cands_of_mem_sv a_in) y y_in,\n    exact defeat_irreflexive prof.voters prof.cands prof.Q y },\n  intros prof cands_card a a_in,\n  simp only [stable_voting, stable_voting', cands_card, exists_prop, \n    exists_and_distrib_right, exists_eq_right, finset.mem_image,\n    finset.mem_filter, finset.filter_congr_decidable, prod.exists, \n    finset.mem_product] at a_in, \n  rcases a_in with \u27e8x,\u27e8\u27e8p_in,hp\u27e9,ha\u27e9\u27e9,\n  have p_in' : (a,x).snd \u2208 prof.cands := by simp only; exact p_in.2,\n  rw dif_pos p_in' at hp,\n  have h_erase_card : (prof.cands.erase x).card = d.succ := \n    by simpa [finset.card_erase_of_mem p_in.2] using nat.pred_eq_of_eq_succ cands_card,\n  set prof' : election_profile \u03c7 \u03c5 := \n    \u27e8(prof.cands.erase x), \n     prof.voters, prof.Q\u27e9 with h_prof', \n  have a_in' : a \u2208 stable_voting prof',\n  { simp only [stable_voting], convert hp.1 },\n  refine undefeated_erase _ hp.2,\n  simpa using (IH prof' (by convert h_erase_card) a a_in'),\nend\n\nlemma sv_winner_undefeated {prof : election_profile \u03c7 \u03c5} :\n  \u2200 a \u2208 stable_voting prof, is_undefeated prof.voters prof.cands prof.Q a :=\nsv_winner_undefeated' prof.cands.card prof rfl\n\nlemma sv_winner_unique' :\n  \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5), prof.cands.card = n \u2192 \n  0 < prof.cands.card \u2192\n  uniquely_weighted prof.voters prof.cands prof.Q \u2192 \n  (stable_voting prof).card = 1 :=\nbegin\n  intro n,\n  cases n with d, \n  { intros prof hn cpos,\n    linarith [hn, cpos], },\n  induction d with d IH,\n  { rintros prof hn - -,\n    rwa sv_singleton prof hn, },\n  set m := d.succ with m_succ,\n  intros prof hm cpos h_uniq,\n  have card_eq_d : prof.cands.card = d + 2 := by rw hm,\n  by_contradiction h_card,\n  have hab : \u2203 a b \u2208 (stable_voting prof), a \u2260 b,\n  { obtain \u27e8a, a_in\u27e9 := exists_sv_winner prof cpos,\n    suffices : 2 \u2264 (stable_voting prof).card,\n    { obtain \u27e8b, b_in, hb\u27e9 := finset.exists_second_distinct_mem this a_in,\n      use \u27e8a, b, a_in, b_in, hb.symm\u27e9, },\n    exact ge_two_of_ne_zero_ne_one \n      (ne_of_gt (finset.card_pos.2 (exists_sv_winner prof cpos))) h_card, },\n  rcases hab with \u27e8a, b, a_in, b_in, a_neq_b\u27e9,\n  simp only [stable_voting, stable_voting', \n    card_eq_d, exists_prop, exists_and_distrib_right, exists_eq_right,\n    finset.mem_image, finset.mem_filter, finset.filter_congr_decidable, \n    prod.exists, finset.mem_product] at a_in b_in,\n  rcases a_in with \u27e8x,\u27e8\u27e8p_in,hp\u27e9,ha\u27e9\u27e9, rcases b_in with \u27e8y,\u27e8\u27e8q_in,hq\u27e9,hb\u27e9\u27e9,\n  rw \u2190 ha at hb,\n  have h_erase_card : \u2200 z \u2208 prof.cands, finset.card (prof.cands.erase z) = m,\n  { intros z  z_in,\n    rw finset.card_erase_of_mem z_in,\n    exact nat.pred_eq_of_eq_succ hm, },\n  suffices : b \u2260 y \u2227 a \u2260 x, \n  { exact (h_uniq b y a x q_in.1 q_in.2 p_in.1 p_in.2 this.1 this.2 \n      (or.inl a_neq_b.symm)) hb },\n  split,\n  { have y_in' : (b, y).snd \u2208 prof.cands := by simp only; exact q_in.2,\n    rw dif_pos y_in' at hq,\n    set prof' : election_profile \u03c7 \u03c5 := \n    \u27e8(prof.cands.erase y), \n     prof.voters, prof.Q\u27e9 with h_prof', \n    have b_in' : b \u2208 stable_voting prof':= \n      by simpa [stable_voting, h_erase_card y q_in.2] using hq.1,\n    exact (finset.mem_erase.1 (mem_cands_of_mem_sv b_in')).1, },\n  { have x_in' : (a, x).snd \u2208 prof.cands := by simp only; exact p_in.2,\n    rw dif_pos x_in' at hp,\n    set prof' : election_profile \u03c7 \u03c5 := \n    \u27e8(prof.cands.erase x), \n     prof.voters, prof.Q\u27e9 with h_prof', \n    have a_in' : a \u2208 stable_voting prof':= \n      by simpa [stable_voting, h_erase_card x p_in.2] using hp.1,\n    exact (finset.mem_erase.1 (mem_cands_of_mem_sv a_in')).1, },  \nend\n\nlemma sv_winner_unique {prof : election_profile \u03c7 \u03c5} \n  (cpos: 0 < prof.cands.card)\n  (h : uniquely_weighted prof.voters prof.cands prof.Q) :\n  (stable_voting prof).card = 1 :=\nsv_winner_unique' prof.cands.card prof rfl cpos h\n\nnoncomputable def stable_voting_alt' (voters : finset \u03c5) (Q : \u03c5 \u2192 \u03c7 \u2192 \u03c7 \u2192 Prop) :\n  \u03a0 (n : \u2115) (cands : finset \u03c7) (hn : cands.card = n), finset \u03c7\n| 0 cands _ := cands\n| 1 cands _  := cands\n| (n+2) cands hn :=\nlet\n  -- whether c wins when candidate rem is removed\n  still_wins (c rem : \u03c7) : Prop :=\n    if rem_prop : rem \u2208 cands \n      then c \u2208 stable_voting_alt' (n+1) (cands.erase rem)\n          (by { rw [finset.card_erase_of_mem, hn]; simp [rem_prop], })\n    else false,\n  viable : finset (\u03c7 \u00d7 \u03c7) := (cands.product cands).filter \n      (\u03bb p, still_wins p.1 p.2 \u2227 is_undefeated voters cands Q p.1)\nin finset.image prod.fst $ viable.filter (\u03bb p, (margin voters Q p.1 p.2 = best_margin voters viable Q))\n\nnoncomputable def stable_voting_alt (prof : election_profile \u03c7 \u03c5)\n  [decidable_eq \u03c7] [\u2200 v, decidable_rel (prof.Q v)] : finset \u03c7 :=\nstable_voting_alt' prof.voters prof.Q prof.cands.card prof.cands rfl \n\nlemma sv_alt_empty  (prof : election_profile \u03c7 \u03c5) (hcands : prof.cands.card = 0) :\n   stable_voting_alt prof = prof.cands :=\n  by simp only [stable_voting_alt, stable_voting_alt', hcands]\n\nlemma sv_alt_singleton (prof : election_profile \u03c7 \u03c5) (hcands : prof.cands.card = 1) : \n  stable_voting_alt prof = prof.cands :=\nby simp only [stable_voting_alt, stable_voting_alt', hcands]\n\nlemma stable_voting_eq_stable_voting_alt' :\n  \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5), prof.cands.card = n \u2192\n  stable_voting prof = stable_voting_alt prof := \nbegin\n  intro n,\n  cases n with d, \n  { intros prof hn,\n     rwa [sv_empty prof hn, sv_alt_empty prof hn], },\n  induction d with d IH,\n  { intros prof hn,\n    rwa [sv_singleton prof hn, sv_alt_singleton prof hn], },\n  set m := d.succ with m_succ,\n  intros prof hm,\n  have card_eq_d : prof.cands.card = d + 2 := by rw hm,\n  have h_erase_card : \u2200 z \u2208 prof.cands, finset.card (prof.cands.erase z) = m,\n  { intros z  z_in,\n    rw finset.card_erase_of_mem z_in,\n    exact nat.pred_eq_of_eq_succ hm, },\n  ext a, split,\n  { intro a_in,\n    have a_undefeated := sv_winner_undefeated a a_in,\n    simp only [stable_voting, stable_voting', stable_voting_alt, \n      stable_voting_alt', card_eq_d, finset.filter_congr_decidable,\n      exists_prop, exists_and_distrib_right, exists_eq_right, \n      finset.mem_image, finset.mem_filter, prod.exists, finset.mem_product] at \u22a2 a_in,\n    rcases a_in with \u27e8x,\u27e8\u27e8p_in,hp\u27e9,ha\u27e9\u27e9,\n    refine \u27e8x,\u27e8\u27e8p_in,\u27e8_,a_undefeated\u27e9\u27e9,_\u27e9\u27e9,\n    { have x_in' : (a, x).snd \u2208 prof.cands := by simp only; exact p_in.2,\n      simp only [x_in', dif_pos] at \u22a2 hp,\n      set prof' : election_profile \u03c7 \u03c5 := \n        \u27e8(prof.cands.erase x), \n         prof.voters, prof.Q\u27e9 with h_prof', \n      suffices : a \u2208 stable_voting_alt prof',\n      { simp only [stable_voting_alt, h_erase_card x p_in.right] at this,\n        exact this, },\n      have prof'_card : prof'.cands.card = m \n        := by simp only [h_erase_card x p_in.2],\n      simpa only [\u2190 IH prof' prof'_card, \n        stable_voting, h_erase_card x p_in.2] using hp.1, },\n    { convert ha, ext p,\n      simp only [eq_self_iff_true] at *,\n      sorry, }, },\n  { sorry, },\nend \n\nclass voting_method {\u03c7 \u03c5 : Type*} (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) : Prop :=\n(winners_subset : \u2200 prof : election_profile \u03c7 \u03c5, M prof \u2286 prof.cands)\n(winners_nonempty : \u2200 prof : election_profile \u03c7 \u03c5, prof.cands.nonempty \u2192 (M prof).nonempty)\n\nexport voting_method (winners_subset winners_nonempty)\n\nlemma voting_method_singleton {prof : election_profile \u03c7 \u03c5} {a : \u03c7} \n  (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) [voting_method M] (h : prof.cands = {a}) : \n  M prof = {a} :=\nbegin\n  have : M prof \u2286 prof.cands := voting_method.winners_subset prof,\n  rw [h, finset.subset_singleton_iff] at this,\n  cases this,\n  { exfalso, refine (voting_method.winners_nonempty prof _).ne_empty this,\n    simpa only [h] using finset.singleton_nonempty a, },\n  { exact this, },\nend\n\ninstance : voting_method (stable_voting : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) := \n\u27e8\u03bb prof a a_in, mem_cands_of_mem_sv a_in, \n \u03bb prof h, exists_sv_winner prof $ finset.card_pos.2 h\u27e9\n\ndef is_stable (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) \n  (prof : election_profile \u03c7 \u03c5) [\u2200 v, decidable_rel (prof.Q v)] (a: \u03c7) : Prop :=\n\u2203 b \u2208 prof.cands, margin_pos prof.voters prof.Q a b \u2227 a \u2208 M (profile_without prof b)\n\ndef is_stable_for_winners_wt (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) : Prop :=  \n\u2200 (prof: election_profile \u03c7 \u03c5),\n(\u2203 x, is_stable M prof x) \u2192 \u2200 a \u2208 (M prof), is_stable M prof a\n\ntheorem sv_stable_for_winners_wt : \n  is_stable_for_winners_wt (stable_voting : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) := \nbegin\n  rintros prof \u27e8x, \u27e8y, y_in, hy\u2081, hy\u2082\u27e9\u27e9 a a_in',\n  have x_in : x \u2208 prof.cands := \n    mem_prof_of_mem_profile_without (mem_cands_of_mem_sv hy\u2082), \n  obtain \u27e8d, hd\u27e9 : \u2203 d : \u2115, prof.cands.card = d + 2,\n  { use (profile_without prof y).cands.card.pred,\n    rw [profile_without_card' prof y_in, \u2190 nat.pred_eq_succ_iff, nat.pred_succ],\n    refine (nat.succ_pred_eq_of_pos _).symm,\n    rw finset.card_pos,\n    exact \u27e8x, mem_cands_of_mem_sv hy\u2082\u27e9, },\n  have h_erase_card : \u2200 z \u2208 prof.cands, finset.card (prof.cands.erase z) = d.succ,\n  { intros z  z_in,\n    rw finset.card_erase_of_mem z_in,\n    exact nat.pred_eq_of_eq_succ hd, },\n  let still_wins : \u03c7 \u2192 \u03c7 \u2192 Prop := \u03bb x\u2081 x\u2082,\n    if x\u2082_in : x\u2082 \u2208 prof.cands \n      then x\u2081 \u2208 stable_voting' prof.voters prof.Q d.succ \n          (prof.cands.erase x\u2082)\n          (by rwa h_erase_card x\u2082 x\u2082_in)\n    else false,\n  set viable_set : finset (\u03c7 \u00d7 \u03c7) := (prof.cands.product prof.cands).filter\n    (\u03bb p, still_wins p.1 p.2 \u2227 \u00ac defeats prof.voters prof.cands prof.Q p.2 p.1) with hvs,\n  simp only [stable_voting, stable_voting', hd, exists_prop,\n    exists_and_distrib_right, exists_eq_right, finset.mem_image,\n    finset.mem_filter, finset.filter_congr_decidable, \n    prod.exists, finset.mem_product] at a_in',\n  rcases a_in' with \u27e8b,\u27e8\u27e8\u27e8a_in,b_in\u27e9,hp\u27e9,hb\u27e9\u27e9,\n  refine \u27e8b, b_in, \u27e8_,_\u27e9\u27e9,\n  { unfold margin_pos, rw hb,\n    refine best_margin_pos_of_exists_pos (x,y) _ (by convert hy\u2081),\n    have xy_in : (x,y).snd \u2208 prof.cands := by simpa,\n    simp only [finset.mem_filter, finset.mem_product, \n      dif_pos, xy_in, and_true, stable_voting, profile_without] at \u22a2 hy\u2082,\n    refine \u27e8x_in, _, not_defeat_of_margin_pos prof.cands hy\u2081,\u27e9,\n    convert hy\u2082, \n    exact (h_erase_card y y_in).symm, },\n  { suffices : (a,b).snd \u2208 prof.cands, \n    { simp only [dif_pos, this, stable_voting] at hp \u22a2,\n      convert hp.1 using 2,\n      rw [cands_erase_eq_profile_without prof b, h_erase_card b b_in], },\n    simpa, },\nend\n\ndef condorcet_winner (prof : election_profile \u03c7 \u03c5) (x : \u03c7) : Prop := \nx \u2208 prof.cands \u2227 \u2200 y \u2208 prof.cands, y \u2260 x \u2192 margin_pos prof.voters prof.Q x y\n\ndef condorcet_criterion (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) : Prop := \n  \u2200 (prof : election_profile \u03c7 \u03c5) (x : \u03c7), \n    condorcet_winner prof x \u2192 M prof = {x}\n\nlemma condorcet_winner_profile_without {prof : election_profile \u03c7 \u03c5} {x b : \u03c7} \n  (h : condorcet_winner prof x) (hb : x \u2260 b): \n  condorcet_winner (profile_without prof b) x := \n\u27e8finset.mem_erase_of_ne_of_mem hb h.1, \u03bb y y_in hxy, h.2 y (finset.mem_of_mem_erase y_in) hxy\u27e9\n\nlemma not_margin_pos_of_condorcet_winner {prof : election_profile \u03c7 \u03c5} {x b: \u03c7} \n  (h : condorcet_winner prof x) (b_in : b \u2208 prof.cands) : \u00ac margin_pos prof.voters prof.Q b x :=\nbegin\n  by_cases hb : b = x, \n  { rw hb, exact not_margin_pos_self prof.voters prof.Q x },\n  { exact not_margin_pos_of_reverse (h.2 b b_in hb) }\nend\n\ntheorem condorcet_of_stability_for_winners_wt (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) \n  [hM : voting_method M] : is_stable_for_winners_wt M \u2192 condorcet_criterion M := \nbegin\n  intros h_stable p y hy,\n  suffices : \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5) (x : \u03c7), prof.cands.card = n \u2192\n    condorcet_winner prof x \u2192 M prof = {x},\n  { exact this p.cands.card p y rfl hy, }, \n  intro n, induction n with n IH,\n  { intros prof x h_card h_cond, exfalso,\n    exact (finset.card_ne_zero_of_mem h_cond.1) h_card, },\n  intros prof x h_card h_cond,\n  by_cases h_cands : prof.cands = {x}, { exact voting_method_singleton M h_cands, },\n  have hx\u2081 : \u2200 a \u2208 prof.cands, x \u2260 a \u2192 M (profile_without prof a) = {x},\n  { intros a a_in hax, \n    refine IH (profile_without prof a) x _ (condorcet_winner_profile_without h_cond hax),\n    rw \u2190 profile_without_card prof a_in,\n    exact nat.pred_eq_of_eq_succ h_card, },\n  have hx\u2082 : \u2200 a \u2208 prof.cands, is_stable M prof a \u2192 a = x,\n  { rintros a a_in \u27e8c, c_in, hc\u2081, hc\u2082\u27e9,\n    have x_ne_c : x \u2260 c,\n    { by_contra h, rw \u2190 h at hc\u2081,\n      exact (not_margin_pos_of_condorcet_winner h_cond a_in) hc\u2081 },\n    rwa [\u2190 finset.mem_singleton, \u2190 hx\u2081 c c_in x_ne_c] },\n  have hx\u2083 : is_stable M prof x,\n  { obtain \u27e8b, b_in, hb\u27e9 := \n      finset.exists_distinct_mem_of_ne_singleton \u27e8x, h_cond.1\u27e9 h_cands,\n    refine \u27e8b,b_in, h_cond.2 b b_in hb, _\u27e9,\n    simp only [hx\u2081 b b_in hb.symm, finset.mem_singleton], },\n  have hx\u2084 : \u2200 a \u2208 M prof, a = x,\n  { intros c c_in,\n    exact hx\u2082 c (voting_method.winners_subset prof c_in) (h_stable prof \u27e8x, hx\u2083\u27e9 c c_in), },\n  rw finset.eq_singleton_iff_unique_mem,\n  refine \u27e8_, hx\u2084\u27e9, \n  obtain \u27e8z, hz\u27e9 : (M prof).nonempty := \n    voting_method.winners_nonempty prof (finset.card_pos.1 (by omega)),\n  rwa \u2190 hx\u2084 z hz,  \nend\n\ntheorem sv_condorcet_criterion : \n  condorcet_criterion (stable_voting : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) :=\ncondorcet_of_stability_for_winners_wt stable_voting sv_stable_for_winners_wt \n\nnoncomputable def smith_set (prof: election_profile \u03c7 \u03c5) : finset \u03c7 := \n{ x \u2208 prof.cands | \u2200 y \u2208 prof.cands,\n   relation.trans_gen (margin_nonneg prof.voters prof.Q) x y }\n\ndef smith_criterion (M : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) : Prop := \n\u2200 (prof : election_profile \u03c7 \u03c5) (x : \u03c7), x \u2208 M prof \u2192 x \u2208 smith_set prof\n\nlemma smith_set_subset (prof : election_profile \u03c7 \u03c5) : smith_set prof \u2286 prof.cands :=\nbegin\n  intros x x_in,\n  simp only [smith_set, finset.sep_def, finset.mem_filter] at x_in,\n  exact x_in.1,\nend\n\nlemma smith_set_singleton {prof : election_profile \u03c7 \u03c5} {a : \u03c7} (h : prof.cands = {a}) : \n  smith_set prof = {a} := \nbegin\n  ext x, split,\n  { intro x_in,\n    simp only [smith_set, finset.sep_def, finset.mem_filter] at x_in,\n    rw h at x_in,\n    exact x_in.1, },\n  { simp only [smith_set, finset.sep_def, finset.mem_singleton, finset.mem_filter],\n    intro hx, split,\n    { rw [h,hx],\n      exact finset.mem_singleton_self a, },\n    { intros y y_in,\n      apply relation.trans_gen.single,\n      have hy : y = a := by rwa [h,finset.mem_singleton] at y_in,\n      rw [hx, hy],\n      exact margin_nonneg_self prof.voters prof.Q a} },\nend\n\nlemma smith_set_of_profile_without_not_condorcet {prof : election_profile \u03c7 \u03c5} {a b : \u03c7}\n  (ha : a \u2208 smith_set (profile_without prof b)) (hb : \u00ac condorcet_winner prof b) :\n  a \u2208 smith_set prof := \nbegin\n  by_cases b_in : b \u2209 prof.cands, { rwa profile_without_eq_cands_of_not_mem b_in at ha, }, \n  simp only [condorcet_winner, exists_prop, not_and, ne.def, not_forall, not_not] at b_in hb,\n  obtain \u27e8x, x_in, hx\u2081, hx\u2082\u27e9 := hb b_in,\n  rw \u2190 margin_nonneg_iff_not_margin_pos prof.voters prof.Q x at hx\u2082,\n  simp only [smith_set, finset.sep_def, finset.mem_filter, \n    voters_eq_profile_without prof b, Q_eq_profile_without prof b] at ha \u22a2,\n  refine \u27e8finset.mem_of_mem_erase ha.1, _\u27e9,\n  intros y y_in,\n  by_cases hyb : y = b,\n  { rw \u2190 hyb at hx\u2082,\n    refine relation.trans_gen.tail (ha.2 x _) hx\u2082,\n    rw [cands_erase_eq_profile_without prof b, finset.mem_erase],\n    exact \u27e8hx\u2081, x_in\u27e9, },\n  { apply ha.2 y,\n    rw [cands_erase_eq_profile_without prof b, finset.mem_erase],\n    exact \u27e8hyb, y_in\u27e9, },\nend\n\ntheorem sv_smith_criterion : \n  smith_criterion (stable_voting : election_profile \u03c7 \u03c5 \u2192 finset \u03c7) :=\nbegin\n  intros p z hz,\n  suffices : \u2200 (n : \u2115) (prof : election_profile \u03c7 \u03c5) (x : \u03c7), prof.cands.card = n \u2192\n    x \u2208 stable_voting prof \u2192 x \u2208 smith_set prof,\n  { exact this p.cands.card p z rfl hz },\n  intro n,\n  cases n with d, \n  { intros prof x h_card x_in', exfalso,\n    exact (finset.nonempty.ne_empty \u27e8x,mem_cands_of_mem_sv x_in'\u27e9)\n      (finset.card_eq_zero.1 h_card), },\n  induction d with d IH,\n  { intros prof x h_card x_in',\n    obtain \u27e8a, ha\u27e9 := finset.card_eq_one.1 h_card,\n    rw [sv_singleton prof h_card, ha] at x_in', \n    rwa smith_set_singleton ha, },\n  set m := d.succ with m_succ,\n  intros prof x hm x_in,\n  have card_eq_d : prof.cands.card = d + 2 := by rw hm,\n  have h_erase_card : \u2200 z \u2208 prof.cands, finset.card (prof.cands.erase z) = m,\n  { intros z  z_in,\n    rw finset.card_erase_of_mem z_in,\n    exact nat.pred_eq_of_eq_succ hm, },\n  let x_in' := x_in,\n  simp only [stable_voting, stable_voting', card_eq_d, exists_prop, \n    exists_and_distrib_right, exists_eq_right, finset.mem_image,\n    finset.mem_filter, finset.filter_congr_decidable, prod.exists, finset.mem_product] at x_in,\n  rcases x_in with \u27e8b,\u27e8\u27e8\u27e8x_in,b_in\u27e9,hp\u27e9,hx\u27e9\u27e9,\n  have xb_in : (x,b).snd \u2208 prof.cands := by simpa,\n  have x_prof_without : x \u2208 stable_voting (profile_without prof b),\n  { simp only [stable_voting, profile_without, h_erase_card b b_in, dif_pos, xb_in] at hp \u22a2,\n    exact hp.1, },\n  refine smith_set_of_profile_without_not_condorcet \n    (IH (profile_without prof b) x _ x_prof_without) _,\n  { rw [cands_erase_eq_profile_without prof b, h_erase_card b b_in], },\n  { by_contra hb,\n    rw [sv_condorcet_criterion prof b hb, finset.mem_singleton] at x_in',\n    exact (ne_of_mem_profile_without (mem_cands_of_mem_sv x_prof_without)) x_in', },\nend", "meta": {"author": "asouther4", "repo": "lean-social-choice", "sha": "9906ade382ace77af4fef1edb70364b84f7afd9c", "save_path": "github-repos/lean/asouther4-lean-social-choice", "path": "github-repos/lean/asouther4-lean-social-choice/lean-social-choice-9906ade382ace77af4fef1edb70364b84f7afd9c/src/stable_voting.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3724036023959944}}
{"text": "import Logic.Predicate.FirstOrder.Calculus\nimport Logic.Predicate.Meta\nopen Qq Lean Elab Meta Tactic\n\nuniverse u v\n\nnamespace FirstOrder\n\nnamespace SubFormula\n\nnamespace Meta\n\nsection lemmata\nvariable {L L' : Language} {\u03bc \u03bc' : Type v} {n n' : \u2115} {f : SubFormula L \u03bc n \u2192L SubFormula L' \u03bc' n'}\n\nlemma hom_and_eq_of_eq {p q : SubFormula L \u03bc n} {p' q'} (hp : f p = p') (hq : f q = q') :\n    f (p \u22cf q) = p' \u22cf q' := by simp[\u2190hp, \u2190hq]\n\nlemma hom_or_eq_of_eq {p q : SubFormula L \u03bc n} {p' q'} (hp : f p = p') (hq : f q = q') :\n    f (p \u22ce q) = p' \u22ce q' := by simp[\u2190hp, \u2190hq]\n\nlemma hom_neg_eq_of_eq {p : SubFormula L \u03bc n} {p'} (h : f p = p') :\n    f (~p) = ~p' := by simp[\u2190h]\n\nlemma free_rel\u2080 (r : L.rel 0) :\n    free (rel r ![] : SyntacticSubFormula L (n + 1)) = rel r ![] := by simp[free_rel]\n\nlemma free_rel\u2081 (r : L.rel 1) {t : SyntacticSubTerm L (n + 1)} {t'} (h : t.free = t') :\n    free (rel r ![t]) = rel r ![t'] := by\n  simp[\u2190h, free_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma free_rel\u2082 (r : L.rel 2) {t\u2081 t\u2082 : SyntacticSubTerm L (n + 1)} {t\u2081' t\u2082'}\n  (h\u2081 : t\u2081.free = t\u2081') (h\u2082 : t\u2082.free = t\u2082') :\n    free (rel r ![t\u2081, t\u2082]) = rel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, free_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma free_nrel\u2080 (r : L.rel 0) :\n    free (nrel r ![] : SyntacticSubFormula L (n + 1)) = nrel r ![] := by simp[free_nrel]\n\nlemma free_nrel\u2081 (r : L.rel 1) {t : SyntacticSubTerm L (n + 1)} {t'} (h : t.free = t') :\n    free (nrel r ![t]) = nrel r ![t'] := by\n  simp[\u2190h, free_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma free_nrel\u2082 (r : L.rel 2) {t\u2081 t\u2082 : SyntacticSubTerm L (n + 1)} {t\u2081' t\u2082'}\n  (h\u2081 : t\u2081.free = t\u2081') (h\u2082 : t\u2082.free = t\u2082') :\n    free (nrel r ![t\u2081, t\u2082]) = nrel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, free_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma free_all_eq_of_eq {p : SyntacticSubFormula L (n + 1 + 1)} {p'} (h : free p = p') :\n    free (\u2200' p) = \u2200' p' := by simp[\u2190h]\n\nlemma free_ex_eq_of_eq {p : SyntacticSubFormula L (n + 1 + 1)} {p'} (h : free p = p') :\n    free (\u2203' p) = \u2203' p' := by simp[\u2190h]\n\nlemma subst_rel\u2080 {s : SubTerm L \u03bc n} (r : L.rel 0) :\n    subst s (rel r ![] : SubFormula L \u03bc (n + 1)) = rel r ![] := by simp[subst_rel]\n\nlemma subst_rel\u2081 {s : SubTerm L \u03bc n} (r : L.rel 1) {t : SubTerm L \u03bc (n + 1)} {t'} (h : SubTerm.subst s t = t') :\n    subst s (rel r ![t]) = rel r ![t'] := by\n  simp[\u2190h, subst_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma subst_rel\u2082 {s : SubTerm L \u03bc n} (r : L.rel 2) {t\u2081 t\u2082 : SubTerm L \u03bc (n + 1)} {t\u2081' t\u2082'}\n  (h\u2081 : SubTerm.subst s t\u2081 = t\u2081') (h\u2082 : SubTerm.subst s t\u2082 = t\u2082') :\n    subst s (rel r ![t\u2081, t\u2082]) = rel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, subst_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma subst_nrel\u2080 {s : SubTerm L \u03bc n} (r : L.rel 0) :\n    subst s (nrel r ![] : SubFormula L \u03bc (n + 1)) = nrel r ![] := by simp[subst_nrel]\n\nlemma subst_nrel\u2081 {s : SubTerm L \u03bc n} (r : L.rel 1) {t : SubTerm L \u03bc (n + 1)} {t'} (h : SubTerm.subst s t = t') :\n    subst s (nrel r ![t]) = nrel r ![t'] := by\n  simp[\u2190h, subst_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma subst_nrel\u2082 {s : SubTerm L \u03bc n} (r : L.rel 2) {t\u2081 t\u2082 : SubTerm L \u03bc (n + 1)} {t\u2081' t\u2082'}\n  (h\u2081 : SubTerm.subst s t\u2081 = t\u2081') (h\u2082 : SubTerm.subst s t\u2082 = t\u2082') :\n    subst s (nrel r ![t\u2081, t\u2082]) = nrel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, subst_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma subst_all_eq_of_eq {s : SubTerm L \u03bc n} {s'} {p : SubFormula L \u03bc (n + 1 + 1)} {p'}\n  (hs : SubTerm.bShift s = s') (hp : subst s' p = p') :\n    subst s (\u2200' p) = \u2200' p' := by simp[\u2190hs, \u2190hp]\n\nlemma subst_ex_eq_of_eq {s : SubTerm L \u03bc n} {s'} {p : SubFormula L \u03bc (n + 1 + 1)} {p'}\n  (hs : SubTerm.bShift s = s') (hp : subst s' p = p') :\n    subst s (\u2203' p) = \u2203' p' := by simp[\u2190hs, \u2190hp]\n\nlemma shift_rel\u2080 (r : L.rel 0) :\n    shift (rel r ![] : SyntacticSubFormula L n) = rel r ![] := by simp[shift_rel]\n\nlemma shift_rel\u2081 (r : L.rel 1) {t : SyntacticSubTerm L n} {t'} (h : t.shift = t') :\n    shift (rel r ![t]) = rel r ![t'] := by\n  simp[\u2190h, shift_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma shift_rel\u2082 (r : L.rel 2) {t\u2081 t\u2082 : SyntacticSubTerm L n} {t\u2081' t\u2082'}\n  (h\u2081 : t\u2081.shift = t\u2081') (h\u2082 : t\u2082.shift = t\u2082') :\n    shift (rel r ![t\u2081, t\u2082]) = rel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, shift_rel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma shift_nrel\u2080 (r : L.rel 0) :\n    shift (nrel r ![] : SyntacticSubFormula L n) = nrel r ![] := by simp[shift_nrel]\n\nlemma shift_nrel\u2081 (r : L.rel 1) {t : SyntacticSubTerm L n} {t'} (h : t.shift = t') :\n    shift (nrel r ![t]) = nrel r ![t'] := by\n  simp[\u2190h, shift_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma shift_nrel\u2082 (r : L.rel 2) {t\u2081 t\u2082 : SyntacticSubTerm L n} {t\u2081' t\u2082'}\n  (h\u2081 : t\u2081.shift = t\u2081') (h\u2082 : t\u2082.shift = t\u2082') :\n    shift (nrel r ![t\u2081, t\u2082]) = nrel r ![t\u2081', t\u2082'] := by\n  simp[\u2190h\u2081, \u2190h\u2082, shift_nrel]; funext i; cases' i using Fin.cases with i <;> simp\n\nlemma shift_all_eq_of_eq {p : SyntacticSubFormula L (n + 1)} {p'} (h : shift p = p') :\n    shift (\u2200' p) = \u2200' p' := by simp[\u2190h]\n\nlemma shift_ex_eq_of_eq {p : SyntacticSubFormula L (n + 1)} {p'} (h : shift p = p') :\n    shift (\u2203' p) = \u2203' p' := by simp[\u2190h]\n\nlemma shift_subst_eq_of_eq {s : SyntacticSubTerm L n} {p : SyntacticSubFormula L (n + 1)} {s' p'}\n  (hs : s.shift = s') (hp : shift p = p') :\n    shift (subst s p) = subst s' p' := by simp[\u2190hs, \u2190hp, shift_subst]\n\nlemma neg_and_eq_of_eq {p q : SubFormula L \u03bc n} {p' q'} (hp : ~p = p') (hq : ~q = q') :\n    ~(p \u22cf q) = p' \u22ce q' := by simp[\u2190hp, \u2190hq]\n\nlemma neg_or_eq_of_eq {p q : SubFormula L \u03bc n} {p' q'} (hp : ~p = p') (hq : ~q = q') :\n    ~(p \u22ce q) = p' \u22cf q' := by simp[\u2190hp, \u2190hq]\n\nlemma neg_rel {k} (r : L.rel k) (v : Fin k \u2192 SubTerm L \u03bc n) :\n    ~(rel r v) = nrel r v := rfl\n\nlemma neg_nrel {k} (r : L.rel k) (v : Fin k \u2192 SubTerm L \u03bc n) :\n    ~(nrel r v) = rel r v := rfl\n\nlemma neg_all_eq_of_eq {p : SyntacticSubFormula L (n + 1)} {p'} (h : ~p = p') :\n    ~(\u2200' p) = \u2203' p' := by simp[\u2190h]\n\nlemma neg_ex_eq_of_eq {p : SyntacticSubFormula L (n + 1)} {p'} (h : ~p = p') :\n    ~(\u2203' p) = \u2200' p' := by simp[\u2190h]\n\nlemma rel\u2081_congr {r : L.rel 1} {t t' : SubTerm L \u03bc n} (h : t = t') :\n    rel r ![t] = rel r ![t'] := congr_arg _ (by simp[h])\n\nlemma rel\u2082_congr {r : L.rel 2} {t\u2081 t\u2082 t\u2081' t\u2082' : SubTerm L \u03bc n} (h\u2081 : t\u2081 = t\u2081') (h\u2082 : t\u2082 = t\u2082') :\n    rel r ![t\u2081, t\u2082] = rel r ![t\u2081', t\u2082'] := congr_arg _ (by simp[h\u2081, h\u2082])\n\nlemma nrel\u2081_congr {r : L.rel 1} {t t' : SubTerm L \u03bc n} (h : t = t') :\n    nrel r ![t] = nrel r ![t'] := congr_arg _ (by simp[h])\n\nlemma nrel\u2082_congr {r : L.rel 2} {t\u2081 t\u2082 t\u2081' t\u2082' : SubTerm L \u03bc n} (h\u2081 : t\u2081 = t\u2081') (h\u2082 : t\u2082 = t\u2082') :\n    nrel r ![t\u2081, t\u2082] = nrel r ![t\u2081', t\u2082'] := congr_arg _ (by simp[h\u2081, h\u2082])\n\nlemma and_congr {p q p' q': SyntacticSubFormula L n} (hp : p = p') (hq : q = q') :\n    p \u22cf q = p' \u22cf q' := congr_arg\u2082 _ hp hq\n\nlemma or_congr {p q p' q': SyntacticSubFormula L n} (hp : p = p') (hq : q = q') :\n    p \u22ce q = p' \u22ce q' := congr_arg\u2082 _ hp hq\n\nlemma all_congr {p p' : SyntacticSubFormula L (n + 1)} (hp : p = p') :\n    \u2200' p = \u2200' p' := congr_arg _ hp\n\nlemma ex_congr {p p' : SyntacticSubFormula L (n + 1)} (hp : p = p') :\n    \u2203' p = \u2203' p' := congr_arg _ hp\n\nlemma neg_congr_eq {p p' : SubFormula L \u03bc n} {q} (e : p = p') (h : ~p' = q) :\n  ~p = q := Eq.trans (congr_arg _ e) h\n\nlemma imp_eq_of_eq {p q p' q' p'' : SubFormula L \u03bc n} (hp : p = p') (hq : q = q') (hp' : ~p' = p'') :\n    p \u27f6 q = p'' \u22ce q' := by simp[\u2190hp, \u2190hq, \u2190hp', imp_eq]\n\nlemma iff_eq_of_eq {p q p' q' p'' q'' : SubFormula L \u03bc n} (hp : p = p') (hq : q = q') (hp' : ~p' = p'') (hq' : ~q' = q'') :\n    (p \u27f7 q) = (p'' \u22ce q') \u22cf (q'' \u22ce p') := by simp[\u2190hp, \u2190hq, \u2190hp', \u2190hq', iff_eq]\n\nlemma free_congr_eq {p p' : SyntacticSubFormula L (n + 1)} {q} (e : p = p') (h : free p' = q) :\n  free p = q := Eq.trans (congr_arg _ e) h\n\nlemma subst_congr_eq {s s' : SubTerm L \u03bc n} {p p' q} (es : s = s') (ep : p = p') (h : subst s' p' = q) :\n  subst s p = q := Eq.trans (congr_arg\u2082 _ (by simp[es]) ep) h\n\nlemma shift_congr_eq {p p' : SyntacticSubFormula L n} {q} (e : p = p') (h : shift p' = q) :\n  shift p = q := Eq.trans (congr_arg _ e) h\n\nend lemmata\n\npartial def resultFree {L : Q(Language.{u})} {n : Q(\u2115)} : (p : Q(SyntacticSubFormula $L ($n + 1))) \u2192\n    MetaM ((res : Q(SyntacticSubFormula $L $n)) \u00d7 Q(free $p = $res))\n  | ~q(\u22a4)                   => pure \u27e8q(\u22a4), q(rfl)\u27e9\n  | ~q(\u22a5)                   => pure \u27e8q(\u22a5), q(rfl)\u27e9\n  | ~q($p \u22cf $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultFree p\n    let \u27e8qn, qe\u27e9 \u2190 resultFree q\n    return \u27e8q($pn \u22cf $qn), q(hom_and_eq_of_eq $pe $qe)\u27e9\n  | ~q($p \u22ce $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultFree p\n    let \u27e8qn, qe\u27e9 \u2190 resultFree q\n    return \u27e8q($pn \u22ce $qn), q(hom_or_eq_of_eq $pe $qe)\u27e9\n  | ~q(~$p)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultFree p\n    return \u27e8q(~$pn), q(hom_neg_eq_of_eq $pe)\u27e9\n  | ~q(\u2200' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultFree p\n    return \u27e8q(\u2200' $pn), q(free_all_eq_of_eq $e)\u27e9\n  | ~q(\u2203' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultFree p\n    return \u27e8q(\u2203' $pn), q(free_ex_eq_of_eq $e)\u27e9\n  | ~q(rel $r ![])          => pure \u27e8q(rel $r ![]), q(free_rel\u2080 $r)\u27e9\n  | ~q(rel $r ![$t])        => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\n    return \u27e8q(rel $r ![$tn]), q(free_rel\u2081 $r $e)\u27e9\n  | ~q(rel $r ![$t\u2081, $t\u2082])  => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\u2082\n    return \u27e8q(rel $r ![$tn\u2081, $tn\u2082]), q(free_rel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q(nrel $r ![])         => pure \u27e8q(nrel $r ![]), q(free_nrel\u2080 $r)\u27e9\n  | ~q(nrel $r ![$t])       => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\n    return \u27e8q(nrel $r ![$tn]), q(free_nrel\u2081 $r $e)\u27e9\n  | ~q(nrel $r ![$t\u2081, $t\u2082]) => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultFree (L := L) (n := n) t\u2082\n    return \u27e8q(nrel $r ![$tn\u2081, $tn\u2082]), q(free_nrel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q($p)                  => pure \u27e8q(free $p), q(rfl)\u27e9\n\npartial def resultSubst {L : Q(Language.{u})} {n : Q(\u2115)} (s : Q(SyntacticSubTerm $L $n)) :\n    (p : Q(SyntacticSubFormula $L ($n + 1))) \u2192 MetaM ((res : Q(SyntacticSubFormula $L $n)) \u00d7 Q(subst $s $p = $res))\n  | ~q(\u22a4)                   => pure \u27e8q(\u22a4), q(rfl)\u27e9\n  | ~q(\u22a5)                   => pure \u27e8q(\u22a5), q(rfl)\u27e9\n  | ~q($p \u22cf $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultSubst s p\n    let \u27e8qn, qe\u27e9 \u2190 resultSubst s q\n    return \u27e8q($pn \u22cf $qn), q(hom_and_eq_of_eq $pe $qe)\u27e9\n  | ~q($p \u22ce $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultSubst s p\n    let \u27e8qn, qe\u27e9 \u2190 resultSubst s q\n    return \u27e8q($pn \u22ce $qn), q(hom_or_eq_of_eq $pe $qe)\u27e9\n  | ~q(~$p)                 => do\n    let \u27e8pn, pe\u27e9 \u2190 resultSubst s p\n    return \u27e8q(~$pn), q(hom_neg_eq_of_eq $pe)\u27e9\n  | ~q(\u2200' $p)               => do\n    let \u27e8sn, se\u27e9 \u2190 SubTerm.Meta.resultBShift s\n    let \u27e8pn, pe\u27e9 \u2190 resultSubst sn p\n    return \u27e8q(\u2200' $pn), q(subst_all_eq_of_eq $se $pe)\u27e9\n  | ~q(\u2203' $p)               => do\n    let \u27e8sn, se\u27e9 \u2190 SubTerm.Meta.resultBShift s\n    let \u27e8pn, pe\u27e9 \u2190 resultSubst sn p\n    return \u27e8q(\u2203' $pn), q(subst_ex_eq_of_eq $se $pe)\u27e9\n  | ~q(rel $r ![])          => pure \u27e8q(rel $r ![]), q(subst_rel\u2080 $r)\u27e9\n  | ~q(rel $r ![$t])        => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\n    return \u27e8q(rel $r ![$tn]), q(subst_rel\u2081 $r $e)\u27e9\n  | ~q(rel $r ![$t\u2081, $t\u2082])  => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\u2082\n    return \u27e8q(rel $r ![$tn\u2081, $tn\u2082]), q(subst_rel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q(nrel $r ![])         => pure \u27e8q(nrel $r ![]), q(subst_nrel\u2080 $r)\u27e9\n  | ~q(nrel $r ![$t])       => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\n    return \u27e8q(nrel $r ![$tn]), q(subst_nrel\u2081 $r $e)\u27e9\n  | ~q(nrel $r ![$t\u2081, $t\u2082]) => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultSubst (L := L) (n := n) s t\u2082\n    return \u27e8q(nrel $r ![$tn\u2081, $tn\u2082]), q(subst_nrel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q($p)                  => pure \u27e8q(subst $s $p), q(rfl)\u27e9\n\npartial def resultShift {L : Q(Language.{u})} {n : Q(\u2115)} : (p : Q(SyntacticSubFormula $L $n)) \u2192\n    MetaM ((res : Q(SyntacticSubFormula $L $n)) \u00d7 Q(shift $p = $res))\n  | ~q(\u22a4)                   => pure \u27e8q(\u22a4), q(rfl)\u27e9\n  | ~q(\u22a5)                   => pure \u27e8q(\u22a5), q(rfl)\u27e9\n  | ~q($p \u22cf $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultShift p\n    let \u27e8qn, qe\u27e9 \u2190 resultShift q\n    return \u27e8q($pn \u22cf $qn), q(hom_and_eq_of_eq $pe $qe)\u27e9\n  | ~q($p \u22ce $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultShift p\n    let \u27e8qn, qe\u27e9 \u2190 resultShift q\n    return \u27e8q($pn \u22ce $qn), q(hom_or_eq_of_eq $pe $qe)\u27e9\n  | ~q(~$p)                 => do\n    let \u27e8pn, pe\u27e9 \u2190 resultShift p\n    return \u27e8q(~$pn), q(hom_neg_eq_of_eq $pe)\u27e9\n  | ~q(\u2200' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultShift p\n    return \u27e8q(\u2200' $pn), q(shift_all_eq_of_eq $e)\u27e9\n  | ~q(\u2203' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultShift p\n    return \u27e8q(\u2203' $pn), q(shift_ex_eq_of_eq $e)\u27e9\n  | ~q(rel $r ![])          => pure \u27e8q(rel $r ![]), q(shift_rel\u2080 $r)\u27e9\n  | ~q(rel $r ![$t])        => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\n    return \u27e8q(rel $r ![$tn]), q(shift_rel\u2081 $r $e)\u27e9\n  | ~q(rel $r ![$t\u2081, $t\u2082])  => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\u2082\n    return \u27e8q(rel $r ![$tn\u2081, $tn\u2082]), q(shift_rel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q(nrel $r ![])         => pure \u27e8q(nrel $r ![]), q(shift_nrel\u2080 $r)\u27e9\n  | ~q(nrel $r ![$t])       => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\n    return \u27e8q(nrel $r ![$tn]), q(shift_nrel\u2081 $r $e)\u27e9\n  | ~q(nrel $r ![$t\u2081, $t\u2082]) => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) t\u2082\n    return \u27e8q(nrel $r ![$tn\u2081, $tn\u2082]), q(shift_nrel\u2082 $r $e\u2081 $e\u2082)\u27e9\n  | ~q(subst $s $p)         => do\n    let \u27e8sn, se\u27e9 \u2190 SubTerm.Meta.resultShift (L := L) (n := n) s\n    let \u27e8pn, pe\u27e9 \u2190 resultShift (L := L) (n := q($n + 1)) p\n    return \u27e8q(subst $sn $pn), q(shift_subst_eq_of_eq $se $pe)\u27e9\n  | ~q($p)                  => pure \u27e8q(shift $p), q(rfl)\u27e9\n\npartial def resultNeg {L : Q(Language.{u})} {n : Q(\u2115)} : (p : Q(SyntacticSubFormula $L $n)) \u2192\n    MetaM ((res : Q(SyntacticSubFormula $L $n)) \u00d7 Q(~$p = $res))\n  | ~q(\u22a4)                   => pure \u27e8q(\u22a5), q(rfl)\u27e9\n  | ~q(\u22a5)                   => pure \u27e8q(\u22a4), q(rfl)\u27e9\n  | ~q($p \u22cf $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultNeg p\n    let \u27e8qn, qe\u27e9 \u2190 resultNeg q\n    return \u27e8q($pn \u22ce $qn), q(neg_and_eq_of_eq $pe $qe)\u27e9\n  | ~q($p \u22ce $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 resultNeg p\n    let \u27e8qn, qe\u27e9 \u2190 resultNeg q\n    return \u27e8q($pn \u22cf $qn), q(neg_or_eq_of_eq $pe $qe)\u27e9\n  | ~q(~$p)                 => do\n    return \u27e8q($p), q(neg_neg' $p)\u27e9\n  | ~q(\u2200' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultNeg p\n    return \u27e8q(\u2203' $pn), q(neg_all_eq_of_eq $e)\u27e9\n  | ~q(\u2203' $p)               => do\n    let \u27e8pn, e\u27e9 \u2190 resultNeg p\n    return \u27e8q(\u2200' $pn), q(neg_ex_eq_of_eq $e)\u27e9\n  | ~q(rel $r ![])          => pure \u27e8q(nrel $r ![]), q(neg_rel $r _)\u27e9\n  | ~q(rel $r ![$t])        => do\n    return \u27e8q(nrel $r ![$t]), q(neg_rel $r _)\u27e9\n  | ~q(rel $r ![$t\u2081, $t\u2082])  => do\n    return \u27e8q(nrel $r ![$t\u2081, $t\u2082]), q(neg_rel $r _)\u27e9\n  | ~q(nrel $r ![])         => pure \u27e8q(rel $r ![]), q(neg_nrel $r _)\u27e9\n  | ~q(nrel $r ![$t])       => do\n    return \u27e8q(rel $r ![$t]), q(neg_nrel $r _)\u27e9\n  | ~q(nrel $r ![$t\u2081, $t\u2082]) => do\n    return \u27e8q(rel $r ![$t\u2081, $t\u2082]), q(neg_nrel $r _)\u27e9\n  | ~q($p)                  => pure \u27e8q(~$p), q(rfl)\u27e9\n\npartial def result {L : Q(Language.{u})} {n : Q(\u2115)} : (p : Q(SyntacticSubFormula $L $n)) \u2192\n    MetaM ((res : Q(SyntacticSubFormula $L $n)) \u00d7 Q($p = $res))\n  | ~q(\u22a4)                   => pure \u27e8q(\u22a4), q(rfl)\u27e9\n  | ~q(\u22a5)                   => pure \u27e8q(\u22a5), q(rfl)\u27e9\n  | ~q($p \u22cf $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 result p\n    let \u27e8qn, qe\u27e9 \u2190 result q\n    return \u27e8q($pn \u22cf $qn), q(and_congr $pe $qe)\u27e9\n  | ~q(~$p)                 => do\n    let \u27e8pn, e\u27e9 \u2190 result p\n    let \u27e8pnn, ee\u27e9 \u2190 resultNeg pn\n    return \u27e8q($pnn), q(neg_congr_eq $e $ee)\u27e9\n  | ~q($p \u27f6 $q)            => do\n    let \u27e8pn, pe\u27e9 \u2190 result (L := L) (n := n) p\n    let \u27e8pnn, pee\u27e9 \u2190 resultNeg pn\n    let \u27e8qn, qe\u27e9 \u2190 result (L := L) (n := n) q\n    return \u27e8q($pnn \u22ce $qn), q(imp_eq_of_eq $pe $qe $pee)\u27e9\n  | ~q($p \u27f7 $q)            => do\n    let \u27e8pn, pe\u27e9 \u2190 result (L := L) (n := n) p\n    let \u27e8pnn, pee\u27e9 \u2190 resultNeg pn\n    let \u27e8qn, qe\u27e9 \u2190 result (L := L) (n := n) q\n    let \u27e8qnn, qee\u27e9 \u2190 resultNeg qn\n    return \u27e8q(($pnn \u22ce $qn) \u22cf ($qnn \u22ce $pn)), q(iff_eq_of_eq $pe $qe $pee $qee)\u27e9\n  | ~q($p \u22ce $q)             => do\n    let \u27e8pn, pe\u27e9 \u2190 result p\n    let \u27e8qn, qe\u27e9 \u2190 result q\n    return \u27e8q($pn \u22ce $qn), q(or_congr $pe $qe)\u27e9\n  | ~q(\u2200' $p)               => do\n    let \u27e8pn, pe\u27e9 \u2190 result p\n    return \u27e8q(\u2200' $pn), q(all_congr $pe)\u27e9\n  | ~q(\u2203' $p)               => do\n    let \u27e8pn, pe\u27e9 \u2190 result p\n    return \u27e8q(\u2203' $pn), q(ex_congr $pe)\u27e9\n  | ~q(rel $r ![])          => pure \u27e8q(rel $r ![]), q(rfl)\u27e9\n  | ~q(rel $r ![$t])        => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\n    return \u27e8q(rel $r ![$tn]), q(rel\u2081_congr $e)\u27e9\n  | ~q(rel $r ![$t\u2081, $t\u2082])  => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\u2082\n    return \u27e8q(rel $r ![$tn\u2081, $tn\u2082]), q(rel\u2082_congr $e\u2081 $e\u2082)\u27e9\n  | ~q(nrel $r ![])         => pure \u27e8q(nrel $r ![]), q(rfl)\u27e9\n  | ~q(nrel $r ![$t])       => do\n    let \u27e8tn, e\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\n    return \u27e8q(nrel $r ![$tn]), q(nrel\u2081_congr $e)\u27e9\n  | ~q(nrel $r ![$t\u2081, $t\u2082]) => do\n    let \u27e8tn\u2081, e\u2081\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\u2081\n    let \u27e8tn\u2082, e\u2082\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := n) t\u2082\n    return \u27e8q(nrel $r ![$tn\u2081, $tn\u2082]), q(nrel\u2082_congr $e\u2081 $e\u2082)\u27e9\n  | ~q(free $p)             => do\n    let \u27e8pn, e\u27e9 \u2190 result (L := L) (n := q($n + 1)) p\n    let \u27e8pnn, ee\u27e9 \u2190 resultFree (L := L) (n := n) pn\n    return \u27e8q($pnn), q(free_congr_eq $e $ee)\u27e9\n  | ~q(subst $s $p)         => do\n    let \u27e8sn, se\u27e9 \u2190 SubTerm.Meta.result (L := L) (n := q($n)) s\n    let \u27e8pn, pe\u27e9 \u2190 result (L := L) (n := q($n + 1)) p\n    let \u27e8n, e\u27e9 \u2190 resultSubst (L := L) (n := n) sn pn\n    return \u27e8q($n), q(subst_congr_eq $se $pe $e)\u27e9\n  | ~q(shift $p)            => do\n    let \u27e8pn, e\u27e9 \u2190 result (L := L) (n := n) p\n    let \u27e8pnn, ee\u27e9 \u2190 resultShift (L := L) (n := n) pn\n    return \u27e8q($pnn), q(shift_congr_eq $e $ee)\u27e9\n  | ~q($p)                  => do\n    -- logInfo m!\"match fail: {p}\"\n    return \u27e8q($p), q(rfl)\u27e9\n\npartial def result' {L : Q(Language.{u})} {n : Q(\u2115)} (p : Q(SyntacticSubFormula $L $n)) :\n    MetaM (Result (u := u) q(SyntacticSubFormula $L $n) p) := do\n    let \u27e8res, e\u27e9 \u2190 result p\n    return \u27e8res, e\u27e9\n\npartial def result\u2080 {L : Q(Language.{u})} (p : Q(SyntacticFormula $L)) :\n    MetaM ((res : Q(SyntacticFormula $L)) \u00d7 Q($p = $res)) :=\n  result (L := L) (n := q(0)) p\n\npartial def result\u2080_res {L : Q(Language.{u})} (p : Q(SyntacticFormula $L)) :\n    MetaM Q(SyntacticFormula $L) := do\n  let \u27e8res, _\u27e9 \u2190 result\u2080 (L := L) p\n  return res\n\npartial def result\u2080List {L : Q(Language.{u})} (l : List Q(SyntacticFormula $L)) :\n    MetaM $ (res : List Q(SyntacticFormula $L)) \u00d7 Q($(toQList (u := u) l) = $(toQList (u := u) res)) :=\n  resultList result\u2080 l\n\npartial def resultShift\u2080 {L : Q(Language.{u})} (p : Q(SyntacticFormula $L)) :\n    MetaM $ (res : Q(SyntacticFormula $L)) \u00d7 Q(shift $p = $res) :=\n  resultShift (L := L) (n := q(0)) p\n\npartial def resultShift\u2080List {L : Q(Language.{u})} (l : List Q(SyntacticFormula $L)) :\n    MetaM $ (res : List Q(SyntacticFormula $L)) \u00d7 Q(List.map shift $(toQList (u := u) l) = $(toQList (u := u) res)) :=\n  funResultList (u := u) (\u03b1 := q(SyntacticFormula $L)) q(shift) resultShift\u2080 l\n\npartial def resultSubst\u2080 {L : Q(Language.{u})} (p : Q(SyntacticSubFormula $L 1)) (s : Q(SyntacticTerm $L)) :\n    MetaM $ (res : Q(SyntacticFormula $L)) \u00d7 Q(subst $s $p = $res) :=\n  resultSubst (L := L) (n := q(0)) s p\n\npartial def resultSubst\u2080List {L : Q(Language.{u})} (v : List Q(SyntacticTerm $L)) (p : Q(SyntacticSubFormula $L 1)) :\n    MetaM $ (res : List Q(SyntacticFormula $L)) \u00d7 Q(List.map (SubFormula.subst \u00b7 $p) $(toQList (u := u) v) = $(toQList (u := u) res)) :=\n  funResultList (u := u) (\u03b1 := q(SyntacticTerm $L)) q((subst \u00b7 $p)) (resultSubst\u2080 p) v\n\nprivate inductive ResultTest (\u03b1 : Type u) : (a : \u03b1) \u2192 Type u\n  | result : (a b : \u03b1) \u2192 a = b \u2192 ResultTest \u03b1 a\n\nelab \"dbg\" : tactic => do\n  let goalType \u2190 Elab.Tactic.getMainTarget\n  let some \u27e8.succ u, ty\u27e9 \u2190 checkSortQ' goalType | throwError \"error: not a type\"\n  let ~q(ResultTest (SyntacticSubFormula $L $n) $p) := ty | throwError \"error: not a type\"\n  logInfo m!\"p = {p} : SyntacticSubFormula {L} {n}\"\n  let p : Q(SyntacticSubFormula $L $n) \u2190 withReducible <| whnf p\n\n  let \u27e8pn, e\u27e9 \u2190 result (L := L) (n := n) p\n  logInfo m!\"pn = {pn}\"\n  logInfo m!\"e = {e}\"\n  let c : Q(ResultTest (SyntacticSubFormula $L $n) $p) := (q(ResultTest.result ($p) $pn $e) : Expr)\n  Lean.Elab.Tactic.closeMainGoal c\n\nexample {t : SyntacticSubTerm Language.oring 14} : ResultTest (SyntacticSubFormula Language.oring 12)\n    $ shift \u201c\u22a4 \u2192 \u2200 !(shift $ subst &99 \u201c(!t) + (#6 * 8) < &7\u201d)\u201d :=\n  by dbg\n\nend Meta\n\nend SubFormula\n\nnamespace DerivationList\nopen Derivation\nvariable {L : Language.{u}} [\u2200 k, DecidableEq (L.func k)] [\u2200 k, DecidableEq (L.rel k)] {G : List (SyntacticFormula L)}\n\ndef congr {G G' : List (SyntacticFormula L)} (e : G = G')\n  (d : DerivationList G) : DerivationList G' := by rw [\u2190e]; exact d\n\ndef head {p} (d : DerivationList (G ++ [p])) : DerivationList (p :: G) := d.cast (by ext; simp[or_comm])\n\ndef headVerum : DerivationList (\u22a4 :: G) := Derivation.verum _ (by simp)\n\ndef verum (h : \u22a4 \u2208 G) : DerivationList G := Derivation.verum _ (by simp[h])\n\ndef tailVerum (p) (h : \u22a4 \u2208 G) : DerivationList (p :: G) := Derivation.verum _ (by simp[h])\n\ndef headEm {p} (h : ~p \u2208 G) : DerivationList (p :: G) := Derivation.em (p := p) (by simp) (by simp[h])\n\ndef headEm' {p np} (e : ~p = np) (h : np \u2208 G) :\n  DerivationList (p :: G) := Derivation.em (p := p) (by simp) (by simp[h, e])\n\ndef rotate {p} (d : DerivationList (G ++ [p])) : DerivationList (p :: G) :=\n  d.cast (by ext; simp[or_comm])\n\ndef headWeakening {p} (d : DerivationList G) : DerivationList (p :: G) :=\n  Derivation.weakening d (by simp; exact Finset.subset_insert  _ _)\n\ndef headWeakeningOfValid {p p'} (h : p = p') (d : Valid p) : DerivationList (p' :: G) :=\n  Derivation.weakening d (by simp[h])\n\ndef headOr {p q} (d : DerivationList (G ++ [p, q])) : DerivationList (p \u22ce q :: G) :=\n  (Derivation.or (\u0394 := G.toFinset) (p := p) (q := q) (d.cast $ by ext; simp; tauto)).cast (by simp)\n\ndef headAnd {p q} (dp : DerivationList (G ++ [p])) (dq : DerivationList (G ++ [q])) : DerivationList (p \u22cf q :: G) :=\n  (Derivation.and (\u0394 := G.toFinset) (p := p) (q := q)\n    (dp.cast $ by ext; simp[or_comm]) (dq.cast $ by ext; simp[or_comm])).cast (by simp)\n\ndef headAll {p : SyntacticSubFormula L 1} (d : DerivationList (G.map SubFormula.shift ++ [SubFormula.free p])) :\n    DerivationList ((\u2200' p) :: G) :=\n  (Derivation.all G.toFinset p (d.cast $ by ext; simp[shifts, SubFormula.shiftEmb, or_comm])).cast (by simp)\n\ndef headAllOfEq {G'} (eG : G.map SubFormula.shift = G') {p : SyntacticSubFormula L 1} {p} (ep : SubFormula.free p = p')\n  (d : DerivationList (G' ++ [p'])) :\n    DerivationList ((\u2200' p) :: G) :=\n  (Derivation.all G.toFinset p (d.cast $ by ext; simp[\u2190eG, \u2190ep, shifts, SubFormula.shiftEmb, or_comm])).cast (by simp)\n\ndef headEx {t} {p : SyntacticSubFormula L 1} (d : DerivationList (G ++ [SubFormula.subst t p])) :\n    DerivationList ((\u2203' p) :: G) :=\n  (Derivation.ex G.toFinset t p (d.cast $ by ext; simp[or_comm])).cast (by simp)\n\ndef headExInstances {v : List (SyntacticTerm L)} {p : SyntacticSubFormula L 1} (d : DerivationList (G ++ v.map (SubFormula.subst \u00b7 p))) :\n    DerivationList ((\u2203' p) :: G) :=\n  (Derivation.exOfInstances (\u0393 := G.toFinset) v p (d.cast $ by ext x; simp[or_comm])).cast (by simp)\n\ndef headExInstancesOfEq {v : List (SyntacticTerm L)} {p : SyntacticSubFormula L 1} {pi : List (SyntacticFormula L)}\n  (ev : v.map (SubFormula.subst \u00b7 p) = pi) (d : DerivationList (G ++ pi)) :\n    DerivationList ((\u2203' p) :: G) :=\n  (Derivation.exOfInstances (\u0393 := G.toFinset) v p (d.cast $ by ext x; simp[\u2190ev, or_comm])).cast (by simp)\n\nend DerivationList\n\nnamespace Valid\nopen Derivation\nvariable {L : Language.{u}} [\u2200 k, DecidableEq (L.func k)] [\u2200 k, DecidableEq (L.rel k)]\n\ndef congr {p p' : SyntacticFormula L} (e : p' = p) (d : Valid p) : Valid p' :=\n  e \u25b8 d\n\nend Valid\n\nset_option linter.unusedVariables false in\nabbrev DerivationListQ (L : Q(Language.{u}))\n  (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k)))\n  (G : List Q(SyntacticFormula $L)) :=\n  Q(DerivationList $(toQList (u := u) G))\n\nnamespace DerivationListQ\nopen SubFormula Derivation\nvariable (L : Q(Language.{u}))\n  (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k))) (G : List Q(SyntacticFormula $L))\n\ndef toValidQ (p : Q(SyntacticFormula $L)) (d : DerivationListQ L dfunc drel [p]) : Q(Valid $p) :=\n  q($d)\n\ndef congrQ {G G' : List Q(SyntacticFormula $L)} (e : Q($(toQList (u := u) G) = $(toQList (u := u) G')))\n  (d : DerivationListQ L dfunc drel G) : DerivationListQ L dfunc drel G' :=\n  q(DerivationList.congr $e $d)\n\ndef verum (h : G.elem q(\u22a4)) : DerivationListQ L dfunc drel G :=\n  (q(DerivationList.verum $(Qq.toQListOfElem (u := u) h)) : Q(DerivationList $(toQList (u := u) G)))\n\n-- assume q(\u22a4) \u2208 G\ndef verumDec : MetaM (DerivationListQ L dfunc drel G) := do\n  let h \u2190 decideTQ q(\u22a4 \u2208 $(toQList (u := u) G))\n  return q(DerivationList.verum $h)\n\ndef headVerum : DerivationListQ L dfunc drel (q(\u22a4) :: G) :=\n  q(DerivationList.headVerum)\n\ndef tailVerum (p : Q(SyntacticFormula $L)) (h : G.elem q(\u22a4)) :\n  DerivationListQ L dfunc drel (p :: G) :=\n  (q(DerivationList.tailVerum $p $(Qq.toQListOfElem (u := u) h)) : Q(DerivationList $(toQList (u := u) (p :: G))))\n\n-- assume q(\u22a4) \u2208 G\ndef tailVerumDec (p : Q(SyntacticFormula $L)) :\n  MetaM $ DerivationListQ L dfunc drel (p :: G) := do\n  let h \u2190 decideTQ q(\u22a4 \u2208 $(toQList (u := u) G))\n  logInfo m!\"h = {h}\"\n  return q(DerivationList.tailVerum $p $h)\n\ndef headWeakening {p} (d : DerivationListQ L dfunc drel G) : DerivationListQ L dfunc drel (p :: G) :=\n  q(DerivationList.headWeakening $d)\n\ndef headWeakeningOfValid (p p' : Q(SyntacticFormula $L)) (h : Q($p = $p')) (d : Q(Valid $p)) : DerivationListQ L dfunc drel (p' :: G) :=\n  q(DerivationList.headWeakeningOfValid $h $d)\n\n-- def headEm {p : Q(SyntacticFormula $L)} (h : G.elem q(~$p)) : DerivationListQ L dfunc drel (p :: G) :=\n--   q(DerivationList.headEm $(Qq.toQListOfElem (u := u) h))\n\ndef headEm {p np : Q(SyntacticFormula $L)} (e : Q(~$p = $np)) (h : G.elem np) : DerivationListQ L dfunc drel (p :: G) :=\n  q(DerivationList.headEm' $e $(Qq.toQListOfElem (u := u) h))\n\n-- assume np \u2208 G\ndef headEmDec {p np : Q(SyntacticFormula $L)} (e : Q(~$p = $np)) : MetaM (DerivationListQ L dfunc drel (p :: G)) := do\n  let h \u2190 decideTQ q($np \u2208 $(toQList (u := u) G))\n  logInfo m!\"h = {h}\"\n  return q(DerivationList.headEm' $e $h)\n\ndef rotate (p : Q(SyntacticFormula $L)) (d : DerivationListQ L dfunc drel (G ++ [p])) :\n  DerivationListQ L dfunc drel (p :: G) :=\n  let x : Q(DerivationList $ $(toQList (u := u) G) ++ [$p]) := d\n  (q(DerivationList.rotate $x) : Q(DerivationList $(toQList (u := u) (p :: G))))\n\ndef headOr {p q : Q(SyntacticFormula $L)}\n  (d : DerivationListQ L dfunc drel (Append.append G [q($p), q($q)])) :\n    DerivationListQ L dfunc drel (q($p \u22ce $q) :: G) :=\n  let x : Q(DerivationList $ Append.append  $(toQList (u := u) G) [$p, $q]) := d\n  (q(DerivationList.headOr $x) : Q(DerivationList ($p \u22ce $q :: $(toQList (u := u) G))))\n\ndef headAnd {p q : Q(SyntacticFormula $L)}\n  (dp : DerivationListQ L dfunc drel (G ++ [p]))\n  (dq : DerivationListQ L dfunc drel (G ++ [q])) :\n    DerivationListQ L dfunc drel (q($p \u22cf $q) :: G) :=\n  let xp : Q(DerivationList $ Append.append  $(toQList (u := u) G) [$p]) := dp\n  let xq : Q(DerivationList $ Append.append  $(toQList (u := u) G) [$q]) := dq\n  (q(DerivationList.headAnd $xp $xq) : Q(DerivationList ($p \u22cf $q :: $(toQList (u := u) G))))\n\ndef headAll (sG : List Q(SyntacticFormula $L)) (eG : Q(List.map shift $(toQList (u := u) G) = $(toQList (u := u) sG)))\n  {p : Q(SyntacticSubFormula $L 1)} {fp : Q(SyntacticFormula $L)} (ep : Q(free $p = $fp))\n  (d : DerivationListQ L dfunc drel (Append.append sG [fp])) :\n    DerivationListQ L dfunc drel (q(\u2200' $p) :: G) :=\n  let x : Q(DerivationList $ $(toQList (u := u) (Append.append sG [fp]))) := d\n  let x : Q(DerivationList $ Append.append $(toQList (u := u) sG) [$fp]) := d\n  (q(DerivationList.headAllOfEq $eG (p := $p) $ep $x) : Q(DerivationList ((\u2200' $p) :: $(toQList (u := u) G))))\n\n/-\ndef headAll (sG : List Q(SyntacticFormula $L)) (eG : Q(List.map shift $(toQList (u := u) G) = $(toQList (u := u) sG)))\n  {p : Q(SyntacticSubFormula $L 1)} {fp : Q(SyntacticFormula $L)} (ep : Q(free $p = $fp))\n  (d : DerivationListQ L dfunc drel (Append.append sG [fp])) :\n    DerivationListQ L dfunc drel (q(\u2200' $p) :: G) :=\n  let x : Q(DerivationList $ $(toQList (u := u) (Append.append sG [fp]))) := d\n  let x : Q(DerivationList $ Append.append $(toQList (u := u) sG) [$fp]) := d\n  let x : Q(DerivationList $ Append.append (List.map shift $(toQList (u := u) G)) [SubFormula.free $p]) :=\n  -- TODO\n    q(by rw[($eG), ($ep)]; exact $x)\n  (q(DerivationList.headAll $x) : Q(DerivationList ((\u2200' $p) :: $(toQList (u := u) G))))\n-/\n\ndef headEx (v : List Q(SyntacticTerm $L)) (p : Q(SyntacticSubFormula $L 1)) (pi : List Q(SyntacticFormula $L))\n  (ev : Q(List.map (SubFormula.subst \u00b7 $p) $(toQList (u := u) v) = $(toQList (u := u) pi)))\n  (d : DerivationListQ L dfunc drel (G ++ pi)) :\n    DerivationListQ L dfunc drel (q(\u2203' $p) :: G) :=\n  -- let x : Q(DerivationList $ $(toQList (u := u) (G ++ pi))) := d\n  let x : Q(DerivationList $ $(toQList (u := u) G) ++ $(toQList (u := u) pi)) := d\n  q(DerivationList.headExInstancesOfEq $ev $x)\n\n/-\ndef headEx (v : List Q(SyntacticTerm $L)) (p : Q(SyntacticSubFormula $L 1)) (pi : List Q(SyntacticFormula $L))\n  (ev : Q(List.map (SubFormula.subst \u00b7 $p) $(toQList (u := u) v) = $(toQList (u := u) pi)))\n  (d : DerivationListQ L dfunc drel (G ++ pi)) :\n    DerivationListQ L dfunc drel (q(\u2203' $p) :: G) :=\n  -- let x : Q(DerivationList $ $(toQList (u := u) (G ++ pi))) := d\n  let x : Q(DerivationList $ $(toQList (u := u) G) ++ $(toQList (u := u) pi)) := d\n  let x : Q(DerivationList $ $(toQList (u := u) G) ++ List.map (SubFormula.subst \u00b7 $p) $(toQList (u := u) v)) :=\n  -- TODO\n    q(by { rw[($ev)]; exact $x })\n  q(DerivationList.headExInstances $x)\n-/\n\ndef getFormula (e : Q(Type u)) : MetaM $ Option Q(SyntacticFormula $L) := do\n  if let ~q(@Valid $L' $dfunc' $drel' $p) := e then\n    if (\u2190 isDefEq (\u2190 whnf L) (\u2190 whnf L')) then\n      return some p\n    else return none\n  else return none\n\nsection tauto\n\ndef tryProveByHyp (L : Q(Language.{u})) (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k)))\n  (p : Q(SyntacticFormula $L)) (G : List Q(SyntacticFormula $L)) : MetaM $ Option (DerivationListQ (u := u) L dfunc drel (p :: G)) := do\n  let ctx \u2190 Lean.MonadLCtx.getLCtx\n    let hyp \u2190 ctx.findDeclM? fun decl: Lean.LocalDecl => do\n      if !decl.isImplementationDetail then\n        let declExpr := decl.toExpr\n        let declType \u2190 Lean.Meta.inferType declExpr\n        let some p' \u2190 getFormula L declType | return none\n        let \u27e8pn', e'\u27e9 \u2190 Meta.result\u2080 p'\n        if \u2190 isDefEq p pn' then\n          let some d \u2190 checkTypeQ (u := .succ u) declExpr q(@Valid $L $dfunc $drel $p') | return none\n            return some (p', e', d)\n        else return none\n      else return none\n    if let some (p', e', d') := hyp then\n      return some $ headWeakeningOfValid L dfunc drel G p p' e' d'\n    else return none\n\ndef proveDerivationListQTauto (hypSearch : Bool)\n  (L : Q(Language.{u})) (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k))) :\n    \u2115 \u2192 (G : List Q(SyntacticFormula $L)) \u2192 MetaM (DerivationListQ (u := u) L dfunc drel G)\n  | 0,     _      => throwError \"failed!\"\n  | _,     []     => throwError \"empty goal\"\n  | s + 1, p :: G => do\n    -- hypothesis search\n    if let some d \u2190 tryProveByHyp L dfunc drel p G then\n      return d\n    else \n    -- proof search\n    let \u27e8npn, npe\u27e9 \u2190 SubFormula.Meta.resultNeg (L := L) (n := q(0)) p\n    if h : G.elem npn then\n      return DerivationListQ.headEm L dfunc drel G npe h\n    else\n    (match p with\n    | ~q(\u22a4)       => pure $ headVerum L dfunc drel G\n    | ~q(\u22a5)       => do\n      let d \u2190 proveDerivationListQTauto hypSearch L dfunc drel s G\n      return headWeakening L dfunc drel G d\n    | ~q($p \u22ce $q) => do\n      let d \u2190 proveDerivationListQTauto hypSearch L dfunc drel s (G ++ [p, q])\n      return (headOr L dfunc drel G d)\n    | ~q($p \u22cf $q) => do\n      let dp \u2190 proveDerivationListQTauto hypSearch L dfunc drel s (G ++ [p])\n      let dq \u2190 proveDerivationListQTauto hypSearch L dfunc drel s (G ++ [q])\n      return (headAnd L dfunc drel G dp dq)\n    | ~q($p)      => do\n      let d \u2190 proveDerivationListQTauto hypSearch L dfunc drel s (G ++ [p])\n      return rotate L dfunc drel G p d\n      : MetaM Q(DerivationList $ $p :: $(toQList (u := u) G)))\n\ndef proveValidTauto (L : Q(Language.{u})) (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k)))\n  (s : \u2115) (p : Q(SyntacticFormula $L)) : MetaM Q(Valid $p) := do\n  let \u27e8pn, e\u27e9 \u2190 SubFormula.Meta.result\u2080 (L := L) p\n  let d \u2190 proveDerivationListQTauto true L dfunc drel s [pn]\n  let h := toValidQ L dfunc drel _ d\n  return q(Valid.congr $e $h)\n\nelab \"proveTauto\" n:(num)? : tactic => do\n  let goalType \u2190 Elab.Tactic.getMainTarget\n  let some \u27e8.succ _, ty\u27e9 \u2190 checkSortQ' goalType | throwError \"not a type\"\n  let ~q(@Valid $L $dfunc $drel $p) := ty | throwError \"not a type: Valid p\"\n  let s : \u2115 :=\n    match n with\n    | some n => n.getNat\n    | none   => 16\n  let b \u2190 proveValidTauto L dfunc drel s p\n  Lean.Elab.Tactic.closeMainGoal b\n\n/-\nsection\nvariable {L : Language.{u}} [\u2200 k, DecidableEq (L.func k)] [\u2200 k, DecidableEq (L.rel k)] (p q r s : SyntacticFormula L)\n\nexample : Valid ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r) := by proveTauto\n\nexample : Valid \u201c((!p \u2192 !q) \u2192 !p) \u2192 !p\u201d := by proveTauto\n\nexample : Valid \u201c!p \u2227 !q \u2227 !r \u2194 !r \u2227 !p \u2227 !q\u201d  := by proveTauto\n\nexample (d : Valid p) : Valid \u201c!p \u2228 !q\u201d  := by proveTauto\n\nexample (_ : Valid \u201c\u00ac(!p \u2227 !q)\u201d) (_ : Valid s) : Valid \u201c!s \u2192 !p \u2227 !q \u2192 !r\u201d  := by proveTauto\n\nexample (_ : Valid \u201c\u00ac(!p \u2227 !q)\u201d) : Valid \u201c\u00ac!p \u2228 \u00ac!q\u201d  := by proveTauto\n\nend\n-/\nend tauto\n\ndef proveDerivationListQ (L : Q(Language.{u})) (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k)))\n  (ts : List Q(SyntacticTerm $L)) :\n    \u2115 \u2192 (G : List Q(SyntacticFormula $L)) \u2192 MetaM (DerivationListQ (u := u) L dfunc drel G)\n  | 0,     _      => throwError \"failed!\"\n  | _,     []     => throwError \"empty goal\"\n  | s + 1, p :: G => do\n   -- hypothesis search\n    if let some d \u2190 tryProveByHyp L dfunc drel p G then\n      return d\n    else \n    -- proof search\n    let \u27e8npn, npe\u27e9 \u2190 SubFormula.Meta.resultNeg (L := L) (n := q(0)) p\n    if h : G.elem npn then\n      return DerivationListQ.headEm L dfunc drel G npe h\n    else\n    (match p with\n    | ~q(\u22a4) => pure $ headVerum L dfunc drel G\n    | ~q(\u22a5) => do\n      let d \u2190 proveDerivationListQ L dfunc drel ts s G\n      return headWeakening L dfunc drel G d\n    | ~q($p \u22ce $q) => do\n      let d \u2190 proveDerivationListQ L dfunc drel ts s (G ++ [p, q])\n      return (headOr L dfunc drel G d)\n     | ~q($p \u22cf $q) => do\n      let dp \u2190 proveDerivationListQ L dfunc drel ts s (G ++ [p])\n      let dq \u2190 proveDerivationListQ L dfunc drel ts s (G ++ [q])\n      return (headAnd L dfunc drel G dp dq)   \n    | ~q(\u2200' $p)  => do\n      let \u27e8fp, fpe\u27e9 \u2190 Meta.resultFree p\n      let \u27e8sG, sGe\u27e9 \u2190 Meta.resultShift\u2080List G\n      let d \u2190 proveDerivationListQ L dfunc drel ts s (Append.append sG [fp])\n      return headAll L dfunc drel G sG sGe fpe d\n    | ~q(\u2203' $p)   => do\n      let \u27e8pi, pie\u27e9 \u2190 Meta.resultSubst\u2080List ts p\n      let d \u2190 proveDerivationListQ L dfunc drel ts s (G ++ pi)\n      return headEx L dfunc drel G ts p pi pie d\n    | ~q($p) => do\n      let d \u2190 proveDerivationListQ L dfunc drel ts s (G ++ [p])\n      return rotate L dfunc drel G p d\n         : MetaM Q(DerivationList $ $p :: $(toQList (u := u) G)))\n\ndef proveValid (L : Q(Language.{u})) (dfunc : Q(\u2200 k, DecidableEq (($L).func k))) (drel : Q(\u2200 k, DecidableEq (($L).rel k)))\n  (ts : List Q(SyntacticTerm $L)) (s : \u2115) (p : Q(SyntacticFormula $L)) : MetaM Q(Valid $p) := do\n  let \u27e8pn, e\u27e9 \u2190 SubFormula.Meta.result\u2080 (L := L) p\n  let d \u2190 proveDerivationListQ L dfunc drel ts s [pn]\n  let h := toValidQ L dfunc drel _ d\n  return q(Valid.congr $e $h)\n\nsyntax termSeq := \" [\" (term,*) \"]\"\n\nelab \"prove\" n:(num)? seq:(termSeq)? : tactic => do\n  let goalType \u2190 Elab.Tactic.getMainTarget\n  let some \u27e8.succ _, ty\u27e9 \u2190 checkSortQ' goalType | throwError \"error: not a type\"\n  let ~q(@Valid $L $dfunc $drel $p) := ty | throwError \"error: not a type 2\"\n  let s : \u2115 :=\n    match n with\n    | some n => n.getNat\n    | none   => 16\n  let ts : Array Q(SyntacticTerm $L) \u2190\n    match seq with\n    | some seq =>\n      match seq with\n      | `(termSeq| [ $ss,* ] ) => do ss.getElems.mapM (Term.elabTerm \u00b7 (some q(SyntacticTerm $L)))\n      | _                      => pure #[]\n    | _        => pure #[q(&0 : SyntacticTerm $L), q(&1 : SyntacticTerm $L)]\n  let b \u2190 proveValid L dfunc drel ts.toList s p\n  Lean.Elab.Tactic.closeMainGoal b\n\n/-\nsection\nvariable {L : Language.{u}} [\u2200 k, DecidableEq (L.func k)] [\u2200 k, DecidableEq (L.rel k)] (p q r s : SyntacticFormula L)\nopen Language\n\nexample (_ : Valid \u201c\u00ac(!p \u2227 !q)\u201d) : Valid \u201c\u00ac!p \u2228 \u00ac!q\u201d  := by proveTauto\n\nexample : Valid (L := oring) \u201c&0 < 3 \u2192 \u2203 &0 < #0\u201d := by prove [T\u201c3\u201d]\n\nexample : Valid (L := oring) \u201c&0 < &1 \u2192 \u2203 \u2203 #0 < #1\u201d := by prove\n\nexample (_ : Valid (L := oring) \u201c0 < 4 + 9\u201d) : Valid (L := oring) \u201c\u22a4 \u2227 (\u2203 0 < 4 + #0)\u201d  := by prove [T\u201c9\u201d]\n\nend\n-/\nend DerivationListQ\n\nend FirstOrder\n", "meta": {"author": "iehality", "repo": "lean4-logic", "sha": "ef518051931fb1ecd0b89e94240b2900cd54d95c", "save_path": "github-repos/lean/iehality-lean4-logic", "path": "github-repos/lean/iehality-lean4-logic/lean4-logic-ef518051931fb1ecd0b89e94240b2900cd54d95c/Logic/Predicate/FirstOrder/Meta.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.37240359441919396}}
{"text": "lemma example2 (x y : mynat) (h : y = x + 7) : 2 * y = 2 * (x + 7) :=\nbegin\nrw \u2190 h,\nrefl,\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/1-tutorial-world/l2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3723767732361381}}
{"text": "import category_theory.limits.limits\nimport category_theory.limits.shapes\nimport category_theory.yoneda\nimport category_theory.opposites\nimport category_theory.types\nimport category_theory.limits.types\n\nrun_cmd mk_simp_attr `PRODUCT    -----  BOF BOF  \nmeta def PRODUCT_CAT  : tactic unit :=\n`[  try {simp only with PRODUCT}]\nrun_cmd add_interactive [`PRODUCT_CAT]\n\nuniverses v u\nopen category_theory\nopen category_theory.limits\nopen category_theory.category\nopen opposite\n\nnamespace lem \nvariables {C : Type u}\nvariables [\ud835\udc9e : category.{v} C]\nvariables  [has_binary_products.{v} C][has_terminal.{v} C]\ninclude \ud835\udc9e\nattribute [PRODUCT] category.assoc category.id_comp category.comp_id \n@[PRODUCT] lemma prod_left_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.left = limits.prod.fst := rfl\n@[PRODUCT] lemma prod_right_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.right = limits.prod.snd := rfl\nlemma prod.hom_ext {A X Y : C} {a b : A \u27f6 X \u2a2f Y} (h1 : a \u226b limits.prod.fst = b \u226b limits.prod.fst) (h2 : a \u226b limits.prod.snd = b \u226b limits.prod.snd) : a = b :=\nbegin\n  apply limit.hom_ext,\n  rintros (_ | _),\n  rw prod_left_def,\n  exact h1,\n  rw prod_right_def,\n  exact h2,\nend\n@[PRODUCT]lemma prod.lift_fst {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : prod.lift f g \u226b category_theory.limits.prod.fst = f :=\nlimit.lift_\u03c0 (binary_fan.mk f g) _\n\n@[PRODUCT]lemma prod.lift_snd {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : prod.lift f g \u226b category_theory.limits.prod.snd = g :=\nlimit.lift_\u03c0 (binary_fan.mk f g) _\n\nend lem\n\nnotation f ` \u2297 `:20 g :20 := category_theory.limits.prod.map f g  ---- 20 \nnotation  `T`C :20 := (terminal C) \nnotation   `T`X : 20 := (terminal.from X)\nnotation f ` | `:20 g :20 :=  prod.lift f g\nnotation `\u03c01` := limits.prod.fst \nnotation `\u03c02` := limits.prod.snd\n\n\nvariables {C : Type u}\nvariables [\ud835\udc9e : category.{v} C]\nvariables [has_binary_products.{v} C][has_terminal.{v} C]\ninclude \ud835\udc9e\nvariables (X :C)\nopen lem\n/-\n     \u03c0 notation for projection \n-/\n@[PRODUCT]lemma ex_1 {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : ( f | g) \u226b \u03c01 = f  :=   prod.lift_fst f g \n/-\n     we can type \u03c0 : A \u2a2f B \u27f6 B if we need \n-/\n@[PRODUCT]lemma ex_2 {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : ( f | g) \u226b (\u03c02 : A \u2a2f B \u27f6 B) = g := prod.lift_snd f g \n\n@[PRODUCT]lemma ex_3 {A X Y : C} {a b : A \u27f6 X \u2a2f Y} (h1 : a \u226b \u03c01  = b \u226b \u03c01 ) (h2 : a \u226b \u03c02  = b \u226b \u03c02)  : a = b :=  prod.hom_ext h1 h2\n\n\n@[PRODUCT]lemma prod.left_composition{Z' Z A B : C}(h : Z' \u27f6 Z)(f : Z \u27f6 A)(g : Z \u27f6 B)  : \n               h \u226b (f | g)  = (h \u226b f | h \u226b g) := \nbegin\n     apply prod.hom_ext,   --- Le right member is of the form ( | )  composition \u03c01 \u03c02 \n     PRODUCT_CAT,  PRODUCT_CAT,  --- here assoc \n     -- rw assoc,\n     -- rw prod.lift_fst,\n     -- rw prod.lift_fst,\n     -- rw prod.lift_snd,\n     -- rw assoc,\n     -- rw prod.lift_snd,\nend\n\n@[PRODUCT]lemma prod.map_first{X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) :  (f \u2297 g) \u226b (\u03c01 : Y \u2a2f W \u27f6 Y) = \u03c01  \u226b f :=  begin \n     exact limit.map_\u03c0 (map_pair f g) walking_pair.left,\nend\n@[PRODUCT]lemma prod.map_second{X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) :  (f \u2297 g) \u226b \u03c02 = \u03c02 \u226b g :=  begin \n     exact limit.map_\u03c0 (map_pair f g) walking_pair.right,\nend\n@[PRODUCT]lemma  prod.otimes_is_prod {X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) : (f \u2297 g) = ( \u03c01  \u226b f | \u03c02 \u226b g ) := begin\n     apply prod.hom_ext,\n     PRODUCT_CAT, PRODUCT_CAT,\n     -- rw prod.lift_fst,\n     -- rw prod.map_first,\n     -- rw prod.lift_snd,\n     -- rw prod.map_second,\nend\n-- notation \u03c01`(`X `x` Y`)` := (limits.prod.fst : X\u2a2fY \u27f6 X)\n@[PRODUCT]lemma prod.map_ext{X Y Z W : C}(f1 f2  : X \u27f6 Y)(g1 g2  : Z \u27f6 W) :  (f1 \u2297 g1) = (f2 \u2297 g2) \u2192 \n(\u03c01 : X \u2a2f Z \u27f6 X) \u226b f1 = (\u03c01 : X \u2a2f Z \u27f6 X)  \u226b f2 := \u03bb certif, begin \n     iterate 2 {rw prod.otimes_is_prod at certif},\n     rw \u2190 prod.map_first ( f1)  (g1),\n     rw \u2190 prod.map_first ( f2)  (g2),\n     iterate 2 {rw prod.otimes_is_prod},\n     rw certif,\nend\n@[PRODUCT]lemma destruction {X Y Z : C} (f :  Y \u27f6 X) (g : X \u27f6 Z ) : \n     (f | \ud835\udfd9 Y) \u226b (g \u2297 (\ud835\udfd9 Y)) = (f \u226b g | \ud835\udfd9 Y) := begin \n     apply prod.hom_ext,\n     -- PRODUCT_CAT,PRODUCT_CAT,     ---------------------- PROBLEME With the tatict HEEEEEEERRRRRRE \n     rw [prod.lift_fst],\n     rw  assoc, \n     rw prod.map_first,\n     rw \u2190 assoc,               ----- \u2190 assoc here  Problem ? \n     rw prod.lift_fst,          \n     tidy, -- super - power tidy \nend\n\n\n\n\n-- def Y (R : C)(A :C) := (yoneda.obj A).obj (op R)\n-- def Y_ (R : C) {A B : C}(\u03c6 : A \u27f6 B) := ((yoneda.map \u03c6).app (op R) : Y R A \u27f6 Y R B)\n-- -- Good notation for yoneda stuff : \n-- -- We fix V : C and we denote by    \n-- -- R[X] := yoneda.obj X).obj (op R) and \u03c6 : A  \u27f6 B (in C) R \u27e6  \u03c6 \u27e7   : R[A] \u2192 R[B]  in type v \n-- local notation R`[`A`]`:20 := Y R A  -- notation ?? \n-- local notation R`<`\u03c6`>` :20   := Y_ R \u03c6  -- \n-- def Yoneda_preserve_product (Y : C)(A B : C) :\n--      Y[A \u2a2f B] \u2245 Y[A] \u2a2f Y[B] :=\n-- { hom := prod.lift\n--     (\u03bb f, f \u226b \u03c01)\n--     (\u03bb f, f \u226b \u03c02),\n--   inv := \u03bb f : (Y \u27f6 A) \u2a2f (Y \u27f6 B),\n--     (prod.lift\n--       ((@category_theory.limits.prod.fst _ _ (Y \u27f6 A) (Y \u27f6 B) _ : ((Y \u27f6 A) \u2a2f (Y \u27f6 B)) \u2192 (Y \u27f6 A)) f)\n--       ((@category_theory.limits.prod.snd _ _ (Y \u27f6 A) _ _ : ((Y \u27f6 A) \u2a2f (Y \u27f6 B)) \u2192 (Y \u27f6 B)) f : Y \u27f6 B)),\n--   hom_inv_id' := begin\n--     ext f,\n--     cases j,\n--     { simp, refl},\n--     { simp, refl}\n--   end,\n--   inv_hom_id' := begin\n--     apply lem.prod.hom_ext,\n--     { rw assoc, rw lem.prod.lift_fst, obviously},\n--     { rw assoc, rw lem.prod.lift_snd, obviously}\n--   end\n-- }\n-- --- Here it just sugar \n-- @[PRODUCT]lemma yoneda_sugar.composition (R : C) {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) : R < f \u226b g > =( R< f >) \u226b (R < g >) \n--  :=  begin \n--      unfold Y_, \n--      simp,\n--  end\n-- def yoneda_sugar.conv {R : C}{A : C}(g : R[A]) : R \u27f6 A := g \n-- def yoneda_sugar.prod (R : C)(A B : C) : R[A \u2a2f B] \u2245 R[A] \u2a2f R[B] := begin \n--      exact Yoneda_preserve_product R A B,\n-- end\n-- @[PRODUCT]lemma yoneda_sugar.prod.hom (R : C)(A B : C) : \n--      (yoneda_sugar.prod R A B).hom =  (R < (limits.prod.fst : A \u2a2f B \u27f6 A) > | R < (limits.prod.snd : A \u2a2f B \u27f6 B)> ) := rfl\n\n-- @[PRODUCT]lemma yoneda_sugar.prod.first (R : C)(A B : C) :\n--  (yoneda_sugar.prod R A B).hom \u226b limits.prod.fst = (R < limits.prod.fst >) := \n--  begin\n--      exact rfl,\n--  end\n--  @[PRODUCT]lemma yoneda_sugar.prod.hom_inv (R : C)(A B : C) : \n--      (yoneda_sugar.prod R A B).hom \u226b (yoneda_sugar.prod R A B).inv = \ud835\udfd9 (R[ A \u2a2f B]) := \n--      (Yoneda_preserve_product R A B).hom_inv_id'\n--  @[PRODUCT]lemma yoneda_sugar.prod.inv_hom (R : C)(A B : C) : \n--      (yoneda_sugar.prod R A B).inv \u226b (yoneda_sugar.prod R A B).hom = \ud835\udfd9 ( R [A]  \u2a2f R[B]) := \n--      (Yoneda_preserve_product R A B).inv_hom_id'\n--  @[PRODUCT]lemma yoneda_sugar.prod.second (R : C)(A B : C) : \n--   (yoneda_sugar.prod R A B).hom \u226b limits.prod.snd = (R < limits.prod.snd >) := rfl\n\n-- @[PRODUCT]lemma yoneda_sugar.id (R : C)(A : C) : R < \ud835\udfd9 A > = \ud835\udfd9 ( R [A] ) := begin \n--      funext,\n--      exact comp_id C g,\n--      -- have T : ((yoneda.map (\ud835\udfd9 A)).app (op R)) g = (g \u226b (\ud835\udfd9 A)),\n-- end \n-- lemma yoneda_sugar_prod (R : C)(A B : C)(X :C)(f : X \u27f6 A)(g : X \u27f6 B) :\n--       R < (f | g) > \u226b (yoneda_sugar.prod R A B).hom  =  (R < f > | R < g > ) :=  -- the  \u226b  is  :/   \n--      begin \n--           PRODUCT_CAT,\n--           -- rw  yoneda_sugar.prod.hom R A B,\n--           -- rw prod.left_composition,\n--           iterate 2 {rw \u2190 yoneda_sugar.composition},   -- rw \u2190 is the problem ? \n--           rw lem.prod.lift_fst,\n--           rw lem.prod.lift_snd,\n--      end\n-- @[PRODUCT]lemma yoneda_sugar_prod_inv (R : C)(A B : C)(X :C)(f : X \u27f6 A)(g : X \u27f6 B) : \n--      R < (f | g) >   =  (R < f > | R < g > ) \u226b (yoneda_sugar.prod R A B).inv :=\n--      begin \n--           PRODUCT_CAT,  -- noting\n--           rw \u2190 yoneda_sugar_prod,\n--           rw assoc,\n--           rw yoneda_sugar.prod.hom_inv,\n--           exact rfl,\n--      end \n-- lemma  yoneda_sugar.otimes (R : C){Y Z K :C}(f : X \u27f6 Y )(g : Z \u27f6 K) : \n--  ( R < (f \u2297 g) > ) = (yoneda_sugar.prod  _ _ _).hom \u226b ((R<f>) \u2297 R<g>) \u226b (yoneda_sugar.prod _ _ _ ).inv := begin \n--      PRODUCT_CAT,\n--      -- iterate 2 {rw prod.otimes_is_prod},\n--      -- rw  yoneda_sugar.prod.hom,\n--      -- iterate 1 {rw yoneda_sugar_prod_inv},\n--      rw \u2190 assoc,\n--      rw prod.left_composition,\n--      rw \u2190 assoc,\n--      rw prod.lift_fst,\n--      rw \u2190 assoc,\n--      rw prod.lift_snd,\n--      -- rw yoneda_sugar.composition,\n--      -- rw yoneda_sugar.composition,\n-- end\n-- @[PRODUCT]lemma yonega_sugar.one_otimes (R :C)(X Y Z: C) (f : X \u27f6 Y) : \n--  (((yoneda_sugar.prod R Z X).inv) \u226b (R <(\ud835\udfd9 Z \u2297 f ) > ) \u226b (yoneda_sugar.prod R Z Y).hom) = (\ud835\udfd9 (R[Z]) \u2297 R < f >) := begin\n--      rw yoneda_sugar.otimes,\n--      iterate 3 {rw \u2190 assoc},\n--      rw yoneda_sugar.prod.inv_hom,\n--      rw id_comp,\n--      rw assoc,\n--      rw yoneda_sugar.prod.inv_hom,\n--      rw \u2190 yoneda_sugar.id,\n--      simp, \n--  end\n-- lemma yonega_sugar.one_otimes' (R :C)(X Y Z: C) (f : X \u27f6 Y) : \n--  ( (R <(\ud835\udfd9 Z \u2297 f ) > ) \u226b (yoneda_sugar.prod R Z Y).hom) = ((yoneda_sugar.prod R Z X).hom) \u226b (\ud835\udfd9 (R[Z]) \u2297 R < f >) := begin\n--      iterate 2{ rw yoneda_sugar.prod.hom},\n--      rw prod.left_composition,\n--      iterate 2{ rw \u2190 yoneda_sugar.composition},\n--      rw prod.map_first,\n--      rw prod.map_second,\n--      rw comp_id,\n--      rw prod.otimes_is_prod,rw prod.left_composition,rw \u2190 assoc, \n--      rw prod.lift_fst,rw \u2190  assoc,rw prod.lift_snd,rw comp_id,\n--      rw yoneda_sugar.composition,\n--  end\n\n\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/geo/group_objet/group_obj_def.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.372376773236138}}
{"text": "/-\nCopyright (c) 2020 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton\n-/\nimport tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\nimport tactic.show_term\nimport topology.basic\n/-!\n# Tactics for topology\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nCurrently we have one domain-specific tactic for topology: `continuity`.\n\n-/\n\n/-!\n### `continuity` tactic\n\nAutomatically solve goals of the form `continuous f`.\n\nMark lemmas with `@[continuity]` to add them to the set of lemmas\nused by `continuity`.\n-/\n\n/-- User attribute used to mark tactics used by `continuity`. -/\n@[user_attribute]\nmeta def continuity : user_attribute :=\n{ name := `continuity,\n  descr := \"lemmas usable to prove continuity\" }\n\n-- Mark some continuity lemmas already defined in `topology.basic`\nattribute [continuity]\n  continuous_id\n  continuous_const\n\n-- As we will be using `apply_rules` with `md := semireducible`,\n-- we need another version of `continuous_id`.\n@[continuity] lemma continuous_id' {\u03b1 : Type*} [topological_space \u03b1] : continuous (\u03bb a : \u03b1, a) :=\ncontinuous_id\n\nnamespace tactic\n\n/--\nTactic to apply `continuous.comp` when appropriate.\n\nApplying `continuous.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove continuous is actually\n  constant, and that constant is a function application `f z`, then\n  continuous.comp would produce new goals `continuous f`, `continuous\n  (\u03bb _, z)`, which is silly. We avoid this by failing if we could\n  apply continuous_const.\n\n* continuous.comp will always succeed on `continuous (\u03bb x, f x)` and\n  produce new goals `continuous (\u03bb x, x)`, `continuous f`. We detect\n  this by failing if a new goal can be closed by applying\n  continuous_id.\n-/\nmeta def apply_continuous.comp : tactic unit :=\n`[fail_if_success { exact continuous_const };\n  refine continuous.comp _ _;\n  fail_if_success { exact continuous_id }]\n\n/-- List of tactics used by `continuity` internally. -/\nmeta def continuity_tactics (md : transparency := reducible) : list (tactic string) :=\n[\n  intros1               >>= \u03bb ns, pure (\"intros \" ++ (\" \".intercalate (ns.map (\u03bb e, e.to_string)))),\n  apply_rules [] [``continuity] 50 { md := md }\n                        >> pure \"apply_rules with continuity\",\n  apply_continuous.comp >> pure \"refine continuous.comp _ _\"\n]\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nSolve goals of the form `continuous f`. `continuity?` reports back the proof term it found.\n-/\nmeta def continuity\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md              := if bang.is_some then semireducible else reducible,\n    continuity_core := tactic.tidy { tactics := continuity_tactics md, ..cfg },\n    trace_fn        := if trace.is_some then show_term else id in\ntrace_fn continuity_core\n\n/-- Version of `continuity` for use with auto_param. -/\nmeta def continuity' : tactic unit := continuity none none {}\n\n/--\n`continuity` solves goals of the form `continuous f` by applying lemmas tagged with the\n`continuity` user attribute.\n\n```\nexample {X Y : Type*} [topological_space X] [topological_space Y]\n  (f\u2081 f\u2082 : X \u2192 Y) (hf\u2081 : continuous f\u2081) (hf\u2082 : continuous f\u2082)\n  (g : Y \u2192 \u211d) (hg : continuous g) : continuous (\u03bb x, (max (g (f\u2081 x)) (g (f\u2082 x))) + 1) :=\nby continuity\n```\nwill discharge the goal, generating a proof term like\n`((continuous.comp hg hf\u2081).max (continuous.comp hg hf\u2082)).add continuous_const`\n\nYou can also use `continuity!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`continuity?` reports back the proof term it found.\n-/\nadd_tactic_doc\n{ name := \"continuity / continuity'\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.continuity, `tactic.interactive.continuity'],\n  tags := [\"lemma application\"] }\n\nend interactive\n\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/topology/tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6859494485880928, "lm_q1q2_score": 0.3723767697509016}}
{"text": "-- Subharmonic / harmonic functions and Hartogs's lemma\n\nimport analysis.convex.integral\nimport analysis.fourier\n\nimport analytic\nimport duals\nimport fubini_ball\nimport holomorphic\nimport max_log\nimport measure\nimport tactics\n\nopen complex (abs exp I log)\nopen filter (tendsto liminf limsup at_top)\nopen function (uncurry)\nopen measure_theory\nopen metric (ball closed_ball sphere)\nopen linear_order (min)\nopen set (Ioc Icc univ)\nopen topological_space (second_countable_topology)\nopen_locale real nnreal ennreal topological_space complex_conjugate\nnoncomputable theory\n\nvariables {S : Type} [is_R_or_C S] [smul_comm_class \u211d S S]\nvariables {T : Type} [is_R_or_C T] [smul_comm_class \u211d T T]\nvariables {E : Type} [normed_add_comm_group E] [complete_space E] [normed_space \u211d E] [second_countable_topology E]\nvariables {F : Type} [normed_add_comm_group F] [complete_space F] [normed_space \u211d F] [second_countable_topology F]\nvariables {H : Type} [normed_add_comm_group H] [complete_space H] [normed_space \u2102 H] [second_countable_topology H]\n\n-- f : \u2102 \u2192 S is harmonic if it is continuous and equal to means on circles.\n-- We require the mean property for large circles because it is easy to prove\n-- for the cases we need, and will be needed for the large submean theorem\n-- for subharmonic functions.\nstructure harmonic_on (f : \u2102 \u2192 E) (s : set \u2102) : Prop :=\n  (cont : continuous_on f s)\n  (mean : \u2200 (c : \u2102) (r : \u211d), r > 0 \u2192 closed_ball c r \u2286 s \u2192 f c = \u2a0d t in Itau, f (circle_map c r t))\n\n-- f : \u2102 \u2192 \u211d is subharmonic if it is upper semicontinuous and is below means on small disks.\n-- We require the submean property only locally, and will prove the global version below.\n-- Out of laziness, we assume continuity as well.  Ideally we'd allow -\u221e as values, but using\n-- ereal instead of \u211d adds a lot of annoying technicalities.\nstructure subharmonic_on (f : \u2102 \u2192 \u211d) (s : set \u2102) : Prop :=\n  (cont : continuous_on f s)\n  (submean' : \u2200 c, c \u2208 interior s \u2192 \u2203 r (rp : r > 0), \u2200 s, 0 < s \u2192 s < r \u2192 f c \u2264 \u2a0d t in Itau, f (circle_map c s t))\n\nlemma subharmonic_on.mono {f : \u2102 \u2192 \u211d} {s t : set \u2102}\n    (fs : subharmonic_on f s) (ts : t \u2286 s) : subharmonic_on f t := {\n  cont := fs.cont.mono ts,\n  submean' := \u03bb c cs, fs.submean' c (interior_mono ts cs),\n}\n\n-- Convex functions of harmonic functions are subharmonic\ntheorem harmonic_on.convex {f : \u2102 \u2192 E} {s : set \u2102} {g : E \u2192 \u211d}\n    (fh : harmonic_on f s) (c : continuous g) (gc : convex_on \u211d set.univ g)\n    : subharmonic_on (\u03bb z, g (f z)) s := {\n  cont := c.comp_continuous_on fh.cont,\n  submean' := begin\n    intros z zs,\n    rcases metric.is_open_iff.mp (is_open_interior) z zs with \u27e8r,rp,rh\u27e9,\n    existsi [r, rp], intros t tp tr,\n    have cs : closed_ball z t \u2286 s := trans (metric.closed_ball_subset_ball tr) (trans rh interior_subset),\n    simp [fh.mean z t tp cs],\n    have n := nice_volume.Itau,\n    apply convex_on.map_set_average_le gc c.continuous_on is_closed_univ n.ne_zero n.ne_top,\n    simp,\n    exact (fh.cont.mono cs).integrable_on_sphere tp,\n    exact ((c.comp_continuous_on fh.cont).mono cs).integrable_on_sphere tp,\n  end,\n}\n\n-- Harmonic functions are subharmonic\nlemma harmonic_on.subharmonic_on {f : \u2102 \u2192 \u211d} {s : set \u2102} (h : harmonic_on f s)\n    : subharmonic_on (\u03bb z, f z) s := begin\n  have e : (\u03bb z, f z) = (\u03bb z, (\u03bb x, x) (f z)) := rfl,\n  rw e, exact h.convex continuous_id (convex_on_id convex_univ),\nend\n\n-- Norms of harmonic functions are subharmonic\nlemma harmonic_on.norm {f : \u2102 \u2192 E} {s : set \u2102} (h : harmonic_on f s)\n    : subharmonic_on (\u03bb z, \u2225f z\u2225) s :=\n  h.convex continuous_norm (convex_on_norm convex_univ)\n\n-- subharmonic_on depends only on values in s (\u2192 version)\ntheorem subharmonic_on.congr {f g : \u2102 \u2192 \u211d} {s : set \u2102}\n    (fs : subharmonic_on f s) (h : set.eq_on g f s) : subharmonic_on g s := {\n  cont := fs.cont.congr h,\n  submean' := begin\n    intros c cs,\n    rcases metric.is_open_iff.mp is_open_interior c cs with \u27e8r0,r0p,r0s\u27e9,\n    rcases fs.submean' c cs with \u27e8r1,r1p,sm\u27e9,\n    have r01p : min r0 r1 > 0 := by bound,\n    existsi [min r0 r1, r01p],\n    intros t tp tr,\n    specialize sm t tp (lt_of_lt_of_le tr (by bound)),\n    have hs : (\u03bb u, f (circle_map c t u)) =\u1d50[volume.restrict Itau] (\u03bb u, g (circle_map c t u)), {\n      rw filter.eventually_eq, rw ae_restrict_iff' measurable_set_Itau, apply filter.eventually_of_forall,\n      intros u us, apply h.symm,\n      apply trans r0s interior_subset,\n      simp [complex.dist_eq, abs_of_pos tp], exact lt_of_lt_of_le tr (by bound),\n    },\n    rw set_average_eq at \u22a2 sm,\n    rwa [\u2190h.symm (interior_subset cs), \u2190integral_congr_ae hs],\n  end\n}\n\n-- subharmonic_at depends only on values near c (\u2194 version)\ntheorem subharmonic_on_congr {f g : \u2102 \u2192 \u211d} {s : set \u2102}\n    (h : set.eq_on f g s) : subharmonic_on f s \u2194 subharmonic_on g s :=\n  \u27e8\u03bb fs, fs.congr h.symm, \u03bb gs, gs.congr h\u27e9\n\n-- Constants are harmonic\nlemma harmonic_on.const (a : E) {s : set \u2102} : harmonic_on (\u03bb _, a) s := {\n  cont := continuous_on_const,\n  mean := begin\n    intros c r rp cs,\n    rw set_average_eq, simp [\u2190smul_assoc, smul_eq_mul],\n    field_simp [ne_of_gt nice_volume.Itau.real_pos],\n  end,\n}\n\n-- Differences are harmonic\nlemma harmonic_on.sub {f g : \u2102 \u2192 E} {s : set \u2102} (fh : harmonic_on f s) (gh : harmonic_on g s)\n    : harmonic_on (f - g) s := {\n  cont := continuous_on.sub fh.cont gh.cont,\n  mean := begin\n    intros c r rp cs, simp [fh.mean c r rp cs, gh.mean c r rp cs],\n    rw set_average.sub ((fh.cont.mono cs).integrable_on_sphere rp) ((gh.cont.mono cs).integrable_on_sphere rp),\n  end,\n}\n\n-- Subharmonic functions add\nlemma subharmonic_on.add {f g : \u2102 \u2192 \u211d} {s : set \u2102} (fs : subharmonic_on f s) (gs : subharmonic_on g s)\n    : subharmonic_on (\u03bb z, f z + g z) s := {\n  cont := fs.cont.add gs.cont,\n  submean' := begin\n    intros c cs,\n    rcases fs.submean' c cs with \u27e8r0,r0p,r0m\u27e9,\n    rcases gs.submean' c cs with \u27e8r1,r1p,r1m\u27e9,\n    rcases metric.is_open_iff.mp is_open_interior c cs with \u27e8r2,r2p,r2s\u27e9,\n    set r := min r0 (min r1 r2),\n    have rr1 : r \u2264 r1 := trans (min_le_right _ _) (by bound),\n    have rr2 : r \u2264 r2 := trans (min_le_right _ _) (by bound),\n    use [r, by bound], intros u up ur,\n    have us : closed_ball c u \u2286 s :=\n      trans (metric.closed_ball_subset_ball (lt_of_lt_of_le ur (by bound))) (trans r2s interior_subset),\n    rw set_average.add ((fs.cont.mono us).integrable_on_sphere up) ((gs.cont.mono us).integrable_on_sphere up),\n    have m0 := r0m u up (lt_of_lt_of_le ur (by bound)),\n    have m1 := r1m u up (lt_of_lt_of_le ur (by bound)),\n    exact add_le_add m0 m1,\n  end,\n}\n\n-- Negations are harmonic\nlemma harmonic_on.neg {f : \u2102 \u2192 E} {s : set \u2102} (fh : harmonic_on f s) : harmonic_on (-f) s := begin\n  have nh := harmonic_on.sub (harmonic_on.const 0) fh,\n  have e : (\u03bb _ : \u2102, (0 : E)) - f = -f, { ext, simp },\n  rwa \u2190e,\nend\n\n-- Additions are harmonic\nlemma harmonic_on.add {f g : \u2102 \u2192 E} {s : set \u2102} (fh : harmonic_on f s) (gh : harmonic_on g s)\n    : harmonic_on (f + g) s := begin\n  have e : f + g = f - (-g), { ext, simp },\n  rw e, exact fh.sub gh.neg,\nend\n\n-- Scalar multiples are harmonic\nlemma harmonic_on.const_mul {f : \u2102 \u2192 S} {s : set \u2102} (fh : harmonic_on f s) (a : S)\n    : harmonic_on (\u03bb z, a * f z) s := {\n  cont := continuous_on.mul (continuous_on_const) fh.cont,\n  mean := begin\n    intros c r rp cs, rw set_average_eq,\n    simp_rw [\u2190smul_eq_mul, integral_smul, smul_comm _ a, \u2190set_average_eq, \u2190fh.mean c r rp cs],\n  end,\n}\n\n-- Scalar multiples are subharmonic\nlemma subharmonic_on.const_mul {f : \u2102 \u2192 \u211d} {s : set \u2102} {a : \u211d} (fs : subharmonic_on f s) (ap : a \u2265 0)\n    : subharmonic_on (\u03bb z, a * f z) s := {\n  cont := continuous_on.mul (continuous_on_const) fs.cont,\n  submean' := begin\n    intros c cs, rcases fs.submean' c cs with \u27e8r,rp,rm\u27e9, use [r,rp], intros s sp sr, specialize rm s sp sr,\n    rw [set_average_eq, smul_eq_mul] at \u22a2 rm,\n    calc a * f c \u2264 a * (((volume Itau).to_real)\u207b\u00b9 * \u222b t in Itau, f (circle_map c s t)) : by bound\n    ... = ((volume Itau).to_real)\u207b\u00b9 * (a * \u222b t in Itau, f (circle_map c s t)) : by ring\n    ... = ((volume Itau).to_real)\u207b\u00b9 * \u222b t in Itau, a * f (circle_map c s t) : by rw integral_mul_left,\n  end,\n}\n\n-- Analytic functions equal circle means\nlemma analytic_on.circle_mean_eq {f : \u2102 \u2192 H} {c : \u2102} {r : \u211d}\n    (fa : analytic_on \u2102 f (closed_ball c r)) (rp : r > 0) : \u2a0d t in Itau, f (circle_map c r t) = f c := begin\n  have h := complex.circle_integral_sub_inv_smul_of_differentiable_on_off_countable\n    set.countable_empty  (metric.mem_ball_self rp) fa.continuous_on _,\n  {\n    simp_rw [circle_integral, deriv_circle_map, circle_map_sub_center, smul_smul, mul_comm _ I] at h,\n    field_simp [circle_map_ne_center (ne_of_gt rp)] at h,\n    rw [\u2190smul_smul, is_unit.smul_left_cancel (ne.is_unit complex.I_ne_zero)] at h,\n    rw [interval_integral.integral_of_le (le_of_lt real.two_pi_pos)] at h,\n    rw [set_average_eq, Itau, h],\n    simp, rw ennreal.to_real_of_real (le_of_lt real.two_pi_pos),\n    rw [\u2190smul_assoc, complex.real_smul], field_simp [real.pi_ne_zero],\n  }, {\n    intros z zs, rw set.diff_empty at zs,\n    exact (fa z (metric.ball_subset_closed_ball zs)).differentiable_at,\n  },\nend\n\n-- Analytic functions are harmonic\ntheorem analytic_on.harmonic_on {f : \u2102 \u2192 H} {s : set \u2102} (fa : analytic_on \u2102 f s) : harmonic_on f s := begin\n  exact {\n    cont := fa.continuous_on,\n    mean := begin intros c r rp cs, rw (fa.mono cs).circle_mean_eq rp end,\n  },\nend\n\n-- Harmonic functions compose with linear maps\ntheorem harmonic_on.linear {f : \u2102 \u2192 E} {s : set \u2102} (fh : harmonic_on f s) (g : E \u2192L[\u211d] F)\n    : harmonic_on (\u03bb z, g (f z)) s := {\n  cont := g.continuous.comp_continuous_on fh.cont,\n  mean := begin\n    intros c r rp cs,\n    rw average_linear_comm ((fh.cont.mono cs).integrable_on_sphere rp),\n    rw fh.mean c r rp cs,\n  end,\n}\n\n-- Real parts of harmonic functions are harmonic\ntheorem harmonic_on.re {f : \u2102 \u2192 \u2102} {s : set \u2102} (fh : harmonic_on f s) : harmonic_on (\u03bb z, (f z).re) s := begin\n  simp only [\u2190complex.re_clm_apply], exact fh.linear _,\nend\n\n-- Complex conjugates of harmonic functions are harmonic\ntheorem harmonic_on.conj {f : \u2102 \u2192 \u2102} {s : set \u2102} (fh : harmonic_on f s) : harmonic_on (\u03bb z, conj (f z)) s := begin\n  simp only [\u2190conj_clm_apply], exact fh.linear _,\nend\n\n-- Real parts of analytic functions are subharmonic\ntheorem analytic_on.re_subharmonic_on {f : \u2102 \u2192 \u2102} {s : set \u2102} (fa : analytic_on \u2102 f s)\n    : subharmonic_on (\u03bb z, (f z).re) s := fa.harmonic_on.re.subharmonic_on\n\n-- The submean property holds at minima\nlemma minimum.submean {f : \u2102 \u2192 \u211d} {s : set \u2102} {c : \u2102} (fc : continuous_on f s) (cs : c \u2208 interior s) (fm : \u2200 z, f c \u2264 f z)\n    : \u2203 r (rp : r > 0), \u2200 s, 0 < s \u2192 s < r \u2192 f c \u2264 \u2a0d t in Itau, f (circle_map c s t) := begin\n  rcases metric.is_open_iff.mp is_open_interior c cs with \u27e8r,rp,rs\u27e9,\n  use [r,rp], intros t tp tr, rw set_average_eq,\n  have fg : \u2200 u (us : u \u2208 Itau), f c \u2264 f (circle_map c t u) := \u03bb _ _, fm _,\n  have ss : closed_ball c t \u2286 s := trans (metric.closed_ball_subset_ball tr) (trans rs interior_subset),\n  have n := nice_volume.Itau,\n  have m := set_integral_ge_of_const_le n.measurable n.ne_top fg ((fc.mono ss).integrable_on_sphere tp),\n  rw [smul_eq_mul, \u2190inv_mul_le_iff (inv_pos.mpr n.real_pos)], simp, rwa mul_comm,\nend\n\n-- max b (log \u2225f z\u2225) is subharmonic for analytic f, \u2102 case\ntheorem analytic_on.max_log_abs_subharmonic_on {f : \u2102 \u2192 \u2102} {s : set \u2102}\n    (fa : analytic_on \u2102 f s) (b : \u211d) : subharmonic_on (\u03bb z, max_log b (f z).abs) s := {\n  cont := fa.continuous_on.max_log_norm b,\n  submean' := begin\n    intros c cs,\n    by_cases bf : b.exp \u2265 (f c).abs, {\n      apply minimum.submean (fa.continuous_on.max_log_norm b) cs,\n      intro z, simp [max_log_eq_b bf, le_max_log],\n    },\n    simp at bf,\n    have anz : \u2225f c\u2225 \u2260 0 := ne_of_gt (trans (real.exp_pos _) bf),\n    have fac : continuous_at f c := fa.continuous_on.continuous_at (mem_interior_iff_mem_nhds.mp cs),\n    -- We define g carefully to avoid the logarithmic branch cut\n    generalize hh : (\u03bb z, complex.log (complex.abs (f c) / f c * f z)) = h,\n    generalize hg : (\u03bb z, (h z).re) = g,\n    have ha : analytic_at \u2102 h c, {\n      rw \u2190hh,\n      apply log_analytic_re_pos (analytic_at.mul (entire.const _ _) (fa c (interior_subset cs))),\n      simp only, field_simp [complex.abs_ne_zero.mp anz],\n    },\n    rcases metric.is_open_iff.mp (is_open_analytic_at \u2102 h) c ha with \u27e8r0,r0p,r0a\u27e9,\n    rcases metric.continuous_at_iff.mp fac ((f c).abs - b.exp) (sub_pos.mpr bf) with \u27e8r1,r1p,r1h\u27e9,\n    set r := min r0 r1,\n    have fg : set.eq_on (\u03bb z, max_log b (complex.abs (f z))) g (ball c r), {\n      intros z zs, simp [complex.dist_eq] at zs r1h, specialize r1h zs.2,\n      have zp : abs (f z) > b.exp, {\n        calc abs (f z) = abs (f c + (f z - f c)) : by ring_nf\n        ... \u2265 abs (f c) - abs (f z - f c) : by bound\n        ... > abs (f c) - (abs (f c) - b.exp) : by bound [sub_lt_sub_left]\n        ... = b.exp : by ring_nf\n      },\n      simp [max_log_eq_log (le_of_lt zp)],\n      rw [\u2190hg, \u2190hh], simp [complex.log_re],\n      field_simp [anz],\n    },\n    have gs : subharmonic_on g (ball c r), {\n      rw \u2190hg, apply analytic_on.re_subharmonic_on, intros z zs,\n      exact r0a (metric.ball_subset_ball (by bound) zs),\n    },\n    rw subharmonic_on_congr fg.symm at gs,\n    refine gs.submean' c _,\n    rw metric.is_open_ball.interior_eq, exact metric.mem_ball_self (by bound),\n  end,\n}\n\n-- If a continuous subharmonic function is maximal at the center of a ball, it is constant on the ball.\ntheorem subharmonic_on.maximum_principle_ball {f : \u2102 \u2192 \u211d} {c : \u2102} {r : \u211d}\n    (fs : subharmonic_on f (closed_ball c r)) (rp : r > 0) \n    : is_max_on f (closed_ball c r) c \u2192 \u2200 z, z \u2208 closed_ball c r \u2192 f c = f z := begin\n  intros cm g gs,\n  by_cases gc : g = c, { rw gc },\n  generalize hu : complex.abs (g - c) = u,\n  have u0 : u > 0, { rw \u2190hu, simp, contrapose gc, simp at \u22a2 gc, rw sub_eq_zero at gc, exact gc },\n  have ur : u \u2264 r, { simp [complex.dist_eq] at gs, simp [\u2190hu, gs] },\n  generalize hy : (g - c) / u = y,\n  have y1 : abs y = 1, { simp [\u2190hy, \u2190hu, gc], field_simp [complex.abs_ne_zero.mpr (sub_ne_zero.mpr gc)] },\n  generalize hs : (\u03bb t : \u211d, f (c + t*y)) \u207b\u00b9' {f c} = s,\n  have s0 : (0 : \u211d) \u2208 s := by simp [\u2190hs],\n  have us : u \u2208 s, {\n    refine is_closed.mem_of_ge_of_forall_exists_gt _ s0 (le_of_lt u0) _, {\n      rw \u2190hs, rw set.inter_comm,\n      refine continuous_on.preimage_closed_of_closed _ is_closed_Icc is_closed_singleton,\n      apply fs.cont.comp (continuous.continuous_on _) _,\n      exact continuous.add continuous_const (continuous.mul complex.continuous_of_real continuous_const),\n      intros t ts, simp at ts, simp [y1, abs_of_nonneg ts.left, trans ts.right ur],\n    }, {\n      intros t ts, rw \u2190hs at ts, simp at ts,\n      set z := c + t*y,\n      rcases ts with \u27e8fz,tp,tu\u27e9,\n      have tz : abs (z - c) = t, { simp [y1, abs_of_nonneg tp] },\n      have zs : z \u2208 ball c r, { simp [y1, abs_of_nonneg tp], exact lt_of_lt_of_le tu ur },\n      rw \u2190interior_closed_ball _ (ne_of_gt rp) at zs,\n      rcases fs.submean' z zs with \u27e8e,ep,lo\u27e9,\n      generalize he' : min (e/2) (u-t) = e',\n      have e'p : e' > 0, { rw \u2190he', bound },\n      have teu : t + e' \u2264 u, { rw \u2190he', transitivity t + (u-t), bound, simp },\n      have e's : e' < e, { rw \u2190he', exact lt_of_le_of_lt (min_le_left _ _) (by bound) },\n      specialize lo e' e'p e's,\n      rw fz at lo,\n      have ss : closed_ball z e' \u2286 closed_ball c r, {\n        apply metric.closed_ball_subset_closed_ball', rw [complex.dist_eq, tz], linarith,\n      },\n      have hi : \u2200 x, x \u2208 Itau \u2192 f (circle_map z e' x) \u2264 f c, {\n        intros x xs, apply is_max_on_iff.mp cm, apply ss,\n        simp [complex.dist_eq], simp [abs_of_pos e'p],\n      },\n      have fcc : continuous_on (\u03bb a, f (circle_map z e' a)) Itau, {\n        apply (fs.cont.mono ss).comp (continuous_circle_map _ _).continuous_on,\n        intros a as, simp [complex.dist_eq, abs_of_pos e'p],\n      },\n      have fw := mean_squeeze nice_volume.Itau local_volume.Itau fcc ((fs.cont.mono ss).integrable_on_sphere e'p) lo hi,\n      have eys : z + e'*y \u2208 sphere z e' := by simp [abs_of_pos e'p, y1],\n      rcases circle_map_Ioc eys with \u27e8a,as,aey\u27e9,\n      specialize fw a as, simp [\u2190aey] at fw,\n      existsi t+e', simp, refine \u27e8_, e'p, teu\u27e9, rw \u2190hs, simp [right_distrib],\n      rw \u2190add_assoc, exact fw, apply_instance,\n    }\n  },\n  simp [\u2190hs, \u2190hy] at us, \n  have unz : (u : \u2102) \u2260 0 := by simp [ne_of_gt u0],\n  field_simp [unz] at us, ring_nf at us, field_simp [unz] at us,\n  exact us.symm,\nend\n\n-- A subharmonic function achieves its maximum on the boundary\ntheorem subharmonic_on.maximum_principle {f : \u2102 \u2192 \u211d} {s : set \u2102}\n    (fs : subharmonic_on f s) (sc : is_compact s) (sn : s.nonempty)\n    : \u2203 w, w \u2208 frontier s \u2227 is_max_on f s w := begin\n  rcases fs.cont.compact_max sc sn with \u27e8x,xs,xm\u27e9,\n  rcases exists_mem_frontier_inf_dist_compl_eq_dist xs sc.ne_univ with \u27e8w,wb,h\u27e9,\n  existsi [w, wb],\n  generalize hr : abs (w - x) = r,\n  by_cases wx : w = x, { rwa wx },\n  have rp : r > 0, { simp [\u2190hr, complex.abs_pos], apply sub_ne_zero.mpr, exact wx },\n  rw [dist_comm, complex.dist_eq, hr] at h,\n  have rs : closed_ball x r \u2286 s, {\n    rw [\u2190closure_ball x (ne_of_gt rp), \u2190sc.is_closed.closure_eq], apply closure_mono, \n    rw \u2190h, apply metric.ball_inf_dist_compl_subset,\n  },\n  have rsi : ball x r \u2286 interior s, {\n    rw \u2190interior_closed_ball _ (ne_of_gt rp), exact interior_mono rs, apply_instance,\n  }, \n  have rm : is_max_on f (closed_ball x r) x, { intros y ys, exact xm (rs ys) },\n  have wx : f x = f w, {\n    apply subharmonic_on.maximum_principle_ball (fs.mono rs) rp rm,\n    simp [complex.dist_eq], rw hr,\n  },\n  intros y ys, rw \u2190wx, exact xm ys,\nend\n\n-- A harmonic function achieves its maximum norm on the boundary.\ntheorem harmonic_on.maximum_principle {f : \u2102 \u2192 E} {s : set \u2102}\n    (fh : harmonic_on f s) (sc : is_compact s) (sn : s.nonempty)\n    : \u2203 w, w \u2208 frontier s \u2227 \u2200 z, z \u2208 s \u2192 \u2225f z\u2225 \u2264 \u2225f w\u2225 := begin\n  rcases fh.norm.maximum_principle sc sn with \u27e8w,wf,wh\u27e9,\n  existsi [w, wf], intros z zs, specialize wh zs, simp at wh, exact wh,\nend\n\n-- Uniform limits of harmonic functions are harmonic\ntheorem uniform_harmonic_lim {f : \u2115 \u2192 \u2102 \u2192 E} {g : \u2102 \u2192 E} {s : set \u2102}\n    (h : \u2200 n, harmonic_on (f n) s) (u : tendsto_uniformly_on f g at_top s)\n    : harmonic_on g s := {\n  cont := u.continuous_on (filter.eventually_of_forall (\u03bb n, (h n).cont)),\n  mean := begin\n    intros c r rp cs,\n    have m := \u03bb n, (h n).mean c r rp cs,\n    simp_rw set_average_eq at \u22a2 m,\n    have se : Itau =\u1d50[volume] Icc 0 (2*\u03c0) := Ioc_ae_eq_Icc,\n    have vp := nice_volume.Itau.real_pos,\n    generalize hv : (volume Itau).to_real = v, simp_rw hv at \u22a2 m vp, clear hv,\n    simp_rw set_integral_congr_set_ae se at \u22a2 m,\n    have cc : Icc 0 (2*\u03c0) \u2286 circle_map c r \u207b\u00b9' s, {\n      rw set.subset_def, intros t ts, simp, apply cs,\n      simp [complex.dist_eq, abs_of_pos rp],\n    },\n    have fu := (u.comp (circle_map c r)).mono cc,\n    have fc : \u2200 n, continuous_on (\u03bb t, f n (circle_map c r t)) (Icc 0 (2*\u03c0)), {\n      intro n, apply continuous.continuous_on,\n      apply ((h n).cont.mono cs).comp_continuous (continuous_circle_map _ _),\n      intro t, simp [complex.dist_eq, abs_of_pos rp],\n    },\n    have ti' := fu.integral_tendsto fc is_compact_Icc,\n    have ti := ti'.const_smul v\u207b\u00b9, clear ti',\n    have ci := u.tendsto_at (cs (metric.mem_closed_ball_self (by bound))),\n    simp_rw \u2190m at ti,\n    exact tendsto_nhds_unique ci ti,\n  end\n}\n\nsection harmonic_extension\n\nvariables {c : \u2102} {r : \u211d}\nlemma rri (rp : r > 0) (z : \u2102) : c + r*(r\u207b\u00b9 * (z - c)) = z := begin ring_nf, field_simp [ne_of_gt rp] end\nlemma rir (rp : r > 0) (z : \u2102) : (\u2191r)\u207b\u00b9 * ((c + r*z) - c) = z := begin ring_nf, field_simp [ne_of_gt rp] end\n\n-- Harmonic extensions inwards the circle\nstructure has_extension (f : C(circle, S)) (g : \u2102 \u2192 S) (c : \u2102) (r : \u211d) : Prop :=\n  (gh : harmonic_on g (closed_ball c r))\n  (b : \u2200 z, f z = g (c + r*z))\n\ndef extendable (f : C(circle, S)) (c : \u2102) (r : \u211d) := \u2203 g : \u2102 \u2192 S, has_extension f g c r\n\n-- has_extension is linear\nlemma has_extension.sub {f0 f1 : C(circle, \u2102)} {g0 g1 : \u2102 \u2192 \u2102} \n    (e0 : has_extension f0 g0 c r) (e1 : has_extension f1 g1 c r)\n    : has_extension (f0 - f1) (g0 - g1) c r := {\n  gh := e0.gh.sub e1.gh,\n  b := by simp [e0.b, e1.b],\n}\n\nlemma extension.maximum_principle {f : C(circle, \u2102)} {g : \u2102 \u2192 \u2102} (e : has_extension f g c r) {b : \u211d}\n    (fb : \u2200 z, \u2225f z\u2225 \u2264 b) (rp : r > 0) : \u2200 z, z \u2208 closed_ball c r \u2192 \u2225g z\u2225 \u2264 b := begin\n  rcases e.gh.maximum_principle (is_compact_closed_ball _ _) _ with \u27e8w,wf,wh\u27e9,\n  intros z zs, specialize wh z zs,\n  rw [frontier_closed_ball _ (ne_of_gt rp)] at wf, simp at wf,\n  set w' := (\u2191r)\u207b\u00b9 * (w - c),\n  have wf' : abs w' = 1, { simp [wf, abs_of_pos rp], field_simp [ne_of_gt rp] },\n  rw [\u2190mem_circle_iff_abs] at wf',\n  have b := e.b \u27e8w',wf'\u27e9, simp [rri rp] at b, rw \u2190b at wh,\n  exact trans wh (fb _),\n  apply_instance,\n  exact \u27e8c, by simp [le_of_lt rp]\u27e9,\nend\n\n-- extendable is closed\nlemma is_closed.extendable {s : set C(circle, \u2102)} (e : \u2200 f, f \u2208 s \u2192 extendable f c r) (rp : r > 0)\n    : \u2200 f, f \u2208 closure s \u2192 extendable f c r := begin\n  intros F Fe,\n  rw \u2190seq_closure_eq_closure at Fe,\n  rcases Fe with \u27e8f,fs,fF\u27e9,\n  rw continuous_map.tendsto_iff_tendsto_locally_uniformly at fF, simp at fF,\n  rw tendsto_locally_uniformly_iff_tendsto_uniformly_of_compact_space at fF,\n  set g := \u03bb n, classical.some (e _ (fs n)),\n  have gs : \u2200 n, has_extension (f n) (g n) c r := \u03bb n, classical.some_spec (e _ (fs n)),\n  have cauchy : uniform_cauchy_seq_on g at_top (closed_ball c r), {\n    rw metric.uniform_cauchy_seq_on_iff,\n    simp_rw [metric.tendsto_uniformly_iff, filter.eventually_at_top] at fF,\n    intros t tp, rcases fF (t/4) (by bound) with \u27e8N,H\u27e9, existsi N,\n    intros a aN b bN z zs,\n    have eab := (gs a).sub (gs b),\n    have fab : \u2200 z : circle, \u2225f a z - f b z\u2225 \u2264 t/2, {\n      intro z,\n      have ta := H a aN z,\n      have tb := H b bN z,\n      rw \u2190dist_eq_norm, rw dist_comm at ta,\n      calc dist (f a z) (f b z) \u2264 dist (f a z) (F z) + dist (F z) (f b z) : by bound\n      ... \u2264 t/4 + t/4 : by bound\n      ... = t/2 : by ring_nf\n    },\n    have m := extension.maximum_principle eab fab rp z zs,\n    simp [complex.dist_eq] at m \u22a2, exact lt_of_le_of_lt m (by bound),\n  },\n  set G := \u03bb z, lim at_top (\u03bb n, g n z),\n  have gG : tendsto_uniformly_on g G at_top (closed_ball c r), {\n    apply uniform_cauchy_seq_on.tendsto_uniformly_on_of_tendsto cauchy,\n    intros z zs, exact (cauchy.cauchy_seq z zs).tendsto_lim,\n  },\n  existsi G, exact {\n    gh := uniform_harmonic_lim (\u03bb n, (gs n).gh) gG,\n    b := begin\n      intros z,\n      refine (filter.tendsto.lim_eq _).symm,\n      simp_rw \u2190(gs _).b,\n      exact fF.tendsto_at z,\n    end,\n  },\nend\n\n-- p is true for all integers if it is true for nonnegative and nonpositive integers\nlemma int.induction_overlap {p : \u2124 \u2192 Prop} (hi : \u2200 n : \u2115, p n) (lo : \u2200 n : \u2115, p (-n)) : \u2200 n : \u2124, p n := begin\n  intro n, induction n with n, exact hi n, exact lo (n+1),\nend\n\n-- Fourier terms extend\nlemma fourier_extend' (rp : r > 0) (n : \u2124) : extendable (fourier n) c r := begin\n  have mh : \u2200 n : \u2115, harmonic_on (\u03bb z, ((\u2191r)\u207b\u00b9 * (z - c))^n) (closed_ball c r), {\n    intro n, apply analytic_on.harmonic_on, refine analytic_on.mono _ (set.subset_univ _),\n    rw \u2190differentiable_iff_analytic (is_open_univ), apply differentiable.differentiable_on,\n    apply differentiable.pow, apply differentiable.mul (differentiable_const _),\n    apply differentiable.sub differentiable_id (differentiable_const _),\n    apply_instance,\n  },\n  induction n using int.induction_overlap, {\n    existsi (\u03bb z : \u2102, ((\u2191r)\u207b\u00b9 * (z - c))^n), exact {\n      gh := mh n,\n      b := begin simp_rw rir rp, simp [fourier] end,\n    },\n  }, {\n    existsi (\u03bb z : \u2102, conj (((\u2191r)\u207b\u00b9 * (z - c))^n)), exact {\n      gh := (mh n).conj,\n      b := begin simp_rw rir rp, simp [fourier, complex.inv_def, complex.norm_sq_eq_abs] end,\n    },\n  },\nend\n\n-- Fourier sums extend\nlemma fourier_extend {f : C(circle, \u2102)} (rp : r > 0) (s : f \u2208 submodule.span \u2102 (set.range fourier)) : extendable f c r := begin\n  apply @submodule.span_induction _ _ _ _ _ f _ (\u03bb f, extendable f c r) s, {\n    intros g gs, simp at gs, rcases gs with \u27e8n,ng\u27e9, rw \u2190ng, exact fourier_extend' rp _,\n  }, {\n    existsi (\u03bb _ : \u2102, (0 : \u2102)), exact { gh := harmonic_on.const _, b := by simp },\n  }, {\n    intros x y xe ye, rcases xe with \u27e8x',xh,xb\u27e9, rcases ye with \u27e8y',yh,yb\u27e9,\n    existsi (\u03bb z, x' z + y' z), exact { gh := xh.add yh, b := by simp [xb,yb] },\n  }, {\n    intros a x xe, rcases xe with \u27e8x',xh,xb\u27e9,\n    existsi (\u03bb z : \u2102, a * x' z), exact { gh := xh.const_mul _, b := by simp [xb] },\n  },\nend\n\n-- All continuous functions extend\nlemma continuous_extend (f : C(circle, \u2102)) (c : \u2102) (rp : r > 0) : extendable f c r := begin\n  set s := submodule.span \u2102 (set.range fourier),\n  have se : \u2200 f, f \u2208 s.carrier \u2192 extendable f c r := \u03bb f fs, fourier_extend rp fs,\n  have ce : \u2200 f, f \u2208 closure s.carrier \u2192 extendable f c r := is_closed.extendable se rp,\n  have e : closure s.carrier = s.topological_closure.carrier := rfl,\n  rw [e, span_fourier_closure_eq_top] at ce,\n  apply ce, simp,\nend\n\nend harmonic_extension\n\n-- Everything is \"harmonic\" on the null set\nlemma harmonic_on.empty {f : \u2102 \u2192 S} : harmonic_on f \u2205 := {\n  cont := continuous_on_empty _,\n  mean := begin intros z s sp zs, have b : z \u2208 \u2205 := zs (metric.mem_closed_ball_self (by bound)), simp at b, finish end,\n}\n\n-- Everything is \"harmonic\" on singletons\nlemma harmonic_on.singleton {f : \u2102 \u2192 S} {c : \u2102} : harmonic_on f {c} := {\n  cont := continuous_on_singleton _ _,\n  mean := begin\n    intros z s sp zs,\n    have zc : z \u2208 {c} := zs (metric.mem_closed_ball_self (by bound)),\n    simp at zc, rw zc at zs,\n    have sc : c + s \u2208 {c} := zs (by simp [abs_of_pos sp]),\n    simp [ne_of_gt sp] at sc, finish,\n  end,\n}\n\n-- Continuous functions on the sphere extend to harmonic functions on the ball (complex case)\nlemma continuous_to_harmonic_complex {f : \u2102 \u2192 \u2102} {c : \u2102} {r : \u211d} (fc : continuous_on f (sphere c r))\n    : \u2203 g : \u2102 \u2192 \u2102, harmonic_on g (closed_ball c r) \u2227 \u2200 z, z \u2208 sphere c r \u2192 f z = g z := begin\n  by_cases r0 : r = 0, { existsi f, simp [r0, harmonic_on.singleton] },\n  by_cases rn : r < 0, { existsi f, simp [metric.closed_ball_eq_empty.mpr rn, harmonic_on.empty] },\n  simp at rn, have rp := lt_of_le_of_ne rn (ne.symm r0), clear r0 rn,\n  generalize hf' : (\u03bb z : circle, f (c + r*z)) = f',\n  have fc' : continuous f', { rw \u2190hf', apply fc.comp_continuous, continuity, simp, bound },\n  rcases continuous_extend \u27e8f',fc'\u27e9 c rp with \u27e8g,e\u27e9,\n  existsi [g, e.gh],\n  intros z zs,\n  set z' := (\u2191r)\u207b\u00b9 * (z - c),\n  have m : z' \u2208 circle, { simp at \u22a2 zs, simp [zs, abs_of_pos rp], field_simp [ne_of_gt rp] },\n  have rr : c + r*(\u27e8_,m\u27e9 : circle) = z := rri rp _,\n  nth_rewrite 1 \u2190rr, rw \u2190e.b \u27e8_,m\u27e9, simp [\u2190hf'], ring_nf, rw [mul_comm, rri rp],\nend\n\n-- Continuous functions on the sphere extend to harmonic functions on the ball (real case)\nlemma continuous_to_harmonic_real {f : \u2102 \u2192 \u211d} {c : \u2102} {r : \u211d} (fc : continuous_on f (sphere c r))\n    : \u2203 g : \u2102 \u2192 \u211d, harmonic_on g (closed_ball c r) \u2227 \u2200 z, z \u2208 sphere c r \u2192 f z = g z := begin\n  set f' := \u03bb z, (f z : \u2102),\n  have fc' : continuous_on f' (sphere c r) := complex.continuous_of_real.comp_continuous_on fc,\n  rcases continuous_to_harmonic_complex fc' with \u27e8g,gh,b\u27e9,\n  existsi [\u03bb z, (g z).re, gh.re],\n  intros z zs, simp [\u2190b z zs],\nend\n\n-- The submean property holds globally\ntheorem subharmonic_on.submean {f : \u2102 \u2192 \u211d} {c : \u2102} {r : \u211d} (fs : subharmonic_on f (closed_ball c r)) (rp : r > 0)\n    : f c \u2264 \u2a0d t in Itau, f (circle_map c r t) := begin \n  rcases continuous_to_harmonic_real (fs.cont.mono metric.sphere_subset_closed_ball) with \u27e8g,gh,fg\u27e9,\n  generalize hd : (\u03bb z, f z - g z) = d,\n  have ds : subharmonic_on d (closed_ball c r), { rw \u2190hd, apply fs.add gh.neg.subharmonic_on },\n  have dz : \u2200 z, z \u2208 sphere c r \u2192 d z = 0, { intros z zs, simp [\u2190hd], rw fg z zs, simp },\n  have dz' : \u2200\u1d50 t, t \u2208 Itau \u2192 d (circle_map c r t) = 0, { apply ae_of_all, intros t ts, apply dz, simp, bound },\n  rcases ds.maximum_principle (is_compact_closed_ball _ _) \u27e8c, metric.mem_closed_ball_self (le_of_lt rp)\u27e9 with \u27e8w,wf,wm\u27e9,\n  rw frontier_closed_ball _ (ne_of_gt rp) at wf, swap, apply_instance,\n  have fd : f = (\u03bb z, d z + g z), { funext z, rw \u2190hd, simp },\n  simp_rw [fd, set_average.add (ds.cont.integrable_on_sphere rp) (gh.cont.integrable_on_sphere rp)],\n  simp [\u2190gh.mean c r rp (subset_refl _)],\n  simp [average_congr_on nice_volume.Itau dz'],\n  rw \u2190dz w wf, apply wm (metric.mem_closed_ball_self (le_of_lt rp)),  \nend\n\n-- A continuous function is subharmonic if it is globally subharmonic.\n-- This is useful since there are sometimes fewer technicalities in proving global subharmonicity.\nlemma subharmonic_on_iff_submean {f : \u2102 \u2192 \u211d} {s : set \u2102} (fc : continuous_on f s)\n    : subharmonic_on f s \u2194 \u2200 (c : \u2102) (r : \u211d), r > 0 \u2192 closed_ball c r \u2286 s \u2192 f c \u2264 \u2a0d t in Itau, f (circle_map c r t) := begin\n  constructor, {\n    intros fs c r rp cs, exact (fs.mono cs).submean rp,\n  }, {\n    intro sm, exact {\n      cont := fc,\n      submean' := begin\n        intros c ci,\n        rcases metric.is_open_iff.mp is_open_interior c ci with \u27e8r,rp,rs\u27e9,\n        use [r,rp], intros t tp tr, apply sm c t tp,\n        exact trans (metric.closed_ball_subset_ball tr) (trans rs interior_subset),\n      end,\n    },\n  },\nend\n\n -- The submean property holds for disks\nlemma subharmonic_on.submean_disk {f : \u2102 \u2192 \u211d} {c : \u2102} {r : \u211d} (fs : subharmonic_on f (closed_ball c r)) (rp : r > 0)\n    : f c \u2264 \u2a0d z in closed_ball c r, f z := begin\n  rw [set_average_eq, complex.volume_closed_ball' (le_of_lt rp), fubini_ball fs.cont],\n  have m : (\u03bb s, (2*\u03c0*s) \u2022 f c) \u2264\u1d50[volume.restrict (Ioc 0 r)] \u03bb s, s \u2022 \u222b (t : \u211d) in set.Ioc 0 (2*\u03c0), f (circle_map c s t), {\n    rw filter.eventually_le, rw ae_restrict_iff' measurable_set_Ioc, apply ae_of_all, intros s sr, simp at sr,\n    have e := (fs.mono (metric.closed_ball_subset_closed_ball sr.2)).submean sr.1,\n    rw smul_eq_mul, rw [set_average_eq, Itau_real_volume, smul_eq_mul] at e,\n    generalize hi : \u222b t in Itau, f (circle_map c s t) = i, rw hi at e,\n    calc 2*\u03c0*s * f c \u2264 2*\u03c0*s * ((2*\u03c0)\u207b\u00b9 * i) : by bound [real.pi_pos, sr.1]\n    ... = s * (2*\u03c0 * (2*\u03c0)\u207b\u00b9) * i : by ring_nf\n    ... \u2264 s * i : by field_simp [ne_of_gt real.two_pi_pos],\n    apply_instance, apply_instance, apply_instance,\n  },\n  have im := integral_mono_ae _ _ m, {\n    generalize hi : \u222b s in Ioc 0 r, s \u2022 \u222b t in Ioc 0 (2*\u03c0), f (circle_map c s t) = i, rw hi at im, clear hi m,\n    simp [\u2190interval_integral.integral_of_le (le_of_lt rp)] at im,\n    ring_nf at im \u22a2, rw smul_eq_mul,\n    calc f c = (r^2 * \u03c0)\u207b\u00b9 * (f c * r^2 * \u03c0) : by { ring_nf, field_simp [ne_of_gt rp, ne_of_gt real.pi_pos], ring_nf }\n    ... \u2264 (r^2 * \u03c0)\u207b\u00b9 * i : by bound [real.pi_pos]\n  }, {\n    apply continuous.integrable_on_Ioc, continuity,\n  }, {\n    refine integrable_on.mono_set _ set.Ioc_subset_Icc_self,\n    apply continuous_on.integrable_on_Icc, apply continuous_on.smul continuous_on_id, swap, apply_instance,\n    simp_rw \u2190interval_integral.integral_of_le (le_of_lt real.two_pi_pos),\n    refine continuous_on.interval_integral _ is_compact_Icc (le_of_lt real.two_pi_pos), simp [uncurry],\n    refine fs.cont.comp (continuous.continuous_on (by continuity)) _,\n    intros t ts, simp at ts, simp [complex.dist_eq, abs_of_nonneg ts.1.1, ts.2.1],\n  },\nend\n\n-- The max of two subharmonic functions is subharmonic\nlemma subharmonic_on.max {f g : \u2102 \u2192 \u211d} {s : set \u2102} (fs : subharmonic_on f s) (gs : subharmonic_on g s)\n    : subharmonic_on (\u03bb z, max (f z) (g z)) s := begin\n  simp_rw \u2190pair_max_eq,\n  have pc : continuous_on (\u03bb z, (f z, g z)) s := fs.cont.prod (gs.cont),\n  have mc : continuous_on (\u03bb z, pair_max (f z, g z)) s := continuous_max.comp_continuous_on pc,\n  rw subharmonic_on_iff_submean mc,\n  intros c r rp cs,\n  have tf : is_finite_measure (volume.restrict Itau), { refine \u27e8_\u27e9, simp, exact nice_volume.Itau.finite },\n  have pi : integrable_on (\u03bb t, (f (circle_map c r t), g (circle_map c r t))) Itau := (pc.mono cs).integrable_on_sphere rp,\n  refine trans _ (@convex_on.map_average_le _ _ _ _ _ _ _ _ _ _ tf convex_on_pair_max\n      continuous_max.continuous_on is_closed_univ _ (by simp) pi _), {\n    simp_rw pair_max, apply max_le_max, {\n      have e : \u2200 p : \u211d \u00d7 \u211d, p.fst = continuous_linear_map.fst \u211d \u211d \u211d p, { intro p, simp [continuous_linear_map.fst] },\n      rw e, rw \u2190average_linear_comm pi, simp [continuous_linear_map.fst], exact (fs.mono cs).submean rp,\n    }, {\n      have e : \u2200 p : \u211d \u00d7 \u211d, p.snd = continuous_linear_map.snd \u211d \u211d \u211d p, { intro p, simp [continuous_linear_map.snd] },\n      rw e, rw \u2190average_linear_comm pi, simp [continuous_linear_map.snd], exact (gs.mono cs).submean rp, apply_instance,\n    },\n  }, {\n    simp, exact nice_volume.Itau.ne_zero,\n  }, {\n    have e : (pair_max \u2218 \u03bb t, (f (circle_map c r t), g (circle_map c r t)))\n        = (\u03bb t, (pair_max \u2218 (\u03bb z, (f z, g z))) (circle_map c r t)), { funext t, simp },\n    rw e, refine (mc.mono cs).integrable_on_sphere rp,\n  },\nend\n\n-- The maxima of a finite set of subharmonic functions is subharmonic\nlemma subharmonic_on.range_max {f : \u2115 \u2192 \u2102 \u2192 \u211d} {s : set \u2102} (fs : \u2200 n, subharmonic_on (f n) s) (n : \u2115)\n    : subharmonic_on (\u03bb z, range_max (\u03bb k, f k z) n) s := begin\n  induction n with n h, simp [fs 0], simp, exact h.max (fs (n+1)),\nend\n\n-- Continuous, monotonic limits of subharmonic functions are subharmonic\ntheorem subharmonic_on.monotone_lim {f : \u2115 \u2192 \u2102 \u2192 \u211d} {g : \u2102 \u2192 \u211d} {s : set \u2102}\n    (fs : \u2200 n, subharmonic_on (f n) s) (fm : monotone f)\n    (ft : \u2200 z, z \u2208 s \u2192 tendsto (\u03bb n, f n z) at_top (\ud835\udcdd (g z))) (gc : continuous_on g s)\n    : subharmonic_on g s := begin\n  rw subharmonic_on_iff_submean gc, intros c r rp cs,\n  have sm := \u03bb n, ((fs n).mono cs).submean rp,\n  have r0 : r \u2265 0 := le_of_lt rp,\n  have cts : \u2200 t, circle_map c r t \u2208 s := \u03bb _, cs (circle_map_mem_closed_ball _ r0 _),\n  have mt : tendsto (\u03bb n, \u2a0d t in Itau, f n (circle_map c r t)) at_top (\ud835\udcdd (\u2a0d t in Itau, g (circle_map c r t))), {\n    simp_rw set_average_eq, apply filter.tendsto.const_smul,\n    set b' := \u03bb z, |f 0 z| + |g z|,\n    set b := \u03bb t, b' (circle_map c r t),\n    have bc' : continuous_on b' (closed_ball c r) := continuous_on.add ((fs 0).mono cs).cont.abs (gc.mono cs).abs,\n    have fcc : \u2200 n, continuous (\u03bb t, f n (circle_map c r t)) :=\n      \u03bb n, ((fs n).cont.mono cs).comp_continuous (continuous_circle_map _ _) (\u03bb t, circle_map_mem_closed_ball _ r0 _),\n    apply tendsto_integral_of_dominated_convergence b, {\n      intro n, exact (fcc n).ae_strongly_measurable,\n    }, {\n      exact bc'.integrable_on_sphere rp,\n    }, {\n      intro n, rw ae_restrict_iff' measurable_set_Itau, apply ae_of_all, intros t ts,\n      generalize hz : circle_map c r t = z,\n      have zs : z \u2208 s, { rw \u2190hz, apply cts },\n      rw real.norm_eq_abs, rw abs_le, constructor, {\n        calc -b t \u2264 -(|f 0 z| + 0) : by { rw \u2190hz, bound [neg_le_neg] }\n        ... = -|f 0 z| : by simp\n        ... \u2264 f 0 z : neg_abs_le_self _\n        ... \u2264 f n z : fm (by simp) _,\n      }, {\n        have mn : monotone (\u03bb n, f n z) := \u03bb _ _ ab, fm ab z,\n        calc f n z \u2264 g z : @monotone.ge_of_tendsto _ _ _ _ _ _ (\u03bb n, f n z) _ mn (ft z zs) n\n        ... \u2264 |g z| : by bound\n        ... = 0 + |g z| : by ring\n        ... \u2264 b t : by { rw \u2190hz, bound },\n      },\n    }, {\n      rw ae_restrict_iff' measurable_set_Itau, apply ae_of_all, intros t ts, exact ft _ (cts _),\n    },\n  },\n  exact le_of_tendsto_of_tendsto' (ft c (cs (metric.mem_closed_ball_self r0))) mt sm,\nend\n\n-- max b (log \u2225f z\u2225) is subharmonic for analytic f.\n-- Some machinery is required to handle general Banach spaces: we rewrite \u2225f z\u2225 as the limit\n-- of norms along larger and larger finite subspaces, and use the fact that linear \u2218 analytic\n-- is analytic to reduce to the case of H = \u2102.\ntheorem analytic_on.max_log_norm_subharmonic_on {f : \u2102 \u2192 H} {s : set \u2102}\n    (fa : analytic_on \u2102 f s) (b : \u211d) : subharmonic_on (\u03bb z, max_log b \u2225f z\u2225) s :=  begin\n  have gc := fa.continuous_on.max_log_norm b,\n  have ft := \u03bb z (zs : z \u2208 s), duals_lim_tendsto_max_log_norm b (f z),\n  refine subharmonic_on.monotone_lim _ _ ft gc, {\n    intro m, apply subharmonic_on.range_max, intro n, simp_rw complex.norm_eq_abs,\n    exact ((duals n).comp_analytic_on fa).max_log_abs_subharmonic_on b,\n  }, {\n    intros a b ab z, simp, apply monotone.range_max ab,\n  },\nend\n\n-- limsup -f = -liminf f\nlemma limsup.neg {f : \u2115 \u2192 \u211d} : at_top.limsup (\u03bb n, f n) = -at_top.liminf (\u03bb n, -f n) := begin\n  rw filter.limsup_eq, rw filter.liminf_eq, rw real.Inf_def,\n  have ns : -{a | \u2200\u1da0 n in at_top, a \u2264 -f n} = {a | \u2200\u1da0 n in at_top, f n \u2264 a}, { apply set.ext, simp },\n  simp_rw \u2190ns, simp,\nend\n\n-- p is true for all ennreals if it is true for \u22a4 and positive reals\nlemma ennreal.induction {p : ennreal \u2192 Prop} (pi : p \u22a4) (pf : \u2200 (x : \u211d) (xp : 0 \u2264 x), p (ennreal.of_real x)) : \u2200 e, p e := begin\n  rw ennreal.forall_ennreal, refine \u27e8_,pi\u27e9, rw nnreal.forall, simpa [\u2190ennreal.of_real_eq_coe_nnreal],\nend\n\n-- positive liminf lower bounds are equivalent over \u211d and ennreal\nlemma le_liminf_real_to_ennreal {f : \u2115 \u2192 \u211d} {c : \u211d} (fp : \u2200 n, f n \u2265 0) (cp : c > 0)\n    (h : c \u2264 at_top.liminf f) : ennreal.of_real c \u2264 at_top.liminf (\u03bb n, ennreal.of_real (f n)) := begin\n  rw filter.liminf_eq, refine le_Sup_iff.mpr _, intros b bs,\n  induction b using ennreal.induction with b, simp,\n  simp [upper_bounds] at bs, apply ennreal.of_real_le_of_real,\n  rw filter.liminf_eq at h, rw real.le_Sup_iff at h, {\n    simp at h, apply le_of_forall_small_le_add cp, intros e ep ec,\n    rcases h (-e) (neg_lt_zero.mpr ep) with \u27e8x,\u27e8n,xb\u27e9,cx\u27e9, specialize @bs (ennreal.of_real x) n,\n    have xb : x \u2264 b, {\n      rw \u2190ennreal.of_real_le_of_real_iff b_xp, apply bs,\n      intros a an, exact ennreal.of_real_le_of_real (xb a an),\n    },\n    simp at cx, rw add_comm at cx, exact trans (le_of_lt cx) (by bound),\n  }, {\n    contrapose h, rw real.Sup_of_not_bdd_above h, simp, exact cp,\n  }, {\n    contrapose h, rw set.not_nonempty_iff_eq_empty at h, rw h, simp, exact cp,\n  }\nend\n\nlemma le_of_lt_imp_le {L : Type} [linear_order L] [densely_ordered L] {a b : L} (h : \u2200 c, c < a \u2192 c \u2264 b) : a \u2264 b := begin\n  contrapose h, simp at \u22a2 h, rcases exists_between h with \u27e8x,bx,xa\u27e9, exact \u27e8x,xa,bx\u27e9,\nend\n\n-- Simple characterization of c \u2264 liminf\nlemma le_liminf.simple {L : Type} [complete_linear_order L] [densely_ordered L] {f : \u2115 \u2192 L} {c : L}\n    : c \u2264 at_top.liminf f \u2194 \u2200 d, d < c \u2192 \u2200\u1da0 n in at_top, d \u2264 f n := begin\n  constructor, {\n    intros h d dc, rw [filter.liminf_eq, le_Sup_iff, upper_bounds] at h, simp at h,\n    specialize h d, contrapose h, simp [dc] at \u22a2 h,\n    intros a n an, rcases h n with \u27e8m,nm,fmd\u27e9,\n    exact trans (an m nm) (le_of_lt fmd),\n  }, {\n    intros h, rw [filter.liminf_eq, le_Sup_iff, upper_bounds], simp,\n    intros a ah, apply le_of_lt_imp_le, intros d dc, \n    rcases filter.eventually_at_top.mp (h d dc) with \u27e8n,hn\u27e9, exact ah n hn,\n  },\nend\n\nlemma ennreal.of_real_neg_lt_of_real_neg {x y : \u211d} (xy : x < y) (xn : x < 0)\n    : ennreal.of_real (-y) < ennreal.of_real (-x) := begin\n  apply (ennreal.of_real_lt_of_real_iff _).mpr, simp [xy], simp [xn],\nend\n\nlemma metric.closed_ball_subset_ball' {M : Type} [pseudo_metric_space M] {x y : M} {r s : \u211d}\n    (h : r + dist x y < s) : closed_ball x r \u2286 metric.ball y s := begin\n  intros z zs, simp at zs \u22a2,\n  calc dist z y \u2264 dist z x + dist x y : by bound\n  ... \u2264 r + dist x y : by bound\n  ... < s : by bound,\nend\n\n-- Superharmonic ennreal functions\nstructure superharmonic_on (f : \u2102 \u2192 ennreal) (s : set \u2102) : Prop :=\n  (ae_measurable : ae_measurable f (volume.restrict s))\n  (supmean : \u2200 (c : \u2102) (r : \u211d), r > 0 \u2192 closed_ball c r \u2286 s \u2192\n               f c \u2265 ennreal.of_real (\u03c0 * r^2)\u207b\u00b9 * \u222b\u207b z in closed_ball c r, f z)\n\n-- ennreal.of_real (-f) is superharmonic if f is negative superharmonic\nlemma subharmonic_on.neg {f : \u2102 \u2192 \u211d} {s : set \u2102}\n    (fs : subharmonic_on f s) (fn : \u2200 z, z \u2208 s \u2192 f z \u2264 0) (sm : measurable_set s)\n    : superharmonic_on (\u03bb z, ennreal.of_real (-f z)) s := {\n  ae_measurable := begin\n    apply ennreal.measurable_of_real.ae_measurable.comp_ae_measurable,\n    apply fs.cont.neg.ae_measurable sm,\n  end,\n  supmean := begin\n    intros c r rp cs,\n    rw \u2190of_real_integral_eq_lintegral_of_real, {\n      rw \u2190ennreal.of_real_mul, apply ennreal.of_real_le_of_real,\n      rw [integral_neg, mul_neg], apply neg_le_neg,\n      rw [\u2190complex.volume_closed_ball' (le_of_lt rp), \u2190smul_eq_mul, \u2190set_average_eq],\n      exact (fs.mono cs).submean_disk rp, bound [real.pi_pos],\n    }, {\n      exact (fs.mono cs).cont.neg.integrable_on_closed_ball,\n    }, {\n      rw filter.eventually_le, rw ae_restrict_iff' measurable_set_closed_ball, apply filter.eventually_of_forall,\n      intros z zs, simp, exact fn z (cs zs), apply_instance,\n    },\n  end,\n}\n\n-- Hartogs's lemma from https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf, superharmonic ennreal case.\n-- Superharmonic functions that are bounded below and liminf bounded pointwise are liminf bounded uniformly.\nlemma superharmonic_on.hartogs {f : \u2115 \u2192 \u2102 \u2192 ennreal} {s k : set \u2102} {c : ennreal}\n    (fs : \u2200 n, superharmonic_on (f n) s) (fc : \u2200 z, z \u2208 s \u2192 at_top.liminf (\u03bb n, f n z) \u2265 c)\n    (ck : is_compact k) (ks : k \u2286 interior s)\n    : \u2200 d, d < c \u2192 \u2200\u1da0 n in at_top, \u2200 z, z \u2208 k \u2192 f n z \u2265 d := begin\n  -- Prepare d and c\n  intros d dc,\n  by_cases dz : d = 0, { simp [dz] },\n  have dp : d > 0 := pos_iff_ne_zero.mpr dz,\n  have df : d \u2260 \u22a4 := ne_top_of_lt dc,\n  have cp : c > 0 := trans dc dp,\n  have drp : d.to_real > 0 := ennreal.to_real_pos dz df,\n  -- Choose e \u2208 (c,d) so that c \u2192 e is due to Fatou, and e \u2192 d is due to area bounding\n  rcases exists_between dc with \u27e8e,de,ec\u27e9,\n  have ep : e > 0 := trans de dp,\n  have ez : e \u2260 0 := pos_iff_ne_zero.mp ep,\n  have ef : e \u2260 \u22a4 := ne_top_of_lt ec,\n  have erp : e.to_real > 0 := ennreal.to_real_pos ez ef,\n  -- Handle induction up from small balls\n  apply is_compact.induction_on ck, {\n    simp,\n  }, {\n    intros k0 k1 k01 h1,\n    refine h1.mp (filter.eventually_of_forall _),\n    exact \u03bb n a1 z z0, a1 z (k01 z0),\n  }, {\n    intros k0 k1 h0 h1,\n    refine (h0.and h1).mp(filter.eventually_of_forall _),\n    intros n h z zs, cases zs, exact h.1 z zs, exact h.2 z zs,\n  },\n  -- Base case: Hartogs's lemma near a point.  We choose radii r1 < r2 within s, apply\n  -- Fatou's lemma at r1, use monotonicity to bound by r2 integrals, and apply the submean\n  -- property with radius r2 to get Hartogs's within radius r2-r1.\n  intros z zs,\n  rcases metric.is_open_iff.mp is_open_interior z (ks zs) with \u27e8r,rp,rs\u27e9,\n  generalize hr2 : r/2 = r2,  -- We'll use the submean property on disks of radius r2 < r\n  generalize hr1 : r2 * real.sqrt (d.to_real / e.to_real) = r1,  -- We'll apply Fatou's lemma to a disk of radius r1 < r2 < r\n  have dep : d.to_real / e.to_real > 0 := div_pos drp erp,\n  have r2p : r2 > 0, { rw \u2190hr2, bound },\n  have r1p : r1 > 0, { rw \u2190hr1, bound [real.sqrt_pos_of_pos] },\n  have r12 : r1 < r2, {\n    rw \u2190hr1, apply mul_lt_of_lt_one_right r2p, rw real.sqrt_lt (le_of_lt dep) zero_le_one, simp,\n    apply (div_lt_one erp).mpr, exact (ennreal.to_real_lt_to_real df ef).mpr de,\n  },\n  have r1r : r1 < r, { apply trans r12, rw \u2190hr2, bound },\n  have r1s : closed_ball z r1 \u2286 s := trans (metric.closed_ball_subset_ball r1r) (trans rs interior_subset),\n  have rde : d = e * (ennreal.of_real (\u03c0*r1^2) * ennreal.of_real (\u03c0*r2^2)\u207b\u00b9), {\n    rw [\u2190ennreal.of_real_mul (by bound [real.pi_pos] : \u03c0*r1^2 \u2265 0), \u2190hr1, mul_pow, real.sq_sqrt (le_of_lt dep)],\n    have smash : (\u03c0 * (r2^2 * (d.to_real / e.to_real)) * (\u03c0 * r2^2)\u207b\u00b9) = d.to_real / e.to_real, {\n      calc (\u03c0 * (r2^2 * (d.to_real / e.to_real)) * (\u03c0 * r2^2)\u207b\u00b9)\n          = (\u03c0 * (r2^2 * (d.to_real / e.to_real)) * (\u03c0\u207b\u00b9 * (r2^2)\u207b\u00b9)) : by simp_rw [mul_inv]\n      ... = d.to_real / e.to_real * (\u03c0 * \u03c0\u207b\u00b9) * (r2^2 * (r2^2)\u207b\u00b9) : by ring_nf\n      ... = d.to_real / e.to_real : by simp [mul_inv_cancel (ne_of_gt real.pi_pos),\n                                             mul_inv_cancel (pow_ne_zero _ (ne_of_gt r2p))]\n    },\n    rw [smash, ennreal.of_real_div_of_pos erp, ennreal.of_real_to_real df, ennreal.of_real_to_real ef],\n    rw ennreal.mul_div_cancel' ez ef,\n  },\n  have s12 : \u2200 w, w \u2208 (closed_ball z (r2-r1)) \u2192 closed_ball z r1 \u2286 closed_ball w r2, {\n    intros w wr, apply metric.closed_ball_subset_closed_ball', simp [dist_comm] at wr, linarith,\n  },\n  have r2s : \u2200 w, w \u2208 closed_ball z (r2-r1) \u2192 closed_ball w r2 \u2286 s, {\n    intros w ws, refine trans _ (trans rs interior_subset), simp [complex.dist_eq, \u2190hr2] at \u22a2 ws,\n    apply metric.closed_ball_subset_ball', simp [complex.dist_eq],\n    calc r/2 + abs (w - z) \u2264 r/2 + (r/2 - r1) : by bound \n    ... = r - r1 : by ring_nf\n    ... < r : sub_lt_self _ r1p\n  },\n  -- Apply Fatou's lemma to closed_ball z (r/2)\n  set fi := \u03bb z, at_top.liminf (\u03bb n, f n z),\n  have fm : \u2200 n, ae_measurable (f n) (volume.restrict (closed_ball z r1)) :=\n    \u03bb n, ae_measurable.mono_set r1s (fs n).ae_measurable,\n  have fatou' := @lintegral_liminf_le' _ _ (volume.restrict (closed_ball z r1)) f fm,\n  have im := @set_lintegral_mono_ae_measurable _ _ _ _ (closed_ball z r1) (\u03bb _, c) _ ae_measurable_const\n    (ae_measurable_liminf fm) measurable_set_closed_ball (\u03bb _ zs, fc _ (r1s zs)),\n  simp at im,\n  have vec : e * volume (closed_ball z r1) < c * volume (closed_ball z r1), {\n    have n := nice_volume.closed_ball z r1p, exact (ennreal.mul_lt_mul_right n.ne_zero n.ne_top).mpr ec,\n  },\n  have fatou := le_liminf.simple.mp (trans im fatou') (e * volume (closed_ball z r1)) vec,\n  rw complex.volume_closed_ball (le_of_lt r1p) at fatou,\n  clear fatou' im fc vec,\n  -- Within radius r2-r1, Fatou's lemma implies local Hartogs's\n  use [closed_ball z (r2-r1), mem_nhds_within_of_mem_nhds (metric.closed_ball_mem_nhds _ (by bound))],\n  refine fatou.mp (filter.eventually_of_forall _),\n  intros n fn w ws,\n  calc d = e * (ennreal.of_real (\u03c0*r1^2) * ennreal.of_real (\u03c0*r2^2)\u207b\u00b9) : by rw rde\n  ... = e * ennreal.of_real (\u03c0*r1^2) * ennreal.of_real (\u03c0*r2^2)\u207b\u00b9 : by rw mul_assoc\n  ... \u2264 (\u222b\u207b v in closed_ball z r1, f n v) * ennreal.of_real (\u03c0*r2^2)\u207b\u00b9 : ennreal.mul_right_mono fn\n  ... \u2264 (\u222b\u207b v in closed_ball w r2, f n v) * ennreal.of_real (\u03c0*r2^2)\u207b\u00b9 : ennreal.mul_right_mono (lintegral_mono_set (s12 w ws))\n  ... = ennreal.of_real (\u03c0*r2^2)\u207b\u00b9 * \u222b\u207b v in closed_ball w r2, f n v : by rw mul_comm\n  ... \u2264 f n w : (fs n).supmean w r2 r2p (r2s w ws),\nend\n\n-- Hartogs's lemma from https://www-users.cse.umn.edu/~garrett/m/complex/hartogs.pdf, real case.\n-- Subharmonic functions that are bounded above and limsup bounded pointwise are limsup bounded uniformly.\n-- I'm going to write out the definition of limsup \u2264 c since \u211d not being complete makes it otherwise complicated.\nlemma subharmonic_on.hartogs {f : \u2115 \u2192 \u2102 \u2192 \u211d} {s k : set \u2102} {c b : \u211d}\n    (fs : \u2200 n, subharmonic_on (f n) s) (fb : \u2200 n z, z \u2208 s \u2192 f n z \u2264 b)\n    (fc : \u2200 z, z \u2208 s \u2192 \u2200 d, d > c \u2192 \u2200\u1da0 n in at_top, f n z \u2264 d)\n    (ck : is_compact k) (ks : k \u2286 interior s)\n    : \u2200 d, d > c \u2192 \u2200\u1da0 n in at_top, \u2200 z, z \u2208 k \u2192 f n z \u2264 d := begin\n  -- Deal with degenerate b \u2264 c case\n  by_cases bc : b \u2264 c, {\n    exact \u03bb d dc, filter.eventually_of_forall (\u03bb n z zk,\n      trans (fb n z (trans ks interior_subset zk)) (trans bc (le_of_lt dc))),\n  },\n  simp at bc,\n  -- Port subharmonic problem to superharmonic ennreal problem\n  generalize hf' : (\u03bb n z, f n z - b) = f',\n  generalize hg : (\u03bb n z, ennreal.of_real (-f' n z)) = g,\n  have fs' : \u2200 n, subharmonic_on (f' n) s, { rw \u2190hf', exact \u03bb n, (fs n).add (harmonic_on.const _).subharmonic_on },\n  have fn' : \u2200 n z, z \u2208 interior s \u2192 f' n z \u2264 0 := \u03bb n z zs, by simp [\u2190hf', fb n z (interior_subset zs)],\n  have gs : \u2200 n, superharmonic_on (g n) (interior s), {\n    rw \u2190hg, exact \u03bb n, ((fs' n).mono interior_subset).neg (fn' n) measurable_set_interior,\n  },\n  have gc : \u2200 z, z \u2208 interior s \u2192 at_top.liminf (\u03bb n, g n z) \u2265 ennreal.of_real (b - c), {\n    intros z zs, specialize fc z (interior_subset zs), refine le_liminf.simple.mpr _,\n    intros d dc,\n    have df : d \u2260 \u22a4 := ne_top_of_lt dc,\n    have dc' : b - d.to_real > c, {\n      calc b - d.to_real > b - (ennreal.of_real (b - c)).to_real\n          : sub_lt_sub_left ((ennreal.to_real_lt_to_real df ennreal.of_real_ne_top).mpr dc) b\n      ... = b - (b - c) : by rw ennreal.to_real_of_real (le_of_lt (sub_pos.mpr bc))\n      ... = c : by ring_nf,\n    },\n    refine (fc _ dc').mp (filter.eventually_of_forall _), intros n fb,\n    calc g n z = ennreal.of_real (b - f n z) : by simp [\u2190hg, \u2190hf']\n    ... \u2265 ennreal.of_real (b - (b - d.to_real)) : by bound [ennreal.of_real_le_of_real]\n    ... = ennreal.of_real (d.to_real) : by ring_nf\n    ... = d : by rw ennreal.of_real_to_real df,\n  },\n  -- Apply Hartogs's lemma to g\n  have ks' := ks, rw \u2190interior_interior at ks',\n  have h := superharmonic_on.hartogs gs gc ck ks',\n  -- Finish up\n  intros d dc,\n  have dc' : ennreal.of_real (b - d) < ennreal.of_real (b - c), {\n    rw ennreal.of_real_lt_of_real_iff (sub_pos.mpr bc), simpa,\n  },\n  refine (h _ dc').mp (filter.eventually_of_forall _),\n  intros n hn z zk, specialize hn z zk,\n  simp [\u2190hg, \u2190hf'] at hn,\n  rw ennreal.of_real_le_of_real_iff (sub_nonneg.mpr (fb n z (interior_subset (ks zk)))) at hn,\n  rwa \u2190sub_le_sub_iff_left,\nend", "meta": {"author": "girving", "repo": "ray", "sha": "e0c501756e067711e2d3667d4b1d18045d83a313", "save_path": "github-repos/lean/girving-ray", "path": "github-repos/lean/girving-ray/ray-e0c501756e067711e2d3667d4b1d18045d83a313/src/subharmonic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.705785040214066, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3721721136589892}}
{"text": "import data.real.basic\n\ninstance notation.sub_eq_add_neg (X : Type) [has_add X] [has_neg X] :\nhas_sub X := \u27e8\u03bb a b, a + -b\u27e9\n\nclass has_field_notation (X : Type) extends has_zero X, has_one X,\nhas_mul X, has_add X, has_neg X -- note: don't use has_sub\n-- because it might cause a diamond now\n\ninstance : has_field_notation \u211d := by refine {};apply_instance\n\nclass complete_ordered_archimedean_field (X : Type) [has_field_notation X] : Prop :=\n(all_the_axioms : \u2200 x y : X, x + y = y + x) -- ...\n(etc : \u2200 x : X, 1 * x = 1)\n\n-- boring proof omitted\ninstance cauchy_reals_are_complete : complete_ordered_archimedean_field \u211d := sorry\n\nstructure Reals :=\n(X : Type)\n[h : has_field_notation X]\n(e : complete_ordered_archimedean_field X)\n\ndef CauchyReals : Reals :=\n{ X := \u211d,\n  h := by apply_instance,\n  e := cauchy_reals_are_complete\n}\n\nstructure ordered_field_equiv (X Y : Type) [has_field_notation X] [has_field_notation Y]\n  [complete_ordered_archimedean_field X] [complete_ordered_archimedean_field Y] \n  extends equiv X Y -- add extra axioms here\n\ntheorem all_reals_are_the_same (X : Type) [has_field_notation X]\n  [complete_ordered_archimedean_field X] :\n\u2203 e : ordered_field_equiv X \u211d, true := sorry -- boring proof omitted\n\nclass platonist_friendly (P : set Reals) : Prop :=\n(e : \u2200 X : Reals, P X \u2194 P CauchyReals)\n\ninstance (P Q : set Reals) [platonist_friendly P] [platonist_friendly Q] :\nplatonist_friendly (P \u2229 Q) := \u27e8\u03bb X, begin\n  show P X \u2227 Q X \u2194 P CauchyReals \u2227 Q CauchyReals,\n  rw [_inst_1.e, _inst_2.e]\nend\u27e9\n\n-- etc\n\nexample (P : Prop) : P \u2228 \u00ac P :=\nbegin\n  induction P with a b c d e f g h i j, -- fails\nend\n\nexample (P : set Reals) : platonist_friendly P :=\nbegin\n  induction (P \u211d)\nend\n", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/canonical_isomorphism/dedekind_reals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.721743206297598, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3721451712048266}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.Match.Match\nimport Lean.Elab.RecAppSyntax\nimport Lean.Elab.PreDefinition.Basic\nimport Lean.Elab.PreDefinition.Structural.Basic\n\nnamespace Lean.Elab.Structural\nopen Meta\n\nprivate def throwToBelowFailed : MetaM \u03b1 :=\n  throwError \"toBelow failed\"\n\n/- See toBelow -/\nprivate partial def toBelowAux (C : Expr) : Expr \u2192 Expr \u2192 Expr \u2192 MetaM Expr\n  | belowDict, arg, F => do\n    let belowDict \u2190 whnf belowDict\n    trace[Elab.definition.structural] \"belowDict: {belowDict}, arg: {arg}\"\n    match belowDict with\n    | Expr.app (Expr.app (Expr.const `PProd _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `PProd.fst #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `PProd.snd #[F]))\n    | Expr.app (Expr.app (Expr.const `And _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `And.left #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `And.right #[F]))\n    | _ => forallTelescopeReducing belowDict fun xs belowDict => do\n      let argArgs := arg.getAppArgs\n      unless argArgs.size >= xs.size do throwToBelowFailed\n      let n := argArgs.size\n      let argTailArgs := argArgs.extract (n - xs.size) n\n      let belowDict := belowDict.replaceFVars xs argTailArgs\n      match belowDict with\n      | Expr.app belowDictFun belowDictArg _ =>\n        unless belowDictFun.getAppFn == C do throwToBelowFailed\n        unless \u2190 isDefEq belowDictArg arg do throwToBelowFailed\n        pure (mkAppN F argTailArgs)\n      | _ => throwToBelowFailed\n\n/- See toBelow -/\nprivate def withBelowDict (below : Expr) (numIndParams : Nat) (k : Expr \u2192 Expr \u2192 MetaM \u03b1) : MetaM \u03b1 := do\n  let belowType \u2190 inferType below\n  trace[Elab.definition.structural] \"belowType: {belowType}\"\n  belowType.withApp fun f args => do\n    let motivePos := numIndParams + 1\n    unless motivePos < args.size do throwError \"unexpected 'below' type{indentExpr belowType}\"\n    let pre := mkAppN f (args.extract 0 numIndParams)\n    let preType \u2190 inferType pre\n    forallBoundedTelescope preType (some 1) fun x _ => do\n      let motiveType \u2190 inferType x[0]\n      withLocalDeclD (\u2190 mkFreshUserName `C) motiveType fun C =>\n        let belowDict := mkApp pre C\n        let belowDict := mkAppN belowDict (args.extract (numIndParams + 1) args.size)\n        k C belowDict\n\n/-\n  `below` is a free variable with type of the form `I.below indParams motive indices major`,\n  where `I` is the name of an inductive datatype.\n\n  For example, when trying to show that the following function terminates using structural recursion\n  ```lean\n  def addAdjacent : List Nat \u2192 List Nat\n  | []       => []\n  | [a]      => [a]\n  | a::b::as => (a+b) :: addAdjacent as\n  ```\n  when we are visiting `addAdjacent as` at `replaceRecApps`, `below` has type\n  `@List.below Nat (fun (x : List Nat) => List Nat) (a::b::as)`\n  The motive `fun (x : List Nat) => List Nat` depends on the actual function we are trying to compute.\n  So, we first replace it with a fresh variable `C` at `withBelowDict`.\n  Recall that `brecOn` implements course-of-values recursion, and `below` can be viewed as a dictionary\n  of the \"previous values\".\n  We search this dictionary using the auxiliary function `toBelowAux`.\n  The dictionary is built using the `PProd` (`And` for inductive predicates).\n  We keep searching it until we find `C recArg`, where `C` is the auxiliary fresh variable created at `withBelowDict`.  -/\nprivate partial def toBelow (below : Expr) (numIndParams : Nat) (recArg : Expr) : MetaM Expr := do\n  withBelowDict below numIndParams fun C belowDict =>\n    toBelowAux C belowDict recArg below\n\n/--\n  This method is used after `matcherApp.addArg arg` to check whether the new type of `arg` has been \"refined/modified\"\n  in at least one alternative.\n-/\ndef refinedArgType (matcherApp : MatcherApp) (arg : Expr) : MetaM Bool := do\n  let argType \u2190 inferType arg\n  (Array.zip matcherApp.alts matcherApp.altNumParams).anyM fun (alt, numParams) =>\n    lambdaTelescope alt fun xs altBody => do\n      if xs.size >= numParams then\n        let refinedArg := xs[numParams - 1]\n        trace[Meta.debug] \"refinedArgType {argType} =?= {\u2190 inferType refinedArg}\"\n        return !(\u2190 isDefEq (\u2190 inferType refinedArg) argType)\n      else\n        return false\n\nprivate partial def replaceRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (below : Expr) (e : Expr) : M Expr :=\n  let rec loop (below : Expr) (e : Expr) : M Expr := do\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkLambdaFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkForallFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (\u2190 loop below type) (\u2190 loop below val) fun x => do\n        mkLetFVars #[x] (\u2190 loop below (body.instantiate1 x)) (usedLetOnly := false)\n    | Expr.mdata d b _   =>\n      if let some stx := getRecAppSyntax? e then\n        loop below b\n      else\n        return mkMData d (\u2190 loop below b)\n    | Expr.proj n i e _  => return mkProj n i (\u2190 loop below e)\n    | Expr.app _ _ _ =>\n      let processApp (e : Expr) : M Expr :=\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let numFixed  := recArgInfo.fixedParams.size\n            let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n            if recArgPos >= args.size then\n              throwError \"insufficient number of parameters at recursive application {indentExpr e}\"\n            let recArg := args[recArgPos]\n            -- For reflexive type, we may have nested recursive applications in recArg\n            let recArg \u2190 loop below recArg\n            let f \u2190 try toBelow below recArgInfo.indParams.size recArg catch  _ => throwError \"failed to eliminate recursive application{indentExpr e}\"\n            -- Recall that the fixed parameters are not in the scope of the `brecOn`. So, we skip them.\n            let argsNonFixed := args.extract numFixed args.size\n            -- The function `f` does not explicitly take `recArg` and its indices as arguments. So, we skip them too.\n            let mut fArgs := #[]\n            for i in [:argsNonFixed.size] do\n              if recArgInfo.pos != i && !recArgInfo.indicesPos.contains i then\n                let arg := argsNonFixed[i]\n                let arg \u2190 replaceRecApps recFnName recArgInfo below arg\n                fArgs := fArgs.push arg\n            return mkAppN f fArgs\n          else\n            return mkAppN (\u2190 loop below f) (\u2190 args.mapM (loop below))\n      let matcherApp? \u2190 matchMatcherApp? e\n      match matcherApp? with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo.recArgPos e then\n          processApp e\n        else\n          /- Here is an example we currently not handle\n             ```\n             def g (xs : List Nat) : Nat :=\n             match xs with\n             | [] => 0\n             | y::ys =>\n               match ys with\n               | []       => 1\n               | _::_::zs => g zs + 1\n               | zs       => g ys + 2\n             ```\n             We are matching on `ys`, but still using `ys` in the third alternative.\n             If we push the `below` argument over the dependent match it will be able to eliminate recursive call using `zs`.\n             To make it work, users have to write the third alternative as `| zs => g zs + 2`\n             If this is too annoying in practice, we may replace `ys` with the matching term, but\n             this may generate weird error messages, when it doesn't work. -/\n          trace[Elab.definition.structural] \"below before matcherApp.addArg: {below} : {\u2190 inferType below}\"\n          let matcherApp \u2190 mapError (matcherApp.addArg below) (fun msg => \"failed to add `below` argument to 'matcher' application\" ++ indentD msg)\n          if !(\u2190 refinedArgType matcherApp below) then\n            processApp e\n          else\n            let altsNew \u2190 (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n              lambdaTelescope alt fun xs altBody => do\n                trace[Elab.definition.structural] \"altNumParams: {numParams}, xs: {xs}\"\n                unless xs.size >= numParams do\n                  throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n                let belowForAlt := xs[numParams - 1]\n                mkLambdaFVars xs (\u2190 loop belowForAlt altBody)\n            pure { matcherApp with alts := altsNew }.toExpr\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop below e\n\ndef mkBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  trace[Elab.definition.structural] \"mkBRecOn: {value}\"\n  let type  := (\u2190 inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive \u2190 mkForallFVars otherArgs type\n  let mut brecOnUniv \u2190 getLevel motive\n  trace[Elab.definition.structural] \"brecOn univ: {brecOnUniv}\"\n  let useBInductionOn := recArgInfo.reflexive && brecOnUniv == levelZero\n  if recArgInfo.reflexive && brecOnUniv != levelZero then\n    brecOnUniv \u2190 decLevel brecOnUniv\n  let motive \u2190 mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n    if useBInductionOn then\n      Lean.mkConst (mkBInductionOnName recArgInfo.indName) recArgInfo.indLevels\n    else\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) (brecOnUniv :: recArgInfo.indLevels)\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType \u2190 inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]\n    let FType \u2190 inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType \u2190 instantiateForall FType recArgInfo.indIndices\n    let FType \u2190 instantiateForall FType #[major]\n    forallBoundedTelescope FType (some 1) fun below _ => do\n      -- TODO: `below` user name is `f`, and it will make a global `f` to be pretty printed as `_root_.f` in error messages.\n      -- We should add an option to `forallBoundedTelescope` to ensure fresh names are used.\n      let below := below[0]\n      let valueNew     \u2190 replaceRecApps recFnName recArgInfo below value\n      let Farg         \u2190 mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n      let brecOn       := mkApp brecOn Farg\n      return mkAppN brecOn otherArgs\n\nend Lean.Elab.Structural\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/src/Lean/Elab/PreDefinition/Structural/BRecOn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757646140788308, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3720809738798834}}
{"text": "import data.rat.denumerable\n\ntheorem t003 : denumerable \u211a := by apply_instance\n", "meta": {"author": "AtnNn", "repo": "lean-sandbox", "sha": "8c68afbdc09213173aef1be195da7a9a86060a97", "save_path": "github-repos/lean/AtnNn-lean-sandbox", "path": "github-repos/lean/AtnNn-lean-sandbox/lean-sandbox-8c68afbdc09213173aef1be195da7a9a86060a97/src/100_theorems/t003.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3720809666890706}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.basic\n\n/-!\n# A computable model of ZFA without infinity\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can be thought of (but aren't implemented) as a list of ZFA lists (not\n  necessarily proper).\n\nFor example, `lists \u2115` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `\u03b1` could be modelled as\n`\u03b1` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-steps definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `lists' \u03b1 ff`: Atoms as ZFA prelists. Basically a copy of `\u03b1`.\n* `lists' \u03b1 tt`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist (`lists'.nil`)\n  and from appending a ZFA prelist to a proper ZFA prelist (`lists'.cons a l`).\n* `lists \u03b1`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n* `finsets`: ZFA sets. Defined as `lists` quotiented by `lists.equiv`, the extensional equivalence.\n-/\n\nvariables {\u03b1 : Type*}\n\n/-- Prelists, helper type to define `lists`. `lists' \u03b1 ff` are the \"atoms\", a copy of `\u03b1`.\n`lists' \u03b1 tt` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and from\nappending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything to an\natom while having only one appending function for appending both atoms and proper ZFC prelists to a\nproper ZFA prelist. -/\n@[derive decidable_eq]\ninductive {u} lists' (\u03b1 : Type u) : bool \u2192 Type u\n| atom : \u03b1 \u2192 lists' ff\n| nil : lists' tt\n| cons' {b} : lists' b \u2192 lists' tt \u2192 lists' tt\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = ff`), corresponding\nto an element of `\u03b1`, or a \"proper\" ZFA list, inductively defined from the empty ZFA list and from\nappending a ZFA list to a proper ZFA list. -/\ndef lists (\u03b1 : Type*) := \u03a3 b, lists' \u03b1 b\n\nnamespace lists'\n\ninstance [inhabited \u03b1] : \u2200 b, inhabited (lists' \u03b1 b)\n| tt := \u27e8nil\u27e9\n| ff := \u27e8atom default\u27e9\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : lists \u03b1 \u2192 lists' \u03b1 tt \u2192 lists' \u03b1 tt\n| \u27e8b, a\u27e9 l := cons' a l\n\n/-- Converts a ZFA prelist to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : \u2200 {b}, lists' \u03b1 b \u2192 list (lists \u03b1)\n| _ (atom a)    := []\n| _ nil         := []\n| _ (cons' a l) := \u27e8_, a\u27e9 :: l.to_list\n\n@[simp] theorem to_list_cons (a : lists \u03b1) (l) :\n  to_list (cons a l) = a :: l.to_list :=\nby cases a; simp [cons]\n\n/-- Converts a `list` of ZFA lists to a proper ZFA prelist. -/\n@[simp] def of_list : list (lists \u03b1) \u2192 lists' \u03b1 tt\n| []       := nil\n| (a :: l) := cons a (of_list l)\n\n@[simp] theorem to_of_list (l : list (lists \u03b1)) : to_list (of_list l) = l :=\nby induction l; simp *\n\n@[simp] theorem of_to_list : \u2200 (l : lists' \u03b1 tt), of_list (to_list l) = l :=\nsuffices \u2200 b (h : tt = b) (l : lists' \u03b1 b),\n  let l' : lists' \u03b1 tt := by rw h; exact l in\n  of_list (to_list l') = l', from this _ rfl,\n\u03bb b h l, begin\n  induction l, {cases h}, {exact rfl},\n  case lists'.cons' : b a l IH\u2081 IH\u2082\n  { intro, change l' with cons' a l,\n    simpa [cons] using IH\u2082 rfl }\nend\n\nend lists'\n\nmutual inductive lists.equiv, lists'.subset\nwith lists.equiv : lists \u03b1 \u2192 lists \u03b1 \u2192 Prop\n| refl (l) : lists.equiv l l\n| antisymm {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  lists'.subset l\u2081 l\u2082 \u2192 lists'.subset l\u2082 l\u2081 \u2192 lists.equiv \u27e8_, l\u2081\u27e9 \u27e8_, l\u2082\u27e9\nwith lists'.subset : lists' \u03b1 tt \u2192 lists' \u03b1 tt \u2192 Prop\n| nil {l} : lists'.subset lists'.nil l\n| cons {a a' l l'} : lists.equiv a a' \u2192 a' \u2208 lists'.to_list l' \u2192\n  lists'.subset l l' \u2192 lists'.subset (lists'.cons a l) l'\nlocal infix ` ~ `:50 := lists.equiv\n\n/-- Equivalence of ZFA lists. Defined inductively. -/\nadd_decl_doc lists.equiv\n\n/-- Subset relation for ZFA lists. Defined inductively. -/\nadd_decl_doc lists'.subset\n\nnamespace lists'\n\ninstance : has_subset (lists' \u03b1 tt) := \u27e8lists'.subset\u27e9\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : has_mem (lists \u03b1) (lists' \u03b1 b) :=\n\u27e8\u03bb a l, \u2203 a' \u2208 l.to_list, a ~ a'\u27e9\n\ntheorem mem_def {b a} {l : lists' \u03b1 b} :\n  a \u2208 l \u2194 \u2203 a' \u2208 l.to_list, a ~ a' := iff.rfl\n\n@[simp] theorem mem_cons {a y l} : a \u2208 @cons \u03b1 y l \u2194 a ~ y \u2228 a \u2208 l :=\nby simp [mem_def, or_and_distrib_right, exists_or_distrib]\n\ntheorem cons_subset {a} {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  lists'.cons a l\u2081 \u2286 l\u2082 \u2194 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082 :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb \u27e8\u27e8a', m, e\u27e9, s\u27e9, subset.cons e m s\u27e9,\n  generalize_hyp h' : lists'.cons a l\u2081 = l\u2081' at h,\n  cases h with l a' a'' l l' e m s, {cases a, cases h'},\n  cases a, cases a', cases h', exact \u27e8\u27e8_, m, e\u27e9, s\u27e9\nend\n\ntheorem of_list_subset {l\u2081 l\u2082 : list (lists \u03b1)} (h : l\u2081 \u2286 l\u2082) :\n  lists'.of_list l\u2081 \u2286 lists'.of_list l\u2082 :=\nbegin\n  induction l\u2081, {exact subset.nil},\n  refine subset.cons (lists.equiv.refl _) _ (l\u2081_ih (list.subset_of_cons_subset h)),\n  simp at h, simp [h]\nend\n\n@[refl] theorem subset.refl {l : lists' \u03b1 tt} : l \u2286 l :=\nby rw \u2190 lists'.of_to_list l; exact\n   of_list_subset (list.subset.refl _)\n\ntheorem subset_nil {l : lists' \u03b1 tt} :\n  l \u2286 lists'.nil \u2192 l = lists'.nil :=\nbegin\n  rw \u2190 of_to_list l,\n  induction to_list l; intro h, {refl},\n  rcases cons_subset.1 h with \u27e8\u27e8_, \u27e8\u27e9, _\u27e9, _\u27e9\nend\n\ntheorem mem_of_subset' {a} {l\u2081 l\u2082 : lists' \u03b1 tt}\n  (s : l\u2081 \u2286 l\u2082) (h : a \u2208 l\u2081.to_list) : a \u2208 l\u2082 :=\nbegin\n  induction s with _ a a' l l' e m s IH, {cases h},\n  simp at h, rcases h with rfl|h,\n  exacts [\u27e8_, m, e\u27e9, IH h]\nend\n\ntheorem subset_def {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  l\u2081 \u2286 l\u2082 \u2194 \u2200 a \u2208 l\u2081.to_list, a \u2208 l\u2082 :=\n\u27e8\u03bb H a, mem_of_subset' H, \u03bb H, begin\n  rw \u2190 of_to_list l\u2081,\n  revert H, induction to_list l\u2081; intro,\n  { exact subset.nil },\n  { simp at H, exact cons_subset.2 \u27e8H.1, ih H.2\u27e9 }\nend\u27e9\n\nend lists'\n\nnamespace lists\n\n/-- Sends `a : \u03b1` to the corresponding atom in `lists \u03b1`. -/\n@[pattern] def atom (a : \u03b1) : lists \u03b1 := \u27e8_, lists'.atom a\u27e9\n\n/-- Converts a proper ZFA prelist to a ZFA list. -/\n@[pattern] def of' (l : lists' \u03b1 tt) : lists \u03b1 := \u27e8_, l\u27e9\n\n/-- Converts a ZFA list to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : lists \u03b1 \u2192 list (lists \u03b1)\n| \u27e8b, l\u27e9 := l.to_list\n\n/-- Predicate stating that a ZFA list is proper. -/\ndef is_list (l : lists \u03b1) : Prop := l.1\n\n/-- Converts a `list` of ZFA lists to a ZFA list. -/\ndef of_list (l : list (lists \u03b1)) : lists \u03b1 := of' (lists'.of_list l)\n\ntheorem is_list_to_list (l : list (lists \u03b1)) : is_list (of_list l) :=\neq.refl _\n\ntheorem to_of_list (l : list (lists \u03b1)) : to_list (of_list l) = l :=\nby simp [of_list, of']\n\ntheorem of_to_list : \u2200 {l : lists \u03b1}, is_list l \u2192 of_list (to_list l) = l\n| \u27e8tt, l\u27e9 _ := by simp [of_list, of']\n\ninstance : inhabited (lists \u03b1) :=\n\u27e8of' lists'.nil\u27e9\n\ninstance [decidable_eq \u03b1] : decidable_eq (lists \u03b1) :=\nby unfold lists; apply_instance\n\ninstance [has_sizeof \u03b1] : has_sizeof (lists \u03b1) :=\nby unfold lists; apply_instance\n\n/-- A recursion principle for pairs of ZFA lists and proper ZFA prelists. -/\ndef induction_mut (C : lists \u03b1 \u2192 Sort*) (D : lists' \u03b1 tt \u2192 Sort*)\n  (C0 : \u2200 a, C (atom a)) (C1 : \u2200 l, D l \u2192 C (of' l))\n  (D0 : D lists'.nil) (D1 : \u2200 a l, C a \u2192 D l \u2192 D (lists'.cons a l)) :\n  pprod (\u2200 l, C l) (\u2200 l, D l) :=\nbegin\n  suffices : \u2200 {b} (l : lists' \u03b1 b),\n    pprod (C \u27e8_, l\u27e9) (match b, l with\n    | tt, l := D l\n    | ff, l := punit\n    end),\n  { exact \u27e8\u03bb \u27e8b, l\u27e9, (this _).1, \u03bb l, (this l).2\u27e9 },\n  intros, induction l with a b a l IH\u2081 IH\u2082,\n  { exact \u27e8C0 _, \u27e8\u27e9\u27e9 },\n  { exact \u27e8C1 _ D0, D0\u27e9 },\n  { suffices, {exact \u27e8C1 _ this, this\u27e9},\n    exact D1 \u27e8_, _\u27e9 _ IH\u2081.1 IH\u2082.2 }\nend\n\n/-- Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a\nproper ZFA prelist. An atom has no members. -/\ndef mem (a : lists \u03b1) : lists \u03b1 \u2192 Prop\n| \u27e8ff, l\u27e9 := false\n| \u27e8tt, l\u27e9 := a \u2208 l\n\ninstance : has_mem (lists \u03b1) (lists \u03b1) := \u27e8mem\u27e9\n\ntheorem is_list_of_mem {a : lists \u03b1} : \u2200 {l : lists \u03b1}, a \u2208 l \u2192 is_list l\n| \u27e8_, lists'.nil\u27e9       _ := rfl\n| \u27e8_, lists'.cons' _ _\u27e9 _ := rfl\n\ntheorem equiv.antisymm_iff {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  of' l\u2081 ~ of' l\u2082 \u2194 l\u2081 \u2286 l\u2082 \u2227 l\u2082 \u2286 l\u2081 :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb \u27e8h\u2081, h\u2082\u27e9, equiv.antisymm h\u2081 h\u2082\u27e9,\n  cases h with _ _ _ h\u2081 h\u2082,\n  { simp [lists'.subset.refl] }, { exact \u27e8h\u2081, h\u2082\u27e9 }\nend\n\nattribute [refl] equiv.refl\n\ntheorem equiv_atom {a} {l : lists \u03b1} : atom a ~ l \u2194 atom a = l :=\n\u27e8\u03bb h, by cases h; refl, \u03bb h, h \u25b8 equiv.refl _\u27e9\n\ntheorem equiv.symm {l\u2081 l\u2082 : lists \u03b1} (h : l\u2081 ~ l\u2082) : l\u2082 ~ l\u2081 :=\nby cases h with _ _ _ h\u2081 h\u2082; [refl, exact equiv.antisymm h\u2082 h\u2081]\n\ntheorem equiv.trans : \u2200 {l\u2081 l\u2082 l\u2083 : lists \u03b1}, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083 :=\nbegin\n  let trans := \u03bb (l\u2081 : lists \u03b1), \u2200 \u2983l\u2082 l\u2083\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083,\n  suffices : pprod (\u2200 l\u2081, trans l\u2081)\n    (\u2200 (l : lists' \u03b1 tt) (l' \u2208 l.to_list), trans l'), {exact this.1},\n  apply induction_mut,\n  { intros a l\u2082 l\u2083 h\u2081 h\u2082,\n    rwa \u2190 equiv_atom.1 h\u2081 at h\u2082 },\n  { intros l\u2081 IH l\u2082 l\u2083 h\u2081 h\u2082,\n    cases h\u2081 with _ _ l\u2082, {exact h\u2082},\n    cases h\u2082 with _ _ l\u2083, {exact h\u2081},\n    cases equiv.antisymm_iff.1 h\u2081 with hl\u2081 hr\u2081,\n    cases equiv.antisymm_iff.1 h\u2082 with hl\u2082 hr\u2082,\n    apply equiv.antisymm_iff.2; split; apply lists'.subset_def.2,\n    { intros a\u2081 m\u2081,\n      rcases lists'.mem_of_subset' hl\u2081 m\u2081 with \u27e8a\u2082, m\u2082, e\u2081\u2082\u27e9,\n      rcases lists'.mem_of_subset' hl\u2082 m\u2082 with \u27e8a\u2083, m\u2083, e\u2082\u2083\u27e9,\n      exact \u27e8a\u2083, m\u2083, IH _ m\u2081 e\u2081\u2082 e\u2082\u2083\u27e9 },\n    { intros a\u2083 m\u2083,\n      rcases lists'.mem_of_subset' hr\u2082 m\u2083 with \u27e8a\u2082, m\u2082, e\u2083\u2082\u27e9,\n      rcases lists'.mem_of_subset' hr\u2081 m\u2082 with \u27e8a\u2081, m\u2081, e\u2082\u2081\u27e9,\n      exact \u27e8a\u2081, m\u2081, (IH _ m\u2081 e\u2082\u2081.symm e\u2083\u2082.symm).symm\u27e9 } },\n  { rintro _ \u27e8\u27e9 },\n  { intros a l IH\u2081 IH\u2082, simpa [IH\u2081] using IH\u2082 }\nend\n\ninstance : setoid (lists \u03b1) :=\n\u27e8(~), equiv.refl, @equiv.symm _, @equiv.trans _\u27e9\n\nsection decidable\n\n@[simp] def equiv.decidable_meas :\n  (psum (\u03a3' (l\u2081 : lists \u03b1), lists \u03b1) $\n   psum (\u03a3' (l\u2081 : lists' \u03b1 tt), lists' \u03b1 tt)\n   \u03a3' (a : lists \u03b1), lists' \u03b1 tt) \u2192 \u2115\n| (psum.inl \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n| (psum.inr $ psum.inl \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n| (psum.inr $ psum.inr \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n\nopen well_founded_tactics\n\ntheorem sizeof_pos {b} (l : lists' \u03b1 b) : 0 < sizeof l :=\nby cases l; unfold_sizeof; trivial_nat_lt\n\ntheorem lt_sizeof_cons' {b} (a : lists' \u03b1 b) (l) :\n  sizeof (\u27e8b, a\u27e9 : lists \u03b1) < sizeof (lists'.cons' a l) :=\nby {unfold_sizeof, apply sizeof_pos}\n\n@[instance] mutual def equiv.decidable, subset.decidable, mem.decidable [decidable_eq \u03b1]\nwith equiv.decidable : \u2200 l\u2081 l\u2082 : lists \u03b1, decidable (l\u2081 ~ l\u2082)\n| \u27e8ff, l\u2081\u27e9 \u27e8ff, l\u2082\u27e9 := decidable_of_iff' (l\u2081 = l\u2082) $\n  by cases l\u2081; refine equiv_atom.trans (by simp [atom])\n| \u27e8ff, l\u2081\u27e9 \u27e8tt, l\u2082\u27e9 := is_false $ by rintro \u27e8\u27e9\n| \u27e8tt, l\u2081\u27e9 \u27e8ff, l\u2082\u27e9 := is_false $ by rintro \u27e8\u27e9\n| \u27e8tt, l\u2081\u27e9 \u27e8tt, l\u2082\u27e9 := begin\n  haveI :=\n    have sizeof l\u2081 + sizeof l\u2082 <\n         sizeof (\u27e8tt, l\u2081\u27e9 : lists \u03b1) + sizeof (\u27e8tt, l\u2082\u27e9 : lists \u03b1),\n    by default_dec_tac,\n    subset.decidable l\u2081 l\u2082,\n  haveI :=\n    have sizeof l\u2082 + sizeof l\u2081 <\n         sizeof (\u27e8tt, l\u2081\u27e9 : lists \u03b1) + sizeof (\u27e8tt, l\u2082\u27e9 : lists \u03b1),\n    by default_dec_tac,\n    subset.decidable l\u2082 l\u2081,\n  exact decidable_of_iff' _ equiv.antisymm_iff,\nend\nwith subset.decidable : \u2200 l\u2081 l\u2082 : lists' \u03b1 tt, decidable (l\u2081 \u2286 l\u2082)\n| lists'.nil l\u2082 := is_true subset.nil\n| (@lists'.cons' _ b a l\u2081) l\u2082 := begin\n  haveI :=\n    have sizeof (\u27e8b, a\u27e9 : lists \u03b1) + sizeof l\u2082 <\n         sizeof (lists'.cons' a l\u2081) + sizeof l\u2082,\n    from add_lt_add_right (lt_sizeof_cons' _ _) _,\n    mem.decidable \u27e8b, a\u27e9 l\u2082,\n  haveI :=\n    have sizeof l\u2081 + sizeof l\u2082 <\n         sizeof (lists'.cons' a l\u2081) + sizeof l\u2082,\n    by default_dec_tac,\n    subset.decidable l\u2081 l\u2082,\n  exact decidable_of_iff' _ (@lists'.cons_subset _ \u27e8_, _\u27e9 _ _)\nend\nwith mem.decidable : \u2200 (a : lists \u03b1) (l : lists' \u03b1 tt), decidable (a \u2208 l)\n| a lists'.nil := is_false $ by rintro \u27e8_, \u27e8\u27e9, _\u27e9\n| a (lists'.cons' b l\u2082) := begin\n  haveI :=\n    have sizeof a + sizeof (\u27e8_, b\u27e9 : lists \u03b1) <\n         sizeof a + sizeof (lists'.cons' b l\u2082),\n    from add_lt_add_left (lt_sizeof_cons' _ _) _,\n    equiv.decidable a \u27e8_, b\u27e9,\n  haveI :=\n    have sizeof a + sizeof l\u2082 <\n         sizeof a + sizeof (lists'.cons' b l\u2082),\n    by default_dec_tac,\n    mem.decidable a l\u2082,\n  refine decidable_of_iff' (a ~ \u27e8_, b\u27e9 \u2228 a \u2208 l\u2082) _,\n  rw \u2190 lists'.mem_cons, refl\nend\nusing_well_founded\n{ rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf equiv.decidable_meas\u27e9],\n  dec_tac := `[assumption] }\n\nend decidable\n\nend lists\n\nnamespace lists'\n\ntheorem mem_equiv_left {l : lists' \u03b1 tt} :\n  \u2200 {a a'}, a ~ a' \u2192 (a \u2208 l \u2194 a' \u2208 l) :=\nsuffices \u2200 {a a'}, a ~ a' \u2192 a \u2208 l \u2192 a' \u2208 l,\n  from \u03bb a a' e, \u27e8this e, this e.symm\u27e9,\n\u03bb a\u2081 a\u2082 e\u2081 \u27e8a\u2083, m\u2083, e\u2082\u27e9, \u27e8_, m\u2083, e\u2081.symm.trans e\u2082\u27e9\n\ntheorem mem_of_subset {a} {l\u2081 l\u2082 : lists' \u03b1 tt}\n  (s : l\u2081 \u2286 l\u2082) : a \u2208 l\u2081 \u2192 a \u2208 l\u2082 | \u27e8a', m, e\u27e9 :=\n(mem_equiv_left e).2 (mem_of_subset' s m)\n\ntheorem subset.trans {l\u2081 l\u2082 l\u2083 : lists' \u03b1 tt}\n  (h\u2081 : l\u2081 \u2286 l\u2082) (h\u2082 : l\u2082 \u2286 l\u2083) : l\u2081 \u2286 l\u2083 :=\nsubset_def.2 $ \u03bb a\u2081 m\u2081, mem_of_subset h\u2082 $ mem_of_subset' h\u2081 m\u2081\n\nend lists'\n\ndef finsets (\u03b1 : Type*) := quotient (@lists.setoid \u03b1)\n\nnamespace finsets\n\ninstance : has_emptyc (finsets \u03b1) := \u27e8\u27e6lists.of' lists'.nil\u27e7\u27e9\n\ninstance : inhabited (finsets \u03b1) := \u27e8\u2205\u27e9\n\ninstance [decidable_eq \u03b1] : decidable_eq (finsets \u03b1) :=\nby unfold finsets; apply_instance\n\nend finsets\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/set_theory/lists.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.37203701313221454}}
{"text": "import category_theory.monoidal.coherence\n\nopen category_theory\n\nuniverses w v u\n\nsection bicategory\nopen_locale bicategory\nvariables {B : Type u} [bicategory.{w v} B] {a b c d e : B}\n\nexample : (\u03bb_ (\ud835\udfd9 a)).hom = (\u03c1_ (\ud835\udfd9 a)).hom := by coherence\nexample : (\u03bb_ (\ud835\udfd9 a)).inv = (\u03c1_ (\ud835\udfd9 a)).inv := by coherence\nexample (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b1_ f g h).inv \u226b (\u03b1_ f g h).hom = \ud835\udfd9 (f \u226b g \u226b h) :=\nby coherence\nexample (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv =\n    (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i :=\nby coherence\nexample (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03bb_ g).inv \u226b (\u03b1_ f (\ud835\udfd9 b) g).inv = (\u03c1_ f).inv \u25b7 g :=\nby coherence\nexample (f g : a \u27f6 a) (\u03b7 : \ud835\udfd9 a \u27f6 f) (\u03b8 : f \u27f6 g) (w : false) :\n  (\u03bb_ (\ud835\udfd9 a)).hom \u226b \u03b7 \u226b \ud835\udfd9 f \u226b \u03b8 = (\u03c1_ (\ud835\udfd9 a)).hom \u226b \u03b7 \u226b \u03b8 :=\nby coherence\n\nexample (f\u2081 : a \u27f6 b) (g\u2081 : b \u27f6 a) (f\u2082 : b \u27f6 c) (g\u2082 : c \u27f6 b) :\n  (\u03b1_ (\ud835\udfd9 a) (\ud835\udfd9 a) (f\u2081 \u226b f\u2082)).hom \u226b\n    \ud835\udfd9 a \u25c1 (\u03b1_ (\ud835\udfd9 a) f\u2081 f\u2082).inv \u226b\n      \ud835\udfd9 a \u25c1 ((\u03bb_ f\u2081).hom \u226b (\u03c1_ f\u2081).inv) \u25b7 f\u2082 \u226b\n        \ud835\udfd9 a \u25c1 (\u03b1_ f\u2081 (\ud835\udfd9 b) f\u2082).hom \u226b\n          (\u03b1_ (\ud835\udfd9 a) f\u2081 (\ud835\udfd9 b \u226b f\u2082)).inv \u226b\n            ((\u03bb_ f\u2081).hom \u226b (\u03c1_ f\u2081).inv) \u25b7 (\ud835\udfd9 b \u226b f\u2082) \u226b\n              (\u03b1_ f\u2081 (\ud835\udfd9 b) (\ud835\udfd9 b \u226b f\u2082)).hom \u226b\n                f\u2081 \u25c1 \ud835\udfd9 b \u25c1 ((\u03bb_ f\u2082).hom \u226b (\u03c1_ f\u2082).inv) \u226b\n                  f\u2081 \u25c1 (\u03b1_ (\ud835\udfd9 b) f\u2082 (\ud835\udfd9 c)).inv \u226b\n                    f\u2081 \u25c1 ((\u03bb_ f\u2082).hom \u226b (\u03c1_ f\u2082).inv) \u25b7 \ud835\udfd9 c \u226b\n                      (f\u2081 \u25c1 (\u03b1_ f\u2082 (\ud835\udfd9 c) (\ud835\udfd9 c)).hom) \u226b\n                        (\u03b1_ f\u2081 f\u2082 (\ud835\udfd9 c \u226b \ud835\udfd9 c)).inv =\n  ((\u03bb_ (\ud835\udfd9 a)).hom \u25b7 (f\u2081 \u226b f\u2082) \u226b (\u03bb_ (f\u2081 \u226b f\u2082)).hom \u226b (\u03c1_ (f\u2081 \u226b f\u2082)).inv) \u226b\n    (f\u2081 \u226b f\u2082) \u25c1 (\u03bb_ (\ud835\udfd9 c)).inv :=\nby coherence\n\nend bicategory\n\nsection monoidal\nvariables {C : Type u} [category.{v} C] [monoidal_category C]\n\nexample : (\u03bb_ (\ud835\udfd9_ C)).hom = (\u03c1_ (\ud835\udfd9_ C)).hom := by coherence\nexample : (\u03bb_ (\ud835\udfd9_ C)).inv = (\u03c1_ (\ud835\udfd9_ C)).inv := by coherence\nexample (X Y Z : C) : (\u03b1_ X Y Z).inv \u226b (\u03b1_ X Y Z).hom = \ud835\udfd9 (X \u2297 Y \u2297 Z) := by coherence\nexample (X Y Z W : C) :\n  (\ud835\udfd9 X \u2297 (\u03b1_ Y Z W).hom) \u226b (\u03b1_ X Y (Z \u2297 W)).inv \u226b (\u03b1_ (X \u2297 Y) Z W).inv =\n    (\u03b1_ X (Y \u2297 Z) W).inv \u226b ((\u03b1_ X Y Z).inv \u2297 \ud835\udfd9 W) :=\nby coherence\nexample (X Y : C) :\n  (\ud835\udfd9 X \u2297 (\u03bb_ Y).inv) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).inv = (\u03c1_ X).inv \u2297 \ud835\udfd9 Y :=\nby coherence\nexample (X Y : C) (f : \ud835\udfd9_ C \u27f6 X) (g : X \u27f6 Y) (w : false) :\n  (\u03bb_ (\ud835\udfd9_ C)).hom \u226b f \u226b \ud835\udfd9 X \u226b g = (\u03c1_ (\ud835\udfd9_ C)).hom \u226b f \u226b g :=\nby coherence\n\nexample (X\u2081 Y\u2081 X\u2082 Y\u2082 : C) :\n  (\u03b1_ (\ud835\udfd9_ C) (\ud835\udfd9_ C) (X\u2081 \u2297 X\u2082)).hom \u226b\n    (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2081 X\u2082).inv) \u226b\n      (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ _).hom \u226b (\u03c1_ X\u2081).inv \u2297 \ud835\udfd9 X\u2082) \u226b\n        (\ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03b1_ X\u2081 (\ud835\udfd9_ C) X\u2082).hom) \u226b\n          (\u03b1_ (\ud835\udfd9_ C) X\u2081 (\ud835\udfd9_ C \u2297 X\u2082)).inv \u226b\n            ((\u03bb_ X\u2081).hom \u226b (\u03c1_ X\u2081).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C \u2297 X\u2082)) \u226b\n              (\u03b1_ X\u2081 (\ud835\udfd9_ C) (\ud835\udfd9_ C \u2297 X\u2082)).hom \u226b\n                (\ud835\udfd9 X\u2081 \u2297 \ud835\udfd9 (\ud835\udfd9_ C) \u2297 (\u03bb_ X\u2082).hom \u226b (\u03c1_ X\u2082).inv) \u226b\n                  (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ (\ud835\udfd9_ C) X\u2082 (\ud835\udfd9_ C)).inv) \u226b\n                    (\ud835\udfd9 X\u2081 \u2297 (\u03bb_ X\u2082).hom \u226b (\u03c1_ X\u2082).inv \u2297 \ud835\udfd9 (\ud835\udfd9_ C)) \u226b\n                      (\ud835\udfd9 X\u2081 \u2297 (\u03b1_ X\u2082 (\ud835\udfd9_ C) (\ud835\udfd9_ C)).hom) \u226b\n                        (\u03b1_ X\u2081 X\u2082 (\ud835\udfd9_ C \u2297 \ud835\udfd9_ C)).inv =\n  (((\u03bb_ (\ud835\udfd9_ C)).hom \u2297 \ud835\udfd9 (X\u2081 \u2297 X\u2082)) \u226b (\u03bb_ (X\u2081 \u2297 X\u2082)).hom \u226b (\u03c1_ (X\u2081 \u2297 X\u2082)).inv) \u226b\n    (\ud835\udfd9 (X\u2081 \u2297 X\u2082) \u2297 (\u03bb_ (\ud835\udfd9_ C)).inv) :=\nby coherence\n\nend monoidal\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/coherence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.37167802004567857}}
{"text": "/-\nCopyright (c) 2019 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.filter.bases\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 \n\nnamespace Mathlib\n\n/-!\n# Lift filters along filter and set functions\n-/\n\nnamespace filter\n\n\n/-- A variant on `bind` using a function `g` taking a set instead of a member of `\u03b1`.\nThis is essentially a push-forward along a function mapping each set to a filter. -/\nprotected def lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : filter \u03b1) (g : set \u03b1 \u2192 filter \u03b2) : filter \u03b2 :=\n  infi fun (s : set \u03b1) => infi fun (H : s \u2208 f) => g s\n\n/-- If `(p : \u03b9 \u2192 Prop, s : \u03b9 \u2192 set \u03b1)` is a basis of a filter `f`, `g` is a monotone function\n`set \u03b1 \u2192 filter \u03b3`, and for each `i`, `(pg : \u03b2 i \u2192 Prop, sg : \u03b2 i \u2192 set \u03b1)` is a basis\nof the filter `g (s i)`, then `(\u03bb (i : \u03b9) (x : \u03b2 i), p i \u2227 pg i x, \u03bb (i : \u03b9) (x : \u03b2 i), sg i x)`\nis a basis of the filter `f.lift g`.\n\nThis basis is parametrized by `i : \u03b9` and `x : \u03b2 i`, so in order to formulate this fact using\n`has_basis` one has to use `\u03a3 i, \u03b2 i` as the index type, see `filter.has_basis.lift`.\nThis lemma states the corresponding `mem_iff` statement without using a sigma type. -/\ntheorem has_basis.mem_lift_iff {\u03b1 : Type u_1} {\u03b3 : Type u_3} {\u03b9 : Type u_2} {p : \u03b9 \u2192 Prop}\n    {s : \u03b9 \u2192 set \u03b1} {f : filter \u03b1} (hf : has_basis f p s) {\u03b2 : \u03b9 \u2192 Type u_4}\n    {pg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop} {sg : (i : \u03b9) \u2192 \u03b2 i \u2192 set \u03b3} {g : set \u03b1 \u2192 filter \u03b3}\n    (hg : \u2200 (i : \u03b9), has_basis (g (s i)) (pg i) (sg i)) (gm : monotone g) :\n    \u2200 {s : set \u03b3},\n        s \u2208 filter.lift f g \u2194 \u2203 (i : \u03b9), \u2203 (hi : p i), \u2203 (x : \u03b2 i), \u2203 (hx : pg i x), sg i x \u2286 s :=\n  sorry\n\n/-- If `(p : \u03b9 \u2192 Prop, s : \u03b9 \u2192 set \u03b1)` is a basis of a filter `f`, `g` is a monotone function\n`set \u03b1 \u2192 filter \u03b3`, and for each `i`, `(pg : \u03b2 i \u2192 Prop, sg : \u03b2 i \u2192 set \u03b1)` is a basis\nof the filter `g (s i)`, then `(\u03bb (i : \u03b9) (x : \u03b2 i), p i \u2227 pg i x, \u03bb (i : \u03b9) (x : \u03b2 i), sg i x)`\nis a basis of the filter `f.lift g`.\n\nThis basis is parametrized by `i : \u03b9` and `x : \u03b2 i`, so in order to formulate this fact using\n`has_basis` one has to use `\u03a3 i, \u03b2 i` as the index type. See also `filter.has_basis.mem_lift_iff`\nfor the corresponding `mem_iff` statement formulated without using a sigma type. -/\ntheorem has_basis.lift {\u03b1 : Type u_1} {\u03b3 : Type u_3} {\u03b9 : Type u_2} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 set \u03b1}\n    {f : filter \u03b1} (hf : has_basis f p s) {\u03b2 : \u03b9 \u2192 Type u_4} {pg : (i : \u03b9) \u2192 \u03b2 i \u2192 Prop}\n    {sg : (i : \u03b9) \u2192 \u03b2 i \u2192 set \u03b3} {g : set \u03b1 \u2192 filter \u03b3}\n    (hg : \u2200 (i : \u03b9), has_basis (g (s i)) (pg i) (sg i)) (gm : monotone g) :\n    has_basis (filter.lift f g)\n        (fun (i : sigma fun (i : \u03b9) => \u03b2 i) => p (sigma.fst i) \u2227 pg (sigma.fst i) (sigma.snd i))\n        fun (i : sigma fun (i : \u03b9) => \u03b2 i) => sg (sigma.fst i) (sigma.snd i) :=\n  sorry\n\ntheorem mem_lift_sets {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2}\n    (hg : monotone g) {s : set \u03b2} : s \u2208 filter.lift f g \u2194 \u2203 (t : set \u03b1), \u2203 (H : t \u2208 f), s \u2208 g t :=\n  sorry\n\ntheorem mem_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2} {s : set \u03b2}\n    {t : set \u03b1} (ht : t \u2208 f) (hs : s \u2208 g t) : s \u2208 filter.lift f g :=\n  iff.mp le_principal_iff\n    ((fun (this : filter.lift f g \u2264 principal s) => this)\n      (infi_le_of_le t (infi_le_of_le ht (iff.mpr le_principal_iff hs))))\n\ntheorem lift_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2} {h : filter \u03b2}\n    {s : set \u03b1} (hs : s \u2208 f) (hg : g s \u2264 h) : filter.lift f g \u2264 h :=\n  infi_le_of_le s (infi_le_of_le hs hg)\n\ntheorem le_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2} {h : filter \u03b2}\n    (hh : \u2200 (s : set \u03b1), s \u2208 f \u2192 h \u2264 g s) : h \u2264 filter.lift f g :=\n  le_infi fun (s : set \u03b1) => le_infi fun (hs : s \u2208 f) => hh s hs\n\ntheorem lift_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f\u2081 : filter \u03b1} {f\u2082 : filter \u03b1}\n    {g\u2081 : set \u03b1 \u2192 filter \u03b2} {g\u2082 : set \u03b1 \u2192 filter \u03b2} (hf : f\u2081 \u2264 f\u2082) (hg : g\u2081 \u2264 g\u2082) :\n    filter.lift f\u2081 g\u2081 \u2264 filter.lift f\u2082 g\u2082 :=\n  infi_le_infi fun (s : set \u03b1) => infi_le_infi2 fun (hs : s \u2208 f\u2082) => Exists.intro (hf hs) (hg s)\n\ntheorem lift_mono' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g\u2081 : set \u03b1 \u2192 filter \u03b2}\n    {g\u2082 : set \u03b1 \u2192 filter \u03b2} (hg : \u2200 (s : set \u03b1), s \u2208 f \u2192 g\u2081 s \u2264 g\u2082 s) :\n    filter.lift f g\u2081 \u2264 filter.lift f g\u2082 :=\n  infi_le_infi fun (s : set \u03b1) => infi_le_infi fun (hs : s \u2208 f) => hg s hs\n\ntheorem tendsto_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} {m : \u03b3 \u2192 \u03b2} {l : filter \u03b3} :\n    tendsto m l (filter.lift f g) \u2194 \u2200 (s : set \u03b1), s \u2208 f \u2192 tendsto m l (g s) :=\n  sorry\n\ntheorem map_lift_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} {m : \u03b2 \u2192 \u03b3} (hg : monotone g) :\n    map m (filter.lift f g) = filter.lift f (map m \u2218 g) :=\n  sorry\n\ntheorem comap_lift_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} {m : \u03b3 \u2192 \u03b2} (hg : monotone g) :\n    comap m (filter.lift f g) = filter.lift f (comap m \u2218 g) :=\n  sorry\n\ntheorem comap_lift_eq2 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1} {m : \u03b2 \u2192 \u03b1}\n    {g : set \u03b2 \u2192 filter \u03b3} (hg : monotone g) :\n    filter.lift (comap m f) g = filter.lift f (g \u2218 set.preimage m) :=\n  sorry\n\ntheorem map_lift_eq2 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b2 \u2192 filter \u03b3} {m : \u03b1 \u2192 \u03b2} (hg : monotone g) :\n    filter.lift (map m f) g = filter.lift f (g \u2218 set.image m) :=\n  sorry\n\ntheorem lift_comm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1} {g : filter \u03b2}\n    {h : set \u03b1 \u2192 set \u03b2 \u2192 filter \u03b3} :\n    (filter.lift f fun (s : set \u03b1) => filter.lift g (h s)) =\n        filter.lift g fun (t : set \u03b2) => filter.lift f fun (s : set \u03b1) => h s t :=\n  sorry\n\ntheorem lift_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} {h : set \u03b2 \u2192 filter \u03b3} (hg : monotone g) :\n    filter.lift (filter.lift f g) h = filter.lift f fun (s : set \u03b1) => filter.lift (g s) h :=\n  sorry\n\ntheorem lift_lift_same_le_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b1 \u2192 filter \u03b2} :\n    (filter.lift f fun (s : set \u03b1) => filter.lift f (g s)) \u2264\n        filter.lift f fun (s : set \u03b1) => g s s :=\n  sorry\n\ntheorem lift_lift_same_eq_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b1 \u2192 filter \u03b2} (hg\u2081 : \u2200 (s : set \u03b1), monotone fun (t : set \u03b1) => g s t)\n    (hg\u2082 : \u2200 (t : set \u03b1), monotone fun (s : set \u03b1) => g s t) :\n    (filter.lift f fun (s : set \u03b1) => filter.lift f (g s)) =\n        filter.lift f fun (s : set \u03b1) => g s s :=\n  sorry\n\ntheorem lift_principal {\u03b1 : Type u_1} {\u03b2 : Type u_2} {g : set \u03b1 \u2192 filter \u03b2} {s : set \u03b1}\n    (hg : monotone g) : filter.lift (principal s) g = g s :=\n  le_antisymm (infi_le_of_le s (infi_le (fun (H : s \u2208 principal s) => g s) (set.subset.refl s)))\n    (le_infi fun (t : set \u03b1) => le_infi fun (hi : t \u2208 principal s) => hg hi)\n\ntheorem monotone_lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [preorder \u03b3] {f : \u03b3 \u2192 filter \u03b1}\n    {g : \u03b3 \u2192 set \u03b1 \u2192 filter \u03b2} (hf : monotone f) (hg : monotone g) :\n    monotone fun (c : \u03b3) => filter.lift (f c) (g c) :=\n  fun (a b : \u03b3) (h : a \u2264 b) => lift_mono (hf h) (hg h)\n\ntheorem lift_ne_bot_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2}\n    (hm : monotone g) : ne_bot (filter.lift f g) \u2194 \u2200 (s : set \u03b1), s \u2208 f \u2192 ne_bot (g s) :=\n  sorry\n\n@[simp] theorem lift_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : filter \u03b2} :\n    (filter.lift f fun (x : set \u03b1) => g) = g :=\n  le_antisymm (lift_le univ_mem_sets (le_refl g))\n    (le_lift fun (s : set \u03b1) (hs : s \u2208 f) => le_refl g)\n\n@[simp] theorem lift_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 filter \u03b2}\n    {h : set \u03b1 \u2192 filter \u03b2} :\n    (filter.lift f fun (x : set \u03b1) => g x \u2293 h x) = filter.lift f g \u2293 filter.lift f h :=\n  sorry\n\n@[simp] theorem lift_principal2 {\u03b1 : Type u_1} {f : filter \u03b1} : filter.lift f principal = f := sorry\n\ntheorem lift_infi {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Sort u_4} {f : \u03b9 \u2192 filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} [h\u03b9 : Nonempty \u03b9] (hg : \u2200 {s t : set \u03b1}, g s \u2293 g t = g (s \u2229 t)) :\n    filter.lift (infi f) g = infi fun (i : \u03b9) => filter.lift (f i) g :=\n  sorry\n\n/-- Specialize `lift` to functions `set \u03b1 \u2192 set \u03b2`. This can be viewed as a generalization of `map`.\nThis is essentially a push-forward along a function mapping each set to a set. -/\nprotected def lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : filter \u03b1) (h : set \u03b1 \u2192 set \u03b2) : filter \u03b2 :=\n  filter.lift f (principal \u2218 h)\n\ntheorem mem_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2} {t : set \u03b1}\n    (ht : t \u2208 f) : h t \u2208 filter.lift' f h :=\n  iff.mp le_principal_iff\n    ((fun (this : filter.lift' f h \u2264 principal (h t)) => this)\n      (infi_le_of_le t (infi_le_of_le ht (le_refl (function.comp principal h t)))))\n\ntheorem tendsto_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {h : set \u03b1 \u2192 set \u03b2} {m : \u03b3 \u2192 \u03b2} {l : filter \u03b3} :\n    tendsto m l (filter.lift' f h) \u2194\n        \u2200 (s : set \u03b1), s \u2208 f \u2192 filter.eventually (fun (a : \u03b3) => m a \u2208 h s) l :=\n  sorry\n\ntheorem has_basis.lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    {\u03b9 : Type u_3} {p : \u03b9 \u2192 Prop} {s : \u03b9 \u2192 set \u03b1} (hf : has_basis f p s) (hh : monotone h) :\n    has_basis (filter.lift' f h) p (h \u2218 s) :=\n  sorry\n\ntheorem mem_lift'_sets {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    (hh : monotone h) {s : set \u03b2} : s \u2208 filter.lift' f h \u2194 \u2203 (t : set \u03b1), \u2203 (H : t \u2208 f), h t \u2286 s :=\n  mem_lift_sets (monotone.comp monotone_principal hh)\n\ntheorem eventually_lift'_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    (hh : monotone h) {p : \u03b2 \u2192 Prop} :\n    filter.eventually (fun (y : \u03b2) => p y) (filter.lift' f h) \u2194\n        \u2203 (t : set \u03b1), \u2203 (H : t \u2208 f), \u2200 (y : \u03b2), y \u2208 h t \u2192 p y :=\n  mem_lift'_sets hh\n\ntheorem lift'_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : set \u03b1 \u2192 set \u03b2} {h : filter \u03b2}\n    {s : set \u03b1} (hs : s \u2208 f) (hg : principal (g s) \u2264 h) : filter.lift' f g \u2264 h :=\n  lift_le hs hg\n\ntheorem lift'_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f\u2081 : filter \u03b1} {f\u2082 : filter \u03b1}\n    {h\u2081 : set \u03b1 \u2192 set \u03b2} {h\u2082 : set \u03b1 \u2192 set \u03b2} (hf : f\u2081 \u2264 f\u2082) (hh : h\u2081 \u2264 h\u2082) :\n    filter.lift' f\u2081 h\u2081 \u2264 filter.lift' f\u2082 h\u2082 :=\n  lift_mono hf fun (s : set \u03b1) => iff.mpr principal_mono (hh s)\n\ntheorem lift'_mono' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h\u2081 : set \u03b1 \u2192 set \u03b2}\n    {h\u2082 : set \u03b1 \u2192 set \u03b2} (hh : \u2200 (s : set \u03b1), s \u2208 f \u2192 h\u2081 s \u2286 h\u2082 s) :\n    filter.lift' f h\u2081 \u2264 filter.lift' f h\u2082 :=\n  infi_le_infi fun (s : set \u03b1) => infi_le_infi fun (hs : s \u2208 f) => iff.mpr principal_mono (hh s hs)\n\ntheorem lift'_cong {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h\u2081 : set \u03b1 \u2192 set \u03b2}\n    {h\u2082 : set \u03b1 \u2192 set \u03b2} (hh : \u2200 (s : set \u03b1), s \u2208 f \u2192 h\u2081 s = h\u2082 s) :\n    filter.lift' f h\u2081 = filter.lift' f h\u2082 :=\n  le_antisymm (lift'_mono' fun (s : set \u03b1) (hs : s \u2208 f) => le_of_eq (hh s hs))\n    (lift'_mono' fun (s : set \u03b1) (hs : s \u2208 f) => le_of_eq (Eq.symm (hh s hs)))\n\ntheorem map_lift'_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    {m : \u03b2 \u2192 \u03b3} (hh : monotone h) : map m (filter.lift' f h) = filter.lift' f (set.image m \u2218 h) :=\n  sorry\n\ntheorem map_lift'_eq2 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b2 \u2192 set \u03b3} {m : \u03b1 \u2192 \u03b2} (hg : monotone g) :\n    filter.lift' (map m f) g = filter.lift' f (g \u2218 set.image m) :=\n  map_lift_eq2 (monotone.comp monotone_principal hg)\n\ntheorem comap_lift'_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {h : set \u03b1 \u2192 set \u03b2} {m : \u03b3 \u2192 \u03b2} (hh : monotone h) :\n    comap m (filter.lift' f h) = filter.lift' f (set.preimage m \u2218 h) :=\n  sorry\n\ntheorem comap_lift'_eq2 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1} {m : \u03b2 \u2192 \u03b1}\n    {g : set \u03b2 \u2192 set \u03b3} (hg : monotone g) :\n    filter.lift' (comap m f) g = filter.lift' f (g \u2218 set.preimage m) :=\n  comap_lift_eq2 (monotone.comp monotone_principal hg)\n\ntheorem lift'_principal {\u03b1 : Type u_1} {\u03b2 : Type u_2} {h : set \u03b1 \u2192 set \u03b2} {s : set \u03b1}\n    (hh : monotone h) : filter.lift' (principal s) h = principal (h s) :=\n  lift_principal (monotone.comp monotone_principal hh)\n\ntheorem lift'_pure {\u03b1 : Type u_1} {\u03b2 : Type u_2} {h : set \u03b1 \u2192 set \u03b2} {a : \u03b1} (hh : monotone h) :\n    filter.lift' (pure a) h = principal (h (singleton a)) :=\n  sorry\n\ntheorem lift'_bot {\u03b1 : Type u_1} {\u03b2 : Type u_2} {h : set \u03b1 \u2192 set \u03b2} (hh : monotone h) :\n    filter.lift' \u22a5 h = principal (h \u2205) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter.lift' \u22a5 h = principal (h \u2205))) (Eq.symm principal_empty)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (filter.lift' (principal \u2205) h = principal (h \u2205)))\n          (lift'_principal hh)))\n      (Eq.refl (principal (h \u2205))))\n\ntheorem principal_le_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    {t : set \u03b2} (hh : \u2200 (s : set \u03b1), s \u2208 f \u2192 t \u2286 h s) : principal t \u2264 filter.lift' f h :=\n  le_infi fun (s : set \u03b1) => le_infi fun (hs : s \u2208 f) => iff.mpr principal_mono (hh s hs)\n\ntheorem monotone_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [preorder \u03b3] {f : \u03b3 \u2192 filter \u03b1}\n    {g : \u03b3 \u2192 set \u03b1 \u2192 set \u03b2} (hf : monotone f) (hg : monotone g) :\n    monotone fun (c : \u03b3) => filter.lift' (f c) (g c) :=\n  fun (a b : \u03b3) (h : a \u2264 b) => lift'_mono (hf h) (hg h)\n\ntheorem lift_lift'_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b2} {h : set \u03b2 \u2192 filter \u03b3} (hg : monotone g) (hh : monotone h) :\n    filter.lift (filter.lift' f g) h = filter.lift f fun (s : set \u03b1) => h (g s) :=\n  sorry\n\ntheorem lift'_lift'_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b2} {h : set \u03b2 \u2192 set \u03b3} (hg : monotone g) (hh : monotone h) :\n    filter.lift' (filter.lift' f g) h = filter.lift' f fun (s : set \u03b1) => h (g s) :=\n  lift_lift'_assoc hg (monotone.comp monotone_principal hh)\n\ntheorem lift'_lift_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} {h : set \u03b2 \u2192 set \u03b3} (hg : monotone g) :\n    filter.lift' (filter.lift f g) h = filter.lift f fun (s : set \u03b1) => filter.lift' (g s) h :=\n  lift_assoc hg\n\ntheorem lift_lift'_same_le_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b1 \u2192 set \u03b2} :\n    (filter.lift f fun (s : set \u03b1) => filter.lift' f (g s)) \u2264\n        filter.lift' f fun (s : set \u03b1) => g s s :=\n  lift_lift_same_le_lift\n\ntheorem lift_lift'_same_eq_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b1 \u2192 set \u03b2} (hg\u2081 : \u2200 (s : set \u03b1), monotone fun (t : set \u03b1) => g s t)\n    (hg\u2082 : \u2200 (t : set \u03b1), monotone fun (s : set \u03b1) => g s t) :\n    (filter.lift f fun (s : set \u03b1) => filter.lift' f (g s)) =\n        filter.lift' f fun (s : set \u03b1) => g s s :=\n  lift_lift_same_eq_lift (fun (s : set \u03b1) => monotone.comp monotone_principal (hg\u2081 s))\n    fun (t : set \u03b1) => monotone.comp monotone_principal (hg\u2082 t)\n\ntheorem lift'_inf_principal_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    {s : set \u03b2} : filter.lift' f h \u2293 principal s = filter.lift' f fun (t : set \u03b1) => h t \u2229 s :=\n  sorry\n\ntheorem lift'_ne_bot_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2}\n    (hh : monotone h) : ne_bot (filter.lift' f h) \u2194 \u2200 (s : set \u03b1), s \u2208 f \u2192 set.nonempty (h s) :=\n  sorry\n\n@[simp] theorem lift'_id {\u03b1 : Type u_1} {f : filter \u03b1} : filter.lift' f id = f := lift_principal2\n\ntheorem le_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {h : set \u03b1 \u2192 set \u03b2} {g : filter \u03b2}\n    (h_le : \u2200 (s : set \u03b1), s \u2208 f \u2192 h s \u2208 g) : g \u2264 filter.lift' f h :=\n  sorry\n\ntheorem lift_infi' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Sort u_4} {f : \u03b9 \u2192 filter \u03b1}\n    {g : set \u03b1 \u2192 filter \u03b2} [Nonempty \u03b9] (hf : directed ge f) (hg : monotone g) :\n    filter.lift (infi f) g = infi fun (i : \u03b9) => filter.lift (f i) g :=\n  sorry\n\ntheorem lift'_infi {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b9 : Sort u_4} {f : \u03b9 \u2192 filter \u03b1}\n    {g : set \u03b1 \u2192 set \u03b2} [Nonempty \u03b9] (hg : \u2200 {s t : set \u03b1}, g s \u2229 g t = g (s \u2229 t)) :\n    filter.lift' (infi f) g = infi fun (i : \u03b9) => filter.lift' (f i) g :=\n  sorry\n\ntheorem lift'_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : filter \u03b1) (g : filter \u03b1) {s : set \u03b1 \u2192 set \u03b2}\n    (hs : \u2200 {t\u2081 t\u2082 : set \u03b1}, s t\u2081 \u2229 s t\u2082 = s (t\u2081 \u2229 t\u2082)) :\n    filter.lift' (f \u2293 g) s = filter.lift' f s \u2293 filter.lift' g s :=\n  sorry\n\ntheorem comap_eq_lift' {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b2} {m : \u03b1 \u2192 \u03b2} :\n    comap m f = filter.lift' f (set.preimage m) :=\n  filter.ext fun (s : set \u03b1) => iff.symm (mem_lift'_sets set.monotone_preimage)\n\ntheorem lift'_infi_powerset {\u03b1 : Type u_1} {\u03b9 : Sort u_4} [Nonempty \u03b9] {f : \u03b9 \u2192 filter \u03b1} :\n    filter.lift' (infi f) set.powerset = infi fun (i : \u03b9) => filter.lift' (f i) set.powerset :=\n  lift'_infi fun (_x _x_1 : set \u03b1) => Eq.symm (set.powerset_inter _x _x_1)\n\ntheorem lift'_inf_powerset {\u03b1 : Type u_1} (f : filter \u03b1) (g : filter \u03b1) :\n    filter.lift' (f \u2293 g) set.powerset = filter.lift' f set.powerset \u2293 filter.lift' g set.powerset :=\n  lift'_inf f g fun (_x _x_1 : set \u03b1) => Eq.symm (set.powerset_inter _x _x_1)\n\ntheorem eventually_lift'_powerset {\u03b1 : Type u_1} {f : filter \u03b1} {p : set \u03b1 \u2192 Prop} :\n    filter.eventually (fun (s : set \u03b1) => p s) (filter.lift' f set.powerset) \u2194\n        \u2203 (s : set \u03b1), \u2203 (H : s \u2208 f), \u2200 (t : set \u03b1), t \u2286 s \u2192 p t :=\n  eventually_lift'_iff set.monotone_powerset\n\ntheorem eventually_lift'_powerset' {\u03b1 : Type u_1} {f : filter \u03b1} {p : set \u03b1 \u2192 Prop}\n    (hp : \u2200 {s t : set \u03b1}, s \u2286 t \u2192 p t \u2192 p s) :\n    filter.eventually (fun (s : set \u03b1) => p s) (filter.lift' f set.powerset) \u2194\n        \u2203 (s : set \u03b1), \u2203 (H : s \u2208 f), p s :=\n  sorry\n\nprotected instance lift'_powerset_ne_bot {\u03b1 : Type u_1} (f : filter \u03b1) :\n    ne_bot (filter.lift' f set.powerset) :=\n  iff.mpr (lift'_ne_bot_iff set.monotone_powerset)\n    fun (_x : set \u03b1) (_x_1 : _x \u2208 f) => set.powerset_nonempty\n\ntheorem tendsto_lift'_powerset_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {la : filter \u03b1} {lb : filter \u03b2}\n    {s : \u03b1 \u2192 set \u03b2} {t : \u03b1 \u2192 set \u03b2} (ht : tendsto t la (filter.lift' lb set.powerset))\n    (hst : filter.eventually (fun (x : \u03b1) => s x \u2286 t x) la) :\n    tendsto s la (filter.lift' lb set.powerset) :=\n  sorry\n\n@[simp] theorem eventually_lift'_powerset_forall {\u03b1 : Type u_1} {f : filter \u03b1} {p : \u03b1 \u2192 Prop} :\n    filter.eventually (fun (s : set \u03b1) => \u2200 (x : \u03b1), x \u2208 s \u2192 p x) (filter.lift' f set.powerset) \u2194\n        filter.eventually (fun (x : \u03b1) => p x) f :=\n  sorry\n\ntheorem eventually.lift'_powerset {\u03b1 : Type u_1} {f : filter \u03b1} {p : \u03b1 \u2192 Prop} :\n    filter.eventually (fun (x : \u03b1) => p x) f \u2192\n        filter.eventually (fun (s : set \u03b1) => \u2200 (x : \u03b1), x \u2208 s \u2192 p x)\n          (filter.lift' f set.powerset) :=\n  iff.mpr eventually_lift'_powerset_forall\n\n@[simp] theorem eventually_lift'_powerset_eventually {\u03b1 : Type u_1} {f : filter \u03b1} {g : filter \u03b1}\n    {p : \u03b1 \u2192 Prop} :\n    filter.eventually (fun (s : set \u03b1) => filter.eventually (fun (x : \u03b1) => x \u2208 s \u2192 p x) g)\n          (filter.lift' f set.powerset) \u2194\n        filter.eventually (fun (x : \u03b1) => p x) (f \u2293 g) :=\n  sorry\n\ntheorem prod_def {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : filter \u03b1} {g : filter \u03b2} :\n    filter.prod f g = filter.lift f fun (s : set \u03b1) => filter.lift' g (set.prod s) :=\n  sorry\n\ntheorem prod_same_eq {\u03b1 : Type u_1} {f : filter \u03b1} :\n    filter.prod f f = filter.lift' f fun (t : set \u03b1) => set.prod t t :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (filter.prod f f = filter.lift' f fun (t : set \u03b1) => set.prod t t))\n        prod_def))\n    (lift_lift'_same_eq_lift' (fun (s : set \u03b1) => set.monotone_prod monotone_const monotone_id)\n      fun (t : set \u03b1) => set.monotone_prod monotone_id monotone_const)\n\ntheorem mem_prod_same_iff {\u03b1 : Type u_1} {f : filter \u03b1} {s : set (\u03b1 \u00d7 \u03b1)} :\n    s \u2208 filter.prod f f \u2194 \u2203 (t : set \u03b1), \u2203 (H : t \u2208 f), set.prod t t \u2286 s :=\n  sorry\n\ntheorem tendsto_prod_self_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u00d7 \u03b1 \u2192 \u03b2} {x : filter \u03b1}\n    {y : filter \u03b2} :\n    tendsto f (filter.prod x x) y \u2194\n        \u2200 (W : set \u03b2) (H : W \u2208 y),\n          \u2203 (U : set \u03b1), \u2203 (H : U \u2208 x), \u2200 (x x' : \u03b1), x \u2208 U \u2192 x' \u2208 U \u2192 f (x, x') \u2208 W :=\n  sorry\n\ntheorem prod_lift_lift {\u03b1\u2081 : Type u_5} {\u03b1\u2082 : Type u_6} {\u03b2\u2081 : Type u_7} {\u03b2\u2082 : Type u_8}\n    {f\u2081 : filter \u03b1\u2081} {f\u2082 : filter \u03b1\u2082} {g\u2081 : set \u03b1\u2081 \u2192 filter \u03b2\u2081} {g\u2082 : set \u03b1\u2082 \u2192 filter \u03b2\u2082}\n    (hg\u2081 : monotone g\u2081) (hg\u2082 : monotone g\u2082) :\n    filter.prod (filter.lift f\u2081 g\u2081) (filter.lift f\u2082 g\u2082) =\n        filter.lift f\u2081\n          fun (s : set \u03b1\u2081) => filter.lift f\u2082 fun (t : set \u03b1\u2082) => filter.prod (g\u2081 s) (g\u2082 t) :=\n  sorry\n\ntheorem prod_lift'_lift' {\u03b1\u2081 : Type u_5} {\u03b1\u2082 : Type u_6} {\u03b2\u2081 : Type u_7} {\u03b2\u2082 : Type u_8}\n    {f\u2081 : filter \u03b1\u2081} {f\u2082 : filter \u03b1\u2082} {g\u2081 : set \u03b1\u2081 \u2192 set \u03b2\u2081} {g\u2082 : set \u03b1\u2082 \u2192 set \u03b2\u2082}\n    (hg\u2081 : monotone g\u2081) (hg\u2082 : monotone g\u2082) :\n    filter.prod (filter.lift' f\u2081 g\u2081) (filter.lift' f\u2082 g\u2082) =\n        filter.lift f\u2081\n          fun (s : set \u03b1\u2081) => filter.lift' f\u2082 fun (t : set \u03b1\u2082) => set.prod (g\u2081 s) (g\u2082 t) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/filter/lift_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529716, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3716278422238082}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johan Commelin, Andrew Yang\n-/\nimport category_theory.limits.preserves.shapes.zero\nimport category_theory.monoidal.End\nimport category_theory.monoidal.discrete\n\n/-!\n# Shift\n\nA `shift` on a category `C` indexed by a monoid `A` is is nothing more than a monoidal functor\nfrom `A` to `C \u2964 C`. A typical example to keep in mind might be the category of\ncomplexes `\u22ef \u2192 C_{n-1} \u2192 C_n \u2192 C_{n+1} \u2192 \u22ef`. It has a shift indexed by `\u2124`, where we assign to\neach `n : \u2124` the functor `C \u2964 C` that re-indexing the terms, so the degree `i` term of `shift n C`\nwould be the degree `i+n`-th term of `C`.\n\n## Main definitions\n* `has_shift`: A typeclass asserting the existence of a shift functor.\n* `shift_equiv`: When the indexing monoid is a group, then the functor indexed by `n` and `-n` forms\n  an self-equivalence of `C`.\n* `shift_comm`: When the indexing monoid is commutative, then shifts commute as well.\n\n## Implementation Notes\n\nMost of the definitions in this file is marked as an `abbreviation` so that the simp lemmas in\n`category_theory/monoidal/End` could apply.\n\n-/\nnamespace category_theory\n\nnoncomputable theory\n\nuniverses v u\n\nvariables (C : Type u) (A : Type*) [category.{v} C]\n\nlocal attribute [instance] endofunctor_monoidal_category\nlocal attribute [reducible] endofunctor_monoidal_category discrete.add_monoidal\n\nsection eq_to_hom\n\nvariables {A C}\n\nvariables [add_monoid A] (F : monoidal_functor (discrete A) (C \u2964 C))\n\n @[simp, reassoc] lemma eq_to_hom_\u03bc_app {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (F.\u03bc i' j').app X =\n     (F.\u03bc i j).app X \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n @[simp, reassoc] lemma \u03bc_inv_app_eq_to_hom {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   (F.\u03bc_iso i j).inv.app X \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n     eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (F.\u03bc_iso i' j').inv.app X :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\nend eq_to_hom\n\nvariables {A C}\n\n/-- A monoidal functor from a group `A` into `C \u2964 C` induces\na self-equivalence of `C` for each `n : A`. -/\n@[simps functor inverse unit_iso_hom unit_iso_inv counit_iso_hom counit_iso_inv]\ndef add_neg_equiv [add_group A] (F : monoidal_functor (discrete A) (C \u2964 C)) (n : A) : C \u224c C :=\nequiv_of_tensor_iso_unit F n (-n : A)\n  (eq_to_iso (add_neg_self n)) (eq_to_iso (neg_add_self n)) (subsingleton.elim _ _)\n\nsection defs\n\nvariables (A C) [add_monoid A]\n\n/-- A category has a shift indexed by an additive monoid `A`\nif there is a monoidal functor from `A` to `C \u2964 C`. -/\nclass has_shift (C : Type u) (A : Type*) [category.{v} C] [add_monoid A] :=\n(shift : monoidal_functor (discrete A) (C \u2964 C))\n\n/-- A helper structure to construct the shift functor `(discrete A) \u2964 (C \u2964 C)`. -/\n@[nolint has_inhabited_instance]\nstructure shift_mk_core :=\n(F : A \u2192 (C \u2964 C))\n(\u03b5 : \ud835\udfed C \u2245 F 0)\n(\u03bc : \u03a0 n m : A, F n \u22d9 F m \u2245 F (n + m))\n(associativity : \u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n  (F m\u2083).map ((\u03bc m\u2081 m\u2082).hom.app X) \u226b (\u03bc (m\u2081 + m\u2082) m\u2083).hom.app X \u226b\n    eq_to_hom (by { congr' 2, exact add_assoc _ _ _ }) =\n    (\u03bc m\u2082 m\u2083).hom.app ((F m\u2081).obj X) \u226b (\u03bc m\u2081 (m\u2082 + m\u2083)).hom.app X . obviously)\n(left_unitality : \u2200 (n : A) (X : C),\n  (F n).map (\u03b5.hom.app X) \u226b (\u03bc 0 n).hom.app X =\n    eq_to_hom (by { dsimp, rw zero_add }) . obviously)\n(right_unitality : \u2200 (n : A) (X : C),\n  \u03b5.hom.app ((F n).obj X) \u226b (\u03bc n 0).hom.app X =\n    eq_to_hom (by { dsimp, rw add_zero }) . obviously)\n\n/-- Constructs a `has_shift C A` instance from `shift_mk_core`. -/\n@[simps]\ndef has_shift_mk (h : shift_mk_core C A) : has_shift C A :=\n\u27e8{ \u03b5 := h.\u03b5.hom,\n   \u03bc := \u03bb m n, (h.\u03bc m n).hom,\n   \u03bc_natural' := by { rintros _ _ _ _ \u27e8\u27e8rfl\u27e9\u27e9 \u27e8\u27e8rfl\u27e9\u27e9, ext, dsimp, simp, dsimp, simp },\n   associativity' := by { introv, ext, dsimp, simpa using h.associativity _ _ _ _, },\n   left_unitality' :=\n    by { introv, ext, dsimp, rw [category.id_comp, \u2190 category.assoc, h.left_unitality], simp },\n   right_unitality' :=\n    by { introv, ext, dsimp, rw [functor.map_id, category.comp_id,\n      \u2190 category.assoc, h.right_unitality], simp },\n ..(discrete.functor h.F) }\u27e9\n\nvariables [has_shift C A]\n\n/-- The monoidal functor from `A` to `C \u2964 C` given a `has_shift` instance. -/\ndef shift_monoidal_functor : monoidal_functor (discrete A) (C \u2964 C) := has_shift.shift\n\nvariable {A}\n\n/-- The shift autoequivalence, moving objects and morphisms 'up'. -/\nabbreviation shift_functor (i : A) : C \u2964 C := (shift_monoidal_functor C A).obj i\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_functor_add (i j : A) :\n  shift_functor C (i + j) \u2245 shift_functor C i \u22d9 shift_functor C j :=\n((shift_monoidal_functor C A).\u03bc_iso i j).symm\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_functor_zero : shift_functor C (0 : A) \u2245 \ud835\udfed C :=\n(shift_monoidal_functor C A).\u03b5_iso.symm\n\n-- Any better notational suggestions?\nnotation X`\u27e6`n`\u27e7`:20 := (shift_functor _ n).obj X\nnotation f`\u27e6`n`\u27e7'`:80 := (shift_functor _ n).map f\n\nend defs\n\nsection examples\nvariables [has_shift C \u2124]\n\nexample {X Y : C} (f : X \u27f6 Y) : X\u27e6(1 : \u2124)\u27e7 \u27f6 Y\u27e61\u27e7 := f\u27e61\u27e7'\nexample {X Y : C} (f : X \u27f6 Y) : X\u27e6(-2 : \u2124)\u27e7 \u27f6 Y\u27e6-2\u27e7 := f\u27e6-2\u27e7'\n\nend examples\n\nsection add_monoid\n\nvariables {C A} [add_monoid A] [has_shift C A] (X Y : C) (f : X \u27f6 Y)\n\n@[simp] lemma has_shift.shift_obj_obj (n : A) (X : C) : (has_shift.shift.obj n).obj X = X\u27e6n\u27e7 := rfl\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_add (i j : A) : X\u27e6i + j\u27e7 \u2245 X\u27e6i\u27e7\u27e6j\u27e7 := (shift_functor_add C i j).app _\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081 (i i' j : A) (h : i = i') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i' j).hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2082 (i j j' : A) (h : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i j').hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n    eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').hom :=\nby { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081 (i i' j : A) (h : i = i') :\n  eq_to_hom (by rw h) \u226b (shift_add X i' j).inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2082 (i j j' : A) (h : j = j') :\n  eq_to_hom (by rw h) \u226b (shift_add X i j').inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2082]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').inv =\n    (shift_add X i j).inv \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081\u2082]\n\nlemma shift_shift' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_add X i j).inv \u226b f\u27e6i + j\u27e7' \u226b (shift_add Y i j).hom :=\nby { symmetry, apply nat_iso.naturality_1 }\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_zero  :\n  X\u27e60\u27e7 \u2245 X := (shift_functor_zero C A).app _\n\nlemma shift_zero' :\n  f\u27e6(0 : A)\u27e7' = (shift_zero A X).hom \u226b f \u226b (shift_zero A Y).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nend add_monoid\n\nsection opaque_eq_to_iso\n\nvariables {\u03b9 : Type*} {i j k : \u03b9}\n\n/-- This definition is used instead of `eq_to_iso` so that the proof of `i = j` is visible\nto the simplifier -/\ndef opaque_eq_to_iso (h : i = j) : @iso (discrete \u03b9) _ i j := eq_to_iso h\n\n@[simp]\nlemma opaque_eq_to_iso_symm (h : i = j) :\n  (opaque_eq_to_iso h).symm = opaque_eq_to_iso h.symm := rfl\n\n@[simp]\nlemma opaque_eq_to_iso_inv (h : i = j) :\n  (opaque_eq_to_iso h).inv = (opaque_eq_to_iso h.symm).hom := rfl\n\n@[simp, reassoc]\nlemma map_opaque_eq_to_iso_comp_app (F : discrete \u03b9 \u2964 C \u2964 C) (h : i = j) (h' : j = k) (X : C) :\n  (F.map (opaque_eq_to_iso h).hom).app X \u226b (F.map (opaque_eq_to_iso h').hom).app X =\n    (F.map (opaque_eq_to_iso $ h.trans h').hom).app X := by { delta opaque_eq_to_iso, simp }\n\nend opaque_eq_to_iso\n\nsection add_group\n\nvariables (C) {A} [add_group A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_functor_comp_shift_functor_neg (i : A) :\n  shift_functor C i \u22d9 shift_functor C (-i) \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) i (-i : A) (opaque_eq_to_iso (add_neg_self i))\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_functor_neg_comp_shift_functor (i : A) :\n  shift_functor C (-i) \u22d9 shift_functor C i \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) (-i : A) i (opaque_eq_to_iso (neg_add_self i))\n\nsection\n\nvariables (C)\n\n/-- Shifting by `n` is a faithful functor. -/\ninstance shift_functor_faithful (i : A) : faithful (shift_functor C i) :=\nfaithful.of_comp_iso (shift_functor_comp_shift_functor_neg C i)\n\n/-- Shifting by `n` is a full functor. -/\ninstance shift_functor_full (i : A) : full (shift_functor C i) :=\nbegin\n  haveI : full (shift_functor C i \u22d9 shift_functor C (-i)) :=\n    full.of_iso (shift_functor_comp_shift_functor_neg C i).symm,\n  exact full.of_comp_faithful _ (shift_functor C (-i))\nend\n\n/-- Shifting by `n` is an essentially surjective functor. -/\ninstance shift_functor_ess_surj (i : A) : ess_surj (shift_functor C i) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y\u27e6-i\u27e7, \u27e8(shift_functor_neg_comp_shift_functor C i).app Y\u27e9\u27e9 }\n\n/-- Shifting by `n` is an equivalence. -/\nnoncomputable instance shift_functor_is_equivalence (n : A) : is_equivalence (shift_functor C n) :=\nequivalence.of_fully_faithfully_ess_surj _\n\nend\n\nvariables {C}\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_shift_neg (i : A) : X\u27e6i\u27e7\u27e6-i\u27e7 \u2245 X :=\n(shift_functor_comp_shift_functor_neg C i).app _\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_neg_shift (i : A) : X\u27e6-i\u27e7\u27e6i\u27e7 \u2245 X :=\n(shift_functor_neg_comp_shift_functor C i).app _\n\nvariables {X Y}\n\nlemma shift_shift_neg' (i : A) :\n  f\u27e6i\u27e7'\u27e6-i\u27e7' = (shift_shift_neg X i).hom \u226b f \u226b (shift_shift_neg Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_neg_shift' (i : A) :\n  f\u27e6-i\u27e7'\u27e6i\u27e7' = (shift_neg_shift X i).hom \u226b f \u226b (shift_neg_shift Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_equiv_triangle (n : A) (X : C) :\n  (shift_shift_neg X n).inv\u27e6n\u27e7' \u226b (shift_neg_shift (X\u27e6n\u27e7) n).hom = \ud835\udfd9 (X\u27e6n\u27e7) :=\n(add_neg_equiv (shift_monoidal_functor C A) n).functor_unit_iso_comp X\n\nlemma shift_shift_neg_hom_shift (n : A) (X : C) :\n  (shift_shift_neg X n).hom \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).hom :=\nby simp\n\nlemma shift_shift_neg_inv_shift (n : A) (X : C) :\n  (shift_shift_neg X n).inv \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).inv :=\nby { ext, rw [\u2190 shift_shift_neg_hom_shift, \u2190 functor.map_comp, iso.hom_inv_id, functor.map_id] }\n\n@[simp]\nlemma shift_shift_neg_shift_eq (n : A) (X : C) :\n  (shift_functor C n).map_iso (shift_shift_neg X n) = shift_neg_shift (X\u27e6n\u27e7) n :=\ncategory_theory.iso.ext $ shift_shift_neg_hom_shift _ _\n\nvariables (C)\n\n/-- Shifting by `n` and shifting by `-n` forms an equivalence. -/\n@[simps]\ndef shift_equiv (n : A) : C \u224c C :=\n{ functor := shift_functor C n,\n  inverse := shift_functor C (-n),\n  ..(add_neg_equiv (shift_monoidal_functor C A) n) }\n\nvariable {C}\n\nopen category_theory.limits\n\nvariables [has_zero_morphisms C]\n\nlemma shift_zero_eq_zero (X Y : C) (n : A) : (0 : X \u27f6 Y)\u27e6n\u27e7' = (0 : X\u27e6n\u27e7 \u27f6 Y\u27e6n\u27e7) :=\ncategory_theory.functor.map_zero _ _ _\n\nend add_group\n\nsection add_comm_monoid\n\nvariables {C A} [add_comm_monoid A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\ndef shift_comm (i j : A) : X\u27e6i\u27e7\u27e6j\u27e7 \u2245 X\u27e6j\u27e7\u27e6i\u27e7 :=\n(shift_add X i j).symm \u226a\u226b ((shift_monoidal_functor C A).to_functor.map_iso\n  (opaque_eq_to_iso $ add_comm i j : _)).app X \u226a\u226b shift_add X j i\n\n@[simp] lemma shift_comm_symm (i j : A) : (shift_comm X i j).symm = shift_comm X j i :=\nbegin\n  ext, dsimp [shift_comm], simpa\nend\n\nvariables {X Y}\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\nlemma shift_comm' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_comm _ _ _).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' \u226b (shift_comm _ _ _).hom :=\nby simp [shift_comm]\n\n@[reassoc] lemma shift_comm_hom_comp (i j : A) :\n  (shift_comm X i j).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' = f\u27e6i\u27e7'\u27e6j\u27e7' \u226b (shift_comm Y i j).hom :=\nby rw [shift_comm', \u2190 shift_comm_symm, iso.symm_hom, iso.inv_hom_id_assoc]\n\nend add_comm_monoid\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3716278422238081}}
{"text": "/- structural subtyping -/\nimport ..phrase\n\nnamespace flow_analysis\n\nopen phrase\nlemma structural_subtyping.base {a b : security_class} (h : a \u2264 b) :\n  \u2203\u03c4 \u03c4', base a = base \u03c4 \u2227 base b = base \u03c4' \u2227 \u03c4 \u2264 \u03c4'\n       \u2228 base a = var \u03c4 \u2227 base a = base b\n       \u2228 base a = cmd \u03c4 \u2227 base b = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4\n:= begin\n  apply exists.intro a,\n  apply exists.intro b,\n  cc,\nend\n\nlemma structural_subtyping.reflex {\u03c1 : phrase} :\n  \u2203\u03c4 \u03c4', \u03c1 = base \u03c4 \u2227 \u03c1 = base \u03c4' \u2227 \u03c4 \u2264 \u03c4'\n       \u2228 \u03c1 = var \u03c4 \u2227 \u03c1 = \u03c1\n       \u2228 \u03c1 = cmd \u03c4 \u2227 \u03c1 = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4\n:= begin\n  cases \u03c1,\n  repeat {apply exists.intro \u03c1},\n  repeat {simp},\nend\n\nlemma structural_subtyping.trans {a b c : phrase}\n    (ihab : \u2203\u03c4 \u03c4', a = base \u03c4 \u2227 b = base \u03c4' \u2227 \u03c4 \u2264 \u03c4'\n                 \u2228 a = var \u03c4 \u2227 a = b\n                 \u2228 a = cmd \u03c4 \u2227 b = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4)\n    (ihbc : \u2203\u03c4 \u03c4', b = base \u03c4 \u2227 c = base \u03c4' \u2227 \u03c4 \u2264 \u03c4'\n                 \u2228 b = var \u03c4 \u2227 b = c\n                 \u2228 b = cmd \u03c4 \u2227 c = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4):\n  \u2203 (\u03c4 \u03c4' : security_class),\n    a = base \u03c4 \u2227 c = base \u03c4' \u2227 \u03c4 \u2264 \u03c4' \u2228\n      a = var \u03c4 \u2227 a = c \u2228 a = cmd \u03c4 \u2227 c = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4\n:= begin\n  cases a,\n  case phrase.base : {\n    cases ihab with x ihab,\n    cases ihab with y ihab,\n    repeat { cases ihab, },\n    cases ihab_right,\n    cases ihbc with y' ihbc,\n    cases ihbc with z ihbc,\n    repeat { cases ihbc, },\n    cases ihbc_right,\n    apply exists.intro x,\n    apply exists.intro z,\n    apply or.inl,\n    apply and.intro ihab_left,\n    apply and.intro ihbc_right_left,\n    simp[ihab_right_left] at ihbc_left,\n    simp[ihbc_left] at *,\n    apply le_trans ihab_right_right ihbc_right_right,\n    repeat { cc, },\n  },\n  case phrase.var : {\n    repeat { cases ihab with _ ihab, cases ihbc with _ ihbc, },\n    repeat { apply exists.intro a },\n    repeat { cc, },\n  },\n  case phrase.cmd : {\n    cases ihab with x ihab,\n    cases ihab with y ihab,\n    repeat { cases ihab, },\n    repeat { cc, },\n    cases ihab_right with ihby ihyx,\n    cases ihbc with y' ihbc,\n    cases ihbc with z ihbc,\n    repeat { cases ihbc with _ ihbc, },\n    repeat { cc, },\n    simp[ihby] at *,\n    apply exists.intro x,\n    apply exists.intro y',\n    cc,\n    apply exists.intro x,\n    apply exists.intro security_class.low,\n    repeat { apply or.inr, },\n    apply and.intro ihab_left,\n    apply and.intro ihbc_left_1,\n    cases x,\n    apply security_class.le.lower,\n    apply security_class.le.refl,\n  },\nend\n\nlemma structural_subtyping.cmd {a b : security_class}\n    (ih : \u2203\u03c4 \u03c4', base a = base \u03c4 \u2227 base b = base \u03c4' \u2227 \u03c4 \u2264 \u03c4'\n               \u2228 base a = var \u03c4 \u2227 base a = base b\n               \u2228 base a = cmd \u03c4 \u2227 base b = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4):\n  \u2203 (\u03c4 \u03c4' : security_class),\n    cmd b = base \u03c4 \u2227 cmd a = base \u03c4' \u2227 \u03c4 \u2264 \u03c4' \u2228\n      cmd b = var \u03c4 \u2227 cmd b = cmd a \u2228 cmd b = cmd \u03c4 \u2227 cmd a = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4\n:= begin\n  cases ih with y ih,\n  cases ih with x ih,\n  apply exists.intro x,\n  apply exists.intro y,\n  cc,\nend\n\n/- Lemma 4.1 (Structural Subtyping)\n - If \u22a2 \u03c1 \u2286 \u03c1', then either\n - (a) \u03c1 is of the form base \u03c4, \u03c1' is of the form base \u03c4' and \u03c4 \u2264 \u03c4'\n - (b) \u03c1 is of the form var \u03c4 and \u03c1' = \u03c1, or\n - (c) \u03c1 is of the form cmd \u03c4, \u03c1' is of the form cmd \u03c4' and \u03c4' \u2264 \u03c4\n-/\nlemma structural_subtyping {\u03c1 \u03c1' : phrase} (h : \u03c1 \u2286\u209b \u03c1') :\n  \u2203\u03c4 \u03c4', (\u03c1 = base \u03c4 \u2227 \u03c1' = base \u03c4' \u2227 \u03c4 \u2264 \u03c4')\n       \u2228 (\u03c1 = var \u03c4 \u2227 \u03c1 = \u03c1')\n       \u2228 (\u03c1 = cmd \u03c4 \u2227 \u03c1' = cmd \u03c4' \u2227 \u03c4' \u2264 \u03c4)\n:= begin\n  induction h,\n  case phrase.ss.base : a b h {\n    apply structural_subtyping.base h,\n  },\n  case phrase.ss.reflex : \u03c1 {\n    apply structural_subtyping.reflex,\n  },\n  case phrase.ss.trans : a b c _ _ ihab ihbc {\n    apply structural_subtyping.trans ihab ihbc,\n  },\n  case phrase.ss.cmd : a b hab ih {\n    apply structural_subtyping.cmd ih,\n  },\nend\n\nend flow_analysis", "meta": {"author": "denismazzucato", "repo": "noninterference-lean", "sha": "a70674cb2af3959bd188b7079868f2505834ac74", "save_path": "github-repos/lean/denismazzucato-noninterference-lean", "path": "github-repos/lean/denismazzucato-noninterference-lean/noninterference-lean-a70674cb2af3959bd188b7079868f2505834ac74/src/lemmata/structural_subtyping.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7549149868676284, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.37156020001406687}}
{"text": "/-\nCopyright (c) 2019 The Flypitch Project. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Jesse Han, Floris van Doorn\n-/\nimport .fol\n\nopen fol set\n\nuniverses u v\n\n/-- Given an xs : list \u03b1, an x : \u03b1, a set T on \u03b1 such that everything in xs which is not x is in T, return the sublist which excludes x, a proof that this list is now a subset of T, and a proof that everything in this list was not the forbidden element x. --/\ndef list_except {\u03b1 : Type u} [decidable_eq \u03b1] (xs : list \u03b1) (x : \u03b1) (T : set \u03b1)\n  (h : \u2200 y \u2208 xs, y \u2260 x \u2192 y \u2208 T) :\n  \u03a3' ys : list \u03b1, ({\u03d5 | \u03d5 \u2208 ys} \u2286 T \u2227 (\u2200 y \u2208 ys, y \u2260 x)) \u2227 (\u2200 y \u2208 xs, y \u2260 x \u2192 y \u2208 ys) :=\nbegin\n  existsi xs.filter (\u2260 x),\n  refine \u27e8\u27e8_, _\u27e9, _\u27e9,\n  { intros y hy, apply h y (list.mem_of_mem_filter hy), apply list.of_mem_filter hy },\n  { intros y hy, apply list.of_mem_filter hy },\n  { intros y hy hxy, apply list.mem_filter_of_mem hy hxy }\nend\n\nopen classical\n\n/- Couldn't find this def in set.basic... sure it's around somewhere-/\n/-- Given x \u2208 f '' S, choose a lift x' in the preimage of x; return x' and a proof that x' is a lift --/\nnoncomputable def image_lift {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {S : set \u03b1} (x \u2208 f '' S) : \u03a3' (x' : \u03b1), x' \u2208 S \u2227 f x' = x :=\nbegin\n  apply psigma_of_exists, apply (set.mem_image _ _ _).mp H\nend\n\n/-- Given a list xs : list \u03b2, a set S : set \u03b1, a proof that {x | x \u2208 xs} \u2286 f '' S, return a list of lifts ys : list \u03b1, a proof that ys \u2286 S and a proof that f '' ys = xs --/\nnoncomputable def image_lift_list {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {S : set \u03b1} {xs : list \u03b2} (h_sub : {x | x \u2208 xs} \u2286 f '' S) : \u03a3' (ys : list \u03b1), ({y' | y' \u2208 ys} \u2286 S) \u2227 f '' {y | y \u2208 ys} = {x | x \u2208 xs} :=\nbegin\n  apply psigma_of_exists,\n  rcases list.exists_of_to_set_subset_image h_sub with \u27e8ys, hys, hys'\u27e9,\n  refine \u27e8ys, hys, _\u27e9, subst hys', ext,\n  apply list.mem_map.symm\nend\n\n/-- Any proof from a set of formulas is provable from a finset of subformulas. --/\nlemma proof_compactness {L : Language.{u}} {\u03c8 : formula L} {T : set $ formula L} :\n  (T \u22a2' \u03c8) \u2192 \u2203\u0393 : finset (formula L), \u2191\u0393 \u22a2' \u03c8 \u2227 \u2191\u0393 \u2286 T :=\nbegin\n  haveI : decidable_eq (formula L) := \u03bbx y, classical.prop_decidable _,\n  intro P, induction P with P, induction P,\n  { exact \u27e8{P_A}, \u27e8axm1\u27e9, set.singleton_subset_iff.mpr P_h\u27e9 },\n  { rcases P_ih with \u27e8\u0393, H, K\u27e9, refine \u27e8\u0393 \\ {P_A}, impI' $ weakening' (by simp) H, by simp [K]\u27e9 },\n  { rcases P_ih_h\u2081 with \u27e8\u0393\u2081, H\u2081, K\u2081\u27e9, rcases P_ih_h\u2082 with \u27e8\u0393\u2082, H\u2082, K\u2082\u27e9,\n    refine \u27e8\u0393\u2081 \u222a \u0393\u2082, impE' _ (weakening' (by simp) H\u2081) (weakening' (by simp) H\u2082), by simp [K\u2081, K\u2082]\u27e9 },\n  { rcases P_ih with \u27e8\u0393, H, K\u27e9, refine \u27e8\u0393 \\ {\u223cP_A}, falsumE' $ weakening' (by simp) H, by simp [K]\u27e9 },\n  { rcases P_ih with \u27e8\u0393, H, K\u27e9, rcases finset.subset_image_iff.mp K with \u27e8\u0393', K', h\u0393\u27e9,\n    subst h\u0393, simp only [finset.coe_image] at H K,\n    exact \u27e8\u0393', allI' H, K'\u27e9 },\n  { rcases P_ih with \u27e8\u0393, H, K\u27e9, exact \u27e8\u0393, allE\u2082' H, K\u27e9 },\n  { exact \u27e8\u2205, ref' _ _, empty_subset _\u27e9 },\n  { rcases P_ih_h\u2081 with \u27e8\u0393\u2081, H\u2081, K\u2081\u27e9, rcases P_ih_h\u2082 with \u27e8\u0393\u2082, H\u2082, K\u2082\u27e9,\n    refine \u27e8\u0393\u2081 \u222a \u0393\u2082, subst\u2082' _ _ _ (weakening' (by simp) H\u2081) (weakening' (by simp) H\u2082), by simp [K\u2081, K\u2082]\u27e9 }\nend\n\nlemma theory_proof_compactness {L : Language} {T : Theory L} {\u03c8 : sentence L} (h\u03c8 : T \u22a2' \u03c8) :\n  \u2203\u0393 : finset (sentence L), \u2191\u0393 \u22a2' \u03c8 \u2227 \u2191\u0393 \u2286 T :=\nbegin\n  haveI : decidable_eq (sentence L) := \u03bbx y, classical.prop_decidable _,\n  haveI : decidable_eq (formula L) := \u03bbx y, classical.prop_decidable _,\n  rcases proof_compactness h\u03c8 with \u27e8\u0393, H, K\u27e9,\n  rcases finset.subset_image_iff.mp K with \u27e8\u0393', K', h\u0393\u27e9,\n  subst h\u0393, simp only [finset.coe_image] at H K,\n  exact \u27e8\u0393', H, K'\u27e9\nend\n\nlemma theory_proof_compactness_iff {L : Language} {T : Theory L} {\u03c8 : sentence L} :\n  T \u22a2' \u03c8 \u2194 \u2203\u0393 : finset (sentence L), \u2191\u0393 \u22a2' \u03c8 \u2227 \u2191\u0393 \u2286 T :=\n\u27e8theory_proof_compactness, \u03bb\u27e8\u0393, H, K\u27e9, weakening' (image_subset _ K) H\u27e9\n\nlemma is_consistent_union {L : Language} {T\u2081 T\u2082 : Theory L} (h\u2081 : is_consistent T\u2081)\n  (h\u2082 : \u2200\u03c8 \u2208 T\u2082, insert (\u223c\u03c8) T\u2081 \u22a2' (\u22a5 : sentence L)) : is_consistent (T\u2081 \u222a T\u2082) :=\nbegin\n  haveI : decidable_eq (sentence L) := \u03bbx y, classical.prop_decidable _,\n  have lem : \u2200(T\u2080 : finset (sentence L)), \u2191T\u2080 \u2286 T\u2082 \u2192 is_consistent (T\u2081 \u222a \u2191T\u2080),\n  { refine finset.induction _ _,\n    { intro hT, rw [finset.coe_empty, union_empty], exact h\u2081 },\n    { intros \u03c8 s h\u03c8 ih hs hT, simp [insert_subset] at hs,\n      apply ih hs.2, apply sprf_by_cases \u03c8,\n      { simp at hT, exact hT },\n      { apply weakening' _ (h\u2082 _ hs.1),\n        apply image_subset, apply insert_subset_insert, apply subset_union_left }}},\n  intro h, rcases theory_proof_compactness h with \u27e8T\u2080, h\u2080, hT\u27e9,\n  have : decidable_pred (\u2208 T\u2081) := \u03bbx, classical.prop_decidable _,\n  let T\u2080' := T\u2080.filter (\u2209 T\u2081),\n  refine lem T\u2080' _ _,\n  { intros x hx, simp [T\u2080'] at hx, exact (hT hx.1).resolve_left hx.2 },\n  { apply weakening' _ h\u2080, apply image_subset, rw [\u2190inter_union_diff (\u2191T\u2080) T\u2081],\n    apply union_subset_union, apply inter_subset_right,\n    intros x hx, rw [finset.mem_coe, finset.mem_filter], exact hx }\nend\n", "meta": {"author": "flypitch", "repo": "flypitch", "sha": "aea5800db1f4cce53fc4a113711454b27388ecf8", "save_path": "github-repos/lean/flypitch-flypitch", "path": "github-repos/lean/flypitch-flypitch/flypitch-aea5800db1f4cce53fc4a113711454b27388ecf8/src/compactness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.6654105653819835, "lm_q1q2_score": 0.3715166907491849}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n\nimport category_theory.limits.shapes.binary_products\n\nuniverses v u\n\nopen category_theory category_theory.category category_theory.limits\nnamespace category_theory\n\nvariables {C : Type u} [\ud835\udc9e : category.{v} C]\ninclude \ud835\udc9e\n\nvariables [has_binary_products.{v} C]\n\nlocal attribute [tidy] tactic.case_bash\n\n@[simps]\ndef prod_functor : C \u2964 C \u2964 C :=\n{ obj := \u03bb X, { obj := \u03bb Y, X \u2a2f Y, map := \u03bb Y Z, limits.prod.map (\ud835\udfd9 X) },\n  map := \u03bb Y Z f, { app := \u03bb T, limits.prod.map f (\ud835\udfd9 T) }}\n\n@[simp] lemma prod_left_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.left = limits.prod.fst := rfl\n@[simp] lemma prod_right_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.right = limits.prod.snd := rfl\n\nlemma prod.hom_ext {A X Y : C} {a b : A \u27f6 X \u2a2f Y} (h1 : a \u226b limits.prod.fst = b \u226b limits.prod.fst) (h2 : a \u226b limits.prod.snd = b \u226b limits.prod.snd) : a = b :=\nbegin\n  apply limit.hom_ext,\n  rintros (_ | _),\n  simpa,\n  simpa\nend\n\nlemma prod_map_comm {A B X Y : C} (f : A \u27f6 B) (g : X \u27f6 Y) :\n  limits.prod.map (\ud835\udfd9 _) f \u226b limits.prod.map g (\ud835\udfd9 _) = limits.prod.map g (\ud835\udfd9 _) \u226b limits.prod.map (\ud835\udfd9 _) f :=\nbegin\n  apply prod.hom_ext, simp, erw id_comp, erw comp_id, simp, erw id_comp, erw comp_id\nend\n\nlemma prod_functorial {X Y Z W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  limits.prod.map (f \u226b g) (\ud835\udfd9 W) = limits.prod.map f (\ud835\udfd9 W) \u226b limits.prod.map g (\ud835\udfd9 W) :=\nbegin\n  apply prod.hom_ext,\n  simp, simp, dsimp, simp\nend\nlemma prod_functorial' {X Y Z W : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  limits.prod.map (\ud835\udfd9 W) (f \u226b g) = limits.prod.map (\ud835\udfd9 W) f \u226b limits.prod.map (\ud835\udfd9 W) g :=\nbegin\n  apply prod.hom_ext,\n  simp, dsimp, simp, simp\nend\n\nend category_theory", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/geo/src/binary_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.37147112796188947}}
{"text": "import Duper.Tactic\nimport Duper.TPTP\n\ndef rec\u2081 : False := by\n  duper [Nat.rec]\n\n#check Nat.rec\n\nnamespace Color1\n\ninductive Color :=\n| red : Color\n\nexample : @Color.rec (fun _ => Nat) a .red = a := by duper [Color.rec]\n\nend Color1\n\nnamespace Color2\n\ninductive Color :=\n| red : Color\n| green : Color\n\nexample : @Color.rec (fun _ => Nat) a b .red = a := by duper [Color.rec]\n\ndef test : Color \u2192 Color\n| .red => .green\n| .green => .red\n\nset_option pp.match false\n#print test\n#print test.match_1\n#print Color.casesOn\n\n-- Not sure why this does not work:\nset_option trace.Prover.saturate true in\nexample : test .red = .green := by\n  duper [test, test.match_1, Color.rec, Color.casesOn]\n\nend Color2\n\n\nset_option simultaneousSuperposition false -- TODO: There is a bug in simultaneous sup that prevents this example from working\nset_option trace.DUnif.result true\nset_option dUnifDbgOn true\nset_option trace.Rule.superposition true\nset_option trace.Prover.saturate true\n-- set_option maxHeartbeats 200\nexample : @Nat.rec (fun _ => Bool) a b Nat.zero = a := by duper [Nat.rec]\n\n", "meta": {"author": "leanprover-community", "repo": "duper", "sha": "96b8f8383363e800976b0fa99830c1b5e8c19b09", "save_path": "github-repos/lean/leanprover-community-duper", "path": "github-repos/lean/leanprover-community-duper/duper-96b8f8383363e800976b0fa99830c1b5e8c19b09/Duper/Tests/test_recursor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.37147112771487956}}
{"text": "-- Copyright (C) 2020 by @ljt12138\n\nimport tactic pal.basics pal.proof \n\nnamespace pal_logic\n\nlemma useless_condition {\u03b1 agent : Type} (\u03c6 \u03c8 : sentence \u03b1 agent) :\n  \u22a2 \u03c8 \u2192 \u22a2 \u03c6\u21a3\u03c8 :=\nbegin\n  intros a,\n  have ht : \u22a2 \u03c8 \u21a3 \u03c6 \u21a3 \u03c8, { existsi proof.ax1 _ _, prover },\n  cases a, cases ht,\n  existsi proof.mp _ _, prover\nend\n\nlemma know_truth {\u03b1 agent : Type} {\u03c6 : sentence \u03b1 agent} (i : agent) : \n  \u22a2 \u03c6 \u2192 \u22a2 \u25a1(i:\u03c6) :=\nbegin\n  intros h, cases h,\n  existsi proof.truth _, prover\nend\n\nlemma id_provable {\u03b1 agent : Type} (\u03c6 : sentence \u03b1 agent) :\n  \u22a2 \u03c6 \u21a3 \u03c6 :=\nbegin\n  existsi (proof.mp (proof.mp (proof.ax2 _ _ _) \n                              (proof.ax1 _ _)) \n                    (proof.ax1 \u03c6 \u03c6)),\n  prover\nend\n\nlemma hs_rule {\u03b1 agent : Type} (\u03c6 \u03c8 \u03b3 : sentence \u03b1 agent) : \n  \u22a2 \u03c6 \u21a3 \u03c8 \u2192 \u22a2 \u03c8 \u21a3 \u03b3 \u2192 \u22a2 \u03c6 \u21a3 \u03b3 :=\nbegin\n  intros a\u2081 a\u2082,\n  have h\u2081 : \u22a2 \u03c6 \u21a3 \u03c8 \u21a3 \u03b3, { apply useless_condition, assumption },\n  cases h\u2081, cases a\u2081,\n  existsi proof.mp (proof.mp (proof.ax2 _ _ _) _) _, prover\nend\n\nlemma explosion {\u03b1 agent : Type} (\u03c6 : sentence \u03b1 agent) :\n  \u22a2 (\u22a5 : sentence \u03b1 agent) \u21a3 \u03c6 :=\nbegin\n  have h\u2081 : \u22a2 (\u22a5 \u21a3 (\u03c6 \u21a3 \u22a5) \u21a3 \u22a5 : sentence \u03b1 agent), \n  { existsi proof.ax1 _ _, prover },\n  have h\u2082 : \u22a2 ((\u03c6 \u21a3 \u22a5) \u21a3 \u22a5) \u21a3 \u03c6, \n  { existsi proof.ax3 _, prover },\n  apply hs_rule, repeat {assumption} \nend\n\n\nlemma double_\u0393_equiv {\u03b1 agent : Type} (\u0393 : list (sentence \u03b1 agent)) :\n  consistent \u0393 \u2192 consistent (\u0393++\u0393) :=  \nbegin\n  intros a\u2081, simp at *, intros a\u2082,\n  apply a\u2081, clear a\u2081, \n  cases a\u2082, \n  existsi proof.conj _ _ _ _ _,\n  apply proof_of.conj, exact a\u2082_h, simp\nend\n\nlemma mp_via_ax2 {\u03b1 agent : Type} (\u03c6 \u03c8 \u03b3 : sentence \u03b1 agent) :\n  \u22a2 \u03c6 \u21a3 \u03c8 \u21a3 \u03b3 \u2192 \u22a2 \u03c6 \u21a3 \u03c8 \u2192 \u22a2 \u03c6 \u21a3 \u03b3 :=\nbegin\n  intros a\u2081 a\u2082, cases a\u2081, cases a\u2082,\n  existsi proof.mp (proof.mp (proof.ax2 _ _ _) _) _, prover\nend\n\nlemma uncurry {\u03b1 agent : Type} (\u03c6 \u03c8 \u03b3 : sentence \u03b1 agent) :\n  \u22a2 \u03c6 \u21a3 \u03c8 \u21a3 \u03b3 \u2192 \u22a2 \u03c6 & \u03c8 \u21a3 \u03b3 :=\nbegin\n  intros a, cases a,\n  existsi proof.uncurry _ _ _, prover\nend\n\nlemma curry {\u03b1 agent : Type} (\u03c6 \u03c8 \u03b3 : sentence \u03b1 agent) :\n  \u22a2 \u03c6 & \u03c8 \u21a3 \u03b3 \u2192 \u22a2 \u03c6 \u21a3 \u03c8 \u21a3 \u03b3 :=\nbegin\n  intros a, cases a,\n  existsi proof.curry _ _ _, prover\nend\n\nlemma dni {\u03b1 agent : Type} (\u03c6 : sentence \u03b1 agent) :\n  \u22a2 \u03c6 \u21a3 (\u03c6\u21a3\u22a5) \u21a3 \u22a5 :=\nbegin\n  have h\u2081 : \u22a2 \u03c6 & (\u03c6 \u21a3 \u22a5) \u21a3 \u03c6,\n  { apply uncurry, existsi proof.ax1 _ _, prover },\n  have h\u2082 : \u22a2 \u03c6 & (\u03c6 \u21a3 \u22a5) \u21a3 (\u03c6 \u21a3 \u22a5),\n  { apply uncurry, apply useless_condition, apply id_provable },\n  apply curry, apply mp_via_ax2, repeat {assumption}\nend\n\nlemma drop_mem {\u03b1 : Type} (\u03c6 : \u03b1) (\u0393 : list \u03b1) : \n  \u2203 \u0393', \u0393' \u2286 \u0393 \u2227 \u03c6 \u2209 \u0393' \u2227 \u0393 \u2286 \u03c6 :: \u0393' :=\nbegin\n  classical, \n  induction \u0393 with \u03c8 \u0393 ih,\n  { existsi list.nil, simp },\n  {\n    cases ih with \u0393' ih\u2081,\n    cases classical.em (\u03c6 = \u03c8), \n    {\n      existsi \u0393', split,\n      { apply list.subset_cons_of_subset, exact ih\u2081.left }, split,\n      { exact ih\u2081.right.left },\n      { simp, split, { left, rewrite h }, { exact ih\u2081.right.right }}\n    },\n    {\n      existsi (list.cons \u03c8 \u0393'), split,\n      { simp, apply list.subset_cons_of_subset, exact ih\u2081.left }, split,\n      { \n        intros contra, simp at contra, cases contra, \n        exact h contra, exact ih\u2081.right.left contra \n      },\n      {\n        simp, intros \u03b3 ht, \n        cases ih\u2081.right.right ht, \n        { rewrite h_1, simp },\n        { simp, right, right, exact h_1 }\n      }\n    }\n  }\nend\n\nlemma add_assumption {\u03b1 agent : Type} {\u03c6 \u03c8 : sentence \u03b1 agent} {\u0393 : list _} :\n  \u22a2 \u03c6 & conjunction \u0393 \u21a3 \u03c8 \u2194 \u22a2 conjunction (\u03c6 :: \u0393) \u21a3 \u03c8 :=\nbegin\n  split,\n  {\n    intros h, cases \u0393 with \u03c8 \u0393',\n    {\n      have ht\u2081 := curry _ _ _ h, simp at ht\u2081, simp, \n      have ht\u2082 : \u22a2 \u03c6 \u21a3 ((\u22a5 : sentence \u03b1 agent) \u21a3 \u22a5),\n      { apply useless_condition, apply id_provable },\n      cases ht\u2081, cases ht\u2082, \n      existsi (proof.mp (proof.mp (proof.ax2 _ _ _) _) _),\n      prover\n    },\n    { unfold conjunction, exact h }\n  },\n  {\n    intros h, cases \u0393 with \u03c8 \u0393',\n    {\n      simp at h, cases h,\n      existsi proof.conjl _ _ _, prover\n    },\n    { unfold conjunction at h, exact h }\n  }\nend\n\nlemma drop_assumption {\u03b1 agent : Type} {\u03c6 \u03c8 : sentence \u03b1 agent} {\u0393 : list _} : \n  \u22a2 \u03c6 & conjunction \u0393 \u21a3 \u03c8 \u2192 \u2203 \u0393', \u0393' \u2286 \u0393 \u2227 \u03c6 \u2209 \u0393' \u2227 \u22a2 \u03c6 & conjunction \u0393' \u21a3 \u03c8 :=\nbegin\n  intros h,\n  cases drop_mem \u03c6 \u0393 with \u0393' h\u2081, \n  existsi \u0393', split,\n  { exact h\u2081.left }, split,\n  { exact h\u2081.right.left },\n  {\n    apply add_assumption.mpr, \n    have ht\u2081 := add_assumption.mp h, \n    have ht\u2082 : (list.cons \u03c6 \u0393) \u2286 \u03c6 :: \u0393', { simp, exact h\u2081.right.right }, \n    cases ht\u2081, existsi proof.conj _ _ _ _ _, \n    apply proof_of.conj, assumption, assumption \n  }\nend \n\nlemma and_swap {\u03b1 agent : Type} {\u03c6 \u03c8 \u03b3 : sentence \u03b1 agent} :\n  \u22a2 \u03c6 & \u03c8 \u21a3 \u03b3 \u2192 \u22a2 \u03c8 & \u03c6 \u21a3 \u03b3 :=\nbegin\n  have h\u2081 : \u2200 \u03c6 \u03c8 : sentence \u03b1 agent, \u03c6 & \u03c8 = conjunction [\u03c6, \u03c8] := by simp, \n  rewrite h\u2081, rewrite h\u2081, \n  intros h\u2082,\n  have h\u2083 : [\u03c6, \u03c8] \u2286 [\u03c8, \u03c6] := by simp, \n  cases h\u2082,\n  existsi proof.conj _ _ _ _ _,\n  apply proof_of.conj, assumption, assumption\nend\n\nlemma contra_imp_conseq {\u03b1 agent : Type} {\u03c6 : sentence \u03b1 agent} {\u0393 : list (sentence \u03b1 agent)} :\n  \u03c6 \u2208 \u0393 \u2192 \u22a2 conjunction \u0393 \u21a3 \u22a5 \u2192 \u2203 \u0393', \u0393' \u2286 \u0393 \u2227 \u03c6 \u2209 \u0393' \u2227 \u22a2 conjunction \u0393' \u21a3 (\u03c6 \u21a3 \u22a5) :=\nbegin\n  intros h\u2081 h\u2082,\n  have h\u2083 : \u22a2 (\u03c6 & conjunction \u0393) \u21a3 \u22a5, \n  { \n    apply add_assumption.mpr, cases h\u2082, \n    existsi proof.conj _ _ _ _ _, apply proof_of.conj, exact h\u2082_h, simp \n  },\n  cases drop_assumption h\u2083 with \u0393' h\u2084, \n  existsi \u0393', split,\n  { exact h\u2084.left }, split,\n  { exact h\u2084.right.left },\n  { apply curry, apply and_swap, exact h\u2084.right.right }\nend\n\nlemma conj_subset {\u03b1 agent : Type} {\u03c6 : sentence \u03b1 agent} {\u0393\u2081 \u0393\u2082 : list (sentence \u03b1 agent)} :\n  \u22a2 conjunction \u0393\u2081 \u21a3 \u03c6 \u2192 \u0393\u2081 \u2286 \u0393\u2082 \u2192 \u22a2 conjunction \u0393\u2082 \u21a3 \u03c6 :=\nbegin\n  intros h\u2081 h\u2082,\n  cases h\u2081, \n  existsi proof.conj _ _ _ _ _,\n  apply proof_of.conj, exact h\u2081_h, exact h\u2082\nend\n\nlemma know_and_intro {\u03b1 agent : Type} {\u03c6 \u03c8 : sentence \u03b1 agent} {i : agent} : \n  \u22a2 \u25a1(i : \u03c6) \u21a3 \u25a1(i : \u03c8) \u21a3 \u25a1(i : \u03c6&\u03c8) :=\nbegin\n  have h\u2081 : \u22a2 \u25a1(i : \u03c6\u21a3\u03c8\u21a3\u03c6&\u03c8),\n  { apply know_truth, apply curry, apply id_provable },\n  have h\u2082 : \u22a2 \u25a1(i : \u03c6) \u21a3 \u25a1(i : \u03c8\u21a3\u03c6&\u03c8),\n  { cases h\u2081, existsi proof.mp (proof.ax4 _ _ _) _, prover },\n  apply hs_rule, exact h\u2082,\n  existsi proof.ax4 _ _ _, prover\nend\n\nend pal_logic\n", "meta": {"author": "ljt12138", "repo": "Formalization-PAL", "sha": "351962172c8e85ec8bdf59421df2acd743cb4e3e", "save_path": "github-repos/lean/ljt12138-Formalization-PAL", "path": "github-repos/lean/ljt12138-Formalization-PAL/Formalization-PAL-351962172c8e85ec8bdf59421df2acd743cb4e3e/src/pal/lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.3713848048322638}}
{"text": "import Lean\n\nnamespace SciLean.ConvIf\n\n@[inline]\nabbrev convIf {\u03b1} (P : Prop) (_ : Decidable P) (x : P \u2192 \u03b1) (y : \u00acP \u2192 \u03b1) : \u03b1 := if h : P then x h else y h\n\ndef convIf.rhs {\u03b1} (P : Prop) [inst : Decidable P] (a : \u03b1) := convIf P inst (\u03bb _ => a) (\u03bb _ => a) \n\ntheorem convIf.id {\u03b1} (P : Prop) [inst : Decidable P] (a : \u03b1) : a = convIf P inst (\u03bb _ => a) (\u03bb _ => a) := \nby\n  simp[convIf]; cases inst; rename_i q; simp[q]; rename_i q; simp[q]; done\n\nopen Lean.Parser.Tactic.Conv \nsyntax (name := conv_if) \"if\" ident \":\" term  \"then\" convSeq \"else\" convSeq : conv\n\nopen Lean.Elab Tactic Conv in\n@[tactic conv_if] \ndef convIfTactic : Tactic \n| `(conv| if $h : $P then $trueConv else $falseConv) => do\n   withMainContext do\n     \n     let p \u2190 elabTerm P none\n     let t' \u2190 Lean.Meta.mkAppM ``convIf.rhs #[p, (\u2190 getLhs)]\n     let h' \u2190 Lean.Meta.mkAppM ``convIf.id  #[p, (\u2190 getLhs)]\n\n     updateLhs t' h'\n     evalTactic (\u2190 \n       `(convSeq| unfold convIf.rhs\n                  conv => enter[3]; intro $h; ($trueConv)\n                  conv => enter[4]; intro $h; ($falseConv)\n                  unfold convIf))\n| _ => throwUnsupportedSyntax\n\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Tactic/ConvIf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.37133290685446835}}
{"text": "import group_theory.group_action.option\nimport mathlib.group_action\nimport mathlib.pointwise\nimport phase1.allowable\n\nnoncomputable theory\n\nopen function set with_bot\nopen_locale cardinal pointwise\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}] [position_data.{}]\n\nopen code Iio_index\n\nvariables (\u03b1 : \u039b) [core_tangle_cumul \u03b1] {\u03b2 : Iio_index \u03b1} {\u03b3 : Iio \u03b1}\n\nabbreviation extensions := \u03a0 \u03b2 : Iio \u03b1, set (tangle \u03b2)\n\nnamespace semitangle\n\nvariables [positioned_tangle_cumul \u03b1] [almost_tangle_cumul \u03b1]\n\n/-- Keeps track of the preferred extension of a semitangle, along with coherence conditions\nrelating each extension of the semitangle. -/\n@[nolint has_nonempty_instance] inductive preference (members : extensions \u03b1)\n| base (atoms : set (tangle (\u22a5 : Iio_index \u03b1))) :\n    (\u2200 \u03b3, A_map bot_ne_coe atoms = members \u03b3) \u2192\n    preference\n| proper (\u03b2 : Iio \u03b1) :\n    (mk \u03b2 (members \u03b2) : code \u03b1).is_even \u2192\n    (\u2200 (\u03b3 : Iio \u03b1) (h\u03b2\u03b3 : Iio_coe \u03b2 \u2260 \u03b3), A_map h\u03b2\u03b3 (members \u03b2) = members \u03b3) \u2192\n    preference\n\nvariables {\u03b1} {members : extensions \u03b1}\n\n/-- The `-1`-extension associated with a given semitangle extension. -/\ndef preference.atoms : preference \u03b1 members \u2192 set atom\n| (preference.base atoms _) := (atoms : set (tangle \u22a5))\n| (preference.proper _ _ _) := \u2205\n\nlemma preference.base_heq_base {m\u2081 m\u2082 : extensions \u03b1} {s\u2081 s\u2082 h\u2081 h\u2082}\n  (hm : m\u2081 = m\u2082) (hs : s\u2081 = s\u2082) :\n  (preference.base s\u2081 h\u2081 : preference \u03b1 m\u2081) == (preference.base s\u2082 h\u2082 : preference \u03b1 m\u2082) :=\nby cases hm; cases hs; refl\n\nlemma preference.proper_heq_proper {m\u2081 m\u2082 : extensions \u03b1} {\u03b2\u2081 \u03b2\u2082 h\u2081 h\u2082 h\u2083 h\u2084}\n  (hm : m\u2081 = m\u2082) (hs : \u03b2\u2081 = \u03b2\u2082) :\n  (preference.proper \u03b2\u2081 h\u2081 h\u2082 : preference \u03b1 m\u2081) ==\n    (preference.proper \u03b2\u2082 h\u2083 h\u2084 : preference \u03b1 m\u2082) :=\nby cases hm; cases hs; refl\n\nend semitangle\n\nopen semitangle\n\nvariables [positioned_tangle_cumul \u03b1] [almost_tangle_cumul \u03b1]\n\n/-- A *semitangle* may become an element of our model of tangled type theory.\nWe keep track of its members, written as tangles of all lower levels `\u03b2 < \u03b1`. -/\n@[nolint has_nonempty_instance]\nstructure semitangle :=\n(members : extensions \u03b1)\n(pref : preference \u03b1 members)\n\nvariables {\u03b1}\nnamespace semitangle\n\n/-- The membership relation for nonempty semitangles. -/\ndef mem (t : tangle \u03b3) (s : semitangle \u03b1) : Prop := t \u2208 s.members \u03b3\n\nnotation t ` \u2208\u209b\u209c `:50 s:50 := mem t s\n\n/-- The even code associated to a nonempty semitangle. -/\ndef repr_code : semitangle \u03b1 \u2192 code \u03b1\n| \u27e8exts, preference.base atoms hA\u27e9 := \u27e8\u22a5, atoms\u27e9\n| \u27e8exts, preference.proper \u03b2 rep hA\u27e9 := \u27e8\u03b2, exts \u03b2\u27e9\n\n@[simp] lemma repr_code_base (exts : extensions \u03b1) (atoms hA) :\n  repr_code \u27e8exts, preference.base atoms hA\u27e9 = \u27e8\u22a5, atoms\u27e9 := rfl\n\n@[simp] lemma repr_code_proper (exts : extensions \u03b1) (\u03b2 rep hA) :\n  repr_code \u27e8exts, preference.proper \u03b2 rep hA\u27e9 = \u27e8\u03b2, exts \u03b2\u27e9 := rfl\n\nlemma repr_code_spec : \u03a0 (s : semitangle \u03b1), (repr_code s : code \u03b1).is_even\n| \u27e8exts, preference.proper \u03b2 rep hA\u27e9 := rep\n| \u27e8exts, preference.base atoms hA\u27e9 := is_even_bot _\n\nlemma repr_code_members_ne :\n  \u03a0 (s : semitangle \u03b1) (\u03b3 : Iio \u03b1) (hc\u03b3 : (repr_code s : code \u03b1).1 \u2260 \u03b3),\n  (A_map_code \u03b3 (repr_code s)).2 = s.members \u03b3\n| \u27e8exts, preference.proper \u03b2 rep hA\u27e9 \u03b3 hc\u03b3 := by rw snd_A_map_code; exact hA _ hc\u03b3\n| \u27e8exts, preference.base atoms hA\u27e9 \u03b3 hc\u03b3 := hA _\n\n-- Remark: This formulation of extensionality holds only for types larger than type zero, since\n-- it doesn't take into account any `-1`-extension.\nlemma ext_core (x y : semitangle \u03b1) : (\u2203 \u03b3, \u03b3 < \u03b1) \u2192 x.members = y.members \u2192 x = y :=\nbegin\n  obtain \u27e8xs, hxs\u27e9 := x,\n  obtain \u27e8ys, hys\u27e9 := y,\n  dsimp,\n  rintro \u27e8\u03b3, h\u03b3\u27e9 rfl,\n  have \u03b3 : Iio \u03b1 := \u27e8\u03b3, h\u03b3\u27e9,\n  refine congr_arg (\u03bb h, \u27e8xs, h\u27e9) _,\n  obtain \u27e8atoms\u2081, hA\u2081\u27e9 | \u27e8\u03b2, even\u2081, hA\u2081\u27e9 := hxs;\n    obtain \u27e8atoms\u2082, hA\u2082\u27e9 | \u27e8\u03b3, even\u2082, hA\u2082\u27e9 := hys,\n  { simp_rw A_map_injective ((hA\u2081 \u03b3).trans (hA\u2082 _).symm) },\n  { cases (is_even_bot _).A_map_code_ne even\u2082 bot_ne_mk_coe (sigma.ext_iff.2 \u27e8rfl, (hA\u2081 \u03b3).heq\u27e9) },\n  { cases (is_even_bot _).A_map_code_ne even\u2081 bot_ne_mk_coe (sigma.ext_iff.2 \u27e8rfl, (hA\u2082 \u03b2).heq\u27e9) },\n  { simp only,\n    refine not_ne_iff.1 (\u03bb h\u03b2\u03b3, even\u2082.A_map_code_ne even\u2081 (Iio.coe_injective.ne h\u03b2\u03b3.symm) $\n      sigma.ext_iff.2 \u27e8rfl, heq_of_eq _\u27e9),\n    rw snd_A_map_code,\n    exact hA\u2082 \u03b2 (\u03bb h, h\u03b2\u03b3.symm (Iio.coe_injective h)) }\nend\n\n/-- One useful form of extensionality in tangled type theory. Two nonempty semitangles are equal if\ntheir even codes are equivalent (and hence equal, by uniqueness). -/\nlemma ext_code : \u2200 {x y : semitangle \u03b1}, (repr_code x : code \u03b1) \u2261 repr_code y \u2192 x = y\n| \u27e8x, preference.base atoms\u2081 hA\u2081\u27e9 \u27e8y, preference.base atoms\u2082 hA\u2082\u27e9 h := begin\n  obtain rfl := code.equiv.bot_bot_iff.1 h,\n  obtain rfl : x = y := funext (\u03bb \u03b3, (hA\u2081 _).symm.trans $ hA\u2082 _),\n  refl,\nend\n| \u27e8x, preference.base s hA\u2081\u27e9 \u27e8y, preference.proper \u03b3 even\u2082 hA\u2082\u27e9 h := begin\n  change code.mk _ _ \u2261 code.mk _ _ at h,\n  obtain \u27e8\u03b4, h\u03b4\u27e9 := (code.equiv.bot_left_iff.1 h).resolve_left\n    (ne_of_apply_ne sigma.fst bot_ne_mk_coe),\n  rw h\u03b4 at even\u2082,\n  cases even\u2082.not_is_odd ((is_even_bot _).A_map_code bot_ne_mk_coe),\nend\n| \u27e8x, preference.proper \u03b3 even\u2081 hA\u2081\u27e9 \u27e8y, preference.base s hA\u2082\u27e9 h := begin\n  change code.mk _ _ \u2261 code.mk _ _ at h,\n  obtain \u27e8\u03b4, h\u03b4\u27e9 := (code.equiv.bot_right_iff.1 h).resolve_left\n    (ne_of_apply_ne sigma.fst mk_coe_ne_bot),\n  rw h\u03b4 at even\u2081,\n  cases even\u2081.not_is_odd ((is_even_bot _).A_map_code bot_ne_mk_coe),\nend\n| \u27e8x, preference.proper \u03b3 even\u2081 hA\u2081\u27e9 \u27e8y, preference.proper \u03b4 even\u2082 hA\u2082\u27e9 h := begin\n  dsimp at h,\n  simp only [code.equiv_iff, sigma.ext_iff, mem_Iio, Iio.coe_inj, ne.def, fst_A_map_code,\n    snd_A_map_code, Iio.coe_mk] at h,\n  obtain \u27e8rfl, h\u27e9 | \u27e8-, \u03b3, h\u03b4\u03b3, rfl, h\u27e9 | \u27e8-, \u03b4, h\u03b3\u03b4, rfl, h\u27e9 |\n    \u27e8c, hc, \u03b3, hc\u03b3, \u03b4, hc\u03b4, \u27e8\u27e8rfl, hx'\u27e9, hx\u27e9, _\u27e9 := h,\n  { suffices : x = y,\n    { subst this },\n    refine funext (\u03bb \u03b5, _),\n    obtain rfl | h\u03b4\u03b5 := eq_or_ne \u03b4 \u03b5,\n    { exact h.eq.symm },\n    refine (hA\u2081 _ (\u03bb h, h\u03b4\u03b5 (Iio.coe_injective h))).symm.trans\n      (eq.trans _ $ hA\u2082 _ (\u03bb h, h\u03b4\u03b5 (Iio.coe_injective h))),\n    dsimp,\n    rw h.eq },\n  { rw h.eq at even\u2081,\n    cases (even\u2082.A_map_code $ Iio.coe_injective.ne h\u03b4\u03b3).not_is_even even\u2081 },\n  { rw h.eq at even\u2082,\n    cases (even\u2081.A_map_code $ Iio.coe_injective.ne h\u03b3\u03b4).not_is_even even\u2082 },\n  { rw hx.eq at even\u2081,\n    cases (hc.A_map_code hc\u03b3).not_is_even even\u2081 }\nend\n\n/-- Extensionality in tangled type theory. Two nonempty semitangles are equal if their\n`\u03b2`-extensions are equal for *any* choice of `\u03b3 < \u03b1`.\nTODO: This proof can be golfed quite a bit just by cleaning up the `simp` calls. -/\nlemma ext (x y : semitangle \u03b1) (h : x.members \u03b3 = y.members \u03b3) : x = y :=\nbegin\n  obtain \u27e8xs, hxs\u27e9 := x,\n  obtain \u27e8ys, hys\u27e9 := y,\n  dsimp only at h,\n  refine ext_code _,\n  obtain \u27e8atoms\u2081, hA\u2081\u27e9 | \u27e8\u03b2, even\u2081, hA\u2081\u27e9 := hxs;\n    obtain \u27e8atoms\u2082, hA\u2082\u27e9 | \u27e8\u03b4, even\u2082, hA\u2082\u27e9 := hys,\n  { refine (code.equiv.A_map_right _ (code.is_even_bot _) \u03b3 bot_ne_mk_coe).trans _,\n    simp only [ne.def, Iio_index.bot_ne_coe, not_false_iff, A_map_code_mk_ne,\n      repr_code_base, subtype.coe_mk],\n    rw [hA\u2081 \u03b3, h, \u2190 hA\u2082 \u03b3],\n    exact code.equiv.A_map_left _ (code.is_even_bot _) \u03b3 bot_ne_mk_coe },\n  { simp only [repr_code_base, subtype.coe_mk, repr_code_proper],\n    obtain rfl | h\u03b4\u03b3 := eq_or_ne \u03b4 \u03b3,\n    { simp only [is_even_bot, mem_Iio, subtype.val_eq_coe, set_coe.forall,\n        ne.def, Iio.coe_inj] at *,\n      have := hA\u2081 \u03b4 \u03b4.prop,\n      rw subtype.coe_eta at this,\n      rw [\u2190 h, \u2190 this],\n      exact code.equiv.A_map_right _ (code.is_even_bot _) _ bot_ne_mk_coe },\n    { refine (code.equiv.A_map_right _ (code.is_even_bot _) \u03b3 bot_ne_mk_coe).trans _,\n      simp only [ne.def, Iio_index.bot_ne_coe, not_false_iff, A_map_code_mk_ne],\n      rw [hA\u2081 \u03b3, h, \u2190hA\u2082 \u03b3 (Iio.coe_injective.ne h\u03b4\u03b3), \u2190 A_map_code_mk_ne],\n      exact code.equiv.A_map_left _ even\u2082 \u03b3 (Iio.coe_injective.ne h\u03b4\u03b3) } },\n  { simp only [repr_code_proper, subtype.coe_mk, repr_code_base],\n    obtain rfl | h\u03b2\u03b3 := eq_or_ne \u03b2 \u03b3,\n    { dsimp only [mem_Iio, ne.def, subtype.val_eq_coe, set_coe.forall] at *,\n      rw [h, \u2190hA\u2082 \u03b2],\n      exact code.equiv.A_map_left _ (code.is_even_bot _) _ bot_ne_mk_coe },\n    { refine (code.equiv.A_map_right _ even\u2081 \u03b3 $ Iio.coe_injective.ne h\u03b2\u03b3).trans _,\n      dsimp only [mem_Iio, ne.def, subtype.val_eq_coe, set_coe.forall] at *,\n      rw A_map_code_mk_ne _ _ (Iio.coe_injective.ne h\u03b2\u03b3),\n      rw [hA\u2081 \u03b3 (Iio.coe_injective.ne h\u03b2\u03b3), h, \u2190hA\u2082 \u03b3],\n      exact code.equiv.A_map_left _ (code.is_even_bot _) \u03b3 bot_ne_mk_coe } },\n  { simp only [repr_code_proper, subtype.coe_mk],\n    obtain rfl | h\u03b2\u03b3 := eq_or_ne \u03b2 \u03b3,\n    { obtain rfl | h\u03b4\u03b2 := eq_or_ne \u03b4 \u03b2,\n      { rw h },\n      { have := A_map_code_ne \u03b2 (code.mk \u03b4 (ys \u03b4)) (Iio.coe_injective.ne h\u03b4\u03b2),\n        dsimp only [mem_Iio, ne.def, subtype.val_eq_coe, set_coe.forall, code.snd_mk] at *,\n        rw [h, \u2190hA\u2082 _ (Iio.coe_injective.ne h\u03b4\u03b2), \u2190 code.mk_def, \u2190 this],\n        exact code.equiv.A_map_left _ even\u2082 _ (Iio.coe_injective.ne h\u03b4\u03b2) } },\n    obtain rfl | h\u03b4\u03b3 := eq_or_ne \u03b4 \u03b3,\n    { have := A_map_code_ne \u03b4 (code.mk \u03b2 (xs \u03b2)) (Iio.coe_injective.ne h\u03b2\u03b3),\n      dsimp only [mem_Iio, ne.def, subtype.val_eq_coe, set_coe.forall, code.snd_mk] at *,\n      simp_rw [\u2190h, \u2190hA\u2081 _ (Iio.coe_injective.ne h\u03b2\u03b3), \u2190 code.mk_def, \u2190 this],\n      exact code.equiv.A_map_right _ even\u2081 _ (Iio.coe_injective.ne h\u03b2\u03b3) },\n    refine (code.equiv.A_map_right _ even\u2081 \u03b3 $ Iio.coe_injective.ne h\u03b2\u03b3).trans _,\n    have := A_map_code_ne \u03b3 (code.mk \u2191\u03b4 (ys \u03b4)) (Iio.coe_injective.ne h\u03b4\u03b3),\n    dsimp only [mem_Iio, ne.def, subtype.val_eq_coe, set_coe.forall, code.snd_mk] at *,\n    rw A_map_code_ne,\n    rw [code.snd_mk, hA\u2081 \u03b3 (Iio.coe_injective.ne h\u03b2\u03b3), h, \u2190hA\u2082 \u03b3 (Iio.coe_injective.ne h\u03b4\u03b3)],\n    rw \u2190 this,\n    exact code.equiv.A_map_left _ even\u2082 \u03b3 (Iio.coe_injective.ne h\u03b4\u03b3) }\nend\n\n/-- Extensionality in tangled type theory. Two nonempty semitangles are equal if their\n`\u03b2`-extensions are equal for *any* choice of `\u03b2 < \u03b1`. -/\nlemma ext' (x y : semitangle \u03b1) (h : \u2200 t : tangle \u03b3, t \u2208\u209b\u209c x \u2194 t \u2208\u209b\u209c y) : x = y :=\next x y $ set.ext h\n\n/-- Extensionality at the lowest level of tangled type theory.\nAt type 0, all nonempty semitangles have a `-1`-extension.\nTherefore, the extensionality principle in this case applies to the `-1`-extensions. -/\nlemma ext_zero (x y : semitangle \u03b1) (\u03b1_zero : is_min \u03b1) (h : x.pref.atoms = y.pref.atoms) :\n  x = y :=\nbegin\n  obtain \u27e8xs, \u27e8atoms\u2081, hA\u2081\u27e9 | \u27e8\u03b3, _, _\u27e9\u27e9 := x, swap,\n  { cases \u03b1_zero.not_lt \u03b3.2 },\n  obtain \u27e8ys, \u27e8atoms\u2082, hA\u2082\u27e9 | \u27e8\u03b3, _, _\u27e9\u27e9 := y, swap,\n  { cases \u03b1_zero.not_lt \u03b3.2 },\n  have : atoms\u2081 = atoms\u2082 := h,\n  subst this,\n  suffices : xs = ys, by subst this,\n  ext \u03b2 -,\n  cases \u03b1_zero.not_lt \u03b2.2,\nend\n\n/-- Construct a semitangle from an even nonempty code. -/\ndef intro (s : set (tangle \u03b2)) (heven : (code.mk \u03b2 s : code \u03b1).is_even) : semitangle \u03b1 :=\n\u27e8extension s, match \u03b2, s, heven with\n  | \u27e8\u22a5, _\u27e9, s, _ := preference.base s $ \u03bb \u03b2, rfl\n  | \u27e8(\u03b3 : \u039b), h\u03b3\u27e9, s, heven := preference.proper \u27e8\u03b3, coe_lt_coe.1 h\u03b3\u27e9\n    (by { convert heven, exact extension_self (show set (tangle $ Iio_coe \u27e8\u03b3, _\u27e9), from s) }) $\n      \u03bb \u03b4 h\u03b4, by { rw extension_ne s \u03b4 h\u03b4, congr,\n        exact extension_self (show set (tangle $ Iio_coe \u27e8\u03b3, _\u27e9), from s) }\n  end\u27e9\n\n@[simp] lemma exts_intro (s : set (tangle \u03b2)) (heven) :\n  (intro s heven).members = extension s := rfl\n\nend semitangle\n\nopen semitangle\n\nvariables [core_tangle_data \u03b1]\n\nnamespace allowable_perm\nvariables {f : allowable_perm \u03b1} {e : extensions \u03b1}\n\n@[simp] lemma smul_extension_apply (f : allowable_perm \u03b1) (s : set (tangle \u03b2)) :\n  f \u2022 extension s \u03b3 = extension (f \u2022 s) \u03b3 :=\nbegin\n  by_cases \u03b2 = \u03b3,\n  { subst h,\n    simp only [extension_eq, cast_eq], },\n  { simp only [extension_ne _ _ h, smul_A_map], },\nend\n\n@[simp] lemma smul_extension (f : allowable_perm \u03b1) (s : set (tangle \u03b2)) :\n  f \u2022 extension s = extension (f \u2022 s) :=\nbegin\n  ext \u03b3 : 1,\n  rw \u2190 smul_extension_apply,\n  refl,\nend\n\nlemma smul_aux\u2081 {s : set (tangle (\u22a5 : Iio_index \u03b1))}\n  (h : \u2200 (\u03b3 : Iio \u03b1), A_map bot_ne_coe s = (e \u03b3 : set (tangle (Iio_coe \u03b3)))) (\u03b3 : Iio \u03b1) :\n  A_map bot_ne_coe (f \u2022 s) = (f \u2022 e) \u03b3 :=\nby simpa only [smul_A_map] using congr_arg (\u03bb c, f \u2022 c) (h \u03b3)\n\nlemma smul_aux\u2082 (h : \u2200 (\u03b4 : Iio \u03b1) (h\u03b3\u03b4 : Iio_coe \u03b3 \u2260 \u03b4),\n  A_map h\u03b3\u03b4 (e \u03b3) = (e \u03b4 : set (tangle (Iio_coe \u03b4)))) (\u03b4 : Iio \u03b1) (h\u03b3\u03b4 : Iio_coe \u03b3 \u2260 \u03b4) :\n  A_map h\u03b3\u03b4 ((f \u2022 e) \u03b3) = (f \u2022 e) \u03b4 :=\nby simpa only [smul_A_map] using congr_arg (\u03bb c, f \u2022 c) (h \u03b4 h\u03b3\u03b4)\n\n/-- Allowable permutations act on nonempty semitangles. -/\nnoncomputable! instance : has_smul (allowable_perm \u03b1) (semitangle \u03b1) :=\n{ smul := \u03bb f t, \u27e8f \u2022 t.members, begin\n    obtain \u27e8members, \u27e8s, h\u27e9 | \u27e8\u03b3, ht, h\u27e9\u27e9 := t,\n    { exact preference.base (f \u2022 s) (smul_aux\u2081 h) },\n    { exact preference.proper _ ht.smul (smul_aux\u2082 h) }\n    end\u27e9 }\n\n@[simp] lemma members_smul (f : allowable_perm \u03b1) (s : semitangle \u03b1) :\n  (f \u2022 s).members = f \u2022 s.members := rfl\n\n@[simp] lemma smul_base (f : allowable_perm \u03b1) (e : extensions \u03b1) (s h) :\n  f \u2022 (\u27e8e, preference.base s h\u27e9 : semitangle \u03b1) =\n    \u27e8f \u2022 e, preference.base (f \u2022 s) (smul_aux\u2081 h)\u27e9 := rfl\n\n@[simp] lemma smul_proper (f : allowable_perm \u03b1) (e : extensions \u03b1) (\u03b3 ht h) :\n  f \u2022 (\u27e8e, preference.proper \u03b3 ht h\u27e9 : semitangle \u03b1) =\n    \u27e8f \u2022 e, preference.proper _ ht.smul (smul_aux\u2082 h)\u27e9 := rfl\n\ninstance mul_action_semitangle : mul_action (allowable_perm \u03b1) (semitangle \u03b1) := {\n  one_smul := begin\n    rintro \u27e8exts, \u27e8s, h\u27e9 | \u27e8\u03b3, ht, h\u27e9\u27e9,\n    { rw smul_base,\n      simp only [one_smul, eq_self_iff_true, true_and],\n      refine preference.base_heq_base _ _,\n      rw one_smul,\n      refl, },\n    { rw smul_proper,\n      simp only [one_smul, eq_self_iff_true, true_and],\n      refine semitangle.preference.proper_heq_proper _ rfl,\n      rw one_smul, },\n  end,\n  mul_smul := begin\n    rintro f g \u27e8exts, \u27e8s, h\u27e9 | \u27e8\u03b3, ht, h\u27e9\u27e9,\n    { simp only [smul_base, mul_smul, eq_self_iff_true, true_and],\n      refine preference.base_heq_base _ _,\n      rw mul_smul,\n      refl, },\n    { simp only [smul_proper, mul_smul, eq_self_iff_true, true_and],\n      refine semitangle.preference.proper_heq_proper _ rfl,\n      rw mul_smul, },\n  end\n}\n\nend allowable_perm\n\nvariables (\u03b1)\n\n/-- A tangle at the new level `\u03b1` is a semitangle supported by a small support.\nThis is `\u03c4_\u03b1` in the blueprint.\nUnlike the type `tangle`, this is not an opaque definition, and we can inspect and unfold it. -/\n@[nolint has_nonempty_instance]\ndef new_tangle := {s : semitangle \u03b1 // supported \u03b1 (allowable_perm \u03b1) s}\n\nvariables {\u03b1} {c d : code \u03b1} {S : set (support_condition \u03b1)}\n\nopen mul_action\n\n/-- If a set of support conditions supports a code, it supports all equivalent codes. -/\nprotected lemma code.equiv.supports (hcd : c \u2261 d) (hS : supports (allowable_perm \u03b1) S c) :\n  supports (allowable_perm \u03b1) S d :=\n\u03bb f h, (hcd.symm.smul.trans $ (code.equiv.of_eq $ hS f h).trans hcd).unique rfl\n\nlemma code.equiv.supports_iff (hcd : c \u2261 d) :\n  supports (allowable_perm \u03b1) S c \u2194 supports (allowable_perm \u03b1) S d :=\n\u27e8hcd.supports, hcd.symm.supports\u27e9\n\n/-- If two codes are equivalent, one is supported if and only if the other is. -/\nlemma code.equiv.small_supported_iff (hcd : c \u2261 d) :\n  supported \u03b1 (allowable_perm \u03b1) c \u2194 supported \u03b1 (allowable_perm \u03b1) d :=\n\u27e8\u03bb \u27e8\u27e8s, hs, h\u27e9\u27e9, \u27e8\u27e8s, hs, hcd.supports h\u27e9\u27e9, \u03bb \u27e8\u27e8s, hs, h\u27e9\u27e9, \u27e8\u27e8s, hs, hcd.symm.supports h\u27e9\u27e9\u27e9\n\n@[simp] lemma smul_intro (f : allowable_perm \u03b1) (s : set (tangle \u03b2)) (hs) :\n  f \u2022 intro s hs = intro (f \u2022 s) hs.smul :=\nbegin\n  cases \u03b2,\n  induction \u03b2_val using with_bot.rec_bot_coe,\n  { simp only [intro, allowable_perm.smul_base, allowable_perm.smul_extension,\n      eq_self_iff_true, true_and],\n    refine preference.base_heq_base _ rfl,\n    rw allowable_perm.smul_extension },\n  { simp only [intro, allowable_perm.smul_proper, allowable_perm.smul_extension,\n      eq_self_iff_true, true_and],\n    refine preference.proper_heq_proper _ rfl,\n    rw allowable_perm.smul_extension }\nend\n\n-- TODO: Move next two lemmas elsewhere.\nlemma allowable_to_struct_perm_bot (\u03c0 : allowable (\u22a5 : Iio_index \u03b1)) :\n  core_tangle_data.allowable_to_struct_perm \u03c0 = struct_perm.to_bot_iso.to_monoid_hom \u03c0 := rfl\n\nlemma _root_.con_nf.semiallowable_perm.to_allowable_bot (\u03c0 : semiallowable_perm \u03b1) :\n  semiallowable_perm.to_allowable \u22a5 \u03c0 = struct_perm.to_near_litter_perm\n    (semiallowable_perm.to_struct_perm \u03c0) :=\nbegin\n  unfold semiallowable_perm.to_allowable semiallowable_perm.to_struct_perm\n    struct_perm.to_near_litter_perm struct_perm.lower allowable.to_struct_perm,\n  rw dif_neg with_bot.bot_ne_coe,\n  simp only [monoid_hom.coe_mk, monoid_hom.coe_comp, mul_equiv.coe_to_monoid_hom,\n    comp_app, struct_perm.of_coe_to_coe, allowable_to_struct_perm_bot, mul_equiv.symm_apply_apply],\n  refl,\nend\n\n/-- For any near-litter `N`, the code `(\u03b1, -1, N)` is a tangle at level `\u03b1`.\nThis is called a *typed near litter*. -/\ndef new_typed_near_litter (N : near_litter) : new_tangle \u03b1 :=\n\u27e8intro (show set (tangle (\u22a5 : Iio_index \u03b1)), from N.2.1) $\n  code.is_even_bot _, \u27e8\u27e8{(sum.inr N, default)}, small_singleton _, \u03bb \u03c0 h, begin\n    simp only [subtype.val_eq_coe, option.smul_some, smul_intro, option.some_inj],\n    have := show (struct_perm.lower (bot_lt_coe \u03b1).le (semiallowable_perm.to_struct_perm \u2191\u03c0)) \u2022\n      sum.inr N = sum.inr N, from congr_arg prod.fst (h rfl),\n    simp only [sum.smul_inr] at this,\n    have : \u03c0 \u2022 N = N := this,\n    conv_rhs { rw \u2190 this },\n    congr' 1,\n    ext : 1,\n    simp only [coe_smul_nonempty, subtype.coe_mk, allowable_perm.snd_smul_near_litter],\n    unfold has_smul.smul has_smul.comp.smul,\n    simp only [semiallowable_perm.to_allowable_bot (allowable_perm.coe_hom \u03c0)],\n  end\u27e9\u27e9\u27e9\n\n/-- For any supported tangle `x`, the code `(\u03b1, \u03b2, {x})` is a tangle at level `\u03b1`. -/\ndef supported_singleton (x : tangle \u03b2) (supp : supported \u03b1 (allowable_perm \u03b1) x) :\n  new_tangle \u03b1 :=\n\u27e8intro {x} (code.is_even_singleton _), begin\n  unfreezingI { obtain \u27e8s, hs\u2081, hs\u2082\u27e9 := supp },\n  refine \u27e8\u27e8s, hs\u2081, \u03bb \u03c0 h, _\u27e9\u27e9,\n  conv_rhs { rw \u2190 hs\u2082 \u03c0 h },\n  simp only [smul_set_singleton, smul_nonempty_mk, option.smul_some, smul_intro],\nend\u27e9\n\n/-- For any small set `B` of supported `\u03b2`-tangles, the code `(\u03b1, \u03b2, B)` is a tangle at level `\u03b1` if\nit is even. -/\ndef supported_set (s : set (tangle \u03b2)) (hs : small s) (hc : (mk \u03b2 s).is_even)\n  (symm : \u2200 b \u2208 s, supported \u03b1 (allowable_perm \u03b1) b) :\n  new_tangle \u03b1 :=\n\u27e8intro s hc, begin\n  have symm : \u03a0 b \u2208 s, support \u03b1 (allowable_perm \u03b1) b,\n  { intros b hb, exact (symm b hb).some },\n  refine \u27e8\u27e8\u22c3 b \u2208 s, symm b \u2039_\u203a, hs.bUnion (\u03bb i hi, (symm _ _).small), \u03bb \u03c0 h, _\u27e9\u27e9,\n  suffices : \u03c0 \u2022 s = s,\n  { simp only [option.smul_some, smul_intro, option.some_inj, this] },\n  have : \u2200 x \u2208 s, \u03c0 \u2022 x = x,\n  { intros x hx,\n    refine (symm x hx).supports \u03c0 _,\n    intros a ha,\n    refine h _,\n    simp only [mem_Union, set_like.mem_coe],\n    refine \u27e8x, hx, ha\u27e9 },\n  ext : 2,\n  refine \u27e8\u03bb hx, _, \u03bb hx, _\u27e9,\n  { have := this (\u03c0\u207b\u00b9 \u2022 x) _,\n    { rw smul_inv_smul at this,\n      rw this,\n      rwa \u2190mem_smul_set_iff_inv_smul_mem },\n    { rwa \u2190mem_smul_set_iff_inv_smul_mem } },\n  { rw \u2190 this x hx,\n    exact smul_mem_smul_set hx }\nend\u27e9\n\nvariables {\u03b1}\n\nnamespace new_tangle\n\ninstance : has_coe (new_tangle \u03b1) (semitangle \u03b1) := coe_subtype\n\nlemma coe_injective : injective (coe : new_tangle \u03b1 \u2192 semitangle \u03b1) := subtype.coe_injective\n\nend new_tangle\n\nnamespace allowable_perm\n\n/-- Allowable permutations act on `\u03b1`-tangles. -/\n--Ya\u00ebl: I suspect we can generalize `supports.smul` so that it applies here\ninstance has_smul_new_tangle : has_smul (allowable_perm \u03b1) (new_tangle \u03b1) :=\n\u27e8\u03bb \u03c0 t, \u27e8\u03c0 \u2022 t, t.2.map $ \u03bb s, { carrier := \u03c0 \u2022 s, small := s.2.image, supports := begin\n  intros \u03c3 h,\n  have := s.supports (\u03c0\u207b\u00b9 * \u03c3 * \u03c0) _,\n  { conv_rhs { rw [\u2190 subtype.val_eq_coe, \u2190 this, \u2190 mul_smul, \u2190 mul_assoc, \u2190 mul_assoc,\n      mul_inv_self, one_mul, mul_smul] },\n    refl },\n  { intros a ha,\n    rw [mul_smul, mul_smul, inv_smul_eq_iff],\n    exact h (smul_mem_smul_set ha) },\nend }\u27e9\u27e9\n\n@[simp, norm_cast] lemma coe_smul_new_tangle (f : allowable_perm \u03b1) (t : new_tangle \u03b1) :\n  (\u2191(f \u2022 t) : semitangle \u03b1) = f \u2022 t := rfl\n\ninstance mul_action_new_tangle : mul_action (allowable_perm \u03b1) (new_tangle \u03b1) :=\nnew_tangle.coe_injective.mul_action _ coe_smul_new_tangle\n\nend allowable_perm\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase1/tangle.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.37133290685446835}}
{"text": "import .sform .pform .norm\n\nnamespace polya.field\n\n--namespace nterm\n--\n--variables {\u03b1 : Type} [discrete_field \u03b1]\n--variables {\u03b3 : Type} [const_space \u03b3]\n--variables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n--\n--instance coe_atom  : has_coe num (nterm \u03b3) := \u27e8atom\u27e9\n--instance coe_const : has_coe \u03b3 (nterm \u03b3) := \u27e8const\u27e9\n--\n--instance : has_zero (nterm \u03b3) := \u27e8const 0\u27e9\n--instance : has_one (nterm \u03b3) := \u27e8const 1\u27e9\n--\n--instance : has_add (nterm \u03b3) := \u27e8sform.add\u27e9\n--instance : has_mul (nterm \u03b3) := \u27e8pform.mul\u27e9\n--instance : has_pow (nterm \u03b3) \u2124 := \u27e8\u03bb (x : nterm \u03b3) (n : \u2124), pow_mul (n : znum) x\u27e9\n--\n--instance : has_neg (nterm \u03b3) := \u27e8scale (-1)\u27e9\n--instance : has_sub (nterm \u03b3) := \u27e8\u03bb x y, x + -y\u27e9\n--instance : has_inv (nterm \u03b3) := \u27e8\u03bb x, x ^ (-1 : \u2124)\u27e9\n--instance : has_div (nterm \u03b3) := \u27e8\u03bb x y, x * y\u207b\u00b9\u27e9\n--\n--instance pow_nat : has_pow (nterm \u03b3) \u2115 := \u27e8\u03bb (x : nterm \u03b3) (n : \u2115), x ^ (n : \u2124)\u27e9\n--\n--section\n--\n--variables {x y : nterm \u03b3} {i : num} {n : \u2124} {c : \u03b3}\n--\n--@[simp] theorem eval_zero  : eval \u03c1 (0 : nterm \u03b3) = 0 := by apply morph.morph_zero'\n--@[simp] theorem eval_one   : eval \u03c1 (1 : nterm \u03b3) = 1 := by apply morph.morph_one'\n--@[simp] theorem eval_const : eval \u03c1 (const c) = c     := rfl\n--@[simp] theorem eval_atom  : eval \u03c1 (atom i : nterm \u03b3) = \u03c1.val i := rfl\n--\n--@[simp] theorem eval_add : eval \u03c1 (x + y) = eval \u03c1 x + eval \u03c1 y := sform.eval_add\n--@[simp] theorem eval_mul : eval \u03c1 (x * y) = eval \u03c1 x * eval \u03c1 y := pform.eval_mul\n--@[simp] theorem eval_pow : eval \u03c1 (x ^ n) = eval \u03c1 x ^ n        := by { convert eval_pow_mul, rw znum.to_of_int }\n--\n--@[simp] theorem eval_neg : eval \u03c1 (-x)    = - x.eval \u03c1          := by { refine eq.trans eval_scale _, rw [morph.morph_neg, morph.morph_one', mul_neg_one] }\n--@[simp] theorem eval_sub : eval \u03c1 (x - y) = x.eval \u03c1 - y.eval \u03c1 := by { refine eq.trans eval_add _, rw [eval_neg, sub_eq_add_neg] }\n--@[simp] theorem eval_inv : eval \u03c1 (x\u207b\u00b9)   = (x.eval \u03c1)\u207b\u00b9        := by { rw [\u2190 fpow_inv, \u2190 eval_pow], refl }\n--@[simp] theorem eval_div : eval \u03c1 (x / y) = x.eval \u03c1 / y.eval \u03c1 := by { rw [division_def, \u2190 eval_inv, \u2190 eval_mul], refl }\n--\n--@[simp] theorem eval_pow_nat {n : \u2115} : eval \u03c1 (x ^ n) = eval \u03c1 x ^ n := eval_pow\n--\n--end\n--\n--end nterm\n\n@[derive decidable_eq, derive has_reflect]\ninductive term : Type\n| atom : num \u2192 term\n| add  : term \u2192 term \u2192 term\n| sub  : term \u2192 term \u2192 term\n| mul  : term \u2192 term \u2192 term\n| div  : term \u2192 term \u2192 term\n| neg  : term \u2192 term\n| inv  : term \u2192 term\n| numeral : \u2115 \u2192 term\n| pow_nat : term \u2192 \u2115 \u2192 term\n| pow_int : term \u2192 \u2124 \u2192 term\n\nnamespace term\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ndef eval (\u03c1 : dict \u03b1) : term \u2192 \u03b1\n| (atom i)  := \u03c1.val i\n| (add x y) := eval x + eval y\n| (sub x y) := eval x - eval y\n| (mul x y) := eval x * eval y\n| (div x y) := (eval x) / (eval y)\n| (neg x)   := - eval x\n| (inv x)   := (eval x)\u207b\u00b9\n| (numeral n)   := (n : \u03b1)\n| (pow_nat x n) := eval x ^ n\n| (pow_int x n) := eval x ^ n\n\ndef to_nterm : term \u2192 nterm \u03b3\n| (atom i)  := \u2191i\n| (add x y) := to_nterm x + to_nterm y\n| (sub x y) := to_nterm x - to_nterm y\n| (mul x y) := to_nterm x * to_nterm y\n| (div x y) := to_nterm x / to_nterm y\n| (neg x)   := - to_nterm x\n| (inv x)   := (to_nterm x)\u207b\u00b9\n| (numeral n)   := \u2191(n : \u03b3)\n| (pow_nat x n) := to_nterm x ^ n\n| (pow_int x n) := to_nterm x ^ n\n\ntheorem correctness {x : term} :\n  nterm.eval \u03c1 (@to_nterm \u03b3 _ x) = eval \u03c1 x :=\nbegin\n  induction x with\n    i           --atom\n    x y ihx ihy --add\n    x y ihx ihy --sub\n    x y ihx ihy --mul\n    x y ihx ihy --div\n    x ihx       --neg\n    x ihx       --inv\n    n           --numeral\n    x n ihx     --pow_nat\n    x n ihx,    --pow_int\n  repeat { unfold to_nterm, unfold eval },\n  repeat { simp [nterm.eval] },\n  repeat { simp [nterm.eval, ihx] },\n  repeat { simp [nterm.eval, ihx, ihy] },\n  --{ rw [fpow_inv, division_def] },\n  --{ rw fpow_inv }\nend\n\nend term\n\ndef norm (\u03b3 : Type) [const_space \u03b3] (x : term) : nterm \u03b3 :=\nnterm.norm $ @term.to_nterm \u03b3 _ x\n\ndef norm_hyps (\u03b3 : Type) [const_space \u03b3] (x : term) : list (nterm \u03b3) :=\nnterm.norm_hyps $ @term.to_nterm \u03b3 _ x\n\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ntheorem correctness {x : term} {\u03c1 : dict \u03b1} :\n  (\u2200 t \u2208 norm_hyps \u03b3 x, nterm.eval \u03c1 t \u2260 0) \u2192\n  term.eval \u03c1 x = nterm.eval \u03c1 (norm \u03b3 x) :=\nbegin\n  intro H,\n  unfold norm,\n  apply eq.symm, apply eq.trans,\n  { apply nterm.correctness, unfold nterm.nonzero,\n    intros t ht, apply H, exact ht },\n  { apply term.correctness }\nend\n\nopen nterm\n\ndef aux1 (t1 t2 : nterm \u03b3) : nterm \u03b3 \u00d7 nterm \u03b3 \u00d7 \u03b3 :=\nif t2.coeff = 0 then\n  (t1.term, 0, t1.coeff)\nelse\n  (t2.term, t1.scale (t2.coeff\u207b\u00b9), -t2.coeff)\n\ndef aux2 (t1 t2 : nterm \u03b3) : nterm \u03b3 \u00d7 nterm \u03b3 \u00d7 \u03b3 :=\nif t2.term < t1.term then\n  aux1 (t2.scale (-1)) (t1.scale (-1))\nelse\n  aux1 t1 t2\n\ntheorem eval_aux1 {t1 t2 t3 t4 : nterm \u03b3} {c : \u03b3} :\n  (t3, t4, c) = aux1 t1 t2 \u2192\n  eval \u03c1 t1 - eval \u03c1 t2 = (eval \u03c1 t3 - eval \u03c1 t4) * c :=\nbegin\n  unfold aux1,\n  by_cases h1 : t2.coeff = 0,\n  { rw if_pos h1, intro h2,\n    rw [prod.mk.inj_iff] at h2, cases h2 with h2 h3,\n    rw [prod.mk.inj_iff] at h3, cases h3 with h3 h4,\n    rw [eval_term_coeff t1, eval_term_coeff t2, h1, h2, h3, h4],\n    simp [morph.morph_neg] },\n  { rw if_neg h1, intro h2,\n    rw [prod.mk.inj_iff] at h2, cases h2 with h2 h3,\n    rw [prod.mk.inj_iff] at h3, cases h3 with h3 h4,\n    rw [h2, h3, h4],\n    rw [morph.morph_neg, mul_neg_eq_neg_mul_symm, neg_mul_eq_neg_mul, neg_sub, sub_mul],\n    rw [\u2190 eval_term_coeff], congr' 1,\n    rw [eval_scale, mul_assoc, \u2190 morph.morph_mul, inv_mul_cancel],\n    rw [morph.morph_one, mul_one], --simp\n    exact h1 }\nend\n\ntheorem eval_aux2 {t1 t2 t3 t4 : nterm \u03b3} {c : \u03b3} :\n  (t3, t4, c) = aux2 t1 t2 \u2192\n  eval \u03c1 t1 - eval \u03c1 t2 = (eval \u03c1 t3 - eval \u03c1 t4) * c :=\nbegin\n  unfold aux2,\n  by_cases h1 : t2.term < t1.term,\n  { rw if_pos h1, intro h2,\n    have : eval \u03c1 (t2.scale (-1)) - eval \u03c1 (t1.scale (-1)) = (eval \u03c1 t3 - eval \u03c1 t4) * \u2191c,\n    { exact eval_aux1 h2 },\n    rw \u2190 this, simp [morph.morph_neg] },\n  { rw if_neg h1, intro h2, exact eval_aux1 h2 }\nend\n\ndef norm2 (\u03b3 : Type) [const_space \u03b3] (t1 t2 : term) : nterm \u03b3 \u00d7 nterm \u03b3 \u00d7 \u03b3 :=\n  aux2 (norm \u03b3 t1) (norm \u03b3 t2)\n\ntheorem eval_norm2 {t1 t2 : term} {nt1 nt2 : nterm \u03b3} {c : \u03b3} :\n  nonzero \u03c1 (norm_hyps \u03b3 t1) \u2192\n  nonzero \u03c1 (norm_hyps \u03b3 t2) \u2192\n  (nt1, nt2, c) = norm2 \u03b3 t1 t2 \u2192\n  term.eval \u03c1 t1 - term.eval \u03c1 t2 =\n    (nterm.eval \u03c1 nt1 - nterm.eval \u03c1 nt2) * c :=\nbegin\n  unfold norm2,\n  intros h1 h2 h3,\n  apply eq.trans,\n  { show _ = eval \u03c1 (norm \u03b3 t1) - eval \u03c1 (norm \u03b3 t2), rw [correctness h1, correctness h2] },\n  { exact eval_aux2 h3 }\nend\n\nend polya.field", "meta": {"author": "lean-forward", "repo": "field", "sha": "7e2127ad485aec25e58a1b9c82a6bb74a599467a", "save_path": "github-repos/lean/lean-forward-field", "path": "github-repos/lean/lean-forward-field/field-7e2127ad485aec25e58a1b9c82a6bb74a599467a/src/data/polya/field/main.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.37133289975367717}}
{"text": "import data.hash_map library_dev.data.list.set\n\nnamespace state\n\n@[inline] def modify {\u03c3 : Type} : (\u03c3 \u2192 \u03c3) \u2192 state \u03c3 unit :=\n\u03bb f s, ((), f s)\n\ndef inc : state \u2115 unit := modify (\u03bb n, n + 1)\ndef dec : state \u2115 unit := modify (\u03bb n, n - 1)\nend state\n\nnamespace list\n\ndef dnth {\u03b1 : Type*} [decidable_eq \u03b1] [inhabited \u03b1] (xs : list \u03b1) (n : \u2115) : \u03b1 :=\nmatch xs^.nth n with\n| (some x) := x\n| none     := default \u03b1\nend\n\ndef at_nth {\u03b1 : Type*} (xs : list \u03b1) (idx : \u2115) (x : \u03b1) : Prop := nth xs idx = some x\n\ndef set_nth {\u03b1 : Type*} : list \u03b1 \u2192 \u2115 \u2192 \u03b1 \u2192 option (list \u03b1)\n| (x::xs) 0     a := some (a :: xs)\n| (x::xs) (i+1) a := do ys \u2190 set_nth xs i a, return (x :: ys)\n| []      _     _ := none\n\nlemma at_nth_of_dnth_lt {\u03b1 : Type*} [decidable_eq \u03b1] [inhabited \u03b1] {xs : list \u03b1} {idx : \u2115} :\n  idx < length xs \u2192 at_nth xs idx (dnth xs idx) := sorry\n\nlemma at_nth_of_len {\u03b1 : Type*} {xs ys : list \u03b1} {x : \u03b1} {k : \u2115} : k = length xs \u2192 at_nth (xs ++ x :: ys) k x := sorry\n\nend list\nnamespace hash_map\n\ndef dfind {\u03b1 : Type*} [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type*} [\u2200 a, inhabited (\u03b2 a)] (m : hash_map \u03b1 \u03b2) (a : \u03b1) : \u03b2 a :=\nmatch m^.find a with\n| (some b) := b\n| none     := default (\u03b2 a)\nend\n\nend hash_map\n\nsection seq\nvariables {\u03b1 : Type*} (rel : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\ninductive star : \u03b1 \u2192 \u03b1 \u2192 Prop\n| rfl    : \u2200 (x : \u03b1), star x x\n| rtrans : \u2200 (x y z : \u03b1), rel x y \u2192 star y z \u2192 star x y\n\nend seq\n\nnamespace star\nvariables {\u03b1 : Type*} (rel : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\nlemma trans (x y z : \u03b1) : star rel x y \u2192 star rel y z \u2192 star rel x z := sorry\n\nend star\n\nnamespace compiler\nopen tactic list\n\nstructure var : Type := (id : \u2115)\n\nnamespace var\ninstance : decidable_eq var := by mk_dec_eq_instance\nend var\n\n@[reducible] def vstate : Type := hash_map var (\u03bb v : var, \u2115)\ndef empty_vstate : vstate := mk_hash_map (\u03bb v : var, v^.id)\n\ninductive aexp : Type\n| aconst : \u2115 \u2192 aexp\n| avar   : var \u2192 aexp\n| aadd   : aexp \u2192 aexp \u2192 aexp\n| asub   : aexp \u2192 aexp \u2192 aexp\n| amul   : aexp \u2192 aexp \u2192 aexp\n\ninductive bexp : Type\n| btrue  : bexp\n| bfalse : bexp\n| bnot   : bexp \u2192 bexp\n| band   : bexp \u2192 bexp \u2192 bexp\n| beq    : aexp \u2192 aexp \u2192 bexp\n| ble    : aexp \u2192 aexp \u2192 bexp\n\ndef aeval (st : vstate) : aexp \u2192 \u2115\n| (aexp.aconst n) := n\n| (aexp.avar v) := st^.dfind v\n| (aexp.aadd e\u2081 e\u2082) := aeval e\u2081 + aeval e\u2082\n| (aexp.asub e\u2081 e\u2082) := aeval e\u2081 - aeval e\u2082\n| (aexp.amul e\u2081 e\u2082) := aeval e\u2081 * aeval e\u2082\n\ndef beval (st : vstate) : bexp \u2192 bool\n| (bexp.btrue)      := tt\n| (bexp.bfalse)     := ff\n| (bexp.bnot b)     := bnot (beval b)\n| (bexp.band b\u2081 b\u2082) := beval b\u2081 && beval b\u2082\n| (bexp.beq e\u2081 e\u2082)  := aeval st e\u2081 = aeval st e\u2082\n| (bexp.ble e\u2081 e\u2082)  := aeval st e\u2081 \u2264 aeval st e\u2082\n\ninductive com : Type\n| cskip  : com\n| cass   : var \u2192 aexp \u2192 com\n| cseq   : com \u2192 com \u2192 com\n| cif    : bexp \u2192 com \u2192 com \u2192 com\n| cwhile : bexp \u2192 com \u2192 com\n\nopen com\n\ninductive ceval : com \u2192 vstate \u2192 vstate \u2192 Prop\n| eskip : \u2200 st, ceval cskip st st\n| eass  : \u2200 st a n x, aeval st a = n \u2192 ceval (cass x a) st (st^.insert x n)\n| eseq : \u2200 c\u2081 c\u2082 st\u2081 st\u2082 st\u2083, ceval c\u2081 st\u2081 st\u2082 \u2192 ceval c\u2082 st\u2082 st\u2083 \u2192 ceval (cseq c\u2081 c\u2082) st\u2081 st\u2083\n| eift : \u2200 st\u2081 st\u2082 b c\u2081 c\u2082, beval st\u2081 b = tt \u2192 ceval c\u2081 st\u2081 st\u2082 \u2192 ceval (cif b c\u2081 c\u2082) st\u2081 st\u2082\n| eiff : \u2200 st\u2081 st\u2082 b c\u2081 c\u2082, beval st\u2081 b = ff \u2192 ceval c\u2082 st\u2081 st\u2082 \u2192 ceval (cif b c\u2081 c\u2082) st\u2081 st\u2082\n| ewhilet : \u2200 st\u2081 st\u2082 st\u2083 b c, beval st\u2081 b = tt \u2192 ceval c st\u2081 st\u2082 \u2192 ceval (cwhile b c) st\u2082 st\u2083 \u2192 ceval (cwhile b c) st\u2081 st\u2083\n| ewhilef : \u2200 st b c, beval st b = ff \u2192 ceval (cwhile b c) st st\n\nopen ceval\n\ninductive instruction : Type\n| iconst : \u2115 \u2192 instruction\n| iget   : \u2115 \u2192 instruction\n| iset   : \u2115 \u2192 instruction\n| iadd   : instruction\n| isub   : instruction\n| imul   : instruction\n| ibf    : \u2115 \u2192 instruction\n| ibb    : \u2115 \u2192 instruction\n| ibeq   : \u2115 \u2192 instruction\n| ibne   : \u2115 \u2192 instruction\n| ible   : \u2115 \u2192 instruction\n| ibgt   : \u2115 \u2192 instruction\n| ihalt  : instruction\n\nopen instruction\n\n@[reducible] def code := list instruction.\n\n@[reducible] def stack : Type := list \u2115\n@[reducible] def config : Type := \u2115 \u00d7 stack\n\ninductive veval (c : code) : config -> config -> Prop\n| vconst : \u2200 pc stk n, at_nth c pc (iconst n) \u2192 veval (pc, stk) (pc + 1, n :: stk)\n| vget   : \u2200 pc stk n v, at_nth c pc (iget n) \u2192 at_nth stk n v \u2192 veval (pc, stk) (pc + 1, v :: stk)\n| vset   : \u2200 pc stk n v stk', at_nth c pc (iset n) \u2192 set_nth stk n v = some stk' \u2192 veval (pc, v :: stk) (pc + 1, stk')\n| vadd   : \u2200 pc stk n n\u2081 n\u2082, at_nth c pc iadd \u2192 n = n\u2081 + n\u2082 \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc + 1, n :: stk)\n| vsub   : \u2200 pc stk n\u2081 n\u2082, at_nth c pc iadd \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc + 1, (n\u2081 - n\u2082) :: stk)\n| vmul   : \u2200 pc stk n\u2081 n\u2082, at_nth c pc iadd \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc + 1, (n\u2081 * n\u2082) :: stk)\n| vbf    : \u2200 pc stk ofs pc', at_nth c pc (ibf ofs) \u2192 pc' = (pc + ofs) + 1 \u2192 veval (pc, stk) (pc', stk)\n| vbb    : \u2200 pc stk ofs pc', at_nth c pc (ibf ofs) \u2192 pc' + ofs = pc + 1 \u2192 veval (pc, stk) (pc', stk)\n| vbeq   : \u2200 pc stk ofs n\u2081 n\u2082 pc', at_nth c pc (ibeq ofs) \u2192 pc' = (if n\u2081 = n\u2082 then (pc + ofs) + 1 else pc + 1) \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc', stk)\n| vbne   : \u2200 pc stk ofs n\u2081 n\u2082 pc', at_nth c pc (ibne ofs) \u2192 pc' = (if n\u2081 = n\u2082 then pc + 1 else (pc + ofs) + 1) \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc', stk)\n| vble   : \u2200 pc stk ofs n\u2081 n\u2082 pc', at_nth c pc (ible ofs) \u2192 pc' = (if n\u2081 \u2264 n\u2082 then (pc + ofs) + 1 else pc + 1) \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc', stk)\n| vbgt   : \u2200 pc stk ofs n\u2081 n\u2082 pc', at_nth c pc (ibgt ofs) \u2192 pc' = (if n\u2081 \u2264 n\u2082 then pc + 1 else (pc + ofs) + 1) \u2192 veval (pc, n\u2082 :: n\u2081 :: stk) (pc', stk)\n\ndef vhalts (c : code) (stk_init stk_fin : stack) : Prop :=\n\u2203 pc, at_nth c pc ihalt \u2227 star (veval c) (0, stk_init) (pc, stk_fin)\n\ndef collect_assigned_vars : com \u2192 list var\n| (cskip)       := []\n| (cass v _)    := [v]\n| (cseq c\u2081 c\u2082)  := collect_assigned_vars c\u2081 \u222a collect_assigned_vars c\u2082\n| (cif b c\u2081 c\u2082) := collect_assigned_vars c\u2081 \u222a collect_assigned_vars c\u2082\n| (cwhile b c)  := collect_assigned_vars c\n\n@[reducible] def stack_offsets : Type := hash_map var (\u03bb v : var, \u2115)\n\ndef compute_stack_offsets_core : list var \u2192 stack_offsets \u2192 stack_offsets\n| []        s := s\n| (v :: vs) s := compute_stack_offsets_core vs (s^.insert v (length vs))\n\ndef compute_stack_offsets (c : com) : stack_offsets :=\ncompute_stack_offsets_core (collect_assigned_vars c) (mk_hash_map (\u03bb (v : var), v^.id))\n\n-- TODO(dhs): not sure if this is the best way to do it\ndef agree (offsets : stack_offsets) (vofs : \u2115) (st : vstate) (stk : stack) : Prop :=\n  \u2200 (v : var), st^.dfind v = dnth stk (offsets^.dfind v + vofs)\n\nlemma agree_push {offsets : stack_offsets} {vofs : \u2115} {st : vstate} {stk : stack} {n : \u2115} :\n  agree offsets vofs st stk \u2192 agree offsets (vofs + 1) st (n :: stk) := sorry\n\nlemma agree_insert {offsets : stack_offsets} {vofs : \u2115} {st : vstate} {stk : stack} :\n  agree offsets vofs st stk \u2192\n  \u2200 v n,\n  agree offsets vofs (hash_map.insert st v n) (update_nth stk (hash_map.dfind offsets v) n) := sorry\n\ninductive codeseq_at : code \u2192 \u2115 \u2192 code \u2192 Prop\n| intro : \u2200 code\u2081 code\u2082 code\u2083 pc, pc = length code\u2081 \u2192 codeseq_at (code\u2081 ++ code\u2082 ++ code\u2083) pc code\u2082\n\ndef compile_aexp_core (offsets : stack_offsets) : aexp \u2192 \u2115 \u2192 code\n| (aexp.aconst n)   vofs := [iconst n]\n| (aexp.avar v)     vofs := [iget $ offsets^.dfind v + vofs]\n| (aexp.aadd e\u2081 e\u2082) vofs := compile_aexp_core e\u2082 vofs ++ compile_aexp_core e\u2081 (vofs + 1) ++ [iadd]\n| (aexp.asub e\u2081 e\u2082) vofs := compile_aexp_core e\u2082 vofs ++ compile_aexp_core e\u2081 (vofs + 1) ++ [isub]\n| (aexp.amul e\u2081 e\u2082) vofs := compile_aexp_core e\u2082 vofs ++ compile_aexp_core e\u2081 (vofs + 1) ++ [imul]\n\ndef compile_aexp (offsets : stack_offsets) (e : aexp) := compile_aexp_core offsets e 0\n\n-- TODO(dhs): weaken the forall?\ndef astack_contains_vars (offsets : stack_offsets) : stack \u2192 \u2115 \u2192 aexp \u2192 Prop\n| stk vofs (aexp.aconst n)   := true\n| stk vofs (aexp.avar v)     := offsets^.dfind v + vofs < length stk\n| stk vofs (aexp.aadd e\u2081 e\u2082) := astack_contains_vars stk vofs e\u2082 \u2227 \u2200 x, astack_contains_vars (x :: stk) (vofs + 1) e\u2081\n| stk vofs (aexp.asub e\u2081 e\u2082) := astack_contains_vars stk vofs e\u2082 \u2227 \u2200 x, astack_contains_vars (x :: stk) (vofs + 1) e\u2081\n| stk vofs (aexp.amul e\u2081 e\u2082) := astack_contains_vars stk vofs e\u2082 \u2227 \u2200 x, astack_contains_vars (x :: stk) (vofs + 1) e\u2081\n\nlemma compile_aexp_core_correct :\n  \u2200 code st e pc stk offsets vofs,\n    codeseq_at code pc (compile_aexp_core offsets e vofs)\n    \u2192 agree offsets vofs st stk\n    \u2192 astack_contains_vars offsets stk vofs e\n    \u2192 star (veval code) (pc, stk) (pc + length (compile_aexp_core offsets e vofs), aeval st e :: stk)\n\n| .(_) st (aexp.aconst n) .(pc) stk offsets vofs (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_astack :=\nbegin\nsimp [compile_aexp_core, length, aeval],\napply star.rtrans,\napply veval.vconst,\napply at_nth_of_len H_pc,\napply star.rfl\nend\n\n| .(_) st (aexp.avar v) .(pc) stk offsets vofs (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_astack :=\nbegin\nsimp [compile_aexp_core, length, aeval],\napply star.rtrans,\napply veval.vget,\napply at_nth_of_len H_pc,\nsimp [agree] at H_agree,\nsimp [astack_contains_vars] at H_astack,\nrw H_agree,\napply at_nth_of_dnth_lt H_astack,\napply star.rfl\nend\n\n| .(_) st (aexp.aadd e\u2081 e\u2082) .(pc) stk offsets vofs (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_astack :=\nbegin\nsimp [compile_aexp_core, length, aeval],\napply star.trans,\n-- Compile e\u2082\napply compile_aexp_core_correct _ st e\u2082 _ _ offsets vofs _ H_agree (and.left H_astack),\nrw \u2190 append_assoc,\napply codeseq_at.intro _ _ _ _ H_pc,\n-- Compile e\u2081\napply star.trans,\napply compile_aexp_core_correct _ st e\u2081 _ _ offsets (vofs+1) _ (agree_push H_agree) (and.right H_astack _),\nhave H_assoc :\n(code\u2081 ++ (compile_aexp_core offsets e\u2082 vofs ++ (compile_aexp_core offsets e\u2081 (vofs + 1) ++ iadd :: code\u2083)))\n=\n(code\u2081 ++ compile_aexp_core offsets e\u2082 vofs) ++ (compile_aexp_core offsets e\u2081 (vofs + 1)) ++ iadd :: code\u2083 := sorry,\nrw H_assoc, clear H_assoc,\napply codeseq_at.intro _ _ _ _,\nsimp [H_pc, length_append],\n\n-- Add instruction\napply star.rtrans,\nhave H_assoc :\ncode\u2081 ++ (compile_aexp_core offsets e\u2082 vofs ++ (compile_aexp_core offsets e\u2081 (vofs + 1) ++ iadd :: code\u2083))\n=\n(code\u2081 ++ compile_aexp_core offsets e\u2082 vofs ++ compile_aexp_core offsets e\u2081 (vofs + 1)) ++ [iadd] ++ code\u2083 := sorry,\nrw H_assoc, clear H_assoc,\n\nhave H_one_at_end :\npc + (1 + (length (compile_aexp_core offsets e\u2082 vofs) + length (compile_aexp_core offsets e\u2081 (vofs + 1))))\n=\n(pc + length (compile_aexp_core offsets e\u2082 vofs) + length (compile_aexp_core offsets e\u2081 (vofs + 1))) + 1 := sorry,\nrw H_one_at_end, clear H_one_at_end,\napply veval.vadd,\n\nhave H_cons :\ncode\u2081 ++ compile_aexp_core offsets e\u2082 vofs ++ compile_aexp_core offsets e\u2081 (vofs + 1) ++ [iadd] ++ code\u2083\n=\n(code\u2081 ++ compile_aexp_core offsets e\u2082 vofs ++ compile_aexp_core offsets e\u2081 (vofs + 1)) ++ (iadd :: code\u2083) := sorry,\nrw H_cons, clear H_cons,\napply at_nth_of_len,\nsimp [H_pc, length_append],\nsimp,\napply star.rfl,\n\nend\n\ndef compile_bexp (offsets : stack_offsets) : bexp \u2192 bool \u2192 \u2115 \u2192 code\n| (bexp.btrue)      cond ofs := if cond then [ibf ofs] else []\n| (bexp.bfalse)     cond ofs := if cond then [] else [ibf ofs]\n| (bexp.bnot b)     cond ofs := compile_bexp b (bnot cond) ofs\n| (bexp.band b\u2081 b\u2082) cond ofs := let code\u2082 := compile_bexp b\u2082 cond ofs,\n                                    code\u2081 := compile_bexp b\u2081 ff (if cond then length code\u2082 else ofs + length code\u2082)\n                                in  code\u2081 ++ code\u2082\n\n| (bexp.beq e\u2081 e\u2082)  cond ofs := compile_aexp_core offsets e\u2082 0 ++ compile_aexp_core offsets e\u2081 1 ++ (if cond then [ibeq ofs] else [ibne ofs])\n| (bexp.ble e\u2081 e\u2082)  cond ofs := compile_aexp_core offsets e\u2082 0 ++ compile_aexp_core offsets e\u2081 1 ++ (if cond then [ible ofs] else [ibgt ofs])\n\n-- TODO(dhs): weaken the forall?\n-- TODO(dhs): is this even right? We'll see soon.\ndef bstack_contains_vars (offsets : stack_offsets) : stack \u2192 bexp \u2192 Prop\n| stk (bexp.btrue)      := true\n| stk (bexp.bfalse)     := true\n| stk (bexp.bnot b)     := bstack_contains_vars stk b\n| stk (bexp.band b\u2081 b\u2082) := bstack_contains_vars stk b\u2081 \u2227 bstack_contains_vars stk b\u2082\n| stk (bexp.beq e\u2081 e\u2082)  := astack_contains_vars offsets stk 0 e\u2082 \u2227 \u2200 x, astack_contains_vars offsets (x::stk) 1 e\u2081\n| stk (bexp.ble e\u2081 e\u2082)  := astack_contains_vars offsets stk 0 e\u2082 \u2227 \u2200 x, astack_contains_vars offsets (x::stk) 1 e\u2081\n\n--set_option pp.all true\n--set_option trace.type_context.is_def_eq true\n--set_option trace.type_context.is_def_eq_detail true\n\nlemma compile_bexp_correct :\n  \u2200 code st b cond ofs pc stk offsets,\n    codeseq_at code pc (compile_bexp offsets b cond ofs)\n    \u2192 agree offsets 0 st stk\n    \u2192 bstack_contains_vars offsets stk b\n    \u2192 star (veval code) (pc, stk)\n                        (pc + (length (compile_bexp offsets b cond ofs) + ite (beval st b = cond) ofs 0), stk)\n/-\n| .(_) st (bexp.btrue) cond ofs .(pc) stk offsets (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_bstack :=\nbegin\nsimp [compile_bexp, compile_aexp_core, length, aeval, beval],\ncases cond,\n{ simp, apply star.rfl },\nsimp,\napply star.rtrans,\napply veval.vbf _ _ ofs,\napply at_nth_of_len H_pc,\nsimp,\napply star.rfl\nend\n\n| .(_) st (bexp.bfalse) cond ofs .(pc) stk offsets (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_bstack :=\nbegin\nsimp [compile_bexp, compile_aexp_core, length, aeval, beval],\ncases cond,\n{ simp, apply star.rtrans, apply veval.vbf _ _ ofs, apply at_nth_of_len H_pc, simp, apply star.rfl },\n{ simp, apply star.rfl },\nend\n\n\n| .(_) st (bexp.bnot b) cond ofs .(pc) stk offsets (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_bstack :=\nbegin\nsimp [compile_bexp, compile_aexp_core, length, aeval, beval],\n-- TODO(dhs): come on, Lean\nhave H_bnot : \u2200 beq, (@ite (bnot (beval st b) = cond) beq _ ofs 0) = (ite (beval st b = bnot cond) ofs 0) := sorry,\nrw H_bnot, clear H_bnot,\n\napply compile_bexp_correct (code\u2081 ++ (compile_bexp offsets b (bnot cond) ofs ++ code\u2083)) st b (bnot cond) ofs pc stk offsets _ H_agree H_bstack,\nrw \u2190 append_assoc,\napply codeseq_at.intro _ _ _ _ H_pc,\n\nend\n-/\n\n| .(_) st (bexp.band b\u2081 b\u2082) cond ofs .(pc) stk offsets (codeseq_at.intro code\u2081 ._ code\u2083 pc H_pc) H_agree H_bstack :=\nbegin\nsimp [compile_bexp, compile_aexp_core, length, aeval, beval],\n-- b\u2081\napply star.trans,\n\nhave H_assoc :\n(code\u2081 ++\n          (compile_bexp offsets b\u2081 ff\n               (ite \u2191cond (length (compile_bexp offsets b\u2082 cond ofs))\n                  (ofs + length (compile_bexp offsets b\u2082 cond ofs))) ++\n             (compile_bexp offsets b\u2082 cond ofs ++ code\u2083)))\n=\n(code\u2081 ++\n          (compile_bexp offsets b\u2081 ff\n               (ite \u2191cond (length (compile_bexp offsets b\u2082 cond ofs))\n                  (ofs + length (compile_bexp offsets b\u2082 cond ofs)))) ++\n             (compile_bexp offsets b\u2082 cond ofs ++ code\u2083)) := sorry,\nrw H_assoc, clear H_assoc,\n\napply compile_bexp_correct _ st b\u2081 ff _ pc stk offsets _ H_agree (and.left H_bstack),\ntactic.rotate 1,\napply codeseq_at.intro _ _ _ _ H_pc,\n\n--\ncases cond,\nsimp,\nhave H_em : (beval st b\u2081 = ff \u2228 beval st b\u2081 = tt) := sorry,\ncases H_em with H_ff H_tt,\nsimp [H_ff],\napply star.rfl,\nsimp [H_tt],\nhave H_assoc :\n(code\u2081 ++\n          (compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs)) ++\n             (compile_bexp offsets b\u2082 ff ofs ++ code\u2083)))\n=\n(code\u2081 ++ compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs)) ++ (compile_bexp offsets b\u2082 ff ofs ++ code\u2083)) := sorry,\nrw H_assoc, clear H_assoc,\n\n-- TODO(dhs): why won't it unify?\nhave H_come_on_lean : \u2200 bdec, @ite (beval st b\u2082 = ff) bdec _ ofs 0 = ite (beval st b\u2082 = ff) ofs 0 := sorry,\n\nhave H_rec\u2082 := compile_bexp_correct (code\u2081 ++ compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs)) ++\n          (compile_bexp offsets b\u2082 ff ofs ++ code\u2083)) st b\u2082 ff ofs (pc + length (compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs)))) stk offsets,\nsimp [H_come_on_lean] at H_rec\u2082,\nsimp [H_come_on_lean],\napply H_rec\u2082,\nclear H_rec\u2082,\nhave H_assoc :\n    (code\u2081 ++\n       (compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs)) ++\n          (compile_bexp offsets b\u2082 ff ofs ++ code\u2083)))\n=\n    (code\u2081 ++ compile_bexp offsets b\u2081 ff (ofs + length (compile_bexp offsets b\u2082 ff ofs))) ++\n          compile_bexp offsets b\u2082 ff ofs ++ code\u2083 := sorry,\nrw H_assoc, clear H_assoc,\napply codeseq_at.intro,\nsimp [H_pc],\nexact H_agree,\nexact and.right H_bstack,\n\n-- cond = tt\nexact sorry\nend\n\n\n-- Example program\n---------------------------\n-- (cass `x 1)\n-- (cass `y (+ x x))\n-- (cass `z (+ x (+ y x))\n\n-- Want\n--------------------------\n-- Initial stack: [x:=0, y:=0, z:=0]\n-- cass `x 1 ==>  push 1, iset 1 ==> [x:=1, y:=0, z:=0]\n-- cass `y (+ x x) ==> iget 0, iget 1, iadd, iset 2 ==> [x:=2, y:=4, z:=0]\n-- cass `z (+ x (+ y x)) ==> iget 0, iget 2, iget 2, iadd, iadd, iset 3\n\ndefinition compile_com (offsets : stack_offsets) : com \u2192 code\n| cskip         := []\n| (cass v e)    := compile_aexp offsets e ++ [iset $ offsets^.dfind v]\n| (cseq c\u2081 c\u2082)  := compile_com c\u2081 ++ compile_com c\u2082\n\n| (cif b c\u2081 c\u2082) := let code\u2081 := compile_com c\u2081,\n                       code\u2082 := compile_com c\u2082\n                   in  compile_bexp offsets b false (length code\u2081 + 1) ++ code\u2081 ++ [ibf (length code\u2082)] ++ code\u2082\n| (cwhile b c)  := let code_body := compile_com c,\n                       code_test := compile_bexp offsets b ff (length code_body + 1)\n                   in  code_test ++ code_body ++ [ibb (length code_test + length code_body + 1)]\n\n\n-- TODO(dhs): is this _strong_ enough, with `offsets` an argument?\n-- TODO(dhs): is this _weak_ enough, to prove?\ntheorem compile_correct_terminating_alt :\n  \u2200 code st c st',\n    ceval c st st' \u2192\n      \u2200 offsets stk pc, codeseq_at code pc (compile_com offsets c) \u2192\n                agree offsets 0 st stk \u2192\n                \u2203 stk', star (veval code) (pc, stk) (pc + length (compile_com offsets c), stk')\n                        \u2227 agree offsets 0 st' stk'\n| code ._ ._ ._ (eskip st) :=\nbegin\nsimp [compile_com, length],\nintros offsets stk pc H_codeseq H_agree,\napply exists.intro stk,\nsplit,\nexact H_agree,\napply star.rfl\nend\n\n| code ._ ._ ._ (eass st a n x H_aeval) :=\nbegin\nsimp [compile_com, length],\nintros offsets stk pc H_codeseq H_agree,\napply exists.intro (update_nth stk (offsets^.dfind x) n),\nsplit,\napply agree_insert H_agree,\napply compile_aexp_core_correct,\nend\n\n| code ._ ._ ._ (eseq c\u2081 c\u2082 st\u2081 st\u2082 st\u2083 H_c\u2081 H_c\u2082) :=\nbegin\n\nend\n\n| code ._ ._ ._ (eift st\u2081 st\u2082 b c\u2081 c\u2082 H_beval_t H_ceval\u2081) :=\nbegin\n\nend\n\n| code ._ ._ ._ (eiff st\u2081 st\u2082 b c\u2081 c\u2082 H_beval_f H_ceval\u2082) :=\nbegin\n\nend\n\n| code ._ ._ ._ (ewhilet st\u2081 st\u2082 st\u2083 b c H_beval_t H_ceval_step H_ceval_loop) :=\nbegin\n\nend\n\n| code ._ ._ ._ (ewhilef st b c H_beval_f) :=\nbegin\n\nend\n\nend compiler\n", "meta": {"author": "dselsam", "repo": "unrealistic_compiler", "sha": "70514de492a6a1ed705ad247333ae5b3f8455a83", "save_path": "github-repos/lean/dselsam-unrealistic_compiler", "path": "github-repos/lean/dselsam-unrealistic_compiler/unrealistic_compiler-70514de492a6a1ed705ad247333ae5b3f8455a83/compiler.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3713328997536771}}
{"text": "/-\nCopyright (c) 2018 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.opposites\nimport Mathlib.PostPort\n\nuniverses u\u2081 v\u2081 v\u2082 u\u2082 \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\n/--\nAn equality `X = Y` gives us a morphism `X \u27f6 Y`.\n\nIt is typically better to use this, rather than rewriting by the equality then using `\ud835\udfd9 _`\nwhich usually leads to dependent type theory hell.\n-/\ndef eq_to_hom {C : Type u\u2081} [category C] {X : C} {Y : C} (p : X = Y) : X \u27f6 Y :=\n  eq.mpr sorry \ud835\udfd9\n\n@[simp] theorem eq_to_hom_refl {C : Type u\u2081} [category C] (X : C) (p : X = X) : eq_to_hom p = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem eq_to_hom_trans {C : Type u\u2081} [category C] {X : C} {Y : C} {Z : C} (p : X = Y) (q : Y = Z) : eq_to_hom p \u226b eq_to_hom q = eq_to_hom (Eq.trans p q) := sorry\n\n/--\nAn equality `X = Y` gives us a morphism `X \u27f6 Y`.\n\nIt is typically better to use this, rather than rewriting by the equality then using `iso.refl _`\nwhich usually leads to dependent type theory hell.\n-/\ndef eq_to_iso {C : Type u\u2081} [category C] {X : C} {Y : C} (p : X = Y) : X \u2245 Y :=\n  iso.mk (eq_to_hom p) (eq_to_hom (Eq.symm p))\n\n@[simp] theorem eq_to_iso.hom {C : Type u\u2081} [category C] {X : C} {Y : C} (p : X = Y) : iso.hom (eq_to_iso p) = eq_to_hom p :=\n  rfl\n\n@[simp] theorem eq_to_iso.inv {C : Type u\u2081} [category C] {X : C} {Y : C} (p : X = Y) : iso.inv (eq_to_iso p) = eq_to_hom (Eq.symm p) :=\n  rfl\n\n@[simp] theorem eq_to_iso_refl {C : Type u\u2081} [category C] {X : C} (p : X = X) : eq_to_iso p = iso.refl X :=\n  rfl\n\n@[simp] theorem eq_to_iso_trans {C : Type u\u2081} [category C] {X : C} {Y : C} {Z : C} (p : X = Y) (q : Y = Z) : eq_to_iso p \u226a\u226b eq_to_iso q = eq_to_iso (Eq.trans p q) := sorry\n\n@[simp] theorem eq_to_hom_op {C : Type u\u2081} [category C] {X : C} {Y : C} (h : X = Y) : has_hom.hom.op (eq_to_hom h) = eq_to_hom (congr_arg opposite.op (Eq.symm h)) := sorry\n\n@[simp] theorem eq_to_hom_unop {C : Type u\u2081} [category C] {X : C\u1d52\u1d56} {Y : C\u1d52\u1d56} (h : X = Y) : has_hom.hom.unop (eq_to_hom h) = eq_to_hom (congr_arg opposite.unop (Eq.symm h)) := sorry\n\nprotected instance eq_to_hom.is_iso {C : Type u\u2081} [category C] {X : C} {Y : C} (h : X = Y) : is_iso (eq_to_hom h) :=\n  is_iso.mk (iso.inv (eq_to_iso h))\n\n@[simp] theorem inv_eq_to_hom {C : Type u\u2081} [category C] {X : C} {Y : C} (h : X = Y) : inv (eq_to_hom h) = eq_to_hom (Eq.symm h) :=\n  rfl\n\nnamespace functor\n\n\n/-- Proving equality between functors. This isn't an extensionality lemma,\n  because usually you don't really want to do this. -/\ntheorem ext {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (h_obj : \u2200 (X : C), obj F X = obj G X) (h_map : \u2200 (X Y : C) (f : X \u27f6 Y), map F f = eq_to_hom (h_obj X) \u226b map G f \u226b eq_to_hom (Eq.symm (h_obj Y))) : F = G := sorry\n\n/-- Proving equality between functors using heterogeneous equality. -/\ntheorem hext {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (h_obj : \u2200 (X : C), obj F X = obj G X) (h_map : \u2200 (X Y : C) (f : X \u27f6 Y), map F f == map G f) : F = G := sorry\n\n-- Using equalities between functors.\n\ntheorem congr_obj {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (h : F = G) (X : C) : obj F X = obj G X :=\n  Eq._oldrec (Eq.refl (obj F X)) h\n\ntheorem congr_hom {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (h : F = G) {X : C} {Y : C} (f : X \u27f6 Y) : map F f = eq_to_hom (congr_obj h X) \u226b map G f \u226b eq_to_hom (Eq.symm (congr_obj h Y)) := sorry\n\nend functor\n\n\n@[simp] theorem eq_to_hom_map {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (p : X = Y) : functor.map F (eq_to_hom p) = eq_to_hom (congr_arg (functor.obj F) p) := sorry\n\n@[simp] theorem eq_to_iso_map {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (F : C \u2964 D) {X : C} {Y : C} (p : X = Y) : functor.map_iso F (eq_to_iso p) = eq_to_iso (congr_arg (functor.obj F) p) := sorry\n\n@[simp] theorem eq_to_hom_app {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (h : F = G) (X : C) : nat_trans.app (eq_to_hom h) X = eq_to_hom (functor.congr_obj h X) :=\n  eq.drec (Eq.refl (nat_trans.app (eq_to_hom (Eq.refl F)) X)) h\n\ntheorem nat_trans.congr {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {F : C \u2964 D} {G : C \u2964 D} (\u03b1 : F \u27f6 G) {X : C} {Y : C} (h : X = Y) : nat_trans.app \u03b1 X = functor.map F (eq_to_hom h) \u226b nat_trans.app \u03b1 Y \u226b functor.map G (eq_to_hom (Eq.symm h)) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/eq_to_hom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3712553773304966}}
{"text": "import Std.Tactic.SimpTrace\nimport Std.Tactic.SqueezeScope\n\nset_option linter.missingDocs false\n\nexample : x + 1 = 1 + x := by simp? [Nat.add_comm, Nat.mul_comm]\nexample : 1 + 1 = 2 := by dsimp?\n\n@[simp] def bar (z : Nat) := 1 + z\n@[simp] def baz (z : Nat) := 1 + z\n\n@[simp] def foo : Nat \u2192 Nat \u2192 Nat\n  | 0, z => bar z\n  | _+1, z => baz z\n\nexample : foo x y = 1 + y := by\n  cases x <;> simp? -- two printouts:\n  -- \"Try this: simp only [foo, bar]\"\n  -- \"Try this: simp only [foo, baz]\"\n\nexample : foo x y = 1 + y := by\n  squeeze_scope\n    cases x <;> simp -- only one printout: \"Try this: simp only [foo, baz, bar]\"\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/test/simp_trace.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.371255368976071}}
{"text": "-- import temporal_logic.tactic\nimport temporal_logic.lemmas\n\nuniverses u u\u2080 u\u2081\n\nopen predicate temporal\n\nnamespace temporal\nnamespace fairness\n\nsection defs\n\nvariables p q A : cpred\n\ndef wf : cpred :=\n\u25c7\u25fbp \u27f6 \u25fb\u25c7A\n\ndef sf : cpred :=\n\u25fb\u25c7p \u27f6 \u25fb\u25c7A\n\ndef sched : cpred :=\n\u25c7\u25fbp \u27f6 \u25fb\u25c7q \u27f6 \u25fb\u25c7(p \u22c0 q \u22c0 A)\n\ninstance persistent_sched : persistent (sched p q A) :=\nby { unfold sched, apply_instance }\n\nend defs\n\nlemma sched_imp_sched {\u0393 p q A A' : cpred}\n  (hA : \u0393 \u22a2 \u25fb(A \u27f6 A'))\n: \u0393 \u22a2 sched p q A \u27f6 sched p q A' :=\nbegin [temporal]\n  dsimp [sched],\n  mono*, apply hA,\nend\n\n-- TODO(Simon) replace ~> with \u25fb\u25c7_ \u27f6 \u25fb\u25c7_\n\nstructure one_to_one_po (S p q A p' q' A' : cpred) : Prop :=\n  (delay : S \u27f9 (p' \u22c0 q' ~> p))\n  (resched : S \u27f9 (p' \u22c0 q' ~> q))\n  (stable : S \u27f9 (\u25fb\u25c7p \u27f6 \u25c7\u25fbp' \u27f6 \u25c7\u25fbp))\n  (sim : S \u27f9 \u25fb(p \u27f6 q \u27f6 A \u27f6 p' \u22c0 q' \u22c0 A'))\n\nstructure event (\u03b1 : Type u\u2080) :=\n(p q : pred' \u03b1) (A : act \u03b1)\n\ndef one_to_one_po' {\u03b1 \u03b2} (S : cpred)\n: event \u03b1 \u2192 event \u03b2 \u2192 tvar \u03b1 \u2192 tvar \u03b2 \u2192 Prop\n| \u27e8p\u2080,q\u2080,A\u2080\u27e9 \u27e8p\u2081,q\u2081,A\u2081\u27e9 v w :=\none_to_one_po S\n  (p\u2080!v) (q\u2080!v) \u27e6 v | A\u2080 \u27e7\n  (p\u2081!w) (q\u2081!w) \u27e6 w | A\u2081 \u27e7\n\nnamespace one_to_one\nsection one_to_one\n\nparameters {S p q A : cpred}\nparameters {p' q' A' : cpred}\nparameters po : one_to_one_po S p q A p' q' A'\nparameters \u0393 : cpred\nparameters hS : \u0393 \u22a2 S\nprivate def H\u2080 : \u0393 \u22a2 p' \u22c0 q' ~> p :=\npo.delay \u0393 hS\n\nprivate def H\u2081 : \u0393 \u22a2 \u25fb\u25c7p \u27f6 \u25c7\u25fbp' \u27f6 \u25c7\u25fbp :=\npo.stable \u0393 hS\n\nprivate def H\u2082 : \u0393 \u22a2 \u25fb(p \u27f6 q \u27f6 A \u27f6 p' \u22c0 q' \u22c0 A') :=\npo.sim \u0393 hS\nprivate def H\u2083 : \u0393 \u22a2 p' \u22c0 q' ~> q :=\npo.resched \u0393 hS\n\ninclude po hS\n\nlemma replacement\n: \u0393 \u22a2 sched p q A \u27f6 sched p' q' A' :=\nbegin [temporal]\n  simp [sched], intros,\n  have swc := coincidence a_1 a_2,\n  have wc : \u25c7\u25fbp,\n  { assume_negation, simp at h,\n    have H\u2081 := H\u2081 po \u0393 hS,\n    have H\u2080 := H\u2080 po \u0393 hS,\n    apply H\u2081 _ a_1,\n    apply inf_often_of_leads_to H\u2080 swc, },\n  have sc : \u25fb\u25c7q,\n  { have H\u2083 := H\u2083 po \u0393 hS,\n    apply inf_often_of_leads_to H\u2083 swc, },\n  replace a := a wc sc, revert a,\n  { have H\u2082 := H\u2082 po \u0393 hS,\n    intros H\u2083, replace H\u2083 := coincidence wc H\u2083,\n    henceforth! at H\u2083 \u22a2, eventually H\u2083 \u22a2,\n    casesm* _ \u22c0 _,\n    apply H\u2082 ; assumption, },\nend\n\nend one_to_one\nend one_to_one\nexport one_to_one (replacement)\n\n\nnamespace splitting\nsection splitting\n\nparameters (t : Sort u)\n-- TODO(Simon) Weaken proof obligation `H\u2080`. We can do with \u25fb\u25c7_ \u27f6 \u25fb\u25c7 _\n-- instead of _ ~> _. Use w i unless -p'\nstructure many_to_many_po (S : cpred) (w p q A : t \u2192 cpred) (p' q' A' : cpred) : Prop :=\n  (delay : S \u27f9 \u2200\u2200 i, p' \u22c0 q' \u22c0 w i ~> p i \u22c0 q i \u22c0 w i)\n  (stable : S \u27f9 \u2200\u2200 i, \u25c7(p i \u22c0 w i) \u27f6 \u25c7\u25fbp' \u27f6 \u25c7\u25fb(p i \u22c0 w i))\n  (wfis : S \u27f9 \u25fb(p' \u22c0 q' \u27f6 \u2203\u2203 i, w i))\n  (sim : S \u27f9 \u2200\u2200 i, \u25fb(p i \u27f6 q i \u27f6 A i \u27f6 p' \u22c0 q' \u22c0 A'))\ndef many_to_many_po' {\u03b1 \u03b2} (S : cpred) (w : t \u2192 cpred)\n: (t \u2192 event \u03b1) \u2192 event \u03b2 \u2192 tvar \u03b1 \u2192 tvar \u03b2 \u2192 Prop\n| e \u27e8p\u2081,q\u2081,A\u2081\u27e9 cv av :=\nmany_to_many_po S w\n  (\u03bb i, (e i).p!cv) (\u03bb i, (e i).q!cv) (\u03bb i, \u27e6 cv | (e i).A \u27e7)\n  (p\u2081!av) (q\u2081!av) \u27e6 av | A\u2081 \u27e7\n\nparameters {t}\nparameters {w p q A : t \u2192 cpred}\nparameters {p' q' A' S : cpred}\nparameters po : many_to_many_po S w p q A p' q' A'\nparameters {\u0393 : cpred}\nparameters hS : \u0393 \u22a2 S\n\n-- variables H\u2080 : \u0393 \u22a2 \u2200\u2200 i, \u25fb\u25c7(p' \u22c0 q' \u22c0 w i) \u27f6 \u25fb\u25c7p i \u22c0 q i \u22c0 w i\ndef H\u2080 : \u0393 \u22a2 \u2200\u2200 i, p' \u22c0 q' \u22c0 w i ~> p i \u22c0 q i \u22c0 w i :=\npo.delay \u0393 hS\n\ndef H\u2081 : \u0393 \u22a2 \u2200\u2200 i, \u25c7(p i \u22c0 w i) \u27f6 \u25c7\u25fbp' \u27f6 \u25c7\u25fb(p i \u22c0 w i) :=\npo.stable \u0393 hS\n\ndef H\u2082 : \u0393 \u22a2 \u2200\u2200 i, \u25fb(p i \u27f6 q i \u27f6 A i \u27f6 p' \u22c0 q' \u22c0 A') :=\npo.sim \u0393 hS\n\ndef H\u2083 : \u0393 \u22a2 \u25fb(p' \u22c0 q' \u27f6 \u2203\u2203 i, w i) :=\npo.wfis \u0393 hS\n\ninclude hS po H\u2080 H\u2081 H\u2082 H\u2083\n\nopen temporal\nlemma splitting\n: \u0393 \u22a2 (\u2200\u2200 i, sched (p i) (q i) (A i)) \u27f6 sched p' q' A' :=\nbegin [temporal]\n  intro H\u2085,\n  simp [sched] at *, intros hp' hq',\n  have H\u2087 := temporal.leads_to_disj_gen temporal.fairness.splitting.H\u2080,\n  replace H\u2087 := inf_often_of_leads_to H\u2087 _,\n  replace H\u2087 : \u2203\u2203 (i : t), \u25c7(p i \u22c0 q i \u22c0 w i),\n  { henceforth at H\u2087, rw eventually_exists at H\u2087, },\n  { cases H\u2087 with i H\u2087,\n    have H\u2089 := temporal.fairness.splitting.H\u2081 i _ hp',\n    have : \u25fb\u25c7q i,\n    { have := inf_often_of_leads_to (temporal.fairness.splitting.H\u2080 i) _, revert this,\n      { mono!*, lifted_pred, show _, { intros, assumption } },\n      rw_using : (p' \u22c0 q' \u22c0 w i) = (p' \u22c0 w i \u22c0 q'),\n      { lifted_pred, tauto },\n      apply coincidence _ hq',\n      { apply stable_and_of_stable_of_stable hp',\n        revert H\u2089, mono! * }, },\n    replace this := H\u2085 i _ this,\n    { have H\u2082 := temporal.fairness.splitting.H\u2082 i,\n      replace this := coincidence H\u2089 this,\n      henceforth! at this \u22a2, eventually this \u22a2,\n      casesm* _ \u22c0 _, apply H\u2082 ; assumption  },\n    { revert H\u2089, mono!, lifted_pred,\n      show _, { intros, assumption } },\n    { revert H\u2087, mono!, lifted_pred }, },\n  { have := coincidence hp' hq', revert this,\n    have H\u2083 := temporal.fairness.splitting.H\u2083,\n    mono!,\n    intros hp,\n    have := H\u2083 hp,\n    revert this, apply p_exists_p_imp_p_exists,\n    tauto, } ,\nend\n\nend splitting\nend splitting\nexport splitting (splitting many_to_many_po many_to_many_po')\nend fairness\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/fairness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.37117737941889456}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Yury Kudryashov\n\n! This file was ported from Lean 3 source module topology.algebra.order.extend_from\n! leanprover-community/mathlib commit 3e32bc908f617039c74c06ea9a897e30c30803c2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Topology.Order.Basic\nimport Mathbin.Topology.ExtendFrom\n\n/-!\n# Lemmas about `extend_from` in an order topology.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nopen Filter Set TopologicalSpace\n\nopen Topology Classical\n\nuniverse u v\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v}\n\n/- warning: continuous_on_Icc_extend_from_Ioo -> continuousOn_Icc_extendFrom_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] [_inst_3 : DenselyOrdered.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))))] [_inst_4 : OrderTopology.{u1} \u03b1 _inst_1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))] [_inst_5 : TopologicalSpace.{u2} \u03b2] [_inst_6 : RegularSpace.{u2} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2} {lb : \u03b2}, (Ne.{succ u1} \u03b1 a b) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b)) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 a (Set.Ioi.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a)) (nhds.{u2} \u03b2 _inst_5 la)) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 b (Set.Iio.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) b)) (nhds.{u2} \u03b2 _inst_5 lb)) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b) f) (Set.Icc.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : LinearOrder.{u2} \u03b1] [_inst_3 : DenselyOrdered.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))))] [_inst_4 : OrderTopology.{u2} \u03b1 _inst_1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))] [_inst_5 : TopologicalSpace.{u1} \u03b2] [_inst_6 : RegularSpace.{u1} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2} {lb : \u03b2}, (Ne.{succ u2} \u03b1 a b) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b)) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 a (Set.Ioi.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a)) (nhds.{u1} \u03b2 _inst_5 la)) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 b (Set.Iio.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) b)) (nhds.{u1} \u03b2 _inst_5 lb)) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b) f) (Set.Icc.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b))\nCase conversion may be inaccurate. Consider using '#align continuous_on_Icc_extend_from_Ioo continuousOn_Icc_extendFrom_Ioo\u2093'. -/\ntheorem continuousOn_Icc_extendFrom_Ioo [TopologicalSpace \u03b1] [LinearOrder \u03b1] [DenselyOrdered \u03b1]\n    [OrderTopology \u03b1] [TopologicalSpace \u03b2] [RegularSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {a b : \u03b1} {la lb : \u03b2}\n    (hab : a \u2260 b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd la))\n    (hb : Tendsto f (\ud835\udcdd[<] b) (\ud835\udcdd lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Icc a b) :=\n  by\n  apply continuousOn_extendFrom\n  \u00b7 rw [closure_Ioo hab]\n  \u00b7 intro x x_in\n    rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl | h)\n    \u00b7 exact \u27e8la, ha.mono_left <| nhdsWithin_mono _ Ioo_subset_Ioi_self\u27e9\n    \u00b7 exact \u27e8lb, hb.mono_left <| nhdsWithin_mono _ Ioo_subset_Iio_self\u27e9\n    \u00b7 use f x, hf x h\n#align continuous_on_Icc_extend_from_Ioo continuousOn_Icc_extendFrom_Ioo\n\n/- warning: eq_lim_at_left_extend_from_Ioo -> eq_lim_at_left_extendFrom_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] [_inst_3 : DenselyOrdered.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))))] [_inst_4 : OrderTopology.{u1} \u03b1 _inst_1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))] [_inst_5 : TopologicalSpace.{u2} \u03b2] [_inst_6 : T2Space.{u2} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2}, (LT.lt.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))) a b) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 a (Set.Ioi.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a)) (nhds.{u2} \u03b2 _inst_5 la)) -> (Eq.{succ u2} \u03b2 (extendFrom.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b) f a) la)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : LinearOrder.{u2} \u03b1] [_inst_3 : DenselyOrdered.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))))] [_inst_4 : OrderTopology.{u2} \u03b1 _inst_1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))] [_inst_5 : TopologicalSpace.{u1} \u03b2] [_inst_6 : T2Space.{u1} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2}, (LT.lt.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))) a b) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 a (Set.Ioi.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a)) (nhds.{u1} \u03b2 _inst_5 la)) -> (Eq.{succ u1} \u03b2 (extendFrom.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b) f a) la)\nCase conversion may be inaccurate. Consider using '#align eq_lim_at_left_extend_from_Ioo eq_lim_at_left_extendFrom_Ioo\u2093'. -/\ntheorem eq_lim_at_left_extendFrom_Ioo [TopologicalSpace \u03b1] [LinearOrder \u03b1] [DenselyOrdered \u03b1]\n    [OrderTopology \u03b1] [TopologicalSpace \u03b2] [T2Space \u03b2] {f : \u03b1 \u2192 \u03b2} {a b : \u03b1} {la : \u03b2} (hab : a < b)\n    (ha : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd la)) : extendFrom (Ioo a b) f a = la :=\n  by\n  apply extendFrom_eq\n  \u00b7 rw [closure_Ioo hab.ne]\n    simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc]\n  \u00b7 simpa [hab]\n#align eq_lim_at_left_extend_from_Ioo eq_lim_at_left_extendFrom_Ioo\n\n/- warning: eq_lim_at_right_extend_from_Ioo -> eq_lim_at_right_extendFrom_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] [_inst_3 : DenselyOrdered.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))))] [_inst_4 : OrderTopology.{u1} \u03b1 _inst_1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))] [_inst_5 : TopologicalSpace.{u2} \u03b2] [_inst_6 : T2Space.{u2} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {lb : \u03b2}, (LT.lt.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))) a b) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 b (Set.Iio.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) b)) (nhds.{u2} \u03b2 _inst_5 lb)) -> (Eq.{succ u2} \u03b2 (extendFrom.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b) f b) lb)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : LinearOrder.{u2} \u03b1] [_inst_3 : DenselyOrdered.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))))] [_inst_4 : OrderTopology.{u2} \u03b1 _inst_1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))] [_inst_5 : TopologicalSpace.{u1} \u03b2] [_inst_6 : T2Space.{u1} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {lb : \u03b2}, (LT.lt.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))) a b) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 b (Set.Iio.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) b)) (nhds.{u1} \u03b2 _inst_5 lb)) -> (Eq.{succ u1} \u03b2 (extendFrom.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b) f b) lb)\nCase conversion may be inaccurate. Consider using '#align eq_lim_at_right_extend_from_Ioo eq_lim_at_right_extendFrom_Ioo\u2093'. -/\ntheorem eq_lim_at_right_extendFrom_Ioo [TopologicalSpace \u03b1] [LinearOrder \u03b1] [DenselyOrdered \u03b1]\n    [OrderTopology \u03b1] [TopologicalSpace \u03b2] [T2Space \u03b2] {f : \u03b1 \u2192 \u03b2} {a b : \u03b1} {lb : \u03b2} (hab : a < b)\n    (hb : Tendsto f (\ud835\udcdd[<] b) (\ud835\udcdd lb)) : extendFrom (Ioo a b) f b = lb :=\n  by\n  apply extendFrom_eq\n  \u00b7 rw [closure_Ioo hab.ne]\n    simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc]\n  \u00b7 simpa [hab]\n#align eq_lim_at_right_extend_from_Ioo eq_lim_at_right_extendFrom_Ioo\n\n/- warning: continuous_on_Ico_extend_from_Ioo -> continuousOn_Ico_extendFrom_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] [_inst_3 : DenselyOrdered.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))))] [_inst_4 : OrderTopology.{u1} \u03b1 _inst_1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))] [_inst_5 : TopologicalSpace.{u2} \u03b2] [_inst_6 : RegularSpace.{u2} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2}, (LT.lt.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))) a b) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b)) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 a (Set.Ioi.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a)) (nhds.{u2} \u03b2 _inst_5 la)) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b) f) (Set.Ico.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : LinearOrder.{u2} \u03b1] [_inst_3 : DenselyOrdered.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))))] [_inst_4 : OrderTopology.{u2} \u03b1 _inst_1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))] [_inst_5 : TopologicalSpace.{u1} \u03b2] [_inst_6 : RegularSpace.{u1} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {la : \u03b2}, (LT.lt.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))) a b) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b)) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 a (Set.Ioi.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a)) (nhds.{u1} \u03b2 _inst_5 la)) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b) f) (Set.Ico.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b))\nCase conversion may be inaccurate. Consider using '#align continuous_on_Ico_extend_from_Ioo continuousOn_Ico_extendFrom_Ioo\u2093'. -/\ntheorem continuousOn_Ico_extendFrom_Ioo [TopologicalSpace \u03b1] [LinearOrder \u03b1] [DenselyOrdered \u03b1]\n    [OrderTopology \u03b1] [TopologicalSpace \u03b2] [RegularSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {a b : \u03b1} {la : \u03b2}\n    (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (\ud835\udcdd[>] a) (\ud835\udcdd la)) :\n    ContinuousOn (extendFrom (Ioo a b) f) (Ico a b) :=\n  by\n  apply continuousOn_extendFrom\n  \u00b7 rw [closure_Ioo hab.ne]\n    exact Ico_subset_Icc_self\n  \u00b7 intro x x_in\n    rcases eq_left_or_mem_Ioo_of_mem_Ico x_in with (rfl | h)\n    \u00b7 use la\n      simpa [hab]\n    \u00b7 use f x, hf x h\n#align continuous_on_Ico_extend_from_Ioo continuousOn_Ico_extendFrom_Ioo\n\n/- warning: continuous_on_Ioc_extend_from_Ioo -> continuousOn_Ioc_extendFrom_Ioo is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : TopologicalSpace.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] [_inst_3 : DenselyOrdered.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))))] [_inst_4 : OrderTopology.{u1} \u03b1 _inst_1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))] [_inst_5 : TopologicalSpace.{u2} \u03b2] [_inst_6 : RegularSpace.{u2} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {lb : \u03b2}, (LT.lt.{u1} \u03b1 (Preorder.toLT.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2))))) a b) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b)) -> (Filter.Tendsto.{u1, u2} \u03b1 \u03b2 f (nhdsWithin.{u1} \u03b1 _inst_1 b (Set.Iio.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) b)) (nhds.{u2} \u03b2 _inst_5 lb)) -> (ContinuousOn.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b) f) (Set.Ioc.{u1} \u03b1 (PartialOrder.toPreorder.{u1} \u03b1 (SemilatticeInf.toPartialOrder.{u1} \u03b1 (Lattice.toSemilatticeInf.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : TopologicalSpace.{u2} \u03b1] [_inst_2 : LinearOrder.{u2} \u03b1] [_inst_3 : DenselyOrdered.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))))] [_inst_4 : OrderTopology.{u2} \u03b1 _inst_1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))] [_inst_5 : TopologicalSpace.{u1} \u03b2] [_inst_6 : RegularSpace.{u1} \u03b2 _inst_5] {f : \u03b1 -> \u03b2} {a : \u03b1} {b : \u03b1} {lb : \u03b2}, (LT.lt.{u2} \u03b1 (Preorder.toLT.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2)))))) a b) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 f (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b)) -> (Filter.Tendsto.{u2, u1} \u03b1 \u03b2 f (nhdsWithin.{u2} \u03b1 _inst_1 b (Set.Iio.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) b)) (nhds.{u1} \u03b2 _inst_5 lb)) -> (ContinuousOn.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (extendFrom.{u2, u1} \u03b1 \u03b2 _inst_1 _inst_5 (Set.Ioo.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b) f) (Set.Ioc.{u2} \u03b1 (PartialOrder.toPreorder.{u2} \u03b1 (SemilatticeInf.toPartialOrder.{u2} \u03b1 (Lattice.toSemilatticeInf.{u2} \u03b1 (DistribLattice.toLattice.{u2} \u03b1 (instDistribLattice.{u2} \u03b1 _inst_2))))) a b))\nCase conversion may be inaccurate. Consider using '#align continuous_on_Ioc_extend_from_Ioo continuousOn_Ioc_extendFrom_Ioo\u2093'. -/\ntheorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace \u03b1] [LinearOrder \u03b1] [DenselyOrdered \u03b1]\n    [OrderTopology \u03b1] [TopologicalSpace \u03b2] [RegularSpace \u03b2] {f : \u03b1 \u2192 \u03b2} {a b : \u03b1} {lb : \u03b2}\n    (hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (\ud835\udcdd[<] b) (\ud835\udcdd lb)) :\n    ContinuousOn (extendFrom (Ioo a b) f) (Ioc a b) :=\n  by\n  have := @continuousOn_Ico_extendFrom_Ioo \u03b1\u1d52\u1d48 _ _ _ _ _ _ _ f _ _ _ hab\n  erw [dual_Ico, dual_Ioi, dual_Ioo] at this\n  exact this hf hb\n#align continuous_on_Ioc_extend_from_Ioo continuousOn_Ioc_extendFrom_Ioo\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Topology/Algebra/Order/ExtendFrom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.37117737941889456}}
{"text": "import tactic.induction\nimport data.nat.parity\nimport data.int.interval\nimport data.set.finite\nimport data.list\nimport data.finset\nimport set_theory.cardinal.basic\n\nnoncomputable theory\nopen_locale classical\n\n@[reducible]\ndef snoc {\u03b1 : Type*} (xs : list \u03b1) (x : \u03b1) : list \u03b1 :=\nxs ++ [x]\n\nlemma snoc_eq_snoc_iff {\u03b1 : Type*} {xs ys : list \u03b1} {x y : \u03b1} :\n  snoc xs x = snoc ys y \u2194 xs = ys \u2227 x = y :=\nbegin\n  split; intro h,\n  { simp_rw [list.append_eq_append_iff] at h, cases h; rcases h with \u27e8zs, h\u2081, h\u2082\u27e9;\n    { have h\u2083 := congr_arg list.length h\u2082, simp at h\u2083,\n      rw list.length_eq_zero at h\u2083, cases h\u2083, simp at h\u2081 h\u2082,\n      exact \u27e8h\u2081.symm, h\u2082\u27e9 <|> exact \u27e8h\u2081, h\u2082.symm\u27e9 }},\n  { rw [h.1, h.2] },\nend\n\nlemma length_snoc {\u03b1 : Type*} {xs : list \u03b1} {x : \u03b1} :\n  (snoc xs x).length = xs.length.succ :=\nby { rw [list.length_append], refl }\n\nlemma length_lt_length_append_cons {\u03b1 : Type*} {xs ys : list \u03b1} {y : \u03b1} :\n  xs.length < (xs ++ y :: ys).length :=\nbegin\n  rw [list.length_append, list.length_cons], change _ + 0 < _,\n  rw add_lt_add_iff_left, exact (list.length ys).succ_pos,\nend\n\nlemma length_lt_length_append_snoc {\u03b1 : Type*} {xs ys : list \u03b1} {y : \u03b1} :\n  xs.length < (xs ++ snoc ys y).length :=\nbegin\n  rw [list.length_append, length_snoc], change _ + 0 < _,\n  rw add_lt_add_iff_left, exact (list.length ys).succ_pos,\nend\n\nlemma length_lt_length_snoc {\u03b1 : Type*} {xs : list \u03b1} {x : \u03b1} :\n  xs.length < (snoc xs x).length :=\nlength_lt_length_append_cons\n\nlemma length_lt_length_snoc\u2082 {\u03b1 : Type*} {xs : list \u03b1} {x y : \u03b1} :\n  xs.length < (snoc (snoc xs x) y).length :=\nby { transitivity (snoc xs x).length; exact length_lt_length_snoc }\n\nlemma exi_ge_of_set_inf {P : \u2115 \u2192 Prop} {n : \u2115}\n  (h : {n : \u2115 | P n}.infinite) : \u2203 (k : \u2115), n \u2264 k \u2227 P k :=\nbegin\n  obtain \u27e8k, h\u2081, h\u2082\u27e9 := h.exists_nat_lt n,\n  exact \u27e8k, nat.le_of_lt h\u2082, h\u2081\u27e9,\nend\n\nlemma exi_set_infinite_of_forall_exi_P {\u03b1 \u03b2 : Type} {P : \u03b1 \u2192 \u03b2 \u2192 Prop}\n  (h\u2081 : (set.univ : set \u03b1).infinite)\n  (h\u2082 : (set.univ : set \u03b2).finite)\n  (h\u2083 : \u2200 (a : \u03b1), \u2203 (b : \u03b2), P a b) :\n  \u2203 (b : \u03b2), {a : \u03b1 | P a b}.infinite :=\nbegin\n  by_contra' h\u2084, simp_rw set.not_infinite at h\u2084,\n  have h\u2085 : {a : \u03b1 | \u2203 (b : \u03b2), P a b}.finite,\n  { convert_to (\u22c3 (b : \u03b2), {a : \u03b1 | P a b}).finite, { ext a, simp, },\n    convert_to (\u22c3 (b : \u03b2) (h : b \u2208 set.univ), {a : \u03b1 | P a b}).finite,\n    { congr, ext b a, simp, intros, apply set.mem_univ },\n    apply set.finite.bUnion h\u2082, intros, apply h\u2084 },\n  replace h\u2085 : (set.univ : set \u03b1).finite,\n  { convert h\u2085, ext a, simp, exact h\u2083 a },\n  contradiction,\nend\n\nlemma exi_set_infinite_of_forall_exi_P_nat {\u03b1 : Type} [fintype \u03b1]\n  {P : \u2115 \u2192 \u03b1 \u2192 Prop}\n  (h : \u2200 (n : \u2115), \u2203 (a : \u03b1), P n a) :\n  \u2203 (a : \u03b1), {n : \u2115 | P n a}.infinite :=\nbegin\n  fapply exi_set_infinite_of_forall_exi_P,\n  { exact set.infinite_univ },\n  { exact set.finite_univ },\n  { exact h },\nend\n\nlemma fintype_subtype_of_set_finite {\u03b1 : Type} {P : \u03b1 \u2192 Prop}\n  (h : {x : \u03b1 | P x}.finite) : fintype {x : \u03b1 // P x} :=\nbegin\n  apply set.fintype_of_univ_finite, rw set.univ_subtype,\n  apply set.finite.bUnion h, simp,\nend\n\nlemma set_finite_of_set_equiv_finite' {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) {P : \u03b1 \u2192 Prop}\n  (h : {x : \u03b1 | P x}.finite) : {y : \u03b2 | P (e.inv_fun y)}.finite :=\nbegin\n  fapply set.finite_of_finite_image,\n  { exact \u03b1 },\n  { exact e.inv_fun },\n  { apply set.inj_on_of_injective,\n    exact function.right_inverse.injective e.right_inv },\n  { convert h, ext a, split; intro h\u2081,\n    { rcases h\u2081 with \u27e8b, h\u2081, h\u2082\u27e9, subst h\u2082, exact h\u2081 },\n    { use e.to_fun a, fsplit,\n      { change P _, convert h\u2081, exact e.left_inv a },\n      { exact e.left_inv a }}},\nend\n\nlemma set_finite_of_set_equiv_finite {\u03b1 \u03b2 : Type} (e : \u03b1 \u2243 \u03b2) {P : \u03b2 \u2192 Prop}\n  (h : {x : \u03b1 | P (e.to_fun x)}.finite) : {y : \u03b2 | P y}.finite :=\nbegin\n  replace h := set_finite_of_set_equiv_finite' e h, dsimp at h, convert h,\n  funext y, congr, exact eq.symm (e.right_inv y),\nend\n\nlemma abs_le_finite {d : \u2124} : {a : \u2124 | |a| \u2264 d}.finite :=\nby { simp_rw abs_le, apply set.finite_Icc }\n\nlemma abs_sub_le_finite {c d : \u2124} : {a : \u2124 | |a - c| \u2264 d}.finite :=\nbegin\n  fapply set.finite.of_preimage,\n  { exact \u2124 },\n  { intro a, exact a + c },\n  { simp, exact abs_le_finite },\n  { intro a, use a - c, simp },\nend\n\nlemma nat_find_le_find_of_imp {P Q : \u2115 \u2192 Prop}\n  {hh\u2081 : \u2203 (n : \u2115), P n}\n  {hh\u2082 : \u2203 (n : \u2115), Q n}\n  (h : \u2200 (n : \u2115), Q n \u2192 P n) :\n  nat.find hh\u2081 \u2264 nat.find hh\u2082 :=\nbegin\n  let n := _, change n \u2264 _, by_cases h\u2081 : Q n,\n  { apply le_of_eq, symmetry, rw nat.find_eq_iff, use h\u2081, rintro k h\u2082,\n    have h\u2083 : nat.find hh\u2081 = n := rfl, rw nat.find_eq_iff at h\u2083,\n    replace h\u2083 := h\u2083.2 k h\u2082, contrapose! h\u2083, exact h _ h\u2083 },\n  { have h\u2082 : \u2200 (k : \u2115), Q k \u2192 n \u2264 k,\n    { rintro k h\u2082, by_contra' hh, have h\u2083 : nat.find hh\u2081 = n := rfl,\n      rw nat.find_eq_iff at h\u2083, apply h\u2083.2 k hh, exact h _ h\u2082 },\n    apply h\u2082, apply nat.find_spec },\nend\n\nlemma nat_Inf_le_Inf_of_subset {P Q : set \u2115}\n  (h\u2081 : Q.nonempty)\n  (h\u2082 : Q \u2286 P) :\n  Inf P \u2264 Inf Q :=\nbegin\n  have h\u2083 : \u2203 n, n \u2208 Q,\n  { cases h\u2081 with n h\u2081, use [n, h\u2081] },\n  have h\u2084 : \u2203 n, n \u2208 P,\n  { cases h\u2083 with n h\u2083, use n, apply h\u2082, exact h\u2083 },\n  simp_rw Inf, split_ifs, exact nat_find_le_find_of_imp h\u2082,\nend\n\nlemma finset_card_insert_erase_eq {\u03b1 : Type} {s : finset \u03b1} {x y : \u03b1}\n  (h\u2081 : x \u2208 s)\n  (h\u2082 : y \u2209 s) :\n  (insert y (s.erase x)).card = s.card :=\nbegin\n  replace h\u2082 : y \u2209 s.erase x,\n  { rw finset.mem_erase, tauto },\n  rw [finset.card_insert_of_not_mem h\u2082, finset.card_erase_of_mem h\u2081],\n  cases h\u2083 : s.card,\n  { rw finset.card_eq_zero at h\u2083, subst h\u2083, cases h\u2081 },\n  { refl },\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/util.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.7090191337850932, "lm_q1q2_score": 0.3711150424083237}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jan-David Salchow, S\u00e9bastien Gou\u00ebzel, Jean Lo, Yury Kudryashov, Fr\u00e9d\u00e9ric Dupuis,\n  Heather Macbeth\n-/\nimport topology.algebra.ring\nimport topology.algebra.mul_action\nimport topology.uniform_space.uniform_embedding\nimport algebra.algebra.basic\nimport linear_algebra.projection\nimport linear_algebra.pi\nimport linear_algebra.determinant\n\n/-!\n# Theory of topological modules and continuous linear maps.\n\nWe use the class `has_continuous_smul` for topological (semi) modules and topological vector spaces.\n\nIn this file we define continuous (semi-)linear maps, as semilinear maps between topological\nmodules which are continuous. The set of continuous semilinear maps between the topological\n`R\u2081`-module `M` and `R\u2082`-module `M\u2082` with respect to the `ring_hom` `\u03c3` is denoted by `M \u2192SL[\u03c3] M\u2082`.\nPlain linear maps are denoted by `M \u2192L[R] M\u2082` and star-linear maps by `M \u2192L\u22c6[R] M\u2082`.\n\nThe corresponding notation for equivalences is `M \u2243SL[\u03c3] M\u2082`, `M \u2243L[R] M\u2082` and `M \u2243L\u22c6[R] M\u2082`.\n-/\n\nopen filter\nopen_locale topological_space big_operators filter\n\nuniverses u v w u'\n\nsection\n\nvariables {R : Type*} {M : Type*}\n[ring R] [topological_space R]\n[topological_space M] [add_comm_group M]\n[module R M]\n\nlemma has_continuous_smul.of_nhds_zero [topological_ring R] [topological_add_group M]\n  (hmul : tendsto (\u03bb p : R \u00d7 M, p.1 \u2022 p.2) (\ud835\udcdd 0 \u00d7\u1da0 (\ud835\udcdd 0)) (\ud835\udcdd 0))\n  (hmulleft : \u2200 m : M, tendsto (\u03bb a : R, a \u2022 m) (\ud835\udcdd 0) (\ud835\udcdd 0))\n  (hmulright : \u2200 a : R, tendsto (\u03bb m : M, a \u2022 m) (\ud835\udcdd 0) (\ud835\udcdd 0)) : has_continuous_smul R M :=\n\u27e8begin\n  rw continuous_iff_continuous_at,\n  rintros \u27e8a\u2080, m\u2080\u27e9,\n  have key : \u2200 p : R \u00d7 M,\n    p.1 \u2022 p.2 = a\u2080 \u2022 m\u2080 + ((p.1 - a\u2080) \u2022 m\u2080 + a\u2080 \u2022 (p.2 - m\u2080) + (p.1 - a\u2080) \u2022 (p.2 - m\u2080)),\n  { rintro \u27e8a, m\u27e9,\n    simp [sub_smul, smul_sub],\n    abel },\n  rw funext key, clear key,\n  refine tendsto_const_nhds.add (tendsto.add (tendsto.add _ _) _),\n  { rw [sub_self, zero_smul],\n    apply (hmulleft m\u2080).comp,\n    rw [show (\u03bb p : R \u00d7 M, p.1 - a\u2080) = (\u03bb a, a - a\u2080) \u2218 prod.fst, by {ext, refl }, nhds_prod_eq],\n    have : tendsto (\u03bb a, a - a\u2080) (\ud835\udcdd a\u2080) (\ud835\udcdd 0),\n    { rw \u2190 sub_self a\u2080,\n      exact tendsto_id.sub tendsto_const_nhds },\n    exact this.comp tendsto_fst  },\n  { rw [sub_self, smul_zero],\n    apply (hmulright a\u2080).comp,\n    rw [show (\u03bb p : R \u00d7 M, p.2 - m\u2080) = (\u03bb m, m - m\u2080) \u2218 prod.snd, by {ext, refl }, nhds_prod_eq],\n    have : tendsto (\u03bb m, m - m\u2080) (\ud835\udcdd m\u2080) (\ud835\udcdd 0),\n    { rw \u2190 sub_self m\u2080,\n      exact tendsto_id.sub tendsto_const_nhds },\n    exact this.comp tendsto_snd },\n  { rw [sub_self, zero_smul, nhds_prod_eq,\n        show (\u03bb p : R \u00d7 M, (p.fst - a\u2080) \u2022 (p.snd - m\u2080)) =\n             (\u03bb  p : R \u00d7 M, p.1 \u2022 p.2) \u2218 (prod.map (\u03bb a, a - a\u2080) (\u03bb m, m - m\u2080)), by { ext, refl }],\n    apply hmul.comp (tendsto.prod_map _ _);\n    { rw \u2190 sub_self ,\n      exact tendsto_id.sub tendsto_const_nhds } },\nend\u27e9\nend\n\nsection\nvariables {R : Type*} {M : Type*}\n[ring R] [topological_space R]\n[topological_space M] [add_comm_group M] [has_continuous_add M]\n[module R M] [has_continuous_smul R M]\n\n/-- If `M` is a topological module over `R` and `0` is a limit of invertible elements of `R`, then\n`\u22a4` is the only submodule of `M` with a nonempty interior.\nThis is the case, e.g., if `R` is a nondiscrete normed field. -/\nlemma submodule.eq_top_of_nonempty_interior'\n  [ne_bot (\ud835\udcdd[{x : R | is_unit x}] 0)]\n  (s : submodule R M) (hs : (interior (s:set M)).nonempty) :\n  s = \u22a4 :=\nbegin\n  rcases hs with \u27e8y, hy\u27e9,\n  refine (submodule.eq_top_iff'.2 $ \u03bb x, _),\n  rw [mem_interior_iff_mem_nhds] at hy,\n  have : tendsto (\u03bb c:R, y + c \u2022 x) (\ud835\udcdd[{x : R | is_unit x}] 0) (\ud835\udcdd (y + (0:R) \u2022 x)),\n    from tendsto_const_nhds.add ((tendsto_nhds_within_of_tendsto_nhds tendsto_id).smul\n      tendsto_const_nhds),\n  rw [zero_smul, add_zero] at this,\n  obtain \u27e8_, hu : y + _ \u2022 _ \u2208 s, u, rfl\u27e9 :=\n    nonempty_of_mem (inter_mem (mem_map.1 (this hy)) self_mem_nhds_within),\n  have hy' : y \u2208 \u2191s := mem_of_mem_nhds hy,\n  rwa [s.add_mem_iff_right hy', \u2190units.smul_def, s.smul_mem_iff' u] at hu,\nend\n\nvariables (R M)\n\n/-- Let `R` be a topological ring such that zero is not an isolated point (e.g., a nondiscrete\nnormed field, see `normed_field.punctured_nhds_ne_bot`). Let `M` be a nontrivial module over `R`\nsuch that `c \u2022 x = 0` implies `c = 0 \u2228 x = 0`. Then `M` has no isolated points. We formulate this\nusing `ne_bot (\ud835\udcdd[\u2260] x)`.\n\nThis lemma is not an instance because Lean would need to find `[has_continuous_smul ?m_1 M]` with\nunknown `?m_1`. We register this as an instance for `R = \u211d` in `real.punctured_nhds_module_ne_bot`.\nOne can also use `haveI := module.punctured_nhds_ne_bot R M` in a proof.\n-/\nlemma module.punctured_nhds_ne_bot [nontrivial M] [ne_bot (\ud835\udcdd[\u2260] (0 : R))]\n  [no_zero_smul_divisors R M] (x : M) :\n  ne_bot (\ud835\udcdd[\u2260] x) :=\nbegin\n  rcases exists_ne (0 : M) with \u27e8y, hy\u27e9,\n  suffices : tendsto (\u03bb c : R, x + c \u2022 y) (\ud835\udcdd[\u2260] 0) (\ud835\udcdd[\u2260] x), from this.ne_bot,\n  refine tendsto.inf _ (tendsto_principal_principal.2 $ _),\n  { convert tendsto_const_nhds.add ((@tendsto_id R _).smul_const y),\n    rw [zero_smul, add_zero] },\n  { intros c hc,\n    simpa [hy] using hc }\nend\n\nend\n\nsection lattice_ops\n\nvariables {\u03b9 R M\u2081 M\u2082 : Type*} [semiring R] [add_comm_monoid M\u2081] [add_comm_monoid M\u2082]\n  [module R M\u2081] [module R M\u2082] [u : topological_space R] {t : topological_space M\u2082}\n  [has_continuous_smul R M\u2082] (f : M\u2081 \u2192\u2097[R] M\u2082)\n\nlemma has_continuous_smul_induced :\n  @has_continuous_smul R M\u2081 _ u (t.induced f) :=\n{ continuous_smul :=\n    begin\n      letI : topological_space M\u2081 := t.induced f,\n      refine continuous_induced_rng _,\n      simp_rw [function.comp, f.map_smul],\n      refine continuous_fst.smul (continuous_induced_dom.comp continuous_snd)\n    end }\n\nend lattice_ops\n\nnamespace submodule\n\nvariables {\u03b1 \u03b2 : Type*} [topological_space \u03b2]\n\ninstance [topological_space \u03b1] [semiring \u03b1] [add_comm_monoid \u03b2] [module \u03b1 \u03b2]\n  [has_continuous_smul \u03b1 \u03b2] (S : submodule \u03b1 \u03b2) :\n  has_continuous_smul \u03b1 S :=\n{ continuous_smul :=\n  begin\n    rw embedding_subtype_coe.to_inducing.continuous_iff,\n    exact continuous_fst.smul\n      (continuous_subtype_coe.comp continuous_snd)\n  end }\n\ninstance [ring \u03b1] [add_comm_group \u03b2] [module \u03b1 \u03b2] [topological_add_group \u03b2] (S : submodule \u03b1 \u03b2) :\n  topological_add_group S :=\nS.to_add_subgroup.topological_add_group\n\nend submodule\n\nsection closure\nvariables {R : Type u} {M : Type v}\n[semiring R] [topological_space R]\n[topological_space M] [add_comm_monoid M]\n[module R M] [has_continuous_smul R M]\n\nlemma submodule.closure_smul_self_subset (s : submodule R M) :\n  (\u03bb p : R \u00d7 M, p.1 \u2022 p.2) '' ((set.univ : set R) \u00d7\u02e2 closure (s : set M))\n  \u2286 closure (s : set M) :=\ncalc\n(\u03bb p : R \u00d7 M, p.1 \u2022 p.2) '' ((set.univ : set R) \u00d7\u02e2 closure (s : set M))\n    = (\u03bb p : R \u00d7 M, p.1 \u2022 p.2) '' (closure ((set.univ : set R) \u00d7\u02e2 (s : set M))) :\n  by simp [closure_prod_eq]\n... \u2286 closure ((\u03bb p : R \u00d7 M, p.1 \u2022 p.2) '' ((set.univ : set R) \u00d7\u02e2 (s : set M))) :\n  image_closure_subset_closure_image continuous_smul\n... = closure s : begin\n  congr,\n  ext x,\n  refine \u27e8_, \u03bb hx, \u27e8\u27e81, x\u27e9, \u27e8set.mem_univ _, hx\u27e9, one_smul R _\u27e9\u27e9,\n  rintros \u27e8\u27e8c, y\u27e9, \u27e8hc, hy\u27e9, rfl\u27e9,\n  simp [s.smul_mem c hy]\nend\n\nlemma submodule.closure_smul_self_eq (s : submodule R M) :\n  (\u03bb p : R \u00d7 M, p.1 \u2022 p.2) '' ((set.univ : set R) \u00d7\u02e2 closure (s : set M))\n  = closure (s : set M) :=\nset.subset.antisymm s.closure_smul_self_subset\n  (\u03bb x hx, \u27e8\u27e81, x\u27e9, \u27e8set.mem_univ _, hx\u27e9, one_smul R _\u27e9)\n\nvariables [has_continuous_add M]\n\n/-- The (topological-space) closure of a submodule of a topological `R`-module `M` is itself\na submodule. -/\ndef submodule.topological_closure (s : submodule R M) : submodule R M :=\n{ carrier := closure (s : set M),\n  smul_mem' := \u03bb c x hx, s.closure_smul_self_subset \u27e8\u27e8c, x\u27e9, \u27e8set.mem_univ _, hx\u27e9, rfl\u27e9,\n  ..s.to_add_submonoid.topological_closure }\n\n@[simp] lemma submodule.topological_closure_coe (s : submodule R M) :\n  (s.topological_closure : set M) = closure (s : set M) :=\nrfl\n\ninstance submodule.topological_closure_has_continuous_smul (s : submodule R M) :\n  has_continuous_smul R (s.topological_closure) :=\n{ continuous_smul :=\n  begin\n    apply continuous_induced_rng,\n    change continuous (\u03bb p : R \u00d7 s.topological_closure, p.1 \u2022 (p.2 : M)),\n    continuity,\n  end,\n  ..s.to_add_submonoid.topological_closure_has_continuous_add }\n\nlemma submodule.submodule_topological_closure (s : submodule R M) :\n  s \u2264 s.topological_closure :=\nsubset_closure\n\nlemma submodule.is_closed_topological_closure (s : submodule R M) :\n  is_closed (s.topological_closure : set M) :=\nby convert is_closed_closure\n\nlemma submodule.topological_closure_minimal\n  (s : submodule R M) {t : submodule R M} (h : s \u2264 t) (ht : is_closed (t : set M)) :\n  s.topological_closure \u2264 t :=\nclosure_minimal h ht\n\nlemma submodule.topological_closure_mono {s : submodule R M} {t : submodule R M} (h : s \u2264 t) :\n  s.topological_closure \u2264 t.topological_closure :=\ns.topological_closure_minimal (h.trans t.submodule_topological_closure)\n  t.is_closed_topological_closure\n\nend closure\n\n/-- Continuous linear maps between modules. We only put the type classes that are necessary for the\ndefinition, although in applications `M` and `M\u2082` will be topological modules over the topological\nring `R`. -/\nstructure continuous_linear_map\n  {R : Type*} {S : Type*} [semiring R] [semiring S] (\u03c3 : R \u2192+* S)\n  (M : Type*) [topological_space M] [add_comm_monoid M]\n  (M\u2082 : Type*) [topological_space M\u2082] [add_comm_monoid M\u2082]\n  [module R M] [module S M\u2082]\n  extends M \u2192\u209b\u2097[\u03c3] M\u2082 :=\n(cont : continuous to_fun . tactic.interactive.continuity')\n\nnotation M ` \u2192SL[`:25 \u03c3 `] ` M\u2082 := continuous_linear_map \u03c3 M M\u2082\nnotation M ` \u2192L[`:25 R `] ` M\u2082 := continuous_linear_map (ring_hom.id R) M M\u2082\nnotation M ` \u2192L\u22c6[`:25 R `] ` M\u2082 := continuous_linear_map (star_ring_end R) M M\u2082\n\n/-- Continuous linear equivalences between modules. We only put the type classes that are necessary\nfor the definition, although in applications `M` and `M\u2082` will be topological modules over the\ntopological ring `R`. -/\n@[nolint has_inhabited_instance]\nstructure continuous_linear_equiv\n  {R : Type*} {S : Type*} [semiring R] [semiring S] (\u03c3 : R \u2192+* S)\n  {\u03c3' : S \u2192+* R} [ring_hom_inv_pair \u03c3 \u03c3'] [ring_hom_inv_pair \u03c3' \u03c3]\n  (M : Type*) [topological_space M] [add_comm_monoid M]\n  (M\u2082 : Type*) [topological_space M\u2082] [add_comm_monoid M\u2082]\n  [module R M] [module S M\u2082]\n  extends M \u2243\u209b\u2097[\u03c3] M\u2082 :=\n(continuous_to_fun  : continuous to_fun . tactic.interactive.continuity')\n(continuous_inv_fun : continuous inv_fun . tactic.interactive.continuity')\n\nnotation M ` \u2243SL[`:50 \u03c3 `] ` M\u2082 := continuous_linear_equiv \u03c3 M M\u2082\nnotation M ` \u2243L[`:50 R `] ` M\u2082 := continuous_linear_equiv (ring_hom.id R) M M\u2082\nnotation M ` \u2243L\u22c6[`:50 R `] ` M\u2082 := continuous_linear_equiv (star_ring_end R) M M\u2082\n\nsection pointwise_limits\n\nvariables\n{M\u2081 M\u2082 \u03b1 R S : Type*}\n[topological_space M\u2082] [t2_space M\u2082] [semiring R] [semiring S]\n[add_comm_monoid M\u2081] [add_comm_monoid M\u2082] [module R M\u2081] [module S M\u2082]\n[topological_space S] [has_continuous_smul S M\u2082]\n\nsection\n\nvariables (M\u2081 M\u2082) (\u03c3 : R \u2192+* S)\n\nlemma is_closed_set_of_map_smul : is_closed {f : M\u2081 \u2192 M\u2082 | \u2200 c x, f (c \u2022 x) = \u03c3 c \u2022 f x} :=\nbegin\n  simp only [set.set_of_forall],\n  exact is_closed_Inter (\u03bb c, is_closed_Inter (\u03bb x, is_closed_eq (continuous_apply _)\n    (continuous_const.smul (continuous_apply _))))\nend\n\nend\n\nvariables [has_continuous_add M\u2082] {\u03c3 : R \u2192+* S} {l : filter \u03b1}\n\n/-- Constructs a bundled linear map from a function and a proof that this function belongs to the\nclosure of the set of linear maps. -/\n@[simps { fully_applied := ff }] def linear_map_of_mem_closure_range_coe (f : M\u2081 \u2192 M\u2082)\n  (hf : f \u2208 closure (set.range (coe_fn : (M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) \u2192 (M\u2081 \u2192 M\u2082)))) :\n  M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082 :=\n{ to_fun := f,\n  map_smul' := (is_closed_set_of_map_smul M\u2081 M\u2082 \u03c3).closure_subset_iff.2\n    (set.range_subset_iff.2 linear_map.map_smul\u209b\u2097) hf,\n  .. add_monoid_hom_of_mem_closure_range_coe f hf }\n\n/-- Construct a bundled linear map from a pointwise limit of linear maps -/\n@[simps { fully_applied := ff }]\ndef linear_map_of_tendsto (f : M\u2081 \u2192 M\u2082) (g : \u03b1 \u2192 M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) [l.ne_bot]\n  (h : tendsto (\u03bb a x, g a x) l (\ud835\udcdd f)) : M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082 :=\nlinear_map_of_mem_closure_range_coe f $ mem_closure_of_tendsto h $\n  eventually_of_forall $ \u03bb a, set.mem_range_self _\n\nvariables (M\u2081 M\u2082 \u03c3)\n\nlemma linear_map.is_closed_range_coe :\n  is_closed (set.range (coe_fn : (M\u2081 \u2192\u209b\u2097[\u03c3] M\u2082) \u2192 (M\u2081 \u2192 M\u2082))) :=\nis_closed_of_closure_subset $ \u03bb f hf, \u27e8linear_map_of_mem_closure_range_coe f hf, rfl\u27e9\n\nend pointwise_limits\n\nnamespace continuous_linear_map\n\nsection semiring\n/-!\n### Properties that hold for non-necessarily commutative semirings.\n-/\n\nvariables\n{R\u2081 : Type*} {R\u2082 : Type*} {R\u2083 : Type*} [semiring R\u2081] [semiring R\u2082] [semiring R\u2083]\n{\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083}\n{M\u2081 : Type*} [topological_space M\u2081] [add_comm_monoid M\u2081]\n{M'\u2081 : Type*} [topological_space M'\u2081] [add_comm_monoid M'\u2081]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083]\n{M\u2084 : Type*} [topological_space M\u2084] [add_comm_monoid M\u2084]\n[module R\u2081 M\u2081] [module R\u2081 M'\u2081] [module R\u2082 M\u2082] [module R\u2083 M\u2083]\n\n/-- Coerce continuous linear maps to linear maps. -/\ninstance : has_coe (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) := \u27e8to_linear_map\u27e9\n\n-- make the coercion the preferred form\n@[simp] lemma to_linear_map_eq_coe (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : f.to_linear_map = f := rfl\n\ntheorem coe_injective : function.injective (coe : (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u2192 (M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)) :=\nby { intros f g H, cases f, cases g, congr' }\n\ninstance : add_monoid_hom_class (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) M\u2081 M\u2082 :=\n{ coe := \u03bb f, f.to_fun,\n  coe_injective' := \u03bb f g h, coe_injective (fun_like.coe_injective h),\n  map_add := \u03bb f, map_add f.to_linear_map,\n  map_zero := \u03bb f, linear_map.map_zero f }\n\n/-- Coerce continuous linear maps to functions. -/\n-- see Note [function coercion]\ninstance to_fun : has_coe_to_fun (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (\u03bb _, M\u2081 \u2192 M\u2082) := \u27e8\u03bb f, f.to_fun\u27e9\n\n@[simp] lemma coe_mk (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) (h) : (mk f h : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f := rfl\n@[simp] lemma coe_mk' (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) (h) : (mk f h : M\u2081 \u2192 M\u2082) = f := rfl\n\n@[continuity]\nprotected lemma continuous (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : continuous f := f.2\n\n@[simp, norm_cast] lemma coe_inj {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} :\n  (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = g \u2194 f = g :=\ncoe_injective.eq_iff\n\ntheorem coe_fn_injective : @function.injective (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192 M\u2082) coe_fn :=\nfun_like.coe_injective\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef simps.apply (h : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082 := h\n\n/-- See Note [custom simps projection]. -/\ndef simps.coe (h : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082 := h\n\ninitialize_simps_projections continuous_linear_map\n  (to_linear_map_to_fun \u2192 apply, to_linear_map \u2192 coe)\n\n@[ext] theorem ext {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : \u2200 x, f x = g x) : f = g :=\nfun_like.ext f g h\n\ntheorem ext_iff {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} : f = g \u2194 \u2200 x, f x = g x :=\nfun_like.ext_iff\n\nvariables (f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2081) (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (x y z : M\u2081) (f\u2097 : M\u2081 \u2192L[R\u2081] M'\u2081)\n\n-- make some straightforward lemmas available to `simp`.\nprotected lemma map_zero : f (0 : M\u2081) = 0 := map_zero f\nprotected lemma map_add  : f (x + y) = f x + f y := map_add f x y\n@[simp] lemma map_smul\u209b\u2097 : f (c \u2022 x) = (\u03c3\u2081\u2082 c) \u2022 f x := (to_linear_map _).map_smul\u209b\u2097 _ _\n\n@[simp] lemma map_smul [module R\u2081 M\u2082] (f : M\u2081 \u2192L[R\u2081] M\u2082)(c : R\u2081) (x : M\u2081) : f (c \u2022 x) = c \u2022 f x :=\nby simp only [ring_hom.id_apply, map_smul\u209b\u2097]\n\n@[simp, priority 900]\nlemma map_smul_of_tower {R S : Type*} [semiring S] [has_scalar R M\u2081]\n  [module S M\u2081] [has_scalar R M\u2082] [module S M\u2082]\n  [linear_map.compatible_smul M\u2081 M\u2082 R S] (f : M\u2081 \u2192L[S] M\u2082) (c : R) (x : M\u2081) :\n  f (c \u2022 x) = c \u2022 f x :=\nlinear_map.compatible_smul.map_smul f c x\n\nprotected lemma map_sum {\u03b9 : Type*} (s : finset \u03b9) (g : \u03b9 \u2192 M\u2081) :\n  f (\u2211 i in s, g i) = \u2211 i in s, f (g i) := f.to_linear_map.map_sum\n\n@[simp, norm_cast] lemma coe_coe : ((f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) : (M\u2081 \u2192 M\u2082)) = (f : M\u2081 \u2192 M\u2082) := rfl\n\n@[ext] theorem ext_ring [topological_space R\u2081] {f g : R\u2081 \u2192L[R\u2081] M\u2081} (h : f 1 = g 1) : f = g :=\ncoe_inj.1 $ linear_map.ext_ring h\n\ntheorem ext_ring_iff [topological_space R\u2081] {f g : R\u2081 \u2192L[R\u2081] M\u2081} : f = g \u2194 f 1 = g 1 :=\n\u27e8\u03bb h, h \u25b8 rfl, ext_ring\u27e9\n\n/-- If two continuous linear maps are equal on a set `s`, then they are equal on the closure\nof the `submodule.span` of this set. -/\nlemma eq_on_closure_span [t2_space M\u2082] {s : set M\u2081} {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : set.eq_on f g s) :\n  set.eq_on f g (closure (submodule.span R\u2081 s : set M\u2081)) :=\n(linear_map.eq_on_span' h).closure f.continuous g.continuous\n\n/-- If the submodule generated by a set `s` is dense in the ambient module, then two continuous\nlinear maps equal on `s` are equal. -/\nlemma ext_on [t2_space M\u2082] {s : set M\u2081} (hs : dense (submodule.span R\u2081 s : set M\u2081))\n  {f g : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (h : set.eq_on f g s) :\n  f = g :=\next $ \u03bb x, eq_on_closure_span h (hs x)\n\n/-- Under a continuous linear map, the image of the `topological_closure` of a submodule is\ncontained in the `topological_closure` of its image. -/\nlemma _root_.submodule.topological_closure_map [ring_hom_surjective \u03c3\u2081\u2082] [topological_space R\u2081]\n  [topological_space R\u2082] [has_continuous_smul R\u2081 M\u2081] [has_continuous_add M\u2081]\n  [has_continuous_smul R\u2082 M\u2082] [has_continuous_add M\u2082] (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (s : submodule R\u2081 M\u2081) :\n  (s.topological_closure.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082))\n  \u2264 (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topological_closure :=\nimage_closure_subset_closure_image f.continuous\n\n/-- Under a dense continuous linear map, a submodule whose `topological_closure` is `\u22a4` is sent to\nanother such submodule.  That is, the image of a dense set under a map with dense range is dense.\n-/\nlemma _root_.dense_range.topological_closure_map_submodule [ring_hom_surjective \u03c3\u2081\u2082]\n  [topological_space R\u2081] [topological_space R\u2082] [has_continuous_smul R\u2081 M\u2081] [has_continuous_add M\u2081]\n  [has_continuous_smul R\u2082 M\u2082] [has_continuous_add M\u2082] {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} (hf' : dense_range f)\n  {s : submodule R\u2081 M\u2081} (hs : s.topological_closure = \u22a4) :\n  (s.map (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)).topological_closure = \u22a4 :=\nbegin\n  rw set_like.ext'_iff at hs \u22a2,\n  simp only [submodule.topological_closure_coe, submodule.top_coe, \u2190 dense_iff_closure_eq] at hs \u22a2,\n  exact hf'.dense_image f.continuous hs\nend\n\n/-- The continuous map that is constantly zero. -/\ninstance: has_zero (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) := \u27e8\u27e80, continuous_zero\u27e9\u27e9\ninstance : inhabited (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) := \u27e80\u27e9\n\n@[simp] lemma default_def : (default : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = 0 := rfl\n@[simp] lemma zero_apply : (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) x = 0 := rfl\n@[simp, norm_cast] lemma coe_zero : ((0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = 0 := rfl\n/- no simp attribute on the next line as simp does not always simplify `0 x` to `0`\nwhen `0` is the zero function, while it does for the zero continuous linear map,\nand this is the most important property we care about. -/\n@[norm_cast] lemma coe_zero' : ((0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082) = 0 := rfl\n\ninstance unique_of_left [subsingleton M\u2081] : unique (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\ncoe_injective.unique\n\ninstance unique_of_right [subsingleton M\u2082] : unique (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\ncoe_injective.unique\n\nsection\n\nvariables (R\u2081 M\u2081)\n\n/-- the identity map as a continuous linear map. -/\ndef id : M\u2081 \u2192L[R\u2081] M\u2081 :=\n\u27e8linear_map.id, continuous_id\u27e9\n\nend\n\ninstance : has_one (M\u2081 \u2192L[R\u2081] M\u2081) := \u27e8id R\u2081 M\u2081\u27e9\n\nlemma one_def : (1 : M\u2081 \u2192L[R\u2081] M\u2081) = id R\u2081 M\u2081 := rfl\nlemma id_apply : id R\u2081 M\u2081 x = x := rfl\n@[simp, norm_cast] lemma coe_id : (id R\u2081 M\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2081) = linear_map.id := rfl\n@[simp, norm_cast] lemma coe_id' : (id R\u2081 M\u2081 : M\u2081 \u2192 M\u2081) = _root_.id := rfl\n\n@[simp, norm_cast] lemma coe_eq_id {f : M\u2081 \u2192L[R\u2081] M\u2081} :\n  (f : M\u2081 \u2192\u2097[R\u2081] M\u2081) = linear_map.id \u2194 f = id _ _ :=\nby rw [\u2190 coe_id, coe_inj]\n\n@[simp] lemma one_apply : (1 : M\u2081 \u2192L[R\u2081] M\u2081) x = x := rfl\n\nsection add\nvariables [has_continuous_add M\u2082]\n\ninstance : has_add (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n\u27e8\u03bb f g, \u27e8f + g, f.2.add g.2\u27e9\u27e9\n\nlemma continuous_nsmul (n : \u2115) : continuous (\u03bb (x : M\u2082), n \u2022 x) :=\nbegin\n  induction n with n ih,\n  { simp [continuous_const] },\n  { simp [nat.succ_eq_add_one, add_smul], exact ih.add continuous_id }\nend\n\n@[continuity]\nlemma continuous.nsmul {\u03b1 : Type*} [topological_space \u03b1] {n : \u2115} {f : \u03b1 \u2192 M\u2082} (hf : continuous f) :\n  continuous (\u03bb (x : \u03b1), n \u2022 (f x)) :=\n(continuous_nsmul n).comp hf\n\n@[simp] lemma add_apply : (f + g) x = f x + g x := rfl\n@[simp, norm_cast] lemma coe_add : (((f + g) : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f + g := rfl\n@[norm_cast] lemma coe_add' : (((f + g) : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082) = (f : M\u2081 \u2192 M\u2082) + g := rfl\n\ninstance : add_comm_monoid (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n{ zero := (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082),\n  add := (+),\n  zero_add := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_zero := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_comm := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  add_assoc := by intros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm],\n  nsmul := \u03bb n f,\n    { to_fun := \u03bb x, n \u2022 (f x),\n      map_add' := by simp,\n      map_smul' := by simp [smul_comm n] },\n  nsmul_zero' := \u03bb f, by { ext, simp },\n  nsmul_succ' := \u03bb n f, by { ext, simp [nat.succ_eq_one_add, add_smul] } }\n\n@[simp, norm_cast] lemma coe_sum {\u03b9 : Type*} (t : finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  \u2191(\u2211 d in t, f d) = (\u2211 d in t, f d : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\n(add_monoid_hom.mk (coe : (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) \u2192 (M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082)) rfl (\u03bb _ _, rfl)).map_sum _ _\n\n@[simp, norm_cast] lemma coe_sum' {\u03b9 : Type*} (t : finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  \u21d1(\u2211 d in t, f d) = \u2211 d in t, f d :=\nby simp only [\u2190 coe_coe, coe_sum, linear_map.coe_fn_sum]\n\nlemma sum_apply {\u03b9 : Type*} (t : finset \u03b9) (f : \u03b9 \u2192 M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) :\n  (\u2211 d in t, f d) b = \u2211 d in t, f d b :=\nby simp only [coe_sum', finset.sum_apply]\n\nend add\n\nvariables {\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083} [ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083]\n\n/-- Composition of bounded linear maps. -/\ndef comp (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192SL[\u03c3\u2081\u2083] M\u2083 :=\n\u27e8(g : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083).comp \u2191f, g.2.comp f.2\u27e9\n\ninfixr ` \u2218L `:80 := @continuous_linear_map.comp _ _ _ _ _ _ (ring_hom.id _) (ring_hom.id _)\n  _ _ _ _ _ _ _ _ _ _ _ _ (ring_hom.id _) ring_hom_comp_triple.ids\n\n@[simp, norm_cast] lemma coe_comp :\n  ((h.comp f) : (M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083)) = (h : M\u2082 \u2192\u209b\u2097[\u03c3\u2082\u2083] M\u2083).comp (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) := rfl\n\ninclude \u03c3\u2081\u2083\n@[simp, norm_cast] lemma coe_comp' : ((h.comp f) : (M\u2081 \u2192 M\u2083)) = (h : M\u2082 \u2192 M\u2083) \u2218 f := rfl\n\nlemma comp_apply (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : (g.comp f) x = g (f x) := rfl\nomit \u03c3\u2081\u2083\n\n@[simp] theorem comp_id : f.comp (id R\u2081 M\u2081) = f :=\next $ \u03bb x, rfl\n\n@[simp] theorem id_comp : (id R\u2082 M\u2082).comp f = f :=\next $ \u03bb x, rfl\n\ninclude \u03c3\u2081\u2083\n@[simp] theorem comp_zero (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) : g.comp (0 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = 0 :=\nby { ext, simp }\n\n@[simp] theorem zero_comp : (0 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp f = 0 :=\nby { ext, simp }\n\n@[simp] lemma comp_add [has_continuous_add M\u2082] [has_continuous_add M\u2083]\n  (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f\u2081 f\u2082 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  g.comp (f\u2081 + f\u2082) = g.comp f\u2081 + g.comp f\u2082 :=\nby { ext, simp }\n\n@[simp] lemma add_comp [has_continuous_add M\u2083]\n  (g\u2081 g\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  (g\u2081 + g\u2082).comp f = g\u2081.comp f + g\u2082.comp f :=\nby { ext, simp }\nomit \u03c3\u2081\u2083\n\ntheorem comp_assoc {R\u2084 : Type*} [semiring R\u2084] [module R\u2084 M\u2084] {\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084} {\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084}\n  {\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084} [ring_hom_comp_triple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084] [ring_hom_comp_triple \u03c3\u2082\u2083 \u03c3\u2083\u2084 \u03c3\u2082\u2084]\n  [ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2084 \u03c3\u2081\u2084] (h : M\u2083 \u2192SL[\u03c3\u2083\u2084] M\u2084) (g : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083)\n  (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  (h.comp g).comp f = h.comp (g.comp f) :=\nrfl\n\ninstance : has_mul (M\u2081 \u2192L[R\u2081] M\u2081) := \u27e8comp\u27e9\n\nlemma mul_def (f g : M\u2081 \u2192L[R\u2081] M\u2081) : f * g = f.comp g := rfl\n\n@[simp] lemma coe_mul (f g : M\u2081 \u2192L[R\u2081] M\u2081) : \u21d1(f * g) = f \u2218 g := rfl\n\nlemma mul_apply (f g : M\u2081 \u2192L[R\u2081] M\u2081) (x : M\u2081) : (f * g) x = f (g x) := rfl\n\n/-- The cartesian product of two bounded linear maps, as a bounded linear map. -/\nprotected def prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) :\n  M\u2081 \u2192L[R\u2081] (M\u2082 \u00d7 M\u2083) :=\n\u27e8(f\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2082).prod f\u2082, f\u2081.2.prod_mk f\u2082.2\u27e9\n\n@[simp, norm_cast] lemma coe_prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082)\n  (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) :\n  (f\u2081.prod f\u2082 : M\u2081 \u2192\u2097[R\u2081] M\u2082 \u00d7 M\u2083) = linear_map.prod f\u2081 f\u2082 :=\nrfl\n\n@[simp, norm_cast] lemma prod_apply [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082)\n  (f\u2082 : M\u2081 \u2192L[R\u2081] M\u2083) (x : M\u2081) :\n  f\u2081.prod f\u2082 x = (f\u2081 x, f\u2082 x) :=\nrfl\n\nsection\n\nvariables (R\u2081 M\u2081 M\u2082)\n\n/-- The left injection into a product is a continuous linear map. -/\ndef inl [module R\u2081 M\u2082] : M\u2081 \u2192L[R\u2081] M\u2081 \u00d7 M\u2082 := (id R\u2081 M\u2081).prod 0\n\n/-- The right injection into a product is a continuous linear map. -/\ndef inr [module R\u2081 M\u2082] : M\u2082 \u2192L[R\u2081] M\u2081 \u00d7 M\u2082 := (0 : M\u2082 \u2192L[R\u2081] M\u2081).prod (id R\u2081 M\u2082)\n\nend\n\n@[simp] lemma inl_apply [module R\u2081 M\u2082] (x : M\u2081) : inl R\u2081 M\u2081 M\u2082 x = (x, 0) := rfl\n@[simp] lemma inr_apply [module R\u2081 M\u2082] (x : M\u2082) : inr R\u2081 M\u2081 M\u2082 x = (0, x) := rfl\n\n@[simp, norm_cast] lemma coe_inl [module R\u2081 M\u2082] :\n  (inl R\u2081 M\u2081 M\u2082 : M\u2081 \u2192\u2097[R\u2081] M\u2081 \u00d7 M\u2082) = linear_map.inl R\u2081 M\u2081 M\u2082 := rfl\n@[simp, norm_cast] lemma coe_inr [module R\u2081 M\u2082] :\n  (inr R\u2081 M\u2081 M\u2082 : M\u2082 \u2192\u2097[R\u2081] M\u2081 \u00d7 M\u2082) = linear_map.inr R\u2081 M\u2081 M\u2082 := rfl\n\n/-- Kernel of a continuous linear map. -/\ndef ker (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : submodule R\u2081 M\u2081 := (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).ker\n\n@[norm_cast] lemma ker_coe : (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).ker = f.ker := rfl\n\n@[simp] lemma mem_ker {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} {x} : x \u2208 f.ker \u2194 f x = 0 := linear_map.mem_ker\n\nlemma is_closed_ker [t1_space M\u2082] : is_closed (f.ker : set M\u2081) :=\ncontinuous_iff_is_closed.1 f.cont _ is_closed_singleton\n\n@[simp] lemma apply_ker (x : f.ker) : f x = 0 := mem_ker.1 x.2\n\nlemma is_complete_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']\n  [module R\u2081 M'] [t1_space M\u2082] (f : M' \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  is_complete (f.ker : set M') :=\nf.is_closed_ker.is_complete\n\ninstance complete_space_ker {M' : Type*} [uniform_space M'] [complete_space M'] [add_comm_monoid M']\n  [module R\u2081 M'] [t1_space M\u2082] (f : M' \u2192SL[\u03c3\u2081\u2082] M\u2082) :\n  complete_space f.ker :=\nf.is_closed_ker.complete_space_coe\n\n@[simp] lemma ker_prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n  ker (f.prod g) = ker f \u2293 ker g :=\nlinear_map.ker_prod f g\n\n/-- Range of a continuous linear map. -/\ndef range [ring_hom_surjective \u03c3\u2081\u2082] (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : submodule R\u2082 M\u2082 :=\n(f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).range\n\nlemma range_coe [ring_hom_surjective \u03c3\u2081\u2082] : (f.range : set M\u2082) = set.range f :=\nlinear_map.range_coe _\nlemma mem_range [ring_hom_surjective \u03c3\u2081\u2082] {f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082} {y} : y \u2208 f.range \u2194 \u2203 x, f x = y :=\nlinear_map.mem_range\n\nlemma mem_range_self [ring_hom_surjective \u03c3\u2081\u2082] (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : f x \u2208 f.range :=\nmem_range.2 \u27e8x, rfl\u27e9\n\nlemma range_prod_le [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n  range (f.prod g) \u2264 (range f).prod (range g) :=\n(f : M\u2081 \u2192\u2097[R\u2081] M\u2082).range_prod_le g\n\n/-- Restrict codomain of a continuous linear map. -/\ndef cod_restrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) :\n  M\u2081 \u2192SL[\u03c3\u2081\u2082] p :=\n{ cont := continuous_subtype_mk h f.continuous,\n  to_linear_map := (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).cod_restrict p h}\n\n@[norm_cast] lemma coe_cod_restrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) :\n  (f.cod_restrict p h : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] p) = (f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).cod_restrict p h :=\nrfl\n\n@[simp] lemma coe_cod_restrict_apply (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p)\n  (x) :\n  (f.cod_restrict p h x : M\u2082) = f x :=\nrfl\n\n@[simp] lemma ker_cod_restrict (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (p : submodule R\u2082 M\u2082) (h : \u2200 x, f x \u2208 p) :\n  ker (f.cod_restrict p h) = ker f :=\n(f : M\u2081 \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082).ker_cod_restrict p h\n\n/-- Embedding of a submodule into the ambient space as a continuous linear map. -/\ndef subtype_val (p : submodule R\u2081 M\u2081) : p \u2192L[R\u2081] M\u2081 :=\n{ cont := continuous_subtype_val,\n  to_linear_map := p.subtype }\n\n@[simp, norm_cast] lemma coe_subtype_val (p : submodule R\u2081 M\u2081) :\n  (subtype_val p : p \u2192\u2097[R\u2081] M\u2081) = p.subtype :=\nrfl\n\n@[simp, norm_cast] lemma subtype_val_apply (p : submodule R\u2081 M\u2081) (x : p) :\n  (subtype_val p : p \u2192 M\u2081) x = x :=\nrfl\n\nvariables (R\u2081 M\u2081 M\u2082)\n\n/-- `prod.fst` as a `continuous_linear_map`. -/\ndef fst [module R\u2081 M\u2082] : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2081 :=\n{ cont := continuous_fst, to_linear_map := linear_map.fst R\u2081 M\u2081 M\u2082 }\n\n/-- `prod.snd` as a `continuous_linear_map`. -/\ndef snd [module R\u2081 M\u2082] : M\u2081 \u00d7 M\u2082 \u2192L[R\u2081] M\u2082 :=\n{ cont := continuous_snd, to_linear_map := linear_map.snd R\u2081 M\u2081 M\u2082 }\n\nvariables {R\u2081 M\u2081 M\u2082}\n\n@[simp, norm_cast] lemma coe_fst [module R\u2081 M\u2082] :\n  (fst R\u2081 M\u2081 M\u2082 : M\u2081 \u00d7 M\u2082 \u2192\u2097[R\u2081] M\u2081) = linear_map.fst R\u2081 M\u2081 M\u2082 := rfl\n\n@[simp, norm_cast] lemma coe_fst' [module R\u2081 M\u2082] : (fst R\u2081 M\u2081 M\u2082 : M\u2081 \u00d7 M\u2082 \u2192 M\u2081) = prod.fst := rfl\n\n@[simp, norm_cast] lemma coe_snd [module R\u2081 M\u2082] :\n  (snd R\u2081 M\u2081 M\u2082 : M\u2081 \u00d7 M\u2082 \u2192\u2097[R\u2081] M\u2082) = linear_map.snd R\u2081 M\u2081 M\u2082 := rfl\n\n@[simp, norm_cast] lemma coe_snd' [module R\u2081 M\u2082] : (snd R\u2081 M\u2081 M\u2082 : M\u2081 \u00d7 M\u2082 \u2192 M\u2082) = prod.snd := rfl\n\n@[simp] lemma fst_prod_snd [module R\u2081 M\u2082] : (fst R\u2081 M\u2081 M\u2082).prod (snd R\u2081 M\u2081 M\u2082) = id R\u2081 (M\u2081 \u00d7 M\u2082) :=\n  ext $ \u03bb \u27e8x, y\u27e9, rfl\n\n@[simp] lemma fst_comp_prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n  (fst R\u2081 M\u2082 M\u2083).comp (f.prod g) = f :=\next $ \u03bb x, rfl\n\n@[simp] lemma snd_comp_prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] (f : M\u2081 \u2192L[R\u2081] M\u2082) (g : M\u2081 \u2192L[R\u2081] M\u2083) :\n  (snd R\u2081 M\u2082 M\u2083).comp (f.prod g) = g :=\next $ \u03bb x, rfl\n\n/-- `prod.map` of two continuous linear maps. -/\ndef prod_map [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) :\n  (M\u2081 \u00d7 M\u2083) \u2192L[R\u2081] (M\u2082 \u00d7 M\u2084) :=\n(f\u2081.comp (fst R\u2081 M\u2081 M\u2083)).prod (f\u2082.comp (snd R\u2081 M\u2081 M\u2083))\n\n@[simp, norm_cast] lemma coe_prod_map [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084]\n  (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) :\n  (f\u2081.prod_map f\u2082 : (M\u2081 \u00d7 M\u2083) \u2192\u2097[R\u2081] (M\u2082 \u00d7 M\u2084))\n  = ((f\u2081 : M\u2081 \u2192\u2097[R\u2081] M\u2082).prod_map (f\u2082 : M\u2083 \u2192\u2097[R\u2081] M\u2084)) :=\nrfl\n\n@[simp, norm_cast] lemma coe_prod_map' [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084]\n  (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2082) (f\u2082 : M\u2083 \u2192L[R\u2081] M\u2084) :\n  \u21d1(f\u2081.prod_map f\u2082) = prod.map f\u2081 f\u2082 :=\nrfl\n\n/-- The continuous linear map given by `(x, y) \u21a6 f\u2081 x + f\u2082 y`. -/\ndef coprod [module R\u2081 M\u2082] [module R\u2081 M\u2083] [has_continuous_add M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n  (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) :\n  (M\u2081 \u00d7 M\u2082) \u2192L[R\u2081] M\u2083 :=\n\u27e8linear_map.coprod f\u2081 f\u2082, (f\u2081.cont.comp continuous_fst).add (f\u2082.cont.comp continuous_snd)\u27e9\n\n@[norm_cast, simp] lemma coe_coprod [module R\u2081 M\u2082] [module R\u2081 M\u2083] [has_continuous_add M\u2083]\n  (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083) (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) :\n  (f\u2081.coprod f\u2082 : (M\u2081 \u00d7 M\u2082) \u2192\u2097[R\u2081] M\u2083) = linear_map.coprod f\u2081 f\u2082 :=\nrfl\n\n@[simp] lemma coprod_apply [module R\u2081 M\u2082] [module R\u2081 M\u2083] [has_continuous_add M\u2083]\n  (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083) (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) (x) :\n  f\u2081.coprod f\u2082 x = f\u2081 x.1 + f\u2082 x.2 := rfl\n\nlemma range_coprod [module R\u2081 M\u2082] [module R\u2081 M\u2083] [has_continuous_add M\u2083] (f\u2081 : M\u2081 \u2192L[R\u2081] M\u2083)\n  (f\u2082 : M\u2082 \u2192L[R\u2081] M\u2083) :\n  (f\u2081.coprod f\u2082).range = f\u2081.range \u2294 f\u2082.range :=\nlinear_map.range_coprod _ _\n\nsection\n\nvariables {R S : Type*} [semiring R] [semiring S] [module R M\u2081] [module R M\u2082] [module R S]\n  [module S M\u2082] [is_scalar_tower R S M\u2082] [topological_space S] [has_continuous_smul S M\u2082]\n\n/-- The linear map `\u03bb x, c x \u2022 f`.  Associates to a scalar-valued linear map and an element of\n`M\u2082` the `M\u2082`-valued linear map obtained by multiplying the two (a.k.a. tensoring by `M\u2082`).\nSee also `continuous_linear_map.smul_right\u2097` and `continuous_linear_map.smul_rightL`. -/\ndef smul_right (c : M\u2081 \u2192L[R] S) (f : M\u2082) : M\u2081 \u2192L[R] M\u2082 :=\n{ cont := c.2.smul continuous_const,\n  ..c.to_linear_map.smul_right f }\n\n@[simp]\nlemma smul_right_apply {c : M\u2081 \u2192L[R] S} {f : M\u2082} {x : M\u2081} :\n  (smul_right c f : M\u2081 \u2192 M\u2082) x = c x \u2022 f :=\nrfl\n\nend\n\nsection pointwise\nopen_locale pointwise\n\n@[simp] lemma image_smul_set\u209b\u2097 (c : R\u2081) (s : set M\u2081) :\n  f '' (c \u2022 s) = (\u03c3\u2081\u2082 c) \u2022 f '' s :=\nf.to_linear_map.image_smul_set\u209b\u2097 c s\n\nlemma image_smul_set (c : R\u2081) (s : set M\u2081) :\n  f\u2097 '' (c \u2022 s) = c \u2022 f\u2097 '' s :=\nf\u2097.to_linear_map.image_smul_set c s\n\nlemma preimage_smul_set\u209b\u2097 {c : R\u2081} (hc : is_unit c) (s : set M\u2082) :\n  f \u207b\u00b9' (\u03c3\u2081\u2082 c \u2022 s) = c \u2022 f \u207b\u00b9' s :=\nf.to_linear_map.preimage_smul_set\u209b\u2097 hc s\n\nlemma preimage_smul_set {c : R\u2081} (hc : is_unit c) (s : set M'\u2081) :\n  f\u2097 \u207b\u00b9' (c \u2022 s) = c \u2022 f\u2097 \u207b\u00b9' s :=\nf\u2097.to_linear_map.preimage_smul_set hc s\n\nend pointwise\n\nvariables [module R\u2081 M\u2082] [topological_space R\u2081] [has_continuous_smul R\u2081 M\u2082]\n\n@[simp]\nlemma smul_right_one_one (c : R\u2081 \u2192L[R\u2081] M\u2082) : smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) (c 1) = c :=\nby ext; simp [\u2190 continuous_linear_map.map_smul_of_tower]\n\n@[simp]\nlemma smul_right_one_eq_iff {f f' : M\u2082} :\n  smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) f = smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) f' \u2194 f = f' :=\nby simp only [ext_ring_iff, smul_right_apply, one_apply, one_smul]\n\nlemma smul_right_comp [has_continuous_mul R\u2081] {x : M\u2082} {c : R\u2081} :\n  (smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) x).comp (smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) c) =\n    smul_right (1 : R\u2081 \u2192L[R\u2081] R\u2081) (c \u2022 x) :=\nby { ext, simp [mul_smul] }\n\nend semiring\n\nsection pi\nvariables\n  {R : Type*} [semiring R]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082] [module R M\u2082]\n  {\u03b9 : Type*} {\u03c6 : \u03b9 \u2192 Type*} [\u2200i, topological_space (\u03c6 i)] [\u2200i, add_comm_monoid (\u03c6 i)]\n  [\u2200i, module R (\u03c6 i)]\n\n/-- `pi` construction for continuous linear functions. From a family of continuous linear functions\nit produces a continuous linear function into a family of topological modules. -/\ndef pi (f : \u03a0i, M \u2192L[R] \u03c6 i) : M \u2192L[R] (\u03a0i, \u03c6 i) :=\n\u27e8linear_map.pi (\u03bb i, f i), continuous_pi (\u03bb i, (f i).continuous)\u27e9\n\n@[simp] lemma coe_pi' (f : \u03a0 i, M \u2192L[R] \u03c6 i) : \u21d1(pi f) = \u03bb c i, f i c := rfl\n@[simp] lemma coe_pi (f : \u03a0 i, M \u2192L[R] \u03c6 i) :\n  (pi f : M \u2192\u2097[R] \u03a0 i, \u03c6 i) = linear_map.pi (\u03bb i, f i) :=\nrfl\n\nlemma pi_apply (f : \u03a0i, M \u2192L[R] \u03c6 i) (c : M) (i : \u03b9) :\n  pi f c i = f i c := rfl\n\nlemma pi_eq_zero (f : \u03a0i, M \u2192L[R] \u03c6 i) : pi f = 0 \u2194 (\u2200i, f i = 0) :=\nby { simp only [ext_iff, pi_apply, function.funext_iff], exact forall_swap }\n\nlemma pi_zero : pi (\u03bbi, 0 : \u03a0i, M \u2192L[R] \u03c6 i) = 0 := ext $ \u03bb _, rfl\n\nlemma pi_comp (f : \u03a0i, M \u2192L[R] \u03c6 i) (g : M\u2082 \u2192L[R] M) : (pi f).comp g = pi (\u03bbi, (f i).comp g) := rfl\n\n/-- The projections from a family of topological modules are continuous linear maps. -/\ndef proj (i : \u03b9) : (\u03a0i, \u03c6 i) \u2192L[R] \u03c6 i :=\n\u27e8linear_map.proj i, continuous_apply _\u27e9\n\n@[simp] lemma proj_apply (i : \u03b9) (b : \u03a0i, \u03c6 i) : (proj i : (\u03a0i, \u03c6 i) \u2192L[R] \u03c6 i) b = b i := rfl\n\nlemma proj_pi (f : \u03a0i, M\u2082 \u2192L[R] \u03c6 i) (i : \u03b9) : (proj i).comp (pi f) = f i :=\next $ assume c, rfl\n\nlemma infi_ker_proj : (\u2a05i, ker (proj i) : submodule R (\u03a0i, \u03c6 i)) = \u22a5 :=\nlinear_map.infi_ker_proj\n\nvariables (R \u03c6)\n\n/-- If `I` and `J` are complementary index sets, the product of the kernels of the `J`th projections\nof `\u03c6` is linearly equivalent to the product over `I`. -/\ndef infi_ker_proj_equiv {I J : set \u03b9} [decidable_pred (\u03bbi, i \u2208 I)]\n  (hd : disjoint I J) (hu : set.univ \u2286 I \u222a J) :\n  (\u2a05i \u2208 J, ker (proj i) : submodule R (\u03a0i, \u03c6 i)) \u2243L[R] (\u03a0i:I, \u03c6 i) :=\n\u27e8 linear_map.infi_ker_proj_equiv R \u03c6 hd hu,\n  continuous_pi (\u03bb i, begin\n    have := @continuous_subtype_coe _ _ (\u03bb x, x \u2208 (\u2a05i \u2208 J, ker (proj i) : submodule R (\u03a0i, \u03c6 i))),\n    have := continuous.comp (by exact continuous_apply i) this,\n    exact this\n  end),\n  continuous_subtype_mk _ (continuous_pi (\u03bb i, begin\n    dsimp, split_ifs; [apply continuous_apply, exact continuous_zero]\n  end)) \u27e9\n\nend pi\n\nsection ring\n\nvariables\n{R : Type*} [ring R] {R\u2082 : Type*} [ring R\u2082]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_group M\u2083]\n{M\u2084 : Type*} [topological_space M\u2084] [add_comm_group M\u2084]\n[module R M] [module R\u2082 M\u2082]\n{\u03c3\u2081\u2082 : R \u2192+* R\u2082}\n\nsection\nvariables (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y : M)\n\nprotected lemma map_neg : f (-x) = - (f x) := (to_linear_map _).map_neg _\nprotected lemma map_sub : f (x - y) = f x - f y := (to_linear_map _).map_sub _ _\n@[simp] lemma sub_apply' (x : M) : ((f : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) - g) x = f x - g x := rfl\nend\n\nsection\nvariables [module R M\u2082] [module R M\u2083] [module R M\u2084]\nvariables (c : R) (f g : M \u2192L[R] M\u2082) (h : M\u2082 \u2192L[R] M\u2083) (x y z : M)\n\nlemma range_prod_eq  {f : M \u2192L[R] M\u2082} {g : M \u2192L[R] M\u2083} (h : ker f \u2294 ker g = \u22a4) :\n  range (f.prod g) = (range f).prod (range g) :=\nlinear_map.range_prod_eq h\n\nlemma ker_prod_ker_le_ker_coprod [has_continuous_add M\u2083]\n  (f : M \u2192L[R] M\u2083) (g : M\u2082 \u2192L[R] M\u2083) :\n  (ker f).prod (ker g) \u2264 ker (f.coprod g) :=\nlinear_map.ker_prod_ker_le_ker_coprod f.to_linear_map g.to_linear_map\n\nlemma ker_coprod_of_disjoint_range [has_continuous_add M\u2083]\n  (f : M \u2192L[R] M\u2083) (g : M\u2082 \u2192L[R] M\u2083) (hd : disjoint f.range g.range) :\n  ker (f.coprod g) = (ker f).prod (ker g) :=\nlinear_map.ker_coprod_of_disjoint_range f.to_linear_map g.to_linear_map hd\nend\n\nsection\nvariables [topological_add_group M\u2082]\nvariables (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y : M)\n\ninstance : has_neg (M \u2192SL[\u03c3\u2081\u2082] M\u2082) := \u27e8\u03bb f, \u27e8-f, f.2.neg\u27e9\u27e9\n\n@[simp] lemma neg_apply : (-f) x = - (f x) := rfl\n\n@[simp, norm_cast] lemma coe_neg :\n  (((-f) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = -(f : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) :=\nrfl\n@[norm_cast] lemma coe_neg' : (((-f) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192 M\u2082) = -(f : M \u2192 M\u2082) := rfl\n\ninstance : has_sub (M \u2192SL[\u03c3\u2081\u2082] M\u2082) := \u27e8\u03bb f g, \u27e8f - g, f.2.sub g.2\u27e9\u27e9\n\nlemma continuous_zsmul : \u2200 (n : \u2124), continuous (\u03bb (x : M\u2082), n \u2022 x)\n| (n : \u2115) := by { simp only [coe_nat_zsmul], exact continuous_nsmul _ }\n| -[1+ n] := by { simp only [zsmul_neg_succ_of_nat], exact (continuous_nsmul _).neg }\n\n@[continuity]\nlemma continuous.zsmul {\u03b1 : Type*} [topological_space \u03b1] {n : \u2124} {f : \u03b1 \u2192 M\u2082} (hf : continuous f) :\n  continuous (\u03bb (x : \u03b1), n \u2022 (f x)) :=\n(continuous_zsmul n).comp hf\n\ninstance : add_comm_group (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\nby refine\n{ zero := 0,\n  add := (+),\n  neg := has_neg.neg,\n  sub := has_sub.sub,\n  sub_eq_add_neg := _,\n  nsmul := \u03bb n f,\n    { to_fun := \u03bb x, n \u2022 (f x),\n      map_add' := by simp,\n      map_smul' := by simp [smul_comm n] },\n  zsmul := \u03bb n f,\n    { to_fun := \u03bb x, n \u2022 (f x),\n      map_add' := by simp,\n      map_smul' := by simp [smul_comm n] },\n  zsmul_zero' := \u03bb f, by { ext, simp },\n  zsmul_succ' := \u03bb n f, by { ext, simp [add_smul, add_comm] },\n  zsmul_neg' := \u03bb n f, by { ext, simp [nat.succ_eq_add_one, add_smul] },\n  .. continuous_linear_map.add_comm_monoid, .. };\nintros; ext; apply_rules [zero_add, add_assoc, add_zero, add_left_neg, add_comm, sub_eq_add_neg]\n\nlemma sub_apply (x : M) : (f - g) x = f x - g x := rfl\n@[simp, norm_cast] lemma coe_sub : (((f - g) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = f - g := rfl\n@[simp, norm_cast] lemma coe_sub' : (((f - g) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192 M\u2082) = (f : M \u2192 M\u2082) - g := rfl\n\nend\n\ninstance [topological_add_group M] : ring (M \u2192L[R] M) :=\n{ mul := (*),\n  one := 1,\n  mul_one := \u03bb _, ext $ \u03bb _, rfl,\n  one_mul := \u03bb _, ext $ \u03bb _, rfl,\n  mul_assoc := \u03bb _ _ _, ext $ \u03bb _, rfl,\n  left_distrib := \u03bb f g h, ext $ \u03bb x, map_add f (g x) (h x),\n  right_distrib := \u03bb _ _ _, ext $ \u03bb _, linear_map.add_apply _ _ _,\n  ..continuous_linear_map.add_comm_group }\n\nlemma smul_right_one_pow [topological_space R] [topological_ring R] (c : R) (n : \u2115) :\n  (smul_right (1 : R \u2192L[R] R) c)^n = smul_right (1 : R \u2192L[R] R) (c^n) :=\nbegin\n  induction n with n ihn,\n  { ext, simp },\n  { rw [pow_succ, ihn, mul_def, smul_right_comp, smul_eq_mul, pow_succ'] }\nend\n\nsection\nvariables {\u03c3\u2082\u2081 : R\u2082 \u2192+* R} [ring_hom_inv_pair \u03c3\u2081\u2082 \u03c3\u2082\u2081]\n\n/-- Given a right inverse `f\u2082 : M\u2082 \u2192L[R] M` to `f\u2081 : M \u2192L[R] M\u2082`,\n`proj_ker_of_right_inverse f\u2081 f\u2082 h` is the projection `M \u2192L[R] f\u2081.ker` along `f\u2082.range`. -/\ndef proj_ker_of_right_inverse [topological_add_group M] (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M)\n  (h : function.right_inverse f\u2082 f\u2081) :\n  M \u2192L[R] f\u2081.ker :=\n(id R M - f\u2082.comp f\u2081).cod_restrict f\u2081.ker $ \u03bb x, by simp [h (f\u2081 x)]\n\n@[simp] lemma coe_proj_ker_of_right_inverse_apply [topological_add_group M]\n  (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : function.right_inverse f\u2082 f\u2081) (x : M) :\n  (f\u2081.proj_ker_of_right_inverse f\u2082 h x : M) = x - f\u2082 (f\u2081 x) :=\nrfl\n\n@[simp] lemma proj_ker_of_right_inverse_apply_idem [topological_add_group M]\n  (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : function.right_inverse f\u2082 f\u2081) (x : f\u2081.ker) :\n  f\u2081.proj_ker_of_right_inverse f\u2082 h x = x :=\nsubtype.ext_iff_val.2 $ by simp\n\n@[simp] lemma proj_ker_of_right_inverse_comp_inv [topological_add_group M]\n  (f\u2081 : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M) (h : function.right_inverse f\u2082 f\u2081) (y : M\u2082) :\n  f\u2081.proj_ker_of_right_inverse f\u2082 h (f\u2082 y) = 0 :=\nsubtype.ext_iff_val.2 $ by simp [h y]\n\nend\n\nend ring\n\nsection smul_monoid\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariables {R R\u2082 R\u2083 S S\u2083 : Type*} [semiring R] [semiring R\u2082] [semiring R\u2083]\n  [monoid S] [monoid S\u2083] [topological_space S] [topological_space S\u2083]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082] [module R\u2082 M\u2082]\n  {M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083] [module R\u2083 M\u2083]\n  {N\u2082 : Type*} [topological_space N\u2082] [add_comm_monoid N\u2082] [module R N\u2082]\n  {N\u2083 : Type*} [topological_space N\u2083] [add_comm_monoid N\u2083] [module R N\u2083]\n  [distrib_mul_action S\u2083 M\u2083] [smul_comm_class R\u2083 S\u2083 M\u2083] [has_continuous_smul S\u2083 M\u2083]\n  [distrib_mul_action S N\u2083] [smul_comm_class R S N\u2083] [has_continuous_smul S N\u2083]\n  {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2081\u2083 : R \u2192+* R\u2083} [ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083]\n\ninstance : mul_action S\u2083 (M \u2192SL[\u03c3\u2081\u2083] M\u2083) :=\n{ smul := \u03bb c f, \u27e8c \u2022 f, (continuous_const.smul f.2 : continuous (\u03bb x, c \u2022 f x))\u27e9,\n  one_smul := \u03bb f, ext $ \u03bb x, one_smul _ _,\n  mul_smul := \u03bb a b f, ext $ \u03bb x, mul_smul _ _ _ }\n\nvariables (c : S\u2083) (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y z : M)\nvariables (h\u2097 : N\u2082 \u2192L[R] N\u2083) (f\u2097 g\u2097 : M \u2192L[R] N\u2082)\n\ninclude \u03c3\u2081\u2083\n@[simp] lemma smul_comp : (c \u2022 h).comp f = c \u2022 (h.comp f) := rfl\nomit \u03c3\u2081\u2083\n\nvariables [distrib_mul_action S\u2083 M\u2082] [has_continuous_smul S\u2083 M\u2082] [smul_comm_class R\u2082 S\u2083 M\u2082]\nvariables [distrib_mul_action S N\u2082] [has_continuous_smul S N\u2082] [smul_comm_class R S N\u2082]\n\nlemma smul_apply : (c \u2022 f) x = c \u2022 (f x) := rfl\n@[simp, norm_cast] lemma coe_smul : (((c \u2022 f) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192\u209b\u2097[\u03c3\u2081\u2082] M\u2082) = c \u2022 f := rfl\n@[simp, norm_cast] lemma coe_smul' : (((c \u2022 f) : M \u2192SL[\u03c3\u2081\u2082] M\u2082) : M \u2192 M\u2082) = c \u2022 f := rfl\n\n@[simp] lemma comp_smul [linear_map.compatible_smul N\u2082 N\u2083 S R] (c : S) :\n  h\u2097.comp (c \u2022 f\u2097) = c \u2022 (h\u2097.comp f\u2097) :=\nby { ext x, exact h\u2097.map_smul_of_tower c (f\u2097 x) }\n\ninclude \u03c3\u2081\u2083\n@[simp] lemma comp_smul\u209b\u2097 (c : R\u2082) [smul_comm_class R\u2082 R\u2082 M\u2082] [smul_comm_class R\u2083 R\u2083 M\u2083]\n  [topological_space R\u2082] [has_continuous_smul R\u2082 M\u2082] [topological_space R\u2083]\n  [has_continuous_smul R\u2083 M\u2083] :\n  h.comp (c \u2022 f) = (\u03c3\u2082\u2083 c) \u2022 (h.comp f) :=\nby { ext x, simp only [coe_smul', coe_comp', function.comp_app, pi.smul_apply, map_smul\u209b\u2097] }\nomit \u03c3\u2081\u2083\n\ninstance {T : Type*} [monoid T] [topological_space T] [distrib_mul_action T M\u2082]\n  [has_continuous_smul T M\u2082] [smul_comm_class R\u2082 T M\u2082] [has_scalar S\u2083 T]\n  [is_scalar_tower S\u2083 T M\u2082] : is_scalar_tower S\u2083 T (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n\u27e8\u03bb a b f, ext $ \u03bb x, smul_assoc a b (f x)\u27e9\n\ninstance {T : Type*} [monoid T] [topological_space T] [distrib_mul_action T M\u2082]\n  [has_continuous_smul T M\u2082] [smul_comm_class R\u2082 T M\u2082] [smul_comm_class S\u2083 T M\u2082] :\n  smul_comm_class S\u2083 T (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n\u27e8\u03bb a b f, ext $ \u03bb x, smul_comm a b (f x)\u27e9\n\ninstance [has_continuous_add M\u2082] : distrib_mul_action S\u2083 (M \u2192SL[\u03c3\u2081\u2082] M\u2082) :=\n{ smul_add := \u03bb a f g, ext $ \u03bb x, smul_add a (f x) (g x),\n  smul_zero := \u03bb a, ext $ \u03bb x, smul_zero _ }\n\nend smul_monoid\n\nsection smul\n\n-- The M's are used for semilinear maps, and the N's for plain linear maps\nvariables {R R\u2082 R\u2083 S S\u2083 : Type*} [semiring R] [semiring R\u2082] [semiring R\u2083]\n  [semiring S] [semiring S\u2083] [topological_space S] [topological_space S\u2083]\n  {M : Type*} [topological_space M] [add_comm_monoid M] [module R M]\n  {M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082] [module R\u2082 M\u2082]\n  {M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083] [module R\u2083 M\u2083]\n  {N\u2082 : Type*} [topological_space N\u2082] [add_comm_monoid N\u2082] [module R N\u2082]\n  {N\u2083 : Type*} [topological_space N\u2083] [add_comm_monoid N\u2083] [module R N\u2083]\n  [module S\u2083 M\u2083] [smul_comm_class R\u2083 S\u2083 M\u2083] [has_continuous_smul S\u2083 M\u2083]\n  [module S N\u2082] [has_continuous_smul S N\u2082] [smul_comm_class R S N\u2082]\n  [module S N\u2083] [smul_comm_class R S N\u2083] [has_continuous_smul S N\u2083]\n  {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2081\u2083 : R \u2192+* R\u2083} [ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083]\n  (c : S) (h : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) (f g : M \u2192SL[\u03c3\u2081\u2082] M\u2082) (x y z : M)\n\n/-- `continuous_linear_map.prod` as an `equiv`. -/\n@[simps apply] def prod_equiv : ((M \u2192L[R] N\u2082) \u00d7 (M \u2192L[R] N\u2083)) \u2243 (M \u2192L[R] N\u2082 \u00d7 N\u2083) :=\n{ to_fun := \u03bb f, f.1.prod f.2,\n  inv_fun := \u03bb f, \u27e8(fst _ _ _).comp f, (snd _ _ _).comp f\u27e9,\n  left_inv := \u03bb f, by ext; refl,\n  right_inv := \u03bb f, by ext; refl }\n\nlemma prod_ext_iff {f g : M \u00d7 N\u2082 \u2192L[R] N\u2083} :\n  f = g \u2194 f.comp (inl _ _ _) = g.comp (inl _ _ _) \u2227 f.comp (inr _ _ _) = g.comp (inr _ _ _) :=\nby { simp only [\u2190 coe_inj, linear_map.prod_ext_iff], refl }\n\n@[ext] lemma prod_ext {f g : M \u00d7 N\u2082 \u2192L[R] N\u2083} (hl : f.comp (inl _ _ _) = g.comp (inl _ _ _))\n  (hr : f.comp (inr _ _ _) = g.comp (inr _ _ _)) : f = g :=\nprod_ext_iff.2 \u27e8hl, hr\u27e9\n\nvariables [has_continuous_add M\u2082] [has_continuous_add M\u2083] [has_continuous_add N\u2082]\n\ninstance : module S\u2083 (M \u2192SL[\u03c3\u2081\u2083] M\u2083) :=\n{ zero_smul := \u03bb _, ext $ \u03bb _, zero_smul _ _,\n  add_smul  := \u03bb _ _ _, ext $ \u03bb _, add_smul _ _ _ }\n\ninstance [module S\u2083\u1d50\u1d52\u1d56 M\u2083] [is_central_scalar S\u2083 M\u2083] : is_central_scalar S\u2083 (M \u2192SL[\u03c3\u2081\u2083] M\u2083) :=\n{ op_smul_eq_smul := \u03bb _ _, ext $ \u03bb _, op_smul_eq_smul _ _ }\n\nvariables (S) [has_continuous_add N\u2083]\n\n/-- `continuous_linear_map.prod` as a `linear_equiv`. -/\n@[simps apply] def prod\u2097 : ((M \u2192L[R] N\u2082) \u00d7 (M \u2192L[R] N\u2083)) \u2243\u2097[S] (M \u2192L[R] N\u2082 \u00d7 N\u2083) :=\n{ map_add' := \u03bb f g, rfl,\n  map_smul' := \u03bb c f, rfl,\n  .. prod_equiv }\n\n/-- The coercion from `M \u2192L[R] M\u2082` to `M \u2192\u2097[R] M\u2082`, as a linear map. -/\n@[simps]\ndef coe_lm : (M \u2192L[R] N\u2083) \u2192\u2097[S] (M \u2192\u2097[R] N\u2083) :=\n{ to_fun := coe,\n  map_add' := \u03bb f g, coe_add f g,\n  map_smul' := \u03bb c f, coe_smul c f }\n\nvariables {S} (\u03c3\u2081\u2083)\n\n/-- The coercion from `M \u2192SL[\u03c3] M\u2082` to `M \u2192\u209b\u2097[\u03c3] M\u2082`, as a linear map. -/\n@[simps]\ndef coe_lm\u209b\u2097 : (M \u2192SL[\u03c3\u2081\u2083] M\u2083) \u2192\u2097[S\u2083] (M \u2192\u209b\u2097[\u03c3\u2081\u2083] M\u2083) :=\n{ to_fun := coe,\n  map_add' := \u03bb f g, coe_add f g,\n  map_smul' := \u03bb c f, coe_smul c f }\n\nvariables {\u03c3\u2081\u2083}\n\nend smul\n\nsection smul_right\u2097\n\nvariables {R S T M M\u2082 : Type*} [ring R] [ring S] [ring T] [module R S]\n  [add_comm_group M\u2082] [module R M\u2082] [module S M\u2082] [is_scalar_tower R S M\u2082]\n  [topological_space S] [topological_space M\u2082] [has_continuous_smul S M\u2082]\n  [topological_space M] [add_comm_group M] [module R M] [topological_add_group M\u2082]\n  [topological_space T] [module T M\u2082] [has_continuous_smul T M\u2082]\n  [smul_comm_class R T M\u2082] [smul_comm_class S T M\u2082]\n\n/-- Given `c : E \u2192L[\ud835\udd5c] \ud835\udd5c`, `c.smul_right\u2097` is the linear map from `F` to `E \u2192L[\ud835\udd5c] F`\nsending `f` to `\u03bb e, c e \u2022 f`. See also `continuous_linear_map.smul_rightL`. -/\ndef smul_right\u2097 (c : M \u2192L[R] S) : M\u2082 \u2192\u2097[T] (M \u2192L[R] M\u2082) :=\n{ to_fun := c.smul_right,\n  map_add' := \u03bb x y, by { ext e, apply smul_add },\n  map_smul' := \u03bb a x, by { ext e, dsimp, apply smul_comm } }\n\n@[simp] lemma coe_smul_right\u2097 (c : M \u2192L[R] S) :\n  \u21d1(smul_right\u2097 c : M\u2082 \u2192\u2097[T] (M \u2192L[R] M\u2082)) = c.smul_right := rfl\n\nend smul_right\u2097\n\nsection comm_ring\n\n/-- The determinant of a continuous linear map, mainly as a convenience device to be able to\nwrite `A.det` instead of `(A : M \u2192\u2097[R] M).det`. -/\n@[reducible] noncomputable def det {R : Type*} [comm_ring R] [is_domain R]\n  {M : Type*} [topological_space M] [add_comm_group M] [module R M] (A : M \u2192L[R] M) : R :=\nlinear_map.det (A : M \u2192\u2097[R] M)\n\nvariables\n{R : Type*} [comm_ring R] [topological_space R]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_group M\u2083]\n[module R M] [module R M\u2082] [module R M\u2083] [has_continuous_smul R M\u2083]\n\nvariables [topological_add_group M\u2082] [has_continuous_smul R M\u2082]\n\ninstance : algebra R (M\u2082 \u2192L[R] M\u2082) :=\nalgebra.of_module smul_comp (\u03bb _ _ _, comp_smul _ _ _)\n\nend comm_ring\n\nsection restrict_scalars\n\nvariables {A M M\u2082 : Type*} [ring A] [add_comm_group M] [add_comm_group M\u2082]\n  [module A M] [module A M\u2082] [topological_space M] [topological_space M\u2082]\n  (R : Type*) [ring R] [module R M] [module R M\u2082] [linear_map.compatible_smul M M\u2082 R A]\n\n/-- If `A` is an `R`-algebra, then a continuous `A`-linear map can be interpreted as a continuous\n`R`-linear map. We assume `linear_map.compatible_smul M M\u2082 R A` to match assumptions of\n`linear_map.map_smul_of_tower`. -/\ndef restrict_scalars (f : M \u2192L[A] M\u2082) : M \u2192L[R] M\u2082 :=\n\u27e8(f : M \u2192\u2097[A] M\u2082).restrict_scalars R, f.continuous\u27e9\n\nvariable {R}\n\n@[simp, norm_cast] lemma coe_restrict_scalars (f : M \u2192L[A] M\u2082) :\n  (f.restrict_scalars R : M \u2192\u2097[R] M\u2082) = (f : M \u2192\u2097[A] M\u2082).restrict_scalars R := rfl\n\n@[simp] lemma coe_restrict_scalars' (f : M \u2192L[A] M\u2082) : \u21d1(f.restrict_scalars R) = f := rfl\n\n@[simp] lemma restrict_scalars_zero : (0 : M \u2192L[A] M\u2082).restrict_scalars R = 0 := rfl\n\nsection\nvariable [topological_add_group M\u2082]\n\n@[simp] lemma restrict_scalars_add (f g : M \u2192L[A] M\u2082) :\n  (f + g).restrict_scalars R = f.restrict_scalars R + g.restrict_scalars R := rfl\n\n@[simp] lemma restrict_scalars_neg (f : M \u2192L[A] M\u2082) :\n  (-f).restrict_scalars R = -f.restrict_scalars R := rfl\nend\n\nvariables {S : Type*} [ring S] [topological_space S] [module S M\u2082] [has_continuous_smul S M\u2082]\n  [smul_comm_class A S M\u2082] [smul_comm_class R S M\u2082]\n\n@[simp] lemma restrict_scalars_smul (c : S) (f : M \u2192L[A] M\u2082) :\n  (c \u2022 f).restrict_scalars R = c \u2022 f.restrict_scalars R := rfl\n\nvariables (A M M\u2082 R S) [topological_add_group M\u2082]\n\n/-- `continuous_linear_map.restrict_scalars` as a `linear_map`. See also\n`continuous_linear_map.restrict_scalarsL`. -/\ndef restrict_scalars\u2097 : (M \u2192L[A] M\u2082) \u2192\u2097[S] (M \u2192L[R] M\u2082) :=\n{ to_fun := restrict_scalars R,\n  map_add' := restrict_scalars_add,\n  map_smul' := restrict_scalars_smul }\n\nvariables {A M M\u2082 R S}\n\n@[simp] lemma coe_restrict_scalars\u2097 : \u21d1(restrict_scalars\u2097 A M M\u2082 R S) = restrict_scalars R := rfl\n\nend restrict_scalars\n\nend continuous_linear_map\n\nnamespace continuous_linear_equiv\n\nsection add_comm_monoid\n\nvariables {R\u2081 : Type*} {R\u2082 : Type*} {R\u2083 : Type*} [semiring R\u2081] [semiring R\u2082] [semiring R\u2083]\n{\u03c3\u2081\u2082 : R\u2081 \u2192+* R\u2082} {\u03c3\u2082\u2081 : R\u2082 \u2192+* R\u2081} [ring_hom_inv_pair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [ring_hom_inv_pair \u03c3\u2082\u2081 \u03c3\u2081\u2082]\n{\u03c3\u2082\u2083 : R\u2082 \u2192+* R\u2083} {\u03c3\u2083\u2082 : R\u2083 \u2192+* R\u2082} [ring_hom_inv_pair \u03c3\u2082\u2083 \u03c3\u2083\u2082] [ring_hom_inv_pair \u03c3\u2083\u2082 \u03c3\u2082\u2083]\n{\u03c3\u2081\u2083 : R\u2081 \u2192+* R\u2083} {\u03c3\u2083\u2081 : R\u2083 \u2192+* R\u2081} [ring_hom_inv_pair \u03c3\u2081\u2083 \u03c3\u2083\u2081] [ring_hom_inv_pair \u03c3\u2083\u2081 \u03c3\u2081\u2083]\n[ring_hom_comp_triple \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083] [ring_hom_comp_triple \u03c3\u2083\u2082 \u03c3\u2082\u2081 \u03c3\u2083\u2081]\n{M\u2081 : Type*} [topological_space M\u2081] [add_comm_monoid M\u2081]\n{M'\u2081 : Type*} [topological_space M'\u2081] [add_comm_monoid M'\u2081]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_monoid M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_monoid M\u2083]\n{M\u2084 : Type*} [topological_space M\u2084] [add_comm_monoid M\u2084]\n[module R\u2081 M\u2081] [module R\u2081 M'\u2081] [module R\u2082 M\u2082] [module R\u2083 M\u2083]\n\ninclude \u03c3\u2082\u2081\n/-- A continuous linear equivalence induces a continuous linear map. -/\ndef to_continuous_linear_map (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082 :=\n{ cont := e.continuous_to_fun,\n  ..e.to_linear_equiv.to_linear_map }\n\n/-- Coerce continuous linear equivs to continuous linear maps. -/\ninstance : has_coe (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) := \u27e8to_continuous_linear_map\u27e9\n\n/-- Coerce continuous linear equivs to maps. -/\n-- see Note [function coercion]\ninstance : has_coe_to_fun (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (\u03bb _, M\u2081 \u2192 M\u2082) := \u27e8\u03bb f, f\u27e9\n\n@[simp] theorem coe_def_rev (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.to_continuous_linear_map = e := rfl\n\ntheorem coe_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) : (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) b = e b := rfl\n\n@[simp] lemma coe_to_linear_equiv (f : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : \u21d1f.to_linear_equiv = f := rfl\n\n@[simp, norm_cast] lemma coe_coe (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : ((e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082) = e := rfl\n\nlemma to_linear_equiv_injective :\n  function.injective (to_linear_equiv : (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) \u2192 (M\u2081 \u2243\u209b\u2097[\u03c3\u2081\u2082] M\u2082))\n| \u27e8e, _, _\u27e9 \u27e8e', _, _\u27e9 rfl := rfl\n\n@[ext] lemma ext {f g : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082} (h : (f : M\u2081 \u2192 M\u2082) = g) : f = g :=\nto_linear_equiv_injective $ linear_equiv.ext $ congr_fun h\n\nlemma coe_injective : function.injective (coe : (M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) \u2192 (M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)) :=\n\u03bb e e' h, ext $ funext $ continuous_linear_map.ext_iff.1 h\n\n@[simp, norm_cast] lemma coe_inj {e e' : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082} : (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = e' \u2194 e = e' :=\ncoe_injective.eq_iff\n\n/-- A continuous linear equivalence induces a homeomorphism. -/\ndef to_homeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2243\u209c M\u2082 := { to_equiv := e.to_linear_equiv.to_equiv, ..e }\n\n@[simp] lemma coe_to_homeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : \u21d1e.to_homeomorph = e := rfl\n\nlemma image_closure (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2081) : e '' closure s = closure (e '' s) :=\ne.to_homeomorph.image_closure s\n\nlemma preimage_closure (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2082) : e \u207b\u00b9' closure s = closure (e \u207b\u00b9' s) :=\ne.to_homeomorph.preimage_closure s\n\n@[simp] lemma is_closed_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : set M\u2081} :\n  is_closed (e '' s) \u2194 is_closed s :=\ne.to_homeomorph.is_closed_image\n\nlemma map_nhds_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : map e (\ud835\udcdd x) = \ud835\udcdd (e x) :=\ne.to_homeomorph.map_nhds_eq x\n\n-- Make some straightforward lemmas available to `simp`.\n@[simp] lemma map_zero (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e (0 : M\u2081) = 0 := (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_zero\n@[simp] lemma map_add (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x y : M\u2081) : e (x + y) = e x + e y :=\n(e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_add x y\n@[simp] lemma map_smul\u209b\u2097 (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2081) (x : M\u2081) : e (c \u2022 x) = \u03c3\u2081\u2082 c \u2022 (e x) :=\n(e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).map_smul\u209b\u2097 c x\nomit \u03c3\u2082\u2081\n\n@[simp] lemma map_smul [module R\u2081 M\u2082] (e : M\u2081 \u2243L[R\u2081] M\u2082) (c : R\u2081) (x : M\u2081) :\n  e (c \u2022 x) = c \u2022 (e x) :=\n(e : M\u2081 \u2192L[R\u2081] M\u2082).map_smul c x\n\ninclude \u03c3\u2082\u2081\n@[simp] lemma map_eq_zero_iff (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x : M\u2081} : e x = 0 \u2194 x = 0 :=\ne.to_linear_equiv.map_eq_zero_iff\n\nattribute [continuity]\n  continuous_linear_equiv.continuous_to_fun continuous_linear_equiv.continuous_inv_fun\n\n@[continuity]\nprotected lemma continuous (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : continuous (e : M\u2081 \u2192 M\u2082) :=\ne.continuous_to_fun\n\nprotected lemma continuous_on (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : set M\u2081} : continuous_on (e : M\u2081 \u2192 M\u2082) s :=\ne.continuous.continuous_on\n\nprotected lemma continuous_at (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x : M\u2081} : continuous_at (e : M\u2081 \u2192 M\u2082) x :=\ne.continuous.continuous_at\n\nprotected lemma continuous_within_at (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {s : set M\u2081} {x : M\u2081} :\n  continuous_within_at (e : M\u2081 \u2192 M\u2082) s x :=\ne.continuous.continuous_within_at\n\nlemma comp_continuous_on_iff\n  {\u03b1 : Type*} [topological_space \u03b1] (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {f : \u03b1 \u2192 M\u2081} {s : set \u03b1} :\n  continuous_on (e \u2218 f) s \u2194 continuous_on f s :=\ne.to_homeomorph.comp_continuous_on_iff _ _\n\nlemma comp_continuous_iff\n  {\u03b1 : Type*} [topological_space \u03b1] (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {f : \u03b1 \u2192 M\u2081} :\n  continuous (e \u2218 f) \u2194 continuous f :=\ne.to_homeomorph.comp_continuous_iff\nomit \u03c3\u2082\u2081\n\n/-- An extensionality lemma for `R \u2243L[R] M`. -/\nlemma ext\u2081 [topological_space R\u2081] {f g : R\u2081 \u2243L[R\u2081] M\u2081} (h : f 1 = g 1) : f = g :=\next $ funext $ \u03bb x, mul_one x \u25b8 by rw [\u2190 smul_eq_mul, map_smul, h, map_smul]\n\nsection\nvariables (R\u2081 M\u2081)\n\n/-- The identity map as a continuous linear equivalence. -/\n@[refl] protected def refl : M\u2081 \u2243L[R\u2081] M\u2081 :=\n{ continuous_to_fun := continuous_id,\n  continuous_inv_fun := continuous_id,\n  .. linear_equiv.refl R\u2081 M\u2081 }\nend\n\n@[simp, norm_cast] lemma coe_refl :\n  (continuous_linear_equiv.refl R\u2081 M\u2081 : M\u2081 \u2192L[R\u2081] M\u2081) = continuous_linear_map.id R\u2081 M\u2081 := rfl\n\n@[simp, norm_cast] lemma coe_refl' :\n  (continuous_linear_equiv.refl R\u2081 M\u2081 : M\u2081 \u2192 M\u2081) = id := rfl\n\n/-- The inverse of a continuous linear equivalence as a continuous linear equivalence-/\n@[symm] protected def symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2082 \u2243SL[\u03c3\u2082\u2081] M\u2081 :=\n{ continuous_to_fun := e.continuous_inv_fun,\n  continuous_inv_fun := e.continuous_to_fun,\n  .. e.to_linear_equiv.symm }\n\ninclude \u03c3\u2082\u2081\n@[simp] lemma symm_to_linear_equiv (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  e.symm.to_linear_equiv = e.to_linear_equiv.symm :=\nby { ext, refl }\n\n@[simp] lemma symm_to_homeomorph (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  e.to_homeomorph.symm = e.symm.to_homeomorph :=\nrfl\n\n/-- See Note [custom simps projection]. We need to specify this projection explicitly in this case,\n  because it is a composition of multiple projections. -/\ndef simps.apply (h : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2081 \u2192 M\u2082 := h\n\n/-- See Note [custom simps projection] -/\ndef simps.symm_apply (h : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : M\u2082 \u2192 M\u2081 := h.symm\n\ninitialize_simps_projections continuous_linear_equiv\n  (to_linear_equiv_to_fun \u2192 apply, to_linear_equiv_inv_fun \u2192 symm_apply)\n\nlemma symm_map_nhds_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : map e.symm (\ud835\udcdd (e x)) = \ud835\udcdd x :=\ne.to_homeomorph.symm_map_nhds_eq x\nomit \u03c3\u2082\u2081\n\ninclude \u03c3\u2082\u2081 \u03c3\u2083\u2082 \u03c3\u2083\u2081\n/-- The composition of two continuous linear equivalences as a continuous linear equivalence. -/\n@[trans] protected def trans (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) : M\u2081 \u2243SL[\u03c3\u2081\u2083] M\u2083 :=\n{ continuous_to_fun := e\u2082.continuous_to_fun.comp e\u2081.continuous_to_fun,\n  continuous_inv_fun := e\u2081.continuous_inv_fun.comp e\u2082.continuous_inv_fun,\n  .. e\u2081.to_linear_equiv.trans e\u2082.to_linear_equiv }\n\ninclude \u03c3\u2081\u2083\n@[simp] lemma trans_to_linear_equiv (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) :\n  (e\u2081.trans e\u2082).to_linear_equiv = e\u2081.to_linear_equiv.trans e\u2082.to_linear_equiv :=\nby { ext, refl }\nomit \u03c3\u2081\u2083 \u03c3\u2082\u2081 \u03c3\u2083\u2082 \u03c3\u2083\u2081\n\n/-- Product of two continuous linear equivalences. The map comes from `equiv.prod_congr`. -/\ndef prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082) (e' : M\u2083 \u2243L[R\u2081] M\u2084) :\n  (M\u2081 \u00d7 M\u2083) \u2243L[R\u2081] (M\u2082 \u00d7 M\u2084) :=\n{ continuous_to_fun := e.continuous_to_fun.prod_map e'.continuous_to_fun,\n  continuous_inv_fun := e.continuous_inv_fun.prod_map e'.continuous_inv_fun,\n  .. e.to_linear_equiv.prod e'.to_linear_equiv }\n\n@[simp, norm_cast] lemma prod_apply [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082)\n  (e' : M\u2083 \u2243L[R\u2081] M\u2084) (x) :\n  e.prod e' x = (e x.1, e' x.2) := rfl\n\n@[simp, norm_cast] lemma coe_prod [module R\u2081 M\u2082] [module R\u2081 M\u2083] [module R\u2081 M\u2084] (e : M\u2081 \u2243L[R\u2081] M\u2082)\n  (e' : M\u2083 \u2243L[R\u2081] M\u2084) :\n  (e.prod e' : (M\u2081 \u00d7 M\u2083) \u2192L[R\u2081] (M\u2082 \u00d7 M\u2084)) = (e : M\u2081 \u2192L[R\u2081] M\u2082).prod_map (e' : M\u2083 \u2192L[R\u2081] M\u2084) :=\nrfl\n\ninclude \u03c3\u2082\u2081\ntheorem bijective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : function.bijective e :=\ne.to_linear_equiv.to_equiv.bijective\ntheorem injective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : function.injective e :=\ne.to_linear_equiv.to_equiv.injective\ntheorem surjective (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : function.surjective e :=\ne.to_linear_equiv.to_equiv.surjective\n\ninclude \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2081\u2083\n@[simp] theorem trans_apply (e\u2081 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2082 : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) (c : M\u2081) :\n  (e\u2081.trans e\u2082) c = e\u2082 (e\u2081 c) :=\nrfl\nomit \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2081\u2083\n\n@[simp] theorem apply_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : M\u2082) : e (e.symm c) = c :=\ne.1.right_inv c\n@[simp] theorem symm_apply_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (b : M\u2081) : e.symm (e b) = b := e.1.left_inv b\n\ninclude \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 \u03c3\u2083\u2081\n@[simp] theorem symm_trans_apply (e\u2081 : M\u2082 \u2243SL[\u03c3\u2082\u2081] M\u2081) (e\u2082 : M\u2083 \u2243SL[\u03c3\u2083\u2082] M\u2082) (c : M\u2081) :\n  (e\u2082.trans e\u2081).symm c = e\u2082.symm (e\u2081.symm c) :=\nrfl\nomit \u03c3\u2081\u2082 \u03c3\u2082\u2083 \u03c3\u2081\u2083 \u03c3\u2083\u2081\n\n@[simp] theorem symm_image_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2081) : e.symm '' (e '' s) = s :=\ne.to_linear_equiv.to_equiv.symm_image_image s\n@[simp] theorem image_symm_image (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2082) : e '' (e.symm '' s) = s :=\ne.symm.symm_image_image s\n\ninclude \u03c3\u2083\u2082 \u03c3\u2083\u2081\n@[simp, norm_cast]\nlemma comp_coe (f : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (f' : M\u2082 \u2243SL[\u03c3\u2082\u2083] M\u2083) :\n  (f' : M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083).comp (f : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = (f.trans f' : M\u2081 \u2192SL[\u03c3\u2081\u2083] M\u2083) :=\nrfl\nomit \u03c3\u2083\u2082 \u03c3\u2083\u2081 \u03c3\u2082\u2081\n\n@[simp] theorem coe_comp_coe_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082).comp (e.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081) = continuous_linear_map.id R\u2082 M\u2082 :=\ncontinuous_linear_map.ext e.apply_symm_apply\n\n@[simp] theorem coe_symm_comp_coe (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  (e.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081).comp (e : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) = continuous_linear_map.id R\u2081 M\u2081 :=\ncontinuous_linear_map.ext e.symm_apply_apply\n\ninclude \u03c3\u2082\u2081\n@[simp] lemma symm_comp_self (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  (e.symm : M\u2082 \u2192 M\u2081) \u2218 (e : M\u2081 \u2192 M\u2082) = id :=\nby{ ext x, exact symm_apply_apply e x }\n\n@[simp] lemma self_comp_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) :\n  (e : M\u2081 \u2192 M\u2082) \u2218 (e.symm : M\u2082 \u2192 M\u2081) = id :=\nby{ ext x, exact apply_symm_apply e x }\n\n@[simp] theorem symm_symm (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) : e.symm.symm = e :=\nby { ext x, refl }\nomit \u03c3\u2082\u2081\n\n@[simp] lemma refl_symm :\n (continuous_linear_equiv.refl R\u2081 M\u2081).symm = continuous_linear_equiv.refl R\u2081 M\u2081 :=\nrfl\n\ninclude \u03c3\u2082\u2081\ntheorem symm_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M\u2081) : e.symm.symm x = e x :=\nrfl\n\nlemma symm_apply_eq (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x y} : e.symm x = y \u2194 x = e y :=\ne.to_linear_equiv.symm_apply_eq\n\nlemma eq_symm_apply (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) {x y} : y = e.symm x \u2194 e y = x :=\ne.to_linear_equiv.eq_symm_apply\n\nprotected lemma image_eq_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2081) : e '' s = e.symm \u207b\u00b9' s :=\ne.to_linear_equiv.to_equiv.image_eq_preimage s\n\nprotected lemma image_symm_eq_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2082) : e.symm '' s = e \u207b\u00b9' s :=\nby rw [e.symm.image_eq_preimage, e.symm_symm]\n\n@[simp] protected lemma symm_preimage_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2082) :\n  e.symm \u207b\u00b9' (e \u207b\u00b9' s) = s := e.to_linear_equiv.to_equiv.symm_preimage_preimage s\n\n@[simp] protected lemma preimage_symm_preimage (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (s : set M\u2081) :\n  e \u207b\u00b9' (e.symm \u207b\u00b9' s) = s := e.symm.symm_preimage_preimage s\n\nomit \u03c3\u2082\u2081\n\n/-- Create a `continuous_linear_equiv` from two `continuous_linear_map`s that are\ninverse of each other. -/\ndef equiv_of_inverse (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081) (h\u2081 : function.left_inverse f\u2082 f\u2081)\n  (h\u2082 : function.right_inverse f\u2082 f\u2081) :\n  M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082 :=\n{ to_fun := f\u2081,\n  continuous_to_fun := f\u2081.continuous,\n  inv_fun := f\u2082,\n  continuous_inv_fun := f\u2082.continuous,\n  left_inv := h\u2081,\n  right_inv := h\u2082,\n  .. f\u2081 }\n\ninclude \u03c3\u2082\u2081\n@[simp] lemma equiv_of_inverse_apply (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 h\u2081 h\u2082 x) :\n  equiv_of_inverse f\u2081 f\u2082 h\u2081 h\u2082 x = f\u2081 x :=\nrfl\n\n@[simp] lemma symm_equiv_of_inverse (f\u2081 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082) (f\u2082 h\u2081 h\u2082) :\n  (equiv_of_inverse f\u2081 f\u2082 h\u2081 h\u2082).symm = equiv_of_inverse f\u2082 f\u2081 h\u2082 h\u2081 :=\nrfl\nomit \u03c3\u2082\u2081\n\nsection pointwise\nopen_locale pointwise\ninclude \u03c3\u2082\u2081\n\n@[simp] lemma image_smul_set\u209b\u2097 (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2081) (s : set M\u2081) :\n  e '' (c \u2022 s) = (\u03c3\u2081\u2082 c) \u2022 e '' s :=\ne.to_linear_equiv.image_smul_set\u209b\u2097 c s\n\n@[simp] lemma preimage_smul_set\u209b\u2097 (e : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (c : R\u2082) (s : set M\u2082) :\n  e \u207b\u00b9' (c \u2022 s) = \u03c3\u2082\u2081 c \u2022 e \u207b\u00b9' s :=\ne.to_linear_equiv.preimage_smul_set\u209b\u2097 c s\nomit \u03c3\u2082\u2081\n\n@[simp] lemma image_smul_set (e : M\u2081 \u2243L[R\u2081] M'\u2081) (c : R\u2081) (s : set M\u2081) :\n  e '' (c \u2022 s) = c \u2022 e '' s :=\ne.to_linear_equiv.image_smul_set c s\n\n@[simp] lemma preimage_smul_set (e : M\u2081 \u2243L[R\u2081] M'\u2081) (c : R\u2081) (s : set M'\u2081) :\n  e \u207b\u00b9' (c \u2022 s) = c \u2022 e \u207b\u00b9' s :=\ne.to_linear_equiv.preimage_smul_set c s\n\nend pointwise\n\nvariable (M\u2081)\n\n/-- The continuous linear equivalences from `M` to itself form a group under composition. -/\ninstance automorphism_group : group (M\u2081 \u2243L[R\u2081] M\u2081) :=\n{ mul          := \u03bb f g, g.trans f,\n  one          := continuous_linear_equiv.refl R\u2081 M\u2081,\n  inv          := \u03bb f, f.symm,\n  mul_assoc    := \u03bb f g h, by {ext, refl},\n  mul_one      := \u03bb f, by {ext, refl},\n  one_mul      := \u03bb f, by {ext, refl},\n  mul_left_inv := \u03bb f, by {ext, exact f.left_inv x} }\n\nvariables {M\u2081} {R\u2084 : Type*} [semiring R\u2084] [module R\u2084 M\u2084]\n  {\u03c3\u2083\u2084 : R\u2083 \u2192+* R\u2084} {\u03c3\u2084\u2083 : R\u2084 \u2192+* R\u2083} [ring_hom_inv_pair \u03c3\u2083\u2084 \u03c3\u2084\u2083] [ring_hom_inv_pair \u03c3\u2084\u2083 \u03c3\u2083\u2084]\n  {\u03c3\u2082\u2084 : R\u2082 \u2192+* R\u2084} {\u03c3\u2081\u2084 : R\u2081 \u2192+* R\u2084}\n  [ring_hom_comp_triple \u03c3\u2082\u2081 \u03c3\u2081\u2084 \u03c3\u2082\u2084] [ring_hom_comp_triple \u03c3\u2082\u2084 \u03c3\u2084\u2083 \u03c3\u2082\u2083]\n  [ring_hom_comp_triple \u03c3\u2081\u2083 \u03c3\u2083\u2084 \u03c3\u2081\u2084]\n\ninclude \u03c3\u2082\u2081 \u03c3\u2083\u2084 \u03c3\u2082\u2083 \u03c3\u2082\u2084 \u03c3\u2081\u2083\n\n/-- A pair of continuous (semi)linear equivalences generates an equivalence between the spaces of\ncontinuous linear maps. -/\n@[simps] def arrow_congr_equiv (e\u2081\u2082 : M\u2081 \u2243SL[\u03c3\u2081\u2082] M\u2082) (e\u2084\u2083 : M\u2084 \u2243SL[\u03c3\u2084\u2083] M\u2083) :\n  (M\u2081 \u2192SL[\u03c3\u2081\u2084] M\u2084) \u2243 (M\u2082 \u2192SL[\u03c3\u2082\u2083] M\u2083) :=\n{ to_fun := \u03bb f, (e\u2084\u2083 : M\u2084 \u2192SL[\u03c3\u2084\u2083] M\u2083).comp (f.comp (e\u2081\u2082.symm : M\u2082 \u2192SL[\u03c3\u2082\u2081] M\u2081)),\n  inv_fun := \u03bb f, (e\u2084\u2083.symm : M\u2083 \u2192SL[\u03c3\u2083\u2084] M\u2084).comp (f.comp (e\u2081\u2082 : M\u2081 \u2192SL[\u03c3\u2081\u2082] M\u2082)),\n  left_inv := \u03bb f, continuous_linear_map.ext $ \u03bb x,\n    by simp only [continuous_linear_map.comp_apply, symm_apply_apply, coe_coe],\n  right_inv := \u03bb f, continuous_linear_map.ext $ \u03bb x,\n    by simp only [continuous_linear_map.comp_apply, apply_symm_apply, coe_coe] }\n\nend add_comm_monoid\n\nsection add_comm_group\n\nvariables {R : Type*} [semiring R]\n{M : Type*} [topological_space M] [add_comm_group M]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082]\n{M\u2083 : Type*} [topological_space M\u2083] [add_comm_group M\u2083]\n{M\u2084 : Type*} [topological_space M\u2084] [add_comm_group M\u2084]\n[module R M] [module R M\u2082] [module R M\u2083] [module R M\u2084]\n\nvariables [topological_add_group M\u2084]\n\n/-- Equivalence given by a block lower diagonal matrix. `e` and `e'` are diagonal square blocks,\n  and `f` is a rectangular block below the diagonal. -/\ndef skew_prod (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) :\n  (M \u00d7 M\u2083) \u2243L[R] M\u2082 \u00d7 M\u2084 :=\n{ continuous_to_fun := (e.continuous_to_fun.comp continuous_fst).prod_mk\n    ((e'.continuous_to_fun.comp continuous_snd).add $ f.continuous.comp continuous_fst),\n  continuous_inv_fun := (e.continuous_inv_fun.comp continuous_fst).prod_mk\n    (e'.continuous_inv_fun.comp $ continuous_snd.sub $ f.continuous.comp $\n      e.continuous_inv_fun.comp continuous_fst),\n.. e.to_linear_equiv.skew_prod e'.to_linear_equiv \u2191f }\n@[simp] lemma skew_prod_apply (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) (x) :\n  e.skew_prod e' f x = (e x.1, e' x.2 + f x.1) := rfl\n\n@[simp] lemma skew_prod_symm_apply (e : M \u2243L[R] M\u2082) (e' : M\u2083 \u2243L[R] M\u2084) (f : M \u2192L[R] M\u2084) (x) :\n  (e.skew_prod e' f).symm x = (e.symm x.1, e'.symm (x.2 - f (e.symm x.1))) := rfl\n\nend add_comm_group\n\nsection ring\n\nvariables {R : Type*} [ring R] {R\u2082 : Type*} [ring R\u2082]\n{M : Type*} [topological_space M] [add_comm_group M] [module R M]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082] [module R\u2082 M\u2082]\nvariables {\u03c3\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2082\u2081 : R\u2082 \u2192+* R} [ring_hom_inv_pair \u03c3\u2081\u2082 \u03c3\u2082\u2081] [ring_hom_inv_pair \u03c3\u2082\u2081 \u03c3\u2081\u2082]\n\ninclude \u03c3\u2082\u2081\n@[simp] lemma map_sub (e : M \u2243SL[\u03c3\u2081\u2082] M\u2082) (x y : M) : e (x - y) = e x - e y :=\n(e : M \u2192SL[\u03c3\u2081\u2082] M\u2082).map_sub x y\n\n@[simp] lemma map_neg (e : M \u2243SL[\u03c3\u2081\u2082] M\u2082) (x : M) : e (-x) = -e x := (e : M \u2192SL[\u03c3\u2081\u2082] M\u2082).map_neg x\nomit \u03c3\u2082\u2081\n\nsection\n/-! The next theorems cover the identification between `M \u2243L[\ud835\udd5c] M`and the group of units of the ring\n`M \u2192L[R] M`. -/\nvariables [topological_add_group M]\n\n/-- An invertible continuous linear map `f` determines a continuous equivalence from `M` to itself.\n-/\ndef of_unit (f : (M \u2192L[R] M)\u02e3) : (M \u2243L[R] M) :=\n{ to_linear_equiv :=\n  { to_fun    := f.val,\n    map_add'  := by simp,\n    map_smul' := by simp,\n    inv_fun   := f.inv,\n    left_inv  := \u03bb x, show (f.inv * f.val) x = x, by {rw f.inv_val, simp},\n    right_inv := \u03bb x, show (f.val * f.inv) x = x, by {rw f.val_inv, simp}, },\n  continuous_to_fun  := f.val.continuous,\n  continuous_inv_fun := f.inv.continuous }\n\n/-- A continuous equivalence from `M` to itself determines an invertible continuous linear map. -/\ndef to_unit (f : (M \u2243L[R] M)) : (M \u2192L[R] M)\u02e3 :=\n{ val     := f,\n  inv     := f.symm,\n  val_inv := by {ext, simp},\n  inv_val := by {ext, simp} }\n\nvariables (R M)\n\n/-- The units of the algebra of continuous `R`-linear endomorphisms of `M` is multiplicatively\nequivalent to the type of continuous linear equivalences between `M` and itself. -/\ndef units_equiv : (M \u2192L[R] M)\u02e3 \u2243* (M \u2243L[R] M) :=\n{ to_fun    := of_unit,\n  inv_fun   := to_unit,\n  left_inv  := \u03bb f, by {ext, refl},\n  right_inv := \u03bb f, by {ext, refl},\n  map_mul'  := \u03bb x y, by {ext, refl} }\n\n@[simp] lemma units_equiv_apply (f : (M \u2192L[R] M)\u02e3) (x : M) :\n  units_equiv R M f x = f x := rfl\n\nend\n\nsection\nvariables (R) [topological_space R] [has_continuous_mul R]\n\n/-- Continuous linear equivalences `R \u2243L[R] R` are enumerated by `R\u02e3`. -/\ndef units_equiv_aut : R\u02e3 \u2243 (R \u2243L[R] R) :=\n{ to_fun := \u03bb u, equiv_of_inverse\n    (continuous_linear_map.smul_right (1 : R \u2192L[R] R) \u2191u)\n    (continuous_linear_map.smul_right (1 : R \u2192L[R] R) \u2191u\u207b\u00b9)\n    (\u03bb x, by simp) (\u03bb x, by simp),\n  inv_fun := \u03bb e, \u27e8e 1, e.symm 1,\n    by rw [\u2190 smul_eq_mul, \u2190 map_smul, smul_eq_mul, mul_one, symm_apply_apply],\n    by rw [\u2190 smul_eq_mul, \u2190 map_smul, smul_eq_mul, mul_one, apply_symm_apply]\u27e9,\n  left_inv := \u03bb u, units.ext $ by simp,\n  right_inv := \u03bb e, ext\u2081 $ by simp }\n\nvariable {R}\n\n@[simp] lemma units_equiv_aut_apply (u : R\u02e3) (x : R) : units_equiv_aut R u x = x * u := rfl\n\n@[simp] lemma units_equiv_aut_apply_symm (u : R\u02e3) (x : R) :\n  (units_equiv_aut R u).symm x = x * \u2191u\u207b\u00b9 := rfl\n\n@[simp] lemma units_equiv_aut_symm_apply (e : R \u2243L[R] R) :\n  \u2191((units_equiv_aut R).symm e) = e 1 :=\nrfl\n\nend\n\nvariables [module R M\u2082] [topological_add_group M]\n\nopen _root_.continuous_linear_map (id fst snd subtype_val mem_ker)\n\n/-- A pair of continuous linear maps such that `f\u2081 \u2218 f\u2082 = id` generates a continuous\nlinear equivalence `e` between `M` and `M\u2082 \u00d7 f\u2081.ker` such that `(e x).2 = x` for `x \u2208 f\u2081.ker`,\n`(e x).1 = f\u2081 x`, and `(e (f\u2082 y)).2 = 0`. The map is given by `e x = (f\u2081 x, x - f\u2082 (f\u2081 x))`. -/\ndef equiv_of_right_inverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M) (h : function.right_inverse f\u2082 f\u2081) :\n  M \u2243L[R] M\u2082 \u00d7 f\u2081.ker :=\nequiv_of_inverse (f\u2081.prod (f\u2081.proj_ker_of_right_inverse f\u2082 h)) (f\u2082.coprod (subtype_val f\u2081.ker))\n  (\u03bb x, by simp)\n  (\u03bb \u27e8x, y\u27e9, by simp [h x])\n\n@[simp] lemma fst_equiv_of_right_inverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n  (h : function.right_inverse f\u2082 f\u2081) (x : M) :\n  (equiv_of_right_inverse f\u2081 f\u2082 h x).1 = f\u2081 x := rfl\n\n@[simp] lemma snd_equiv_of_right_inverse (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n  (h : function.right_inverse f\u2082 f\u2081) (x : M) :\n  ((equiv_of_right_inverse f\u2081 f\u2082 h x).2 : M) = x - f\u2082 (f\u2081 x) := rfl\n\n@[simp] lemma equiv_of_right_inverse_symm_apply (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n  (h : function.right_inverse f\u2082 f\u2081) (y : M\u2082 \u00d7 f\u2081.ker) :\n  (equiv_of_right_inverse f\u2081 f\u2082 h).symm y = f\u2082 y.1 + y.2 := rfl\n\nend ring\n\nsection\n\nvariables (\u03b9 R M : Type*) [unique \u03b9] [semiring R] [add_comm_monoid M] [module R M]\n  [topological_space M]\n\n/-- If `\u03b9` has a unique element, then `\u03b9 \u2192 M` is continuously linear equivalent to `M`. -/\ndef fun_unique : (\u03b9 \u2192 M) \u2243L[R] M :=\n{ to_linear_equiv := linear_equiv.fun_unique \u03b9 R M,\n  .. homeomorph.fun_unique \u03b9 M }\n\nvariables {\u03b9 R M}\n\n@[simp] lemma coe_fun_unique : \u21d1(fun_unique \u03b9 R M) = function.eval default := rfl\n@[simp] lemma coe_fun_unique_symm : \u21d1(fun_unique \u03b9 R M).symm = function.const \u03b9 := rfl\n\nvariables (R M)\n\n/-- Continuous linear equivalence between dependent functions `\u03a0 i : fin 2, M i` and `M 0 \u00d7 M 1`. -/\n@[simps { fully_applied := ff }]\ndef pi_fin_two (M : fin 2 \u2192 Type*) [\u03a0 i, add_comm_monoid (M i)] [\u03a0 i, module R (M i)]\n  [\u03a0 i, topological_space (M i)] :\n  (\u03a0 i, M i) \u2243L[R] M 0 \u00d7 M 1 :=\n{ to_linear_equiv := linear_equiv.pi_fin_two R M, .. homeomorph.pi_fin_two M }\n\n/-- Continuous linear equivalence between vectors in `M\u00b2 = fin 2 \u2192 M` and `M \u00d7 M`. -/\n@[simps { fully_applied := ff }]\ndef fin_two_arrow : (fin 2 \u2192 M) \u2243L[R] M \u00d7 M :=\n{ to_linear_equiv := linear_equiv.fin_two_arrow R M, .. pi_fin_two R (\u03bb _, M) }\n\nend\n\nend continuous_linear_equiv\n\nnamespace continuous_linear_map\n\nopen_locale classical\n\nvariables {R : Type*} {M : Type*} {M\u2082 : Type*} [topological_space M] [topological_space M\u2082]\n\nsection\nvariables [semiring R]\nvariables [add_comm_monoid M\u2082] [module R M\u2082]\nvariables [add_comm_monoid M] [module R M]\n\n/-- Introduce a function `inverse` from `M \u2192L[R] M\u2082` to `M\u2082 \u2192L[R] M`, which sends `f` to `f.symm` if\n`f` is a continuous linear equivalence and to `0` otherwise.  This definition is somewhat ad hoc,\nbut one needs a fully (rather than partially) defined inverse function for some purposes, including\nfor calculus. -/\nnoncomputable def inverse : (M \u2192L[R] M\u2082) \u2192 (M\u2082 \u2192L[R] M) :=\n\u03bb f, if h : \u2203 (e : M \u2243L[R] M\u2082), (e : M \u2192L[R] M\u2082) = f then ((classical.some h).symm : M\u2082 \u2192L[R] M)\nelse 0\n\n/-- By definition, if `f` is invertible then `inverse f = f.symm`. -/\n@[simp] lemma inverse_equiv (e : M \u2243L[R] M\u2082) : inverse (e : M \u2192L[R] M\u2082) = e.symm :=\nbegin\n  have h : \u2203 (e' : M \u2243L[R] M\u2082), (e' : M \u2192L[R] M\u2082) = \u2191e := \u27e8e, rfl\u27e9,\n  simp only [inverse, dif_pos h],\n  congr,\n  exact_mod_cast (classical.some_spec h)\nend\n\n/-- By definition, if `f` is not invertible then `inverse f = 0`. -/\n@[simp] lemma inverse_non_equiv (f : M \u2192L[R] M\u2082) (h : \u00ac\u2203 (e' : M \u2243L[R] M\u2082), \u2191e' = f) :\n  inverse f = 0 :=\ndif_neg h\n\nend\n\nsection\nvariables [ring R]\nvariables [add_comm_group M] [topological_add_group M] [module R M]\nvariables [add_comm_group M\u2082] [module R M\u2082]\n\n@[simp] lemma ring_inverse_equiv (e : M \u2243L[R] M) :\n  ring.inverse \u2191e = inverse (e : M \u2192L[R] M) :=\nbegin\n  suffices :\n    ring.inverse ((((continuous_linear_equiv.units_equiv _ _).symm e) : M \u2192L[R] M)) = inverse \u2191e,\n  { convert this },\n  simp,\n  refl,\nend\n\n/-- The function `continuous_linear_equiv.inverse` can be written in terms of `ring.inverse` for the\nring of self-maps of the domain. -/\nlemma to_ring_inverse (e : M \u2243L[R] M\u2082) (f : M \u2192L[R] M\u2082) :\n  inverse f = (ring.inverse ((e.symm : (M\u2082 \u2192L[R] M)).comp f)) \u2218L \u2191e.symm :=\nbegin\n  by_cases h\u2081 : \u2203 (e' : M \u2243L[R] M\u2082), \u2191e' = f,\n  { obtain \u27e8e', he'\u27e9 := h\u2081,\n    rw \u2190 he',\n    change _ = (ring.inverse \u2191(e'.trans e.symm)) \u2218L \u2191e.symm,\n    ext,\n    simp },\n  { suffices : \u00acis_unit ((e.symm : M\u2082 \u2192L[R] M).comp f),\n    { simp [this, h\u2081] },\n    contrapose! h\u2081,\n    rcases h\u2081 with \u27e8F, hF\u27e9,\n    use (continuous_linear_equiv.units_equiv _ _ F).trans e,\n    ext,\n    dsimp, rw [coe_fn_coe_base' F, hF], simp }\nend\n\nlemma ring_inverse_eq_map_inverse : ring.inverse = @inverse R M M _ _ _ _ _ _ _ :=\nbegin\n  ext,\n  simp [to_ring_inverse (continuous_linear_equiv.refl R M)],\nend\n\nend\n\nend continuous_linear_map\n\nnamespace submodule\n\nvariables\n{R : Type*} [ring R]\n{M : Type*} [topological_space M] [add_comm_group M] [module R M]\n{M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082] [module R M\u2082]\n\nopen continuous_linear_map\n\n/-- A submodule `p` is called *complemented* if there exists a continuous projection `M \u2192\u2097[R] p`. -/\ndef closed_complemented (p : submodule R M) : Prop := \u2203 f : M \u2192L[R] p, \u2200 x : p, f x = x\n\nlemma closed_complemented.has_closed_complement {p : submodule R M} [t1_space p]\n  (h : closed_complemented p) :\n  \u2203 (q : submodule R M) (hq : is_closed (q : set M)), is_compl p q :=\nexists.elim h $ \u03bb f hf, \u27e8f.ker, f.is_closed_ker, linear_map.is_compl_of_proj hf\u27e9\n\nprotected lemma closed_complemented.is_closed [topological_add_group M] [t1_space M]\n  {p : submodule R M} (h : closed_complemented p) :\n  is_closed (p : set M) :=\nbegin\n  rcases h with \u27e8f, hf\u27e9,\n  have : ker (id R M - (subtype_val p).comp f) = p := linear_map.ker_id_sub_eq_of_proj hf,\n  exact this \u25b8 (is_closed_ker _)\nend\n\n@[simp] lemma closed_complemented_bot : closed_complemented (\u22a5 : submodule R M) :=\n\u27e80, \u03bb x, by simp only [zero_apply, eq_zero_of_bot_submodule x]\u27e9\n\n@[simp] lemma closed_complemented_top : closed_complemented (\u22a4 : submodule R M) :=\n\u27e8(id R M).cod_restrict \u22a4 (\u03bb x, trivial), \u03bb x, subtype.ext_iff_val.2 $ by simp\u27e9\n\nend submodule\n\nlemma continuous_linear_map.closed_complemented_ker_of_right_inverse {R : Type*} [ring R]\n  {M : Type*} [topological_space M] [add_comm_group M]\n  {M\u2082 : Type*} [topological_space M\u2082] [add_comm_group M\u2082] [module R M] [module R M\u2082]\n  [topological_add_group M] (f\u2081 : M \u2192L[R] M\u2082) (f\u2082 : M\u2082 \u2192L[R] M)\n  (h : function.right_inverse f\u2082 f\u2081) :\n  f\u2081.ker.closed_complemented :=\n\u27e8f\u2081.proj_ker_of_right_inverse f\u2082 h, f\u2081.proj_ker_of_right_inverse_apply_idem f\u2082 h\u27e9\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/topology/algebra/module/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37108288726693794}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.products.bifunctor\n\n/-!\n# Curry and uncurry, as functors.\n\nWe define `curry : ((C \u00d7 D) \u2964 E) \u2964 (C \u2964 (D \u2964 E))` and `uncurry : (C \u2964 (D \u2964 E)) \u2964 ((C \u00d7 D) \u2964 E)`,\nand verify that they provide an equivalence of categories\n`currying : (C \u2964 (D \u2964 E)) \u224c ((C \u00d7 D) \u2964 E)`.\n\n-/\nnamespace category_theory\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n          {D : Type u\u2082} [category.{v\u2082} D]\n          {E : Type u\u2083} [category.{v\u2083} E]\n\n/--\nThe uncurrying functor, taking a functor `C \u2964 (D \u2964 E)` and producing a functor `(C \u00d7 D) \u2964 E`.\n-/\ndef uncurry : (C \u2964 (D \u2964 E)) \u2964 ((C \u00d7 D) \u2964 E) :=\n{ obj := \u03bb F,\n  { obj := \u03bb X, (F.obj X.1).obj X.2,\n    map := \u03bb X Y f, (F.map f.1).app X.2 \u226b (F.obj Y.1).map f.2,\n    map_comp' := \u03bb X Y Z f g,\n    begin\n      simp only [prod_comp_fst, prod_comp_snd, functor.map_comp,\n                 nat_trans.comp_app, category.assoc],\n      slice_lhs 2 3 { rw \u2190 nat_trans.naturality },\n      rw category.assoc,\n    end },\n  map := \u03bb F G T,\n  { app := \u03bb X, (T.app X.1).app X.2,\n    naturality' := \u03bb X Y f,\n    begin\n      simp only [prod_comp_fst, prod_comp_snd, category.comp_id, category.assoc,\n        functor.map_id, functor.map_comp, nat_trans.id_app, nat_trans.comp_app],\n      slice_lhs 2 3 { rw nat_trans.naturality },\n      slice_lhs 1 2 { rw [\u2190nat_trans.comp_app, nat_trans.naturality, nat_trans.comp_app] },\n      rw category.assoc,\n    end } }.\n\n/--\nThe object level part of the currying functor. (See `curry` for the functorial version.)\n-/\ndef curry_obj (F : (C \u00d7 D) \u2964 E) : C \u2964 (D \u2964 E) :=\n{ obj := \u03bb X,\n    { obj := \u03bb Y, F.obj (X, Y),\n      map := \u03bb Y Y' g, F.map (\ud835\udfd9 X, g) },\n    map := \u03bb X X' f, { app := \u03bb Y, F.map (f, \ud835\udfd9 Y) } }\n\n/--\nThe currying functor, taking a functor `(C \u00d7 D) \u2964 E` and producing a functor `C \u2964 (D \u2964 E)`.\n-/\ndef curry : ((C \u00d7 D) \u2964 E) \u2964 (C \u2964 (D \u2964 E)) :=\n{ obj := \u03bb F, curry_obj F,\n  map := \u03bb F G T,\n  { app := \u03bb X,\n    { app := \u03bb Y, T.app (X, Y),\n      naturality' := \u03bb Y Y' g,\n      begin\n        dsimp [curry_obj],\n        rw nat_trans.naturality,\n      end },\n    naturality' := \u03bb X X' f,\n    begin\n      ext, dsimp [curry_obj],\n      rw nat_trans.naturality,\n    end } }.\n\n@[simp] lemma uncurry.obj_obj {F : C \u2964 (D \u2964 E)} {X : C \u00d7 D} :\n  (uncurry.obj F).obj X = (F.obj X.1).obj X.2 := rfl\n@[simp] lemma uncurry.obj_map {F : C \u2964 (D \u2964 E)} {X Y : C \u00d7 D} {f : X \u27f6 Y} :\n  (uncurry.obj F).map f = ((F.map f.1).app X.2) \u226b ((F.obj Y.1).map f.2) := rfl\n@[simp] lemma uncurry.map_app {F G : C \u2964 (D \u2964 E)} {\u03b1 : F \u27f6 G} {X : C \u00d7 D} :\n  (uncurry.map \u03b1).app X = (\u03b1.app X.1).app X.2 := rfl\n@[simp] lemma curry.obj_obj_obj\n  {F : (C \u00d7 D) \u2964 E} {X : C} {Y : D} :\n  ((curry.obj F).obj X).obj Y = F.obj (X, Y) := rfl\n@[simp] lemma curry.obj_obj_map\n  {F : (C \u00d7 D) \u2964 E} {X : C} {Y Y' : D} {g : Y \u27f6 Y'} :\n  ((curry.obj F).obj X).map g = F.map (\ud835\udfd9 X, g) := rfl\n@[simp] lemma curry.obj_map_app {F : (C \u00d7 D) \u2964 E} {X X' : C} {f : X \u27f6 X'} {Y} :\n  ((curry.obj F).map f).app Y = F.map (f, \ud835\udfd9 Y) := rfl\n@[simp] lemma curry.map_app_app {F G : (C \u00d7 D) \u2964 E} {\u03b1 : F \u27f6 G} {X} {Y} :\n  ((curry.map \u03b1).app X).app Y = \u03b1.app (X, Y) := rfl\n\n/--\nThe equivalence of functor categories given by currying/uncurrying.\n-/\n@[simps] -- create projection simp lemmas even though this isn't a `{ .. }`.\ndef currying : (C \u2964 (D \u2964 E)) \u224c ((C \u00d7 D) \u2964 E) :=\nequivalence.mk uncurry curry\n  (nat_iso.of_components (\u03bb F, nat_iso.of_components\n    (\u03bb X, nat_iso.of_components (\u03bb Y, iso.refl _) (by tidy)) (by tidy)) (by tidy))\n  (nat_iso.of_components (\u03bb F, nat_iso.of_components\n    (\u03bb X, eq_to_iso (by simp)) (by tidy)) (by tidy))\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/currying.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37108288726693794}}
{"text": "import Mathlib.Tactic.Constructor\n\nstructure Foo where\n  a : Type\n  x : List a\n\n-- fconstructor\nexample : Foo := by\n  fconstructor\n  exact Nat\n  exact [0,1,2]\n\n-- econstructor\nexample : Foo := by\n  econstructor\n  exact [0,1,2]\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/Constructor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.37108287899745385}}
{"text": "import lambda_calculus.utlc.beta.distance\nimport lambda_calculus.utlc.beta.encoding.basic\nimport lambda_calculus.utlc.beta.encoding.core\nimport lambda_calculus.utlc.beta.encoding.nat\nimport lambda_calculus.utlc.beta.encoding.utlc\nimport lambda_calculus.utlc.beta.complexity.core\nimport lambda_calculus.utlc.beta.complexity.nat\nimport complexity.basic\nimport complexity.nat\n\nopen complexity\nopen lambda_calculus.utlc.\u03b2.encoding\n\nnamespace lambda_calculus\nnamespace utlc\nnamespace \u03b2\nnamespace complexity\nnamespace utlc\n\nnamespace rec_complexity\n\ndef cost {\u03b1: Type}\n  (rf: \u03a0 (f: utlc), (\u03bb _: utlc, \u03b1) f)\n  (c_down: \u2115 \u2192 \u2115) (c_lambda: utlc \u2192 \u03b1 \u2192 \u2115)\n  (c_dot: utlc \u2192 utlc \u2192 \u03b1 \u2192 \u03b1 \u2192 \u2115): utlc \u2192 \u2115\n| (\u2193n) := c_down n + 8\n| (\u039b f) := c_lambda f (rf f) + cost f + 9\n| (f\u00b7g) := c_dot f g (rf f) (rf g) + cost f + cost g + 10\n\nend rec_complexity\n\nlocal attribute [simp] closed closed_below\nlocal attribute [simp] \u03b2.normal_iteration \u03b2.strategic_reduction_step\nlocal attribute [simp] substitution down_shift head_reduced\nlocal attribute [simp] complexity.cast_unwrap distance_model\nlocal attribute [simp] encoding.utlc.encode_utlc\n\ndef handle_down (f y: utlc): utlc := f\ndef handle_lambda (f y: utlc): utlc := \u039b (f \u2191\u00b9 0)\u00b7\u21930\u00b7((y \u2191\u00b9 0)\u00b7\u21930)\ndef handle_dot (f y: utlc): utlc := \u039b \u039b (f \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21931\u00b7\u21930\u00b7((y \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21931)\u00b7((y \u2191\u00b9 0 \u2191\u00b9 1)\u00b7\u21930)\ndef rec_utlc (y g f\u2080 f\u2081 f\u2082: utlc): utlc := g\u00b7handle_down f\u2080 y\u00b7handle_lambda f\u2081 y\u00b7handle_dot f\u2082 y\n\nlocal notation a `[` b `:=` c  `]` : 70 := has_substitution.substitution a b c\n\ntheorem rec_utlc_sub {y g f\u2080 f\u2081 f\u2082: utlc} (n: \u2115) (x: utlc):\n  (rec_utlc y g f\u2080 f\u2081 f\u2082)[n:=x] = (rec_utlc (y[n:=x]) (g[n:=x]) (f\u2080[n:=x]) (f\u2081[n:=x]) (f\u2082[n:=x])) :=\nbegin\n  simp [rec_utlc, handle_down, handle_lambda, handle_dot, substitution_shift_ge],\n  repeat { rw [substitution_shift_ge] },\n  all_goals { linarith },\nend\n\ntheorem rec_utlc_down (et: encoding_type) [\u2115_en: has_encoding (distance_model et) \u2115] (y: utlc) (n: \u2115) (f\u2080 f\u2081 f\u2082: utlc):\n  distance_le 3\n    (rec_utlc y (encoding.utlc.encode_utlc et (\u2193n)).value f\u2080 f\u2081 f\u2082)\n    (f\u2080\u00b7(complexity.encode (distance_model et) n).value) :=\nbegin\n  rw [rec_utlc, utlc.encode_utlc, \u03b2.encoding.alternative],\n  simp,\n  apply distance_le_of_normal_iteration,\n  simp [handle_down],\nend\n\ntheorem rec_utlc_lambda (et: encoding_type) [\u2115_en: has_encoding (distance_model et) \u2115] (y g f\u2080 f\u2081 f\u2082: utlc):\n  distance_le 4\n    (rec_utlc y (encoding.utlc.encode_utlc et (\u039b g)).value f\u2080 f\u2081 f\u2082)\n    (f\u2081\u00b7(complexity.encode (distance_model et) g).value\u00b7(y\u00b7(complexity.encode (distance_model et) g).value)) :=\nbegin\n  rw [rec_utlc, utlc.encode_utlc, \u03b2.encoding.alternative],\n  simp,\n  apply distance_le_of_normal_iteration 4,\n  simp [handle_lambda, utlc.encode_utlc, encode],\n  refl,\nend\n\ntheorem rec_utlc_dot (et: encoding_type) [\u2115_en: has_encoding (distance_model et) \u2115] (y g\u2080 g\u2081 f\u2080 f\u2081 f\u2082: utlc):\n  distance_le 5\n    (rec_utlc y (encoding.utlc.encode_utlc et (g\u2080\u00b7g\u2081)).value f\u2080 f\u2081 f\u2082)\n    (f\u2082\u00b7(complexity.encode (distance_model et) g\u2080).value\u00b7(complexity.encode (distance_model et) g\u2081).value\u00b7(y\u00b7(complexity.encode (distance_model et) g\u2080).value)\u00b7(y\u00b7(complexity.encode (distance_model et) g\u2081).value)) :=\nbegin\n  rw [rec_utlc, utlc.encode_utlc, \u03b2.encoding.alternative],\n  simp,\n  apply distance_le_of_normal_iteration 5,\n  simp [handle_dot, and_assoc],\n  refine \u27e8 rfl, rfl, rfl, rfl \u27e9,\nend\n\ninstance rec_complexity\n  (\u03b1: Type) (et: encoding_type) [\u03b1_en: has_encoding (distance_model et) \u03b1]\n  [\u2115_en: has_encoding (distance_model et) \u2115]\n  (f\u2080: \u2115 \u2192 \u03b1) [cf\u2080: has_complexity (distance_model et) f\u2080]\n  (f\u2081: utlc \u2192 \u03b1 \u2192 \u03b1) [cf\u2081: has_complexity (distance_model et) f\u2081]\n  (f\u2082: utlc \u2192 utlc \u2192 \u03b1 \u2192 \u03b1 \u2192 \u03b1) [cf\u2082: has_complexity (distance_model et) f\u2082]:\n  has_complexity (distance_model et) (simp_rec f\u2080 f\u2081 f\u2082) :=\nbegin\n  fconstructor,\n  fconstructor,\n  exact rec_complexity.cost (simp_rec f\u2080 f\u2081 f\u2082) cf\u2080.value.cost cf\u2081.value.cost cf\u2082.value.cost,\n  rcases cf\u2080.value with \u27e8 cfc\u2080, fp\u2080, cfp\u2080 \u27e9,\n  rcases cf\u2081.value with \u27e8 cfc\u2081, fp\u2081, cfp\u2081 \u27e9,\n  rcases cf\u2082.value with \u27e8 cfc\u2082, fp\u2082, cfp\u2082 \u27e9,\n  fconstructor,\n  fconstructor,\n  exact yrec (\u039b \u039b rec_utlc (\u21931:utlc) (\u21930:utlc) fp\u2080.value fp\u2081.value fp\u2082.value),\n  simp [ycomb, yrec, rec_utlc, handle_down, handle_lambda, handle_dot],\n  simp [cast_unwrap],\n  intro a,\n  simp [has_encoding.value, distance_model, complexity.encode],\n  induction a,\n  all_goals { apply distance_le_trans',\n    apply dot_distance_le_dot_left,\n    apply yrec_apply,\n    apply distance_le_trans',\n    apply distance_le_of_normal_iteration 2,\n    simp [rec_utlc_sub] },\n  { apply distance_le_trans',\n    apply rec_utlc_down,\n    simp,\n    apply cfp\u2080,\n    refl },\n  refl,\n  simp [rec_complexity.cost],\n  ring,\n  { apply distance_le_trans',\n    apply rec_utlc_lambda,\n    apply distance_le_trans',\n    apply dot_distance_le_dot_right,\n    apply a_ih,\n    simp,\n    apply cfp\u2081,\n    refl,\n    refl },\n  refl,\n  simp [rec_complexity.cost],\n  ring,\n  { apply distance_le_trans',\n    apply rec_utlc_dot,\n    apply distance_le_trans',\n    apply dot_distance_le_dot_right,\n    apply a_ih_g,\n    apply distance_le_trans',\n    apply dot_distance_le_dot_left,\n    apply dot_distance_le_dot_right,\n    apply a_ih_f,\n    apply cfp\u2082,\n    refl,\n    refl,\n    refl },\n  refl,\n  simp [rec_complexity.cost],\n  ring_nf,\nend\n\ninstance size_complexity:\n  has_complexity church_model utlc.size :=\nbegin\n  fconstructor,\n  fconstructor,\n  exact (\u03bb f, 67 * f.size: utlc \u2192 \u2115),\n  apply omega_equiv,\n  rotate 2,\n  exact (simp_rec\n    (@const \u2115 \u2115 1)\n    (curry (compose (nat.add (1:\u2115)) prod.snd))\n    (curry (curry (curry (compose (uncurry nat.add) (fork (compose prod.snd prod.fst) (compose (nat.add 1) prod.snd))))))),\n  apply complexity_of_instance,\n  intro f,\n  simp [complexity, has_complexity.value],\n  norm_num,\n  unfold_coes,\n  unfold has_add.add,\n  unfold cost_function.add,\n  unfold cost_function.less_than_or_equal,\n  conv {\n    to_lhs,\n    congr,\n    skip,\n    skip,\n    skip,\n    funext,\n    whnf,\n    simp [nat.succ_eq_add_one],\n    ring_nf,\n    rw [nat.add_comm],\n    whnf,\n    simp [nat.succ_eq_add_one],\n    norm_num },\n  induction f,\n  { simp [rec_complexity.cost],\n    linarith },\n  { simp [rec_complexity.cost, add_mul],\n    ring_nf,\n    apply add_le_add,\n    apply f_ih,\n    linarith },\n  { simp [rec_complexity.cost, add_mul],\n    ring_nf,\n    apply add_le_add,\n    apply f_ih_f,\n    apply add_le_add,\n    apply f_ih_g,\n    refl },\n  ext1,\n  simp [const, curry, compose, uncurry, fork],\n  induction x,\n  { simp },\n  { simp [x_ih, nat.add_comm _ 1] },\n  { simp [x_ih_f, x_ih_g, add_comm 1 x_g.size, add_assoc x_f.size] }\nend\n\nend utlc\nend complexity\nend \u03b2\nend utlc\nend lambda_calculus\n", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/lambda_calculus/utlc/beta/complexity/utlc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3709281565761754}}
{"text": "/-\nCopyright (c) 2022 Moritz Doll. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Moritz Doll\n\n! This file was ported from Lean 3 source module linear_algebra.basis.bilinear\n! leanprover-community/mathlib commit 832f7b9162039c28b9361289c8681f155cae758f\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.LinearAlgebra.Basis\nimport Mathbin.LinearAlgebra.BilinearMap\n\n/-!\n# Lemmas about bilinear maps with a basis over each argument\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nnamespace LinearMap\n\nvariable {\u03b9\u2081 \u03b9\u2082 : Type _}\n\nvariable {R R\u2082 S S\u2082 M N P : Type _}\n\nvariable {M\u2097 N\u2097 P\u2097 : Type _}\n\nvariable [CommSemiring R] [CommSemiring S] [CommSemiring R\u2082] [CommSemiring S\u2082]\n\nsection AddCommMonoid\n\nvariable [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P]\n\nvariable [AddCommMonoid M\u2097] [AddCommMonoid N\u2097] [AddCommMonoid P\u2097]\n\nvariable [Module R M] [Module S N] [Module R\u2082 P] [Module S\u2082 P]\n\nvariable [Module R M\u2097] [Module R N\u2097] [Module R P\u2097]\n\nvariable [SMulCommClass S\u2082 R\u2082 P]\n\nvariable {\u03c1\u2081\u2082 : R \u2192+* R\u2082} {\u03c3\u2081\u2082 : S \u2192+* S\u2082}\n\nvariable (b\u2081 : Basis \u03b9\u2081 R M) (b\u2082 : Basis \u03b9\u2082 S N) (b\u2081' : Basis \u03b9\u2081 R M\u2097) (b\u2082' : Basis \u03b9\u2082 R N\u2097)\n\n/- warning: linear_map.ext_basis -> LinearMap.ext_basis is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9\u2081 : Type.{u1}} {\u03b9\u2082 : Type.{u2}} {R : Type.{u3}} {R\u2082 : Type.{u4}} {S : Type.{u5}} {S\u2082 : Type.{u6}} {M : Type.{u7}} {N : Type.{u8}} {P : Type.{u9}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommSemiring.{u5} S] [_inst_3 : CommSemiring.{u4} R\u2082] [_inst_4 : CommSemiring.{u6} S\u2082] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u8} N] [_inst_7 : AddCommMonoid.{u9} P] [_inst_11 : Module.{u3, u7} R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5] [_inst_12 : Module.{u5, u8} S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6] [_inst_13 : Module.{u4, u9} R\u2082 P (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_7] [_inst_14 : Module.{u6, u9} S\u2082 P (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u6, u4, u9} S\u2082 R\u2082 P (SMulZeroClass.toHasSmul.{u6, u9} S\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u6, u9} S\u2082 P (MulZeroClass.toHasZero.{u6} S\u2082 (MulZeroOneClass.toMulZeroClass.{u6} S\u2082 (MonoidWithZero.toMulZeroOneClass.{u6} S\u2082 (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u6, u9} S\u2082 P (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u6, u9} S\u2082 P (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_7 _inst_14)))) (SMulZeroClass.toHasSmul.{u4, u9} R\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u4, u9} R\u2082 P (MulZeroClass.toHasZero.{u4} R\u2082 (MulZeroOneClass.toMulZeroClass.{u4} R\u2082 (MonoidWithZero.toMulZeroOneClass.{u4} R\u2082 (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u4, u9} R\u2082 P (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u4, u9} R\u2082 P (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_7 _inst_13))))] {\u03c1\u2081\u2082 : RingHom.{u3, u4} R R\u2082 (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))} {\u03c3\u2081\u2082 : RingHom.{u5, u6} S S\u2082 (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))} (b\u2081 : Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (b\u2082 : Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) {B : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)} {B' : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)}, (forall (i : \u03b9\u2081) (j : \u03b9\u2082), Eq.{succ u9} P (coeFn.{max (succ u8) (succ u9), max (succ u8) (succ u9)} (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (fun (_x : LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) => N -> P) (LinearMap.hasCoeToFun.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (coeFn.{max (succ u7) (succ (max u8 u9)), max (succ u7) (succ (max u8 u9))} (LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) (fun (_x : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) => M -> (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14)) (LinearMap.hasCoeToFun.{u3, u4, u7, max u8 u9} R R\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B (coeFn.{max (succ u1) (succ u3) (succ u7), max (succ u1) (succ u7)} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (fun (_x : Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) => \u03b9\u2081 -> M) (FunLike.hasCoeToFun.{max (succ u1) (succ u3) (succ u7), succ u1, succ u7} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => M) (Basis.funLike.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11)) b\u2081 i)) (coeFn.{max (succ u2) (succ u5) (succ u8), max (succ u2) (succ u8)} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) (fun (_x : Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) => \u03b9\u2082 -> N) (FunLike.hasCoeToFun.{max (succ u2) (succ u5) (succ u8), succ u2, succ u8} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => N) (Basis.funLike.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12)) b\u2082 j)) (coeFn.{max (succ u8) (succ u9), max (succ u8) (succ u9)} (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (fun (_x : LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) => N -> P) (LinearMap.hasCoeToFun.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (coeFn.{max (succ u7) (succ (max u8 u9)), max (succ u7) (succ (max u8 u9))} (LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) (fun (_x : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) => M -> (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14)) (LinearMap.hasCoeToFun.{u3, u4, u7, max u8 u9} R R\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B' (coeFn.{max (succ u1) (succ u3) (succ u7), max (succ u1) (succ u7)} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (fun (_x : Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) => \u03b9\u2081 -> M) (FunLike.hasCoeToFun.{max (succ u1) (succ u3) (succ u7), succ u1, succ u7} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => M) (Basis.funLike.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11)) b\u2081 i)) (coeFn.{max (succ u2) (succ u5) (succ u8), max (succ u2) (succ u8)} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) (fun (_x : Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) => \u03b9\u2082 -> N) (FunLike.hasCoeToFun.{max (succ u2) (succ u5) (succ u8), succ u2, succ u8} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => N) (Basis.funLike.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12)) b\u2082 j))) -> (Eq.{max (succ u7) (succ (max u8 u9))} (LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) B B')\nbut is expected to have type\n  forall {\u03b9\u2081 : Type.{u1}} {\u03b9\u2082 : Type.{u2}} {R : Type.{u9}} {R\u2082 : Type.{u8}} {S : Type.{u4}} {S\u2082 : Type.{u3}} {M : Type.{u7}} {N : Type.{u5}} {P : Type.{u6}} [_inst_1 : CommSemiring.{u9} R] [_inst_2 : CommSemiring.{u4} S] [_inst_3 : CommSemiring.{u8} R\u2082] [_inst_4 : CommSemiring.{u3} S\u2082] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u5} N] [_inst_7 : AddCommMonoid.{u6} P] [_inst_11 : Module.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5] [_inst_12 : Module.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6] [_inst_13 : Module.{u8, u6} R\u2082 P (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_7] [_inst_14 : Module.{u3, u6} S\u2082 P (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u3, u8, u6} S\u2082 R\u2082 P (SMulZeroClass.toSMul.{u3, u6} S\u2082 P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u3, u6} S\u2082 P (CommMonoidWithZero.toZero.{u3} S\u2082 (CommSemiring.toCommMonoidWithZero.{u3} S\u2082 _inst_4)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (MulActionWithZero.toSMulWithZero.{u3, u6} S\u2082 P (Semiring.toMonoidWithZero.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (Module.toMulActionWithZero.{u3, u6} S\u2082 P (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_7 _inst_14)))) (SMulZeroClass.toSMul.{u8, u6} R\u2082 P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u8, u6} R\u2082 P (CommMonoidWithZero.toZero.{u8} R\u2082 (CommSemiring.toCommMonoidWithZero.{u8} R\u2082 _inst_3)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (MulActionWithZero.toSMulWithZero.{u8, u6} R\u2082 P (Semiring.toMonoidWithZero.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (Module.toMulActionWithZero.{u8, u6} R\u2082 P (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_7 _inst_13))))] {\u03c1\u2081\u2082 : RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))} {\u03c3\u2081\u2082 : RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))} (b\u2081 : Basis.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) (b\u2082 : Basis.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) {B : LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)} {B' : LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)}, (forall (i : \u03b9\u2081) (j : \u03b9\u2082), Eq.{succ u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u2) (succ u4)) (succ u5), succ u2, succ u5} (Basis.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (FunLike.coe.{max (succ u5) (succ u6), succ u5, succ u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (FunLike.coe.{max (max (succ u1) (succ u9)) (succ u7), succ u1, succ u7} (Basis.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (a : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) a) (Basis.funLike.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (FunLike.coe.{max (max (succ u7) (succ u5)) (succ u6), succ u7, max (succ u5) (succ u6)} (LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _x) (LinearMap.instFunLikeLinearMap.{u9, u8, u7, max u5 u6} R R\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B (FunLike.coe.{max (max (succ u1) (succ u9)) (succ u7), succ u1, succ u7} (Basis.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) _x) (Basis.funLike.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) (FunLike.coe.{max (max (succ u2) (succ u4)) (succ u5), succ u2, succ u5} (Basis.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) _x) (Basis.funLike.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (FunLike.coe.{max (succ u5) (succ u6), succ u5, succ u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (FunLike.coe.{max (max (succ u1) (succ u9)) (succ u7), succ u1, succ u7} (Basis.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (a : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) a) (Basis.funLike.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (FunLike.coe.{max (max (succ u7) (succ u5)) (succ u6), succ u7, max (succ u5) (succ u6)} (LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _x) (LinearMap.instFunLikeLinearMap.{u9, u8, u7, max u5 u6} R R\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B' (FunLike.coe.{max (max (succ u1) (succ u9)) (succ u7), succ u1, succ u7} (Basis.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) _x) (Basis.funLike.{u1, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) (FunLike.coe.{max (max (succ u2) (succ u4)) (succ u5), succ u2, succ u5} (Basis.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) _x) (Basis.funLike.{u2, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j))) -> (Eq.{max (max (succ u7) (succ u5)) (succ u6)} (LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)) B B')\nCase conversion may be inaccurate. Consider using '#align linear_map.ext_basis LinearMap.ext_basis\u2093'. -/\n/-- Two bilinear maps are equal when they are equal on all basis vectors. -/\ntheorem ext_basis {B B' : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P} (h : \u2200 i j, B (b\u2081 i) (b\u2082 j) = B' (b\u2081 i) (b\u2082 j)) :\n    B = B' :=\n  b\u2081.ext fun i => b\u2082.ext fun j => h i j\n#align linear_map.ext_basis LinearMap.ext_basis\n\n/- warning: linear_map.sum_repr_mul_repr_mul\u209b\u2097 -> LinearMap.sum_repr_mul_repr_mul\u209b\u2097 is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9\u2081 : Type.{u1}} {\u03b9\u2082 : Type.{u2}} {R : Type.{u3}} {R\u2082 : Type.{u4}} {S : Type.{u5}} {S\u2082 : Type.{u6}} {M : Type.{u7}} {N : Type.{u8}} {P : Type.{u9}} [_inst_1 : CommSemiring.{u3} R] [_inst_2 : CommSemiring.{u5} S] [_inst_3 : CommSemiring.{u4} R\u2082] [_inst_4 : CommSemiring.{u6} S\u2082] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u8} N] [_inst_7 : AddCommMonoid.{u9} P] [_inst_11 : Module.{u3, u7} R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5] [_inst_12 : Module.{u5, u8} S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6] [_inst_13 : Module.{u4, u9} R\u2082 P (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_7] [_inst_14 : Module.{u6, u9} S\u2082 P (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u6, u4, u9} S\u2082 R\u2082 P (SMulZeroClass.toHasSmul.{u6, u9} S\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u6, u9} S\u2082 P (MulZeroClass.toHasZero.{u6} S\u2082 (MulZeroOneClass.toMulZeroClass.{u6} S\u2082 (MonoidWithZero.toMulZeroOneClass.{u6} S\u2082 (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u6, u9} S\u2082 P (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u6, u9} S\u2082 P (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_7 _inst_14)))) (SMulZeroClass.toHasSmul.{u4, u9} R\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u4, u9} R\u2082 P (MulZeroClass.toHasZero.{u4} R\u2082 (MulZeroOneClass.toMulZeroClass.{u4} R\u2082 (MonoidWithZero.toMulZeroOneClass.{u4} R\u2082 (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u4, u9} R\u2082 P (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u4, u9} R\u2082 P (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_7 _inst_13))))] {\u03c1\u2081\u2082 : RingHom.{u3, u4} R R\u2082 (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))} {\u03c3\u2081\u2082 : RingHom.{u5, u6} S S\u2082 (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))} (b\u2081 : Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (b\u2082 : Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) {B : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)} (x : M) (y : N), Eq.{succ u9} P (Finsupp.sum.{u1, u3, u9} \u03b9\u2081 R P (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_7 (coeFn.{max (succ u7) (succ (max u1 u3)), max (succ u7) (succ (max u1 u3))} (LinearEquiv.{u3, u3, u7, max u1 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) M (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_5 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_11 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (fun (_x : LinearEquiv.{u3, u3, u7, max u1 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) M (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_5 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_11 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) => M -> (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))))) (LinearEquiv.hasCoeToFun.{u3, u3, u7, max u1 u3} R R M (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_11 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (Basis.repr.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11 b\u2081) x) (fun (i : \u03b9\u2081) (xi : R) => Finsupp.sum.{u2, u5, u9} \u03b9\u2082 S P (MulZeroClass.toHasZero.{u5} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))))) _inst_7 (coeFn.{max (succ u8) (succ (max u2 u5)), max (succ u8) (succ (max u2 u5))} (LinearEquiv.{u5, u5, u8, max u2 u5} S S (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u5} S _inst_2) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) N (Finsupp.{u2, u5} \u03b9\u2082 S (MulZeroClass.toHasZero.{u5} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))))) _inst_6 (Finsupp.addCommMonoid.{u2, u5} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))))) _inst_12 (Finsupp.module.{u2, u5, u5} \u03b9\u2082 S S (CommSemiring.toSemiring.{u5} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))) (Semiring.toModule.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))) (fun (_x : LinearEquiv.{u5, u5, u8, max u2 u5} S S (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u5} S _inst_2) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) N (Finsupp.{u2, u5} \u03b9\u2082 S (MulZeroClass.toHasZero.{u5} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))))) _inst_6 (Finsupp.addCommMonoid.{u2, u5} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))))) _inst_12 (Finsupp.module.{u2, u5, u5} \u03b9\u2082 S S (CommSemiring.toSemiring.{u5} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))) (Semiring.toModule.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))) => N -> (Finsupp.{u2, u5} \u03b9\u2082 S (MulZeroClass.toHasZero.{u5} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))))))) (LinearEquiv.hasCoeToFun.{u5, u5, u8, max u2 u5} S S N (Finsupp.{u2, u5} \u03b9\u2082 S (MulZeroClass.toHasZero.{u5} S (NonUnitalNonAssocSemiring.toMulZeroClass.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))))) (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 (Finsupp.addCommMonoid.{u2, u5} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))))) _inst_12 (Finsupp.module.{u2, u5, u5} \u03b9\u2082 S S (CommSemiring.toSemiring.{u5} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u5} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)))) (Semiring.toModule.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHom.id.{u5} S (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (RingHomInvPair.ids.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2))) (Basis.repr.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12 b\u2082) y) (fun (j : \u03b9\u2082) (yj : S) => SMul.smul.{u4, u9} R\u2082 P (SMulZeroClass.toHasSmul.{u4, u9} R\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u4, u9} R\u2082 P (MulZeroClass.toHasZero.{u4} R\u2082 (MulZeroOneClass.toMulZeroClass.{u4} R\u2082 (MonoidWithZero.toMulZeroOneClass.{u4} R\u2082 (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u4, u9} R\u2082 P (Semiring.toMonoidWithZero.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u4, u9} R\u2082 P (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_7 _inst_13)))) (coeFn.{max (succ u3) (succ u4), max (succ u3) (succ u4)} (RingHom.{u3, u4} R R\u2082 (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))) (fun (_x : RingHom.{u3, u4} R R\u2082 (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))) => R -> R\u2082) (RingHom.hasCoeToFun.{u3, u4} R R\u2082 (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (Semiring.toNonAssocSemiring.{u4} R\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3))) \u03c1\u2081\u2082 xi) (SMul.smul.{u6, u9} S\u2082 P (SMulZeroClass.toHasSmul.{u6, u9} S\u2082 P (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (SMulWithZero.toSmulZeroClass.{u6, u9} S\u2082 P (MulZeroClass.toHasZero.{u6} S\u2082 (MulZeroOneClass.toMulZeroClass.{u6} S\u2082 (MonoidWithZero.toMulZeroOneClass.{u6} S\u2082 (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))))) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (MulActionWithZero.toSMulWithZero.{u6, u9} S\u2082 P (Semiring.toMonoidWithZero.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4)) (AddZeroClass.toHasZero.{u9} P (AddMonoid.toAddZeroClass.{u9} P (AddCommMonoid.toAddMonoid.{u9} P _inst_7))) (Module.toMulActionWithZero.{u6, u9} S\u2082 P (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_7 _inst_14)))) (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (RingHom.{u5, u6} S S\u2082 (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))) (fun (_x : RingHom.{u5, u6} S S\u2082 (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))) => S -> S\u2082) (RingHom.hasCoeToFun.{u5, u6} S S\u2082 (Semiring.toNonAssocSemiring.{u5} S (CommSemiring.toSemiring.{u5} S _inst_2)) (Semiring.toNonAssocSemiring.{u6} S\u2082 (CommSemiring.toSemiring.{u6} S\u2082 _inst_4))) \u03c3\u2081\u2082 yj) (coeFn.{max (succ u8) (succ u9), max (succ u8) (succ u9)} (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (fun (_x : LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) => N -> P) (LinearMap.hasCoeToFun.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (coeFn.{max (succ u7) (succ (max u8 u9)), max (succ u7) (succ (max u8 u9))} (LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) (fun (_x : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) => M -> (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14)) (LinearMap.hasCoeToFun.{u3, u4, u7, max u8 u9} R R\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B (coeFn.{max (succ u1) (succ u3) (succ u7), max (succ u1) (succ u7)} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) (fun (_x : Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) => \u03b9\u2081 -> M) (FunLike.hasCoeToFun.{max (succ u1) (succ u3) (succ u7), succ u1, succ u7} (Basis.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => M) (Basis.funLike.{u1, u3, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u3} R _inst_1) _inst_5 _inst_11)) b\u2081 i)) (coeFn.{max (succ u2) (succ u5) (succ u8), max (succ u2) (succ u8)} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) (fun (_x : Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) => \u03b9\u2082 -> N) (FunLike.hasCoeToFun.{max (succ u2) (succ u5) (succ u8), succ u2, succ u8} (Basis.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => N) (Basis.funLike.{u2, u5, u8} \u03b9\u2082 S N (CommSemiring.toSemiring.{u5} S _inst_2) _inst_6 _inst_12)) b\u2082 j)))))) (coeFn.{max (succ u8) (succ u9), max (succ u8) (succ u9)} (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (fun (_x : LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) => N -> P) (LinearMap.hasCoeToFun.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (coeFn.{max (succ u7) (succ (max u8 u9)), max (succ u7) (succ (max u8 u9))} (LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) (fun (_x : LinearMap.{u3, u4, u7, max u8 u9} R R\u2082 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18)) => M -> (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14)) (LinearMap.hasCoeToFun.{u3, u4, u7, max u8 u9} R R\u2082 M (LinearMap.{u5, u6, u8, u9} S S\u2082 (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u5, u6, u8, u9} S S\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.module.{u5, u6, u4, u8, u9} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u5} S _inst_2) (CommSemiring.toSemiring.{u6} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u4} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B x) y)\nbut is expected to have type\n  forall {\u03b9\u2081 : Type.{u2}} {\u03b9\u2082 : Type.{u1}} {R : Type.{u9}} {R\u2082 : Type.{u8}} {S : Type.{u4}} {S\u2082 : Type.{u3}} {M : Type.{u7}} {N : Type.{u5}} {P : Type.{u6}} [_inst_1 : CommSemiring.{u9} R] [_inst_2 : CommSemiring.{u4} S] [_inst_3 : CommSemiring.{u8} R\u2082] [_inst_4 : CommSemiring.{u3} S\u2082] [_inst_5 : AddCommMonoid.{u7} M] [_inst_6 : AddCommMonoid.{u5} N] [_inst_7 : AddCommMonoid.{u6} P] [_inst_11 : Module.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5] [_inst_12 : Module.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6] [_inst_13 : Module.{u8, u6} R\u2082 P (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_7] [_inst_14 : Module.{u3, u6} S\u2082 P (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_7] [_inst_18 : SMulCommClass.{u3, u8, u6} S\u2082 R\u2082 P (SMulZeroClass.toSMul.{u3, u6} S\u2082 P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u3, u6} S\u2082 P (CommMonoidWithZero.toZero.{u3} S\u2082 (CommSemiring.toCommMonoidWithZero.{u3} S\u2082 _inst_4)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (MulActionWithZero.toSMulWithZero.{u3, u6} S\u2082 P (Semiring.toMonoidWithZero.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (Module.toMulActionWithZero.{u3, u6} S\u2082 P (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_7 _inst_14)))) (SMulZeroClass.toSMul.{u8, u6} R\u2082 P (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (SMulWithZero.toSMulZeroClass.{u8, u6} R\u2082 P (CommMonoidWithZero.toZero.{u8} R\u2082 (CommSemiring.toCommMonoidWithZero.{u8} R\u2082 _inst_3)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (MulActionWithZero.toSMulWithZero.{u8, u6} R\u2082 P (Semiring.toMonoidWithZero.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3)) (AddMonoid.toZero.{u6} P (AddCommMonoid.toAddMonoid.{u6} P _inst_7)) (Module.toMulActionWithZero.{u8, u6} R\u2082 P (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_7 _inst_13))))] {\u03c1\u2081\u2082 : RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))} {\u03c3\u2081\u2082 : RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))} (b\u2081 : Basis.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) (b\u2082 : Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) {B : LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)} (x : M) (y : N), Eq.{succ u6} P (Finsupp.sum.{u2, u9, u6} \u03b9\u2081 R P (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) _inst_7 (FunLike.coe.{max (max (succ u2) (succ u9)) (succ u7), succ u7, max (succ u2) (succ u9)} (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : M) => Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _x) (SMulHomClass.toFunLike.{max (max u2 u9) u7, u9, u7, max u2 u9} (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) R M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (SMulZeroClass.toSMul.{u9, u7} R M (AddMonoid.toZero.{u7} M (AddCommMonoid.toAddMonoid.{u7} M _inst_5)) (DistribSMul.toSMulZeroClass.{u9, u7} R M (AddMonoid.toAddZeroClass.{u7} M (AddCommMonoid.toAddMonoid.{u7} M _inst_5)) (DistribMulAction.toDistribSMul.{u9, u7} R M (MonoidWithZero.toMonoid.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (AddCommMonoid.toAddMonoid.{u7} M _inst_5) (Module.toDistribMulAction.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11)))) (SMulZeroClass.toSMul.{u9, max u2 u9} R (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (AddMonoid.toZero.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (DistribSMul.toSMulZeroClass.{u9, max u2 u9} R (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (AddMonoid.toAddZeroClass.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (DistribMulAction.toDistribSMul.{u9, max u2 u9} R (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (MonoidWithZero.toMonoid.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (AddCommMonoid.toAddMonoid.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))))) (Module.toDistribMulAction.{u9, max u2 u9} R (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u9) u7, u9, u7, max u2 u9} (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) R M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (MonoidWithZero.toMonoid.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (AddCommMonoid.toAddMonoid.{u7} M _inst_5) (AddCommMonoid.toAddMonoid.{max u2 u9} (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))))) (Module.toDistribMulAction.{u9, u7} R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) (Module.toDistribMulAction.{u9, max u2 u9} R (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (SemilinearMapClass.distribMulActionHomClass.{u9, u7, max u2 u9, max (max u2 u9) u7} R M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (SemilinearEquivClass.instSemilinearMapClass.{u9, u9, u7, max u2 u9, max (max u2 u9) u7} R R M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (LinearEquiv.{u9, u9, u7, max u9 u2} R R (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u9, u9, u7, max u2 u9} R R M (Finsupp.{u2, u9} \u03b9\u2081 R (MonoidWithZero.toZero.{u9} R (Semiring.toMonoidWithZero.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 (Finsupp.addCommMonoid.{u2, u9} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))) _inst_11 (Finsupp.module.{u2, u9, u9} \u03b9\u2081 R R (CommSemiring.toSemiring.{u9} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (Semiring.toModule.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHom.id.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (RingHomInvPair.ids.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))))))) (Basis.repr.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11 b\u2081) x) (fun (i : \u03b9\u2081) (xi : R) => Finsupp.sum.{u1, u4, u6} \u03b9\u2082 S P (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) _inst_7 (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u5, max (succ u1) (succ u4)} (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : N) => Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _x) (SMulHomClass.toFunLike.{max (max u1 u4) u5, u4, u5, max u1 u4} (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) S N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (SMulZeroClass.toSMul.{u4, u5} S N (AddMonoid.toZero.{u5} N (AddCommMonoid.toAddMonoid.{u5} N _inst_6)) (DistribSMul.toSMulZeroClass.{u4, u5} S N (AddMonoid.toAddZeroClass.{u5} N (AddCommMonoid.toAddMonoid.{u5} N _inst_6)) (DistribMulAction.toDistribSMul.{u4, u5} S N (MonoidWithZero.toMonoid.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (AddCommMonoid.toAddMonoid.{u5} N _inst_6) (Module.toDistribMulAction.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12)))) (SMulZeroClass.toSMul.{u4, max u1 u4} S (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (AddMonoid.toZero.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (AddCommMonoid.toAddMonoid.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))))) (DistribSMul.toSMulZeroClass.{u4, max u1 u4} S (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (AddMonoid.toAddZeroClass.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (AddCommMonoid.toAddMonoid.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))))) (DistribMulAction.toDistribSMul.{u4, max u1 u4} S (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (MonoidWithZero.toMonoid.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (AddCommMonoid.toAddMonoid.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))))) (Module.toDistribMulAction.{u4, max u1 u4} S (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (CommSemiring.toSemiring.{u4} S _inst_2) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u1 u4) u5, u4, u5, max u1 u4} (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) S N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (MonoidWithZero.toMonoid.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (AddCommMonoid.toAddMonoid.{u5} N _inst_6) (AddCommMonoid.toAddMonoid.{max u1 u4} (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))))) (Module.toDistribMulAction.{u4, u5} S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) (Module.toDistribMulAction.{u4, max u1 u4} S (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (CommSemiring.toSemiring.{u4} S _inst_2) (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (SemilinearMapClass.distribMulActionHomClass.{u4, u5, max u1 u4, max (max u1 u4) u5} S N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (SemilinearEquivClass.instSemilinearMapClass.{u4, u4, u5, max u1 u4, max (max u1 u4) u5} S S N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (LinearEquiv.{u4, u4, u5, max u4 u1} S S (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u4, u4, u5, max u1 u4} S S N (Finsupp.{u1, u4} \u03b9\u2082 S (MonoidWithZero.toZero.{u4} S (Semiring.toMonoidWithZero.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 (Finsupp.addCommMonoid.{u1, u4} \u03b9\u2082 S (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))) _inst_12 (Finsupp.module.{u1, u4, u4} \u03b9\u2082 S S (CommSemiring.toSemiring.{u4} S _inst_2) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (Semiring.toModule.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHom.id.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (RingHomInvPair.ids.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))))))) (Basis.repr.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12 b\u2082) y) (fun (j : \u03b9\u2082) (yj : S) => HSMul.hSMul.{u8, u6, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (instHSMul.{u8, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (SMulZeroClass.toSMul.{u8, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (SMulWithZero.toSMulZeroClass.{u8, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (CommMonoidWithZero.toZero.{u8} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) (CommSemiring.toCommMonoidWithZero.{u8} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) _inst_3)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (MulActionWithZero.toSMulWithZero.{u8, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (Semiring.toMonoidWithZero.{u8} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) (CommSemiring.toSemiring.{u8} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) _inst_3)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (Module.toMulActionWithZero.{u8, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (CommSemiring.toSemiring.{u8} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) xi) _inst_3) _inst_7 _inst_13))))) (FunLike.coe.{max (succ u9) (succ u8), succ u9, succ u8} (RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))) R (fun (_x : R) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : R) => R\u2082) _x) (MulHomClass.toFunLike.{max u9 u8, u9, u8} (RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))) R R\u2082 (NonUnitalNonAssocSemiring.toMul.{u9} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)))) (NonUnitalNonAssocSemiring.toMul.{u8} R\u2082 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u8} R\u2082 (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3)))) (NonUnitalRingHomClass.toMulHomClass.{max u9 u8, u9, u8} (RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))) R R\u2082 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u9} R (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u8} R\u2082 (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))) (RingHomClass.toNonUnitalRingHomClass.{max u9 u8, u9, u8} (RingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3))) R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3)) (RingHom.instRingHomClassRingHom.{u9, u8} R R\u2082 (Semiring.toNonAssocSemiring.{u9} R (CommSemiring.toSemiring.{u9} R _inst_1)) (Semiring.toNonAssocSemiring.{u8} R\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3)))))) \u03c1\u2081\u2082 xi) (HSMul.hSMul.{u3, u6, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (instHSMul.{u3, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (SMulZeroClass.toSMul.{u3, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (SMulWithZero.toSMulZeroClass.{u3, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (CommMonoidWithZero.toZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) (CommSemiring.toCommMonoidWithZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) _inst_4)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (MulActionWithZero.toSMulWithZero.{u3, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (Semiring.toMonoidWithZero.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) (CommSemiring.toSemiring.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) _inst_4)) (AddMonoid.toZero.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (AddCommMonoid.toAddMonoid.{u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) _inst_7)) (Module.toMulActionWithZero.{u3, u6} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) a) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)) (CommSemiring.toSemiring.{u3} ((fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) yj) _inst_4) _inst_7 _inst_14))))) (FunLike.coe.{max (succ u4) (succ u3), succ u4, succ u3} (RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))) S (fun (_x : S) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : S) => S\u2082) _x) (MulHomClass.toFunLike.{max u4 u3, u4, u3} (RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))) S S\u2082 (NonUnitalNonAssocSemiring.toMul.{u4} S (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)))) (NonUnitalNonAssocSemiring.toMul.{u3} S\u2082 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S\u2082 (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4)))) (NonUnitalRingHomClass.toMulHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))) S S\u2082 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u4} S (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} S\u2082 (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))) (RingHomClass.toNonUnitalRingHomClass.{max u4 u3, u4, u3} (RingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4))) S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4)) (RingHom.instRingHomClassRingHom.{u4, u3} S S\u2082 (Semiring.toNonAssocSemiring.{u4} S (CommSemiring.toSemiring.{u4} S _inst_2)) (Semiring.toNonAssocSemiring.{u3} S\u2082 (CommSemiring.toSemiring.{u3} S\u2082 _inst_4)))))) \u03c3\u2081\u2082 yj) (FunLike.coe.{max (succ u5) (succ u6), succ u5, succ u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (FunLike.coe.{max (max (succ u2) (succ u9)) (succ u7), succ u2, succ u7} (Basis.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (a : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) a) (Basis.funLike.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (FunLike.coe.{max (max (succ u7) (succ u5)) (succ u6), succ u7, max (succ u5) (succ u6)} (LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _x) (LinearMap.instFunLikeLinearMap.{u9, u8, u7, max u5 u6} R R\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B (FunLike.coe.{max (max (succ u2) (succ u9)) (succ u7), succ u2, succ u7} (Basis.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) \u03b9\u2081 (fun (_x : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M) _x) (Basis.funLike.{u2, u9, u7} \u03b9\u2081 R M (CommSemiring.toSemiring.{u9} R _inst_1) _inst_5 _inst_11) b\u2081 i)) (FunLike.coe.{max (max (succ u1) (succ u4)) (succ u5), succ u1, succ u5} (Basis.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) \u03b9\u2082 (fun (_x : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N) _x) (Basis.funLike.{u1, u4, u5} \u03b9\u2082 S N (CommSemiring.toSemiring.{u4} S _inst_2) _inst_6 _inst_12) b\u2082 j)))))) (FunLike.coe.{max (succ u5) (succ u6), succ u5, succ u6} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) x) N (fun (_x : N) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N) => P) _x) (LinearMap.instFunLikeLinearMap.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) (FunLike.coe.{max (max (succ u7) (succ u5)) (succ u6), succ u7, max (succ u5) (succ u6)} (LinearMap.{u9, u8, u7, max u6 u5} R R\u2082 (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) \u03c1\u2081\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18)) M (fun (_x : M) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M) => LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) _x) (LinearMap.instFunLikeLinearMap.{u9, u8, u7, max u5 u6} R R\u2082 M (LinearMap.{u4, u3, u5, u6} S S\u2082 (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) \u03c3\u2081\u2082 N P _inst_6 _inst_7 _inst_12 _inst_14) (CommSemiring.toSemiring.{u9} R _inst_1) (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_5 (LinearMap.addCommMonoid.{u4, u3, u5, u6} S S\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082) _inst_11 (LinearMap.instModuleLinearMapAddCommMonoid.{u4, u3, u8, u5, u6} S S\u2082 R\u2082 N P (CommSemiring.toSemiring.{u4} S _inst_2) (CommSemiring.toSemiring.{u3} S\u2082 _inst_4) _inst_6 _inst_7 _inst_12 _inst_14 \u03c3\u2081\u2082 (CommSemiring.toSemiring.{u8} R\u2082 _inst_3) _inst_13 _inst_18) \u03c1\u2081\u2082) B x) y)\nCase conversion may be inaccurate. Consider using '#align linear_map.sum_repr_mul_repr_mul\u209b\u2097 LinearMap.sum_repr_mul_repr_mul\u209b\u2097\u2093'. -/\n/-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis.\n\nVersion for semi-bilinear maps, see `sum_repr_mul_repr_mul` for the bilinear version. -/\ntheorem sum_repr_mul_repr_mul\u209b\u2097 {B : M \u2192\u209b\u2097[\u03c1\u2081\u2082] N \u2192\u209b\u2097[\u03c3\u2081\u2082] P} (x y) :\n    ((b\u2081.repr x).Sum fun i xi => (b\u2082.repr y).Sum fun j yj => \u03c1\u2081\u2082 xi \u2022 \u03c3\u2081\u2082 yj \u2022 B (b\u2081 i) (b\u2082 j)) =\n      B x y :=\n  by\n  conv_rhs => rw [\u2190 b\u2081.total_repr x, \u2190 b\u2082.total_repr y]\n  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum\u2082, map_sum, LinearMap.map_smul\u209b\u2097\u2082,\n    LinearMap.map_smul\u209b\u2097]\n#align linear_map.sum_repr_mul_repr_mul\u209b\u2097 LinearMap.sum_repr_mul_repr_mul\u209b\u2097\n\n/- warning: linear_map.sum_repr_mul_repr_mul -> LinearMap.sum_repr_mul_repr_mul is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9\u2081 : Type.{u1}} {\u03b9\u2082 : Type.{u2}} {R : Type.{u3}} {M\u2097 : Type.{u4}} {N\u2097 : Type.{u5}} {P\u2097 : Type.{u6}} [_inst_1 : CommSemiring.{u3} R] [_inst_8 : AddCommMonoid.{u4} M\u2097] [_inst_9 : AddCommMonoid.{u5} N\u2097] [_inst_10 : AddCommMonoid.{u6} P\u2097] [_inst_15 : Module.{u3, u4} R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8] [_inst_16 : Module.{u3, u5} R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9] [_inst_17 : Module.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10] (b\u2081' : Basis.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15) (b\u2082' : Basis.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16) {B : LinearMap.{u3, u3, u4, max u5 u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))} (x : M\u2097) (y : N\u2097), Eq.{succ u6} P\u2097 (Finsupp.sum.{u1, u3, u6} \u03b9\u2081 R P\u2097 (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_10 (coeFn.{max (succ u4) (succ (max u1 u3)), max (succ u4) (succ (max u1 u3))} (LinearEquiv.{u3, u3, u4, max u1 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) M\u2097 (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_8 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_15 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (fun (_x : LinearEquiv.{u3, u3, u4, max u1 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) M\u2097 (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_8 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_15 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) => M\u2097 -> (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))))) (LinearEquiv.hasCoeToFun.{u3, u3, u4, max u1 u3} R R M\u2097 (Finsupp.{u1, u3} \u03b9\u2081 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u1, u3} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_15 (Finsupp.module.{u1, u3, u3} \u03b9\u2081 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (Basis.repr.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15 b\u2081') x) (fun (i : \u03b9\u2081) (xi : R) => Finsupp.sum.{u2, u3, u6} \u03b9\u2082 R P\u2097 (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_10 (coeFn.{max (succ u5) (succ (max u2 u3)), max (succ u5) (succ (max u2 u3))} (LinearEquiv.{u3, u3, u5, max u2 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) N\u2097 (Finsupp.{u2, u3} \u03b9\u2082 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_9 (Finsupp.addCommMonoid.{u2, u3} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_16 (Finsupp.module.{u2, u3, u3} \u03b9\u2082 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (fun (_x : LinearEquiv.{u3, u3, u5, max u2 u3} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) N\u2097 (Finsupp.{u2, u3} \u03b9\u2082 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) _inst_9 (Finsupp.addCommMonoid.{u2, u3} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_16 (Finsupp.module.{u2, u3, u3} \u03b9\u2082 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) => N\u2097 -> (Finsupp.{u2, u3} \u03b9\u2082 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))))) (LinearEquiv.hasCoeToFun.{u3, u3, u5, max u2 u3} R R N\u2097 (Finsupp.{u2, u3} \u03b9\u2082 R (MulZeroClass.toHasZero.{u3} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))))) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 (Finsupp.addCommMonoid.{u2, u3} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) _inst_16 (Finsupp.module.{u2, u3, u3} \u03b9\u2082 R R (CommSemiring.toSemiring.{u3} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u3} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (Semiring.toModule.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (RingHomInvPair.ids.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (Basis.repr.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16 b\u2082') y) (fun (j : \u03b9\u2082) (yj : R) => SMul.smul.{u3, u6} R P\u2097 (SMulZeroClass.toHasSmul.{u3, u6} R P\u2097 (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (SMulWithZero.toSmulZeroClass.{u3, u6} R P\u2097 (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (MulActionWithZero.toSMulWithZero.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17)))) xi (SMul.smul.{u3, u6} R P\u2097 (SMulZeroClass.toHasSmul.{u3, u6} R P\u2097 (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (SMulWithZero.toSmulZeroClass.{u3, u6} R P\u2097 (MulZeroClass.toHasZero.{u3} R (MulZeroOneClass.toMulZeroClass.{u3} R (MonoidWithZero.toMulZeroOneClass.{u3} R (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))))) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (MulActionWithZero.toSMulWithZero.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17)))) yj (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (fun (_x : LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) => N\u2097 -> P\u2097) (LinearMap.hasCoeToFun.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (coeFn.{max (succ u4) (succ (max u5 u6)), max (succ u4) (succ (max u5 u6))} (LinearMap.{u3, u3, u4, max u5 u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))) (fun (_x : LinearMap.{u3, u3, u4, max u5 u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))) => M\u2097 -> (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17)) (LinearMap.hasCoeToFun.{u3, u3, u4, max u5 u6} R R M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17)))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) B (coeFn.{max (succ u1) (succ u3) (succ u4), max (succ u1) (succ u4)} (Basis.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15) (fun (_x : Basis.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15) => \u03b9\u2081 -> M\u2097) (FunLike.hasCoeToFun.{max (succ u1) (succ u3) (succ u4), succ u1, succ u4} (Basis.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15) \u03b9\u2081 (fun (_x : \u03b9\u2081) => M\u2097) (Basis.funLike.{u1, u3, u4} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 _inst_15)) b\u2081' i)) (coeFn.{max (succ u2) (succ u3) (succ u5), max (succ u2) (succ u5)} (Basis.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16) (fun (_x : Basis.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16) => \u03b9\u2082 -> N\u2097) (FunLike.hasCoeToFun.{max (succ u2) (succ u3) (succ u5), succ u2, succ u5} (Basis.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (_x : \u03b9\u2082) => N\u2097) (Basis.funLike.{u2, u3, u5} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_16)) b\u2082' j)))))) (coeFn.{max (succ u5) (succ u6), max (succ u5) (succ u6)} (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (fun (_x : LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) => N\u2097 -> P\u2097) (LinearMap.hasCoeToFun.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) (coeFn.{max (succ u4) (succ (max u5 u6)), max (succ u4) (succ (max u5 u6))} (LinearMap.{u3, u3, u4, max u5 u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))) (fun (_x : LinearMap.{u3, u3, u4, max u5 u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17))))) => M\u2097 -> (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17)) (LinearMap.hasCoeToFun.{u3, u3, u4, max u5 u6} R R M\u2097 (LinearMap.{u3, u3, u5, u6} R R (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_8 (LinearMap.addCommMonoid.{u3, u3, u5, u6} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) _inst_15 (LinearMap.module.{u3, u3, u3, u5, u6} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1))) (CommSemiring.toSemiring.{u3} R _inst_1) _inst_17 (smulCommClass_self.{u3, u6} R P\u2097 (CommSemiring.toCommMonoid.{u3} R _inst_1) (MulActionWithZero.toMulAction.{u3, u6} R P\u2097 (Semiring.toMonoidWithZero.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)) (AddZeroClass.toHasZero.{u6} P\u2097 (AddMonoid.toAddZeroClass.{u6} P\u2097 (AddCommMonoid.toAddMonoid.{u6} P\u2097 _inst_10))) (Module.toMulActionWithZero.{u3, u6} R P\u2097 (CommSemiring.toSemiring.{u3} R _inst_1) _inst_10 _inst_17)))) (RingHom.id.{u3} R (Semiring.toNonAssocSemiring.{u3} R (CommSemiring.toSemiring.{u3} R _inst_1)))) B x) y)\nbut is expected to have type\n  forall {\u03b9\u2081 : Type.{u2}} {\u03b9\u2082 : Type.{u1}} {R : Type.{u6}} {M\u2097 : Type.{u5}} {N\u2097 : Type.{u3}} {P\u2097 : Type.{u4}} [_inst_1 : CommSemiring.{u6} R] [_inst_8 : AddCommMonoid.{u5} M\u2097] [_inst_9 : AddCommMonoid.{u3} N\u2097] [_inst_10 : AddCommMonoid.{u4} P\u2097] [_inst_15 : Module.{u6, u5} R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8] [_inst_16 : Module.{u6, u3} R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9] [_inst_17 : Module.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10] (b\u2081' : Basis.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) (b\u2082' : Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) {B : LinearMap.{u6, u6, u5, max u4 u3} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) M\u2097 (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R P\u2097 (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R P\u2097 (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} P\u2097 (AddCommMonoid.toAddMonoid.{u4} P\u2097 _inst_10)) (Module.toMulActionWithZero.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))} (x : M\u2097) (y : N\u2097), Eq.{succ u4} P\u2097 (Finsupp.sum.{u2, u6, u4} \u03b9\u2081 R P\u2097 (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) _inst_10 (FunLike.coe.{max (max (succ u2) (succ u6)) (succ u5), succ u5, max (succ u2) (succ u6)} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) M\u2097 (fun (_x : M\u2097) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : M\u2097) => Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _x) (SMulHomClass.toFunLike.{max (max u2 u6) u5, u6, u5, max u2 u6} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SMulZeroClass.toSMul.{u6, u5} R M\u2097 (AddMonoid.toZero.{u5} M\u2097 (AddCommMonoid.toAddMonoid.{u5} M\u2097 _inst_8)) (DistribSMul.toSMulZeroClass.{u6, u5} R M\u2097 (AddMonoid.toAddZeroClass.{u5} M\u2097 (AddCommMonoid.toAddMonoid.{u5} M\u2097 _inst_8)) (DistribMulAction.toDistribSMul.{u6, u5} R M\u2097 (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u5} M\u2097 _inst_8) (Module.toDistribMulAction.{u6, u5} R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15)))) (SMulZeroClass.toSMul.{u6, max u2 u6} R (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toZero.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribSMul.toSMulZeroClass.{u6, max u2 u6} R (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toAddZeroClass.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulAction.toDistribSMul.{u6, max u2 u6} R (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, max u2 u6} R (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u2 u6) u5, u6, u5, max u2 u6} (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u5} M\u2097 _inst_8) (AddCommMonoid.toAddMonoid.{max u2 u6} (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, u5} R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) (Module.toDistribMulAction.{u6, max u2 u6} R (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SemilinearMapClass.distribMulActionHomClass.{u6, u5, max u2 u6, max (max u2 u6) u5} R M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (SemilinearEquivClass.instSemilinearMapClass.{u6, u6, u5, max u2 u6, max (max u2 u6) u5} R R M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u5, max u6 u2} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u6, u6, u5, max u2 u6} R R M\u2097 (Finsupp.{u2, u6} \u03b9\u2081 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (Finsupp.addCommMonoid.{u2, u6} \u03b9\u2081 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_15 (Finsupp.module.{u2, u6, u6} \u03b9\u2081 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (Basis.repr.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15 b\u2081') x) (fun (i : \u03b9\u2081) (xi : R) => Finsupp.sum.{u1, u6, u4} \u03b9\u2082 R P\u2097 (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) _inst_10 (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u3, max (succ u1) (succ u6)} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) N\u2097 (fun (_x : N\u2097) => (fun (x._@.Mathlib.Algebra.Hom.GroupAction._hyg.2186 : N\u2097) => Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _x) (SMulHomClass.toFunLike.{max (max u1 u6) u3, u6, u3, max u1 u6} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SMulZeroClass.toSMul.{u6, u3} R N\u2097 (AddMonoid.toZero.{u3} N\u2097 (AddCommMonoid.toAddMonoid.{u3} N\u2097 _inst_9)) (DistribSMul.toSMulZeroClass.{u6, u3} R N\u2097 (AddMonoid.toAddZeroClass.{u3} N\u2097 (AddCommMonoid.toAddMonoid.{u3} N\u2097 _inst_9)) (DistribMulAction.toDistribSMul.{u6, u3} R N\u2097 (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} N\u2097 _inst_9) (Module.toDistribMulAction.{u6, u3} R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16)))) (SMulZeroClass.toSMul.{u6, max u1 u6} R (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toZero.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribSMul.toSMulZeroClass.{u6, max u1 u6} R (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddMonoid.toAddZeroClass.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulAction.toDistribSMul.{u6, max u1 u6} R (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, max u1 u6} R (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (DistribMulActionHomClass.toSMulHomClass.{max (max u1 u6) u3, u6, u3, max u1 u6} (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) R N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (MonoidWithZero.toMonoid.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (AddCommMonoid.toAddMonoid.{u3} N\u2097 _inst_9) (AddCommMonoid.toAddMonoid.{max u1 u6} (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))))) (Module.toDistribMulAction.{u6, u3} R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) (Module.toDistribMulAction.{u6, max u1 u6} R (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (SemilinearMapClass.distribMulActionHomClass.{u6, u3, max u1 u6, max (max u1 u6) u3} R N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (SemilinearEquivClass.instSemilinearMapClass.{u6, u6, u3, max u1 u6, max (max u1 u6) u3} R R N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (LinearEquiv.{u6, u6, u3, max u6 u1} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (LinearEquiv.instSemilinearEquivClassLinearEquiv.{u6, u6, u3, max u1 u6} R R N\u2097 (Finsupp.{u1, u6} \u03b9\u2082 R (MonoidWithZero.toZero.{u6} R (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 (Finsupp.addCommMonoid.{u1, u6} \u03b9\u2082 R (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))) _inst_16 (Finsupp.module.{u1, u6, u6} \u03b9\u2082 R R (CommSemiring.toSemiring.{u6} R _inst_1) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u6} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (Semiring.toModule.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (RingHomInvPair.ids.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))))))) (Basis.repr.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16 b\u2082') y) (fun (j : \u03b9\u2082) (yj : R) => HSMul.hSMul.{u6, u4, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (instHSMul.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (SMulZeroClass.toSMul.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (SMulWithZero.toSMulZeroClass.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (CommMonoidWithZero.toZero.{u6} R (CommSemiring.toCommMonoidWithZero.{u6} R _inst_1)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (MulActionWithZero.toSMulWithZero.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (Module.toMulActionWithZero.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))) xi (HSMul.hSMul.{u6, u4, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (instHSMul.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (SMulZeroClass.toSMul.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (SMulWithZero.toSMulZeroClass.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (CommMonoidWithZero.toZero.{u6} R (CommSemiring.toCommMonoidWithZero.{u6} R _inst_1)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (MulActionWithZero.toSMulWithZero.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (AddCommMonoid.toAddMonoid.{u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) _inst_10)) (Module.toMulActionWithZero.{u6, u4} R ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (a : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) a) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))) yj (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M\u2097) => LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (FunLike.coe.{max (max (succ u2) (succ u6)) (succ u5), succ u2, succ u5} (Basis.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) \u03b9\u2081 (fun (a : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M\u2097) a) (Basis.funLike.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) b\u2081' i)) N\u2097 (fun (_x : N\u2097) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) _x) (LinearMap.instFunLikeLinearMap.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (FunLike.coe.{max (max (succ u5) (succ u3)) (succ u4), succ u5, max (succ u3) (succ u4)} (LinearMap.{u6, u6, u5, max u4 u3} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) M\u2097 (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R P\u2097 (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R P\u2097 (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} P\u2097 (AddCommMonoid.toAddMonoid.{u4} P\u2097 _inst_10)) (Module.toMulActionWithZero.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))) M\u2097 (fun (_x : M\u2097) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M\u2097) => LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _x) (LinearMap.instFunLikeLinearMap.{u6, u6, u5, max u3 u4} R R M\u2097 (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R P\u2097 (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R P\u2097 (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} P\u2097 (AddCommMonoid.toAddMonoid.{u4} P\u2097 _inst_10)) (Module.toMulActionWithZero.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17)))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) B (FunLike.coe.{max (max (succ u2) (succ u6)) (succ u5), succ u2, succ u5} (Basis.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) \u03b9\u2081 (fun (_x : \u03b9\u2081) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2081) => M\u2097) _x) (Basis.funLike.{u2, u6, u5} \u03b9\u2081 R M\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 _inst_15) b\u2081' i)) (FunLike.coe.{max (max (succ u1) (succ u6)) (succ u3), succ u1, succ u3} (Basis.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) \u03b9\u2082 (fun (_x : \u03b9\u2082) => (fun (x._@.Mathlib.LinearAlgebra.Basis._hyg.548 : \u03b9\u2082) => N\u2097) _x) (Basis.funLike.{u1, u6, u3} \u03b9\u2082 R N\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_16) b\u2082' j)))))) (FunLike.coe.{max (succ u3) (succ u4), succ u3, succ u4} ((fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M\u2097) => LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) x) N\u2097 (fun (_x : N\u2097) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : N\u2097) => P\u2097) _x) (LinearMap.instFunLikeLinearMap.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) (FunLike.coe.{max (max (succ u5) (succ u3)) (succ u4), succ u5, max (succ u3) (succ u4)} (LinearMap.{u6, u6, u5, max u4 u3} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) M\u2097 (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R P\u2097 (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R P\u2097 (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} P\u2097 (AddCommMonoid.toAddMonoid.{u4} P\u2097 _inst_10)) (Module.toMulActionWithZero.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17))))) M\u2097 (fun (_x : M\u2097) => (fun (x._@.Mathlib.Algebra.Module.LinearMap._hyg.6190 : M\u2097) => LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) _x) (LinearMap.instFunLikeLinearMap.{u6, u6, u5, max u3 u4} R R M\u2097 (LinearMap.{u6, u6, u3, u4} R R (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) N\u2097 P\u2097 _inst_9 _inst_10 _inst_16 _inst_17) (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_8 (LinearMap.addCommMonoid.{u6, u6, u3, u4} R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) _inst_15 (LinearMap.instModuleLinearMapAddCommMonoid.{u6, u6, u6, u3, u4} R R R N\u2097 P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_9 _inst_10 _inst_16 _inst_17 (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1))) (CommSemiring.toSemiring.{u6} R _inst_1) _inst_17 (smulCommClass_self.{u6, u4} R P\u2097 (CommSemiring.toCommMonoid.{u6} R _inst_1) (MulActionWithZero.toMulAction.{u6, u4} R P\u2097 (Semiring.toMonoidWithZero.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)) (AddMonoid.toZero.{u4} P\u2097 (AddCommMonoid.toAddMonoid.{u4} P\u2097 _inst_10)) (Module.toMulActionWithZero.{u6, u4} R P\u2097 (CommSemiring.toSemiring.{u6} R _inst_1) _inst_10 _inst_17)))) (RingHom.id.{u6} R (Semiring.toNonAssocSemiring.{u6} R (CommSemiring.toSemiring.{u6} R _inst_1)))) B x) y)\nCase conversion may be inaccurate. Consider using '#align linear_map.sum_repr_mul_repr_mul LinearMap.sum_repr_mul_repr_mul\u2093'. -/\n/-- Write out `B x y` as a sum over `B (b i) (b j)` if `b` is a basis.\n\nVersion for bilinear maps, see `sum_repr_mul_repr_mul\u209b\u2097` for the semi-bilinear version. -/\ntheorem sum_repr_mul_repr_mul {B : M\u2097 \u2192\u2097[R] N\u2097 \u2192\u2097[R] P\u2097} (x y) :\n    ((b\u2081'.repr x).Sum fun i xi => (b\u2082'.repr y).Sum fun j yj => xi \u2022 yj \u2022 B (b\u2081' i) (b\u2082' j)) =\n      B x y :=\n  by\n  conv_rhs => rw [\u2190 b\u2081'.total_repr x, \u2190 b\u2082'.total_repr y]\n  simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum\u2082, map_sum, LinearMap.map_smul\u2082,\n    LinearMap.map_smul]\n#align linear_map.sum_repr_mul_repr_mul LinearMap.sum_repr_mul_repr_mul\n\nend AddCommMonoid\n\nend LinearMap\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/LinearAlgebra/Basis/Bilinear.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.37087435629044263}}
{"text": "/-\n  Copyright (c) 2022 Arthur Paulino. All rights reserved.\n  Released under Apache 2.0 license as described in the file LICENSE.\n  Authors: Arthur Paulino\n-/\n\nimport FxyLang.Reasoning.Defs\n\nopen Continuation.extends\n\ntheorem Continuation.depthOfExtends {k k' : Continuation} (h : k.extends k') :\n    k.depth \u2264 k'.depth := by\n  induction h with\n  | byId          => simp\n  | bySeq    _ hi => exact Nat.le_step hi\n  | byDecl   _ hi => exact Nat.le_step hi\n  | byFork   _ hi => exact Nat.le_step hi\n  | byLoop   _ hi => exact Nat.le_step hi\n  | byUnOp   _ hi => exact Nat.le_step hi\n  | byBinOp\u2081 _ hi => exact Nat.le_step hi\n  | byBinOp\u2082 _ hi => exact Nat.le_step hi\n  | byApp    _ hi => exact Nat.le_step hi\n  | byBlock  _ hi => exact Nat.le_step hi\n  | byPrint  _ hi => exact Nat.le_step hi\n\ntheorem State.skip : \u27e6c, .skip\u27e7 \u00bb \u27e6c, .nil\u27e7 := by\n  intro k\n  refine \u27e81, ?_\u27e9\n  simp [stepN, step]\n  intro i h\u1d62\n  cases i with\n  | zero => rw [stepN]; exact byId\n  | succ i =>\n    cases i with\n    | zero => rw [stepN]; exact byId\n    | succ i =>\n    have : i.succ.succ > 1 := by\n      by_cases h : i.succ.succ \u2264 1\n      \u00b7 by_cases h' : i.succ.succ > 1\n        \u00b7 exact h'\n        \u00b7 contradiction\n      \u00b7 exact Nat.gt_of_not_le h  \n    contradiction\n\n-- set_option hygiene false in\nmacro \"big_step \" h:ident \" with \"\n    k:ident n:ident h\u2081:ident h\u2082:ident : tactic => do\n  `(tactic| have $k:ident : Continuation := default;\n            have $h\u2081:ident := $h:ident $k:ident;\n            cases $h\u2081:ident with | intro $n:ident FOO__ => ?_;\n            have $h\u2081:ident := FOO__.1;\n            have $h\u2082:ident := FOO__.2;\n            clear FOO__)\n\nopen Lean.Elab.Tactic in\nset_option hygiene false in\nelab \"small_step \" n:ident \" at \" h:ident : tactic => do\n  evalTactic $ \u2190`(tactic| cases $n:ident with\n                          | zero => simp [step, stepN] at $h:ident\n                          | succ $n:ident => ?_)\n  evalTactic $ \u2190`(tactic| simp [step, stepN] at $h:ident)\n\ntheorem State.eval (h : \u27e6c, .eval e\u27e7 \u00bb \u27e6c', v\u27e7) : c = c' := by\n  big_step h with k n h\u2081 h\u2082\n  cases e with\n  | lit l =>\n    small_step n at h\u2081\n    small_step n at h\u2081\n    sorry\n  | _ => sorry\n\ntheorem State.decl (h : \u27e6c, p\u27e7 \u00bb \u27e6c', v\u27e7) :\n    \u27e6c, .decl nm p\u27e7 \u00bb \u27e6c.insert nm v, .nil\u27e7 := by\n  big_step h with k n h\u2081 h\u2082\n  sorry\n", "meta": {"author": "arthurpaulino", "repo": "FxyLang", "sha": "fe1c1df2af522bb3f5c7f4e5b895715e27671df0", "save_path": "github-repos/lean/arthurpaulino-FxyLang", "path": "github-repos/lean/arthurpaulino-FxyLang/FxyLang-fe1c1df2af522bb3f5c7f4e5b895715e27671df0/FxyLang/Reasoning/Consistency.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.37087434165108674}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.products.bifunctor\n\n/-!\n# Curry and uncurry, as functors.\n\nWe define `curry : ((C \u00d7 D) \u2964 E) \u2964 (C \u2964 (D \u2964 E))` and `uncurry : (C \u2964 (D \u2964 E)) \u2964 ((C \u00d7 D) \u2964 E)`,\nand verify that they provide an equivalence of categories\n`currying : (C \u2964 (D \u2964 E)) \u224c ((C \u00d7 D) \u2964 E)`.\n\n-/\nnamespace category_theory\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n          {D : Type u\u2082} [category.{v\u2082} D]\n          {E : Type u\u2083} [category.{v\u2083} E]\n\n/--\nThe uncurrying functor, taking a functor `C \u2964 (D \u2964 E)` and producing a functor `(C \u00d7 D) \u2964 E`.\n-/\ndef uncurry : (C \u2964 (D \u2964 E)) \u2964 ((C \u00d7 D) \u2964 E) :=\n{ obj := \u03bb F,\n  { obj := \u03bb X, (F.obj X.1).obj X.2,\n    map := \u03bb X Y f, (F.map f.1).app X.2 \u226b (F.obj Y.1).map f.2,\n    map_comp' := \u03bb X Y Z f g,\n    begin\n      simp only [prod_comp_fst, prod_comp_snd, functor.map_comp,\n                 nat_trans.comp_app, category.assoc],\n      slice_lhs 2 3 { rw \u2190 nat_trans.naturality },\n      rw category.assoc,\n    end },\n  map := \u03bb F G T,\n  { app := \u03bb X, (T.app X.1).app X.2,\n    naturality' := \u03bb X Y f,\n    begin\n      simp only [prod_comp_fst, prod_comp_snd, category.comp_id, category.assoc,\n        functor.map_id, functor.map_comp, nat_trans.id_app, nat_trans.comp_app],\n      slice_lhs 2 3 { rw nat_trans.naturality },\n      slice_lhs 1 2 {\n        rw [\u2190nat_trans.comp_app, nat_trans.naturality,\n            nat_trans.comp_app],\n      },\n      rw category.assoc,\n    end } }.\n\n/--\nThe object level part of the currying functor. (See `curry` for the functorial version.)\n-/\ndef curry_obj (F : (C \u00d7 D) \u2964 E) : C \u2964 (D \u2964 E) :=\n{ obj := \u03bb X,\n    { obj := \u03bb Y, F.obj (X, Y),\n      map := \u03bb Y Y' g, F.map (\ud835\udfd9 X, g) },\n    map := \u03bb X X' f, { app := \u03bb Y, F.map (f, \ud835\udfd9 Y) } }\n\n/--\nThe currying functor, taking a functor `(C \u00d7 D) \u2964 E` and producing a functor `C \u2964 (D \u2964 E)`.\n-/\ndef curry : ((C \u00d7 D) \u2964 E) \u2964 (C \u2964 (D \u2964 E)) :=\n{ obj := \u03bb F, curry_obj F,\n  map := \u03bb F G T,\n  { app := \u03bb X,\n    { app := \u03bb Y, T.app (X, Y),\n      naturality' := \u03bb Y Y' g,\n      begin\n        dsimp [curry_obj],\n        rw nat_trans.naturality,\n      end },\n    naturality' := \u03bb X X' f,\n    begin\n      ext, dsimp [curry_obj],\n      rw nat_trans.naturality,\n    end } }.\n\n@[simp] lemma uncurry.obj_obj {F : C \u2964 (D \u2964 E)} {X : C \u00d7 D} :\n  (uncurry.obj F).obj X = (F.obj X.1).obj X.2 := rfl\n@[simp] lemma uncurry.obj_map {F : C \u2964 (D \u2964 E)} {X Y : C \u00d7 D} {f : X \u27f6 Y} :\n  (uncurry.obj F).map f = ((F.map f.1).app X.2) \u226b ((F.obj Y.1).map f.2) := rfl\n@[simp] lemma uncurry.map_app {F G : C \u2964 (D \u2964 E)} {\u03b1 : F \u27f6 G} {X : C \u00d7 D} :\n  (uncurry.map \u03b1).app X = (\u03b1.app X.1).app X.2 := rfl\n@[simp] lemma curry.obj_obj_obj\n  {F : (C \u00d7 D) \u2964 E} {X : C} {Y : D} :\n  ((curry.obj F).obj X).obj Y = F.obj (X, Y) := rfl\n@[simp] lemma curry.obj_obj_map\n  {F : (C \u00d7 D) \u2964 E} {X : C} {Y Y' : D} {g : Y \u27f6 Y'} :\n  ((curry.obj F).obj X).map g = F.map (\ud835\udfd9 X, g) := rfl\n@[simp] lemma curry.obj_map_app {F : (C \u00d7 D) \u2964 E} {X X' : C} {f : X \u27f6 X'} {Y} :\n  ((curry.obj F).map f).app Y = F.map (f, \ud835\udfd9 Y) := rfl\n@[simp] lemma curry.map_app_app {F G : (C \u00d7 D) \u2964 E} {\u03b1 : F \u27f6 G} {X} {Y} :\n  ((curry.map \u03b1).app X).app Y = \u03b1.app (X, Y) := rfl\n\n/--\nThe equivalence of functor categories given by currying/uncurrying.\n-/\n@[simps] -- create projection simp lemmas even though this isn't a `{ .. }`.\ndef currying : (C \u2964 (D \u2964 E)) \u224c ((C \u00d7 D) \u2964 E) :=\nequivalence.mk uncurry curry\n  (nat_iso.of_components (\u03bb F, nat_iso.of_components\n    (\u03bb X, nat_iso.of_components (\u03bb Y, iso.refl _) (by tidy)) (by tidy)) (by tidy))\n  (nat_iso.of_components (\u03bb F, nat_iso.of_components\n    (\u03bb X, eq_to_iso (by simp)) (by tidy)) (by tidy))\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/currying.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044135, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3708674211340377}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport tactic.pi_instances\nimport algebra.group.pi\nimport algebra.hom.ring\n\n/-!\n# Pi instances for ring\n\nThis file defines instances for ring, semiring and related structures on Pi Types\n-/\n\nnamespace pi\nuniverses u v w\nvariable {I : Type u}     -- The indexing type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\n\ninstance distrib [\u03a0 i, distrib $ f i] : distrib (\u03a0 i : I, f i) :=\nby refine_struct { add := (+), mul := (*), .. }; tactic.pi_instance_derive_field\n\ninstance non_unital_non_assoc_semiring [\u2200 i, non_unital_non_assoc_semiring $ f i] :\n  non_unital_non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_unital_semiring [\u2200 i, non_unital_semiring $ f i] :\n  non_unital_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_assoc_semiring [\u2200 i, non_assoc_semiring $ f i] :\n  non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance semiring [\u2200 i, semiring $ f i] : semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance non_unital_comm_semiring [\u2200 i, non_unital_comm_semiring $ f i] :\n  non_unital_comm_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), nsmul := add_monoid.nsmul };\ntactic.pi_instance_derive_field\n\ninstance comm_semiring [\u2200 i, comm_semiring $ f i] : comm_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance non_unital_non_assoc_ring [\u2200 i, non_unital_non_assoc_ring $ f i] :\n  non_unital_non_assoc_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };\ntactic.pi_instance_derive_field\n\ninstance non_unital_ring [\u2200 i, non_unital_ring $ f i] :\n  non_unital_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };\ntactic.pi_instance_derive_field\n\ninstance non_assoc_ring [\u2200 i, non_assoc_ring $ f i] :\n  non_assoc_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };\ntactic.pi_instance_derive_field\n\ninstance ring [\u2200 i, ring $ f i] : ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance non_unital_comm_ring [\u2200 i, non_unital_comm_ring $ f i] :\n  non_unital_comm_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), neg := has_neg.neg,\n  nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul };\ntactic.pi_instance_derive_field\n\ninstance comm_ring [\u2200 i, comm_ring $ f i] : comm_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\n/-- A family of ring homomorphisms `f a : \u03b3 \u2192+* \u03b2 a` defines a ring homomorphism\n`pi.ring_hom f : \u03b3 \u2192+* \u03a0 a, \u03b2 a` given by `pi.ring_hom f x b = f b x`. -/\n@[simps]\nprotected def ring_hom {\u03b3 : Type w} [\u03a0 i, non_assoc_semiring (f i)] [non_assoc_semiring \u03b3]\n  (g : \u03a0 i, \u03b3 \u2192+* f i) : \u03b3 \u2192+* \u03a0 i, f i :=\n{ to_fun := \u03bb x b, g b x,\n  map_add' := \u03bb x y, funext $ \u03bb z, (g z).map_add x y,\n  map_mul' := \u03bb x y, funext $ \u03bb z, (g z).map_mul x y,\n  map_one' := funext $ \u03bb z, (g z).map_one,\n  map_zero' := funext $ \u03bb z, (g z).map_zero }\n\nlemma ring_hom_injective {\u03b3 : Type w} [nonempty I] [\u03a0 i, non_assoc_semiring (f i)]\n  [non_assoc_semiring \u03b3] (g : \u03a0 i, \u03b3 \u2192+* f i) (hg : \u2200 i, function.injective (g i)) :\n  function.injective (pi.ring_hom g) :=\n\u03bb x y h, let \u27e8i\u27e9 := \u2039nonempty I\u203a in hg i ((function.funext_iff.mp h : _) i)\n\nend pi\n\nsection ring_hom\n\nuniverses u v\nvariable {I : Type u}\n\n/-- Evaluation of functions into an indexed collection of rings at a point is a ring\nhomomorphism. This is `function.eval` as a `ring_hom`. -/\n@[simps]\ndef pi.eval_ring_hom (f : I \u2192 Type v) [\u03a0 i, non_assoc_semiring (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192+* f i :=\n{ ..(pi.eval_monoid_hom f i),\n  ..(pi.eval_add_monoid_hom f i) }\n\n/-- `function.const` as a `ring_hom`. -/\n@[simps]\ndef pi.const_ring_hom (\u03b1 \u03b2 : Type*) [non_assoc_semiring \u03b2] : \u03b2 \u2192+* (\u03b1 \u2192 \u03b2) :=\n{ to_fun := function.const _,\n  .. pi.ring_hom (\u03bb _, ring_hom.id \u03b2) }\n\n/-- Ring homomorphism between the function spaces `I \u2192 \u03b1` and `I \u2192 \u03b2`, induced by a ring\nhomomorphism `f` between `\u03b1` and `\u03b2`. -/\n@[simps] protected def ring_hom.comp_left {\u03b1 \u03b2 : Type*} [non_assoc_semiring \u03b1]\n  [non_assoc_semiring \u03b2] (f : \u03b1 \u2192+* \u03b2) (I : Type*) :\n  (I \u2192 \u03b1) \u2192+* (I \u2192 \u03b2) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  .. f.to_monoid_hom.comp_left I,\n  .. f.to_add_monoid_hom.comp_left I }\n\nend ring_hom\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/algebra/ring/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891307678321, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.37086741295004955}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.basic\nimport data.seq.seq\n\nnamespace stream\nopen function\nuniverses u v w\n\n/-\ncoinductive wseq (\u03b1 : Type u) : Type u\n| nil : wseq \u03b1\n| cons : \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1\n| think : wseq \u03b1 \u2192 wseq \u03b1\n-/\n\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef wseq (\u03b1) := seq (option \u03b1)\n\nnamespace wseq\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n\n/-- Turn a sequence into a weak sequence -/\ndef of_seq : seq \u03b1 \u2192 wseq \u03b1 := (<$>) some\n\n/-- Turn a list into a weak sequence -/\ndef of_list (l : list \u03b1) : wseq \u03b1 := of_seq l\n\n/-- Turn a stream into a weak sequence -/\ndef of_stream (l : stream \u03b1) : wseq \u03b1 := of_seq l\n\ninstance coe_seq : has_coe (seq \u03b1) (wseq \u03b1) := \u27e8of_seq\u27e9\ninstance coe_list : has_coe (list \u03b1) (wseq \u03b1) := \u27e8of_list\u27e9\ninstance coe_stream : has_coe (stream \u03b1) (wseq \u03b1) := \u27e8of_stream\u27e9\n\n/-- The empty weak sequence -/\ndef nil : wseq \u03b1 := seq.nil\n\ninstance : inhabited (wseq \u03b1) := \u27e8nil\u27e9\n\n/-- Prepend an element to a weak sequence -/\ndef cons (a : \u03b1) : wseq \u03b1 \u2192 wseq \u03b1 := seq.cons (some a)\n\n/-- Compute for one tick, without producing any elements -/\ndef think : wseq \u03b1 \u2192 wseq \u03b1 := seq.cons none\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct : wseq \u03b1 \u2192 computation (option (\u03b1 \u00d7 wseq \u03b1)) :=\ncomputation.corec (\u03bb s, match seq.destruct s with\n  | none              := sum.inl none\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := sum.inl (some (a, s'))\n  end)\n\n\n/-- Recursion principle for weak sequences, compare with `list.rec_on`. -/\ndef rec_on {C : wseq \u03b1 \u2192 Sort v} (s : wseq \u03b1) (h1 : C nil)\n  (h2 : \u2200 x s, C (cons x s)) (h3 : \u2200 s, C (think s)) : C s :=\nseq.rec_on s h1 (\u03bb o, option.rec_on o h3 h2)\n\n/-- membership for weak sequences-/\nprotected def mem (a : \u03b1) (s : wseq \u03b1) := seq.mem (some a) s\n\ninstance : has_mem \u03b1 (wseq \u03b1) :=\n\u27e8wseq.mem\u27e9\n\ntheorem not_mem_nil (a : \u03b1) : a \u2209 @nil \u03b1 := seq.not_mem_nil a\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head (s : wseq \u03b1) : computation (option \u03b1) :=\ncomputation.map ((<$>) prod.fst) (destruct s)\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten : computation (wseq \u03b1) \u2192 wseq \u03b1 :=\nseq.corec (\u03bb c, match computation.destruct c with\n  | sum.inl s := seq.omap return (seq.destruct s)\n  | sum.inr c' := some (none, c')\n  end)\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail (s : wseq \u03b1) : wseq \u03b1 :=\nflatten $ (\u03bb o, option.rec_on o nil prod.snd) <$> destruct s\n\n/-- drop the first `n` elements from `s`. -/\ndef drop (s : wseq \u03b1) : \u2115 \u2192 wseq \u03b1\n| 0     := s\n| (n+1) := tail (drop n)\nattribute [simp] drop\n\n/-- Get the nth element of `s`. -/\ndef nth (s : wseq \u03b1) (n : \u2115) : computation (option \u03b1) := head (drop s n)\n\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef to_list (s : wseq \u03b1) : computation (list \u03b1) :=\n@computation.corec (list \u03b1) (list \u03b1 \u00d7 wseq \u03b1) (\u03bb \u27e8l, s\u27e9,\n  match seq.destruct s with\n  | none              := sum.inl l.reverse\n  | some (none, s')   := sum.inr (l, s')\n  | some (some a, s') := sum.inr (a::l, s')\n  end) ([], s)\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length (s : wseq \u03b1) : computation \u2115 :=\n@computation.corec \u2115 (\u2115 \u00d7 wseq \u03b1) (\u03bb \u27e8n, s\u27e9,\n  match seq.destruct s with\n  | none              := sum.inl n\n  | some (none, s')   := sum.inr (n, s')\n  | some (some a, s') := sum.inr (n+1, s')\n  end) (0, s)\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\nclass is_finite (s : wseq \u03b1) : Prop := (out : (to_list s).terminates)\n\ninstance to_list_terminates (s : wseq \u03b1) [h : is_finite s] : (to_list s).terminates := h.out\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get (s : wseq \u03b1) [is_finite s] : list \u03b1 := (to_list s).get\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\nclass productive (s : wseq \u03b1) : Prop := (nth_terminates : \u2200 n, (nth s n).terminates)\n\ntheorem productive_iff (s : wseq \u03b1) : productive s \u2194 \u2200 n, (nth s n).terminates :=\n\u27e8\u03bb h, h.1, \u03bb h, \u27e8h\u27e9\u27e9\n\ninstance nth_terminates (s : wseq \u03b1) [h : productive s] :\n  \u2200 n, (nth s n).terminates := h.nth_terminates\n\ninstance head_terminates (s : wseq \u03b1) [productive s] :\n  (head s).terminates := s.nth_terminates 0\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef update_nth (s : wseq \u03b1) (n : \u2115) (a : \u03b1) : wseq \u03b1 :=\n@seq.corec (option \u03b1) (\u2115 \u00d7 wseq \u03b1) (\u03bb \u27e8n, s\u27e9,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (some a, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef remove_nth (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n@seq.corec (option \u03b1) (\u2115 \u00d7 wseq \u03b1) (\u03bb \u27e8n, s\u27e9,\n  match seq.destruct s, n with\n  | none,               n     := none\n  | some (none, s'),    n     := some (none, n, s')\n  | some (some a', s'), 0     := some (some a', 0, s')\n  | some (some a', s'), 1     := some (none, 0, s')\n  | some (some a', s'), (n+2) := some (some a', n+1, s')\n  end) (n+1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filter_map (f : \u03b1 \u2192 option \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 :=\nseq.corec (\u03bb s, match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, s')\n  | some (some a, s') := some (f a, s')\n  end)\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter (p : \u03b1 \u2192 Prop) [decidable_pred p] : wseq \u03b1 \u2192 wseq \u03b1 :=\nfilter_map (\u03bb a, if p a then some a else none)\n\n-- example of infinite list manipulations\n/-- Get the first element of `s` satisfying `p`. -/\ndef find (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation (option \u03b1) :=\nhead $ filter p s\n\n/-- Zip a function over two weak sequences -/\ndef zip_with (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (s1 : wseq \u03b1) (s2 : wseq \u03b2) : wseq \u03b3 :=\n@seq.corec (option \u03b3) (wseq \u03b1 \u00d7 wseq \u03b2) (\u03bb \u27e8s1, s2\u27e9,\n  match seq.destruct s1, seq.destruct s2 with\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (none, s1, s2')\n  | some (none, s1'),    some (some a2, s2') := some (none, s1', s2)\n  | some (some a1, s1'), some (some a2, s2') := some (some (f a1 a2), s1', s2')\n  | _,                   _                   := none\n  end) (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip : wseq \u03b1 \u2192 wseq \u03b2 \u2192 wseq (\u03b1 \u00d7 \u03b2) := zip_with prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef find_indexes (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : wseq \u2115 :=\n(zip s (stream.nats : wseq \u2115)).filter_map\n  (\u03bb \u27e8a, n\u27e9, if p a then some n else none)\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef find_index (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation \u2115 :=\n(\u03bb o, option.get_or_else o 0) <$> head (find_indexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef index_of [decidable_eq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 computation \u2115 := find_index (eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexes_of [decidable_eq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 wseq \u2115 := find_indexes (eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union (s1 s2 : wseq \u03b1) : wseq \u03b1 :=\n@seq.corec (option \u03b1) (wseq \u03b1 \u00d7 wseq \u03b1) (\u03bb \u27e8s1, s2\u27e9,\n  match seq.destruct s1, seq.destruct s2 with\n  | none,                none                := none\n  | some (a1, s1'),      none                := some (a1, s1', nil)\n  | none,                some (a2, s2')      := some (a2, nil, s2')\n  | some (none, s1'),    some (none, s2')    := some (none, s1', s2')\n  | some (some a1, s1'), some (none, s2')    := some (some a1, s1', s2')\n  | some (none, s1'),    some (some a2, s2') := some (some a2, s1', s2')\n  | some (some a1, s1'), some (some a2, s2') := some (some a1, cons a2 s1', s2')\n  end) (s1, s2)\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef is_empty (s : wseq \u03b1) : computation bool :=\ncomputation.map option.is_none $ head s\n\n/-- Calculate one step of computation -/\ndef compute (s : wseq \u03b1) : wseq \u03b1 :=\nmatch seq.destruct s with\n| some (none, s') := s'\n| _               := s\nend\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n@seq.corec (option \u03b1) (\u2115 \u00d7 wseq \u03b1) (\u03bb \u27e8n, s\u27e9,\n  match n, seq.destruct s with\n  | 0,   _                 := none\n  | m+1, none              := none\n  | m+1, some (none, s')   := some (none, m+1, s')\n  | m+1, some (some a, s') := some (some a, m, s')\n  end) (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef split_at (s : wseq \u03b1) (n : \u2115) : computation (list \u03b1 \u00d7 wseq \u03b1) :=\n@computation.corec (list \u03b1 \u00d7 wseq \u03b1) (\u2115 \u00d7 list \u03b1 \u00d7 wseq \u03b1) (\u03bb \u27e8n, l, s\u27e9,\n  match n, seq.destruct s with\n  | 0,   _                 := sum.inl (l.reverse, s)\n  | m+1, none              := sum.inl (l.reverse, s)\n  | m+1, some (none, s')   := sum.inr (n, l, s')\n  | m+1, some (some a, s') := sum.inr (m, a::l, s')\n  end) (n, [], s)\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any (s : wseq \u03b1) (p : \u03b1 \u2192 bool) : computation bool :=\ncomputation.corec (\u03bb s : wseq \u03b1,\n  match seq.destruct s with\n  | none              := sum.inl ff\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inl tt else sum.inr s'\n  end) s\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all (s : wseq \u03b1) (p : \u03b1 \u2192 bool) : computation bool :=\ncomputation.corec (\u03bb s : wseq \u03b1,\n  match seq.destruct s with\n  | none              := sum.inl tt\n  | some (none, s')   := sum.inr s'\n  | some (some a, s') := if p a then sum.inr s' else sum.inl ff\n  end) s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (s : wseq \u03b2) : wseq \u03b1 :=\ncons a $ @seq.corec (option \u03b1) (\u03b1 \u00d7 wseq \u03b2) (\u03bb \u27e8a, s\u27e9,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, a, s')\n  | some (some b, s') := let a' := f a b in some (some a', a', s')\n  end) (a, s)\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits (s : wseq \u03b1) : wseq (list \u03b1) :=\ncons [] $ @seq.corec (option (list \u03b1)) (dlist \u03b1 \u00d7 wseq \u03b1) (\u03bb \u27e8l, s\u27e9,\n  match seq.destruct s with\n  | none              := none\n  | some (none, s')   := some (none, l, s')\n  | some (some a, s') := let l' := l.concat a in\n                         some (some l'.to_list, l', s')\n  end) (dlist.empty, s)\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect (s : wseq \u03b1) (n : \u2115) : list \u03b1 :=\n(seq.take n s).filter_map id\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append : wseq \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1 := seq.append\n\n/-- Map a function over a weak sequence -/\ndef map (f : \u03b1 \u2192 \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 := seq.map (option.map f)\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join (S : wseq (wseq \u03b1)) : wseq \u03b1 :=\nseq.join ((\u03bb o : option (wseq \u03b1), match o with\n  | none := seq1.ret none\n  | some s := (none, s)\n  end) <$> S)\n\n/-- Monadic bind operator for weak sequences -/\ndef bind (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2) : wseq \u03b2 :=\njoin (map f s)\n\n/-- lift a relation to a relation over weak sequences -/\n@[simp] def lift_rel_o (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop) :\n  option (\u03b1 \u00d7 wseq \u03b1) \u2192 option (\u03b2 \u00d7 wseq \u03b2) \u2192 Prop\n| none          none          := true\n| (some (a, s)) (some (b, t)) := R a b \u2227 C s t\n| _             _             := false\n\ntheorem lift_rel_o.imp {R S : \u03b1 \u2192 \u03b2 \u2192 Prop} {C D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop}\n  (H1 : \u2200 a b, R a b \u2192 S a b) (H2 : \u2200 s t, C s t \u2192 D s t) :\n  \u2200 {o p}, lift_rel_o R C o p \u2192 lift_rel_o S D o p\n| none          none          h := trivial\n| (some (a, s)) (some (b, t)) h := and.imp (H1 _ _) (H2 _ _) h\n| none          (some _)      h := false.elim h\n| (some (_, _)) none          h := false.elim h\n\ntheorem lift_rel_o.imp_right (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {C D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop}\n  (H : \u2200 s t, C s t \u2192 D s t) {o p} : lift_rel_o R C o p \u2192 lift_rel_o R D o p :=\nlift_rel_o.imp (\u03bb _ _, id) H\n\n/-- Definitino of bisimilarity for weak sequences-/\n@[simp] def bisim_o (R : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop) :\n  option (\u03b1 \u00d7 wseq \u03b1) \u2192 option (\u03b1 \u00d7 wseq \u03b1) \u2192 Prop := lift_rel_o (=) R\n\ntheorem bisim_o.imp {R S : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop} (H : \u2200 s t, R s t \u2192 S s t) {o p} :\n  bisim_o R o p \u2192 bisim_o S o p :=\nlift_rel_o.imp_right _ H\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef lift_rel (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1) (t : wseq \u03b2) : Prop :=\n\u2203 C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop, C s t \u2227\n\u2200 {s t}, C s t \u2192 computation.lift_rel (lift_rel_o R C) (destruct s) (destruct t)\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef equiv : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop := lift_rel (=)\n\ntheorem lift_rel_destruct {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} :\n  lift_rel R s t \u2192\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t)\n| \u27e8R, h1, h2\u27e9 :=\n  by refine computation.lift_rel.imp _ _ _ (h2 h1);\n     apply lift_rel_o.imp_right; exact \u03bb s' t' h', \u27e8R, h', @h2\u27e9\n\ntheorem lift_rel_destruct_iff {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} :\n  lift_rel R s t \u2194\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n\u27e8lift_rel_destruct, \u03bb h, \u27e8\u03bb s t, lift_rel R s t \u2228\n  computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n  or.inr h, \u03bb s t h, begin\n    have h : computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t),\n    { cases h with h h, exact lift_rel_destruct h, assumption },\n    apply computation.lift_rel.imp _ _ _ h,\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end\u27e9\u27e9\n\ninfix (name := equiv) ` ~ `:50 := equiv\n\ntheorem destruct_congr {s t : wseq \u03b1} :\n  s ~ t \u2192 computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct\n\ntheorem destruct_congr_iff {s t : wseq \u03b1} :\n  s ~ t \u2194 computation.lift_rel (bisim_o (~)) (destruct s) (destruct t) :=\nlift_rel_destruct_iff\n\ntheorem lift_rel.refl (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : reflexive R) : reflexive (lift_rel R) :=\n\u03bb s, begin\n  refine \u27e8(=), rfl, \u03bb s t (h : s = t), _\u27e9,\n  rw \u2190h, apply computation.lift_rel.refl,\n  intro a, cases a with a, simp, cases a; simp, apply H\nend\n\ntheorem lift_rel_o.swap (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C) :\n  swap (lift_rel_o R C) = lift_rel_o (swap R) (swap C) :=\nby funext x y; cases x with x; [skip, cases x]; { cases y with y; [skip, cases y]; refl }\n\ntheorem lift_rel.swap_lem {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s1 s2} (h : lift_rel R s1 s2) :\n  lift_rel (swap R) s2 s1 :=\nbegin\n  refine \u27e8swap (lift_rel R), h, \u03bb s t (h : lift_rel R t s), _\u27e9,\n  rw [\u2190lift_rel_o.swap, computation.lift_rel.swap],\n  apply lift_rel_destruct h\nend\n\ntheorem lift_rel.swap (R : \u03b1 \u2192 \u03b2 \u2192 Prop) :\n  swap (lift_rel R) = lift_rel (swap R) :=\nfunext $ \u03bb x, funext $ \u03bb y, propext \u27e8lift_rel.swap_lem, lift_rel.swap_lem\u27e9\n\ntheorem lift_rel.symm (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : symmetric R) : symmetric (lift_rel R) :=\n\u03bb s1 s2 (h : swap (lift_rel R) s2 s1),\nby rwa [lift_rel.swap, show swap R = R, from\n        funext $ \u03bb a, funext $ \u03bb b, propext $ by constructor; apply H] at h\n\ntheorem lift_rel.trans (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : transitive R) : transitive (lift_rel R) :=\n\u03bb s t u h1 h2, begin\n  refine \u27e8\u03bb s u, \u2203 t, lift_rel R s t \u2227 lift_rel R t u, \u27e8t, h1, h2\u27e9, \u03bb s u h, _\u27e9,\n  rcases h with \u27e8t, h1, h2\u27e9,\n  have h1 := lift_rel_destruct h1,\n  have h2 := lift_rel_destruct h2,\n  refine computation.lift_rel_def.2\n    \u27e8(computation.terminates_of_lift_rel h1).trans\n     (computation.terminates_of_lift_rel h2), \u03bb a c ha hc, _\u27e9,\n  rcases h1.left ha with \u27e8b, hb, t1\u27e9,\n  have t2 := computation.rel_of_lift_rel h2 hb hc,\n  cases a with a; cases c with c,\n  { trivial },\n  { cases b, {cases t2}, {cases t1} },\n  { cases a, cases b with b, {cases t1}, {cases b, cases t2} },\n  { cases a with a s, cases b with b, {cases t1},\n    cases b with b t, cases c with c u,\n    cases t1 with ab st, cases t2 with bc tu,\n    exact \u27e8H ab bc, t, st, tu\u27e9 }\nend\n\ntheorem lift_rel.equiv (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : equivalence R \u2192 equivalence (lift_rel R)\n| \u27e8refl, symm, trans\u27e9 :=\n  \u27e8lift_rel.refl R refl, lift_rel.symm R symm, lift_rel.trans R trans\u27e9\n\n@[refl] theorem equiv.refl : \u2200 (s : wseq \u03b1), s ~ s :=\nlift_rel.refl (=) eq.refl\n\n@[symm] theorem equiv.symm : \u2200 {s t : wseq \u03b1}, s ~ t \u2192 t ~ s :=\nlift_rel.symm (=) (@eq.symm _)\n\n@[trans] \n\ntheorem equiv.equivalence : equivalence (@equiv \u03b1) :=\n\u27e8@equiv.refl _, @equiv.symm _, @equiv.trans _\u27e9\n\nopen computation\nlocal notation `return` := computation.return\n\n@[simp] theorem destruct_nil : destruct (nil : wseq \u03b1) = return none :=\ncomputation.destruct_eq_ret rfl\n\n@[simp] theorem destruct_cons (a : \u03b1) (s) : destruct (cons a s) = return (some (a, s)) :=\ncomputation.destruct_eq_ret $ by simp [destruct, cons, computation.rmap]\n\n@[simp] theorem destruct_think (s : wseq \u03b1) : destruct (think s) = (destruct s).think :=\ncomputation.destruct_eq_think $ by simp [destruct, think, computation.rmap]\n\n@[simp] theorem seq_destruct_nil : seq.destruct (nil : wseq \u03b1) = none :=\nseq.destruct_nil\n\n@[simp] theorem seq_destruct_cons (a : \u03b1) (s) : seq.destruct (cons a s) = some (some a, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem seq_destruct_think (s : wseq \u03b1) : seq.destruct (think s) = some (none, s) :=\nseq.destruct_cons _ _\n\n@[simp] theorem head_nil : head (nil : wseq \u03b1) = return none := by simp [head]; refl\n@[simp] theorem head_cons (a : \u03b1) (s) : head (cons a s) = return (some a) := by simp [head]; refl\n@[simp] theorem head_think (s : wseq \u03b1) : head (think s) = (head s).think := by simp [head]; refl\n\n@[simp] theorem flatten_ret (s : wseq \u03b1) : flatten (return s) = s :=\nbegin\n  refine seq.eq_of_bisim (\u03bb s1 s2, flatten (return s2) = s1) _ rfl,\n  intros s' s h, rw \u2190h, simp [flatten],\n  cases seq.destruct s, { simp },\n  { cases val with o s', simp }\nend\n\n@[simp] theorem flatten_think (c : computation (wseq \u03b1)) : flatten c.think = think (flatten c) :=\nseq.destruct_eq_cons $ by simp [flatten, think]\n\n@[simp]\ntheorem destruct_flatten (c : computation (wseq \u03b1)) : destruct (flatten c) = c >>= destruct :=\nbegin\n  refine computation.eq_of_bisim (\u03bb c1 c2, c1 = c2 \u2228\n    \u2203 c, c1 = destruct (flatten c) \u2227 c2 = computation.bind c destruct) _ (or.inr \u27e8c, rfl, rfl\u27e9),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, (or.inr \u27e8c, rfl, rfl\u27e9) := begin\n    apply c.rec_on (\u03bb a, _) (\u03bb c', _); repeat {simp},\n    { cases (destruct a).destruct; simp },\n    { exact or.inr \u27e8c', rfl, rfl\u27e9 }\n  end end\nend\n\ntheorem head_terminates_iff (s : wseq \u03b1) : terminates (head s) \u2194 terminates (destruct s) :=\nterminates_map_iff _ (destruct s)\n\n@[simp] theorem tail_nil : tail (nil : wseq \u03b1) = nil := by simp [tail]\n@[simp] theorem tail_cons (a : \u03b1) (s) : tail (cons a s) = s := by simp [tail]\n@[simp] theorem tail_think (s : wseq \u03b1) : tail (think s) = (tail s).think := by simp [tail]\n\n@[simp] theorem dropn_nil (n) :\n  drop (nil : wseq \u03b1) n = nil := by induction n; simp [*, drop]\n@[simp] theorem dropn_cons (a : \u03b1) (s) (n) :\n  drop (cons a s) (n+1) = drop s n := by induction n; simp [*, drop]\n@[simp] theorem dropn_think (s : wseq \u03b1) (n) :\n  drop (think s) n = (drop s n).think := by induction n; simp [*, drop]\n\ntheorem dropn_add (s : wseq \u03b1) (m) : \u2200 n, drop s (m + n) = drop (drop s m) n\n| 0     := rfl\n| (n+1) := congr_arg tail (dropn_add n)\n\ntheorem dropn_tail (s : wseq \u03b1) (n) : drop (tail s) n = drop s (n + 1) :=\nby rw add_comm; symmetry; apply dropn_add\n\ntheorem nth_add (s : wseq \u03b1) (m n) : nth s (m + n) = nth (drop s m) n :=\ncongr_arg head (dropn_add _ _ _)\n\ntheorem nth_tail (s : wseq \u03b1) (n) : nth (tail s) n = nth s (n + 1) :=\ncongr_arg head (dropn_tail _ _)\n\n@[simp] theorem join_nil : join nil = (nil : wseq \u03b1) := seq.join_nil\n\n@[simp] theorem join_think (S : wseq (wseq \u03b1)) :\n  join (think S) = think (join S) :=\nby { simp [think, join], unfold functor.map, simp [join, seq1.ret] }\n\n@[simp] theorem join_cons (s : wseq \u03b1) (S) :\n  join (cons s S) = think (append s (join S)) :=\nby { simp [think, join], unfold functor.map, simp [join, cons, append] }\n\n@[simp] theorem nil_append (s : wseq \u03b1) : append nil s = s := seq.nil_append _\n\n@[simp] theorem cons_append (a : \u03b1) (s t) :\n  append (cons a s) t = cons a (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem think_append (s t : wseq \u03b1) :\n  append (think s) t = think (append s t) := seq.cons_append _ _ _\n\n@[simp] theorem append_nil (s : wseq \u03b1) : append s nil = s := seq.append_nil _\n\n@[simp] theorem append_assoc (s t u : wseq \u03b1) :\n  append (append s t) u = append s (append t u) := seq.append_assoc _ _ _\n\n/-- auxilary defintion of tail over weak sequences-/\n@[simp] def tail.aux : option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (option (\u03b1 \u00d7 wseq \u03b1))\n| none          := return none\n| (some (a, s)) := destruct s\n\ntheorem destruct_tail (s : wseq \u03b1) :\n  destruct (tail s) = destruct s >>= tail.aux :=\nbegin\n  simp [tail], rw [\u2190 bind_pure_comp_eq_map, is_lawful_monad.bind_assoc],\n  apply congr_arg, ext1 (_|\u27e8a, s\u27e9);\n  apply (@pure_bind computation _ _ _ _ _ _).trans _; simp\nend\n\n/-- auxilary defintion of drop over weak sequences-/\n@[simp] def drop.aux : \u2115 \u2192 option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (option (\u03b1 \u00d7 wseq \u03b1))\n| 0     := return\n| (n+1) := \u03bb a, tail.aux a >>= drop.aux n\n\ntheorem drop.aux_none : \u2200 n, @drop.aux \u03b1 n none = return none\n| 0     := rfl\n| (n+1) := show computation.bind (return none) (drop.aux n) = return none,\n           by rw [ret_bind, drop.aux_none]\n\ntheorem destruct_dropn :\n  \u2200 (s : wseq \u03b1) n, destruct (drop s n) = destruct s >>= drop.aux n\n| s 0     := (bind_ret' _).symm\n| s (n+1) := by rw [\u2190 dropn_tail, destruct_dropn _ n,\n  destruct_tail, is_lawful_monad.bind_assoc]; refl\n\ntheorem head_terminates_of_head_tail_terminates (s : wseq \u03b1) [T : terminates (head (tail s))] :\n  terminates (head s) :=\n(head_terminates_iff _).2 $ begin\n  rcases (head_terminates_iff _).1 T with \u27e8\u27e8a, h\u27e9\u27e9,\n  simp [tail] at h,\n  rcases exists_of_mem_bind h with \u27e8s', h1, h2\u27e9,\n  unfold functor.map at h1,\n  exact let \u27e8t, h3, h4\u27e9 := computation.exists_of_mem_map h1 in computation.terminates_of_mem h3\nend\n\ntheorem destruct_some_of_destruct_tail_some {s : wseq \u03b1} {a}\n  (h : some a \u2208 destruct (tail s)) : \u2203 a', some a' \u2208 destruct s :=\nbegin\n  unfold tail functor.map at h, simp at h,\n  rcases exists_of_mem_bind h with \u27e8t, tm, td\u27e9, clear h,\n  rcases computation.exists_of_mem_map tm with \u27e8t', ht', ht2\u27e9, clear tm,\n  cases t' with t'; rw \u2190ht2 at td; simp at td,\n  { have := mem_unique td (ret_mem _), contradiction },\n  { exact \u27e8_, ht'\u27e9 }\nend\n\ntheorem head_some_of_head_tail_some {s : wseq \u03b1} {a}\n  (h : some a \u2208 head (tail s)) : \u2203 a', some a' \u2208 head s :=\nbegin\n  unfold head at h,\n  rcases computation.exists_of_mem_map h with \u27e8o, md, e\u27e9, clear h,\n  cases o with o; injection e with h', clear e h',\n  cases destruct_some_of_destruct_tail_some md with a am,\n  exact \u27e8_, computation.mem_map ((<$>) (@prod.fst \u03b1 (wseq \u03b1))) am\u27e9\nend\n\ntheorem head_some_of_nth_some {s : wseq \u03b1} {a n}\n  (h : some a \u2208 nth s n) : \u2203 a', some a' \u2208 head s :=\nbegin\n  revert a, induction n with n IH; intros,\n  exacts [\u27e8_, h\u27e9, let \u27e8a', h'\u27e9 := head_some_of_head_tail_some h in IH h']\nend\n\ninstance productive_tail (s : wseq \u03b1) [productive s] : productive (tail s) :=\n\u27e8\u03bb n, by rw [nth_tail]; apply_instance\u27e9\n\ninstance productive_dropn (s : wseq \u03b1) [productive s] (n) : productive (drop s n) :=\n\u27e8\u03bb m, by rw [\u2190nth_add]; apply_instance\u27e9\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef to_seq (s : wseq \u03b1) [productive s] : seq \u03b1 :=\n\u27e8\u03bb n, (nth s n).get, \u03bb n h,\nbegin\n  cases e : computation.get (nth s (n + 1)), {assumption},\n  have := mem_of_get_eq _ e,\n  simp [nth] at this h, cases head_some_of_head_tail_some this with a' h',\n  have := mem_unique h' (@mem_of_get_eq _ _ _ _ h),\n  contradiction\nend\u27e9\n\ntheorem nth_terminates_le {s : wseq \u03b1} {m n} (h : m \u2264 n) :\n  terminates (nth s n) \u2192 terminates (nth s m) :=\nby induction h with m' h IH; [exact id,\n  exact \u03bb T, IH (@head_terminates_of_head_tail_terminates _ _ T)]\n\ntheorem head_terminates_of_nth_terminates {s : wseq \u03b1} {n} :\n  terminates (nth s n) \u2192 terminates (head s) :=\nnth_terminates_le (nat.zero_le n)\n\ntheorem destruct_terminates_of_nth_terminates {s : wseq \u03b1} {n} (T : terminates (nth s n)) :\n  terminates (destruct s) :=\n(head_terminates_iff _).1 $ head_terminates_of_nth_terminates T\n\ntheorem mem_rec_on {C : wseq \u03b1 \u2192 Prop} {a s} (M : a \u2208 s)\n  (h1 : \u2200 b s', (a = b \u2228 C s') \u2192 C (cons b s'))\n  (h2 : \u2200 s, C s \u2192 C (think s)) : C s :=\nbegin\n  apply seq.mem_rec_on M,\n  intros o s' h, cases o with b,\n  { apply h2, cases h, {contradiction}, {assumption} },\n  { apply h1, apply or.imp_left _ h, intro h, injection h }\nend\n\n@[simp] theorem mem_think (s : wseq \u03b1) (a) : a \u2208 think s \u2194 a \u2208 s :=\nbegin\n  cases s with f al,\n  change some (some a) \u2208 some none :: f \u2194 some (some a) \u2208 f,\n  constructor; intro h,\n  { apply (stream.eq_or_mem_of_mem_cons h).resolve_left,\n    intro, injections },\n  { apply stream.mem_cons_of_mem _ h }\nend\n\ntheorem eq_or_mem_iff_mem {s : wseq \u03b1} {a a' s'} :\n  some (a', s') \u2208 destruct s \u2192 (a \u2208 s \u2194 a = a' \u2228 a \u2208 s') :=\nbegin\n  generalize e : destruct s = c, intro h,\n  revert s, apply computation.mem_rec_on h _ (\u03bb c IH, _); intro s;\n  apply s.rec_on _ (\u03bb x s, _) (\u03bb s, _); intros m;\n  have := congr_arg computation.destruct m; simp at this;\n  cases this with i1 i2,\n  { rw [i1, i2],\n    cases s' with f al,\n    unfold cons has_mem.mem wseq.mem seq.mem seq.cons, simp,\n    have h_a_eq_a' : a = a' \u2194 some (some a) = some (some a'), {simp},\n    rw [h_a_eq_a'],\n    refine \u27e8stream.eq_or_mem_of_mem_cons, \u03bb o, _\u27e9,\n    { cases o with e m,\n      { rw e, apply stream.mem_cons },\n      { exact stream.mem_cons_of_mem _ m } } },\n  { simp, exact IH this }\nend\n\n@[simp] theorem mem_cons_iff (s : wseq \u03b1) (b) {a} : a \u2208 cons b s \u2194 a = b \u2228 a \u2208 s :=\neq_or_mem_iff_mem $ by simp [ret_mem]\n\ntheorem mem_cons_of_mem {s : wseq \u03b1} (b) {a} (h : a \u2208 s) : a \u2208 cons b s :=\n(mem_cons_iff _ _).2 (or.inr h)\n\ntheorem mem_cons (s : wseq \u03b1) (a) : a \u2208 cons a s :=\n(mem_cons_iff _ _).2 (or.inl rfl)\n\ntheorem mem_of_mem_tail {s : wseq \u03b1} {a} : a \u2208 tail s \u2192 a \u2208 s :=\nbegin\n  intro h, have := h, cases h with n e, revert s, simp [stream.nth],\n  induction n with n IH; intro s; apply s.rec_on _ (\u03bb x s, _) (\u03bb s, _);\n    repeat{simp}; intros m e; injections,\n  { exact or.inr m },\n  { exact or.inr m },\n  { apply IH m, rw e, cases tail s, refl }\nend\n\ntheorem mem_of_mem_dropn {s : wseq \u03b1} {a} : \u2200 {n}, a \u2208 drop s n \u2192 a \u2208 s\n| 0     h := h\n| (n+1) h := @mem_of_mem_dropn n (mem_of_mem_tail h)\n\ntheorem nth_mem {s : wseq \u03b1} {a n} : some a \u2208 nth s n \u2192 a \u2208 s :=\nbegin\n  revert s, induction n with n IH; intros s h,\n  { rcases computation.exists_of_mem_map h with \u27e8o, h1, h2\u27e9,\n    cases o with o; injection h2 with h',\n    cases o with a' s',\n    exact (eq_or_mem_iff_mem h1).2 (or.inl h'.symm) },\n  { have := @IH (tail s), rw nth_tail at this,\n    exact mem_of_mem_tail (this h) }\nend\n\ntheorem exists_nth_of_mem {s : wseq \u03b1} {a} (h : a \u2208 s) : \u2203 n, some a \u2208 nth s n :=\nbegin\n  apply mem_rec_on h,\n  { intros a' s' h, cases h with h h,\n    { existsi 0, simp [nth], rw h, apply ret_mem },\n    { cases h with n h, existsi n+1,\n      simp [nth], exact h } },\n  { intros s' h, cases h with n h,\n    existsi n, simp [nth], apply think_mem h }\nend\n\ntheorem exists_dropn_of_mem {s : wseq \u03b1} {a} (h : a \u2208 s) :\n  \u2203 n s', some (a, s') \u2208 destruct (drop s n) :=\nlet \u27e8n, h\u27e9 := exists_nth_of_mem h in \u27e8n, begin\n  rcases (head_terminates_iff _).1 \u27e8\u27e8_, h\u27e9\u27e9 with \u27e8\u27e8o, om\u27e9\u27e9,\n  have := computation.mem_unique (computation.mem_map _ om) h,\n  cases o with o; injection this with i,\n  cases o with a' s', dsimp at i,\n  rw i at om, exact \u27e8_, om\u27e9\nend\u27e9\n\ntheorem lift_rel_dropn_destruct {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (H : lift_rel R s t) :\n  \u2200 n, computation.lift_rel (lift_rel_o R (lift_rel R))\n    (destruct (drop s n)) (destruct (drop t n))\n| 0     := lift_rel_destruct H\n| (n+1) := begin\n  simp [destruct_tail],\n  apply lift_rel_bind,\n  apply lift_rel_dropn_destruct n,\n  exact \u03bb a b o, match a, b, o with\n  | none,       none,         _        := by simp\n  | some (a, s), some (b, t), \u27e8h1, h2\u27e9 := by simp [tail.aux]; apply lift_rel_destruct h2\n  end\nend\n\ntheorem exists_of_lift_rel_left {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t}\n  (H : lift_rel R s t) {a} (h : a \u2208 s) : \u2203 {b}, b \u2208 t \u2227 R a b :=\nlet \u27e8n, h\u27e9 := exists_nth_of_mem h,\n    \u27e8some (._, s'), sd, rfl\u27e9 := computation.exists_of_mem_map h,\n    \u27e8some (b, t'), td, \u27e8ab, _\u27e9\u27e9 := (lift_rel_dropn_destruct H n).left sd in\n\u27e8b, nth_mem (computation.mem_map ((<$>) prod.fst.{v v}) td), ab\u27e9\n\ntheorem exists_of_lift_rel_right {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t}\n  (H : lift_rel R s t) {b} (h : b \u2208 t) : \u2203 {a}, a \u2208 s \u2227 R a b :=\nby rw \u2190lift_rel.swap at H; exact exists_of_lift_rel_left H h\n\ntheorem head_terminates_of_mem {s : wseq \u03b1} {a} (h : a \u2208 s) : terminates (head s) :=\nlet \u27e8n, h\u27e9 := exists_nth_of_mem h in head_terminates_of_nth_terminates \u27e8\u27e8_, h\u27e9\u27e9\n\ntheorem of_mem_append {s\u2081 s\u2082 : wseq \u03b1} {a : \u03b1} : a \u2208 append s\u2081 s\u2082 \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\nseq.of_mem_append\n\ntheorem mem_append_left {s\u2081 s\u2082 : wseq \u03b1} {a : \u03b1} : a \u2208 s\u2081 \u2192 a \u2208 append s\u2081 s\u2082 :=\nseq.mem_append_left\n\ntheorem exists_of_mem_map {f} {b : \u03b2} : \u2200 {s : wseq \u03b1}, b \u2208 map f s \u2192 \u2203 a, a \u2208 s \u2227 f a = b\n| \u27e8g, al\u27e9 h := let \u27e8o, om, oe\u27e9 := seq.exists_of_mem_map h in\n  by cases o with a; injection oe with h'; exact \u27e8a, om, h'\u27e9\n\n@[simp] theorem lift_rel_nil (R : \u03b1 \u2192 \u03b2 \u2192 Prop) : lift_rel R nil nil :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_cons (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a b s t) :\n  lift_rel R (cons a s) (cons b t) \u2194 R a b \u2227 lift_rel R s t :=\nby rw [lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_left (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s t) :\n  lift_rel R (think s) t \u2194 lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\n@[simp] theorem lift_rel_think_right (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s t) :\n  lift_rel R s (think t) \u2194 lift_rel R s t :=\nby rw [lift_rel_destruct_iff, lift_rel_destruct_iff]; simp\n\ntheorem cons_congr {s t : wseq \u03b1} (a : \u03b1) (h : s ~ t) : cons a s ~ cons a t :=\nby unfold equiv; simp; exact h\n\ntheorem think_equiv (s : wseq \u03b1) : think s ~ s :=\nby unfold equiv; simp; apply equiv.refl\n\ntheorem think_congr {s t : wseq \u03b1} (h : s ~ t) : think s ~ think t :=\nby unfold equiv; simp; exact h\n\ntheorem head_congr : \u2200 {s t : wseq \u03b1}, s ~ t \u2192 head s ~ head t :=\nsuffices \u2200 {s t : wseq \u03b1}, s ~ t \u2192 \u2200 {o}, o \u2208 head s \u2192 o \u2208 head t, from\n\u03bb s t h o, \u27e8this h, this h.symm\u27e9,\nbegin\n  intros s t h o ho,\n  rcases @computation.exists_of_mem_map _ _ _ _ (destruct s) ho with \u27e8ds, dsm, dse\u27e9,\n  rw \u2190dse,\n  cases destruct_congr h with l r,\n  rcases l dsm with \u27e8dt, dtm, dst\u27e9,\n  cases ds with a; cases dt with b,\n  { apply computation.mem_map _ dtm },\n  { cases b, cases dst },\n  { cases a, cases dst },\n  { cases a with a s', cases b with b t', rw dst.left,\n    exact @computation.mem_map _ _ (@functor.map _ _ (\u03b1 \u00d7 wseq \u03b1) _ prod.fst)\n      _ (destruct t) dtm }\nend\n\ntheorem flatten_equiv {c : computation (wseq \u03b1)} {s} (h : s \u2208 c) : flatten c ~ s :=\nbegin\n  apply computation.mem_rec_on h, { simp },\n  { intro s', apply equiv.trans, simp [think_equiv] }\nend\n\ntheorem lift_rel_flatten {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {c1 : computation (wseq \u03b1)} {c2 : computation (wseq \u03b2)}\n  (h : c1.lift_rel (lift_rel R) c2) : lift_rel R (flatten c1) (flatten c2) :=\nlet S := \u03bb s t,\n  \u2203 c1 c2, s = flatten c1 \u2227 t = flatten c2 \u2227 computation.lift_rel (lift_rel R) c1 c2 in\n\u27e8S, \u27e8c1, c2, rfl, rfl, h\u27e9, \u03bb s t h,\n  match s, t, h with ._, ._, \u27e8c1, c2, rfl, rfl, h\u27e9 := begin\n    simp, apply lift_rel_bind _ _ h,\n    intros a b ab, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct ab),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t h, refine \u27e8return s, return t, _, _, _\u27e9; simp [h]\n  end end\u27e9\n\ntheorem flatten_congr {c1 c2 : computation (wseq \u03b1)} :\n  computation.lift_rel equiv c1 c2 \u2192 flatten c1 ~ flatten c2 := lift_rel_flatten\n\ntheorem tail_congr {s t : wseq \u03b1} (h : s ~ t) : tail s ~ tail t :=\nbegin\n  apply flatten_congr,\n  unfold functor.map, rw [\u2190bind_ret, \u2190bind_ret],\n  apply lift_rel_bind _ _ (destruct_congr h),\n  intros a b h, simp,\n  cases a with a; cases b with b,\n  { trivial },\n  { cases h },\n  { cases a, cases h },\n  { cases a with a s', cases b with b t', exact h.right }\nend\n\ntheorem dropn_congr {s t : wseq \u03b1} (h : s ~ t) (n) : drop s n ~ drop t n :=\nby induction n; simp [*, tail_congr]\n\ntheorem nth_congr {s t : wseq \u03b1} (h : s ~ t) (n) : nth s n ~ nth t n :=\nhead_congr (dropn_congr h _)\n\ntheorem mem_congr {s t : wseq \u03b1} (h : s ~ t) (a) : a \u2208 s \u2194 a \u2208 t :=\nsuffices \u2200 {s t : wseq \u03b1}, s ~ t \u2192 a \u2208 s \u2192 a \u2208 t, from \u27e8this h, this h.symm\u27e9,\n\u03bb s t h as, let \u27e8n, hn\u27e9 := exists_nth_of_mem as in\nnth_mem ((nth_congr h _ _).1 hn)\n\ntheorem productive_congr {s t : wseq \u03b1} (h : s ~ t) : productive s \u2194 productive t :=\nby simp only [productive_iff]; exact\n  forall_congr (\u03bb n, terminates_congr $ nth_congr h _)\n\ntheorem equiv.ext {s t : wseq \u03b1} (h : \u2200 n, nth s n ~ nth t n) : s ~ t :=\n\u27e8\u03bb s t, \u2200 n, nth s n ~ nth t n, h, \u03bb s t h, begin\n  refine lift_rel_def.2 \u27e8_, _\u27e9,\n  { rw [\u2190head_terminates_iff, \u2190head_terminates_iff],\n    exact terminates_congr (h 0) },\n  { intros a b ma mb,\n    cases a with a; cases b with b,\n    { trivial },\n    { injection mem_unique (computation.mem_map _ ma) ((h 0 _).2 (computation.mem_map _ mb)) },\n    { injection mem_unique (computation.mem_map _ ma) ((h 0 _).2 (computation.mem_map _ mb)) },\n    { cases a with a s', cases b with b t',\n      injection mem_unique\n        (computation.mem_map _ ma) ((h 0 _).2 (computation.mem_map _ mb)) with ab,\n      refine \u27e8ab, \u03bb n, _\u27e9,\n      refine (nth_congr (flatten_equiv (computation.mem_map _ ma)) n).symm.trans\n        ((_ : nth (tail s) n ~ nth (tail t) n).trans\n        (nth_congr (flatten_equiv (computation.mem_map _ mb)) n)),\n      rw [nth_tail, nth_tail], apply h } }\nend\u27e9\n\ntheorem length_eq_map (s : wseq \u03b1) : length s = computation.map list.length (to_list s) :=\nbegin\n  refine computation.eq_of_bisim\n    (\u03bb c1 c2, \u2203 (l : list \u03b1) (s : wseq \u03b1),\n      c1 = computation.corec length._match_2 (l.length, s) \u2227\n      c2 = computation.map list.length (computation.corec to_list._match_2 (l, s)))\n    _ \u27e8[], s, rfl, rfl\u27e9,\n  intros s1 s2 h, rcases h with \u27e8l, s, h\u27e9, rw [h.left, h.right],\n  apply s.rec_on _ (\u03bb a s, _) (\u03bb s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine \u27e8a::l, s, _, _\u27e9; simp },\n  { refine \u27e8l, s, _, _\u27e9; simp }\nend\n\n@[simp] theorem of_list_nil : of_list [] = (nil : wseq \u03b1) := rfl\n\n@[simp] theorem of_list_cons (a : \u03b1) (l) :\n  of_list (a :: l) = cons a (of_list l) :=\nshow seq.map some (seq.of_list (a :: l)) =\n     seq.cons (some a) (seq.map some (seq.of_list l)), by simp\n\n@[simp] theorem to_list'_nil (l : list \u03b1) :\n  computation.corec to_list._match_2 (l, nil) = return l.reverse :=\ndestruct_eq_ret rfl\n\n@[simp] theorem to_list'_cons (l : list \u03b1) (s : wseq \u03b1) (a : \u03b1) :\n  computation.corec to_list._match_2 (l, cons a s) =\n  (computation.corec to_list._match_2 (a::l, s)).think :=\ndestruct_eq_think $ by simp [to_list, cons]\n\n@[simp] theorem to_list'_think (l : list \u03b1) (s : wseq \u03b1) :\n  computation.corec to_list._match_2 (l, think s) =\n  (computation.corec to_list._match_2 (l, s)).think :=\ndestruct_eq_think $ by simp [to_list, think]\n\ntheorem to_list'_map (l : list \u03b1) (s : wseq \u03b1) :\n  computation.corec to_list._match_2 (l, s) =\n  ((++) l.reverse) <$> to_list s :=\nbegin\n  refine computation.eq_of_bisim\n    (\u03bb c1 c2, \u2203 (l' : list \u03b1) (s : wseq \u03b1),\n      c1 = computation.corec to_list._match_2 (l' ++ l, s) \u2227\n      c2 = computation.map ((++) l.reverse) (computation.corec to_list._match_2 (l', s)))\n    _ \u27e8[], s, rfl, rfl\u27e9,\n  intros s1 s2 h, rcases h with \u27e8l', s, h\u27e9, rw [h.left, h.right],\n  apply s.rec_on _ (\u03bb a s, _) (\u03bb s, _);\n    repeat {simp [to_list, nil, cons, think, length]},\n  { refine \u27e8a::l', s, _, _\u27e9; simp },\n  { refine \u27e8l', s, _, _\u27e9; simp }\nend\n\n@[simp] theorem to_list_cons (a : \u03b1) (s) :\n  to_list (cons a s) = (list.cons a <$> to_list s).think :=\ndestruct_eq_think $ by unfold to_list; simp; rw to_list'_map; simp; refl\n\n@[simp] theorem to_list_nil : to_list (nil : wseq \u03b1) = return [] :=\ndestruct_eq_ret rfl\n\ntheorem to_list_of_list (l : list \u03b1) : l \u2208 to_list (of_list l) :=\nby induction l with a l IH; simp [ret_mem]; exact think_mem (computation.mem_map _ IH)\n\n@[simp] theorem destruct_of_seq (s : seq \u03b1) :\n  destruct (of_seq s) = return (s.head.map $ \u03bb a, (a, of_seq s.tail)) :=\ndestruct_eq_ret $ begin\n  simp [of_seq, head, destruct, seq.destruct, seq.head],\n  rw [show seq.nth (some <$> s) 0 = some <$> seq.nth s 0, by apply seq.map_nth],\n  cases seq.nth s 0 with a, { refl },\n  unfold functor.map,\n  simp [destruct]\nend\n\n@[simp] theorem head_of_seq (s : seq \u03b1) : head (of_seq s) = return s.head :=\nby simp [head]; cases seq.head s; refl\n\n@[simp] theorem tail_of_seq (s : seq \u03b1) : tail (of_seq s) = of_seq s.tail :=\nbegin\n  simp [tail], apply s.rec_on _ (\u03bb x s, _); simp [of_seq], {refl},\n  rw [seq.head_cons, seq.tail_cons], refl\nend\n\n@[simp] theorem dropn_of_seq (s : seq \u03b1) : \u2200 n, drop (of_seq s) n = of_seq (s.drop n)\n| 0 := rfl\n| (n+1) := by dsimp [drop]; rw [dropn_of_seq, tail_of_seq]\n\ntheorem nth_of_seq (s : seq \u03b1) (n) : nth (of_seq s) n = return (seq.nth s n) :=\nby dsimp [nth]; rw [dropn_of_seq, head_of_seq, seq.head_dropn]\n\ninstance productive_of_seq (s : seq \u03b1) : productive (of_seq s) :=\n\u27e8\u03bb n, by rw nth_of_seq; apply_instance\u27e9\n\ntheorem to_seq_of_seq (s : seq \u03b1) : to_seq (of_seq s) = s :=\nbegin\n  apply subtype.eq, funext n,\n  dsimp [to_seq], apply get_eq_of_mem,\n  rw nth_of_seq, apply ret_mem\nend\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret (a : \u03b1) : wseq \u03b1 := of_list [a]\n\n@[simp] theorem map_nil (f : \u03b1 \u2192 \u03b2) : map f nil = nil := rfl\n\n@[simp] theorem map_cons (f : \u03b1 \u2192 \u03b2) (a s) :\n  map f (cons a s) = cons (f a) (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_think (f : \u03b1 \u2192 \u03b2) (s) :\n  map f (think s) = think (map f s) := seq.map_cons _ _ _\n\n@[simp] theorem map_id (s : wseq \u03b1) : map id s = s := by simp [map]\n\n@[simp] theorem map_ret (f : \u03b1 \u2192 \u03b2) (a) : map f (ret a) = ret (f a) := by simp [ret]\n\n@[simp] theorem map_append (f : \u03b1 \u2192 \u03b2) (s t) : map f (append s t) = append (map f s) (map f t) :=\nseq.map_append _ _ _\n\ntheorem map_comp (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (s : wseq \u03b1) :\n  map (g \u2218 f) s = map g (map f s) :=\nbegin\n  dsimp [map], rw \u2190seq.map_comp,\n  apply congr_fun, apply congr_arg,\n  ext \u27e8\u27e9; refl\nend\n\ntheorem mem_map (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : wseq \u03b1} : a \u2208 s \u2192 f a \u2208 map f s :=\nseq.mem_map (option.map f)\n\n-- The converse is not true without additional assumptions\ntheorem exists_of_mem_join {a : \u03b1} : \u2200 {S : wseq (wseq \u03b1)}, a \u2208 join S \u2192 \u2203 s, s \u2208 S \u2227 a \u2208 s :=\nsuffices \u2200 ss : wseq \u03b1, a \u2208 ss \u2192 \u2200 s S, append s (join S) = ss \u2192\n  a \u2208 append s (join S) \u2192 a \u2208 s \u2228 \u2203 s, s \u2208 S \u2227 a \u2208 s, from \u03bb S h,\n  (this _ h nil S (by simp) (by simp [h])).resolve_left (not_mem_nil _),\nbegin\n  intros ss h, apply mem_rec_on h (\u03bb b ss o, _) (\u03bb ss IH, _); intros s S,\n  { refine s.rec_on (S.rec_on _ (\u03bb s S, _) (\u03bb S, _)) (\u03bb b' s, _) (\u03bb s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try {cases this}; try {contradiction},\n    substs b' ss,\n    simp at m \u22a2,\n    cases o with e IH, { simp [e] },\n    cases m with e m, { simp [e] },\n    exact or.imp_left or.inr (IH _ _ rfl m) },\n  { refine s.rec_on (S.rec_on _ (\u03bb s S, _) (\u03bb S, _)) (\u03bb b' s, _) (\u03bb s, _);\n    intros ej m; simp at ej;\n    have := congr_arg seq.destruct ej; simp at this;\n    try { try {have := this.1}, contradiction }; subst ss,\n    { apply or.inr, simp at m \u22a2,\n      cases IH s S rfl m with as ex,\n      { exact \u27e8s, or.inl rfl, as\u27e9 },\n      { rcases ex with \u27e8s', sS, as\u27e9,\n        exact \u27e8s', or.inr sS, as\u27e9 } },\n    { apply or.inr, simp at m,\n      rcases (IH nil S (by simp) (by simp [m])).resolve_left (not_mem_nil _) with \u27e8s, sS, as\u27e9,\n      exact \u27e8s, by simp [sS], as\u27e9 },\n    { simp at m IH \u22a2, apply IH _ _ rfl m } }\nend\n\ntheorem exists_of_mem_bind {s : wseq \u03b1} {f : \u03b1 \u2192 wseq \u03b2} {b}\n  (h : b \u2208 bind s f) : \u2203 a \u2208 s, b \u2208 f a :=\nlet \u27e8t, tm, bt\u27e9 := exists_of_mem_join h,\n    \u27e8a, as, e\u27e9 := exists_of_mem_map tm in \u27e8a, as, by rwa e\u27e9\n\ntheorem destruct_map (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) :\n  destruct (map f s) = computation.map (option.map (prod.map f (map f))) (destruct s) :=\nbegin\n  apply computation.eq_of_bisim (\u03bb c1 c2, \u2203 s, c1 = destruct (map f s) \u2227\n    c2 = computation.map (option.map (prod.map f (map f))) (destruct s)),\n  { intros c1 c2 h, cases h with s h, rw [h.left, h.right],\n    apply s.rec_on _ (\u03bb a s, _) (\u03bb s, _); simp,\n    exact \u27e8s, rfl, rfl\u27e9 },\n  { exact \u27e8s, rfl, rfl\u27e9 }\nend\n\ntheorem lift_rel_map {\u03b4} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop)\n  {s1 : wseq \u03b1} {s2 : wseq \u03b2}\n  {f1 : \u03b1 \u2192 \u03b3} {f2 : \u03b2 \u2192 \u03b4}\n  (h1 : lift_rel R s1 s2) (h2 : \u2200 {a b}, R a b \u2192 S (f1 a) (f2 b))\n  : lift_rel S (map f1 s1) (map f2 s2) :=\n\u27e8\u03bb s1 s2, \u2203 s t, s1 = map f1 s \u2227 s2 = map f2 t \u2227 lift_rel R s t,\n\u27e8s1, s2, rfl, rfl, h1\u27e9,\n\u03bb s1 s2 h, match s1, s2, h with ._, ._, \u27e8s, t, rfl, rfl, h\u27e9 := begin\n  simp [destruct_map], apply computation.lift_rel_map _ _ (lift_rel_destruct h),\n  intros o p h,\n  cases o with a; cases p with b; simp,\n  { cases b; cases h },\n  { cases a; cases h },\n  { cases a with a s; cases b with b t, cases h with r h,\n    exact \u27e8h2 r, s, rfl, t, rfl, h\u27e9 }\nend end\u27e9\n\ntheorem map_congr (f : \u03b1 \u2192 \u03b2) {s t : wseq \u03b1} (h : s ~ t) : map f s ~ map f t :=\nlift_rel_map _ _ h (\u03bb _ _, congr_arg _)\n\n/-- auxilary defintion of `destruct_append` over weak sequences-/\n@[simp] def destruct_append.aux (t : wseq \u03b1) :\n  option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (option (\u03b1 \u00d7 wseq \u03b1))\n| none          := destruct t\n| (some (a, s)) := return (some (a, append s t))\n\ntheorem destruct_append (s t : wseq \u03b1) :\n  destruct (append s t) = (destruct s).bind (destruct_append.aux t) :=\nbegin\n  apply computation.eq_of_bisim (\u03bb c1 c2, \u2203 s t, c1 = destruct (append s t) \u2227\n    c2 = (destruct s).bind (destruct_append.aux t)) _ \u27e8s, t, rfl, rfl\u27e9,\n  intros c1 c2 h, rcases h with \u27e8s, t, h\u27e9, rw [h.left, h.right],\n  apply s.rec_on _ (\u03bb a s, _) (\u03bb s, _); simp,\n  { apply t.rec_on _ (\u03bb b t, _) (\u03bb t, _); simp,\n    { refine \u27e8nil, t, _, _\u27e9; simp } },\n  { exact \u27e8s, t, rfl, rfl\u27e9 }\nend\n\n/-- auxilary defintion of `destruct_join` over weak sequences-/\n@[simp] def destruct_join.aux : option (wseq \u03b1 \u00d7 wseq (wseq \u03b1)) \u2192 computation (option (\u03b1 \u00d7 wseq \u03b1))\n| none          := return none\n| (some (s, S)) := (destruct (append s (join S))).think\n\ntheorem destruct_join (S : wseq (wseq \u03b1)) :\n  destruct (join S) = (destruct S).bind destruct_join.aux :=\nbegin\n  apply computation.eq_of_bisim (\u03bb c1 c2, c1 = c2 \u2228 \u2203 S, c1 = destruct (join S) \u2227\n    c2 = (destruct S).bind destruct_join.aux) _ (or.inr \u27e8S, rfl, rfl\u27e9),\n  intros c1 c2 h, exact match c1, c2, h with\n  | _, _, (or.inl $ eq.refl c) := by cases c.destruct; simp\n  | _, _, or.inr \u27e8S, rfl, rfl\u27e9 := begin\n    apply S.rec_on _ (\u03bb s S, _) (\u03bb S, _); simp,\n    { refine or.inr \u27e8S, rfl, rfl\u27e9 }\n  end end\nend\n\ntheorem lift_rel_append (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {s1 s2 : wseq \u03b1} {t1 t2 : wseq \u03b2}\n  (h1 : lift_rel R s1 t1) (h2 : lift_rel R s2 t2) :\n  lift_rel R (append s1 s2) (append t1 t2) :=\n\u27e8\u03bb s t, lift_rel R s t \u2228 \u2203 s1 t1, s = append s1 s2 \u2227 t = append t1 t2 \u2227 lift_rel R s1 t1,\nor.inr \u27e8s1, t1, rfl, rfl, h1\u27e9,\n\u03bb s t h, match s, t, h with\n| s, t, or.inl h := begin\n    apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h),\n    intros a b, apply lift_rel_o.imp_right,\n    intros s t, apply or.inl\n  end\n| ._, ._, or.inr \u27e8s1, t1, rfl, rfl, h\u27e9 := begin\n    simp [destruct_append],\n    apply computation.lift_rel_bind _ _ (lift_rel_destruct h),\n    intros o p h,\n    cases o with a; cases p with b,\n    { simp, apply computation.lift_rel.imp _ _ _ (lift_rel_destruct h2),\n      intros a b, apply lift_rel_o.imp_right,\n      intros s t, apply or.inl },\n    { cases b; cases h },\n    { cases a; cases h },\n    { cases a with a s; cases b with b t, cases h with r h,\n      simp, exact \u27e8r, or.inr \u27e8s, rfl, t, rfl, h\u27e9\u27e9 }\n  end\nend\u27e9\n\ntheorem lift_rel_join.lem (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S T} {U : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop}\n  (ST : lift_rel (lift_rel R) S T) (HU : \u2200 s1 s2, (\u2203 s t S T,\n      s1 = append s (join S) \u2227 s2 = append t (join T) \u2227\n      lift_rel R s t \u2227 lift_rel (lift_rel R) S T) \u2192 U s1 s2) {a} (ma : a \u2208 destruct (join S)) :\n  \u2203 {b}, b \u2208 destruct (join T) \u2227 lift_rel_o R U a b :=\nbegin\n  cases exists_results_of_mem ma with n h, clear ma, revert a S T,\n  apply nat.strong_induction_on n _,\n  intros n IH a S T ST ra, simp [destruct_join] at ra, exact\n  let \u27e8o, m, k, rs1, rs2, en\u27e9 := of_results_bind ra,\n      \u27e8p, mT, rop\u27e9 := computation.exists_of_lift_rel_left (lift_rel_destruct ST) rs1.mem in\n  by exact match o, p, rop, rs1, rs2, mT with\n  | none, none, _, rs1, rs2, mT := by simp only [destruct_join]; exact\n    \u27e8none, mem_bind mT (ret_mem _), by rw eq_of_ret_mem rs2.mem; trivial\u27e9\n  | some (s, S'), some (t, T'), \u27e8st, ST'\u27e9, rs1, rs2, mT :=\n    by simp [destruct_append] at rs2; exact\n    let \u27e8k1, rs3, ek\u27e9 := of_results_think rs2,\n        \u27e8o', m1, n1, rs4, rs5, ek1\u27e9 := of_results_bind rs3,\n        \u27e8p', mt, rop'\u27e9 := computation.exists_of_lift_rel_left (lift_rel_destruct st) rs4.mem in\n    by exact match o', p', rop', rs4, rs5, mt with\n    | none, none, _, rs4, rs5', mt :=\n      have n1 < n, begin\n        rw [en, ek, ek1],\n        apply lt_of_lt_of_le _ (nat.le_add_right _ _),\n        apply nat.lt_succ_of_le (nat.le_add_right _ _)\n      end,\n      let \u27e8ob, mb, rob\u27e9 := IH _ this ST' rs5' in by refine \u27e8ob, _, rob\u27e9;\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, exact mb }\n    | some (a, s'), some (b, t'), \u27e8ab, st'\u27e9, rs4, rs5, mt := begin\n      simp at rs5,\n      refine \u27e8some (b, append t' (join T')), _, _\u27e9,\n      { simp [destruct_join], apply mem_bind mT, simp [destruct_append],\n        apply think_mem, apply mem_bind mt, apply ret_mem },\n      rw eq_of_ret_mem rs5.mem,\n      exact \u27e8ab, HU _ _ \u27e8s', t', S', T', rfl, rfl, st', ST'\u27e9\u27e9\n    end end\n  end\nend\n\ntheorem lift_rel_join (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S : wseq (wseq \u03b1)} {T : wseq (wseq \u03b2)}\n  (h : lift_rel (lift_rel R) S T) : lift_rel R (join S) (join T) :=\n\u27e8\u03bb s1 s2, \u2203 s t S T,\n  s1 = append s (join S) \u2227 s2 = append t (join T) \u2227\n  lift_rel R s t \u2227 lift_rel (lift_rel R) S T,\n  \u27e8nil, nil, S, T, by simp, by simp, by simp, h\u27e9,\n\u03bb s1 s2 \u27e8s, t, S, T, h1, h2, st, ST\u27e9, begin\n  clear _fun_match _x,\n  rw [h1, h2], rw [destruct_append, destruct_append],\n  apply computation.lift_rel_bind _ _ (lift_rel_destruct st),\n  exact \u03bb o p h, match o, p, h with\n  | some (a, s), some (b, t), \u27e8h1, h2\u27e9 :=\n    by simp; exact \u27e8h1, s, t, S, rfl, T, rfl, h2, ST\u27e9\n  | none, none, _ := begin\n    dsimp [destruct_append.aux, computation.lift_rel], constructor,\n    { intro, apply lift_rel_join.lem _ ST (\u03bb _ _, id) },\n    { intros b mb,\n      rw [\u2190lift_rel_o.swap], apply lift_rel_join.lem (swap R),\n      { rw [\u2190lift_rel.swap R, \u2190lift_rel.swap], apply ST },\n      { rw [\u2190lift_rel.swap R, \u2190lift_rel.swap (lift_rel R)],\n        exact \u03bb s1 s2 \u27e8s, t, S, T, h1, h2, st, ST\u27e9,\n                      \u27e8t, s, T, S, h2, h1, st, ST\u27e9 },\n      { exact mb } }\n  end end\nend\u27e9\n\ntheorem join_congr {S T : wseq (wseq \u03b1)} (h : lift_rel equiv S T) : join S ~ join T :=\nlift_rel_join _ h\n\ntheorem lift_rel_bind {\u03b4} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop)\n  {s1 : wseq \u03b1} {s2 : wseq \u03b2}\n  {f1 : \u03b1 \u2192 wseq \u03b3} {f2 : \u03b2 \u2192 wseq \u03b4}\n  (h1 : lift_rel R s1 s2) (h2 : \u2200 {a b}, R a b \u2192 lift_rel S (f1 a) (f2 b))\n  : lift_rel S (bind s1 f1) (bind s2 f2) :=\nlift_rel_join _ (lift_rel_map _ _ h1 @h2)\n\ntheorem bind_congr {s1 s2 : wseq \u03b1} {f1 f2 : \u03b1 \u2192 wseq \u03b2}\n  (h1 : s1 ~ s2) (h2 : \u2200 a, f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=\nlift_rel_bind _ _ h1 (\u03bb a b h, by rw h; apply h2)\n\n@[simp] theorem join_ret (s : wseq \u03b1) : join (ret s) ~ s :=\nby simp [ret]; apply think_equiv\n\n@[simp] theorem join_map_ret (s : wseq \u03b1) : join (map ret s) ~ s :=\nbegin\n  refine \u27e8\u03bb s1 s2, join (map ret s2) = s1, rfl, _\u27e9,\n  intros s' s h, rw \u2190h,\n  apply lift_rel_rec\n    (\u03bb c1 c2, \u2203 s,\n      c1 = destruct (join (map ret s)) \u2227 c2 = destruct s),\n  { exact \u03bb c1 c2 h, match c1, c2, h with\n    | ._, ._, \u27e8s, rfl, rfl\u27e9 := begin\n      clear h _match,\n      have : \u2200 s, \u2203 s' : wseq \u03b1, (map ret s).join.destruct = (map ret s').join.destruct \u2227\n        destruct s = s'.destruct, from \u03bb s, \u27e8s, rfl, rfl\u27e9,\n      apply s.rec_on _ (\u03bb a s, _) (\u03bb s, _); simp [ret, ret_mem, this, option.exists]\n    end end },\n  { exact \u27e8s, rfl, rfl\u27e9 }\nend\n\n@[simp] theorem join_append (S T : wseq (wseq \u03b1)) :\n  join (append S T) ~ append (join S) (join T) :=\nbegin\n  refine \u27e8\u03bb s1 s2, \u2203 s S T,\n    s1 = append s (join (append S T)) \u2227\n    s2 = append s (append (join S) (join T)), \u27e8nil, S, T, by simp, by simp\u27e9, _\u27e9,\n  intros s1 s2 h,\n  apply lift_rel_rec (\u03bb c1 c2, \u2203 (s : wseq \u03b1) S T,\n    c1 = destruct (append s (join (append S T))) \u2227\n    c2 = destruct (append s (append (join S) (join T)))) _ _ _\n    (let \u27e8s, S, T, h1, h2\u27e9 := h in\n         \u27e8s, S, T, congr_arg destruct h1, congr_arg destruct h2\u27e9),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, \u27e8s, S, T, rfl, rfl\u27e9 := begin\n    clear _match h h,\n    apply wseq.rec_on s _ (\u03bb a s, _) (\u03bb s, _); simp,\n    { apply wseq.rec_on S _ (\u03bb s S, _) (\u03bb S, _); simp,\n      { apply wseq.rec_on T _ (\u03bb s T, _) (\u03bb T, _); simp,\n        { refine \u27e8s, nil, T, _, _\u27e9; simp },\n        { refine \u27e8nil, nil, T, _, _\u27e9; simp } },\n      { exact \u27e8s, S, T, rfl, rfl\u27e9 },\n      { refine \u27e8nil, S, T, _, _\u27e9; simp } },\n    { exact \u27e8s, S, T, rfl, rfl\u27e9 },\n    { exact \u27e8s, S, T, rfl, rfl\u27e9 }\n  end end\nend\n\n@[simp] theorem bind_ret (f : \u03b1 \u2192 \u03b2) (s) : bind s (ret \u2218 f) ~ map f s :=\nbegin\n  dsimp [bind], change (\u03bb x, ret (f x)) with (ret \u2218 f),\n  rw [map_comp], apply join_map_ret\nend\n\n@[simp] theorem ret_bind (a : \u03b1) (f : \u03b1 \u2192 wseq \u03b2) :\n  bind (ret a) f ~ f a := by simp [bind]\n\n@[simp] theorem map_join (f : \u03b1 \u2192 \u03b2) (S) :\n  map f (join S) = join (map (map f) S) :=\nbegin\n  apply seq.eq_of_bisim (\u03bb s1 s2,\n    \u2203 s S, s1 = append s (map f (join S)) \u2227\n      s2 = append s (join (map (map f) S))),\n  { intros s1 s2 h,\n    exact match s1, s2, h with ._, ._, \u27e8s, S, rfl, rfl\u27e9 := begin\n      apply wseq.rec_on s _ (\u03bb a s, _) (\u03bb s, _); simp,\n      { apply wseq.rec_on S _ (\u03bb s S, _) (\u03bb S, _); simp,\n        { exact \u27e8map f s, S, rfl, rfl\u27e9 },\n        { refine \u27e8nil, S, _, _\u27e9; simp } },\n      { exact \u27e8_, _, rfl, rfl\u27e9 },\n      { exact \u27e8_, _, rfl, rfl\u27e9 }\n    end end },\n  { refine \u27e8nil, S, _, _\u27e9; simp }\nend\n\n@[simp] theorem join_join (SS : wseq (wseq (wseq \u03b1))) :\n  join (join SS) ~ join (map join SS) :=\nbegin\n  refine \u27e8\u03bb s1 s2, \u2203 s S SS,\n    s1 = append s (join (append S (join SS))) \u2227\n    s2 = append s (append (join S) (join (map join SS))),\n    \u27e8nil, nil, SS, by simp, by simp\u27e9, _\u27e9,\n  intros s1 s2 h,\n  apply lift_rel_rec (\u03bb c1 c2, \u2203 s S SS,\n      c1 = destruct (append s (join (append S (join SS)))) \u2227\n      c2 = destruct (append s (append (join S) (join (map join SS)))))\n    _ (destruct s1) (destruct s2)\n    (let \u27e8s, S, SS, h1, h2\u27e9 := h in \u27e8s, S, SS, by simp [h1], by simp [h2]\u27e9),\n  intros c1 c2 h,\n  exact match c1, c2, h with ._, ._, \u27e8s, S, SS, rfl, rfl\u27e9 := begin\n    clear _match h h,\n    apply wseq.rec_on s _ (\u03bb a s, _) (\u03bb s, _); simp,\n    { apply wseq.rec_on S _ (\u03bb s S, _) (\u03bb S, _); simp,\n      { apply wseq.rec_on SS _ (\u03bb S SS, _) (\u03bb SS, _); simp,\n        { refine \u27e8nil, S, SS, _, _\u27e9; simp },\n        { refine \u27e8nil, nil, SS, _, _\u27e9; simp } },\n      { exact \u27e8s, S, SS, rfl, rfl\u27e9 },\n      { refine \u27e8nil, S, SS, _, _\u27e9; simp } },\n    { exact \u27e8s, S, SS, rfl, rfl\u27e9 },\n    { exact \u27e8s, S, SS, rfl, rfl\u27e9 }\n  end end\nend\n\n@[simp] theorem bind_assoc (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2) (g : \u03b2 \u2192 wseq \u03b3) :\n  bind (bind s f) g ~ bind s (\u03bb (x : \u03b1), bind (f x) g) :=\nbegin\n  simp [bind], rw [\u2190 map_comp f (map g), map_comp (map g \u2218 f) join],\n  apply join_join\nend\n\ninstance : monad wseq :=\n{ map  := @map,\n  pure := @ret,\n  bind := @bind }\n\n/-\n  Unfortunately, wseq is not a lawful monad, because it does not satisfy\n  the monad laws exactly, only up to sequence equivalence.\n  Furthermore, even quotienting by the equivalence is not sufficient,\n  because the join operation involves lists of quotient elements,\n  with a lifted equivalence relation, and pure quotients cannot handle\n  this type of construction.\n\ninstance : is_lawful_monad wseq :=\n{ id_map := @map_id,\n  bind_pure_comp_eq_map := @bind_ret,\n  pure_bind := @ret_bind,\n  bind_assoc := @bind_assoc }\n-/\n\nend wseq\nend stream\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/seq/wseq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3706825552400729}}
{"text": "import ..mcrl2_basic.mcrl2_basic\n\nopen mcrl2\n\nvariable {\u03b1 : Type}\nvariable [comm_semigroup_with_zero \u03b1]\n\n/- Now we do the same for |_. -/\ninductive R_parl {x\u2081 x\u2082 y\u2081 y\u2082 : mcrl2 \u03b1} (R\u2081 R\u2082 : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop) :\nmcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop\n| R\u2081 {x y} (h : R\u2081 x y) : R_parl x y\n| R\u2082 {x y} (h : R\u2082 x y) : R_parl x y\n| basel : R_parl (x\u2081 |_ y\u2081) (x\u2082 |_ y\u2082)\n| baser : R_parl (x\u2082 |_ y\u2082) (x\u2081 |_ y\u2081)\n| stepl {a x\u2081' x\u2082'} (hR : R\u2081 x\u2081' x\u2082') (h\u2081 : transition x\u2081 a x\u2081') (h\u2082 : transition x\u2082 a x\u2082') :\nR_parl (x\u2081' || y\u2081) (x\u2082' || y\u2082) \n| stepr {a x\u2081' x\u2082'} (hR : R\u2081 x\u2081' x\u2082') (h\u2081 : transition x\u2081 a x\u2081') (h\u2082 : transition x\u2082 a x\u2082') :\nR_parl (x\u2082' || y\u2082) (x\u2081' || y\u2081) \n| par_l {x x' y y' a z z'} (hR : R_parl (x || y) (x' || y')) (hR' : R_parl (x' || y') (x || y))\n  (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n  (hRz : R\u2081 z z')\n  (ht : transition x a z) (ht' : transition x' a z') :\n  R_parl (z || y) (z' || y') \n| par_r {x x' y y' a z z'} (hR : R_parl (x || y) (x' || y')) (hR' : R_parl (x' || y') (x || y))\n  (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n  (hRz : R\u2082 z z')\n  (ht : transition y a z) (ht' : transition y' a z') :\n  R_parl (x || z) (x' || z')\n| comm {x x' y y' a b z\u2081 z\u2081' z\u2082 z\u2082'} (hR : R_parl (x || y) (x' || y')) (hR' : R_parl (x' || y') (x || y))\n  (hRz\u2081 : R\u2081 z\u2081 z\u2081') (hRz\u2082 : R\u2082 z\u2082 z\u2082')\n  (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n  (htx : transition x a z\u2081) (htx' : transition x' a z\u2081')\n  (hty : transition y b z\u2082) (hty' : transition y' b z\u2082') :\n  R_parl (z\u2081 || z\u2082) (z\u2081' || z\u2082')\n\nlemma R_parl.symm {x\u2081 x\u2082 y\u2081 y\u2082 : mcrl2 \u03b1} {R\u2081 R\u2082 : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop}\n(h\u2081 : symmetric R\u2081) (h\u2082 : symmetric R\u2082):\nsymmetric (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) :=\nbegin\n  intros x y h,\n  cases h,\n  { apply R_parl.R\u2081,\n    apply h\u2081,\n    assumption},\n  { apply R_parl.R\u2082,\n    apply h\u2082,\n    assumption},\n  { exact R_parl.baser},\n  { exact R_parl.basel},\n  { apply R_parl.stepr; assumption},\n  { apply R_parl.stepl; assumption},\n  { apply R_parl.par_l,\n    repeat {assumption},\n    exact h\u2081 h_hR\u2081,\n    exact h\u2082 h_hR\u2082,\n    exact h\u2081 h_hRz},\n  { apply R_parl.par_r,\n    repeat {assumption},\n    exact h\u2081 h_hR\u2081,\n    exact h\u2082 h_hR\u2082,\n    apply h\u2082 h_hRz},\n  { apply R_parl.comm,\n    exact h_hR',\n    repeat {assumption},\n    exact h\u2081 h_hRz\u2081,\n    exact h\u2082 h_hRz\u2082,\n    exact h\u2081 h_hR\u2081,\n    exact h\u2082 h_hR\u2082}\nend\n\nlemma bisim_exists_lift_parl_l {x\u2081 x\u2082 y\u2081 y\u2082 x x' y y' a z} {R\u2081 R\u2082  : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop} \n (hxz : transition x a z)\n (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n (hR : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x || y) (x' || y' )) \n (hR' : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x' || y' ) (x || y)) :\n(\u2203z', transition x' a z' \u2227 option.rel R\u2081 z z') \u2192 (\u2203z', transition x' a z' \u2227 option.rel (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (par' z y) (par' z' y')) :=\nbegin \n  intro h,\n  rcases h with \u27e8w, hwa, hRw\u27e9,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_parl.par_l,\n    exact hR,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_parl.R\u2082,\n    assumption}\nend\n\nlemma bisim_exists_lift_parl_r {x\u2081 x\u2082 y\u2081 y\u2082 x x' y y' a z} {R\u2081 R\u2082  : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop} \n (hxz : transition y a z)\n (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n (hR : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x || y) (x' || y' )) \n (hR' : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x' || y' ) (x || y)) :\n(\u2203z', transition y' a z' \u2227 option.rel R\u2082 z z') \u2192 (\u2203z', transition y' a z' \u2227 option.rel (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (par' x z) (par' x' z')) :=\nbegin \n  intro h,\n  rcases h with \u27e8w, hwa, hRw\u27e9,\n  apply exists.intro w,\n  apply and.intro,\n  assumption,\n  cases hRw,\n  { apply option.rel.some,\n    apply R_parl.par_r,\n    exact hR,\n    repeat {assumption}},\n  { apply option.rel.some,\n    apply R_parl.R\u2081,\n    assumption}\nend\n\nlemma bisim_exists_lift_parl_comm {x\u2081 x\u2082 y\u2081 y\u2082 x x' y y' a b z\u2081 z\u2082} {R\u2081 R\u2082  : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop} \n (hxz : transition x a z\u2081) (hyz' : transition y b z\u2082) (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y')\n (hab : a * b \u2260 0)\n (hR : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x || y) (x' || y' )) \n (hR' : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x' || y' ) (x || y)) :\n(\u2203z' , transition x' a z' \u2227 option.rel R\u2081 z\u2081 z') \u2192 (\u2203z' , transition y' b z' \u2227 option.rel R\u2082 z\u2082 z') \u2192 \n(\u2203z', transition (x' || y') (a * b) z' \u2227 option.rel (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (par' z\u2081 z\u2082) z') :=\nbegin \n  intros h\u2081 h\u2082,\n  rcases h\u2081 with \u27e8w\u2081, hw\u2081a, hRw\u2081\u27e9,\n  rcases h\u2082 with \u27e8w\u2082, hw\u2082b, hRw\u2082\u27e9,\n  simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, \u2190exists_and_distrib_right, and_assoc, exists_eq_left],\n  apply or.inr,\n  apply or.inr,\n  apply exists.intro (par' w\u2081 w\u2082),\n  apply exists.intro w\u2081,\n  apply exists.intro w\u2082,\n  apply and.intro,\n  refl,\n  apply exists.intro a,\n  apply exists.intro b,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  refl,\n  apply and.intro,\n  assumption,\n  apply and.intro,\n  assumption,\n  cases hRw\u2082,\n  { cases hRw\u2081,\n    { apply option.rel.some,\n      apply R_parl.comm,\n      repeat {assumption}},\n    { apply option.rel.some,\n      cases hRw\u2082,\n      apply R_parl.R\u2082,\n      assumption}},\n  { cases hRw\u2081,\n    { apply option.rel.some,\n      apply R_parl.R\u2081,\n      assumption},\n    { apply option.rel.none}}\nend\n\nlemma bisim_exists_lift_parl {x\u2081 x\u2082 y\u2081 y\u2082 x x' y y' a z} {R\u2081 R\u2082 : mcrl2 \u03b1 \u2192 mcrl2 \u03b1 \u2192 Prop} \n (R\u2081_bisim : is_bisimulation R\u2081) (R\u2082_bisim : is_bisimulation R\u2082) (hR\u2081 : R\u2081 x x') (hR\u2082 : R\u2082 y y') (haz : transition (x || y) a z)\n (hR : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x || y) (x' || y' )) \n (hR' : (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) (x' || y' ) (x || y)) :\n (\u2203z', transition (x' || y') a z' \u2227 option.rel (@R_parl _ _ x\u2081 x\u2082 y\u2081 y\u2082 R\u2081 R\u2082) z z') :=\nbegin \n  cases R\u2081_bisim with R\u2081_bisim R\u2081_symm,\n  cases R\u2082_bisim with R\u2082_bisim R\u2082_symm,\n  cases haz,\n  { specialize R\u2081_bisim x x' haz_x' a hR\u2081 haz_h,\n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, \u2190exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_parl_l haz_h hR\u2081 hR\u2082 hR hR' R\u2081_bisim,},\n  { specialize R\u2082_bisim y y' haz_y' a hR\u2082 haz_h, \n    simp only [transition.par_iff, exists_or_distrib, or_and_distrib_right, \u2190exists_and_distrib_right, and_assoc, exists_eq_left],\n    apply or.inr,\n    apply or.inl,\n    simp only [exists_comm, exists_eq_left],\n    exact bisim_exists_lift_parl_r haz_h hR\u2081 hR\u2082 hR hR' R\u2082_bisim},\n  { specialize R\u2081_bisim x x' haz_x' haz_a hR\u2081 haz_h\u2081, \n    specialize R\u2082_bisim y y' haz_y' haz_b hR\u2082 haz_h\u2082,\n    exact bisim_exists_lift_parl_comm haz_h\u2081 haz_h\u2082 hR\u2081 hR\u2082 haz_h\u2083 hR hR' R\u2081_bisim R\u2082_bisim}\nend\n\ntheorem bisim.parl {x\u2081 x\u2082 y\u2081 y\u2082: mcrl2 \u03b1} (h\u2081 : x\u2081 \u2248 x\u2082) (h\u2082 : y\u2081 \u2248 y\u2082) : \nx\u2081 |_ y\u2081 \u2248 x\u2082 |_ y\u2082 :=\nbegin \n  rcases h\u2081 with \u27e8R\u2081, R\u2081x, R\u2081_bisim\u27e9,\n  rcases h\u2082 with \u27e8R\u2082, R\u2082y, R\u2082_bisim\u27e9,\n  apply exists.intro (R_parl R\u2081 R\u2082),\n  apply and.intro,\n  apply R_parl.basel,\n  apply and.intro,\n  { intros x y x' a Rxy xax',\n    cases Rxy,\n    { apply bisim_exists_lift,\n      { intros v w, exact R_parl.R\u2081},\n      { apply bisim_lift; assumption}},\n    { apply bisim_exists_lift,\n      { intros v w, exact R_parl.R\u2082},\n      { apply bisim_lift; assumption}},\n    { simp [transition.parl_iff, \u2190exists_and_distrib_right, and_assoc, exists_comm, exists_eq_left],\n      cases xax',\n      cases R\u2081_bisim with R\u2081_bisim R\u2081_symm,\n      specialize R\u2081_bisim x\u2081 x\u2082 xax'_x' a R\u2081x xax'_h,\n      rcases R\u2081_bisim with \u27e8w, haw, Rw\u27e9,\n      apply exists.intro w,\n      apply and.intro,\n      assumption,\n      cases Rw,\n      { apply option.rel.some,\n        apply R_parl.stepl; assumption},\n      { apply option.rel.some,\n        apply R_parl.R\u2082,\n        assumption}},\n    { simp [transition.parl_iff, \u2190exists_and_distrib_right, and_assoc, exists_comm, exists_eq_left],\n      cases xax',\n      cases R\u2081_bisim with R\u2081_bisim R\u2081_symm,\n      specialize R\u2081_bisim x\u2082 x\u2081 xax'_x' a (R\u2081_symm R\u2081x) xax'_h,\n      rcases R\u2081_bisim with \u27e8w, haw, Rw\u27e9,\n      apply exists.intro w,\n      apply and.intro,\n      assumption,\n      cases Rw,\n      { apply option.rel.some,\n        apply R_parl.stepr,\n        exact R\u2081_symm Rw_\u1fb0,\n        repeat {assumption}},\n      { apply option.rel.some,\n        apply R_parl.R\u2082,\n        apply R\u2082_bisim.right,\n        assumption}},\n    { apply bisim_exists_lift_parl,\n      repeat {assumption},\n      exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right Rxy},\n    { apply bisim_exists_lift_parl,\n      repeat {assumption},\n      exact R\u2081_bisim.right Rxy_hR,\n      exact R\u2082_bisim.right R\u2082y,\n      exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right Rxy},\n    { apply bisim_exists_lift_parl,\n      repeat {assumption},\n      exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right Rxy},\n    { apply bisim_exists_lift_parl,\n      repeat {assumption},\n      exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right Rxy},\n    { apply bisim_exists_lift_parl,\n      repeat {assumption},\n      exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right Rxy}},\n  { exact R_parl.symm R\u2081_bisim.right R\u2082_bisim.right}\nend", "meta": {"author": "Wolfb34", "repo": "mucrl2lean_public", "sha": "0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39", "save_path": "github-repos/lean/Wolfb34-mucrl2lean_public", "path": "github-repos/lean/Wolfb34-mucrl2lean_public/mucrl2lean_public-0d687d0ad00a6f276f1c1e9acbfc3dd4c0b2ce39/Lean/mcrl2_mrg/parl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947155710234, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.37068254761517816}}
{"text": "example (hp : p) (hq : p \u2192 q) (hr : p \u2192 r) : (s \u2228 q) \u2227 (r \u2228 s) := by\n  constructor\n  case' left  => apply Or.inr\n  case' right => apply Or.inl\n  case' left  => apply hq\n  case' right => apply hr\n  all_goals assumption\n\nexample (hp : p) (hq : p \u2192 q) (hr : p \u2192 r) : (p \u2227 q) \u2227 (r \u2227 p) := by\n  constructor\n  case' left | right => constructor\n  case' right.left => apply hr\n  case' left.right => apply hq\n  all_goals assumption\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/casePrime.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3706090844600527}}
{"text": "/-\n  Testing out the tactic.\n-/\n\nimport tactic.ring2\nimport tactic.ring\nimport tactic.show_term\nimport data.real.basic\nimport data.mv_polynomial.basic\nimport lib.poly lib.psd .sos float.basic\n\nopen mv_polynomial poly\n\n-- Test intermediate steps.\n\n-- noncomputable def p : mv_polynomial \u2115 \u211d := (X 1) * (X 1)\n-- noncomputable def ms : fin 1 \u2192 mv_polynomial \u2115 \u211d := \u03bb _, X 1\n-- def Q : fin 1 \u2192 fin 1 \u2192 \u211d := \u03bb _ _, 1\n\n-- lemma Qsymmetric : matrix.symmetric Q := \n-- by prove_symmetric\n\n-- lemma Qsymmetric2 : matrix.symmetric (\u03bb _ _, 1 : matrix (fin 1) (fin 1) \u211d) :=\n-- by prove_symmetric \n\n-- lemma Qmsp : p = matrix.dot_product ms (matrix.mul_vec (matrix.to_poly Q) ms) :=\n-- by prove_poly_eq\n\n-- lemma Qcholesky : @cholesky_decomposition (fin 1) _ (fin 1) _ \u211d _ Q Qsymmetric :=\n-- by prove_cholesky ``(\u03bb _ _, 1) \n\n-- -- Experiments with the ring tactic.\n\n-- example : (C 1) + (C 1) = ((C 2) : mv_polynomial \u2115 \u211d) := \n-- begin \n--   simp; ring,\n-- end\n\n-- example : (X 1) + (X 1) = ((C 2) * (X 1) : mv_polynomial \u2115 \u211d) := \n-- begin \n--   simp; ring,\n-- end\n\n-- example : finset.univ.sum (\u03bb x : fin 2, (C 1 : mv_polynomial \u2115 \u211a)) = C 2 :=\n-- begin \n--   simp,\n-- end  \n\n-- example : \n--   ((X 1) * (X 1) + (C 2) * (X 1) * (X 2) + (X 2) * (X 2) : mv_polynomial \u2115 \u211a)\n--   = matrix.dot_product \n--     (list_to_vector 2 (list_to_monomials [[1], [2]]) (by simp)) \n--     ((matrix.to_poly \n--       (list_to_matrix 2 2 [[1, 1], [1, 1]] (by simp) (\u03bb i, by fin_cases i; simp)) \n--       : matrix (fin 2) (fin 2) (mv_polynomial \u2115 \u211a)).mul_vec \n--     (list_to_vector 2 (list_to_monomials [[1], [2]]) (by simp))) :=\n-- begin \n--   prove_poly_eq,\n-- end \n\n#check real.of_cauchy\n\nset_option timeout 1000000\n\n#eval if ((C (float.mk2 1 0)) = (1 : mv_polynomial \u2115 float)) then 0 else 1\n\n#eval if ((C (1 : \u211d)) = (1 : mv_polynomial \u2115 \u211d)) then 0 else 1\n\nexample : (C (rat.of_int 1 : \u211a)) = (1 : mv_polynomial \u2115 \u211a) :=\nbegin \n  refl,\nend \n\nexample : (C (float.mk \u27e81, 0\u27e9)) = (1 : mv_polynomial \u2115 float) :=\nbegin \n  show_term { refl, }\nend \n\nexample : (C (1 : \u211d)) * (X 1) = (X 1 : mv_polynomial \u2115 \u211d) :=\nbegin \n  simp,\nend \n\nexample {\u03b1} [linear_ordered_comm_ring \u03b1] \n: (C (1 : \u03b1)) * (X 1) = (X 1 : mv_polynomial \u2115 \u03b1) :=\nbegin \n  show_term { simp, }\nend \n\nexample : (C (rat.of_int 1 : \u211a)) * (X 1) = (X 1 : mv_polynomial \u2115 \u211a) :=\nbegin \n  simp, sorry,\nend \n\nexample : (C (1 : float)) * (X 1) = (X 1 : mv_polynomial \u2115 float) :=\nbegin \n  simp,\nend \n\nexample : (1 : float) \u2022 (X 1) = (X 1 : mv_polynomial \u2115 float) :=\nbegin \n  show_term {\n  simp, }\nend \n\n#check one_smul\n\nexample : C (1 + 0) * (X 1) = (X 1 : mv_polynomial \u2115 \u211a) :=\nbegin\n  simp,\nend \n\n@[simp] lemma avo : float.mk \u27e81, 0\u27e9 = 1 := rfl\n\nexample : (C (float.mk \u27e81, 0\u27e9)) * (X 1) = (X 1 : mv_polynomial \u2115 float) :=\nbegin \n  simp,\nend \n\nexample : (rat.of_int 1) + (rat.of_int 2) = rat.of_int 3 :=\nbegin \n  simp [rat.of_int], ring,\nend \n\n@[simp] theorem add_def (a b c d : \u2124)\n: (float.mk \u27e8a, b\u27e9) + (float.mk \u27e8c, d\u27e9) =\nfloat.mk (if b \u2264 d \nthen \u27e8a + c * 2 ^ int.to_nat (d - b), b\u27e9 \nelse \u27e8c + a * 2 ^ int.to_nat (b - d), d\u27e9 ) :=\nbegin \n  sorry,\nend  \n\nexample : (float.mk \u27e81, -1\u27e9) + (float.mk \u27e83, -1\u27e9) = float.mk \u27e82, 0\u27e9 :=\nbegin \n  simp [add_def], split_ifs; try {contradiction}, simp [float.mk], show to_rat _ = _,\n  simp [to_rat], norm_num, \nend \n\nexample : \n(C (float.mk \u27e81, 1\u27e9)) * (X 1) + (C (float.mk \u27e83, 1\u27e9)) * (X 1) \n= (C (float.mk \u27e82, 0\u27e9)) * (X 1 : mv_polynomial \u2115 float) :=\nbegin \n  simp, ring,\nend \n\n-- Test whole thing.\n\n-- set_option trace.app_builder true\n-- set_option timeout 1000000\n\n#eval if (1 : \u2115) = 2 then 1 else 0\n#eval if (((C (float.mk2 1 0)) : mv_polynomial \u2115 float) = (C (1 : float))) then 1 else 0\n\n\n\n-- 0 \u2264 x^2\nexample : (C (0 : float)) \u2264 (X 1) * (X 1) := \nbegin \n  sos,\n  { simp [matrix.dot_product, matrix.mul_vec, matrix.to_poly],\n    simp only [matrix.map, list.map, list.foldl, float.mk, list_to_vector, \n    list_to_monomials, list_to_matrix, list_to_monomial, fin.sum_univ_succ], dsimp,\n    ring, sorry, },\n  sorry,\nend \n\n-- 0 \u2264 x^2 + 2xy + y^2\nexample : (C (0 : \u211a)) \u2264 (((X 1) * (X 1)) + ((C 2) * (X 1) * (X 2)) + ((X 2) * (X 2)) : mv_polynomial \u2115 \u211a) :=\nbegin\n  --sos,\n  sorry, \nend \n\n-- 0 \u2264 1 + x + x^2 ([[1,1/2],[1,1/2]])\n\n", "meta": {"author": "ramonfmir", "repo": "leanSOS", "sha": "14b50713dc887f6d408b7b2bce1f8af5bb619958", "save_path": "github-repos/lean/ramonfmir-leanSOS", "path": "github-repos/lean/ramonfmir-leanSOS/leanSOS-14b50713dc887f6d408b7b2bce1f8af5bb619958/lean3SOS/src/test/test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.37060908446005264}}
{"text": "-- import the definition of the non-euclidean maze\nimport mazes.noneuclidean_maze.solutions.definition\nopen maze direction\n\n/-\n\n# Non-euclidean maze.\n\nYou are in a maze of twisty passages, all distinct. \n\nYou can go north, south east or west.\n\nIf you hit the wall there's an error.\n\nWhen you're at the exit (room `J`), type `out`.\n\nSolver remark : there are 10 rooms.\n-/\n\n/- Lemma : no-side-bar\nCan you escape from this non-Euclidean maze?\n-/\nlemma solve : can_escape A :=\nbegin\n  s,s,e,e,w,w,out,\n\n\n\n\n\n\n\nend\n", "meta": {"author": "kbuzzard", "repo": "lean-game-skeleton", "sha": "098454dd6acc4c06beccf52b6547bf4cd99cc581", "save_path": "github-repos/lean/kbuzzard-lean-game-skeleton", "path": "github-repos/lean/kbuzzard-lean-game-skeleton/lean-game-skeleton-098454dd6acc4c06beccf52b6547bf4cd99cc581/src/mazes/noneuclidean_maze/level.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.37060907636207724}}
{"text": "example (p q r s: Prop) (h1: p -> s) (h2: q -> s) (h3: r -> s)\n    : ((p \\/ q) -> s) /\\ (r -> s) := by {\n  constructor <;> intro h <;>\n  (try (apply Or.elim h <;> intro h)) <;>\n  revert h <;> assumption;\n}\n\nexample (p q r s: Prop) (h1: p -> s) (h2: q -> s) (h3: r -> s)\n    : ((p \\/ q) -> s) /\\ (r -> s) := by {\n  constructor <;> intro h <;>\n  (try (apply h.elim <;> intro h)) <;>\n  revert h <;> assumption;\n}\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1037.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737473266736, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.37054422548111804}}
{"text": "import lib tree reducibility\n\nopen encodable denumerable\n\nattribute [simp] set.set_of_app_iff\n\nstructure strategy (n : \u2115) :=\n(priority (k : \u2115) : omega_ordering (Tree k \u00d7 \u2115))\n(effective : \u2200 k, computable (priority k).ordering)\n\nnamespace strategy\n\nprotected def default (n : \u2115) : strategy n :=\n{ priority := (\u03bb k, omega_ordering.default (Tree k \u00d7 \u2115)),\n  effective := \u03bb k, computable.encode }\n\ninstance (n) : inhabited (strategy n) := \u27e8strategy.default n\u27e9\n\nvariables {n\u2080 : \u2115} (S : strategy n\u2080)\n\nnamespace approx\nvariables {k : \u2115}\n\ndef derivative (\u03b7 : Tree (k + 1)) {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : list (ancestor \u03bc) :=\n\u03bc.ancestors.filter (\u03bb a, \u03c5 a = \u03b7)\n\nlemma derivative_ordered (\u03b7 : Tree (k + 1)) {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) :\n  (derivative \u03b7 \u03c5).ordered (<) :=\nby simp[derivative]; exact list.ordered_filter _ (ancestor.ancestors_ordered \u03bc)\n\ndef initial_derivative\n  (\u03b7 : Tree (k + 1)) {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : option (ancestor \u03bc) :=\n(derivative \u03b7 \u03c5).nth 0\n\ndef pi_derivative\n  (\u03b7 : Tree (k + 1)) {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : list (ancestor \u03bc) :=\n(derivative \u03b7 \u03c5).filter (\u03bb \u03bc\u2080, (out \u03bc\u2080).is_pi)\n\ndef principal_derivative\n  (\u03b7 : Tree (k + 1)) {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : option (ancestor \u03bc) :=\n((pi_derivative \u03b7 \u03c5).nth 0).cases_on' (initial_derivative \u03b7 \u03c5) some\n\ndef lambda : \u03a0 {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)), Tree (k + 1)\n| []       _ := []\n| (x :: \u03bc) \u03c5 := let ih := lambda (ancestor.extend_fn \u03c5 \u03bc (by simp)) in \n    if \u03c5 \u27e8\u03bc, by simp\u27e9 = ih \u2228\n    (x.is_pi \u2227 pi_derivative (\u03c5 \u27e8\u03bc, by simp\u27e9) (ancestor.extend_fn \u03c5 \u03bc (by simp)) = [])\n    then (x :: \u03bc) :: (\u03c5 \u27e8\u03bc, by simp\u27e9) else ih\n\ndef assignment {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : Tree (k + 1) \u00d7 \u2115 :=\n(S.priority (k + 1)).Min_le\n  ((lambda \u03c5, 0) :: \n    ((lambda \u03c5).ancestors.filter (\u03bb \u03b7, (out \u03b7).is_pi)).map (\u03bb \u03b7, (\u03b7.val, (derivative \u03b7.val \u03c5).length))) (by simp)\n\ndef up {\u03bc : Tree k} (\u03c5 : ancestor \u03bc \u2192 Tree (k + 1)) : Tree (k + 1) :=\n(assignment S \u03c5).1\n\nend approx\nvariables {k : \u2115}\n\ndef up' : \u03a0 (\u03b7 : Tree k) (\u03bc : ancestor \u03b7), Tree (k + 1)\n| []       \u27e8\u03bc, \u03bc_p\u27e9 := by exfalso; simp* at*\n| (_ :: \u03b7) \u27e8\u03bc, _\u27e9   := if h : \u03bc \u2282\u1d62 \u03b7 then up' \u03b7 \u27e8\u03bc, h\u27e9 else approx.up S (up' \u03b7)\n\ndef assignment (\u03b7 : Tree k) : Tree (k + 1) \u00d7 \u2115 := approx.assignment S (up' S \u03b7)\n\ndef up (\u03b7 : Tree k) : Tree (k + 1) := approx.up S (up' S \u03b7)\n\nnotation `up[` S `]` := up S\n\n@[simp] lemma up'_up_consistent {\u03b7 : Tree k} : \u2200 (\u03bc : ancestor \u03b7), S.up' \u03b7 \u03bc = up[S] \u03bc.val :=\nbegin\n  induction \u03b7 with \u03bd \u03b7 IH,\n  { intros \u03bc, have := \u03bc.property, simp* at* },\n  { intros \u03bc, cases \u03bc with \u03bc \u03bc_p, \n    have : \u03bc = \u03b7 \u2228 \u03bc \u2282\u1d62 \u03b7, from list.is_initial_cons_iff.mp \u03bc_p,\n    cases this; simp[this, up'],\n    { refl }, { exact IH _ } }\nend\n\nlemma up'_up_consistent' {\u03b7 : Tree k} : S.up' \u03b7 = \u03bb \u03bc, up[S] \u03bc.val :=\nfunext (\u03bb x, by simp)\n\ndef derivative (\u03b7 : Tree (k + 1)) (\u03bc : Tree k) : list (ancestor \u03bc) := approx.derivative \u03b7 (S.up' \u03bc)\n\nlemma derivative_cons (\u03b7 : Tree (k + 1)) (\u03bd) (\u03bc : Tree k) :\n  S.derivative \u03b7 (\u03bd :: \u03bc) = if up[S] \u03bc = \u03b7 then \u27e8\u03bc, by simp\u27e9 :: (S.derivative \u03b7 \u03bc).map (ancestor.extend (by simp)) else \n    (S.derivative \u03b7 \u03bc).map (ancestor.extend (by simp)) :=\nby { simp[derivative, approx.derivative, list.filter, list.map_filter, function.comp], congr }\n\ndef pi_derivative (\u03b7 : Tree (k + 1)) (\u03bc : Tree k) : list (ancestor \u03bc) := approx.pi_derivative \u03b7 (S.up' \u03bc)\n\ndef is_link_free (\u03b7 : Tree (k + 1)) (\u03bc : Tree k) (\u03bc\u2080 : ancestor \u03bc) : bool :=\n((S.derivative \u03b7 \u03bc).filter (\u03bb \u03bd, \u03bd \u2264 \u03bc\u2080) = []) || ((S.pi_derivative \u03b7 \u03bc).filter (\u03bb \u03bd, \u03bc\u2080 \u2264 \u03bd) = [])\n\ndef lambda (\u03b7 : Tree k) : Tree (k + 1) := approx.lambda (S.up' \u03b7)\n\nnotation `\u03bb[` S `]` := lambda S\n\n\n@[simp] lemma up_extend {\u03bc\u2081 \u03bc\u2082 : Tree k} {h : \u03bc\u2082 <:+ \u03bc\u2081} : ancestor.extend_fn (S.up' \u03bc\u2081) \u03bc\u2082 h = S.up' \u03bc\u2082 :=\nby { simp[ancestor.extend_fn], exact eq.symm S.up'_up_consistent' }\n\n@[simp] lemma extend_lambda {\u03bc \u03bc\u2080 : Tree k} (h : \u03bc\u2080 <:+ \u03bc) :\n  approx.lambda (ancestor.extend_fn (S.up' \u03bc) \u03bc\u2080 h) = \u03bb[S] \u03bc\u2080 :=\nby { simp[ancestor.extend_fn, lambda], congr, funext x, simp}\n\nlemma assignment_fst_eq_up (\u03bc : Tree k) : (S.assignment \u03bc).1 = up[S] \u03bc :=\nby simp[assignment, up, approx.up]\n\nlemma up_eq_lambda_or_pi (\u03bc : Tree k) : up[S] \u03bc = \u03bb[S] \u03bc \u2228 \u2203 \u03b7 : ancestor (\u03bb[S] \u03bc), (out \u03b7).is_pi \u2227 up[S] \u03bc = \u03b7 :=\nby { have : S.assignment \u03bc \u2208 _, from omega_ordering.Min_le_mem _ _, simp at this,\n     cases this,\n     { left, simp[\u2190assignment_fst_eq_up, this], refl },\n     { right, rcases this with \u27e8\u03b7, pi, eqn\u27e9, refine \u27e8\u03b7, pi, _\u27e9, simp[\u2190assignment_fst_eq_up, \u2190eqn] } }\n\nlemma up_eq_or_lt (\u03bc : Tree k) : up[S] \u03bc = \u03bb[S] \u03bc \u2228 \u2203 lt : up[S] \u03bc \u2282\u1d62 \u03bb[S] \u03bc, (out \u27e8up[S] \u03bc, lt\u27e9).is_pi :=\nby { have : S.assignment \u03bc \u2208 _, from omega_ordering.Min_le_mem _ _, simp at this,\n     cases this,\n     { left, simp[\u2190assignment_fst_eq_up, this], refl },\n     { right, rcases this with \u27e8\u03b7, pi, eqn\u27e9, simp[\u2190assignment_fst_eq_up, \u2190eqn], exact \u27e8\u03b7.property, pi\u27e9 } }\n\n@[simp] lemma lambda_nil_eq : \u03bb[S] ([] : Tree k) = [] :=\nby simp[lambda, approx.lambda]\n\nlemma lambda_cons_eq (x) (\u03bc : Tree k) : \u03bb[S] (x :: \u03bc) = (x :: \u03bc) :: up[S] \u03bc \u2228 \u03bb[S] (x :: \u03bc) = \u03bb[S] \u03bc :=\nby { unfold lambda, simp[approx.lambda],\n     by_cases C : up[S] \u03bc = approx.lambda (S.up' \u03bc) \u2228 \u21a5(x.is_pi) \u2227 approx.pi_derivative (up[S] \u03bc) (S.up' \u03bc) = [];\n     simp[C] }\n\n@[simp] lemma up_nil_eq : up[S] ([] : Tree k) = [] :=\nby { have := S.up_eq_or_lt ([] : Tree k), simp at this, exact this }\n\n-- Consistency 1\n\nlemma up_le_lambda (\u03bc : Tree k) : up[S] \u03bc <:+ \u03bb[S] \u03bc :=\nby { rcases S.up_eq_or_lt \u03bc with (eqn | \u27e8lt, eqn\u27e9), { simp[eqn] }, { exact list.suffix_of_is_initial lt } }\n\nlemma eq_lambda_of_le_lambda {\u03bc : Tree k} {\u03b7 : Tree (k + 1)} (le : \u03b7 <:+ \u03bb[S] \u03bc) :\n  \u03b7 = [] \u2228 \u2203 \u03bc\u2080 : ancestor \u03bc, \u03b7 = \u03bb[S] ((out \u03bc\u2080) :: \u03bc\u2080.val) \u2227 \n  (up[S] \u03bc\u2080.val = \u03bb[S] \u03bc\u2080.val \u2228\n    (out \u03bc\u2080).is_pi \u2227 (\u2200 (a : ancestor \u03bc\u2080.val), a \u2208 S.derivative (up[S] \u03bc\u2080.val) \u03bc\u2080.val \u2192 (out a).is_sigma)) \u2227\n    \u03b7 = ((out \u03bc\u2080) :: \u03bc\u2080.val) :: up[S] \u03bc\u2080 :=\nbegin\n  induction \u03bc with x \u03bc IH,\n  { left, simp[lambda, approx.lambda] at le, exact le },\n  { by_cases C :\n      up[S] \u03bc = \u03bb[S] \u03bc \u2228 x.is_pi \u2227 approx.pi_derivative (up[S] \u03bc) (S.up' \u03bc) = list.nil,\n    { have eqn : \u03bb[S] (x :: \u03bc) = (x :: \u03bc) :: up[S] \u03bc, { unfold lambda at*, simp[approx.lambda, C] },\n      have C\u2082 : \u03b7 = (x :: \u03bc) :: up[S] \u03bc \u2228 \u03b7 <:+ up[S] \u03bc,\n      { simp [eqn] at le, exact list.suffix_cons_iff.mp le },\n      rcases C\u2082 with (rfl | C\u2082),\n      { refine or.inr \u27e8\u27e8\u03bc, by simp\u27e9, _\u27e9, simp[eqn, C],\n        simp[approx.pi_derivative, list.filter_eq_nil] at C, exact C },\n      { have := IH (C\u2082.trans (S.up_le_lambda \u03bc)),\n        rcases this with (rfl | \u27e8\u03bc\u2080, rfl, eqn\u27e9), { simp },\n        { refine or.inr \u27e8\u03bc\u2080.extend (by simp), _\u27e9, simp, exact eqn } } },\n    { have eqn : \u03bb[S] (x :: \u03bc) = \u03bb[S] \u03bc,\n      { unfold lambda, simp[approx.lambda, C, show approx.lambda (S.up' \u03bc) = \u03bb[S] \u03bc, by refl] },\n      have := IH (by { simp[\u2190eqn, le] }),\n      rcases this with (rfl | \u27e8\u03bc\u2080, rfl, eqn\u27e9), { simp },\n      refine or.inr \u27e8\u03bc\u2080.extend (by simp), _\u27e9, simp, exact eqn } }\nend\n\nlemma eq_lambda_of_lt_lambda {\u03bc : Tree k} (\u03b7 : ancestor (\u03bb[S] \u03bc)) :\n  \u2203 \u03bc\u2080 : ancestor \u03bc, out \u03b7 :: \u03b7.val = \u03bb[S] ((out \u03bc\u2080) :: \u03bc\u2080.val) \u2227\n  ( up[S] \u03bc\u2080.val = \u03bb[S] \u03bc\u2080 \u2228\n    (out \u03bc\u2080).is_pi \u2227 \u2200 (\u03bd : ancestor \u03bc\u2080.val), \u03bd \u2208 S.derivative (up[S] \u2191\u03bc\u2080) \u03bc\u2080.val \u2192 (out \u03bd).is_sigma ) \u2227\n  out \u03b7 = (out \u03bc\u2080) :: \u03bc\u2080.val \u2227 \u03b7.val = up[S] \u03bc\u2080 :=\nby { have := S.eq_lambda_of_le_lambda (suffix_out_cons \u03b7), simp at this,\n     rcases this with \u27e8\u03bc\u2080, eqn\u2081, h, eqn\u2082\u27e9,\n     exact \u27e8\u03bc\u2080, eqn\u2081, h, list.head_eq_of_cons_eq eqn\u2082, list.tail_eq_of_cons_eq eqn\u2082\u27e9 }\n\nlemma eq_lambda_of_le_lambda' {\u03bc : Tree k} {\u03b7 : Tree (k + 1)} (le : \u03b7 <:+ \u03bb[S] \u03bc) :\n\u2203 \u03bc\u2080 : Tree k, \u03bc\u2080 <:+ \u03bc \u2227 \u03b7 = \u03bb[S] \u03bc\u2080 :=\nbegin\n  rcases S.eq_lambda_of_le_lambda le with (rfl | \u27e8\u03bc\u2080, eqn\u2081, _, eqn\u2082\u27e9),\n  { refine \u27e8[], by simp[lambda, approx.lambda, list.nil_suffix]\u27e9 },\n  { refine \u27e8out \u03bc\u2080 :: \u03bc\u2080.val, by simp[eqn\u2082]; exact suffix_out_cons _, eqn\u2081\u27e9 }\nend\n\nlemma eq_lambda_of_pred {\u03bc \u03bd : Tree k} {\u03b7 : Tree (k + 1)} (eqn : \u03bb[S] \u03bc = \u03bd :: \u03b7) : \u03bb[S] \u03bd = \u03bb[S] \u03bc :=\nbegin\n  have lt : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc, { simp[eqn] },\n  rcases S.eq_lambda_of_lt_lambda \u27e8\u03b7, lt\u27e9 with \u27e8\u03bc\u2080, eqn_lam, _, eqn_out, eqn_up\u27e9, simp at*,\n  have : out \u27e8\u03b7, lt\u27e9 = \u03bd, { simp[out_eq_iff, eqn] },\n  simp[\u2190eqn_out, this] at eqn_lam, simp[eqn, eqn_lam]\nend\n\nlemma initial_of_mem_lambda {\u03c1 \u03bc : Tree k}\n  (h : \u03bc \u2208 \u03bb[S] \u03c1) : \u2203 \u03bc\u2080 : ancestor \u03c1, \u03bc = out \u03bc\u2080 :: \u03bc\u2080.val :=\nbegin\n  rcases list.mem_iff_rnth.mp h with \u27e8n, eqn\u27e9,\n  have le\u2081 : \u03bc :: \u03bb[S] \u03c1\u21be*n <:+ \u03bb[S] \u03c1, from list.rnth_eq_iff_suffix_cons_initial.mp eqn,\n  have lt : \u03bb[S] \u03c1\u21be*n \u2282\u1d62 \u03bb[S] \u03c1, from list.suffix_cons_iff_is_initial.mp \u27e8_, le\u2081\u27e9,\n  rcases S.eq_lambda_of_lt_lambda \u27e8_, lt\u27e9 with \u27e8\u03bc\u2080, _, _, out_eq, _\u27e9,\n  have : \u03bc = out \u03bc\u2080 :: \u03bc\u2080.val,\n  { have := list.suffix_or_suffix_of_suffix le\u2081 (out_eq_iff.mp out_eq), simp at this,\n    cases this; simp [this] },\n  exact \u27e8\u03bc\u2080, this\u27e9\nend\n\nlemma suffix_of_mem_lambda {\u03c1 \u03bc : Tree k}\n  (h : \u03bc \u2208 \u03bb[S] \u03c1) : \u03bc <:+ \u03c1 :=\nby rcases S.initial_of_mem_lambda h with \u27e8\u03bc\u2080, rfl\u27e9; exact suffix_out_cons \u03bc\u2080\n\nlemma out_eq_out {\u03c1 : Tree k}\n  (\u03b7 : ancestor (\u03bb[S] \u03c1)) : \u2203 \u03bc\u2080 : ancestor \u03c1, out \u03b7 = out \u03bc\u2080 :: \u03bc\u2080.val :=\nS.initial_of_mem_lambda (by { rcases suffix_out_cons \u03b7 with \u27e8l, eqn\u27e9, simp[\u2190eqn] })\n\nlemma suffix_out {\u03c1 : Tree k}\n  (\u03b7 : ancestor (\u03bb[S] \u03c1)) : out \u03b7 <:+ \u03c1 :=\nS.suffix_of_mem_lambda (by { rcases suffix_out_cons \u03b7 with \u27e8l, eqn\u27e9, simp[\u2190eqn] })\n\nlemma noninitial_of_suffix {\u03bc\u2081 \u03bc\u2082 : Tree k}\n  (lt : \u03bc\u2081 <:+ \u03bc\u2082) : \u00ac\u03bb[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] \u03bc\u2081 :=\nbegin\n  rcases lt with \u27e8l, rfl\u27e9,\n  induction l with x \u03bd IH,\n  { simp },\n  { by_cases C : up[S] (\u03bd ++ \u03bc\u2081) = approx.lambda (S.up' (\u03bd ++ \u03bc\u2081)) \u2228\n      (x.is_pi) \u2227 approx.pi_derivative (up[S] (\u03bd ++ \u03bc\u2081)) (S.up' (\u03bd ++ \u03bc\u2081)) = [],\n    { intros h,\n      have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2081)) = (x :: (\u03bd ++ \u03bc\u2081)) :: up[S] (\u03bd ++ \u03bc\u2081),\n      { simp[lambda, approx.lambda, C] },\n      simp[lambda_eqn] at h,\n      have : x :: (\u03bd ++ \u03bc\u2081) <:+ \u03bc\u2081, from S.suffix_of_mem_lambda (by { rcases h with \u27e8l, a, eqn\u27e9, simp[\u2190eqn] }),\n      have : \u03bc\u2081 <:+ \u03bc\u2081 \u2227 \u03bc\u2081 \u2260 \u03bc\u2081, from list.is_initial_iff_suffix.mp\n          (by rw [\u2190list.cons_append] at this; exact list.is_initial_of_pos_suffix this (by simp)),\n      simp at this, contradiction },\n    have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2081)) = \u03bb[S] (\u03bd ++ \u03bc\u2081),\n      { simp[lambda, approx.lambda, C] }, simp[lambda_eqn], exact IH }\nend\n\n@[simp] lemma noninitial_of_suffix' (\u03bc \u03bd : Tree k) : \u00ac\u03bb[S] (\u03bd ++ \u03bc) \u2282\u1d62 \u03bb[S] \u03bc :=\nS.noninitial_of_suffix (by simp)\n\nlemma incomparable_of_incomparable {\u03bc\u2081 \u03bc\u2082 \u03bc\u2083 : Tree k}\n  (le\u2081 : \u03bc\u2081 <:+ \u03bc\u2082) (le\u2082 : \u03bc\u2082 <:+ \u03bc\u2083) (h : \u03bb[S] \u03bc\u2081 \u2225 \u03bb[S] \u03bc\u2082) : \u03bb[S] \u03bc\u2081 \u2225 \u03bb[S] \u03bc\u2083 :=\nbegin\n  rcases le\u2082 with \u27e8l, rfl\u27e9,\n  induction l with x \u03bd IH,\n  { simp[h] },\n  { by_cases C : up[S] (\u03bd ++ \u03bc\u2082) = approx.lambda (S.up' (\u03bd ++ \u03bc\u2082)) \u2228\n      (x.is_pi) \u2227 approx.pi_derivative (up[S] (\u03bd ++ \u03bc\u2082)) (S.up' (\u03bd ++ \u03bc\u2082)) = list.nil; simp[C],\n    { have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2082)) = (x :: (\u03bd ++ \u03bc\u2082)) :: up[S] (\u03bd ++ \u03bc\u2082),\n      { simp[lambda, approx.lambda, C] },\n      refine list.incomparable_iff_suffix_is_initial.mpr \u27e8\u03bb A, _, \u03bb A, _\u27e9,\n      { have C\u2082 : \u03bb[S] \u03bc\u2081 <:+ up[S] (\u03bd ++ \u03bc\u2082),\n        { rw [lambda_eqn] at A, exact list.is_initial_cons_iff_suffix.mp A },\n        { have := IH.1 (C\u2082.trans (S.up_le_lambda (\u03bd ++ \u03bc\u2082))), contradiction } },\n      { rw [lambda_eqn] at A,\n        have : x :: (\u03bd ++ \u03bc\u2082) <:+ \u03bc\u2081, from S.suffix_of_mem_lambda (by rcases A with \u27e8l, eqn\u27e9; simp[\u2190eqn]),\n        have : \u03bc\u2082 <:+ \u03bc\u2081 \u2227 \u03bc\u2082 \u2260 \u03bc\u2081, from list.is_initial_iff_suffix.mp\n          (by rw [\u2190list.cons_append] at this; exact list.is_initial_of_pos_suffix this (by simp)),\n        rcases list.suffix_antisymm le\u2081 this.1 with rfl, simp at this, contradiction } },\n    have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2082)) = \u03bb[S] (\u03bd ++ \u03bc\u2082),\n      { simp[lambda, approx.lambda, C] },\n    simp[lambda_eqn], exact IH }\nend\n\nlemma suffix_of_suffix {\u03bc\u2081 \u03bc\u2082 \u03bc\u2083 : Tree k}\n  (le\u2081 : \u03bc\u2081 <:+ \u03bc\u2082) (le\u2082 : \u03bc\u2082 <:+ \u03bc\u2083) (h : \u03bb[S] \u03bc\u2081 <:+ \u03bb[S] \u03bc\u2083) : \u03bb[S] \u03bc\u2081 <:+ \u03bb[S] \u03bc\u2082 :=\nby { have := mt (S.incomparable_of_incomparable le\u2081 le\u2082) (\u03bb nonle, nonle.1 h),\n     simp[list.incomparable_iff_is_initial_suffix, S.noninitial_of_suffix le\u2081] at this, exact this }\n\nlemma sigma_preserve {\u03bc\u2081 : Tree k} {\u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082)\n  {\u03b7 : ancestor (\u03bb[S] \u03bc\u2081)} (sigma : (out \u03b7).is_sigma) (lt : \u03b7.val \u2282\u1d62 \u03bb[S] \u03bc\u2082) :\n  out \u03b7 :: \u03b7.val <:+ \u03bb[S] \u03bc\u2082 :=\nbegin\n  rcases le with \u27e8l, rfl\u27e9,\n  induction l with x \u03bd IH,\n  { simp, exact suffix_out_cons \u03b7 },\n  { by_cases C : up[S] (\u03bd ++ \u03bc\u2081) = approx.lambda (S.up' (\u03bd ++ \u03bc\u2081)) \u2228\n      (x.is_pi) \u2227 approx.pi_derivative (up[S] (\u03bd ++ \u03bc\u2081)) (S.up' (\u03bd ++ \u03bc\u2081)) = [],\n    { have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2081)) = (x :: (\u03bd ++ \u03bc\u2081)) :: up[S] (\u03bd ++ \u03bc\u2081),\n      { simp[lambda, approx.lambda, C] },\n      have le : \u03b7.val <:+ up[S] (\u03bd ++ \u03bc\u2081), { simp[lambda_eqn] at lt, exact list.is_initial_cons_iff_suffix.mp lt },      \n      have lt : \u03b7.val \u2282\u1d62 \u03bb[S] (\u03bd ++ \u03bc\u2081),\n      { have := le.trans (S.up_le_lambda _),\n        have C\u2082 := list.suffix_iff_is_initial.mp this, rcases C\u2082, exact C\u2082,\n        have := \u03b7.property, simp[C\u2082] at this, contradiction },\n      have IH' : out \u03b7 :: \u03b7.val <:+ \u03bb[S] (\u03bd ++ \u03bc\u2081), from IH lt,\n      have C\u2082 : \u03b7.val \u2282\u1d62 up[S] (\u03bd ++ \u03bc\u2081) \u2228 \u03b7.val = up[S] (\u03bd ++ \u03bc\u2081), from list.suffix_iff_is_initial.mp le,\n      cases C\u2082,\n      { rcases list.suffix_cons_iff_is_initial.mpr C\u2082 with \u27e8y, eqn\u27e9,\n        have : out \u03b7 = y,\n        { have := list.suffix_of_suffix_length_le IH' (eqn.trans (S.up_le_lambda _)) (by simp),\n          simp at this, exact this },\n        simp[lambda_eqn, this], exact eqn.trans (by simp) },\n      { have C\u2083 := S.up_eq_or_lt (\u03bd ++ \u03bc\u2081), rcases C\u2083 with (eqn | \u27e8lt_up, pi\u27e9),\n        { exfalso, simp[eqn] at C\u2082, simp[C\u2082] at lt, contradiction },\n        { exfalso,\n          have : out \u27e8\u03b7.val, lt\u27e9 = out \u03b7, from out_eq_iff.mpr IH',\n          have : out \u27e8up[S] (\u03bd ++ \u03bc\u2081), lt_up\u27e9 = out \u03b7, rw\u2190this, from suffix_out_eq (by simp[C\u2082]) (by refl),\n          simp[this] at pi, exact neg_is_pi_iff.mpr sigma pi } } },\n    { have lambda_eqn : \u03bb[S] (x :: (\u03bd ++ \u03bc\u2081)) = \u03bb[S] (\u03bd ++ \u03bc\u2081),\n      { simp[lambda, approx.lambda, C] },\n      simp[lambda_eqn] at lt \u22a2, exact IH lt } }\nend\n\nlemma eq_out_of_sigma {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082) {\u03b7 : Tree (k + 1)}\n  (lt\u2081 : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2081) (lt\u2082 : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2082) (sigma : (out \u27e8\u03b7, lt\u2081\u27e9).is_sigma) :\n  out \u27e8\u03b7, lt\u2081\u27e9 = out \u27e8\u03b7, lt\u2082\u27e9 :=\nbegin\n  have lmm\u2081 : out \u27e8\u03b7, lt\u2081\u27e9 :: \u03b7 <:+ \u03bb[S] \u03bc\u2082, from S.sigma_preserve le sigma lt\u2082,\n  have lmm\u2082 : out \u27e8\u03b7, lt\u2082\u27e9 :: \u03b7 <:+ \u03bb[S] \u03bc\u2082, from suffix_out_cons \u27e8\u03b7, lt\u2082\u27e9,\n  have := list.suffix_of_suffix_length_le lmm\u2081 lmm\u2082 (by simp), simp at this, exact this\nend\n\nlemma up_eq_lambda_of_pi {\u03bc : Tree k} {\u03b7 : ancestor (\u03bb[S] \u03bc)} (pi : (out \u03b7).is_pi) :\n  \u2203 \u03bd : ancestor \u03bc, out \u03b7 = out \u03bd :: \u03bd.val \u2227 \u03bb[S] \u03bd.val = \u03b7 \u2227 \u03bb[S] (out \u03bd :: \u03bd.val) = out \u03b7 :: \u03b7.val :=\nbegin\n  rcases S.eq_lambda_of_lt_lambda \u03b7 with \u27e8\u03bd, eqn_lam, (eqn_up\u2081 | \u27e8pi', _\u27e9), eqn_out, eqn_up\u2082\u27e9,\n  { refine \u27e8\u03bd, eqn_out, by simp[\u2190eqn_up\u2081, \u2190eqn_up\u2082], _\u27e9, rw [eqn_lam] },\n  { exfalso, simp[eqn_out] at pi, exact not_pi_sigma pi' pi }\nend\n\nlemma eq_out_of_pi {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082) {\u03b7 : Tree (k + 1)}\n  (lt\u2081 : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2081) (lt\u2082 : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2082) (pi : (out \u27e8\u03b7, lt\u2082\u27e9).is_pi) :\n  out \u27e8\u03b7, lt\u2081\u27e9 = out \u27e8\u03b7, lt\u2082\u27e9 :=\nbegin\n  have C\u2081 : (out \u27e8\u03b7, lt\u2081\u27e9).is_pi \u2228 (out \u27e8\u03b7, lt\u2081\u27e9).is_sigma, from pi_or_sigma (out \u27e8\u03b7, lt\u2081\u27e9),\n  cases C\u2081,\n  { rcases S.up_eq_lambda_of_pi pi with \u27e8\u27e8\u03bd\u2081, lt_\u03bd\u2081\u27e9, eqn_out\u2081, eqn_lam\u2081, eqn_lam\u2081'\u27e9, simp at eqn_lam\u2081 eqn_out\u2081 eqn_lam\u2081',\n    rcases S.up_eq_lambda_of_pi C\u2081 with \u27e8\u27e8\u03bd\u2082, lt_\u03bd\u2082\u27e9, eqn_out\u2082, eqn_lam\u2082, eqn_lam\u2082'\u27e9, simp at eqn_lam\u2082 eqn_out\u2082 eqn_lam\u2082',\n    have lt_\u03bd\u2082' : \u03bd\u2082 \u2282\u1d62 \u03bc\u2082, from list.is_initial.is_initial_of_suffix lt_\u03bd\u2082 le,\n    have eqn_out_out : out \u27e8\u03bd\u2082, lt_\u03bd\u2082'\u27e9 = out \u27e8\u03bd\u2082, lt_\u03bd\u2082\u27e9, from suffix_out_eq (by simp) le,\n    suffices : \u03bd\u2081 = \u03bd\u2082,\n    { rcases this with rfl, \n      simp[eqn_out\u2081, eqn_out\u2082, eqn_out_out] },\n    have C : \u03bd\u2081 \u2282\u1d62 \u03bd\u2082 \u2228 \u03bd\u2081 = \u03bd\u2082 \u2228 \u03bd\u2082 \u2282\u1d62 \u03bd\u2081,\n    { have : ancestor.mk' lt_\u03bd\u2081 < ancestor.mk' lt_\u03bd\u2082' \u2228\n        ancestor.mk' lt_\u03bd\u2081 = ancestor.mk' lt_\u03bd\u2082' \u2228 ancestor.mk' lt_\u03bd\u2082' < ancestor.mk' lt_\u03bd\u2081,\n      exact trichotomous (ancestor.mk' lt_\u03bd\u2081) (ancestor.mk' lt_\u03bd\u2082'), simp[ancestor.lt_iff] at this,\n      exact this },\n    cases C,\n    { exfalso,\n      have : out \u27e8\u03bd\u2081, lt_\u03bd\u2081\u27e9 :: \u03bd\u2081 <:+ \u03bd\u2082,\n      { have eqn : out \u27e8\u03bd\u2081, lt_\u03bd\u2081\u27e9 = out \u27e8\u03bd\u2081, C\u27e9, refine suffix_out_eq (by simp) (list.suffix_of_is_initial lt_\u03bd\u2082'),\n        have := suffix_out_cons \u27e8\u03bd\u2081, C\u27e9, simp at this, simp[eqn, this] },\n      have := S.noninitial_of_suffix this, simp[eqn_lam\u2081', eqn_lam\u2082] at this, contradiction }, cases C,\n    { exact C },\n    { exfalso,\n      have : \u03bd\u2081 \u2282\u1d62 \u03bc\u2081,\n      { have := lt_or_le_of_le_of_le (list.suffix_of_is_initial lt_\u03bd\u2081) le,\n        cases this, { exact this }, { exfalso, simp[\u2190eqn_lam\u2081] at lt\u2081, exact noninitial_of_suffix S this lt\u2081 } },\n      have : out \u27e8\u03bd\u2082, lt_\u03bd\u2082\u27e9 :: \u03bd\u2082 <:+ \u03bd\u2081,\n      { have eqn : out \u27e8\u03bd\u2082, lt_\u03bd\u2082\u27e9 = out \u27e8\u03bd\u2082, C\u27e9, refine suffix_out_eq (by simp) (list.suffix_of_is_initial this),\n        have := suffix_out_cons \u27e8\u03bd\u2082, C\u27e9, simp at this, simp[eqn, this] },\n      have := S.noninitial_of_suffix this, simp[eqn_lam\u2082', eqn_lam\u2081] at this, contradiction } },\n  { exact S.eq_out_of_sigma le lt\u2081 lt\u2082 C\u2081 }\nend\n\nlemma lt_lambda_of_lt_le {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082)\n  {\u03b7 : Tree (k + 1)} (lt : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2081) (le_\u03b7 : \u03b7 <:+ \u03bb[S] \u03bc\u2082) : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2082 :=\nbegin\n  have C\u2081 : \u03b7 \u2282\u1d62 \u03bb[S] \u03bc\u2082 \u2228 \u03b7 = \u03bb[S] \u03bc\u2082, from list.suffix_iff_is_initial.mp le_\u03b7,\n  rcases C\u2081 with (C\u2081 | rfl),\n  { exact C\u2081 },\n  { exfalso, exact S.noninitial_of_suffix le lt }\nend\n\nprivate lemma sigma_outcome_of_eq_up (\u03bc) {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt\u2081 : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082) (lt\u2082 : \u03bc\u2082 \u2282\u1d62 \u03bc)\n  (eqn : up[S] \u03bc\u2081 = up[S] \u03bc\u2082) (up_lt : up[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] \u03bc\u2082) : (out \u27e8\u03bc\u2081, lt\u2081\u27e9).is_sigma :=\nbegin\n  suffices : \u00ac(out \u27e8\u03bc\u2081, lt\u2081\u27e9).is_pi,\n  { simp[Tree'.is_sigma, this] },\n  intros A,\n  induction \u03bc with x \u03bc IH generalizing \u03bc\u2081 \u03bc\u2082,\n  { simp at lt\u2082, contradiction },\n  { have up_lt\u2081 : up[S] \u03bc\u2081 \u2282\u1d62 \u03bb[S] \u03bc\u2082, { simp[eqn, up_lt] },\n    have C\u2081 : \u03bc\u2082 \u2282\u1d62 \u03bc \u2228 \u03bc\u2082 = \u03bc, from list.suffix_iff_is_initial.mp (list.is_initial_cons_iff_suffix.mp lt\u2082),\n    rcases C\u2081 with (C\u2081 | rfl),\n    { exact IH lt\u2081 C\u2081 eqn up_lt A },\n    { have eqn_lam\u2081 : \u03bb[S] (out \u27e8\u03bc\u2081, lt\u2081\u27e9 :: \u03bc\u2081) = (out \u27e8\u03bc\u2081, lt\u2081\u27e9 :: \u03bc\u2081) :: up[S] \u03bc\u2081,\n      { have C\u2082 : up[S] \u03bc\u2081 \u2282\u1d62 \u03bb[S] \u03bc\u2081 \u2228 up[S] \u03bc\u2081 = \u03bb[S] \u03bc\u2081, from list.suffix_iff_is_initial.mp (S.up_le_lambda \u03bc\u2081),\n        cases C\u2082,\n        { have : approx.pi_derivative (up[S] \u03bc\u2081) (S.up' \u03bc\u2081) = [],\n          { simp[approx.pi_derivative, approx.derivative, list.filter_eq_nil],\n            rintros \u27e8\u03bd, lt_\u03bd\u27e9 pi_\u03bd eqn_\u03bd, exact IH lt_\u03bd lt\u2081 eqn_\u03bd C\u2082 pi_\u03bd },\n          unfold lambda, simp[approx.lambda, A, this] },\n        { unfold lambda at C\u2082 \u22a2, simp[approx.lambda, C\u2082] } },\n      have out_eq : out (\u27e8up[S] \u03bc\u2081, by simp[eqn_lam\u2081]\u27e9 : ancestor (\u03bb[S] (out \u27e8\u03bc\u2081, lt\u2081\u27e9 :: \u03bc\u2081))) = out \u27e8\u03bc\u2081, lt\u2081\u27e9 :: \u03bc\u2081,\n        from out_eq_iff.mpr (by simp[eqn_lam\u2081]),      \n      have : out \u27e8up[S] \u03bc\u2081, _\u27e9 = out \u27e8up[S] \u03bc\u2081, up_lt\u2081\u27e9,\n        from @eq_out_of_sigma _ S _ (out \u27e8\u03bc\u2081, lt\u2081\u27e9 :: \u03bc\u2081) \u03bc\u2082 (suffix_out_cons \u27e8\u03bc\u2081, lt\u2081\u27e9)\n        (up[S] \u03bc\u2081) (by simp[eqn_lam\u2081]) up_lt\u2081 (by simp[out_eq, Tree'.is_sigma, A]),\n      have sigma : (out \u27e8up[S] \u03bc\u2081, up_lt\u2081\u27e9).is_sigma,\n      { simp[\u2190this, out_eq, Tree'.is_sigma, A] },\n      have C\u2082 := S.up_eq_or_lt \u03bc\u2082, rcases C\u2082 with (eqn | \u27e8lt', pi\u27e9),\n      { simp[eqn] at up_lt, contradiction },\n      { simp[\u2190eqn] at pi lt', exact neg_is_pi_iff.mpr sigma pi } } }\nend\n\n-- Consistency 2\n\nlemma sigma_outcome_of_eq_up {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082)\n  (eqn : up[S] \u03bc\u2081 = up[S] \u03bc\u2082) (up_lt : up[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] \u03bc\u2082) : (out \u27e8\u03bc\u2081, lt\u27e9).is_sigma :=\nsigma_outcome_of_eq_up S ((default _) :: \u03bc\u2082) lt (by simp) eqn up_lt\n\nlemma sigma_outcome_of_pi {\u03bc : Tree k} {\u03bc\u2080 : ancestor \u03bc} (pi : (out \u03bc\u2080).is_pi) :\n  \u03bb[S] (out \u03bc\u2080 :: \u03bc\u2080.val) = (out \u03bc\u2080 :: \u03bc\u2080.val) :: up[S] \u03bc\u2080 :=\nbegin\n  simp[lambda, approx.lambda],\n  have : up[S] \u2191\u03bc\u2080 \u2282\u1d62 \u03bb[S] \u2191\u03bc\u2080 \u2228 up[S] \u2191\u03bc\u2080 = \u03bb[S] \u2191\u03bc\u2080, from list.suffix_iff_is_initial.mp (S.up_le_lambda \u03bc\u2080.val),\n  rcases this with (lt_up | eq_up),\n  { have : approx.pi_derivative (up[S] \u2191\u03bc\u2080) (S.up' \u2191\u03bc\u2080) = [],\n    { simp[approx.pi_derivative, approx.derivative, list.filter_eq_nil],\n      rintros \u27e8\u03bc\u2081, lt_\u03bc\u2081\u27e9 pi' eq_up,\n      have := S.sigma_outcome_of_eq_up lt_\u03bc\u2081 eq_up lt_up, exact not_pi_sigma pi' this },\n    simp [this, pi] },\n  { simp[eq_up, lambda] }\nend\n\nlemma sigma_outcome_of_eq_up' {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082)\n  (eqn : up[S] \u03bc\u2081 = up[S] \u03bc\u2082) : (out \u27e8\u03bc\u2081, lt\u27e9).is_sigma :=\nbegin\n  have C : (out \u27e8\u03bc\u2081, lt\u27e9).is_pi \u2228 (out \u27e8\u03bc\u2081, lt\u27e9).is_sigma, from pi_or_sigma (out \u27e8\u03bc\u2081, lt\u27e9),\n  cases C,  \n  have C\u2082 : up[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] \u03bc\u2082 \u2228 up[S] \u03bc\u2082 = \u03bb[S] \u03bc\u2082, from list.suffix_iff_is_initial.mp (S.up_le_lambda \u03bc\u2082),\n  cases C\u2082,\n  { exact S.sigma_outcome_of_eq_up lt eqn C\u2082 },\n  { have : \u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081) = (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081) :: up[S] \u03bc\u2081, from S.sigma_outcome_of_pi C,\n    have : \u03bb[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081), { simp[this, eqn, C\u2082] },\n    have : \u00ac\u03bb[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081), from S.noninitial_of_suffix (suffix_out_cons \u27e8\u03bc\u2081, lt\u27e9),\n    contradiction },\n  exact C\nend\n\nvariables (\u039b : Path k)\n\ntheorem finite_injury (n : \u2115) :\n  \u2203 s\u2080, \u2200 s, s\u2080 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = \u03bb[S] (\u039b s\u2080)\u21be*n :=\nbegin\n  induction n with n IH,\n  { simp },\n  { rcases IH with \u27e8s\u2080, IH\u27e9,\n    suffices :\n      \u2203 s\u2081, s\u2080 \u2264 s\u2081 \u2227 \u2200 s, s\u2081 \u2264 s \u2192 (\u03bb[S] (\u039b s)).rnth n = (\u03bb[S] (\u039b s\u2081)).rnth n,\n    { rcases this with \u27e8s\u2081, eqn, hyp_s\u2081\u27e9, refine \u27e8s\u2081, \u03bb s eqn_s, _\u27e9,\n      apply list.rnth_ext',\n      { intros m a, simp[list.initial_rnth_some_iff], intros eqn_m,\n        have C : m < n \u2228 m = n, from lt_or_eq_of_le (nat.lt_succ_iff.mp eqn_m),\n        cases C,\n        { have : (\u03bb[S] (\u039b s)\u21be*n).rnth m = (\u03bb[S] (\u039b s\u2080)\u21be*n).rnth m,\n            from congr_arg (\u03bb l : list _, l.rnth m) (IH s (eqn.trans eqn_s)),\n          simp [list.initial_rnth_of_lt C] at this, simp[this],\n          have : (\u03bb[S] (\u039b s\u2081)\u21be*n).rnth m = (\u03bb[S] (\u039b s\u2080)\u21be*n).rnth m,\n            from congr_arg (\u03bb l : list _, l.rnth m) (IH s\u2081 eqn),\n          simp [list.initial_rnth_of_lt C] at this, simp[this] },\n        { simp[C], have := hyp_s\u2081 s eqn_s, simp[this] } } },\n    by_cases C\u2081 : \u2200 s, s\u2080 \u2264 s \u2192 (\u03bb[S] (\u039b s)).rnth n = none,\n    { refine \u27e8s\u2080, by refl, \u03bb s eqn_s, _\u27e9, simp[C\u2081 s eqn_s, C\u2081 s\u2080 (by refl)] },\n    { have : \u2203 s\u2081, s\u2080 \u2264 s\u2081 \u2227 \u2203 x, (\u03bb[S] (\u039b s\u2081)).rnth n = some x,\n      { simp at C\u2081, rcases C\u2081 with \u27e8s\u2081, eqn, C\u2081\u27e9, exact \u27e8s\u2081, eqn, option.ne_none_iff_exists'.mp C\u2081\u27e9 },\n      rcases this with \u27e8s\u2081, eqn_s\u2081, \u03bd, C\u2081\u27e9,\n      have IH' : \u2200 (s : \u2115), s\u2081 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = \u03bb[S] (\u039b s\u2081)\u21be*n, \n      { intros s le, simp[IH s\u2081 eqn_s\u2081, IH s (eqn_s\u2081.trans le)] },\n      have lt : \u03bb[S] (\u039b s\u2081)\u21be*n \u2282\u1d62 \u03bb[S] (\u039b s\u2081),\n        from list.suffix_cons_iff_is_initial.mp \u27e8_, list.rnth_eq_iff_suffix_cons_initial.mp C\u2081\u27e9,\n      have lt' : \u2200 s, s\u2081 \u2264 s \u2192 \u03bb[S] (\u039b s\u2081)\u21be*n \u2282\u1d62 \u03bb[S] (\u039b s),\n      { intros s eqn_s, refine S.lt_lambda_of_lt_le (\u039b.mono' eqn_s) lt\n        (by { simp[\u2190IH' s eqn_s], exact list.suffix_initial _ _}) },\n      have rnth_eqn : \u2200 s (le : s\u2081 \u2264 s), (\u03bb[S] (\u039b s)).rnth n = some (out \u27e8\u03bb[S] (\u039b s\u2081)\u21be*n, lt' s le\u27e9),\n      { intros s le, refine list.rnth_eq_iff_suffix_cons_initial.mpr _, have := suffix_out_cons \u27e8_, lt' s le\u27e9,\n        simp[IH' s le], exact this },\n      by_cases C\u2082 : \u2200 (s : \u2115) (le : s\u2081 \u2264 s), (out \u27e8\u03bb[S] (\u039b s\u2081)\u21be*n, lt' s le\u27e9).is_pi,\n      { refine \u27e8s\u2081, eqn_s\u2081, \u03bb s eqn_s, _\u27e9, simp[rnth_eqn, rnth_eqn _ eqn_s],\n        refine eq.symm (S.eq_out_of_pi (\u039b.mono' eqn_s) _ _ (by simp[C\u2082])) },\n      { have : \u2203 s\u2082 (h : s\u2081 \u2264 s\u2082), \u21a5((out \u27e8\u03bb[S] (\u039b s\u2081)\u21be*n, lt' s\u2082 h\u27e9).is_sigma),\n        { simp at C\u2082, exact C\u2082 },\n        rcases this with \u27e8s\u2082, eqn_s\u2082, C\u2082\u27e9,\n        refine \u27e8s\u2082, eqn_s\u2081.trans eqn_s\u2082, \u03bb s eqn_s, _\u27e9, simp[rnth_eqn _ eqn_s\u2082, rnth_eqn _ (eqn_s\u2082.trans eqn_s)], \n        refine eq.symm (S.eq_out_of_sigma (\u039b.mono' eqn_s) _ _ C\u2082) } } }\nend\n\ntheorem Path_exists :\n  \u2203 \u039b' : Path (k + 1), \u2200 n, \u2203 s\u2080, \u2200 s, s\u2080 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = \u039b' n :=\nbegin\n  let P : \u2115 \u2192 \u2115 \u2192 Prop := \u03bb n s\u2080, (\u2200 s, s\u2080 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = \u03bb[S] (\u039b s\u2080)\u21be*n),\n  have : \u2200 n, \u2203 s\u2080, P n s\u2080, from \u03bb n, S.finite_injury \u039b n,\n  have : \u2200 n, \u2203 s\u2080, (\u2200 s, s < s\u2080 \u2192 \u00acP n s) \u2227 P n s\u2080,\n  { intros n, exact nat.least_number (this n) },\n  have : \u2203 (f : \u2115 \u2192 \u2115), \u2200 x, (\u2200 s, s < f x \u2192 \u00acP x s) \u2227 P x (f x),\n    from classical.skolem.mp this,\n  rcases this with \u27e8f, h_f\u27e9,\n  let path : \u2115 \u2192 Tree (k + 1) := \u03bb n, \u03bb[S] (\u039b (f n))\u21be*n,\n  have mono : \u2200 n, path n <:+ path (n + 1),\n  { intros n, simp[path],\n    have mono : f n \u2264 f (n + 1),\n    { suffices : \u00acf (n + 1) < f n, { simp* at* },\n      intros A,\n      have min : \u2203 x, f (n + 1) \u2264 x \u2227 \u00ac\u03bb[S] (\u039b x)\u21be*n = \u03bb[S] (\u039b (f (n + 1)))\u21be*n,\n      { have := (h_f n).1 _ A, simp[P] at this, exact this },\n      rcases min with \u27e8m, le_m, neq\u27e9,\n      have : \u03bb[S] (\u039b m)\u21be*(n + 1) = \u03bb[S] (\u039b (f (n + 1)))\u21be*(n + 1),\n      { have := (h_f (n + 1)).2 _ le_m, exact this },\n      have := (congr_arg (\u03bb l : list _, l\u21be*n) this), simp at this, exact neq this },\n    have : \u03bb[S] (\u039b (f (n + 1)))\u21be*n = \u03bb[S] (\u039b (f n))\u21be*n, from (h_f n).2 _ mono, \n    simp[\u2190this],\n    rw (show \u03bb[S] (\u039b (f (n + 1)))\u21be*n = \u03bb[S] (\u039b (f (n + 1)))\u21be*(n + 1)\u21be*n, by simp),\n    exact list.suffix_initial _ _ },\n  refine \u27e8\u27e8path, mono\u27e9, \u03bb n, \u27e8f n, \u03bb s eqn_s, _\u27e9\u27e9,\n  exact (h_f n).2 _ eqn_s\nend\n\nnoncomputable def Lambda (\u039b : Path k) : Path (k + 1) := classical.epsilon\n(\u03bb \u039b',  \u2200 n, \u2203 s\u2080, \u2200 s, s\u2080 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = \u039b' n)\n\nnotation `\u039b[` S `]` := Lambda S\n\ntheorem Lambda_spec : \u2200 n, \u2203 s\u2080, \u2200 s, s\u2080 \u2264 s \u2192 \u03bb[S] (\u039b s)\u21be*n = (\u039b[S] \u039b) n :=\nclassical.epsilon_spec (S.Path_exists \u039b)\n\nlemma lt_Lambda_iff {\u039b : Path k} {\u03b7 : Tree (k + 1)} :\n  \u03b7 \u2282' \u039b[S] \u039b \u2194 \u2203 s\u2080, \u2200 s, s\u2080 \u2264 s \u2192 \u03b7 \u2282\u1d62 \u03bb[S] (\u039b s) :=\n\u27e8\u03bb \u27e8n, h\u27e9, by {\n    rcases S.Lambda_spec \u039b n with \u27e8s\u2080, eqn\u27e9,\n    refine \u27e8s\u2080, \u03bb s eqn_s, _\u27e9,\n    have : \u03bb[S] (\u039b s)\u21be*n = (\u039b[S] \u039b) n, from eqn s eqn_s, simp[\u2190this] at h,\n    have : \u03bb[S] (\u039b s)\u21be*n <:+ \u03bb[S] (\u039b s), from list.suffix_initial _ _,\n    exact list.is_initial.is_initial_of_suffix h this }, \n  \u03bb \u27e8s\u2080, h\u27e9, by { \n    rcases S.Lambda_spec \u039b (\u03b7.length + 1) with \u27e8s\u2081, eqn\u27e9,\n    refine \u27e8\u03b7.length + 1, _\u27e9,\n    have := eqn (max s\u2080 s\u2081) (le_max_right s\u2080 s\u2081), rw \u2190this,\n    rcases h (max s\u2080 s\u2081) (le_max_left s\u2080 s\u2081) with \u27e8l, a, eqn\u27e9,\n    simp[\u2190eqn, list.initial] }\u27e9\n\nlemma le_lamvda_of_lt_Lambda' {\u039b : Path k} {\u03b7 : Tree (k + 1)} {s\u2080} (lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080) :\n  \u2203 s\u2081, \u2200 s, s\u2081 \u2264 s \u2192 out \u27e8\u03b7, lt\u27e9 :: \u03b7 <:+ \u03bb[S] (\u039b s) :=\nbegin\n  rcases S.Lambda_spec \u039b s\u2080 with \u27e8s\u2081, eqn\u27e9, refine \u27e8s\u2081, \u03bb s le_s, _\u27e9,\n  have : out \u27e8\u03b7, lt\u27e9 :: \u03b7 <:+ \u03bb[S] (\u039b s)\u21be*s\u2080, simp[eqn s le_s], from suffix_out_cons \u27e8\u03b7, lt\u27e9,\n  exact this.trans (list.suffix_initial (lambda S (\u039b s)) s\u2080)\nend\n\nlemma le_Lambda_of_thick {\u039b : Path k} (thick : \u039b.thick)\n  {\u03b7 : Tree (k + 1)} {s\u2080} (le : \u03b7 <:+ (\u039b[S] \u039b) s\u2080) : lim s, \u03b7 =\u2264 \u03bb[S] (\u039b s) :=\nbegin\n  rcases S.Lambda_spec \u039b s\u2080 with \u27e8t, eqn\u27e9,\n  have le' : \u2200 s, t \u2264 s \u2192 \u03b7 <:+ \u03bb[S] (\u039b s),\n  { intros s le_s,\n    have : (\u039b[S] \u039b) s\u2080 <:+ \u03bb[S] (\u039b s), { simp[\u2190eqn s le_s], exact list.suffix_initial _ _ },\n    exact le.trans this },\n  rcases S.eq_lambda_of_le_lambda' (le' t (by refl)) with \u27e8\u03bc\u2081, le_\u03bc\u2081, rfl\u27e9,\n  rcases thick.ssubset.mp \u27e8t, le_\u03bc\u2081\u27e9 with \u27e8s\u2081, rfl\u27e9,\n  have : \u2200 s, s\u2081 \u2264 s \u2192 \u03bb[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b s),\n  { intros s le_s, have C : s \u2264 t \u2228 t \u2264 s, exact le_total s t,\n    cases C,\n    { exact S.suffix_of_suffix (\u039b.mono' le_s) (\u039b.mono' C) (le' t (by refl)) },\n    { exact le' s C } },\n  refine \u27e8s\u2081, rfl, this\u27e9\nend\n\nlemma le_Lambda_of_thick' {\u039b : Path k} (thick : \u039b.thick)\n  {\u03b7 : Tree (k + 1)} (le : \u03b7 \u2286' \u039b[S] \u039b) : lim s, \u03b7 =\u2264 \u03bb[S] (\u039b s) :=\nby { rcases le with \u27e8s\u2081, le\u27e9, exact S.le_Lambda_of_thick thick le }\n\nlemma up_eq_lambda_of_thick {\u039b : Path k} (thick : \u039b.thick) {s}\n  {\u03b7 : Tree (k + 1)} (le : up[S] (\u039b s) \u2286' \u039b[S] \u039b) : \u2203 s\u2080 \u2264 s, up[S] (\u039b s) = \u03bb[S] (\u039b s\u2080) :=\nbegin\n  have C : up[S] (\u039b s) \u2282\u1d62 \u03bb[S] (\u039b s) \u2228 up[S] (\u039b s) = \u03bb[S] (\u039b s), from list.suffix_iff_is_initial.mp (S.up_le_lambda (\u039b s)),\n  cases C,\n  { rcases S.le_Lambda_of_thick' thick le with \u27e8t, up_eq, _\u27e9, simp[up_eq] at C,\n    have : t \u2264 s,\n    { by_contradiction, have : s \u2264 t, exact le_of_not_ge h,\n      have : \u00ac\u03bb[S] (\u039b t) \u2282\u1d62 \u03bb[S] (\u039b s), from S.noninitial_of_suffix (thick.le_mono_iff.mpr (le_of_not_ge h)),\n      contradiction },\n    exact \u27e8t, this, up_eq\u27e9 },\n  exact \u27e8s, by refl, C\u27e9\nend\n\nlemma eq_lt_lambda_of_lt_Lambda_of_pi {\u039b : Path k} (thick : \u039b.thick)\n  {\u03b7 : Tree (k + 1)} {s\u2080} (lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080) (pi : (out \u27e8\u03b7, lt\u27e9).is_pi) :\n  lim s, \u27e8\u03b7, lt\u27e9 =< \u03bb[S] (\u039b s) :=\nbegin\n  rcases S.le_Lambda_of_thick thick (suffix_out_cons \u27e8\u03b7, lt\u27e9) with \u27e8s\u2081, eqn\u2081, le\u2081\u27e9, simp at eqn\u2081 le\u2081,\n  rcases S.le_Lambda_of_thick thick (lt.suffix) with \u27e8s\u2082, rfl, le\u2082\u27e9, \n  have : \u2203 s, s\u2082 \u2264 s \u2227 \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b s) \u2227 \u03bb[S] (\u039b s\u2082) \u2282\u1d62 \u03bb[S] (\u039b (s + 1)),\n  { by_contradiction,\n    simp at h,\n    have : \u2200 s, s\u2082 \u2264 s \u2192 \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b s) \u2192 \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b (s + 1)),\n    { intros s le eqn,\n      have : \u00ac\u03bb[S] (\u039b s\u2082) \u2282\u1d62 \u03bb[S] (\u039b (s + 1)), from h s le eqn,\n      rcases list.suffix_iff_is_initial.mp (le\u2082 (s + 1) (le_add_right le)) with (lt_lam | eq_lam),\n      { exfalso, exact this lt_lam }, { exact eq_lam } },\n    have eq_lam' : \u2200 s, s\u2082 \u2264 s \u2192 \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b s),\n    { suffices : \u2200 s, \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b (s\u2082 + s)),\n      { intros s le,\n        simp[this (s - s\u2082), show  s\u2082 + (s - s\u2082) = s, from nat.add_sub_of_le le] },\n      intros s, induction s with s IH,\n      { refl }, { simp[\u2190nat.add_one, \u2190add_assoc], exact this (s\u2082 + s) (le_self_add) IH } },\n    have lt_lam : \u03bb[S] (\u039b s\u2082) \u2282\u1d62 \u03bb[S] (\u039b (max s\u2081 s\u2082)),\n      from list.suffix_cons_iff_is_initial.mp \u27e8_, le\u2081 (max s\u2081 s\u2082) (le_max_left s\u2081 s\u2082)\u27e9,\n    have eq_lam : \u03bb[S] (\u039b s\u2082) = \u03bb[S] (\u039b (max s\u2081 s\u2082)), from eq_lam' (max s\u2081 s\u2082) (le_max_right s\u2081 s\u2082),\n    simp[eq_lam] at lt_lam, contradiction },\n  rcases this with \u27e8s\u2083, le_s\u2083, eq_lam_s\u2082, lt_lam_s\u2082\u27e9,\n  have : \u2200 s, s\u2083 \u2264 s \u2192 out \u27e8\u03bb[S] (\u039b s\u2082), lt\u27e9 :: \u03bb[S] (\u039b s\u2082) <:+ \u03bb[S] (\u039b (s + 1)),\n  { intros s le_s, have C : s\u2081 \u2264 s \u2228 s < s\u2081, exact le_or_lt s\u2081 s, cases C,\n    { exact le\u2081 (s + 1) (le_add_right C) }, \n    { have lt\u2081 : \u03bb[S] (\u039b s\u2082) \u2282\u1d62 \u03bb[S] (\u039b s.succ),\n      { rcases list.suffix_iff_is_initial.mp (le\u2082 (s + 1) (le_add_right (le_s\u2083.trans le_s))) with (C | C); simp at C,\n        { exact C }, { exfalso, simp[C] at lt_lam_s\u2082, refine S.noninitial_of_suffix (\u039b.mono' (by simp[le_s])) lt_lam_s\u2082 } },\n      have lt\u2082 : \u03bb[S] (\u039b s\u2082) \u2282\u1d62 \u03bb[S] (\u039b s\u2081), from list.suffix_cons_iff_is_initial.mp \u27e8_, le\u2081 s\u2081 (by refl)\u27e9,\n      have eqn\u2081 : out \u27e8\u03bb[S] (\u039b s\u2082), lt\u2082\u27e9 = out \u27e8\u03bb[S] (\u039b s\u2082), lt\u27e9, { simp[out_eq_iff], exact le\u2081 s\u2081 (by refl) },      \n      have eqn\u2082 : out \u27e8\u03bb[S] (\u039b s\u2082), lt\u2081\u27e9 = out \u27e8\u03bb[S] (\u039b s\u2082), lt\u2082\u27e9, \n        from S.eq_out_of_pi (\u039b.mono' (nat.succ_le_iff.mpr C)) lt\u2081 lt\u2082 (by simp[eqn\u2081]; exact pi),\n      simp[\u2190eqn\u2081, \u2190eqn\u2082], exact suffix_out_cons \u27e8\u03bb[S] (\u039b s\u2082), lt\u2081\u27e9 } },\n  refine \u27e8s\u2083, eq_lam_s\u2082, this\u27e9  \nend\n\nlemma equiv_lambda {\u039b\u2081 \u039b\u2082 : Path k} (equiv : \u039b\u2081 \u2243\u209a \u039b\u2082) (thick : \u039b\u2082.thick):\n  \u039b[S] \u039b\u2081 = \u039b[S] \u039b\u2082 :=\nPath.ext (\u03bb s, begin\n  rcases S.Lambda_spec \u039b\u2081 s with \u27e8t\u2081, eqn\u2081\u27e9,\n  rcases S.Lambda_spec \u039b\u2082 s with \u27e8t\u2082, eqn\u2082\u27e9,\n  rcases equiv with \u27e8le\u2081, le\u2082\u27e9,\n  sorry\nend)\n\n@[simp] def lambda_itr : \u2200 (\u03bc : Tree k) (i : \u2115), Tree (k + i)\n| \u03bc 0       := \u03bc\n| \u03bc (i + 1) := \u03bb[S] (lambda_itr \u03bc i)\n\n@[simp] def up_itr : \u2200 (\u03bc : Tree k) (i : \u2115), Tree (k + i)\n| \u03bc 0       := \u03bc\n| \u03bc (i + 1) := up[S] (up_itr \u03bc i)\n\nlemma lambda_proper {\u03bc : Tree k} (proper : \u03bc.proper) : (\u03bb[S] \u03bc).proper :=\nbegin\n  induction \u03bc with \u03bd \u03bc IH,\n  { simp },\n  { have C : \u03bb[S] (\u03bd :: \u03bc) = (\u03bd :: \u03bc) :: up[S] \u03bc \u2228 \u03bb[S] (\u03bd :: \u03bc) = \u03bb[S] \u03bc, from S.lambda_cons_eq \u03bd \u03bc,\n    have proper' : @Tree'.proper (k + 1) \u03bc, exact Tree'.proper.proper_of_cons proper,\n    cases C,\n    { simp[C, Tree'.proper],\n      have lt_of_mem : \u2200 \u03b7, \u03b7 \u2208 up[S] \u03bc \u2192 \u03b7 \u2282\u1d62 \u03bd :: \u03bc,\n      { intros \u03b7 mem,\n        have mem' : \u03b7 \u2208 \u03bb[S] \u03bc, { rcases S.up_le_lambda \u03bc with \u27e8l, h\u27e9,simp[\u2190h, mem] },\n        exact list.is_initial_cons_iff_suffix.mpr (S.suffix_of_mem_lambda mem') },\n      have proper_of_mem : \u2200 \u03b7 : Tree' (k + 1), \u03b7 \u2208 up[S] \u03bc \u2192 \u03b7.proper,\n      { intros \u03b7 mem,\n        have mem' : \u03b7 \u2208 \u03bb[S] \u03bc, { rcases S.up_le_lambda \u03bc with \u27e8l, h\u27e9,simp[\u2190h, mem] },\n        exact (IH proper').2 mem' },      \n      refine \u27e8\u27e8list.ordered_suffix (S.up_le_lambda \u03bc) ((IH proper').1), lt_of_mem\u27e9, proper, proper_of_mem\u27e9 },\n    { simp[C], exact IH proper' } }\nend\n\nlemma up_proper {\u03bc : Tree k} (proper : \u03bc.proper) : (up[S] \u03bc).proper :=\nTree'.proper.proper_of_le (S.up_le_lambda \u03bc) (S.lambda_proper proper)\n\nlemma weight_lambda_mono {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082) (ne : \u03bb[S] \u03bc\u2081 \u2260 \u03bb[S] \u03bc\u2082) :\n  (\u03bb[S] \u03bc\u2081).weight < (\u03bb[S] \u03bc\u2082).weight :=\nbegin\n  cases eqn\u2081 : \u03bb[S] \u03bc\u2081 with \u03c0 \u03b7\u2081;\n  cases eqn\u2082 : \u03bb[S] \u03bc\u2082 with \u03c3 \u03b7\u2082,\n  { simp [eqn\u2081, eqn\u2082] at ne, contradiction },\n  { simp },\n  { exfalso, have := S.noninitial_of_suffix (list.suffix_of_is_initial lt), simp[eqn\u2081, eqn\u2082] at this, contradiction },\n  have le\u2081 : \u03c0 <:+ \u03bc\u2081, from  S.suffix_of_mem_lambda (by simp[eqn\u2081]),\n  have le\u2082 : \u03c3 <:+ \u03bc\u2082, from  S.suffix_of_mem_lambda (by simp[eqn\u2082]),\n  have le\u2081' : \u03c0 <:+ \u03bc\u2082, from list.suffix_of_is_initial (list.is_suffix.is_initial_of_is_initial le\u2081 lt),\n  have eqn_lam\u2081 : \u03bb[S] \u03c0 = \u03bb[S] \u03bc\u2081, from S.eq_lambda_of_pred eqn\u2081,\n  have eqn_lam\u2082 : \u03bb[S] \u03c3 = \u03bb[S] \u03bc\u2082, from S.eq_lambda_of_pred eqn\u2082,\n  have C : \u03c0 \u2282\u1d62 \u03c3 \u2228 \u03c0 = \u03c3 \u2228 \u03c3 \u2282\u1d62 \u03c0, from trichotomous_of_le_of_le le\u2081' le\u2082,\n  cases C,\n  { exact lt_weight_cons_of_lt C }, exfalso, rcases C with \u27e8rfl, C\u27e9,\n  { simp[\u2190eqn_lam\u2081, \u2190eqn_lam\u2082] at ne, contradiction },\n  { have := S.suffix_of_suffix (list.suffix_of_is_initial C) le\u2081' (by simp[eqn_lam\u2082]),\n    simp[eqn_lam\u2081, eqn_lam\u2082] at this, \n    have C\u2081 : \u03bb[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] \u03bc\u2081 \u2228 \u03bb[S] \u03bc\u2082 = \u03bb[S] \u03bc\u2081,\n      from list.suffix_iff_is_initial.mp this,\n    cases C\u2081, { exact S.noninitial_of_suffix (list.suffix_of_is_initial lt) C\u2081 },\n    { exact ne (eq.symm C\u2081)} }\nend\n\nlemma lt_weight_lambda_of_incomparable {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082) (ne : \u03bb[S] \u03bc\u2081 \u2225 \u03bb[S] \u03bc\u2082) :\n  (\u03bb[S] \u03bc\u2081).weight < (\u03bb[S] \u03bc\u2082).weight :=\nS.weight_lambda_mono lt (\u03bb eq, by { simp [eq] at ne, contradiction })\n\nlemma weight_lambda_le_mono {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082) :\n  (\u03bb[S] \u03bc\u2081).weight \u2264 (\u03bb[S] \u03bc\u2082).weight :=\nbegin\n  have : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082 \u2228 \u03bc\u2081 = \u03bc\u2082, from list.suffix_iff_is_initial.mp le,\n  rcases this with (lt | rfl),\n  { by_cases C : \u03bb[S] \u03bc\u2081 = \u03bb[S] \u03bc\u2082, { simp[C] }, { exact le_of_lt (S.weight_lambda_mono lt C) } },\n  { refl }\nend\n\nlemma weight_lambda_inj_of_thick {\u039b : Path k} (thick : \u039b.thick) {s\u2081 s\u2082} :\n  (\u03bb[S] (\u039b s\u2081)).weight = (\u03bb[S] (\u039b s\u2082)).weight \u2192 \u03bb[S] (\u039b s\u2081) = \u03bb[S] (\u039b s\u2082) := \u03bb eq,\nbegin\n  by_contradiction A,\n  have : s\u2081 < s\u2082 \u2228 s\u2081 = s\u2082 \u2228 s\u2082 < s\u2081, from trichotomous s\u2081 s\u2082,\n  rcases this with (lt | rfl | lt),\n  { have : (\u03bb[S] (\u039b s\u2081)).weight < (\u03bb[S] (\u039b s\u2082)).weight, from S.weight_lambda_mono (thick.lt_mono_iff.mpr lt) A,\n    simp [eq] at this, contradiction },\n  { simp at A, contradiction },\n  { have : (\u03bb[S] (\u039b s\u2082)).weight < (\u03bb[S] (\u039b s\u2081)).weight, from S.weight_lambda_mono (thick.lt_mono_iff.mpr lt) (ne.symm A),\n    simp [eq] at this, contradiction  }\nend\n\nlemma Lambda_pi_outcome\n  {\u03b7 : Tree (k + 1)} {s\u2080} {lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080} (pi : (out \u27e8\u03b7, lt\u27e9).is_pi)\n  {\u03bc : Tree k} {t\u2080} (lt' : \u03bc \u2282\u1d62 \u039b t\u2080) (up_eq : up[S] \u03bc = \u03b7) : (out \u27e8\u03bc, lt'\u27e9).is_sigma :=\nbegin\n  rcases up_eq with rfl,\n  rcases S.le_lamvda_of_lt_Lambda' lt with \u27e8s\u2081, le\u27e9,\n  by_contradiction A, simp at A,\n  have eq_lam : \u03bb[S] (out \u27e8\u03bc, lt'\u27e9 :: \u03bc) = (out \u27e8\u03bc, lt'\u27e9 :: \u03bc) :: up[S] \u03bc, \n    from S.sigma_outcome_of_pi A,\n  have : out \u27e8up[S] \u03bc, lt\u27e9 = out \u27e8\u03bc, lt'\u27e9 :: \u03bc,\n  { have lt\u2081 : up[S] \u03bc \u2282\u1d62 \u03bb[S] (out \u27e8\u03bc, lt'\u27e9 :: \u03bc), { simp[eq_lam] },\n    have lt\u2082 : up[S] \u03bc \u2282\u1d62 \u03bb[S] (\u039b (max s\u2081 t\u2080)),\n      from list.suffix_cons_iff_is_initial.mp \u27e8_, le (max s\u2081 t\u2080) (le_max_left s\u2081 t\u2080)\u27e9,\n    have eq_out\u2081 : out \u27e8up[S] \u03bc, lt\u2081\u27e9 = out \u27e8\u03bc, lt'\u27e9 :: \u03bc, { simp[out_eq_iff, eq_lam] },\n    have eq_out\u2082 : out \u27e8up[S] \u03bc, lt\u2081\u27e9 = out \u27e8up[S] \u03bc, lt\u2082\u27e9,\n      from S.eq_out_of_sigma ((suffix_out_cons \u27e8\u03bc, lt'\u27e9).trans (\u039b.mono' (le_max_right s\u2081 t\u2080))) lt\u2081 lt\u2082 (by simp[eq_out\u2081, A]),\n    have : out \u27e8up[S] \u03bc, lt\u2082\u27e9 = out \u27e8up[S] \u03bc, lt\u27e9,\n    { simp[out_eq_iff], exact le (max s\u2081 t\u2080) (le_max_left s\u2081 t\u2080) },\n    rw[\u2190eq_out\u2081, eq_out\u2082, this] },\n  simp[this] at pi, exact not_pi_sigma A pi\nend\n\nlemma Lambda_pi_outcome_of_thick {\u039b : Path k} (thick : \u039b.thick)\n  {\u03b7 : Tree (k + 1)} {s\u2080} {lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080} (pi : (out \u27e8\u03b7, lt\u27e9).is_pi)\n  (s) (up_eq : up[S] (\u039b s) = \u03b7) : (thick.out s).is_sigma :=\nbegin\n  have := S.Lambda_pi_outcome \u039b pi (thick.lt_mono_iff.mpr (lt_add_one s)) up_eq,\n  simp[thick.out_eq_out] at*, exact this\nend\n\nlemma Lambda_sigma_outcome\n  {\u03b7 : Tree (k + 1)} {s\u2080} (lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080) (sigma : (out \u27e8\u03b7, lt\u27e9).is_sigma) :\n  \u2203 {\u03bc : Tree k} {t\u2080} (lt' : \u03bc \u2282\u1d62 \u039b t\u2080) (up_eq : up[S] \u03bc = \u03b7), out \u27e8\u03b7, lt\u27e9 = out \u27e8\u03bc, lt'\u27e9 :: \u03bc \u2227 (out \u27e8\u03bc, lt'\u27e9).is_pi :=\nbegin\n  rcases S.le_lamvda_of_lt_Lambda' lt with \u27e8s\u2081, le\u27e9,\n  have lt' : \u03b7 \u2282\u1d62 \u03bb[S] (\u039b s\u2081), from list.suffix_cons_iff_is_initial.mp \u27e8_, le s\u2081 (by refl)\u27e9,\n  rcases S.out_eq_out \u27e8\u03b7, lt'\u27e9 with \u27e8\u27e8\u03bc, lt_\u03bc\u27e9, eqn_\u03bc\u27e9,\n  have eq_out : out \u27e8\u03b7, lt'\u27e9 = out \u27e8\u03b7, lt\u27e9, { simp[out_eq_iff], exact le s\u2081 (by refl) },\n  have pi : (out \u27e8\u03bc, lt_\u03bc\u27e9).is_pi, { simp[\u2190eq_out, eqn_\u03bc] at sigma, exact sigma },\n  have : up[S] \u03bc = \u03b7,\n  { rcases S.eq_lambda_of_lt_lambda \u27e8\u03b7, lt'\u27e9 with \u27e8\u27e8\u03bc\u2080, lt_\u03bc\u2080\u27e9, _, _, eqn_\u03bc\u2080, eq_up\u2080\u27e9, \n    have : \u03bc = \u03bc\u2080, { simp[eqn_\u03bc] at eqn_\u03bc\u2080, exact list.tail_eq_of_cons_eq eqn_\u03bc\u2080 },\n    rcases this with rfl, exact eq.symm eq_up\u2080 },\n  exact \u27e8\u03bc, s\u2081, lt_\u03bc, this, by simp[\u2190eq_out, eqn_\u03bc], pi\u27e9\nend\n\nlemma Lambda_sigma_outcome_of_thick (thick : \u039b.thick)\n  {\u03b7 : Tree (k + 1)} {s\u2080} (lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080) (sigma : (out \u27e8\u03b7, lt\u27e9).is_sigma) :\n  \u2203 {s : \u2115}, up[S] (\u039b s) = \u03b7 \u2227 out \u27e8\u03b7, lt\u27e9 = thick.out s :: \u039b s \u2227 (thick.out s).is_pi :=\nbegin\n  rcases S.Lambda_sigma_outcome \u039b lt sigma with \u27e8\u03bc, t\u2080, lt_\u03bc, rfl, eq_out, pi\u27e9,\n  rcases thick.ssubset.mp \u27e8t\u2080, lt_\u03bc.suffix\u27e9 with \u27e8s\u2081, rfl\u27e9,\n  simp[thick.out_eq_out] at*,\n  refine \u27e8s\u2081, rfl, eq_out, pi\u27e9\nend\n\nlemma up_sigma_semimono {\u039b : Path k} (thick : \u039b.thick)\n  {s\u2081 s\u2082 : \u2115} (le : s\u2081 \u2264 s\u2082) (pi\u2082 : (thick.out s\u2082).is_pi) (le\u2081 : up[S] (\u039b s\u2081) \u2286' \u039b[S] \u039b) :\n  up[S] (\u039b s\u2081) <:+ up[S] (\u039b s\u2082) :=\nbegin\n  rcases S.le_Lambda_of_thick' thick le\u2081 with \u27e8t\u2081, eq_up, le_up\u27e9, simp at eq_up le_up,\n  have C : up[S] (\u039b s\u2081) <:+ up[S] (\u039b s\u2082) \u2228 up[S] (\u039b s\u2082) \u2282\u1d62 up[S] (\u039b s\u2081) \u2228 up[S] (\u039b s\u2081) \u2225 up[S] (\u039b s\u2082),\n    from list.incomparable_trichotomy (up[S] (\u039b s\u2081)) (up[S] (\u039b s\u2082)),\n  cases C,\n  { exact C },\n  exfalso,\n  { have C\u2082 : up[S] (\u039b s\u2081) \u2282\u1d62 \u03bb[S] (\u039b s\u2081) \u2228 up[S] (\u039b s\u2081) = \u03bb[S] (\u039b s\u2081), from list.suffix_iff_is_initial.mp (S.up_le_lambda (\u039b s\u2081)), \n    cases C\u2082,\n    { have le_t\u2081 : t\u2081 \u2264 s\u2081,\n      { by_contradiction nle,\n        have : \u00acup[S] (\u039b s\u2081) \u2282\u1d62 \u03bb[S] (\u039b s\u2081), simp[eq_up], from S.noninitial_of_suffix (thick.le_mono_iff.mpr (le_of_not_ge nle)),\n        contradiction },\n      cases C,\n      { have : \u03bb[S] (\u039b (s\u2082 + 1)) = \u039b (s\u2082 + 1) :: up[S] (\u039b s\u2082), simp[thick.succ_eq], from S.sigma_outcome_of_pi pi\u2082,\n        have : \u03bb[S] (\u039b (s\u2082 + 1)) \u2225 up[S] (\u039b s\u2081), simp[this],\n          from list.incomparable_of_lt C (\u03bb A, by { have : \u039b (s\u2082 + 1) <:+ \u039b s\u2081, from S.suffix_of_mem_lambda (list.mem_of_suffix (S.up_le_lambda (\u039b s\u2081)) A),\n          exact nat.lt_le_antisymm (nat.succ_le_iff.mp (thick.le_mono_iff.mp this)) le }),\n        have : \u00acup[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (s\u2082 + 1)), from this.2,\n        have : up[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (s\u2082 + 1)), from le_up (s\u2082 + 1) (le_t\u2081.trans (le.trans (nat.le_succ s\u2082))),\n        contradiction },\n      { have : up[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b s\u2082), from le_up s\u2082 (le_t\u2081.trans le),\n        have : \u00acup[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b s\u2082), from (list.incomparable_of_le_of_le C (by refl) (S.up_le_lambda _)).1,\n        contradiction } },\n    { cases C,\n      { have : \u03bb[S] (\u039b (s\u2082 + 1)) = \u039b (s\u2082 + 1) :: up[S] (\u039b s\u2082), simp[thick.succ_eq], from S.sigma_outcome_of_pi pi\u2082,\n        have : \u03bb[S] (\u039b (s\u2082 + 1)) \u2225 \u03bb[S] (\u039b s\u2081), simp[this, \u2190C\u2082],\n          from list.incomparable_of_lt C (\u03bb A, by { have : \u039b (s\u2082 + 1) <:+ \u039b s\u2081, from S.suffix_of_mem_lambda (list.mem_of_suffix (S.up_le_lambda (\u039b s\u2081)) A),\n          exact nat.lt_le_antisymm (nat.succ_le_iff.mp (thick.le_mono_iff.mp this)) le }),\n        have : \u00ac\u03bb[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (max (s\u2082 + 1) t\u2081)),\n          from (S.incomparable_of_incomparable (thick.le_mono_iff.mpr (le.trans (nat.le_succ s\u2082)))\n          (thick.le_mono_iff.mpr (le_max_left (s\u2082 + 1) t\u2081)) this.symm).1,\n        have : \u03bb[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (max (s\u2082 + 1) t\u2081)), simp[\u2190C\u2082], from le_up (max (s\u2082 + 1) t\u2081) (le_max_right (s\u2082 + 1) t\u2081),\n        contradiction },\n      { have : \u03bb[S] (\u039b s\u2081) \u2225 \u03bb[S] (\u039b s\u2082), from list.incomparable_of_le_of_le C (by simp[C\u2082]) (S.up_le_lambda _),\n        have : \u00ac\u03bb[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (max s\u2082 t\u2081)),\n          from (S.incomparable_of_incomparable (thick.le_mono_iff.mpr le) (thick.le_mono_iff.mpr (le_max_left s\u2082 t\u2081)) this).1,\n        have : \u03bb[S] (\u039b s\u2081) <:+ \u03bb[S] (\u039b (max s\u2082 t\u2081)), simp[\u2190C\u2082], from le_up (max s\u2082 t\u2081) (le_max_right s\u2082 t\u2081),\n        contradiction } } }\nend\n\nlemma lt_weight_lambda_up {\u039b : Path k} (thick : \u039b.thick) (proper : \u039b.proper)\n  {s\u2081 s\u2082 : \u2115} (lt : s\u2081 < s\u2082) (pi\u2081 : (thick.out s\u2081).is_pi) (pi\u2082 : (thick.out s\u2082).is_pi) (lt\u2081 : up[S] (\u039b s\u2081) \u2286' \u039b[S] \u039b) :\n  (\u03bb[S] (\u039b s\u2081)).weight < (up[S] (\u039b s\u2082)).weight :=\nbegin\n  have eq_lam\u2081 : \u03bb[S] (\u039b (s\u2081 + 1)) = \u039b (s\u2081 + 1) :: up[S] (\u039b s\u2081), simp[thick.succ_eq], from S.sigma_outcome_of_pi pi\u2081,\n  have eq_lam\u2082 : \u03bb[S] (\u039b (s\u2082 + 1)) = \u039b (s\u2082 + 1) :: up[S] (\u039b s\u2082), simp[thick.succ_eq], from S.sigma_outcome_of_pi pi\u2082,\n  have lt_weight : (\u03bb[S] (\u039b s\u2081)).weight < (\u03bb[S] (\u039b (s\u2081 + 1))).weight,\n  { have C : up[S] (\u039b s\u2081) \u2282\u1d62 \u03bb[S] (\u039b s\u2081) \u2228 up[S] (\u039b s\u2081) = \u03bb[S] (\u039b s\u2081), from list.suffix_iff_is_initial.mp (S.up_le_lambda (\u039b s\u2081)),\n    cases C,\n    { have : \u03bb[S] (\u039b (s\u2081 + 1)) \u2225 \u03bb[S] (\u039b s\u2081), simp[eq_lam\u2081],\n        from list.incomparable_of_lt C (\u03bb h, nat.not_succ_le_self s\u2081 (thick.le_mono_iff.mp (S.suffix_of_mem_lambda h))),\n      exact S.lt_weight_lambda_of_incomparable (thick.lt_mono_iff.mpr (lt_add_one s\u2081)) this.symm },\n    { exact lt_weight_of_lt (S.lambda_proper (proper (s\u2081 + 1))) (by simp[\u2190C, eq_lam\u2081]) } },\n  have : \u03bb[S] (\u039b (s\u2081 + 1)) <:+ up[S] (\u039b s\u2082),\n  { have C : up[S] (\u039b s\u2081) \u2282\u1d62 up[S] (\u039b s\u2082) \u2228 up[S] (\u039b s\u2081) = up[S] (\u039b s\u2082),\n    from list.suffix_iff_is_initial.mp (S.up_sigma_semimono thick (le_of_lt lt) pi\u2082 lt\u2081),\n    cases C,\n    { simp[eq_lam\u2081],\n      have lt\u2081 : up[S] (\u039b s\u2081) \u2282\u1d62 \u03bb[S] (\u039b (s\u2081 + 1)), { simp[eq_lam\u2081] },\n      have lt\u2082 : up[S] (\u039b s\u2081) \u2282\u1d62 \u03bb[S] (\u039b (s\u2082 + 1)), { refine C.trans (by simp[eq_lam\u2082]) },\n      have eq_out\u2081 : out \u27e8up[S] (\u039b s\u2081), lt\u2081\u27e9 = \u039b (s\u2081 + 1), { simp[out_eq_iff, eq_lam\u2081] },\n      have : \u039b (s\u2081 + 1) = out \u27e8up[S] (\u039b s\u2081), lt\u2082\u27e9, rw \u2190eq_out\u2081,\n        from S.eq_out_of_sigma (thick.lt_mono_iff.mpr (nat.succ_lt_succ lt)).suffix lt\u2081 lt\u2082\n        (by simp[eq_out\u2081, thick.out_sigma, pi\u2081]), \n      have : \u039b (s\u2081 + 1) = out \u27e8up[S] (\u039b s\u2081), C\u27e9, simp[this], from suffix_out_eq rfl (by simp[eq_lam\u2082]),\n      exact out_eq_iff.mp (eq.symm this) },\n    { exfalso,\n      have : (out \u27e8\u039b s\u2081, _\u27e9).is_sigma, from S.sigma_outcome_of_eq_up' (thick.lt_mono_iff.mpr lt) C, simp[thick.out_eq_out] at this,\n      exact not_pi_sigma pi\u2081 this } },\n  have : (\u03bb[S] (\u039b (s\u2081 + 1))).weight \u2264 (up[S] (\u039b s\u2082)).weight, from le_weight_of_le (S.up_proper (proper s\u2082)) this,\n  exact lt_of_lt_of_le lt_weight this,\nend\n\n-- derivatives \u306e\u5019\u88dc\ndef antiderivatives (\u03bc : Tree k) : list (Tree (k + 1) \u00d7 \u2115) :=\n(\u03bb[S] \u03bc, 0) :: ((\u03bb[S] \u03bc).ancestors.filter (\u03bb \u03b7, (out \u03b7).is_pi)).map (\u03bb \u03b7, (\u03b7, (S.derivative \u2191\u03b7 \u03bc).length))\n\nlemma Min_antiderivative_eq_assignment (\u03bc : Tree k) :\n  (S.priority (k + 1)).Min_le (S.antiderivatives \u03bc) (by simp[antiderivatives]) = S.assignment \u03bc :=\nby simp[antiderivatives, assignment, approx.assignment]; refl\n\n@[simp] lemma mem_assignment_antiderivatives (\u03bc : Tree k) :\n  S.assignment \u03bc \u2208 S.antiderivatives \u03bc := (S.priority (k + 1)).Min_le_mem _\n\nlemma nonsuffix_of_scons (\u03bc\u2081 \u03bc\u2082 : Tree k) (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082) : \u00ac\u03bb[S] \u03bc\u2082 <:+ up[S] \u03bc\u2081 := \u03bb le,\nbegin\n  have C : \u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081) = (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081) :: up[S] \u03bc\u2081 \u2228\n    (\u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081) = \u03bb[S] \u03bc\u2081 \u2227 up[S] \u03bc\u2081 \u2282\u1d62 \u03bb[S] \u03bc\u2081),\n  { simp[lambda, approx.lambda],\n    by_cases C :\n      up[S] \u03bc\u2081 = approx.lambda (S.up' \u03bc\u2081) \u2228 (out \u27e8\u03bc\u2081, lt\u27e9).is_pi \u2227 approx.pi_derivative (up[S] \u03bc\u2081) (S.up' \u03bc\u2081) = []; simp[C],\n    { simp[not_or_distrib] at C, right, refine list.is_initial_iff_suffix.mpr \u27e8up_le_lambda S \u03bc\u2081, C.1\u27e9 } },\n  rcases C with (C | \u27e8C, lt'\u27e9),\n  { have le_out : out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081 <:+ \u03bc\u2082, from suffix_out_cons \u27e8\u03bc\u2081, lt\u27e9,\n    have lt : \u03bb[S] \u03bc\u2082 \u2282\u1d62 \u03bb[S] (out \u27e8\u03bc\u2081, lt\u27e9 :: \u03bc\u2081), simp[C], from list.is_initial_cons_iff_suffix.mpr le,\n    exact noninitial_of_suffix S le_out lt },\n  { have := le.is_initial_of_is_initial lt', exact S.noninitial_of_suffix lt.suffix this }\nend\n\nlemma le_of_mem_antiderivatives {\u03bc : Tree k} {\u03b7 : Tree (k + 1)} {n : \u2115} (mem : (\u03b7, n) \u2208 S.antiderivatives \u03bc) :\n  \u03b7 <:+ \u03bb[S] \u03bc \u2227 n = (S.derivative \u03b7 \u03bc).length :=\nbegin\n  simp[antiderivatives] at mem,\n  rcases mem with (\u27e8rfl, rfl\u27e9 | \u27e8\u27e8\u03bc\u2081, lt\u27e9, pi, rfl, rfl\u27e9),\n  { have : S.derivative (\u03bb[S] \u03bc) \u03bc = [],\n    { simp[derivative, approx.derivative, list.filter_eq_nil], rintros \u27e8\u03bc\u2081, lt\u27e9 eqn,\n      have := S.nonsuffix_of_scons _ _ lt (by simp[\u2190eqn]), contradiction },\n    simp[this] },\n  { simp, exact list.is_initial.suffix lt }\nend\n\nlemma assignment_snd_eq (\u03bc : Tree k) : (S.assignment \u03bc).2 = (S.derivative (up[S] \u03bc) \u03bc).length :=\nbegin\n  have : S.assignment \u03bc \u2208 _, from omega_ordering.Min_le_mem _ _, simp at this, rcases this with (eqn| \u27e8\u03b7, pi, eqn\u27e9),\n  { have eqn_lam : up[S] \u03bc = \u03bb[S] \u03bc, from congr_arg prod.fst eqn,\n    have : S.derivative (up[S] \u03bc) \u03bc = [],\n    { simp[derivative, approx.derivative, list.filter_eq_nil],\n      rintros \u27e8\u03bc\u2081, lt\u27e9 eqn_\u03bc\u2081,\n      exact S.nonsuffix_of_scons _ _ lt (by simp[\u2190eqn_lam, \u2190eqn_\u03bc\u2081]) },\n    rw[this, eqn], simp },\n  have : \u2191\u03b7 = up[S] \u03bc,\n  { have : (S.assignment \u03bc).fst = up[S] \u03bc, from S.assignment_fst_eq_up \u03bc, simp[\u2190eqn] at this, exact this },\n  simp[\u2190eqn, this], refl \nend\n\nlemma assignment_eq (\u03bc : Tree k) : S.assignment \u03bc = (up[S] \u03bc, (S.derivative (up[S] \u03bc) \u03bc).length) :=\nprod.ext (by simp[S.assignment_fst_eq_up \u03bc]) (by simp[assignment_snd_eq])\n\nlemma derivative_mono (\u03b7 : Tree (k + 1)) {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082) :\n  (S.derivative \u03b7 \u03bc\u2081).map subtype.val <:+ (S.derivative \u03b7 \u03bc\u2082).map subtype.val :=\nby { have : \u2200 \u03bc, (S.derivative \u03b7 \u03bc).map subtype.val = ((\u03bc.ancestors.map subtype.val).filter (\u03bb a, up[S] a = \u03b7)),\n    { intros \u03bc, simp[derivative, approx.derivative], simp[list.map_filter, function.comp], congr },\n    simp[this], exact (list.is_suffix.filter _ (ancestor.ancestors'_suffix_of_suffix le)) }\n\nlemma length_derivative_mono (\u03b7 : Tree (k + 1)) {\u03bc\u2081 \u03bc\u2082 : Tree k} (le : \u03bc\u2081 <:+ \u03bc\u2082) :\n  (S.derivative \u03b7 \u03bc\u2081).length \u2264 (S.derivative \u03b7 \u03bc\u2082).length :=\nby { have := (S.derivative_mono \u03b7 le).le_length, simp at this, exact this }\n\nlemma nonmem_antiderivatives {\u03bc\u2081 \u03bc\u2082 : Tree k} (lt : \u03bc\u2081 \u2282\u1d62 \u03bc\u2082) : S.assignment \u03bc\u2081 \u2209 S.antiderivatives \u03bc\u2082 := \u03bb A,\nbegin\n  simp [antiderivatives] at A, rcases A with (A | \u27e8\u27e8\u03b7, lt_\u03b7\u27e9, pi, A\u27e9),\n  { have : up[S] \u03bc\u2081 = \u03bb[S] \u03bc\u2082, from congr_arg prod.fst A,\n    exact nonsuffix_of_scons S \u03bc\u2081 \u03bc\u2082 lt (by simp[this]) },\n  { have : \u03b7 = up[S] \u03bc\u2081, { have := S.assignment_fst_eq_up \u03bc\u2081, simp [\u2190A] at this, exact this }, rcases this with rfl,\n    have eq : ((S.derivative (up[S] \u03bc\u2081) \u03bc\u2082).map subtype.val).length = ((S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val).length,\n    { have := S.assignment_snd_eq \u03bc\u2081, simp [\u2190A] at this, simp[this] },\n    have neq : (S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val \u2260 (S.derivative (up[S] \u03bc\u2081) \u03bc\u2082).map subtype.val,\n    { intros eqn_der,\n      have : \u03bc\u2081 \u2208 (S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val,\n      { rw eqn_der, simp[derivative, approx.derivative], exact \u27e8\u27e8\u03bc\u2081, lt\u27e9, rfl, rfl\u27e9 },\n      have : \u03bc\u2081 \u2209 (S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val,\n      { simp[derivative, approx.derivative], rintros \u27e8\u03bc\u2082, lt\u27e9 _ eqn, simp[\u2190eqn] at lt, contradiction },\n      contradiction },\n    have : (S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val \u2282\u1d62 (S.derivative (up[S] \u03bc\u2081) \u03bc\u2082).map subtype.val,\n      from list.is_initial_iff_suffix.mpr \u27e8S.derivative_mono (up[S] \u03bc\u2081) lt.suffix, neq\u27e9,\n    have : ((S.derivative (up[S] \u03bc\u2081) \u03bc\u2081).map subtype.val).length < ((S.derivative (up[S] \u03bc\u2081) \u03bc\u2082).map subtype.val).length,\n      from list.is_initial_length this,\n    simp[eq] at this, contradiction }\nend\n\nlemma case_mem_antiderivative {\u03bc : Tree k} {\u03bd : Tree' k} (\u03b7 : Tree (k + 1)) (n : \u2115) (lt : \u03b7 \u2282\u1d62 \u03bb[S] (\u03bd :: \u03bc))\n  (pi : (out \u27e8\u03b7, lt\u27e9).is_pi)\n  (mem : (\u03b7, n) \u2208 S.antiderivatives \u03bc) :\n  (\u03b7, n) = S.assignment \u03bc \u2227 (\u03b7, n + 1) \u2208 S.antiderivatives (\u03bd :: \u03bc) \u2228 \n  (\u03b7, n) \u2260 S.assignment \u03bc \u2227 (\u03b7, n) \u2208 S.antiderivatives (\u03bd :: \u03bc) :=\nbegin\n  have der : n = (S.derivative \u03b7 \u03bc).length, from (S.le_of_mem_antiderivatives mem).2,\n  simp[antiderivatives] at mem \u22a2, rcases mem with (\u27e8rfl, rfl\u27e9 | \u27e8\u27e8\u03b7\u2081, lt'\u27e9, pi, rfl, rfl\u27e9),\n  { by_cases C :\n      up[S] \u03bc = \u03bb[S] \u03bc \u2228 \u03bd.is_pi \u2227 approx.pi_derivative (up[S] \u03bc) (S.up' \u03bc) = list.nil,\n    { have eqn : \u03bb[S] (\u03bd :: \u03bc) = (\u03bd :: \u03bc) :: up[S] \u03bc, { unfold lambda at*, simp[approx.lambda, C] },\n      cases C,\n      { left, simp[assignment_eq, C, der],\n        have lt : \u03bb[S] \u03bc \u2282\u1d62 \u03bb[S] (\u03bd :: \u03bc), { simp[eqn, C] },\n        refine \u27e8\u27e8\u03bb[S] \u03bc, lt\u27e9, _\u27e9, simp,\n        have eqn_der : S.derivative (\u03bb[S] \u03bc) (\u03bd :: \u03bc) = [\u27e8\u03bc, by simp\u27e9],\n        { simp[derivative_cons, C], exact list.length_eq_zero.mp (eq.symm der) },\n        simp[eqn_der, pi, \u2190der] },\n      { exfalso,\n        have :  \u03bb[S] \u03bc \u2282\u1d62 (\u03bd :: \u03bc) :: up[S] \u03bc, simp[\u2190eqn, lt],\n        have : \u03bb[S] \u03bc = up[S] \u03bc,\n        { have C\u2081 := list.is_initial_cons_iff.mp this, \n          cases C\u2081, { exact C\u2081 }, { exfalso, exact list.is_initial_suffix_antisymm C\u2081 (S.up_le_lambda _) } },\n        have : out \u27e8\u03bb[S] \u03bc, lt\u27e9 = \u03bd :: \u03bc, { simp[out_eq_iff, this, eqn] },\n        simp[this] at pi, exact not_pi_sigma C.1 pi } },\n    { have eqn : \u03bb[S] (\u03bd :: \u03bc) = \u03bb[S] \u03bc, { unfold lambda at*, simp[approx.lambda, C] },\n      have ne_up : \u00acup[S] \u03bc = \u03bb[S] \u03bc, { simp[not_or_distrib] at C, exact C.1 },\n      right,\n      simp[assignment_eq, eqn], intros h, have := eq.symm h, contradiction } },\n  { by_cases C : up[S] \u03bc = \u03b7\u2081,\n    { left, simp[assignment_eq, C], refine \u27e8\u27e8\u03b7\u2081, lt\u27e9, pi, rfl, by simp[derivative_cons, C]\u27e9 },\n    { right, simp[assignment_eq, C],\n      exact \u27e8\u03bb h, by exfalso; exact C (eq.symm h), or.inr \u27e8\u27e8\u03b7\u2081, lt\u27e9, pi, rfl, by simp[derivative_cons, C]\u27e9\u27e9 } }\nend\n\nvariables {\u039b}\n\nlemma infinite_substrategy_of_pi\n  (thick : \u039b.thick) {\u03b7 : Tree (k + 1)} (s\u2080) (lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080) (pi : (out \u27e8\u03b7, lt\u27e9).is_pi) (n) :\n  \u2203 s, (\u03b7, n) = S.assignment (\u039b s) :=\nbegin\n  rcases S.eq_lt_lambda_of_lt_Lambda_of_pi thick lt pi with \u27e8s\u2081, eqn_\u03b7, le\u27e9, simp at eqn_\u03b7,\n  suffices : \u2203 s, s\u2081 \u2264 s \u2227 (\u03b7, n) = S.assignment (\u039b s), { rcases this with \u27e8s, _, eqn\u27e9, exact \u27e8s, eqn\u27e9 },\n  induction n with n IH,\n  { have mem : (\u03b7, 0) \u2208 S.antiderivatives (\u039b s\u2081), { simp[antiderivatives, eqn_\u03b7], },\n    have mem_of_ne : \u2200 s, \n      (\u03b7, 0) \u2260 S.assignment (\u039b (s\u2081 + s)) \u2192\n      (\u03b7, 0) \u2208 S.antiderivatives (\u039b (s\u2081 + s)) \u2192\n      (\u03b7, 0) \u2208 S.antiderivatives (\u039b (s\u2081 + s + 1)),\n    { intros s neq mem,   \n      rcases thick.2 (s\u2081 + s) with \u27e8\u03bd, eqn_path\u27e9,\n      have lt' : \u03b7 \u2282\u1d62 \u03bb[S] (\u03bd :: \u039b (s\u2081 + s)),\n      { simp[\u2190eqn_path], exact list.suffix_cons_iff_is_initial.mp \u27e8_, le (s\u2081 + s) (le_self_add)\u27e9 },\n      have : out \u27e8\u03b7, lt'\u27e9 = out \u27e8\u03b7, lt\u27e9, { simp[out_eq_iff, \u2190eqn_path], exact le (s\u2081 + s) (le_self_add) },\n      have := S.case_mem_antiderivative \u03b7 0 lt' (by simp[this]; exact pi) mem,\n      simp[neq, \u2190eqn_path] at this, exact this },\n    have : \u2203 s, (\u03b7, 0) = S.assignment (\u039b (s\u2081 + s)),\n      from (S.priority (k + 1)).eq_Min_sequence (\u03bb s, S.antiderivatives (\u039b (s\u2081 + s))) (by simp[antiderivatives])\n        (\u03bb s t lt, S.nonmem_antiderivatives (thick.is_initial_of_lt (add_lt_add_left lt s\u2081))) mem mem_of_ne,\n    rcases this with \u27e8s, eqn\u27e9, exact \u27e8s\u2081 + s, le_self_add, eqn\u27e9 },\n  { rcases IH with \u27e8s\u2082, le_s\u2082, eqn_IH\u27e9,\n    have mem : (\u03b7, n + 1) \u2208 S.antiderivatives (\u039b (s\u2082 + 1)),\n    { rcases thick.2 s\u2082 with \u27e8\u03bd, eqn_path\u27e9,\n      have lt' : \u03b7 \u2282\u1d62 \u03bb[S] (\u03bd :: \u039b s\u2082),\n      { simp[\u2190eqn_path], exact list.suffix_cons_iff_is_initial.mp \u27e8_, le s\u2082 le_s\u2082\u27e9 },\n      have : out \u27e8\u03b7, lt'\u27e9 = out \u27e8\u03b7, lt\u27e9, { simp[out_eq_iff, \u2190eqn_path], exact le s\u2082 le_s\u2082 },\n      have : (\u03b7, n) = S.assignment (\u039b s\u2082) \u2227 (\u03b7, n + 1) \u2208 S.antiderivatives (\u03bd :: \u039b s\u2082) \u2228\n             (\u03b7, n) \u2260 S.assignment (\u039b s\u2082) \u2227 (\u03b7, n)     \u2208 S.antiderivatives (\u03bd :: \u039b s\u2082),\n        from S.case_mem_antiderivative \u03b7 n lt' (by simp[this]; exact pi) (by simp[eqn_IH]),\n      simp[eqn_IH, \u2190eqn_path] at this, exact this },\n    have mem_of_ne : \u2200 s, \n      (\u03b7, n + 1) \u2260 S.assignment (\u039b (s\u2082 + 1 + s)) \u2192\n      (\u03b7, n + 1) \u2208 S.antiderivatives (\u039b (s\u2082 + 1 + s)) \u2192\n      (\u03b7, n + 1) \u2208 S.antiderivatives (\u039b (s\u2082 + 1 + s + 1)),\n    { intros s ne mem,\n      rcases thick.2 (s\u2082 + 1 + s) with \u27e8\u03bd, eqn_path\u27e9,\n      have lt' : \u03b7 \u2282\u1d62 \u03bb[S] (\u03bd :: \u039b (s\u2082 + 1 + s)),\n      { simp[\u2190eqn_path], exact list.suffix_cons_iff_is_initial.mp \u27e8_, le (s\u2082 + 1 + s) (by simp[add_assoc]; exact le_add_right le_s\u2082)\u27e9 },\n      have : out \u27e8\u03b7, lt'\u27e9 = out \u27e8\u03b7, lt\u27e9,\n      { simp[out_eq_iff, \u2190eqn_path], exact le (s\u2082 + 1 + s) (by simp[add_assoc]; exact le_add_right le_s\u2082) },\n      have := S.case_mem_antiderivative \u03b7 (n + 1) lt' (by simp[this]; exact pi) mem,\n      simp[ne, \u2190eqn_path] at this, exact this },    \n    have : \u2203 s, (\u03b7, n + 1) = S.assignment (\u039b (s\u2082 + 1 + s)),\n      from (S.priority (k + 1)).eq_Min_sequence (\u03bb s, S.antiderivatives (\u039b (s\u2082 + 1 + s))) (by simp[antiderivatives])\n        (\u03bb s t lt, S.nonmem_antiderivatives (thick.is_initial_of_lt (add_lt_add_left lt (s\u2082 + 1)))) mem mem_of_ne,\n    rcases this with \u27e8s, eqn\u27e9,\n    exact \u27e8s\u2082 + 1 + s, (by simp[add_assoc]; exact le_add_right le_s\u2082), eqn\u27e9 }\nend\n\nlemma infinite_substrategy_of_pi'\n  (thick : \u039b.thick) {\u03b7 : Tree (k + 1)} {s\u2080} {lt : \u03b7 \u2282\u1d62 (\u039b[S] \u039b) s\u2080} (pi : (out \u27e8\u03b7, lt\u27e9).is_pi)\n  (s\u2081 : \u2115) : \u2203 s > s\u2081, up[S] (\u039b s) = \u03b7 :=\nbegin\n  have : \u2203 s, (\u03b7, (S.derivative \u03b7 (\u039b s\u2081)).length + 1) = S.assignment (\u039b s),\n    from S.infinite_substrategy_of_pi thick s\u2080 lt pi ((S.derivative \u03b7 (\u039b s\u2081)).length + 1),\n  rcases this with \u27e8s, eq_assn\u27e9,\n  \n  have : \u03b7 = up[S] (\u039b s),\n  { have := S.assignment_fst_eq_up (\u039b s), simp[\u2190eq_assn] at this, exact this },\n  rcases this with rfl,\n  have lt_der : (S.derivative (up[S] (\u039b s)) (\u039b s\u2081)).length < (S.derivative (up[S] (\u039b s)) (\u039b s)).length,\n  { have := S.assignment_snd_eq (\u039b s), simp[\u2190eq_assn] at this, simp [\u2190this] },\nhave : s\u2081 < s,\n{ by_contradiction,\n  have : (S.derivative (up[S] (\u039b s)) (\u039b s)).length \u2264 (S.derivative (up[S] (\u039b s)) (\u039b s\u2081)).length,\n    from S.length_derivative_mono (up[S] (\u039b s)) (thick.le_mono_iff.mpr (not_lt.mp h)),\n  exact nat.lt_le_antisymm lt_der this },\nexact \u27e8s, this, rfl\u27e9\nend\n\nlemma Lambda_infinite\n  (thick : \u039b.thick) : (\u039b[S] \u039b).infinite := \u03bb s\u2080,\nbegin\n  rcases S.Lambda_spec \u039b s\u2080 with \u27e8s\u2081, eqn\u2081\u27e9,\n  rcases S.Lambda_spec \u039b (s\u2080 + 1) with \u27e8s\u2082, eqn\u2082\u27e9,\n  by_contradiction A, simp at A,\n  have eq_Lambda : \u2200 s, (\u039b[S] \u039b) s\u2080 = (\u039b[S] \u039b) (s\u2080 + s),\n  { intros s,\n    have : (\u039b[S] \u039b) s\u2080 <:+ (\u039b[S] \u039b) (s\u2080 + s), from (\u039b[S] \u039b).mono' (le_self_add),\n    rcases list.suffix_iff_is_initial.mp this with (lt | eq), { exfalso, exact A s lt }, { exact eq } },\n  have le_length : \u2200 s, s\u2082 \u2264 s \u2192 (\u03bb[S] (\u039b s)).length \u2264 s\u2080 + 1,\n  { intros s le_s,\n    have C : (\u03bb[S] (\u039b s)).length \u2264 s\u2080 + 1 \u2228 s\u2080 + 1 < (\u03bb[S] (\u039b s)).length, from le_or_lt _ _,\n    rcases C, { exact C },\n    { exfalso, have := list.initial_length C,\n      have eqn_s\u2080 : ((\u039b[S] \u039b) s\u2080).length = s\u2080 + 1, { simp[eq_Lambda 1, \u2190eqn\u2082 s le_s, this] },\n      have le_s\u2080  : ((\u039b[S] \u039b) s\u2080).length \u2264 s\u2080, { simp[\u2190eqn\u2081 s\u2081 (by refl)] },\n      simp[eqn_s\u2080] at le_s\u2080, contradiction } },\n  have eq_lam : \u2200 s, \u03bb[S] (\u039b (s\u2082 + s)) = \u03bb[S] (\u039b s\u2082),\n  { intros s, have := eqn\u2082 s\u2082 (by refl),\n    have : \u2200 s, \u03bb[S] (\u039b (s\u2082 + s)) = (\u039b[S] \u039b) (s\u2080 + 1),\n    { intros s, have := list.initial_elim (le_length (s\u2082 + s) (le_self_add)),\n      simp[\u2190eqn\u2082 (s\u2082 + s) (le_self_add), this] },\n    simp[this], exact eq.symm (this 0) },\n  have mem : (\u03bb[S] (\u039b s\u2082), 0) \u2208 S.antiderivatives (\u039b s\u2082), { simp[antiderivatives] },\n  have mem_of_ne : \u2200 s,\n    (\u03bb[S] (\u039b s\u2082), 0) \u2260 S.assignment (\u039b (s\u2082 + s)) \u2192\n    (\u03bb[S] (\u039b s\u2082), 0) \u2208 S.antiderivatives (\u039b (s\u2082 + s)) \u2192\n    (\u03bb[S] (\u039b s\u2082), 0) \u2208 S.antiderivatives (\u039b (s\u2082 + s + 1)),\n  { intros s ne mem, simp[antiderivatives], exact or.inl (eq.symm (eq_lam (s + 1))) },\n  have : \u2203 s, (\u03bb[S] (\u039b s\u2082), 0) = S.assignment (\u039b (s\u2082 + s)),\n    from (S.priority (k + 1)).eq_Min_sequence (\u03bb s, S.antiderivatives (\u039b (s\u2082 + s)))\n      (by simp[antiderivatives])\n      (\u03bb s t lt, S.nonmem_antiderivatives (thick.is_initial_of_lt (add_lt_add_left lt s\u2082))) mem mem_of_ne,\n  rcases this with \u27e8s\u2083, eqn_assn\u27e9,\n  have eq_up: \u03bb[S] (\u039b (s\u2082 + s\u2083)) = up[S] (\u039b (s\u2082 + s\u2083)),\n  { have := S.assignment_fst_eq_up (\u039b (s\u2082 + s\u2083)), rw[\u2190eqn_assn, \u2190eq_lam s\u2083] at this, exact this },\n  rcases thick.2 (s\u2082 + s\u2083) with \u27e8\u03bd, eqn_path\u27e9,\n  have : \u03bb[S] (\u039b (s\u2082 + s\u2083 + 1)) = \u039b (s\u2082 + s\u2083 + 1) :: \u03bb[S] (\u039b (s\u2082 + s\u2083)),\n  { simp[eqn_path, lambda, approx.lambda, \u2190eq_up] },\n  simp[show \u03bb[S] (\u039b (s\u2082 + s\u2083 + 1)) = \u03bb[S] (\u039b s\u2082), from eq_lam (s\u2083 + 1),\n    eq_lam s\u2083] at this, contradiction\nend\n\nlemma lambda_infinitely (thick : \u039b.thick) (proper : \u039b.proper) (n : \u2115) :\n  \u2203 s, n < (\u03bb[S] (\u039b s)).weight :=\nbegin\n  have : \u2203 m, n < ((\u039b[S] \u039b) m).length, from (S.Lambda_infinite thick).length n,\n  rcases this with \u27e8m, lt_length\u27e9,\n  rcases S.Lambda_spec \u039b m with \u27e8s\u2082, eqn\u2081\u27e9,\n  have : n < (\u03bb[S] (\u039b s\u2082)).length,\n  { have := (list.suffix_initial (\u03bb[S] (\u039b s\u2082)) m).le_length, simp[eqn\u2081 s\u2082 (by refl)] at this,\n    exact gt_of_ge_of_gt this lt_length },\n  have : n < (\u03bb[S] (\u039b s\u2082)).weight, from lt_of_lt_of_le this (le_weight_length (S.lambda_proper (proper s\u2082))),\n  refine \u27e8s\u2082, this\u27e9\nend\n\nstructure construction (\u03b1 : Type*) [primcodable \u03b1] :=\n(requirements : \u2115)\n(directing_sentence : fin requirements \u2192 \u03b1 \u2192 Tree 0 \u2192 bool)\n(action : fin requirements \u2192 \u03b1 \u2192 Tree 0 \u2192 \u03b1)\n(directing_sentence_computable : computable (prod.unpaired3 directing_sentence))\n(action_computable : computable (prod.unpaired3 action))\n(initial : \u03b1)\n\nnamespace construction\n\nvariables {\u03b1 : Type*} [primcodable \u03b1] (C : construction \u03b1)\n\ndef generator : \u2115 \u2192 (Tree 0 \u00d7 \u03b1)\n| 0       := ([], C.initial)\n| (s + 1) :=\n    let \u03bc  : Tree 0 := (generator s).1, \n        G  : \u03b1      := (generator s).2 in\n    if C.directing_sentence G \u03bc then (\u221e :: \u03bc, C.action G \u03bc) else (\ud835\udfd8 :: \u03bc, G)\n\ndef gen (s : \u2115) : \u03b1 := (C.generator s).2\n\ndef \u039b : Path 0 := \u27e8\u03bb s, (C.generator s).1, \u03bb s,\n  by { cases C : C.directing_sentence (C.generator s).2 (C.generator s).1; simp[generator, C] }\u27e9\n\nlemma \u039b_thick : C.\u039b.thick :=\n\u27e8by simp[\u039b, generator], \u03bb s, by { cases C : C.directing_sentence (C.generator s).2 (C.generator s).1; simp[\u039b, generator, C],\n  { refine \u27e8_, rfl\u27e9 }, { refine \u27e8_, rfl\u27e9 } }\u27e9\n\n\n\nend construction\n\nend strategy\n", "meta": {"author": "iehality", "repo": "lean-reducibility", "sha": "82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7", "save_path": "github-repos/lean/iehality-lean-reducibility", "path": "github-repos/lean/iehality-lean-reducibility/lean-reducibility-82a7e3ec0fcedfb0d69c25e77bcd24c9b29626b7/src/its.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.37045571400368565}}
{"text": "import .dlo \n\nvariables {\u03b1 \u03b2 : Type}\n\ndef dlo_qe_aux (as : list adlo) := -- (HZ) (H) := \nlet prs := list.product (dlo_qe_lbs as) (dlo_qe_ubs as) in \nlist_conj $ @list.map adlo (fm adlo) (@fm.atom adlo) (list.map (\u03bb pr, (prod.fst pr <' prod.snd pr)) prs)\n\ndef dlo_qe (\u03b2 : Type) [atom_eq_type adlo \u03b2] (as : list adlo) : fm adlo := \n@ite (adlo.lt 0 0 \u2208 as) \n  (@list.decidable_mem adlo (atom_type.dec_eq _ \u03b2) (0 <' 0) as) _\n  \u22a5'\n  (@ite (list.allp is_b_atm as) \n    (dec_allp _) _ \n    (dlo_qe_aux as)\n    (\u22a5')\n  )\n\ndef dlo_qelim (\u03b2 : Type) [atom_eq_type adlo \u03b2] : fm adlo \u2192 fm adlo :=  \n@lift_dnfeq_qe _ \u03b2 _ (dlo_qe \u03b2)\n\n-- Q : Why doesn't `cases (dlo_dec_mem (0 <' 0) as)` simplify things here? \nlemma dlo_qe_qfree [HA : atom_eq_type adlo \u03b2] : \n  \u2200 (as : list adlo) (Has : allp (@atom_type.dep0 adlo \u03b2 _) as), \n    qfree (dlo_qe \u03b2 as) := \nbegin\n  intros as Has, unfold dlo_qe, \n  apply cases_ite, intro H, trivial, \n  intro HM, apply cases_ite, intro Hlt,\n  unfold dlo_qe_aux, simp, apply qfree_list_conj,\n  intros x Hx, \n  cases (list.exists_of_mem_map Hx) with y Hy,\n  cases Hy with Hyl Hyr, rewrite Hyr,\n  cases y, unfold function.comp, \n  intro H, trivial\nend\n\nlemma btw_of_lt [dlo \u03b2] {m n} {bs : list \u03b2} (H : atom_type.val bs (m <' n)) :\n  \u2203 b, ((tval m bs < b) \u2227 (b < tval n bs)) :=  \nbegin\n  cases (dlo.btw H) with b Hb, \n  existsi b, apply Hb\nend\n\n#exit\n\nlemma is_b_atm_of [dlo \u03b2] : \n  \u2200 (a), (\u03bb a', atom_type.dep0 \u03b2 a' \u2227 \u00acatom_eq_type.solv0 \u03b2 a' \u2227 a' \u2260 (0 <' 0)) a \u2192 is_b_atm a \n| (0   =' n  ) h := \n  begin exfalso, apply h^.elim_right^.elim_left, apply or.inl rfl end\n| (m   =' 0  ) h := \n  begin exfalso, apply h^.elim_right^.elim_left, apply or.inr rfl end\n| (m+1 =' n+1) h := \n  begin exfalso, cases h^.elim_left with h h; cases h end\n| (0   <' 0  ) h := \n  begin exfalso, apply h^.elim_right^.elim_right, refl end\n| (m+1 <' 0  ) h := \n  begin existsi m, apply or.inl rfl end\n| (0   <' n+1) h := \n  begin existsi n, apply or.inr rfl end\n| (m+1 <' n+1) h := \n  begin exfalso, cases h^.elim_left with h h; cases h end\n\nlemma ex_high_lb_of_ex_lb [dlo \u03b2]   \n  {as : list adlo} (hlb : \u2203 m, is_lb m as) (bs : list \u03b2) :\n\u2203 k, (is_lb k as \u2227 \u2200 j, is_lb j as \u2192 dle j k bs) := \nbegin\n  cases hlb with m hm, \n  have hi : list.map (\u03bb n, tval n bs) (dlo_qe_lbs as) \u2260 [],\n  intro hc, have he := list.eq_nil_of_map_eq_nil hc,\n  have hmm := mem_lbs_of_is_lb hm, \n  rewrite he at hmm, cases hmm,\n  cases (exists_maximum _ hi) with b hb,\n  cases hb with hb1 hb2, \n  cases (list.exists_of_mem_map hb1) with k hk,\n  cases hk with hk1 hk2, \n  existsi k, apply and.intro (is_lb_of_mem_lbs hk1),\n  intros j hj, simp at hk2, unfold dle,  \n  rewrite eq.symm hk2, apply hb2, \n  apply mem_map_of_mem, apply mem_lbs_of_is_lb hj\nend\n\nlemma ex_low_ub_of_ex_ub [dlo \u03b2] \n  {as : list adlo} (hub : \u2203 n, is_ub n as) (bs : list \u03b2) :\n\u2203 k, (is_ub k as \u2227 \u2200 j, is_ub j as \u2192 dle k j bs) := \nbegin\n  cases hub with n hn, \n  have hi : list.map (\u03bb n, tval n bs) (dlo_qe_ubs as) \u2260 [],\n  intro hc, have he := list.eq_nil_of_map_eq_nil hc,\n  have hnm := mem_ubs_of_is_ub hn, \n  rewrite he at hnm, cases hnm,\n  cases (exists_minimum _ hi) with b hb,\n  cases hb with hb1 hb2, \n  cases (list.exists_of_mem_map hb1) with k hk,\n  cases hk with hk1 hk2, \n  existsi k, apply and.intro (is_ub_of_mem_ubs hk1),\n  intros j hj, simp at hk2, unfold dle,  \n  rewrite eq.symm hk2, apply hb2, \n  apply mem_map_of_mem, apply mem_ubs_of_is_ub hj\nend\n\nlemma dlo_qe_is_dnf [HD : dlo \u03b2] : \u2200 (as : list adlo), \n  (\u2200 (a : adlo), a \u2208 as \u2192 atom_type.dep0 \u03b2 a \u2227 \u00ac atom_eq_type.solv0 \u03b2 a) \n  \u2192 qe_prsv \u03b2 (dlo_qe \u03b2) as := \nbegin\n  intros as Has,\n  unfold qe_prsv, intro bs, \n  unfold dlo_qe, unfold dlo_qe_aux, simp, \n  cases (@list.decidable_mem adlo (atom_type.dec_eq _ \u03b2) (0 <' 0) as) with Hc Hc,\n  rewrite (exp_ite_eq_of_not), \n \n  have HW : allp is_b_atm as := \n  begin\n    apply allp_of_allp is_b_atm_of, intros a' Ha', \n    cases Has a' Ha' with Ha1' Ha2',\n    apply and.intro Ha1',  \n    apply and.intro Ha2', intro HN, \n    apply Hc, subst HN, apply Ha'  \n  end, clear Has, clear Hc, \n\n  rewrite (exp_ite_eq_of), \n  unfold function.comp, rewrite exp_I_list_conj,  \n\n  apply @classical.by_cases (\u2203 m, is_lb m as) ; intro Hlb,\n  cases (ex_high_lb_of_ex_lb Hlb bs) with m Hm, clear Hlb,\n  cases Hm with Hm1 Hm2, \n\n  apply @classical.by_cases (\u2203 n, is_ub n as) ; intro Hub,\n  cases (ex_low_ub_of_ex_ub Hub bs) with n Hn, clear Hub,\n  cases Hn with Hn1 Hn2, \n\n  apply @iff.trans _ (I (A' (m <' n)) bs),\n  rewrite map_compose,\n  apply iff.intro, intro HL, apply HL, \n  apply @mem_map_of_mem _ _ (\u03bb (x : \u2115 \u00d7 \u2115), I ((fm.atom \u2218 \u03bb (pr : \u2115 \u00d7 \u2115), pr.fst<' pr.snd) x) bs) _ (m,n),  \n  apply mem_product_of_mem_and_mem,\n  apply mem_omap _ Hm1, refl, \n  apply mem_omap _ Hn1, refl, \n  intro HR, intros a Ha, \n  cases (list.exists_of_mem_map Ha) with pr Hpr,\n  cases Hpr with Hpr1 Hpr2, subst Hpr2,\n  cases pr with x y, simp, simp at Ha, \n  cases (list.exists_of_mem_map Ha) with xy Hxy, \n  cases xy with x' y', simp at Hxy,\n  cases Hxy with Hxy1 Hxy2, rewrite Hxy2,\n  apply lt_of_le_of_lt (Hm2 _ _), \n  apply lt_of_lt_of_le _ (Hn2 _ _), \n  apply HR, \n  have Hy := snd_mem_of_mem_product Hxy1, simp at Hy,\n  apply is_ub_of_mem_ubs, apply Hy, \n  have Hx := fst_mem_of_mem_product Hxy1, simp at Hx,\n  apply is_lb_of_mem_lbs, apply Hx, \n  apply iff.intro; intro H, \n  cases (btw_of_lt H) with b Hb, cases Hb with Hb1 Hb2,\n  existsi b, intros a Ha, \n\n  cases (HW a Ha) with k Ha', cases Ha' with Ha' Ha'; subst Ha',\n  unfold I, unfold interp, rewrite exp_val_lt,\n  rewrite nth_dft_succ, rewrite nth_dft_head, \n  apply lt_of_le_of_lt, \n  apply (Hm2 k Ha), apply Hb1,\n  unfold I, unfold interp, rewrite exp_val_lt,\n  rewrite nth_dft_succ, rewrite nth_dft_head, \n  apply lt_of_lt_of_le, \n  apply Hb2, apply (Hn2 k Ha), \n  cases H with b Hb, unfold I, unfold interp,\n  have HE := (atom_type.decr_prsv (m+1 <' n+1) _ b bs),\n  rewrite (exp_decr_lt (m+1) (n+1)) at HE, simp at HE,\n  rewrite HE, clear HE, \n  apply lt_trans, \n  let Hbm := Hb _ Hm1, \n  unfold I at Hbm, unfold interp at Hbm, apply Hbm,\n  let Hbn := Hb _ Hn1, \n  unfold I at Hbn, unfold interp at Hbn, apply Hbn,\n  intro Hc', cases Hc' with Hc' Hc'; cases Hc', \n\n  apply true_iff_true,  \n  rewrite (ubs_eq_nil_of_none_is_ub Hub), \n  rewrite product_nil, simp, apply all_true_nil,\n  cases (dlo.abv (tval m bs)) with u Hu, \n  existsi u, intros a Ha, \n  cases (HW a Ha) with k Ha', cases Ha' with Ha' Ha'; subst Ha',\n  unfold I, unfold interp, rewrite exp_val_lt,\n  rewrite nth_dft_succ, rewrite nth_dft_head, \n  apply lt_of_le_of_lt, \n  apply (Hm2 k Ha), apply Hu, \n  exfalso, apply Hub, existsi k, apply Ha,\n\n  apply @classical.by_cases (\u2203 n, is_ub n as) ; intro Hub,\n  cases (ex_low_ub_of_ex_ub Hub bs) with n Hn, clear Hub,\n  cases Hn with Hn1 Hn2, \n\n  apply true_iff_true,  \n  rewrite (lbs_eq_nil_of_none_is_lb Hlb), \n  simp, apply all_true_nil, \n  cases (dlo.blw (tval n bs)) with l Hl, \n  existsi l, intros a Ha, \n  cases (HW a Ha) with k Ha', cases Ha' with Ha' Ha'; subst Ha',\n  exfalso, apply Hlb, existsi k, apply Ha,\n  unfold I, unfold interp, rewrite exp_val_lt,\n  rewrite nth_dft_succ, rewrite nth_dft_head, \n  apply lt_of_lt_of_le, apply Hl,\n  apply (Hn2 k Ha), \n  \n  cases as with a, simp, unfold dlo_qe_lbs, \n  unfold list.omap, unfold list.product, \n  unfold list.map, apply true_iff_true,\n  intros _ Hx, cases Hx, existsi (@dlo.inh \u03b2 HD), \n  trivial, cases (HW _ (or.inl rfl)) with k Hk,\n  cases Hk with Hk Hk, \n  exfalso, apply Hlb, existsi k, subst Hk, apply or.inl rfl, \n  exfalso, apply Hub, existsi k, subst Hk, apply or.inl rfl, \n  apply HW, apply Hc, \n\n  rewrite exp_ite_eq_of, apply false_iff_false,\n  intro h, apply h, \n  intro h, cases h with b hb, \n  apply (lt_irrefl b), \n  apply (hb _ Hc), apply Hc\n\nend\n\nlemma dlo_qelim_prsv [dlo \u03b2] : \n  \u2200 (p : fm adlo) (bs : list \u03b2), I (dlo_qelim \u03b2 p) bs \u2194 I p bs := \nldeq_prsv (dlo_qe \u03b2) dlo_qe_qfree dlo_qe_is_dnf \n", "meta": {"author": "avigad", "repo": "qelim", "sha": "b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60", "save_path": "github-repos/lean/avigad-qelim", "path": "github-repos/lean/avigad-qelim/qelim-b7d22864f1f0a2d21adad0f4fb3fc7ba665f8e60/dlo/dlo_qelim.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3704557074352874}}
{"text": "/-\nCopyright (c) 2021 Yuma Mizuno. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yuma Mizuno\n-/\nimport category_theory.isomorphism\nimport tactic.slice\n\n/-!\n# Bicategories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we define typeclass for bicategories.\n\nA bicategory `B` consists of\n* objects `a : B`,\n* 1-morphisms `f : a \u27f6 b` between objects `a b : B`, and\n* 2-morphisms `\u03b7 : f \u27f6 g` beween 1-morphisms `f g : a \u27f6 b` between objects `a b : B`.\n\nWe use `u`, `v`, and `w` as the universe variables for objects, 1-morphisms, and 2-morphisms,\nrespectively.\n\nA typeclass for bicategories extends `category_theory.category_struct` typeclass. This means that\nwe have\n* a composition `f \u226b g : a \u27f6 c` for each 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c`, and\n* a identity `\ud835\udfd9 a : a \u27f6 a` for each object `a : B`.\n\nFor each object `a b : B`, the collection of 1-morphisms `a \u27f6 b` has a category structure. The\n2-morphisms in the bicategory are implemented as the morphisms in this family of categories.\n\nThe composition of 1-morphisms is in fact a object part of a functor\n`(a \u27f6 b) \u2964 (b \u27f6 c) \u2964 (a \u27f6 c)`. The definition of bicategories in this file does not\nrequire this functor directly. Instead, it requires the whiskering functions. For a 1-morphism\n`f : a \u27f6 b` and a 2-morphism `\u03b7 : g \u27f6 h` between 1-morphisms `g h : b \u27f6 c`, there is a\n2-morphism `whisker_left f \u03b7 : f \u226b g \u27f6 f \u226b h`. Similarly, for a 2-morphism `\u03b7 : f \u27f6 g`\nbetween 1-morphisms `f g : a \u27f6 b` and a 1-morphism `f : b \u27f6 c`, there is a 2-morphism\n`whisker_right \u03b7 h : f \u226b h \u27f6 g \u226b h`. These satisfy the exchange law\n`whisker_left f \u03b8 \u226b whisker_right \u03b7 i = whisker_right \u03b7 h \u226b whisker_left g \u03b8`,\nwhich is required as an axiom in the definition here.\n-/\n\nnamespace category_theory\n\nuniverses w v u\n\nopen category iso\n\n/--\nIn a bicategory, we can compose the 1-morphisms `f : a \u27f6 b` and `g : b \u27f6 c` to obtain\na 1-morphism `f \u226b g : a \u27f6 c`. This composition does not need to be strictly associative,\nbut there is a specified associator, `\u03b1_ f g h : (f \u226b g) \u226b h \u2245 f \u226b (g \u226b h)`.\nThere is an identity 1-morphism `\ud835\udfd9 a : a \u27f6 a`, with specified left and right unitor\nisomorphisms `\u03bb_ f : \ud835\udfd9 a \u226b f \u2245 f` and `\u03c1_ f : f \u226b \ud835\udfd9 a \u2245 f`.\nThese associators and unitors satisfy the pentagon and triangle equations.\n\nSee https://ncatlab.org/nlab/show/bicategory.\n-/\n@[nolint check_univs] -- intended to be used with explicit universe parameters\nclass bicategory (B : Type u) extends category_struct.{v} B :=\n-- category structure on the collection of 1-morphisms:\n(hom_category : \u2200 (a b : B), category.{w} (a \u27f6 b) . tactic.apply_instance)\n-- left whiskering:\n(whisker_left {a b c : B} (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) : f \u226b g \u27f6 f \u226b h)\n(infixr ` \u25c1 `:81 := whisker_left)\n-- right whiskering:\n(whisker_right {a b c : B} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) : f \u226b h \u27f6 g \u226b h)\n(infixl ` \u25b7 `:81 := whisker_right)\n-- associator:\n(associator {a b c d : B} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) :\n  (f \u226b g) \u226b h \u2245 f \u226b (g \u226b h))\n(notation `\u03b1_` := associator)\n-- left unitor:\n(left_unitor {a b : B} (f : a \u27f6 b) : \ud835\udfd9 a \u226b f \u2245 f)\n(notation `\u03bb_` := left_unitor)\n-- right unitor:\n(right_unitor {a b : B} (f : a \u27f6 b) : f \u226b \ud835\udfd9 b \u2245 f)\n(notation `\u03c1_` := right_unitor)\n-- axioms for left whiskering:\n(whisker_left_id' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c),\n  f \u25c1 \ud835\udfd9 g = \ud835\udfd9 (f \u226b g) . obviously)\n(whisker_left_comp' : \u2200 {a b c} (f : a \u27f6 b) {g h i : b \u27f6 c} (\u03b7 : g \u27f6 h) (\u03b8 : h \u27f6 i),\n  f \u25c1 (\u03b7 \u226b \u03b8) = f \u25c1 \u03b7 \u226b f \u25c1 \u03b8 . obviously)\n(id_whisker_left' : \u2200 {a b} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g),\n  \ud835\udfd9 a \u25c1 \u03b7 = (\u03bb_ f).hom \u226b \u03b7 \u226b (\u03bb_ g).inv . obviously)\n(comp_whisker_left' : \u2200 {a b c d} (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h'),\n  (f \u226b g) \u25c1 \u03b7 = (\u03b1_ f g h).hom \u226b f \u25c1 g \u25c1 \u03b7 \u226b (\u03b1_ f g h').inv . obviously)\n-- axioms for right whiskering:\n(id_whisker_right' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c),\n  \ud835\udfd9 f \u25b7 g = \ud835\udfd9 (f \u226b g) . obviously)\n(comp_whisker_right' : \u2200 {a b c} {f g h : a \u27f6 b} (\u03b7 : f \u27f6 g) (\u03b8 : g \u27f6 h) (i : b \u27f6 c),\n  (\u03b7 \u226b \u03b8) \u25b7 i = \u03b7 \u25b7 i \u226b \u03b8 \u25b7 i . obviously)\n(whisker_right_id' : \u2200 {a b} {f g : a \u27f6 b} (\u03b7 : f \u27f6 g),\n  \u03b7 \u25b7 \ud835\udfd9 b = (\u03c1_ f).hom \u226b \u03b7 \u226b (\u03c1_ g).inv . obviously)\n(whisker_right_comp' : \u2200 {a b c d} {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d),\n  \u03b7 \u25b7 (g \u226b h) = (\u03b1_ f g h).inv \u226b \u03b7 \u25b7 g \u25b7 h \u226b (\u03b1_ f' g h).hom . obviously)\n-- associativity of whiskerings:\n(whisker_assoc' : \u2200 {a b c d} (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d),\n  (f \u25c1 \u03b7) \u25b7 h = (\u03b1_ f g h).hom \u226b f \u25c1 (\u03b7 \u25b7 h) \u226b (\u03b1_ f g' h).inv . obviously)\n-- exchange law of left and right whiskerings:\n(whisker_exchange' : \u2200 {a b c} {f g : a \u27f6 b} {h i : b \u27f6 c} (\u03b7 : f \u27f6 g) (\u03b8 : h \u27f6 i),\n  f \u25c1 \u03b8 \u226b \u03b7 \u25b7 i = \u03b7 \u25b7 h \u226b g \u25c1 \u03b8 . obviously)\n-- pentagon identity:\n(pentagon' : \u2200 {a b c d e} (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e),\n  (\u03b1_ f g h).hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).hom \u226b f \u25c1 (\u03b1_ g h i).hom =\n    (\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom . obviously)\n-- triangle identity:\n(triangle' : \u2200 {a b c} (f : a \u27f6 b) (g : b \u27f6 c),\n  (\u03b1_ f (\ud835\udfd9 b) g).hom \u226b f \u25c1 (\u03bb_ g).hom = (\u03c1_ f).hom \u25b7 g . obviously)\n\n-- The precedence of the whiskerings is higher than that of the composition `\u226b`.\nlocalized \"infixr (name := bicategory.whisker_left) ` \u25c1 `:81 := bicategory.whisker_left\"\n  in bicategory\nlocalized \"infixl (name := bicategory.whisker_right) ` \u25b7 `:81 := bicategory.whisker_right\"\n  in bicategory\nlocalized \"notation (name := bicategory.associator) `\u03b1_` := bicategory.associator\"\n  in bicategory\nlocalized \"notation (name := bicategory.left_unitor) `\u03bb_` := bicategory.left_unitor\"\n  in bicategory\nlocalized \"notation (name := bicategory.right_unitor) `\u03c1_` := bicategory.right_unitor\"\n  in bicategory\n\nnamespace bicategory\n\n/-!\n### Simp-normal form for 2-morphisms\n\nRewriting involving associators and unitors could be very complicated. We try to ease this\ncomplexity by putting carefully chosen simp lemmas that rewrite any 2-morphisms into simp-normal\nform defined below. Rewriting into simp-normal form is also useful when applying (forthcoming)\n`coherence` tactic.\n\nThe simp-normal form of 2-morphisms is defined to be an expression that has the minimal number of\nparentheses. More precisely,\n1. it is a composition of 2-morphisms like `\u03b7\u2081 \u226b \u03b7\u2082 \u226b \u03b7\u2083 \u226b \u03b7\u2084 \u226b \u03b7\u2085` such that each `\u03b7\u1d62` is\n  either a structural 2-morphisms (2-morphisms made up only of identities, associators, unitors)\n  or non-structural 2-morphisms, and\n2. each non-structural 2-morphism in the composition is of the form `f\u2081 \u25c1 f\u2082 \u25c1 f\u2083 \u25c1 \u03b7 \u25b7 f\u2084 \u25b7 f\u2085`,\n  where each `f\u1d62` is a 1-morphism that is not the identity or a composite and `\u03b7` is a\n  non-structural 2-morphisms that is also not the identity or a composite.\n\nNote that `f\u2081 \u25c1 f\u2082 \u25c1 f\u2083 \u25c1 \u03b7 \u25b7 f\u2084 \u25b7 f\u2085` is actually `f\u2081 \u25c1 (f\u2082 \u25c1 (f\u2083 \u25c1 ((\u03b7 \u25b7 f\u2084) \u25b7 f\u2085)))`.\n-/\n\nrestate_axiom whisker_left_id'\nrestate_axiom whisker_left_comp'\nrestate_axiom id_whisker_left'\nrestate_axiom comp_whisker_left'\nrestate_axiom id_whisker_right'\nrestate_axiom comp_whisker_right'\nrestate_axiom whisker_right_id'\nrestate_axiom whisker_right_comp'\nrestate_axiom whisker_assoc'\nrestate_axiom whisker_exchange'\nrestate_axiom pentagon'\nrestate_axiom triangle'\n\nattribute [simp]  pentagon triangle\nattribute [reassoc]\n  whisker_left_comp id_whisker_left comp_whisker_left\n  comp_whisker_right whisker_right_id whisker_right_comp\n  whisker_assoc whisker_exchange pentagon triangle\n/-\nThe following simp attributes are put in order to rewrite any 2-morphisms into normal forms. There\nare associators and unitors in the RHS in the several simp lemmas here (e.g. `id_whisker_left`),\nwhich at first glance look more complicated than the LHS, but they will be eventually reduced by the\npentagon or the triangle identities, and more generally, (forthcoming) `coherence` tactic.\n-/\nattribute [simp]\n  whisker_left_id whisker_left_comp id_whisker_left comp_whisker_left\n  id_whisker_right comp_whisker_right whisker_right_id whisker_right_comp\n  whisker_assoc\nattribute [instance] hom_category\n\nvariables {B : Type u} [bicategory.{w v} B] {a b c d e : B}\n\n@[simp, reassoc]\nlemma hom_inv_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  f \u25c1 \u03b7.hom \u226b f \u25c1 \u03b7.inv = \ud835\udfd9 (f \u226b g) :=\nby rw [\u2190whisker_left_comp, hom_inv_id, whisker_left_id]\n\n@[simp, reassoc]\nlemma hom_inv_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  \u03b7.hom \u25b7 h \u226b \u03b7.inv \u25b7 h = \ud835\udfd9 (f \u226b h) :=\nby rw [\u2190comp_whisker_right, hom_inv_id, id_whisker_right]\n\n@[simp, reassoc]\nlemma inv_hom_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  f \u25c1 \u03b7.inv \u226b f \u25c1 \u03b7.hom = \ud835\udfd9 (f \u226b h) :=\nby rw [\u2190whisker_left_comp, inv_hom_id, whisker_left_id]\n\n@[simp, reassoc]\nlemma inv_hom_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  \u03b7.inv \u25b7 h \u226b \u03b7.hom \u25b7 h = \ud835\udfd9 (g \u226b h) :=\nby rw [\u2190comp_whisker_right, inv_hom_id, id_whisker_right]\n\n/-- The left whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whisker_left_iso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u2245 h) :\n  f \u226b g \u2245 f \u226b h :=\n{ hom := f \u25c1 \u03b7.hom,\n  inv := f \u25c1 \u03b7.inv }\n\ninstance whisker_left_is_iso (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [is_iso \u03b7] :\n  is_iso (f \u25c1 \u03b7) :=\nis_iso.of_iso (whisker_left_iso f (as_iso \u03b7))\n\n@[simp]\nlemma inv_whisker_left (f : a \u27f6 b) {g h : b \u27f6 c} (\u03b7 : g \u27f6 h) [is_iso \u03b7] :\n  inv (f \u25c1 \u03b7) = f \u25c1 (inv \u03b7) :=\nby { ext, simp only [\u2190whisker_left_comp, whisker_left_id, is_iso.hom_inv_id] }\n\n/-- The right whiskering of a 2-isomorphism is a 2-isomorphism. -/\n@[simps]\ndef whisker_right_iso {f g : a \u27f6 b} (\u03b7 : f \u2245 g) (h : b \u27f6 c) :\n  f \u226b h \u2245 g \u226b h :=\n{ hom := \u03b7.hom \u25b7 h,\n  inv := \u03b7.inv \u25b7 h }\n\ninstance whisker_right_is_iso {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [is_iso \u03b7] :\n  is_iso (\u03b7 \u25b7 h) :=\nis_iso.of_iso (whisker_right_iso (as_iso \u03b7) h)\n\n@[simp]\nlemma inv_whisker_right {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) (h : b \u27f6 c) [is_iso \u03b7] :\n  inv (\u03b7 \u25b7 h) = (inv \u03b7) \u25b7 h :=\nby { ext, simp only [\u2190comp_whisker_right, id_whisker_right, is_iso.hom_inv_id] }\n\n@[simp, reassoc]\nlemma pentagon_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i =\n    (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv :=\neq_of_inv_eq_inv (by simp)\n\n@[simp, reassoc]\nlemma pentagon_inv_inv_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom =\n    f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv :=\nby { rw [\u2190cancel_epi (f \u25c1 (\u03b1_ g h i).inv), \u2190cancel_mono (\u03b1_ (f \u226b g) h i).inv], simp }\n\n@[simp, reassoc]\nlemma pentagon_inv_hom_hom_hom_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i \u226b (\u03b1_ f (g \u226b h) i).hom =\n    (\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv :=\neq_of_inv_eq_inv (by simp)\n\n@[simp, reassoc]\nlemma pentagon_hom_inv_inv_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv =\n    (\u03b1_ f (g \u226b h) i).inv \u226b (\u03b1_ f g h).inv \u25b7 i :=\nby simp [\u2190cancel_epi (f \u25c1 (\u03b1_ g h i).inv)]\n\n@[simp, reassoc]\n\n\n@[simp, reassoc]\nlemma pentagon_hom_inv_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f g (h \u226b i)).hom \u226b f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv =\n    (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i :=\nby { rw [\u2190cancel_epi (\u03b1_ f g (h \u226b i)).inv, \u2190cancel_mono ((\u03b1_ f g h).inv \u25b7 i)], simp }\n\n@[simp, reassoc]\nlemma pentagon_hom_hom_inv_inv_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f (g \u226b h) i).hom \u226b f \u25c1 (\u03b1_ g h i).hom \u226b (\u03b1_ f g (h \u226b i)).inv =\n    (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom :=\neq_of_inv_eq_inv (by simp)\n\n@[simp, reassoc]\nlemma pentagon_inv_hom_hom_hom_hom (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f g h).inv \u25b7 i \u226b (\u03b1_ (f \u226b g) h i).hom \u226b (\u03b1_ f g (h \u226b i)).hom =\n    (\u03b1_ f (g \u226b h) i).hom \u226b f \u25c1 (\u03b1_ g h i).hom :=\nby simp [\u2190cancel_epi ((\u03b1_ f g h).hom \u25b7 i)]\n\n@[simp, reassoc]\nlemma pentagon_inv_inv_hom_inv_inv (f : a \u27f6 b) (g : b \u27f6 c) (h : c \u27f6 d) (i : d \u27f6 e) :\n  (\u03b1_ f g (h \u226b i)).inv \u226b (\u03b1_ (f \u226b g) h i).inv \u226b (\u03b1_ f g h).hom \u25b7 i =\n    f \u25c1 (\u03b1_ g h i).inv \u226b (\u03b1_ f (g \u226b h) i).inv :=\neq_of_inv_eq_inv (by simp)\n\nlemma triangle_assoc_comp_left (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03b1_ f (\ud835\udfd9 b) g).hom \u226b f \u25c1 (\u03bb_ g).hom = (\u03c1_ f).hom \u25b7 g :=\ntriangle f g\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03b1_ f (\ud835\udfd9 b) g).inv \u226b (\u03c1_ f).hom \u25b7 g = f \u25c1 (\u03bb_ g).hom :=\nby rw [\u2190triangle, inv_hom_id_assoc]\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_right_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03c1_ f).inv \u25b7 g \u226b (\u03b1_ f (\ud835\udfd9 b) g).hom = f \u25c1 (\u03bb_ g).inv :=\nby simp [\u2190cancel_mono (f \u25c1 (\u03bb_ g).hom)]\n\n@[simp, reassoc]\nlemma triangle_assoc_comp_left_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03bb_ g).inv \u226b (\u03b1_ f (\ud835\udfd9 b) g).inv = (\u03c1_ f).inv \u25b7 g :=\nby simp [\u2190cancel_mono ((\u03c1_ f).hom \u25b7 g)]\n\n@[reassoc]\nlemma associator_naturality_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b7 \u25b7 g) \u25b7 h \u226b (\u03b1_ f' g h).hom = (\u03b1_ f g h).hom \u226b \u03b7 \u25b7 (g \u226b h) :=\nby simp\n\n@[reassoc]\nlemma associator_inv_naturality_left {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv = (\u03b1_ f g h).inv \u226b (\u03b7 \u25b7 g) \u25b7 h :=\nby simp\n\n@[reassoc]\nlemma whisker_right_comp_symm {f f' : a \u27f6 b} (\u03b7 : f \u27f6 f') (g : b \u27f6 c) (h : c \u27f6 d) :\n  (\u03b7 \u25b7 g) \u25b7 h = (\u03b1_ f g h).hom \u226b \u03b7 \u25b7 (g \u226b h) \u226b (\u03b1_ f' g h).inv :=\nby simp\n\n@[reassoc]\nlemma associator_naturality_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).hom = (\u03b1_ f g h).hom \u226b f \u25c1 (\u03b7 \u25b7 h) :=\nby simp\n\n@[reassoc]\nlemma associator_inv_naturality_middle (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  f \u25c1 (\u03b7 \u25b7 h) \u226b (\u03b1_ f g' h).inv = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h :=\nby simp\n\n@[reassoc]\nlemma whisker_assoc_symm (f : a \u27f6 b) {g g' : b \u27f6 c} (\u03b7 : g \u27f6 g') (h : c \u27f6 d) :\n  f \u25c1 (\u03b7 \u25b7 h) = (\u03b1_ f g h).inv \u226b (f \u25c1 \u03b7) \u25b7 h \u226b (\u03b1_ f g' h).hom :=\nby simp\n\n@[reassoc]\nlemma associator_naturality_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').hom = (\u03b1_ f g h).hom \u226b f \u25c1 (g \u25c1 \u03b7) :=\nby simp\n\n@[reassoc]\nlemma associator_inv_naturality_right (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  f \u25c1 (g \u25c1 \u03b7) \u226b (\u03b1_ f g h').inv = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7 :=\nby simp\n\n@[reassoc]\nlemma comp_whisker_left_symm (f : a \u27f6 b) (g : b \u27f6 c) {h h' : c \u27f6 d} (\u03b7 : h \u27f6 h') :\n  f \u25c1 (g \u25c1 \u03b7) = (\u03b1_ f g h).inv \u226b (f \u226b g) \u25c1 \u03b7 \u226b (\u03b1_ f g h').hom :=\nby simp\n\n@[reassoc]\nlemma left_unitor_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).hom = (\u03bb_ f).hom \u226b \u03b7 :=\nby simp\n\n@[reassoc]\nlemma left_unitor_inv_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \u03b7 \u226b (\u03bb_ g).inv = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7 :=\nby simp\n\nlemma id_whisker_left_symm {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \u03b7 = (\u03bb_ f).inv \u226b \ud835\udfd9 a \u25c1 \u03b7 \u226b (\u03bb_ g).hom :=\nby simp\n\n@[reassoc]\nlemma right_unitor_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).hom = (\u03c1_ f).hom \u226b \u03b7 :=\nby simp\n\n@[reassoc]\nlemma right_unitor_inv_naturality {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \u03b7 \u226b (\u03c1_ g).inv = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b :=\nby simp\n\nlemma whisker_right_id_symm {f g : a \u27f6 b} (\u03b7 : f \u27f6 g) :\n  \u03b7 = (\u03c1_ f).inv \u226b \u03b7 \u25b7 \ud835\udfd9 b \u226b (\u03c1_ g).hom :=\nby simp\n\nlemma whisker_left_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) :\n  (\ud835\udfd9 a \u25c1 \u03b7 = \ud835\udfd9 a \u25c1 \u03b8) \u2194 (\u03b7 = \u03b8) :=\nby simp\n\nlemma whisker_right_iff {f g : a \u27f6 b} (\u03b7 \u03b8 : f \u27f6 g) :\n  (\u03b7 \u25b7 \ud835\udfd9 b = \u03b8 \u25b7 \ud835\udfd9 b) \u2194 (\u03b7 = \u03b8) :=\nby simp\n\n/--\nWe state it as a simp lemma, which is regarded as an involved version of\n`id_whisker_right f g : \ud835\udfd9 f \u25b7 g = \ud835\udfd9 (f \u226b g)`.\n-/\n@[reassoc, simp]\nlemma left_unitor_whisker_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ f).hom \u25b7 g = (\u03b1_ (\ud835\udfd9 a) f g).hom \u226b (\u03bb_ (f \u226b g)).hom :=\nby rw [\u2190whisker_left_iff, whisker_left_comp, \u2190cancel_epi (\u03b1_ _ _ _).hom,\n  \u2190cancel_epi ((\u03b1_ _ _ _).hom \u25b7 _), pentagon_assoc, triangle,\n  \u2190associator_naturality_middle, \u2190comp_whisker_right_assoc, triangle,\n  associator_naturality_left]; apply_instance\n\n@[reassoc, simp]\nlemma left_unitor_inv_whisker_right (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ f).inv \u25b7 g = (\u03bb_ (f \u226b g)).inv \u226b (\u03b1_ (\ud835\udfd9 a) f g).inv :=\neq_of_inv_eq_inv (by simp)\n\n@[reassoc, simp]\nlemma whisker_left_right_unitor (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03c1_ g).hom = (\u03b1_ f g (\ud835\udfd9 c)).inv \u226b (\u03c1_ (f \u226b g)).hom :=\nby rw [\u2190whisker_right_iff, comp_whisker_right, \u2190cancel_epi (\u03b1_ _ _ _).inv,\n  \u2190cancel_epi (f \u25c1 (\u03b1_ _ _ _).inv), pentagon_inv_assoc, triangle_assoc_comp_right,\n  \u2190associator_inv_naturality_middle, \u2190whisker_left_comp_assoc, triangle_assoc_comp_right,\n  associator_inv_naturality_right]; apply_instance\n\n@[reassoc, simp]\nlemma whisker_left_right_unitor_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  f \u25c1 (\u03c1_ g).inv = (\u03c1_ (f \u226b g)).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).hom :=\neq_of_inv_eq_inv (by simp)\n\n/-\nIt is not so obvious whether `left_unitor_whisker_right` or `left_unitor_comp` should be a simp\nlemma. Our choice is the former. One reason is that the latter yields the following loop:\n[id_whisker_left]   : \ud835\udfd9 a \u25c1 (\u03c1_ f).hom ==> (\u03bb_ (f \u226b \ud835\udfd9 b)).hom \u226b (\u03c1_ f).hom \u226b (\u03bb_ f).inv\n[left_unitor_comp]  : (\u03bb_ (f \u226b \ud835\udfd9 b)).hom ==> (\u03b1_ (\ud835\udfd9 a) f (\ud835\udfd9 b)).inv \u226b (\u03bb_ f).hom \u25b7 \ud835\udfd9 b\n[whisker_right_id]  : (\u03bb_ f).hom \u25b7 \ud835\udfd9 b ==> (\u03c1_ (\ud835\udfd9 a \u226b f)).hom \u226b (\u03bb_ f).hom \u226b (\u03c1_ f).inv\n[right_unitor_comp] : (\u03c1_ (\ud835\udfd9 a \u226b f)).hom ==> (\u03b1_ (\ud835\udfd9 a) f (\ud835\udfd9 b)).hom \u226b \ud835\udfd9 a \u25c1 (\u03c1_ f).hom\n-/\n@[reassoc]\nlemma left_unitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ (f \u226b g)).hom = (\u03b1_ (\ud835\udfd9 a) f g).inv \u226b (\u03bb_ f).hom \u25b7 g :=\nby simp\n\n@[reassoc]\nlemma left_unitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03bb_ (f \u226b g)).inv = (\u03bb_ f).inv \u25b7 g \u226b (\u03b1_ (\ud835\udfd9 a) f g).hom :=\nby simp\n\n@[reassoc]\nlemma right_unitor_comp (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03c1_ (f \u226b g)).hom = (\u03b1_ f g (\ud835\udfd9 c)).hom \u226b f \u25c1 (\u03c1_ g).hom :=\nby simp\n\n@[reassoc]\nlemma right_unitor_comp_inv (f : a \u27f6 b) (g : b \u27f6 c) :\n  (\u03c1_ (f \u226b g)).inv = f \u25c1 (\u03c1_ g).inv \u226b (\u03b1_ f g (\ud835\udfd9 c)).inv :=\nby simp\n\n@[simp]\nlemma unitors_equal : (\u03bb_ (\ud835\udfd9 a)).hom = (\u03c1_ (\ud835\udfd9 a)).hom :=\nby rw [\u2190whisker_left_iff, \u2190cancel_epi (\u03b1_ _ _ _).hom, \u2190cancel_mono (\u03c1_ _).hom, triangle,\n  \u2190right_unitor_comp, right_unitor_naturality]; apply_instance\n\n@[simp]\nlemma unitors_inv_equal : (\u03bb_ (\ud835\udfd9 a)).inv = (\u03c1_ (\ud835\udfd9 a)).inv :=\nby simp [iso.inv_eq_inv]\n\nend bicategory\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/bicategory/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7772998611746912, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3704452966104781}}
{"text": "import data.int.parity\nimport data.real.irrational\n\nopen real\n\n#print irrational\n#check set.range (coe : \u211a \u2192 \u211d)\n\nlemma eq_mul_div_of_nonneg {a b : \u211d} (h : b \u2260 0) : a = a / b * b :=\nbegin\n  calc\n  a   = 1 * a     : by ring\n  ... = b / b * a : by rw \u2190div_self h\n  ... = a / b * b : by exact div_mul_comm' b b a\nend\n\n#check div_self\n#check mul_div_right_comm\n\nlemma eq_of_mul_eq_mul_left' {a b t : \u2115} (ht : t \u2260 0) :\n  t * a = t * b \u2192 a = b :=\nbegin\nend\n\nexample : irrational (sqrt 2) :=\nbegin\n  rintro \u27e8\u27e8a, b, pos, cop\u27e9, h\u27e9,\n  have : (b : \u211d) \u2260 0,\n  { norm_cast,\n    apply ne_of_gt,\n    assumption },\n  have h\u2081 : (a : \u211d) = sqrt 2 * b,\n  { apply_fun (\u03bb x, x * b) at h,\n    calc\n    (a : \u211d) = 1 * a      : by ring\n    ...     = b / b * a  : by rwa \u2190div_self\n    ...     = a / b * b  : by ring\n    ...     = sqrt 2 * b : by exact h },\n  have h\u2082 : (a : \u211d)^2 = 2 * b^2,\n  { apply_fun (\u03bb x, x * x) at h\u2081,\n    ring at *,\n    rw sqr_sqrt at *; linarith },\n  have a_even : a.even,\n  { suffices : (a^2).even,\n    { rw int.even_pow at this, cc },\n    use b^2, norm_cast at *, assumption },\n  have a_even : b.even,\n  { suffices : (b^2).even,\n    { rw nat.even_pow at this, cc },\n    cases a_even with c hc,\n    use (c^2).to_nat,\n    apply eq_of_mul_eq_mul_left' (show 2 \u2260 0, by linarith),\n    rw hc at h\u2082,\n    norm_cast at *,\n    ring,\n    calc\n    2 * b ^ 2 = 2 * b * b          : by ring\n    ...       = ((2 * c)^2).to_nat : by { rw h\u2082, ring }\n    ...       = (4 * c^2).to_nat   : by ring\n    -- Is at this point that I stop. This is not worthwhile. My strategy is\n    -- probably really bad. I should just be working with `\u211a`.\n    ...       = 4 * (c ^ 2).to_nat : by { sorry } },\n  sorry\nend", "meta": {"author": "pedrominicz", "repo": "learn", "sha": "b79b802a9846c86c21d4b6f3e17af36e7382f0ef", "save_path": "github-repos/lean/pedrominicz-learn", "path": "github-repos/lean/pedrominicz-learn/learn-b79b802a9846c86c21d4b6f3e17af36e7382f0ef/src/logic_and_proof/01_introduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7772998611746911, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.37044529661047804}}
{"text": "import tactic.rename_var\n\nexample (P : \u2115 \u2192  \u2115 \u2192 Prop) (h : \u2200 n, \u2203 m, P n m) : true :=\nbegin\n  rename_var n q at h,\n  guard_hyp_strict h : \u2200 (q : \u2115), \u2203 (m : \u2115), P q m,\n  rename_var m z at h,\n  guard_hyp_strict h : \u2200 (q : \u2115), \u2203 (z : \u2115), P q z,\n  trivial\nend\n\nexample (P : \u2115 \u2192  \u2115 \u2192 Prop) (h : \u2200 n, \u2203 m, P n m) : \u2200 n, \u2203 m, P n m :=\nbegin\n  rename_var n q,\n  guard_target_strict \u2200 (q : \u2115), \u2203 (m : \u2115), P q m,\n  rename_var m l,\n  guard_target_strict \u2200 (q : \u2115), \u2203 (l : \u2115), P q l,\n  exact h\nend\n\nexample (h : (\u03bb n : \u2115, n) = id) : true :=\nbegin\n  rename_var n m at h,\n  guard_hyp_strict h : (\u03bb (m : \u2115), m) = id,\n  trivial\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/rename_var.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514763, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.3703079708970749}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n-/\nimport Mathlib.PrePort\n\nuniverses u l v w u_1 u_2 u\u2081 u\u2082 u\u2083 \n\nnamespace Mathlib\n\n/- Reserving notation. We do this so that the precedence of all of the operators\ncan be seen in one place and to prevent core notation being accidentally overloaded later.  -/\n\n/- Notation for logical operations and relations -/\n\n/- types and type constructors -/\n\n/- arithmetic operations -/\n\n/- boolean operations -/\n\n/- set operations -/\n\n/- other symbols -/\n\n/--\nThe kernel definitional equality test (t =?= s) has special support for id_delta applications.\nIt implements the following rules\n\n   1)   (id_delta t) =?= t\n   2)   t =?= (id_delta t)\n   3)   (id_delta t) =?= s  IF (unfold_of t) =?= s\n   4)   t =?= id_delta s    IF t =?= (unfold_of s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use id_delta applications to address performance problems when type checking\nlemmas generated by the equation compiler.\n-/\ndef id_delta {\u03b1 : Sort u} (a : \u03b1) : \u03b1 :=\n  a\n\n/-- Gadget for optional parameter support. -/\ndef opt_param (\u03b1 : Sort u) (default : \u03b1) :=\n  \u03b1\n\n/-- Gadget for marking output parameters in type classes. -/\ndef out_param (\u03b1 : Sort u) :=\n  \u03b1\n\n/-\n  id_rhs is an auxiliary declaration used in the equation compiler to address performance\n  issues when proving equational lemmas. The equation compiler uses it as a marker.\n-/\n\ndef id_rhs (\u03b1 : Sort u) (a : \u03b1) : \u03b1 :=\n  a\n\n/-- An abbreviation for `punit.{0}`, its most common instantiation.\n    This type should be preferred over `punit` where possible to avoid\n    unnecessary universe parameters. -/\ndef unit :=\n  PUnit\n\ndef unit.star : Unit :=\n  PUnit.unit\n\n/--\nGadget for defining thunks, thunk parameters have special treatment.\nExample: given\n      def f (s : string) (t : thunk nat) : nat\nan application\n     f \"hello\" 10\n is converted into\n     f \"hello\" (\u03bb _, 10)\n-/\ndef thunk (\u03b1 : Type u) :=\n  Unit \u2192 \u03b1\n\ninductive empty \nwhere\n\n/--\nLogical not.\n\n`not P`, with notation `\u00ac P`, is the `Prop` which is true if and only if `P` is false. It is\ninternally represented as `P \u2192 false`, so one way to prove a goal `\u22a2 \u00ac P` is to use `intro h`,\nwhich gives you a new hypothesis `h : P` and the goal `\u22a2 false`.\n\nA hypothesis `h : \u00ac P` can be used in term mode as a function, so if `w : P` then `h w : false`.\n\nRelated mathlib tactic: `contrapose`.\n-/\ndef not (a : Prop) :=\n  a \u2192 False\n\nprefix:40 \"\u00ac\" => Mathlib.not\n\n/-\nInitialize the quotient module, which effectively adds the following definitions:\n\nconstant quot {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Sort u\n\nconstant quot.mk {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) : quot r\n\nconstant quot.lift {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  (\u2200 a b : \u03b1, r a b \u2192 eq (f a) (f b)) \u2192 quot r \u2192 \u03b2\n\nconstant quot.ind {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : quot r \u2192 Prop} :\n  (\u2200 a : \u03b1, \u03b2 (quot.mk r a)) \u2192 \u2200 q : quot r, \u03b2 q\n\nAlso the reduction rule:\n\nquot.lift f _ (quot.mk a) ~~> f a\n\n-/\n\n/--\nHeterogeneous equality.\n\nIts purpose is to write down equalities between terms whose types are not definitionally equal.\nFor example, given `x : vector \u03b1 n` and `y : vector \u03b1 (0+n)`, `x = y` doesn't typecheck but `x == y` does.\n\nIf you have a goal `\u22a2 x == y`, \nyour first instinct should be to ask (either yourself, or on [zulip](https://leanprover.zulipchat.com/))\nif something has gone wrong already.\nIf you really do need to follow this route, \nyou may find the lemmas `eq_rec_heq` and `eq_mpr_heq` useful.\n-/\n/-- Similar to `prod`, but \u03b1 and \u03b2 can be propositions.\n   We use this type internally to automatically generate the brec_on recursor. -/\n/--\nLogical and.\n\n`and P Q`, with notation `P \u2227 Q`, is the `Prop` which is true precisely when `P` and `Q` are\nboth true. \n\nTo prove a goal `\u22a2 P \u2227 Q`, you can use the tactic `split`,\nwhich gives two separate goals `\u22a2 P` and `\u22a2 Q`.\n\nGiven a hypothesis `h : P \u2227 Q`, you can use the tactic `cases h with hP hQ`\nto obtain two new hypotheses `hP : P` and `hQ : Q`. See also the `obtain` or `rcases` tactics in\nmathlib.\n-/\ndef and.elim_left {a : Prop} {b : Prop} (h : a \u2227 b) : a :=\n  and.left h\n\ndef and.elim_right {a : Prop} {b : Prop} (h : a \u2227 b) : b :=\n  and.right h\n\ninfixl:50 \" = \" => Mathlib.eq\n\n/- eq basic support -/\n\ndef rfl {\u03b1 : Sort u} {a : \u03b1} : a = a :=\n  Eq.refl a\n\ntheorem eq.subst {\u03b1 : Sort u} {P : \u03b1 \u2192 Prop} {a : \u03b1} {b : \u03b1} (h\u2081 : a = b) (h\u2082 : P a) : P b :=\n  Eq._oldrec h\u2082 h\u2081\n\ninfixr:75 \" \u25b8 \" => Mathlib.eq.subst\n\ntheorem eq.trans {\u03b1 : Sort u} {a : \u03b1} {b : \u03b1} {c : \u03b1} (h\u2081 : a = b) (h\u2082 : b = c) : a = c :=\n  h\u2082 \u25b8 h\u2081\n\ntheorem eq.symm {\u03b1 : Sort u} {a : \u03b1} {b : \u03b1} (h : a = b) : b = a :=\n  h \u25b8 rfl\n\ninfixl:50 \" == \" => Mathlib.heq\n\ndef heq.rfl {\u03b1 : Sort u} {a : \u03b1} : a == a :=\n  HEq.refl a\n\ntheorem eq_of_heq {\u03b1 : Sort u} {a : \u03b1} {a' : \u03b1} (h : a == a') : a = a' :=\n  (fun (this : \u2200 (\u03b1' : Sort u) (a' : \u03b1'), a == a' \u2192 \u2200 (h\u2082 : \u03b1 = \u03b1'), eq.rec_on h\u2082 a = a') =>\n      (fun (this : eq.rec_on (Eq.refl \u03b1) a = a') => this) (this \u03b1 a' h (Eq.refl \u03b1)))\n    fun (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : a == a') => heq.rec_on h\u2081 fun (h\u2082 : \u03b1 = \u03b1) => rfl\n\n/- The following four lemmas could not be automatically generated when the\n   structures were declared, so we prove them manually here. -/\n\ntheorem prod.mk.inj {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 x\u2081 = x\u2082 \u2227 y\u2081 = y\u2082 :=\n  fun (h : (x\u2081, y\u2081) = (x\u2082, y\u2082)) => prod.no_confusion h fun (h\u2081 : x\u2081 = x\u2082) (h\u2082 : y\u2081 = y\u2082) => { left := h\u2081, right := h\u2082 }\n\ntheorem prod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 {P : Sort w} \u2192 (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n  fun (h\u2081 : (x\u2081, y\u2081) = (x\u2082, y\u2082)) (_x : Sort w) (h\u2082 : x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 _x) => prod.no_confusion h\u2081 h\u2082\n\ntheorem pprod.mk.inj {\u03b1 : Sort u} {\u03b2 : Sort v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} : { fst := x\u2081, snd := y\u2081 } = { fst := x\u2082, snd := y\u2082 } \u2192 x\u2081 = x\u2082 \u2227 y\u2081 = y\u2082 :=\n  fun (h : { fst := x\u2081, snd := y\u2081 } = { fst := x\u2082, snd := y\u2082 }) =>\n    pprod.no_confusion h fun (h\u2081 : x\u2081 = x\u2082) (h\u2082 : y\u2081 = y\u2082) => { left := h\u2081, right := h\u2082 }\n\ntheorem pprod.mk.inj_arrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} : (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 {P : Sort w} \u2192 (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P :=\n  fun (h\u2081 : (x\u2081, y\u2081) = (x\u2082, y\u2082)) (_x : Sort w) (h\u2082 : x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 _x) => prod.no_confusion h\u2081 h\u2082\n\ninductive sum (\u03b1 : Type u) (\u03b2 : Type v) \nwhere\n| inl : \u03b1 \u2192 sum \u03b1 \u03b2\n| inr : \u03b2 \u2192 sum \u03b1 \u03b2\n\ninductive psum (\u03b1 : Sort u) (\u03b2 : Sort v) \nwhere\n| inl : \u03b1 \u2192 psum \u03b1 \u03b2\n| inr : \u03b2 \u2192 psum \u03b1 \u03b2\n\n/--\nLogical or.\n\n`or P Q`, with notation `P \u2228 Q`, is the proposition which is true if and only if `P` or `Q` is\ntrue.\n\nTo prove a goal `\u22a2 P \u2228 Q`, if you know which alternative you want to prove,\nyou can use the tactics `left` (which gives the goal `\u22a2 P`)\nor `right` (which gives the goal `\u22a2 Q`).\n\nGiven a hypothesis `h : P \u2228 Q` and goal `\u22a2 R`,\nthe tactic `cases h` will give you two copies of the goal `\u22a2 R`,\nwith the hypothesis `h : P` in the first, and the hypothesis `h : Q` in the second.\n-/\ndef or.intro_left {a : Prop} (b : Prop) (ha : a) : a \u2228 b :=\n  Or.inl ha\n\ndef or.intro_right (a : Prop) {b : Prop} (hb : b) : a \u2228 b :=\n  Or.inr hb\n\nstructure sigma {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) \nwhere\n  fst : \u03b1\n  snd : \u03b2 fst\n\nstructure psigma {\u03b1 : Sort u} (\u03b2 : \u03b1 \u2192 Sort v) \nwhere\n  fst : \u03b1\n  snd : \u03b2 fst\n\n/- Remark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description. -/\n\ndef decidable_pred {\u03b1 : Sort u} (r : \u03b1 \u2192 Prop) :=\n  (a : \u03b1) \u2192 Decidable (r a)\n\ndef decidable_rel {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\n  (a b : \u03b1) \u2192 Decidable (r a b)\n\ndef decidable_eq (\u03b1 : Sort u) :=\n  DecidableRel Eq\n\ninfixr:67 \" :: \" => Mathlib.list.cons\n\nstructure unification_constraint \nwhere\n  \u03b1 : Type u\n  lhs : \u03b1\n  rhs : \u03b1\n\ninfixl:50 \" \u225f \" => Mathlib.unification_constraint.mk\n\ninfixl:50 \" =?= \" => Mathlib.unification_constraint.mk\n\nstructure unification_hint \nwhere\n  pattern : unification_constraint\n  constraints : List unification_constraint\n\n/- Declare builtin and reserved notation -/\n\nclass has_inv (\u03b1 : Type u) \nwhere\n  inv : \u03b1 \u2192 \u03b1\n\nclass has_dvd (\u03b1 : Type u) \nwhere\n  dvd : \u03b1 \u2192 \u03b1 \u2192 Prop\n\nclass has_andthen (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : outParam (Type w)) \nwhere\n  andthen : \u03b1 \u2192 \u03b2 \u2192 \u03c3\n\nclass has_union (\u03b1 : Type u) \nwhere\n  union : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_inter (\u03b1 : Type u) \nwhere\n  inter : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_sdiff (\u03b1 : Type u) \nwhere\n  sdiff : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\nclass has_equiv (\u03b1 : Sort u) \nwhere\n  equiv : \u03b1 \u2192 \u03b1 \u2192 Prop\n\nclass has_subset (\u03b1 : Type u) \nwhere\n  subset : \u03b1 \u2192 \u03b1 \u2192 Prop\n\n/- Type classes has_emptyc and has_insert are\nclass has_ssubset (\u03b1 : Type u) \nwhere\n  ssubset : \u03b1 \u2192 \u03b1 \u2192 Prop\n\n   used to implement polymorphic notation for collections.\n   Example: {a, b, c}. -/\n\nclass has_emptyc (\u03b1 : Type u) \nwhere\n  emptyc : \u03b1\n\nclass has_insert (\u03b1 : outParam (Type u)) (\u03b3 : Type v) \nwhere\n  insert : \u03b1 \u2192 \u03b3 \u2192 \u03b3\n\n/- Type class used to implement the notation { a \u2208 c | p a } -/\n\nclass has_singleton (\u03b1 : outParam (Type u)) (\u03b2 : Type v) \nwhere\n  singleton : \u03b1 \u2192 \u03b2\n\nclass has_sep (\u03b1 : outParam (Type u)) (\u03b3 : Type v) \nwhere\n  sep : (\u03b1 \u2192 Prop) \u2192 \u03b3 \u2192 \u03b3\n\n/- Type class for set-like membership -/\n\nclass has_mem (\u03b1 : outParam (Type u)) (\u03b3 : Type v) \nwhere\n  mem : \u03b1 \u2192 \u03b3 \u2192 Prop\n\nclass has_pow (\u03b1 : Type u) (\u03b2 : Type v) \nwhere\n  pow : \u03b1 \u2192 \u03b2 \u2192 \u03b1\n\ninfixl:50 \" \u2208 \" => Mathlib.has_mem.mem\n\ninfixl:65 \" + \" => Mathlib.has_add.add\n\ninfixl:70 \" * \" => Mathlib.has_mul.mul\n\ninfixl:65 \" - \" => Mathlib.has_sub.sub\n\ninfixl:70 \" / \" => Mathlib.has_div.div\n\ninfixl:50 \" \u2223 \" => Mathlib.has_dvd.dvd\n\ninfixl:70 \" % \" => Mathlib.has_mod.mod\n\nprefix:75 \"-\" => Mathlib.has_neg.neg\n\ninfixl:50 \" <= \" => Mathlib.has_le.le\n\ninfixl:50 \" \u2264 \" => Mathlib.has_le.le\n\ninfixl:50 \" < \" => Mathlib.has_lt.lt\n\ninfixl:65 \" ++ \" => Mathlib.has_append.append\n\ninfixl:1 \"; \" => Mathlib.has_andthen.andthen\n\nnotation:1024 \"\u2205\" => Mathlib.has_emptyc.emptyc\n\ninfixl:65 \" \u222a \" => Mathlib.has_union.union\n\ninfixl:70 \" \u2229 \" => Mathlib.has_inter.inter\n\ninfixl:50 \" \u2286 \" => Mathlib.has_subset.subset\n\ninfixl:50 \" \u2282 \" => Mathlib.has_ssubset.ssubset\n\ninfixl:70 \" \\ \" => Mathlib.has_sdiff.sdiff\n\ninfixl:50 \" \u2248 \" => Mathlib.has_equiv.equiv\n\ninfixr:80 \" ^ \" => Mathlib.has_pow.pow\n\ndef ge {\u03b1 : Type u} [HasLessEq \u03b1] (a : \u03b1) (b : \u03b1) :=\n  b \u2264 a\n\ndef gt {\u03b1 : Type u} [HasLess \u03b1] (a : \u03b1) (b : \u03b1) :=\n  b < a\n\ninfixl:50 \" >= \" => Mathlib.ge\n\ninfixl:50 \" \u2265 \" => Mathlib.ge\n\ninfixl:50 \" > \" => Mathlib.gt\n\ndef superset {\u03b1 : Type u} [has_subset \u03b1] (a : \u03b1) (b : \u03b1) :=\n  b \u2286 a\n\ndef ssuperset {\u03b1 : Type u} [has_ssubset \u03b1] (a : \u03b1) (b : \u03b1) :=\n  b \u2282 a\n\ninfixl:50 \" \u2287 \" => Mathlib.superset\n\ninfixl:50 \" \u2283 \" => Mathlib.ssuperset\n\ndef bit0 {\u03b1 : Type u} [s : Add \u03b1] (a : \u03b1) : \u03b1 :=\n  a + a\n\ndef bit1 {\u03b1 : Type u} [s\u2081 : HasOne \u03b1] [s\u2082 : Add \u03b1] (a : \u03b1) : \u03b1 :=\n  bit0 a + 1\n\nclass is_lawful_singleton (\u03b1 : Type u) (\u03b2 : Type v) [has_emptyc \u03b2] [has_insert \u03b1 \u03b2] [has_singleton \u03b1 \u03b2] \nwhere\n  insert_emptyc_eq : \u2200 (x : \u03b1), insert x \u2205 = has_singleton.singleton x\n\n/- nat basic instances -/\n\nnamespace nat\n\n\nprotected def add : Nat \u2192 Nat \u2192 Nat :=\n  Nat.add\n\nend nat\n\n\nprotected instance nat.has_zero : HasZero Nat :=\n  { zero := 0 }\n\nprotected instance nat.has_one : HasOne Nat :=\n  { one := 1 }\n\nprotected instance nat.has_add : Add Nat :=\n  { add := Nat.add }\n\ndef std.priority.default : Nat :=\n  bit0 (bit0 (bit0 (bit1 (bit0 (bit1 (bit1 (bit1 (bit1 1))))))))\n\ndef std.priority.max : Nat :=\n  bit1\n    (bit1\n      (bit1\n        (bit1\n          (bit1\n            (bit1\n              (bit1\n                (bit1\n                  (bit1\n                    (bit1\n                      (bit1\n                        (bit1\n                          (bit1\n                            (bit1\n                              (bit1\n                                (bit1\n                                  (bit1\n                                    (bit1\n                                      (bit1\n                                        (bit1\n                                          (bit1\n                                            (bit1\n                                              (bit1\n                                                (bit1\n                                                  (bit1\n                                                    (bit1 (bit1 (bit1 (bit1 (bit1 (bit1 1))))))))))))))))))))))))))))))\n\nnamespace nat\n\n\nend nat\n\n\nprotected def nat.prio : Nat :=\n  std.priority.default + bit0 (bit0 (bit1 (bit0 (bit0 (bit1 1)))))\n\n/-\n  Global declarations of right binding strength\n\n  If a module reassigns these, it will be incompatible with other modules that adhere to these\n  conventions.\n\n  When hovering over a symbol, use \"C-c C-k\" to see how to input it.\n-/\n\ndef std.prec.max : Nat :=\n  bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 (bit0 1)))))))))\n\ndef std.prec.arrow : Nat :=\n  bit1 (bit0 (bit0 (bit1 1)))\n\n/-\nThe next def is \"max + 10\". It can be used e.g. for postfix operations that should\nbe stronger than application.\n-/\n\ndef std.prec.max_plus : Nat :=\n  std.prec.max + bit0 (bit1 (bit0 1))\n\npostfix:0 \"\u207b\u00b9\" => Mathlib.has_inv.inv\n\ninfixr:35 \" \u00d7 \" => Mathlib.prod\n\n-- notation for n-ary tuples\n\n/- sizeof -/\n\ndef sizeof {\u03b1 : Sort u} [s : SizeOf \u03b1] : \u03b1 \u2192 Nat :=\n  has_sizeof.sizeof\n\n/-\nDeclare sizeof instances and lemmas for types declared before has_sizeof.\nFrom now on, the inductive compiler will automatically generate sizeof instances and lemmas.\n-/\n\n/- Every type `\u03b1` has a default has_sizeof instance that just returns 0 for every element of `\u03b1` -/\n\nprotected def default.sizeof (\u03b1 : Sort u) : \u03b1 \u2192 Nat :=\n  sorry\n\nprotected instance default_has_sizeof (\u03b1 : Sort u) : SizeOf \u03b1 :=\n  { sizeOf := default.sizeof \u03b1 }\n\nprotected def nat.sizeof : Nat \u2192 Nat :=\n  sorry\n\nprotected instance nat.has_sizeof : SizeOf Nat :=\n  { sizeOf := nat.sizeof }\n\nprotected def prod.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : \u03b1 \u00d7 \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance prod.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (\u03b1 \u00d7 \u03b2) :=\n  { sizeOf := prod.sizeof }\n\nprotected def sum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : sum \u03b1 \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance sum.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (sum \u03b1 \u03b2) :=\n  { sizeOf := sum.sizeof }\n\nprotected def psum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : psum \u03b1 \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance psum.has_sizeof (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (psum \u03b1 \u03b2) :=\n  { sizeOf := psum.sizeof }\n\nprotected def sigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : sigma \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance sigma.has_sizeof (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : SizeOf (sigma \u03b2) :=\n  { sizeOf := sigma.sizeof }\n\nprotected def psigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : psigma \u03b2 \u2192 Nat :=\n  sorry\n\nprotected instance psigma.has_sizeof (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1] [(a : \u03b1) \u2192 SizeOf (\u03b2 a)] : SizeOf (psigma \u03b2) :=\n  { sizeOf := psigma.sizeof }\n\nprotected def punit.sizeof : PUnit \u2192 Nat :=\n  sorry\n\nprotected instance punit.has_sizeof : SizeOf PUnit :=\n  { sizeOf := punit.sizeof }\n\nprotected def bool.sizeof : Bool \u2192 Nat :=\n  sorry\n\nprotected instance bool.has_sizeof : SizeOf Bool :=\n  { sizeOf := bool.sizeof }\n\nprotected def option.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : Option \u03b1 \u2192 Nat :=\n  sorry\n\nprotected instance option.has_sizeof (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (Option \u03b1) :=\n  { sizeOf := option.sizeof }\n\nprotected def list.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : List \u03b1 \u2192 Nat :=\n  sorry\n\nprotected instance list.has_sizeof (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (List \u03b1) :=\n  { sizeOf := list.sizeof }\n\nprotected def subtype.sizeof {\u03b1 : Type u} [SizeOf \u03b1] {p : \u03b1 \u2192 Prop} : Subtype p \u2192 Nat :=\n  sorry\n\nprotected instance subtype.has_sizeof {\u03b1 : Type u} [SizeOf \u03b1] (p : \u03b1 \u2192 Prop) : SizeOf (Subtype p) :=\n  { sizeOf := subtype.sizeof }\n\ntheorem nat_add_zero (n : Nat) : n + 0 = n :=\n  rfl\n\n/- Combinator calculus -/\n\nnamespace combinator\n\n\ndef I {\u03b1 : Type u\u2081} (a : \u03b1) : \u03b1 :=\n  a\n\ndef K {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} (a : \u03b1) (b : \u03b2) : \u03b1 :=\n  a\n\nend combinator\n\n\ndef combinator.S {\u03b1 : Type u\u2081} {\u03b2 : Type u\u2082} {\u03b3 : Type u\u2083} (x : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (y : \u03b1 \u2192 \u03b2) (z : \u03b1) : \u03b3 :=\n  x z (y z)\n\n/-- Auxiliary datatype for #[ ... ] notation.\n    #[1, 2, 3, 4] is notation for\n\n    bin_tree.node\n      (bin_tree.node (bin_tree.leaf 1) (bin_tree.leaf 2))\n      (bin_tree.node (bin_tree.leaf 3) (bin_tree.leaf 4))\n\n    We use this notation to input long sequences without exhausting the system stack space.\n    Later, we define a coercion from `bin_tree` into `list`.\n-/\ninductive bin_tree (\u03b1 : Type u) \nwhere\n| empty : bin_tree \u03b1\n| leaf : \u03b1 \u2192 bin_tree \u03b1\n| node : bin_tree \u03b1 \u2192 bin_tree \u03b1 \u2192 bin_tree \u03b1\n\n/- Basic unification hints -/\n\ndef add_succ_defeq_succ_add_hint (x : Nat) (y : Nat) (z : Nat) : unification_hint :=\n  unification_hint.mk (x + Nat.succ y =?= Nat.succ z) [z =?= x + y]\n\n/-- Like `by apply_instance`, but not dependent on the tactic framework. -/\ndef infer_instance {\u03b1 : Sort u} [i : \u03b1] : \u03b1 :=\n  i\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/init/core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3703079616555655}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johan Commelin, Andrew Yang\n-/\nimport category_theory.limits.preserves.shapes.zero\nimport category_theory.monoidal.End\nimport category_theory.monoidal.discrete\n\n/-!\n# Shift\n\nA `shift` on a category `C` indexed by a monoid `A` is nothing more than a monoidal functor\nfrom `A` to `C \u2964 C`. A typical example to keep in mind might be the category of\ncomplexes `\u22ef \u2192 C_{n-1} \u2192 C_n \u2192 C_{n+1} \u2192 \u22ef`. It has a shift indexed by `\u2124`, where we assign to\neach `n : \u2124` the functor `C \u2964 C` that re-indexes the terms, so the degree `i` term of `shift n C`\nwould be the degree `i+n`-th term of `C`.\n\n## Main definitions\n* `has_shift`: A typeclass asserting the existence of a shift functor.\n* `shift_equiv`: When the indexing monoid is a group, then the functor indexed by `n` and `-n` forms\n  an self-equivalence of `C`.\n* `shift_comm`: When the indexing monoid is commutative, then shifts commute as well.\n\n## Implementation Notes\n\nMany of the definitions in this file are marked as an `abbreviation` so that the simp lemmas in\n`category_theory/monoidal/End` can apply.\n\n-/\nnamespace category_theory\n\nnoncomputable theory\n\nuniverses v u\n\nvariables (C : Type u) (A : Type*) [category.{v} C]\n\nlocal attribute [instance] endofunctor_monoidal_category\n\nsection eq_to_hom\n\nvariables {A C}\n\nvariables [add_monoid A] (F : monoidal_functor (discrete A) (C \u2964 C))\n\n @[simp, reassoc] lemma eq_to_hom_\u03bc_app {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   eq_to_hom (by rw [h\u2081, h\u2082] : (F.obj \u27e8i\u27e9 \u2297 F.obj \u27e8j\u27e9).obj X =\n       (F.obj \u27e8i'\u27e9 \u2297 F.obj \u27e8j'\u27e9).obj X) \u226b (F.\u03bc \u27e8i'\u27e9 \u27e8j'\u27e9).app X =\n     (F.\u03bc \u27e8i\u27e9 \u27e8j\u27e9).app X \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n @[simp, reassoc] lemma \u03bc_inv_app_eq_to_hom {i j i' j' : A} (h\u2081 : i = i') (h\u2082 : j = j') (X : C) :\n   inv ((F.\u03bc \u27e8i\u27e9 \u27e8j\u27e9).app X) \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n     eq_to_hom (by rw [h\u2081, h\u2082]) \u226b inv ((F.\u03bc \u27e8i'\u27e9 \u27e8j'\u27e9).app X) :=\n by { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\nend eq_to_hom\n\nvariables {A C}\n\n/-- A monoidal functor from a group `A` into `C \u2964 C` induces\na self-equivalence of `C` for each `n : A`. -/\n@[simps functor inverse unit_iso_hom unit_iso_inv counit_iso_hom counit_iso_inv]\ndef add_neg_equiv [add_group A] (F : monoidal_functor (discrete A) (C \u2964 C)) (n : A) : C \u224c C :=\nequiv_of_tensor_iso_unit F \u27e8n\u27e9 \u27e8(-n : A)\u27e9\n  (discrete.eq_to_iso (add_neg_self n)) (discrete.eq_to_iso (neg_add_self n))\n  (subsingleton.elim _ _)\n\nsection defs\n\nvariables (A C) [add_monoid A]\n\n/-- A category has a shift indexed by an additive monoid `A`\nif there is a monoidal functor from `A` to `C \u2964 C`. -/\nclass has_shift (C : Type u) (A : Type*) [category.{v} C] [add_monoid A] :=\n(shift : monoidal_functor (discrete A) (C \u2964 C))\n\n/-- A helper structure to construct the shift functor `(discrete A) \u2964 (C \u2964 C)`. -/\n@[nolint has_inhabited_instance]\nstructure shift_mk_core :=\n(F : A \u2192 (C \u2964 C))\n(\u03b5 : \ud835\udfed C \u2245 F 0)\n(\u03bc : \u03a0 n m : A, F n \u22d9 F m \u2245 F (n + m))\n(associativity : \u2200 (m\u2081 m\u2082 m\u2083 : A) (X : C),\n  (F m\u2083).map ((\u03bc m\u2081 m\u2082).hom.app X) \u226b (\u03bc (m\u2081 + m\u2082) m\u2083).hom.app X \u226b\n    eq_to_hom (by { congr' 2, exact add_assoc _ _ _ }) =\n    (\u03bc m\u2082 m\u2083).hom.app ((F m\u2081).obj X) \u226b (\u03bc m\u2081 (m\u2082 + m\u2083)).hom.app X . obviously)\n(left_unitality : \u2200 (n : A) (X : C),\n  (F n).map (\u03b5.hom.app X) \u226b (\u03bc 0 n).hom.app X =\n    eq_to_hom (by { dsimp, rw zero_add }) . obviously)\n(right_unitality : \u2200 (n : A) (X : C),\n  \u03b5.hom.app ((F n).obj X) \u226b (\u03bc n 0).hom.app X =\n    eq_to_hom (by { dsimp, rw add_zero }) . obviously)\n\nsection\nlocal attribute [simp] eq_to_hom_map\nlocal attribute [reducible] endofunctor_monoidal_category discrete.add_monoidal\n\n/-- Constructs a `has_shift C A` instance from `shift_mk_core`. -/\n@[simps]\ndef has_shift_mk (h : shift_mk_core C A) : has_shift C A :=\n\u27e8{ \u03b5 := h.\u03b5.hom,\n   \u03bc := \u03bb m n, (h.\u03bc m.as n.as).hom,\n   \u03bc_natural' := by { rintros \u27e8X\u27e9 \u27e8Y\u27e9 \u27e8X'\u27e9 \u27e8Y'\u27e9 \u27e8\u27e8\u27e8rfl\u27e9\u27e9\u27e9 \u27e8\u27e8\u27e8rfl\u27e9\u27e9\u27e9, ext,\n     dsimp, simp, dsimp, simp },\n   associativity' := by { introv, ext, dsimp, simpa using h.associativity _ _ _ _, },\n   left_unitality' :=\n    by { rintro \u27e8X\u27e9, ext, dsimp, rw [category.id_comp, \u2190 category.assoc, h.left_unitality], simp },\n   right_unitality' :=\n    by { rintro \u27e8X\u27e9, ext, dsimp, rw [functor.map_id, category.comp_id,\n      \u2190 category.assoc, h.right_unitality], simp },\n ..(discrete.functor h.F) }\u27e9\n\nend\n\nvariables [has_shift C A]\n\n/-- The monoidal functor from `A` to `C \u2964 C` given a `has_shift` instance. -/\ndef shift_monoidal_functor : monoidal_functor (discrete A) (C \u2964 C) := has_shift.shift\n\nvariable {A}\n\n/-- The shift autoequivalence, moving objects and morphisms 'up'. -/\nabbreviation shift_functor (i : A) : C \u2964 C := (shift_monoidal_functor C A).obj \u27e8i\u27e9\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_functor_add (i j : A) :\n  shift_functor C (i + j) \u2245 shift_functor C i \u22d9 shift_functor C j :=\n((shift_monoidal_functor C A).\u03bc_iso \u27e8i\u27e9 \u27e8j\u27e9).symm\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_functor_zero : shift_functor C (0 : A) \u2245 \ud835\udfed C :=\n(shift_monoidal_functor C A).\u03b5_iso.symm\n\n-- Any better notational suggestions?\nnotation X`\u27e6`n`\u27e7`:20 := (shift_functor _ n).obj X\nnotation f`\u27e6`n`\u27e7'`:80 := (shift_functor _ n).map f\n\nend defs\n\nsection add_monoid\n\nvariables {C A} [add_monoid A] [has_shift C A] (X Y : C) (f : X \u27f6 Y)\n\n@[simp] lemma has_shift.shift_obj_obj (n : A) (X : C) : (has_shift.shift.obj \u27e8n\u27e9).obj X = X\u27e6n\u27e7 :=\nrfl\n\n/-- Shifting by `i + j` is the same as shifting by `i` and then shifting by `j`. -/\nabbreviation shift_add (i j : A) : X\u27e6i + j\u27e7 \u2245 X\u27e6i\u27e7\u27e6j\u27e7 := (shift_functor_add C i j).app _\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081 (i i' j : A) (h : i = i') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i' j).hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2082 (i j j' : A) (h : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw h) = eq_to_hom (by rw h) \u226b (shift_add X i j').hom :=\nby { cases h, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma shift_add_hom_comp_eq_to_hom\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  (shift_add X i j).hom \u226b eq_to_hom (by rw [h\u2081, h\u2082]) =\n    eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').hom :=\nby { cases h\u2081, cases h\u2082, rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id] }\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081 (i i' j : A) (h : i = i') :\n  eq_to_hom (by rw h) \u226b (shift_add X i' j).inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2082 (i j j' : A) (h : j = j') :\n  eq_to_hom (by rw h) \u226b (shift_add X i j').inv = (shift_add X i j).inv \u226b eq_to_hom (by rw h) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2082]\n\n@[reassoc] lemma eq_to_hom_comp_shift_add_inv\u2081\u2082 (i j i' j' : A) (h\u2081 : i = i') (h\u2082 : j = j') :\n  eq_to_hom (by rw [h\u2081, h\u2082]) \u226b (shift_add X i' j').inv =\n    (shift_add X i j).inv \u226b eq_to_hom (by rw [h\u2081, h\u2082]) :=\nby rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, shift_add_hom_comp_eq_to_hom\u2081\u2082]\n\nlemma shift_shift' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_add X i j).inv \u226b f\u27e6i + j\u27e7' \u226b (shift_add Y i j).hom :=\nby { symmetry, apply nat_iso.naturality_1 }\n\nvariables (A)\n\n/-- Shifting by zero is the identity functor. -/\nabbreviation shift_zero  :\n  X\u27e60\u27e7 \u2245 X := (shift_functor_zero C A).app _\n\nlemma shift_zero' :\n  f\u27e6(0 : A)\u27e7' = (shift_zero A X).hom \u226b f \u226b (shift_zero A Y).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nend add_monoid\n\nsection add_group\n\nvariables (C) {A} [add_group A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n/-- Shifting by `i` is an equivalence. -/\ninstance (i : A) : is_equivalence (shift_functor C i) :=\nbegin\n  change is_equivalence (add_neg_equiv (shift_monoidal_functor C A) i).functor,\n  apply_instance,\nend\n\n@[simp] lemma shift_functor_inv (i : A) :\n  (shift_functor C i).inv = shift_functor C (-i) :=\nrfl\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_functor_comp_shift_functor_neg (i : A) :\n  shift_functor C i \u22d9 shift_functor C (-i) \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) \u27e8i\u27e9 \u27e8(-i : A)\u27e9\n  (discrete.eq_to_iso (add_neg_self i))\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_functor_neg_comp_shift_functor (i : A) :\n  shift_functor C (-i) \u22d9 shift_functor C i \u2245 \ud835\udfed C :=\nunit_of_tensor_iso_unit (shift_monoidal_functor C A) \u27e8(-i : A)\u27e9 \u27e8i\u27e9\n  (discrete.eq_to_iso (neg_add_self i))\n\nsection\n\nvariables (C)\n\n/-- Shifting by `n` is a faithful functor. -/\ninstance shift_functor_faithful (i : A) : faithful (shift_functor C i) :=\nfaithful.of_comp_iso (shift_functor_comp_shift_functor_neg C i)\n\n/-- Shifting by `n` is a full functor. -/\ninstance shift_functor_full (i : A) : full (shift_functor C i) :=\nbegin\n  haveI : full (shift_functor C i \u22d9 shift_functor C (-i)) :=\n    full.of_iso (shift_functor_comp_shift_functor_neg C i).symm,\n  exact full.of_comp_faithful _ (shift_functor C (-i))\nend\n\n/-- Shifting by `n` is an essentially surjective functor. -/\ninstance shift_functor_ess_surj (i : A) : ess_surj (shift_functor C i) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y\u27e6-i\u27e7, \u27e8(shift_functor_neg_comp_shift_functor C i).app Y\u27e9\u27e9 }\n\nend\n\nvariables {C}\n\n/-- Shifting by `i` and then shifting by `-i` is the identity. -/\nabbreviation shift_shift_neg (i : A) : X\u27e6i\u27e7\u27e6-i\u27e7 \u2245 X :=\n(shift_functor_comp_shift_functor_neg C i).app _\n\n/-- Shifting by `-i` and then shifting by `i` is the identity. -/\nabbreviation shift_neg_shift (i : A) : X\u27e6-i\u27e7\u27e6i\u27e7 \u2245 X :=\n(shift_functor_neg_comp_shift_functor C i).app _\n\nvariables {X Y}\n\nlemma shift_shift_neg' (i : A) :\n  f\u27e6i\u27e7'\u27e6-i\u27e7' = (shift_shift_neg X i).hom \u226b f \u226b (shift_shift_neg Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_neg_shift' (i : A) :\n  f\u27e6-i\u27e7'\u27e6i\u27e7' = (shift_neg_shift X i).hom \u226b f \u226b (shift_neg_shift Y i).inv :=\nby { symmetry, apply nat_iso.naturality_2 }\n\nlemma shift_equiv_triangle (n : A) (X : C) :\n  (shift_shift_neg X n).inv\u27e6n\u27e7' \u226b (shift_neg_shift (X\u27e6n\u27e7) n).hom = \ud835\udfd9 (X\u27e6n\u27e7) :=\n(add_neg_equiv (shift_monoidal_functor C A) n).functor_unit_iso_comp X\n\nsection\nlocal attribute [reducible] discrete.add_monoidal\n\nlemma shift_shift_neg_hom_shift (n : A) (X : C) :\n  (shift_shift_neg X n).hom \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).hom :=\nbegin\n  -- This is just `simp, simp [eq_to_hom_map]`.\n  simp only [iso.app_hom, unit_of_tensor_iso_unit_hom_app, eq_to_iso.hom, functor.map_comp,\n    obj_\u03bc_app, eq_to_iso.inv, obj_\u03b5_inv_app, \u03bc_naturality\u2097_assoc, category.assoc,\n    \u03bc_inv_hom_app_assoc, \u03b5_inv_app_obj, \u03bc_naturality\u1d63_assoc],\n  simp only [eq_to_hom_map, eq_to_hom_app, eq_to_hom_trans],\nend\n\nend\n\nlemma shift_shift_neg_inv_shift (n : A) (X : C) :\n  (shift_shift_neg X n).inv \u27e6n\u27e7' = (shift_neg_shift (X\u27e6n\u27e7) n).inv :=\nby { ext, rw [\u2190 shift_shift_neg_hom_shift, \u2190 functor.map_comp, iso.hom_inv_id, functor.map_id] }\n\n@[simp]\nlemma shift_shift_neg_shift_eq (n : A) (X : C) :\n  (shift_functor C n).map_iso (shift_shift_neg X n) = shift_neg_shift (X\u27e6n\u27e7) n :=\ncategory_theory.iso.ext $ shift_shift_neg_hom_shift _ _\n\nvariables (C)\n\n/-- Shifting by `n` and shifting by `-n` forms an equivalence. -/\n@[simps]\ndef shift_equiv (n : A) : C \u224c C :=\n{ functor := shift_functor C n,\n  inverse := shift_functor C (-n),\n  ..(add_neg_equiv (shift_monoidal_functor C A) n) }\n\nvariable {C}\n\nopen category_theory.limits\n\nvariables [has_zero_morphisms C]\n\nlemma shift_zero_eq_zero (X Y : C) (n : A) : (0 : X \u27f6 Y)\u27e6n\u27e7' = (0 : X\u27e6n\u27e7 \u27f6 Y\u27e6n\u27e7) :=\ncategory_theory.functor.map_zero _ _ _\n\nend add_group\n\nsection add_comm_monoid\n\nvariables {C A} [add_comm_monoid A] [has_shift C A]\nvariables (X Y : C) (f : X \u27f6 Y)\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\ndef shift_comm (i j : A) : X\u27e6i\u27e7\u27e6j\u27e7 \u2245 X\u27e6j\u27e7\u27e6i\u27e7 :=\n(shift_add X i j).symm \u226a\u226b ((shift_monoidal_functor C A).to_functor.map_iso\n  (discrete.eq_to_iso $ add_comm i j : (\u27e8i+j\u27e9 : discrete A) \u2245 \u27e8j+i\u27e9)).app X \u226a\u226b shift_add X j i\n\n@[simp] lemma shift_comm_symm (i j : A) : (shift_comm X i j).symm = shift_comm X j i :=\nbegin\n  ext, dsimp [shift_comm], simpa [eq_to_hom_map],\nend\n\nvariables {X Y}\n\n/-- When shifts are indexed by an additive commutative monoid, then shifts commute. -/\nlemma shift_comm' (i j : A) :\n  f\u27e6i\u27e7'\u27e6j\u27e7' = (shift_comm _ _ _).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' \u226b (shift_comm _ _ _).hom :=\nbegin\n  -- This is just `simp, simp [eq_to_hom_map]`.\n  simp only [shift_comm, iso.trans_hom, iso.symm_hom, iso.app_inv, iso.symm_inv,\n    monoidal_functor.\u03bc_iso_hom, iso.app_hom, functor.map_iso_hom, eq_to_iso.hom, \u03bc_naturality_assoc,\n    nat_trans.naturality_assoc, nat_trans.naturality, functor.comp_map, category.assoc,\n    \u03bc_inv_hom_app_assoc],\n  simp only [eq_to_hom_map, eq_to_hom_app, eq_to_hom_trans_assoc, eq_to_hom_refl, category.id_comp,\n    \u03bc_hom_inv_app_assoc],\nend\n\n@[reassoc] lemma shift_comm_hom_comp (i j : A) :\n  (shift_comm X i j).hom \u226b f\u27e6j\u27e7'\u27e6i\u27e7' = f\u27e6i\u27e7'\u27e6j\u27e7' \u226b (shift_comm Y i j).hom :=\nby rw [shift_comm', \u2190 shift_comm_symm, iso.symm_hom, iso.inv_hom_id_assoc]\n\nend add_comm_monoid\n\nvariables {D : Type*} [category D] [add_monoid A] [has_shift D A]\nvariables (F : C \u2964 D) [full F] [faithful F]\n\nsection\nlocal attribute [reducible] discrete.add_monoidal\n\n/-- Given a family of endomorphisms of `C` which are interwined by a fully faithful `F : C \u2964 D`\nwith shift functors on `D`, we can promote that family to shift functors on `C`. -/\ndef has_shift_of_fully_faithful\n  (s : A \u2192 C \u2964 C) (i : \u2200 i, s i \u22d9 F \u2245 F \u22d9 shift_functor D i) : has_shift C A :=\nhas_shift_mk C A\n{ F := s,\n  \u03b5 := nat_iso_of_comp_fully_faithful F\n    (calc \ud835\udfed C \u22d9 F \u2245 F                                   : functor.left_unitor _\n      ... \u2245 F \u22d9 \ud835\udfed D                                     : (functor.right_unitor _).symm\n      ... \u2245 F \u22d9 shift_functor D (0 : A)                 :\n              iso_whisker_left F (shift_functor_zero D A).symm\n      ... \u2245 s 0 \u22d9 F                                     : (i 0).symm),\n  \u03bc := \u03bb a b, nat_iso_of_comp_fully_faithful F\n    (calc (s a \u22d9 s b) \u22d9 F \u2245 s a \u22d9 s b \u22d9 F             : functor.associator _ _ _\n     ... \u2245 s a \u22d9 F \u22d9 shift_functor D b                 : iso_whisker_left _ (i b)\n     ... \u2245 (s a \u22d9 F) \u22d9 shift_functor D b               : (functor.associator _ _ _).symm\n     ... \u2245 (F \u22d9 shift_functor D a) \u22d9 shift_functor D b : iso_whisker_right (i a) _\n     ... \u2245 F \u22d9 shift_functor D a \u22d9 shift_functor D b   : functor.associator _ _ _\n     ... \u2245 F \u22d9 shift_functor D (a + b)                  :\n             iso_whisker_left _ (shift_functor_add D a b).symm\n     ... \u2245 s (a + b) \u22d9 F                                : (i (a + b)).symm),\n  associativity := begin\n    intros, apply F.map_injective, dsimp,\n    simp only [category.comp_id, category.id_comp, category.assoc,\n      category_theory.functor.map_comp, functor.image_preimage,\n       eq_to_hom_map, iso.inv_hom_id_app_assoc],\n    erw (i m\u2083).hom.naturality_assoc,\n    congr' 1,\n    dsimp,\n    simp only [eq_to_iso.inv, eq_to_hom_app, eq_to_hom_map, obj_\u03bc_app, \u03bc_naturality_assoc,\n      category.assoc, category_theory.functor.map_comp, functor.image_preimage],\n    congr' 3,\n    dsimp,\n    simp only [\u2190(shift_functor D m\u2083).map_comp_assoc, iso.inv_hom_id_app],\n    erw [(shift_functor D m\u2083).map_id, category.id_comp],\n    erw [((shift_monoidal_functor D A).\u03bc_iso \u27e8m\u2081 + m\u2082\u27e9 \u27e8m\u2083\u27e9).inv_hom_id_app_assoc],\n    congr' 1,\n    have := dcongr_arg (\u03bb a, (i a).inv.app X) (add_assoc m\u2081 m\u2082 m\u2083),\n    dsimp at this,\n    simp [this],\n  end,\n  left_unitality := begin\n    intros, apply F.map_injective, dsimp,\n    simp only [category.comp_id, category.id_comp, category.assoc, category_theory.functor.map_comp,\n      eq_to_hom_app, eq_to_hom_map, functor.image_preimage],\n    erw (i n).hom.naturality_assoc,\n    dsimp,\n    simp only [eq_to_iso.inv, eq_to_hom_app, category.assoc, category_theory.functor.map_comp,\n      eq_to_hom_map, obj_\u03b5_app, functor.image_preimage],\n    simp only [\u2190(shift_functor D n).map_comp_assoc, iso.inv_hom_id_app],\n    dsimp,\n    simp only [category.id_comp, \u03bc_inv_hom_app_assoc, category_theory.functor.map_id],\n    have := dcongr_arg (\u03bb a, (i a).inv.app X) (zero_add n),\n    dsimp at this,\n    simp [this],\n  end,\n  right_unitality := begin\n    intros, apply F.map_injective, dsimp,\n    simp only [category.comp_id, category.id_comp, category.assoc,\n      iso.inv_hom_id_app_assoc, eq_to_iso.inv, eq_to_hom_app, eq_to_hom_map,\n      category_theory.functor.map_comp, functor.image_preimage,\n      obj_zero_map_\u03bc_app, \u03b5_hom_inv_app_assoc],\n    have := dcongr_arg (\u03bb a, (i a).inv.app X) (add_zero n),\n    dsimp at this,\n    simp [this],\n  end, }\n\nend\n\n/-- When we construct shifts on a subcategory from shifts on the ambient category,\nthe inclusion functor intertwines the shifts. -/\n@[nolint unused_arguments] -- incorrectly reports that `[full F]` and `[faithful F]` are unused.\ndef has_shift_of_fully_faithful_comm\n  (s : A \u2192 C \u2964 C) (i : \u2200 i, s i \u22d9 F \u2245 F \u22d9 shift_functor D i) (m : A) :\n  begin\n    haveI := has_shift_of_fully_faithful F s i,\n    exact (shift_functor C m) \u22d9 F \u2245 F \u22d9 shift_functor D m\n  end :=\ni m\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.370190792880969}}
{"text": "import for_mathlib.endomorphisms.basic\nimport for_mathlib.derived.les_facts\nimport for_mathlib.additive_functor\nimport for_mathlib.exact_functor\nimport for_mathlib.homotopy_category_op\nimport for_mathlib.has_homology_aux\nimport for_mathlib.hom_single_iso2\nimport for_mathlib.nat_trans\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.limits opposite\nopen bounded_homotopy_category\n\nnamespace homological_complex\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0]\nvariables {\u03b9 : Type*} {c : complex_shape \u03b9}\n\ndef e (X : homological_complex (endomorphisms \ud835\udcd0) c) :\n  End (((endomorphisms.forget \ud835\udcd0).map_homological_complex c).obj X) :=\n{ f := \u03bb i, (X.X i).e,\n  comm' := \u03bb i j hij, (X.d i j).comm }\n\ndef mk_end (X : homological_complex \ud835\udcd0 c) (f : X \u27f6 X) :\n  homological_complex (endomorphisms \ud835\udcd0) c :=\n{ X := \u03bb i, \u27e8X.X i, f.f i\u27e9,\n  d := \u03bb i j, \u27e8X.d i j, f.comm i j\u27e9,\n  shape' := by { intros i j h, ext, apply X.shape i j h },\n  d_comp_d' := by { intros, ext, apply X.d_comp_d } }\n\nend homological_complex\n\nnamespace homotopy_category\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0]\nvariables {\ud835\udcd1 : Type*} [category \ud835\udcd1] [abelian \ud835\udcd1]\nvariables (F : \ud835\udcd0 \u2964 \ud835\udcd1) [functor.additive F]\n\ninstance map_homotopy_category_is_bounded_above\n  (X : homotopy_category \ud835\udcd0 $ complex_shape.up \u2124) [X.is_bounded_above] :\n  ((F.map_homotopy_category _).obj X).is_bounded_above :=\nbegin\n  obtain \u27e8b, hb\u27e9 := is_bounded_above.cond X,\n  exact \u27e8\u27e8b, \u03bb i hi, category_theory.functor.map_is_zero _ (hb i hi)\u27e9\u27e9,\n end\n\nend homotopy_category\n\nnamespace bounded_homotopy_category\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0]\nvariables [has_coproducts_of_shape (ulift.{v} \u2115) \ud835\udcd0]\nvariables [has_products_of_shape (ulift.{v} \u2115) \ud835\udcd0]\n\nvariables (X : bounded_homotopy_category (endomorphisms \ud835\udcd0))\n\n/-- `unEnd` is the \"forget the endomorphism\" map from the category whose objects are complexes\nof pairs `(A\u2071,e\u2071)` with morphisms defined up to homotopy, to the category whose objects are\ncomplexes of objects `A\u2071` with morphisms defined up to homotopy.  -/\ndef unEnd : bounded_homotopy_category \ud835\udcd0 :=\nof $ ((endomorphisms.forget _).map_homotopy_category _).obj X.val\n\ndef e : End X.unEnd := (homotopy_category.quotient _ _).map $ X.val.as.e\n\nend bounded_homotopy_category\n\nnamespace category_theory\n\nsection\nvariables {C : Type*} [category C] {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\nlemma is_iso.comp_right_iff [is_iso g] : is_iso (f \u226b g) \u2194 is_iso f :=\nbegin\n  split; introI h,\n  { have : is_iso ((f \u226b g) \u226b inv g), { apply_instance },\n    simpa only [category.assoc, is_iso.hom_inv_id, category.comp_id] },\n  { apply_instance }\nend\n\nlemma is_iso.comp_left_iff [is_iso f] : is_iso (f \u226b g) \u2194 is_iso g :=\nbegin\n  split; introI h,\n  { have : is_iso (inv f \u226b (f \u226b g)), { apply_instance },\n    simpa only [category.assoc, is_iso.inv_hom_id_assoc] },\n  { apply_instance }\nend\n\nend\n\nnamespace endomorphisms\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0] [enough_projectives \ud835\udcd0]\nvariables [has_coproducts_of_shape (ulift.{v} \u2115) \ud835\udcd0]\nvariables [has_products_of_shape (ulift.{v} \u2115) \ud835\udcd0]\n\ndef mk_bo_ho_ca' (X : cochain_complex \ud835\udcd0 \u2124)\n  [((homotopy_category.quotient \ud835\udcd0 (complex_shape.up \u2124)).obj X).is_bounded_above] (f : X \u27f6 X) :\n  bounded_homotopy_category (endomorphisms \ud835\udcd0) :=\n{ val := { as :=\n  { X := \u03bb i, \u27e8X.X i, f.f i\u27e9,\n    d := \u03bb i j, \u27e8X.d i j, f.comm _ _\u27e9,\n    shape' := \u03bb i j h, by { ext, exact X.shape i j h, },\n    d_comp_d' := \u03bb i j k hij hjk, by { ext, apply homological_complex.d_comp_d } } },\n  bdd := begin\n    obtain \u27e8a, ha\u27e9 := homotopy_category.is_bounded_above.cond ((homotopy_category.quotient \ud835\udcd0 (complex_shape.up \u2124)).obj X),\n    refine \u27e8\u27e8a, \u03bb i hi, _\u27e9\u27e9,\n    rw is_zero_iff_id_eq_zero, ext, dsimp, rw \u2190 is_zero_iff_id_eq_zero,\n    exact ha i hi,\n  end }\n\ndef mk_bo_ho_ca (X : bounded_homotopy_category \ud835\udcd0) (f : X \u27f6 X) :\n  bounded_homotopy_category (endomorphisms \ud835\udcd0) :=\n@mk_bo_ho_ca' _ _ _ _ _ _ X.val.as (by { cases X with X hX, cases X, exact hX }) f.out\n.\n\nlemma quot_out_single_map {X Y : \ud835\udcd0} (f : X \u27f6 Y) (i : \u2124) :\n  ((homotopy_category.single \ud835\udcd0 i).map f).out = (homological_complex.single \ud835\udcd0 _ i).map f :=\nbegin\n  have h := homotopy_category.homotopy_out_map\n    ((homological_complex.single \ud835\udcd0 (complex_shape.up \u2124) i).map f),\n  ext k,\n  erw h.comm k,\n  suffices : (d_next k) h.hom + (prev_d k) h.hom = 0, { rw [this, zero_add] },\n  obtain (hki|rfl) := ne_or_eq k i,\n  { apply limits.is_zero.eq_of_src,\n    show is_zero (ite (k = i) X _), rw [if_neg hki], apply is_zero_zero },\n  { have hk1 : (complex_shape.up \u2124).rel (k-1) k := sub_add_cancel _ _,\n    have hk2 : (complex_shape.up \u2124).rel k (k+1) := rfl,\n    rw [prev_d_eq _ hk1, d_next_eq _ hk2],\n    have aux1 : h.hom (k + 1) k = 0,\n    { apply limits.is_zero.eq_of_src, show is_zero (ite _ X _), rw if_neg, apply is_zero_zero,\n      linarith },\n    have aux2 : h.hom k (k - 1) = 0,\n    { apply limits.is_zero.eq_of_tgt, show is_zero (ite _ Y _), rw if_neg, apply is_zero_zero,\n      linarith },\n    rw [aux1, aux2, comp_zero, zero_comp, add_zero], }\nend\n\ndef mk_bo_ha_ca'_single (X : \ud835\udcd0) (f : X \u27f6 X) :\n  mk_bo_ho_ca' ((homological_complex.single _ _ 0).obj X) (functor.map _ f) \u2245 (single _ 0).obj \u27e8X, f\u27e9 :=\nbounded_homotopy_category.mk_iso\nbegin\n  refine (homotopy_category.quotient _ _).map_iso _,\n  refine homological_complex.hom.iso_of_components _ _,\n  { intro i,\n    refine endomorphisms.mk_iso _ _,\n    { dsimp, split_ifs, { exact iso.refl _ },\n      { refine (is_zero_zero _).iso _, apply endomorphisms.is_zero_X,\n        exact is_zero_zero (endomorphisms \ud835\udcd0), } },\n    { dsimp, split_ifs with hi,\n      { subst i, dsimp, erw [iso.refl_hom], simp only [category.id_comp, category.comp_id],\n        convert rfl, },\n      { apply is_zero.eq_of_src, rw [if_neg hi], exact is_zero_zero _ } } },\n  { rintro i j (rfl : _ = _),\n    by_cases hi : i = 0,\n    { apply is_zero.eq_of_tgt, dsimp, rw [if_neg], exact is_zero_zero _, linarith only [hi] },\n    { apply is_zero.eq_of_src, dsimp, rw [is_zero_iff_id_eq_zero], ext, dsimp, rw [if_neg hi],\n      apply (is_zero_zero _).eq_of_src } }\nend\n\ndef mk_bo_ha_ca_single (X : \ud835\udcd0) (f : X \u27f6 X) :\n  mk_bo_ho_ca ((single _ 0).obj X) ((single _ 0).map f) \u2245 (single _ 0).obj \u27e8X, f\u27e9 :=\nbounded_homotopy_category.mk_iso\nbegin\n  dsimp only [mk_bo_ho_ca, single],\n  refine (homotopy_category.quotient _ _).map_iso _,\n  refine homological_complex.hom.iso_of_components _ _,\n  { intro i,\n    refine endomorphisms.mk_iso _ _,\n    { dsimp, split_ifs, { exact iso.refl _ },\n      { refine (is_zero_zero _).iso _, apply endomorphisms.is_zero_X,\n        exact is_zero_zero (endomorphisms \ud835\udcd0), } },\n    { dsimp, erw quot_out_single_map, dsimp, split_ifs with hi,\n      { subst i, dsimp, erw [iso.refl_hom], simp only [category.id_comp, category.comp_id],\n        convert rfl, },\n      { apply is_zero.eq_of_src, rw [if_neg hi], exact is_zero_zero _ } } },\n  { rintro i j (rfl : _ = _),\n    by_cases hi : i = 0,\n    { apply is_zero.eq_of_tgt, dsimp, rw [if_neg], exact is_zero_zero _, linarith only [hi] },\n    { apply is_zero.eq_of_src, dsimp, rw [is_zero_iff_id_eq_zero], ext, dsimp, rw [if_neg hi],\n      apply (is_zero_zero _).eq_of_src } }\nend\n.\n\ninstance {P Q : bounded_homotopy_category (endomorphisms \ud835\udcd0)} (f : P \u27f6 Q)\n  [homotopy_category.is_quasi_iso f] :\nhomotopy_category.is_quasi_iso (((endomorphisms.forget _).map_bounded_homotopy_category).map f) :=\n(endomorphisms.forget \ud835\udcd0).map_quasi_iso_on_homotopy_category f\n\ninstance forget_preserves_is_quasi_iso {P Q : bounded_homotopy_category \ud835\udcd0} (f : P \u27f6 Q)\n  [hf : homotopy_category.is_quasi_iso f] :\n  homotopy_category.is_quasi_iso ((bounded_homotopy_category.forget \ud835\udcd0).map f) := hf\n\ninstance forget_of_termwise_projective_is_termwise_projective\n  {P : bounded_homotopy_category (endomorphisms \ud835\udcd0)} (k : \u2124)\n   [hP : projective (P.val.as.X k)] :\n  projective (((endomorphisms.forget \ud835\udcd0).map_bounded_homotopy_category.obj P).val.as.X k) :=\nby { dsimp [of], apply_instance, }\n\ndef forget_mk_end (X : chain_complex \ud835\udcd0 \u2115) (f : X \u27f6 X) :\n  (endomorphisms.forget \ud835\udcd0).map_bounded_homotopy_category.obj\n    (chain_complex.to_bounded_homotopy_category.obj (homological_complex.mk_end X f)) \u2245\n  chain_complex.to_bounded_homotopy_category.obj X :=\nbounded_homotopy_category.mk_iso $ (homotopy_category.quotient _ _).map_iso $\nhomological_complex.hom.iso_of_components\n(\u03bb m,\nmatch m with\n| int.of_nat 0 := iso.refl _\n| int.of_nat (i+1) := is_zero.iso (functor.map_is_zero _ $ is_zero_zero _) (is_zero_zero _)\n| -[1+i] := iso.refl _\nend)\nbegin\n  rintros (i|(_|i)) j (rfl : _ = _),\n  { apply (is_zero_zero _).eq_of_tgt, },\n  { erw [iso.refl_hom, iso.refl_hom, category.id_comp, category.comp_id], refl },\n  { erw [iso.refl_hom, iso.refl_hom, category.id_comp, category.comp_id], refl },\nend\n\n\n/-\n\nMathematical summary of the `Ext_is_zero_iff` proof according to kmb's\npossibly flawed understanding:\n\nThe lemma will follow from the following things:\n\n1) If X is a complex in the bounded homotopy category\nand Y is an object, thought of as a `single`\ncomplex, then Ext\u2071(X,Y) is the homology of the complex\n(C\u1d62) whose i'th term is Hom(P\u2071,Y), where P is a projective\nreplacement of X. This applies to both the category \ud835\udcd0\nand to the endomorphism category. The reason is\nthat Ext\u2071(X,Y)=Hom(P,Y\u27e6i\u27e7).\n\n2) For a cleverly chosen choice of P\u2071 (see `exists_K_projective_endomorphism_replacement`)\nwe have a short exact sequence of complexes\n0 -> Hom_{endos}(P\u2071,Y) -> Hom(P\u2071,Y) -> Hom(P\u2071,Y)->0\nwhere the surjection is e(P) - e(Y), with e the endomorphism.\nThis can be checked to be surjective via an explicit construction;\nthe trick is that P\u2071 is going to be `free Q` for some object `Q : \ud835\udcd0`\n\n-/\n\nvariables (Y : \ud835\udcd0) (g : Y \u27f6 Y) (P : bounded_homotopy_category (endomorphisms \ud835\udcd0))\n\ndef C\u2081 (Y : endomorphisms \ud835\udcd0) (P : bounded_homotopy_category (endomorphisms \ud835\udcd0)) :=\n((preadditive_yoneda.obj Y).map_homological_complex _).obj P.val.as.op\n\ndef C\u2082  (Y : \ud835\udcd0) (P : bounded_homotopy_category (endomorphisms \ud835\udcd0)) :=\n((preadditive_yoneda.obj Y).map_homological_complex _).obj P.unEnd.val.as.op\n\ndef map\u2081 : C\u2081 \u27e8Y,g\u27e9 P \u27f6 C\u2082 Y P :=\n{ f := \u03bb i,\n  { to_fun := endomorphisms.hom.f,\n    map_zero' := rfl,\n    map_add' := \u03bb _ _, rfl },\n  comm' := \u03bb i j h, rfl }\n\nopen category_theory.preadditive\n\n@[simps]\ndef map\u2082_left : C\u2082 Y P \u27f6 C\u2082 Y P :=\n((preadditive_yoneda.obj Y).map_homological_complex _).map\n  (homological_complex.op_functor.map (quiver.hom.op P.val.as.e))\n\n@[simps]\ndef map\u2082_right : C\u2082 Y P \u27f6 C\u2082 Y P :=\n(nat_trans.map_homological_complex (preadditive_yoneda.map g) _).app P.unEnd.val.as.op\n\ndef map\u2082 : C\u2082 Y P \u27f6 C\u2082 Y P :=\nmap\u2082_left Y P - map\u2082_right Y g P\n\nlemma map\u2081_mono (n : \u2124) : mono ((map\u2081 Y g P).f n) :=\nbegin\n  rw [AddCommGroup.mono_iff_injective, injective_iff_map_eq_zero],\n  intros \u03c6 h\u03c6, ext, exact h\u03c6\nend\n\nlemma map\u2082_epi {n : \u2124} (h : projective (P.val.as.X n)) : epi ((map\u2082 Y g P).f n) :=\nbegin\n  -- this is Joel Riou's argument, reduce to `free` and do an explicit calculation\n  let F : (endomorphisms \ud835\udcd0)\u1d52\u1d56 \u2964 AddCommGroup :=\n    (endomorphisms.forget \ud835\udcd0).op \u22d9 (preadditive_yoneda.obj Y),\n  let \u03c4 : F \u27f6 F :=\n  { app := \u03bb Q, add_monoid_hom.mk' (\u03bb \u03c8, Q.unop.e \u226b \u03c8 - \u03c8 \u226b g)\n    begin\n      intros a b,\n      simp only [comp_add, add_comp, sub_eq_add_neg, neg_add, add_assoc],\n      congr' 1, apply add_left_comm,\n    end,\n    naturality' := \u03bb Q\u2081 Q\u2082 f, begin\n      ext,\n      simp only [functor.comp_map, functor.op_map, forget_map, comp_apply,\n        add_monoid_hom.mk'_apply, map_sub],\n      congr' 1,\n      { dsimp, apply f.unop.comm_assoc, },\n      { dsimp, rw category.assoc, },\n    end, },\n  suffices : \u2200 (Q : endomorphisms \ud835\udcd0) (hQ : projective Q), epi (\u03c4.app (op Q)),\n  { exact this _ h, },\n  suffices : \u2200 (A : \ud835\udcd0) (hA : projective A), epi (\u03c4.app (op (free A))),\n  { intro Q,\n    introI,\n    haveI := this Q.X (infer_instance),\n    let \u03c0 := free.desc (\ud835\udfd9 (Q.X)),\n    haveI : epi \u03c0.f := begin\n      refine @epi_of_epi _ _ _ _ _ (sigma.\u03b9 _ (ulift.up 0)) _ (id _),\n      simp only [free.desc_f, category.id_comp, colimit.\u03b9_desc, cofan.mk_\u03b9_app, pow_zero,\n        End.one_def],\n      apply_instance,\n    end,\n    haveI : epi \u03c0 := epi_of_epi_f _,\n    let s := projective.factor_thru (\ud835\udfd9 Q) \u03c0,\n    have hs : s \u226b \u03c0 = \ud835\udfd9 Q := projective.factor_thru_comp _ _,\n    constructor,\n    intros Z q\u2081 q\u2082 hq\u2080,\n    have hq\u2081 := congr_arg (\u03bb (q : _ \u27f6 _), (F.map s.op) \u226b q) hq\u2080,\n    simp only at hq\u2081,\n    rw [\u03c4.naturality_assoc, \u03c4.naturality_assoc, cancel_epi (\u03c4.app (op (free Q.X)))] at hq\u2081,\n    simpa only [\u2190 F.map_comp_assoc, \u2190 op_comp, hs, op_id, F.map_id, category.id_comp]\n      using congr_arg (\u03bb (q : _ \u27f6 _), (F.map \u03c0.op) \u226b q) hq\u2081, },\n  introsI A hA,\n  rw AddCommGroup.epi_iff_surjective,\n  intros \u03c6,\n  let \u03c8 : (free A).X \u27f6 Y,\n  { refine sigma.desc _, rintro \u27e8i\u27e9, refine nat.rec_on i 0 (\u03bb j \u03c8, _),\n    refine ((sigma.\u03b9 (\u03bb (i : ulift \u2115), A) (ulift.up j)) \u226b \u03c6) + (\u03c8 \u226b g) },\n  refine \u27e8\u03c8, _\u27e9,\n  ext \u27e8\u27e8i\u27e9\u27e9,\n  dsimp only [\u03c4, add_monoid_hom.mk'_apply, free, unop_op, op_unop, \u03c8],\n  rw [comp_sub, sub_eq_iff_eq_add],\n  simp only [colimit.\u03b9_desc_assoc, colimit.\u03b9_desc, limits.cofan.mk_\u03b9_app],\n  refl,\nend\n\nlemma map\u2081\u2082_exact {n : \u2124} (h : projective (P.val.as.X n)) :\n  exact ((map\u2081 Y g P).f n) ((map\u2082 Y g P).f n) :=\nbegin\n  rw AddCommGroup.exact_iff', split,\n  { rw [map\u2082, homological_complex.sub_f_apply, comp_sub],\n    ext \u03c6,\n    simp only [map\u2081, map\u2082_left_f_apply, map\u2082_right_f_apply, add_monoid_hom.sub_apply, comp_apply,\n      AddCommGroup.zero_apply],\n    erw [endomorphisms.hom.comm], apply sub_self },\n  intros \u03c6 h\u03c6,\n  refine \u27e8\u27e8\u03c6, _\u27e9, _\u27e9,\n  { simpa only [add_monoid_hom.mem_ker, map\u2082, map\u2082_left_f_apply, map\u2082_right_f_apply,\n      homological_complex.sub_f_apply, add_monoid_hom.sub_apply, sub_eq_zero] using h\u03c6, },\n  { refl },\nend\n\nlemma map\u2081\u2082_short_exact {n : \u2124} (h : projective (P.val.as.X n)) :\n  short_exact ((map\u2081 Y g P).f n) ((map\u2082 Y g P).f n) :=\n{ mono := map\u2081_mono _ _ _ _,\n  epi := map\u2082_epi _ _ _ h,\n  exact := map\u2081\u2082_exact _ _ _ h }\n\nlemma homology_is_zero_iff_is_iso (h : \u2200 n, projective (P.val.as.X n)) :\n  (\u2200 i, is_zero ((homology_functor _ _ i).obj (C\u2081 \u27e8Y, g\u27e9 P))) \u2194\n  (\u2200 j, is_iso ((homology_functor _ _ j).map (map\u2082 Y g P))) :=\nbegin\n  have LES := homological_complex.six_term_exact_seq (map\u2081 Y g P) (map\u2082 Y g P)\n    (\u03bb n, map\u2081\u2082_short_exact _ _ _ (h n)),\n  split,\n  { intros H i,\n    obtain \u27e8i, rfl\u27e9 : \u2203 j, j + 1 = i := \u27e8i-1, sub_add_cancel _ _\u27e9,\n    apply ((LES (i+1) i rfl)).is_iso_of_zero_of_zero,\n    { refine (H (i+1)).eq_of_src _ _, },\n    { refine (H i).eq_of_tgt _ _, }, },\n  { intros H i,\n    refine is_zero_of_exact_zero_zero' _ _ ((LES (i+1) i rfl).drop 2).pair _ _,\n    { refine ((LES (i+1) i rfl).drop 1).pair.epi_iff_eq_zero.mp _,\n      exact @is_iso.epi_of_iso _ _ _ _ _ (H _), },\n    { refine ((LES (i+1) i rfl).drop 3).pair.mono_iff_eq_zero.mp _,\n      exact @is_iso.mono_of_iso _ _ _ _ _ (H _), } }\nend\n\n@[reassoc]\nlemma flip_obj_map_comm {C D E : Type*} [category C] [category D] [category E]\n  (F : C \u2964 D \u2964 E) {c : C} {d : D} (f : c \u27f6 c) (g : d \u27f6 d) :\n  (F.flip.obj d).map f \u226b (F.obj c).map g = (F.obj c).map g \u226b (F.flip.obj d).map f :=\n(nat_trans.naturality _ _).symm\n\n@[reassoc]\nlemma flip_map_app_comm {C D E : Type*} [category C] [category D] [category E]\n  (F : C \u2964 D \u2964 E) {c : C} {d : D} (f : c \u27f6 c) (g : d \u27f6 d) :\n  (F.flip.map g).app c \u226b (F.map f).app d = (F.map f).app d \u226b (F.flip.map g).app c  :=\n(nat_trans.naturality _ _).symm\n\nlemma lift_unop_op {X Y : bounded_homotopy_category \ud835\udcd0}\n  [homotopy_category.is_K_projective X.val] (g : X \u27f6 Y) (f : Y \u27f6 Y) :\n  bounded_homotopy_category.lift g Y.\u03c0 \u226b\n    bounded_homotopy_category.lift ((unop (op Y)).\u03c0 \u226b f.op.unop) (unop (op Y)).\u03c0 =\n  lift (g \u226b f) Y.\u03c0 \u226b eq_to_hom (by refl) :=\nbegin\n  erw lift_comp_lift_comp,\n  symmetry,\n  apply category.comp_id,\nend\n\nlemma compatibility\u2081_aux {Y : \ud835\udcd0} {P\u2081 P\u2082 : bounded_homotopy_category \ud835\udcd0} (f : P\u2081 \u27f6 P\u2082) :\n  (homotopy_category.quotient AddCommGroup (complex_shape.up \u2124).symm).map\n    (homological_complex.unop_functor.map\n      (((preadditive_yoneda.obj Y).right_op.map_homological_complex (complex_shape.up \u2124)).map (quot.out f)).op) =\n  (homotopy_category.quotient AddCommGroup (complex_shape.up \u2124).symm).map (\n      -- show ((preadditive_yoneda.obj Y).map_homological_complex _).obj P\u2082.val.as.op \u27f6\n      --     ((preadditive_yoneda.obj Y).map_homological_complex _).obj P\u2081.val.as.op, from\n    ((preadditive_yoneda.obj Y).map_homological_complex (complex_shape.up \u2124).symm).map $\n      -- show homological_complex.op_functor.obj (op P\u2082.val.as) \u27f6\n      --      homological_complex.op_functor.obj (op P\u2081.val.as), from\n           homological_complex.op_functor.map (f.out.op)) :=\nrfl\n\nlemma compatibility\u2081 {Y : \ud835\udcd0} {P\u2081 P\u2082 P\u2083 : bounded_homotopy_category \ud835\udcd0} (g : P\u2081 \u27f6 P\u2083) (f : P\u2082 \u27f6 P\u2083)\n  (h : P\u2081.val.as \u27f6 P\u2082.val.as)\n  (H : (homotopy_category.quotient _ _).map h \u226b f = g)\n  (i : \u2124) :\n  (preadditive_yoneda.obj ((single \ud835\udcd0 i).obj Y)).map f.op \u226b (P\u2082.hom_single_iso Y i).hom \u226b\n    (homology_functor AddCommGroup _ i).map\n      (((preadditive_yoneda.obj Y).map_homological_complex _).map\n        (homological_complex.op_functor.map (quiver.hom.op h)))\n      = (preadditive_yoneda.obj ((single \ud835\udcd0 i).obj Y)).map g.op \u226b\n        (P\u2081.hom_single_iso Y i).hom :=\nbegin\n  rw hom_single_iso_naturality,\n  slice_lhs 1 2 { rw hom_single_iso_naturality, },\n  simp only [category.assoc],\n  congr' 1,\n  dsimp only [functor.comp_map, functor.op],\n  simp only [quiver.hom.unop_op, \u2190 functor.map_comp,\n    homotopy_category.homology_functor_map_factors],\n  congr' 1,\n  simp only [functor.right_op_map, quiver.hom.unop_op, functor.map_comp, compatibility\u2081_aux],\n  simp only [\u2190 functor.map_comp],\n  apply homotopy_category.eq_of_homotopy,\n  apply functor.map_homotopy,\n  rw [\u2190 op_comp],\n  apply homological_complex.op_functor_map_homotopy,\n  apply homotopy_category.homotopy_of_eq,\n  simp only [functor.map_comp, homotopy_category.quotient_map_out],\n  exact H,\nend\n\ninstance preadditive_yoneda_flip_additive :\n  (preadditive_yoneda : \ud835\udcd0 \u2964 _).flip.additive := { }\n\nvariable (\ud835\udcd0)\n\n@[simps]\ndef congr_single_functor {p q : \u2124} (h : p = q) :\n  single \ud835\udcd0 p \u2245 single \ud835\udcd0 q := eq_to_iso (by subst h)\n\nvariable {\ud835\udcd0}\n\n@[reassoc]\nlemma preadditive_yoneda_bifunctor_comm {C : Type*} [category C] [preadditive C]\n  {X\u2081 X\u2082 : C} {Y\u2081 Y\u2082 : C\u1d52\u1d56} (f : X\u2081 \u27f6 X\u2082) (g : Y\u2081 \u27f6 Y\u2082) :\n  (preadditive_yoneda.obj X\u2081).map g \u226b (preadditive_yoneda.map f).app Y\u2082 =\n  ((preadditive_yoneda.map f).app Y\u2081) \u226b (preadditive_yoneda.obj X\u2082).map g:=\nnat_trans.naturality _ _\n\nlemma compatibility\u2082 {Y\u2081 Y\u2082 : \ud835\udcd0} (g : Y\u2081 \u27f6 Y\u2082) {P\u2081 P\u2082 : bounded_homotopy_category \ud835\udcd0} (\u03c0 : P\u2081 \u27f6 P\u2082)\n  (i : \u2124) :\n  (preadditive_yoneda.map (((shift_single_iso 0 (-i)).app Y\u2081).hom \u226b\n    eq_to_hom (show (single \ud835\udcd0 (0 - -i)).obj Y\u2081 = (single \ud835\udcd0 i).obj Y\u2081,\n    by { congr, linarith, }))).app (op P\u2082) \u226b\n    (preadditive_yoneda.obj ((single \ud835\udcd0 i).obj Y\u2081)).map \u03c0.op \u226b\n    (hom_single_iso P\u2081 Y\u2081 i).hom \u226b\n    (homology_functor _ _ i).map ((nat_trans.map_homological_complex\n      (preadditive_yoneda.map g) _).app P\u2081.val.as.op) =\n  (preadditive_yoneda.map ((shift_functor (bounded_homotopy_category \ud835\udcd0) (-i)).map ((single \ud835\udcd0 0).map g))).app (op P\u2082) \u226b\n    (preadditive_yoneda.map (((shift_single_iso 0 (-i)).app Y\u2082).hom \u226b\n    eq_to_hom (show (single \ud835\udcd0 (0 - -i)).obj Y\u2082 = (single \ud835\udcd0 i).obj Y\u2082,\n    by { congr, linarith, }))).app (op P\u2082) \u226b\n    (preadditive_yoneda.obj ((single \ud835\udcd0 i).obj Y\u2082)).map \u03c0.op \u226b\n    (hom_single_iso P\u2081 Y\u2082 i).hom :=\nbegin\n  let \u03c4iso := shift_single_iso 0 (-i) \u226a\u226b congr_single_functor \ud835\udcd0 (show 0 - -i = i, by linarith),\n  let \u03c4 := (shift_single_iso 0 (-i) \u226a\u226b congr_single_functor \ud835\udcd0 (show 0 - -i = i, by linarith)).hom,\n  have eq\u2081 : \u03a0 Y, eq_to_hom (show (single \ud835\udcd0 (0 - -i)).obj Y = (single \ud835\udcd0 i).obj Y,\n    by { congr, linarith, }) = ((congr_single_functor _ (by linarith)).app Y).hom :=\n    \u03bb Y, by simp only [iso.app_hom, congr_single_functor_hom, eq_to_hom_app],\n  have eq\u2082 : shift_single_iso 0 (-i) \u226a\u226b\n    congr_single_functor \ud835\udcd0 (show 0 - -i = i, by linarith) = \u03c4iso := rfl,\n  have eq\u2083 : \u03a0 Y, (\u03c4iso.app Y).hom = \u03c4.app Y := \u03bb Y, rfl,\n  simp only [eq\u2081, \u2190 iso.trans_hom, \u2190 nat_iso.trans_app, eq\u2082, eq\u2083],\n  have eq\u2084 := preadditive_yoneda.congr_map (\u03c4.naturality g),\n  simp only [functor.map_comp] at eq\u2084,\n  have eq\u2085 := nat_trans.congr_app eq\u2084 (op P\u2082),\n  simp only [nat_trans.comp_app] at eq\u2085,\n  slice_rhs 1 2 { erw eq\u2085, },\n  simp only [category.assoc],\n  congr' 1,\n  rw \u2190 preadditive_yoneda_bifunctor_comm_assoc,\n  congr' 1,\n  apply concrete_category.hom_ext,\n  intro x,\n  apply hom_single_iso_naturality_snd_var,\nend\n\n--attribute [reassoc] nat_trans.comp_app\n\nlemma Ext_is_zero_iff (X : chain_complex \ud835\udcd0 \u2115) (Y : \ud835\udcd0)\n  (f : X \u27f6 X) (g : Y \u27f6 Y) :\n  (\u2200 i, is_zero (((Ext i).obj (op $ chain_complex.to_bounded_homotopy_category.obj\n    (X.mk_end f))).obj $ (single _ 0).obj \u27e8Y, g\u27e9)) \u2194\n  (\u2200 i, is_iso $ ((Ext i).map (chain_complex.to_bounded_homotopy_category.map f).op).app _ -\n                 ((Ext i).obj (op _)).map ((single _ 0).map g)) :=\nbegin\n  obtain \u27e8P, _inst, fP, h1, h2\u27e9 := exists_K_projective_replacement\n    (chain_complex.to_bounded_homotopy_category.obj (X.mk_end f)),\n  let fP' := (bounded_homotopy_category.forget _).map\n    (((endomorphisms.forget _).map_bounded_homotopy_category).map fP \u226b (forget_mk_end X f).hom),\n  /- use that fP commutes with the given endomorphisms... -/\n  have fP'_eq : fP' \u226b chain_complex.to_bounded_homotopy_category.map f =\n    (homotopy_category.quotient _ _).map P.val.as.e \u226b fP',\n  { dsimp only [chain_complex.to_bounded_homotopy_category_map, functor.comp_map],\n    erw [\u2190 (homotopy_category.quotient _ _).map_comp],\n    erw [\u2190 (homotopy_category.quotient _ _).map_comp],\n    congr' 1,\n    ext ((_|i)|i),\n    { dsimp only [forget_mk_end, homological_complex.comp_f, endomorphisms.forget,\n        functor.map_homological_complex_map_f, homological_complex.hom.iso_of_components_hom_f],\n      erw [iso.refl_hom, category.comp_id],\n      erw ((quot.out fP).f 0).comm, refl },\n    { apply limits.is_zero.eq_of_tgt, exact is_zero_zero _ },\n    { dsimp only [forget_mk_end, homological_complex.comp_f, endomorphisms.forget,\n        functor.map_homological_complex_map_f, homological_complex.hom.iso_of_components_hom_f],\n      erw [iso.refl_hom, category.comp_id],\n      erw ((quot.out fP).f _).comm, refl }, },\n  resetI,\n  have foo : \u2200 (h : \u2124 \u2192 Prop), (\u2200 i, h i) \u2194 (\u2200 i, h (-i)),\n  { intro h, split,\n    { intros h1 i, apply h1 (-i) },\n    { intros h1 i, specialize h1 (-i), rwa neg_neg at h1, } },\n  convert homology_is_zero_iff_is_iso Y g P h2,\n  { apply propext,\n    rw foo,\n    apply forall_congr,\n    intro i,\n    let := Ext_iso (-i) P\n      (chain_complex.to_bounded_homotopy_category.obj (homological_complex.mk_end X f))\n      ((single (endomorphisms \ud835\udcd0) 0).obj {X := Y, e := g}) fP,\n    rw iso.is_zero_iff this, clear this,\n    delta C\u2081,\n    apply iso.is_zero_iff,\n    let := hom_single_iso P \u27e8Y, g\u27e9 i,\n    refine iso.trans _ this, clear this,\n    let := (shift_single_iso 0 (-i) : single (endomorphisms \ud835\udcd0) 0 \u22d9 _ \u2245 _),\n    change (preadditive_yoneda.flip.obj (op P)).obj _ \u2245\n      (preadditive_yoneda.flip.obj (op P)).obj _,\n    apply (preadditive_yoneda.flip.obj (op P)).map_iso,\n    convert iso.app this \u27e8Y, g\u27e9, -- I \u2665 you Lean, this just worked first time\n    ring, },\n  { apply propext,\n    rw foo,\n    apply forall_congr,\n    intro i,\n    let j : (((Ext (-i)).obj (op (chain_complex.to_bounded_homotopy_category.obj X))).obj ((single \ud835\udcd0 0).obj Y))\n    \u2245 ((homology_functor AddCommGroup (complex_shape.up \u2124).symm i).obj (C\u2082 Y P)),\n    { -- need that post-composing with an iso sends quasi-isos to quasi-isos! More precisely:\n      -- Above I sorried that if fP is a quasi-iso then so is\n      -- ((endomorphisms.forget _).map_bounded_homotopy_category).map fP,\n      -- however unfortunately we now need to post-compose with something\n      -- which is close to, but not equal to, \ud835\udfd9.\n      -- This should hopefully be straightforward\n      haveI : homotopy_category.is_quasi_iso fP',\n      { dsimp only [fP'], rw functor.map_comp, apply_instance, },\n      refine iso.trans (Ext_iso (-i) _ _ ((single \ud835\udcd0 0).obj Y) fP') _,\n--      delta C\u2082,\n      refine iso.trans _ (hom_single_iso ((endomorphisms.forget \ud835\udcd0).map_bounded_homotopy_category.obj P) Y i),\n      let := (shift_single_iso 0 (-i) : single \ud835\udcd0 0 \u22d9 _ \u2245 _),\n      refine (preadditive_yoneda.flip.obj (op ((endomorphisms.forget \ud835\udcd0).map_bounded_homotopy_category.obj P))).map_iso _,\n      let e := this.app Y,\n      refine e \u226a\u226b eq_to_iso _,\n      ring_nf, },\n    -- Goal is `is_iso f : A \u27f6 A` iff `is_iso f' : A' \u27f6 A'` and we have an\n    -- iso `j : A \u27f6 A'` so it suffices to prove that the square\n    -- (with `j` on two sides) commutes.\n    suffices : j.hom \u226b ((homology_functor AddCommGroup (complex_shape.up \u2124).symm i).map (map\u2082 Y g P))\n      = (((Ext (-i)).map (chain_complex.to_bounded_homotopy_category.map f).op).app ((single \ud835\udcd0 0).obj Y) -\n      ((Ext (-i)).obj (op (chain_complex.to_bounded_homotopy_category.obj X))).map ((single \ud835\udcd0 0).map g)) \u226b j.hom,\n    { rw [\u2190 is_iso_iff_is_iso_comp_left j.hom, this, is_iso_iff_is_iso_comp_right], },\n    delta map\u2082,\n    rw [functor.map_sub, comp_sub, sub_comp],\n    refine congr_arg2 _ _ _,\n    { dsimp only [j, iso.trans_hom, Ext_iso, Ext, Ext0, functor.map_iso_hom, functor.comp_map,\n        whiskering_left_obj_map, whisker_left_app, functor.flip_obj_map, functor.flip_map_app,\n        iso.op_hom], clear j,\n      simp only [nat_trans.naturality, nat_trans.naturality_assoc],\n      erw [nat_trans.naturality_assoc, \u2190 functor.map_comp_assoc],\n      simp only [\u2190 op_comp, category.assoc],\n      congr' 1,\n      dsimp only [bounded_homotopy_category.replacement_iso],\n      rw lift_unop_op,\n      apply compatibility\u2081,\n      simp only [eq_to_hom_refl, category.comp_id],\n      erw fP'_eq,\n      apply lift_unique,\n      erw category.assoc,\n      erw bounded_homotopy_category.lift_lifts, },\n    { dsimp only [j, iso.trans_hom, Ext_iso, Ext, Ext0, functor.map_iso_hom, functor.comp_map,\n        whiskering_left_obj_map, whisker_left_app, functor.flip_obj_map, functor.flip_map_app,\n        iso.op_hom, functor.comp_obj, whiskering_left_obj_obj, unop_op, op_unop], clear j,\n      simp only [nat_trans.naturality, nat_trans.naturality_assoc,\n        category.assoc, map\u2082_right],\n      apply compatibility\u2082, } }\nend\n\nopen_locale zero_object\n\ndef single_unEnd (X : endomorphisms \ud835\udcd0) : ((single _ 0).obj X).unEnd \u2245 (single _ 0).obj X.X :=\n{ hom := quot.mk _\n  { f := \u03bb i, show (ite (i = 0) X 0).X \u27f6 ite (i = 0) X.X 0,\n    from if hi : i = 0 then eq_to_hom (by { simp only [if_pos hi] })\n      else 0,\n    comm' := begin\n      rintros i j _,\n      change _ \u226b 0 = 0 \u226b _, simp, end },\n  inv := quot.mk _ {\n    f := \u03bb i, show ite (i = 0) X.X 0 \u27f6 (ite (i = 0) X 0).X,\n    from if hi : i = 0 then eq_to_hom (by { simp only [if_pos hi] })\n      else 0,\n    comm' := begin\n      rintros i j (rfl : _ = _),\n      change _ \u226b 0 = 0 \u226b _, simp, end },\n  hom_inv_id' := begin\n    change quot.mk _ (_ \u226b _) = quot.mk _ _,\n    apply congr_arg,\n    ext i,\n    simp only [homological_complex.comp_f, homological_complex.id_f],\n    split_ifs,\n    { simp },\n    { rw [comp_zero, eq_comm, \u2190 limits.is_zero.iff_id_eq_zero],\n      change is_zero (ite (i = 0) X 0).X,\n      rw if_neg h,\n      apply is_zero_X,\n      apply is_zero_zero,\n    },\n  end,\n  inv_hom_id' := begin\n    change quot.mk _ (_ \u226b _) = quot.mk _ _,\n    apply congr_arg,\n    ext i,\n    simp only [homological_complex.comp_f, homological_complex.id_f],\n    split_ifs,\n    { simp },\n    { rw [comp_zero, eq_comm, \u2190 limits.is_zero.iff_id_eq_zero],\n      change is_zero (ite (i = 0) X.X 0),\n      rw if_neg h,\n      apply is_zero_zero, },\n  end }\n\nlemma single_unEnd_e (X : endomorphisms \ud835\udcd0) :\n  (single_unEnd X).hom \u226b (single _ 0).map X.e = ((single _ 0).obj X).e \u226b (single_unEnd X).hom :=\nbegin\n  change quot.mk _ (_ \u226b _) = quot.mk _ _,\n  apply congr_arg,\n  ext i,\n  change dite _ _ _ \u226b dite _ _ _ = _ \u226b dite _ _ _,\n  split_ifs,\n  { subst h,\n    rw [eq_to_hom_trans_assoc, \u2190 category.assoc],\n    congr',\n    simp,\n    refl, },\n  { simp, },\nend\n\nlemma single_e (X : endomorphisms \ud835\udcd0) :\n  (single_unEnd X).hom \u226b (single _ 0).map X.e \u226b (single_unEnd X).inv = ((single _ 0).obj X).e :=\nby rw [\u2190 category.assoc, iso.comp_inv_eq, single_unEnd_e]\n\nopen category_theory.preadditive\n\ndef embed_single (X : \ud835\udcd0) :\n  (homological_complex.embed complex_shape.embedding.nat_down_int_up).obj\n    ((homological_complex.single \ud835\udcd0 (complex_shape.down \u2115) 0).obj X) \u2245\n  (homological_complex.single \ud835\udcd0 (complex_shape.up \u2124) 0).obj X :=\nhomological_complex.hom.iso_of_components (by rintro ((_|i)|i); exact iso.refl _)\nbegin\n  rintro (i|i) j (rfl : _ = _),\n  { apply is_zero.eq_of_tgt, exact is_zero_zero _ },\n  { apply is_zero.eq_of_src, exact is_zero_zero _ },\nend\n\ndef to_bounded_homotopy_category_single (X : \ud835\udcd0) :\n  chain_complex.to_bounded_homotopy_category.obj ((homological_complex.single _ _ 0).obj X) \u2245\n  (single _ 0).obj X :=\nbounded_homotopy_category.mk_iso $ (homotopy_category.quotient _ _).map_iso $\nembed_single X\n\nlemma to_bounded_homotopy_category_single_naturality (X : \ud835\udcd0) (f : X \u27f6 X) :\n  (to_bounded_homotopy_category_single X).op.hom \u226b\n  (chain_complex.to_bounded_homotopy_category.map\n       ((homological_complex.single \ud835\udcd0 (complex_shape.down \u2115) 0).map f)).op \u226b\n    (to_bounded_homotopy_category_single X).op.inv = ((single _ 0).map f).op :=\nbegin\n  dsimp only [iso.op], simp only [\u2190 op_comp], congr' 1,\n  dsimp only [to_bounded_homotopy_category_single, chain_complex.to_bounded_homotopy_category,\n    bounded_homotopy_category.mk_iso, functor.comp_map, functor.map_iso, single,\n    homotopy_category.single],\n  erw [\u2190 functor.map_comp, \u2190 functor.map_comp], congr' 1,\n  ext ((_|i)|i),\n  { simp only [category.assoc, homological_complex.comp_f, homological_complex.single_map_f,\n      int.of_nat_eq_coe, zmod.nat_cast_self, eq_self_iff_true, eq_to_hom_refl, category.comp_id,\n      category.id_comp, dite_eq_ite, if_true],\n    dsimp only [embed_single,\n      homological_complex.hom.iso_of_components_hom_f, iso.refl_hom,\n      homological_complex.hom.iso_of_components_inv_f, iso.refl_inv],\n    erw [category.comp_id, category.id_comp, homological_complex.embed_nat_obj_down_up_zero_f],\n    dsimp only [homological_complex.single_map_f],\n    simp only [eq_self_iff_true, eq_to_hom_refl, category.comp_id, category.id_comp, dite_eq_ite, if_true], },\n  { apply is_zero.eq_of_src, apply is_zero_zero },\n  { apply is_zero.eq_of_src, apply is_zero_zero },\nend\n\ndef to_bounded_homotopy_category_mk_end_single (X : \ud835\udcd0) (f : X \u27f6 X) :\n  chain_complex.to_bounded_homotopy_category.obj\n    (((homological_complex.single \ud835\udcd0 _ 0).obj X).mk_end\n       ((homological_complex.single \ud835\udcd0 _ 0).map f)) \u2245\n  (single (endomorphisms \ud835\udcd0) 0).obj (\u27e8X,f\u27e9) :=\nbegin\n  refine _ \u226a\u226b to_bounded_homotopy_category_single _,\n  apply functor.map_iso,\n  refine homological_complex.hom.iso_of_components _ _,\n  { rintro (_|i); refine endomorphisms.mk_iso _ _,\n    { exact iso.refl _ },\n    { dsimp [homological_complex.mk_end],\n      simp only [category.id_comp, category.comp_id, if_pos rfl], refl, },\n    { apply (is_zero_zero _).iso, apply is_zero_X, apply is_zero_zero },\n    { apply (is_zero_zero _).eq_of_src }, },\n  { rintro _ i (rfl : _ = _), apply is_zero.eq_of_src, rw is_zero_iff_id_eq_zero, ext, }\nend\n.\n\nlemma Ext'_is_zero_iff (X Y : \ud835\udcd0) (f : X \u27f6 X) (g : Y \u27f6 Y) :\n  (\u2200 i, is_zero (((Ext' i).obj (op $ endomorphisms.mk X f)).obj $ endomorphisms.mk Y g)) \u2194\n  (\u2200 i, is_iso $ ((Ext' i).map f.op).app _ - ((Ext' i).obj _).map g) :=\nbegin\n  convert (Ext_is_zero_iff ((homological_complex.single _ _ 0).obj X) Y (functor.map _ f) g)\n    using 1,\n  { apply propext, apply forall_congr, intro i,\n    apply iso.is_zero_iff, dsimp only [Ext', functor.comp_obj, functor.flip_obj_obj],\n    apply iso.app, apply functor.map_iso, dsimp only [functor.op_obj], apply iso.op,\n    apply to_bounded_homotopy_category_mk_end_single },\n  { apply propext, apply forall_congr, intro i,\n    let e := ((Ext i).map_iso (to_bounded_homotopy_category_single X).op).app ((single _ 0).obj Y),\n    rw [\u2190 is_iso.comp_left_iff e.hom, \u2190 is_iso.comp_right_iff _ e.inv],\n    simp only [comp_sub, sub_comp, iso.app_hom, iso.app_inv, category.assoc,\n      functor.map_iso_hom, functor.map_iso_inv, \u2190 nat_trans.comp_app, \u2190 functor.map_comp,\n      to_bounded_homotopy_category_single_naturality],\n    clear e,\n    dsimp only [Ext', functor.comp_obj, functor.comp_map],\n    congr' 3,\n    rw [nat_trans.naturality, \u2190 nat_trans.comp_app_assoc, \u2190 functor.map_comp, iso.hom_inv_id,\n      functor.map_id, nat_trans.id_app, category.id_comp],\n    refl },\nend\n\nend endomorphisms\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/endomorphisms/Ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7520125737597971, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3701316667174318}}
{"text": "import combinatorics.simple_graph.connectivity data.finset.basic\n\nvariables {V : Type*} {G : simple_graph V}\n\nnamespace finset\n@[simp] lemma singleton_inter_nonempty [decidable_eq V] {a : V} {X : finset V} :\n  ({a} \u2229 X).nonempty \u2194 a \u2208 X :=\n{ mp := not_not.mp \u2218 mt singleton_inter_of_not_mem \u2218 nonempty_iff_ne_empty.mp,\n  mpr := eq.rec (singleton_nonempty a) \u2218 eq.symm \u2218 singleton_inter_of_mem }\nend finset\n\nnamespace simple_graph\nlemma reachable.step {x y : V} : G.adj x y \u2192 reachable G x y :=\n\u03bb h, \u27e8walk.cons h walk.nil\u27e9\nend simple_graph\n", "meta": {"author": "vbeffara", "repo": "lean", "sha": "0004b1d502ac3f4ccd213dbd23589d4c4f9fece8", "save_path": "github-repos/lean/vbeffara-lean", "path": "github-repos/lean/vbeffara-lean/lean-0004b1d502ac3f4ccd213dbd23589d4c4f9fece8/src/graph_theory/to_mathlib.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3700197995459541}}
{"text": "import category_theory.preadditive.additive_functor\nimport algebra.category.Group.preadditive\nimport algebra.category.Group.limits\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.category\n\nnamespace AddCommGroup\n\nopen category_theory.limits\n\nvariables\n\ndef pi' {I : Type v} (X : I \u2192 AddCommGroup.{u}) := (AddCommGroup.of (\u03a0 i, (X i).\u03b1))\n\n@[simps]\ndef cone_pi {I : Type} (X : I \u2192 AddCommGroup.{u}) : fan X := fan.mk (pi' X) (\u03bb i,\n  { to_fun := \u03bb x, x i,\n    map_zero' := by tidy,\n    map_add' := by tidy, })\n\nlemma cone_pi_is_limit {I : Type} (X : I \u2192 AddCommGroup.{u}) : is_limit (cone_pi X) :=\nmk_fan_limit _\n  (\u03bb s,\n    { to_fun := \u03bb x i, s.proj i x,\n      map_zero' := by tidy,\n      map_add' := by tidy}) (by tidy)\n  (\u03bb s m hm, begin\n    ext x i,\n    dsimp,\n    simpa only [\u2190 hm],\n  end)\n\ndef pi_iso_pi' {I : Type} (X : I \u2192 AddCommGroup.{u}) : \u220f X \u2245 pi' X :=\nis_limit.cone_point_unique_up_to_iso (limit.is_limit _) (cone_pi_is_limit X)\n\nvariables {I : Type v} {X Y Z : I \u2192 AddCommGroup.{u}} (f : \u03a0 i, X i \u27f6 Y i) (g : \u03a0 i, Y i \u27f6 Z i)\n\n@[simps]\ndef pi'_map : pi' X \u27f6 pi' Y :=\n{ to_fun := \u03bb x i, f i (x i),\n  map_zero' := by tidy,\n  map_add' := by tidy, }\n\n@[simp, reassoc]\nlemma pi'_map_comp : pi'_map (\u03bb i, f i \u226b g i) = pi'_map f \u226b pi'_map g := rfl\n\n@[simps]\ndef pi'_lift {I : Type} {X : AddCommGroup.{u}} {Y : I \u2192 AddCommGroup.{u}} (\u03c6 : \u03a0 i, X \u27f6 Y i) :\n  X \u27f6 pi' Y :=\n{ to_fun := \u03bb x i, \u03c6 i x,\n  map_zero' := by tidy,\n  map_add' := by tidy, }\n\nend AddCommGroup\n\nnamespace algebra\n\nnamespace homology\n\ndef concrete_exact {X\u2081 X\u2082 X\u2083 : AddCommGroup.{u}} (f\u2081 : X\u2081 \u27f6 X\u2082) (f\u2082 : X\u2082 \u27f6 X\u2083) : Prop :=\n\u2200 (x\u2082 : X\u2082) (h : f\u2082 x\u2082 = 0), \u2203 (x\u2081 : X\u2081), f\u2081 x\u2081 = x\u2082\n\ndef concrete_exact.lift {X\u2081 X\u2082 X\u2083 : AddCommGroup.{u}} {f\u2081 : X\u2081 \u27f6 X\u2082} {f\u2082 : X\u2082 \u27f6 X\u2083}\n  (h : concrete_exact f\u2081 f\u2082) {x\u2082 : X\u2082} (zero : f\u2082 x\u2082 = 0) : X\u2081 :=\n(h x\u2082 zero).some\n\n@[simp]\nlemma concrete_exact.lift_spec {X\u2081 X\u2082 X\u2083 : AddCommGroup.{u}} {f\u2081 : X\u2081 \u27f6 X\u2082} {f\u2082 : X\u2082 \u27f6 X\u2083}\n  (h : concrete_exact f\u2081 f\u2082) {x\u2082 : X\u2082} (zero : f\u2082 x\u2082 = 0) :\n  f\u2081 (h.lift zero) = x\u2082 := (h x\u2082 zero).some_spec\n\nlemma concrete_exact.of_iso {X\u2081 Y\u2081 X\u2082 Y\u2082 X\u2083 Y\u2083 : AddCommGroup.{u}} {g\u2081 : Y\u2081 \u27f6 Y\u2082} {g\u2082 : Y\u2082 \u27f6 Y\u2083}\n  (h : concrete_exact g\u2081 g\u2082) (f\u2081 : X\u2081 \u27f6 X\u2082) (f\u2082 : X\u2082 \u27f6 X\u2083) (e\u2081 : X\u2081 \u2245 Y\u2081) (e\u2082 : X\u2082 \u2245 Y\u2082)\n  (e\u2083 : X\u2083 \u2245 Y\u2083) (comm\u2081 : f\u2081 \u226b e\u2082.hom = e\u2081.hom \u226b g\u2081) (comm\u2082 : f\u2082 \u226b e\u2083.hom = e\u2082.hom \u226b g\u2082) :\n  concrete_exact f\u2081 f\u2082 :=\nbegin\n  intros x\u2082 hx\u2082,\n  have hy\u2082 : g\u2082 (e\u2082.hom x\u2082) = 0,\n  { rw [\u2190 comp_apply, \u2190 comm\u2082, comp_apply, hx\u2082, map_zero], },\n  obtain \u27e8y\u2081, hy\u2081\u27e9 := h _ hy\u2082,\n  refine \u27e8e\u2081.inv y\u2081, _\u27e9,\n  have comm\u2081' : e\u2081.inv \u226b f\u2081 = g\u2081 \u226b e\u2082.inv,\n  { rw [\u2190 cancel_mono e\u2082.hom, \u2190 cancel_epi e\u2081.hom, assoc, assoc, e\u2082.inv_hom_id, comp_id,\n      e\u2081.hom_inv_id_assoc, comm\u2081], },\n  rw [\u2190 comp_apply, comm\u2081', comp_apply, hy\u2081, \u2190 comp_apply, e\u2082.hom_inv_id, id_apply],\nend\n\nlemma injective_iff {X\u2081 X\u2082 : AddCommGroup.{u}} (f : X\u2081 \u27f6 X\u2082) :\n  function.injective f \u2194 \u2200 (x\u2081 : X\u2081) (h : f x\u2081 = 0), x\u2081 = 0 :=\nbegin\n  split,\n  { intros h x\u2081 hx\u2081,\n    apply h,\n    rw [hx\u2081, map_zero], },\n  { intros h x\u2081 x\u2082 hx,\n    rw \u2190 sub_eq_zero,\n    apply h,\n    rw [map_sub, hx, sub_self], },\nend\n\nlemma concrete_exact.pi' {I : Type v} {X\u2081 X\u2082 X\u2083 : I \u2192 AddCommGroup.{u}}\n  (f\u2081 : \u03a0 i, X\u2081 i \u27f6 X\u2082 i) (f\u2082 : \u03a0 i, X\u2082 i \u27f6 X\u2083 i) (h : \u2200 i, concrete_exact (f\u2081 i) (f\u2082 i)) :\n  concrete_exact (AddCommGroup.pi'_map f\u2081) (AddCommGroup.pi'_map f\u2082) :=\nbegin\n  intros x\u2082 hx\u2082,\n  have h : \u2200 (i : I), \u2203 (x\u2081 : X\u2081 i), (f\u2081 i) x\u2081 = x\u2082 i := \u03bb i, h i (x\u2082 i) (congr_fun hx\u2082 i),\n  exact \u27e8\u03bb i, (h i).some, by { ext i, exact (h i).some_spec, }\u27e9,\nend\n\nvariables (A : Type*) [category A] [preadditive A]\n\nstructure five_complex :=\n(X\u2081 X\u2082 X\u2083 X\u2084 X\u2085 : A)\n(f\u2081 : X\u2081 \u27f6 X\u2082)\n(f\u2082 : X\u2082 \u27f6 X\u2083)\n(f\u2083 : X\u2083 \u27f6 X\u2084)\n(f\u2084 : X\u2084 \u27f6 X\u2085)\n(h\u2081\u2082 : f\u2081 \u226b f\u2082 = 0)\n(h\u2082\u2083 : f\u2082 \u226b f\u2083 = 0)\n(h\u2083\u2084 : f\u2083 \u226b f\u2084 = 0)\n\nnamespace five_complex\n\nvariable {A}\n\n@[ext]\nstructure hom (E E' : five_complex A) :=\n(\u03c4\u2081 : E.X\u2081 \u27f6 E'.X\u2081)\n(\u03c4\u2082 : E.X\u2082 \u27f6 E'.X\u2082)\n(\u03c4\u2083 : E.X\u2083 \u27f6 E'.X\u2083)\n(\u03c4\u2084 : E.X\u2084 \u27f6 E'.X\u2084)\n(\u03c4\u2085 : E.X\u2085 \u27f6 E'.X\u2085)\n(comm\u2081 : E.f\u2081 \u226b \u03c4\u2082 = \u03c4\u2081 \u226b E'.f\u2081)\n(comm\u2082 : E.f\u2082 \u226b \u03c4\u2083 = \u03c4\u2082 \u226b E'.f\u2082)\n(comm\u2083 : E.f\u2083 \u226b \u03c4\u2084 = \u03c4\u2083 \u226b E'.f\u2083)\n(comm\u2084 : E.f\u2084 \u226b \u03c4\u2085 = \u03c4\u2084 \u226b E'.f\u2084)\n\n@[simps]\ninstance : category (five_complex A) :=\n{ hom := hom,\n  id := \u03bb E, hom.mk (\ud835\udfd9 _) (\ud835\udfd9 _) (\ud835\udfd9 _) (\ud835\udfd9 _) (\ud835\udfd9 _) (by rw [id_comp,comp_id])\n    (by rw [id_comp,comp_id]) (by rw [id_comp,comp_id]) (by rw [id_comp,comp_id]),\n  comp := \u03bb E E' E'' \u03c6 \u03c6', hom.mk (\u03c6.\u03c4\u2081 \u226b \u03c6'.\u03c4\u2081) (\u03c6.\u03c4\u2082 \u226b \u03c6'.\u03c4\u2082) (\u03c6.\u03c4\u2083 \u226b \u03c6'.\u03c4\u2083)\n    (\u03c6.\u03c4\u2084 \u226b \u03c6'.\u03c4\u2084) (\u03c6.\u03c4\u2085 \u226b \u03c6'.\u03c4\u2085) (by rw [assoc, reassoc_of (\u03c6.comm\u2081), \u03c6'.comm\u2081])\n    (by rw [assoc, reassoc_of (\u03c6.comm\u2082), \u03c6'.comm\u2082])\n    (by rw [assoc, reassoc_of (\u03c6.comm\u2083), \u03c6'.comm\u2083])\n    (by rw [assoc, reassoc_of (\u03c6.comm\u2084), \u03c6'.comm\u2084]), }\n\nvariable (A)\n\n@[simps]\ndef eval\u2081 : five_complex A \u2964 A :=\n{ obj := \u03bb E, E.X\u2081,\n  map := \u03bb E E' \u03c6, \u03c6.\u03c4\u2081, }\n\n@[simps]\ndef eval\u2082 : five_complex A \u2964 A :=\n{ obj := \u03bb E, E.X\u2082,\n  map := \u03bb E E' \u03c6, \u03c6.\u03c4\u2082, }\n\n@[simps]\ndef eval\u2083 : five_complex A \u2964 A :=\n{ obj := \u03bb E, E.X\u2083,\n  map := \u03bb E E' \u03c6, \u03c6.\u03c4\u2083, }\n\n@[simps]\ndef eval\u2084 : five_complex A \u2964 A :=\n{ obj := \u03bb E, E.X\u2084,\n  map := \u03bb E E' \u03c6, \u03c6.\u03c4\u2084, }\n\n@[simps]\ndef eval\u2085 : five_complex A \u2964 A :=\n{ obj := \u03bb E, E.X\u2085,\n  map := \u03bb E E' \u03c6, \u03c6.\u03c4\u2085, }\n\n\nlemma is_iso_of_isos {E E' : five_complex A} (\u03c6 : E \u27f6 E') (h\u2081 : is_iso \u03c6.\u03c4\u2081)\n  (h\u2082 : is_iso \u03c6.\u03c4\u2082) (h\u2083 : is_iso \u03c6.\u03c4\u2083) (h\u2084 : is_iso \u03c6.\u03c4\u2084) (h\u2085 : is_iso \u03c6.\u03c4\u2085) :\n  is_iso \u03c6 :=\nbegin\n  let \u03c8 : E' \u27f6 E :=\n  { \u03c4\u2081 := inv \u03c6.\u03c4\u2081,\n    \u03c4\u2082 := inv \u03c6.\u03c4\u2082,\n    \u03c4\u2083 := inv \u03c6.\u03c4\u2083,\n    \u03c4\u2084 := inv \u03c6.\u03c4\u2084,\n    \u03c4\u2085 := inv \u03c6.\u03c4\u2085,\n    comm\u2081 := by { simp only [\u2190 cancel_mono (\u03c6.\u03c4\u2082), \u03c6.comm\u2081, assoc, is_iso.inv_hom_id, comp_id,\n      is_iso.inv_hom_id_assoc], },\n    comm\u2082 := by { simp only [\u2190 cancel_mono (\u03c6.\u03c4\u2083), \u03c6.comm\u2082, assoc, is_iso.inv_hom_id, comp_id,\n      is_iso.inv_hom_id_assoc], },\n    comm\u2083 := by { simp only [\u2190 cancel_mono (\u03c6.\u03c4\u2084), \u03c6.comm\u2083, assoc, is_iso.inv_hom_id, comp_id,\n      is_iso.inv_hom_id_assoc], },\n    comm\u2084 := by { simp only [\u2190 cancel_mono (\u03c6.\u03c4\u2085), \u03c6.comm\u2084, assoc, is_iso.inv_hom_id, comp_id,\n      is_iso.inv_hom_id_assoc], }, },\n  exact \u27e8\u27e8\u03c8, \u27e8by tidy, by tidy\u27e9\u27e9\u27e9,\nend\n\nvariable {A}\n\nstructure exact (E : five_complex (AddCommGroup.{u})) : Prop :=\n(ex\u2082 : concrete_exact E.f\u2081 E.f\u2082)\n(ex\u2083 : concrete_exact E.f\u2082 E.f\u2083)\n(ex\u2084 : concrete_exact E.f\u2083 E.f\u2084)\n\nlemma concrete_comm\u2081 {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (x\u2081 : E.X\u2081) : \u03c6.\u03c4\u2082 (E.f\u2081 x\u2081) = E'.f\u2081 (\u03c6.\u03c4\u2081 x\u2081) :=\nby simp only [\u2190 comp_apply, \u03c6.comm\u2081]\n\nlemma concrete_comm\u2082 {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (x\u2082 : E.X\u2082) : \u03c6.\u03c4\u2083 (E.f\u2082 x\u2082) = E'.f\u2082 (\u03c6.\u03c4\u2082 x\u2082) :=\nby simp only [\u2190 comp_apply, \u03c6.comm\u2082]\n\nlemma concrete_comm\u2083 {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (x\u2083 : E.X\u2083) : \u03c6.\u03c4\u2084 (E.f\u2083 x\u2083) = E'.f\u2083 (\u03c6.\u03c4\u2083 x\u2083) :=\nby simp only [\u2190 comp_apply, \u03c6.comm\u2083]\n\nlemma concrete_comm\u2084 {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (x\u2084 : E.X\u2084) : \u03c6.\u03c4\u2085 (E.f\u2084 x\u2084) = E'.f\u2084 (\u03c6.\u03c4\u2084 x\u2084) :=\nby simp only [\u2190 comp_apply, \u03c6.comm\u2084]\n\nlemma exact.of_iso {E E' : five_complex AddCommGroup.{u}} (\u03c6 : E \u27f6 E') (h\u03c6 : is_iso \u03c6)\n  (hE' : E'.exact) : E.exact :=\nbegin\n  let e := as_iso \u03c6,\n  constructor,\n  { exact concrete_exact.of_iso hE'.ex\u2082 E.f\u2081 E.f\u2082 ((eval\u2081 _).map_iso e)\n      ((eval\u2082 _).map_iso e) ((eval\u2083 _).map_iso e) \u03c6.comm\u2081 \u03c6.comm\u2082, },\n  { exact concrete_exact.of_iso hE'.ex\u2083 E.f\u2082 E.f\u2083 ((eval\u2082 _).map_iso e)\n      ((eval\u2083 _).map_iso e) ((eval\u2084 _).map_iso e) \u03c6.comm\u2082 \u03c6.comm\u2083, },\n  { exact concrete_exact.of_iso hE'.ex\u2084 E.f\u2083 E.f\u2084 ((eval\u2083 _).map_iso e)\n      ((eval\u2084 _).map_iso e) ((eval\u2085 _).map_iso e) \u03c6.comm\u2083 \u03c6.comm\u2084, },\nend\n\nlemma five_lemma_injective {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (hE : E.exact) (hE' : E'.exact)\n  (h\u2081 : function.surjective \u03c6.\u03c4\u2081)\n  (h\u2082 : function.injective \u03c6.\u03c4\u2082)\n  (h\u2084 : function.injective \u03c6.\u03c4\u2084) :\n  function.injective \u03c6.\u03c4\u2083 :=\nbegin\n  rw injective_iff at \u22a2 h\u2084,\n  intros x\u2083 hx\u2083,\n  have eq\u2081 : E.f\u2083 x\u2083 = 0,\n  { apply h\u2084,\n    rw [concrete_comm\u2083, hx\u2083, map_zero], },\n  let x\u2082 := hE.ex\u2083.lift eq\u2081,\n  have hx\u2082 : E.f\u2082 x\u2082 = x\u2083 := hE.ex\u2083.lift_spec eq\u2081,\n  let x\u2082' := \u03c6.\u03c4\u2082 x\u2082,\n  have eq\u2082 : E'.f\u2082 x\u2082' = 0,\n  { dsimp only [x\u2082'],\n    rw [\u2190 concrete_comm\u2082, concrete_exact.lift_spec, hx\u2083], },\n  let x\u2081' := hE'.ex\u2082.lift eq\u2082,\n  obtain \u27e8x\u2081, hx\u2081\u27e9 := h\u2081 x\u2081',\n  have eq\u2083 : E.f\u2081 x\u2081 = x\u2082,\n  { apply h\u2082,\n    rw [concrete_comm\u2081, hx\u2081, concrete_exact.lift_spec], },\n  rw [\u2190 hx\u2082, \u2190 eq\u2083, \u2190 comp_apply, E.h\u2081\u2082, AddCommGroup.zero_apply],\nend\n\nlemma five_lemma_surjective {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (hE : E.exact) (hE' : E'.exact)\n  (h\u2082 : function.surjective \u03c6.\u03c4\u2082)\n  (h\u2084 : function.surjective \u03c6.\u03c4\u2084)\n  (h\u2085 : function.injective \u03c6.\u03c4\u2085) :\n  function.surjective \u03c6.\u03c4\u2083 :=\nbegin\n  intro x\u2083',\n  obtain \u27e8x\u2084, hx\u2084\u27e9 := h\u2084 (E'.f\u2083 x\u2083'),\n  have eq\u2081 : E.f\u2084 x\u2084 = 0,\n  { apply h\u2085,\n    rw [concrete_comm\u2084, hx\u2084, \u2190 comp_apply, E'.h\u2083\u2084, AddCommGroup.zero_apply, map_zero], },\n  let x\u2083 := hE.ex\u2084.lift eq\u2081,\n  have hx\u2083 : E.f\u2083 x\u2083 = x\u2084 := by simp only [concrete_exact.lift_spec],\n  let \u03b4 := x\u2083' - \u03c6.\u03c4\u2083 x\u2083,\n  have eq\u2082 : E'.f\u2083 \u03b4 = 0,\n  { dsimp only [\u03b4],\n    simp only [map_sub, \u2190 concrete_comm\u2083, hx\u2083, hx\u2084, sub_self], },\n  let \u03b5 := hE'.ex\u2083.lift eq\u2082,\n  have h\u03b5 : E'.f\u2082 \u03b5 = \u03b4 := by simp only [concrete_exact.lift_spec],\n  obtain \u27e8x\u2082, hx\u2082\u27e9 := h\u2082 \u03b5,\n  refine \u27e8x\u2083 + E.f\u2082 x\u2082, _\u27e9,\n  rw [map_add, concrete_comm\u2082, hx\u2082, h\u03b5, add_sub_cancel'_right],\nend\n\nlemma five_lemma_bijective {E E' : five_complex (AddCommGroup.{u})} (\u03c6 : E \u27f6 E')\n  (hE : E.exact) (hE' : E'.exact)\n  (h\u2081 : function.bijective \u03c6.\u03c4\u2081)\n  (h\u2082 : function.bijective \u03c6.\u03c4\u2082)\n  (h\u2084 : function.bijective \u03c6.\u03c4\u2084)\n  (h\u2085 : function.bijective \u03c6.\u03c4\u2085) :\n  function.bijective \u03c6.\u03c4\u2083 :=\n\u27e8five_lemma_injective \u03c6 hE hE' h\u2081.2 h\u2082.1 h\u2084.1, five_lemma_surjective \u03c6 hE hE' h\u2082.2 h\u2084.2 h\u2085.1\u27e9\n\n@[simps]\ndef pi {I : Type} (E : I \u2192 five_complex AddCommGroup.{u}) : five_complex AddCommGroup.{u} :=\n{ X\u2081 := \u220f (\u03bb i, (E i).X\u2081),\n  X\u2082 := \u220f (\u03bb i, (E i).X\u2082),\n  X\u2083 := \u220f (\u03bb i, (E i).X\u2083),\n  X\u2084 := \u220f (\u03bb i, (E i).X\u2084),\n  X\u2085 := \u220f (\u03bb i, (E i).X\u2085),\n  f\u2081 := limits.pi.map (\u03bb i, (E i).f\u2081),\n  f\u2082 := limits.pi.map (\u03bb i, (E i).f\u2082),\n  f\u2083 := limits.pi.map (\u03bb i, (E i).f\u2083),\n  f\u2084 := limits.pi.map (\u03bb i, (E i).f\u2084),\n  h\u2081\u2082 := limits.limit.hom_ext begin\n    rintro \u27e8i\u27e9,\n    simp only [assoc, limits.lim_map_\u03c0, discrete.nat_trans_app, limits.lim_map_\u03c0_assoc,\n      limits.zero_comp, (E i).h\u2081\u2082, limits.comp_zero],\n  end,\n  h\u2082\u2083 := limits.limit.hom_ext begin\n    rintro \u27e8i\u27e9,\n    simp only [assoc, limits.lim_map_\u03c0, discrete.nat_trans_app, limits.lim_map_\u03c0_assoc,\n      limits.zero_comp, (E i).h\u2082\u2083, limits.comp_zero],\n  end,\n  h\u2083\u2084 := limits.limit.hom_ext begin\n    rintro \u27e8i\u27e9,\n    simp only [assoc, limits.lim_map_\u03c0, discrete.nat_trans_app, limits.lim_map_\u03c0_assoc,\n      limits.zero_comp, (E i).h\u2083\u2084, limits.comp_zero],\n  end, }\n\n@[simps]\ndef pi' {I : Type v} (E : I \u2192 five_complex AddCommGroup.{u}) :\n  five_complex AddCommGroup.{max u v} :=\n{ X\u2081 := AddCommGroup.pi'.{v u} (\u03bb i, (E i).X\u2081),\n  X\u2082 := AddCommGroup.pi'.{v u} (\u03bb i, (E i).X\u2082),\n  X\u2083 := AddCommGroup.pi'.{v u} (\u03bb i, (E i).X\u2083),\n  X\u2084 := AddCommGroup.pi'.{v u} (\u03bb i, (E i).X\u2084),\n  X\u2085 := AddCommGroup.pi'.{v u} (\u03bb i, (E i).X\u2085),\n  f\u2081 := AddCommGroup.pi'_map (\u03bb i, (E i).f\u2081),\n  f\u2082 := AddCommGroup.pi'_map (\u03bb i, (E i).f\u2082),\n  f\u2083 := AddCommGroup.pi'_map (\u03bb i, (E i).f\u2083),\n  f\u2084 := AddCommGroup.pi'_map (\u03bb i, (E i).f\u2084),\n  h\u2081\u2082 := by simpa only [\u2190 AddCommGroup.pi'_map_comp, (E _).h\u2081\u2082],\n  h\u2082\u2083 := by simpa only [\u2190 AddCommGroup.pi'_map_comp, (E _).h\u2082\u2083],\n  h\u2083\u2084 := by simpa only [\u2190 AddCommGroup.pi'_map_comp, (E _).h\u2083\u2084], }\n\nlemma pi'_exact {I : Type v} (E : I \u2192 five_complex AddCommGroup.{u})\n  (h : \u2200 i, (E i).exact) : (pi' E).exact :=\n\u27e8concrete_exact.pi' _ _ (\u03bb i, (h i).ex\u2082),\n  concrete_exact.pi' _ _ (\u03bb i, (h i).ex\u2083),\n  concrete_exact.pi' _ _ (\u03bb i, (h i).ex\u2084)\u27e9\n\n@[simps]\ndef pi'_lift {I : Type} {E : five_complex AddCommGroup.{u}}\n  {E' : I \u2192 five_complex AddCommGroup.{u}} (\u03c6 : \u03a0 i, E \u27f6 E' i) :\n  E \u27f6 pi' E' :=\n{ \u03c4\u2081 := AddCommGroup.pi'_lift (\u03bb i, (\u03c6 i).\u03c4\u2081),\n  \u03c4\u2082 := AddCommGroup.pi'_lift (\u03bb i, (\u03c6 i).\u03c4\u2082),\n  \u03c4\u2083 := AddCommGroup.pi'_lift (\u03bb i, (\u03c6 i).\u03c4\u2083),\n  \u03c4\u2084 := AddCommGroup.pi'_lift (\u03bb i, (\u03c6 i).\u03c4\u2084),\n  \u03c4\u2085 := AddCommGroup.pi'_lift (\u03bb i, (\u03c6 i).\u03c4\u2085),\n  comm\u2081 := begin\n    ext x i,\n    simp only [comp_apply, AddCommGroup.pi'_lift_apply, pi'_f\u2081, AddCommGroup.pi'_map_apply],\n    simp only [\u2190 comp_apply, (\u03c6 i).comm\u2081],\n  end,\n  comm\u2082 := begin\n    ext x i,\n    simp only [comp_apply, AddCommGroup.pi'_lift_apply, pi'_f\u2082, AddCommGroup.pi'_map_apply],\n    simp only [\u2190 comp_apply, (\u03c6 i).comm\u2082],\n  end,\n  comm\u2083 := begin\n    ext x i,\n    simp only [comp_apply, AddCommGroup.pi'_lift_apply, pi'_f\u2083, AddCommGroup.pi'_map_apply],\n    simp only [\u2190 comp_apply, (\u03c6 i).comm\u2083],\n  end,\n  comm\u2084 := begin\n    ext x i,\n    simp only [comp_apply, AddCommGroup.pi'_lift_apply, pi'_f\u2084, AddCommGroup.pi'_map_apply],\n    simp only [\u2190 comp_apply, (\u03c6 i).comm\u2084],\n  end, }\n\nend five_complex\n\nend homology\n\nend algebra\n\nnamespace category_theory\n\nnamespace functor\n\nopen algebra.homology\n\n@[simps]\ndef map_five_complex {C D : Type*} [category C] [category D] [preadditive C]\n  [preadditive D] (F : C \u2964 D) [F.additive] :\n  five_complex C \u2964 five_complex D :=\n{ obj := \u03bb E,\n  { X\u2081 := F.obj E.X\u2081,\n    X\u2082 := F.obj E.X\u2082,\n    X\u2083 := F.obj E.X\u2083,\n    X\u2084 := F.obj E.X\u2084,\n    X\u2085 := F.obj E.X\u2085,\n    f\u2081 := F.map E.f\u2081,\n    f\u2082 := F.map E.f\u2082,\n    f\u2083 := F.map E.f\u2083,\n    f\u2084 := F.map E.f\u2084,\n    h\u2081\u2082 := by { rw [\u2190 F.map_comp, E.h\u2081\u2082, F.map_zero], },\n    h\u2082\u2083 := by { rw [\u2190 F.map_comp, E.h\u2082\u2083, F.map_zero], },\n    h\u2083\u2084 := by { rw [\u2190 F.map_comp, E.h\u2083\u2084, F.map_zero], }, },\n  map := \u03bb E E' \u03c6,\n  { \u03c4\u2081 := F.map \u03c6.\u03c4\u2081,\n    \u03c4\u2082 := F.map \u03c6.\u03c4\u2082,\n    \u03c4\u2083 := F.map \u03c6.\u03c4\u2083,\n    \u03c4\u2084 := F.map \u03c6.\u03c4\u2084,\n    \u03c4\u2085 := F.map \u03c6.\u03c4\u2085,\n    comm\u2081 := by simp only [\u2190 F.map_comp, \u03c6.comm\u2081],\n    comm\u2082 := by simp only [\u2190 F.map_comp, \u03c6.comm\u2082],\n    comm\u2083 := by simp only [\u2190 F.map_comp, \u03c6.comm\u2083],\n    comm\u2084 := by simp only [\u2190 F.map_comp, \u03c6.comm\u2084], }, }\n\nend functor\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/basic_five_lemma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3699642290173419}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 \u03b5 > 0, \u2203 n : \u2124, |\u03b1 * n - (\u03b1 * n)%\u2124| < \u03b5 :=\nbegin\n  assume \u03b5 h\u03b5,\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1,\n    have h3 : \u03b1 = (j - i)\u207b\u00b9 * (\u230a(i : \u211d) * \u03b1\u230b - \u230a(j : \u211d) * \u03b1\u230b), from by {\n      rw [\u2190 h2, mul_sub, mul_sub, mul_assoc, mul_comm \u03b1 (j - i), mul_inv_cancel h],\n      ring,\n    },\n    exact h\u03b1 h3,\n  },\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h9 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h10 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h11 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h12 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1,\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw [\u2190 h2, sub_eq_zero],\n      ring,\n    },\n    exact h1 i j h h3,\n  },\n  have h13 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 % 1 \u2260 (j : \u211d) * \u03b1 % 1, from by {\n    assume i j h,\n    assume h2 : (i : \u211d) * \u03b1 % 1 = (j : \u211d\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac (\u03b1 \u2208 \u211a)) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, |y - x| < 1 :=\nbegin\n  assume y h,\n  have h1 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    use (i + 1),\n    split,\n    assume h2 : i = i + 1,\n    have h3 : i * \u03b1 = (i + 1) * \u03b1, from by {rw h2, ring},\n    have h4 : i * \u03b1 \u2208 \u211a, from by {apply exists_rat_btwn h\u03b1 h3},\n    have h5 : i * \u03b1 \u2208 \u2124, from by {apply exists_int_btwn h4},\n    have h6 : i \u2208 \u2124, from by {apply exists_int_btwn h5},\n    have h7 : i + 1 \u2208 \u2124, from by {apply exists_int_btwn h6},\n    have h8 : i + 1 = i, from by {rw \u2190 h7, ring},\n    exact absurd h8 dec_trivial,\n    obviously,\n  },\n  cases h1 0 with j h2,\n  have h3 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h1 i with j h4,\n    use j,\n    split,\n    exact h4.left,\n    have h5 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h4.right},\n    exact h5,\n  },\n  have h4 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h3 i with j h5,\n    use j,\n    split,\n    exact h5.left,\n    have h6 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h5.right},\n    exact h6,\n  },\n  have h5 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h4 i with j h6,\n    use j,\n    split,\n    exact h6.left,\n    have h7 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h6.right},\n    exact h7,\n  },\n  have h6 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h5 i with j h7,\n    use j,\n    split,\n    exact h7.left,\n    have h8 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h7.right},\n    exact h8,\n  },\n  have h7 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h6 i with j h8,\n    use j,\n    split,\n    exact h8.left,\n    have h9 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h8.right},\n    exact h9,\n  },\n  have h8 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h7 i with j h9,\n    use j,\n    split,\n    exact h9.left,\n    have h10 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h9.right},\n    exact h10,\n  },\n  have h9 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h8 i with j h10,\n    use j,\n    split,\n    exact h10.left,\n    have h11 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h10.right},\n    exact h11,\n  },\n  have h10 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h9 i with j h11,\n    use j,\n    split,\n    exact h11.left,\n    have h12 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h11.right},\n    exact h12,\n  },\n  have h11 : \u2200 i : \u2124, \u2203 j : \u2124, i \u2260 j \u2227 (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {\n    assume i : \u2124,\n    cases h10 i with j h12,\n    use j,\n    split,\n    exact h12.left,\n    have h13 : (set.Icc 0 1).Icc (i * \u03b1) (i * \u03b1 + 1) = (set.Icc 0 1).Icc (j * \u03b1) (j * \u03b1 + 1), from by {rw h12.right},\n    exact h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2260 y \u2227 x - y < 1 :=\nbegin\n  assume (y : \u211d) (h1 : y \u2208 Icc 0 1),\n  have h2 : \u2200 n : \u2115, \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n    assume n : \u2115,\n    have h3 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n      have h4 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n        have h5 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n          have h6 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n            have h7 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n              have h8 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                have h9 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                  have h10 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                    have h11 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                      have h12 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                        have h13 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                          have h14 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                            have h15 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                              have h16 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                have h17 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                  have h18 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                    have h19 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                      have h20 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                        have h21 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                          have h22 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                            have h23 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                              have h24 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                have h25 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                  have h26 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                    have h27 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                      have h28 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                        have h29 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                          have h30 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                            have h31 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                              have h32 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                have h33 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                  have h34 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                    have h35 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                      have h36 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                        have h37 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                          have h38 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                            have h39 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                              have h40 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                have h41 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                  have h42 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                    have h43 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                      have h44 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                        have h45 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                          have h46 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                            have h47 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                              have h48 : \u2203 x : \u211d, x \u2208 Icc 0 1 \u2227 x \u2260 y \u2227 x - y < 1, from by {\n                                                                                                have h49 : \u2203 x : \u211d,\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < \u03b5 \u2194 ((x0 - l) < \u03b5) \u2227 ((l - x0) < \u03b5), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro \u27e8 S_1, S_2 \u27e9, \n    split; linarith, \n    rintro \u27e8 S_3, S_4 \u27e9, \n    split; linarith,\n    },\n  \n  assume (h7 : \u03b5 > 0),\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/lean_proof-4_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7956581000631542, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.36990270099532513}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Adam Topaz\n\n! This file was ported from Lean 3 source module category_theory.limits.kan_extension\n! leanprover-community/mathlib commit c9c9fa15fec7ca18e9ec97306fb8764bfe988a7e\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.CategoryTheory.Limits.Shapes.Terminal\nimport Mathlib.CategoryTheory.PUnit\nimport Mathlib.CategoryTheory.StructuredArrow\n\n/-!\n\n# Kan extensions\n\nThis file defines the right and left Kan extensions of a functor.\nThey exist under the assumption that the target category has enough limits\nresp. colimits.\n\nThe main definitions are `Ran \u03b9` and `Lan \u03b9`, where `\u03b9 : S \u2964 L` is a functor.\nNamely, `Ran \u03b9` is the right Kan extension, while `Lan \u03b9` is the left Kan extension,\nboth as functors `(S \u2964 D) \u2964 (L \u2964 D)`.\n\nTo access the right resp. left adjunction associated to these, use `Ran.adjunction`\nresp. `Lan.adjunction`.\n\n# Projects\n\nA lot of boilerplate could be generalized by defining and working with pseudofunctors.\n\n-/\n\n\nnoncomputable section\n\nnamespace CategoryTheory\n\nopen Limits\n\nuniverse v v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nvariable {S : Type u\u2081} {L : Type u\u2082} {D : Type u\u2083}\n\nvariable [Category.{v\u2081} S] [Category.{v\u2082} L] [Category.{v\u2083} D]\n\nvariable (\u03b9 : S \u2964 L)\n\nnamespace Ran\n\nattribute [local simp] StructuredArrow.proj\n\n/-- The diagram indexed by `Ran.index \u03b9 x` used to define `Ran`. -/\nabbrev diagram (F : S \u2964 D) (x : L) : StructuredArrow x \u03b9 \u2964 D :=\n  StructuredArrow.proj x \u03b9 \u22d9 F\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.diagram CategoryTheory.Ran.diagram\n\nvariable {\u03b9}\n\n/-- A cone over `Ran.diagram \u03b9 F x` used to define `Ran`. -/\n@[simp]\ndef cone {F : S \u2964 D} {G : L \u2964 D} (x : L) (f : \u03b9 \u22d9 G \u27f6 F) : Cone (diagram \u03b9 F x)\n    where\n  pt := G.obj x\n  \u03c0 :=\n    { app := fun i => G.map i.hom \u226b f.app i.right\n      naturality := by\n        rintro \u27e8\u27e8il\u27e9, ir, i\u27e9 \u27e8\u27e8jl\u27e9, jr, j\u27e9 \u27e8\u27e8\u27e8fl\u27e9\u27e9, fr, ff\u27e9\n        dsimp at *\n        dsimp at ff\n        simp only [Category.id_comp, Category.assoc] at *\n        rw [ff]\n        have := f.naturality\n        aesop_cat }\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.cone CategoryTheory.Ran.cone\n\nvariable (\u03b9)\n\n/-- An auxiliary definition used to define `Ran`. -/\n@[simps]\ndef loc (F : S \u2964 D) [h : \u2200 x, HasLimit (diagram \u03b9 F x)] : L \u2964 D\n    where\n  obj x := limit (diagram \u03b9 F x)\n  map {X Y} f :=\n    haveI : HasLimit <| StructuredArrow.map f \u22d9 diagram \u03b9 F X := h Y\n    limit.pre (diagram \u03b9 F X) (StructuredArrow.map f)\n  map_id := by\n    intro l\n    haveI : HasLimit (StructuredArrow.map (\ud835\udfd9 _) \u22d9 diagram \u03b9 F l) := h _\n    dsimp\n    ext j\n    simp only [Category.id_comp, limit.pre_\u03c0]\n    congr 1\n    simp\n  map_comp := by\n    intro x y z f g\n    apply limit.hom_ext\n    intro j\n    -- Porting note: The fact that we need to add these instances all over the place\n    -- is certainly not ideal.\n    haveI : HasLimit (StructuredArrow.map f \u22d9 diagram \u03b9 F _) := h _\n    haveI : HasLimit (StructuredArrow.map g \u22d9 diagram \u03b9 F _) := h _\n    haveI : HasLimit (StructuredArrow.map (f \u226b g) \u22d9 diagram \u03b9 F _) := h _\n    haveI : HasLimit (StructuredArrow.map g \u22d9 StructuredArrow.map f \u22d9 diagram \u03b9 F _) := h _\n    haveI : HasLimit ((StructuredArrow.map g \u22d9 StructuredArrow.map f) \u22d9 diagram \u03b9 F _) := h _\n    erw [limit.pre_pre, limit.pre_\u03c0, limit.pre_\u03c0]\n    congr 1\n    aesop_cat\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.loc CategoryTheory.Ran.loc\n\n/-- An auxiliary definition used to define `Ran` and `Ran.adjunction`. -/\n@[simps]\ndef equiv (F : S \u2964 D) [h : \u2200 x, HasLimit (diagram \u03b9 F x)] (G : L \u2964 D) :\n    (G \u27f6 loc \u03b9 F) \u2243 (((whiskeringLeft _ _ _).obj \u03b9).obj G \u27f6 F)\n    where\n  toFun f :=\n    { app := fun x => f.app _ \u226b limit.\u03c0 (diagram \u03b9 F (\u03b9.obj x)) (StructuredArrow.mk (\ud835\udfd9 _))\n      naturality := by\n        intro x y ff\n        dsimp only [whiskeringLeft]\n        simp only [Functor.comp_map, NatTrans.naturality_assoc, loc_map, Category.assoc]\n        congr 1\n        haveI : HasLimit (StructuredArrow.map (\u03b9.map ff) \u22d9 diagram \u03b9 F (\u03b9.obj x)) := h _\n        erw [limit.pre_\u03c0]\n        let t : StructuredArrow.mk (\ud835\udfd9 (\u03b9.obj x)) \u27f6\n          (StructuredArrow.map (\u03b9.map ff)).obj (StructuredArrow.mk (\ud835\udfd9 (\u03b9.obj y))) :=\n          StructuredArrow.homMk ff ?_\n        convert (limit.w (diagram \u03b9 F (\u03b9.obj x)) t).symm using 1\n        simp }\n  invFun f :=\n    { app := fun x => limit.lift (diagram \u03b9 F x) (cone _ f)\n      naturality := by\n        intro x y ff\n        apply limit.hom_ext\n        intros j\n        haveI : HasLimit (StructuredArrow.map ff \u22d9 diagram \u03b9 F x) := h _\n        erw [limit.lift_pre, limit.lift_\u03c0, Category.assoc, limit.lift_\u03c0 (cone _ f) j]\n        simp }\n  left_inv := by\n    intro x\n    ext k\n    apply limit.hom_ext\n    intros j\n    dsimp only [cone]\n    rw [limit.lift_\u03c0]\n    simp only [NatTrans.naturality_assoc, loc_map]\n    haveI : HasLimit (StructuredArrow.map j.hom \u22d9 diagram \u03b9 F k) := h _\n    erw [limit.pre_\u03c0]\n    congr\n    rcases j with \u27e8\u27e8\u27e9, _, _\u27e9\n    aesop_cat\n  right_inv := by aesop_cat\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.equiv CategoryTheory.Ran.equiv\n\nend Ran\n\n/-- The right Kan extension of a functor. -/\n@[simps!]\ndef ran [\u2200 X, HasLimitsOfShape (StructuredArrow X \u03b9) D] : (S \u2964 D) \u2964 L \u2964 D :=\n  Adjunction.rightAdjointOfEquiv (fun F G => (Ran.equiv \u03b9 G F).symm) (by {\n    -- Porting note: was `tidy`\n    intros X' X Y f g\n    ext t\n    apply limit.hom_ext\n    intros j\n    dsimp [Ran.equiv]\n    simp })\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran CategoryTheory.ran\n\nnamespace Ran\n\nvariable (D)\n\n/-- The adjunction associated to `Ran`. -/\ndef adjunction [\u2200 X, HasLimitsOfShape (StructuredArrow X \u03b9) D] :\n    (whiskeringLeft _ _ D).obj \u03b9 \u22a3 ran \u03b9 :=\n  Adjunction.adjunctionOfEquivRight _ _\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.adjunction CategoryTheory.Ran.adjunction\n\ntheorem reflective [Full \u03b9] [Faithful \u03b9] [\u2200 X, HasLimitsOfShape (StructuredArrow X \u03b9) D] :\n    IsIso (adjunction D \u03b9).counit := by\n  suffices : \u2200 (X : S \u2964 D), IsIso (NatTrans.app (adjunction D \u03b9).counit X)\n  \u00b7 apply NatIso.isIso_of_isIso_app\n  intro F\n  suffices : \u2200 (X : S), IsIso (NatTrans.app (NatTrans.app (adjunction D \u03b9).counit F) X)\n  \u00b7 apply NatIso.isIso_of_isIso_app\n  intro X\n  dsimp [adjunction, equiv]\n  simp only [Category.id_comp]\n  exact\n    IsIso.of_iso\n      ((limit.isLimit _).conePointUniqueUpToIso\n        (limitOfDiagramInitial StructuredArrow.mkIdInitial _))\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Ran.reflective CategoryTheory.Ran.reflective\n\nend Ran\n\nnamespace Lan\n\nattribute [local simp] CostructuredArrow.proj\n\n/-- The diagram indexed by `Lan.index \u03b9 x` used to define `Lan`. -/\nabbrev diagram (F : S \u2964 D) (x : L) : CostructuredArrow \u03b9 x \u2964 D :=\n  CostructuredArrow.proj \u03b9 x \u22d9 F\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.diagram CategoryTheory.Lan.diagram\n\nvariable {\u03b9}\n\n/-- A cocone over `Lan.diagram \u03b9 F x` used to define `Lan`. -/\n@[simp]\ndef cocone {F : S \u2964 D} {G : L \u2964 D} (x : L) (f : F \u27f6 \u03b9 \u22d9 G) : Cocone (diagram \u03b9 F x)\n    where\n  pt := G.obj x\n  \u03b9 :=\n    { app := fun i => f.app i.left \u226b G.map i.hom\n      naturality := by\n        rintro \u27e8ir, \u27e8il\u27e9, i\u27e9 \u27e8jl, \u27e8jr\u27e9, j\u27e9 \u27e8fl, \u27e8\u27e8fl\u27e9\u27e9, ff\u27e9\n        dsimp at *\n        simp only [Functor.comp_map, Category.comp_id, NatTrans.naturality_assoc]\n        rw [\u2190 G.map_comp, ff]\n        aesop_cat }\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.cocone CategoryTheory.Lan.cocone\n\nvariable (\u03b9)\n\n/-- An auxiliary definition used to define `Lan`. -/\n@[simps]\ndef loc (F : S \u2964 D) [I : \u2200 x, HasColimit (diagram \u03b9 F x)] : L \u2964 D\n    where\n  obj x := colimit (diagram \u03b9 F x)\n  map {x y} f :=\n    haveI : HasColimit (CostructuredArrow.map f \u22d9 diagram \u03b9 F y) := I _\n    colimit.pre (diagram \u03b9 F y) (CostructuredArrow.map f)\n  map_id := by\n    intro l\n    dsimp\n    haveI : HasColimit (CostructuredArrow.map (\ud835\udfd9 l) \u22d9 diagram \u03b9 F l) := I _\n    ext j\n    erw [colimit.\u03b9_pre, Category.comp_id]\n    congr 1\n    simp\n  map_comp := by\n    intro x y z f g\n    dsimp\n    haveI : HasColimit (CostructuredArrow.map (f \u226b g) \u22d9 diagram \u03b9 F z) := I _\n    ext j\n    let ff : CostructuredArrow \u03b9 _ \u2964 _ := CostructuredArrow.map f\n    let gg : CostructuredArrow \u03b9 _ \u2964 _ := CostructuredArrow.map g\n    let dd := diagram \u03b9 F z\n    -- Porting note: It seems that even Lean3 had some trouble with instances in this case.\n    -- I don't know why lean can't deduce the following three instances...\n    haveI : HasColimit (ff \u22d9 gg \u22d9 dd) := I _\n    haveI : HasColimit ((ff \u22d9 gg) \u22d9 dd) := I _\n    haveI : HasColimit (gg \u22d9 dd) := I _\n    change _ = colimit.\u03b9 ((ff \u22d9 gg) \u22d9 dd) j \u226b _ \u226b _\n    erw [colimit.pre_pre dd gg ff, colimit.\u03b9_pre, colimit.\u03b9_pre]\n    congr 1\n    simp\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.loc CategoryTheory.Lan.loc\n\n/-- An auxiliary definition used to define `Lan` and `Lan.adjunction`. -/\n@[simps]\ndef equiv (F : S \u2964 D) [I : \u2200 x, HasColimit (diagram \u03b9 F x)] (G : L \u2964 D) :\n    (loc \u03b9 F \u27f6 G) \u2243 (F \u27f6 ((whiskeringLeft _ _ _).obj \u03b9).obj G)\n    where\n  toFun f :=\n    { app := fun x => colimit.\u03b9 (diagram \u03b9 F (\u03b9.obj x)) (CostructuredArrow.mk (\ud835\udfd9 _)) \u226b f.app _\n      naturality := by\n        intro x y ff\n        dsimp only [whiskeringLeft]\n        simp only [Functor.comp_map, Category.assoc]\n        rw [\u2190 f.naturality (\u03b9.map ff), \u2190 Category.assoc, \u2190 Category.assoc]\n        let fff : CostructuredArrow \u03b9 _ \u2964 _ := CostructuredArrow.map (\u03b9.map ff)\n        -- same issue :-(\n        haveI : HasColimit (fff \u22d9 diagram \u03b9 F (\u03b9.obj y)) := I _\n        erw [colimit.\u03b9_pre (diagram \u03b9 F (\u03b9.obj y)) fff (CostructuredArrow.mk (\ud835\udfd9 _))]\n        let xx : CostructuredArrow \u03b9 (\u03b9.obj y) := CostructuredArrow.mk (\u03b9.map ff)\n        let yy : CostructuredArrow \u03b9 (\u03b9.obj y) := CostructuredArrow.mk (\ud835\udfd9 _)\n        let fff : xx \u27f6 yy :=\n          CostructuredArrow.homMk ff\n            (by\n              simp only [CostructuredArrow.mk_hom_eq_self]\n              erw [Category.comp_id])\n        erw [colimit.w (diagram \u03b9 F (\u03b9.obj y)) fff]\n        congr\n        simp }\n  invFun f :=\n    { app := fun x => colimit.desc (diagram \u03b9 F x) (cocone _ f)\n      naturality := by\n        intro x y ff\n        apply colimit.hom_ext\n        intros j\n        haveI : HasColimit (CostructuredArrow.map ff \u22d9 diagram \u03b9 F y) := I _\n        erw [colimit.pre_desc, \u2190 Category.assoc, colimit.\u03b9_desc, colimit.\u03b9_desc]\n        simp }\n  left_inv := by\n    intros x\n    dsimp\n    ext k\n    dsimp\n    apply colimit.hom_ext\n    intros j\n    rw [colimit.\u03b9_desc]\n    dsimp only [cocone]\n    rw [Category.assoc, \u2190 x.naturality j.hom, \u2190 Category.assoc]\n    congr 1\n    dsimp [loc]\n    haveI : HasColimit (CostructuredArrow.map j.hom \u22d9 diagram \u03b9 F k) := I _\n    erw [colimit.\u03b9_pre (diagram \u03b9 F k) (CostructuredArrow.map j.hom)]\n    congr\n    rcases j with \u27e8_, \u27e8\u27e9, _\u27e9\n    simp only [CostructuredArrow.map_mk, Category.id_comp]\n    rfl\n  right_inv := by aesop_cat\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.equiv CategoryTheory.Lan.equiv\n\nend Lan\n\n/-- The left Kan extension of a functor. -/\n@[simps!]\ndef lan [\u2200 X, HasColimitsOfShape (CostructuredArrow \u03b9 X) D] : (S \u2964 D) \u2964 L \u2964 D :=\n  Adjunction.leftAdjointOfEquiv (fun F G => Lan.equiv \u03b9 F G) (by {\n    intros X' X Y f g\n    ext\n    simp [Lan.equiv] })\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan CategoryTheory.lan\n\nnamespace Lan\n\nvariable (D)\n\n/-- The adjunction associated to `Lan`. -/\ndef adjunction [\u2200 X, HasColimitsOfShape (CostructuredArrow \u03b9 X) D] :\n    lan \u03b9 \u22a3 (whiskeringLeft _ _ D).obj \u03b9 :=\n  Adjunction.adjunctionOfEquivLeft _ _\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.adjunction CategoryTheory.Lan.adjunction\n\ntheorem coreflective [Full \u03b9] [Faithful \u03b9] [\u2200 X, HasColimitsOfShape (CostructuredArrow \u03b9 X) D] :\n    IsIso (adjunction D \u03b9).unit := by\n  suffices : \u2200 (X : S \u2964 D), IsIso (NatTrans.app (adjunction D \u03b9).unit X)\n  \u00b7 apply NatIso.isIso_of_isIso_app\n  intro F\n  suffices : \u2200 (X : S), IsIso (NatTrans.app (NatTrans.app (adjunction D \u03b9).unit F) X)\n  \u00b7 apply NatIso.isIso_of_isIso_app\n  intro X\n  dsimp [adjunction, equiv]\n  simp only [Category.comp_id]\n  exact\n    IsIso.of_iso\n      ((colimit.isColimit _).coconePointUniqueUpToIso\n          (colimitOfDiagramTerminal CostructuredArrow.mkIdTerminal _)).symm\nset_option linter.uppercaseLean3 false in\n  #align category_theory.Lan.coreflective CategoryTheory.Lan.coreflective\n\nend Lan\n\nend CategoryTheory\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/CategoryTheory/Limits/KanExtension.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3698163297345373}}
{"text": "import for_mathlib.universal_delta_functor.basic\nimport for_mathlib.derived.les3\nimport for_mathlib.derived.les_facts\nimport for_mathlib.derived.Ext_lemmas\n\nopen category_theory\nuniverses v u\nvariables {A : Type u} [category.{v} A] [abelian A]\n\nnoncomputable theory\n\n-- move me\n/-- Get an SES in `A` from a SES in `A\u1d52\u1d56`. -/\ndef short_exact_sequence.unop (S : short_exact_sequence A\u1d52\u1d56) : short_exact_sequence A :=\n{ fst := S.trd.unop,\n  snd := S.snd.unop,\n  trd := S.fst.unop,\n  f := S.g.unop,\n  g := S.f.unop,\n  mono' := infer_instance,\n  epi' := infer_instance,\n  exact' := S.exact'.unop }\n\n-- move me\nlemma short_exact_sequence.short_exact (S : short_exact_sequence A) :\n  short_exact S.f S.g :=\n{ exact := S.exact' }\n\nvariables (A) [enough_projectives A]\n\n/-- `Ext' i (-, B)` as a \u03b4-functor. -/\ndef Ext_\u03b4_functor (B : A) : A\u1d52\u1d56 \u2964\u03b4 Ab.{v} :=\n{ F := \u03bb i, (Ext' i).flip.obj B,\n  additive := infer_instance,\n  \u03b4 := \u03bb n,\n  { app := \u03bb S, by apply\n      Ext'_\u03b4 B (short_exact_sequence.short_exact $ short_exact_sequence.unop S) n,\n    naturality' := begin\n      intros S T f,\n      dsimp only [functor.comp_map],\n      have := Ext'_\u03b4_natural _ _ _ _ ((short_exact_sequence.Trd A\u1d52\u1d56).map f).unop _\n        ((short_exact_sequence.Fst A\u1d52\u1d56).map f).unop _ _ B\n        (short_exact_sequence.short_exact $ short_exact_sequence.unop T)\n        (short_exact_sequence.short_exact $ short_exact_sequence.unop S) n,\n      dsimp only [quiver.hom.op_unop] at this,\n      exact this,\n      exact f.snd.unop,\n      dsimp only [short_exact_sequence.unop, \u2190 unop_comp], rw f.sq2, refl,\n      dsimp only [short_exact_sequence.unop, \u2190 unop_comp], rw \u2190 f.sq1, refl,\n    end },\n  mono := begin\n    intros S,\n    let e := bounded_derived_category.Ext'_zero_flip_iso A B,\n    let t := _, change mono t,\n    have ht : t = (e.hom.app _) \u226b category_theory.functor.map _ S.f \u226b (e.inv.app _),\n    { dsimp [t],\n      erw \u2190 e.hom.naturality_assoc,\n      rw [\u2190 nat_trans.comp_app, iso.hom_inv_id, nat_trans.id_app, category.comp_id],\n      refl },\n    rw ht, clear ht t,\n    apply_with mono_comp { instances := ff }, apply_instance,\n    apply_with mono_comp { instances := ff }, swap, apply_instance,\n    apply concrete_category.mono_of_injective, intros a b h,\n    dsimp at h,\n    rwa \u2190 cancel_epi S.f.unop,\n  end,\n  exact' := begin\n    intros n S,\n    have := (short_exact.Ext'_five_term_exact_seq B\n      (short_exact_sequence.short_exact $ short_exact_sequence.unop S) n).extract 0 2,\n    rw exact_iff_exact_seq, exact this,\n  end,\n  exact_\u03b4 := begin\n    intros n S,\n    have := (short_exact.Ext'_five_term_exact_seq B\n      (short_exact_sequence.short_exact $ short_exact_sequence.unop S) n).extract 1 2,\n    rw exact_iff_exact_seq, exact this,\n  end,\n  \u03b4_exact := begin\n    intros n S,\n    have := (short_exact.Ext'_five_term_exact_seq B\n      (short_exact_sequence.short_exact $ short_exact_sequence.unop S) n).extract 2 3,\n    rw exact_iff_exact_seq, exact this,\n  end }\n\n.\n\ninstance (B : A) : delta_functor.tohoku.effaceable (Ext_\u03b4_functor A B) :=\nbegin\n  constructor, intros X n,\n  let P := projective.over X.unop,\n  let \u03c0 : P \u27f6 X.unop := projective.\u03c0 _,\n  constructor,\n  refine \u27e8_,\u03c0.op,_\u27e9,\n  apply limits.is_zero.eq_of_tgt,\n  apply bounded_derived_category.Ext'_is_zero_of_projective,\n  apply_instance,\n  norm_cast,\n  exact nat.succ_pos n,\nend\n\ninstance (B : A) : delta_functor.universal (Ext_\u03b4_functor A B) :=\ndelta_functor.universal_of_effaceable _\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/universal_delta_functor/Ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7634837635542924, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3698163297345373}}
{"text": "import category_theory.limits.limits\nimport category_theory.limits.shapes\nimport category_theory.yoneda\nimport category_theory.opposites\nimport category_theory.types\nimport category_theory.limits.types\n\nrun_cmd mk_simp_attr `PRODUCT    -----  BOF BOF  \nmeta def PRODUCT_CAT  : tactic unit :=\n`[  try {simp only with PRODUCT}]\nrun_cmd add_interactive [`PRODUCT_CAT]\n\nuniverses v u\nopen category_theory\nopen category_theory.limits\nopen category_theory.category\nopen opposite\nnamespace lem \nvariables {C : Type u}\nvariables [\ud835\udc9e : category.{v} C]\nvariables  [has_binary_products.{v} C][has_terminal.{v} C]\ninclude \ud835\udc9e\nattribute [PRODUCT] category.assoc category.id_comp category.comp_id \n@[PRODUCT] lemma prod_left_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.left = limits.prod.fst := rfl\n@[PRODUCT] lemma prod_right_def {X Y : C} : limit.\u03c0 (pair X Y) walking_pair.right = limits.prod.snd := rfl\nlemma prod.hom_ext {A X Y : C} {a b : A \u27f6 X \u2a2f Y} (h1 : a \u226b limits.prod.fst = b \u226b limits.prod.fst) (h2 : a \u226b limits.prod.snd = b \u226b limits.prod.snd) : a = b :=\nbegin\n  apply limit.hom_ext,\n  rintros (_ | _),\n  rw prod_left_def,\n  exact h1,\n  rw prod_right_def,\n  exact h2,\nend\n@[PRODUCT]lemma prod.lift_fst {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : prod.lift f g \u226b category_theory.limits.prod.fst = f :=\nlimit.lift_\u03c0 (binary_fan.mk f g) _\n\n@[PRODUCT]lemma prod.lift_snd {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : prod.lift f g \u226b category_theory.limits.prod.snd = g :=\nlimit.lift_\u03c0 (binary_fan.mk f g) _\n\nend lem\n\nnotation f ` \u2297 `:20 g :20 := category_theory.limits.prod.map f g  ---- 20 \nnotation  `T`C :20 := (terminal C) \nnotation   `T`X : 20 := (terminal.from X)\nnotation f ` | `:20 g :20 :=  prod.lift f g\nnotation `\u03c01` := limits.prod.fst \nnotation `\u03c02` := limits.prod.snd\n\n\nvariables {C : Type u}\nvariables [\ud835\udc9e : category.{v} C]\nvariables [has_binary_products.{v} C][has_terminal.{v} C]\ninclude \ud835\udc9e\nvariables (X :C)\nopen lem\n/-\n     \u03c0 notation for projection \n-/\n@[PRODUCT]lemma ex_1 {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : ( f | g) \u226b \u03c01 = f  :=   prod.lift_fst f g \n/-\n     we can type \u03c0 : A \u2a2f B \u27f6 B if we need \n-/\n@[PRODUCT]lemma ex_2 {Y A B : C} (f : Y \u27f6 A) (g : Y \u27f6 B) : ( f | g) \u226b (\u03c02 : A \u2a2f B \u27f6 B) = g := prod.lift_snd f g \n\n@[PRODUCT]lemma ex_3 {A X Y : C} {a b : A \u27f6 X \u2a2f Y} (h1 : a \u226b \u03c01  = b \u226b \u03c01 ) (h2 : a \u226b \u03c02  = b \u226b \u03c02)  : a = b :=  prod.hom_ext h1 h2\n\n\n@[PRODUCT]lemma prod.left_composition{Z' Z A B : C}(h : Z' \u27f6 Z)(f : Z \u27f6 A)(g : Z \u27f6 B)  : \n               h \u226b (f | g)  = (h \u226b f | h \u226b g) := \nbegin\n     apply prod.hom_ext,   --- Le right member is of the form ( | )  composition \u03c01 \u03c02 \n     PRODUCT_CAT,  PRODUCT_CAT,  --- here assoc \n     -- rw assoc,\n     -- rw prod.lift_fst,\n     -- rw prod.lift_fst,\n     -- rw prod.lift_snd,\n     -- rw assoc,\n     -- rw prod.lift_snd,\nend\n\n@[PRODUCT]lemma prod.map_first{X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) :  (f \u2297 g) \u226b (\u03c01 : Y \u2a2f W \u27f6 Y) = \u03c01  \u226b f :=  begin \n     exact limit.map_\u03c0 (map_pair f g) walking_pair.left,\nend\n@[PRODUCT]lemma prod.map_second{X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) :  (f \u2297 g) \u226b \u03c02 = \u03c02 \u226b g :=  begin \n     exact limit.map_\u03c0 (map_pair f g) walking_pair.right,\nend\n@[PRODUCT]lemma  prod.otimes_is_prod {X Y Z W : C}(f  : X \u27f6 Y)(g  : Z \u27f6 W) : (f \u2297 g) = ( \u03c01  \u226b f | \u03c02 \u226b g ) := begin\n     apply prod.hom_ext,\n     PRODUCT_CAT, PRODUCT_CAT,\n     -- rw prod.lift_fst,\n     -- rw prod.map_first,\n     -- rw prod.lift_snd,\n     -- rw prod.map_second,\nend\n-- notation \u03c01`(`X `x` Y`)` := (limits.prod.fst : X\u2a2fY \u27f6 X)\n@[PRODUCT]lemma prod.map_ext{X Y Z W : C}(f1 f2  : X \u27f6 Y)(g1 g2  : Z \u27f6 W) :  (f1 \u2297 g1) = (f2 \u2297 g2) \u2192 \n(\u03c01 : X \u2a2f Z \u27f6 X) \u226b f1 = (\u03c01 : X \u2a2f Z \u27f6 X)  \u226b f2 := \u03bb certif, begin \n     iterate 2 {rw prod.otimes_is_prod at certif},\n     rw \u2190 prod.map_first ( f1)  (g1),\n     rw \u2190 prod.map_first ( f2)  (g2),\n     iterate 2 {rw prod.otimes_is_prod},\n     rw certif,\nend\n@[PRODUCT]lemma destruction {X Y Z : C} (f :  Y \u27f6 X) (g : X \u27f6 Z ) : \n     (f | \ud835\udfd9 Y) \u226b (g \u2297 (\ud835\udfd9 Y)) = (f \u226b g | \ud835\udfd9 Y) := begin \n     apply prod.hom_ext,\n     -- PRODUCT_CAT,PRODUCT_CAT,     ---------------------- PROBLEME With the tatict HEEEEEEERRRRRRE \n     rw [prod.lift_fst],\n     rw  assoc, \n     rw prod.map_first,\n     rw \u2190 assoc,               ----- \u2190 assoc here  Problem ? \n     rw prod.lift_fst,          \n     tidy, -- super - power tidy \nend\n\n\n\n\ndef Y (R : C)(A :C) := (yoneda.obj A).obj (op R)\ndef Y_ (R : C) {A B : C}(\u03c6 : A \u27f6 B) := ((yoneda.map \u03c6).app (op R) : Y R A \u27f6 Y R B)\n-- Good notation for yoneda stuff : \n-- We fix V : C and we denote by    \n-- R[X] := yoneda.obj X).obj (op R) and \u03c6 : A  \u27f6 B (in C) R \u27e6  \u03c6 \u27e7   : R[A] \u2192 R[B]  in type v \nnotation R`[`A`]`:20 := Y R A  -- notation ?? \nnotation R`<`\u03c6`>` :20   := Y_ R \u03c6  -- \ndef Yoneda_preserve_product (Y : C)(A B : C) :\n     Y[A \u2a2f B] \u2245 Y[A] \u2a2f Y[B] :=\n{ hom := prod.lift\n    (\u03bb f, f \u226b \u03c01)\n    (\u03bb f, f \u226b \u03c02),\n  inv := \u03bb f : (Y \u27f6 A) \u2a2f (Y \u27f6 B),\n    (prod.lift\n      ((@category_theory.limits.prod.fst _ _ (Y \u27f6 A) (Y \u27f6 B) _ : ((Y \u27f6 A) \u2a2f (Y \u27f6 B)) \u2192 (Y \u27f6 A)) f)\n      ((@category_theory.limits.prod.snd _ _ (Y \u27f6 A) _ _ : ((Y \u27f6 A) \u2a2f (Y \u27f6 B)) \u2192 (Y \u27f6 B)) f : Y \u27f6 B)),\n  hom_inv_id' := begin\n    ext f,\n    cases j,\n    { simp, refl},\n    { simp, refl}\n  end,\n  inv_hom_id' := begin\n    apply lem.prod.hom_ext,\n    { rw assoc, rw lem.prod.lift_fst, obviously},\n    { rw assoc, rw lem.prod.lift_snd, obviously}\n  end\n}\n--- Here it just sugar \n@[PRODUCT]lemma yoneda_sugar.composition (R : C) {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) : R < f \u226b g > =( R< f >) \u226b (R < g >) \n :=  begin \n     unfold Y_, \n     simp,\n end\ndef yoneda_sugar.conv {R : C}{A : C}(g : R[A]) : R \u27f6 A := g \ndef yoneda_sugar.prod (R : C)(A B : C) : R[A \u2a2f B] \u2245 R[A] \u2a2f R[B] := begin \n     exact Yoneda_preserve_product R A B,\nend\n@[PRODUCT]lemma yoneda_sugar.prod.hom (R : C)(A B : C) : \n     (yoneda_sugar.prod R A B).hom =  (R < (limits.prod.fst : A \u2a2f B \u27f6 A) > | R < (limits.prod.snd : A \u2a2f B \u27f6 B)> ) := rfl\n\n@[PRODUCT]lemma yoneda_sugar.prod.first (R : C)(A B : C) :\n (yoneda_sugar.prod R A B).hom \u226b limits.prod.fst = (R < limits.prod.fst >) := \n begin\n     exact rfl,\n end\n @[PRODUCT]lemma yoneda_sugar.prod.hom_inv (R : C)(A B : C) : \n     (yoneda_sugar.prod R A B).hom \u226b (yoneda_sugar.prod R A B).inv = \ud835\udfd9 (R[ A \u2a2f B]) := \n     (Yoneda_preserve_product R A B).hom_inv_id'\n @[PRODUCT]lemma yoneda_sugar.prod.inv_hom (R : C)(A B : C) : \n     (yoneda_sugar.prod R A B).inv \u226b (yoneda_sugar.prod R A B).hom = \ud835\udfd9 ( R [A]  \u2a2f R[B]) := \n     (Yoneda_preserve_product R A B).inv_hom_id'\n @[PRODUCT]lemma yoneda_sugar.prod.second (R : C)(A B : C) : \n  (yoneda_sugar.prod R A B).hom \u226b limits.prod.snd = (R < limits.prod.snd >) := rfl\n\n@[PRODUCT]lemma yoneda_sugar.id (R : C)(A : C) : R < \ud835\udfd9 A > = \ud835\udfd9 ( R [A] ) := begin \n     funext,\n     exact comp_id C g,\n     -- have T : ((yoneda.map (\ud835\udfd9 A)).app (op R)) g = (g \u226b (\ud835\udfd9 A)),\nend \nlemma yoneda_sugar_prod (R : C)(A B : C)(X :C)(f : X \u27f6 A)(g : X \u27f6 B) :\n      R < (f | g) > \u226b (yoneda_sugar.prod R A B).hom  =  (R < f > | R < g > ) :=  -- the  \u226b  is  :/   \n     begin \n          PRODUCT_CAT,\n          -- rw  yoneda_sugar.prod.hom R A B,\n          -- rw prod.left_composition,\n          iterate 2 {rw \u2190 yoneda_sugar.composition},   -- rw \u2190 is the problem ? \n          rw lem.prod.lift_fst,\n          rw lem.prod.lift_snd,\n     end\n@[PRODUCT]lemma yoneda_sugar_prod_inv (R : C)(A B : C)(X :C)(f : X \u27f6 A)(g : X \u27f6 B) : \n     R < (f | g) >   =  (R < f > | R < g > ) \u226b (yoneda_sugar.prod R A B).inv :=\n     begin \n          PRODUCT_CAT,  -- noting\n          rw \u2190 yoneda_sugar_prod,\n          rw assoc,\n          rw yoneda_sugar.prod.hom_inv,\n          exact rfl,\n     end \nlemma  yoneda_sugar.otimes (R : C){Y Z K :C}(f : X \u27f6 Y )(g : Z \u27f6 K) : \n ( R < (f \u2297 g) > ) = (yoneda_sugar.prod  _ _ _).hom \u226b ((R<f>) \u2297 R<g>) \u226b (yoneda_sugar.prod _ _ _ ).inv := begin \n     PRODUCT_CAT,\n     -- iterate 2 {rw prod.otimes_is_prod},\n     -- rw  yoneda_sugar.prod.hom,\n     -- iterate 1 {rw yoneda_sugar_prod_inv},\n     rw \u2190 assoc,\n     rw prod.left_composition,\n     rw \u2190 assoc,\n     rw prod.lift_fst,\n     rw \u2190 assoc,\n     rw prod.lift_snd,\n     -- rw yoneda_sugar.composition,\n     -- rw yoneda_sugar.composition,\nend\n@[PRODUCT]lemma yonega_sugar.one_otimes (R :C)(X Y Z: C) (f : X \u27f6 Y) : \n (((yoneda_sugar.prod R Z X).inv) \u226b (R <(\ud835\udfd9 Z \u2297 f ) > ) \u226b (yoneda_sugar.prod R Z Y).hom) = (\ud835\udfd9 (R[Z]) \u2297 R < f >) := begin\n     rw yoneda_sugar.otimes,\n     iterate 3 {rw \u2190 assoc},\n     rw yoneda_sugar.prod.inv_hom,\n     rw id_comp,\n     rw assoc,\n     rw yoneda_sugar.prod.inv_hom,\n     rw \u2190 yoneda_sugar.id,\n     simp, \n end\nlemma yonega_sugar.one_otimes' (R :C)(X Y Z: C) (f : X \u27f6 Y) : \n ( (R <(\ud835\udfd9 Z \u2297 f ) > ) \u226b (yoneda_sugar.prod R Z Y).hom) = ((yoneda_sugar.prod R Z X).hom) \u226b (\ud835\udfd9 (R[Z]) \u2297 R < f >) := begin\n     iterate 2{ rw yoneda_sugar.prod.hom},\n     rw prod.left_composition,\n     iterate 2{ rw \u2190 yoneda_sugar.composition},\n     rw prod.map_first,\n     rw prod.map_second,\n     rw comp_id,\n     rw prod.otimes_is_prod,rw prod.left_composition,rw \u2190 assoc, \n     rw prod.lift_fst,rw \u2190  assoc,rw prod.lift_snd,rw comp_id,\n     rw yoneda_sugar.composition,\n end\n\n\n", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/geo/src/group_objet/group_obj_def.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3697152464892773}}
{"text": "import .core\n\nnamespace tts ------------------------------------------------------------------\nnamespace sch ------------------------------------------------------------------\nvariables {V : Type} [_root_.decidable_eq V] -- Type of variable names\nvariables {ts : list (typ V)} -- Lists of types\nvariables {s : sch V} -- Type schemes\n\n-- A locally-closed schema opened with a list of types is locally-closed if all\n-- types are locally-closed.\ntheorem lc_open_typs\n  (ls : lc s)\n  (ln_s_eq_ts : s.arity = ts.length)\n  (lts : \u2200 t \u2208 ts, typ.lc t) :\n  typ.lc (open_typs ts s) :=\nbegin\n  cases ls with L ls,\n  let L' := fv s \u222a typ.fv_list ts \u222a L,\n  let FL' := ((freshL.tagged s.vars_nodup).pgen L').1,\n  have ndL' : FL'.nodup := (freshL.tagged s.vars_nodup).pgen_nodup L',\n  have ln_FL'_eq_s : FL'.length = s.arity :=\n    freshL.tagged_length_eq L' s.vars_nodup,\n  have nm_s_ts : \u2200 x \u2208 FL', x \u2209 typ.fv s.type \u222a typ.fv_list ts :=\n    \u03bb _ h, ((freshL.tagged s.vars_nodup).pgen_mem_union h).1,\n  have nm_L : \u2200 x \u2208 FL', x \u2209 L :=\n    \u03bb _ h, ((freshL.tagged s.vars_nodup).pgen_mem_union h).2,\n  have ln_FL'_eq_ts : FL'.length = ts.length := ln_FL'_eq_s.trans ln_s_eq_ts,\n  rw [open_typs, typ.subst_list_intro ndL' ln_FL'_eq_ts nm_s_ts lts],\n  exact typ.subst_list_lc ln_FL'_eq_ts lts (ls ndL' ln_FL'_eq_s nm_L)\nend\n\nend /- namespace -/ sch --------------------------------------------------------\nend /- namespace -/ tts --------------------------------------------------------\n", "meta": {"author": "spl", "repo": "tts", "sha": "b65298fea68ce47c8ed3ba3dbce71c1a20dd3481", "save_path": "github-repos/lean/spl-tts", "path": "github-repos/lean/spl-tts/tts-b65298fea68ce47c8ed3ba3dbce71c1a20dd3481/src/sch/open.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3697152395686251}}
{"text": "namespace Nat\n\ntheorem example1 (x y z : Nat) : x * y + z = x * y + z :=\n  rfl\n\ntheorem example2 (x y : Nat) (h : y = x + 7) : 2 * y = 2 * (x + 7) :=\n  rw\n\nend Nat\n\n\n", "meta": {"author": "mothematician", "repo": "lean-tutorial-imperial", "sha": "1e68a6341616450c7f5d3e2b35a56fc95bdfced4", "save_path": "github-repos/lean/mothematician-lean-tutorial-imperial", "path": "github-repos/lean/mothematician-lean-tutorial-imperial/lean-tutorial-imperial-1e68a6341616450c7f5d3e2b35a56fc95bdfced4/tutorial world/l1_refl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.36969958573526934}}
{"text": "def twice : Nat \u2192 Nat := \u03bb n => 2*n\n\ndef foo1 : (\u03bb x : Nat => id (twice (id x))) = twice := by\n  conv in (id _) =>\n    trace_state\n    conv =>\n      enter [1,1]\n      trace_state\n      simp\n      trace_state\n    trace_state  -- `id (twice x)`\n\n\ntheorem foo2 (y : Nat) : (fun x => x + y = 0) = (fun x => False) := by\n  conv =>\n    trace_state\n    conv =>\n      lhs\n      trace_state\n      intro x\n      rw [Nat.add_comm]\n      trace_state\n    trace_state\n  trace_state\n  sorry\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/convInConv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754489059775, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.36967456985843516}}
{"text": "abbrev M := ExceptT String $ StateT Nat Id\n\ndef add (n : Nat) : M Unit :=\n  modify (\u00b7 + 0)\n\n@[simp] theorem addZero : add (natLit! 0) = pure () :=\n  rfl\n\nsyntax \"bigAdd0Seq! \" num : term\n\nmacro_rules\n  | `(bigAdd0Seq! $n) =>\n    let n := n.toNat\n    if n == 0 then\n      `(pure ())\n    else\n      `(add (natLit! 0) >>= fun _ => bigAdd0Seq! $(Lean.quote (n - 1)))\n\nset_option maxRecDepth 10000\n\ntheorem ex : bigAdd0Seq! 10 = pure () := by\n  simp\n\n-- set_option pp.explicit true\n-- set_option pp.notation false\n-- #print ex\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/playground/seq2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.36957803313002396}}
{"text": "/-\nCopyright (c) 2021 Kalle Kyt\u00f6l\u00e4. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kalle Kyt\u00f6l\u00e4\n-/\nimport tactic \nimport measure_theory.measurable_space\nimport measure_theory.integration\nimport measure_theory.borel_space\nimport measure_theory.lebesgue_measure\nimport topology.metric_space.basic\nimport topology.instances.real\nimport topology.instances.ennreal\nimport order.liminf_limsup\nimport portmanteau_limsup_lemmas\nimport portmanteau_proba_lemmas\nimport portmanteau_topological_lemmas\nimport portmanteau_metric_lemmas\nimport portmanteau_definitions\n\n\n\nnoncomputable theory\nopen set \nopen classical\nopen measure_theory\nopen measurable_space\nopen metric_space\nopen metric\nopen real\nopen borel_space\nopen filter\nopen order\nopen tactic.interactive\nopen_locale topological_space ennreal big_operators classical\n\n\nnamespace portmanteau\n\nsection portmanteau_borel_condition_implies_closed_condition\n\n\n\nvariables {\u03b1 : Type} [metric_space \u03b1]\n\nnotation `borel_measure`(\u03b1) := @measure_theory.measure \u03b1 (borel \u03b1)\nnotation `borel_set`(\u03b1) E := (borel \u03b1).measurable_set' E\n\n\nlemma exists_infdist_level_sets_of_zero_measure_with_small_level\n  (a b : \u211d) (a_pos : 0 < a) (a_lt_b : a < b)\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) :\n    {\u03b4 : \u211d | \u03b4 \u2208 Ioo a b \u2227 \u03bc {x : \u03b1 | ((inf_dist x F) = \u03b4) } = 0 }.nonempty :=\nbegin\n  -- TODO: It seems more appropriate to do this with uncountable cardinality,\n  -- but I didn't find the relevant lemmas... This is now done instead using the\n  -- stronger condition of positive Lebesgue measure. Either one works, but the\n  -- former would arguably be more natural and elegant.\n  have bad_small := countably_many_infdist_level_sets_of_positive_measure \u03bc F ,\n  set B' := {\u03b4 : \u211d | \u03b4 > 0 \u2227 \u03bc {x : \u03b1 | ((inf_dist x F) = \u03b4) } > 0 } with hB' ,\n  set B := B' \u2229 (Ioo a b) with hB ,\n  have sub1 : B \u2286 B' := inter_subset_left B' (Ioo a b) ,\n  have sub2 : B \u2286 (Ioo a b) := inter_subset_right B' (Ioo a b) ,\n  have ctble_B : B.countable := countable.mono sub1 bad_small ,\n  have mble_B : measurable_set B := countable.measurable_set ctble_B ,\n  have null_B : (volume : measure \u211d) B = 0 ,\n  { apply set.countable.measure_zero ctble_B ,\n    exact real.has_no_atoms_volume , } ,  \n  have Ioo_large : (volume : measure \u211d) (Ioo a b) > 0 ,\n  { rw @volume_Ioo a b ,\n    simp only [a_lt_b, ennreal.of_real_pos, gt_iff_lt, sub_pos] , } ,\n  have compl : {\u03b4 : \u211d | \u03b4 \u2208 Ioo a b \u2227 \u03bc {x : \u03b1 | ((inf_dist x F) = \u03b4) } = 0 } = (Ioo a b) \\ B ,\n  { apply le_antisymm ,\n    { intros \u03b4 h\u03b4 ,\n      have notbad : \u03b4 \u2209 B ,\n      { by_contradiction pretendbad ,\n        exact ne_of_gt pretendbad.1.2 h\u03b4.2 , } ,\n      exact \u27e8 h\u03b4.1 , notbad \u27e9 , } ,\n    { intros \u03b4 h\u03b4 ,\n      have \u03b4_in_Ioo : \u03b4 \u2208 Ioo a b := mem_of_mem_diff h\u03b4 ,\n      have \u03b4_pos : 0 < \u03b4 := lt_trans a_pos \u03b4_in_Ioo.1 ,\n      have good : \u03b4 \u2209 B := not_mem_of_mem_diff h\u03b4 ,\n      have good' : \u03b4 \u2209 B' ,\n      { by_contradiction nogood' ,\n        have in_B : \u03b4 \u2208 B := mem_inter nogood' \u03b4_in_Ioo ,\n        contradiction , } ,\n      have key : \u03bc {x : \u03b1 | inf_dist x F = \u03b4} = 0 ,\n      { by_contradiction hcontra ,\n        have pos_meas : 0 < \u03bc {x : \u03b1 | inf_dist x F = \u03b4} ,\n        { have meas_ne_zero : \u03bc {x : \u03b1 | inf_dist x F = \u03b4} \u2260 0 := hcontra ,\n          have meas_ge_zero : 0 \u2264 \u03bc {x : \u03b1 | inf_dist x F = \u03b4} := zero_le _ ,\n          apply lt_iff_le_and_ne.mpr \u27e8meas_ge_zero , meas_ne_zero.symm\u27e9 , } ,\n        exact good' \u27e8\u03b4_pos , pos_meas\u27e9 , } ,\n      exact \u27e8\u03b4_in_Ioo , key\u27e9 , } ,\n  } ,\n  rw compl ,\n  clear compl hB ,\n  have Ioo_minus_large : (volume : measure \u211d) ((Ioo a b) \\ B) > 0 ,\n  { suffices : (volume : measure \u211d) ((Ioo a b) \\ B) = (volume : measure \u211d) (Ioo a b) ,\n    { rwa this , } ,\n    have mdiff := @measure_diff \u211d _ (volume : measure \u211d) _ _ sub2 (measurable_set_Ioo) mble_B (by simp [null_B]) ,\n    rwa [mdiff, null_B] ,\n    simp only [ennreal.sub_zero] , } , \n  by_contradiction hcontra ,\n  have emp : ((Ioo a b) \\ B) = \u2205 := set.not_nonempty_iff_eq_empty.mp hcontra ,\n  rw emp at Ioo_minus_large ,\n  simp only [measure_empty, ennreal.not_lt_zero, gt_iff_lt] at Ioo_minus_large ,\n  contradiction ,\nend\n\n\nprivate lemma reciprocal_lt (n : \u2115) : (1/(n+2) : \u211d) < (1/(n+1) : \u211d) :=\nbegin\n  have decr : \u2200 (x y : \u211d) , 0 < x \u2192 x < y \u2192 1/y < 1/x ,\n  { intros x y ,\n    exact one_div_lt_one_div_of_lt , } ,\n  have pos : 0 < ((n+1) : \u211d) := nat.cast_add_one_pos n ,\n  have lt' : (n+1) < (n+2) := lt_add_one (n+1) ,\n  have lt : (n+1 : \u211d) < (n+2 : \u211d) ,\n  {simp at * , norm_cast , exact dec_trivial , } ,\n  exact decr (n+1) (n+2) pos lt ,\nend\n\n\nprivate def seq_of_good_radii\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) : \u2115 \u2192 \u211d :=\n    \u03bb n , classical.some (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(n+2) : \u211d) ((1/(n+1)) : \u211d) (by tidy) (reciprocal_lt n) \u03bc F)\n\n\nprivate lemma seq_of_good_radii_decr\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) : is_decreasing_seq (seq_of_good_radii \u03bc F) :=\nbegin\n  set s := (seq_of_good_radii \u03bc F) with hs ,\n  intros n m hnm ,\n  by_cases h : n = m ,\n  { rw h , } ,\n  { have n_succ_le_m : n+1 \u2264 m := nat.succ_le_iff.mpr ((ne.le_iff_lt h).mp hnm) ,\n    have key_le : (1/(m+1) : \u211d) \u2264 (1/(n+2) : \u211d) ,\n    { have n_succ_le_m' : (n+2 : \u211d) \u2264 (m+1 : \u211d) ,\n      { norm_cast , \n        exact nat.succ_le_succ n_succ_le_m , } ,\n      apply one_div_le_one_div_of_le _ n_succ_le_m' ,\n      norm_cast , \n      exact dec_trivial , } , \n    have lbn : (1/(n+2) : \u211d) < s n\n      := (some_spec (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(n+2) : \u211d) ((1/(n+1)) : \u211d) (by tidy) (reciprocal_lt n) \u03bc F)).1.1 ,\n    have ubm : s m < (1/(m+1) : \u211d) \n      := (some_spec (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(m+2) : \u211d) ((1/(m+1)) : \u211d) (by tidy) (reciprocal_lt m) \u03bc F)).1.2 ,\n    have key := lt_trans (lt_of_lt_of_le ubm key_le) lbn ,\n    exact le_of_lt key , } ,\nend\n\n\nprivate lemma seq_of_good_radii_pos\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) : \n    \u2200 (n : \u2115) , 0 < (seq_of_good_radii \u03bc F n) :=\nbegin\n  set s := (seq_of_good_radii \u03bc F) with hs ,\n  intros n ,\n  have lbn : (1/(n+2) : \u211d) < s n\n    := (some_spec (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(n+2) : \u211d) ((1/(n+1)) : \u211d) (by tidy) (reciprocal_lt n) \u03bc F)).1.1 ,\n  have pos : 0 < (1/(n+2) : \u211d) ,\n  { simp only [one_div , inv_pos] ,\n    norm_cast, \n    exact dec_trivial , } ,\n  linarith ,\nend\n\n\nprivate lemma seq_of_good_radii_tendsto\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) : lim_R (seq_of_good_radii \u03bc F) 0 :=\nbegin\n  have posseq := seq_of_good_radii_pos \u03bc F ,\n  set s := (seq_of_good_radii \u03bc F) with hs ,\n  have ub : \u2200 (n : \u2115) , s(n) < 1/(n+1)\n    := \u03bb n , (some_spec (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(n+2) : \u211d) ((1/(n+1)) : \u211d) (by tidy) (reciprocal_lt n) \u03bc F)).1.2 ,\n  apply squeeze_zero (\u03bb n , le_of_lt (posseq n)) (\u03bb n , le_of_lt (ub n)) ,\n  exact tendsto_one_div_add_at_top_nhds_0_nat ,\nend\n\n\nprivate lemma seq_of_good_radii_null\n  (\u03bc : borel_measure(\u03b1)) [hfin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc]\n  (F : set \u03b1) : \n  \u2200 (n : \u2115) , \u03bc {x : \u03b1 | inf_dist x F = seq_of_good_radii \u03bc F n } = 0 :=\nbegin\n  set s := (seq_of_good_radii \u03bc F) with hs ,\n  intros n ,\n  exact (some_spec (exists_infdist_level_sets_of_zero_measure_with_small_level (1/(n+2) : \u211d) ((1/(n+1)) : \u211d) (by tidy) (reciprocal_lt n) \u03bc F)).2 ,\nend\n\n\nlemma portmanteau_borel_imp_closed\n  (\u03bcseq : \u2115 \u2192 @measure_theory.measure \u03b1 (borel \u03b1)) \n  (\u03bcseq_fin : \u2200 (n : \u2115) , @probability_measure \u03b1 (borel(\u03b1)) (\u03bcseq(n)))\n  (\u03bc : @measure_theory.measure \u03b1 (borel \u03b1)) (\u03bc_fin : @probability_measure \u03b1 (borel(\u03b1)) \u03bc) : \n  portmanteau_borel \u03bcseq \u03bc \u2192 portmanteau_closed \u03bcseq \u03bc :=\nbegin\n  intros hborcond F hFclos ,\n  by_cases emp : F = \u2205 ,\n  { rw emp ,\n    simp only [measure_empty, nonpos_iff_eq_zero] ,\n    exact limsup_const 0 , } ,\n  have nonemp : F.nonempty := ne_empty_iff_nonempty.mp emp ,\n  set \u03b4seq := (seq_of_good_radii \u03bc F) with hs ,\n  suffices : \u2200 (c : ennreal) , \u03bc(F) < c \u2192 limsup_enn (\u03bb n , (\u03bcseq n)(F)) \u2264 c ,\n  { exact le_of_forall_le_of_dense this , } , \n  intros c hc ,\n  set thick := \u03bb (j : \u2115) , thickening_o (\u03b4seq(j)) F with hthick ,\n  have closeenough : \u2203 (j : \u2115) , \u03bc (thick(j)) \u2264 c ,\n  { have approx := closed_set_borel_proba_by_thickenings \u03bc F hFclos nonemp \n      \u03b4seq (seq_of_good_radii_pos \u03bc F) (seq_of_good_radii_decr \u03bc F) (seq_of_good_radii_tendsto \u03bc F) ,\n    have near := approx (Iic_mem_nhds hc) ,\n    simp at near ,\n    cases near with j hj ,\n    use j ,\n    apply hj j (by refl) , } ,\n  cases closeenough with j hj ,\n  have \u03b4pos : \u03b4seq(j) > 0 := seq_of_good_radii_pos \u03bc F j ,\n  have nullfrontier : \u03bc (frontier (thick(j))) = 0 ,\n  { have meas_mono := @measure_mono \u03b1 (borel(\u03b1)) \u03bc _ _ (frontier_thickening_o F (\u03b4seq(j)) \u03b4pos) ,\n    rw (seq_of_good_radii_null \u03bc F j) at meas_mono ,\n    rw hthick ,\n    apply le_antisymm ,\n    { exact meas_mono , } ,\n    simp only [zero_le] , } ,\n  have openthick : is_open (thick j) := is_open_thickening_o , \n  have limthick := hborcond (thick j) (open_imp_borel openthick) (nullfrontier) ,\n  have limsupthick := lim_eq_limsup_ennreal (limthick) ,\n  have key_le : \u2200 (n : \u2115) , (\u03bcseq(n))(F) \u2264 (\u03bcseq(n))(thick j) ,\n  { intros n,\n    rw hthick ,\n    have sub := closure_subset_thickening_o (\u03b4seq(j)) \u03b4pos F ,\n    rw closure_eq_iff_is_closed.mpr hFclos at sub ,\n    exact @measure_mono \u03b1 (borel(\u03b1)) (\u03bcseq(n)) _ _ sub , } ,\n  have limsup_le := limsup_enn_mono key_le ,\n  rw limsupthick at limsup_le ,\n  exact le_trans limsup_le hj ,\nend\n\n\n\nend portmanteau_borel_condition_implies_closed_condition\n\nend portmanteau\n\n", "meta": {"author": "kkytola", "repo": "lean_portmanteau", "sha": "ac55eb4e24be43032cbc082e2b68d8fb8bd63f22", "save_path": "github-repos/lean/kkytola-lean_portmanteau", "path": "github-repos/lean/kkytola-lean_portmanteau/lean_portmanteau-ac55eb4e24be43032cbc082e2b68d8fb8bd63f22/portmanteau_borel_imp_closed_cond.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3695780252928127}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.closed.cartesian\nimport Mathlib.category_theory.limits.preserves.shapes.binary_products\nimport Mathlib.category_theory.adjunction.fully_faithful\nimport Mathlib.PostPort\n\nuniverses v u u' l \n\nnamespace Mathlib\n\n/-!\n# Cartesian closed functors\n\nDefine the exponential comparison morphisms for a functor which preserves binary products, and use\nthem to define a cartesian closed functor: one which (naturally) preserves exponentials.\n\nDefine the Frobenius morphism, and show it is an isomorphism iff the exponential comparison is an\nisomorphism.\n\n## TODO\nSome of the results here are true more generally for closed objects and for closed monoidal\ncategories, and these could be generalised.\n\n## References\nhttps://ncatlab.org/nlab/show/cartesian+closed+functor\nhttps://ncatlab.org/nlab/show/Frobenius+reciprocity\n\n## Tags\nFrobenius reciprocity, cartesian closed functor\n\n-/\n\nnamespace category_theory\n\n\n/--\nThe Frobenius morphism for an adjunction `L \u22a3 F` at `A` is given by the morphism\n\n    L(FA \u2a2f B) \u27f6 LFA \u2a2f LB \u27f6 A \u2a2f LB\n\nnatural in `B`, where the first morphism is the product comparison and the latter uses the counit\nof the adjunction.\n\nWe will show that if `C` and `D` are cartesian closed, then this morphism is an isomorphism for all\n`A` iff `F` is a cartesian closed functor, i.e. it preserves exponentials.\n-/\ndef frobenius_morphism {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} (h : L \u22a3 F) (A : C) : functor.obj limits.prod.functor (functor.obj F A) \u22d9 L \u27f6 L \u22d9 functor.obj limits.prod.functor A :=\n  limits.prod_comparison_nat_trans L (functor.obj F A) \u226b\n    whisker_left L (functor.map limits.prod.functor (nat_trans.app (adjunction.counit h) A))\n\n/--\nIf `F` is full and faithful and has a left adjoint `L` which preserves binary products, then the\nFrobenius morphism is an isomorphism.\n-/\nprotected instance frobenius_morphism_iso_of_preserves_binary_products {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} (h : L \u22a3 F) (A : C) [limits.preserves_limits_of_shape (discrete limits.walking_pair) L] [full F] [faithful F] : is_iso (frobenius_morphism F h A) :=\n  nat_iso.is_iso_of_is_iso_app (frobenius_morphism F h A)\n\n/--\nThe exponential comparison map.\n`F` is a cartesian closed functor if this is an iso for all `A`.\n-/\ndef exp_comparison {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (A : C) : exp A \u22d9 F \u27f6 F \u22d9 exp (functor.obj F A) :=\n  coe_fn (transfer_nat_trans (exp.adjunction A) (exp.adjunction (functor.obj F A)))\n    (iso.inv (limits.prod_comparison_nat_iso F A))\n\ntheorem exp_comparison_ev {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (A : C) (B : C) : limits.prod.map \ud835\udfd9 (nat_trans.app (exp_comparison F A) B) \u226b nat_trans.app (ev (functor.obj F A)) (functor.obj F B) =\n  inv (limits.prod_comparison F A (functor.obj (exp A) B)) \u226b functor.map F (nat_trans.app (ev A) B) := sorry\n\ntheorem coev_exp_comparison {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (A : C) (B : C) : functor.map F (nat_trans.app (coev A) B) \u226b nat_trans.app (exp_comparison F A) (A \u2a2f B) =\n  nat_trans.app (coev (functor.obj F A)) (functor.obj F B) \u226b\n    functor.map (exp (functor.obj F A)) (inv (limits.prod_comparison F A B)) := sorry\n\ntheorem uncurry_exp_comparison {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (A : C) (B : C) : cartesian_closed.uncurry (nat_trans.app (exp_comparison F A) B) =\n  inv (limits.prod_comparison F A (functor.obj (exp A) B)) \u226b functor.map F (nat_trans.app (ev A) B) := sorry\n\n/-- The exponential comparison map is natural in `A`. -/\ntheorem exp_comparison_whisker_left {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] {A : C} {A' : C} (f : A' \u27f6 A) : exp_comparison F A \u226b whisker_left F (pre (functor.map F f)) = whisker_right (pre f) F \u226b exp_comparison F A' := sorry\n\n/--\nThe functor `F` is cartesian closed (ie preserves exponentials) if each natural transformation\n`exp_comparison F A` is an isomorphism\n-/\nclass cartesian_closed_functor {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] \nwhere\n  comparison_iso : (A : C) \u2192 is_iso (exp_comparison F A)\n\ntheorem frobenius_morphism_mate {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (h : L \u22a3 F) (A : C) : coe_fn\n    (transfer_nat_trans_self (adjunction.comp (functor.obj limits.prod.functor A) (exp A) h (exp.adjunction A))\n      (adjunction.comp L F (exp.adjunction (functor.obj F A)) h))\n    (frobenius_morphism F h A) =\n  exp_comparison F A := sorry\n\n/--\nIf the exponential comparison transformation (at `A`) is an isomorphism, then the Frobenius morphism\nat `A` is an isomorphism.\n-/\ndef frobenius_morphism_iso_of_exp_comparison_iso {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (h : L \u22a3 F) (A : C) [i : is_iso (exp_comparison F A)] : is_iso (frobenius_morphism F h A) :=\n  transfer_nat_trans_self_of_iso (adjunction.comp (functor.obj limits.prod.functor A) (exp A) h (exp.adjunction A))\n    (adjunction.comp L F (exp.adjunction (functor.obj F A)) h) (frobenius_morphism F h A)\n\n/--\nIf the Frobenius morphism at `A` is an isomorphism, then the exponential comparison transformation\n(at `A`) is an isomorphism.\n-/\ndef exp_comparison_iso_of_frobenius_morphism_iso {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (h : L \u22a3 F) (A : C) [i : is_iso (frobenius_morphism F h A)] : is_iso (exp_comparison F A) :=\n  eq.mpr sorry\n    (category_theory.transfer_nat_trans_self_iso\n      (adjunction.comp (functor.obj limits.prod.functor A) (exp A) h (exp.adjunction A))\n      (adjunction.comp L F (exp.adjunction (functor.obj F A)) h) (frobenius_morphism F h A))\n\n/--\nIf `F` is full and faithful, and has a left adjoint which preserves binary products, then it is\ncartesian closed.\n\nTODO: Show the converse, that if `F` is cartesian closed and its left adjoint preserves binary\nproducts, then it is full and faithful.\n-/\ndef cartesian_closed_functor_of_left_adjoint_preserves_binary_products {C : Type u} [category C] {D : Type u'} [category D] [limits.has_finite_products C] [limits.has_finite_products D] (F : C \u2964 D) {L : D \u2964 C} [cartesian_closed C] [cartesian_closed D] [limits.preserves_limits_of_shape (discrete limits.walking_pair) F] (h : L \u22a3 F) [full F] [faithful F] [limits.preserves_limits_of_shape (discrete limits.walking_pair) L] : cartesian_closed_functor F :=\n  cartesian_closed_functor.mk fun (A : C) => exp_comparison_iso_of_frobenius_morphism_iso F h A\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/closed/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499942, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3695694097551544}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\nPorted by: Heather Macbeth\n-/\nimport Mathlib.Tactic.Monotonicity.Attr\nimport Mathlib.Tactic.SolveByElim\n\n/-! # Monotonicity tactic\n\nThe tactic `mono` applies monotonicity rules (collected through the library by being tagged\n`@[mono]`).\n\nThe version of the tactic here is a cheap partial port of the `mono` tactic from Lean 3, which had\nmany more options and features.  It is implemented as a wrapper on top of `solve_by_elim`.\n\nTemporary syntax change: Lean 3 `mono` applied a single monotonicity rule, then applied local\nhypotheses and the `rfl` tactic as many times as it could.  This is hard to implement on top of\n`solve_by_elim` because the counting system used in the `maxDepth` field of its configuration would\ncount these as separate steps, throwing off the count in the desired configuration\n`maxDepth := 1`.  So instead we just implement a version of `mono` in which monotonicity rules,\nlocal hypotheses and `rfl` are all applied repeatedly until nothing more is applicable.  The syntax\nfor this in Lean 3 was `mono*`. Both `mono` and `mono*` implement this behavior for now.\n-/\n\nopen Lean Elab Tactic Parser Tactic\nopen Mathlib Tactic SolveByElim\n\nnamespace Mathlib.Tactic.Monotonicity\n\n/--\n`mono` applies monotonicity rules and local hypotheses repetitively.  For example,\n```lean\nexample (x y z k : \u2124)\n    (h : 3 \u2264 (4 : \u2124))\n    (h' : z \u2264 y) :\n    (k + 3 + x) - y \u2264 (k + 4 + x) - z := by\n  mono\n```\n-/\nsyntax (name := mono) \"mono\" \"*\"? (ppSpace mono.side)?\n  (\" with \" (colGt term),+)? (\" using \" (colGt simpArg),+)? : tactic\n\nelab_rules : tactic\n| `(tactic| mono $[*]? $[$h:mono.side]? $[ with%$w $a:term,*]? $[ using%$u $s,*]? ) => do\n  let msg (s : String) := s ++ \" syntax is not yet supported in 'mono'\"\n  if let some h := h then throwErrorAt h (msg \"'left'/'right'/'both'\")\n  if let some w := w then throwErrorAt w (msg \"'with'\")\n  if let some u := u then throwErrorAt u (msg \"'using'\")\n  let cfg \u2190 elabApplyRulesConfig <| mkNullNode #[]\n  let cfg := { cfg.noBackTracking with\n    transparency := .reducible\n    failAtMaxDepth := false\n    exfalso := false }\n  liftMetaTactic fun g => do solveByElim.processSyntax cfg false false [] [] #[mkIdent `mono] [g]\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/Monotonicity/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6619228691808012, "lm_q1q2_score": 0.36956940603156396}}
{"text": "/-\nCopyright (c) 2019 Reid Barton. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Reid Barton, Johan Commelin\n-/\nimport category_theory.adjunction.basic\nimport category_theory.limits.creates\n\n/-!\n# Adjunctions and limits\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA left adjoint preserves colimits (`category_theory.adjunction.left_adjoint_preserves_colimits`),\nand a right adjoint preserves limits (`category_theory.adjunction.right_adjoint_preserves_limits`).\n\nEquivalences create and reflect (co)limits.\n(`category_theory.adjunction.is_equivalence_creates_limits`,\n`category_theory.adjunction.is_equivalence_creates_colimits`,\n`category_theory.adjunction.is_equivalence_reflects_limits`,\n`category_theory.adjunction.is_equivalence_reflects_colimits`,)\n\nIn `category_theory.adjunction.cocones_iso` we show that\nwhen `F \u22a3 G`,\nthe functor associating to each `Y` the cocones over `K \u22d9 F` with cone point `Y`\nis naturally isomorphic to\nthe functor associating to each `Y` the cocones over `K` with cone point `G.obj Y`.\n-/\n\nopen opposite\n\nnamespace category_theory.adjunction\nopen category_theory\nopen category_theory.functor\nopen category_theory.limits\n\nuniverses v u v\u2081 v\u2082 v\u2080 u\u2081 u\u2082\n\nsection arbitrary_universe\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n\nvariables {F : C \u2964 D} {G : D \u2964 C} (adj : F \u22a3 G)\ninclude adj\n\nsection preservation_colimits\nvariables {J : Type u} [category.{v} J] (K : J \u2964 C)\n\n/--\nThe right adjoint of `cocones.functoriality K F : cocone K \u2964 cocone (K \u22d9 F)`.\n\nAuxiliary definition for `functoriality_is_left_adjoint`.\n-/\ndef functoriality_right_adjoint : cocone (K \u22d9 F) \u2964 cocone K :=\n(cocones.functoriality _ G) \u22d9\n  (cocones.precompose (K.right_unitor.inv \u226b (whisker_left K adj.unit) \u226b (associator _ _ _).inv))\n\nlocal attribute [reducible] functoriality_right_adjoint\n\n/--\nThe unit for the adjunction for `cocones.functoriality K F : cocone K \u2964 cocone (K \u22d9 F)`.\n\nAuxiliary definition for `functoriality_is_left_adjoint`.\n-/\n@[simps] def functoriality_unit :\n  \ud835\udfed (cocone K) \u27f6 cocones.functoriality _ F \u22d9 functoriality_right_adjoint adj K :=\n{ app := \u03bb c, { hom := adj.unit.app c.X } }\n\n/--\nThe counit for the adjunction for `cocones.functoriality K F : cocone K \u2964 cocone (K \u22d9 F)`.\n\nAuxiliary definition for `functoriality_is_left_adjoint`.\n-/\n@[simps] def functoriality_counit :\n  functoriality_right_adjoint adj K \u22d9 cocones.functoriality _ F \u27f6 \ud835\udfed (cocone (K \u22d9 F)) :=\n{ app := \u03bb c, { hom := adj.counit.app c.X } }\n\n/-- The functor `cocones.functoriality K F : cocone K \u2964 cocone (K \u22d9 F)` is a left adjoint. -/\ndef functoriality_is_left_adjoint :\n  is_left_adjoint (cocones.functoriality K F) :=\n{ right := functoriality_right_adjoint adj K,\n  adj := mk_of_unit_counit\n  { unit := functoriality_unit adj K,\n    counit := functoriality_counit adj K } }\n\n/--\nA left adjoint preserves colimits.\n\nSee <https://stacks.math.columbia.edu/tag/0038>.\n-/\ndef left_adjoint_preserves_colimits : preserves_colimits_of_size.{v u} F :=\n{ preserves_colimits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_colimit := \u03bb F,\n    by exactI\n    { preserves := \u03bb c hc, is_colimit.iso_unique_cocone_morphism.inv\n        (\u03bb s, @equiv.unique _ _ (is_colimit.iso_unique_cocone_morphism.hom hc _)\n          (((adj.functoriality_is_left_adjoint _).adj).hom_equiv _ _)) } } }.\n\nomit adj\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_preserves_colimits (E : C \u2964 D) [is_equivalence E] :\n  preserves_colimits_of_size.{v u} E :=\nleft_adjoint_preserves_colimits E.adjunction\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_reflects_colimits (E : D \u2964 C) [is_equivalence E] :\n  reflects_colimits_of_size.{v u} E :=\n{ reflects_colimits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { reflects_colimit := \u03bb K,\n    { reflects := \u03bb c t,\n      begin\n        have l := (is_colimit_of_preserves E.inv t).map_cocone_equiv E.as_equivalence.unit_iso.symm,\n        refine (((is_colimit.precompose_inv_equiv K.right_unitor _).symm) l).of_iso_colimit _,\n        tidy,\n      end } } }\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_creates_colimits (H : D \u2964 C) [is_equivalence H] :\n  creates_colimits_of_size.{v u} H :=\n{ creates_colimits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { creates_colimit := \u03bb F,\n    { lifts := \u03bb c t,\n      { lifted_cocone := H.map_cocone_inv c,\n        valid_lift := H.map_cocone_map_cocone_inv c } } } }\n\n-- verify the preserve_colimits instance works as expected:\nexample (E : C \u2964 D) [is_equivalence E]\n  (c : cocone K) (h : is_colimit c) : is_colimit (E.map_cocone c) :=\npreserves_colimit.preserves h\n\nlemma has_colimit_comp_equivalence (E : C \u2964 D) [is_equivalence E] [has_colimit K] :\n  has_colimit (K \u22d9 E) :=\nhas_colimit.mk\n{ cocone := E.map_cocone (colimit.cocone K),\n  is_colimit := preserves_colimit.preserves (colimit.is_colimit K) }\n\nlemma has_colimit_of_comp_equivalence (E : C \u2964 D) [is_equivalence E] [has_colimit (K \u22d9 E)] :\n  has_colimit K :=\n@has_colimit_of_iso _ _ _ _ (K \u22d9 E \u22d9 inv E) K\n(@has_colimit_comp_equivalence _ _ _ _ _ _ (K \u22d9 E) (inv E) _ _)\n((functor.right_unitor _).symm \u226a\u226b iso_whisker_left K (E.as_equivalence.unit_iso))\n\n/-- Transport a `has_colimits_of_shape` instance across an equivalence. -/\nlemma has_colimits_of_shape_of_equivalence (E : C \u2964 D) [is_equivalence E]\n  [has_colimits_of_shape J D] : has_colimits_of_shape J C :=\n\u27e8\u03bb F, by exactI has_colimit_of_comp_equivalence F E\u27e9\n\n/-- Transport a `has_colimits` instance across an equivalence. -/\nlemma has_colimits_of_equivalence (E : C \u2964 D) [is_equivalence E] [has_colimits_of_size.{v u} D] :\n  has_colimits_of_size.{v u} C :=\n\u27e8\u03bb J hJ, by { exactI has_colimits_of_shape_of_equivalence E }\u27e9\n\nend preservation_colimits\n\nsection preservation_limits\nvariables {J : Type u} [category.{v} J] (K : J \u2964 D)\n\n/--\nThe left adjoint of `cones.functoriality K G : cone K \u2964 cone (K \u22d9 G)`.\n\nAuxiliary definition for `functoriality_is_right_adjoint`.\n-/\ndef functoriality_left_adjoint : cone (K \u22d9 G) \u2964 cone K :=\n(cones.functoriality _ F) \u22d9 (cones.postcompose\n    ((associator _ _ _).hom \u226b (whisker_left K adj.counit) \u226b K.right_unitor.hom))\n\nlocal attribute [reducible] functoriality_left_adjoint\n\n/--\nThe unit for the adjunction for`cones.functoriality K G : cone K \u2964 cone (K \u22d9 G)`.\n\nAuxiliary definition for `functoriality_is_right_adjoint`.\n-/\n@[simps] def functoriality_unit' :\n  \ud835\udfed (cone (K \u22d9 G)) \u27f6 functoriality_left_adjoint adj K \u22d9 cones.functoriality _ G :=\n{ app := \u03bb c, { hom := adj.unit.app c.X, } }\n\n/--\nThe counit for the adjunction for`cones.functoriality K G : cone K \u2964 cone (K \u22d9 G)`.\n\nAuxiliary definition for `functoriality_is_right_adjoint`.\n-/\n@[simps] def functoriality_counit' :\n  cones.functoriality _ G \u22d9 functoriality_left_adjoint adj K \u27f6 \ud835\udfed (cone K) :=\n{ app := \u03bb c, { hom := adj.counit.app c.X, } }\n\n/-- The functor `cones.functoriality K G : cone K \u2964 cone (K \u22d9 G)` is a right adjoint. -/\ndef functoriality_is_right_adjoint :\n  is_right_adjoint (cones.functoriality K G) :=\n{ left := functoriality_left_adjoint adj K,\n  adj := mk_of_unit_counit\n  { unit := functoriality_unit' adj K,\n    counit := functoriality_counit' adj K } }\n\n/--\nA right adjoint preserves limits.\n\nSee <https://stacks.math.columbia.edu/tag/0038>.\n-/\ndef right_adjoint_preserves_limits : preserves_limits_of_size.{v u} G :=\n{ preserves_limits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_limit := \u03bb K,\n    by exactI\n    { preserves := \u03bb c hc, is_limit.iso_unique_cone_morphism.inv\n        (\u03bb s, @equiv.unique _ _ (is_limit.iso_unique_cone_morphism.hom hc _)\n          (((adj.functoriality_is_right_adjoint _).adj).hom_equiv _ _).symm) } } }.\n\nomit adj\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_preserves_limits (E : D \u2964 C) [is_equivalence E] :\n  preserves_limits_of_size.{v u} E :=\nright_adjoint_preserves_limits E.inv.adjunction\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_reflects_limits (E : D \u2964 C) [is_equivalence E] :\n  reflects_limits_of_size.{v u} E :=\n{ reflects_limits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { reflects_limit := \u03bb K,\n    { reflects := \u03bb c t,\n      begin\n        have := (is_limit_of_preserves E.inv t).map_cone_equiv E.as_equivalence.unit_iso.symm,\n        refine (((is_limit.postcompose_hom_equiv K.left_unitor _).symm) this).of_iso_limit _,\n        tidy,\n      end } } }\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_equivalence_creates_limits (H : D \u2964 C) [is_equivalence H] :\n  creates_limits_of_size.{v u} H :=\n{ creates_limits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { creates_limit := \u03bb F,\n    { lifts := \u03bb c t,\n      { lifted_cone := H.map_cone_inv c,\n        valid_lift := H.map_cone_map_cone_inv c } } } }\n\n-- verify the preserve_limits instance works as expected:\nexample (E : D \u2964 C) [is_equivalence E]\n  (c : cone K) [h : is_limit c] : is_limit (E.map_cone c) :=\npreserves_limit.preserves h\n\nlemma has_limit_comp_equivalence (E : D \u2964 C) [is_equivalence E] [has_limit K] :\n  has_limit (K \u22d9 E) :=\nhas_limit.mk\n{ cone := E.map_cone (limit.cone K),\n  is_limit := preserves_limit.preserves (limit.is_limit K) }\n\nlemma has_limit_of_comp_equivalence (E : D \u2964 C) [is_equivalence E] [has_limit (K \u22d9 E)] :\n  has_limit K :=\n@has_limit_of_iso _ _ _ _ (K \u22d9 E \u22d9 inv E) K\n(@has_limit_comp_equivalence _ _ _ _ _ _ (K \u22d9 E) (inv E) _ _)\n((iso_whisker_left K E.as_equivalence.unit_iso.symm) \u226a\u226b (functor.right_unitor _))\n\n/-- Transport a `has_limits_of_shape` instance across an equivalence. -/\nlemma has_limits_of_shape_of_equivalence (E : D \u2964 C) [is_equivalence E] [has_limits_of_shape J C] :\n  has_limits_of_shape J D :=\n\u27e8\u03bb F, by exactI has_limit_of_comp_equivalence F E\u27e9\n\n/-- Transport a `has_limits` instance across an equivalence. -/\nlemma has_limits_of_equivalence (E : D \u2964 C) [is_equivalence E] [has_limits_of_size.{v u} C] :\n  has_limits_of_size.{v u} D :=\n\u27e8\u03bb J hJ, by exactI has_limits_of_shape_of_equivalence E\u27e9\n\nend preservation_limits\n\n/-- auxiliary construction for `cocones_iso` -/\n@[simps]\ndef cocones_iso_component_hom {J : Type u} [category.{v} J] {K : J \u2964 C}\n  (Y : D) (t : ((cocones J D).obj (op (K \u22d9 F))).obj Y) :\n  (G \u22d9 (cocones J C).obj (op K)).obj Y :=\n{ app := \u03bb j, (adj.hom_equiv (K.obj j) Y) (t.app j),\n  naturality' := \u03bb j j' f, by { erw [\u2190 adj.hom_equiv_naturality_left, t.naturality], dsimp, simp } }\n\n/-- auxiliary construction for `cocones_iso` -/\n@[simps]\ndef cocones_iso_component_inv {J : Type u} [category.{v} J] {K : J \u2964 C}\n  (Y : D) (t : (G \u22d9 (cocones J C).obj (op K)).obj Y) :\n  ((cocones J D).obj (op (K \u22d9 F))).obj Y :=\n{ app := \u03bb j, (adj.hom_equiv (K.obj j) Y).symm (t.app j),\n  naturality' := \u03bb j j' f,\n  begin\n    erw [\u2190 adj.hom_equiv_naturality_left_symm, \u2190 adj.hom_equiv_naturality_right_symm, t.naturality],\n    dsimp, simp\n  end }\n\n/-- auxiliary construction for `cones_iso` -/\n@[simps]\ndef cones_iso_component_hom {J : Type u} [category.{v} J] {K : J \u2964 D}\n  (X : C\u1d52\u1d56) (t : (functor.op F \u22d9 (cones J D).obj K).obj X) :\n  ((cones J C).obj (K \u22d9 G)).obj X :=\n{ app := \u03bb j, (adj.hom_equiv (unop X) (K.obj j)) (t.app j),\n  naturality' := \u03bb j j' f,\n  begin\n    erw [\u2190 adj.hom_equiv_naturality_right, \u2190 t.naturality, category.id_comp, category.id_comp],\n    refl\n  end }\n\n/-- auxiliary construction for `cones_iso` -/\n@[simps]\ndef cones_iso_component_inv {J : Type u} [category.{v} J] {K : J \u2964 D}\n  (X : C\u1d52\u1d56) (t : ((cones J C).obj (K \u22d9 G)).obj X) :\n  (functor.op F \u22d9 (cones J D).obj K).obj X :=\n{ app := \u03bb j, (adj.hom_equiv (unop X) (K.obj j)).symm (t.app j),\n  naturality' := \u03bb j j' f,\n  begin\n    erw [\u2190 adj.hom_equiv_naturality_right_symm, \u2190 t.naturality, category.id_comp, category.id_comp]\n  end }\n\nend arbitrary_universe\n\nvariables {C : Type u\u2081} [category.{v\u2080} C] {D : Type u\u2082} [category.{v\u2080} D]\n{F : C \u2964 D} {G : D \u2964 C} (adj : F \u22a3 G)\n\n/--\nWhen `F \u22a3 G`,\nthe functor associating to each `Y` the cocones over `K \u22d9 F` with cone point `Y`\nis naturally isomorphic to\nthe functor associating to each `Y` the cocones over `K` with cone point `G.obj Y`.\n-/\n-- Note: this is natural in K, but we do not yet have the tools to formulate that.\ndef cocones_iso {J : Type u} [category.{v} J] {K : J \u2964 C} :\n  (cocones J D).obj (op (K \u22d9 F)) \u2245 G \u22d9 (cocones J C).obj (op K) :=\nnat_iso.of_components (\u03bb Y,\n{ hom := cocones_iso_component_hom adj Y,\n  inv := cocones_iso_component_inv adj Y, })\n(by tidy)\n\n-- Note: this is natural in K, but we do not yet have the tools to formulate that.\n/--\nWhen `F \u22a3 G`,\nthe functor associating to each `X` the cones over `K` with cone point `F.op.obj X`\nis naturally isomorphic to\nthe functor associating to each `X` the cones over `K \u22d9 G` with cone point `X`.\n-/\ndef cones_iso {J : Type u} [category.{v} J] {K : J \u2964 D} :\n  F.op \u22d9 (cones J D).obj K \u2245 (cones J C).obj (K \u22d9 G) :=\nnat_iso.of_components (\u03bb X,\n{ hom := cones_iso_component_hom adj X,\n  inv := cones_iso_component_inv adj X, } )\n(by tidy)\n\nend category_theory.adjunction\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/adjunction/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3695694023079734}}
{"text": "import data.bitvec\nimport data.stream\nimport util.data.list\nimport util.data.fin\nimport util.meta.tactic\n\nimport tactic.norm_num\n\nimport algebra.group_power\nimport test.slim_check.liftable\nimport system.io\nimport system.random\n\nopen list io applicative\n\nuniverses u v w\n\n-- instance {g : Type v} [random_gen g] : random_gen (ulift.{u} g) :=\n-- by refine_struct { .. }; have_field; rintro \u27e8 g \u27e9;\n--      apply prod.map _ _ (field g);\n--      { apply id <|> apply ulift.up }\n\n-- structure generator :=\n--   (seed1 : unsigned)\n--   (seed2 : unsigned)\n\n-- instance : has_to_string generator :=\n-- \u27e8 \u03bb \u27e8x,y\u27e9, to_string (x,y) \u27e9\n\nnamespace generator\n/- ported from\n    https://hackage.haskell.org/package/random\n    -/\n\n-- def from_nat32 (s : unsigned) : generator :=\n-- let s\u2080 := s.to_nat,\n--     q  := s\u2080 / 2147483562,\n--     s\u2081 : unsigned := fin.of_nat $ s\u2080 % 2147483562,\n--     s\u2082 : unsigned := fin.of_nat $ q % 2147483398 in\n-- \u27e8 s\u2081 + 1, s\u2082 + 1 \u27e9\n\n-- def next : generator \u2192 unsigned \u00d7 generator\n--   | \u27e8 \u27e8 s1, _\u27e9, \u27e8 s2, _ \u27e9 \u27e9 :=\n-- let k : \u2124 := s1 / 53668,\n--     s1'  := 40014 * ((s1 : \u2124) - k * 53668) - k * 12211,\n--     s1'' := if s1' < 0 then s1' + 2147483563 else s1',\n\n--     k' : \u2124 := s2 / 52774,\n--     s2'  := 40692 * ((s2 : \u2124) - k' * 52774) - k' * 3791,\n--     s2'' := if s2' < 0 then s2' + 2147483399 else s2',\n--     z    := s1'' - s2'',\n--     z'   := if z < 1 then z + 2147483562 else z\n-- in\n-- \u27e8 fin.of_nat z'.to_nat, fin.of_nat s1''.to_nat, fin.of_nat s2''.to_nat \u27e9\n\n-- def split : generator \u2192 generator \u00d7 generator\n--  | \u27e8 s1, s2 \u27e9 :=\n-- let    nx := (next \u27e8s1,s2\u27e9).2,\n--        \u27e8t1,t2\u27e9 := nx,\n--        new_s1 := if s1 = 2147483562\n--                  then  1\n--                  else s1 + 1,\n\n--        new_s2 := if s2 = 1\n--                  then 2147483398\n--                  else s2 - 1,\n\n--        -- no statistical foundation for this!\n--        left  : generator := \u27e8new_s1, t2\u27e9,\n--        right : generator := \u27e8t1, new_s2\u27e9\n-- in \u27e8 left, right \u27e9\n\nend generator\n\n@[reducible]\ndef rand_g (g : Type) := state (ulift g)\n@[reducible]\ndef rand := rand_g std_gen\n\ninstance (g : Type) : liftable (rand_g.{u} g) (rand_g.{v} g) :=\n@state_t.liftable' _ _ _ _ _ _ _ _ _  (equiv.ulift.trans.{u u u u u} equiv.ulift.symm)\n\n-- def rand (\u03b1 : Type u) := generator \u2192 \u03b1 \u00d7 generator\n\n-- namespace rand\n\n-- variables {\u03b1 \u03b2 \u03b3 : Type u}\n\n-- protected def pure (x : \u03b1) : rand \u03b1 :=\n-- \u03bb g, (x,g)\n\n-- protected def bind (x : rand \u03b1) (f : \u03b1 \u2192 rand \u03b2) : rand \u03b2 :=\n-- \u03bb g,\n-- let r := x g in\n-- f r.1 r.2\n\n-- instance : has_bind rand :=\n-- \u27e8 @rand.bind \u27e9\n\n-- instance : has_pure rand :=\n-- \u27e8 @rand.pure \u27e9\n\n-- lemma bind_assoc (x : rand \u03b1) (f : \u03b1 \u2192 rand \u03b2) (g : \u03b2 \u2192 rand \u03b3)\n-- : x >>= f >>= g = x >>= (\u03bb i, f i >>= g) :=\n-- begin\n--   funext sz,\n--   simp [has_bind.bind,rand.bind],\n-- end\n\n-- lemma pure_bind (x : \u03b1) (f : \u03b1 \u2192 rand \u03b2)\n-- : pure x >>= f = f x :=\n-- by refl\n\n-- lemma id_map (x : rand \u03b1)\n-- : x >>= pure \u2218 id = x :=\n-- begin\n--   funext i,\n--   simp [function.comp,has_bind.bind,rand.bind],\n--   cases x i, refl\n-- end\n\n-- end rand\n\n-- instance : monad rand :=\n-- { pure := @rand.pure\n-- , bind := @rand.bind\n-- , bind_assoc := @rand.bind_assoc\n-- , pure_bind  := @rand.pure_bind\n-- , id_map := @rand.id_map }\n-- #check @random_gen.next\n\nopen ulift\n\ndef random.next {gen : Type} [random_gen gen] : rand_g gen \u2115 :=\n\u27e8 prod.map id up \u2218 random_gen.next \u2218 down \u27e9\n\ndef range {\u03b1 : Type u} [has_le \u03b1] (i j : \u03b1) :=\n{ x : \u03b1 // i \u2264 x \u2227 x \u2264 j }\n\ninfix ` .. `:41 := range\n\nopen stream\n\nclass random (\u03b1 : Type u) extends has_le \u03b1 :=\n(random : \u03a0 (g : Type) [random_gen g], rand_g g \u03b1)\n(random_r : \u03a0 g [random_gen g] (x y : \u03b1),\n              x \u2264 y \u2192\n              rand_g g (x .. y))\n(random_series : \u03a0 (gen : Type) [random_gen gen], gen \u2192 stream \u03b1 :=\nby { introsI,\n     exact corec prod.fst ((random gen).run \u2218 prod.snd) ( (random gen).run \u27e8 a \u27e9 ) } )\n(random_series_r : \u03a0 (g : Type) [random_gen g] (x y : \u03b1)\n                        (h : x \u2264 y),\n                        g \u2192\n                        stream (x .. y) :=\nby { introsI,\n     exact corec prod.fst ((random_r g x y h).run \u2218 prod.snd) ((random_r g x y h).run \u27e8 a \u27e9) } )\n\nnamespace tactic.interactive\n\nmeta def time_in_nanos : tactic \u2115 :=\ndo time \u2190 tactic.unsafe_run_io (@io.cmd { cmd := \"gdate\", args := [ \"+%s%N\" ] } ),\n   pure time.to_nat\n\nmeta def check_range : tactic unit :=\nassumption <|> do\n`[apply of_as_true, trivial]\n\nend tactic.interactive\n\nexport tactic.interactive (check_range)\n\nnamespace io\n\n-- variable [io.interface]\n\ndef read_dev_random (n : \u2115) : io (array n char) := do\nfh \u2190 mk_file_handle \"/dev/random\" mode.read tt,\nbuf \u2190 fs.read fh n,\nfs.close fh,\nif h : buf.size = n\nthen return (cast (by rw h) buf.to_array)\nelse io.fail \"wrong number of bytes read from /dev/random\"\n\ndef accum_char (w : \u2115) (c : char) : \u2115 :=\nc.to_nat + 256 * w\n\ndef mk_generator : io std_gen := do\nx \u2190 io.read_dev_random 8,\nreturn $ mk_std_gen (foldl accum_char 0 $ x.to_list : \u2115)\n\nvariables {\u03b1 : Type}\n\ndef run_rand (cmd : _root_.rand \u03b1) : io \u03b1 :=\ndo g \u2190 io.mk_generator,\n   return $ (cmd.run \u27e8g\u27e9).1\n\nvariable [random \u03b1]\n\ndef random : io \u03b1 :=\nio.run_rand (random.random \u03b1 _)\n\ndef random_r (x y : \u03b1) (p : x \u2264 y . check_range) : io (x .. y) :=\nio.run_rand (random.random_r _ x y p)\n\ndef random_series : io (stream \u03b1) := do\ng \u2190 io.mk_generator,\nreturn $ random.random_series _ _ g\n\ndef random_series_r (x y : \u03b1) (h : x \u2264 y . check_range) : io (stream $ x .. y) := do\ng \u2190 io.mk_generator,\nreturn $ random.random_series_r _ x y h g\n\nend io\n\nnamespace tactic.interactive\n\nmeta def mk_generator : tactic std_gen := do\ntactic.unsafe_run_io @io.mk_generator\n\nmeta def tactic' (\u03b1 : Type u) : Type (max u 1) :=\n\u03a0 (\u03b2 : Type), (\u03b1 \u2192 tactic \u03b2) \u2192 tactic \u03b2\n\nmeta def run_rand' {\u03b1 : Type u} (cmd : rand \u03b1) (\u03b2 : Type) (tac : \u03b1 \u2192 tactic \u03b2)\n: tactic \u03b2 := do\ng \u2190 mk_generator,\ntac (cmd.run \u27e8g\u27e9).1\n\nsection random'\n\nvariables {\u03b1 : Type u}\nvariable [random \u03b1]\n\nmeta def random' : tactic' \u03b1 :=\nrun_rand' (random.random _ _)\n\nmeta def random_r' (x y : \u03b1) (p : x \u2264 y . check_range) : tactic' (x .. y) :=\nrun_rand' (random.random_r _ x y p)\n\nmeta def random_series' : tactic' (stream \u03b1)\n | \u03b2 cmd := do\ng \u2190 mk_generator,\ncmd $ random.random_series _ std_gen g\n\nmeta def random_series_r' (x y : \u03b1) (h : x \u2264 y . check_range) : tactic' (stream $ x .. y)\n | \u03b2 cmd := do\ng \u2190 mk_generator,\ncmd $ random.random_series_r std_gen x y h g\n\nend random'\n\nsection random\n\nvariable {\u03b1 : Type}\nvariable [random \u03b1]\n\nmeta def random : tactic \u03b1 :=\nrandom' _ return\n\nmeta def random_r (x y : \u03b1) (p : x \u2264 y . check_range) : tactic (x .. y) :=\nrandom_r' _ _ p _ return\n\nmeta def random_series : tactic (stream \u03b1) :=\nrandom_series' _ return\n\nmeta def random_series_r (x y : \u03b1) (h : x \u2264 y . check_range) : tactic (stream $ x .. y) :=\nrandom_series_r' _ _ h _ return\n\nend random\n\nend tactic.interactive\n\ninstance : preorder bool :=\n{ le := \u03bb p q, p \u2192 q\n, le_refl := by { introv h, apply h }\n, le_trans := by { introv ha hb h, apply hb, apply ha h } }\n\nnamespace bool\n\ndef coerce (x y : bool) (p : x \u2264 y) (i : bool) : x .. y := do\n  if hx : x \u2264 i \u2227 i \u2264 y\n  then \u27e8 i, hx \u27e9\n  else \u27e8 x , le_refl x , p \u27e9\nopen ulift\nvariables {gen : Type} [random_gen gen]\nprotected def get_random : rand_g gen bool :=\n\u27e8 prod.map id up \u2218 @rand_bool gen _ \u2218 down \u27e9\n\nstructure bool_generator (g : Type) :=\n  (next : bool)\n  (queue : unsigned \u00d7 \u2115)\n  (gen : g)\n\nprotected def first (g : gen) : bool_generator gen  :=\nlet (r,g') := random_gen.next g in\n{ next := r % 2 = 1\n, queue := (r / 2,30)\n, gen := g' }\n\nprotected def next : bool_generator gen \u2192 bool_generator gen\n | \u27e8_,(_,0),g\u27e9 := bool.first g\n | \u27e8_,(n,k),g\u27e9 := \u27e8(n%2 = 1),(n/2,k-1),g\u27e9\n\ndef random_series' (g : gen) : stream (bool_generator gen) :=\nstream.iterate bool.next (bool.first g)\n\ndef random_series (g : gen) : stream bool :=\nstream.map bool.bool_generator.next $ random_series' g\n\nend bool\n\ninstance : random bool :=\n{ to_has_le := by apply_instance\n, random   := \u03bb g, @bool.get_random _\n, random_r := \u03bb g _inst x y p, bool.coerce _ _ p <$> (@bool.get_random g _inst)\n, random_series   := @bool.random_series\n, random_series_r := \u03bb gen _inst x y p g, stream.map (bool.coerce _ _ p) $ @bool.random_series _ _inst g }\n\ninstance (n : \u2115) : preorder (bitvec n) :=\n{ le := \u03bb x y, x.to_nat \u2264 y.to_nat\n, le_refl  := by { introv, apply nat.le_refl }\n, le_trans := by { introv ha hb, apply nat.le_trans ha hb } }\n\nlemma bitvec.le_def {n : \u2115} (x y : bitvec n)\n: x \u2264 y \u2194 x.to_nat \u2264 y.to_nat :=\nby refl\n\nopen nat (succ one_add mod_eq_of_lt zero_lt_succ add_one succ_le_succ)\n\nnamespace stream\n\nvariable {\u03b1 : Type u}\n\nopen list (length) stream (approx)\n\nlemma length_approx\n: \u2200 (s : stream \u03b1) (n : \u2115), length (approx n s) = n\n | s 0 := rfl\n | s (succ n) := by simp [approx,length,one_add,length_approx]\n\nend stream\n\nvariables {gen : Type} [random_gen gen]\n\ndef bitvec.random (n : \u2115) : rand_g gen (bitvec n) :=\n\u27e8 \u03bb \u27e8 g \u27e9,\nlet r := bool.random_series' g,\n    v := map bool.bool_generator.next $ stream.approx n r in\nhave Hv : length v = n,\n     by { simp [stream.length_approx _ _], },\n\u27e8 \u27e8 v, Hv \u27e9 , \u27e8 (r.nth $ succ n).gen \u27e9 \u27e9 \u27e9\n\nsection coerce\n\nparameters {i' n : \u2115}\nparameters {x y : bitvec n}\n\nparameters P' : x.to_nat \u2264 y.to_nat\ninclude P'\n\nlocal infix ^ := nat.pow\n\nlemma bitvec.interval_fits_in_word_size\n: x.to_nat + i' % (1 + (y.to_nat - x.to_nat)) < 2^n :=\nbegin\n  let x' := x.to_nat,\n  let y' := y.to_nat,\n  apply @lt_of_lt_of_le _ _ _ (x' + (y' - x' + 1)),\n  { apply add_lt_add_left, simp,\n    apply nat.mod_lt,\n    rw one_add, apply zero_lt_succ },\n  { rw [\u2190 add_assoc,\u2190 nat.add_sub_assoc P',nat.add_sub_cancel_left,add_one],\n    clear P' i',\n    cases y with y Hy,\n    unfold bitvec.to_nat vector.to_list subtype.val bitvec.bits_to_nat,\n    rw [\u2190 reverse_reverse y, foldl_reverse,\u2190 Hy,\u2190 length_reverse],\n    generalize : reverse y = z,\n    clear x' x y' Hy y n,\n    induction z with x xs,\n    { rw [list.length,list.foldr,nat.pow] },\n    { simp [foldr,length,one_add,pow_succ,flip,bitvec.add_lsb],\n      transitivity succ (1 +\n       (foldr (\u03bb (b : bool) (a : \u2115), a + (a + cond b 1 0)) 0 xs +\n          foldr (\u03bb (b : bool) (a : \u2115), a + (a + cond b 1 0)) 0 xs)),\n      { apply succ_le_succ, apply add_le_add_right,\n        cases x, apply nat.zero_le, refl, },\n      { simp!,\n        rw [\u2190 nat.add_succ,\u2190 nat.add_succ,one_add,\u2190 nat.succ_add,mul_comm,\u2190 two_mul],\n        apply nat.mul_le_mul_left,\n        simp [flip,bitvec.add_lsb] at z_ih,\n        apply z_ih } }, },\nend\nend coerce\n\nopen nat\n\ndef bitvec.coerce {n : \u2115} (x y : bitvec n) (P : x \u2264 y)\n  (i : bitvec n)\n: (x .. y) :=\nlet x' := x.to_nat,\n    y' := y.to_nat,\n    i' := i.to_nat,\n    r := i' % (y' - x' + 1) + x' in\nhave Hx : x \u2264 bitvec.of_nat n r,\n  begin\n    unfold_local r,\n    simp [bitvec.le_def,bitvec.to_nat_of_nat],\n    rw [mod_eq_of_lt],\n    { apply nat.le_add_right },\n    unfold_locals x' y' i',\n    apply bitvec.interval_fits_in_word_size,\n    apply P\n  end,\nhave Hy : bitvec.of_nat n r \u2264 y,\n  begin\n    unfold_local r,\n    rw [bitvec.le_def,bitvec.to_nat_of_nat,mod_eq_of_lt],\n    transitivity (y' - x') + x',\n    { apply add_le_add_right,\n      apply le_of_lt_succ,\n      rw \u2190 add_one,\n      apply mod_lt,\n      rw add_one, apply zero_lt_succ },\n    { transitivity x' + (y' - x'),\n      apply le_of_eq, ac_refl,\n      rw [\u2190 nat.add_sub_assoc P,nat.add_sub_cancel_left], },\n    simp, apply bitvec.interval_fits_in_word_size P,\n  end,\n\u27e8 bitvec.of_nat _ r , Hx , Hy \u27e9\n\ndef bitvec.random_series (n : \u2115) (g : gen) : stream (bitvec n) :=\nstream.corec\n(\u03bb s, \u27e8 stream.approx n s, stream.length_approx _ _ \u27e9)\n(stream.drop n)\n(@random.random_series bool _ gen _ g)\n\ninstance random_bitvec (n : \u2115) : random (bitvec n) :=\n{ to_has_le := by apply_instance\n, random := \u03bb _ inst, @bitvec.random _ inst n\n, random_r := \u03bb _ inst x y p, bitvec.coerce _ _ p <$> @bitvec.random _ inst n\n, random_series := \u03bb _ inst, @bitvec.random_series _ inst n\n, random_series_r := \u03bb _ inst x y p g, bitvec.coerce _ _ p \u2218 @bitvec.random_series _ inst n g }\n\n-- example : true :=\n-- begin\n-- tactic.trace \"\\n\\n\",\n-- (do x \u2190 (tactic.interactive.random : tactic (bitvec 16)),\n--     tactic.trace (x : bitvec 16).to_nat),\n-- (do x \u2190 (tactic.interactive.random_series),\n--     tactic.trace $ map bitvec.to_nat (stream.approx 10 x : list (bitvec 16))),\n-- (do x \u2190 (tactic.interactive.random_series_r (25 : bitvec 15) 100),\n--     tactic.trace $ map (bitvec.to_nat \u2218 subtype.val) (stream.approx 10 x)),\n-- trivial\n-- end\n\n-- meta def main [io.interface] : io unit := do\n-- print_ln \"\\n\\n\",\n-- x \u2190 (io.random : io (bitvec 16)),\n-- print_ln (x : bitvec 16).to_nat,\n-- x \u2190 io.random_series,\n-- print_ln $ map bitvec.to_nat (stream.approx 10 x : list (bitvec 16)),\n-- x \u2190 (io.random_series_r (25 : bitvec 15) 100),\n-- print_ln $ map (bitvec.to_nat \u2218 subtype.val) (stream.approx 10 x)\n\n-- run_cmd tactic.run_io @main\n\nopen nat\n\nlemma div_two_round_up\n  (x : \u2115)\n  (h\u2080 : 1 < x)\n: (x + 1) / 2 < x :=\nbegin\n  rw [div_lt_iff_lt_mul,norm_num.mul_bit0,mul_one,bit0],\n  apply add_lt_add_left, apply h\u2080,\n  apply of_as_true, trivial\nend\n\ndef word_size : \u2115 \u2192 \u2115\n | x :=\nif h : 1 < x then\n  have (x + 1) / 2 < x,\n    from div_two_round_up _ h,\n  succ $ word_size ((x + 1) / 2)\nelse 0\n\nlocal infix ^ := nat.pow\n\nlemma word_size_le_two_pow (n : \u2115)\n: n \u2264 2^word_size n :=\nbegin\n  apply nat.strong_induction_on n,\n  clear n, intros n IH,\n  by_cases h : 1 < n,\n  { rw [word_size,if_pos h,nat.pow],\n    cases n with n, { cases not_lt_zero _ h },\n    change n < _,\n    rw \u2190 @div_lt_iff_lt_mul _ _ 2 dec_trivial,\n    have h' := div_two_round_up (succ n) h,\n    specialize IH ((succ n + 1) / 2) h', clear h h',\n    rw [succ_add,\u2190 add_succ] at *,\n    simp only [-succ_pos', add_zero] at *,\n    have h : (n + 2 * 1) / 2 = n / 2 + 1 :=\n       add_mul_div_left _ _ dec_trivial,\n    rw [mul_one] at h,\n    rw h at IH \u22a2,\n    apply IH },\n  { rw [word_size,if_neg h,nat.pow],\n    apply le_of_not_gt h }\nend\n\nnamespace fin\nsection fin\nparameter {n : \u2115}\n\ndef shift_31l : \u2115 :=\nby apply_normed 2^31\n\nprotected def random_aux : \u2115 \u2192 \u2115 \u2192 rand_g gen (fin (succ n))\n | 0 k := return $ fin.of_nat k\n | (succ n) k :=\ndo x \u2190 random.next,\n   random_aux n $ x + (k * shift_31l)\n\nprotected def random : rand_g gen (fin (succ n)) :=\nlet m := word_size n / 31 + 1 in\nrandom_aux m 0\n\nsection coerce\n\nparameters {i' r k : \u2115}\nparameters {y : fin k}\n\n-- def x' := x.val\n-- def y' := y.val\nparameters {x' : \u2115}\n\nparameters P' : x' \u2264 y.val\ninclude P'\n\nlemma fin.interval_fits_in_word_size\n: x' + i' % (1 + (y.val - x')) < k :=\nbegin\n  apply @lt_of_lt_of_le _ _ _ (x' + (y.val - x' + 1)),\n  { apply add_lt_add_left, simp,\n    apply nat.mod_lt,\n    rw one_add, apply zero_lt_succ },\n  { rw [\u2190 add_assoc,\u2190 nat.add_sub_assoc P',nat.add_sub_cancel_left,add_one],\n    apply y.is_lt }\nend\nend coerce\n\nprotected def coerce {n : \u2115} (x y : fin (succ n)) (P : x \u2264 y)\n  (i : fin (succ n))\n: (x .. y) :=\nlet x' := x.val,\n    i' := i.val,\n    r := i' % (y.val - x' + 1) + x' in\nhave P' : x.val \u2264 y.val,\n  by { rw \u2190 le_def, apply P },\nhave Hx : x \u2264 fin.of_nat r,\n  begin\n    unfold_local r,\n    simp [fin.le_def,fin.val_of_nat_eq_mod],\n    rw [mod_eq_of_lt],\n    { apply nat.le_add_right },\n    apply fin.interval_fits_in_word_size,\n    unfold_locals x',\n    rw \u2190 fin.le_def, apply P\n  end,\nhave Hy : fin.of_nat r \u2264 y,\n  begin\n    unfold_local r,\n    rw [fin.le_def,fin.val_of_nat_eq_mod,mod_eq_of_lt],\n    transitivity (y.val - x') + x',\n    { apply add_le_add_right,\n      apply le_of_lt_succ,\n      rw add_one,\n      apply mod_lt,\n      apply zero_lt_succ },\n    { transitivity x' + (y.val - x'),\n      apply le_of_eq, ac_refl,\n      rw [\u2190 nat.add_sub_assoc P',nat.add_sub_cancel_left], },\n    simp, apply fin.interval_fits_in_word_size P',\n  end,\n\u27e8 fin.of_nat r , Hx , Hy \u27e9\n\nprotected def random_r (x y : fin (succ n)) (p : x \u2264 y) : rand_g gen (x .. y) :=\nfin.coerce _ _ p <$> fin.random\n\nend fin\nend fin\n\ninstance fin_random (n : \u2115) : random (fin (succ n)) :=\n{ to_has_le := by apply_instance\n, random := \u03bb g, @fin.random _ g\n, random_r := \u03bb x y p, @fin.random_r n x y p }\n\nopen nat\n\ndef random_fin_of_pos : \u2200 (n : \u2115) (h : 0 < n), random (fin n)\n | (succ n) _ := fin_random _\n | 0 h := false.elim (not_lt_zero _ h)\n\nsection\nopen stream\n\ndef try_bitvec_random : io unit := do\nput_str_ln \"\",\ny \u2190 io.mk_generator,\nlet w := succ $ 31,\nlet i : bitvec w := 2,\nlet j : bitvec w := 10000,\nprint_ln (repr y),\nx \u2190 (io.random : io (bitvec w)), return x.to_nat,\nx \u2190 (io.random_r (i) j : io _), return (x.val.to_nat),\nx \u2190 (io.random_series : io (stream (bitvec w))),\n    return $ map bitvec.to_nat $ approx 10 x,\nx \u2190 (io.random_series_r i j),\n    return $ map (bitvec.to_nat \u2218 subtype.val) $ approx 10 x,\nreturn ()\n\n\ndef try_fin_random : io unit := do\nput_str_ln \"\",\nlet n := by { apply_normed 2^31 },\nlet i : fin (succ n) := fin.of_nat 2,\nlet j : fin (succ n) := fin.of_nat 100000,\nhave dp : i \u2264 j,\nby { unfold_locals i j n,\n     rw [fin.le_def,fin.val_of_nat,fin.val_of_nat]\n     ; [ skip, apply succ_le_succ, apply succ_le_succ ]\n     ; norm_num ; trivial, }, do\nx \u2190 (io.random : io (fin 10)), print_ln x.val,\nx \u2190 (io.random : io (fin (succ n))), return x,\nx \u2190 (io.random_r i j : io _), print_ln (x.val),\nx \u2190 (io.random_series : io (stream (fin (succ n)))), return $ approx 10 x,\nx \u2190 (io.random_series_r i j), return $ map subtype.val $ approx 10 x,\nreturn ()\n\nend\n\n-- run_cmd do\n--   tactic.timetac \"fin_random:    \" (tactic.unsafe_run_io @try_fin_random ),\n--   tactic.timetac \"bitvec_random: \" (tactic.unsafe_run_io @try_bitvec_random),\n--   tactic.trace \"do\"\n\n-- fin_random:     46.1ms\n-- bitvec_random:  785ms\n", "meta": {"author": "cipher1024", "repo": "slim_check", "sha": "5969b7f72e01fdd46f2502ed0cbf69c0699061d4", "save_path": "github-repos/lean/cipher1024-slim_check", "path": "github-repos/lean/cipher1024-slim_check/slim_check-5969b7f72e01fdd46f2502ed0cbf69c0699061d4/src/test/random.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.36946629627805566}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport algebra.homology.homological_complex\nimport for_mathlib.algebra.homology.homological_complex_biprod\nimport for_mathlib.algebra.homology.trunc_le\n\nopen category_theory\nopen category_theory.category\n\nvariables {C : Type*} [category C]\n\nnamespace category_theory\n\nnamespace limits\n\nlemma is_zero.iff_of_biprod [preadditive C] (A B : C) [has_binary_biproduct A B] :\n  is_zero (A \u229e B) \u2194 is_zero A \u2227 is_zero B :=\nbegin\n  simp only [is_zero.iff_id_eq_zero],\n  split,\n  { intro h,\n    split,\n    { suffices : \ud835\udfd9 A = biprod.inl \u226b (\ud835\udfd9 (A \u229e B)) \u226b biprod.fst,\n      { rw [this, h, zero_comp, comp_zero], },\n      rw [id_comp, biprod.inl_fst], },\n    { suffices : \ud835\udfd9 B = biprod.inr \u226b (\ud835\udfd9 (A \u229e B)) \u226b biprod.snd,\n      { rw [this, h, zero_comp, comp_zero], },\n      rw [id_comp, biprod.inr_snd], }, },\n  { intro h,\n    ext,\n    { simpa only [comp_id, biprod.inl_fst, comp_zero, zero_comp] using h.left, },\n    { simp only [comp_id, biprod.inl_snd, comp_zero, zero_comp], },\n    { simp only [comp_id, biprod.inr_fst, comp_zero, zero_comp], },\n    { simpa only [comp_id, biprod.inr_snd, comp_zero, zero_comp] using h.right, } },\nend\n\nend limits\n\nend category_theory\n\nopen category_theory.limits\n\nnamespace cochain_complex\n\ndef is_bounded_above [has_zero_morphisms C] (K : cochain_complex C \u2124) : Prop :=\n\u2203 (r : \u2124), \u2200 (i : \u2124) (hi : r < i), is_zero (K.X i)\n\nnamespace is_bounded_above\n\nlemma of_biprod [preadditive C] (K L : cochain_complex C \u2124)\n  (hK : K.is_bounded_above) (hL : L.is_bounded_above)\n  [\u2200 i, has_binary_biproduct (K.X i) (L.X i)] :\n  is_bounded_above (homological_complex.biprod K L) :=\nbegin\n  cases hK with k hk,\n  cases hL with l hl,\n  use max k l,\n  intros i hi,\n  dsimp,\n  rw is_zero.iff_of_biprod,\n  split,\n  { apply hk,\n    exact lt_of_le_of_lt (le_max_left _ _) hi, },\n  { apply hl,\n    exact lt_of_le_of_lt (le_max_right _ _) hi, },\nend\n\nlemma of_is_strictly_le [abelian C] (K : cochain_complex C \u2124) (n : \u2124)\n  [K.is_strictly_le n] : is_bounded_above K :=\n\u27e8n, \u03bb i hi, is_strictly_le.is_zero K n i hi\u27e9\n\nend is_bounded_above\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/bounded_above.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.369466289137774}}
{"text": "\nimport tactic\nimport category.functor\nimport category.applicative\n\nuniverses u v\n\nnamespace monad\n\n@[simp]\nlemma bind_pure_star {m} [monad m] [is_lawful_monad m] (x : m punit) :\n  x >>= (\u03bb (_x : punit), pure punit.star : punit \u2192 m punit) = x :=\nby { transitivity,\n     { apply congr_arg, ext z, cases z, refl },\n     { simp } }\n\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\nvariables {m : Type u \u2192 Type v} [monad m]\n\n@[reducible]\ndef pipe (a : \u03b1 \u2192 m \u03b2) (b : \u03b2 \u2192 m \u03b3) : \u03b1 \u2192 m \u03b3 :=\n\u03bb x, a x >>= b\n\ninfixr ` >=> `:55 := pipe\n\n@[functor_norm]\nlemma map_bind_eq_bind_comp {\u03b1 \u03b2 \u03b3} {m} [monad m] [is_lawful_monad m]\n  (f : \u03b1 \u2192 \u03b2) (cmd : m \u03b1) (g : \u03b2 \u2192 m \u03b3) :\n  (f <$> cmd) >>= g = cmd >>= g \u2218 f :=\nby rw [\u2190 bind_pure_comp_eq_map,bind_assoc,(\u2218)]; simp\n\n@[functor_norm]\nlemma bind_map {\u03b1 \u03b2 \u03b3} {m} [monad m] [is_lawful_monad m]\n  (f : \u03b1 \u2192 \u03b3 \u2192 \u03b2) (cmd : m \u03b1) (g : \u03b1 \u2192 m \u03b3) :\n  cmd >>= (\u03bb x, f x <$> g x) = do { x \u2190 cmd, y \u2190 g x, pure $ f x y }  :=\nby congr; ext; rw [\u2190 bind_pure (g x),map_bind]; simp\n\n@[functor_norm]\nlemma bind_seq {\u03b1 \u03b2 \u03b3 : Type u} {m} [monad m] [is_lawful_monad m]\n  (f : \u03b1 \u2192 m (\u03b3 \u2192 \u03b2)) (cmd : m \u03b1) (g : \u03b1 \u2192 m \u03b3) :\n  cmd >>= (\u03bb x, f x <*> g x) = do { x \u2190 cmd, h \u2190 f x, y \u2190 g x, pure $ h y }  :=\nby congr; ext; simp [seq_eq_bind_map] with functor_norm\n\nend monad\n\nattribute [functor_norm] bind_assoc has_bind.and_then map_bind seq_left_eq seq_right_eq\n\nnamespace sum\n\nvariables {e : Type v} {\u03b1 \u03b2 : Type u}\n\nprotected def seq : \u03a0 (x : sum e (\u03b1 \u2192 \u03b2)) (f : sum e \u03b1), sum e \u03b2\n| (sum.inl e) _ := sum.inl e\n| (sum.inr f) x := f <$> x\n\ninstance : applicative (sum e) :=\n{ seq := @sum.seq e,\n  pure := @sum.inr e }\n\ninstance : is_lawful_applicative (sum e) :=\nby constructor; intros;\n   casesm* _ \u2295 _; simp [(<*>),sum.seq,pure,(<$>)];\n   refl\n\nend sum\n", "meta": {"author": "leanprover-community", "repo": "mathlib-nursery", "sha": "0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec", "save_path": "github-repos/lean/leanprover-community-mathlib-nursery", "path": "github-repos/lean/leanprover-community-mathlib-nursery/mathlib-nursery-0479b31fa5b4d39f41e89b8584c9f5bf5271e8ec/src/category/nursery.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3694222455444816}}
{"text": "\nimport tactic\n\nuniverse variables u v\n\nnotation `\u03b5` binder `, ` r:(scoped p, classical.epsilon p) := r\n\nnamespace classical\n\nvariables {\u03b1 : Sort u}\nvariables {\u03b2 : Sort v}\nvariables {p q : \u03b1 \u2192 Prop}\nvariables Hpq : \u2200 x, p x \u2192 q x\nvariables Hex : \u2203 x, p x\n\ninclude Hpq Hex\n\nlemma some_spec'\n: q (some Hex) :=\nbegin\n  apply Hpq,\n  apply some_spec\nend\n\nvariable [nonempty \u03b1]\n\nlemma epsilon_spec'\n: q (epsilon p) :=\nbegin\n  apply Hpq,\n  apply epsilon_spec Hex,\nend\n\nend classical\n\nopen tactic interactive interactive.types lean.parser\nopen tactic.interactive (tauto)\n\nmeta def apply_some_spec (id : parse $ optional (tk \"with\" *> ident_)) : tactic unit :=\ndo t \u2190 target,\n   (l,_) \u2190 solve_aux t (do\n     e \u2190 to_expr ``(@classical.some _ _ _),\n     v \u2190 mk_fresh_name,\n     generalize e v,\n     (expr.pi v bi t e) \u2190 target,\n     return (expr.lam v bi t e)),\n   refine ``(@classical.some_spec' _ _ %%l _ _),\n   `[simp only],\n   interactive.intro id,\n   `[intros h, apply h] <|>\n      tauto (some ()) <|>\n      smt_tactic.execute smt_tactic.eblast <|>\n      return ()\n\nmeta def apply_epsilon_spec (ex : parse $ optional texpr)\n  (id : parse $ optional (tk \"with\" *> ident_)) : tactic unit :=\nfocus1 $\ndo t \u2190 target,\n   (l,_) \u2190 solve_aux t (do\n     e \u2190 to_expr ``(classical.epsilon _),\n     v \u2190 mk_fresh_name,\n     generalize e v,\n     (expr.pi v bi t e) \u2190 target,\n     return (expr.lam v bi t e)),\n   refine ``(@classical.epsilon_spec' _ _ %%l _ _ _),\n   `[simp only],\n   interactive.intro id,\n   try `[intros h, apply h],\n   all_goals (solve_by_elim <|> \u2191ex >>= tactic.refine <|>\n              tauto (some ()) <|>\n              smt_tactic.execute (smt_tactic.intros >> smt_tactic.eblast) <|>\n              return ())\n\nrun_cmd add_interactive [`apply_some_spec,`apply_epsilon_spec]\n", "meta": {"author": "unitb", "repo": "lean-lib", "sha": "439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9", "save_path": "github-repos/lean/unitb-lean-lib", "path": "github-repos/lean/unitb-lean-lib/lean-lib-439b80e606b4ebe4909a08b1d77f4f5c0ee3dee9/src/util/classical.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.36925452797135777}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.limits.constructions.binary_products\n\n/-!\n# Limits involving zero objects\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nBinary products and coproducts with a zero object always exist,\nand pullbacks/pushouts over a zero object are products/coproducts.\n-/\n\nnoncomputable theory\n\nopen category_theory\n\nvariables {C : Type*} [category C]\n\nnamespace category_theory.limits\n\nvariables [has_zero_object C] [has_zero_morphisms C]\nopen_locale zero_object\n\n/-- The limit cone for the product with a zero object. -/\ndef binary_fan_zero_left (X : C) : binary_fan (0 : C) X :=\nbinary_fan.mk 0 (\ud835\udfd9 X)\n\n/-- The limit cone for the product with a zero object is limiting. -/\ndef binary_fan_zero_left_is_limit (X : C) : is_limit (binary_fan_zero_left X) :=\nbinary_fan.is_limit_mk (\u03bb s, binary_fan.snd s) (by tidy) (by tidy) (by tidy)\n\ninstance has_binary_product_zero_left (X : C) : has_binary_product (0 : C) X :=\nhas_limit.mk \u27e8_, binary_fan_zero_left_is_limit X\u27e9\n\n/-- A zero object is a left unit for categorical product. -/\ndef zero_prod_iso (X : C) : (0 : C) \u2a2f X \u2245 X :=\nlimit.iso_limit_cone \u27e8_, binary_fan_zero_left_is_limit X\u27e9\n\n@[simp] lemma zero_prod_iso_hom (X : C) : (zero_prod_iso X).hom = prod.snd :=\nrfl\n@[simp] lemma zero_prod_iso_inv_snd (X : C) : (zero_prod_iso X).inv \u226b prod.snd = \ud835\udfd9 X :=\nby { dsimp [zero_prod_iso, binary_fan_zero_left], simp, }\n\n/-- The limit cone for the product with a zero object. -/\ndef binary_fan_zero_right (X : C) : binary_fan X (0 : C) :=\nbinary_fan.mk (\ud835\udfd9 X) 0\n\n/-- The limit cone for the product with a zero object is limiting. -/\ndef binary_fan_zero_right_is_limit (X : C) : is_limit (binary_fan_zero_right X) :=\nbinary_fan.is_limit_mk (\u03bb s, binary_fan.fst s) (by tidy) (by tidy) (by tidy)\n\ninstance has_binary_product_zero_right (X : C) : has_binary_product X (0 : C) :=\nhas_limit.mk \u27e8_, binary_fan_zero_right_is_limit X\u27e9\n\n/-- A zero object is a right unit for categorical product. -/\ndef prod_zero_iso (X : C) : X \u2a2f (0 : C) \u2245 X :=\nlimit.iso_limit_cone \u27e8_, binary_fan_zero_right_is_limit X\u27e9\n\n@[simp] lemma prod_zero_iso_hom (X : C) : (prod_zero_iso X).hom = prod.fst :=\nrfl\n@[simp] lemma prod_zero_iso_iso_inv_snd (X : C) : (prod_zero_iso X).inv \u226b prod.fst = \ud835\udfd9 X :=\nby { dsimp [prod_zero_iso, binary_fan_zero_right], simp, }\n\n/-- The colimit cocone for the coproduct with a zero object. -/\ndef binary_cofan_zero_left (X : C) : binary_cofan (0 : C) X :=\nbinary_cofan.mk 0 (\ud835\udfd9 X)\n\n/-- The colimit cocone for the coproduct with a zero object is colimiting. -/\ndef binary_cofan_zero_left_is_colimit (X : C) : is_colimit (binary_cofan_zero_left X) :=\nbinary_cofan.is_colimit_mk (\u03bb s, binary_cofan.inr s) (by tidy) (by tidy) (by tidy)\n\ninstance has_binary_coproduct_zero_left (X : C) : has_binary_coproduct (0 : C) X :=\nhas_colimit.mk \u27e8_, binary_cofan_zero_left_is_colimit X\u27e9\n\n/-- A zero object is a left unit for categorical coproduct. -/\ndef zero_coprod_iso (X : C) : (0 : C) \u2a3f X \u2245 X :=\ncolimit.iso_colimit_cocone \u27e8_, binary_cofan_zero_left_is_colimit X\u27e9\n\n@[simp] lemma inr_zero_coprod_iso_hom (X : C) : coprod.inr \u226b (zero_coprod_iso X).hom = \ud835\udfd9 X :=\nby { dsimp [zero_coprod_iso, binary_cofan_zero_left], simp, }\n@[simp] lemma zero_coprod_iso_inv (X : C) : (zero_coprod_iso X).inv = coprod.inr :=\nrfl\n\n/-- The colimit cocone for the coproduct with a zero object. -/\ndef binary_cofan_zero_right (X : C) : binary_cofan X (0 : C) :=\nbinary_cofan.mk (\ud835\udfd9 X) 0\n\n/-- The colimit cocone for the coproduct with a zero object is colimiting. -/\ndef binary_cofan_zero_right_is_colimit (X : C) : is_colimit (binary_cofan_zero_right X) :=\nbinary_cofan.is_colimit_mk (\u03bb s, binary_cofan.inl s) (by tidy) (by tidy) (by tidy)\n\ninstance has_binary_coproduct_zero_right (X : C) : has_binary_coproduct X (0 : C) :=\nhas_colimit.mk \u27e8_, binary_cofan_zero_right_is_colimit X\u27e9\n\n/-- A zero object is a right unit for categorical coproduct. -/\ndef coprod_zero_iso (X : C) : X \u2a3f (0 : C) \u2245 X :=\ncolimit.iso_colimit_cocone \u27e8_, binary_cofan_zero_right_is_colimit X\u27e9\n\n@[simp] lemma inr_coprod_zeroiso_hom (X : C) : coprod.inl \u226b (coprod_zero_iso X).hom = \ud835\udfd9 X :=\nby { dsimp [coprod_zero_iso, binary_cofan_zero_right], simp, }\n@[simp] lemma coprod_zero_iso_inv (X : C) : (coprod_zero_iso X).inv = coprod.inl :=\nrfl\n\ninstance has_pullback_over_zero\n  (X Y : C) [has_binary_product X Y] : has_pullback (0 : X \u27f6 0) (0 : Y \u27f6 0) :=\nhas_limit.mk \u27e8_, is_pullback_of_is_terminal_is_product _ _ _ _\n  has_zero_object.zero_is_terminal (prod_is_prod X Y)\u27e9\n\n/-- The pullback over the zeron object is the product. -/\ndef pullback_zero_zero_iso (X Y : C) [has_binary_product X Y] :\n  pullback (0 : X \u27f6 0) (0 : Y \u27f6 0) \u2245 X \u2a2f Y :=\nlimit.iso_limit_cone \u27e8_, is_pullback_of_is_terminal_is_product _ _ _ _\n  has_zero_object.zero_is_terminal (prod_is_prod X Y)\u27e9\n\n@[simp] lemma pullback_zero_zero_iso_inv_fst (X Y : C) [has_binary_product X Y] :\n  (pullback_zero_zero_iso X Y).inv \u226b pullback.fst = prod.fst :=\nby { dsimp [pullback_zero_zero_iso], simp, }\n@[simp] lemma pullback_zero_zero_iso_inv_snd (X Y : C) [has_binary_product X Y] :\n  (pullback_zero_zero_iso X Y).inv \u226b pullback.snd = prod.snd :=\nby { dsimp [pullback_zero_zero_iso], simp, }\n@[simp] lemma pullback_zero_zero_iso_hom_fst (X Y : C) [has_binary_product X Y] :\n  (pullback_zero_zero_iso X Y).hom \u226b prod.fst = pullback.fst :=\nby { simp [\u2190iso.eq_inv_comp], }\n@[simp] lemma pullback_zero_zero_iso_hom_snd (X Y : C) [has_binary_product X Y] :\n  (pullback_zero_zero_iso X Y).hom \u226b prod.snd = pullback.snd :=\nby { simp [\u2190iso.eq_inv_comp], }\n\ninstance has_pushout_over_zero\n  (X Y : C) [has_binary_coproduct X Y] : has_pushout (0 : 0 \u27f6 X) (0 : 0 \u27f6 Y) :=\nhas_colimit.mk \u27e8_, is_pushout_of_is_initial_is_coproduct _ _ _ _\n  has_zero_object.zero_is_initial (coprod_is_coprod X Y)\u27e9\n\n/-- The pushout over the zero object is the coproduct. -/\ndef pushout_zero_zero_iso\n  (X Y : C) [has_binary_coproduct X Y] : pushout (0 : 0 \u27f6 X) (0 : 0 \u27f6 Y) \u2245 X \u2a3f Y :=\ncolimit.iso_colimit_cocone \u27e8_, is_pushout_of_is_initial_is_coproduct _ _ _ _\n  has_zero_object.zero_is_initial (coprod_is_coprod X Y)\u27e9\n\n@[simp] lemma inl_pushout_zero_zero_iso_hom (X Y : C) [has_binary_coproduct X Y] :\n  pushout.inl \u226b (pushout_zero_zero_iso X Y).hom = coprod.inl :=\nby { dsimp [pushout_zero_zero_iso], simp, }\n@[simp] lemma inr_pushout_zero_zero_iso_hom (X Y : C) [has_binary_coproduct X Y] :\n  pushout.inr \u226b (pushout_zero_zero_iso X Y).hom = coprod.inr :=\nby { dsimp [pushout_zero_zero_iso], simp, }\n@[simp] lemma inl_pushout_zero_zero_iso_inv (X Y : C) [has_binary_coproduct X Y] :\n  coprod.inl \u226b (pushout_zero_zero_iso X Y).inv = pushout.inl :=\nby { simp [iso.comp_inv_eq], }\n@[simp] lemma inr_pushout_zero_zero_iso_inv (X Y : C) [has_binary_coproduct X Y] :\n  coprod.inr \u226b (pushout_zero_zero_iso X Y).inv = pushout.inr :=\nby { simp [iso.comp_inv_eq], }\n\nend category_theory.limits\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/constructions/zero_objects.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.36925451933630454}}
{"text": "/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.balanced\nimport category_theory.limits.essentially_small\nimport category_theory.limits.opposites\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.subobject.lattice\nimport category_theory.subobject.well_powered\nimport data.set.opposite\n\n/-!\n# Separating and detecting sets\n\nThere are several non-equivalent notions of a generator of a category. Here, we consider two of\nthem:\n\n* We say that `\ud835\udca2` is a separating set if the functors `C(G, -)` for `G \u2208 \ud835\udca2` are collectively\n    faithful, i.e., if `h \u226b f = h \u226b g` for all `h` with domain in `\ud835\udca2` implies `f = g`.\n* We say that `\ud835\udca2` is a detecting set if the functors `C(G, -)` collectively reflect isomorphisms,\n    i.e., if any `h` with domain in `\ud835\udca2` uniquely factors through `f`, then `f` is an isomorphism.\n\nThere are, of course, also the dual notions of coseparating and codetecting sets.\n\n## Main results\n\nWe\n* define predicates `is_separating`, `is_coseparating`, `is_detecting` and `is_codetecting` on\n  sets of objects;\n* show that separating and coseparating are dual notions;\n* show that detecting and codetecting are dual notions;\n* show that if `C` has equalizers, then detecting implies separating;\n* show that if `C` has coequalizers, then codetecting implies separating;\n* show that if `C` is balanced, then separating implies detecting and coseparating implies\n  codetecting;\n* show that `\u2205` is separating if and only if `\u2205` is coseparating if and only if `C` is thin;\n* show that `\u2205` is detecting if and only if `\u2205` is codetecting if and only if `C` is a groupoid;\n* define predicates `is_separator`, `is_coseparator`, `is_detector` and `is_codetector` as the\n  singleton counterparts to the definitions for sets above and restate the above results in this\n  situation;\n* show that `G` is a separator if and only if `coyoneda.obj (op G)` is faithful (and the dual);\n* show that `G` is a detector if and only if `coyoneda.obj (op G)` reflects isomorphisms (and the\n  dual).\n\n## Future work\n\n* We currently don't have any examples yet.\n* We will want typeclasses `has_separator C` and similar.\n\n-/\n\nuniverses w v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category_theory.limits opposite\n\nnamespace category_theory\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n\n/-- We say that `\ud835\udca2` is a separating set if the functors `C(G, -)` for `G \u2208 \ud835\udca2` are collectively\n    faithful, i.e., if `h \u226b f = h \u226b g` for all `h` with domain in `\ud835\udca2` implies `f = g`. -/\ndef is_separating (\ud835\udca2 : set C) : Prop :=\n\u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 (G \u2208 \ud835\udca2) (h : G \u27f6 X), h \u226b f = h \u226b g) \u2192 f = g\n\n/-- We say that `\ud835\udca2` is a coseparating set if the functors `C(-, G)` for `G \u2208 \ud835\udca2` are collectively\n    faithful, i.e., if `f \u226b h = g \u226b h` for all `h` with codomain in `\ud835\udca2` implies `f = g`. -/\ndef is_coseparating (\ud835\udca2 : set C) : Prop :=\n\u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 (G \u2208 \ud835\udca2) (h : Y \u27f6 G), f \u226b h = g \u226b h) \u2192 f = g\n\n/-- We say that `\ud835\udca2` is a detecting set if the functors `C(G, -)` collectively reflect isomorphisms,\n    i.e., if any `h` with domain in `\ud835\udca2` uniquely factors through `f`, then `f` is an isomorphism. -/\ndef is_detecting (\ud835\udca2 : set C) : Prop :=\n\u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 (G \u2208 \ud835\udca2) (h : G \u27f6 Y), \u2203! (h' : G \u27f6 X), h' \u226b f = h) \u2192 is_iso f\n\n/-- We say that `\ud835\udca2` is a codetecting set if the functors `C(-, G)` collectively reflect\n    isomorphisms, i.e., if any `h` with codomain in `G` uniquely factors through `f`, then `f` is\n    an isomorphism. -/\ndef is_codetecting (\ud835\udca2 : set C) : Prop :=\n\u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 (G \u2208 \ud835\udca2) (h : X \u27f6 G), \u2203! (h' : Y \u27f6 G), f \u226b h' = h) \u2192 is_iso f\n\nsection dual\n\nlemma is_separating_op_iff (\ud835\udca2 : set C) : is_separating \ud835\udca2.op \u2194 is_coseparating \ud835\udca2 :=\nbegin\n  refine \u27e8\u03bb h\ud835\udca2 X Y f g hfg, _, \u03bb h\ud835\udca2 X Y f g hfg, _\u27e9,\n  { refine quiver.hom.op_inj (h\ud835\udca2 _ _ (\u03bb G hG h, quiver.hom.unop_inj _)),\n    simpa only [unop_comp, quiver.hom.unop_op] using hfg _ (set.mem_op.1 hG) _ },\n  { refine quiver.hom.unop_inj (h\ud835\udca2 _ _ (\u03bb G hG h, quiver.hom.op_inj _)),\n    simpa only [op_comp, quiver.hom.op_unop] using hfg _ (set.op_mem_op.2 hG) _ }\nend\n\nlemma is_coseparating_op_iff (\ud835\udca2 : set C) : is_coseparating \ud835\udca2.op \u2194 is_separating \ud835\udca2 :=\nbegin\n  refine \u27e8\u03bb h\ud835\udca2 X Y f g hfg, _, \u03bb h\ud835\udca2 X Y f g hfg, _\u27e9,\n  { refine quiver.hom.op_inj (h\ud835\udca2 _ _ (\u03bb G hG h, quiver.hom.unop_inj _)),\n    simpa only [unop_comp, quiver.hom.unop_op] using hfg _ (set.mem_op.1 hG) _ },\n  { refine quiver.hom.unop_inj (h\ud835\udca2 _ _ (\u03bb G hG h, quiver.hom.op_inj _)),\n    simpa only [op_comp, quiver.hom.op_unop] using hfg _ (set.op_mem_op.2 hG) _ }\nend\n\nlemma is_coseparating_unop_iff (\ud835\udca2 : set C\u1d52\u1d56) : is_coseparating \ud835\udca2.unop \u2194 is_separating \ud835\udca2 :=\nby rw [\u2190 is_separating_op_iff, set.unop_op]\n\nlemma is_separating_unop_iff (\ud835\udca2 : set C\u1d52\u1d56) : is_separating \ud835\udca2.unop \u2194 is_coseparating \ud835\udca2 :=\nby rw [\u2190 is_coseparating_op_iff, set.unop_op]\n\nlemma is_detecting_op_iff (\ud835\udca2 : set C) : is_detecting \ud835\udca2.op \u2194 is_codetecting \ud835\udca2 :=\nbegin\n  refine \u27e8\u03bb h\ud835\udca2 X Y f hf, _, \u03bb h\ud835\udca2 X Y f hf, _\u27e9,\n  { refine (is_iso_op_iff _).1 (h\ud835\udca2 _ (\u03bb G hG h, _)),\n    obtain \u27e8t, ht, ht'\u27e9 := hf (unop G) (set.mem_op.1 hG) h.unop,\n    exact \u27e8t.op, quiver.hom.unop_inj ht, \u03bb y hy,\n      quiver.hom.unop_inj (ht' _ (quiver.hom.op_inj hy))\u27e9 },\n  { refine (is_iso_unop_iff _).1 (h\ud835\udca2 _ (\u03bb G hG h, _)),\n    obtain \u27e8t, ht, ht'\u27e9 := hf (op G) (set.op_mem_op.2 hG) h.op,\n    refine \u27e8t.unop, quiver.hom.op_inj ht, \u03bb y hy, quiver.hom.op_inj (ht' _ _)\u27e9,\n    exact quiver.hom.unop_inj (by simpa only using hy) }\nend\n\nlemma is_codetecting_op_iff (\ud835\udca2 : set C) : is_codetecting \ud835\udca2.op \u2194 is_detecting \ud835\udca2 :=\nbegin\n  refine \u27e8\u03bb h\ud835\udca2 X Y f hf, _, \u03bb h\ud835\udca2 X Y f hf, _\u27e9,\n  { refine (is_iso_op_iff _).1 (h\ud835\udca2 _ (\u03bb G hG h, _)),\n    obtain \u27e8t, ht, ht'\u27e9 := hf (unop G) (set.mem_op.1 hG) h.unop,\n    exact \u27e8t.op, quiver.hom.unop_inj ht, \u03bb y hy,\n      quiver.hom.unop_inj (ht' _ (quiver.hom.op_inj hy))\u27e9 },\n  { refine (is_iso_unop_iff _).1 (h\ud835\udca2 _ (\u03bb G hG h, _)),\n    obtain \u27e8t, ht, ht'\u27e9 := hf (op G) (set.op_mem_op.2 hG) h.op,\n    refine \u27e8t.unop, quiver.hom.op_inj ht, \u03bb y hy, quiver.hom.op_inj (ht' _ _)\u27e9,\n    exact quiver.hom.unop_inj (by simpa only using hy) }\nend\n\nlemma is_detecting_unop_iff (\ud835\udca2 : set C\u1d52\u1d56) : is_detecting \ud835\udca2.unop \u2194 is_codetecting \ud835\udca2 :=\nby rw [\u2190 is_codetecting_op_iff, set.unop_op]\n\nlemma is_codetecting_unop_iff {\ud835\udca2 : set C\u1d52\u1d56} : is_codetecting \ud835\udca2.unop \u2194 is_detecting \ud835\udca2 :=\nby rw [\u2190 is_detecting_op_iff, set.unop_op]\n\nend dual\n\nlemma is_detecting.is_separating [has_equalizers C] {\ud835\udca2 : set C} (h\ud835\udca2 : is_detecting \ud835\udca2) :\n  is_separating \ud835\udca2 :=\n\u03bb X Y f g hfg,\n  have is_iso (equalizer.\u03b9 f g), from h\ud835\udca2 _ (\u03bb G hG h, equalizer.exists_unique _ (hfg _ hG _)),\n  by exactI eq_of_epi_equalizer\n\nsection\n\nlemma is_codetecting.is_coseparating [has_coequalizers C] {\ud835\udca2 : set C} :\n  is_codetecting \ud835\udca2 \u2192 is_coseparating \ud835\udca2 :=\nby simpa only [\u2190 is_separating_op_iff, \u2190 is_detecting_op_iff] using is_detecting.is_separating\n\nend\n\nlemma is_separating.is_detecting [balanced C] {\ud835\udca2 : set C} (h\ud835\udca2 : is_separating \ud835\udca2) :\n  is_detecting \ud835\udca2 :=\nbegin\n  intros X Y f hf,\n  refine (is_iso_iff_mono_and_epi _).2 \u27e8\u27e8\u03bb Z g h hgh, h\ud835\udca2 _ _ (\u03bb G hG i, _)\u27e9, \u27e8\u03bb Z g h hgh, _\u27e9\u27e9,\n  { obtain \u27e8t, -, ht\u27e9 := hf G hG (i \u226b g \u226b f),\n    rw [ht (i \u226b g) (category.assoc _ _ _), ht (i \u226b h) (hgh.symm \u25b8 category.assoc _ _ _)] },\n  { refine h\ud835\udca2 _ _ (\u03bb G hG i, _),\n    obtain \u27e8t, rfl, -\u27e9 := hf G hG i,\n    rw [category.assoc, hgh, category.assoc] }\nend\n\nsection\nlocal attribute [instance] balanced_opposite\n\nlemma is_coseparating.is_codetecting [balanced C] {\ud835\udca2 : set C} :\n  is_coseparating \ud835\udca2 \u2192 is_codetecting \ud835\udca2 :=\nby simpa only [\u2190 is_detecting_op_iff, \u2190 is_separating_op_iff] using is_separating.is_detecting\n\nend\n\nlemma is_detecting_iff_is_separating [has_equalizers C] [balanced C] (\ud835\udca2 : set C) :\n  is_detecting \ud835\udca2 \u2194 is_separating \ud835\udca2 :=\n\u27e8is_detecting.is_separating, is_separating.is_detecting\u27e9\n\nlemma is_codetecting_iff_is_coseparating [has_coequalizers C] [balanced C] {\ud835\udca2 : set C} :\n  is_codetecting \ud835\udca2 \u2194 is_coseparating \ud835\udca2 :=\n\u27e8is_codetecting.is_coseparating, is_coseparating.is_codetecting\u27e9\n\nsection mono\n\nlemma is_separating.mono {\ud835\udca2 : set C} (h\ud835\udca2 : is_separating \ud835\udca2) {\u210b : set C} (h\ud835\udca2\u210b : \ud835\udca2 \u2286 \u210b) :\n  is_separating \u210b :=\n\u03bb X Y f g hfg, h\ud835\udca2 _ _ $ \u03bb G hG h, hfg _ (h\ud835\udca2\u210b hG) _\n\nlemma is_coseparating.mono {\ud835\udca2 : set C} (h\ud835\udca2 : is_coseparating \ud835\udca2) {\u210b : set C} (h\ud835\udca2\u210b : \ud835\udca2 \u2286 \u210b) :\n  is_coseparating \u210b :=\n\u03bb X Y f g hfg, h\ud835\udca2 _ _ $ \u03bb G hG h, hfg _ (h\ud835\udca2\u210b hG) _\n\nlemma is_detecting.mono {\ud835\udca2 : set C} (h\ud835\udca2 : is_detecting \ud835\udca2) {\u210b : set C} (h\ud835\udca2\u210b : \ud835\udca2 \u2286 \u210b) :\n  is_detecting \u210b :=\n\u03bb X Y f hf, h\ud835\udca2 _ $ \u03bb G hG h, hf _ (h\ud835\udca2\u210b hG) _\n\nlemma is_codetecting.mono {\ud835\udca2 : set C} (h\ud835\udca2 : is_codetecting \ud835\udca2) {\u210b : set C} (h\ud835\udca2\u210b : \ud835\udca2 \u2286 \u210b) :\n  is_codetecting \u210b :=\n\u03bb X Y f hf, h\ud835\udca2 _ $ \u03bb G hG h, hf _ (h\ud835\udca2\u210b hG) _\n\nend mono\n\nsection empty\n\nlemma thin_of_is_separating_empty (h : is_separating (\u2205 : set C)) : quiver.is_thin C :=\n\u03bb _ _, \u27e8\u03bb f g, h _ _ $ \u03bb G, false.elim\u27e9\n\nlemma is_separating_empty_of_thin [quiver.is_thin C] : is_separating (\u2205 : set C) :=\n\u03bb X Y f g hfg, subsingleton.elim _ _\n\n\n\nlemma is_coseparating_empty_of_thin [quiver.is_thin C] :\n  is_coseparating (\u2205 : set C) :=\n\u03bb X Y f g hfg, subsingleton.elim _ _\n\nlemma groupoid_of_is_detecting_empty (h : is_detecting (\u2205 : set C)) {X Y : C} (f : X \u27f6 Y) :\n  is_iso f :=\nh _ $ \u03bb G, false.elim\n\nlemma is_detecting_empty_of_groupoid [\u2200 {X Y : C} (f : X \u27f6 Y), is_iso f] :\n  is_detecting (\u2205 : set C) :=\n\u03bb X Y f hf, infer_instance\n\nlemma groupoid_of_is_codetecting_empty (h : is_codetecting (\u2205 : set C)) {X Y : C} (f : X \u27f6 Y) :\n  is_iso f :=\nh _ $ \u03bb G, false.elim\n\nlemma is_codetecting_empty_of_groupoid [\u2200 {X Y : C} (f : X \u27f6 Y), is_iso f] :\n  is_codetecting (\u2205 : set C) :=\n\u03bb X Y f hf, infer_instance\n\nend empty\n\nlemma is_separating_iff_epi (\ud835\udca2 : set C)\n  [\u03a0 (A : C), has_coproduct (\u03bb f : \u03a3 G : \ud835\udca2, (G : C) \u27f6 A, (f.1 : C))] :\n  is_separating \ud835\udca2 \u2194 \u2200 A : C, epi (sigma.desc (@sigma.snd \ud835\udca2 (\u03bb G, (G : C) \u27f6 A))) :=\nbegin\n  refine \u27e8\u03bb h A, \u27e8\u03bb Z u v huv, h _ _ (\u03bb G hG f, _)\u27e9, \u03bb h X Y f g hh, _\u27e9,\n  { simpa using (sigma.\u03b9 (\u03bb f : \u03a3 G : \ud835\udca2, (G : C) \u27f6 A, (f.1 : C)) \u27e8\u27e8G, hG\u27e9, f\u27e9) \u226b= huv },\n  { haveI := h X,\n    refine (cancel_epi (sigma.desc (@sigma.snd \ud835\udca2 (\u03bb G, (G : C) \u27f6 X)))).1 (colimit.hom_ext (\u03bb j, _)),\n    simpa using hh j.as.1.1 j.as.1.2 j.as.2 }\nend\n\nlemma is_coseparating_iff_mono (\ud835\udca2 : set C)\n  [\u03a0 (A : C), has_product (\u03bb f : \u03a3 G : \ud835\udca2, A \u27f6 (G : C), (f.1 : C))] :\n  is_coseparating \ud835\udca2 \u2194 \u2200 A : C, mono (pi.lift (@sigma.snd \ud835\udca2 (\u03bb G, A \u27f6 (G : C)))) :=\nbegin\n  refine \u27e8\u03bb h A, \u27e8\u03bb Z u v huv, h _ _ (\u03bb G hG f, _)\u27e9, \u03bb h X Y f g hh, _\u27e9,\n  { simpa using huv =\u226b (pi.\u03c0 (\u03bb f : \u03a3 G : \ud835\udca2, A \u27f6 (G : C), (f.1 : C)) \u27e8\u27e8G, hG\u27e9, f\u27e9) },\n  { haveI := h Y,\n    refine (cancel_mono (pi.lift (@sigma.snd \ud835\udca2 (\u03bb G, Y \u27f6 (G : C))))).1 (limit.hom_ext (\u03bb j, _)),\n    simpa using hh j.as.1.1 j.as.1.2 j.as.2 }\nend\n\n/-- An ingredient of the proof of the Special Adjoint Functor Theorem: a complete well-powered\n    category with a small coseparating set has an initial object.\n\n    In fact, it follows from the Special Adjoint Functor Theorem that `C` is already cocomplete,\n    see `has_colimits_of_has_limits_of_is_coseparating`. -/\nlemma has_initial_of_is_coseparating [well_powered C] [has_limits C] {\ud835\udca2 : set C} [small.{v\u2081} \ud835\udca2]\n  (h\ud835\udca2 : is_coseparating \ud835\udca2) : has_initial C :=\nbegin\n  haveI := has_products_of_shape_of_small C \ud835\udca2,\n  haveI := \u03bb A, has_products_of_shape_of_small.{v\u2081} C (\u03a3 G : \ud835\udca2, A \u27f6 (G : C)),\n  letI := complete_lattice_of_complete_semilattice_Inf (subobject (pi_obj (coe : \ud835\udca2 \u2192 C))),\n  suffices : \u2200 A : C, unique (((\u22a5 : subobject (pi_obj (coe : \ud835\udca2 \u2192 C))) : C) \u27f6 A),\n  { exactI has_initial_of_unique ((\u22a5 : subobject (pi_obj (coe : \ud835\udca2 \u2192 C))) : C) },\n  refine \u03bb A, \u27e8\u27e8_\u27e9, \u03bb f, _\u27e9,\n  { let s := pi.lift (\u03bb f : \u03a3 G : \ud835\udca2, A \u27f6 (G : C), id (pi.\u03c0 (coe : \ud835\udca2 \u2192 C)) f.1),\n    let t := pi.lift (@sigma.snd \ud835\udca2 (\u03bb G, A \u27f6 (G : C))),\n    haveI : mono t := (is_coseparating_iff_mono \ud835\udca2).1 h\ud835\udca2 A,\n    exact subobject.of_le_mk _ (pullback.fst : pullback s t \u27f6 _) bot_le \u226b pullback.snd },\n  { generalize : default = g,\n    suffices : is_split_epi (equalizer.\u03b9 f g),\n    { exactI eq_of_epi_equalizer },\n    exact is_split_epi.mk' \u27e8subobject.of_le_mk _ (equalizer.\u03b9 f g \u226b subobject.arrow _)\n      bot_le, by { ext, simp }\u27e9 }\nend\n\n/-- An ingredient of the proof of the Special Adjoint Functor Theorem: a cocomplete well-copowered\n    category with a small separating set has a terminal object.\n\n    In fact, it follows from the Special Adjoint Functor Theorem that `C` is already complete, see\n    `has_limits_of_has_colimits_of_is_separating`. -/\nlemma has_terminal_of_is_separating [well_powered C\u1d52\u1d56] [has_colimits C] {\ud835\udca2 : set C} [small.{v\u2081} \ud835\udca2]\n  (h\ud835\udca2 : is_separating \ud835\udca2) : has_terminal C :=\nbegin\n  haveI : small.{v\u2081} \ud835\udca2.op := small_of_injective (set.op_equiv_self \ud835\udca2).injective,\n  haveI : has_initial C\u1d52\u1d56 := has_initial_of_is_coseparating ((is_coseparating_op_iff _).2 h\ud835\udca2),\n  exact has_terminal_of_has_initial_op\nend\n\nsection well_powered\n\nnamespace subobject\n\nlemma eq_of_le_of_is_detecting {\ud835\udca2 : set C} (h\ud835\udca2 : is_detecting \ud835\udca2) {X : C} (P Q : subobject X)\n  (h\u2081 : P \u2264 Q) (h\u2082 : \u2200 (G \u2208 \ud835\udca2) {f : G \u27f6 X}, Q.factors f \u2192 P.factors f) : P = Q :=\nbegin\n  suffices : is_iso (of_le _ _ h\u2081),\n  { exactI le_antisymm h\u2081 (le_of_comm (inv (of_le _ _ h\u2081)) (by simp)) },\n  refine h\ud835\udca2 _ (\u03bb G hG f, _),\n  have : P.factors (f \u226b Q.arrow) := h\u2082 _ hG ((factors_iff _ _).2 \u27e8_, rfl\u27e9),\n  refine \u27e8factor_thru _ _ this, _, \u03bb g (hg : g \u226b _ = f), _\u27e9,\n  { simp only [\u2190 cancel_mono Q.arrow, category.assoc, of_le_arrow, factor_thru_arrow] },\n  { simp only [\u2190 cancel_mono (subobject.of_le _ _ h\u2081), \u2190 cancel_mono Q.arrow, hg,\n      category.assoc, of_le_arrow, factor_thru_arrow] }\nend\n\nlemma inf_eq_of_is_detecting [has_pullbacks C] {\ud835\udca2 : set C} (h\ud835\udca2 : is_detecting \ud835\udca2) {X : C}\n  (P Q : subobject X) (h : \u2200 (G \u2208 \ud835\udca2) {f : G \u27f6 X}, P.factors f \u2192 Q.factors f) : P \u2293 Q = P :=\neq_of_le_of_is_detecting h\ud835\udca2 _ _ _root_.inf_le_left (\u03bb G hG f hf, (inf_factors _).2 \u27e8hf, h _ hG hf\u27e9)\n\nlemma eq_of_is_detecting [has_pullbacks C] {\ud835\udca2 : set C} (h\ud835\udca2 : is_detecting \ud835\udca2) {X : C}\n  (P Q : subobject X) (h : \u2200 (G \u2208 \ud835\udca2) {f : G \u27f6 X}, P.factors f \u2194 Q.factors f) : P = Q :=\ncalc P = P \u2293 Q : eq.symm $ inf_eq_of_is_detecting h\ud835\udca2 _ _ $ \u03bb G hG f hf, (h G hG).1 hf\n   ... = Q \u2293 P : inf_comm\n   ... = Q     : inf_eq_of_is_detecting h\ud835\udca2 _ _ $ \u03bb G hG f hf, (h G hG).2 hf\n\nend subobject\n\n/-- A category with pullbacks and a small detecting set is well-powered. -/\nlemma well_powered_of_is_detecting [has_pullbacks C] {\ud835\udca2 : set C} [small.{v\u2081} \ud835\udca2]\n  (h\ud835\udca2 : is_detecting \ud835\udca2) : well_powered C :=\n\u27e8\u03bb X, @small_of_injective _ _ _ (\u03bb P : subobject X, { f : \u03a3 G : \ud835\udca2, G.1 \u27f6 X | P.factors f.2 }) $\n  \u03bb P Q h, subobject.eq_of_is_detecting h\ud835\udca2 _ _ (by simpa [set.ext_iff] using h)\u27e9\n\nend well_powered\n\nnamespace structured_arrow\nvariables (S : D) (T : C \u2964 D)\n\nlemma is_coseparating_proj_preimage {\ud835\udca2 : set C} (h\ud835\udca2 : is_coseparating \ud835\udca2) :\n  is_coseparating ((proj S T).obj \u207b\u00b9' \ud835\udca2) :=\nbegin\n  refine \u03bb X Y f g hfg, ext _ _ (h\ud835\udca2 _ _ (\u03bb G hG h, _)),\n  exact congr_arg comma_morphism.right (hfg (mk (Y.hom \u226b T.map h)) hG (hom_mk h rfl))\nend\n\nend structured_arrow\n\nnamespace costructured_arrow\nvariables (S : C \u2964 D) (T : D)\n\nlemma is_separating_proj_preimage {\ud835\udca2 : set C} (h\ud835\udca2 : is_separating \ud835\udca2) :\n  is_separating ((proj S T).obj \u207b\u00b9' \ud835\udca2) :=\nbegin\n  refine \u03bb X Y f g hfg, ext _ _ (h\ud835\udca2 _ _ (\u03bb G hG h, _)),\n  convert congr_arg comma_morphism.left (hfg (mk (S.map h \u226b X.hom)) hG (hom_mk h rfl))\nend\n\nend costructured_arrow\n\n/-- We say that `G` is a separator if the functor `C(G, -)` is faithful. -/\ndef is_separator (G : C) : Prop :=\nis_separating ({G} : set C)\n\n/-- We say that `G` is a coseparator if the functor `C(-, G)` is faithful. -/\ndef is_coseparator (G : C) : Prop :=\nis_coseparating ({G} : set C)\n\n/-- We say that `G` is a detector if the functor `C(G, -)` reflects isomorphisms. -/\ndef is_detector (G : C) : Prop :=\nis_detecting ({G} : set C)\n\n/-- We say that `G` is a codetector if the functor `C(-, G)` reflects isomorphisms. -/\ndef is_codetector (G : C) : Prop :=\nis_codetecting ({G} : set C)\n\nsection dual\n\nlemma is_separator_op_iff (G : C) : is_separator (op G) \u2194 is_coseparator G :=\nby rw [is_separator, is_coseparator, \u2190 is_separating_op_iff, set.singleton_op]\n\nlemma is_coseparator_op_iff (G : C) : is_coseparator (op G) \u2194 is_separator G :=\nby rw [is_separator, is_coseparator, \u2190 is_coseparating_op_iff, set.singleton_op]\n\nlemma is_coseparator_unop_iff (G : C\u1d52\u1d56) : is_coseparator (unop G) \u2194 is_separator G :=\nby rw [is_separator, is_coseparator, \u2190 is_coseparating_unop_iff, set.singleton_unop]\n\nlemma is_separator_unop_iff (G : C\u1d52\u1d56) : is_separator (unop G) \u2194 is_coseparator G :=\nby rw [is_separator, is_coseparator, \u2190 is_separating_unop_iff, set.singleton_unop]\n\nlemma is_detector_op_iff (G : C) : is_detector (op G) \u2194 is_codetector G :=\nby rw [is_detector, is_codetector, \u2190 is_detecting_op_iff, set.singleton_op]\n\nlemma is_codetector_op_iff (G : C) : is_codetector (op G) \u2194 is_detector G :=\nby rw [is_detector, is_codetector, \u2190 is_codetecting_op_iff, set.singleton_op]\n\nlemma is_codetector_unop_iff (G : C\u1d52\u1d56) : is_codetector (unop G) \u2194 is_detector G :=\nby rw [is_detector, is_codetector, \u2190 is_codetecting_unop_iff, set.singleton_unop]\n\nlemma is_detector_unop_iff (G : C\u1d52\u1d56) : is_detector (unop G) \u2194 is_codetector G :=\nby rw [is_detector, is_codetector, \u2190 is_detecting_unop_iff, set.singleton_unop]\n\nend dual\n\nlemma is_detector.is_separator [has_equalizers C] {G : C} : is_detector G \u2192 is_separator G :=\nis_detecting.is_separating\n\nlemma is_codetector.is_coseparator [has_coequalizers C] {G : C} :\n  is_codetector G \u2192 is_coseparator G :=\nis_codetecting.is_coseparating\n\nlemma is_separator.is_detector [balanced C] {G : C} : is_separator G \u2192 is_detector G :=\nis_separating.is_detecting\n\nlemma is_cospearator.is_codetector [balanced C] {G : C} : is_coseparator G \u2192 is_codetector G :=\nis_coseparating.is_codetecting\n\nlemma is_separator_def (G : C) :\n  is_separator G \u2194 \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 h : G \u27f6 X, h \u226b f = h \u226b g) \u2192 f = g :=\n\u27e8\u03bb hG X Y f g hfg, hG _ _ $ \u03bb H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hfg h },\n \u03bb hG X Y f g hfg, hG _ _ $ \u03bb h, hfg _ (set.mem_singleton _) _\u27e9\n\nlemma is_separator.def {G : C} :\n  is_separator G \u2192 \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 h : G \u27f6 X, h \u226b f = h \u226b g) \u2192 f = g :=\n(is_separator_def _).1\n\nlemma is_coseparator_def (G : C) :\n  is_coseparator G \u2194 \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 h : Y \u27f6 G, f \u226b h = g \u226b h) \u2192 f = g :=\n\u27e8\u03bb hG X Y f g hfg, hG _ _ $ \u03bb H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hfg h },\n \u03bb hG X Y f g hfg, hG _ _ $ \u03bb h, hfg _ (set.mem_singleton _) _\u27e9\n\nlemma is_coseparator.def {G : C} :\n  is_coseparator G \u2192 \u2200 \u2983X Y : C\u2984 (f g : X \u27f6 Y), (\u2200 h : Y \u27f6 G, f \u226b h = g \u226b h) \u2192 f = g :=\n(is_coseparator_def _).1\n\nlemma is_detector_def (G : C) :\n  is_detector G \u2194 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 h : G \u27f6 Y, \u2203! h', h' \u226b f = h) \u2192 is_iso f :=\n\u27e8\u03bb hG X Y f hf, hG _ $ \u03bb H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hf h },\n \u03bb hG X Y f hf, hG _ $ \u03bb h, hf _ (set.mem_singleton _) _\u27e9\n\nlemma is_detector.def {G : C} :\n  is_detector G \u2192 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 h : G \u27f6 Y, \u2203! h', h' \u226b f = h) \u2192 is_iso f :=\n(is_detector_def _).1\n\nlemma is_codetector_def (G : C) :\n  is_codetector G \u2194 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 h : X \u27f6 G, \u2203! h', f \u226b h' = h) \u2192 is_iso f :=\n\u27e8\u03bb hG X Y f hf, hG _ $ \u03bb H hH h, by { obtain rfl := set.mem_singleton_iff.1 hH, exact hf h },\n \u03bb hG X Y f hf, hG _ $ \u03bb h, hf _ (set.mem_singleton _) _\u27e9\n\nlemma is_codetector.def {G : C} :\n  is_codetector G \u2192 \u2200 \u2983X Y : C\u2984 (f : X \u27f6 Y), (\u2200 h : X \u27f6 G, \u2203! h', f \u226b h' = h) \u2192 is_iso f :=\n(is_codetector_def _).1\n\nlemma is_separator_iff_faithful_coyoneda_obj (G : C) :\n  is_separator G \u2194 faithful (coyoneda.obj (op G)) :=\n\u27e8\u03bb hG, \u27e8\u03bb X Y f g hfg, hG.def _ _ (congr_fun hfg)\u27e9,\n \u03bb h, (is_separator_def _).2 $ \u03bb X Y f g hfg,\n  by exactI (coyoneda.obj (op G)).map_injective (funext hfg)\u27e9\n\nlemma is_coseparator_iff_faithful_yoneda_obj (G : C) :\n  is_coseparator G \u2194 faithful (yoneda.obj G) :=\n\u27e8\u03bb hG, \u27e8\u03bb X Y f g hfg, quiver.hom.unop_inj (hG.def _ _ (congr_fun hfg))\u27e9,\n \u03bb h, (is_coseparator_def _).2 $ \u03bb X Y f g hfg, quiver.hom.op_inj $\n  by exactI (yoneda.obj G).map_injective (funext hfg)\u27e9\n\nlemma is_separator_iff_epi (G : C) [\u03a0 A : C, has_coproduct (\u03bb (f : G \u27f6 A), G)] :\n  is_separator G \u2194 \u2200 (A : C), epi (sigma.desc (\u03bb (f : G \u27f6 A), f)) :=\nbegin\n  rw is_separator_def,\n  refine \u27e8\u03bb h A, \u27e8\u03bb Z u v huv, h _ _ (\u03bb i, _)\u27e9, \u03bb h X Y f g hh, _\u27e9,\n  { simpa using (sigma.\u03b9 _ i) \u226b= huv },\n  { haveI := h X,\n    refine (cancel_epi (sigma.desc (\u03bb (f : G \u27f6 X), f))).1 (colimit.hom_ext (\u03bb j, _)),\n    simpa using hh j.as }\nend\n\nlemma is_coseparator_iff_mono (G : C) [\u03a0 A : C, has_product (\u03bb (f : A \u27f6 G), G)] :\n  is_coseparator G \u2194 \u2200 (A : C), mono (pi.lift (\u03bb (f : A \u27f6 G), f)) :=\nbegin\n  rw is_coseparator_def,\n  refine \u27e8\u03bb h A, \u27e8\u03bb Z u v huv, h _ _ (\u03bb i, _)\u27e9, \u03bb h X Y f g hh, _\u27e9,\n  { simpa using huv =\u226b (pi.\u03c0 _ i) },\n  { haveI := h Y,\n    refine (cancel_mono (pi.lift (\u03bb (f : Y \u27f6 G), f))).1 (limit.hom_ext (\u03bb j, _)),\n    simpa using hh j.as }\nend\n\nsection zero_morphisms\nvariables [has_zero_morphisms C]\n\nlemma is_separator_coprod (G H : C) [has_binary_coproduct G H] :\n  is_separator (G \u2a3f H) \u2194 is_separating ({G, H} : set C) :=\nbegin\n  refine \u27e8\u03bb h X Y u v huv, _, \u03bb h, (is_separator_def _).2 (\u03bb X Y u v huv, h _ _ (\u03bb Z hZ g, _))\u27e9,\n  { refine h.def _ _ (\u03bb g, coprod.hom_ext _ _),\n    { simpa using huv G (by simp) (coprod.inl \u226b g) },\n    { simpa using huv H (by simp) (coprod.inr \u226b g) } },\n  { simp only [set.mem_insert_iff, set.mem_singleton_iff] at hZ,\n    unfreezingI { rcases hZ with rfl|rfl },\n    { simpa using coprod.inl \u226b= huv (coprod.desc g 0) },\n    { simpa using coprod.inr \u226b= huv (coprod.desc 0 g) } }\nend\n\nlemma is_separator_coprod_of_is_separator_left (G H : C) [has_binary_coproduct G H]\n  (hG : is_separator G) : is_separator (G \u2a3f H) :=\n(is_separator_coprod _ _).2 $ is_separating.mono hG $ by simp\n\nlemma is_separator_coprod_of_is_separator_right (G H : C) [has_binary_coproduct G H]\n  (hH : is_separator H) : is_separator (G \u2a3f H) :=\n(is_separator_coprod _ _).2 $ is_separating.mono hH $ by simp\n\nlemma is_separator_sigma {\u03b2 : Type w} (f : \u03b2 \u2192 C) [has_coproduct f] :\n  is_separator (\u2210 f) \u2194 is_separating (set.range f) :=\nbegin\n  refine \u27e8\u03bb h X Y u v huv, _, \u03bb h, (is_separator_def _).2 (\u03bb X Y u v huv, h _ _ (\u03bb Z hZ g, _))\u27e9,\n  { refine h.def _ _ (\u03bb g, colimit.hom_ext (\u03bb b, _)),\n    simpa using huv (f b.as) (by simp) (colimit.\u03b9 (discrete.functor f) _ \u226b g) },\n  { obtain \u27e8b, rfl\u27e9 := set.mem_range.1 hZ,\n    classical,\n    simpa using sigma.\u03b9 f b \u226b= huv (sigma.desc (pi.single b g)) }\nend\n\nlemma is_separator_sigma_of_is_separator {\u03b2 : Type w} (f : \u03b2 \u2192 C) [has_coproduct f]\n  (b : \u03b2) (hb : is_separator (f b)) : is_separator (\u2210 f) :=\n(is_separator_sigma _).2 $ is_separating.mono hb $ by simp\n\nlemma is_coseparator_prod (G H : C) [has_binary_product G H] :\n  is_coseparator (G \u2a2f H) \u2194 is_coseparating ({G, H} : set C) :=\nbegin\n  refine \u27e8\u03bb h X Y u v huv, _, \u03bb h, (is_coseparator_def _).2 (\u03bb X Y u v huv, h _ _ (\u03bb Z hZ g, _))\u27e9,\n  { refine h.def _ _ (\u03bb g, prod.hom_ext _ _),\n    { simpa using huv G (by simp) (g \u226b limits.prod.fst) },\n    { simpa using huv H (by simp) (g \u226b limits.prod.snd) } },\n  { simp only [set.mem_insert_iff, set.mem_singleton_iff] at hZ,\n    unfreezingI { rcases hZ with rfl|rfl },\n    { simpa using huv (prod.lift g 0) =\u226b limits.prod.fst },\n    { simpa using huv (prod.lift 0 g) =\u226b limits.prod.snd } }\nend\n\nlemma is_coseparator_prod_of_is_coseparator_left (G H : C) [has_binary_product G H]\n  (hG : is_coseparator G) : is_coseparator (G \u2a2f H) :=\n(is_coseparator_prod _ _).2 $ is_coseparating.mono hG $ by simp\n\nlemma is_coseparator_prod_of_is_coseparator_right (G H : C) [has_binary_product G H]\n  (hH : is_coseparator H) : is_coseparator (G \u2a2f H) :=\n(is_coseparator_prod _ _).2 $ is_coseparating.mono hH $ by simp\n\nlemma is_coseparator_pi {\u03b2 : Type w} (f : \u03b2 \u2192 C) [has_product f] :\n  is_coseparator (\u220f f) \u2194 is_coseparating (set.range f) :=\nbegin\n  refine \u27e8\u03bb h X Y u v huv, _, \u03bb h, (is_coseparator_def _).2 (\u03bb X Y u v huv, h _ _ (\u03bb Z hZ g, _))\u27e9,\n  { refine h.def _ _ (\u03bb g, limit.hom_ext (\u03bb b, _)),\n    simpa using huv (f b.as) (by simp) (g \u226b limit.\u03c0 (discrete.functor f) _ ) },\n  { obtain \u27e8b, rfl\u27e9 := set.mem_range.1 hZ,\n    classical,\n    simpa using huv (pi.lift (pi.single b g)) =\u226b pi.\u03c0 f b }\nend\n\nlemma is_coseparator_pi_of_is_coseparator {\u03b2 : Type w} (f : \u03b2 \u2192 C) [has_product f]\n  (b : \u03b2) (hb : is_coseparator (f b)) : is_coseparator (\u220f f) :=\n(is_coseparator_pi _).2 $ is_coseparating.mono hb $ by simp\n\nend zero_morphisms\n\nlemma is_detector_iff_reflects_isomorphisms_coyoneda_obj (G : C) :\n  is_detector G \u2194 reflects_isomorphisms (coyoneda.obj (op G)) :=\nbegin\n  refine \u27e8\u03bb hG, \u27e8\u03bb X Y f hf, hG.def _ (\u03bb h, _)\u27e9, \u03bb h, (is_detector_def _).2 (\u03bb X Y f hf, _)\u27e9,\n  { rw [is_iso_iff_bijective, function.bijective_iff_exists_unique] at hf,\n    exact hf h },\n  { suffices : is_iso ((coyoneda.obj (op G)).map f),\n    { exactI @is_iso_of_reflects_iso _ _ _ _ _ _ _ (coyoneda.obj (op G)) _ h },\n    rwa [is_iso_iff_bijective, function.bijective_iff_exists_unique] }\nend\n\nlemma is_codetector_iff_reflects_isomorphisms_yoneda_obj (G : C) :\n  is_codetector G \u2194 reflects_isomorphisms (yoneda.obj G) :=\nbegin\n  refine \u27e8\u03bb hG, \u27e8\u03bb X Y f hf, _ \u27e9, \u03bb h, (is_codetector_def _).2 (\u03bb X Y f hf, _)\u27e9,\n  { refine (is_iso_unop_iff _).1 (hG.def _ _),\n    rwa [is_iso_iff_bijective, function.bijective_iff_exists_unique] at hf },\n  { rw \u2190 is_iso_op_iff,\n    suffices : is_iso ((yoneda.obj G).map f.op),\n    { exactI @is_iso_of_reflects_iso _ _ _ _ _ _ _ (yoneda.obj G) _ h },\n    rwa [is_iso_iff_bijective, function.bijective_iff_exists_unique] }\nend\n\nlemma well_powered_of_is_detector [has_pullbacks C] (G : C) (hG : is_detector G) :\n  well_powered C :=\nwell_powered_of_is_detecting hG\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/generator.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832805, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.36923304495520126}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport algebra.algebra.operations\nimport algebra.ring.equiv\nimport data.nat.choose.sum\nimport ring_theory.coprime.lemmas\nimport ring_theory.ideal.quotient\nimport ring_theory.non_zero_divisors\n/-!\n# More operations on modules and ideals\n-/\nuniverses u v w x\n\nopen_locale big_operators pointwise\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v} {F : Type*} {G : Type*}\n\nsection comm_semiring\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\nopen_locale pointwise\n\ninstance has_smul' : has_smul (ideal R) (submodule R M) :=\n\u27e8submodule.map\u2082 (linear_map.lsmul R M)\u27e9\n\n/-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to\napply. -/\nprotected lemma _root_.ideal.smul_eq_mul (I J : ideal R) : I \u2022 J = I * J := rfl\n\n/-- `N.annihilator` is the ideal of all elements `r : R` such that `r \u2022 N = 0`. -/\ndef annihilator (N : submodule R M) : ideal R :=\n(linear_map.lsmul R N).ker\n\nvariables {I J : ideal R} {N P : submodule R M}\n\ntheorem mem_annihilator {r} : r \u2208 N.annihilator \u2194 \u2200 n \u2208 N, r \u2022 n = (0:M) :=\n\u27e8\u03bb hr n hn, congr_arg subtype.val (linear_map.ext_iff.1 (linear_map.mem_ker.1 hr) \u27e8n, hn\u27e9),\n\u03bb h, linear_map.mem_ker.2 $ linear_map.ext $ \u03bb n, subtype.eq $ h n.1 n.2\u27e9\n\ntheorem mem_annihilator' {r} : r \u2208 N.annihilator \u2194 N \u2264 comap (r \u2022 linear_map.id) \u22a5 :=\nmem_annihilator.trans \u27e8\u03bb H n hn, (mem_bot R).2 $ H n hn, \u03bb H n hn, (mem_bot R).1 $ H hn\u27e9\n\nlemma mem_annihilator_span (s : set M) (r : R) :\n  r \u2208 (submodule.span R s).annihilator \u2194 \u2200 n : s, r \u2022 (n : M) = 0 :=\nbegin\n  rw submodule.mem_annihilator,\n  split,\n  { intros h n, exact h _ (submodule.subset_span n.prop) },\n  { intros h n hn,\n    apply submodule.span_induction hn,\n    { intros x hx, exact h \u27e8x, hx\u27e9 },\n    { exact smul_zero _ },\n    { intros x y hx hy, rw [smul_add, hx, hy, zero_add] },\n    { intros a x hx, rw [smul_comm, hx, smul_zero] } }\nend\n\nlemma mem_annihilator_span_singleton (g : M) (r : R) :\n  r \u2208 (submodule.span R ({g} : set M)).annihilator \u2194 r \u2022 g = 0 :=\nby simp [mem_annihilator_span]\n\ntheorem annihilator_bot : (\u22a5 : submodule R M).annihilator = \u22a4 :=\n(ideal.eq_top_iff_one _).2 $ mem_annihilator'.2 bot_le\n\ntheorem annihilator_eq_top_iff : N.annihilator = \u22a4 \u2194 N = \u22a5 :=\n\u27e8\u03bb H, eq_bot_iff.2 $ \u03bb (n:M) hn, (mem_bot R).2 $\n  one_smul R n \u25b8 mem_annihilator.1 ((ideal.eq_top_iff_one _).1 H) n hn,\n  \u03bb H, H.symm \u25b8 annihilator_bot\u27e9\n\ntheorem annihilator_mono (h : N \u2264 P) : P.annihilator \u2264 N.annihilator :=\n\u03bb r hrp, mem_annihilator.2 $ \u03bb n hn, mem_annihilator.1 hrp n $ h hn\n\ntheorem annihilator_supr (\u03b9 : Sort w) (f : \u03b9 \u2192 submodule R M) :\n  (annihilator \u2a06 i, f i) = \u2a05 i, annihilator (f i) :=\nle_antisymm (le_infi $ \u03bb i, annihilator_mono $ le_supr _ _)\n(\u03bb r H, mem_annihilator'.2 $ supr_le $ \u03bb i,\n  have _ := (mem_infi _).1 H i, mem_annihilator'.1 this)\n\ntheorem smul_mem_smul {r} {n} (hr : r \u2208 I) (hn : n \u2208 N) : r \u2022 n \u2208 I \u2022 N := apply_mem_map\u2082 _ hr hn\n\ntheorem smul_le {P : submodule R M} : I \u2022 N \u2264 P \u2194 \u2200 (r \u2208 I) (n \u2208 N), r \u2022 n \u2208 P := map\u2082_le\n\n@[elab_as_eliminator]\ntheorem smul_induction_on {p : M \u2192 Prop} {x} (H : x \u2208 I \u2022 N)\n  (Hb : \u2200 (r \u2208 I) (n \u2208 N), p (r \u2022 n))\n  (H1 : \u2200 x y, p x \u2192 p y \u2192 p (x + y)) : p x :=\nbegin\n  have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem,\n  refine submodule.supr_induction _ H _ H0 H1,\n  rintros \u27e8i, hi\u27e9 m \u27e8j, hj, (rfl : i \u2022 _ = m) \u27e9,\n  exact Hb _ hi _ hj,\nend\n\ntheorem mem_smul_span_singleton {I : ideal R} {m : M} {x : M} :\n  x \u2208 I \u2022 span R ({m} : set M) \u2194 \u2203 y \u2208 I, y \u2022 m = x :=\n\u27e8\u03bb hx, smul_induction_on hx\n  (\u03bb r hri n hnm,\n    let \u27e8s, hs\u27e9 := mem_span_singleton.1 hnm in \u27e8r * s, I.mul_mem_right _ hri, hs \u25b8 mul_smul r s m\u27e9)\n  (\u03bb m1 m2 \u27e8y1, hyi1, hy1\u27e9 \u27e8y2, hyi2, hy2\u27e9,\n    \u27e8y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]\u27e9),\n\u03bb \u27e8y, hyi, hy\u27e9, hy \u25b8 smul_mem_smul hyi (subset_span $ set.mem_singleton m)\u27e9\n\ntheorem smul_le_right : I \u2022 N \u2264 N :=\nsmul_le.2 $ \u03bb r hr n, N.smul_mem r\n\ntheorem smul_mono (hij : I \u2264 J) (hnp : N \u2264 P) : I \u2022 N \u2264 J \u2022 P := map\u2082_le_map\u2082 hij hnp\n\ntheorem smul_mono_left (h : I \u2264 J) : I \u2022 N \u2264 J \u2022 N := map\u2082_le_map\u2082_left h\n\ntheorem smul_mono_right (h : N \u2264 P) : I \u2022 N \u2264 I \u2022 P := map\u2082_le_map\u2082_right h\n\nlemma map_le_smul_top (I : ideal R) (f : R \u2192\u2097[R] M) :\n  submodule.map f I \u2264 I \u2022 (\u22a4 : submodule R M) :=\nbegin\n  rintros _ \u27e8y, hy, rfl\u27e9,\n  rw [\u2190 mul_one y, \u2190 smul_eq_mul, f.map_smul],\n  exact smul_mem_smul hy mem_top\nend\n\n@[simp] theorem annihilator_smul (N : submodule R M) : annihilator N \u2022 N = \u22a5 :=\neq_bot_iff.2 (smul_le.2 (\u03bb r, mem_annihilator.1))\n\n@[simp] theorem annihilator_mul (I : ideal R) : annihilator I * I = \u22a5 :=\nannihilator_smul I\n\n@[simp] theorem mul_annihilator (I : ideal R) : I * annihilator I = \u22a5 :=\nby rw [mul_comm, annihilator_mul]\n\nvariables (I J N P)\n@[simp] theorem smul_bot : I \u2022 (\u22a5 : submodule R M) = \u22a5 := map\u2082_bot_right _ _\n\n@[simp] theorem bot_smul : (\u22a5 : ideal R) \u2022 N = \u22a5 := map\u2082_bot_left _ _\n\n@[simp] theorem top_smul : (\u22a4 : ideal R) \u2022 N = N :=\nle_antisymm smul_le_right $ \u03bb r hri, one_smul R r \u25b8 smul_mem_smul mem_top hri\n\ntheorem smul_sup : I \u2022 (N \u2294 P) = I \u2022 N \u2294 I \u2022 P := map\u2082_sup_right _ _ _ _\n\ntheorem sup_smul : (I \u2294 J) \u2022 N = I \u2022 N \u2294 J \u2022 N := map\u2082_sup_left _ _ _ _\n\nprotected theorem smul_assoc : (I \u2022 J) \u2022 N = I \u2022 (J \u2022 N) :=\nle_antisymm (smul_le.2 $ \u03bb rs hrsij t htn,\n  smul_induction_on hrsij\n  (\u03bb r hr s hs,\n    (@smul_eq_mul R _ r s).symm \u25b8 smul_smul r s t \u25b8 smul_mem_smul hr (smul_mem_smul hs htn))\n  (\u03bb x y, (add_smul x y t).symm \u25b8 submodule.add_mem _))\n(smul_le.2 $ \u03bb r hr sn hsn, suffices J \u2022 N \u2264 submodule.comap (r \u2022 linear_map.id) ((I \u2022 J) \u2022 N),\n  from this hsn,\nsmul_le.2 $ \u03bb s hs n hn, show r \u2022 (s \u2022 n) \u2208 (I \u2022 J) \u2022 N,\n  from mul_smul r s n \u25b8 smul_mem_smul (smul_mem_smul hr hs) hn)\n\nlemma smul_inf_le (M\u2081 M\u2082 : submodule R M) : I \u2022 (M\u2081 \u2293 M\u2082) \u2264 I \u2022 M\u2081 \u2293 I \u2022 M\u2082 :=\nle_inf (submodule.smul_mono_right inf_le_left) (submodule.smul_mono_right inf_le_right)\n\nvariables (S : set R) (T : set M)\n\ntheorem span_smul_span : (ideal.span S) \u2022 (span R T) =\n  span R (\u22c3 (s \u2208 S) (t \u2208 T), {s \u2022 t}) :=\n(map\u2082_span_span _ _ _ _).trans $ congr_arg _ $ set.image2_eq_Union _ _ _\n\nlemma ideal_span_singleton_smul (r : R) (N : submodule R M) :\n  (ideal.span {r} : ideal R) \u2022 N = r \u2022 N :=\nbegin\n  have : span R (\u22c3 (t : M) (x : t \u2208 N), {r \u2022 t}) = r \u2022 N,\n  { convert span_eq _, exact (set.image_eq_Union _ (N : set M)).symm },\n  conv_lhs { rw [\u2190 span_eq N, span_smul_span] },\n  simpa\nend\n\nlemma span_smul_eq (r : R) (s : set M) : span R (r \u2022 s) = r \u2022 span R s :=\nby rw [\u2190 ideal_span_singleton_smul, span_smul_span, \u2190set.image2_eq_Union,\n    set.image2_singleton_left, set.image_smul]\n\nlemma mem_of_span_top_of_smul_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M) (H : \u2200 r : s, (r : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  suffices : (\u22a4 : ideal R) \u2022 (span R ({x} : set M)) \u2264 M',\n  { rw top_smul at this, exact this (subset_span (set.mem_singleton x)) },\n  rw [\u2190 hs, span_smul_span, span_le],\n  simpa using H\nend\n\n/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a\nsubmodule `M'` of `x`, we only need to show that `r ^ n \u2022 x \u2208 M'` for some `n` for each `r : s`. -/\nlemma mem_of_span_eq_top_of_smul_pow_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M)\n  (H : \u2200 r : s, \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  obtain \u27e8s', hs\u2081, hs\u2082\u27e9 := (ideal.span_eq_top_iff_finite _).mp hs,\n  replace H : \u2200 r : s', \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M' := \u03bb r, H \u27e8_, hs\u2081 r.prop\u27e9,\n  choose n\u2081 n\u2082 using H,\n  let N := s'.attach.sup n\u2081,\n  have hs' := ideal.span_pow_eq_top (s' : set R) hs\u2082 N,\n  apply M'.mem_of_span_top_of_smul_mem _ hs',\n  rintro \u27e8_, r, hr, rfl\u27e9,\n  convert M'.smul_mem (r ^ (N - n\u2081 \u27e8r, hr\u27e9)) (n\u2082 \u27e8r, hr\u27e9) using 1,\n  simp only [subtype.coe_mk, smul_smul, \u2190 pow_add],\n  rw tsub_add_cancel_of_le (finset.le_sup (s'.mem_attach _) : n\u2081 \u27e8r, hr\u27e9 \u2264 N),\nend\n\nvariables {M' : Type w} [add_comm_monoid M'] [module R M']\n\ntheorem map_smul'' (f : M \u2192\u2097[R] M') : (I \u2022 N).map f = I \u2022 N.map f :=\nle_antisymm (map_le_iff_le_comap.2 $ smul_le.2 $ \u03bb r hr n hn, show f (r \u2022 n) \u2208 I \u2022 N.map f,\n    from (f.map_smul r n).symm \u25b8 smul_mem_smul hr (mem_map_of_mem hn)) $\nsmul_le.2 $ \u03bb r hr n hn, let \u27e8p, hp, hfp\u27e9 := mem_map.1 hn in\nhfp \u25b8 f.map_smul r p \u25b8 mem_map_of_mem (smul_mem_smul hr hp)\n\nvariables {I}\n\nlemma mem_smul_span {s : set M} {x : M} :\n  x \u2208 I \u2022 submodule.span R s \u2194 x \u2208 submodule.span R (\u22c3 (a \u2208 I) (b \u2208 s), ({a \u2022 b} : set M)) :=\nby rw [\u2190 I.span_eq, submodule.span_smul_span, I.span_eq]; refl\n\nvariables (I)\n\n/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,\nthen we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/\nlemma exists_sum_of_mem_ideal_smul_span {\u03b9 : Type*} (s : set \u03b9) (f : \u03b9 \u2192 M) (x : M)\n  (hx : x \u2208 I \u2022 span R (f '' s)) :\n  \u2203 (a : s \u2192\u2080 R) (ha : \u2200 i, a i \u2208 I), a.sum (\u03bb i c, c \u2022 f i) = x :=\nbegin\n  refine span_induction (mem_smul_span.mp hx) _ _ _ _,\n  { simp only [set.mem_Union, set.mem_range, set.mem_singleton_iff],\n    rintros x \u27e8y, hy, x, \u27e8i, hi, rfl\u27e9, rfl\u27e9,\n    refine \u27e8finsupp.single \u27e8i, hi\u27e9 y, \u03bb j, _, _\u27e9,\n    { letI := classical.dec_eq s,\n      rw finsupp.single_apply, split_ifs, { assumption }, { exact I.zero_mem } },\n    refine @finsupp.sum_single_index s R M _ _ \u27e8i, hi\u27e9 _ (\u03bb i y, y \u2022 f i) _,\n    simp },\n  { exact \u27e80, \u03bb i, I.zero_mem, finsupp.sum_zero_index\u27e9 },\n  { rintros x y \u27e8ax, hax, rfl\u27e9 \u27e8ay, hay, rfl\u27e9,\n    refine \u27e8ax + ay, \u03bb i, I.add_mem (hax i) (hay i), finsupp.sum_add_index _ _\u27e9;\n      intros; simp only [zero_smul, add_smul] },\n  { rintros c x \u27e8a, ha, rfl\u27e9,\n    refine \u27e8c \u2022 a, \u03bb i, I.mul_mem_left c (ha i), _\u27e9,\n    rw [finsupp.sum_smul_index, finsupp.smul_sum];\n      intros; simp only [zero_smul, mul_smul] },\nend\n\n@[simp] lemma smul_comap_le_comap_smul (f : M \u2192\u2097[R] M') (S : submodule R M') (I : ideal R) :\n  I \u2022 S.comap f \u2264 (I \u2022 S).comap f :=\nbegin\n  refine (submodule.smul_le.mpr (\u03bb r hr x hx, _)),\n  rw [submodule.mem_comap] at \u22a2 hx,\n  rw f.map_smul,\n  exact submodule.smul_mem_smul hr hx\nend\n\nend comm_semiring\n\nsection comm_ring\n\nvariables [comm_ring R] [add_comm_group M] [module R M]\nvariables {N N\u2081 N\u2082 P P\u2081 P\u2082 : submodule R M}\n\n/-- `N.colon P` is the ideal of all elements `r : R` such that `r \u2022 P \u2286 N`. -/\ndef colon (N P : submodule R M) : ideal R :=\nannihilator (P.map N.mkq)\n\ntheorem mem_colon {r} : r \u2208 N.colon P \u2194 \u2200 p \u2208 P, r \u2022 p \u2208 N :=\nmem_annihilator.trans \u27e8\u03bb H p hp, (quotient.mk_eq_zero N).1 (H (quotient.mk p) (mem_map_of_mem hp)),\n\u03bb H m \u27e8p, hp, hpm\u27e9, hpm \u25b8 (N.mkq).map_smul r p \u25b8 (quotient.mk_eq_zero N).2 $ H p hp\u27e9\n\ntheorem mem_colon' {r} : r \u2208 N.colon P \u2194 P \u2264 comap (r \u2022 linear_map.id) N :=\nmem_colon\n\ntheorem colon_mono (hn : N\u2081 \u2264 N\u2082) (hp : P\u2081 \u2264 P\u2082) : N\u2081.colon P\u2082 \u2264 N\u2082.colon P\u2081 :=\n\u03bb r hrnp, mem_colon.2 $ \u03bb p\u2081 hp\u2081, hn $ mem_colon.1 hrnp p\u2081 $ hp hp\u2081\n\ntheorem infi_colon_supr (\u03b9\u2081 : Sort w) (f : \u03b9\u2081 \u2192 submodule R M)\n  (\u03b9\u2082 : Sort x) (g : \u03b9\u2082 \u2192 submodule R M) :\n  (\u2a05 i, f i).colon (\u2a06 j, g j) = \u2a05 i j, (f i).colon (g j) :=\nle_antisymm (le_infi $ \u03bb i, le_infi $ \u03bb j, colon_mono (infi_le _ _) (le_supr _ _))\n(\u03bb r H, mem_colon'.2 $ supr_le $ \u03bb j, map_le_iff_le_comap.1 $ le_infi $ \u03bb i,\n  map_le_iff_le_comap.2 $ mem_colon'.1 $ have _ := ((mem_infi _).1 H i),\n  have _ := ((mem_infi _).1 this j), this)\n\nend comm_ring\n\nend submodule\n\nnamespace ideal\n\nsection mul_and_radical\nvariables {R : Type u} {\u03b9 : Type*} [comm_semiring R]\nvariables {I J K L : ideal R}\n\ninstance : has_mul (ideal R) := \u27e8(\u2022)\u27e9\n\n@[simp] lemma add_eq_sup : I + J = I \u2294 J := rfl\n@[simp] lemma zero_eq_bot : (0 : ideal R) = \u22a5 := rfl\n@[simp] lemma one_eq_top : (1 : ideal R) = \u22a4 :=\nby erw [submodule.one_eq_range, linear_map.range_id]\n\ntheorem mul_mem_mul {r s} (hr : r \u2208 I) (hs : s \u2208 J) : r * s \u2208 I * J :=\nsubmodule.smul_mem_smul hr hs\n\ntheorem mul_mem_mul_rev {r s} (hr : r \u2208 I) (hs : s \u2208 J) : s * r \u2208 I * J :=\nmul_comm r s \u25b8 mul_mem_mul hr hs\n\nlemma pow_mem_pow {x : R} (hx : x \u2208 I) (n : \u2115) : x ^ n \u2208 I ^ n :=\nbegin\n  induction n with n ih, { simp only [pow_zero, ideal.one_eq_top], },\n  simpa only [pow_succ] using mul_mem_mul hx ih,\nend\n\nlemma prod_mem_prod {\u03b9 : Type*} {s : finset \u03b9} {I : \u03b9 \u2192 ideal R} {x : \u03b9 \u2192 R} :\n  (\u2200 i \u2208 s, x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i :=\nbegin\n  classical,\n  apply finset.induction_on s,\n  { intro _, rw [finset.prod_empty, finset.prod_empty, one_eq_top], exact submodule.mem_top },\n  { intros a s ha IH h,\n    rw [finset.prod_insert ha, finset.prod_insert ha],\n    exact mul_mem_mul (h a $ finset.mem_insert_self a s)\n      (IH $ \u03bb i hi, h i $ finset.mem_insert_of_mem hi) }\nend\n\ntheorem mul_le : I * J \u2264 K \u2194 \u2200 (r \u2208 I) (s \u2208 J), r * s \u2208 K :=\nsubmodule.smul_le\n\nlemma mul_le_left : I * J \u2264 J :=\nideal.mul_le.2 (\u03bb r hr s, J.mul_mem_left _)\n\nlemma mul_le_right : I * J \u2264 I :=\nideal.mul_le.2 (\u03bb r hr s hs, I.mul_mem_right _ hr)\n\n@[simp] lemma sup_mul_right_self : I \u2294 (I * J) = I :=\nsup_eq_left.2 ideal.mul_le_right\n\n@[simp] lemma sup_mul_left_self : I \u2294 (J * I) = I :=\nsup_eq_left.2 ideal.mul_le_left\n\n@[simp] lemma mul_right_self_sup : (I * J) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_right\n\n@[simp] lemma mul_left_self_sup : (J * I) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_left\n\nvariables (I J K)\nprotected theorem mul_comm : I * J = J * I :=\nle_antisymm (mul_le.2 $ \u03bb r hrI s hsJ, mul_mem_mul_rev hsJ hrI)\n  (mul_le.2 $ \u03bb r hrJ s hsI, mul_mem_mul_rev hsI hrJ)\n\nprotected theorem mul_assoc : (I * J) * K = I * (J * K) :=\nsubmodule.smul_assoc I J K\n\ntheorem span_mul_span (S T : set R) : span S * span T =\n  span \u22c3 (s \u2208 S) (t \u2208 T), {s * t} :=\nsubmodule.span_smul_span S T\nvariables {I J K}\n\nlemma span_mul_span' (S T : set R) : span S * span T = span (S*T) :=\nby { unfold span, rw submodule.span_mul_span, }\n\nlemma span_singleton_mul_span_singleton (r s : R) :\n  span {r} * span {s} = (span {r * s} : ideal R) :=\nby { unfold span, rw [submodule.span_mul_span, set.singleton_mul_singleton], }\n\nlemma span_singleton_pow (s : R) (n : \u2115):\n  span {s} ^ n = (span {s ^ n} : ideal R) :=\nbegin\n  induction n with n ih, { simp [set.singleton_one], },\n  simp only [pow_succ, ih, span_singleton_mul_span_singleton],\nend\n\nlemma mem_mul_span_singleton {x y : R} {I : ideal R} :\n  x \u2208 I * span {y} \u2194 \u2203 z \u2208 I, z * y = x :=\nsubmodule.mem_smul_span_singleton\n\nlemma mem_span_singleton_mul {x y : R} {I : ideal R} :\n  x \u2208 span {y} * I \u2194 \u2203 z \u2208 I, y * z = x :=\nby simp only [mul_comm, mem_mul_span_singleton]\n\nlemma le_span_singleton_mul_iff {x : R} {I J : ideal R} :\n  I \u2264 span {x} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI :=\nshow (\u2200 {zI} (hzI : zI \u2208 I), zI \u2208 span {x} * J) \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI,\nby simp only [mem_span_singleton_mul]\n\nlemma span_singleton_mul_le_iff {x : R} {I J : ideal R} :\n  span {x} * I \u2264 J \u2194 \u2200 z \u2208 I, x * z \u2208 J :=\nbegin\n  simp only [mul_le, mem_span_singleton_mul, mem_span_singleton],\n  split,\n  { intros h zI hzI,\n    exact h x (dvd_refl x) zI hzI },\n  { rintros h _ \u27e8z, rfl\u27e9 zI hzI,\n    rw [mul_comm x z, mul_assoc],\n    exact J.mul_mem_left _ (h zI hzI) },\nend\n\nlemma span_singleton_mul_le_span_singleton_mul {x y : R} {I J : ideal R} :\n  span {x} * I \u2264 span {y} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ :=\nby simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm]\n\nlemma eq_span_singleton_mul {x : R} (I J : ideal R) :\n  I = span {x} * J \u2194 ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI) \u2227 (\u2200 z \u2208 J, x * z \u2208 I)) :=\nby simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff]\n\nlemma span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : ideal R) :\n  span {x} * I = span {y} * J \u2194\n    ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ) \u2227\n     (\u2200 zJ \u2208 J, \u2203 zI \u2208 I, x * zI = y * zJ)) :=\nby simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]\n\nlemma prod_span {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 set R) :\n  (\u220f i in s, ideal.span (I i)) = ideal.span (\u220f i in s, I i) :=\nsubmodule.prod_span s I\n\nlemma prod_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R) :\n  (\u220f i in s, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nsubmodule.prod_span_singleton s I\n\nlemma finset_inf_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R)\n  (hI : set.pairwise \u2191s (is_coprime on I)) :\n  (s.inf $ \u03bb i, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nbegin\n  ext x,\n  simp only [submodule.mem_finset_inf, ideal.mem_span_singleton],\n  exact \u27e8finset.prod_dvd_of_coprime hI,\n    \u03bb h i hi, (finset.dvd_prod_of_mem _ hi).trans h\u27e9\nend\n\nlemma infi_span_singleton {\u03b9 : Type*} [fintype \u03b9] (I : \u03b9 \u2192 R)\n  (hI : \u2200 i j (hij : i \u2260 j), is_coprime (I i) (I j)):\n  (\u2a05 i, ideal.span ({I i} : set R)) = ideal.span {\u220f i, I i} :=\nbegin\n  rw [\u2190 finset.inf_univ_eq_infi, finset_inf_span_singleton],\n  rwa [finset.coe_univ, set.pairwise_univ]\nend\n\nlemma sup_eq_top_iff_is_coprime {R : Type*} [comm_semiring R] (x y : R) :\n  span ({x} : set R) \u2294 span {y} = \u22a4 \u2194 is_coprime x y :=\nbegin\n  rw [eq_top_iff_one, submodule.mem_sup],\n  split,\n  { rintro \u27e8u, hu, v, hv, h1\u27e9,\n    rw mem_span_singleton' at hu hv,\n    rw [\u2190 hu.some_spec, \u2190 hv.some_spec] at h1,\n    exact \u27e8_, _, h1\u27e9 },\n  { exact \u03bb \u27e8u, v, h1\u27e9,\n      \u27e8_, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, _, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, h1\u27e9 },\nend\n\ntheorem mul_le_inf : I * J \u2264 I \u2293 J :=\nmul_le.2 $ \u03bb r hri s hsj, \u27e8I.mul_mem_right s hri, J.mul_mem_left r hsj\u27e9\n\ntheorem multiset_prod_le_inf {s : multiset (ideal R)} :\n  s.prod \u2264 s.inf :=\nbegin\n  classical, refine s.induction_on _ _,\n  { rw [multiset.inf_zero], exact le_top },\n  intros a s ih,\n  rw [multiset.prod_cons, multiset.inf_cons],\n  exact le_trans mul_le_inf (inf_le_inf le_rfl ih)\nend\n\ntheorem prod_le_inf {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} : s.prod f \u2264 s.inf f :=\nmultiset_prod_le_inf\n\ntheorem mul_eq_inf_of_coprime (h : I \u2294 J = \u22a4) : I * J = I \u2293 J :=\nle_antisymm mul_le_inf $ \u03bb r \u27e8hri, hrj\u27e9,\nlet \u27e8s, hsi, t, htj, hst\u27e9 := submodule.mem_sup.1 ((eq_top_iff_one _).1 h) in\nmul_one r \u25b8 hst \u25b8 (mul_add r s t).symm \u25b8 ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj)\n  (mul_mem_mul hri htj)\n\nlemma sup_mul_eq_of_coprime_left (h : I \u2294 J = \u22a4) : I \u2294 (J * K) = I \u2294 K :=\nle_antisymm (sup_le_sup_left mul_le_left _) $ \u03bb i hi,\nbegin\n  rw eq_top_iff_one at h, rw submodule.mem_sup at h hi \u22a2,\n  obtain \u27e8i1, hi1, j, hj, h\u27e9 := h, obtain \u27e8i', hi', k, hk, hi\u27e9 := hi,\n  refine \u27e8_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _\u27e9,\n  rw [add_assoc, \u2190 add_mul, h, one_mul, hi]\nend\n\nlemma sup_mul_eq_of_coprime_right (h : I \u2294 K = \u22a4) : I \u2294 (J * K) = I \u2294 J :=\nby { rw mul_comm, exact sup_mul_eq_of_coprime_left h }\n\nlemma mul_sup_eq_of_coprime_left (h : I \u2294 J = \u22a4) : (I * K) \u2294 J = K \u2294 J :=\nby { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] }\n\nlemma mul_sup_eq_of_coprime_right (h : K \u2294 J = \u22a4) : (I * K) \u2294 J = I \u2294 J :=\nby { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] }\n\nlemma sup_prod_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 I \u2294 J i = \u22a4) :\n  I \u2294 \u220f i in s, J i = \u22a4 :=\nfinset.prod_induction _ (\u03bb J, I \u2294 J = \u22a4) (\u03bb J K hJ hK, (sup_mul_eq_of_coprime_left hJ).trans hK)\n(by rw [one_eq_top, sup_top_eq]) h\n\nlemma sup_infi_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 I \u2294 J i = \u22a4) :\n  I \u2294 (\u2a05 i \u2208 s, J i) = \u22a4 :=\neq_top_iff.mpr $ le_of_eq_of_le (sup_prod_eq_top h).symm $ sup_le_sup_left\n  (le_of_le_of_eq prod_le_inf $ finset.inf_eq_infi _ _) _\n\nlemma prod_sup_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 J i \u2294 I = \u22a4) :\n  (\u220f i in s, J i) \u2294 I = \u22a4 :=\nsup_comm.trans (sup_prod_eq_top $ \u03bb i hi, sup_comm.trans $ h i hi)\n\nlemma infi_sup_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 J i \u2294 I = \u22a4) :\n  (\u2a05 i \u2208 s, J i) \u2294 I = \u22a4 :=\nsup_comm.trans (sup_infi_eq_top $ \u03bb i hi, sup_comm.trans $ h i hi)\n\nlemma sup_pow_eq_top {n : \u2115} (h : I \u2294 J = \u22a4) : I \u2294 (J ^ n) = \u22a4 :=\nby { rw [\u2190 finset.card_range n, \u2190 finset.prod_const], exact sup_prod_eq_top (\u03bb _ _, h) }\n\nlemma pow_sup_eq_top {n : \u2115} (h : I \u2294 J = \u22a4) : (I ^ n) \u2294 J = \u22a4 :=\nby { rw [\u2190 finset.card_range n, \u2190 finset.prod_const], exact prod_sup_eq_top (\u03bb _ _, h) }\n\nlemma pow_sup_pow_eq_top {m n : \u2115} (h : I \u2294 J = \u22a4) : (I ^ m) \u2294 (J ^ n) = \u22a4 :=\nsup_pow_eq_top (pow_sup_eq_top h)\n\nvariables (I)\n@[simp] theorem mul_bot : I * \u22a5 = \u22a5 :=\nsubmodule.smul_bot I\n\n@[simp] theorem bot_mul : \u22a5 * I = \u22a5 :=\nsubmodule.bot_smul I\n\n@[simp] theorem mul_top : I * \u22a4 = I :=\nideal.mul_comm \u22a4 I \u25b8 submodule.top_smul I\n\n@[simp] theorem top_mul : \u22a4 * I = I :=\nsubmodule.top_smul I\nvariables {I}\n\ntheorem mul_mono (hik : I \u2264 K) (hjl : J \u2264 L) : I * J \u2264 K * L :=\nsubmodule.smul_mono hik hjl\n\ntheorem mul_mono_left (h : I \u2264 J) : I * K \u2264 J * K :=\nsubmodule.smul_mono_left h\n\ntheorem mul_mono_right (h : J \u2264 K) : I * J \u2264 I * K :=\nsubmodule.smul_mono_right h\n\nvariables (I J K)\ntheorem mul_sup : I * (J \u2294 K) = I * J \u2294 I * K :=\nsubmodule.smul_sup I J K\n\ntheorem sup_mul : (I \u2294 J) * K = I * K \u2294 J * K :=\nsubmodule.sup_smul I J K\nvariables {I J K}\n\nlemma pow_le_pow {m n : \u2115} (h : m \u2264 n) :\n  I^n \u2264 I^m :=\nbegin\n  cases nat.exists_eq_add_of_le h with k hk,\n  rw [hk, pow_add],\n  exact le_trans (mul_le_inf) (inf_le_left)\nend\n\nlemma pow_le_self {n : \u2115} (hn : n \u2260 0) : I^n \u2264 I :=\ncalc I^n \u2264 I ^ 1 : pow_le_pow (nat.pos_of_ne_zero hn)\n     ... = I : pow_one _\n\nlemma mul_eq_bot {R : Type*} [comm_ring R] [is_domain R] {I J : ideal R} :\n  I * J = \u22a5 \u2194 I = \u22a5 \u2228 J = \u22a5 :=\n\u27e8\u03bb hij, or_iff_not_imp_left.mpr (\u03bb I_ne_bot, J.eq_bot_iff.mpr (\u03bb j hj,\n  let \u27e8i, hi, ne0\u27e9 := I.ne_bot_iff.mp I_ne_bot in\n    or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0)),\n \u03bb h, by cases h; rw [\u2190 ideal.mul_bot, h, ideal.mul_comm]\u27e9\n\ninstance {R : Type*} [comm_ring R] [is_domain R] : no_zero_divisors (ideal R) :=\n{ eq_zero_or_eq_zero_of_mul_eq_zero := \u03bb I J, mul_eq_bot.1 }\n\n/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/\nlemma prod_eq_bot {R : Type*} [comm_ring R] [is_domain R]\n  {s : multiset (ideal R)} : s.prod = \u22a5 \u2194 \u2203 I \u2208 s, I = \u22a5 :=\nprod_zero_iff_exists_zero\n\n/-- The radical of an ideal `I` consists of the elements `r` such that `r^n \u2208 I` for some `n`. -/\ndef radical (I : ideal R) : ideal R :=\n{ carrier := { r | \u2203 n : \u2115, r ^ n \u2208 I },\n  zero_mem' := \u27e81, (pow_one (0:R)).symm \u25b8 I.zero_mem\u27e9,\n  add_mem' := \u03bb x y \u27e8m, hxmi\u27e9 \u27e8n, hyni\u27e9, \u27e8m + n,\n    (add_pow x y (m + n)).symm \u25b8 I.sum_mem $\n    show \u2200 c \u2208 finset.range (nat.succ (m + n)),\n      x ^ c * y ^ (m + n - c) * (nat.choose (m + n) c) \u2208 I,\n    from \u03bb c hc, or.cases_on (le_total c m)\n      (\u03bb hcm, I.mul_mem_right _ $ I.mul_mem_left _ $ nat.add_comm n m \u25b8\n        (add_tsub_assoc_of_le hcm n).symm \u25b8\n        (pow_add y n (m-c)).symm \u25b8 I.mul_mem_right _ hyni)\n      (\u03bb hmc, I.mul_mem_right _ $ I.mul_mem_right _ $ add_tsub_cancel_of_le hmc \u25b8\n        (pow_add x m (c-m)).symm \u25b8 I.mul_mem_right _ hxmi)\u27e9,\n  smul_mem' := \u03bb r s \u27e8n, hsni\u27e9, \u27e8n, (mul_pow r s n).symm \u25b8 I.mul_mem_left (r^n) hsni\u27e9 }\n\ntheorem le_radical : I \u2264 radical I :=\n\u03bb r hri, \u27e81, (pow_one r).symm \u25b8 hri\u27e9\n\nvariables (R)\ntheorem radical_top : (radical \u22a4 : ideal R) = \u22a4 :=\n(eq_top_iff_one _).2 \u27e80, submodule.mem_top\u27e9\nvariables {R}\n\ntheorem radical_mono (H : I \u2264 J) : radical I \u2264 radical J :=\n\u03bb r \u27e8n, hrni\u27e9, \u27e8n, H hrni\u27e9\n\nvariables (I)\n@[simp] theorem radical_idem : radical (radical I) = radical I :=\nle_antisymm (\u03bb r \u27e8n, k, hrnki\u27e9, \u27e8n * k, (pow_mul r n k).symm \u25b8 hrnki\u27e9) le_radical\nvariables {I}\n\ntheorem radical_le_radical_iff : radical I \u2264 radical J \u2194 I \u2264 radical J :=\n\u27e8\u03bb h, le_trans le_radical h, \u03bb h, radical_idem J \u25b8 radical_mono h\u27e9\n\ntheorem radical_eq_top : radical I = \u22a4 \u2194 I = \u22a4 :=\n\u27e8\u03bb h, (eq_top_iff_one _).2 $ let \u27e8n, hn\u27e9 := (eq_top_iff_one _).1 h in\n  @one_pow R _ n \u25b8 hn, \u03bb h, h.symm \u25b8 radical_top R\u27e9\n\ntheorem is_prime.radical (H : is_prime I) : radical I = I :=\nle_antisymm (\u03bb r \u27e8n, hrni\u27e9, H.mem_of_pow_mem n hrni) le_radical\n\nvariables (I J)\ntheorem radical_sup : radical (I \u2294 J) = radical (radical I \u2294 radical J) :=\nle_antisymm (radical_mono $ sup_le_sup le_radical le_radical) $\n\u03bb r \u27e8n, hrnij\u27e9, let \u27e8s, hs, t, ht, hst\u27e9 := submodule.mem_sup.1 hrnij in\n@radical_idem _ _ (I \u2294 J) \u25b8 \u27e8n, hst \u25b8 ideal.add_mem _\n  (radical_mono le_sup_left hs) (radical_mono le_sup_right ht)\u27e9\n\ntheorem radical_inf : radical (I \u2293 J) = radical I \u2293 radical J :=\nle_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 I.mul_mem_right _ hrm,\n(pow_add r m n).symm \u25b8 J.mul_mem_left _ hrn\u27e9)\n\ntheorem radical_mul : radical (I * J) = radical I \u2293 radical J :=\nle_antisymm (radical_inf I J \u25b8 radical_mono $ @mul_le_inf _ _ I J)\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 mul_mem_mul hrm hrn\u27e9)\nvariables {I J}\n\ntheorem is_prime.radical_le_iff (hj : is_prime J) :\n  radical I \u2264 J \u2194 I \u2264 J :=\n\u27e8le_trans le_radical, \u03bb hij r \u27e8n, hrni\u27e9, hj.mem_of_pow_mem n $ hij hrni\u27e9\n\ntheorem radical_eq_Inf (I : ideal R) :\n  radical I = Inf { J : ideal R | I \u2264 J \u2227 is_prime J } :=\nle_antisymm (le_Inf $ \u03bb J hJ, hJ.2.radical_le_iff.2 hJ.1) $\n\u03bb r hr, classical.by_contradiction $ \u03bb hri,\nlet \u27e8m, (hrm : r \u2209 radical m), him, hm\u27e9 := zorn_nonempty_partial_order\u2080\n  {K : ideal R | r \u2209 radical K}\n  (\u03bb c hc hcc y hyc, \u27e8Sup c, \u03bb \u27e8n, hrnc\u27e9, let \u27e8y, hyc, hrny\u27e9 :=\n      (submodule.mem_Sup_of_directed \u27e8y, hyc\u27e9 hcc.directed_on).1 hrnc in hc hyc \u27e8n, hrny\u27e9,\n    \u03bb z, le_Sup\u27e9) I hri in\nhave \u2200 x \u2209 m, r \u2208 radical (m \u2294 span {x}) := \u03bb x hxm, classical.by_contradiction $ \u03bb hrmx, hxm $\n  hm (m \u2294 span {x}) hrmx le_sup_left \u25b8 (le_sup_right : _ \u2264 m \u2294 span {x})\n    (subset_span $ set.mem_singleton _),\nhave is_prime m, from \u27e8by rintro rfl; rw radical_top at hrm; exact hrm trivial,\n  \u03bb x y hxym, or_iff_not_imp_left.2 $ \u03bb hxm, classical.by_contradiction $ \u03bb hym,\n  let \u27e8n, hrn\u27e9 := this _ hxm,\n      \u27e8p, hpm, q, hq, hpqrn\u27e9 := submodule.mem_sup.1 hrn,\n      \u27e8c, hcxq\u27e9 := mem_span_singleton'.1 hq in\n  let \u27e8k, hrk\u27e9 := this _ hym,\n      \u27e8f, hfm, g, hg, hfgrk\u27e9 := submodule.mem_sup.1 hrk,\n      \u27e8d, hdyg\u27e9 := mem_span_singleton'.1 hg in\n  hrm \u27e8n + k, by rw [pow_add, \u2190 hpqrn, \u2190 hcxq, \u2190 hfgrk, \u2190 hdyg, add_mul, mul_add (c*x),\n                     mul_assoc c x (d*y), mul_left_comm x, \u2190 mul_assoc];\n    refine m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm)\n      (m.mul_mem_left _ hxym))\u27e9\u27e9,\nhrm $ this.radical.symm \u25b8 (Inf_le \u27e8him, this\u27e9 : Inf {J : ideal R | I \u2264 J \u2227 is_prime J} \u2264 m) hr\n\n@[simp] lemma radical_bot_of_is_domain {R : Type u} [comm_ring R] [is_domain R] :\n  radical (\u22a5 : ideal R) = \u22a5 :=\neq_bot_iff.2 (\u03bb x hx, hx.rec_on (\u03bb n hn, pow_eq_zero hn))\n\ninstance : comm_semiring (ideal R) := submodule.comm_semiring\n\nvariables (R)\ntheorem top_pow (n : \u2115) : (\u22a4 ^ n : ideal R) = \u22a4 :=\nnat.rec_on n one_eq_top $ \u03bb n ih, by rw [pow_succ, ih, top_mul]\nvariables {R}\n\nvariables (I)\ntheorem radical_pow (n : \u2115) (H : n > 0) : radical (I^n) = radical I :=\nnat.rec_on n (not.elim dec_trivial) (\u03bb n ih H,\nor.cases_on (lt_or_eq_of_le $ nat.le_of_lt_succ H)\n  (\u03bb H, calc radical (I^(n+1))\n           = radical I \u2293 radical (I^n) : by { rw pow_succ, exact radical_mul _ _ }\n       ... = radical I \u2293 radical I : by rw ih H\n       ... = radical I : inf_idem)\n  (\u03bb H, H \u25b8 (pow_one I).symm \u25b8 rfl)) H\n\ntheorem is_prime.mul_le {I J P : ideal R} (hp : is_prime P) :\n  I * J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hip j hj, let \u27e8i, hi, hip\u27e9 := set.not_subset.1 hip in\n  (hp.mem_or_mem $ h $ mul_mem_mul hi hj).resolve_left hip,\n\u03bb h, or.cases_on h (le_trans $ le_trans mul_le_inf inf_le_left)\n  (le_trans $ le_trans mul_le_inf inf_le_right)\u27e9\n\ntheorem is_prime.inf_le {I J P : ideal R} (hp : is_prime P) :\n  I \u2293 J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, hp.mul_le.1 $ le_trans mul_le_inf h,\n\u03bb h, or.cases_on h (le_trans inf_le_left) (le_trans inf_le_right)\u27e9\n\ntheorem is_prime.multiset_prod_le {s : multiset (ideal R)} {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  s.prod \u2264 P \u2194 \u2203 I \u2208 s, I \u2264 P :=\nsuffices s.prod \u2264 P \u2192 \u2203 I \u2208 s, I \u2264 P,\n  from \u27e8this, \u03bb \u27e8i, his, hip\u27e9, le_trans multiset_prod_le_inf $\n    le_trans (multiset.inf_le his) hip\u27e9,\nbegin\n  classical,\n  obtain \u27e8b, hb\u27e9 : \u2203 b, b \u2208 s := multiset.exists_mem_of_ne_zero hne,\n  obtain \u27e8t, rfl\u27e9 : \u2203 t, s = b ::\u2098 t,\n  from \u27e8s.erase b, (multiset.cons_erase hb).symm\u27e9,\n  refine t.induction_on _ _,\n  { simp only [exists_prop, \u2190multiset.singleton_eq_cons, multiset.prod_singleton,\n      multiset.mem_singleton, exists_eq_left, imp_self] },\n  intros a s ih h,\n  rw [multiset.cons_swap, multiset.prod_cons, hp.mul_le] at h,\n  rw multiset.cons_swap,\n  cases h,\n  { exact \u27e8a, multiset.mem_cons_self a _, h\u27e9 },\n  obtain \u27e8I, hI, ih\u27e9 : \u2203 I \u2208 b ::\u2098 s, I \u2264 P := ih h,\n  exact \u27e8I, multiset.mem_cons_of_mem hI, ih\u27e9\nend\n\ntheorem is_prime.multiset_prod_map_le {s : multiset \u03b9} (f : \u03b9 \u2192 ideal R) {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  (s.map f).prod \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nbegin\n  rw hp.multiset_prod_le (mt multiset.map_eq_zero.mp hne),\n  simp_rw [exists_prop, multiset.mem_map, exists_exists_and_eq_and],\nend\n\ntheorem is_prime.prod_le {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R}\n  (hp : is_prime P) (hne : s.nonempty) :\n  s.prod f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nhp.multiset_prod_map_le f (mt finset.val_eq_zero.mp hne.ne_empty)\n\ntheorem is_prime.inf_le' {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R} (hp : is_prime P)\n  (hsne: s.nonempty) :\n  s.inf f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\n\u27e8\u03bb h, (hp.prod_le hsne).1 $ le_trans prod_le_inf h,\n  \u03bb \u27e8i, his, hip\u27e9, le_trans (finset.inf_le his) hip\u27e9\n\ntheorem subset_union {R : Type u} [comm_ring R] {I J K : ideal R} :\n  (I : set R) \u2286 J \u222a K \u2194 I \u2264 J \u2228 I \u2264 K :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hij s hsi,\n  let \u27e8r, hri, hrj\u27e9 := set.not_subset.1 hij in classical.by_contradiction $ \u03bb hsk,\n  or.cases_on (h $ I.add_mem hri hsi)\n    (\u03bb hj, hrj $ add_sub_cancel r s \u25b8 J.sub_mem hj ((h hsi).resolve_right hsk))\n    (\u03bb hk, hsk $ add_sub_cancel' r s \u25b8 K.sub_mem hk ((h hri).resolve_left hrj)),\n\u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset_union_left J K)\n  (\u03bb h, set.subset.trans h $ set.subset_union_right J K)\u27e9\n\ntheorem subset_union_prime' {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {a b : \u03b9}\n  (hp : \u2200 i \u2208 s, is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192\n  I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i,\n  from \u27e8this, \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_left _ _) (set.subset_union_left _ _)) $\n    \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_right _ _) (set.subset_union_left _ _)) $\n    \u03bb \u27e8i, his, hi\u27e9, by refine (set.subset.trans hi $ set.subset.trans _ $\n        set.subset_union_right _ _);\n      exact set.subset_bUnion_of_mem (finset.mem_coe.2 his)\u27e9,\nbegin\n  generalize hn : s.card = n, intros h,\n  unfreezingI { induction n with n ih generalizing a b s },\n  { clear hp,\n    rw finset.card_eq_zero at hn, subst hn,\n    rw [finset.coe_empty, set.bUnion_empty, set.union_empty, subset_union] at h,\n    simpa only [exists_prop, finset.not_mem_empty, false_and, exists_false, or_false] },\n  classical,\n  replace hn : \u2203 (i : \u03b9) (t : finset \u03b9), i \u2209 t \u2227 insert i t = s \u2227 t.card = n :=\n  finset.card_eq_succ.1 hn,\n  unfreezingI { rcases hn with \u27e8i, t, hit, rfl, hn\u27e9 },\n  replace hp : is_prime (f i) \u2227 \u2200 x \u2208 t, is_prime (f x) := (t.forall_mem_insert _ _).1 hp,\n  by_cases Ht : \u2203 j \u2208 t, f j \u2264 f i,\n  { obtain \u27e8j, hjt, hfji\u27e9 : \u2203 j \u2208 t, f j \u2264 f i := Ht,\n    obtain \u27e8u, hju, rfl\u27e9 : \u2203 u, j \u2209 u \u2227 insert j u = t,\n    { exact \u27e8t.erase j, t.not_mem_erase j, finset.insert_erase hjt\u27e9 },\n    have hp' : \u2200 k \u2208 insert i u, is_prime (f k),\n    { rw finset.forall_mem_insert at hp \u22a2, exact \u27e8hp.1, hp.2.2\u27e9 },\n    have hiu : i \u2209 u := mt finset.mem_insert_of_mem hit,\n    have hn' : (insert i u).card = n,\n    { rwa finset.card_insert_of_not_mem at hn \u22a2, exacts [hiu, hju] },\n    have h' : (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 k \u2208 (\u2191(insert i u) : set \u03b9), f k),\n    { rw finset.coe_insert at h \u22a2, rw finset.coe_insert at h,\n      simp only [set.bUnion_insert] at h \u22a2,\n      rw [\u2190 set.union_assoc \u2191(f i)] at h,\n      erw [set.union_eq_self_of_subset_right hfji] at h,\n      exact h },\n    specialize @ih a b (insert i u) hp' hn' h',\n    refine ih.imp id (or.imp id (exists_imp_exists $ \u03bb k, _)), simp only [exists_prop],\n    exact and.imp (\u03bb hk, finset.insert_subset_insert i (finset.subset_insert j u) hk) id },\n  by_cases Ha : f a \u2264 f i,\n  { have h' : (I : set R) \u2286 f i \u222a f b \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc,\n          set.union_right_comm \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Ha] at h,\n      exact h },\n    specialize @ih i b t hp.2 hn h', right,\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9 },\n    { exact or.inl ih },\n    { exact or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  by_cases Hb : f b \u2264 f i,\n  { have h' : (I : set R) \u2286 f a \u222a f i \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc, set.union_assoc \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Hb] at h,\n      exact h },\n    specialize @ih a i t hp.2 hn h',\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inl ih },\n    { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9) },\n    { exact or.inr (or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9) } },\n  by_cases Hi : I \u2264 f i,\n  { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, Hi\u27e9) },\n  have : \u00acI \u2293 f a \u2293 f b \u2293 t.inf f \u2264 f i,\n  { rcases t.eq_empty_or_nonempty with (rfl | hsne),\n    { rw [finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or_distrib, not_or_distrib],\n      exact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9 },\n    simp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or_distrib],\n    exact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9 },\n  rcases set.not_subset.1 this with \u27e8r, \u27e8\u27e8\u27e8hrI, hra\u27e9, hrb\u27e9, hr\u27e9, hri\u27e9,\n  by_cases HI : (I : set R) \u2286 f a \u222a f b \u222a \u22c3 j \u2208 (\u2191t : set \u03b9), f j,\n  { specialize ih hp.2 hn HI, rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { left, exact ih }, { right, left, exact ih },\n    { right, right, exact \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  exfalso, rcases set.not_subset.1 HI with \u27e8s, hsI, hs\u27e9,\n  rw [finset.coe_insert, set.bUnion_insert] at h,\n  have hsi : s \u2208 f i := ((h hsI).resolve_left (mt or.inl hs)).resolve_right (mt or.inr hs),\n  rcases h (I.add_mem hrI hsI) with \u27e8ha | hb\u27e9 | hi | ht,\n  { exact hs (or.inl $ or.inl $ add_sub_cancel' r s \u25b8 (f a).sub_mem ha hra) },\n  { exact hs (or.inl $ or.inr $ add_sub_cancel' r s \u25b8 (f b).sub_mem hb hrb) },\n  { exact hri (add_sub_cancel r s \u25b8 (f i).sub_mem hi hsi) },\n  { rw set.mem_Union\u2082 at ht, rcases ht with \u27e8j, hjt, hj\u27e9,\n    simp only [finset.inf_eq_infi, set_like.mem_coe, submodule.mem_infi] at hr,\n    exact hs (or.inr $ set.mem_bUnion hjt $ add_sub_cancel' r s \u25b8 (f j).sub_mem hj $ hr j hjt) }\nend\n\n/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/\ntheorem subset_union_prime {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} (a b : \u03b9)\n  (hp : \u2200 i \u2208 s, i \u2260 a \u2192 i \u2260 b \u2192 is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i,\n  from \u27e8\u03bb h, bex_def.2 $ this h, \u03bb \u27e8i, his, hi\u27e9, set.subset.trans hi $ set.subset_bUnion_of_mem $\n    show i \u2208 (\u2191s : set \u03b9), from his\u27e9,\nassume h : (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i),\nbegin\n  classical,\n  by_cases has : a \u2208 s,\n  { unfreezingI { obtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s :=\n      \u27e8s.erase a, finset.not_mem_erase a s, finset.insert_erase has\u27e9 },\n    by_cases hbt : b \u2208 t,\n    { unfreezingI { obtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t :=\n        \u27e8t.erase b, finset.not_mem_erase b t, finset.insert_erase hbt\u27e9 },\n      have hp' : \u2200 i \u2208 u, is_prime (f i),\n      { intros i hiu, refine hp i (finset.mem_insert_of_mem (finset.mem_insert_of_mem hiu)) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, finset.coe_insert, set.bUnion_insert, set.bUnion_insert,\n          \u2190 set.union_assoc, subset_union_prime' hp', bex_def] at h,\n      rwa [finset.exists_mem_insert, finset.exists_mem_insert] },\n    { have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f a : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } },\n  { by_cases hbs : b \u2208 s,\n    { unfreezingI { obtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s :=\n        \u27e8s.erase b, finset.not_mem_erase b s, finset.insert_erase hbs\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f b : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert },\n    cases s.eq_empty_or_nonempty with hse hsne,\n    { substI hse, rw [finset.coe_empty, set.bUnion_empty, set.subset_empty_iff] at h,\n      have : (I : set R) \u2260 \u2205 := set.nonempty.ne_empty (set.nonempty_of_mem I.zero_mem),\n      exact absurd h this },\n    { cases hsne.bex with i his,\n      unfreezingI { obtain \u27e8t, hit, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s :=\n        \u27e8s.erase i, finset.not_mem_erase i s, finset.insert_erase his\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f i : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } }\nend\n\nsection dvd\n\n/-- If `I` divides `J`, then `I` contains `J`.\n\nIn a Dedekind domain, to divide and contain are equivalent, see `ideal.dvd_iff_le`.\n-/\nlemma le_of_dvd {I J : ideal R} : I \u2223 J \u2192 J \u2264 I\n| \u27e8K, h\u27e9 := h.symm \u25b8 le_trans mul_le_inf inf_le_left\n\nlemma is_unit_iff {I : ideal R} :\n  is_unit I \u2194 I = \u22a4 :=\nis_unit_iff_dvd_one.trans ((@one_eq_top R _).symm \u25b8\n \u27e8\u03bb h, eq_top_iff.mpr (ideal.le_of_dvd h), \u03bb h, \u27e8\u22a4, by rw [mul_top, h]\u27e9\u27e9)\n\ninstance unique_units : unique ((ideal R)\u02e3) :=\n{ default := 1,\n  uniq := \u03bb u, units.ext\n    (show (u : ideal R) = 1, by rw [is_unit_iff.mp u.is_unit, one_eq_top]) }\n\nend dvd\n\nend mul_and_radical\n\nsection map_and_comap\n\nvariables {R : Type u} {S : Type v}\n\nsection semiring\nvariables {F : Type*} [semiring R] [semiring S]\nvariables [rc : ring_hom_class F R S]\nvariables (f : F)\nvariables {I J : ideal R} {K L : ideal S}\n\ninclude rc\n/-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than\n  the image itself. -/\ndef map (I : ideal R) : ideal S :=\nspan (f '' I)\n\n/-- `I.comap f` is the preimage of `I` under `f`. -/\ndef comap (I : ideal S) : ideal R :=\n{ carrier := f \u207b\u00b9' I,\n  add_mem' := \u03bb x y hx hy, by simp only [set.mem_preimage, set_like.mem_coe,\n                                         map_add, add_mem hx hy] at *,\n  zero_mem' := by simp only [set.mem_preimage, map_zero, set_like.mem_coe, submodule.zero_mem],\n  smul_mem' := \u03bb c x hx, by { simp only [smul_eq_mul, set.mem_preimage, map_mul,\n                                         set_like.mem_coe] at *,\n                              exact mul_mem_left I _ hx } }\n\nvariables {f}\ntheorem map_mono (h : I \u2264 J) : map f I \u2264 map f J :=\nspan_mono $ set.image_subset _ h\n\ntheorem mem_map_of_mem (f : F) {I : ideal R} {x : R} (h : x \u2208 I) : f x \u2208 map f I :=\nsubset_span \u27e8x, h, rfl\u27e9\n\nlemma apply_coe_mem_map (f : F) (I : ideal R) (x : I) : f x \u2208 I.map f :=\nmem_map_of_mem f x.prop\n\ntheorem map_le_iff_le_comap :\n  map f I \u2264 K \u2194 I \u2264 comap f K :=\nspan_le.trans set.image_subset_iff\n\n@[simp] theorem mem_comap {x} : x \u2208 comap f K \u2194 f x \u2208 K := iff.rfl\n\ntheorem comap_mono (h : K \u2264 L) : comap f K \u2264 comap f L :=\nset.preimage_mono (\u03bb x hx, h hx)\nvariables (f)\n\ntheorem comap_ne_top (hK : K \u2260 \u22a4) : comap f K \u2260 \u22a4 :=\n(ne_top_iff_one _).2 $ by rw [mem_comap, map_one];\n  exact (ne_top_iff_one _).1 hK\n\nvariables {G : Type*} [rcg : ring_hom_class G S R]\n\ninclude rcg\nlemma map_le_comap_of_inv_on (g : G) (I : ideal R) (hf : set.left_inv_on g f I) :\n  I.map f \u2264 I.comap g :=\nbegin\n  refine ideal.span_le.2 _,\n  rintros x \u27e8x, hx, rfl\u27e9,\n  rw [set_like.mem_coe, mem_comap, hf hx],\n  exact hx,\nend\n\nlemma comap_le_map_of_inv_on (g : G) (I : ideal S) (hf : set.left_inv_on g f (f \u207b\u00b9' I)) :\n  I.comap f \u2264 I.map g :=\n\u03bb x (hx : f x \u2208 I), hf hx \u25b8 ideal.mem_map_of_mem g hx\n\n/-- The `ideal` version of `set.image_subset_preimage_of_inverse`. -/\nlemma map_le_comap_of_inverse (g : G) (I : ideal R) (h : function.left_inverse g f) :\n  I.map f \u2264 I.comap g :=\nmap_le_comap_of_inv_on _ _ _ $ h.left_inv_on _\n\n/-- The `ideal` version of `set.preimage_subset_image_of_inverse`. -/\nlemma comap_le_map_of_inverse (g : G) (I : ideal S) (h : function.left_inverse g f) :\n  I.comap f \u2264 I.map g :=\ncomap_le_map_of_inv_on _ _ _ $ h.left_inv_on _\nomit rcg\n\ninstance is_prime.comap [hK : K.is_prime] : (comap f K).is_prime :=\n\u27e8comap_ne_top _ hK.1, \u03bb x y,\n  by simp only [mem_comap, map_mul]; apply hK.2\u27e9\n\nvariables (I J K L)\n\ntheorem map_top : map f \u22a4 = \u22a4 :=\n(eq_top_iff_one _).2 $ subset_span \u27e81, trivial, map_one f\u27e9\n\nvariable (f)\nlemma gc_map_comap : galois_connection (ideal.map f) (ideal.comap f) :=\n\u03bb I J, ideal.map_le_iff_le_comap\nomit rc\n\n@[simp] lemma comap_id : I.comap (ring_hom.id R) = I :=\nideal.ext $ \u03bb _, iff.rfl\n\n@[simp] lemma map_id : I.map (ring_hom.id R) = I :=\n(gc_map_comap (ring_hom.id R)).l_unique galois_connection.id comap_id\n\nlemma comap_comap {T : Type*} [semiring T] {I : ideal T} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl\n\nlemma map_map {T : Type*} [semiring T] {I : ideal R} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.map f).map g = I.map (g.comp f) :=\n((gc_map_comap f).compose (gc_map_comap g)).l_unique\n  (gc_map_comap (g.comp f)) (\u03bb _, comap_comap _ _)\n\ninclude rc\nlemma map_span (f : F) (s : set R) :\n  map f (span s) = span (f '' s) :=\nsymm $ submodule.span_eq_of_le _\n  (\u03bb y \u27e8x, hy, x_eq\u27e9, x_eq \u25b8 mem_map_of_mem f (subset_span hy))\n  (map_le_iff_le_comap.2 $ span_le.2 $ set.image_subset_iff.1 subset_span)\n\nvariables {f I J K L}\n\nlemma map_le_of_le_comap : I \u2264 K.comap f \u2192 I.map f \u2264 K :=\n(gc_map_comap f).l_le\n\nlemma le_comap_of_map_le : I.map f \u2264 K \u2192 I \u2264 K.comap f :=\n(gc_map_comap f).le_u\n\nlemma le_comap_map : I \u2264 (I.map f).comap f :=\n(gc_map_comap f).le_u_l _\n\nlemma map_comap_le : (K.comap f).map f \u2264 K :=\n(gc_map_comap f).l_u_le _\n\n@[simp] lemma comap_top : (\u22a4 : ideal S).comap f = \u22a4 :=\n(gc_map_comap f).u_top\n\n@[simp] lemma comap_eq_top_iff {I : ideal S} : I.comap f = \u22a4 \u2194 I = \u22a4 :=\n\u27e8 \u03bb h, I.eq_top_iff_one.mpr (map_one f \u25b8 mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)),\n  \u03bb h, by rw [h, comap_top] \u27e9\n\n@[simp] lemma map_bot : (\u22a5 : ideal R).map f = \u22a5 :=\n(gc_map_comap f).l_bot\n\nvariables (f I J K L)\n\n@[simp] lemma map_comap_map : ((I.map f).comap f).map f = I.map f :=\n(gc_map_comap f).l_u_l_eq_l I\n\n@[simp] lemma comap_map_comap : ((K.comap f).map f).comap f = K.comap f :=\n(gc_map_comap f).u_l_u_eq_u K\n\nlemma map_sup : (I \u2294 J).map f = I.map f \u2294 J.map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_sup\n\ntheorem comap_inf : comap f (K \u2293 L) = comap f K \u2293 comap f L := rfl\n\nvariables {\u03b9 : Sort*}\n\nlemma map_supr (K : \u03b9 \u2192 ideal R) : (supr K).map f = \u2a06 i, (K i).map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_supr\n\nlemma comap_infi (K : \u03b9 \u2192 ideal S) : (infi K).comap f = \u2a05 i, (K i).comap f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).u_infi\n\nlemma map_Sup (s : set (ideal R)): (Sup s).map f = \u2a06 I \u2208 s, (I : ideal R).map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_Sup\n\nlemma comap_Inf (s : set (ideal S)): (Inf s).comap f = \u2a05 I \u2208 s, (I : ideal S).comap f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).u_Inf\n\nlemma comap_Inf' (s : set (ideal S)) : (Inf s).comap f = \u2a05 I \u2208 (comap f '' s), I :=\ntrans (comap_Inf f s) (by rw infi_image)\n\ntheorem comap_is_prime [H : is_prime K] : is_prime (comap f K) :=\n\u27e8comap_ne_top f H.ne_top,\n  \u03bb x y h, H.mem_or_mem $ by rwa [mem_comap, map_mul] at h\u27e9\n\nvariables {I J K L}\n\ntheorem map_inf_le : map f (I \u2293 J) \u2264 map f I \u2293 map f J :=\n(gc_map_comap f : galois_connection (map f) (comap f)).monotone_l.map_inf_le _ _\n\ntheorem le_comap_sup : comap f K \u2294 comap f L \u2264 comap f (K \u2294 L) :=\n(gc_map_comap f : galois_connection (map f) (comap f)).monotone_u.le_map_sup _ _\nomit rc\n\n@[simp] lemma smul_top_eq_map {R S : Type*} [comm_semiring R] [comm_semiring S] [algebra R S]\n  (I : ideal R) : I \u2022 (\u22a4 : submodule R S) = (I.map (algebra_map R S)).restrict_scalars R :=\nbegin\n  refine le_antisymm (submodule.smul_le.mpr (\u03bb r hr y _, _) )\n      (\u03bb x hx, submodule.span_induction hx _ _ _ _),\n  { rw algebra.smul_def,\n     exact mul_mem_right _ _ (mem_map_of_mem _ hr) },\n\n  { rintros _ \u27e8x, hx, rfl\u27e9,\n    rw [\u2190 mul_one (algebra_map R S x), \u2190 algebra.smul_def],\n    exact submodule.smul_mem_smul hx submodule.mem_top },\n  { exact submodule.zero_mem _ },\n  { intros x y, exact submodule.add_mem _ },\n  intros a x hx,\n  refine submodule.smul_induction_on hx _ _,\n  { intros r hr s hs,\n    rw smul_comm,\n    exact submodule.smul_mem_smul hr submodule.mem_top },\n  { intros x y hx hy,\n    rw smul_add, exact submodule.add_mem _ hx hy },\nend\n\nsection surjective\nvariables (hf : function.surjective f)\ninclude hf\n\nopen function\n\ntheorem map_comap_of_surjective (I : ideal S) :\n  map f (comap f I) = I :=\nle_antisymm (map_le_iff_le_comap.2 le_rfl)\n(\u03bb s hsi, let \u27e8r, hfrs\u27e9 := hf s in\n  hfrs \u25b8 (mem_map_of_mem f $ show f r \u2208 I, from hfrs.symm \u25b8 hsi))\n\n/-- `map` and `comap` are adjoint, and the composition `map f \u2218 comap f` is the\n  identity -/\ndef gi_map_comap : galois_insertion (map f) (comap f) :=\ngalois_insertion.monotone_intro\n  ((gc_map_comap f).monotone_u)\n  ((gc_map_comap f).monotone_l)\n  (\u03bb _, le_comap_map)\n  (map_comap_of_surjective _ hf)\n\nlemma map_surjective_of_surjective : surjective (map f) :=\n(gi_map_comap f hf).l_surjective\n\nlemma comap_injective_of_surjective : injective (comap f) :=\n(gi_map_comap f hf).u_injective\n\nlemma map_sup_comap_of_surjective (I J : ideal S) : (I.comap f \u2294 J.comap f).map f = I \u2294 J :=\n(gi_map_comap f hf).l_sup_u _ _\n\nlemma map_supr_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a06i, (K i).comap f).map f = supr K :=\n(gi_map_comap f hf).l_supr_u _\n\nlemma map_inf_comap_of_surjective (I J : ideal S) : (I.comap f \u2293 J.comap f).map f = I \u2293 J :=\n(gi_map_comap f hf).l_inf_u _ _\n\nlemma map_infi_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a05i, (K i).comap f).map f = infi K :=\n(gi_map_comap f hf).l_infi_u _\n\ntheorem mem_image_of_mem_map_of_surjective {I : ideal R} {y}\n  (H : y \u2208 map f I) : y \u2208 f '' I :=\nsubmodule.span_induction H (\u03bb _, id) \u27e80, I.zero_mem, map_zero f\u27e9\n(\u03bb y1 y2 \u27e8x1, hx1i, hxy1\u27e9 \u27e8x2, hx2i, hxy2\u27e9,\n  \u27e8x1 + x2, I.add_mem hx1i hx2i, hxy1 \u25b8 hxy2 \u25b8 map_add f _ _\u27e9)\n(\u03bb c y \u27e8x, hxi, hxy\u27e9,\n  let \u27e8d, hdc\u27e9 := hf c in \u27e8d * x, I.mul_mem_left _ hxi, hdc \u25b8 hxy \u25b8 map_mul f _ _\u27e9)\n\nlemma mem_map_iff_of_surjective {I : ideal R} {y} :\n  y \u2208 map f I \u2194 \u2203 x, x \u2208 I \u2227 f x = y :=\n\u27e8\u03bb h, (set.mem_image _ _ _).2 (mem_image_of_mem_map_of_surjective f hf h),\n  \u03bb \u27e8x, hx\u27e9, hx.right \u25b8 (mem_map_of_mem f hx.left)\u27e9\n\nlemma le_map_of_comap_le_of_surjective : comap f K \u2264 I \u2192 K \u2264 map f I :=\n\u03bb h, (map_comap_of_surjective f hf K) \u25b8 map_mono h\n\nend surjective\n\nsection injective\nvariables (hf : function.injective f)\ninclude hf\n\nlemma comap_bot_le_of_injective : comap f \u22a5 \u2264 I :=\nbegin\n  refine le_trans (\u03bb x hx, _) bot_le,\n  rw [mem_comap, submodule.mem_bot, \u2190 map_zero f] at hx,\n  exact eq.symm (hf hx) \u25b8 (submodule.zero_mem \u22a5)\nend\n\nend injective\n\nend semiring\n\nsection ring\nvariables {F : Type*} [ring R] [ring S]\nvariables [ring_hom_class F R S] (f : F) {I : ideal R}\n\nsection surjective\n\nvariables (hf : function.surjective f)\ninclude hf\n\ntheorem comap_map_of_surjective (I : ideal R) : comap f (map f I) = I \u2294 comap f \u22a5 :=\nle_antisymm (assume r h, let \u27e8s, hsi, hfsr\u27e9 := mem_image_of_mem_map_of_surjective f hf h in\n  submodule.mem_sup.2 \u27e8s, hsi, r - s, (submodule.mem_bot S).2 $ by rw [map_sub, hfsr, sub_self],\n  add_sub_cancel'_right s r\u27e9)\n(sup_le (map_le_iff_le_comap.1 le_rfl) (comap_mono bot_le))\n\n\n/-- Correspondence theorem -/\ndef rel_iso_of_surjective : ideal S \u2243o { p : ideal R // comap f \u22a5 \u2264 p } :=\n{ to_fun := \u03bb J, \u27e8comap f J, comap_mono bot_le\u27e9,\n  inv_fun := \u03bb I, map f I.1,\n  left_inv := \u03bb J, map_comap_of_surjective f hf J,\n  right_inv := \u03bb I, subtype.eq $ show comap f (map f I.1) = I.1,\n    from (comap_map_of_surjective f hf I).symm \u25b8 le_antisymm\n      (sup_le le_rfl I.2) le_sup_left,\n  map_rel_iff' := \u03bb I1 I2, \u27e8\u03bb H, map_comap_of_surjective f hf I1 \u25b8\n    map_comap_of_surjective f hf I2 \u25b8 map_mono H, comap_mono\u27e9 }\n\n/-- The map on ideals induced by a surjective map preserves inclusion. -/\ndef order_embedding_of_surjective : ideal S \u21aao ideal R :=\n(rel_iso_of_surjective f hf).to_rel_embedding.trans (subtype.rel_embedding _ _)\n\ntheorem map_eq_top_or_is_maximal_of_surjective {I : ideal R} (H : is_maximal I) :\n  (map f I) = \u22a4 \u2228 is_maximal (map f I) :=\nbegin\n  refine or_iff_not_imp_left.2 (\u03bb ne_top, \u27e8\u27e8\u03bb h, ne_top h, \u03bb J hJ, _\u27e9\u27e9),\n  { refine (rel_iso_of_surjective f hf).injective\n      (subtype.ext_iff.2 (eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm)),\n    { exact (map_le_iff_le_comap).1 (le_of_lt hJ) },\n    { exact \u03bb h, hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm)) } }\nend\n\ntheorem comap_is_maximal_of_surjective {K : ideal S} [H : is_maximal K] : is_maximal (comap f K) :=\nbegin\n  refine \u27e8\u27e8comap_ne_top _ H.1.1, \u03bb J hJ, _\u27e9\u27e9,\n  suffices : map f J = \u22a4,\n  { replace this := congr_arg (comap f) this,\n    rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this,\n    rw eq_top_iff,\n    exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono (bot_le)) (le_of_lt hJ))) },\n  refine H.1.2 (map f J) (lt_of_le_of_ne (le_map_of_comap_le_of_surjective _ hf (le_of_lt hJ))\n    (\u03bb h, ne_of_lt hJ (trans (congr_arg (comap f) h) _))),\n  rw [comap_map_of_surjective _ hf, sup_eq_left],\n  exact le_trans (comap_mono bot_le) (le_of_lt hJ)\nend\n\ntheorem comap_le_comap_iff_of_surjective (I J : ideal S) : comap f I \u2264 comap f J \u2194 I \u2264 J :=\n\u27e8\u03bb h, (map_comap_of_surjective f hf I).symm.le.trans (map_le_of_le_comap h),\n  \u03bb h, le_comap_of_map_le ((map_comap_of_surjective f hf I).le.trans h)\u27e9\n\nend surjective\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f (map f.symm) = I`. -/\n@[simp]\nlemma map_of_equiv (I : ideal R) (f : R \u2243+* S) : (I.map (f : R \u2192+* S)).map (f.symm : S \u2192+* R) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, map_map]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `comap f.symm (comap f) = I`. -/\n@[simp]\nlemma comap_of_equiv (I : ideal R) (f : R \u2243+* S) :\n  (I.comap (f.symm : S \u2192+* R)).comap (f : R \u2192+* S) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, comap_comap]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f I = comap f.symm I`. -/\nlemma map_comap_of_equiv (I : ideal R) (f : R \u2243+* S) : I.map (f : R \u2192+* S) = I.comap f.symm :=\nle_antisymm (le_comap_of_map_le (map_of_equiv I f).le)\n  (le_map_of_comap_le_of_surjective _ f.surjective (comap_of_equiv I f).le)\n\nsection bijective\nvariables (hf : function.bijective f)\ninclude hf\n\n/-- Special case of the correspondence theorem for isomorphic rings -/\ndef rel_iso_of_bijective : ideal S \u2243o ideal R :=\n{ to_fun := comap f,\n  inv_fun := map f,\n  left_inv := (rel_iso_of_surjective f hf.right).left_inv,\n  right_inv := \u03bb J, subtype.ext_iff.1\n    ((rel_iso_of_surjective f hf.right).right_inv \u27e8J, comap_bot_le_of_injective f hf.left\u27e9),\n  map_rel_iff' := (rel_iso_of_surjective f hf.right).map_rel_iff' }\n\nlemma comap_le_iff_le_map {I : ideal R} {K : ideal S} : comap f K \u2264 I \u2194 K \u2264 map f I :=\n\u27e8\u03bb h, le_map_of_comap_le_of_surjective f hf.right h,\n \u03bb h, ((rel_iso_of_bijective f hf).right_inv I) \u25b8 comap_mono h\u27e9\n\ntheorem map.is_maximal {I : ideal R} (H : is_maximal I) : is_maximal (map f I) :=\nby refine or_iff_not_imp_left.1\n  (map_eq_top_or_is_maximal_of_surjective f hf.right H) (\u03bb h, H.1.1 _);\ncalc I = comap f (map f I) : ((rel_iso_of_bijective f hf).right_inv I).symm\n   ... = comap f \u22a4 : by rw h\n   ... = \u22a4 : by rw comap_top\n\nend bijective\n\nlemma ring_equiv.bot_maximal_iff (e : R \u2243+* S) :\n  (\u22a5 : ideal R).is_maximal \u2194 (\u22a5 : ideal S).is_maximal :=\n\u27e8\u03bb h, (@map_bot _ _ _ _ _ _ e.to_ring_hom) \u25b8 map.is_maximal e.to_ring_hom e.bijective h,\n  \u03bb h, (@map_bot _ _ _ _ _ _ e.symm.to_ring_hom) \u25b8 map.is_maximal e.symm.to_ring_hom\n          e.symm.bijective h\u27e9\n\nend ring\n\nsection comm_ring\n\nvariables {F : Type*} [comm_ring R] [comm_ring S]\nvariables [rc : ring_hom_class F R S]\nvariables (f : F)\nvariables {I J : ideal R} {K L : ideal S}\n\nvariables (I J K L)\n\ninclude rc\ntheorem map_mul : map f (I * J) = map f I * map f J :=\nle_antisymm (map_le_iff_le_comap.2 $ mul_le.2 $ \u03bb r hri s hsj,\n  show f (r * s) \u2208 _, by rw map_mul;\n  exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))\n(trans_rel_right _ (span_mul_span _ _) $ span_le.2 $\n  set.Union\u2082_subset $ \u03bb i \u27e8r, hri, hfri\u27e9,\n  set.Union\u2082_subset $ \u03bb j \u27e8s, hsj, hfsj\u27e9,\n  set.singleton_subset_iff.2 $ hfri \u25b8 hfsj \u25b8\n  by rw [\u2190 map_mul];\n  exact mem_map_of_mem f (mul_mem_mul hri hsj))\n\n/-- The pushforward `ideal.map` as a monoid-with-zero homomorphism. -/\n@[simps]\ndef map_hom : ideal R \u2192*\u2080 ideal S :=\n{ to_fun := map f,\n  map_mul' := \u03bb I J, ideal.map_mul f I J,\n  map_one' := by convert ideal.map_top f; exact one_eq_top,\n  map_zero' := ideal.map_bot }\n\nprotected theorem map_pow (n : \u2115) : map f (I^n) = (map f I)^n :=\nmap_pow (map_hom f) I n\n\ntheorem comap_radical : comap f (radical K) = radical (comap f K) :=\nle_antisymm (\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, show f (r ^ n) \u2208 K,\n  from (map_pow f r n).symm \u25b8 hfrnk\u27e9)\n(\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, map_pow f r n \u25b8 hfrnk\u27e9)\nomit rc\n\n@[simp] lemma map_quotient_self :\n  map (quotient.mk I) I = \u22a5 :=\neq_bot_iff.2 $ ideal.map_le_iff_le_comap.2 $ \u03bb x hx,\n(submodule.mem_bot (R \u29f8 I)).2 $ ideal.quotient.eq_zero_iff_mem.2 hx\n\nvariables {I J K L}\n\ninclude rc\ntheorem map_radical_le : map f (radical I) \u2264 radical (map f I) :=\nmap_le_iff_le_comap.2 $ \u03bb r \u27e8n, hrni\u27e9, \u27e8n, map_pow f r n \u25b8 mem_map_of_mem f hrni\u27e9\n\ntheorem le_comap_mul : comap f K * comap f L \u2264 comap f (K * L) :=\nmap_le_iff_le_comap.1 $ (map_mul f (comap f K) (comap f L)).symm \u25b8\nmul_mono (map_le_iff_le_comap.2 $ le_rfl) (map_le_iff_le_comap.2 $ le_rfl)\nomit rc\n\nend comm_ring\n\nend map_and_comap\n\nsection is_primary\nvariables {R : Type u} [comm_semiring R]\n\n/-- A proper ideal `I` is primary iff `xy \u2208 I` implies `x \u2208 I` or `y \u2208 radical I`. -/\ndef is_primary (I : ideal R) : Prop :=\nI \u2260 \u22a4 \u2227 \u2200 {x y : R}, x * y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 radical I\n\ntheorem is_prime.is_primary {I : ideal R} (hi : is_prime I) : is_primary I :=\n\u27e8hi.1, \u03bb x y hxy, (hi.mem_or_mem hxy).imp id $ \u03bb hyi, le_radical hyi\u27e9\n\ntheorem mem_radical_of_pow_mem {I : ideal R} {x : R} {m : \u2115} (hx : x ^ m \u2208 radical I) :\n  x \u2208 radical I :=\nradical_idem I \u25b8 \u27e8m, hx\u27e9\n\ntheorem is_prime_radical {I : ideal R} (hi : is_primary I) : is_prime (radical I) :=\n\u27e8mt radical_eq_top.1 hi.1, \u03bb x y \u27e8m, hxy\u27e9, begin\n  rw mul_pow at hxy, cases hi.2 hxy,\n  { exact or.inl \u27e8m, h\u27e9 },\n  { exact or.inr (mem_radical_of_pow_mem h) }\nend\u27e9\n\ntheorem is_primary_inf {I J : ideal R} (hi : is_primary I) (hj : is_primary J)\n  (hij : radical I = radical J) : is_primary (I \u2293 J) :=\n\u27e8ne_of_lt $ lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1), \u03bb x y \u27e8hxyi, hxyj\u27e9,\nbegin\n  rw [radical_inf, hij, inf_idem],\n  cases hi.2 hxyi with hxi hyi, cases hj.2 hxyj with hxj hyj,\n  { exact or.inl \u27e8hxi, hxj\u27e9 },\n  { exact or.inr hyj },\n  { rw hij at hyi, exact or.inr hyi }\nend\u27e9\n\nend is_primary\n\nend ideal\n\nlemma associates.mk_ne_zero' {R : Type*} [comm_ring R] {r : R} :\n  (associates.mk (ideal.span {r} : ideal R)) \u2260 0 \u2194 (r \u2260 0):=\nby rw [associates.mk_ne_zero, ideal.zero_eq_bot, ne.def, ideal.span_singleton_eq_bot]\n\nnamespace ring_hom\n\nvariables {R : Type u} {S : Type v} {T : Type v}\n\nsection semiring\nvariables {F : Type*} {G : Type*} [semiring R] [semiring S] [semiring T]\nvariables [rcf : ring_hom_class F R S] [rcg : ring_hom_class G T S]\n(f : F) (g : G)\n\ninclude rcf\n/-- Kernel of a ring homomorphism as an ideal of the domain. -/\ndef ker : ideal R := ideal.comap f \u22a5\n\n/-- An element is in the kernel if and only if it maps to zero.-/\nlemma mem_ker {r} : r \u2208 ker f \u2194 f r = 0 :=\nby rw [ker, ideal.mem_comap, submodule.mem_bot]\n\nlemma ker_eq : ((ker f) : set R) = set.preimage f {0} := rfl\n\nlemma ker_eq_comap_bot (f : F) : ker f = ideal.comap f \u22a5 := rfl\nomit rcf\n\nlemma comap_ker (f : S \u2192+* R) (g : T \u2192+* S) : f.ker.comap g = (f.comp g).ker :=\nby rw [ring_hom.ker_eq_comap_bot, ideal.comap_comap, ring_hom.ker_eq_comap_bot]\n\ninclude rcf\n/-- If the target is not the zero ring, then one is not in the kernel.-/\nlemma not_one_mem_ker [nontrivial S] (f : F) : (1:R) \u2209 ker f :=\nby { rw [mem_ker, map_one], exact one_ne_zero }\n\nlemma ker_ne_top [nontrivial S] (f : F) : ker f \u2260 \u22a4 :=\n(ideal.ne_top_iff_one _).mpr $ not_one_mem_ker f\nomit rcf\n\nend semiring\n\nsection ring\nvariables {F : Type*} [ring R] [semiring S] [rc : ring_hom_class F R S] (f : F)\n\ninclude rc\nlemma injective_iff_ker_eq_bot : function.injective f \u2194 ker f = \u22a5 :=\nby { rw [set_like.ext'_iff, ker_eq, set.ext_iff], exact injective_iff_map_eq_zero' f }\n\nlemma ker_eq_bot_iff_eq_zero : ker f = \u22a5 \u2194 \u2200 x, f x = 0 \u2192 x = 0 :=\nby { rw [\u2190 injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] }\nomit rc\n\n@[simp] lemma ker_coe_equiv (f : R \u2243+* S) :\n  ker (f : R \u2192+* S) = \u22a5 :=\nby simpa only [\u2190injective_iff_ker_eq_bot] using equiv_like.injective f\n\n@[simp] lemma ker_equiv {F' : Type*} [ring_equiv_class F' R S] (f : F') :\n  ker f = \u22a5 :=\nby simpa only [\u2190injective_iff_ker_eq_bot] using equiv_like.injective f\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S] (f : R \u2192+* S)\n\n/-- The induced map from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_equiv_of_surjective`).\n-/\ndef ker_lift (f : R \u2192+* S) : R \u29f8 f.ker \u2192+* S :=\nideal.quotient.lift _ f $ \u03bb r, f.mem_ker.mp\n\n@[simp]\nlemma ker_lift_mk (f : R \u2192+* S) (r : R) : ker_lift f (ideal.quotient.mk f.ker r) = f r :=\nideal.quotient.lift_mk _ _ _\n\n/-- The induced map from the quotient by the kernel is injective. -/\nlemma ker_lift_injective (f : R \u2192+* S) : function.injective (ker_lift f) :=\nassume a b, quotient.induction_on\u2082' a b $\n  assume a b (h : f a = f b), ideal.quotient.eq.2 $\nshow a - b \u2208 ker f, by rw [mem_ker, map_sub, h, sub_self]\n\nvariable {f}\n\n/-- The **first isomorphism theorem** for commutative rings, computable version. -/\ndef quotient_ker_equiv_of_right_inverse\n  {g : S \u2192 R} (hf : function.right_inverse g f) :\n  R \u29f8 f.ker \u2243+* S :=\n{ to_fun := ker_lift f,\n  inv_fun := (ideal.quotient.mk f.ker) \u2218 g,\n  left_inv := begin\n    rintro \u27e8x\u27e9,\n    apply ker_lift_injective,\n    simp [hf (f x)],\n  end,\n  right_inv := hf,\n  ..ker_lift f}\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : R \u29f8 f.ker) : quotient_ker_equiv_of_right_inverse hf x = ker_lift f x := rfl\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.symm.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : S) : (quotient_ker_equiv_of_right_inverse hf).symm x = ideal.quotient.mk f.ker (g x) := rfl\n\n/-- The **first isomorphism theorem** for commutative rings. -/\nnoncomputable def quotient_ker_equiv_of_surjective (hf : function.surjective f) :\n  R \u29f8 f.ker \u2243+* S :=\nquotient_ker_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\nend comm_ring\n\n/-- The kernel of a homomorphism to a domain is a prime ideal. -/\nlemma ker_is_prime {F : Type*} [ring R] [ring S] [is_domain S] [ring_hom_class F R S]\n  (f : F) : (ker f).is_prime :=\n\u27e8by { rw [ne.def, ideal.eq_top_iff_one], exact not_one_mem_ker f },\n\u03bb x y, by simpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _\u27e9\n\n/-- The kernel of a homomorphism to a field is a maximal ideal. -/\nlemma ker_is_maximal_of_surjective {R K F : Type*} [ring R] [field K] [ring_hom_class F R K]\n  (f : F) (hf : function.surjective f) :\n  (ker f).is_maximal :=\nbegin\n  refine ideal.is_maximal_iff.mpr\n    \u27e8\u03bb h1, @one_ne_zero K _ _ $ map_one f \u25b8 (mem_ker f).mp h1,\n    \u03bb J x hJ hxf hxJ, _\u27e9,\n  obtain \u27e8y, hy\u27e9 := hf (f x)\u207b\u00b9,\n  have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm,\n  rw H,\n  refine J.sub_mem (J.mul_mem_left _ hxJ) (hJ _),\n  rw mem_ker,\n  simp only [hy, map_sub, map_one, map_mul,\n    inv_mul_cancel (mt (mem_ker f).mpr hxf), sub_self],\nend\n\nend ring_hom\n\nnamespace ideal\n\nvariables {R : Type*} {S : Type*} {F : Type*}\n\nsection semiring\nvariables [semiring R] [semiring S] [rc : ring_hom_class F R S]\n\ninclude rc\nlemma map_eq_bot_iff_le_ker {I : ideal R} (f : F) : I.map f = \u22a5 \u2194 I \u2264 (ring_hom.ker f) :=\nby rw [ring_hom.ker, eq_bot_iff, map_le_iff_le_comap]\n\nlemma ker_le_comap {K : ideal S} (f : F) : ring_hom.ker f \u2264 comap f K :=\n\u03bb x hx, mem_comap.2 (((ring_hom.mem_ker f).1 hx).symm \u25b8 K.zero_mem)\n\nend semiring\n\nsection ring\nvariables [ring R] [ring S] [rc : ring_hom_class F R S]\n\ninclude rc\nlemma map_Inf {A : set (ideal R)} {f : F} (hf : function.surjective f) :\n  (\u2200 J \u2208 A, ring_hom.ker f \u2264 J) \u2192 map f (Inf A) = Inf (map f '' A) :=\nbegin\n  refine \u03bb h, le_antisymm (le_Inf _) _,\n  { intros j hj y hy,\n    cases (mem_map_iff_of_surjective f hf).1 hy with x hx,\n    cases (set.mem_image _ _ _).mp hj with J hJ,\n    rw [\u2190 hJ.right, \u2190 hx.right],\n    exact mem_map_of_mem f (Inf_le_of_le hJ.left (le_of_eq rfl) hx.left) },\n  { intros y hy,\n    cases hf y with x hx,\n    refine hx \u25b8 (mem_map_of_mem f _),\n    have : \u2200 I \u2208 A, y \u2208 map f I, by simpa using hy,\n    rw [submodule.mem_Inf],\n    intros J hJ,\n    rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with \u27e8x', hx', rfl\u27e9,\n    have : x - x' \u2208 J,\n    { apply h J hJ,\n      rw [ring_hom.mem_ker, map_sub, hx, sub_self] },\n    simpa only [sub_add_cancel] using J.add_mem this hx' }\nend\n\ntheorem map_is_prime_of_surjective {f : F} (hf : function.surjective f) {I : ideal R}\n  [H : is_prime I] (hk : ring_hom.ker f \u2264 I) : is_prime (map f I) :=\nbegin\n  refine \u27e8\u03bb h, H.ne_top (eq_top_iff.2 _), \u03bb x y, _\u27e9,\n  { replace h := congr_arg (comap f) h,\n    rw [comap_map_of_surjective _ hf, comap_top] at h,\n    exact h \u25b8 sup_le (le_of_eq rfl) hk },\n  { refine \u03bb hxy, (hf x).rec_on (\u03bb a ha, (hf y).rec_on (\u03bb b hb, _)),\n    rw [\u2190 ha, \u2190 hb, \u2190 _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy,\n    rcases hxy with \u27e8c, hc, hc'\u27e9,\n    rw [\u2190 sub_eq_zero, \u2190 map_sub] at hc',\n    have : a * b \u2208 I,\n    { convert I.sub_mem hc (hk (hc' : c - a * b \u2208 ring_hom.ker f)),\n      abel },\n    exact (H.mem_or_mem this).imp (\u03bb h, ha \u25b8 mem_map_of_mem f h) (\u03bb h, hb \u25b8 mem_map_of_mem f h) }\nend\nomit rc\n\ntheorem map_is_prime_of_equiv {F' : Type*} [ring_equiv_class F' R S]\n  (f : F') {I : ideal R} [is_prime I] :\n  is_prime (map f I) :=\nmap_is_prime_of_surjective (equiv_like.surjective f) $ by simp only [ring_hom.ker_equiv, bot_le]\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S]\n\n@[simp] lemma mk_ker {I : ideal R} : (quotient.mk I).ker = I :=\nby ext; rw [ring_hom.ker, mem_comap, submodule.mem_bot, quotient.eq_zero_iff_mem]\n\nlemma map_mk_eq_bot_of_le {I J : ideal R} (h : I \u2264 J) : I.map (J^.quotient.mk) = \u22a5 :=\nby { rw [map_eq_bot_iff_le_ker, mk_ker], exact h }\n\nlemma ker_quotient_lift {S : Type v} [comm_ring S] {I : ideal R} (f : R \u2192+* S) (H : I \u2264 f.ker) :\n  (ideal.quotient.lift I f H).ker = (f.ker).map I^.quotient.mk :=\nbegin\n  ext x,\n  split,\n  { intro hx,\n    obtain \u27e8y, hy\u27e9 := quotient.mk_surjective x,\n    rw [ring_hom.mem_ker, \u2190 hy, ideal.quotient.lift_mk, \u2190 ring_hom.mem_ker] at hx,\n    rw [\u2190 hy, mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective],\n    exact \u27e8y, hx, rfl\u27e9 },\n  { intro hx,\n    rw mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective at hx,\n    obtain \u27e8y, hy\u27e9 := hx,\n    rw [ring_hom.mem_ker, \u2190 hy.right, ideal.quotient.lift_mk, \u2190 (ring_hom.mem_ker f)],\n    exact hy.left },\nend\n\ntheorem map_eq_iff_sup_ker_eq_of_surjective {I J : ideal R} (f : R \u2192+* S)\n  (hf : function.surjective f) : map f I = map f J \u2194 I \u2294 f.ker = J \u2294 f.ker :=\nby rw [\u2190 (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf,\n  comap_map_of_surjective f hf, ring_hom.ker_eq_comap_bot]\n\ntheorem map_radical_of_surjective {f : R \u2192+* S} (hf : function.surjective f) {I : ideal R}\n  (h : ring_hom.ker f \u2264 I) : map f (I.radical) = (map f I).radical :=\nbegin\n  rw [radical_eq_Inf, radical_eq_Inf],\n  have : \u2200 J \u2208 {J : ideal R | I \u2264 J \u2227 J.is_prime}, f.ker \u2264 J := \u03bb J hJ, le_trans h hJ.left,\n  convert map_Inf hf this,\n  refine funext (\u03bb j, propext \u27e8_, _\u27e9),\n  { rintros \u27e8hj, hj'\u27e9,\n    haveI : j.is_prime := hj',\n    exact \u27e8comap f j, \u27e8\u27e8map_le_iff_le_comap.1 hj, comap_is_prime f j\u27e9,\n      map_comap_of_surjective f hf j\u27e9\u27e9 },\n  { rintro \u27e8J, \u27e8hJ, hJ'\u27e9\u27e9,\n    haveI : J.is_prime := hJ.right,\n    refine \u27e8hJ' \u25b8 map_mono hJ.left, hJ' \u25b8 map_is_prime_of_surjective hf (le_trans h hJ.left)\u27e9 },\nend\n\n@[simp] lemma bot_quotient_is_maximal_iff (I : ideal R) :\n  (\u22a5 : ideal (R \u29f8 I)).is_maximal \u2194 I.is_maximal :=\n\u27e8\u03bb hI, (@mk_ker _ _ I) \u25b8\n  @comap_is_maximal_of_surjective _ _ _ _ _ _ (quotient.mk I) quotient.mk_surjective \u22a5 hI,\n \u03bb hI, @bot_is_maximal _ (@field.to_division_ring _ (@quotient.field _ _ I hI)) \u27e9\n\n/-- See also `ideal.mem_quotient_iff_mem` in case `I \u2264 J`. -/\n@[simp]\nlemma mem_quotient_iff_mem_sup {I J : ideal R} {x : R} :\n  quotient.mk I x \u2208 J.map (quotient.mk I) \u2194 x \u2208 J \u2294 I :=\nby rw [\u2190 mem_comap, comap_map_of_surjective (quotient.mk I) quotient.mk_surjective,\n       \u2190 ring_hom.ker_eq_comap_bot, mk_ker]\n\n/-- See also `ideal.mem_quotient_iff_mem_sup` if the assumption `I \u2264 J` is not available. -/\nlemma mem_quotient_iff_mem {I J : ideal R} (hIJ : I \u2264 J) {x : R} :\n  quotient.mk I x \u2208 J.map (quotient.mk I) \u2194 x \u2208 J :=\nby rw [mem_quotient_iff_mem_sup, sup_eq_left.mpr hIJ]\n\nsection quotient_algebra\n\nvariables (R\u2081 R\u2082 : Type*) {A B : Type*}\nvariables [comm_semiring R\u2081] [comm_semiring R\u2082] [comm_ring A] [comm_ring B]\nvariables [algebra R\u2081 A] [algebra R\u2082 A] [algebra R\u2081 B]\n\n/-- The `R\u2081`-algebra structure on `A/I` for an `R\u2081`-algebra `A` -/\ninstance quotient.algebra {I : ideal A} : algebra R\u2081 (A \u29f8 I) :=\n{ to_fun := \u03bb x, ideal.quotient.mk I (algebra_map R\u2081 A x),\n  smul := (\u2022),\n  smul_def' := \u03bb r x, quotient.induction_on' x $ \u03bb x,\n      ((quotient.mk I).congr_arg $ algebra.smul_def _ _).trans (ring_hom.map_mul _ _ _),\n  commutes' := \u03bb _ _, mul_comm _ _,\n  .. ring_hom.comp (ideal.quotient.mk I) (algebra_map R\u2081 A) }\n\n-- Lean can struggle to find this instance later if we don't provide this shortcut\ninstance quotient.is_scalar_tower [has_smul R\u2081 R\u2082] [is_scalar_tower R\u2081 R\u2082 A] (I : ideal A) :\n  is_scalar_tower R\u2081 R\u2082 (A \u29f8 I) :=\nby apply_instance\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] A \u29f8 I` as morphism of `R\u2081`-algebras, for `I` an ideal of\n`A`, where `A` is an `R\u2081`-algebra. -/\ndef quotient.mk\u2090 (I : ideal A) : A \u2192\u2090[R\u2081] A \u29f8 I :=\n\u27e8\u03bb a, submodule.quotient.mk a, rfl, \u03bb _ _, rfl, rfl, \u03bb _ _, rfl, \u03bb _, rfl\u27e9\n\nlemma quotient.alg_map_eq (I : ideal A) :\n  algebra_map R\u2081 (A \u29f8 I) = (algebra_map A (A \u29f8 I)).comp (algebra_map R\u2081 A) :=\nrfl\n\nlemma quotient.mk\u2090_to_ring_hom (I : ideal A) :\n  (quotient.mk\u2090 R\u2081 I).to_ring_hom = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.mk\u2090_eq_mk (I : ideal A) :\n  \u21d1(quotient.mk\u2090 R\u2081 I) = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.algebra_map_eq (I : ideal R) :\n  algebra_map R (R \u29f8 I) = I^.quotient.mk :=\nrfl\n\n@[simp] lemma quotient.mk_comp_algebra_map (I : ideal A) :\n  (quotient.mk I).comp (algebra_map R\u2081 A) = algebra_map R\u2081 (A \u29f8 I) :=\nrfl\n\n@[simp] lemma quotient.mk_algebra_map (I : ideal A) (x : R\u2081) :\n  quotient.mk I (algebra_map R\u2081 A x) = algebra_map R\u2081 (A \u29f8 I) x :=\nrfl\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] I.quotient` is surjective. -/\nlemma quotient.mk\u2090_surjective (I : ideal A) : function.surjective (quotient.mk\u2090 R\u2081 I) :=\nsurjective_quot_mk _\n\n/-- The kernel of `A \u2192\u2090[R\u2081] I.quotient` is `I`. -/\n@[simp]\nlemma quotient.mk\u2090_ker (I : ideal A) : (quotient.mk\u2090 R\u2081 I : A \u2192+* A \u29f8 I).ker = I :=\nideal.mk_ker\n\nvariables {R\u2081}\n\nlemma ker_lift.map_smul (f : A \u2192\u2090[R\u2081] B) (r : R\u2081) (x : A \u29f8 f.to_ring_hom.ker) :\n  f.to_ring_hom.ker_lift (r \u2022 x) = r \u2022 f.to_ring_hom.ker_lift x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := quotient.mk\u2090_surjective R\u2081 _ x,\n  rw [\u2190 alg_hom.map_smul, quotient.mk\u2090_eq_mk, ring_hom.ker_lift_mk],\n  exact f.map_smul _ _\nend\n\n/-- The induced algebras morphism from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_alg_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_alg_equiv_of_surjective`).\n-/\ndef ker_lift_alg (f : A \u2192\u2090[R\u2081] B) : (A \u29f8 f.to_ring_hom.ker) \u2192\u2090[R\u2081] B :=\nalg_hom.mk' f.to_ring_hom.ker_lift (\u03bb _ _, ker_lift.map_smul f _ _)\n\n@[simp]\nlemma ker_lift_alg_mk (f : A \u2192\u2090[R\u2081] B) (a : A) :\n  ker_lift_alg f (quotient.mk f.to_ring_hom.ker a) = f a := rfl\n\n@[simp]\nlemma ker_lift_alg_to_ring_hom (f : A \u2192\u2090[R\u2081] B) :\n  (ker_lift_alg f).to_ring_hom = ring_hom.ker_lift f := rfl\n\n/-- The induced algebra morphism from the quotient by the kernel is injective. -/\nlemma ker_lift_alg_injective (f : A \u2192\u2090[R\u2081] B) : function.injective (ker_lift_alg f) :=\nring_hom.ker_lift_injective f\n\n/-- The **first isomorphism** theorem for algebras, computable version. -/\ndef quotient_ker_alg_equiv_of_right_inverse\n  {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A} (hf : function.right_inverse g f) :\n  (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\n{ ..ring_hom.quotient_ker_equiv_of_right_inverse (\u03bb x, show f.to_ring_hom (g x) = x, from hf x),\n  ..ker_lift_alg f}\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : A \u29f8 f.to_ring_hom.ker) :\n  quotient_ker_alg_equiv_of_right_inverse hf x = ker_lift_alg f x := rfl\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse_symm.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : B) :\n  (quotient_ker_alg_equiv_of_right_inverse hf).symm x = quotient.mk\u2090 R\u2081 f.to_ring_hom.ker (g x) :=\n  rfl\n\n/-- The **first isomorphism theorem** for algebras. -/\nnoncomputable def quotient_ker_alg_equiv_of_surjective\n  {f : A \u2192\u2090[R\u2081] B} (hf : function.surjective f) : (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\nquotient_ker_alg_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\n/-- The ring hom `R/I \u2192+* S/J` induced by a ring hom `f : R \u2192+* S` with `I \u2264 f\u207b\u00b9(J)` -/\ndef quotient_map {I : ideal R} (J : ideal S) (f : R \u2192+* S) (hIJ : I \u2264 J.comap f) :\n  R \u29f8 I \u2192+* S \u29f8 J :=\n(quotient.lift I ((quotient.mk J).comp f) (\u03bb _ ha,\n  by simpa [function.comp_app, ring_hom.coe_comp, quotient.eq_zero_iff_mem] using hIJ ha))\n\n@[simp]\nlemma quotient_map_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R} : quotient_map I f H (quotient.mk J x) = quotient.mk I (f x) :=\nquotient.lift_mk J _ _\n\n@[simp]\nlemma quotient_map_algebra_map {J : ideal A} {I : ideal S} {f : A \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R\u2081} :\n  quotient_map I f H (algebra_map R\u2081 (A \u29f8 J) x) = quotient.mk I (f (algebra_map _ _ x)) :=\nquotient.lift_mk J _ _\n\nlemma quotient_map_comp_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} (H : J \u2264 I.comap f) :\n  (quotient_map I f H).comp (quotient.mk J) = (quotient.mk I).comp f :=\nring_hom.ext (\u03bb x, by simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient_map_mk])\n\n/-- The ring equiv `R/I \u2243+* S/J` induced by a ring equiv `f : R \u2243+** S`,  where `J = f(I)`. -/\n@[simps]\ndef quotient_equiv (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S)) :\n  R \u29f8 I \u2243+* S \u29f8 J :=\n{ inv_fun := quotient_map I \u2191f.symm (by {rw hIJ, exact le_of_eq (map_comap_of_equiv I f)}),\n  left_inv := by {rintro \u27e8r\u27e9, simp },\n  right_inv := by {rintro \u27e8s\u27e9, simp },\n  ..quotient_map J \u2191f (by {rw hIJ, exact @le_comap_map _ S _ _ _ _ _ _}) }\n\n@[simp]\nlemma quotient_equiv_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S))\n  (x : R) : quotient_equiv I J f hIJ (ideal.quotient.mk I x) = ideal.quotient.mk J (f x) := rfl\n\n@[simp]\nlemma quotient_equiv_symm_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S)\n  (hIJ : J = I.map (f : R \u2192+* S)) (x : S) :\n  (quotient_equiv I J f hIJ).symm (ideal.quotient.mk J x) = ideal.quotient.mk I (f.symm x) := rfl\n\n/-- `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. -/\nlemma quotient_map_injective' {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (h : I.comap f \u2264 J) : function.injective (quotient_map I f H) :=\nbegin\n  refine (injective_iff_map_eq_zero (quotient_map I f H)).2 (\u03bb a ha, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  rw [quotient_map_mk, quotient.eq_zero_iff_mem] at ha,\n  exact (quotient.eq_zero_iff_mem).mpr (h ha),\nend\n\n/-- If we take `J = I.comap f` then `quotient_map` is injective automatically. -/\nlemma quotient_map_injective {I : ideal S} {f : R \u2192+* S} :\n  function.injective (quotient_map I f le_rfl) :=\nquotient_map_injective' le_rfl\n\nlemma quotient_map_surjective {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (hf : function.surjective f) : function.surjective (quotient_map I f H) :=\n\u03bb x, let \u27e8x, hx\u27e9 := quotient.mk_surjective x in\n  let \u27e8y, hy\u27e9 := hf x in \u27e8(quotient.mk J) y, by simp [hx, hy]\u27e9\n\n/-- Commutativity of a square is preserved when taking quotients by an ideal. -/\nlemma comp_quotient_map_eq_of_comp_eq {R' S' : Type*} [comm_ring R'] [comm_ring S']\n  {f : R \u2192+* S} {f' : R' \u2192+* S'} {g : R \u2192+* R'} {g' : S \u2192+* S'} (hfg : f'.comp g = g'.comp f)\n  (I : ideal S') : (quotient_map I g' le_rfl).comp (quotient_map (I.comap g') f le_rfl) =\n    (quotient_map I f' le_rfl).comp (quotient_map (I.comap f') g\n      (le_of_eq (trans (comap_comap f g') (hfg \u25b8 (comap_comap g f'))))) :=\nbegin\n  refine ring_hom.ext (\u03bb a, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  simp only [ring_hom.comp_apply, quotient_map_mk],\n  exact congr_arg (quotient.mk I) (trans (g'.comp_apply f r).symm (hfg \u25b8 (f'.comp_apply g r))),\nend\n\n/-- The algebra hom `A/I \u2192+* B/J` induced by an algebra hom `f : A \u2192\u2090[R\u2081] B` with `I \u2264 f\u207b\u00b9(J)`. -/\ndef quotient_map\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (hIJ : I \u2264 J.comap f) :\n  A \u29f8 I \u2192\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_map J (f : A \u2192+* B) hIJ }\n\n@[simp]\nlemma quotient_map_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f)\n  {x : A} : quotient_map\u2090 J f H (quotient.mk I x) = quotient.mk\u2090 R\u2081 J (f x) := rfl\n\nlemma quotient_map_comp_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f) :\n  (quotient_map\u2090 J f H).comp (quotient.mk\u2090 R\u2081 I) = (quotient.mk\u2090 R\u2081 J).comp f :=\nalg_hom.ext (\u03bb x, by simp only [quotient_map_mk\u2090, quotient.mk\u2090_eq_mk, alg_hom.comp_apply])\n\n/-- The algebra equiv `A/I \u2243\u2090[R] B/J` induced by an algebra equiv `f : A \u2243\u2090[R] B`,\nwhere`J = f(I)`. -/\ndef quotient_equiv_alg (I : ideal A) (J : ideal B) (f : A \u2243\u2090[R\u2081] B)\n  (hIJ : J = I.map (f : A \u2192+* B)) :\n  (A \u29f8 I) \u2243\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_equiv I J (f : A \u2243+* B) hIJ }\n\n@[priority 100]\ninstance quotient_algebra {I : ideal A} [algebra R A] :\n  algebra (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I) :=\n(quotient_map I (algebra_map R A) (le_of_eq rfl)).to_algebra\n\nlemma algebra_map_quotient_injective {I : ideal A} [algebra R A]:\n  function.injective (algebra_map (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I)) :=\nbegin\n  rintros \u27e8a\u27e9 \u27e8b\u27e9 hab,\n  replace hab := quotient.eq.mp hab,\n  rw \u2190 ring_hom.map_sub at hab,\n  exact quotient.eq.mpr hab\nend\n\nend quotient_algebra\n\nend comm_ring\n\nend ideal\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v}\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\n-- TODO: show `[algebra R A] : algebra (ideal R) A` too\n\ninstance module_submodule : module (ideal R) (submodule R M) :=\n{ smul_add := smul_sup,\n  add_smul := sup_smul,\n  mul_smul := submodule.smul_assoc,\n  one_smul := by simp,\n  zero_smul := bot_smul,\n  smul_zero := smul_bot }\n\nend submodule\n\nnamespace ring_hom\nvariables {A B C : Type*} [ring A] [ring B] [ring C]\nvariables (f : A \u2192+* B) (f_inv : B \u2192 A)\n\n/-- Auxiliary definition used to define `lift_of_right_inverse` -/\ndef lift_of_right_inverse_aux\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) :\n  B \u2192+* C :=\n{ to_fun := \u03bb b, g (f_inv b),\n  map_one' :=\n  begin\n    rw [\u2190 g.map_one, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_one],\n    exact hf 1\n  end,\n  map_mul' :=\n  begin\n    intros x y,\n    rw [\u2190 g.map_mul, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_mul],\n    simp only [hf _],\n  end,\n  .. add_monoid_hom.lift_of_right_inverse f.to_add_monoid_hom f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 }\n\n@[simp] lemma lift_of_right_inverse_aux_comp_apply\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) (a : A) :\n  (f.lift_of_right_inverse_aux f_inv hf g hg) (f a) = g a :=\nf.to_add_monoid_hom.lift_of_right_inverse_comp_apply f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 a\n\n/-- `lift_of_right_inverse f hf g hg` is the unique ring homomorphism `\u03c6`\n\n* such that `\u03c6.comp f = g` (`ring_hom.lift_of_right_inverse_comp`),\n* where `f : A \u2192+* B` is has a right_inverse `f_inv` (`hf`),\n* and `g : B \u2192+* C` satisfies `hg : f.ker \u2264 g.ker`.\n\nSee `ring_hom.eq_lift_of_right_inverse` for the uniqueness lemma.\n\n```\n   A .\n   |  \\\n f |   \\ g\n   |    \\\n   v     \\\u231f\n   B ----> C\n      \u2203!\u03c6\n```\n-/\ndef lift_of_right_inverse\n  (hf : function.right_inverse f_inv f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\n{ to_fun := \u03bb g, f.lift_of_right_inverse_aux f_inv hf g.1 g.2,\n  inv_fun := \u03bb \u03c6, \u27e8\u03c6.comp f, \u03bb x hx, (mem_ker _).mpr $ by simp [(mem_ker _).mp hx]\u27e9,\n  left_inv := \u03bb g, by\n  { ext,\n    simp only [comp_apply, lift_of_right_inverse_aux_comp_apply, subtype.coe_mk,\n      subtype.val_eq_coe], },\n  right_inv := \u03bb \u03c6, by\n  { ext b,\n    simp [lift_of_right_inverse_aux, hf b], } }\n\n/-- A non-computable version of `ring_hom.lift_of_right_inverse` for when no computable right\ninverse is available, that uses `function.surj_inv`. -/\n@[simp]\nnoncomputable abbreviation lift_of_surjective\n  (hf : function.surjective f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\nf.lift_of_right_inverse (function.surj_inv hf) (function.right_inverse_surj_inv hf)\n\nlemma lift_of_right_inverse_comp_apply\n  (hf : function.right_inverse f_inv f) (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) (x : A) :\n  (f.lift_of_right_inverse f_inv hf g) (f x) = g x :=\nf.lift_of_right_inverse_aux_comp_apply f_inv hf g.1 g.2 x\n\nlemma lift_of_right_inverse_comp (hf : function.right_inverse f_inv f)\n  (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) :\n  (f.lift_of_right_inverse f_inv hf g).comp f = g :=\nring_hom.ext $ f.lift_of_right_inverse_comp_apply f_inv hf g\n\nlemma eq_lift_of_right_inverse (hf : function.right_inverse f_inv f) (g : A \u2192+* C)\n  (hg : f.ker \u2264 g.ker) (h : B \u2192+* C) (hh : h.comp f = g) :\n  h = (f.lift_of_right_inverse f_inv hf \u27e8g, hg\u27e9) :=\nbegin\n  simp_rw \u2190hh,\n  exact ((f.lift_of_right_inverse f_inv hf).apply_symm_apply _).symm,\nend\n\nend ring_hom\n\nnamespace double_quot\nopen ideal\nvariables {R : Type u} [comm_ring R] (I J : ideal R)\n\n/-- The obvious ring hom `R/I \u2192 R/(I \u2294 J)` -/\ndef quot_left_to_quot_sup : R \u29f8 I \u2192+* R \u29f8 (I \u2294 J) :=\nideal.quotient.factor I (I \u2294 J) le_sup_left\n\n/-- The kernel of `quot_left_to_quot_sup` -/\nlemma ker_quot_left_to_quot_sup :\n  (quot_left_to_quot_sup I J).ker = J.map (ideal.quotient.mk I) :=\nby simp only [mk_ker, sup_idem, sup_comm, quot_left_to_quot_sup, quotient.factor, ker_quotient_lift,\n    map_eq_iff_sup_ker_eq_of_surjective I^.quotient.mk quotient.mk_surjective, \u2190 sup_assoc]\n\n/-- The ring homomorphism `(R/I)/J' -> R/(I \u2294 J)` induced by `quot_left_to_quot_sup` where `J'`\n  is the image of `J` in `R/I`-/\ndef quot_quot_to_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2192+* R \u29f8 I \u2294 J :=\nby exact ideal.quotient.lift (J.map (ideal.quotient.mk I)) (quot_left_to_quot_sup I J)\n  (ker_quot_left_to_quot_sup I J).symm.le\n\n/-- The composite of the maps `R \u2192 (R/I)` and `(R/I) \u2192 (R/I)/J'` -/\ndef quot_quot_mk : R \u2192+* ((R \u29f8 I) \u29f8 J.map I^.quotient.mk) :=\nby exact ((J.map I^.quotient.mk)^.quotient.mk).comp I^.quotient.mk\n\n/-- The kernel of `quot_quot_mk` -/\nlemma ker_quot_quot_mk : (quot_quot_mk I J).ker = I \u2294 J :=\nby rw [ring_hom.ker_eq_comap_bot, quot_quot_mk, \u2190 comap_comap, \u2190 ring_hom.ker, mk_ker,\n  comap_map_of_surjective (ideal.quotient.mk I) (quotient.mk_surjective), \u2190 ring_hom.ker, mk_ker,\n  sup_comm]\n\n/-- The ring homomorphism `R/(I \u2294 J) \u2192 (R/I)/J' `induced by `quot_quot_mk` -/\ndef lift_sup_quot_quot_mk (I J : ideal R) :\n  R \u29f8 (I \u2294 J) \u2192+* (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) :=\nideal.quotient.lift (I \u2294 J) (quot_quot_mk I J) (ker_quot_quot_mk I J).symm.le\n\n/-- `quot_quot_to_quot_add` and `lift_sup_double_qot_mk` are inverse isomorphisms -/\ndef quot_quot_equiv_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2243+* R \u29f8 I \u2294 J :=\nring_equiv.of_hom_inv (quot_quot_to_quot_sup I J) (lift_sup_quot_quot_mk I J)\n  (by { ext z, refl }) (by { ext z, refl })\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_quot_quot_mk (x : R) :\n  quot_quot_equiv_quot_sup I J (quot_quot_mk I J x) = ideal.quotient.mk (I \u2294 J) x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_symm_quot_quot_mk (x : R) :\n  (quot_quot_equiv_quot_sup I J).symm (ideal.quotient.mk (I \u2294 J) x) = quot_quot_mk I J x :=\nrfl\n\n/-- The obvious isomorphism `(R/I)/J' \u2192 (R/J)/I' `   -/\ndef quot_quot_equiv_comm :\n  (R \u29f8 I) \u29f8 J.map I^.quotient.mk \u2243+* (R \u29f8 J) \u29f8 I.map J^.quotient.mk :=\n((quot_quot_equiv_quot_sup I J).trans (quot_equiv_of_eq sup_comm)).trans\n  (quot_quot_equiv_quot_sup J I).symm\n\n@[simp]\nlemma quot_quot_equiv_comm_quot_quot_mk (x : R) :\n  quot_quot_equiv_comm I J (quot_quot_mk I J x) = quot_quot_mk J I x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_comm_comp_quot_quot_mk :\n  ring_hom.comp \u2191(quot_quot_equiv_comm I J) (quot_quot_mk I J) = quot_quot_mk J I :=\nring_hom.ext $ quot_quot_equiv_comm_quot_quot_mk I J\n\n@[simp]\nlemma quot_quot_equiv_comm_symm :\n  (quot_quot_equiv_comm I J).symm = quot_quot_equiv_comm J I :=\nrfl\n\nend double_quot\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/ring_theory/ideal/operations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3692330447912386}}
{"text": "/-\nCopyright (c) 2018 Michael Jendrusch. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer\n\n! This file was ported from Lean 3 source module category_theory.monoidal.coherence_lemmas\n! leanprover-community/mathlib commit b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Monoidal.Coherence\n\n/-!\n# Lemmas which are consequences of monoidal coherence\n\nThese lemmas are all proved `by coherence`.\n\n## Future work\nInvestigate whether these lemmas are really needed,\nor if they can be replaced by use of the `coherence` tactic.\n-/\n\n\nopen CategoryTheory\n\nopen CategoryTheory.Category\n\nopen CategoryTheory.Iso\n\nnamespace CategoryTheory.MonoidalCategory\n\nvariable {C : Type _} [Category C] [MonoidalCategory C]\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n-- See Proposition 2.2.4 of <http://www-math.mit.edu/~etingof/egnobookfinal.pdf>\n@[reassoc.1]\ntheorem leftUnitor_tensor' (X Y : C) : (\u03b1_ (\ud835\udfd9_ C) X Y).Hom \u226b (\u03bb_ (X \u2297 Y)).Hom = (\u03bb_ X).Hom \u2297 \ud835\udfd9 Y :=\n  by coherence\n#align category_theory.monoidal_category.left_unitor_tensor' CategoryTheory.MonoidalCategory.leftUnitor_tensor'\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1, simp]\ntheorem leftUnitor_tensor (X Y : C) : (\u03bb_ (X \u2297 Y)).Hom = (\u03b1_ (\ud835\udfd9_ C) X Y).inv \u226b ((\u03bb_ X).Hom \u2297 \ud835\udfd9 Y) :=\n  by coherence\n#align category_theory.monoidal_category.left_unitor_tensor CategoryTheory.MonoidalCategory.leftUnitor_tensor\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem leftUnitor_tensor_inv (X Y : C) :\n    (\u03bb_ (X \u2297 Y)).inv = ((\u03bb_ X).inv \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ (\ud835\udfd9_ C) X Y).Hom := by coherence\n#align category_theory.monoidal_category.left_unitor_tensor_inv CategoryTheory.MonoidalCategory.leftUnitor_tensor_inv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem id_tensor_rightUnitor_inv (X Y : C) : \ud835\udfd9 X \u2297 (\u03c1_ Y).inv = (\u03c1_ _).inv \u226b (\u03b1_ _ _ _).Hom := by\n  coherence\n#align category_theory.monoidal_category.id_tensor_right_unitor_inv CategoryTheory.MonoidalCategory.id_tensor_rightUnitor_inv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem leftUnitor_inv_tensor_id (X Y : C) : (\u03bb_ X).inv \u2297 \ud835\udfd9 Y = (\u03bb_ _).inv \u226b (\u03b1_ _ _ _).inv := by\n  coherence\n#align category_theory.monoidal_category.left_unitor_inv_tensor_id CategoryTheory.MonoidalCategory.leftUnitor_inv_tensor_id\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem pentagon_inv_inv_hom (W X Y Z : C) :\n    (\u03b1_ W (X \u2297 Y) Z).inv \u226b ((\u03b1_ W X Y).inv \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ (W \u2297 X) Y Z).Hom =\n      (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).Hom) \u226b (\u03b1_ W X (Y \u2297 Z)).inv :=\n  by coherence\n#align category_theory.monoidal_category.pentagon_inv_inv_hom CategoryTheory.MonoidalCategory.pentagon_inv_inv_hom\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[simp, reassoc.1]\ntheorem triangle_assoc_comp_right_inv (X Y : C) :\n    ((\u03c1_ X).inv \u2297 \ud835\udfd9 Y) \u226b (\u03b1_ X (\ud835\udfd9_ C) Y).Hom = \ud835\udfd9 X \u2297 (\u03bb_ Y).inv := by coherence\n#align category_theory.monoidal_category.triangle_assoc_comp_right_inv CategoryTheory.MonoidalCategory.triangle_assoc_comp_right_inv\n\ntheorem unitors_equal : (\u03bb_ (\ud835\udfd9_ C)).Hom = (\u03c1_ (\ud835\udfd9_ C)).Hom := by coherence\n#align category_theory.monoidal_category.unitors_equal CategoryTheory.MonoidalCategory.unitors_equal\n\ntheorem unitors_inv_equal : (\u03bb_ (\ud835\udfd9_ C)).inv = (\u03c1_ (\ud835\udfd9_ C)).inv := by coherence\n#align category_theory.monoidal_category.unitors_inv_equal CategoryTheory.MonoidalCategory.unitors_inv_equal\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem pentagon_hom_inv {W X Y Z : C} :\n    (\u03b1_ W X (Y \u2297 Z)).Hom \u226b (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) =\n      (\u03b1_ (W \u2297 X) Y Z).inv \u226b ((\u03b1_ W X Y).Hom \u2297 \ud835\udfd9 Z) \u226b (\u03b1_ W (X \u2297 Y) Z).Hom :=\n  by coherence\n#align category_theory.monoidal_category.pentagon_hom_inv CategoryTheory.MonoidalCategory.pentagon_hom_inv\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n@[reassoc.1]\ntheorem pentagon_inv_hom (W X Y Z : C) :\n    (\u03b1_ (W \u2297 X) Y Z).inv \u226b ((\u03b1_ W X Y).Hom \u2297 \ud835\udfd9 Z) =\n      (\u03b1_ W X (Y \u2297 Z)).Hom \u226b (\ud835\udfd9 W \u2297 (\u03b1_ X Y Z).inv) \u226b (\u03b1_ W (X \u2297 Y) Z).inv :=\n  by coherence\n#align category_theory.monoidal_category.pentagon_inv_hom CategoryTheory.MonoidalCategory.pentagon_inv_hom\n\nend CategoryTheory.MonoidalCategory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Monoidal/CoherenceLemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3692330362385316}}
{"text": "\nnamespace SciLean\n\n-- This is like `ExactSolution` but it is intended to be used in automation.\ninductive AutoExactSolution {\u03b1 : Type _} : (\u03b1 \u2192 Prop) \u2192 Type _ where\n| exact {spec : \u03b1 \u2192 Prop} (a : \u03b1) (h : spec a) : AutoExactSolution spec\n\ndef AutoImpl {\u03b1} (a : \u03b1) := AutoExactSolution \u03bb x => x = a\n\n@[inline]\ndef AutoImpl.val {\u03b1} {a : \u03b1} (x : AutoImpl a) : \u03b1 :=\nmatch x with\n| .exact val _ => val\n\ndef AutoImpl.finish {\u03b1} {a : \u03b1} : AutoImpl a := .exact a rfl\n\ntheorem AutoImpl.impl_eq_spec (x : AutoImpl a) : a = x.val :=\nby\n  cases x; rename_i a' h; \n  simp[AutoImpl.val, val, h]\n  done\n\n-- I don't think think this can be proven. Can it lead to contradiction?\naxiom AutoImpl.injectivity_axiom {\u03b1} (a b : \u03b1) : (AutoImpl a = AutoImpl b) \u2192 (a = b)\n\n-- Do we really need AutoImpl.injectivity_axiom?\n@[simp] theorem AutoImpl.normalize_val {\u03b1 : Type u} (a b : \u03b1) (h : (AutoImpl a = AutoImpl b)) \n  : AutoImpl.val (Eq.mpr h (AutoImpl.finish (a:=b))) = b := \nby\n  have h' : a = b := by apply AutoImpl.injectivity_axiom; apply h\n  revert h; rw[h']\n  simp[val,finish,Eq.mpr]\n  done\n\n-- This is a new version of `AutoImpl.normalize_val`, some tactic uses `cast` instead of `Eq.mpr` now\n-- TODO: clean this up\n@[simp] theorem AutoImpl.normalize_val' {\u03b1 : Type u} (a b : \u03b1) (h : (AutoImpl a = AutoImpl b)) \n  : AutoImpl.val (cast h (AutoImpl.finish (a:=a))) = a := \nby sorry\n  -- have h' : a = b := by apply AutoImpl.injectivity_axiom; apply h\n  -- revert h; rw[h']\n  -- simp[val,finish,Eq.mpr]\n  -- done\n\n\nexample {\u03b1 : Type} (a b : \u03b1) (A : (\u03a3' x, x = a)) (h : (\u03a3' x, x = a) = (\u03a3' x, x = b))\n  : (a = b) \u2194 (h \u25b8 A).1 = A.1 := \nby\n  constructor\n  {\n    intro eq; rw[A.2]; conv => rhs; rw [eq]\n    apply (h \u25b8 A).2\n  }\n  {\n    intro eq; rw[\u2190 A.2]; rw[\u2190 eq]\n    apply (h \u25b8 A).2\n  }\n\nopen Lean.Parser.Tactic.Conv\n\nsyntax term:max \"rewrite_by\" convSeq : term\n\nmacro_rules\n  | `($x rewrite_by $rw:convSeq) =>\n    `((by (conv => enter[1]; ($rw)); (apply AutoImpl.finish) : AutoImpl $x).val)\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/AutoImpl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.369168570803066}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.traversable.basic\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Control.Functor\nimport Mathbin.Tactic.Ext\n\n/-!\n# Traversable type class\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nType classes for traversing collections. The concepts and laws are taken from\n<http://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html>\n\nTraversable collections are a generalization of functors. Whereas\nfunctors (such as `list`) allow us to apply a function to every\nelement, it does not allow functions which external effects encoded in\na monad. Consider for instance a functor `invite : email \u2192 io response`\nthat takes an email address, sends an email and waits for a\nresponse. If we have a list `guests : list email`, using calling\n`invite` using `map` gives us the following: `map invite guests : list\n(io response)`.  It is not what we need. We need something of type `io\n(list response)`. Instead of using `map`, we can use `traverse` to\nsend all the invites: `traverse invite guests : io (list response)`.\n`traverse` applies `invite` to every element of `guests` and combines\nall the resulting effects. In the example, the effect is encoded in the\nmonad `io` but any applicative functor is accepted by `traverse`.\n\nFor more on how to use traversable, consider the Haskell tutorial:\n<https://en.wikibooks.org/wiki/Haskell/Traversable>\n\n## Main definitions\n  * `traversable` type class - exposes the `traverse` function\n  * `sequence` - based on `traverse`,\n    turns a collection of effects into an effect returning a collection\n  * `is_lawful_traversable` - laws for a traversable functor\n  * `applicative_transformation` - the notion of a natural transformation for applicative functors\n\n## Tags\n\ntraversable iterator functor applicative\n\n## References\n\n * \"Applicative Programming with Effects\", by Conor McBride and Ross Paterson,\n   Journal of Functional Programming 18:1 (2008) 1-13, online at\n   <http://www.soi.city.ac.uk/~ross/papers/Applicative.html>\n * \"The Essence of the Iterator Pattern\", by Jeremy Gibbons and Bruno Oliveira,\n   in Mathematically-Structured Functional Programming, 2006, online at\n   <http://web.comlab.ox.ac.uk/oucl/work/jeremy.gibbons/publications/#iterator>\n * \"An Investigation of the Laws of Traversals\", by Mauro Jaskelioff and Ondrej Rypacek,\n   in Mathematically-Structured Functional Programming, 2012,\n   online at <http://arxiv.org/pdf/1202.2919>\n-/\n\n\nopen Function hiding comp\n\nuniverse u v w\n\nsection ApplicativeTransformation\n\nvariable (F : Type u \u2192 Type v) [Applicative F] [LawfulApplicative F]\n\nvariable (G : Type u \u2192 Type w) [Applicative G] [LawfulApplicative G]\n\n/- warning: applicative_transformation -> ApplicativeTransformation is a dubious translation:\nlean 3 declaration is\n  forall (F : Type.{u1} -> Type.{u2}) [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] (G : Type.{u1} -> Type.{u3}) [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3], Type.{max (succ u1) u2 u3}\nbut is expected to have type\n  forall (F : Type.{u1} -> Type.{u2}) [_inst_1 : Applicative.{u1, u2} F] (_inst_2 : Type.{u1} -> Type.{u3}) [G : Applicative.{u1, u3} _inst_2], Type.{max (succ u1) u2 u3}\nCase conversion may be inaccurate. Consider using '#align applicative_transformation ApplicativeTransformation\u2093'. -/\n/-- A transformation between applicative functors.  It is a natural\ntransformation such that `app` preserves the `has_pure.pure` and\n`functor.map` (`<*>`) operations. See\n`applicative_transformation.preserves_map` for naturality. -/\nstructure ApplicativeTransformation : Type max (u + 1) v w where\n  app : \u2200 \u03b1 : Type u, F \u03b1 \u2192 G \u03b1\n  preserves_pure' : \u2200 {\u03b1 : Type u} (x : \u03b1), app _ (pure x) = pure x\n  preserves_seq' : \u2200 {\u03b1 \u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), app _ (x <*> y) = app _ x <*> app _ y\n#align applicative_transformation ApplicativeTransformation\n\nend ApplicativeTransformation\n\nnamespace ApplicativeTransformation\n\nvariable (F : Type u \u2192 Type v) [Applicative F] [LawfulApplicative F]\n\nvariable (G : Type u \u2192 Type w) [Applicative G] [LawfulApplicative G]\n\ninstance : CoeFun (ApplicativeTransformation F G) fun _ => \u2200 {\u03b1}, F \u03b1 \u2192 G \u03b1 :=\n  \u27e8ApplicativeTransformation.app\u27e9\n\nvariable {F G}\n\n/- warning: applicative_transformation.app_eq_coe -> ApplicativeTransformation.app_eq_coe is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4), Eq.{max (succ (succ u1)) (succ u2) (succ u3)} (forall (\u03b1 : Type.{u1}), (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4 \u03b7) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G), Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (forall (\u03b1 : Type.{u1}), (F \u03b1) -> (_inst_2 \u03b1)) (ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3) (fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243)\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.app_eq_coe ApplicativeTransformation.app_eq_coe\u2093'. -/\n@[simp]\ntheorem app_eq_coe (\u03b7 : ApplicativeTransformation F G) : \u03b7.app = \u03b7 :=\n  rfl\n#align applicative_transformation.app_eq_coe ApplicativeTransformation.app_eq_coe\n\n/- warning: applicative_transformation.coe_mk -> ApplicativeTransformation.coe_mk is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (f : forall (\u03b1 : Type.{u1}), (F \u03b1) -> (G \u03b1)) (pp : forall {\u03b1 : Type.{u1}} (x : \u03b1), Eq.{succ u3} (G \u03b1) (f \u03b1 (Pure.pure.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) (Applicative.toHasPure.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1) \u03b1 x)) (Pure.pure.{u1, u3} (fun (\u03b1 : Type.{u1}) => G \u03b1) (Applicative.toHasPure.{u1, u3} (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3) \u03b1 x)) (ps : forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (x : F (\u03b1 -> \u03b2)) (y : F \u03b1), Eq.{succ u3} (G \u03b2) (f \u03b2 (Seq.seq.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) (Applicative.toHasSeq.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1) \u03b1 \u03b2 x y)) (Seq.seq.{u1, u3} (fun (\u03b1 : Type.{u1}) => G \u03b1) (Applicative.toHasSeq.{u1, u3} (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3) \u03b1 \u03b2 (f (\u03b1 -> \u03b2) x) (f \u03b1 y))), Eq.{max (succ (succ u1)) (succ u2) (succ u3)} (forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 _inst_2 (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 _inst_2 (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 _inst_2 (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3 _inst_4) (ApplicativeTransformation.mk.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 _inst_2 (fun (\u03b1 : Type.{u1}) => G \u03b1) _inst_3 _inst_4 f pp ps)) f\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : forall (\u03b1 : Type.{u1}), (F \u03b1) -> (_inst_2 \u03b1)) (_inst_4 : forall {\u03b1 : Type.{u1}} (x : \u03b1), Eq.{succ u3} (_inst_2 \u03b1) (_inst_3 \u03b1 (Pure.pure.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) (Applicative.toPure.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1) \u03b1 x)) (Pure.pure.{u1, u3} (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) (Applicative.toPure.{u1, u3} (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) G) \u03b1 x)) (f : forall {\u03b1 : Type.{u1}} {\u1fb0 : Type.{u1}} (x : F (\u03b1 -> \u1fb0)) (y : F \u03b1), Eq.{succ u3} (_inst_2 \u1fb0) (_inst_3 \u1fb0 (Seq.seq.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) (Applicative.toSeq.{u1, u2} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1) \u03b1 \u1fb0 x (fun (x._@.Mathlib.Control.Traversable.Basic._hyg.113 : Unit) => y))) (Seq.seq.{u1, u3} (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) (Applicative.toSeq.{u1, u3} (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) G) \u03b1 \u1fb0 (_inst_3 (\u03b1 -> \u1fb0) x) (fun (x._@.Mathlib.Control.Traversable.Basic._hyg.125 : Unit) => _inst_3 \u03b1 y))), Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (forall {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}}, (F \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) -> (_inst_2 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243)) (fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) G (ApplicativeTransformation.mk.{u1, u2, u3} (fun (\u03b1 : Type.{u1}) => F \u03b1) _inst_1 (fun (\u03b1 : Type.{u1}) => _inst_2 \u03b1) G _inst_3 _inst_4 f) \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_3\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.coe_mk ApplicativeTransformation.coe_mk\u2093'. -/\n@[simp]\ntheorem coe_mk (f : \u2200 \u03b1 : Type u, F \u03b1 \u2192 G \u03b1) (pp ps) :\n    \u21d1(ApplicativeTransformation.mk f pp ps) = f :=\n  rfl\n#align applicative_transformation.coe_mk ApplicativeTransformation.coe_mk\n\n/- warning: applicative_transformation.congr_fun -> ApplicativeTransformation.congr_fun is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (\u03b7' : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4), (Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b7') -> (forall {\u03b1 : Type.{u1}} (x : F \u03b1), Eq.{succ u3} (G \u03b1) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 x) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7' \u03b1 x))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) (_inst_4 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G), (Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) _inst_3 _inst_4) -> (forall {\u03b7' : Type.{u1}} (h : F \u03b7'), Eq.{succ u3} (_inst_2 \u03b7') ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b7' h) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_4 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b7' h))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.congr_fun ApplicativeTransformation.congr_fun\u2093'. -/\nprotected theorem congr_fun (\u03b7 \u03b7' : ApplicativeTransformation F G) (h : \u03b7 = \u03b7') {\u03b1 : Type u}\n    (x : F \u03b1) : \u03b7 x = \u03b7' x :=\n  congr_arg (fun \u03b7'' : ApplicativeTransformation F G => \u03b7'' x) h\n#align applicative_transformation.congr_fun ApplicativeTransformation.congr_fun\n\n/- warning: applicative_transformation.congr_arg -> ApplicativeTransformation.congr_arg is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) {\u03b1 : Type.{u1}} {x : F \u03b1} {y : F \u03b1}, (Eq.{succ u2} (F \u03b1) x y) -> (Eq.{succ u3} (G \u03b1) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 x) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 y))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) {_inst_4 : Type.{u1}} {\u03b7 : F _inst_4} {\u03b1 : F _inst_4}, (Eq.{succ u2} (F _inst_4) \u03b7 \u03b1) -> (Eq.{succ u3} (_inst_2 _inst_4) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_4 \u03b7) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_4 \u03b1))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.congr_arg ApplicativeTransformation.congr_arg\u2093'. -/\nprotected theorem congr_arg (\u03b7 : ApplicativeTransformation F G) {\u03b1 : Type u} {x y : F \u03b1}\n    (h : x = y) : \u03b7 x = \u03b7 y :=\n  congr_arg (fun z : F \u03b1 => \u03b7 z) h\n#align applicative_transformation.congr_arg ApplicativeTransformation.congr_arg\n\n/- warning: applicative_transformation.coe_inj -> ApplicativeTransformation.coe_inj is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {{\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4}} {{\u03b7' : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4}}, (Eq.{max (succ (succ u1)) (succ u2) (succ u3)} ((fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) \u03b7) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7')) -> (Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b7')\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {{_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G}} {{_inst_4 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G}}, (Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (forall {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}}, (F \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) -> (_inst_2 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243)) (fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) (fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_4 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243)) -> (Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) _inst_3 _inst_4)\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.coe_inj ApplicativeTransformation.coe_inj\u2093'. -/\ntheorem coe_inj \u2983\u03b7 \u03b7' : ApplicativeTransformation F G\u2984 (h : (\u03b7 : \u2200 \u03b1, F \u03b1 \u2192 G \u03b1) = \u03b7') : \u03b7 = \u03b7' :=\n  by\n  cases \u03b7\n  cases \u03b7'\n  congr\n  exact h\n#align applicative_transformation.coe_inj ApplicativeTransformation.coe_inj\n\n/- warning: applicative_transformation.ext -> ApplicativeTransformation.ext is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {{\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4}} {{\u03b7' : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4}}, (forall (\u03b1 : Type.{u1}) (x : F \u03b1), Eq.{succ u3} (G \u03b1) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 x) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7' \u03b1 x)) -> (Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b7')\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {{_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G}} {{_inst_4 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G}}, (forall (\u03b1 : Type.{u1}) (x : F \u03b1), Eq.{succ u3} (_inst_2 \u03b1) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b1 x) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_4 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b1 x)) -> (Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) _inst_3 _inst_4)\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.ext ApplicativeTransformation.ext\u2093'. -/\n@[ext]\ntheorem ext \u2983\u03b7 \u03b7' : ApplicativeTransformation F G\u2984 (h : \u2200 (\u03b1 : Type u) (x : F \u03b1), \u03b7 x = \u03b7' x) :\n    \u03b7 = \u03b7' := by\n  apply coe_inj\n  ext1 \u03b1\n  exact funext (h \u03b1)\n#align applicative_transformation.ext ApplicativeTransformation.ext\n\n/- warning: applicative_transformation.ext_iff -> ApplicativeTransformation.ext_iff is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4} {\u03b7' : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4}, Iff (Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b7') (forall (\u03b1 : Type.{u1}) (x : F \u03b1), Eq.{succ u3} (G \u03b1) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 x) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7' \u03b1 x))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G} {_inst_4 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G}, Iff (Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) _inst_3 _inst_4) (forall (\u03b1 : Type.{u1}) (x : F \u03b1), Eq.{succ u3} (_inst_2 \u03b1) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b1 x) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_4 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b1 x))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.ext_iff ApplicativeTransformation.ext_iff\u2093'. -/\ntheorem ext_iff {\u03b7 \u03b7' : ApplicativeTransformation F G} :\n    \u03b7 = \u03b7' \u2194 \u2200 (\u03b1 : Type u) (x : F \u03b1), \u03b7 x = \u03b7' x :=\n  \u27e8fun h \u03b1 x => h \u25b8 rfl, fun h => ext h\u27e9\n#align applicative_transformation.ext_iff ApplicativeTransformation.ext_iff\n\nsection Preserves\n\nvariable (\u03b7 : ApplicativeTransformation F G)\n\n/- warning: applicative_transformation.preserves_pure -> ApplicativeTransformation.preserves_pure is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) {\u03b1 : Type.{u1}} (x : \u03b1), Eq.{succ u3} (G \u03b1) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 (Pure.pure.{u1, u2} F (Applicative.toHasPure.{u1, u2} F _inst_1) \u03b1 x)) (Pure.pure.{u1, u3} G (Applicative.toHasPure.{u1, u3} G _inst_3) \u03b1 x)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) {_inst_4 : Type.{u1}} (\u03b7 : _inst_4), Eq.{succ u3} (_inst_2 _inst_4) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_4 (Pure.pure.{u1, u2} F (Applicative.toPure.{u1, u2} F _inst_1) _inst_4 \u03b7)) (Pure.pure.{u1, u3} _inst_2 (Applicative.toPure.{u1, u3} _inst_2 G) _inst_4 \u03b7)\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.preserves_pure ApplicativeTransformation.preserves_pure\u2093'. -/\n@[functor_norm]\ntheorem preserves_pure {\u03b1} : \u2200 x : \u03b1, \u03b7 (pure x) = pure x :=\n  \u03b7.preserves_pure'\n#align applicative_transformation.preserves_pure ApplicativeTransformation.preserves_pure\n\n/- warning: applicative_transformation.preserves_seq -> ApplicativeTransformation.preserves_seq is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} (x : F (\u03b1 -> \u03b2)) (y : F \u03b1), Eq.{succ u3} (G \u03b2) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b2 (Seq.seq.{u1, u2} F (Applicative.toHasSeq.{u1, u2} F _inst_1) \u03b1 \u03b2 x y)) (Seq.seq.{u1, u3} G (Applicative.toHasSeq.{u1, u3} G _inst_3) \u03b1 \u03b2 (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 (\u03b1 -> \u03b2) x) (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 y))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) {_inst_4 : Type.{u1}} {\u03b7 : Type.{u1}} (\u03b1 : F (_inst_4 -> \u03b7)) (\u03b2 : F _inst_4), Eq.{succ u3} (_inst_2 \u03b7) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) \u03b7 (Seq.seq.{u1, u2} F (Applicative.toSeq.{u1, u2} F _inst_1) _inst_4 \u03b7 \u03b1 (fun (x._@.Mathlib.Control.Traversable.Basic._hyg.768 : Unit) => \u03b2))) (Seq.seq.{u1, u3} _inst_2 (Applicative.toSeq.{u1, u3} _inst_2 G) _inst_4 \u03b7 ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) (_inst_4 -> \u03b7) \u03b1) (fun (x._@.Mathlib.Control.Traversable.Basic._hyg.780 : Unit) => (fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_4 \u03b2))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.preserves_seq ApplicativeTransformation.preserves_seq\u2093'. -/\n@[functor_norm]\ntheorem preserves_seq {\u03b1 \u03b2 : Type u} : \u2200 (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), \u03b7 (x <*> y) = \u03b7 x <*> \u03b7 y :=\n  \u03b7.preserves_seq'\n#align applicative_transformation.preserves_seq ApplicativeTransformation.preserves_seq\n\n#print ApplicativeTransformation.preserves_map /-\n@[functor_norm]\ntheorem preserves_map {\u03b1 \u03b2} (x : \u03b1 \u2192 \u03b2) (y : F \u03b1) : \u03b7 (x <$> y) = x <$> \u03b7 y := by\n  rw [\u2190 pure_seq_eq_map, \u03b7.preserves_seq] <;> simp [functor_norm]\n#align applicative_transformation.preserves_map ApplicativeTransformation.preserves_map\n-/\n\n#print ApplicativeTransformation.preserves_map' /-\ntheorem preserves_map' {\u03b1 \u03b2} (x : \u03b1 \u2192 \u03b2) : @\u03b7 _ \u2218 Functor.map x = Functor.map x \u2218 @\u03b7 _ :=\n  by\n  ext y\n  exact preserves_map \u03b7 x y\n#align applicative_transformation.preserves_map' ApplicativeTransformation.preserves_map'\n-/\n\nend Preserves\n\n#print ApplicativeTransformation.idTransformation /-\n/-- The identity applicative transformation from an applicative functor to itself. -/\ndef idTransformation : ApplicativeTransformation F F\n    where\n  app \u03b1 := id\n  preserves_pure' := by simp\n  preserves_seq' \u03b1 \u03b2 x y := by simp\n#align applicative_transformation.id_transformation ApplicativeTransformation.idTransformation\n-/\n\ninstance : Inhabited (ApplicativeTransformation F F) :=\n  \u27e8idTransformation\u27e9\n\nuniverse s t\n\nvariable {H : Type u \u2192 Type s} [Applicative H] [LawfulApplicative H]\n\n/- warning: applicative_transformation.comp -> ApplicativeTransformation.comp is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {H : Type.{u1} -> Type.{u4}} [_inst_5 : Applicative.{u1, u4} H] [_inst_6 : LawfulApplicative.{u1, u4} H _inst_5], (ApplicativeTransformation.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) -> (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) -> (ApplicativeTransformation.{u1, u2, u4} F _inst_1 _inst_2 H _inst_5 _inst_6)\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {_inst_3 : Type.{u1} -> Type.{u4}} [_inst_4 : Applicative.{u1, u4} _inst_3], (ApplicativeTransformation.{u1, u3, u4} _inst_2 G _inst_3 _inst_4) -> (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) -> (ApplicativeTransformation.{u1, u2, u4} F _inst_1 _inst_3 _inst_4)\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.comp ApplicativeTransformation.comp\u2093'. -/\n/-- The composition of applicative transformations. -/\ndef comp (\u03b7' : ApplicativeTransformation G H) (\u03b7 : ApplicativeTransformation F G) :\n    ApplicativeTransformation F H where\n  app \u03b1 x := \u03b7' (\u03b7 x)\n  preserves_pure' \u03b1 x := by simp [functor_norm]\n  preserves_seq' \u03b1 \u03b2 x y := by simp [functor_norm]\n#align applicative_transformation.comp ApplicativeTransformation.comp\n\n/- warning: applicative_transformation.comp_apply -> ApplicativeTransformation.comp_apply is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {H : Type.{u1} -> Type.{u4}} [_inst_5 : Applicative.{u1, u4} H] [_inst_6 : LawfulApplicative.{u1, u4} H _inst_5] (\u03b7' : ApplicativeTransformation.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) {\u03b1 : Type.{u1}} (x : F \u03b1), Eq.{succ u4} (H \u03b1) (coeFn.{succ (max (succ u1) u2 u4), max (succ (succ u1)) (succ u2) (succ u4)} (ApplicativeTransformation.{u1, u2, u4} F _inst_1 _inst_2 H _inst_5 _inst_6) (fun (_x : ApplicativeTransformation.{u1, u2, u4} F _inst_1 _inst_2 H _inst_5 _inst_6) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (H \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u4} F _inst_1 _inst_2 H _inst_5 _inst_6) (ApplicativeTransformation.comp.{u1, u2, u3, u4} F _inst_1 _inst_2 G _inst_3 _inst_4 H _inst_5 _inst_6 \u03b7' \u03b7) \u03b1 x) (coeFn.{succ (max (succ u1) u3 u4), max (succ (succ u1)) (succ u3) (succ u4)} (ApplicativeTransformation.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) (fun (_x : ApplicativeTransformation.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) => forall {\u03b1 : Type.{u1}}, (G \u03b1) -> (H \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) \u03b7' \u03b1 (coeFn.{succ (max (succ u1) u2 u3), max (succ (succ u1)) (succ u2) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (fun (_x : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) => forall {\u03b1 : Type.{u1}}, (F \u03b1) -> (G \u03b1)) (ApplicativeTransformation.hasCoeToFun.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) \u03b7 \u03b1 x))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {_inst_3 : Type.{u1} -> Type.{u4}} [_inst_4 : Applicative.{u1, u4} _inst_3] (H : ApplicativeTransformation.{u1, u3, u4} _inst_2 G _inst_3 _inst_4) (_inst_5 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) {_inst_6 : Type.{u1}} (\u03b7' : F _inst_6), Eq.{succ u4} (_inst_3 _inst_6) ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u4} F _inst_1 _inst_3 _inst_4 (ApplicativeTransformation.comp.{u1, u2, u3, u4} F _inst_1 _inst_2 G _inst_3 _inst_4 H _inst_5) \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_6 \u03b7') ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u3, u4} _inst_2 G _inst_3 _inst_4 H \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_6 ((fun {\u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243 : Type.{u1}} => ApplicativeTransformation.app.{u1, u2, u3} F _inst_1 _inst_2 G _inst_5 \u03b1._@.Mathlib.Control.Traversable.Basic._hyg.243) _inst_6 \u03b7'))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.comp_apply ApplicativeTransformation.comp_apply\u2093'. -/\n@[simp]\ntheorem comp_apply (\u03b7' : ApplicativeTransformation G H) (\u03b7 : ApplicativeTransformation F G)\n    {\u03b1 : Type u} (x : F \u03b1) : \u03b7'.comp \u03b7 x = \u03b7' (\u03b7 x) :=\n  rfl\n#align applicative_transformation.comp_apply ApplicativeTransformation.comp_apply\n\n/- warning: applicative_transformation.comp_assoc -> ApplicativeTransformation.comp_assoc is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] {H : Type.{u1} -> Type.{u4}} [_inst_5 : Applicative.{u1, u4} H] [_inst_6 : LawfulApplicative.{u1, u4} H _inst_5] {I : Type.{u1} -> Type.{u5}} [_inst_7 : Applicative.{u1, u5} I] [_inst_8 : LawfulApplicative.{u1, u5} I _inst_7] (\u03b7'' : ApplicativeTransformation.{u1, u4, u5} H _inst_5 _inst_6 I _inst_7 _inst_8) (\u03b7' : ApplicativeTransformation.{u1, u3, u4} G _inst_3 _inst_4 H _inst_5 _inst_6) (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4), Eq.{succ (max (succ u1) u2 u5)} (ApplicativeTransformation.{u1, u2, u5} F _inst_1 _inst_2 I _inst_7 _inst_8) (ApplicativeTransformation.comp.{u1, u2, u3, u5} F _inst_1 _inst_2 G _inst_3 _inst_4 I _inst_7 _inst_8 (ApplicativeTransformation.comp.{u1, u3, u4, u5} G _inst_3 _inst_4 H _inst_5 _inst_6 I _inst_7 _inst_8 \u03b7'' \u03b7') \u03b7) (ApplicativeTransformation.comp.{u1, u2, u4, u5} F _inst_1 _inst_2 H _inst_5 _inst_6 I _inst_7 _inst_8 \u03b7'' (ApplicativeTransformation.comp.{u1, u2, u3, u4} F _inst_1 _inst_2 G _inst_3 _inst_4 H _inst_5 _inst_6 \u03b7' \u03b7))\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] {_inst_3 : Type.{u1} -> Type.{u4}} [_inst_4 : Applicative.{u1, u4} _inst_3] {H : Type.{u1} -> Type.{u5}} [_inst_5 : Applicative.{u1, u5} H] (_inst_6 : ApplicativeTransformation.{u1, u4, u5} _inst_3 _inst_4 H _inst_5) (I : ApplicativeTransformation.{u1, u3, u4} _inst_2 G _inst_3 _inst_4) (_inst_7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G), Eq.{max (max (succ u5) (succ (succ u1))) (succ u2)} (ApplicativeTransformation.{u1, u2, u5} F _inst_1 H _inst_5) (ApplicativeTransformation.comp.{u1, u2, u3, u5} F _inst_1 _inst_2 G H _inst_5 (ApplicativeTransformation.comp.{u1, u3, u4, u5} _inst_2 G _inst_3 _inst_4 H _inst_5 _inst_6 I) _inst_7) (ApplicativeTransformation.comp.{u1, u2, u4, u5} F _inst_1 _inst_3 _inst_4 H _inst_5 _inst_6 (ApplicativeTransformation.comp.{u1, u2, u3, u4} F _inst_1 _inst_2 G _inst_3 _inst_4 I _inst_7))\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.comp_assoc ApplicativeTransformation.comp_assoc\u2093'. -/\ntheorem comp_assoc {I : Type u \u2192 Type t} [Applicative I] [LawfulApplicative I]\n    (\u03b7'' : ApplicativeTransformation H I) (\u03b7' : ApplicativeTransformation G H)\n    (\u03b7 : ApplicativeTransformation F G) : (\u03b7''.comp \u03b7').comp \u03b7 = \u03b7''.comp (\u03b7'.comp \u03b7) :=\n  rfl\n#align applicative_transformation.comp_assoc ApplicativeTransformation.comp_assoc\n\n/- warning: applicative_transformation.comp_id -> ApplicativeTransformation.comp_id is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4), Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (ApplicativeTransformation.comp.{u1, u2, u2, u3} F _inst_1 _inst_2 F _inst_1 _inst_2 G _inst_3 _inst_4 \u03b7 (ApplicativeTransformation.idTransformation.{u1, u2} F _inst_1 _inst_2)) \u03b7\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G), Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) (ApplicativeTransformation.comp.{u1, u2, u2, u3} F _inst_1 F _inst_1 _inst_2 G _inst_3 (ApplicativeTransformation.idTransformation.{u1, u2} F _inst_1)) _inst_3\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.comp_id ApplicativeTransformation.comp_id\u2093'. -/\n@[simp]\ntheorem comp_id (\u03b7 : ApplicativeTransformation F G) : \u03b7.comp idTransformation = \u03b7 :=\n  ext fun \u03b1 x => rfl\n#align applicative_transformation.comp_id ApplicativeTransformation.comp_id\n\n/- warning: applicative_transformation.id_comp -> ApplicativeTransformation.id_comp is a dubious translation:\nlean 3 declaration is\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] [_inst_2 : LawfulApplicative.{u1, u2} F _inst_1] {G : Type.{u1} -> Type.{u3}} [_inst_3 : Applicative.{u1, u3} G] [_inst_4 : LawfulApplicative.{u1, u3} G _inst_3] (\u03b7 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4), Eq.{succ (max (succ u1) u2 u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G _inst_3 _inst_4) (ApplicativeTransformation.comp.{u1, u2, u3, u3} F _inst_1 _inst_2 G _inst_3 _inst_4 G _inst_3 _inst_4 (ApplicativeTransformation.idTransformation.{u1, u3} G _inst_3 _inst_4) \u03b7) \u03b7\nbut is expected to have type\n  forall {F : Type.{u1} -> Type.{u2}} [_inst_1 : Applicative.{u1, u2} F] {_inst_2 : Type.{u1} -> Type.{u3}} [G : Applicative.{u1, u3} _inst_2] (_inst_3 : ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G), Eq.{max (max (succ (succ u1)) (succ u2)) (succ u3)} (ApplicativeTransformation.{u1, u2, u3} F _inst_1 _inst_2 G) (ApplicativeTransformation.comp.{u1, u2, u3, u3} F _inst_1 _inst_2 G _inst_2 G (ApplicativeTransformation.idTransformation.{u1, u3} _inst_2 G) _inst_3) _inst_3\nCase conversion may be inaccurate. Consider using '#align applicative_transformation.id_comp ApplicativeTransformation.id_comp\u2093'. -/\n@[simp]\ntheorem id_comp (\u03b7 : ApplicativeTransformation F G) : idTransformation.comp \u03b7 = \u03b7 :=\n  ext fun \u03b1 x => rfl\n#align applicative_transformation.id_comp ApplicativeTransformation.id_comp\n\nend ApplicativeTransformation\n\nopen ApplicativeTransformation\n\n#print Traversable /-\n/-- A traversable functor is a functor along with a way to commute\nwith all applicative functors (see `sequence`).  For example, if `t`\nis the traversable functor `list` and `m` is the applicative functor\n`io`, then given a function `f : \u03b1 \u2192 io \u03b2`, the function `functor.map f` is\n`list \u03b1 \u2192 list (io \u03b2)`, but `traverse f` is `list \u03b1 \u2192 io (list \u03b2)`. -/\nclass Traversable (t : Type u \u2192 Type u) extends Functor t where\n  traverse : \u2200 {m : Type u \u2192 Type u} [Applicative m] {\u03b1 \u03b2}, (\u03b1 \u2192 m \u03b2) \u2192 t \u03b1 \u2192 m (t \u03b2)\n#align traversable Traversable\n-/\n\nopen Functor\n\nexport Traversable (traverse)\n\nsection Functions\n\nvariable {t : Type u \u2192 Type u}\n\nvariable {m : Type u \u2192 Type v} [Applicative m]\n\nvariable {\u03b1 \u03b2 : Type u}\n\nvariable {f : Type u \u2192 Type u} [Applicative f]\n\n#print sequence /-\n/-- A traversable functor commutes with all applicative functors. -/\ndef sequence [Traversable t] : t (f \u03b1) \u2192 f (t \u03b1) :=\n  traverse id\n#align sequence sequence\n-/\n\nend Functions\n\n#print IsLawfulTraversable /-\n/-- A traversable functor is lawful if its `traverse` satisfies a\nnumber of additional properties.  It must send `id.mk` to `id.mk`,\nsend the composition of applicative functors to the composition of the\n`traverse` of each, send each function `f` to `\u03bb x, f <$> x`, and\nsatisfy a naturality condition with respect to applicative\ntransformations. -/\nclass IsLawfulTraversable (t : Type u \u2192 Type u) [Traversable t] extends LawfulFunctor t :\n  Type (u + 1) where\n  id_traverse : \u2200 {\u03b1} (x : t \u03b1), traverse id.mk x = x\n  comp_traverse :\n    \u2200 {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G] {\u03b1 \u03b2 \u03b3}\n      (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 (map (traverse f) (traverse g x))\n  traverse_eq_map_id : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (x : t \u03b1), traverse (id.mk \u2218 f) x = id.mk (f <$> x)\n  naturality :\n    \u2200 {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G]\n      (\u03b7 : ApplicativeTransformation F G) {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) (x : t \u03b1),\n      \u03b7 (traverse f x) = traverse (@\u03b7 _ \u2218 f) x\n#align is_lawful_traversable IsLawfulTraversable\n-/\n\ninstance : Traversable id :=\n  \u27e8fun _ _ _ _ => id\u27e9\n\ninstance : IsLawfulTraversable id := by refine' { .. } <;> intros <;> rfl\n\nsection\n\nvariable {F : Type u \u2192 Type v} [Applicative F]\n\ninstance : Traversable Option :=\n  \u27e8@Option.traverse\u27e9\n\ninstance : Traversable List :=\n  \u27e8@List.traverse\u27e9\n\nend\n\nnamespace Sum\n\nvariable {\u03c3 : Type u}\n\nvariable {F : Type u \u2192 Type u}\n\nvariable [Applicative F]\n\n#print Sum.traverse /-\n/-- Defines a `traverse` function on the second component of a sum type.\nThis is used to give a `traversable` instance for the functor `\u03c3 \u2295 -`. -/\nprotected def traverse {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) : Sum \u03c3 \u03b1 \u2192 F (Sum \u03c3 \u03b2)\n  | Sum.inl x => pure (Sum.inl x)\n  | Sum.inr x => Sum.inr <$> f x\n#align sum.traverse Sum.traverse\n-/\n\nend Sum\n\ninstance {\u03c3 : Type u} : Traversable.{u} (Sum \u03c3) :=\n  \u27e8@Sum.traverse _\u27e9\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Traversable/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160664, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3691685708030659}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport group_theory.group_action.defs\n\n/-!\n# Sigma instances for additive and multiplicative actions\n\nThis file defines instances for arbitrary sum of additive and multiplicative actions.\n\n## See also\n\n* `group_theory.group_action.pi`\n* `group_theory.group_action.prod`\n* `group_theory.group_action.sum`\n-/\n\nvariables {\u03b9 : Type*} {M N : Type*} {\u03b1 : \u03b9 \u2192 Type*}\n\nnamespace sigma\n\nsection has_smul\nvariables [\u03a0 i, has_smul M (\u03b1 i)] [\u03a0 i, has_smul N (\u03b1 i)] (a : M) (i : \u03b9) (b : \u03b1 i)\n  (x : \u03a3 i, \u03b1 i)\n\n@[to_additive sigma.has_vadd] instance : has_smul M (\u03a3 i, \u03b1 i) := \u27e8\u03bb a, sigma.map id $ \u03bb i, (\u2022) a\u27e9\n\n@[to_additive] lemma smul_def : a \u2022 x = x.map id (\u03bb i, (\u2022) a) := rfl\n@[simp, to_additive] lemma smul_mk : a \u2022 mk i b = \u27e8i, a \u2022 b\u27e9 := rfl\n\ninstance [has_smul M N] [\u03a0 i, is_scalar_tower M N (\u03b1 i)] : is_scalar_tower M N (\u03a3 i, \u03b1 i) :=\n\u27e8\u03bb a b x, by { cases x, rw [smul_mk, smul_mk, smul_mk, smul_assoc] }\u27e9\n\n@[to_additive] instance [\u03a0 i, smul_comm_class M N (\u03b1 i)] : smul_comm_class M N (\u03a3 i, \u03b1 i) :=\n\u27e8\u03bb a b x, by { cases x, rw [smul_mk, smul_mk, smul_mk, smul_mk, smul_comm] }\u27e9\n\ninstance [\u03a0 i, has_smul M\u1d50\u1d52\u1d56 (\u03b1 i)] [\u03a0 i, is_central_scalar M (\u03b1 i)] :\n  is_central_scalar M (\u03a3 i, \u03b1 i) :=\n\u27e8\u03bb a x, by { cases x, rw [smul_mk, smul_mk, op_smul_eq_smul] }\u27e9\n\n/-- This is not an instance because `i` becomes a metavariable. -/\n@[to_additive \"This is not an instance because `i` becomes a metavariable.\"]\nprotected lemma has_faithful_smul' [has_faithful_smul M (\u03b1 i)] : has_faithful_smul M (\u03a3 i, \u03b1 i) :=\n\u27e8\u03bb x y h, eq_of_smul_eq_smul $ \u03bb a : \u03b1 i, heq_iff_eq.1 (ext_iff.1 $ h $ mk i a).2\u27e9\n\n@[to_additive] instance [nonempty \u03b9] [\u03a0 i, has_faithful_smul M (\u03b1 i)] :\n  has_faithful_smul M (\u03a3 i, \u03b1 i) :=\nnonempty.elim \u2039_\u203a $ \u03bb i, sigma.has_faithful_smul' i\n\nend has_smul\n\n@[to_additive] instance {m : monoid M} [\u03a0 i, mul_action M (\u03b1 i)] : mul_action M (\u03a3 i, \u03b1 i) :=\n{ mul_smul := \u03bb a b x, by { cases x, rw [smul_mk, smul_mk, smul_mk, mul_smul] },\n  one_smul := \u03bb x, by { cases x, rw [smul_mk, one_smul] } }\n\nend sigma\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/group_theory/group_action/sigma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.36916856233339684}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Scott Morrison, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.unbundled_hom\nimport Mathlib.topology.continuous_map\nimport Mathlib.topology.opens\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-- The category of topological spaces and continuous maps. -/\ndef Top := category_theory.bundled topological_space\n\nnamespace Top\n\n\nprotected instance bundled_hom : category_theory.bundled_hom continuous_map :=\n  category_theory.bundled_hom.mk continuous_map.to_fun continuous_map.id continuous_map.comp\n\nprotected instance has_coe_to_sort : has_coe_to_sort Top := category_theory.bundled.has_coe_to_sort\n\nprotected instance topological_space_unbundled (x : Top) : topological_space \u21a5x :=\n  category_theory.bundled.str x\n\n@[simp] theorem id_app (X : Top) (x : \u21a5X) : coe_fn \ud835\udfd9 x = x := rfl\n\n@[simp] theorem comp_app {X : Top} {Y : Top} {Z : Top} (f : X \u27f6 Y) (g : Y \u27f6 Z) (x : \u21a5X) :\n    coe_fn (f \u226b g) x = coe_fn g (coe_fn f x) :=\n  rfl\n\n/-- Construct a bundled `Top` from the underlying type and the typeclass. -/\ndef of (X : Type u) [topological_space X] : Top := category_theory.bundled.mk X\n\nprotected instance topological_space (X : Top) : topological_space \u21a5X :=\n  category_theory.bundled.str X\n\n@[simp] theorem coe_of (X : Type u) [topological_space X] : \u21a5(of X) = X := rfl\n\nprotected instance inhabited : Inhabited Top := { default := of empty }\n\n/-- The discrete topology on any type. -/\ndef discrete : Type u \u2964 Top :=\n  category_theory.functor.mk (fun (X : Type u) => category_theory.bundled.mk X)\n    fun (X Y : Type u) (f : X \u27f6 Y) => continuous_map.mk f\n\n/-- The trivial topology on any type. -/\ndef trivial : Type u \u2964 Top :=\n  category_theory.functor.mk (fun (X : Type u) => category_theory.bundled.mk X)\n    fun (X Y : Type u) (f : X \u27f6 Y) => continuous_map.mk f\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/category/Top/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.369168562005546}}
{"text": "import Std.Data.AssocList\n\nnamespace MWE6\n\ninductive Syntax where\n  | leaf\n  | withDeclaration (d: String) (s: Syntax) \n  | withSpecialization (sub: String) (sup: String) (s: Syntax)\n  deriving Repr\n\ndef s1 := Syntax.leaf |>.withDeclaration \"a\" |>.withSpecialization \"a\" \"b\"\n#eval s1\n\nabbrev Strings := List String\n\nabbrev Sups := Std.AssocList String Strings\ninstance : Repr Sups where reprPrec s n := s.toList.repr n\n\nstructure State where\n  decls: Strings := .nil\n  specializations: Sups := .nil\n  deriving Repr\n\ndef State.wff(s: State) : Prop :=\n  s.specializations.all \n    fun (sub sups) => \n      s.decls.contains sub &&\n      sups.all (fun sup => s.decls.contains sup)\n\ndef State.empty: State := {}\n\ntheorem State.empty.wff : State.empty.wff := by rfl\n\ndef State.withDecl (s: State) (d: String) : State :=\n  match s.decls.contains d with\n  | true    => s\n  | false   => { s with decls := d :: s.decls }\n\n\ntheorem State.withDecl.noChange (s: State) (d: String) (h: s.decls.contains d) : s = s.withDecl d := by {\n  sorry\n}\n\ntheorem State.withDecl.added (s: State) (d: String) (h: ! s.decls.contains d) : (s.withDecl d).decls.contains d := by {\n  sorry\n}\n\ntheorem State.withDecl.wff (s: State) (d: String) (h: s.wff) : s.withDecl d |>.wff := by {\n  sorry\n}\n\n-- To ensure WFF, make sure sub and sup are declared.\ndef State.withSpecialization (s: State) (sub: String) (sup: String) : State :=\n  let s' := s.withDecl sub |>.withDecl sup\n  let sups : Strings := match s'.specializations.find? sub with \n    | none => [sup]\n    | some xs => if xs.contains sup then xs else xs.cons sup\n  match s'.specializations.contains sub with\n  | false => { s' with specializations := s'.specializations.cons sub .nil }\n  | true => { s' with specializations := s'.specializations.replace sub sups }\n\ndef Syntax.toState (s: Syntax): State :=\n  match s with \n  | .leaf =>\n    .empty\n  | .withDeclaration d s =>\n    s.toState.withDecl d\n  | .withSpecialization sub sup s => \n    s.toState.withSpecialization sub sup\n\ndef st1 := s1.toState\n#eval st1\n\nend MWE6\n", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/MWE6.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3691495320813761}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport data.int.basic\nimport category_theory.shift\nimport category_theory.concrete_category.basic\n\n/-!\n# Differential objects in a category.\n\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n\nWe build the category of differential objects, and some basic constructions\nsuch as the forgetful functor, zero morphisms and zero objects, and the shift functor\non differential objects.\n-/\n\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\n-- TODO: generaize to `has_shift C A` for an arbitrary `[add_monoid A]` `[has_one A]`.\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\n/--\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n-/\n@[nolint has_inhabited_instance]\nstructure differential_object :=\n(X : C)\n(d : X \u27f6 X\u27e61\u27e7)\n(d_squared' : d \u226b d\u27e6(1:\u2124)\u27e7' = 0 . obviously)\n\nrestate_axiom differential_object.d_squared'\nattribute [simp] differential_object.d_squared\n\nvariables {C}\n\nnamespace differential_object\n\n/--\nA morphism of differential objects is a morphism commuting with the differentials.\n-/\n@[ext, nolint has_inhabited_instance]\nstructure hom (X Y : differential_object C) :=\n(f : X.X \u27f6 Y.X)\n(comm' : X.d \u226b f\u27e61\u27e7' = f \u226b Y.d . obviously)\n\nrestate_axiom hom.comm'\nattribute [simp, reassoc] hom.comm\n\nnamespace hom\n\n/-- The identity morphism of a differential object. -/\n@[simps]\ndef id (X : differential_object C) : hom X X :=\n{ f := \ud835\udfd9 X.X }\n\n/-- The composition of morphisms of differential objects. -/\n@[simps]\ndef comp {X Y Z : differential_object C} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ f := f.f \u226b g.f, }\n\nend hom\n\ninstance category_of_differential_objects : category (differential_object C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z f g, hom.comp f g, }\n\n@[simp]\n\n\n@[simp]\nlemma comp_f {X Y Z : differential_object C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  (f \u226b g).f = f.f \u226b g.f :=\nrfl\n\n@[simp]\nlemma eq_to_hom_f {X Y : differential_object C} (h : X = Y) :\n  hom.f (eq_to_hom h) = eq_to_hom (congr_arg _ h) :=\nby { subst h, rw [eq_to_hom_refl, eq_to_hom_refl], refl }\n\nvariables (C)\n\n/-- The forgetful functor taking a differential object to its underlying object. -/\ndef forget : (differential_object C) \u2964 C :=\n{ obj := \u03bb X, X.X,\n  map := \u03bb X Y f, f.f, }\n\ninstance forget_faithful : faithful (forget C) :=\n{ }\n\ninstance has_zero_morphisms : has_zero_morphisms (differential_object C) :=\n{ has_zero := \u03bb X Y,\n  \u27e8{ f := 0 }\u27e9}\n\nvariables {C}\n\n@[simp]\nlemma zero_f (P Q : differential_object C) : (0 : P \u27f6 Q).f = 0 := rfl\n\n/--\nAn isomorphism of differential objects gives an isomorphism of the underlying objects.\n-/\n@[simps] def iso_app {X Y : differential_object C} (f : X \u2245 Y) : X.X \u2245 Y.X :=\n\u27e8f.hom.f, f.inv.f, by { dsimp, rw [\u2190 comp_f, iso.hom_inv_id, id_f] },\n  by { dsimp, rw [\u2190 comp_f, iso.inv_hom_id, id_f] }\u27e9\n\n@[simp] lemma iso_app_refl (X : differential_object C) : iso_app (iso.refl X) = iso.refl X.X := rfl\n@[simp] lemma iso_app_symm {X Y : differential_object C} (f : X \u2245 Y) :\n  iso_app f.symm = (iso_app f).symm := rfl\n@[simp] lemma iso_app_trans {X Y Z : differential_object C} (f : X \u2245 Y) (g : Y \u2245 Z) :\n  iso_app (f \u226a\u226b g) = iso_app f \u226a\u226b iso_app g := rfl\n\n/-- An isomorphism of differential objects can be constructed\nfrom an isomorphism of the underlying objects that commutes with the differentials. -/\n@[simps] def mk_iso {X Y : differential_object C}\n  (f : X.X \u2245 Y.X) (hf : X.d \u226b f.hom\u27e61\u27e7' = f.hom \u226b Y.d) : X \u2245 Y :=\n{ hom := \u27e8f.hom, hf\u27e9,\n  inv := \u27e8f.inv, by { dsimp, rw [\u2190 functor.map_iso_inv, iso.comp_inv_eq, category.assoc,\n    iso.eq_inv_comp, functor.map_iso_hom, hf] }\u27e9,\n  hom_inv_id' := by { ext1, dsimp, exact f.hom_inv_id },\n  inv_hom_id' := by { ext1, dsimp, exact f.inv_hom_id } }\n\nend differential_object\n\nnamespace functor\n\nuniverses v' u'\nvariables (D : Type u') [category.{v'} D]\nvariables [has_zero_morphisms D] [has_shift D \u2124]\n\n/--\nA functor `F : C \u2964 D` which commutes with shift functors on `C` and `D` and preserves zero morphisms\ncan be lifted to a functor `differential_object C \u2964 differential_object D`.\n-/\n@[simps]\ndef map_differential_object (F : C \u2964 D)\n  (\u03b7 : (shift_functor C (1:\u2124)).comp F \u27f6 F.comp (shift_functor D (1:\u2124)))\n  (hF : \u2200 c c', F.map (0 : c \u27f6 c') = 0) :\n  differential_object C \u2964 differential_object D :=\n{ obj := \u03bb X, { X := F.obj X.X,\n    d := F.map X.d \u226b \u03b7.app X.X,\n    d_squared' := begin\n      rw [functor.map_comp, \u2190 functor.comp_map F (shift_functor D (1:\u2124))],\n      slice_lhs 2 3 { rw [\u2190 \u03b7.naturality X.d] },\n      rw [functor.comp_map],\n      slice_lhs 1 2 { rw [\u2190 F.map_comp, X.d_squared, hF] },\n      rw [zero_comp, zero_comp],\n    end },\n  map := \u03bb X Y f, { f := F.map f.f,\n    comm' := begin\n      dsimp,\n      slice_lhs 2 3 { rw [\u2190 functor.comp_map F (shift_functor D (1:\u2124)), \u2190 \u03b7.naturality f.f] },\n      slice_lhs 1 2 { rw [functor.comp_map, \u2190 F.map_comp, f.comm, F.map_comp] },\n      rw [category.assoc]\n    end },\n  map_id' := by { intros, ext, simp },\n  map_comp' := by { intros, ext, simp }, }\n\nend functor\n\nend category_theory\n\nnamespace category_theory\n\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\n\nvariables [has_zero_object C] [has_zero_morphisms C] [has_shift C \u2124]\n\nopen_locale zero_object\n\ninstance has_zero_object : has_zero_object (differential_object C) :=\nby { refine \u27e8\u27e8\u27e80, 0\u27e9, \u03bb X, \u27e8\u27e8\u27e8\u27e80\u27e9\u27e9, \u03bb f, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e8\u27e80\u27e9\u27e9, \u03bb f, _\u27e9\u27e9\u27e9\u27e9; ext, }\n\nend differential_object\n\nnamespace differential_object\n\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_zero_morphisms C] [has_shift C \u2124]\n\ninstance concrete_category_of_differential_objects :\n  concrete_category (differential_object C) :=\n{ forget := forget C \u22d9 category_theory.forget C }\n\ninstance : has_forget\u2082 (differential_object C) C :=\n{ forget\u2082 := forget C }\n\nend differential_object\n\n/-! The category of differential objects itself has a shift functor. -/\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\nnoncomputable theory\n\n/-- The shift functor on `differential_object C`. -/\n@[simps]\ndef shift_functor (n : \u2124) : differential_object C \u2964 differential_object C :=\n{ obj := \u03bb X,\n  { X := X.X\u27e6n\u27e7,\n    d := X.d\u27e6n\u27e7' \u226b (shift_comm _ _ _).hom,\n    d_squared' := by rw [functor.map_comp, category.assoc, shift_comm_hom_comp_assoc,\n        \u2190functor.map_comp_assoc, X.d_squared, functor.map_zero, zero_comp] },\n  map := \u03bb X Y f,\n  { f := f.f\u27e6n\u27e7',\n    comm' := by { dsimp, rw [category.assoc, shift_comm_hom_comp, \u2190 functor.map_comp_assoc,\n      f.comm, functor.map_comp_assoc], }, },\n  map_id' := by { intros X, ext1, dsimp, rw functor.map_id },\n  map_comp' := by { intros X Y Z f g, ext1, dsimp, rw functor.map_comp } }\n\nlocal attribute [simp] eq_to_hom_map\nlocal attribute [reducible] discrete.add_monoidal shift_comm\n\n/-- The shift functor on `differential_object C` is additive. -/\n@[simps] def shift_functor_add (m n : \u2124) :\n  shift_functor C (m + n) \u2245 shift_functor C m \u22d9 shift_functor C n :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso (shift_add X.X _ _) _) _,\n  { dsimp,\n    -- This is just `simp, simp [eq_to_hom_map]`.\n    simp_rw [category.assoc, obj_\u03bc_inv_app, \u03bc_inv_hom_app_assoc, functor.map_comp, obj_\u03bc_app,\n      category.assoc, \u03bc_naturality_assoc, \u03bc_inv_hom_app_assoc, obj_\u03bc_inv_app, category.assoc,\n      \u03bc_naturality\u2097_assoc, \u03bc_inv_hom_app_assoc, \u03bc_inv_naturality\u1d63_assoc],\n    simp only [eq_to_hom_map, eq_to_hom_app, eq_to_iso.hom, eq_to_hom_trans_assoc,\n      eq_to_iso.inv], },\n  { intros X Y f, ext, dsimp, exact nat_trans.naturality _ _ }\nend\n\nlocal attribute [reducible] endofunctor_monoidal_category\n\nsection\nlocal attribute [instance] endofunctor_monoidal_category\n\n/-- The shift by zero is naturally isomorphic to the identity. -/\n@[simps]\ndef shift_\u03b5 : \ud835\udfed (differential_object C) \u2245 shift_functor C 0 :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso ((shift_monoidal_functor C \u2124).\u03b5_iso.app X.X) _) _,\n  { dsimp, simp, dsimp, simp },\n  { introv, ext, dsimp, simp }\nend\n\nend\n\nlocal attribute [simp] eq_to_hom_map\n\ninstance : has_shift (differential_object C) \u2124 :=\nhas_shift_mk _ _\n{ F := shift_functor C,\n  \u03b5 := shift_\u03b5 C,\n  \u03bc := \u03bb m n, (shift_functor_add C m n).symm }\n\nend differential_object\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/differential_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3691495320813761}}
{"text": "/-\nCopyright (c) 2022 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\n\nimport Std.Tactic.RCases\n\nnamespace Char\n\nprivate theorem csize_eq (c) :\n    String.csize c = 1 \u2228 String.csize c = 2 \u2228 String.csize c = 3 \u2228\n    String.csize c = 4 := by\n  simp only [String.csize, utf8Size]\n  repeat (first | split | (solve | simp))\n\ntheorem csize_pos (c) : 0 < String.csize c := by\n  rcases csize_eq c with _|_|_|_ <;> simp_all\n\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/Char.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6187804337438502, "lm_q1q2_score": 0.36906116094483415}}
{"text": "/-\nCopyright (c) 2023 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.simulation_semantics.is_tracking\n\n/-!\n# Tracking Simulation Oracles\n\nThis file defines a typeclass `sim_oracle.is_stateless` for oracles in which there is no\nmeaningful internal state, represented by the state type being `subsingleton`.\nThis is a special case of `sim_oracle.is_tracking`, see `is_stateless.is_tracking`.\nThis allows for a number of very general lemmas that simplify the process of working\nwith simulated computations, by automatically removing states.\n-/\n\nvariables {\u03b1 \u03b2 \u03b3 : Type} {spec spec' spec'' : oracle_spec} {S S' : Type}\n\nopen_locale big_operators ennreal\nopen oracle_comp oracle_spec\n\nnamespace sim_oracle\n\n/-- Class to represent oracles that make no use of their internal state.\nThis class is introduced rather than using a `subsingleton` hypothesis directly\nin order to create a unified `is_tracking` instance based on this fact.\nThis also allows typeclass resolution to work \"backwords\", in that subsingleton instances don't\nhave to be defined on the type parameter `S` explicitly, but on the actual oracle instead. -/\nclass is_stateless (so : sim_oracle spec spec' S) :=\n(state_subsingleton : subsingleton S)\n\nvariables (so : sim_oracle spec spec' S) (i : spec.\u03b9)\n  (t t' : spec.domain i) (s s' : S) (u u' : spec.range i)\n\n/-- Specialize `subsingleton.elim` to simplify the state to the default oracle state.\nUsefull for giving a unified convergence point for state values. -/\nlemma state_elim [hso : so.is_stateless] (s : S) : s = so.default_state :=\n@subsingleton.elim S hso.state_subsingleton s so.default_state\n\ninstance is_stateless.is_tracking [hso : so.is_stateless] : so.is_tracking :=\n{\n  query_f := \u03bb i t, prod.fst <$> so i (t, so.default_state),\n  state_f := \u03bb s i t u, so.default_state,\n  apply_equiv_state_f_map_query_f :=\n    begin\n      sorry,\n      -- refine \u03bb i t s, trans ((eval_dist_map_id $ (so i (t, s))).symm.trans\n      --   (map_equiv_congr (\u03bb x, _) (by rw so.state_elim s))) (eval_dist_map_comp' _ _ _).symm,\n      -- simp only [prod.eq_iff_fst_eq_snd_eq, so.state_elim x.2, id.def, eq_self_iff_true, and_self]\n    end\n}\n\nnamespace is_stateless\n\nvariable [is_stateless so]\n\n@[simp] lemma answer_query.def : so.answer_query =\n  \u03bb i t, prod.fst <$> so i (t, so.default_state) := rfl\n\n@[simp] lemma update_state.def : so.update_state =\n  \u03bb s i t u, so.default_state := rfl\n\nsection support\n\nlemma support_apply' : (so i (t, s)).support =\n  ((\u03bb u, (u, so.default_state)) <$> so.answer_query i t).support :=\nby simp only [is_tracking.support_apply', update_state.def]\n\nlemma support_apply : (so i (t, s)).support =\n  (\u03bb u, (u, so.default_state)) '' (so.answer_query i t).support :=\nby rw [support_apply', support_map]\n\nend support\n\nend is_stateless\n\nend sim_oracle", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/simulation_semantics/is_stateless.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.36906116094483404}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sebastian Ullrich\n-/\nprelude\nimport init.data.bool.lemmas\nimport init.data.string.basic\nimport init.meta.well_founded_tactics\n\nnamespace string\n\nnamespace iterator\n\n@[simp] lemma next_to_string_mk_iterator (s : string) : s.mk_iterator.next_to_string = s :=\nby induction s; refl\n\n@[simp] lemma length_next_to_string_next (it : iterator) :\n  it.next.next_to_string.length = it.next_to_string.length - 1 :=\nby cases it; cases it_snd;\n  simp [iterator.next, iterator.next_to_string, string.length, nat.add_sub_cancel_left]\n\nlemma zero_lt_length_next_to_string_of_has_next {it : iterator} :\n  it.has_next \u2192 0 < it.next_to_string.length :=\nby cases it; cases it_snd; simp [iterator.has_next, iterator.next_to_string, string.length,\n  nat.zero_lt_one_add, nat.add_comm, false_implies_iff]\n\nend iterator\n\n-- TODO(Sebastian): generalize to something like\n-- https://doc.rust-lang.org/std/primitive.str.html#method.split\nprivate def split_core (p : char \u2192 bool) : iterator \u2192 iterator \u2192 list string\n| start stop :=\nif h : stop.has_next then\n  -- wf hint\n  have stop.next_to_string.length - 1 < stop.next_to_string.length,\n    from nat.sub_lt (iterator.zero_lt_length_next_to_string_of_has_next h) dec_trivial,\n  if p stop.curr then\n    let rest := stop.next.next_to_string in\n    (start.extract stop).get_or_else \"\" :: split_core stop.next stop.next\n  else\n    split_core start stop.next\nelse\n  [start.next_to_string]\nusing_well_founded { rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb e, e.2.next_to_string.length)\u27e9] }\n\ndef split (p : char \u2192 bool) (s : string) : list string :=\nsplit_core p s.mk_iterator s.mk_iterator\n\nend string\n", "meta": {"author": "subfish-zhou", "repo": "N2Lean", "sha": "8e858cc5b01f1ad921094dc355db3cb9473a42fd", "save_path": "github-repos/lean/subfish-zhou-N2Lean", "path": "github-repos/lean/subfish-zhou-N2Lean/N2Lean-8e858cc5b01f1ad921094dc355db3cb9473a42fd/library/init/data/string/ops.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.36906116094483404}}
{"text": "import tactic\nimport row_bump\n\n/-\n\nRecording tableau\n\nThe recording tableau \"records\" the outer corner resulting from [ssyt.row_bump]\non an associated ssyt of the same shape (called the \"insertion tableau\").\nA specified value is placed in the outer corner of the recording tableau.\n\nThis is defined using [ssyt.legal.add].\n\nThis file contains:\n  - basic definitions and weight of the tableau\n    [ssyt.rec_cert]\n    [ssyt.rec_cert.to_legal]\n    [ssyt.rec_cert.rec_step]\n    [ssyt.rec_cert.rec_wt]\n    [ssyt.rec_cert.rec_wt]\n\n  - The key fact: \n    after bumping/recording a legal pair (recval, bumpval), any other \n    pair (recval', bumpval') will then be legal as long as\n        (recval, bumpval) \u2264\u2097 (recval', bumpval')\n    where \u2264\u2097 refers to lexicographic order on \u2115 \u00d7 \u2115. That is,\n      recval < recval' \u2228 (recval = recval' \u2227 bumpval \u2264 bumpval').\n    \n    [ssyt.rec_cert.next_cert]\n    This fact uses the key lemma [ssyt.rbs_cert.rbwf_pieri].\n\n-/\n\nsection recording_tableau\n\nsection rec_cert\n\nstructure ssyt.rec_cert {\u03bc : young_diagram} (R B : ssyt \u03bc) :=\n  (recval bumpval : \u2115)\n  (rec_le : \u2200 i j, R i j \u2264 recval)\n  (rec_eq_left : \u2200 i j (cell : (i, j) \u2208 \u03bc),\n      R i j = recval \u2192 j < (B.row_bump bumpval).1.j)\n\ndef ssyt.rec_cert.to_legal\n  {\u03bc : young_diagram} {R B : ssyt \u03bc} (rcert : ssyt.rec_cert R B) : R.legal :=\n{ i := (B.row_bump rcert.bumpval).1.i,\n  j := (B.row_bump rcert.bumpval).1.j,\n  val := rcert.recval,\n  cell_left := (B.row_bump rcert.bumpval).1.cell_left,\n  cell_up := (B.row_bump rcert.bumpval).1.cell_up,\n  left := \u03bb _ _, rcert.rec_le _ _,\n  right := \u03bb j hj cell, absurd (\u03bc.nw_of (le_refl _) (le_of_lt hj) cell) \n                               (B.row_bump rcert.bumpval).1.not_cell,\n  up := \u03bb i hi, \n    lt_of_le_of_ne (rcert.rec_le _ _)\n      (\u03bb h, (lt_self_iff_false _).mp $\n        rcert.rec_eq_left _ _ ((B.row_bump rcert.bumpval).1.cell_up hi) h),\n  down := \u03bb i hi cell, absurd (\u03bc.nw_of (le_of_lt hi) (le_refl _) cell) \n                               (B.row_bump rcert.bumpval).1.not_cell\n}\n\nend rec_cert\n\nsection rec_step\n\ndef ssyt.rec_cert.rec_step\n  {\u03bc : young_diagram} {R B : ssyt \u03bc} (rcert : ssyt.rec_cert R B) :\n  ssyt (B.row_bump rcert.bumpval).1.add :=\nrcert.to_legal.add (B.row_bump rcert.bumpval).1.not_cell\n\nlemma ssyt.rec_cert.rec_entry\n  {\u03bc : young_diagram} {R B : ssyt \u03bc} (rcert : ssyt.rec_cert R B) (i j : \u2115) :\n  rcert.rec_step i j =\n  ite ((i, j) = (rcert.to_legal.i, rcert.to_legal.j)) rcert.to_legal.val (R i j) := rfl\n\nlemma ssyt.rec_cert.rec_wt\n  {\u03bc : young_diagram} {R B : ssyt \u03bc} (rcert : ssyt.rec_cert R B) (val : \u2115) :\n  rcert.rec_step.wt val = R.wt val + ite (val = rcert.recval) 1 0 :=\nby apply ssyt.wt_add\n\ndef ssyt.rec_cert_of_gt {\u03bc : young_diagram} (R B : ssyt \u03bc) (recval bumpval : \u2115)\n  (h_lt : \u2200 i j (cell : (i, j) \u2208 \u03bc), R i j < recval) : ssyt.rec_cert R B :=\n{ bumpval := bumpval, recval := recval,\n  rec_le := \u03bb i j, dite ((i, j) \u2208 \u03bc) \n    (\u03bb cell, le_of_lt $ h_lt _ _ cell)\n    (\u03bb not_cell, (R.zeros not_cell).symm \u25b8 nat.zero_le recval),\n  rec_eq_left := \u03bb _ _ cell h_eq, false.rec _ $ ne_of_lt (h_lt _ _ cell) h_eq }\n\ndef ssyt.rec_cert.next_cert\n  {\u03bc : young_diagram} {R B : ssyt \u03bc} (rcert : ssyt.rec_cert R B)\n  (recval' bumpval' : \u2115) \n  (h : rcert.recval < recval' \u2228 \n       (rcert.recval = recval' \u2227 rcert.bumpval \u2264 bumpval')) :\nssyt.rec_cert rcert.rec_step (B.row_bump rcert.bumpval).2 :=\n{ bumpval := bumpval', recval := recval',\n  rec_le := \u03bb i j, begin\n    apply @le_trans _ _ _ rcert.recval,\n      rw ssyt.rec_cert.rec_entry,\n      split_ifs, refl, apply rcert.rec_le,\n    cases h, exact le_of_lt h, exact le_of_eq h.1,\n  end,\n  rec_eq_left := \u03bb i j cell h_eq, begin\n    cases h,\n      { exfalso, apply ne_of_lt (lt_of_le_of_lt _ h) h_eq,\n        rw rcert.rec_entry, split_ifs, refl, apply rcert.rec_le },\n      { cases h.1,\n        apply lt_of_le_of_lt _,\n        apply ssyt.rbs_cert.rbwf_pieri _ _, refl, exact h.2,\n        rw rcert.rec_entry at h_eq, split_ifs at h_eq,\n          { cases h_1, refl },\n          { apply le_of_lt (rcert.rec_eq_left i j _ h_eq),\n            rw young_diagram.outer_corner.mem_add at cell,\n            exact cell.resolve_left h_1 }\n      }\n  end\n}\n\nend rec_step\n\nend recording_tableau", "meta": {"author": "jakelev", "repo": "lean-rsk", "sha": "dbd97f8fe9fc2ba13d080d37e298ae87d03ff541", "save_path": "github-repos/lean/jakelev-lean-rsk", "path": "github-repos/lean/jakelev-lean-rsk/lean-rsk-dbd97f8fe9fc2ba13d080d37e298ae87d03ff541/src/recording_tableau.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6187804267137442, "lm_q1q2_score": 0.3690611567518459}}
{"text": "import data.set.basic\nimport logic.embedding\nimport pfun_to_fun\nimport polytime\n\nopen ptree (pencodable)\nopen ptree.pencodable (encode decode)\n\ndef polytime_fun {\u03b1 \u03b2 : Type*} [pencodable \u03b1] [pencodable \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n\u2203 (c : code) (pc : polytime c), \u2200 x, c.eval (encode x) = part.some (encode (f x))\n\nsection\nvariables {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Type*} [pencodable \u03b1] [pencodable \u03b2] [pencodable \u03b3]\n  [pencodable \u03b4] [pencodable \u03b5]\n\nlemma polytime_fun.encode {\u03b1 : Type*} [pencodable \u03b1] : polytime_fun (@encode \u03b1 _) :=\n\u27e8code.id, polytime_id, \u03bb x, by simp\u27e9\n\nlemma polytime_fun.decode {f : \u03b1 \u2192 \u03b2} (hf : polytime_fun (encode \u2218 f)) : polytime_fun f := hf\n\nlemma polytime_fun.id : polytime_fun (@id \u03b1) := \u27e8code.id, polytime_id, \u03bb x, by simp\u27e9\n\nlemma polytime_fun.const (x : \u03b1) : polytime_fun (function.const \u03b2 x) := \u27e8code.const (encode x), polytime_const _, \u03bb x, by simp\u27e9\n\nlemma polytime_fun.comp {f : \u03b2 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b2} : polytime_fun f \u2192 polytime_fun g \u2192 polytime_fun (f \u2218 g)\n| \u27e8c\u2081, pc\u2081, s\u2081\u27e9 \u27e8c\u2082, pc\u2082, s\u2082\u27e9 := \u27e8c\u2081.comp c\u2082, polytime_comp pc\u2081 pc\u2082, \u03bb x, by simp [s\u2081, s\u2082]\u27e9\n\n\nsection pair\n\nlemma polytime_fun.fst : polytime_fun (@prod.fst \u03b1 \u03b2) :=\n\u27e8code.left, polytime_left, \u03bb \u27e8a, b\u27e9, by simp [ptree.pencodable.encode_pair_def]\u27e9\n\nlemma polytime_fun.snd : polytime_fun (@prod.snd \u03b1 \u03b2) :=\n\u27e8code.right, polytime_left, \u03bb \u27e8a, b\u27e9, by simp [ptree.pencodable.encode_pair_def]\u27e9\n\nlemma polytime_fun.pair {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} : polytime_fun f \u2192 polytime_fun g \u2192 polytime_fun (\u03bb x, (f x, g x))\n| \u27e8c\u2081, pc\u2081, s\u2081\u27e9 \u27e8c\u2082, pc\u2082, s\u2082\u27e9 := \u27e8code.node c\u2081 c\u2082, polytime_node pc\u2081 pc\u2082, \u03bb x, by { simp [s\u2081, s\u2082], refl, }\u27e9\n\ndef polytime_fun\u2082 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : Prop := polytime_fun (function.uncurry f)\n\ndef polytime_fun\u2083 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) : Prop :=\npolytime_fun (\u03bb x : \u03b1 \u00d7 \u03b2 \u00d7 \u03b3, f x.1 x.2.1 x.2.2)\n\nlemma polytime_fun.comp\u2082 {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {g : \u03b4 \u2192 \u03b1} {h : \u03b4 \u2192 \u03b2} \n  (hf : polytime_fun\u2082 f) (hg : polytime_fun g) (hh : polytime_fun h) :\n  polytime_fun (\u03bb x, f (g x) (h x)) :=\npolytime_fun.comp hf (polytime_fun.pair hg hh)\n\nlemma polytime_fun.comp\u2083 {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4} {g\u2081 : \u03b5 \u2192 \u03b1} {g\u2082 : \u03b5 \u2192 \u03b2} {g\u2083 : \u03b5 -> \u03b3} \n  (hf : polytime_fun\u2083 f) (hg\u2081 : polytime_fun g\u2081) (hg\u2082 : polytime_fun g\u2082) (hg\u2083 : polytime_fun g\u2083) :\n  polytime_fun (\u03bb x, f (g\u2081 x) (g\u2082 x) (g\u2083 x)) :=\npolytime_fun.comp hf (polytime_fun.pair hg\u2081 (polytime_fun.pair hg\u2082 hg\u2083))\n\nend pair\n\nend\n\nclass polycodable (\u03b1 : Type*) extends ptree.pencodable \u03b1 :=\n(polytime_decode [] : polytime_fun (\u03bb x : ptree, encode (decode x)))\n\ninstance : polycodable ptree :=\n{ polytime_decode := \u27e8_, polytime_id, \u03bb x, by simp\u27e9 }\n\nvariables {\u03b1 \u03b2 \u03b3 : Type*} [polycodable \u03b1] [polycodable \u03b2] [polycodable \u03b3]\n\nlemma polytime_fun_iff (f : \u03b1 \u2192 \u03b2) :\n  polytime_fun f \u2194 \u2203 (c : code) (pc : polytime c), \u2200 x, c.eval x = part.some (encode (f (decode x))) :=\nbegin\n  split,\n  { rintro \u27e8c, pc, s\u27e9, rcases polycodable.polytime_decode \u03b1 with \u27e8c\u2082, pc\u2082, s\u2082\u27e9,\n    use [c.comp c\u2082, polytime_comp pc pc\u2082], intro x, simp at s\u2082, simp [s\u2082, s], },\n  rintro \u27e8c, pc, s\u27e9, use [c, pc], intro x, simp [s],\nend\n\nlemma polytime_fun_iff' (f : \u03b1 \u2192 \u03b2) :\n  polytime_fun f \u2194 \u2203 (c : code) (pc : polytime c), \u2200 x, (c.eval (encode x)).map decode = part.some (f x) :=\nbegin\n  split, { rintro \u27e8c, pc, s\u27e9, use [c, pc], simp [s], },\n  rintro \u27e8c, pc, s\u27e9, rcases polycodable.polytime_decode \u03b2 with \u27e8c\u2082, pc\u2082, s\u2082\u27e9,\n  use [c\u2082.comp c, polytime_comp pc\u2082 pc], intro x, specialize s x,\n  simp [part.eq_some_iff] at s \u22a2, rcases s with \u27e8a, h\u2081, h\u2082\u27e9, use [a, h\u2081],\n  simp at s\u2082, simp [s\u2082, h\u2082],\nend\n\nlemma polytime_fun.decode' : polytime_fun (@decode \u03b1 _) :=\nby { rw polytime_fun_iff, exact polycodable.polytime_decode _, }\n\nlemma polytime_fun.ptree_left : polytime_fun ptree.left := \u27e8code.left, polytime_left, \u03bb x, by simp\u27e9\nlemma polytime_fun.ptree_right : polytime_fun ptree.right := \u27e8code.right, polytime_right, \u03bb x, by simp\u27e9\n\nlemma polytime_fun.polytime_code {c : code} (pc : polytime c) : polytime_fun pc.to_fun := \u27e8c, pc, \u03bb x, by simp\u27e9\n\nprivate lemma polytime_fun.node_aux {f : ptree \u2192 ptree} {g : ptree \u2192 ptree} : polytime_fun f \u2192 polytime_fun g \u2192 polytime_fun (\u03bb x, (f x).node (g x))\n| \u27e8cf, pf, sf\u27e9 \u27e8cg, pg, sg\u27e9 := \u27e8cf.node cg, polytime_node pf pg, \u03bb x, by { simp at sf sg, simp [sf, sg], }\u27e9 \n\ninstance : polycodable (\u03b1 \u00d7 \u03b2) :=\n{ polytime_decode :=\nbegin\n  simp only [ptree.pencodable.decode_pair_def, ptree.pencodable.encode_pair_def],\n  apply polytime_fun.node_aux,\n  all_goals { apply polytime_fun.comp polytime_fun.encode, apply polytime_fun.comp polytime_fun.decode', },\n  exacts [polytime_fun.ptree_left, polytime_fun.ptree_right],\nend }\n\nlemma polytime_fun.node : polytime_fun\u2082 ptree.node := polytime_fun.id\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/polycodable_init.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.36895049453991247}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport tactic.pi_instances\nimport algebra.group.pi\nimport algebra.ring.basic\n\n/-!\n# Pi instances for ring\n\nThis file defines instances for ring, semiring and related structures on Pi Types\n-/\n\nnamespace pi\nuniverses u v w\nvariable {I : Type u}     -- The indexing type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\n\ninstance distrib [\u03a0 i, distrib $ f i] : distrib (\u03a0 i : I, f i) :=\nby refine_struct { add := (+), mul := (*), .. }; tactic.pi_instance_derive_field\n\ninstance non_unital_non_assoc_semiring [\u2200 i, non_unital_non_assoc_semiring $ f i] :\n  non_unital_non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_unital_semiring [\u2200 i, non_unital_semiring $ f i] :\n  non_unital_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance non_assoc_semiring [\u2200 i, non_assoc_semiring $ f i] :\n  non_assoc_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*), .. };\n  tactic.pi_instance_derive_field\n\ninstance semiring [\u2200 i, semiring $ f i] : semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance comm_semiring [\u2200 i, comm_semiring $ f i] : comm_semiring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  nsmul := add_monoid.nsmul, npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance ring [\u2200 i, ring $ f i] : ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\ninstance comm_ring [\u2200 i, comm_ring $ f i] : comm_ring (\u03a0 i : I, f i) :=\nby refine_struct { zero := (0 : \u03a0 i, f i), one := 1, add := (+), mul := (*),\n  neg := has_neg.neg, nsmul := add_monoid.nsmul, zsmul := sub_neg_monoid.zsmul,\n  npow := monoid.npow };\ntactic.pi_instance_derive_field\n\n/-- A family of ring homomorphisms `f a : \u03b3 \u2192+* \u03b2 a` defines a ring homomorphism\n`pi.ring_hom f : \u03b3 \u2192+* \u03a0 a, \u03b2 a` given by `pi.ring_hom f x b = f b x`. -/\n@[simps]\nprotected def ring_hom {\u03b3 : Type w} [\u03a0 i, non_assoc_semiring (f i)] [non_assoc_semiring \u03b3]\n  (g : \u03a0 i, \u03b3 \u2192+* f i) : \u03b3 \u2192+* \u03a0 i, f i :=\n{ to_fun := \u03bb x b, g b x,\n  map_add' := \u03bb x y, funext $ \u03bb z, (g z).map_add x y,\n  map_mul' := \u03bb x y, funext $ \u03bb z, (g z).map_mul x y,\n  map_one' := funext $ \u03bb z, (g z).map_one,\n  map_zero' := funext $ \u03bb z, (g z).map_zero }\n\nlemma ring_hom_injective {\u03b3 : Type w} [nonempty I] [\u03a0 i, non_assoc_semiring (f i)]\n  [non_assoc_semiring \u03b3] (g : \u03a0 i, \u03b3 \u2192+* f i) (hg : \u2200 i, function.injective (g i)) :\n  function.injective (pi.ring_hom g) :=\n\u03bb x y h, let \u27e8i\u27e9 := \u2039nonempty I\u203a in hg i ((function.funext_iff.mp h : _) i)\n\nend pi\n\nsection ring_hom\n\nuniverses u v\nvariable {I : Type u}\n\n/-- Evaluation of functions into an indexed collection of monoids at a point is a monoid\nhomomorphism. This is `function.eval` as a `ring_hom`. -/\n@[simps]\ndef pi.eval_ring_hom (f : I \u2192 Type v) [\u03a0 i, non_assoc_semiring (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192+* f i :=\n{ ..(pi.eval_monoid_hom f i),\n  ..(pi.eval_add_monoid_hom f i) }\n\n/-- `function.const` as a `ring_hom`. -/\n@[simps]\ndef pi.const_ring_hom (\u03b1 \u03b2 : Type*) [non_assoc_semiring \u03b2] : \u03b2 \u2192+* (\u03b1 \u2192 \u03b2) :=\n{ to_fun := function.const _,\n  .. pi.ring_hom (\u03bb _, ring_hom.id \u03b2) }\n\n/-- Ring homomorphism between the function spaces `I \u2192 \u03b1` and `I \u2192 \u03b2`, induced by a ring\nhomomorphism `f` between `\u03b1` and `\u03b2`. -/\n@[simps] protected def ring_hom.comp_left {\u03b1 \u03b2 : Type*} [non_assoc_semiring \u03b1]\n  [non_assoc_semiring \u03b2] (f : \u03b1 \u2192+* \u03b2) (I : Type*) :\n  (I \u2192 \u03b1) \u2192+* (I \u2192 \u03b2) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  .. f.to_monoid_hom.comp_left I,\n  .. f.to_add_monoid_hom.comp_left I }\n\nend ring_hom\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/ring/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593452091672, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.36891086373475374}}
{"text": "/-\nCopyright (c) 2020 Yury G. Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury G. Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.dynamics.fixed_points.basic\nimport Mathlib.data.set.lattice\nimport Mathlib.data.pnat.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Periodic points\n\nA point `x : \u03b1` is a periodic point of `f : \u03b1 \u2192 \u03b1` of period `n` if `f^[n] x = x`.\n\n## Main definitions\n\n* `is_periodic_pt f n x` : `x` is a periodic point of `f` of period `n`, i.e. `f^[n] x = x`.\n  We do not require `n > 0` in the definition.\n* `pts_of_period f n` : the set `{x | is_periodic_pt f n x}`. Note that `n` is not required to\n  be the minimal period of `x`.\n* `periodic_pts f` : the set of all periodic points of `f`.\n* `minimal_period f x` : the minimal period of a point `x` under an endomorphism `f` or zero\n  if `x` is not a periodic point of `f`.\n\n## Main statements\n\nWe provide \u201cdot syntax\u201d-style operations on terms of the form `h : is_periodic_pt f n x` including\narithmetic operations on `n` and `h.map (hg : semiconj_by g f f')`. We also prove that `f`\nis bijective on each set `pts_of_period f n` and on `periodic_pts f`. Finally, we prove that `x`\nis a periodic point of `f` of period `n` if and only if `minimal_period f x | n`.\n\n## References\n\n* https://en.wikipedia.org/wiki/Periodic_point\n\n-/\n\nnamespace function\n\n\n/-- A point `x` is a periodic point of `f : \u03b1 \u2192 \u03b1` of period `n` if `f^[n] x = x`.\nNote that we do not require `0 < n` in this definition. Many theorems about periodic points\nneed this assumption. -/\ndef is_periodic_pt {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (x : \u03b1) := is_fixed_pt (nat.iterate f n) x\n\n/-- A fixed point of `f` is a periodic point of `f` of any prescribed period. -/\ntheorem is_fixed_pt.is_periodic_pt {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} (hf : is_fixed_pt f x)\n    (n : \u2115) : is_periodic_pt f n x :=\n  is_fixed_pt.iterate hf n\n\n/-- For the identity map, all points are periodic. -/\ntheorem is_periodic_id {\u03b1 : Type u_1} (n : \u2115) (x : \u03b1) : is_periodic_pt id n x :=\n  is_fixed_pt.is_periodic_pt (is_fixed_pt_id x) n\n\n/-- Any point is a periodic point of period `0`. -/\ntheorem is_periodic_pt_zero {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) (x : \u03b1) : is_periodic_pt f 0 x :=\n  is_fixed_pt_id x\n\nnamespace is_periodic_pt\n\n\nprotected instance decidable {\u03b1 : Type u_1} [DecidableEq \u03b1] {f : \u03b1 \u2192 \u03b1} {n : \u2115} {x : \u03b1} :\n    Decidable (is_periodic_pt f n x) :=\n  is_fixed_pt.decidable\n\nprotected theorem is_fixed_pt {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115}\n    (hf : is_periodic_pt f n x) : is_fixed_pt (nat.iterate f n) x :=\n  hf\n\nprotected theorem map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {fa : \u03b1 \u2192 \u03b1} {fb : \u03b2 \u2192 \u03b2} {x : \u03b1} {n : \u2115}\n    (hx : is_periodic_pt fa n x) {g : \u03b1 \u2192 \u03b2} (hg : semiconj g fa fb) : is_periodic_pt fb n (g x) :=\n  is_fixed_pt.map hx (semiconj.iterate_right hg n)\n\ntheorem apply_iterate {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hx : is_periodic_pt f n x)\n    (m : \u2115) : is_periodic_pt f n (nat.iterate f m x) :=\n  is_periodic_pt.map hx (commute.iterate_self f m)\n\nprotected theorem apply {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hx : is_periodic_pt f n x) :\n    is_periodic_pt f n (f x) :=\n  apply_iterate hx 1\n\nprotected theorem add {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115} (hn : is_periodic_pt f n x)\n    (hm : is_periodic_pt f m x) : is_periodic_pt f (n + m) x :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (is_periodic_pt f (n + m) x)) (equations._eqn_1 f (n + m) x)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (is_fixed_pt (nat.iterate f (n + m)) x)) (iterate_add f n m)))\n      (is_fixed_pt.comp hn hm))\n\ntheorem left_of_add {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115}\n    (hn : is_periodic_pt f (n + m) x) (hm : is_periodic_pt f m x) : is_periodic_pt f n x :=\n  sorry\n\ntheorem right_of_add {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115}\n    (hn : is_periodic_pt f (n + m) x) (hm : is_periodic_pt f n x) : is_periodic_pt f m x :=\n  left_of_add (eq.mp (Eq._oldrec (Eq.refl (is_periodic_pt f (n + m) x)) (add_comm n m)) hn) hm\n\nprotected theorem sub {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115} (hm : is_periodic_pt f m x)\n    (hn : is_periodic_pt f n x) : is_periodic_pt f (m - n) x :=\n  sorry\n\nprotected theorem mul_const {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} (hm : is_periodic_pt f m x)\n    (n : \u2115) : is_periodic_pt f (m * n) x :=\n  sorry\n\nprotected theorem const_mul {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} (hm : is_periodic_pt f m x)\n    (n : \u2115) : is_periodic_pt f (n * m) x :=\n  sorry\n\ntheorem trans_dvd {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} (hm : is_periodic_pt f m x) {n : \u2115}\n    (hn : m \u2223 n) : is_periodic_pt f n x :=\n  sorry\n\nprotected theorem iterate {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hf : is_periodic_pt f n x)\n    (m : \u2115) : is_periodic_pt (nat.iterate f m) n x :=\n  sorry\n\nprotected theorem mod {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115} (hm : is_periodic_pt f m x)\n    (hn : is_periodic_pt f n x) : is_periodic_pt f (m % n) x :=\n  left_of_add\n    (eq.mp (Eq._oldrec (Eq.refl (is_periodic_pt f m x)) (Eq.symm (nat.mod_add_div m n))) hm)\n    (is_periodic_pt.mul_const hn (m / n))\n\nprotected theorem gcd {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {m : \u2115} {n : \u2115} (hm : is_periodic_pt f m x)\n    (hn : is_periodic_pt f n x) : is_periodic_pt f (nat.gcd m n) x :=\n  sorry\n\n/-- If `f` sends two periodic points `x` and `y` of the same positive period to the same point,\nthen `x = y`. For a similar statement about points of different periods see `eq_of_apply_eq`. -/\ntheorem eq_of_apply_eq_same {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {y : \u03b1} {n : \u2115}\n    (hx : is_periodic_pt f n x) (hy : is_periodic_pt f n y) (hn : 0 < n) (h : f x = f y) : x = y :=\n  sorry\n\n/-- If `f` sends two periodic points `x` and `y` of positive periods to the same point,\nthen `x = y`. -/\ntheorem eq_of_apply_eq {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {y : \u03b1} {m : \u2115} {n : \u2115}\n    (hx : is_periodic_pt f m x) (hy : is_periodic_pt f n y) (hm : 0 < m) (hn : 0 < n)\n    (h : f x = f y) : x = y :=\n  eq_of_apply_eq_same (is_periodic_pt.mul_const hx n) (is_periodic_pt.const_mul hy m)\n    (mul_pos hm hn) h\n\nend is_periodic_pt\n\n\n/-- The set of periodic points of a given (possibly non-minimal) period. -/\ndef pts_of_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) (n : \u2115) : set \u03b1 :=\n  set_of fun (x : \u03b1) => is_periodic_pt f n x\n\n@[simp] theorem mem_pts_of_period {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} :\n    x \u2208 pts_of_period f n \u2194 is_periodic_pt f n x :=\n  iff.rfl\n\ntheorem semiconj.maps_to_pts_of_period {\u03b1 : Type u_1} {\u03b2 : Type u_2} {fa : \u03b1 \u2192 \u03b1} {fb : \u03b2 \u2192 \u03b2}\n    {g : \u03b1 \u2192 \u03b2} (h : semiconj g fa fb) (n : \u2115) :\n    set.maps_to g (pts_of_period fa n) (pts_of_period fb n) :=\n  semiconj.maps_to_fixed_pts (semiconj.iterate_right h n)\n\ntheorem bij_on_pts_of_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) {n : \u2115} (hn : 0 < n) :\n    set.bij_on f (pts_of_period f n) (pts_of_period f n) :=\n  sorry\n\ntheorem directed_pts_of_period_pnat {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) :\n    directed has_subset.subset fun (n : \u2115+) => pts_of_period f \u2191n :=\n  sorry\n\n/-- The set of periodic points of a map `f : \u03b1 \u2192 \u03b1`. -/\ndef periodic_pts {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) : set \u03b1 :=\n  set_of fun (x : \u03b1) => \u2203 (n : \u2115), \u2203 (H : n > 0), is_periodic_pt f n x\n\ntheorem mk_mem_periodic_pts {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hn : 0 < n)\n    (hx : is_periodic_pt f n x) : x \u2208 periodic_pts f :=\n  Exists.intro n (Exists.intro hn hx)\n\ntheorem mem_periodic_pts {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} :\n    x \u2208 periodic_pts f \u2194 \u2203 (n : \u2115), \u2203 (H : n > 0), is_periodic_pt f n x :=\n  iff.rfl\n\ntheorem bUnion_pts_of_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) :\n    (set.Union fun (n : \u2115) => set.Union fun (H : n > 0) => pts_of_period f n) = periodic_pts f :=\n  sorry\n\ntheorem Union_pnat_pts_of_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) :\n    (set.Union fun (n : \u2115+) => pts_of_period f \u2191n) = periodic_pts f :=\n  Eq.trans supr_subtype (bUnion_pts_of_period f)\n\ntheorem bij_on_periodic_pts {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) :\n    set.bij_on f (periodic_pts f) (periodic_pts f) :=\n  Union_pnat_pts_of_period f \u25b8\n    set.bij_on_Union_of_directed (directed_pts_of_period_pnat f)\n      fun (i : \u2115+) => bij_on_pts_of_period f (pnat.pos i)\n\ntheorem semiconj.maps_to_periodic_pts {\u03b1 : Type u_1} {\u03b2 : Type u_2} {fa : \u03b1 \u2192 \u03b1} {fb : \u03b2 \u2192 \u03b2}\n    {g : \u03b1 \u2192 \u03b2} (h : semiconj g fa fb) : set.maps_to g (periodic_pts fa) (periodic_pts fb) :=\n  sorry\n\n/-- Minimal period of a point `x` under an endomorphism `f`. If `x` is not a periodic point of `f`,\nthen `minimal_period f x = 0`. -/\ndef minimal_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) (x : \u03b1) : \u2115 :=\n  dite (x \u2208 periodic_pts f) (fun (h : x \u2208 periodic_pts f) => nat.find h)\n    fun (h : \u00acx \u2208 periodic_pts f) => 0\n\ntheorem is_periodic_pt_minimal_period {\u03b1 : Type u_1} (f : \u03b1 \u2192 \u03b1) (x : \u03b1) :\n    is_periodic_pt f (minimal_period f x) x :=\n  sorry\n\ntheorem minimal_period_pos_of_mem_periodic_pts {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1}\n    (hx : x \u2208 periodic_pts f) : 0 < minimal_period f x :=\n  sorry\n\ntheorem is_periodic_pt.minimal_period_pos {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hn : 0 < n)\n    (hx : is_periodic_pt f n x) : 0 < minimal_period f x :=\n  minimal_period_pos_of_mem_periodic_pts (mk_mem_periodic_pts hn hx)\n\ntheorem minimal_period_pos_iff_mem_periodic_pts {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} :\n    0 < minimal_period f x \u2194 x \u2208 periodic_pts f :=\n  sorry\n\ntheorem is_periodic_pt.minimal_period_le {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} (hn : 0 < n)\n    (hx : is_periodic_pt f n x) : minimal_period f x \u2264 n :=\n  sorry\n\ntheorem is_periodic_pt.eq_zero_of_lt_minimal_period {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115}\n    (hx : is_periodic_pt f n x) (hn : n < minimal_period f x) : n = 0 :=\n  Eq.symm\n    (or.resolve_right (eq_or_lt_of_le (nat.zero_le n))\n      fun (hn0 : 0 < n) => iff.mpr not_lt (is_periodic_pt.minimal_period_le hn0 hx) hn)\n\ntheorem is_periodic_pt.minimal_period_dvd {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115}\n    (hx : is_periodic_pt f n x) : minimal_period f x \u2223 n :=\n  sorry\n\ntheorem is_periodic_pt_iff_minimal_period_dvd {\u03b1 : Type u_1} {f : \u03b1 \u2192 \u03b1} {x : \u03b1} {n : \u2115} :\n    is_periodic_pt f n x \u2194 minimal_period f x \u2223 n :=\n  { mp := is_periodic_pt.minimal_period_dvd,\n    mpr :=\n      fun (h : minimal_period f x \u2223 n) =>\n        is_periodic_pt.trans_dvd (is_periodic_pt_minimal_period f x) h }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/dynamics/periodic_pts_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3689108554330874}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.limits.functor_category\nimport Mathlib.PostPort\n\nuniverses v u \n\nnamespace Mathlib\n\n/-!\n# Limit properties relating to the (co)yoneda embedding.\n\nWe calculate the colimit of `Y \u21a6 (X \u27f6 Y)`, which is just `punit`.\n(This is used in characterising cofinal functors.)\n\nWe also show the (co)yoneda embeddings preserve limits and jointly reflect them.\n-/\n\nnamespace category_theory\n\n\nnamespace coyoneda\n\n\n/--\nThe colimit cocone over `coyoneda.obj X`, with cocone point `punit`.\n-/\n@[simp] theorem colimit_cocone_\u03b9_app {C : Type v} [small_category C] (X : C\u1d52\u1d56) (X_1 : C) : \u2200 (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1),\n  nat_trans.app (limits.cocone.\u03b9 (colimit_cocone X)) X_1 \u1fb0 =\n    id\n      (fun (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1) =>\n        id (fun (X : C\u1d52\u1d56) (X_1 : C) (\u1fb0 : opposite.unop X \u27f6 X_1) => PUnit.unit) X X_1 \u1fb0)\n      \u1fb0 :=\n  fun (\u1fb0 : functor.obj (functor.obj coyoneda X) X_1) => Eq.refl (nat_trans.app (limits.cocone.\u03b9 (colimit_cocone X)) X_1 \u1fb0)\n\n/--\nThe proposed colimit cocone over `coyoneda.obj X` is a colimit cocone.\n-/\ndef colimit_cocone_is_colimit {C : Type v} [small_category C] (X : C\u1d52\u1d56) : limits.is_colimit (colimit_cocone X) :=\n  limits.is_colimit.mk\n    fun (s : limits.cocone (functor.obj coyoneda X)) (x : limits.cocone.X (colimit_cocone X)) =>\n      nat_trans.app (limits.cocone.\u03b9 s) (opposite.unop X) \ud835\udfd9\n\nprotected instance obj.category_theory.limits.has_colimit {C : Type v} [small_category C] (X : C\u1d52\u1d56) : limits.has_colimit (functor.obj coyoneda X) :=\n  limits.has_colimit.mk (limits.colimit_cocone.mk (colimit_cocone X) (colimit_cocone_is_colimit X))\n\n/--\nThe colimit of `coyoneda.obj X` is isomorphic to `punit`.\n-/\ndef colimit_coyoneda_iso {C : Type v} [small_category C] (X : C\u1d52\u1d56) : limits.colimit (functor.obj coyoneda X) \u2245 PUnit :=\n  limits.colimit.iso_colimit_cocone (limits.colimit_cocone.mk (colimit_cocone X) (colimit_cocone_is_colimit X))\n\nend coyoneda\n\n\n/-- The yoneda embedding `yoneda.obj X : C\u1d52\u1d56 \u2964 Type v` for `X : C` preserves limits. -/\nprotected instance yoneda_preserves_limits {C : Type u} [category C] (X : C) : limits.preserves_limits (functor.obj yoneda X) :=\n  limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : small_category J) =>\n      limits.preserves_limits_of_shape.mk\n        fun (K : J \u2964 (C\u1d52\u1d56)) =>\n          limits.preserves_limit.mk\n            fun (c : limits.cone K) (t : limits.is_limit c) =>\n              limits.is_limit.mk\n                fun (s : limits.cone (K \u22d9 functor.obj yoneda X)) (x : limits.cone.X s) =>\n                  has_hom.hom.unop\n                    (limits.is_limit.lift t\n                      (limits.cone.mk (opposite.op X)\n                        (nat_trans.mk fun (j : J) => has_hom.hom.op (nat_trans.app (limits.cone.\u03c0 s) j x))))\n\n/-- The coyoneda embedding `coyoneda.obj X : C \u2964 Type v` for `X : C\u1d52\u1d56` preserves limits. -/\nprotected instance coyoneda_preserves_limits {C : Type u} [category C] (X : C\u1d52\u1d56) : limits.preserves_limits (functor.obj coyoneda X) :=\n  limits.preserves_limits.mk\n    fun (J : Type v) (\ud835\udca5 : small_category J) =>\n      limits.preserves_limits_of_shape.mk\n        fun (K : J \u2964 C) =>\n          limits.preserves_limit.mk\n            fun (c : limits.cone K) (t : limits.is_limit c) =>\n              limits.is_limit.mk\n                fun (s : limits.cone (K \u22d9 functor.obj coyoneda X)) (x : limits.cone.X s) =>\n                  limits.is_limit.lift t\n                    (limits.cone.mk (opposite.unop X) (nat_trans.mk fun (j : J) => nat_trans.app (limits.cone.\u03c0 s) j x))\n\n/-- The yoneda embeddings jointly reflect limits. -/\ndef yoneda_jointly_reflects_limits {C : Type u} [category C] (J : Type v) [small_category J] (K : J \u2964 (C\u1d52\u1d56)) (c : limits.cone K) (t : (X : C) \u2192 limits.is_limit (functor.map_cone (functor.obj yoneda X) c)) : limits.is_limit c :=\n  let s' : (s : limits.cone K) \u2192 limits.cone (K \u22d9 functor.obj yoneda (opposite.unop (limits.cone.X s))) :=\n    fun (s : limits.cone K) =>\n      limits.cone.mk PUnit\n        (nat_trans.mk\n          fun (j : J) (_x : functor.obj (functor.obj (functor.const J) PUnit) j) =>\n            has_hom.hom.unop (nat_trans.app (limits.cone.\u03c0 s) j));\n  limits.is_limit.mk\n    fun (s : limits.cone K) =>\n      has_hom.hom.op (limits.is_limit.lift (t (opposite.unop (limits.cone.X s))) (s' s) PUnit.unit)\n\n/-- The coyoneda embeddings jointly reflect limits. -/\ndef coyoneda_jointly_reflects_limits {C : Type u} [category C] (J : Type v) [small_category J] (K : J \u2964 C) (c : limits.cone K) (t : (X : C\u1d52\u1d56) \u2192 limits.is_limit (functor.map_cone (functor.obj coyoneda X) c)) : limits.is_limit c :=\n  let s' : (s : limits.cone K) \u2192 limits.cone (K \u22d9 functor.obj coyoneda (opposite.op (limits.cone.X s))) :=\n    fun (s : limits.cone K) =>\n      limits.cone.mk PUnit\n        (nat_trans.mk\n          fun (j : J) (_x : functor.obj (functor.obj (functor.const J) PUnit) j) => nat_trans.app (limits.cone.\u03c0 s) j);\n  limits.is_limit.mk fun (s : limits.cone K) => limits.is_limit.lift (t (opposite.op (limits.cone.X s))) (s' s) PUnit.unit\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3687284168802605}}
{"text": "import abstraction.statespaces\nuniverse u\nopen set topological_space classical\nlocal attribute [instance] prop_decidable\n\n-- ALL FILES IN THIS FOLDER ARE A WORK IN PROGRESS.\n-- THE TERMINOLOGY AND IDEAS DEFINED HERE ALSO DIFFER\n-- MARKEDLY FROM ONTOLOGY.LEAN. \n\n\nnamespace ontology\n\n  variables {\u03c9 : ontology}\n\n  -- We can define virtual substances as products, \n  -- coproducts, subtypes and quotients of \n  -- state spaces of real substances.\n  -- These spaces so generated represent the would be state spaces\n  -- of entities which perhaps are not possible to exist.\n  \n  -- Concepts are virtual substances abstracted away from the state\n  -- spaces of particulars substances. \n  -- Universals are concepts defined up to isomorphism.\n  \n  -- The process of abstraction, telling whether a Type has been\n  -- abstracted away from particular substances.\n  -- An element of this type is one possible way the topological\n  -- space could be abstracted from the particulars, i.e.\n  -- it is an abstraction of the space.\n  class inductive abstraction (\u03c9 : ontology) : \u03a0 (\u03b1 : Type u) [topological_space \u03b1], Type (u+1)\n  | particular (s : \u03c9.substance) : abstraction s.State\n  | pi           {I : Type u} (get : I \u2192 Type u)\n  [get_top : \u03a0 i : I, topological_space (get i)]\n  (h : \u2200 i : I, abstraction (get i) ) \n  : abstraction (\u03a0 i : I, (get i))\n  \n  | sigma        {I : Type u} (get : I \u2192 Type u)\n  [get_top : \u03a0 i : I, topological_space (get i)]\n  (h : \u2200 i : I, abstraction (get i) ) \n  : abstraction (\u03a3 i : I, (get i))\n  \n  | subtype      {\u03b1 : Type u} [topological_space \u03b1]\n  (p : \u03b1 \u2192 Prop)\n  (h : abstraction \u03b1)\n  : abstraction (subtype p)\n  \n  | quotient     {\u03b1 : Type u} [topological_space \u03b1]\n  (s : setoid \u03b1)\n  (h : abstraction \u03b1)\n  : abstraction (quotient s)\n  \n  instance particular (s : \u03c9.substance) : \u03c9.abstraction s.State := abstraction.particular s \n  \n  -- CONCEPTS\n  \n  -- A concept for a Type is a topology\n  -- for that Type that was abstracted away\n  -- from the particular substances.\n  -- It is a \"concrete universal\", an\n  -- universal not defined up to isomorphism.\n  structure concept (\u03c9 : ontology) : Type (u+1) :=\n    (state : Type u)\n    [t : topological_space state]\n    [abs : \u03c9.abstraction state]\n  \n  def substance.concept (s : \u03c9.substance) : \u03c9.concept := \u27e8s.State\u27e9\n\n  instance concept_top (c : \u03c9.concept) : topological_space c.state := c.t\n  \n  instance concept_abs (c : \u03c9.concept) : \u03c9.abstraction c.state := c.abs\n  \n  -- concepts are inhabited by the concept of a\n  -- necessary being.\n  instance concept_inhabited : inhabited \u03c9.concept := \u27e8\u03c9.nb.concept\u27e9\n  \n  variable c : \u03c9.concept\n  \n  @[reducible]\n  def concept.event := set c.state\n  \n  -- Concepts can have world indexed states just as substances\n  -- but the concept doesnt just make sense for just any concept,\n  -- so occasionally we can associate a nonempty set of states to\n  -- a concept and occasionally it will be empty \n  -- (namely, if the concept is defined as a subconcept of another concept).\n  def concept.state_at (w : \u03c9.world) : c.event :=\n    begin\n      cases c,\n      induction c_abs,\n      exact {c_abs.State_at w},\n      all_goals {\n      dunfold concept.event concept.state,\n      -- simp,\n      },\n      repeat {\n      have ih : \u03a0 (i : c_abs_I), set (c_abs_get i) := c_abs_ih,\n      intro x,\n      },\n      exact \u2200i, x i \u2208 ih i,\n      exact x.snd \u2208 ih x.fst,\n      exact {x | x.val \u2208 c_abs_ih},\n      exact (@quotient.mk _ c_abs_s) '' c_abs_ih,\n    end\n  \n  -- Given this we can define the notion of\n  -- a map between state spaces preserving states.\n  def state_preserving {c\u2081 c\u2082 : \u03c9.concept} (f : c\u2081.state \u2192 c\u2082.state) :=\n  \u2200 w, f '' (c\u2081.state_at w) \u2286 c\u2082.state_at w\n  \n  -- An abstract quality is the analogue for concepts\n  -- of the perfections of substances.\n  structure concept.quality :=\n  (exist : c.event)\n  (is_open : is_open exist)\n  (ne : exist.nonempty)\n  (nuniv : exist \u2260 univ)\n  \n  -- check whether an event is a quality\n  def concept.is_quality (e : c.event):= is_open e \u2227 e.nonempty \u2227 e \u2260 univ\n  \n  -- We can generate the set of substances which grounds a concept\n  -- in reality. That is the set of substances from which the concept\n  -- was abstracted.\n  def concept.grounded (c : \u03c9.concept) : set \u03c9.substance :=\n    begin\n      cases c,\n      induction c_abs,\n      exact {c_abs},\n      repeat{exact \u22c3 i, c_abs_ih i},\n      repeat{assumption},\n    end\n      \n  -- Given a set of substances it is possible to\n  -- construct different types of wholes/totalities\n  -- which contain the set\n  def integral_whole (s : set \u03c9.substance) : \u03c9.concept :=\n    let get := \u03bbx : subtype s, x.val.State,\n    type := \u03a0 i : subtype s, (get i)\n    in begin\n      set abs := @abstraction.pi \u03c9 _ get (by apply_instance) (by apply_instance),\n      exact @concept.mk \u03c9 type _ abs,\n    end\n  \n  def abstract_whole (s : set \u03c9.substance) : \u03c9.concept :=\n    let get := \u03bbx : subtype s, x.val.State,\n    type := \u03a3 i : subtype s, (get i)\n    in begin\n      set abs := @abstraction.sigma \u03c9 _ get (by apply_instance) (by apply_instance),\n      exact @concept.mk \u03c9 type _ abs,\n    end \n  \n  -- UNIVERSALS\n  \n  -- Since universals will \n  -- be equivalence classes of concepts, \n  -- we need to define a setoid of concepts.\n  \n  -- Two concepts are equivalent if they are homeomorphic\n  @[reducible]\n  def concept_equiv (c\u2081 c\u2082 : \u03c9.concept) :=\n  nonempty (c\u2081.state \u2243\u209c c\u2082.state)\n  \n  instance concept_setoid : setoid \u03c9.concept :=\n  begin\n  fconstructor,\n  exact concept_equiv,\n  repeat{constructor};\n  simp [reflexive, symmetric, transitive],\n  intro x,\n  exact \u27e8homeomorph.refl x.state\u27e9,\n  intros x y h,\n  constructor,\n  exact homeomorph.symm h,\n  intros x y z h\u2081 h\u2082,\n  constructor,\n  exact homeomorph.trans h\u2081 h\u2082,\n  end\n  \n  -- finally\n  def universal (\u03c9 : ontology) := @quotient \u03c9.concept (by apply_instance)\n  \n  -- In order to the define the concept of `essence` and that of\n  -- `definition` we require the definition of an invariant property.\n  @[reducible]\n  def property := \u03c9.concept \u2192 Prop\n  def property.invariant (p : property) :=\n  \u2200 c\u2081 c\u2082 : \u03c9.concept, \n  c\u2081 \u2248 c\u2082 \u2192 (p c\u2081 \u2194 p c\u2082)\n  \n  -- The essence of an universal is a property which defines\n  -- its concepts up to homeomorphism.\n  def universal.is_essence (u : \u03c9.universal) (p : property) :=\n  p.invariant \u2227\n  \u2203 c\u2081, \u27e6c\u2081\u27e7 = u \u2227 p c\u2081 \u2227\n  (\u2200 c\u2082, p c\u2082 \u2192 c\u2081 \u2248 c\u2082)\n  \n  @[reducible]\n  def universal.essence (u : \u03c9.universal) := subtype u.is_essence\n  \n  -- every universal has an essence\n  theorem essentialism : \u2200 u : \u03c9.universal, nonempty u.essence :=\n  begin\n  intro u,\n  -- Take a representative concept c,\n  -- then equivalence with c is the essence of u.\n  -- Do note however that this essence is noncomputable,\n  -- because u.out depends on classical.choice.\n  -- So even though we know that every universal has an\n  -- essence we do not know (in more concrete terms)\n  -- the essence of every universal.\n  -- Now, if instead we defined this in terms of concepts we\n  -- would be able to construct the essence, therefore it makes\n  -- more sense philosophically to only define essence for universals.\n  let c := u.out,\n  repeat{fconstructor},\n  exact (\u2248) c,\n  simp [property.invariant],\n  suffices h : \u2200 (c\u2081 c\u2082 : \u03c9.concept), c\u2081 \u2248 c\u2082 \u2192 (c \u2248 c\u2081 \u2192 c \u2248 c\u2082),\n  intros c\u2081 c\u2082 h\u2082,\n  have h\u2083 : c\u2082 \u2248 c\u2081 := setoid.symm h\u2082,\n  exact \u27e8h c\u2081 c\u2082 h\u2082, h c\u2082 c\u2081 h\u2083\u27e9,\n  intros c\u2081 c\u2082 h\u2081 h\u2082,\n  exact setoid.trans h\u2082 h\u2081,\n  exact c,\n  simp [c],\n  simp,\n  end\n  \n  \n  -- In another sense u.out, for u an universal, might be considered to be\n  -- the essence of u, since it is an \"abstract\" representative of u.\n  -- In this sense we could consider (e.g.) \"the\" natural numbers \n  -- to be an \"essence\" of sorts, because given the class of all models of\n  -- second order arithmetic, neither a particular model nor the class itself\n  -- appears to be a good candidate for \"the\" natural numbers, but an abstract\n  -- representative of the class appears to be it. In this sense we can\n  -- somewhat avoid the \"up to isomorphism\" restriction placed upon mathematical\n  -- concepts. A restriction which, if we were to be consistent with it, should\n  -- preclude us from talking about \"the\" natural numbers at all.\n  \n  -- It appears to be the same thing for most purposes to either take\n  -- u.out as the essence or the relation of \"being homeomorphic to u.out\"\n  -- as the essence. Although it would look like the first point of view\n  -- is the traditional essentialist one, while the second appears to be\n  -- some form of similarity nominalism. Arguably even this similarity \n  -- view is a pretty essentialistic one insofar as the representative u.out\n  -- is totally abstract and impossible to concretely construct or to.\n  -- concretely compare with anything except by means of essential invariant\n  -- properties which are necessarilly true for all instances of an universal.\n  \n  -- In this the noncomputable nature of classical.choice can be given \n  -- a philosophical interpretation, since we cannot compute or construct\n  -- abstract essences, otherwise they would be concrete.\n  \n  -- The representation u.out also acts as a generic instance of the universal.\n  -- For any given invariant property, suffices to show that it is valid\n  -- for u.out to conclude it is valid for any representation.\n  -- An example of this is given below:\n  \n  -- A concept is instantiable if it is homeomorphic to the state\n  -- space of a substance, so that it could be thought as being an\n  -- entity of the same species as that substance.\n  def instantiable (\u03c9 : ontology) : property := \u03bbc, \u2203 s : \u03c9.substance, s.concept \u2248 c\n  \n  -- Instantiability is of course an invariant\n  lemma concept_instantiable_invariant : \u03c9.instantiable.invariant :=\n  begin\n  dunfold instantiable property.invariant,\n  intros c\u2081 c\u2082 h,\n  suffices c : (\u2203 (s : \u03c9.substance), substance.concept s \u2248 c\u2081) \u2192 \u2203 (s : \u03c9.substance), substance.concept s \u2248 c\u2082,\n  constructor,\n  exact c,\n  all_goals{\n  intro hs,\n  obtain \u27e8s, hs\u27e9 := hs,\n  existsi s,\n  },\n  exact setoid.trans hs (setoid.symm h),\n  exact setoid.trans hs h,\n  end\n  \n  -- Therefore we define instantiablity for universals via quotient.out\n  def universal.instantiable (u : \u03c9.universal) := \u03c9.instantiable u.out\n  \n  -- TODO: argue that Aquinas defended the u.out point of view in\n  -- De ente et essentia.\n  \n  -- The nb as an universal.\n  @[reducible]\n  def nbu (\u03c9 : ontology) : \u03c9.universal := \u27e6\u03c9.nb.concept\u27e7\n  instance universal_inhabited : inhabited \u03c9.universal := \u27e8\u03c9.nbu\u27e9\n  \n  -- A notion is a quality defined abstractly in the representation\n  def universal.notion (u : \u03c9.universal) := u.out.quality \n  \n  \nend ontology", "meta": {"author": "maxd13", "repo": "topological_ontology", "sha": "68d21c9a00024fba3aed301e16c31e05733c1786", "save_path": "github-repos/lean/maxd13-topological_ontology", "path": "github-repos/lean/maxd13-topological_ontology/topological_ontology-68d21c9a00024fba3aed301e16c31e05733c1786/src/abstraction/concepts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3687284168802605}}
{"text": "import Smt\n\ntheorem triv (p : Bool) : p \u2192 p := by\n  smt\n  simp_all\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Bool/Triv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.36872841027049647}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nMultisets.\n-/\nimport logic.function order.boolean_algebra\n  data.list.basic data.list.perm data.list.sort data.quot data.string\n  algebra.order_functions algebra.group_power algebra.ordered_group\n  category.traversable.lemmas tactic.interactive\n  category.traversable.instances category.basic\n\nopen list subtype nat lattice\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n\nlocal infix ` \u2022 ` := add_monoid.smul\n\ninstance list.perm.setoid (\u03b1 : Type*) : setoid (list \u03b1) :=\nsetoid.mk perm \u27e8perm.refl, @perm.symm _, @perm.trans _\u27e9\n\n/-- `multiset \u03b1` is the quotient of `list \u03b1` by list permutation. The result\n  is a type of finite sets with duplicates allowed.  -/\ndef {u} multiset (\u03b1 : Type u) : Type u :=\nquotient (list.perm.setoid \u03b1)\n\nnamespace multiset\n\ninstance : has_coe (list \u03b1) (multiset \u03b1) := \u27e8quot.mk _\u27e9\n\n@[simp] theorem quot_mk_to_coe (l : list \u03b1) : @eq (multiset \u03b1) \u27e6l\u27e7 l := rfl\n\n@[simp] theorem quot_mk_to_coe' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk (\u2248) l) l := rfl\n\n@[simp] theorem quot_mk_to_coe'' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk setoid.r l) l := rfl\n\n@[simp] theorem coe_eq_coe {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) = l\u2082 \u2194 l\u2081 ~ l\u2082 := quotient.eq\n\ninstance has_decidable_eq [decidable_eq \u03b1] : decidable_eq (multiset \u03b1)\n| s\u2081 s\u2082 := quotient.rec_on_subsingleton\u2082 s\u2081 s\u2082 $ \u03bb l\u2081 l\u2082,\n  decidable_of_iff' _ quotient.eq\n\n/- empty multiset -/\n\n/-- `0 : multiset \u03b1` is the empty set -/\nprotected def zero : multiset \u03b1 := @nil \u03b1\n\ninstance : has_zero (multiset \u03b1)   := \u27e8multiset.zero\u27e9\ninstance : has_emptyc (multiset \u03b1) := \u27e80\u27e9\ninstance : inhabited (multiset \u03b1)  := \u27e80\u27e9\n\n@[simp] theorem coe_nil_eq_zero : (@nil \u03b1 : multiset \u03b1) = 0 := rfl\n@[simp] theorem empty_eq_zero : (\u2205 : multiset \u03b1) = 0 := rfl\n\ntheorem coe_eq_zero (l : list \u03b1) : (l : multiset \u03b1) = 0 \u2194 l = [] :=\niff.trans coe_eq_coe perm_nil\n\n/- cons -/\n\n/-- `cons a s` is the multiset which contains `s` plus one more\n  instance of `a`. -/\ndef cons (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (a :: l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound ((perm_cons a).2 p))\n\nnotation a :: b := cons a b\n\ninstance : has_insert \u03b1 (multiset \u03b1) := \u27e8cons\u27e9\n\n@[simp] theorem insert_eq_cons (a : \u03b1) (s : multiset \u03b1) :\n  insert a s = a::s := rfl\n\n@[simp] theorem cons_coe (a : \u03b1) (l : list \u03b1) :\n  (a::l : multiset \u03b1) = (a::l : list \u03b1) := rfl\n\ntheorem singleton_coe (a : \u03b1) : (a::0 : multiset \u03b1) = ([a] : list \u03b1) := rfl\n\n@[simp] theorem cons_inj_left {a b : \u03b1} (s : multiset \u03b1) :\n  a::s = b::s \u2194 a = b :=\n\u27e8quot.induction_on s $ \u03bb l e,\n  have [a] ++ l ~ [b] ++ l, from quotient.exact e,\n  eq_singleton_of_perm $ (perm_app_right_iff _).1 this, congr_arg _\u27e9\n\n@[simp] theorem cons_inj_right (a : \u03b1) : \u2200{s t : multiset \u03b1}, a::s = a::t \u2194 s = t :=\nby rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9; simp [perm_cons]\n\n@[recursor 5] protected theorem induction {p : multiset \u03b1 \u2192 Prop}\n  (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : \u2200s, p s :=\nby rintros \u27e8l\u27e9; induction l with _ _ ih; [exact h\u2081, exact h\u2082 ih]\n\n@[elab_as_eliminator] protected theorem induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : p s :=\nmultiset.induction h\u2081 h\u2082 s\n\ntheorem cons_swap (a b : \u03b1) (s : multiset \u03b1) : a :: b :: s = b :: a :: s :=\nquot.induction_on s $ \u03bb l, quotient.sound $ perm.swap _ _ _\n\nsection rec\nvariables {C : multiset \u03b1 \u2192 Sort*}\n\n/-- Dependent recursor on multisets.\n\nTODO: should be @[recursor 6], but then the definition of `multiset.pi` failes with a stack\noverflow in `whnf`.\n-/\nprotected def rec\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b))\n  (m : multiset \u03b1) : C m :=\nquotient.hrec_on m (@list.rec \u03b1 (\u03bbl, C \u27e6l\u27e7) C_0 (\u03bba l b, C_cons a \u27e6l\u27e7 b)) $\n  assume l l' h,\n  list.rec_heq_of_perm h\n    (assume a l l' b b' hl, have \u27e6l\u27e7 = \u27e6l'\u27e7, from quot.sound hl, by cc)\n    (assume a a' l, C_cons_heq a a' \u27e6l\u27e7)\n\n@[elab_as_eliminator]\nprotected def rec_on (m : multiset \u03b1)\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)) :\n  C m :=\nmultiset.rec C_0 C_cons C_cons_heq m\n\nvariables {C_0 : C 0} {C_cons : \u03a0a m, C m \u2192 C (a::m)}\n  {C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)}\n\n@[simp] lemma rec_on_0 : @multiset.rec_on \u03b1 C (0:multiset \u03b1) C_0 C_cons C_cons_heq = C_0 :=\nrfl\n\n@[simp] lemma rec_on_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).rec_on C_0 C_cons C_cons_heq = C_cons a m (m.rec_on C_0 C_cons C_cons_heq) :=\nquotient.induction_on m $ assume l, rfl\n\nend rec\n\nsection mem\n\n/-- `a \u2208 s` means that `a` has nonzero multiplicity in `s`. -/\ndef mem (a : \u03b1) (s : multiset \u03b1) : Prop :=\nquot.lift_on s (\u03bb l, a \u2208 l) (\u03bb l\u2081 l\u2082 (e : l\u2081 ~ l\u2082), propext $ mem_of_perm e)\n\ninstance : has_mem \u03b1 (multiset \u03b1) := \u27e8mem\u27e9\n\n@[simp] lemma mem_coe {a : \u03b1} {l : list \u03b1} : a \u2208 (l : multiset \u03b1) \u2194 a \u2208 l := iff.rfl\n\ninstance decidable_mem [decidable_eq \u03b1] (a : \u03b1) (s : multiset \u03b1) : decidable (a \u2208 s) :=\nquot.rec_on_subsingleton s $ list.decidable_mem a\n\n@[simp] theorem mem_cons {a b : \u03b1} {s : multiset \u03b1} : a \u2208 b :: s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, iff.rfl\n\nlemma mem_cons_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 b :: s :=\nmem_cons.2 $ or.inr h\n\n@[simp] theorem mem_cons_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 a :: s :=\nmem_cons.2 (or.inl rfl)\n\ntheorem exists_cons_of_mem {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 \u2203 t, s = a :: t :=\nquot.induction_on s $ \u03bb l (h : a \u2208 l),\nlet \u27e8l\u2081, l\u2082, e\u27e9 := mem_split h in\ne.symm \u25b8 \u27e8(l\u2081++l\u2082 : list \u03b1), quot.sound perm_middle\u27e9\n\n@[simp] theorem not_mem_zero (a : \u03b1) : a \u2209 (0 : multiset \u03b1) := id\n\ntheorem eq_zero_of_forall_not_mem {s : multiset \u03b1} : (\u2200x, x \u2209 s) \u2192 s = 0 :=\nquot.induction_on s $ \u03bb l H, by rw eq_nil_iff_forall_not_mem.mpr H; refl\n\ntheorem exists_mem_of_ne_zero {s : multiset \u03b1} : s \u2260 0 \u2192 \u2203 a : \u03b1, a \u2208 s :=\nquot.induction_on s $ assume l hl,\n  match l, hl with\n  | [] := assume h, false.elim $ h rfl\n  | (a :: l) := assume _, \u27e8a, by simp\u27e9\n  end\n\n@[simp] lemma zero_ne_cons {a : \u03b1} {m : multiset \u03b1} : 0 \u2260 a :: m :=\nassume h, have a \u2208 (0:multiset \u03b1), from h.symm \u25b8 mem_cons_self _ _, not_mem_zero _ this\n\n@[simp] lemma cons_ne_zero {a : \u03b1} {m : multiset \u03b1} : a :: m \u2260 0 := zero_ne_cons.symm\n\nlemma cons_eq_cons {a b : \u03b1} {as bs : multiset \u03b1} :\n  a :: as = b :: bs \u2194 ((a = b \u2227 as = bs) \u2228 (a \u2260 b \u2227 \u2203cs, as = b :: cs \u2227 bs = a :: cs)) :=\nbegin\n  haveI : decidable_eq \u03b1 := classical.dec_eq \u03b1,\n  split,\n  { assume eq,\n    by_cases a = b,\n    { subst h, simp * at * },\n    { have : a \u2208 b :: bs, from eq \u25b8 mem_cons_self _ _,\n      have : a \u2208 bs, by simpa [h],\n      rcases exists_cons_of_mem this with \u27e8cs, hcs\u27e9,\n      simp [h, hcs],\n      have : a :: as = b :: a :: cs, by simp [eq, hcs],\n      have : a :: as = a :: b :: cs, by rwa [cons_swap],\n      simpa using this } },\n  { assume h,\n    rcases h with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n    { simp * },\n    { simp [*, cons_swap a b] } }\nend\n\nend mem\n\n/- subset -/\nsection subset\n\n/-- `s \u2286 t` is the lift of the list subset relation. It means that any\n  element with nonzero multiplicity in `s` has nonzero multiplicity in `t`,\n  but it does not imply that the multiplicity of `a` in `s` is less or equal than in `t`;\n  see `s \u2264 t` for this relation. -/\nprotected def subset (s t : multiset \u03b1) : Prop := \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 a \u2208 t\n\ninstance : has_subset (multiset \u03b1) := \u27e8multiset.subset\u27e9\n\n@[simp] theorem coe_subset {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2286 l\u2082 \u2194 l\u2081 \u2286 l\u2082 := iff.rfl\n\n@[simp] theorem subset.refl (s : multiset \u03b1) : s \u2286 s := \u03bb a h, h\n\ntheorem subset.trans {s t u : multiset \u03b1} : s \u2286 t \u2192 t \u2286 u \u2192 s \u2286 u :=\n\u03bb h\u2081 h\u2082 a m, h\u2082 (h\u2081 m)\n\ntheorem subset_iff {s t : multiset \u03b1} : s \u2286 t \u2194 (\u2200\u2983x\u2984, x \u2208 s \u2192 x \u2208 t) := iff.rfl\n\ntheorem mem_of_subset {s t : multiset \u03b1} {a : \u03b1} (h : s \u2286 t) : a \u2208 s \u2192 a \u2208 t := @h _\n\n@[simp] theorem zero_subset (s : multiset \u03b1) : 0 \u2286 s :=\n\u03bb a, (not_mem_nil a).elim\n\n@[simp] theorem cons_subset {a : \u03b1} {s t : multiset \u03b1} : (a :: s) \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t :=\nby simp [subset_iff, or_imp_distrib, forall_and_distrib]\n\ntheorem eq_zero_of_subset_zero {s : multiset \u03b1} (h : s \u2286 0) : s = 0 :=\neq_zero_of_forall_not_mem h\n\ntheorem subset_zero {s : multiset \u03b1} : s \u2286 0 \u2194 s = 0 :=\n\u27e8eq_zero_of_subset_zero, \u03bb xeq, xeq.symm \u25b8 subset.refl 0\u27e9\n\nend subset\n\n/- multiset order -/\n\n/-- `s \u2264 t` means that `s` is a sublist of `t` (up to permutation).\n  Equivalently, `s \u2264 t` means that `count a s \u2264 count a t` for all `a`. -/\nprotected def le (s t : multiset \u03b1) : Prop :=\nquotient.lift_on\u2082 s t (<+~) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  propext (p\u2082.subperm_left.trans p\u2081.subperm_right)\n\ninstance : partial_order (multiset \u03b1) :=\n{ le          := multiset.le,\n  le_refl     := by rintros \u27e8l\u27e9; exact subperm.refl _,\n  le_trans    := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 \u27e8l\u2083\u27e9; exact @subperm.trans _ _ _ _,\n  le_antisymm := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 h\u2081 h\u2082; exact quot.sound (subperm.antisymm h\u2081 h\u2082) }\n\ntheorem subset_of_le {s t : multiset \u03b1} : s \u2264 t \u2192 s \u2286 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subset_of_subperm\n\ntheorem mem_of_le {s t : multiset \u03b1} {a : \u03b1} (h : s \u2264 t) : a \u2208 s \u2192 a \u2208 t :=\nmem_of_subset (subset_of_le h)\n\n@[simp] theorem coe_le {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2264 l\u2082 \u2194 l\u2081 <+~ l\u2082 := iff.rfl\n\n@[elab_as_eliminator] theorem le_induction_on {C : multiset \u03b1 \u2192 multiset \u03b1 \u2192 Prop}\n  {s t : multiset \u03b1} (h : s \u2264 t)\n  (H : \u2200 {l\u2081 l\u2082 : list \u03b1}, l\u2081 <+ l\u2082 \u2192 C l\u2081 l\u2082) : C s t :=\nquotient.induction_on\u2082 s t (\u03bb l\u2081 l\u2082 \u27e8l, p, s\u27e9,\n  (show \u27e6l\u27e7 = \u27e6l\u2081\u27e7, from quot.sound p) \u25b8 H s) h\n\ntheorem zero_le (s : multiset \u03b1) : 0 \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ nil_sublist l\n\ntheorem le_zero {s : multiset \u03b1} : s \u2264 0 \u2194 s = 0 :=\n\u27e8\u03bb h, le_antisymm h (zero_le _), le_of_eq\u27e9\n\ntheorem lt_cons_self (s : multiset \u03b1) (a : \u03b1) : s < a :: s :=\nquot.induction_on s $ \u03bb l,\nsuffices l <+~ a :: l \u2227 (\u00acl ~ a :: l),\n  by simpa [lt_iff_le_and_ne],\n\u27e8subperm_of_sublist (sublist_cons _ _),\n \u03bb p, ne_of_lt (lt_succ_self (length l)) (perm_length p)\u27e9\n\n\ntheorem le_cons_self (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s :=\nle_of_lt $ lt_cons_self _ _\n\ntheorem cons_le_cons_iff (a : \u03b1) {s t : multiset \u03b1} : a :: s \u2264 a :: t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_cons a\n\ntheorem cons_le_cons (a : \u03b1) {s t : multiset \u03b1} : s \u2264 t \u2192 a :: s \u2264 a :: t :=\n(cons_le_cons_iff a).2\n\ntheorem le_cons_of_not_mem {a : \u03b1} {s t : multiset \u03b1} (m : a \u2209 s) : s \u2264 a :: t \u2194 s \u2264 t :=\nbegin\n  refine \u27e8_, \u03bb h, le_trans h $ le_cons_self _ _\u27e9,\n  suffices : \u2200 {t'} (_ : s \u2264 t') (_ : a \u2208 t'), a :: s \u2264 t',\n  { exact \u03bb h, (cons_le_cons_iff a).1 (this h (mem_cons_self _ _)) },\n  introv h, revert m, refine le_induction_on h _,\n  introv s m\u2081 m\u2082,\n  rcases mem_split m\u2082 with \u27e8r\u2081, r\u2082, rfl\u27e9,\n  exact perm_middle.subperm_left.2 ((subperm_cons _).2 $ subperm_of_sublist $\n    (sublist_or_mem_of_sublist s).resolve_right m\u2081)\nend\n\n/- cardinality -/\n\n/-- The cardinality of a multiset is the sum of the multiplicities\n  of all its elements, or simply the length of the underlying list. -/\ndef card (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s length $ \u03bb l\u2081 l\u2082, perm_length\n\n@[simp] theorem coe_card (l : list \u03b1) : card (l : multiset \u03b1) = length l := rfl\n\n@[simp] theorem card_zero : @card \u03b1 0 = 0 := rfl\n\n@[simp] theorem card_cons (a : \u03b1) (s : multiset \u03b1) : card (a :: s) = card s + 1 :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem card_singleton (a : \u03b1) : card (a::0) = 1 := by simp\n\ntheorem card_le_of_le {s t : multiset \u03b1} (h : s \u2264 t) : card s \u2264 card t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, length_le_of_sublist\n\ntheorem eq_of_le_of_card_le {s t : multiset \u03b1} (h : s \u2264 t) : card t \u2264 card s \u2192 s = t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 s h\u2082, congr_arg coe $ eq_of_sublist_of_length_le s h\u2082\n\ntheorem card_lt_of_lt {s t : multiset \u03b1} (h : s < t) : card s < card t :=\nlt_of_not_ge $ \u03bb h\u2082, ne_of_lt h $ eq_of_le_of_card_le (le_of_lt h) h\u2082\n\ntheorem lt_iff_cons_le {s t : multiset \u03b1} : s < t \u2194 \u2203 a, a :: s \u2264 t :=\n\u27e8quotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\n  subperm.exists_of_length_lt (le_of_lt h) (card_lt_of_lt h),\n\u03bb \u27e8a, h\u27e9, lt_of_lt_of_le (lt_cons_self _ _) h\u27e9\n\n@[simp] theorem card_eq_zero {s : multiset \u03b1} : card s = 0 \u2194 s = 0 :=\n\u27e8\u03bb h, (eq_of_le_of_card_le (zero_le _) (le_of_eq h)).symm, \u03bb e, by simp [e]\u27e9\n\ntheorem card_pos {s : multiset \u03b1} : 0 < card s \u2194 s \u2260 0 :=\npos_iff_ne_zero.trans $ not_congr card_eq_zero\n\ntheorem card_pos_iff_exists_mem {s : multiset \u03b1} : 0 < card s \u2194 \u2203 a, a \u2208 s :=\nquot.induction_on s $ \u03bb l, length_pos_iff_exists_mem\n\n@[elab_as_eliminator] def strong_induction_on {p : multiset \u03b1 \u2192 Sort*} :\n  \u2200 (s : multiset \u03b1), (\u2200 s, (\u2200t < s, p t) \u2192 p s) \u2192 p s\n| s := \u03bb ih, ih s $ \u03bb t h,\n  have card t < card s, from card_lt_of_lt h,\n  strong_induction_on t ih\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf card\u27e9]}\n\ntheorem strong_induction_eq {p : multiset \u03b1 \u2192 Sort*}\n  (s : multiset \u03b1) (H) : @strong_induction_on _ p s H =\n    H s (\u03bb t h, @strong_induction_on _ p t H) :=\nby rw [strong_induction_on]\n\n@[elab_as_eliminator] lemma case_strong_induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2080 : p 0) (h\u2081 : \u2200 a s, (\u2200t \u2264 s, p t) \u2192 p (a :: s)) : p s :=\nmultiset.strong_induction_on s $ assume s,\nmultiset.induction_on s (\u03bb _, h\u2080) $ \u03bb a s _ ih, h\u2081 _ _ $\n\u03bb t h, ih _ $ lt_of_le_of_lt h $ lt_cons_self _ _\n\n/- singleton -/\n@[simp] theorem singleton_eq_singleton (a : \u03b1) : singleton a = a::0 := rfl\n\n@[simp] theorem mem_singleton {a b : \u03b1} : b \u2208 a::0 \u2194 b = a := by simp\n\ntheorem mem_singleton_self (a : \u03b1) : a \u2208 (a::0 : multiset \u03b1) := mem_cons_self _ _\n\ntheorem singleton_inj {a b : \u03b1} : a::0 = b::0 \u2194 a = b := cons_inj_left _\n\n@[simp] theorem singleton_ne_zero (a : \u03b1) : a::0 \u2260 0 :=\nne_of_gt (lt_cons_self _ _)\n\n@[simp] theorem singleton_le {a : \u03b1} {s : multiset \u03b1} : a::0 \u2264 s \u2194 a \u2208 s :=\n\u27e8\u03bb h, mem_of_le h (mem_singleton_self _),\n \u03bb h, let \u27e8t, e\u27e9 := exists_cons_of_mem h in e.symm \u25b8 cons_le_cons _ (zero_le _)\u27e9\n\ntheorem card_eq_one {s : multiset \u03b1} : card s = 1 \u2194 \u2203 a, s = a::0 :=\n\u27e8quot.induction_on s $ \u03bb l h,\n  (list.length_eq_one.1 h).imp $ \u03bb a, congr_arg coe,\n \u03bb \u27e8a, e\u27e9, e.symm \u25b8 rfl\u27e9\n\n/- add -/\n\n/-- The sum of two multisets is the lift of the list append operation.\n  This adds the multiplicities of each element,\n  i.e. `count a (s + t) = count a s + count a t`. -/\nprotected def add (s\u2081 s\u2082 : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s\u2081 s\u2082 (\u03bb l\u2081 l\u2082, ((l\u2081 ++ l\u2082 : list \u03b1) : multiset \u03b1)) $\n  \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082, quot.sound $ perm_app p\u2081 p\u2082\n\ninstance : has_add (multiset \u03b1) := \u27e8multiset.add\u27e9\n\n@[simp] theorem coe_add (s t : list \u03b1) : (s + t : multiset \u03b1) = (s ++ t : list \u03b1) := rfl\n\nprotected theorem add_comm (s t : multiset \u03b1) : s + t = t + s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quot.sound perm_app_comm\n\nprotected theorem zero_add (s : multiset \u03b1) : 0 + s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\ntheorem singleton_add (a : \u03b1) (s : multiset \u03b1) : \u2191[a] + s = a::s := rfl\n\nprotected theorem add_le_add_left (s) {t u : multiset \u03b1} : s + t \u2264 s + u \u2194 t \u2264 u :=\nquotient.induction_on\u2083 s t u $ \u03bb l\u2081 l\u2082 l\u2083, subperm_app_left _\n\nprotected theorem add_left_cancel (s) {t u : multiset \u03b1} (h : s + t = s + u) : t = u :=\nle_antisymm ((multiset.add_le_add_left _).1 (le_of_eq h))\n  ((multiset.add_le_add_left _).1 (le_of_eq h.symm))\n\ninstance : ordered_cancel_comm_monoid (multiset \u03b1) :=\n{ zero                  := 0,\n  add                   := (+),\n  add_comm              := multiset.add_comm,\n  add_assoc             := \u03bb s\u2081 s\u2082 s\u2083, quotient.induction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb l\u2081 l\u2082 l\u2083,\n    congr_arg coe $ append_assoc l\u2081 l\u2082 l\u2083,\n  zero_add              := multiset.zero_add,\n  add_zero              := \u03bb s, by rw [multiset.add_comm, multiset.zero_add],\n  add_left_cancel       := multiset.add_left_cancel,\n  add_right_cancel      := \u03bb s\u2081 s\u2082 s\u2083 h, multiset.add_left_cancel s\u2082 $\n    by simpa [multiset.add_comm] using h,\n  add_le_add_left       := \u03bb s\u2081 s\u2082 h s\u2083, (multiset.add_le_add_left _).2 h,\n  le_of_add_le_add_left := \u03bb s\u2081 s\u2082 s\u2083, (multiset.add_le_add_left _).1,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem cons_add (a : \u03b1) (s t : multiset \u03b1) : a :: s + t = a :: (s + t) :=\nby rw [\u2190 singleton_add, \u2190 singleton_add, add_assoc]\n\n@[simp] theorem add_cons (a : \u03b1) (s t : multiset \u03b1) : s + a :: t = a :: (s + t) :=\nby rw [add_comm, cons_add, add_comm]\n\ntheorem le_add_right (s t : multiset \u03b1) : s \u2264 s + t :=\nby simpa using add_le_add_left (zero_le t) s\n\ntheorem le_add_left (s t : multiset \u03b1) : s \u2264 t + s :=\nby simpa using add_le_add_right (zero_le t) s\n\n@[simp] theorem card_add (s t : multiset \u03b1) : card (s + t) = card s + card t :=\nquotient.induction_on\u2082 s t length_append\n\nlemma card_smul (s : multiset \u03b1) (n : \u2115) :\n  (n \u2022 s).card = n * s.card :=\nby induction n; simp [succ_smul, *, nat.succ_mul]\n\n@[simp] theorem mem_add {a : \u03b1} {s t : multiset \u03b1} : a \u2208 s + t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, mem_append\n\ntheorem le_iff_exists_add {s t : multiset \u03b1} : s \u2264 t \u2194 \u2203 u, t = s + u :=\n\u27e8\u03bb h, le_induction_on h $ \u03bb l\u2081 l\u2082 s,\n  let \u27e8l, p\u27e9 := exists_perm_append_of_sublist s in \u27e8l, quot.sound p\u27e9,\n\u03bb\u27e8u, e\u27e9, e.symm \u25b8 le_add_right s u\u27e9\n\ninstance : canonically_ordered_monoid (multiset \u03b1) :=\n{ lt_of_add_lt_add_left := @lt_of_add_lt_add_left _ _,\n  le_iff_exists_add     := @le_iff_exists_add _,\n  bot                   := 0,\n  bot_le                := multiset.zero_le,\n  ..multiset.ordered_cancel_comm_monoid }\n\n/- repeat -/\n\n/-- `repeat a n` is the multiset containing only `a` with multiplicity `n`. -/\ndef repeat (a : \u03b1) (n : \u2115) : multiset \u03b1 := repeat a n\n\n@[simp] lemma repeat_zero (a : \u03b1) : repeat a 0 = 0 := rfl\n\n@[simp] lemma repeat_succ (a : \u03b1) (n) : repeat a (n+1) = a :: repeat a n := by simp [repeat]\n\n@[simp] lemma repeat_one (a : \u03b1) : repeat a 1 = a :: 0 := by simp\n\n@[simp] lemma card_repeat : \u2200 (a : \u03b1) n, card (repeat a n) = n := length_repeat\n\ntheorem eq_of_mem_repeat {a b : \u03b1} {n} : b \u2208 repeat a n \u2192 b = a := eq_of_mem_repeat\n\ntheorem eq_repeat' {a : \u03b1} {s : multiset \u03b1} : s = repeat a s.card \u2194 \u2200 b \u2208 s, b = a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  (perm_repeat.1 $ (quotient.exact h).symm).symm, congr_arg coe\u27e9 eq_repeat'\n\ntheorem eq_repeat_of_mem {a : \u03b1} {s : multiset \u03b1} : (\u2200 b \u2208 s, b = a) \u2192 s = repeat a s.card :=\neq_repeat'.2\n\ntheorem eq_repeat {a : \u03b1} {n} {s : multiset \u03b1} : s = repeat a n \u2194 card s = n \u2227 \u2200 b \u2208 s, b = a :=\n\u27e8\u03bb h, h.symm \u25b8 \u27e8card_repeat _ _, \u03bb b, eq_of_mem_repeat\u27e9,\n \u03bb \u27e8e, al\u27e9, e \u25b8 eq_repeat_of_mem al\u27e9\n\ntheorem repeat_subset_singleton : \u2200 (a : \u03b1) n, repeat a n \u2286 a::0 := repeat_subset_singleton\n\ntheorem repeat_le_coe {a : \u03b1} {n} {l : list \u03b1} : repeat a n \u2264 l \u2194 list.repeat a n <+ l :=\n\u27e8\u03bb \u27e8l', p, s\u27e9, (perm_repeat.1 p.symm).symm \u25b8 s, subperm_of_sublist\u27e9\n\n/- range -/\n\n/-- `range n` is the multiset lifted from the list `range n`,\n  that is, the set `{0, 1, ..., n-1}`. -/\ndef range (n : \u2115) : multiset \u2115 := range n\n\n@[simp] theorem range_zero : range 0 = 0 := rfl\n\n@[simp] theorem range_succ (n : \u2115) : range (succ n) = n :: range n :=\nby rw [range, range_concat, \u2190 coe_add, add_comm]; refl\n\n@[simp] theorem card_range (n : \u2115) : card (range n) = n := length_range _\n\ntheorem range_subset {m n : \u2115} : range m \u2286 range n \u2194 m \u2264 n := range_subset\n\n@[simp] theorem mem_range {m n : \u2115} : m \u2208 range n \u2194 m < n := mem_range\n\n@[simp] theorem not_mem_range_self {n : \u2115} : n \u2209 range n := not_mem_range_self\n\n/- erase -/\nsection erase\nvariables [decidable_eq \u03b1] {s t : multiset \u03b1} {a b : \u03b1}\n\n/-- `erase s a` is the multiset that subtracts 1 from the\n  multiplicity of `a`. -/\ndef erase (s : multiset \u03b1) (a : \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase a : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (erase_perm_erase a p))\n\n@[simp] theorem coe_erase (l : list \u03b1) (a : \u03b1) :\n  erase (l : multiset \u03b1) a = l.erase a := rfl\n\n@[simp] theorem erase_zero (a : \u03b1) : (0 : multiset \u03b1).erase a = 0 := rfl\n\n@[simp] theorem erase_cons_head (a : \u03b1) (s : multiset \u03b1) : (a :: s).erase a = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_head a l\n\n@[simp] theorem erase_cons_tail {a b : \u03b1} (s : multiset \u03b1) (h : b \u2260 a) : (b::s).erase a = b :: s.erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_tail l h\n\n@[simp] theorem erase_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 s.erase a = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ erase_of_not_mem h\n\n@[simp] theorem cons_erase {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 a :: s.erase a = s :=\nquot.induction_on s $ \u03bb l h, quot.sound (perm_erase h).symm\n\ntheorem le_cons_erase (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s.erase a :=\nif h : a \u2208 s then le_of_eq (cons_erase h).symm\nelse by rw erase_of_not_mem h; apply le_cons_self\n\n@[simp] theorem card_erase_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 card (s.erase a) = pred (card s) :=\nquot.induction_on s $ \u03bb l, length_erase_of_mem\n\ntheorem erase_add_left_pos {a : \u03b1} {s : multiset \u03b1} (t) : a \u2208 s \u2192 (s + t).erase a = s.erase a + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_left l\u2082 h\n\ntheorem erase_add_right_pos {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2208 t) : (s + t).erase a = s + t.erase a :=\nby rw [add_comm, erase_add_left_pos s h, add_comm]\n\ntheorem erase_add_right_neg {a : \u03b1} {s : multiset \u03b1} (t) : a \u2209 s \u2192 (s + t).erase a = s + t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_right l\u2082 h\n\ntheorem erase_add_left_neg {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2209 t) : (s + t).erase a = s.erase a + t :=\nby rw [add_comm, erase_add_right_neg s h, add_comm]\n\ntheorem erase_le (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist (erase_sublist a l)\n\n@[simp] theorem erase_lt {a : \u03b1} {s : multiset \u03b1} : s.erase a < s \u2194 a \u2208 s :=\n\u27e8\u03bb h, not_imp_comm.1 erase_of_not_mem (ne_of_lt h),\n \u03bb h, by simpa [h] using lt_cons_self (s.erase a) a\u27e9\n\ntheorem erase_subset (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2286 s :=\nsubset_of_le (erase_le a s)\n\ntheorem mem_erase_of_ne {a b : \u03b1} {s : multiset \u03b1} (ab : a \u2260 b) : a \u2208 s.erase b \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, list.mem_erase_of_ne ab\n\ntheorem mem_of_mem_erase {a b : \u03b1} {s : multiset \u03b1} : a \u2208 s.erase b \u2192 a \u2208 s :=\nmem_of_subset (erase_subset _ _)\n\ntheorem erase_comm (s : multiset \u03b1) (a b : \u03b1) : (s.erase a).erase b = (s.erase b).erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ l.erase_comm a b\n\ntheorem erase_le_erase {s t : multiset \u03b1} (a : \u03b1) (h : s \u2264 t) : s.erase a \u2264 t.erase a :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist (erase_sublist_erase _ h)\n\ntheorem erase_le_iff_le_cons {s t : multiset \u03b1} {a : \u03b1} : s.erase a \u2264 t \u2194 s \u2264 a :: t :=\n\u27e8\u03bb h, le_trans (le_cons_erase _ _) (cons_le_cons _ h),\n \u03bb h, if m : a \u2208 s\n  then by rw \u2190 cons_erase m at h; exact (cons_le_cons_iff _).1 h\n  else le_trans (erase_le _ _) ((le_cons_of_not_mem m).1 h)\u27e9\n\nend erase\n\n@[simp] theorem coe_reverse (l : list \u03b1) : (reverse l : multiset \u03b1) = l :=\nquot.sound $ reverse_perm _\n\n/- map -/\n\n/-- `map f s` is the lift of the list `map` operation. The multiplicity\n  of `b` in `map f s` is the number of `a \u2208 s` (counting multiplicity)\n  such that `f a = b`. -/\ndef map (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l : list \u03b1, (l.map f : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (perm_map f p))\n\n@[simp] theorem coe_map (f : \u03b1 \u2192 \u03b2) (l : list \u03b1) : map f \u2191l = l.map f := rfl\n\n@[simp] theorem map_zero (f : \u03b1 \u2192 \u03b2) : map f 0 = 0 := rfl\n\n@[simp] theorem map_cons (f : \u03b1 \u2192 \u03b2) (a s) : map f (a::s) = f a :: map f s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] lemma map_singleton (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : ({a} : multiset \u03b1).map f = {f a} := rfl\n\n@[simp] theorem map_add (f : \u03b1 \u2192 \u03b2) (s t) : map f (s + t) = map f s + map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ map_append _ _ _\n\ninstance (f : \u03b1 \u2192 \u03b2) : is_add_monoid_hom (map f) :=\nby refine_struct {..}; simp\n\n@[simp] theorem mem_map {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : multiset \u03b1} :\n  b \u2208 map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = b :=\nquot.induction_on s $ \u03bb l, mem_map\n\n@[simp] theorem card_map (f : \u03b1 \u2192 \u03b2) (s) : card (map f s) = card s :=\nquot.induction_on s $ \u03bb l, length_map _ _\n\ntheorem mem_map_of_mem (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : f a \u2208 map f s :=\nmem_map.2 \u27e8_, h, rfl\u27e9\n\n@[simp] theorem mem_map_of_inj {f : \u03b1 \u2192 \u03b2} (H : function.injective f) {a : \u03b1} {s : multiset \u03b1} :\n  f a \u2208 map f s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_map_of_inj H\n\n@[simp] theorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : map g (map f s) = map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ list.map_map _ _ _\n\n@[simp] theorem map_id (s : multiset \u03b1) : map id s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_id _\n\n@[simp] lemma map_id' (s : multiset \u03b1) : map (\u03bbx, x) s = s := map_id s\n\n@[simp] theorem map_const (s : multiset \u03b1) (b : \u03b2) : map (function.const \u03b1 b) s = repeat b s.card :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_const _ _\n\n@[congr] theorem map_congr {f g : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200 x \u2208 s, f x = g x) \u2192 map f s = map g s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_congr H\n\nlemma map_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : map f m == map f' m :=\nbegin subst h, simp at hf, simp [map_congr hf] end\n\ntheorem eq_of_mem_map_const {b\u2081 b\u2082 : \u03b2} {l : list \u03b1} (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) l) : b\u2081 = b\u2082 :=\neq_of_mem_repeat $ by rwa map_const at h\n\n@[simp] theorem map_le_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (h : s \u2264 t) : map f s \u2264 map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ map_sublist_map f h\n\n@[simp] theorem map_subset_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (H : s \u2286 t) : map f s \u2286 map f t :=\n\u03bb b m, let \u27e8a, h, e\u27e9 := mem_map.1 m in mem_map.2 \u27e8a, H h, e\u27e9\n\n/- fold -/\n\n/-- `foldl f H b s` is the lift of the list operation `foldl f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is right-commutative,\n  that is, `f (f b a\u2081) a\u2082 = f (f b a\u2082) a\u2081`. -/\ndef foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldl f b l)\n  (\u03bb l\u2081 l\u2082 p, foldl_eq_of_perm H p b)\n\n@[simp] theorem foldl_zero (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b) : foldl f H b 0 = b := rfl\n\n@[simp] theorem foldl_cons (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b a s) : foldl f H b (a :: s) = foldl f H (f b a) s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldl_add (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b s t) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldl_append _ _ _ _\n\n/-- `foldr f H b s` is the lift of the list operation `foldr f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is left-commutative,\n  that is, `f a\u2081 (f a\u2082 b) = f a\u2082 (f a\u2081 b)`. -/\ndef foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldr f b l)\n  (\u03bb l\u2081 l\u2082 p, foldr_eq_of_perm H p b)\n\n@[simp] theorem foldr_zero (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b) : foldr f H b 0 = b := rfl\n\n@[simp] theorem foldr_cons (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b a s) : foldr f H b (a :: s) = f a (foldr f H b s) :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldr_add (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b s t) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldr_append _ _ _ _\n\n@[simp] theorem coe_foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldr f b := rfl\n\n@[simp] theorem coe_foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldl f H b l = l.foldl f b := rfl\n\ntheorem coe_foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldl (\u03bb x y, f y x) b :=\n(congr_arg (foldr f H b) (coe_reverse l)).symm.trans $ foldr_reverse _ _ _\n\ntheorem foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldr f H b s = foldl (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\nquot.induction_on s $ \u03bb l, coe_foldr_swap _ _ _ _\n\ntheorem foldl_swap (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldl f H b s = foldr (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\n(foldr_swap _ _ _ _).symm\n\n/-- Product of a multiset given a commutative monoid structure on `\u03b1`.\n  `prod {a, b, c} = a * b * c` -/\ndef prod [comm_monoid \u03b1] : multiset \u03b1 \u2192 \u03b1 :=\nfoldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1\nattribute [to_additive multiset.sum._proof_1] prod._proof_1\nattribute [to_additive multiset.sum] prod\n\n@[to_additive multiset.sum_eq_foldr]\ntheorem prod_eq_foldr [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1 s := rfl\n\n@[to_additive multiset.sum_eq_foldl]\ntheorem prod_eq_foldl [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldl (*) (\u03bb x y z, by simp [mul_right_comm]) 1 s :=\n(foldr_swap _ _ _ _).trans (by simp [mul_comm])\n\n@[simp, to_additive multiset.coe_sum]\ntheorem coe_prod [comm_monoid \u03b1] (l : list \u03b1) : prod \u2191l = l.prod :=\nprod_eq_foldl _\n\n@[simp, to_additive multiset.sum_zero]\ntheorem prod_zero [comm_monoid \u03b1] : @prod \u03b1 _ 0 = 1 := rfl\n\n@[simp, to_additive multiset.sum_cons]\ntheorem prod_cons [comm_monoid \u03b1] (a : \u03b1) (s) : prod (a :: s) = a * prod s :=\nfoldr_cons _ _ _ _ _\n\n@[to_additive multiset.sum_singleton]\ntheorem prod_singleton [comm_monoid \u03b1] (a : \u03b1) : prod (a :: 0) = a := by simp\n\n@[simp, to_additive multiset.sum_add]\ntheorem prod_add [comm_monoid \u03b1] (s t : multiset \u03b1) : prod (s + t) = prod s * prod t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, by simp\n\ninstance sum.is_add_monoid_hom [add_comm_monoid \u03b1] : is_add_monoid_hom (sum : multiset \u03b1 \u2192 \u03b1) :=\nby refine_struct {..}; simp\n\nlemma prod_smul {\u03b1 : Type*} [comm_monoid \u03b1] (m : multiset \u03b1) :\n  \u2200n, (add_monoid.smul n m).prod = m.prod ^ n\n| 0       := rfl\n| (n + 1) :=\n  by rw [add_monoid.add_smul, add_monoid.one_smul, _root_.pow_add, _root_.pow_one, prod_add, prod_smul n]\n\n@[simp] theorem prod_repeat [comm_monoid \u03b1] (a : \u03b1) (n : \u2115) : prod (multiset.repeat a n) = a ^ n :=\nby simp [repeat, list.prod_repeat]\n@[simp] theorem sum_repeat [add_comm_monoid \u03b1] : \u2200 (a : \u03b1) (n : \u2115), sum (multiset.repeat a n) = n \u2022 a :=\n@prod_repeat (multiplicative \u03b1) _\nattribute [to_additive multiset.sum_repeat] prod_repeat\n\n@[simp] lemma prod_map_one [comm_monoid \u03b3] {m : multiset \u03b1} :\n  prod (m.map (\u03bba, (1 : \u03b3))) = (1 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\n@[simp] lemma sum_map_zero [add_comm_monoid \u03b3] {m : multiset \u03b1} :\n  sum (m.map (\u03bba, (0 : \u03b3))) = (0 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\nattribute [to_additive multiset.sum_map_zero] prod_map_one\n\n@[simp, to_additive multiset.sum_map_add]\nlemma prod_map_mul [comm_monoid \u03b3] {m : multiset \u03b1} {f g : \u03b1 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, f a * g a) = prod (m.map f) * prod (m.map g) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih]; cc)\n\nlemma prod_map_prod_map [comm_monoid \u03b3] (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, prod $ n.map $ \u03bbb, f a b) = prod (n.map $ \u03bbb, prod $ m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih])\n\nlemma sum_map_sum_map [add_comm_monoid \u03b3] : \u2200 (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3},\n  sum (m.map $ \u03bba, sum $ n.map $ \u03bbb, f a b) = sum (n.map $ \u03bbb, sum $ m.map $ \u03bba, f a b) :=\n@prod_map_prod_map _ _ (multiplicative \u03b3) _\nattribute [to_additive multiset.sum_map_sum_map] prod_map_prod_map\n\nlemma sum_map_mul_left [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, b * f a)) = b * sum (s.map f) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, mul_add])\n\nlemma sum_map_mul_right [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, f a * b)) = sum (s.map f) * b :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, add_mul])\n\nlemma prod_hom [comm_monoid \u03b1] [comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_monoid_hom f] (s : multiset \u03b1) :\n  (s.map f).prod = f s.prod :=\nmultiset.induction_on s (by simp [is_monoid_hom.map_one f])\n  (by simp [is_monoid_hom.map_mul f] {contextual := tt})\n\nlemma dvd_prod [comm_semiring \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 a \u2223 s.prod :=\nquotient.induction_on s (\u03bb l a h, by simpa using list.dvd_prod h) a\n\nlemma sum_hom [add_comm_monoid \u03b1] [add_comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) [is_add_monoid_hom f] (s : multiset \u03b1) :\n  (s.map f).sum = f s.sum :=\nmultiset.induction_on s (by simp [is_add_monoid_hom.map_zero f])\n  (by simp [is_add_monoid_hom.map_add f] {contextual := tt})\nattribute [to_additive multiset.sum_hom] multiset.prod_hom\n\nlemma le_sum_of_subadditive [add_comm_monoid \u03b1] [ordered_comm_monoid \u03b2]\n  (f : \u03b1 \u2192 \u03b2) (h_zero : f 0 = 0) (h_add : \u2200x y, f (x + y) \u2264 f x + f y) (s : multiset \u03b1) :\n  f s.sum \u2264 (s.map f).sum :=\nmultiset.induction_on s (le_of_eq h_zero) $\n  assume a s ih, by rw [sum_cons, map_cons, sum_cons];\n    from le_trans (h_add a s.sum) (add_le_add_left' ih)\n\nlemma abs_sum_le_sum_abs [discrete_linear_ordered_field \u03b1] {s : multiset \u03b1} :\n  abs s.sum \u2264 (s.map abs).sum :=\nle_sum_of_subadditive _ abs_zero abs_add s\n\n/- join -/\n\n/-- `join S`, where `S` is a multiset of multisets, is the lift of the list join\n  operation, that is, the union of all the sets.\n\n     join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2} -/\ndef join : multiset (multiset \u03b1) \u2192 multiset \u03b1 := sum\n\ntheorem coe_join : \u2200 L : list (list \u03b1),\n  join (L.map (@coe _ (multiset \u03b1) _) : multiset (multiset \u03b1)) = L.join\n| []       := rfl\n| (l :: L) := congr_arg (\u03bb s : multiset \u03b1, \u2191l + s) (coe_join L)\n\n@[simp] theorem join_zero : @join \u03b1 0 = 0 := rfl\n\n@[simp] theorem join_cons (s S) : @join \u03b1 (s :: S) = s + join S :=\nsum_cons _ _\n\n@[simp] theorem join_add (S T) : @join \u03b1 (S + T) = join S + join T :=\nsum_add _ _\n\n@[simp] theorem mem_join {a S} : a \u2208 @join \u03b1 S \u2194 \u2203 s \u2208 S, a \u2208 s :=\nmultiset.induction_on S (by simp) $\n  by simp [or_and_distrib_right, exists_or_distrib] {contextual := tt}\n\n@[simp] theorem card_join (S) : card (@join \u03b1 S) = sum (map card S) :=\nmultiset.induction_on S (by simp) (by simp)\n\n/- bind -/\n\n/-- `bind s f` is the monad bind operation, defined as `join (map f s)`.\n  It is the union of `f a` as `a` ranges over `s`. -/\ndef bind (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : multiset \u03b2 :=\njoin (map f s)\n\n@[simp] theorem coe_bind (l : list \u03b1) (f : \u03b1 \u2192 list \u03b2) :\n  @bind \u03b1 \u03b2 l (\u03bb a, f a) = l.bind f :=\nby rw [list.bind, \u2190 coe_join, list.map_map]; refl\n\n@[simp] theorem zero_bind (f : \u03b1 \u2192 multiset \u03b2) : bind 0 f = 0 := rfl\n\n@[simp] theorem cons_bind (a s) (f : \u03b1 \u2192 multiset \u03b2) : bind (a::s) f = f a + bind s f :=\nby simp [bind]\n\n@[simp] theorem add_bind (s t) (f : \u03b1 \u2192 multiset \u03b2) : bind (s + t) f = bind s f + bind t f :=\nby simp [bind]\n\n@[simp] theorem bind_zero (s : multiset \u03b1) : bind s (\u03bba, 0 : \u03b1 \u2192 multiset \u03b2) = 0 :=\nby simp [bind, -map_const, join]\n\n@[simp] theorem bind_add (s : multiset \u03b1) (f g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a + g a) = bind s f + bind s g :=\nby simp [bind, join]\n\n@[simp] theorem bind_cons (s : multiset \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a :: g a) = map f s + bind s g :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\n@[simp] theorem mem_bind {b s} {f : \u03b1 \u2192 multiset \u03b2} : b \u2208 bind s f \u2194 \u2203 a \u2208 s, b \u2208 f a :=\nby simp [bind]; simp [-exists_and_distrib_right, exists_and_distrib_right.symm];\n   rw exists_swap; simp [and_assoc]\n\n@[simp] theorem card_bind (s) (f : \u03b1 \u2192 multiset \u03b2) : card (bind s f) = sum (map (card \u2218 f) s) :=\nby simp [bind]\n\nlemma bind_congr {f g : \u03b1 \u2192 multiset \u03b2} {m : multiset \u03b1} : (\u2200a\u2208m, f a = g a) \u2192 bind m f = bind m g :=\nby simp [bind] {contextual := tt}\n\nlemma bind_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {f' : \u03b1 \u2192 multiset \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : bind m f == bind m f' :=\nbegin subst h, simp at hf, simp [bind_congr hf] end\n\nlemma map_bind (m : multiset \u03b1) (n : \u03b1 \u2192 multiset \u03b2) (f : \u03b2 \u2192 \u03b3) :\n  map f (bind m n) = bind m (\u03bba, map f (n a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map (m : multiset \u03b1) (n : \u03b2 \u2192 multiset \u03b3) (f : \u03b1 \u2192 \u03b2) :\n  bind (map f m) n = bind m (\u03bba, n (f a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_assoc {s : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b2 \u2192 multiset \u03b3} :\n  (s.bind f).bind g = s.bind (\u03bba, (f a).bind g) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma bind_bind (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 multiset \u03b3} :\n  (bind m $ \u03bba, bind n $ \u03bbb, f a b) = (bind n $ \u03bbb, bind m $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map_comm (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  (bind m $ \u03bba, n.map $ \u03bbb, f a b) = (bind n $ \u03bbb, m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\n@[simp, to_additive multiset.sum_bind]\nlemma prod_bind [comm_monoid \u03b2] (s : multiset \u03b1) (t : \u03b1 \u2192 multiset \u03b2) :\n  prod (bind s t) = prod (s.map $ \u03bba, prod (t a)) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, cons_bind])\n\n/- product -/\n\n/-- The multiplicity of `(a, b)` in `product s t` is\n  the product of the multiplicity of `a` in `s` and `b` in `t`. -/\ndef product (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u00d7 \u03b2) :=\ns.bind $ \u03bb a, t.map $ prod.mk a\n\n@[simp] theorem coe_product (l\u2081 : list \u03b1) (l\u2082 : list \u03b2) :\n  @product \u03b1 \u03b2 l\u2081 l\u2082 = l\u2081.product l\u2082 :=\nby rw [product, list.product, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_product (t) : @product \u03b1 \u03b2 0 t = 0 := rfl\n\n@[simp] theorem cons_product (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b2) :\n  product (a :: s) t = map (prod.mk a) t + product s t :=\nby simp [product]\n\n@[simp] theorem product_singleton (a : \u03b1) (b : \u03b2) : product (a::0) (b::0) = (a,b)::0 := rfl\n\n@[simp] theorem add_product (s t : multiset \u03b1) (u : multiset \u03b2) :\n  product (s + t) u = product s u + product t u :=\nby simp [product]\n\n@[simp] theorem product_add (s : multiset \u03b1) : \u2200 t u : multiset \u03b2,\n  product s (t + u) = product s t + product s u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_product, IH]; simp\n\n@[simp] theorem mem_product {s t} : \u2200 {p : \u03b1 \u00d7 \u03b2}, p \u2208 @product \u03b1 \u03b2 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t\n| (a, b) := by simp [product, and.left_comm]\n\n@[simp] theorem card_product (s : multiset \u03b1) (t : multiset \u03b2) : card (product s t) = card s * card t :=\nby simp [product, repeat, (\u2218), mul_comm]\n\n/- sigma -/\nsection\nvariable {\u03c3 : \u03b1 \u2192 Type*}\n\n/-- `sigma s t` is the dependent version of `product`. It is the sum of\n  `(a, b)` as `a` ranges over `s` and `b` ranges over `t a`. -/\nprotected def sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) : multiset (\u03a3 a, \u03c3 a) :=\ns.bind $ \u03bb a, (t a).map $ sigma.mk a\n\n@[simp] theorem coe_sigma (l\u2081 : list \u03b1) (l\u2082 : \u03a0 a, list (\u03c3 a)) :\n  @multiset.sigma \u03b1 \u03c3 l\u2081 (\u03bb a, l\u2082 a) = l\u2081.sigma l\u2082 :=\nby rw [multiset.sigma, list.sigma, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_sigma (t) : @multiset.sigma \u03b1 \u03c3 0 t = 0 := rfl\n\n@[simp] theorem cons_sigma (a : \u03b1) (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  (a :: s).sigma t = map (sigma.mk a) (t a) + s.sigma t :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_singleton (a : \u03b1) (b : \u03b1 \u2192 \u03b2) :\n  (a::0).sigma (\u03bb a, b a::0) = \u27e8a, b a\u27e9::0 := rfl\n\n@[simp] theorem add_sigma (s t : multiset \u03b1) (u : \u03a0 a, multiset (\u03c3 a)) :\n  (s + t).sigma u = s.sigma u + t.sigma u :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_add (s : multiset \u03b1) : \u2200 t u : \u03a0 a, multiset (\u03c3 a),\n  s.sigma (\u03bb a, t a + u a) = s.sigma t + s.sigma u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_sigma, IH]; simp\n\n@[simp] theorem mem_sigma {s t} : \u2200 {p : \u03a3 a, \u03c3 a},\n  p \u2208 @multiset.sigma \u03b1 \u03c3 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t p.1\n| \u27e8a, b\u27e9 := by simp [multiset.sigma, and_assoc, and.left_comm]\n\n@[simp] theorem card_sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  card (s.sigma t) = sum (map (\u03bb a, card (t a)) s) :=\nby simp [multiset.sigma, (\u2218)]\n\nend\n\n/- map for partial functions -/\n\n/-- Lift of the list `pmap` operation. Map a partial function `f` over a multiset\n  `s` whose elements are all in the domain of `f`. -/\ndef pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (s : multiset \u03b1) : (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2 :=\nquot.rec_on s (\u03bb l H, \u2191(pmap f l H)) $ \u03bb l\u2081 l\u2082 (pp : l\u2081 ~ l\u2082),\nfunext $ \u03bb (H\u2082 : \u2200 a \u2208 l\u2082, p a),\nhave H\u2081 : \u2200 a \u2208 l\u2081, p a, from \u03bb a h, H\u2082 a ((mem_of_perm pp).1 h),\nhave \u2200 {s\u2082 e H}, @eq.rec (multiset \u03b1) l\u2081\n  (\u03bb s, (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2) (\u03bb _, \u2191(pmap f l\u2081 H\u2081))\n  s\u2082 e H = \u2191(pmap f l\u2081 H\u2081), by intros s\u2082 e _; subst e,\nthis.trans $ quot.sound $ perm_pmap f pp\n\n@[simp] theorem coe_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (l : list \u03b1) (H : \u2200 a \u2208 l, p a) : pmap f l H = l.pmap f H := rfl\n\n@[simp] lemma pmap_zero {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (h : \u2200a\u2208(0:multiset \u03b1), p a) :\n  pmap f 0 h = 0 := rfl\n\n@[simp] lemma pmap_cons {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (a : \u03b1) (m : multiset \u03b1) :\n  \u2200(h : \u2200b\u2208a::m, p b), pmap f (a :: m) h =\n    f a (h a (mem_cons_self a m)) :: pmap f m (\u03bba ha, h a $ mem_cons_of_mem ha) :=\nquotient.induction_on m $ assume l h, rfl\n\n/-- \"Attach\" a proof that `a \u2208 s` to each element `a` in `s` to produce\n  a multiset on `{x // x \u2208 s}`. -/\ndef attach (s : multiset \u03b1) : multiset {x // x \u2208 s} := pmap subtype.mk s (\u03bb a, id)\n\n@[simp] theorem coe_attach (l : list \u03b1) :\n @eq (multiset {x // x \u2208 l}) (@attach \u03b1 l) l.attach := rfl\n\ntheorem pmap_eq_map (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  \u2200 H, @pmap _ _ p (\u03bb a _, f a) s H = map f s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map p f l H\n\ntheorem pmap_congr {p q : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {g : \u03a0 a, q a \u2192 \u03b2}\n  (s : multiset \u03b1) {H\u2081 H\u2082} (h : \u2200 a h\u2081 h\u2082, f a h\u2081 = g a h\u2082) :\n  pmap f s H\u2081 = pmap g s H\u2082 :=\nquot.induction_on s (\u03bb l H\u2081 H\u2082, congr_arg coe $ pmap_congr l h) H\u2081 H\u2082\n\ntheorem map_pmap {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3) (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, map g (pmap f s H) = pmap (\u03bb a h, g (f a h)) s H :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_pmap g f l H\n\ntheorem pmap_eq_map_attach {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, pmap f s H = s.attach.map (\u03bb x, f x.1 (H _ x.2)) :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map_attach f l H\n\ntheorem attach_map_val (s : multiset \u03b1) : s.attach.map subtype.val = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ attach_map_val l\n\n@[simp] theorem mem_attach (s : multiset \u03b1) : \u2200 x, x \u2208 s.attach :=\nquot.induction_on s $ \u03bb l, mem_attach _\n\n@[simp] theorem mem_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2}\n  {s H b} : b \u2208 pmap f s H \u2194 \u2203 a (h : a \u2208 s), f a (H a h) = b :=\nquot.induction_on s (\u03bb l H, mem_pmap) H\n\n@[simp] theorem card_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s H) : card (pmap f s H) = card s :=\nquot.induction_on s (\u03bb l H, length_pmap) H\n\n@[simp] theorem card_attach {m : multiset \u03b1} : card (attach m) = card m := card_pmap _ _ _\n\n@[simp] lemma attach_zero : (0 : multiset \u03b1).attach = 0 := rfl\n\nlemma attach_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).attach = \u27e8a, mem_cons_self a m\u27e9 :: (m.attach.map $ \u03bbp, \u27e8p.1, mem_cons_of_mem p.2\u27e9) :=\nquotient.induction_on m $ assume l, congr_arg coe $ congr_arg (list.cons _) $\n  by rw [list.map_pmap]; exact list.pmap_congr _ (assume a' h\u2081 h\u2082, subtype.eq rfl)\n\nsection decidable_pi_exists\nvariables {m : multiset \u03b1}\n\nprotected def decidable_forall_multiset {p : \u03b1 \u2192 Prop} [hp : \u2200a, decidable (p a)] :\n  decidable (\u2200a\u2208m, p a) :=\nquotient.rec_on_subsingleton m (\u03bbl, decidable_of_iff (\u2200a\u2208l, p a) $ by simp)\n\ninstance decidable_dforall_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2200a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_forall_multiset {a // a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (assume h a ha, h \u27e8a, ha\u27e9 (mem_attach _ _)) (assume h \u27e8a, ha\u27e9 _, h _ _))\n\n/-- decidable equality for functions whose domain is bounded by multisets -/\ninstance decidable_eq_pi_multiset {\u03b2 : \u03b1 \u2192 Type*} [h : \u2200a, decidable_eq (\u03b2 a)] :\n  decidable_eq (\u03a0a\u2208m, \u03b2 a) :=\nassume f g, decidable_of_iff (\u2200a (h : a \u2208 m), f a h = g a h) (by simp [function.funext_iff])\n\ndef decidable_exists_multiset {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  decidable (\u2203 x \u2208 m, p x) :=\nquotient.rec_on_subsingleton m list.decidable_exists_mem\n\ninstance decidable_dexists_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2203a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_exists_multiset {a // a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (\u03bb \u27e8\u27e8a, ha\u2081\u27e9, _, ha\u2082\u27e9, \u27e8a, ha\u2081, ha\u2082\u27e9)\n    (\u03bb \u27e8a, ha\u2081, ha\u2082\u27e9, \u27e8\u27e8a, ha\u2081\u27e9, mem_attach _ _, ha\u2082\u27e9))\n\nend decidable_pi_exists\n\n/- subtraction -/\nsection\nvariables [decidable_eq \u03b1] {s t u : multiset \u03b1} {a b : \u03b1}\n\n/-- `s - t` is the multiset such that\n  `count a (s - t) = count a s - count a t` for all `a`. -/\nprotected def sub (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.diff l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_diff_right w\u2081 p\u2082 \u25b8 perm_diff_left _ p\u2081\n\ninstance : has_sub (multiset \u03b1) := \u27e8multiset.sub\u27e9\n\n@[simp] theorem coe_sub (s t : list \u03b1) : (s - t : multiset \u03b1) = (s.diff t : list \u03b1) := rfl\n\ntheorem sub_eq_fold_erase (s t : multiset \u03b1) : s - t = foldl erase erase_comm s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nshow \u2191(l\u2081.diff l\u2082) = foldl erase erase_comm \u2191l\u2081 \u2191l\u2082,\nby rw diff_eq_foldl l\u2081 l\u2082; exact foldl_hom _ _ _ _ (\u03bb x y, rfl) _\n\n@[simp] theorem sub_zero (s : multiset \u03b1) : s - 0 = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem sub_cons (a : \u03b1) (s t : multiset \u03b1) : s - a::t = s.erase a - t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ diff_cons _ _ _\n\ntheorem add_sub_of_le (h : s \u2264 t) : s + (t - s) = t :=\nbegin\n  revert t,\n  refine multiset.induction_on s (by simp) (\u03bb a s IH t h, _),\n  have := cons_erase (mem_of_le h (mem_cons_self _ _)),\n  rw [cons_add, sub_cons, IH, this],\n  exact (cons_le_cons_iff a).1 (this.symm \u25b8 h)\nend\n\ntheorem sub_add' : s - (t + u) = s - t - u :=\nquotient.induction_on\u2083 s t u $\n\u03bb l\u2081 l\u2082 l\u2083, congr_arg coe $ diff_append _ _ _\n\ntheorem sub_add_cancel (h : t \u2264 s) : s - t + t = s :=\nby rw [add_comm, add_sub_of_le h]\n\n@[simp] theorem add_sub_cancel_left (s : multiset \u03b1) : \u2200 t, s + t - s = t :=\nmultiset.induction_on s (by simp)\n  (\u03bb a s IH t, by rw [cons_add, sub_cons, erase_cons_head, IH])\n\n@[simp] theorem add_sub_cancel (s t : multiset \u03b1) : s + t - t = s :=\nby rw [add_comm, add_sub_cancel_left]\n\ntheorem sub_le_sub_right (h : s \u2264 t) (u) : s - u \u2264 t - u :=\nby revert s t h; exact\nmultiset.induction_on u (by simp {contextual := tt})\n  (\u03bb a u IH s t h, by simp [IH, erase_le_erase a h])\n\ntheorem sub_le_sub_left (h : s \u2264 t) : \u2200 u, u - t \u2264 u - s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, begin\n  induction h with l\u2081 l\u2082 a s IH l\u2081 l\u2082 a s IH; intro u,\n  { refl },\n  { rw [\u2190 cons_coe, sub_cons],\n    exact le_trans (sub_le_sub_right (erase_le _ _) _) (IH u) },\n  { rw [\u2190 cons_coe, sub_cons, \u2190 cons_coe, sub_cons],\n    exact IH _ }\nend\n\ntheorem sub_le_iff_le_add : s - t \u2264 u \u2194 s \u2264 u + t :=\nby revert s; exact\nmultiset.induction_on t (by simp)\n  (\u03bb a t IH s, by simp [IH, erase_le_iff_le_cons])\n\ntheorem le_sub_add (s t : multiset \u03b1) : s \u2264 s - t + t :=\nsub_le_iff_le_add.1 (le_refl _)\n\ntheorem sub_le_self (s t : multiset \u03b1) : s - t \u2264 s :=\nsub_le_iff_le_add.2 (le_add_right _ _)\n\n@[simp] theorem card_sub {s t : multiset \u03b1} (h : t \u2264 s) : card (s - t) = card s - card t :=\n(nat.sub_eq_of_eq_add $ by rw [add_comm, \u2190 card_add, sub_add_cancel h]).symm\n\n/- union -/\n\n/-- `s \u222a t` is the lattice join operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u222a t` is the maximum\n  of the multiplicities in `s` and `t`. -/\ndef union (s t : multiset \u03b1) : multiset \u03b1 := s - t + t\n\ninstance : has_union (multiset \u03b1) := \u27e8union\u27e9\n\ntheorem union_def (s t : multiset \u03b1) : s \u222a t = s - t + t := rfl\n\ntheorem le_union_left (s t : multiset \u03b1) : s \u2264 s \u222a t := le_sub_add _ _\n\ntheorem le_union_right (s t : multiset \u03b1) : t \u2264 s \u222a t := le_add_left _ _\n\ntheorem eq_union_left : t \u2264 s \u2192 s \u222a t = s := sub_add_cancel\n\ntheorem union_le_union_right (h : s \u2264 t) (u) : s \u222a u \u2264 t \u222a u :=\nadd_le_add_right (sub_le_sub_right h _) u\n\ntheorem union_le (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u :=\nby rw \u2190 eq_union_left h\u2082; exact union_le_union_right h\u2081 t\n\n@[simp] theorem mem_union : a \u2208 s \u222a t \u2194 a \u2208 s \u2228 a \u2208 t :=\n\u27e8\u03bb h, (mem_add.1 h).imp_left (mem_of_le $ sub_le_self _ _),\n or.rec (mem_of_le $ le_union_left _ _) (mem_of_le $ le_union_right _ _)\u27e9\n\n@[simp] theorem map_union [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : function.injective f) {s t : multiset \u03b1} :\n  map f (s \u222a t) = map f s \u222a map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\ncongr_arg coe (by rw [list.map_append f, list.map_diff finj])\n\n/- inter -/\n\n/-- `s \u2229 t` is the lattice meet operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u2229 t` is the minimum\n  of the multiplicities in `s` and `t`. -/\ndef inter (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.bag_inter l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_bag_inter_right w\u2081 p\u2082 \u25b8 perm_bag_inter_left _ p\u2081\n\ninstance : has_inter (multiset \u03b1) := \u27e8inter\u27e9\n\n@[simp] theorem inter_zero (s : multiset \u03b1) : s \u2229 0 = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.bag_inter_nil\n\n@[simp] theorem zero_inter (s : multiset \u03b1) : 0 \u2229 s = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.nil_bag_inter\n\n@[simp] theorem cons_inter_of_pos {a} (s : multiset \u03b1) {t} :\n  a \u2208 t \u2192 (a :: s) \u2229 t = a :: s \u2229 t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_pos _ h\n\n@[simp] theorem cons_inter_of_neg {a} (s : multiset \u03b1) {t} :\n  a \u2209 t \u2192 (a :: s) \u2229 t = s \u2229 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_neg _ h\n\ntheorem inter_le_left (s t : multiset \u03b1) : s \u2229 t \u2264 s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ bag_inter_sublist_left _ _\n\ntheorem inter_le_right (s : multiset \u03b1) : \u2200 t, s \u2229 t \u2264 t :=\nmultiset.induction_on s (\u03bb t, (zero_inter t).symm \u25b8 zero_le _) $\n\u03bb a s IH t, if h : a \u2208 t\n  then by simpa [h] using cons_le_cons a (IH (t.erase a))\n  else by simp [h, IH]\n\ntheorem le_inter (h\u2081 : s \u2264 t) (h\u2082 : s \u2264 u) : s \u2264 t \u2229 u :=\nbegin\n  revert s u, refine multiset.induction_on t _ (\u03bb a t IH, _); intros,\n  { simp [h\u2081] },\n  by_cases a \u2208 u,\n  { rw [cons_inter_of_pos _ h, \u2190 erase_le_iff_le_cons],\n    exact IH (erase_le_iff_le_cons.2 h\u2081) (erase_le_erase _ h\u2082) },\n  { rw cons_inter_of_neg _ h,\n    exact IH ((le_cons_of_not_mem $ mt (mem_of_le h\u2082) h).1 h\u2081) h\u2082 }\nend\n\n@[simp] theorem mem_inter : a \u2208 s \u2229 t \u2194 a \u2208 s \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (inter_le_left _ _) h, mem_of_le (inter_le_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, by rw [\u2190 cons_erase h\u2081, cons_inter_of_pos _ h\u2082]; apply mem_cons_self\u27e9\n\ninstance : lattice (multiset \u03b1) :=\n{ sup          := (\u222a),\n  sup_le       := @union_le _ _,\n  le_sup_left  := le_union_left,\n  le_sup_right := le_union_right,\n  inf          := (\u2229),\n  le_inf       := @le_inter _ _,\n  inf_le_left  := inter_le_left,\n  inf_le_right := inter_le_right,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem sup_eq_union (s t : multiset \u03b1) : s \u2294 t = s \u222a t := rfl\n@[simp] theorem inf_eq_inter (s t : multiset \u03b1) : s \u2293 t = s \u2229 t := rfl\n\n@[simp] theorem le_inter_iff : s \u2264 t \u2229 u \u2194 s \u2264 t \u2227 s \u2264 u := le_inf_iff\n@[simp] theorem union_le_iff : s \u222a t \u2264 u \u2194 s \u2264 u \u2227 t \u2264 u := sup_le_iff\n\ninstance : semilattice_inf_bot (multiset \u03b1) :=\n{ bot := 0, bot_le := zero_le, ..multiset.lattice.lattice }\n\ntheorem union_comm (s t : multiset \u03b1) : s \u222a t = t \u222a s := sup_comm\ntheorem inter_comm (s t : multiset \u03b1) : s \u2229 t = t \u2229 s := inf_comm\n\ntheorem eq_union_right (h : s \u2264 t) : s \u222a t = t :=\nby rw [union_comm, eq_union_left h]\n\ntheorem union_le_union_left (h : s \u2264 t) (u) : u \u222a s \u2264 u \u222a t :=\nsup_le_sup_left h _\n\ntheorem union_le_add (s t : multiset \u03b1) : s \u222a t \u2264 s + t :=\nunion_le (le_add_right _ _) (le_add_left _ _)\n\ntheorem union_add_distrib (s t u : multiset \u03b1) : (s \u222a t) + u = (s + u) \u222a (t + u) :=\nby simpa [(\u222a), union, eq_comm] using show s + u - (t + u) = s - t,\nby rw [add_comm t, sub_add', add_sub_cancel]\n\ntheorem add_union_distrib (s t u : multiset \u03b1) : s + (t \u222a u) = (s + t) \u222a (s + u) :=\nby rw [add_comm, union_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_union_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u222a t) = (a :: s) \u222a (a :: t) :=\nby simpa using add_union_distrib (a::0) s t\n\ntheorem inter_add_distrib (s t u : multiset \u03b1) : (s \u2229 t) + u = (s + u) \u2229 (t + u) :=\nbegin\n  by_contra h,\n  cases lt_iff_cons_le.1 (lt_of_le_of_ne (le_inter\n    (add_le_add_right (inter_le_left s t) u)\n    (add_le_add_right (inter_le_right s t) u)) h) with a hl,\n  rw \u2190 cons_add at hl,\n  exact not_le_of_lt (lt_cons_self (s \u2229 t) a) (le_inter\n    (le_of_add_le_add_right (le_trans hl (inter_le_left _ _)))\n    (le_of_add_le_add_right (le_trans hl (inter_le_right _ _))))\nend\n\ntheorem add_inter_distrib (s t u : multiset \u03b1) : s + (t \u2229 u) = (s + t) \u2229 (s + u) :=\nby rw [add_comm, inter_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_inter_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u2229 t) = (a :: s) \u2229 (a :: t) :=\nby simp\n\ntheorem union_add_inter (s t : multiset \u03b1) : s \u222a t + s \u2229 t = s + t :=\nbegin\n  apply le_antisymm,\n  { rw union_add_distrib,\n    refine union_le (add_le_add_left (inter_le_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (inter_le_left _ _) _ },\n  { rw [add_comm, add_inter_distrib],\n    refine le_inter (add_le_add_right (le_union_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (le_union_left _ _) _ }\nend\n\ntheorem sub_add_inter (s t : multiset \u03b1) : s - t + s \u2229 t = s :=\nbegin\n  rw [inter_comm],\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  by_cases a \u2208 s,\n  { rw [cons_inter_of_pos _ h, sub_cons, add_cons, IH, cons_erase h] },\n  { rw [cons_inter_of_neg _ h, sub_cons, erase_of_not_mem h, IH] }\nend\n\ntheorem sub_inter (s t : multiset \u03b1) : s - (s \u2229 t) = s - t :=\nadd_right_cancel $\nby rw [sub_add_inter s t, sub_add_cancel (inter_le_left _ _)]\n\nend\n\n\n/- filter -/\nsection\nvariables {p : \u03b1 \u2192 Prop} [decidable_pred p]\n\n/-- `filter p s` returns the elements in `s` (with the same multiplicities)\n  which satisfy `p`, and removes the rest. -/\ndef filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p] (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (filter p l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $ perm_filter p h)\n\n@[simp] theorem coe_filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p]\n  (l : list \u03b1) : filter p (\u2191l) = l.filter p := rfl\n\n@[simp] theorem filter_zero (p : \u03b1 \u2192 Prop) [h : decidable_pred p] : filter p 0 = 0 := rfl\n\n@[simp] theorem filter_cons_of_pos {a : \u03b1} (s) : p a \u2192 filter p (a::s) = a :: filter p s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_cons_of_pos l h\n\n@[simp] theorem filter_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 filter p (a::s) = filter p s :=\nquot.induction_on s $ \u03bb l h, @congr_arg _ _ _ _ coe $ filter_cons_of_neg l h\n\nlemma filter_congr {p q : \u03b1 \u2192 Prop} [decidable_pred p] [decidable_pred q]\n  {s : multiset \u03b1} : (\u2200 x \u2208 s, p x \u2194 q x) \u2192 filter p s = filter q s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_congr h\n\n@[simp] theorem filter_add (s t : multiset \u03b1) :\n  filter p (s + t) = filter p s + filter p t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ filter_append _ _\n\n@[simp] theorem filter_le (s : multiset \u03b1) : filter p s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ filter_sublist _\n\n@[simp] theorem filter_subset (s : multiset \u03b1) : filter p s \u2286 s :=\nsubset_of_le $ filter_le _\n\n@[simp] theorem mem_filter {a : \u03b1} {s} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a :=\nquot.induction_on s $ \u03bb l, mem_filter\n\ntheorem of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : p a :=\n(mem_filter.1 h).2\n\ntheorem mem_of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : a \u2208 s :=\n(mem_filter.1 h).1\n\ntheorem mem_filter_of_mem {a : \u03b1} {l} (m : a \u2208 l) (h : p a) : a \u2208 filter p l :=\nmem_filter.2 \u27e8m, h\u27e9\n\ntheorem filter_eq_self {s} : filter p s = s \u2194 \u2200 a \u2208 s, p a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_of_sublist_of_length_eq (filter_sublist _) (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_self\n\ntheorem filter_eq_nil {s} : filter p s = 0 \u2194 \u2200 a \u2208 s, \u00acp a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_nil_of_length_eq_zero (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_nil\n\ntheorem filter_le_filter {s t} (h : s \u2264 t) : filter p s \u2264 filter p t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ filter_sublist_filter h\n\ntheorem le_filter {s t} : s \u2264 filter p t \u2194 s \u2264 t \u2227 \u2200 a \u2208 s, p a :=\n\u27e8\u03bb h, \u27e8le_trans h (filter_le _), \u03bb a m, of_mem_filter (mem_of_le h m)\u27e9,\n \u03bb \u27e8h, al\u27e9, filter_eq_self.2 al \u25b8 filter_le_filter h\u27e9\n\n@[simp] theorem filter_sub [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s - t) = filter p s - filter p t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  rw [sub_cons, IH],\n  by_cases p a,\n  { rw [filter_cons_of_pos _ h, sub_cons], congr,\n    by_cases m : a \u2208 s,\n    { rw [\u2190 cons_inj_right a, \u2190 filter_cons_of_pos _ h,\n          cons_erase (mem_filter_of_mem m h), cons_erase m] },\n    { rw [erase_of_not_mem m, erase_of_not_mem (mt mem_of_mem_filter m)] } },\n  { rw [filter_cons_of_neg _ h],\n    by_cases m : a \u2208 s,\n    { rw [(by rw filter_cons_of_neg _ h : filter p (erase s a) = filter p (a :: erase s a)),\n          cons_erase m] },\n    { rw [erase_of_not_mem m] } }\nend\n\n@[simp] theorem filter_union [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u222a t) = filter p s \u222a filter p t :=\nby simp [(\u222a), union]\n\n@[simp] theorem filter_inter [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u2229 t) = filter p s \u2229 filter p t :=\nle_antisymm (le_inter\n    (filter_le_filter $ inter_le_left _ _)\n    (filter_le_filter $ inter_le_right _ _)) $ le_filter.2\n\u27e8inf_le_inf (filter_le _) (filter_le _),\n  \u03bb a h, of_mem_filter (mem_of_le (inter_le_left _ _) h)\u27e9\n\n@[simp] theorem filter_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p (filter q s) = filter (\u03bb a, p a \u2227 q a) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter l\n\ntheorem filter_add_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p s + filter q s = filter (\u03bb a, p a \u2228 q a) s + filter (\u03bb a, p a \u2227 q a) s :=\nmultiset.induction_on s rfl $ \u03bb a s IH,\nby by_cases p a; by_cases q a; simp *\n\ntheorem filter_add_not (s : multiset \u03b1) :\n  filter p s + filter (\u03bb a, \u00ac p a) s = s :=\nby rw [filter_add_filter, filter_eq_self.2, filter_eq_nil.2]; simp [decidable.em]\n\n/- filter_map -/\n\n/-- `filter_map f s` is a combination filter/map operation on `s`.\n  The function `f : \u03b1 \u2192 option \u03b2` is applied to each element of `s`;\n  if `f a` is `some b` then `b` is added to the result, otherwise\n  `a` is removed from the resulting multiset. -/\ndef filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l, (filter_map f l : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $perm_filter_map f h)\n\n@[simp] theorem coe_filter_map (f : \u03b1 \u2192 option \u03b2) (l : list \u03b1) : filter_map f l = l.filter_map f := rfl\n\n@[simp] theorem filter_map_zero (f : \u03b1 \u2192 option \u03b2) : filter_map f 0 = 0 := rfl\n\n@[simp] theorem filter_map_cons_none {f : \u03b1 \u2192 option \u03b2} (a : \u03b1) (s : multiset \u03b1) (h : f a = none) :\n  filter_map f (a :: s) = filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_none a l h\n\n@[simp] theorem filter_map_cons_some (f : \u03b1 \u2192 option \u03b2)\n  (a : \u03b1) (s : multiset \u03b1) {b : \u03b2} (h : f a = some b) :\n  filter_map f (a :: s) = b :: filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_some f a l h\n\ntheorem filter_map_eq_map (f : \u03b1 \u2192 \u03b2) : filter_map (some \u2218 f) = map f :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_map f) l\n\ntheorem filter_map_eq_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  filter_map (option.guard p) = filter p :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_filter p) l\n\ntheorem filter_map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (filter_map f s) = filter_map (\u03bb x, (f x).bind g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter_map f g l\n\ntheorem map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b3) (s : multiset \u03b1) :\n  map g (filter_map f s) = filter_map (\u03bb x, (f x).map g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map f g l\n\ntheorem filter_map_map (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (map f s) = filter_map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_map f g l\n\ntheorem filter_filter_map (f : \u03b1 \u2192 option \u03b2) (p : \u03b2 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n  filter p (filter_map f s) = filter_map (\u03bb x, (f x).filter p) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter_map f p l\n\ntheorem filter_map_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) :\n  filter_map f (filter p s) = filter_map (\u03bb x, if p x then f x else none) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter p f l\n\n@[simp] theorem filter_map_some (s : multiset \u03b1) : filter_map some s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_some l\n\n@[simp] theorem mem_filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) {b : \u03b2} :\n  b \u2208 filter_map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = some b :=\nquot.induction_on s $ \u03bb l, mem_filter_map f l\n\ntheorem map_filter_map_of_inv (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b1)\n  (H : \u2200 x : \u03b1, (f x).map g = some x) (s : multiset \u03b1) :\n  map g (filter_map f s) = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map_of_inv f g H l\n\ntheorem filter_map_le_filter_map (f : \u03b1 \u2192 option \u03b2) {s t : multiset \u03b1}\n  (h : s \u2264 t) : filter_map f s \u2264 filter_map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h,\nsubperm_of_sublist $ filter_map_sublist_filter_map _ h\n\n/- powerset -/\n\ndef powerset_aux (l : list \u03b1) : list (multiset \u03b1) :=\n0 :: sublists_aux l (\u03bb x y, x :: y)\n\ntheorem powerset_aux_eq_map_coe {l : list \u03b1} :\n  powerset_aux l = (sublists l).map coe :=\nby simp [powerset_aux, sublists];\n   rw [\u2190 show @sublists_aux\u2081 \u03b1 (multiset \u03b1) l (\u03bb x, [\u2191x]) =\n              sublists_aux l (\u03bb x, list.cons \u2191x),\n         from sublists_aux\u2081_eq_sublists_aux _ _,\n       sublists_aux_cons_eq_sublists_aux\u2081,\n       \u2190 bind_ret_eq_map, sublists_aux\u2081_bind]; refl\n\n@[simp] theorem mem_powerset_aux {l : list \u03b1} {s} :\n  s \u2208 powerset_aux l \u2194 s \u2264 \u2191l :=\nquotient.induction_on s $\nby simp [powerset_aux_eq_map_coe, subperm, and.comm]\n\ndef powerset_aux' (l : list \u03b1) : list (multiset \u03b1) := (sublists' l).map coe\n\ntheorem powerset_aux_perm_powerset_aux' {l : list \u03b1} :\n  powerset_aux l ~ powerset_aux' l :=\nby rw powerset_aux_eq_map_coe; exact\nperm_map _ (sublists_perm_sublists' _)\n\n@[simp] theorem powerset_aux'_nil : powerset_aux' (@nil \u03b1) = [0] := rfl\n\n@[simp] theorem powerset_aux'_cons (a : \u03b1) (l : list \u03b1) :\n  powerset_aux' (a::l) = powerset_aux' l ++ list.map (cons a) (powerset_aux' l) :=\nby simp [powerset_aux']; refl\n\ntheorem powerset_aux'_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux' l\u2081 ~ powerset_aux' l\u2082 :=\nbegin\n  induction p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 p\u2082 IH\u2081 IH\u2082, {simp},\n  { simp, exact perm_app IH (perm_map _ IH) },\n  { simp, apply perm_app_right,\n    rw [\u2190 append_assoc, \u2190 append_assoc,\n        (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)],\n    exact perm_app_left _ perm_app_comm },\n  { exact IH\u2081.trans IH\u2082 }\nend\n\ntheorem powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux l\u2081 ~ powerset_aux l\u2082 :=\npowerset_aux_perm_powerset_aux'.trans $\n(powerset_aux'_perm p).trans powerset_aux_perm_powerset_aux'.symm\n\ndef powerset (s : multiset \u03b1) : multiset (multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (powerset_aux l : multiset (multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (powerset_aux_perm h))\n\ntheorem powerset_coe (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists l).map coe : list (multiset \u03b1)) :=\ncongr_arg coe powerset_aux_eq_map_coe\n\n@[simp] theorem powerset_coe' (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists' l).map coe : list (multiset \u03b1)) :=\nquot.sound powerset_aux_perm_powerset_aux'\n\n@[simp] theorem powerset_zero : @powerset \u03b1 0 = 0::0 := rfl\n\n@[simp] theorem powerset_cons (a : \u03b1) (s) :\n  powerset (a::s) = powerset s + map (cons a) (powerset s) :=\nquotient.induction_on s $ \u03bb l, by simp; refl\n\n@[simp] theorem mem_powerset {s t : multiset \u03b1} :\n  s \u2208 powerset t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ by simp [subperm, and.comm]\n\ntheorem map_single_le_powerset (s : multiset \u03b1) :\n  s.map (\u03bb a, a::0) \u2264 powerset s :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [powerset_coe],\n  show l.map (coe \u2218 list.ret) <+~ (sublists l).map coe,\n  rw \u2190 list.map_map,\n  exact subperm_of_sublist\n    (map_sublist_map _ (map_ret_sublist_sublists _))\nend\n\n@[simp] theorem card_powerset (s : multiset \u03b1) :\n  card (powerset s) = 2 ^ card s :=\nquotient.induction_on s $ by simp\n\n/- diagonal -/\n\ntheorem revzip_powerset_aux {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux_eq_map_coe, \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists _ _ _ h)\nend\n\ntheorem revzip_powerset_aux' {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux' l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux', \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists' _ _ _ h)\nend\n\ntheorem revzip_powerset_aux_lemma [decidable_eq \u03b1] (l : list \u03b1)\n  {l' : list (multiset \u03b1)} (H : \u2200 \u2983s t\u2984, (s, t) \u2208 revzip l' \u2192 s + t = \u2191l) :\n  revzip l' = l'.map (\u03bb x, (x, \u2191l - x)) :=\nbegin\n  have : forall\u2082 (\u03bb (p : multiset \u03b1 \u00d7 multiset \u03b1) (s : multiset \u03b1), p = (s, \u2191l - s))\n    (revzip l') ((revzip l').map prod.fst),\n  { rw forall\u2082_map_right_iff,\n    apply forall\u2082_same, rintro \u27e8s, t\u27e9 h,\n    dsimp, rw [\u2190 H h, add_sub_cancel_left] },\n  rw [\u2190 forall\u2082_eq_eq_eq, forall\u2082_map_right_iff], simpa\nend\n\ntheorem revzip_powerset_aux_perm_aux' {l : list \u03b1} :\n  revzip (powerset_aux l) ~ revzip (powerset_aux' l) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  rw [revzip_powerset_aux_lemma l revzip_powerset_aux,\n      revzip_powerset_aux_lemma l revzip_powerset_aux'],\n  exact perm_map _ powerset_aux_perm_powerset_aux',\nend\n\ntheorem revzip_powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  revzip (powerset_aux l\u2081) ~ revzip (powerset_aux l\u2082) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  simp [\u03bb l:list \u03b1, revzip_powerset_aux_lemma l revzip_powerset_aux, coe_eq_coe.2 p],\n  exact perm_map _ (powerset_aux_perm p)\nend\n\ndef diagonal (s : multiset \u03b1) : multiset (multiset \u03b1 \u00d7 multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (revzip (powerset_aux l) : multiset (multiset \u03b1 \u00d7 multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (revzip_powerset_aux_perm h))\n\ntheorem diagonal_coe (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux l) := rfl\n\n@[simp] theorem diagonal_coe' (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux' l) :=\nquot.sound revzip_powerset_aux_perm_aux'\n\n@[simp] theorem mem_diagonal {s\u2081 s\u2082 t : multiset \u03b1} :\n  (s\u2081, s\u2082) \u2208 diagonal t \u2194 s\u2081 + s\u2082 = t :=\nquotient.induction_on t $ \u03bb l, begin\n  simp [diagonal_coe], refine \u27e8\u03bb h, revzip_powerset_aux h, \u03bb h, _\u27e9,\n  haveI := classical.dec_eq \u03b1,\n  simp [revzip_powerset_aux_lemma l revzip_powerset_aux, h.symm],\n  exact \u27e8_, le_add_right _ _, rfl, add_sub_cancel_left _ _\u27e9\nend\n\n@[simp] theorem diagonal_map_fst (s : multiset \u03b1) :\n  (diagonal s).map prod.fst = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_map_snd (s : multiset \u03b1) :\n  (diagonal s).map prod.snd = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_zero : @diagonal \u03b1 0 = (0, 0)::0 := rfl\n\n@[simp] theorem diagonal_cons (a : \u03b1) (s) : diagonal (a::s) =\n  map (prod.map id (cons a)) (diagonal s) +\n  map (prod.map (cons a) id) (diagonal s) :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [revzip, reverse_append],\n  rw [\u2190 zip_map, \u2190 zip_map, zip_append, (_ : _++_=_)],\n  {congr; simp}, {simp}\nend\n\n@[simp] theorem card_diagonal (s : multiset \u03b1) :\n  card (diagonal s) = 2 ^ card s :=\nby have := card_powerset s;\n   rwa [\u2190 diagonal_map_fst, card_map] at this\n\nlemma prod_map_add [comm_semiring \u03b2] {s : multiset \u03b1} {f g : \u03b1 \u2192 \u03b2} :\n  prod (s.map (\u03bba, f a + g a)) = sum ((diagonal s).map (\u03bbp, (p.1.map f).prod * (p.2.map g).prod)) :=\nbegin\n  refine s.induction_on _ _,\n  { simp },\n  { assume a s ih, simp [ih, add_mul, mul_comm, mul_left_comm, mul_assoc, sum_map_mul_left.symm] },\nend\n\n/- countp -/\n\n/-- `countp p s` counts the number of elements of `s` (with multiplicity) that\n  satisfy `p`. -/\ndef countp (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s (countp p) (\u03bb l\u2081 l\u2082, perm_countp p)\n\n@[simp] theorem coe_countp (l : list \u03b1) : countp p l = l.countp p := rfl\n\n@[simp] theorem countp_zero (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p 0 = 0 := rfl\n\n@[simp] theorem countp_cons_of_pos {a : \u03b1} (s) : p a \u2192 countp p (a::s) = countp p s + 1 :=\nquot.induction_on s countp_cons_of_pos\n\n@[simp] theorem countp_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 countp p (a::s) = countp p s :=\nquot.induction_on s countp_cons_of_neg\n\ntheorem countp_eq_card_filter (s) : countp p s = card (filter p s) :=\nquot.induction_on s $ \u03bb l, countp_eq_length_filter _\n\n@[simp] theorem countp_add (s t) : countp p (s + t) = countp p s + countp p t :=\nby simp [countp_eq_card_filter]\n\ninstance countp.is_add_monoid_hom : is_add_monoid_hom (countp p : multiset \u03b1 \u2192 \u2115) :=\nby refine_struct {..}; simp\n\ntheorem countp_pos {s} : 0 < countp p s \u2194 \u2203 a \u2208 s, p a :=\nby simp [countp_eq_card_filter, card_pos_iff_exists_mem]\n\n@[simp] theorem countp_sub [decidable_eq \u03b1] {s t : multiset \u03b1} (h : t \u2264 s) :\n  countp p (s - t) = countp p s - countp p t :=\nby simp [countp_eq_card_filter, h, filter_le_filter]\n\ntheorem countp_pos_of_mem {s a} (h : a \u2208 s) (pa : p a) : 0 < countp p s :=\ncountp_pos.2 \u27e8_, h, pa\u27e9\n\ntheorem countp_le_of_le {s t} (h : s \u2264 t) : countp p s \u2264 countp p t :=\nby simpa [countp_eq_card_filter] using card_le_of_le (filter_le_filter h)\n\n@[simp] theorem countp_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  countp p (filter q s) = countp (\u03bb a, p a \u2227 q a) s :=\nby simp [countp_eq_card_filter]\n\nend\n\n/- count -/\n\nsection\nvariable [decidable_eq \u03b1]\n\n/-- `count a s` is the multiplicity of `a` in `s`. -/\ndef count (a : \u03b1) : multiset \u03b1 \u2192 \u2115 := countp (eq a)\n\n@[simp] theorem coe_count (a : \u03b1) (l : list \u03b1) : count a (\u2191l) = l.count a := coe_countp _\n\n@[simp] theorem count_zero (a : \u03b1) : count a 0 = 0 := rfl\n\n@[simp] theorem count_cons_self (a : \u03b1) (s : multiset \u03b1) : count a (a::s) = succ (count a s) :=\ncountp_cons_of_pos _ rfl\n\n@[simp] theorem count_cons_of_ne {a b : \u03b1} (h : a \u2260 b) (s : multiset \u03b1) : count a (b::s) = count a s :=\ncountp_cons_of_neg _ h\n\ntheorem count_le_of_le (a : \u03b1) {s t} : s \u2264 t \u2192 count a s \u2264 count a t :=\ncountp_le_of_le\n\ntheorem count_le_count_cons (a b : \u03b1) (s : multiset \u03b1) : count a s \u2264 count a (b :: s) :=\ncount_le_of_le _ (le_cons_self _ _)\n\ntheorem count_singleton (a : \u03b1) : count a (a::0) = 1 :=\nby simp\n\n@[simp] theorem count_add (a : \u03b1) : \u2200 s t, count a (s + t) = count a s + count a t :=\ncountp_add\n\ninstance count.is_add_monoid_hom (a : \u03b1) : is_add_monoid_hom (count a : multiset \u03b1 \u2192 \u2115) :=\ncountp.is_add_monoid_hom\n\n@[simp] theorem count_smul (a : \u03b1) (n s) : count a (n \u2022 s) = n * count a s :=\nby induction n; simp [*, succ_smul', succ_mul]\n\ntheorem count_pos {a : \u03b1} {s : multiset \u03b1} : 0 < count a s \u2194 a \u2208 s :=\nby simp [count, countp_pos]\n\n@[simp] theorem count_eq_zero_of_not_mem {a : \u03b1} {s : multiset \u03b1} (h : a \u2209 s) : count a s = 0 :=\nby_contradiction $ \u03bb h', h $ count_pos.1 (nat.pos_of_ne_zero h')\n\ntheorem count_eq_zero {a : \u03b1} {s : multiset \u03b1} : count a s = 0 \u2194 a \u2209 s :=\niff_not_comm.1 $ count_pos.symm.trans pos_iff_ne_zero\n\n@[simp] theorem count_repeat (a : \u03b1) (n : \u2115) : count a (repeat a n) = n :=\nby simp [repeat]\n\n@[simp] theorem count_erase_self (a : \u03b1) (s : multiset \u03b1) : count a (erase s a) = pred (count a s) :=\nbegin\n  by_cases a \u2208 s,\n  { rw [(by rw cons_erase h : count a s = count a (a::erase s a)),\n        count_cons_self]; refl },\n  { rw [erase_of_not_mem h, count_eq_zero.2 h]; refl }\nend\n\n@[simp] theorem count_erase_of_ne {a b : \u03b1} (ab : a \u2260 b) (s : multiset \u03b1) : count a (erase s b) = count a s :=\nbegin\n  by_cases b \u2208 s,\n  { rw [\u2190 count_cons_of_ne ab, cons_erase h] },\n  { rw [erase_of_not_mem h] }\nend\n\n@[simp] theorem count_sub (a : \u03b1) (s t : multiset \u03b1) : count a (s - t) = count a s - count a t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb b t IH s, _),\n  rw [sub_cons, IH],\n  by_cases ab : a = b,\n  { subst b, rw [count_erase_self, count_cons_self, sub_succ, pred_sub] },\n  { rw [count_erase_of_ne ab, count_cons_of_ne ab] }\nend\n\n@[simp] theorem count_union (a : \u03b1) (s t : multiset \u03b1) : count a (s \u222a t) = max (count a s) (count a t) :=\nby simp [(\u222a), union, sub_add_eq_max, -add_comm]\n\n@[simp] theorem count_inter (a : \u03b1) (s t : multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t) :=\nbegin\n  apply @nat.add_left_cancel (count a (s - t)),\n  rw [\u2190 count_add, sub_add_inter, count_sub, sub_add_min],\nend\n\nlemma count_bind {m : multiset \u03b2} {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} :\n  count a (bind m f) = sum (m.map $ \u03bbb, count a $ f b) :=\nmultiset.induction_on m (by simp) (by simp)\n\ntheorem le_count_iff_repeat_le {a : \u03b1} {s : multiset \u03b1} {n : \u2115} : n \u2264 count a s \u2194 repeat a n \u2264 s :=\nquot.induction_on s $ \u03bb l, le_count_iff_repeat_sublist.trans repeat_le_coe.symm\n\n@[simp] theorem count_filter {p} [decidable_pred p]\n  {a} {s : multiset \u03b1} (h : p a) : count a (filter p s) = count a s :=\nquot.induction_on s $ \u03bb l, count_filter h\n\ntheorem ext {s t : multiset \u03b1} : s = t \u2194 \u2200 a, count a s = count a t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quotient.eq.trans perm_iff_count\n\n@[extensionality]\ntheorem ext' {s t : multiset \u03b1} : (\u2200 a, count a s = count a t) \u2192 s = t :=\next.2\n\n@[simp] theorem coe_inter (s t : list \u03b1) : (s \u2229 t : multiset \u03b1) = (s.bag_inter t : list \u03b1) :=\nby ext; simp\n\ntheorem le_iff_count {s t : multiset \u03b1} : s \u2264 t \u2194 \u2200 a, count a s \u2264 count a t :=\n\u27e8\u03bb h a, count_le_of_le a h, \u03bb al,\n by rw \u2190 (ext.2 (\u03bb a, by simp [max_eq_right (al a)]) : s \u222a t = t);\n    apply le_union_left\u27e9\n\ninstance : distrib_lattice (multiset \u03b1) :=\n{ le_sup_inf := \u03bb s t u, le_of_eq $ eq.symm $\n    ext.2 $ \u03bb a, by simp [max_min_distrib_left],\n  ..multiset.lattice.lattice }\n\ninstance : semilattice_sup_bot (multiset \u03b1) :=\n{ bot := 0,\n  bot_le := zero_le,\n  ..multiset.lattice.lattice }\n\nend\n\n/- relator -/\n\nsection rel\n\n/-- `rel r s t` -- lift the relation `r` between two elements to a relation between `s` and `t`,\ns.t. there is a one-to-one mapping betweem elements in `s` and `t` following `r`. -/\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : multiset \u03b1 \u2192 multiset \u03b2 \u2192 Prop\n| zero {} : rel 0 0\n| cons {a b as bs} : r a b \u2192 rel as bs \u2192 rel (a :: as) (b :: bs)\n\nrun_cmd tactic.mk_iff_of_inductive_prop `multiset.rel `multiset.rel_iff\n\nvariables {\u03b4 : Type*} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop}\n\nprivate lemma rel_flip_aux {s t} (h : rel r s t) : rel (flip r) t s :=\nrel.rec_on h rel.zero (assume _ _ _ _ h\u2080 h\u2081 ih, rel.cons h\u2080 ih)\n\nlemma rel_flip {s t} : rel (flip r) s t \u2194 rel r t s :=\n\u27e8rel_flip_aux, rel_flip_aux\u27e9\n\nlemma rel_eq_refl {s : multiset \u03b1} : rel (=) s s :=\nmultiset.induction_on s rel.zero (assume a s, rel.cons rfl)\n\nlemma rel_eq {s t : multiset \u03b1} : rel (=) s t \u2194 s = t :=\nbegin\n  split,\n  { assume h, induction h; simp * },\n  { assume h, subst h, exact rel_eq_refl }\nend\n\nlemma rel.mono {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (h : \u2200a b, r a b \u2192 p a b) (hst : rel r s t) : rel p s t :=\nbegin\n  induction hst,\n  case rel.zero { exact rel.zero },\n  case rel.cons : a b s t hab hst ih { exact ih.cons (h a b hab) }\nend\n\nlemma rel.add {s t u v} (hst : rel r s t) (huv : rel r u v) : rel r (s + u) (t + v) :=\nbegin\n  induction hst,\n  case rel.zero { simpa using huv },\n  case rel.cons : a b s t hab hst ih { simpa using ih.cons hab }\nend\n\nlemma rel_flip_eq  {s t : multiset \u03b1} : rel (\u03bba b, b = a) s t \u2194 s = t :=\nshow rel (flip (=)) s t \u2194 s = t, by rw [rel_flip, rel_eq, eq_comm]\n\n@[simp] lemma rel_zero_left {b : multiset \u03b2} : rel r 0 b \u2194 b = 0 :=\nby rw [rel_iff]; simp\n\n@[simp] lemma rel_zero_right {a : multiset \u03b1} : rel r a 0 \u2194 a = 0 :=\nby rw [rel_iff]; simp\n\nlemma rel_cons_left {a as bs} :\n  rel r (a :: as) bs \u2194 (\u2203b bs', r a b \u2227 rel r as bs' \u2227 bs = b :: bs') :=\nbegin\n  split,\n  { generalize hm : a :: as = m,\n    assume h,\n    induction h generalizing as,\n    case rel.zero { simp at hm, contradiction },\n    case rel.cons : a' b as' bs ha'b h ih {\n      rcases cons_eq_cons.1 hm with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n      { subst eq\u2081, subst eq\u2082, exact \u27e8b, bs, ha'b, h, rfl\u27e9 },\n      { rcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9,\n        exact \u27e8b', b::bs', h\u2081, eq\u2081.symm \u25b8 rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9  }\n    } },\n  { exact assume \u27e8b, bs', hab, h, eq\u27e9, eq.symm \u25b8 rel.cons hab h }\nend\n\nlemma rel_cons_right {as b bs} :\n  rel r as (b :: bs) \u2194 (\u2203a as', r a b \u2227 rel r as' bs \u2227 as = a :: as') :=\nbegin\n  rw [\u2190 rel_flip, rel_cons_left],\n  apply exists_congr, assume a,\n  apply exists_congr, assume as',\n  rw [rel_flip, flip]\nend\n\nlemma rel_add_left {as\u2080 as\u2081} :\n  \u2200{bs}, rel r (as\u2080 + as\u2081) bs \u2194 (\u2203bs\u2080 bs\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081) :=\nmultiset.induction_on as\u2080 (by simp)\n  begin\n    assume a s ih bs,\n    simp only [ih, cons_add, rel_cons_left],\n    split,\n    { assume h,\n      rcases h with \u27e8b, bs', hab, h, rfl\u27e9,\n      rcases h with \u27e8bs\u2080, bs\u2081, h\u2080, h\u2081, rfl\u27e9,\n      exact \u27e8b :: bs\u2080, bs\u2081, \u27e8b, bs\u2080, hab, h\u2080, rfl\u27e9, h\u2081, by simp\u27e9 },\n    { assume h,\n      rcases h with \u27e8bs\u2080, bs\u2081, h, h\u2081, rfl\u27e9,\n      rcases h with \u27e8b, bs, hab, h\u2080, rfl\u27e9,\n      exact \u27e8b, bs + bs\u2081, hab, \u27e8bs, bs\u2081, h\u2080, h\u2081, rfl\u27e9, by simp\u27e9 }\n  end\n\nlemma rel_add_right {as bs\u2080 bs\u2081} :\n  rel r as (bs\u2080 + bs\u2081) \u2194 (\u2203as\u2080 as\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081) :=\nby rw [\u2190 rel_flip, rel_add_left]; simp [rel_flip]\n\nlemma rel_map_left {s : multiset \u03b3} {f : \u03b3 \u2192 \u03b1} :\n  \u2200{t}, rel r (s.map f) t \u2194 rel (\u03bba b, r (f a) b) s t :=\nmultiset.induction_on s (by simp) (by simp [rel_cons_left] {contextual := tt})\n\nlemma rel_map_right {s : multiset \u03b1} {t : multiset \u03b3} {f : \u03b3 \u2192 \u03b2} :\n  rel r s (t.map f) \u2194 rel (\u03bba b, r a (f b)) s t :=\nby rw [\u2190 rel_flip, rel_map_left, \u2190 rel_flip]; refl\n\nlemma rel_join {s t} (h : rel (rel r) s t) : rel r s.join t.join :=\nbegin\n  induction h,\n  case rel.zero { simp },\n  case rel.cons : a b s t hab hst ih { simpa using hab.add ih }\nend\n\nlemma rel_map {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} (h : (r \u21d2 p) f g) (hst : rel r s t) :\n  rel p (s.map f) (t.map g) :=\nby rw [rel_map_left, rel_map_right]; exact hst.mono (assume a b, h)\n\nlemma rel_bind {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 multiset \u03b3} {g : \u03b2 \u2192 multiset \u03b4}\n  (h : (r \u21d2 rel p) f g) (hst : rel r s t) :\n  rel p (s.bind f) (t.bind g) :=\nby apply rel_join; apply rel_map; assumption\n\nlemma card_eq_card_of_rel {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) :\n  card s = card t :=\nby induction h; simp [*]\n\nlemma exists_mem_of_rel_of_mem {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) :\n  \u2200 {a : \u03b1} (ha : a \u2208 s), \u2203 b \u2208 t, r a b :=\nbegin\n  induction h with x y s t hxy hst ih,\n  { simp },\n  { assume a ha,\n    cases mem_cons.1 ha with ha ha,\n    { exact \u27e8y, mem_cons_self _ _, ha.symm \u25b8 hxy\u27e9 },\n    { rcases ih ha with \u27e8b, hbt, hab\u27e9,\n      exact \u27e8b, mem_cons.2 (or.inr hbt), hab\u27e9 } }\nend\n\nend rel\n\nsection map\n\ntheorem map_eq_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) {s t : multiset \u03b1} :\n  s.map f = t.map f \u2194 s = t :=\nby rw [\u2190 rel_eq, \u2190 rel_eq, rel_map_left, rel_map_right]; simp [hf.eq_iff]\n\ntheorem injective_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  function.injective (multiset.map f) :=\nassume x y, (map_eq_map hf).1\n\nend map\n\nsection quot\n\ntheorem map_mk_eq_map_mk_of_rel {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s t : multiset \u03b1} (hst : s.rel r t) :\n s.map (quot.mk r) = t.map (quot.mk r) :=\nrel.rec_on hst rfl $ assume a b s t hab hst ih, by simp [ih, quot.sound hab]\n\ntheorem exists_multiset_eq_map_quot_mk {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : multiset (quot r)) :\n  \u2203t:multiset \u03b1, s = t.map (quot.mk r) :=\nmultiset.induction_on s \u27e80, rfl\u27e9 $\n  assume a s \u27e8t, ht\u27e9, quot.induction_on a $ assume a, ht.symm \u25b8 \u27e8a::t, (map_cons _ _ _).symm\u27e9\n\ntheorem induction_on_multiset_quot\n  {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {p : multiset (quot r) \u2192 Prop} (s : multiset (quot r)) :\n  (\u2200s:multiset \u03b1, p (s.map (quot.mk r))) \u2192 p s :=\nmatch s, exists_multiset_eq_map_quot_mk s with _, \u27e8t, rfl\u27e9 := assume h, h _ end\n\nend quot\n\n/- disjoint -/\n\n/-- `disjoint s t` means that `s` and `t` have no elements in common. -/\ndef disjoint (s t : multiset \u03b1) : Prop := \u2200 \u2983a\u2984, a \u2208 s \u2192 a \u2208 t \u2192 false\n\n@[simp] theorem coe_disjoint (l\u2081 l\u2082 : list \u03b1) : @disjoint \u03b1 l\u2081 l\u2082 \u2194 l\u2081.disjoint l\u2082 := iff.rfl\n\ntheorem disjoint.symm {s t : multiset \u03b1} (d : disjoint s t) : disjoint t s\n| a i\u2082 i\u2081 := d i\u2081 i\u2082\n\n@[simp] theorem disjoint_comm {s t : multiset \u03b1} : disjoint s t \u2194 disjoint t s :=\n\u27e8disjoint.symm, disjoint.symm\u27e9\n\ntheorem disjoint_left {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 s \u2192 a \u2209 t := iff.rfl\n\ntheorem disjoint_right {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 t \u2192 a \u2209 s :=\ndisjoint_comm\n\ntheorem disjoint_iff_ne {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 a \u2208 s, \u2200 b \u2208 t, a \u2260 b :=\nby simp [disjoint_left, imp_not_comm]\n\ntheorem disjoint_of_subset_left {s t u : multiset \u03b1} (h : s \u2286 u) (d : disjoint u t) : disjoint s t\n| x m\u2081 := d (h m\u2081)\n\ntheorem disjoint_of_subset_right {s t u : multiset \u03b1} (h : t \u2286 u) (d : disjoint s u) : disjoint s t\n| x m m\u2081 := d m (h m\u2081)\n\ntheorem disjoint_of_le_left {s t u : multiset \u03b1} (h : s \u2264 u) : disjoint u t \u2192 disjoint s t :=\ndisjoint_of_subset_left (subset_of_le h)\n\ntheorem disjoint_of_le_right {s t u : multiset \u03b1} (h : t \u2264 u) : disjoint s u \u2192 disjoint s t :=\ndisjoint_of_subset_right (subset_of_le h)\n\n@[simp] theorem zero_disjoint (l : multiset \u03b1) : disjoint 0 l\n| a := (not_mem_nil a).elim\n\n@[simp] theorem singleton_disjoint {l : multiset \u03b1} {a : \u03b1} : disjoint (a::0) l \u2194 a \u2209 l :=\nby simp [disjoint]; refl\n\n@[simp] theorem disjoint_singleton {l : multiset \u03b1} {a : \u03b1} : disjoint l (a::0) \u2194 a \u2209 l :=\nby rw disjoint_comm; simp\n\n@[simp] theorem disjoint_add_left {s t u : multiset \u03b1} :\n  disjoint (s + t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_add_right {s t u : multiset \u03b1} :\n  disjoint s (t + u) \u2194 disjoint s t \u2227 disjoint s u :=\ndisjoint_comm.trans $ by simp [disjoint_append_left]\n\n@[simp] theorem disjoint_cons_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (a::s) t \u2194 a \u2209 t \u2227 disjoint s t :=\n(@disjoint_add_left _ (a::0) s t).trans $ by simp\n\n@[simp] theorem disjoint_cons_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (a::t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp [disjoint_cons_left]\n\ntheorem inter_eq_zero_iff_disjoint [decidable_eq \u03b1] {s t : multiset \u03b1} : s \u2229 t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\n@[simp] theorem disjoint_union_left [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint (s \u222a t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_union_right [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint s (t \u222a u) \u2194 disjoint s t \u2227 disjoint s u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\nlemma disjoint_map_map {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {s : multiset \u03b1} {t : multiset \u03b2} :\n  disjoint (s.map f) (t.map g) \u2194 (\u2200a\u2208s, \u2200b\u2208t, f a \u2260 g b) :=\nbegin\n  simp [disjoint],\n  split,\n  from assume h a ha b hb eq, h _ ha rfl _ hb eq.symm,\n  from assume h c a ha eq\u2081 b hb eq\u2082, h _ ha _ hb (eq\u2082.symm \u25b8 eq\u2081)\nend\n\n/-- `pairwise r m` states that there exists a list of the elements s.t. `r` holds pairwise on this list. -/\ndef pairwise (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (m : multiset \u03b1) : Prop :=\n\u2203l:list \u03b1, m = l \u2227 l.pairwise r\n\nlemma pairwise_coe_iff_pairwise {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (hr : symmetric r) {l : list \u03b1} :\n  multiset.pairwise r l \u2194 l.pairwise r :=\niff.intro\n  (assume \u27e8l', eq, h\u27e9, (list.perm_pairwise hr (quotient.exact eq)).2 h)\n  (assume h, \u27e8l, rfl, h\u27e9)\n\n/- nodup -/\n\n/-- `nodup s` means that `s` has no duplicates, i.e. the multiplicity of\n  any element is at most 1. -/\ndef nodup (s : multiset \u03b1) : Prop :=\nquot.lift_on s nodup (\u03bb s t p, propext $ perm_nodup p)\n\n@[simp] theorem coe_nodup {l : list \u03b1} : @nodup \u03b1 l \u2194 l.nodup := iff.rfl\n\n@[simp] theorem forall_mem_ne {a : \u03b1} {l : list \u03b1} : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 \u00aca = a') \u2194 a \u2209 l :=\n\u27e8\u03bb h m, h _ m rfl, \u03bb h a' m e, h (e.symm \u25b8 m)\u27e9\n\n@[simp] theorem nodup_zero : @nodup \u03b1 0 := pairwise.nil\n\n@[simp] theorem nodup_cons {a : \u03b1} {s : multiset \u03b1} : nodup (a::s) \u2194 a \u2209 s \u2227 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_cons\n\ntheorem nodup_cons_of_nodup {a : \u03b1} {s : multiset \u03b1} (m : a \u2209 s) (n : nodup s) : nodup (a::s) :=\nnodup_cons.2 \u27e8m, n\u27e9\n\ntheorem nodup_singleton : \u2200 a : \u03b1, nodup (a::0) := nodup_singleton\n\ntheorem nodup_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : nodup s :=\n(nodup_cons.1 h).2\n\ntheorem not_mem_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : a \u2209 s :=\n(nodup_cons.1 h).1\n\ntheorem nodup_of_le {s t : multiset \u03b1} (h : s \u2264 t) : nodup t \u2192 nodup s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, nodup_of_sublist\n\ntheorem not_nodup_pair : \u2200 a : \u03b1, \u00ac nodup (a::a::0) := not_nodup_pair\n\ntheorem nodup_iff_le {s : multiset \u03b1} : nodup s \u2194 \u2200 a : \u03b1, \u00ac a::a::0 \u2264 s :=\nquot.induction_on s $ \u03bb l, nodup_iff_sublist.trans $ forall_congr $ \u03bb a,\nnot_congr (@repeat_le_coe _ a 2 _).symm\n\ntheorem nodup_iff_count_le_one [decidable_eq \u03b1] {s : multiset \u03b1} : nodup s \u2194 \u2200 a, count a s \u2264 1 :=\nquot.induction_on s $ \u03bb l, nodup_iff_count_le_one\n\n@[simp] theorem count_eq_one_of_mem [decidable_eq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n  (d : nodup s) (h : a \u2208 s) : count a s = 1 :=\nle_antisymm (nodup_iff_count_le_one.1 d a) (count_pos.2 h)\n\nlemma pairwise_of_nodup {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : multiset \u03b1} :\n  (\u2200a\u2208s, \u2200b\u2208s, a \u2260 b \u2192 r a b) \u2192 nodup s \u2192 pairwise r s :=\nquotient.induction_on s $ assume l h hl, \u27e8l, rfl, hl.imp_of_mem $ assume a b ha hb, h a ha b hb\u27e9\n\nlemma forall_of_pairwise {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (H : symmetric r) {s : multiset \u03b1}\n   (hs : pairwise r s) : (\u2200a\u2208s, \u2200b\u2208s, a \u2260 b \u2192 r a b) :=\nlet \u27e8l, hl\u2081, hl\u2082\u27e9 := hs in hl\u2081.symm \u25b8 list.forall_of_pairwise H hl\u2082\n\ntheorem nodup_add {s t : multiset \u03b1} : nodup (s + t) \u2194 nodup s \u2227 nodup t \u2227 disjoint s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, nodup_append\n\ntheorem disjoint_of_nodup_add {s t : multiset \u03b1} (d : nodup (s + t)) : disjoint s t :=\n(nodup_add.1 d).2.2\n\ntheorem nodup_add_of_nodup {s t : multiset \u03b1} (d\u2081 : nodup s) (d\u2082 : nodup t) : nodup (s + t) \u2194 disjoint s t :=\nby simp [nodup_add, d\u2081, d\u2082]\n\ntheorem nodup_of_nodup_map (f : \u03b1 \u2192 \u03b2) {s : multiset \u03b1} : nodup (map f s) \u2192 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_of_nodup_map f\n\ntheorem nodup_map_on {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200x\u2208s, \u2200y\u2208s, f x = f y \u2192 x = y) \u2192\n  nodup s \u2192 nodup (map f s) :=\nquot.induction_on s $ \u03bb l, nodup_map_on\n\ntheorem nodup_map {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} (hf : function.injective f) : nodup s \u2192 nodup (map f s) :=\nnodup_map_on (\u03bb x _ y _ h, hf h)\n\ntheorem nodup_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] {s} : nodup s \u2192 nodup (filter p s) :=\nquot.induction_on s $ \u03bb l, nodup_filter p\n\n@[simp] theorem nodup_attach {s : multiset \u03b1} : nodup (attach s) \u2194 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_attach\n\ntheorem nodup_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {s : multiset \u03b1} {H}\n  (hf : \u2200 a ha b hb, f a ha = f b hb \u2192 a = b) : nodup s \u2192 nodup (pmap f s H) :=\nquot.induction_on s (\u03bb l H, nodup_pmap hf) H\n\ninstance nodup_decidable [decidable_eq \u03b1] (s : multiset \u03b1) : decidable (nodup s) :=\nquotient.rec_on_subsingleton s $ \u03bb l, l.nodup_decidable\n\ntheorem nodup_erase_eq_filter [decidable_eq \u03b1] (a : \u03b1) {s} : nodup s \u2192 s.erase a = filter (\u2260 a) s :=\nquot.induction_on s $ \u03bb l d, congr_arg coe $ nodup_erase_eq_filter a d\n\ntheorem nodup_erase_of_nodup [decidable_eq \u03b1] (a : \u03b1) {l} : nodup l \u2192 nodup (l.erase a) :=\nnodup_of_le (erase_le _ _)\n\ntheorem mem_erase_iff_of_nodup [decidable_eq \u03b1] {a b : \u03b1} {l} (d : nodup l) :\n  a \u2208 l.erase b \u2194 a \u2260 b \u2227 a \u2208 l :=\nby rw nodup_erase_eq_filter b d; simp [and_comm]\n\ntheorem mem_erase_of_nodup [decidable_eq \u03b1] {a : \u03b1} {l} (h : nodup l) : a \u2209 l.erase a :=\nby rw mem_erase_iff_of_nodup h; simp\n\ntheorem nodup_product {s : multiset \u03b1} {t : multiset \u03b2} : nodup s \u2192 nodup t \u2192 nodup (product s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, by simp [nodup_product d\u2081 d\u2082]\n\ntheorem nodup_sigma {\u03c3 : \u03b1 \u2192 Type*} {s : multiset \u03b1} {t : \u03a0 a, multiset (\u03c3 a)} :\n  nodup s \u2192 (\u2200 a, nodup (t a)) \u2192 nodup (s.sigma t) :=\nquot.induction_on s $ assume l\u2081,\nbegin\n  choose f hf using assume a, quotient.exists_rep (t a),\n  rw show t = \u03bb a, f a, from (eq.symm $ funext $ \u03bb a, hf a),\n  simpa using nodup_sigma\nend\n\ntheorem nodup_filter_map (f : \u03b1 \u2192 option \u03b2) {s : multiset \u03b1}\n  (H : \u2200 (a a' : \u03b1) (b : \u03b2), b \u2208 f a \u2192 b \u2208 f a' \u2192 a = a') :\n  nodup s \u2192 nodup (filter_map f s) :=\nquot.induction_on s $ \u03bb l, nodup_filter_map H\n\ntheorem nodup_range (n : \u2115) : nodup (range n) := nodup_range _\n\ntheorem nodup_inter_left [decidable_eq \u03b1] {s : multiset \u03b1} (t) : nodup s \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_left _ _\n\ntheorem nodup_inter_right [decidable_eq \u03b1] (s) {t : multiset \u03b1} : nodup t \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_right _ _\n\n@[simp] theorem nodup_union [decidable_eq \u03b1] {s t : multiset \u03b1} : nodup (s \u222a t) \u2194 nodup s \u2227 nodup t :=\n\u27e8\u03bb h, \u27e8nodup_of_le (le_union_left _ _) h, nodup_of_le (le_union_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, nodup_iff_count_le_one.2 $ \u03bb a, by rw [count_union]; exact\n   max_le (nodup_iff_count_le_one.1 h\u2081 a) (nodup_iff_count_le_one.1 h\u2082 a)\u27e9\n\n@[simp] theorem nodup_powerset {s : multiset \u03b1} : nodup (powerset s) \u2194 nodup s :=\n\u27e8\u03bb h, nodup_of_nodup_map _ (nodup_of_le (map_single_le_powerset _) h),\n  quotient.induction_on s $ \u03bb l h,\n  by simp; refine list.nodup_map_on _ (nodup_sublists'.2 h); exact\n  \u03bb x sx y sy e,\n    (perm_ext_sublist_nodup h (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1\n      (quotient.exact e)\u27e9\n\n@[simp] lemma nodup_bind {s : multiset \u03b1} {t : \u03b1 \u2192 multiset \u03b2} :\n  nodup (bind s t) \u2194 ((\u2200a\u2208s, nodup (t a)) \u2227 (s.pairwise (\u03bba b, disjoint (t a) (t b)))) :=\nhave h\u2081 : \u2200a, \u2203l:list \u03b2, t a = l, from\n  assume a, quot.induction_on (t a) $ assume l, \u27e8l, rfl\u27e9,\nlet \u27e8t', h'\u27e9 := classical.axiom_of_choice h\u2081 in\nhave t = \u03bba, t' a, from funext h',\nhave hd : symmetric (\u03bba b, list.disjoint (t' a) (t' b)), from assume a b h, h.symm,\nquot.induction_on s $ by simp [this, list.nodup_bind, pairwise_coe_iff_pairwise hd]\n\ntheorem nodup_ext {s t : multiset \u03b1} : nodup s \u2192 nodup t \u2192 (s = t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, quotient.eq.trans $ perm_ext d\u2081 d\u2082\n\ntheorem le_iff_subset {s t : multiset \u03b1} : nodup s \u2192 (s \u2264 t \u2194 s \u2286 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d, \u27e8subset_of_le, subperm_of_subset_nodup d\u27e9\n\ntheorem range_le {m n : \u2115} : range m \u2264 range n \u2194 m \u2264 n :=\n(le_iff_subset (nodup_range _)).trans range_subset\n\ntheorem mem_sub_of_nodup [decidable_eq \u03b1] {a : \u03b1} {s t : multiset \u03b1} (d : nodup s) :\n  a \u2208 s - t \u2194 a \u2208 s \u2227 a \u2209 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (sub_le_self _ _) h, \u03bb h',\n  by refine count_eq_zero.1 _ h; rw [count_sub a s t, nat.sub_eq_zero_iff_le];\n     exact le_trans (nodup_iff_count_le_one.1 d _) (count_pos.2 h')\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, or.resolve_right (mem_add.1 $ mem_of_le (le_sub_add _ _) h\u2081) h\u2082\u27e9\n\nsection\nvariable [decidable_eq \u03b1]\n\n/- erase_dup -/\n\n/-- `erase_dup s` removes duplicates from `s`, yielding a `nodup` multiset. -/\ndef erase_dup (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase_dup : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_erase_dup_of_perm p))\n\n@[simp] theorem coe_erase_dup (l : list \u03b1) : @erase_dup \u03b1 _ l = l.erase_dup := rfl\n\n@[simp] theorem erase_dup_zero : @erase_dup \u03b1 _ 0 = 0 := rfl\n\n@[simp] theorem mem_erase_dup {a : \u03b1} {s : multiset \u03b1} : a \u2208 erase_dup s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_erase_dup\n\n@[simp] theorem erase_dup_cons_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192\n  erase_dup (a::s) = erase_dup s :=\nquot.induction_on s $ \u03bb l m, @congr_arg _ _ _ _ coe $ erase_dup_cons_of_mem m\n\n@[simp] theorem erase_dup_cons_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192\n  erase_dup (a::s) = a :: erase_dup s :=\nquot.induction_on s $ \u03bb l m, congr_arg coe $ erase_dup_cons_of_not_mem m\n\ntheorem erase_dup_le (s : multiset \u03b1) : erase_dup s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ erase_dup_sublist _\n\ntheorem erase_dup_subset (s : multiset \u03b1) : erase_dup s \u2286 s :=\nsubset_of_le $ erase_dup_le _\n\ntheorem subset_erase_dup (s : multiset \u03b1) : s \u2286 erase_dup s :=\n\u03bb a, mem_erase_dup.2\n\n@[simp] theorem erase_dup_subset' {s t : multiset \u03b1} : erase_dup s \u2286 t \u2194 s \u2286 t :=\n\u27e8subset.trans (subset_erase_dup _), subset.trans (erase_dup_subset _)\u27e9\n\n@[simp] theorem subset_erase_dup' {s t : multiset \u03b1} : s \u2286 erase_dup t \u2194 s \u2286 t :=\n\u27e8\u03bb h, subset.trans h (erase_dup_subset _), \u03bb h, subset.trans h (subset_erase_dup _)\u27e9\n\n@[simp] theorem nodup_erase_dup (s : multiset \u03b1) : nodup (erase_dup s) :=\nquot.induction_on s nodup_erase_dup\n\ntheorem erase_dup_eq_self {s : multiset \u03b1} : erase_dup s = s \u2194 nodup s :=\n\u27e8\u03bb e, e \u25b8 nodup_erase_dup s,\n quot.induction_on s $ \u03bb l h, congr_arg coe $ erase_dup_eq_self.2 h\u27e9\n\ntheorem erase_dup_eq_zero {s : multiset \u03b1} : erase_dup s = 0 \u2194 s = 0 :=\n\u27e8\u03bb h, eq_zero_of_subset_zero $ h \u25b8 subset_erase_dup _,\n \u03bb h, h.symm \u25b8 erase_dup_zero\u27e9\n\n@[simp] theorem erase_dup_singleton {a : \u03b1} : erase_dup (a :: 0) = a :: 0 :=\nerase_dup_eq_self.2 $ nodup_singleton _\n\ntheorem le_erase_dup {s t : multiset \u03b1} : s \u2264 erase_dup t \u2194 s \u2264 t \u2227 nodup s :=\n\u27e8\u03bb h, \u27e8le_trans h (erase_dup_le _), nodup_of_le h (nodup_erase_dup _)\u27e9,\n \u03bb \u27e8l, d\u27e9, (le_iff_subset d).2 $ subset.trans (subset_of_le l) (subset_erase_dup _)\u27e9\n\ntheorem erase_dup_ext {s t : multiset \u03b1} : erase_dup s = erase_dup t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t :=\nby simp [nodup_ext]\n\ntheorem erase_dup_map_erase_dup_eq [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  erase_dup (map f (erase_dup s)) = erase_dup (map f s) := by simp [erase_dup_ext]\n\n/- finset insert -/\n\n/-- `ndinsert a s` is the lift of the list `insert` operation. This operation\n  does not respect multiplicities, unlike `cons`, but it is suitable as\n  an insert operation on `finset`. -/\ndef ndinsert (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.insert a : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_insert a p))\n\n@[simp] theorem coe_ndinsert (a : \u03b1) (l : list \u03b1) : ndinsert a l = (insert a l : list \u03b1) := rfl\n\n@[simp] theorem ndinsert_zero (a : \u03b1) : ndinsert a 0 = a::0 := rfl\n\n@[simp] theorem ndinsert_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 ndinsert a s = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_mem h\n\n@[simp] theorem ndinsert_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 ndinsert a s = a :: s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_not_mem h\n\n@[simp] theorem mem_ndinsert {a b : \u03b1} {s : multiset \u03b1} : a \u2208 ndinsert b s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_insert_iff\n\n@[simp] theorem le_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : s \u2264 ndinsert a s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ sublist_of_suffix $ suffix_insert _ _\n\n@[simp] theorem mem_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 ndinsert a s :=\nmem_ndinsert.2 (or.inl rfl)\n\n@[simp] theorem mem_ndinsert_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 ndinsert b s :=\nmem_ndinsert.2 (or.inr h)\n\n@[simp] theorem length_ndinsert_of_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2208 s) :\n  card (ndinsert a s) = card s :=\nby simp [h]\n\n@[simp] theorem length_ndinsert_of_not_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2209 s) :\n  card (ndinsert a s) = card s + 1 :=\nby simp [h]\n\ntheorem erase_dup_cons {a : \u03b1} {s : multiset \u03b1} :\n  erase_dup (a::s) = ndinsert a (erase_dup s) :=\nby by_cases a \u2208 s; simp [h]\n\ntheorem nodup_ndinsert (a : \u03b1) {s : multiset \u03b1} : nodup s \u2192 nodup (ndinsert a s) :=\nquot.induction_on s $ \u03bb l, nodup_insert\n\ntheorem ndinsert_le {a : \u03b1} {s t : multiset \u03b1} : ndinsert a s \u2264 t \u2194 s \u2264 t \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8le_trans (le_ndinsert_self _ _) h, mem_of_le h (mem_ndinsert_self _ _)\u27e9,\n \u03bb \u27e8l, m\u27e9, if h : a \u2208 s then by simp [h, l] else\n   by rw [ndinsert_of_not_mem h, \u2190 cons_erase m, cons_le_cons_iff,\n          \u2190 le_cons_of_not_mem h, cons_erase m]; exact l\u27e9\n\nlemma attach_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (s.ndinsert a).attach =\n    ndinsert \u27e8a, mem_ndinsert_self a s\u27e9 (s.attach.map $ \u03bbp, \u27e8p.1, mem_ndinsert_of_mem p.2\u27e9) :=\nhave eq : \u2200h : \u2200(p : {x // x \u2208 s}), p.1 \u2208 s,\n    (\u03bb (p : {x // x \u2208 s}), \u27e8p.val, h p\u27e9 : {x // x \u2208 s} \u2192 {x // x \u2208 s}) = id, from\n  assume h, funext $ assume p, subtype.eq rfl,\nhave \u2200t (eq : s.ndinsert a = t), t.attach = ndinsert \u27e8a, eq \u25b8 mem_ndinsert_self a s\u27e9\n  (s.attach.map $ \u03bbp, \u27e8p.1, eq \u25b8 mem_ndinsert_of_mem p.2\u27e9),\nbegin\n  intros t ht,\n  by_cases a \u2208 s,\n  { rw [ndinsert_of_mem h] at ht,\n    subst ht,\n    rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)] },\n  { rw [ndinsert_of_not_mem h] at ht,\n    subst ht,\n    simp [attach_cons, h] }\nend,\nthis _ rfl\n\n@[simp] theorem disjoint_ndinsert_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (ndinsert a s) t \u2194 a \u2209 t \u2227 disjoint s t :=\niff.trans (by simp [disjoint]) disjoint_cons_left\n\n@[simp] theorem disjoint_ndinsert_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (ndinsert a t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp\n\n/- finset union -/\n\n/-- `ndunion s t` is the lift of the list `union` operation. This operation\n  does not respect multiplicities, unlike `s \u222a t`, but it is suitable as\n  a union operation on `finset`. (`s \u222a t` would also work as a union operation\n  on finset, but this is more efficient.) -/\ndef ndunion (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.union l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_union p\u2081 p\u2082\n\n@[simp] theorem coe_ndunion (l\u2081 l\u2082 : list \u03b1) : @ndunion \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u222a l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndunion (s : multiset \u03b1) : ndunion 0 s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem cons_ndunion (s t : multiset \u03b1) (a : \u03b1) : ndunion (a :: s) t = ndinsert a (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, rfl\n\n@[simp] theorem mem_ndunion {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndunion s t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.mem_union\n\ntheorem le_ndunion_right (s t : multiset \u03b1) : t \u2264 ndunion s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ sublist_of_suffix $ suffix_union_right _ _\n\ntheorem ndunion_le_add (s t : multiset \u03b1) : ndunion s t \u2264 s + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_of_sublist $ union_sublist_append _ _\n\ntheorem ndunion_le {s t u : multiset \u03b1} : ndunion s t \u2264 u \u2194 s \u2286 u \u2227 t \u2264 u :=\nmultiset.induction_on s (by simp) (by simp [ndinsert_le, and_comm, and.left_comm] {contextual := tt})\n\ntheorem subset_ndunion_left (s t : multiset \u03b1) : s \u2286 ndunion s t :=\n\u03bb a h, mem_ndunion.2 $ or.inl h\n\ntheorem le_ndunion_left {s} (t : multiset \u03b1) (d : nodup s) : s \u2264 ndunion s t :=\n(le_iff_subset d).2 $ subset_ndunion_left _ _\n\ntheorem ndunion_le_union (s t : multiset \u03b1) : ndunion s t \u2264 s \u222a t :=\nndunion_le.2 \u27e8subset_of_le (le_union_left _ _), le_union_right _ _\u27e9\n\ntheorem nodup_ndunion (s : multiset \u03b1) {t : multiset \u03b1} : nodup t \u2192 nodup (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.nodup_union _\n\n@[simp] theorem ndunion_eq_union {s t : multiset \u03b1} (d : nodup s) : ndunion s t = s \u222a t :=\nle_antisymm (ndunion_le_union _ _) $ union_le (le_ndunion_left _ d) (le_ndunion_right _ _)\n\ntheorem erase_dup_add (s t : multiset \u03b1) : erase_dup (s + t) = ndunion s (erase_dup t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ erase_dup_append _ _\n\n/- finset inter -/\n\n/-- `ndinter s t` is the lift of the list `\u2229` operation. This operation\n  does not respect multiplicities, unlike `s \u2229 t`, but it is suitable as\n  an intersection operation on `finset`. (`s \u2229 t` would also work as a union operation\n  on finset, but this is more efficient.) -/\ndef ndinter (s t : multiset \u03b1) : multiset \u03b1 := filter (\u2208 t) s\n\n@[simp] theorem coe_ndinter (l\u2081 l\u2082 : list \u03b1) : @ndinter \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u2229 l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndinter (s : multiset \u03b1) : ndinter 0 s = 0 := rfl\n\n@[simp] theorem cons_ndinter_of_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2208 t) :\n  ndinter (a::s) t = a :: (ndinter s t) := by simp [ndinter, h]\n\n@[simp] theorem ndinter_cons_of_not_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2209 t) :\n  ndinter (a::s) t = ndinter s t := by simp [ndinter, h]\n\n@[simp] theorem mem_ndinter {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndinter s t \u2194 a \u2208 s \u2227 a \u2208 t :=\nmem_filter\n\ntheorem nodup_ndinter {s : multiset \u03b1} (t : multiset \u03b1) : nodup s \u2192 nodup (ndinter s t) :=\nnodup_filter _\n\ntheorem le_ndinter {s t u : multiset \u03b1} : s \u2264 ndinter t u \u2194 s \u2264 t \u2227 s \u2286 u :=\nby simp [ndinter, le_filter, subset_iff]\n\ntheorem ndinter_le_left (s t : multiset \u03b1) : ndinter s t \u2264 s :=\n(le_ndinter.1 (le_refl _)).1\n\ntheorem ndinter_subset_right (s t : multiset \u03b1) : ndinter s t \u2286 t :=\n(le_ndinter.1 (le_refl _)).2\n\ntheorem ndinter_le_right {s} (t : multiset \u03b1) (d : nodup s) : ndinter s t \u2264 t :=\n(le_iff_subset $ nodup_ndinter _ d).2 (ndinter_subset_right _ _)\n\ntheorem inter_le_ndinter (s t : multiset \u03b1) : s \u2229 t \u2264 ndinter s t :=\nle_ndinter.2 \u27e8inter_le_left _ _, subset_of_le $ inter_le_right _ _\u27e9\n\n@[simp] theorem ndinter_eq_inter {s t : multiset \u03b1} (d : nodup s) : ndinter s t = s \u2229 t :=\nle_antisymm (le_inter (ndinter_le_left _ _) (ndinter_le_right _ d)) (inter_le_ndinter _ _)\n\ntheorem ndinter_eq_zero_iff_disjoint {s t : multiset \u03b1} : ndinter s t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\nend\n\n/- fold -/\nsection fold\nvariables (op : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [hc : is_commutative \u03b1 op] [ha : is_associative \u03b1 op]\nlocal notation a * b := op a b\ninclude hc ha\n\n/-- `fold op b s` folds a commutative associative operation `op` over\n  the multiset `s`. -/\ndef fold : \u03b1 \u2192 multiset \u03b1 \u2192 \u03b1 := foldr op (left_comm _ hc.comm ha.assoc)\n\ntheorem fold_eq_foldr (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldr op (left_comm _ hc.comm ha.assoc) b s := rfl\n\n@[simp] theorem coe_fold_r (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldr op b := rfl\n\ntheorem coe_fold_l (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldl op b :=\n(coe_foldr_swap op _ b l).trans $ by simp [hc.comm]\n\ntheorem fold_eq_foldl (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldl op (right_comm _ hc.comm ha.assoc) b s :=\nquot.induction_on s $ \u03bb l, coe_fold_l _ _ _\n\n@[simp] theorem fold_zero (b : \u03b1) : (0 : multiset \u03b1).fold op b = b := rfl\n\n@[simp] theorem fold_cons_left : \u2200 (b a : \u03b1) (s : multiset \u03b1),\n  (a :: s).fold op b = a * s.fold op b := foldr_cons _ _\n\ntheorem fold_cons_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op b * a :=\nby simp [hc.comm]\n\ntheorem fold_cons'_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (b * a) :=\nby rw [fold_eq_foldl, foldl_cons, \u2190 fold_eq_foldl]\n\ntheorem fold_cons'_left (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (a * b) :=\nby rw [fold_cons'_right, hc.comm]\n\ntheorem fold_add (b\u2081 b\u2082 : \u03b1) (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).fold op (b\u2081 * b\u2082) = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nmultiset.induction_on s\u2082\n  (by rw [add_zero, fold_zero, \u2190 fold_cons'_right, \u2190 fold_cons_right op])\n  (by simp {contextual := tt}; cc)\n\ntheorem fold_singleton (b a : \u03b1) : (a::0 : multiset \u03b1).fold op b = a * b := by simp\n\ntheorem fold_distrib {f g : \u03b2 \u2192 \u03b1} (u\u2081 u\u2082 : \u03b1) (s : multiset \u03b2) :\n  (s.map (\u03bbx, f x * g x)).fold op (u\u2081 * u\u2082) = (s.map f).fold op u\u2081 * (s.map g).fold op u\u2082 :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt}; cc)\n\ntheorem fold_hom {op' : \u03b2 \u2192 \u03b2 \u2192 \u03b2} [is_commutative \u03b2 op'] [is_associative \u03b2 op']\n  {m : \u03b1 \u2192 \u03b2} (hm : \u2200x y, m (op x y) = op' (m x) (m y)) (b : \u03b1) (s : multiset \u03b1) :\n  (s.map m).fold op' (m b) = m (s.fold op b) :=\nmultiset.induction_on s (by simp) (by simp [hm] {contextual := tt})\n\ntheorem fold_union_inter [decidable_eq \u03b1] (s\u2081 s\u2082 : multiset \u03b1) (b\u2081 b\u2082 : \u03b1) :\n  (s\u2081 \u222a s\u2082).fold op b\u2081 * (s\u2081 \u2229 s\u2082).fold op b\u2082 = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nby rw [\u2190 fold_add op, union_add_inter, fold_add op]\n\n@[simp] theorem fold_erase_dup_idem [decidable_eq \u03b1] [hi : is_idempotent \u03b1 op] (s : multiset \u03b1) (b : \u03b1) :\n  (erase_dup s).fold op b = s.fold op b :=\nmultiset.induction_on s (by simp) $ \u03bb a s IH, begin\n  by_cases a \u2208 s; simp [IH, h],\n  show fold op b s = op a (fold op b s),\n  rw [\u2190 cons_erase h, fold_cons_left, \u2190 ha.assoc, hi.idempotent],\nend\n\nend fold\n\ntheorem le_smul_erase_dup [decidable_eq \u03b1] (s : multiset \u03b1) :\n  \u2203 n : \u2115, s \u2264 n \u2022 erase_dup s :=\n\u27e8(s.map (\u03bb a, count a s)).fold max 0, le_iff_count.2 $ \u03bb a, begin\n  rw count_smul, by_cases a \u2208 s,\n  { refine le_trans _ (mul_le_mul_left _ $ count_pos.2 $ mem_erase_dup.2 h),\n    have : count a s \u2264 fold max 0 (map (\u03bb a, count a s) (a :: erase s a));\n    [simp [le_max_left], simpa [cons_erase h]] },\n  { simp [count_eq_zero.2 h, nat.zero_le] }\nend\u27e9\n\nsection sup\nvariables [semilattice_sup_bot \u03b1]\n\n/-- Supremum of a multiset: `sup {a, b, c} = a \u2294 b \u2294 c` -/\ndef sup (s : multiset \u03b1) : \u03b1 := s.fold (\u2294) \u22a5\n\n@[simp] lemma sup_zero : (0 : multiset \u03b1).sup = \u22a5 :=\nfold_zero _ _\n\n@[simp] lemma sup_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).sup = a \u2294 s.sup :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma sup_singleton {a : \u03b1} : (a::0).sup = a := by simp\n\n@[simp] lemma sup_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\neq.trans (by simp [sup]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma sup_erase_dup (s : multiset \u03b1) : (erase_dup s).sup = s.sup :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma sup_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).sup = a \u2294 s.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_cons]; simp\n\nlemma sup_le {s : multiset \u03b1} {a : \u03b1} : s.sup \u2264 a \u2194 (\u2200b \u2208 s, b \u2264 a) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma le_sup {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : a \u2264 s.sup :=\nsup_le.1 (le_refl _) _ h\n\nlemma sup_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2081.sup \u2264 s\u2082.sup :=\nsup_le.2 $ assume b hb, le_sup (h hb)\n\nend sup\n\nsection inf\nvariables [semilattice_inf_top \u03b1]\n\n/-- Infimum of a multiset: `inf {a, b, c} = a \u2293 b \u2293 c` -/\ndef inf (s : multiset \u03b1) : \u03b1 := s.fold (\u2293) \u22a4\n\n@[simp] lemma inf_zero : (0 : multiset \u03b1).inf = \u22a4 :=\nfold_zero _ _\n\n@[simp] lemma inf_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).inf = a \u2293 s.inf :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma inf_singleton {a : \u03b1} : (a::0).inf = a := by simp\n\n@[simp] lemma inf_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\neq.trans (by simp [inf]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma inf_erase_dup (s : multiset \u03b1) : (erase_dup s).inf = s.inf :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma inf_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).inf = a \u2293 s.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_cons]; simp\n\nlemma le_inf {s : multiset \u03b1} {a : \u03b1} : a \u2264 s.inf \u2194 (\u2200b \u2208 s, a \u2264 b) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma inf_le {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : s.inf \u2264 a :=\nle_inf.1 (le_refl _) _ h\n\nlemma inf_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2082.inf \u2264 s\u2081.inf :=\nle_inf.2 $ assume b hb, inf_le (h hb)\n\nend inf\n\nsection sort\nvariables (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [decidable_rel r]\n  [is_trans \u03b1 r] [is_antisymm \u03b1 r] [is_total \u03b1 r]\n\n/-- `sort s` constructs a sorted list from the multiset `s`.\n  (Uses merge sort algorithm.) -/\ndef sort (s : multiset \u03b1) : list \u03b1 :=\nquot.lift_on s (merge_sort r) $ \u03bb a b h,\neq_of_sorted_of_perm\n  ((perm_merge_sort _ _).trans $ h.trans (perm_merge_sort _ _).symm)\n  (sorted_merge_sort r _)\n  (sorted_merge_sort r _)\n\n@[simp] theorem coe_sort (l : list \u03b1) : sort r l = merge_sort r l := rfl\n\n@[simp] theorem sort_sorted (s : multiset \u03b1) : sorted r (sort r s) :=\nquot.induction_on s $ \u03bb l, sorted_merge_sort r _\n\n@[simp] theorem sort_eq (s : multiset \u03b1) : \u2191(sort r s) = s :=\nquot.induction_on s $ \u03bb l, quot.sound $ perm_merge_sort _ _\n\n@[simp] theorem mem_sort {s : multiset \u03b1} {a : \u03b1} : a \u2208 sort r s \u2194 a \u2208 s :=\nby rw [\u2190 mem_coe, sort_eq]\n\nend sort\n\ninstance [has_repr \u03b1] : has_repr (multiset \u03b1) :=\n\u27e8\u03bb s, \"{\" ++ string.intercalate \", \" ((s.map repr).sort (\u2264)) ++ \"}\"\u27e9\n\nsection sections\n\ndef sections (s : multiset (multiset \u03b1)) : multiset (multiset \u03b1) :=\nmultiset.rec_on s {0} (\u03bbs _ c, s.bind $ \u03bba, c.map ((::) a))\n  (assume a\u2080 a\u2081 s pi, by simp [map_bind, bind_bind a\u2080 a\u2081, cons_swap])\n\n@[simp] lemma sections_zero : sections (0 : multiset (multiset \u03b1)) = 0::0 :=\nrfl\n\n@[simp] lemma sections_cons (s : multiset (multiset \u03b1)) (m : multiset \u03b1) :\n  sections (m :: s) = m.bind (\u03bba, (sections s).map ((::) a)) :=\nrec_on_cons m s\n\nlemma coe_sections : \u2200(l : list (list \u03b1)),\n  sections ((l.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1)) =\n    ((l.sections.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1))\n| [] := rfl\n| (a :: l) :=\n  begin\n    simp,\n    rw [\u2190 cons_coe, sections_cons, bind_map_comm, coe_sections l],\n    simp [list.sections, (\u2218), list.bind]\n  end\n\n@[simp] lemma sections_add (s t : multiset (multiset \u03b1)) :\n  sections (s + t) = (sections s).bind (\u03bbm, (sections t).map ((+) m)) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, bind_assoc, map_bind, bind_map, -add_comm])\n\nlemma mem_sections {s : multiset (multiset \u03b1)} :\n  \u2200{a}, a \u2208 sections s \u2194 s.rel (\u03bbs a, a \u2208 s) a :=\nmultiset.induction_on s (by simp)\n  (assume a s ih a',\n    by simp [ih, rel_cons_left, -exists_and_distrib_left, exists_and_distrib_left.symm, eq_comm])\n\nlemma card_sections {s : multiset (multiset \u03b1)} : card (sections s) = prod (s.map card) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma prod_map_sum [comm_semiring \u03b1] {s : multiset (multiset \u03b1)} :\n  prod (s.map sum) = sum ((sections s).map prod) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, map_bind, sum_map_mul_left, sum_map_mul_right])\n\nend sections\n\nsection pi\nvariables [decidable_eq \u03b1] {\u03b4 : \u03b1 \u2192 Type*}\nopen function\n\ndef pi.cons (m : multiset \u03b1) (a : \u03b1) (b : \u03b4 a) (f : \u03a0a\u2208m, \u03b4 a) : \u03a0a'\u2208a::m, \u03b4 a' :=\n\u03bba' ha', if h : a' = a then eq.rec b h.symm else f a' $ (mem_cons.1 ha').resolve_left h\n\ndef pi.empty (\u03b4 : \u03b1 \u2192 Type*) : (\u03a0a\u2208(0:multiset \u03b1), \u03b4 a) .\n\nlemma pi.cons_same {m : multiset \u03b1} {a : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2208 a :: m) :\n  pi.cons m a b f a h = b :=\ndif_pos rfl\n\nlemma pi.cons_ne {m : multiset \u03b1} {a a' : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h' : a' \u2208 a :: m) (h : a' \u2260 a) :\n  pi.cons m a b f a' h' = f a' ((mem_cons.1 h').resolve_left h) :=\ndif_neg h\n\nlemma pi.cons_swap {a a' : \u03b1} {b : \u03b4 a} {b' : \u03b4 a'} {m : multiset \u03b1} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2260 a') :\n  pi.cons (a' :: m) a b (pi.cons m a' b' f) == pi.cons (a :: m) a' b' (pi.cons m a b f) :=\nbegin\n  apply hfunext, { refl }, intros a'' _ h, subst h,\n  apply hfunext, { rw [cons_swap] }, intros ha\u2081 ha\u2082 h,\n  by_cases h\u2081 : a'' = a; by_cases h\u2082 : a'' = a';\n    simp [*, pi.cons_same, pi.cons_ne] at *,\n  { subst h\u2081, rw [pi.cons_same, pi.cons_same] },\n  { subst h\u2082, rw [pi.cons_same, pi.cons_same] }\nend\n\n/-- `pi m t` constructs the Cartesian product over `t` indexed by `m`. -/\ndef pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) : multiset (\u03a0a\u2208m, \u03b4 a) :=\nm.rec_on {pi.empty \u03b4} (\u03bba m (p : multiset (\u03a0a\u2208m, \u03b4 a)), (t a).bind $ \u03bbb, p.map $ pi.cons m a b)\nbegin\n  intros a a' m n,\n  by_cases eq : a = a',\n  { subst eq },\n  { simp [map_bind, bind_bind (t a') (t a)],\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b hb,\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b' hb',\n    apply map_hcongr, { rw [cons_swap a a'] },\n    intros f hf,\n    exact pi.cons_swap eq }\nend\n\n@[simp] lemma pi_zero (t : \u03a0a, multiset (\u03b4 a)) : pi 0 t = pi.empty \u03b4 :: 0 := rfl\n\n@[simp] lemma pi_cons (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) (a : \u03b1) :\n  pi (a :: m) t = ((t a).bind $ \u03bbb, (pi m t).map $ pi.cons m a b) :=\nrec_on_cons a m\n\nlemma injective_pi_cons {a : \u03b1} {b : \u03b4 a} {s : multiset \u03b1} (hs : a \u2209 s) :\n  function.injective (pi.cons s a b) :=\nassume f\u2081 f\u2082 eq, funext $ assume a', funext $ assume h',\nhave ne : a \u2260 a', from assume h, hs $ h.symm \u25b8 h',\nhave a' \u2208 a :: s, from mem_cons_of_mem h',\ncalc f\u2081 a' h' = pi.cons s a b f\u2081 a' this : by rw [pi.cons_ne this ne.symm]\n  ... = pi.cons s a b f\u2082 a' this : by rw [eq]\n  ... = f\u2082 a' h' : by rw [pi.cons_ne this ne.symm]\n\nlemma card_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  card (pi m t) = prod (m.map $ \u03bba, card (t a)) :=\nmultiset.induction_on m (by simp) (by simp [mul_comm] {contextual := tt})\n\nlemma nodup_pi {s : multiset \u03b1} {t : \u03a0a, multiset (\u03b4 a)} :\n  nodup s \u2192 (\u2200a\u2208s, nodup (t a)) \u2192 nodup (pi s t) :=\nmultiset.induction_on s (assume _ _, nodup_singleton _)\nbegin\n  assume a s ih hs ht,\n  have has : a \u2209 s, by simp at hs; exact hs.1,\n  have hs : nodup s, by simp at hs; exact hs.2,\n  simp,\n  split,\n  { assume b hb,\n    from nodup_map (injective_pi_cons has) (ih hs $ assume a' h', ht a' $ mem_cons_of_mem h') },\n  { apply pairwise_of_nodup _ (ht a $ mem_cons_self _ _),\n    from assume b\u2081 hb\u2081 b\u2082 hb\u2082 neb, disjoint_map_map.2 (assume f hf g hg 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 _ _),\n        by rw [eq],\n      neb $ show b\u2081 = b\u2082, by rwa [pi.cons_same, pi.cons_same] at this) }\nend\n\nlemma mem_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  \u2200f:\u03a0a\u2208m, \u03b4 a, (f \u2208 pi m t) \u2194 (\u2200a (h : a \u2208 m), f a h \u2208 t a) :=\nbegin\n  refine multiset.induction_on m (\u03bb f, _) (\u03bb a m ih f, _),\n  { simpa using show f = pi.empty \u03b4, by funext a ha; exact ha.elim },\n  simp, split,\n  { rintro \u27e8b, hb, f', hf', rfl\u27e9 a' ha',\n    rw [ih] at hf',\n    by_cases a' = a,\n    { subst h, rwa [pi.cons_same] },\n    { rw [pi.cons_ne _ h], apply hf' } },\n  { intro hf,\n    refine \u27e8_, hf a (mem_cons_self a _), \u03bba ha, f a (mem_cons_of_mem ha),\n      (ih _).2 (\u03bb a' h', hf _ _), _\u27e9,\n    funext a' h',\n    by_cases a' = a,\n    { subst h, rw [pi.cons_same] },\n    { rw [pi.cons_ne _ h] } }\nend\n\nend pi\nend multiset\n\nnamespace multiset\n\ninstance : functor multiset :=\n{ map := @map }\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u_1 \u2192 Type u_1} [applicative F] [is_comm_applicative F]\nvariables {\u03b1' \u03b2' : Type u_1} (f : \u03b1' \u2192 F \u03b2')\n\ndef traverse : multiset \u03b1' \u2192 F (multiset \u03b2') :=\nquotient.lift (functor.map coe \u2218 traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.skip {\n    have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2081) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2082),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap {\n    have : (\u03bba b (l:list \u03b2'), (\u2191(a :: b :: l) : multiset \u03b2')) <$> f p_y <*> f p_x =\n      (\u03bba b l, \u2191(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(\u2218), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\ninstance : monad multiset :=\n{ pure := \u03bb \u03b1 x, x::0,\n  bind := @bind,\n  .. multiset.functor }\n\ninstance : is_lawful_monad multiset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, multiset.induction_on s rfl $ \u03bb a s ih,\n    by rw [bind_cons, map_cons, bind_zero, add_zero],\n  pure_bind := \u03bb \u03b1 \u03b2 x f, by simp only [cons_bind, zero_bind, add_zero],\n  bind_assoc := @bind_assoc }\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_beta {\u03b1 \u03b2 : Type*} (x : list \u03b1) (f : list \u03b1 \u2192 \u03b2)\n  (h : \u2200 a b : list \u03b1, a \u2248 b \u2192 f a = f b) :\n  quotient.lift f h (x : multiset \u03b1) = f x :=\nquotient.lift_beta _ _ _\n\n@[simp]\nlemma map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 coe = (coe \u2218 functor.map h : list \u03b1 \u2192 multiset \u03b2) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {\u03b1 : Type*} (x : multiset \u03b1) :\n  traverse id.mk x = x :=\nquotient.induction_on x\n(by { intro, rw [traverse,quotient.lift_beta,function.comp],\n      simp, congr })\n\nlemma comp_traverse {G H : Type* \u2192 Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : multiset \u03b1) :\n  traverse (comp.mk \u2218 functor.map h \u2218 g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(\u2218)] with functor_norm)\n\nlemma map_traverse {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3)\n               (x : multiset \u03b1) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [comp_map,map_traverse])\n\nlemma traverse_map {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3)\n               (x : multiset \u03b1) :\n  traverse h (map g x) =\n  traverse (h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [\u2190 traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* \u2192 Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 G \u03b2) (x : multiset \u03b1) :\n  eta (traverse f x) = traverse (@eta _ \u2218 f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nsection choose\nvariables (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n\ndef choose_x : \u03a0 hp : (\u2203! a, a \u2208 l \u2227 p a), { a // a \u2208 l \u2227 p a } :=\nquotient.rec_on l (\u03bb l' ex_unique, list.choose_x p l' (exists_of_exists_unique ex_unique)) begin\n  intros,\n  funext hp,\n  suffices all_equal : \u2200 x y : { t // t \u2208 b \u2227 p t }, x = y,\n  { apply all_equal },\n  { rintros \u27e8x, px\u27e9 \u27e8y, py\u27e9,\n    rcases hp with \u27e8z, \u27e8z_mem_l, pz\u27e9, z_unique\u27e9,\n    congr,\n    calc x = z : z_unique x px\n    ...    = y : (z_unique y py).symm }\nend\n\ndef choose (hp : \u2203! a, a \u2208 l \u2227 p a) : \u03b1 := choose_x p l hp\n\nlemma choose_spec (hp : \u2203! a, a \u2208 l \u2227 p a) : choose p l hp \u2208 l \u2227 p (choose p l hp) :=\n(choose_x p l hp).property\n\nlemma choose_mem (hp : \u2203! a, a \u2208 l \u2227 p a) : choose p l hp \u2208 l := (choose_spec _ _ _).1\n\nlemma choose_property (hp : \u2203! a, a \u2208 l \u2227 p a) : p (choose p l hp) := (choose_spec _ _ _).2\n\nend choose\n\n/- Ico -/\n\n/-- `Ico n m` is the multiset lifted from the list `Ico n m`, e.g. the set `{n, n+1, ..., m-1}`. -/\ndef Ico (n m : \u2115) : multiset \u2115 := Ico n m\n\nnamespace Ico\n\ntheorem map_add (n m k : \u2115) : (Ico n m).map ((+) k) = Ico (n + k) (m + k) :=\ncongr_arg coe $ list.Ico.map_add _ _ _\n\ntheorem map_sub (n m k : \u2115) (h : k \u2264 n) : (Ico n m).map (\u03bb x, x - k) = Ico (n - k) (m - k) :=\ncongr_arg coe $ list.Ico.map_sub _ _ _ h\n\ntheorem zero_bot (n : \u2115) : Ico 0 n = range n :=\ncongr_arg coe $ list.Ico.zero_bot _\n\n@[simp] theorem card (n m : \u2115) : (Ico n m).card = m - n :=\nlist.Ico.length _ _\n\ntheorem nodup (n m : \u2115) : nodup (Ico n m) := Ico.nodup _ _\n\n@[simp] theorem mem {n m l : \u2115} : l \u2208 Ico n m \u2194 n \u2264 l \u2227 l < m :=\nlist.Ico.mem\n\ntheorem eq_zero_of_le {n m : \u2115} (h : m \u2264 n) : Ico n m = 0 :=\ncongr_arg coe $ list.Ico.eq_nil_of_le h\n\n@[simp] theorem self_eq_zero {n : \u2115} : Ico n n = 0 :=\neq_zero_of_le $ le_refl n\n\n@[simp] theorem eq_zero_iff {n m : \u2115} : Ico n m = 0 \u2194 m \u2264 n :=\niff.trans (coe_eq_zero _) list.Ico.eq_empty_iff\n\nlemma add_consecutive {n m l : \u2115} (hnm : n \u2264 m) (hml : m \u2264 l) :\n  Ico n m + Ico m l = Ico n l :=\ncongr_arg coe $ list.Ico.append_consecutive hnm hml\n\n@[simp] lemma inter_consecutive (n m l : \u2115) : Ico n m \u2229 Ico m l = 0 :=\ncongr_arg coe $ list.Ico.bag_inter_consecutive n m l\n\n@[simp] theorem succ_singleton {n : \u2115} : Ico n (n+1) = {n} :=\ncongr_arg coe $ list.Ico.succ_singleton\n\ntheorem succ_top {n m : \u2115} (h : n \u2264 m) : Ico n (m + 1) = m :: Ico n m :=\nby rw [Ico, list.Ico.succ_top h, \u2190 coe_add, add_comm]; refl\n\ntheorem eq_cons {n m : \u2115} (h : n < m) : Ico n m = n :: Ico (n + 1) m :=\ncongr_arg coe $ list.Ico.eq_cons h\n\n@[simp] theorem pred_singleton {m : \u2115} (h : m > 0) : Ico (m - 1) m = {m - 1} :=\ncongr_arg coe $ list.Ico.pred_singleton h\n\n@[simp] theorem not_mem_top {n m : \u2115} : m \u2209 Ico n m :=\nlist.Ico.not_mem_top\n\nlemma filter_lt_of_top_le {n m l : \u2115} (hml : m \u2264 l) : (Ico n m).filter (\u03bb x, x < l) = Ico n m :=\ncongr_arg coe $ list.Ico.filter_lt_of_top_le hml\n\nlemma filter_lt_of_le_bot {n m l : \u2115} (hln : l \u2264 n) : (Ico n m).filter (\u03bb x, x < l) = \u2205 :=\ncongr_arg coe $ list.Ico.filter_lt_of_le_bot hln\n\nlemma filter_lt_of_ge {n m l : \u2115} (hlm : l \u2264 m) : (Ico n m).filter (\u03bb x, x < l) = Ico n l :=\ncongr_arg coe $ list.Ico.filter_lt_of_ge hlm\n\n@[simp] lemma filter_lt (n m l : \u2115) : (Ico n m).filter (\u03bb x, x < l) = Ico n (min m l) :=\ncongr_arg coe $ list.Ico.filter_lt n m l\n\nlemma filter_ge_of_le_bot {n m l : \u2115} (hln : l \u2264 n) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico n m :=\ncongr_arg coe $ list.Ico.filter_ge_of_le_bot hln\n\nlemma filter_ge_of_top_le {n m l : \u2115} (hml : m \u2264 l) : (Ico n m).filter (\u03bb x, x \u2265 l) = \u2205 :=\ncongr_arg coe $ list.Ico.filter_ge_of_top_le hml\n\nlemma filter_ge_of_ge {n m l : \u2115} (hnl : n \u2264 l) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico l m :=\ncongr_arg coe $ list.Ico.filter_ge_of_ge hnl\n\n@[simp] lemma filter_ge (n m l : \u2115) : (Ico n m).filter (\u03bb x, x \u2265 l) = Ico (max n l) m :=\ncongr_arg coe $ list.Ico.filter_ge n m l\n\nend Ico\n\nend multiset\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/data/multiset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6992544210587585, "lm_q1q2_score": 0.36872841027049647}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes H\u00f6lzl, Simon Hudon, Kenny Lau\n-/\nimport data.multiset.bind\nimport control.traversable.lemmas\nimport control.traversable.instances\n\n/-!\n# Functoriality of `multiset`.\n-/\n\nuniverses u\n\nnamespace multiset\n\nopen list\n\ninstance : functor multiset :=\n{ map := @map }\n\n@[simp] lemma fmap_def {\u03b1' \u03b2'} {s : multiset \u03b1'} (f : \u03b1' \u2192 \u03b2') : f <$> s = s.map f := rfl\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u \u2192 Type u} [applicative F] [is_comm_applicative F]\nvariables {\u03b1' \u03b2' : Type u} (f : \u03b1' \u2192 F \u03b2')\n\ndef traverse : multiset \u03b1' \u2192 F (multiset \u03b2') :=\nquotient.lift (functor.map coe \u2218 traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.cons\n  { have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2081) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2082),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap\n  { have : (\u03bba b (l:list \u03b2'), (\u2191(a :: b :: l) : multiset \u03b2')) <$> f p_y <*> f p_x =\n      (\u03bba b l, \u2191(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(\u2218), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\ninstance : monad multiset :=\n{ pure := \u03bb \u03b1 x, {x},\n  bind := @bind,\n  .. multiset.functor }\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 multiset \u03b1) = singleton := rfl\n@[simp] lemma bind_def {\u03b1 \u03b2} : (>>=) = @bind \u03b1 \u03b2 := rfl\n\ninstance : is_lawful_monad multiset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, multiset.induction_on s rfl $ \u03bb a s ih, by simp,\n  pure_bind := \u03bb \u03b1 \u03b2 x f, by simp [pure],\n  bind_assoc := @bind_assoc }\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_coe {\u03b1 \u03b2 : Type*} (x : list \u03b1) (f : list \u03b1 \u2192 \u03b2)\n  (h : \u2200 a b : list \u03b1, a \u2248 b \u2192 f a = f b) :\n  quotient.lift f h (x : multiset \u03b1) = f x :=\nquotient.lift_mk _ _ _\n\n@[simp]\nlemma map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 coe = (coe \u2218 functor.map h : list \u03b1 \u2192 multiset \u03b2) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {\u03b1 : Type*} (x : multiset \u03b1) :\n  traverse id.mk x = x :=\nquotient.induction_on x begin intro, simp [traverse], refl end\n\nlemma comp_traverse {G H : Type* \u2192 Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : multiset \u03b1) :\n  traverse (comp.mk \u2218 functor.map h \u2218 g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(\u2218)] with functor_norm)\n\nlemma map_traverse {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3)\n               (x : multiset \u03b1) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [is_lawful_functor.comp_map, map_traverse])\n\nlemma traverse_map {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3)\n               (x : multiset \u03b1) :\n  traverse h (map g x) =\n  traverse (h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [\u2190 traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* \u2192 Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 G \u03b2) (x : multiset \u03b1) :\n  eta (traverse f x) = traverse (@eta _ \u2218 f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nend multiset\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/multiset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.3687177306974765}}
{"text": "-- Copyright (c) 2017 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Stephen Morgan, Scott Morrison\n\nimport category_theory.functor\nimport .initial\n\nopen category_theory\nopen category_theory.initial\n\nnamespace category_theory.universal\n\nuniverses u v\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [\ud835\udc9e : category.{u v} C]\ninclude \ud835\udc9e \n\nstructure Cone (F : J \u219d C) : Type (max u v) :=\n(cone_point    : C)\n(cone_maps     : \u03a0 j : J, cone_point \u27f6 (F j))\n(commutativity : \u03a0 {j k : J}, \u03a0 f : j \u27f6 k, (cone_maps j) \u226b (F.map f) = cone_maps k . obviously)\n\nrestate_axiom Cone.commutativity\nattribute [simp,ematch] Cone.commutativity_lemma\n\nvariable {F : J \u219d C}\n\nstructure ConeMorphism (X Y : Cone F) : Type v :=\n(cone_morphism : X.cone_point \u27f6 Y.cone_point)\n(commutativity : \u03a0 j : J, cone_morphism \u226b (Y.cone_maps j) = (X.cone_maps j) . obviously)\n\nrestate_axiom ConeMorphism.commutativity\nattribute [simp,ematch] ConeMorphism.commutativity_lemma\n\nnamespace ConeMorphism\n\n@[simp,ematch] def commutativity_lemma_assoc {X Y : Cone F} (c : ConeMorphism X Y) (j : J) {Z : C} (z : (F j) \u27f6 Z): c.cone_morphism \u226b Y.cone_maps j \u226b z = X.cone_maps j \u226b z :=\nbegin\n  /- obviously' say: -/\n  rw \u2190 category.assoc,\n  simp,\nend\n\n@[extensionality] lemma ext {X Y : Cone F} {f g : ConeMorphism X Y} (w : f.cone_morphism = g.cone_morphism) : f = g :=\nbegin\n  /- obviously' say: -/\n  induction f,\n  induction g,\n  dsimp at w,\n  induction w,\n  refl,\nend\n\nend ConeMorphism\n\ninstance Cones (F : J \u219d C) : category.{(max u v) v} (Cone F) :=\n{ hom      := \u03bb X Y, ConeMorphism X Y,\n  comp    := \u03bb X Y Z f g, { cone_morphism := f.cone_morphism \u226b g.cone_morphism,\n                            commutativity := begin /- `obviously'` says: -/ intros, simp end },\n  id      := \u03bb X, { cone_morphism := \ud835\udfd9 X.cone_point, \n                    commutativity := begin /- `obviously'` says: -/ intros, simp end },\n  id_comp := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end,\n  comp_id := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end,\n  assoc   := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end }\n\n-- TODO rename or namespace?\n@[simp] lemma Cones.identity.cone_morphism {F : J \u219d C} (c : Cone F) : (\ud835\udfd9 c : ConeMorphism c c).cone_morphism = \ud835\udfd9 (c.cone_point) := rfl\n@[simp] lemma Cones.compose.cone_morphism {F : J \u219d C} {c d e : Cone F} (f : c \u27f6 d) (g : d \u27f6 e) : ((f \u226b g) : ConeMorphism c e).cone_morphism = (f : ConeMorphism c d).cone_morphism \u226b (g : ConeMorphism d e).cone_morphism := rfl\n\nsection\nvariables {D : Type u} [\ud835\udc9f : category.{u v} D]\ninclude \ud835\udc9f\n\ndef Cones_functoriality (F : J \u219d C) (G : C \u219d D) : (Cone F) \u219d (Cone (F \u22d9 G)) := \n{ obj      := \u03bb X, { cone_point    := G X.cone_point,\n                     cone_maps     := \u03bb j, G.map (X.cone_maps j), \n                     commutativity := begin /- `obviously'` says: -/ intros, simp, erw [\u2190functor.map_comp_lemma, Cone.commutativity_lemma] end },\n  map      := \u03bb X Y f, { cone_morphism := G.map f.cone_morphism,\n                         commutativity := begin /- `obviously'` says: -/ intros, dsimp, erw [\u2190functor.map_comp_lemma, ConeMorphism.commutativity_lemma] end },\n  map_id   := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end,\n  map_comp := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end }\nend\n\nstructure Cocone (F : J \u219d C) :=\n(cocone_point  : C)\n(cocone_maps   : \u03a0 j : J, (F j) \u27f6 cocone_point)\n(commutativity : \u03a0 {j k : J}, \u03a0 f : j \u27f6 k, (F.map f) \u226b (cocone_maps k) = cocone_maps j . obviously)\n\nrestate_axiom Cocone.commutativity\nattribute [simp,ematch] Cocone.commutativity_lemma\n\nstructure CoconeMorphism (X Y : Cocone F) :=\n(cocone_morphism : X.cocone_point \u27f6 Y.cocone_point)\n(commutativity   : \u03a0 j : J, (X.cocone_maps j) \u226b cocone_morphism = (Y.cocone_maps j) . obviously)\n\nrestate_axiom CoconeMorphism.commutativity\nattribute [simp,ematch] CoconeMorphism.commutativity_lemma\n\nnamespace CoconeMorphism\n@[simp,ematch] def commutativity_lemma_assoc {X Y : Cocone F} (c : CoconeMorphism X Y) (j : J) {Z : C} (z : Y.cocone_point \u27f6 Z): (X.cocone_maps j) \u226b c.cocone_morphism \u226b z = (Y.cocone_maps j) \u226b z :=\nbegin\n  -- `obviously'` says:\n  erw [\u2190category.assoc_lemma, CoconeMorphism.commutativity_lemma]\nend\n\n@[extensionality] lemma ext {X Y : Cocone F} {f g : CoconeMorphism X Y} (w : f.cocone_morphism = g.cocone_morphism) : f = g :=\nbegin \n  induction f,\n  induction g,\n  -- `obviously'` says:\n  dsimp at *,\n  induction w,\n  refl,\nend\nend CoconeMorphism\n\ninstance Cocones (F : J \u219d C) : category.{(max u v) v} (Cocone F) := \n{ hom     := \u03bb X Y, CoconeMorphism X Y,\n  comp    := \u03bb X Y Z f g, { cocone_morphism := f.cocone_morphism \u226b g.cocone_morphism,\n                            commutativity   := begin /- `obviously'` says: -/ intros, simp end },\n  id      := \u03bb X,         { cocone_morphism := \ud835\udfd9 X.cocone_point,\n                            commutativity   := begin /- `obviously'` says: -/ intros, simp end },\n  id_comp := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end,\n  comp_id := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end,\n  assoc   := begin /- `obviously'` says: -/ intros, ext, dsimp, simp end }\n\n-- TODO rename or namespace?\n@[simp] lemma Cocones.identity.cone_morphism {F : J \u219d C} (c : Cocone F) : (\ud835\udfd9 c : CoconeMorphism c c).cocone_morphism = \ud835\udfd9 (c.cocone_point) := rfl\n@[simp] lemma Cocones.compose.cone_morphism {F : J \u219d C} {c d e : Cocone F} (f : c \u27f6 d) (g : d \u27f6 e) : ((f \u226b g) : CoconeMorphism c e).cocone_morphism = (f : CoconeMorphism c d).cocone_morphism \u226b (g : CoconeMorphism d e).cocone_morphism := rfl\n\nsection\nvariables {D : Type u} [\ud835\udc9f : category.{u v} D]\ninclude \ud835\udc9f\n\ndef Cocones_functoriality (F : J \u219d C) (G : C \u219d D) : (Cocone F) \u219d (Cocone (F \u22d9 G)) := \n{ obj      := \u03bb X,     { cocone_point    := G X.cocone_point,\n                         cocone_maps     := \u03bb j, G.map (X.cocone_maps j),\n                         commutativity   := begin /- `obviously'` says: -/ intros, simp, erw [\u2190functor.map_comp_lemma, Cocone.commutativity_lemma] end },\n  map      := \u03bb X Y f, { cocone_morphism := G.map f.cocone_morphism,\n                         commutativity   := begin /- `obviously'` says: -/ intros, dsimp, erw [\u2190functor.map_comp_lemma, CoconeMorphism.commutativity_lemma] end },\n  map_id   := begin /- `obviously'` says -/ intros, ext, dsimp, simp end,\n  map_comp := begin /- `obviously'` says -/ intros, ext, dsimp, simp end }\nend\n\ndef LimitCone     (F : J \u219d C) := terminal_object (Cone F)\ndef ColimitCocone (F : J \u219d C) := initial_object (Cocone F)\n\nend category_theory.universal\n\nnamespace category_theory.functor\n\nuniverses u v\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [category.{u v} C] {D : Type u} [category.{u v} D]\nvariable {F : J \u219d C}\n\nopen category_theory.universal\n\ndef on_cone   (G : C \u219d D) (c : Cone F)   : Cone (F \u22d9 G)   := (Cones_functoriality F G) c\ndef on_cocone (G : C \u219d D) (c : Cocone F) : Cocone (F \u22d9 G) := (Cocones_functoriality F G) c\n\nend category_theory.functor", "meta": {"author": "semorrison", "repo": "lean-category-theory-pr", "sha": "7adc8d91835e883db0fe75aa33661bc1480dbe55", "save_path": "github-repos/lean/semorrison-lean-category-theory-pr", "path": "github-repos/lean/semorrison-lean-category-theory-pr/lean-category-theory-pr-7adc8d91835e883db0fe75aa33661bc1480dbe55/src/categories/universal/cones.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.36871772248069956}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\nimport measure_theory.set_integral\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.core\nimport formal_ml.measurable_space\nimport formal_ml.semiring\nimport formal_ml.real_measurable_space\nimport formal_ml.set\nimport formal_ml.filter_util\nimport topology.instances.ennreal\nimport formal_ml.int\nimport formal_ml.with_density_compose_eq_multiply\nimport formal_ml.classical\nimport formal_ml.restrict\nimport formal_ml.with_top\n\n/-\n  This does not prove the classic Hahn decomposition theorem for signed measures.\n  Instead, it proves a similar theorem for unsigned measures. To understand various\n  operations on unsigned measures (like the supremum of two measures, the infimum of\n  two measures, or the \"difference\" of two measures), it is important to be able\n  to consider two non-negative measures and partition the space into a set where\n  the first measure is less than or equal to the second, and a set where the second\n  is less than or equal to the first. So, given measures \u03bc and \u03bd, we want a measurable\n  set S such that:\n\n  \u03bc.restrict S \u2264 \u03bd.restrict S and  \u03bd.restrict S\u1d9c \u2264 \u03bc.restrict S\u1d9c.\n\n  This means that for any set T \u2286 S, \u03bc T \u2264 \u03bd T, and for any U \u2286 S\u1d9c, \u03bd U \u2264 \u03bc U.\n\n  By partitioning the space like so, we can separately consider supremum, infimum,\n  and difference in each half, and combine them back together. This in turn\n  helps to prove the Lebesgue-Radon-Nikodym theorem.\n\n  Note that {S|measurable_set S \u2227 \u03bc.restrict S \u2264 \u03bd.restrict S} is a ring of sets.\n\n -/\n\n\n\n\nlemma nonnegative_fn {\u03b1 \u03b2:Type*} [canonically_ordered_add_monoid \u03b2] {f:\u03b1 \u2192 \u03b2}:0 \u2264 f :=\nbegin\n  intro x,\n  simp,\nend\n\nlemma ennreal.tsum {\u03b1:Type*} {f:\u03b1 \u2192 ennreal}:tsum f = supr (\u03bb s:finset \u03b1,s.sum f) :=\nbegin\n  rw \u2190 summable.has_sum_iff ennreal.summable,\n  apply @ennreal.has_sum \u03b1 f,\nend\n\nlemma ennreal.tsum_le {f:\u2115 \u2192 ennreal} {x:ennreal}:(\u2200 n:\u2115,\n(finset.range n).sum f \u2264 x) \u2194\n(tsum f \u2264 x) :=\nbegin\n  rw ennreal.tsum,\n\n  split;intros A1,\n  {\n    apply @supr_le ennreal (finset \u2115) _ (\u03bb s:finset \u2115,s.sum f),\n    intro s,\n    have B1:=finset_range_bound s,\n    cases B1 with n B1,\n    apply le_trans _ (A1 n),\n    simp,\n    apply @finset.sum_le_sum_of_subset \u2115 ennreal s (finset.range n) f _ B1,\n  },\n  {\n    intro n,\n    apply le_trans _ A1,\n    apply @le_supr ennreal (finset \u2115) _ (\u03bb s, s.sum f),\n  },\nend\n\n\nlemma ennreal.finset_sum_le_tsum {f:\u2115 \u2192 ennreal} {n:\u2115}:\n(finset.range n).sum f \u2264 tsum f :=\nbegin\n  rw ennreal.tsum,\n  apply @le_supr ennreal (finset \u2115) _ (\u03bb s, s.sum f),\nend \n\n\nlemma ennreal.lim_finset_sum_eq_tsum {f:\u2115 \u2192 ennreal}:\n(\u2a06  n, (finset.range n).sum f)  = tsum f :=\nbegin\n  apply le_antisymm,\n  apply @supr_le ennreal _ _,\n  intro n,\n  apply ennreal.finset_sum_le_tsum,\n  rw \u2190 ennreal.tsum_le,\n  intro n,\n  apply @le_supr ennreal _ _,\nend\n\nlemma ennreal.Sup_eq_supr {S:set ennreal}:S.nonempty \u2192\n   (\u2203 f:\u2115 \u2192 ennreal, \n      (\u2200 n, f n \u2208 S) \u2227\n      (supr f = Sup S)) :=\nbegin\n  intro A1,\n  cases (classical.em (Sup S = \u22a4)) with A2 A2, \n  {\n    -- If Sup S = \u22a4, then we have a sequence that is larger than every natural.\n    have B1:\u2200 n:\u2115, \u2203 s:ennreal,  (s\u2208 S) \u2227 (n:ennreal) \u2264 s,\n    {\n      intro n,\n      rw Sup_eq_top at A2,\n      have B1A:=A2 n _,\n      cases B1A with s B1A,\n      cases B1A with B1A B1B,\n      apply exists.intro s,\n      apply and.intro B1A (@le_of_lt ennreal _ _ _ B1B),\n      have B1C:(n:ennreal) = ((n:nnreal):ennreal),\n      {\n        simp,\n      },\n      rw B1C,\n      apply ennreal.coe_lt_top,\n    },\n    have B2:=classical.some_func B1,\n    cases B2 with f B2,\n    apply exists.intro f,\n    simp at B2,\n    split,\n    {\n      intro n,\n      apply (B2 n).left,\n    },\n    {\n      rw A2,\n      rw \u2190 top_le_iff,\n      rw \u2190 ennreal.supr_coe_nat,\n      apply @supr_le_supr ennreal \u2115 _,\n      intro n,\n      apply (B2 n).right,\n    },\n  },\n  {\n    have C1:\u2200 n:\u2115, \u2203 s:ennreal, (s\u2208S) \u2227 (Sup S - (1/(n.succ:ennreal)) \u2264 s),\n    {\n      intro n,\n      have C1A:=@ennreal.Sup_elim S (1/(n.succ:nnreal)) _ A1 A2,\n      cases C1A with s C1A,\n      cases C1A with C1A C1B,\n      apply exists.intro s,\n      apply and.intro C1A _,\n      rw ennreal.coe_div at C1B,\n      simp at C1B,\n      simp,\n      apply C1B,\n      simp,\n      simp,\n      apply zero_lt_one,  \n    },\n    have C2:=classical.some_func C1,\n    cases C2 with f C2,\n    apply exists.intro f,\n    split,\n    {\n      intro n,\n      apply (C2 n).left,\n    },\n    apply le_antisymm,\n    {\n      apply @supr_le ennreal _ _,\n      intro i,\n      have C3 := C2 i,\n      apply @le_Sup ennreal _ _,\n      apply C3.left,\n    },\n    apply @ennreal.le_of_forall_pos_le_add,\n    intros \u03b5 C4 C5,\n    have C6 := nnreal.exists_unit_frac_lt_pos C4,\n    cases C6 with n C6,\n    have C7 := C2 n,  \n    have C8:Sup S \u2264 f n + \u03b5,\n    {\n      have C8A:1/ ((nat.succ n):ennreal) \u2264 (\u03b5:ennreal),\n      {\n        simp,\n        rw \u2190 ennreal.coe_one,\n        have C8A1:(n:ennreal) = ((n:nnreal):ennreal),\n        {\n          simp,\n        },\n        rw C8A1,\n        rw \u2190 ennreal.coe_add,\n        rw \u2190 ennreal.coe_inv,\n        rw ennreal.coe_le_coe,\n        apply le_of_lt,\n        rw \u2190 one_div,\n        apply C6,\n        simp,\n      },\n      have C8B:Sup S - (\u03b5:ennreal) \u2264\n               Sup S - 1/ ((nat.succ n):ennreal),\n      {\n        apply ennreal.sub_le_sub,\n        apply le_refl (Sup S),\n        apply C8A,    \n      },\n      have C8C:Sup S - (\u03b5:ennreal) \u2264 f n,\n      {\n        apply le_trans C8B C7.right,\n      },\n      rw add_comm,\n      rw \u2190 ennreal.sub_le_iff_le_add',\n      apply C8C,   \n    },\n    apply le_trans C8,\n    have C9 := le_supr f n,\n    apply add_le_add_right C9,\n  },\nend\n\n\ndef Sup_so_far {\u03b1:Type*} [semilattice_sup_bot \u03b1] (f:\u2115 \u2192 \u03b1) (n:\u2115):\u03b1 :=\n  (finset.range (n.succ)).sup f\n\n\nlemma Sup_so_far_def {\u03b1:Type*} [semilattice_sup_bot \u03b1] {f:\u2115 \u2192 \u03b1} \n    {n:\u2115}:\n    Sup_so_far f n = (finset.range (n.succ)).sup f := rfl\n\n\nlemma le_Sup_so_far {\u03b1:Type*} [semilattice_sup_bot \u03b1] (f:\u2115 \u2192 \u03b1):\n   f \u2264 (Sup_so_far f) := \nbegin\n  rw le_func_def2,\n  intro n,\n  rw Sup_so_far_def,\n  apply finset.le_sup,\n  simp,\nend\n\nlemma finset.sup_closed_nonempty {\u03b1 \u03b2:Type*} [decidable_eq \u03b1] \n   [semilattice_sup_bot \u03b2]\n   (S:finset \u03b1) \n   (f:\u03b1 \u2192 \u03b2) (T:set \u03b2):\n   (S \u2260 \u2205) \u2192\n   (\u2200 a\u2208 S, f a \u2208 T) \u2192\n   (\u2200 a\u2208 T, \u2200 b\u2208 T, a\u2294b \u2208 T) \u2192\n   (S.sup f \u2208 T) :=\nbegin\n  intros A2 A4 A3,\n  revert A2,\n  revert A4,\n  apply finset.induction_on S,\n  {\n    intros B1,\n    simp,\n  },\n  {\n    intros a S C1 C2 C3,\n    simp,\n    cases classical.em (S = \u2205) with C5 C5,\n    {\n      subst S,\n      simp,\n      apply C3,\n      simp,\n    },\n    apply A3,\n    {\n      apply C3,\n      simp,  \n    },\n    {\n      apply C2,\n      intro a2,\n      intro C4,\n      apply C3,\n      simp,\n      apply or.inr C4,  \n      apply C5,\n    },\n  },\nend\n\nlemma Sup_so_far_of_closed {\u03b1:Type*} [semilattice_sup_bot \u03b1] (f:\u2115 \u2192 \u03b1) (S:set \u03b1):\n   (\u2200 n:\u2115, f n \u2208 S) \u2192\n   (\u2200 a\u2208 S, \u2200 b\u2208 S, a\u2294b \u2208 S) \u2192\n   (\u2200 n:\u2115, (Sup_so_far f n \u2208 S)) :=\nbegin\n  intros A1 A2 n,\n  rw Sup_so_far_def,\n  apply finset.sup_closed_nonempty,\n  {\n    rw finset.range_eq_Ico,\n    intro B1,\n    rw finset.Ico.eq_empty_iff at B1,\n    simp at B1,\n    apply nat.succ_ne_zero n B1,\n  },\n  {\n    intros n2 C1,\n    apply A1,\n  },\n  {\n    apply A2,\n  },\nend\n\n\nlemma monotone_Sup_so_far {\u03b1:Type*} [semilattice_sup_bot \u03b1] (f:\u2115 \u2192 \u03b1):\n   monotone (Sup_so_far f) := \nbegin\n  intros a b A1,\n  rw Sup_so_far_def,\n  rw Sup_so_far_def,\n  apply finset.sup_mono,\n  simp,\n  apply nat.succ_le_succ A1,\nend\n\n\n\n------------------------------------------------\n\n--No such (sane) set exists, certainly not if \u03bc and \u03bd are finite.\n--Note that we don't make the system inconsistent\n--by constructing a definition that is possibly unsatisfiable.\n--The general idea that we want to prove is that if \u03bc X < \u03bd X,\n--then there must be some pure set X'\u2286 X, where for all subsets\n-- X''\u2286 X', \u03bc X'' \u2264 \u03bd X'', and \u03bc X' < \u03bd X'. A hahn crazy set\n--is a counterexample to this theorem.\ndef hahn_crazy_set {\u03b1:Type*} [M:measurable_space \u03b1] \n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1):Prop :=\n    (\u03bc X < \u03bd X) \u2227 \n    measurable_set X \u2227 \n    (\u2200 X':set \u03b1,  (X' \u2286 X) \u2192 measurable_set X' \u2192 \n    (\u03bc.restrict X' \u2264 \u03bd.restrict X')\u2192 (\u03bd X' \u2264 \u03bc X') )\n\nlemma hahn_crazy_set_def' {\u03b1:Type*} [M:measurable_space \u03b1] \n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1):hahn_crazy_set \u03bc \u03bd X =\n    ((\u03bc X < \u03bd X) \u2227 \n    measurable_set X \u2227 \n    (\u2200 X':set \u03b1,  (X' \u2286 X) \u2192 measurable_set X' \u2192 \n    (\u03bc.restrict X' \u2264 \u03bd.restrict X')\u2192 (\u03bd X' \u2264 \u03bc X') )) := rfl\n\n\n\n--The contradiction proofs chip away at this set. This\n--theorem shows that when you chip a piece off a crazy set,\n--a crazy set remains.\nlemma hahn_crazy_set_subset {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1)\n  (X:set \u03b1) (X':set \u03b1):\n  hahn_crazy_set \u03bc \u03bd X \u2192 \n  X' \u2286 X \u2192\n  measurable_set X' \u2192\n  \u03bd X' \u2264 \u03bc X' \u2192\n  hahn_crazy_set \u03bc \u03bd (X\\X') :=\nbegin\n  intros A1 A2 A3 A5,\n  rw hahn_crazy_set_def' at A1,\n  have A4:\u03bc X' < \u22a4,\n  {\n    have A4A:\u03bc X' \u2264 \u03bc X,\n    {\n      apply measure_theory.measure_mono A2,\n    },\n    apply lt_of_le_of_lt A4A,\n    apply lt_of_lt_of_le A1.left,\n    apply @le_top ennreal _,\n  },\n  rw hahn_crazy_set_def',\n  split,\n  {\n    rw measure_theory.measure_diff A2 A1.right.left A3 A4, \n    rw measure_theory.measure_diff A2 A1.right.left A3\n       (lt_of_le_of_lt A5 A4),\n    -- \u22a2 \u21d1\u03bc X - \u21d1\u03bc X' < \u21d1\u03bd X - \u21d1\u03bd X'\n    apply ennreal.sub_lt_sub_of_lt_of_le,\n    { -- \u22a2 \u21d1\u03bc X < \u21d1\u03bd X\n      apply A1.left,\n    },\n    { -- \u22a2 \u21d1\u03bd X' \u2264 \u21d1\u03bc X'\n      apply A5,\n    },\n    { -- \u22a2 \u21d1\u03bc X' \u2264 \u21d1\u03bc X\n      apply (measure_theory.measure_mono A2),\n    },\n  },\n  apply and.intro (measurable_set.diff A1.right.left A3),    \n  intros X'' C1 C2,\n  apply A1.right.right,\n  apply @set.subset.trans \u03b1 X'' (X \\ X') X C1,\n  apply set.diff_subset,\n  apply C2,\nend\n\n\n--In a hahn_crazy_set \u03bc \u03bd X, you will find sets X'\u2286 X where \u03bd X' < \u03bc X',\n--even though \u03bc X < \u03bd X. So, by creating X - X', you get an even crazier\n--set. In hahn_crazy_diff_big below, we show how we can select an element\n--that is big ENOUGH.\ndef hahn_crazy_diff_set {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1)\n  (X:set \u03b1):set (set \u03b1) := { X':set \u03b1|X' \u2286 X \u2227 measurable_set X' \u2227 \n                      \u03bc X' < \u03bd X'}\n\nlemma hahn_crazy_diff_set_def {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1):hahn_crazy_diff_set \u03bc \u03bd X = \n  { X':set \u03b1|X' \u2286 X \u2227 measurable_set X' \u2227 \u03bc X' < \u03bd X'} := rfl\n\n\n\nlemma not_restrict_le_elim {\u03b1:Type*} [M:measurable_space \u03b1]\n  {\u03bc \u03bd:measure_theory.measure \u03b1} {X:set \u03b1}: \n  measurable_set X \u2192\n  \u00ac(\u03bc.restrict X \u2264 \u03bd.restrict X) \u2192\n  (\u2203 X':set \u03b1, X'\u2286 X \u2227 measurable_set X' \u2227 \u03bd X' < \u03bc X') :=\nbegin\n  intros A1 A2,\n  rw \u2190 @decidable.not_forall_not _ _ (@classical.prop_decidable _),\n  intro A3,\n  apply A2,\n  rw measure_theory.measure.le_iff,\n  intros s A4,\n  repeat {rw measure_theory.measure.restrict_apply},\n  have A5 := A3 (s \u2229 X),\n  simp at A5,\n  apply A5,\n  repeat {simp [A4,A1]},\nend\n\nlemma hahn_crazy_diff_set_nonempty' {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1): \n  hahn_crazy_set \u03bd \u03bc X \u2192\n  (hahn_crazy_diff_set \u03bc \u03bd X).nonempty :=\nbegin\n  intros A1,\n  rw hahn_crazy_set_def' at A1,\n  cases A1 with A1 A2,\n  cases A2 with A2 A3,\n  rw set.nonempty_def,\n  have A4:\u00ac (\u03bd.restrict X \u2264 \u03bc.restrict X),\n  { intro A4A, rw lt_iff_not_ge at A1,\n    apply A1, apply A3 X (set.subset.refl X) A2 A4A},\n  have A5 := not_restrict_le_elim A2 A4,\n  cases A5 with X' A5,\n  apply exists.intro X',\n  rw hahn_crazy_diff_set_def,\n  simp only [set.mem_set_of_eq],\n  apply A5,\nend\n\n--Too trivial, used once.\nlemma pred_eq_set {\u03b1:Type*} {S:set \u03b1} {a:\u03b1}:\n    (S a) = (a\u2208 S) := rfl\n\nlemma nat.find_spec_set {S : set \u2115} [_inst_1 : decidable_pred S] (H : \u2203 (n : \u2115), S n):(nat.find H) \u2208 S :=\nbegin\n  rw \u2190 pred_eq_set,\n  apply nat.find_spec,\nend\n\nlemma nat.Inf_of_nonempty {S:set \u2115}:S.nonempty \u2192 Inf S \u2208 S :=\nbegin\n  intro A1,\n  have A2:=set.nonempty_def.mp A1,\n  rw nat.Inf_def A2,\n  have A3:decidable_pred S := classical_set \u2115 S,\n  apply nat.find_spec_set,\nend\n\nlemma nat.exists_le_Inf_map \n  {\u03b1:Type*} {f:\u03b1 \u2192 \u2115} {S:set \u03b1}:S.nonempty \u2192 \u2203 s\u2208 S,\n  \u2200 a\u2208 S, f s \u2264 f a :=\nbegin\n  intro A1,\n  let n := Inf (f '' S),\n  begin\n    have A2:(f '' S).nonempty,\n    {\n      apply set.nonempty_image_iff.mpr A1,\n    },\n    have A3 := nat.Inf_of_nonempty A2,\n    simp at A3,\n    cases A3 with a A3,\n    apply exists.intro a,\n    apply exists.intro A3.left,\n    intros b B1,\n    rw A3.right,\n    apply nat.Inf_le,\n    simp,\n    apply exists.intro b,\n    apply and.intro B1,\n    refl,\n  end\nend\n\nlemma nat.le_Inf  {S:set \u2115} {a:\u2115}:S.nonempty \u2192 (\u2200 s\u2208 S,\n  a \u2264 s) \u2192 a \u2264 Inf S := \nbegin\n  intros A1 A2,\n  apply A2,\n  apply nat.Inf_of_nonempty A1,\nend\n\nlemma nat.exists_eq_Inf_map {\u03b1:Type*} {f:\u03b1 \u2192 \u2115} {S:set \u03b1}:S.nonempty \u2192 \n    \u2203 s\u2208 S, f s = Inf (f '' S) :=\nbegin\n  intros A1,\n  have B1:(f '' S).nonempty,\n  {\n    apply set.nonempty_image_iff.mpr A1,\n  },\n\n  have B2 := nat.Inf_of_nonempty B1,\n  simp at B2,\n  cases B2 with a B2,\n  cases B2 with B2 B3,\n  apply exists.intro a,\n  apply exists.intro B2,\n  apply B3,\nend\n\n/--Notice that this function chooses the minimum n that is greater than or equal to the inverse.\nPut another way, this chooses the maximum 1/n that is less than or equal to the value.-/\nnoncomputable def floor_simple_fraction (x:ennreal):\u2115  := Inf {n:\u2115|(n:ennreal) \u2265 x\u207b\u00b9}\n\n/--This is a way of selecting a \"big\" input of a function when it is hard to select a maximum\ninput. Because we are mapping the extended non-negative reals onto the naturals with a\nmonotonically decreasing function, and for any nonempty set of natural numbers there is\na minimum, we get a set of \"big\" inputs to the function, and we apply classical.some, we \nget one of these values.\n \nThis is good for showing progress: if we whittle away something, and get smaller and\nsmaller results, we eventually grab every nonzero result remaining.-/\nlemma hahn_infi_ennreal {\u03b1:Type*} {f:\u03b1\u2192 ennreal} (S:set \u03b1):S.nonempty \u2192 \n  \u2203 a\u2208 S, \n  (floor_simple_fraction \u2218 f) a = Inf ((floor_simple_fraction \u2218 f) '' S) :=\nbegin\n  intro A1,\n  apply nat.exists_eq_Inf_map,\n  apply A1,\nend \n\n/-- When we start with some hahn_crazy_set \u03bc \u03bd X, we know \u03bc X < \u03bd X. We want to\ngrab a big chunk X' where \u03bd X' < \u03bc X'. This can only go on for so long.\n-/\ndef hahn_crazy_diff_big {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):set \u03b1 :=\n  classical.some\n  (@hahn_infi_ennreal (set \u03b1) (\u03bb X':set \u03b1, \u03bc X' - \u03bd X')\n  (hahn_crazy_diff_set \u03bd \u03bc X) (hahn_crazy_diff_set_nonempty' \u03bd \u03bc X H))\n\n\n\nlemma hahn_crazy_diff_big_def {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):hahn_crazy_diff_big \u03bc \u03bd X H =\n  classical.some\n  (@hahn_infi_ennreal (set \u03b1) (\u03bb X':set \u03b1, \u03bc X' - \u03bd X')\n  (hahn_crazy_diff_set \u03bd \u03bc X) (hahn_crazy_diff_set_nonempty' \u03bd \u03bc X H)) := rfl\n\n\nlemma hahn_crazy_diff_big_spec {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):\n (hahn_crazy_diff_big \u03bc \u03bd X H) \u2208 \n  (hahn_crazy_diff_set \u03bd \u03bc X) \u2227\n  (floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X')) (hahn_crazy_diff_big \u03bc \u03bd X H)\n  = Inf  ((floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'))''(hahn_crazy_diff_set \u03bd \u03bc X)) :=\nbegin\n  let f :=  (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'),\n  let S := (hahn_crazy_diff_set \u03bd \u03bc X),\n  let P := (\u03bb (a:set \u03b1), \n       (\u2203 (H:a\u2208 hahn_crazy_diff_set  \u03bd \u03bc X),\n      (floor_simple_fraction \u2218 f)\n       a =\n      Inf ((floor_simple_fraction \u2218 f) '' S))),\n  begin\n  have A1: f = (\u03bb X':set \u03b1, \u03bc X' - \u03bd X') := rfl,\n  have A2: S = (hahn_crazy_diff_set \u03bd \u03bc X) := rfl,\n  have A3: P = (\u03bb (a:set \u03b1), \n       (\u2203 (H:a\u2208 hahn_crazy_diff_set  \u03bd \u03bc X),\n      (floor_simple_fraction \u2218 f)\n       a =\n      Inf ((floor_simple_fraction \u2218 f) '' S))) := rfl,\n  have B1:(\u03bb (a:set \u03b1), \n       (\u2203 (H:a\u2208 hahn_crazy_diff_set  \u03bd \u03bc X),\n      (floor_simple_fraction \u2218 f)\n       a =\n      Inf ((floor_simple_fraction \u2218 f) '' S))) \n       (hahn_crazy_diff_big \u03bc \u03bd X H),\n  {  \n    rw hahn_crazy_diff_big_def,\n\n    have B1A:\u2203 y:set \u03b1, P y,\n    {\n      rw A3,\n      apply hahn_infi_ennreal,\n      apply hahn_crazy_diff_set_nonempty',\n      apply H,\n    },\n    apply @classical.some_spec (set \u03b1) P B1A,\n  },\n  cases B1 with B1 B2,\n  split,\n  apply B1,\n  apply B2\n  end\nend\n\n\n\n\nlemma hahn_crazy_diff_big_Inf {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):\n  (floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X')) (hahn_crazy_diff_big \u03bc \u03bd X H)\n  = Inf  ((floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'))''(hahn_crazy_diff_set \u03bd \u03bc X)) :=\nbegin\n  have A1 := hahn_crazy_diff_big_spec \u03bc \u03bd X H,\n  apply A1.right,\nend\n\nlemma hahn_crazy_diff_big_mem {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):\n  (hahn_crazy_diff_big \u03bc \u03bd X H) \u2208 (hahn_crazy_diff_set \u03bd \u03bc X) := \nbegin\n  have A1 := hahn_crazy_diff_big_spec \u03bc \u03bd X H,\n  apply A1.left,\nend\n\n\nlemma lt_of_hahn_crazy_diff_big {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):\n  \u03bd (hahn_crazy_diff_big \u03bc \u03bd X H) < \u03bc (hahn_crazy_diff_big \u03bc \u03bd X H) := \nbegin\n  have A1 := hahn_crazy_diff_big_mem \u03bc \u03bd X H,\n  rw hahn_crazy_diff_set_def at A1,\n  simp at A1,\n  apply A1.right.right,\nend\n\n\nlemma hahn_crazy_set_of_hahn_crazy_diff_big {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd X):\n  (hahn_crazy_set \u03bc \u03bd (X \\ (hahn_crazy_diff_big \u03bc \u03bd X H))) :=\nbegin\n  have B1:=hahn_crazy_diff_big_mem \u03bc \u03bd X H,\n  rw hahn_crazy_diff_set_def at B1,\n  simp at B1,\n  apply hahn_crazy_set_subset,\n  {\n    apply H,\n  },\n  {\n    -- \u22a2 hahn_crazy_diff_big \u03bc \u03bd X H \u2286 X\n    apply B1.left,\n  },\n  {\n    apply B1.right.left,\n  },\n  {\n    apply le_of_lt B1.right.right,\n  },\nend\n\ndef next_hahn_crazy_set {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:subtype (hahn_crazy_set \u03bc \u03bd)):\n  subtype (hahn_crazy_set \u03bc \u03bd) := subtype.mk \n      (X.1 \\ (hahn_crazy_diff_big \u03bc \u03bd X.1 X.2))\n      (hahn_crazy_set_of_hahn_crazy_diff_big \u03bc \u03bd X.1 X.2)\n\nlemma next_hahn_crazy_set_val {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (S:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd S):\n  ((next_hahn_crazy_set \u03bc \u03bd (subtype.mk S H)):set \u03b1) =\n      (S \\ (hahn_crazy_diff_big \u03bc \u03bd S H)) := rfl\n\nlemma set.diff_diff_eq_of_subset {\u03b1:Type*} (S T:set \u03b1):S \u2286 T \u2192\n  T \\ (T \\ S) = S :=\nbegin\n  intro A1,\n  ext a,split;intros B1,\n  {\n    rw set.mem_diff at B1,\n    cases B1 with B1 B2,\n    rw set.mem_diff at B2,\n    simp at B2,\n    apply B2 B1,\n  },\n  {\n    simp,\n    rw set.subset_def at A1,\n    apply and.intro (A1 a B1), \n    intro C1,\n    apply B1,\n  },\nend\n\nlemma next_hahn_crazy_set_diff {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (S:set \u03b1) (H:hahn_crazy_set \u03bc \u03bd S):\n  S \\ (next_hahn_crazy_set \u03bc \u03bd (subtype.mk S H)) =\n      (hahn_crazy_diff_big \u03bc \u03bd S H) := \nbegin\n  rw next_hahn_crazy_set_val,\n  apply set.diff_diff_eq_of_subset (hahn_crazy_diff_big \u03bc \u03bd S H) S,\n  have B1:=hahn_crazy_diff_big_mem \u03bc \u03bd S H,\n  rw hahn_crazy_diff_set_def at B1,\n  cases B1 with B1 B2,\n  apply B1,\nend\n\n/-- Given a crazy set, we can repeatedly cut away big elements of\nhahn_crazy_diff_sets.-/\ndef nth_hahn_crazy_set {\u03b1:Type*} [measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:subtype (hahn_crazy_set \u03bc \u03bd)):\n  \u2115 \u2192 subtype (hahn_crazy_set \u03bc \u03bd)\n  | 0 := X\n  | (nat.succ n) := next_hahn_crazy_set \u03bc \u03bd (nth_hahn_crazy_set n)\n\nlemma neg_monotone_of_succ_le {\u03b1:Type*} [partial_order \u03b1] {f:\u2115 \u2192 \u03b1}:\n  (\u2200 n:\u2115, f (n.succ) \u2264 f n) \u2192\n  (\u2200 i j, i \u2264 j \u2192 f j \u2264 f i) :=\nbegin\n  intros A1,\n  intros i j,\n  induction j,\n  {\n    intro B1,\n    simp at B1,\n    subst i,\n  },\n  {\n    intro C1,\n    cases C1 with C1 C1,\n    apply le_refl _,\n    have C2 := j_ih C1,\n    apply le_trans (A1 j_n) C2,\n  },\nend\n\nlemma directed_superset_of_monotone {\u03b1:Type*} {f:\u2115 \u2192 set \u03b1}:\n  (\u2200 n:\u2115,  f n.succ \u2286 f n) \u2192 directed superset f :=\nbegin\n  intros A1,\n  unfold directed superset,\n  intros x y,\n  apply exists.intro (max x y),\n  split,\n  {\n    rw \u2190 set.le_eq_subset,\n    apply @neg_monotone_of_succ_le (set \u03b1) _ f  A1,\n    apply le_max_left,\n  },\n  {\n    rw \u2190 set.le_eq_subset,\n    apply @neg_monotone_of_succ_le (set \u03b1) _ f  A1,\n    apply le_max_right,\n  },\nend\n\nlemma measure_Inter_eq_infi_nat' {\u03b1:Type*} [measurable_space \u03b1]\n  {\u03bc:measure_theory.measure \u03b1} {f:\u2115 \u2192 set \u03b1}:\n  (\u2200 n:\u2115,  f n.succ \u2286 f n) \u2192\n  (\u2200 n, measurable_set (f n)) \u2192\n  (\u03bc (f 0) < \u22a4) \u2192 \n  \u03bc (\u22c2 n, f n) =  \u2a05 n, \u03bc (f n)  :=\nbegin\n  intros A1 A2 A3,\n  apply measure_theory.measure_Inter_eq_infi,\n  apply A2,\n  {\n    apply directed_superset_of_monotone A1,\n  },\n  apply exists.intro 0,\n  apply A3,\nend\n\nlemma measure_monotone_finite {\u03b1:Type*} [measurable_space \u03b1]\n  {\u03bc:measure_theory.measure \u03b1} {f:\u2115 \u2192 set \u03b1}:\n  (\u2200 n:\u2115,  f n.succ \u2286 f n) \u2192\n  (\u2200 n, measurable_set (f n)) \u2192\n  (\u03bc (f 0) < \u22a4) \u2192\n  (\u2200 n, \u03bc (f n) < \u22a4) :=\nbegin\n  intros A1 A2 A3,\n  intro n,\n  induction n,\n  {\n    apply A3,\n  },\n  {\n    have B1:\u03bc (f (n_n.succ)) \u2264 \u03bc (f (n_n)),\n    {\n      apply measure_theory.measure_mono,\n      apply A1,\n    },\n    apply lt_of_le_of_lt B1,\n    apply n_ih,\n  },\nend\n\n\n\n--\u2200\u1da0\n--All of the variants of has_classical_limit follow easily from this.\nlemma has_classical_limit_nonneg {\u03b2:Type*} [ordered_add_comm_monoid \u03b2] [topological_space \u03b2]\n  [order_topology \u03b2]\n  [t2_space \u03b2] (f:\u2115 \u2192 \u03b2) (v:\u03b2):(0\u2264 f) \u2192\n  (\u2200 z<v, \u2203 m,  z < (finset.sum (finset.range m) f))  \u2192\n  (\u2200 m', (finset.sum (finset.range m') f) \u2264 v) \u2192\n  has_sum f v\n  :=\nbegin\n  intros A1 A2 A3,\n  unfold has_sum,\n  rw tendsto_order,\n  split,\n  {\n    intros x B1,\n    rw filter.eventually_iff,\n    simp,\n    have B2 := A2 x B1,\n    cases B2 with m B2,\n    apply exists.intro (finset.range m),\n    intros T B3,\n    apply lt_of_lt_of_le B2,\n    apply finset.sum_monotone_of_nonnegative A1 B3,\n  },\n  {\n    intros b C1,\n    rw filter.eventually_iff,\n    simp,\n    apply exists.intro finset.empty,\n    intros T C2,\n    apply lt_of_le_of_lt _ C1,\n    have C3:\u2203 m, T \u2286 (finset.range m) := finset_range_bound T,\n    cases C3 with m C3,\n    apply le_trans _ (A3 m),\n    apply finset.sum_monotone_of_nonnegative A1 C3,\n  },\nend\n\n\n\n\n--Not quite the classical way of thinking about a limit, but\n--more practical for nonnegative sums.\n--See has_classical_limit_nnreal' \nlemma has_classical_limit_ennreal (f:\u2115 \u2192 ennreal) (v:ennreal):\n  (\u2200 z<v, \u2203 m,  z < (finset.sum (finset.range m) f))  \u2192\n  (\u2200 m', (finset.sum (finset.range m') f) \u2264 v) \u2192\n  has_sum f v\n  :=\nbegin\n  intros A1 A2,\n  apply has_classical_limit_nonneg,\n  {\n    apply @nonnegative_fn \u2115 ennreal _,\n  },\n  {\n    apply A1,\n  },\n  {\n    apply A2,\n  },\nend\n\n\n\n\nlemma ennreal_telescope_helper2 {f:\u2115 \u2192 ennreal} {m:\u2115}:\n  (\u2200 n:\u2115,  f n.succ \u2264 f n) \u2192\n  (f 0) - (f m) = (finset.range m).sum (\u03bb n, (f n) - (f n.succ)) :=\nbegin\n  intros A1,\n  induction m,\n  {\n    simp,\n  },\n  {\n    rw finset.range_succ,\n    simp,\n    rw \u2190 m_ih,\n    rw add_comm,\n    symmetry,\n    apply ennreal.sub_add_sub,\n    apply A1,\n    apply neg_monotone_of_succ_le A1,\n    simp,\n  },\nend\n\n\n\n\nlemma Inf_lt {\u03b1:Type*} [complete_linear_order \u03b1]\n   {S:set \u03b1} {z:\u03b1}:S.nonempty \u2192\n  Inf S < z \u2192 \u2203 s\u2208 S, s < z :=\nbegin\n  intros A1 A2,\n  apply classical.exists_of_not_forall_not,\n  intro A3,\n  have A4:z \u2264 Inf S,\n  {\n    apply @le_Inf \u03b1 _ _,\n    intros b A4A,\n    have A4B := A3 b,\n    rw not_exists_iff_forall_not at A4B,\n    have A4C := A4B A4A,\n    apply le_of_not_lt A4C,\n  },\n  apply not_lt_of_le A4,\n  apply A2,\nend \n\n\n\nlemma infi_lt {\u03b1 \u03b2:Type*} [nonempty \u03b1] [complete_linear_order \u03b2] \n  {f:\u03b1 \u2192 \u03b2} {z:\u03b2}:\n  infi f < z \u2192 \u2203 a, f a < z :=\nbegin\n  intro A1,\n  have B1:(set.range f).nonempty,\n  {\n    apply set.range_nonempty,\n  },\n  have B2:(Inf (set.range f)) < z,\n  {\n    unfold infi at A1,\n    apply A1,\n  },\n  have B3:=Inf_lt B1 B2,\n  cases B3 with x B3,\n  cases B3 with B3 B4,\n  cases B3 with a B3,\n  apply exists.intro a,\n  subst x,\n  apply B4,\nend \n\n\nlemma ennreal_infi_diff {f:\u2115 \u2192 ennreal} {z:ennreal}:z < f 0 - infi f \u2192\n  \u2203 n, z < f 0 - f n := \nbegin\n  intros A1,\n  have B1:infi f \u2264 f 0 := @infi_le ennreal _ _ f 0,\n  have B2:= ennreal.add_lt_of_lt_sub A1,\n  rw add_comm at B2,\n  have B3 := ennreal.lt_sub_of_add_lt B2,\n  have B4 := infi_lt B3,\n  cases B4 with n B4,\n  apply exists.intro n,\n  apply ennreal.lt_sub_of_add_lt,\n  rw add_comm,\n  apply ennreal.add_lt_of_lt_sub,\n  apply B4,\nend\n\n\nlemma ennreal_telescope_helper {f:\u2115 \u2192 ennreal}:\n  (\u2200 n:\u2115,  f n.succ \u2264 f n) \u2192\n  (f 0) - (infi f) = (\u2211' (n:\u2115), (f n - (f (nat.succ n)))) :=\nbegin\n  intro A1,\n  symmetry,\n  rw \u2190 summable.has_sum_iff (ennreal.summable),\n  apply has_classical_limit_ennreal,\n  {\n    intros z B1,\n    have B2 := ennreal_infi_diff B1,\n    cases B2 with n B2,\n    apply exists.intro n,\n    rw \u2190 ennreal_telescope_helper2 A1,\n    apply B2,\n  },\n  {\n    intro m,\n    rw \u2190 ennreal_telescope_helper2 A1,\n    simp,\n    apply ennreal.le_sub_add,\n    apply @infi_le ennreal \u2115 _,\n    apply neg_monotone_of_succ_le A1,\n    simp,\n  },\nend\n\n\nlemma ennreal_telescope {f:\u2115 \u2192 ennreal}:\n  (\u2200 n:\u2115,  f n.succ \u2264 f n) \u2192\n  (f 0) = (\u2211' (n:\u2115), ((f n) - (f (nat.succ n)))) + (infi f) :=\nbegin\n  intros A1,\n  have B1 := ennreal_telescope_helper A1,\n  rw add_comm,\n  apply ennreal.eq_add_of_sub_eq _ B1,   \n  apply @infi_le ennreal _ _,\nend\n\n\nlemma measure_Inter_telescope {\u03b1:Type*} [measurable_space \u03b1]\n  {\u03bc:measure_theory.measure \u03b1} {f:\u2115 \u2192 set \u03b1}:\n  (\u2200 n:\u2115,  f n.succ \u2286 f n) \u2192\n  (\u2200 n, measurable_set (f n)) \u2192\n  (\u03bc (f 0) < \u22a4) \u2192\n  \u03bc (f 0) = (\u2211' n, \u03bc (f n \\ (f (n.succ)))) + \u03bc (\u22c2 n, f n) := \nbegin\n  intros A1 A2 A3,\n  rw measure_Inter_eq_infi_nat' A1 A2 A3,\n  have B1:(\u03bb n, \u03bc (f n \\ (f (n.succ))))=(\u03bb n, \u03bc (f n) - \u03bc (f (n.succ))),\n  {\n    apply funext,\n    intro n,\n    have B1A:\u03bc (f n \\ f (n.succ))=\u03bc (f n \\ (f (n.succ))) := rfl,\n    rw B1A,\n    rw measure_theory.measure_diff,\n    apply A1,\n    apply A2,\n    apply A2,\n    apply measure_monotone_finite A1 A2 A3,\n  },\n  rw B1,\n  apply @ennreal_telescope (\u03bc \u2218 f),\n  intro n,\n  apply measure_theory.measure_mono,\n  apply A1,\nend\n\n\nlemma measure_Inter_telescope' {\u03b1:Type*} [measurable_space \u03b1]\n  {\u03bc:measure_theory.measure \u03b1} {f:\u2115 \u2192 set \u03b1}:\n  (\u2200 n:\u2115,  f n.succ \u2286 f n) \u2192\n  (\u2200 n, measurable_set (f n)) \u2192\n  (\u03bc (f 0) < \u22a4) \u2192\n  \u03bc (\u22c2 n, f n) =\n  \u03bc (f 0) - (\u2211' n, \u03bc (f n \\ (f (n.succ)))) := \nbegin\n  intros A1 A2 A3,\n  have B1 := measure_Inter_telescope A1 A2 A3,\n  rw B1,\n  rw add_comm,\n  simp,\n  rw ennreal.add_sub_cancel,\n  rw B1 at A3,\n  apply lt_of_le_of_lt _ A3,\n  apply @le_add_nonnegative ennreal _ _ _,\nend\n\n\nlemma lt_Sup {\u03b1:Type*} [complete_linear_order \u03b1]\n   {S:set \u03b1} {z:\u03b1}:S.nonempty \u2192\n  z < Sup S  \u2192 \u2203 s\u2208 S,  z < s :=\nbegin\n  intros A1 A2,\n  apply classical.exists_of_not_forall_not,\n  intro A3,\n  have A4:Sup S \u2264 z,\n  {\n    apply @Sup_le \u03b1 _ _,\n    intros b A4A,\n    have A4B := A3 b,\n    rw not_exists_iff_forall_not at A4B,\n    have A4C := A4B A4A,\n    apply le_of_not_lt A4C,\n  },\n  apply not_lt_of_le A4,\n  apply A2,\nend \n\n\n\nlemma lt_supr {\u03b1 \u03b2:Type*} [nonempty \u03b1] [complete_linear_order \u03b2] \n  {f:\u03b1 \u2192 \u03b2} {z:\u03b2}:\n  z < supr f \u2192 \u2203 a, z < f a :=\nbegin\n  intro A1,\n  have B1:(set.range f).nonempty,\n  {\n    apply set.range_nonempty,\n  },\n  have B2:z < (Sup (set.range f)),\n  {\n    unfold supr at A1,\n    apply A1,\n  },\n  have B3:= lt_Sup B1 B2,\n  cases B3 with x B3,\n  cases B3 with B3 B4,\n  cases B3 with a B3,\n  apply exists.intro a,\n  subst x,\n  apply B4,\nend \n\n\n--This is true WITHOUT the non-negative part. So long as the sum is \n--well-defined, this should be true.\n--Revisit later (and search mathlib).\nlemma ennreal.tendsto_zero_of_finite_sum (f:\u2115 \u2192 ennreal):tsum f < \u22a4 \u2192\n  filter.tendsto f filter.at_top (nhds 0) :=\nbegin\n  intro A1,\n  rw tendsto_order,\n  split,\n  {\n    intros a B1,\n    exfalso,\n    simp at B1,\n    apply B1,\n  },\n  {\n    intros x C1,\n    rw filter.eventually_iff,\n    simp,\n    have C2:=ennreal.exists_coe A1,\n    cases C2 with v C2,\n    have C5:(\u2a06  n, (finset.range n).sum f)  = (v:ennreal),\n    {\n      rw ennreal.lim_finset_sum_eq_tsum,\n      apply C2,\n    },\n    cases (lt_or_le x (v:ennreal)) with C4 C4,\n    { -- C4 : a < \u2191v\n      have D1:(v:ennreal) - x < (\u2a06 (n : \u2115), (finset.range n).sum f),\n      {\n        rw C5,\n        simp,\n        apply ennreal.coe_sub_lt_self,\n        have D1A:0 < (v:ennreal),\n        {\n          apply lt_trans C1 C4,\n        },\n        simp at D1A,\n        apply D1A,\n        apply C1,\n      },\n      have D2 := lt_supr D1,\n      cases D2 with b D2,\n      apply exists.intro b,\n      intros c D3,\n      have D4:(finset.range (c.succ)).sum f \u2264 (v:ennreal),\n      {\n        rw \u2190 C5,\n        apply @le_supr ennreal \u2115 _,\n      },\n      have D5:0 \u2264 b,\n      {\n        simp,\n      },\n      have D6:b \u2264 c.succ,\n      {\n        apply le_trans D3,\n        apply nat.le_succ,\n      },\n      rw \u2190 finset.Ico.zero_bot at D4,\n      have D7:disjoint (finset.Ico 0 b) (finset.Ico b c.succ),\n      {\n        apply finset.Ico.disjoint_consecutive,\n      },\n      rw \u2190 finset.Ico.union_consecutive D5 D6 at D4,\n      rw finset.sum_union D7 at D4,\n      rw finset.Ico.zero_bot at D4,\n      -- b' + c'' \u2264 v\n      -- c' \u2264 c''\n      -- v - x < b' \n      -- \u22a2 c' < x\n      -- Working here.\n      have D8:f c \u2264 (finset.Ico b c.succ).sum f,\n      {\n        apply @finset.element_le_sum \u2115 ennreal _ _,\n        {\n          simp [D3,lt_succ], \n        }, \n      },\n      have D9:(v:ennreal) < \u22a4,\n      {\n        simp,\n      },\n      apply ennreal.lt_of_add_le_of_le_of_sub_lt D9 D4 D8 D2,      \n    },\n\n    cases (@eq_or_ne nnreal _ v 0) with C6 C6,\n    {\n      subst v,\n      apply exists.intro 0,\n      intros n E1,\n      simp at C5,\n      have E2:(finset.range (n.succ)).sum f \u2264 0,\n      {\n        rw \u2190 C5,\n        apply @le_supr ennreal \u2115 _,\n      },\n      simp at C1,\n      apply lt_of_le_of_lt _ C1,\n      apply le_trans _ E2,  \n      apply @finset.element_le_sum \u2115 ennreal _ _ _ n f,\n      {\n        simp,\n      },\n    },\n    {\n      have F1:0 < v,\n      {\n        rw lt_iff_le_and_ne,\n        split,\n        simp,\n        symmetry,\n        apply C6,\n      },\n      have F2:0 < (\u2a06 (n : \u2115), (finset.range n).sum f),\n      {\n        rw C5,\n        simp,\n        apply F1,\n      },\n      have F3 := lt_supr F2,\n      cases F3 with a F3,\n      apply exists.intro a,\n      intros b F4,\n      apply lt_of_lt_of_le _ C4,\n      have F5:(finset.range (b.succ)).sum f \u2264 (v:ennreal),\n      {\n        rw \u2190 C5,\n        apply @le_supr ennreal \u2115 _,\n      },\n      have F6:0 \u2264 a,\n      {\n        simp,\n      },\n      have F7:a \u2264 b.succ,\n      {\n        apply le_trans F4,\n        apply nat.le_succ,\n      },\n      rw \u2190 finset.Ico.zero_bot at F5,\n      have F8:disjoint (finset.Ico 0 a) (finset.Ico a b.succ),\n      {\n        apply finset.Ico.disjoint_consecutive,\n      },\n      rw \u2190 finset.Ico.union_consecutive F6 F7 at F5,\n      rw finset.sum_union F8 at F5,\n      rw finset.Ico.zero_bot at F5,\n      have F9:f b \u2264 (finset.Ico a b.succ).sum f,\n      {\n        apply @finset.element_le_sum \u2115 ennreal _ _,\n        {\n          simp [F4,lt_succ], \n        }, \n      },\n      have F10:(v:ennreal) < \u22a4,\n      {\n        simp,\n      },\n      have F11: (finset.range a).sum (\u03bb (x : \u2115), f x) + f b \u2264 \u2191v,\n      {\n        apply le_trans _ F5,\n        apply add_le_add_left F9,\n      },\n      apply ennreal.lt_of_lt_top_of_add_lt_of_pos F10 F11 F3,      \n    },\n  },\nend\n\nlemma filter.tendsto_const {\u03b1 \u03b2:Type*} [topological_space \u03b2]\n    {v:\u03b2} {F:filter \u03b1}:\n    filter.tendsto (\u03bb n:\u03b1, v) F (nhds v) :=\nbegin\n  apply filter.tendsto.mono_right (filter.tendsto_const_pure) (pure_le_nhds v),\nend\n\n\n/-- If a function is bracketed by two functions that converge to the same value,\nthen it too converges to that value. -/\nlemma filter.tendsto_le {\u03b1 \u03b2:Type*} [topological_space \u03b2]\n    [partial_order \u03b2] [order_topology \u03b2] {F:filter \u03b1} \n    {f g h:\u03b1  \u2192 \u03b2} {v:\u03b2}:\n    f \u2264 g \u2192\n    g \u2264 h \u2192\n    filter.tendsto f F (nhds v) \u2192\n    filter.tendsto h F (nhds v) \u2192\n    filter.tendsto g F (nhds v) :=\nbegin\n  intros A1 A2 A3 A4,\n  rw tendsto_order,\n  split,\n  {\n    intros x B1,\n    rw filter.eventually_iff,\n    rw tendsto_order at A3,\n    have B2 := A3.left x B1,\n    rw filter.eventually_iff at B2,\n    have B4: {x_1 : \u03b1 | x < f x_1}\u2286 {x_1 : \u03b1 | x < g x_1},\n    {\n      rw set.subset_def,\n      intros a B4A,\n      simp at B4A,\n      simp,\n      apply lt_of_lt_of_le B4A (A1 a),\n    },\n    apply filter.sets_of_superset F B2 B4,\n  },\n  {\n    intros x B1,\n    rw filter.eventually_iff,\n    rw tendsto_order at A4,\n    have B2 := A4.right x B1,\n    rw filter.eventually_iff at B2,\n    have B4: {x_1 : \u03b1 | h x_1 < x}\u2286 {x_1 : \u03b1 | g x_1 < x},\n    {\n      rw set.subset_def,\n      intros a B4A,\n      simp at B4A,\n      simp,\n      apply lt_of_le_of_lt (A2 a) B4A,\n    },\n    apply filter.sets_of_superset F B2 B4,\n  },\nend\n\n\n--Extend to nonnegative type (well, nnreal).\nlemma ennreal.tendsto_le {\u03b1:Type*} {F:filter \u03b1} {g h:\u03b1  \u2192 ennreal}:\n    g \u2264 h \u2192\n    filter.tendsto h F (nhds 0) \u2192\n    filter.tendsto g F (nhds 0) :=\nbegin\n  intros A1 A2,\n  let f:\u03b1 \u2192 ennreal := (\u03bb a:\u03b1, 0),\n  begin\n    have B1:f = (\u03bb a:\u03b1, 0) := rfl,\n    have B2:f \u2264 g,\n    {\n      rw B1,\n      intros a,\n      simp,\n    },\n    have B3:filter.tendsto f F (nhds 0),\n    {\n      apply filter.tendsto_const,\n    },\n    apply filter.tendsto_le B2 A1 B3 A2,\n  end\nend\n\nlemma set.preimage_subset_preimage_of_subset {\u03b1 \u03b2:Type*} {S T:set \u03b1} {f:\u03b2  \u2192 \u03b1}:\n  (S\u2286 T)\u2192 set.preimage f S \u2286  set.preimage f T :=\nbegin\n--{x:\u03b2 |f x \u2208 S} \u2286 {x:\u03b2 |f x \u2208 T} :=\n  intro A1,\n  have B1:set.preimage f S = {x:\u03b2 |f x \u2208 S} := rfl,\n  have B2:set.preimage f T = {x:\u03b2 |f x \u2208 T} := rfl,\n  rw B1,\n  rw B2,\n  apply set.preimage_mono,\n  apply A1,\nend\n\n--\u2200\u1da0 (b : \u2115) in filter.at_top, f b < x\nlemma tendsto_top {\u03b1 \u03b2:Type*} [NE:nonempty \u03b2] [SL:semilattice_sup \u03b2] {g:\u03b1 \u2192 \u03b2} {F:filter \u03b1}:filter.tendsto g F filter.at_top \u2194 (\u2200 b:\u03b2,  \u2200\u1da0 (c:\u03b1) in F, b \u2264 g c) :=\nbegin\n  split;intros A1,\n  {\n    rw filter.tendsto_iff_eventually at A1,\n    intro b,apply A1,\n    rw filter.eventually_iff,\n    simp,\n    apply exists.intro b,\n    intros b_1,\n    simp,\n  },\n  {\n    rw filter.tendsto_def,\n    intros S C1,\n    rw filter.mem_at_top_sets at C1,\n    cases C1 with a C1,\n    have C2 := A1 a,\n    rw filter.eventually_iff at C2,\n    apply filter.mem_sets_of_superset C2,\n    apply set.preimage_subset_preimage_of_subset,\n    rw set.subset_def,\n    intros b C3,\n    apply C1,\n    apply C3,\n  },\nend\n\nlemma eventually_at_top_iff {\u03b1:Type*} [nonempty \u03b1] [semilattice_sup \u03b1] {P:\u03b1 \u2192 Prop}:\n  (\u2200\u1da0 (c:\u03b1) in filter.at_top,  P c) \u2194 \u2203 (a : \u03b1), \u2200 (b : \u03b1), b \u2265 a \u2192 b \u2208 {x : \u03b1 | P x}:=\nbegin\n  rw filter.eventually_iff,\n  split;intros A1,\n  {\n    rw filter.mem_at_top_sets at A1,\n    ---cases A1 with a A1,\n    apply A1,\n  },\n  {\n    rw filter.mem_at_top_sets,\n    apply A1,\n  },\nend\n\n\nlemma floor_simple_fraction_def (x:ennreal):floor_simple_fraction x = Inf {n:\u2115|(n:ennreal) \u2265 x\u207b\u00b9} := rfl\n\n\n\nlemma floor_simple_fraction_bound (b:\u2115) (x:ennreal):0 < x \u2192\nx < (1/(b:ennreal)) \u2192\nb \u2264 floor_simple_fraction x := \nbegin\n  intros A1 A2,\n  cases b,\n  {\n    simp,\n  },\n  rw floor_simple_fraction_def,\n  \n  apply @nat.le_Inf,\n  {\n    simp,\n    cases x,\n    {\n      simp,\n    },\n    simp,\n    simp at A1,\n    have A3 := nnreal.exists_unit_frac_lt_pos A1,\n    cases A3 with a A3,\n    have A4 := le_of_lt A3,\n    rw nnreal.inv_as_fraction at A4,\n    have A6 := nnreal.inverse_le_of_le _ A4,\n    rw inv_inv' at A6,\n    rw set.nonempty_def,\n    apply exists.intro a.succ,\n    simp,\n    rw \u2190 ennreal.coe_inv,\n    have A7:((a + 1:nnreal):ennreal) = (a:ennreal) + 1,\n    {\n      simp,\n    },\n    rw \u2190 A7,\n    rw ennreal.coe_le_coe,\n    apply A6,\n    {\n      intro B1,\n      subst x,\n      simp at A1,\n      apply A1,\n    },\n    {\n      rw nnreal.inv_pos,\n      rw add_comm,\n      have B2:(0:nnreal) < (1:nnreal) := zero_lt_one,\n      apply lt_of_lt_of_le B2,\n      apply le_add_nonnegative _ _,\n    },\n  },\n  {\n    intros c C1,\n    simp at C1,\n    simp at A2,\n    have C2:(1:ennreal)/(c:ennreal) \u2264 x,\n    {\n      rw ennreal.inv_as_fraction,\n      rw \u2190 @ennreal.inv_inv x,\n      apply ennreal.inverse_le_of_le,\n      have C2A:x < \u22a4,\n      {\n        apply lt_trans A2,\n        simp,\n        rw add_comm,\n        \n        apply @lt_of_lt_of_le ennreal _ 0 1 _\n              (ennreal.zero_lt_one),\n        apply le_add_nonnegative 1 (b:ennreal),\n      },\n      rw lt_top_iff_ne_top at C2A,\n      rw \u2190 ennreal.inv_pos at C2A,\n      apply C1,\n    },\n    have C3 := lt_of_le_of_lt C2 A2,\n    have C4 := le_of_lt C3,\n    rw ennreal.inv_as_fraction at C4,\n    --rw ennreal.inv_as_fraction at C4,\n    have C5 := ennreal.inverse_le_of_le C4,\n    rw ennreal.inv_inv at C5,\n    rw ennreal.inv_inv at C5,\n    have C6:((1:nat):ennreal) = (1:ennreal),\n    {\n      simp,\n    },\n    rw \u2190 C6 at C5,    \n    rw ennreal.nat_coe_add at C5,\n    rw ennreal.nat_coe_le_coe at C5,\n    apply C5,\n  },\nend\n\n/-- If positive g approaches zero, then (floor_simple_fraction \u2218 g )approaches infinity. -/\nlemma floor_simple_fraction_limit_top {g:\u2115  \u2192 ennreal}:\n    (\u2200 n, 0 < g n) \u2192\n    filter.tendsto g filter.at_top (nhds 0) \u2192\n    filter.tendsto (floor_simple_fraction\u2218 g) filter.at_top filter.at_top :=\nbegin\n  intros AX A1,\n  rw tendsto_order at A1,\n  cases A1 with A1 A2,\n  clear A1,\n  rw tendsto_top,\n  intro b,\n  rw eventually_at_top_iff,\n  have B1:((1:ennreal)/(b.succ:ennreal)) > 0,\n  {\n    simp,\n  },\n  have B2 := A2 ((1:ennreal)/(b.succ:ennreal)) B1,\n  rw eventually_at_top_iff at B2,\n  cases B2 with a B2,\n  apply exists.intro a,\n  intros c B3,\n  have B4 := B2 c B3,\n  simp,\n  simp at B4,\n  have B5:b \u2264 b.succ,\n  {\n    apply nat.le_succ,\n  },\n  apply le_trans B5,\n  apply floor_simple_fraction_bound,\n  apply AX,\n  rw one_div,\n  apply B4\nend\n\n/-\n  This is the crux of the hahn decomposition theorem, the key of a proof by induction by\n  contradiction. We assume that there is a set X where \u03bc X < \u03bd X and v X < \u22a4, and there \n  does not exista a subset X' \u2286 X where \u03bc X' < \u03bd X', where for all X''\u2286 X', \u03bc X'' \u2264 \u03bd X''.\n\n  The proof follows the contradiction part in An Epsilon of Room. If such a hahn crazy set\n  existed, then we could find a set Y \u2286 X where \u03bd Y < \u03bc Y. And if we subtracted this set\n  off, we would be back where we started with X-Y being a set where \u03bc (X - Y) < \u03bd (X - Y) \n  and no subset X' \u2286 X - Y where \u03bc X' < \u03bd X' and \u03bc.restrict X' \u2264 \u03bd.restrict X'. \n\n  What if we want to grab a set Y which maximizes \u03bc Y - \u03bd Y?\n  Unfortunately, we find this as hard as the entire Hahn decomposition\n  problem itself. But we don't need to find the biggest one, just one \n  that is big enough. What follows is one of the most unusual mathematical \n  tricks I have seen. We basically chunk the reals into (1,\u221e],(1/2,1],(1/3,1/2],\n  et cetera. Instead of grabbing the absolute largest element, we grab an\n  element in the first populated range. Thus, if we do this an infinite number\n  of times, either the values we get sum to infinity (which they can't),\n  or each range gets eventually depopulated. Thus, after this point, any remaining\n  set must have \u03bc Y - \u03bd Y=0, a contradiction.\n -/\nlemma hahn_crazy_set_not_finite {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1):\n  (hahn_crazy_set \u03bc \u03bd X) \u2192\n  \u00ac(measure_theory.finite_measure \u03bd) :=\nbegin\n  intros A1 A2,\n  let h:\u2115 \u2192 set \u03b1 :=\n      (\u03bb n, (nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) n).val),\n  let d:\u2115 \u2192 (set \u03b1) :=\n      \u03bb n, h n \\ h (n.succ),\n  let Z:=\u22c2 n, (h n),\n  begin\n    have B1:h =\u03bb n, (nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) n).val := rfl,\n    have B2:d = \u03bb n, h (n) \\ h (n.succ) := rfl,\n    have B3:Z = \u22c2 n, (h n) := rfl,\n    have B4:(h 0) = X,\n    {\n      rw B1,\n      refl,\n    },\n    have B5:\u2200 n:\u2115, nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) (nat.succ n)\n           = next_hahn_crazy_set \u03bc \u03bd (nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) (n)),\n    {\n      intro n,\n      refl,\n    },\n    have B6:\u2200 n:\u2115, h (n.succ) = next_hahn_crazy_set \u03bc \u03bd (nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) (n)),\n    {\n      intro n,\n      rw B1,\n      refl,\n    },\n    have J0:\u2200 n:\u2115, (hahn_crazy_set \u03bc \u03bd (h n)),\n    {\n      intros n,\n      rw B1,\n      apply (nth_hahn_crazy_set \u03bc \u03bd (subtype.mk X A1) n).property,\n    },\n    have J0B:\u2200 n:\u2115, h (n.succ) = next_hahn_crazy_set \u03bc \u03bd  \n             (subtype.mk (h n) (J0 n)),\n    {\n      intro n,\n      rw B6,\n      refl,\n    },\n    have J1:\u2200 n:\u2115, h (n.succ) \u2286 h n,\n    {\n      intro n,\n      rw J0B,\n      unfold next_hahn_crazy_set,\n      apply set.diff_subset,\n    },\n    have J2:\u2200 n:\u2115, measurable_set (h n),\n    {\n      intro n,\n      have J2A := J0 n,\n      rw hahn_crazy_set_def' at J2A,\n      apply J2A.right.left,  \n    },\n    have C1A:\u03bd  (h 0) < \u22a4,\n    {\n      rw B4,\n      apply @measure_theory.measure_lt_top \u03b1 M \u03bd A2 X,\n    },\n    have J4:\u03bc (h 0) < \u03bd (h 0),\n    {\n       rw B4,\n       rw hahn_crazy_set_def' at A1,\n       apply A1.left,\n    },\n    have C1:\u03bd Z = \u03bd (h 0) - (\u2211' n, \u03bd (d n)),\n    {\n      rw B3,\n      rw B2,\n      simp,\n      apply @measure_Inter_telescope' \u03b1 M \u03bd h J1 J2 C1A,\n    },\n      have C2A:\u03bc  (h 0) < \u22a4,\n      {\n        apply lt_trans J4 C1A,\n      },\n\n    have C2:\u03bc Z = \u03bc (h 0) - (\u2211' n, \u03bc (d n)),\n    {\n      rw B3,\n      rw B2,\n      apply @measure_Inter_telescope' \u03b1 M \u03bc h J1 J2 C2A,\n    },\n    have C3C:(\u2211' (n : \u2115), \u03bc (d n)) \u2264 \u03bc (h 0),\n    {  \n      rw measure_Inter_telescope J1 J2 C2A,\n      have C3B1:(\u2211' (n : \u2115), \u03bc (d n)) =\n                (\u2211' (n : \u2115), \u03bc (h n \\ h n.succ)) := rfl,\n      rw \u2190 C3B1,\n      apply le_add_nonnegative _ _,\n    },\n    have C3X:(\u2211' (n : \u2115), \u03bd (d n)) \u2264 \u03bd (h 0),\n    {  \n      rw measure_Inter_telescope J1 J2 C1A,\n      have C3B1:(\u2211' (n : \u2115), \u03bd (d n)) =\n                (\u2211' (n : \u2115), \u03bd (h n \\ h n.succ)) := rfl,\n      rw \u2190 C3B1,\n      apply le_add_nonnegative _ _,\n    },\n\n    have C3:\u03bc Z < \u03bd Z,\n    {\n      rw C1,\n      rw C2,\n      apply ennreal.sub_lt_sub_of_lt_of_le,\n      {\n        rw B4,\n        rw hahn_crazy_set_def' at A1,\n        apply A1.left,\n      },\n      {\n        apply tsum_le_tsum _ ennreal.summable ennreal.summable,\n        intro n,\n        rw B2,\n        simp,\n        rw J0B,\n        rw next_hahn_crazy_set_diff,\n        have C3A1:= hahn_crazy_diff_big_mem \u03bc \u03bd (h n) (J0 n),\n        rw hahn_crazy_diff_set_def at C3A1,\n        apply le_of_lt (C3A1.right.right),\n      },\n      {\n        apply C3C,\n      },\n    },\n    have D1:measurable_set Z,\n    {\n      apply measurable_set.Inter,\n      apply J2,\n    },\n    have D3:Z \u2286 X,\n    {\n      rw B3,\n      rw \u2190 B4,\n      simp,\n      apply set.Inter_subset,\n    },\n    have D2:hahn_crazy_set \u03bc \u03bd Z,\n    {\n      rw hahn_crazy_set_def',\n      apply and.intro C3,\n      apply and.intro D1,\n      intros X' D2A D2B,\n      rw hahn_crazy_set_def' at A1,\n      apply A1.right.right,\n      apply set.subset.trans D2A D3,\n      apply D2B,\n    },\n    have D3:filter.tendsto \n      (\u03bb n:\u2115, \u03bc (d n)) filter.at_top (nhds 0),\n    {\n      --There is a sequence of positive numbers with a finite sum.\n      --Thus, their limit must be zero.\n      apply ennreal.tendsto_zero_of_finite_sum,\n      apply lt_of_le_of_lt C3C C2A,\n    },\n    have D3B:filter.tendsto \n      (\u03bb n:\u2115, \u03bd (d n)) filter.at_top (nhds 0),\n    {\n      --This is definitely true, but I am not sure if I need it,\n      --or D3 above is what is needed. I need to walk through the\n      --rest of this proof.\n      apply ennreal.tendsto_zero_of_finite_sum,\n      apply lt_of_le_of_lt C3X C1A,\n    },\n    have D4:filter.tendsto \n      (\u03bb n:\u2115, Inf ((floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'))\n       ''(hahn_crazy_diff_set \u03bd \u03bc (h n))))  filter.at_top filter.at_top,\n    {\n      --I reversed this: I need to figure out if I can make the rest of the proof work.\n      --Now, I need to reverse it back.\n      have D4A:(\u03bb n:\u2115, Inf ((floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'))\n       ''(hahn_crazy_diff_set \u03bd \u03bc (h n)))) = \n         (\u03bb n:\u2115, (floor_simple_fraction \u2218 (\u03bb X':set \u03b1,  \u03bc X' - \u03bd X'))\n                (hahn_crazy_diff_big \u03bc \u03bd (h n) (J0 n))),\n      {\n        -- J0 n:hahn_crazy_set \u03bc \u03bd (h n)\n        apply funext,\n        intro n,\n        symmetry,\n        apply @hahn_crazy_diff_big_Inf \u03b1 _ \u03bc \u03bd (h n) (J0 n),\n        \n      },\n      have D4B:\u2200 n, (hahn_crazy_diff_big \u03bc \u03bd (h n) (J0 n)) = d n,\n      {\n        intro n,\n        rw B2,\n        simp,\n        rw \u2190 next_hahn_crazy_set_diff,\n        rw J0B,\n      },\n      have D4C:(\u03bb n:\u2115, Inf ((floor_simple_fraction \u2218 (\u03bb X':set \u03b1, \u03bc X' - \u03bd X'))\n       ''(hahn_crazy_diff_set \u03bd \u03bc (h n)))) = \n         (\u03bb n:\u2115, (floor_simple_fraction \u2218 (\u03bb X':set \u03b1,  \u03bc X' - \u03bd X'))\n                (d n)),\n      {\n        rw D4A,\n        apply funext,\n        intro n,\n        rw D4B n,\n      },\n      have D4E: (\u03bb n:\u2115, (\u03bb X':set \u03b1,  \u03bc X' - \u03bd X')\n                (d n)) \u2264 (\u03bb n:\u2115, \u03bc (d n)),\n      {\n        intro n,\n        simp,\n        apply le_add_nonnegative _ _,\n      },\n  \n      have D4G:\u2200 n, \u03bd (d n) < \u03bc (d n),\n      {\n        intro n,\n        rw \u2190 D4B,\n        apply @lt_of_hahn_crazy_diff_big \u03b1 _ \u03bc \u03bd (h n),\n      },\n      have D4F:filter.tendsto (\u03bb n:\u2115,  (\u03bb X':set \u03b1,  \u03bc X' - \u03bd X')\n                (d n)) filter.at_top (nhds (0:ennreal)),\n      {\n        apply ennreal.tendsto_le D4E D3,  \n      },\n      rw D4C,\n      apply floor_simple_fraction_limit_top,\n      {\n        intro n,\n        simp,\n        rw \u2190 D4B,\n        apply @lt_of_hahn_crazy_diff_big \u03b1 _ \u03bc \u03bd (h n),\n      },\n      apply D4F,\n    },\n    have E1:(hahn_crazy_diff_set \u03bd \u03bc Z).nonempty,\n    {\n      apply hahn_crazy_diff_set_nonempty' \u03bd \u03bc Z D2,\n    },\n    have E2:\u2203 S, S\u2208(hahn_crazy_diff_set \u03bd \u03bc Z),\n    {\n      apply set.nonempty_def.mp E1,\n    },\n    cases E2 with S E2,\n    rw hahn_crazy_diff_set_def at E2,\n    simp at E2,\n    let n := floor_simple_fraction (\u03bc S - \u03bd S),\n    begin\n      have G1:n = floor_simple_fraction (\u03bc S - \u03bd S) := rfl,\n      have H1: {m:\u2115| n.succ \u2264 m} \u2208 filter.at_top,\n      {\n        apply filter.mem_at_top,\n      },\n      have H2 := filter_tendsto_elim D4 H1,\n      simp at H2,\n      cases H2 with n2 H2,\n      have H3 := H2 n2 (le_refl n2),\n      have H4:Inf ((\u03bb (a : set \u03b1), floor_simple_fraction (\u03bc a - \u03bd a)) '' hahn_crazy_diff_set \u03bd \u03bc (h n2))\n              \u2264 n,\n      {\n        apply nat.Inf_le,\n        simp,\n        apply exists.intro S,\n        split,\n        {\n          rw hahn_crazy_diff_set_def,\n          simp,\n          split,\n          {\n             apply @set.subset.trans \u03b1 S Z (h n2) E2.left,\n             rw B3,\n             apply set.Inter_subset,\n           },\n           apply (E2.right),\n         },\n         rw G1,\n      },\n      have H5 := le_trans H3 H4,\n      apply not_lt_of_le H5,\n      apply nat.lt.base,\n    end\n  end\nend\n\n\n\nlemma finite_set_not_hahn_crazy_set {\u03b1:Type*} [M:measurable_space \u03b1]\n  (\u03bc \u03bd:measure_theory.measure \u03b1) (X:set \u03b1)\n  [F:measure_theory.finite_measure \u03bd]: \n  \u00ac (hahn_crazy_set \u03bc \u03bd X)  :=\nbegin\n intros A2,\n apply hahn_crazy_set_not_finite \u03bc \u03bd X A2 F, \nend\n\n/-- This theorem is a weak variant of hahn_unsigned_inequality_decomp.\nHowever, it probably has uses in its own right, beyond that of\nits parent theorem.\n -/\nlemma hahn_unsigned_inequality_decomp_junior' {\u03b1:Type*} [M:measurable_space \u03b1]\n    (\u03bc \u03bd:measure_theory.measure \u03b1) {X:set \u03b1} [A1:measure_theory.finite_measure \u03bd]:\n    (measurable_set X) \u2192\n    (\u03bc X < \u03bd X) \u2192 \n    (\u2203 X':set \u03b1, \n      X' \u2286 X \u2227\n      \u03bc X' < \u03bd X' \u2227\n      measurable_set X' \u2227\n      \u03bc.restrict X' \u2264 \u03bd.restrict X') :=\nbegin\n  intros A2 A3,\n  have B1:= @finite_set_not_hahn_crazy_set _ _ \u03bc \u03bd X A1,\n  rw hahn_crazy_set_def' at B1,\n  simp at B1,\n  have B2 := B1 A3 A2,\n  cases B2 with X' B2,\n  apply exists.intro X',\n  simp [B2],\nend\n\n\n--TODO: Unify with Sup_apply_eq_supr_apply_of_closed'\nlemma Sup_apply_eq_supr_apply_of_closed'' {\u03b1:Type*}\n  [complete_lattice \u03b1] {S:set \u03b1} (g:\u03b1 \u2192 ennreal):\n  (\u2200 (a\u2208 S) (b\u2208 S), a \u2264 b \u2192 g a \u2264 g b) \u2192\n  (\u2200 f:\u2115 \u2192 \u03b1, set.range f \u2286 S \u2192 monotone f \u2192 (supr f)\u2208 S) \u2192\n  (S.nonempty) \u2192\n  (\u2200 a \u2208 S, \u2200 b \u2208 S, a \u2294 b \u2208 S)\u2192\n  (\u2203 f:\u2115 \u2192 \u03b1,\n            (\u2200 n, f n \u2208 S) \u2227 \n            (monotone f) \u2227\n            g (supr f) = Sup (g '' S)) :=\nbegin\n  intros A1 AX A2 A3,\n  have B1:(g '' S).nonempty,\n  {\n    apply set.nonempty_image_iff.mpr A2,\n  },\n  have B1X := ennreal.Sup_eq_supr B1,\n  cases B1X with f' B1X,\n  have B2:\u2203 f'':\u2115 \u2192 \u03b1, \u2200 n:\u2115, \n          (f'' n)\u2208 S \u2227 g (f'' n) = f' n, \n  {\n    apply @classical.some_func \u2115 \u03b1 (\u03bb (n:\u2115) (a:\u03b1), \n        a\u2208 S \u2227 g a = f' n),\n    intro n,\n    have B2A:=(B1X.left) n,\n    simp at B2A,\n    cases B2A with a B2A,\n    apply exists.intro a,\n    simp,\n    apply B2A,\n  },\n  cases B2 with f'' B2,\n  have C1:\u2200 (n : \u2115), Sup_so_far f'' n \u2208 S,\n  {\n    apply Sup_so_far_of_closed,\n    intro n,\n    apply (B2 n).left,\n    apply A3,  \n  },\n  apply exists.intro (Sup_so_far f''),\n  split,\n  {\n    apply C1,\n  },\n  split,\n  {\n    apply monotone_Sup_so_far,\n  },\n  {\n    --rw \u2190 AX,\n      have D1:(supr (Sup_so_far f''))\u2208 S,\n      {\n        apply AX,\n        {\n          rw set.subset_def,\n          intros x D1A,\n          --apply C1,\n          simp at D1A,\n          cases D1A with y D1A,\n          subst x,\n          apply C1,\n        },\n        apply monotone_Sup_so_far,\n      },\n    apply le_antisymm,\n    {\n      apply @le_Sup ennreal _ _,\n      simp,\n      apply exists.intro (supr (Sup_so_far f'')),\n      apply and.intro D1,\n      refl,   \n    },\n    {\n      rw \u2190 B1X.right,\n      apply @supr_le ennreal _ _,\n      intro i,\n      rw \u2190 (B2 i).right,\n      apply A1,\n      apply (B2 i).left,\n      apply D1,\n      have D2:f'' i \u2264 (Sup_so_far f'') i,\n      {\n        apply le_Sup_so_far,\n      },\n      apply le_trans D2,\n      apply @le_supr _ \u2115 _ (Sup_so_far f'') i,\n   },\n  },\nend\n\n--Replacing hahn_unsigned_inequality_decomp' (and hahn_unsigned_inequality_decomp).\nlemma hahn_unsigned_inequality_decomp' {\u03b1:Type*} [M:measurable_space \u03b1]\n    (\u03bc \u03bd:measure_theory.measure \u03b1) [A1:measure_theory.finite_measure \u03bd]: \n    (\u2203 X:set \u03b1, measurable_set X \u2227  \u03bc.restrict X \u2264 \u03bd.restrict X \u2227 \u03bd.restrict (X\u1d9c) \u2264 \u03bc.restrict (X\u1d9c)) :=\nbegin\n  /-\n    What we want is the argmax of f on S: this is our candidate for X.\n    However, we must first establish that such an argmax exists.\n     \n    First, we construct an  M that is our candidate for X.\n    It is the supremum of \n   -/\n  let S:set (set \u03b1) := {X:set \u03b1|measurable_set X \u2227  \u03bc.restrict X \u2264 \u03bd.restrict X},\n  let f:set \u03b1 \u2192 ennreal := (\u03bb T:set \u03b1, (\u03bd T) - (\u03bc T)),\n  -- M is unused.\n  let M:ennreal := Sup (f '' S),\n  begin\n    -- S is a ring of sets (closed under countable union).\n    have A2:S = {X:set \u03b1|measurable_set X \u2227  \u03bc.restrict X \u2264 \u03bd.restrict X} := rfl,\n    have A3:f = (\u03bb T:set \u03b1, (\u03bd T) - (\u03bc T)) := rfl,\n    have A5:\u2200 X, measurable_set X \u2192 \u03bc.restrict X \u2264 \u03bd.restrict X \u2192 \u03bc X < \u22a4,\n    {\n      intros X A5A A5B,\n      apply lt_of_le_of_lt (measure_theory.measure.le_of_restrict_le_restrict_self _ _ A5A A5B),\n      apply measure_theory.measure_lt_top,\n    },\n    have A6:\u2200 T, f T = \u03bd T - \u03bc T,\n    {\n      intro T,\n      refl,\n    },\n    have B1:\u2200 (a\u2208 S) (b\u2208 S), a \u2264 b \u2192 f a \u2264 f b,\n    {\n      intros T1 B1A T2 B1B B1C,\n      rw A2 at B1A,\n      simp at B1A,      \n      rw A2 at B1B,\n      simp at B1B,\n      repeat {rw A6},\n      have B1F:\u03bc.restrict (T2 \\ T1) \u2264 \u03bd.restrict (T2 \\ T1),\n      {\n        apply restrict_le_restrict_of_restrict_le_restrict_of_subset B1B.right,\n        apply set.diff_subset,\n        repeat {simp [B1A.left,B1B.left]},\n      },\n      have E1:measurable_set (T2 \\ T1),\n      {\n        simp [B1A.left,B1B.left],\n      },\n      have B1G:T2 = T1 \u222a (T2 \\ T1),\n      { \n        rw set.union_diff_cancel,\n        apply B1C,\n      },\n      rw B1G,\n      rw restrict_le_restrict_add,\n      apply @le_add_of_nonneg_right ennreal _,\n      simp only [zero_le],\n      {\n        apply A5 T1 B1A.left B1A.right,\n      },\n      {\n        apply A5 (T2 \\ T1) E1 B1F,\n      },\n      apply B1A.left,\n      apply E1,\n      apply B1A.right,\n      apply B1F, \n      apply set.disjoint_diff,\n    },\n    \n    have B2B:(\u2200 h:\u2115 \u2192 set \u03b1, set.range h \u2286 S \u2192 monotone h \u2192 (supr h)\u2208 S),\n    {\n      intros h B2C B2D,\n      have B2BG:\u2200 n, measurable_set (h n) \u2227  \u03bc.restrict (h n) \u2264 \u03bd.restrict (h n),\n      {\n        intro n,\n        apply B2C,\n        simp,\n      }, \n      rw A2,\n      rw supr_eq_Union,\n      simp only [set.mem_set_of_eq],\n      split,\n      apply measurable_set.Union,\n      intros b,\n      --simp at B2C,\n      have B2BA:h b \u2208 S,\n      {apply B2C, simp},\n      rw A2 at B2BA,\n      simp at B2BA,\n      apply B2BA.left,\n      apply restrict_le_restrict_m_Union,\n      apply B2D,\n      intro n,\n      have B2E := B2BG n,\n      --simp at B2E, \n      apply (B2BG n).left,\n      intro n,\n      apply (B2BG n).right,\n    },\n    have B3B:\u2205 \u2208 S,\n    {\n      simp [le_refl _],\n    },\n    have B3:S.nonempty,\n    {\n      apply set.nonempty_of_mem B3B,\n    },\n    have B4:(\u2200 (a \u2208 S) (b \u2208 S), a \u2294 b \u2208 S),\n    {\n      rw A2,\n      simp,\n      intros a B4A B4B b B4D B4E,\n      have B4C:a \u2294 b = a \u222a b := rfl,\n      split,\n      {simp [B4A,B4D]},\n      apply restrict_le_restrict_union B4B B4E,\n      repeat {assumption},\n    },\n    have C1:=@Sup_apply_eq_supr_apply_of_closed'' (set \u03b1) _ S f B1 B2B B3 B4,\n    cases C1 with g C1,\n    apply exists.intro (supr g),\n    have E1:\u2200 n, measurable_set (g n) \u2227 \u03bc.restrict (g n) \u2264 \u03bd.restrict (g n),\n    {\n      intro n,\n      apply (C1.left n),\n    },\n    have E2:=\u03bb n, (E1 n).left,\n    have E3 := \u03bb n, (E1 n).right,\n    have E4:measurable_set (supr g),\n    {\n      rw supr_eq_Union,\n      apply measurable_set.Union,\n      apply E2,\n    },\n    have C2:\u03bc.restrict (supr g) \u2264 \u03bd.restrict (supr g),\n    {\n      rw supr_eq_Union,\n      apply restrict_le_restrict_m_Union,\n      apply C1.right.left,\n      apply E2,\n      apply E3,\n    },\n    apply and.intro E4,\n    apply and.intro C2,\n    -- \u22a2 \u03bd.restrict (supr g)\u1d9c \u2264 \u03bc.restrict (supr g)\u1d9c\n    {\n      --intros X' D1 D2,\n      apply restrict_le_restrict_of_le_subset,\n      apply measurable_set.compl E4,\n      intros X' D1 D2, \n      apply le_of_not_lt _,\n      intro D3,\n      have D4:= hahn_unsigned_inequality_decomp_junior' \u03bc \u03bd D2 D3,\n      cases D4 with X'' D4,\n      have D5:f (X'' \u222a supr g) \u2264  f (supr g),\n      {\n        rw C1.right.right,\n        apply @le_Sup ennreal _ _,\n        simp,\n        apply exists.intro (X'' \u222a supr g),\n        simp only [D4, E4, true_and, measurable_set.union, and_true, eq_self_iff_true],\n        --squeeze_simp [D4,D2,E4],\n        apply restrict_le_restrict_union,\n        repeat {simp [D4,C2,E4]},\n      },\n      repeat {rw A6 at D5},\n      rw restrict_le_restrict_add at D5,\n      repeat {rw \u2190 A6 at D5},\n      rw add_comm at D5,\n      apply @ennreal.not_add_le_of_lt_of_lt_top (f (supr g)) (f X'') _ _ _,\n      {\n        rw A6,\n        simp,\n        apply D4.right.left,\n      },\n      {\n        rw A6,\n        have D6:\u03bd (supr g) < \u22a4,\n        {\n          apply measure_theory.measure_lt_top,\n        },\n        apply lt_of_le_of_lt _ D6,\n        simp,\n        apply ennreal.le_add,\n        apply le_refl _,\n      },\n      apply D5,\n      apply A5 X'' D4.right.right.left,\n      apply D4.right.right.right,\n      apply A5 (supr g) E4 C2,\n      apply D4.right.right.left,\n      repeat {simp [D4,E4,C2]},\n      {\n        apply @set.disjoint_of_subset_left _ _ _ ((supr g)\u1d9c),\n        apply set.subset.trans D4.left D1,\n        apply set.disjoint.symm,\n        apply set.disjoint_compl_right,\n      },\n    },\n  end\nend\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/hahn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.36871772248069956}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n\nimport category_theory.limits.shapes\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.finite_limits\nimport category_theory.limits.types\nimport category_theory.monad.monadicity\nimport category_theory.adjunction.limits\nimport category_theory.adjunction.lifting\nimport category_theory.monad.limits\nimport category_theory.limits.opposites\nimport category_theory.limits.over\nimport category_theory.epi_mono\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.constructions.limits_of_products_and_equalizers\nimport category_theory.limits.preserves.shapes.binary_products\nimport category_theory.limits.preserves.shapes.equalizers\nimport locally_cartesian_closed\nimport subobject_classifier\n\n/-!\n# Power objects\n\nDefine power objects.\nShow that power objects induce a (contravariant) functor `P_functor`.\nShow that this is self-adjoint on the right.\nDefine the singleton arrow {} : B \u27f6 PB and internal image (for monos only)\nand show the latter is functorial too.\nShow the existence of a subobject classifier given power objects and show\n\n-/\nuniverses v u v\u2082 u\u2082\nnoncomputable theory\n\nnamespace category_theory\n\nopen category_theory category limits\n\nattribute [instance] has_finite_wide_pullbacks_of_has_finite_limits\n\nvariables {C : Type u} [category.{v} C]\n\nvariables [has_finite_limits.{v} C]\n\nabbreviation powerises {A PA niA B R : C} (memA : niA \u27f6 PA \u2a2f A) (m : R \u27f6 B \u2a2f A) (mhat : B \u27f6 PA) :=\nhas_pullback_top m (limits.prod.map mhat (\ud835\udfd9 A)) memA\n\ninstance subsingleton_powerises {A PA niA B R : C} (memA : niA \u27f6 PA \u2a2f A) [mono memA] (m : R \u27f6 B \u2a2f A) (mhat : B \u27f6 PA) :\n  subsingleton (powerises memA m mhat) :=\n\u27e8by { intros P Q, cases P, cases Q, congr, rw [\u2190 cancel_mono memA, P_comm, Q_comm] }\u27e9\n\nstructure is_power_object {A PA niA : C} (memA : niA \u27f6 PA \u2a2f A) :=\n(hat : \u2200 {B R} (m : R \u27f6 B \u2a2f A) [mono.{v} m], B \u27f6 PA)\n(powerises' : \u2200 {B R} (m : R \u27f6 B \u2a2f A) [hm : mono m], powerises memA m (hat m))\n(uniquely' : \u2200 {B R} {m : R \u27f6 B \u2a2f A} [hm : mono m] {hat' : B \u27f6 PA}, powerises memA m hat' \u2192 hat m = hat')\n\nclass has_power_object (A : C) :=\n(PA niA : C)\n(memA : niA \u27f6 PA \u2a2f A)\n[is_mono : mono memA]\n(is_power : is_power_object memA)\n\nattribute [instance] has_power_object.is_mono\n\nvariable (C)\n\nclass has_power_objects :=\n(has_power_object : \u03a0 (A : C), has_power_object.{v} A)\n\nvariable {C}\n\nattribute [instance, priority 990] has_power_objects.has_power_object\nattribute [simp] pullback.condition\n\nsection convenience\n\nvariables (A : C) [has_power_object.{v} A]\n\ndef P : C := has_power_object.PA.{v} A\ndef ni : C := has_power_object.niA.{v} A\ndef mem : ni A \u27f6 P A \u2a2f A := has_power_object.memA\ndef power_is_power : is_power_object (mem A) := has_power_object.is_power\ninstance mono_mem : mono (mem A) := has_power_object.is_mono\n\ndef mem_sub : sub (P A \u2a2f A) := sub.mk' (mem A)\ndef mem_subq : subq (P A \u2a2f A) := \u27e6mem_sub A\u27e7\n\nvariables {A} {B R : C} (m : R \u27f6 B \u2a2f A) [mono m]\n\ndef hat : B \u27f6 P A := (power_is_power A).hat m\ndef hat_powerises : powerises (mem A) m (hat m) := (power_is_power A).powerises' m\n\nvariable {m}\nlemma unique_hat {hat' : B \u27f6 P A} (hp : powerises (mem A) m hat') : hat m = hat' := (power_is_power A).uniquely' hp\nend convenience\n\nlemma P_unique_aux {A : C} {PA\u2081 niA\u2081 PA\u2082 niA\u2082 : C}\n  (memA\u2081 : niA\u2081 \u27f6 PA\u2081 \u2a2f A) (memA\u2082 : niA\u2082 \u27f6 PA\u2082 \u2a2f A) [mono memA\u2081] [mono memA\u2082]\n  (h\u2081 : is_power_object memA\u2081) (h\u2082 : is_power_object memA\u2082) :\nh\u2081.hat memA\u2082 \u226b h\u2082.hat memA\u2081 = \ud835\udfd9 PA\u2082 :=\nbegin\n  have: h\u2082.hat memA\u2082 = \ud835\udfd9 _,\n  { apply h\u2082.uniquely',\n    change has_pullback_top _ _ _,\n    rw prod.map_id_id,\n    apply top_iso_has_pullback_top (\ud835\udfd9 _),\n    rw [id_comp, comp_id] },\n  rw \u2190 this,\n  symmetry,\n  apply h\u2082.uniquely',\n  change has_pullback_top _ _ _,\n  rw prod.map_comp_id,\n  apply left_right_hpb_to_both_hpb _ (h\u2081.powerises' memA\u2082) (h\u2082.powerises' memA\u2081),\nend\n\ndef P_unique_up_to_iso {A : C} {PA\u2081 niA\u2081 PA\u2082 niA\u2082 : C}\n  {memA\u2081 : niA\u2081 \u27f6 PA\u2081 \u2a2f A} {memA\u2082 : niA\u2082 \u27f6 PA\u2082 \u2a2f A} [mono memA\u2081] [mono memA\u2082]\n  (h\u2081 : is_power_object memA\u2081) (h\u2082 : is_power_object memA\u2082) :\nPA\u2081 \u2245 PA\u2082 :=\n{ hom := h\u2082.hat memA\u2081,\n  inv := h\u2081.hat memA\u2082,\n  hom_inv_id' := P_unique_aux memA\u2082 memA\u2081 h\u2082 h\u2081,\n  inv_hom_id' := P_unique_aux memA\u2081 memA\u2082 h\u2081 h\u2082 }\n\nvariables {A B : C} [has_power_object.{v} A]\n\nlemma hat_lift_of_is_iso {B R\u2081 R\u2082 : C} {g\u2081 : R\u2081 \u27f6 B \u2a2f A} {g\u2082 : R\u2082 \u27f6 B \u2a2f A} [mono g\u2081] [mono g\u2082] (hom : R\u2081 \u27f6 R\u2082) [is_iso hom] (k : hom \u226b g\u2082 = g\u2081) :\n  hat g\u2081 = hat g\u2082 :=\nbegin\n  apply unique_hat,\n  change has_pullback_top _ _ _,\n  rw [\u2190 id_comp (limits.prod.map _ _)],\n  refine left_right_hpb_to_both_hpb g\u2082 (top_iso_has_pullback_top hom _ _ _ (by simp [k])) (hat_powerises g\u2082),\nend\n\nlemma hat_lift_of_iso {B R\u2081 R\u2082 : C} {g\u2081 : R\u2081 \u27f6 B \u2a2f A} {g\u2082 : R\u2082 \u27f6 B \u2a2f A} [mono g\u2081] [mono g\u2082] (h : R\u2081 \u2245 R\u2082) (k : h.hom \u226b g\u2082 = g\u2081) :\n  hat g\u2081 = hat g\u2082 :=\nhat_lift_of_is_iso h.hom k\n\n-- We need to assume g\u2081 = hom \u226b g\u2082. From here if we know that hom,inv cancel then we get g\u2082 = inv \u226b g\u2081.\n-- Instead we assume this and derive that hom,inv cancel\nlemma lifting {A B R\u2081 R\u2082 : C} [has_power_object.{v} A] {g\u2081 : R\u2081 \u27f6 B \u2a2f A} {g\u2082 : R\u2082 \u27f6 B \u2a2f A} [mono g\u2081] [mono g\u2082] (hom : R\u2081 \u27f6 R\u2082) (inv : R\u2082 \u27f6 R\u2081) :\n  hom \u226b g\u2082 = g\u2081 \u2192 inv \u226b g\u2081 = g\u2082 \u2192 hat g\u2081 = hat g\u2082 :=\nbegin\n  intros k l,\n  apply hat_lift_of_iso \u27e8hom, inv, _, _\u27e9 k;\n  simp [\u2190 cancel_mono g\u2081, \u2190 cancel_mono g\u2082, l, k],\nend\n\nlemma liftable {B : C} (a b : sub (B \u2a2f A)) (i : a \u2248 b) : hat a.arrow = hat b.arrow :=\nnonempty.elim i (\u03bb i, lifting _ _ (sub.w i.hom) (sub.w i.inv))\n\ndef get_named_object {B : C} (k : B \u27f6 P A) : C := pullback (mem A) (limits.prod.map k (\ud835\udfd9 _))\ndef get_named_arrow {B : C} (k : B \u27f6 P A) : get_named_object k \u27f6 B \u2a2f A := pullback.snd\ninstance get_named_mono {B : C} (k : B \u27f6 P A) : mono (get_named_arrow k) := pullback.snd_of_mono\nlemma hat_get_named_arrow {B : C} (k : B \u27f6 P A) : hat (get_named_arrow k) = k :=\nunique_hat has_pullback_top_of_pb\n\ndef hat_natural_left {B B' R : C} (k : R \u27f6 B \u2a2f A) [mono k] (g : B' \u27f6 B) :\n  hat (pullback.snd : pullback k (limits.prod.map g (\ud835\udfd9 A)) \u27f6 B' \u2a2f A) = g \u226b hat k :=\nbegin\n  apply unique_hat,\n  change has_pullback_top _ _ _,\n  rw prod.map_comp_id,\n  apply left_right_hpb_to_both_hpb _ has_pullback_top_of_pb (hat_powerises k),\nend\n\n@[simps]\ndef name_bijection {A B : C} [has_power_object.{v} A] : (B \u27f6 P A) \u2243 subq (B \u2a2f A) :=\n{ to_fun := \u03bb k, \u27e6sub.mk' (get_named_arrow k)\u27e7,\n  inv_fun := quotient.lift (\u03bb (f : sub (B \u2a2f A)), hat f.arrow) liftable,\n  left_inv := hat_get_named_arrow,\n  right_inv := quotient.ind\n  begin\n    intro g,\n    apply quotient.sound,\n    exact equiv_of_both_ways\n      (sub.hom_mk _ ((hat_powerises g.arrow).is_pb.fac _ walking_cospan.right))\n      (sub.hom_mk _ (pullback.lift_snd _ _ (hat_powerises g.arrow).comm)),\n  end }\n\nabbreviation name_subobject {B : C} : subq (B \u2a2f A) \u2192 (B \u27f6 P A) := name_bijection.symm\n\nlemma get_named_subobject_eq_pullback_mem {B : C} (k : B \u27f6 P A) :\n  name_bijection k = (subq.pullback (limits.prod.map k (\ud835\udfd9 _))).obj (mem_subq A) := rfl\n\ndef get_named_subobject_natural_left {B B' : C} (k : B \u27f6 P A) (g : B' \u27f6 B) :\n  name_bijection (g \u226b k) = (subq.pullback (limits.prod.map g (\ud835\udfd9 A))).obj (name_bijection k) :=\nby { rw [get_named_subobject_eq_pullback_mem, prod.map_comp_id, subq.pullback_comp], refl }\n\nlemma name_pullback {B' : C} (g : subq (B \u2a2f A)) (f : B' \u27f6 B) :\n  name_subobject ((subq.pullback (limits.prod.map f (\ud835\udfd9 _))).obj g) = f \u226b name_subobject g :=\nquotient.induction_on g (\u03bb a, hat_natural_left a.arrow _)\n\nlemma pullback_along_hat_eq_self {R : C} (m : R \u27f6 B \u2a2f A) [mono m] :\n  (subq.pullback (limits.prod.map (hat m) (\ud835\udfd9 A))).obj (mem_subq A) = \u27e6sub.mk' m\u27e7 :=\nbegin\n  rw \u2190 get_named_subobject_eq_pullback_mem,\n  erw name_bijection.apply_eq_iff_eq_symm_apply,\n  refl\nend\n\nsection functor_setup\n\nvariables (f : A \u27f6 B) [has_power_object.{v} B]\ndef E : C := pullback (mem B) (limits.prod.map (\ud835\udfd9 _) f)\ndef Emap : E f \u27f6 P B \u2a2f A := pullback.snd\ninstance Emap_mono : mono (Emap f) := pullback.snd_of_mono\ndef Esubq : subq (P B \u2a2f A) := (subq.pullback (limits.prod.map (\ud835\udfd9 _) f)).obj (mem_subq B)\nlemma Esquare : (pullback.fst : E f \u27f6 _) \u226b mem B = Emap f \u226b limits.prod.map (\ud835\udfd9 _) f := pullback.condition\nlemma Epb : is_limit (pullback_cone.mk _ _ (Esquare f)) :=\ncone_is_pullback _ _\n\nvariable [has_power_object.{v} A]\ndef P_map : P B \u27f6 P A :=\nname_subobject (Esubq f)\n\nlemma hat_natural_right {D R : C} (m : R \u27f6 D \u2a2f B) [hm : mono m] :\n  hat (pullback.snd : pullback m (limits.prod.map (\ud835\udfd9 D) f) \u27f6 D \u2a2f A) = hat m \u226b P_map f :=\nbegin\n  apply unique_hat,\n  change has_pullback_top _ _ _,\n  rw prod.map_comp_id,\n  apply left_right_hpb_to_both_hpb _ _ (hat_powerises _),\n  apply right_both_hpb_to_left_hpb _ _ _ has_pullback_top_of_pb,\n  rw \u2190 prod.map_swap,\n  apply left_right_hpb_to_both_hpb m has_pullback_top_of_pb (hat_powerises _),\nend\n\nlemma name_other_pullback {D : C} :\n  \u2200 m, name_subobject ((subq.pullback (limits.prod.map (\ud835\udfd9 D) f)).obj m) = name_subobject m \u226b P_map f :=\nquotient.ind (by { intro a, apply hat_natural_right })\n\n@[simp] lemma lift'_right {X Y Z : C} {f : X \u27f6 Z} {g : Y \u27f6 Z} {t : pullback_cone f g} (ht : is_limit t) {W : C} (h : W \u27f6 X) (k : W \u27f6 Y) (w : h \u226b f = k \u226b g) :\n  (pullback_cone.is_limit.lift' ht h k w).val \u226b t.snd = k :=\n(pullback_cone.is_limit.lift' ht h k w).2.2\n\ndef how_inj_is_hat {B R\u2081 R\u2082 : C} {f\u2081 : R\u2081 \u27f6 B \u2a2f A} {f\u2082 : R\u2082 \u27f6 B \u2a2f A} [mono f\u2081] [mono f\u2082] (h : hat f\u2081 = hat f\u2082) :\n  R\u2081 \u2245 R\u2082 :=\n{ hom := (pullback_cone.is_limit.lift' (hat_powerises f\u2082).is_pb (hat_powerises f\u2081).top f\u2081 (h \u25b8 (hat_powerises f\u2081).comm)).1,\n  inv := (pullback_cone.is_limit.lift' (hat_powerises f\u2081).is_pb (hat_powerises f\u2082).top f\u2082 (h.symm \u25b8 (hat_powerises f\u2082).comm)).1,\n  hom_inv_id' := by erw [\u2190 cancel_mono_id f\u2081, assoc, lift'_right, lift'_right],\n  inv_hom_id' := by erw [\u2190 cancel_mono_id f\u2082, assoc, lift'_right, lift'_right] }\n\nlemma very_inj {B R\u2081 R\u2082 : C} {f\u2081 : R\u2081 \u27f6 B \u2a2f A} {f\u2082 : R\u2082 \u27f6 B \u2a2f A} [mono f\u2081] [mono f\u2082] (h : hat f\u2081 = hat f\u2082) :\n  (how_inj_is_hat h).hom \u226b f\u2082 = f\u2081 :=\nlift'_right _ _ _ _\n\nlemma P_map_id (X : C) [has_power_object.{v} X] : P_map (\ud835\udfd9 X) = \ud835\udfd9 (P X) :=\nhat_get_named_arrow _\n\nlemma P_map_comp {X Y Z : C} [has_power_object.{v} X] [has_power_object.{v} Y] [has_power_object.{v} Z] (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  P_map (f \u226b g) = P_map g \u226b P_map f :=\nby { erw [\u2190 name_other_pullback, Esubq, \u2190 subq.pullback_comp, \u2190 prod.map_id_comp], refl }\n\n@[simps]\ndef P_functor [has_power_objects.{v} C] : C\u1d52\u1d56 \u2964 C :=\n{ obj := \u03bb X, P X.unop,\n  map := \u03bb X Y f, P_map f.unop,\n  map_id' := \u03bb X, P_map_id _,\n  map_comp' := \u03bb X Y Z f g, P_map_comp _ _ }\n\nend functor_setup\n\ninstance self_adj [has_power_objects.{v} C] : is_right_adjoint (P_functor : C\u1d52\u1d56 \u2964 C) :=\n{ left := P_functor.right_op,\n  adj := adjunction.mk_of_hom_equiv\n  { hom_equiv := \u03bb A B,\n    begin\n      apply equiv.trans (op_equiv (opposite.op (P A)) B),\n      apply equiv.trans name_bijection,\n      apply equiv.trans _ name_bijection.symm,\n      apply postcompose_sub_equiv_of_iso (limits.prod.braiding _ _),\n    end,\n    hom_equiv_naturality_left_symm' := \u03bb X' X Y f g,\n    begin\n      rw \u2190 has_hom.hom.unop_inj.eq_iff,\n      change name_subobject ((subq.post _).obj (name_bijection (f \u226b g))) =\n             name_subobject ((subq.post _).obj (name_bijection g)) \u226b P_map f,\n      rw [\u2190 name_other_pullback, get_named_subobject_natural_left],\n      congr' 1,\n      exact postcompose_pullback_comm _ (pullback_square_iso _ _ _ _ (braid_natural _ _)) _,\n    end,\n    hom_equiv_naturality_right' := \u03bb X Y Y' f g,\n    begin\n      change name_subobject ((subq.post _).obj (name_bijection (g.unop \u226b f.unop))) =\n             name_subobject ((subq.post _).obj (name_bijection f.unop)) \u226b P_map g.unop,\n      rw [\u2190 name_other_pullback, get_named_subobject_natural_left],\n      congr' 1,\n      exact postcompose_pullback_comm _ (pullback_square_iso _ _ _ _ (braid_natural _ _)) _,\n    end } }\n\ndef diagonal (A : C) : A \u27f6 A \u2a2f A := limits.prod.lift (\ud835\udfd9 A) (\ud835\udfd9 A)\n\ninstance mono_diagonal (A : C) : mono.{v} (diagonal A) := category_theory.mono_prod_lift_of_left _ _\n\ndef diagonal_sub (A : C) : sub (A \u2a2f A) := sub.mk' (diagonal A)\ndef diagonal_subq (A : C) : subq (A \u2a2f A) := \u27e6diagonal_sub A\u27e7\n\n-- @[reducible]\ndef singleton_arrow (A : C) [has_power_object.{v} A] : A \u27f6 P A := hat (diagonal A)\n\nlemma seven_six_one {A B : C} [has_power_object.{v} B] (f : A \u27f6 B) :\n  hat (limits.prod.lift (\ud835\udfd9 A) f) = f \u226b singleton_arrow B :=\nbegin\n  rw [singleton_arrow, \u2190 hat_natural_left],\n  apply lifting (pullback.lift f _ _) (pullback.snd \u226b limits.prod.fst) (pullback.lift_snd _ _ _) _,\n  { rw [prod.lift_map, diagonal],\n    apply prod.hom_ext; simp },\n  { apply prod.hom_ext,\n    { simp only [prod.lift_fst, assoc, comp_id] },\n    { rw [assoc, prod.lift_snd, assoc, \u2190 limits.prod.map_fst f (\ud835\udfd9 _), \u2190 comp_id limits.prod.snd,\n          \u2190 limits.prod.map_snd f _, \u2190 pullback.condition_assoc, \u2190 pullback.condition_assoc, diagonal],\n      simp } }\nend\n\nlemma seven_six_two {A B : C} [has_power_object.{v} A] [has_power_object.{v} B] (f : A \u27f6 B) :\n  hat (limits.prod.lift f (\ud835\udfd9 A)) = singleton_arrow B \u226b P_map f :=\nbegin\n  rw [singleton_arrow, \u2190 hat_natural_right],\n  apply lifting (pullback.lift f _ _) (pullback.snd \u226b limits.prod.snd) (pullback.lift_snd _ _ _) _,\n  { rw [prod.lift_map, diagonal],\n    apply prod.hom_ext; simp },\n  { apply prod.hom_ext,\n    { rw [assoc, prod.lift_fst, assoc,  \u2190 limits.prod.map_snd (\ud835\udfd9 _) f, \u2190 pullback.condition_assoc,\n          \u2190 comp_id limits.prod.fst, \u2190 limits.prod.map_fst (\ud835\udfd9 _) f, \u2190 pullback.condition_assoc, diagonal],\n      simp },\n    { simp only [assoc, comp_id, prod.lift_snd] } },\nend\n\ninstance singleton_mono (A : C) [has_power_object.{v} A] : mono (singleton_arrow A) :=\n\u27e8\u03bb Z g h w,\nbegin\n  rw [\u2190 seven_six_one, \u2190 seven_six_one] at w,\n  have q := very_inj w =\u226b limits.prod.fst,\n  have r := very_inj w =\u226b limits.prod.snd,\n  simp only [prod.lift_fst, assoc, comp_id] at q,\n  simpa [q] using r.symm,\nend\u27e9\n\nlemma p_faithful {A B : C} [has_power_object.{v} A] [has_power_object.{v} B] {f g : A \u27f6 B} (k : P_map f = P_map g) :\n  f = g :=\nbegin\n  have w := singleton_arrow _ \u226b= k,\n  rw [\u2190 seven_six_two, \u2190 seven_six_two] at w,\n  have q := very_inj w =\u226b limits.prod.fst,\n  have r := very_inj w =\u226b limits.prod.snd,\n  simp only [prod.lift_snd, assoc, comp_id] at r,\n  simpa [r] using q.symm,\nend\n\ninstance pfaithful [has_power_objects.{v} C] : faithful (P_functor : C\u1d52\u1d56 \u2964 C) :=\n\u27e8\u03bb A B f g k, has_hom.hom.unop_inj (p_faithful k)\u27e9\n\ndef internal_image {A B : C} [has_power_object.{v} A] [has_power_object.{v} B] (f : A \u27f6 B) [mono f] : P A \u27f6 P B :=\nhat (mem A \u226b limits.prod.map (\ud835\udfd9 (P A)) f)\n\n-- TODO: this doesn't use pasting so it's super long. can we make it nicer by using pasting?\n-- TODO: if not, it's still a horribly long proof which desperately needs a cleanup\nlemma naturalish {A B : C} [has_power_object.{v} A] [has_power_object.{v} B] (f : A \u27f6 B) [mono f] {R D : C} (m : R \u27f6 D \u2a2f A) [mono m] :\n  hat m \u226b internal_image f = hat (m \u226b limits.prod.map (\ud835\udfd9 D) f) :=\nbegin\n  have comm : limits.prod.map (hat m) (\ud835\udfd9 _) \u226b limits.prod.map (\ud835\udfd9 _) f = limits.prod.map (\ud835\udfd9 _) f \u226b limits.prod.map (hat m) (\ud835\udfd9 _),\n    rw prod.map_swap,\n  change hat m \u226b name_bijection.symm ((subq.post (limits.prod.map _ _)).obj (mem_subq A)) = name_bijection.symm ((subq.post _).obj \u27e6sub.mk' m\u27e7),\n  rw [\u2190 name_pullback, \u2190 postcompose_pullback_comm comm _, pullback_along_hat_eq_self], refl,\n  refine is_limit.mk''' _ _ _,\n    exact (category_theory.mono_prod_map (\ud835\udfd9 D) f),\n  intro s,\n  refine \u27e8_, _\u27e9,\n  apply prod.lift (s.snd \u226b limits.prod.fst) (s.fst \u226b limits.prod.snd),\n  erw [prod.lift_map, comp_id, assoc, \u2190 limits.prod.map_snd (\ud835\udfd9 _), s.condition_assoc,\n       limits.prod.map_snd, comp_id],\n  apply prod.hom_ext; simp,\nend\n\nlemma internal_image_map_comp {X Y Z : C} [has_power_object.{v} X] [has_power_object.{v} Y] [has_power_object.{v} Z]\n  (f : X \u27f6 Y) (g : Y \u27f6 Z) [mono f] [mono g] :\n  internal_image (f \u226b g) = internal_image f \u226b internal_image g :=\nbegin\n  erw [naturalish, internal_image],\n  congr' 1,\n  rw [assoc, prod.map_id_comp],\nend\n\nlemma internal_image_map_id {X : C} [has_power_object.{v} X] : internal_image (\ud835\udfd9 X) = \ud835\udfd9 (P X) :=\nbegin\n  change name_subobject ((subq.post (limits.prod.map _ _)).obj (mem_subq _)) = _,\n  rw [name_bijection.symm_apply_eq, get_named_subobject_eq_pullback_mem],\n  conv { for (limits.prod.map _ _) [1, 2] { rw prod.map_id_id } },\n  rw [subq.post_id, subq.pullback_id],\nend\n\ntheorem beck_chevalley {A B C' D : C}\n  [has_power_object.{v} A] [has_power_object.{v} B]\n  [has_power_object.{v} C'] [has_power_object.{v} D]\n  {h : D \u27f6 A} {f : A \u27f6 C'} {k : D \u27f6 B} {g : B \u27f6 C'} (comm : h \u226b f = k \u226b g) [mono f] [mono k]\n  (t : is_limit (pullback_cone.mk h k comm)) :\n  internal_image f \u226b P_map g = P_map h \u226b internal_image k :=\nbegin\n  erw [\u2190 hat_natural_right, naturalish],\n  change name_subobject ((subq.pullback _).obj ((subq.post (limits.prod.map _ _)).obj (mem_subq A))) =\n         name_subobject ((subq.post (limits.prod.map _ _)).obj ((subq.pullback _).obj (mem_subq A))),\n  rw equiv.apply_eq_iff_eq,\n  symmetry,\n  apply postcompose_pullback_comm _ _,\n  rw [\u2190 prod.map_id_comp, comm, prod.map_id_comp],\n  haveI : preserves_limits_of_shape walking_cospan _ := prod_preserves_connected_limits (P A),\n  apply preserves_pullback_cone (prod.functor.obj (P A)) _ _ _ _ comm t,\nend\n\nvariable (C)\ndef weak_topos_has_subobj [has_power_object.{v} (\u22a4_ C)] : has_subobject_classifier.{v} C :=\n{ \u03a9 := P (\u22a4_ C),\n  \u03a9\u2080 := ni (\u22a4_ C),\n  truth := mem (\u22a4_ C) \u226b (prod.right_unitor _).hom,\n  is_subobj_classifier :=\n  { classifier_of := \u03bb U X f hf, by exactI hat (f \u226b (prod.right_unitor _).inv),\n    classifies' := \u03bb U X f hf, by exactI\n    begin\n      change has_pullback_top _ _ _,\n      conv {congr, rw [\u2190 comp_id f, \u2190 (prod.right_unitor X).inv_hom_id, \u2190 assoc] },\n      apply stretch_hpb_down _ _ _ limits.prod.fst _ _ (hat_powerises _) (limits.prod.map_fst _ _),\n      apply pullback_flip (pullback_prod _ _),\n    end,\n    uniquely' := \u03bb U X f hf \u03c7\u2081 k,\n    begin\n      apply unique_hat,\n      apply cut_hpb_up _ _ _ (prod.right_unitor _).hom (prod.right_unitor _).hom _ _ _ (pullback_flip (pullback_prod _ _)),\n      { apply_instance },\n      { rw [assoc, (prod.right_unitor X).inv_hom_id, comp_id],\n        exact k },\n      { apply limits.prod.map_fst }\n    end } }\nvariable {C}\n\ninstance p_reflects_iso [has_power_objects.{v} C] : reflects_isomorphisms (P_functor : C\u1d52\u1d56 \u2964 C) :=\n{ reflects := \u03bb A B f i, by exactI\nbegin\n  suffices : is_iso f.unop,\n    resetI,\n    refine \u27e8this.inv.op,\n            has_hom.hom.unop_inj (is_iso.inv_hom_id f.unop),\n            has_hom.hom.unop_inj (is_iso.hom_inv_id f.unop)\u27e9,\n  haveI : has_subobject_classifier.{v} C := weak_topos_has_subobj _,\n  haveI := reflects_isos _ (P_functor.right_op : C \u2964 _),\n  haveI : is_iso (P_functor.right_op.map f.unop) :=\n    \u27e8i.inv.op, has_hom.hom.unop_inj (is_iso.inv_hom_id _),\n               has_hom.hom.unop_inj (is_iso.hom_inv_id _)\u27e9,\n  refine is_iso_of_reflects_iso f.unop P_functor.right_op,\nend }\n\ndef exists_power {A B : C} [has_power_object.{v} A] [has_power_object.{v} B] (f : A \u27f6 B) [mono f] :\n  internal_image f \u226b P_map f = \ud835\udfd9 (P A) :=\nby rw [beck_chevalley _ (pullback_of_mono f), P_map_id, internal_image_map_id, comp_id]\n\ninstance fin_category_op (J : Type v) [small_category J] [fcj : fin_category J] : fin_category J\u1d52\u1d56 :=\n{ decidable_eq_obj := \u03bb x y, decidable_of_decidable_of_iff infer_instance opposite.unop_injective.eq_iff,\n  fintype_obj :=\n    { elems := finset.map \u27e8opposite.op, opposite.op_injective\u27e9 _,\n      complete := \u03bb x, finset.mem_map_of_mem _ (fintype.complete x.unop) },\n  decidable_eq_hom := \u03bb x y f g, decidable_of_decidable_of_iff infer_instance has_hom.hom.unop_inj.eq_iff,\n  fintype_hom := \u03bb X Y,\n  { elems := (@fin_category.fintype_hom J _ fcj Y.unop X.unop).elems.map \u27e8has_hom.hom.op, has_hom.hom.op_inj\u27e9,\n    complete := \u03bb f, finset.mem_map_of_mem _ (fintype.complete f.unop) } }\n\nlocal attribute [instance] has_colimits_of_shape_op_of_has_limits_of_shape\n\ninstance pare [has_power_objects.{v} C] : monadic_right_adjoint (P_functor : C\u1d52\u1d56 \u2964 C) :=\nbegin\n  apply monad.monadic_of_has_preserves_reflexive_coequalizers_of_reflects_isomorphisms _,\n  { apply_instance },\n  { apply_instance },\n  { apply_instance },\n  { introsI B' A' f' g' i,\n    apply preserves_colimit_of_preserves_colimit_cocone (coequalizer_is_coequalizer f' g'),\n    apply (is_colimit_map_cocone_cofork_equiv _ _).symm _,\n    let e' := coequalizer.\u03c0 f' g',\n    let e : (coequalizer f' g').unop \u27f6 A'.unop := e'.unop,\n    haveI : split_mono g'.unop := \u27e8(common_section f' g').unop, by { simp [\u2190unop_comp] }\u27e9,\n    change is_colimit (cofork.of_\u03c0 (P_map e) _),\n    have : is_split_coequalizer (P_functor.map f') (P_functor.map g') (P_functor.map e'),\n    { refine \u27e8internal_image e, internal_image g'.unop, _, exists_power _, exists_power _, _\u27e9,\n      { rw [\u2190 functor.map_comp, coequalizer.condition, functor.map_comp] },\n      { apply beck_chevalley _ _,\n        { rw [\u2190 unop_comp, \u2190 coequalizer.condition, unop_comp] },\n        { refine is_limit.mk''' _ _ (\u03bb s, _),\n          { dsimp, apply_instance },\n          { have equal_legs : s.fst = s.snd,\n            { simpa [\u2190 unop_comp] using s.condition =\u226b (common_section f' g').unop },\n            refine \u27e8(coequalizer.desc s.fst.op _).unop, _\u27e9,\n            { rw [\u2190 has_hom.hom.unop_inj.eq_iff, unop_comp, unop_comp, has_hom.hom.unop_op,\n                  s.condition, equal_legs] },\n            { change (e' \u226b _).unop = _,\n              rw coequalizer.\u03c0_desc,\n              apply equal_legs } } } } },\n    apply this.is_coequalizer }\nend\n\ndef some_colims (J : Type v) [small_category J] [has_power_objects.{v} C] [has_limits_of_shape J\u1d52\u1d56 C] : has_colimits_of_shape J C :=\n{ has_colimit := \u03bb F, by exactI\n  begin\n    suffices: has_colimit (F \u22d9 (op_op_equivalence _).inverse),\n    { resetI,\n      apply adjunction.has_colimit_of_comp_equivalence F (op_op_equivalence _).inverse },\n    let F'' : J\u1d52\u1d56 \u2964 C\u1d52\u1d56 := (F \u22d9 (op_op_equivalence _).inverse).left_op,\n    suffices : has_limit F'',\n    { resetI,\n      apply limits.has_colimit_of_has_limit_left_op },\n    haveI := monadic_creates_limits (P_functor : C\u1d52\u1d56 \u2964 C),\n    suffices : has_limit (F'' \u22d9 P_functor),\n      refine has_limit_of_created _ P_functor,\n    apply_instance\n  end }\n\nnamespace intersect\n\nvariables {A} [has_power_object.{v} A]\n\ndef intersect_names {B : C} (m n : B \u27f6 P A) : B \u27f6 P A :=\nname_subobject $ name_bijection m \u2293 name_bijection n\n\ndef intersect_names_natural {B B' : C} (f : B' \u27f6 B) (m n : B \u27f6 P A) :\n  f \u226b intersect_names m n = intersect_names (f \u226b m) (f \u226b n) :=\nbegin\n  dunfold intersect_names,\n  rw [get_named_subobject_natural_left, get_named_subobject_natural_left, \u2190 inf_pullback,\n      name_bijection.eq_symm_apply, get_named_subobject_natural_left, name_bijection.apply_symm_apply],\nend\n\ndef intersect (A : C) [has_power_object.{v} A] : P A \u2a2f P A \u27f6 P A := intersect_names limits.prod.fst limits.prod.snd\n\nend intersect\n\n@[priority 10000] instance [has_finite_limits.{v} C] {B : C} : has_finite_limits.{v} (over B) :=\nbegin\n  haveI := has_finite_wide_pullbacks_of_has_finite_limits C,\n  apply over.has_finite_limits,\nend\n\ndef P\u2081_obj (A : C) [has_power_object.{v} A] : C := equalizer (intersect.intersect A) limits.prod.fst\ndef P\u2081_arrow (A : C) [has_power_object.{v} A] : P\u2081_obj A \u27f6 P A \u2a2f P A := equalizer.\u03b9 (intersect.intersect A) limits.prod.fst\ninstance P\u2081_arrow_mono (A : C) [has_power_object.{v} A] : mono (P\u2081_arrow A) := equalizer.\u03b9_mono\ndef P\u2081_sub (A : C) [has_power_object.{v} A] : subq (P A \u2a2f P A) := \u27e6sub.mk' (P\u2081_arrow A)\u27e7\n\nlemma leq_prop' (A B : C) (m n : subq (B \u2a2f A)) [has_power_object.{v} A] :\n  m \u2264 n \u2194 limits.prod.lift (name_subobject m) (name_subobject n) \u226b intersect.intersect A = limits.prod.lift (name_subobject m) (name_subobject n) \u226b limits.prod.fst :=\nbegin\n  rw [\u2190 inf_eq_left, intersect.intersect, intersect.intersect_names_natural, prod.lift_fst,\n      prod.lift_snd, intersect.intersect_names, name_bijection.eq_symm_apply],\n  simp only [name_bijection.apply_symm_apply],\nend\n\nlemma leq_prop (A B R\u2081 R\u2082 : C) [has_power_object.{v} A] (m : R\u2081 \u27f6 B \u2a2f A) (n : R\u2082 \u27f6 B \u2a2f A) [mono m] [mono n] :\n  factors_through m n \u2194 limits.prod.lift (hat m) (hat n) \u226b intersect.intersect A = limits.prod.lift (hat m) (hat n) \u226b limits.prod.fst :=\nleq_prop' _ _ \u27e6sub.mk' m\u27e7 \u27e6sub.mk' n\u27e7\n\n-- lemma leq_iff_factor (A B R\u2081 R\u2082 : C) [has_power_object.{v} A] (m : R\u2081 \u27f6 B \u2a2f A) (n : R\u2082 \u27f6 B \u2a2f A) [mono m] [mono n] :\n--   factors_through m n \u2194 factors_through (prod.lift (hat m) (hat n)) (P\u2081_arrow A) :=\n-- begin\n--   rw [leq_prop, factors_through],\n\n--   -- refine \u27e8\u03bb k, \u27e8_, (equalizer.lift' _ k).2\u27e9, _\u27e9,\n--   -- rintro \u27e8k, hk\u27e9,\n--   -- simp [\u2190hk, P\u2081_arrow, equalizer.condition],\n-- end\n\nnamespace slicing\n\n-- EVERYTHING FROM HERE DOWN NEEDS TIDYING!!\n\n-- def lift_exists_of_regular {X Y : C} {r : X \u27f6 Y} [hr : regular_mono r] {Z : C} {l : Z \u27f6 Y} (h : \u2203 (q : Z \u27f6 X), q \u226b r = l) : {q // q \u226b r = l} :=\n-- begin\n--   apply fork.is_limit.lift' hr.is_limit l,\n--   cases h,\n--   simp [\u2190 h_h, hr.w],\n-- end\n\n-- def power_object_of_hats {A PA : C} (mem : sub'.{v} (PA \u2a2f A)) (hats : \u03a0 {B} (f : sub'.{v} (B \u2a2f A)), B \u27f6 PA)\n--   [regular_mono mem.arrow.hom]\n--   (mediate : \u03a0 {B} (f : sub'.{v} (B \u2a2f A)), { k : pullback mem.arrow.hom (limits.prod.map (hats f) (\ud835\udfd9 _)) \u2245 f.arrow.left // k.hom \u226b f.arrow.hom = pullback.snd }) :\n-- is_power_object.{v} mem.arrow.hom :=\n-- { hat := \u03bb B R m hm, by exactI hats (sub'.mk' m),\n--   powerises' := \u03bb B R m hm, by exactI\n--   begin\n--     change has_pullback_top _ _ _,\n--     obtain \u27e8\u27e8hom, inv, hom_inv_id, inv_hom_id\u27e9, hq\u27e9 := mediate (sub'.mk' m),\n--     dsimp at hom inv hom_inv_id inv_hom_id hq,\n--     -- let q' : R \u27f6 pullback mem.arrow.hom (limits.prod.map (hats (sub'.mk' m)) (\ud835\udfd9 A)) := pullback.lift _ m _,\n--     -- sorry,\n--     -- refine \u27e8_, _, _\u27e9,\n--   end\n\n-- }\n\nvariables {B} (f g : over B)\n\n-- def reflect_pullback (P Q R S : over B) (f : P \u27f6 Q) (g : Q \u27f6 S) (h : P \u27f6 R) (k : R \u27f6 S)\n--   (comm : f \u226b g = h \u226b k) (t : is_limit (pullback_cone.mk f.left h.left (begin exact congr_arg comma_morphism.left comm end))) :\n-- is_limit (pullback_cone.mk f h comm) :=\n-- begin\n--   apply is_limit.mk',\n--   intro s,\n--   let s' : pullback_cone g.left k.left := pullback_cone.mk (pullback_cone.fst s).left (pullback_cone.snd s).left (congr_arg comma_morphism.left (pullback_cone.condition s)),\n--   refine \u27e8over.hom_mk (t.lift s') _, _, _, _\u27e9,\n--   dsimp, change t.lift s' \u226b P.hom = _, rw \u2190 over.w f, slice_lhs 1 2 {erw t.fac _ walking_cospan.left}, exact over.w (pullback_cone.fst s),\n--   ext1, dsimp, exact t.fac _ walking_cospan.left,\n--   ext1, dsimp, exact t.fac _ walking_cospan.right,\n--   intros m m\u2081 m\u2082,\n--   ext1,\n--   dsimp,\n--   refine t.hom_ext _,\n--   apply pullback_cone.equalizer_ext (pullback_cone.mk f.left h.left _),\n--   erw t.fac _ walking_cospan.left,\n--   exact congr_arg comma_morphism.left m\u2081,\n--   erw t.fac _ walking_cospan.right,\n--   exact congr_arg comma_morphism.left m\u2082,\n-- end\n\n-- def preserve_pullback {P Q R S : over B} {f : P \u27f6 Q} {g : Q \u27f6 S} {h : P \u27f6 R} {k : R \u27f6 S}\n--   {comm : f \u226b g = h \u226b k} (t : is_limit (pullback_cone.mk f h comm)) :\n-- is_limit (pullback_cone.mk f.left h.left (begin exact congr_arg comma_morphism.left comm end)) :=\n-- begin\n--   apply is_limit.mk',\n--   intro s,\n--   let sX' : over B := over.mk (pullback_cone.snd s \u226b R.hom),\n--   have: pullback_cone.fst s \u226b Q.hom = pullback_cone.snd s \u226b R.hom,\n--     rw [\u2190 over.w g, pullback_cone.condition_assoc s, over.w k],\n--   let fst' : sX' \u27f6 Q := over.hom_mk (pullback_cone.fst s) (by assumption),\n--   let snd' : sX' \u27f6 R := over.hom_mk (pullback_cone.snd s),\n--   have comm': fst' \u226b g = snd' \u226b k,\n--     ext, dsimp, apply pullback_cone.condition s,\n--   let q : sX' \u27f6 P := t.lift (pullback_cone.mk fst' snd' comm'),\n--   have qf : q \u226b f = fst' := t.fac _ walking_cospan.left,\n--   have qh : q \u226b h = snd' := t.fac _ walking_cospan.right,\n--   refine \u27e8q.left, congr_arg comma_morphism.left qf, congr_arg comma_morphism.left qh, _\u27e9,\n--   intros m m\u2081 m\u2082,\n--   have z: m \u226b P.hom = pullback_cone.snd s \u226b R.hom,\n--   { rw [\u2190 over.w h, \u2190 m\u2082, assoc], refl },\n--   let m' : sX' \u27f6 P := over.hom_mk m (by apply z),\n--   have: m' = q,\n--     apply t.hom_ext,\n--     refine pullback_cone.equalizer_ext (pullback_cone.mk f h comm) _ _,\n--     { erw qf,\n--       ext,\n--       dsimp,\n--       erw m\u2081 },\n--     { erw qh,\n--       ext,\n--       dsimp,\n--       erw m\u2082 },\n--   apply congr_arg comma_morphism.left this,\n-- end\n\nvariables [has_power_object.{v} B] [has_power_object.{v} f.left]\n\n-- @[reducible]\ndef bottom : P f.left \u2a2f B \u27f6 P f.left \u2a2f P f.left := limits.prod.map (\ud835\udfd9 _) (singleton_arrow B \u226b P_map f.hom)\n\ndef Q : C := pullback (P\u2081_arrow f.left) (bottom f)\ndef hk : Q f \u27f6 P f.left \u2a2f B := pullback.snd\ndef k : Q f \u27f6 B        := hk f \u226b limits.prod.snd\ndef h : Q f \u27f6 P f.left := hk f \u226b limits.prod.fst\ndef over_pow : over B  := over.mk (k f)\n\ndef up : C := pullback (mem f.left) (limits.prod.map (h f) (\ud835\udfd9 f.left))\ndef h' : up f \u27f6 Q f \u2a2f f.left := pullback.snd\ninstance mono_h' : mono (h' f) := pullback.snd_of_mono\ninstance mono_hk : mono (hk f) := pullback.snd_of_mono\n\ndef hat_h' : hat (h' f) = h f :=\nunique_hat has_pullback_top_of_pb\n\ndef over.ni (f : over B) [has_power_object.{v} B] [has_power_object.{v} f.left] : over B :=\nover.mk (h' f \u226b limits.prod.snd \u226b f.hom)\n\n-- fix me.\ndef prop (f : over B) [has_power_object.{v} B] [has_power_object.{v} f.left] :\n  \u2203 q, q \u226b (pullback.snd : pullback (prod.lift f.hom (\ud835\udfd9 f.left)) (limits.prod.map ((k f) : _ \u27f6 B) (\ud835\udfd9 f.left)) \u27f6 _) = h' f :=\nbegin\n  have: pullback.fst \u226b P\u2081_arrow f.left = limits.prod.lift (h f) (k f \u226b singleton_arrow B \u226b P_map f.hom),\n    rw [pullback.condition],\n    dunfold bottom,\n    apply prod.hom_ext,\n    { rw [assoc, prod.lift_fst, h, hk, limits.prod.map_fst, comp_id] },\n    { rw [assoc, prod.lift_snd, k, hk, limits.prod.map_snd, assoc] },\n  rw [\u2190 seven_six_two, \u2190 hat_natural_left, \u2190 hat_h' f] at this,\n  have: limits.prod.lift (hat (h' f)) (hat pullback.snd) \u226b intersect.intersect f.left = limits.prod.lift (hat (h' f)) (hat pullback.snd) \u226b limits.prod.fst,\n    rw \u2190 this,\n    erw [assoc, assoc, equalizer.condition], refl,\n  rw \u2190 leq_prop at this,\n  cases this with a,\n  refine \u27e8_, over.w a\u27e9,\nend\n\n-- @[reducible]\ndef over.mem : over.ni f \u27f6 over_pow f \u2a2f f :=\nbegin\n  apply prod.lift _ _,\n  apply over.hom_mk _ _,\n  exact h' f \u226b limits.prod.fst,\n  dsimp [over_pow, over.ni, hk],\n  change (((h' f : up f \u27f6 _) \u226b (limits.prod.fst : Q f \u2a2f f.left \u27f6 Q f)) : _ \u27f6 Q f) \u226b (k f : Q f \u27f6 B) =\n    (h' f : up f \u27f6 Q f \u2a2f f.left) \u226b (limits.prod.snd : Q f \u2a2f f.left \u27f6 f.left) \u226b (f.hom : f.left \u27f6 B),\n  obtain \u27e8q, hq\u27e9 := prop f,\n  have z : (pullback.fst : pullback (prod.lift f.hom (\ud835\udfd9 f.left)) (limits.prod.map ((k f) : _ \u27f6 B) (\ud835\udfd9 f.left)) \u27f6 _) \u226b _ = _ \u226b _ := pullback.condition,\n    have z\u2081 := z =\u226b limits.prod.fst,\n    rw [assoc, prod.lift_fst, assoc, limits.prod.map_fst] at z\u2081,\n    have z\u2082 := z =\u226b limits.prod.snd,\n    erw [assoc, assoc, prod.lift_snd, limits.prod.map_snd, comp_id, comp_id] at z\u2082,\n    rw [\u2190 hq, assoc, assoc, \u2190 z\u2081, z\u2082, assoc, assoc],\n  apply over.hom_mk _ _,\n  exact h' f \u226b limits.prod.snd,\n  simp only [assoc, auto_param_eq], refl,\nend\n-- pretty gross.\ninstance over.mem_mono : mono (over.mem f) :=\nbegin\n  refine \u27e8\u03bb Z p q eq, _\u27e9,\n  ext1,\n  rw \u2190 cancel_mono (h' f),\n  apply prod.hom_ext,\n  rw [assoc, assoc],\n  have e\u2081 := eq =\u226b limits.prod.fst,\n  rw [over.mem, assoc, assoc, prod.lift_fst] at e\u2081,\n  exact congr_arg comma_morphism.left e\u2081,\n  have e\u2082 := eq =\u226b limits.prod.snd,\n  rw [over.mem, assoc, assoc, prod.lift_snd] at e\u2082,\n  rw [assoc, assoc],\n  exact congr_arg comma_morphism.left e\u2082,\nend\n\nsection hat\n\nvariables {f g}\nvariables {r : over B} (m : r \u27f6 g \u2a2f f) [mono m]\n\ndef m' : r.left \u27f6 (g \u2a2f f).left := m.left\ninstance m'_mono : mono (m' m) := category_theory.over_mono m\n\ndef m'' : r.left \u27f6 g.left \u2a2f f.left := m' m \u226b magic_arrow f g\ninstance m''_mono : mono (m'' m) := mono_comp _ _\n\n-- @[reducible]\ndef mhat : g.left \u27f6 P f.left := hat (m'' m)\n-- @[reducible]\ndef l : g.left \u27f6 P f.left \u2a2f P f.left := prod.lift (mhat m) g.hom \u226b bottom f\nlemma l_eq : l m = prod.lift (hat (m'' m)) (g.hom \u226b (singleton_arrow B : B \u27f6 P B) \u226b P_map f.hom) :=\nbegin\n  rw [l, bottom, prod.lift_map, comp_id], refl,\nend\n\nlemma llem : l m \u226b intersect.intersect f.left = l m \u226b limits.prod.fst :=\nbegin\n  have := l_eq m,\n  erw [\u2190 seven_six_two, \u2190 hat_natural_left] at this,\n  rw [this, \u2190 leq_prop],\n  refine \u27e8_\u27e9,\n  apply over.hom_mk _ _,\n  { apply pullback.lift (m'' m \u226b limits.prod.snd) (m'' m) _,\n    apply prod.hom_ext,\n    { erw [assoc, assoc, assoc, assoc, m'', assoc, prod.lift_fst, limits.prod.map_fst],\n      slice_lhs 2 3 {rw prod.lift_snd},\n      slice_rhs 2 3 {rw prod.lift_fst},\n      rw over.w (limits.prod.fst : g \u2a2f f \u27f6 g),\n      rw over.w (limits.prod.snd : g \u2a2f f \u27f6 f) },\n    { erw [assoc, assoc, assoc, assoc, assoc, prod.lift_snd, comp_id, limits.prod.map_snd, comp_id] } },\n  { dsimp, rw limit.lift_\u03c0, refl }\nend\n-- @[reducible]\ndef top : g.left \u27f6 P\u2081_obj f.left := equalizer.lift (l m) (llem m)\n-- @[reducible]\ndef h'' : g.left \u27f6 Q f := pullback.lift (top m) (prod.lift (mhat m) g.hom) (limit.lift_\u03c0 _ _)\n-- @[reducible]\ndef make_arrow : g \u27f6 over_pow f := over.hom_mk (h'' m) $ by { dsimp [over_pow, hk, k, h''], simp }\n-- @[reducible]\ndef square_top (m : r \u27f6 g \u2a2f f) [mono m] : r \u27f6 over.ni f :=\nbegin\n  refine over.hom_mk (pullback.lift (hat_powerises (m'' m)).top _ _) _,\n  { apply (m'' m) \u226b limits.prod.map (h'' m) (\ud835\udfd9 _) },\n  { rw (hat_powerises (m'' m)).comm, conv_rhs {rw [assoc, \u2190 prod.map_comp_id]}, congr' 2,\n    erw [h, hk, h'', limit.lift_\u03c0_assoc, prod.lift_fst, mhat] },\n  { dsimp [h'], erw [limit.lift_\u03c0_assoc, assoc, limits.prod.map_snd_assoc, id_comp],\n    erw [\u2190 over.w m, assoc, prod.lift_snd_assoc, over.w (limits.prod.snd : g \u2a2f f \u27f6 f)], refl }\nend\ndef alt_square_commutes : square_top m \u226b over.mem f \u226b limits.prod.fst = (m \u226b limits.prod.fst) \u226b make_arrow m :=\nbegin\n  rw [assoc, over.mem, prod.lift_fst, make_arrow],\n  ext1,\n  dsimp [h', m'', magic_arrow, h'', square_top],\n  rw limit.lift_\u03c0_assoc,\n  dsimp,\n  rw [assoc, limits.prod.map_fst, assoc, prod.lift_fst_assoc], refl\nend\ndef square_commutes : square_top m \u226b over.mem f = m \u226b limits.prod.map (make_arrow m) (\ud835\udfd9 _) :=\nbegin\n  apply prod.hom_ext,\n  { rw [assoc, alt_square_commutes, assoc, assoc, limits.prod.map_fst] },\n  { rw [assoc, over.mem, prod.lift_snd, assoc, limits.prod.map_snd, comp_id],\n    ext1,\n    dsimp [h', square_top],\n    rw limit.lift_\u03c0_assoc,\n    dsimp,\n    rw [assoc, limits.prod.map_snd, comp_id],\n    simp [m'', m'] }\nend\n\ndef alt_square_pb : is_limit (pullback_cone.mk _ _ (alt_square_commutes m)) :=\nbegin\n  apply reflects_pullback_cone (over.forget _),\n  -- apply reflect_pullback,\n  dsimp [square_top],\n  refine is_limit.mk' _ _,\n  intro s,\n  have lem : (pullback_cone.fst s \u226b pullback.fst) \u226b mem f.left =\n    prod.lift (pullback_cone.snd s) (pullback_cone.fst s \u226b pullback.fst \u226b mem f.left \u226b limits.prod.snd) \u226b\n      limits.prod.map (hat (m'' m)) (\ud835\udfd9 f.left),\n  { apply prod.hom_ext,\n    { rw [assoc, assoc, assoc, limits.prod.map_fst, prod.lift_fst_assoc,\n          category_theory.limits.pullback.condition_assoc, limits.prod.map_fst],\n      have : pullback_cone.fst s \u226b (over.mem f \u226b _).left = _ := pullback_cone.condition s,\n      dsimp only [over.mem] at this,\n      rw [prod.lift_fst] at this,\n      dsimp [h'] at this,\n      slice_lhs 1 3 {rw this},\n      dsimp [h, hk, make_arrow, h''],\n      rw [assoc],\n      rw [pullback.lift_snd_assoc, prod.lift_fst], refl },\n    { rw [assoc, assoc, assoc, limits.prod.map_snd, comp_id, prod.lift_snd] } },\n  let t : s.X \u27f6 r.left := (hat_powerises (m'' m)).is_pb.lift (pullback_cone.mk _ _ lem),\n  have t\u2083 : t \u226b m'' m \u226b limits.prod.fst = pullback_cone.snd s,\n    rw \u2190 assoc,\n    erw (hat_powerises (m'' m)).is_pb.fac (pullback_cone.mk _ _ lem) walking_cospan.right,\n    dsimp,\n    rw prod.lift_fst,\n  have t\u2082 : t \u226b m'' m \u226b limits.prod.snd = pullback_cone.fst s \u226b pullback.fst \u226b mem f.left \u226b limits.prod.snd,\n    rw \u2190 assoc,\n    erw (hat_powerises (m'' m)).is_pb.fac (pullback_cone.mk _ _ lem) walking_cospan.right,\n    dsimp,\n    rw prod.lift_snd,\n  have t\u2081: t \u226b (hat_powerises (m'' m)).top = pullback_cone.fst s \u226b pullback.fst,\n    erw (hat_powerises (m'' m)).is_pb.fac (pullback_cone.mk _ _ lem) walking_cospan.left,\n    refl,\n  refine \u27e8t, _, _, _\u27e9,\n  { change t \u226b pullback.lift (hat_powerises (m'' m)).top (m'' m \u226b limits.prod.map (h'' m) (\ud835\udfd9 f.left)) _ = s.\u03c0.app walking_cospan.left,\n    apply pullback.hom_ext,\n    { rw \u2190 t\u2081, simp },\n    { rw [assoc], slice_lhs 2 3 {rw limit.lift_\u03c0},\n      dsimp,\n      apply prod.hom_ext,\n      { rw [assoc, assoc, limits.prod.map_fst],\n        slice_lhs 1 3 {rw t\u2083},\n        rw [h''],\n        erw \u2190 pullback_cone.condition s,\n        rw assoc,\n        change pullback_cone.fst s \u226b (over.mem f \u226b limits.prod.fst).left = s.\u03c0.app walking_cospan.left \u226b pullback.snd \u226b limits.prod.fst,\n        erw prod.lift_fst,\n        refl },\n      { rw [assoc, assoc, limits.prod.map_snd, comp_id, t\u2082,\n            category_theory.limits.pullback.condition_assoc, limits.prod.map_snd,\n            comp_id, assoc] } } },\n  { dunfold pullback_cone.snd,\n    dsimp,\n    rw [m'', assoc, magic_arrow, prod.lift_fst] at t\u2083,\n    exact t\u2083 },\n  { intros t' m\u2081 m\u2082,\n    have m\u2081' : t' \u226b pullback.lift (hat_powerises (m'' m)).top (m'' m \u226b limits.prod.map (h'' m) (\ud835\udfd9 f.left)) _ =\n    pullback_cone.fst s := m\u2081,\n    have m\u2082' : t' \u226b m.left \u226b _ = pullback_cone.snd s := m\u2082,\n    clear m\u2081 m\u2082,\n    rw \u2190 cancel_mono (m'' m),\n    change t' \u226b m' m \u226b magic_arrow f g = t \u226b m' m \u226b magic_arrow f g,\n    apply prod.hom_ext,\n    { rw [assoc, assoc],\n      slice_lhs 3 4 {rw prod.lift_fst},\n      rw m',\n      rw m\u2082',\n      rw \u2190 t\u2083,\n      rw assoc, refl },\n    { conv_rhs {erw [assoc, t\u2082, \u2190 m\u2081']},\n      rw [assoc, assoc, assoc],\n      slice_rhs 2 3 {rw limit.lift_\u03c0},\n      dsimp,\n      rw (hat_powerises (m'' m)).comm,\n      rw [assoc, limits.prod.map_snd, comp_id],\n      simp [m''] } },\n  refine \u27e8\u03bb K, by apply_instance\u27e9,\nend\n\nend hat\n\ndef main' (f : over B) [has_power_object.{v} f.left] : is_power_object (over.mem f) :=\n{ hat := \u03bb b r m hm, by exactI make_arrow m,\n  powerises' := \u03bb g r m hm, by exactI\n  begin\n    refine \u27e8square_top m, square_commutes m, _\u27e9,\n    apply is_limit.mk' _ _,\n    intro s,\n    have comm : pullback_cone.fst s \u226b over.mem f \u226b limits.prod.fst = (pullback_cone.snd s \u226b limits.prod.fst) \u226b make_arrow m,\n      rw [pullback_cone.condition_assoc s, limits.prod.map_fst, assoc],\n    let s' := pullback_cone.mk _ _ comm,\n    let t := (alt_square_pb m).lift s',\n    have t\u2081 : t \u226b square_top m = pullback_cone.fst s' := (alt_square_pb m).fac s' walking_cospan.left,\n    have t\u2082 : t \u226b m \u226b limits.prod.fst = pullback_cone.snd s' := (alt_square_pb m).fac s' walking_cospan.right,\n    have t\u2083 := t\u2081 =\u226b over.mem f,\n      rw [assoc, square_commutes m] at t\u2083,\n    replace t\u2083 := t\u2083 =\u226b limits.prod.snd,\n    rw [assoc, assoc, assoc, limits.prod.map_snd, comp_id] at t\u2083,\n    refine \u27e8(alt_square_pb m).lift s', (alt_square_pb m).fac s' walking_cospan.left, _, _\u27e9,\n    { change t \u226b m = pullback_cone.snd s,\n      apply prod.hom_ext,\n      { rw [assoc, t\u2082], refl },\n      { rw [assoc, t\u2083], erw [pullback_cone.condition_assoc s, limits.prod.map_snd, comp_id] } },\n    { intros t' m\u2081 m\u2082,\n      apply (alt_square_pb m).hom_ext,\n      apply pullback_cone.equalizer_ext (pullback_cone.mk (square_top m) (m \u226b limits.prod.fst) _),\n      erw t\u2081,\n      exact m\u2081,\n      erw [t\u2082, \u2190 assoc, m\u2082], refl }\n  end,\n  uniquely' := \u03bb g r m hm hat' p, by exactI\n  begin\n    ext1,\n    rw \u2190 cancel_mono (hk f),\n    have z\u2081: hat'.left \u226b k f = g.hom := over.w hat',\n    suffices z\u2082: hat'.left \u226b h f = (make_arrow m).left \u226b h f,\n      apply prod.hom_ext,\n      { erw [assoc, assoc, z\u2082], refl },\n      { rw [assoc, assoc],\n        change _ \u226b k f = _ \u226b k f,\n        rw [z\u2081, make_arrow, over.hom_mk_left, h'', k, hk, pullback.lift_snd_assoc, prod.lift_snd] },\n    erw [make_arrow, limit.lift_\u03c0_assoc, prod.lift_fst],\n    symmetry,\n    apply unique_hat,\n    change has_pullback_top _ _ _,\n    rw prod.map_comp_id,\n    apply left_right_hpb_to_both_hpb (h' f) _ has_pullback_top_of_pb,\n    have: h' f = (over.mem f).left \u226b magic_arrow f (over_pow f),\n    { apply prod.hom_ext,\n      { rw [assoc, prod.lift_fst, \u2190 over.comp_left, over.mem, prod.lift_fst], refl },\n      { rw [assoc, prod.lift_snd, \u2190 over.comp_left, over.mem, prod.lift_snd], refl } },\n    rw this,\n    apply stretch_hpb_down _ _ (limits.prod.map hat' (\ud835\udfd9 f)).left _ _ _ _ _ (magic_pb _ _ f hat'),\n    apply over_forget_preserves_hpb _ _ _ p,\n  end }\n\ndef main (f : over B) [has_power_object.{v} f.left] : has_power_object.{v} f :=\n{ PA := over_pow f,\n  niA := over.ni f,\n  memA := over.mem f,\n  is_power := main' f }\n\nend slicing\n\ninstance fundamental_theorem (B : C) [has_power_objects.{v} C] : has_power_objects.{v} (over B) :=\n{ has_power_object := \u03bb f, slicing.main f }\n\ndef comparison [has_power_objects.{v} C]\n  {D : Type u\u2082} [category.{v} D] [has_finite_limits.{v} D] [has_power_objects.{v} D]\n  (F : C \u2964 D) (h : \u03a0 (J : Type v) [\ud835\udca5\u2081 : small_category J] [@fin_category J \ud835\udca5\u2081], @preserves_limits_of_shape _ _ _ _ J \ud835\udca5\u2081 F)\n  (A : C) : F.obj (P A) \u27f6 P (F.obj A) :=\nbegin\n  let m := F.map (mem A) \u226b (prod_comparison F (P A) A),\n  letI : mono (F.map (mem A)) := preserves_mono_of_preserves_pullback F _ _ _,\n  exact hat m,\nend\n\ndef comp_natural' [has_power_objects.{v} C]\n  {D : Type u\u2082} [category.{v} D] [has_finite_limits.{v} D] [has_power_objects.{v} D]\n  (F : C \u2964 D) (h : \u03a0 (J : Type v) [\ud835\udca5\u2081 : small_category J] [@fin_category J \ud835\udca5\u2081], @preserves_limits_of_shape _ _ _ _ J \ud835\udca5\u2081 F)\n  (A B : C) (f : B \u27f6 A) :\n  F.map (P_map f) \u226b comparison F h B = comparison F h A \u226b P_map (F.map f) :=\nbegin\n  dsimp [comparison],\n  rw [\u2190 hat_natural_left, \u2190 hat_natural_right],\n  let m\u2081 := F.map (mem A) \u226b (prod_comparison F (P A) A),\n  let m\u2082 := F.map (mem B) \u226b (prod_comparison F (P B) B),\n  letI : mono (F.map (mem A)) := preserves_mono_of_preserves_pullback F _ _ _,\n  letI : mono (F.map (mem B)) := preserves_mono_of_preserves_pullback F _ _ _,\n  letI : mono (F.map (Emap f)) := preserves_mono_of_preserves_pullback F _ _ _,\n  let P\u2081 := pullback (F.map (mem B) \u226b (prod_comparison F (P B) B)) (limits.prod.map (F.map (P_map f)) (\ud835\udfd9 (F.obj B))),\n  let P\u2082 := pullback (F.map (mem A) \u226b (prod_comparison F (P A) A)) (limits.prod.map (\ud835\udfd9 _) (F.map f)),\n  let h\u2081 : P\u2081 \u27f6 _ := pullback.snd,\n  let h\u2082 : P\u2082 \u27f6 _ := pullback.snd,\n  change hat h\u2081 = hat h\u2082,\n  let s\u2081 := (hat_powerises (Emap f)).is_pb,\n  let s\u2082 := Epb f,\n  let Fs\u2081 := preserves_pullback_cone F _ _ _ _ _ s\u2081,\n  let Fs\u2082 := preserves_pullback_cone F _ _ _ _ _ s\u2082,\n  have s\u2083comm : F.map (limits.prod.map (P_map f) (\ud835\udfd9 B)) \u226b (prod_comparison F (P B) B) = (prod_comparison F (P A) B) \u226b limits.prod.map (F.map (P_map f)) (\ud835\udfd9 (F.obj B)),\n    rw [prod_comparison, prod_comparison],\n    apply prod.hom_ext,\n    { erw [assoc, prod.lift_fst, assoc, limits.prod.map_fst, \u2190 F.map_comp, limits.prod.map_fst, prod.lift_fst_assoc, F.map_comp] },\n    { erw [assoc, prod.lift_snd, assoc, limits.prod.map_snd, comp_id, \u2190 F.map_comp, limits.prod.map_snd, comp_id, prod.lift_snd] },\n  let s\u2083 := pullback_square_iso (F.map (limits.prod.map (P_map f) (\ud835\udfd9 _))) (prod_comparison F (P A) B) (prod_comparison F (P B) B) (limits.prod.map (F.map (P_map f)) (\ud835\udfd9 _)) s\u2083comm,\n  let Fs\u2081s\u2083 := vpaste _ _ _ _ _ _ _ _ _ s\u2083 Fs\u2081,\n  have eq\u2081: hat h\u2081 = hat (F.map (Emap f) \u226b (prod_comparison F (P A) B)),\n  { apply lifting _ _ _ _,\n    { apply Fs\u2081s\u2083.lift (limit.cone _) },\n    { apply limit.lift _ (pullback_cone.mk (F.map (hat_powerises (Emap f)).top) (F.map (Emap f) \u226b (prod_comparison F (P A) B)) _),\n      rw [assoc, \u2190 s\u2083comm, \u2190 assoc, \u2190 F.map_comp, (hat_powerises (Emap f)).comm, F.map_comp, assoc], refl },\n    { exact (Fs\u2081s\u2083.fac (limit.cone _) walking_cospan.right) },\n    { rw limit.lift_\u03c0, refl } },\n  have s\u2084comm : F.map (limits.prod.map (\ud835\udfd9 (P A)) f) \u226b (prod_comparison F (P A) A) = (prod_comparison F (P A) B) \u226b limits.prod.map (\ud835\udfd9 (F.obj (P A))) (F.map f),\n    rw [prod_comparison, prod_comparison],\n    apply prod.hom_ext,\n    { rw [assoc, prod.lift_fst, assoc, limits.prod.map_fst, \u2190 F.map_comp, limits.prod.map_fst, comp_id, comp_id, prod.lift_fst] },\n    { rw [assoc, prod.lift_snd, assoc, limits.prod.map_snd, \u2190 F.map_comp, limits.prod.map_snd, prod.lift_snd_assoc, F.map_comp] },\n  let s\u2084 := pullback_square_iso (F.map (limits.prod.map (\ud835\udfd9 _) f)) (prod_comparison F (P A) B) (prod_comparison F (P A) A) (limits.prod.map (\ud835\udfd9 _) (F.map f)) s\u2084comm,\n  let Fs\u2082s\u2084 := vpaste _ _ _ _ _ _ _ _ _ s\u2084 Fs\u2082,\n  have eq\u2082: hat h\u2082 = hat (F.map (Emap f) \u226b (prod_comparison F (P A) B)),\n  { apply lifting _ _ _ _,\n    { apply Fs\u2082s\u2084.lift (limit.cone _) },\n    { apply limit.lift _ (pullback_cone.mk (F.map pullback.fst) (F.map (Emap f) \u226b (prod_comparison F (P A) B)) _),\n      rw [assoc, \u2190 s\u2084comm, \u2190 assoc, \u2190 F.map_comp, pullback.condition, F.map_comp, assoc], refl },\n    { exact (Fs\u2082s\u2084.fac (limit.cone _) walking_cospan.right) },\n    { rw limit.lift_\u03c0, refl } },\n  rw [eq\u2081, eq\u2082],\nend\n\n-- Define F as a logical functor if this is an iso.\ndef comp_natural [has_power_objects.{v} C]\n  {D : Type u\u2082} [category.{v} D] [has_finite_limits.{v} D] [has_power_objects.{v} D]\n  (F : C \u2964 D) [h : \u03a0 (J : Type v) [\ud835\udca5\u2081 : small_category J] [@fin_category J \ud835\udca5\u2081], @preserves_limits_of_shape _ _ _ _ J \ud835\udca5\u2081 F] :\n  (P_functor \u22d9 F) \u27f6 (F.op \u22d9 P_functor) :=\n{ app := \u03bb A, comparison F h A.unop,\n  naturality' := \u03bb A B g, comp_natural' F h A.unop B.unop g.unop }\n\ndef star_power (A B : C) [has_power_object.{v} A] : (star B).obj (ni A) \u27f6 (star B).obj (P A) \u2a2f (star B).obj A :=\nbegin\n  haveI := adjunction.right_adjoint_preserves_limits (forget_adj_star B),\n  exact (star B).map (mem A) \u226b (prod_comparison (star B) (P A) A)\nend\ninstance star_mono (A B : C) [has_power_object.{v} A] : mono (star_power A B) :=\nbegin\n  haveI : mono ((star B).map (mem A)) := right_adjoint_preserves_mono (forget_adj_star B) (by apply_instance),\n  haveI := adjunction.right_adjoint_preserves_limits (forget_adj_star B),\n  rw star_power,\n  haveI : is_iso (prod_comparison (star B) (P A) A) := by apply_instance,\n  apply_instance\nend\n\ndef alt_prod (A : C) {B : C} (g : over B) : over B := over.mk ((limits.prod.fst : g.left \u2a2f A \u27f6 g.left) \u226b g.hom)\n\n@[simps]\ndef the_iso (A : C) {B : C} (g : over B) : g \u2a2f (star B).obj A \u2245 alt_prod A g :=\n{ hom :=\n  begin\n    apply over.hom_mk _ _,\n    apply prod.lift (limits.prod.fst : g \u2a2f _ \u27f6 _).left _,\n    refine (limits.prod.snd : g \u2a2f _ \u27f6 _).left \u226b limits.prod.snd,\n    erw limit.lift_\u03c0_assoc,\n    exact over.w (limits.prod.fst : g \u2a2f (star B).obj A \u27f6 _),\n  end,\n  inv :=\n  begin\n    apply prod.lift,\n    refine over.hom_mk limits.prod.fst rfl,\n    refine over.hom_mk (limits.prod.map g.hom (\ud835\udfd9 _)) (limits.prod.map_fst _ _),\n  end,\n  hom_inv_id' :=\n  begin\n    ext1,\n    dsimp,\n    rw \u2190 cancel_mono (magic_arrow ((star B).obj A) g),\n    rw id_comp,\n    apply prod.hom_ext,\n    rw [prod.lift_fst, assoc, prod.lift_fst, assoc, \u2190 over.comp_left, prod.lift_fst, over.hom_mk_left, prod.lift_fst],\n    rw [prod.lift_snd, assoc, prod.lift_snd, assoc, \u2190 over.comp_left, prod.lift_snd, over.hom_mk_left],\n    apply prod.hom_ext,\n    rw [assoc, limits.prod.map_fst, prod.lift_fst_assoc, over.w (limits.prod.fst : g \u2a2f (star B).obj A \u27f6 _)],\n    exact (over.w (limits.prod.snd : g \u2a2f (star B).obj A \u27f6 _)).symm,\n    rw [assoc, limits.prod.map_snd, prod.lift_snd_assoc, comp_id],\n  end,\n  inv_hom_id' :=\n  begin\n    ext,\n    dsimp,\n    rw [assoc, prod.lift_fst, \u2190 over.comp_left, prod.lift_fst, id_comp], refl,\n    rw [over.comp_left, assoc, over.hom_mk_left, prod.lift_snd, \u2190 assoc, \u2190 over.comp_left,\n        prod.lift_snd, over.hom_mk_left, limits.prod.map_snd, over.id_left, id_comp, comp_id],\n  end }\n\n\ndef star_hat {A B : C} [has_power_object.{v} A] {g r : over B} (m : r \u27f6 g \u2a2f (star B).obj A) (k : g.left \u27f6 P A) [mono m] : g \u27f6 (star B).obj (P A):=\nover.hom_mk (prod.lift g.hom k) (limit.lift_\u03c0 _ _)\n\ndef seven_eleven_r_comm (A B : C) [has_power_object.{v} A] :\n  \ud835\udfd9 (B \u2a2f _) \u226b limits.prod.map (\ud835\udfd9 _) (mem A) = (star_power A B \u226b (the_iso A ((star B).obj (P A))).hom).left \u226b (prod.associator B (P A) A).hom :=\nbegin\n  dsimp [star_power, the_iso, prod_comparison],\n  rw [assoc, assoc, id_comp],\n  apply prod.hom_ext,\n  rw [assoc, assoc, assoc, prod.lift_fst, prod.lift_fst_assoc, limits.prod.map_fst, comp_id],\n  slice_rhs 2 3 {rw \u2190 over.comp_left},\n  rw [prod.lift_fst, over.hom_mk_left, \u2190 assoc, \u2190 prod.map_id_comp, limits.prod.map_fst, comp_id],\n  rw [assoc, assoc, assoc, prod.lift_snd, limits.prod.map_snd],\n  apply prod.hom_ext,\n  rw [assoc, assoc, assoc, assoc, prod.lift_fst, prod.lift_fst_assoc],\n  slice_rhs 2 3 {rw \u2190 over.comp_left},\n  rw [prod.lift_fst, over.hom_mk_left, \u2190 prod.map_id_comp_assoc, limits.prod.map_snd],\n  rw [assoc, assoc, assoc, assoc, prod.lift_snd, prod.lift_snd],\n  slice_rhs 2 3 {rw \u2190 over.comp_left},\n  rw [prod.lift_snd, over.hom_mk_left, \u2190 prod.map_id_comp_assoc, limits.prod.map_snd],\nend\n\ndef seven_eleven_aux (A B : C) [has_power_object.{v} A] (g r : over B) (m : r \u27f6 g \u2a2f (star B).obj A) [mono m] (k : g.left \u27f6 P A) :\n  powerises (mem A) (m \u226b (the_iso A g).hom).left k \u2245 powerises (star_power A B) m (star_hat m k) :=\nbegin\n  have bottom_comm :\n    limits.prod.map (star_hat m k) (\ud835\udfd9 _) \u226b (the_iso A _).hom =\n    (the_iso A g).hom \u226b over.hom_mk (limits.prod.map (prod.lift g.hom k) (\ud835\udfd9 A))\n      (by { dsimp, erw [limits.prod.map_fst_assoc, limits.prod.lift_fst], refl }),\n  { dsimp [the_iso], ext : 2,\n    { rw [over.comp_left, over.comp_left, over.hom_mk_left, assoc, prod.lift_fst,\n          \u2190 over.comp_left, limits.prod.map_fst, over.comp_left, star_hat, over.hom_mk_left,\n          over.hom_mk_left, over.hom_mk_left, prod.lift_map, prod.lift_fst] },\n    { rw [over.comp_left, over.comp_left, over.hom_mk_left, assoc, prod.lift_snd, \u2190 assoc,\n          \u2190 over.comp_left, limits.prod.map_snd, comp_id, over.hom_mk_left, over.hom_mk_left,\n          prod.lift_map, comp_id, prod.lift_snd] } },\n\n  have b_pb := pullback_square_iso _ _ _ _ bottom_comm,\n  have right\u2081_comm : \ud835\udfd9 (B \u2a2f _) \u226b limits.prod.map (\ud835\udfd9 _) (mem A) = (star_power A B \u226b (the_iso A ((star B).obj (P A))).hom).left \u226b (prod.associator B (P A) A).hom,\n    apply seven_eleven_r_comm,\n  have r\u2081_pb := pullback_square_iso' _ _ _ _ right\u2081_comm,\n  have r\u2082_pb := pullback_prod' (mem A) B,\n  have r_pb := (left_pb_to_both_pb _ _ _ _ _ _ _ _ _ r\u2081_pb) r\u2082_pb,\n  have p : limits.prod.map (prod.lift g.hom k) (\ud835\udfd9 A) \u226b (prod.associator B (P A) A).hom \u226b limits.prod.snd = limits.prod.map k (\ud835\udfd9 A),\n    rw [prod.associator_hom, prod.lift_snd],\n    apply prod.hom_ext,\n    { rw [assoc, prod.lift_fst, limits.prod.map_fst, prod.map_fst_assoc, prod.lift_snd] },\n    { rw [assoc, prod.lift_snd, limits.prod.map_snd, limits.prod.map_snd] },\n  refine \u27e8_, _, subsingleton.elim _ _, subsingleton.elim _ _\u27e9,\n  { intro q,\n    refine cut_hpb_up _ _ _ _ _ _ _ _ b_pb,\n    apply over_forget_reflects_hpb,\n    refine right_both_hpb_to_left_hpb _ _ _ (has_pullback_top_of_is_pb r_pb),\n    convert q },\n  { intro q,\n    have := stretch_hpb_down _ _ _ _ _ _ q _ b_pb,\n    have := over_forget_preserves_hpb _ _ _ this,\n    change has_pullback_top _ _ _,\n    have p' := p.symm,\n    convert left_hpb_right_pb_to_both_hpb _ _ _ _ _ _ this _ r_pb }\nend\n\ndef seven_eleven (A B : C) [has_power_object.{v} A] : is_power_object (star_power A B) :=\n{ hat := \u03bb g r m hm, by exactI over.hom_mk (prod.lift g.hom (hat (m \u226b (the_iso A g).hom).left)) (limit.lift_\u03c0 _ _),\n  powerises' := \u03bb g r m hm, by exactI\n  begin\n    apply (seven_eleven_aux A B g r m (hat (m \u226b (the_iso A g).hom).left)).hom,\n    exact hat_powerises (m \u226b (the_iso A g).hom).left,\n  end,\n  uniquely' := \u03bb g r m hm hat' pow, by exactI\n  begin\n    ext,\n    rw [over.hom_mk_left, prod.lift_fst, \u2190 over.w hat'], refl,\n    rw [over.hom_mk_left, prod.lift_snd],\n    apply unique_hat,\n    apply (seven_eleven_aux A B g r m (hat'.left \u226b limits.prod.snd)).inv,\n    convert pow,\n    rw [star_hat],\n    ext,\n    rw [over.hom_mk_left, prod.lift_fst, \u2190 over.w hat'], refl,\n    rw [over.hom_mk_left, prod.lift_snd]\n  end }\n\ndef logical_star [has_power_objects.{v} C] (B : C) : P_functor \u22d9 star B \u2245 (star B).op \u22d9 P_functor :=\nbegin\n  apply nat_iso.of_components _ _,\n  intro A,\n  exact P_unique_up_to_iso (seven_eleven A.unop B) (power_is_power _),\n  intros X Y g,\n  haveI := adjunction.right_adjoint_preserves_limits (forget_adj_star B),\n  apply comp_natural' (star B) infer_instance X.unop Y.unop g.unop,\nend\n\nlocal attribute [instance] has_finite_products_of_has_finite_limits\n\ndef cc_of_pow [has_power_objects.{v} C] : cartesian_closed.{v} C :=\n{ closed := \u03bb B,\n  begin\n    haveI : is_right_adjoint (star B) := \u27e8over.forget _, forget_adj_star B\u27e9,\n    haveI := monadic_adjoint_square_lift _ _ _ _ (logical_star B),\n    refine exponentiable_of_star_is_left_adj B _,\n    apply left_adjoint_of_right_adjoint_op,\n  end }\n\ndef lcc_of_pow [has_power_objects.{v} C] : is_locally_cartesian_closed.{v} C :=\n{ overs_cc := \u03bb B, cc_of_pow }\n\ndef subobj_hat {A B R : C} [exponentiable A] [has_subobject_classifier.{v} C] (m : R \u27f6 B \u2a2f A) [mono m] :\n  B \u27f6 A \u27f9 classifier.\u03a9 C :=\ncartesian_closed.curry ((limits.prod.braiding _ _).inv \u226b classifier.classifier_of m)\n\ndef power_of_subobj (A : C) [exponentiable A] [has_subobject_classifier.{v} C] : has_power_object.{v} A :=\n{ PA := A \u27f9 classifier.\u03a9 C,\n  niA := pullback (classifier.truth C) ((limits.prod.braiding _ _).hom \u226b (ev A).app _),\n  memA := pullback.snd,\n  is_power :=\n  { hat := \u03bb B R m hm, by exactI subobj_hat m,\n    powerises' := \u03bb B R m hm,\n    begin\n      haveI := hm,\n      apply right_both_hpb_to_left_hpb _ _ _ has_pullback_top_of_pb,\n      erw [braid_natural_assoc, subobj_hat, curry_eq, prod.map_id_comp, assoc, (ev _).naturality,\n           ev_coev_assoc, iso.hom_inv_id_assoc],\n      apply classifier.classifies m,\n    end,\n    uniquely' := \u03bb B R m hm hat' p,\n    begin\n      rw [subobj_hat, curry_eq_iff, iso.inv_comp_eq],\n      apply classifier.uniquely,\n      change has_pullback_top _ _ _,\n      rw [uncurry_eq, \u2190 braid_natural_assoc],\n      apply left_right_hpb_to_both_hpb pullback.snd p has_pullback_top_of_pb,\n    end } }\n\ninstance topos_has_power [has_subobject_classifier.{v} C] [cartesian_closed.{v} C] : has_power_objects.{v} C :=\n\u27e8\u03bb A, power_of_subobj A\u27e9\n\ninstance topos_has_some_colims (J : Type v) [small_category J] [has_subobject_classifier.{v} C] [cartesian_closed.{v} C] [has_limits_of_shape J\u1d52\u1d56 C] :\n  has_colimits_of_shape J C :=\nsome_colims J\n\ninstance topos_has_finite_colimits [has_subobject_classifier.{v} C] [cartesian_closed.{v} C] : has_finite_colimits.{v} C :=\n\u03bb _ _ _, by {resetI, apply_instance}\n\ninstance topos_is_lcc [has_subobject_classifier.{v} C] [cartesian_closed.{v} C] : is_locally_cartesian_closed.{v} C :=\nlcc_of_pow\n\nend category_theory", "meta": {"author": "b-mehta", "repo": "topos", "sha": "c9032b11789e36038bc841a1e2b486972421b983", "save_path": "github-repos/lean/b-mehta-topos", "path": "github-repos/lean/b-mehta-topos/topos-c9032b11789e36038bc841a1e2b486972421b983/src/power.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.36871772248069956}}
{"text": "/-\nCopyright (c) 2021 Chris Birkbeck. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Birkbeck\n-/\nimport data.setoid.basic\nimport group_theory.subgroup.basic\nimport group_theory.coset\nimport group_theory.subgroup.pointwise\nimport data.set.basic\nimport tactic.group\n\n/-!\n# Double cosets\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines double cosets for two subgroups `H K` of a group `G` and the quotient of `G` by\nthe double coset relation, i.e. `H \\ G / K`. We also prove that `G` can be writen as a disjoint\nunion of the double cosets and that if one of `H` or `K` is the trivial group (i.e. `\u22a5` ) then\nthis is the usual left or right quotient of a group by a subgroup.\n\n## Main definitions\n\n* `rel`: The double coset relation defined by two subgroups `H K` of `G`.\n* `double_coset.quotient`: The quotient of `G` by the double coset relation, i.e, ``H \\ G / K`.\n-/\n\n\nvariables {G : Type*} [group G] {\u03b1 : Type*} [has_mul \u03b1] (J: subgroup G) (g : G)\n\nnamespace doset\n\nopen_locale pointwise\n\n/--The double_coset as an element of `set \u03b1` corresponding to `s a t` -/\ndef _root_.doset (a : \u03b1) (s t : set \u03b1) : set \u03b1 := s * {a} * t\n\nlemma mem_doset {s t : set \u03b1} {a b : \u03b1} : b \u2208 doset a s t \u2194 \u2203 (x \u2208 s) (y \u2208 t), b = x * a * y :=\n\u27e8\u03bb \u27e8_, y, \u27e8x, _, hx, rfl, rfl\u27e9, hy, h\u27e9, \u27e8x, hx, y, hy, h.symm\u27e9,\n  \u03bb \u27e8x, hx, y, hy, h\u27e9, \u27e8x * a, y, \u27e8x, a, hx, rfl, rfl\u27e9, hy, h.symm\u27e9\u27e9\n\nlemma mem_doset_self (H K : subgroup G) (a : G) : a \u2208 doset a H K :=\nmem_doset.mpr \u27e81, H.one_mem, 1, K.one_mem, (one_mul a).symm.trans (mul_one (1 * a)).symm\u27e9\n\nlemma doset_eq_of_mem {H K : subgroup G} {a b : G} (hb : b \u2208 doset a H K) :\n  doset b H K = doset a H K :=\nbegin\n  obtain \u27e8_, k, \u27e8h, a, hh, (rfl : _ = _), rfl\u27e9, hk, rfl\u27e9 := hb,\n  rw [doset, doset, \u2190set.singleton_mul_singleton, \u2190set.singleton_mul_singleton, mul_assoc,\n    mul_assoc, subgroup.singleton_mul_subgroup hk, \u2190mul_assoc, \u2190mul_assoc,\n    subgroup.subgroup_mul_singleton hh],\nend\n\nlemma mem_doset_of_not_disjoint {H K : subgroup G} {a b : G}\n  (h : \u00ac disjoint (doset a H K) (doset b H K)) : b \u2208 doset a H K :=\nbegin\n  rw set.not_disjoint_iff at h,\n  simp only [mem_doset] at *,\n  obtain \u27e8x, \u27e8l, hl, r, hr, hrx\u27e9, y, hy, \u27e8r', hr', rfl\u27e9\u27e9 := h,\n  refine \u27e8y\u207b\u00b9 * l, H.mul_mem (H.inv_mem hy) (hl), r * r'\u207b\u00b9, K.mul_mem hr (K.inv_mem hr'), _\u27e9,\n  rwa [mul_assoc, mul_assoc, eq_inv_mul_iff_mul_eq, \u2190mul_assoc, \u2190mul_assoc, eq_mul_inv_iff_mul_eq],\nend\n\nlemma eq_of_not_disjoint {H K : subgroup G} {a b : G} (h: \u00ac disjoint (doset a H K) (doset b H K)) :\n  doset a H K = doset b H K :=\nbegin\n  rw disjoint.comm at h,\n  have ha : a \u2208 doset b H K := mem_doset_of_not_disjoint h,\n  apply doset_eq_of_mem ha,\nend\n\n/-- The setoid defined by the double_coset relation -/\ndef setoid (H K : set G) : setoid G :=\nsetoid.ker (\u03bb x, doset x H K)\n\n/-- Quotient of `G` by the double coset relation, i.e. `H \\ G / K` -/\ndef quotient (H K : set G) : Type* :=\nquotient (setoid H K)\n\nlemma rel_iff {H K : subgroup G} {x y : G} :\n  (setoid \u2191H \u2191K).rel x y \u2194 \u2203 (a \u2208 H) (b \u2208 K), y = a * x * b :=\niff.trans \u27e8\u03bb hxy, (congr_arg _ hxy).mpr (mem_doset_self H K y),\n  \u03bb hxy, (doset_eq_of_mem hxy).symm\u27e9 mem_doset\n\nlemma bot_rel_eq_left_rel (H : subgroup G) :\n  (setoid \u2191(\u22a5 : subgroup G) \u2191H).rel = (quotient_group.left_rel H).rel :=\nbegin\n  ext a b,\n  rw [rel_iff, setoid.rel, quotient_group.left_rel_apply],\n  split,\n  { rintros \u27e8a, (rfl : a = 1), b, hb, rfl\u27e9,\n    change a\u207b\u00b9 * (1 * a * b) \u2208 H,\n    rwa [one_mul, inv_mul_cancel_left] },\n  { rintro (h : a\u207b\u00b9 * b \u2208 H),\n    exact \u27e81, rfl, a\u207b\u00b9 * b, h, by rw [one_mul, mul_inv_cancel_left]\u27e9 },\nend\n\nlemma rel_bot_eq_right_group_rel (H : subgroup G) :\n  (setoid \u2191H \u2191(\u22a5 : subgroup G)).rel = (quotient_group.right_rel H).rel :=\nbegin\n  ext a b,\n  rw [rel_iff, setoid.rel, quotient_group.right_rel_apply],\n  split,\n  { rintros \u27e8b, hb, a, (rfl : a = 1), rfl\u27e9,\n    change b * a * 1 * a\u207b\u00b9 \u2208 H,\n    rwa [mul_one, mul_inv_cancel_right] },\n  { rintro (h : b * a\u207b\u00b9 \u2208 H),\n    exact \u27e8b * a\u207b\u00b9, h, 1, rfl, by rw [mul_one, inv_mul_cancel_right]\u27e9 },\nend\n\n/--Create a doset out of an element of `H \\ G / K`-/\ndef quot_to_doset (H K : subgroup G) (q : quotient \u2191H \u2191K) : set G := (doset q.out' H K)\n\n/--Map from `G` to `H \\ G / K`-/\nabbreviation mk (H K : subgroup G) (a : G) : quotient \u2191H \u2191K :=\nquotient.mk' a\n\ninstance (H K : subgroup G) : inhabited (quotient \u2191H \u2191K) := \u27e8mk H K (1 : G)\u27e9\n\nlemma eq (H K : subgroup G) (a b : G) : mk H K a = mk H K b \u2194 \u2203 (h \u2208 H) (k \u2208 K), b = h * a * k :=\nby { rw quotient.eq', apply rel_iff, }\n\nlemma out_eq' (H K : subgroup G) (q : quotient \u2191H \u2191K) : mk H K q.out' = q :=\nquotient.out_eq' q\n\nlemma mk_out'_eq_mul (H K : subgroup G) (g : G) :\n  \u2203 (h k : G), (h \u2208 H) \u2227 (k \u2208 K) \u2227 (mk H K g : quotient \u2191H \u2191K).out' = h * g * k :=\nbegin\nhave := eq H K (mk H K g : quotient \u2191H \u2191K).out' g,\n  rw out_eq' at this,\n  obtain \u27e8h, h_h, k, hk, T\u27e9 := this.1 rfl,\n  refine \u27e8h\u207b\u00b9, k\u207b\u00b9, (H.inv_mem h_h), K.inv_mem hk, eq_mul_inv_of_mul_eq (eq_inv_mul_of_mul_eq _)\u27e9,\n  rw [\u2190 mul_assoc, \u2190 T]\nend\n\nlemma mk_eq_of_doset_eq {H K : subgroup G} {a b : G} (h : doset a H K = doset b H K) :\n  mk H K a = mk H K b :=\nbegin\n  rw eq,\n  exact mem_doset.mp (h.symm \u25b8 mem_doset_self H K b)\nend\n\nlemma disjoint_out' {H K : subgroup G} {a b : quotient H.1 K} :\n  a \u2260 b \u2192 disjoint (doset a.out' H K) (doset b.out' H K) :=\nbegin\n  contrapose!,\n  intro h,\n  simpa [out_eq'] using mk_eq_of_doset_eq (eq_of_not_disjoint  h),\nend\n\nlemma union_quot_to_doset (H K : subgroup G) : (\u22c3 q, quot_to_doset H K q) = set.univ :=\nbegin\n  ext x,\n  simp only [set.mem_Union, quot_to_doset, mem_doset, set_like.mem_coe, exists_prop,\n    set.mem_univ, iff_true],\n  use mk H K x,\n  obtain \u27e8h, k, h3, h4, h5\u27e9 := mk_out'_eq_mul H K x,\n  refine \u27e8h\u207b\u00b9, H.inv_mem h3, k\u207b\u00b9, K.inv_mem h4, _\u27e9,\n  simp only [h5, subgroup.coe_mk, \u2190mul_assoc, one_mul, mul_left_inv, mul_inv_cancel_right],\nend\n\nlemma doset_union_right_coset (H K : subgroup G) (a : G) :\n  (\u22c3 (k : K), right_coset \u2191H (a * k)) = doset a H K :=\nbegin\n  ext x,\n  simp only [mem_right_coset_iff, exists_prop, mul_inv_rev, set.mem_Union, mem_doset,\n  subgroup.mem_carrier, set_like.mem_coe],\n  split,\n  {rintro \u27e8y, h_h\u27e9,\n    refine \u27e8x * (y\u207b\u00b9 * a\u207b\u00b9), h_h, y, y.2, _\u27e9,\n    simp only [\u2190 mul_assoc, subgroup.coe_mk, inv_mul_cancel_right]},\n  {rintros \u27e8x, hx, y, hy, hxy\u27e9,\n    refine \u27e8\u27e8y,hy\u27e9,_\u27e9,\n    simp only [hxy, \u2190mul_assoc, hx, mul_inv_cancel_right, subgroup.coe_mk]},\nend\n\nlemma doset_union_left_coset (H K : subgroup G) (a : G) :\n  (\u22c3 (h : H), left_coset (h * a : G) K) = doset a H K :=\nbegin\n  ext x,\n  simp only [mem_left_coset_iff, mul_inv_rev, set.mem_Union, mem_doset],\n  split,\n  { rintro \u27e8y, h_h\u27e9,\n    refine \u27e8y, y.2, a\u207b\u00b9 * y\u207b\u00b9 * x, h_h, _\u27e9,\n    simp only [\u2190mul_assoc, one_mul, mul_right_inv, mul_inv_cancel_right]},\n  { rintros \u27e8x, hx, y, hy, hxy\u27e9,\n    refine \u27e8\u27e8x, hx\u27e9, _\u27e9,\n    simp only [hxy, \u2190mul_assoc, hy, one_mul, mul_left_inv, subgroup.coe_mk, inv_mul_cancel_right]},\n  end\n\nlemma left_bot_eq_left_quot (H : subgroup G) :\n  quotient (\u22a5 : subgroup G).1 H = (G \u29f8 H) :=\nbegin\n  unfold quotient,\n  congr,\n  ext,\n  simp_rw \u2190 bot_rel_eq_left_rel H,\n  refl,\nend\n\nlemma right_bot_eq_right_quot (H : subgroup G) :\n  quotient H.1 (\u22a5 : subgroup G) = _root_.quotient (quotient_group.right_rel H) :=\nbegin\n  unfold quotient,\n  congr,\n  ext,\n  simp_rw \u2190 rel_bot_eq_right_group_rel H,\n  refl,\nend\n\nend doset\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/group_theory/double_coset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.3687011749699521}}
{"text": "import Lean\nimport Init.Classical\n\nimport SciLean.Core.Attributes\nimport SciLean.Core.HasAdjoint\nimport SciLean.Core.Defs\n\nimport SciLean.Tactic.CustomSimp.DebugSimp\n\n-- import SciLean.Tactic.CustomSimp.SimpGuard\nimport SciLean.Tactic.AutoDiff\nimport SciLean.Core.AutoDiffSimps\n\nnamespace SciLean\n\nvariable {\u03b1 \u03b2 \u03b3 : Type}\nvariable {X Y Z U V : Type} [Vec X] [Vec Y] [Vec Z] [Vec U] [Vec V]\nvariable {Y\u2081 Y\u2082 : Type} [Vec Y\u2081] [Vec Y\u2082]\n\n\n--------------------------------------------------------------------------------\n-- Differential --\n--------------------------------------------------------------------------------\n\n-- maybe provide notation  `\u2202[dx] (x:=x\u2080), f x = \u2202 f x\u2080 dx` and its variants\n-- Variants\n--     1. \u2202[dx] (x:=x\u2080), f x          -- `\u2202[dx]` would be directional derivative operator\n--     2. \u2202 (x:=x\u2080,dx), f x           -- this has weird version without `x\u2080` \u2202 (x:=;dx), f x \n--     3. \u2202_dx (x:=x\u2080), f x           -- Can we parse this properly? What if `dx` is complicated, do we allow `\u2202_(dx)` ?\n--     4. ??\n-- macro \"\u2202\" x:Lean.Parser.Term.funBinder \",\" f:term:66 : term => `(\u2202 \u03bb $x => $f)\nsyntax diffBinderType  := \":\" term\nsyntax diffBinderValue := \":=\" term\nsyntax diffBinder := ident (diffBinderType <|> diffBinderValue)?\nsyntax \"\u2202\" diffBinder \",\" term:66 : term\nsyntax \"\u2202\" \"(\" diffBinder \")\" \",\" term:66 : term\nmacro_rules\n| `(\u2202 $x:ident, $f) =>\n  `(\u2202 \u03bb $x => $f)\n| `(\u2202 $x:ident : $type:term, $f) =>\n  `(\u2202 \u03bb $x : $type => $f)\n| `(\u2202 $x:ident := $val:term, $f) =>\n  `((\u2202 \u03bb $x => $f) $val)\n| `(\u2202 ($b:diffBinder), $f) =>\n  `(\u2202 $b, $f)\n\n\n--------------------------------------------------------------------------------\n-- Smooth Differential --\n--------------------------------------------------------------------------------\n\n-- instance differential.arg_dx.isSmooth (f : X \u2192 Y) [IsSmoothT f] (x : X) \n--   : IsSmoothT (\u03bb dx => \u2202 f x dx) := by (try infer_instance); sorry_proof\n-- instance differential.arg_dx.isLin    (f : X \u2192 Y) [IsSmoothT f] (x : X) \n--   : IsLinT (\u03bb dx => \u2202 f x dx) := by (try infer_instance); sorry_proof\n-- instance differential.arg_x.isSmooth  (f : X \u2192 Y) [IsSmoothT f] \n--   : IsSmoothT (\u03bb x => \u03bb dx \u22b8 \u2202 f x dx) := by (try infer_instance); sorry_proof\n-- instance differential.arg_x.isSmooth' (f : X \u2192 Y) [IsSmoothT f] \n--   : IsSmoothT (\u03bb x => \u03bb dx \u27ff \u2202 f x dx) := by (try infer_instance); sorry_proof\n\n\n-- instance differential.arg_y.isLin \n--   (f : X \u2192 Y \u2192 Z) [IsSmoothT f] [\u2200 x, IsLinT (f x)] (x dx) \n--   : IsLinT (\u03bb y => \u2202 f x dx y) := by (try infer_instance); sorry_proof\n-- instance differential.arg_y.isSmooth (f : X \u2192 Y \u2192 Z) [IsSmoothNT 2 f] (x dx) \n--   : IsSmoothT (\u03bb y => \u2202 f x dx y) := by (try infer_instance); sorry_proof\n\n-- instance differential.arg_x.comp.isSmooth {X Y Z} [Vec X] [Vec Y] [Vec Z] [Vec W]\n--   (f : Y \u2192 Z \u2192 W) [IsSmoothNT 2 f]\n--   (g : X \u2192 Y) [IsSmoothT g]\n--   : IsSmoothT (\u03bb x => \u2202 (f (g x))) := by (try infer_instance); sorry_proof\n\n\n-- instance SmoothMap.mk'.arg_f.diff_simp {X Y W} [Vec X] [Vec Y] [Vec W]\n--   (f : W \u2192 X \u2192 Y) [IsSmoothNT 2 f]\n--   : \u2202 (\u03bb w => \u03bb x \u27ff f w x)\n--     =\n--     \u03bb w dw => \u03bb x \u27ff \u2202 f w dw x := by simp; sorry_proof\n\n\n-- instance LinMap.mk'.arg_f.diff_simp {X Y W} [Vec X] [Vec Y] [Vec W]\n--   (f : W \u2192 X \u2192 Y) [IsSmoothNT 2 f] [\u2200 w, IsLinT (f w)]\n--   : \u2202 (\u03bb w => \u03bb x \u22b8 f w x)\n--     =\n--     \u03bb w dw => \u03bb x \u22b8 \u2202 f w dw x := by sorry_proof\n\n-- noncomputable\n-- def Smooth.differential (f : X \u27ff Y) : (X \u27ff X \u22b8 Y) := fun x \u27ff fun dx \u22b8 \u2202 f.1 x dx\n\n-- instance (f : X \u27ff Y) : Partial f (Smooth.differential f) := \u27e8\u27e9\n\n\n-- --------------------------------------------------------------------------------\n-- -- Scalar Differential --\n-- --------------------------------------------------------------------------------\n\n-- noncomputable\n-- abbrev differentialScalar (f : \u211d \u2192 X) (t : \u211d) : X := \u2202 f t 1\n\n-- noncomputable\n-- abbrev Smooth.differentialScalar (f : \u211d \u27ff X) : \u211d \u27ff X := \u03bb t \u27ff ((\u2202 f t) 1)\n\n-- @[default_instance] \n-- instance differentialScalar.instDifferentialNotation (f : \u211d \u2192 X) \n--   : Differential f (differentialScalar f) := \u27e8\u27e9\n\n-- instance Smooth.differentialScalar.instDifferentialNotation (f : \u211d \u27ff X) \n--   : Differential f (Smooth.differentialScalar f) := \u27e8\u27e9\n\n \n-- Notation \n-- \u2146 s, f s         --> \u2146 \u03bb s => f s\n-- \u2146 s : \u211d, f s     --> \u2146 \u03bb s : \u211d => f s\n-- \u2146 s := t, f s    --> (\u2146 \u03bb s => f s) t\nsyntax \"\u2146\" diffBinder \",\" term:66 : term\nsyntax \"\u2146\" \"(\" diffBinder \")\" \",\" term:66 : term\nmacro_rules\n| `(\u2146 $x:ident, $f) =>\n  `(\u2146 \u03bb $x => $f)\n| `(\u2146 $x:ident : $type:term, $f) =>\n  `(\u2146 \u03bb $x : $type => $f)\n| `(\u2146 $x:ident := $val:term, $f) =>\n  `((\u2146 \u03bb $x => $f) $val)\n| `(\u2146 ($b:diffBinder), $f) =>\n  `(\u2146 $b, $f)\n\n\n-- --------------------------------------------------------------------------------\n-- -- Dual Number Differential --\n-- --------------------------------------------------------------------------------\n\n-- noncomputable\n-- def tangentMap (f : X \u2192 Y) : X\u00d7X \u2192 Y\u00d7Y := \u03bb (x,dx) => (f x, \u2202 f x dx)\n\n-- instance Prod.mk.arg_xy.isSmooth : IsSmoothN 2 (Prod.mk : X \u2192 Y \u2192 X\u00d7Y) := sorry_proof\n\n-- instance (f : X \u2192 Y) : IsSmooth (\u03bb (x,dx) => \u2202 f x dx) := sorry_proof\n-- instance (f : X \u27ff Y) : IsSmooth (\u03bb (x,dx) => \u2202 f x dx) := sorry_proof\n\n-- noncomputable\n-- def Smooth.tangentMap (f : X \u27ff Y) : X\u00d7X \u27ff Y\u00d7Y := \u03bb xdx \u27ff (f xdx.1, \u2202 f xdx.1 xdx.2)\n\n-- @[default_instance]\n-- instance (f : X \u2192 Y) : TangentMap f (tangentMap f) := \u27e8\u27e9\n\n-- instance (f : X \u27ff Y) : TangentMap f (Smooth.tangentMap f) := \u27e8\u27e9\n\n\n\ninstance differential.arg_dx.isLin (f : X \u2192 Y) [IsSmoothT f] (x : X)\n  : IsLinT (\u03bb dx => \u2202 f x dx) := sorry_proof\n\ninstance differential.arg_dx.isSmooth (f : X \u2192 Y) [IsSmoothT f] (x : X)\n  : IsSmoothT (\u03bb dx => \u2202 f x dx) := sorry_proof\n\n\ninstance (f : X \u2192 Y \u2192 Z) [\u2200 x, IsLin (f x)] [IsSmoothT \u03bb x => \u03bb y \u22b8 f x y]\n  : IsSmoothT (\u03bb x => \u03bb y \u27ff f x y) := show_smoothness_via (Smooth.comp (\u03bb (L : Y\u22b8Z) \u27ff \u03bb y \u27ff L y) (\u03bb x \u27ff \u03bb y \u22b8 f x y)) (by ext x y; simp)\n\n-- instance differential.arg_x_dx.isSmooth' (f : X \u2192 Y) [IsSmoothT f]\n--   : IsSmoothT (\u03bb x => \u03bb dx \u22b8 \u2202 f x dx) := sorry_proof\n\ninstance differential.arg_x_dx.isSmooth (f : X \u2192 Y) [IsSmoothT f]\n  : IsSmoothT (\u03bb x => \u03bb dx \u27ff \u2202 f x dx) := sorry_proof\n\n-- instance differential.arg_f_xdx.isSmooth' (f : U \u2192 X \u2192 Y) [\u2200 u, IsSmoothT (f u)] [IsSmoothT \u03bb u => \u03bb x \u27ff f u x]\n--   : IsSmoothT (\u03bb u => \u03bb x \u27ff \u03bb dx \u22b8 \u2202 (f u) x dx) := sorry_proof\n\ninstance differential.arg_f_xdx.isSmooth (f : U \u2192 X \u2192 Y) [\u2200 u, IsSmoothT (f u)] [IsSmoothT \u03bb u => \u03bb x \u27ff f u x]\n  : IsSmoothT (\u03bb u => \u03bb x dx \u27ff \u2202 (f u) x dx) := sorry_proof\n\ninstance differential.arg_y.isSmooth (f : X \u2192 Y \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT (\u03bb x => \u03bb y \u27ff f x y)] (x dx : X)\n  : IsSmoothT (\u03bb y => \u2202 f x dx y) := by (try infer_instance); sorry_proof\ninstance differential.arg_dx_y.isSmooth (f : X \u2192 Y \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT (\u03bb x => \u03bb y \u27ff f x y)] (x : X)\n  : IsSmoothT (\u03bb dx => \u03bb y \u27ff \u2202 f x dx y) := by (try infer_instance); sorry_proof\ninstance differential.arg_x_dxy.isSmooth (f : X \u2192 Y \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT (\u03bb x => \u03bb y \u27ff f x y)]\n  : IsSmoothT (\u03bb x => \u03bb dx y \u27ff \u2202 f x dx y) := by (try infer_instance); sorry_proof\ninstance differential.arg_f_xdxy.isSmooth (f : U \u2192 X \u2192 Y \u2192 Z) [\u2200 u x, IsSmoothT (f u x)] [\u2200 u, IsSmoothT (\u03bb x => \u03bb y \u27ff f u x y)] [IsSmoothT (\u03bb u => \u03bb x y \u27ff f u x y)]\n  : IsSmoothT (\u03bb u => \u03bb x dx y \u27ff \u2202 (f u) x dx y) := by (try infer_instance); sorry_proof\n\n  \n--------------------------------------------------------------------------------\n-- Differential Rules --\n--------------------------------------------------------------------------------\n\n-- -- I: X\u27ffX\n\n-- @[diff]\n-- theorem differential_rule_I \n--   : \u2202 (\u03bb x : X => x) = \u03bb _ dx => dx := sorry_proof\n\n\n-- -- K: X\u27ffY\u27ffX\n\n-- @[diff]\n-- theorem differential_rule_K\u2082 (x : X) \n--   : \u2202 (\u03bb _ : Y => x) = \u03bb _ _ => 0 := sorry_proof\n\n-- set_option trace.Meta.Tactic.simp.rewrite true in\n-- @[diff]\n-- theorem differential_rule_K\u2081 \n--   : \u2202 (\u03bb (x : X) (_ : Y) => x) = \u03bb _ dx _ => dx := sorry_proof\n\n\n-- -- S: (X\u27ffY\u27ffZ)\u27ff(X\u27ffY)\u27ffX\u27ffZ\n\n-- @[diff]\n-- theorem differential_rule_S\u2083\n--   (f : X \u2192 Y \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y] -- [IsSmoothN 2 f]\n--   (g : X \u2192 Y)  [IsSmoothT g]\n--   : \u2202 (\u03bb x => f x (g x)) \n--     = \n--     \u03bb x dx => \n--       let (y,dy) := \ud835\udcaf g x dx\n--       \u2202 f x dx y + \u2202 (f x) y dy\n--   := sorry_proof\n\n-- instance (f : U \u2192 X \u2192 Y \u2192 Z) [\u2200 u x, IsSmoothT (f u x)] [\u2200 u, IsSmoothT (\u03bb x => \u03bb y \u27ff f u x y)] [IsSmoothT (\u03bb u => \u03bb x y \u27ff f u x y)]\n--   (g : U \u2192 X) [IsSmoothT g]\n--   : IsSmoothT \u03bb u => \u03bb y \u27ff f u (g u) y := \n-- by \n--   try infer_instance\n--   have : IsSmoothT fun u => \u03bb u' y \u27ff f u (g u') y := by (try infer_instance); apply IsSmoothT_rule_S\u2081 (\u03bb u x y => f u y x) (\u03bb v _ => g v)\n--   apply IsSmoothT_duplicate_argument (\u03bb u u' => \u03bb y \u27ff f u (g u') y)\n\n-- @[diff]\n-- theorem differential_rule_S\u2082\n--   (f : X \u2192 Y \u2192 Z)   [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y] -- [IsSmoothN 2 f]\n--   (g : V \u2192 (X \u2192 Y)) [\u2200 v, IsSmoothT (g v)] [IsSmoothT \u03bb v => \u03bb x \u27ff g v x] -- [IsSmoothN 2 g]\n--   : \u2202 (\u03bb v => \u03bb x \u27ff f x (g v x))\n--     =\n--     \u03bb v dv => \u03bb x \u27ff \u2202 (f x) (g v x) (\u2202 g v dv x)\n--   := sorry_proof\n\n\n--------------------------\n\n@[simp \u2193, diff]\ntheorem differential.of_id\n  : \u2202 (\u03bb x : X => x) = \u03bb x dx => dx := sorry_proof\n\n@[simp \u2193, diff]\ntheorem differential.of_const (x : X)\n  : \u2202 (\u03bb y : Y => x) = \u03bb y dy => (0 : X) := sorry_proof\n\n@[simp \u2193 low-3, diff low-3]\ntheorem differential.of_swap (f : \u03b1 \u2192 X \u2192 Y) [\u2200 i, IsSmoothT (f i)]\n  : \u2202 (\u03bb x a => f a x) = \u03bb x dx a => \u2202 (f a) x dx := sorry_proof\n\n@[simp \u2193 low-1, diff low-1, simp_guard g (\u03bb x => x)]\ntheorem differential.of_comp\n  (f : Y \u2192 Z) [IsSmoothT f] \n  (g : X \u2192 Y) [IsSmoothT g]\n  : \u2202 (\u03bb x => f (g x)) \n    = \n    \u03bb x dx => \n      let (y,dy) := (\ud835\udcaf g) x dx\n      -- let y := g x\n      -- let dy := \u2202 g x dx\n      \u2202 f y dy \n  := sorry_proof\n\n@[simp \u2193 low-2, diff low-2, simp_guard g\u2081 Prod.fst, g\u2082 Prod.snd]\ntheorem differential.of_diag\n  (f : Y\u2081 \u2192 Y\u2082 \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y] \n  (g\u2081 : X \u2192 Y\u2081) [IsSmoothT g\u2081]\n  (g\u2082 : X \u2192 Y\u2082) [IsSmoothT g\u2082]\n  : \u2202 (\u03bb x => f (g\u2081 x) (g\u2082 x)) \n    = \n    \u03bb x dx => \n      let (y\u2081,dy\u2081) := \ud835\udcaf g\u2081 x dx\n      let (y\u2082,dy\u2082) := \ud835\udcaf g\u2082 x dx\n      let df := \u2202 (uncurryN 2 f)\n      -- let y\u2081 := g\u2081 x\n      -- let dy\u2081 := \u2202 g\u2081 x dx\n      -- let y\u2082 := g\u2082 x\n      -- let dy\u2082 := \u2202 g\u2082 x dx\n      df (y\u2081,y\u2082) (dy\u2081,dy\u2082)\n      -- \u2202 f y\u2081 dy\u2081 y\u2082 +  \u2202 (f y\u2081) y\u2082 dy\u2082 \n  := sorry_proof\n\n/-- Last resort theorem that changes tangent map to normal differential \n\nBilinear maps should usually provide a rewrite rule for `\ud835\udcaf (uncurryN 2 f)`\n-/\n@[simp \u2193 low-5, diff low-5]\ntheorem differential.of_uncurryN (f : Y\u2081 \u2192 Y\u2082 \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y]\n  : \u2202 (uncurryN 2 f) \n    =\n    \u03bb (y\u2081,y\u2082) (dy\u2081,dy\u2082) =>\n    \u2202 f y\u2081 dy\u2081 y\u2082 + \u2202 (f y\u2081) y\u2082 dy\u2082\n  := sorry_proof\n\n@[simp \u2193 low, diff low]\ntheorem differential.of_parm\n  (f : X \u2192 \u03b1 \u2192 Y) [IsSmoothT f] (a : \u03b1)\n  : \u2202 (\u03bb x => f x a) = \u03bb x dx => \u2202 f x dx a := \nby\n  rw[differential.of_swap (\u03bb a x => f x a)]\n\n@[simp \u2193, diff]\ntheorem differential.of_eval\n  (a : \u03b1)\n  : \u2202 (\u03bb f : \u03b1 \u2192 Y => f a) = \u03bb _ df => df a := by simp\n\n\n--------------------------------------------------------------------------------\n-- Tangent Map Rules --\n--------------------------------------------------------------------------------\n\n@[simp \u2193, diff]\ntheorem tangentMap.of_id\n  : \ud835\udcaf (\u03bb x : X => x) = \u03bb x dx => (x,dx)\n  := by symdiff; done\n\n@[simp \u2193, diff]\ntheorem tangentMap.of_const (x : X)\n  : \ud835\udcaf (\u03bb y : Y => x) = \u03bb y dy => (x,0) \n  := by symdiff; done\n\n@[simp \u2193 low-3, diff]\ntheorem tangentMap.of_swap (f : \u03b1 \u2192 X \u2192 Y) [\u2200 i, IsSmoothT (f i)]\n  : \ud835\udcaf (\u03bb x a => f a x) = \u03bb x dx => (\u03bb a => f a x, \u03bb a => \u2202 (f a) x dx) \n  := by symdiff; done\n\nset_option trace.Meta.Tactic.simp true in\nset_option trace.Meta.Tactic.simp.unify false in\n@[simp \u2193 low-1, diff, simp_guard g (\u03bb x => x)]\ntheorem tangentMap.of_comp\n  (f : Y \u2192 Z) [IsSmoothT f] \n  (g : X \u2192 Y) [IsSmoothT g] \n  : \ud835\udcaf (\u03bb x => f (g x)) \n    = \n    \u03bb x dx =>\n      let (y,dy) := \ud835\udcaf g x dx\n      \ud835\udcaf f y dy\n  := by unfold tangentMap; simp[tangentMap] --  debug_simp; symdiff_core; done\n\n\n@[simp \u2193 low-2, diff, simp_guard g\u2081 Prod.fst, g\u2082 Prod.snd]\ntheorem tangentMap.of_diag\n  (f : Y\u2081 \u2192 Y\u2082 \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y]\n  (g\u2081 : X \u2192 Y\u2081) [IsSmoothT g\u2081]\n  (g\u2082 : X \u2192 Y\u2082) [IsSmoothT g\u2082]\n  : \ud835\udcaf (\u03bb x => f (g\u2081 x) (g\u2082 x))\n    = \n    \u03bb x dx => \n      let (y\u2081,dy\u2081) := \ud835\udcaf g\u2081 x dx\n      let (y\u2082,dy\u2082) := \ud835\udcaf g\u2082 x dx\n      -- (f y\u2081 y\u2082, \u2202 f y\u2081 dy\u2081 y\u2082 + \u2202 (f y\u2081) y\u2082 dy\u2082)\n      \ud835\udcaf (uncurryN 2 f) (y\u2081,y\u2082) (dy\u2081,dy\u2082)\n  := by simp[tangentMap]; done\n\n/-- Last resort theorem that changes tangent map to normal differential \n\nBilinear maps should usually provide a rewrite rule for `\ud835\udcaf (uncurryN 2 f)`\n-/\n@[simp \u2193 low-5, diff low-5]\ntheorem tangentMap.of_uncurryN (f : Y\u2081 \u2192 Y\u2082 \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y]\n  : \ud835\udcaf (uncurryN 2 f) \n    =\n    \u03bb (y\u2081,y\u2082) (dy\u2081,dy\u2082) =>\n    (f y\u2081 y\u2082, \u2202 f y\u2081 dy\u2081 y\u2082 + \u2202 (f y\u2081) y\u2082 dy\u2082)\n  := by simp[tangentMap]; done\n\n@[simp \u2193 low, diff]\ntheorem tangentMap.of_parm\n  (f : X \u2192 \u03b1 \u2192 Y) [IsSmoothT f] (a : \u03b1)\n  : \ud835\udcaf (\u03bb x => f x a) = \u03bb x dx => let (f',df') := \ud835\udcaf f x dx; (f' a, df' a) \n  := by simp[tangentMap]; done\n\n@[simp \u2193, diff]\ntheorem tangentMap.of_eval\n  (a : \u03b1)\n  : \ud835\udcaf (\u03bb f : \u03b1 \u2192 Y => f a) = \u03bb f df => (f a, df a) := by simp\n\n-- @[simp \u2193 low, diff]\n-- theorem uncurry.arg_xy.diff_simp\n--   (f : X \u2192 Y \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y]\n--   : \u2202 (\u03bb (xy : (X\u00d7Y)) => f xy.1 xy.2) = \u03bb (x,y) (dx,dy) => \u2202 f x dx y + \u2202 (f x) y dy := sorry_proof\n\n--   -- : \u2202 (\u03bb ((x,y) : (X\u00d7Y)) => f x y) = \u03bb (x,y) (dx,dy) => \u2202 f x dx y + \u2202 (f x) y dy := sorry_proof \n\n-- @[simp \u2193 low, diff]\n-- theorem uncurry.arg_xy.parm1.diff_simp\n--   (a : \u03b1)\n--   (f : X \u2192 Y \u2192 \u03b1 \u2192 Z) [\u2200 x, IsSmoothT (f x)] [IsSmoothT \u03bb x => \u03bb y \u27ff f x y]\n--   : \u2202 (\u03bb (xy : (X\u00d7Y)) => f xy.1 xy.2 a) = \u03bb (x,y) (dx,dy) => \u2202 f x dx y a + \u2202 (f x) y dy a := sorry_proof\n\n\n\n--------------------------------------------------------------------------------\n\n/-- Differential of linear function is the function itself.\n\nThis theorem is too general and we do not want to try to apply it \nevery time we try to differentiate something. That is why it it has \nlow priority and more importantly it asks for `IsLin` and not for `IsLinT`.\nOnly elementary functions(that are not composite composite) are allowed\nto be differentiated with this theorem. -/\n\n@[simp low, diff] \ntheorem tangentMap_of_linear (f : X \u2192 Y) [IsLin f]\n  : \ud835\udcaf f = \u03bb x dx => (f x, f dx) := by simp[tangentMap]; done\n\n\n@[simp low, diff] \ntheorem diff_of_linear_2_1 (f : X \u2192 Y \u2192 Z) [IsLinN 2 f] : \u2202 f = \u03bb _ dx _ => f dx 0 := sorry_proof\n@[simp low, diff] \ntheorem diff_of_linear_2_2 (f : X \u2192 Y \u2192 Z) [IsLinN 2 f] (x : X) : \u2202 (\u03bb y => f x y) = \u03bb _ dy => f 0 dy := sorry_proof\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Core/Differential.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3686789340526035}}
{"text": "import data.set\nimport tactic.linarith\n\n/-!\n# First-order predicate logic\n\nIn this file we define the syntax of intuitionistic first-order logic and a natural deduction\nproof calculus. \n\n## Main result\n\n- `formula \u03c3`      : the definition of first-order formulas over a signature \u03c3\n- `proof_term \u03c3`   : the definition of proof terms of natural deduction over a signature \u03c3\n\n## Notations\n\nWe define the following notations for lifts and substitutions:\n\n- `X \u2191 m \uff20 i` for `lift X s k`  where `X` can be a term or a formula .\n- `X[ s \u2044 k ]` for `subst X s k` where `X` can be a term or a formula .\n- `\u0393 \u22a2 \u03c6` for `proof_term \u0393 \u03c6` \n\nWe use the following local notations\n\n- `>>` for `set.insert`\n\n## Notes\n\nWe wrote comments whenever we felt like a topic wasn't really covered by the literature referenced.\nThis includes some definitions that are not part of the actual implementation but simplify talking about it.\n\n## References\n\n* [N.G. de Bruijn, *Lambda calculus notation with nameless dummies*] [DB72]\n  -- the original paper describing de Bruijn indices\n* [J.M. Han, F.van Doorn, *A Formal Proof of the Independence of the Continuum Hypothesis*] [HD20]\n  -- we followed their implementation of first-order logic using \"partially applied\" terms and formulas \n  -- See also: https://flypitch.github.io/\n* [I. Chiswell, W. Hodges,*Mathematical Logic*] [CH04]\n  -- first order logic and natural deduction\n* [M. Huth, M. Ryan, *Logic in computer science*] [HR04]\n  -- first order logic and natural deduction\n* [S. Berghofer, C. Urban, *A Head-to-Head Comparison of de Bruijn Indices and Names*] [BH07]\n  -- for a good breakdown of the proof of the substitution lemma `subst_subst`\n* https://github.com/coq-community/dblib/blob/master/src/DeBruijn.v \n  -- as a good reference on lifting and substitution lemmas for de Bruijn indices\n-/\n\n-- use with `simp only with tls`\nmk_simp_attribute tls\n\"Collection of definitions and lemmas for simplifying trivial  combinations of lifts and substitutions.\"\n\nopen nat set\nuniverse variable u\n\nnamespace fol\n/-- A signature of a first-order logic defining its function and predicate symbols with arities. -/\nstructure signature : Type (u+1) :=\n(func_symb : \u2115 \u2192 Type u) (pred_symb : \u2115 \u2192 Type u)\n\ndef signature.constants (\u03c3 : signature) := \u03c3.func_symb 0\n\ninductive sorry_nothing : \u2115 \u2192 Type u\n\ndef trivial_signature : signature :=\n{ func_symb := sorry_nothing, pred_symb := sorry_nothing}\n\nvariable (\u03c3 : signature.{u})\n\n/-! ### terms -/\n/--\n  `preterm \u03c3 a` is partially applied term of first-order logic over the signature `\u03c3`.\n  If applied to `a` terms it becomes a well-formed term.\n-/\ninductive preterm : \u2115 \u2192 Type u\n| var  (index : \u2115) : preterm 0\n| func {arity : \u2115} (f : \u03c3.func_symb arity) : preterm arity\n| fapp {arity : \u2115} (t : preterm (arity+1))  (s : preterm 0) : preterm arity\n\nexport preterm\n\nprefix `#`:max := preterm.var\n@[reducible] def term := preterm \u03c3 0\n\nvariable {\u03c3}\n\nnamespace term\n\n/-- `lift t m i` increases the index of each `i`-free variable in `t` by `m`. -/\ndef lift : \u2200{a} , preterm \u03c3 a \u2192 \u2115 \u2192 \u2115 \u2192 preterm \u03c3 a \n| _ #x          m i := #(if i \u2264 x then x+m else x)\n| _ (func f)    m i := func f\n| _ (fapp t s)   m i := fapp (lift t m i) (lift s m i)\n\nend term\n\n-- we use \uff20 (U+FF20) instead of the regular @ (U+0040)\nnotation t ` \u2191 `:90 m ` \uff20 `:90 i:90  := term.lift t m i\n\nnamespace term\n\n-- unfolding lemmas for the simplifier\n@[simp, tls] lemma lift_fapp {a} (t : preterm \u03c3 (a+1)) (s : preterm \u03c3 0) (m i : \u2115) \n  : (fapp t s) \u2191 m \uff20 i = fapp (t \u2191 m \uff20 i) (s \u2191 m \uff20 i) := by refl\n@[simp, tls] lemma lift_func {a} (f : \u03c3.func_symb a) (m i : \u2115) \n  : (func f) \u2191 m \uff20 i = func f := by refl\n\n-- lifting of variables by cases for the simplifier\n@[simp] lemma lift_var_lt {x m i : \u2115} (H : x < i) : #x \u2191 m \uff20 i = (#x : term \u03c3) \n  := begin unfold lift, rw if_neg (not_le.mpr H), end\n@[simp, tls] lemma lift_var_eq {x m}: ((#x \u2191 m \uff20 x) = (#(x+m) : term \u03c3)) \n  := begin rw lift, rw if_pos x.le_refl, end\n@[simp] lemma lift_var_gt {x m i} (H : i < x) : #x \u2191 m \uff20 i = (#(x+m) : term \u03c3)\n  := begin rw lift, rw if_pos (le_of_lt H), end\n@[simp] lemma lift_var_ge {x m i} (H : i \u2264 x) : #x \u2191 m \uff20 i = (#(x+m) : term \u03c3)\n  := begin rw lift, rw if_pos H, end\n@[simp] lemma lift_var_nge {x m i : \u2115} (H : \u00ac i \u2264 x) : #x \u2191 m \uff20 i = (#x : term \u03c3) \n  := begin unfold lift, rw if_neg H, end\n\n\n@[simp, tls] lemma lift_by_0: \u2200 {a} (t : preterm \u03c3 a) {i}, t \u2191 0 \uff20 i = t\n| _ #x         i  := by simp[lift]\n| _ (func f)   _  := by refl\n| _ (fapp f t) _  := begin unfold lift, congr; apply lift_by_0, end\n\n\n/- Various lifting lemmas. -/\n\nlemma lift_lift: \u2200 {a} (t : preterm \u03c3 a) (m) {i} (n) {j} (H : j \u2264 i), \n  (t \u2191 m \uff20 i) \u2191 n \uff20 j = (t \u2191 n \uff20 j) \u2191 m \uff20 (i+n) \n| _ #x          m i n j H   := \n  begin by_cases h\u2080 : i \u2264 x,\n    { have h\u2081 : j \u2264 x := le_trans H h\u2080, \n      have h\u2082 : j \u2264 x + m := le_trans h\u2081 (x.le_add_right m),\n      simp[*, add_right_comm], },\n    { have h\u2081 : \u00ac(i + n \u2264  x + n) := begin intro h, exact  h\u2080 (le_of_add_le_add_right h) end,\n      have h\u2082 : \u00ac(i + n \u2264 x)      := begin intro h, exact h\u2081 (le_trans h (x.le_add_right n)) end,\n      by_cases j \u2264 x; simp[*], }, \n  end\n| _ (func f)    _ _ _ _ _      := by refl\n| _ (fapp f t)  _ _ _ _ _   := by simp*\n    \nlemma lift_lift_reverse {a} (t : preterm \u03c3 a) {m i} (n) {j} (H : i + m \u2264 j) : \n  (t \u2191 m \uff20 i) \u2191 n \uff20 j = (t \u2191 n \uff20 (j-m)) \u2191 m \uff20 i :=\nbegin\n  have h : i \u2264 (j-m) := nat.le_sub_right_of_add_le H,\n  have h': m \u2264 j := (le_trans (m.le_add_left i) H),\n  rw [lift_lift t n m h, nat.sub_add_cancel h'],\nend\n\nlemma lift_lift_merge: \u2200 {a} (t : preterm \u03c3 a) {m i} (n) {j} (H : i \u2264 j) (H' : j \u2264 i + m), \n  (t \u2191 m \uff20 i) \u2191 n \uff20 j = t \u2191 (m+n) \uff20 i\n| _ #x         m i n j H H' :=  \n  begin by_cases h\u2080 : i \u2264 x,\n    { have h\u2081 : j \u2264 x + m  := le_trans H' (add_le_add_right h\u2080 m),\n      simp[*, add_assoc], },\n    { have h\u2081 : \u00ac (j \u2264 x) := (\u03bb h,  h\u2080 (le_trans H h)),\n      simp[*], }, \n  end\n| _ (func f)   _ _ _ _ _ _  := by refl\n| _ (fapp t s) _ _ _ _ _ _  := by simp*\n\nlemma lift_by_succ {a} (t : preterm \u03c3 a) {m i} : t \u2191 (m+1) \uff20 i  = (t \u2191 1 \uff20 i) \u2191 m \uff20 i\n  := begin rw[lift_lift_merge, one_add], apply le_refl, apply le_succ, end\n\n\n\n/-- \n  `subst t s k` substitutes `s \u2191 k \uff20 0` for each variable at `k` in `t` and \n  reduces the index of all `k+1`-free variables by `1`.\n-/\ndef subst: \u2200{a}, preterm \u03c3 a \u2192 term \u03c3 \u2192 \u2115 \u2192 preterm \u03c3 a\n| _ #x           s k := if x < k then #x else if k < x then #(x-1) else (s \u2191 k \uff20 0)\n| _ (func f)     s k := func f\n| _ (fapp t\u2081 t\u2082) s k := fapp (subst t\u2081 s k) (subst t\u2082 s k)\n\nend term\n\n-- we use \u2044 (U+2044) instead of the usual slash / (U+002F) to avoid conflict with the division operator\nnotation t `[`:max s ` \u2044 `:95 n `]`:0 := term.subst t s n \n\nnamespace term\n\n-- lemmas for the simplifier\n@[simp, tls] lemma subst_fapp {a} (t\u2081 : preterm \u03c3 (a+1)) (t\u2082 s : preterm \u03c3 0) (k : \u2115) : \n  (fapp t\u2081 t\u2082) [s \u2044 k] = fapp (t\u2081 [s \u2044 k]) (t\u2082 [s \u2044 k]) := by refl\n@[simp, tls] lemma subst_func {a} (f : \u03c3.func_symb a) (s k) : \n  (func f) [s \u2044 k] = func f := by refl\n\n@[simp] lemma subst_var_lt (s : term \u03c3) {x k : \u2115} (H : x < k) : #x[s \u2044 k] = #x := \n    begin rw subst, rw if_pos H, end\n@[simp, tls] lemma subst_var_eq (s : term \u03c3) {k : \u2115}  : #k[s \u2044 k] =  s \u2191 k \uff20 0 := \n    begin rw subst, repeat{ rw if_neg (lt_irrefl k) }, end\n@[simp] lemma subst_var_gt (s : term \u03c3) {x k  : \u2115} (H : k < x) : #x[s \u2044 k] = #(x-1) := \n    begin rw subst, rw if_neg (lt_asymm H), rw if_pos H, end\n@[simp] lemma subst_var_nle (s : term \u03c3) {x k : \u2115} (H : \u00ac (x \u2264 k)) : #x[s \u2044 k] = #(x-1) := \n  subst_var_gt s (not_le.mp H)\n\n@[simp, tls] lemma subst_var0 (s : term \u03c3): #0[ s \u2044 0 ] = s := \n    begin rw subst_var_eq, exact lift_by_0 s, end\n\n\n/- Various substitution lemmas -/\n\nlemma lift_subst : \u2200 {a}  (t : preterm \u03c3 a) (s: term \u03c3) (m) {i} (k) (H: i \u2264 k),\n    t [ s \u2044 k ] \u2191 m \uff20 i = (t \u2191 m \uff20 i)[ s \u2044 k+m ] \n| _ #x s m i k H := \n  begin \n    apply decidable.lt_by_cases x k; intro h\u2081,\n    { -- x < k\n      have h\u2082 : x < k + m, from nat.lt_add_right x k m h\u2081, by_cases i\u2264x;\n      simp* , },\n    { -- x = k\n      subst h\u2081, \n      simp[*, lift_lift_merge] , } ,\n    { -- x > k\n      have h\u2082 : i < x, by linarith,\n      have : i \u2264 x-1, from nat.le_sub_right_of_add_le (succ_le_of_lt h\u2082),\n      have : i \u2264 x, by linarith,\n      have : 1 \u2264 x, by linarith,\n      simp[*, nat.sub_add_comm] },\n  end\n| _ (func f)   _ _ _ _ _ := by refl\n| _ (fapp f t) _ _ _ _ _ := by simp* \n\nlemma subst_lift: \u2200  {a}  (t : preterm \u03c3 a) (s: term \u03c3) {m i k : \u2115} (H: i \u2264 k) (H' : k \u2264 i + m),\n    (t \u2191 (m+1) \uff20 i) [s \u2044 k] = t \u2191 m \uff20 i \n| _ #x s m i k H H' := \n  begin by_cases h: i \u2264 x,\n    { have h\u2081 : k < x + (m + 1), from lt_succ_of_le (le_trans H' (add_le_add_right h m)), simp[*] , },\n    { have h\u2081 : x < k, from lt_of_lt_of_le (lt_of_not_ge h) H, simp[*] , } \n  end\n| _ (func f)   _ _ _ _ _ _ := by refl\n| _ (fapp f t) _ _ _ _ _ _ := by simp* \n\nlemma subst_subst: \u2200 {a} (t : preterm \u03c3 a) (s\u2081) {k\u2081}  (s\u2082) {k\u2082} (H : k\u2081 \u2264 k\u2082), \n    t[s\u2081 \u2044 k\u2081][s\u2082 \u2044 k\u2082] = t[s\u2082 \u2044 k\u2082 + 1][(s\u2081 [s\u2082 \u2044 k\u2082 - k\u2081]) \u2044 k\u2081] \n| _ #x s\u2081 k\u2081 s\u2082 k\u2082 H := \n  begin apply decidable.lt_by_cases x k\u2081; intro h\u2081,\n    { have h\u2082 : x < k\u2082, from lt_of_lt_of_le h\u2081 H, \n      have h\u2083 : x < k\u2082 + 1, from lt.step h\u2082,\n      simp[*] , },\n    { subst h\u2081,\n      have h\u2082 : x < k\u2082 + 1, from lt_succ_iff.mpr H,\n      simp[*, lift_subst, nat.sub_add_cancel] , },\n    { apply decidable.lt_by_cases (x-1) k\u2082; intro h\u2082,\n      { have : x < k\u2082 + 1, from nat.lt_add_of_sub_lt_right h\u2082, simp*, },\n      { have h\u2083: 1 \u2264  x , from by linarith,\n        have h\u2084: x  = k\u2082 + 1, from (nat.sub_eq_iff_eq_add h\u2083).mp h\u2082,\n        subst h\u2084, clear h\u2083,\n        simp[*, subst_lift, lt_irrefl] },\n      { have: k\u2082+1 < x,   from nat.add_lt_of_lt_sub_right h\u2082, \n        have: k\u2081 < x - 1, from gt_of_gt_of_ge h\u2082 H,\n        simp[*], }, }, \n  end\n| _ (func f) _ _ _ _ _ := by refl\n| _ (fapp t s) _ _ _ _ _ := by simp*\n\nlemma subst_lift_by_lift : \u2200{a} (t : preterm \u03c3 a) (s : term \u03c3) (m i k : \u2115),\n    (t \u2191 m \uff20 (i + k + 1)) [ (s \u2191 m \uff20 i) \u2044 k] = (t [ s \u2044 k ]) \u2191 m \uff20 (i+k)\n| _ #x s m i k :=\n  begin by_cases h\u2081 : i + k + 1 \u2264 x,\n    { -- i + k + 1 \u2264 x\n      have h\u2082 : k < x := lt_of_le_of_lt (le_add_left k i) (lt_of_succ_le h\u2081),\n      have : k < x + m := lt_add_right k x m h\u2082,\n      have : i + k \u2264 x - 1 := nat.le_sub_right_of_add_le h\u2081,\n      have : 1 \u2264 x := one_le_of_lt h\u2082,\n      simp [*, nat.sub_add_comm] , },\n    { -- \u00ac i + k + 1 \u2264 x\n      apply decidable.lt_by_cases x k; intro h\u2082,\n      { -- x < k\n        have : \u00ac i + k \u2264 x := not_le_of_lt (lt_add_left x k i h\u2082), simp[*] , },\n      { -- x = l\n        subst h\u2082, simp[*, lift_lift] , },\n      { -- k < x\n        have h\u2081: \u00ac i+k \u2264 x - 1, \n        begin\n          intro h, \n          have h\u2083 : i + k + 1 \u2264 x - 1 + 1, from succ_le_succ h,\n          rw nat.sub_add_cancel (one_le_of_lt h\u2082) at h\u2083, \n          exact h\u2081 h\u2083,\n        end, \n        simp[*] , }, }, \n  end\n| _ (func f)     _ _ _ _ := by refl\n| _ (fapp t\u2081 t\u2082) _ _ _ _ := by simp* \n\nlemma subst_var0_lift : \u2200{a} (t : preterm \u03c3 a) (m i : \u2115), (t \u2191 (m+1) \uff20 (i+1))[ #0 \u2044 i] = t \u2191 m \uff20 (i+1)\n| _ #x m i := begin apply decidable.lt_by_cases i x; intro h\u2080,\n                { have: i+1 \u2264 x, by linarith,\n                  have: \u00ac (x + (m + 1) < i), by linarith, \n                  have: i < x + (m + 1), by linarith,\n                  simp* , },\n                { subst h\u2080, simp , },\n                { have: \u00ac (i + 1 \u2264 x), by linarith, simp* , },\n            end\n| _ (func f) _ _ := by refl\n| _ (fapp t s) _ _ := by simp* \n\n@[simp, tls] lemma subst_var0_lift_by_1 {a} (t : preterm \u03c3 a) (i : \u2115) : (t \u2191 (1) \uff20 (i+1))[#0 \u2044 i] = t := \nbegin\n    have h:= subst_var0_lift t 0 i,\n    rw lift_by_0 at h,\n    exact h,\nend\n\n@[simp, tls] lemma subst_for_0_lift_by_1: \u2200 {a} (t : preterm \u03c3 a) (s : term \u03c3) , (t \u2191 1 \uff20 0)[s \u2044 0] = t \n| _ #x _         := by refl\n| _ (func f) _   := by refl\n| _ (fapp t s) _ := by simp* \n\n/--\n  Biggest (deepest) reference depth of variables occurring in a term (plus one).\n\n  Examples:\n  * `max_free_var #k = k+1` by definition.\n  * `max_free_var t = 0` means no variables occur in `t`.\n-/\ndef max_free_var: \u2200 {a} (t : preterm \u03c3 a), \u2115 \n| _ #x := x+1\n| _ (func f)  := 0\n| _ (fapp t s) := max (max_free_var t) (max_free_var s)\n\n/- If `t` is a fixed point for lifting at `i`, then its a fixed point for lifting at `j` for all `i\u2264j`  -/\nlemma lift_fixed_points_monotone {a} {t:preterm \u03c3 a} {i j: \u2115} (h: i \u2264 j) (H: t \u2191 1 \uff20 i = t) : t \u2191 1 \uff20 j = t :=\nbegin\n  induction j with j,\n  { rwa[le_zero_iff.mp h] at H,},\n  {  by_cases h': i = j+1, \n    { rwa h' at H, },\n    { have h\u2081: i\u2264j, from lt_succ_iff.mp (lt_of_le_of_ne h h'),\n      have h\u2082 := j_ih h\u2081,\n      rw [\u2190H, \u2190lift_lift t 1 1 h\u2081, h\u2082], },\n  },\nend\n\n@[simp, tls] lemma lift_at_max_free_var {a} (t : preterm \u03c3 a) : t \u2191 1 \uff20 (max_free_var t) = t :=\nbegin\n  induction t with T,\n  { simp[max_free_var], },\n  { refl },\n  { unfold lift max_free_var,\n    congr,\n    { have t_h := le_max_left (max_free_var t_t) (max_free_var t_s),\n      exact lift_fixed_points_monotone t_h t_ih_t, },\n    { have s_h := le_max_right (max_free_var t_t) (max_free_var t_s),\n      exact lift_fixed_points_monotone s_h t_ih_s, } }\nend \n\nend term\n\n/-! ### formulas -/\nsection formulas\n\nvariable (\u03c3)\n/--\n  `preformula \u03c3 a` is a partially applied formula of first-order logic over the signature `\u03c3`. \n  If applied to `a` terms it becomes a well-formed formula.\n-/\ninductive preformula : \u2115 \u2192 Type u\n| bot                           : preformula 0\n| eq   (t s : term \u03c3)         : preformula 0\n| imp  (\u03c6 \u03c8 : preformula 0)   : preformula 0\n| and  (\u03c6 \u03c8 : preformula 0)   : preformula 0\n| or   (\u03c6 \u03c8 : preformula 0)   : preformula 0\n| all  (\u03c6 : preformula 0)     : preformula 0\n| ex   (\u03c6 : preformula 0)     : preformula 0 \n| pred {arity : \u2115} (P : \u03c3.pred_symb arity)                  : preformula arity \n| papp {arity : \u2115} (\u03c6 : preformula (arity+1)) (t : term \u03c3) : preformula arity\n\n@[reducible] def formula := preformula \u03c3 0\n\nvariable {\u03c3}\n\nnotation `\u22a5'`     := preformula.bot \ninfix ` =' `:100  := preformula.eq  \n\ninfixr ` \u2192' `:80  := preformula.imp\ninfixr ` \u2228' `:85  := preformula.or\ninfixr ` \u2227' `:90  := preformula.and \n\nprefix `\u2200'`:110   := preformula.all\nprefix `\u2203'`:110   := preformula.ex \n\n@[simp] def preformula.iff (\u03c6 \u03c8 : formula \u03c3) : formula \u03c3 := (\u03c6 \u2192' \u03c8) \u2227' (\u03c8  \u2192' \u03c6)\ninfix ` \u2194' `:70 := preformula.iff -- input \\<=>\n\n@[simp] def preformula.not (\u03c6 : formula \u03c3) : formula \u03c3 := (\u03c6 \u2192' \u22a5') \nprefix `\u00ac'`:115 := preformula.not\n\ndef preformula.top : formula \u03c3  := \u00ac' \u22a5'\nnotation `\u22a4'` := preformula.top\n\nexport preformula\n\nsection lifts_and_substitutions\n\nnamespace formula\n\n/-- `lift \u03c6 m i` increases the index of `i`-free variables in `\u03c6` by `m`. -/\n@[simp, reducible] def lift : \u2200{a} , preformula \u03c3 a \u2192 \u2115 \u2192 \u2115 \u2192 preformula \u03c3 a \n| _ \u22a5'  _ _         := \u22a5'  \n| _ (t =' s) m i    := (term.lift t m i) =' (term.lift s m i)\n| _ (\u03c6 \u2192' \u03c8) m i  := (lift \u03c6 m i) \u2192' (lift \u03c8  m i)\n| _ (\u03c6 \u2227' \u03c8) m i  := (lift \u03c6 m i) \u2227' (lift \u03c8  m i)\n| _ (\u03c6 \u2228' \u03c8) m i  := (lift \u03c6 m i) \u2228' (lift \u03c8  m i)\n| _ (\u2200' \u03c6) m i    := \u2200' (lift \u03c6 m (i+1))\n| _ (\u2203' \u03c6) m i    := \u2203' (lift \u03c6 m (i+1))\n| _ (pred P) _ _    := pred P\n| _ (papp \u03c6 t) m i  := papp (lift \u03c6 m i) (term.lift t m i)\n\n/-- `subst t s k` substitutes `s \u2191 k \uff20 0` for each variable at `k` in `t` -/\n@[simp, tls] def subst :  \u2200{a} , preformula \u03c3 a \u2192 term \u03c3 \u2192 \u2115 \u2192 preformula \u03c3 a\n| _ \u22a5'  _ _         := \u22a5'  \n| _ (t\u2081 =' t\u2082) s k := (term.subst t\u2081 s k) =' (term.subst t\u2082 s k)\n| _ (\u03c6 \u2192' \u03c8) s k  := (subst \u03c6 s k) \u2192' (subst \u03c8  s k)\n| _ (\u03c6 \u2227' \u03c8) s k  := (subst \u03c6 s k) \u2227' (subst \u03c8  s k)\n| _ (\u03c6 \u2228' \u03c8) s k  := (subst \u03c6 s k) \u2228' (subst \u03c8  s k)\n| _ (\u2200' \u03c6) s k    := \u2200' (subst \u03c6  s (k+1))\n| _ (\u2203' \u03c6) s k    := \u2203' (subst \u03c6  s (k+1))\n| _ (pred P) _ _    := pred P\n| _ (papp \u03c6 t) s k  := papp (subst \u03c6 s k) (term.subst t s k)\n\nend formula\n\nnotation f ` \u2191 `:90 m ` \uff20 `:90 i :90 := formula.lift f m i\nnotation \u03c6 `[`:max t ` \u2044 `:95 n `]`:0 := formula.subst \u03c6 t n\n\n-- #reduce #3 \u2191 3 \uff20 1\n-- #reduce (#3 =' #0) \u2191 3 \uff20 1\n-- #reduce #5[#2 \u2044 1]\n-- #reduce (#5 =' #4)[#0 \u2044 5] \n\nnamespace formula\nopen preformula\n\n-- lift and substitution lemmas for formulas\n@[simp, tls] lemma lift_by_0: \u2200 {a} (\u03c6 : preformula \u03c3 a) {i}, \u03c6 \u2191 0 \uff20 i = \u03c6\n| _ \u22a5' _          := by refl\n| _ (t =' s) _    := by simp\n| _ (\u03c6 \u2192' \u03c8) _    := begin rw lift, congr; exact lift_by_0 _, end\n| _ (\u03c6 \u2227' \u03c8) _    := begin rw lift, congr; exact lift_by_0 _, end\n| _ (\u03c6 \u2228' \u03c8) _    := begin rw lift, congr; exact lift_by_0 _, end\n| _ (\u2200' \u03c6) _      := begin rw lift, congr, exact lift_by_0 \u03c6, end\n| _ (\u2203' \u03c6) _      := begin rw lift, congr, exact lift_by_0 \u03c6, end\n| _ (pred P) _    := by refl\n| _ (papp \u03c6 t) _  := begin rw lift, congr, exact lift_by_0 \u03c6, exact term.lift_by_0 t, end\n\nlemma lift_lift: \u2200{a} (\u03c6 : preformula \u03c3 a) (m) {i} (n) {j} (H : j \u2264 i),\n    (\u03c6 \u2191 m \uff20 i) \u2191 n \uff20 j = (\u03c6 \u2191 n \uff20 j) \u2191 m \uff20 (i+n)   \n| _ \u22a5' _ _ _ _ _           := by refl\n| _ (t =' s) _ _ _ _ _     := by simp[*, term.lift_lift]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _ _     := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _ _     := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _ _     := by simp[*]\n| _ (\u2200' \u03c6) _ _ _ _ _       := by simp[*, add_right_comm]\n| _ (\u2203' \u03c6) _ _ _ _ _       := by simp[*, add_right_comm]\n| _ (pred P) _ _ _ _ _     := by refl\n| _ (papp \u03c6 t) _ _ _ _ _   := by simp[*, term.lift_lift]\n\nlemma lift_lift_reverse {a} (\u03c6 : preformula \u03c3 a) {m i} (n) {j} (H : i + m \u2264 j) :\n    (\u03c6 \u2191 m \uff20 i) \u2191 n \uff20 j = (\u03c6 \u2191 n \uff20 (j-m)) \u2191 m \uff20 i := \nbegin\n    have h : i \u2264 (j-m), from nat.le_sub_right_of_add_le H,\n    have h': m \u2264 j, from (le_trans (m.le_add_left i) H),\n    rw [lift_lift \u03c6 n m h, nat.sub_add_cancel h'],\nend\n\nlemma lift_lift_merge: \u2200 {a} (\u03c6 : preformula \u03c3 a) {m i} (n) {j} (H : i \u2264 j) (H' : j \u2264 i + m),\n    (\u03c6 \u2191 m \uff20 i) \u2191 n \uff20 j = \u03c6 \u2191 (m+n) \uff20 i\n| _ \u22a5' _ _ _ _ _ _         := by refl \n| _ (t =' s) _ _ _ _ _ _   := by simp[*, term.lift_lift_merge]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _ _ _   := by simp[*]\n| _ (\u2200' \u03c6) _ _ _ _ _ _     := by simp[*, add_right_comm]\n| _ (\u2203' \u03c6) _ _ _ _ _ _     := by simp[*, add_right_comm]\n| _ (pred P) _ _ _ _ _ _   := by refl\n| _ (papp \u03c6 t) _ _ _ _ _ _ := by simp[*, term.lift_lift_merge]\n\n@[simp, tls] lemma lift_at_lift_merge {a} (\u03c6 : preformula \u03c3 a) (m i n):\n    (\u03c6 \u2191 m \uff20 i) \u2191 n \uff20 i = \u03c6 \u2191 (m+n) \uff20 i := lift_lift_merge \u03c6 n (le_rfl) (i.le_add_right m)\n\nlemma lambda_lift_lift {a} (m) {i} (n) {j} (H : j\u2264i) : \n    (\u03bb (\u03c6 :preformula \u03c3 a),  (\u03c6 \u2191 m \uff20 i) \u2191 n \uff20 j) = (\u03bb \u03c6, (\u03c6 \u2191 n \uff20 j) \u2191 m \uff20 (i+n)) :=\nbegin funext, apply lift_lift, exact H, end\n\nlemma lift_subst: \u2200 {a}  (\u03c6 : preformula \u03c3 a) (s: term \u03c3) (m i k : \u2115) (h': i \u2264 k),\n     \u03c6[s \u2044 k] \u2191 m \uff20 i = (\u03c6 \u2191 m \uff20 i)[s \u2044 (k+m)]  \n| _ \u22a5' _ _ _ _ _          := by refl\n| _ (t\u2081 =' t\u2082) _ _ _ _ _  := by simp[*, term.lift_subst]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _ _    := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _ _    := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _ _    := by simp[*]\n| _ (\u2200' \u03c6) _ _ _ _ _      := by simp[*, add_right_comm]\n| _ (\u2203' \u03c6) _ _ _ _ _      := by simp[*, add_right_comm]\n| _ (pred P) _ _ _ _ _    := by refl\n| _ (papp \u03c6 t) _ _ _ _ _  := by simp[*, term.lift_subst]\n\nlemma lambda_lift_subst_formula {a} {s: term \u03c3} { m i k : \u2115 } (h': i \u2264 k) :\n  (\u03bb (\u03d5: preformula \u03c3 a), lift (subst \u03d5 s k) m i) = (\u03bb \u03d5, subst (lift \u03d5 m i) s (k+m)) :=\nbegin funext, apply lift_subst, assumption, end\n\nlemma subst_lift : \u2200 {a}  (\u03c6 : preformula \u03c3 a) (s: term \u03c3) {m i k : \u2115 } (H: i \u2264 k) (H' : k \u2264 i + m),\n    (\u03c6 \u2191 (m+1) \uff20 i)[s \u2044 k] = \u03c6 \u2191 m \uff20 i \n| _ \u22a5' _ _ _ _ _ _          := by refl\n| _ (t\u2081 =' t\u2082) _ _ _ _ _ _  := by simp[*, term.subst_lift]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _ _ _  := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _ _ _  := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _ _ _  := by simp[*]\n| _ (\u2200' \u03c6) _ _ _ _ _ _    := by simp[*, add_right_comm]\n| _ (\u2203' \u03c6) _ _ _ _ _ _    := by simp[*, add_right_comm]\n| _ (pred P) _ _ _ _ _ _    := by refl\n| _ (papp \u03c6 t) _ _ _ _ _ _ := by simp[*, term.subst_lift]\n\nlemma subst_lift_in_lift : \u2200{a} (\u03c6 : preformula \u03c3 a) (s : term \u03c3) (m i k),\n    (\u03c6 \u2191 m \uff20 (i + k + 1)) [ (s \u2191 m \uff20 i) \u2044 k] = \u03c6[s \u2044 k] \u2191 m \uff20 (i+k) \n| _ \u22a5' _ _ _ _         := by refl\n| _ (t\u2081 =' t\u2082) _ _ _ _ := by simp[*, term.subst_lift_by_lift]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _   := by simp[*]\n| _ (\u2200' \u03c6) s m i k     := begin have := subst_lift_in_lift \u03c6 s m i (k+1), rw[add_succ i k] at this, simp[*, add_right_comm], end\n| _ (\u2203' \u03c6) s m i k     := begin have := subst_lift_in_lift \u03c6 s m i (k+1), rw[add_succ i k] at this, simp[*, add_right_comm], end\n| _ (pred P) _ _ _ _   := by refl\n| _ (papp \u03c6 t) _ _ _ _ := by simp[*, term.subst_lift_by_lift]\n\n\n@[tls] lemma subst0_lift_by_lift {a}  (\u03c6 : preformula \u03c3 a) {s : term \u03c3} {m i:\u2115 } :\n  (\u03c6 \u2191 m \uff20 (i + 1)) [(s \u2191 m \uff20 i) \u2044 0] = \u03c6[s \u2044 0] \u2191 m \uff20 i := subst_lift_in_lift \u03c6 s m i 0\n\n@[tls] lemma subst_at_lift {a} (\u03c6 : preformula \u03c3 a) (m) (s : term \u03c3) (k) : \n  (\u03c6 \u2191 (m+1) \uff20 k)[s \u2044 k] =  \u03c6 \u2191 m \uff20 k := subst_lift \u03c6 s (le_refl k) (le.intro rfl)\n\n@[tls] lemma subst_var0_lift : \u2200{a} (\u03c6 : preformula \u03c3 a) (m i : \u2115), \n  (\u03c6 \u2191 (m+1) \uff20 (i+1))[#0 \u2044 i] = \u03c6 \u2191 m \uff20 (i+1)\n| _ \u22a5'  _ _        := by refl \n| _ (t\u2081 =' t\u2082) m i := by simp[term.subst_var0_lift]\n| _ (\u03c6 \u2192' \u03c8) m i   := by simp*\n| _ (\u03c6 \u2227' \u03c8) m i   := by simp*\n| _ (\u03c6 \u2228' \u03c8) m i   := by simp*\n| _ (\u2200' \u03c6) m i     := by simp*\n| _ (\u2203' \u03c6) m i     := by simp*\n| _ (pred P) _ _   := by refl\n| _ (papp \u03c6 t) m i := by simp[*, term.subst_var0_lift]\n\n@[tls] lemma subst_var0_lift_by_1 {a}  (\u03c6 : preformula \u03c3 a)  (i : \u2115) : \n  (\u03c6 \u2191 1 \uff20 (i+1))[#0 \u2044 i] = \u03c6 := \nbegin\n  have h:= subst_var0_lift \u03c6 0 i,\n  rwa lift_by_0 at h,\nend\n\n@[tls] lemma subst_var0_for_0_lift_by_1 {a} (\u03c6 : preformula \u03c3 a) : \n  (\u03c6 \u2191 1 \uff20 1)[#0 \u2044 0] = \u03c6 := subst_var0_lift_by_1 \u03c6 0\n\n@[simp, tls] lemma subst_for_0_lift_by_1: \u2200 {a} (\u03c6 : preformula \u03c3 a) (s : term \u03c3), \n  (\u03c6 \u2191 1 \uff20 0)[s \u2044 0] = \u03c6 \n| _ \u22a5' _         := by refl\n| _ (t\u2081 =' t\u2082) _ := by simp[*, term.subst_for_0_lift_by_1]\n| _ (\u03c6 \u2192' \u03c8) _   := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _   := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _   := by simp[*]\n| _ (\u2200'\u03c6) s      := begin dsimp, congr, have h:= subst_at_lift \u03c6 0 s (0+1), rw lift_by_0 at h, exact h, end\n| _ (\u2203'\u03c6) s      := begin dsimp, congr, have h:= subst_at_lift \u03c6 0 s (0+1), rw lift_by_0 at h, exact h, end\n| _ (pred P) _   := by refl\n| _ (papp \u03c6 t) _ := by simp[*, term.subst_lift_by_lift]\n\nlemma subst_subst : \u2200 {a} (\u03c6 : preformula \u03c3 a) (s\u2081) {k\u2081} (s\u2082) {k\u2082} (H : k\u2081 \u2264 k\u2082), \n    \u03c6 [ s\u2081 \u2044 k\u2081] [ s\u2082 \u2044 k\u2082] = \u03c6 [ s\u2082 \u2044 k\u2082 + 1] [ (s\u2081 [s\u2082 \u2044 k\u2082 - k\u2081]) \u2044 k\u2081 ] \n| _ \u22a5' _ _ _ _ _         := by refl\n| _ (t\u2081 =' t\u2082) _ _ _ _ _ := by simp[*, term.subst_subst]\n| _ (\u03c6 \u2192' \u03c8) _ _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2227' \u03c8) _ _ _ _ _   := by simp[*]\n| _ (\u03c6 \u2228' \u03c8) _ _ _ _ _   := by simp[*]\n| _ (\u2200' \u03c6) _ _ _ _ _     := by simp[*, add_right_comm]\n| _ (\u2203' \u03c6) _ _ _ _ _     := by simp[*, add_right_comm]\n| _ (pred P) _ _ _ _ _   := by refl\n| _ (papp \u03c6 t) _ _ _ _ _ := by simp[*, term.subst_subst]\n\n\n/- If `\u03c6` is a fixed point for lifting at `i`, then its a fixed point for lifting at `j` for all `i\u2264j`  -/\nlemma lift_fixed_points_monotone {a} {\u03c6 : preformula \u03c3 a} {i j} \n  (H : \u03c6 \u2191 1 \uff20 i = \u03c6) (h : i \u2264 j) : \u03c6 \u2191 1 \uff20 j = \u03c6 :=\nbegin\n  induction j with j,\n  { rwa[le_zero_iff.mp h] at H, },\n  { by_cases h': i = j+1, \n    { rwa h' at H, },  \n    { have h\u2081: i\u2264j, from lt_succ_iff.mp (lt_of_le_of_ne h h'),\n      have h\u2082 := j_ih h\u2081,\n      rw [\u2190H, \u2190lift_lift \u03c6 1 1 h\u2081, h\u2082], }, },\nend \n\n-- We can give improve this lemma a  bit:\n-- - we can state this for fixed points of lifts by m\n-- - we can state this for fixed points of lifts by m \u2265 1 at i\n--   and conclude they are fixed points of lifts by n at j for j\u2265i\n-- Note that we do not place any conditions on n.\n\n\n/--\n  `alls k \u03c6` is the formula obtained by binding the the first `k` free variables in `\u03c6` \n  with universal quantifiers. \n  \n  In other words, we add `k` universal quantifier in from of `\u03c6`\n-/\ndef alls : \u2200 (k:\u2115) (\u03c6: formula \u03c3) , formula \u03c3\n| 0     \u03c6   := \u03c6 \n| (k+1) \u03c6   := \u2200' (alls k \u03c6)\n\n-- lemmas about alls\nlemma all_alls: \u2200 (\u03c6: formula \u03c3) (k:\u2115) , \u2200' (alls k \u03c6) = alls k (\u2200'\u03c6) \n| \u03c6 0 := by refl\n| \u03c6 (k+1) := begin unfold alls, congr' 1, apply all_alls, end\n\nlemma alls_succ (k) (\u03c6 : formula \u03c3) : alls (k+1) \u03c6 = alls k (\u2200' \u03c6) := begin rw [alls, all_alls], end\n\nlemma alls_alls: \u2200 (\u03c6: formula \u03c3) (m n:\u2115) , alls n (alls m \u03c6) = alls m (alls n \u03c6)\n| \u03c6 0 n := by refl\n| \u03c6 (m+1) n := begin rw alls, rw \u2190all_alls _ _, rw alls_alls _ m n, refl, end\n\nlemma alls_lift : \u2200  (\u03c6: formula \u03c3) (m i n:\u2115), alls n (\u03c6 \u2191 m \uff20 (i+n)) = (alls n \u03c6) \u2191 m \uff20 i\n| \u03c6 m i 0 := by refl\n| \u03c6 m i (n+1) := begin dsimp[alls], congr, rw \u2190succ_add_eq_succ_add i n, apply alls_lift,end\n\nlemma alls_at_lift  (\u03c6: formula \u03c3) (m n:\u2115) : alls n (\u03c6 \u2191 m \uff20 n) = (alls n \u03c6) \u2191 m \uff20 0 :=\nbegin \n  let h := alls_lift \u03c6 m 0 n, \n  rwa zero_add at h, \nend\n\n/--\n  `substs k i j \u03c6` is the formula `\u03c6[#(k+i) \u2044 k+j]...[#(1+i) \u2044 1+j][#i \u2044 j]`.\n-/\ndef substs : \u2200(k i j: \u2115) (\u03c6: formula \u03c3), formula \u03c3\n| 0 i j \u03c6  := \u03c6 \n| (k+1) i j \u03c6 := substs k i j (\u03c6 [#(k+i) \u2044 (k+j)])\n\n-- lemmas about substs\nlemma substs_succ (k i j: \u2115) (\u03c6 : formula \u03c3): substs (k+1) i j \u03c6 = (substs k (i+1) (j+1) \u03c6) [ #i \u2044 j] :=\nbegin\n  induction k generalizing \u03c6,\n  { simp[substs] },\n  { simp[*,substs, succ_add_eq_succ_add] }\nend \n\nlemma all_substs {k i j}{\u03c6 : formula \u03c3} : \n  \u2200'(substs k i (j+1) \u03c6) = substs k i j \u2200'\u03c6 :=\nbegin\n  induction k generalizing \u03c6,\n  { dsimp[substs], refl },\n  { simp[*,substs, succ_add_eq_succ_add, add_assoc] }\nend\n\n/--\n  A formula `\u03c6` is `k`-closed if it has no `k`-free variables,\n  i.e. if lifting at `k` does not change the formula.\n-/\n@[simp, reducible] def closed {a} (k : \u2115) (\u03c6 : preformula \u03c3 a) := \u03c6 \u2191 1 \uff20 k = \u03c6 \n\n/-- A sentence is a `0`-closed formula, i.e. a formula without free variables. -/\n@[simp, reducible] def sentence (\u03c6 : formula \u03c3) := closed 0 \u03c6 \npostfix ` is_sentence`:max := sentence\n\n/- Various lemmas involving lifts and substitutions of closed formulas -/\n\nlemma closed_all {\u03c6 : formula \u03c3} {k} (H : closed (k+1) \u03c6) : closed k (\u2200' \u03c6) :=\nbegin dsimp, congr, exact H, end\n\nlemma closed_ex {\u03c6 : formula \u03c3} {k} (H : closed (k+1) \u03c6) : closed k (\u2203' \u03c6) :=\nbegin dsimp, congr, exact H end\n\nlemma lift_closed_id_h { \u03c6 : formula \u03c3} {k} (H : closed k \u03c6) (m i) : \n  \u03c6 \u2191 m \uff20 (k+i) = \u03c6 :=\nbegin\n  induction m generalizing \u03c6,\n  { apply lift_by_0, },\n  { rw [succ_eq_add_one, \u2190lift_lift_merge \u03c6 1 (le_refl _) (le.intro rfl), m_ih H],\n    apply lift_fixed_points_monotone H (le.intro rfl) },\nend\n\n-- `k`-closed formulas are fixed points for lifts at reference depth `\u2265k`\nlemma lift_closed_id { \u03c6 : formula \u03c3} {k} (H : closed k \u03c6) (m) {l} (h : k \u2264 l): \n  (\u03c6 \u2191 m \uff20 l) = \u03c6 :=\nbegin\n  cases le_iff_exists_add.mp h with i h_i,\n  subst h_i, \n  exact lift_closed_id_h H m i,\nend\n\n-- sentences are fixed points of all lifts \nlemma lift_sentence_id {\u03c6 : formula \u03c3} (H: sentence \u03c6) { m i } : \n  (\u03c6 \u2191 m \uff20 i) = \u03c6 := lift_closed_id H m (i.zero_le)\n\nlemma lift_set_of_sentences_id {\u0393 : set $ formula \u03c3} (H : \u2200 \u03d5 \u2208 \u0393, sentence \u03d5) {m i} \n  : (\u03bb \u03d5: formula \u03c3, \u03d5 \u2191 m \uff20 i) '' \u0393 = \u0393 :=\nbegin\n  apply ext, intro x,\n  apply iff.intro,\n  { intro h_x, rw mem_image_eq at h_x, \n    cases h_x with y h', \n    have yx:= h'.right, \n    have y_h := h'.left, \n    subst yx, rwa lift_sentence_id (H y y_h), }, \n  { intro h, rw mem_image_eq, use x, exact \u27e8h, lift_sentence_id (H x h)\u27e9, },\nend\n\nlemma subst_closed_id_h { \u03c6 : formula \u03c3} (t:term \u03c3) {k} (i) (H : closed k \u03c6) : \n  (\u03c6 [t \u2044 k+i]) = \u03c6 :=\nbegin\n  have h := subst_at_lift \u03c6 0 t (k+i),\n  repeat {rwa lift_closed_id_h H _ _ at h,},\nend\n\n-- `k`-closed formulas are fixed points for substitutions at reference depth `\u2265k`\nlemma subst_closed_id {\u03c6 : formula \u03c3}{i}  (H : closed i \u03c6)  (t:term \u03c3) {k} (h : i\u2264k) : \n  (\u03c6 [t \u2044 k]) = \u03c6 :=\nbegin\n  cases le_iff_exists_add.mp h with j h_j,\n  subst h_j, exact subst_closed_id_h t j H,\nend\n\nlemma subst_sentence_id { \u03c6 : formula \u03c3} (H : sentence \u03c6)  {t: term \u03c3} {k:\u2115} :  (\u03c6 [t \u2044 k]) = \u03c6 \n  := subst_closed_id H t (k.zero_le)\n\nlemma subst_set_of_sentences_id {\u0393 : set $ formula \u03c3} {t k} (H : \u2200f \u2208 \u0393, sentence f) : \n  (\u03bb (\u03d5: formula \u03c3), \u03d5[t \u2044 k]) '' \u0393 = \u0393 :=\nbegin\n  apply ext, intro x,\n  apply iff.intro,\n  { intro h_x, rw mem_image_eq at h_x, \n    cases h_x with y h', \n    have yx := h'.right, \n    have h_y := h'.left, \n    subst yx, rwa subst_sentence_id (H y h_y), }, \n  { intro h, rw mem_image_eq, use x, exact \u27e8h, subst_sentence_id (H x h)\u27e9, },\nend\n\n/--\n  Biggest (deepest) reference depth of variables occurring in a formula (plus one).\n\n  If equal to `0` the formula has no free variables.\n-/\ndef max_free_var :  \u2200 {a} (\u03c6: preformula \u03c3 a), \u2115 \n| _ \u22a5'         := 0\n| _ (t\u2081 =' t\u2082) := max (term.max_free_var t\u2081) (term.max_free_var t\u2082)\n| _ (\u2200'\u03c6)      := (max_free_var \u03c6) - 1\n| _ (\u2203'\u03c6)      := (max_free_var \u03c6) - 1\n| _ (\u03c6 \u2192' \u03c8)   := max (max_free_var \u03c6) (max_free_var \u03c8)\n| _ (\u03c6 \u2227' \u03c8)   := max (max_free_var \u03c6) (max_free_var \u03c8)\n| _ (\u03c6 \u2228' \u03c8)   := max (max_free_var \u03c6) (max_free_var \u03c8)\n| _ (pred P)   := 0\n| _ (papp \u03c6 t) := max (max_free_var \u03c6) (term.max_free_var t)\n\n/- This lemma shows that our definition of closed is exactly what our intuition tells us. -/\nlemma closed_max_free_var {a} (\u03c6 : preformula \u03c3 a) : closed (max_free_var \u03c6) \u03c6 :=\nbegin\n  unfold closed,\n  induction \u03c6,\n  { refl },\n  { have h\u2081 := term.lift_fixed_points_monotone (le_max_left (term.max_free_var \u03c6_t)  (term.max_free_var \u03c6_s)) (term.lift_at_max_free_var \u03c6_t),\n    have h\u2082 := term.lift_fixed_points_monotone (le_max_right (term.max_free_var \u03c6_t) (term.max_free_var \u03c6_s)) (term.lift_at_max_free_var \u03c6_s),\n    rw[max_free_var, formula.lift, h\u2081,h\u2082] },\n  { have h\u2081:= lift_fixed_points_monotone \u03c6_ih_\u03c6 (le_max_left  (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    have h\u2082:= lift_fixed_points_monotone \u03c6_ih_\u03c8 (le_max_right (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    rw[max_free_var, formula.lift, h\u2081,h\u2082] },\n  { have h\u2081:= lift_fixed_points_monotone \u03c6_ih_\u03c6 (le_max_left  (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    have h\u2082:= lift_fixed_points_monotone \u03c6_ih_\u03c8 (le_max_right (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    rw[max_free_var, formula.lift, h\u2081,h\u2082] },\n  { have h\u2081:= lift_fixed_points_monotone \u03c6_ih_\u03c6 (le_max_left  (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    have h\u2082:= lift_fixed_points_monotone \u03c6_ih_\u03c8 (le_max_right (max_free_var \u03c6_\u03c6) (max_free_var \u03c6_\u03c8)),\n    rw[max_free_var, formula.lift, h\u2081,h\u2082] },\n  { have h := lift_fixed_points_monotone \u03c6_ih (nat.le_sub_add (max_free_var \u03c6_\u03c6) 1),\n    rw[formula.lift, max_free_var, h], },\n  { have h := lift_fixed_points_monotone \u03c6_ih (nat.le_sub_add (max_free_var \u03c6_\u03c6) 1),\n    rw[formula.lift, max_free_var, h] },\n  { refl },\n  { have h\u2081:= lift_fixed_points_monotone \u03c6_ih (le_max_left (max_free_var \u03c6_\u03c6) (term.max_free_var \u03c6_t)),\n    have h\u2082:= term.lift_fixed_points_monotone (le_max_right (max_free_var \u03c6_\u03c6) (term.max_free_var \u03c6_t)) (term.lift_at_max_free_var \u03c6_t) ,\n    rw[max_free_var, formula.lift, h\u2081, h\u2082] }\nend \n\n/-- The (universal) closure of a `k`-closed formula, binding up to the `k`-th free variable -/\n@[reducible] def closure (\u03c6 : formula \u03c3) {k} (H: closed k \u03c6) := alls k \u03c6\n\nlemma closure_is_sentence  {\u03c6 : formula \u03c3} {k} (H : closed k \u03c6) : (closure \u03c6 H) is_sentence :=\nbegin\n  induction k generalizing \u03c6,\n  { exact H, },\n  { unfold closure,\n    rw[alls, all_alls],\n    exact k_ih (closed_all H), },\nend\n\ndef not_free (k) (\u03c6 : formula \u03c3) : Prop := \u2203\u03d5, \u03c6 = \u03d5 \u2191 1 \uff20 k\n\nlemma not_free_trival_witness (k)  (\u03c6 : formula \u03c3) (h : not_free k \u03c6) : \u03c6 = \u03c6[#0 \u2044 k] \u2191 1 \uff20 k :=\nbegin\n  cases h with \u03c8 \u03c8_h,\n  subst \u03c8_h,\n  rw [subst_at_lift, lift_by_0],\nend \n\n-- /-- Lift operation on sets of formulas. -/\n-- @[simp] def lift_set (\u0393 : set $ formula \u03c3) (m i) : set $ formula \u03c3 := ((\u03bb (\u03d5 : formula \u03c3), \u03d5 \u2191 m \uff20 i) '' \u0393)\n-- /-- Substitution operation on sets of formulas. -/\n-- @[simp] def subst_set  (\u0393 : set $ formula \u03c3) (s k) : set $ formula \u03c3 := ((\u03bb (\u03d5 : formula \u03c3), \u03d5 [s \u2044 k]) '' \u0393)\n\nend formula\n\nend lifts_and_substitutions\n\nend formulas\n\nexport formula\n\n/-!### Proof terms of natural deduction -/\nsection proof_terms\n\nlocal notation \u03c6 >> \u0393 := insert \u03c6 \u0393 \n\n/--\n  An intuitionistic natural deduction proof calculus \n  for first order predicate logic with rules for equality \n\n  Fresh variables for universal quantifier introduction and existential quantifier elimination\n  are introduced by lifting. \n-/\ninductive proof_term : (set $ formula \u03c3) \u2192 formula \u03c3 \u2192 Type u\n| hypI {\u0393} {\u03c6} (h : \u03c6 \u2208 \u0393) : proof_term \u0393 \u03c6 \n| botE {\u0393} {\u03c6} (H : proof_term \u0393  \u22a5') : proof_term \u0393 \u03c6\n-- implication\n| impI {\u0393} {\u03c6 \u03c8}   (H : proof_term (\u03c6>>\u0393) \u03c8) : proof_term \u0393 (\u03c6 \u2192' \u03c8)\n| impE {\u0393} (\u03c6) {\u03c8} (H\u2081 : proof_term \u0393 \u03c6) (H\u2082 : proof_term \u0393 (\u03c6 \u2192' \u03c8)) : proof_term \u0393 \u03c8\n-- conjunction\n| andI  {\u0393} {\u03c6 \u03c8} (H\u2081 : proof_term \u0393 \u03c6) \n                  (H\u2082 : proof_term \u0393 \u03c8) : proof_term \u0393 (\u03c6 \u2227' \u03c8) \n| andE\u2081 {\u0393} {\u03c6} (\u03c8) (H : proof_term \u0393 (\u03c6 \u2227' \u03c8)) : proof_term \u0393 \u03c6\n| andE\u2082 {\u0393} (\u03c6) {\u03c8} (H : proof_term \u0393 (\u03c6 \u2227' \u03c8)) : proof_term \u0393 \u03c8\n-- disjunction\n| orI\u2081 {\u0393} {\u03c6 \u03c8} (H : proof_term \u0393 \u03c6) : proof_term \u0393 (\u03c6 \u2228' \u03c8)\n| orI\u2082 {\u0393} {\u03c6 \u03c8} (H : proof_term \u0393 \u03c8) : proof_term \u0393 (\u03c6 \u2228' \u03c8)\n| orE  {\u0393}  (\u03c6 \u03c8) {\u03c7} (H  : proof_term \u0393 (\u03c6 \u2228' \u03c8))  \n                      (H\u2081 : proof_term (\u03c6 >> \u0393) \u03c7) \n                      (H\u2082 : proof_term (\u03c8 >> \u0393) \u03c7) : proof_term \u0393 \u03c7\n-- quantification\n| allI  {\u0393} {\u03c6} (H : proof_term ((\u03bb \u03d5, \u03d5 \u2191 1 \uff20 0) '' \u0393) \u03c6) : proof_term \u0393 (\u2200'\u03c6)\n| allE  {\u0393} (\u03c6) {t} (H : proof_term \u0393 (\u2200'\u03c6)) : proof_term \u0393 (\u03c6 [t \u2044 0])\n| exI   {\u0393 \u03c6} (t) (H : proof_term \u0393 (\u03c6[t \u2044 0])) : proof_term \u0393  (\u2203'\u03c6)\n| exE   {\u0393 \u03c8} (\u03c6) (H\u2081 : proof_term \u0393 (\u2203'\u03c6)) \n  (H\u2082 : proof_term  (\u03c6 >> (\u03bb \u03d5, \u03d5 \u2191 1 \uff20 0) '' \u0393) (\u03c8 \u2191 1 \uff20 0)) : proof_term \u0393 \u03c8\n-- equality\n| eqI {\u0393} (t) : proof_term \u0393 (t =' t)\n| eqE {\u0393} {s t \u03c6 } (H\u2081 : proof_term \u0393 (s =' t)) (H\u2082 : proof_term \u0393 (\u03c6[s \u2044 0])) : proof_term \u0393 (\u03c6 [t \u2044 0])\ninfix ` \u22a2 `:55 := proof_term \n\n/-- \n  `provable \u0393 \u03c6` says that there exists a proof_term of `\u03c6` under the hypotheses in `\u0393`,\n  i.e. it is a fancy way to say that the type `\u0393 \u22a2 \u03c6` is non-empty. \n-/\ndef provable (\u03c6 : formula \u03c3) (\u0393)  : Prop := nonempty (\u0393 \u22a2 \u03c6)\ninfix ` is_provable_within `:100 := provable\n\n/--\n  The law of excluded middle for when we want to argue in classical logic.\n-/\ndef lem : set $ formula \u03c3 := { (\u03c6 \u2228' \u00ac'\u03c6) | (\u03c6: formula \u03c3) (h: \u03c6 is_sentence) } -- do we need the extra condition?\n\nnamespace proof_term\n/--\n  Rule for weakening the context of a proof_term by allowing more premises.\n-/\ndef weak {\u0394 \u03c6}  (\u0393: set $ formula \u03c3) (H : \u0393 \u22a2 \u03c6) (h: \u0393 \u2286 \u0394): (\u0394 \u22a2 \u03c6) :=\nbegin\n  induction H generalizing \u0394,\n  { apply hypI (h H_h) },\n  { apply botE, apply H_ih, assumption },\n\n  { apply impI, apply H_ih, apply insert_subset_insert, assumption },\n  { apply impE, apply H_ih_H\u2081, assumption, \n    apply H_ih_H\u2082, assumption },\n\n  { apply andI,  apply H_ih_H\u2081, exact h, \n    apply H_ih_H\u2082, exact h},\n  { apply andE\u2081, apply H_ih, exact h },\n  { apply andE\u2082, apply H_ih, exact h },\n\n  { apply orI\u2081, apply H_ih, exact h, },\n  { apply orI\u2082, apply H_ih, exact h, },\n  { apply orE,  apply H_ih_H, exact h, \n    apply H_ih_H\u2081, apply insert_subset_insert, exact h, \n    apply H_ih_H\u2082, apply insert_subset_insert, exact h},\n\n  { apply allI, apply H_ih, exact image_subset _ h,},\n  { apply allE, apply H_ih, exact h},\n\n  { apply exI, apply H_ih, exact h},\n  { apply exE, apply H_ih_H\u2081, exact h, \n    apply H_ih_H\u2082, apply insert_subset_insert, exact image_subset _ h,},\n\n  { apply eqI, },\n  { apply eqE, apply H_ih_H\u2081 h, apply H_ih_H\u2082 h, },\nend\n\n/--\n  Proof rule for weakening the context of a proof_term by inserting a single premise.\n-/\ndef weak1 {\u0393} {\u03c6 \u03c8: formula \u03c3} (H: \u0393 \u22a2 \u03c8) :  (\u03c6>>\u0393) \u22a2 \u03c8 := weak \u0393 H (subset_insert \u03c6 \u0393)\n\n/--\n  Proof rule for weakening the context of a proof_term from a single premise.\n-/\ndef weak_singleton {\u0393} (\u03c6) {\u03c8: formula \u03c3} (H: { \u03c6 } \u22a2 \u03c8) (h: \u03c6 \u2208 \u0393) :  \u0393 \u22a2 \u03c8 :=\nbegin\n  apply weak {\u03c6} H,\n  assume x xh,\n  rw mem_singleton_iff at xh,\n  subst xh,\n  assumption,\nend\n\n-- QoL rules for hypothesis\ndef hypI1 {\u0393} (\u03c6: formula \u03c3)  : (\u03c6 >> \u0393) \u22a2 \u03c6 := hypI (mem_insert \u03c6 \u0393)\n\ndef hypI2 {\u0393} (\u03c6 \u03c8: formula \u03c3)  : \u03c6 >> (\u03c8 >> \u0393) \u22a2 \u03c8 := \nbegin\n  apply hypI, right, exact mem_insert \u03c8 \u0393,\nend\n/--\n  Rule for top introduction.\n-/\ndef topI {\u0393: set $ formula \u03c3} : \u0393 \u22a2 \u22a4' := begin apply impI, apply hypI1, end\n\n\n-- rules for implications\ndef impE_insert {\u0393} {\u03c6 \u03c8: formula \u03c3} (H\u2081 : \u0393 \u22a2 (\u03c6 \u2192' \u03c8)) : \u03c6 >> \u0393 \u22a2 \u03c8  :=\nbegin\n  apply impE \u03c6, \n  apply hypI1, \n  apply weak1,\n  assumption,\nend\n\n/--\n  Proof rule for reflexivity of implications.\n-/\ndef impI_refl {\u0393} (\u03c6 : formula \u03c3) : \u0393 \u22a2 (\u03c6 \u2192' \u03c6) := \nbegin\n    apply impI, apply hypI1,\nend\n\n/--\n  Proof rule for transitivity of implications.\n-/\ndef impI_trans  {\u0393} (\u03c6 \u03c8 \u03c7 : formula \u03c3) (H\u2081: \u0393 \u22a2 (\u03c6 \u2192' \u03c8)) (H\u2082 : \u0393 \u22a2 (\u03c8  \u2192' \u03c7)) : \u0393 \u22a2 (\u03c6  \u2192' \u03c7) :=\nbegin\n  apply impI, \n  apply impE \u03c8, \n  apply impE_insert H\u2081,\n  apply weak1 H\u2082,\nend\n\n/--\n  QoL proof_term rule for universal quantification elimination.\n-/\ndef allE' {\u0393} (\u03c6) (t: term \u03c3) {\u03c8}  (H : \u0393 \u22a2 (\u2200'\u03c6)) (h: \u03c8  = \u03c6[t \u2044 0]) : \u0393 \u22a2 \u03c8 :=\nbegin subst h, apply allE, assumption, end\n\n/--\n  Proof rule for a common case of universal quantification elimination.\n-/\ndef allE_var0 {\u0393} {\u03c6: formula \u03c3}  (H : \u0393 \u22a2 (\u2200'\u03c6) \u2191 1 \uff20 0) : \u0393 \u22a2 \u03c6  :=\nbegin\n  apply allE' (\u03c6 \u2191 1 \uff20 1) #0,\n  { exact H, }, \n  { symmetry, exact subst_var0_lift_by_1 \u03c6 0, } \nend\n\n/--\n  Proof rule for equality elimination. _(QoL)_\n-/\ndef eqE' {\u0393} {\u03c8}  (s t) (\u03c6 : formula \u03c3) (H\u2081 : \u0393 \u22a2 (s =' t)) (H\u2082 : \u0393 \u22a2 (\u03c6 [s \u2044 0])) (h:  \u03c8 = \u03c6[t \u2044 0]) : \u0393 \u22a2 \u03c8 :=\nbegin rw h, apply eqE H\u2081 H\u2082, end\n\n/-- Proof rule for congruence introduction. -/\ndef congrI {\u0393} {t s\u2081 s\u2082: term \u03c3} (H :  \u0393 \u22a2 (s\u2081 =' s\u2082)) :  \u0393 \u22a2 (t[s\u2081 \u2044 0] =' t[s\u2082 \u2044 0]):=\nbegin\n  apply eqE' s\u2081 s\u2082 (((t[s\u2081\u2044 0] \u2191 1 \uff20 0)=' t)) H;\n  rw [subst, term.subst_for_0_lift_by_1 (term.subst t _ 0) _],\n  apply eqI,\nend\n\n/-- Proof rule for congruence introduction. -/\ndef congrI' {\u0393} {t\u2081 s\u2081 t\u2082 s\u2082 : term \u03c3} (t) (H: \u0393 \u22a2 s\u2081 =' s\u2082) \n  (h\u2081: t\u2081 = t[s\u2081 \u2044 0]) (h\u2082: t\u2082 = t[s\u2082 \u2044 0]) : \u0393 \u22a2 (t\u2081 =' t\u2082) := \nbegin rw [h\u2081, h\u2082], apply congrI H, end\n\n/-- Proof rule for reflexivity of equality. -/\ndef eqI_refl {\u0393} (t: term \u03c3): \u0393 \u22a2 (t =' t) := @eqI \u03c3 \u0393 t\n\n/-- Proof rule for symmetry of equality. -/\ndef eqI_symm {\u0393} (s t : term \u03c3) (H : \u0393 \u22a2 (s =' t)) : \u0393 \u22a2 (t =' s) :=\nbegin\n  apply eqE' s t (#0 =' (s \u2191 1 \uff20 0)) H;\n  rw [subst, term.subst_var0, term.subst_for_0_lift_by_1],\n  apply eqI, \nend\n/-- Proof rule for transitivity of equality. -/\ndef eqI_trans {\u0393} (s t u : term \u03c3) (H\u2081 : \u0393 \u22a2 (s =' t)) (H\u2082 : \u0393 \u22a2 (t =' u)) : proof_term \u0393 (s =' u) :=\nbegin\n  apply eqE' t u ((s \u2191 1 \uff20 0) =' #0) H\u2082;\n  rw[subst, term.subst_for_0_lift_by_1, term.subst_var0], \n  assumption,\nend\n\n/- biconditionals -/\n\n/-- Proof rule for introducing a biconditional. -/\ndef iffI {\u0393} {\u03c6 \u03c8 : formula \u03c3} (H\u2081 : \u0393 \u22a2 \u03c6 \u2192' \u03c8) (H\u2082 : \u0393 \u22a2 \u03c8 \u2192' \u03c6)  : \u0393 \u22a2 (\u03c6  \u2194' \u03c8) :=\nbegin apply andI; assumption, end\n\ndef iffE_r {\u0393} {\u03c6 \u03c8 : formula \u03c3} (H : \u0393 \u22a2 \u03c6 \u2194' \u03c8)  : (\u0393 \u22a2 \u03c6 \u2192' \u03c8) := andE\u2081 _ H\n\ndef iffE_l {\u0393} {\u03c6 \u03c8 : formula \u03c3} (H : \u0393 \u22a2 \u03c6 \u2194' \u03c8)  : (\u0393 \u22a2 \u03c8 \u2192' \u03c6) := andE\u2082 _ H\n\n/--\n  Proof rule for eliminating the right part of a biconditional.\n-/\ndef iffE\u2081 {\u0393} {\u03c6: formula \u03c3} (\u03c8 : formula \u03c3) (H\u2081 : \u0393 \u22a2 \u03c8) (H\u2082 : \u0393 \u22a2 \u03c6 \u2194' \u03c8)  : \u0393 \u22a2 \u03c6 :=\nbegin\n  apply impE \u03c8,\n  { exact H\u2081, },\n  { apply andE\u2082, exact H\u2082, },\nend\n\n/-- Proof rule for eliminating the left part of a biconditional. -/\ndef iffE\u2082 {\u0393} (\u03c6) {\u03c8 : formula \u03c3} (H\u2081 : \u0393 \u22a2 \u03c6) (H\u2082 : \u0393 \u22a2 \u03c6 \u2194' \u03c8)  : (\u0393 \u22a2  \u03c8) :=\nbegin\n  apply impE \u03c6,\n  { exact H\u2081, },\n  { apply andE\u2081, exact H\u2082, },\nend\n\n/-- Proof rule for reflexivity of biconditionals.-/\ndef iffI_refl {\u0393} (\u03c6 : formula \u03c3) : \u0393 \u22a2 (\u03c6 \u2194' \u03c6) := begin apply iffI; apply impI_refl,end\n\n/-- Proof rule for transitivity of biconditionals. -/\ndef iffI_trans {\u0393} {\u03c6} (\u03c8: formula \u03c3) {\u03c7}  (H\u2081: \u0393 \u22a2 (\u03c6 \u2194' \u03c8)) (H\u2082 : \u0393 \u22a2 (\u03c8 \u2194' \u03c7)) : \u0393 \u22a2 (\u03c6 \u2194' \u03c7) :=\nbegin\n    apply andI;\n    apply impI_trans _ \u03c8 _,\n    apply andE\u2081 _ H\u2081, apply andE\u2081 _ H\u2082,\n    apply andE\u2082 _ H\u2082, apply andE\u2082 _ H\u2081,\nend\n\n/-- Proof rule for symmetry of biconditionals. -/\ndef iffI_symm {\u0393} {\u03c6 \u03c8: formula \u03c3}  (H: \u0393 \u22a2 (\u03c6 \u2194' \u03c8)) : \u0393 \u22a2 (\u03c8 \u2194' \u03c6) := \nbegin apply iffI, apply andE\u2082, exact H, apply andE\u2081, exact H, end\n\n\n/-- Proof rule for substituting a term for free variable. -/\ndef substI {\u0393} {\u03c6 : formula \u03c3} (t k) (H: \u0393 \u22a2 \u03c6) : (\u03bb \u03d5, \u03d5[t \u2044 k])'' \u0393 \u22a2 \u03c6[t \u2044 k] :=\nbegin\n  induction H generalizing k,\n  { apply hypI, exact mem_image_of_mem (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 [t \u2044 k]) H_h, },\n  { apply botE, apply H_ih, },\n\n  { apply impI, rw \u2190 (@image_insert_eq _ _ (\u03bb (x : preformula \u03c3 0), x[t \u2044 k])), exact H_ih k, },\n  { apply impE (H_\u03c6 [t \u2044 k]), exact H_ih_H\u2081 k, exact H_ih_H\u2082 k, },\n\n  { apply andI, exact H_ih_H\u2081 k, exact H_ih_H\u2082 k, },\n  { apply andE\u2081, exact H_ih k, },\n  { apply andE\u2082, exact H_ih k, },\n\n  { apply orI\u2081, exact H_ih k, },\n  { apply orI\u2082, exact H_ih k, },\n  { apply orE (H_\u03c6 [t \u2044 k]) (H_\u03c8 [t \u2044 k]), \n    apply H_ih_H k, \n    have H\u2081:= H_ih_H\u2081 k, rw image_insert_eq at H\u2081, exact H\u2081,\n    have H\u2082:= H_ih_H\u2082 k, rw image_insert_eq at H\u2082, exact H\u2082, },\n\n  { apply allI, rw [image_image, lambda_lift_subst_formula(k.zero_le)], \n    have H := H_ih (k+1), rw[image_image] at H, exact H, },\n  { apply allE' _ (H_t[t \u2044 k]) (H_ih k), apply subst_subst, exact (k.zero_le), },\n\n  { apply exI (H_t [t\u2044 k]), \n      have h:= subst_subst H_\u03c6 H_t t (k.zero_le), \n      rw nat.sub_zero at h, rw \u2190h, exact H_ih k,},\n  { apply exE (H_\u03c6 [t\u2044(k+1)]), apply H_ih_H\u2081 k, rw lift_subst H_\u03c8 t 1 0 k (k.zero_le),\n      have h:= H_ih_H\u2082 (k+1),\n      rw [image_insert_eq, image_image, \u2190lambda_lift_subst_formula(k.zero_le)] at h,\n      rw [image_image], exact h, },\n  \n  { apply eqI_refl, },\n  { apply eqE', apply H_ih_H\u2081 k,\n    have h:= H_ih_H\u2082 k, rwa [subst_subst H_\u03c6 H_s t (k.zero_le), nat.sub_zero] at h, \n    exact subst_subst H_\u03c6 H_t t (k.zero_le), }\nend\n\n/-- Proof rule for introducing `m` fresh variables at `i`. -/\ndef liftI {\u0393} {\u03c6 : formula \u03c3} (m i : \u2115) (H: \u0393 \u22a2 \u03c6) \n  : (\u03bb (\u03d5 :formula \u03c3), \u03d5 \u2191 m \uff20 i) '' \u0393 \u22a2 (\u03c6 \u2191 m \uff20 i) :=\nbegin\n  induction H generalizing i,\n  { apply hypI, exact mem_image_of_mem (\u03bb (\u03d5 : preformula \u03c3 0),  \u03d5 \u2191 m \uff20 i) H_h, },\n  { apply botE, exact H_ih i, },\n\n  { apply impI, have:= H_ih i, rwa image_insert_eq at this, },\n  { apply impE (H_\u03c6 \u2191 m \uff20 i) , exact H_ih_H\u2081 i, exact H_ih_H\u2082 i,},\n\n  { apply andI, apply H_ih_H\u2081 i, apply H_ih_H\u2082 i, },\n  { apply andE\u2081, apply H_ih i, },\n  { apply andE\u2082, apply H_ih i, },\n\n  { apply orI\u2081, apply H_ih i, },\n  { apply orI\u2082, apply H_ih i, },\n  { apply orE, apply H_ih_H i,\n    have H\u2081 := H_ih_H\u2081 i, rw image_insert_eq at H\u2081, exact H\u2081,\n    have H\u2082 := H_ih_H\u2082 i, rw image_insert_eq at H\u2082, exact H\u2082, },\n  \n  { apply allI, rw[image_image, lambda_lift_lift _ _ (i.zero_le)],\n    have h:= H_ih (i+1), rw[image_image] at h, exact h, },\n  { apply allE' _ (H_t \u2191 m \uff20 i) (H_ih i), \n    have h := eq.symm (subst_lift_in_lift H_\u03c6 H_t m i 0), exact h,},\n\n  { apply exI (H_t \u2191 m \uff20 i), \n    rw subst0_lift_by_lift H_\u03c6, \n    exact H_ih i,  },\n  { apply exE (H_\u03c6 \u2191 m \uff20 (i+1)), apply H_ih_H\u2081 i, \n    rw[image_image, lift_lift H_\u03c8 m 1 (i.zero_le), lambda_lift_lift _ _ (i.zero_le)],\n    have h := H_ih_H\u2082 (i+1), rw[image_insert_eq, image_image] at h, exact h, },\n  \n  { apply eqI_refl, },\n  { apply eqE' _ _ _ (H_ih_H\u2081 i),\n    have h\u2081:= symm (subst0_lift_by_lift H_\u03c6),\n    have h\u2082 := H_ih_H\u2082 i, rw h\u2081 at h\u2082, exact h\u2082,\n    exact symm (subst0_lift_by_lift _), },\nend\n\n/-- Proof rule for removing a single fresh variables at `0`. -/\ndef liftE_h {\u0393} {\u03c6 : formula \u03c3} (m i : \u2115) (H:  (\u03bb (\u03d5 :formula \u03c3), \u03d5 \u2191 1 \uff20 0) '' \u0393 \u22a2 (\u03c6 \u2191 1 \uff20 0)) \n  : \u0393 \u22a2 \u03c6 :=\nbegin\n  rw \u2190subst_for_0_lift_by_1 \u03c6 #0,\n  apply allE,\n  apply allI,\n  exact H,\nend\n\n/-- Proof rule for binding the first `n` variables with universal quantifiers. -/\ndef allsI {\u0393} {\u03c6: formula \u03c3}  (n) (H: (\u03bb \u03d5 , \u03d5 \u2191 n \uff20 0) '' \u0393 \u22a2 \u03c6) :  \u0393 \u22a2 alls n \u03c6  :=\nbegin\n  induction n generalizing \u03c6 \u0393,\n  { simp [lift_by_0] at H, assumption,},\n  { rw[alls], \n    apply allI,\n    have h : (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 n_n.succ \uff20 0) \n      = (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 n_n \uff20 0) \u2218 (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 1\uff20 0),\n    begin funext, dsimp, rw lift_at_lift_merge, rw add_comm 1 n_n, end,\n    rw [h, image_comp] at H,\n    exact n_ih H, },\nend\n/-- Proof rule unbinding the `n` universal quantifiers. -/\ndef allsE  {\u0393} {\u03c6: formula \u03c3}  (n i) (H :  \u0393 \u22a2 (alls n \u03c6)) :  \u0393 \u22a2 substs n i 0 \u03c6 :=\nbegin\n  induction n generalizing \u03c6 i,\n  { exact H,},\n  { rw substs_succ, \n    apply allE, \n    rw all_substs, \n    rw [alls, all_alls] at H, \n    exact n_ih (i+1) H, },\nend\n\n/-- Proof rule unbinding the `n` universal quantifiers. -/\ndef allsE' {\u0393} (n) {\u03c6  : formula \u03c3} (H : \u0393 \u22a2 (alls n \u03c6)) : (\u03bb \u03d5 , \u03d5 \u2191 n \uff20 0) '' \u0393 \u22a2 \u03c6  :=\nbegin\n  induction n generalizing \u03c6 \u0393,\n  { have h :  (\u03bb (\u03d5: formula \u03c3) , \u03d5 \u2191 0 \uff20 0) = id, from begin funext, rw lift_by_0, refl, end,\n    rw [h, image_id] at *, \n    rwa alls at H, },\n  { have h: (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 n_n.succ \uff20 0) \n          = (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 1 \uff20 0) \u2218 (\u03bb (\u03d5 : preformula \u03c3 0), \u03d5 \u2191 n_n \uff20 0),\n    begin funext, dsimp, rw lift_at_lift_merge, end,\n    rw [alls_succ] at H,\n    apply allE_var0,\n    rw [h,image_comp],\n    apply liftI, \n    exact n_ih H, },\nend\n\n-- def modus_tollens {\u0393} {\u03c6} (\u03c8: formula \u03c3) (H\u2081: \u0393 \u22a2 (\u03c6 \u2192' \u03c8)) (H\u2082: \u0393 \u22a2 \u00ac'\u03c8) : \u0393 \u22a2 \u00ac'\u03c6  :=\n-- begin\n--   apply impI,\n--   apply impE \u03c8,\n--   { apply impE_insert,\n--    assumption, },\n--   { apply weak1,\n--     assumption, },\n-- end\nend proof_term\n\nexport proof_term\n\n/-- Formal proof that there always exists an object of discourse. -/\ndef let_there_be_light : (\u2205 : set $ formula \u03c3) \u22a2 \u2203'(#0 =' #0) :=\nbegin\n  apply exI #0,\n  apply eqI,\nend\n\n/- Two variants of\n  \"All men are mortal.\n   Socrates is a man.\n   Therefore, Socrates is mortal.\" .   \n-/\n\nexample {\u0393:set $ formula \u03c3}{\u03c6 \u03c8 \u03c7}  (H\u2081: \u0393 \u22a2 \u2200'(\u03c6 \u2192' \u03c8))  (H\u2082: \u0393 \u22a2 \u2200'(\u03c8 \u2192' \u03c7)) : \u0393 \u22a2 \u2200' (\u03c6 \u2192' \u03c7) :=\nbegin\n  apply allI,\n  apply impI,\n  apply impE \u03c8,\n  { apply impE_insert,\n    apply allE' ((\u03c6 \u2192' \u03c8) \u2191 1 \uff20 1) #0,\n    rw \u2190formula.lift,\n    apply liftI,\n    exact H\u2081,\n    rw subst_var0_lift_by_1, },\n  { apply weak1,\n    apply allE' ((\u03c8 \u2192' \u03c7) \u2191 1 \uff20 1) #0,\n    rw \u2190formula.lift,\n    apply liftI,\n    exact H\u2082,\n    rw subst_var0_lift_by_1, },\nend\n\nexample {\u0393:set $ formula \u03c3}{\u03c6 \u03c8 \u03c7}  (H\u2081: \u0393 \u22a2 \u2200'(\u03c6 \u2192' \u03c8))  (H\u2082: \u0393 \u22a2 \u2200'(\u03c8 \u2192' \u03c7)) : \u0393 \u22a2 \u2200' (\u03c6 \u2192' \u03c7) :=\nbegin\n  apply allI,\n  apply impI,\n  apply impE \u03c8,\n  apply impE_insert,\n  swap,\n  apply weak1,\n  all_goals \n  { apply allE' (_ \u2191 1 \uff20 1) #0,\n    rw \u2190formula.lift,\n    apply liftI,\n    swap,\n    rw subst_var0_lift_by_1,\n    assumption, },\nend\n\nend proof_terms\n\nend fol", "meta": {"author": "maxkaske", "repo": "zfolean", "sha": "b74bb7accf01b25a6efb3af6b06538f98e7e5a6c", "save_path": "github-repos/lean/maxkaske-zfolean", "path": "github-repos/lean/maxkaske-zfolean/zfolean-b74bb7accf01b25a6efb3af6b06538f98e7e5a6c/src/fol.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.368678926391135}}
{"text": "import entanglement.basic\n\n/-\n/-- Thm. 1 in https://www.nature.com/articles/s41567-022-01873-9-/\ntheorem entanglement_theory_irreversible :\n\u2203 \u03c1, distillable_entanglement \u03c1 \u2260 entanglement_cost \u03c1 := sorry\n-/", "meta": {"author": "BassemSafieldeen", "repo": "qvantum-shannon-theory", "sha": "bc44469ed27b0ad0335b75a670fe7b76feea4550", "save_path": "github-repos/lean/BassemSafieldeen-qvantum-shannon-theory", "path": "github-repos/lean/BassemSafieldeen-qvantum-shannon-theory/qvantum-shannon-theory-bc44469ed27b0ad0335b75a670fe7b76feea4550/src/entanglement/entanglement_transformations_are_irreversible.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581626286833, "lm_q2_score": 0.5, "lm_q1q2_score": 0.36857908131434164}}
{"text": "variables x : Type \n\nconstant duh : \u2115\nconstant doh : \u2115 \nconstant dah : \u2115 \n\nlemma exercise : x = x := by refl  \n\nlemma another_exercise : 0 = 1 := sorry\n\nlemma yet_another_exercise: \u2115 \u00d7 \u2115 \u00d7 \u2115 := \u27e8 duh, doh, dah \u27e9\n\n", "meta": {"author": "mattrobball", "repo": "lean-autograding", "sha": "c004f4539968eaa5b3a42624308df9e9c9eeeca8", "save_path": "github-repos/lean/mattrobball-lean-autograding", "path": "github-repos/lean/mattrobball-lean-autograding/lean-autograding-c004f4539968eaa5b3a42624308df9e9c9eeeca8/src/assignment.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581510799252, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3685790755399626}}
{"text": "inductive Fam2 : Type \u2192 Type \u2192 Type 1 where\n  | any : Fam2 \u03b1 \u03b1\n  | nat : Nat \u2192 Fam2 Nat Nat\n\nexample (a : \u03b1) (x : Fam2 \u03b1 \u03b2) : \u03b2 :=\n  match \u03b1, \u03b2, a, x with\n  | ?\u03b1, ?\u03b2, ?a, Fam2.any   => _\n  | ?\u03b1, ?\u03b2, ?a, Fam2.nat n => _\n\nexample (a : \u03b1) (x : Fam2 \u03b1 \u03b2) : \u03b2 :=\n  match x with\n  | Fam2.any   => _\n  | Fam2.nat n => _\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/syntheticHolesAsPatterns.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3683505576733595}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\n\nimport category_theory.sites.grothendieck\n\n/-!\n# Grothendieck pretopologies\n\nDefinition and lemmas about Grothendieck pretopologies.\nA Grothendieck pretopology for a category `C` is a set of families of morphisms with fixed codomain,\nsatisfying certain closure conditions.\n\nWe show that a pretopology generates a genuine Grothendieck topology, and every topology has\na maximal pretopology which generates it.\n\nThe pretopology associated to a topological space is defined in `spaces.lean`.\n\n## Tags\n\ncoverage, pretopology, site\n\n## References\n\n* [nLab, *Grothendieck pretopology*](https://ncatlab.org/nlab/show/Grothendieck+pretopology)\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n* [Stacks, *00VG*](https://stacks.math.columbia.edu/tag/00VG)\n-/\n\nuniverses v u\nnoncomputable theory\n\nnamespace category_theory\n\nopen category_theory category limits presieve\n\nvariables {C : Type u} [category.{v} C] [has_pullbacks C]\n\nvariables (C)\n\n/--\nA (Grothendieck) pretopology on `C` consists of a collection of families of morphisms with a fixed\ntarget `X` for every object `X` in `C`, called \"coverings\" of `X`, which satisfies the following\nthree axioms:\n1. Every family consisting of a single isomorphism is a covering family.\n2. The collection of covering families is stable under pullback.\n3. Given a covering family, and a covering family on each domain of the former, the composition\n   is a covering family.\n\nIn some sense, a pretopology can be seen as Grothendieck topology with weaker saturation conditions,\nin that each covering is not necessarily downward closed.\n\nSee: https://ncatlab.org/nlab/show/Grothendieck+pretopology, or\nhttps://stacks.math.columbia.edu/tag/00VH, or [MM92] Chapter III, Section 2, Definition 2.\nNote that Stacks calls a category together with a pretopology a site, and [MM92] calls this\na basis for a topology.\n-/\n@[ext]\nstructure pretopology :=\n(coverings : \u03a0 (X : C), set (presieve X))\n(has_isos : \u2200 \u2983X Y\u2984 (f : Y \u27f6 X) [is_iso f], presieve.singleton f \u2208 coverings X)\n(pullbacks : \u2200 \u2983X Y\u2984 (f : Y \u27f6 X) S, S \u2208 coverings X \u2192 pullback_arrows f S \u2208 coverings Y)\n(transitive : \u2200 \u2983X : C\u2984 (S : presieve X)\n               (Ti : \u03a0 \u2983Y\u2984 (f : Y \u27f6 X), S f \u2192 presieve Y), S \u2208 coverings X \u2192\n               (\u2200 \u2983Y\u2984 f (H : S f), Ti f H \u2208 coverings Y) \u2192 S.bind Ti \u2208 coverings X)\n\nnamespace pretopology\n\ninstance : has_coe_to_fun (pretopology C) (\u03bb _, \u03a0 X : C, set (presieve X)) := \u27e8coverings\u27e9\n\nvariable {C}\n\ninstance : has_le (pretopology C) :=\n{ le := \u03bb K\u2081 K\u2082, (K\u2081 : \u03a0 (X : C), set (presieve X)) \u2264 K\u2082 }\n\nlemma le_def {K\u2081 K\u2082 : pretopology C} : K\u2081 \u2264 K\u2082 \u2194 (K\u2081 : \u03a0 (X : C), set (presieve X)) \u2264 K\u2082 := iff.rfl\n\nvariable (C)\n\ninstance : partial_order (pretopology C) :=\n{ le_refl := \u03bb K, le_def.mpr le_rfl,\n  le_trans := \u03bb K\u2081 K\u2082 K\u2083 h\u2081\u2082 h\u2082\u2083, le_def.mpr (le_trans h\u2081\u2082 h\u2082\u2083),\n  le_antisymm := \u03bb K\u2081 K\u2082 h\u2081\u2082 h\u2082\u2081, pretopology.ext _ _ (le_antisymm h\u2081\u2082 h\u2082\u2081),\n  ..pretopology.has_le }\n\ninstance : order_top (pretopology C) :=\n{ top :=\n  { coverings := \u03bb _, set.univ,\n    has_isos := \u03bb _ _ _ _, set.mem_univ _,\n    pullbacks := \u03bb _ _ _ _ _, set.mem_univ _,\n    transitive := \u03bb _ _ _ _ _, set.mem_univ _ },\n  le_top := \u03bb K X S hS, set.mem_univ _ }\n\ninstance : inhabited (pretopology C) := \u27e8\u22a4\u27e9\n\n/--\nA pretopology `K` can be completed to a Grothendieck topology `J` by declaring a sieve to be\n`J`-covering if it contains a family in `K`.\n\nSee <https://stacks.math.columbia.edu/tag/00ZC>, or [MM92] Chapter III, Section 2, Equation (2).\n-/\ndef to_grothendieck (K : pretopology C) : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2203 R \u2208 K X, R \u2264 (S : presieve _),\n  top_mem' := \u03bb X, \u27e8presieve.singleton (\ud835\udfd9 _), K.has_isos _, \u03bb _ _ _, \u27e8\u27e9\u27e9,\n  pullback_stable' := \u03bb X Y S g,\n  begin\n    rintro \u27e8R, hR, RS\u27e9,\n    refine \u27e8_, K.pullbacks g _ hR, _\u27e9,\n    rw [\u2190 sieve.sets_iff_generate, sieve.pullback_arrows_comm],\n    apply sieve.pullback_monotone,\n    rwa sieve.gi_generate.gc,\n  end,\n  transitive' :=\n  begin\n    rintro X S \u27e8R', hR', RS\u27e9 R t,\n    choose t\u2081 t\u2082 t\u2083 using t,\n    refine \u27e8_, K.transitive _ _ hR' (\u03bb _ f hf, t\u2082 (RS _ hf)), _\u27e9,\n    rintro Y _ \u27e8Z, g, f, hg, hf, rfl\u27e9,\n    apply t\u2083 (RS _ hg) _ hf,\n  end }\n\nlemma mem_to_grothendieck (K : pretopology C) (X S) :\n  S \u2208 to_grothendieck C K X \u2194 \u2203 R \u2208 K X, R \u2264 (S : presieve X) :=\niff.rfl\n\n/--\nThe largest pretopology generating the given Grothendieck topology.\n\nSee [MM92] Chapter III, Section 2, Equations (3,4).\n-/\ndef of_grothendieck (J : grothendieck_topology C) : pretopology C :=\n{ coverings := \u03bb X R, sieve.generate R \u2208 J X,\n  has_isos := \u03bb X Y f i, by exactI J.covering_of_eq_top (by simp),\n  pullbacks := \u03bb X Y f R hR,\n  begin\n    rw [set.mem_def, sieve.pullback_arrows_comm],\n    apply J.pullback_stable f hR,\n  end,\n  transitive := \u03bb X S Ti hS hTi,\n  begin\n    apply J.transitive hS,\n    intros Y f,\n    rintros \u27e8Z, g, f, hf, rfl\u27e9,\n    rw sieve.pullback_comp,\n    apply J.pullback_stable g,\n    apply J.superset_covering _ (hTi _ hf),\n    rintro Y g \u27e8W, h, g, hg, rfl\u27e9,\n    exact \u27e8_, h, _, \u27e8_, _, _, hf, hg, rfl\u27e9, by simp\u27e9,\n  end }\n\n/-- We have a galois insertion from pretopologies to Grothendieck topologies. -/\ndef gi : galois_insertion (to_grothendieck C) (of_grothendieck C) :=\n{ gc :=\n  \u03bb K J,\n  begin\n    split,\n    { intros h X R hR,\n      exact h _ \u27e8_, hR, sieve.le_generate R\u27e9 },\n    { rintro h X S \u27e8R, hR, RS\u27e9,\n      apply J.superset_covering _ (h _ hR),\n      rwa sieve.gi_generate.gc }\n  end,\n  le_l_u := \u03bb J X S hS, \u27e8S, J.superset_covering S.le_generate hS, le_rfl\u27e9,\n  choice := \u03bb x hx, to_grothendieck C x,\n  choice_eq := \u03bb _ _, rfl }\n\n/--\nThe trivial pretopology, in which the coverings are exactly singleton isomorphisms. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee <https://stacks.math.columbia.edu/tag/07GE>\n-/\ndef trivial : pretopology C :=\n{ coverings := \u03bb X S, \u2203 Y (f : Y \u27f6 X) (h : is_iso f), S = presieve.singleton f,\n  has_isos := \u03bb X Y f i, \u27e8_, _, i, rfl\u27e9,\n  pullbacks := \u03bb X Y f S,\n  begin\n    rintro \u27e8Z, g, i, rfl\u27e9,\n    refine \u27e8pullback g f, pullback.snd, _, _\u27e9,\n    { resetI, refine \u27e8\u27e8pullback.lift (f \u226b inv g) (\ud835\udfd9 _) (by simp), \u27e8_, by tidy\u27e9\u27e9\u27e9,\n      apply pullback.hom_ext,\n      { rw [assoc, pullback.lift_fst, \u2190pullback.condition_assoc],\n        simp },\n      { simp } },\n    { apply pullback_singleton },\n  end,\n  transitive :=\n  begin\n    rintro X S Ti \u27e8Z, g, i, rfl\u27e9 hS,\n    rcases hS g (singleton_self g) with \u27e8Y, f, i, hTi\u27e9,\n    refine \u27e8_, f \u226b g, _, _\u27e9,\n    { resetI, apply_instance },\n    ext W k,\n    split,\n    { rintro \u27e8V, h, k, \u27e8_\u27e9, hh, rfl\u27e9,\n      rw hTi at hh,\n      cases hh,\n      apply singleton.mk },\n    { rintro \u27e8_\u27e9,\n      refine bind_comp g presieve.singleton.mk _,\n      rw hTi,\n      apply presieve.singleton.mk }\n  end }\n\ninstance : order_bot (pretopology C) :=\n{ bot := trivial C,\n  bot_le := \u03bb K X R,\n  begin\n    rintro \u27e8Y, f, hf, rfl\u27e9,\n    exactI K.has_isos f,\n  end }\n\n/-- The trivial pretopology induces the trivial grothendieck topology. -/\nlemma to_grothendieck_bot : to_grothendieck C \u22a5 = \u22a5 :=\n(gi C).gc.l_bot\n\nend pretopology\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/pretopology.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3683505576733595}}
{"text": "import data.bitvec\nimport data.dlist\nimport util.logic\nimport util.data.option\nimport util.control.applicative\nimport util.control.monad.non_termination\nimport util.control.monad.state\nimport util.meta.tactic\nimport util.predicate\nimport separation.heap\nimport separation.program\n\nimport logic.basic\n\nuniverses u v w w'\n\nopen nat list function\n\nnamespace separation\nopen heap predicate\ndef hprop := pred' heap\n\ndef embed (p : Prop) : hprop :=\n\u27e8 \u03bb h, p \u2227 h = heap.emp \u27e9\n\nnotation `[| `p` |]` := embed p\n\ndef s_and (p q : hprop) : hprop :=\n\u27e8 \u03bb h, \u2203 hp\u2080 hp\u2081,\n         some h = part (some hp\u2080) (some hp\u2081) \u2227\n         p.apply hp\u2080 \u2227\n         q.apply hp\u2081 \u27e9\n\ninfix ` :*: `:55 := s_and\n\ndef emp : hprop :=\n\u27e8 \u03bb h, h = heap.emp \u27e9\n\n@[simp]\nprotected lemma apply_emp (hp : heap)\n: emp.apply hp \u2194 hp = heap.emp :=\nby refl\n\nlemma exists_congr_elim {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} {q : Prop} (x : \u03b1)\n  (h : \u2200 y, p y \u2194 y = x \u2227 q)\n: (\u2203 y, p y) \u2194 q :=\nby simp [h]\n\n@[simp]\nlemma s_and_emp (p : hprop)\n: p :*: emp = p :=\nby { cases p, funext i, simp_one_point [s_and], }\n\nlemma s_and_comm (p q : hprop)\n: p :*: q = q :*: p :=\nbegin\n  cases p, cases q, simp [s_and],\n  ext ihp,\n  split ; simp ; intros hp\u2080 hp\u2081 hh h\u2080 h\u2081\n  ; existsi [hp\u2081,hp\u2080]\n  ; rw hh\n  ; repeat { split <|> assumption <|> ac_refl },\nend\n\n@[simp]\nlemma emp_s_and (p : hprop)\n: emp :*: p = p :=\nby rw [s_and_comm,s_and_emp]\n\ndef points_to (p : \u2115) (val : word) : hprop :=\n\u27e8 \u03bb h, h = maplet p val \u27e9\n\ninfix ` \u21a6 `:60 := points_to\n\ndef points_to_multiple : \u2200 (p : \u2115), list word \u2192 hprop\n | _ [] := emp\n | p (x :: xs) := p \u21a6 x :*: points_to_multiple (p+1) xs\n\ninfix ` \u21a6* `:60 := points_to_multiple\n\nlemma points_to_multiple_iff_eq_heap_mk\n  (p : pointer) (vs : list word) (hp : heap)\n: (p \u21a6* vs).apply hp \u2194 hp = heap.mk p vs :=\nbegin\n  revert p hp,\n  induction vs with v vs ; intros p hp,\n  { simp [points_to_multiple], },\n  { simp [points_to_multiple,heap.mk],\n    split,\n    { simp [s_and],\n      intros hp\u2080 hp\u2081 H\u2080 H\u2081 H\u2082,\n      rw vs_ih at H\u2082,\n      change _ = _ at H\u2081,\n      subst hp\u2081, subst hp\u2080,\n      apply eq_part'_of_some_eq_part _ _ _ H\u2080, },\n    { intros h\u2080,\n      simp_one_point [s_and,(\u21a6)],\n      existsi [heap.mk (p + 1) vs],\n      rw vs_ih, split,\n      subst hp, rw [\u2190 some_part'],\n      refl, apply maplet_disjoint_heap_mk,\n      refl } }\nend\n\nstructure spec (r : Type u) :=\n  (pre : hprop)\n  (post : r \u2192 hprop)\n\ndef sat {\u03b1} (p : program \u03b1) (s : spec \u03b1) : Prop :=\n\u2200 (\u03c3 : hstate) hp\u2080 hp\u2081,\n   some \u03c3.heap = part (some hp\u2080) (some hp\u2081) \u2192\n   s.pre.apply hp\u2080 \u2192\n(\u2203 r \u03c3' hp', p.run \u03c3 ~> (r, \u03c3') \u2227\n            some \u03c3'.heap = part (some hp') (some hp\u2081) \u2227\n            (s.post r).apply hp')\n\nlemma s_and_part {hp\u2080 hp\u2081 : heap} {p\u2080 p\u2081 : hprop}\n  (h : hp\u2080 ## hp\u2081)\n  (Hp\u2080 : p\u2080.apply hp\u2080)\n  (Hp\u2081 : p\u2081.apply hp\u2081)\n: (p\u2080 :*: p\u2081).apply (part' hp\u2080 hp\u2081) :=\nbegin\n  cases p\u2080, cases p\u2081, simp [s_and],\n  split, split, split,\n  rw [part], ite_cases, contradiction,\n  split ; assumption,\nend\n\nlemma embed_eq_emp {p : Prop}\n  (hp : p)\n: [| p |] = emp :=\nby simp [embed,emp,eq_true_intro hp]\n\n@[simp]\nlemma embed_s_and_apply {p : Prop} {q : hprop}\n  (hp : heap)\n: ([| p |] :*: q).apply hp \u2194 p \u2227 q.apply hp :=\nby simp_one_point [s_and,embed]\n\ndef h_imp (p q : hprop) : Prop :=\np_entails p q\n\ninfix ` =*> `:41 := h_imp\n\nlemma s_and_assoc (p q r : hprop)\n: (p :*: q) :*: r = p :*: (q :*: r) :=\nbegin\n  cases p with p, cases q with q, cases r with r,\n  simp [s_and],\n  ext hp,\n  split ; simp,\n  { intros hp_pq hp_r Hd_pqr ,\n    intros hp_p hp_q Hpq Hp Hq,\n    intros Hr,\n    have Hd_qr : hp_q ## hp_r,\n    { apply disjoint_of_is_some_part,\n      apply is_some_of_is_some_part_right (some hp_p),\n      apply is_some_of_eq_some hp,\n      simp [Hd_pqr,Hpq], ac_refl },\n    existsi [hp_p,part' hp_q hp_r Hd_qr], -- ],\n    simp [Hd_pqr,Hpq],\n    split, ac_refl,\n    let hp_qr := part' hp_q hp_r,\n    existsi [Hp,hp_q,hp_r,rfl],\n    split ; assumption, },\n  { intros hp_p hp_qr Hd_pqr Hp,\n    intros hp_q hp_r H_qr Hq Hr,\n    have Hd_pq : hp_p ## hp_q,\n    { apply disjoint_of_is_some_part,\n      apply is_some_of_is_some_part_right (some hp_r),\n      apply is_some_of_eq_some hp,\n      simp [Hd_pqr,H_qr], ac_refl },\n    let hp_pq := part' hp_p hp_q,\n    existsi [hp_pq,hp_r],\n    split, simp [Hd_pqr,H_qr], ac_refl,\n    split,\n    { existsi [hp_p,hp_q], simp,\n      split ; assumption, },\n    assumption }\nend\n\n\nlemma embed_s_and_embed (p q : Prop)\n: [| p |] :*: [| q |] = [| p \u2227 q |] :=\nbegin\n  unfold embed emp s_and, apply congr_arg,\n  apply funext, intro, simp_one_point,\n  ac_refl\nend\n\n@[simp]\nlemma embed_s_and_self (p : Prop)\n: [| p |] :*: [| p |] = [| p |] :=\nby simp [embed_s_and_embed]\n\n@[refl]\nlemma s_imp_refl (p : hprop)\n: p =*> p :=\nby { lifted_pred }\n\n@[trans]\nlemma s_imp_trans {p : hprop} (q : hprop) {r : hprop}\n  (h\u2080 : p =*> q)\n  (h\u2081 : q =*> r)\n: p =*> r :=\nby { lifted_pred using h\u2080 h\u2081, intro ; solve_by_elim }\n\nlemma s_exists_intro  {\u03b1 : Type u}\n  {p : hprop} {q : \u03b1 \u2192 hprop} (x : \u03b1)\n  (h : p =*> q x)\n: p =*> \u2203\u2203 x, q x :=\nby { transitivity, apply h, apply p_exists_intro x, }\n\nlemma s_exists_elim  {\u03b1 : Type u}\n  {p : \u03b1 \u2192 hprop} {q : hprop} (x : \u03b1)\n  (h : \u2200 x, p x =*> q)\n: (\u2203\u2203 x, p x) =*> q :=\nbegin\n  simp [h_imp,p_exists_entails_eq_p_forall_entails],\n  apply h,\nend\n\nlemma s_imp_of_eq {p q : hprop}\n  (h : p = q)\n: p =*> q :=\nby rw h\n\n-- @[congr]\n-- lemma s_exists_congr {\u03b1 : Type u}\n--   {p q : \u03b1 \u2192 hprop}\n--   (h : \u2200 x, p x = q x)\n-- : hexists p = hexists q :=\n-- begin\n--   unfold hexists,\n--   congr,\n--   simp [h],\n-- end\n\nlemma s_exists_s_and_distr {\u03b1 : Type u}\n  (p : \u03b1 \u2192 hprop) (q : hprop)\n: (\u2203\u2203 x, p x) :*: q = (\u2203\u2203 x, p x :*: q) :=\nbegin\n  simp [s_and,p_exists],\n  congr, funext hp,\n  apply iff.to_eq,\n  split ; simp ; intros,\n  { split, split, split,\n    existsi a,\n    existsi a_1,\n    assumption },\n  { split, split,\n    existsi a,\n    split, split, assumption,\n    assumption },\nend\n\nlemma s_and_s_exists_distr {\u03b1 : Type u}\n  (p : \u03b1 \u2192 hprop) (q : hprop)\n: q :*: (\u2203\u2203 x, p x) = (\u2203\u2203 x, q :*: p x) :=\nbegin\n  rw [s_and_comm,s_exists_s_and_distr], congr,\n  funext hp, apply s_and_comm,\nend\n\nlemma s_and_s_imp_s_and\n  {p q r s : hprop}\n  (h\u2080 : p =*> r)\n  (h\u2081 : q =*> s)\n: p :*: q =*> r :*: s :=\nbegin\n  lifted_pred ,\n  simp only [s_and],\n  intros_mono,\n  apply and.imp\n  ; apply entails_of_forall_impl _ _ ; assumption,\nend\n\nlemma s_and_s_imp_s_and_left\n  (p : hprop) {q s : hprop}\n  (h\u2081 : q =*> s)\n: p :*: q =*> p :*: s :=\nby { apply s_and_s_imp_s_and _ h\u2081, refl }\n\n@[monotonic]\nlemma s_and_s_imp_s_and_right\n  {p : hprop} (q : hprop) {r : hprop}\n  (h\u2080 : p =*> r)\n: p :*: q =*> r :*: q :=\nby { apply s_and_s_imp_s_and h\u2080, refl }\n\ninstance : comm_monoid hprop :=\n{ mul := s_and\n, one := emp\n, mul_one := by { intros, simp [has_mul.mul] }\n, one_mul := by { intros, simp [has_mul.mul] }\n, mul_assoc := by { intros, simp [s_and_assoc] }\n, mul_comm := by { intros, apply s_and_comm, } }\n\n-- -- @[priority 1000]\n-- instance : is_left_id hprop s_and emp :=\n-- show is_left_id hprop has_mul.mul 1, by apply_instance\n\n-- -- @[priority 1000]\n-- instance : is_right_id hprop s_and emp :=\n-- show is_right_id hprop has_mul.mul 1, by apply_instance\n\nsection\n\nvariables {\u03b1 \u03b2 : Type}\nvariables {P : program \u03b1}\nvariable {P' : \u03b1 \u2192 program \u03b2}\nvariables {p p\u2080 p\u2081 q : hprop}\nvariables {r r\u2081 : \u03b1 \u2192 hprop}\nvariables {r' : \u03b2 \u2192 hprop}\nvariable {s : spec \u03b1}\n\nlemma framing_right (q : hprop)\n  (h : sat P { pre := p, post := r })\n: sat P { pre := p :*: q, post := \u03bb x, r x :*: q } :=\nbegin\n  unfold sat spec.pre spec.post,\n  introv Hpart Hpre,\n  cases Hpre with hp\u2082 Hpre, cases Hpre with hp\u2083 Hpre,\n  rw Hpre.left at Hpart,\n  cases Hpre with Hpre\u2080 Hpre\u2081, cases Hpre\u2081 with Hpre\u2081 Hpre\u2082,\n  simp [Hpart],\n  have h' : hp\u2083 ## hp\u2081,\n  { apply disjoint_of_is_some_part,\n    apply is_some_of_is_some_part_right (some hp\u2082),\n    apply is_some_of_eq_some \u03c3.heap,\n    simp [Hpart], ac_refl },\n  specialize h \u03c3 hp\u2082 (part' hp\u2083 hp\u2081),\n  simp [Hpart] at h, specialize h (by ac_refl) Hpre\u2081,\n  revert h,\n  intros_mono rr \u03c3',\n  simp, intros H_yield hp' H_p' Hr,\n  have Hd_p'_p\u2083 : hp' ## hp\u2083,\n  { apply disjoint_of_is_some_part,\n    apply is_some_of_is_some_part_right (some hp\u2081),\n    apply is_some_of_eq_some \u03c3'.heap,\n    simp [H_p'], ac_refl },\n  existsi [H_yield,part' hp' hp\u2083],-- [s_and_part _ Hr Hpre\u2082,H_yield],\n  simp [H_p'], split, ac_refl, apply s_and_part _ Hr Hpre\u2082,\nend\n\nlemma framing_left (q : hprop)\n  (h : sat P { pre := p, post := r })\n: sat P { pre := q :*: p, post := \u03bb x, q :*: r x } :=\nbegin\n  have h := framing_right q h,\n  apply eq.mp _ h,\n  apply congr_arg,\n  apply congr, ac_refl,\n  funext x,\n  ac_refl,\nend\n\nlemma bind_spec (r : \u03b1 \u2192 hprop)\n  (h  : sat P { pre := p, post := r })\n  (h' : \u2200 x, sat (P' x) { pre := r x, post := r' })\n: sat (P >>= P') { pre := p, post := r' } :=\nbegin\n  unfold sat, introv h\u2082 h\u2083,\n  specialize h \u03c3 hp\u2080 hp\u2081 h\u2082 h\u2083,\n  cases h with x h,\n  cases h with \u03c3' h,\n  cases h with hp' h,\n  cases h with Hspec\u2080 h,\n  cases h with Hspec\u2081 Hspec\u2082, simp at Hspec\u2082,\n  specialize h' x \u03c3' hp' hp\u2081 Hspec\u2081 Hspec\u2082,\n  revert h',\n  intros_mono r'' \u03c3'' hp'',\n  apply and.imp, intros h\u2080,\n  apply nonterm.yields_bind _ _ Hspec\u2080,\n  apply h\u2080, apply id,\nend\n\nlemma postcondition (r : \u03b1 \u2192 hprop)\n (Hspec : sat P { pre := p, post := r })\n (Hside : \u2200 x, r x = r\u2081 x)\n: sat P { pre := p, post := r\u2081 } :=\nbegin\n  apply eq.mp _ Hspec,\n  congr, funext x,\n  apply Hside,\nend\n\nlemma postcondition' (r : \u03b1 \u2192 hprop)\n (Hspec : sat P { pre := p, post := r })\n (Hside : \u2200 x, r x =*> r\u2081 x)\n: sat P { pre := p, post := r\u2081 } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  specialize Hspec _ _ _ H\u2080 H\u2081,\n  revert Hspec,\n  intros_mono x hp\u2080 hp\u2081 _ _,\n  apply entails_of_forall_impl,\n  apply Hside _,\nend\n\nlemma precondition (p : hprop)\n (Hspec : sat P { pre := p, post := r })\n (Hside : p = q)\n: sat P { pre := q, post := r } :=\nby { subst q, apply Hspec }\n\nlemma precondition' (p : hprop)\n (Hspec : sat P { pre := p, post := r })\n (Hside : q =*> p)\n: sat P { pre := q, post := r } :=\nbegin\n  revert Hspec, unfold sat,\n  intros_mono \u03c3 hp\u2080 hp\u2081 h _,\n  apply entails_of_forall_impl,\n  apply Hside ,\nend\n\nlemma bind_framing_left (p\u2081 : hprop)\n  (H\u2080 : sat P { pre := p\u2080, post := r })\n  (H\u2082 : p = p\u2080 :*: p\u2081)\n  (H\u2081 : \u2200 x, sat (P' x) { pre := r x :*: p\u2081, post := r' } )\n: sat (P >>= P') { pre := p, post := r' } :=\nbegin\n  apply precondition _ _ H\u2082.symm,\n  apply bind_spec (\u03bb x, r x :*: p\u2081),\n  { apply framing_right _ H\u2080, },\n  apply H\u2081,\nend\n\nlemma bind_framing_right (p\u2080 : hprop)\n  (H\u2080 : sat P { pre := p\u2081, post := r })\n  (H\u2082 : p = p\u2080 :*: p\u2081)\n  (H\u2081 : \u2200 x, sat (P' x) { pre := p\u2080 :*: r x, post := r' } )\n: sat (P >>= P') { pre := p, post := r' } :=\nbegin\n  simp [s_and_comm p\u2080 _] at H\u2081,\n  apply bind_framing_left p\u2080 H\u2080 _ H\u2081,\n  rw H\u2082, ac_refl\nend\n\nlemma s_exists_intro_pre {\u03b1 : Type u} {P : program \u03b2} {r : \u03b1 \u2192 hprop}\n  (H : \u2200 x, sat P { pre := r x, post := r' })\n: sat P { pre := (\u2203\u2203 x, r x), post := r' } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  cases H\u2081 with x H\u2081,\n  apply H x _ _ _ H\u2080 H\u2081,\nend\n\nlemma s_exists_elim_pre {\u03b1 : Type u} {P : program \u03b2} {r : \u03b1 \u2192 hprop} (x : \u03b1)\n  (H : sat P { pre := (\u2203\u2203 x, r x), post := r' })\n: sat P { pre := r x, post := r' } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  apply H _ _ _ H\u2080 \u27e8_,H\u2081\u27e9,\nend\n\nlemma s_exists_replace_pre {t : Type w} {\u03b1 : Type w'} {r : \u03b1 \u2192 hprop} {P : program \u03b2}\n  (f : t \u2192 \u03b1) (g : \u03b1 \u2192 t) (I : left_inverse f g)\n  (H : sat P { pre := (\u2203\u2203 x, r (f x)), post := r' })\n: sat P { pre := (\u2203\u2203 x, r x), post := r' } :=\nbegin\n  apply s_exists_intro_pre,\n  intro x,\n  rw \u2190 I x,\n  apply s_exists_elim_pre (g x) H\nend\n\nlemma s_exists_intro_post {P : program \u03b2} {b : \u03b1 \u2192 \u03b2 \u2192 hprop} (x : \u03b1)\n  (H : sat P { pre := p, post := b x })\n: sat P { pre := p, post := \u03bb r, \u2203\u2203 x, b x r } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  specialize H _ _ _ H\u2080 H\u2081,\n  revert H,\n  intros_mono y hp hp' _ _,\n  apply Exists.intro x\nend\n\nlemma adapt_spec\n  (h : sat P { pre := p\u2081, post := r\u2081 })\n  (Hpre : p = p\u2081)\n  (Hpost : \u2200 x, r x = r\u2081 x)\n: sat P { pre := p, post := \u03bb x, r x } :=\nby simp [Hpre,Hpost,h]\n\nlemma adapt_spec'\n  (h : sat P { pre := p\u2081, post := r\u2081 })\n  (Hpre : p =*> p\u2081)\n  (Hpost : \u2200 x, r x = r\u2081 x)\n: sat P { pre := p, post := \u03bb x, r x } :=\nbegin\n  apply precondition' _ _ Hpre,\n  apply postcondition r\u2081 h,\n  intro, symmetry, apply Hpost\nend\n\nlemma framing_spec'  (q : hprop)\n  (h : sat P { pre := p\u2081, post := r\u2081 })\n  (Hpre : p =*> p\u2081 :*: q)\n  (Hpost : \u2200 x, r\u2081 x :*: q =*> r x)\n: sat P { pre := p, post := \u03bb x, r x } :=\nbegin\n  apply precondition' _ _ Hpre,\n  apply postcondition' _ _ Hpost,\n  apply framing_right _ h,\nend\n\nlemma framing_spec  (q : hprop)\n  (h : sat P { pre := p\u2081, post := r\u2081 })\n  (Hpre : p = p\u2081 :*: q)\n  (Hpost : \u2200 x, r x = r\u2081 x :*: q)\n: sat P { pre := p, post := \u03bb x, r x } :=\nbegin\n  simp [Hpre,Hpost],\n  apply framing_right _ h\nend\n\nlemma context_left (p : Prop)\n (H : p \u2192 sat P { pre := q, post := r })\n: sat P { pre := [| p |] :*: q, post := r } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  simp at H\u2081,\n  apply H H\u2081.1 _ _ _ H\u2080 H\u2081.2,\nend\n\nlemma context_right (p : Prop)\n (H : p \u2192 sat P { pre := q, post := r })\n: sat P { pre := q :*: [| p |], post := r } :=\nbegin\n  rw s_and_comm,\n  apply context_left _ H\nend\n\nlemma return.spec {\u03b1 : Type} (x : \u03b1) (p : \u03b1 \u2192 hprop)\n: sat (return x) { pre := p x, post := \u03bb y, p y } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  existsi [x,\u03c3,hp\u2080,nonterm.pure_yields _,H\u2080],\n  assumption\nend\n\nlemma return.spec' {\u03b1 : Type} (x : \u03b1) (p : hprop)\n: sat (return x) { pre := p, post := \u03bb _, p } :=\nreturn.spec _ _\n\nlemma read.spec (p : pointer) (v : word)\n: sat (read p) { pre := p \u21a6 v\n               , post := \u03bb r, [| r = v |] :*: p \u21a6 v } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  existsi [v,\u03c3,hp\u2080],\n  simp at *, simp [H\u2080,H\u2081],\n  simp [read,return,has_bind.bind,state_t.bind],\n  apply nonterm.yields_bind,\n  apply nonterm.pure_yields,\n  simp [points_to] at H\u2081,\n  simp [state_t.bind._match_1,state_t.lift],\n  simp [H\u2081] at H\u2080,\n  rw [\u2190 opt_apl_some (\u03c3.heap) p,H\u2080,opt_apl_part_maplet],\n  simp [pure_bind],\n  apply nonterm.pure_yields,\n  apply disjoint_of_is_some_part,\n  apply is_some_of_eq_some \u03c3.heap H\u2080,\nend\n\nlemma read_head.spec (p : pointer) (v : word) (vs : list word)\n: sat (read p) { pre := p \u21a6* v :: vs\n               , post := \u03bb r, [| r = v |] :*: p \u21a6* v :: vs } :=\nbegin\n  simp [points_to_multiple,s_and_assoc],\n  apply framing_spec (p + 1 \u21a6* vs) (read.spec p v),\n  { ac_refl },\n  intro, ac_refl,\nend\n\nlemma read_nth.spec (p : pointer) (i : \u2115) (vs : list word)\n  (H : i < vs.length)\n: sat (read_nth p i _ H)\n      { pre := p \u21a6* vs,\n        post := \u03bb r, [| r = nth_le vs i H |] :*: p \u21a6* vs } :=\nbegin\n  revert p i,\n  induction vs with x xs ; introv,\n  { cases not_lt_zero _ H },\n  cases i with i,\n  { simp [read_nth,points_to_multiple],\n    apply framing_spec (p+1 \u21a6* xs),\n    apply read.spec _ x, refl,\n    intros r,\n    rw [\u2190 s_and_assoc], },\n  { simp [read_nth,points_to_multiple],\n    rw [add_succ,\u2190 succ_add],\n    apply framing_spec (p \u21a6 x),\n    { specialize vs_ih (p+1) i (lt_of_succ_lt_succ H),\n      simp [add_one,read_nth] at vs_ih,\n      rw [add_comm] at vs_ih, apply vs_ih },\n    { rw s_and_comm },\n    intro r, rw [s_and_comm  (p \u21a6 x), \u2190 s_and_assoc],\n    congr, }\nend\n\nlemma write.spec (p : pointer) (v v' : word)\n: sat (write p v') { pre := p \u21a6 v, post := \u03bb r, p \u21a6 v' } :=\nbegin\n  intros _ _ _ H\u2080 H\u2081,\n  unfold write,\n  existsi (),\n  split, existsi (hp\u2080.insert p v'),\n  simp ,\n  constructor_matching* (_ \u2227 _),\n  show heap.insert hp\u2080 p v' \u22a8 p \u21a6 v',\n  { change _ = _ at H\u2081,\n    change _ = _, rw H\u2081,\n    funext x,\n    by_cases h : p = x ;\n    simp [heap.insert,maplet,h], },\n  show _ ~> _,\n  { rw [dif_pos,state_t.put],\n    apply nonterm.pure_yields,\n    simp [points_to] at H\u2081,\n    rw [\u2190 opt_apl_some \u03c3.heap,H\u2080,H\u2081,opt_apl_part_maplet ],\n    exact rfl,\n    subst hp\u2080,\n    apply disjoint_of_part_eq_some H\u2080 },\n  { have H := disjoint_of_part_eq_some H\u2080,\n    rw [\u2190 some_part',part'_insert _ _ _ _ _ H],\n    rw \u2190 some_part' _ _ H at H\u2080, injection H\u2080,\n    simp, rw [h_1],\n    { change _ = _ at H\u2081, subst hp\u2080,\n      rw maplet_insert_disjoint_iff,\n      assumption }, }\nend\n\ndef replace {\u03b1} (f : \u03b1 \u2192 \u03b1) : \u2115 \u2192 list \u03b1 \u2192 list \u03b1\n  | i [] := []\n  | 0 (x :: xs) := f x :: xs\n  | (succ i) (x :: xs) := x :: replace i xs\n\nlemma write_head.spec (p : pointer) (v v' : word) (vs : list word)\n: sat (write p v') { pre := p \u21a6* v :: vs, post := \u03bb _, p \u21a6* v' :: vs } :=\nbegin\n  simp [(\u21a6*)],\n  apply framing_right,\n  apply write.spec,\nend\n\nlemma write_nth.spec (p : pointer) (v' : word) (i : \u2115) (vs : list word)\n  (H : i < vs.length)\n: sat (write_nth p i _ v' H)\n   { pre := p \u21a6* vs\n   , post := \u03bb _, p \u21a6* replace (const _ v') i vs } :=\nbegin\n  revert p i,\n  induction vs with v vs ; intros p i H,\n  { cases not_lt_zero _ H },\n  cases i with i,\n  { simp [write_nth],\n    apply write_head.spec },\n  { simp [write_nth,add_succ,replace,(\u21a6*)],\n    rw \u2190 succ_add,\n    apply framing_left,\n    apply vs_ih,\n    apply lt_of_succ_lt_succ H, }\nend\n\nlemma modify.spec (p : pointer) (f : word \u2192 word) (v : word)\n: sat (modify p f) { pre := p \u21a6 v, post := \u03bb _, p \u21a6 f v } :=\nbegin\n  unfold modify,\n  apply bind_spec _ (read.spec p v),\n  intro x, simp [function.comp],\n  apply context_left,\n  intro, subst x,\n  apply write.spec\nend\n\nlemma modify_head.spec (p : pointer) (f : word \u2192 word) (v : word) (vs : list word)\n: sat (modify p f) { pre := p \u21a6* v :: vs, post := \u03bb _, p \u21a6* f v :: vs } :=\nbegin\n  unfold points_to_multiple,\n  apply framing_right,\n  apply modify.spec,\nend\n\nlemma modify_nth.spec (p : pointer) (f : word \u2192 word) (i : \u2115) (vs : list word)\n  (H : i < vs.length)\n: sat (modify (p+i) f) { pre := p \u21a6* vs, post := \u03bb _, p \u21a6* replace f i vs } :=\nbegin\n  revert i p,\n  induction vs with v vs\n  ; intros i p H,\n  { cases nat.not_lt_zero _ H, },\n  cases i,\n  case zero\n  { simp [replace],\n    apply modify_head.spec },\n  case succ : i\n  { simp [replace,points_to_multiple],\n    apply framing_left,\n    rw [add_succ,add_succ,\u2190 succ_add,add_zero],\n    apply vs_ih i (succ p),\n    apply lt_of_succ_lt_succ H, }\nend\n\nlemma alloc.spec (vs : list word)\n: sat (alloc vs) { pre := emp, post := \u03bb r, [| r \u2260 0 |] :*: r \u21a6* vs } :=\nbegin\n  simp [alloc,sat],\n  intros _ _ _ H\u2080 H\u2081,\n  -- simp [state_t.get_bind],\n  existsi \u03c3.next+1,\n  existsi { hstate . .. }, split,\n  { apply nonterm.pure_yields, },\n  have : \u00ac\u03c3.next+1 = 0,\n  { apply ne_of_gt,\n    apply zero_lt_succ, },\n  { simp [hstate.heap,this], simp [H\u2081] at H\u2080,\n    let p := \u03c3.next+1,\n    have h' : heap.mk p vs ## \u03c3.heap,\n    { apply is_free, apply le_succ, },\n    subst hp\u2081,\n    simp_one_point [points_to_multiple_iff_eq_heap_mk,some_insert_left_eq_part h'] },\nend\n\nlemma alloc1.spec (v : word)\n: sat (alloc1 v) { pre := emp, post := \u03bb r, [| r \u2260 0 |] :*: r \u21a6 v } :=\nbegin\n  have h := alloc.spec [v],\n  unfold points_to_multiple at h,\n  simp [s_and_emp] at h,\n  apply h\nend\n\nlemma free.spec (p : pointer) (n : \u2115) (vs : list word)\n  (h : n = length vs)\n: sat (free p n) { pre := p \u21a6* vs, post := \u03bb r, emp } :=\nbegin\n  unfold sat,\n  introv H\u2080 H\u2081,\n  existsi [()],\n  split, existsi heap.emp,\n  simp [free,nonterm.pure_yields],\n  split, apply nonterm.pure_yields,\n  simp, subst n,\n  rw points_to_multiple_iff_eq_heap_mk at H\u2081,\n  rw H\u2081 at H\u2080,\n  have H\u2082 := eq_part'_of_some_eq_part _ _ _ H\u2080,\n  rw [H\u2082,delete_part'_heap_mk],\nend\n\nlemma free1.spec (p : pointer) (v : word)\n: sat (free1 p) { pre := p \u21a6 v, post := \u03bb r, emp } :=\nbegin\n  have h := free.spec p 1 [v] rfl,\n  simp [points_to_multiple] at h,\n  apply h,\nend\n\nlemma copy.spec (p q : pointer) (v\u2080 v\u2081 : word)\n: sat (copy p q) { pre := p \u21a6 v\u2080 :*: q \u21a6 v\u2081\n                 , post := \u03bb _, p \u21a6 v\u2081 :*: q \u21a6 v\u2081 } :=\nbegin\n  apply bind_spec (\u03bb r, p \u21a6 v\u2080 :*: ([| r = v\u2081 |] :*: q \u21a6 v\u2081)),\n  { apply framing_left, apply read.spec },\n  { intro r, simp,\n    apply precondition (p \u21a6 v\u2080 :*: q \u21a6 v\u2081 :*: [| r = v\u2081 |]),\n    { apply context_right, intro, subst r,\n      apply framing_right,\n      apply write.spec },\n    { ac_refl } }\nend\n\nend\n\nend separation\n", "meta": {"author": "unitb", "repo": "separation-logic", "sha": "bdde6fc8f16fd43932aea9827d6c63cadd91c2e8", "save_path": "github-repos/lean/unitb-separation-logic", "path": "github-repos/lean/unitb-separation-logic/separation-logic-bdde6fc8f16fd43932aea9827d6c63cadd91c2e8/src/separation/specification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.36829040529008694}}
{"text": "import logic.function.basic\nimport tactic\n\n\nlemma aux_1\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (g : \u03b1 \u2192 \u03b2)\n  (f f' : \u03b1 \u2192 \u03b1)\n  (x : \u03b1)\n  (a : \u03b2)\n  (h1 : f' \u2218 f = id) :\n  (function.update g (f x) a) \u2218 f = function.update (g \u2218 f) x a :=\nbegin\n  have s1 : function.left_inverse f' f,\n  exact congr_fun h1,\n\n  apply function.update_comp_eq_of_injective,\n  exact function.left_inverse.injective s1,\nend\n\n\nlemma aux_2\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (g : \u03b1 \u2192 \u03b2)\n  (f f' : \u03b1 \u2192 \u03b1)\n  (x : \u03b1)\n  (a : \u03b2)\n  (h1 : f' \u2218 f = id)\n  (h2 : f \u2218 f' = id) :\n  (function.update g x a) \u2218 f = function.update (g \u2218 f) (f' x) a :=\nbegin\n  rewrite <- aux_1 g f f' (f' x) a h1,\n  congr,\n  rewrite <- function.comp_app f f' x,\n  rewrite h2,\n  exact id.def x,\nend\n\n\nlemma aux_3\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f g : \u03b1 \u2192 \u03b2)\n  (x : \u03b1)\n  (h1 : \u2200 (y : \u03b1), \u00ac y = x \u2192 f y = g y) :\n  function.update f x (g x) = g :=\nbegin\n  apply funext, intros y,\n  by_cases c1 : y = x,\n  {\n    rewrite c1,\n    simp only [function.update_same],\n  },\n  {\n    simp only [function.update_noteq c1],\n    exact h1 y c1,\n  },\nend\n\n\nlemma nodup_eq_len_imp_eqv\n  {\u03b1 : Type}\n  (l1 l2 : list \u03b1)\n  (h1 : l1.length = l2.length)\n  (h2 : l1.nodup)\n  (h3 : l2.nodup) :\n  \u2203 (f : \u03b1 \u2243 \u03b1), l1.map f = l2 :=\nbegin\n  classical,\n  induction l1 with x l1 ih generalizing l2,\n  { rw [list.length, eq_comm] at h1,\n    cases list.eq_nil_of_length_eq_zero h1,\n    use equiv.refl _,\n    refl, },\n  cases l2 with y l2,\n  { cases h1 },\n  simp only [list.length, add_left_inj] at h1,\n  simp only [list.nodup_cons] at h2 h3,\n  obtain \u27e8f, hf\u27e9 := ih h2.2 l2 h1 h3.2,\n  have : f x \u2209 l2 := by simp [\u2190 hf, h2],\n  use f.trans (equiv.swap y (f x)),\n  simp only [list.map, equiv.coe_trans, function.comp_app, equiv.swap_apply_right,\n    eq_self_iff_true, true_and],\n  rw \u2190 hf,\n  refine list.map_congr (\u03bb z hz, _),\n  simp only [function.comp_app],\n  apply equiv.swap_apply_of_ne_of_ne,\n  { rintro rfl,\n    apply h3.1,\n    simpa only [\u2190hf, list.mem_map, embedding_like.apply_eq_iff_eq, exists_eq_right] using hz },\n  { intro h,\n    apply this,\n    rw [\u2190 h, \u2190 hf],\n    simpa using hz }\nend\n\n\nlemma list.nth_le_mem_zip\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (l1 : list \u03b1)\n  (l2 : list \u03b2)\n  (n : \u2115)\n  (h1 : n < l1.length)\n  (h2 : n < l2.length) :\n  ((l1.nth_le n h1, l2.nth_le n h2) \u2208 l1.zip l2) :=\nbegin\n  have s1 : n < (l1.zip l2).length,\n  simp only [list.length_zip, lt_min_iff],\n  split,\n  {\n    exact h1,\n  },\n  {\n    exact h2,\n  },\n\n  have s2 : (list.zip l1 l2).nth_le n s1 = (l1.nth_le n h1, l2.nth_le n h2),\n  exact list.nth_le_zip,\n\n  rewrite <- s2,\n  exact (list.zip l1 l2).nth_le_mem n s1,\nend\n\n\nlemma list.map_fst_zip_is_prefix\n  {\u03b1 \u03b2 : Type}\n  (l1 : list \u03b1)\n  (l2 : list \u03b2) :\n  list.map prod.fst (l1.zip l2) <+: l1 :=\nbegin\n  induction l1 generalizing l2,\n  case list.nil : l2\n  {\n    simp only [list.zip_nil_left, list.map_nil],\n  },\n  case list.cons : l1_hd l1_tl l1_ih l2\n  {\n    induction l2,\n    case list.nil\n    {\n      unfold list.is_prefix,\n      apply exists.intro (l1_hd :: l1_tl),\n      simp only [list.zip_nil_right, list.map_nil, list.nil_append, eq_self_iff_true, and_self],\n    },\n    case list.cons : l2_hd l2_tl l2_ih\n    {\n      simp only [list.map, list.zip_cons_cons],\n      rewrite list.prefix_cons_inj,\n      exact l1_ih l2_tl,\n    },\n  },\nend\n\n\nlemma list.map_fst_zip_nodup\n  {\u03b1 \u03b2 : Type}\n  (l1 : list \u03b1)\n  (l2 : list \u03b2)\n  (h1 : l1.nodup) :\n  (list.map prod.fst (l1.zip l2)).nodup :=\nbegin\n  have s1 : list.map prod.fst (l1.zip l2) <+ l1,\n  apply list.is_prefix.sublist,\n  exact l1.map_fst_zip_is_prefix l2,\n\n  exact list.nodup.sublist s1 h1,\nend\n\n\ndef function.update_list\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f : \u03b1 \u2192 \u03b2) :\n  list (\u03b1 \u00d7 \u03b2) \u2192 \u03b1 \u2192 \u03b2\n| [] := f\n| (hd :: tl) := function.update (function.update_list tl) hd.fst hd.snd\n\n#eval function.update_list (fun (n : \u2115), n) [(0,1), (3,2), (0,2)] 0\n\n\nlemma function.update_list_mem\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f : \u03b1 \u2192 \u03b2)\n  (l : list (\u03b1 \u00d7 \u03b2))\n  (x : \u03b1 \u00d7 \u03b2)\n  (h1 : list.nodup (list.map prod.fst l))\n  (h2 : x \u2208 l) :\n  function.update_list f l x.fst = x.snd :=\nbegin\n  induction l,\n  case list.nil\n  {\n    simp only [list.not_mem_nil] at h2,\n    contradiction,\n  },\n  case list.cons : hd tl ih\n  {\n    simp only [list.map, list.nodup_cons, list.mem_map, prod.exists,\n      exists_and_distrib_right, exists_eq_right, not_exists] at h1,\n    cases h1,\n\n    simp only [list.mem_cons_iff] at h2,\n\n    unfold function.update_list,\n    cases h2,\n    {\n      rewrite h2,\n      simp only [function.update_same],\n    },\n    {\n      have s1 : \u00ac x.fst = hd.fst,\n      intro contra,\n      apply h1_left x.snd,\n      rewrite <- contra,\n      simp only [prod.mk.eta],\n      exact h2,\n\n      simp only [function.update_noteq s1],\n      exact ih h1_right h2,\n    }\n  },\nend\n\n\nlemma function.update_list_not_mem\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f : \u03b1 \u2192 \u03b2)\n  (l : list (\u03b1 \u00d7 \u03b2))\n  (x : \u03b1)\n  (h1 : x \u2209 list.map prod.fst l) :\n  function.update_list f l x = f x :=\nbegin\n  induction l,\n  case list.nil\n  {\n    unfold function.update_list,\n  },\n  case list.cons : hd tl ih\n  {\n    simp only [list.map, list.mem_cons_iff, list.mem_map, prod.exists,\n      exists_and_distrib_right, exists_eq_right] at h1,\n    push_neg at h1,\n    cases h1,\n\n    unfold function.update_list,\n    simp only [function.update_noteq h1_left],\n    apply ih,\n    simp only [list.mem_map, prod.exists, exists_and_distrib_right, exists_eq_right, not_exists],\n    exact h1_right,\n  },\nend\n\n\nlemma function.update_list_mem_ext\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f g : \u03b1 \u2192 \u03b2)\n  (l : list (\u03b1 \u00d7 \u03b2))\n  (x : \u03b1)\n  (h1 : x \u2208 list.map prod.fst l) :\n  function.update_list f l x = function.update_list g l x :=\nbegin\n  induction l,\n  case list.nil\n  {\n    simp only [list.map_nil, list.not_mem_nil] at h1,\n    contradiction,\n  },\n  case list.cons : hd tl ih\n  {\n    simp only [list.map, list.mem_cons_iff] at h1,\n\n    unfold function.update_list,\n    by_cases c1 : x = hd.fst,\n    {\n      rewrite c1,\n      simp only [function.update_same],\n    },\n    {\n      simp only [function.update_noteq c1],\n      cases h1,\n      {\n        contradiction,\n      },\n      {\n        exact ih h1,\n      }\n    },\n  },\nend\n\n\nlemma function.update_list_zip_mem_ext\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f g : \u03b1 \u2192 \u03b2)\n  (l1 : list \u03b1)\n  (l2 : list \u03b2)\n  (x : \u03b1)\n  (h1 : l1.length \u2264 l2.length)\n  (h2 : x \u2208 l1) :\n  function.update_list f (l1.zip l2) x =\n    function.update_list g (l1.zip l2) x :=\nbegin\n  have s1 : x \u2208 list.map prod.fst (l1.zip l2),\n  rewrite list.map_fst_zip l1 l2 h1,\n  exact h2,\n\n  exact function.update_list_mem_ext f g (list.zip l1 l2) x s1,\nend\n\n\nlemma function.update_list_zip_map_mem_ext\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (l1 l2 : list \u03b1)\n  (f g h : \u03b1 \u2192 \u03b2)\n  (x : \u03b1)\n  (h1 : l1.length \u2264 l2.length)\n  (h2 : x \u2208 l1) :\n  function.update_list f (l1.zip (list.map h l2)) x =\n    function.update_list g (l1.zip (list.map h l2)) x :=\nbegin\n  have s1 : l1.length \u2264 (list.map h l2).length,\n  simp only [list.length_map],\n  exact h1,\n\n  exact function.update_list_zip_mem_ext f g l1 (list.map h l2) x s1 h2,\nend\n\n\nlemma function.update_list_zip_map_mem_ext'\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (l1 l2 : list \u03b1)\n  (f g h h' : \u03b1 \u2192 \u03b2)\n  (x : \u03b1)\n  (h1 : \u2200 (y : \u03b1), y \u2208 l2 \u2192 h y = h' y)\n  (h2 : l1.length \u2264 l2.length)\n  (h3 : x \u2208 l1) :\n  function.update_list f (l1.zip (list.map h l2)) x =\n    function.update_list g (l1.zip (list.map h' l2)) x :=\nbegin\n  have s1 : list.map h l2 = list.map h' l2,\n  rewrite list.map_eq_map_iff,\n  exact h1,\n\n  rewrite s1,\n  exact function.update_list_zip_map_mem_ext l1 l2 f g h' x h2 h3,\nend\n\n\nlemma function.update_list_zip_map_mem\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f g : \u03b1 \u2192 \u03b2)\n  (l : list \u03b1)\n  (x : \u03b1)\n  (h1 : x \u2208 l) :\n  function.update_list f (l.zip (list.map g l)) x = g x :=\nbegin\n  induction l,\n  case list.nil\n  {\n    simp only [list.not_mem_nil] at h1,\n    contradiction,\n  },\n  case list.cons : hd tl ih\n  {\n    simp only [list.mem_cons_iff] at h1,\n\n    simp only [list.map, list.zip_cons_cons],\n    unfold function.update_list,\n    by_cases c1 : x = hd,\n    {\n      rewrite c1,\n      simp only [function.update_same],\n    },\n    {\n      cases h1,\n      {\n        contradiction,\n      },\n      {\n        simp only [function.update_noteq c1],\n        exact ih h1,\n      }\n    }\n  },\nend\n\n\nlemma function.update_list_update\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f g : \u03b1 \u2192 \u03b2)\n  (l1 l2 : list \u03b1)\n  (v : \u03b1)\n  (a : \u03b2)\n  (x : \u03b1)\n  (h1 : \u2200 (y : \u03b1), y \u2208 l2 \u2192 \u00ac y = v)\n  (h2 : l1.length \u2264 l2.length)\n  (h3 : x \u2208 l1) :\n  function.update_list g (l1.zip (list.map (function.update f v a) l2)) x =\n    function.update_list f (l1.zip (list.map f l2)) x:=\nbegin\n  have s1 : \u2200 (y : \u03b1), y \u2208 l2 \u2192 function.update f v a y = f y,\n  intros y a1,\n  exact function.update_noteq (h1 y a1) a f,\n\n  exact function.update_list_zip_map_mem_ext' l1 l2 g f (function.update f v a) f x s1 h2 h3,\nend\n\n\nlemma function.update_list_nth_le_zip\n  {\u03b1 \u03b2 : Type}\n  [decidable_eq \u03b1]\n  (f : \u03b1 \u2192 \u03b2)\n  (l1 : list \u03b1)\n  (l2 : list \u03b2)\n  (n : \u2115)\n  (h1 : n < l1.length)\n  (h2 : n < l2.length)\n  (h3 : l1.nodup) :\n  (function.update_list f (l1.zip l2)) (l1.nth_le n h1) = l2.nth_le n h2 :=\nbegin\n  have s1 : (list.map prod.fst (l1.zip l2)).nodup,\n  exact list.map_fst_zip_nodup l1 l2 h3,\n\n  have s2 : (l1.nth_le n h1, l2.nth_le n h2) \u2208 l1.zip l2,\n  exact list.nth_le_mem_zip l1 l2 n h1 h2,\n\n  exact function.update_list_mem f (l1.zip l2) (l1.nth_le n h1, l2.nth_le n h2) s1 s2,\nend\n\n\ndef list.option_to_option_list {\u03b1 : Type} [decidable_eq \u03b1] (l : list (option \u03b1)) : option (list \u03b1) :=\n  if none \u2208 l then none else some l.reduce_option\n", "meta": {"author": "pthomas505", "repo": "lean3", "sha": "eb449be2b9a92becda4be38aac76e080194e3f7c", "save_path": "github-repos/lean/pthomas505-lean3", "path": "github-repos/lean/pthomas505-lean3/lean3-eb449be2b9a92becda4be38aac76e080194e3f7c/src/metalogic/mm0/aux.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.36829040529008694}}
{"text": "/-\nCopyright (c) 2021 Anne Baanen. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Anne Baanen\n-/\n\nimport data.fun_like.basic\n\n/-!\n# Typeclass for a type `F` with an injective map to `A \u21aa B`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis typeclass is primarily for use by embeddings such as `rel_embedding`.\n\n## Basic usage of `embedding_like`\n\nA typical type of embedding should be declared as:\n```\nstructure my_embedding (A B : Type*) [my_class A] [my_class B] :=\n(to_fun : A \u2192 B)\n(injective' : function.injective to_fun)\n(map_op' : \u2200 {x y : A}, to_fun (my_class.op x y) = my_class.op (to_fun x) (to_fun y))\n\nnamespace my_embedding\n\nvariables (A B : Type*) [my_class A] [my_class B]\n\n-- This instance is optional if you follow the \"Embedding class\" design below:\ninstance : embedding_like (my_embedding A B) A B :=\n{ coe := my_embedding.to_fun,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective' }\n\n/-- Helper instance for when there's too many metavariables to directly\napply `fun_like.to_coe_fn`. -/\ninstance : has_coe_to_fun (my_embedding A B) (\u03bb _, A \u2192 B) := \u27e8my_embedding.to_fun\u27e9\n\n@[simp] lemma to_fun_eq_coe {f : my_embedding A B} : f.to_fun = (f : A \u2192 B) := rfl\n\n@[ext] theorem ext {f g : my_embedding A B} (h : \u2200 x, f x = g x) : f = g := fun_like.ext f g h\n\n/-- Copy of a `my_embedding` with a new `to_fun` equal to the old one. Useful to fix definitional\nequalities. -/\nprotected def copy (f : my_embedding A B) (f' : A \u2192 B) (h : f' = \u21d1f) : my_embedding A B :=\n{ to_fun := f',\n  injective' := h.symm \u25b8 f.injective',\n  map_op' := h.symm \u25b8 f.map_op' }\n\nend my_embedding\n```\n\nThis file will then provide a `has_coe_to_fun` instance and various\nextensionality and simp lemmas.\n\n## Embedding classes extending `embedding_like`\n\nThe `embedding_like` design provides further benefits if you put in a bit more work.\nThe first step is to extend `embedding_like` to create a class of those types satisfying\nthe axioms of your new type of morphisms.\nContinuing the example above:\n\n```\nsection\nset_option old_structure_cmd true\n\n/-- `my_embedding_class F A B` states that `F` is a type of `my_class.op`-preserving embeddings.\nYou should extend this class when you extend `my_embedding`. -/\nclass my_embedding_class (F : Type*) (A B : out_param $ Type*) [my_class A] [my_class B]\n  extends embedding_like F A B :=\n(map_op : \u2200 (f : F) (x y : A), f (my_class.op x y) = my_class.op (f x) (f y))\n\nend\n\n@[simp] lemma map_op {F A B : Type*} [my_class A] [my_class B] [my_embedding_class F A B]\n  (f : F) (x y : A) : f (my_class.op x y) = my_class.op (f x) (f y) :=\nmy_embedding_class.map_op\n\n-- You can replace `my_embedding.embedding_like` with the below instance:\ninstance : my_embedding_class (my_embedding A B) A B :=\n{ coe := my_embedding.to_fun,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective',\n  map_op := my_embedding.map_op' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThe second step is to add instances of your new `my_embedding_class` for all types extending\n`my_embedding`.\nTypically, you can just declare a new class analogous to `my_embedding_class`:\n\n```\nstructure cooler_embedding (A B : Type*) [cool_class A] [cool_class B]\n  extends my_embedding A B :=\n(map_cool' : to_fun cool_class.cool = cool_class.cool)\n\nsection\nset_option old_structure_cmd true\n\nclass cooler_embedding_class (F : Type*) (A B : out_param $ Type*) [cool_class A] [cool_class B]\n  extends my_embedding_class F A B :=\n(map_cool : \u2200 (f : F), f cool_class.cool = cool_class.cool)\n\nend\n\n@[simp] lemma map_cool {F A B : Type*} [cool_class A] [cool_class B] [cooler_embedding_class F A B]\n  (f : F) : f cool_class.cool = cool_class.cool :=\nmy_embedding_class.map_op\n\n-- You can also replace `my_embedding.embedding_like` with the below instance:\ninstance : cool_embedding_class (cool_embedding A B) A B :=\n{ coe := cool_embedding.to_fun,\n  coe_injective' := \u03bb f g h, by cases f; cases g; congr',\n  injective' := my_embedding.injective',\n  map_op := cool_embedding.map_op',\n  map_cool := cool_embedding.map_cool' }\n\n-- [Insert `has_coe_to_fun`, `to_fun_eq_coe`, `ext` and `copy` here]\n```\n\nThen any declaration taking a specific type of morphisms as parameter can instead take the\nclass you just defined:\n```\n-- Compare with: lemma do_something (f : my_embedding A B) : sorry := sorry\nlemma do_something {F : Type*} [my_embedding_class F A B] (f : F) : sorry := sorry\n```\n\nThis means anything set up for `my_embedding`s will automatically work for `cool_embedding_class`es,\nand defining `cool_embedding_class` only takes a constant amount of effort,\ninstead of linearly increasing the work per `my_embedding`-related declaration.\n\n-/\n\nset_option old_structure_cmd true\n\n/-- The class `embedding_like F \u03b1 \u03b2` expresses that terms of type `F` have an\ninjective coercion to injective functions `\u03b1 \u21aa \u03b2`.\n-/\nclass embedding_like (F : Sort*) (\u03b1 \u03b2 : out_param Sort*)\n  extends fun_like F \u03b1 (\u03bb _, \u03b2) :=\n(injective' : \u2200 (f : F), @function.injective \u03b1 \u03b2 (coe f))\n\nnamespace embedding_like\n\nvariables {F \u03b1 \u03b2 \u03b3 : Sort*} [i : embedding_like F \u03b1 \u03b2]\n\ninclude i\n\nprotected lemma injective (f : F) : function.injective f := injective' f\n\n@[simp] lemma apply_eq_iff_eq (f : F) {x y : \u03b1} : f x = f y \u2194 x = y :=\n(embedding_like.injective f).eq_iff\n\nomit i\n\n@[simp] lemma comp_injective {F : Sort*} [embedding_like F \u03b2 \u03b3] (f : \u03b1 \u2192 \u03b2) (e : F) :\n  function.injective (e \u2218 f) \u2194 function.injective f :=\n(embedding_like.injective e).of_comp_iff f\n\nend embedding_like\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/fun_like/embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6334102705979902, "lm_q1q2_score": 0.36820334794446247}}
{"text": "import Std\n\nclass OMonad (W) extends Monad W where\n  le : \u2200 {A}, W A \u2192 W A \u2192 Prop\n\ninstance [OMonad w] : LE (w \u03b1) where\n  le := OMonad.le\n\nclass LawfulOMonad (W) [OMonad W] extends LawfulMonad W where\n  le_refl : \u2200 a : W \u03b1, a \u2264 a\n  le_trans : \u2200 a b c : W \u03b1, a \u2264 b \u2192 b \u2264 c \u2192 a \u2264 c\n  le_antisymm : \u2200 a b : W \u03b1, a \u2264 b \u2192 b \u2264 a \u2192 a = b\n  bind_mono : \u2200 {A B} (w1 w2 : W A) (f1 f2 : A \u2192 W B),\n    w1 \u2264 w2 \u2192 (\u2200 a, (f1 a) \u2264 (f2 a)) \u2192 (bind w1 f1) \u2264 (bind w2 f2)\n\n/-! Dijkstra Monad -/\nclass DijkstraMonad (M W) [Monad M] [OMonad W] [LawfulOMonad W] where\n  obs : \u2200 {A}, M A \u2192 W A\n  obsPure : obs (pure a) = pure a\n  obsBind : obs (bind m f) = bind (obs m) (fun a => obs (f a))\n\n\ndef Cont (\u03c1 \u03b1) := (\u03b1 \u2192 \u03c1) \u2192 \u03c1\ndef IDSpec (A) := { w : Cont Prop A // \u2200 {p1 p2}, (\u2200 a, p1 a \u2192 p2 a) \u2192 w p1 \u2192 w p2}\n\ninstance : OMonad IDSpec where\n  pure a := \u27e8(\u00b7 a), (\u00b7 a)\u27e9\n  bind\n  | \u27e8w,hw\u27e9, f => \u27e8fun P => w (fun x => (f x).val P),\n    by\n      intro p1 p2 mono h; refine hw ?_ h; intro a; generalize f a = w'\n      match w' with | \u27e8w',hw'\u27e9 => simp; apply hw'; exact mono\u27e9\n  le w1 w2 := \u2200 p, w2.val p \u2192 w1.val p\n\n@[ext] theorem Subtype.ext (s1 s2 : Subtype p) (h : s1.val = s2.val) : s1 = s2 := by\n  cases s1; cases s2; simp at h \u22a2; assumption\n@[ext] theorem IDSpec.ext (s1 s2 : IDSpec p) (h : s1.val = s2.val) : s1 = s2 := Subtype.ext _ _ h\n\ninstance : LawfulOMonad IDSpec where\n  map_const       := by intros; congr\n  id_map          := by intros; congr\n  seqLeft_eq      := by intros; congr\n  seqRight_eq     := by intros; congr\n  pure_seq        := by intros; congr\n  bind_pure_comp  := by intros; congr\n  bind_map        := by intros; congr\n  pure_bind       := by intros; congr\n  le_refl := by simp [LE.le, OMonad.le]\n  le_trans := by simp (config := {contextual := true}) [LE.le, OMonad.le]\n  le_antisymm := by simp only [LE.le, OMonad.le]; intros; ext; funext p; simp; constructor <;> (intro; simp [*])\n  bind_assoc := by\n    intros; simp [bind]\n    split\n    next h =>\n    split at h\n    cases h\n    congr\n  bind_mono\n  | _w1, w2, f1, f2, h1, h2, p, h =>\n    h1 _ (w2.property (\u03bb a hf2 => h2 a _ hf2) h)\n\ninstance : DijkstraMonad Id IDSpec where\n  obs a := \u27e8(\u00b7 a), (\u00b7 a)\u27e9\n  obsPure := rfl\n  obsBind := rfl\n\n\nclass MonadTrans (T : (Type _ \u2192 Type _) \u2192 (Type _ \u2192 Type _)) where\n  pure : [Monad M] \u2192 \u03b1 \u2192 T M \u03b1\n  bind : [Monad M] \u2192 T M \u03b1 \u2192 (\u03b1 \u2192 T M \u03b2) \u2192 T M \u03b2\n  lift : [Monad M] \u2192 M \u03b1 \u2192 T M \u03b1\n\ninstance [Monad m] [MonadTrans T] : Monad (T m) where\n  pure := MonadTrans.pure\n  bind := MonadTrans.bind\n\nclass LawfulMonadTrans (T) [MonadTrans T] where\n  lift_pure : [Monad M] \u2192 [LawfulMonad M] \u2192 \u2200 {a : \u03b1},\n    MonadTrans.lift (T := T) (M := M) (pure a) = pure a\n  lift_bind : [Monad M] \u2192 [LawfulMonad M] \u2192 \u2200 {m : M \u03b1} {k : \u03b1 \u2192 M \u03b2},\n    MonadTrans.lift (T := T) (M := M) (m >>= k) = (MonadTrans.lift m) >>= (fun x => (MonadTrans.lift <| k x) >>= fun kx => pure kx)\n\nclass OrderedMonadTrans (T) extends MonadTrans T where\n  le : [OMonad m] \u2192 \u2200 {\u03b1}, T m \u03b1 \u2192 T m \u03b1 \u2192 Prop\n\ninstance [OMonad m] [OrderedMonadTrans T] : OMonad (T m) where\n  le := OrderedMonadTrans.le\n\nclass LawfulOrderedMonadTrans (T) [OrderedMonadTrans T] extends LawfulMonadTrans T where\n  lift_mono : [OMonad M] \u2192 [LawfulOMonad M] \u2192 \u2200 {m1 m2 : M \u03b1}, m1 \u2264 m2 \u2192 MonadTrans.lift (T := T) m1 \u2264 MonadTrans.lift m2\n\n\ninstance : OrderedMonadTrans (StateT \u03c3) where\n  pure := pure\n  bind := bind\n  lift := liftM\n  le m1 m2 := \u2200 s, m1 s \u2264 m2 s\n\ninstance : LawfulOrderedMonadTrans (StateT \u03c3) where\n  lift_pure := by intros; simp [MonadTrans.lift, MonadTrans.pure, liftM, pure, monadLift, MonadLift.monadLift, StateT.lift, StateT.pure]\n  lift_bind := by intros; simp [MonadTrans.lift, MonadTrans.bind, liftM, bind, monadLift, MonadLift.monadLift, StateT.lift, StateT.bind]; rfl\n  lift_mono := by intros; simp [MonadTrans.lift, liftM, monadLift, MonadLift.monadLift, StateT.lift, LE.le, OMonad.le, OrderedMonadTrans.le]\n                  intro s; apply LawfulOMonad.bind_mono; assumption; simp [LawfulOMonad.le_refl]\n\ninstance : OrderedMonadTrans (ExceptT \u03b5) where\n  pure := pure\n  bind := bind\n  lift := liftM\n  le {m _ \u03b1} (m1 m2 : m (Except \u03b5 \u03b1)) := m1 \u2264 m2\n\n-- instance : @LawfulOrderedMonadTrans _ (ExceptT \u03b5) where\n--   lift_pure := sorry\n--   lift_bind := sorry\n--   lift_mono := sorry\n\n-- instance {\u03b5 : Type} : DijkstraMonad (ExceptT \u03b5 Id) (ExceptT \u03b5 IDSpec) where\n--   obs m := pure (f := IDSpec) m\n--   obsPure := by sorry\n--   obsBind := by sorry\n\ninstance : LawfulOMonad (StateT \u03c3 IDSpec) where\n  le_refl := by simp [LE.le, OMonad.le, OrderedMonadTrans.le]\n  le_trans := by simp (config := {contextual := true}) [LE.le, OMonad.le, OrderedMonadTrans.le]\n  le_antisymm := by \n    intro _ a b ha hb; funext x; ext; funext p; ext; constructor; apply hb; apply ha\n  bind_mono := by\n    intro _ _ w1 w2 f1 f2 hw hf s p h\n    simp [bind, MonadTrans.bind, StateT.bind] at h \u22a2\n    have := hw s\n    -- variable finagling\n    split; split at h\n    next a b c d e f g i j k l m =>\n    simp at h \u22a2\n    generalize w1 s = w1' at *\n    generalize w2 s = w2' at *\n    subst_vars\n    clear hw w1 w2 s c d i j\n    -- back to meaningful proof\n    apply this\n    simp\n    refine l ?_ h\n    rintro \u27e8x,y\u27e9\n    simp\n    apply hf\n\n\ninstance {\u03c3 : Type} : DijkstraMonad (StateT \u03c3 Id) (StateT \u03c3 IDSpec) where\n  obs m := fun s => DijkstraMonad.obs (m s)\n  obsPure := rfl\n  obsBind := rfl\n\n\ninstance [OrderedMonadTrans T] [LawfulOrderedMonadTrans T] : DijkstraMonad (T Id) (T IDSpec) where\n  obs t := sorry\n  obsPure := sorry\n  obsBind := sorry\n\nvariable (\u03c3 : Type)\n\ndef DijkstraVerify M [Monad M] W [OMonad W] [LawfulOMonad W] [D : DijkstraMonad M W] A (w : W A) (m : M A) : Prop :=\n  D.obs m \u2264 w\n\n\ntheorem IDSpec.pure_iff_eq : DijkstraVerify Id IDSpec A (pure x) a \u2194 x = a\n  := by simp [DijkstraVerify, DijkstraMonad.obs, LE.le, OMonad.le, pure]\n        constructor\n        intro h; apply h; rfl\n        intro h; cases h; intro; apply id\n\ndef foldSpec (inv : \u03b1 \u2192 Prop) : IDSpec \u03b1 :=\n  \u27e8 fun (p : \u03b1 \u2192 Prop) => \u2200 a, inv a \u2192 p a, by\n    intro p1 p2 hp hinv a ha\n    apply hp; apply hinv; exact ha \u27e9\n\ntheorem foldlInv' (L : List \u03c4) (inv : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03c4 \u2192 \u03b1) (init : \u03b1)\n    (h_init : inv init) (h_f : \u2200 {a t}, t \u2208 L \u2192 inv a \u2192 inv (f a t))\n  : DijkstraVerify Id IDSpec \u03b1 (foldSpec inv) (L.foldl f init) := by\n  intro post h\n  induction L generalizing init with\n  | nil =>\n    simp; apply h; assumption\n  | cons x xs ih =>\n    simp; apply ih; apply h_f <;> simp [h_init, h_f]; \n    intro a t ht inv_a\n    specialize @h_f a t \n    simp [ht] at h_f\n    exact h_f inv_a\n\ntheorem foldlInv (L : List \u03c4) (inv : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03c4 \u2192 \u03b1) (init : \u03b1)\n    (h_init : inv init) (h_f : \u2200 {a t}, inv a \u2192 inv (f a t))\n  : DijkstraVerify Id IDSpec \u03b1 (foldSpec inv) (L.foldl f init) := \n    foldlInv' L inv f init h_init (@fun _ _ _ inv => h_f inv)\n\n\n#check List.foldlM\n\ntheorem foldlMInv [Monad M] [LawfulMonad M] (L : List \u03c4) (inv : M \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03c4 \u2192 M \u03b1) (init : \u03b1)\n    (h_init : inv (pure init)) (h_f : \u2200 {ma t}, inv ma \u2192 inv (ma >>= (fun a => f a t))) \n    : DijkstraVerify Id IDSpec (M \u03b1) (foldSpec inv) (@List.foldlM M _ \u03b1 \u03c4 f init L) := by\n    intro post h\n    induction L generalizing init with\n    | nil => simp; apply h; exact h_init\n    | cons x xs ih => \n      simp [DijkstraMonad.obs] at *\n      sorry\n\n\ntheorem foldlMInvArr [Monad M] (A : Array \u03c4) (inv : M \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03c4 \u2192 M \u03b1) (init : \u03b1) (start := 0) (stop := A.size) \n    (stop_h : stop \u2264 A.size) (h_init : inv (pure init)) (h_f : \u2200 {ma t}, inv ma \u2192 inv (ma >>= (fun a => f a t))) :\n    DijkstraVerify Id IDSpec (M \u03b1) (foldSpec inv) (@Array.foldlM \u03c4 \u03b1 M _ f init A start stop) := by \n    intro post h\n    simp [DijkstraMonad.obs, Array.foldlM]\n    split <;> apply h <;> cases A\n    case inl L h_stop =>\n      induction L generalizing init with\n      | nil => \n        simp [Array.size] at h_stop\n        rw [Array.foldlM.loop]\n        simp [h_stop]\n        assumption\n      | cons x xs ih => \n        rw [Array.foldlM.loop]\n        split\n        case inl start_lt_stop => \n          have : \u2203 i', stop - start = Nat.succ i' := by sorry\n          cases this with\n          | _ i h_i =>\n            simp [*] at *\n            sorry\n        case inr _ => exact h_init\n    case inr L h_stop => exfalso; exact h_stop stop_h\n    \n      \n\ntheorem foldrInv' (L : List \u03c4) (inv : \u03b1 \u2192 Prop) (f : \u03c4 \u2192 \u03b1 \u2192 \u03b1) (init : \u03b1)\n    (h_init : inv init) (h_f : \u2200 {a t}, t \u2208 L \u2192 inv a \u2192 inv (f t a))\n  : DijkstraVerify Id IDSpec \u03b1 (foldSpec inv) (L.foldr f init) := by\n  intro post h\n  induction L generalizing post with\n  | nil =>\n    simp; apply h; assumption\n  | cons x xs ih =>\n    simp [foldSpec] at h\n    apply h; apply h_f; \n    simp only [List.mem_cons, true_or]; apply ih;\n    intro a t txs inva\n    apply h_f (by simp only [List.mem_cons, txs, or_true]) inva;\n    simp [foldSpec]\n    \ntheorem foldrInv (L : List \u03c4) (inv : \u03b1 \u2192 Prop) (f : \u03c4 \u2192 \u03b1 \u2192 \u03b1) (init : \u03b1)\n    (h_init : inv init) (h_f : \u2200 {a t}, inv a \u2192 inv (f t a))\n  : DijkstraVerify Id IDSpec \u03b1 (foldSpec inv) (L.foldr f init) := \n  foldrInv' L inv f init h_init (@fun _ _ _ inv => h_f inv)\n\n", "meta": {"author": "zaxioms", "repo": "Dijkstra", "sha": "81bf939b505c5b49d6c0e31030ad3d204c226214", "save_path": "github-repos/lean/zaxioms-Dijkstra", "path": "github-repos/lean/zaxioms-Dijkstra/Dijkstra-81bf939b505c5b49d6c0e31030ad3d204c226214/Djikstra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3682033439217579}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.comma\nimport Mathlib.category_theory.punit\nimport Mathlib.category_theory.reflects_isomorphisms\nimport Mathlib.category_theory.epi_mono\nimport Mathlib.PostPort\n\nuniverses v\u2081 u\u2081 v\u2082 u\u2082 \n\nnamespace Mathlib\n\n/-!\n# Over and under categories\n\nOver (and under) categories are special cases of comma categories.\n* If `L` is the identity functor and `R` is a constant functor, then `comma L R` is the \"slice\" or\n  \"over\" category over the object `R` maps to.\n* Conversely, if `L` is a constant functor and `R` is the identity functor, then `comma L R` is the\n  \"coslice\" or \"under\" category under the object `L` maps to.\n\n## Tags\n\ncomma, slice, coslice, over, under\n-/\n\nnamespace category_theory\n\n\n/--\nThe over category has as objects arrows in `T` with codomain `X` and as morphisms commutative\ntriangles.\n\nSee https://stacks.math.columbia.edu/tag/001G.\n-/\ndef over {T : Type u\u2081} [category T] (X : T) :=\n  comma \ud835\udfed (functor.from_punit X)\n\n-- Satisfying the inhabited linter\n\nprotected instance over.inhabited {T : Type u\u2081} [category T] [Inhabited T] : Inhabited (over Inhabited.default) :=\n  { default := comma.mk \ud835\udfd9 }\n\nnamespace over\n\n\ntheorem over_morphism.ext {T : Type u\u2081} [category T] {X : T} {U : over X} {V : over X} {f : U \u27f6 V} {g : U \u27f6 V} (h : comma_morphism.left f = comma_morphism.left g) : f = g := sorry\n\n@[simp] theorem over_right {T : Type u\u2081} [category T] {X : T} (U : over X) : comma.right U = PUnit.unit :=\n  of_as_true trivial\n\n@[simp] theorem id_left {T : Type u\u2081} [category T] {X : T} (U : over X) : comma_morphism.left \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem comp_left {T : Type u\u2081} [category T] {X : T} (a : over X) (b : over X) (c : over X) (f : a \u27f6 b) (g : b \u27f6 c) : comma_morphism.left (f \u226b g) = comma_morphism.left f \u226b comma_morphism.left g :=\n  rfl\n\n@[simp] theorem w {T : Type u\u2081} [category T] {X : T} {A : over X} {B : over X} (f : A \u27f6 B) : comma_morphism.left f \u226b comma.hom B = comma.hom A := sorry\n\n/-- To give an object in the over category, it suffices to give a morphism with codomain `X`. -/\n@[simp] theorem mk_left {T : Type u\u2081} [category T] {X : T} {Y : T} (f : Y \u27f6 X) : comma.left (mk f) = Y :=\n  Eq.refl (comma.left (mk f))\n\n/-- We can set up a coercion from arrows with codomain `X` to `over X`. This most likely should not\n    be a global instance, but it is sometimes useful. -/\ndef coe_from_hom {T : Type u\u2081} [category T] {X : T} {Y : T} : has_coe (Y \u27f6 X) (over X) :=\n  has_coe.mk mk\n\n@[simp] theorem coe_hom {T : Type u\u2081} [category T] {X : T} {Y : T} (f : Y \u27f6 X) : comma.hom \u2191f = f :=\n  rfl\n\n/-- To give a morphism in the over category, it suffices to give an arrow fitting in a commutative\n    triangle. -/\ndef hom_mk {T : Type u\u2081} [category T] {X : T} {U : over X} {V : over X} (f : comma.left U \u27f6 comma.left V) (w : autoParam (f \u226b comma.hom V = comma.hom U)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])) : U \u27f6 V :=\n  comma_morphism.mk\n\n/--\nConstruct an isomorphism in the over category given isomorphisms of the objects whose forward\ndirection gives a commutative triangle.\n-/\n@[simp] theorem iso_mk_inv_left {T : Type u\u2081} [category T] {X : T} {f : over X} {g : over X} (hl : comma.left f \u2245 comma.left g) (hw : autoParam (iso.hom hl \u226b comma.hom g = comma.hom f)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])) : comma_morphism.left (iso.inv (iso_mk hl)) = iso.inv hl :=\n  Eq.refl (iso.inv hl)\n\n/--\nThe forgetful functor mapping an arrow to its domain.\n\nSee https://stacks.math.columbia.edu/tag/001G.\n-/\ndef forget {T : Type u\u2081} [category T] (X : T) : over X \u2964 T :=\n  comma.fst \ud835\udfed (functor.from_punit X)\n\n@[simp] theorem forget_obj {T : Type u\u2081} [category T] {X : T} {U : over X} : functor.obj (forget X) U = comma.left U :=\n  rfl\n\n@[simp] theorem forget_map {T : Type u\u2081} [category T] {X : T} {U : over X} {V : over X} {f : U \u27f6 V} : functor.map (forget X) f = comma_morphism.left f :=\n  rfl\n\n/--\nA morphism `f : X \u27f6 Y` induces a functor `over X \u2964 over Y` in the obvious way.\n\nSee https://stacks.math.columbia.edu/tag/001G.\n-/\ndef map {T : Type u\u2081} [category T] {X : T} {Y : T} (f : X \u27f6 Y) : over X \u2964 over Y :=\n  comma.map_right \ud835\udfed (discrete.nat_trans fun (_x : discrete PUnit) => f)\n\n@[simp] theorem map_obj_left {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : over X} : comma.left (functor.obj (map f) U) = comma.left U :=\n  rfl\n\n@[simp] theorem map_obj_hom {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : over X} : comma.hom (functor.obj (map f) U) = comma.hom U \u226b f :=\n  rfl\n\n@[simp] theorem map_map_left {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : over X} {V : over X} {g : U \u27f6 V} : comma_morphism.left (functor.map (map f) g) = comma_morphism.left g :=\n  rfl\n\n/-- Mapping by the identity morphism is just the identity functor. -/\ndef map_id {T : Type u\u2081} [category T] {Y : T} : map \ud835\udfd9 \u2245 \ud835\udfed :=\n  nat_iso.of_components (fun (X : over Y) => iso_mk (iso.refl (comma.left (functor.obj (map \ud835\udfd9) X)))) sorry\n\n/-- Mapping by the composite morphism `f \u226b g` is the same as mapping by `f` then by `g`. -/\ndef map_comp {T : Type u\u2081} [category T] {X : T} {Y : T} {Z : T} (f : X \u27f6 Y) (g : Y \u27f6 Z) : map (f \u226b g) \u2245 map f \u22d9 map g :=\n  nat_iso.of_components (fun (X_1 : over X) => iso_mk (iso.refl (comma.left (functor.obj (map (f \u226b g)) X_1)))) sorry\n\nprotected instance forget_reflects_iso {T : Type u\u2081} [category T] {X : T} : reflects_isomorphisms (forget X) :=\n  reflects_isomorphisms.mk\n    fun (Y Z : over X) (f : Y \u27f6 Z) (t : is_iso (functor.map (forget X) f)) =>\n      is_iso.mk (hom_mk (inv (functor.map (forget X) f)))\n\nprotected instance forget_faithful {T : Type u\u2081} [category T] {X : T} : faithful (forget X) :=\n  faithful.mk\n\n/--\nIf `k.left` is an epimorphism, then `k` is an epimorphism. In other words, `over.forget X` reflects\nepimorphisms.\nThe converse does not hold without additional assumptions on the underlying category.\n-/\n-- TODO: Show the converse holds if `T` has binary products or pushouts.\n\ntheorem epi_of_epi_left {T : Type u\u2081} [category T] {X : T} {f : over X} {g : over X} (k : f \u27f6 g) [hk : epi (comma_morphism.left k)] : epi k :=\n  faithful_reflects_epi (forget X) hk\n\n/--\nIf `k.left` is a monomorphism, then `k` is a monomorphism. In other words, `over.forget X` reflects\nmonomorphisms.\nThe converse of `category_theory.over.mono_left_of_mono`.\n\nThis lemma is not an instance, to avoid loops in type class inference.\n-/\ntheorem mono_of_mono_left {T : Type u\u2081} [category T] {X : T} {f : over X} {g : over X} (k : f \u27f6 g) [hk : mono (comma_morphism.left k)] : mono k :=\n  faithful_reflects_mono (forget X) hk\n\n/--\nIf `k` is a monomorphism, then `k.left` is a monomorphism. In other words, `over.forget X` preserves\nmonomorphisms.\nThe converse of `category_theory.over.mono_of_mono_left`.\n-/\nprotected instance mono_left_of_mono {T : Type u\u2081} [category T] {X : T} {f : over X} {g : over X} (k : f \u27f6 g) [mono k] : mono (comma_morphism.left k) :=\n  mono.mk\n    fun (Y : T) (l m : Y \u27f6 comma.left f) (a : l \u226b comma_morphism.left k = m \u226b comma_morphism.left k) =>\n      let l' : mk (m \u226b comma.hom f) \u27f6 f := hom_mk l;\n      congr_arg comma_morphism.left\n        (eq.mpr (id (Eq._oldrec (Eq.refl (l' = hom_mk m)) (Eq.symm (propext (cancel_mono k))))) (over_morphism.ext a))\n\n/-- Given f : Y \u27f6 X, this is the obvious functor from (T/X)/f to T/Y -/\n@[simp] theorem iterated_slice_forward_obj {T : Type u\u2081} [category T] {X : T} (f : over X) (\u03b1 : over f) : functor.obj (iterated_slice_forward f) \u03b1 = mk (comma_morphism.left (comma.hom \u03b1)) :=\n  Eq.refl (functor.obj (iterated_slice_forward f) \u03b1)\n\n/-- Given f : Y \u27f6 X, this is the obvious functor from T/Y to (T/X)/f -/\n@[simp] theorem iterated_slice_backward_map {T : Type u\u2081} [category T] {X : T} (f : over X) (g : over (comma.left f)) (h : over (comma.left f)) (\u03b1 : g \u27f6 h) : functor.map (iterated_slice_backward f) \u03b1 = hom_mk (hom_mk (comma_morphism.left \u03b1)) :=\n  Eq.refl (functor.map (iterated_slice_backward f) \u03b1)\n\n/-- Given f : Y \u27f6 X, we have an equivalence between (T/X)/f and T/Y -/\n@[simp] theorem iterated_slice_equiv_counit_iso {T : Type u\u2081} [category T] {X : T} (f : over X) : equivalence.counit_iso (iterated_slice_equiv f) =\n  nat_iso.of_components\n    (fun (g : over (comma.left f)) =>\n      iso_mk (iso.refl (comma.left (functor.obj (iterated_slice_backward f \u22d9 iterated_slice_forward f) g))))\n    (iterated_slice_equiv._proof_5 f) :=\n  Eq.refl (equivalence.counit_iso (iterated_slice_equiv f))\n\ntheorem iterated_slice_forward_forget {T : Type u\u2081} [category T] {X : T} (f : over X) : iterated_slice_forward f \u22d9 forget (comma.left f) = forget f \u22d9 forget X :=\n  rfl\n\ntheorem iterated_slice_backward_forget_forget {T : Type u\u2081} [category T] {X : T} (f : over X) : iterated_slice_backward f \u22d9 forget f \u22d9 forget X = forget (comma.left f) :=\n  rfl\n\n/-- A functor `F : T \u2964 D` induces a functor `over X \u2964 over (F.obj X)` in the obvious way. -/\n@[simp] theorem post_map_right {T : Type u\u2081} [category T] {X : T} {D : Type u\u2082} [category D] (F : T \u2964 D) (Y\u2081 : over X) (Y\u2082 : over X) (f : Y\u2081 \u27f6 Y\u2082) : comma_morphism.right (functor.map (post F) f) =\n  id (fun (F : T \u2964 D) (Y\u2081 Y\u2082 : over X) (f : Y\u2081 \u27f6 Y\u2082) => ulift.up (eq.mpr post._proof_1 (plift.up (of_as_true trivial))))\n    F Y\u2081 Y\u2082 f :=\n  Eq.refl (comma_morphism.right (functor.map (post F) f))\n\nend over\n\n\n/-- The under category has as objects arrows with domain `X` and as morphisms commutative\n    triangles. -/\ndef under {T : Type u\u2081} [category T] (X : T) :=\n  comma (functor.from_punit X) \ud835\udfed\n\n-- Satisfying the inhabited linter\n\nprotected instance under.inhabited {T : Type u\u2081} [category T] [Inhabited T] : Inhabited (under Inhabited.default) :=\n  { default := comma.mk \ud835\udfd9 }\n\nnamespace under\n\n\ntheorem under_morphism.ext {T : Type u\u2081} [category T] {X : T} {U : under X} {V : under X} {f : U \u27f6 V} {g : U \u27f6 V} (h : comma_morphism.right f = comma_morphism.right g) : f = g := sorry\n\n@[simp] theorem under_left {T : Type u\u2081} [category T] {X : T} (U : under X) : comma.left U = PUnit.unit :=\n  of_as_true trivial\n\n@[simp] theorem id_right {T : Type u\u2081} [category T] {X : T} (U : under X) : comma_morphism.right \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem comp_right {T : Type u\u2081} [category T] {X : T} (a : under X) (b : under X) (c : under X) (f : a \u27f6 b) (g : b \u27f6 c) : comma_morphism.right (f \u226b g) = comma_morphism.right f \u226b comma_morphism.right g :=\n  rfl\n\n@[simp] theorem w {T : Type u\u2081} [category T] {X : T} {A : under X} {B : under X} (f : A \u27f6 B) : comma.hom A \u226b comma_morphism.right f = comma.hom B := sorry\n\n/-- To give an object in the under category, it suffices to give an arrow with domain `X`. -/\n@[simp] theorem mk_left {T : Type u\u2081} [category T] {X : T} {Y : T} (f : X \u27f6 Y) : comma.left (mk f) = PUnit.unit :=\n  Eq.refl (comma.left (mk f))\n\n/-- To give a morphism in the under category, it suffices to give a morphism fitting in a\n    commutative triangle. -/\n@[simp] theorem hom_mk_left {T : Type u\u2081} [category T] {X : T} {U : under X} {V : under X} (f : comma.right U \u27f6 comma.right V) (w : autoParam (comma.hom U \u226b f = comma.hom V)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])) : comma_morphism.left (hom_mk f) =\n  id\n    (fun {X : T} {U V : under X} (f : comma.right U \u27f6 comma.right V) (w : comma.hom U \u226b f = comma.hom V) =>\n      eq.mpr hom_mk._proof_1 (ulift.up (eq.mpr hom_mk._proof_2 (plift.up (of_as_true trivial)))))\n    X U V f w :=\n  Eq.refl (comma_morphism.left (hom_mk f))\n\n/--\nConstruct an isomorphism in the over category given isomorphisms of the objects whose forward\ndirection gives a commutative triangle.\n-/\ndef iso_mk {T : Type u\u2081} [category T] {X : T} {f : under X} {g : under X} (hr : comma.right f \u2245 comma.right g) (hw : comma.hom f \u226b iso.hom hr = comma.hom g) : f \u2245 g :=\n  comma.iso_mk (eq_to_iso sorry) hr sorry\n\n@[simp] theorem iso_mk_hom_right {T : Type u\u2081} [category T] {X : T} {f : under X} {g : under X} (hr : comma.right f \u2245 comma.right g) (hw : comma.hom f \u226b iso.hom hr = comma.hom g) : comma_morphism.right (iso.hom (iso_mk hr hw)) = iso.hom hr :=\n  rfl\n\n@[simp] theorem iso_mk_inv_right {T : Type u\u2081} [category T] {X : T} {f : under X} {g : under X} (hr : comma.right f \u2245 comma.right g) (hw : comma.hom f \u226b iso.hom hr = comma.hom g) : comma_morphism.right (iso.inv (iso_mk hr hw)) = iso.inv hr :=\n  rfl\n\n/-- The forgetful functor mapping an arrow to its domain. -/\ndef forget {T : Type u\u2081} [category T] (X : T) : under X \u2964 T :=\n  comma.snd (functor.from_punit X) \ud835\udfed\n\n@[simp] theorem forget_obj {T : Type u\u2081} [category T] {X : T} {U : under X} : functor.obj (forget X) U = comma.right U :=\n  rfl\n\n@[simp] theorem forget_map {T : Type u\u2081} [category T] {X : T} {U : under X} {V : under X} {f : U \u27f6 V} : functor.map (forget X) f = comma_morphism.right f :=\n  rfl\n\n/-- A morphism `X \u27f6 Y` induces a functor `under Y \u2964 under X` in the obvious way. -/\ndef map {T : Type u\u2081} [category T] {X : T} {Y : T} (f : X \u27f6 Y) : under Y \u2964 under X :=\n  comma.map_left \ud835\udfed (discrete.nat_trans fun (_x : discrete PUnit) => f)\n\n@[simp] theorem map_obj_right {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : under Y} : comma.right (functor.obj (map f) U) = comma.right U :=\n  rfl\n\n@[simp] theorem map_obj_hom {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : under Y} : comma.hom (functor.obj (map f) U) = f \u226b comma.hom U :=\n  rfl\n\n@[simp] theorem map_map_right {T : Type u\u2081} [category T] {X : T} {Y : T} {f : X \u27f6 Y} {U : under Y} {V : under Y} {g : U \u27f6 V} : comma_morphism.right (functor.map (map f) g) = comma_morphism.right g :=\n  rfl\n\n/-- Mapping by the identity morphism is just the identity functor. -/\ndef map_id {T : Type u\u2081} [category T] {Y : T} : map \ud835\udfd9 \u2245 \ud835\udfed :=\n  nat_iso.of_components (fun (X : under Y) => iso_mk (iso.refl (comma.right (functor.obj (map \ud835\udfd9) X))) sorry) sorry\n\n/-- Mapping by the composite morphism `f \u226b g` is the same as mapping by `f` then by `g`. -/\ndef map_comp {T : Type u\u2081} [category T] {X : T} {Y : T} {Z : T} (f : X \u27f6 Y) (g : Y \u27f6 Z) : map (f \u226b g) \u2245 map g \u22d9 map f :=\n  nat_iso.of_components (fun (X_1 : under Z) => iso_mk (iso.refl (comma.right (functor.obj (map (f \u226b g)) X_1))) sorry)\n    sorry\n\n/-- A functor `F : T \u2964 D` induces a functor `under X \u2964 under (F.obj X)` in the obvious way. -/\n@[simp] theorem post_map_left {T : Type u\u2081} [category T] {D : Type u\u2082} [category D] {X : T} (F : T \u2964 D) (Y\u2081 : under X) (Y\u2082 : under X) (f : Y\u2081 \u27f6 Y\u2082) : comma_morphism.left (functor.map (post F) f) =\n  id\n    (fun {X : T} (F : T \u2964 D) (Y\u2081 Y\u2082 : under X) (f : Y\u2081 \u27f6 Y\u2082) =>\n      ulift.up (eq.mpr post._proof_1 (plift.up (of_as_true trivial))))\n    X F Y\u2081 Y\u2082 f :=\n  Eq.refl (comma_morphism.left (functor.map (post F) f))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.3682033358763488}}
{"text": "/-\nCopyright (c) 2022 Jun Yoshida. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n-/\n\n/-!\n\n# Miscellaneous lemmas on `LawfulMonad`\n\n-/\n\nuniverse u v\nvariable {m : Type u \u2192 Type v}\n\ntheorem map_bind [Monad m] [LawfulMonad m] {\u03b1 \u03b2 \u03b3 : Type u} (f : \u03b1 \u2192 \u03b2) (x : m \u03b1) (g : \u03b2 \u2192 m \u03b3) : (f <$> x) >>= g = x >>= (g \u2218 f) := by\n  rw [map_eq_pure_bind, bind_assoc]\n  apply bind_congr\n  intro a; rw [pure_bind]\n  rfl\n", "meta": {"author": "Junology", "repo": "dijkstra", "sha": "19ff3ddd7ff112c69848fa9c643f773008cdd5ff", "save_path": "github-repos/lean/Junology-dijkstra", "path": "github-repos/lean/Junology-dijkstra/dijkstra-19ff3ddd7ff112c69848fa9c643f773008cdd5ff/Dijkstra/Control/Lawful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.36820333587634874}}
{"text": "import Mathlib.Tactic.Alias\nimport Mathlib.Tactic.RunCmd\nimport Std.Tactic.GuardExpr\n\nopen Lean Meta\nnamespace Alias\nnamespace A\n\n/-- doc string for foo -/\ntheorem foo : 1 + 1 = 2 := rfl\n\n/-- doc string for `alias foo` -/\nalias foo \u2190 foo1 foo2 foo3 _root_.B.foo4\n\nexample : 1 + 1 = 2 := foo1\nexample : 1 + 1 = 2 := foo2\nexample : 1 + 1 = 2 := foo3\n\ndef bar : Nat := 5\nalias bar \u2190 bar1 bar2\nexample : bar1 = 5 := rfl\nexample : bar2 = 5 := rfl\n\ntheorem baz (x : Nat) : x = x := rfl\nalias baz \u2190 baz1\nexample : 3 = 3 := baz1 3\n\ntheorem ab_iff_ba {t : Type} {a b : t} : a = b \u2194 b = a := Iff.intro Eq.symm Eq.symm\nalias ab_iff_ba \u2194 ba_of_ab ab_of_ba\nexample {a b : Nat} : a = b \u2192 b = a := ba_of_ab\nexample {t : Type} {a b : t} : b = a \u2192 a = b := ab_of_ba\n\ntheorem a_iff_a_and_a (a : Prop) : a \u2194 a \u2227 a :=\n  Iff.intro (\u03bb x => \u27e8x,x\u27e9) (\u03bb x => x.1)\n\nalias a_iff_a_and_a \u2194 forward _\nalias a_iff_a_and_a \u2194 _ backward\n\nexample : True \u2192 True \u2227 True := forward True\nexample : True \u2227 True \u2192 True := backward True\n\n/-- doc string for `alias a_iff_a_and_a` -/\nalias a_iff_a_and_a \u2194 ..\nexample : True \u2192 True \u2227 True := a_and_a_of_a True\nexample : True \u2227 True \u2192 True := a_of_a_and_a True\n\nend A\n\n-- test namespacing\nexample : 1 + 1 = 2 := A.foo1\nexample : 1 + 1 = 2 := B.foo4\nexample : True \u2192 True \u2227 True := A.a_and_a_of_a True\nexample : True \u2192 True \u2227 True := A.forward True\nexample : True \u2227 True \u2192 True := A.backward True\n\nnamespace C\n\nalias A.a_iff_a_and_a \u2194 _root_.B.forward2 _\nalias A.a_iff_a_and_a \u2194 _ _root_.B.backward2\n\nend C\n\nexample : True \u2192 True \u2227 True := B.forward2 True\nexample : True \u2227 True \u2192 True := B.backward2 True\n\ntheorem checkType : 1 + 1 = 2 \u2194 2 = 2 := .rfl\nalias checkType \u2194 forward backward\n\nexample : True := by\n  have h1 := forward\n  have h2 := backward\n  guard_hyp h1 :\u209b 1 + 1 = 2 \u2192 2 = 2\n  guard_hyp h2 :\u209b 2 = 2 \u2192 1 + 1 = 2\n  trivial\n\nend Alias\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/Alias.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.36816643100802515}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monad.adjunction\nimport Mathlib.category_theory.adjunction.limits\nimport Mathlib.category_theory.limits.preserves.shapes.terminal\nimport Mathlib.PostPort\n\nuniverses u\u2081 v\u2081 u\u2082 \n\nnamespace Mathlib\n\nnamespace category_theory\n\n\nnamespace monad\n\n\nnamespace forget_creates_limits\n\n\n/-- (Impl) The natural transformation used to define the new cone -/\n@[simp] theorem \u03b3_app {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) (j : J) : nat_trans.app (\u03b3 D) j = algebra.a (functor.obj D j) :=\n  Eq.refl (nat_trans.app (\u03b3 D) j)\n\n/-- (Impl) This new cone is used to construct the algebra structure -/\n@[simp] theorem new_cone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T)) : limits.cone.X (new_cone D c) = functor.obj T (limits.cone.X c) :=\n  Eq.refl (limits.cone.X (new_cone D c))\n\n/-- The algebra structure which will be the apex of the new limit cone for `D`. -/\n@[simp] theorem cone_point_A {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T)) (t : limits.is_limit c) : algebra.A (cone_point D c t) = limits.cone.X c :=\n  Eq.refl (algebra.A (cone_point D c t))\n\n/-- (Impl) Construct the lifted cone in `algebra T` which will be limiting. -/\n@[simp] theorem lifted_cone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T)) (t : limits.is_limit c) : limits.cone.X (lifted_cone D c t) = cone_point D c t :=\n  Eq.refl (limits.cone.X (lifted_cone D c t))\n\n/-- (Impl) Prove that the lifted cone is limiting. -/\ndef lifted_cone_is_limit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) (c : limits.cone (D \u22d9 forget T)) (t : limits.is_limit c) : limits.is_limit (lifted_cone D c t) :=\n  limits.is_limit.mk fun (s : limits.cone D) => algebra.hom.mk (limits.is_limit.lift t (functor.map_cone (forget T) s))\n\nend forget_creates_limits\n\n\n-- Theorem 5.6.5 from [Riehl][riehl2017]\n\n/-- The forgetful functor from the Eilenberg-Moore category creates limits. -/\nprotected instance forget_creates_limits {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] : creates_limits (forget T) :=\n  creates_limits.mk\n    fun (J : Type v\u2081) (\ud835\udca5 : small_category J) =>\n      creates_limits_of_shape.mk\n        fun (D : J \u2964 algebra T) =>\n          creates_limit_of_reflects_iso\n            fun (c : limits.cone (D \u22d9 forget T)) (t : limits.is_limit c) =>\n              lifts_to_limit.mk\n                (liftable_cone.mk sorry\n                  (limits.cones.ext (iso.refl (limits.cone.X (functor.map_cone (forget T) sorry))) sorry))\n                sorry\n\n/-- `D \u22d9 forget T` has a limit, then `D` has a limit. -/\ntheorem has_limit_of_comp_forget_has_limit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) [limits.has_limit (D \u22d9 forget T)] : limits.has_limit D :=\n  has_limit_of_created D (forget T)\n\nnamespace forget_creates_colimits\n\n\n-- Let's hide the implementation details in a namespace\n\n-- We have a diagram D of shape J in the category of algebras, and we assume that we are given a\n\n-- colimit for its image D \u22d9 forget T under the forgetful functor, say its apex is L.\n\n-- We'll construct a colimiting coalgebra for D, whose carrier will also be L.\n\n-- To do this, we must find a map TL \u27f6 L. Since T preserves colimits, TL is also a colimit.\n\n-- In particular, it is a colimit for the diagram `(D \u22d9 forget T) \u22d9 T`\n\n-- so to construct a map TL \u27f6 L it suffices to show that L is the apex of a cocone for this diagram.\n\n-- In other words, we need a natural transformation from const L to `(D \u22d9 forget T) \u22d9 T`.\n\n-- But we already know that L is the apex of a cocone for the diagram `D \u22d9 forget T`, so it\n\n-- suffices to give a natural transformation `((D \u22d9 forget T) \u22d9 T) \u27f6 (D \u22d9 forget T)`:\n\n/--\n(Impl)\nThe natural transformation given by the algebra structure maps, used to construct a cocone `c` with\napex `colimit (D \u22d9 forget T)`.\n -/\n@[simp] theorem \u03b3_app {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (j : J) : nat_trans.app \u03b3 j = algebra.a (functor.obj D j) :=\n  Eq.refl (nat_trans.app \u03b3 j)\n\n/--\n(Impl)\nA cocone for the diagram `(D \u22d9 forget T) \u22d9 T` found by composing the natural transformation `\u03b3`\nwith the colimiting cocone for `D \u22d9 forget T`.\n-/\n@[simp] theorem new_cocone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) : limits.cocone.X (new_cocone c) = limits.cocone.X c :=\n  Eq.refl (limits.cocone.X (new_cocone c))\n\n/--\n(Impl)\nDefine the map `\u03bb : TL \u27f6 L`, which will serve as the structure of the coalgebra on `L`, and\nwe will show is the colimiting object. We use the cocone constructed by `c` and the fact that\n`T` preserves colimits to produce this morphism.\n-/\ndef lambda {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T] : limits.cocone.X (functor.map_cocone T c) \u27f6 limits.cocone.X c :=\n  limits.is_colimit.desc (limits.preserves_colimit.preserves t) (new_cocone c)\n\n/-- (Impl) The key property defining the map `\u03bb : TL \u27f6 L`. -/\ntheorem commuting {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T] (j : J) : functor.map T (nat_trans.app (limits.cocone.\u03b9 c) j) \u226b lambda c t =\n  algebra.a (functor.obj D j) \u226b nat_trans.app (limits.cocone.\u03b9 c) j :=\n  limits.is_colimit.fac (limits.preserves_colimit.preserves t) (new_cocone c) j\n\n/--\n(Impl)\nConstruct the colimiting algebra from the map `\u03bb : TL \u27f6 L` given by `lambda`. We are required to\nshow it satisfies the two algebra laws, which follow from the algebra laws for the image of `D` and\nour `commuting` lemma.\n-/\n@[simp] theorem cocone_point_A {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T] [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] : algebra.A (cocone_point c t) = limits.cocone.X c :=\n  Eq.refl (algebra.A (cocone_point c t))\n\n/-- (Impl) Construct the lifted cocone in `algebra T` which will be colimiting. -/\n@[simp] theorem lifted_cocone_X {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T] [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] : limits.cocone.X (lifted_cocone c t) = cocone_point c t :=\n  Eq.refl (limits.cocone.X (lifted_cocone c t))\n\n/-- (Impl) Prove that the lifted cocone is colimiting. -/\n@[simp] theorem lifted_cocone_is_colimit_desc_f {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] {D : J \u2964 algebra T} (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) [limits.preserves_colimit (D \u22d9 forget T) T] [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] (s : limits.cocone D) : algebra.hom.f (limits.is_colimit.desc (lifted_cocone_is_colimit c t) s) =\n  limits.is_colimit.desc t (functor.map_cocone (forget T) s) :=\n  Eq.refl (algebra.hom.f (limits.is_colimit.desc (lifted_cocone_is_colimit c t) s))\n\nend forget_creates_colimits\n\n\n-- TODO: the converse of this is true as well\n\n/--\nThe forgetful functor from the Eilenberg-Moore category for a monad creates any colimit\nwhich the monad itself preserves.\n-/\nprotected instance forget_creates_colimit {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] (D : J \u2964 algebra T) [limits.preserves_colimit (D \u22d9 forget T) T] [limits.preserves_colimit ((D \u22d9 forget T) \u22d9 T) T] : creates_colimit D (forget T) :=\n  creates_colimit_of_reflects_iso\n    fun (c : limits.cocone (D \u22d9 forget T)) (t : limits.is_colimit c) =>\n      lifts_to_colimit.mk\n        (liftable_cocone.mk\n          (limits.cocone.mk (forget_creates_colimits.cocone_point c t)\n            (nat_trans.mk fun (j : J) => algebra.hom.mk (nat_trans.app (limits.cocone.\u03b9 c) j)))\n          (limits.cocones.ext\n            (iso.refl\n              (limits.cocone.X\n                (functor.map_cocone (forget T)\n                  (limits.cocone.mk (forget_creates_colimits.cocone_point c t)\n                    (nat_trans.mk fun (j : J) => algebra.hom.mk (nat_trans.app (limits.cocone.\u03b9 c) j))))))\n            sorry))\n        (forget_creates_colimits.lifted_cocone_is_colimit c t)\n\nprotected instance forget_creates_colimits_of_shape {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] [limits.preserves_colimits_of_shape J T] : creates_colimits_of_shape J (forget T) :=\n  creates_colimits_of_shape.mk fun (K : J \u2964 algebra T) => monad.forget_creates_colimit K\n\nprotected instance forget_creates_colimits {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] [limits.preserves_colimits T] : creates_colimits (forget T) :=\n  creates_colimits.mk fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) => monad.forget_creates_colimits_of_shape\n\n/--\nFor `D : J \u2964 algebra T`, `D \u22d9 forget T` has a colimit, then `D` has a colimit provided colimits\nof shape `J` are preserved by `T`.\n-/\ntheorem forget_creates_colimits_of_monad_preserves {C : Type u\u2081} [category C] {T : C \u2964 C} [monad T] {J : Type v\u2081} [small_category J] [limits.preserves_colimits_of_shape J T] (D : J \u2964 algebra T) [limits.has_colimit (D \u22d9 forget T)] : limits.has_colimit D :=\n  has_colimit_of_created D (forget T)\n\nend monad\n\n\nprotected instance comp_comparison_forget_has_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (F : J \u2964 D) (R : D \u2964 C) [monadic_right_adjoint R] [limits.has_limit (F \u22d9 R)] : limits.has_limit ((F \u22d9 monad.comparison R) \u22d9 monad.forget (left_adjoint R \u22d9 R)) :=\n  limits.has_limit_of_iso (iso_whisker_left F (iso.symm (monad.comparison_forget R)))\n\nprotected instance comp_comparison_has_limit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (F : J \u2964 D) (R : D \u2964 C) [monadic_right_adjoint R] [limits.has_limit (F \u22d9 R)] : limits.has_limit (F \u22d9 monad.comparison R) :=\n  monad.has_limit_of_comp_forget_has_limit (F \u22d9 monad.comparison R)\n\n/-- Any monadic functor creates limits. -/\ndef monadic_creates_limits {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C) [monadic_right_adjoint R] : creates_limits R :=\n  creates_limits_of_nat_iso (monad.comparison_forget R)\n\n/--\nThe forgetful functor from the Eilenberg-Moore category for a monad creates any colimit\nwhich the monad itself preserves.\n-/\ndef monadic_creates_colimit_of_preserves_colimit {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (R : D \u2964 C) (K : J \u2964 D) [monadic_right_adjoint R] [limits.preserves_colimit (K \u22d9 R) (left_adjoint R \u22d9 R)] [limits.preserves_colimit ((K \u22d9 R) \u22d9 left_adjoint R \u22d9 R) (left_adjoint R \u22d9 R)] : creates_colimit K R :=\n  creates_colimit_of_nat_iso (monad.comparison_forget R)\n\n/-- A monadic functor creates any colimits of shapes it preserves. -/\ndef monadic_creates_colimits_of_shape_of_preserves_colimits_of_shape {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (R : D \u2964 C) [monadic_right_adjoint R] [limits.preserves_colimits_of_shape J R] : creates_colimits_of_shape J R :=\n  creates_colimits_of_shape_of_nat_iso (monad.comparison_forget R)\n\n/-- A monadic functor creates colimits if it preserves colimits. -/\ndef monadic_creates_colimits_of_preserves_colimits {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C) [monadic_right_adjoint R] [limits.preserves_colimits R] : creates_colimits R :=\n  creates_colimits.mk\n    fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) => monadic_creates_colimits_of_shape_of_preserves_colimits_of_shape R\n\ntheorem has_limit_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] (F : J \u2964 D) (R : D \u2964 C) [limits.has_limit (F \u22d9 R)] [reflective R] : limits.has_limit F :=\n  has_limit_of_created F R\n\n/-- If `C` has limits of shape `J` then any reflective subcategory has limits of shape `J`. -/\ntheorem has_limits_of_shape_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] {J : Type v\u2081} [small_category J] [limits.has_limits_of_shape J C] (R : D \u2964 C) [reflective R] : limits.has_limits_of_shape J D :=\n  limits.has_limits_of_shape.mk fun (F : J \u2964 D) => has_limit_of_reflective F R\n\n/-- If `C` has limits then any reflective subcategory has limits. -/\ntheorem has_limits_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C) [limits.has_limits C] [reflective R] : limits.has_limits D :=\n  limits.has_limits.mk fun (J : Type v\u2081) (\ud835\udca5\u2081 : small_category J) => has_limits_of_shape_of_reflective R\n\n/--\nThe reflector always preserves terminal objects. Note this in general doesn't apply to any other\nlimit.\n-/\ndef left_adjoint_preserves_terminal_of_reflective {C : Type u\u2081} [category C] {D : Type u\u2082} [category D] (R : D \u2964 C) [reflective R] [limits.has_terminal C] : limits.preserves_limits_of_shape (discrete pempty) (left_adjoint R) :=\n  limits.preserves_limits_of_shape.mk\n    fun (K : discrete pempty \u2964 C) =>\n      let _inst : limits.has_terminal D := sorry;\n      let _inst_3 : creates_limits R := monadic_creates_limits R;\n      let _inst_6 : limits.preserves_limit (functor.empty D) R :=\n        category_theory.preserves_limit_of_creates_limit_and_has_limit (functor.empty D) R;\n      let _inst_7 : limits.preserves_limit (functor.empty C) (left_adjoint R) :=\n        limits.preserves_terminal_of_iso (left_adjoint R)\n          (functor.map_iso (left_adjoint R) (iso.symm (limits.preserves_terminal.iso R)) \u226a\u226b\n            as_iso (nat_trans.app (adjunction.counit (adjunction.of_right_adjoint R)) (\u22a4_D)));\n      limits.preserves_limit_of_iso_diagram (left_adjoint R) (iso.symm (functor.unique_from_empty K))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monad/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.36816642722145615}}
{"text": "/-\nCopyright (c) 2021 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov, Ya\u00ebl Dillies\n-/\n\nimport algebra.group.defs\nimport order.synonym\n\n/-!\n# Group structure on the order type synonyms\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nTransfer algebraic instances from `\u03b1` to `\u03b1\u1d52\u1d48` and `lex \u03b1`.\n-/\n\nopen order_dual\n\nvariables {\u03b1 \u03b2 : Type*}\n\n/-! ### `order_dual` -/\n\n@[to_additive] instance [h : has_one \u03b1] : has_one \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : has_mul \u03b1] : has_mul \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : has_inv \u03b1] : has_inv \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : has_div \u03b1] : has_div \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : has_smul \u03b1 \u03b2] : has_smul \u03b1 \u03b2\u1d52\u1d48 := h\n@[to_additive] instance order_dual.has_smul' [h : has_smul \u03b1 \u03b2] : has_smul \u03b1\u1d52\u1d48 \u03b2 := h\n@[to_additive order_dual.has_smul]\ninstance order_dual.has_pow [h : has_pow \u03b1 \u03b2] : has_pow \u03b1\u1d52\u1d48 \u03b2 := h\n@[to_additive order_dual.has_smul']\ninstance order_dual.has_pow' [h : has_pow \u03b1 \u03b2] : has_pow \u03b1 \u03b2\u1d52\u1d48 := h\n@[to_additive] instance [h : semigroup \u03b1] : semigroup \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : comm_semigroup \u03b1] : comm_semigroup \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : left_cancel_semigroup \u03b1] : left_cancel_semigroup \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : right_cancel_semigroup \u03b1] : right_cancel_semigroup \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : mul_one_class \u03b1] : mul_one_class \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : monoid \u03b1] : monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : comm_monoid \u03b1] : comm_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : left_cancel_monoid \u03b1] : left_cancel_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : right_cancel_monoid \u03b1] : right_cancel_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : cancel_monoid \u03b1] : cancel_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : cancel_comm_monoid \u03b1] : cancel_comm_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : has_involutive_inv \u03b1] : has_involutive_inv \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : div_inv_monoid \u03b1] : div_inv_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive order_dual.subtraction_monoid]\ninstance [h : division_monoid \u03b1] : division_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive order_dual.subtraction_comm_monoid]\ninstance [h : division_comm_monoid \u03b1] : division_comm_monoid \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : group \u03b1] : group \u03b1\u1d52\u1d48 := h\n@[to_additive] instance [h : comm_group \u03b1] : comm_group \u03b1\u1d52\u1d48 := h\n\n@[simp, to_additive] lemma to_dual_one [has_one \u03b1] : to_dual (1 : \u03b1) = 1 := rfl\n@[simp, to_additive] lemma of_dual_one [has_one \u03b1] : (of_dual 1 : \u03b1) = 1 := rfl\n@[simp, to_additive]\nlemma to_dual_mul [has_mul \u03b1] (a b : \u03b1) : to_dual (a * b) = to_dual a * to_dual b := rfl\n@[simp, to_additive]\nlemma of_dual_mul [has_mul \u03b1] (a b : \u03b1\u1d52\u1d48) : of_dual (a * b) = of_dual a * of_dual b := rfl\n@[simp, to_additive] lemma to_dual_inv [has_inv \u03b1] (a : \u03b1) : to_dual a\u207b\u00b9 = (to_dual a)\u207b\u00b9 := rfl\n@[simp, to_additive] lemma of_dual_inv [has_inv \u03b1] (a : \u03b1\u1d52\u1d48) : of_dual a\u207b\u00b9 = (of_dual a)\u207b\u00b9 := rfl\n@[simp, to_additive]\nlemma to_dual_div [has_div \u03b1] (a b : \u03b1) : to_dual (a / b) = to_dual a / to_dual b := rfl\n@[simp, to_additive]\nlemma of_dual_div [has_div \u03b1] (a b : \u03b1\u1d52\u1d48) : of_dual (a / b) = of_dual a / of_dual b := rfl\n@[simp, to_additive]\nlemma to_dual_smul [has_smul \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_dual (a \u2022 b) = a \u2022 to_dual b := rfl\n@[simp, to_additive]\nlemma of_dual_smul [has_smul \u03b1 \u03b2] (a : \u03b1) (b : \u03b2\u1d52\u1d48) : of_dual (a \u2022 b) = a \u2022 of_dual b := rfl\n@[simp, to_additive]\nlemma to_dual_smul' [has_smul \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_dual a \u2022 b = a \u2022 b := rfl\n@[simp, to_additive]\nlemma of_dual_smul' [has_smul \u03b1 \u03b2] (a : \u03b1\u1d52\u1d48) (b : \u03b2) : of_dual a \u2022 b = a \u2022 b := rfl\n@[simp, to_additive to_dual_smul, to_additive_reorder 1 4]\nlemma to_dual_pow [has_pow \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_dual (a ^ b) = to_dual a ^ b := rfl\n@[simp, to_additive of_dual_smul, to_additive_reorder 1 4]\nlemma of_dual_pow [has_pow \u03b1 \u03b2] (a : \u03b1\u1d52\u1d48) (b : \u03b2) : of_dual (a ^ b) = of_dual a ^ b := rfl\n@[simp, to_additive to_dual_smul', to_additive_reorder 1 4]\nlemma pow_to_dual [has_pow \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : a ^ to_dual b = a ^ b := rfl\n@[simp, to_additive of_dual_smul', to_additive_reorder 1 4]\nlemma pow_of_dual [has_pow \u03b1 \u03b2] (a : \u03b1) (b : \u03b2\u1d52\u1d48) : a ^ of_dual b = a ^ b := rfl\n\n/-! ### Lexicographical order -/\n\n@[to_additive] instance [h : has_one \u03b1] : has_one (lex \u03b1) := h\n@[to_additive] instance [h : has_mul \u03b1] : has_mul (lex \u03b1) := h\n@[to_additive] instance [h : has_inv \u03b1] : has_inv (lex \u03b1) := h\n@[to_additive] instance [h : has_div \u03b1] : has_div (lex \u03b1) := h\n@[to_additive] instance [h : has_smul \u03b1 \u03b2] : has_smul \u03b1 (lex \u03b2) := h\n@[to_additive] instance lex.has_smul' [h : has_smul \u03b1 \u03b2] : has_smul (lex \u03b1) \u03b2 := h\n@[to_additive lex.has_smul] instance lex.has_pow [h : has_pow \u03b1 \u03b2] : has_pow (lex \u03b1) \u03b2 := h\n@[to_additive lex.has_smul'] instance lex.has_pow' [h : has_pow \u03b1 \u03b2] : has_pow \u03b1 (lex \u03b2) := h\n@[to_additive] instance [h : semigroup \u03b1] : semigroup (lex \u03b1) := h\n@[to_additive] instance [h : comm_semigroup \u03b1] : comm_semigroup (lex \u03b1) := h\n@[to_additive] instance [h : left_cancel_semigroup \u03b1] : left_cancel_semigroup (lex \u03b1) := h\n@[to_additive] instance [h : right_cancel_semigroup \u03b1] : right_cancel_semigroup (lex \u03b1) := h\n@[to_additive] instance [h : mul_one_class \u03b1] : mul_one_class (lex \u03b1) := h\n@[to_additive] instance [h : monoid \u03b1] : monoid (lex \u03b1) := h\n@[to_additive] instance [h : comm_monoid \u03b1] : comm_monoid (lex \u03b1) := h\n@[to_additive] instance [h : left_cancel_monoid \u03b1] : left_cancel_monoid (lex \u03b1) := h\n@[to_additive] instance [h : right_cancel_monoid \u03b1] : right_cancel_monoid (lex \u03b1) := h\n@[to_additive] instance [h : cancel_monoid \u03b1] : cancel_monoid (lex \u03b1) := h\n@[to_additive] instance [h : cancel_comm_monoid \u03b1] : cancel_comm_monoid (lex \u03b1) := h\n@[to_additive] instance [h : has_involutive_inv \u03b1] : has_involutive_inv (lex \u03b1) := h\n@[to_additive] instance [h : div_inv_monoid \u03b1] : div_inv_monoid (lex \u03b1) := h\n@[to_additive order_dual.subtraction_monoid]\ninstance [h : division_monoid \u03b1] : division_monoid (lex \u03b1) := h\n@[to_additive order_dual.subtraction_comm_monoid]\ninstance [h : division_comm_monoid \u03b1] : division_comm_monoid (lex \u03b1) := h\n@[to_additive] instance [h : group \u03b1] : group (lex \u03b1) := h\n@[to_additive] instance [h : comm_group \u03b1] : comm_group (lex \u03b1) := h\n\n@[simp, to_additive] lemma to_lex_one [has_one \u03b1] : to_lex (1 : \u03b1) = 1 := rfl\n@[simp, to_additive] lemma of_lex_one [has_one \u03b1] : (of_lex 1 : \u03b1) = 1 := rfl\n@[simp, to_additive]\nlemma to_lex_mul [has_mul \u03b1] (a b : \u03b1) : to_lex (a * b) = to_lex a * to_lex b := rfl\n@[simp, to_additive]\nlemma of_lex_mul [has_mul \u03b1] (a b : lex \u03b1) : of_lex (a * b) = of_lex a * of_lex b := rfl\n@[simp, to_additive] lemma to_lex_inv [has_inv \u03b1] (a : \u03b1) : to_lex a\u207b\u00b9 = (to_lex a)\u207b\u00b9 := rfl\n@[simp, to_additive] lemma of_lex_inv [has_inv \u03b1] (a : lex \u03b1) : of_lex a\u207b\u00b9 = (of_lex a)\u207b\u00b9 := rfl\n@[simp, to_additive]\nlemma to_lex_div [has_div \u03b1] (a b : \u03b1) : to_lex (a / b) = to_lex a / to_lex b := rfl\n@[simp, to_additive]\nlemma of_lex_div [has_div \u03b1] (a b : lex \u03b1) : of_lex (a / b) = of_lex a / of_lex b := rfl\n@[simp, to_additive]\nlemma to_lex_smul [has_smul \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_lex (a \u2022 b) = a \u2022 to_lex b := rfl\n@[simp, to_additive]\nlemma of_lex_smul [has_smul \u03b1 \u03b2] (a : \u03b1) (b : lex \u03b2) : of_lex (a \u2022 b) = a \u2022 of_lex b := rfl\n@[simp, to_additive]\nlemma to_lex_smul' [has_smul \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_lex a \u2022 b = a \u2022 b := rfl\n@[simp, to_additive]\nlemma of_lex_smul' [has_smul \u03b1 \u03b2] (a : lex \u03b1) (b : \u03b2) : of_lex a \u2022 b = a \u2022 b := rfl\n@[simp, to_additive to_lex_smul, to_additive_reorder 1 4]\nlemma to_lex_pow [has_pow \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : to_lex (a ^ b) = to_lex a ^ b := rfl\n@[simp, to_additive of_lex_smul, to_additive_reorder 1 4]\nlemma of_lex_pow [has_pow \u03b1 \u03b2] (a : lex \u03b1) (b : \u03b2) : of_lex (a ^ b) = of_lex a ^ b := rfl\n@[simp, to_additive to_lex_smul, to_additive_reorder 1 4]\nlemma pow_to_lex [has_pow \u03b1 \u03b2] (a : \u03b1) (b : \u03b2) : a ^ to_lex b = a ^ b := rfl\n@[simp, to_additive of_lex_smul, to_additive_reorder 1 4]\nlemma pow_of_lex [has_pow \u03b1 \u03b2] (a : \u03b1) (b : lex \u03b2) : a ^ of_lex b = a ^ b := rfl\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebra/group/order_synonym.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.36809772630593524}}
{"text": "import Kenny.sites.basic order.complete_boolean_algebra\n\nuniverses u v\n\nnamespace lattice\n\nclass Sup_lattice (X : Type u) extends lattice X, has_Sup X :=\n(le_Sup : \u2200 {s : set X} {a : X}, a \u2208 s \u2192 a \u2264 Sup s)\n(Sup_le : \u2200 {s : set X} {a : X}, (\u2200 (b : X), b \u2208 s \u2192 b \u2264 a) \u2192 Sup s \u2264 a)\n\nclass Sup_distrib_lattice (X : Type u) extends Sup_lattice X :=\n(inf_Sup_le {} : \u2200 {x : X} {s : set X}, x \u2293 lattice.Sup s \u2264 lattice.Sup ((\u2293) x '' s))\n\nsection Sup_lattice\n\ninstance complete_lattice.to_Sup_lattice {X : Type u} [complete_lattice X] : Sup_lattice X :=\n{ .. (infer_instance : complete_lattice X) }\n\nvariables {X : Type u} [Sup_lattice X]\n\ntheorem le_Sup' {s : set X} {a : X} : a \u2208 s \u2192 a \u2264 Sup s :=\nSup_lattice.le_Sup\n\ntheorem Sup_le' {s : set X} {a : X} : (\u2200 (b : X), b \u2208 s \u2192 b \u2264 a) \u2192 Sup s \u2264 a :=\nSup_lattice.Sup_le\n\ntheorem Sup_singleton' (x : X) : Sup {x} = x :=\nle_antisymm (Sup_le' $ \u03bb b hb, set.eq_of_mem_singleton hb \u25b8 le_refl _) $\nle_Sup' $ set.mem_singleton x\n\nend Sup_lattice\n\nsection Sup_discrete_lattice\n\ninstance complete_distrib_lattice.to_Sup_distrib_lattice {X : Type u} [complete_distrib_lattice X] : Sup_distrib_lattice X :=\n{ inf_Sup_le := \u03bb x s, by rw [inf_Sup_eq, Sup_image],\n  .. (infer_instance : complete_distrib_lattice X) }\n\nvariables {X : Type u} [Sup_distrib_lattice X]\n\ntheorem inf_Sup {x : X} {s : set X} : x \u2293 lattice.Sup s = lattice.Sup ((\u2293) x '' s) :=\nle_antisymm Sup_distrib_lattice.inf_Sup_le $ Sup_le' $ \u03bb b \u27e8c, hcs, hxcb\u27e9, hxcb \u25b8 inf_le_inf (le_refl x) (le_Sup' hcs)\n\nend Sup_discrete_lattice\n\nend lattice\n\nnamespace category_theory\n\nopen lattice\n\nvariables {X : Type u}\n\nclass is_univalent (X : Type u) [category.{v} X] : Prop :=\n(univalent : \u2200 x y : X, \u2200 e : x \u2245 y, x = y)\n\ntheorem eq_of_iso [category.{v} X] [is_univalent X] {x y : X} (e : x \u2245 y) : x = y :=\nis_univalent.univalent x y e\n\ninstance is_univalent_partial_order [partial_order X] : is_univalent X :=\n\u27e8\u03bb x y e, le_antisymm e.1.1.1 e.2.1.1\u27e9\n\ninstance semilattice_inf.has_pullback [semilattice_inf X] : has_pullback X :=\n\u27e8\u03bb F,\n{ cone :=\n  { X := F.obj pullback_diagram.base_left \u2293 F.obj pullback_diagram.base_right,\n    \u03c0 :=\n    { app := \u03bb p, pullback_diagram.rec_on p \u27e8\u27e8inf_le_left\u27e9\u27e9 \u27e8\u27e8inf_le_right\u27e9\u27e9\n        \u27e8\u27e8le_trans inf_le_left (F.map pullback_diagram.hom.to_target_left).down.down\u27e9\u27e9,\n      naturality' := by intros; ext } },\n  is_limit :=\n  { lift := \u03bb c, \u27e8\u27e8le_inf (c.\u03c0.app pullback_diagram.base_left).down.down (c.\u03c0.app pullback_diagram.base_right).down.down\u27e9\u27e9,\n    fac' := by intros; ext,\n    uniq' := by intros; ext } }\u27e9\n\ninstance Sup_lattice.has_site [Sup_distrib_lattice X] : has_site X :=\n{ cov := \u03bb U, { c | U \u2264 Sup (sigma.fst '' c) },\n  iso_mem := \u03bb U V e, show U \u2264 _, by rw [set.image_singleton, Sup_singleton']; exact e.2.1.1,\n  comp_mem := \u03bb U S HS F HF, le_trans HS $ Sup_le' $ \u03bb x hx, let \u27e8m, hmS, hmx\u27e9 := hx in\n    hmx \u25b8 le_trans (HF m hmS) (Sup_le' $ \u03bb y hy, let \u27e8n, hnFS, hny\u27e9 := hy in\n      le_Sup' \u27e8\u27e8n.1, \u27e8\u27e8le_trans n.2.1.1 m.2.1.1\u27e9\u27e9\u27e9, \u27e8m, hmS, n, hnFS, rfl\u27e9, hny\u27e9),\n  pullback_mem := \u03bb U S HS V f,\n  calc  V\n      \u2264 V \u2293 Sup (sigma.fst '' S) : le_inf (le_refl V) (le_trans f.1.1 HS)\n  ... = Sup ((\u2293) V '' (sigma.fst '' S)) : inf_Sup\n  ... = Sup ((\u2293) V \u2218 sigma.fst '' S) : congr_arg Sup (set.image_comp _ _ S).symm\n  ... \u2264 Sup (sigma.fst '' {m | \u2203 t \u2208 S, (\u27e8_, pullback.fst f t.2\u27e9 : \u03a3 W, W \u27f6 V) = m}) :\n    Sup_le' (\u03bb b \u27e8c, hcs, hb\u27e9, le_Sup' \u27e8\u27e8V \u2293 c.1, \u27e8\u27e8inf_le_left\u27e9\u27e9\u27e9, \u27e8c, hcs, rfl\u27e9, hb\u27e9) }\n\nend category_theory\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/Kenny/sites/lattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7981867777396212, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.3679774975631955}}
{"text": "import analysis.calculus.local_extr\nimport analysis.calculus.times_cont_diff\nimport analysis.calculus.iterated_deriv\nimport tactic data.fin\nopen set\n\nnamespace rolle_general\n\nlemma shing (n : \u2115) (i j : fin (n+1)) (h : (j.val : fin (n+2)) < (i.val.succ : fin (n+2))) : \n    j.val < i.val.succ :=\nbegin\n  have h1 : (j.val : fin (n+2)).val = j.val,\n  { rw fin.coe_val_of_lt (show j.1 < n + 2, by linarith [j.2]) },\n  have h2 : (i.val.succ : fin (n+2)).val = i.val.succ,\n  { rw fin.coe_val_of_lt (show i.1 + 1 < n + 2, by linarith [i.2]) },\n  change (j.val : fin (n+2)).val < (i.val.succ : fin (n+2)).val at h,\n  rwa [h1, h2] at h,\nend\n\nexample (n : \u2115) (i j : fin n) (h : (j.val : fin (n+2)) < (i.val.succ : fin (n+2))) : j.val < i.val.succ :=\nbegin\n  change (j.val : fin (n+2)).val < (i.val.succ : fin (n+2)).val at h,\n  rwa [fin.coe_val_of_lt (show j.1 < n + 2, by linarith [j.2]),\n       fin.coe_val_of_lt (show i.1 + 1 < n + 2, by linarith [i.2])] at h,\nend\n\nlemma one_step (n : \u2115) (a b : \u211d) (f : \u211d \u2192 \u211d) (x : fin (n+2) \u2192 \u211d) (hx : strict_mono x) :\n    \u2200 (f : \u211d \u2192 \u211d), continuous_on f (Icc a b) \u2192 \n    (\u2200 i, x i \u2208 (Icc a b) \u2227 f (x i) = 0)  \u2192\n    \u2203 (xp : fin(n+1) \u2192 \u211d), strict_mono xp \u2227 \u2200 (i : fin (n+1)), xp i \u2208 (Icc a b) \u2227 deriv f (xp i) = 0 :=\nbegin\n    intros f hf hxi,\n    have h1 : \u2200 (i : fin (n+1)), \u2203 y \u2208 (Ioo (x i) (x (i+1))), deriv f y = 0,\n        sorry,\n    choose xp hxp using h1, \n    use xp, split,\n    intros i j hij,\n    have hi := (hxp i).1, have hj := (hxp j).1,\n    cases hi with hi1 hi2, cases hj with hj1 hj2,\n    rcases lt_trichotomy ((i+1) : fin (n+2) ) (j : fin (n+2)) with h1|h2|h3,\n    --rcases lt_trichotomy (fin.cast_succ (i+1)) (fin.cast_succ j) with h1|h2|h3,\n    -- case (i+1) < j\n    have hii1 := hx h1, linarith, \n    -- case (i+1) = j\n    rw h2 at hi2, linarith,\n    -- case j < (i+1) is not possible because i < j\n    exfalso, \n        have h3n : (j : \u2115) < ((i + 1) : \u2115), \n            norm_num at h3,\n            have m3 := shing n i j h3, exact m3,\n        have gf1 := nat.lt_succ_iff.mp h3n,\n        have hijn : (i : \u2115) < (j : \u2115), exact hij, --strange as it looks, linarith needs this\n        linarith,\n    intro i, split,\n    swap, exact (hxp i).2,\n    have g0 := (hxp i).1,\n    split,\n    have g1 := (hxi i).1, cases g1 with g11 g12, cases g0 with g01 g02,\n    linarith,\n    have g1 := (hxi (i+1)).1, cases g1 with g11 g12, cases g0 with g01 g02,\n    linarith,\nend\n\nexample (a b : \u211d) (hab : a < b) (f : \u211d \u2192 \u211d) (n : \u2115) (hf : times_cont_diff_on \u211d (n+1) f (Ioo a b) ) :\n  times_cont_diff_on \u211d n (deriv f) (Ioo a b) :=\nbegin\n  have : deriv f = (\u03bb u : \u211d \u2192L[\u211d] \u211d, u 1) \u2218 (fderiv \u211d f), by { ext x, refl },\n  simp only [this],\n  have : times_cont_diff_on \u211d n (fderiv \u211d f) (Ioo a b),\n  { apply ((times_cont_diff_on_succ_iff_fderiv_within (unique_diff_on_Ioo a b)).1 hf).2.congr,\n    assume x hx,\n    calc fderiv \u211d f x = fderiv_within \u211d f univ x : by simp\n    ... = fderiv_within \u211d f (univ \u2229 Ioo a b) x :\n      (fderiv_within_inter (Ioo_mem_nhds hx.1 hx.2) unique_diff_within_at_univ).symm\n    ... = fderiv_within \u211d f (Ioo a b) x : by simp },\n  apply times_cont_diff.comp_times_cont_diff_on _ this,\n  exact (is_bounded_bilinear_map_apply.is_bounded_linear_map_left _).times_cont_diff\nend\n\n\n--This proof will unfortunately **not** work for Icc:\n/-\n\nexample (a b : \u211d) (hab : a < b) (f : \u211d \u2192 \u211d) (n : \u2115) (hf : times_cont_diff_on \u211d (n+1) f (Icc a b) ) :\n  times_cont_diff_on \u211d n (deriv f) (Icc a b) :=\nbegin\n  have : deriv f = (\u03bb u : \u211d \u2192L[\u211d] \u211d, u 1) \u2218 (fderiv \u211d f), by { ext x, refl },\n  simp only [this],\n  have : times_cont_diff_on \u211d n (fderiv \u211d f) (Icc a b),\n  { apply ((times_cont_diff_on_succ_iff_fderiv_within (unique_diff_on_Icc hab)).1 hf).2.congr,\n    assume x hx,\n    calc fderiv \u211d f x = fderiv_within \u211d f univ x : by simp\n    ... = fderiv_within \u211d f (univ \u2229 Icc a b) x :\n      (fderiv_within_inter (Ioo_mem_nhds hx.1 hx.2) unique_diff_within_at_univ).symm\n    ... = fderiv_within \u211d f (Icc a b) x : by simp },\n  apply times_cont_diff.comp_times_cont_diff_on _ this,\n  exact (is_bounded_bilinear_map_apply.is_bounded_linear_map_left _).times_cont_diff\nend\n\n-/\n\n\ntheorem general_rolle (n : \u2115) (A B : \u211d) (hAB : A < B) (x : fin (n+2) \u2192 \u211d) (hx : strict_mono x) :\n    \u2200 (f : \u211d \u2192 \u211d), times_cont_diff_on \u211d n f (Icc A B) \u2192 \n    (\u2200 i, x i \u2208 (Icc A B) \u2227 f (x i) = 0)  \u2192 \n    \u2203 c \u2208 Ioo A B, iterated_deriv (n+1) f c = 0 :=\nbegin\n    induction n with d hd,\n    { -- base case, just plain Rolle `exists_deriv_eq_zero`\n        intros f hf hi,\n        norm_cast at hf,\n        rw times_cont_diff_on_zero at hf,\n        --unfold strict_mono at hx,\n        have h001 : 0 < 1, linarith,\n        -- The above was needed because linarith fails on next one:\n        have h002 : (0 : fin 2) < (1 : fin 2), exact h001, -- linarith fails !!!???\n        have hx01 := hx h002, clear h001, clear h002,\n        have hx0 := hi 0,\n        have hx1 := hi 1,\n        cases hx0 with h11 h121, cases hx1 with h21 h221, \n        have h3 : 0 < 1, linarith,\n        have h41 : continuous_on f (Icc (x 0) (x 1)), \n            have h412 : (Icc (x 0) (x 1)) \u2286 Icc A B, \n            intros z hz, cases hz with hz1 hz2,\n            cases h11 with h11z h12z,\n            split, linarith,\n            cases h21 with h21z h22z,\n            linarith,\n            exact continuous_on.mono hf h412,\n        have h42 : f (x 0) = f (x 1), rw [h121, h221], \n        have h5 := exists_deriv_eq_zero f hx01 h41 h42, \n        cases h5 with c hc, cases hc with hc1 hc2,\n        have h6 : c \u2208 Ioo A B, \n            cases hc1 with h61 h62, cases h11 with h111 h112,\n            have h71: A < c, linarith,\n            cases h21 with h211 h212,\n            have h72 : c < B, linarith,\n            split, exact h71, exact h72,\n        rw iterated_deriv_one,\n        use [c, h6], exact hc2,\n    },\n    { -- induction step\n        -- the derivative is in C\u1d48\n        intros f hf hi,\n        have hfc := times_cont_diff_on.continuous_on hf,\n        have H := one_step d.succ A B f x hx f hfc hi,\n        cases H with xp hxp, cases hxp with hxpx hxpi,\n        set g := deriv f with hg,\n        --have h1 := times_cont_diff_on_succ_iff_has_fderiv_within_at.mp hf (x 0) (hi 0).1, \n        -- above is not immediately useful\n        have h0 := unique_diff_on_Icc hAB,\n        have h00 : ((d + 1) : with_top \u2115) \u2264 d.succ, norm_cast, \n        have h1 := times_cont_diff_on.fderiv_within hf h0 h00, \n        simp only [] at h1,\n        have h000 : (1 : with_top \u2115) \u2264 d.succ, norm_cast, sorry,\n        have h01 := times_cont_diff_on.continuous_on_iterated_deriv_within hf h000 h0,\n        have hder : times_cont_diff_on \u211d d g (Icc A B), -- should come from hf\n            rw hg,\n            sorry, -- this seems much harder to get than it should!!!\n        have hdg := hd xp hxpx g hder, clear hd,\n        have G := hdg hxpi,\n        have K : iterated_deriv (d.succ + 1) f = iterated_deriv d.succ g,\n            apply iterated_deriv_succ',\n        rw \u2190 K at G,\n        exact G,\n    },\n    done\nend\n\nend rolle_general\n\n------------------ Scratch space below here ------------------------------\n\n#check deriv\n#check times_cont_diff_on.continuous_on_iterated_deriv_within\n#check times_cont_diff_on.differentiable_on_iterated_deriv_within\nvariables (f : \u211d \u2192 \u211d)\nexample (a b : \u211d) (hab : a < b) (f : \u211d \u2192 \u211d) (n : \u2115) (hf : times_cont_diff_on \u211d (n+1) f (Ioo a b) ) :\n    times_cont_diff_on \u211d n (deriv f) (Ioo a b) :=\nbegin \n    refine times_cont_diff_on_of_differentiable_on_deriv _,\n    have h0 := unique_diff_on_Ioo a b,\n    have h := (times_cont_diff_on_iff_continuous_on_differentiable_on_deriv h0).mp hf,\n    cases h with h1 h2,\n    intros m hm, \n    have g := h2 m,\n    have g1 : (m : with_top \u2115) < n + 1, sorry,\n    have g2 := g g1,\n    sorry,\n    --refine times_cont_diff.times_cont_diff_on _\nend\n\n#check unique_diff_on_Ioo\n#check times_cont_diff_iff_continuous_differentiable.mp  \n#check iterated_deriv_within_succ\n#check differentiable \u211d f\n#check times_cont_diff_on_iff_continuous_on_differentiable_on_deriv\n#check times_cont_diff_on \u211d 3 f\n#check times_cont_diff_zero\n#check  times_cont_diff_on_succ_iff_has_fderiv_within_at\n", "meta": {"author": "stanescuUW", "repo": "numerical-analysis-with-Lean", "sha": "98e6974f8b68cc5232ceff40535d776a33444c73", "save_path": "github-repos/lean/stanescuUW-numerical-analysis-with-Lean", "path": "github-repos/lean/stanescuUW-numerical-analysis-with-Lean/numerical-analysis-with-Lean-98e6974f8b68cc5232ceff40535d776a33444c73/src/Interpolation/try-rolle-Icc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3678823312930567}}
{"text": "/- Copyright (c) 2020 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: ...\n-/\nimport linear_algebra.basic\nimport algebra.group.hom\n/- we will probably need to import other files soon too, like `linear_algebra.finite_dimensional`. -/\n-- set_option trace.simplify true\nrun_cmd mk_simp_attr `RW_REP\nmeta def rw_simp  : tactic unit :=\n`[  try {simp only with RW_REP}, try {exact rfl}]\nrun_cmd add_interactive [`rw_simp]\n\nuniverse variables u v w w'\nopen group\nopen linear_map   \nopen linear_equiv\nopen submodule \nopen linear_map.general_linear_group\n/-\n     I thinck i add three  lemma into  mathlib  line 1922 at the file linear_algebra.bassic \ndef to_linear_map (f : general_linear_group R M) : M \u2192\u2097[R] M :=(to_linear_equiv f).to_linear_map \ndef to_linear_map_inv (f : general_linear_group R M) :  \nM \u2192\u2097[R] M := (to_linear_equiv f\u207b\u00b9).to_linear_map\ndef to_fun (f : general_linear_group R M) : M \u2192 M :=  f.val \n-/\nattribute [RW_REP] coe_add coe_neg coe_smul coe_mk subtype.eta   eq.symm\nnamespace NOTATION\nnotation  `GL`                := general_linear_group\nnotation  `L`:80 f :80        := general_linear_group.to_linear_map f\nnotation  `F`:80 f :82        := general_linear_group.to_fun f   --- add in mathlib \nnotation  a` \u229a `:80 b:80      := linear_map.comp a b  \nend NOTATION\n/- maybe needs a shorter name -/\n/-- A representation of a group `G` is an `R`-module `M` with a group homomorphisme from  `G` to\n  `GL(M)`. Normally `M` is a vector space, but we don't need that for the definition. -/\n\ndef group_representation (G R M : Type*) [group G] [ring R] [add_comm_group M] [module R M] :\n  Type* :=  G \u2192* GL R M\n-- open NOTATION\nvariables {G : Type u} {R : Type v} {M : Type w} {M' : Type w'}\n  [group G] [ring R] [add_comm_group M] [module R M] [add_comm_group M'] [module R M']\n\ninstance : has_coe_to_fun (group_representation G R M) := \u27e8_, \u03bb g, g.to_fun\u27e9   --- \ndef has_coe_to (\u03c1 : group_representation G R M) : G \u2192 (M \u2192\u2097[R] M ) := \u03bb g, L \u03c1 g  \nnotation  `\u27e6` \u03c1 `\u27e7` :=  has_coe_to \u03c1  \nnamespace MY_TEST\nvariables (f  : M \u2192\u2097[R] M) ( x y : M) \nvariables (\u03c1 : group_representation G R M)\ninclude \u03c1 \nexample : f (x+ y) = f(x)+f(y) :=  f.add x y  \n\n@[RW_REP] theorem   linearity (g : G ) (x y : M): \u27e6 \u03c1 \u27e7 g (x+y) = \u27e6\u03c1\u27e7 g (x) + \u27e6\u03c1\u27e7 g (y) := \nbegin exact (\u27e6\u03c1\u27e7 g).add x y end\n\n@[RW_REP] theorem   smul' (g : G)  (r : R)(m : M) : (\u27e6 \u03c1\u27e7  g) (r \u2022 m) = r \u2022 ((\u27e6 \u03c1 \u27e7  g) m) := begin \n     exact ( \u27e6 \u03c1 \u27e7 g).smul r m,\nend\nvariables  (g g' : G) \n\nvariables (p : submodule R M)\n\n@[RW_REP]lemma F_linearity (x y : M) (g : G) : (F \u03c1 g) (x+y) = (F \u03c1 g) x + (F \u03c1 g) y := begin \n     exact (L \u03c1 g).add x y,   --- the same for L !  \nend \n\nvariables (h : \u2200 x : M, \u2200 g : G,  (L \u03c1  g) x \u2208 p)\n\n--- le m\u00e9canisme est le suivant : \n--- \u03c1 : G \u2192* GL R M \n--- le probl\u00e8me \u00e9tant que :: c'est un morphsime de groupe  mais c'est une structure\n--- GL R M donc convertion merdique via L \n--- Il y a \u21d1 \n\ntheorem L_to_F  (g : G) : (F \u03c1 g)  = \u27e6\u03c1\u27e7 g  := rfl \n\nexample (\u03c1 : group_representation G R M) ( g g' : G) : \u03c1 (g * g') = \u03c1 g * \u03c1 g' := \u03c1.map_mul g g' \n@[RW_REP]lemma rmap_mul (\u03c1 : group_representation G R M) ( g g' : G) :\n      \u27e6 \u03c1 \u27e7  (g * g')  =  \u27e6\u03c1\u27e7  g   \u229a  \u27e6 \u03c1 \u27e7  g' := \nbegin \n     ext, rw comp_apply, iterate 3{rw \u2190 L_to_F}, rw \u03c1.map_mul,exact rfl,\nend\n-- @[RW_REP]lemma rmap_mul' (\u03c1 : group_representation G R M) ( g g' : G) :\n--           \u27e6\u03c1\u27e7  g   \u229a  \u27e6 \u03c1 \u27e7  g' = \u27e6 \u03c1 \u27e7  (g * g') := \n-- begin \n--      ext, rw comp_apply, iterate 3{rw \u2190 L_to_F}, rw \u03c1.map_mul,exact rfl,\n-- end\n@[RW_REP] lemma rmap_one (\u03c1 : group_representation G R M)  :\n      \u27e6 \u03c1 \u27e7  (1)  =  linear_map.id := begin \n      ext,rw \u2190 L_to_F,rw \u03c1.map_one,exact rfl,\n      end \n@[RW_REP] lemma rmap_inv_mul (\u03c1 : group_representation G R M)(g: G)  :\n      \u27e6 \u03c1 \u27e7  (g * g\u207b\u00b9 )  =  linear_map.id := begin \n      rw mul_inv_self,exact rmap_one \u03c1,\n      end   \n@[RW_REP] lemma rmap_mul_inv (\u03c1 : group_representation G R M)(g: G)  :\n      \u27e6 \u03c1 \u27e7  (g\u207b\u00b9  * g )  =  linear_map.id := begin \n      rw inv_mul_self,exact rmap_one \u03c1,\n      end  \n@[RW_REP] lemma rmap_inv' (\u03c1 : group_representation G R M) (g : G) : \n     \u27e6\u03c1\u27e7  g   \u229a  \u27e6 \u03c1 \u27e7  g\u207b\u00b9  = linear_map.id := begin \n          rw \u2190 rmap_mul,exact rmap_inv_mul \u03c1 g,\n     end\n@[RW_REP] lemma rmap_inv''(\u03c1 : group_representation G R M) (g : G) : \n     \u27e6\u03c1\u27e7  g\u207b\u00b9    \u229a  \u27e6 \u03c1 \u27e7  g  = linear_map.id := begin \n          rw \u2190 rmap_mul,exact rmap_mul_inv \u03c1 g ,\n     end\n@[RW_REP] lemma rmap_inv_apply'' (\u03c1 : group_representation G R M) (g : G)(x : M) : \n     (\u27e6\u03c1\u27e7  g\u207b\u00b9    \u229a  \u27e6 \u03c1 \u27e7  g ) x = x := begin \n          rw rmap_inv'',exact rfl,\n     end\n@[RW_REP] lemma rmap_inv_apply' (\u03c1 : group_representation G R M) (g : G)(x : M) : \n     (\u27e6\u03c1\u27e7  g    \u229a  \u27e6 \u03c1 \u27e7  g\u207b\u00b9  ) x = x := begin \n          rw rmap_inv',exact rfl,\n     end\ndef has_inv (\u03c1 : group_representation G R M)(g : G) :  M \u2243\u2097[R]  M :=  { \n     to_fun := \u27e6 \u03c1 \u27e7 g , \n     add := linearity \u03c1 g , \n     smul :=  smul' \u03c1 g,\n     inv_fun :=  \u27e6 \u03c1 \u27e7 g\u207b\u00b9, \n     left_inv :=  rmap_inv_apply'' \u03c1 g,  \n     right_inv :=  rmap_inv_apply' \u03c1 g\n}\n-- @[RW_REP] lemma rmap_inv (\u03c1 : group_representation G R M)(g : G) :  -- 207 \n--      \u27e6 \u03c1 \u27e7 g\u207b\u00b9 =  to_linear_map_inv (has_inv' \u03c1 g )   := begin \n--           ext, \n--           sorry, \n--      end \n@[RW_REP]lemma star_is_oo (\u03c1 : group_representation G R M) ( g g' : G) :\n      \u27e6\u03c1\u27e7  g   \u229a  \u27e6 \u03c1 \u27e7  g'  =  \u27e6\u03c1\u27e7  g   *  \u27e6 \u03c1 \u27e7  g' :=  by rw_simp\n\n\n@[RW_REP]lemma rmap_map_assoc (\u03c1 : group_representation G R M)( g1 g2 g3 : G) : \u27e6\u03c1\u27e7 (g1 * g2 *g3)  =\n     \u27e6\u03c1\u27e7 (g1) \u229a  (\u27e6\u03c1\u27e7  g2 \u229a   \u27e6\u03c1\u27e7 g3)  := \nbegin\n      rw  group.mul_assoc,rw rmap_mul,rw rmap_mul,\n     -- rw_simp,\nend\nexample (\u03c1 : group_representation G R M)( g1 g2 g3 g4 : G) : \u27e6\u03c1\u27e7 (g1 * (g2 *g3 * g4))  =\n     \u27e6\u03c1\u27e7 (g1) \u229a  (\u27e6\u03c1\u27e7  g2 \u229a   \u27e6\u03c1\u27e7 g3) * \u27e6 \u03c1 \u27e7 g4  := \nbegin\n\n     rw_simp,\nend\n@[RW_REP]lemma times_to_oo (\u03c1 : group_representation G R M)( g g' : G) :  \u27e6 \u03c1 \u27e7  (g * g')  =  \u27e6\u03c1\u27e7  g  *  \u27e6 \u03c1 \u27e7  g' := begin \n     rw_simp, \nend\nexample (\u03c1 : group_representation G R M)( g1 g2 g3 : G) : \u27e6\u03c1\u27e7 (g1 * g2 *g3)  =\n     \u27e6\u03c1\u27e7 (g1) \u229a  \u27e6\u03c1\u27e7  g2  *   \u27e6\u03c1\u27e7 g3  := \nbegin\n     rw_simp,  \nend\n@[RW_REP]lemma mul_to_composition_of_function (\u03c1 : group_representation G R M) ( g g' : G) :\n \u27e6 \u03c1\u27e7  (g * g')  = ( \u27e6 \u03c1\u27e7  g )  *  (\u27e6  \u03c1 \u27e7  g') := begin \n  rw_simp, \nend\n@[RW_REP]lemma mixte_linearity (\u03c1 : group_representation G R M) ( g g' : G) (x y : M) (r : R): \n       \u27e6 \u03c1\u27e7  (g * g') (x+r \u2022 y) = \u27e6 \u03c1 \u27e7 g ( \u27e6 \u03c1 \u27e7 g' x )+ r \u2022 \u27e6 \u03c1 \u27e7 g ( \u27e6 \u03c1 \u27e7 g' y ) := begin\n          iterate 2 {rw \u2190 comp_apply},rw_simp, rw \u2190 rmap_mul,\nend\n-- @[RW_REP]lemma L_to_F (g : G) :  (L \u03c1 g).to_fun = (F \u03c1 g) := rfl\n\n\nexample :    \u27e6 \u03c1 \u27e7  (g * g') = ( \u27e6 \u03c1\u27e7  g) \u229a   ( \u27e6\u03c1\u27e7 g')  := by rw_simp\nlemma mul_one (\u03c1 : group_representation G R M) : (L \u03c1 1) = 1 := begin \nrw \u03c1.map_one, exact rfl,\nend\nend MY_TEST\n\nnamespace group_representation \n/- do we want this instance? Then we don't have to write `(\u03c1 g).1 x` instead of `\u03c1 g x`. -/\ninstance : has_coe (general_linear_group R M) (M \u2192\u2097[R] M) := \u27e8\u03bb x, x.1\u27e9\nprotected structure morphism (\u03c1 : group_representation G R M) (\u03c0 : group_representation G R M') :\n  Type (max w w') :=\n  (linear_map : M \u2192\u2097[R] M')\n  (commute : \u2200(g : G), linear_map \u2218 \u03c1 g  = \u03c0 g \u2218 linear_map)\nprotected structure equiv (\u03c1 : group_representation G R M) (\u03c0 : group_representation G R M') :\n  Type (max w w') :=\n  (f : M \u2243\u2097[R] M')\n  (commute : \u2200(g : G), f \u2218  \u03c1 g = \u03c0 g \u2218  f)\n\nvariables (\u03c1 : group_representation G R M)\nvariables (g g' : G)(x : M)\n-- example (x y : M) (g : G) :  \u03c1  g (x+y)= \u03c1 g x + \u03c1 g y := begin rw (L \u03c1 g).add x y, end \n\n\n\n\nnamespace stability  \n/-\n     We define the notion of stable submodule. \n     We make a sub-representation.\n     ligne  384 algebra module submodule (conduit)\n     il y a des lemmes de convertions.\n -/\nvariables {\u03c11 : group_representation G R M}{p : submodule R M}\n/-\n     Strategy maths : We have \u03c1 g x \u2208 p for x \u2208 p so \n     you have a map : \u03c1' g : p \u2192 p ... linear_map, invertible (restriction of \u03c1 g\u207b\u00b9 ) and trivial trivial trivial \n     For lean : this is not trivial. We have to verify some stuff. \n     Lemma to try to deal with convertion  \n-/\n@[RW_REP]lemma sub_module.eq_trans (x y : M) (hx : x \u2208 p)(hy : y \u2208 p) :  \n(x : M) = (y : M) \u2192  (\u27e8x,hx\u27e9 : p)   = (\u27e8 y,hy\u27e9 : p )  := begin \n     intros,congr ; try { assumption },\nend \n@[RW_REP] lemma sub_module.eq_trans' (x y : p) : (x : M) = (y : M) \u2192 x = y := begin\n          intros,rcases x,rcases y,congr; try {assumption},\nend\n-- lemma sub_module_val (x : M) (hx : x \u2208 p) : (\u27e8x,hx \u27e9 : p).val = (x : M) := rfl\n\ndef stable_sub_module (\u03c1 : group_representation G R M)(p : submodule R M) :=  \n                     \u2200 g : G, \u2200 x : p, (\u27e6\u03c1\u27e7 \u03c1 g) x \u2208 p \n/-\n     First Step : we define G \u2192 (p \u2192\u2097[R] p)\n-/\n\ndef restriction (h : stable_sub_module \u03c1 p) : G \u2192 (p \u2192\u2097[R] p)  := \u03bb g, begin\n     exact {\n          to_fun    := \u03bb x, \u27e8( \u27e6 \u03c1\u27e7  g ) x, h g x\u27e9,  \n          add       := begin  intros x y, rw_simp, end,  \n          smul      := begin intro c, intros x, rw_simp, end\n     }, \nend\nopen MY_TEST\n@[RW_REP]lemma restriction_ext (h : stable_sub_module \u03c1 p) (y : p) \n: (( \u27e6\u03c1\u27e7  g) y : M ) = (restriction \u03c1 h g y : M) := rfl \n@[RW_REP]lemma restriction_ext' (h : stable_sub_module \u03c1 p) (y : p) \n:   (restriction \u03c1 h g y : M) = (( \u27e6\u03c1\u27e7  g) y : M ) := rfl \ndef  restriction_equiv (h : stable_sub_module \u03c1 p) (g : G) :  (p \u2243 p) :=   \n{ to_fun := (restriction \u03c1 h g),\n  inv_fun := (restriction \u03c1 h g\u207b\u00b9),\n  left_inv := begin \n               intros x,\n                    apply sub_module.eq_trans',\n                    iterate 2 {rw \u2190 restriction_ext},\n                    rw \u2190 comp_apply,\n                    apply rmap_inv_apply'',\n                    -- rw [\u2190 comp_apply,\u2190  mul_to_composition_of_linear_map, inv_mul_self, \u03c1.map_one],\n                    -- exact rfl,end,\n               end\n  , right_inv := begin \n               intros x,\n               apply sub_module.eq_trans',  \n               iterate 2 {rw \u2190 restriction_ext}, \n               rw  \u2190 comp_apply, \n               apply rmap_inv_apply',\n  end }\ndef Restriction (h : stable_sub_module \u03c1 p) (g : G) : p \u2243\u2097[R] p :=\n { .. restriction \u03c1 h g, .. restriction_equiv \u03c1 h g}\n\n\ndef sub_representation (h : stable_sub_module \u03c1 p) : group_representation G R p := \n{ to_fun := \u03bb g, of_linear_equiv (Restriction \u03c1 h g),\n  map_one' := begin \n               rw units.ext_iff, --- Creer un helper pour la sous structure car c'est chiant\n               ext,rcases x,\n               apply sub_module.eq_trans,\n                rw rmap_one,exact rfl,\n               end,\n  map_mul' := begin intros g1 g2,rw units.ext_iff, ext, rcases x,\n                    apply sub_module.eq_trans,rw_simp,rw of_linear_equiv_val, \n                    rw rmap_mul,\n                    rw comp_apply, exact rfl,\n end }\n variables (h : stable_sub_module \u03c1 p)\n #check sub_representation \u03c1 h\n notation \u03c1 `/`h := sub_representation \u03c1 h \n #check  \u27e6 \u03c1 / h\u27e7  g\n @[RW_REP]lemma sub_representation.val (\u03c1 : group_representation G R M) (h : stable_sub_module \u03c1 p)( x : p) : \n     (\u27e6\u03c1\u27e7 g ) x.val = ( \u27e6 \u03c1 / h \u27e7  g) x    := rfl\n \n@[RW_REP] lemma rw_sub_module_action_to (\u03c1 : group_representation G R M) (h : stable_sub_module \u03c1 p) (x : p) : \n     (\u27e6\u03c1 \u27e7 g) x = \u27e6\u03c1 / h\u27e7 g x :=   \n     begin \n          -- rw_simp, --- joke :D\n          exact rfl, \n     end\nexample (\u03c1 : group_representation G R M) (h : stable_sub_module \u03c1 p) (x y : p)(r : R) (g g' : G): true := \n     begin \n     \n          have R : \u27e6 \u03c1 \u27e7 g ( \u27e6 \u03c1 \u27e7 g' x )+ r \u2022 \u27e6 \u03c1 \u27e7 g ( \u27e6 \u03c1 \u27e7 g' y ) =   \u27e6 \u03c1 / h \u27e7 (g * g') (x+ r \u2022 y),  \n               swap, trivial,\n               iterate 2 {rw \u2190 comp_apply}, rw \u2190 rmap_mul,rw \u2190 smul',rw \u2190 linearity,rw_simp,  \n          end \nend stability\nend group_representation", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/group_representation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3678823312930567}}
{"text": "variables {A: Type*} {B : Type*} {C : Type*} {D : Type*} {h : A \u2192 B} {g : B \u2192 C} {f : C \u2192 D}\n\ntheorem Q1007 : (f \u2218 g) \u2218 h = f \u2218 (g \u2218 h) := \nbegin\ntrivial,\nend", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/M1F/problem_bank/PB1008/S1008.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7185944046238982, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3677166888935524}}
{"text": "import main\nimport split_cycle\nimport data.list.nodup\nimport tactic\n\nopen_locale classical\nnoncomputable theory\n\nvariables {V X : Type}\n\ndef minus_candidate (P : Prof V X) (b : X) : Prof V {x : X // x \u2260 b} := \u03bb v x y, P v x y\n\nvariables (P : Prof V X) (c : X) (D : set {x : X // x \u2260 c})\n\ndef clones : Prop := D.nonempty \u2227 (\u2200 (c' \u2208 D) (x : {x : X // x \u2260 c}) (i : V), x \u2209 D \u2192 ((P i c x \u2194 P i c' x) \u2227 (P i x c \u2194 P i x c')))\n\ndef non_clone_choice_ind_clones : VSCC \u2192 Prop := \u03bb F, clones P c D \u2192 (\u2200 a : {x : X // x \u2260 c}, a \u2209 D \u2192 (a.val \u2208 (F V X P) \u2194 a \u2208 (F V {x : X // x \u2260 c} (minus_candidate P c))))\n\ndef clone_choice_ind_clones : VSCC \u2192 Prop := \u03bb F, clones P c D \u2192 ((c \u2209 (F V X P) \u2227 (\u2200 c' : {x : X // x \u2260 c}, c' \u2208 D \u2192 c'.val \u2209 (F V X P))) \u2194 (\u2200 c' \u2208 D, c' \u2209 (F V {x : X // x \u2260 c} (minus_candidate P c))))\n\nlemma margin_eq_margin_minus_candidate [fintype V] {a b : {x : X // x \u2260 c}} : margin P a b = margin (minus_candidate P c) a b :=\nbegin\n  obviously,\nend\n\nlemma margin_eq_clone_non_clone [fintype V] (b : {x : X // x \u2260 c}) (e \u2208 D) (p : b \u2209 D) : clones P c D \u2192 margin P c b = margin (minus_candidate P c) e b :=\nbegin\n  intro clone,\n  unfold margin,\n  have m1 : ((finset.filter (\u03bb (x_1 : V), P x_1 c \u2191b) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), minus_candidate P c x_1 e b) finset.univ).card),\n  apply finset.card_congr _ _ _ _,\n  intros a a_spec, exact a,\n  intro a, simp, unfold minus_candidate, intro p1,\n  unfold clones at clone,\n  cases clone with n clone,\n  have clone := clone e H b a p,\n  exact clone.left.mp p1,\n  intros v1 v2 hv1 hv2, simp,\n  intros a ha,\n  use a,\n  simp, simp at ha,\n  cases clone with n clone,\n  have clone := clone e H b a p,\n  exact clone.left.mpr ha,\n\n  have m2 : ((finset.filter (\u03bb (x_1 : V), P x_1 \u2191b c) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), minus_candidate P c x_1 b e) finset.univ).card),\n  apply finset.card_congr _ _ _ _,\n  intros a a_spec, exact a,\n  intro a, simp, unfold minus_candidate, intro p1,\n  unfold clones at clone,\n  cases clone with n clone,\n  have clone := clone e H b a p,\n  exact clone.right.mp p1,\n  intros v1 v2 hv1 hv2, simp,\n  intros a ha,\n  use a,\n  simp, simp at ha,\n  cases clone with n clone,\n  have clone := clone e H b a p,\n  exact clone.right.mpr ha,\n\n  rw m1, rw m2,\nend\n\nlemma margin_eq_clone_non_clone' [fintype V] {b : {x : X // x \u2260 c}} {e : {x : X // x \u2260 c}} (H : e \u2208 D) (p : b \u2209 D) : clones P c D \u2192 margin P b c = margin (minus_candidate P c) b e :=\nbegin\n  intro clone,\n  have x := margin_eq_clone_non_clone P c D b e H p clone,\n  rw margin_antisymmetric,\n  rw x,\n  rw margin_antisymmetric,\n  rw int.neg_neg,\nend\n\n-- y \u2192 d \u2192 d' \u2192 x\ndef remove_clones : list X \u2192 list X := \u03bb l, to_path (list.map (\u03bb x : X, (ite (\u2200p : (x \u2260 c), (\u27e8x,p\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x)) l)\n\nlemma replace_clones_helper {x : X} (h : \u00ac\u2200 (p : x \u2260 c), (\u27e8x, p\u27e9 : { x : X // x \u2260 c}) \u2208 D) : x \u2260 c :=\nbegin\n  obviously,\nend\n\ndef replace_clones (d \u2208 D) : list X \u2192 list {x : X // x \u2260 c} := \u03bb l, to_path (list.map (\u03bb x, (dite (\u2200p : (x \u2260 c), (\u27e8x,p\u27e9 : {x : X // x \u2260 c}) \u2208 D) (\u03bb h, d) (\u03bb h, \u27e8x, replace_clones_helper c D h\u27e9)) ) l)\n\ndef remove_clones' : list {x : X // x \u2260 c} \u2192 list X := \u03bb l, to_path (l.map (\u03bb x, (ite (x \u2208 D) c x))) \n\nlemma remove_clones_nil_iff (l : list X) : l = list.nil \u2194 remove_clones c D l = list.nil :=\nbegin\n  split,\n  intro n,\n  rw n,\n  unfold remove_clones,\n  rw list.map_nil _,\n  rw to_path_eq_nil_iff,\n  intro n,\n  unfold remove_clones at n,\n  rw to_path_eq_nil_iff at n,\n  rw list.map_eq_nil at n,\n  exact n,\nend\n\nlemma remove_clones_ne_nil_iff (l : list X) : l \u2260 list.nil \u2194 remove_clones c D l \u2260 list.nil := not_iff_not.mpr (remove_clones_nil_iff c D l)\n\nlemma remove_clones_nodup (l : list X) : (remove_clones c D l).nodup :=\nbegin\n  unfold remove_clones,\n  apply to_path_nodup,\nend\n\nlemma remove_clones_first_elem (l : list X) (a : X) (e : \u2200(p : a \u2260 c), (\u27e8a, p\u27e9 : {x : X // x \u2260 c}) \u2209 D) (n : l \u2260 list.nil) : ((l.nth_le 0 (list.length_pos_of_ne_nil n)) = a) \u2192 (((remove_clones c D l).nth_le 0 (list.length_pos_of_ne_nil ((remove_clones_ne_nil_iff c D l).mp n))) = a) :=\nbegin\n  unfold remove_clones,\n  intro m,\n  rw to_path_first_elem,\n  rw list.nth_le_map _ _ (list.length_pos_of_ne_nil n),\n  rw m,\n  simp,\n  intro p,\n  \n  apply eq.symm,\n  contrapose m,\n  exfalso,\n  specialize e m,\n  specialize p m,\n  exact e p,\n\n  intro con,\n  rw list.map_eq_nil at con,\n  exact n con,\nend \n\nlemma remove_clones_last_elem (l : list X) (a : X) (e : \u2200(p : a \u2260 c), (\u27e8a, p\u27e9 : {x : X // x \u2260 c}) \u2209 D) (n : l \u2260 list.nil) : ((l.last n) = a) \u2192 (((remove_clones c D l).last ((remove_clones_ne_nil_iff c D l).mp n)) = a) :=\nbegin\n  unfold remove_clones,\n  intro m,\n  rw to_path_last_elem,\n  rw list.last_map,\n  rw m,\n  simp,\n  intro con,\n  apply eq.symm,\n  contrapose m,\n  exfalso,\n  specialize e m,\n  specialize con m,\n  exact e con,\nend\n\nlemma remove_clones'_nil_iff (l : list {x : X // x \u2260 c}) : l = list.nil \u2194 remove_clones' c D l = list.nil :=\nbegin\n  split,\n  intro n,\n  rw n,\n  unfold remove_clones',\n  rw list.map_nil _,\n  rw to_path_eq_nil_iff,\n  intro n,\n  unfold remove_clones' at n,\n  rw to_path_eq_nil_iff at n,\n  rw list.map_eq_nil at n,\n  exact n,\nend\n\nlemma remove_clones'_ne_nil_iff (l : list {x : X // x \u2260 c}) : l \u2260 list.nil \u2194 remove_clones' c D l \u2260 list.nil := not_iff_not.mpr (remove_clones'_nil_iff c D l)\n\nlemma remove_clones'_nodup (l : list {x : X // x \u2260 c}) : (remove_clones' c D l).nodup :=\nbegin\n  unfold remove_clones',\n  apply to_path_nodup,\nend\n\nlemma remove_clones'_first_elem (l : list {x : X // x \u2260 c}) (a : {x : X // x \u2260 c}) (e : a \u2209 D) (n : l \u2260 list.nil) : ((l.nth_le 0 (list.length_pos_of_ne_nil n)) = a) \u2192 (((remove_clones' c D l).nth_le 0 (list.length_pos_of_ne_nil ((remove_clones'_ne_nil_iff c D l).mp n))) = a) :=\nbegin\n  unfold remove_clones',\n  intro m,\n  rw to_path_first_elem,\n  rw list.nth_le_map _ _ (list.length_pos_of_ne_nil n),\n  rw m,\n  simp,\n  intro p,\n  exfalso,\n  exact e p,\n\n  intro con,\n  rw list.map_eq_nil at con,\n  exact n con,\nend \n\nlemma remove_clones'_last_elem (l : list {x : X // x \u2260 c}) (a : {x : X // x \u2260 c}) (e : a \u2209 D) (n : l \u2260 list.nil) : ((l.last n) = a) \u2192 (((remove_clones' c D l).last ((remove_clones'_ne_nil_iff c D l).mp n)) = a) :=\nbegin\n  unfold remove_clones',\n  intro m,\n  rw to_path_last_elem,\n  rw list.last_map,\n  rw m,\n  simp,\n  intro con,\n  exfalso,\n  exact e con,\nend\n\nlemma replace_clones_nil_iff (l : list X) {d \u2208 D} : l = list.nil \u2194 replace_clones c D d H l = list.nil :=\nbegin\n  split,\n  intro n,\n  rw n,\n  unfold replace_clones,\n  rw list.map_nil _,\n  rw to_path_eq_nil_iff,\n  intro n,\n  unfold replace_clones at n,\n  rw to_path_eq_nil_iff at n,\n  rw list.map_eq_nil at n,\n  exact n,\nend\n\nlemma replace_clones_ne_nil_iff (d \u2208 D) (l : list X) : l \u2260 list.nil \u2194 replace_clones c D d H l \u2260 list.nil := not_iff_not.mpr (replace_clones_nil_iff c D l)\n\nlemma replace_clones_nodup (l : list X) {d \u2208 D} : (replace_clones c D d H l).nodup :=\nbegin\n  unfold replace_clones,\n  apply to_path_nodup,\nend\n\nlemma replace_clones_first_elem (l : list X) (a : X) (e : \u00ac \u2200(p : a \u2260 c), (\u27e8a, p\u27e9 : {x : X // x \u2260 c}) \u2208 D) (n : l \u2260 list.nil) {d \u2208 D} : ((l.nth_le 0 (list.length_pos_of_ne_nil n)) = a) \u2192 (((replace_clones c D d H l).nth_le 0 (list.length_pos_of_ne_nil ((replace_clones_ne_nil_iff c D d H l).mp n))) = \u27e8a, replace_clones_helper c D e\u27e9) :=\nbegin\n  unfold replace_clones,\n  intro m,\n  rw to_path_first_elem,\n  rw list.nth_le_map _ _ (list.length_pos_of_ne_nil n),\n  rw m,\n  split_ifs,\n  refl,\n  \n  apply (not_iff_not.mpr list.map_eq_nil).mpr,\n  exact n,\nend \n\nlemma replace_clones_last_elem (l : list X) (a : X) (e : \u00ac\u2200(p : a \u2260 c), (\u27e8a, p\u27e9 : {x : X // x \u2260 c}) \u2208 D) (n : l \u2260 list.nil) {d \u2208 D} : ((l.last n) = a) \u2192 (((replace_clones c D d H l).last ((replace_clones_ne_nil_iff c D d H l).mp n)) = \u27e8a, replace_clones_helper c D e\u27e9) :=\nbegin\n  unfold replace_clones,\n  intro m,\n  rw to_path_last_elem,\n  rw list.last_map,\n  rw m,\n  split_ifs,\n  refl,\nend\n\nlemma replace_clones_last_c (l : list X) (n : l \u2260 list.nil) {d \u2208 D} : ((l.last n) = c) \u2192 (((replace_clones c D d H l).last ((replace_clones_ne_nil_iff c D d H l).mp n)) = d) :=\nbegin\n  unfold replace_clones,\n  intro m,\n  rw to_path_last_elem,\n  rw list.last_map,\n  rw m,\n  split_ifs,\n  refl,\n  push_neg at h,\n  cases h with contr h,\n  exfalso,\n  exact false_of_ne contr,\nend\n\nlemma remove_clones_chain'_of_chain' [fintype V] (l : list X) (b : {x : X // x \u2260 c}) (e : b \u2209 D) (d \u2208 D) (clone : clones P c D) : \n  list.chain' (\u03bb (a b_1 : X), margin P \u2191d \u2191b \u2264 margin P a b_1) l \u2192 list.chain' (\u03bb (a b_1 : X), margin P c \u2191b \u2264 margin P a b_1) (remove_clones c D l) :=\nbegin\n  intro a,\n  induction l,\n   {obviously,},\n  unfold remove_clones,\n  simp [to_path],\n  specialize l_ih (and.right (list.chain'_cons'.mp a)),\n  by_cases (list.index_of (ite (\u2200 (h : \u00acl_hd = c), (\u27e8l_hd, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c l_hd) (to_path (list.map (\u03bb (x : X), ite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x) l_tl)) < (to_path (list.map (\u03bb (x : X), ite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x) l_tl)).length),\n  simp_rw (ite_left_if h),\n  exact drop_chain'_of_chain' l_ih,\n\n  simp_rw (ite_right_if h),\n  by_cases j : l_tl = list.nil,\n  rw j,\n  rw list.map_nil,\n  rw \u2190j,\n  rw (to_path_eq_nil_iff l_tl).mpr j,\n  exact list.chain'_singleton (ite (\u2200 (h : \u00acl_hd = c), (\u27e8l_hd, _\u27e9 : {x : X // x \u2260 c}) \u2208 D) c l_hd),\n\n  rw list.chain'_iff_nth_le,\n  rw list.chain'_iff_nth_le at l_ih,\n  rw list.chain'_iff_nth_le at a,\n  intros i i_bounds,\n  have nodup := h,\n  by_cases i = 0,\n  have i_eq := h,\n  simp_rw h,\n  rw [list.nth_le], rw [list.nth_le],\n  specialize a 0,\n  have a_proof : 0 < (l_hd :: l_tl).length - 1,\n  rw list.length_cons,\n  simp only [nat.add_succ_sub_one, add_zero],\n  exact (list.length_pos_of_ne_nil j),\n  specialize a a_proof,\n  rw [list.nth_le] at a, rw [list.nth_le] at a,\n  have x := (to_path_first_elem X l_tl j),\n  rw to_path_first_elem',\n  simp,\n  \n  by_cases (\u2200 (h : \u00acl_hd = c), (\u27e8l_hd, h\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  rw ite_left_if h,\n  have h' := h,\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  exfalso,\n  contrapose nodup,\n  push_neg,\n  rw list.index_of_lt_length,\n  rw ite_left_if h',\n  rw list.mem_iff_nth_le,\n  use 0,\n  have len : 0 < (to_path (list.map (\u03bb (x : X), ite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x) l_tl)).length,\n  change (0 < (remove_clones c D l_tl).length),\n  apply list.length_pos_of_ne_nil,\n  rw \u2190remove_clones_ne_nil_iff,\n  exact j,\n  use len,\n  rw to_path_first_elem',\n  rw list.nth_le_map (\u03bb (x : X), ite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x) _ (list.length_pos_of_ne_nil j),\n  simp,\n  intro w,\n  specialize h w,\n  intro contr,\n  exfalso, exact contr h,\n\n  rw ite_right_if h,\n  push_neg at h,\n  cases h with not_c not_d,\n  by_cases l_hd = c,\n  rw h at a,\n  rw margin_eq_clone_non_clone P c D b d H e clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  specialize h' h,\n  rw margin_eq_clone_non_clone P c D b d H e clone,\n  have r := margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 \u27e8l_hd, h\u27e9 h' not_d clone,\n  simp at r,\n  rw r,\n  rw \u2190margin_eq_margin_minus_candidate, rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  rw ite_right_if h,\n  have h' := h,\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  rw ite_left_if h,\n  push_neg at h',\n  cases h' with not_c not_d,\n  have h'' := h,\n  by_cases (l_tl.nth_le 0 (list.nth_le._main._proof_1 l_hd l_tl 0 (nat.lt_pred_iff.mp a_proof))) = c,\n  rw h at a,\n  rw margin_eq_clone_non_clone P c D b d H e clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  specialize h'' h,\n  rw margin_eq_clone_non_clone P c D b d H e clone,\n  have r := margin_eq_clone_non_clone' P c D h'' not_d clone,\n  simp at r,\n  rw r,\n  rw \u2190margin_eq_margin_minus_candidate, rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  rw ite_right_if h,\n  rw margin_eq_clone_non_clone P c D b d H e clone,\n  exact a,\n\n  rw [list.nth_le],\n  specialize l_ih (i - 1),\n  rw list.length_cons at i_bounds,\n  simp only [nat.add_succ_sub_one, add_zero] at i_bounds,\n  have o : \u2200 i n, \u00ac i = 0 \u2192 i < n \u2192 i - 1 < n - 1 := by omega,\n  specialize l_ih (o i (to_path (list.map (\u03bb (x : X), ite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c x) l_tl)).length h i_bounds),\n  --change (margin P c \u2191b \u2264 margin P ((((ite (\u2200 (h : \u00acl_hd = c), (\u27e8l_hd, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c l_hd)) :: (remove_clones c D l_tl)).nth_le i (nat.lt_of_lt_pred i_bounds)) ((remove_clones c D l_tl).nth_le i _)),\n  rw nth_le_cons h,\n  have o : \u2200 i, \u00ac i = 0 \u2192 i - 1 + 1 = i := by omega,\n  simp_rw (o i h) at l_ih,\n  exact l_ih,\nend\n\nlemma nodup_lift_of_nodup {l : list {x : X // x \u2260 c}} : l.nodup \u2192 (lift l : list X).nodup :=\nbegin\n  intro n,\n  unfold lift,\n  unfold has_lift.lift,\n  rw list.nodup_map_iff_inj_on,\n  obviously,\nend\n\nlemma lift_ne_nil_iff {l : list {x : X // x \u2260 c}} : (lift l : list X) \u2260 list.nil \u2194 l \u2260 list.nil:=\nbegin\n  unfold lift, unfold has_lift.lift, change (\u00ac list.map coe l = list.nil \u2194 l \u2260 list.nil),\n   rw list.map_eq_nil,\nend\n\nlemma dite_left_if {p : Prop} {\u03b1 : Type} [decidable p] {a : p \u2192 \u03b1} {b : \u00ac p \u2192 \u03b1} (proof : p) : dite p a b = a proof := \nbegin\n  split_ifs, refl,\nend\n\nlemma dite_right_if {p : Prop} {\u03b1 : Type} [decidable p] {a : p \u2192 \u03b1} {b : \u00ac p \u2192 \u03b1} (proof : \u00acp) : \u00ac p \u2192 dite p a b = b proof :=\nbegin\n  split_ifs, intro unused, refl,\nend\n\n--set_option pp.all true \n\nlemma A6_chain [fintype V] (l : list X) {b : {x : X // x \u2260 c}} {d : {x : X // x \u2260 c}} {d' : {x : X // x \u2260 c}} (H : d \u2208 D) (clone : clones P c D) : list.chain' (\u03bb (a b_1 : X), margin P \u2191b \u2191d' \u2264 margin P a b_1) l\n\u2192 list.chain' (\u03bb (a b_1 : {x // x \u2260 c}), margin (minus_candidate P c) b d' \u2264 margin (minus_candidate P c) a b_1) (replace_clones c D d H l) :=\nbegin\n  intro a,\n  induction l,\n    {obviously,},\n  unfold replace_clones,\n  simp [to_path],\n  specialize l_ih (and.right (list.chain'_cons'.mp a)),\n  by_cases @has_lt.lt.{0} nat nat.has_lt\n  (@list.index_of.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (\u03bb (a b : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)),\n        classical.prop_decidable (@eq.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) a b))\n     (@dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (\u2200 (h : not (@eq.{1} X l_hd c)),\n           @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n             (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                   (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                      (iff.refl (@ne.{1} X l_hd c))\n                      (not (@eq.{1} X l_hd c))\n                      (@ne.def.{1} X l_hd c))\n                   h))\n             D)\n        (@forall_prop_decidable (not (@eq.{1} X l_hd c))\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))\n                D)\n           (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) l_hd c)\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))))\n        (\u03bb\n         (h :\n           \u2200 (h : not (@eq.{1} X l_hd c)),\n             @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n               (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                  (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                     (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                        (iff.refl (@ne.{1} X l_hd c))\n                        (not (@eq.{1} X l_hd c))\n                        (@ne.def.{1} X l_hd c))\n                     h))\n               D), d)\n        (\u03bb\n         (h :\n           not\n             (\u2200 (h : not (@eq.{1} X l_hd c)),\n                @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                  (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                     (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                        (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                           (iff.refl (@ne.{1} X l_hd c))\n                           (not (@eq.{1} X l_hd c))\n                           (@ne.def.{1} X l_hd c))\n                        h))\n                  D)),\n           @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n             (@replace_clones_helper X c D l_hd\n                (@iff.mpr\n                   (not\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D))\n                   (not\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D))\n                   (@not_iff_not_of_iff\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D)\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D)\n                      (@forall_prop_congr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            iff.refl\n                              (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                 (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                                 D))\n                         (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                            (iff.refl (@ne.{1} X l_hd c))\n                            (not (@eq.{1} X l_hd c))\n                            (@ne.def.{1} X l_hd c))))\n                   h))))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl)))\n  (@list.length.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl))),\n\n  simp_rw (ite_left_if h),\n  exact drop_chain'_of_chain' l_ih,\n\n  simp_rw (ite_right_if h),\n  by_cases j : l_tl = list.nil,\n  rw j,\n  simp [to_path],\n  \n  rw list.chain'_iff_nth_le,\n  rw list.chain'_iff_nth_le at l_ih,\n  rw list.chain'_iff_nth_le at a,\n  intros i i_bounds,\n  rename h nodup,\n  by_cases i = 0,\n  rename h i_eq,\n  simp_rw i_eq,\n  rw [list.nth_le], rw [list.nth_le],\n  specialize a 0,\n  have a_proof : 0 < (l_hd :: l_tl).length - 1,\n  rw list.length_cons,\n  simp only [nat.add_succ_sub_one, add_zero],\n  exact (list.length_pos_of_ne_nil j),\n  specialize a a_proof,\n  rw [list.nth_le] at a, rw [list.nth_le] at a,\n  have x := (to_path_first_elem X l_tl j),\n  rw to_path_first_elem',\n  simp,\n  \n  split_ifs with h',\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  exfalso,\n  contrapose nodup,\n  push_neg,\n  rw list.index_of_lt_length,\n  rw dite_left_if h',\n  rw list.mem_iff_nth_le,\n  use 0,\n  have len : (0 < (replace_clones c D d H l_tl).length),\n  apply list.length_pos_of_ne_nil,\n  rw \u2190replace_clones_ne_nil_iff,\n  exact j,\n  use len,\n  rw to_path_first_elem',\n  rw list.nth_le_map (\u03bb (x : X), dite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) (\u03bb (h : \u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), d) (\u03bb (h : \u00ac\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), \u27e8x, replace_clones_helper c D h\u27e9)) _ (list.length_pos_of_ne_nil j),\n  simp,\n  rw dite_left_if h,\n\n  rw dite_right_if h,\n\n  push_neg at h,\n  cases h with not_c not_d,\n  by_cases l_hd = c,\n  rw h at a,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n\n  specialize h' h,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H not_d clone,\n  rw margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 \u27e8l_hd, h\u27e9 h' not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate, rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n  exact h,\n\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  rw dite_left_if h,\n  push_neg at h',\n  cases h' with not_c not_d,\n  rename h h'',\n  by_cases (l_tl.nth_le 0 (list.nth_le._main._proof_1 l_hd l_tl 0 (nat.lt_pred_iff.mp a_proof))) = c,\n  rw h at a,\n  rw \u2190margin_eq_clone_non_clone' P c D H not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  specialize h'' h,\n  rw \u2190margin_eq_clone_non_clone' P c D H not_d clone,\n  rw margin_eq_clone_non_clone' P c D h'' not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate, rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  rw dite_right_if h,\n  rw \u2190margin_eq_margin_minus_candidate, rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n  exact h,\n\n  rw [list.nth_le],\n  specialize l_ih (i - 1),\n  rw list.length_cons at i_bounds,\n  simp only [nat.add_succ_sub_one, add_zero] at i_bounds,\n  have o : \u2200 i n, \u00ac i = 0 \u2192 i < n \u2192 i - 1 < n - 1 := by omega,\n  specialize l_ih (o i (replace_clones c D d H l_tl).length h i_bounds),\n  rw nth_le_cons h,\n  have o : \u2200 i, \u00ac i = 0 \u2192 i - 1 + 1 = i := by omega,\n  simp_rw (o i h) at l_ih,\n  exact l_ih,\nend\n\n-- A.5 might not be true if a and b are in D?\nlemma clone_maintains_defeat (a b : {x : X // x \u2260 c}) (H : b \u2209 D) : clones P c D \u2192 ((split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c)) a b \u2194 (split_cycle_VCCR V X P) a b) :=\nbegin\n  intro clone,\n  rw split_cycle_definitions,\n  unfold split_cycle_VCCR',\n  simp,\n  split,\n  intro f,\n  introI fV,\n  cases f with m f,\n  rw margin_eq_margin_minus_candidate,\n  use m,\n  intros l n ne_nil b_mem a_mem,\n  have clone' := clone,\n  unfold clones at clone,\n  cases clone with n clone,\n  let d := (ite (a \u2208 D) a n.some),\n  have d_spec : d \u2208 D,\n    {change (ite (a \u2208 D) a n.some) \u2208 D, split_ifs, exact h, exact n.some_spec, },\n  specialize f (replace_clones c D d d_spec l),\n  specialize f (replace_clones_nodup c D l),\n  specialize f ((replace_clones_ne_nil_iff c D d d_spec l).mp ne_nil),\n  have b_not : \u00ac\u2200 (p : \u2191b \u2260 c), (\u27e8\u2191b, p\u27e9 : {x : X // x \u2260 c}) \u2208 D,\n   {push_neg, use b.property, simp, exact H,},\n  rw (replace_clones_first_elem c D l b b_not ne_nil b_mem) at f,\n  simp at f,\n  have last_elem : (replace_clones c D d d_spec l).last ((replace_clones_ne_nil_iff c D d d_spec l).mp ne_nil) = a,\n    {change (replace_clones c D (ite (a \u2208 D) a n.some) d_spec l).last ((replace_clones_ne_nil_iff c D d d_spec l).mp ne_nil) = a,\n    split_ifs, unfold replace_clones, rw to_path_last_elem, rw list.last_map, split_ifs, \n    refl, push_neg at h_1, simp_rw a_mem, simp, exact ne_nil, rw replace_clones_last_elem c D l _ _ ne_nil a_mem, simp, \n    push_neg, use a.property, simp, exact h, },\n  specialize f last_elem,\n  contrapose f,\n  push_neg at f, push_neg, \n  exact A6_chain P c D l d_spec clone' f,\n\n\n  intro f,\n  introI fV,\n  cases f with m f,\n  rw \u2190margin_eq_margin_minus_candidate,\n  use m,\n  intros l l_nodup ne_nil b_mem a_mem,\n  specialize f \u2191l,\n  contrapose f,\n  push_neg,\n  split,\n  exact nodup_lift_of_nodup c l_nodup,\n  use (lift_ne_nil_iff c).mpr ne_nil,\n  unfold coe, unfold lift_t, unfold has_lift_t.lift,\n  unfold lift, unfold has_lift.lift, unfold coe_t, \n  unfold has_coe_t.coe, unfold coe_b, unfold has_coe.coe,\n  split,\n  rw list.nth_le_map, rw b_mem, refl,\n  split, \n  have test := list.last_map (coe : {x // x \u2260 c} \u2192 X) ne_nil,\n  rw a_mem at test,\n  change (list.map coe l).last _ = \u2191a,\n  rw \u2190test,\n  refl,\n  push_neg at f,\n  apply (list.chain'_map (coe : {x // x \u2260 c} \u2192 X)).mpr,\n  refine list.chain'.imp _ f,\n  intros a_1 b_1 m_1,\n  rw \u2190margin_eq_margin_minus_candidate at m_1,\n  exact m_1,\nend\n\nlemma clone_maintains_defeat' (a b : {x : X // x \u2260 c}) (H : a \u2209 D) (H' : b \u2208 D) : clones P c D \u2192 ((split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c)) a b \u2194 (split_cycle_VCCR V X P) a b) :=\nbegin\n  intro clone,\n  rw split_cycle_definitions,\n  unfold split_cycle_VCCR',\n  simp,\n  split,\n  intro f,\n  introI fV,\n  cases f with m f,\n  rw margin_eq_margin_minus_candidate,\n  use m,\n  intros l n ne_nil b_mem a_mem,\n  have clone' := clone,\n  unfold clones at clone,\n  cases clone with n clone,\n  let d := b,\n  let d_spec := H',\n  specialize f (replace_clones c D d d_spec l),\n  specialize f (replace_clones_nodup c D l),\n  specialize f ((replace_clones_ne_nil_iff c D d d_spec l).mp ne_nil),\n  have b_not : \u00ac\u2200 (p : \u2191a \u2260 c), (\u27e8\u2191a, p\u27e9 : {x : X // x \u2260 c}) \u2208 D,\n   {push_neg, use a.property, simp, exact H,},\n\n  contrapose f, push_neg, push_neg at f,\n  split,\n  unfold replace_clones,\n  rw to_path_first_elem',\n  rw list.nth_le_map,\n  split_ifs,\n  refl,\n  simp_rw b_mem, \n  simp,\n  exact list.length_pos_of_ne_nil ne_nil,\n  split,\n  rw (replace_clones_last_elem c D l a b_not ne_nil a_mem),\n  simp,\n  exact A6_chain P c D l d_spec clone' f,\n\n\n  intro f,\n  introI fV,\n  cases f with m f,\n  rw \u2190margin_eq_margin_minus_candidate,\n  use m,\n  intros l l_nodup ne_nil b_mem a_mem,\n  specialize f \u2191l,\n  contrapose f,\n  push_neg at f,\n  push_neg,\n  split,\n  exact nodup_lift_of_nodup c l_nodup,\n  use (lift_ne_nil_iff c).mpr ne_nil,\n  unfold coe, unfold lift_t, unfold has_lift_t.lift,\n  unfold lift, unfold has_lift.lift, unfold coe_t, \n  unfold has_coe_t.coe, unfold coe_b, unfold has_coe.coe,\n  split,\n  rw list.nth_le_map, rw b_mem, refl,\n  split, \n  have test := list.last_map (coe : {x // x \u2260 c} \u2192 X) ne_nil,\n  rw a_mem at test,\n  change (list.map coe l).last _ = \u2191a,\n  rw \u2190test,\n  refl,\n  apply (list.chain'_map (coe : {x // x \u2260 c} \u2192 X)).mpr,\n  refine list.chain'.imp _ f,\n  intros a_1 b_1 m_1,\n  rw \u2190margin_eq_margin_minus_candidate at m_1,\n  exact m_1,\nend\n\nlemma A5_chain {l : list X} [fintype V] {d : {x : X // x \u2260 c}} {b : {x : X // x \u2260 c}} (H : d \u2208 D) (H' : b \u2209 D) (clone : clones P c D) : list.chain' (\u03bb (a b_1 : X), margin P c \u2191b \u2264 margin P a b_1) l \u2192 list.chain' (\u03bb (a b_1 : {x // x \u2260 c}), margin P \u2191d \u2191b \u2264 margin P \u2191a \u2191b_1) (replace_clones c D d H l) :=\nbegin\n  intro a,\n  induction l,\n    {obviously,},\n  unfold replace_clones,\n  simp [to_path],\n  specialize l_ih (and.right (list.chain'_cons'.mp a)),\n  by_cases @has_lt.lt.{0} nat nat.has_lt\n  (@list.index_of.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (\u03bb (a b : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)),\n        classical.prop_decidable (@eq.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) a b))\n     (@dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (\u2200 (h : not (@eq.{1} X l_hd c)),\n           @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n             (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                   (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                      (iff.refl (@ne.{1} X l_hd c))\n                      (not (@eq.{1} X l_hd c))\n                      (@ne.def.{1} X l_hd c))\n                   h))\n             D)\n        (@forall_prop_decidable (not (@eq.{1} X l_hd c))\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))\n                D)\n           (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) l_hd c)\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))))\n        (\u03bb\n         (h :\n           \u2200 (h : not (@eq.{1} X l_hd c)),\n             @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n               (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                  (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                     (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                        (iff.refl (@ne.{1} X l_hd c))\n                        (not (@eq.{1} X l_hd c))\n                        (@ne.def.{1} X l_hd c))\n                     h))\n               D), d)\n        (\u03bb\n         (h :\n           not\n             (\u2200 (h : not (@eq.{1} X l_hd c)),\n                @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                  (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                     (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                        (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                           (iff.refl (@ne.{1} X l_hd c))\n                           (not (@eq.{1} X l_hd c))\n                           (@ne.def.{1} X l_hd c))\n                        h))\n                  D)),\n           @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n             (@replace_clones_helper X c D l_hd\n                (@iff.mpr\n                   (not\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D))\n                   (not\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D))\n                   (@not_iff_not_of_iff\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D)\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D)\n                      (@forall_prop_congr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            iff.refl\n                              (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                 (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                                 D))\n                         (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                            (iff.refl (@ne.{1} X l_hd c))\n                            (not (@eq.{1} X l_hd c))\n                            (@ne.def.{1} X l_hd c))))\n                   h))))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl)))\n  (@list.length.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl))),\n\n  simp_rw (ite_left_if h),\n  exact drop_chain'_of_chain' l_ih,\n\n  simp_rw (ite_right_if h),\n  by_cases j : l_tl = list.nil,\n  rw j,\n  simp [to_path],\n  \n  rw list.chain'_iff_nth_le,\n  rw list.chain'_iff_nth_le at l_ih,\n  rw list.chain'_iff_nth_le at a,\n  intros i i_bounds,\n  rename h nodup,\n  by_cases i = 0,\n  rename h i_eq,\n  simp_rw i_eq,\n  rw [list.nth_le], rw [list.nth_le],\n  specialize a 0,\n  have a_proof : 0 < (l_hd :: l_tl).length - 1,\n  rw list.length_cons,\n  simp only [nat.add_succ_sub_one, add_zero],\n  exact (list.length_pos_of_ne_nil j),\n  specialize a a_proof,\n  rw [list.nth_le] at a, rw [list.nth_le] at a,\n  have x := (to_path_first_elem X l_tl j),\n  rw to_path_first_elem',\n  simp,\n  \n  split_ifs with h',\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  exfalso,\n  contrapose nodup,\n  push_neg,\n  rw list.index_of_lt_length,\n  rw dite_left_if h',\n  rw list.mem_iff_nth_le,\n  use 0,\n  have len : (0 < (replace_clones c D d H l_tl).length),\n  apply list.length_pos_of_ne_nil,\n  rw \u2190replace_clones_ne_nil_iff,\n  exact j,\n  use len,\n  rw to_path_first_elem',\n  rw list.nth_le_map (\u03bb (x : X), dite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) (\u03bb (h : \u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), d) (\u03bb (h : \u00ac\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), \u27e8x, replace_clones_helper c D h\u27e9)) _ (list.length_pos_of_ne_nil j),\n  simp,\n  rw dite_left_if h,\n\n  rw dite_right_if h,\n\n  push_neg at h,\n  cases h with not_c not_d,\n  by_cases l_hd = c,\n  rw h at a, simp,\n  have test := @margin_eq_margin_minus_candidate V X P c _inst_1 d \u27e8(l_tl.nth_le 0 _), not_c\u27e9,\n  simp at test,\n  rw margin_eq_margin_minus_candidate,\n  rw test,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H not_d clone,\n  simp,\n  rw \u2190margin_eq_clone_non_clone P c D b d H H' clone,\n  exact a,\n\n\n  specialize h' h,\n  simp,\n  have test := @margin_eq_margin_minus_candidate V X P c _inst_1 d \u27e8(l_tl.nth_le 0 _), not_c\u27e9,\n  simp at test,\n  rw margin_eq_margin_minus_candidate,\n  rw test,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H not_d clone,\n  rw margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 \u27e8l_hd, h\u27e9 h' not_d clone,\n  rw \u2190margin_eq_clone_non_clone P c D b d H H' clone,\n  rw \u2190margin_eq_margin_minus_candidate, \n  simp,\n  exact a,\n  exact h,\n\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  rw dite_left_if h,\n  push_neg at h',\n  cases h' with not_c not_d,\n  rename h h'',\n  by_cases (l_tl.nth_le 0 (list.nth_le._main._proof_1 l_hd l_tl 0 (nat.lt_pred_iff.mp a_proof))) = c,\n  rw h at a,\n  rw margin_eq_margin_minus_candidate,\n  rw \u2190margin_eq_clone_non_clone P c D b d H H' clone,\n  rw margin_eq_margin_minus_candidate, \n  rw \u2190margin_eq_clone_non_clone' P c D H not_d clone,\n  exact a,\n\n  specialize h'' h,\n  rw margin_eq_margin_minus_candidate, rw margin_eq_margin_minus_candidate,\n  rw \u2190margin_eq_clone_non_clone' P c D H not_d clone,\n  rw margin_eq_clone_non_clone' P c D h'' not_d clone,\n  rw \u2190margin_eq_clone_non_clone P c D b d H H' clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  rw dite_right_if h,\n  rw margin_eq_margin_minus_candidate,\n  rw \u2190margin_eq_clone_non_clone P c D b d H H' clone,\n  exact a,\n  exact h,\n\n  rw [list.nth_le],\n  specialize l_ih (i - 1),\n  rw list.length_cons at i_bounds,\n  simp only [nat.add_succ_sub_one, add_zero] at i_bounds,\n  have o : \u2200 i n, \u00ac i = 0 \u2192 i < n \u2192 i - 1 < n - 1 := by omega,\n  specialize l_ih (o i (replace_clones c D d H l_tl).length h i_bounds),\n  rw nth_le_cons h,\n  have o : \u2200 i, \u00ac i = 0 \u2192 i - 1 + 1 = i := by omega,\n  simp_rw (o i h) at l_ih,\n  exact l_ih,\nend\n\n-- A.5, would hold both directions. Added condition\nlemma every_clone_defeats (b : {x : X // x \u2260 c}) (e : b \u2209 D) (d \u2208 D) : clones P c D \u2192 ((split_cycle_VCCR V X P) c b \u2194 (split_cycle_VCCR V X P) d b) :=\nbegin\n  intro clone,\n  rw split_cycle_definitions,\n  unfold split_cycle_VCCR',\n  split,\n  intro w,\n  introI f,\n  cases w with m w,\n  unfold margin_pos at m,\n  rw margin_eq_clone_non_clone P c D b d H e clone at m,\n  rw \u2190margin_eq_margin_minus_candidate at m,\n  use m,\n  push_neg, push_neg at w,\n  intro l,\n  specialize w (remove_clones c D l),\n  intro ne_nil,\n  specialize w ((remove_clones_ne_nil_iff c D l).mp ne_nil),\n  contrapose w,\n  push_neg, push_neg at w,\n  use remove_clones_nodup c D l,\n  cases w with l_nodup w,\n  cases w with first w,\n  cases w with last w,\n\n  have f : \u2200(p : \u2191b \u2260 c), (\u27e8\u2191b, p\u27e9 : {x : X // x \u2260 c}) \u2209 D,\n  simp, intro _, exact e,\n\n  use remove_clones_first_elem c D l \u2191b f ne_nil first, \n  unfold remove_clones,\n  rw to_path_last_elem,\n  rw list.last_map,\n  rw last,\n  simp,\n  have t : (\u00ac\u2191d = c \u2192 d \u2209 D \u2192 \u2191d = c),\n  intros x y,\n  exfalso, exact y H,\n  use t,\n  exact remove_clones_chain'_of_chain' P c D l b e d H clone w,\n\n\n\n  intro w,\n  introI f,\n  cases w with m w,\n  unfold margin_pos at m,\n  rw margin_eq_margin_minus_candidate at m,\n  rw \u2190margin_eq_clone_non_clone P c D b d H e clone at m,\n  use m,\n  push_neg, push_neg at w,\n  intros l ne_nil,\n  specialize w (lift (replace_clones c D d H l)),\n  have n1: (replace_clones c D d H l) \u2260 list.nil,\n  rw \u2190replace_clones_ne_nil_iff,\n  exact ne_nil,\n\n  have n : (lift (replace_clones c D d H l)) \u2260 list.nil,\n  rw lift_ne_nil_iff,\n  exact n1,\n  specialize w n,\n  contrapose w,\n  push_neg, push_neg at w,\n  split,\n  apply nodup_lift_of_nodup,\n  apply replace_clones_nodup,\n  cases w with nd w,\n  cases w with first w,\n  cases w with last w,\n\n  split,\n  unfold lift, unfold has_lift.lift,\n  rw list.nth_le_map,\n  rw replace_clones_first_elem c D l b,\n  simp,\n  push_neg, use b.property, simp, exact e,\n  exact first,\n\n  split,\n  unfold lift, unfold has_lift.lift,\n  rw list.last_eq_nth_le,\n  rw list.nth_le_map,\n  simp_rw list.length_map,\n  rw \u2190list.last_eq_nth_le, \n  rw replace_clones_last_c c D l ne_nil,\n  exact last,\n  rw list.length_map, \n  have o : \u2200 x : \u2115, 0 < x \u2192 x - 1 < x := by omega,\n  apply o, apply list.length_pos_of_ne_nil, exact n1,\n  have imp : \u2200 (a b_1 : {x : X // x \u2260 c}), (\u03bb a b_1, margin P \u2191d \u2191b \u2264 margin P (coe a) (coe b_1)) a b_1 \u2192 (\u03bb a b_1, margin P \u2191d \u2191b \u2264 margin P a b_1) (coe a) (coe b_1),\n  intros a b_1 e, exact e,\n  apply list.chain'_map_of_chain' coe imp,\n  simp, \n  \n  exact A5_chain P c D H e clone w,\nend\n\nlemma remove_clones'_cycle1 [fintype V] {l : list {x : X // x \u2260 c}} {a' : {x : X // x \u2260 c}} (H : a' \u2209 D) {d : {x : X // x \u2260 c}} (H' : d \u2208 D) (clone : clones P c D) : list.chain' (\u03bb (a_1 b : {x // x \u2260 c}), margin (minus_candidate P c) a' d \u2264 margin (minus_candidate P c) a_1 b) l\n\u2192 list.chain' (\u03bb (a_1 b : X), margin P \u2191a' c \u2264 margin P a_1 b) (remove_clones' c D l) :=\nbegin\n  intro a,\n  induction l,\n   {obviously,},\n  unfold remove_clones',\n  simp [to_path],\n  specialize l_ih (and.right (list.chain'_cons'.mp a)),\n  by_cases (list.index_of (ite (l_hd \u2208 D) c \u2191l_hd) (to_path (list.map (\u03bb (x : {x // x \u2260 c}), ite (x \u2208 D) c \u2191x) l_tl)) < (to_path (list.map (\u03bb (x : {x // x \u2260 c}), ite (x \u2208 D) c \u2191x) l_tl)).length),\n  simp_rw (ite_left_if h),\n  exact drop_chain'_of_chain' l_ih,\n\n  simp_rw (ite_right_if h),\n  by_cases j : l_tl = list.nil,\n  rw j,\n  rw list.map_nil,\n  rw (to_path_eq_nil_iff list.nil).mpr,\n  exact list.chain'_singleton (ite (l_hd \u2208 D) c \u2191l_hd),\n  refl,\n\n  rw list.chain'_iff_nth_le,\n  rw list.chain'_iff_nth_le at l_ih,\n  rw list.chain'_iff_nth_le at a,\n  intros i i_bounds,\n  have nodup := h,\n  by_cases i = 0,\n  have i_eq := h,\n  simp_rw h,\n  rw [list.nth_le], rw [list.nth_le],\n  specialize a 0,\n  have a_proof : 0 < (l_hd :: l_tl).length - 1,\n  rw list.length_cons,\n  simp only [nat.add_succ_sub_one, add_zero],\n  exact (list.length_pos_of_ne_nil j),\n  specialize a a_proof,\n  rw [list.nth_le] at a, rw [list.nth_le] at a,\n  have x := (to_path_first_elem {x : X // x \u2260 c} l_tl j),\n  rw to_path_first_elem',\n  simp,\n  \n  by_cases (l_hd \u2208 D),\n  rw ite_left_if h,\n  have h' := h,\n  by_cases (l_tl.nth_le 0 _ \u2208 D),\n  exfalso,\n  contrapose nodup,\n  push_neg,\n  rw list.index_of_lt_length,\n  rw ite_left_if h',\n  rw list.mem_iff_nth_le,\n  use 0,\n  have len : 0 < (to_path (list.map (\u03bb (x : {x // x \u2260 c}), ite (x \u2208 D) c \u2191x) l_tl)).length,\n  change (0 < (remove_clones' c D l_tl).length),\n  apply list.length_pos_of_ne_nil,\n  rw \u2190remove_clones'_ne_nil_iff,\n  exact j,\n  use len,\n  rw to_path_first_elem',\n  rw list.nth_le_map (\u03bb (x : {x // x \u2260 c}), ite (x \u2208 D) c \u2191x) _ (list.length_pos_of_ne_nil j),\n  simp,\n  intro w,\n  exfalso, exact w h,\n\n  rw ite_right_if h,\n  rw margin_eq_clone_non_clone' P c D H' H clone, \n  rw margin_eq_clone_non_clone P c D (l_tl.nth_le 0 _) l_hd h' h clone, \n  exact a,\n\n  rw ite_right_if h,\n  have h' := h,\n  by_cases (l_tl.nth_le 0 _ \u2208 D),\n  rw ite_left_if h,\n  rw margin_eq_clone_non_clone' P c D H' H clone, \n  rw margin_eq_clone_non_clone' P c D h h' clone, \n  exact a,\n\n  rw ite_right_if h,\n  rw margin_eq_clone_non_clone' P c D H' H clone, \n  rw margin_eq_margin_minus_candidate P,\n  exact a,\n\n  rw [list.nth_le],\n  specialize l_ih (i - 1),\n  rw list.length_cons at i_bounds,\n  simp only [nat.add_succ_sub_one, add_zero] at i_bounds,\n  have o : \u2200 i n, \u00ac i = 0 \u2192 i < n \u2192 i - 1 < n - 1 := by omega,\n  specialize l_ih (o i (to_path (list.map (\u03bb (x : {x // x \u2260 c}), ite (x \u2208 D) c \u2191x) l_tl)).length h i_bounds),\n  --change (margin P c \u2191b \u2264 margin P ((((ite (\u2200 (h : \u00acl_hd = c), (\u27e8l_hd, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) c l_hd)) :: (remove_clones c D l_tl)).nth_le i (nat.lt_of_lt_pred i_bounds)) ((remove_clones c D l_tl).nth_le i _)),\n  rw nth_le_cons h,\n  have o : \u2200 i, \u00ac i = 0 \u2192 i - 1 + 1 = i := by omega,\n  simp_rw (o i h) at l_ih,\n  exact l_ih,\nend\n\nlemma remove_clones'_cycle2 [fintype V] {l : list X} {a' : {x : X // x \u2260 c}} (H : a' \u2209 D) {d : {x : X // x \u2260 c}} (H' : d \u2208 D) (clone : clones P c D) : list.chain' (\u03bb (a_1 b : X), margin P \u2191a' c \u2264 margin P a_1 b) l\n\u2192 list.chain' (\u03bb (a_1 b : {x // x \u2260 c}), margin (minus_candidate P c) a' d \u2264 margin (minus_candidate P c) a_1 b) (replace_clones c D d H' l) :=\nbegin\n  intro a,\n  induction l,\n    {obviously,},\n  unfold replace_clones,\n  simp [to_path],\n  specialize l_ih (and.right (list.chain'_cons'.mp a)),\n  by_cases @has_lt.lt.{0} nat nat.has_lt\n  (@list.index_of.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (\u03bb (a b : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)),\n        classical.prop_decidable (@eq.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) a b))\n     (@dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (\u2200 (h : not (@eq.{1} X l_hd c)),\n           @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n             (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n             (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                   (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                      (iff.refl (@ne.{1} X l_hd c))\n                      (not (@eq.{1} X l_hd c))\n                      (@ne.def.{1} X l_hd c))\n                   h))\n             D)\n        (@forall_prop_decidable (not (@eq.{1} X l_hd c))\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))\n                D)\n           (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) l_hd c)\n           (\u03bb (h : not (@eq.{1} X l_hd c)),\n              @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                   (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                      (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                         (iff.refl (@ne.{1} X l_hd c))\n                         (not (@eq.{1} X l_hd c))\n                         (@ne.def.{1} X l_hd c))\n                      h))))\n        (\u03bb\n         (h :\n           \u2200 (h : not (@eq.{1} X l_hd c)),\n             @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n               (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n               (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                  (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                     (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                        (iff.refl (@ne.{1} X l_hd c))\n                        (not (@eq.{1} X l_hd c))\n                        (@ne.def.{1} X l_hd c))\n                     h))\n               D), d)\n        (\u03bb\n         (h :\n           not\n             (\u2200 (h : not (@eq.{1} X l_hd c)),\n                @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                  (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                  (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                     (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                        (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                           (iff.refl (@ne.{1} X l_hd c))\n                           (not (@eq.{1} X l_hd c))\n                           (@ne.def.{1} X l_hd c))\n                        h))\n                  D)),\n           @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n             (@replace_clones_helper X c D l_hd\n                (@iff.mpr\n                   (not\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D))\n                   (not\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D))\n                   (@not_iff_not_of_iff\n                      (\u2200 (p : @ne.{1} X l_hd c),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd p)\n                           D)\n                      (\u2200 (h : not (@eq.{1} X l_hd c)),\n                         @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                           (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                           (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd\n                              (@iff.mpr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                                    (iff.refl (@ne.{1} X l_hd c))\n                                    (not (@eq.{1} X l_hd c))\n                                    (@ne.def.{1} X l_hd c))\n                                 h))\n                           D)\n                      (@forall_prop_congr (@ne.{1} X l_hd c) (not (@eq.{1} X l_hd c))\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                              (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                              (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                              D)\n                         (\u03bb (h : @ne.{1} X l_hd c),\n                            iff.refl\n                              (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                 (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                 (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) l_hd h)\n                                 D))\n                         (@eq.rec.{0 1} Prop (@ne.{1} X l_hd c) (\u03bb (A : Prop), iff (@ne.{1} X l_hd c) A)\n                            (iff.refl (@ne.{1} X l_hd c))\n                            (not (@eq.{1} X l_hd c))\n                            (@ne.def.{1} X l_hd c))))\n                   h))))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl)))\n  (@list.length.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n     (@to_path (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n        (@list.map.{0 0} X (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n           (\u03bb (x : X),\n              @dite.{1} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                (\u2200 (h : not (@eq.{1} X x c)),\n                   @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                     (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                     (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                        (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                           (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                              (iff.refl (@ne.{1} X x c))\n                              (not (@eq.{1} X x c))\n                              (@ne.def.{1} X x c))\n                           h))\n                     D)\n                (@forall_prop_decidable (not (@eq.{1} X x c))\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                        (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))\n                        D)\n                   (@ne.decidable.{1} X (\u03bb (a b : X), classical.prop_decidable (@eq.{1} X a b)) x c)\n                   (\u03bb (h : not (@eq.{1} X x c)),\n                      @set.decidable_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)) D\n                        (\u03bb (a : @subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)), classical.prop_decidable (D a))\n                        (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                           (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                              (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                 (iff.refl (@ne.{1} X x c))\n                                 (not (@eq.{1} X x c))\n                                 (@ne.def.{1} X x c))\n                              h))))\n                (\u03bb\n                 (h :\n                   \u2200 (h : not (@eq.{1} X x c)),\n                     @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                       (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                       (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                          (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                             (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                (iff.refl (@ne.{1} X x c))\n                                (not (@eq.{1} X x c))\n                                (@ne.def.{1} X x c))\n                             h))\n                       D), d)\n                (\u03bb\n                 (h :\n                   not\n                     (\u2200 (h : not (@eq.{1} X x c)),\n                        @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                          (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                          (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                             (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                   (iff.refl (@ne.{1} X x c))\n                                   (not (@eq.{1} X x c))\n                                   (@ne.def.{1} X x c))\n                                h))\n                          D)),\n                   @subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                     (@replace_clones_helper X c D x\n                        (@iff.mpr\n                           (not\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D))\n                           (not\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D))\n                           (@not_iff_not_of_iff\n                              (\u2200 (p : @ne.{1} X x c),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x p)\n                                   D)\n                              (\u2200 (h : not (@eq.{1} X x c)),\n                                 @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                   (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                   (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x\n                                      (@iff.mpr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                         (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                            (iff.refl (@ne.{1} X x c))\n                                            (not (@eq.{1} X x c))\n                                            (@ne.def.{1} X x c))\n                                         h))\n                                   D)\n                              (@forall_prop_congr (@ne.{1} X x c) (not (@eq.{1} X x c))\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    @has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                      (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                      (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                      D)\n                                 (\u03bb (h : @ne.{1} X x c),\n                                    iff.refl\n                                      (@has_mem.mem.{0 0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c))\n                                         (set.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@set.has_mem.{0} (@subtype.{1} X (\u03bb (x : X), @ne.{1} X x c)))\n                                         (@subtype.mk.{1} X (\u03bb (x : X), @ne.{1} X x c) x h)\n                                         D))\n                                 (@eq.rec.{0 1} Prop (@ne.{1} X x c) (\u03bb (A : Prop), iff (@ne.{1} X x c) A)\n                                    (iff.refl (@ne.{1} X x c))\n                                    (not (@eq.{1} X x c))\n                                    (@ne.def.{1} X x c))))\n                           h))))\n           l_tl))),\n\n  simp_rw (ite_left_if h),\n  exact drop_chain'_of_chain' l_ih,\n\n  simp_rw (ite_right_if h),\n  by_cases j : l_tl = list.nil,\n  rw j,\n  simp [to_path],\n  \n  rw list.chain'_iff_nth_le,\n  rw list.chain'_iff_nth_le at l_ih,\n  rw list.chain'_iff_nth_le at a,\n  intros i i_bounds,\n  rename h nodup,\n  by_cases i = 0,\n  rename h i_eq,\n  simp_rw i_eq,\n  rw [list.nth_le], rw [list.nth_le],\n  specialize a 0,\n  have a_proof : 0 < (l_hd :: l_tl).length - 1,\n  rw list.length_cons,\n  simp only [nat.add_succ_sub_one, add_zero],\n  exact (list.length_pos_of_ne_nil j),\n  specialize a a_proof,\n  rw [list.nth_le] at a, rw [list.nth_le] at a,\n  have x := (to_path_first_elem X l_tl j),\n  rw to_path_first_elem',\n  simp,\n  \n  split_ifs with h',\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  exfalso,\n  contrapose nodup,\n  push_neg,\n  rw list.index_of_lt_length,\n  rw dite_left_if h',\n  rw list.mem_iff_nth_le,\n  use 0,\n  have len : (0 < (replace_clones c D d H' l_tl).length),\n  apply list.length_pos_of_ne_nil,\n  rw \u2190replace_clones_ne_nil_iff,\n  exact j,\n  use len,\n  rw to_path_first_elem',\n  rw list.nth_le_map (\u03bb (x : X), dite (\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D) (\u03bb (h : \u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), d) (\u03bb (h : \u00ac\u2200 (h : \u00acx = c), (\u27e8x, h\u27e9 : {x : X // x \u2260 c}) \u2208 D), \u27e8x, replace_clones_helper c D h\u27e9)) _ (list.length_pos_of_ne_nil j),\n  simp,\n  rw dite_left_if h,\n\n  rw dite_right_if h,\n\n  push_neg at h,\n  cases h with not_c not_d,\n  by_cases l_hd = c,\n  rw h at a,\n  rw \u2190margin_eq_clone_non_clone' P c D H' H clone,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H' not_d clone,\n  exact a,\n\n\n  specialize h' h,\n  rw \u2190margin_eq_clone_non_clone' P c D H' H clone,\n  rw \u2190margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 d H' not_d clone,\n  rw margin_eq_clone_non_clone P c D \u27e8l_tl.nth_le 0 _, not_c\u27e9 \u27e8l_hd, h\u27e9 h' not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n  exact h,\n\n  by_cases (\u2200 (h_1 : \u00acl_tl.nth_le 0 (list.length_pos_of_ne_nil j) = c), (\u27e8l_tl.nth_le 0 (list.length_pos_of_ne_nil j), h_1\u27e9 : {x : X // x \u2260 c}) \u2208 D),\n  rw dite_left_if h,\n  push_neg at h',\n  cases h' with not_c not_d,\n  rename h h'',\n  by_cases (l_tl.nth_le 0 (list.nth_le._main._proof_1 l_hd l_tl 0 (nat.lt_pred_iff.mp a_proof))) = c,\n  rw h at a,\n  rw \u2190margin_eq_clone_non_clone' P c D H' H clone,\n  rw \u2190margin_eq_clone_non_clone' P c D H' not_d clone,\n  exact a,\n\n  specialize h'' h,\n  rw \u2190margin_eq_clone_non_clone' P c D H' H clone,\n  rw \u2190margin_eq_clone_non_clone' P c D H' not_d clone,\n  rw margin_eq_clone_non_clone' P c D h'' not_d clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n\n  rw dite_right_if h,\n  rw \u2190margin_eq_clone_non_clone' P c D H' H clone,\n  rw \u2190margin_eq_margin_minus_candidate,\n  exact a,\n  exact h,\n\n  rw [list.nth_le],\n  specialize l_ih (i - 1),\n  rw list.length_cons at i_bounds,\n  simp only [nat.add_succ_sub_one, add_zero] at i_bounds,\n  have o : \u2200 i n, \u00ac i = 0 \u2192 i < n \u2192 i - 1 < n - 1 := by omega,\n  specialize l_ih (o i (replace_clones c D d H' l_tl).length h i_bounds),\n  rw nth_le_cons h,\n  have o : \u2200 i, \u00ac i = 0 \u2192 i - 1 + 1 = i := by omega,\n  simp_rw (o i h) at l_ih,\n  exact l_ih,\nend\n\nlemma every_clone_defeated' (a : {x : X // x \u2260 c}) (e : a \u2209 D) (d \u2208 D) : clones P c D \u2192 ((split_cycle_VCCR V X P) a c \u2194 (split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c)) a d) :=\nbegin\n  intro clone,\n  rw split_cycle_definitions,\n  unfold split_cycle_VCCR',\n  split,\n  intro w,\n  introI f,\n  cases w with m w,\n  unfold margin_pos at m,\n  rw margin_eq_clone_non_clone' P c D H e clone at m,\n  rw \u2190margin_eq_margin_minus_candidate at m,\n  use m,\n  push_neg, push_neg at w,\n  intro l,\n  specialize w (remove_clones' c D l),\n  intro ne_nil,\n  specialize w ((remove_clones'_ne_nil_iff c D l).mp ne_nil),\n  contrapose w,\n  push_neg, push_neg at w,\n  use remove_clones'_nodup c D l,\n  cases w with l_nodup w,\n  cases w with first w,\n  cases w with last w,\n\n  split,\n  unfold remove_clones',\n  rw to_path_first_elem',\n  rw list.nth_le_map,\n  simp_rw first,\n  split_ifs,\n  refl,\n  exact list.length_pos_of_ne_nil ne_nil,\n  use remove_clones'_last_elem c D l a e ne_nil last, \n  exact remove_clones'_cycle1 P c D e H clone w,\n\n  intro w,\n  introI f,\n  cases w with m w,\n  unfold margin_pos at m,\n  rw \u2190margin_eq_clone_non_clone' P c D H e clone at m,\n  use m,\n  push_neg, push_neg at w,\n  intros l ne_nil,\n  specialize w (replace_clones c D d H l),\n  have n: (replace_clones c D d H l) \u2260 list.nil,\n  rw \u2190replace_clones_ne_nil_iff,\n  exact ne_nil,\n\n  specialize w n,\n  contrapose w,\n  push_neg, push_neg at w,\n  split,\n  apply replace_clones_nodup,\n  cases w with nd w,\n  cases w with first w,\n  cases w with last w,\n\n  split,\n  unfold replace_clones,\n  rw to_path_first_elem,\n  rw list.nth_le_map,\n  split_ifs,\n  refl,\n  push_neg at h,\n  cases h with contr h,\n  exfalso,\n  exact contr first,\n  simp,\n  exact ne_nil,\n\n  rw replace_clones_last_elem c D l a,\n  simp,\n  \n  exact remove_clones'_cycle2 P c D e H clone w,\n  push_neg,\n  use a.property,\n  simp, exact e,\n  exact last,\nend\n--set_option pp.all true \n\ntheorem non_clone_choice_ind_clones_split_cyle [fintype V] : non_clone_choice_ind_clones P c D split_cycle :=\nbegin\n  unfold non_clone_choice_ind_clones,\n  intro clone,\n  intro a, intro not_d,\n\n  unfold split_cycle,\n  unfold max_el_VSCC,\n  simp,\n  split,\n  intro d,\n  contrapose d,\n  push_neg at d, push_neg,\n  cases d with b d,\n  cases d with b_c d,\n  use b,\n  have test := (clone_maintains_defeat P c D \u27e8b, b_c\u27e9 a not_d clone).mp, \n  simp at test,\n  apply test,\n  introI _inst_1,\n  exact d,\n\n  intro d,\n  contrapose d,\n  push_neg at d, push_neg,\n  cases d with b d,\n  by_cases b = c,\n  have clone' := clone,\n  unfold clones at clone',\n  cases clone' with n clone',\n  use n.some,\n  use n.some.property,\n  simp,\n  apply (clone_maintains_defeat P c D n.some a not_d clone).mpr, \n  have test := every_clone_defeats P c D a not_d n.some n.some_spec clone, \n  apply test.mp,\n  rw h at d,\n  introI _inst_1,\n  exact d,\n\n  use b,\n  use h,\n  have test := (clone_maintains_defeat P c D \u27e8b, h\u27e9 a not_d clone).mpr, \n  simp at test,\n  apply test,\n  introI _inst_1,\n  exact d,\nend\n\ntheorem clone_choice_ind_clones_split_cyle [fintype V] [fintype X] : clone_choice_ind_clones P c D split_cycle :=\nbegin\n   unfold clone_choice_ind_clones,\n   intro clone, \n   split,\n   intro a,\n   cases a with a1 a2,\n\n   have spec : \u2200 h : c \u2260 c, (\u27e8c, h\u27e9 : {x // x \u2260 c}) \u2208 D,\n      {intro contr, exfalso, exact contr (eq.refl c), },\n   by_cases \u2200 c' : {x // \u2200 h : x \u2260 c, (\u27e8x, h\u27e9 : {x // x \u2260 c}) \u2208 D}, \u2203 d : {x // \u2200 h : x \u2260 c, (\u27e8x, h\u27e9 : {x // x \u2260 c}) \u2208 D}, split_cycle_VCCR V X P d c',\n   have clone' := clone,\n   cases clone' with n clone,\n   let f : {x // \u2200 h : x \u2260 c, (\u27e8x, h\u27e9 : {x // x \u2260 c}) \u2208 D} \u2192 {x // \u2200 h : x \u2260 c, (\u27e8x, h\u27e9 : {x // x \u2260 c}) \u2208 D} := \u03bb x, (h x).some,\n   let seq := stream.iterate f \u27e8c, spec\u27e9,\n   have property : \u2200 x : \u2115, split_cycle_VCCR V X P (seq.nth x.succ) (seq.nth x),\n      {intro i,\n      change split_cycle_VCCR V X P ((h (seq.nth i)).some) (seq.nth i),\n      have test := (h (seq.nth i)).some_spec,\n\n      unfold split_cycle_VCCR, intro f, \n      unfold margin_pos,\n      simp_rw margin_eq_margin f _inst_1,\n      exact test,},\n\n   let seq2 : stream X := \u03bb (x : \u2115), (seq.nth x).val,\n   exfalso, \n   exact false_of_sequence_acyclic_vccr (split_cycle_VCCR_acyclic P) seq2 property,\n\n   push_neg at h,\n   cases h with d h,\n   unfold split_cycle at a1,\n   unfold max_el_VSCC at a1,\n   unfold split_cycle at a2, unfold max_el_VSCC at a2,\n   simp at a2,\n   simp at a1,\n   have some_a : \u2203 a, (\u2203 h: a \u2260 c, (\u27e8a, h\u27e9 : {x // x \u2260 c}) \u2209 D) \u2227 split_cycle_VCCR V X P a d, \n      {by_cases h': \u2191d = c, cases a1 with x a1, use x, \n      split, contrapose h, push_neg, simp at h,\n      use \u27e8x, h\u27e9, rw h', simp, intro f, \n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      exact a1, rw h', intro f, \n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      exact a1,\n\n      specialize a2 d, specialize a2 h', specialize a2 (d.property h'),\n      cases a2 with a a2,\n      use a,\n      split, contrapose h, simp, simp at h,\n      use a, use h, intro f, \n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      exact a2, intro f, \n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      exact a2,\n      },\n\n   intros e e_D, \n   unfold split_cycle, unfold max_el_VSCC, simp,\n   cases some_a with a some_a,\n   use a,\n   cases some_a with some_a defeats,\n   cases some_a with not_c some_a,\n   use not_c,\n   by_cases \u2191d = c,\n   rw \u2190every_clone_defeated' P c D \u27e8a, not_c\u27e9 some_a e e_D clone,\n   simp_rw \u2190h, intro f,\n   unfold margin_pos,\n   simp_rw margin_eq_margin f _inst_1, \n   exact defeats,\n   rw \u2190every_clone_defeated' P c D \u27e8a, not_c\u27e9 some_a e e_D clone,\n   rw every_clone_defeated' P c D \u27e8a, not_c\u27e9 some_a \u27e8\u2191d, h\u27e9 (d.property h) clone,\n   rw clone_maintains_defeat' P c D \u27e8a, not_c\u27e9 \u27e8\u2191d, h\u27e9 some_a (d.property h) clone,\n   intro f,\n   unfold margin_pos,\n   simp_rw margin_eq_margin f _inst_1, \n   exact defeats,\n   intro a,\n\n   by_cases \u2200 c' : {x // x \u2208 D}, \u2203 d : {x // x \u2208 D}, split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c) d c',\n   have clone' := clone,\n   cases clone' with n clone,\n   let f : {x // x \u2208 D} \u2192 {x // x \u2208 D} := \u03bb x, (h x).some,\n   let seq := stream.iterate f \u27e8n.some, n.some_spec\u27e9,\n   have property : \u2200 x : \u2115, split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c) (seq.nth x.succ) (seq.nth x),\n      {intro i,\n      change split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c) ((h (seq.nth i)).some) (seq.nth i),\n      have test := (h (seq.nth i)).some_spec,\n\n      unfold split_cycle_VCCR, intro f, \n      unfold margin_pos,\n      simp_rw margin_eq_margin f _inst_1,\n      exact test,},\n\n   let seq2 : stream {x : X // x \u2260 c} := \u03bb (x : \u2115), (seq.nth x).val,\n   exfalso, \n   exact false_of_sequence_acyclic_vccr (split_cycle_VCCR_acyclic (minus_candidate P c)) seq2 property,\n\n   push_neg at h,\n   cases h with d h,\n   unfold split_cycle at a,\n   unfold max_el_VSCC at a,\n   simp at a,\n   have some_a : \u2203 a \u2209 D, split_cycle_VCCR V {x : X // x \u2260 c} (minus_candidate P c) a d, \n      {\n      specialize a d, specialize a d.val.property, simp at a,\n      cases a with a' a,\n      use a', cases a with not_c a,\n      exact not_c,\n      split, contrapose h, simp, simp at h,\n      use a', cases a with not_c a, use not_c, use h, intro f,\n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      exact a, intro f, \n      unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n      cases a with not_c a,\n      exact a,\n      },\n\n\n   unfold split_cycle, unfold max_el_VSCC, simp,\n   split,\n   cases some_a with a some_a,\n   cases some_a with H some_a,\n   use \u2191a,\n   rw every_clone_defeated' P c D a H d.val d.property clone,\n   intro f, unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n   exact some_a,\n\n   intros a not_c in_d, \n   cases some_a with a' some_a,\n   use a',\n   cases some_a with not_d defeats,\n   have test := clone_maintains_defeat' P c D a' \u27e8a, not_c\u27e9 not_d in_d clone,\n   simp at test,\n   rw \u2190test,\n   rw \u2190every_clone_defeated' P c D a' not_d \u27e8a, not_c\u27e9 in_d clone,\n   rw every_clone_defeated' P c D a' not_d d.val d.property clone,\n   intro f, unfold margin_pos, simp_rw margin_eq_margin f _inst_1,\n   exact defeats,\nend", "meta": {"author": "chasenorman", "repo": "Formalized-Voting", "sha": "de04e630b83525b042db166670ba97f9952b5691", "save_path": "github-repos/lean/chasenorman-Formalized-Voting", "path": "github-repos/lean/chasenorman-Formalized-Voting/Formalized-Voting-de04e630b83525b042db166670ba97f9952b5691/src/clones.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3676122715909644}}
{"text": "/-\nCopyright (c) 2018 Sean Leather. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sean Leather, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.alist\nimport Mathlib.data.finset.basic\nimport Mathlib.data.pfun\nimport Mathlib.PostPort\n\nuniverses u v l u_1 w \n\nnamespace Mathlib\n\n/-!\n# Finite maps over `multiset`\n-/\n\n/-! ### multisets of sigma types-/\n\nnamespace multiset\n\n\n/-- Multiset of keys of an association multiset. -/\ndef keys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : multiset (sigma \u03b2)) : multiset \u03b1 :=\n  map sigma.fst s\n\n@[simp] theorem coe_keys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {l : List (sigma \u03b2)} : keys \u2191l = \u2191(list.keys l) :=\n  rfl\n\n/-- `nodupkeys s` means that `s` has no duplicate keys. -/\ndef nodupkeys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : multiset (sigma \u03b2)) :=\n  quot.lift_on s list.nodupkeys sorry\n\n@[simp] theorem coe_nodupkeys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {l : List (sigma \u03b2)} : nodupkeys \u2191l \u2194 list.nodupkeys l :=\n  iff.rfl\n\nend multiset\n\n\n/-! ### finmap -/\n\n/-- `finmap \u03b2` is the type of finite maps over a multiset. It is effectively\n  a quotient of `alist \u03b2` by permutation of the underlying list. -/\nstructure finmap {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) \nwhere\n  entries : multiset (sigma \u03b2)\n  nodupkeys : multiset.nodupkeys entries\n\n/-- The quotient map from `alist` to `finmap`. -/\ndef alist.to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : alist \u03b2) : finmap \u03b2 :=\n  finmap.mk (\u2191(alist.entries s)) (alist.nodupkeys s)\n\ntheorem alist.to_finmap_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {s\u2081 : alist \u03b2} {s\u2082 : alist \u03b2} : alist.to_finmap s\u2081 = alist.to_finmap s\u2082 \u2194 alist.entries s\u2081 ~ alist.entries s\u2082 := sorry\n\n@[simp] theorem alist.to_finmap_entries {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : alist \u03b2) : finmap.entries (alist.to_finmap s) = \u2191(alist.entries s) :=\n  rfl\n\n/-- Given `l : list (sigma \u03b2)`, create a term of type `finmap \u03b2` by removing\nentries with duplicate keys. -/\ndef list.to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (s : List (sigma \u03b2)) : finmap \u03b2 :=\n  alist.to_finmap (list.to_alist s)\n\nnamespace finmap\n\n\n/-! ### lifting from alist -/\n\n/-- Lift a permutation-respecting function on `alist` to `finmap`. -/\ndef lift_on {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {\u03b3 : Type u_1} (s : finmap \u03b2) (f : alist \u03b2 \u2192 \u03b3) (H : \u2200 (a b : alist \u03b2), alist.entries a ~ alist.entries b \u2192 f a = f b) : \u03b3 :=\n  roption.get\n    (quotient.lift_on (entries s)\n      (fun (l : List (sigma \u03b2)) => roption.mk (list.nodupkeys l) fun (nd : list.nodupkeys l) => f (alist.mk l nd)) sorry)\n    sorry\n\n@[simp] theorem lift_on_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {\u03b3 : Type u_1} (s : alist \u03b2) (f : alist \u03b2 \u2192 \u03b3) (H : \u2200 (a b : alist \u03b2), alist.entries a ~ alist.entries b \u2192 f a = f b) : lift_on (alist.to_finmap s) f H = f s :=\n  alist.cases_on s\n    fun (s_entries : List (sigma \u03b2)) (s_nodupkeys : list.nodupkeys s_entries) =>\n      Eq.refl (lift_on (alist.to_finmap (alist.mk s_entries s_nodupkeys)) f H)\n\n/-- Lift a permutation-respecting function on 2 `alist`s to 2 `finmap`s. -/\ndef lift_on\u2082 {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {\u03b3 : Type u_1} (s\u2081 : finmap \u03b2) (s\u2082 : finmap \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3) (H : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : alist \u03b2), alist.entries a\u2081 ~ alist.entries a\u2082 \u2192 alist.entries b\u2081 ~ alist.entries b\u2082 \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082) : \u03b3 :=\n  lift_on s\u2081 (fun (l\u2081 : alist \u03b2) => lift_on s\u2082 (f l\u2081) sorry) sorry\n\n@[simp] theorem lift_on\u2082_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {\u03b3 : Type u_1} (s\u2081 : alist \u03b2) (s\u2082 : alist \u03b2) (f : alist \u03b2 \u2192 alist \u03b2 \u2192 \u03b3) (H : \u2200 (a\u2081 b\u2081 a\u2082 b\u2082 : alist \u03b2), alist.entries a\u2081 ~ alist.entries a\u2082 \u2192 alist.entries b\u2081 ~ alist.entries b\u2082 \u2192 f a\u2081 b\u2081 = f a\u2082 b\u2082) : lift_on\u2082 (alist.to_finmap s\u2081) (alist.to_finmap s\u2082) f H = f s\u2081 s\u2082 := sorry\n\n/-! ### induction -/\n\ntheorem induction_on {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {C : finmap \u03b2 \u2192 Prop} (s : finmap \u03b2) (H : \u2200 (a : alist \u03b2), C (alist.to_finmap a)) : C s := sorry\n\ntheorem induction_on\u2082 {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {C : finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop} (s\u2081 : finmap \u03b2) (s\u2082 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 : alist \u03b2), C (alist.to_finmap a\u2081) (alist.to_finmap a\u2082)) : C s\u2081 s\u2082 :=\n  induction_on s\u2081 fun (l\u2081 : alist \u03b2) => induction_on s\u2082 fun (l\u2082 : alist \u03b2) => H l\u2081 l\u2082\n\ntheorem induction_on\u2083 {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {C : finmap \u03b2 \u2192 finmap \u03b2 \u2192 finmap \u03b2 \u2192 Prop} (s\u2081 : finmap \u03b2) (s\u2082 : finmap \u03b2) (s\u2083 : finmap \u03b2) (H : \u2200 (a\u2081 a\u2082 a\u2083 : alist \u03b2), C (alist.to_finmap a\u2081) (alist.to_finmap a\u2082) (alist.to_finmap a\u2083)) : C s\u2081 s\u2082 s\u2083 :=\n  induction_on\u2082 s\u2081 s\u2082 fun (l\u2081 l\u2082 : alist \u03b2) => induction_on s\u2083 fun (l\u2083 : alist \u03b2) => H l\u2081 l\u2082 l\u2083\n\n/-! ### extensionality -/\n\ntheorem ext {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {s : finmap \u03b2} {t : finmap \u03b2} : entries s = entries t \u2192 s = t := sorry\n\n@[simp] theorem ext_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {s : finmap \u03b2} {t : finmap \u03b2} : entries s = entries t \u2194 s = t :=\n  { mp := ext, mpr := congr_arg fun {s : finmap \u03b2} => entries s }\n\n/-! ### mem -/\n\n/-- The predicate `a \u2208 s` means that `s` has a value associated to the key `a`. -/\nprotected instance has_mem {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} : has_mem \u03b1 (finmap \u03b2) :=\n  has_mem.mk fun (a : \u03b1) (s : finmap \u03b2) => a \u2208 multiset.keys (entries s)\n\ntheorem mem_def {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {a : \u03b1} {s : finmap \u03b2} : a \u2208 s \u2194 a \u2208 multiset.keys (entries s) :=\n  iff.rfl\n\n@[simp] theorem mem_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {a : \u03b1} {s : alist \u03b2} : a \u2208 alist.to_finmap s \u2194 a \u2208 s :=\n  iff.rfl\n\n/-! ### keys -/\n\n/-- The set of keys of a finite map. -/\ndef keys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : finmap \u03b2) : finset \u03b1 :=\n  finset.mk (multiset.keys (entries s)) sorry\n\n@[simp] theorem keys_val {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s : alist \u03b2) : finset.val (keys (alist.to_finmap s)) = \u2191(alist.keys s) :=\n  rfl\n\n@[simp] theorem keys_ext {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {s\u2081 : alist \u03b2} {s\u2082 : alist \u03b2} : keys (alist.to_finmap s\u2081) = keys (alist.to_finmap s\u2082) \u2194 alist.keys s\u2081 ~ alist.keys s\u2082 := sorry\n\ntheorem mem_keys {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {a : \u03b1} {s : finmap \u03b2} : a \u2208 keys s \u2194 a \u2208 s :=\n  induction_on s fun (s : alist \u03b2) => alist.mem_keys\n\n/-! ### empty -/\n\n/-- The empty map. -/\nprotected instance has_emptyc {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} : has_emptyc (finmap \u03b2) :=\n  has_emptyc.mk (mk 0 list.nodupkeys_nil)\n\nprotected instance inhabited {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} : Inhabited (finmap \u03b2) :=\n  { default := \u2205 }\n\n@[simp] theorem empty_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} : alist.to_finmap \u2205 = \u2205 :=\n  rfl\n\n@[simp] theorem to_finmap_nil {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] : list.to_finmap [] = \u2205 :=\n  rfl\n\ntheorem not_mem_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {a : \u03b1} : \u00aca \u2208 \u2205 :=\n  multiset.not_mem_zero a\n\n@[simp] theorem keys_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} : keys \u2205 = \u2205 :=\n  rfl\n\n/-! ### singleton -/\n\n/-- The singleton map. -/\ndef singleton {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (a : \u03b1) (b : \u03b2 a) : finmap \u03b2 :=\n  alist.to_finmap (alist.singleton a b)\n\n@[simp] theorem keys_singleton {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (a : \u03b1) (b : \u03b2 a) : keys (singleton a b) = singleton a :=\n  rfl\n\n@[simp] theorem mem_singleton {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (x : \u03b1) (y : \u03b1) (b : \u03b2 y) : x \u2208 singleton y b \u2194 x = y := sorry\n\nprotected instance has_decidable_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] [(a : \u03b1) \u2192 DecidableEq (\u03b2 a)] : DecidableEq (finmap \u03b2) :=\n  sorry\n\n/-! ### lookup -/\n\n/-- Look up the value associated to a key in a map. -/\ndef lookup {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : Option (\u03b2 a) :=\n  lift_on s (alist.lookup a) sorry\n\n@[simp] theorem lookup_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : alist \u03b2) : lookup a (alist.to_finmap s) = alist.lookup a s :=\n  rfl\n\n@[simp] theorem lookup_list_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : List (sigma \u03b2)) : lookup a (list.to_finmap s) = list.lookup a s := sorry\n\n@[simp] theorem lookup_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) : lookup a \u2205 = none :=\n  rfl\n\ntheorem lookup_is_some {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s : finmap \u03b2} : \u21a5(option.is_some (lookup a s)) \u2194 a \u2208 s :=\n  induction_on s fun (s : alist \u03b2) => alist.lookup_is_some\n\ntheorem lookup_eq_none {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s : finmap \u03b2} : lookup a s = none \u2194 \u00aca \u2208 s :=\n  induction_on s fun (s : alist \u03b2) => alist.lookup_eq_none\n\n@[simp] theorem lookup_singleton_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} : lookup a (singleton a b) = some b := sorry\n\nprotected instance has_mem.mem.decidable {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : Decidable (a \u2208 s) :=\n  decidable_of_iff \u21a5(option.is_some (lookup a s)) sorry\n\ntheorem mem_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s : finmap \u03b2} : a \u2208 s \u2194 \u2203 (b : \u03b2 a), lookup a s = some b :=\n  induction_on s\n    fun (s : alist \u03b2) =>\n      iff.trans list.mem_keys (exists_congr fun (b : \u03b2 a) => iff.symm (list.mem_lookup_iff (alist.nodupkeys s)))\n\ntheorem mem_of_lookup_eq_some {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} (h : lookup a s = some b) : a \u2208 s :=\n  iff.mpr mem_iff (Exists.intro b h)\n\ntheorem ext_lookup {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : (\u2200 (x : \u03b1), lookup x s\u2081 = lookup x s\u2082) \u2192 s\u2081 = s\u2082 := sorry\n\n/-! ### replace -/\n\n/-- Replace a key with a given value in a finite map.\n  If the key is not present it does nothing. -/\ndef replace {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\n  lift_on s (fun (t : alist \u03b2) => alist.to_finmap (alist.replace a b t)) sorry\n\n@[simp] theorem replace_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) : replace a b (alist.to_finmap s) = alist.to_finmap (alist.replace a b s) := sorry\n\n@[simp] theorem keys_replace {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : keys (replace a b s) = keys s := sorry\n\n@[simp] theorem mem_replace {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} : a' \u2208 replace a b s \u2194 a' \u2208 s := sorry\n\n/-! ### foldl -/\n\n/-- Fold a commutative function over the key-value pairs in the map -/\ndef foldl {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} {\u03b4 : Type w} (f : \u03b4 \u2192 (a : \u03b1) \u2192 \u03b2 a \u2192 \u03b4) (H : \u2200 (d : \u03b4) (a\u2081 : \u03b1) (b\u2081 : \u03b2 a\u2081) (a\u2082 : \u03b1) (b\u2082 : \u03b2 a\u2082), f (f d a\u2081 b\u2081) a\u2082 b\u2082 = f (f d a\u2082 b\u2082) a\u2081 b\u2081) (d : \u03b4) (m : finmap \u03b2) : \u03b4 :=\n  multiset.foldl (fun (d : \u03b4) (s : sigma \u03b2) => f d (sigma.fst s) (sigma.snd s)) sorry d (entries m)\n\n/-- `any f s` returns `tt` iff there exists a value `v` in `s` such that `f v = tt`. -/\ndef any {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (f : (x : \u03b1) \u2192 \u03b2 x \u2192 Bool) (s : finmap \u03b2) : Bool :=\n  foldl (fun (x : Bool) (y : \u03b1) (z : \u03b2 y) => to_bool (\u21a5x \u2228 \u21a5(f y z))) sorry false s\n\n/-- `all f s` returns `tt` iff `f v = tt` for all values `v` in `s`. -/\ndef all {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (f : (x : \u03b1) \u2192 \u03b2 x \u2192 Bool) (s : finmap \u03b2) : Bool :=\n  foldl (fun (x : Bool) (y : \u03b1) (z : \u03b2 y) => to_bool (\u21a5x \u2227 \u21a5(f y z))) sorry false s\n\n/-! ### erase -/\n\n/-- Erase a key from the map. If the key is not present it does nothing. -/\ndef erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : finmap \u03b2 :=\n  lift_on s (fun (t : alist \u03b2) => alist.to_finmap (alist.erase a t)) sorry\n\n@[simp] theorem erase_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : alist \u03b2) : erase a (alist.to_finmap s) = alist.to_finmap (alist.erase a s) := sorry\n\n@[simp] theorem keys_erase_to_finset {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : alist \u03b2) : keys (alist.to_finmap (alist.erase a s)) = finset.erase (keys (alist.to_finmap s)) a := sorry\n\n@[simp] theorem keys_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : keys (erase a s) = finset.erase (keys s) a := sorry\n\n@[simp] theorem mem_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {s : finmap \u03b2} : a' \u2208 erase a s \u2194 a' \u2260 a \u2227 a' \u2208 s := sorry\n\ntheorem not_mem_erase_self {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s : finmap \u03b2} : \u00aca \u2208 erase a s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (\u00aca \u2208 erase a s)) (propext mem_erase)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (\u00ac(a \u2260 a \u2227 a \u2208 s))) (propext not_and_distrib)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (\u00aca \u2260 a \u2228 \u00aca \u2208 s)) (propext not_not))) (Or.inl (Eq.refl a))))\n\n@[simp] theorem lookup_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : lookup a (erase a s) = none :=\n  induction_on s (alist.lookup_erase a)\n\n@[simp] theorem lookup_erase_ne {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {s : finmap \u03b2} (h : a \u2260 a') : lookup a (erase a' s) = lookup a s :=\n  induction_on s fun (s : alist \u03b2) => alist.lookup_erase_ne h\n\ntheorem erase_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {s : finmap \u03b2} : erase a (erase a' s) = erase a' (erase a s) := sorry\n\n/-! ### sdiff -/\n\n/-- `sdiff s s'` consists of all key-value pairs from `s` and `s'` where the keys are in `s` or\n`s'` but not both. -/\ndef sdiff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (s : finmap \u03b2) (s' : finmap \u03b2) : finmap \u03b2 :=\n  foldl (fun (s : finmap \u03b2) (x : \u03b1) (_x : \u03b2 x) => erase x s) sorry s s'\n\nprotected instance has_sdiff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] : has_sdiff (finmap \u03b2) :=\n  has_sdiff.mk sdiff\n\n/-! ### insert -/\n\n/-- Insert a key-value pair into a finite map, replacing any existing pair with\n  the same key. -/\ndef insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) : finmap \u03b2 :=\n  lift_on s (fun (t : alist \u03b2) => alist.to_finmap (alist.insert a b t)) sorry\n\n@[simp] theorem insert_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : alist \u03b2) : insert a b (alist.to_finmap s) = alist.to_finmap (alist.insert a b s) := sorry\n\ntheorem insert_entries_of_neg {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {s : finmap \u03b2} : \u00aca \u2208 s \u2192 entries (insert a b s) = sigma.mk a b ::\u2098 entries s := sorry\n\n@[simp] theorem mem_insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {b' : \u03b2 a'} {s : finmap \u03b2} : a \u2208 insert a' b' s \u2194 a = a' \u2228 a \u2208 s :=\n  induction_on s alist.mem_insert\n\n@[simp] theorem lookup_insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} (s : finmap \u03b2) : lookup a (insert a b s) = some b := sorry\n\n@[simp] theorem lookup_insert_of_ne {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {b : \u03b2 a} (s : finmap \u03b2) (h : a' \u2260 a) : lookup a' (insert a b s) = lookup a' s := sorry\n\n@[simp] theorem insert_insert {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {b' : \u03b2 a} (s : finmap \u03b2) : insert a b' (insert a b s) = insert a b' s := sorry\n\ntheorem insert_insert_of_ne {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {a' : \u03b1} {b : \u03b2 a} {b' : \u03b2 a'} (s : finmap \u03b2) (h : a \u2260 a') : insert a' b' (insert a b s) = insert a b (insert a' b' s) := sorry\n\ntheorem to_finmap_cons {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (xs : List (sigma \u03b2)) : list.to_finmap (sigma.mk a b :: xs) = insert a b (list.to_finmap xs) :=\n  rfl\n\ntheorem mem_list_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (xs : List (sigma \u03b2)) : a \u2208 list.to_finmap xs \u2194 \u2203 (b : \u03b2 a), sigma.mk a b \u2208 xs := sorry\n\n@[simp] theorem insert_singleton_eq {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {b' : \u03b2 a} : insert a b (singleton a b') = singleton a b := sorry\n\n/-! ### extract -/\n\n/-- Erase a key from the map, and return the corresponding value, if found. -/\ndef extract {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : Option (\u03b2 a) \u00d7 finmap \u03b2 :=\n  lift_on s (fun (t : alist \u03b2) => prod.map id alist.to_finmap (alist.extract a t)) sorry\n\n@[simp] theorem extract_eq_lookup_erase {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (s : finmap \u03b2) : extract a s = (lookup a s, erase a s) := sorry\n\n/-! ### union -/\n\n/-- `s\u2081 \u222a s\u2082` is the key-based union of two finite maps. It is left-biased: if\nthere exists an `a \u2208 s\u2081`, `lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081`. -/\ndef union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (s\u2081 : finmap \u03b2) (s\u2082 : finmap \u03b2) : finmap \u03b2 :=\n  lift_on\u2082 s\u2081 s\u2082 (fun (s\u2081 s\u2082 : alist \u03b2) => alist.to_finmap (s\u2081 \u222a s\u2082)) sorry\n\nprotected instance has_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] : has_union (finmap \u03b2) :=\n  has_union.mk union\n\n@[simp] theorem mem_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : a \u2208 s\u2081 \u222a s\u2082 \u2194 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\n  induction_on\u2082 s\u2081 s\u2082 fun (_x _x_1 : alist \u03b2) => alist.mem_union\n\n@[simp] theorem union_to_finmap {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (s\u2081 : alist \u03b2) (s\u2082 : alist \u03b2) : alist.to_finmap s\u2081 \u222a alist.to_finmap s\u2082 = alist.to_finmap (s\u2081 \u222a s\u2082) := sorry\n\ntheorem keys_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : keys (s\u2081 \u222a s\u2082) = keys s\u2081 \u222a keys s\u2082 := sorry\n\n@[simp] theorem lookup_union_left {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : a \u2208 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 :=\n  induction_on\u2082 s\u2081 s\u2082 fun (s\u2081 s\u2082 : alist \u03b2) => alist.lookup_union_left\n\n@[simp] theorem lookup_union_right {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : \u00aca \u2208 s\u2081 \u2192 lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2082 :=\n  induction_on\u2082 s\u2081 s\u2082 fun (s\u2081 s\u2082 : alist \u03b2) => alist.lookup_union_right\n\ntheorem lookup_union_left_of_not_in {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} (h : \u00aca \u2208 s\u2082) : lookup a (s\u2081 \u222a s\u2082) = lookup a s\u2081 := sorry\n\n@[simp] theorem mem_lookup_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : b \u2208 lookup a (s\u2081 \u222a s\u2082) \u2194 b \u2208 lookup a s\u2081 \u2228 \u00aca \u2208 s\u2081 \u2227 b \u2208 lookup a s\u2082 :=\n  induction_on\u2082 s\u2081 s\u2082 fun (s\u2081 s\u2082 : alist \u03b2) => alist.mem_lookup_union\n\ntheorem mem_lookup_union_middle {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} {s\u2083 : finmap \u03b2} : b \u2208 lookup a (s\u2081 \u222a s\u2083) \u2192 \u00aca \u2208 s\u2082 \u2192 b \u2208 lookup a (s\u2081 \u222a s\u2082 \u222a s\u2083) :=\n  induction_on\u2083 s\u2081 s\u2082 s\u2083 fun (s\u2081 s\u2082 s\u2083 : alist \u03b2) => alist.mem_lookup_union_middle\n\ntheorem insert_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b2 a} {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : insert a b (s\u2081 \u222a s\u2082) = insert a b s\u2081 \u222a s\u2082 := sorry\n\ntheorem union_assoc {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} {s\u2083 : finmap \u03b2} : s\u2081 \u222a s\u2082 \u222a s\u2083 = s\u2081 \u222a (s\u2082 \u222a s\u2083) := sorry\n\n@[simp] theorem empty_union {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} : \u2205 \u222a s\u2081 = s\u2081 := sorry\n\n@[simp] theorem union_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} : s\u2081 \u222a \u2205 = s\u2081 := sorry\n\ntheorem erase_union_singleton {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b2 a) (s : finmap \u03b2) (h : lookup a s = some b) : erase a s \u222a singleton a b = s := sorry\n\n/-! ### disjoint -/\n\n/-- `disjoint s\u2081 s\u2082` holds if `s\u2081` and `s\u2082` have no keys in common. -/\ndef disjoint {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (s\u2081 : finmap \u03b2) (s\u2082 : finmap \u03b2) :=\n  \u2200 (x : \u03b1), x \u2208 s\u2081 \u2192 \u00acx \u2208 s\u2082\n\ntheorem disjoint_empty {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (x : finmap \u03b2) : disjoint \u2205 x :=\n  fun (x_1 : \u03b1) (H : x_1 \u2208 \u2205) (\u1fb0 : x_1 \u2208 x) => false.dcases_on (fun (H : x_1 \u2208 \u2205) => False) H\n\ntheorem disjoint.symm {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (x : finmap \u03b2) (y : finmap \u03b2) (h : disjoint x y) : disjoint y x :=\n  fun (p : \u03b1) (hy : p \u2208 y) (hx : p \u2208 x) => h p hx hy\n\ntheorem disjoint.symm_iff {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (x : finmap \u03b2) (y : finmap \u03b2) : disjoint x y \u2194 disjoint y x :=\n  { mp := disjoint.symm x y, mpr := disjoint.symm y x }\n\nprotected instance disjoint.decidable_rel {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] : DecidableRel disjoint :=\n  fun (x y : finmap \u03b2) => id multiset.decidable_dforall_multiset\n\ntheorem disjoint_union_left {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (x : finmap \u03b2) (y : finmap \u03b2) (z : finmap \u03b2) : disjoint (x \u222a y) z \u2194 disjoint x z \u2227 disjoint y z := sorry\n\ntheorem disjoint_union_right {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] (x : finmap \u03b2) (y : finmap \u03b2) (z : finmap \u03b2) : disjoint x (y \u222a z) \u2194 disjoint x y \u2227 disjoint x z := sorry\n\ntheorem union_comm_of_disjoint {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} : disjoint s\u2081 s\u2082 \u2192 s\u2081 \u222a s\u2082 = s\u2082 \u222a s\u2081 := sorry\n\ntheorem union_cancel {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1] {s\u2081 : finmap \u03b2} {s\u2082 : finmap \u03b2} {s\u2083 : finmap \u03b2} (h : disjoint s\u2081 s\u2083) (h' : disjoint s\u2082 s\u2083) : s\u2081 \u222a s\u2083 = s\u2082 \u222a s\u2083 \u2194 s\u2081 = s\u2082 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/finmap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.36761224694419087}}
{"text": "universe u\naxiom f {\u03b1 : Sort u} (a : \u03b1) : \u03b1\naxiom f_eq {\u03b1 : Sort u} (a : \u03b1) : f a = a\n\nexample (a : Nat) : f id a = a := by\n  simp only [f_eq]\n  trace_state\n  rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/simpPrefixIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3675893835507342}}
{"text": "import polytime.size\nimport stack_rec\nimport misc\n\nopen tree function polysize (size)\nopen_locale tree\n\nnamespace tree\n\nsection\n\nvariables {\u03b1 : Type} {\u03b2 : Type} [tencodable \u03b1] [tencodable \u03b2]\n  {base : \u03b1 \u2192 \u03b2} {pre\u2081 pre\u2082 : tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b1}\n  {post : \u03b2 \u2192 \u03b2 \u2192 tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b2}\n\n@[elab_as_eliminator]\ntheorem stack_step_induction\n  (P : tree unit \u2192 \u03b1 \u2192 iterator_stack \u03b1 \u03b2 \u2192 Prop)\n  (hb : \u2200 t arg, P t arg (sum.inl (t, arg, none))) \n  (hb' : \u2200 \u2983l r arg res\u2984, res = stack_rec base pre\u2081 pre\u2082 post l (pre\u2081 l r arg) \u2192 P l (pre\u2081 l r arg) (sum.inr res)\n           \u2192 P (l \u25b3 r) arg (sum.inl (l \u25b3 r, arg, some res)))\n  (hbr : \u2200 arg, P tree.nil arg (sum.inr (base arg))) \n  (hi\u2081 : \u2200 \u2983l r arg e\u2984, P l (pre\u2081 l r arg) e \u2192 P (l \u25b3 r) arg e)\n  (hi\u2082 : \u2200 \u2983l r arg e\u2984, P r (pre\u2082 l r arg) e \u2192 P (l \u25b3 r) arg e)\n  (hp : \u2200 \u2983l r arg res\u2081 res\u2082\u2984,\n    res\u2081 = stack_rec base pre\u2081 pre\u2082 post l (pre\u2081 l r arg) \u2192\n    res\u2082 = stack_rec base pre\u2081 pre\u2082 post r (pre\u2082 l r arg) \u2192\n    P l (pre\u2081 l r arg) (sum.inr res\u2081) \u2192\n    P r (pre\u2082 l r arg) (sum.inr res\u2082) \u2192\n    P (l \u25b3 r) arg (sum.inr $ post res\u2081 res\u2082 l r arg))\n  :\n  \u2200 (x : tree unit) (arg : \u03b1) (n : \u2115)\n    (e : iterator_stack \u03b1 \u03b2) (he : e \u2208 ((stack_step base pre\u2081 pre\u2082 post)^[n] [sum.inl (x, arg, none)])),\n    P x arg e :=\nbegin\n  -- For the inductive hypothesis, we add an excess `xs`\n  suffices : \u2200 (xs : list (iterator_stack \u03b1 \u03b2)) (x : tree unit) (arg : \u03b1) (n : \u2115) (hn : n \u2264 x.time_steps)\n    (e : iterator_stack \u03b1 \u03b2) (he : e \u2208 ((stack_step base pre\u2081 pre\u2082 post)^[n] (sum.inl (x, arg, none) :: xs))),\n    P x arg e \u2228 e \u2208 xs,\n  { intros x arg n e he, refine (this [] x arg (min n x.time_steps) (min_le_right _ _) e _).resolve_right not_false, rwa stack_step_iterate_min, },\n  intros,\n  induction x using tree.unit_rec_on with l r ih\u2081 ih\u2082 generalizing arg xs n e,\n  -- In the case `n=0`, no iteration happens, so the only\n  -- nontrivial possibility is `e = sum.inl (t, arg, none)` (the base case)\n  all_goals { cases n, { refine or.imp_left _ he, rintro rfl, exact hb _ arg, }, },\n  { obtain rfl : n = 0 := by simpa [nat.succ_eq_add_one] using hn,\n    refine or.imp_left _ he, rintro rfl, exact hbr arg, },\n  -- In the inductive step, suppose we take `n+1` steps; we first show that `P` is true of the previous results (used several times later)\n  have P_l_res : P l (pre\u2081 l r arg) (sum.inr $ stack_rec base pre\u2081 pre\u2082 post l (pre\u2081 l r arg)),\n  { refine (ih\u2081 _ [] _ _ rfl.le _).resolve_right not_false, simp, },\n  have P_r_res : P r (pre\u2082 l r arg) (sum.inr $ stack_rec base pre\u2081 pre\u2082 post r (pre\u2082 l r arg)),\n  { refine (ih\u2082 _ [] _ _ rfl.le _).resolve_right not_false, simp, },\n  -- Now we consider intermediate stages `0 \u2264 n \u2264 l.time_steps`\n  by_cases H : n \u2264 l.time_steps,\n  { clear ih\u2082, rcases ih\u2081 (pre\u2081 l r arg) _ n e H he with (ih|rfl|ih),\n    { left, exact hi\u2081 ih, }, { left, exact hb _ _, }, { right, exact ih, } },\n  -- Now we consider the cases `n = l.time_steps + 1` i.e. `c=1` letting `c = n - l.time_steps`\n  rcases lt_iff_exists_add.mp (lt_of_not_le H) with \u27e8c, hc, rfl\u27e9, clear H,\n  rw [iterate_succ_apply, add_comm l.time_steps, iterate_add_apply, stack_step, stack_step_iterate] at he,\n  cases c, { cases not_lt_zero' hc, }, clear hc,\n  cases c, { refine or.imp_left _ he, rintro rfl, exact hb' rfl P_l_res, },\n  -- Next, we consider the cases `l.time_steps + 2 \u2264 n \u2264 l.time_steps + 2 + r.time_steps`\n  by_cases H : c \u2264 r.time_steps,\n  { rcases ih\u2082 (pre\u2082 l r arg) _ c e H he with (ih|rfl|ih),\n    { left, exact hi\u2082 ih, }, { left, exact hb' rfl P_l_res, }, { right, exact ih, }, },\n  -- Finally, we consider the last step `n = l.time_steps + 2 + r.time_steps + 1`\n  obtain rfl : c = 1 + r.time_steps,\n  { simp [time_steps_node, nat.succ_eq_add_one] at hn, \n    linarith only [H, hn], },\n  simp [iterate_add] at he,\n  refine he.imp_left _, rintro rfl,\n  exact hp rfl rfl P_l_res P_r_res,\nend\n\n-- First we show the length is bounded\n-- Note this is inefficient because it does branching in many cases where we do not need to branch\nlemma stack_step_len_le (ls : list (iterator_stack \u03b1 \u03b2)) :\n  (stack_step base pre\u2081 pre\u2082 post ls).length \u2264 ls.length + 1 :=\nby rcases ls with (_|\u27e8(\u27e8(_|\u27e8\u27e8\u27e9, l, r\u27e9), arg, (_|_)\u27e9|res), (_|\u27e8(\u27e8_,_,(_|_)\u27e9|_), _\u27e9)\u27e9); simp [add_assoc]\n\nlemma stack_step_iter_len_le (ls : list (iterator_stack \u03b1 \u03b2)) (n : \u2115) :\n  ((stack_step base pre\u2081 pre\u2082 post)^[n] ls).length \u2264 ls.length + n :=\nby { induction n with n ih, { refl, }, rw [iterate_succ_apply'], exact (stack_step_len_le _).trans (nat.succ_le_succ ih), }\n\nlemma stack_step_iter_len_le' (x : tree unit) (arg : \u03b1) (n : \u2115) :\n  ((stack_step base pre\u2081 pre\u2082 post)^[n] [sum.inl (x, arg, none)]).length \u2264 x.time_steps + 1 :=\nby { rw [\u2190 stack_step_iterate_min, add_comm x.time_steps], refine (stack_step_iter_len_le _ _).trans _, simp, }\n\nvariables [polysize \u03b1] [polysize \u03b2] (bb bpr\u2081 bpr\u2082 bpo : polynomial \u2115)\n  (hbb : \u2200 x, size (base x) \u2264 bb.eval (size x))\n  (hbpr\u2081 : \u2200 x y a, size (pre\u2081 x y a) \u2264 (size a) + bpr\u2081.eval (size (x \u25b3 y)))\n  (hbpr\u2082 : \u2200 x y a, size (pre\u2082 x y a) \u2264 (size a) + bpr\u2082.eval (size (x \u25b3 y)))\n  (hbpo : \u2200 ih\u2081 ih\u2082 x y a, size (post ih\u2081 ih\u2082 x y a) \u2264 (size ih\u2081) + (size ih\u2082) + bpo.eval (size (x \u25b3 y, a)))\n\ndef max_arg_size (x\u2080 : tree unit) (arg\u2080 : \u03b1) : \u2115 :=\nsize arg\u2080 + x\u2080.height * ((bpr\u2081.eval x\u2080.num_nodes) + (bpr\u2082.eval x\u2080.num_nodes))\n\ndef max_res_size (x\u2080 : tree unit) (arg\u2080 : \u03b1) : \u2115 :=\nx\u2080.num_leaves * bb.eval (max_arg_size bpr\u2081 bpr\u2082 x\u2080 arg\u2080) + x\u2080.num_nodes * bpo.eval (x\u2080.num_nodes + (max_arg_size bpr\u2081 bpr\u2082 x\u2080 arg\u2080))\n\ndef bdd_size (x\u2080 : tree unit) (arg\u2080 : \u03b1) : iterator_stack \u03b1 \u03b2 \u2192 Prop\n| (sum.inl (x, arg, res)) := size x \u2264 size x\u2080 \u2227\n      size arg \u2264 max_arg_size bpr\u2081 bpr\u2082 x\u2080 arg\u2080 \u2227 \n      size res \u2264 max_res_size bb bpr\u2081 bpr\u2082 bpo x\u2080 arg\u2080 \n| (sum.inr res) := size res \u2264 max_res_size bb bpr\u2081 bpr\u2082 bpo x\u2080 arg\u2080\n\nvariables {bb bpr\u2081 bpr\u2082 bpo}\n\nlemma max_res_size_node (l r : tree unit) (arg : \u03b1) :\n  max_res_size bb bpr\u2081 bpr\u2082 bpo (l \u25b3 r) arg =\n  (l.num_leaves * bb.eval (max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg) + l.num_nodes * bpo.eval ((l \u25b3 r).num_nodes + (max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg))) +\n  (r.num_leaves * bb.eval (max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg) + r.num_nodes * bpo.eval ((l \u25b3 r).num_nodes + (max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg))) +\n  bpo.eval ((l \u25b3 r).num_nodes + (max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg)) :=\nby { simp [max_res_size], ring, }\n\nlemma max_arg_size_le\u2081 (hbpr\u2081 : \u2200 x y a, size (pre\u2081 x y a) \u2264 (size a) + bpr\u2081.eval (size (x \u25b3 y))) (l r : tree unit) (arg : \u03b1) :\n  max_arg_size bpr\u2081 bpr\u2082 l (pre\u2081 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg :=\nbegin\n  simp only [max_arg_size, tree.height, add_comm (max l.height r.height), add_mul, one_mul, \u2190 add_assoc],\n  mono*,\n  exacts [(hbpr\u2081 _ _ _).trans le_self_add, le_max_left _ _,\n    (l \u25b3 r).left_num_nodes_le, (l \u25b3 r).left_num_nodes_le, zero_le', zero_le'],\nend\n\nlemma max_arg_size_le\u2082 (hbpr\u2082 : \u2200 x y a, size (pre\u2082 x y a) \u2264 (size a) + bpr\u2082.eval (size (x \u25b3 y))) (l r : tree unit) (arg : \u03b1) :\n  max_arg_size bpr\u2081 bpr\u2082 r (pre\u2082 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg :=\nbegin\n  simp only [max_arg_size, tree.height, add_comm (max l.height r.height), add_mul, one_mul, \u2190 add_assoc],\n  mono*,\n  exacts [(hbpr\u2082 _ _ _).trans (add_le_add_right le_self_add _),\n    le_max_right _ _, (l \u25b3 r).right_num_nodes_le, (l \u25b3 r).right_num_nodes_le, zero_le', zero_le'],\nend\n\nlemma max_res_size_le\u2081 (hbpr\u2081 : \u2200 x y a, size (pre\u2081 x y a) \u2264 (size a) + bpr\u2081.eval (size (x \u25b3 y))) (l r : tree unit) (arg : \u03b1) :\n  max_res_size bb bpr\u2081 bpr\u2082 bpo l (pre\u2081 l r arg) \u2264 max_res_size bb bpr\u2081 bpr\u2082 bpo (l \u25b3 r) arg :=\nlet h : max_arg_size bpr\u2081 bpr\u2082 l (pre\u2081 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg := max_arg_size_le\u2081 hbpr\u2081 _ _ _ in\n(add_le_add (mul_le_mul' le_self_add (bb.eval_mono h)) (mul_le_mul' (l \u25b3 r).left_num_nodes_le (bpo.eval_mono $ add_le_add (l \u25b3 r).left_num_nodes_le h)))\n\nlemma max_res_size_le\u2082 (hbpr\u2082 : \u2200 x y a, size (pre\u2082 x y a) \u2264 (size a) + bpr\u2082.eval (size (x \u25b3 y))) (l r : tree unit) (arg : \u03b1) :\n  max_res_size bb bpr\u2081 bpr\u2082 bpo r (pre\u2082 l r arg) \u2264 max_res_size bb bpr\u2081 bpr\u2082 bpo (l \u25b3 r) arg :=\nlet h : max_arg_size bpr\u2081 bpr\u2082 r (pre\u2082 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg := max_arg_size_le\u2082 hbpr\u2082 _ _ _ in\nadd_le_add (mul_le_mul' le_add_self (bb.eval_mono h)) (mul_le_mul' (l \u25b3 r).right_num_nodes_le (bpo.eval_mono $ add_le_add (l \u25b3 r).right_num_nodes_le h))\n\ninclude hbpr\u2081 hbpr\u2082\n\nlemma add_max_res_size_le (l r : tree unit) (arg : \u03b1) :\n  max_res_size bb bpr\u2081 bpr\u2082 bpo l (pre\u2081 l r arg) + max_res_size bb bpr\u2081 bpr\u2082 bpo r (pre\u2082 l r arg) + bpo.eval (size (l \u25b3 r, arg)) \u2264\n    max_res_size bb bpr\u2081 bpr\u2082 bpo (l \u25b3 r) arg :=\nbegin\n  have h\u2081 : max_arg_size bpr\u2081 bpr\u2082 l (pre\u2081 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg := max_arg_size_le\u2081 hbpr\u2081 _ _ _,\n  have h\u2082 : max_arg_size bpr\u2081 bpr\u2082 r (pre\u2082 l r arg) \u2264 max_arg_size bpr\u2081 bpr\u2082 (l \u25b3 r) arg := max_arg_size_le\u2082 hbpr\u2082 _ _ _,\n  conv_rhs { rw max_res_size_node, }, simp only [max_res_size, polysize.prod_size],\n  mono*,\n  exacts [zero_le', (l \u25b3 r).left_num_nodes_le, zero_le', zero_le', (l \u25b3 r).right_num_nodes_le, zero_le', le_self_add],\nend\n\ninclude hbb hbpo\n\ntheorem stack_step_iter_le (x : tree unit) (arg : \u03b1) (n : \u2115)\n  (e : iterator_stack \u03b1 \u03b2) (he : e \u2208 ((stack_step base pre\u2081 pre\u2082 post)^[n] [sum.inl (x, arg, none)])) :\n  bdd_size bb bpr\u2081 bpr\u2082 bpo x arg e :=\nbegin\n  refine stack_step_induction _ _ _ _ _ _ _ _ _ _ _ he; clear he e x arg n,\n  { intros t arg, refine \u27e8rfl.le, le_self_add, zero_le'\u27e9, },\n  { intros l r arg res _ h, refine \u27e8rfl.le, le_self_add, trans h (max_res_size_le\u2081 hbpr\u2081 _ _ _)\u27e9, },\n  { intros arg, refine (hbb _).trans (le_add_right _), simpa using bb.eval_mono le_self_add, },\n  { intros l r arg e he, \n    rcases e with (\u27e8x, arg', res\u27e9|res),\n    { exact \u27e8he.1.trans (l \u25b3 r).left_num_nodes_le, he.2.1.trans (max_arg_size_le\u2081 hbpr\u2081 _ _ _), he.2.2.trans (max_res_size_le\u2081 hbpr\u2081 _ _ _)\u27e9, },\n    exact trans he (max_res_size_le\u2081 hbpr\u2081 _ _ _), },\n  { intros l r arg e he,\n    rcases e with (\u27e8x, arg', res\u27e9|res),\n    { exact \u27e8he.1.trans (l \u25b3 r).right_num_nodes_le, he.2.1.trans (max_arg_size_le\u2082 hbpr\u2082 _ _ _), he.2.2.trans (max_res_size_le\u2082 hbpr\u2082 _ _ _)\u27e9, },\n    exact trans he (max_res_size_le\u2082 hbpr\u2082 _ _ _), },\n  { intros l r arg res\u2081 res\u2082 _ _ ih\u2081 ih\u2082, \n    rw [bdd_size] at ih\u2081 ih\u2082 \u22a2,\n    refine (hbpo _ _ _ _ _).trans _,\n    refine trans _ (add_max_res_size_le hbpr\u2081 hbpr\u2082 _ _ _), \n    mono*, },\nend\n\nend\n\nsection\n\nvariables {\u03b1 \u03b2 \u03b3 : Type} [tencodable \u03b1] [tencodable \u03b2] [tencodable \u03b3]\n  [polysize \u03b1] [polysize \u03b2] [polysize \u03b3]\n  {st : \u03b3 \u2192 tree unit} {arg : \u03b3 \u2192 \u03b1}\n  {base : \u03b3 \u2192 \u03b1 \u2192 \u03b2} {pre\u2081 pre\u2082 : \u03b3 \u2192 tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b1}\n  {post : \u03b3 \u2192 \u03b2 \u2192 \u03b2 \u2192 tree unit \u2192 tree unit \u2192 \u03b1 \u2192 \u03b2}\n  (hst : polysize_fun st) (harg : polysize_fun arg)\n  (hb : polysize_fun base)\n  (hpr\u2081 : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit) (a : \u03b1), pre\u2081 usf.1 usf.2.1 usf.2.2 a))\n  (hpr\u2082 : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit) (sf : \u03b1), pre\u2082 usf.1 usf.2.1 usf.2.2 sf))\n  (hpo : polysize_safe (\u03bb (usf : \u03b3 \u00d7 tree unit \u00d7 tree unit \u00d7 \u03b1) (sf : \u03b2 \u00d7 \u03b2), post usf.1 sf.1 sf.2 usf.2.1 usf.2.2.1 usf.2.2.2))\n\n@[reducible] noncomputable def _root_.polynomial.peval {\u03b1 : Type*} [semiring \u03b1] (p : polynomial \u03b1) (x : \u03b1) : polynomial \u03b1 :=\np.comp ((polynomial.C x) + polynomial.X)\n\nnoncomputable def max_arg_size_poly : polynomial \u2115 :=\nharg.poly + hst.poly * ((hpr\u2081.poly.comp (polynomial.X + hst.poly)) + (hpr\u2082.poly.comp (polynomial.X + hst.poly)))\n\nnoncomputable def max_res_size_poly : polynomial \u2115 :=\n(hst.poly + 1) * hb.poly.comp (polynomial.X + max_arg_size_poly hst harg hpr\u2081 hpr\u2082) +\n  hst.poly * hpo.poly.comp (polynomial.X + hst.poly + max_arg_size_poly hst harg hpr\u2081 hpr\u2082)\n\nnoncomputable def bdd_size_poly : polynomial \u2115 :=\nhst.poly + (max_arg_size_poly hst harg hpr\u2081 hpr\u2082) + (max_res_size_poly hst harg hb hpr\u2081 hpr\u2082 hpo)\n\nlemma arg_poly_le (x : \u03b3) :\n  max_arg_size (hpr\u2081.poly.peval (size x)) (hpr\u2082.poly.peval (size x)) (st x) (arg x) \u2264\n   (max_arg_size_poly hst harg hpr\u2081 hpr\u2082).eval (size x) :=\nbegin\n  have := hst.spec x, have := harg.spec x,\n  simp only [max_arg_size_poly, max_arg_size, polynomial.eval_add, polynomial.eval_X,\n    polynomial.eval_mul, polysize.prod_size, polysize_tree_unit, polynomial.eval_comp, polynomial.eval_C],\n  mono*,\n  exacts [(st x).height_le_num_nodes.trans (hst.spec x), zero_le', zero_le'],\nend\n\nlemma res_poly_le (x : \u03b3) :\n  max_res_size (hb.poly.peval (size x)) (hpr\u2081.poly.peval (size x)) (hpr\u2082.poly.peval (size x)) (hpo.poly.peval (size x)) (st x) (arg x) \u2264\n    (max_res_size_poly hst harg hb hpr\u2081 hpr\u2082 hpo).eval (size x) :=\nbegin\n  have := hst.spec, have := arg_poly_le hst harg hpr\u2081 hpr\u2082 x,\n  simp only [max_res_size, max_res_size_poly, polynomial.peval, \u2190add_assoc, eq_nat_cast,\n    polynomial.eval_comp, polynomial.eval_add, polynomial.eval_nat_cast, nat.cast_id, polynomial.eval_X,\n    polynomial.eval_mul, polynomial.eval_one, (st x).num_leaves_eq_num_nodes_succ],\n  mono*; exact zero_le',\nend\n\nlemma bdd_size_poly_le (x : \u03b3) (e : iterator_stack \u03b1 \u03b2)\n  (he : bdd_size (hb.poly.peval (size x)) (hpr\u2081.poly.peval (size x)) (hpr\u2082.poly.peval (size x)) \n    (hpo.poly.peval (size x)) (st x) (arg x) e) : size e \u2264 (bdd_size_poly hst harg hb hpr\u2081 hpr\u2082 hpo).eval (size x) :=\nbegin\n  rcases e with (\u27e8t, a, res\u27e9|res),\n  { conv_lhs { simp [\u2190 add_assoc], },\n    simp [bdd_size, bdd_size_poly] at \u22a2 he,\n    exact add_le_add (add_le_add (he.1.trans $ hst.spec x) \n      (he.2.1.trans $ arg_poly_le _ _ _ _ x))\n      (he.2.2.trans $ res_poly_le _ _ _ _ _ _ x), },\n  { simp [bdd_size, bdd_size_poly] at he \u22a2,\n    exact le_add_left (he.trans $ res_poly_le _ _ _ _ _ _ x), }\nend\n\ninclude hst harg hb hpr\u2081 hpr\u2082 hpo\n\ntheorem stack_step_iter_le' : \u2203 (p : polynomial \u2115),\n  \u2200 (x : \u03b3) (n : \u2115)\n  (e : iterator_stack \u03b1 \u03b2) (he : e \u2208 ((stack_step (base x) (pre\u2081 x) (pre\u2082 x) (post x))^[n] [sum.inl (st x, arg x, none)])),\n  size e \u2264 p.eval (size x) :=\nbegin\n  use (bdd_size_poly hst harg hb hpr\u2081 hpr\u2082 hpo),\n  intros x n e he,\n  apply bdd_size_poly_le, apply stack_step_iter_le _ _ _ _ _ _ n e he,\n  { intro a, simpa using hb.spec (x, a), },\n  { intros l r a, refine (hpr\u2081.spec (x, l, r) a).trans _, simp [add_assoc], mono*, exact nat.le_succ _, },\n  { intros l r a, refine (hpr\u2082.spec (x, l, r) a).trans _, simp [add_assoc], mono*, exact nat.le_succ _, },\n  { intros ih\u2081 ih\u2082 l r a, refine (hpo.spec (x, l, r, a) (ih\u2081, ih\u2082)).trans _, simp [add_assoc], mono*, exact le_add_self, }\nend\n\ntheorem stack_step_polysize : \u2203 (p : polynomial \u2115), \u2200 (x : \u03b3) (n : \u2115),\n  size ((stack_step (base x) (pre\u2081 x) (pre\u2082 x) (post x))^[n] [sum.inl (st x, arg x, none)]) \u2264\n    p.eval (size x) :=\nbegin\n  cases stack_step_iter_le' hst harg hb hpr\u2081 hpr\u2082 hpo with p hp,\n  use (5 * hst.poly + 2) * (p + 1),\n  intros x n,\n  simp only [polynomial.eval_add, polynomial.eval_mul, polynomial.eval_one],\n  apply list.size_le_mul_of_le,\n  { refine (stack_step_iter_len_le' _ _ _).trans _, simpa [tree.time_steps, add_assoc] using hst.spec x, },\n  { exact hp x n, },\nend\n\nend\n\nend tree\n", "meta": {"author": "prakol16", "repo": "circuits", "sha": "cdf4ce1e019d6817e4abe0d082d8d379539fddca", "save_path": "github-repos/lean/prakol16-circuits", "path": "github-repos/lean/prakol16-circuits/circuits-cdf4ce1e019d6817e4abe0d082d8d379539fddca/src/polytime/stack_rec_size.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.3675189569615299}}
{"text": "import number_theory.bernoulli\n\n\n--OUTPUT 1\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  induction n with d hd,\n  {\n    simp,\n  },\n  {\n    have h1 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h2 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h3 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h4 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h5 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h6 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h7 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h8 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h9 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h10 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h11 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h12 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h13 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h14 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h15 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h16 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h17 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h18 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h19 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h20 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h21 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h22 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h23 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h24 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h25 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h26 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h27 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h28 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h29 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h30 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h31 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h32 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h33 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have h34 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (1 + x)^d - (d + 1) * x^d, from rfl,\n    have\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nBernoulli polynomial evaluation\nGiven a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$\n`proof`\nWe apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\nMultiplying both sides by $d + 1$, and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n$$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\nThe conclusion then follows easily.\n\nQED\n\n-/\ntheorem  bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Bernoulli polynomial evaluation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7772998508568416, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.36741679587955584}}
{"text": "/-\nCopyright (c) 2018 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Patrick Stevens\n\n! This file was ported from Lean 3 source module data.nat.choose.sum\n! leanprover-community/mathlib commit 3e32bc908f617039c74c06ea9a897e30c30803c2\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Nat.Choose.Basic\nimport Mathbin.Tactic.Linarith.Default\nimport Mathbin.Algebra.BigOperators.Ring\nimport Mathbin.Algebra.BigOperators.Intervals\nimport Mathbin.Algebra.BigOperators.Order\nimport Mathbin.Algebra.BigOperators.NatAntidiagonal\n\n/-!\n# Sums of binomial coefficients\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file includes variants of the binomial theorem and other results on sums of binomial\ncoefficients. Theorems whose proofs depend on such sums may also go in this file for import\nreasons.\n\n-/\n\n\nopen Nat\n\nopen Finset\n\nopen BigOperators\n\nvariable {R : Type _}\n\nnamespace Commute\n\nvariable [Semiring R] {x y : R} (h : Commute x y) (n : \u2115)\n\ninclude h\n\n/- warning: commute.add_pow -> Commute.add_pow is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {x : R} {y : R}, (Commute.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y) -> (forall (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y) n) (Finset.sum.{u1, 0} R Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (m : Nat) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) x m) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) y (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) n m))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTC\u2093.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))))) (Nat.choose n m)))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {x : R} {y : R}, (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (forall (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y) n) (Finset.sum.{u1, 0} R Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (m : Nat) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) x m) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) y (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n m))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R _inst_1) (Nat.choose n m)))))\nCase conversion may be inaccurate. Consider using '#align commute.add_pow Commute.add_pow\u2093'. -/\n/-- A version of the **binomial theorem** for commuting elements in noncommutative semirings. -/\ntheorem add_pow : (x + y) ^ n = \u2211 m in range (n + 1), x ^ m * y ^ (n - m) * choose n m :=\n  by\n  let t : \u2115 \u2192 \u2115 \u2192 R := fun n m => x ^ m * y ^ (n - m) * choose n m\n  change (x + y) ^ n = \u2211 m in range (n + 1), t n m\n  have h_first : \u2200 n, t n 0 = y ^ n := fun n =>\n    by\n    dsimp [t]\n    rw [choose_zero_right, pow_zero, Nat.cast_one, mul_one, one_mul]\n  have h_last : \u2200 n, t n n.succ = 0 := fun n =>\n    by\n    dsimp [t]\n    rw [choose_succ_self, Nat.cast_zero, MulZeroClass.mul_zero]\n  have h_middle :\n    \u2200 n i : \u2115, i \u2208 range n.succ \u2192 (t n.succ \u2218 Nat.succ) i = x * t n i + y * t n i.succ :=\n    by\n    intro n i h_mem\n    have h_le : i \u2264 n := Nat.le_of_lt_succ (mem_range.mp h_mem)\n    dsimp [t]\n    rw [choose_succ_succ, Nat.cast_add, mul_add]\n    congr 1\n    \u00b7 rw [pow_succ x, succ_sub_succ, mul_assoc, mul_assoc, mul_assoc]\n    \u00b7 rw [\u2190 mul_assoc y, \u2190 mul_assoc y, (h.symm.pow_right i.succ).Eq]\n      by_cases h_eq : i = n\n      \u00b7 rw [h_eq, choose_succ_self, Nat.cast_zero, MulZeroClass.mul_zero, MulZeroClass.mul_zero]\n      \u00b7 rw [succ_sub (lt_of_le_of_ne h_le h_eq)]\n        rw [pow_succ y, mul_assoc, mul_assoc, mul_assoc, mul_assoc]\n  induction' n with n ih\n  \u00b7 rw [pow_zero, sum_range_succ, range_zero, sum_empty, zero_add]\n    dsimp [t]\n    rw [pow_zero, pow_zero, choose_self, Nat.cast_one, mul_one, mul_one]\n  \u00b7 rw [sum_range_succ', h_first]\n    rw [sum_congr rfl (h_middle n), sum_add_distrib, add_assoc]\n    rw [pow_succ (x + y), ih, add_mul, mul_sum, mul_sum]\n    congr 1\n    rw [sum_range_succ', sum_range_succ, h_first, h_last, MulZeroClass.mul_zero, add_zero, pow_succ]\n#align commute.add_pow Commute.add_pow\n\n/- warning: commute.add_pow' -> Commute.add_pow' is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {x : R} {y : R}, (Commute.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) x y) -> (forall (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y) n) (Finset.sum.{u1, 0} R (Prod.{0, 0} Nat Nat) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Finset.Nat.antidiagonal n) (fun (m : Prod.{0, 0} Nat Nat) => SMul.smul.{0, u1} Nat R (AddMonoid.SMul.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (Nat.choose n (Prod.fst.{0, 0} Nat Nat m)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) x (Prod.fst.{0, 0} Nat Nat m)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) y (Prod.snd.{0, 0} Nat Nat m))))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Semiring.{u1} R] {x : R} {y : R}, (Commute.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) x y) -> (forall (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))))) x y) n) (Finset.sum.{u1, 0} R (Prod.{0, 0} Nat Nat) (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1))) (Finset.Nat.antidiagonal n) (fun (m : Prod.{0, 0} Nat Nat) => HSMul.hSMul.{0, u1, u1} Nat R R (instHSMul.{0, u1} Nat R (AddMonoid.SMul.{u1} R (AddMonoidWithOne.toAddMonoid.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))))) (Nat.choose n (Prod.fst.{0, 0} Nat Nat m)) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R _inst_1)))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) x (Prod.fst.{0, 0} Nat Nat m)) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R _inst_1)))) y (Prod.snd.{0, 0} Nat Nat m))))))\nCase conversion may be inaccurate. Consider using '#align commute.add_pow' Commute.add_pow'\u2093'. -/\n/-- A version of `commute.add_pow` that avoids \u2115-subtraction by summing over the antidiagonal and\nalso with the binomial coefficient applied via scalar action of \u2115. -/\ntheorem add_pow' :\n    (x + y) ^ n = \u2211 m in Nat.antidiagonal n, choose n m.fst \u2022 (x ^ m.fst * y ^ m.snd) := by\n  simp_rw [Finset.Nat.sum_antidiagonal_eq_sum_range_succ fun m p => choose n m \u2022 (x ^ m * y ^ p),\n    _root_.nsmul_eq_mul, cast_comm, h.add_pow]\n#align commute.add_pow' Commute.add_pow'\n\nend Commute\n\n/- warning: add_pow -> add_pow is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (x : R) (y : R) (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toHasAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) n) (Finset.sum.{u1, 0} R Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (m : Nat) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (Distrib.toHasMul.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x m) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) n m))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat R (HasLiftT.mk.{1, succ u1} Nat R (CoeTC\u2093.coe.{1, succ u1} Nat R (Nat.castCoe.{u1} R (AddMonoidWithOne.toNatCast.{u1} R (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} R (NonAssocSemiring.toAddCommMonoidWithOne.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))))) (Nat.choose n m))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : CommSemiring.{u1} R] (x : R) (y : R) (n : Nat), Eq.{succ u1} R (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HAdd.hAdd.{u1, u1, u1} R R R (instHAdd.{u1} R (Distrib.toAdd.{u1} R (NonUnitalNonAssocSemiring.toDistrib.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))))) x y) n) (Finset.sum.{u1, 0} R Nat (NonUnitalNonAssocSemiring.toAddCommMonoid.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)))) (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (m : Nat) => HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (NonUnitalNonAssocSemiring.toMul.{u1} R (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} R (Semiring.toNonAssocSemiring.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) x m) (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1))))) y (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) n m))) (Nat.cast.{u1} R (Semiring.toNatCast.{u1} R (CommSemiring.toSemiring.{u1} R _inst_1)) (Nat.choose n m))))\nCase conversion may be inaccurate. Consider using '#align add_pow add_pow\u2093'. -/\n/-- The **binomial theorem** -/\ntheorem add_pow [CommSemiring R] (x y : R) (n : \u2115) :\n    (x + y) ^ n = \u2211 m in range (n + 1), x ^ m * y ^ (n - m) * choose n m :=\n  (Commute.all x y).add_pow n\n#align add_pow add_pow\n\nnamespace Nat\n\n#print Nat.sum_range_choose /-\n/-- The sum of entries in a row of Pascal's triangle -/\ntheorem sum_range_choose (n : \u2115) : (\u2211 m in range (n + 1), choose n m) = 2 ^ n := by\n  simpa using (add_pow 1 1 n).symm\n#align nat.sum_range_choose Nat.sum_range_choose\n-/\n\n#print Nat.sum_range_choose_halfway /-\ntheorem sum_range_choose_halfway (m : Nat) : (\u2211 i in range (m + 1), choose (2 * m + 1) i) = 4 ^ m :=\n  have :\n    (\u2211 i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i)) =\n      \u2211 i in range (m + 1), choose (2 * m + 1) i :=\n    sum_congr rfl fun i hi => choose_symm <| by linarith [mem_range.1 hi]\n  mul_right_injective\u2080 two_ne_zero <|\n    calc\n      (2 * \u2211 i in range (m + 1), choose (2 * m + 1) i) =\n          (\u2211 i in range (m + 1), choose (2 * m + 1) i) +\n            \u2211 i in range (m + 1), choose (2 * m + 1) (2 * m + 1 - i) :=\n        by rw [two_mul, this]\n      _ =\n          (\u2211 i in range (m + 1), choose (2 * m + 1) i) +\n            \u2211 i in Ico (m + 1) (2 * m + 2), choose (2 * m + 1) i :=\n        by\n        rw [range_eq_Ico, sum_Ico_reflect]\n        \u00b7 congr\n          have A : m + 1 \u2264 2 * m + 1 := by linarith\n          rw [add_comm, add_tsub_assoc_of_le A, \u2190 add_comm]\n          congr\n          rw [tsub_eq_iff_eq_add_of_le A]\n          ring\n        \u00b7 linarith\n      _ = \u2211 i in range (2 * m + 2), choose (2 * m + 1) i := (sum_range_add_sum_Ico _ (by linarith))\n      _ = 2 ^ (2 * m + 1) := (sum_range_choose (2 * m + 1))\n      _ = 2 * 4 ^ m := by\n        rw [pow_succ, pow_mul]\n        rfl\n      \n#align nat.sum_range_choose_halfway Nat.sum_range_choose_halfway\n-/\n\n#print Nat.choose_middle_le_pow /-\ntheorem choose_middle_le_pow (n : \u2115) : choose (2 * n + 1) n \u2264 4 ^ n :=\n  by\n  have t : choose (2 * n + 1) n \u2264 \u2211 i in range (n + 1), choose (2 * n + 1) i :=\n    single_le_sum (fun x _ => by linarith) (self_mem_range_succ n)\n  simpa [sum_range_choose_halfway n] using t\n#align nat.choose_middle_le_pow Nat.choose_middle_le_pow\n-/\n\n#print Nat.four_pow_le_two_mul_add_one_mul_central_binom /-\ntheorem four_pow_le_two_mul_add_one_mul_central_binom (n : \u2115) :\n    4 ^ n \u2264 (2 * n + 1) * choose (2 * n) n :=\n  calc\n    4 ^ n = (1 + 1) ^ (2 * n) := by norm_num [pow_mul]\n    _ = \u2211 m in range (2 * n + 1), choose (2 * n) m := by simp [add_pow]\n    _ \u2264 \u2211 m in range (2 * n + 1), choose (2 * n) (2 * n / 2) :=\n      (sum_le_sum fun i hi => choose_le_middle i (2 * n))\n    _ = (2 * n + 1) * choose (2 * n) n := by simp\n    \n#align nat.four_pow_le_two_mul_add_one_mul_central_binom Nat.four_pow_le_two_mul_add_one_mul_central_binom\n-/\n\nend Nat\n\n/- warning: int.alternating_sum_range_choose -> Int.alternating_sum_range_choose is a dubious translation:\nlean 3 declaration is\n  forall {n : Nat}, Eq.{1} Int (Finset.sum.{0, 0} Int Nat Int.addCommMonoid (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (m : Nat) => HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.hasMul) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.monoid)) (Neg.neg.{0} Int Int.hasNeg (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne)))) m) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Int (HasLiftT.mk.{1, 1} Nat Int (CoeTC\u2093.coe.{1, 1} Nat Int (coeBase.{1, 1} Nat Int Int.hasCoe))) (Nat.choose n m)))) (ite.{1} Int (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (Nat.decidableEq n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne))) (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {n : Nat}, Eq.{1} Int (Finset.sum.{0, 0} Int Nat Int.instAddCommMonoidInt (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (m : Nat) => HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMulInt) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.instMonoidInt)) (Neg.neg.{0} Int Int.instNegInt (OfNat.ofNat.{0} Int 1 (instOfNatInt 1))) m) (Nat.cast.{0} Int instNatCastInt (Nat.choose n m)))) (ite.{1} Int (Eq.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (instDecidableEqNat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) (OfNat.ofNat.{0} Int 1 (instOfNatInt 1)) (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align int.alternating_sum_range_choose Int.alternating_sum_range_choose\u2093'. -/\ntheorem Int.alternating_sum_range_choose {n : \u2115} :\n    (\u2211 m in range (n + 1), ((-1) ^ m * \u2191(choose n m) : \u2124)) = if n = 0 then 1 else 0 :=\n  by\n  cases n; \u00b7 simp\n  have h := add_pow (-1 : \u2124) 1 n.succ\n  simp only [one_pow, mul_one, add_left_neg] at h\n  rw [\u2190 h, zero_pow (Nat.succ_pos n), if_neg (Nat.succ_ne_zero n)]\n#align int.alternating_sum_range_choose Int.alternating_sum_range_choose\n\n/- warning: int.alternating_sum_range_choose_of_ne -> Int.alternating_sum_range_choose_of_ne is a dubious translation:\nlean 3 declaration is\n  forall {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Eq.{1} Int (Finset.sum.{0, 0} Int Nat Int.addCommMonoid (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (m : Nat) => HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.hasMul) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.monoid)) (Neg.neg.{0} Int Int.hasNeg (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne)))) m) ((fun (a : Type) (b : Type) [self : HasLiftT.{1, 1} a b] => self.0) Nat Int (HasLiftT.mk.{1, 1} Nat Int (CoeTC\u2093.coe.{1, 1} Nat Int (coeBase.{1, 1} Nat Int Int.hasCoe))) (Nat.choose n m)))) (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{1} Int (Finset.sum.{0, 0} Int Nat Int.instAddCommMonoidInt (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (m : Nat) => HMul.hMul.{0, 0, 0} Int Int Int (instHMul.{0} Int Int.instMulInt) (HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.instMonoidInt)) (Neg.neg.{0} Int Int.instNegInt (OfNat.ofNat.{0} Int 1 (instOfNatInt 1))) m) (Nat.cast.{0} Int instNatCastInt (Nat.choose n m)))) (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align int.alternating_sum_range_choose_of_ne Int.alternating_sum_range_choose_of_ne\u2093'. -/\ntheorem Int.alternating_sum_range_choose_of_ne {n : \u2115} (h0 : n \u2260 0) :\n    (\u2211 m in range (n + 1), ((-1) ^ m * \u2191(choose n m) : \u2124)) = 0 := by\n  rw [Int.alternating_sum_range_choose, if_neg h0]\n#align int.alternating_sum_range_choose_of_ne Int.alternating_sum_range_choose_of_ne\n\nnamespace Finset\n\n/- warning: finset.sum_powerset_apply_card -> Finset.sum_powerset_apply_card is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : AddCommMonoid.{u1} \u03b1] (f : Nat -> \u03b1) {x : Finset.{u2} \u03b2}, Eq.{succ u1} \u03b1 (Finset.sum.{u1, u2} \u03b1 (Finset.{u2} \u03b2) _inst_1 (Finset.powerset.{u2} \u03b2 x) (fun (m : Finset.{u2} \u03b2) => f (Finset.card.{u2} \u03b2 m))) (Finset.sum.{u1, 0} \u03b1 Nat _inst_1 (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (Finset.card.{u2} \u03b2 x) (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (m : Nat) => SMul.smul.{0, u1} Nat \u03b1 (AddMonoid.SMul.{u1} \u03b1 (AddCommMonoid.toAddMonoid.{u1} \u03b1 _inst_1)) (Nat.choose (Finset.card.{u2} \u03b2 x) m) (f m)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : AddCommMonoid.{u2} \u03b1] (f : Nat -> \u03b1) {x : Finset.{u1} \u03b2}, Eq.{succ u2} \u03b1 (Finset.sum.{u2, u1} \u03b1 (Finset.{u1} \u03b2) _inst_1 (Finset.powerset.{u1} \u03b2 x) (fun (m : Finset.{u1} \u03b2) => f (Finset.card.{u1} \u03b2 m))) (Finset.sum.{u2, 0} \u03b1 Nat _inst_1 (Finset.range (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (Finset.card.{u1} \u03b2 x) (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (m : Nat) => HSMul.hSMul.{0, u2, u2} Nat \u03b1 \u03b1 (instHSMul.{0, u2} Nat \u03b1 (AddMonoid.SMul.{u2} \u03b1 (AddCommMonoid.toAddMonoid.{u2} \u03b1 _inst_1))) (Nat.choose (Finset.card.{u1} \u03b2 x) m) (f m)))\nCase conversion may be inaccurate. Consider using '#align finset.sum_powerset_apply_card Finset.sum_powerset_apply_card\u2093'. -/\ntheorem sum_powerset_apply_card {\u03b1 \u03b2 : Type _} [AddCommMonoid \u03b1] (f : \u2115 \u2192 \u03b1) {x : Finset \u03b2} :\n    (\u2211 m in x.powerset, f m.card) = \u2211 m in range (x.card + 1), x.card.choose m \u2022 f m :=\n  by\n  trans \u2211 m in range (x.card + 1), \u2211 j in x.powerset.filter fun z => z.card = m, f j.card\n  \u00b7 refine' (sum_fiberwise_of_maps_to _ _).symm\n    intro y hy\n    rw [mem_range, Nat.lt_succ_iff]\n    rw [mem_powerset] at hy\n    exact card_le_of_subset hy\n  \u00b7 refine' sum_congr rfl fun y hy => _\n    rw [\u2190 card_powerset_len, \u2190 sum_const]\n    refine' sum_congr powerset_len_eq_filter.symm fun z hz => _\n    rw [(mem_powerset_len.1 hz).2]\n#align finset.sum_powerset_apply_card Finset.sum_powerset_apply_card\n\n/- warning: finset.sum_powerset_neg_one_pow_card -> Finset.sum_powerset_neg_one_pow_card is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {x : Finset.{u1} \u03b1}, Eq.{1} Int (Finset.sum.{0, u1} Int (Finset.{u1} \u03b1) Int.addCommMonoid (Finset.powerset.{u1} \u03b1 x) (fun (m : Finset.{u1} \u03b1) => HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.monoid)) (Neg.neg.{0} Int Int.hasNeg (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne)))) (Finset.card.{u1} \u03b1 m))) (ite.{1} Int (Eq.{succ u1} (Finset.{u1} \u03b1) x (EmptyCollection.emptyCollection.{u1} (Finset.{u1} \u03b1) (Finset.hasEmptyc.{u1} \u03b1))) (Finset.decidableEq.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) x (EmptyCollection.emptyCollection.{u1} (Finset.{u1} \u03b1) (Finset.hasEmptyc.{u1} \u03b1))) (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne))) (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {x : Finset.{u1} \u03b1}, Eq.{1} Int (Finset.sum.{0, u1} Int (Finset.{u1} \u03b1) Int.instAddCommMonoidInt (Finset.powerset.{u1} \u03b1 x) (fun (m : Finset.{u1} \u03b1) => HPow.hPow.{0, 0, 0} Int Nat Int Int.instHPowIntNat (Neg.neg.{0} Int Int.instNegInt (OfNat.ofNat.{0} Int 1 (instOfNatInt 1))) (Finset.card.{u1} \u03b1 m))) (ite.{1} Int (Eq.{succ u1} (Finset.{u1} \u03b1) x (EmptyCollection.emptyCollection.{u1} (Finset.{u1} \u03b1) (Finset.instEmptyCollectionFinset.{u1} \u03b1))) (Finset.decidableEq.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) x (EmptyCollection.emptyCollection.{u1} (Finset.{u1} \u03b1) (Finset.instEmptyCollectionFinset.{u1} \u03b1))) (OfNat.ofNat.{0} Int 1 (instOfNatInt 1)) (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align finset.sum_powerset_neg_one_pow_card Finset.sum_powerset_neg_one_pow_card\u2093'. -/\ntheorem sum_powerset_neg_one_pow_card {\u03b1 : Type _} [DecidableEq \u03b1] {x : Finset \u03b1} :\n    (\u2211 m in x.powerset, (-1 : \u2124) ^ m.card) = if x = \u2205 then 1 else 0 :=\n  by\n  rw [sum_powerset_apply_card]\n  simp only [nsmul_eq_mul', \u2190 card_eq_zero, Int.alternating_sum_range_choose]\n#align finset.sum_powerset_neg_one_pow_card Finset.sum_powerset_neg_one_pow_card\n\n/- warning: finset.sum_powerset_neg_one_pow_card_of_nonempty -> Finset.sum_powerset_neg_one_pow_card_of_nonempty is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {x : Finset.{u1} \u03b1}, (Finset.Nonempty.{u1} \u03b1 x) -> (Eq.{1} Int (Finset.sum.{0, u1} Int (Finset.{u1} \u03b1) Int.addCommMonoid (Finset.powerset.{u1} \u03b1 x) (fun (m : Finset.{u1} \u03b1) => HPow.hPow.{0, 0, 0} Int Nat Int (instHPow.{0, 0} Int Nat (Monoid.Pow.{0} Int Int.monoid)) (Neg.neg.{0} Int Int.hasNeg (OfNat.ofNat.{0} Int 1 (OfNat.mk.{0} Int 1 (One.one.{0} Int Int.hasOne)))) (Finset.card.{u1} \u03b1 m))) (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {x : Finset.{u1} \u03b1}, (Finset.Nonempty.{u1} \u03b1 x) -> (Eq.{1} Int (Finset.sum.{0, u1} Int (Finset.{u1} \u03b1) Int.instAddCommMonoidInt (Finset.powerset.{u1} \u03b1 x) (fun (m : Finset.{u1} \u03b1) => HPow.hPow.{0, 0, 0} Int Nat Int Int.instHPowIntNat (Neg.neg.{0} Int Int.instNegInt (OfNat.ofNat.{0} Int 1 (instOfNatInt 1))) (Finset.card.{u1} \u03b1 m))) (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align finset.sum_powerset_neg_one_pow_card_of_nonempty Finset.sum_powerset_neg_one_pow_card_of_nonempty\u2093'. -/\ntheorem sum_powerset_neg_one_pow_card_of_nonempty {\u03b1 : Type _} {x : Finset \u03b1} (h0 : x.Nonempty) :\n    (\u2211 m in x.powerset, (-1 : \u2124) ^ m.card) = 0 := by\n  classical\n    rw [sum_powerset_neg_one_pow_card, if_neg]\n    rw [\u2190 Ne.def, \u2190 nonempty_iff_ne_empty]\n    apply h0\n#align finset.sum_powerset_neg_one_pow_card_of_nonempty Finset.sum_powerset_neg_one_pow_card_of_nonempty\n\nend Finset\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Nat/Choose/Sum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.36724109881250505}}
{"text": "import .Borel_Isomorphism\n\nclass standard_borel (\u03b1 : Type*) [measurable_space \u03b1] : Prop :=\n  (is_polishable : \u2203 \u03c4 : topological_space \u03b1, polish_space \u03b1 \u2227 borel_space \u03b1)\n\nclass upgraded_standard_borel (\u03b1 : Type*) extends measurable_space \u03b1, \n  topological_space \u03b1, borel_space \u03b1, polish_space \u03b1\n\nnoncomputable\ndef upgrade_standard_borel (\u03b1 : Type*) [hm : measurable_space \u03b1] [hs : standard_borel \u03b1] :\n  upgraded_standard_borel \u03b1 :=\nbegin\n  choose \u03c4 hp hb using hs.is_polishable,\n  letI := \u03c4,\n  apply upgraded_standard_borel.mk,\nend\n\nsection clopenable\n\nopen measurable_space\nvariables {\u03b1 : Type*}\n\nset_option pp.implicit true\ntheorem eq_borel_of_polish_le_polish {s t : topological_space \u03b1} \n  (hs : @polish_space _ s) (ht : @polish_space _ t) (h : s \u2264 t) : @borel _ s = @borel _ t :=\nbegin\n  dsimp[borel],\n  apply le_antisymm; apply generate_from_le; intros X hX,  {\n    have := @is_open.is_clopenable _ s _ X hX,\n    rcases this with \u27e8r,rs,hr\u27e9,\n    rw \u2190 @measure_theory.is_clopenable_iff_measurable_set _ t _ (@borel _ t) (by {\n      exact @borel_space.mk \u03b1 t (@borel \u03b1 t) (@rfl (measurable_space \u03b1) (@borel \u03b1 t)),\n    }),\n    refine \u27e8r,_,hr\u27e9,\n    exact le_trans rs h,\n  },\n  apply measurable_set_generate_from,\n  apply h,\n  exact hX,\nend\n\nlemma clopenable_same_borel [t : topological_space \u03b1] [ht : polish_space \u03b1] \n  [m : measurable_space \u03b1] [hb : borel_space \u03b1]\n  {X : set \u03b1} (hX : polish_space.is_clopenable X) : \u2203 s : topological_space \u03b1,\n  (@polish_space _ s) \u2227 @is_closed _ s X \u2227 @is_open _ s X \u2227 @borel_space _ s _ :=\nbegin\n  rcases hX with  \u27e8s,st,hs1,hs2,hs3\u27e9,\n  refine \u27e8s,hs1,hs2,hs3,_\u27e9,\n  constructor,\n  rw hb.measurable_eq,\n  symmetry,\n  apply eq_borel_of_polish_le_polish; assumption,\nend\n\nend clopenable\n\nsection sb_instances\n\nvariables {\u03b1 \u03b2: Type*} \nvariables [measurable_space \u03b1] [standard_borel \u03b1] [measurable_space \u03b2] [standard_borel \u03b2]\n\n--set_option pp.implicit true\ntheorem measurable_set.standard_borel {s : set \u03b1} (hs : measurable_set s) : standard_borel s :=\nbegin\n  letI := upgrade_standard_borel \u03b1,\n  rw \u2190 measure_theory.is_clopenable_iff_measurable_set at hs,\n  rcases clopenable_same_borel hs with \u27e8t,tpolish,sclosed,-,ht\u27e9,\n  letI := t,\n  constructor,\n  use infer_instance,\n  split, {\n    apply sclosed.polish_space,\n  },\n  apply_instance,\nend \n\ninstance prod_standard_borel : standard_borel (\u03b1 \u00d7 \u03b2) :=\nbegin\n  letI := upgrade_standard_borel \u03b1,\n  letI := upgrade_standard_borel \u03b2,\n  constructor,\n  use[infer_instance],\n  sorry, --no polish_space.prod!\nend\n\n\n\nend sb_instances\n", "meta": {"author": "Felix-Weilacher", "repo": "dst", "sha": "29702f743a4752ba31c0973b0167521f194d2045", "save_path": "github-repos/lean/Felix-Weilacher-dst", "path": "github-repos/lean/Felix-Weilacher-dst/dst-29702f743a4752ba31c0973b0167521f194d2045/src/SBS.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3672410925134762}}
{"text": "import for_mathlib.AddCommGroup.explicit_products\n\nopen category_theory\nopen category_theory.limits\n\nnamespace AddCommGroup\n\nuniverses u\nvariables {J : Type u} [small_category J] (F : J \u2964 AddCommGroup.{u})\n\nopen_locale classical\nnoncomputable theory\n\ndef explicit_cocone_point_kernel :\n  add_subgroup (direct_sum J (\u03bb i, F.obj i)) :=\nadd_subgroup.closure\n{ x | \u2203 (i j : J) (f : i \u27f6 j) (t : F.obj i),\n  x = direct_sum.of _ j (F.map f t) - direct_sum.of _ i t  }\n\ndef as_small_succ (i : as_small.{u} \u2115) : as_small \u2115 :=\n  as_small.up.obj (as_small.down.obj i + 1)\n\ndef to_as_small_succ (i : as_small.{u} \u2115) : i \u27f6 as_small_succ i :=\nas_small.up.map (hom_of_le $ nat.le_succ _)\n\nlemma explicit_cocone_point_kernel_eq_of_as_small_nat\n  (F : as_small.{u} \u2115 \u2964 AddCommGroup.{u}) :\n  explicit_cocone_point_kernel F =\n  add_subgroup.closure { x | \u2203 i (t : F.obj i), x =\n    direct_sum.of (\u03bb i, F.obj i) (as_small_succ i) (F.map (to_as_small_succ i) t) -\n    direct_sum.of _ i t } :=\nbegin\n  apply le_antisymm,\n  { erw add_subgroup.closure_le,\n    rintros x \u27e8\u27e8i\u27e9,\u27e8j\u27e9,f,t,rfl\u27e9,\n    obtain \u27e8k,rfl\u27e9 : \u2203 k : \u2115, j = i + k,\n    { have : i \u2264 j := le_of_hom (as_small.down.map f),\n      exact le_iff_exists_add.mp this },\n    induction k with k hk,\n    { have : f = \ud835\udfd9 _, ext, rw this,\n      simp only [category_theory.functor.map_id, id_apply, set_like.mem_coe],\n      erw sub_self,\n      exact add_subgroup.zero_mem _, },\n    { let f\u2081 : as_small.up.obj i \u27f6 as_small.up.obj (i + k) := as_small.up.map\n        (hom_of_le $ le_self_add),\n      let f\u2082 : as_small.up.obj (i + k) \u27f6 as_small.up.obj (i + (k + 1)) :=\n        as_small.up.map (hom_of_le $ by nlinarith),\n      have hf : f = f\u2081 \u226b f\u2082, by ext, rw hf, clear hf,\n      specialize hk f\u2081,\n      let t' := _, change t' \u2208 _, let s := _, change s \u2208 _ at hk,\n      rw (show t' = (t' - s) + s, by simp),\n      let A := add_subgroup.closure {x :\n        direct_sum (as_small \u2115) (\u03bb (i : as_small \u2115), \u21a5(F.obj i)) |\n          \u2203 (i : as_small \u2115) (t : \u21a5(F.obj i)), x =\n            (direct_sum.of (\u03bb (i : as_small \u2115), \u21a5(F.obj i)) (as_small_succ i))\n              ((F.map (to_as_small_succ i)) t) -\n            (direct_sum.of (\u03bb (i : as_small \u2115), \u21a5(F.obj i)) i) t},\n      change _ \u2208 A,\n      suffices : (t' - s) \u2208 A, by exact A.add_mem this hk,\n      dsimp [t', s], simp only [functor.map_comp, comp_apply, sub_sub_sub_cancel_right],\n      apply add_subgroup.subset_closure,\n      use as_small.up.obj (i + k),\n      let tt : F.obj (as_small.up.obj (i + k)) := F.map f\u2081 t,\n      use tt,\n      congr } },\n  { rw add_subgroup.closure_le,\n    rintros x \u27e8i,t,rfl\u27e9,\n    apply add_subgroup.subset_closure,\n    refine \u27e8i,as_small_succ i, to_as_small_succ i, t, _\u27e9,\n    congr }\nend\n\ndef explicit_cocone_point : AddCommGroup.{u} :=\nAddCommGroup.of\n((direct_sum J (\u03bb i, F.obj i)) \u29f8 explicit_cocone_point_kernel F)\n\ndef explicit_cocone : cocone F :=\n{ X := explicit_cocone_point F,\n  \u03b9 :=\n  { app := \u03bb j, add_monoid_hom.comp (quotient_add_group.mk' _)\n      (direct_sum.of _ j),\n    naturality' := begin\n      intros i j f, ext t,\n      dsimp,\n      simp only [comp_apply, add_monoid_hom.coe_comp, quotient_add_group.coe_mk',\n        function.comp_app, id_apply],\n      rw quotient_add_group.eq_iff_sub_mem,\n      apply add_subgroup.subset_closure,\n      dsimp, refine \u27e8i, j, f, t, rfl\u27e9,\n    end } }\n\ndef is_colimit_explicit_cocone : is_colimit (explicit_cocone F) :=\n{ desc := \u03bb S, quotient_add_group.lift _\n    (direct_sum.to_add_monoid $ \u03bb i, S.\u03b9.app _)\n    begin\n      intros t ht,\n      apply add_subgroup.closure_induction ht,\n      { rintros x \u27e8i,j,f,t,rfl\u27e9,\n        simp only [map_sub, direct_sum.to_add_monoid_of, cocone.w_apply, sub_self] },\n      { simp only [map_zero], },\n      { intros x y hx hy, simp only [hx, hy, map_add, add_zero] },\n      { intros x hx, simp only [hx, map_neg, neg_zero] },\n    end,\n  fac' := begin\n    intros S j, ext t, dsimp [explicit_cocone],\n    simp only [direct_sum.to_add_monoid_of, comp_apply, add_monoid_hom.coe_comp,\n      quotient_add_group.coe_mk', quotient_add_group.lift_mk],\n  end,\n  uniq' := begin\n    intros S m hm, ext j t,\n    simp only [direct_sum.to_add_monoid_of, add_monoid_hom.coe_comp, quotient_add_group.coe_mk',\n      function.comp_app, quotient_add_group.lift_mk],\n    rw \u2190 hm, refl,\n  end }\n\nend AddCommGroup\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/AddCommGroup/direct_sum_colimit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.752012562644147, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3671952473930057}}
{"text": "/-\nCopyright (c) 2019 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes, Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.ring_theory.integral_closure\nimport Mathlib.data.polynomial.field_division\nimport Mathlib.ring_theory.polynomial.gauss_lemma\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Minimal polynomials\n\nThis file defines the minimal polynomial of an element `x` of an `A`-algebra `B`,\nunder the assumption that x is integral over `A`.\n\nAfter stating the defining property we specialize to the setting of field extensions\nand derive some well-known properties, amongst which the fact that minimal polynomials\nare irreducible, and uniquely determined by their defining property.\n\n-/\n\n/-- Let `B` be an `A`-algebra, and `x` an element of `B` that is integral over `A`\nso we have some term `hx : is_integral A x`.\nThe minimal polynomial `minpoly A x` of `x` is a monic polynomial of smallest degree\nthat has `x` as its root.\nFor instance, if `V` is a `K`-vector space for some field `K`, and `f : V \u2192\u2097[K] V` then\nthe minimal polynomial of `f` is `minpoly f.is_integral`. -/\ndef minpoly (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) : polynomial A :=\n  dite (is_integral A x)\n    (fun (hx : is_integral A x) =>\n      well_founded.min sorry (fun (p : polynomial A) => polynomial.monic p \u2227 polynomial.eval\u2082 (algebra_map A B) x p = 0)\n        hx)\n    fun (hx : \u00acis_integral A x) => 0\n\nnamespace minpoly\n\n\n/--A minimal polynomial is monic.-/\ntheorem monic {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B} (hx : is_integral A x) : polynomial.monic (minpoly A x) := sorry\n\n/-- A minimal polynomial is nonzero. -/\ntheorem ne_zero {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B} [nontrivial A] (hx : is_integral A x) : minpoly A x \u2260 0 :=\n  polynomial.ne_zero_of_monic (monic hx)\n\ntheorem eq_zero {A : Type u_1} {B : Type u_2} [comm_ring A] [ring B] [algebra A B] {x : B} (hx : \u00acis_integral A x) : minpoly A x = 0 :=\n  dif_neg hx\n\n/--An element is a root of its minimal polynomial.-/\n@[simp] theorem aeval (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) : coe_fn (polynomial.aeval x) (minpoly A x) = 0 := sorry\n\ntheorem mem_range_of_degree_eq_one (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) (hx : polynomial.degree (minpoly A x) = 1) : x \u2208 ring_hom.range (algebra_map A B) := sorry\n\n/--The defining property of the minimal polynomial of an element x:\nit is the monic polynomial with smallest degree that has x as its root.-/\ntheorem min (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) {p : polynomial A} (pmonic : polynomial.monic p) (hp : coe_fn (polynomial.aeval x) p = 0) : polynomial.degree (minpoly A x) \u2264 polynomial.degree p := sorry\n\n-- TODO(Commelin, Brasca): this is a duplicate\n\n/-- If an element `x` is a root of a nonzero monic polynomial `p`,\nthen the degree of `p` is at least the degree of the minimal polynomial of `x`. -/\ntheorem degree_le_of_monic (A : Type u_1) {B : Type u_2} [comm_ring A] [ring B] [algebra A B] (x : B) {p : polynomial A} (hmonic : polynomial.monic p) (hp : coe_fn (polynomial.aeval x) p = 0) : polynomial.degree (minpoly A x) \u2264 polynomial.degree p := sorry\n\n/-- The degree of a minimal polynomial is positive. -/\ntheorem degree_pos {A : Type u_1} {B : Type u_2} [integral_domain A] [ring B] [algebra A B] [nontrivial B] {x : B} [nontrivial A] (hx : is_integral A x) : 0 < polynomial.degree (minpoly A x) := sorry\n\n/-- If `B/A` is an injective ring extension, and `a` is an element of `A`,\nthen the minimal polynomial of `algebra_map A B a` is `X - C a`. -/\ntheorem eq_X_sub_C_of_algebra_map_inj {A : Type u_1} {B : Type u_2} [integral_domain A] [ring B] [algebra A B] [nontrivial B] [nontrivial A] (a : A) (hf : function.injective \u21d1(algebra_map A B)) : minpoly A (coe_fn (algebra_map A B) a) = polynomial.X - coe_fn polynomial.C a := sorry\n\n/-- A minimal polynomial is not a unit. -/\ntheorem not_is_unit (A : Type u_1) {B : Type u_2} [integral_domain A] [ring B] [algebra A B] [nontrivial B] (x : B) : \u00acis_unit (minpoly A x) := sorry\n\n/-- If `a` strictly divides the minimal polynomial of `x`, then `x` cannot be a root for `a`. -/\ntheorem aeval_ne_zero_of_dvd_not_unit_minpoly {A : Type u_1} {B : Type u_2} [integral_domain A] [domain B] [algebra A B] {x : B} {a : polynomial A} (hx : is_integral A x) (hamonic : polynomial.monic a) (hdvd : dvd_not_unit a (minpoly A x)) : coe_fn (polynomial.aeval x) a \u2260 0 := sorry\n\n/--A minimal polynomial is irreducible.-/\ntheorem irreducible {A : Type u_1} {B : Type u_2} [integral_domain A] [domain B] [algebra A B] {x : B} (hx : is_integral A x) : irreducible (minpoly A x) := sorry\n\n/-- If an element `x` is a root of a nonzero polynomial `p`,\nthen the degree of `p` is at least the degree of the minimal polynomial of `x`. -/\ntheorem degree_le_of_ne_zero (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (pnz : p \u2260 0) (hp : coe_fn (polynomial.aeval x) p = 0) : polynomial.degree (minpoly A x) \u2264 polynomial.degree p := sorry\n\n/-- The minimal polynomial of an element x is uniquely characterized by its defining property:\nif there is another monic polynomial of minimal degree that has x as a root,\nthen this polynomial is equal to the minimal polynomial of x. -/\ntheorem unique (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (pmonic : polynomial.monic p) (hp : coe_fn (polynomial.aeval x) p = 0) (pmin : \u2200 (q : polynomial A), polynomial.monic q \u2192 coe_fn (polynomial.aeval x) q = 0 \u2192 polynomial.degree p \u2264 polynomial.degree q) : p = minpoly A x := sorry\n\n/-- If an element x is a root of a polynomial p,\nthen the minimal polynomial of x divides p. -/\ntheorem dvd (A : Type u_1) {B : Type u_2} [field A] [ring B] [algebra A B] (x : B) {p : polynomial A} (hp : coe_fn (polynomial.aeval x) p = 0) : minpoly A x \u2223 p := sorry\n\ntheorem dvd_map_of_is_scalar_tower (A : Type u_1) (K : Type u_2) {R : Type u_3} [comm_ring A] [field K] [comm_ring R] [algebra A K] [algebra A R] [algebra K R] [is_scalar_tower A K R] (x : R) : minpoly K x \u2223 polynomial.map (algebra_map A K) (minpoly A x) := sorry\n\ntheorem unique' {A : Type u_1} {B : Type u_2} [field A] [ring B] [algebra A B] {x : B} [nontrivial B] {p : polynomial A} (hx : is_integral A x) (hp1 : irreducible p) (hp2 : coe_fn (polynomial.aeval x) p = 0) (hp3 : polynomial.monic p) : p = minpoly A x := sorry\n\n/-- If `y` is the image of `x` in an extension, their minimal polynomials coincide.\n\nWe take `h : y = algebra_map L T x` as an argument because `rw h` typically fails\nsince `is_integral R y` depends on y.\n-/\ntheorem eq_of_algebra_map_eq {K : Type u_1} {S : Type u_2} {T : Type u_3} [field K] [comm_ring S] [comm_ring T] [algebra K S] [algebra K T] [algebra S T] [is_scalar_tower K S T] (hST : function.injective \u21d1(algebra_map S T)) {x : S} {y : T} (hx : is_integral K x) (h : y = coe_fn (algebra_map S T) x) : minpoly K x = minpoly K y := sorry\n\n/-- For GCD domains, the minimal polynomial over the ring is the same as the minimal polynomial\nover the fraction field. -/\ntheorem gcd_domain_eq_field_fractions {A : Type u_1} {K : Type u_2} {R : Type u_3} [integral_domain A] [gcd_monoid A] [field K] [integral_domain R] (f : fraction_map A K) [algebra (localization_map.codomain f) R] [algebra A R] [is_scalar_tower A (localization_map.codomain f) R] {x : R} (hx : is_integral A x) : minpoly (localization_map.codomain f) x = polynomial.map (localization_map.to_ring_hom f) (minpoly A x) := sorry\n\n/-- The minimal polynomial over `\u2124` is the same as the minimal polynomial over `\u211a`. -/\n--TODO use `gcd_domain_eq_field_fractions` directly when localizations are defined\n\n-- in terms of algebras instead of `ring_hom`s\n\ntheorem over_int_eq_over_rat {A : Type u_1} [integral_domain A] {x : A} [h\u211aA : algebra \u211a A] (hx : is_integral \u2124 x) : minpoly \u211a x = polynomial.map (int.cast_ring_hom \u211a) (minpoly \u2124 x) := sorry\n\n/-- For GCD domains, the minimal polynomial divides any primitive polynomial that has the integral\nelement as root. -/\ntheorem gcd_domain_dvd {A : Type u_1} {K : Type u_2} {R : Type u_3} [integral_domain A] [gcd_monoid A] [field K] [integral_domain R] (f : fraction_map A K) [algebra (localization_map.codomain f) R] [algebra A R] [is_scalar_tower A (localization_map.codomain f) R] {x : R} (hx : is_integral A x) {P : polynomial A} (hprim : polynomial.is_primitive P) (hroot : coe_fn (polynomial.aeval x) P = 0) : minpoly A x \u2223 P := sorry\n\n/-- The minimal polynomial over `\u2124` divides any primitive polynomial that has the integral element\nas root. -/\n-- TODO use `gcd_domain_dvd` directly when localizations are defined in terms of algebras\n\n-- instead of `ring_hom`s\n\ntheorem integer_dvd {A : Type u_1} [integral_domain A] [algebra \u211a A] {x : A} (hx : is_integral \u2124 x) {P : polynomial \u2124} (hprim : polynomial.is_primitive P) (hroot : coe_fn (polynomial.aeval x) P = 0) : minpoly \u2124 x \u2223 P := sorry\n\n/-- If `B/K` is a nontrivial algebra over a field, and `x` is an element of `K`,\nthen the minimal polynomial of `algebra_map K B x` is `X - C x`. -/\ntheorem eq_X_sub_C {A : Type u_1} (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] (a : A) : minpoly A (coe_fn (algebra_map A B) a) = polynomial.X - coe_fn polynomial.C a :=\n  eq_X_sub_C_of_algebra_map_inj a (ring_hom.injective (algebra_map A B))\n\ntheorem eq_X_sub_C' {A : Type u_1} [field A] (a : A) : minpoly A a = polynomial.X - coe_fn polynomial.C a :=\n  eq_X_sub_C A a\n\n/-- The minimal polynomial of `0` is `X`. -/\n@[simp] theorem zero (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] : minpoly A 0 = polynomial.X := sorry\n\n/-- The minimal polynomial of `1` is `X - 1`. -/\n@[simp] theorem one (A : Type u_1) (B : Type u_2) [field A] [ring B] [algebra A B] [nontrivial B] : minpoly A 1 = polynomial.X - 1 := sorry\n\n/-- A minimal polynomial is prime. -/\ntheorem prime {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B} (hx : is_integral A x) : prime (minpoly A x) := sorry\n\n/-- If `L/K` is a field extension and an element `y` of `K` is a root of the minimal polynomial\nof an element `x \u2208 L`, then `y` maps to `x` under the field embedding. -/\ntheorem root {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B} (hx : is_integral A x) {y : A} (h : polynomial.is_root (minpoly A x) y) : coe_fn (algebra_map A B) y = x := sorry\n\n/--The constant coefficient of the minimal polynomial of `x` is `0` if and only if `x = 0`. -/\n@[simp] theorem coeff_zero_eq_zero {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B} (hx : is_integral A x) : polynomial.coeff (minpoly A x) 0 = 0 \u2194 x = 0 := sorry\n\n/--The minimal polynomial of a nonzero element has nonzero constant coefficient. -/\ntheorem coeff_zero_ne_zero {A : Type u_1} {B : Type u_2} [field A] [domain B] [algebra A B] {x : B} (hx : is_integral A x) (h : x \u2260 0) : polynomial.coeff (minpoly A x) 0 \u2260 0 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/field_theory/minpoly.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.36711338503882546}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.AppBuilder\nimport Lean.Meta.Instances\n\nnamespace Lean.Meta\n\n/-- Create `SizeOf` local instances for applicable parameters, and execute `k` using them. -/\nprivate partial def mkLocalInstances {\u03b1} (params : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  loop 0 #[]\nwhere\n  loop (i : Nat) (insts : Array Expr) : MetaM \u03b1 := do\n    if i < params.size then\n      let param := params[i]\n      let paramType \u2190 inferType param\n      let instType? \u2190 forallTelescopeReducing paramType fun xs _ => do\n        let type \u2190 mkAppN param xs\n        try\n          let sizeOf \u2190 mkAppM `SizeOf #[type]\n          let instType \u2190 mkForallFVars xs sizeOf\n          return some instType\n        catch _ =>\n          return none\n      match instType? with\n      | none => loop (i+1) insts\n      | some instType =>\n        let instName \u2190 mkFreshUserName `inst\n        withLocalDecl instName BinderInfo.instImplicit instType fun inst =>\n          loop (i+1) (insts.push inst)\n    else\n      k insts\n\n/--\n  Return `some x` if `fvar` has type of the form `... -> motive ... fvar` where `motive` in `motiveFVars`.\n  That is, `x` \"produces\" one of the recursor motives.\n-/\nprivate def isInductiveHypothesis? (motiveFVars : Array Expr) (fvar : Expr) : MetaM (Option Expr) := do\n  forallTelescopeReducing (\u2190 inferType fvar) fun _ type =>\n    if type.isApp && motiveFVars.contains type.getAppFn then\n      return some type.appArg!\n    else\n      return none\n\nprivate def isInductiveHypothesis (motiveFVars : Array Expr) (fvar : Expr) : MetaM Bool :=\n  return (\u2190 isInductiveHypothesis? motiveFVars fvar).isSome\n\n/--\n  Let `motiveFVars` be free variables for each motive in a kernel recursor, and `minorFVars` the free variables for a minor premise.\n  Then, return `some idx` if `minorFVars[idx]` has a type of the form `... -> motive ... fvar` for some `motive` in `motiveFVars`.\n-/\nprivate def isRecField? (motiveFVars : Array Expr) (minorFVars : Array Expr) (fvar : Expr) : MetaM (Option Nat) := do\n  let mut idx := 0\n  for minorFVar in minorFVars do\n    if let some fvar' \u2190 isInductiveHypothesis? motiveFVars minorFVar then\n      if fvar == fvar' then\n        return some idx\n    idx := idx + 1\n  return none\n\nprivate partial def mkSizeOfMotives {\u03b1} (motiveFVars : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  loop 0 #[]\nwhere\n  loop (i : Nat) (motives : Array Expr) : MetaM \u03b1 := do\n    if i < motiveFVars.size then\n      let type \u2190 inferType motiveFVars[i]\n      let motive \u2190 forallTelescopeReducing type fun xs _ => do\n        mkLambdaFVars xs <| mkConst ``Nat\n      trace[Meta.sizeOf] \"motive: {motive}\"\n      loop (i+1) (motives.push motive)\n    else\n      k motives\n\nprivate partial def mkSizeOfMinors {\u03b1} (motiveFVars : Array Expr) (minorFVars : Array Expr) (minorFVars' : Array Expr) (k : Array Expr \u2192 MetaM \u03b1) : MetaM \u03b1 :=\n  assert! minorFVars.size == minorFVars'.size\n  loop 0 #[]\nwhere\n  loop (i : Nat) (minors : Array Expr) : MetaM \u03b1 := do\n    if i < minorFVars.size then\n      forallTelescopeReducing (\u2190 inferType minorFVars[i]) fun xs _ => do\n      forallBoundedTelescope (\u2190 inferType minorFVars'[i]) xs.size fun xs' _ => do\n        let mut minor \u2190 mkNumeral (mkConst ``Nat) 1\n        for x in xs, x' in xs' do\n          unless (\u2190 isInductiveHypothesis motiveFVars x) do\n          unless (\u2190 whnf (\u2190 inferType x)).isForall do -- we suppress higher-order fields\n            match (\u2190 isRecField? motiveFVars xs x) with\n            | some idx => minor \u2190 mkAdd minor xs'[idx]\n            | none     => minor \u2190 mkAdd minor (\u2190 mkAppM ``SizeOf.sizeOf #[x'])\n        minor \u2190 mkLambdaFVars xs' minor\n        trace[Meta.sizeOf] \"minor: {minor}\"\n        loop (i+1) (minors.push minor)\n    else\n      k minors\n\n/--\n  Create a \"sizeOf\" function with name `declName` using the recursor `recName`.\n-/\npartial def mkSizeOfFn (recName : Name) (declName : Name): MetaM Unit := do\n  trace[Meta.sizeOf] \"recName: {recName}\"\n  let recInfo : RecursorVal \u2190 getConstInfoRec recName\n  forallTelescopeReducing recInfo.type fun xs type =>\n    let levelParams := recInfo.levelParams.tail! -- universe parameters for declaration being defined\n    let params := xs[:recInfo.numParams]\n    let motiveFVars := xs[recInfo.numParams : recInfo.numParams + recInfo.numMotives]\n    let minorFVars := xs[recInfo.getFirstMinorIdx : recInfo.getFirstMinorIdx + recInfo.numMinors]\n    let indices := xs[recInfo.getFirstIndexIdx : recInfo.getFirstIndexIdx + recInfo.numIndices]\n    let major := xs[recInfo.getMajorIdx]\n    let nat := mkConst ``Nat\n    mkLocalInstances params fun localInsts =>\n    mkSizeOfMotives motiveFVars fun motives => do\n      let us := levelOne :: levelParams.map mkLevelParam -- universe level parameters for `rec`-application\n      let recFn := mkConst recName us\n      let val := mkAppN recFn (params ++ motives)\n      forallBoundedTelescope (\u2190 inferType val) recInfo.numMinors fun minorFVars' _ =>\n      mkSizeOfMinors motiveFVars minorFVars minorFVars' fun minors => do\n        let sizeOfParams := params ++ localInsts ++ indices ++ #[major]\n        let sizeOfType \u2190 mkForallFVars sizeOfParams nat\n        let val := mkAppN val (minors ++ indices ++ #[major])\n        trace[Meta.sizeOf] \"val: {val}\"\n        let sizeOfValue \u2190 mkLambdaFVars sizeOfParams val\n        addDecl <| Declaration.defnDecl {\n          name        := declName\n          levelParams := levelParams\n          type        := sizeOfType\n          value       := sizeOfValue\n          safety      := DefinitionSafety.safe\n          hints       := ReducibilityHints.abbrev\n        }\n\n/--\n  Create `sizeOf` functions for all inductive datatypes in the mutual inductive declaration containing `typeName`\n  The resulting array contains the generated functions names. The `NameMap` maps recursor names into the generated function names.\n  There is a function for each element of the mutual inductive declaration, and for auxiliary recursors for nested inductive types.\n-/\ndef mkSizeOfFns (typeName : Name) : MetaM (Array Name \u00d7 NameMap Name) := do\n  let indInfo \u2190 getConstInfoInduct typeName\n  let recInfo \u2190 getConstInfoRec (mkRecName typeName)\n  let numExtra := recInfo.numMotives - indInfo.all.length -- numExtra > 0 for nested inductive types\n  let mut result := #[]\n  let baseName := indInfo.all.head! ++ `_sizeOf -- we use the first inductive type as the base name for `sizeOf` functions\n  let mut i := 1\n  let mut recMap : NameMap Name := {}\n  for indTypeName in indInfo.all do\n    let sizeOfName := baseName.appendIndexAfter i\n    let recName := mkRecName indTypeName\n    mkSizeOfFn recName sizeOfName\n    recMap := recMap.insert recName sizeOfName\n    result := result.push sizeOfName\n    i := i + 1\n  for j in [:numExtra] do\n    let recName := (mkRecName indInfo.all.head!).appendIndexAfter (j+1)\n    let sizeOfName := baseName.appendIndexAfter i\n    mkSizeOfFn recName sizeOfName\n    recMap := recMap.insert recName sizeOfName\n    result := result.push sizeOfName\n    i := i + 1\n  return (result, recMap)\n\ndef mkSizeOfSpecLemmaName (ctorName : Name) : Name :=\n  ctorName ++ `sizeOf_spec\n\ndef mkSizeOfSpecLemmaInstance (ctorApp : Expr) : MetaM Expr :=\n  matchConstCtor ctorApp.getAppFn (fun _ => throwError \"failed to apply 'sizeOf' spec, constructor expected{indentExpr ctorApp}\") fun ctorInfo ctorLevels => do\n    let ctorArgs     := ctorApp.getAppArgs\n    let ctorFields   := ctorArgs[ctorArgs.size - ctorInfo.numFields:]\n    let lemmaName  := mkSizeOfSpecLemmaName ctorInfo.name\n    let lemmaInfo  \u2190 getConstInfo lemmaName\n    let lemmaArity \u2190 forallTelescopeReducing lemmaInfo.type fun xs _ => return xs.size\n    let lemmaArgMask := mkArray (lemmaArity - ctorInfo.numFields) (none (\u03b1 := Expr))\n    let lemmaArgMask := lemmaArgMask ++ ctorFields.toArray.map some\n    mkAppOptM lemmaName lemmaArgMask\n\n/- SizeOf spec theorem for nested inductive types -/\nnamespace SizeOfSpecNested\n\nstructure Context where\n  indInfo    : InductiveVal\n  sizeOfFns  : Array Name\n  ctorName   : Name\n  params     : Array Expr\n  localInsts : Array Expr\n  recMap     : NameMap Name -- mapping from recursor name into `_sizeOf_<idx>` function name (see `mkSizeOfFns`)\n\nabbrev M := ReaderT Context MetaM\n\ndef throwUnexpected {\u03b1} (msg : MessageData) : M \u03b1 := do\n  throwError \"failed to generate sizeOf theorem for {(\u2190 read).ctorName} (use `set_option genSizeOfSpec false` to disable theorem generation), {msg}\"\n\ndef throwFailed {\u03b1} : M \u03b1 := do\n  throwError \"failed to generate sizeOf theorem for {(\u2190 read).ctorName}, (use `set_option genSizeOfSpec false` to disable theorem generation)\"\n\n/-- Convert a recursor application into a `_sizeOf_<idx>` application. -/\nprivate def recToSizeOf (e : Expr) : M Expr := do\n  matchConstRec e.getAppFn (fun _ => throwFailed) fun info us => do\n    match (\u2190 read).recMap.find? info.name with\n    | none => throwUnexpected m!\"expected recursor application {indentExpr e}\"\n    | some sizeOfName =>\n      let args    := e.getAppArgs\n      let indices := args[info.getFirstIndexIdx : info.getFirstIndexIdx + info.numIndices]\n      let major   := args[info.getMajorIdx]\n      return mkAppN (mkConst sizeOfName us.tail!) ((\u2190 read).params ++ (\u2190 read).localInsts ++ indices ++ #[major])\n\nmutual\n  /-- Construct minor premise proof for `mkSizeOfAuxLemmaProof`. `ys` contains fields and inductive hypotheses for the minor premise. -/\n  private partial def mkMinorProof (ys : Array Expr) (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.minor] \"{lhs} =?= {rhs}\"\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      match (\u2190 whnfI lhs).natAdd?, (\u2190 whnfI rhs).natAdd? with\n      | some (a\u2081, b\u2081), some (a\u2082, b\u2082) =>\n        let p\u2081 \u2190 mkMinorProof ys a\u2081 a\u2082\n        let p\u2082 \u2190 mkMinorProofStep ys b\u2081 b\u2082\n        mkCongr (\u2190 mkCongrArg (mkConst ``Nat.add) p\u2081) p\u2082\n      | _, _ =>\n        throwUnexpected m!\"expected 'Nat.add' application, lhs is {indentExpr lhs}\\nrhs is{indentExpr rhs}\"\n\n  /--\n    Helper method for `mkMinorProof`. The proof step is one of the following\n    - Reflexivity\n    - Assumption (i.e., using an inductive hypotheses from `ys`)\n    - `mkSizeOfAuxLemma` application. This case happens when we have multiple levels of nesting\n  -/\n  private partial def mkMinorProofStep (ys : Array Expr) (lhs rhs : Expr) : M Expr := do\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      let lhs \u2190 recToSizeOf lhs\n      trace[Meta.sizeOf.minor.step] \"{lhs} =?= {rhs}\"\n      let target \u2190 mkEq lhs rhs\n      for y in ys do\n        if (\u2190 isDefEq (\u2190 inferType y) target) then\n          return y\n      mkSizeOfAuxLemma lhs rhs\n\n  /-- Construct proof of auxiliary lemma. See `mkSizeOfAuxLemma` -/\n  private partial def mkSizeOfAuxLemmaProof (info : InductiveVal) (lhs rhs : Expr) : M Expr := do\n    let lhsArgs := lhs.getAppArgs\n    let sizeOfBaseArgs := lhsArgs[:lhsArgs.size - info.numIndices - 1]\n    let indicesMajor := lhsArgs[lhsArgs.size - info.numIndices - 1:]\n    let sizeOfLevels := lhs.getAppFn.constLevels!\n    /- Auxiliary function for constructing an `_sizeOf_<idx>` for `ys`,\n       where `ys` are the indices + major.\n       Recall that if `info.name` is part of a mutually inductive declaration, then the resulting application\n       is not necessarily a `lhs.getAppFn` application.\n       The result is an application of one of the `(\u2190 read),sizeOfFns` functions.\n       We use this auxiliary function to builtin the motive of the recursor. -/\n    let rec mkSizeOf (ys : Array Expr) : M Expr := do\n      for sizeOfFn in (\u2190 read).sizeOfFns do\n        let candidate := mkAppN (mkAppN (mkConst sizeOfFn sizeOfLevels) sizeOfBaseArgs) ys\n        if (\u2190 isTypeCorrect candidate) then\n          return candidate\n      throwFailed\n    let major := lhs.appArg!\n    let majorType \u2190 whnf (\u2190 inferType major)\n    let majorTypeArgs := majorType.getAppArgs\n    match majorType.getAppFn.const? with\n    | none => throwFailed\n    | some (_, us) =>\n      let recName := mkRecName info.name\n      let recInfo \u2190 getConstInfoRec recName\n      let r := mkConst recName (levelZero :: us)\n      let r := mkAppN r majorTypeArgs[:info.numParams]\n      forallBoundedTelescope (\u2190 inferType r) recInfo.numMotives fun motiveFVars _ => do\n        let mut r := r\n        -- Add motives\n        for motiveFVar in motiveFVars do\n          let motive \u2190 forallTelescopeReducing (\u2190 inferType motiveFVar) fun ys _ => do\n            let lhs \u2190 mkSizeOf ys\n            let rhs \u2190 mkAppM ``SizeOf.sizeOf #[ys.back]\n            mkLambdaFVars ys (\u2190 mkEq lhs rhs)\n          r := mkApp r motive\n        forallBoundedTelescope (\u2190 inferType r) recInfo.numMinors fun minorFVars _ => do\n          let mut r := r\n          -- Add minors\n          for minorFVar in minorFVars do\n            let minor \u2190 forallTelescopeReducing (\u2190 inferType minorFVar) fun ys target => do\n              let target \u2190 whnf target\n              match target.eq? with\n              | none => throwFailed\n              | some (_, lhs, rhs) =>\n                if (\u2190 isDefEq lhs rhs) then\n                  mkLambdaFVars ys (\u2190 mkEqRefl rhs)\n                else\n                  let lhs \u2190 unfoldDefinition lhs -- Unfold `_sizeOf_<idx>`\n                  -- rhs is of the form `sizeOf (ctor ...)`\n                  let ctorApp := rhs.appArg!\n                  let specLemma \u2190 mkSizeOfSpecLemmaInstance ctorApp\n                  let specEq \u2190 whnf (\u2190 inferType specLemma)\n                  match specEq.eq? with\n                  | none => throwFailed\n                  | some (_, rhs, rhsExpanded) =>\n                    let lhs_eq_rhsExpanded \u2190 mkMinorProof ys lhs rhsExpanded\n                    let rhsExpanded_eq_rhs \u2190 mkEqSymm specLemma\n                    mkLambdaFVars ys (\u2190 mkEqTrans lhs_eq_rhsExpanded rhsExpanded_eq_rhs)\n            r := mkApp r minor\n          -- Add indices and major\n          return mkAppN r indicesMajor\n\n  /--\n    Generate proof for `C._sizeOf_<idx> t = sizeOf t` where `C._sizeOf_<idx>` is a auxiliary function\n    generated for a nested inductive type in `C`.\n    For example, given\n    ```lean\n    inductive Expr where\n      | app (f : String) (args : List Expr)\n    ```\n    We generate the auxiliary function `Expr._sizeOf_1 : List Expr \u2192 Nat`.\n    To generate the `sizeOf` spec lemma\n    ```\n    sizeOf (Expr.app f args) = 1 + sizeOf f + sizeOf args\n    ```\n    we need an auxiliary lemma for showing `Expr._sizeOf_1 args = sizeOf args`.\n    Recall that `sizeOf (Expr.app f args)` is definitionally equal to `1 + sizeOf f + Expr._sizeOf_1 args`, but\n    `Expr._sizeOf_1 args` is **not** definitionally equal to `sizeOf args`. We need a proof by induction.\n  -/\n  private partial def mkSizeOfAuxLemma (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.aux] \"{lhs} =?= {rhs}\"\n    match lhs.getAppFn.const? with\n    | none => throwFailed\n    | some (fName, us) =>\n      let thmLevelParams \u2190 us.mapM fun\n        | Level.param n _ => return n\n        | _ => throwFailed\n      let thmName  := fName.appendAfter \"_eq\"\n      if (\u2190 getEnv).contains thmName then\n        -- Auxiliary lemma has already been defined\n        return mkAppN (mkConst thmName us) lhs.getAppArgs\n      else\n        -- Define auxiliary lemma\n        -- First, generalize indices\n        let x := lhs.appArg!\n        let xType \u2190 whnf (\u2190 inferType x)\n        matchConstInduct xType.getAppFn (fun _ => throwFailed) fun info _ => do\n          let params := xType.getAppArgs[:info.numParams]\n          forallTelescopeReducing (\u2190 inferType (mkAppN xType.getAppFn params)) fun indices _ => do\n            let majorType := mkAppN (mkAppN xType.getAppFn params) indices\n            withLocalDeclD `x majorType fun major => do\n              let lhsArgs := lhs.getAppArgs\n              let lhsArgsNew := lhsArgs[:lhsArgs.size - 1 - indices.size] ++ indices ++ #[major]\n              let lhsNew := mkAppN lhs.getAppFn lhsArgsNew\n              let rhsNew \u2190 mkAppM ``SizeOf.sizeOf #[major]\n              let eq \u2190 mkEq lhsNew rhsNew\n              let thmParams := lhsArgsNew\n              let thmType \u2190 mkForallFVars thmParams eq\n              let thmValue \u2190 mkSizeOfAuxLemmaProof info lhsNew rhsNew\n              let thmValue \u2190 mkLambdaFVars thmParams thmValue\n              trace[Meta.sizeOf] \"thmValue: {thmValue}\"\n              addDecl <| Declaration.thmDecl {\n                name        := thmName\n                levelParams := thmLevelParams\n                type        := thmType\n                value       := thmValue\n              }\n              return mkAppN (mkConst thmName us) lhs.getAppArgs\n\nend\n\n/- Prove SizeOf spec lemma of the form `sizeOf <ctor-application> = 1 + sizeOf <field_1> + ... + sizeOf <field_n> -/\npartial def main (lhs rhs : Expr) : M Expr := do\n  if (\u2190 isDefEq lhs rhs) then\n    mkEqRefl rhs\n  else\n    /- Expand lhs and rhs to obtain `Nat.add` applications -/\n    let lhs \u2190 whnfI lhs            -- Expand `sizeOf (ctor ...)` into `_sizeOf_<idx>` application\n    let lhs \u2190 unfoldDefinition lhs -- Unfold `_sizeOf_<idx>` application into `HAdd.hAdd` application\n    loop lhs rhs\nwhere\n  loop (lhs rhs : Expr) : M Expr := do\n    trace[Meta.sizeOf.loop] \"{lhs} =?= {rhs}\"\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      match (\u2190 whnfI lhs).natAdd?, (\u2190 whnfI rhs).natAdd? with\n      | some (a\u2081, b\u2081), some (a\u2082, b\u2082) =>\n        let p\u2081 \u2190 loop a\u2081 a\u2082\n        let p\u2082 \u2190 step b\u2081 b\u2082\n        mkCongr (\u2190 mkCongrArg (mkConst ``Nat.add) p\u2081) p\u2082\n      | _, _ =>\n        throwUnexpected m!\"expected 'Nat.add' application, lhs is {indentExpr lhs}\\nrhs is{indentExpr rhs}\"\n\n  step (lhs rhs : Expr) : M Expr := do\n    if (\u2190 isDefEq lhs rhs) then\n      mkEqRefl rhs\n    else\n      let lhs \u2190 recToSizeOf lhs\n      mkSizeOfAuxLemma lhs rhs\n\nend SizeOfSpecNested\n\nprivate def mkSizeOfSpecTheorem (indInfo : InductiveVal) (sizeOfFns : Array Name) (recMap : NameMap Name) (ctorName : Name) : MetaM Unit := do\n  let ctorInfo \u2190 getConstInfoCtor ctorName\n  let us := ctorInfo.levelParams.map mkLevelParam\n  forallTelescopeReducing ctorInfo.type fun xs _ => do\n    let params := xs[:ctorInfo.numParams]\n    let fields := xs[ctorInfo.numParams:]\n    let ctorApp := mkAppN (mkConst ctorName us) xs\n    mkLocalInstances params fun localInsts => do\n      let lhs \u2190 mkAppM ``SizeOf.sizeOf #[ctorApp]\n      let mut rhs \u2190 mkNumeral (mkConst ``Nat) 1\n      for field in fields do\n        unless (\u2190 whnf (\u2190 inferType field)).isForall do\n          rhs \u2190 mkAdd rhs (\u2190 mkAppM ``SizeOf.sizeOf #[field])\n      let target \u2190 mkEq lhs rhs\n      let thmName   := mkSizeOfSpecLemmaName ctorName\n      let thmParams := params ++ localInsts ++ fields\n      let thmType \u2190 mkForallFVars thmParams target\n      let thmValue \u2190\n        if indInfo.isNested then\n          SizeOfSpecNested.main lhs rhs |>.run {\n            indInfo := indInfo, sizeOfFns := sizeOfFns, ctorName := ctorName, params := params, localInsts := localInsts, recMap := recMap\n          }\n        else\n          mkEqRefl rhs\n      let thmValue \u2190 mkLambdaFVars thmParams thmValue\n      addDecl <| Declaration.thmDecl {\n        name        := thmName\n        levelParams := ctorInfo.levelParams\n        type        := thmType\n        value       := thmValue\n      }\n\nprivate def mkSizeOfSpecTheorems (indTypeNames : Array Name) (sizeOfFns : Array Name) (recMap : NameMap Name) : MetaM Unit := do\n  for indTypeName in indTypeNames do\n    let indInfo \u2190 getConstInfoInduct indTypeName\n    for ctorName in indInfo.ctors do\n      mkSizeOfSpecTheorem indInfo sizeOfFns recMap ctorName\n  return ()\n\nregister_builtin_option genSizeOf : Bool := {\n  defValue := true\n  descr    := \"generate `SizeOf` instance for inductive types and structures\"\n}\n\nregister_builtin_option genSizeOfSpec : Bool := {\n  defValue := true\n  descr    := \"generate `SizeOf` specificiation theorems for automatically generated instances\"\n}\n\ndef mkSizeOfInstances (typeName : Name) : MetaM Unit := do\n  if (\u2190 getEnv).contains ``SizeOf && genSizeOf.get (\u2190 getOptions) && !(\u2190 isInductivePredicate typeName) then\n    let indInfo \u2190 getConstInfoInduct typeName\n    unless indInfo.isUnsafe do\n      let (fns, recMap) \u2190 mkSizeOfFns typeName\n      for indTypeName in indInfo.all, fn in fns do\n        let indInfo \u2190 getConstInfoInduct indTypeName\n        forallTelescopeReducing indInfo.type fun xs _ =>\n          let params := xs[:indInfo.numParams]\n          let indices := xs[indInfo.numParams:]\n          mkLocalInstances params fun localInsts => do\n            let us := indInfo.levelParams.map mkLevelParam\n            let indType := mkAppN (mkConst indTypeName us) xs\n            let sizeOfIndType \u2190 mkAppM ``SizeOf #[indType]\n            withLocalDeclD `m indType fun m => do\n              let v \u2190 mkLambdaFVars #[m] <| mkAppN (mkConst fn us) (params ++ localInsts ++ indices ++ #[m])\n              let sizeOfMk \u2190 mkAppM ``SizeOf.mk #[v]\n              let instDeclName := indTypeName ++ `_sizeOf_inst\n              let instDeclType \u2190 mkForallFVars (xs ++ localInsts) sizeOfIndType\n              let instDeclValue \u2190 mkLambdaFVars (xs ++ localInsts) sizeOfMk\n              addDecl <| Declaration.defnDecl {\n                name        := instDeclName\n                levelParams := indInfo.levelParams\n                type        := instDeclType\n                value       := instDeclValue\n                safety      := DefinitionSafety.safe\n                hints       := ReducibilityHints.abbrev\n              }\n              addInstance instDeclName AttributeKind.global (eval_prio default)\n      if genSizeOfSpec.get (\u2190 getOptions) then\n        mkSizeOfSpecTheorems indInfo.all.toArray fns recMap\n\nbuiltin_initialize\n  registerTraceClass `Meta.sizeOf\n\nend Lean.Meta\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/stage0/src/Lean/Meta/SizeOf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3670504754921644}}
{"text": "-- lemmas about validity of logical propositions\n\nimport .definitions2 .freevars .substitution .evaluation .bindings\n\nlemma valid.not_false: \u22a8 vc.not value.false :=\n  begin\n    have : \u22a8 value.false \u22c1 vc.not value.false, from valid.em,\n    cases valid.or.elim this with h1 h1,\n    have h2, from valid.eq.true.mp h1,\n    have h3, from valid.eq.true.mp h2,\n    have h4, from valid.binop.mpr h3,\n    unfold binop.apply at h4,\n    unfold binop.apply at h4,\n    have h5: \u00ac (value.true = value.false), by begin\n      assume h5,\n      contradiction\n    end,\n    simp[h5] at h4,\n    have h6, from option.some.inj h4,\n    contradiction,\n    from h1\n  end\n\nlemma valid.false.elim {P: vc}: \u22a8 vc.implies value.false P :=\n  have h1: \u22a8 value.true, from valid.tru,\n  have unop.apply unop.not value.false = value.true, by unfold unop.apply,\n  have \u22a8 value.true \u2261 term.unop unop.not value.false, from valid.unop.mp this,\n  have \u22a8 term.unop unop.not value.false, from valid.eq.true.mpr this,\n  have \u22a8 vc.not value.false, from valid.not_false,\n  have \u22a8 vc.not value.false \u22c1 P, from valid.or.left this,\n  show \u22a8 vc.implies value.false P, from this\n\nlemma valid.implies.mp {P Q: vc}: ((\u22a8 P) \u2192 (\u22a8 Q)) \u2192 \u22a8 vc.implies P Q :=\n  assume h1: (\u22a8 P) \u2192 (\u22a8 Q),\n  have \u22a8 P \u22c1 P.not, from valid.em,\n  or.elim (valid.or.elim this) (\n    assume : \u22a8 P,\n    have \u22a8 Q, from h1 this,\n    show \u22a8 P.not \u22c1 Q, from valid.or.right this\n  ) (\n    assume : \u22a8 P.not,\n    show \u22a8 P.not \u22c1 Q, from valid.or.left this\n  )\n\nlemma valid.implies.mpr {P Q: vc}: (\u22a8 vc.implies P Q) \u2192 (\u22a8 P) \u2192 (\u22a8 Q) :=\n  assume h1: (\u22a8 P.not \u22c1 Q),\n  assume h2: (\u22a8 P),\n  or.elim (valid.or.elim h1) (\n    assume : \u22a8 P.not,\n    have \u22a8 P \u22c0 P.not, from valid.and.mp \u27e8h2, this\u27e9,\n    show \u22a8 Q, from false.elim (valid.contradiction this)\n  ) id\n\nlemma valid.not.mp {P: vc}: (\u00ac \u22a8 P) \u2192 \u22a8 P.not :=\n  assume h1: \u00ac (\u22a8 P),\n  have \u22a8 P \u22c1 P.not, from valid.em,\n  or.elim (valid.or.elim this) (\n    assume : \u22a8 P,\n    show \u22a8 P.not, from absurd this h1\n  ) id\n\nlemma valid.not.mpr {P: vc}: (\u22a8 P.not) \u2192 \u00ac \u22a8 P :=\n  assume h2: \u22a8 P.not,\n  assume h3: \u22a8 P,\n  have \u22a8 (P \u22c0 P.not), from valid.and.mp \u27e8h3, h2\u27e9,\n  show \u00abfalse\u00bb, from valid.contradiction this\n\nlemma valid.not_not {P: vc}: (\u22a8 P.not.not) \u2194 \u22a8 P :=\n  iff.intro (\n    assume h1: \u22a8 P.not.not,\n    have h3: \u00ac \u22a8 P.not, from valid.not.mpr h1,\n    have h4: \u00ac \u00ac \u22a8 P, from (\n      assume : \u00ac \u22a8 P,\n      have \u22a8 P.not, from valid.not.mp this,\n      show \u00abfalse\u00bb, from h3 this\n    ),\n    or.elim (valid.or.elim valid.em) id (\n      assume : \u22a8 P.not,\n      have \u00ac \u22a8 P, from valid.not.mpr this,\n      show \u22a8 P, from absurd this h4\n    )\n  ) (\n    assume h1: \u22a8 P,\n    have h2: \u00ac \u22a8 P.not, from (\n      assume : \u22a8 P.not,\n      have \u00ac \u22a8 P, from valid.not.mpr this,\n      show \u00abfalse\u00bb, from this h1\n    ),\n    show \u22a8 P.not.not, from valid.not.mp h2\n  )\n\nlemma valid.mt {P Q: vc}: (\u22a8 vc.implies P Q) \u2192 (\u22a8 Q.not) \u2192 \u22a8 P.not :=\n  assume h1: \u22a8 vc.implies P Q,\n  assume : \u22a8 Q.not,\n  have h2: \u00ac \u22a8 Q, from valid.not.mpr this,\n  have \u00ac \u22a8 P, from (\n    assume : \u22a8 P,\n    have \u22a8 Q, from valid.implies.mpr h1 this,\n    show \u00abfalse\u00bb, from h2 this\n  ),\n  show \u22a8 P.not, from valid.not.mp this\n\nlemma valid.refl {v: value}: \u22a8 (v \u2261 v) :=\n  have binop.apply binop.eq v v = value.true, from binop.eq_of_equal_values,\n  have \u22a8 (value.true \u2261 (v \u2261 v)), from valid.binop.mp this,\n  show \u22a8 (v \u2261 v), from valid.eq.true.mpr this\n\nlemma valid.implies.trans {P\u2081 P\u2082 P\u2083: vc}:\n      (\u22a8 vc.implies P\u2081 P\u2082) \u2192 (\u22a8 vc.implies P\u2082 P\u2083) \u2192 \u22a8 vc.implies P\u2081 P\u2083 :=\n  assume h1: \u22a8 vc.implies P\u2081 P\u2082,\n  assume h2: \u22a8 vc.implies P\u2082 P\u2083,\n  show \u22a8 vc.implies P\u2081 P\u2083, from valid.implies.mp (\n    assume : \u22a8 P\u2081,\n    have \u22a8 P\u2082, from valid.implies.mpr h1 this,\n    show \u22a8 P\u2083, from valid.implies.mpr h2 this\n  )\n\nlemma valid_env.true {\u03c3: env}: \u03c3 \u22a8 value.true :=\n  have h1: \u22a8 value.true, from valid.tru,\n  have term.subst_env \u03c3 value.true = value.true, from term.subst_env.value,\n  have h2: \u22a8 term.subst_env \u03c3 value.true, from this.symm \u25b8 h1,\n  have vc.subst_env \u03c3 value.true = vc.term (term.subst_env \u03c3 value.true), from vc.subst_env.term,\n  show \u03c3 \u22a8 value.true, from this.symm \u25b8 h2\n\nlemma valid_env.mt {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 vc.implies P Q) \u2192 (\u03c3 \u22a8 Q.not) \u2192 \u03c3 \u22a8 P.not :=\n  assume h1: \u03c3 \u22a8 vc.implies P Q,\n  have vc.subst_env \u03c3 (vc.implies P Q) = vc.implies (vc.subst_env \u03c3 P) (vc.subst_env \u03c3 Q),\n  from vc.subst_env.implies,\n  have h2: \u22a8 vc.implies (vc.subst_env \u03c3 P) (vc.subst_env \u03c3 Q), from this \u25b8 h1,\n  assume h3: \u03c3 \u22a8 Q.not,\n  have vc.subst_env \u03c3 Q.not = (vc.subst_env \u03c3 Q).not, from vc.subst_env.not,\n  have h4: \u22a8 (vc.subst_env \u03c3 Q).not, from this \u25b8 h3,\n  have h5: \u22a8 (vc.subst_env \u03c3 P).not, from valid.mt h2 h4,\n  have vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n  show \u03c3 \u22a8 P.not, from this.symm \u25b8 h5\n\nlemma valid_env.eq.true {\u03c3: env} {t: term}: \u03c3 \u22a8 t \u2194 \u03c3 \u22a8 (value.true \u2261 t) :=\n  iff.intro (\n    assume t_valid: \u22a8 vc.subst_env \u03c3 t,\n    have vc.subst_env \u03c3 t = vc.term (term.subst_env \u03c3 t), from vc.subst_env.term,\n    have \u22a8 vc.term (term.subst_env \u03c3 t), from this \u25b8 t_valid,\n    have h: \u22a8 vc.term (value.true \u2261 (term.subst_env \u03c3 t)), from valid.eq.true.mp this,\n    have term.subst_env \u03c3 value.true = value.true, from term.subst_env.value,\n    have h2: \u22a8 vc.term ((term.subst_env \u03c3 value.true) \u2261 (term.subst_env \u03c3 t)),\n    from this.symm \u25b8 h,\n    have (term.subst_env \u03c3 (value.true \u2261 t)) = ((term.subst_env \u03c3 value.true) \u2261 (term.subst_env \u03c3 t)),\n    from term.subst_env.binop,\n    have h3: \u22a8 term.subst_env \u03c3 (value.true \u2261 t), from this.symm \u25b8 h2,\n    have vc.subst_env \u03c3 (value.true \u2261 t) = vc.term (term.subst_env \u03c3 (value.true \u2261 t)), from vc.subst_env.term,\n    show \u03c3 \u22a8 (value.true \u2261 t), from this.symm \u25b8 h3\n  ) (\n    assume t_valid: \u22a8 vc.subst_env \u03c3 (value.true \u2261 t),\n    have vc.subst_env \u03c3 (value.true \u2261 t) = vc.term (term.subst_env \u03c3 (value.true \u2261 t)), from vc.subst_env.term,\n    have h: \u22a8 vc.term (term.subst_env \u03c3 (value.true \u2261 t)),\n    from this \u25b8 t_valid,\n    have (term.subst_env \u03c3 (value.true \u2261 t)) = ((term.subst_env \u03c3 value.true) \u2261 (term.subst_env \u03c3 t)),\n    from term.subst_env.binop,\n    have h2: \u22a8 vc.term ((term.subst_env \u03c3 value.true) \u2261 (term.subst_env \u03c3 t)),\n    from this \u25b8 h,\n    have term.subst_env \u03c3 value.true = value.true, from term.subst_env.value,\n    have \u22a8 vc.term (value.true \u2261 (term.subst_env \u03c3 t)), from this \u25b8 h2,\n    have h3: \u22a8 vc.term (term.subst_env \u03c3 t), from valid.eq.true.mpr this,\n    have vc.subst_env \u03c3 t = vc.term (term.subst_env \u03c3 t), from vc.subst_env.term,\n    show \u22a8 vc.subst_env \u03c3 t, from this.symm \u25b8 h3\n  )\n\nlemma valid.eq.terms {v\u2081 v\u2082: value}: (\u22a8 v\u2081 \u2261 v\u2082) \u2192 (v\u2081 = v\u2082) :=\n  begin\n    assume h1,\n    have h2, from valid.eq.true.mp h1,\n    have h3, from valid.binop.mpr h2,\n    from binop.eq.inv h3\n  end\n\nlemma valid_env.not_not {\u03c3: env} {P: vc}: (\u03c3 \u22a8 P.not.not) \u2194 \u03c3 \u22a8 P :=\n  iff.intro (\n    assume h1: \u03c3 \u22a8 P.not.not,\n    have vc.subst_env \u03c3 P.not.not = (vc.subst_env \u03c3 P.not).not, from vc.subst_env.not,\n    have h2: \u22a8 (vc.subst_env \u03c3 P.not).not, from this \u25b8 h1,\n    have vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n    have  \u22a8 (vc.subst_env \u03c3 P).not.not, from this \u25b8 h2,\n    show \u03c3 \u22a8 P, from valid.not_not.mp this\n  ) (\n    assume : \u03c3 \u22a8 P,\n    have h1: \u22a8 (vc.subst_env \u03c3 P).not.not, from valid.not_not.mpr this,\n    have vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n    have h2: \u22a8 (vc.subst_env \u03c3 P.not).not, from this.symm \u25b8 h1,\n    have vc.subst_env \u03c3 P.not.not = (vc.subst_env \u03c3 P.not).not, from vc.subst_env.not,\n    show \u03c3 \u22a8 P.not.not, from this.symm \u25b8 h2\n  )\n\nlemma valid_env.and {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 P) \u2192 (\u03c3 \u22a8 Q) \u2192 \u03c3 \u22a8 (P \u22c0 Q) :=\n  assume p_valid: \u22a8 vc.subst_env \u03c3 P,\n  assume q_valid: \u22a8 vc.subst_env \u03c3 Q,\n  have vc.subst_env \u03c3 (P \u22c0 Q) = (vc.subst_env \u03c3 P \u22c0 vc.subst_env \u03c3 Q), from vc.subst_env.and,\n  show \u03c3 \u22a8 (P \u22c0 Q), from this.symm \u25b8 valid.and.mp \u27e8p_valid, q_valid\u27e9\n\nlemma valid_env.and.elim {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 P \u22c0 Q) \u2192 (\u03c3 \u22a8 P) \u2227 \u03c3 \u22a8 Q :=\n  assume p_and_q_valid: \u22a8 vc.subst_env \u03c3 (P \u22c0 Q),\n  have vc.subst_env \u03c3 (P \u22c0 Q) = (vc.subst_env \u03c3 P \u22c0 vc.subst_env \u03c3 Q), from vc.subst_env.and,\n  have \u22a8 (vc.subst_env \u03c3 P \u22c0 vc.subst_env \u03c3 Q), from this \u25b8 p_and_q_valid,\n  show (\u03c3 \u22a8 P) \u2227 (\u03c3 \u22a8 Q), from valid.and.mpr this\n\nlemma valid_env.or\u2081 {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 P) \u2192 \u03c3 \u22a8 (P \u22c1 Q) :=\n  assume h1: \u22a8 vc.subst_env \u03c3 P,\n  have h: \u22a8 vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q, from valid.or.left h1,\n  have vc.subst_env \u03c3 (P \u22c1 Q) = (vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q), from vc.subst_env.or,\n  show \u03c3 \u22a8 (P \u22c1 Q), from this.symm \u25b8 h\n\nlemma valid_env.or\u2082 {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 Q) \u2192 \u03c3 \u22a8 (P \u22c1 Q) :=\n  assume h1: \u22a8 vc.subst_env \u03c3 Q,\n  have h: \u22a8 vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q, from valid.or.right h1,\n  have vc.subst_env \u03c3 (P \u22c1 Q) = (vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q), from vc.subst_env.or,\n  show \u03c3 \u22a8 (P \u22c1 Q), from this.symm \u25b8 h\n\nlemma valid_env.or.elim {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 P \u22c1 Q) \u2192 (\u03c3 \u22a8 P) \u2228 \u03c3 \u22a8 Q :=\n  assume p_or_q_valid: \u22a8 vc.subst_env \u03c3 (P \u22c1 Q),\n  have vc.subst_env \u03c3 (P \u22c1 Q) = (vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q), from vc.subst_env.or,\n  have \u22a8 (vc.subst_env \u03c3 P \u22c1 vc.subst_env \u03c3 Q), from this \u25b8 p_or_q_valid,\n  show (\u03c3 \u22a8 P) \u2228 (\u03c3 \u22a8 Q), from valid.or.elim this\n\nlemma valid_env.not.mp {\u03c3: env} {P: vc}: \u00ac (\u03c3 \u22a8 P) \u2192 (\u03c3 \u22a8 P.not) :=\n  assume h1: \u00ac (\u03c3 \u22a8 P),\n  have h2: vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n  have \u00ac \u22a8 (vc.subst_env \u03c3 P), from h2 \u25b8 h1,\n  have \u22a8 (vc.subst_env \u03c3 P).not, from valid.not.mp this,\n  show \u03c3 \u22a8 P.not, from h2.symm \u25b8 this\n\nlemma valid_env.not.mpr {\u03c3: env} {P: vc}: (\u03c3 \u22a8 P.not) \u2192 \u00ac (\u03c3 \u22a8 P) :=\n  assume h1: \u03c3 \u22a8 P.not,\n  have h2: vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n  have \u22a8 (vc.subst_env \u03c3 P).not, from h2 \u25b8 h1,\n  have \u00ac \u22a8 (vc.subst_env \u03c3 P), from valid.not.mpr this,\n  show \u00ac (\u03c3 \u22a8 P), from h2.symm \u25b8 this\n\nlemma valid_env.mp {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 vc.implies P Q) \u2192 (\u03c3 \u22a8 P) \u2192 \u03c3 \u22a8 Q :=\n  assume impl: \u03c3 \u22a8 (vc.implies P Q),\n  assume p: \u03c3 \u22a8 P,\n  have vc.subst_env \u03c3 (vc.implies P Q) = (vc.subst_env \u03c3 P.not \u22c1 vc.subst_env \u03c3 Q), from vc.subst_env.or,\n  have h: \u22a8 (vc.subst_env \u03c3 P.not \u22c1 vc.subst_env \u03c3 Q), from this \u25b8 impl,\n  have vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n  have \u22a8 ((vc.subst_env \u03c3 P).not \u22c1 vc.subst_env \u03c3 Q), from this \u25b8 h,\n  have \u22a8 vc.implies (vc.subst_env \u03c3 P) (vc.subst_env \u03c3 Q), from this,\n  show \u03c3 \u22a8 Q, from valid.implies.mpr this p\n\nlemma valid_env.mpr {\u03c3: env} {P Q: vc}: ((\u03c3 \u22a8 P) \u2192 (\u03c3 \u22a8 Q)) \u2192 \u03c3 \u22a8 vc.implies P Q :=\n  assume : ((\u03c3 \u22a8 P) \u2192 \u03c3 \u22a8 Q),\n  have \u22a8 vc.implies (vc.subst_env \u03c3 P) (vc.subst_env \u03c3 Q), from valid.implies.mp this,\n  have h1: \u22a8 vc.or (vc.subst_env \u03c3 P).not (vc.subst_env \u03c3 Q), from this,\n  have vc.subst_env \u03c3 P.not = (vc.subst_env \u03c3 P).not, from vc.subst_env.not,\n  have h2: \u22a8 vc.or (vc.subst_env \u03c3 P.not) (vc.subst_env \u03c3 Q), from this.symm \u25b8 h1,\n  have vc.subst_env \u03c3 (P.not \u22c1 Q) = (vc.subst_env \u03c3 P.not \u22c1 vc.subst_env \u03c3 Q),\n  from vc.subst_env.or,\n  have \u22a8 vc.subst_env \u03c3 (P.not \u22c1 Q), from this.symm \u25b8 h2,\n  show \u03c3 \u22a8 vc.implies P Q, from this\n\nlemma valid_env.nmt {\u03c3: env} {P Q: vc}: (\u03c3 \u22a8 vc.implies P.not Q.not) \u2192 (\u03c3 \u22a8 vc.implies Q P) :=\n  begin\n    assume h1,\n    apply valid_env.mpr,\n    assume h2,\n    unfold vc.implies at h1,\n    cases valid_env.or.elim h1 with h3 h4,\n    from valid_env.not_not.mp h3,\n    have h5, from valid_env.not.mpr h4,\n    contradiction\n  end\n\nlemma valid_env.implies.trans {\u03c3: env} {P\u2081 P\u2082 P\u2083: vc}:\n      (\u03c3 \u22a8 vc.implies P\u2081 P\u2082) \u2192 (\u03c3 \u22a8 vc.implies P\u2082 P\u2083) \u2192 \u03c3 \u22a8 vc.implies P\u2081 P\u2083 :=\n  assume h1: \u03c3 \u22a8 vc.implies P\u2081 P\u2082,\n  assume h2: \u03c3 \u22a8 vc.implies P\u2082 P\u2083,\n  show \u03c3 \u22a8 vc.implies P\u2081 P\u2083, from valid_env.mpr (\n    assume : \u03c3 \u22a8 P\u2081,\n    have \u03c3 \u22a8 P\u2082, from valid_env.mp h1 this,\n    show \u03c3 \u22a8 P\u2083, from valid_env.mp h2 this\n  )\n\nlemma vc.implies.trans {\u03c3: env} {P\u2081 P\u2082 P\u2083: vc}:\n      (\u03c3 \u22a8 vc.implies P\u2081 P\u2082) \u2192 (\u03c3 \u22a8 vc.implies P\u2082 P\u2083) \u2192 \u03c3 \u22a8 vc.implies P\u2081 P\u2083 := valid_env.implies.trans\n\nlemma valid_env.univ.mp {\u03c3: env} {x: var} {P: vc}: (\u2200v, \u03c3 \u22a8 vc.subst x v P) \u2192 \u03c3 \u22a8 vc.univ x P :=\n  assume h1: \u2200v, \u03c3 \u22a8 vc.subst x v P,\n  have h2: \u22a8 vc.univ x (vc.subst_env (\u03c3.without x) P), from valid.univ.mp (\n    assume v: value,\n    have h3: \u22a8 vc.subst_env \u03c3 (vc.subst x v P), from h1 v,\n    have vc.subst_env \u03c3 (vc.subst x v P) = vc.subst x v (vc.subst_env (\u03c3.without x) P),\n    from vc.subst_env.reorder,\n    show \u22a8 vc.subst x v (vc.subst_env (\u03c3.without x) P), from this \u25b8 h3\n  ),\n  have vc.subst_env \u03c3 (vc.univ x P) = vc.univ x (vc.subst_env (\u03c3.without x) P),\n  from vc.subst_env.univ,\n  have \u22a8 vc.subst_env \u03c3 (vc.univ x P), from this.symm \u25b8 h2,\n  show \u03c3 \u22a8 vc.univ x P, from this\n\nlemma env.contains_of_valid_env_term {\u03c3: env} {x: var} {t: term}:\n      x \u2208 FV t \u2192 closed_subst \u03c3 t \u2192 (x \u2208 \u03c3) :=\n  assume x_free_in_t: x \u2208 FV t,\n  assume t_closed: closed_subst \u03c3 t,\n  show x \u2208 \u03c3, from t_closed x_free_in_t\n\nlemma valid_env.subst_of_eq {\u03c3: env} {x: var} {v: value}:\n      (\u03c3 \u22a8 x \u2261 v) \u2192 (\u03c3 x = v) :=\n  assume h1: \u03c3 \u22a8 vc.term (x \u2261 v),\n  have h2: \u22a8 vc.subst_env \u03c3 (vc.term (x \u2261 v)), from h1,\n  have vc.subst_env \u03c3 (vc.term (x \u2261 v)) = vc.term (term.subst_env \u03c3 (x \u2261 v)),\n  from vc.subst_env.term,\n  have h3: \u22a8 vc.term (term.subst_env \u03c3 (x \u2261 v)), from this \u25b8 h2,\n  have term.subst_env \u03c3 (x \u2261 v) = (term.subst_env \u03c3 x \u2261 term.subst_env \u03c3 v),\n  from term.subst_env.binop,\n  have h4: \u22a8 (term.subst_env \u03c3 x \u2261 term.subst_env \u03c3 v), from this \u25b8 h3,\n  have term.subst_env \u03c3 v = v, from term.subst_env.value,\n  have h5: \u22a8 (term.subst_env \u03c3 x \u2261 v), from this \u25b8 h4,\n  have x \u2208 \u03c3, by begin\n    by_contradiction h6,\n    have h7: x \u2208 FV (x \u2261 v), from free_in_term.binop\u2081 (free_in_term.var x),\n    have h8, from free_in_vc.term h7,\n    have h9, from vc.free_of_subst_env h8 h6,\n    have h10, from valid.univ.free \u27e8h9, h2\u27e9,\n    have h11: (\u2200v': value, v \u2260 v' \u2192 \u00abfalse\u00bb), by begin\n      assume v',\n      assume h11,\n\n      have h12, from valid.univ.mpr h10 v',\n      have h13: (vc.substt x v' (vc.subst_env \u03c3 \u2191(\u2191x \u2261 \u2191v))\n              = vc.subst x v' (vc.subst_env \u03c3 \u2191(\u2191x \u2261 \u2191v))),\n      from vc.substt_value_eq_subst,\n      rw[h13] at h12,\n      have h14: (vc.subst_env \u03c3 (vc.subst x v' \u2191(\u2191x \u2261 \u2191v))\n              = vc.subst x v' (vc.subst_env \u03c3 \u2191(\u2191x \u2261 \u2191v))),\n      from vc.subst_env.order (or.inl h6),\n      rw[h14.symm] at h12,\n      have h15: \u03c3 \u22a8 vc.subst x v' (vc.term (\u2191x \u2261 \u2191v)), from h12,\n      unfold vc.subst at h15,\n      unfold term.subst at h15,\n      have h16: \u03c3 \u22a8 \u2191(term.subst x v' (term.var x) \u2261 term.subst x v' (term.value v)),\n      from h15,\n      unfold term.subst at h16,\n      simp at h16,\n      have h17: closed (vc.term (\u2191v' \u2261 \u2191v)), by begin\n        assume z,\n        assume h18,\n        have h19, from free_in_vc.term.inv h18,\n        cases (free_in_term.binop.inv h19) with h20 h21,\n        have h22: \u00ac free_in_term z \u2191v', from free_in_term.value.inv,\n        contradiction,\n        have h22: \u00ac free_in_term z \u2191v, from free_in_term.value.inv,\n        contradiction\n      end,\n      have h18: (vc.subst_env \u03c3 (vc.term (\u2191v' \u2261 \u2191v)) = vc.term (\u2191v' \u2261 \u2191v)),\n      from unchanged_of_subst_env_nonfree_vc h17 \u03c3,\n      have h19: \u22a8 vc.subst_env \u03c3 (vc.term (\u2191v' \u2261 \u2191v)), from h16,\n      rw[h18] at h19,\n      have h20, from (valid.eq.terms h19).symm,\n      contradiction\n    end,\n    \n    by_cases (v = value.true) with h12,\n\n    have h13: value.true \u2260 value.false, by { assume h14, contradiction },\n    from h11 value.false (h12.symm \u25b8 h13),\n\n    have h13: value.true \u2260 v, by { assume h14, have h15, from h14.symm, contradiction },\n    from h11 value.true h13.symm,\n  end,\n  have \u2203v', \u03c3 x = some v', from env.contains_apply_equiv.right.mpr this,\n  let \u27e8v', h6\u27e9 := this in\n  have term.subst_env \u03c3 x = v', from (term.subst_env.var.right v').mp h6,\n  have \u22a8 (v' \u2261 v), from this \u25b8 h5,\n  have \u22a8 value.true \u2261 (v' \u2261 v), from valid.eq.true.mp this,\n  have binop.apply binop.eq v' v = some value.true, from valid.binop.mpr this,\n  have v' = v, from binop.eq.inv this,\n  show \u03c3 x = some v, from h6.symm \u25b8 (some.inj.inv this)\n\nlemma valid.alpha_equiv {x y: var} {P: vc}: (\u22a8 P) \u2192 \u22a8 vc.substt x y P :=\n  begin\n    assume h1,\n    by_cases (free_in_vc x P) with h2,\n\n    have h3, from valid.univ.free \u27e8h2, h1\u27e9,\n    from valid.univ.mpr h3 y,\n\n    have h4: (vc.substt x y P = P),\n    from unchanged_of_substt_nonfree_vc h2,\n    rw[h4],\n    from h1\n  end\n\nlemma valid.to_vc_and {P Q: prop}: (\u22a8 P.to_vc) \u2192 (\u22a8 Q.to_vc) \u2192 \u22a8 (P \u22c0 Q).to_vc :=\n  begin\n    assume h1: \u22a8 P.to_vc,\n    assume h2: \u22a8 Q.to_vc,\n    change \u22a8 prop.to_vc (prop.and P Q),\n    unfold prop.to_vc,\n    from valid.and.mp \u27e8h1, h2\u27e9\n  end\n\nlemma valid_env.to_vc_and {P Q: prop} {\u03c3: env}: (\u03c3 \u22a8 P.to_vc) \u2192 (\u03c3 \u22a8 Q.to_vc) \u2192 \u03c3 \u22a8 (P \u22c0 Q).to_vc :=\n  begin\n    assume h1: \u03c3 \u22a8 P.to_vc,\n    assume h2: \u03c3 \u22a8 Q.to_vc,\n    change \u03c3 \u22a8 prop.to_vc (prop.and P Q),\n    unfold prop.to_vc,\n    from valid_env.and h1 h2\n  end\n\nlemma valid_env.to_vc_and.elim {P Q: prop} {\u03c3: env}: (\u03c3 \u22a8 (P \u22c0 Q).to_vc) \u2192 ((\u03c3 \u22a8 P.to_vc) \u2227 (\u03c3 \u22a8 Q.to_vc)) :=\n  begin\n    assume h1: \u03c3 \u22a8 (P \u22c0 Q).to_vc,\n    have h2: \u03c3 \u22a8 (prop.and P Q).to_vc, from h1,\n    unfold prop.to_vc at h2,\n    from valid_env.and.elim h2\n  end\n\nlemma valid_env.to_vc_or_elim {P Q: prop} {\u03c3: env}: (\u03c3 \u22a8 (P \u22c1 Q).to_vc) \u2192 (\u03c3 \u22a8 P.to_vc \u22c1 Q.to_vc) :=\n  begin\n    assume h1: \u03c3 \u22a8 (P \u22c1 Q).to_vc,\n    have h2: \u03c3 \u22a8 (prop.or P Q).to_vc, from h1,\n    unfold prop.to_vc at h2,\n    cases valid_env.or.elim h2 with h3 h4,\n    apply valid_env.or\u2081,\n    from h3,\n    apply valid_env.or\u2082,\n    from h4\n  end\n\nlemma val_of_free_in_env {P: prop} {\u03c3: env} {x: var}: (\u22a9 \u03c3 : P) \u2192 x \u2208 FV P \u2192 \u2203v, \u03c3 x = some v :=\n  assume env_verified: \u22a9 \u03c3: P,\n  assume x_free_in_P: x \u2208 FV P,\n  have x \u2208 \u03c3, from contains_of_free env_verified x_free_in_P,\n  show \u2203v, \u03c3 x = some v, from env.contains_apply_equiv.right.mpr this\n\nlemma val_of_free_in_pre_env {R: spec} {\u03c3: env} {P: prop} {x: var}:\n                              (\u22a9 \u03c3 : P) \u2192 FV R.to_prop \u2286 FV P \u2192 x \u2208 FV (R.to_prop \u22c0 P) \u2192 \u2203v, \u03c3 x = some v :=\n  assume \u03c3_verified: \u22a9 \u03c3 : P,\n  assume fv_R: FV R.to_prop \u2286 FV P,\n  assume x_free_in_R_P: x \u2208 FV (R.to_prop \u22c0 P),\n  have free_in_prop x R.to_prop \u2228 free_in_prop x P, from free_in_prop.and.inv x_free_in_R_P,\n  have x \u2208 FV P, from or.elim this.symm id (\n    assume : free_in_prop x R.to_prop,\n    show x \u2208 FV P, from set.mem_of_mem_of_subset this fv_R\n  ),\n  show \u2203v, \u03c3 x = some v, from val_of_free_in_env \u03c3_verified this\n\nlemma to_vc_implies {P Q: prop}: (prop.implies P Q).to_vc = vc.implies P.to_vc Q.to_vc :=\n  begin\n    unfold prop.implies,\n    unfold vc.implies,\n    unfold prop.to_vc,\n    congr\n  end\n\nlemma valid.to_vc_implies {P Q: prop}: (\u22a8 (prop.implies P Q).to_vc) \u2194 \u22a8 vc.implies P.to_vc Q.to_vc :=\n  begin\n    have h1: ((prop.implies P Q).to_vc = vc.implies P.to_vc Q.to_vc), from to_vc_implies,\n    rw[h1]\n  end\n\nlemma valid_env.to_vc_implies {P Q: prop} {\u03c3: env}: (\u03c3 \u22a8 (prop.implies P Q).to_vc) \u2194 \u03c3 \u22a8 vc.implies P.to_vc Q.to_vc :=\n  begin\n    have h1: ((prop.implies P Q).to_vc = vc.implies P.to_vc Q.to_vc), from to_vc_implies,\n    rw[h1]\n  end\n\nlemma simple_equality_valid {\u03c3: env} {x: var} {v: value}:\n  x \u2209 \u03c3 \u2192 (\u03c3[x\u21a6v]) \u22a8 (prop.term (x \u2261 v)).to_vc :=\n  assume x_not_free_in_\u03c3: x \u2209 \u03c3,\n  have \u03c3.apply x = none, from env.contains_apply_equiv.left.mpr x_not_free_in_\u03c3,\n  have h1: term.subst_env \u03c3 x = x, from term.subst_env.var.left.mp this,\n  have (term.subst_env (\u03c3[x\u21a6v]) x = term.subst x v (term.subst_env \u03c3 x)),\n  by unfold term.subst_env,\n  have h2: term.subst_env (\u03c3[x\u21a6v]) x = term.subst x v x,\n  from @eq.subst term (\u03bba, term.subst_env (\u03c3[x\u21a6v]) x = term.subst x v a) (term.subst_env \u03c3 x) x h1 this,\n  have term.subst x v (term.var x) = (if x = x then v else x), by unfold term.subst,\n  have term.subst x v (term.var x) = v, by simp[this],\n  have h3: term.subst_env (\u03c3[x\u21a6v]) x = v, from eq.trans h2 this,\n  have h4: term.subst_env (\u03c3[x\u21a6v]) v = v, from term.subst_env.value,\n  have term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = (term.subst_env (\u03c3[x\u21a6v]) x \u2261 term.subst_env (\u03c3[x\u21a6v]) v),\n  from term.subst_env.binop,\n  have term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = (v \u2261 term.subst_env (\u03c3[x\u21a6v]) v),\n  from @eq.subst term (\u03bba, term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = (a \u2261 term.subst_env (\u03c3[x\u21a6v]) v))\n                      (term.subst_env (\u03c3[x\u21a6v]) x) v h3 this,\n  have h5: term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = (v \u2261 v),\n  from @eq.subst term (\u03bba, term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = (v \u2261 a))\n                      (term.subst_env (\u03c3[x\u21a6v]) v) v h4 this,\n  have h6: vc.term (term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v)) = vc.term (v \u2261 v), by simp[h5],\n  have vc.subst_env (\u03c3[x\u21a6v]) (x \u2261 v) = vc.term (term.subst_env (\u03c3[x\u21a6v]) (x \u2261 v)), from vc.subst_env.term,\n  have h7: vc.subst_env (\u03c3[x\u21a6v]) (vc.term (x \u2261 v)) = vc.term (v \u2261 v), from eq.trans this h6,\n  have prop.to_vc (prop.term (x \u2261 v)) = vc.term (x \u2261 v), by unfold prop.to_vc,\n  have h8: vc.subst_env (\u03c3[x\u21a6v]) (prop.term (x \u2261 v)).to_vc = vc.term (v \u2261 v), from this.symm \u25b8 h7,\n  have \u22a8 vc.term (v \u2261 v), from valid.refl,\n  show (\u03c3[x\u21a6v]) \u22a8 prop.to_vc (x \u2261 v), from h8.symm \u25b8 this\n\nlemma simple_equality_env_valid {P: prop} {\u03c3: env} {x: var} {v: value}:\n                                     (\u22a9 \u03c3: P) \u2192 x \u2209 \u03c3 \u2192 (\u03c3 \u22a8 P.to_vc) \u2192 (\u03c3[x\u21a6v]) \u22a8 (P \u22c0 x \u2261 v).to_vc :=\n  assume \u03c3_verified: \u22a9 \u03c3: P,\n  assume x_not_free_in_\u03c3: x \u2209 \u03c3,\n  assume ih: \u03c3 \u22a8 P.to_vc,\n  have \u03c3.apply x = none, from env.contains_apply_equiv.left.mpr x_not_free_in_\u03c3,\n  have h1: \u22a8 vc.subst_env \u03c3 P.to_vc, from ih,\n  have x_not_in_P: x \u2209 FV (vc.subst_env \u03c3 P.to_vc), from (\n    assume : x \u2208 FV (vc.subst_env \u03c3 P.to_vc),\n    have x \u2208 FV P.to_vc, from free_in_vc.subst_env this,\n    have x \u2208 FV P, from set.mem_of_mem_of_subset this free_in_prop_of_free_in_to_vc,\n    have \u2203v, \u03c3 x = some v, from val_of_free_in_env \u03c3_verified this,\n    have x \u2208 \u03c3, from env.contains_apply_equiv.right.mp this,\n    show \u00abfalse\u00bb, from x_not_free_in_\u03c3 this\n  ),\n  have vc.subst x v (vc.subst_env \u03c3 P.to_vc) = vc.subst_env \u03c3 P.to_vc,\n  from unchanged_of_subst_nonfree_vc x_not_in_P,\n  have h2: \u22a8 vc.subst x v (vc.subst_env \u03c3 P.to_vc),\n  from @eq.subst vc (\u03bba, \u22a8 a) (vc.subst_env \u03c3 P.to_vc)\n          (vc.subst x v (vc.subst_env \u03c3 P.to_vc)) this.symm h1,\n  have vc.subst x v (vc.subst_env \u03c3 P.to_vc)\n      = vc.subst_env (\u03c3[x\u21a6v]) P.to_vc, by unfold vc.subst_env, \n  have h3: \u22a8 vc.subst_env (\u03c3[x\u21a6v]) P.to_vc, from this \u25b8 h2,\n  have h4: (\u03c3[x\u21a6v]) \u22a8 (prop.term (x \u2261 v)).to_vc,\n  from simple_equality_valid x_not_free_in_\u03c3,\n  have h5: (\u03c3[x\u21a6v]) \u22a8 (P.to_vc \u22c0 (prop.term (x \u2261 v)).to_vc),\n  from valid_env.and h3 h4,\n  have (P.to_vc \u22c0 (prop.term (x \u2261 v)).to_vc) = prop.to_vc (prop.and P (prop.term (x \u2261 v))),\n  by unfold prop.to_vc,\n  have (\u03c3[x\u21a6v]) \u22a8 prop.to_vc (prop.and P (prop.term (x \u2261 v))), from this \u25b8 h5,\n  show (\u03c3[x\u21a6v]) \u22a8 (P \u22c0 x \u2261 v).to_vc, from this\n\nlemma env_translation_valid {P: prop} {\u03c3: env}: (\u22a9 \u03c3: P) \u2192 \u03c3 \u22a8 P.to_vc :=\n  assume env_verified: (\u22a9 \u03c3 : P),\n  begin\n    induction env_verified,\n    case env.dvcgen.empty {\n      unfold vc.subst_env,\n      change \u22a8 prop.to_vc (prop.term value.true),\n      unfold prop.to_vc,\n      from valid.tru\n    },\n    case env.dvcgen.tru \u03c3' x' Q x_not_free_in_\u03c3' \u03c3'_verified ih {\n      from simple_equality_env_valid \u03c3'_verified x_not_free_in_\u03c3' ih\n    },\n    case env.dvcgen.fls \u03c3' x' Q x_not_free_in_\u03c3' \u03c3'_verified ih {\n      from simple_equality_env_valid \u03c3'_verified x_not_free_in_\u03c3' ih\n    },\n    case env.dvcgen.num n \u03c3' x' Q x_not_free_in_\u03c3' \u03c3'_verified ih {\n      from simple_equality_env_valid \u03c3'_verified x_not_free_in_\u03c3' ih\n    },\n    case env.dvcgen.func \u03c3\u2081 \u03c3\u2082 f g gx R S e Q\u2081 Q\u2082 Q\u2083\n      f_not_free_in_\u03c3\u2081 g_not_free_in_\u03c3\u2082 gx_not_free_in_\u03c3\u2082 g_neq_gx \u03c3\u2081_verified \u03c3\u2082_verified gx_free_in_R R_fv S_fv func_verified\n      S_valid ih\u2081 ih\u2082 { from (\n      let vf := value.func g gx R S e \u03c3\u2082 in\n      have h1: ((\u03c3\u2081[f\u21a6vf]) \u22a8 (Q\u2081 \u22c0 f \u2261 vf).to_vc),\n      from simple_equality_env_valid \u03c3\u2081_verified f_not_free_in_\u03c3\u2081 ih\u2081,\n      have h1a: (\u03c3\u2081[f\u21a6vf]) \u22a8 Q\u2081.to_vc,\n      from (valid_env.to_vc_and.elim h1).left,\n      have h1b: (\u03c3\u2081[f\u21a6vf]) \u22a8 (prop.term (f \u2261 vf)).to_vc,\n      from (valid_env.to_vc_and.elim h1).right,\n\n      have g_subst: term.subst_env (\u03c3\u2082[g\u21a6vf]) g = vf, from (\n        have h1: term.subst g vf g = vf, from term.subst.var.same,\n        have \u03c3\u2082 g = none, from env.contains_apply_equiv.left.mpr g_not_free_in_\u03c3\u2082,\n        have term.subst_env \u03c3\u2082 g = g, from term.subst_env.var.left.mp this,\n        have h2: term.subst g vf (term.subst_env \u03c3\u2082 g) = vf, from this.symm \u25b8 h1,\n        have term.subst_env (\u03c3\u2082[g\u21a6vf]) g = term.subst g vf (term.subst_env \u03c3\u2082 g), by unfold term.subst_env,\n        show term.subst_env (\u03c3\u2082[g\u21a6vf]) g = vf, from eq.trans this h2\n      ),\n\n      have h2: \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)), from (\n        have unop.apply unop.isFunc vf = value.true, by unfold unop.apply,\n        have \u22a8 (value.true \u2261 term.unop unop.isFunc vf), from valid.unop.mp this,\n        have \u22a8 term.unop unop.isFunc vf, from valid.eq.true.mpr this,\n        have h3: \u22a8 term.unop unop.isFunc (term.subst_env (\u03c3\u2082[g\u21a6vf]) g), from g_subst.symm \u25b8 this,\n        have term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g) = term.unop unop.isFunc (term.subst_env (\u03c3\u2082[g\u21a6vf]) g),\n        from term.subst_env.unop,\n        have h4: \u22a8 vc.term (term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)), from this.symm \u25b8 h3,\n        have prop.to_vc (prop.term (term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)))\n           = vc.term (term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)),\n        by unfold prop.to_vc,\n        have h5: \u22a8 prop.to_vc (prop.term (term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g))), from this.symm \u25b8 h4,\n        have prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.term (term.unop unop.isFunc g))\n           = prop.term (term.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)), from prop.subst_env.term,\n        show \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g)), from this.symm \u25b8 h5\n      ),\n\n      let forallp := prop.implies R.to_prop (prop.pre g gx)\n                  \u22c0 prop.implies (prop.post g gx) (Q\u2083 (term.app g gx) \u22c0 S.to_prop) in\n      let pfunc: prop := prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.func g gx R (Q\u2083 (term.app g gx) \u22c0 S)) in\n\n      have h4: \u2200v, \u22a8 vc.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp).to_vc, from (\n        assume v: value,\n\n        have h5: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies R.to_prop (prop.pre g gx))).to_vc, from (\n\n          have h51: (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom = \u03c3\u2082.dom \u222a {g, gx}, from env.dom.two_elems,\n          have \u03c3\u2082.dom = FV Q\u2082, from free_iff_contains \u03c3\u2082_verified,\n          have h52: (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom = FV Q\u2082 \u222a {g, gx}, from this \u25b8 h51,\n          have FV R.to_prop \u2286 (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom, from h52.symm \u25b8 R_fv,\n          have closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop), from prop.closed_of_closed_subst this,\n          have h53: closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop).to_vc,\n          from to_vc_closed_of_closed this,\n\n          have FV (prop.pre g gx) \u2286 FV Q\u2082 \u222a {g, gx}, from (\n            assume x: var,\n            assume : x \u2208 FV (prop.pre g gx),\n            or.elim (free_in_prop.pre.inv this) (\n              assume : free_in_term x g,\n              have x = g, from free_in_term.var.inv this,\n              have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n              show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n            ) (\n              assume : free_in_term x gx,\n              have x = gx, from free_in_term.var.inv this,\n              have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inr this),\n              show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n            )\n          ),\n          have FV (prop.pre g gx) \u2286 (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom, from h52.symm \u25b8 this,\n          have closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)), from prop.closed_of_closed_subst this,\n          have h54: closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)).to_vc,\n          from to_vc_closed_of_closed this,\n\n          have h6: \u22a8 vc.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop).to_vc\n                                (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)).to_vc,\n          from valid.implies.mp (\n            assume h8: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop).to_vc,\n            have vc.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop.to_vc\n                = (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop).to_vc,\n            from subst_env_distrib_to_vc,\n            have \u22a8 vc.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop.to_vc, from this.symm \u25b8 h8,\n            have h9: \u22a8 vc.pre vf v, from valid.pre.mp this,\n            have term.subst gx v gx = v, from term.subst.var.same,\n            have h10: \u22a8 vc.pre vf (term.subst gx v gx), from this.symm \u25b8 h9,\n            have \u00ac(gx = g \u2228 gx \u2208 \u03c3\u2082), from not_or_distrib.mpr \u27e8g_neq_gx.symm, gx_not_free_in_\u03c3\u2082\u27e9,\n            have gx \u2209 (\u03c3\u2082[g\u21a6vf]), from (mt env.contains.inv) this,\n            have (\u03c3\u2082[g\u21a6vf]) gx = none, from env.contains_apply_equiv.left.mpr this,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf]) gx = gx, from term.subst_env.var.left.mp this,\n            have h11: \u22a8 vc.pre vf (term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) gx)),\n            from this.symm \u25b8 h10,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx = term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) gx),\n            by unfold term.subst_env,\n            have h12: \u22a8 vc.pre vf (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from this.symm \u25b8 h11,\n            have term.subst gx v (term.value vf) = vf, by unfold term.subst,\n            have \u22a8 vc.pre (term.subst gx v vf) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from this.symm \u25b8 h12,\n            have h13: \u22a8 vc.pre (term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) g)) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from g_subst.symm \u25b8 this,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g = term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) g),\n            by unfold term.subst_env,\n            have h14: \u22a8 vc.pre (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from this.symm \u25b8 h13,\n            have prop.to_vc (prop.pre (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx))\n               = (vc.pre (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx)),\n            by unfold prop.to_vc,\n            have h15: \u22a8 (prop.pre (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx)).to_vc,\n            from this.symm \u25b8 h14,\n            have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)\n               = prop.pre (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from prop.subst_env.pre,\n            show \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)).to_vc, from this.symm \u25b8 h15\n          ),\n          have h8: \u22a8 (prop.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop)\n                                   (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx))).to_vc,\n          from valid.to_vc_implies.mp h6,\n          have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies R.to_prop (prop.pre g gx))\n             = prop.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) R.to_prop)\n                            (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.pre g gx)),\n          from prop.subst_env.implies,\n          show \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies R.to_prop (prop.pre g gx))).to_vc,\n          from this.symm \u25b8 h8\n        ),\n\n        have h6: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies (prop.post g gx)\n                                                     (Q\u2083 (term.app g gx) \u22c0 S.to_prop))).to_vc, from (\n\n          have h61: (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom = \u03c3\u2082.dom \u222a {g, gx}, from env.dom.two_elems,\n          have \u03c3\u2082.dom = FV Q\u2082, from free_iff_contains \u03c3\u2082_verified,\n          have h62: (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom = FV Q\u2082 \u222a {g, gx}, from this \u25b8 h61,\n\n          have FV (Q\u2083 (term.app g gx) \u22c0 S.to_prop) \u2286 FV Q\u2082 \u222a {g, gx}, from (\n            assume x: var,\n            assume : x \u2208 FV (Q\u2083 (term.app g gx) \u22c0 S.to_prop),\n            or.elim (free_in_prop.and.inv this) (\n              assume : x \u2208 FV (Q\u2083 (term.app g gx)),\n              have x \u2208 FV (term.app g gx) \u2228 x \u2208 FV (Q\u2082 \u22c0 spec.func \u2191g gx R S \u22c0 R),\n              from exp.post_free func_verified (term.app g gx) this,\n              or.elim this (\n                assume : x \u2208 FV (term.app g gx),\n                or.elim (free_in_term.app.inv this) (\n                  assume : free_in_term x g,\n                  have x = g, from free_in_term.var.inv this,\n                  have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n                  show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                ) (\n                  assume : free_in_term x gx,\n                  have x = gx, from free_in_term.var.inv this,\n                  have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inr this),\n                  show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                )\n              ) (\n                assume : x \u2208 FV (Q\u2082 \u22c0 spec.func \u2191g gx R S \u22c0 R),\n                or.elim (free_in_prop.and.inv this) (\n                  assume : x \u2208 FV Q\u2082,\n                  show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_left {g, gx} this\n                ) (\n                  assume : free_in_prop x (spec.func \u2191g gx R S \u22c0 R),\n                  have free_in_prop x (spec.func \u2191g gx R S \u22c0 R), from this,\n                  or.elim (free_in_prop.and.inv this) (\n                    assume : free_in_prop x (spec.func \u2191g gx R S),\n                    have h63: free_in_prop x (spec.func \u2191g gx R S).to_prop, from this,\n                    have spec.to_prop (spec.func \u2191g gx R S) = (prop.func \u2191g gx R.to_prop S.to_prop),\n                    by unfold spec.to_prop,\n                    have h64: free_in_prop x (prop.func \u2191g gx R S), from this \u25b8 h63,\n                    let forallp := prop.implies R.to_prop (prop.pre g gx)\n                                \u22c0 prop.implies (prop.post g gx) S.to_prop in\n                    have prop.func g gx R.to_prop S.to_prop\n                      = (term.unop unop.isFunc g \u22c0 prop.forallc gx forallp),\n                    by unfold prop.func,\n                    have free_in_prop x (term.unop unop.isFunc g \u22c0 prop.forallc gx forallp),\n                    from this \u25b8 h64,\n                    or.elim (free_in_prop.and.inv this) (\n                      assume : free_in_prop x (term.unop unop.isFunc g),\n                      have free_in_term x (term.unop unop.isFunc g), from free_in_prop.term.inv this,\n                      have free_in_term x g, from free_in_term.unop.inv this,\n                      have x = g, from free_in_term.var.inv this,\n                      have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n                      show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                    ) (\n                      assume : free_in_prop x (prop.forallc gx forallp),\n                      have x_neq_gx: x \u2260 gx, from (free_in_prop.forallc.inv this).left,\n                      have free_in_prop x forallp, from (free_in_prop.forallc.inv this).right,\n                      or.elim (free_in_prop.and.inv this) (\n                        assume : free_in_prop x (prop.implies R.to_prop (prop.pre g gx)),\n                        or.elim (free_in_prop.implies.inv this) (\n                          assume : free_in_prop x R.to_prop,\n                          show x \u2208 FV Q\u2082 \u222a {g, gx}, from R_fv this\n                        ) (\n                          assume : x \u2208 FV (prop.pre g gx),\n                          or.elim (free_in_prop.pre.inv this) (\n                            assume : free_in_term x g,\n                            have x = g, from free_in_term.var.inv this,\n                            have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n                            show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                          ) (\n                            assume : free_in_term x gx,\n                            have x = gx, from free_in_term.var.inv this,\n                            have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inr this),\n                            show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                          )\n                        )\n                      ) (\n                        assume : free_in_prop x (prop.implies (prop.post g gx) S.to_prop),\n                        or.elim (free_in_prop.implies.inv this) (\n                          assume : x \u2208 FV (prop.post g gx),\n                          or.elim (free_in_prop.post.inv this) (\n                            assume : free_in_term x g,\n                            have x = g, from free_in_term.var.inv this,\n                            have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n                            show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                          ) (\n                            assume : free_in_term x gx,\n                            have x = gx, from free_in_term.var.inv this,\n                            have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inr this),\n                            show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n                          )\n                        ) (\n                          assume : free_in_prop x S.to_prop,\n                          show x \u2208 FV Q\u2082 \u222a {g, gx}, from S_fv this\n                        )\n                      )\n                    )\n                  ) (\n                    assume : free_in_prop x R,\n                    show x \u2208 FV Q\u2082 \u222a {g, gx}, from R_fv this\n                  )\n                )\n              )\n            ) (\n              assume : free_in_prop x S.to_prop,\n              show x \u2208 FV Q\u2082 \u222a {g, gx}, from S_fv this\n            )\n          ),\n\n          have FV (Q\u2083 (term.app g gx) \u22c0 S.to_prop) \u2286 (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom, from h62.symm \u25b8 this,\n          have closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)),\n          from prop.closed_of_closed_subst this,\n          have h63: closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)).to_vc,\n          from to_vc_closed_of_closed this,\n\n          have FV (prop.post g gx) \u2286 FV Q\u2082 \u222a {g, gx}, from (\n            assume x: var,\n            assume : x \u2208 FV (prop.post g gx),\n            or.elim (free_in_prop.post.inv this) (\n              assume : free_in_term x g,\n              have x = g, from free_in_term.var.inv this,\n              have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inl this),\n              show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n            ) (\n              assume : free_in_term x gx,\n              have x = gx, from free_in_term.var.inv this,\n              have x \u2208 {g, gx}, from set.two_elems_mem.inv (or.inr this),\n              show x \u2208 FV Q\u2082 \u222a {g, gx}, from set.mem_union_right (FV Q\u2082) this\n            )\n          ),\n          have FV (prop.post g gx) \u2286 (\u03c3\u2082[g\u21a6vf][gx\u21a6v]).dom, from h62.symm \u25b8 this,\n          have closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx)), from prop.closed_of_closed_subst this,\n          have h64: closed (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx)).to_vc,\n          from to_vc_closed_of_closed this,\n\n          have h7: \u22a8 vc.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx)).to_vc\n                                (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)).to_vc,\n          from valid.implies.mp (\n            assume h8: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx)).to_vc,\n            have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx)\n               = prop.post (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from prop.subst_env.post,\n            have h9: \u22a8 (prop.post (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g)\n                                  (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx)).to_vc,\n            from this \u25b8 h8,\n\n            have (prop.post (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx)).to_vc\n                = vc.post (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            by unfold prop.to_vc,\n            have h10: \u22a8 vc.post (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from this \u25b8 h9,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) g = term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) g),\n            by unfold term.subst_env,\n            have \u22a8 vc.post (term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) g)) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx),\n            from this \u25b8 h10,\n            have h11: \u22a8 vc.post (term.subst gx v vf) (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx), from g_subst \u25b8 this,\n            have term.subst gx v (term.value vf) = vf, by unfold term.subst,\n            have h12: \u22a8 vc.post vf (term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx), from this \u25b8 h11,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) gx = term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) gx),\n            by unfold term.subst_env,\n            have h13: \u22a8 vc.post vf (term.subst gx v (term.subst_env (\u03c3\u2082[g\u21a6vf]) gx)), from this \u25b8 h12,\n            have \u00ac(gx = g \u2228 gx \u2208 \u03c3\u2082), from not_or_distrib.mpr \u27e8g_neq_gx.symm, gx_not_free_in_\u03c3\u2082\u27e9,\n            have gx \u2209 (\u03c3\u2082[g\u21a6vf]), from (mt env.contains.inv) this,\n            have (\u03c3\u2082[g\u21a6vf]) gx = none, from env.contains_apply_equiv.left.mpr this,\n            have term.subst_env (\u03c3\u2082[g\u21a6vf]) gx = gx, from term.subst_env.var.left.mp this,\n            have h14: \u22a8 vc.post vf (term.subst gx v gx), from this \u25b8 h13,\n            have term.subst gx v gx = v, from term.subst.var.same,\n            have \u22a8 vc.post vf v, from this \u25b8 h14,\n            have (\u03c3\u2082[g\u21a6vf][gx\u21a6v] \u22a8 (Q\u2083 (term.app g gx)).to_vc \u22c0 S.to_prop.to_vc),\n            from valid.post.mpr \u03c3\u2082_verified func_verified this,\n            have h15: (\u03c3\u2082[g\u21a6vf][gx\u21a6v] \u22a8 (Q\u2083 (term.app g gx) \u22c0 S.to_prop).to_vc),\n            from valid_env.to_vc_and (valid_env.and.elim this).left (valid_env.and.elim this).right,\n            have vc.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop).to_vc\n              = (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)).to_vc,\n            from subst_env_distrib_to_vc,\n            show \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)).to_vc,\n            from this \u25b8 h15\n          ),\n          have h8: \u22a8 (prop.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx))\n                                   (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop))).to_vc,\n          from valid.to_vc_implies.mp h7,\n          have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies (prop.post g gx) (Q\u2083 (term.app g gx) \u22c0 S.to_prop))\n             = prop.implies (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.post g gx))\n                            (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (Q\u2083 (term.app g gx) \u22c0 S.to_prop)),\n          from prop.subst_env.implies,\n          show \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies (prop.post g gx)\n                                                      (Q\u2083 (term.app g gx) \u22c0 S.to_prop))).to_vc,\n          from this.symm \u25b8 h8\n        ),\n\n        have h7: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies R.to_prop (prop.pre g gx)) \u22c0\n                    prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies (prop.post g gx)\n                                                                   (Q\u2083 (term.app g gx) \u22c0 S.to_prop))).to_vc,\n        from valid.to_vc_and h5 h6,\n        have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) forallp\n           = (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies R.to_prop (prop.pre g gx)) \u22c0\n             prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) (prop.implies (prop.post g gx) (Q\u2083 (term.app g gx) \u22c0 S.to_prop))),\n        from prop.subst_env.and,\n        have h8: \u22a8 (prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) forallp).to_vc, from this.symm \u25b8 h7,\n        have prop.subst_env (\u03c3\u2082[g\u21a6vf][gx\u21a6v]) forallp = prop.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp),\n        by unfold prop.subst_env,\n        have h9: \u22a8 (prop.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp)).to_vc, from this \u25b8 h8,\n        have vc.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp).to_vc\n           = (prop.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp)).to_vc,\n        from subst_distrib_to_vc,\n        show \u22a8 vc.subst gx v (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp).to_vc, from this.symm \u25b8 h9\n      ),\n\n      have h5: \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.forallc gx forallp)), from (\n        have h6: \u22a8 vc.univ gx (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp).to_vc, from valid.univ.mp h4,\n        have prop.to_vc (prop.forallc gx (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp))\n           = vc.univ gx (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp).to_vc,\n        by unfold prop.to_vc,\n        have h7: \u22a8 prop.to_vc (prop.forallc gx (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp)), from this.symm \u25b8 h6,\n        have \u00ac(gx = g \u2228 gx \u2208 \u03c3\u2082), from not_or_distrib.mpr \u27e8g_neq_gx.symm, gx_not_free_in_\u03c3\u2082\u27e9,\n        have gx \u2209 (\u03c3\u2082[g\u21a6vf]), from (mt env.contains.inv) this,\n        have (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.forallc gx forallp)\n            = prop.forallc gx (prop.subst_env (\u03c3\u2082[g\u21a6vf]) forallp)),\n        from prop.subst_env.forallc_not_in this,\n        show \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.forallc gx forallp)), from this.symm \u25b8 h7\n      ),\n\n      have h7: \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g) \u22c0\n                             prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.forallc gx forallp)),\n      from valid.to_vc_and h2 h5,\n      have prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g \u22c0 prop.forallc gx forallp)\n         = (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g) \u22c0 prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.forallc gx forallp)),\n      from prop.subst_env.and,\n      have h8: \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (term.unop unop.isFunc g \u22c0 prop.forallc gx forallp)),\n      from this.symm \u25b8 h7,\n      have prop.func g gx R.to_prop (Q\u2083 (term.app g gx) \u22c0 S.to_prop)\n         = (term.unop unop.isFunc g \u22c0 prop.forallc gx forallp),\n      by unfold prop.func,\n      have \u22a8 prop.to_vc (prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.func g gx R (Q\u2083 (term.app g gx) \u22c0 S))),\n      from this.symm \u25b8 h8,\n      have h9: \u22a8 prop.to_vc pfunc, from this,\n\n      have h10: (\u2200x, x \u2209 FV pfunc), from (\n        have \u22a9 (\u03c3\u2082[g\u21a6vf]) : Q\u2082\n          \u22c0 (g \u2261 (vf))\n          \u22c0 prop.subst_env (\u03c3\u2082[g\u21a6vf]) (prop.func g gx R (Q\u2083 (term.app g gx) \u22c0 S)),\n        from env.dvcgen.func g_not_free_in_\u03c3\u2082 g_not_free_in_\u03c3\u2082 gx_not_free_in_\u03c3\u2082 g_neq_gx\n             \u03c3\u2082_verified \u03c3\u2082_verified gx_free_in_R R_fv S_fv func_verified S_valid,\n        prop_func_closed this\n      ),\n\n      have h11: (\u2200x, x \u2209 FV pfunc.to_vc), from (\n        assume x: var,\n        assume : x \u2208 FV pfunc.to_vc,\n        have x \u2208 FV pfunc, from set.mem_of_mem_of_subset this free_in_prop_of_free_in_to_vc,\n        show \u00abfalse\u00bb, from (h10 x) this\n      ),\n\n      have vc.subst_env (\u03c3\u2081[f\u21a6vf]) pfunc.to_vc = pfunc.to_vc,\n      from unchanged_of_subst_env_nonfree_vc h11 (\u03c3\u2081[f\u21a6vf]),\n      have (\u03c3\u2081[f\u21a6vf]) \u22a8 pfunc.to_vc, from this.symm \u25b8 h9,\n\n      have (\u03c3\u2081[f\u21a6vf]) \u22a8 ((prop.term (f \u2261 vf)) \u22c0 pfunc).to_vc,\n      from valid_env.to_vc_and h1b this,\n      show (\u03c3\u2081[f\u21a6vf]) \u22a8 (Q\u2081 \u22c0 (f \u2261 vf) \u22c0 pfunc).to_vc,\n      from valid_env.to_vc_and h1a this\n    )}\n  end\n\nlemma consequent_of_pre_P {R: spec} {\u03c3: env} {P Q: prop}:\n      (\u22a9 \u03c3: P) \u2192 closed_subst \u03c3 R.to_prop \u2192 (\u03c3 \u22a8 R.to_prop.to_vc) \u2192\n     closed_subst \u03c3 Q \u2192 \u2983 prop.implies (R \u22c0 P) Q \u2984 \u2192 \u03c3 \u22a8 Q.to_vc :=\n  assume env_verified: (\u22a9 \u03c3 : P),\n  assume R_closed: closed_subst \u03c3 R.to_prop,\n  assume R_valid: (\u03c3 \u22a8 R.to_prop.to_vc),\n  assume Q_closed: closed_subst \u03c3 Q,\n  assume vc_valid: \u2983 prop.implies (R \u22c0 P) Q \u2984,\n\n  have closed_subst \u03c3 P, from env_translation_closed_subst env_verified,\n  have closed_subst \u03c3 (\u2191R \u22c0 P), from prop.closed_subst.and R_closed this,\n  have closed_subst \u03c3 (prop.implies (\u2191R \u22c0 P) Q), from prop.closed_subst.implies this Q_closed,\n  have impl: \u03c3 \u22a8 (prop.implies (\u2191R \u22c0 P) Q).to_vc, from vc_valid \u03c3 this,\n  have (prop.implies (\u2191R \u22c0 P) Q).to_vc = vc.implies (\u2191R \u22c0 P).to_vc Q.to_vc,\n  by { unfold prop.implies, unfold vc.implies, unfold prop.to_vc, congr },\n  have impl2: \u03c3 \u22a8 vc.implies (\u2191R \u22c0 P).to_vc Q.to_vc, from this \u25b8 impl,\n  have \u03c3 \u22a8 P.to_vc, from env_translation_valid env_verified,\n  have \u03c3 \u22a8 (\u2191R \u22c0 P).to_vc, from valid_env.to_vc_and R_valid this,\n  show \u03c3 \u22a8 Q.to_vc, from valid_env.mp impl2 this\n\nlemma consequent_of_pre_P_call {R: spec} {\u03c3: env} {P Q: prop} {x: var}:\n      (\u22a9 \u03c3: P) \u2192 closed_subst \u03c3 R.to_prop \u2192 (\u03c3 \u22a8 R.to_prop.to_vc) \u2192 x \u2208 \u03c3 \u2192\n     closed_subst \u03c3 Q \u2192 \u2983 prop.implies ((R \u22c0 P) \u22c0 prop.call x) Q \u2984 \u2192 \u03c3 \u22a8 Q.to_vc :=\n  assume env_verified: (\u22a9 \u03c3 : P),\n  assume R_closed: closed_subst \u03c3 R.to_prop,\n  assume R_valid: (\u03c3 \u22a8 R.to_prop.to_vc),\n  assume env_has_x: x \u2208 \u03c3,\n  assume Q_closed: closed_subst \u03c3 Q,\n  assume vc_valid: \u2983 prop.implies ((R \u22c0 P) \u22c0 prop.call x) Q \u2984,\n\n  have closed_subst \u03c3 P, from env_translation_closed_subst env_verified,\n  have h1: closed_subst \u03c3 (\u2191R \u22c0 P), from prop.closed_subst.and R_closed this,\n  have closed_subst \u03c3 (prop.call x), by begin\n    assume y: var,\n    assume h1,\n    have h2, from free_in_prop.call.inv h1,\n    have h3: (y = x), from free_in_term.var.inv h2,\n    rw[\u2190h3] at env_has_x,\n    from env_has_x\n  end,\n  have closed_subst \u03c3 ((\u2191R \u22c0 P) \u22c0 prop.call x), from prop.closed_subst.and h1 this,\n  have closed_subst \u03c3 (prop.implies ((\u2191R \u22c0 P) \u22c0 prop.call x) Q),\n  from prop.closed_subst.implies this Q_closed,\n  have impl: \u03c3 \u22a8 (prop.implies ((\u2191R \u22c0 P) \u22c0 prop.call x) Q).to_vc, from vc_valid \u03c3 this,\n  have (prop.implies (\u2191R \u22c0 P) Q).to_vc = vc.implies (\u2191R \u22c0 P).to_vc Q.to_vc,\n  by { unfold prop.implies, unfold vc.implies, unfold prop.to_vc, congr },\n  have impl2: \u03c3 \u22a8 vc.implies ((\u2191R \u22c0 P) \u22c0 prop.call x).to_vc Q.to_vc, from this \u25b8 impl,\n  have \u03c3 \u22a8 P.to_vc, from env_translation_valid env_verified,\n  have h2: \u03c3 \u22a8 (\u2191R \u22c0 P).to_vc, from valid_env.to_vc_and R_valid this,\n  have h3: \u03c3 \u22a8 value.true, from valid_env.true,\n  have prop.to_vc (prop.call x) = value.true, by { unfold prop.to_vc, congr },\n  have \u03c3 \u22a8 (prop.call x).to_vc, from this.symm \u25b8 h3,\n  have \u03c3 \u22a8 ((\u2191R \u22c0 P) \u22c0 prop.call x).to_vc, from valid_env.to_vc_and h2 this,\n  show \u03c3 \u22a8 Q.to_vc, from valid_env.mp impl2 this\n\nlemma vc.implies.self {\u03c3: env} {P: vc}: \u03c3 \u22a8 vc.implies P P :=\n  begin\n    apply valid_env.mpr,\n    from id\n  end\n\nlemma vc.implies.and_left_intro {P Q: prop} {\u03c3: env}:\n      ((\u03c3 \u22a8 Q.to_vc) \u2192 \u03c3 \u22a8 P.to_vc) \u2192 \u03c3 \u22a8 vc.implies Q.to_vc (P \u22c0 Q).to_vc :=\n  begin\n    assume h1,\n    apply valid_env.mpr,\n    assume h2,\n    apply valid_env.to_vc_and,\n    from h1 h2,\n    from h2\n  end\n\nlemma vc.implies.and_right_intro {P Q: prop} {\u03c3: env}:\n      ((\u03c3 \u22a8 P.to_vc) \u2192 \u03c3 \u22a8 Q.to_vc) \u2192 \u03c3 \u22a8 vc.implies P.to_vc (P \u22c0 Q).to_vc :=\n  begin\n    assume h1,\n    apply valid_env.mpr,\n    assume h2,\n    apply valid_env.to_vc_and,\n    from h2,\n    from h1 h2\n  end\n\nlemma vc.implies.and_intro {\u03c3: env} {P P' Q Q': prop}:\n      (\u03c3 \u22a8 vc.implies P.to_vc P'.to_vc) \u2192 ((\u03c3 \u22a8 P.to_vc) \u2192 \u03c3 \u22a8 vc.implies Q.to_vc Q'.to_vc) \u2192\n      (\u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (P' \u22c0 Q').to_vc) :=\n  begin\n    assume h1,\n    assume h2,\n    apply valid_env.mpr,\n    assume h3,\n    have h4, from (valid_env.to_vc_and.elim h3).left,\n    have h5, from (valid_env.to_vc_and.elim h3).right,\n    have h6, from h2 h4,\n    apply valid_env.to_vc_and,\n    from valid_env.mp h1 h4,\n    from valid_env.mp h6 h5\n  end\n\nlemma vc.implies.and_symm {\u03c3: env} {P Q: prop}: (\u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (Q \u22c0 P).to_vc) :=\n  begin\n    apply valid_env.mpr,\n    assume h1,\n    apply valid_env.to_vc_and,\n    from (valid_env.to_vc_and.elim h1).right,\n    from (valid_env.to_vc_and.elim h1).left\n  end\n\nlemma vc.implies.and_elim_left {\u03c3: env} {P\u2081 P\u2082 P\u2083: prop}:\n      (\u03c3 \u22a8 vc.implies P\u2081.to_vc (P\u2082 \u22c0 P\u2083).to_vc) \u2192 (\u03c3 \u22a8 vc.implies P\u2081.to_vc P\u2082.to_vc) :=\n  begin\n    assume h1,\n    apply valid_env.mpr,\n    assume h2,\n    have h3, from valid_env.mp h1 h2,\n    from (valid_env.to_vc_and.elim h3).left\n  end\n\nlemma vc.implies.and_assoc {\u03c3: env} {P\u2081 P\u2082 P\u2083: prop}:\n      \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082 \u22c0 P\u2083).to_vc ((P\u2081 \u22c0 P\u2082) \u22c0 P\u2083).to_vc :=\n  begin\n    apply valid_env.mpr,\n    assume h1,\n    have h2, from (valid_env.to_vc_and.elim h1).right,\n\n    apply valid_env.to_vc_and,\n    apply valid_env.to_vc_and,\n    from (valid_env.to_vc_and.elim h1).left,\n    from (valid_env.to_vc_and.elim h2).left,\n    from (valid_env.to_vc_and.elim h2).right\n  end\n\nlemma vc.implies.subst {\u03c3: env} {x: var} {v: value} {P: prop}:\n      (\u03c3 x = v) \u2192 (\u03c3 \u22a8 vc.implies (prop.subst x v P).to_vc P.to_vc) :=\n  begin\n    assume h1,\n    apply valid_env.mpr,\n    assume h2,\n    have h3: (vc.subst x v (prop.to_vc P) = prop.to_vc (prop.subst x v P)),\n    from subst_distrib_to_vc,\n    rw[h3.symm] at h2,\n    have h4: (vc.subst_env \u03c3 (vc.subst x v (prop.to_vc P)) = vc.subst_env \u03c3 (prop.to_vc P)),\n    from vc.subst_env_inner h1,\n    rw[h4] at h2,\n    from h2\n  end\n\nlemma valid_with_additional_var {P: vc} {x: var} {v: value} {\u03c3: env}:\n      (\u03c3 \u22a8 P) \u2192 ((\u03c3[x\u21a6v]) \u22a8 P) :=\n  begin\n    assume h1,\n\n    by_cases (x \u2208 \u03c3) with h4,\n    unfold vc.subst_env,\n    have h7: x \u2209 FV (vc.subst_env \u03c3 P), from vc.not_free_of_subst_env h4,\n    have h8: (vc.subst x v (vc.subst_env \u03c3 P) = vc.subst_env \u03c3 P),\n    from unchanged_of_subst_nonfree_vc h7,\n    rw[\u2190h8] at h1,\n    from h1,\n\n    by_cases (free_in_vc x P) with h5,\n\n    have h8: x \u2208 FV (vc.subst_env \u03c3 P),\n    from vc.free_of_subst_env h5 h4,\n    have h9, from valid.univ.free \u27e8h8, h1\u27e9,\n    have h10, from valid.univ.mpr h9 v,\n\n    have h11: (vc.substt x \u2191v (vc.subst_env \u03c3 P) = vc.subst x v (vc.subst_env \u03c3 P)),\n    from vc.substt_value_eq_subst,\n    rw[h11] at h10,\n    unfold vc.subst_env,\n    from h10,\n\n    have h9: x \u2209 FV (vc.subst_env \u03c3 P), by begin\n      assume h10,\n      have h11, from vc.free_of_free_subst_env h10,\n      contradiction\n    end,\n    have h10: (vc.substt x v (vc.subst_env \u03c3 P) = vc.subst_env \u03c3 P),\n    from unchanged_of_substt_nonfree_vc h9,\n    unfold vc.subst_env,\n    have h11: (vc.substt x \u2191v (vc.subst_env \u03c3 P) = vc.subst x v (vc.subst_env \u03c3 P)),\n    from vc.substt_value_eq_subst,\n    have h12: (vc.subst_env \u03c3 P = vc.subst x v (vc.subst_env \u03c3 P)), from eq.trans h10.symm h11,\n    rw[\u2190h12],\n    from h1\n  end\n\nlemma valid_with_additional_vars {P: vc} {\u03c3: env}: (\u22a8 P) \u2192 (\u03c3 \u22a8 P) :=\n  begin\n    assume h1,\n    \n    induction \u03c3 with \u03c3' x v ih,\n\n    show env.empty \u22a8 P, by begin\n      unfold vc.subst_env,\n      from h1\n    end,\n\n    show (\u03c3'[x\u21a6v]) \u22a8 P, by begin\n      from valid_with_additional_var ih\n    end\n  end\n\nlemma vc.implies.exis {\u03c3: env} {x: var} {P: prop}:\n      \u03c3 \u22a8 vc.implies P.to_vc (prop.exis x P).to_vc :=\n  begin\n    apply valid_env.nmt,\n    apply valid_env.mpr,\n    assume h1,\n    apply valid_env.not.mp,\n    by_contradiction h2,\n    unfold prop.to_vc at h1,\n    have h3, from valid_env.not_not.mp h1,\n    have h4: (vc.subst_env \u03c3 (vc.univ x (vc.not (prop.to_vc P)))\n           = vc.univ x (vc.subst_env (\u03c3.without x) (vc.not (prop.to_vc P)))),\n    from vc.subst_env.univ,\n    rw[h4] at h3,\n\n    by_cases (x \u2208 \u03c3) with h4,\n    have h5, from env.contains_apply_equiv.right.mpr h4,\n    cases h5 with v h6,\n    have h7, from valid.univ.mpr h3 v,\n    have h8: (vc.substt x \u2191v (vc.subst_env (env.without \u03c3 x) (vc.not (prop.to_vc P)))\n            = vc.subst x v (vc.subst_env (env.without \u03c3 x) (vc.not (prop.to_vc P)))),\n    from vc.substt_value_eq_subst,\n    rw[h8] at h7,\n    have h9: (vc.subst x v (vc.subst_env (env.without \u03c3 x) (vc.not (prop.to_vc P)))\n     = vc.subst_env ((env.without \u03c3 x)[x\u21a6v]) (vc.not (prop.to_vc P))),\n    by unfold vc.subst_env,\n    rw[h9] at h7,\n    have h10: (vc.subst_env ((env.without \u03c3 x)[x\u21a6v]) (vc.not (prop.to_vc P)) = vc.subst_env \u03c3 (vc.not (prop.to_vc P))),\n    from vc.subst_env_with_without_equivalent h6,\n    rw[h10] at h7,\n    have h11, from valid_env.not.mpr h7,\n    contradiction,\n\n    have h5: (env.without \u03c3 x = \u03c3), from env.without_nonexisting h4,\n    rw[h5] at h3,\n    by_cases (free_in_vc x P.to_vc) with h6,\n\n    have h7: x \u2208 FV P.to_vc, from h6,\n    have h8: x \u2208 FV (vc.subst_env \u03c3 P.to_vc),\n    from vc.free_of_subst_env h7 h4,\n    have h9, from valid.univ.free \u27e8h8, h2\u27e9,\n    have h10, from valid.univ.mpr h3 value.true,\n    have h11, from valid.univ.mpr h9 value.true,\n    have h12: (vc.subst_env \u03c3 (vc.substt x value.true (vc.not (prop.to_vc P)))\n             = vc.substt x value.true (vc.subst_env \u03c3 (vc.not (prop.to_vc P)))),\n    from vc.substt_env.order (\u03bbx, free_in_term.value.inv) h4,\n    rw[h12.symm] at h10,\n    have h13: (vc.subst_env \u03c3 (vc.substt x value.true (prop.to_vc P))\n             = vc.substt x value.true (vc.subst_env \u03c3 (prop.to_vc P))),\n    from vc.substt_env.order (\u03bbx, free_in_term.value.inv) h4,\n    rw[h13.symm] at h11,\n    unfold vc.substt at h10,\n    have h14, from valid_env.not.mpr h10,\n    contradiction,\n\n    have h7, from valid.univ.mpr h3 value.true,\n    have h8: x \u2209 FV (vc.not P.to_vc), by begin\n      assume h9,\n      have h10, from free_in_vc.not.inv h9,\n      contradiction\n    end,\n    have h9: x \u2209 FV (vc.subst_env \u03c3 (vc.not (prop.to_vc P))), by begin\n      assume h10,\n      have h11, from vc.free_of_free_subst_env h10,\n      contradiction\n    end,\n    have h10: (vc.substt x \u2191value.true (vc.subst_env \u03c3 (vc.not (prop.to_vc P)))\n            = (vc.subst_env \u03c3 (vc.not (prop.to_vc P)))),\n    from unchanged_of_substt_nonfree_vc h9,\n    rw[h10] at h7,\n    have h11, from valid_env.not.mpr h7,\n    contradiction\n  end\n\nlemma vc.implies.same_right {\u03c3: env} {P P' Q: prop}:\n  ((\u03c3 \u22a8 Q.to_vc) \u2192 \u03c3 \u22a8 vc.implies P.to_vc P'.to_vc) \u2192 (\u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (P' \u22c0 Q).to_vc) :=\n  begin\n    assume h1: (\u03c3 \u22a8 Q.to_vc) \u2192 \u03c3 \u22a8 vc.implies P.to_vc P'.to_vc,\n    apply valid_env.mpr,\n    assume h2: \u03c3 \u22a8 (P \u22c0 Q).to_vc,\n    apply valid_env.to_vc_and,\n    have h3, from (valid_env.to_vc_and.elim h2).left,\n    from valid_env.mp (h1 (valid_env.to_vc_and.elim h2).right) h3,\n    from (valid_env.to_vc_and.elim h2).right\n  end\n\nlemma vc.implies.and_assoc.symm {P\u2081 P\u2082 P\u2083: prop} {\u03c3: env}:\n      \u03c3 \u22a8 vc.implies ((P\u2081 \u22c0 P\u2082) \u22c0 P\u2083).to_vc (P\u2081 \u22c0 P\u2082 \u22c0 P\u2083).to_vc :=\n  have h1: \u03c3 \u22a8 vc.implies ((P\u2081 \u22c0 P\u2082) \u22c0 P\u2083).to_vc (P\u2083 \u22c0 P\u2081 \u22c0 P\u2082).to_vc, from vc.implies.and_symm,\n  have h2: \u03c3 \u22a8 vc.implies (P\u2083 \u22c0 P\u2081 \u22c0 P\u2082).to_vc ((P\u2083 \u22c0 P\u2081) \u22c0 P\u2082).to_vc, from vc.implies.and_assoc,\n  have h3: \u03c3 \u22a8 vc.implies ((P\u2083 \u22c0 P\u2081) \u22c0 P\u2082).to_vc (P\u2082 \u22c0 P\u2083 \u22c0 P\u2081).to_vc, from vc.implies.and_symm,\n  have h4: \u03c3 \u22a8 vc.implies (P\u2082 \u22c0 P\u2083 \u22c0 P\u2081).to_vc ((P\u2082 \u22c0 P\u2083) \u22c0 P\u2081).to_vc, from vc.implies.and_assoc,\n  have h5: \u03c3 \u22a8 vc.implies ((P\u2082 \u22c0 P\u2083) \u22c0 P\u2081).to_vc (P\u2081 \u22c0 P\u2082 \u22c0 P\u2083).to_vc , from vc.implies.and_symm,\n  show \u03c3 \u22a8 vc.implies ((P\u2081 \u22c0 P\u2082) \u22c0 P\u2083).to_vc (P\u2081 \u22c0 P\u2082 \u22c0 P\u2083).to_vc,\n  from vc.implies.trans h1 (vc.implies.trans h2 (vc.implies.trans h3 (vc.implies.trans h4 h5)))\n\nlemma vc.implies.shuffle {P Q R S: prop} {\u03c3: env}:\n      \u03c3 \u22a8 vc.implies (P \u22c0 Q \u22c0 R \u22c0 S).to_vc ((P \u22c0 Q \u22c0 R) \u22c0 S).to_vc :=\n  have h1: \u03c3 \u22a8 vc.implies (P \u22c0 Q \u22c0 R \u22c0 S).to_vc ((Q \u22c0 R \u22c0 S) \u22c0 P).to_vc, from vc.implies.and_symm,\n  have h2: \u03c3 \u22a8 vc.implies ((Q \u22c0 R \u22c0 S) \u22c0 P).to_vc (((Q \u22c0 R) \u22c0 S) \u22c0 P).to_vc,\n  from vc.implies.same_right (\u03bb_, vc.implies.and_assoc),\n  have h3: \u03c3 \u22a8 vc.implies  (((Q \u22c0 R) \u22c0 S) \u22c0 P).to_vc ((Q \u22c0 R) \u22c0 S \u22c0 P).to_vc, from vc.implies.and_assoc.symm,\n  have h4: \u03c3 \u22a8 vc.implies ((Q \u22c0 R) \u22c0 S \u22c0 P).to_vc ((S \u22c0 P) \u22c0 Q \u22c0 R).to_vc, from vc.implies.and_symm,\n  have h5: \u03c3 \u22a8 vc.implies ((S \u22c0 P) \u22c0 Q \u22c0 R).to_vc (S \u22c0 P \u22c0 Q \u22c0 R).to_vc, from vc.implies.and_assoc.symm,\n  have h6: \u03c3 \u22a8 vc.implies (S \u22c0 P \u22c0 Q \u22c0 R).to_vc ((P \u22c0 Q \u22c0 R) \u22c0 S).to_vc, from vc.implies.and_symm,\n  show \u03c3 \u22a8 vc.implies  (P \u22c0 Q \u22c0 R \u22c0 S).to_vc ((P \u22c0 Q \u22c0 R) \u22c0 S).to_vc,\n  from vc.implies.trans h1 (vc.implies.trans h2 (vc.implies.trans h3 (vc.implies.trans h4 (vc.implies.trans h5 h6))))\n\nlemma vc.implies.same_left {\u03c3: env} {P Q Q': prop}:\n      ((\u03c3 \u22a8 P.to_vc) \u2192 \u03c3 \u22a8 vc.implies Q.to_vc Q'.to_vc) \u2192 \u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (P \u22c0 Q').to_vc :=\n  assume h1: (\u03c3 \u22a8 P.to_vc) \u2192 \u03c3 \u22a8 vc.implies Q.to_vc Q'.to_vc,\n  have h2: \u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (Q \u22c0 P).to_vc, from vc.implies.and_symm,\n  have h3: \u03c3 \u22a8 vc.implies (Q \u22c0 P).to_vc (Q' \u22c0 P).to_vc, from vc.implies.same_right h1,\n  have h4: \u03c3 \u22a8 vc.implies (Q' \u22c0 P).to_vc (P \u22c0 Q').to_vc, from vc.implies.and_symm,\n  show \u03c3 \u22a8 vc.implies (P \u22c0 Q).to_vc (P \u22c0 Q').to_vc,\n  from vc.implies.trans h2 (vc.implies.trans h3 h4)\n\nlemma vc.implies.and_elim_right {\u03c3: env} {P\u2081 P\u2082 P\u2083: prop}:\n      (\u03c3 \u22a8 vc.implies P\u2081.to_vc (P\u2082 \u22c0 P\u2083).to_vc) \u2192 \u03c3 \u22a8 vc.implies P\u2081.to_vc P\u2083.to_vc :=\n  assume h1: \u03c3 \u22a8 vc.implies P\u2081.to_vc (P\u2082 \u22c0 P\u2083).to_vc,\n  have h2: \u03c3 \u22a8 vc.implies (P\u2082 \u22c0 P\u2083).to_vc (P\u2083 \u22c0 P\u2082).to_vc, from vc.implies.and_symm,\n  have h3: \u03c3 \u22a8 vc.implies P\u2081.to_vc (P\u2083 \u22c0 P\u2082).to_vc, from vc.implies.trans h1 h2,\n  show \u03c3 \u22a8 vc.implies P\u2081.to_vc P\u2083.to_vc, from vc.implies.and_elim_left h3\n\nlemma vc.implies.left_elim {P\u2081 P\u2082 P\u2083: prop} {\u03c3: env}:\n      ((\u03c3 \u22a8 P\u2081.to_vc) \u2192 \u03c3 \u22a8 vc.implies P\u2082.to_vc P\u2083.to_vc) \u2192 \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2083.to_vc :=\n  assume h1: (\u03c3 \u22a8 P\u2081.to_vc) \u2192 \u03c3 \u22a8 vc.implies P\u2082.to_vc P\u2083.to_vc,\n  have h2: \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc (P\u2081 \u22c0 P\u2083).to_vc, from vc.implies.same_left h1,\n  show \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2083.to_vc, from vc.implies.and_elim_right h2\n\nlemma vc.implies.right_elim {P\u2081 P\u2082 P\u2083: prop} {\u03c3: env}:\n      ((\u03c3 \u22a8 P\u2082.to_vc) \u2192 \u03c3 \u22a8 vc.implies P\u2081.to_vc P\u2083.to_vc) \u2192 \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2083.to_vc :=\n  assume h1: (\u03c3 \u22a8 P\u2082.to_vc) \u2192 \u03c3 \u22a8 vc.implies P\u2081.to_vc P\u2083.to_vc,\n  have h2: \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc (P\u2083 \u22c0 P\u2082).to_vc, from vc.implies.same_right h1,\n  show \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2083.to_vc, from vc.implies.and_elim_left h2\n\nlemma vc.implies.of_and_left {P\u2081 P\u2082: prop} {\u03c3: env}: \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2081.to_vc :=\n  have \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc (P\u2081 \u22c0 P\u2082).to_vc, from vc.implies.self,\n  show \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2081.to_vc, from vc.implies.and_elim_left this\n\nlemma vc.implies.of_and_right {P\u2081 P\u2082: prop} {\u03c3: env}: \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2082.to_vc :=\n  have h1: \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc (P\u2082 \u22c0 P\u2081).to_vc, from vc.implies.and_symm,\n  have h2: \u03c3 \u22a8 vc.implies (P\u2082 \u22c0 P\u2081).to_vc P\u2082.to_vc, from vc.implies.of_and_left,\n  show \u03c3 \u22a8 vc.implies (P\u2081 \u22c0 P\u2082).to_vc P\u2082.to_vc, from vc.implies.trans h1 h2\n\nlemma vc.implies.equiv_subst {\u03c3\u2081 \u03c3\u2082: env} {P: prop}:\n  (\u2200y, y \u2208 \u03c3\u2081 \u2192 (\u03c3\u2081 y = \u03c3\u2082 y)) \u2192 \u03c3\u2082 \u22a8 vc.implies (prop.subst_env \u03c3\u2081 P).to_vc P.to_vc :=\n  begin\n    assume env_equiv,\n    \n    induction \u03c3\u2081 with \u03c3' x v ih,\n\n    show \u03c3\u2082 \u22a8 vc.implies (prop.to_vc (prop.subst_env env.empty P)) (prop.to_vc P), by begin\n      unfold prop.subst_env,\n      from vc.implies.self\n    end,\n\n    unfold prop.subst_env,\n    have h2: \u03c3\u2082 \u22a8 vc.implies (prop.subst x v (prop.subst_env \u03c3' P)).to_vc (prop.subst_env \u03c3' P).to_vc, by begin\n      by_cases (x \u2208 \u03c3') with h,\n\n      have h3: x \u2209 FV (prop.subst_env \u03c3' P), from prop.not_free_of_subst_env h,\n      have h4: (prop.subst x v (prop.subst_env \u03c3' P) = prop.subst_env \u03c3' P),\n      from unchanged_of_subst_nonfree_prop h3,\n      have h5: \u03c3\u2082 \u22a8 vc.implies (prop.subst_env \u03c3' P).to_vc (prop.subst_env \u03c3' P).to_vc, from vc.implies.self,\n      from h4.symm \u25b8 h5,\n\n      have h2, from env_equiv x env.contains.same,\n      have h3: ((\u03c3'[x\u21a6v]) x = v), from env.apply_of_contains h,\n      have h4: (\u03c3\u2082 x = v), from eq.trans h2.symm h3,\n      show \u03c3\u2082 \u22a8 vc.implies (prop.subst x v (prop.subst_env \u03c3' P)).to_vc (prop.subst_env \u03c3' P).to_vc,\n      from vc.implies.subst h4\n    end,\n    have h3: (\u2200 (y : var), y \u2208 \u03c3' \u2192 (\u03c3' y = \u03c3\u2082 y)), by begin\n      assume y,\n      assume h3,\n      have h4: y \u2208 (\u03c3'[x\u21a6v]), from env.contains.rest h3,\n      have h5, from env_equiv y h4,\n      have h6: (\u2203 (v : value), env.apply \u03c3' y = some v), from env.contains_apply_equiv.right.mpr h3,\n      have h7, from option.is_some_iff_exists.mpr h6,\n      have h8, from option.some_iff_not_none.mp h7,\n      have h9: (x \u2260 y \u2228 \u00ac (option.is_none (env.apply \u03c3' y))), from or.inr h8,\n      have h10: \u00ac (x = y \u2227 (option.is_none (env.apply \u03c3' y))), from not_and_distrib.mpr h9,\n      have h11: (env.apply (\u03c3'[x\u21a6v]) y = (\u03c3' y)), by { unfold env.apply, simp[h10], refl },\n      from eq.trans h11.symm h5\n    end,\n    have h4, from ih h3,\n    from vc.implies.trans h2 h4\n  end\n\nlemma valid_env.equiv_env {\u03c3\u2081 \u03c3\u2082: env} {P: prop}: (\u2200y, y \u2208 \u03c3\u2081 \u2192 (\u03c3\u2081 y = \u03c3\u2082 y)) \u2192 (\u03c3\u2081 \u22a8 P.to_vc) \u2192 \u03c3\u2082 \u22a8 P.to_vc :=\n  begin\n    assume h1,\n    have h2: \u03c3\u2082 \u22a8 vc.implies (prop.subst_env \u03c3\u2081 P).to_vc P.to_vc, from vc.implies.equiv_subst h1,\n    have h3, from valid_env.mp h2,\n    assume h4,\n    have h5: (\u03c3\u2082 \u22a8 prop.to_vc (prop.subst_env \u03c3\u2081 P)), by begin\n      have h6: (vc.subst_env \u03c3\u2081 (prop.to_vc P) = prop.to_vc (prop.subst_env \u03c3\u2081 P)),\n      from subst_env_distrib_to_vc,\n      rw[h6] at h4,\n      from valid_with_additional_vars h4\n    end,\n    from h3 h5\n  end\n", "meta": {"author": "levjj", "repo": "esverify-theory", "sha": "8565b123c87b0113f83553d7732cd6696c9b5807", "save_path": "github-repos/lean/levjj-esverify-theory", "path": "github-repos/lean/levjj-esverify-theory/esverify-theory-8565b123c87b0113f83553d7732cd6696c9b5807/src/logic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3670504754921644}}
{"text": "/-\nCopyright (c) 2023 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.constructions.repeat\n\n/-!\n# Repeated Computation Until a Condition\n\nThis file defines a construction `try_until oa p n` that repeats a computation until `p` holds.\nThe parameter `n` gives a bound on the number of runs (called \"gas\" in some formulations).\nThis solves the problem of an unbounded computation, by giving a finite computation depth.\nBecause this may not always produce a final result, we use an option type to represent failure.\n\nWe implement this as a mapping of `oracle_comp.repeat` for simplicity in deriving lemmas.\nThis means that the computation always \"runs\" `n` times even if a result is found before that.\nHowever it isn't clear that this is commonly a problem, so we take this approach for now.\n-/\n\nnamespace oracle_comp\n\nopen oracle_spec\n\nvariables {\u03b1 \u03b2 \u03b3 : Type} {spec spec' : oracle_spec}\n\n/-- Computation that repeats `oa` until `p` holds on the result, with at most `n` attempts. -/\ndef try_until (oa : oracle_comp spec \u03b1) (p : \u03b1 \u2192 Prop) [decidable_pred p]\n  (n : \u2115) : oracle_comp spec (option \u03b1) :=\n(\u03bb xs, (vector.to_list xs).find p) <$> (repeat oa n)\n\nvariables (oa : oracle_comp spec \u03b1) (p : \u03b1 \u2192 Prop) [decidable_pred p] (n : \u2115)\n\nlemma try_until_zero : oa.try_until p 0 =\n  (\u03bb xs, (vector.to_list xs).find p) <$> (return vector.nil) := rfl\n\nlemma try_until_succ : oa.try_until p n.succ =\n  (\u03bb xs, (vector.to_list xs).find p) <$> do {a \u2190 oa, as \u2190 oa.repeat n, return (a ::\u1d65 as)} := rfl\n\ninstance try_until.decidable [decidable_eq \u03b1] [decidable oa] : decidable (oa.try_until p n) :=\noracle_comp.decidable_map _ _\n\nsection support\n\n/-- Any positive result of `oa.try_until p n` will be some output of `oa`. -/\nlemma mem_support_of_some_mem_support_try_until (x : \u03b1)\n  (hx : some x \u2208 (oa.try_until p n).support) : x \u2208 oa.support :=\nbegin\n  simp_rw [try_until, support_map, set.mem_image, mem_support_repeat_iff_forall] at hx,\n  exact let \u27e8xs, hxs, hxs'\u27e9 := hx in hxs x (list.find_mem hxs')\nend\n\n/-- Any positive result of `oa.try_until p n` will satisfy the predicate `p`. -/\nlemma pos_of_some_mem_support_try_until (x : \u03b1)\n  (hx : some x \u2208 (oa.try_until p n).support) : p x :=\nbegin\n  simp_rw [try_until, support_map, set.mem_image, mem_support_repeat_iff_forall] at hx,\n  exact let \u27e8xs, hxs, hxs'\u27e9 := hx in list.find_some hxs',\nend\n\n/-- Running a computation zero times will never return a positive result. -/\n@[simp] lemma support_try_until_zero : (oa.try_until p 0).support = {none} :=\nby rw [try_until_zero, support_map, support_return, set.image_singleton,\n  vector.to_list_nil, list.find_nil]\n\nlemma mem_support_try_until_zero_iff (x : option \u03b1) : x \u2208 (oa.try_until p 0).support \u2194 x = none :=\nby rw [support_try_until_zero, set.mem_singleton_iff]\n\n/-- `oa.try_until p n` can fail to find a result iff there's an output `x` of `oa` with `\u00ac p x`. -/\nlemma none_mem_support_try_until_succ_iff :\n  none \u2208 (oa.try_until p n.succ).support \u2194 \u2203 x \u2208 oa.support, \u00ac p x :=\nbegin\n  simp only [try_until, mem_support_map_iff, list.find_eq_none],\n  exact \u27e8\u03bb h, let \u27e8xs, hxs, hp\u27e9 := h in \u27e8xs.head, mem_support_of_mem_of_support_repeat hxs\n    xs.head_mem, hp _ xs.head_mem\u27e9, \u03bb h, let \u27e8x, hx, hp\u27e9 := h in \u27e8vector.repeat x n.succ,\n      repeat_mem_support_repeat n.succ hx, \u03bb y hy, (list.eq_of_mem_repeat hy).symm \u25b8 hp\u27e9\u27e9\nend\n\nlemma none_not_mem_support_try_until (hx : \u2200 x \u2208 oa.support, p x) :\n  none \u2209 (oa.try_until p n.succ).support :=\nmt (none_mem_support_try_until_succ_iff oa p n).1 (by simpa only [not_exists, not_not] using hx)\n\n/-- The possible successful results of `oa.try_until p n` are outputs `x` of `oa` with `p x`. -/\nlemma some_mem_support_try_until_succ_iff (x : \u03b1) :\n  some x \u2208 (oa.try_until p n.succ).support \u2194 x \u2208 oa.support \u2227 p x :=\nbegin\n  simp only [try_until, mem_support_map_iff],\n  refine \u27e8\u03bb h, let \u27e8xs, hxs, hp\u27e9 := h in \u27e8mem_support_of_mem_of_support_repeat hxs\n    (list.find_mem hp), list.find_some hp\u27e9, \u03bb h, \u27e8vector.repeat x n.succ, repeat_mem_support_repeat\n      _ h.1, _\u27e9,\n\n      \u27e9,\n  simp only [vector.repeat, list.find_repeat, vector.to_list, h.2],\n  simp only [nat.succ_pos', and_self, if_true]\nend\n\nlemma some_mem_support_try_until_succ {x : \u03b1} (hx : x \u2208 oa.support) (h : p x) :\n  some x \u2208 (oa.try_until p n.succ).support :=\n(some_mem_support_try_until_succ_iff oa p n x).2 \u27e8hx, h\u27e9\n\n/-- If at least one result of `oa` doesn't satisfy `p` then the result of `oa.try_until p n.succ`\nis either `none` (in the case of failure) or `some x` for some output `x` of `oa` with `p x`. -/\nlemma support_try_until_succ_of_exists_neg (h : \u2203 x \u2208 oa.support, \u00ac p x) :\n  (oa.try_until p n.succ).support = insert none (option.some '' {x | x \u2208 oa.support \u2227 p x}) :=\nbegin\n  obtain \u27e8x, hx, hpx\u27e9 := h,\n  refine set.ext (\u03bb y, _),\n  rw [try_until, support_map, support_repeat_eq_forall],\n  cases y with y,\n  { simp only [set.mem_image, set.mem_set_of_eq, list.find_eq_none, set.mem_insert_iff,\n      eq_self_iff_true, and_false, exists_false, or_false, iff_true],\n    refine \u27e8vector.repeat x n.succ, \u03bb y hy, _, \u03bb y hy, _\u27e9;\n    { rw [vector.repeat, vector.to_list, list.mem_repeat_succ_iff] at hy,\n      simpa only [hy] } },\n  { simp only [set.mem_image, set.mem_set_of_eq, set.mem_insert_iff, exists_eq_right, false_or],\n    refine \u27e8\u03bb h, let \u27e8xs, hxs\u27e9 := h in \u27e8hxs.1 _ (list.find_mem hxs.2), list.find_some hxs.2\u27e9,\n    \u03bb h, \u27e8vector.repeat y n.succ, \u03bb z hz, _, list.find_cons_of_pos _ h.2\u27e9\u27e9,\n    rw [vector.repeat, vector.to_list, list.mem_repeat_succ_iff] at hz,\n    exact hz.symm \u25b8 h.1 }\nend\n\n/-- If all results of `oa` satisfy `p`, then `oa.try_until p n.succ` will just return `some x`,\nfor some `x \u2208 oa.support`(in particular the result of the first of the `n.succ` runs). -/\nlemma support_try_until_succ_of_forall_pos (hp : \u2200 x \u2208 oa.support, p x) :\n  (oa.try_until p n.succ).support = option.some '' oa.support :=\nbegin\n  refine set.ext (\u03bb y, \u27e8\u03bb h, _, \u03bb h, _\u27e9),\n  { cases y with y,\n    { exact false.elim (none_not_mem_support_try_until oa p n hp h) },\n    { exact \u27e8y, mem_support_of_some_mem_support_try_until oa p _ y h, rfl\u27e9 } },\n  { exact let \u27e8x, hx\u27e9 := h in hx.2 \u25b8 some_mem_support_try_until_succ oa p _ hx.1 (hp x hx.1) }\nend\n\nend support\n\n\nend oracle_comp", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/constructions/try_until.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.36701667734474835}}
{"text": "import set_theory.cardinal\nimport fol\nimport completeness\nimport Rings.Notation\nimport Rings.ToMathlib.fol\n\nnoncomputable theory\n\nnamespace fol\nopen_locale cardinal\n\nvariables {L : Language}\n\nlemma realize_lift_bounded_term (M : Structure L) {n} :\n  \u03a0 {l} (t : bounded_preterm L n l) (xs : dvector M n) (v : dvector M l) (a : M),\n  realize_bounded_term (dvector.cons a xs) (t \u2191 1) v =\n    realize_bounded_term xs t v\n| _ (bd_var k) _ _ _ :=\nbegin\n  have hle : 0 \u2264 (k : \u2115) := nat.zero_le _,\n  dsimp [lift_bounded_term],\n  rw [if_pos hle],\n  congr',\nend\n| _ (bd_func f) _ _ _ := by { dsimp only [lift_bounded_term], simp }\n| _ (bd_app t s) xs v _ :=\nbegin\n  dsimp only [lift_bounded_term],\n  simp only [lift_bounded_term_at, realize_bounded_term_bd_app],\n  have hinds := realize_lift_bounded_term s xs,\n  rw hinds,\n  have hind := realize_lift_bounded_term t xs\n    (dvector.cons (realize_bounded_term xs s dvector.nil) v),\n  rw hind,\nend\n\n/-- The formula that will give rise to a witness for function symbols -/\nabbreviation wit_bd_func {n : \u2115} (f : L.functions n) (xs : dvector L.constants n) :\n  bounded_formula L 1 := bd_apps (bd_func f) (dvector.map bd_const xs) \u2243 x_ 0\n\n-- /-- The formula that will give rise to a witness for terms -/\n-- abbreviation wit_bd_preterm {n l} (t : bounded_preterm L n l) (v : dvector ) :\n--   bounded_preformula L (n+1) l := bd_equal (lift_bounded_term1 t) x_ 0\n\n/-- The formula that will give rise to a witness for terms -/\nabbreviation wit_bd_term (t : bounded_term L 0) :\n  bounded_formula L 1 := (lift_bounded_term1 t : bounded_term L 1) \u2243 x_ 0\n\nvariables (T : Theory L) [hcompl : fact (is_complete T)]\n\ninclude hcompl\n\n-- namespace is_complete\n\n-- def M : Structure L := classical.some $ (model_existence _).1 hcompl.1.1\n\n-- def hM0 : nonempty (M T) := (classical.some_spec $ (model_existence _).1 hcompl.1.1).1\n\n-- def hMT : (M T) \u22a8 T := (classical.some_spec $ (model_existence _).1 hcompl.1.1).2\n\n-- end is_complete\n\ninstance equality_of_constants : setoid L.constants :=\n{ r := \u03bb c d, (bd_const c \u2243 bd_const d) \u2208 T,\n  iseqv :=\n  begin\n    have hconsis := hcompl.1.1,\n    have hcompl1 := hcompl.1,\n    obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hconsis,\n    repeat {split},\n    { intro c,\n      apply or.resolve_right (hcompl1.2 (bd_const c \u2243 bd_const c)),\n      intro hmem,\n      specialize hMT hmem,\n      apply hMT,\n      simp },\n    { intros c d hcd,\n      apply or.resolve_right (hcompl1.2 (bd_const d \u2243 bd_const c)),\n      intro hmem,\n      apply hMT hmem,\n      specialize hMT hcd,\n      simp only [realize_sentence_equal, realize_closed_term] at hMT,\n      simp [hMT] },\n    { intros c d e hcd hde,\n      apply or.resolve_right (hcompl1.2 (bd_const c \u2243 bd_const e)),\n      intro hmem,\n      apply hMT hmem,\n      have Hcd := hMT hcd,\n      have Hde := hMT hde,\n      simp only [realize_sentence_equal, realize_closed_term] at Hcd Hde,\n      simp [Hcd, Hde] },\n  end }\n\nnamespace bounded_model_of_infinite_model\n\nvariables (T)\n\ndef model_carrier := @quotient L.constants (@fol.equality_of_constants _ T _)\n\nvariables [hwit : fact (has_enough_constants T)]\ninclude hwit\n\ndef w : bounded_formula L 1 \u2192 L.constants := classical.some hwit.1\n\nlemma hw : \u2200 (f : bounded_formula L 1), T \u22a2' (\u2203' f) \u27f9 f[bd_const ((w T) f) /0] :=\nclassical.some_spec hwit.1\n\nvariable {T}\n\nlemma all_realize_wit_of_mem (\u03d5 : bounded_formula L 1) : (\u2203' \u03d5) \u2208 T \u2192 T \u22a8 \u03d5[bd_const (w T \u03d5) /0] :=\nbegin\n  intro hmem,\n  have hT\u03d5 := hw T \u03d5,\n  rw completeness at hT\u03d5,\n  intros M hM0 hMT,\n  exact hT\u03d5 hM0 hMT (hMT hmem),\nend\n\n\nvariable (T)\n\n/-- The action of a function symbol on the set of constant symbols is given by the witness property:\n  if `xs` is a list of constant symbols then we take the formula `\u2203 v, f(xs) = v \u2208 T`.\n  The witness property gives us `c : L.constants` such that `f(xs) = c \u2208 T`.\n  Proving these formulas are in `T` repeatedly uses completeness and (hence) consistency of `T`.\n-/\nabbreviation fun_map_on_L_constants {n : \u2115} (f : L.functions n) (xs : dvector L.constants n) :\n  L.constants := w T (wit_bd_func f xs)\n\nlemma fun_map_on_L_constants_all_realize_sentence {n : \u2115} (f : L.functions n)\n  (xs : dvector L.constants n) : T \u22a8 (wit_bd_func f xs)[bd_const (w T (wit_bd_func f xs)) /0] :=\nbegin\n  obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  set x := realize_bounded_term (dvector.nil : dvector M 0)\n    (bd_apps (bd_func f) (dvector.map bd_const xs)) dvector.nil with hx,\n  apply all_realize_wit_of_mem,\n  apply or.resolve_right (hcompl.1.2 _),\n  intro hmem,\n  have hbot := hMT hmem,\n  simp only [realize_bounded_term, realize_sentence_not, realize_bounded_formula,\n    realize_sentence_ex, dvector.nth, fin.val_zero', not_exists] at hbot,\n  apply hbot x,\n  simpa [realize_bounded_term_bd_apps, hx],\nend\n\nlemma bd_func_on_L_constants_mem {n : \u2115} (f : L.functions n) (xs : dvector L.constants n) :\n  -- T \u22a8 (wit_bd_func f xs)[bd_const (w T (wit_bd_func f xs)) /0] \u2192\n  ((bd_apps (bd_func f) (dvector.map bd_const xs)) \u2243 (bd_const (w T (wit_bd_func f xs)))) \u2208 T :=\nbegin\n  obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  apply or.resolve_right (hcompl.1.2 _),\n  intro hmem,\n  have hbot := hMT hmem,\n  apply hbot,\n  simp [realize_sentence_equal, realize_closed_term, bd_const, realize_bounded_term_bd_apps],\n  have hM\u03d5 := fun_map_on_L_constants_all_realize_sentence T f xs hM0 hMT,\n  simp [realize_subst_formula0] at hM\u03d5, -- cannot fix\n  convert hM\u03d5,\n  simp [bd_const, realize_bounded_term_bd_apps],\nend\n\n\n/-- The map on functions uses choice twice, once to find representatives from the\n  quotient terms, once to find constant a symbol using witness property / henkin / enough constants\n  (since `fol.bounded_model_of_infinite_model.w` uses choice)\n-/\ndef fun_map_on_M : \u03a0 {n : \u2115}, L.functions n \u2192 dvector (model_carrier T) n \u2192\n  model_carrier T :=\n\u03bb n f xs, quotient.mk (fun_map_on_L_constants T f (dvector.map quotient.out xs))\n\nvariable [L.is_algebraic]\n\ndef model :\n  Structure L :=\n{ carrier := model_carrier T,\n  fun_map := \u03bb _, fun_map_on_M T,\n  rel_map := \u03bb _ r, false.elim $ Language.is_algebraic.empty_relations _ r }\n\nlemma realize_term_all_realize_sentence (t : bounded_term L 0) :\n  T \u22a8 (wit_bd_term t)[bd_const (w T (wit_bd_term t)) /0] :=\nbegin\n  apply all_realize_wit_of_mem,\n  obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  apply or.resolve_right (hcompl.1.2 _),\n  intro hmem,\n  apply hMT hmem,\n  simp only [realize_bounded_formula, lift_bounded_term1,\n    realize_bounded_term, realize_bounded_formula_ex, dvector.nth, fin.val_zero'],\n  use (realize_closed_term M t),\n  rw realize_lift_bounded_term,\nend\n\n-- lemma realize_bounded_term {n l} (xs : dvector (model T) n) (t : bounded_preterm L n l)\n--   (v : dvector (model T) l) :\n--   realize_bounded_term xs t v\n--   = realize_closed_term (model T) (bd_const (w T (wit_bd_term t))) :=\n\n\nlemma realize_term' (t : bounded_term L 0) : realize_closed_term (model T) t\n  = realize_closed_term (model T) (bd_const (w T (wit_bd_term t))) :=\nbegin\n  sorry\n  -- let \u03d5 := (lift_bounded_term1 t : bounded_term L 1) \u2243 -- x_ 0,\n  -- have hex : (\u2203' \u03d5) \u2208 T,\n  -- {\n  --   obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  --   apply or.resolve_right (hcompl.1.2 _),\n  --   intro hmem,\n  --   apply hMT hmem,\n  --   simp only [realize_bounded_formula, lift_bounded_term1,\n  --     realize_bounded_term, realize_bounded_formula_ex, dvector.nth, fin.val_zero'],\n  --   use (realize_closed_term M t),\n  --   rw realize_lift_bounded_term },\n  -- use w T \u03d5,\n  -- simp only [realize_closed_term],\n  -- have hT\u03d5 := all_realize_wit_of_mem \u03d5 hex,\n\n  -- cases t with k,\n  -- { apply fin_zero_elim k },\n  -- {\n    -- simp [bd_const],\n    -- sorry\n  -- },\n  -- {sorry},\nend\n\nlemma realize_sentence_iff (\u03d5 : sentence L) : model T \u22a8 \u03d5 \u2194 \u03d5 \u2208 T :=\nbegin\n  obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  cases \u03d5,\n  { simp only [false_of_satisfied_false, false_iff],\n    intro hbot,\n    apply hMT hbot },\n  {\n    simp [realize_sentence_equal, realize_closed_term],\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\nend\n\nlemma all_realize_sentence : model T \u22a8 T :=\nbegin\n  obtain \u27e8 M , hM0 , hMT \u27e9 := (model_existence _).1 hcompl.1.1,\n  intro \u03d5,\n  cases \u03d5,\n  { intro hbot,\n    exfalso,\n    apply hMT hbot },\n  {\n    intro hmem,\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\n  {\n    sorry\n  },\nend\n\nlemma cardinality : #(model T) \u2264 #L.constants := sorry\n\nend bounded_model_of_infinite_model\n\n/-- If a theory `T` is complete (i.e. maximal) and consistent\n  then `T` has a model `M` with `#M \u2264 |constant symbols of the language|` -/\ntheorem bounded_model_of_infinite_model\n  [hwit : fact (has_enough_constants T)] [L.is_algebraic] :\n  \u2203 M : Structure L, nonempty M \u2227 M \u22a8 T \u2227 #M \u2264 #L.constants :=\n\u27e8 bounded_model_of_infinite_model.model T,\n    sorry,\n    bounded_model_of_infinite_model.all_realize_sentence T ,\n    bounded_model_of_infinite_model.cardinality T \u27e9\n\n\n#check Language.constants\n\nend fol\n", "meta": {"author": "Jlh18", "repo": "ModelTheoryInLean8", "sha": "fbda7d869d4169b6e739bb74165e99ee03ca63d6", "save_path": "github-repos/lean/Jlh18-ModelTheoryInLean8", "path": "github-repos/lean/Jlh18-ModelTheoryInLean8/ModelTheoryInLean8-fbda7d869d4169b6e739bb74165e99ee03ca63d6/Trash/upward_lownheim_skolem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.366990324373707}}
{"text": "import tactic \n\nimport data.stream.defs \nimport data.set.basic \nimport data.set.lattice \nimport data.list.basic \nimport data.list.nodup \nimport data.finset.basic\nimport justification\nimport tactic \n\n@[reducible]\ndef Val : Type := \u2115 \ninstance : has_lt Val := by {unfold Val, apply_instance}\ninstance : has_add Val := by {unfold Val, apply_instance}\ndef State (Var : Type) [fintype Var] [decidable_eq Var] : Type := Var \u2192 Val\n\ndef Trace (Var : Type) [fintype Var] [decidable_eq Var] : Type := stream (State Var)\n\nclass AssertionLang (Assertion : Type) (Var : Type) [fintype Var] [decidable_eq Var] \nextends has_top Assertion := \n(compl : Assertion \u2192 Assertion)\n(conj : Assertion \u2192 Assertion \u2192 Assertion)\n(disj : Assertion \u2192 Assertion \u2192 Assertion)\n(sem : Assertion \u2192 set (Trace Var))\n(T_def : sem \u22a4 = set.univ)\n(compl_def : \u2200 A : Assertion, sem (compl A) = set.univ \\ (sem A))\n(conj_def :  \u2200 A B : Assertion, sem (conj A B) =  (sem A) \u2229 (sem B))\n(disj_def :  \u2200 A B : Assertion,  sem (disj A B) =  (sem A) \u222a (sem B))\n\n\nnotation `\u2983` \u03a6 `;;`Var `;;` A `\u2984` := (AssertionLang.sem \u03a6 Var _ _ _ A)\n\nnamespace AssertionLang \n\nvariables {\u03a6 Var : Type} [fintype Var] [decidable_eq Var] [AssertionLang \u03a6 Var]\n\ndef Impl : \u03a6 \u2192 \u03a6 \u2192 \u03a6 := \u03bb P Q, AssertionLang.disj Var (AssertionLang.compl Var P) Q\n\nlemma impl_def {P Q : \u03a6} {\u03c3 : Trace Var} : \n  \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ (@Impl \u03a6 Var _ _ _ P Q)) \u2194 \n  \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ P) \u2192 \u03c3 \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ Q) := \nbegin \nsplit,\nintros h\u2081 h\u2082,\nrw Impl at h\u2081,\nsimp at h\u2081,\nrw disj_def at h\u2081,\ncases h\u2081,\nrw compl_def at h\u2081,\nexfalso, exact set.not_mem_of_mem_diff h\u2081 h\u2082, assumption,\n\nintros h,\nrw Impl, rw disj_def, simp,\nhave : \u03c3 \u2208  (@AssertionLang.sem \u03a6 Var _ _ _ P) \u2228 \u03c3 \u2209  (@AssertionLang.sem \u03a6 Var _ _ _ P), from or_not,\ncases this,\nright, apply h this,\nleft,rw compl_def,\nexact set.mem_sep trivial this,\nend \n\nend AssertionLang \n\n\n\n\n@[ext, derive decidable_eq]\nstructure Component (Var : Type) [fintype Var] [decidable_eq Var] := \n(ports : finset Var)\n\n\n\n\nvariables {Var : Type} [fintype Var] [decidable_eq Var]\n\n\n@[reducible]\ndef Impl (Var : Type) [fintype Var] [decidable_eq Var] : Type := set (Trace Var)\n\n@[reducible]\ndef Env (Var : Type) [fintype Var] [decidable_eq Var]: Type := set (Trace Var)\n\n@[derive decidable_eq]\nstructure Contract (\u03a6 : Type) (Var : Type) \n[fintype Var] [decidable_eq Var] [AssertionLang \u03a6 Var] := \n(A : \u03a6)\n(G : \u03a6)\n\n\n\nvariables {\u03a6 : Type} [fintype Var] [decidable_eq Var] [AssertionLang \u03a6 Var]\n\ninstance : inhabited (Contract \u03a6 Var) := \u27e8{A := \u22a4 , G := \u22a4}\u27e9 \n\nnamespace Contract \n@[reducible]\ndef nf (C : Contract \u03a6 Var) : \u03a6 := AssertionLang.disj Var (AssertionLang.compl Var C.A) C.G \nend Contract \n\ntheorem nf_def {C : Contract \u03a6 Var} {x : Trace Var} : x \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ C.nf) \u2194 \nx \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ C.A) \u2192 x \u2208 (@AssertionLang.sem \u03a6 Var _ _ _ C.G) := \nbegin\nrw AssertionLang.disj_def,\nrw AssertionLang.compl_def,\nrw imp_iff_not_or, simp,\nend \n\nstructure ComponentWithContract (\u03a6 : Type) (Var : Type) \n[fintype Var] [decidable_eq Var] [AssertionLang \u03a6 Var] \nextends Component Var := \n(C : Contract \u03a6 Var)\n\ninstance : has_coe (ComponentWithContract \u03a6 Var) (Component Var) := \u27e8\u03bb C, {..C}\u27e9 \n\nnamespace Impl \n\nvariables {S : Component Var}\n\n\ndef satisfiesContract (C : Contract \u03a6 Var) : Impl Var \u2192 Prop := \n\u03bb I, I \u2229 (AssertionLang.sem C.A) \u2286 (AssertionLang.sem C.G)\n\nend Impl\n\nnamespace Env \n\nvariables {S : Component Var}\n\n\ndef satisfiesContract (C : Contract \u03a6 Var) : Env Var \u2192 Prop := \n\u03bb En, En \u2286 (AssertionLang.sem C.A)\n\nend Env\n\n\n\n\n@[reducible]\ndef ImplC (S : ComponentWithContract \u03a6 Var) : Type := set (Trace Var)\n\n@[reducible]\ndef EnvC (S : ComponentWithContract \u03a6 Var) : Type := set (Trace Var)\n\nnamespace ImplC\n\nvariables {S : ComponentWithContract \u03a6 Var}\n\ndef toSet : ImplC S \u2192 set (Trace Var) := id \n\ndef satisfiesContract (C : Contract \u03a6 Var) : ImplC S \u2192 Prop := \n\u03bb I, (I.toSet) \u2229 (AssertionLang.sem C.A) \u2286 (AssertionLang.sem C.G)\n\nend ImplC\n\nnamespace EnvC \n\nvariables {S : ComponentWithContract \u03a6 Var}\n\n\n@[reducible, simp]\ndef toSet  : EnvC S \u2192 set (Trace Var) := id \n\ninstance : has_coe (set (Trace Var)) (EnvC S) := { coe := \u03bb a, a}\n\ndef satisfiesContract (C : Contract \u03a6 Var) : EnvC S \u2192 Prop := \n\u03bb En, En.toSet \u2286 (AssertionLang.sem C.A)\n\nend EnvC\n\nnamespace Contract \n\nvariables {S : Component Var} (C : Contract \u03a6 Var) \n\ndef satisfyingImplentations : set (Impl Var) := \n{I | I.satisfiesContract C}\n\ndef satisfyingEnvironments : set (Env Var) := \n{E | E.satisfiesContract C}\n\nend Contract \n\n@[derive decidable_eq]\nstructure Architecture (S : Component Var) := \n(subs : list (Component Var))\n(delegation : list (Var \u00d7 Var))\n\n\nvariables {S : Component Var} \n\ndef Implementations (A : Architecture S) : Type := \n  \u03a0 (S' : Component Var) (h : S' \u2208 A.subs), Impl Var\n\n\ndef CompositeImplementation \n(A : Architecture S)\n(Impls : Implementations A) : Impl Var \n:= \u22c2 (S' : Component Var) {S' \u2208 A.subs} , Impls S' H\n\nopen list \ndef CompositeEnvironment \n(A : Architecture S)\n(U : Component Var) \n(Impls : Implementations A) \n(E : Env Var) : Env Var := \nE \u2229 \u22c2 (S' : Component Var) {S' \u2208 A.subs.erase U}, Impls S' (mem_of_mem_erase H)\n\n", "meta": {"author": "loganrjmurphy", "repo": "ForeMoSt", "sha": "c7affc7c8971562520d2775ac48fe4f188f84b02", "save_path": "github-repos/lean/loganrjmurphy-ForeMoSt", "path": "github-repos/lean/loganrjmurphy-ForeMoSt/ForeMoSt-c7affc7c8971562520d2775ac48fe4f188f84b02/src/Architectural/Architecture.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3669903177229996}}
{"text": "    \nopen classical\n\nvariables (\u03b1 : Type) (p q : \u03b1 \u2192 Prop)\nvariable a : \u03b1\nvariable r : Prop\n\n-- This is required to add a to tactics\ninclude a\n\nexample : (\u2203 x : \u03b1, r) \u2192 r :=\nbegin\n  intros h,\n  cases h,\n  assumption\nend\nexample : r \u2192 (\u2203 x : \u03b1, r) :=\nbegin\n  intros,\n  apply exists.intro,\n  repeat { assumption }\nend\nexample : (\u2203 x, p x \u2227 r) \u2194 (\u2203 x, p x) \u2227 r :=\nbegin\n  apply iff.intro,\n    intros,\n    cases a_1 with x hpx,\n    simp *,\n    existsi x,\n    simp *,\n  intros,\n  cases a_1.left with x hx,\n  existsi x,\n  simp *\nend\nexample : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\nbegin\n  apply iff.intro,\n    intros,\n    cases a_1 with w hw,\n    cases hw with h,\n      left,\n      existsi w,\n      assumption,\n    right,\n    existsi w,\n    assumption,\n  intros,\n  cases a_1 with hp hq,\n    cases hp with w hw,\n    existsi w,\n    simp *,\n  cases hq with w hw,\n  existsi w,\n  simp *\nend\n\nexample : (\u2200 x, p x) \u2194 \u00ac (\u2203 x, \u00ac p x) :=\nbegin\n  apply iff.intro,\n    intros h hn,\n    cases hn with w hw,\n    simp * at *,\n  intros,\n  apply by_contradiction,\n  intros,\n  have : \u2203 x, \u00ac p x,\n    apply exists.intro,\n    apply a_2,\n  contradiction\nend\nexample : (\u2203 x, p x) \u2194 \u00ac (\u2200 x, \u00ac p x) :=\nbegin\n  apply iff.intro,\n    intros h hn,\n    simp * at *,\n    cases h,\n    contradiction,\n  intros h,\n  apply by_contradiction,\n  intros hn,\n  have : \u2200 x, \u00ac p x,\n    intros y,\n    apply not.intro,\n    intros hy,\n    have : \u2203 x, p x,\n      apply exists.intro,\n      assumption,\n    contradiction,\n  contradiction\nend\nexample : (\u00ac \u2203 x, p x) \u2194 (\u2200 x, \u00ac p x) :=\nbegin\n  apply iff.intro,\n    intros h x hp,\n    have : \u2203 x, p x,\n      apply exists.intro,\n      assumption,\n    contradiction,\n  intros h hp,\n  cases hp with x hx,\n  simp * at *\nend\nexample : (\u00ac \u2200 x, p x) \u2194 (\u2203 x, \u00ac p x) :=\nbegin\n  apply iff.intro,\n    intros h,\n    apply by_contradiction,\n    intros hn,\n    have : \u2200 x, p x,\n      intros y,\n      apply by_contradiction,\n      intro hny,\n      have : \u2203 x, \u00ac p x,\n        apply exists.intro,\n        assumption,\n      contradiction,\n    contradiction,\n  intros h hn,\n  cases h with w hw,\n  simp * at *\nend\n\nexample : (\u2200 x, p x \u2192 r) \u2194 (\u2203 x, p x) \u2192 r :=\nbegin\n  apply iff.intro,\n    intros h1 h2,\n    cases h2 with w hw,\n    apply h1,\n    exact hw,\n  intros h x hx,\n  apply h,\n  existsi x,\n  assumption\nend\nexample : (\u2203 x, p x \u2192 r) \u2194 (\u2200 x, p x) \u2192 r :=\nbegin\n  apply iff.intro,\n    intros h1 h2,\n    cases h1 with w hw,\n    apply hw,\n    simp [h2],\n  intros h,\n  apply by_cases,\n    intros h1,\n    have : r,\n      apply h,\n      assumption,\n    existsi a,\n    simp [h1 a, this],\n  intros h1,\n  have : \u2203 x, \u00ac p x,\n    apply by_contradiction,\n    intros h2,\n    have : \u2200 x, p x,\n      intros,\n        intros,\n        apply by_contradiction,\n        intro h3,\n        have : \u2203 x, \u00ac p x,\n          apply exists.intro,\n          assumption,\n        contradiction,  \n      intros,\n      apply by_contradiction,\n      intro h3,\n      contradiction,\n  cases this with w hw,\n  existsi w,\n  simp * at *,\nend\nexample : (\u2203 x, r \u2192 p x) \u2194 (r \u2192 \u2203 x, p x) :=\nbegin\n  apply iff.intro,\n    intros h hr,\n    simp [hr] at h,\n    assumption,\n  intros h,\n  apply by_cases,\n    intros hr,\n    have hp : \u2203 x, p x,\n      apply h,\n      assumption,\n    cases hp with w hw,\n    apply exists.intro,\n    intros,\n    exact hw,\n  intros hnr,\n  simp * at *,\n  existsi a,\n  assumption\nend\n", "meta": {"author": "zeptometer", "repo": "LearnLean", "sha": "bb84d5dbe521127ba134d4dbf9559b294a80b9f7", "save_path": "github-repos/lean/zeptometer-LearnLean", "path": "github-repos/lean/zeptometer-LearnLean/LearnLean-bb84d5dbe521127ba134d4dbf9559b294a80b9f7/bluejam/chap5_exercise4.5.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631840431539, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.36698714102693647}}
{"text": "import YatimaStdLib.Bit\n\nnamespace UInt8\n\ndef showBits (u : UInt8) : String :=\n  let numStr := u.toNat |> Nat.toDigits 2\n  \"\".pushn '0' (8 - numStr.length) ++ \u27e8numStr\u27e9\n\ndef toHexString (n : UInt8) : String :=\n  let toLetter\n    | 10 => \"a\"\n    | 11 => \"b\"\n    | 12 => \"c\"\n    | 13 => \"d\"\n    | 14 => \"e\"\n    | 15 => \"f\"\n    | n => toString n\n  \"0x\" ++ toLetter (n / 16) ++ toLetter (n % 16)\n\ndef getBit (u : UInt8) (n : Nat) : Bit :=\n  if u &&& (1 <<< (7 - n)).toUInt8 == 0 then .zero else .one\n\n/-- sums up two u8's and returns the result and the cout -/\ndef sum2 (a b : UInt8) : UInt8 \u00d7 UInt8 :=\n  (a + b, if b <= 255 - a then 0 else 1)\n\n/-- sums up three u8's and returns the result and the cout -/\ndef sum3 (i a b : UInt8) : UInt8 \u00d7 UInt8 :=\n  let (a', o\u2081) := sum2 i a\n  let (b', o) := sum2 a' b\n  (b', o + o\u2081)\n\nend UInt8\n\ndef UInt16.toByteArrayL (n : UInt16) : ByteArray :=\n  \u27e8#[n.toUInt8, (n / 256) % 65536 |>.toUInt8]\u27e9\n\n@[extern \"lean_uint16_to_byte_array\"]\ndef UInt16.toByteArray : UInt16 \u2192 ByteArray :=\n  UInt16.toByteArrayL\n\ntheorem UInt16.toByteArray_size_2 : (UInt16.toByteArray n).size = 2 := by\n  simp [ByteArray.size, Array.size, List.length]\n\ndef UInt32.toByteArrayL (n : UInt32) : ByteArray :=\n  let a\u2080 := n.toUInt8\n  let n := n / 256\n  let a\u2081 := n % 65536 |>.toUInt8\n  let n := n / 256\n  let a\u2082 := n % 16777216 |>.toUInt8\n  let n := n / 256\n  let a\u2083 := n % 4294967296 |>.toUInt8\n  \u27e8#[a\u2080, a\u2081, a\u2082, a\u2083]\u27e9\n\n@[extern \"lean_uint32_to_byte_array\"]\ndef UInt32.toByteArray : UInt32 \u2192 ByteArray :=\n  UInt32.toByteArrayL\n\ntheorem UInt32.toByteArray_size_4 : (UInt32.toByteArray n).size = 4 := by\n  simp [ByteArray.size, Array.size, List.length]\n\ndef UInt64.toByteArrayL (n : UInt64) : ByteArray :=\n  let a\u2080 := n.toUInt8\n  let n := n / 256\n  let a\u2081 := n % 65536 |>.toUInt8\n  let n := n / 256\n  let a\u2082 := n % 16777216 |>.toUInt8\n  let n := n / 256\n  let a\u2083 := n % 4294967296 |>.toUInt8\n  let n := n / 256\n  let a\u2084 := n % 1099511627776 |>.toUInt8\n  let n := n / 256\n  let a\u2085 := n % 281474976710656 |>.toUInt8\n  let n := n / 256\n  let a\u2086 := n % 72057594037927936 |>.toUInt8\n  let n := n / 256\n  let a\u2087 := n % 18446744073709551616 |>.toUInt8\n  \u27e8#[a\u2080, a\u2081, a\u2082, a\u2083, a\u2084, a\u2085, a\u2086, a\u2087]\u27e9\n\n@[extern \"lean_uint64_to_byte_array\"]\ndef UInt64.toByteArray : UInt64 \u2192 ByteArray :=\n  UInt64.toByteArrayL\n\ntheorem UInt64.toByteArray_size_8 : (UInt64.toByteArray n).size = 8 := by\n  simp [ByteArray.size, Array.size, List.length]\n", "meta": {"author": "lurk-lab", "repo": "YatimaStdLib.lean", "sha": "f39dca7a0815ee65e71776d46337f0240037ff6d", "save_path": "github-repos/lean/lurk-lab-YatimaStdLib.lean", "path": "github-repos/lean/lurk-lab-YatimaStdLib.lean/YatimaStdLib.lean-f39dca7a0815ee65e71776d46337f0240037ff6d/YatimaStdLib/UInt.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3669871324449067}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nA model of ZFC in Lean.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 l u_2 u_3 v \n\nnamespace Mathlib\n\n/-- The type of `n`-ary functions `\u03b1 \u2192 \u03b1 \u2192 ... \u2192 \u03b1`. -/\ndef arity (\u03b1 : Type u) : \u2115 \u2192 Type u := sorry\n\nnamespace arity\n\n\n/-- Constant `n`-ary function with value `a`. -/\ndef const {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : arity \u03b1 n := sorry\n\nprotected instance arity.inhabited {\u03b1 : Type u_1} {n : \u2115} [Inhabited \u03b1] : Inhabited (arity \u03b1 n) :=\n  { default := const Inhabited.default n }\n\nend arity\n\n\n/-- The type of pre-sets in universe `u`. A pre-set\n  is a family of pre-sets indexed by a type in `Type u`.\n  The ZFC universe is defined as a quotient of this\n  to ensure extensionality. -/\ninductive pSet where\n| mk : (\u03b1 : Type u) \u2192 (\u03b1 \u2192 pSet) \u2192 pSet\n\nnamespace pSet\n\n\n/-- The underlying type of a pre-set -/\ndef type : pSet \u2192 Type u := sorry\n\n/-- The underlying pre-set family of a pre-set -/\ndef func (x : pSet) : type x \u2192 pSet := sorry\n\ntheorem mk_type_func (x : pSet) : mk (type x) (func x) = x :=\n  pSet.cases_on x\n    fun (x_\u03b1 : Type u_1) (x_A : x_\u03b1 \u2192 pSet) =>\n      idRhs\n        (mk (type (mk x_\u03b1 x_A)) (func (mk x_\u03b1 x_A)) = mk (type (mk x_\u03b1 x_A)) (func (mk x_\u03b1 x_A)))\n        rfl\n\n/-- Two pre-sets are extensionally equivalent if every\n  element of the first family is extensionally equivalent to\n  some element of the second family and vice-versa. -/\ndef equiv (x : pSet) (y : pSet) :=\n  pSet.rec (fun (\u03b1 : Type u_1) (z : \u03b1 \u2192 pSet) (m : \u03b1 \u2192 pSet \u2192 Prop) (_x : pSet) => sorry) x y\n\ntheorem equiv.refl (x : pSet) : equiv x x :=\n  pSet.rec_on x\n    fun (\u03b1 : Type u_1) (A : \u03b1 \u2192 pSet) (IH : \u2200 (\u1fb0 : \u03b1), equiv (A \u1fb0) (A \u1fb0)) =>\n      { left := fun (a : \u03b1) => Exists.intro a (IH a),\n        right := fun (a : \u03b1) => Exists.intro a (IH a) }\n\ntheorem equiv.euc {x : pSet} {y : pSet} {z : pSet} : equiv x y \u2192 equiv z y \u2192 equiv x z := sorry\n\ntheorem equiv.symm {x : pSet} {y : pSet} : equiv x y \u2192 equiv y x := equiv.euc (equiv.refl y)\n\ntheorem equiv.trans {x : pSet} {y : pSet} {z : pSet} (h1 : equiv x y) (h2 : equiv y z) :\n    equiv x z :=\n  equiv.euc h1 (equiv.symm h2)\n\nprotected instance setoid : setoid pSet := setoid.mk equiv sorry\n\nprotected def subset : pSet \u2192 pSet \u2192 Prop := sorry\n\nprotected instance has_subset : has_subset pSet := has_subset.mk pSet.subset\n\ntheorem equiv.ext (x : pSet) (y : pSet) : equiv x y \u2194 x \u2286 y \u2227 y \u2286 x := sorry\n\ntheorem subset.congr_left {x : pSet} {y : pSet} {z : pSet} : equiv x y \u2192 (x \u2286 z \u2194 y \u2286 z) := sorry\n\ntheorem subset.congr_right {x : pSet} {y : pSet} {z : pSet} : equiv x y \u2192 (z \u2286 x \u2194 z \u2286 y) := sorry\n\n/-- `x \u2208 y` as pre-sets if `x` is extensionally equivalent to a member\n  of the family `y`. -/\ndef mem : pSet \u2192 pSet \u2192 Prop := sorry\n\nprotected instance has_mem : has_mem pSet pSet := has_mem.mk mem\n\ntheorem mem.mk {\u03b1 : Type u} (A : \u03b1 \u2192 pSet) (a : \u03b1) : A a \u2208 mk \u03b1 A :=\n  (fun (this : mem (A a) (mk \u03b1 A)) => this) (Exists.intro a (equiv.refl (A a)))\n\ntheorem mem.ext {x : pSet} {y : pSet} : (\u2200 (w : pSet), w \u2208 x \u2194 w \u2208 y) \u2192 equiv x y := sorry\n\ntheorem mem.congr_right {x : pSet} {y : pSet} : equiv x y \u2192 \u2200 {w : pSet}, w \u2208 x \u2194 w \u2208 y := sorry\n\ntheorem equiv_iff_mem {x : pSet} {y : pSet} : equiv x y \u2194 \u2200 {w : pSet}, w \u2208 x \u2194 w \u2208 y := sorry\n\ntheorem mem.congr_left {x : pSet} {y : pSet} : equiv x y \u2192 \u2200 {w : pSet}, x \u2208 w \u2194 y \u2208 w := sorry\n\n/-- Convert a pre-set to a `set` of pre-sets. -/\ndef to_set (u : pSet) : set pSet := set_of fun (x : pSet) => x \u2208 u\n\n/-- Two pre-sets are equivalent iff they have the same members. -/\ntheorem equiv.eq {x : pSet} {y : pSet} : equiv x y \u2194 to_set x = to_set y :=\n  iff.trans equiv_iff_mem (iff.symm set.ext_iff)\n\nprotected instance set.has_coe : has_coe pSet (set pSet) := has_coe.mk to_set\n\n/-- The empty pre-set -/\nprotected def empty : pSet := mk (ulift empty) fun (e : ulift empty) => sorry\n\nprotected instance has_emptyc : has_emptyc pSet := has_emptyc.mk pSet.empty\n\nprotected instance inhabited : Inhabited pSet := { default := \u2205 }\n\ntheorem mem_empty (x : pSet) : \u00acx \u2208 \u2205 := sorry\n\n/-- Insert an element into a pre-set -/\nprotected def insert : pSet \u2192 pSet \u2192 pSet := sorry\n\nprotected instance has_insert : has_insert pSet pSet := has_insert.mk pSet.insert\n\nprotected instance has_singleton : has_singleton pSet pSet :=\n  has_singleton.mk fun (s : pSet) => insert s \u2205\n\nprotected instance is_lawful_singleton : is_lawful_singleton pSet pSet :=\n  is_lawful_singleton.mk fun (_x : pSet) => rfl\n\n/-- The n-th von Neumann ordinal -/\ndef of_nat : \u2115 \u2192 pSet := sorry\n\n/-- The von Neumann ordinal \u03c9 -/\ndef omega : pSet := mk (ulift \u2115) fun (n : ulift \u2115) => of_nat (ulift.down n)\n\n/-- The separation operation `{x \u2208 a | p x}` -/\nprotected def sep (p : set pSet) : pSet \u2192 pSet := sorry\n\nprotected instance has_sep : has_sep pSet pSet := has_sep.mk pSet.sep\n\n/-- The powerset operator -/\ndef powerset : pSet \u2192 pSet := sorry\n\ntheorem mem_powerset {x : pSet} {y : pSet} : y \u2208 powerset x \u2194 y \u2286 x := sorry\n\n/-- The set union operator -/\ndef Union : pSet \u2192 pSet := sorry\n\ntheorem mem_Union {x : pSet} {y : pSet} : y \u2208 Union x \u2194 \u2203 (z : pSet), \u2203 (_x : z \u2208 x), y \u2208 z := sorry\n\n/-- The image of a function -/\ndef image (f : pSet \u2192 pSet) : pSet \u2192 pSet := sorry\n\ntheorem mem_image {f : pSet \u2192 pSet} (H : \u2200 {x y : pSet}, equiv x y \u2192 equiv (f x) (f y)) {x : pSet}\n    {y : pSet} : y \u2208 image f x \u2194 \u2203 (z : pSet), \u2203 (H : z \u2208 x), equiv y (f z) :=\n  sorry\n\n/-- Universe lift operation -/\nprotected def lift : pSet \u2192 pSet := sorry\n\n/-- Embedding of one universe in another -/\ndef embed : pSet := mk (ulift pSet) fun (_x : ulift pSet) => sorry\n\ntheorem lift_mem_embed (x : pSet) : pSet.lift x \u2208 embed :=\n  Exists.intro (ulift.up x) (equiv.refl (pSet.lift x))\n\n/-- Function equivalence is defined so that `f ~ g` iff\n  `\u2200 x y, x ~ y \u2192 f x ~ g y`. This extends to equivalence of n-ary\n  functions. -/\ndef arity.equiv {n : \u2115} : arity pSet n \u2192 arity pSet n \u2192 Prop := sorry\n\ntheorem arity.equiv_const {a : pSet} (n : \u2115) : arity.equiv (arity.const a n) (arity.const a n) :=\n  sorry\n\n/-- `resp n` is the collection of n-ary functions on `pSet` that respect\n  equivalence, i.e. when the inputs are equivalent the output is as well. -/\ndef resp (n : \u2115) := Subtype fun (x : arity pSet n) => arity.equiv x x\n\nprotected instance resp.inhabited {n : \u2115} : Inhabited (resp n) :=\n  { default := { val := arity.const Inhabited.default n, property := sorry } }\n\ndef resp.f {n : \u2115} (f : resp (n + 1)) (x : pSet) : resp n :=\n  { val := subtype.val f x, property := sorry }\n\ndef resp.equiv {n : \u2115} (a : resp n) (b : resp n) := arity.equiv (subtype.val a) (subtype.val b)\n\ntheorem resp.refl {n : \u2115} (a : resp n) : resp.equiv a a := subtype.property a\n\ntheorem resp.euc {n : \u2115} {a : resp n} {b : resp n} {c : resp n} :\n    resp.equiv a b \u2192 resp.equiv c b \u2192 resp.equiv a c :=\n  sorry\n\nprotected instance resp.setoid {n : \u2115} : setoid (resp n) := setoid.mk resp.equiv sorry\n\nend pSet\n\n\n/-- The ZFC universe of sets consists of the type of pre-sets,\n  quotiented by extensional equivalence. -/\ndef Set := quotient pSet.setoid\n\nnamespace pSet\n\n\nnamespace resp\n\n\ndef eval_aux {n : \u2115} :\n    Subtype fun (f : resp n \u2192 arity Set n) => \u2200 (a b : resp n), equiv a b \u2192 f a = f b :=\n  sorry\n\n/-- An equivalence-respecting function yields an n-ary Set function. -/\ndef eval (n : \u2115) : resp n \u2192 arity Set n := subtype.val eval_aux\n\ntheorem eval_val {n : \u2115} {f : resp (n + 1)} {x : pSet} :\n    eval (n + 1) f (quotient.mk x) = eval n (f f x) :=\n  rfl\n\nend resp\n\n\n/-- A set function is \"definable\" if it is the image of some n-ary pre-set\n  function. This isn't exactly definability, but is useful as a sufficient\n  condition for functions that have a computable image. -/\nclass inductive definable (n : \u2115) : arity Set n \u2192 Type (u + 1) where\n| mk : (f : resp n) \u2192 definable n (resp.eval n f)\n\ndef definable.eq_mk {n : \u2115} (f : resp n) {s : arity Set n} (H : resp.eval n f = s) :\n    definable n s :=\n  sorry\n\ndef definable.resp {n : \u2115} (s : arity Set n) [definable n s] : resp n := sorry\n\ntheorem definable.eq {n : \u2115} (s : arity Set n) [H : definable n s] :\n    resp.eval n (definable.resp s) = s :=\n  sorry\n\nend pSet\n\n\nnamespace classical\n\n\ndef all_definable {n : \u2115} (F : arity Set n) : pSet.definable n F := sorry\n\nend classical\n\n\nnamespace Set\n\n\ndef mk : pSet \u2192 Set := quotient.mk\n\n@[simp] theorem mk_eq (x : pSet) : quotient.mk x = mk x := rfl\n\n@[simp] theorem eval_mk {n : \u2115} {f : pSet.resp (n + 1)} {x : pSet} :\n    pSet.resp.eval (n + 1) f (mk x) = pSet.resp.eval n (pSet.resp.f f x) :=\n  rfl\n\ndef mem : Set \u2192 Set \u2192 Prop := quotient.lift\u2082 pSet.mem sorry\n\nprotected instance has_mem : has_mem Set Set := has_mem.mk mem\n\n/-- Convert a ZFC set into a `set` of sets -/\ndef to_set (u : Set) : set Set := set_of fun (x : Set) => x \u2208 u\n\nprotected def subset (x : Set) (y : Set) := \u2200 {z : Set}, z \u2208 x \u2192 z \u2208 y\n\nprotected instance has_subset : has_subset Set := has_subset.mk Set.subset\n\ntheorem subset_def {x : Set} {y : Set} : x \u2286 y \u2194 \u2200 {z : Set}, z \u2208 x \u2192 z \u2208 y := iff.rfl\n\ntheorem subset_iff (x : pSet) (y : pSet) : mk x \u2286 mk y \u2194 x \u2286 y := sorry\n\ntheorem ext {x : Set} {y : Set} : (\u2200 (z : Set), z \u2208 x \u2194 z \u2208 y) \u2192 x = y :=\n  quotient.induction_on\u2082 x y\n    fun (u v : pSet) (h : \u2200 (z : Set), z \u2208 quotient.mk u \u2194 z \u2208 quotient.mk v) =>\n      quotient.sound (pSet.mem.ext fun (w : pSet) => h (quotient.mk w))\n\ntheorem ext_iff {x : Set} {y : Set} : (\u2200 (z : Set), z \u2208 x \u2194 z \u2208 y) \u2194 x = y := sorry\n\n/-- The empty set -/\ndef empty : Set := mk \u2205\n\nprotected instance has_emptyc : has_emptyc Set := has_emptyc.mk empty\n\nprotected instance inhabited : Inhabited Set := { default := \u2205 }\n\n@[simp] theorem mem_empty (x : Set) : \u00acx \u2208 \u2205 := quotient.induction_on x pSet.mem_empty\n\ntheorem eq_empty (x : Set) : x = \u2205 \u2194 \u2200 (y : Set), \u00acy \u2208 x := sorry\n\n/-- `insert x y` is the set `{x} \u222a y` -/\nprotected def insert : Set \u2192 Set \u2192 Set :=\n  pSet.resp.eval (bit0 1) { val := pSet.insert, property := sorry }\n\nprotected instance has_insert : has_insert Set Set := has_insert.mk Set.insert\n\nprotected instance has_singleton : has_singleton Set Set :=\n  has_singleton.mk fun (x : Set) => insert x \u2205\n\nprotected instance is_lawful_singleton : is_lawful_singleton Set Set :=\n  is_lawful_singleton.mk fun (x : Set) => rfl\n\n@[simp] theorem mem_insert {x : Set} {y : Set} {z : Set} : x \u2208 insert y z \u2194 x = y \u2228 x \u2208 z := sorry\n\n@[simp] theorem mem_singleton {x : Set} {y : Set} : x \u2208 singleton y \u2194 x = y :=\n  iff.trans mem_insert\n    { mp :=\n        fun (o : x = y \u2228 x \u2208 \u2205) =>\n          Or._oldrec (fun (h : x = y) => h) (fun (n : x \u2208 \u2205) => absurd n (mem_empty x)) o,\n      mpr := Or.inl }\n\n@[simp] theorem mem_pair {x : Set} {y : Set} {z : Set} :\n    x \u2208 insert y (singleton z) \u2194 x = y \u2228 x = z :=\n  iff.trans mem_insert (or_congr iff.rfl mem_singleton)\n\n/-- `omega` is the first infinite von Neumann ordinal -/\ndef omega : Set := mk pSet.omega\n\n@[simp] theorem omega_zero : \u2205 \u2208 omega :=\n  (fun (this : pSet.mem \u2205 pSet.omega) => this) (Exists.intro (ulift.up 0) (pSet.equiv.refl \u2205))\n\n@[simp] theorem omega_succ {n : Set} : n \u2208 omega \u2192 insert n n \u2208 omega := sorry\n\n/-- `{x \u2208 a | p x}` is the set of elements in `a` satisfying `p` -/\nprotected def sep (p : Set \u2192 Prop) : Set \u2192 Set :=\n  pSet.resp.eval 1 { val := pSet.sep fun (y : pSet) => p (quotient.mk y), property := sorry }\n\nprotected instance has_sep : has_sep Set Set := has_sep.mk Set.sep\n\n@[simp] theorem mem_sep {p : Set \u2192 Prop} {x : Set} {y : Set} :\n    y \u2208 has_sep.sep (fun (y : Set) => p y) x \u2194 y \u2208 x \u2227 p y :=\n  sorry\n\n/-- The powerset operation, the collection of subsets of a set -/\ndef powerset : Set \u2192 Set := pSet.resp.eval 1 { val := pSet.powerset, property := sorry }\n\n@[simp] theorem mem_powerset {x : Set} {y : Set} : y \u2208 powerset x \u2194 y \u2286 x := sorry\n\ntheorem Union_lem {\u03b1 : Type u} {\u03b2 : Type u} (A : \u03b1 \u2192 pSet) (B : \u03b2 \u2192 pSet)\n    (\u03b1\u03b2 : \u2200 (a : \u03b1), \u2203 (b : \u03b2), pSet.equiv (A a) (B b)) (a : pSet.type (pSet.Union (pSet.mk \u03b1 A))) :\n    \u2203 (b : pSet.type (pSet.Union (pSet.mk \u03b2 B))),\n        pSet.equiv (pSet.func (pSet.Union (pSet.mk \u03b1 A)) a)\n          (pSet.func (pSet.Union (pSet.mk \u03b2 B)) b) :=\n  sorry\n\n/-- The union operator, the collection of elements of elements of a set -/\ndef Union : Set \u2192 Set := pSet.resp.eval 1 { val := pSet.Union, property := sorry }\n\nnotation:1024 \"\u22c3\" => Mathlib.Set.Union\n\n@[simp] theorem mem_Union {x : Set} {y : Set} : y \u2208 \u22c3 \u2194 \u2203 (z : Set), \u2203 (H : z \u2208 x), y \u2208 z := sorry\n\n@[simp] theorem Union_singleton {x : Set} : \u22c3 = x := sorry\n\ntheorem singleton_inj {x : Set} {y : Set} (H : singleton x = singleton y) : x = y :=\n  let this : \u22c3 = \u22c3 := congr_arg \u22c3 H;\n  eq.mp (Eq._oldrec (Eq.refl (x = \u22c3)) Union_singleton)\n    (eq.mp (Eq._oldrec (Eq.refl (\u22c3 = \u22c3)) Union_singleton) this)\n\n/-- The binary union operation -/\nprotected def union (x : Set) (y : Set) : Set := \u22c3\n\n/-- The binary intersection operation -/\nprotected def inter (x : Set) (y : Set) : Set := has_sep.sep (fun (z : Set) => z \u2208 y) x\n\n/-- The set difference operation -/\nprotected def diff (x : Set) (y : Set) : Set := has_sep.sep (fun (z : Set) => \u00acz \u2208 y) x\n\nprotected instance has_union : has_union Set := has_union.mk Set.union\n\nprotected instance has_inter : has_inter Set := has_inter.mk Set.inter\n\nprotected instance has_sdiff : has_sdiff Set := has_sdiff.mk Set.diff\n\n@[simp] theorem mem_union {x : Set} {y : Set} {z : Set} : z \u2208 x \u222a y \u2194 z \u2208 x \u2228 z \u2208 y := sorry\n\n@[simp] theorem mem_inter {x : Set} {y : Set} {z : Set} : z \u2208 x \u2229 y \u2194 z \u2208 x \u2227 z \u2208 y := mem_sep\n\n@[simp] theorem mem_diff {x : Set} {y : Set} {z : Set} : z \u2208 x \\ y \u2194 z \u2208 x \u2227 \u00acz \u2208 y := mem_sep\n\ntheorem induction_on {p : Set \u2192 Prop} (x : Set)\n    (h : \u2200 (x : Set), (\u2200 (y : Set), y \u2208 x \u2192 p y) \u2192 p x) : p x :=\n  sorry\n\ntheorem regularity (x : Set) (h : x \u2260 \u2205) : \u2203 (y : Set), \u2203 (H : y \u2208 x), x \u2229 y = \u2205 := sorry\n\n/-- The image of a (definable) set function -/\ndef image (f : Set \u2192 Set) [H : pSet.definable 1 f] : Set \u2192 Set :=\n  let r : pSet.resp 1 := pSet.definable.resp f;\n  pSet.resp.eval 1 { val := pSet.image (subtype.val r), property := sorry }\n\ntheorem image.mk (f : Set \u2192 Set) [H : pSet.definable 1 f] (x : Set) {y : Set} (h : y \u2208 x) :\n    f y \u2208 image f x :=\n  sorry\n\n@[simp] theorem mem_image {f : Set \u2192 Set} [H : pSet.definable 1 f] {x : Set} {y : Set} :\n    y \u2208 image f x \u2194 \u2203 (z : Set), \u2203 (H : z \u2208 x), f z = y :=\n  sorry\n\n/-- Kuratowski ordered pair -/\ndef pair (x : Set) (y : Set) : Set := insert (singleton x) (singleton (insert x (singleton y)))\n\n/-- A subset of pairs `{(a, b) \u2208 x \u00d7 y | p a b}` -/\ndef pair_sep (p : Set \u2192 Set \u2192 Prop) (x : Set) (y : Set) : Set :=\n  has_sep.sep\n    (fun (z : Set) => \u2203 (a : Set), \u2203 (H : a \u2208 x), \u2203 (b : Set), \u2203 (H : b \u2208 y), z = pair a b \u2227 p a b)\n    (powerset (powerset (x \u222a y)))\n\n@[simp] theorem mem_pair_sep {p : Set \u2192 Set \u2192 Prop} {x : Set} {y : Set} {z : Set} :\n    z \u2208 pair_sep p x y \u2194\n        \u2203 (a : Set), \u2203 (H : a \u2208 x), \u2203 (b : Set), \u2203 (H : b \u2208 y), z = pair a b \u2227 p a b :=\n  sorry\n\ntheorem pair_inj {x : Set} {y : Set} {x' : Set} {y' : Set} (H : pair x y = pair x' y') :\n    x = x' \u2227 y = y' :=\n  sorry\n\n/-- The cartesian product, `{(a, b) | a \u2208 x, b \u2208 y}` -/\ndef prod : Set \u2192 Set \u2192 Set := pair_sep fun (a b : Set) => True\n\n@[simp] theorem mem_prod {x : Set} {y : Set} {z : Set} :\n    z \u2208 prod x y \u2194 \u2203 (a : Set), \u2203 (H : a \u2208 x), \u2203 (b : Set), \u2203 (H : b \u2208 y), z = pair a b :=\n  sorry\n\n@[simp] theorem pair_mem_prod {x : Set} {y : Set} {a : Set} {b : Set} :\n    pair a b \u2208 prod x y \u2194 a \u2208 x \u2227 b \u2208 y :=\n  sorry\n\n/-- `is_func x y f` is the assertion `f : x \u2192 y` where `f` is a ZFC function\n  (a set of ordered pairs) -/\ndef is_func (x : Set) (y : Set) (f : Set) :=\n  f \u2286 prod x y \u2227 \u2200 (z : Set), z \u2208 x \u2192 exists_unique fun (w : Set) => pair z w \u2208 f\n\n/-- `funs x y` is `y ^ x`, the set of all set functions `x \u2192 y` -/\ndef funs (x : Set) (y : Set) : Set :=\n  has_sep.sep (fun (f : Set) => is_func x y f) (powerset (prod x y))\n\n@[simp] theorem mem_funs {x : Set} {y : Set} {f : Set} : f \u2208 funs x y \u2194 is_func x y f := sorry\n\n-- TODO(Mario): Prove this computably\n\nprotected instance map_definable_aux (f : Set \u2192 Set) [H : pSet.definable 1 f] :\n    pSet.definable 1 fun (y : Set) => pair y (f y) :=\n  classical.all_definable fun (y : Set) => pair y (f y)\n\n/-- Graph of a function: `map f x` is the ZFC function which maps `a \u2208 x` to `f a` -/\ndef map (f : Set \u2192 Set) [H : pSet.definable 1 f] : Set \u2192 Set := image fun (y : Set) => pair y (f y)\n\n@[simp] theorem mem_map {f : Set \u2192 Set} [H : pSet.definable 1 f] {x : Set} {y : Set} :\n    y \u2208 map f x \u2194 \u2203 (z : Set), \u2203 (H : z \u2208 x), pair z (f z) = y :=\n  mem_image\n\ntheorem map_unique {f : Set \u2192 Set} [H : pSet.definable 1 f] {x : Set} {z : Set} (zx : z \u2208 x) :\n    exists_unique fun (w : Set) => pair z w \u2208 map f x :=\n  sorry\n\n@[simp] theorem map_is_func {f : Set \u2192 Set} [H : pSet.definable 1 f] {x : Set} {y : Set} :\n    is_func x y (map f x) \u2194 \u2200 (z : Set), z \u2208 x \u2192 f z \u2208 y :=\n  sorry\n\nend Set\n\n\ndef Class := set Set\n\nnamespace Class\n\n\nprotected instance has_subset : has_subset Class := has_subset.mk set.subset\n\nprotected instance has_sep : has_sep Set Class := has_sep.mk set.sep\n\nprotected instance has_emptyc : has_emptyc Class := has_emptyc.mk fun (a : Set) => False\n\nprotected instance inhabited : Inhabited Class := { default := \u2205 }\n\nprotected instance has_insert : has_insert Set Class := has_insert.mk set.insert\n\nprotected instance has_union : has_union Class := has_union.mk set.union\n\nprotected instance has_inter : has_inter Class := has_inter.mk set.inter\n\nprotected instance has_neg : Neg Class := { neg := set.compl }\n\nprotected instance has_sdiff : has_sdiff Class := has_sdiff.mk set.diff\n\n/-- Coerce a set into a class -/\ndef of_Set (x : Set) : Class := set_of fun (y : Set) => y \u2208 x\n\nprotected instance has_coe : has_coe Set Class := has_coe.mk of_Set\n\n/-- The universal class -/\ndef univ : Class := set.univ\n\n/-- Assert that `A` is a set satisfying `p` -/\ndef to_Set (p : Set \u2192 Prop) (A : Class) := \u2203 (x : Set), \u2191x = A \u2227 p x\n\n/-- `A \u2208 B` if `A` is a set which is a member of `B` -/\nprotected def mem (A : Class) (B : Class) := to_Set B A\n\nprotected instance has_mem : has_mem Class Class := has_mem.mk Class.mem\n\ntheorem mem_univ {A : Class} : A \u2208 univ \u2194 \u2203 (x : Set), \u2191x = A :=\n  exists_congr fun (x : Set) => and_true (\u2191x = A)\n\n/-- Convert a conglomerate (a collection of classes) into a class -/\ndef Cong_to_Class (x : set Class) : Class := set_of fun (y : Set) => \u2191y \u2208 x\n\n/-- Convert a class into a conglomerate (a collection of classes) -/\ndef Class_to_Cong (x : Class) : set Class := set_of fun (y : Class) => y \u2208 x\n\n/-- The power class of a class is the class of all subclasses that are sets -/\ndef powerset (x : Class) : Class := Cong_to_Class (\ud835\udcab x)\n\n/-- The union of a class is the class of all members of sets in the class -/\ndef Union (x : Class) : Class := \u22c3\u2080Class_to_Cong x\n\nnotation:1024 \"\u22c3\" => Mathlib.Class.Union\n\ntheorem of_Set.inj {x : Set} {y : Set} (h : \u2191x = \u2191y) : x = y := sorry\n\n@[simp] theorem to_Set_of_Set (p : Set \u2192 Prop) (x : Set) : to_Set p \u2191x \u2194 p x := sorry\n\n@[simp] theorem mem_hom_left (x : Set) (A : Class) : \u2191x \u2208 A \u2194 A x :=\n  to_Set_of_Set (fun (x : Set) => A x) x\n\n@[simp] theorem mem_hom_right (x : Set) (y : Set) : coe y x \u2194 x \u2208 y := iff.rfl\n\n@[simp] theorem subset_hom (x : Set) (y : Set) : \u2191x \u2286 \u2191y \u2194 x \u2286 y := iff.rfl\n\n@[simp] theorem sep_hom (p : Set \u2192 Prop) (x : Set) :\n    \u2191(has_sep.sep (fun (y : Set) => p y) x) = has_sep.sep (fun (y : Set) => p y) \u2191x :=\n  set.ext fun (y : Set) => Set.mem_sep\n\n@[simp] theorem empty_hom : \u2191\u2205 = \u2205 :=\n  set.ext\n    fun (y : Set) =>\n      (fun (this : y \u2208 \u2191\u2205 \u2194 False) => this)\n        (eq.mpr (id (propext (iff_false (y \u2208 \u2191\u2205)))) (Set.mem_empty y))\n\n@[simp] theorem insert_hom (x : Set) (y : Set) : insert x \u2191y = \u2191(insert x y) :=\n  set.ext fun (z : Set) => iff.symm Set.mem_insert\n\n@[simp] theorem union_hom (x : Set) (y : Set) : \u2191x \u222a \u2191y = \u2191(x \u222a y) :=\n  set.ext fun (z : Set) => iff.symm Set.mem_union\n\n@[simp] theorem inter_hom (x : Set) (y : Set) : \u2191x \u2229 \u2191y = \u2191(x \u2229 y) :=\n  set.ext fun (z : Set) => iff.symm Set.mem_inter\n\n@[simp] theorem diff_hom (x : Set) (y : Set) : \u2191x \\ \u2191y = \u2191(x \\ y) :=\n  set.ext fun (z : Set) => iff.symm Set.mem_diff\n\n@[simp] theorem powerset_hom (x : Set) : powerset \u2191x = \u2191(Set.powerset x) :=\n  set.ext fun (z : Set) => iff.symm Set.mem_powerset\n\n@[simp] theorem Union_hom (x : Set) : \u22c3 = \u2191\u22c3 := sorry\n\n/-- The definite description operator, which is {x} if `{a | p a} = {x}`\n  and \u2205 otherwise -/\ndef iota (p : Set \u2192 Prop) : Class := \u22c3\n\ntheorem iota_val (p : Set \u2192 Prop) (x : Set) (H : \u2200 (y : Set), p y \u2194 y = x) : iota p = \u2191x := sorry\n\n/-- Unlike the other set constructors, the `iota` definite descriptor\n  is a set for any set input, but not constructively so, so there is no\n  associated `(Set \u2192 Prop) \u2192 Set` function. -/\ntheorem iota_ex (p : Set \u2192 Prop) : iota p \u2208 univ := sorry\n\n/-- Function value -/\ndef fval (F : Class) (A : Class) : Class :=\n  iota fun (y : Set) => to_Set (fun (x : Set) => F (Set.pair x y)) A\n\ninfixl:100 \"\u2032\" => Mathlib.Class.fval\n\ntheorem fval_ex (F : Class) (A : Class) : F\u2032A \u2208 univ :=\n  iota_ex fun (y : Set) => to_Set (fun (x : Set) => F (Set.pair x y)) A\n\nend Class\n\n\nnamespace Set\n\n\n@[simp] theorem map_fval {f : Set \u2192 Set} [H : pSet.definable 1 f] {x : Set} {y : Set} (h : y \u2208 x) :\n    \u2191(map f x)\u2032\u2191y = \u2191(f y) :=\n  sorry\n\n/-- A choice function on the set of nonempty sets `x` -/\ndef choice (x : Set) : Set := map (fun (y : Set) => classical.epsilon fun (z : Set) => z \u2208 y) x\n\ntheorem choice_mem_aux (x : Set) (h : \u00ac\u2205 \u2208 x) (y : Set) (yx : y \u2208 x) :\n    (classical.epsilon fun (z : Set) => z \u2208 y) \u2208 y :=\n  sorry\n\ntheorem choice_is_func (x : Set) (h : \u00ac\u2205 \u2208 x) : is_func x \u22c3 (choice x) := sorry\n\ntheorem choice_mem (x : Set) (h : \u00ac\u2205 \u2208 x) (y : Set) (yx : y \u2208 x) : \u2191(choice x)\u2032\u2191y \u2208 \u2191y := sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/set_theory/zfc_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3669871238628768}}
{"text": "/-\nCopyright (c) 2019 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.default\nimport Mathlib.category_theory.discrete_category\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.PostPort\n\nuniverses v u u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Category of categories\n\nThis file contains the definition of the category `Cat` of all categories.\nIn this category objects are categories and\nmorphisms are functors between these categories.\n\n## Implementation notes\n\nThough `Cat` is not a concrete category, we use `bundled` to define\nits carrier type.\n-/\n\nnamespace category_theory\n\n\n/-- Category of categories. -/\ndef Cat := bundled category\n\nnamespace Cat\n\n\nprotected instance inhabited : Inhabited Cat := { default := bundled.mk (Type u) }\n\nprotected instance has_coe_to_sort : has_coe_to_sort Cat := has_coe_to_sort.mk (Type u) bundled.\u03b1\n\nprotected instance str (C : Cat) : category \u21a5C := bundled.str C\n\n/-- Construct a bundled `Cat` from the underlying type and the typeclass. -/\ndef of (C : Type u) [category C] : Cat := bundled.of C\n\n/-- Category structure on `Cat` -/\nprotected instance category : large_category Cat := category.mk\n\n/-- Functor that gets the set of objects of a category. It is not\ncalled `forget`, because it is not a faithful functor. -/\ndef objects : Cat \u2964 Type u :=\n  functor.mk (fun (C : Cat) => \u21a5C) fun (C D : Cat) (F : C \u27f6 D) => functor.obj F\n\n/-- Any isomorphism in `Cat` induces an equivalence of the underlying categories. -/\ndef equiv_of_iso {C : Cat} {D : Cat} (\u03b3 : C \u2245 D) : \u21a5C \u224c \u21a5D :=\n  equivalence.mk' (iso.hom \u03b3) (iso.inv \u03b3) (eq_to_iso sorry) (eq_to_iso (iso.inv_hom_id \u03b3))\n\nend Cat\n\n\n/--\nEmbedding `Type` into `Cat` as discrete categories.\n\nThis ought to be modelled as a 2-functor!\n-/\n@[simp] theorem Type_to_Cat_obj (X : Type u) : functor.obj Type_to_Cat X = Cat.of (discrete X) :=\n  Eq.refl (functor.obj Type_to_Cat X)\n\nprotected instance Type_to_Cat.faithful : faithful Type_to_Cat := faithful.mk\n\nprotected instance Type_to_Cat.full : full Type_to_Cat :=\n  full.mk\n    fun (X Y : Type (max (max (max u_1 u_2 u_3 u_4) u_1 u_2 u_3) (max u_1 u_2 u_3 u_4) u_1 u_2))\n      (F : functor.obj Type_to_Cat X \u27f6 functor.obj Type_to_Cat Y) => functor.obj F\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/category/Cat_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.366944351932376}}
{"text": "import Days\nimport Days.Common\nimport Std\nimport Lean.Data.Parsec\n\nopen Days\nopen Days.Common\nnamespace Days.Day10\ndef day: ProblemNumber := 10\n\ndef smallExample := \"noop\naddx 3\naddx -5\"\n\ndef largerExample := \"addx 15\naddx -11\naddx 6\naddx -3\naddx 5\naddx -1\naddx -8\naddx 13\naddx 4\nnoop\naddx -1\naddx 5\naddx -1\naddx 5\naddx -1\naddx 5\naddx -1\naddx 5\naddx -1\naddx -35\naddx 1\naddx 24\naddx -19\naddx 1\naddx 16\naddx -11\nnoop\nnoop\naddx 21\naddx -15\nnoop\nnoop\naddx -3\naddx 9\naddx 1\naddx -3\naddx 8\naddx 1\naddx 5\nnoop\nnoop\nnoop\nnoop\nnoop\naddx -36\nnoop\naddx 1\naddx 7\nnoop\nnoop\nnoop\naddx 2\naddx 6\nnoop\nnoop\nnoop\nnoop\nnoop\naddx 1\nnoop\nnoop\naddx 7\naddx 1\nnoop\naddx -13\naddx 13\naddx 7\nnoop\naddx 1\naddx -33\nnoop\nnoop\nnoop\naddx 2\nnoop\nnoop\nnoop\naddx 8\nnoop\naddx -1\naddx 2\naddx 1\nnoop\naddx 17\naddx -9\naddx 1\naddx 1\naddx -3\naddx 11\nnoop\nnoop\naddx 1\nnoop\naddx 1\nnoop\nnoop\naddx -13\naddx -19\naddx 1\naddx 3\naddx 26\naddx -30\naddx 12\naddx -1\naddx 3\naddx 1\nnoop\nnoop\nnoop\naddx -9\naddx 18\naddx 1\naddx 2\nnoop\nnoop\naddx 9\nnoop\nnoop\nnoop\naddx -1\naddx 2\naddx -37\naddx 1\naddx 3\nnoop\naddx 15\naddx -21\naddx 22\naddx -6\naddx 1\nnoop\naddx 2\naddx 1\nnoop\naddx -10\nnoop\nnoop\naddx 20\naddx 1\naddx 2\naddx 2\naddx -6\naddx -11\nnoop\nnoop\nnoop\"\n\nopen Std (RBSet)\nopen Lean.Parsec\nopen Lean (Parsec)\n\ninductive Instruction where\n  | noop: Instruction\n  | addx: Int -> Instruction\n  deriving Repr, Ord, Hashable, BEq, DecidableEq, Inhabited\n\ndef Instruction.parse: Parsec Instruction := do\n  match \u2190 (pstring \"noop\" <|> pstring \"addx\") with\n  | \"noop\" => return .noop\n  | \"addx\" => return .addx ((\u2190 ws *> many1Chars (pchar '-' <|> digit)) |> String.toInt!)\n  | i => fail s!\"unknown instruction: {i}\"\n\ndef Instruction.parseMany [ToString \u03b1] (input: \u03b1) : Except String $ Array Instruction :=\n  match parseInstructions (toString input).iter with \n  | .success _ res => .ok res\n  | .error _ msg => .error msg\n  where\n    parseInstructions: Parsec $ Array Instruction := many (ws *> Instruction.parse <* ws)\n\n#eval Instruction.parseMany smallExample\n--theorem verify_parse : Instruction.parseMany smallExample = .ok #[.noop, .addx 3, .addx $ -5] := by simp\n\nstructure Registers where\n  x: Int\n  deriving Repr, BEq, DecidableEq\n\ninstance : Inhabited Registers where\n  default := \u27e8 1 \u27e9\n\nabbrev Cycle := Nat\n\ndef run (instructions: Array Instruction): List $ Cycle \u00d7 Registers :=\n  process $ instructions\n  where \n    init: Cycle \u00d7 Registers := (1, default)\n    process (insts: Array Instruction) : List $ Cycle \u00d7 Registers :=\n    insts.foldl (init:=[init]) (\u03bb \n    | [], _ => [init]\n    | (cycle, registers)::rest, .noop => (cycle + 1, registers)::(cycle, registers)::rest\n    | (cycle, r)::rest, .addx x\u2082 => \n      (cycle + 2, {x:= r.x + x\u2082})::\n      (cycle + 1, r)::\n      (cycle, r)::rest)\n\ndef parseAndRun! [ToString \u03b1] (input: \u03b1) : List $ Cycle \u00d7 Registers :=\n  List.reverse <| match Instruction.parseMany input with \n  | .ok insts => run insts\n  | .error msg => panic! s!\"Error: {msg}\"\n\n\n#eval parseAndRun! smallExample\n\n/--\nboo\n-/\ndef part\u2081 (input: Input) : String := \n  parseAndRun! input\n  |>.filter (\u03bb \n    | (20, _) => true\n    | (c, _) => if c > 20 \u2227 (c - 20) % 40 = 0 then true else false) \n  |>.map (\u03bb | (c, r) => c * r.x)\n  |> sum\n  |> toString\n\n/--\n-/\ndef part\u2082 (input: Input) : String :=\n  let crt := Array.mkArray 6 (mkArray 40 \".\")\n  parseAndRun! input\n  |>.map (\u03bb \n    | (cycle, {x}) => \n      let char := if (((cycle-1)%40) - x).natAbs \u2264 1\n        then \"#\" \n        else \".\"\n      (cycle, char)\n  )\n  |>.foldl (init:=crt) (\u03bb \n    | crt, (cycle, char) => \n      let rowNum := ((cycle / 40) % 6)\n      let row := crt.get! rowNum\n      row.set! (cycle % 40) char\n      |> crt.set! rowNum\n  )\n  |>.map (String.intercalate \"\" \u2218 Array.toList)\n  |>.toList\n  |> String.intercalate \"\\n\"\n  |> (\"\\n\" ++ \u00b7)\n\n\ndef solution : Problem String := \u27e8 day, part\u2081, part\u2082 \u27e9 \n\n\n#eval parseAndRun! largerExample\n  |>.map (\u03bb \n    | (cycle, {x}) => \n      let char := if (((cycle-1)%40) - x).natAbs \u2264 1\n        then \"#\" \n        else \".\"\n      (cycle, char, x)\n  )\n\n\n#eval testPart\u2081 solution largerExample (expect:=\"13140\")\n\n#eval testPart\u2082 solution largerExample (expect:=\"\n..#..##..##..##..##..##..##..##..##..##.\n.###...###...###...###...###...###...###\n.####....####....####....####....####...\n.#####.....#####.....#####.....#####....\n.######......######......######......###\n########.......#######.......#######....\") ()\n\n/-\nSolution for day 10 part 2\n-/\nexample : String := \"\n..###...##..##..####.###...##..#....#..#\n.#.......#.#..#.#....#..#.#..#.#....#..#\n.###.....#.#....###..#..#.#....#....####\n.#.......#.#....#....###..#.##.#....#..#\n.#....#..#.#..#.#....#....#..#.#....#..#\n.####..##...##..#....#.....###.####.#..#\n\"", "meta": {"author": "jakeswenson", "repo": "advent2022", "sha": "af941092292ff0bc5552bce9c145d6b5b173c20d", "save_path": "github-repos/lean/jakeswenson-advent2022", "path": "github-repos/lean/jakeswenson-advent2022/advent2022-af941092292ff0bc5552bce9c145d6b5b173c20d/Days/Day10.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.366944343432683}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.morphisms.ring_hom_properties\nimport ring_theory.ring_hom.finite_type\n\n/-!\n# Morphisms of finite type\n\nA morphism of schemes `f : X \u27f6 Y` is locally of finite type if for each affine `U \u2286 Y` and\n`V \u2286 f \u207b\u00b9' U`, The induced map `\u0393(Y, U) \u27f6 \u0393(X, V)` is of finite type.\n\nA morphism of schemes is of finite type if it is both locally of finite type and quasi-compact.\n\nWe show that these properties are local, and are stable under compositions.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverses v u\n\nnamespace algebraic_geometry\n\nvariables {X Y : Scheme.{u}} (f : X \u27f6 Y)\n\n/--\nA morphism of schemes `f : X \u27f6 Y` is locally of finite type if for each affine `U \u2286 Y` and\n`V \u2286 f \u207b\u00b9' U`, The induced map `\u0393(Y, U) \u27f6 \u0393(X, V)` is of finite type.\n-/\n@[mk_iff]\nclass locally_of_finite_type (f : X \u27f6 Y) : Prop :=\n(finite_type_of_affine_subset :\n  \u2200 (U : Y.affine_opens) (V : X.affine_opens) (e : V.1 \u2264 (opens.map f.1.base).obj U.1),\n  (f.app_le e).finite_type)\n\nlemma locally_of_finite_type_eq :\n  @locally_of_finite_type = affine_locally @ring_hom.finite_type :=\nbegin\n  ext X Y f,\n  rw [locally_of_finite_type_iff, affine_locally_iff_affine_opens_le],\n  exact ring_hom.finite_type_respects_iso\nend\n\n@[priority 900]\ninstance locally_of_finite_type_of_is_open_immersion {X Y : Scheme} (f : X \u27f6 Y)\n  [is_open_immersion f] : locally_of_finite_type f :=\nlocally_of_finite_type_eq.symm \u25b8 ring_hom.finite_type_is_local.affine_locally_of_is_open_immersion f\n\nlemma locally_of_finite_type_stable_under_composition :\n  morphism_property.stable_under_composition @locally_of_finite_type :=\nlocally_of_finite_type_eq.symm \u25b8\nring_hom.finite_type_is_local.affine_locally_stable_under_composition\n\ninstance locally_of_finite_type_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [hf : locally_of_finite_type f] [hg : locally_of_finite_type g] :\n  locally_of_finite_type (f \u226b g) :=\nlocally_of_finite_type_stable_under_composition f g hf hg\n\nlemma locally_of_finite_type_of_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [hf : locally_of_finite_type (f \u226b g)] :\n  locally_of_finite_type f :=\nbegin\n  unfreezingI { revert hf },\n  rw [locally_of_finite_type_eq],\n  apply ring_hom.finite_type_is_local.affine_locally_of_comp,\n  introv H,\n  exactI ring_hom.finite_type.of_comp_finite_type H,\nend\n\nlemma locally_of_finite_type.affine_open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)]\n  (\ud835\udcb0' : \u2200 i, Scheme.open_cover.{u} ((\ud835\udcb0.pullback_cover f).obj i))\n  [\u2200 i j, is_affine ((\ud835\udcb0' i).obj j)] :\n  locally_of_finite_type f \u2194\n    (\u2200 i j, (Scheme.\u0393.map ((\ud835\udcb0' i).map j \u226b pullback.snd).op).finite_type) :=\nlocally_of_finite_type_eq.symm \u25b8 ring_hom.finite_type_is_local.affine_open_cover_iff f \ud835\udcb0 \ud835\udcb0'\n\nlemma locally_of_finite_type.source_open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} X) :\n  locally_of_finite_type f \u2194 (\u2200 i, locally_of_finite_type (\ud835\udcb0.map i \u226b f)) :=\nlocally_of_finite_type_eq.symm \u25b8 ring_hom.finite_type_is_local.source_open_cover_iff f \ud835\udcb0\n\nlemma locally_of_finite_type.open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} Y) :\n  locally_of_finite_type f \u2194\n    (\u2200 i, locally_of_finite_type (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _)) :=\nlocally_of_finite_type_eq.symm \u25b8\n  ring_hom.finite_type_is_local.is_local_affine_locally.open_cover_iff f \ud835\udcb0\n\nlemma locally_of_finite_type_respects_iso :\n  morphism_property.respects_iso @locally_of_finite_type :=\nlocally_of_finite_type_eq.symm \u25b8 target_affine_locally_respects_iso\n  (source_affine_locally_respects_iso ring_hom.finite_type_respects_iso)\n\nend algebraic_geometry\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/morphisms/finite_type.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3669443349329899}}
{"text": "/-\nCopyright (c) 2018 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.category.Mon.basic\nimport Mathlib.category_theory.endomorphism\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Category instances for group, add_group, comm_group, and add_comm_group.\n\nWe introduce the bundled categories:\n* `Group`\n* `AddGroup`\n* `CommGroup`\n* `AddCommGroup`\nalong with the relevant forgetful functors between them, and to the bundled monoid categories.\n-/\n\n/-- The category of groups and group morphisms. -/\ndef AddGroup :=\n  category_theory.bundled add_group\n\n/-- The category of additive groups and group morphisms -/\nnamespace Group\n\n\nprotected instance Mathlib.AddGroup.group.to_monoid.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection add_group.to_add_monoid :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance has_coe_to_sort : has_coe_to_sort Group :=\n  category_theory.bundled.has_coe_to_sort\n\n/-- Construct a bundled `Group` from the underlying type and typeclass. -/\ndef of (X : Type u) [group X] : Group :=\n  category_theory.bundled.of X\n\n/-- Construct a bundled `AddGroup` from the underlying type and typeclass. -/\nprotected instance group (G : Group) : group \u21a5G :=\n  category_theory.bundled.str G\n\n@[simp] theorem coe_of (R : Type u) [group R] : \u21a5(of R) = R :=\n  rfl\n\nprotected instance Mathlib.AddGroup.has_zero : HasZero AddGroup :=\n  { zero := AddGroup.of PUnit }\n\nprotected instance inhabited : Inhabited Group :=\n  { default := 1 }\n\nprotected instance one.unique : unique \u21a51 :=\n  unique.mk { default := 1 } sorry\n\n@[simp] theorem one_apply (G : Group) (H : Group) (g : \u21a5G) : coe_fn 1 g = 1 :=\n  rfl\n\ntheorem ext (G : Group) (H : Group) (f\u2081 : G \u27f6 H) (f\u2082 : G \u27f6 H) (w : \u2200 (x : \u21a5G), coe_fn f\u2081 x = coe_fn f\u2082 x) : f\u2081 = f\u2082 :=\n  monoid_hom.ext fun (x : \u21a5G) => w x\n\n-- should to_additive do this automatically?\n\nprotected instance Mathlib.AddGroup.has_forget_to_AddMon : category_theory.has_forget\u2082 AddGroup AddMon :=\n  category_theory.bundled_hom.forget\u2082 add_monoid_hom add_group.to_add_monoid\n\nend Group\n\n\n/-- The category of commutative groups and group morphisms. -/\ndef AddCommGroup :=\n  category_theory.bundled add_comm_group\n\n/-- The category of additive commutative groups and group morphisms. -/\n/-- `Ab` is an abbreviation for `AddCommGroup`, for the sake of mathematicians' sanity. -/\ndef Ab :=\n  AddCommGroup\n\nnamespace CommGroup\n\n\nprotected instance comm_group.to_group.category_theory.bundled_hom.parent_projection : category_theory.bundled_hom.parent_projection comm_group.to_group :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance large_category : category_theory.large_category CommGroup :=\n  category_theory.bundled_hom.category\n    (category_theory.bundled_hom.map_hom (category_theory.bundled_hom.map_hom monoid_hom group.to_monoid)\n      comm_group.to_group)\n\n/-- Construct a bundled `CommGroup` from the underlying type and typeclass. -/\ndef of (G : Type u) [comm_group G] : CommGroup :=\n  category_theory.bundled.of G\n\n/-- Construct a bundled `AddCommGroup` from the underlying type and typeclass. -/\nprotected instance Mathlib.AddCommGroup.add_comm_group_instance (G : AddCommGroup) : add_comm_group \u21a5G :=\n  category_theory.bundled.str G\n\n@[simp] theorem coe_of (R : Type u) [comm_group R] : \u21a5(of R) = R :=\n  rfl\n\nprotected instance Mathlib.AddCommGroup.has_zero : HasZero AddCommGroup :=\n  { zero := AddCommGroup.of PUnit }\n\nprotected instance inhabited : Inhabited CommGroup :=\n  { default := 1 }\n\nprotected instance one.unique : unique \u21a51 :=\n  unique.mk { default := 1 } sorry\n\n@[simp] theorem one_apply (G : CommGroup) (H : CommGroup) (g : \u21a5G) : coe_fn 1 g = 1 :=\n  rfl\n\ntheorem ext (G : CommGroup) (H : CommGroup) (f\u2081 : G \u27f6 H) (f\u2082 : G \u27f6 H) (w : \u2200 (x : \u21a5G), coe_fn f\u2081 x = coe_fn f\u2082 x) : f\u2081 = f\u2082 :=\n  monoid_hom.ext fun (x : \u21a5G) => w x\n\nprotected instance Mathlib.AddCommGroup.has_forget_to_AddGroup : category_theory.has_forget\u2082 AddCommGroup AddGroup :=\n  category_theory.bundled_hom.forget\u2082 (category_theory.bundled_hom.map_hom add_monoid_hom add_group.to_add_monoid)\n    add_comm_group.to_add_group\n\nprotected instance Mathlib.AddCommGroup.has_forget_to_AddCommMon : category_theory.has_forget\u2082 AddCommGroup AddCommMon :=\n  category_theory.induced_category.has_forget\u2082 fun (G : AddCommGroup) => AddCommMon.of \u21a5G\n\nend CommGroup\n\n\n-- This example verifies an improvement possible in Lean 3.8.\n\n-- Before that, to have `monoid_hom.map_map` usable by `simp` here,\n\n-- we had to mark all the concrete category `has_coe_to_sort` instances reducible.\n\n-- Now, it just works.\n\nnamespace AddCommGroup\n\n\n/-- Any element of an abelian group gives a unique morphism from `\u2124` sending\n`1` to that element. -/\n-- Note that because `\u2124 : Type 0`, this forces `G : AddCommGroup.{0}`,\n\n-- so we write this explicitly to be clear.\n\n-- TODO generalize this, requiring a `ulift_instances.lean` file\n\ndef as_hom {G : AddCommGroup} (g : \u21a5G) : of \u2124 \u27f6 G :=\n  coe_fn (gmultiples_hom \u21a5G) g\n\n@[simp] theorem as_hom_apply {G : AddCommGroup} (g : \u21a5G) (i : \u2124) : coe_fn (as_hom g) i = i \u2022 g :=\n  rfl\n\ntheorem as_hom_injective {G : AddCommGroup} : function.injective as_hom := sorry\n\ntheorem int_hom_ext {G : AddCommGroup} (f : of \u2124 \u27f6 G) (g : of \u2124 \u27f6 G) (w : coe_fn f 1 = coe_fn g 1) : f = g :=\n  add_monoid_hom.ext_int w\n\n-- TODO: this argument should be generalised to the situation where\n\n-- the forgetful functor is representable.\n\ntheorem injective_of_mono {G : AddCommGroup} {H : AddCommGroup} (f : G \u27f6 H) [category_theory.mono f] : function.injective \u21d1f := sorry\n\nend AddCommGroup\n\n\n/-- Build an isomorphism in the category `Group` from a `mul_equiv` between `group`s. -/\ndef mul_equiv.to_Group_iso {X : Type u} {Y : Type u} [group X] [group Y] (e : X \u2243* Y) : Group.of X \u2245 Group.of Y :=\n  category_theory.iso.mk (mul_equiv.to_monoid_hom e) (mul_equiv.to_monoid_hom (mul_equiv.symm e))\n\n/-- Build an isomorphism in the category `AddGroup` from an `add_equiv` between `add_group`s. -/\n/-- Build an isomorphism in the category `CommGroup` from a `mul_equiv` between `comm_group`s. -/\ndef add_equiv.to_AddCommGroup_iso {X : Type u} {Y : Type u} [add_comm_group X] [add_comm_group Y] (e : X \u2243+ Y) : AddCommGroup.of X \u2245 AddCommGroup.of Y :=\n  category_theory.iso.mk (add_equiv.to_add_monoid_hom e) (add_equiv.to_add_monoid_hom (add_equiv.symm e))\n\n/-- Build an isomorphism in the category `AddCommGroup` from a `add_equiv` between\n`add_comm_group`s. -/\nnamespace category_theory.iso\n\n\n/-- Build a `mul_equiv` from an isomorphism in the category `Group`. -/\n@[simp] theorem Group_iso_to_add_equiv_apply {X : AddGroup} {Y : AddGroup} (i : X \u2245 Y) : \u2200 (\u1fb0 : \u21a5X), coe_fn (AddGroup_iso_to_add_equiv i) \u1fb0 = coe_fn (hom i) \u1fb0 :=\n  fun (\u1fb0 : \u21a5X) => Eq.refl (coe_fn (hom i) \u1fb0)\n\n/-- Build a `mul_equiv` from an isomorphism in the category `CommGroup`. -/\n@[simp] theorem CommGroup_iso_to_add_equiv_apply {X : AddCommGroup} {Y : AddCommGroup} (i : X \u2245 Y) : \u2200 (\u1fb0 : \u21a5X), coe_fn (AddCommGroup_iso_to_add_equiv i) \u1fb0 = coe_fn (hom i) \u1fb0 :=\n  fun (\u1fb0 : \u21a5X) => Eq.refl (coe_fn (hom i) \u1fb0)\n\nend category_theory.iso\n\n\n/-- multiplicative equivalences between `group`s are the same as (isomorphic to) isomorphisms\nin `Group` -/\ndef add_equiv_iso_AddGroup_iso {X : Type u} {Y : Type u} [add_group X] [add_group Y] : X \u2243+ Y \u2245 AddGroup.of X \u2245 AddGroup.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243+ Y) => add_equiv.to_AddGroup_iso e)\n    fun (i : AddGroup.of X \u2245 AddGroup.of Y) => category_theory.iso.AddGroup_iso_to_add_equiv i\n\n/-- multiplicative equivalences between `comm_group`s are the same as (isomorphic to) isomorphisms\nin `CommGroup` -/\ndef mul_equiv_iso_CommGroup_iso {X : Type u} {Y : Type u} [comm_group X] [comm_group Y] : X \u2243* Y \u2245 CommGroup.of X \u2245 CommGroup.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243* Y) => mul_equiv.to_CommGroup_iso e)\n    fun (i : CommGroup.of X \u2245 CommGroup.of Y) => category_theory.iso.CommGroup_iso_to_mul_equiv i\n\nnamespace category_theory.Aut\n\n\n/-- The (bundled) group of automorphisms of a type is isomorphic to the (bundled) group\nof permutations. -/\ndef iso_perm {\u03b1 : Type u} : Group.of (Aut \u03b1) \u2245 Group.of (equiv.perm \u03b1) :=\n  iso.mk (monoid_hom.mk (fun (g : \u21a5(Group.of (Aut \u03b1))) => iso.to_equiv g) sorry sorry)\n    (monoid_hom.mk (fun (g : \u21a5(Group.of (equiv.perm \u03b1))) => equiv.to_iso g) sorry sorry)\n\n/-- The (unbundled) group of automorphisms of a type is `mul_equiv` to the (unbundled) group\nof permutations. -/\ndef mul_equiv_perm {\u03b1 : Type u} : Aut \u03b1 \u2243* equiv.perm \u03b1 :=\n  iso.Group_iso_to_mul_equiv iso_perm\n\nend category_theory.Aut\n\n\nprotected instance Group.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget Group) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : Group) (f : X \u27f6 Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget Group) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget Group) X \u2245\n          category_theory.functor.obj (category_theory.forget Group) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget Group) f);\n      let e : \u21a5X \u2243* \u21a5Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_Group_iso e))\n\nprotected instance CommGroup.forget_reflects_isos : category_theory.reflects_isomorphisms (category_theory.forget CommGroup) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : CommGroup) (f : X \u27f6 Y)\n      (_x : category_theory.is_iso (category_theory.functor.map (category_theory.forget CommGroup) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget CommGroup) X \u2245\n          category_theory.functor.obj (category_theory.forget CommGroup) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget CommGroup) f);\n      let e : \u21a5X \u2243* \u21a5Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_CommGroup_iso e))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Group/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3669443346881678}}
{"text": "import Table.API\n\nuniverse u u_\u03b7\n\nvariable {\u03b7 : Type u_\u03b7} [dec_\u03b7 : DecidableEq \u03b7] {sch : @Schema \u03b7}\n\ntheorem emptyTable_spec1 : @schema \u03b7 dec_\u03b7 _ emptyTable = [] := rfl\n\ntheorem emptyTable_spec2 : @nrows \u03b7 dec_\u03b7 _ emptyTable = 0 := rfl\n\n-- We omit the precondition because it is enforced by the type system\ntheorem addRows_spec1 :\n  \u2200 (t : Table sch) (rs : List (Row sch)), schema (addRows t rs) = schema t :=\n\u03bb t rs => rfl\n\ntheorem addRows_spec2 :\n  \u2200 (t : Table sch) (rs : List (Row sch)),\n    nrows (addRows t rs) = nrows t + rs.length :=\n\u03bb t rs => List.length_append (t.rows) rs\n\n-- TODO: deal with precondition 1\n-- We must enforce decidable equality of \u03c4 to state this theorem\n-- We omit the precondition because it is not required for this portion of the\n-- spec\ntheorem addColumn_spec1 :\n  \u2200 {\u03c4 : Type u} [DecidableEq \u03c4]\n    (t : Table sch) (c : \u03b7) (vs : List $ Option \u03c4),\n    header (addColumn t c vs) = List.append (header t) [c]\n:= by\n  intros \u03c4 inst t c vs\n  simp [header, addColumn, Schema.names]\n  induction sch with\n  | nil => simp [List.map, List.append]\n  | cons s ss ih =>\n    simp only [List.map]\n    -- TODO: after updating to Lean 4 m4, these unfolds become necessary...\n    unfold HAppend.hAppend at ih\n    unfold instHAppend at ih\n    unfold Append.append at ih\n    unfold List.instAppendList at ih\n    simp only at ih\n    rw [ih]\n    simp [List.append]\n    -- TODO: Can we avoid this somehow? (Arises because of induction on schema\n    -- used as index in Table type. Could use empty table instead, but this\n    -- seems more suggestive.)\n    exact dropColumn t \u27e8_, Schema.HasName.hd\u27e9\n\n-- \u27e8cc.val,\n-- by cases cc with | mk val prop =>\n--    induction prop with\n--    | hd => apply Schema.HasName.hd\n--    | @tl r c' rs h ih =>\n--     apply Schema.HasName.tl\n--     simp only [schema, addColumn] at ih\n-- \u27e9\n\n-- TODO: maybe use `lookupType` for this and `buildColumn_spec3`?\ntheorem addColumn_spec2 :\n  \u2200 {\u03c4 : Type u}\n    (t : Table sch)\n    (c : \u03b7)\n    (vs : List $ Option \u03c4)\n    (c' : \u03b7)\n    -- This is equivalent to `c \u2208 header t`. (Unfortunately, we can't take the\n    -- hypothesis in that form beecause creating a `HasName` therefrom would\n    -- require large elimination from `Prop`)\n    (h' : sch.HasName c'),\n      (schema t).lookup \u27e8c', h'\u27e9 =\n      (schema (addColumn t c vs)).lookup \u27e8c', Schema.hasNameOfAppend h'\u27e9 :=\n\u03bb t c vs c' h' => Schema.lookup_eq_lookup_append _ _ _ _\n\ntheorem addColumn_spec3 {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : \u03b7) (vs : List $ Option \u03c4),\n    (schema (addColumn t c vs)).lookupType\n      \u27e8c, sch.hasAppendedSingletonName c \u03c4\u27e9 = \u03c4 := by\n  intros t c vs\n  induction sch with\n  | nil =>\n    simp only [Schema.hasAppendedSingletonName, Schema.lookupType]\n  | cons s ss ih =>\n    simp only [Schema.hasAppendedSingletonName, Schema.lookupType]\n    -- TODO: again, could use `Table.mk []`, but this is more suggestive?\n    apply ih (dropColumn t \u27e8_, Schema.HasName.hd\u27e9)\n\ntheorem addColumn_spec4 :\n  \u2200 {\u03c4 : Type u} [DecidableEq \u03c4] (t : Table sch) (c : \u03b7) (vs : List $ Option \u03c4),\n    vs.length = nrows t \u2192\n    nrows (addColumn t c vs) = nrows t := by\n  intros \u03c4 inst t c vs h\n  simp only [nrows, addColumn, List.length_map]\n  rw [List.zip_length_eq_of_length_eq]\n  exact Eq.symm h\n\n-- theorem addColumn_spec1' :\n--   \u2200 {\u03c4 : Type u} [DecidableEq \u03c4] (t : Table sch) (c : \u03b7) (vs : List \u03c4),\n--     header (addColumn t c vs) = List.append (header t) [c] :=\n-- \u03bb t c vs =>\n--   match sch with\n--   | [] => by cases t with | mk rows => cases rows with | nil => simp [header, addColumn, Schema.names, List.map, List.append]\n--                                                        | cons s ss => simp [header, addColumn, Schema.names, List.map, List.append]\n--   | s :: ss =>\n--     have h := @addColumn_spec1' _ _ ss _ _ t c vs\n--     sorry\n\n-- Spec 1 is enforced by the type system\ntheorem buildColumn_spec2 :\n  \u2200 {\u03c4 : Type u} (t : Table sch) (c : \u03b7) (f : Row sch \u2192 Option \u03c4),\n    header (buildColumn t c f) = List.append (header t) [c] :=\nby intros \u03c4 t c f\n   simp [header, Schema.names]\n   induction sch with\n  | nil => simp [List.map, List.append]\n  | cons s ss ih =>\n    simp only [List.map]\n    -- TODO: same issue with the ih\n    unfold HAppend.hAppend at ih\n    unfold instHAppend at ih\n    unfold Append.append at ih\n    unfold List.instAppendList at ih\n    simp only at ih\n    rw [ih]\n    simp [List.append]\n    exact Table.mk []\n    exact (\u03bb x => f (Row.cons Cell.emp x))\n\ntheorem buildColumn_spec3 :\n  \u2200 {\u03c4 : Type u}\n    (t : Table sch)\n    (c : \u03b7)\n    (f : Row sch \u2192 Option \u03c4)\n    (c' : \u03b7)\n    (h' : sch.HasName c'),\n      (schema t).lookup \u27e8c', h'\u27e9 =\n      (schema (buildColumn t c f)).lookup \u27e8c', Schema.hasNameOfAppend h'\u27e9 :=\n\u03bb t c vs c' h' => Schema.lookup_eq_lookup_append _ _ _ _\n\ntheorem buildColumn_spec4 {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : \u03b7) (f : Row sch \u2192 Option \u03c4),\n    (schema (buildColumn t c f)).lookupType\n      \u27e8c, sch.hasAppendedSingletonName c \u03c4\u27e9 = \u03c4 := by\n  intros t c f\n  induction sch with\n  | nil =>\n    simp only [Schema.hasAppendedSingletonName, Schema.lookupType]\n  | cons s ss ih =>\n    simp only [Schema.hasAppendedSingletonName, Schema.lookupType]\n    apply ih\n    . exact (dropColumn t \u27e8_, Schema.HasName.hd\u27e9)\n    . exact f \u2218 (Row.cons Cell.emp)\n\ntheorem buildColumn_spec5 :\n  \u2200 {\u03c4 : Type u} (t : Table sch) (c : \u03b7) (f : Row sch \u2192 Option \u03c4),\n    nrows (buildColumn t c f) = nrows t :=\nby intros \u03c4 t c f\n   simp only [nrows, buildColumn, addColumn]\n   rw [List.length_map, List.zip_length_eq_of_length_eq]\n   apply Eq.symm\n   apply List.length_map\n\ntheorem vcat_spec1 :\n  \u2200 (t1 : Table sch) (t2 : Table sch),\n    schema (vcat t1 t2) = schema t1 :=\n\u03bb _ _ => rfl\n\ntheorem vcat_spec2 :\n  \u2200 (t1 : Table sch) (t2 : Table sch),\n    nrows (vcat t1 t2) = nrows t1 + nrows t2 :=\n\u03bb t1 t2 => List.length_append t1.rows t2.rows\n\n-- This is precisely the type signature, but we can state it anyway\ntheorem hcat_spec1 :\n  \u2200 {sch\u2081 : @Schema \u03b7} {sch\u2082 : @Schema \u03b7} (t\u2081 : Table sch\u2081) (t\u2082 : Table sch\u2082),\n    schema (hcat t\u2081 t\u2082) = List.append (schema t\u2081) (schema t\u2082) :=\n\u03bb _ _ => rfl\n\ntheorem hcat_spec2 :\n  \u2200 {sch\u2081 : @Schema \u03b7} {sch\u2082 : @Schema \u03b7} (t\u2081 : Table sch\u2081) (t\u2082 : Table sch\u2082),\n    nrows t\u2081 = nrows t\u2082 \u2192 nrows (hcat t\u2081 t\u2082) = nrows t\u2081 :=\nby intros sch\u2081 sch\u2082 t\u2081 t\u2082 h\n   simp only [nrows, hcat]\n   rw [List.length_map]\n   apply List.zip_length_eq_of_length_eq _ _ h\n\ntheorem values_spec1 :\n  \u2200 (rs : List (Row sch)),\n    (h : rs \u2260 []) \u2192 schema (values rs) = Row.schema (rs.head h) :=\n\u03bb rs h => rfl\n\ntheorem values_spec2 :\n  \u2200 (rs : List (Row sch)), nrows (values rs) = rs.length :=\n\u03bb rs => rfl\n\ntheorem crossJoin_spec1 :\n  \u2200 {sch\u2081 : @Schema \u03b7} {sch\u2082 : @Schema \u03b7} (t\u2081 : Table sch\u2081) (t\u2082 : Table sch\u2082),\n    schema (crossJoin t\u2081 t\u2082) = List.append (schema t\u2081) (schema t\u2082) :=\n\u03bb _ _ => rfl\n\ntheorem crossJoin_spec2 :\n  \u2200 {sch\u2081 : @Schema \u03b7} {sch\u2082 : @Schema \u03b7} (t\u2081 : Table sch\u2081) (t\u2082 : Table sch\u2082),\n    nrows (crossJoin t\u2081 t\u2082) = nrows t\u2081 * nrows t\u2082 :=\nby intros sch\u2081 sch\u2082 t\u2081 t\u2082\n   simp only [nrows, crossJoin, List.length_map]\n   apply List.length_prod\n\n-- This is the closest we can approximate the spec given uniqueness issues\ntheorem leftJoin_spec1 {s\u2081 s\u2082 : @Schema \u03b7} :\n  \u2200 (t\u2081 : Table s\u2081) (t\u2082 : Table s\u2082) \n    (cs : ActionList (Schema.removeOtherDecCH s\u2081) s\u2082),\n  schema (leftJoin t\u2081 t\u2082 cs) =\n  List.append (schema t\u2081)\n              (Schema.removeOtherDecCHs (schema t\u2081) (schema t\u2082) cs) :=\n\u03bb _ _ _ => rfl\n\n-- TODO: again, should we use `lookupType`?\ntheorem leftJoin_spec2 {s\u2081 s\u2082 : @Schema \u03b7} :\n  \u2200 (t\u2081 : Table s\u2081) (t\u2082 : Table s\u2082) \n    (cs : ActionList (Schema.removeOtherDecCH s\u2081) s\u2082)\n    (c : \u03b7)\n    (h : s\u2081.HasName c),\n      (schema t\u2081).lookup \u27e8c, h\u27e9 =\n      (schema (leftJoin t\u2081 t\u2082 cs)).lookup \u27e8c, Schema.hasNameOfAppend h\u27e9 :=\n\u03bb _ _ _ _ _ => Schema.lookup_eq_lookup_append _ _ _ _\n\n-- TODO: spec 3\n\n-- TODO: Spec 4 appears to be wrong. Consider the following SQLite queries:\n/-\nCREATE TABLE demo (\n  id INTEGER NOT NULL PRIMARY KEY AUTOINCREMENT,\n  name VARCHAR(100) NOT NULL,\n  age INTEGER NOT NULL\n);\n\nCREATE TABLE demo2 (\n  id INTEGER NOT NULL PRIMARY KEY AUTOINCREMENT,\n  name VARCHAR(100) NOT NULL,\n  location VARCHAR(100) NOT NULL\n);\n\nINSERT INTO demo VALUES (NULL, \"Bob\", 18);\nINSERT INTO demo2 VALUES (NULL, \"Bob\", \"USA\");\nINSERT INTO demo2 VALUES (NULL, \"Bob\", \"UK\");\n\nSELECT *\nFROM demo\nLEFT JOIN demo2\nON demo.name=demo2.name;\n-/\n-- theorem leftJoin_spec4 {s\u2081 s\u2082 : @Schema \u03b7} :\n--   \u2200 (t\u2081 : Table s\u2081) (t\u2082 : Table s\u2082) \n--     (cs : ActionList (Schema.removeOtherDecCH s\u2081) s\u2082),\n--   nrows (leftJoin t\u2081 t\u2082 cs) = nrows t\u2081 := by\n--   intros t\u2081 t\u2082 cs\n--   simp only [leftJoin, nrows]\n\n-- The spec for `getValue` (as nearly as it can be approximated up to uniqueness\n-- issues) is enforced by types\n\ntheorem getColumn1_spec1 :\n  \u2200 (t : Table sch) (n : Nat) (h : n < ncols t),\n    List.length (getColumn1 t n h) = nrows t :=\n\u03bb t n h => List.length_map _ _\n\n-- Spec 2 is encoded in the return type of `getColumn1`\n\n-- TODO: gC2 spec 1\ntheorem getColumn2_spec2 :\n  \u2200 {\u03c4 : Type u} (t : Table sch) (c : \u03b7) (h : sch.HasCol (c, \u03c4)),\n    List.length (getColumn2 t c h) = nrows t :=\n\u03bb t c h => List.length_map _ _\n\n-- Precondition is enforced by subtype\ntheorem selectRows1_spec1 :\n  \u2200 (t : Table sch) (ns : List {n // n < nrows t}),\n    schema (selectRows1 t ns) = schema t :=\n\u03bb t ns => rfl\n\ntheorem selectRows1_spec2 :\n  \u2200 (t : Table sch) (ns : List {n // n < nrows t}),\n    nrows (selectRows1 t ns) = ns.length :=\n\u03bb t ns => List.length_map _ _\n\n-- Precondition is enforced by `h`\ntheorem selectRows2_spec1 :\n  \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = nrows t),\n    schema (selectRows2 t bs h) = schema t :=\n\u03bb t bs h => rfl\n\ntheorem selectRows2_spec2 :\n  \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = nrows t),\n    nrows (selectRows2 t bs h) = (bs.removeAll [false]).length :=\n\u03bb t bs h => List.sieve_removeAll _ _ h\n\ntheorem selectColumns1_spec1 :\n  \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = ncols t),\n    List.Sublist (header (selectColumns1 t bs h)) (header t) :=\n\u03bb t bs h => List.sublist_of_map_sublist _ _ Prod.fst $ List.sieve_sublist bs sch\n\n-- TODO: sC1 spec 2 (I don't think this is actually currently true due to\n-- uniqueness issues -- in particular, the same value may appear later in the\n-- header. The failed proof below illustrates where this goes wrong more\n-- clearly.)\n\n-- theorem List.sieve_mem_iff_true :\n--   List.get xs \u27e8i, pf1\u27e9 \u2208 List.sieve bs xs \u2194 List.get bs \u27e8i, pf2\u27e9 = true :=\n-- by apply Iff.intro\n--    . intro hf\n--      cases xs with\n--      | nil => contradiction\n--      | cons x xs =>\n--      cases bs with\n--      | nil => contradiction\n--      | cons b bs =>\n--      induction i with\n--      | zero =>\n--       simp only [get] at *\n--       cases b with\n--       | false =>\n--         simp only [sieve] at hf\n\n-- The original failed proof\n-- theorem ncols_eq_header_length :\n--   \u2200 (t : Table sch), ncols t = (header t).length :=\n-- \u03bb t => Eq.symm (List.length_map _ _)\n-- theorem selectColumns1_spec2 :\n--   \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = ncols t) (i : Nat) (h' : i < ncols t),\n--   (List.get (header t) \u27e8i, (ncols_eq_header_length t).subst h'\u27e9) \u2208 (header (selectColumns1 t bs h)) \u2194\n--    List.get bs \u27e8i, Eq.subst h.symm h'\u27e9 = true := sorry\n-- by intros t bs h i h'\n--    apply Iff.intro\n--    . intros hforward\n--      unfold Membership.mem at hforward\n--      unfold List.instMembershipList at hforward\n--      simp only [header, Schema.names] at hforward\n--      cases sch with\n--      | nil => contradiction\n--      | cons hdr sch' =>\n--      cases bs with\n--      | nil => contradiction\n--      | cons b bs' =>\n--      simp only [List.sieve, List.map] at hforward\n--      admit\n--     . admit\n\ntheorem selectColumns1_spec3 :\n  \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = ncols t),\n    List.Sublist (schema (selectColumns1 t bs h)) (schema t) :=\n\u03bb t bs h => List.sieve_sublist _ _\n\ntheorem selectColumns1_spec4 :\n  \u2200 (t : Table sch) (bs : List Bool) (h : bs.length = ncols t),\n    nrows (selectColumns1 t bs h) = nrows t :=\n\u03bb t bs h => List.length_map _ _\n\ntheorem selectColumns2_spec1 :\n  \u2200 (t : Table sch) (ns : List {n // n < ncols t}),\n    ncols (selectColumns2 t ns) = ns.length :=\n\u03bb t ns => List.length_map _ _\n\n-- TODO: sc2 specs 2 and 3\n\ntheorem selectColumns2_spec4 :\n  \u2200 (t : Table sch) (ns : List {n // n < ncols t}),\n    nrows (selectColumns2 t ns) = nrows t :=\n\u03bb t ns => List.length_map _ _\n\ntheorem selectColumns3_spec1 :\n  \u2200 (t : Table sch) (cs : List (CertifiedHeader sch)),\n    header (selectColumns3 t cs) = cs.map (Prod.fst \u2218 Sigma.fst) :=\nby intros t cs\n   simp only [header, selectColumns3, Schema.names]\n   induction cs with\n   | nil => simp only [Schema.pick, List.map]\n   | cons c cs ih =>\n     simp only [Schema.pick, List.map, List.cons.injEq]\n     apply And.intro\n     . simp only [Function.comp, Schema.lookup_fst_eq_nm, CertifiedName.val]\n     . exact ih\n\n-- TODO: sc3 spec 2\n\ntheorem selectColumns3_spec3 :\n  \u2200 (t : Table sch) (cs : List (CertifiedHeader sch)),\n    nrows (selectColumns3 t cs) = nrows t :=\n\u03bb t cs => List.length_map _ _\n\ntheorem head_spec1 : \u2200 (t : Table sch) (z : {z : Int // z.abs < nrows t}),\n  schema (head t z) = schema t :=\n\u03bb _ _ => rfl\n\ntheorem head_spec2 : \u2200 (t : Table sch) (z : {z : Int // z.abs < nrows t}),\n  z.val \u2265 0 \u2192 nrows (head t z) = z.val :=\nby intros t z h\n   cases z with | mk z prop =>\n   simp only [head]\n   have h_not_neg : \u00ac (z < 0) := by\n     intro contra\n     cases z with\n     | ofNat n => contradiction\n     | negSucc n => contradiction\n   simp only [ite_false, h_not_neg]\n   simp only [List.take, nrows]\n   rw [List.length_take]\n   . exact Int.toNat_of_ofNat_inj z h\n   . unfold nrows at prop\n     rw [Int.abs_of_nonneg_eq_toNat] at prop\n     . exact prop\n     . exact h\n\n-- TODO: changed slightly from B2T2 b/c casting is a pain (should this be redone\n-- to match the spec exactly?)\ntheorem head_spec3 : \u2200 (t : Table sch) (z : {z : Int // z.abs < nrows t}),\n  z.val < 0 \u2192 nrows (head t z) = nrows t - z.val.abs :=\nby intros t z h\n   cases z with | mk z prop =>\n   simp only [head, nrows, h, ite_true, List.dropLastN, Function.comp]\n   rw [List.length_reverse, List.length_drop, List.length_reverse]\n   -- Need separate `rw`s here so that the equality proof can be auto-generated\n   rw [List.length_reverse]\n   exact prop\n\n-- theorem head_spec3' : \u2200 (t : Table sch) (z : {z : Int // z.abs < nrows t}),\n--   z.val < 0 \u2192 nrows (head t z) = nrows t + z.val :=\n-- by intros t z h\n--    rw [Int.add_neg_eq_sub]\n--    cases z with | mk z prop =>\n--    cases z with\n--    | ofNat n => contradiction\n--    | negSucc n =>\n--      simp only [Int.abs]\n\ntheorem distinct_spec : \u2200 (t : Table sch) [DecidableEq $ Row sch],\n  schema (distinct t) = schema t :=\n\u03bb t => rfl\n\ntheorem dropColumn_spec1 : \u2200 (t : Table sch) (c : CertifiedName sch),\n  nrows (dropColumn t c) = nrows t :=\n\u03bb t c => List.length_map _ _\n\n-- dC spec 2 is not currently true because of duplicate issues. This is the best\n-- approximation we can get instead:\ntheorem dropColumn_spec2 : \u2200 (t : Table sch) (c : CertifiedName sch),\n  header (dropColumn t c) = Schema.names (sch.removeName c.2) :=\n\u03bb t c => rfl\n\ntheorem dropColumn_spec3 : \u2200 (t : Table sch) (c : CertifiedName sch),\n  List.Sublist (schema (dropColumn t c)) (schema t) :=\n\u03bb _ c => Schema.removeName_sublist sch c.val c.property\n\ntheorem dropColumns_spec1 :\n  \u2200 (t : Table sch) (cs : ActionList Schema.removeCertifiedName sch),\n  nrows (dropColumns t cs) = nrows t :=\n\u03bb t cs => List.length_map _ _\n\n-- dCs spec 2 has the same issue as dC spec 2.\ntheorem dropColumns_spec2 :\n  \u2200 (t : Table sch) (cs : ActionList Schema.removeCertifiedName sch),\n  header (dropColumns t cs) = Schema.names (sch.removeNames cs) :=\n\u03bb t cs => rfl\n\ntheorem dropColumns_spec3 :\n  \u2200 (t : Table sch) (cs : ActionList Schema.removeCertifiedName sch),\n  List.Sublist (schema $ dropColumns t cs) (schema t) :=\n\u03bb _ cs => Schema.removeNames_sublist sch cs\n\n-- Spec 1 is enforced by types\ntheorem tfilter_spec2 : \u2200 (t : Table sch) (f : Row sch \u2192 Bool),\n  schema (tfilter t f) = schema t :=\n\u03bb t f => rfl\n\ntheorem tsort_spec1 : \u2200 {\u03c4 : Type u} [inst : Ord \u03c4]\n                        (t : Table sch)\n                        (c : ((c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)))\n                        (b : Bool),\n  nrows (tsort t c b) = nrows t :=\n\u03bb t c b => List.length_mergeSortWith _ t.rows\n\ntheorem tsort_spec2 : \u2200 {\u03c4 : Type u} [Ord \u03c4]\n                        (t : Table sch)\n                        (c : ((c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)))\n                        (b : Bool),\n  schema (tsort t c b) = schema t :=\n\u03bb t c b => rfl\n\ntheorem sortByColumns_spec1 :\n  \u2200 (t : Table sch) (hs : List ((h : Header) \u00d7 sch.HasCol h \u00d7 Ord h.snd)),\n    nrows (sortByColumns t hs) = nrows t :=\nby intros t hs\n   simp only [nrows, sortByColumns]\n   apply List.foldr_invariant (\u03bb x => nrows x = nrows t)\n   -- Initialization\n   . rfl\n   -- Preservation\n   . intros x acc h\n     rw [\u2190h]\n     apply tsort_spec1 (inst := x.snd.snd)\n\ntheorem sortByColumns_spec2 :\n  \u2200 (t : Table sch) (hs : List ((h : Header) \u00d7 sch.HasCol h \u00d7 Ord h.snd)),\n    schema (sortByColumns t hs) = schema t :=\n\u03bb t hs => rfl\n\n-- Spec 1 is enforced by types\ntheorem orderBy_spec2 :\n  \u2200 (t : Table sch)\n    (cmps : List ((\u03ba : Type u) \u00d7 (Row sch \u2192 \u03ba) \u00d7 (\u03ba \u2192 \u03ba \u2192 Bool))),\n    schema (orderBy t cmps) = schema t :=\n\u03bb _ _ => rfl\n\ntheorem orderBy_spec3 :\n  \u2200 (t : Table sch)\n    (cmps : List ((\u03ba : Type u) \u00d7 (Row sch \u2192 \u03ba) \u00d7 (\u03ba \u2192 \u03ba \u2192 Bool))),\n    nrows (orderBy t cmps) = nrows t :=\n\u03bb t _ => List.length_mergeSortWith _ t.rows\n\ntheorem count_spec1 :\n  \u2200 {\u03c4} [DecidableEq \u03c4]\n    (t : Table sch) (c : ((c : \u03b7) \u00d7 sch.HasCol (c, \u03c4))),\n    header (count t c) = [\"value\", \"count\"] :=\n\u03bb t c => rfl\n\n-- This can't yet be in tactic mode because the `induction` tactic doesn't\n-- support `(nm, \u03c4)` as an index\ntheorem count_spec2 :\n  \u2200 {sch : @Schema \u03b7} {\u03c4} [DecidableEq \u03c4]\n    (t : Table sch) (c : ((c : \u03b7) \u00d7 sch.HasCol (c, \u03c4))),\n  (schema (count t c)).lookupType \u27e8\"value\", Schema.HasName.hd\u27e9 =\n  Option (sch.lookupType \u27e8c.1, Schema.colImpliesName c.2\u27e9)\n| _ :: _, _, _, t, \u27e8_, Schema.HasCol.hd\u27e9 => rfl\n  -- As with prior proofs, the table in the IH doesn't matter\n| _ :: _, \u03c4, _, t, \u27e8nm, Schema.HasCol.tl h\u27e9 => count_spec2 (Table.mk []) \u27e8nm, h\u27e9\n\ntheorem count_spec3 {\u03c4} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (schema (count t c)).lookupType \u27e8\"count\", .tl .hd\u27e9 = Nat :=\n\u03bb _ _ => rfl\n\n-- TODO: move this somewhere\ntheorem length_count_pairsToRow : \u2200 (xs : List (Option \u03c4 \u00d7 Nat)),\n  List.length (count.pairsToRow xs) = xs.length\n| [] => rfl\n| x :: xs => congrArg (\u00b7+1) (length_count_pairsToRow xs)\n\ntheorem count_spec4 {\u03c4} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  nrows (count t c) = (getColumn2 t c.1 c.2).unique.length :=\n\u03bb t c => Eq.trans (length_count_pairsToRow _) (List.length_counts _)      \n\ntheorem bin_spec1 [ToString \u03b7] :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 Schema.HasCol (c, Nat) sch)\n    (n : { n // n > 0 }),\n    header (bin t c n) = [\"group\", \"count\"] :=\n\u03bb _ _ _ => rfl\n\ntheorem bin_spec2 [ToString \u03b7] :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 Schema.HasCol (c, Nat) sch)\n    (n : { n // n > 0 }),\n    (schema (bin t c n)).lookupType \u27e8\"group\", Schema.HasName.hd\u27e9 = String :=\n\u03bb _ _ _ => rfl\n\ntheorem bin_spec3 [ToString \u03b7] :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 Schema.HasCol (c, Nat) sch)\n    (n : { n // n > 0 }),\n    (schema (bin t c n)).lookupType\n      \u27e8\"count\", Schema.HasName.tl Schema.HasName.hd\u27e9 = Nat :=\n\u03bb _ _ _ => rfl\n\n-- Spec 1 is enforced by types\ntheorem pivotTable_spec2 :\n  \u2200 (t : Table sch)\n    (cs : List $ CertifiedHeader sch)\n    (inst : DecidableEq (Row (Schema.fromCHeaders cs)))\n    (aggs : List ((c' : Header) \u00d7\n                  (c : CertifiedHeader sch) \u00d7\n                  (List (Option c.fst.snd) \u2192 Option c'.snd))),\n  header (pivotTable t cs inst aggs) =\n  List.append (cs.map (\u00b7.1.1)) (aggs.map (\u00b7.1.1)) :=\n\u03bb t cs inst aggs => List.map_map_append cs aggs Prod.fst Sigma.fst Sigma.fst\n\n-- TODO: get rid of `Classical.choice` (termination proof)\ntheorem pivotTable_spec3_aux :\n  \u2200 (cs : List $ CertifiedHeader sch) \n    (aggs : List ((c' : Header) \u00d7 (c : CertifiedHeader sch) \u00d7\n                  (List (Option c.fst.snd) \u2192 Option c'.snd)))\n    (cn : CertifiedName (Schema.fromCHeaders cs)),\n  Schema.lookup\n    (List.append (Schema.fromCHeaders cs) (aggs.map (fun a => a.fst)))\n    \u27e8cn.fst, Schema.hasNameOfAppend cn.snd\u27e9 =\n  Schema.lookup sch \u27e8cn.fst, Schema.hasNameOfFromCHeaders cn.snd\u27e9\n| \u27e8(.(nm), \u03c4), _\u27e9 :: cs, aggs, \u27e8nm, .hd\u27e9 => by\n  simp only [Schema.fromCHeaders, List.map, List.append, Schema.hasNameOfAppend]\n  rw [Schema.lookup_eq_1,\n      Schema.hasNameOfFromCHeaders_eq_1,\n      Schema.lookup_of_colImpliesName]\n| \u27e8_, _\u27e9 :: cs, aggs, \u27e8nm, .tl _\u27e9 => by\n  simp only [Schema.fromCHeaders, List.map, List.append, Schema.hasNameOfAppend]\n  rw [Schema.lookup_eq_2,\n      Schema.hasNameOfFromCHeaders_eq_2]\n  apply pivotTable_spec3_aux cs aggs \u27e8nm, _\u27e9\n\ntheorem pivotTable_spec3 :\n  \u2200 (t : Table sch)\n    (cs : List $ CertifiedHeader sch)\n    (inst : DecidableEq (Row (Schema.fromCHeaders cs)))\n    (aggs : List ((c' : Header) \u00d7\n                  (c : CertifiedHeader sch) \u00d7\n                  (List (Option c.fst.snd) \u2192 Option c'.snd)))\n    (cn : CertifiedName (Schema.fromCHeaders cs)),\n    (schema $ pivotTable t cs inst aggs).lookup\n      \u27e8cn.1, Schema.hasNameOfAppend cn.2\u27e9 =\n    (schema t).lookup \u27e8cn.1, Schema.hasNameOfFromCHeaders cn.2\u27e9 :=\n\u03bb t cs inst => pivotTable_spec3_aux cs\n\n-- Spec 4 is enforced by types\n\n-- Specs 1 and 2 are enforced by types\n-- Spec 3 is also enforced by types, but since it is actually expressible as an\n-- (albeit trivial) proof, we state it here for completeness\ntheorem groupBy_spec3  {\u03b7'} [DecidableEq \u03b7'] {sch' : @Schema \u03b7'}\n                       {\u03ba \u03bd} [DecidableEq \u03ba] :\n  \u2200 (t : Table sch)\n    (key : Row sch \u2192 \u03ba)\n    (project : Row sch \u2192 \u03bd)\n    (aggregate : \u03ba \u2192 List \u03bd \u2192 Row sch')\n    (k : \u03ba) (vs : List \u03bd),\n  schema (groupBy t key project aggregate) = (aggregate k vs).schema :=\n\u03bb _ _ _ _ _ _ => rfl\n\ntheorem groupBy_spec4 {\u03b7'} [DecidableEq \u03b7'] {sch' : @Schema \u03b7'}\n                      {\u03ba \u03bd} [DecidableEq \u03ba] :\n  \u2200 (t : Table sch)\n    (key : Row sch \u2192 \u03ba)\n    (project : Row sch \u2192 \u03bd)\n    (aggregate : \u03ba \u2192 List \u03bd \u2192 Row sch'),\n  nrows (groupBy t key project aggregate) = (t.rows.map key).unique.length :=\nby intros t key proj agg\n   simp only [nrows, groupBy]\n   rw [List.length_map, List.length_groupByKey]\n   apply congrArg\n   apply congrArg\n   rw [List.map_map]\n   apply congr _ rfl\n   apply congrArg\n   simp only [Function.comp]\n\ntheorem completeCases_spec {\u03c4 : Type u} :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (completeCases t c).length = nrows t :=\n\u03bb t c => Eq.trans (List.length_map _ _) (List.length_map _ _)\n\ntheorem dropna_spec : \u2200 (t : Table sch), schema (dropna t) = schema t :=\n\u03bb t => rfl\n\ntheorem fillna_spec1 {\u03c4 : Type u} :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4))\n    (v : \u03c4),\n    schema (fillna t c v) = schema t :=\n\u03bb _ _ _ => rfl\n\ntheorem fillna_spec2 {\u03c4 : Type u} :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4))\n    (v : \u03c4),\n    nrows (fillna t c v) = nrows t :=\n\u03bb _ _ _ => List.length_map _ _\n\n-- TODO: `pivotLonger` and `pivotWider`\n-- Specs 1 don't hold because of uniqueness issues, I think\n\ntheorem flatten_spec1 :\n  \u2200 (t : Table sch) (cs : ActionList Schema.flattenList sch),\n  header (flatten t cs) = header t := by\n  intros t cs\n  simp only [flatten, header, Schema.names]\n  induction cs with\n  | nil => simp only [Schema.flattenLists]\n  | cons c cs ih =>\n    simp only [Schema.flattenLists]\n    rw [ih]\n    simp only [Schema.flattenList]\n    apply Schema.retypeColumn_preserves_names\n    -- TODO: this shouldn't be necessary with more careful induction\n    exact Table.mk []\n\n-- TODO: `flatten` spec 2\n\n-- TODO: `transformColumn` spec 3\n\ntheorem transformColumn_spec1 {\u03c4\u2081 \u03c4\u2082} :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4\u2081))\n    (f : Option \u03c4\u2081 \u2192 Option \u03c4\u2082),\n  sch.lookupType \u27e8c.1, Schema.colImpliesName c.2\u27e9 = \u03c4\u2081 :=\n\u03bb t c f => Eq.trans (Schema.lookupType_eq_snd_lookup sch\n                        \u27e8c.1, Schema.colImpliesName c.2\u27e9)\n                    (Eq.subst (motive := \u03bb a => a.snd = \u03c4\u2081)\n                        (Eq.symm $ Schema.lookup_of_colImpliesName sch c.2)\n                        rfl)\n\ntheorem transformColumn_spec2 {\u03c4\u2081 \u03c4\u2082} :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4\u2081))\n    (f : Option \u03c4\u2081 \u2192 Option \u03c4\u2082),\n  header (transformColumn t c f) = header t :=\n\u03bb t c f => sch.retypeColumn_preserves_names _ _\n\ntheorem transformColumn_spec4 :\n  \u2200 (t : Table sch)\n    (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4\u2081))\n    (f : Option \u03c4\u2081 \u2192 Option \u03c4\u2082),\n  nrows (transformColumn t c f) = nrows t :=\n\u03bb t c f => List.length_map _ _\n\n-- TODO: `renameColumns` specs 1 and 2\n\ntheorem renameColumns_spec3 :\n  \u2200 (t : Table sch)\n    (ccs : ActionList Schema.renameColumnCN sch),\n  nrows (renameColumns t ccs) = nrows t :=\n\u03bb _ _ => List.length_map _ _\n\n-- The specification for `find` is contained in its type (`Option` corresponds\n-- to \"Error,\" and `Fin` restricts the range of the output)\n\n-- TODO: `groupByRetentive` specs 2\u20136 (in progress)\ntheorem groupByRetentive_spec1 [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  header (groupByRetentive t c) = [\"key\", \"groups\"] :=\n\u03bb _ _ => rfl\n\ntheorem groupByRetentive_spec2\n  {\u03b7 : Type u_\u03b7} [DecidableEq \u03b7] {sch : @Schema \u03b7}\n  {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (schema (groupByRetentive t c)).lookupType \u27e8\"key\", Schema.HasName.hd\u27e9\n    = ULift.{max (u+1) u_\u03b7} \u03c4 :=\n\u03bb _ _ => rfl\n\ntheorem groupByRetentive_spec3\n  {\u03b7 : Type u_\u03b7} [DecidableEq \u03b7] {sch : @Schema \u03b7}\n  {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (schema (groupByRetentive t c)).lookupType \u27e8\"groups\", .tl .hd\u27e9 = Table sch :=\n\u03bb _ _ => rfl\n\n-- Need decidable equality of `ULift`s for `groupBy{Retentive,Subtractive}`\nderiving instance DecidableEq for ULift\n\ndef Function.injective (f : \u03b1 \u2192 \u03b2) := \u2200 {x y}, f x = f y \u2192 x = y\ndef Function.biInjective (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) := \u2200 x\u2081 x\u2082 y\u2081 y\u2082, f x\u2081 x\u2082 = f y\u2081 y\u2082 \u2192 x\u2081 = y\u2081 \u2227 x\u2082 = y\u2082\n\ntheorem List.groupByKey_fsts_no_duplicates [DecidableEq \u03ba] (xs : List (\u03ba \u00d7 \u03bd)) :\n  NoDuplicates $ (groupByKey xs).map Prod.fst := sorry\n\ntheorem Cell.toOption_fromOption {nm : \u03b7} :\n  \u2200 (v : Option \u03c4), toOption (fromOption (nm := nm) v) = v\n| none => rfl\n| some x => rfl\n\n-- TODO: this cannot possibly be useful... delete?\ntheorem List.mem_exists {x : \u03b1} {xs : List \u03b1} :\n  x \u2208 xs \u2194 \u2203 y, y \u2208 xs \u2227 x = y := by\n  apply Iff.intro\n  . intros hf\n    induction hf with\n    | head x xs =>\n      apply Exists.intro x\n      apply And.intro (List.Mem.head _ _) rfl\n    | tail x' xs ih =>\n      apply Exists.elim ih\n      intros yih hyih\n      apply Exists.intro yih\n      apply And.intro\n      . apply List.Mem.tail _ hyih.left\n      . apply hyih.right\n  . intros hb\n    apply Exists.elim hb\n    intros y hy\n    cases hy with | intro hmem heq =>\n    rw [heq]\n    apply hmem\n\ntheorem List.no_dups_of_cons : List.NoDuplicates (x :: xs) \u2192 List.NoDuplicates xs := sorry\n\ntheorem List.mem_of_mem_injective_map (f : \u03b1 \u2192 \u03b2) (hf : f.injective) :\n  \u2200 (x : \u03b1) (xs : List \u03b1),\n  f x \u2208 map f xs \u2192 x \u2208 xs := \u03bb x xs h => sorry\n  -- List.Mem.recOn (motive := (\u03bb a as hmem => x \u2208 xs\n  --   -- hmem.casesOn (\u03bb b bs => b \u2208 bs) (\u03bb b c bs mtv => b \u2208 bs)\n  --   -- match hmem with\n  --   -- | Mem.head .(a) .(as) => sorry\n  --   -- | Mem.tail _ _ => sorry\n  -- )) h\n  --   (\u03bb a as => sorry)\n  --   sorry\n\n  -- := by\n  -- intros x xs hin\n  -- cases xs with | nil => contradiction | cons x' xs =>\n  -- simp only [map] at hin\n\ntheorem List.no_dups_map_injective\n  (f : \u03b1 \u2192 \u03b2) (hf : f.injective) : \u2200 (xs : List \u03b1) (hxs : NoDuplicates xs),\n  NoDuplicates $ map f xs\n| [], hxs => NoDuplicates.nil\n| x :: xs, NoDuplicates.cons _ _ hxnin hndxs =>\n  NoDuplicates.cons (f x) (map f xs)\n    (\u03bb hneg => absurd (mem_of_mem_injective_map f hf x xs hneg) hxnin)\n    (no_dups_map_injective f hf xs hndxs)\n\n-- TODO: this should be an interesting challenge...\n-- set_option pp.explicit true\ntheorem groupByRetentive_spec4 [inst : DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (getColumn2 (groupByRetentive t c) \"key\" Schema.HasCol.hd).NoDuplicates := by\n  intros t c\n  simp only [groupByRetentive, groupBy, getColumn2]\n  rw [List.map_map]\n  simp only [Function.comp, getValue, Row.getCell]\n  conv =>\n    rhs\n    lhs\n    apply funext (f\u2082 := _) _\n    apply (\u03bb x => Option.map ULift.up x.fst)\n    -- `intros` doesn't seem to be working in `conv` mode?\n    apply (\u03bb x => Cell.toOption_fromOption _)\n  conv =>\n    rhs\n    lhs\n    apply funext (f\u2082 := _) _\n    apply (\u03bb x => Option.map ULift.up \u2218 Prod.fst)\n    apply (\u03bb x => rfl)\n  rw [\u2190List.map_map]\n  apply List.no_dups_map_injective\n  -- Show `Option.map ULift.up` is injective\n  . intros x y hxy\n    cases x\n    . cases y\n      . rfl\n      . contradiction\n    . cases y\n      . contradiction\n      . cases hxy; rfl\n  . apply List.groupByKey_fsts_no_duplicates\n\n  -- simp only [getColumn2]\n  -- simp only [groupByRetentive]\n  -- cases c with | mk c pf =>\n  -- simp only\n  -- apply groupBy_specPlus\n  -- intros x\u2081 x\u2082 y\u2081 y\u2082 heq\n  -- simp at heq\n  -- cases heq with | intro left right =>\n  -- apply And.intro\n  -- . have fromOpt_inj {\u03b7} [i : DecidableEq \u03b7] {nm : \u03b7} {\u03b4} : Function.injective (Cell.fromOption (nm := nm) (\u03c4 := \u03b4))\n  --   | some x, some y, heq => by cases heq; rfl\n  --   | none, none, heq => rfl\n  --   have optMap_inj {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (h : f.injective) : Function.injective (Option.map f)\n  --   | some x, some y, heq => by\n  --     simp only [Option.map, Option.bind, Function.comp] at heq\n  --     injection heq with heq'\n  --     apply congrArg _ (h heq')\n  --   | none, none, _ => rfl\n  --   have := fromOpt_inj left\n  --   have := optMap_inj ULift.up (by intros x y hxy; cases hxy; rfl) this\n  --   exact this\n  -- . exact right\n  \ntheorem groupByRetentive_spec5\n  {\u03b7 : Type u_\u03b7} {\u03c4 : Type u} [dec_\u03b7 : DecidableEq \u03b7]\n  {sch : @Schema \u03b7} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  \u2200 t', t' \u2208 (getColumn2 (groupByRetentive t c) \"groups\" (.tl .hd)).somes \u2192\n  schema t' = sch :=\n\u03bb _ _ _ _ => rfl\n\ntheorem groupByRetentive_spec6\n  {\u03b7 : Type u_\u03b7} {\u03c4 : Type u} [dec_\u03b7 : DecidableEq \u03b7]\n  {sch : @Schema \u03b7} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  nrows (groupByRetentive t c) = (getColumn2 t c.1 c.2).unique.length :=\n\u03bb t c =>\n  groupBy_spec4 (sch' := [(\"key\", ULift.{max (u+1) u_\u03b7} \u03c4),\n                          (\"groups\", Table sch)])\n    t\n    (\u03bb r => getValue r c.1 c.2)\n    (\u03bb r => r)\n    (\u03bb k vs => Row.cons (Cell.fromOption (Option.map ULift.up k))\n                        (Row.cons (Cell.val (Table.mk vs)) Row.nil))\n\ntheorem groupBySubtractive_spec1 [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  header (groupBySubtractive t c) = [\"key\", \"groups\"] :=\n\u03bb _ _ => rfl\n\ntheorem groupBySubtractive_spec2\n  {\u03b7 : Type u_\u03b7} [DecidableEq \u03b7] {sch : @Schema \u03b7}\n  {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (schema (groupBySubtractive t c)).lookupType \u27e8\"key\", Schema.HasName.hd\u27e9\n    = ULift.{max (u+1) u_\u03b7} \u03c4 :=\n\u03bb _ _ => rfl\n\ntheorem groupBySubtractive_spec3\n  {\u03b7 : Type u_\u03b7} [DecidableEq \u03b7] {sch : @Schema \u03b7}\n  {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  (schema (groupBySubtractive t c)).lookupType \u27e8\"groups\", .tl .hd\u27e9 =\n  Table (sch.removeName (Schema.colImpliesName c.snd)) :=\n\u03bb _ _ => rfl\n\n-- TODO: `groupBySubtractive` spec 4\n\n-- Closest approximation possible given uniqueness issues\ntheorem groupBySubtractive_spec5\n  {\u03b7 : Type u_\u03b7} [DecidableEq \u03b7] {sch : @Schema \u03b7}\n  {\u03c4 : Type u} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  \u2200 t', t' \u2208 (getColumn2 (groupBySubtractive t c) \"groups\" (.tl .hd)).somes \u2192\n  header t' = Schema.names (sch.removeName (Schema.colImpliesName c.2)) :=\n\u03bb _ _ _ _ => rfl\n\ntheorem groupBySubtractive_spec6\n  {\u03b7 : Type u_\u03b7} {\u03c4 : Type u} [dec_\u03b7 : DecidableEq \u03b7]\n  {sch : @Schema \u03b7} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  \u2200 t', t' \u2208 (getColumn2 (groupBySubtractive t c) \"groups\" (.tl .hd)).somes \u2192\n  List.Sublist (schema t') sch :=\n\u03bb _ c _ _ => Schema.removeName_sublist sch c.1 (Schema.colImpliesName c.2)\n\ntheorem groupBySubtractive_spec7\n  {\u03b7 : Type u_\u03b7} {\u03c4 : Type u} [dec_\u03b7 : DecidableEq \u03b7]\n  {sch : @Schema \u03b7} [DecidableEq \u03c4] :\n  \u2200 (t : Table sch) (c : (c : \u03b7) \u00d7 sch.HasCol (c, \u03c4)),\n  nrows (groupBySubtractive t c) = (getColumn2 t c.1 c.2).unique.length :=\n\u03bb t c =>\n  groupBy_spec4\n    t\n    (\u03bb r => getValue r c.1 c.2)\n    (\u03bb r => r)\n    (\u03bb k vs => Row.cons (Cell.fromOption (Option.map ULift.up k))\n                        (Row.cons (Cell.val (Table.mk (vs.map (\u03bb r =>\n                          Row.removeColumn (Schema.colImpliesName c.snd) r))))\n                        Row.nil))\n\n-- TODO: `update` (once correctly implemented)\n\n-- Specs 1, 2, and 3 are enforced by types\ntheorem select_spec4 {sch' : @Schema \u03b7} :\n  \u2200 (t : Table sch) (f : Row sch \u2192 Fin (nrows t) \u2192 Row sch'),\n  nrows (select t f) = nrows t :=\n\u03bb t f => Eq.trans (List.length_map _ _) (List.length_verifiedEnum _)\n\n-- All `selectMany` specifications are enforced by types\n\n-- Specs 1 through 5 are enforced by types\ntheorem groupJoin_spec6\n  {\u03ba} [DecidableEq \u03ba] {\u03bd : Type _} {s\u2081 s\u2082 s\u2083 : @Schema \u03b7} :\n  \u2200 (t\u2081 : Table s\u2081) (t\u2082 : Table s\u2082)\n    (getKey\u2081 : Row s\u2081 \u2192 \u03ba) (getKey\u2082 : Row s\u2082 \u2192 \u03ba)\n    (aggregate : Row s\u2081 \u2192 Table s\u2082 \u2192 Row s\u2083),\n  nrows (groupJoin t\u2081 t\u2082 getKey\u2081 getKey\u2082 aggregate) =\n  nrows t\u2081 :=\n\u03bb _ _ _ _ _ => select_spec4 _ _\n\n-- All `join` specifications are enforced by types\n", "meta": {"author": "jrr6", "repo": "lean-tables", "sha": "4eb550d12b6e68639c0c0ae6451bcd55cf8a52d0", "save_path": "github-repos/lean/jrr6-lean-tables", "path": "github-repos/lean/jrr6-lean-tables/lean-tables-4eb550d12b6e68639c0c0ae6451bcd55cf8a52d0/Table/Proofs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3668587933112855}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro, Patrick Massot\n-/\nimport topology.order\n\n/-!\n# Specific classes of maps between topological spaces\n\nThis file introduces the following properties of a map `f : X \u2192 Y` between topological spaces:\n\n* `is_open_map f` means the image of an open set under `f` is open.\n* `is_closed_map f` means the image of a closed set under `f` is closed.\n\n(Open and closed maps need not be continuous.)\n\n* `inducing f` means the topology on `X` is the one induced via `f` from the topology on `Y`.\n  These behave like embeddings except they need not be injective. Instead, points of `X` which\n  are identified by `f` are also inseparable in the topology on `X`.\n* `embedding f` means `f` is inducing and also injective. Equivalently, `f` identifies `X` with\n  a subspace of `Y`.\n* `open_embedding f` means `f` is an embedding with open image, so it identifies `X` with an\n  open subspace of `Y`. Equivalently, `f` is an embedding and an open map.\n* `closed_embedding f` similarly means `f` is an embedding with closed image, so it identifies\n  `X` with a closed subspace of `Y`. Equivalently, `f` is an embedding and a closed map.\n\n* `quotient_map f` is the dual condition to `embedding f`: `f` is surjective and the topology\n  on `Y` is the one coinduced via `f` from the topology on `X`. Equivalently, `f` identifies\n  `Y` with a quotient of `X`. Quotient maps are also sometimes known as identification maps.\n\n## References\n\n* <https://en.wikipedia.org/wiki/Open_and_closed_maps>\n* <https://en.wikipedia.org/wiki/Embedding#General_topology>\n* <https://en.wikipedia.org/wiki/Quotient_space_(topology)#Quotient_map>\n\n## Tags\n\nopen map, closed map, embedding, quotient map, identification map\n\n-/\n\nopen set filter\nopen_locale topological_space filter\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*} {\u03b4 : Type*}\n\nsection inducing\n\n/-- A function `f : \u03b1 \u2192 \u03b2` between topological spaces is inducing if the topology on `\u03b1` is induced\nby the topology on `\u03b2` through `f`, meaning that a set `s : set \u03b1` is open iff it is the preimage\nunder `f` of some open set `t : set \u03b2`. -/\nstructure inducing [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) : Prop :=\n(induced : t\u03b1 = t\u03b2.induced f)\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n\nlemma inducing_id : inducing (@id \u03b1) :=\n\u27e8induced_id.symm\u27e9\n\nprotected lemma inducing.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : inducing g) (hf : inducing f) :\n  inducing (g \u2218 f) :=\n\u27e8by rw [hf.induced, hg.induced, induced_compose]\u27e9\n\nlemma inducing_of_inducing_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : inducing (g \u2218 f)) : inducing f :=\n\u27e8le_antisymm\n    (by rwa \u2190 continuous_iff_le_induced)\n    (by { rw [hgf.induced, \u2190 continuous_iff_le_induced], apply hg.comp continuous_induced_dom })\u27e9\n\nlemma inducing.nhds_eq_comap {f : \u03b1 \u2192 \u03b2} (hf : inducing f) :\n  \u2200 (a : \u03b1), \ud835\udcdd a = comap f (\ud835\udcdd $ f a) :=\n(induced_iff_nhds_eq f).1 hf.induced\n\nlemma inducing.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_eq a\n\nlemma inducing.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) :\n  (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.induced.symm \u25b8 map_nhds_induced_of_mem h\n\nlemma inducing.image_mem_nhds_within {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {a : \u03b1} {s : set \u03b1}\n  (hs : s \u2208 \ud835\udcdd a) : f '' s \u2208 \ud835\udcdd[range f] (f a) :=\nhf.map_nhds_eq a \u25b8 image_mem_map hs\n\nlemma inducing.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : inducing g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nby rw [hg.nhds_eq_comap, tendsto_comap_iff]\n\nlemma inducing.continuous_at_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) {x : \u03b1} :\n  continuous_at f x \u2194 continuous_at (g \u2218 f) x :=\nby simp_rw [continuous_at, inducing.tendsto_nhds_iff hg]\n\nlemma inducing.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\nby simp_rw [continuous_iff_continuous_at, hg.continuous_at_iff]\n\nlemma inducing.continuous_at_iff' {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : inducing f) {x : \u03b1}\n  (h : range f \u2208 \ud835\udcdd (f x)) : continuous_at (g \u2218 f) x \u2194 continuous_at g (f x) :=\nby { simp_rw [continuous_at, filter.tendsto, \u2190 hf.map_nhds_of_mem _ h, filter.map_map] }\n\nprotected lemma inducing.continuous {f : \u03b1 \u2192 \u03b2} (hf : inducing f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nprotected lemma inducing.inducing_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : inducing g) :\n  inducing f \u2194 inducing (g \u2218 f) :=\nbegin\n  refine \u27e8\u03bb h, hg.comp h, \u03bb hgf, inducing_of_inducing_compose _ hg.continuous hgf\u27e9,\n  rw hg.continuous_iff,\n  exact hgf.continuous\nend\n\nlemma inducing.closure_eq_preimage_closure_image {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (s : set \u03b1) :\n  closure s = f \u207b\u00b9' closure (f '' s) :=\nby { ext x, rw [set.mem_preimage, \u2190 closure_induced, hf.induced] }\n\nlemma inducing.is_closed_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_closed s \u2194 \u2203 t, is_closed t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_closed_induced_iff]\n\nlemma inducing.is_closed_iff' {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_closed s \u2194 \u2200 x, f x \u2208 closure (f '' s) \u2192 x \u2208 s :=\nby rw [hf.induced, is_closed_induced_iff']\n\nlemma inducing.is_open_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  is_open s \u2194 \u2203 t, is_open t \u2227 f \u207b\u00b9' t = s :=\nby rw [hf.induced, is_open_induced_iff]\n\nlemma inducing.dense_iff {f : \u03b1 \u2192 \u03b2} (hf : inducing f) {s : set \u03b1} :\n  dense s \u2194 \u2200 x, f x \u2208 closure (f '' s) :=\nby simp only [dense, hf.closure_eq_preimage_closure_image, mem_preimage]\n\nend inducing\n\nsection embedding\n\n/-- A function between topological spaces is an embedding if it is injective,\n  and for all `s : set \u03b1`, `s` is open iff it is the preimage of an open set. -/\nstructure embedding [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2] (f : \u03b1 \u2192 \u03b2)\n  extends inducing f : Prop :=\n(inj : function.injective f)\n\nlemma function.injective.embedding_induced [t : topological_space \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  @embedding \u03b1 \u03b2 (t.induced f) t f :=\n{ induced := rfl,\n  inj := hf }\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\nlemma embedding.mk' (f : \u03b1 \u2192 \u03b2) (inj : function.injective f)\n  (induced : \u2200a, comap f (\ud835\udcdd (f a)) = \ud835\udcdd a) : embedding f :=\n\u27e8\u27e8(induced_iff_nhds_eq f).2 (\u03bb a, (induced a).symm)\u27e9, inj\u27e9\n\nlemma embedding_id : embedding (@id \u03b1) :=\n\u27e8inducing_id, assume a\u2081 a\u2082 h, h\u27e9\n\nlemma embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : embedding g) (hf : embedding f) :\n  embedding (g \u2218 f) :=\n{ inj:= assume a\u2081 a\u2082 h, hf.inj $ hg.inj h,\n  ..hg.to_inducing.comp hf.to_inducing }\n\nlemma embedding_of_embedding_compose {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : continuous f) (hg : continuous g)\n  (hgf : embedding (g \u2218 f)) : embedding f :=\n{ induced := (inducing_of_inducing_compose hf hg hgf.to_inducing).induced,\n  inj := assume a\u2081 a\u2082 h, hgf.inj $ by simp [h, (\u2218)] }\n\nprotected lemma function.left_inverse.embedding {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b1}\n  (h : function.left_inverse f g) (hf : continuous f) (hg : continuous g) :\n  embedding g :=\nembedding_of_embedding_compose hg hf $ h.comp_eq_id.symm \u25b8 embedding_id\n\nlemma embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : embedding f) (a : \u03b1) :\n  (\ud835\udcdd a).map f = \ud835\udcdd[range f] (f a) :=\nhf.1.map_nhds_eq a\n\nlemma embedding.map_nhds_of_mem {f : \u03b1 \u2192 \u03b2}\n  (hf : embedding f) (a : \u03b1) (h : range f \u2208 \ud835\udcdd (f a)) : (\ud835\udcdd a).map f = \ud835\udcdd (f a) :=\nhf.1.map_nhds_of_mem a h\n\nlemma embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : embedding g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nhg.to_inducing.tendsto_nhds_iff\n\nlemma embedding.continuous_iff {f : \u03b1 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : embedding g) :\n  continuous f \u2194 continuous (g \u2218 f) :=\ninducing.continuous_iff hg.1\n\nlemma embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : embedding f) : continuous f :=\ninducing.continuous hf.1\n\nlemma embedding.closure_eq_preimage_closure_image {e : \u03b1 \u2192 \u03b2} (he : embedding e) (s : set \u03b1) :\n  closure s = e \u207b\u00b9' closure (e '' s) :=\nhe.1.closure_eq_preimage_closure_image s\n\nend embedding\n\n/-- A function between topological spaces is a quotient map if it is surjective,\n  and for all `s : set \u03b2`, `s` is open iff its preimage is an open set. -/\ndef quotient_map {\u03b1 : Type*} {\u03b2 : Type*} [t\u03b1 : topological_space \u03b1] [t\u03b2 : topological_space \u03b2]\n  (f : \u03b1 \u2192 \u03b2) : Prop :=\nfunction.surjective f \u2227 t\u03b2 = t\u03b1.coinduced f\n\nlemma quotient_map_iff {\u03b1 \u03b2 : Type*} [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  quotient_map f \u2194 function.surjective f \u2227 \u2200 s : set \u03b2, is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nand_congr iff.rfl topological_space_eq_iff\n\nnamespace quotient_map\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] [topological_space \u03b4]\n  {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n\nprotected lemma id : quotient_map (@id \u03b1) :=\n\u27e8assume a, \u27e8a, rfl\u27e9, coinduced_id.symm\u27e9\n\nprotected lemma comp (hg : quotient_map g) (hf : quotient_map f) :\n  quotient_map (g \u2218 f) :=\n\u27e8hg.left.comp hf.left, by rw [hg.right, hf.right, coinduced_compose]\u27e9\n\nprotected lemma of_quotient_map_compose (hf : continuous f) (hg : continuous g)\n  (hgf : quotient_map (g \u2218 f)) : quotient_map g :=\n\u27e8hgf.1.of_comp,\n  le_antisymm\n    (by { rw [hgf.right, \u2190 continuous_iff_coinduced_le], apply continuous_coinduced_rng.comp hf })\n    (by rwa \u2190 continuous_iff_coinduced_le)\u27e9\n\nprotected lemma continuous_iff (hf : quotient_map f) :\n  continuous g \u2194 continuous (g \u2218 f) :=\nby rw [continuous_iff_coinduced_le, continuous_iff_coinduced_le, hf.right, coinduced_compose]\n\nprotected lemma continuous (hf : quotient_map f) : continuous f :=\nhf.continuous_iff.mp continuous_id\n\nprotected lemma surjective (hf : quotient_map f) : function.surjective f := hf.1\n\nprotected lemma is_open_preimage (hf : quotient_map f) {s : set \u03b2} :\n  is_open (f \u207b\u00b9' s) \u2194 is_open s :=\n((quotient_map_iff.1 hf).2 s).symm\n\nprotected \n\nend quotient_map\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be an *open map*, if the image of any open `U : set \u03b1`\nis open in `\u03b2`. -/\ndef is_open_map [topological_space \u03b1] [topological_space \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n\u2200 U : set \u03b1, is_open U \u2192 is_open (f '' U)\n\nnamespace is_open_map\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3] {f : \u03b1 \u2192 \u03b2}\nopen function\n\nprotected lemma id : is_open_map (@id \u03b1) := assume s hs, by rwa [image_id]\n\nprotected lemma comp\n  {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_open_map g) (hf : is_open_map f) : is_open_map (g \u2218 f) :=\nby intros s hs; rw [image_comp]; exact hg _ (hf _ hs)\n\nlemma is_open_range (hf : is_open_map f) : is_open (range f) :=\nby { rw \u2190 image_univ, exact hf _ is_open_univ }\n\nlemma image_mem_nhds (hf : is_open_map f) {x : \u03b1} {s : set \u03b1} (hx : s \u2208 \ud835\udcdd x) :\n  f '' s \u2208 \ud835\udcdd (f x) :=\nlet \u27e8t, hts, ht, hxt\u27e9 := mem_nhds_iff.1 hx in\nmem_of_superset (is_open.mem_nhds (hf t ht) (mem_image_of_mem _ hxt)) (image_subset _ hts)\n\nlemma maps_to_interior (hf : is_open_map f) {s : set \u03b1} {t : set \u03b2} (h : maps_to f s t) :\n  maps_to f (interior s) (interior t) :=\nmaps_to'.2 $ interior_maximal (h.mono interior_subset subset.rfl).image_subset\n  (hf _ is_open_interior)\n\nlemma image_interior_subset (hf : is_open_map f) (s : set \u03b1) :\n  f '' interior s \u2286 interior (f '' s) :=\n(hf.maps_to_interior (maps_to_image f s)).image_subset\n\nlemma nhds_le (hf : is_open_map f) (a : \u03b1) : \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\nle_map $ \u03bb s, hf.image_mem_nhds\n\nlemma of_nhds_le (hf : \u2200 a, \ud835\udcdd (f a) \u2264 map f (\ud835\udcdd a)) : is_open_map f :=\n\u03bb s hs, is_open_iff_mem_nhds.2 $ \u03bb b \u27e8a, has, hab\u27e9,\n  hab \u25b8 hf _ (image_mem_map $ is_open.mem_nhds hs has)\n\nlemma of_sections {f : \u03b1 \u2192 \u03b2}\n  (h : \u2200 x, \u2203 g : \u03b2 \u2192 \u03b1, continuous_at g (f x) \u2227 g (f x) = x \u2227 right_inverse g f) :\n  is_open_map f :=\nof_nhds_le $ \u03bb x, let \u27e8g, hgc, hgx, hgf\u27e9 := h x in\ncalc \ud835\udcdd (f x) = map f (map g (\ud835\udcdd (f x))) : by rw [map_map, hgf.comp_eq_id, map_id]\n... \u2264 map f (\ud835\udcdd (g (f x))) : map_mono hgc\n... = map f (\ud835\udcdd x) : by rw hgx\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_open_map f :=\nof_sections $ \u03bb x, \u27e8f', h.continuous_at, r_inv _, l_inv\u27e9\n\n/-- A continuous surjective open map is a quotient map. -/\nlemma to_quotient_map {f : \u03b1 \u2192 \u03b2}\n  (open_map : is_open_map f) (cont : continuous f) (surj : surjective f) :\n  quotient_map f :=\nquotient_map_iff.2 \u27e8surj, \u03bb s, \u27e8\u03bb h, h.preimage cont, \u03bb h, surj.image_preimage s \u25b8 open_map _ h\u27e9\u27e9\n\nlemma interior_preimage_subset_preimage_interior (hf : is_open_map f) {s : set \u03b2} :\n  interior (f\u207b\u00b9' s) \u2286 f\u207b\u00b9' (interior s) :=\nhf.maps_to_interior (maps_to_preimage _ _)\n\nlemma preimage_interior_eq_interior_preimage (hf\u2081 : is_open_map f) (hf\u2082 : continuous f)\n  (s : set \u03b2) :\n  f\u207b\u00b9' (interior s) = interior (f\u207b\u00b9' s) :=\nsubset.antisymm\n  (preimage_interior_subset_interior_preimage hf\u2082)\n  (interior_preimage_subset_preimage_interior hf\u2081)\n\nlemma preimage_closure_subset_closure_preimage (hf : is_open_map f) {s : set \u03b2} :\n  f \u207b\u00b9' (closure s) \u2286 closure (f \u207b\u00b9' s) :=\nbegin\n  rw \u2190 compl_subset_compl,\n  simp only [\u2190 interior_compl, \u2190 preimage_compl, hf.interior_preimage_subset_preimage_interior]\nend\n\nlemma preimage_closure_eq_closure_preimage (hf : is_open_map f) (hfc : continuous f) (s : set \u03b2) :\n  f \u207b\u00b9' (closure s) = closure (f \u207b\u00b9' s) :=\nhf.preimage_closure_subset_closure_preimage.antisymm (hfc.closure_preimage_subset s)\n\nlemma preimage_frontier_subset_frontier_preimage (hf : is_open_map f) {s : set \u03b2} :\n  f \u207b\u00b9' (frontier s) \u2286 frontier (f \u207b\u00b9' s) :=\nby simpa only [frontier_eq_closure_inter_closure, preimage_inter]\n  using inter_subset_inter hf.preimage_closure_subset_closure_preimage\n    hf.preimage_closure_subset_closure_preimage\n\nlemma preimage_frontier_eq_frontier_preimage (hf : is_open_map f) (hfc : continuous f) (s : set \u03b2) :\n  f \u207b\u00b9' (frontier s) = frontier (f \u207b\u00b9' s) :=\nby simp only [frontier_eq_closure_inter_closure, preimage_inter, preimage_compl,\n  hf.preimage_closure_eq_closure_preimage hfc]\n\nend is_open_map\n\nlemma is_open_map_iff_nhds_le [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_open_map f \u2194 \u2200(a:\u03b1), \ud835\udcdd (f a) \u2264 (\ud835\udcdd a).map f :=\n\u27e8\u03bb hf, hf.nhds_le, is_open_map.of_nhds_le\u27e9\n\nlemma is_open_map_iff_interior [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_open_map f \u2194 \u2200 s, f '' (interior s) \u2286 interior (f '' s) :=\n\u27e8is_open_map.image_interior_subset, \u03bb hs u hu, subset_interior_iff_open.mp $\n  calc f '' u = f '' (interior u) : by rw hu.interior_eq\n          ... \u2286 interior (f '' u) : hs u\u27e9\n\n/-- An inducing map with an open range is an open map. -/\nprotected lemma inducing.is_open_map [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2}\n  (hi : inducing f) (ho : is_open (range f)) :\n  is_open_map f :=\nis_open_map.of_nhds_le $ \u03bb x, (hi.map_nhds_of_mem _ $ is_open.mem_nhds ho $ mem_range_self _).ge\n\nsection is_closed_map\nvariables [topological_space \u03b1] [topological_space \u03b2]\n\n/-- A map `f : \u03b1 \u2192 \u03b2` is said to be a *closed map*, if the image of any closed `U : set \u03b1`\nis closed in `\u03b2`. -/\ndef is_closed_map (f : \u03b1 \u2192 \u03b2) := \u2200 U : set \u03b1, is_closed U \u2192 is_closed (f '' U)\n\nend is_closed_map\n\nnamespace is_closed_map\n\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\nopen function\n\nprotected lemma id : is_closed_map (@id \u03b1) := assume s hs, by rwa image_id\n\nprotected lemma comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (hg : is_closed_map g) (hf : is_closed_map f) :\n  is_closed_map (g \u2218 f) :=\nby { intros s hs, rw image_comp, exact hg _ (hf _ hs) }\n\nlemma closure_image_subset {f : \u03b1 \u2192 \u03b2} (hf : is_closed_map f) (s : set \u03b1) :\n  closure (f '' s) \u2286 f '' closure s :=\nclosure_minimal (image_subset _ subset_closure) (hf _ is_closed_closure)\n\nlemma of_inverse {f : \u03b1 \u2192 \u03b2} {f' : \u03b2 \u2192 \u03b1}\n  (h : continuous f') (l_inv : left_inverse f f') (r_inv : right_inverse f f') :\n  is_closed_map f :=\nassume s hs,\nhave f' \u207b\u00b9' s = f '' s, by ext x; simp [mem_image_iff_of_inverse r_inv l_inv],\nthis \u25b8 hs.preimage h\n\nlemma of_nonempty {f : \u03b1 \u2192 \u03b2} (h : \u2200 s, is_closed s \u2192 s.nonempty \u2192 is_closed (f '' s)) :\n  is_closed_map f :=\nbegin\n  intros s hs, cases eq_empty_or_nonempty s with h2s h2s,\n  { simp_rw [h2s, image_empty, is_closed_empty] },\n  { exact h s hs h2s }\nend\n\nlemma closed_range {f : \u03b1 \u2192 \u03b2} (hf : is_closed_map f) : is_closed (range f) :=\n@image_univ _ _ f \u25b8 hf _ is_closed_univ\n\nend is_closed_map\n\nlemma inducing.is_closed_map [topological_space \u03b1] [topological_space \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : inducing f) (h : is_closed (range f)) : is_closed_map f :=\nbegin\n  intros s hs,\n  rcases hf.is_closed_iff.1 hs with \u27e8t, ht, rfl\u27e9,\n  rw image_preimage_eq_inter_range,\n  exact ht.inter h\nend\n\nlemma is_closed_map_iff_closure_image [topological_space \u03b1] [topological_space \u03b2] {f : \u03b1 \u2192 \u03b2} :\n  is_closed_map f \u2194 \u2200 s, closure (f '' s) \u2286 f '' closure s :=\n\u27e8is_closed_map.closure_image_subset, \u03bb hs c hc, is_closed_of_closure_subset $\n  calc closure (f '' c) \u2286 f '' (closure c) : hs c\n                    ... = f '' c : by rw hc.closure_eq\u27e9\n\nsection open_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- An open embedding is an embedding with open image. -/\nstructure open_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(open_range : is_open $ range f)\n\nlemma open_embedding.is_open_map {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : is_open_map f :=\nhf.to_embedding.to_inducing.is_open_map hf.open_range\n\nlemma open_embedding.map_nhds_eq {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) (a : \u03b1) :\n  map f (\ud835\udcdd a) = \ud835\udcdd (f a) :=\nhf.to_embedding.map_nhds_of_mem _ $ hf.open_range.mem_nhds $ mem_range_self _\n\nlemma open_embedding.open_iff_image_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b1} : is_open s \u2194 is_open (f '' s) :=\n\u27e8hf.is_open_map s,\n \u03bb h, begin\n   convert \u2190 h.preimage hf.to_embedding.continuous,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma open_embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {f : \u03b9 \u2192 \u03b2} {g : \u03b2 \u2192 \u03b3} {a : filter \u03b9} {b : \u03b2} (hg : open_embedding g) :\n  tendsto f a (\ud835\udcdd b) \u2194 tendsto (g \u2218 f) a (\ud835\udcdd (g b)) :=\nhg.to_embedding.tendsto_nhds_iff\n\nlemma open_embedding.continuous {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma open_embedding.open_iff_preimage_open {f : \u03b1 \u2192 \u03b2} (hf : open_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_open s \u2194 is_open (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.open_iff_image_open.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma open_embedding_of_embedding_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f)\n  (h\u2082 : is_open_map f) : open_embedding f :=\n\u27e8h\u2081, h\u2082.is_open_range\u27e9\n\nlemma open_embedding_of_continuous_injective_open {f : \u03b1 \u2192 \u03b2} (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_open_map f) : open_embedding f :=\nbegin\n  refine open_embedding_of_embedding_open \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s,\n  change is_open _ \u2192 is_open _,\n  rw is_open_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma open_embedding_id : open_embedding (@id \u03b1) :=\n\u27e8embedding_id, is_open_map.id.is_open_range\u27e9\n\nlemma open_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : open_embedding g) (hf : open_embedding f) : open_embedding (g \u2218 f) :=\n\u27e8hg.1.comp hf.1, (hg.is_open_map.comp hf.is_open_map).is_open_range\u27e9\n\nlemma open_embedding_of_open_embedding_compose {\u03b1 \u03b2 \u03b3 : Type*} [topological_space \u03b1]\n  [topological_space \u03b2] [topological_space \u03b3] (f : \u03b1 \u2192 \u03b2) {g : \u03b2 \u2192 \u03b3} (hg : open_embedding g)\n    (h : open_embedding (g \u2218 f)) : open_embedding f :=\nbegin\n  have hf := hg.to_embedding.continuous_iff.mpr h.continuous,\n  split,\n  { exact embedding_of_embedding_compose hf hg.continuous h.to_embedding },\n  { rw [hg.open_iff_image_open, \u2190 set.image_univ, \u2190 set.image_comp, \u2190 h.open_iff_image_open],\n    exact is_open_univ }\nend\n\nlemma open_embedding_iff_open_embedding_compose {\u03b1 \u03b2 \u03b3 : Type*} [topological_space \u03b1]\n  [topological_space \u03b2] [topological_space \u03b3] (f : \u03b1 \u2192 \u03b2) {g : \u03b2 \u2192 \u03b3} (hg : open_embedding g) :\n    open_embedding (g \u2218 f) \u2194 open_embedding f :=\n\u27e8open_embedding_of_open_embedding_compose f hg, hg.comp\u27e9\n\nend open_embedding\n\nsection closed_embedding\nvariables [topological_space \u03b1] [topological_space \u03b2] [topological_space \u03b3]\n\n/-- A closed embedding is an embedding with closed image. -/\nstructure closed_embedding (f : \u03b1 \u2192 \u03b2) extends embedding f : Prop :=\n(closed_range : is_closed $ range f)\n\nvariables {f : \u03b1 \u2192 \u03b2}\n\nlemma closed_embedding.tendsto_nhds_iff {\u03b9 : Type*}\n  {g : \u03b9 \u2192 \u03b1} {a : filter \u03b9} {b : \u03b1} (hf : closed_embedding f) :\n  tendsto g a (\ud835\udcdd b) \u2194 tendsto (f \u2218 g) a (\ud835\udcdd (f b)) :=\nhf.to_embedding.tendsto_nhds_iff\n\nlemma closed_embedding.continuous (hf : closed_embedding f) : continuous f :=\nhf.to_embedding.continuous\n\nlemma closed_embedding.is_closed_map (hf : closed_embedding f) : is_closed_map f :=\nhf.to_embedding.to_inducing.is_closed_map hf.closed_range\n\nlemma closed_embedding.closed_iff_image_closed (hf : closed_embedding f)\n  {s : set \u03b1} : is_closed s \u2194 is_closed (f '' s) :=\n\u27e8hf.is_closed_map s,\n \u03bb h, begin\n   convert \u2190continuous_iff_is_closed.mp hf.continuous _ h,\n   apply preimage_image_eq _ hf.inj\n end\u27e9\n\nlemma closed_embedding.closed_iff_preimage_closed (hf : closed_embedding f)\n  {s : set \u03b2} (hs : s \u2286 range f) : is_closed s \u2194 is_closed (f \u207b\u00b9' s) :=\nbegin\n  convert \u2190hf.closed_iff_image_closed.symm,\n  rwa [image_preimage_eq_inter_range, inter_eq_self_of_subset_left]\nend\n\nlemma closed_embedding_of_embedding_closed (h\u2081 : embedding f)\n  (h\u2082 : is_closed_map f) : closed_embedding f :=\n\u27e8h\u2081, by convert h\u2082 univ is_closed_univ; simp\u27e9\n\nlemma closed_embedding_of_continuous_injective_closed (h\u2081 : continuous f)\n  (h\u2082 : function.injective f) (h\u2083 : is_closed_map f) : closed_embedding f :=\nbegin\n  refine closed_embedding_of_embedding_closed \u27e8\u27e8_\u27e9, h\u2082\u27e9 h\u2083,\n  apply le_antisymm (continuous_iff_le_induced.mp h\u2081) _,\n  intro s',\n  change is_open _ \u2264 is_open _,\n  rw [\u2190is_closed_compl_iff, \u2190is_closed_compl_iff],\n  generalize : s'\u1d9c = s,\n  rw is_closed_induced_iff,\n  refine \u03bb hs, \u27e8f '' s, h\u2083 s hs, _\u27e9,\n  rw preimage_image_eq _ h\u2082\nend\n\nlemma closed_embedding_id : closed_embedding (@id \u03b1) :=\n\u27e8embedding_id, by convert is_closed_univ; apply range_id\u27e9\n\nlemma closed_embedding.comp {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2}\n  (hg : closed_embedding g) (hf : closed_embedding f) : closed_embedding (g \u2218 f) :=\n\u27e8hg.to_embedding.comp hf.to_embedding, show is_closed (range (g \u2218 f)),\n by rw [range_comp, \u2190hg.closed_iff_image_closed]; exact hf.closed_range\u27e9\n\nlemma closed_embedding.closure_image_eq {f : \u03b1 \u2192 \u03b2} (hf : closed_embedding f) (s : set \u03b1) :\n  closure (f '' s) = f '' closure s :=\nle_antisymm (is_closed_map_iff_closure_image.mp hf.is_closed_map _)\n  (image_closure_subset_closure_image hf.continuous)\n\nend closed_embedding\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/topology/maps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.542863297964157, "lm_q2_score": 0.6757645879592642, "lm_q1q2_score": 0.36684779286695585}}
{"text": "import semantics.syntacticCat_cartesian\n\n\nnamespace synPoset_monadic\n\nopen synPoset\nopen deduction_basic\nopen deduction_cart\nopen deduction_monadic\n\nvariable {Form : Type}\n\n\nlemma diamond_liftable [Der : has_diamond Form] : \u2200 \u03c6 \u03c6' : Form,\n  (\u03c6 \u22a3\u22a2 \u03c6') \u2192 \u2983 \u25c7\u03c6 \u2984 = \u2983\u25c7\u03c6'\u2984 :=\n  begin\n    assume \u03c6 \u03c6' h,\n    apply quotient.sound,\n    cases h with \u03c6\u03c6' \u03c6'\u03c6,\n    split,\n    apply Der.dmap,\n    exact \u03c6\u03c6',\n    apply Der.dmap,\n    exact \u03c6'\u03c6,\n  end\n\ndef diamond_eq [Der : has_diamond Form] : Form _eq \u2192 Form _eq :=\n  quot.lift (\u03bb \u03c6 , \u2983Der.diamond \u03c6\u2984) diamond_liftable\n\nend synPoset_monadic\n", "meta": {"author": "lean-catLogic", "repo": "formalization", "sha": "bf4eda918a90ed04b8867bfcc587cc0dda1c4347", "save_path": "github-repos/lean/lean-catLogic-formalization", "path": "github-repos/lean/lean-catLogic-formalization/formalization-bf4eda918a90ed04b8867bfcc587cc0dda1c4347/src/semantics/syntacticPoset_monadic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7279754607093178, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3668313266451754}}
{"text": "universe u v\n\ntheorem eqLitOfSize0 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 0) : a = #[] :=\na.toArrayLit_eq 0 hsz\n\ntheorem eqLitOfSize1 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 1) : a = #[a.getLit 0 hsz (of_decide_eq_true rfl)] :=\na.toArrayLit_eq 1 hsz\n\ntheorem eqLitOfSize2 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 2) : a = #[a.getLit 0 hsz (of_decide_eq_true rfl), a.getLit 1 hsz (of_decide_eq_true rfl)] :=\na.toArrayLit_eq 2 hsz\n\ntheorem eqLitOfSize3 {\u03b1 : Type u} (a : Array \u03b1) (hsz : a.size = 3) :\n  a = #[a.getLit 0 hsz (of_decide_eq_true rfl), a.getLit 1 hsz (of_decide_eq_true rfl), a.getLit 2 hsz (of_decide_eq_true rfl)] :=\na.toArrayLit_eq 3 hsz\n\n/-\nMatcher for the following patterns\n```\n| #[]           => _\n| #[a\u2081]         => _\n| #[a\u2081, a\u2082, a\u2083] => _\n| a             => _\n``` -/\ndef matchArrayLit {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v) (a : Array \u03b1)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    : C a :=\nif h : a.size = 0 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2081 ()) _ (a.toArrayLit_eq 0 h).symm\nelse if h : a.size = 1 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2082 (a.getLit 0 h (of_decide_eq_true rfl))) _ (a.toArrayLit_eq 1 h).symm\nelse if h : a.size = 3 then\n  @Eq.rec _ _ (fun x _ => C x) (h\u2083 (a.getLit 0 h (of_decide_eq_true rfl)) (a.getLit 1 h (of_decide_eq_true rfl)) (a.getLit 2 h (of_decide_eq_true rfl))) _ (a.toArrayLit_eq 3 h).symm\nelse\n  h\u2084 a\n\n/- Equational lemmas that should be generated automatically. -/\ntheorem matchArrayLit.eq1 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    : matchArrayLit C #[] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2081 () :=\nrfl\n\ntheorem matchArrayLit.eq2 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    (a\u2081 : \u03b1)\n    : matchArrayLit C #[a\u2081] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2082 a\u2081 :=\nrfl\n\ntheorem matchArrayLit.eq3 {\u03b1 : Type u} (C : Array \u03b1 \u2192 Sort v)\n    (h\u2081 : Unit \u2192      C #[])\n    (h\u2082 : \u2200 a\u2081,       C #[a\u2081])\n    (h\u2083 : \u2200 a\u2081 a\u2082 a\u2083, C #[a\u2081, a\u2082, a\u2083])\n    (h\u2084 : \u2200 a,        C a)\n    (a\u2081 a\u2082 a\u2083 : \u03b1)\n    : matchArrayLit C #[a\u2081, a\u2082, a\u2083] h\u2081 h\u2082 h\u2083 h\u2084 = h\u2083 a\u2081 a\u2082 a\u2083 :=\nrfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/matchArrayLit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.36673049578507216}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.terminal\nimport Mathlib.category_theory.limits.shapes.binary_products\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.images\nimport Mathlib.PostPort\n\nuniverses v u l u_1 v' u' \n\nnamespace Mathlib\n\n/-!\n# Zero morphisms and zero objects\n\nA category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. (Notice this is extra\nstructure, not merely a property.)\n\nA category \"has a zero object\" if it has an object which is both initial and terminal. Having a\nzero object provides zero morphisms, as the unique morphisms factoring through the zero object.\n\n## References\n\n* https://en.wikipedia.org/wiki/Zero_morphism\n* [F. Borceux, *Handbook of Categorical Algebra 2*][borceux-vol2]\n-/\n\nnamespace category_theory.limits\n\n\n/-- A category \"has zero morphisms\" if there is a designated \"zero morphism\" in each morphism space,\nand compositions of zero morphisms with anything give the zero morphism. -/\nclass has_zero_morphisms (C : Type u) [category C] \nwhere\n  has_zero : (X Y : C) \u2192 HasZero (X \u27f6 Y)\n  comp_zero' : autoParam (\u2200 {X Y : C} (f : X \u27f6 Y), C \u2192 f \u226b 0 = 0)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  zero_comp' : autoParam (C \u2192 \u2200 {Y Z : C} (f : Y \u27f6 Z), 0 \u226b f = 0)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem has_zero_morphisms.comp_zero {C : Type u} [category C] [c : has_zero_morphisms C] {X : C} {Y : C} (f : X \u27f6 Y) (Z : C) : f \u226b 0 = 0 := sorry\n\ntheorem has_zero_morphisms.zero_comp {C : Type u} [category C] [c : has_zero_morphisms C] (X : C) {Y : C} {Z : C} (f : Y \u27f6 Z) : 0 \u226b f = 0 := sorry\n\n@[simp] theorem comp_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {f : X \u27f6 Y} {Z : C} : f \u226b 0 = 0 :=\n  has_zero_morphisms.comp_zero f Z\n\n@[simp] theorem zero_comp {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C} {f : Y \u27f6 Z} : 0 \u226b f = 0 :=\n  has_zero_morphisms.zero_comp X f\n\nprotected instance has_zero_morphisms_pempty : has_zero_morphisms (discrete pempty) :=\n  has_zero_morphisms.mk\n\nprotected instance has_zero_morphisms_punit : has_zero_morphisms (discrete PUnit) :=\n  has_zero_morphisms.mk\n\nnamespace has_zero_morphisms\n\n\n/-- This lemma will be immediately superseded by `ext`, below. -/\n/--\nIf you're tempted to use this lemma \"in the wild\", you should probably\ncarefully consider whether you've made a mistake in allowing two\ninstances of `has_zero_morphisms` to exist at all.\n\nSee, particularly, the note on `zero_morphisms_of_zero_object` below.\n-/\ntheorem ext {C : Type u} [category C] (I : has_zero_morphisms C) (J : has_zero_morphisms C) : I = J := sorry\n\nprotected instance subsingleton {C : Type u} [category C] : subsingleton (has_zero_morphisms C) :=\n  subsingleton.intro ext\n\nend has_zero_morphisms\n\n\ntheorem zero_of_comp_mono {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C} {f : X \u27f6 Y} (g : Y \u27f6 Z) [mono g] (h : f \u226b g = 0) : f = 0 :=\n  eq.mp (Eq._oldrec (Eq.refl (f \u226b g = 0 \u226b g)) (propext (cancel_mono g)))\n    (eq.mp (Eq._oldrec (Eq.refl (f \u226b g = 0)) (Eq.symm zero_comp)) h)\n\ntheorem zero_of_epi_comp {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C} (f : X \u27f6 Y) {g : Y \u27f6 Z} [epi f] (h : f \u226b g = 0) : g = 0 :=\n  eq.mp (Eq._oldrec (Eq.refl (f \u226b g = f \u226b 0)) (propext (cancel_epi f)))\n    (eq.mp (Eq._oldrec (Eq.refl (f \u226b g = 0)) (Eq.symm comp_zero)) h)\n\ntheorem eq_zero_of_image_eq_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f] (w : image.\u03b9 f = 0) : f = 0 := sorry\n\ntheorem nonzero_image_of_nonzero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {f : X \u27f6 Y} [has_image f] (w : f \u2260 0) : image.\u03b9 f \u2260 0 :=\n  fun (h : image.\u03b9 f = 0) => w (eq_zero_of_image_eq_zero h)\n\ntheorem equivalence_preserves_zero_morphisms {C : Type u} [category C] (D : Type u') [category D] [has_zero_morphisms C] [has_zero_morphisms D] (F : C \u224c D) (X : C) (Y : C) : functor.map (equivalence.functor F) 0 = 0 := sorry\n\n@[simp] theorem is_equivalence_preserves_zero_morphisms {C : Type u} [category C] (D : Type u') [category D] [has_zero_morphisms C] [has_zero_morphisms D] (F : C \u2964 D) [is_equivalence F] (X : C) (Y : C) : functor.map F 0 = 0 := sorry\n\n/-- A category \"has a zero object\" if it has an object which is both initial and terminal. -/\nclass has_zero_object (C : Type u) [category C] \nwhere\n  zero : C\n  unique_to : (X : C) \u2192 unique (zero \u27f6 X)\n  unique_from : (X : C) \u2192 unique (X \u27f6 zero)\n\nprotected instance has_zero_object_punit : has_zero_object (discrete PUnit) :=\n  has_zero_object.mk PUnit.unit (fun (X : discrete PUnit) => punit.cases_on X (unique.mk sorry sorry))\n    fun (X : discrete PUnit) => punit.cases_on X (unique.mk sorry sorry)\n\nnamespace has_zero_object\n\n\n/--\nConstruct a `has_zero C` for a category with a zero object.\nThis can not be a global instance as it will trigger for every `has_zero C` typeclass search.\n-/\nprotected def has_zero {C : Type u} [category C] [has_zero_object C] : HasZero C :=\n  { zero := zero }\n\ntheorem to_zero_ext {C : Type u} [category C] [has_zero_object C] {X : C} (f : X \u27f6 0) (g : X \u27f6 0) : f = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f = g)) (unique.uniq (unique_from X) f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (Inhabited.default = g)) (unique.uniq (unique_from X) g)))\n      (Eq.refl Inhabited.default))\n\ntheorem from_zero_ext {C : Type u} [category C] [has_zero_object C] {X : C} (f : 0 \u27f6 X) (g : 0 \u27f6 X) : f = g :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f = g)) (unique.uniq (unique_to X) f)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (Inhabited.default = g)) (unique.uniq (unique_to X) g))) (Eq.refl Inhabited.default))\n\nprotected instance category_theory.iso.subsingleton {C : Type u} [category C] [has_zero_object C] (X : C) : subsingleton (X \u2245 0) :=\n  subsingleton.intro fun (a b : X \u2245 0) => iso.ext (of_as_true trivial)\n\nprotected instance category_theory.mono {C : Type u} [category C] [has_zero_object C] {X : C} (f : 0 \u27f6 X) : mono f :=\n  mono.mk fun (Z : C) (g h : Z \u27f6 0) (w : g \u226b f = h \u226b f) => to_zero_ext g h\n\nprotected instance category_theory.epi {C : Type u} [category C] [has_zero_object C] {X : C} (f : X \u27f6 0) : epi f :=\n  epi.mk fun (Z : C) (g h : 0 \u27f6 Z) (w : f \u226b g = f \u226b h) => from_zero_ext g h\n\n/-- A category with a zero object has zero morphisms.\n\n    It is rarely a good idea to use this. Many categories that have a zero object have zero\n    morphisms for some other reason, for example from additivity. Library code that uses\n    `zero_morphisms_of_zero_object` will then be incompatible with these categories because\n    the `has_zero_morphisms` instances will not be definitionally equal. For this reason library\n    code should generally ask for an instance of `has_zero_morphisms` separately, even if it already\n    asks for an instance of `has_zero_objects`. -/\ndef zero_morphisms_of_zero_object {C : Type u} [category C] [has_zero_object C] : has_zero_morphisms C :=\n  has_zero_morphisms.mk\n\n/-- A zero object is in particular initial. -/\ntheorem has_initial {C : Type u} [category C] [has_zero_object C] : has_initial C :=\n  has_initial_of_unique 0\n\n/-- A zero object is in particular terminal. -/\ntheorem has_terminal {C : Type u} [category C] [has_zero_object C] : has_terminal C :=\n  has_terminal_of_unique 0\n\nend has_zero_object\n\n\n@[simp] theorem id_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] : \ud835\udfd9 = 0 :=\n  has_zero_object.from_zero_ext \ud835\udfd9 0\n\n/--  An arrow ending in the zero object is zero -/\n-- This can't be a `simp` lemma because the left hand side would be a metavariable.\n\ntheorem zero_of_to_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} (f : X \u27f6 0) : f = 0 :=\n  has_zero_object.to_zero_ext f 0\n\ntheorem zero_of_target_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} {Y : C} (f : X \u27f6 Y) (i : Y \u2245 0) : f = 0 := sorry\n\n/-- An arrow starting at the zero object is zero -/\ntheorem zero_of_from_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} (f : 0 \u27f6 X) : f = 0 :=\n  has_zero_object.from_zero_ext f 0\n\ntheorem zero_of_source_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} {Y : C} (f : X \u27f6 Y) (i : X \u2245 0) : f = 0 := sorry\n\ntheorem mono_of_source_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} {Y : C} (f : X \u27f6 Y) (i : X \u2245 0) : mono f := sorry\n\ntheorem epi_of_target_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] {X : C} {Y : C} (f : X \u27f6 Y) (i : Y \u2245 0) : epi f := sorry\n\n/--\nAn object `X` has `\ud835\udfd9 X = 0` if and only if it is isomorphic to the zero object.\n\nBecause `X \u2245 0` contains data (even if a subsingleton), we express this `\u2194` as an `\u2243`.\n-/\ndef id_zero_equiv_iso_zero {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] (X : C) : \ud835\udfd9 = 0 \u2243 (X \u2245 0) :=\n  equiv.mk (fun (h : \ud835\udfd9 = 0) => iso.mk 0 0) sorry sorry sorry\n\n@[simp] theorem id_zero_equiv_iso_zero_apply_hom {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] (X : C) (h : \ud835\udfd9 = 0) : iso.hom (coe_fn (id_zero_equiv_iso_zero X) h) = 0 :=\n  rfl\n\n@[simp] theorem id_zero_equiv_iso_zero_apply_inv {C : Type u} [category C] [has_zero_object C] [has_zero_morphisms C] (X : C) (h : \ud835\udfd9 = 0) : iso.inv (coe_fn (id_zero_equiv_iso_zero X) h) = 0 :=\n  rfl\n\n/--\nA zero morphism `0 : X \u27f6 Y` is an isomorphism if and only if\nthe identities on both `X` and `Y` are zero.\n-/\ndef is_iso_zero_equiv {C : Type u} [category C] [has_zero_morphisms C] (X : C) (Y : C) : is_iso 0 \u2243 \ud835\udfd9 = 0 \u2227 \ud835\udfd9 = 0 :=\n  equiv.mk sorry (fun (h : \ud835\udfd9 = 0 \u2227 \ud835\udfd9 = 0) => is_iso.mk 0) sorry sorry\n\n/--\nA zero morphism `0 : X \u27f6 X` is an isomorphism if and only if\nthe identity on `X` is zero.\n-/\ndef is_iso_zero_self_equiv {C : Type u} [category C] [has_zero_morphisms C] (X : C) : is_iso 0 \u2243 \ud835\udfd9 = 0 :=\n  eq.mpr sorry (eq.mp sorry (is_iso_zero_equiv X X))\n\n/--\nA zero morphism `0 : X \u27f6 Y` is an isomorphism if and only if\n`X` and `Y` are isomorphic to the zero object.\n-/\ndef is_iso_zero_equiv_iso_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] (X : C) (Y : C) : is_iso 0 \u2243 (X \u2245 0) \u00d7 (Y \u2245 0) :=\n  equiv.trans (is_iso_zero_equiv X Y)\n    (equiv.symm\n      (equiv.mk sorry\n        (fun (\u1fb0 : \ud835\udfd9 = 0 \u2227 \ud835\udfd9 = 0) =>\n          and.dcases_on \u1fb0\n            fun (hX : \ud835\udfd9 = 0) (hY : \ud835\udfd9 = 0) => (coe_fn (id_zero_equiv_iso_zero X) hX, coe_fn (id_zero_equiv_iso_zero Y) hY))\n        sorry sorry))\n\n/--\nA zero morphism `0 : X \u27f6 X` is an isomorphism if and only if\n`X` is isomorphic to the zero object.\n-/\ndef is_iso_zero_self_equiv_iso_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] (X : C) : is_iso 0 \u2243 (X \u2245 0) :=\n  equiv.trans (is_iso_zero_equiv_iso_zero X X) subsingleton_prod_self_equiv\n\n/-- If there are zero morphisms, any initial object is a zero object. -/\nprotected instance has_zero_object_of_has_initial_object {C : Type u} [category C] [has_zero_morphisms C] [has_initial C] : has_zero_object C :=\n  has_zero_object.mk (\u22a5_C) (fun (X : C) => unique.mk { default := 0 } sorry)\n    fun (X : C) => unique.mk { default := 0 } sorry\n\n/-- If there are zero morphisms, any terminal object is a zero object. -/\nprotected instance has_zero_object_of_has_terminal_object {C : Type u} [category C] [has_zero_morphisms C] [has_terminal C] : has_zero_object C :=\n  has_zero_object.mk (\u22a4_C) (fun (X : C) => unique.mk { default := 0 } sorry)\n    fun (X : C) => unique.mk { default := 0 } sorry\n\ntheorem image_\u03b9_comp_eq_zero {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} {Z : C} {f : X \u27f6 Y} {g : Y \u27f6 Z} [has_image f] [epi (factor_thru_image f)] (h : f \u226b g = 0) : image.\u03b9 f \u226b g = 0 := sorry\n\n/--\nThe zero morphism has a `mono_factorisation` through the zero object.\n-/\n@[simp] theorem mono_factorisation_zero_e {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] (X : C) (Y : C) : mono_factorisation.e (mono_factorisation_zero X Y) = 0 :=\n  Eq.refl (mono_factorisation.e (mono_factorisation_zero X Y))\n\n/--\nThe factorisation through the zero object is an image factorisation.\n-/\ndef image_factorisation_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] (X : C) (Y : C) : image_factorisation 0 :=\n  image_factorisation.mk (mono_factorisation_zero X Y) (is_image.mk fun (F' : mono_factorisation 0) => 0)\n\nprotected instance has_image_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C} {Y : C} : has_image 0 :=\n  has_image.mk (image_factorisation_zero X Y)\n\n/-- The image of a zero morphism is the zero object. -/\ndef image_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C} {Y : C} : image 0 \u2245 0 :=\n  is_image.iso_ext (image.is_image 0) (image_factorisation.is_image (image_factorisation_zero X Y))\n\n/-- The image of a morphism which is equal to zero is the zero object. -/\ndef image_zero' {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C} {Y : C} {f : X \u27f6 Y} (h : f = 0) [has_image f] : image f \u2245 0 :=\n  image.eq_to_iso h \u226a\u226b image_zero\n\n@[simp] theorem image.\u03b9_zero {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] {X : C} {Y : C} [has_image 0] : image.\u03b9 0 = 0 := sorry\n\n/--\nIf we know `f = 0`,\nit requires a little work to conclude `image.\u03b9 f = 0`,\nbecause `f = g` only implies `image f \u2245 image g`.\n-/\n@[simp] theorem image.\u03b9_zero' {C : Type u} [category C] [has_zero_morphisms C] [has_zero_object C] [has_equalizers C] {X : C} {Y : C} {f : X \u27f6 Y} (h : f = 0) [has_image f] : image.\u03b9 f = 0 := sorry\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_sigma_\u03b9 {C : Type u} [category C] {\u03b2 : Type v} [DecidableEq \u03b2] [has_zero_morphisms C] (f : \u03b2 \u2192 C) [has_colimit (discrete.functor f)] (b : \u03b2) : split_mono (sigma.\u03b9 f b) :=\n  split_mono.mk\n    (sigma.desc fun (b' : \u03b2) => dite (b' = b) (fun (h : b' = b) => eq_to_hom (congr_arg f h)) fun (h : \u00acb' = b) => 0)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_pi_\u03c0 {C : Type u} [category C] {\u03b2 : Type v} [DecidableEq \u03b2] [has_zero_morphisms C] (f : \u03b2 \u2192 C) [has_limit (discrete.functor f)] (b : \u03b2) : split_epi (pi.\u03c0 f b) :=\n  split_epi.mk\n    (pi.lift fun (b' : \u03b2) => dite (b = b') (fun (h : b = b') => eq_to_hom (congr_arg f h)) fun (h : \u00acb = b') => 0)\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_coprod_inl {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} [has_colimit (pair X Y)] : split_mono coprod.inl :=\n  split_mono.mk (coprod.desc \ud835\udfd9 0)\n\n/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/\nprotected instance split_mono_coprod_inr {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} [has_colimit (pair X Y)] : split_mono coprod.inr :=\n  split_mono.mk (coprod.desc 0 \ud835\udfd9)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_prod_fst {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} [has_limit (pair X Y)] : split_epi prod.fst :=\n  split_epi.mk (prod.lift \ud835\udfd9 0)\n\n/-- In the presence of zero morphisms, projections into a product are (split) epimorphisms. -/\nprotected instance split_epi_prod_snd {C : Type u} [category C] [has_zero_morphisms C] {X : C} {Y : C} [has_limit (pair X Y)] : split_epi prod.snd :=\n  split_epi.mk (prod.lift 0 \ud835\udfd9)\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/shapes/zero.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6187804267137442, "lm_q1q2_score": 0.36673049161856325}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.bifunctor\nimport Mathlib.control.traversable.basic\nimport Mathlib.PostPort\n\nuniverses u l u_1 l_1 \n\nnamespace Mathlib\n\n/-!\n# Bitraversable type class\n\nType class for traversing bifunctors. The concepts and laws are taken from\n<https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\nSimple examples of `bitraversable` are `prod` and `sum`. A more elaborate example is\nto define an a-list as:\n\n```\ndef alist (key val : Type) := list (key \u00d7 val)\n```\n\nThen we can use `f : key \u2192 io key'` and `g : val \u2192 io val'` to manipulate the `alist`'s key\nand value respectively with `bitraverse f g : alist key val \u2192 io (alist key' val')`\n\n## Main definitions\n  * bitraversable - exposes the `bitraverse` function\n  * is_lawful_bitraversable - laws similar to is_lawful_traversable\n\n## Tags\n\ntraversable bitraversable iterator functor bifunctor applicative\n\n-/\n\nclass bitraversable (t : Type u \u2192 Type u \u2192 Type u) extends bifunctor t where\n  bitraverse :\n    {m : Type u \u2192 Type u} \u2192\n      [_inst_1 : Applicative m] \u2192\n        {\u03b1 \u03b1' \u03b2 \u03b2' : Type u} \u2192 (\u03b1 \u2192 m \u03b1') \u2192 (\u03b2 \u2192 m \u03b2') \u2192 t \u03b1 \u03b2 \u2192 m (t \u03b1' \u03b2')\n\ndef bisequence {t : Type u_1 \u2192 Type u_1 \u2192 Type u_1} {m : Type u_1 \u2192 Type u_1} [bitraversable t]\n    [Applicative m] {\u03b1 : Type u_1} {\u03b2 : Type u_1} : t (m \u03b1) (m \u03b2) \u2192 m (t \u03b1 \u03b2) :=\n  bitraverse id id\n\nclass is_lawful_bitraversable (t : Type u \u2192 Type u \u2192 Type u) [bitraversable t]\n    extends is_lawful_bifunctor t where\n  id_bitraverse : \u2200 {\u03b1 \u03b2 : Type u} (x : t \u03b1 \u03b2), bitraverse id.mk id.mk x = id.mk x\n  comp_bitraverse :\n    \u2200 {F G : Type u \u2192 Type u} [_inst_1_1 : Applicative F] [_inst_2 : Applicative G]\n      [_inst_3 : is_lawful_applicative F] [_inst_4 : is_lawful_applicative G]\n      {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3' : Type u} (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3') (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2')\n      (x : t \u03b1 \u03b1'),\n      bitraverse (functor.comp.mk \u2218 Functor.map f \u2218 g) (functor.comp.mk \u2218 Functor.map f' \u2218 g') x =\n        functor.comp.mk (bitraverse f f' <$> bitraverse g g' x)\n  bitraverse_eq_bimap_id :\n    \u2200 {\u03b1 \u03b1' \u03b2 \u03b2' : Type u} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') (x : t \u03b1 \u03b1'),\n      bitraverse (id.mk \u2218 f) (id.mk \u2218 f') x = id.mk (bimap f f' x)\n  binaturality :\n    \u2200 {F G : Type u \u2192 Type u} [_inst_1_1 : Applicative F] [_inst_2 : Applicative G]\n      [_inst_3 : is_lawful_applicative F] [_inst_4 : is_lawful_applicative G]\n      (\u03b7 : applicative_transformation F G) {\u03b1 \u03b1' \u03b2 \u03b2' : Type u} (f : \u03b1 \u2192 F \u03b2) (f' : \u03b1' \u2192 F \u03b2')\n      (x : t \u03b1 \u03b1'),\n      coe_fn \u03b7 (t \u03b2 \u03b2') (bitraverse f f' x) = bitraverse (coe_fn \u03b7 \u03b2 \u2218 f) (coe_fn \u03b7 \u03b2' \u2218 f') x\n\ntheorem is_lawful_bitraversable.bitraverse_id_id {t : Type l_1 \u2192 Type l_1 \u2192 Type l_1}\n    [bitraversable t] [c : is_lawful_bitraversable t] {\u03b1 : Type l_1} {\u03b2 : Type l_1} :\n    bitraverse id.mk id.mk = id.mk :=\n  funext fun (x : t \u03b1 \u03b2) => id_bitraverse x\n\ntheorem is_lawful_bitraversable.bitraverse_comp {t : Type l_1 \u2192 Type l_1 \u2192 Type l_1}\n    [bitraversable t] [c : is_lawful_bitraversable t] {F : Type l_1 \u2192 Type l_1}\n    {G : Type l_1 \u2192 Type l_1} :\n    \u2200 [_inst_1_1 : Applicative F] [_inst_2 : Applicative G] [_inst_3 : is_lawful_applicative F]\n        [_inst_4 : is_lawful_applicative G] {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3' : Type l_1} (f : \u03b2 \u2192 F \u03b3)\n        (f' : \u03b2' \u2192 F \u03b3') (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2'),\n        bitraverse (functor.comp.mk \u2218 Functor.map f \u2218 g) (functor.comp.mk \u2218 Functor.map f' \u2218 g') =\n          functor.comp.mk \u2218 Functor.map (bitraverse f f') \u2218 bitraverse g g' :=\n  fun (_inst_1_1 : Applicative F) (_inst_2 : Applicative G) (_inst_3 : is_lawful_applicative F)\n    (_inst_4 : is_lawful_applicative G) (\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3' : Type l_1) (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3')\n    (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2') => funext fun (x : t \u03b1 \u03b1') => comp_bitraverse f f' g g' x\n\ntheorem is_lawful_bitraversable.bitraverse_eq_bimap_id' {t : Type l_1 \u2192 Type l_1 \u2192 Type l_1}\n    [bitraversable t] [c : is_lawful_bitraversable t] {\u03b1 : Type l_1} {\u03b1' : Type l_1} {\u03b2 : Type l_1}\n    {\u03b2' : Type l_1} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') :\n    bitraverse (id.mk \u2218 f) (id.mk \u2218 f') = id.mk \u2218 bimap f f' :=\n  funext fun (x : t \u03b1 \u03b1') => bitraverse_eq_bimap_id f f' x\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/bitraversable/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3667304874520543}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport algebra.algebra.tower\n\n/-!\n\n# The `restrict_scalars` type alias\n\nSee the documentation attached to the `restrict_scalars` definition for advice on how and when to\nuse this type alias. As described there, it is often a better choice to use the `is_scalar_tower`\ntypeclass instead.\n\n## Main definitions\n\n* `restrict_scalars R S M`: the `S`-module `M` viewed as an `R` module when `S` is an `R`-algebra.\n  Note that by default we do *not* have a `module S (restrict_scalars R S M)` instance\n  for the original action.\n  This is available as a def `restrict_scalars.module_orig` if really needed.\n* `restrict_scalars.add_equiv : restrict_scalars R S M \u2243+ M`: the additive equivalence\n  between the restricted and original space (in fact, they are definitionally equal,\n  but sometimes it is helpful to avoid using this fact, to keep instances from leaking).\n* `restrict_scalars.ring_equiv : restrict_scalars R S A \u2243+* A`: the ring equivalence\n   between the restricted and original space when the module is an algebra.\n\n## See also\n\nThere are many similarly-named definitions elsewhere which do not refer to this type alias. These\nrefer to restricting the scalar type in a bundled type, such as from `A \u2192\u2097[R] B` to `A \u2192\u2097[S] B`:\n\n* `linear_map.restrict_scalars`\n* `linear_equiv.restrict_scalars`\n* `alg_hom.restrict_scalars`\n* `alg_equiv.restrict_scalars`\n* `submodule.restrict_scalars`\n* `subalgebra.restrict_scalars`\n-/\n\nvariables (R S M A : Type*)\n\n/-- If we put an `R`-algebra structure on a semiring `S`, we get a natural equivalence from the\ncategory of `S`-modules to the category of representations of the algebra `S` (over `R`). The type\nsynonym `restrict_scalars` is essentially this equivalence.\n\nWarning: use this type synonym judiciously! Consider an example where we want to construct an\n`R`-linear map from `M` to `S`, given:\n```lean\nvariables (R S M : Type*)\nvariables [comm_semiring R] [semiring S] [algebra R S] [add_comm_monoid M] [module S M]\n```\nWith the assumptions above we can't directly state our map as we have no `module R M` structure, but\n`restrict_scalars` permits it to be written as:\n```lean\n-- an `R`-module structure on `M` is provided by `restrict_scalars` which is compatible\nexample : restrict_scalars R S M \u2192\u2097[R] S := sorry\n```\nHowever, it is usually better just to add this extra structure as an argument:\n```lean\n-- an `R`-module structure on `M` and proof of its compatibility is provided by the user\nexample [module R M] [is_scalar_tower R S M] : M \u2192\u2097[R] S := sorry\n```\nThe advantage of the second approach is that it defers the duty of providing the missing typeclasses\n`[module R M] [is_scalar_tower R S M]`. If some concrete `M` naturally carries these (as is often\nthe case) then we have avoided `restrict_scalars` entirely. If not, we can pass\n`restrict_scalars R S M` later on instead of `M`.\n\nNote that this means we almost always want to state definitions and lemmas in the language of\n`is_scalar_tower` rather than `restrict_scalars`.\n\nAn example of when one might want to use `restrict_scalars` would be if one has a vector space\nover a field of characteristic zero and wishes to make use of the `\u211a`-algebra structure. -/\n@[nolint unused_arguments]\ndef restrict_scalars (R S M : Type*) : Type* := M\n\ninstance [I : inhabited M] : inhabited (restrict_scalars R S M) := I\n\ninstance [I : add_comm_monoid M] : add_comm_monoid (restrict_scalars R S M) := I\n\ninstance [I : add_comm_group M] : add_comm_group (restrict_scalars R S M) := I\n\nsection module\n\nsection\nvariables [semiring S] [add_comm_monoid M]\n\n/-- We temporarily install an action of the original ring on `restrict_sclars R S M`. -/\ndef restrict_scalars.module_orig [I : module S M] :\n  module S (restrict_scalars R S M) := I\n\nvariables [comm_semiring R] [algebra R S]\nsection\nlocal attribute [instance] restrict_scalars.module_orig\n\n/--\nWhen `M` is a module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nmodule structure over `R`.\n\nThe preferred way of setting this up is `[module R M] [module S M] [is_scalar_tower R S M]`.\n-/\ninstance [module S M] : module R (restrict_scalars R S M) :=\nmodule.comp_hom M (algebra_map R S)\n\n/--\nThis instance is only relevant when `restrict_scalars.module_orig` is available as an instance.\n-/\ninstance [module S M] : is_scalar_tower R S (restrict_scalars R S M) :=\n\u27e8\u03bb r S M, by { rw [algebra.smul_def, mul_smul], refl }\u27e9\n\nend\n\n/--\nWhen `M` is a right-module over a ring `S`, and `S` is an algebra over `R`, then `M` inherits a\nright-module structure over `R`.\nThe preferred way of setting this up is\n`[module R\u1d50\u1d52\u1d56 M] [module S\u1d50\u1d52\u1d56 M] [is_scalar_tower R\u1d50\u1d52\u1d56 S\u1d50\u1d52\u1d56 M]`.\n-/\ninstance restrict_scalars.op_module [module S\u1d50\u1d52\u1d56 M] : module R\u1d50\u1d52\u1d56 (restrict_scalars R S M) :=\nbegin\n  letI : module S\u1d50\u1d52\u1d56 (restrict_scalars R S M) := \u2039module S\u1d50\u1d52\u1d56 M\u203a,\n  exact module.comp_hom M (algebra_map R S).op\nend\n\ninstance restrict_scalars.is_central_scalar [module S M] [module S\u1d50\u1d52\u1d56 M] [is_central_scalar S M] :\n  is_central_scalar R (restrict_scalars R S M) :=\n{ op_smul_eq_smul := \u03bb r x, (op_smul_eq_smul (algebra_map R S r) (_ : M) : _)}\n\n/--\nThe `R`-algebra homomorphism from the original coefficient algebra `S` to endomorphisms\nof `restrict_scalars R S M`.\n-/\ndef restrict_scalars.lsmul [module S M] : S \u2192\u2090[R] module.End R (restrict_scalars R S M) :=\nbegin\n  -- We use `restrict_scalars.module_orig` in the implementation,\n  -- but not in the type.\n  letI : module S (restrict_scalars R S M) := restrict_scalars.module_orig R S M,\n  exact algebra.lsmul R (restrict_scalars R S M),\nend\n\nend\n\nvariables [add_comm_monoid M]\n\n/-- `restrict_scalars.add_equiv` is the additive equivalence with the original module. -/\n@[simps] def restrict_scalars.add_equiv : restrict_scalars R S M \u2243+ M :=\nadd_equiv.refl M\n\nvariables [comm_semiring R] [semiring S] [algebra R S] [module S M]\n\nlemma restrict_scalars_smul_def (c : R) (x : restrict_scalars R S M) :\n  c \u2022 x = ((algebra_map R S c) \u2022 x : M) := rfl\n\n@[simp] lemma restrict_scalars.add_equiv_map_smul (t : R) (x : restrict_scalars R S M) :\n  restrict_scalars.add_equiv R S M (t \u2022 x)\n  = (algebra_map R S t) \u2022 restrict_scalars.add_equiv R S M x :=\nrfl\n\nend module\n\nsection algebra\n\ninstance [I : semiring A] : semiring (restrict_scalars R S A) := I\ninstance [I : ring A] : ring (restrict_scalars R S A) := I\ninstance [I : comm_semiring A] : comm_semiring (restrict_scalars R S A) := I\ninstance [I : comm_ring A] : comm_ring (restrict_scalars R S A) := I\n\nvariables [semiring A]\n\n/-- Tautological ring isomorphism `restrict_scalars R S A \u2243+* A`. -/\ndef restrict_scalars.ring_equiv : restrict_scalars R S A \u2243+* A := ring_equiv.refl _\n\nvariables [comm_semiring S] [algebra S A] [comm_semiring R] [algebra R S]\n\n@[simp] lemma restrict_scalars.ring_equiv_map_smul (r : R) (x : restrict_scalars R S A) :\n  restrict_scalars.ring_equiv R S A (r \u2022 x)\n  = (algebra_map R S r) \u2022 restrict_scalars.ring_equiv R S A x :=\nrfl\n\n/-- `R \u27f6 S` induces `S-Alg \u2964 R-Alg` -/\ninstance : algebra R (restrict_scalars R S A) :=\n{ smul := (\u2022),\n  commutes' := \u03bb r x, algebra.commutes _ _,\n  smul_def' := \u03bb _ _, algebra.smul_def _ _,\n  .. (algebra_map S A).comp (algebra_map R S) }\n\n@[simp] lemma restrict_scalars.ring_equiv_algebra_map (r : R) :\n  restrict_scalars.ring_equiv R S A (algebra_map R (restrict_scalars R S A) r) =\n    algebra_map S A (algebra_map R S r) :=\nrfl\n\nend algebra\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/algebra/algebra/restrict_scalars.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599563, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3666703642457349}}
{"text": "import tactic.abel\n\nimport topology.algebra.group\nimport topology.algebra.uniform_ring\nimport ring_theory.subring\n\nimport for_mathlib.topology\nimport for_mathlib.filter\nimport for_mathlib.data.set.basic\nimport algebra.pointwise\n\n/-\nopen filter function\n\nuniverse u\n\nclass filter_at_one (\u03b1 : Type u) [group \u03b1] :=\n(F : filter \u03b1)\n(one_in : pure 1 \u2264 F) -- we could ask instead that Z \u2260 \u22a5\n(mul {} : tendsto (uncurry' ((*) : \u03b1 \u2192 \u03b1 \u2192 \u03b1)) (F.prod F) F)\n(inv {} : tendsto (\u03bb x: \u03b1, x\u207b\u00b9) F F)\n(conj {} : \u2200 x\u2080 : \u03b1, tendsto (\u03bb x: \u03b1, x\u2080*x*x\u2080\u207b\u00b9) F F).\n\nnamespace filter_at_one\n\ndef topology (\u03b1 : Type u) [group \u03b1] [filter_at_one \u03b1] : topological_space \u03b1 :=\ntopological_space.mk_of_nhds $ \u03bba, map (\u03bbx, a*x) (F \u03b1)\n\nlocal attribute [instance] topology\nlemma topological_group (\u03b1 : Type u) [group \u03b1] [filter_at_one \u03b1]: topological_group \u03b1 :=\nsorry\n\nend filter_at_one\n\nUsing the above setup, we get :\n\nfilter_at_one.topological_group :\n  \u2200 (\u03b1 : Type u_1) [_inst_1 : group \u03b1] [_inst_2 : @filter_at_one \u03b1 _inst_1],\n    @topological_group \u03b1 (@filter_at_one.topology \u03b1 _inst_1 _inst_2) _inst_1\n\nSo this lemma (which could be then turned into a local instance) is only about the\ntopology built by `filter_at_one.topology`. It doesn't not say anything about\na group endowed by a random topology that happens to satisfies the axioms of filter_at_one.\n-/\nuniverse u\nopen filter function set topological_space\nlocal infixr ` \u00d7\u1da0 `:51 := filter.prod\nlocal prefix \ud835\udcdd:100 := nhds\n\n@[to_additive]\nlemma topological_monoid.of_comm_of_nice_nhds_one (\u03b1 : Type u) [comm_monoid \u03b1] [topological_space \u03b1]\n  (hmul : tendsto (uncurry' ((*) : \u03b1 \u2192 \u03b1 \u2192 \u03b1)) (\ud835\udcdd 1 \u00d7\u1da0 \ud835\udcdd 1) \ud835\udcdd 1)\n  (hleft : \u2200 x\u2080 : \u03b1, \ud835\udcdd x\u2080 = map (\u03bb x, x\u2080*x) \ud835\udcdd 1) : topological_monoid \u03b1 :=\n{ continuous_mul := begin\n    rw continuous_iff_continuous_at,\n    rintros \u27e8x\u2080, y\u2080\u27e9,\n    have key : (\u03bb (p : \u03b1 \u00d7 \u03b1), x\u2080 * p.1 * (y\u2080 * p.2)) =\n      ((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*))),\n    { ext,\n      change x\u2080 * x.1 * (y\u2080 * x.2) = x\u2080 * y\u2080 * (x.1 *  x.2),\n      ac_refl },\n\n    calc map (\u03bb (p : \u03b1 \u00d7 \u03b1), p.1 * p.2) \ud835\udcdd (x\u2080, y\u2080)\n        = map (\u03bb (p : \u03b1 \u00d7 \u03b1), p.1 * p.2) (\ud835\udcdd x\u2080 \u00d7\u1da0 \ud835\udcdd y\u2080)\n            : by rw nhds_prod_eq\n    ... = map (\u03bb (p : \u03b1 \u00d7 \u03b1), x\u2080 * p.1 * (y\u2080 * p.2)) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1))\n            : by rw [hleft x\u2080, hleft y\u2080, prod_map_map_eq, filter.map_map]\n    ... = map ((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*))) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1)) : by rw key\n    ... = map (\u03bb x, x\u2080*y\u2080*x) (map (uncurry' (*)) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1)))   : by rw filter.map_map\n    ... \u2264 map (\u03bb x, x\u2080*y\u2080*x) (\ud835\udcdd 1)   : map_mono hmul\n    ... = \ud835\udcdd (x\u2080*y\u2080)   : (hleft _).symm\n  end }\n\nprotected meta def prove_conj : tactic unit :=\n`[ intro x\u2080,\n   convert continuous_id.continuous_at,\n   simpa [mul_comm, inv_mul_cancel_left]]\n\n@[to_additive]\nlemma topological_group.of_nice_nhds_one (\u03b1 : Type u) [group \u03b1] [topological_space \u03b1]\n  (hmul : tendsto (uncurry' ((*) : \u03b1 \u2192 \u03b1 \u2192 \u03b1)) ((\ud835\udcdd 1).prod \ud835\udcdd 1) \ud835\udcdd 1)\n  (hinv : tendsto (\u03bb x : \u03b1, x\u207b\u00b9) \ud835\udcdd 1 \ud835\udcdd 1)\n  (hleft : \u2200 x\u2080 : \u03b1, \ud835\udcdd x\u2080 = map (\u03bb x, x\u2080*x) \ud835\udcdd 1)\n  (hconj : \u2200 x\u2080 : \u03b1, tendsto (\u03bb x: \u03b1, x\u2080*x*x\u2080\u207b\u00b9) \ud835\udcdd 1 \ud835\udcdd 1 . prove_conj) : topological_group \u03b1 :=\n{ continuous_mul := begin\n    rw continuous_iff_continuous_at,\n    rintros \u27e8x\u2080, y\u2080\u27e9,\n    have key : (\u03bb (p : \u03b1 \u00d7 \u03b1), x\u2080 * p.1 * (y\u2080 * p.2)) =\n      ((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*)) \u2218 ((\u03bb x, y\u2080\u207b\u00b9*x*y\u2080) \u2a2f id)),\n      by { ext, simp [uncurry', prod.map', mul_assoc] },\n    specialize hconj y\u2080\u207b\u00b9, rw inv_inv at hconj,\n    calc map (\u03bb (p : \u03b1 \u00d7 \u03b1), p.1 * p.2) \ud835\udcdd (x\u2080, y\u2080)\n        = map (\u03bb (p : \u03b1 \u00d7 \u03b1), p.1 * p.2) (\ud835\udcdd x\u2080 \u00d7\u1da0 \ud835\udcdd y\u2080)\n            : by rw nhds_prod_eq\n    ... = map (\u03bb (p : \u03b1 \u00d7 \u03b1), x\u2080 * p.1 * (y\u2080 * p.2)) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1))\n            : by rw [hleft x\u2080, hleft y\u2080, prod_map_map_eq, filter.map_map]\n    ... = map (((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*))) \u2218 ((\u03bb x, y\u2080\u207b\u00b9*x*y\u2080) \u2a2f id))((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1))\n            : by rw key\n    ... = map ((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*))) ((map  (\u03bb x, y\u2080\u207b\u00b9*x*y\u2080) \ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1))\n            : by rw [\u2190 filter.map_map, filter.map_prod_prod, map_id]\n    ... \u2264 map ((\u03bb x, x\u2080*y\u2080*x) \u2218 (uncurry' (*))) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1))\n            : map_mono (filter.prod_mono hconj $ le_refl _)\n    ... = map (\u03bb x, x\u2080*y\u2080*x) (map (uncurry' (*)) ((\ud835\udcdd 1) \u00d7\u1da0 (\ud835\udcdd 1)))   : by rw filter.map_map\n    ... \u2264 map (\u03bb x, x\u2080*y\u2080*x) (\ud835\udcdd 1)   : map_mono hmul\n    ... = \ud835\udcdd (x\u2080*y\u2080)   : (hleft _).symm\n  end,\n  continuous_inv := begin\n    rw continuous_iff_continuous_at,\n    rintros x\u2080,\n    have key : (\u03bb x, (x\u2080*x)\u207b\u00b9) = (\u03bb x, x\u2080\u207b\u00b9*x) \u2218 (\u03bb x, x\u2080*x*x\u2080\u207b\u00b9) \u2218 (\u03bb x, x\u207b\u00b9),\n      by {ext ; simp[mul_assoc] },\n    calc map (\u03bb x, x\u207b\u00b9) (\ud835\udcdd x\u2080)\n        = map (\u03bb x, x\u207b\u00b9) (map (\u03bb x, x\u2080*x) \ud835\udcdd 1) : by rw hleft\n    ... = map (\u03bb x, (x\u2080*x)\u207b\u00b9) \ud835\udcdd 1 : by rw filter.map_map\n    ... = map (((\u03bb x, x\u2080\u207b\u00b9*x) \u2218 (\u03bb x, x\u2080*x*x\u2080\u207b\u00b9)) \u2218 (\u03bb x, x\u207b\u00b9)) \ud835\udcdd 1 : by rw key\n    ... = map ((\u03bb x, x\u2080\u207b\u00b9*x) \u2218 (\u03bb x, x\u2080*x*x\u2080\u207b\u00b9)) _ : by rw \u2190 filter.map_map\n    ... \u2264 map ((\u03bb x, x\u2080\u207b\u00b9 * x) \u2218 \u03bb x, x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1) : map_mono hinv\n    ... = map (\u03bb x, x\u2080\u207b\u00b9 * x) (map (\u03bb x, x\u2080 * x * x\u2080\u207b\u00b9) (\ud835\udcdd 1)) : filter.map_map\n    ... \u2264 map (\u03bb x, x\u2080\u207b\u00b9 * x) \ud835\udcdd 1 : map_mono (hconj x\u2080)\n    ... = \ud835\udcdd x\u2080\u207b\u00b9 : (hleft _).symm\n  end }\n\n\n@[to_additive]\nlemma topological_group.of_comm_of_nice_nhds_one (\u03b1 : Type u) [comm_group \u03b1] [topological_space \u03b1]\n  (hmul : tendsto (uncurry' ((*) : \u03b1 \u2192 \u03b1 \u2192 \u03b1)) ((\ud835\udcdd 1).prod \ud835\udcdd 1) \ud835\udcdd 1)\n  (hinv : tendsto (\u03bb x : \u03b1, x\u207b\u00b9) \ud835\udcdd 1 \ud835\udcdd 1)\n  (hleft : \u2200 x\u2080 : \u03b1, \ud835\udcdd x\u2080 = map (\u03bb x, x\u2080*x) \ud835\udcdd 1) : topological_group \u03b1 :=\ntopological_group.of_nice_nhds_one \u03b1 hmul hinv hleft\n\nopen set\nlocal attribute [instance] pointwise_mul pointwise_add\n\nclass group_filter_basis (\u03b1 : Type u) [group \u03b1] extends filter_basis \u03b1 :=\n(one : \u2200 {U}, U \u2208 sets \u2192 (1 : \u03b1) \u2208 U)\n(mul : \u2200 {U}, U \u2208 sets \u2192 \u2203 V \u2208 sets, V * V \u2286 U)\n(inv : \u2200 {U}, U \u2208 sets \u2192 \u2203 V \u2208 sets, V \u2286 (\u03bb x, x\u207b\u00b9) \u207b\u00b9' U)\n(conj : \u2200 x\u2080, \u2200 U \u2208 sets, \u2203 V \u2208 sets, V \u2286 (\u03bb x, x\u2080*x*x\u2080\u207b\u00b9) \u207b\u00b9' U)\n\nclass add_group_filter_basis (\u03b1 : Type u) [add_group \u03b1] extends filter_basis \u03b1 :=\n(zero : \u2200 {U}, U \u2208 sets \u2192 (0 : \u03b1) \u2208 U)\n(add : \u2200 {U}, U \u2208 sets \u2192 \u2203 V \u2208 sets, V + V \u2286 U)\n(neg : \u2200 {U}, U \u2208 sets \u2192 \u2203 V \u2208 sets, V \u2286 (\u03bb x, -x) \u207b\u00b9' U)\n(conj : \u2200 x\u2080, \u2200 U \u2208 sets, \u2203 V \u2208 sets, V \u2286 (\u03bb x, x\u2080+x-x\u2080) \u207b\u00b9' U)\n\nattribute [to_additive add_group_filter_basis] group_filter_basis\nattribute [to_additive] group_filter_basis.one\nattribute [to_additive] group_filter_basis.mul\nattribute [to_additive] group_filter_basis.inv\nattribute [to_additive] group_filter_basis.conj\nattribute [to_additive] group_filter_basis.to_filter_basis\n\n\n/- -- We didn't use class directly because we still want \u03b1 to be an explicit argument of projections\nattribute [class] group_filter_basis\nattribute [class] add_group_filter_basis\n -/\ninstance group_filter_basis.has_mem {\u03b1 : Type*} [group \u03b1] : has_mem (set \u03b1) (group_filter_basis \u03b1) := \u27e8\u03bb s f, s \u2208 f.sets\u27e9\ninstance add_group_filter_basis.has_mem {\u03b1 : Type*} [add_group \u03b1] : has_mem (set \u03b1) (add_group_filter_basis \u03b1) := \u27e8\u03bb s f, s \u2208 f.sets\u27e9\n\nattribute [to_additive] group_filter_basis.has_mem\n\nnamespace group_filter_basis\nvariables {\u03b1 : Type*} [group \u03b1]\n\n@[to_additive]\nlemma prod_subset_self (f : group_filter_basis \u03b1) {U : set \u03b1} (h : U \u2208 f) : U \u2286 U*U :=\n\u03bb x x_in, (mul_one x) \u25b8 mul_mem_pointwise_mul x_in $ group_filter_basis.one h\n\n/-- The neighborhood function of a `group_filter_basis` -/\n@[to_additive]\ndef N (f : group_filter_basis \u03b1) : \u03b1 \u2192 filter \u03b1 :=\n\u03bb x, map (\u03bb y, x*y) f.to_filter_basis.filter\n\n@[simp, to_additive]\nlemma N_one (f : group_filter_basis \u03b1) : f.N 1 = f.to_filter_basis.filter :=\nby simpa [N, map_id]\n\n@[to_additive]\nlemma mem_N (f : group_filter_basis \u03b1) (x : \u03b1) (U : set \u03b1) :\n  U \u2208 f.N x \u2194 \u2203 V \u2208 f, (\u03bb y, x*y) '' V \u2286 U :=\nby simpa [N, mem_map, filter_basis.mem_filter, image_subset_iff]\n\n@[to_additive]\nlemma mem_N_of_mem (f : group_filter_basis \u03b1) (x : \u03b1) {U : set \u03b1} (h : U \u2208 f) :\n(\u03bb y, x*y) '' U \u2208 f.N x :=\nby { rw mem_N, use [U, h] }\n\n@[to_additive]\nlemma N_is_nice (f : group_filter_basis \u03b1) :\n  (pure \u2264 f.N) \u2227\n  \u2200 {a s}, s \u2208 f.N a \u2192 \u2203 t \u2208 f.N a, t \u2286 s \u2227 \u2200 a' \u2208 t, s \u2208 f.N a' :=\nbegin\n  split,\n  { intros x U U_in,\n    rw f.mem_N at U_in,\n    rcases U_in with \u27e8V, V_in, H\u27e9,\n    simpa using H (mem_image_of_mem _ (group_filter_basis.one V_in)) },\n  { intros x U U_in,\n    rw f.mem_N at U_in,\n    rcases U_in with \u27e8V, V_in, H\u27e9,\n    rcases group_filter_basis.mul V_in with \u27e8W, W_in, hW\u27e9,\n    use [(\u03bb y, x*y) '' W, image_mem_map (filter_basis.mem_filter_of_mem W_in)],\n    split,\n    { rw image_subset_iff at H \u22a2,\n      exact subset.trans (subset.trans (f.prod_subset_self W_in) hW) H},\n    { rintros y \u27e8t, tW, rfl\u27e9,\n      rw f.mem_N,\n      use [W, W_in],\n      apply subset.trans _ H, clear H,\n      rintros z \u27e8w, wW, rfl\u27e9,\n      exact \u27e8t*w, hW (mul_mem_pointwise_mul tW wW), by simp [mul_assoc]\u27e9 } },\nend\n\n@[to_additive]\nlemma is_top_group {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1) [topological_space \u03b1]\n  (hnhds : \u2200 x\u2080 : \u03b1, \ud835\udcdd x\u2080 = basis.N x\u2080) : topological_group \u03b1 :=\nbegin\n  have hnhds1 : \ud835\udcdd 1 = basis.to_filter_basis.filter, by rw [hnhds 1, N_one],\n  apply topological_group.of_nice_nhds_one,\n  { rw [hnhds1, \u2190 basis.to_filter_basis.prod_filter, filter_basis.tendsto_both],\n    intros V V_in,\n    rcases group_filter_basis.mul V_in with \u27e8W, W_in, hW\u27e9,\n    use [set.prod W W, filter_basis.mem_prod_of_mem W_in W_in],\n    rwa [pointwise_mul_eq_image, image_subset_iff] at hW },\n  { rw [hnhds1, basis.to_filter_basis.tendsto_both],\n    exact basis.inv },\n  { exact hnhds1.symm \u25b8 hnhds },\n  { intro x\u2080,\n    rw [hnhds1, basis.to_filter_basis.tendsto_both],\n    exact  group_filter_basis.conj x\u2080 }\nend\n\n/-- The topological space structure coming a group filter basis. -/\n@[to_additive]\ndef topology {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1) : topological_space \u03b1 :=\ntopological_space.mk_of_nhds basis.N\n\n/-- The topological space structure coming a group filter basis. Version using tc resolution -/\n@[to_additive]\ndef to_topological_space {\u03b1 : Type u} [group \u03b1] [basis : group_filter_basis \u03b1] : topological_space \u03b1 :=\nbasis.topology\n\n@[to_additive]\nlemma nhds_eq {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1)\n  [t : topological_space \u03b1] (h : t = basis.topology) {x\u2080 : \u03b1} :\n  \ud835\udcdd x\u2080 = basis.N x\u2080 :=\nby rw [h, nhds_mk_of_nhds _ x\u2080 basis.N_is_nice.1 basis.N_is_nice.2]\n\n@[to_additive]\nlemma nhds_one_eq {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1)\n  [t : topological_space \u03b1] (h : t = basis.topology) :\n  \ud835\udcdd (1 : \u03b1) = basis.to_filter_basis.filter :=\nby { rw basis.nhds_eq h, simp only [N, one_mul], exact map_id }\n\n@[to_additive]\nlemma mem_nhds {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1)\n  [t : topological_space \u03b1] (h : t = basis.topology) {x\u2080 : \u03b1} {U : set \u03b1} :\n  U \u2208 \ud835\udcdd x\u2080 \u2194 \u2203 V \u2208 basis, V \u2286 (\u03bb x, x\u2080 * x) \u207b\u00b9' U :=\nbegin\n  rw basis.nhds_eq h,\n  exact filter_basis.mem_filter basis.to_filter_basis\nend\n\n@[to_additive]\nlemma is_topological_group {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1)\n  [t : topological_space \u03b1] (h : t = basis.topology) : topological_group \u03b1 :=\nbegin\n  apply basis.is_top_group,\n  rw h,\n  exact \u03bb x, nhds_mk_of_nhds _ x basis.N_is_nice.1 basis.N_is_nice.2\nend\n\n\n/-- The neighborhood basis on a group coming from a group filter basis -/\n@[to_additive]\ndef nhds_basis {\u03b1 : Type u} [group \u03b1] (basis : group_filter_basis \u03b1)\n  [t : topological_space \u03b1] (h : t = basis.topology) : nhds_basis \u03b1 :=\n{ B := \u03bb x\u2080, filter_basis.map (\u03bb x, x\u2080*x) basis.to_filter_basis,\n  is_nhds := \u03bb x\u2080, by rw [\u2190 filter_basis.map_filter, h,\n                          nhds_mk_of_nhds _ x\u2080 basis.N_is_nice.1 basis.N_is_nice.2, N] }\n\n\nlocal attribute [instance] group_filter_basis.to_topological_space\n\n-- The following can be made an instance when needed\n@[to_additive]\ndef to_nhds_basis {\u03b1 : Type u} [group \u03b1] [basis : group_filter_basis \u03b1]\n   : _root_.nhds_basis \u03b1 := basis.nhds_basis rfl\n\n\nlocal attribute [instance] group_filter_basis.to_nhds_basis add_group_filter_basis.to_nhds_basis\n\n@[to_additive]\nlemma mem_nhds_basis {\u03b1 : Type u} [group \u03b1] [basis : group_filter_basis \u03b1] {s : set \u03b1} {x\u2080 : \u03b1} :\ns \u2208 nhds_basis.B x\u2080 \u2194 (\u03bb x, x\u2080*x) \u207b\u00b9' s \u2208 basis.to_filter_basis.sets :=\nbegin\n  change s \u2208 filter_basis.map (\u03bb x, x\u2080*x) basis.to_filter_basis \u2194 _,\n  rw filter_basis.mem_map,\n  split ; intro h,\n  { rcases h with \u27e8U, h, rfl\u27e9,\n    rw preimage_image_eq,\n    exact h,\n    intros x y, simp },\n  { use [(\u03bb (x : \u03b1), x\u2080 * x) \u207b\u00b9' s, h],\n    rw image_preimage_eq,\n    intros y,\n    use [x\u2080\u207b\u00b9*y], simp }\nend\nend group_filter_basis\n\n\n\nsection\nvariables {G : Type u} [add_comm_group G]\n\ndef add_group_with_zero_nhd.of_open_add_subgroup\n  (H : set G) [is_add_subgroup H] (t : topological_space H) (h : @topological_add_group H t _) :\n  add_group_with_zero_nhd G :=\n{ Z := (nhds (0 : H)).map $ (subtype.val : H \u2192 G),\n  zero_Z := calc pure ((0 : H) : G) = map subtype.val (pure 0) : (filter.map_pure _ _).symm\n                                ... \u2264 _ : map_mono (pure_le_nhds _),\n  sub_Z :=\n  begin\n    let \u03b4_G := \u03bb (p : G \u00d7 G), p.1 - p.2,\n    let \u03b4_H := \u03bb (p : H \u00d7 H), p.1 - p.2,\n    let \u03b9 : H \u2192 G := subtype.val,\n    let N := \ud835\udcdd (0 : H),\n    let Z := map subtype.val N,\n    change map \u03b4_G (filter.prod Z Z) \u2264 Z,\n    have key\u2081: map \u03b4_H (\ud835\udcdd (0, 0)) \u2264 N,\n    { rw [show N = nhds (\u03b4_H (0, 0)), by simp [*]],\n      exact continuous_sub.tendsto _ },\n    have key\u2082 : \u03b4_G \u2218 \u03b9\u2a2f\u03b9 = \u03b9 \u2218 \u03b4_H,\n    { ext p,\n      change (p.1 : G) - (p.2 : G) = (p.1 - p.2 : G),\n      simp [is_add_subgroup.coe_neg, is_add_submonoid.coe_add] },\n\n    calc map \u03b4_G (filter.prod Z Z)\n          = map \u03b4_G (map (\u03b9 \u2a2f \u03b9) $ filter.prod N N) : by rw prod_map_map_eq;refl\n      ... = map \u03b9 (map \u03b4_H $ filter.prod N N)       : map_comm key\u2082 _\n      ... = map \u03b9 (map \u03b4_H $ \ud835\udcdd (0, 0))           : by rw \u2190 nhds_prod_eq\n      ... \u2264 map \u03b9 N : map_mono key\u2081\n  end,\n  ..\u2039add_comm_group G\u203a }\n\ndef of_open_add_subgroup {G : Type u} [str : add_comm_group G] (H : set G) [is_add_subgroup H]\n  (t : topological_space H) (h : @topological_add_group H t _) : topological_space G :=\n@add_group_with_zero_nhd.topological_space G\n  (add_group_with_zero_nhd.of_open_add_subgroup H t h)\n\nend\n\nnamespace add_group_with_zero_nhd\n\nlocal attribute [instance] add_group_with_zero_nhd.topological_space\nlocal notation `Z` := add_group_with_zero_nhd.Z\n\nvariables {\u03b1 : Type*}\nvariables {G : Type*} [add_group_with_zero_nhd G]\n\nlemma nhds_eq_comap (g : G) : nhds g = comap (\u03bb g', g' + -g) (Z G) :=\nby rw [\u2190 nhds_zero_eq_Z, nhds_translation_add_neg g]\nend add_group_with_zero_nhd\n\nnamespace topological_group\nvariables {G : Type*} {H : Type*}\nvariables [group G] [topological_space G] [topological_group G]\nvariables [group H] [topological_space H] [topological_group H]\nvariables (f : G \u2192 H) [is_group_hom f]\n\n\n-- TODO when PR'ing to mathlib, make sure to include _right in the name\n-- of this and nhds_translation_mul_inv\n@[to_additive]\nlemma nhds_translation_mul (g : G) :\n  map (\u03bb h, h*g) (nhds 1) = nhds g :=\nbegin\n  rw \u2190 nhds_translation_mul_inv g,\n  apply map_eq_comap_of_inverse ; ext ; simp\nend\n\n\n@[to_additive]\nlemma nhds_translation_mul_inv_left (g : G) :\n  comap (\u03bb h, g\u207b\u00b9*h) (nhds 1) = nhds g :=\nbegin\n  refine comap_eq_of_inverse (\u03bb h, g*h) _ _ _,\n  { funext x; simp },\n  { suffices : tendsto (\u03bb h,g\u207b\u00b9*h) (nhds g) (nhds (g\u207b\u00b9 * g)), by simpa,\n    exact tendsto_const_nhds.mul tendsto_id },\n  { suffices : tendsto (\u03bb h, g*h) (nhds 1) (nhds (g*1)), by simpa,\n    exact tendsto_const_nhds.mul tendsto_id }\nend\n\n@[to_additive]\nlemma nhds_translation_mul_left (g : G) :\n  map (\u03bb h, g*h) (nhds 1) = nhds g :=\nbegin\n  rw \u2190 nhds_translation_mul_inv_left g,\n  apply map_eq_comap_of_inverse ; ext ; simp\nend\n\n@[to_additive]\nlemma continuous_of_continuous_at_one (h : continuous_at f 1) :\n  continuous f :=\nbegin\n  replace h : map f (nhds 1) \u2264 nhds 1, by rw \u2190 is_group_hom.map_one f ; exact h,\n  rw continuous_iff_continuous_at,\n  intro g,\n  have key : (f \u2218 \u03bb (h : G), g * h) = (\u03bb (h : H), (f g) * h) \u2218 f,\n    by ext ; simp [is_mul_hom.map_mul f],\n  change map f (nhds g) \u2264 nhds (f g),\n  rw [\u2190 nhds_translation_mul_left g, \u2190 nhds_translation_mul_left (f g),\n      filter.map_comm key],\n  exact map_mono h\nend\n\n@[to_additive topological_add_group.tendsto_nhds_iff']\nlemma tendsto_nhds_iff {\u03b1 : Type*} (f : \u03b1 \u2192 H) (F : filter \u03b1) (h : H) :\n  tendsto f F (nhds h) \u2194 \u2200 V \u2208 nhds (1 : H), {a | f a * h\u207b\u00b9 \u2208 V} \u2208 F :=\nlet R := \u03bb h', h' * h\u207b\u00b9,\n    N := nhds (1 : H) in\ncalc tendsto f F (nhds h) \u2194 map f F \u2264 (nhds h) : iff.rfl\n  ... \u2194 map f F \u2264 comap R N : by rw nhds_translation_mul_inv\n  ... \u2194 map R (map f F) \u2264 N : map_le_iff_le_comap.symm\n  ... \u2194 map (\u03bb a, f a * h\u207b\u00b9) F \u2264 N : by rw filter.map_map\n\n@[to_additive topological_add_group.tendsto_nhds_nhds_iff']\nlemma tendsto_nhds_nhds_iff (f : G \u2192 H) (g : G) (h : H) :\n  tendsto f (nhds g) (nhds h) \u2194\n  \u2200 V \u2208 nhds (1 : H), \u2203 U \u2208 nhds (1 : G), \u2200 g', g'*g\u207b\u00b9 \u2208 U \u2192 f g' * h\u207b\u00b9 \u2208 V :=\nby rw [tendsto_nhds_iff f, \u2190 nhds_translation_mul_inv g] ; exact iff.rfl\nend topological_group\n\nnamespace topological_add_group\n-- `to_additive` generates statements using `g + -h` instead of `g-h`, let's fix that\n\nvariables {G : Type*} [add_group G] [topological_space G] [topological_add_group G]\nvariables {H : Type*} [add_group H] [topological_space H] [topological_add_group H]\n\nlemma tendsto_nhds_iff {\u03b1 : Type*} (f : \u03b1 \u2192 H) (F : filter \u03b1) (h : H) :\n    tendsto f F (nhds h) \u2194 \u2200 (V : set H), V \u2208 nhds (0 : H) \u2192 {a : \u03b1 | f a - h \u2208 V} \u2208 F :=\ntopological_add_group.tendsto_nhds_iff' _ _ _\n\nlemma tendsto_nhds_nhds_iff (f : G \u2192 H) (g : G) (h : H) :\n  tendsto f (nhds g) (nhds h) \u2194\n  \u2200 V \u2208 nhds (0 : H), \u2203 U \u2208 nhds (0 : G), \u2200 g', g' - g \u2208 U \u2192 f g' - h \u2208 V :=\ntopological_add_group.tendsto_nhds_nhds_iff' _ _ _\nend topological_add_group\n\nnamespace add_group_with_zero_nhd\nvariables {\u03b1 : Type*} [add_group_with_zero_nhd \u03b1]\nopen filter\n\nlemma nhds_eq' (a : \u03b1) : nhds a = map (\u03bbx, a + x) (Z \u03b1) :=\nby convert nhds_eq a ; ext ; simp\n\nend add_group_with_zero_nhd\n\n\n\nsection\nopen topological_add_group\nvariables (G : Type u) [add_comm_group G] [topological_space G] [topological_add_group G]\n\nlocal attribute [instance] topological_add_group.to_uniform_space\nlocal attribute [instance] topological_add_group_is_uniform\n\nlemma topological_add_group.separated_iff_zero_closed : separated G \u2194 is_closed ({0} : set G) :=\nbegin\n  unfold separated,\n  rw \u2190 closure_eq_iff_is_closed,\n  split ; intro h,\n  { apply subset.antisymm,\n    { intros x x_in,\n      have := group_separation_rel x 0,\n      rw sub_zero at this,\n      rw [\u2190 this, h] at x_in,\n      change x = 0 at x_in,\n      simp [x_in] },\n    { exact subset_closure  } },\n  { ext p,\n    cases p with x y,\n    rw [group_separation_rel x, h, mem_singleton_iff, sub_eq_zero_iff_eq],\n    refl }\nend\n\nlemma topological_add_group.separated_of_zero_sep\n  (H : \u2200 x : G, x \u2260 0 \u2192 \u2203 U \u2208 nhds (0 : G), x \u2209 U) : separated G:=\nbegin\n  rw topological_add_group.separated_iff_zero_closed,\n  rw [\u2190 is_open_compl_iff, is_open_iff_mem_nhds],\n  intros x x_not,\n  have : x \u2260 0, from mem_compl_singleton_iff.mp x_not,\n  rcases H x this with \u27e8U, U_in, xU\u27e9,\n  rw \u2190 nhds_zero_symm G at U_in,\n  rcases U_in with \u27e8W, W_in, UW\u27e9,\n  rw \u2190 nhds_translation_add_neg_left x,\n  use [W, W_in],\n  rw subset_compl_comm,\n  suffices : -x \u2209 W, by simp[this],\n  intro h,\n  exact xU (UW h)\nend\n\nlemma topological_add_group.discrete_iff_open_zero {G : Type*} [add_group G] [topological_space G]\n  [topological_add_group G] : discrete_topology G \u2194 is_open ({0} : set G) :=\nbegin\n  rw discrete_iff_open_singletons,\n  split ; intro h,\n  { exact h 0 },\n  { intro x,\n    convert (continuous_add_right (-x) : continuous (\u03bb y, y - x)) _ h,\n    ext z,\n    erw [mem_singleton_iff, mem_preimage, mem_singleton_iff, sub_eq_zero_iff_eq] },\nend\n\n-- Wedhorn Definition 5.31 page 38\ndefinition is_complete_hausdorff : Prop := is_complete (univ : set G) \u2227 is_hausdorff G\nend\n\n\n-- I used to think I would need the next section soon, but I no longer do.\n-- I keep it because we'll want some form of this in mathlib at some point\nsection top_mul_equiv\nvariables (G : Type*) [group G] [topological_space G] [topological_group G]\nvariables (H : Type*) [group H] [topological_space H] [topological_group H]\n\nstructure top_mul_equiv extends homeomorph G H :=\n(hom : is_group_hom to_fun)\n\ninfix ` \u2243*\u209c `:50 := top_mul_equiv\n\ninstance top_mul_equiv.is_group_hom (h : G \u2243*\u209c H) : is_group_hom h.to_homeomorph :=\nh.hom\nend top_mul_equiv\n\nnamespace top_mul_equiv\nvariables (G : Type*) [group G] [topological_space G] [topological_group G]\nvariables (H : Type*) [group H] [topological_space H] [topological_group H]\nvariables (K : Type*) [group K] [topological_space K] [topological_group K]\n\n@[refl] def refl : G \u2243*\u209c G :=\n{ hom := is_group_hom.id,\n  continuous_to_fun := continuous_id,\n  continuous_inv_fun := continuous_id,\n  ..equiv.refl _}\n\n@[symm] def symm (h : G \u2243*\u209c H) : H \u2243*\u209c G :=\n{ hom := is_group_hom.mk' $ \u03bb n\u2081 n\u2082, function.injective_of_left_inverse h.left_inv begin\n   rw h.hom.map_mul, unfold equiv.symm, rw [h.right_inv, h.right_inv, h.right_inv], end,\n  continuous_to_fun := h.continuous_inv_fun,\n  continuous_inv_fun := h.continuous_to_fun,\n  ..h.to_equiv.symm}\n\n@[trans] def trans (h1 : G \u2243*\u209c H) (h2 : H \u2243*\u209c K) : (G \u2243*\u209c K) :=\n{ hom := is_group_hom.comp h1.to_homeomorph.to_equiv.to_fun h2.to_homeomorph.to_equiv.to_fun,\n  continuous_to_fun := h2.continuous_to_fun.comp h1.continuous_to_fun,\n  continuous_inv_fun := h1.continuous_inv_fun.comp h2.continuous_inv_fun,\n  ..equiv.trans h1.to_equiv h2.to_equiv }\n\nend top_mul_equiv\n\n-- Next secton will move to topology/basic.lean\nsection\nvariables {\u03b1 : Type*} {\u03b2 : Type*} [topological_space \u03b2]\n\n/-- If a function is constant on some set of a proper filter then it converges along this filter -/\nlemma exists_limit_of_ultimately_const {\u03c6 : \u03b1 \u2192 \u03b2} {f : filter \u03b1} (hf : f \u2260 \u22a5)\n{U : set \u03b1} (hU : U \u2208 f) (h : \u2200 x y \u2208 U,  \u03c6 x = \u03c6 y) : \u2203 b, tendsto \u03c6 f (nhds b) :=\nbegin\n  obtain \u27e8x\u2080, x\u2080_in\u27e9 : \u2203 x\u2080, x\u2080 \u2208 U, from nonempty_of_mem_sets hf hU,\n  use \u03c6 x\u2080,\n  have : U \u2286 \u03c6 \u207b\u00b9' {\u03c6 x\u2080},\n  { intros x x_in,\n    simp [h x x\u2080 x_in x\u2080_in] },\n  calc\n   map \u03c6 f \u2264 pure (\u03c6 x\u2080) : by { rw [le_pure_iff, mem_map], exact mem_sets_of_superset hU this}\n       ... \u2264 \ud835\udcdd \u03c6 x\u2080 : pure_le_nhds _\nend\nend\n\n-- The next section will be used to extend a valuation to the completion of a field (for the\n-- valuation induced topology). The group \u0393 will be the value group, G = K^* and H = \\hat{K}^*\n-- (units of the completed field). \u03c6 will be the valuation restricted to K^*\nsection\nopen is_group_hom\nvariables {G : Type*} [group G] [topological_space G] [topological_group G]\nvariables {H : Type*} [group H] [topological_space H] [topological_group H]\nvariables {\u0393 : Type*} [group \u0393] [topological_space \u0393] [topological_group \u0393] [regular_space \u0393]\n\nvariables {\u03b9 : G \u2192 H} [is_group_hom \u03b9] (d\u03b9 : dense_inducing \u03b9)\nvariables {\u03c6 : G \u2192 \u0393} [is_group_hom \u03c6]\n\n-- misc missing lemma, nothing to do with extensions of stuff\n\nlemma mul_right_nhds_one {U : set G} (U_in : U \u2208 nhds (1 : G)) (g : G) :\n  (\u03bb x, x*g) '' U \u2208 nhds g :=\nbegin\n  have l : function.left_inverse (\u03bb (x : G), x * g\u207b\u00b9) (\u03bb (x : G), x * g), from \u03bb x, by simp,\n  have r : function.right_inverse (\u03bb (x : G), x * g\u207b\u00b9) (\u03bb (x : G), x * g), from \u03bb x, by simp,\n  rw image_eq_preimage_of_inverse l r,\n  have : continuous (\u03bb (x : G), x * g\u207b\u00b9), from continuous_id.mul continuous_const,\n  apply this.tendsto g,\n  simpa,\nend\n\n\nlemma continuous_extend_of_open_kernel (op_ker : is_open (ker \u03c6)) : continuous (d\u03b9.extend \u03c6) :=\nbegin\n  have : \u2203 V, V \u2208 nhds (1 : H) \u2227 \u03b9 \u207b\u00b9' V \u2286 ker \u03c6,\n  { have : ker \u03c6 \u2208 nhds (1 : G),\n      from mem_nhds_sets op_ker (is_submonoid.one_mem (ker \u03c6)),\n    rw [d\u03b9.nhds_eq_comap, mem_comap_sets] at this,\n    rcases this with \u27e8V, V_in, hV\u27e9,\n    rw map_one \u03b9 at V_in,\n    use [V, V_in, hV] },\n  rcases this with \u27e8V, V_in, hV\u27e9,\n  have : \u2203 V' \u2208 nhds (1 : H), \u2200 x y \u2208 V', x*y\u207b\u00b9 \u2208 V,\n    from exists_nhds_split_inv V_in,\n  rcases this with \u27e8V', V'_in, hV'\u27e9,\n  refine d\u03b9.continuous_extend _,\n  intro h,\n  have : \u03b9 \u207b\u00b9' ((\u03bb x, x*h) '' V') \u2208 comap \u03b9 (nhds h),\n    from \u27e8(\u03bb (x : H), x * h) '' V', mul_right_nhds_one V'_in h, subset.refl _\u27e9,\n  apply exists_limit_of_ultimately_const d\u03b9.comap_nhds_ne_bot this, clear this,\n  intros x y x_in y_in,\n  rw mem_preimage at x_in y_in,\n  rcases x_in with \u27e8v\u2093, v\u2093_in, hx\u27e9,\n  rcases y_in with \u27e8vy, vy_in, hy\u27e9,\n  change v\u2093 * h = \u03b9 x at hx,\n  change vy * h = \u03b9 y at hy,\n  rw inv_iff_ker \u03c6,\n  apply hV,\n  rw [mem_preimage, is_mul_hom.map_mul \u03b9, map_inv \u03b9, \u2190 hx, \u2190 hy, mul_assoc, mul_inv_rev, mul_inv_cancel_left],\n  simp only [hV', *],\nend\nend\n\ninstance discrete_top_group {G : Type*} [group G] [topological_space G] [discrete_topology G] :\n  topological_group G :=\n{ continuous_mul := continuous_of_discrete_topology,\n  continuous_inv := continuous_of_discrete_topology }\n\n/- section top_group_extend\nopen is_group_hom\nvariables {G : Type*} [group G] [topological_space G] [topological_group G]\nvariables {H : Type*} [group H] [topological_space H] [topological_group H]\nvariables {L : Type*} [group L] [topological_space L] [topological_group L]\n[t2_space L]\n\nvariables {\u03b9 : G \u2192 H} [is_group_hom \u03b9] (de : dense_inducing \u03b9)\nvariables {\u03c6 : G \u2192 L} [is_group_hom \u03c6]\n\nlemma topological_group.extend_is_group_hom (h\u03c6 : continuous \u03c6) (h : continuous (de.extend \u03c6)) :\n  is_group_hom (de.extend \u03c6) :=\nsorry\n-- TODO: Fix is_closed_property2 in mathlib. It has nothing to do with dense embedding. Need\n-- dense_range.prod etc.\n/- \u27e8begin\n  let \u03a6 := de.extend \u03c6,\n  let P := \u03bb x y : H, \u03a6 (x*y) = \u03a6 x*\u03a6 y,\n  have closed : is_closed { q : H \u00d7 H | P q.1 q.2 } :=\n    have c1 : continuous (\u03bb q : H \u00d7 H, \u03a6 (q.1 * q.2)), from h.comp continuous_mul',\n    have c2 : continuous (\u03bb q : H \u00d7 H, \u03a6 q.1 * \u03a6 q.2),\n      from continuous_mul (h.comp continuous_fst) (h.comp continuous_snd),\n  is_closed_eq c1 c2,\n\n  apply is_closed_property2 de closed,\n  intros x y,\n  dsimp [P, \u03a6],\n  rw \u2190 is_group_hom.map_mul \u03b9,\n  repeat { rw dense_embedding.extend_e_eq },\n  rw is_group_hom.map_mul \u03c6\nend\u27e9 -/\nend top_group_extend\n -/\n", "meta": {"author": "leanprover-community", "repo": "lean-perfectoid-spaces", "sha": "95a6520ce578b30a80b4c36e36ab2d559a842690", "save_path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces", "path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces/lean-perfectoid-spaces-95a6520ce578b30a80b4c36e36ab2d559a842690/src/for_mathlib/topological_groups.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.36666580612526284}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.morphisms.quasi_compact\nimport topology.quasi_separated\n\n/-!\n# Quasi-separated morphisms\n\nA morphism of schemes `f : X \u27f6 Y` is quasi-separated if the diagonal morphism `X \u27f6 X \u00d7[Y] X` is\nquasi-compact.\n\nA scheme is quasi-separated if the intersections of any two affine open sets is quasi-compact.\n(`algebraic_geometry.quasi_separated_space_iff_affine`)\n\nWe show that a morphism is quasi-separated if the preimage of every affine open is quasi-separated.\n\nWe also show that this property is local at the target,\nand is stable under compositions and base-changes.\n\n## Main result\n- `is_localization_basic_open_of_qcqs` (**Qcqs lemma**):\n  If `U` is qcqs, then `\u0393(X, D(f)) \u2243 \u0393(X, U)_f` for every `f : \u0393(X, U)`.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverse u\n\nopen_locale algebraic_geometry\n\nnamespace algebraic_geometry\n\nvariables {X Y : Scheme.{u}} (f : X \u27f6 Y)\n\n/-- A morphism is `quasi_separated` if diagonal map is quasi-compact. -/\n@[mk_iff]\nclass quasi_separated (f : X \u27f6 Y) : Prop :=\n(diagonal_quasi_compact : quasi_compact (pullback.diagonal f))\n\n/-- The `affine_target_morphism_property` corresponding to `quasi_separated`, asserting that the\ndomain is a quasi-separated scheme. -/\ndef quasi_separated.affine_property : affine_target_morphism_property :=\n(\u03bb X Y f _, quasi_separated_space X.carrier)\n\nlemma quasi_separated_space_iff_affine (X : Scheme) :\n  quasi_separated_space X.carrier \u2194 \u2200 (U V : X.affine_opens), is_compact (U \u2229 V : set X.carrier) :=\nbegin\n  rw quasi_separated_space_iff,\n  split,\n  { intros H U V, exact H U V U.1.2 U.2.is_compact V.1.2 V.2.is_compact },\n  { intros H,\n    suffices : \u2200 (U : opens X.carrier) (hU : is_compact U.1) (V : opens X.carrier)\n      (hV : is_compact V.1), is_compact (U \u2293 V).1,\n    { intros U V hU hU' hV hV', exact this \u27e8U, hU\u27e9 hU' \u27e8V, hV\u27e9 hV' },\n    intros U hU V hV,\n    apply compact_open_induction_on V hV,\n    { simp },\n    { intros S hS V hV,\n      change is_compact (U.1 \u2229 (S.1 \u222a V.1)),\n      rw set.inter_union_distrib_left,\n      apply hV.union,\n      clear hV,\n      apply compact_open_induction_on U hU,\n      { simp },\n      { intros S hS W hW,\n      change is_compact ((S.1 \u222a W.1) \u2229 V.1),\n        rw set.union_inter_distrib_right,\n        apply hW.union,\n        apply H } } }\nend\n\nlemma quasi_compact_affine_property_iff_quasi_separated_space {X Y : Scheme} [is_affine Y]\n  (f : X \u27f6 Y) :\n  quasi_compact.affine_property.diagonal f \u2194 quasi_separated_space X.carrier :=\nbegin\n  delta affine_target_morphism_property.diagonal,\n  rw quasi_separated_space_iff_affine,\n  split,\n  { intros H U V,\n    haveI : is_affine _ := U.2,\n    haveI : is_affine _ := V.2,\n    let g : pullback (X.of_restrict U.1.open_embedding) (X.of_restrict V.1.open_embedding) \u27f6 X :=\n      pullback.fst \u226b X.of_restrict _,\n    have : is_open_immersion g := infer_instance,\n    have e := homeomorph.of_embedding _ this.base_open.to_embedding,\n    rw is_open_immersion.range_pullback_to_base_of_left at e,\n    erw [subtype.range_coe, subtype.range_coe] at e,\n    rw is_compact_iff_compact_space,\n    exact @@homeomorph.compact_space _ _ (H _ _) e },\n  { introv H h\u2081 h\u2082,\n    resetI,\n    let g : pullback f\u2081 f\u2082 \u27f6 X := pullback.fst \u226b f\u2081,\n    have : is_open_immersion g := infer_instance,\n    have e := homeomorph.of_embedding _ this.base_open.to_embedding,\n    rw is_open_immersion.range_pullback_to_base_of_left at e,\n    simp_rw is_compact_iff_compact_space at H,\n    exact @@homeomorph.compact_space _ _\n      (H \u27e8\u27e8_, h\u2081.base_open.open_range\u27e9, range_is_affine_open_of_open_immersion _\u27e9\n        \u27e8\u27e8_, h\u2082.base_open.open_range\u27e9, range_is_affine_open_of_open_immersion _\u27e9) e.symm },\nend\n\nlemma quasi_separated_eq_diagonal_is_quasi_compact :\n  @quasi_separated = morphism_property.diagonal @quasi_compact :=\nby { ext, exact quasi_separated_iff _ }\n\nlemma quasi_compact_affine_property_diagonal_eq :\n  quasi_compact.affine_property.diagonal = quasi_separated.affine_property :=\nby { ext, rw quasi_compact_affine_property_iff_quasi_separated_space, refl }\n\nlemma quasi_separated_eq_affine_property_diagonal :\n  @quasi_separated =\n    target_affine_locally quasi_compact.affine_property.diagonal :=\nbegin\n  rw [quasi_separated_eq_diagonal_is_quasi_compact, quasi_compact_eq_affine_property],\n  exact diagonal_target_affine_locally_eq_target_affine_locally\n    _ quasi_compact.affine_property_is_local\nend\n\nlemma quasi_separated_eq_affine_property :\n  @quasi_separated =\n    target_affine_locally quasi_separated.affine_property :=\nby rw [quasi_separated_eq_affine_property_diagonal, quasi_compact_affine_property_diagonal_eq]\n\nlemma quasi_separated.affine_property_is_local :\n  quasi_separated.affine_property.is_local :=\nquasi_compact_affine_property_diagonal_eq \u25b8\nquasi_compact.affine_property_is_local.diagonal\n\n@[priority 900]\ninstance quasi_separated_of_mono {X Y : Scheme} (f : X \u27f6 Y) [mono f] : quasi_separated f :=\n\u27e8infer_instance\u27e9\n\nlemma quasi_separated_stable_under_composition :\n  morphism_property.stable_under_composition @quasi_separated :=\nquasi_separated_eq_diagonal_is_quasi_compact.symm \u25b8\n  quasi_compact_stable_under_composition.diagonal\n    quasi_compact_respects_iso\n    quasi_compact_stable_under_base_change\n\nlemma quasi_separated_stable_under_base_change :\n  morphism_property.stable_under_base_change @quasi_separated :=\nquasi_separated_eq_diagonal_is_quasi_compact.symm \u25b8\n  quasi_compact_stable_under_base_change.diagonal\n    quasi_compact_respects_iso\n\ninstance quasi_separated_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [quasi_separated f] [quasi_separated g] : quasi_separated (f \u226b g) :=\nquasi_separated_stable_under_composition f g infer_instance infer_instance\n\nlemma quasi_separated_respects_iso : morphism_property.respects_iso @quasi_separated :=\nquasi_separated_eq_diagonal_is_quasi_compact.symm \u25b8\n  quasi_compact_respects_iso.diagonal\n\nlemma quasi_separated.affine_open_cover_tfae {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [quasi_separated f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)],\n      \u2200 (i : \ud835\udcb0.J), quasi_separated_space (pullback f (\ud835\udcb0.map i)).carrier,\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J),\n      quasi_separated_space (pullback f (\ud835\udcb0.map i)).carrier,\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_affine U] [is_open_immersion g],\n      quasi_separated_space (pullback f g).carrier,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)]\n      (\ud835\udcb0' : \u03a0 (i : \ud835\udcb0.J), Scheme.open_cover.{u} (pullback f (\ud835\udcb0.map i)))\n      [\u2200 i j, is_affine ((\ud835\udcb0' i).obj j)], by exactI \u2200 (i : \ud835\udcb0.J) (j k : (\ud835\udcb0' i).J),\n        compact_space (pullback ((\ud835\udcb0' i).map j) ((\ud835\udcb0' i).map k)).carrier] :=\nbegin\n  have := quasi_compact.affine_property_is_local.diagonal_affine_open_cover_tfae f,\n  simp_rw [\u2190 quasi_compact_eq_affine_property,\n    \u2190 quasi_separated_eq_diagonal_is_quasi_compact,\n    quasi_compact_affine_property_diagonal_eq] at this,\n  exact this\nend\n\nlemma quasi_separated.is_local_at_target :\n  property_is_local_at_target @quasi_separated :=\nquasi_separated_eq_affine_property_diagonal.symm \u25b8\n  quasi_compact.affine_property_is_local.diagonal.target_affine_locally_is_local\n\nlemma quasi_separated.open_cover_tfae {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [quasi_separated f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y), \u2200 (i : \ud835\udcb0.J),\n      quasi_separated (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) (i : \ud835\udcb0.J),\n      quasi_separated (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (U : opens Y.carrier), quasi_separated (f \u2223_ U),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_open_immersion g],\n      quasi_separated (pullback.snd : pullback f g \u27f6 _),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4),\n      \u2200 i, quasi_separated (f \u2223_ (U i))] :=\nquasi_separated.is_local_at_target.open_cover_tfae f\n\nlemma quasi_separated_over_affine_iff {X Y : Scheme} (f : X \u27f6 Y) [is_affine Y] :\n  quasi_separated f \u2194 quasi_separated_space X.carrier :=\nby rw [quasi_separated_eq_affine_property,\n  quasi_separated.affine_property_is_local.affine_target_iff f,\n  quasi_separated.affine_property]\n\nlemma quasi_separated_space_iff_quasi_separated (X : Scheme) :\n  quasi_separated_space X.carrier \u2194 quasi_separated (terminal.from X) :=\n(quasi_separated_over_affine_iff _).symm\n\nlemma quasi_separated.affine_open_cover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.open_cover.{u} Y)\n  [\u2200 i, is_affine (\ud835\udcb0.obj i)] (f : X \u27f6 Y) :\n  quasi_separated f \u2194 \u2200 i, quasi_separated_space (pullback f (\ud835\udcb0.map i)).carrier :=\nbegin\n  rw [quasi_separated_eq_affine_property,\n    quasi_separated.affine_property_is_local.affine_open_cover_iff f \ud835\udcb0],\n  refl,\nend\n\nlemma quasi_separated.open_cover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.open_cover.{u} Y)\n  (f : X \u27f6 Y) :\n  quasi_separated f \u2194 \u2200 i, quasi_separated (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) :=\nquasi_separated.is_local_at_target.open_cover_iff f \ud835\udcb0\n\ninstance {X Y S : Scheme} (f : X \u27f6 S) (g : Y \u27f6 S) [quasi_separated g] :\n  quasi_separated (pullback.fst : pullback f g \u27f6 X) :=\nquasi_separated_stable_under_base_change.fst f g infer_instance\n\ninstance {X Y S : Scheme} (f : X \u27f6 S) (g : Y \u27f6 S) [quasi_separated f] :\n  quasi_separated (pullback.snd : pullback f g \u27f6 Y) :=\nquasi_separated_stable_under_base_change.snd f g infer_instance\n\ninstance {X Y Z: Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z) [quasi_separated f] [quasi_separated g] :\n  quasi_separated (f \u226b g) :=\nquasi_separated_stable_under_composition f g infer_instance infer_instance\n\nlemma quasi_separated_space_of_quasi_separated {X Y : Scheme} (f : X \u27f6 Y)\n  [hY : quasi_separated_space Y.carrier] [quasi_separated f] : quasi_separated_space X.carrier :=\nbegin\n  rw quasi_separated_space_iff_quasi_separated at hY \u22a2,\n  have : f \u226b terminal.from Y = terminal.from X := terminal_is_terminal.hom_ext _ _,\n  rw \u2190 this,\n  resetI, apply_instance\nend\n\ninstance quasi_separated_space_of_is_affine (X : Scheme) [is_affine X] :\n  quasi_separated_space X.carrier :=\nbegin\n  constructor,\n  intros U V hU hU' hV hV',\n  obtain \u27e8s, hs, e\u27e9 := (is_compact_open_iff_eq_basic_open_union _).mp \u27e8hU', hU\u27e9,\n  obtain \u27e8s', hs', e'\u27e9 := (is_compact_open_iff_eq_basic_open_union _).mp \u27e8hV', hV\u27e9,\n  rw [e, e', set.Union\u2082_inter],\n  simp_rw [set.inter_Union\u2082],\n  apply hs.is_compact_bUnion,\n  { intros i hi,\n    apply hs'.is_compact_bUnion,\n    intros i' hi',\n    change is_compact (X.basic_open i \u2293 X.basic_open i').1,\n    rw \u2190 Scheme.basic_open_mul,\n    exact ((top_is_affine_open _).basic_open_is_affine _).is_compact }\nend\n\nlemma is_affine_open.is_quasi_separated {X : Scheme} {U : opens X.carrier} (hU : is_affine_open U) :\n  is_quasi_separated (U : set X.carrier)  :=\nbegin\n  rw is_quasi_separated_iff_quasi_separated_space,\n  exacts [@@algebraic_geometry.quasi_separated_space_of_is_affine _ hU, U.is_open],\nend\n\nlemma quasi_separated_of_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [H : quasi_separated (f \u226b g)] : quasi_separated f :=\nbegin\n  rw (quasi_separated.affine_open_cover_tfae f).out 0 1,\n  rw (quasi_separated.affine_open_cover_tfae (f \u226b g)).out 0 2 at H,\n  use (Z.affine_cover.pullback_cover g).bind (\u03bb x, Scheme.affine_cover _),\n  split, { intro i, dsimp, apply_instance },\n  rintro \u27e8i, j\u27e9, dsimp at *,\n  specialize H _ i,\n  refine @@quasi_separated_space_of_quasi_separated _ H _,\n  { exact pullback.map _ _ _ _ (\ud835\udfd9 _) _ _ (by simp) (category.comp_id _) \u226b\n      (pullback_right_pullback_fst_iso g (Z.affine_cover.map i) f).hom },\n  { apply algebraic_geometry.quasi_separated_of_mono }\nend\n\nlemma exists_eq_pow_mul_of_is_affine_open (X : Scheme) (U : opens X.carrier) (hU : is_affine_open U)\n  (f : X.presheaf.obj (op U)) (x : X.presheaf.obj (op $ X.basic_open f)) :\n  \u2203 (n : \u2115) (y : X.presheaf.obj (op U)),\n    y |_ X.basic_open f = (f |_ X.basic_open f) ^ n * x :=\nbegin\n  have := (is_localization_basic_open hU f).2,\n  obtain \u27e8\u27e8y, _, n, rfl\u27e9, d\u27e9 := this x,\n  use [n, y],\n  delta Top.presheaf.restrict_open Top.presheaf.restrict,\n  simpa [mul_comm x] using d.symm,\nend\n\nlemma exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux (X : Scheme)\n  (S : X.affine_opens) (U\u2081 U\u2082 : opens X.carrier)\n  {n\u2081 n\u2082 : \u2115} {y\u2081 : X.presheaf.obj (op U\u2081)}\n  {y\u2082 : X.presheaf.obj (op U\u2082)} {f : X.presheaf.obj (op $ U\u2081 \u2294 U\u2082)}\n  {x : X.presheaf.obj (op $ X.basic_open f)}\n  (h\u2081 : S.1 \u2264 U\u2081) (h\u2082 : S.1 \u2264 U\u2082)\n  (e\u2081 : X.presheaf.map (hom_of_le $ X.basic_open_le\n    (X.presheaf.map (hom_of_le le_sup_left).op f) : _ \u27f6 U\u2081).op y\u2081 =\n      X.presheaf.map (hom_of_le (by { erw X.basic_open_res, exact inf_le_left })).op\n        (X.presheaf.map (hom_of_le le_sup_left).op f) ^ n\u2081 *\n      (X.presheaf.map (hom_of_le (by { erw X.basic_open_res, exact inf_le_right })).op) x)\n  (e\u2082 : X.presheaf.map (hom_of_le $ X.basic_open_le\n    (X.presheaf.map (hom_of_le le_sup_right).op f) : _ \u27f6 U\u2082).op y\u2082 =\n      X.presheaf.map (hom_of_le (by { rw X.basic_open_res, exact inf_le_left })).op\n        (X.presheaf.map (hom_of_le le_sup_right).op f) ^ n\u2082 *\n      (X.presheaf.map (hom_of_le (by { rw X.basic_open_res, exact inf_le_right })).op) x) :\n  \u2203 n : \u2115, X.presheaf.map (hom_of_le $ h\u2081).op\n    ((X.presheaf.map (hom_of_le le_sup_left).op f) ^ (n + n\u2082) * y\u2081) =\n    X.presheaf.map (hom_of_le $ h\u2082).op\n      ((X.presheaf.map (hom_of_le le_sup_right).op f) ^ (n + n\u2081) * y\u2082) :=\nbegin\n  have := (is_localization_basic_open S.2\n    (X.presheaf.map (hom_of_le $ le_trans h\u2081 le_sup_left).op f)),\n  obtain \u27e8\u27e8_, n, rfl\u27e9, e\u27e9 :=\n    (@is_localization.eq_iff_exists _ _ _ _ _ _ this (X.presheaf.map (hom_of_le $ h\u2081).op\n      ((X.presheaf.map (hom_of_le le_sup_left).op f) ^ n\u2082 * y\u2081))\n    (X.presheaf.map (hom_of_le $ h\u2082).op\n      ((X.presheaf.map (hom_of_le le_sup_right).op f) ^ n\u2081 * y\u2082))).mp _,\n  swap,\n  { simp only [map_pow, ring_hom.algebra_map_to_algebra, map_mul, \u2190 comp_apply,\n      \u2190 functor.map_comp, \u2190 op_comp, hom_of_le_comp],\n    have h\u2083 : X.basic_open ((X.presheaf.map (hom_of_le (h\u2081.trans le_sup_left)).op) f) \u2264 S.val,\n    { simpa only [X.basic_open_res] using inf_le_left, },\n    transitivity\n      X.presheaf.map (hom_of_le $ h\u2083.trans $ h\u2081.trans le_sup_left).op f ^ (n\u2082 + n\u2081) *\n      X.presheaf.map (hom_of_le $ (X.basic_open_res f _).trans_le inf_le_right).op x,\n    { rw [pow_add, mul_assoc], congr' 1,\n      convert congr_arg (X.presheaf.map (hom_of_le _).op) e\u2081,\n      { simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp], congr },\n      { simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp], congr },\n      { rw [X.basic_open_res, X.basic_open_res], rintros x \u27e8H\u2081, H\u2082\u27e9, exact \u27e8h\u2081 H\u2081, H\u2082\u27e9 } },\n    { rw [add_comm, pow_add, mul_assoc], congr' 1,\n      convert congr_arg (X.presheaf.map (hom_of_le _).op) e\u2082.symm,\n      { simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp], congr },\n      { simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp], congr },\n      { simp only [X.basic_open_res],\n        rintros x \u27e8H\u2081, H\u2082\u27e9, exact \u27e8h\u2082 H\u2081, H\u2082\u27e9 } } },\n  use n,\n  simp only [pow_add, map_pow, map_mul, \u2190 comp_apply, \u2190 mul_assoc,\n    \u2190 functor.map_comp, subtype.coe_mk] at e \u22a2,\n  exact e\nend\n\nlemma exists_eq_pow_mul_of_is_compact_of_is_quasi_separated (X : Scheme)\n  (U : opens X.carrier) (hU : is_compact U.1) (hU' : is_quasi_separated U.1)\n  (f : X.presheaf.obj (op U)) (x : X.presheaf.obj (op $ X.basic_open f)) :\n  \u2203 (n : \u2115) (y : X.presheaf.obj (op U)), y |_ X.basic_open f = (f |_ X.basic_open f) ^ n * x :=\nbegin\n  delta Top.presheaf.restrict_open Top.presheaf.restrict,\n  revert hU' f x,\n  apply compact_open_induction_on U hU,\n  { intros hU' f x,\n    use [0, f],\n    refine @@subsingleton.elim (CommRing.subsingleton_of_is_terminal\n      (X.sheaf.is_terminal_of_eq_empty _)) _ _,\n    erw eq_bot_iff,\n    exact X.basic_open_le f },\n  { -- Given `f : \ud835\udcaa(S \u222a U), x : \ud835\udcaa(X_f)`, we need to show that `f ^ n * x` is the restriction of\n    -- some `y : \ud835\udcaa(S \u222a U)` for some `n : \u2115`.\n    intros S hS U hU hSU f x,\n    -- We know that such `y\u2081, n\u2081` exists on `S` by the induction hypothesis.\n    obtain \u27e8n\u2081, y\u2081, hy\u2081\u27e9 := hU (hSU.of_subset $ set.subset_union_left _ _)\n      (X.presheaf.map (hom_of_le le_sup_left).op f) (X.presheaf.map (hom_of_le _).op x),\n    swap, { rw X.basic_open_res, exact inf_le_right },\n    -- We know that such `y\u2082, n\u2082` exists on `U` since `U` is affine.\n    obtain \u27e8n\u2082, y\u2082, hy\u2082\u27e9 := exists_eq_pow_mul_of_is_affine_open X _ U.2\n      (X.presheaf.map (hom_of_le le_sup_right).op f) (X.presheaf.map (hom_of_le _).op x),\n    delta Top.presheaf.restrict_open Top.presheaf.restrict at hy\u2082,\n    swap, { rw X.basic_open_res, exact inf_le_right },\n    -- Since `S \u222a U` is quasi-separated, `S \u2229 U` can be covered by finite affine opens.\n    obtain \u27e8s, hs', hs\u27e9 := (is_compact_open_iff_eq_finset_affine_union _).mp\n      \u27e8hSU _ _ (set.subset_union_left _ _) S.2 hS\n        (set.subset_union_right _ _) U.1.2 U.2.is_compact, (S \u2293 U.1).2\u27e9,\n    haveI := hs'.to_subtype,\n    casesI nonempty_fintype s,\n    replace hs : S \u2293 U.1 = supr (\u03bb i : s, (i : opens X.carrier)) := by { ext1, simpa using hs },\n    have hs\u2081 : \u2200 i : s, i.1.1 \u2264 S,\n    { intro i, change (i : opens X.carrier) \u2264 S,\n      refine le_trans _ inf_le_left, use U.1, erw hs, exact le_supr _ _ },\n    have hs\u2082 : \u2200 i : s, i.1.1 \u2264 U.1,\n    { intro i, change (i : opens X.carrier) \u2264 U,\n      refine le_trans _ inf_le_right, use S, erw hs, exact le_supr _ _ },\n    -- On each affine open in the intersection, we have `f ^ (n + n\u2082) * y\u2081 = f ^ (n + n\u2081) * y\u2082`\n    -- for some `n` since `f ^ n\u2082 * y\u2081 = f ^ (n\u2081 + n\u2082) * x = f ^ n\u2081 * y\u2082` on `X_f`.\n    have : \u2200 i : s, \u2203 n : \u2115,\n      X.presheaf.map (hom_of_le $ hs\u2081 i).op\n        ((X.presheaf.map (hom_of_le le_sup_left).op f) ^ (n + n\u2082) * y\u2081) =\n      X.presheaf.map (hom_of_le $ hs\u2082 i).op\n        ((X.presheaf.map (hom_of_le le_sup_right).op f) ^ (n + n\u2081) * y\u2082),\n    { intro i,\n      exact exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux X i.1 S U (hs\u2081 i) (hs\u2082 i)\n        hy\u2081 hy\u2082 },\n    choose n hn using this,\n    -- We can thus choose a big enough `n` such that `f ^ (n + n\u2082) * y\u2081 = f ^ (n + n\u2081) * y\u2082`\n    -- on `S \u2229 U`.\n    have : X.presheaf.map (hom_of_le $ inf_le_left).op\n      ((X.presheaf.map (hom_of_le le_sup_left).op f) ^ (finset.univ.sup n + n\u2082) * y\u2081) =\n        X.presheaf.map (hom_of_le $ inf_le_right).op\n          ((X.presheaf.map (hom_of_le le_sup_right).op f) ^ (finset.univ.sup n + n\u2081) * y\u2082),\n    { fapply X.sheaf.eq_of_locally_eq' (\u03bb i : s, i.1.1),\n      { refine \u03bb i, hom_of_le _, erw hs, exact le_supr _ _ },\n      { exact le_of_eq hs },\n      { intro i,\n        replace hn := congr_arg (\u03bb x, X.presheaf.map (hom_of_le\n          (le_trans (hs\u2081 i) le_sup_left)).op f ^ (finset.univ.sup n - n i) * x) (hn i),\n        dsimp only at hn,\n        delta Scheme.sheaf SheafedSpace.sheaf,\n        simp only [\u2190 map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp, \u2190 mul_assoc]\n          at hn \u22a2,\n        erw [\u2190 map_mul, \u2190 map_mul] at hn,\n        rw [\u2190 pow_add, \u2190 pow_add, \u2190 add_assoc, \u2190 add_assoc, tsub_add_cancel_of_le] at hn,\n        convert hn,\n        exact finset.le_sup (finset.mem_univ _) } },\n    use finset.univ.sup n + n\u2081 + n\u2082,\n    -- By the sheaf condition, since `f ^ (n + n\u2082) * y\u2081 = f ^ (n + n\u2081) * y\u2082`, it can be glued into\n    -- the desired section on `S \u222a U`.\n    use (X.sheaf.obj_sup_iso_prod_eq_locus S U.1).inv \u27e8\u27e8_ * _, _ * _\u27e9, this\u27e9,\n    refine X.sheaf.eq_of_locally_eq\u2082\n      (hom_of_le (_ : X.basic_open (X.presheaf.map (hom_of_le le_sup_left).op f) \u2264 _))\n      (hom_of_le (_ : X.basic_open (X.presheaf.map (hom_of_le le_sup_right).op f) \u2264 _)) _ _ _ _ _,\n    { rw X.basic_open_res, exact inf_le_right },\n    { rw X.basic_open_res, exact inf_le_right },\n    { rw [X.basic_open_res, X.basic_open_res],\n      erw \u2190 inf_sup_right,\n      refine le_inf_iff.mpr \u27e8X.basic_open_le f, le_of_eq rfl\u27e9 },\n    { convert congr_arg (X.presheaf.map (hom_of_le _).op)\n        (X.sheaf.obj_sup_iso_prod_eq_locus_inv_fst S U.1 \u27e8\u27e8_ * _, _ * _\u27e9, this\u27e9) using 1,\n      { delta Scheme.sheaf SheafedSpace.sheaf,\n        simp only [\u2190 comp_apply (X.presheaf.map _) (X.presheaf.map _),\n          \u2190 functor.map_comp, \u2190 op_comp],\n        congr },\n      { delta Scheme.sheaf SheafedSpace.sheaf,\n        simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp, mul_assoc,\n          pow_add], erw hy\u2081, congr' 1, rw [\u2190 mul_assoc, \u2190 mul_assoc], congr' 1,\n        rw [mul_comm, \u2190 comp_apply, \u2190 functor.map_comp], congr } },\n    { convert congr_arg (X.presheaf.map (hom_of_le _).op)\n        (X.sheaf.obj_sup_iso_prod_eq_locus_inv_snd S U.1 \u27e8\u27e8_ * _, _ * _\u27e9, this\u27e9) using 1,\n      { delta Scheme.sheaf SheafedSpace.sheaf,\n        simp only [\u2190 comp_apply (X.presheaf.map _) (X.presheaf.map _),\n          \u2190 functor.map_comp, \u2190 op_comp],\n        congr },\n      { delta Scheme.sheaf SheafedSpace.sheaf,\n        simp only [map_pow, map_mul, \u2190 comp_apply, \u2190 functor.map_comp, \u2190 op_comp, mul_assoc,\n          pow_add], erw hy\u2082, rw [\u2190 comp_apply, \u2190 functor.map_comp], congr } } }\nend\n\n/-- If `U` is qcqs, then `\u0393(X, D(f)) \u2243 \u0393(X, U)_f` for every `f : \u0393(X, U)`.\nThis is known as the **Qcqs lemma** in [R. Vakil, *The rising sea*][RisingSea]. -/\nlemma is_localization_basic_open_of_qcqs {X : Scheme} {U : opens X.carrier}\n  (hU : is_compact U.1) (hU' : is_quasi_separated U.1)\n  (f : X.presheaf.obj (op U)) :\n  is_localization.away f (X.presheaf.obj (op $ X.basic_open f)) :=\nbegin\n  constructor,\n  { rintro \u27e8_, n, rfl\u27e9,\n    simp only [map_pow, subtype.coe_mk, ring_hom.algebra_map_to_algebra],\n    exact is_unit.pow _ (RingedSpace.is_unit_res_basic_open _ f), },\n  { intro z,\n    obtain \u27e8n, y, e\u27e9 := exists_eq_pow_mul_of_is_compact_of_is_quasi_separated X U hU hU' f z,\n    refine \u27e8\u27e8y, _, n, rfl\u27e9, _\u27e9,\n    simpa only [map_pow, subtype.coe_mk, ring_hom.algebra_map_to_algebra, mul_comm z]\n      using e.symm },\n  { intros x y,\n    rw [\u2190 sub_eq_zero, \u2190 map_sub, ring_hom.algebra_map_to_algebra],\n    simp_rw [\u2190 @sub_eq_zero _ _ (_ * x) (_ * y), \u2190 mul_sub],\n    generalize : x - y = z,\n    split,\n    { intro H,\n      obtain \u27e8n, e\u27e9 := exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_compact X hU _ _ H,\n      refine \u27e8\u27e8_, n, rfl\u27e9, _\u27e9,\n      simpa [mul_comm z] using e },\n    { rintro \u27e8\u27e8_, n, rfl\u27e9, e : f ^ n * z = 0\u27e9,\n      rw [\u2190 ((RingedSpace.is_unit_res_basic_open _ f).pow n).mul_right_inj, mul_zero, \u2190 map_pow,\n        \u2190 map_mul, e, map_zero] } }\nend\n\nend algebraic_geometry\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/morphisms/quasi_separated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521102, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3666657984278983}}
{"text": "import AutograderTests.Util\n\nin_exercise\n\n  inductive Example\n    | ctor (h : False)\n\n  theorem exercise : Example :=\n    sorry\n\nin_solution\n\n  inductive Example\n    | ctor (h : True) -- clever change\n\n  theorem exercise : Example :=\n    \u27e8\u27e8\u27e9\u27e9\n", "meta": {"author": "gebner", "repo": "autograder", "sha": "9d23bfc346c672e93d0b4ee11453925ed15cd091", "save_path": "github-repos/lean/gebner-autograder", "path": "github-repos/lean/gebner-autograder/autograder-9d23bfc346c672e93d0b4ee11453925ed15cd091/AutograderTests/TodoFail/ChangedCtor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.36666579073053374}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.number_theory.pell\nimport Mathlib.data.pfun\nimport Mathlib.data.fin2\nimport Mathlib.PostPort\n\nuniverses u u_1 u_2 u_3 \n\nnamespace Mathlib\n\nnamespace int\n\n\ntheorem eq_nat_abs_iff_mul (x : \u2124) (n : \u2115) : nat_abs x = n \u2194 (x - \u2191n) * (x + \u2191n) = 0 := sorry\n\nend int\n\n\n/-- Alternate definition of `vector` based on `fin2`. -/\ndef vector3 (\u03b1 : Type u) (n : \u2115) := fin2 n \u2192 \u03b1\n\nnamespace vector3\n\n\n/-- The empty vector -/\ndef nil {\u03b1 : Type u_1} : vector3 \u03b1 0 := sorry\n\n/-- The vector cons operation -/\ndef cons {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) : vector3 \u03b1 (Nat.succ n) :=\n  fun (i : fin2 (Nat.succ n)) => fin2.cases' a v i\n\ninfixr:67 \" :: \" => Mathlib.vector3.cons\n\n/- We do not want to make the following notation global, because then these expressions will be\noverloaded, and only the expected type will be able to disambiguate the meaning. Worse: Lean will\ntry to insert a coercion from `vector3 \u03b1 _` to `list \u03b1`, if a list is expected. -/\n\n@[simp] theorem cons_fz {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) : cons a v fin2.fz = a :=\n  rfl\n\n@[simp] theorem cons_fs {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) (i : fin2 n) :\n    cons a v (fin2.fs i) = v i :=\n  rfl\n\n/-- Get the `i`th element of a vector -/\ndef nth {\u03b1 : Type u_1} {n : \u2115} (i : fin2 n) (v : vector3 \u03b1 n) : \u03b1 := v i\n\n/-- Construct a vector from a function on `fin2`. -/\ndef of_fn {\u03b1 : Type u_1} {n : \u2115} (f : fin2 n \u2192 \u03b1) : vector3 \u03b1 n := f\n\n/-- Get the head of a nonempty vector. -/\ndef head {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : \u03b1 := v fin2.fz\n\n/-- Get the tail of a nonempty vector. -/\ndef tail {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : vector3 \u03b1 n :=\n  fun (i : fin2 n) => v (fin2.fs i)\n\ntheorem eq_nil {\u03b1 : Type u_1} (v : vector3 \u03b1 0) : v = nil := sorry\n\ntheorem cons_head_tail {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : head v :: tail v = v :=\n  funext fun (i : fin2 (Nat.succ n)) => fin2.cases' rfl (fun (_x : fin2 n) => rfl) i\n\ndef nil_elim {\u03b1 : Type u_1} {C : vector3 \u03b1 0 \u2192 Sort u} (H : C nil) (v : vector3 \u03b1 0) : C v :=\n  eq.mpr sorry H\n\ndef cons_elim {\u03b1 : Type u_1} {n : \u2115} {C : vector3 \u03b1 (Nat.succ n) \u2192 Sort u}\n    (H : (a : \u03b1) \u2192 (t : vector3 \u03b1 n) \u2192 C (a :: t)) (v : vector3 \u03b1 (Nat.succ n)) : C v :=\n  eq.mpr sorry (H (head v) (tail v))\n\n@[simp] theorem cons_elim_cons {\u03b1 : Type u_1} {n : \u2115} {C : vector3 \u03b1 (Nat.succ n) \u2192 Sort u_2}\n    {H : (a : \u03b1) \u2192 (t : vector3 \u03b1 n) \u2192 C (a :: t)} {a : \u03b1} {t : vector3 \u03b1 n} :\n    cons_elim H (a :: t) = H a t :=\n  rfl\n\nprotected def rec_on {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u} {n : \u2115} (v : vector3 \u03b1 n)\n    (H0 : C nil) (Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)) : C v :=\n  nat.rec_on n (fun (v : vector3 \u03b1 0) => nil_elim H0 v)\n    (fun (n : \u2115) (IH : (_a : vector3 \u03b1 n) \u2192 C _a) (v : vector3 \u03b1 (Nat.succ n)) =>\n      cons_elim (fun (a : \u03b1) (t : vector3 \u03b1 n) => Hs a t (IH t)) v)\n    v\n\n@[simp] theorem rec_on_nil {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u_2} {H0 : C nil}\n    {Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)} :\n    vector3.rec_on nil H0 Hs = H0 :=\n  rfl\n\n@[simp] theorem rec_on_cons {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u_2} {H0 : C nil}\n    {Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)} {n : \u2115} {a : \u03b1}\n    {v : vector3 \u03b1 n} : vector3.rec_on (a :: v) H0 Hs = Hs a v (vector3.rec_on v H0 Hs) :=\n  rfl\n\n/-- Append two vectors -/\ndef append {\u03b1 : Type u_1} {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n) : vector3 \u03b1 (n + m) :=\n  nat.rec_on m (fun (_x : vector3 \u03b1 0) => w)\n    (fun (m : \u2115) (IH : vector3 \u03b1 m \u2192 vector3 \u03b1 (n + m)) (v : vector3 \u03b1 (Nat.succ m)) =>\n      cons_elim (fun (a : \u03b1) (t : vector3 \u03b1 m) => fin2.cases' a (IH t)) v)\n    v\n\n@[simp] theorem append_nil {\u03b1 : Type u_1} {n : \u2115} (w : vector3 \u03b1 n) : append nil w = w := rfl\n\n@[simp] theorem append_cons {\u03b1 : Type u_1} (a : \u03b1) {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115}\n    (w : vector3 \u03b1 n) : append (a :: v) w = a :: append v w :=\n  rfl\n\n@[simp] theorem append_left {\u03b1 : Type u_1} {m : \u2115} (i : fin2 m) (v : vector3 \u03b1 m) {n : \u2115}\n    (w : vector3 \u03b1 n) : append v w (fin2.left n i) = v i :=\n  sorry\n\n@[simp] theorem append_add {\u03b1 : Type u_1} {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n)\n    (i : fin2 n) : append v w (fin2.add i m) = w i :=\n  sorry\n\n/-- Insert `a` into `v` at index `i`. -/\ndef insert {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (v : vector3 \u03b1 n) (i : fin2 (Nat.succ n)) :\n    vector3 \u03b1 (Nat.succ n) :=\n  fun (j : fin2 (Nat.succ n)) => cons a v (fin2.insert_perm i j)\n\n@[simp] theorem insert_fz {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (v : vector3 \u03b1 n) :\n    insert a v fin2.fz = a :: v :=\n  sorry\n\n@[simp] theorem insert_fs {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (b : \u03b1) (v : vector3 \u03b1 n)\n    (i : fin2 (Nat.succ n)) : insert a (b :: v) (fin2.fs i) = b :: insert a v i :=\n  sorry\n\ntheorem append_insert {\u03b1 : Type u_1} (a : \u03b1) {k : \u2115} (t : vector3 \u03b1 k) {n : \u2115} (v : vector3 \u03b1 n)\n    (i : fin2 (Nat.succ n)) (e : Nat.succ n + k = Nat.succ (n + k)) :\n    insert a (append t v) (eq.rec_on e (fin2.add i k)) = eq.rec_on e (append t (insert a v i)) :=\n  sorry\n\nend vector3\n\n\n/-- \"Curried\" exists, i.e. \u2203 x1 ... xn, f [x1, ..., xn] -/\ndef vector_ex {\u03b1 : Type u_1} (k : \u2115) : (vector3 \u03b1 k \u2192 Prop) \u2192 Prop := sorry\n\n/-- \"Curried\" forall, i.e. \u2200 x1 ... xn, f [x1, ..., xn] -/\ndef vector_all {\u03b1 : Type u_1} (k : \u2115) : (vector3 \u03b1 k \u2192 Prop) \u2192 Prop := sorry\n\ntheorem exists_vector_zero {\u03b1 : Type u_1} (f : vector3 \u03b1 0 \u2192 Prop) : Exists f \u2194 f vector3.nil :=\n  sorry\n\ntheorem exists_vector_succ {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 (Nat.succ n) \u2192 Prop) :\n    Exists f \u2194 \u2203 (x : \u03b1), \u2203 (v : vector3 \u03b1 n), f (x :: v) :=\n  sorry\n\ntheorem vector_ex_iff_exists {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 n \u2192 Prop) :\n    vector_ex n f \u2194 Exists f :=\n  sorry\n\ntheorem vector_all_iff_forall {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 n \u2192 Prop) :\n    vector_all n f \u2194 \u2200 (v : vector3 \u03b1 n), f v :=\n  sorry\n\n/-- `vector_allp p v` is equivalent to `\u2200 i, p (v i)`, but unfolds directly to a conjunction,\n  i.e. `vector_allp p [0, 1, 2] = p 0 \u2227 p 1 \u2227 p 2`. -/\ndef vector_allp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (v : vector3 \u03b1 n) :=\n  vector3.rec_on v True\n    fun (n : \u2115) (a : \u03b1) (v : vector3 \u03b1 n) (IH : Prop) =>\n      vector3.rec_on v (p a) fun (n : \u2115) (b : \u03b1) (v' : vector3 \u03b1 n) (_x : Prop) => p a \u2227 IH\n\n@[simp] theorem vector_allp_nil {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) : vector_allp p vector3.nil = True :=\n  rfl\n\n@[simp] theorem vector_allp_singleton {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (x : \u03b1) :\n    vector_allp p (x :: vector3.nil) = p x :=\n  rfl\n\n@[simp] theorem vector_allp_cons {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (x : \u03b1) (v : vector3 \u03b1 n) :\n    vector_allp p (x :: v) \u2194 p x \u2227 vector_allp p v :=\n  vector3.rec_on v (iff.symm (and_true (vector_allp p (x :: vector3.nil))))\n    fun (n : \u2115) (a : \u03b1) (v : vector3 \u03b1 n) (IH : vector_allp p (x :: v) \u2194 p x \u2227 vector_allp p v) =>\n      iff.rfl\n\ntheorem vector_allp_iff_forall {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (v : vector3 \u03b1 n) :\n    vector_allp p v \u2194 \u2200 (i : fin2 n), p (v i) :=\n  sorry\n\ntheorem vector_allp.imp {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2192 q x)\n    {n : \u2115} {v : vector3 \u03b1 n} (al : vector_allp p v) : vector_allp q v :=\n  iff.mpr (vector_allp_iff_forall q v)\n    fun (i : fin2 n) => h (v i) (iff.mp (vector_allp_iff_forall p v) al i)\n\n/-- `list_all p l` is equivalent to `\u2200 a \u2208 l, p a`, but unfolds directly to a conjunction,\n  i.e. `list_all p [0, 1, 2] = p 0 \u2227 p 1 \u2227 p 2`. -/\n@[simp] def list_all {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) : List \u03b1 \u2192 Prop := sorry\n\n@[simp] theorem list_all_cons {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (x : \u03b1) (l : List \u03b1) :\n    list_all p (x :: l) \u2194 p x \u2227 list_all p l :=\n  list.cases_on l\n    (idRhs (list_all p [x] \u2194 list_all p [x] \u2227 True) (iff.symm (and_true (list_all p [x]))))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) =>\n      idRhs (list_all p (x :: l_hd :: l_tl) \u2194 list_all p (x :: l_hd :: l_tl)) iff.rfl\n\ntheorem list_all_iff_forall {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (l : List \u03b1) :\n    list_all p l \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 p x :=\n  sorry\n\ntheorem list_all.imp {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2192 q x)\n    {l : List \u03b1} : list_all p l \u2192 list_all q l :=\n  sorry\n\n@[simp] theorem list_all_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} :\n    list_all p (list.map f l) \u2194 list_all (p \u2218 f) l :=\n  sorry\n\ntheorem list_all_congr {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2194 q x)\n    {l : List \u03b1} : list_all p l \u2194 list_all q l :=\n  { mp := list_all.imp fun (x : \u03b1) => iff.mp (h x),\n    mpr := list_all.imp fun (x : \u03b1) => iff.mpr (h x) }\n\nprotected instance decidable_list_all {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p]\n    (l : List \u03b1) : Decidable (list_all p l) :=\n  decidable_of_decidable_of_iff (list.decidable_ball (fun (x : \u03b1) => p x) l) sorry\n\n/- poly -/\n\n/-- A predicate asserting that a function is a multivariate integer polynomial.\n  (We are being a bit lazy here by allowing many representations for multiplication,\n  rather than only allowing monomials and addition, but the definition is equivalent\n  and this is easier to use.) -/\ninductive is_poly {\u03b1 : Sort u_1} : ((\u03b1 \u2192 \u2115) \u2192 \u2124) \u2192 Prop where\n| proj : \u2200 (i : \u03b1), is_poly fun (x : \u03b1 \u2192 \u2115) => \u2191(x i)\n| const : \u2200 (n : \u2124), is_poly fun (x : \u03b1 \u2192 \u2115) => n\n| sub : \u2200 {f g : (\u03b1 \u2192 \u2115) \u2192 \u2124}, is_poly f \u2192 is_poly g \u2192 is_poly fun (x : \u03b1 \u2192 \u2115) => f x - g x\n| mul : \u2200 {f g : (\u03b1 \u2192 \u2115) \u2192 \u2124}, is_poly f \u2192 is_poly g \u2192 is_poly fun (x : \u03b1 \u2192 \u2115) => f x * g x\n\n/-- The type of multivariate integer polynomials -/\ndef poly (\u03b1 : Type u) := Subtype fun (f : (\u03b1 \u2192 \u2115) \u2192 \u2124) => is_poly f\n\nnamespace poly\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u} : has_coe_to_fun (poly \u03b1) :=\n  has_coe_to_fun.mk (fun (f : poly \u03b1) => (\u03b1 \u2192 \u2115) \u2192 \u2124) fun (f : poly \u03b1) => subtype.val f\n\n/-- The underlying function of a `poly` is a polynomial -/\ntheorem isp {\u03b1 : Type u} (f : poly \u03b1) : is_poly \u21d1f := subtype.property f\n\n/-- Extensionality for `poly \u03b1` -/\ntheorem ext {\u03b1 : Type u} {f : poly \u03b1} {g : poly \u03b1} (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = coe_fn g x) :\n    f = g :=\n  subtype.eq (funext e)\n\n/-- Construct a `poly` given an extensionally equivalent `poly`. -/\ndef subst {\u03b1 : Type u} (f : poly \u03b1) (g : (\u03b1 \u2192 \u2115) \u2192 \u2124) (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = g x) :\n    poly \u03b1 :=\n  { val := g, property := sorry }\n\n@[simp] theorem subst_eval {\u03b1 : Type u} (f : poly \u03b1) (g : (\u03b1 \u2192 \u2115) \u2192 \u2124)\n    (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = g x) (x : \u03b1 \u2192 \u2115) : coe_fn (subst f g e) x = g x :=\n  rfl\n\n/-- The `i`th projection function, `x_i`. -/\ndef proj {\u03b1 : Type u} (i : \u03b1) : poly \u03b1 :=\n  { val := fun (x : \u03b1 \u2192 \u2115) => \u2191(x i), property := is_poly.proj i }\n\n@[simp] theorem proj_eval {\u03b1 : Type u} (i : \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (proj i) x = \u2191(x i) := rfl\n\n/-- The constant function with value `n : \u2124`. -/\ndef const {\u03b1 : Type u} (n : \u2124) : poly \u03b1 :=\n  { val := fun (x : \u03b1 \u2192 \u2115) => n, property := is_poly.const n }\n\n@[simp] theorem const_eval {\u03b1 : Type u} (n : \u2124) (x : \u03b1 \u2192 \u2115) : coe_fn (const n) x = n := rfl\n\n/-- The zero polynomial -/\ndef zero {\u03b1 : Type u} : poly \u03b1 := const 0\n\nprotected instance has_zero {\u03b1 : Type u} : HasZero (poly \u03b1) := { zero := zero }\n\n@[simp] theorem zero_eval {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) : coe_fn 0 x = 0 := rfl\n\n/-- The zero polynomial -/\ndef one {\u03b1 : Type u} : poly \u03b1 := const 1\n\nprotected instance has_one {\u03b1 : Type u} : HasOne (poly \u03b1) := { one := one }\n\n@[simp] theorem one_eval {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) : coe_fn 1 x = 1 := rfl\n\n/-- Subtraction of polynomials -/\ndef sub {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 := sorry\n\nprotected instance has_sub {\u03b1 : Type u} : Sub (poly \u03b1) := { sub := sub }\n\n@[simp] theorem sub_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) :\n    coe_fn (f - g) x = coe_fn f x - coe_fn g x :=\n  sorry\n\n/-- Negation of a polynomial -/\ndef neg {\u03b1 : Type u} (f : poly \u03b1) : poly \u03b1 := 0 - f\n\nprotected instance has_neg {\u03b1 : Type u} : Neg (poly \u03b1) := { neg := neg }\n\n@[simp] theorem neg_eval {\u03b1 : Type u} (f : poly \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (-f) x = -coe_fn f x :=\n  sorry\n\n/-- Addition of polynomials -/\ndef add {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 := sorry\n\nprotected instance has_add {\u03b1 : Type u} : Add (poly \u03b1) := { add := add }\n\n@[simp] theorem add_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) :\n    coe_fn (f + g) x = coe_fn f x + coe_fn g x :=\n  sorry\n\n/-- Multiplication of polynomials -/\ndef mul {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 := sorry\n\nprotected instance has_mul {\u03b1 : Type u} : Mul (poly \u03b1) := { mul := mul }\n\n@[simp] theorem mul_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) :\n    coe_fn (f * g) x = coe_fn f x * coe_fn g x :=\n  sorry\n\nprotected instance comm_ring {\u03b1 : Type u} : comm_ring (poly \u03b1) :=\n  comm_ring.mk Add.add sorry 0 sorry sorry Neg.neg Sub.sub sorry sorry Mul.mul sorry 1 sorry sorry\n    sorry sorry sorry\n\ntheorem induction {\u03b1 : Type u} {C : poly \u03b1 \u2192 Prop} (H1 : \u2200 (i : \u03b1), C (proj i))\n    (H2 : \u2200 (n : \u2124), C (const n)) (H3 : \u2200 (f g : poly \u03b1), C f \u2192 C g \u2192 C (f - g))\n    (H4 : \u2200 (f g : poly \u03b1), C f \u2192 C g \u2192 C (f * g)) (f : poly \u03b1) : C f :=\n  sorry\n\n/-- The sum of squares of a list of polynomials. This is relevant for\n  Diophantine equations, because it means that a list of equations\n  can be encoded as a single equation: `x = 0 \u2227 y = 0 \u2227 z = 0` is\n  equivalent to `x^2 + y^2 + z^2 = 0`. -/\ndef sumsq {\u03b1 : Type u} : List (poly \u03b1) \u2192 poly \u03b1 := sorry\n\ntheorem sumsq_nonneg {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) (l : List (poly \u03b1)) : 0 \u2264 coe_fn (sumsq l) x := sorry\n\ntheorem sumsq_eq_zero {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) (l : List (poly \u03b1)) :\n    coe_fn (sumsq l) x = 0 \u2194 list_all (fun (a : poly \u03b1) => coe_fn a x = 0) l :=\n  sorry\n\n/-- Map the index set of variables, replacing `x_i` with `x_(f i)`. -/\ndef remap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (g : poly \u03b1) : poly \u03b2 :=\n  { val := fun (v : \u03b2 \u2192 \u2115) => coe_fn g (v \u2218 f), property := sorry }\n\n@[simp] theorem remap_eval {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (g : poly \u03b1) (v : \u03b2 \u2192 \u2115) :\n    coe_fn (remap f g) v = coe_fn g (v \u2218 f) :=\n  rfl\n\nend poly\n\n\nnamespace sum\n\n\n/-- combine two functions into a function on the disjoint union -/\ndef join {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Sort u_3} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 :=\n  sum.rec f g\n\nend sum\n\n\nnamespace option\n\n\n/-- Functions from `option` can be combined similarly to `vector.cons` -/\ndef cons {\u03b1 : Type u_1} {\u03b2 : Sort u_2} (a : \u03b2) (v : \u03b1 \u2192 \u03b2) : Option \u03b1 \u2192 \u03b2 := Option.rec a v\n\ninfixr:67 \" :: \" => Mathlib.option.cons\n\n@[simp] theorem cons_head_tail {\u03b1 : Type u_1} {\u03b2 : Sort u_2} (v : Option \u03b1 \u2192 \u03b2) :\n    v none :: v \u2218 some = v :=\n  sorry\n\nend option\n\n\n/- dioph -/\n\n/-- A set `S \u2286 \u2115^\u03b1` is diophantine if there exists a polynomial on\n  `\u03b1 \u2295 \u03b2` such that `v \u2208 S` iff there exists `t : \u2115^\u03b2` with `p (v, t) = 0`. -/\ndef dioph {\u03b1 : Type u} (S : set (\u03b1 \u2192 \u2115)) :=\n  Exists\n    fun {\u03b2 : Type u} =>\n      \u2203 (p : poly (\u03b1 \u2295 \u03b2)), \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0\n\nnamespace dioph\n\n\ntheorem ext {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S)\n    (H : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 S' v) : dioph S' :=\n  Eq._oldrec d ((fun (this : S = S') => this) (set.ext H))\n\ntheorem of_no_dummies {\u03b1 : Type u} (S : set (\u03b1 \u2192 \u2115)) (p : poly \u03b1)\n    (h : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 coe_fn p v = 0) : dioph S :=\n  sorry\n\ntheorem inject_dummies_lem {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} (f : \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 Option \u03b2)\n    (inv : \u2200 (x : \u03b2), g (f x) = some x) (p : poly (\u03b1 \u2295 \u03b2)) (v : \u03b1 \u2192 \u2115) :\n    (\u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0) \u2194\n        \u2203 (t : \u03b3 \u2192 \u2115), coe_fn (poly.remap (sum.join sum.inl (sum.inr \u2218 f)) p) (sum.join v t) = 0 :=\n  sorry\n\ntheorem inject_dummies {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {S : set (\u03b1 \u2192 \u2115)} (f : \u03b2 \u2192 \u03b3)\n    (g : \u03b3 \u2192 Option \u03b2) (inv : \u2200 (x : \u03b2), g (f x) = some x) (p : poly (\u03b1 \u2295 \u03b2))\n    (h : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0) :\n    \u2203 (q : poly (\u03b1 \u2295 \u03b3)), \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b3 \u2192 \u2115), coe_fn q (sum.join v t) = 0 :=\n  Exists.intro (poly.remap (sum.join sum.inl (sum.inr \u2218 f)) p)\n    fun (v : \u03b1 \u2192 \u2115) => iff.trans (h v) (inject_dummies_lem f g inv p v)\n\ntheorem reindex_dioph {\u03b1 : Type u} {\u03b2 : Type u} {S : set (\u03b1 \u2192 \u2115)} (d : dioph S) (f : \u03b1 \u2192 \u03b2) :\n    dioph fun (v : \u03b2 \u2192 \u2115) => S (v \u2218 f) :=\n  sorry\n\ntheorem dioph_list_all {\u03b1 : Type u} (l : List (set (\u03b1 \u2192 \u2115))) (d : list_all dioph l) :\n    dioph fun (v : \u03b1 \u2192 \u2115) => list_all (fun (S : set (\u03b1 \u2192 \u2115)) => S v) l :=\n  sorry\n\ntheorem and_dioph {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S) (d' : dioph S') :\n    dioph fun (v : \u03b1 \u2192 \u2115) => S v \u2227 S' v :=\n  dioph_list_all [S, S'] { left := d, right := d' }\n\ntheorem or_dioph {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S) (d' : dioph S') :\n    dioph fun (v : \u03b1 \u2192 \u2115) => S v \u2228 S' v :=\n  sorry\n\n/-- A partial function is Diophantine if its graph is Diophantine. -/\ndef dioph_pfun {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192. \u2115) :=\n  dioph fun (v : Option \u03b1 \u2192 \u2115) => pfun.graph f (v \u2218 some, v none)\n\n/-- A function is Diophantine if its graph is Diophantine. -/\ndef dioph_fn {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192 \u2115) := dioph fun (v : Option \u03b1 \u2192 \u2115) => f (v \u2218 some) = v none\n\ntheorem reindex_dioph_fn {\u03b1 : Type u} {\u03b2 : Type u} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} (d : dioph_fn f) (g : \u03b1 \u2192 \u03b2) :\n    dioph_fn fun (v : \u03b2 \u2192 \u2115) => f (v \u2218 g) :=\n  reindex_dioph d (Functor.map g)\n\ntheorem ex_dioph {\u03b1 : Type u} {\u03b2 : Type u} {S : set (\u03b1 \u2295 \u03b2 \u2192 \u2115)} :\n    dioph S \u2192 dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (x : \u03b2 \u2192 \u2115), S (sum.join v x) :=\n  sorry\n\ntheorem ex1_dioph {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} :\n    dioph S \u2192 dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (x : \u2115), S (x :: v) :=\n  sorry\n\ntheorem dom_dioph {\u03b1 : Type u} {f : (\u03b1 \u2192 \u2115) \u2192. \u2115} (d : dioph_pfun f) : dioph (pfun.dom f) :=\n  cast (congr_arg dioph (set.ext fun (v : \u03b1 \u2192 \u2115) => iff.symm (pfun.dom_iff_graph f v)))\n    (ex1_dioph d)\n\ntheorem dioph_fn_iff_pfun {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192 \u2115) : dioph_fn f = dioph_pfun \u2191f :=\n  congr_arg dioph (set.ext fun (v : Option \u03b1 \u2192 \u2115) => iff.symm pfun.lift_graph)\n\ntheorem abs_poly_dioph {\u03b1 : Type u} (p : poly \u03b1) :\n    dioph_fn fun (v : \u03b1 \u2192 \u2115) => int.nat_abs (coe_fn p v) :=\n  of_no_dummies\n    (fun (v : Option \u03b1 \u2192 \u2115) => (fun (v : \u03b1 \u2192 \u2115) => int.nat_abs (coe_fn p v)) (v \u2218 some) = v none)\n    ((poly.remap some p - poly.proj none) * (poly.remap some p + poly.proj none))\n    fun (v : Option \u03b1 \u2192 \u2115) => int.eq_nat_abs_iff_mul (coe_fn p (v \u2218 some)) (v none)\n\ntheorem proj_dioph {\u03b1 : Type u} (i : \u03b1) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => v i :=\n  abs_poly_dioph (poly.proj i)\n\ntheorem dioph_pfun_comp1 {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} (d : dioph S) {f : (\u03b1 \u2192 \u2115) \u2192. \u2115}\n    (df : dioph_pfun f) : dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (h : pfun.dom f v), S (pfun.fn f v h :: v) :=\n  sorry\n\ntheorem dioph_fn_comp1 {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} (d : dioph S) {f : (\u03b1 \u2192 \u2115) \u2192 \u2115}\n    (df : dioph_fn f) : dioph fun (v : \u03b1 \u2192 \u2115) => S (f v :: v) :=\n  sorry\n\ntheorem dioph_fn_vec_comp1 {n : \u2115} {S : set (vector3 \u2115 (Nat.succ n))} (d : dioph S)\n    {f : vector3 \u2115 n \u2192 \u2115} (df : dioph_fn f) : dioph fun (v : vector3 \u2115 n) => S (f v :: v) :=\n  sorry\n\ntheorem vec_ex1_dioph (n : \u2115) {S : set (vector3 \u2115 (Nat.succ n))} (d : dioph S) :\n    dioph fun (v : vector3 \u2115 n) => \u2203 (x : \u2115), S (x :: v) :=\n  sorry\n\ntheorem dioph_fn_vec {n : \u2115} (f : vector3 \u2115 n \u2192 \u2115) :\n    dioph_fn f \u2194 dioph fun (v : vector3 \u2115 (Nat.succ n)) => f (v \u2218 fin2.fs) = v fin2.fz :=\n  { mp := fun (h : dioph_fn f) => reindex_dioph h (fin2.fz :: fin2.fs),\n    mpr :=\n      fun (h : dioph fun (v : vector3 \u2115 (Nat.succ n)) => f (v \u2218 fin2.fs) = v fin2.fz) =>\n        reindex_dioph h (none :: some) }\n\ntheorem dioph_pfun_vec {n : \u2115} (f : vector3 \u2115 n \u2192. \u2115) :\n    dioph_pfun f \u2194\n        dioph fun (v : vector3 \u2115 (Nat.succ n)) => pfun.graph f (v \u2218 fin2.fs, v fin2.fz) :=\n  sorry\n\ntheorem dioph_fn_compn {\u03b1 : Type} {n : \u2115} {S : set (\u03b1 \u2295 fin2 n \u2192 \u2115)} (d : dioph S)\n    {f : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n} (df : vector_allp dioph_fn f) :\n    dioph fun (v : \u03b1 \u2192 \u2115) => S (sum.join v fun (i : fin2 n) => f i v) :=\n  sorry\n\ntheorem dioph_comp {\u03b1 : Type} {n : \u2115} {S : set (vector3 \u2115 n)} (d : dioph S)\n    (f : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n) (df : vector_allp dioph_fn f) :\n    dioph fun (v : \u03b1 \u2192 \u2115) => S fun (i : fin2 n) => f i v :=\n  dioph_fn_compn (reindex_dioph d sum.inr) df\n\ntheorem dioph_fn_comp {\u03b1 : Type} {n : \u2115} {f : vector3 \u2115 n \u2192 \u2115} (df : dioph_fn f)\n    (g : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n) (dg : vector_allp dioph_fn g) :\n    dioph_fn fun (v : \u03b1 \u2192 \u2115) => f fun (i : fin2 n) => g i v :=\n  sorry\n\ntheorem proj_dioph_of_nat {n : \u2115} (m : \u2115) [fin2.is_lt m n] :\n    dioph_fn fun (v : vector3 \u2115 n) => v (fin2.of_nat' m) :=\n  proj_dioph (fin2.of_nat' m)\n\ntheorem const_dioph {\u03b1 : Type} (n : \u2115) : dioph_fn (function.const (\u03b1 \u2192 \u2115) n) :=\n  abs_poly_dioph (poly.const \u2191n)\n\ntheorem dioph_comp2 {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) {S : \u2115 \u2192 \u2115 \u2192 Prop}\n    (d : dioph fun (v : vector3 \u2115 (bit0 1)) => S (v (fin2.of_nat' 0)) (v (fin2.of_nat' 1))) :\n    dioph fun (v : \u03b1 \u2192 \u2115) => S (f v) (g v) :=\n  dioph_comp d (f :: g :: vector3.nil) { left := df, right := dg }\n\ntheorem dioph_fn_comp2 {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) {h : \u2115 \u2192 \u2115 \u2192 \u2115}\n    (d : dioph_fn fun (v : vector3 \u2115 (bit0 1)) => h (v (fin2.of_nat' 0)) (v (fin2.of_nat' 1))) :\n    dioph_fn fun (v : \u03b1 \u2192 \u2115) => h (f v) (g v) :=\n  dioph_fn_comp d (f :: g :: vector3.nil) { left := df, right := dg }\n\ntheorem eq_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v = g v :=\n  sorry\n\ntheorem add_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v + g v :=\n  dioph_fn_comp2 df dg (abs_poly_dioph (poly.proj (fin2.of_nat' 0) + poly.proj (fin2.of_nat' 1)))\n\ntheorem mul_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v * g v :=\n  dioph_fn_comp2 df dg (abs_poly_dioph (poly.proj (fin2.of_nat' 0) * poly.proj (fin2.of_nat' 1)))\n\ntheorem le_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2264 g v :=\n  sorry\n\ntheorem lt_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v < g v :=\n  le_dioph (add_dioph df (const_dioph 1)) dg\n\ntheorem ne_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2260 g v :=\n  ext (or_dioph (lt_dioph df dg) (lt_dioph dg df)) fun (v : \u03b1 \u2192 \u2115) => iff.symm ne_iff_lt_or_gt\n\ntheorem sub_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v - g v :=\n  sorry\n\ntheorem dvd_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2223 g v :=\n  sorry\n\ntheorem mod_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v % g v :=\n  sorry\n\ntheorem modeq_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) {h : (\u03b1 \u2192 \u2115) \u2192 \u2115} (dh : dioph_fn h) :\n    dioph fun (v : \u03b1 \u2192 \u2115) => nat.modeq (h v) (f v) (g v) :=\n  eq_dioph (mod_dioph df dh) (mod_dioph dg dh)\n\ntheorem div_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v / g v :=\n  sorry\n\ntheorem pell_dioph :\n    dioph\n        fun (v : vector3 \u2115 (bit0 (bit0 1))) =>\n          \u2203 (h : 1 < v (fin2.of_nat' 0)),\n            pell.xn h (v (fin2.of_nat' 1)) = v (fin2.of_nat' (bit0 1)) \u2227\n              pell.yn h (v (fin2.of_nat' 1)) = v (fin2.of_nat' (bit1 1)) :=\n  sorry\n\ntheorem xn_dioph :\n    dioph_pfun\n        fun (v : vector3 \u2115 (bit0 1)) =>\n          roption.mk (1 < v (fin2.of_nat' 0))\n            fun (h : 1 < v (fin2.of_nat' 0)) => pell.xn h (v (fin2.of_nat' 1)) :=\n  sorry\n\ntheorem pow_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f)\n    (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v ^ g v :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/number_theory/dioph_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6477982043529715, "lm_q1q2_score": 0.36666579073053374}}
{"text": "/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.limits.preserves.shapes.terminal\nimport category_theory.limits.shapes.zero\n\n/-!\n# Preservation of zero objects and zero morphisms\n\nWe define the class `preserves_zero_morphisms` and show basic properties.\n\n## Main results\n\nWe provide the following results:\n* Left adjoints and right adjoints preserve zero morphisms;\n* full functors preserve zero morphisms;\n* if both categories involved have a zero object, then a functor preserves zero morphisms if and\n  only if it preserves the zero object;\n* functors which preserve initial or terminal objects preserve zero morphisms.\n\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nnamespace category_theory.functor\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n\nsection zero_morphisms\nvariables [has_zero_morphisms C] [has_zero_morphisms D]\n\n/-- A functor preserves zero morphisms if it sends zero morphisms to zero morphisms. -/\nclass preserves_zero_morphisms (F : C \u2964 D) : Prop :=\n(map_zero' : \u2200 (X Y : C), F.map (0 : X \u27f6 Y) = 0 . obviously)\n\n@[simp]\nprotected lemma map_zero (F : C \u2964 D) [preserves_zero_morphisms F] (X Y : C) :\n  F.map (0 : X \u27f6 Y) = 0 :=\npreserves_zero_morphisms.map_zero' _ _\n\nlemma zero_of_map_zero (F : C \u2964 D) [preserves_zero_morphisms F] [faithful F] {X Y : C}\n  (f : X \u27f6 Y) (h : F.map f = 0) : f = 0 :=\nF.map_injective $ h.trans $ eq.symm $ F.map_zero _ _\n\nlemma map_eq_zero_iff (F : C \u2964 D) [preserves_zero_morphisms F] [faithful F] {X Y : C} {f : X \u27f6 Y} :\n  F.map f = 0 \u2194 f = 0 :=\n\u27e8F.zero_of_map_zero _, by { rintro rfl, exact F.map_zero _ _ }\u27e9\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_is_left_adjoint (F : C \u2964 D) [is_left_adjoint F] :\n  preserves_zero_morphisms F :=\n{ map_zero' := \u03bb X Y, let adj := adjunction.of_left_adjoint F in\n  begin\n    calc F.map (0 : X \u27f6 Y) = F.map 0 \u226b F.map (adj.unit.app Y) \u226b adj.counit.app (F.obj Y) : _\n    ... = F.map 0 \u226b F.map ((right_adjoint F).map (0 : F.obj X \u27f6 _)) \u226b adj.counit.app (F.obj Y) : _\n    ... = 0 : _,\n    { rw adjunction.left_triangle_components, exact (category.comp_id _).symm },\n    { simp only [\u2190 category.assoc, \u2190 F.map_comp, zero_comp] },\n    { simp only [adjunction.counit_naturality, comp_zero] }\n  end }\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_is_right_adjoint (G : C \u2964 D) [is_right_adjoint G] :\n  preserves_zero_morphisms G :=\n{ map_zero' := \u03bb X Y, let adj := adjunction.of_right_adjoint G in\n  begin\n    calc G.map (0 : X \u27f6 Y) = adj.unit.app (G.obj X) \u226b G.map (adj.counit.app X) \u226b G.map 0 : _\n    ... = adj.unit.app (G.obj X) \u226b G.map ((left_adjoint G).map (0 : _ \u27f6 G.obj X)) \u226b G.map 0 : _\n    ... = 0 : _,\n    { rw adjunction.right_triangle_components_assoc },\n    { simp only [\u2190 G.map_comp, comp_zero] },\n    { simp only [adjunction.unit_naturality_assoc, zero_comp] }\n  end }\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_full (F : C \u2964 D) [full F] : preserves_zero_morphisms F :=\n{ map_zero' := \u03bb X Y, calc\n  F.map (0 : X \u27f6 Y) = F.map (0 \u226b (F.preimage (0 : F.obj Y \u27f6 F.obj Y))) : by rw zero_comp\n                ... = 0 : by rw [F.map_comp, F.image_preimage, comp_zero] }\n\nend zero_morphisms\n\nsection zero_object\nvariables [has_zero_object C] [has_zero_object D]\n\nopen_locale zero_object\n\nvariables [has_zero_morphisms C] [has_zero_morphisms D] (F : C \u2964 D)\n\n/-- A functor that preserves zero morphisms also preserves the zero object. -/\n@[simps] def map_zero_object [preserves_zero_morphisms F] : F.obj 0 \u2245 0 :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' := by rw [\u2190 F.map_id, id_zero, F.map_zero, zero_comp],\n  inv_hom_id' := by rw [id_zero, comp_zero] }\n\nvariables {F}\n\nlemma preserves_zero_morphisms_of_map_zero_object (i : F.obj 0 \u2245 0) : preserves_zero_morphisms F :=\n{ map_zero' := \u03bb X Y, calc\n  F.map (0 : X \u27f6 Y) = F.map (0 : X \u27f6 0) \u226b F.map 0 : by rw [\u2190 functor.map_comp, comp_zero]\n                ... = F.map 0 \u226b (i.hom \u226b i.inv) \u226b F.map 0\n                        : by rw [iso.hom_inv_id, category.id_comp]\n                ... = 0 : by simp only [zero_of_to_zero i.hom, zero_comp, comp_zero] }\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_preserves_initial_object\n  [preserves_colimit (functor.empty.{v\u2081} C) F] : preserves_zero_morphisms F :=\npreserves_zero_morphisms_of_map_zero_object $ (F.map_iso has_zero_object.zero_iso_initial).trans $\n  (preserves_initial.iso F).trans has_zero_object.zero_iso_initial.symm\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_preserves_terminal_object\n  [preserves_limit (functor.empty.{v\u2081} C) F] : preserves_zero_morphisms F :=\npreserves_zero_morphisms_of_map_zero_object $ (F.map_iso has_zero_object.zero_iso_terminal).trans $\n    (preserves_terminal.iso F).trans has_zero_object.zero_iso_terminal.symm\n\nend zero_object\n\nend category_theory.functor\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/limits/preserves/shapes/zero.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3665595344897862}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau, Yury Kudryashov\n-/\nimport algebra.module.basic\nimport linear_algebra.basic\nimport tactic.abel\nimport data.equiv.ring_aut\n\n/-!\n# Algebras over commutative semirings\n\nIn this file we define associative unital `algebra`s over commutative (semi)rings, algebra\nhomomorphisms `alg_hom`, and algebra equivalences `alg_equiv`.\n\n`subalgebra`s are defined in `algebra.algebra.subalgebra`.\n\nFor the category of `R`-algebras, denoted `Algebra R`, see the file\n`algebra/category/Algebra/basic.lean`.\n\nSee the implementation notes for remarks about non-associative and non-unital algebras.\n\n## Main definitions:\n\n* `algebra R A`: the algebra typeclass.\n* `alg_hom R A B`: the type of `R`-algebra morphisms from `A` to `B`.\n* `alg_equiv R A B`: the type of `R`-algebra isomorphisms between `A` to `B`.\n* `algebra_map R A : R \u2192+* A`: the canonical map from `R` to `A`, as a `ring_hom`. This is the\n  preferred spelling of this map.\n* `algebra.linear_map R A : R \u2192\u2097[R] A`: the canonical map from `R` to `A`, as a `linear_map`.\n* `algebra.of_id R A : R \u2192\u2090[R] A`: the canonical map from `R` to `A`, as n `alg_hom`.\n* Instances of `algebra` in this file:\n  * `algebra.id`\n  * `pi.algebra`\n  * `prod.algebra`\n  * `algebra_nat`\n  * `algebra_int`\n  * `algebra_rat`\n  * `mul_opposite.algebra`\n  * `module.End.algebra`\n\n## Notations\n\n* `A \u2192\u2090[R] B` : `R`-algebra homomorphism from `A` to `B`.\n* `A \u2243\u2090[R] B` : `R`-algebra equivalence from `A` to `B`.\n\n## Implementation notes\n\nGiven a commutative (semi)ring `R`, there are two ways to define an `R`-algebra structure on a\n(possibly noncommutative) (semi)ring `A`:\n* By endowing `A` with a morphism of rings `R \u2192+* A` denoted `algebra_map R A` which lands in the\n  center of `A`.\n* By requiring `A` be an `R`-module such that the action associates and commutes with multiplication\n  as `r \u2022 (a\u2081 * a\u2082) = (r \u2022 a\u2081) * a\u2082 = a\u2081 * (r \u2022 a\u2082)`.\n\nWe define `algebra R A` in a way that subsumes both definitions, by extending `has_scalar R A` and\nrequiring that this scalar action `r \u2022 x` must agree with left multiplication by the image of the\nstructure morphism `algebra_map R A r * x`.\n\nAs a result, there are two ways to talk about an `R`-algebra `A` when `A` is a semiring:\n1. ```lean\n   variables [comm_semiring R] [semiring A]\n   variables [algebra R A]\n   ```\n2. ```lean\n   variables [comm_semiring R] [semiring A]\n   variables [module R A] [smul_comm_class R A A] [is_scalar_tower R A A]\n   ```\n\nThe first approach implies the second via typeclass search; so any lemma stated with the second set\nof arguments will automatically apply to the first set. Typeclass search does not know that the\nsecond approach implies the first, but this can be shown with:\n```lean\nexample {R A : Type*} [comm_semiring R] [semiring A]\n  [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] : algebra R A :=\nalgebra.of_module smul_mul_assoc mul_smul_comm\n```\n\nThe advantage of the first approach is that `algebra_map R A` is available, and `alg_hom R A B` and\n`subalgebra R A` can be used. For concrete `R` and `A`, `algebra_map R A` is often definitionally\nconvenient.\n\nThe advantage of the second approach is that `comm_semiring R`, `semiring A`, and `module R A` can\nall be relaxed independently; for instance, this allows us to:\n* Replace `semiring A` with `non_unital_non_assoc_semiring A` in order to describe non-unital and/or\n  non-associative algebras.\n* Replace `comm_semiring R` and `module R A` with `comm_group R'` and `distrib_mul_action R' A`,\n  which when `R' = R\u02e3` lets us talk about the \"algebra-like\" action of `R\u02e3` on an\n  `R`-algebra `A`.\n\nWhile `alg_hom R A B` cannot be used in the second approach, `non_unital_alg_hom R A B` still can.\n\nYou should always use the first approach when working with associative unital algebras, and mimic\nthe second approach only when you need to weaken a condition on either `R` or `A`.\n\n-/\n\nuniverses u v w u\u2081 v\u2081\n\nopen_locale big_operators\n\nsection prio\n-- We set this priority to 0 later in this file\nset_option extends_priority 200 /- control priority of\n`instance [algebra R A] : has_scalar R A` -/\n\n/--\nAn associative unital `R`-algebra is a semiring `A` equipped with a map into its center `R \u2192 A`.\n\nSee the implementation notes in this file for discussion of the details of this definition.\n-/\n@[nolint has_inhabited_instance]\nclass algebra (R : Type u) (A : Type v) [comm_semiring R] [semiring A]\n  extends has_scalar R A, R \u2192+* A :=\n(commutes' : \u2200 r x, to_fun r * x = x * to_fun r)\n(smul_def' : \u2200 r x, r \u2022 x = to_fun r * x)\nend prio\n\n/-- Embedding `R \u2192+* A` given by `algebra` structure. -/\ndef algebra_map (R : Type u) (A : Type v) [comm_semiring R] [semiring A] [algebra R A] : R \u2192+* A :=\nalgebra.to_ring_hom\n\n/-- Creating an algebra from a morphism to the center of a semiring. -/\ndef ring_hom.to_algebra' {R S} [comm_semiring R] [semiring S] (i : R \u2192+* S)\n  (h : \u2200 c x, i c * x = x * i c) :\n  algebra R S :=\n{ smul := \u03bb c x, i c * x,\n  commutes' := h,\n  smul_def' := \u03bb c x, rfl,\n  to_ring_hom := i}\n\n/-- Creating an algebra from a morphism to a commutative semiring. -/\ndef ring_hom.to_algebra {R S} [comm_semiring R] [comm_semiring S] (i : R \u2192+* S) :\n  algebra R S :=\ni.to_algebra' $ \u03bb _, mul_comm _\n\nlemma ring_hom.algebra_map_to_algebra {R S} [comm_semiring R] [comm_semiring S]\n  (i : R \u2192+* S) :\n  @algebra_map R S _ _ i.to_algebra = i :=\nrfl\n\nnamespace algebra\n\nvariables {R : Type u} {S : Type v} {A : Type w} {B : Type*}\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 1) * x = x * (r \u2022 1) = r \u2022 x` for all `r : R` and `x : A`, then `A` is an `algebra`\nover `R`.\n\nSee note [reducible non-instances]. -/\n@[reducible]\ndef of_module' [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x : A), (r \u2022 1) * x = r \u2022 x)\n  (h\u2082 : \u2200 (r : R) (x : A), x * (r \u2022 1) = r \u2022 x) : algebra R A :=\n{ to_fun := \u03bb r, r \u2022 1,\n  map_one' := one_smul _ _,\n  map_mul' := \u03bb r\u2081 r\u2082, by rw [h\u2081, mul_smul],\n  map_zero' := zero_smul _ _,\n  map_add' := \u03bb r\u2081 r\u2082, add_smul r\u2081 r\u2082 1,\n  commutes' := \u03bb r x, by simp only [h\u2081, h\u2082],\n  smul_def' := \u03bb r x, by simp only [h\u2081] }\n\n/-- Let `R` be a commutative semiring, let `A` be a semiring with a `module R` structure.\nIf `(r \u2022 x) * y = x * (r \u2022 y) = r \u2022 (x * y)` for all `r : R` and `x y : A`, then `A`\nis an `algebra` over `R`.\n\nSee note [reducible non-instances]. -/\n@[reducible]\ndef of_module [comm_semiring R] [semiring A] [module R A]\n  (h\u2081 : \u2200 (r : R) (x y : A), (r \u2022 x) * y = r \u2022 (x * y))\n  (h\u2082 : \u2200 (r : R) (x y : A), x * (r \u2022 y) = r \u2022 (x * y)) : algebra R A :=\nof_module' (\u03bb r x, by rw [h\u2081, one_mul]) (\u03bb r x, by rw [h\u2082, mul_one])\n\nsection semiring\n\nvariables [comm_semiring R] [comm_semiring S]\nvariables [semiring A] [algebra R A] [semiring B] [algebra R B]\n\n/-- We keep this lemma private because it picks up the `algebra.to_has_scalar` instance\nwhich we set to priority 0 shortly. See `smul_def` below for the public version. -/\nprivate lemma smul_def'' (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\n/--\nTo prove two algebra structures on a fixed `[comm_semiring R] [semiring A]` agree,\nit suffices to check the `algebra_map`s agree.\n-/\n-- We'll later use this to show `algebra \u2124 M` is a subsingleton.\n@[ext]\nlemma algebra_ext {R : Type*} [comm_semiring R] {A : Type*} [semiring A] (P Q : algebra R A)\n  (w : \u2200 (r : R), by { haveI := P, exact algebra_map R A r } =\n    by { haveI := Q, exact algebra_map R A r }) :\n  P = Q :=\nbegin\n  unfreezingI { rcases P with \u27e8\u27e8P\u27e9\u27e9, rcases Q with \u27e8\u27e8Q\u27e9\u27e9 },\n  congr,\n  { funext r a,\n    replace w := congr_arg (\u03bb s, s * a) (w r),\n    simp only [\u2190smul_def''] at w,\n    apply w, },\n  { ext r,\n    exact w r, },\n  { apply proof_irrel_heq, },\n  { apply proof_irrel_heq, },\nend\n\n@[priority 200] -- see Note [lower instance priority]\ninstance to_module : module R A :=\n{ one_smul := by simp [smul_def''],\n  mul_smul := by simp [smul_def'', mul_assoc],\n  smul_add := by simp [smul_def'', mul_add],\n  smul_zero := by simp [smul_def''],\n  add_smul := by simp [smul_def'', add_mul],\n  zero_smul := by simp [smul_def''] }\n\n-- From now on, we don't want to use the following instance anymore.\n-- Unfortunately, leaving it in place causes deterministic timeouts later in mathlib.\nattribute [instance, priority 0] algebra.to_has_scalar\n\nlemma smul_def (r : R) (x : A) : r \u2022 x = algebra_map R A r * x :=\nalgebra.smul_def' r x\n\nlemma algebra_map_eq_smul_one (r : R) : algebra_map R A r = r \u2022 1 :=\ncalc algebra_map R A r = algebra_map R A r * 1 : (mul_one _).symm\n                   ... = r \u2022 1                 : (algebra.smul_def r 1).symm\n\nlemma algebra_map_eq_smul_one' : \u21d1(algebra_map R A) = \u03bb r, r \u2022 (1 : A) :=\nfunext algebra_map_eq_smul_one\n\n/-- `mul_comm` for `algebra`s when one element is from the base ring. -/\ntheorem commutes (r : R) (x : A) : algebra_map R A r * x = x * algebra_map R A r :=\nalgebra.commutes' r x\n\n/-- `mul_left_comm` for `algebra`s when one element is from the base ring. -/\ntheorem left_comm (x : A) (r : R) (y : A) :\n  x * (algebra_map R A r * y) = algebra_map R A r * (x * y) :=\nby rw [\u2190 mul_assoc, \u2190 commutes, mul_assoc]\n\n/-- `mul_right_comm` for `algebra`s when one element is from the base ring. -/\ntheorem right_comm (x : A) (r : R) (y : A) :\n  (x * algebra_map R A r) * y = (x * y) * algebra_map R A r :=\nby rw [mul_assoc, commutes, \u2190mul_assoc]\n\ninstance _root_.is_scalar_tower.right : is_scalar_tower R A A :=\n\u27e8\u03bb x y z, by rw [smul_eq_mul, smul_eq_mul, smul_def, smul_def, mul_assoc]\u27e9\n\n/-- This is just a special case of the global `mul_smul_comm` lemma that requires less typeclass\nsearch (and was here first). -/\n@[simp] protected lemma mul_smul_comm (s : R) (x y : A) :\n  x * (s \u2022 y) = s \u2022 (x * y) :=\n-- TODO: set up `is_scalar_tower.smul_comm_class` earlier so that we can actually prove this using\n-- `mul_smul_comm s x y`.\nby rw [smul_def, smul_def, left_comm]\n\n/-- This is just a special case of the global `smul_mul_assoc` lemma that requires less typeclass\nsearch (and was here first). -/\n@[simp] protected lemma smul_mul_assoc (r : R) (x y : A) :\n  (r \u2022 x) * y = r \u2022 (x * y) :=\nsmul_mul_assoc r x y\n\nsection\nvariables {r : R} {a : A}\n\n@[simp] lemma bit0_smul_one : bit0 r \u2022 (1 : A) = bit0 (r \u2022 (1 : A)) :=\nby simp [bit0, add_smul]\nlemma bit0_smul_one' : bit0 r \u2022 (1 : A) = r \u2022 2 :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit0 : bit0 r \u2022 bit0 a = r \u2022 (bit0 (bit0 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit0_smul_bit1 : bit0 r \u2022 bit1 a = r \u2022 (bit0 (bit1 a)) :=\nby simp [bit0, add_smul, smul_add]\n@[simp] lemma bit1_smul_one : bit1 r \u2022 (1 : A) = bit1 (r \u2022 (1 : A)) :=\nby simp [bit1, add_smul]\n\n\nend\n\nvariables (R A)\n\n/--\nThe canonical ring homomorphism `algebra_map R A : R \u2192* A` for any `R`-algebra `A`,\npackaged as an `R`-linear map.\n-/\nprotected def linear_map : R \u2192\u2097[R] A :=\n{ map_smul' := \u03bb x y, by simp [algebra.smul_def],\n  ..algebra_map R A }\n\n@[simp]\nlemma linear_map_apply (r : R) : algebra.linear_map R A r = algebra_map R A r := rfl\n\nlemma coe_linear_map : \u21d1(algebra.linear_map R A) = algebra_map R A := rfl\n\ninstance id : algebra R R := (ring_hom.id R).to_algebra\n\nvariables {R A}\n\nnamespace id\n\n@[simp] lemma map_eq_id : algebra_map R R = ring_hom.id _ := rfl\n\nlemma map_eq_self (x : R) : algebra_map R R x = x := rfl\n\n@[simp] lemma smul_eq_mul (x y : R) : x \u2022 y = x * y := rfl\n\nend id\n\nsection prod\nvariables (R A B)\n\ninstance _root_.prod.algebra : algebra R (A \u00d7 B) :=\n{ commutes' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [commutes r a, commutes r b] },\n  smul_def' := by { rintro r \u27e8a, b\u27e9, dsimp, rw [smul_def r a, smul_def r b] },\n  .. prod.module,\n  .. ring_hom.prod (algebra_map R A) (algebra_map R B) }\n\nvariables {R A B}\n\n@[simp] lemma algebra_map_prod_apply (r : R) :\n  algebra_map R (A \u00d7 B) r = (algebra_map R A r, algebra_map R B r) := rfl\n\nend prod\n\n/-- Algebra over a subsemiring. This builds upon `subsemiring.module`. -/\ninstance of_subsemiring (S : subsemiring R) : algebra S A :=\n{ smul := (\u2022),\n  commutes' := \u03bb r x, algebra.commutes r x,\n  smul_def' := \u03bb r x, algebra.smul_def r x,\n  .. (algebra_map R A).comp S.subtype }\n\n/-- Algebra over a subring. This builds upon `subring.module`. -/\ninstance of_subring {R A : Type*} [comm_ring R] [ring A] [algebra R A]\n  (S : subring R) : algebra S A :=\n{ smul := (\u2022),\n  .. algebra.of_subsemiring S.to_subsemiring,\n  .. (algebra_map R A).comp S.subtype }\n\nlemma algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192+* R) = subring.subtype S := rfl\n\nlemma coe_algebra_map_of_subring {R : Type*} [comm_ring R] (S : subring R) :\n  (algebra_map S R : S \u2192 R) = subtype.val := rfl\n\nlemma algebra_map_of_subring_apply {R : Type*} [comm_ring R] (S : subring R) (x : S) :\n  algebra_map S R x = x := rfl\n\n/-- Explicit characterization of the submonoid map in the case of an algebra.\n`S` is made explicit to help with type inference -/\ndef algebra_map_submonoid (S : Type*) [semiring S] [algebra R S]\n  (M : submonoid R) : (submonoid S) :=\nsubmonoid.map (algebra_map R S : R \u2192* S) M\n\nlemma mem_algebra_map_submonoid_of_mem [algebra R S] {M : submonoid R} (x : M) :\n  (algebra_map R S x) \u2208 algebra_map_submonoid S M :=\nset.mem_image_of_mem (algebra_map R S) x.2\n\nend semiring\n\nsection ring\nvariables [comm_ring R]\n\nvariables (R)\n\n/-- A `semiring` that is an `algebra` over a commutative ring carries a natural `ring` structure.\nSee note [reducible non-instances]. -/\n@[reducible]\ndef semiring_to_ring [semiring A] [algebra R A] : ring A :=\n{ ..module.add_comm_monoid_to_add_comm_group R,\n  ..(infer_instance : semiring A) }\n\nvariables {R}\n\nlemma mul_sub_algebra_map_commutes [ring A] [algebra R A] (x : A) (r : R) :\n  x * (x - algebra_map R A r) = (x - algebra_map R A r) * x :=\nby rw [mul_sub, \u2190commutes, sub_mul]\n\nlemma mul_sub_algebra_map_pow_commutes [ring A] [algebra R A] (x : A) (r : R) (n : \u2115) :\n  x * (x - algebra_map R A r) ^ n = (x - algebra_map R A r) ^ n * x :=\nbegin\n  induction n with n ih,\n  { simp },\n  { rw [pow_succ, \u2190mul_assoc, mul_sub_algebra_map_commutes,\n      mul_assoc, ih, \u2190mul_assoc], }\nend\n\nend ring\n\nend algebra\n\nnamespace no_zero_smul_divisors\n\nvariables {R A : Type*}\n\nopen algebra\n\nsection ring\n\nvariables [comm_ring R]\n\n/-- If `algebra_map R A` is injective and `A` has no zero divisors,\n`R`-multiples in `A` are zero only if one of the factors is zero.\n\nCannot be an instance because there is no `injective (algebra_map R A)` typeclass.\n-/\nlemma of_algebra_map_injective\n  [semiring A] [algebra R A] [no_zero_divisors A]\n  (h : function.injective (algebra_map R A)) : no_zero_smul_divisors R A :=\n\u27e8\u03bb c x hcx, (mul_eq_zero.mp ((smul_def c x).symm.trans hcx)).imp_left\n  ((algebra_map R A).injective_iff.mp h _)\u27e9\n\nvariables (R A)\nlemma algebra_map_injective [ring A] [nontrivial A]\n  [algebra R A] [no_zero_smul_divisors R A] :\n  function.injective (algebra_map R A) :=\nsuffices function.injective (\u03bb (c : R), c \u2022 (1 : A)),\nby { convert this, ext, rw [algebra.smul_def, mul_one] },\nsmul_left_injective R one_ne_zero\n\nvariables {R A}\nlemma iff_algebra_map_injective [ring A] [is_domain A] [algebra R A] :\n  no_zero_smul_divisors R A \u2194 function.injective (algebra_map R A) :=\n\u27e8@@no_zero_smul_divisors.algebra_map_injective R A _ _ _ _,\n no_zero_smul_divisors.of_algebra_map_injective\u27e9\n\nend ring\n\nsection field\n\nvariables [field R] [semiring A] [algebra R A]\n\n@[priority 100] -- see note [lower instance priority]\ninstance algebra.no_zero_smul_divisors [nontrivial A] [no_zero_divisors A] :\n  no_zero_smul_divisors R A :=\nno_zero_smul_divisors.of_algebra_map_injective (algebra_map R A).injective\n\nend field\n\nend no_zero_smul_divisors\n\nnamespace mul_opposite\n\nvariables {R A : Type*} [comm_semiring R] [semiring A] [algebra R A]\n\ninstance : algebra R A\u1d50\u1d52\u1d56 :=\n{ to_ring_hom := (algebra_map R A).to_opposite $ \u03bb x y, algebra.commutes _ _,\n  smul_def' := \u03bb c x, unop_injective $\n    by { dsimp, simp only [op_mul, algebra.smul_def, algebra.commutes, op_unop] },\n  commutes' := \u03bb r, mul_opposite.rec $ \u03bb x, by dsimp; simp only [\u2190 op_mul, algebra.commutes],\n  .. mul_opposite.has_scalar A R }\n\n@[simp] lemma algebra_map_apply (c : R) : algebra_map R A\u1d50\u1d52\u1d56 c = op (algebra_map R A c) := rfl\n\nend mul_opposite\n\nnamespace module\nvariables (R : Type u) (M : Type v) [comm_semiring R] [add_comm_monoid M] [module R M]\n\ninstance : algebra R (module.End R M) :=\nalgebra.of_module smul_mul_assoc (\u03bb r f g, (smul_comm r f g).symm)\n\nlemma algebra_map_End_eq_smul_id (a : R) :\n  (algebra_map R (End R M)) a = a \u2022 linear_map.id := rfl\n\n@[simp] lemma algebra_map_End_apply (a : R) (m : M) :\n  (algebra_map R (End R M)) a m = a \u2022 m := rfl\n\n@[simp] lemma ker_algebra_map_End (K : Type u) (V : Type v)\n  [field K] [add_comm_group V] [module K V] (a : K) (ha : a \u2260 0) :\n  ((algebra_map K (End K V)) a).ker = \u22a5 :=\nlinear_map.ker_smul _ _ ha\n\nend module\n\nset_option old_structure_cmd true\n/-- Defining the homomorphism in the category R-Alg. -/\n@[nolint has_inhabited_instance]\nstructure alg_hom (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B] extends ring_hom A B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nrun_cmd tactic.add_doc_string `alg_hom.to_ring_hom \"Reinterpret an `alg_hom` as a `ring_hom`\"\n\ninfixr ` \u2192\u2090 `:25 := alg_hom _\nnotation A ` \u2192\u2090[`:25 R `] ` B := alg_hom R A B\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {C : Type u\u2081} {D : Type v\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A] [semiring B] [semiring C] [semiring D]\nvariables [algebra R A] [algebra R B] [algebra R C] [algebra R D]\n\ninstance : has_coe_to_fun (A \u2192\u2090[R] B) (\u03bb _, A \u2192 B) := \u27e8alg_hom.to_fun\u27e9\n\ninitialize_simps_projections alg_hom (to_fun \u2192 apply)\n\n@[simp] lemma to_fun_eq_coe (f : A \u2192\u2090[R] B) : f.to_fun = f := rfl\n\ninstance : ring_hom_class (A \u2192\u2090[R] B) A B :=\n{ coe := to_fun,\n  coe_injective' := \u03bb f g h, by { cases f, cases g, congr' },\n  map_add := map_add',\n  map_zero := map_zero',\n  map_mul := map_mul',\n  map_one := map_one' }\n\ninstance coe_ring_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+* B) := \u27e8alg_hom.to_ring_hom\u27e9\n\ninstance coe_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192* B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\ninstance coe_add_monoid_hom : has_coe (A \u2192\u2090[R] B) (A \u2192+ B) := \u27e8\u03bb f, \u2191(f : A \u2192+* B)\u27e9\n\n@[simp, norm_cast] lemma coe_mk {f : A \u2192 B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  \u21d1(\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := rfl\n\n-- make the coercion the simp-normal form\n@[simp] lemma to_ring_hom_eq_coe (f : A \u2192\u2090[R] B) : f.to_ring_hom = f := rfl\n\n@[simp, norm_cast] lemma coe_to_ring_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+* B) = f := rfl\n\n@[simp, norm_cast] lemma coe_to_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192* B) = f := rfl\n\n@[simp, norm_cast] lemma coe_to_add_monoid_hom (f : A \u2192\u2090[R] B) : \u21d1(f : A \u2192+ B) = f := rfl\n\nvariables (\u03c6 : A \u2192\u2090[R] B)\n\ntheorem coe_fn_injective : @function.injective (A \u2192\u2090[R] B) (A \u2192 B) coe_fn := fun_like.coe_injective\n\ntheorem coe_fn_inj {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : (\u03c6\u2081 : A \u2192 B) = \u03c6\u2082 \u2194 \u03c6\u2081 = \u03c6\u2082 := fun_like.coe_fn_eq\n\ntheorem coe_ring_hom_injective : function.injective (coe : (A \u2192\u2090[R] B) \u2192 (A \u2192+* B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 H, coe_fn_injective $ show ((\u03c6\u2081 : (A \u2192+* B)) : A \u2192 B) = ((\u03c6\u2082 : (A \u2192+* B)) : A \u2192 B),\n  from congr_arg _ H\n\ntheorem coe_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192* B)) :=\nring_hom.coe_monoid_hom_injective.comp coe_ring_hom_injective\n\ntheorem coe_add_monoid_hom_injective : function.injective (coe : (A \u2192\u2090[R] B)  \u2192 (A \u2192+ B)) :=\nring_hom.coe_add_monoid_hom_injective.comp coe_ring_hom_injective\n\nprotected lemma congr_fun {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u03c6\u2081 = \u03c6\u2082) (x : A) : \u03c6\u2081 x = \u03c6\u2082 x :=\nfun_like.congr_fun H x\nprotected lemma congr_arg (\u03c6 : A \u2192\u2090[R] B) {x y : A} (h : x = y) : \u03c6 x = \u03c6 y :=\nfun_like.congr_arg \u03c6 h\n\n@[ext]\ntheorem ext {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} (H : \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x) : \u03c6\u2081 = \u03c6\u2082 := fun_like.ext _ _ H\n\ntheorem ext_iff {\u03c6\u2081 \u03c6\u2082 : A \u2192\u2090[R] B} : \u03c6\u2081 = \u03c6\u2082 \u2194 \u2200 x, \u03c6\u2081 x = \u03c6\u2082 x := fun_like.ext_iff\n\n@[simp] theorem mk_coe {f : A \u2192\u2090[R] B} (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A \u2192\u2090[R] B) = f := ext $ \u03bb _, rfl\n\n@[simp]\ntheorem commutes (r : R) : \u03c6 (algebra_map R A r) = algebra_map R B r := \u03c6.commutes' r\n\ntheorem comp_algebra_map : (\u03c6 : A \u2192+* B).comp (algebra_map R A) = algebra_map R B :=\nring_hom.ext $ \u03c6.commutes\n\nlemma map_add (r s : A) : \u03c6 (r + s) = \u03c6 r + \u03c6 s := map_add _ _ _\nlemma map_zero : \u03c6 0 = 0 := map_zero _\nlemma map_mul (x y) : \u03c6 (x * y) = \u03c6 x * \u03c6 y := map_mul _ _ _\nlemma map_one : \u03c6 1 = 1 := map_one _\nlemma map_pow (x : A) (n : \u2115) : \u03c6 (x ^ n) = (\u03c6 x) ^ n :=\nmap_pow _ _ _\n\n@[simp] lemma map_smul (r : R) (x : A) : \u03c6 (r \u2022 x) = r \u2022 \u03c6 x :=\nby simp only [algebra.smul_def, map_mul, commutes]\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u2211 x in s, f x) = \u2211 x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.sum g) = f.sum (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_sum _ _\n\nlemma map_bit0 (x) : \u03c6 (bit0 x) = bit0 (\u03c6 x) := map_bit0 _ _\nlemma map_bit1 (x) : \u03c6 (bit1 x) = bit1 (\u03c6 x) := map_bit1 _ _\n\n/-- If a `ring_hom` is `R`-linear, then it is an `alg_hom`. -/\ndef mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : A \u2192\u2090[R] B :=\n{ to_fun := f,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, h, f.map_one],\n  .. f }\n\n@[simp] lemma coe_mk' (f : A \u2192+* B) (h : \u2200 (c : R) x, f (c \u2022 x) = c \u2022 f x) : \u21d1(mk' f h) = f := rfl\n\nsection\n\nvariables (R A)\n/-- Identity map as an `alg_hom`. -/\nprotected def id : A \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl,\n  ..ring_hom.id A }\n\n@[simp] lemma coe_id : \u21d1(alg_hom.id R A) = id := rfl\n\n@[simp] lemma id_to_ring_hom : (alg_hom.id R A : A \u2192+* A) = ring_hom.id _ := rfl\n\nend\n\nlemma id_apply (p : A) : alg_hom.id R A p = p := rfl\n\n/-- Composition of algebra homeomorphisms. -/\ndef comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : A \u2192\u2090[R] C :=\n{ commutes' := \u03bb r : R, by rw [\u2190 \u03c6\u2081.commutes, \u2190 \u03c6\u2082.commutes]; refl,\n  .. \u03c6\u2081.to_ring_hom.comp \u2191\u03c6\u2082 }\n\n@[simp] lemma coe_comp (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) : \u21d1(\u03c6\u2081.comp \u03c6\u2082) = \u03c6\u2081 \u2218 \u03c6\u2082 := rfl\n\nlemma comp_apply (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) (p : A) : \u03c6\u2081.comp \u03c6\u2082 p = \u03c6\u2081 (\u03c6\u2082 p) := rfl\n\nlemma comp_to_ring_hom (\u03c6\u2081 : B \u2192\u2090[R] C) (\u03c6\u2082 : A \u2192\u2090[R] B) :\n  \u21d1(\u03c6\u2081.comp \u03c6\u2082 : A \u2192+* C) = (\u03c6\u2081 : B \u2192+* C).comp \u2191\u03c6\u2082 := rfl\n\n@[simp] theorem comp_id : \u03c6.comp (alg_hom.id R A) = \u03c6 :=\next $ \u03bb x, rfl\n\n@[simp] theorem id_comp : (alg_hom.id R B).comp \u03c6 = \u03c6 :=\next $ \u03bb x, rfl\n\ntheorem comp_assoc (\u03c6\u2081 : C \u2192\u2090[R] D) (\u03c6\u2082 : B \u2192\u2090[R] C) (\u03c6\u2083 : A \u2192\u2090[R] B) :\n  (\u03c6\u2081.comp \u03c6\u2082).comp \u03c6\u2083 = \u03c6\u2081.comp (\u03c6\u2082.comp \u03c6\u2083) :=\next $ \u03bb x, rfl\n\n/-- R-Alg \u2964 R-Mod -/\ndef to_linear_map : A \u2192\u2097[R] B :=\n{ to_fun := \u03c6,\n  map_add' := \u03c6.map_add,\n  map_smul' := \u03c6.map_smul }\n\n@[simp] lemma to_linear_map_apply (p : A) : \u03c6.to_linear_map p = \u03c6 p := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A \u2192\u2097[R] B)) :=\n\u03bb \u03c6\u2081 \u03c6\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma comp_to_linear_map (f : A \u2192\u2090[R] B) (g : B \u2192\u2090[R] C) :\n  (g.comp f).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_id : to_linear_map (alg_hom.id R A) = linear_map.id :=\nlinear_map.ext $ \u03bb _, rfl\n\n/-- Promote a `linear_map` to an `alg_hom` by supplying proofs about the behavior on `1` and `*`. -/\n@[simps]\ndef of_linear_map (f : A \u2192\u2097[R] B) (map_one : f 1 = 1) (map_mul : \u2200 x y, f (x * y) = f x * f y) :\n  A \u2192\u2090[R] B :=\n{ to_fun := f,\n  map_one' := map_one,\n  map_mul' := map_mul,\n  commutes' := \u03bb c, by simp only [algebra.algebra_map_eq_smul_one, f.map_smul, map_one],\n  .. f.to_add_monoid_hom }\n\n@[simp] lemma of_linear_map_to_linear_map (map_one) (map_mul) :\n  of_linear_map \u03c6.to_linear_map map_one map_mul = \u03c6 :=\nby { ext, refl }\n\n@[simp] lemma to_linear_map_of_linear_map (f : A \u2192\u2097[R] B) (map_one) (map_mul) :\n  to_linear_map (of_linear_map f map_one map_mul) = f :=\nby { ext, refl }\n\n@[simp] lemma of_linear_map_id (map_one) (map_mul) :\n  of_linear_map linear_map.id map_one map_mul = alg_hom.id R A :=\next $ \u03bb _, rfl\n\nlemma map_list_prod (s : list A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_list_prod s\n\nsection prod\n\n/-- First projection as `alg_hom`. -/\ndef fst : A \u00d7 B \u2192\u2090[R] A :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.fst A B}\n\n/-- Second projection as `alg_hom`. -/\ndef snd : A \u00d7 B \u2192\u2090[R] B :=\n{ commutes' := \u03bb r, rfl, .. ring_hom.snd A B}\n\nend prod\n\nlemma algebra_map_eq_apply (f : A \u2192\u2090[R] B) {y : R} {x : A} (h : algebra_map R A y = x) :\n  algebra_map R B y = f x :=\nh \u25b8 (f.commutes _).symm\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A] [comm_semiring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\nlemma map_multiset_prod (s : multiset A) :\n  \u03c6 s.prod = (s.map \u03c6).prod :=\n\u03c6.to_ring_hom.map_multiset_prod s\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A) (s : finset \u03b9) :\n  \u03c6 (\u220f x in s, f x) = \u220f x in s, \u03c6 (f x) :=\n\u03c6.to_ring_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A) :\n  \u03c6 (f.prod g) = f.prod (\u03bb i a, \u03c6 (g i a)) :=\n\u03c6.map_prod _ _\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_semiring R] [ring A] [ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\nlemma map_neg (x) : \u03c6 (-x) = -\u03c6 x := map_neg _ _\nlemma map_sub (x y) : \u03c6 (x - y) = \u03c6 x - \u03c6 y := map_sub _ _ _\n\n@[simp] lemma map_int_cast (n : \u2124) : \u03c6 n = n :=\n\u03c6.to_ring_hom.map_int_cast n\n\nend ring\n\nsection division_ring\n\nvariables [comm_ring R] [division_ring A] [division_ring B]\nvariables [algebra R A] [algebra R B] (\u03c6 : A \u2192\u2090[R] B)\n\n@[simp] lemma map_inv (x) : \u03c6 (x\u207b\u00b9) = (\u03c6 x)\u207b\u00b9 :=\n\u03c6.to_ring_hom.map_inv x\n\n@[simp] lemma map_div (x y) : \u03c6 (x / y) = \u03c6 x / \u03c6 y :=\n\u03c6.to_ring_hom.map_div x y\n\nend division_ring\n\ntheorem injective_iff {R A B : Type*} [comm_semiring R] [ring A] [semiring B]\n  [algebra R A] [algebra R B] (f : A \u2192\u2090[R] B) :\n  function.injective f \u2194 (\u2200 x, f x = 0 \u2192 x = 0) :=\nring_hom.injective_iff (f : A \u2192+* B)\n\nend alg_hom\n\n@[simp] lemma rat.smul_one_eq_coe {A : Type*} [division_ring A] [algebra \u211a A] (m : \u211a) :\n  m \u2022 (1 : A) = \u2191m :=\nby rw [algebra.smul_def, mul_one, ring_hom.eq_rat_cast]\n\nset_option old_structure_cmd true\n/-- An equivalence of algebras is an equivalence of rings commuting with the actions of scalars. -/\nstructure alg_equiv (R : Type u) (A : Type v) (B : Type w)\n  [comm_semiring R] [semiring A] [semiring B] [algebra R A] [algebra R B]\n  extends A \u2243 B, A \u2243* B, A \u2243+ B, A \u2243+* B :=\n(commutes' : \u2200 r : R, to_fun (algebra_map R A r) = algebra_map R B r)\n\nattribute [nolint doc_blame] alg_equiv.to_ring_equiv\nattribute [nolint doc_blame] alg_equiv.to_equiv\nattribute [nolint doc_blame] alg_equiv.to_add_equiv\nattribute [nolint doc_blame] alg_equiv.to_mul_equiv\n\nnotation A ` \u2243\u2090[`:50 R `] ` A' := alg_equiv R A A'\n\nnamespace alg_equiv\n\nvariables {R : Type u} {A\u2081 : Type v} {A\u2082 : Type w} {A\u2083 : Type u\u2081}\n\nsection semiring\n\nvariables [comm_semiring R] [semiring A\u2081] [semiring A\u2082] [semiring A\u2083]\nvariables [algebra R A\u2081] [algebra R A\u2082] [algebra R A\u2083]\nvariables (e : A\u2081 \u2243\u2090[R] A\u2082)\n\ninstance : has_coe_to_fun (A\u2081 \u2243\u2090[R] A\u2082) (\u03bb _, A\u2081 \u2192 A\u2082) := \u27e8alg_equiv.to_fun\u27e9\n\n@[ext]\nlemma ext {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : \u2200 a, f a = g a) : f = g :=\nbegin\n  have h\u2081 : f.to_equiv = g.to_equiv := equiv.ext h,\n  cases f, cases g, congr,\n  { exact (funext h) },\n  { exact congr_arg equiv.inv_fun h\u2081 }\nend\n\nprotected lemma congr_arg {f : A\u2081 \u2243\u2090[R] A\u2082} : \u03a0 {x x' : A\u2081}, x = x' \u2192 f x = f x'\n| _ _ rfl := rfl\n\nprotected lemma congr_fun {f g : A\u2081 \u2243\u2090[R] A\u2082} (h : f = g) (x : A\u2081) : f x = g x := h \u25b8 rfl\n\nlemma ext_iff {f g : A\u2081 \u2243\u2090[R] A\u2082} : f = g \u2194 \u2200 x, f x = g x :=\n\u27e8\u03bb h x, h \u25b8 rfl, ext\u27e9\n\nlemma coe_fun_injective : @function.injective (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192 A\u2082) (\u03bb e, (e : A\u2081 \u2192 A\u2082)) :=\nbegin\n  intros f g w,\n  ext,\n  exact congr_fun w a,\nend\n\ninstance has_coe_to_ring_equiv : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2243+* A\u2082) := \u27e8alg_equiv.to_ring_equiv\u27e9\n\n@[simp] lemma coe_mk {to_fun inv_fun left_inv right_inv map_mul map_add commutes} :\n  \u21d1(\u27e8to_fun, inv_fun, left_inv, right_inv, map_mul, map_add, commutes\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = to_fun :=\nrfl\n\n@[simp] theorem mk_coe (e : A\u2081 \u2243\u2090[R] A\u2082) (e' h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8e, e', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082) = e := ext $ \u03bb _, rfl\n\n@[simp] lemma to_fun_eq_coe (e : A\u2081 \u2243\u2090[R] A\u2082) : e.to_fun = e := rfl\n\n@[simp] lemma to_ring_equiv_eq_coe : e.to_ring_equiv = e := rfl\n\n@[simp, norm_cast] lemma coe_ring_equiv : ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192 A\u2082) = e := rfl\nlemma coe_ring_equiv' : (e.to_ring_equiv : A\u2081 \u2192 A\u2082) = e := rfl\n\nlemma coe_ring_equiv_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2243+* A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ ring_equiv.congr_fun h\n\n@[simp] lemma map_add : \u2200 x y, e (x + y) = e x + e y := e.to_add_equiv.map_add\n\n@[simp] lemma map_zero : e 0 = 0 := e.to_add_equiv.map_zero\n\n@[simp] lemma map_mul : \u2200 x y, e (x * y) = (e x) * (e y) := e.to_mul_equiv.map_mul\n\n@[simp] lemma map_one : e 1 = 1 := e.to_mul_equiv.map_one\n\n@[simp] lemma commutes : \u2200 (r : R), e (algebra_map R A\u2081 r) = algebra_map R A\u2082 r :=\n  e.commutes'\n\n@[simp] lemma map_smul (r : R) (x : A\u2081) : e (r \u2022 x) = r \u2022 e x :=\nby simp only [algebra.smul_def, map_mul, commutes]\n\nlemma map_sum {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u2211 x in s, f x) = \u2211 x in s, e (f x) :=\ne.to_add_equiv.map_sum f s\n\nlemma map_finsupp_sum {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.sum g) = f.sum (\u03bb i b, e (g i b)) :=\ne.map_sum _ _\n\n/-- Interpret an algebra equivalence as an algebra homomorphism.\n\nThis definition is included for symmetry with the other `to_*_hom` projections.\nThe `simp` normal form is to use the coercion of the `has_coe_to_alg_hom` instance. -/\ndef to_alg_hom : A\u2081 \u2192\u2090[R] A\u2082 :=\n{ map_one' := e.map_one, map_zero' := e.map_zero, ..e }\n\ninstance has_coe_to_alg_hom : has_coe (A\u2081 \u2243\u2090[R] A\u2082) (A\u2081 \u2192\u2090[R] A\u2082) :=\n\u27e8to_alg_hom\u27e9\n\n@[simp] lemma to_alg_hom_eq_coe : e.to_alg_hom = e := rfl\n\n@[simp, norm_cast] lemma coe_alg_hom : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192 A\u2082) = e :=\nrfl\n\nlemma coe_alg_hom_injective : function.injective (coe : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2081 \u2192\u2090[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ alg_hom.congr_fun h\n\n/-- The two paths coercion can take to a `ring_hom` are equivalent -/\nlemma coe_ring_hom_commutes : ((e : A\u2081 \u2192\u2090[R] A\u2082) : A\u2081 \u2192+* A\u2082) = ((e : A\u2081 \u2243+* A\u2082) : A\u2081 \u2192+* A\u2082) :=\nrfl\n\n@[simp] lemma map_pow : \u2200 (x : A\u2081) (n : \u2115), e (x ^ n) = (e x) ^ n := e.to_alg_hom.map_pow\n\nlemma injective : function.injective e := e.to_equiv.injective\n\nlemma surjective : function.surjective e := e.to_equiv.surjective\n\nlemma bijective : function.bijective e := e.to_equiv.bijective\n\n/-- Algebra equivalences are reflexive. -/\n@[refl] def refl : A\u2081 \u2243\u2090[R] A\u2081 := {commutes' := \u03bb r, rfl, ..(1 : A\u2081 \u2243+* A\u2081)}\n\ninstance : inhabited (A\u2081 \u2243\u2090[R] A\u2081) := \u27e8refl\u27e9\n\n@[simp] lemma refl_to_alg_hom : \u2191(refl : A\u2081 \u2243\u2090[R] A\u2081) = alg_hom.id R A\u2081 := rfl\n\n@[simp] lemma coe_refl : \u21d1(refl : A\u2081 \u2243\u2090[R] A\u2081) = id := rfl\n\n/-- Algebra equivalences are symmetric. -/\n@[symm]\ndef symm (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2243\u2090[R] A\u2081 :=\n{ commutes' := \u03bb r, by { rw \u2190e.to_ring_equiv.symm_apply_apply (algebra_map R A\u2081 r), congr,\n                         change _ = e _, rw e.commutes, },\n  ..e.to_ring_equiv.symm, }\n\n/-- See Note [custom simps projection] -/\ndef simps.symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2082 \u2192 A\u2081 := e.symm\n\ninitialize_simps_projections alg_equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\n\n@[simp] lemma inv_fun_eq_symm {e : A\u2081 \u2243\u2090[R] A\u2082} : e.inv_fun = e.symm := rfl\n\n@[simp] lemma symm_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : e.symm.symm = e :=\nby { ext, refl, }\n\nlemma symm_bijective : function.bijective (symm : (A\u2081 \u2243\u2090[R] A\u2082) \u2192 (A\u2082 \u2243\u2090[R] A\u2081)) :=\nequiv.bijective \u27e8symm, symm, symm_symm, symm_symm\u27e9\n\n@[simp] lemma mk_coe' (e : A\u2081 \u2243\u2090[R] A\u2082) (f h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, e, h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2082 \u2243\u2090[R] A\u2081) = e.symm :=\nsymm_bijective.injective $ ext $ \u03bb x, rfl\n\n@[simp] theorem symm_mk (f f') (h\u2081 h\u2082 h\u2083 h\u2084 h\u2085) :\n  (\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm =\n  { to_fun := f', inv_fun := f,\n    ..(\u27e8f, f', h\u2081, h\u2082, h\u2083, h\u2084, h\u2085\u27e9 : A\u2081 \u2243\u2090[R] A\u2082).symm } := rfl\n\n/-- Algebra equivalences are transitive. -/\n@[trans]\ndef trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) : A\u2081 \u2243\u2090[R] A\u2083 :=\n{ commutes' := \u03bb r, show e\u2082.to_fun (e\u2081.to_fun _) = _, by rw [e\u2081.commutes', e\u2082.commutes'],\n  ..(e\u2081.to_ring_equiv.trans e\u2082.to_ring_equiv), }\n\n@[simp] lemma apply_symm_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e (e.symm x) = x :=\n  e.to_equiv.apply_symm_apply\n\n@[simp] lemma symm_apply_apply (e : A\u2081 \u2243\u2090[R] A\u2082) : \u2200 x, e.symm (e x) = x :=\n  e.to_equiv.symm_apply_apply\n\n@[simp] lemma symm_trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2083) :\n  (e\u2081.trans e\u2082).symm x = e\u2081.symm (e\u2082.symm x) := rfl\n\n@[simp] lemma coe_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  \u21d1(e\u2081.trans e\u2082) = e\u2082 \u2218 e\u2081 := rfl\n\nlemma trans_apply (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (x : A\u2081) :\n  (e\u2081.trans e\u2082) x = e\u2082 (e\u2081 x) := rfl\n\n@[simp] lemma comp_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp (e : A\u2081 \u2192\u2090[R] A\u2082) \u2191e.symm = alg_hom.id R A\u2082 :=\nby { ext, simp }\n\n@[simp] lemma symm_comp (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  alg_hom.comp \u2191e.symm (e : A\u2081 \u2192\u2090[R] A\u2082) = alg_hom.id R A\u2081 :=\nby { ext, simp }\n\ntheorem left_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.left_inverse e.symm e := e.left_inv\n\ntheorem right_inverse_symm (e : A\u2081 \u2243\u2090[R] A\u2082) : function.right_inverse e.symm e := e.right_inv\n\n/-- If `A\u2081` is equivalent to `A\u2081'` and `A\u2082` is equivalent to `A\u2082'`, then the type of maps\n`A\u2081 \u2192\u2090[R] A\u2082` is equivalent to the type of maps `A\u2081' \u2192\u2090[R] A\u2082'`. -/\ndef arrow_congr {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082'] [algebra R A\u2081'] [algebra R A\u2082']\n  (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') : (A\u2081 \u2192\u2090[R] A\u2082) \u2243 (A\u2081' \u2192\u2090[R] A\u2082') :=\n{ to_fun := \u03bb f, (e\u2082.to_alg_hom.comp f).comp e\u2081.symm.to_alg_hom,\n  inv_fun := \u03bb f, (e\u2082.symm.to_alg_hom.comp f).comp e\u2081.to_alg_hom,\n  left_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, symm_comp],\n    simp only [\u2190alg_hom.comp_assoc, symm_comp, alg_hom.id_comp, alg_hom.comp_id] },\n  right_inv := \u03bb f, by { simp only [alg_hom.comp_assoc, to_alg_hom_eq_coe, comp_symm],\n    simp only [\u2190alg_hom.comp_assoc, comp_symm, alg_hom.id_comp, alg_hom.comp_id] } }\n\nlemma arrow_congr_comp {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082')\n  (e\u2083 : A\u2083 \u2243\u2090[R] A\u2083') (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2083) :\n  arrow_congr e\u2081 e\u2083 (g.comp f) = (arrow_congr e\u2082 e\u2083 g).comp (arrow_congr e\u2081 e\u2082 f) :=\nby { ext, simp only [arrow_congr, equiv.coe_fn_mk, alg_hom.comp_apply],\n  congr, exact (e\u2082.symm_apply_apply _).symm }\n\n@[simp] lemma arrow_congr_refl :\n  arrow_congr alg_equiv.refl alg_equiv.refl = equiv.refl (A\u2081 \u2192\u2090[R] A\u2082) :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_trans {A\u2081' A\u2082' A\u2083' : Type*} [semiring A\u2081'] [semiring A\u2082'] [semiring A\u2083']\n  [algebra R A\u2081'] [algebra R A\u2082'] [algebra R A\u2083'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2081' : A\u2081' \u2243\u2090[R] A\u2082')\n  (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) (e\u2082' : A\u2082' \u2243\u2090[R] A\u2083') :\n  arrow_congr (e\u2081.trans e\u2082) (e\u2081'.trans e\u2082') = (arrow_congr e\u2081 e\u2081').trans (arrow_congr e\u2082 e\u2082') :=\nby { ext, refl }\n\n@[simp] lemma arrow_congr_symm {A\u2081' A\u2082' : Type*} [semiring A\u2081'] [semiring A\u2082']\n  [algebra R A\u2081'] [algebra R A\u2082'] (e\u2081 : A\u2081 \u2243\u2090[R] A\u2081') (e\u2082 : A\u2082 \u2243\u2090[R] A\u2082') :\n  (arrow_congr e\u2081 e\u2082).symm = arrow_congr e\u2081.symm e\u2082.symm :=\nby { ext, refl }\n\n/-- If an algebra morphism has an inverse, it is a algebra isomorphism. -/\ndef of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 : f.comp g = alg_hom.id R A\u2082)\n  (h\u2082 : g.comp f = alg_hom.id R A\u2081) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun    := f,\n  inv_fun   := g,\n  left_inv  := alg_hom.ext_iff.1 h\u2082,\n  right_inv := alg_hom.ext_iff.1 h\u2081,\n  ..f }\n\nlemma coe_alg_hom_of_alg_hom (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  \u2191(of_alg_hom f g h\u2081 h\u2082) = f := alg_hom.ext $ \u03bb _, rfl\n\n@[simp]\nlemma of_alg_hom_coe_alg_hom (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  of_alg_hom \u2191f g h\u2081 h\u2082 = f := ext $ \u03bb _, rfl\n\nlemma of_alg_hom_symm (f : A\u2081 \u2192\u2090[R] A\u2082) (g : A\u2082 \u2192\u2090[R] A\u2081) (h\u2081 h\u2082) :\n  (of_alg_hom f g h\u2081 h\u2082).symm = of_alg_hom g f h\u2082 h\u2081 := rfl\n\n/-- Promotes a bijective algebra homomorphism to an algebra equivalence. -/\nnoncomputable def of_bijective (f : A\u2081 \u2192\u2090[R] A\u2082) (hf : function.bijective f) : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ .. ring_equiv.of_bijective (f : A\u2081 \u2192+* A\u2082) hf, .. f }\n\n@[simp] lemma coe_of_bijective {f : A\u2081 \u2192\u2090[R] A\u2082} {hf : function.bijective f} :\n  (alg_equiv.of_bijective f hf : A\u2081 \u2192 A\u2082) = f := rfl\n\nlemma of_bijective_apply {f : A\u2081 \u2192\u2090[R] A\u2082} {hf : function.bijective f} (a : A\u2081) :\n  (alg_equiv.of_bijective f hf) a = f a := rfl\n\n/-- Forgetting the multiplicative structures, an equivalence of algebras is a linear equivalence. -/\n@[simps apply] def to_linear_equiv (e : A\u2081 \u2243\u2090[R] A\u2082) : A\u2081 \u2243\u2097[R] A\u2082 :=\n{ to_fun    := e,\n  map_smul' := e.map_smul,\n  inv_fun   := e.symm,\n  .. e }\n\n@[simp] lemma to_linear_equiv_refl :\n  (alg_equiv.refl : A\u2081 \u2243\u2090[R] A\u2081).to_linear_equiv = linear_equiv.refl R A\u2081 := rfl\n\n@[simp] lemma to_linear_equiv_symm (e : A\u2081 \u2243\u2090[R] A\u2082) :\n  e.to_linear_equiv.symm = e.symm.to_linear_equiv := rfl\n\n@[simp] lemma to_linear_equiv_trans (e\u2081 : A\u2081 \u2243\u2090[R] A\u2082) (e\u2082 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (e\u2081.trans e\u2082).to_linear_equiv = e\u2081.to_linear_equiv.trans e\u2082.to_linear_equiv := rfl\n\ntheorem to_linear_equiv_injective : function.injective (to_linear_equiv : _ \u2192 (A\u2081 \u2243\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_equiv.congr_fun h\n\n/-- Interpret an algebra equivalence as a linear map. -/\ndef to_linear_map : A\u2081 \u2192\u2097[R] A\u2082 :=\ne.to_alg_hom.to_linear_map\n\n@[simp] lemma to_alg_hom_to_linear_map :\n  (e : A\u2081 \u2192\u2090[R] A\u2082).to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_equiv_to_linear_map :\n  e.to_linear_equiv.to_linear_map = e.to_linear_map := rfl\n\n@[simp] lemma to_linear_map_apply (x : A\u2081) : e.to_linear_map x = e x := rfl\n\ntheorem to_linear_map_injective : function.injective (to_linear_map : _ \u2192 (A\u2081 \u2192\u2097[R] A\u2082)) :=\n\u03bb e\u2081 e\u2082 h, ext $ linear_map.congr_fun h\n\n@[simp] lemma trans_to_linear_map (f : A\u2081 \u2243\u2090[R] A\u2082) (g : A\u2082 \u2243\u2090[R] A\u2083) :\n  (f.trans g).to_linear_map = g.to_linear_map.comp f.to_linear_map := rfl\n\nsection of_linear_equiv\n\nvariables (l : A\u2081 \u2243\u2097[R] A\u2082)\n  (map_mul : \u2200 x y : A\u2081, l (x * y) = l x * l y)\n  (commutes : \u2200 r : R, l (algebra_map R A\u2081 r) = algebra_map R A\u2082 r)\n\n/--\nUpgrade a linear equivalence to an algebra equivalence,\ngiven that it distributes over multiplication and action of scalars.\n-/\n@[simps apply]\ndef of_linear_equiv : A\u2081 \u2243\u2090[R] A\u2082 :=\n{ to_fun := l,\n  inv_fun := l.symm,\n  map_mul' := map_mul,\n  commutes' := commutes,\n  ..l }\n\n@[simp]\nlemma of_linear_equiv_symm :\n  (of_linear_equiv l map_mul commutes).symm = of_linear_equiv l.symm\n    ((of_linear_equiv l map_mul commutes).symm.map_mul)\n    ((of_linear_equiv l map_mul commutes).symm.commutes) :=\nrfl\n\n@[simp] lemma of_linear_equiv_to_linear_equiv (map_mul) (commutes) :\n  of_linear_equiv e.to_linear_equiv map_mul commutes = e :=\nby { ext, refl }\n\n@[simp] lemma to_linear_equiv_of_linear_equiv :\n  to_linear_equiv (of_linear_equiv l map_mul commutes) = l :=\nby { ext, refl }\n\nend of_linear_equiv\n\n@[simps mul one {attrs := []}] instance aut : group (A\u2081 \u2243\u2090[R] A\u2081) :=\n{ mul := \u03bb \u03d5 \u03c8, \u03c8.trans \u03d5,\n  mul_assoc := \u03bb \u03d5 \u03c8 \u03c7, rfl,\n  one := refl,\n  one_mul := \u03bb \u03d5, ext $ \u03bb x, rfl,\n  mul_one := \u03bb \u03d5, ext $ \u03bb x, rfl,\n  inv := symm,\n  mul_left_inv := \u03bb \u03d5, ext $ symm_apply_apply \u03d5 }\n\n@[simp] lemma one_apply (x : A\u2081) : (1 : A\u2081 \u2243\u2090[R] A\u2081) x = x := rfl\n\n@[simp] lemma mul_apply (e\u2081 e\u2082 : A\u2081 \u2243\u2090[R] A\u2081) (x : A\u2081) : (e\u2081 * e\u2082) x = e\u2081 (e\u2082 x) := rfl\n\n/-- An algebra isomorphism induces a group isomorphism between automorphism groups -/\n@[simps apply]\ndef aut_congr (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (A\u2081 \u2243\u2090[R] A\u2081) \u2243* (A\u2082 \u2243\u2090[R] A\u2082) :=\n{ to_fun := \u03bb \u03c8, \u03d5.symm.trans (\u03c8.trans \u03d5),\n  inv_fun := \u03bb \u03c8, \u03d5.trans (\u03c8.trans \u03d5.symm),\n  left_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, symm_apply_apply] },\n  right_inv := \u03bb \u03c8, by { ext, simp_rw [trans_apply, apply_symm_apply] },\n  map_mul' := \u03bb \u03c8 \u03c7, by { ext, simp only [mul_apply, trans_apply, symm_apply_apply] } }\n\n@[simp] lemma aut_congr_refl : aut_congr (alg_equiv.refl) = mul_equiv.refl (A\u2081 \u2243\u2090[R] A\u2081) :=\nby { ext, refl }\n\n@[simp] lemma aut_congr_symm (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) : (aut_congr \u03d5).symm = aut_congr \u03d5.symm := rfl\n\n@[simp] lemma aut_congr_trans (\u03d5 : A\u2081 \u2243\u2090[R] A\u2082) (\u03c8 : A\u2082 \u2243\u2090[R] A\u2083) :\n  (aut_congr \u03d5).trans (aut_congr \u03c8) = aut_congr (\u03d5.trans \u03c8) := rfl\n\n/-- The tautological action by `A\u2081 \u2243\u2090[R] A\u2081` on `A\u2081`.\n\nThis generalizes `function.End.apply_mul_action`. -/\ninstance apply_mul_semiring_action : mul_semiring_action (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul := ($),\n  smul_zero := alg_equiv.map_zero,\n  smul_add := alg_equiv.map_add,\n  smul_one := alg_equiv.map_one,\n  smul_mul := alg_equiv.map_mul,\n  one_smul := \u03bb _, rfl,\n  mul_smul := \u03bb _ _ _, rfl }\n\n@[simp] protected lemma smul_def (f : A\u2081 \u2243\u2090[R] A\u2081) (a : A\u2081) : f \u2022 a = f a := rfl\n\ninstance apply_has_faithful_scalar : has_faithful_scalar (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n\u27e8\u03bb _ _, alg_equiv.ext\u27e9\n\ninstance apply_smul_comm_class : smul_comm_class R (A\u2081 \u2243\u2090[R] A\u2081) A\u2081 :=\n{ smul_comm := \u03bb r e a, (e.map_smul r a).symm }\n\ninstance apply_smul_comm_class' : smul_comm_class (A\u2081 \u2243\u2090[R] A\u2081) R A\u2081 :=\n{ smul_comm := \u03bb e r a, (e.map_smul r a) }\n\n@[simp] lemma algebra_map_eq_apply (e : A\u2081 \u2243\u2090[R] A\u2082) {y : R} {x : A\u2081} :\n  (algebra_map R A\u2082 y = e x) \u2194 (algebra_map R A\u2081 y = x) :=\n\u27e8\u03bb h, by simpa using e.symm.to_alg_hom.algebra_map_eq_apply h,\n \u03bb h, e.to_alg_hom.algebra_map_eq_apply h\u27e9\n\nend semiring\n\nsection comm_semiring\n\nvariables [comm_semiring R] [comm_semiring A\u2081] [comm_semiring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\nlemma map_prod {\u03b9 : Type*} (f : \u03b9 \u2192 A\u2081) (s : finset \u03b9) :\n  e (\u220f x in s, f x) = \u220f x in s, e (f x) :=\ne.to_alg_hom.map_prod f s\n\nlemma map_finsupp_prod {\u03b1 : Type*} [has_zero \u03b1] {\u03b9 : Type*} (f : \u03b9 \u2192\u2080 \u03b1) (g : \u03b9 \u2192 \u03b1 \u2192 A\u2081) :\n  e (f.prod g) = f.prod (\u03bb i a, e (g i a)) :=\ne.to_alg_hom.map_finsupp_prod f g\n\nend comm_semiring\n\nsection ring\n\nvariables [comm_ring R] [ring A\u2081] [ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\n@[simp] lemma map_neg (x) : e (-x) = -e x :=\ne.to_alg_hom.map_neg x\n\n@[simp] lemma map_sub (x y) : e (x - y) = e x - e y :=\ne.to_alg_hom.map_sub x y\n\nend ring\n\nsection division_ring\n\nvariables [comm_ring R] [division_ring A\u2081] [division_ring A\u2082]\nvariables [algebra R A\u2081] [algebra R A\u2082] (e : A\u2081 \u2243\u2090[R] A\u2082)\n\n@[simp] lemma map_inv (x) : e (x\u207b\u00b9) = (e x)\u207b\u00b9 :=\ne.to_alg_hom.map_inv x\n\n@[simp] lemma map_div (x y) : e (x / y) = e x / e y :=\ne.to_alg_hom.map_div x y\n\nend division_ring\n\nend alg_equiv\n\nnamespace mul_semiring_action\n\nvariables {M G : Type*} (R A : Type*) [comm_semiring R] [semiring A] [algebra R A]\n\nsection\nvariables [monoid M] [mul_semiring_action M A] [smul_comm_class M R A]\n\n/-- Each element of the monoid defines a algebra homomorphism.\n\nThis is a stronger version of `mul_semiring_action.to_ring_hom` and\n`distrib_mul_action.to_linear_map`. -/\n@[simps]\ndef to_alg_hom (m : M) : A \u2192\u2090[R] A :=\nalg_hom.mk' (mul_semiring_action.to_ring_hom _ _ m) (smul_comm _)\n\ntheorem to_alg_hom_injective [has_faithful_scalar M A] :\n  function.injective (mul_semiring_action.to_alg_hom R A : M \u2192 A \u2192\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_hom.ext_iff.1 h r\n\nend\n\nsection\nvariables [group G] [mul_semiring_action G A] [smul_comm_class G R A]\n\n/-- Each element of the group defines a algebra equivalence.\n\nThis is a stronger version of `mul_semiring_action.to_ring_equiv` and\n`distrib_mul_action.to_linear_equiv`. -/\n@[simps]\ndef to_alg_equiv (g : G) : A \u2243\u2090[R] A :=\n{ .. mul_semiring_action.to_ring_equiv _ _ g,\n  .. mul_semiring_action.to_alg_hom R A g }\n\ntheorem to_alg_equiv_injective [has_faithful_scalar G A] :\n  function.injective (mul_semiring_action.to_alg_equiv R A : G \u2192 A \u2243\u2090[R] A) :=\n\u03bb m\u2081 m\u2082 h, eq_of_smul_eq_smul $ \u03bb r, alg_equiv.ext_iff.1 h r\n\nend\n\nend mul_semiring_action\n\nsection nat\n\nvariables {R : Type*} [semiring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2115`-algebras. This is only an issue since `algebra.id` and `algebra_nat` are not yet defeq.\n-- TODO: fix this by adding an `of_nat` field to semirings.\n/-- Semiring \u2964 \u2115-Alg -/\n@[priority 99] instance algebra_nat : algebra \u2115 R :=\n{ commutes' := nat.cast_commute,\n  smul_def' := \u03bb _ _, nsmul_eq_mul _ _,\n  to_ring_hom := nat.cast_ring_hom R }\n\ninstance nat_algebra_subsingleton : subsingleton (algebra \u2115 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend nat\n\nnamespace ring_hom\n\nvariables {R S : Type*}\n\n/-- Reinterpret a `ring_hom` as an `\u2115`-algebra homomorphism. -/\ndef to_nat_alg_hom [semiring R] [semiring S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2115] S :=\n{ to_fun := f, commutes' := \u03bb n, by simp, .. f }\n\n/-- Reinterpret a `ring_hom` as a `\u2124`-algebra homomorphism. -/\ndef to_int_alg_hom [ring R] [ring S] [algebra \u2124 R] [algebra \u2124 S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u2124] S :=\n{ commutes' := \u03bb n, by simp, .. f }\n\n@[simp] lemma map_rat_algebra_map [ring R] [ring S] [algebra \u211a R] [algebra \u211a S] (f : R \u2192+* S)\n  (r : \u211a) :\n  f (algebra_map \u211a R r) = algebra_map \u211a S r :=\nring_hom.ext_iff.1 (subsingleton.elim (f.comp (algebra_map \u211a R)) (algebra_map \u211a S)) r\n\n/-- Reinterpret a `ring_hom` as a `\u211a`-algebra homomorphism. -/\ndef to_rat_alg_hom [ring R] [ring S] [algebra \u211a R] [algebra \u211a S] (f : R \u2192+* S) :\n  R \u2192\u2090[\u211a] S :=\n{ commutes' := f.map_rat_algebra_map, .. f }\n\nend ring_hom\n\nsection rat\n\ninstance algebra_rat {\u03b1} [division_ring \u03b1] [char_zero \u03b1] : algebra \u211a \u03b1 :=\n(rat.cast_hom \u03b1).to_algebra' $ \u03bb r x, r.cast_commute x\n\n@[simp] theorem algebra_map_rat_rat : algebra_map \u211a \u211a = ring_hom.id \u211a :=\nsubsingleton.elim _ _\n\n-- TODO[gh-6025]: make this an instance once safe to do so\nlemma algebra_rat_subsingleton {\u03b1} [semiring \u03b1] :\n  subsingleton (algebra \u211a \u03b1) :=\n\u27e8\u03bb x y, algebra.algebra_ext x y $ ring_hom.congr_fun $ subsingleton.elim _ _\u27e9\n\nend rat\n\nnamespace algebra\nopen module\n\nvariables (R : Type u) (A : Type v)\n\nvariables [comm_semiring R] [semiring A] [algebra R A]\n\n/-- `algebra_map` as an `alg_hom`. -/\ndef of_id : R \u2192\u2090[R] A :=\n{ commutes' := \u03bb _, rfl, .. algebra_map R A }\nvariables {R}\n\ntheorem of_id_apply (r) : of_id R A r = algebra_map R A r := rfl\n\nend algebra\n\nsection int\n\nvariables (R : Type*) [ring R]\n\n-- Lower the priority so that `algebra.id` is picked most of the time when working with\n-- `\u2124`-algebras. This is only an issue since `algebra.id \u2124` and `algebra_int \u2124` are not yet defeq.\n-- TODO: fix this by adding an `of_int` field to rings.\n/-- Ring \u2964 \u2124-Alg -/\n@[priority 99] instance algebra_int : algebra \u2124 R :=\n{ commutes' := int.cast_commute,\n  smul_def' := \u03bb _ _, zsmul_eq_mul _ _,\n  to_ring_hom := int.cast_ring_hom R }\n\n/-- A special case of `ring_hom.eq_int_cast'` that happens to be true definitionally -/\n@[simp] lemma algebra_map_int_eq : algebra_map \u2124 R = int.cast_ring_hom R := rfl\n\nvariables {R}\n\ninstance int_algebra_subsingleton : subsingleton (algebra \u2124 R) :=\n\u27e8\u03bb P Q, by { ext, simp, }\u27e9\n\nend int\n\n/-!\nThe R-algebra structure on `\u03a0 i : I, A i` when each `A i` is an R-algebra.\n\nWe couldn't set this up back in `algebra.pi_instances` because this file imports it.\n-/\nnamespace pi\n\nvariable {I : Type u}     -- The indexing type\nvariable {R : Type*}      -- The scalar type\nvariable {f : I \u2192 Type v} -- The family of types already equipped with instances\nvariables (x y : \u03a0 i, f i) (i : I)\nvariables (I f)\n\ninstance algebra {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] :\n  algebra R (\u03a0 i : I, f i) :=\n{ commutes' := \u03bb a f, begin ext, simp [algebra.commutes], end,\n  smul_def' := \u03bb a f, begin ext, simp [algebra.smul_def], end,\n  ..(pi.ring_hom (\u03bb i, algebra_map R (f i)) : R \u2192+* \u03a0 i : I, f i) }\n\n@[simp] lemma algebra_map_apply {r : comm_semiring R}\n  [s : \u2200 i, semiring (f i)] [\u2200 i, algebra R (f i)] (a : R) (i : I) :\n  algebra_map R (\u03a0 i, f i) a i = algebra_map R (f i) a := rfl\n\n-- One could also build a `\u03a0 i, R i`-algebra structure on `\u03a0 i, A i`,\n-- when each `A i` is an `R i`-algebra, although I'm not sure that it's useful.\n\nvariables {I} (R) (f)\n\n/-- `function.eval` as an `alg_hom`. The name matches `pi.eval_ring_hom`, `pi.eval_monoid_hom`,\netc. -/\n@[simps]\ndef eval_alg_hom {r : comm_semiring R} [\u03a0 i, semiring (f i)] [\u03a0 i, algebra R (f i)] (i : I) :\n  (\u03a0 i, f i) \u2192\u2090[R] f i :=\n{ to_fun := \u03bb f, f i, commutes' := \u03bb r, rfl, .. pi.eval_ring_hom f i}\n\nvariables (A B : Type*) [comm_semiring R] [semiring B] [algebra R B]\n\n/-- `function.const` as an `alg_hom`. The name matches `pi.const_ring_hom`, `pi.const_monoid_hom`,\netc. -/\n@[simps]\ndef const_alg_hom : B \u2192\u2090[R] (A \u2192 B) :=\n{ to_fun := function.const _,\n  commutes' := \u03bb r, rfl,\n  .. pi.const_ring_hom A B}\n\n/-- When `R` is commutative and permits an `algebra_map`, `pi.const_ring_hom` is equal to that\nmap. -/\n@[simp] lemma const_ring_hom_eq_algebra_map : const_ring_hom A R = algebra_map R (A \u2192 R) :=\nrfl\n\n@[simp] lemma const_alg_hom_eq_algebra_of_id : const_alg_hom R A R = algebra.of_id R (A \u2192 R) :=\nrfl\n\nend pi\n\nsection is_scalar_tower\n\nvariables {R : Type*} [comm_semiring R]\nvariables (A : Type*) [semiring A] [algebra R A]\nvariables {M : Type*} [add_comm_monoid M] [module A M] [module R M] [is_scalar_tower R A M]\nvariables {N : Type*} [add_comm_monoid N] [module A N] [module R N] [is_scalar_tower R A N]\n\nlemma algebra_compatible_smul (r : R) (m : M) : r \u2022 m = ((algebra_map R A) r) \u2022 m :=\nby rw [\u2190(one_smul A m), \u2190smul_assoc, algebra.smul_def, mul_one, one_smul]\n\n@[simp] lemma algebra_map_smul (r : R) (m : M) : ((algebra_map R A) r) \u2022 m = r \u2022 m :=\n(algebra_compatible_smul A r m).symm\n\nvariable {A}\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class : smul_comm_class R A M :=\n\u27e8\u03bb r a m, by rw [algebra_compatible_smul A r (a \u2022 m), smul_smul, algebra.commutes, mul_smul,\n  \u2190algebra_compatible_smul]\u27e9\n\n@[priority 100] -- see Note [lower instance priority]\ninstance is_scalar_tower.to_smul_comm_class' : smul_comm_class A R M :=\nsmul_comm_class.symm _ _ _\n\nlemma smul_algebra_smul_comm (r : R) (a : A) (m : M) : a \u2022 r \u2022 m = r \u2022 a \u2022 m :=\nsmul_comm _ _ _\n\nnamespace linear_map\n\ninstance coe_is_scalar_tower : has_coe (M \u2192\u2097[A] N) (M \u2192\u2097[R] N) :=\n\u27e8restrict_scalars R\u27e9\n\nvariables (R) {A M N}\n\n@[simp, norm_cast squash] lemma coe_restrict_scalars_eq_coe (f : M \u2192\u2097[A] N) :\n  (f.restrict_scalars R : M \u2192 N) = f := rfl\n\n@[simp, norm_cast squash] lemma coe_coe_is_scalar_tower (f : M \u2192\u2097[A] N) :\n  ((f : M \u2192\u2097[R] N) : M \u2192 N) = f := rfl\n\n/-- `A`-linearly coerce a `R`-linear map from `M` to `A` to a function, given an algebra `A` over\na commutative semiring `R` and `M` a module over `R`. -/\ndef lto_fun (R : Type u) (M : Type v) (A : Type w)\n  [comm_semiring R] [add_comm_monoid M] [module R M] [comm_ring A] [algebra R A] :\n  (M \u2192\u2097[R] A) \u2192\u2097[A] (M \u2192 A) :=\n{ to_fun := linear_map.to_fun,\n  map_add' := \u03bb f g, rfl,\n  map_smul' := \u03bb c f, rfl }\n\nend linear_map\n\nend is_scalar_tower\n\n/-! TODO: The following lemmas no longer involve `algebra` at all, and could be moved closer\nto `algebra/module/submodule.lean`. Currently this is tricky because `ker`, `range`, `\u22a4`, and `\u22a5`\nare all defined in `linear_algebra/basic.lean`. -/\nsection module\nopen module\n\nvariables (R S M N : Type*) [semiring R] [semiring S] [has_scalar R S]\nvariables [add_comm_monoid M] [module R M] [module S M] [is_scalar_tower R S M]\nvariables [add_comm_monoid N] [module R N] [module S N] [is_scalar_tower R S N]\n\nvariables {S M N}\n\n@[simp]\nlemma linear_map.ker_restrict_scalars (f : M \u2192\u2097[S] N) :\n  (f.restrict_scalars R).ker = f.ker.restrict_scalars R :=\nrfl\n\nend module\n\nnamespace submodule\n\nvariables (R A M : Type*)\nvariables [comm_semiring R] [semiring A] [algebra R A] [add_comm_monoid M]\nvariables [module R M] [module A M] [is_scalar_tower R A M]\n\n/-- If `A` is an `R`-algebra such that the induced morhpsim `R \u2192+* A` is surjective, then the\n`R`-module generated by a set `X` equals the `A`-module generated by `X`. -/\nlemma span_eq_restrict_scalars (X : set M) (hsur : function.surjective (algebra_map R A)) :\n  span R X = restrict_scalars R (span A X) :=\nbegin\n  apply (span_le_restrict_scalars R A X).antisymm (\u03bb m hm, _),\n  refine span_induction hm subset_span (zero_mem _) (\u03bb _ _, add_mem _) (\u03bb a m hm, _),\n  obtain \u27e8r, rfl\u27e9 := hsur a,\n  simpa [algebra_map_smul] using smul_mem _ r hm\nend\n\nend submodule\n\nnamespace alg_hom\n\nvariables {R : Type u} {A : Type v} {B : Type w} {I : Type*}\n\nvariables [comm_semiring R] [semiring A] [semiring B]\nvariables [algebra R A] [algebra R B]\n\n/-- `R`-algebra homomorphism between the function spaces `I \u2192 A` and `I \u2192 B`, induced by an\n`R`-algebra homomorphism `f` between `A` and `B`. -/\n@[simps] protected def comp_left (f : A \u2192\u2090[R] B) (I : Type*) : (I \u2192 A) \u2192\u2090[R] (I \u2192 B) :=\n{ to_fun := \u03bb h, f \u2218 h,\n  commutes' := \u03bb c, by { ext, exact f.commutes' c },\n  .. f.to_ring_hom.comp_left I }\n\nend alg_hom\n\nexample {R A} [comm_semiring R] [semiring A]\n  [module R A] [smul_comm_class R A A] [is_scalar_tower R A A] : algebra R A :=\nalgebra.of_module smul_mul_assoc mul_smul_comm\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebra/algebra/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3665595344897862}}
{"text": "example : 1 + 1 = 2 :=\nby simp * at *\n", "meta": {"author": "zeptometer", "repo": "LearnLean", "sha": "bb84d5dbe521127ba134d4dbf9559b294a80b9f7", "save_path": "github-repos/lean/zeptometer-LearnLean", "path": "github-repos/lean/zeptometer-LearnLean/LearnLean-bb84d5dbe521127ba134d4dbf9559b294a80b9f7/bluejam/topprover/03.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.3665595255075682}}
{"text": "variables (p q r : Prop)\n\nexample (hp : p) : p \u2227 q \u2194 q :=\n  by simp *\n\nexample (hp : p) : p \u2228 q :=\n  by simp *\n\nexample (hp : p) (hq : q) : p \u2227 (q \u2228 r) :=\n  by simp *\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0711.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.36637995475091556}}
{"text": "import algebra.homology.exact\nimport category_theory.abelian.opposite\nimport category_theory.abelian.exact\nimport category_theory.limits.constructions.epi_mono\nimport category_theory.abelian.pseudoelements\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses w v u\n\nnamespace list\n\nvariables {\u03b1 : Type*} (a : \u03b1) (L : list \u03b1) (m n : \u2115)\n\n/-- Returns the sublist of `L` starting at index `m` of length `n`\n(or shorter, if `L` is too short). -/\ndef extract := (L.drop m).take n\n\n@[simp] lemma extract_nil : [].extract m n = ([] : list \u03b1) :=\nby { cases n, refl, cases m, refl, refl }\n\n@[simp] lemma extract_zero_right : L.extract m 0 = [] := rfl\n\n@[simp] lemma extract_cons_succ_left : (a :: L).extract m.succ n = L.extract m n := rfl\n\nend list\n\nexample : [0,1,2,3,4,5,6,7,8,9].extract 4 3 = [4,5,6] := rfl\n\nnamespace category_theory\nvariables (\ud835\udc9e : Type u) [category.{v} \ud835\udc9e]\nvariables [has_zero_morphisms \ud835\udc9e] [has_images \ud835\udc9e] [has_kernels \ud835\udc9e]\nvariables {\ud835\udc9c : Type u} [category.{v} \ud835\udc9c] [abelian \ud835\udc9c]\n\nnamespace exact -- move this\n\nvariables {A B C : \ud835\udc9c} (f : A \u27f6 B) (g : B \u27f6 C)\n\ndef kernel_op_iso : (kernel f.op).unop \u2245 cokernel f :=\n{ hom := (kernel.lift _ (cokernel.\u03c0 f).op begin\n    simp [\u2190 op_comp, limits.cokernel.condition],\n  end).unop \u226b eq_to_hom (opposite.unop_op (cokernel f)),\n  inv := cokernel.desc _ (eq_to_hom (opposite.unop_op B).symm \u226b (kernel.\u03b9 f.op).unop) begin\n    dsimp,\n    rw [category.id_comp, \u2190 f.unop_op, \u2190 unop_comp, f.unop_op, kernel.condition],\n    refl,\n  end,\n  hom_inv_id' := begin\n    dsimp,\n    simp,\n    rw [\u2190 unop_id, \u2190 (cokernel.desc f (kernel.\u03b9 f.op).unop _).unop_op, \u2190 unop_comp],\n    congr' 1,\n    apply limits.equalizer.hom_ext,\n    dsimp,\n    simp [\u2190 op_comp],\n  end,\n  inv_hom_id' := begin\n    apply limits.coequalizer.hom_ext,\n    dsimp,\n    simp [\u2190 unop_comp],\n  end }\n\ndef cokernel_op_iso : (cokernel f.op).unop \u2245 kernel f :=\n{ hom := kernel.lift _ ((cokernel.\u03c0 f.op).unop \u226b eq_to_hom (opposite.unop_op _)) begin\n    simp only [eq_to_hom_refl, category.comp_id],\n    rw [\u2190 f.unop_op, \u2190 unop_comp, f.op.op_unop, cokernel.condition],\n    refl,\n  end,\n  inv := eq_to_hom (opposite.unop_op _).symm \u226b (cokernel.desc _ (kernel.\u03b9 f).op (by simp [\u2190 op_comp])).unop,\n  hom_inv_id' := begin\n    simp only [category.id_comp, eq_to_hom_refl, category.comp_id, \u2190 unop_id, \u2190 unop_comp],\n    rw [\u2190 (kernel.lift f (cokernel.\u03c0 f.op).unop _).unop_op, \u2190 unop_comp],\n    congr' 1,\n    apply limits.coequalizer.hom_ext,\n    dsimp,\n    simp [\u2190 op_comp],\n  end,\n  inv_hom_id' := begin\n    apply limits.equalizer.hom_ext,\n    dsimp,\n    simp [\u2190 unop_comp]\n  end } .\n\n@[simp]\nlemma kernel.\u03b9_op : (kernel.\u03b9 f.op).unop =\n  eq_to_hom (opposite.unop_op _) \u226b cokernel.\u03c0 f \u226b (kernel_op_iso f).inv :=\nbegin\n  dsimp [kernel_op_iso],\n  simp,\nend\n\n@[simp]\nlemma cokernel.\u03c0_op : (cokernel.\u03c0 f.op).unop =\n  (cokernel_op_iso f).hom \u226b kernel.\u03b9 f \u226b eq_to_hom (opposite.unop_op _).symm :=\nbegin\n  dsimp [cokernel_op_iso],\n  simp,\nend\n\nvariables {f g}\n\nlemma op (h : exact f g) : exact g.op f.op :=\nbegin\n  rw abelian.exact_iff,\n  refine \u27e8_, _\u27e9,\n  { simp only [\u2190 op_comp, h.w, op_zero], },\n  apply_fun quiver.hom.unop,\n  swap, { exact quiver.hom.unop_inj },\n  simp only [h, unop_comp, cokernel.\u03c0_op, eq_to_hom_refl, kernel.\u03b9_op, category.id_comp,\n    category.assoc, kernel_comp_cokernel_assoc, zero_comp, comp_zero, unop_zero],\nend\n\nvariables (f g)\n\ndef kernel_unop_iso {C B : \ud835\udc9c\u1d52\u1d56} (f : C \u27f6 B) : opposite.op (kernel f.unop) \u2245 cokernel f :=\n{ hom := (kernel.lift _ (cokernel.\u03c0 f).unop (by simp [\u2190 unop_comp])).op \u226b\n    eq_to_hom (opposite.op_unop (cokernel f)),\n  inv := cokernel.desc _ (eq_to_hom (opposite.op_unop _).symm \u226b (kernel.\u03b9 f.unop).op) begin\n    dsimp,\n    rw [\u2190 f.op_unop, category.id_comp, \u2190 op_comp, f.op_unop, kernel.condition],\n    refl,\n  end,\n  hom_inv_id' := begin\n    dsimp,\n    simp,\n    rw [\u2190 (cokernel.desc f (kernel.\u03b9 f.unop).op _).op_unop, \u2190 op_comp, \u2190 op_id],\n    congr' 1,\n    apply limits.equalizer.hom_ext,\n    dsimp,\n    simp [\u2190 unop_comp],\n  end,\n  inv_hom_id' := begin\n    apply limits.coequalizer.hom_ext,\n    dsimp,\n    simp [\u2190 op_comp],\n  end }\n\ndef cokernel_unop_iso {C B : \ud835\udc9c\u1d52\u1d56} (f : C \u27f6 B) : opposite.op (cokernel f.unop) \u2245 kernel f :=\n{ hom := kernel.lift _ ((cokernel.\u03c0 f.unop).op \u226b eq_to_hom (opposite.op_unop _)) begin\n    dsimp,\n    rw [\u2190 f.op_unop, category.comp_id, \u2190 op_comp, f.op_unop, cokernel.condition],\n    refl,\n  end,\n  inv := eq_to_hom (opposite.op_unop _).symm \u226b\n    (cokernel.desc _ (kernel.\u03b9 f).unop (by simp [\u2190 unop_comp])).op,\n  hom_inv_id' := begin\n    dsimp,\n    rw category.id_comp,\n    rw [\u2190 (kernel.lift f ((cokernel.\u03c0 f.unop).op \u226b \ud835\udfd9 C) _).op_unop, \u2190 op_comp, \u2190 op_id],\n    congr' 1,\n    apply limits.coequalizer.hom_ext,\n    dsimp,\n    simp [\u2190 unop_comp],\n  end,\n  inv_hom_id' := begin\n    apply limits.equalizer.hom_ext,\n    dsimp,\n    simp [\u2190 op_comp]\n  end }\n\n@[simp]\nlemma cokernel.\u03c0_unop {C B : \ud835\udc9c\u1d52\u1d56} (f : C \u27f6 B) : (cokernel.\u03c0 f.unop).op =\n  (cokernel_unop_iso f).hom \u226b kernel.\u03b9 f \u226b eq_to_hom (opposite.op_unop _).symm :=\nbegin\n  dsimp [cokernel_unop_iso],\n  simp,\nend\n\n@[simp]\nlemma kernel.\u03b9_unop {C B : \ud835\udc9c\u1d52\u1d56} (f : C \u27f6 B) : (kernel.\u03b9 f.unop).op =\n  eq_to_hom (opposite.op_unop _) \u226b cokernel.\u03c0 f \u226b (kernel_unop_iso f).inv :=\nbegin\n  dsimp [kernel_unop_iso],\n  simp,\nend\n\nlemma unop {C B A : \ud835\udc9c\u1d52\u1d56} {g : C \u27f6 B} {f : B \u27f6 A} (h : exact g f) : exact f.unop g.unop :=\nbegin\n  rw abelian.exact_iff,\n  refine \u27e8by simp only [\u2190 unop_comp, h.w, unop_zero], _\u27e9,\n  apply_fun quiver.hom.op,\n  swap, { exact quiver.hom.op_inj },\n  simp [h],\nend\n\nend exact\n\n/-- A sequence `[f, g, ...]` of morphisms is exact if the pair `(f,g)` is exact,\nand the sequence `[g, ...]` is exact.\n\nRecall that the pair `(f,g)` is exact if `f \u226b g = 0`\nand the natural map from the image of `f` to the kernel of `g` is an epimorphism\n(equivalently, in abelian categories: isomorphism). -/\ninductive exact_seq : list (arrow \ud835\udc9e) \u2192 Prop\n| nil    : exact_seq []\n| single : \u2200 f, exact_seq [f]\n| cons   : \u2200 {A B C : \ud835\udc9e} (f : A \u27f6 B) (g : B \u27f6 C) (hfg : exact f g) (L) (hgL : exact_seq (g :: L)),\n              exact_seq (f :: g :: L)\n\nvariable {\ud835\udc9e}\n\nlemma exact_iff_exact_seq {A B C : \ud835\udc9e} (f : A \u27f6 B) (g : B \u27f6 C) :\n  exact f g \u2194 exact_seq \ud835\udc9e [f, g] :=\nbegin\n  split,\n  { intro h, exact exact_seq.cons f g h _ (exact_seq.single _), },\n  { rintro (_ | _ | \u27e8A, B, C, f, g, hfg, _, _ | _ | _\u27e9), exact hfg, }\nend\n\nnamespace exact_seq\n\nlemma extract : \u2200 {L : list (arrow \ud835\udc9e)} (h : exact_seq \ud835\udc9e L) (m n : \u2115),\n  exact_seq \ud835\udc9e (L.extract m n)\n| L (nil)               m     n     := by { rw list.extract_nil, exact nil }\n| L (single f)          m     0     := nil\n| L (single f)          0     (n+1) := by { cases n; exact single f }\n| L (single f)          (m+1) (n+1) := by { cases m; exact nil }\n| _ (cons f g hfg L hL) (m+1) n     := extract hL m n\n| _ (cons f g hfg L hL) 0     0     := nil\n| _ (cons f g hfg L hL) 0     1     := single f\n| _ (cons f g hfg L hL) 0     (n+2) := cons f g hfg (L.take n) (extract hL 0 (n+1))\n\ninductive arrow_congr : \u03a0 (L L' : list (arrow \ud835\udc9e)), Prop\n| nil  : arrow_congr [] []\n| cons : \u2200 {A B : \ud835\udc9e} {f f' : A \u27f6 B} {L L' : list (arrow \ud835\udc9e)} (h : f = f') (H : arrow_congr L L'),\n         arrow_congr (f :: L) (f' :: L')\n\nlemma congr : \u2200 {L L' : list (arrow \ud835\udc9e)}, exact_seq \ud835\udc9e L \u2192 arrow_congr L L' \u2192 exact_seq \ud835\udc9e L'\n| _ _ h arrow_congr.nil                                 := exact_seq.nil\n| _ _ h (arrow_congr.cons h\u2081 arrow_congr.nil)           := exact_seq.single _\n| _ _ h (arrow_congr.cons h\u2081 ((arrow_congr.cons h\u2082 H))) :=\nbegin\n  substs h\u2081 h\u2082,\n  rcases h with _ | _ | \u27e8A, B, C, f, g, hfg, _, hL\u27e9,\n  refine exact_seq.cons _ _ hfg _ (congr hL (arrow_congr.cons rfl H)),\nend\n\nlemma append : \u2200 {L\u2081 L\u2082 L\u2083 : list (arrow \ud835\udc9e)}\n  (h\u2081\u2082 : exact_seq \ud835\udc9e (L\u2081 ++ L\u2082)) (h\u2082\u2083 : exact_seq \ud835\udc9e (L\u2082 ++ L\u2083)) (h\u2082 : L\u2082 \u2260 []),\n  exact_seq \ud835\udc9e (L\u2081 ++ L\u2082 ++ L\u2083)\n| L\u2081         []      L\u2083 h\u2081\u2082                 h\u2082\u2083 h := (h rfl).elim\n| []         L\u2082      L\u2083 h\u2081\u2082                 h\u2082\u2083 h := by rwa list.nil_append\n| (_::[])    (_::L\u2082) L\u2083 (cons f g hfg L hL) h\u2082\u2083 h := cons f g hfg _ h\u2082\u2083\n| (_::_::L\u2081) L\u2082      L\u2083 (cons f g hfg L hL) h\u2082\u2083 h :=\nsuffices exact_seq \ud835\udc9e ([f] ++ ([g] ++ L\u2081 ++ L\u2082) ++ L\u2083), { simpa only [list.append_assoc] },\ncons _ _ hfg _ $\nsuffices exact_seq \ud835\udc9e ((g :: L\u2081) ++ L\u2082 ++ L\u2083), { simpa only [list.append_assoc] },\nappend (by simpa only using hL) h\u2082\u2083 h\n\nend exact_seq\n\nnamespace arrow\n\nopen _root_.opposite\n\nvariables {C : Type*} [category C]\n\n@[simps]\ndef op (f : arrow C) : arrow C\u1d52\u1d56 :=\n{ left := op f.right,\n  right := op f.left,\n  hom := f.hom.op }\n\n@[simps]\ndef unop (f : arrow C\u1d52\u1d56) : arrow C :=\n{ left := unop f.right,\n  right := unop f.left,\n  hom := f.hom.unop }\n\n@[simp] lemma op_unop (f : arrow C)   : f.op.unop = f := by { cases f, dsimp [op, unop], refl }\n@[simp] lemma unop_op (f : arrow C\u1d52\u1d56) : f.unop.op = f := by { cases f, dsimp [op, unop], refl }\n\n@[simp] lemma op_comp_unop : (op \u2218 unop : arrow C\u1d52\u1d56 \u2192 arrow C\u1d52\u1d56) = id := by { ext, exact unop_op _ }\n@[simp] lemma unop_comp_op : (unop \u2218 op : arrow C   \u2192 arrow C  ) = id := by { ext, exact op_unop _ }\n\nend arrow\n\nnamespace exact_seq\n\nlemma op : \u2200 {L : list (arrow \ud835\udc9c)}, exact_seq \ud835\udc9c L \u2192 exact_seq \ud835\udc9c\u1d52\u1d56 (L.reverse.map arrow.op)\n| _ nil                 := nil\n| _ (single f)          := single f.op\n| _ (cons f g hfg L hL) :=\nbegin\n  have := op hL,\n  simp only [list.reverse_cons, list.map_append] at this \u22a2,\n  refine this.append _ (list.cons_ne_nil _ _),\n  exact cons _ _ hfg.op _ (single _),\nend\n\nlemma unop : \u2200 {L : list (arrow \ud835\udc9c\u1d52\u1d56)}, exact_seq \ud835\udc9c\u1d52\u1d56 L \u2192 exact_seq \ud835\udc9c (L.reverse.map arrow.unop)\n| _ nil                 := nil\n| _ (single f)          := single f.unop\n| _ (cons f g hfg L hL) :=\nbegin\n  have := unop hL,\n  simp only [list.reverse_cons, list.map_append] at this \u22a2,\n  refine this.append _ (list.cons_ne_nil _ _),\n  exact cons _ _ hfg.unop _ (single _),\nend\n\nlemma of_op {L : list (arrow \ud835\udc9c)} (h : exact_seq \ud835\udc9c\u1d52\u1d56 (L.reverse.map arrow.op)) : exact_seq \ud835\udc9c L :=\nby simpa only [list.map_reverse, list.reverse_reverse, list.map_map,\n  arrow.unop_comp_op, list.map_id] using h.unop\n\nlemma of_unop {L : list (arrow \ud835\udc9c\u1d52\u1d56)} (h : exact_seq \ud835\udc9c (L.reverse.map arrow.unop)) :\n  exact_seq \ud835\udc9c\u1d52\u1d56 L :=\nby simpa only [list.map_reverse, list.reverse_reverse, list.map_map,\n  arrow.op_comp_unop, list.map_id] using h.op\n\nend exact_seq\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/exact_seq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056295505783, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3661656115365653}}
{"text": "import localization\n\nuniverse u\n\n--local infix ^ := monoid.pow\n\nnamespace localization_away\n\nvariables (R : Type u) [comm_ring R] (f : R)\n\ndef r : R \u00d7 \u2115 \u2192 R \u00d7 \u2115 \u2192 Prop :=\n\u03bb x y, \u2203 t : \u2115, (f ^ x.2 * y.1 - f ^ y.2 * x.1) * f ^ t = 0\n\nlocal infix \u2248 := r R f\n\ntheorem refl : \u2200 (x : R \u00d7 \u2115), x \u2248 x :=\n\u03bb \u27e8r\u2081, s\u2081\u27e9, \u27e80, by simp\u27e9\n\ntheorem symm : \u2200 (x y : R \u00d7 \u2115), x \u2248 y \u2192 y \u2248 x :=\n\u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8t, ht\u27e9, \u27e8t, calc\n        (f ^ s\u2082 * r\u2081 - f ^ s\u2081 * r\u2082) * f ^ t\n      = -((f ^ s\u2081 * r\u2082 - f ^ s\u2082 * r\u2081) * f ^ t) : by simp [add_mul]\n  ... = 0 : by rw ht; simp\u27e9\n\ntheorem trans : \u2200 (x y z : R \u00d7 \u2115), x \u2248 y \u2192 y \u2248 z \u2192 x \u2248 z :=\n\u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8r\u2083, s\u2083\u27e9 \u27e8t, ht\u27e9 \u27e8t', ht'\u27e9,\n\u27e8s\u2082 + t' + t, calc\n         (f ^ s\u2081 * r\u2083 - f ^ s\u2083 * r\u2081) * f ^ (s\u2082 + t' + t)\n       = f ^ t' * f ^ s\u2083 * ((f ^ s\u2081 * r\u2082 - f ^ s\u2082 * r\u2081) * f ^ t) + f ^ t * f ^ s\u2081 * ((f ^ s\u2082 * r\u2083 - f ^ s\u2083 * r\u2082) * f ^ t') :\n           by simp [pow_add, mul_left_comm, mul_add, mul_comm]\n   ... = 0 : by rw [ht, ht']; simp\u27e9\n\ninstance : setoid (R \u00d7 \u2115) :=\n\u27e8r R f, refl R f, symm R f, trans R f\u27e9\n\ndef loc := quotient $ localization_away.setoid R f\n\n@[reducible] def mk : R \u2192 \u2115 \u2192 loc R f :=\n\u03bb a r, @quotient.mk _ (localization_away.setoid R f) (a, r)\n\nprivate def add_aux : R \u00d7 \u2115 \u2192 R \u00d7 \u2115 \u2192 loc R f :=\n\u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9, mk R f (f ^ s\u2081 * r\u2082 + f ^ s\u2082 * r\u2081) (s\u2081 + s\u2082)\n\ninstance : has_add (loc R f) :=\n\u27e8@quotient.lift\u2082 _ _ _\n (localization_away.setoid R f) (localization_away.setoid R f)\n (add_aux R f) $\n \u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8r\u2083, s\u2083\u27e9 \u27e8r\u2084, s\u2084\u27e9 \u27e8t\u2085, ht\u2085\u27e9 \u27e8t\u2086, ht\u2086\u27e9,\n @quotient.sound _ (localization_away.setoid R f) _ _\n   \u27e8t\u2086 + t\u2085, by dsimp; from calc\n         (f ^ (s\u2081 + s\u2082) * (f ^ s\u2083 * r\u2084 + f ^ s\u2084 * r\u2083) - f ^ (s\u2083 + s\u2084) * (f ^ s\u2081 * r\u2082 + f ^ s\u2082 * r\u2081)) * (f ^ (t\u2086 + t\u2085))\n       = f ^ s\u2081 * f ^ s\u2083 * ((f ^ s\u2082 * r\u2084 - f ^ s\u2084 * r\u2082) * f ^ t\u2086) * f ^ t\u2085 + f ^ s\u2082 * f ^ s\u2084 * ((f ^ s\u2081 * r\u2083 - f ^ s\u2083 * r\u2081) * f ^ t\u2085) * f ^ t\u2086 : by rw [pow_add, pow_add, pow_add]; ring\n   ... = 0 : by rw [ht\u2086, ht\u2085]; simp\u27e9\u27e9\n\nprivate def neg_aux : R \u00d7 \u2115 \u2192 loc R f :=\n\u03bb \u27e8r, s\u27e9, mk R f (-r) s\n\ninstance : has_neg (loc R f) :=\n\u27e8@quotient.lift _ _ (localization_away.setoid R f) (neg_aux R f) $\n \u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8t, ht\u27e9,\n @quotient.sound _ (localization_away.setoid R f) _ _\n \u27e8t, by dsimp; from calc\n         (f ^ s\u2081 * -r\u2082 - f ^ s\u2082 * -r\u2081) * f ^ t\n       = -((f ^ s\u2081 * r\u2082 - f ^ s\u2082 * r\u2081) * f ^ t) : by ring\n   ... = 0 : by rw ht; simp\u27e9\u27e9\n\nprivate def mul_aux : R \u00d7 \u2115 \u2192 R \u00d7 \u2115 \u2192 loc R f :=\n\u03bb x y, mk R f (x.1 * y.1) (x.2 + y.2)\n\ninstance : has_mul (loc R f) :=\n\u27e8@quotient.lift\u2082 _ _ _\n (localization_away.setoid R f) (localization_away.setoid R f)\n (mul_aux R f) $\n \u03bb \u27e8r\u2081, s\u2081\u27e9 \u27e8r\u2082, s\u2082\u27e9 \u27e8r\u2083, s\u2083\u27e9 \u27e8r\u2084, s\u2084\u27e9 \u27e8t\u2085, ht\u2085\u27e9 \u27e8t\u2086, ht\u2086\u27e9,\n @quotient.sound _ (localization_away.setoid R f) _ _\n \u27e8t\u2086 + t\u2085, by dsimp; from calc\n         (f ^ (s\u2081 + s\u2082) * (r\u2083 * r\u2084) - f ^ (s\u2083 + s\u2084) * (r\u2081 * r\u2082)) * f ^ (t\u2086 + t\u2085)\n       = f ^ t\u2086 * ((f ^ s\u2081 * r\u2083 - f ^ s\u2083 * r\u2081) * f ^ t\u2085) * r\u2082 * f ^ s\u2084 + f ^ t\u2085 * ((f ^ s\u2082 * r\u2084 - f ^ s\u2084 * r\u2082) * f ^ t\u2086) * r\u2083 * f ^ s\u2081 :\n           by rw [pow_add, pow_add, pow_add]; simp [mul_left_comm, mul_add, mul_comm,mul_assoc]\n   ... = 0 : by rw [ht\u2085, ht\u2086]; simp\u27e9\u27e9\n\ninstance : comm_ring (loc R f) :=\nby letI := localization_away.setoid R f; refine\n{ add            := has_add.add,\n  add_assoc      := \u03bb m n k, quotient.induction_on\u2083 m n k _,\n  zero           := \u27e6\u27e80, 0\u27e9\u27e7,\n  zero_add       := quotient.ind _,\n  add_zero       := quotient.ind _,\n  neg            := has_neg.neg,\n  add_left_neg   := quotient.ind _,\n  add_comm       := quotient.ind\u2082 _,\n  mul            := has_mul.mul,\n  mul_assoc      := \u03bb m n k, quotient.induction_on\u2083 m n k _,\n  one            := \u27e6\u27e81, 0\u27e9\u27e7,\n  one_mul        := quotient.ind _,\n  mul_one        := quotient.ind _,\n  left_distrib   := \u03bb m n k, quotient.induction_on\u2083 m n k _,\n  right_distrib  := \u03bb m n k, quotient.induction_on\u2083 m n k _,\n  mul_comm       := quotient.ind\u2082 _ };\n{ intros,\n  try {cases a with r\u2081 s\u2081},\n  try {cases b with r\u2082 s\u2082},\n  try {cases c with r\u2083 s\u2083},\n  apply quotient.sound,\n  existsi 0,\n  simp [pow_add, mul_left_comm, mul_add, mul_comm],\n  try { ring } }\n\ndef of_comm_ring : R \u2192 loc R f :=\n\u03bb r, mk R f r 0\n\ninstance : is_ring_hom (of_comm_ring R f) :=\n{ map_add := \u03bb x y, by apply quotient.sound; simp,\n  map_mul := \u03bb x y, by apply quotient.sound; simp,\n  map_one := rfl }\n\ndef to_localization : loc R f \u2192 localization.away f :=\n@quotient.lift _ _ (localization_away.setoid R f)\n  (\u03bb x, \u27e6(\u27e8x.1, f ^ x.2, x.2, rfl\u27e9 : R \u00d7 powers f)\u27e7)\n  (\u03bb x y \u27e8t, ht\u27e9, by apply quotient.sound; existsi f^t;\n    existsi (\u27e8t, rfl\u27e9 : f ^ t \u2208 powers f); exact ht)\n\nvariables {R} (g : R)\n\ndef more_left : loc R f \u2192 loc R (f * g) :=\n@quotient.lift _ _ (localization_away.setoid R f)\n  (\u03bb x, mk R (f * g) (g ^ x.2 * x.1) x.2)\n  (\u03bb x y \u27e8t, ht\u27e9, by apply quotient.sound; existsi t; clear _x _fun_match;\n     cases x with r\u2081 s\u2081; cases y with r\u2082 s\u2082; from calc\n          ((f * g) ^ s\u2081 * (g ^ s\u2082 * r\u2082) + -((f * g) ^ s\u2082 * (g ^ s\u2081 * r\u2081))) * (f * g) ^ t\n        = g ^ s\u2081 * g ^ s\u2082 * ((f ^ s\u2081 * r\u2082 - f ^ s\u2082 * r\u2081) * f ^ t) * g ^ t : by simp [mul_pow]; ring\n    ... = 0 : by rw ht; simp )\n\ndef more_right : loc R g \u2192 loc R (f * g) :=\n@quotient.lift _ _ (localization_away.setoid R g)\n  (\u03bb x, mk R (f * g) (f ^ x.2 * x.1) x.2)\n  (\u03bb x y \u27e8t, ht\u27e9, by apply quotient.sound; existsi t; clear _x _fun_match;\n     cases x with r\u2081 s\u2081; cases y with r\u2082 s\u2082; from calc\n          ((f * g) ^ s\u2081 * (f ^ s\u2082 * r\u2082) + -((f * g) ^ s\u2082 * (f ^ s\u2081 * r\u2081))) * (f * g) ^ t\n        = f ^ s\u2081 * f ^ s\u2082 * ((g ^ s\u2081 * r\u2082 - g ^ s\u2082 * r\u2081) * g ^ t) * f ^ t : by simp [mul_pow]; ring\n    ... = 0 : by rw ht; simp )\n\nend localization_away", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/src/localization_away.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3661656047556074}}
{"text": "import tactic\nimport row_bump\nimport inverse_row_bump\n\n/-\n\nVery preliminary -- see comments in inverse_row_bump.lean\n\n-/\n\nsection irb_rbwf\n\n-- irb_inductive' \u2218 rbwf = id (start and end in row i)\n\ndef ssyt.rbs_cert.del_rbwf {\u03bc : young_diagram} {T : ssyt \u03bc} (h : T.rbs_cert) :\n  ssyt \u03bc := \n(h.rbwf.2.del h.rbwf.1.to_inner).copy (young_diagram.inner_outer _)\n\ndef ssyt.irb_with_stopping :\n  \u03a0 {\u03bc : young_diagram} (T : ssyt \u03bc) (c : \u03bc.inner_corner)\n  {end_after : \u2115} {i : \u2115} (hi : c.i = end_after + i),\n  ssyt c.del \u00d7 \u2115\n| \u03bc T c end_after 0 hi := \u27e8T.del c, T c.i c.j\u27e9\n| \u03bc T c end_after (nat.succ n) hi := begin\n  have key : c.i \u2260 0 := by convert nat.succ_ne_zero (end_after + n),\n  have key2 : c.i.pred = end_after + n := \n    by rw [hi, nat.add_succ, nat.pred_succ],\n  set out := (T.irbs_cert_of_inner_corner c key).irb_inductive' key2,\n  exact \u27e8out.2.1.irbs, out.2.1.out\u27e9,\nend\n\ndef ssyt.irb_with_stopping'\n  {\u03bc : young_diagram} {c : \u03bc.outer_corner} (T : ssyt c.add)\n  {end_after : \u2115} {i : \u2115} (hi : c.i = end_after + i) :\n  ssyt \u03bc \u00d7 \u2115 :=\nlet out := T.irb_with_stopping c.to_inner hi in\n\u27e8ssyt.copy out.1 (young_diagram.inner_outer c), out.2\u27e9\n\nlemma ssyt.irb_rbwf :\n  \u03a0 {\u03bc : young_diagram} (T : ssyt \u03bc) (h : T.rbs_cert)\n  (hi : h.rbwf.1.i = h.i + (h.rbwf.1.i - h.i)),\n    -- begin\n    -- by rw [\u2190 nat.add_sub_assoc h.i_le_rbwf_corner,\n    --       nat.add_sub_cancel_left]\n    -- end),\n  h.rbwf.2.irb_with_stopping' --h.i (h.rbwf.1.i - h.i)\n    hi =\n  -- (by rw [\u2190 nat.add_sub_assoc h.i_le_rbwf_corner,\n  --         nat.add_sub_cancel_left] : h.rbwf.1.i = h.i + (h.rbwf.1.i - h.i)) =\n  (T, h.val)\n| \u03bc T h hi :=\ndite ((h.i, h.j) \u2208 \u03bc)\n(\u03bb cell, begin sorry\nend)\n(\u03bb not_cell, begin sorry\nend)\n\nend irb_rbwf", "meta": {"author": "jakelev", "repo": "lean-rsk", "sha": "dbd97f8fe9fc2ba13d080d37e298ae87d03ff541", "save_path": "github-repos/lean/jakelev-lean-rsk", "path": "github-repos/lean/jakelev-lean-rsk/lean-rsk-dbd97f8fe9fc2ba13d080d37e298ae87d03ff541/src/bump_inverse_lemmas.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3661656047556074}}
{"text": "example (a b : Nat) (h : Nat.le a b) : Nat.le a (b + 1) := by\n  induction h with\n  | step x? hStep ih => admit\n  | refl => admit\n\nexample (a b : Nat) (h : Nat.le a b) : Nat.le a (b + 1) := by\n  induction h with\n  | step hStep ih => trace_state; admit\n  | refl => admit\n\nexample (a b : Nat) (h : Nat.le a b) : Nat.le a (b + 1) := by\n  induction h with\n  | @step x hStep ih => trace_state; admit\n  | refl => admit\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/690.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3661655979746493}}
{"text": "import Std.Tactic.ByCases\n\nexample : True := by\n  if 1 + 1 = 2 then _ else ?_\n  case pos => trivial\n  fail_if_success case neg => contradiction\n  \u00b7 contradiction\n\nexample (p : Prop) : True := by\n  if p then ?foo else trivial\n  case foo => trivial\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/test/by_cases.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.36616559797464926}}
{"text": "import to_mathlib_maybe.Hilbert_space\nimport to_qShannon_theory_maybe.state\n\nvariables\n{\u210b : Type} [complex_hilbert_space \u210b]\n{\u210b\u2081 : Type} [complex_hilbert_space \u210b\u2081]\n{\u210b\u2082 : Type} [complex_hilbert_space \u210b\u2082]\n{U : module.End \u2102 \u210b} [unitary U]\n{\u03c1 : module.End \u2102 \u210b} [quantum_state \u03c1]\n\n/--\nA quantum channel is a linear map between linear operators \nthat satisifies certain axioms.\n-/\nclass quantum_channel (\ud835\udca9 : (module.End \u2102 \u210b\u2081) \u2192\u2097[\u2102] (module.End \u2102 \u210b\u2082)) := \n(quantum_channel_ness : 1=1)\n\nvariables\n{\ud835\udca9 : (module.End \u2102 \u210b\u2081) \u2192\u2097[\u2102] (module.End \u2102 \u210b\u2082)} [quantum_channel \ud835\udca9]\n{\u03c3 \u03c3' : (\u210b\u2081 \u2192\u2097[\u2102] \u210b\u2081)} [quantum_state \u03c3] [quantum_state \u03c3']\n\nexample : \ud835\udca9(\u03c3 + \u03c3') = \ud835\udca9(\u03c3) + \ud835\udca9(\u03c3') := \nbegin\n    rw linear_map.map_add,\nend\n\n-- d\u00d7d matrix with 1 at (i j) and 0 otherwise\ndef E (i j d : \u2115) : module.End \u2102 \u210b := sorry\n-- cyclic shift operators\ndef shift (k n : \u2115) := \u2211 i \u2208 finset.range n, (E i (i+k)%n n)\n-- shift by 1 on S1, ..., Sn\ndef cycle_s (d n : \u2115) := (Id d) \u2297 (shift 1 n) \u2297 (Id n)\n-- shift by one on A\ndef cycle_ancilla (d n : \u2115) := 1^\u2297n \u2297 (shift 1 n)\n-- Dephasing channel sending \u03c7 to its diagonal in the eigenbasis of \u03c1'\ndef dephasing_channel (\u03c1') := \u03bb \u03c1, \u2211 v \u2208 eigenvectors \u03c1', inner v (\u03c1 v)\n-- Unitary and ancilla on Naimark's dilated system corresponding to an arbitrary quantum channel\ndef naimark_unitary (C : module.End \u2102 \u210b \u2192\u2097[\u2102] module.End \u2102 \u210b) [quantum_channel C] : module.End \u2102 \u210b := sorry\ndef naimark_ancilla (C : module.End \u2102 \u210b \u2192\u2097[\u2102] module.End \u2102 \u210b) [quantum_channel C] : module.End \u2102 \u210b := sorry", "meta": {"author": "BassemSafieldeen", "repo": "Entropy_and_reversible_catalysis", "sha": "5dd6ee062f61e26bbcf254477e3e24aa3fc489af", "save_path": "github-repos/lean/BassemSafieldeen-Entropy_and_reversible_catalysis", "path": "github-repos/lean/BassemSafieldeen-Entropy_and_reversible_catalysis/Entropy_and_reversible_catalysis-5dd6ee062f61e26bbcf254477e3e24aa3fc489af/src/to_qShannon_theory_maybe/channel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7745833737577159, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.36613276204018125}}
{"text": "import data.cpi.process data.cpi.transition\nimport data.multiset2\n\nnamespace cpi\n/-- Given two equivalent species, there is some isomorphism between species of\n    the same  kind and label, such that isomorphic transitions have equivalent\n    productions. -/\n@[nolint has_inhabited_instance]\ndef has_iso {\u210d : Type} {\u03c9 \u0393 : context} [\u2200 \u0393, setoid (species \u210d \u03c9 \u0393)] [\u2200 \u0393 b y, setoid (concretion \u210d \u03c9 \u0393 b y)]\n    {A B : species \u210d \u03c9 \u0393} (\u2113 : lookup \u210d \u03c9 \u0393)\n  : A \u2248 B \u2192 Type\n| c := \u2200 k (\u03b1 : label \u210d \u0393 k)\n       , \u03a3' (iso : (\u03a3 E, A [\u2113, \u03b1]\u27f6 E) \u2243 (\u03a3 E, B [\u2113, \u03b1]\u27f6 E))\n         , \u2200 E (t : A [\u2113, \u03b1]\u27f6 E), E \u2248 (iso.to_fun \u27e8 E, t \u27e9).1\n\n/-- An equivalence class over species and concretions, which allows for a notion of \"prime\n    decomposition\". -/\nclass cpi_equiv (\u210d : Type) (\u03c9 : context) :=\n  [species_equiv {} : \u2200 \u0393, setoid (species \u210d \u03c9 \u0393)]\n  [concretion_equiv {} : \u2200 \u0393 b y, setoid (concretion \u210d \u03c9 \u0393 b y)]\n  [decide_species {} : \u2200 \u0393, decidable_rel (species_equiv \u0393).r]\n  [decide_concretion {} : \u2200 \u0393 b y, decidable_rel (concretion_equiv \u0393 b y).r]\n\n  /- Decompose a species into primes. -/\n  (prime_decompose {\u0393} : species \u210d \u03c9 \u0393 \u2192 multiset (prime_species \u210d \u03c9 \u0393))\n\n  ( prime_decompose_equiv {\u0393} {A B : species \u210d \u03c9 \u0393}\n  : A \u2248 B\n  \u2192 multiset.map quotient.mk (prime_decompose A)\n  = multiset.map quotient.mk (prime_decompose B) )\n\n  /- Prime decomposition of nil, returns an empty set. -/\n  (prime_decompose_nil {\u0393} : prime_decompose (@species.nil \u210d \u03c9 \u0393) = 0)\n\n  ( prime_decompose_parallel {\u0393} (A B : species \u210d \u03c9 \u0393)\n  : prime_decompose (A |\u209b B) = prime_decompose A + prime_decompose B )\n\n  ( prime_decompose_prime {\u0393} (A : prime_species \u210d \u03c9 \u0393)\n  : prime_decompose A.val = [ A ] )\n\n  ( pseudo_apply {\u0393} {a b : \u2115}\n  : concretion' \u210d \u03c9 \u0393 a b \u2192 concretion' \u210d \u03c9 \u0393 b a\n  \u2192 species' \u210d \u03c9 \u0393 )\n\nnamespace cpi_equiv\n  instance to_species (\u210d : Type) (\u03c9 \u0393 : context) [r : cpi_equiv \u210d \u03c9]\n    : setoid (species \u210d \u03c9 \u0393)\n    := species_equiv \u0393\n\n  instance to_concretion (\u210d : Type) (\u03c9 \u0393 : context) (b y : \u2115) [r : cpi_equiv \u210d \u03c9]\n    : setoid (concretion \u210d \u03c9 \u0393 b y)\n    := concretion_equiv \u0393 b y\n\n  variables {\u210d : Type} {\u03c9 : context} [cpi_equiv \u210d \u03c9]\n\n  /-- `prime_decompose` lifted to quotients. -/\n  def prime_decompose' {\u0393} :\n    species' \u210d \u03c9 \u0393 \u2192 multiset (prime_species' \u210d \u03c9 \u0393)\n  | A := quot.lift_on A (multiset.map quotient.mk \u2218 prime_decompose)\n    (\u03bb A B eq, prime_decompose_equiv eq)\n\n  lemma prime_decompose_nil' {\u0393} : prime_decompose' \u27e6 @species.nil \u210d \u03c9 \u0393 \u27e7 = 0 := begin\n      show multiset.map quotient.mk (prime_decompose nil) = 0,\n      rw [prime_decompose_nil, multiset.map_zero],\n    end\n\n  lemma prime_decompose_parallel' {\u0393} (A B : species \u210d \u03c9 \u0393)\n    : prime_decompose' \u27e6 A |\u209b B \u27e7 = prime_decompose' \u27e6 A \u27e7 + prime_decompose' \u27e6 B \u27e7\n    := begin\n      show multiset.map quotient.mk (prime_decompose (A |\u209b B))\n         = multiset.map quotient.mk (prime_decompose A)\n         + multiset.map quotient.mk (prime_decompose B),\n      rw [prime_decompose_parallel, multiset.map_add],\n    end\n\n  lemma prime_decompose_prime' {\u0393} (A : prime_species' \u210d \u03c9 \u0393)\n    : prime_decompose' (prime_species.unwrap A) = [ A ]\n    := quot.rec_on A (\u03bb A, begin\n      show multiset.map quotient.mk (prime_decompose A.val) = [ \u27e6 A \u27e7 ],\n      simp only [prime_decompose_prime, multiset.coe_map, list.map_nil, multiset.coe_eq_coe, list.map],\n    end) (\u03bb a b _, rfl)\nend cpi_equiv\n\n/-- Additional properties that we need for some lemmas, but nothing else./ -/\nclass cpi_equiv_prop (\u210d : Type) (\u03c9 : context) extends cpi_equiv \u210d \u03c9 :=\n  /- Show our equivalence relation holds over transitions. Namely the transition\n     sets are isomorphic, and have equivalent productions. -/\n  ( transition_iso {\u0393} (\u2113 : lookup \u210d \u03c9 \u0393) {A B : species \u210d \u03c9 \u0393} (eq : A \u2248 B)\n  : nonempty (has_iso \u2113 eq) )\n\n  ( pseudo_apply_symm {\u0393} {a b : \u2115} (F : concretion' \u210d \u03c9 \u0393 a b) (G : concretion' \u210d \u03c9 \u0393 b a)\n    : pseudo_apply F G = pseudo_apply G F )\n\n/-- Build an equivalent transition in the forward direction. -/\ndef cpi_equiv.transition_from_fwd {\u210d : Type} {\u03c9 \u0393 : context} [r : cpi_equiv \u210d \u03c9] {A B : species \u210d \u03c9 \u0393}\n    {\u2113 : lookup \u210d \u03c9 \u0393} {eq : A \u2248 B}\n  : has_iso \u2113 eq\n  \u2192 transition.transition_from \u2113 A \u2192 transition.transition_from \u2113 B\n| iso \u27e8 k, \u03b1, p \u27e9 := \u27e8 k, \u03b1, (iso k \u03b1).1.to_fun p \u27e9\n\n/-- Build an equivalent transition in the reverse direction. -/\ndef cpi_equiv.transition_from_inv {\u210d : Type} {\u03c9 \u0393 : context} [r : cpi_equiv \u210d \u03c9] {A B : species \u210d \u03c9 \u0393}\n    {\u2113 : lookup \u210d \u03c9 \u0393} {eq : A \u2248 B}\n  : has_iso \u2113 eq\n  \u2192 transition.transition_from \u2113 B \u2192 transition.transition_from \u2113 A\n| iso \u27e8 k, \u03b1, p \u27e9 := \u27e8 k, \u03b1, (iso k \u03b1).1.inv_fun p \u27e9\n\n/-- cpi_equiv.transition_iso, lifted to transition_from -/\ndef cpi_equiv_prop.transition_from_iso {\u210d : Type} {\u03c9 \u0393 : context} [r : cpi_equiv \u210d \u03c9] {A B : species \u210d \u03c9 \u0393}\n    {\u2113 : lookup \u210d \u03c9 \u0393} {eq : A \u2248 B}\n  : has_iso \u2113 eq\n  \u2192 transition.transition_from \u2113 A \u2243 transition.transition_from \u2113 B\n| iso :=\n  { to_fun    := cpi_equiv.transition_from_fwd iso,\n    inv_fun   := cpi_equiv.transition_from_inv iso,\n    left_inv  := \u03bb \u27e8 k, \u03b1, p \u27e9, begin\n      simp only [cpi_equiv.transition_from_fwd, cpi_equiv.transition_from_inv],\n      rw (iso k \u03b1).1.left_inv p\n    end,\n    right_inv := \u03bb \u27e8 k, \u03b1, p \u27e9, begin\n      simp only [cpi_equiv.transition_from_fwd, cpi_equiv.transition_from_inv],\n      rw (iso k \u03b1).1.right_inv p\n    end }\n\ninstance species'.decidable_eq {\u210d \u03c9 \u0393} [r : cpi_equiv \u210d \u03c9] : decidable_eq (species' \u210d \u03c9 \u0393)\n  := @quotient.decidable_eq _ _ (cpi_equiv.decide_species \u0393)\n\ninstance concretion'.decidable_eq {\u210d \u03c9 \u0393 b y} [r : cpi_equiv \u210d \u03c9] : decidable_eq (concretion' \u210d \u03c9 \u0393 b y)\n  := @quotient.decidable_eq _ _ (cpi_equiv.decide_concretion \u0393 b y)\n\n/-- Determine if two prime species are equal. Effectively a decision procedure\n    structural congruence. -/\ninstance prime'.decidable_eq {\u210d \u03c9 \u0393} [r : cpi_equiv \u210d \u03c9] : decidable_eq (prime_species' \u210d \u03c9 \u0393)\n| A B := quotient.rec_on_subsingleton\u2082 A B\n  (\u03bb \u27e8 a, _ \u27e9 \u27e8 b, _ \u27e9,\n    match cpi_equiv.decide_species \u0393 a b with\n    | is_true h := is_true (quot.sound h)\n    | is_false h := is_false (\u03bb h', absurd (quotient.exact h') h)\n    end)\n\nend cpi\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/semantics/relation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.36608993377205684}}
{"text": "inductive Foo (n : Nat) : Type\n| foo (t: Foo n): Foo n\n\nnamespace Foo\n  inductive Bar: Foo n \u2192 Prop\n\n  theorem ex\u2081 {s: Foo n} (H: s.Bar): True := by\n    cases h\u2081 : s\n    case foo s' =>\n      cases h\u2082 : n; sorry\n      have: Bar s' := sorry\n      exact ex\u2081 this\n  termination_by _ => sizeOf s\n\n  theorem ex\u2082\n    {s: Foo n}\n    (H: s.Bar):\n    True := by\n    generalize hs': s = s'\n    match s' with\n    | foo s' =>\n      have: Bar s' := sorry\n      have hterm: sizeOf s' < sizeOf s := by simp_all_arith\n      exact ex\u2082 this\n  termination_by _ => sizeOf s\n\n  theorem ex\u2083 {s: Foo n} (H: s.Bar): True := by\n    cases h\u2081 : s\n    case foo s' =>\n      match n with\n      | 0 => sorry\n      | _ =>\n        have: Bar s' := sorry\n        exact ex\u2083 this\n  termination_by _ => sizeOf s\n\n  -- it works\n  theorem ex\u2084 {s: Foo n} (H: s.Bar): True := by\n    match s with\n    | foo s' =>\n    match n with\n    | 0 => sorry\n    | _ =>\n      have: Bar s' := sorry\n      exact ex\u2084 this\n  termination_by _ => sizeOf s\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1228.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3660659297040985}}
{"text": "-- Copyright (c) 2018 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Scott Morrison\n\nimport category_theory.isomorphism\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083 -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nnamespace category_theory\n\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C] {D : Type u\u2082} [\ud835\udc9f : category.{v\u2082} D]\ninclude \ud835\udc9e \ud835\udc9f\n\nclass full (F : C \u2964 D) :=\n(preimage : \u2200 {X Y : C} (f : (F.obj X) \u27f6 (F.obj Y)), X \u27f6 Y)\n(witness' : \u2200 {X Y : C} (f : (F.obj X) \u27f6 (F.obj Y)), F.map (preimage f) = f . obviously)\n\nrestate_axiom full.witness'\nattribute [simp] full.witness\n\nclass faithful (F : C \u2964 D) : Prop :=\n(injectivity' : \u2200 {X Y : C} {f g : X \u27f6 Y} (p : F.map f = F.map g), f = g . obviously)\n\nrestate_axiom faithful.injectivity'\n\nnamespace functor\ndef injectivity (F : C \u2964 D) [faithful F] {X Y : C} {f g : X \u27f6 Y} (p : F.map f = F.map g) : f = g :=\nfaithful.injectivity F p\n\ndef preimage (F : C \u2964 D) [full F] {X Y : C} (f : F.obj X \u27f6 F.obj Y) : X \u27f6 Y :=\nfull.preimage.{v\u2081 v\u2082} f\n@[simp] lemma image_preimage (F : C \u2964 D) [full F] {X Y : C} (f : F.obj X \u27f6 F.obj Y) :\n  F.map (preimage F f) = f :=\nby unfold preimage; obviously\nend functor\n\n\nsection\nvariables {F : C \u2964 D} [full F] [faithful F] {X Y : C}\ndef preimage_iso (f : (F.obj X) \u2245 (F.obj Y)) : X \u2245 Y :=\n{ hom := F.preimage f.hom,\n  inv := F.preimage f.inv,\n  hom_inv_id' := begin apply @faithful.injectivity _ _ _ _ F, obviously, end,\n  inv_hom_id' := begin apply @faithful.injectivity _ _ _ _ F, obviously, end, }\n\n@[simp] lemma preimage_iso_hom (f : (F.obj X) \u2245 (F.obj Y)) :\n  (preimage_iso f).hom = F.preimage f.hom := rfl\n@[simp] lemma preimage_iso_inv (f : (F.obj X) \u2245 (F.obj Y)) :\n  (preimage_iso f).inv = F.preimage (f.inv) := rfl\nend\n\nclass fully_faithful (F : C \u2964 D) extends (full F), (faithful F).\n\n@[simp] lemma preimage_id (F : C \u2964 D) [fully_faithful F] (X : C) : F.preimage (\ud835\udfd9 (F.obj X)) = \ud835\udfd9 X :=\nF.injectivity (by simp)\n\nend category_theory\n\nnamespace category_theory\n\nvariables {C : Type u\u2081} [\ud835\udc9e : category.{v\u2081} C]\ninclude \ud835\udc9e\n\ninstance full.id : full (functor.id C) :=\n{ preimage := \u03bb _ _ f, f }\n\ninstance : faithful (functor.id C) := by obviously\n\ninstance : fully_faithful (functor.id C) := { ((by apply_instance) : full (functor.id C)) with }\n\nvariables {D : Type u\u2082} [\ud835\udc9f : category.{v\u2082} D] {E : Type u\u2083} [\u2130 : category.{v\u2083} E]\ninclude \ud835\udc9f \u2130\nvariables (F : C \u2964 D) (G : D \u2964 E)\n\ninstance faithful.comp [faithful F] [faithful G] : faithful (F \u22d9 G) :=\n{ injectivity' := \u03bb _ _ _ _ p, F.injectivity (G.injectivity p) }\ninstance full.comp [full F] [full G] : full (F \u22d9 G) :=\n{ preimage := \u03bb _ _ f, F.preimage (G.preimage f) }\n\nend category_theory\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/fully_faithful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629465, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.36606592970409846}}
{"text": "/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nnamespace Lean\nuniverse u v w w'\n\nnamespace PersistentHashMap\n\ninductive Entry (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : Type w) where\n  | entry (key : \u03b1) (val : \u03b2) : Entry \u03b1 \u03b2 \u03c3\n  | ref   (node : \u03c3) : Entry \u03b1 \u03b2 \u03c3\n  | null  : Entry \u03b1 \u03b2 \u03c3\n\ninstance {\u03b1 \u03b2 \u03c3} : Inhabited (Entry \u03b1 \u03b2 \u03c3) := \u27e8Entry.null\u27e9\n\ninductive Node (\u03b1 : Type u) (\u03b2 : Type v) : Type (max u v) where\n  | entries   (es : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : Node \u03b1 \u03b2\n  | collision (ks : Array \u03b1) (vs : Array \u03b2) (h : ks.size = vs.size) : Node \u03b1 \u03b2\n\ninstance {\u03b1 \u03b2} : Inhabited (Node \u03b1 \u03b2) := \u27e8Node.entries #[]\u27e9\n\nabbrev shift         : USize  := 5\nabbrev branching     : USize  := USize.ofNat (2 ^ shift.toNat)\nabbrev maxDepth      : USize  := 7\nabbrev maxCollisions : Nat    := 4\n\ndef mkEmptyEntriesArray {\u03b1 \u03b2} : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2)) :=\n  (Array.mkArray PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)\n\nend PersistentHashMap\n\nstructure PersistentHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] where\n  root    : PersistentHashMap.Node \u03b1 \u03b2 := PersistentHashMap.Node.entries PersistentHashMap.mkEmptyEntriesArray\n  size    : Nat                        := 0\n\nabbrev PHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] := PersistentHashMap \u03b1 \u03b2\n\nnamespace PersistentHashMap\n\ndef empty [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 := {}\n\ndef isEmpty [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) : Bool :=\n  m.size == 0\n\ninstance [BEq \u03b1] [Hashable \u03b1] : Inhabited (PersistentHashMap \u03b1 \u03b2) := \u27e8{}\u27e9\n\ndef mkEmptyEntries {\u03b1 \u03b2} : Node \u03b1 \u03b2 :=\n  Node.entries mkEmptyEntriesArray\n\nabbrev mul2Shift (i : USize) (shift : USize) : USize := i.shiftLeft shift\nabbrev div2Shift (i : USize) (shift : USize) : USize := i.shiftRight shift\nabbrev mod2Shift (i : USize) (shift : USize) : USize := USize.land i ((USize.shiftLeft 1 shift) - 1)\n\ninductive IsCollisionNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (keys : Array \u03b1) (vals : Array \u03b2) (h : keys.size = vals.size) : IsCollisionNode (Node.collision keys vals h)\n\nabbrev CollisionNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsCollisionNode n }\n\ninductive IsEntriesNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (entries : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : IsEntriesNode (Node.entries entries)\n\nabbrev EntriesNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsEntriesNode n }\n\nprivate theorem size_set {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (i : Fin ks.size) (j : Fin vs.size) (k : \u03b1) (v : \u03b2)\n                           : (ks.set i k).size = (vs.set j v).size := by\n  simp [h]\n\nprivate theorem size_push {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (k : \u03b1) (v : \u03b2) : (ks.push k).size = (vs.push v).size := by\n  simp [h]\n\npartial def insertAtCollisionNodeAux [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 Nat \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2\n  | n@\u27e8Node.collision keys vals heq, _\u27e9, i, k, v =>\n    if h : i < keys.size then\n      let idx : Fin keys.size := \u27e8i, h\u27e9;\n      let k' := keys.get idx;\n      if k == k' then\n         let j : Fin vals.size := \u27e8i, by rw [\u2190heq]; assumption\u27e9\n         \u27e8Node.collision (keys.set idx k) (vals.set j v) (size_set heq idx j k v), IsCollisionNode.mk _ _ _\u27e9\n      else insertAtCollisionNodeAux n (i+1) k v\n    else\n      \u27e8Node.collision (keys.push k) (vals.push v) (size_push heq k v), IsCollisionNode.mk _ _ _\u27e9\n  | \u27e8Node.entries _, h\u27e9, _, _, _ => False.elim (nomatch h)\n\ndef insertAtCollisionNode [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2 :=\n  fun n k v => insertAtCollisionNodeAux n 0 k v\n\ndef getCollisionNodeSize : CollisionNode \u03b1 \u03b2 \u2192 Nat\n  | \u27e8Node.collision keys _ _, _\u27e9 => keys.size\n  | \u27e8Node.entries _, h\u27e9          => False.elim (nomatch h)\n\ndef mkCollisionNode (k\u2081 : \u03b1) (v\u2081 : \u03b2) (k\u2082 : \u03b1) (v\u2082 : \u03b2) : Node \u03b1 \u03b2 :=\n  let ks : Array \u03b1 := Array.mkEmpty maxCollisions\n  let ks := (ks.push k\u2081).push k\u2082\n  let vs : Array \u03b2 := Array.mkEmpty maxCollisions\n  let vs := (vs.push v\u2081).push v\u2082\n  Node.collision ks vs rfl\n\npartial def insertAux [BEq \u03b1] [Hashable \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 USize \u2192 \u03b1 \u2192 \u03b2 \u2192 Node \u03b1 \u03b2\n  | Node.collision keys vals heq, _, depth, k, v =>\n    let newNode := insertAtCollisionNode \u27e8Node.collision keys vals heq, IsCollisionNode.mk _ _ _\u27e9 k v\n    if depth >= maxDepth || getCollisionNodeSize newNode < maxCollisions then newNode.val\n    else match newNode with\n      | \u27e8Node.entries _, h\u27e9 => False.elim (nomatch h)\n      | \u27e8Node.collision keys vals heq, _\u27e9 =>\n        let rec traverse (i : Nat) (entries : Node \u03b1 \u03b2) : Node \u03b1 \u03b2 :=\n          if h : i < keys.size then\n            let k := keys[i]\n            have : i < vals.size := heq \u25b8 h\n            let v := vals[i]\n            let h := hash k |>.toUSize\n            let h := div2Shift h (shift * (depth - 1))\n            traverse (i+1) (insertAux entries h depth k v)\n          else\n            entries\n        traverse 0 mkEmptyEntries\n  | Node.entries entries, h, depth, k, v =>\n    let j     := (mod2Shift h shift).toNat\n    Node.entries $ entries.modify j fun entry =>\n      match entry with\n      | Entry.null        => Entry.entry k v\n      | Entry.ref node    => Entry.ref $ insertAux node (div2Shift h shift) (depth+1) k v\n      | Entry.entry k' v' =>\n        if k == k' then Entry.entry k v\n        else Entry.ref $ mkCollisionNode k' v' k v\n\ndef insert {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k, v => { root := insertAux n (hash k |>.toUSize) 1 k v, size := sz + 1 }\n\npartial def findAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option \u03b2 :=\n  if h : i < keys.size then\n    let k' := keys[i]\n    have : i < vals.size := by rw [\u2190heq]; assumption\n    if k == k' then some vals[i]\n    else findAtAux keys vals heq (i+1) k\n  else none\n\npartial def findAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option \u03b2\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some v else none\n  | Node.collision keys vals heq, _, k => findAtAux keys vals heq 0 k\n\ndef find? {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option \u03b2\n  | { root := n, .. }, k => findAux n (hash k |>.toUSize) k\n\ninstance {_ : BEq \u03b1} {_ : Hashable \u03b1} : GetElem (PersistentHashMap \u03b1 \u03b2) \u03b1 (Option \u03b2) fun _ _ => True where\n  getElem m i _ := m.find? i\n\n@[inline] def findD {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) (b\u2080 : \u03b2) : \u03b2 :=\n  (m.find? a).getD b\u2080\n\n@[inline] def find! {_ : BEq \u03b1} {_ : Hashable \u03b1} [Inhabited \u03b2] (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) : \u03b2 :=\n  match m.find? a with\n  | some b => b\n  | none   => panic! \"key is not in the map\"\n\npartial def findEntryAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < keys.size then\n    let k' := keys[i]\n    have : i < vals.size := by rw [\u2190heq]; assumption\n    if k == k' then some (k', vals[i])\n    else findEntryAtAux keys vals heq (i+1) k\n  else none\n\npartial def findEntryAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findEntryAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some (k', v) else none\n  | Node.collision keys vals heq, _, k => findEntryAtAux keys vals heq 0 k\n\ndef findEntry? {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | { root := n, .. }, k => findEntryAux n (hash k |>.toUSize) k\n\npartial def containsAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Bool :=\n  if h : i < keys.size then\n    let k' := keys[i]\n    if k == k' then true\n    else containsAtAux keys vals heq (i+1) k\n  else false\n\npartial def containsAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Bool\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => false\n    | Entry.ref node   => containsAux node (div2Shift h shift) k\n    | Entry.entry k' _ => k == k'\n  | Node.collision keys vals heq, _, k => containsAtAux keys vals heq 0 k\n\ndef contains [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Bool\n  | { root := n, .. }, k => containsAux n (hash k |>.toUSize) k\n\npartial def isUnaryEntries (a : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) (i : Nat) (acc : Option (\u03b1 \u00d7 \u03b2)) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < a.size then\n    match a[i] with\n    | Entry.null      => isUnaryEntries a (i+1) acc\n    | Entry.ref _     => none\n    | Entry.entry k v =>\n      match acc with\n      | none   => isUnaryEntries a (i+1) (some (k, v))\n      | some _ => none\n  else acc\n\ndef isUnaryNode : Node \u03b1 \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries         => isUnaryEntries entries 0 none\n  | Node.collision keys vals heq =>\n    if h : 1 = keys.size then\n      have : 0 < keys.size := by rw [\u2190h]; decide\n      have : 0 < vals.size := by rw [\u2190heq]; assumption\n      some (keys[0], vals[0])\n    else\n      none\n\npartial def eraseAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Node \u03b1 \u03b2 \u00d7 Bool\n  | n@(Node.collision keys vals heq), _, k =>\n    match keys.indexOf? k with\n    | some idx =>\n      let \u27e8keys', keq\u27e9 := keys.eraseIdx' idx\n      let \u27e8vals', veq\u27e9 := vals.eraseIdx' (Eq.ndrec idx heq)\n      have : keys.size - 1 = vals.size - 1 := by rw [heq]\n      (Node.collision keys' vals' (keq.trans (this.trans veq.symm)), true)\n    | none     => (n, false)\n  | n@(Node.entries entries), h, k =>\n    let j       := (mod2Shift h shift).toNat\n    let entry   := entries.get! j\n    match entry with\n    | Entry.null       => (n, false)\n    | Entry.entry k' _ =>\n      if k == k' then (Node.entries (entries.set! j Entry.null), true) else (n, false)\n    | Entry.ref node   =>\n      let entries := entries.set! j Entry.null\n      let (newNode, deleted) := eraseAux node (div2Shift h shift) k\n      if !deleted then (n, false)\n      else match isUnaryNode newNode with\n        | none        => (Node.entries (entries.set! j (Entry.ref newNode)), true)\n        | some (k, v) => (Node.entries (entries.set! j (Entry.entry k v)), true)\n\ndef erase {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k =>\n    let h := hash k |>.toUSize\n    let (n, del) := eraseAux n h k\n    { root := n, size := if del then sz - 1 else sz }\n\nsection\nvariable {m : Type w \u2192 Type w'} [Monad m]\nvariable {\u03c3 : Type w}\n\npartial def foldlMAux (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) : Node \u03b1 \u03b2 \u2192 \u03c3 \u2192 m \u03c3\n  | Node.collision keys vals heq, acc =>\n    let rec traverse (i : Nat) (acc : \u03c3) : m \u03c3 := do\n      if h : i < keys.size then\n        let k := keys[i]\n        have : i < vals.size := heq \u25b8 h\n        let v := vals[i]\n        traverse (i+1) (\u2190 f acc k v)\n      else\n        pure acc\n    traverse 0 acc\n  | Node.entries entries, acc => entries.foldlM (fun acc entry =>\n    match entry with\n    | Entry.null      => pure acc\n    | Entry.entry k v => f acc k v\n    | Entry.ref node  => foldlMAux f node acc)\n    acc\n\ndef foldlM {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) (init : \u03c3) : m \u03c3 :=\n  foldlMAux f map.root init\n\ndef forM {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03b1 \u2192 \u03b2 \u2192 m PUnit) : m PUnit :=\n  map.foldlM (fun _ => f) \u27e8\u27e9\n\ndef foldl {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03c3) (init : \u03c3) : \u03c3 :=\n  Id.run <| map.foldlM f init\n\nprotected def forIn {_ : BEq \u03b1} {_ : Hashable \u03b1} [Monad m]\n    (map : PersistentHashMap \u03b1 \u03b2) (init : \u03c3) (f : \u03b1 \u00d7 \u03b2 \u2192 \u03c3 \u2192 m (ForInStep \u03c3)) : m \u03c3 := do\n  let intoError : ForInStep \u03c3 \u2192 Except \u03c3 \u03c3\n  | .done s => .error s\n  | .yield s => .ok s\n  let result \u2190 foldlM (m := ExceptT \u03c3 m) map (init := init) fun s a b =>\n    (intoError <$> f (a, b) s : m _)\n  match result with\n  | .ok s | .error s => pure s\n\ninstance {_ : BEq \u03b1} {_ : Hashable \u03b1} : ForIn m (PersistentHashMap \u03b1 \u03b2) (\u03b1 \u00d7 \u03b2) where\n  forIn := PersistentHashMap.forIn\n\nend\n\npartial def mapMAux {\u03b1 : Type u} {\u03b2 : Type v} {\u03c3 : Type u} {m : Type u \u2192 Type w} [Monad m] (f : \u03b2 \u2192 m \u03c3) (n : Node \u03b1 \u03b2) : m (Node \u03b1 \u03c3) := do\n  match n with\n  | .collision keys vals heq =>\n    let \u27e8vals', h\u27e9 \u2190 vals.mapM' f\n    return .collision keys vals' (h \u25b8 heq)\n  | .entries entries =>\n    let entries' \u2190 entries.mapM fun\n      | .null      => return .null\n      | .entry k v => return .entry k (\u2190 f v)\n      | .ref node  => return .ref (\u2190 mapMAux f node)\n    return .entries entries'\n\ndef mapM {\u03b1 : Type u} {\u03b2 : Type v} {\u03c3 : Type u} {m : Type u \u2192 Type w} [Monad m] {_ : BEq \u03b1} {_ : Hashable \u03b1} (pm : PersistentHashMap \u03b1 \u03b2) (f : \u03b2 \u2192 m \u03c3) : m (PersistentHashMap \u03b1 \u03c3) := do\n  let root \u2190 mapMAux f pm.root\n  return { pm with root }\n\ndef map {\u03b1 : Type u} {\u03b2 : Type v} {\u03c3 : Type u} {_ : BEq \u03b1} {_ : Hashable \u03b1} (pm : PersistentHashMap \u03b1 \u03b2) (f : \u03b2 \u2192 \u03c3) : PersistentHashMap \u03b1 \u03c3 :=\n  Id.run <| pm.mapM f\n\ndef toList {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) : List (\u03b1 \u00d7 \u03b2) :=\n  m.foldl (init := []) fun ps k v => (k, v) :: ps\n\nstructure Stats where\n  numNodes      : Nat := 0\n  numNull       : Nat := 0\n  numCollisions : Nat := 0\n  maxDepth      : Nat := 0\n\npartial def collectStats : Node \u03b1 \u03b2 \u2192 Stats \u2192 Nat \u2192 Stats\n  | Node.collision keys _ _, stats, depth =>\n    { stats with\n      numNodes      := stats.numNodes + 1,\n      numCollisions := stats.numCollisions + keys.size - 1,\n      maxDepth      := Nat.max stats.maxDepth depth }\n  | Node.entries entries, stats, depth =>\n    let stats :=\n      { stats with\n        numNodes      := stats.numNodes + 1,\n        maxDepth      := Nat.max stats.maxDepth depth }\n    entries.foldl (fun stats entry =>\n      match entry with\n      | Entry.null      => { stats with numNull := stats.numNull + 1 }\n      | Entry.ref node  => collectStats node stats (depth + 1)\n      | Entry.entry _ _ => stats)\n      stats\n\ndef stats {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) : Stats :=\n  collectStats m.root {} 1\n\ndef Stats.toString (s : Stats) : String :=\n  s!\"\\{ nodes := {s.numNodes}, null := {s.numNull}, collisions := {s.numCollisions}, depth := {s.maxDepth}}\"\n\ninstance : ToString Stats := \u27e8Stats.toString\u27e9\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Lean/Data/PersistentHashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.3660659157039514}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Bhavik Mehta\n\n! This file was ported from Lean 3 source module combinatorics.simple_graph.regularity.equitabilise\n! leanprover-community/mathlib commit b6da1a0b3e7cd83b1f744c49ce48ef8c6307d2f6\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Order.Partition.Equipartition\n\n/-!\n# Equitabilising a partition\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file allows to blow partitions up into parts of controlled size. Given a partition `P` and\n`a b m : \u2115`, we want to find a partition `Q` with `a` parts of size `m` and `b` parts of size\n`m + 1` such that all parts of `P` are \"as close as possible\" to unions of parts of `Q`. By\n\"as close as possible\", we mean that each part of `P` can be written as the union of some parts of\n`Q` along with at most `m` other elements.\n\n## Main declarations\n\n* `finpartition.equitabilise`: `P.equitabilise h` where `h : a * m + b * (m + 1)` is a partition\n  with `a` parts of size `m` and `b` parts of size `m + 1` which almost refines `P`.\n* `finpartition.exists_equipartition_card_eq`: We can find equipartitions of arbitrary size.\n-/\n\n\nopen Finset Nat\n\nnamespace Finpartition\n\nvariable {\u03b1 : Type _} [DecidableEq \u03b1] {s t : Finset \u03b1} {m n a b : \u2115} {P : Finpartition s}\n\n/- warning: finpartition.equitabilise_aux -> Finpartition.equitabilise_aux is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s), (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)) -> (Exists.{succ u1} (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (fun (Q : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) => And (forall (x : Finset.{u1} \u03b1), (Membership.Mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.hasMem.{u1} (Finset.{u1} \u03b1)) x (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s Q)) -> (Or (Eq.{1} Nat (Finset.card.{u1} \u03b1 x) m) (Eq.{1} Nat (Finset.card.{u1} \u03b1 x) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))) (And (forall (x : Finset.{u1} \u03b1), (Membership.Mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.hasMem.{u1} (Finset.{u1} \u03b1)) x (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s P)) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} \u03b1 (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.hasSdiff.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x (Finset.bunion\u1d62.{u1, u1} (Finset.{u1} \u03b1) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (y : Finset.{u1} \u03b1) => HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.hasSubset.{u1} \u03b1) y x) (fun (a : Finset.{u1} \u03b1) => Finset.decidableDforallFinset.{u1} \u03b1 a (fun (a_1 : \u03b1) (\u1fb0 : Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 a) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 x) (fun (a_1 : \u03b1) (h : Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 a) => Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) a_1 x)) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s Q)) (id.{succ u1} (Finset.{u1} \u03b1))))) m)) (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (i : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 i) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (a : Finset.{u1} \u03b1) => Nat.decidableEq (Finset.card.{u1} \u03b1 a) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s Q))) b))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s}, (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)) -> (Exists.{succ u1} (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (fun (Q : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) => And (forall (x : Finset.{u1} \u03b1), (Membership.mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.instMembershipFinset.{u1} (Finset.{u1} \u03b1)) x (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s Q)) -> (Or (Eq.{1} Nat (Finset.card.{u1} \u03b1 x) m) (Eq.{1} Nat (Finset.card.{u1} \u03b1 x) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))) (And (forall (x : Finset.{u1} \u03b1), (Membership.mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.instMembershipFinset.{u1} (Finset.{u1} \u03b1)) x (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s P)) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} \u03b1 (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.instSDiffFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) x (Finset.bunion\u1d62.{u1, u1} (Finset.{u1} \u03b1) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (y : Finset.{u1} \u03b1) => HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.instHasSubsetFinset.{u1} \u03b1) y x) (fun (a : Finset.{u1} \u03b1) => Finset.decidableSubsetFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) a x) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s Q)) (id.{succ u1} (Finset.{u1} \u03b1))))) m)) (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (i : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 i) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (a : Finset.{u1} \u03b1) => instDecidableEqNat (Finset.card.{u1} \u03b1 a) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s Q))) b))))\nCase conversion may be inaccurate. Consider using '#align finpartition.equitabilise_aux Finpartition.equitabilise_aux\u2093'. -/\n/-- Given a partition `P` of `s`, as well as a proof that `a * m + b * (m + 1) = s.card`, we can\nfind a new partition `Q` of `s` where each part has size `m` or `m + 1`, every part of `P` is the\nunion of parts of `Q` plus at most `m` extra elements, there are `b` parts of size `m + 1` and\n(provided `m > 0`, because a partition does not have parts of size `0`) there are `a` parts of size\n`m` and hence `a + b` parts in total. -/\ntheorem equitabilise_aux (P : Finpartition s) (hs : a * m + b * (m + 1) = s.card) :\n    \u2203 Q : Finpartition s,\n      (\u2200 x : Finset \u03b1, x \u2208 Q.parts \u2192 x.card = m \u2228 x.card = m + 1) \u2227\n        (\u2200 x, x \u2208 P.parts \u2192 (x \\ (Q.parts.filter\u2093 fun y => y \u2286 x).bunion\u1d62 id).card \u2264 m) \u2227\n          (Q.parts.filter\u2093 fun i => card i = m + 1).card = b :=\n  by\n  -- Get rid of the easy case `m = 0`\n  obtain rfl | m_pos := m.eq_zero_or_pos\n  \u00b7 refine' \u27e8\u22a5, by simp, _, by simpa using hs.symm\u27e9\n    simp only [le_zero_iff, card_eq_zero, mem_bUnion, exists_prop, mem_filter, id.def, and_assoc',\n      sdiff_eq_empty_iff_subset, subset_iff]\n    exact 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  induction' s using Finset.strongInduction with s ih generalizing P a b\n  -- If `a = b = 0`, then `s = \u2205` and we can partition into zero parts\n  by_cases hab : a = 0 \u2227 b = 0\n  \u00b7 simp only [hab.1, hab.2, add_zero, MulZeroClass.zero_mul, eq_comm, card_eq_zero] at hs\n    subst hs\n    exact \u27e8Finpartition.empty _, by simp, by simp [Unique.eq_default P], by simp [hab.2]\u27e9\n  simp_rw [not_and_or, \u2190 Ne.def, \u2190 pos_iff_ne_zero] at hab\n  -- `n` will be the size of the smallest part\n  set n := if 0 < a then m else m + 1 with hn\n  -- Some easy facts about it\n  obtain \u27e8hn\u2080, hn\u2081, hn\u2082, hn\u2083\u27e9 :\n    0 < n \u2227\n      n \u2264 m + 1 \u2227\n        n \u2264 a * m + b * (m + 1) \u2227\n          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 <;> 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 least\n    one part `u` of `P` has size `m + 1` (in which case we take `t` to be an arbitrary subset of `u`\n    of size `n`). The rest of each branch is just tedious calculations to satisfy the induction\n    hypothesis. -/\n  by_cases \u2200 u \u2208 P.parts, card u < m + 1\n  \u00b7 obtain \u27e8t, hts, htn\u27e9 := exists_smaller_set s n (hn\u2082.trans_eq hs)\n    have ht : t.nonempty := by rwa [\u2190 card_pos, htn]\n    have 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    obtain \u27e8R, hR\u2081, hR\u2082, hR\u2083\u27e9 :=\n      @ih (s \\ t) (sdiff_ssubset hts \u2039t.nonempty\u203a) (P.avoid t) (if 0 < a then a - 1 else a)\n        (if 0 < a then b else b - 1) hcard\n    refine' \u27e8R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel hts), _, _, _\u27e9\n    \u00b7 simp only [extend_parts, mem_insert, forall_eq_or_imp, and_iff_left hR\u2081, htn, hn]\n      exact ite_eq_or_eq _ _ _\n    \u00b7 exact fun x hx => (card_le_of_subset <| sdiff_subset _ _).trans (lt_succ_iff.1 <| h _ hx)\n    simp_rw [extend_parts, filter_insert, htn, hn, m.succ_ne_self.symm.ite_eq_right_iff]\n    split_ifs with ha\n    \u00b7 rw [hR\u2083, if_pos ha]\n    rw [card_insert_of_not_mem fun H => _, hR\u2083, if_neg ha, tsub_add_cancel_of_le]\n    \u00b7 exact hab.resolve_left ha\n    \u00b7 exact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)\n  push_neg  at h\n  obtain \u27e8u, hu\u2081, hu\u2082\u27e9 := h\n  obtain \u27e8t, htu, htn\u27e9 := exists_smaller_set _ _ (hn\u2081.trans hu\u2082)\n  have ht : t.nonempty := by rwa [\u2190 card_pos, htn]\n  have 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  obtain \u27e8R, hR\u2081, hR\u2082, hR\u2083\u27e9 :=\n    @ih (s \\ t) (sdiff_ssubset (htu.trans <| P.le hu\u2081) ht) (P.avoid t) (if 0 < a then a - 1 else a)\n      (if 0 < a then b else b - 1) hcard\n  refine' \u27e8R.extend ht.ne_empty sdiff_disjoint (sdiff_sup_cancel <| htu.trans <| P.le hu\u2081), _, _, _\u27e9\n  \u00b7 simp only [mem_insert, forall_eq_or_imp, extend_parts, and_iff_left hR\u2081, htn, hn]\n    exact ite_eq_or_eq _ _ _\n  \u00b7 conv in _ \u2208 _ => rw [\u2190 insert_erase hu\u2081]\n    simp only [and_imp, mem_insert, forall_eq_or_imp, Ne.def, extend_parts]\n    refine' \u27e8_, fun x hx => (card_le_of_subset _).trans <| hR\u2082 x _\u27e9\n    \u00b7 simp only [filter_insert, if_pos htu, bUnion_insert, mem_erase, id.def]\n      obtain rfl | hut := eq_or_ne u t\n      \u00b7 rw [sdiff_eq_empty_iff_subset.2 (subset_union_left _ _)]\n        exact bot_le\n      refine'\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      simp only [not_exists, mem_bUnion, and_imp, mem_union, mem_filter, mem_sdiff, id.def, not_or]\n      exact fun hi\u2081 hi\u2082 hi\u2083 =>\n        \u27e8\u27e8hi\u2081, hi\u2082\u27e9, fun x hx hx' => hi\u2083 _ hx <| hx'.trans <| sdiff_subset _ _\u27e9\n    \u00b7 apply sdiff_subset_sdiff subset.rfl (bUnion_subset_bUnion_of_subset_left _ _)\n      exact filter_subset_filter _ (subset_insert _ _)\n    simp only [avoid, of_erase, mem_erase, mem_image, bot_eq_empty]\n    exact\n      \u27e8(nonempty_of_mem_parts _ <| mem_of_mem_erase hx).ne_empty, _, mem_of_mem_erase hx,\n        (disjoint_of_subset_right htu <|\n            P.disjoint (mem_of_mem_erase hx) hu\u2081 <| ne_of_mem_erase hx).sdiff_eq_left\u27e9\n  simp only [extend_parts, filter_insert, htn, hn, m.succ_ne_self.symm.ite_eq_right_iff]\n  split_ifs\n  \u00b7 rw [hR\u2083, if_pos h]\n  \u00b7 rw [card_insert_of_not_mem fun H => _, hR\u2083, if_neg h, Nat.sub_add_cancel (hab.resolve_left h)]\n    exact ht.ne_empty (le_sdiff_iff.1 <| R.le <| filter_subset _ _ H)\n#align finpartition.equitabilise_aux Finpartition.equitabilise_aux\n\nvariable (P) (h : a * m + b * (m + 1) = s.card)\n\n/- warning: finpartition.equitabilise -> Finpartition.equitabilise is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat}, (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) -> (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)) -> (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s}, (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)) -> (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s)\nCase conversion may be inaccurate. Consider using '#align finpartition.equitabilise Finpartition.equitabilise\u2093'. -/\n/-- Given a partition `P` of `s`, as well as a proof that `a * m + b * (m + 1) = s.card`, build a\nnew partition `Q` of `s` where each part has size `m` or `m + 1`, every part of `P` is the union of\nparts of `Q` plus at most `m` extra elements, there are `b` parts of size `m + 1` and (provided\n`m > 0`, because a partition does not have parts of size `0`) there are `a` parts of size `m` and\nhence `a + b` parts in total. -/\nnoncomputable def equitabilise : Finpartition s :=\n  (P.equitabilise_aux h).some\n#align finpartition.equitabilise Finpartition.equitabilise\n\nvariable {P h}\n\n/- warning: finpartition.card_eq_of_mem_parts_equitabilise -> Finpartition.card_eq_of_mem_parts_equitabilise is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {t : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s} {h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)}, (Membership.Mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.hasMem.{u1} (Finset.{u1} \u03b1)) t (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) -> (Or (Eq.{1} Nat (Finset.card.{u1} \u03b1 t) m) (Eq.{1} Nat (Finset.card.{u1} \u03b1 t) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {t : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s} {h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)}, (Membership.mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.instMembershipFinset.{u1} (Finset.{u1} \u03b1)) t (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) -> (Or (Eq.{1} Nat (Finset.card.{u1} \u03b1 t) m) (Eq.{1} Nat (Finset.card.{u1} \u03b1 t) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))))\nCase conversion may be inaccurate. Consider using '#align finpartition.card_eq_of_mem_parts_equitabilise Finpartition.card_eq_of_mem_parts_equitabilise\u2093'. -/\ntheorem card_eq_of_mem_parts_equitabilise :\n    t \u2208 (P.equitabilise h).parts \u2192 t.card = m \u2228 t.card = m + 1 :=\n  (P.equitabilise_aux h).choose_spec.1 _\n#align finpartition.card_eq_of_mem_parts_equitabilise Finpartition.card_eq_of_mem_parts_equitabilise\n\n/- warning: finpartition.equitabilise_is_equipartition -> Finpartition.equitabilise_isEquipartition is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s} {h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)}, Finpartition.IsEquipartition.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} {P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s} {h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)}, Finpartition.IsEquipartition.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)\nCase conversion may be inaccurate. Consider using '#align finpartition.equitabilise_is_equipartition Finpartition.equitabilise_isEquipartition\u2093'. -/\ntheorem equitabilise_isEquipartition : (P.equitabilise h).IsEquipartition :=\n  Set.equitableOn_iff_exists_eq_eq_add_one.2 \u27e8m, fun u => card_eq_of_mem_parts_equitabilise\u27e9\n#align finpartition.equitabilise_is_equipartition Finpartition.equitabilise_isEquipartition\n\nvariable (P h)\n\n/- warning: finpartition.card_filter_equitabilise_big -> Finpartition.card_filter_equitabilise_big is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)), Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 u) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (fun (a : Finset.{u1} \u03b1) => Nat.decidableEq (Finset.card.{u1} \u03b1 a) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)))) b\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)), Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 u) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (fun (a : Finset.{u1} \u03b1) => instDecidableEqNat (Finset.card.{u1} \u03b1 a) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)))) b\nCase conversion may be inaccurate. Consider using '#align finpartition.card_filter_equitabilise_big Finpartition.card_filter_equitabilise_big\u2093'. -/\ntheorem card_filter_equitabilise_big :\n    ((P.equitabilise h).parts.filter\u2093 fun u : Finset \u03b1 => u.card = m + 1).card = b :=\n  (P.equitabilise_aux h).choose_spec.2.2\n#align finpartition.card_filter_equitabilise_big Finpartition.card_filter_equitabilise_big\n\n/- warning: finpartition.card_filter_equitabilise_small -> Finpartition.card_filter_equitabilise_small is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)), (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 u) m) (fun (a : Finset.{u1} \u03b1) => Nat.decidableEq (Finset.card.{u1} \u03b1 a) m) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)))) a)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)), (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => Eq.{1} Nat (Finset.card.{u1} \u03b1 u) m) (fun (a : Finset.{u1} \u03b1) => instDecidableEqNat (Finset.card.{u1} \u03b1 a) m) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h)))) a)\nCase conversion may be inaccurate. Consider using '#align finpartition.card_filter_equitabilise_small Finpartition.card_filter_equitabilise_small\u2093'. -/\ntheorem card_filter_equitabilise_small (hm : m \u2260 0) :\n    ((P.equitabilise h).parts.filter\u2093 fun u : Finset \u03b1 => u.card = m).card = a :=\n  by\n  refine' (mul_eq_mul_right_iff.1 <| (add_left_inj (b * (m + 1))).1 _).resolve_right hm\n  rw [h, \u2190 (P.equitabilise h).sum_card_parts]\n  have hunion :\n    (P.equitabilise h).parts =\n      ((P.equitabilise h).parts.filter\u2093 fun u => u.card = m) \u222a\n        (P.equitabilise h).parts.filter\u2093 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  nth_rw 2 [hunion]\n  rw [sum_union, sum_const_nat fun x hx => (mem_filter.1 hx).2,\n    sum_const_nat fun x hx => (mem_filter.1 hx).2, P.card_filter_equitabilise_big]\n  refine' disjoint_filter_filter' _ _ _\n  intro x ha hb i h\n  apply succ_ne_self m _\n  exact (hb i h).symm.trans (ha i h)\n#align finpartition.card_filter_equitabilise_small Finpartition.card_filter_equitabilise_small\n\n/- warning: finpartition.card_parts_equitabilise -> Finpartition.card_parts_equitabilise is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)), (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) a b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)), (Ne.{1} Nat m (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) a b))\nCase conversion may be inaccurate. Consider using '#align finpartition.card_parts_equitabilise Finpartition.card_parts_equitabilise\u2093'. -/\ntheorem card_parts_equitabilise (hm : m \u2260 0) : (P.equitabilise h).parts.card = a + b :=\n  by\n  rw [\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  exact disjoint_filter.2 fun x _ h\u2080 h\u2081 => Nat.succ_ne_self m <| h\u2081.symm.trans h\u2080\n  infer_instance\n#align finpartition.card_parts_equitabilise Finpartition.card_parts_equitabilise\n\n/- warning: finpartition.card_parts_equitabilise_subset_le -> Finpartition.card_parts_equitabilise_subset_le is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {t : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat Nat.hasMul) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) m (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (Finset.card.{u1} \u03b1 s)), (Membership.Mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.hasMem.{u1} (Finset.{u1} \u03b1)) t (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s P)) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u1} \u03b1 (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.hasSdiff.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) t (Finset.bunion\u1d62.{u1, u1} (Finset.{u1} \u03b1) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.hasSubset.{u1} \u03b1) u t) (fun (a : Finset.{u1} \u03b1) => Finset.decidableDforallFinset.{u1} \u03b1 a (fun (a_1 : \u03b1) (\u1fb0 : Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 a) => Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 t) (fun (a_1 : \u03b1) (h : Membership.Mem.{u1, u1} \u03b1 (Finset.{u1} \u03b1) (Finset.hasMem.{u1} \u03b1) a_1 a) => Finset.decidableMem.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) a_1 t)) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) (id.{succ u1} (Finset.{u1} \u03b1))))) m)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] {s : Finset.{u1} \u03b1} {t : Finset.{u1} \u03b1} {m : Nat} {a : Nat} {b : Nat} (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (h : Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) a m) (HMul.hMul.{0, 0, 0} Nat Nat Nat (instHMul.{0} Nat instMulNat) b (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) m (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (Finset.card.{u1} \u03b1 s)), (Membership.mem.{u1, u1} (Finset.{u1} \u03b1) (Finset.{u1} (Finset.{u1} \u03b1)) (Finset.instMembershipFinset.{u1} (Finset.{u1} \u03b1)) t (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s P)) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} \u03b1 (SDiff.sdiff.{u1} (Finset.{u1} \u03b1) (Finset.instSDiffFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) t (Finset.bunion\u1d62.{u1, u1} (Finset.{u1} \u03b1) \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Finset.filter.{u1} (Finset.{u1} \u03b1) (fun (u : Finset.{u1} \u03b1) => HasSubset.Subset.{u1} (Finset.{u1} \u03b1) (Finset.instHasSubsetFinset.{u1} \u03b1) u t) (fun (a : Finset.{u1} \u03b1) => Finset.decidableSubsetFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) a t) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s (Finpartition.equitabilise.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s m a b P h))) (id.{succ u1} (Finset.{u1} \u03b1))))) m)\nCase conversion may be inaccurate. Consider using '#align finpartition.card_parts_equitabilise_subset_le Finpartition.card_parts_equitabilise_subset_le\u2093'. -/\ntheorem card_parts_equitabilise_subset_le :\n    t \u2208 P.parts \u2192 (t \\ ((P.equitabilise h).parts.filter\u2093 fun u => u \u2286 t).bunion\u1d62 id).card \u2264 m :=\n  (Classical.choose_spec <| P.equitabilise_aux h).2.1 t\n#align finpartition.card_parts_equitabilise_subset_le Finpartition.card_parts_equitabilise_subset_le\n\nvariable (s)\n\n/- warning: finpartition.exists_equipartition_card_eq -> Finpartition.exists_equipartition_card_eq is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (OfNat.mk.{0} Nat 0 (Zero.zero.{0} Nat Nat.hasZero)))) -> (LE.le.{0} Nat Nat.hasLe n (Finset.card.{u1} \u03b1 s)) -> (Exists.{succ u1} (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) (fun (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s) => And (Finpartition.IsEquipartition.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s P) (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.lattice.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.orderBot.{u1} \u03b1) s P)) n)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DecidableEq.{succ u1} \u03b1] (s : Finset.{u1} \u03b1) {n : Nat}, (Ne.{1} Nat n (OfNat.ofNat.{0} Nat 0 (instOfNatNat 0))) -> (LE.le.{0} Nat instLENat n (Finset.card.{u1} \u03b1 s)) -> (Exists.{succ u1} (Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) (fun (P : Finpartition.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s) => And (Finpartition.IsEquipartition.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) s P) (Eq.{1} Nat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finpartition.parts.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b)) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) s P)) n)))\nCase conversion may be inaccurate. Consider using '#align finpartition.exists_equipartition_card_eq Finpartition.exists_equipartition_card_eq\u2093'. -/\n/-- We can find equipartitions of arbitrary size. -/\ntheorem exists_equipartition_card_eq (hn : n \u2260 0) (hs : n \u2264 s.card) :\n    \u2203 P : Finpartition s, P.IsEquipartition \u2227 P.parts.card = n :=\n  by\n  rw [\u2190 pos_iff_ne_zero] at hn\n  have : (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,\n      tsub_add_cancel_of_le (Nat.mul_le_mul_right _ (mod_lt _ hn).le), mul_one, add_comm,\n      mod_add_div]\n  refine'\n    \u27e8(indiscrete (card_pos.1 <| hn.trans_le hs).ne_empty).equitabilise this,\n      equitabilise_is_equipartition, _\u27e9\n  rw [card_parts_equitabilise _ _ (Nat.div_pos hs hn).ne', tsub_add_cancel_of_le (mod_lt _ hn).le]\n#align finpartition.exists_equipartition_card_eq Finpartition.exists_equipartition_card_eq\n\nend Finpartition\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Combinatorics/SimpleGraph/Regularity/Equitabilise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.36600399962676916}}
{"text": "import littleendian\nimport category_theory.category.basic\nimport data.nat.digits\n\nopen littleendian\n\nopen category_theory\nopen nat\n\nnamespace littleendian_examples\n\nvariables [category (bitvec params.word_len)] [semiring (list \u2115)]\n\n/- \n  Examples from the spec.\n\n  https://cr.yp.to/snuffle/spec.pdf\n\n  ## Straight examples\n-/\n\n/-- `littleendian(0, 0, 0, 0) = 0x00000000` -/\nlemma example1 : littleendian [0, 0, 0, 0] = 0 := \nbegin\n  unfold littleendian,\n  unfold of_digits,\n  norm_num,\nend\n\n/-- `littleendian(86, 75, 30, 9) = 0x091e4b56` -/\nlemma example2 : littleendian [86, 75, 30, 9] = 0x091e4b56 := \nbegin\n  unfold littleendian,\n  unfold of_digits,\n  norm_num,  \nend\n\n/-- `littleendian(255, 255, 255, 250) = 0xfaffffff` -/\nlemma example3 : littleendian [255, 255, 255, 250] = 0xfaffffff := \nbegin\n  unfold littleendian,\n  unfold of_digits,\n  norm_num,\nend\n\n/- \n  ## Inverse examples \n-/\n\n/-- `littleendian\u207b\u00b9 0 = (0, 0, 0, 0)` -/\nlemma example1_inv : littleendian_inv [0] = list.nil := \nbegin\n  unfold littleendian_inv,\n  norm_num,\nend\n\n/-- `littleendian\u207b\u00b9 152980310 = (86, 75, 30, 9)` -/\nlemma example2_inv : littleendian_inv [0x091e4b56] = [86, 75, 30, 9] := \nbegin\n  unfold littleendian_inv,\n  norm_num,\nend\n\n/-- `littleendian\u207b\u00b9 4211081215 = (255, 255, 255, 250)` -/\nlemma example3_inv : littleendian_inv [0xfaffffff] = [255, 255, 255, 250] := \nbegin\n  unfold littleendian_inv,\n  norm_num,\nend\n\nend littleendian_examples\n", "meta": {"author": "oxarbitrage", "repo": "salsa20", "sha": "12d0ebb3c27801931e61d470fb2ed548a5562578", "save_path": "github-repos/lean/oxarbitrage-salsa20", "path": "github-repos/lean/oxarbitrage-salsa20/salsa20-12d0ebb3c27801931e61d470fb2ed548a5562578/src/examples/littleendian.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.36579149153507556}}
{"text": "/-\n## `arith` dialect\n\nThis file formalises part of the `arith` dialect. The goal is to showcase\noperations on multiple types (with overloading) and basic reasoning. `arith`\ndoes not have new datatypes, but it supports operations on tensors and vectors,\nwhich are some of the most complex builtin types.\n\nTODO: This file uses shorter operation names (without \"arith.\") to work around\n      a normalization performance issue that is affected by the string length\nSee https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/unfold.20essentially.20loops\n-/\n\nimport MLIR.Semantics.Fitree\nimport MLIR.Semantics.Semantics\nimport MLIR.Semantics.SSAEnv\nimport MLIR.Semantics.UB\nimport MLIR.Dialects.BuiltinModel\nimport MLIR.Util.Metagen\nimport MLIR.AST\nimport MLIR.EDSL\nimport MLIR.Semantics.Rewriting\nopen MLIR.AST\n\n/-\n### Dialect extensions\n\n`arith` has no extended types or attributes.\n-/\n\ninstance arith: Dialect Void Void (fun _ => Unit) where\n  name := \"arith\"\n  i\u03b1 := inferInstance\n  i\u03b5 := inferInstance\n\n/-\n### Dialect operations\n\nIn order to support type overloads while keeping reasonably-strong typing on\noperands and disallowing incorrect types in the operation arguments, we define\nscalar, tensor, and vector overloads of each operation.\n-/\n\ninductive ComparisonPred :=\n  | eq  | ne\n  | slt | sle | sgt | sge\n  | ult | ule | ugt | uge\n\ndef ComparisonPred.ofInt: Int \u2192 Option ComparisonPred\n  | 0 => some eq\n  | 1 => some ne\n  | 2 => some slt\n  | 3 => some sle\n  | 4 => some sgt\n  | 5 => some sge\n  | 6 => some ult\n  | 7 => some ule\n  | 8 => some ugt\n  | 9 => some uge\n  | _ => none\n\n-- inductive ArithE: Type \u2192 Type :=\n--   | CmpI: (sz: Nat) \u2192 (pred: ComparisonPred) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt 1)\n--   | CmpIndex: (pred: ComparisonPred) \u2192 (lhs rhs: Int) \u2192\n--           ArithE (FinInt 1)\n--   | AddI: (sz: Nat) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | AddT: (sz: Nat) \u2192 (D: DimList) \u2192 (lhs rhs: RankedTensor D (.int sgn sz)) \u2192\n--           ArithE (RankedTensor D (.int sgn sz))\n--   | AddV: (sz: Nat) \u2192 (sc fx: List Nat) \u2192\n--           (lhs rhs: Vector sc fx (.int sgn sz)) \u2192\n--           ArithE (Vector sc fx (.int sgn sz))\n--   | SubI: (sz: Nat) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | NegI: (sz: Nat) \u2192 (op: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | AndI: (sz: Nat) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | OrI: (sz: Nat) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | XorI: (sz: Nat) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n--   | Zext: (sz\u2081: Nat) \u2192 (sz\u2082: Nat) \u2192 (FinInt sz\u2081) \u2192\n--           ArithE (FinInt sz\u2082)\n--   | Select: (sz: Nat) \u2192 (b: FinInt 1) \u2192 (lhs rhs: FinInt sz) \u2192\n--           ArithE (FinInt sz)\n\ndef unary_semantics_op (op: IOp \u0394)\n      (ctor: {sz: Nat} \u2192 FinInt sz \u2192 FinInt sz): OpM \u0394 (TypedArgs \u0394) :=\n  match op with\n  | IOp.mk _ _ [\u27e8.int sgn sz, arg\u27e9]  [] _ => do\n      let r := ctor arg\n      return [\u27e8.int sgn sz, r\u27e9]\n  | IOp.mk name .. =>  OpM.Unhandled s!\"unary_semantics_op: unhandled {name}\"\n\ndef binary_semantics_op {\u0394: Dialect \u03b1' \u03c3' \u03b5'}\n      (name: String) (args: List ((\u03c4: MLIRType \u0394) \u00d7 \u03c4.eval))\n      (ctor: {sz: Nat} \u2192 FinInt sz \u2192 FinInt sz \u2192 FinInt sz): OpM \u0394 (TypedArgs \u0394) :=\n  match args with\n  | [\u27e8.int sgn sz, lhs\u27e9, \u27e8.int sgn' sz', rhs\u27e9] =>\n      if EQ: sgn = sgn' /\\ sz = sz' then  do\n        let r := ctor lhs (EQ.2 \u25b8 rhs)\n        return [\u27e8.int sgn sz, r\u27e9]\n      else OpM.Unhandled s!\"binary_semantics_op: sgn != sgn' || sz != sz': {name}\"\n  | _ => OpM.Unhandled s!\"binary_semantics_op: unhandled {name}\"\n\ndef cmpIndex (pred : ComparisonPred) (lhs rhs: Int): FinInt 1 :=\n      let b: Bool :=\n        match pred with\n        | .eq  => lhs = rhs\n        | .ne  => lhs != rhs\n        | .slt => lhs <  rhs\n        | .sle => lhs <= rhs\n        | .sgt => lhs >  rhs\n        | .sge => lhs >= rhs\n        | .ult => lhs <  rhs\n        | .ule => lhs <= rhs\n        | .ugt => lhs >  rhs\n        | .uge => lhs >= rhs\n      FinInt.ofInt 1 (if b then 1 else 0)\n\n\ndef cmpI (sz : \u2115) (pred : ComparisonPred) (lhs: FinInt sz) (rhs: FinInt sz): FinInt 1 :=\n      let b: Bool :=\n        match pred with\n        | .eq  => lhs = rhs\n        | .ne  => lhs != rhs\n        | .slt => lhs.toSint <  rhs.toSint\n        | .sle => lhs.toSint <= rhs.toSint\n        | .sgt => lhs.toSint >  rhs.toSint\n        | .sge => lhs.toSint >= rhs.toSint\n        | .ult => lhs.toUint <  rhs.toUint\n        | .ule => lhs.toUint <= rhs.toUint\n        | .ugt => lhs.toUint >  rhs.toUint\n        | .uge => lhs.toUint >= rhs.toUint\n      FinInt.ofInt 1 (if b then 1 else 0)\n\ndef arith_semantics_op (o: IOp \u0394): OpM \u0394 (TypedArgs \u0394) :=\n  match o with\n  | IOp.mk \"arith.constant\" [\u03c4\u2081] [] [] attrs =>\n      match AttrDict.find attrs \"value\" with\n      | some (.int value \u03c4\u2082) =>\n          if \u03c4\u2081 = \u03c4\u2082 then\n            match \u03c4\u2082 with\n            | .int sgn sz => do\n                -- TODO: Check range of constants\n                let v := FinInt.ofInt sz value\n                return [\u27e8.int sgn sz, v\u27e9]\n            | .index => do\n                return [\u27e8.index, value\u27e9]\n            | _ => OpM.Error s! \"arith.constant: unknown type {\u03c4\u2082}\"\n          else OpM.Error \"arith.constant: retty not equal to value ty\"\n      | some _\n      | nofne => OpM.Error \"arith.constant: cannot find value\"\n\n  | IOp.mk \"arith.cmpi\" _ [ \u27e8(.int sgn sz), lhs\u27e9, \u27e8(.int sgn' sz'), rhs\u27e9 ] []\n    attrs =>\n      if EQ: sgn = sgn' /\\ sz = sz' then\n            match attrs.find \"predicate\" with\n            | some (.int n (.int .Signless 64)) =>\n                match (ComparisonPred.ofInt n) with\n                | some pred => do\n                  let r := cmpI sz pred lhs (EQ.2 \u25b8 rhs)\n                  return [\u27e8.i1, r\u27e9]\n                | none => OpM.Unhandled \"arith.cmpi\"\n            | some _\n            | none => OpM.Unhandled \"arith.cmpi\"\n      else OpM.Unhandled \"arith.cmpi\"\n\n  | IOp.mk \"arith.cmpi\" _ [ \u27e8.index, lhs\u27e9, \u27e8.index, rhs\u27e9 ] [] attrs =>\n      match attrs.find \"predicate\" with\n      | some (.int n (.int .Signless 64)) =>\n          match (ComparisonPred.ofInt n) with\n          | some pred => do\n            let r := cmpIndex pred lhs rhs\n            return [\u27e8.i1, r\u27e9]\n          | none => OpM.Unhandled \"arith.cmpi\"\n      | some _\n      | none => OpM.Unhandled \"arith.cmpi\"\n\n  | IOp.mk \"arith.zext\" [.int sgn\u2082 sz\u2082] [\u27e8.int sgn\u2081 sz\u2081, value\u27e9]  [] _ =>\n      if sgn\u2081 = sgn\u2082 then do\n        let r :=  FinInt.zext sz\u2082 value\n        return [\u27e8.int sgn\u2082 sz\u2082, r\u27e9]\n      else OpM.Unhandled \"arith.zext\"\n\n  | IOp.mk \"arith.select\" _ [\u27e8.i1, b\u27e9, \u27e8.int sgn sz, lhs\u27e9, \u27e8.int sgn' sz', rhs\u27e9]  [] _ =>\n      if EQ: sgn = sgn' /\\ sz = sz' then  do\n        let r := if b.toUint = 1 then lhs else (EQ.2 \u25b8 rhs)\n        return [\u27e8.int sgn sz, r\u27e9]\n      else OpM.Unhandled \"arith.select\"\n  | IOp.mk \"arith.negi\" .. =>\n      unary_semantics_op o FinInt.neg\n  | IOp.mk name _ args _ _  =>\n      if name = \"arith.addi\" then\n        binary_semantics_op name args FinInt.add\n      else if name = \"arith.subi\" then\n        binary_semantics_op name args FinInt.sub\n      else if name = \"arith.andi\" then\n        binary_semantics_op name args FinInt.and\n      else if name = \"arith.ori\" then\n        binary_semantics_op name args FinInt.or\n      else if name = \"arith.xori\" then\n        binary_semantics_op name args FinInt.xor\n      else\n        OpM.Unhandled (s!\"generic_check {name}\")\n\ninstance: Semantics arith where\n  semantics_op := arith_semantics_op\n\n\n/-\n### Semantics of individual operations\n\nIn principle we would compute the semantics of entire programs simply by\nunfolding the definitions. But simp and dsimp have many problems which makes\nthis extremely slow, buggy, or infeasible even for programs with only a couple\nof operations. We work around this issue by precomputing the semantics of\nindividual operations and then substituting them as needed.\n-/\n\nprivate abbrev ops.constant (output: SSAVal) (value: Int):\n    Op arith :=\n  Op.mk \"arith.constant\" [\u27e8output, .i32\u27e9] [] [] (.mk [.mk \"value\" (.int value .i32)])\n\nprivate abbrev ops.negi (output input: SSAVal): Op arith :=\n  .mk \"arith.negi\" [\u27e8output, .i32\u27e9] [\u27e8input, .i32\u27e9] [] (.mk [])\n\nprivate abbrev ops.zext (sz\u2081 sz\u2082: Nat) (output input: SSAVal): Op arith :=\n  .mk \"arith.zext\" [\u27e8output, .int .Signless sz\u2082\u27e9] [\u27e8input, .int .Signless sz\u2081\u27e9] [] (.mk [])\n\nprivate abbrev ops.select (output cond t f: SSAVal): Op arith :=\n  .mk \"arith.select\" [\u27e8output, .i32\u27e9] [\u27e8cond, .i1\u27e9, \u27e8t, .i32\u27e9, \u27e8f, .i32\u27e9] [] (.mk [])\n\nprivate abbrev ops._binary (name: String) (output lhs rhs: SSAVal):\n    Op arith :=\n  .mk name [\u27e8output, .i32\u27e9] [\u27e8lhs, .i32\u27e9, \u27e8rhs, .i32\u27e9] [] (.mk [])\n\nprivate abbrev ops.addi := ops._binary \"arith.addi\"\nprivate abbrev ops.subi := ops._binary \"arith.subi\"\nprivate abbrev ops.andi := ops._binary \"arith.andi\"\nprivate abbrev ops.ori  := ops._binary \"arith.ori\"\nprivate abbrev ops.xori := ops._binary \"arith.xori\"\n\n/-\n\nThe great Commenting\n====================\n\nEverything below assumes we have a handle + semantics, which makes\nthings quite complex. Now that we have removed the handle, the hope\nis that we can recover the semantics proofs in a much easier fashion.\n\n\nprivate theorem ops.constant.sem output value:\n    denoteOp arith (ops.constant output value) =\n  Fitree.Vis (E := SSAEnvE arith +' Semantics.E arith +' UBE)\n    (Sum.inl <| SSAEnvE.Set .i32 output (FinInt.ofInt 32 value)) fun _ =>\n  Fitree.ret (TypedArgs.Next (\u03b4 := arith)\n    \u27e8.i32, FinInt.ofInt 32 value\u27e9) := by\n  simp [ops.constant, denoteOp, denoteOpBase, Semantics.semantics_op]\n  simp_itree\n  simp [arith_semantics_op]\n  simp [List.map]\n\nprivate theorem ops.negi.sem output input:\n    denoteOp arith (ops.negi output input) =\n  Fitree.Vis (E := SSAEnvE arith +' Semantics.E arith +' UBE)\n    (Sum.inl <| SSAEnvE.Get .i32 input) fun r =>\n  Fitree.Vis (Sum.inr <| Sum.inl <| ArithE.NegI 32 r) fun r =>\n  Fitree.Vis (Sum.inl <| SSAEnvE.Set .i32 output r) fun _ =>\n  Fitree.ret (TypedArgs.Next \u27e8.i32, r\u27e9) := by\n  simp [ops.negi, denoteOp, denoteOpBase, Semantics.semantics_op]\n  simp_itree\n\nprivate theorem ops.zext.sem sz\u2081 sz\u2082 output input:\n    denoteOp arith (ops.zext sz\u2081 sz\u2082 output input) =\n  Fitree.Vis (E := SSAEnvE arith +' Semantics.E arith +' UBE)\n    (Sum.inl <| @SSAEnvE.Get _ _ _ _ (.int .Signless sz\u2081) (instInhabitedEval _) input) fun r =>\n  Fitree.Vis (Sum.inr <| Sum.inl <| ArithE.Zext sz\u2081 sz\u2082 r) fun r =>\n  Fitree.Vis (Sum.inl <| SSAEnvE.Set (.int .Signless sz\u2082) output r) fun _ =>\n  Fitree.ret (TypedArgs.Next \u27e8.int .Signless sz\u2082, r\u27e9) := by\n  simp [ops.zext, denoteOp, denoteOpBase, Semantics.semantics_op]\n  simp_itree\n\n\nprivate theorem ops.select.sem output cond t f:\n    denoteOp arith (ops.select output cond t f) =\n  Fitree.Vis (E := SSAEnvE arith +' UBE)\n    (Sum.inl <| SSAEnvE.Get .i1 cond) fun cond =>\n  Fitree.Vis (E := SSAEnvE arith +' UBE)\n    (Sum.inl <| SSAEnvE.Get .i32 t) fun t =>\n  Fitree.Vis (E := SSAEnvE arith +' UBE)\n    (Sum.inl <| SSAEnvE.Get .i32 f) fun f =>\n  Fitree.Vis (Sum.inr <| Sum.inl <| ArithE.Select 32 cond t f) fun r =>\n  Fitree.Vis (Sum.inl <| SSAEnvE.Set .i32 output r) fun _ =>\n  Fitree.ret (TypedArgs.Next \u27e8.i32, r\u27e9) := by\n  simp [ops.select, denoteOp, denoteOpBase, Semantics.semantics_op]\n  simp_itree\n\nprivate theorem ops._binary.sem name ctor output lhs rhs:\n    (forall (n m: FinInt 32),\n      arith_semantics_op (\u0394 := arith)\n        (IOp.mk name [.i32] [\u27e8.i32, n\u27e9, \u27e8.i32, m\u27e9] [] 0 (.mk []))  =\n      binary_semantics_op name [\u27e8.i32, n\u27e9, \u27e8.i32, m\u27e9] ctor) \u2192\n    denoteOp arith (ops._binary name output lhs rhs) =\n  Fitree.Vis (E := SSAEnvE arith +' Semantics.E arith +' UBE)\n    (Sum.inl <| SSAEnvE.Get .i32 lhs) fun lhs =>\n  Fitree.Vis (Sum.inl <| SSAEnvE.Get .i32 rhs) fun rhs =>\n  Fitree.Vis (Sum.inr <| Sum.inl <| ctor 32 lhs rhs) fun r =>\n  Fitree.Vis (Sum.inl <| SSAEnvE.Set .i32 output r) fun _ =>\n  Fitree.ret (TypedArgs.Next \u27e8.i32, r\u27e9) := by\n  intro h\n  simp [denoteOp, denoteOpBase, Semantics.semantics_op]\n  simp [List.zip, List.zipWith, List.mapM, List.map]\n  simp [h];\n  sorry -- the proof broke when updating to the new Lean version.\n\nprivate abbrev ops.addi.sem output lhs rhs :=\n  ops._binary.sem \"arith.addi\" ArithE.AddI output lhs rhs (fun _ _ => rfl)\nprivate abbrev ops.subi.sem output lhs rhs :=\n  ops._binary.sem \"arith.subi\" ArithE.SubI output lhs rhs (fun _ _ => rfl)\nprivate abbrev ops.andi.sem output lhs rhs :=\n  ops._binary.sem \"arith.andi\" ArithE.AndI output lhs rhs (fun _ _ => rfl)\nprivate abbrev ops.ori.sem output lhs rhs :=\n  ops._binary.sem \"arith.ori\" ArithE.OrI output lhs rhs (fun _ _ => rfl)\nprivate abbrev ops.xori.sem output lhs rhs :=\n  ops._binary.sem \"arith.xori\" ArithE.XorI output lhs rhs (fun _ _ => rfl)\n\n/-\n### Basic examples\n-/\n\nprivate def cst1: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %true = \"arith.constant\" () {value = 1: i1}: () -> i1\n    %false = \"arith.constant\" () {value = 0: i1}: () -> i1\n    %r1 = \"arith.constant\" () {value = 25: i32}: () -> i32\n    %r2 = \"arith.constant\" () {value = 17: i32}: () -> i32\n    %r = \"arith.addi\" (%r1, %r2): (i32, i32) -> i32\n    %s = \"arith.subi\" (%r2, %r): (i32, i32) -> i32\n    %b1 = \"arith.cmpi\" (%r, %r1) {predicate = 5 /- sge -/}: (i32, i32) -> i1\n    %b2 = \"arith.cmpi\" (%r2, %r) {predicate = 8 /- ugt -/}: (i32, i32) -> i1\n]\n\n#eval run (\u0394 := arith) \u27e6cst1\u27e7 (SSAEnv.empty (\u03b4 := arith))\n-/\n\n/-\n### Rewriting heorems\n-/\n\n/- Commutativity of addition -/\n\nnamespace th1\ndef LHS: Op arith := [mlir_op|\n  %r = \"arith.addi\"(%n, %m): (i32, i32) -> i32\n]\ndef RHS: Op arith := [mlir_op|\n  %r = \"arith.addi\"(%m, %n): (i32, i32) -> i32\n]\n\ntheorem equivalent (n m: FinInt 32):\n    run \u27e6LHS\u27e7 (SSAEnv.One [ (\"n\", \u27e8.i32, n\u27e9), (\"m\", \u27e8.i32, m\u27e9) ]) =\n    run \u27e6RHS\u27e7 (SSAEnv.One [ (\"n\", \u27e8.i32, n\u27e9), (\"m\", \u27e8.i32, m\u27e9) ]) := by\n  simp [LHS, RHS,\n        run, StateT.run,\n        denoteOp, bind, List.mapM, StateT.bind, denoteOpArgs, List.mapM,\n        List.mapM.loop, Except.bind, TopM.get, StateT.get, pure, Except.pure,\n        StateT.pure, TopM.mapDenoteRegion, OpM.toTopM, TopM.set, StateT.set, MLIRType.eval,\n        SSAEnv.get, SSAEnv.getT, cast];\n  simp [FinInt.add_comm']\n\ndef th1 : PeepholeRewriteOp arith := \n{\n  findRoot := MTerm.buildOp \"arith.addi\" \n        [MTerm.buildOperand \"n\" .i32, MTerm.buildOperand \"m\" .i32]\n        [MTerm.buildOperand \"r\" .i32]\n  , findSubtree := []\n  , replaceSubtree := [MTerm.buildOp \"arith.addi\" \n        [MTerm.buildOperand \"m\" .i32, MTerm.buildOperand \"n\" .i32]\n        [MTerm.buildOperand \"r\" .i32]]\n  , wellformed := by {\n     intros toplevelProg _prog matchCtx replacedProg matchctx domctx \n     intros MATCH FIND SUBST DOMFIND\n     simp [List.append] at *;\n     sorry\n  } \n  , correct := by {\n     intros toplevelProg _prog matchCtx replacedProg matchctx domctx\n     intros MATCH FIND SUBST DOMFIND\n     simp [List.append] at *;\n     simp [MTerm.concretizeProg, List.mapM, List.mapM.loop] at FIND;\n     simp [MTerm.concretizeOp, MTerm.buildOp, MTerm.concretizeOperands, MTerm.concretizeOperand, MTerm.buildOperand,\n        MTerm.concretizeVariable, List.mapM, List.mapM.loop] at FIND;\n      -- cases on the MTerm.getVariable and show that we must have such a variable.\n      -- then generalize on this.\n      sorry\n\n  }\n}\n\nend th1\n\n/- LLVM InstCombine: `C-(X+C2) --> (C-C2)-X`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L1794 -/\n\ntheorem FinInt.sub_add_dist: forall (C X C2: FinInt sz),\n    C - (X + C2) = (C - C2) - X := by\n  intros C X C2\n  apply eq_of_toUint_cong2\n  simp [cong2, FinInt.sub_toUint, FinInt.add_toUint]\n  apply FinInt.mod2_fequal\n  simp [Int.sub_add_dist, Int.sub_assoc]\n\n/-\n\nnamespace th2\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %t = \"arith.addi\"(%X, %C2): (i32, i32) -> i32\n    %r = \"arith.subi\"(%C, %t): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %t = \"arith.subi\"(%C, %C2): (i32, i32) -> i32\n    %r = \"arith.subi\"(%t, %X): (i32, i32) -> i32\n]\ndef INPUT (C X C2: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"C\", \u27e8.i32, C\u27e9), (\"X\", \u27e8.i32, X\u27e9), (\"C2\", \u27e8.i32, C2\u27e9)\n]\n\ntheorem equivalent (C X C2: FinInt 32):\n    semanticPostCondition\u2082\n      (run \u27e6LHS\u27e7 (INPUT C X C2))\n      (run \u27e6RHS\u27e7 (INPUT C X C2))\n    fun _ env\u2081 _ env\u2082 =>\n      env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  apply FinInt.sub_add_dist\n\n/-\n-- This proof works, but triggers a Lean performance issue and basically loops.\n-- During testing, the trigger was unfolding `denoteTypedArgs`, though the\n-- culprit is specifically the combination of conditions that led to reducing\n-- WF-induction proofs within the terms. Changing `denoteTypedArgs` to a simple\n-- event `SSAEnvE.SetMultiple` did not eliminate the problem.\ntheorem equivalent2 (C X C2: FinInt 32):\n    (run \u27e6LHS\u27e7 (INPUT C X C2) |>.snd.get \"r\" .i32) =\n    (run \u27e6RHS\u27e7 (INPUT C X C2) |>.snd.get \"r\" .i32) := by\n  simp [LHS, RHS, INPUT, run, Semantics.handle]\n  simp [denoteBB, denoteBBStmts]\n  rw [ops.addi.sem]\n  rw [ops.subi.sem]\n  rw [ops.subi.sem]\n  rw [ops.subi.sem]\n  simp [interpUB'_bind]\n  simp [interpSSA'_bind]\n  repeat conv in SSAEnvE.handle _ _ => simp [SSAEnvE.handle]\n  simp [Fitree.interp_bind]\n  repeat conv in ArithE.handle _ _ => simp [ArithE.handle]\n  simp [cast_eq]\n  repeat conv in SSAEnvE.handle _ _ => simp [SSAEnvE.handle]\n  simp [cast_eq]\n  apply FinInt.sub_add_dist\n-/\nend th2\n\n/- LLVM InstCombine: `~X + C --> (C-1) - X`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L882 -/\n\ntheorem FinInt.comp_add: sz > 0 \u2192 forall (X C: FinInt sz),\n    (X ^^^ -1) + C = (C - 1) - X := by\n  intros h_sz X C\n  simp [\u2190FinInt.comp_eq_xor_minusOne]\n  apply eq_of_toUint_cong2\n  simp [cong2, FinInt.add_toUint, FinInt.comp_toUint, FinInt.sub_toUint]\n  simp [toUint_ofNat]\n  have h: Int.ofNat 1 = 1 := by decide\n  simp [h, mod2_idem \u27e8by decide, Int.one_lt_two_pow h_sz\u27e9]\n  simp [Int.sub_eq_add_neg, Int.add_assoc, FinInt.mod2_add_left]\n  rw [\u2190@Int.add_assoc _ (-1) _, @Int.add_comm _ (-1)]\n  simp [@Int.add_comm (-X.toUint), Int.add_assoc]\n\nnamespace th3\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.xori\"(%X, %_2): (i32, i32) -> i32\n    %r = \"arith.addi\"(%_3, %C): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %o = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %t = \"arith.subi\"(%C, %o): (i32, i32) -> i32\n    %r = \"arith.subi\"(%t, %X): (i32, i32) -> i32\n]\ndef INPUT (C X: FinInt 32): SSAEnv arith := SSAEnv.One [\n    (\"C\", \u27e8.i32, C\u27e9), (\"X\", \u27e8.i32, X\u27e9)\n]\n\ntheorem equivalent (C X: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT C X))\n    (run \u27e6RHS\u27e7 (INPUT C X))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  -- TODO: If we could simplify `denoteTypedArgs` we could save a lot of work\n  -- TODO: here. But it triggers the WF-induction bug, so we instead we unfold\n  -- TODO: the SSA stuff later. Investigate.\n  simp [INPUT, LHS, RHS, run, denoteOps, denoteBB]\n  rw [ops.constant.sem]\n  rw [ops.negi.sem]\n  rw [ops.xori.sem]\n  rw [ops.addi.sem]\n  rw [ops.constant.sem]\n  rw [ops.subi.sem]\n  rw [ops.subi.sem]\n  simp [interpSSA', Fitree.interpState, SSAEnvE.handle]; simp_itree\n  apply FinInt.comp_add (by decide)\nend th3\n\n/- LLVM InstCombine: `-A + -B --> -(A + B)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L1316 -/\n\ntheorem FinInt.neg_add_dist (A B: FinInt sz):\n    -(A + B) = -A + -B := by\n  apply eq_of_toUint_cong2\n  simp [cong2, neg_toUint, add_toUint]\n  apply mod2_fequal\n  simp [Int.neg_add]\n\nnamespace th4\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.negi\"(%A): (i32) -> i32\n    %_2 = \"arith.negi\"(%B): (i32) -> i32\n    %r = \"arith.addi\"(%_1, %_2): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.addi\"(%A, %B): (i32, i32) -> i32\n    %r = \"arith.negi\"(%_1): (i32) -> i32\n]\ndef INPUT (A B: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"A\", \u27e8.i32, A\u27e9), (\"B\", \u27e8.i32, B\u27e9)\n]\n\ntheorem equivalent (A B: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT A B))\n    (run \u27e6RHS\u27e7 (INPUT A B))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  rw [FinInt.neg_add_dist]\nend th4\n\n/- LLVM InstCombine: `-(X - Y) --> (Y - X)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L2290 -/\n\ntheorem FinInt.neg_sub_dist (X Y: FinInt sz):\n    -(X - Y) = Y - X := by\n  apply eq_of_toUint_cong2\n  simp [cong2, neg_toUint, sub_toUint]\n  apply mod2_fequal\n  simp [Int.neg_sub]\n\nnamespace th5\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.subi\"(%X, %Y): (i32, i32) -> i32\n    %r = \"arith.negi\"(%_1): (i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %r = \"arith.subi\"(%Y, %X): (i32, i32) -> i32\n]\ndef INPUT (X Y: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"X\", \u27e8.i32, X\u27e9), (\"Y\", \u27e8.i32, Y\u27e9)\n]\n\ntheorem equivalent (X Y: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT X Y))\n    (run \u27e6RHS\u27e7 (INPUT X Y))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  apply FinInt.neg_sub_dist\nend th5\n\n/- LLVM InstCombine: `(A + 1) + ~B --> A - B`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L1331 -/\n\ntheorem FinInt.plus_one_plus_comp (A B: FinInt sz):\n    (A + 1) + (B ^^^ -1) = A - B := by\n  simp [\u2190FinInt.comp_eq_xor_minusOne]\n  apply eq_of_toUint_cong2\n  simp [cong2, neg_toUint, add_toUint, sub_toUint, comp_toUint]\n  simp [toUint_ofNat, (by decide: Int.ofNat 1 = 1)]\n  -- Rearranging terms without the powerful Mathlib tactics is quite tedious\n  simp [Int.add_comm _ (mod2 _ _), Int.add_assoc]\n  simp [Int.sub_eq_add_neg]\n  simp [\u2190@Int.add_assoc A.toUint _, Int.add_comm A.toUint _]\n  simp [\u2190@Int.add_assoc 1 _, Int.add_comm 1 _]\n  simp [Int.add_assoc, mod2_add_left]\n  simp [Int.add_comm _ 1, \u2190Int.add_assoc _ 1 _, Int.add_assoc 1 _ _]\n  simp [\u2190Int.add_assoc, Int.add_right_neg, Int.zero_add]\n\nnamespace th6\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.addi\"(%A, %_1): (i32, i32) -> i32\n    %_3 = \"arith.negi\"(%_1): (i32) -> i32\n    %_4 = \"arith.xori\"(%B, %_3): (i32, i32) -> i32\n    %r = \"arith.addi\"(%_2, %_4): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %r = \"arith.subi\"(%A, %B): (i32, i32) -> i32\n]\ndef INPUT (A B: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"A\", \u27e8.i32, A\u27e9), (\"B\", \u27e8.i32, B\u27e9)\n]\n\ntheorem equivalent (A B: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT A B))\n    (run \u27e6RHS\u27e7 (INPUT A B))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [INPUT, LHS, RHS, run, denoteOps, denoteBB]\n  rw [ops.constant.sem]\n  rw [ops.addi.sem]\n  rw [ops.negi.sem]\n  rw [ops.xori.sem]\n  rw [ops.addi.sem]\n  rw [ops.subi.sem]\n  simp [interpSSA', Fitree.interpState, SSAEnvE.handle]; simp_itree\n  apply FinInt.plus_one_plus_comp\nend th6\n\n/- LLVM InstCombine: `(~X) - (~Y) --> Y - X`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L1867 -/\n\ntheorem FinInt.comp_sub_comp (X Y: FinInt sz):\n    (X ^^^ -1) - (Y ^^^ -1) = Y - X := by\n  simp [\u2190FinInt.comp_eq_xor_minusOne]\n  apply eq_of_toUint_cong2\n  simp [cong2, sub_toUint, comp_toUint]\n  simp [Int.sub_eq_add_neg, Int.add_assoc, Int.neg_add, Int.neg_neg]\n  simp [mod2_add_left]\n  simp [\u2190Int.add_assoc]\n  rw [Int.add_comm _ (-_)]\n  have h x: mod2 (-(2^sz) + x) sz = mod2 x sz := by sorry_arith\n  simp [Int.add_assoc, h]\n  rw [\u2190Int.add_assoc (-1) _ _, Int.add_comm (-1) _]\n  rw [Int.add_assoc _ (-1) _, \u2190Int.add_assoc _ 1 _]\n  rw [Int.add_left_neg, Int.zero_add, Int.add_comm]\n\nnamespace th7\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.xori\"(%X, %_2): (i32, i32) -> i32\n    %_4 = \"arith.xori\"(%Y, %_2): (i32, i32) -> i32\n    %r = \"arith.subi\"(%_3, %_4): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %r = \"arith.subi\"(%Y, %X): (i32, i32) -> i32\n]\ndef INPUT (X Y: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"X\", \u27e8.i32, X\u27e9), (\"Y\", \u27e8.i32, Y\u27e9)\n]\n\ntheorem equivalent (X Y: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT X Y))\n    (run \u27e6RHS\u27e7 (INPUT X Y))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [INPUT, LHS, RHS, run, denoteBB, denoteOps]\n  rw [ops.constant.sem]\n  rw [ops.negi.sem]\n  rw [ops.xori.sem]\n  rw [ops.xori.sem]\n  rw [ops.subi.sem]\n  rw [ops.subi.sem]\n  simp [interpSSA', Fitree.interpState, SSAEnvE.handle]; simp_itree\n  apply FinInt.comp_sub_comp\nend th7\n\n/- LLVM InstCombine: `(add (xor A, B) (and A, B)) --> (or A, B)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L1411 -/\n\ntheorem FinInt.addfull_xor_and (A B: FinInt sz):\n    addfull (A ^^^ B) (A &&& B) = .next false (A ||| B) := by\n  induction sz with\n  | zero => cases A; cases B; decide\n  | succ sz ih =>\n    match A, B with\n    | next bA A', next bB B' =>\n      simp [HXor.hXor, HAnd.hAnd, HOr.hOr, xor, and, or, logic2] at *\n      simp [addfull, ih]\n      cases bA <;> cases bB <;> decide;\n\n\ntheorem FinInt.add_xor_and (A B: FinInt sz):\n    (A ^^^ B) + (A &&& B) = (A ||| B) := by\n  simp [HAdd.hAdd, add, addfull_xor_and]\n\nnamespace th8\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.xori\"(%A, %B): (i32, i32) -> i32\n    %_2 = \"arith.andi\"(%A, %B): (i32, i32) -> i32\n    %r = \"arith.addi\"(%_1, %_2): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %r = \"arith.ori\"(%A, %B): (i32, i32) -> i32\n]\ndef INPUT (A B: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"A\", \u27e8.i32, A\u27e9), (\"B\", \u27e8.i32, B\u27e9)\n]\n\ntheorem equivalent (A B: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT A B))\n    (run \u27e6RHS\u27e7 (INPUT A B))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  apply FinInt.add_xor_and\nend th8\n\n/- LLVM InstCombine: `zext(bool) + C --> bool ? C + 1 : C`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAddSub.cpp#L873 -/\n\ntheorem FinInt.add_bool_eq_select (B: FinInt 1) (C: FinInt 32):\n    zext 32 B + C = select B (C + 1) C := by\n  apply eq_of_toUint_cong2\n  simp [cong2, add_toUint]\n  rw [zext_toUint' (by decide)]\n  cases bool_cases B <;> subst B <;> simp [select, Int.add_assoc, Int.zero_add]\n  rw [add_toUint]\n  simp [toUint, (by decide: 2^0 = 1), Int.add_zero]\n  simp [Int.add_comm]\n\nnamespace th9\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.zext\"(%B): (i1) -> i32\n    %r = \"arith.addi\"(%_1, %C): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.addi\"(%C, %_1): (i32, i32) -> i32\n    %r = \"arith.select\"(%B, %_2, %C): (i1, i32, i32) -> i32\n]\ndef INPUT (B: FinInt 1) (C: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"B\", \u27e8.i1, B\u27e9), (\"C\", \u27e8.i32, C\u27e9)\n]\n\ntheorem equivalent (B: FinInt 1) (C: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT B C))\n    (run \u27e6RHS\u27e7 (INPUT B C))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  simp [List.map, Semantics.semantics_op, arith_semantics_op]; simp_itree\n  apply FinInt.add_bool_eq_select\nend th9\n\n/- LLVM InstCombine: `(A & ~B) & ~C --> A & ~(B | C)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAndOrXor.cpp#L1340 -/\n\ntheorem FinInt.and_not_and_not (A B C: FinInt sz):\n    (A &&& (B ^^^ -1)) &&& (C ^^^ -1) = A &&& ((B ||| C) ^^^ -1) := by\n  simp [\u2190comp_eq_xor_minusOne]\n  induction sz with\n  | zero => cases A; cases B; cases C; decide\n  | succ sz ih =>\n      match A, B, C with\n      | .next bA A', .next bB B', .next bC C' =>\n          simp [HAnd.hAnd, HOr.hOr, and, or, comp] at *\n          simp [logic2, ih]\n          cases bA <;> cases bB <;> cases bC <;> decide\n\nnamespace th10\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.xori\"(%B, %_2): (i32, i32) -> i32\n    %_4 = \"arith.andi\"(%A, %_3): (i32, i32) -> i32\n    %_5 = \"arith.xori\"(%C, %_2): (i32, i32) -> i32\n    %r = \"arith.andi\"(%_4, %_5): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.ori\"(%B, %C): (i32, i32) -> i32\n    %_4 = \"arith.xori\"(%_3, %_2): (i32, i32) -> i32\n    %r = \"arith.andi\"(%A, %_4): (i32, i32) -> i32\n]\ndef INPUT (A B C: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"A\", \u27e8.i32, A\u27e9), (\"B\", \u27e8.i32, B\u27e9), (\"C\", \u27e8.i32, C\u27e9)\n]\n\ntheorem equivalent (A B C: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT A B C))\n    (run \u27e6RHS\u27e7 (INPUT A B C))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [INPUT, LHS, RHS, run, denoteBB, denoteOps]\n  rw [ops.constant.sem]\n  rw [ops.negi.sem]\n  rw [ops.xori.sem]\n  rw [ops.andi.sem]\n  rw [ops.xori.sem]\n  rw [ops.andi.sem]\n  rw [ops.ori.sem]\n  rw [ops.xori.sem]\n  rw [ops.andi.sem]\n  simp [interpSSA', Fitree.interpState, SSAEnvE.handle]; simp_itree\n  apply FinInt.and_not_and_not\nend th10\n\n/- LLVM InstCombine: `(A & B) | ~(A | B) --> ~(A ^ B)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAndOrXor.cpp#L1510 -/\n\ntheorem FinInt.and_or_not_or (A B: FinInt sz):\n    (A &&& B) ||| ((A ||| B) ^^^ -1) = ((A ^^^ B) ^^^ -1) := by\n  simp [\u2190comp_eq_xor_minusOne]\n  induction sz with\n  | zero => cases A; cases B; decide\n  | succ sz ih =>\n      match A, B with\n      | .next bA A', .next bB B' =>\n          simp [HXor.hXor, HAnd.hAnd, HOr.hOr, xor, and, or, comp] at *\n          simp [logic1, logic2, ih]\n          cases bA <;> cases bB <;> decide\n\nnamespace th11\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.ori\"(%A, %B): (i32, i32) -> i32\n    %_4 = \"arith.xori\"(%_3, %_2): (i32, i32) -> i32\n    %_5 = \"arith.andi\"(%A, %B): (i32, i32) -> i32\n    %r = \"arith.ori\"(%_5, %_4): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.constant\"() {value = 1: i32}: () -> i32\n    %_2 = \"arith.negi\"(%_1): (i32) -> i32\n    %_3 = \"arith.xori\"(%A, %B): (i32, i32) -> i32\n    %r = \"arith.xori\"(%_3, %_2): (i32, i32) -> i32\n]\ndef INPUT (A B: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"A\", \u27e8.i32, A\u27e9), (\"B\", \u27e8.i32, B\u27e9)\n]\n\ntheorem equivalent (A B: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT A B))\n    (run \u27e6RHS\u27e7 (INPUT A B))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [INPUT, LHS, RHS, run, denoteBB, denoteOps]\n  rw [ops.constant.sem]\n  rw [ops.negi.sem]\n  rw [ops.ori.sem]\n  rw [ops.xori.sem]\n  rw [ops.andi.sem]\n  rw [ops.ori.sem]\n  rw [ops.xori.sem]\n  rw [ops.xori.sem]\n  simp [interpSSA', Fitree.interpState, SSAEnvE.handle]; simp_itree\n  apply FinInt.and_or_not_or\nend th11\n\n/- LLVM InstCombine: `(X ^ C1) & C2 --> (X & C2) ^ (C1&C2)`\n   https://github.com/llvm/llvm-project/blob/291e3a85658e264a2918298e804972bd68681af8/llvm/lib/Transforms/InstCombine/InstCombineAndOrXor.cpp#L1778 -/\n\ntheorem FinInt.xor_and (X C\u2081 C\u2082: FinInt sz):\n    (X ^^^ C\u2081) &&& C\u2082 = (X &&& C\u2082) ^^^ (C\u2081 &&& C\u2082) := by\n  induction sz with\n  | zero => cases X; cases C\u2081; cases C\u2082; decide\n  | succ sz ih =>\n      match X, C\u2081, C\u2082 with\n      | .next bX X', .next bC\u2081 C\u2081', .next bC\u2082 C\u2082' =>\n          simp [HXor.hXor, HAnd.hAnd, HOr.hOr, xor, and, or] at *\n          simp [logic2, ih]\n          cases bX <;> cases bC\u2081 <;> cases bC\u2082 <;> decide\n\nnamespace th12\ndef LHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.xori\"(%X, %C1): (i32, i32) -> i32\n    %r = \"arith.andi\"(%_1, %C2): (i32, i32) -> i32\n]\ndef RHS: BasicBlock arith := [mlir_bb|\n  ^bb:\n    %_1 = \"arith.andi\"(%X, %C2): (i32, i32) -> i32\n    %_2 = \"arith.andi\"(%C1, %C2): (i32, i32) -> i32\n    %r = \"arith.xori\"(%_1, %_2): (i32, i32) -> i32\n]\ndef INPUT (X C\u2081 C\u2082: FinInt 32): SSAEnv arith := SSAEnv.One [\n  (\"X\", \u27e8.i32, X\u27e9), (\"C1\", \u27e8.i32, C\u2081\u27e9), (\"C2\", \u27e8.i32, C\u2082\u27e9)\n]\n\ntheorem equivalent (X C\u2081 C\u2082: FinInt 32):\n  semanticPostCondition\u2082\n    (run \u27e6LHS\u27e7 (INPUT X C\u2081 C\u2082))\n    (run \u27e6RHS\u27e7 (INPUT X C\u2081 C\u2082))\n  fun _ env\u2081 _ env\u2082 =>\n    env\u2081.get \"r\" .i32 = env\u2082.get \"r\" .i32 := by\n  simp [LHS, RHS, INPUT]\n  simp [run, denoteBB, denoteOps, denoteOp, denoteOpBase]; simp_itree\n  apply FinInt.xor_and\nend th12\n-/\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Dialects/ArithSemantics.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3657914835423665}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.model_category\nimport category_theory.limits.comma\n\nnoncomputable theory\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {C J : Type*} [category C] [category J]\n\nnamespace under\n\n@[simp]\ndef nat_trans {X : C} (F : J \u2964 under X) : (functor.const J).obj X \u27f6 F \u22d9 under.forget X :=\nnat_trans.hcomp (\ud835\udfd9 F) (comma.nat_trans _ _)\n\n@[simp]\ndef arrow\u2081 {X : C} (F : J \u2964 under X) [has_colimit (F \u22d9 under.forget _)]\n  [has_colimit ((functor.const J).obj X)] : colimit ((functor.const J).obj X) \u27f6 colimit (F \u22d9 under.forget _) :=\ncolim_map (under.nat_trans F)\n\n@[simp]\ndef arrow\u2082 {X : C} (F : J \u2964 under X) [has_colimit ((functor.const J).obj X)] :\n  colimit ((functor.const J).obj X) \u27f6 X := colimit.desc _ (cocone.mk X (\ud835\udfd9 _))\n\n@[simps]\ndef cocone {X : C} (F : J \u2964 under X) [has_colimit (F \u22d9 under.forget _)]\n  [has_colimit ((functor.const J).obj X)] [has_pushout (under.arrow\u2081 F) (under.arrow\u2082 F)] : cocone F :=\nbegin\n  apply cocone.mk (under.mk (pushout.inr : X \u27f6 pushout (under.arrow\u2081 F) (under.arrow\u2082 F))),\n  exact\n  { app := \u03bb j, under.hom_mk ((colimit.\u03b9 (F \u22d9 under.forget X) j) \u226b pushout.inl) begin\n      dsimp,\n      have eq : (F.obj j).hom = (under.nat_trans F).app j := (category.comp_id _).symm,\n      erw [eq, \u2190 category.assoc, \u2190 \u03b9_colim_map, category.assoc, pushout.condition, \u2190 category.assoc,\n        colimit.\u03b9_desc, nat_trans.id_app, category.id_comp],\n    end,\n    naturality' := \u03bb j j' f, begin\n      ext,\n      simp only [category.assoc, under.comp_right, under.hom_mk_right, functor.const_obj_map],\n      erw [category.comp_id, \u2190 category.assoc],\n      congr,\n      exact colimit.w (F \u22d9 under.forget X) f,\n    end },\nend\n\ndef cocone_is_colimit {X : C} (F : J \u2964 under X) [has_colimit (F \u22d9 under.forget _)]\n  [has_colimit ((functor.const J).obj X)] [has_pushout (under.arrow\u2081 F) (under.arrow\u2082 F)] :\n  is_colimit (under.cocone F) :=\n{ desc := \u03bb s, begin\n    refine under.hom_mk (pushout.desc (colimit.desc _ (cocone.mk s.X.right (s.\u03b9 \u25eb (\ud835\udfd9 (under.forget X))))) s.X.hom _) _,\n    { ext j,\n      simp only [under.arrow\u2081, under.arrow\u2082, under.nat_trans, \u03b9_colim_map_assoc,\n        nat_trans.hcomp_app, comma.nat_trans_app, nat_trans.id_app, functor.comp_map,\n        comma.snd_map, under.id_right, functor.id_map, colimit.\u03b9_desc, category.id_comp,\n        nat_trans.hcomp_id_app, under.forget_map, category.assoc, colimit.\u03b9_desc_assoc],\n      erw [category.id_comp, under.w],\n      refl, },\n    { dsimp,\n      simp only [pushout.inr_desc], },\n  end,\n  fac' := \u03bb s j, begin\n    ext,\n    simp only [colimit.\u03b9_desc, nat_trans.hcomp_id_app, under.forget_map, category.assoc,\n      colimit.\u03b9_desc_assoc, category.id_comp, under.w, pushout.inr_desc, under.cocone_\u03b9_app,\n      under.comp_right, under.hom_mk_right, pushout.inl_desc],\n  end,\n  uniq' := \u03bb s m h, begin\n    ext j,\n    { simp only [colimit.\u03b9_desc, nat_trans.hcomp_id_app, under.forget_map, category.assoc,\n        colimit.\u03b9_desc_assoc, category.id_comp, under.w, pushout.inr_desc, under.hom_mk_right,\n        pushout.inl_desc, \u2190 h j, under.cocone_\u03b9_app, under.comp_right], },\n    { simpa only [pushout.inr_desc, under.hom_mk_right] using under.w m, },\n  end, }\n\ndef colimit_cocone {X : C} (F : J \u2964 under X) [has_colimit (F \u22d9 under.forget _)]\n  [has_colimit ((functor.const J).obj X)] [has_pushout (under.arrow\u2081 F) (under.arrow\u2082 F)] : colimit_cocone F :=\n{ cocone := under.cocone F,\n  is_colimit := under.cocone_is_colimit F, }\n\ninstance {X : C} (F : J \u2964 under X) [has_colimit (F \u22d9 under.forget _)]\n  [has_colimit ((functor.const J).obj X)] [has_pushout (under.arrow\u2081 F) (under.arrow\u2082 F)] : has_colimit F :=\n\u27e8nonempty.intro (under.colimit_cocone F)\u27e9\n\nlemma has_colimits_of_shape (X : C) [has_colimits_of_shape J C] [has_pushouts C] :\n  has_colimits_of_shape J (under X) := {}\n\ninstance (X : C) [has_finite_colimits C] : has_finite_colimits (under X) :=\n\u27e8by { introsI J hJ hJ', apply under.has_colimits_of_shape, }\u27e9\n\ninstance (X : C) [has_finite_limits C] : has_finite_limits (under X) :=\n\u27e8by { introsI J hJ hJ', apply comma.has_limits_of_shape, }\u27e9\n\nend under\n\nend limits\n\nend category_theory\n\nopen category_theory\n.\nnamespace algebraic_topology\n\nnamespace model_category\n\nvariables {C : Type*} [category C]\n\ninstance model_category_under [M : model_category C] (X : C) : model_category (under X) :=\n{ to_category_with_fib_cof_weq := M.to_category_with_fib_cof_weq.inverse_image (under.forget X),\n  CM1axiom := \u27e8infer_instance, infer_instance\u27e9,\n  CM2axiom := CM2axiom.inverse_image (under.forget _),\n  CM3axiom := CM3.inverse_image (under.forget _),\n  CM4axiom := \u27e8CM4a.under X, CM4b.under X\u27e9,\n  CM5axiom := \u27e8CM5a.under X, CM5b.under X\u27e9, }\n\nend model_category\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3657914835423665}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro\n-/\nimport data.list.join\n\n/-!\n# Permutations of a list\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we prove properties about `list.permutations`, a list of all permutations of a list. It\nis defined in `data.list.defs`.\n\n## Order of the permutations\n\nDesigned for performance, the order in which the permutations appear in `list.permutations` is\nrather intricate and not very amenable to induction. That's why we also provide `list.permutations'`\nas a less efficient but more straightforward way of listing permutations.\n\n### `list.permutations`\n\nTODO. In the meantime, you can try decrypting the docstrings.\n\n### `list.permutations'`\n\nThe list of partitions is built by recursion. The permutations of `[]` are `[[]]`. Then, the\npermutations of `a :: l` are obtained by taking all permutations of `l` in order and adding `a` in\nall positions. Hence, to build `[0, 1, 2, 3].permutations'`, it does\n* `[[]]`\n* `[[3]]`\n* `[[2, 3], [3, 2]]]`\n* `[[1, 2, 3], [2, 1, 3], [2, 3, 1], [1, 3, 2], [3, 1, 2], [3, 2, 1]]`\n* `[[0, 1, 2, 3], [1, 0, 2, 3], [1, 2, 0, 3], [1, 2, 3, 0],`\n   `[0, 2, 1, 3], [2, 0, 1, 3], [2, 1, 0, 3], [2, 1, 3, 0],`\n   `[0, 2, 3, 1], [2, 0, 3, 1], [2, 3, 0, 1], [2, 3, 1, 0],`\n   `[0, 1, 3, 2], [1, 0, 3, 2], [1, 3, 0, 2], [1, 3, 2, 0],`\n   `[0, 3, 1, 2], [3, 0, 1, 2], [3, 1, 0, 2], [3, 1, 2, 0],`\n   `[0, 3, 2, 1], [3, 0, 2, 1], [3, 2, 0, 1], [3, 2, 1, 0]]`\n\n## TODO\n\nShow that `l.nodup \u2192 l.permutations.nodup`. See `data.fintype.list`.\n-/\n\nopen nat\n\nvariables {\u03b1 \u03b2 : Type*}\n\nnamespace list\n\nlemma permutations_aux2_fst (t : \u03b1) (ts : list \u03b1) (r : list \u03b2) : \u2200 (ys : list \u03b1) (f : list \u03b1 \u2192 \u03b2),\n  (permutations_aux2 t ts r ys f).1 = ys ++ ts\n| []      f := rfl\n| (y::ys) f := match _, permutations_aux2_fst ys _ : \u2200 o : list \u03b1 \u00d7 list \u03b2, o.1 = ys ++ ts \u2192\n      (permutations_aux2._match_1 t y f o).1 = y :: ys ++ ts with\n  | \u27e8_, zs\u27e9, rfl := rfl\n  end\n\n@[simp] lemma permutations_aux2_snd_nil (t : \u03b1) (ts : list \u03b1) (r : list \u03b2) (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t ts r [] f).2 = r := rfl\n\n@[simp] lemma permutations_aux2_snd_cons (t : \u03b1) (ts : list \u03b1) (r : list \u03b2) (y : \u03b1) (ys : list \u03b1)\n  (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t ts r (y::ys) f).2 = f (t :: y :: ys ++ ts) ::\n    (permutations_aux2 t ts r ys (\u03bbx : list \u03b1, f (y::x))).2 :=\nmatch _, permutations_aux2_fst t ts r _ _ : \u2200 o : list \u03b1 \u00d7 list \u03b2, o.1 = ys ++ ts \u2192\n   (permutations_aux2._match_1 t y f o).2 = f (t :: y :: ys ++ ts) :: o.2 with\n| \u27e8_, zs\u27e9, rfl := rfl\nend\n\n/-- The `r` argument to `permutations_aux2` is the same as appending. -/\nlemma permutations_aux2_append (t : \u03b1) (ts : list \u03b1) (r : list \u03b2) (ys : list \u03b1) (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t ts nil ys f).2 ++ r = (permutations_aux2 t ts r ys f).2 :=\nby induction ys generalizing f; simp *\n\n/-- The `ts` argument to `permutations_aux2` can be folded into the `f` argument. -/\nlemma permutations_aux2_comp_append {t : \u03b1} {ts ys : list \u03b1} {r : list \u03b2} (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t [] r ys $ \u03bb x, f (x ++ ts)).2 = (permutations_aux2 t ts r ys f).2 :=\nbegin\n  induction ys generalizing f,\n  { simp },\n  { simp [ys_ih (\u03bb xs, f (ys_hd :: xs))] },\nend\n\nlemma map_permutations_aux2' {\u03b1 \u03b2 \u03b1' \u03b2'} (g : \u03b1 \u2192 \u03b1') (g' : \u03b2 \u2192 \u03b2')\n  (t : \u03b1) (ts ys : list \u03b1) (r : list \u03b2) (f : list \u03b1 \u2192 \u03b2) (f' : list \u03b1' \u2192 \u03b2')\n  (H : \u2200 a, g' (f a) = f' (map g a)) :\n  map g' (permutations_aux2 t ts r ys f).2 =\n  (permutations_aux2 (g t) (map g ts) (map g' r) (map g ys) f').2 :=\nbegin\n  induction ys generalizing f f'; simp *,\n  apply ys_ih, simp [H],\nend\n\n/-- The `f` argument to `permutations_aux2` when `r = []` can be eliminated. -/\nlemma map_permutations_aux2 (t : \u03b1) (ts : list \u03b1) (ys : list \u03b1) (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t ts [] ys id).2.map f = (permutations_aux2 t ts [] ys f).2 :=\nbegin\n  rw [map_permutations_aux2' id, map_id, map_id], refl,\n  simp\nend\n\n/-- An expository lemma to show how all of `ts`, `r`, and `f` can be eliminated from\n`permutations_aux2`.\n\n`(permutations_aux2 t [] [] ys id).2`, which appears on the RHS, is a list whose elements are\nproduced by inserting `t` into every non-terminal position of `ys` in order. As an example:\n```lean\n#eval permutations_aux2 1 [] [] [2, 3, 4] id\n-- [[1, 2, 3, 4], [2, 1, 3, 4], [2, 3, 1, 4]]\n```\n-/\nlemma permutations_aux2_snd_eq (t : \u03b1) (ts : list \u03b1) (r : list \u03b2) (ys : list \u03b1) (f : list \u03b1 \u2192 \u03b2) :\n  (permutations_aux2 t ts r ys f).2 =\n    (permutations_aux2 t [] [] ys id).2.map (\u03bb x, f (x ++ ts)) ++ r :=\nby rw [\u2190 permutations_aux2_append, map_permutations_aux2, permutations_aux2_comp_append]\n\nlemma map_map_permutations_aux2 {\u03b1 \u03b1'} (g : \u03b1 \u2192 \u03b1') (t : \u03b1) (ts ys : list \u03b1) :\n  map (map g) (permutations_aux2 t ts [] ys id).2 =\n  (permutations_aux2 (g t) (map g ts) [] (map g ys) id).2 :=\nmap_permutations_aux2' _ _ _ _ _ _ _ _ (\u03bb _, rfl)\n\nlemma map_map_permutations'_aux (f : \u03b1 \u2192 \u03b2) (t : \u03b1) (ts : list \u03b1) :\n  map (map f) (permutations'_aux t ts) = permutations'_aux (f t) (map f ts) :=\nby induction ts with a ts ih; [refl, {simp [\u2190 ih], refl}]\n\nlemma permutations'_aux_eq_permutations_aux2 (t : \u03b1) (ts : list \u03b1) :\n  permutations'_aux t ts = (permutations_aux2 t [] [ts ++ [t]] ts id).2 :=\nbegin\n  induction ts with a ts ih, {refl},\n  simp [permutations'_aux, permutations_aux2_snd_cons, ih],\n  simp only [\u2190 permutations_aux2_append] {single_pass := tt},\n  simp [map_permutations_aux2],\nend\n\nlemma mem_permutations_aux2 {t : \u03b1} {ts : list \u03b1} {ys : list \u03b1} {l l' : list \u03b1} :\n  l' \u2208 (permutations_aux2 t ts [] ys (append l)).2 \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 :=\nbegin\n  induction ys with y ys ih generalizing l,\n  { simp {contextual := tt} },\n  rw [permutations_aux2_snd_cons, show (\u03bb (x : list \u03b1), l ++ y :: x) = append (l ++ [y]),\n      by funext; simp, mem_cons_iff, ih], split,\n  { rintro (rfl | \u27e8l\u2081, l\u2082, l0, rfl, rfl\u27e9),\n    { exact \u27e8[], y::ys, by simp\u27e9 },\n    { exact \u27e8y::l\u2081, l\u2082, l0, by simp\u27e9 } },\n  { rintro \u27e8_ | \u27e8y', l\u2081\u27e9, l\u2082, l0, ye, rfl\u27e9,\n    { simp [ye] },\n    { simp only [cons_append] at ye, rcases ye with \u27e8rfl, rfl\u27e9,\n      exact or.inr \u27e8l\u2081, l\u2082, l0, by simp\u27e9 } }\nend\n\nlemma mem_permutations_aux2' {t : \u03b1} {ts : list \u03b1} {ys : list \u03b1} {l : list \u03b1} :\n  l \u2208 (permutations_aux2 t ts [] ys id).2 \u2194\n    \u2203 l\u2081 l\u2082, l\u2082 \u2260 [] \u2227 ys = l\u2081 ++ l\u2082 \u2227 l = l\u2081 ++ t :: l\u2082 ++ ts :=\nby rw [show @id (list \u03b1) = append nil, by funext; refl]; apply mem_permutations_aux2\n\nlemma length_permutations_aux2 (t : \u03b1) (ts : list \u03b1) (ys : list \u03b1) (f : list \u03b1 \u2192 \u03b2) :\n  length (permutations_aux2 t ts [] ys f).2 = length ys :=\nby induction ys generalizing f; simp *\n\nlemma foldr_permutations_aux2 (t : \u03b1) (ts : list \u03b1) (r L : list (list \u03b1)) :\n  foldr (\u03bby r, (permutations_aux2 t ts r y id).2) r L =\n    L.bind (\u03bb y, (permutations_aux2 t ts [] y id).2) ++ r :=\nby induction L with l L ih; [refl, {simp [ih], rw \u2190 permutations_aux2_append}]\n\nlemma mem_foldr_permutations_aux2 {t : \u03b1} {ts : list \u03b1} {r L : list (list \u03b1)} {l' : list \u03b1} :\n  l' \u2208 foldr (\u03bby r, (permutations_aux2 t ts r y id).2) 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 :=\nhave (\u2203 (a : list \u03b1), a \u2208 L \u2227\n    \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),\nfrom \u27e8\u03bb \u27e8a, aL, l\u2081, l\u2082, l0, e, h\u27e9, \u27e8l\u2081, l\u2082, l0, e \u25b8 aL, h\u27e9,\n      \u03bb \u27e8l\u2081, l\u2082, l0, aL, h\u27e9, \u27e8_, aL, l\u2081, l\u2082, l0, rfl, h\u27e9\u27e9,\nby rw foldr_permutations_aux2; simp [mem_permutations_aux2', this,\n  or.comm, or.left_comm, or.assoc, and.comm, and.left_comm, and.assoc]\n\nlemma length_foldr_permutations_aux2 (t : \u03b1) (ts : list \u03b1) (r L : list (list \u03b1)) :\n  length (foldr (\u03bby r, (permutations_aux2 t ts r y id).2) r L) = sum (map length L) + length r :=\nby simp [foldr_permutations_aux2, (\u2218), length_permutations_aux2]\n\nlemma length_foldr_permutations_aux2' (t : \u03b1) (ts : list \u03b1) (r L : list (list \u03b1))\n  (n) (H : \u2200 l \u2208 L, length l = n) :\n  length (foldr (\u03bby r, (permutations_aux2 t ts r y id).2) r L) = n * length L + length r :=\nbegin\n  rw [length_foldr_permutations_aux2, (_ : sum (map length L) = n * length L)],\n  induction L with l L ih, {simp},\n  have sum_map : sum (map length L) = n * length L :=\n    ih (\u03bb l m, H l (mem_cons_of_mem _ m)),\n  have length_l : length l = n := H _ (mem_cons_self _ _),\n  simp [sum_map, length_l, mul_add, add_comm]\nend\n\n@[simp] lemma permutations_aux_nil (is : list \u03b1) : permutations_aux [] is = [] :=\nby rw [permutations_aux, permutations_aux.rec]\n\n@[simp] lemma permutations_aux_cons (t : \u03b1) (ts is : list \u03b1) :\n  permutations_aux (t :: ts) is = foldr (\u03bby r, (permutations_aux2 t ts r y id).2)\n    (permutations_aux ts (t::is)) (permutations is) :=\nby rw [permutations_aux, permutations_aux.rec]; refl\n\n@[simp] lemma permutations_nil : permutations ([] : list \u03b1) = [[]] :=\nby rw [permutations, permutations_aux_nil]\n\nlemma map_permutations_aux (f : \u03b1 \u2192 \u03b2) : \u2200 (ts is : list \u03b1),\n  map (map f) (permutations_aux ts is) = permutations_aux (map f ts) (map f is) :=\nbegin\n  refine permutations_aux.rec (by simp) _,\n  introv IH1 IH2, rw map at IH2,\n  simp only [foldr_permutations_aux2, map_append, map, map_map_permutations_aux2, permutations,\n    bind_map, IH1, append_assoc, permutations_aux_cons, cons_bind, \u2190 IH2, map_bind],\nend\n\nlemma map_permutations (f : \u03b1 \u2192 \u03b2) (ts : list \u03b1) :\n  map (map f) (permutations ts) = permutations (map f ts) :=\nby rw [permutations, permutations, map, map_permutations_aux, map]\n\nlemma map_permutations' (f : \u03b1 \u2192 \u03b2) (ts : list \u03b1) :\n  map (map f) (permutations' ts) = permutations' (map f ts) :=\nby induction ts with t ts ih; [refl, simp [\u2190 ih, map_bind, \u2190 map_map_permutations'_aux, bind_map]]\n\n\n\nlemma permutations_append (is ts : list \u03b1) :\n  permutations (is ++ ts) = (permutations is).map (++ ts) ++ permutations_aux ts is.reverse :=\nby simp [permutations, permutations_aux_append]\n\nend list\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/list/permutation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.3657023646718346}}
{"text": "\nimport for_mathlib.derived.les_facts\nimport laurent_measures.ses2\nimport invpoly.ses\nimport Lbar.ses\nimport Lbar.ext\nimport free_pfpng.acyclic\nimport challenge_notations\n\n.\n\nuniverse u\n\nopen category_theory\nopen_locale nnreal\n\nnamespace laurent_measures\n\nvariables (p' p : \u211d\u22650) [fact (0 < p')] [fact (p' < p)] [fact (p \u2264 1)]\nvariables (S : Profinite.{0})\n\nlocal notation `r'` := @r p'\nlocal notation `r` := @r p\n\n-- move me\ninstance fact_half_pos : fact ((0:\u211d\u22650) < 2\u207b\u00b9) := \u27e8by simp\u27e9\n\nlemma epi_and_is_iso\n  (V : SemiNormedGroup.{0}) [normed_with_aut r V] [complete_space V] [separated_space V]\n  (hV : \u2200 (v : V), (normed_with_aut.T.inv v) = 2 \u2022 v) :\n  epi (((Ext' 0).map ((condensify_Tinv2 (Fintype_LaurentMeasures r')).app S).op).app\n    (Condensed.of_top_ab V)) \u2227\n  \u2200 i > 0, is_iso (((Ext' i).map ((condensify_Tinv2 (Fintype_LaurentMeasures r')).app S).op).app\n    (Condensed.of_top_ab V)) :=\nbegin\n  have SES := Lbar.short_exact.{0 0} r' S,\n  haveI : fact (r < r'),\n  { refine \u27e8nnreal.rpow_lt_rpow_of_exponent_gt _ _ _\u27e9,\n    { exact fact.out _ },\n    { apply nnreal.two_inv_lt_one },\n    { norm_cast, exact fact.out _ },\n  },\n  haveI : fact (r < 1) := \u27e8(fact.out _ : r < r').trans (fact.out _)\u27e9,\n  haveI : fact (p' \u2264 1) := \u27e8(fact.out _ : p' < p).le.trans (fact.out _)\u27e9,\n  haveI : fact (p' < 1) := \u27e8lt_of_lt_of_le (fact.out _ : p' < p) (fact.out _ : p \u2264 1)\u27e9,\n  rw \u2190 epi_and_is_iso_iff_of_is_iso _ _ _ _\n    ((condensify_Tinv2 _).app S) ((condensify_Tinv2 _).app S) ((condensify_Tinv2 _).app S)\n    _ _ (Condensed.of_top_ab V) SES SES (Lbar.is_iso_Tinv2 r r' S V hV),\n  { rw \u2190 is_zero_iff_epi_and_is_iso _ _ (Condensed.of_top_ab V) (invpoly.short_exact p' S),\n    intros i hi,\n    apply (free_pfpng_acyclic S V i hi).of_iso _,\n    apply iso.app _ _,\n    refine (Ext' i).map_iso _,\n    exact (as_iso ((cond_free_pfpng_to_normed_free_pfpng.{0 0} p').app S)).op, },\n  { rw [\u2190 nat_trans.comp_app, condensify_map_comp_Tinv2, nat_trans.comp_app], },\n  { rw [\u2190 nat_trans.comp_app, condensify_map_comp_Tinv2, nat_trans.comp_app], }\nend\n\n\nend laurent_measures\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/laurent_measures/ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7931059609645724, "lm_q2_score": 0.4610167793123159, "lm_q1q2_score": 0.3656351557772865}}
{"text": "/-\nCopyright (c) 2020 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n-/\nimport category_theory.natural_isomorphism\nimport category_theory.equivalence\nimport category_theory.eq_to_hom\n\n/-!\n# Quotient category\n\nConstructs the quotient of a category by an arbitrary family of relations on its hom-sets,\nby introducing a type synonym for the objects, and identifying homs as necessary.\n\nThis is analogous to 'the quotient of a group by the normal closure of a subset', rather\nthan 'the quotient of a group by a normal subgroup'. When taking the quotient by a congruence\nrelation, `functor_map_eq_iff` says that no unnecessary identifications have been made.\n-/\n\n/-- A `hom_rel` on `C` consists of a relation on every hom-set. -/\n@[derive inhabited]\ndef hom_rel (C) [quiver C] := \u03a0 \u2983X Y : C\u2984, (X \u27f6 Y) \u2192 (X \u27f6 Y) \u2192 Prop\n\nnamespace category_theory\n\nvariables {C : Type*} [category C] (r : hom_rel C)\n\ninclude r\n\n/-- A `hom_rel` is a congruence when it's an equivalence on every hom-set, and it can be composed\nfrom left and right. -/\nclass congruence : Prop :=\n(is_equiv : \u2200 {X Y}, is_equiv _ (@r X Y))\n(comp_left : \u2200 {X Y Z} (f : X \u27f6 Y) {g g' : Y \u27f6 Z}, r g g' \u2192 r (f \u226b g) (f \u226b g'))\n(comp_right : \u2200 {X Y Z} {f f' : X \u27f6 Y} (g : Y \u27f6 Z), r f f' \u2192 r (f \u226b g) (f' \u226b g))\n\nattribute [instance] congruence.is_equiv\n\n/-- A type synonym for `C`, thought of as the objects of the quotient category. -/\n@[ext]\nstructure quotient := (as : C)\n\ninstance [inhabited C] : inhabited (quotient r) := \u27e8 { as := default C } \u27e9\n\nnamespace quotient\n\n/-- Generates the closure of a family of relations w.r.t. composition from left and right. -/\ninductive comp_closure \u2983s t : C\u2984 : (s \u27f6 t) \u2192 (s \u27f6 t) \u2192 Prop\n| intro {a b} (f : s \u27f6 a) (m\u2081 m\u2082 : a \u27f6 b) (g : b \u27f6 t) (h : r m\u2081 m\u2082) :\n  comp_closure (f \u226b m\u2081 \u226b g) (f \u226b m\u2082 \u226b g)\n\nlemma comp_left {a b c : C} (f : a \u27f6 b) : \u03a0 (g\u2081 g\u2082 : b \u27f6 c) (h : comp_closure r g\u2081 g\u2082),\n  comp_closure r (f \u226b g\u2081) (f \u226b g\u2082)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro (f \u226b x) m\u2081 m\u2082 y h\n\nlemma comp_right {a b c : C} (g : b \u27f6 c) : \u03a0 (f\u2081 f\u2082 : a \u27f6 b) (h : comp_closure r f\u2081 f\u2082),\n  comp_closure r (f\u2081 \u226b g) (f\u2082 \u226b g)\n| _ _ \u27e8x, m\u2081, m\u2082, y, h\u27e9 := by simpa using comp_closure.intro x m\u2081 m\u2082 (y \u226b g) h\n\n/-- Hom-sets of the quotient category. -/\ndef hom (s t : quotient r) := quot $ @comp_closure C _ r s.as t.as\n\ninstance (a : quotient r) : inhabited (hom r a a) := \u27e8quot.mk _ (\ud835\udfd9 a.as)\u27e9\n\n/-- Composition in the quotient category. -/\ndef comp \u2983a b c : quotient r\u2984 : hom r a b \u2192 hom r b c \u2192 hom r a c :=\n\u03bb hf hg, quot.lift_on hf ( \u03bb f, quot.lift_on hg (\u03bb g, quot.mk _ (f \u226b g))\n  (\u03bb g\u2081 g\u2082 h, quot.sound $ comp_left r f g\u2081 g\u2082 h) )\n  (\u03bb f\u2081 f\u2082 h, quot.induction_on hg $ \u03bb g, quot.sound $ comp_right r g f\u2081 f\u2082 h)\n\n@[simp]\nlemma comp_mk {a b c : quotient r} (f : a.as \u27f6 b.as) (g : b.as \u27f6 c.as) :\n  comp r (quot.mk _ f) (quot.mk _ g) = quot.mk _ (f \u226b g) := rfl\n\ninstance category : category (quotient r) :=\n{ hom := hom r,\n  id := \u03bb a, quot.mk _ (\ud835\udfd9 a.as),\n  comp := comp r }\n\n/-- The functor from a category to its quotient. -/\n@[simps]\ndef functor : C \u2964 quotient r :=\n{ obj := \u03bb a, { as := a },\n  map := \u03bb _ _ f, quot.mk _ f }\n\nnoncomputable instance : full (functor r) :=\n{ preimage := \u03bb X Y f, quot.out f, }\n\ninstance : ess_surj (functor r) :=\n{ mem_ess_image := \u03bb Y, \u27e8Y.as, \u27e8eq_to_iso (by { ext, refl, })\u27e9\u27e9 }\n\nprotected lemma induction {P : \u03a0 {a b : quotient r}, (a \u27f6 b) \u2192 Prop}\n  (h : \u2200 {x y : C} (f : x \u27f6 y), P ((functor r).map f)) :\n  \u2200 {a b : quotient r} (f : a \u27f6 b), P f :=\nby { rintros \u27e8x\u27e9 \u27e8y\u27e9 \u27e8f\u27e9, exact h f, }\n\nprotected lemma sound {a b : C} {f\u2081 f\u2082 : a \u27f6 b} (h : r f\u2081 f\u2082) :\n  (functor r).map f\u2081 = (functor r).map f\u2082 :=\nby simpa using quot.sound (comp_closure.intro (\ud835\udfd9 a) f\u2081 f\u2082 (\ud835\udfd9 b) h)\n\n\n\nvariables {D : Type*} [category D]\n  (F : C \u2964 D)\n  (H : \u2200 (x y : C) (f\u2081 f\u2082 : x \u27f6 y), r f\u2081 f\u2082 \u2192 F.map f\u2081 = F.map f\u2082)\ninclude H\n\n/-- The induced functor on the quotient category. -/\n@[simps]\ndef lift : quotient r \u2964 D :=\n{ obj := \u03bb a, F.obj a.as,\n  map := \u03bb a b hf, quot.lift_on hf (\u03bb f, F.map f)\n    (by { rintros _ _ \u27e8_, _, _, _, _, _, h\u27e9, simp [H _ _ _ _ h], }),\n  map_id' := \u03bb a, F.map_id a.as,\n  map_comp' := by { rintros a b c \u27e8f\u27e9 \u27e8g\u27e9, exact F.map_comp f g, } }\n\n/-- The original functor factors through the induced functor. -/\ndef lift.is_lift : (functor r) \u22d9 lift r F H \u2245 F :=\nnat_iso.of_components (\u03bb X, iso.refl _) (by tidy)\n\n@[simp]\nlemma lift.is_lift_hom (X : C) : (lift.is_lift r F H).hom.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n@[simp]\nlemma lift.is_lift_inv (X : C) : (lift.is_lift r F H).inv.app X = \ud835\udfd9 (F.obj X) :=\nrfl\n\nlemma lift_map_functor_map {X Y : C} (f : X \u27f6 Y) :\n  (lift r F H).map ((functor r).map f) = F.map f :=\nby { rw \u2190(nat_iso.naturality_1 (lift.is_lift r F H)), dsimp, simp, }\n\nend quotient\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/quotient.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.36545081167571325}}
{"text": "import spaces.cont_diff_map_support_in\nimport measure_theory.function.l1_space\nimport analysis.locally_convex.with_seminorms\n\nopen filter topological_space set measure_theory\nopen_locale topological_space filter pointwise bounded_cont_diff_map ennreal\n\nsection prelim\n\nvariables {\u03b1 : Type*} (S : set \u03b1) [topological_space \u03b1]\n\ndef cocompact_in : filter \u03b1 :=\n\u2a05 (s : set S) (h : is_compact s), \ud835\udcdf (coe '' s : set \u03b1)\u1d9c\n\nlemma has_basis_cocompact_in' : (cocompact_in S).has_basis (is_compact : set S \u2192 Prop) \n  (compl \u2218 image coe) :=\nhas_basis_binfi_principal'\n  (\u03bb s hs t ht, \u27e8s \u222a t, hs.union ht, compl_subset_compl.2 \n    (image_subset _ $ subset_union_left s t),\n    compl_subset_compl.2 (image_subset _ $ subset_union_right s t)\u27e9)\n  \u27e8\u2205, is_compact_empty\u27e9\n\nlemma has_basis_cocompact_in : (cocompact_in S).has_basis (\u03bb K : set \u03b1, is_compact K \u2227 K \u2286 S) \n  compl :=\nsorry\n\nlemma cocompact_le_cocompact_in : cocompact \u03b1 \u2264 cocompact_in S :=\n\u03bb s hs, let \u27e8t, ht, hts\u27e9 := (has_basis_cocompact_in' S).mem_iff.mp hs in \n  mem_cocompact.mpr \u27e8coe '' t, ht.image continuous_subtype_coe, hts\u27e9\n\nend prelim\n\nprivate def test_function_submodule (\ud835\udd5c E F : Type*) [nondiscrete_normed_field \ud835\udd5c] \n  [normed_group E] [normed_group F] [normed_space \ud835\udd5c E] [normed_space \ud835\udd5c F] (\u03a9 : set E) \n  (n : with_top \u2115) : submodule \ud835\udd5c (E \u2192 F) :=\n{ carrier := {f | cont_diff \ud835\udd5c n f \u2227 f =\u1da0[cocompact_in \u03a9] 0}, -- TODO !!!!!!\n  zero_mem' := \u27e8cont_diff_zero_fun, by refl\u27e9,\n  add_mem' := \u03bb f g hf hg, \u27e8hf.1.add hg.1, \n    by filter_upwards [hf.2, hg.2] using \u03bb x hfx hgx, \n      by rw [pi.add_apply, hfx, hgx, pi.zero_apply, add_zero]\u27e9,\n  smul_mem' := \u03bb c f hf, \u27e8cont_diff_const.smul hf.1, \n    by filter_upwards [hf.2] using \u03bb x hfx, \n      by rw [pi.smul_apply, hfx, pi.zero_apply, smul_zero]\u27e9 }\n\ndef test_function (\ud835\udd5c E F : Type*) [nondiscrete_normed_field \ud835\udd5c] [normed_group E] \n  [normed_group F] [normed_space \ud835\udd5c E] [normed_space \ud835\udd5c F] (\u03a9 : set E) (n : with_top \u2115) :=\n\u21a5(test_function_submodule \ud835\udd5c E F \u03a9 n)\n\nlocalized \"notation `Cc^`n`\u27ee`\u03a9`,`E`,`F`;`\ud835\udd5c`\u27ef` := test_function \ud835\udd5c E F \u03a9 n\" in \n  test_function\n\nnamespace test_function\n\nsection general\n\nvariables {\ud835\udd5c E F : Type*} [nondiscrete_normed_field \ud835\udd5c] [normed_group E] [normed_group F]\n  [normed_space \ud835\udd5c E] [normed_space \ud835\udd5c F] {\u03a9 : set E} {n : with_top \u2115} \n  {f g : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef} {x : E}\n\ninstance : add_comm_group (Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) := submodule.add_comm_group _\ninstance : module \ud835\udd5c (Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) := submodule.module _\ninstance : has_coe_to_fun (Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) (\u03bb _, E \u2192 F) := \u27e8\u03bb f, f.1\u27e9\n\n@[ext] lemma ext (H : \u2200x, f x = g x) : f = g :=\nby {ext, exact H x}\n\nlemma cont_diff (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  cont_diff \ud835\udd5c n f :=\nf.2.1\n\nprotected lemma continuous (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  continuous f :=\nf.cont_diff.continuous\n\nlemma eventually_eq_cocompact_in (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) : \n  f =\u1da0[cocompact_in \u03a9] 0 :=\nf.2.2\n\nlemma eventually_eq_cocompact (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) : \n  f =\u1da0[cocompact E] 0 :=\ncocompact_le_cocompact_in \u03a9 f.2.2\n\nlemma has_compact_support (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  has_compact_support f :=\nbegin\n  rw [has_compact_support_iff_eventually_eq, coclosed_compact_eq_cocompact],\n  exact f.eventually_eq_cocompact\nend\n\nprotected def tsupport (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  compacts E :=\n\u27e8tsupport f, f.has_compact_support\u27e9\n\nprotected lemma tsupport_subset (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  \u2191f.tsupport \u2286 \u03a9 :=\nbegin\n  sorry\nend\n\nvariables (\ud835\udd5c) (F) (n)\n\ndef of_support_in (K : compacts E) (hK : \u2191K \u2286 \u03a9)\n  (f : cont_diff_map_supported_in \ud835\udd5c E F K n) : \nCc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef :=\n\u27e8f, f.cont_diff, (has_basis_cocompact_in \u03a9).mem_iff.mpr \u27e8K, \u27e8K.2, hK\u27e9, f.supported_in\u27e9\u27e9\n\ndef to_support_in {K : set E} (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) (hK : \u2200 x \u2209 K, f x = 0) :\n  cont_diff_map_supported_in \ud835\udd5c E F K n :=\n\u27e8f, f.cont_diff, hK\u27e9\n\ndef to_support_in_tsupport (f : Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef) :\n  cont_diff_map_supported_in \ud835\udd5c E F f.tsupport n :=\n\u27e8f, f.cont_diff, \u03bb x, image_eq_zero_of_nmem_tsupport\u27e9\n\ndef of_support_in\u2097 (K : compacts E) (hK : \u2191K \u2286 \u03a9) :\n  cont_diff_map_supported_in \ud835\udd5c E F K n \n    \u2192\u2097[\ud835\udd5c] Cc^n\u27ee\u03a9, E, F; \ud835\udd5c\u27ef :=\n{ to_fun := of_support_in \ud835\udd5c F n K hK,\n  map_add' := \u03bb f g, by ext; refl,\n  map_smul' := \u03bb f g, by ext; refl }\n\nend general\n\nsection real\n\nvariables {E F : Type*} [normed_group E] [normed_group F]\n  [normed_space \u211d E] [normed_space \u211d F] {n : with_top \u2115} {\u03a9 : set E}\n  {f g : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef} {x : E}\n\nprivate noncomputable def tmp_topology : topological_space (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) := \n  \u2a06 (K : compacts E) (hK : \u2191K \u2286 \u03a9), coinduced (of_support_in\u2097 \u211d F n K hK) infer_instance\n\nlocal notation `\ud835\udce3\u2080` := tmp_topology\n\nprotected def topology : topological_space (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) := \nInf { t | \ud835\udce3\u2080 \u2264 t \u2227 @topological_add_group _ t _ \u2227 @has_continuous_smul \u211d _ _ _ t \u2227 \n          @locally_convex_space \u211d _ _ _ _ t }\n\nlocal notation `\ud835\udce3` := test_function.topology\n\nprivate lemma tmp_topology_le_topology : \n  (\ud835\udce3\u2080 : topological_space (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef)) \u2264 \ud835\udce3 := \nle_Inf (\u03bb t ht, ht.1)\n\nprivate lemma topology_le_iff {t : topological_space (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef)} \n  (h\u2081 : @topological_add_group _ t _) (h\u2082 : @has_continuous_smul \u211d _ _ _ t) \n  (h\u2083 : @locally_convex_space \u211d _ _ _ _ t) :\n\ud835\udce3 \u2264 t \u2194 \ud835\udce3\u2080 \u2264 t :=\nsorry -- consequence of general lemma\n\nprivate def basis_zero : filter_basis Cc^n\u27ee\u03a9, E, F; \u211d\u27ef :=\n{ sets := { s | s \u2208 (\u2a06 (K : compacts E) (hK : \u2191K \u2286 \u03a9), (\ud835\udcdd 0).map (of_support_in \u211d F n K hK)) \u2227 \n                convex \u211d s \u2227 absorbent \u211d s \u2227 s = -s },\n  nonempty := \u27e8univ, univ_mem, convex_univ, absorbent_univ, neg_univ.symm\u27e9,\n  inter_sets := \u03bb s t hs ht, \u27e8s \u2229 t, \u27e8inter_mem hs.1 ht.1, hs.2.1.inter ht.2.1, \n    sorry /- absorbent_inter doesn't exist -/, by rw [inter_neg, \u2190 hs.2.2.2, \u2190 ht.2.2.2]\u27e9, subset_refl _\u27e9 }\n\nprivate noncomputable def add_group_basis_zero : add_group_filter_basis Cc^n\u27ee\u03a9, E, F; \u211d\u27ef := \n{ zero' := sorry,\n  add' := sorry,\n  neg' := sorry,\n  conj' := sorry,\n  ..basis_zero }\n\nprivate noncomputable def module_basis_zero : module_filter_basis \u211d Cc^n\u27ee\u03a9, E, F; \u211d\u27ef :=\n{ smul' := sorry,\n  smul_left' := sorry,\n  smul_right' := sorry,\n  ..add_group_basis_zero }\n\nprivate noncomputable def basis_topology : topological_space (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) := \nmodule_basis_zero.topology\n\nlocal notation `\ud835\udce3\u2081` := basis_topology\n\nprivate lemma topology_eq_basis_topology : (\ud835\udce3 : topological_space Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) = \ud835\udce3\u2081 :=\nsorry\n\nprivate lemma continuous_of_support_in {K : compacts E} {hK : \u2191K \u2286 \u03a9} : \n  @continuous _ _ _ \ud835\udce3 (of_support_in \u211d F n K hK) :=\n@continuous.comp _ _ _ _ \ud835\udce3\u2080 \ud835\udce3 _ _ (continuous_id_of_le tmp_topology_le_topology) \n  (continuous_supr_rng $ continuous_supr_rng $ continuous_coinduced_rng)\n\nattribute [instance] test_function.topology\n\ninstance : topological_add_group Cc^n\u27ee\u03a9, E, F; \u211d\u27ef := \ntopological_add_group_Inf (\u03bb t ht, ht.2.1)\n\ninstance : has_continuous_smul \u211d Cc^n\u27ee\u03a9, E, F; \u211d\u27ef := \nhas_continuous_smul_Inf (\u03bb t ht, ht.2.2.1)\n\ninstance : locally_convex_space \u211d Cc^n\u27ee\u03a9, E, F; \u211d\u27ef := \nsorry\n\nvariables (F n)\n\nnoncomputable def of_support_inL (K : compacts E) (hK : \u2191K \u2286 \u03a9) :\n  cont_diff_map_supported_in \u211d E F K n \u2192L[\u211d] Cc^n\u27ee\u03a9, E, F; \u211d\u27ef :=\n{ to_linear_map := of_support_in\u2097 \u211d F n K hK,\n  cont := continuous_of_support_in } \n\nvariables {F n}\n\nlemma continuous_iff_of_linear {G : Type*} [tG : topological_space G] [add_comm_group G] [module \u211d G] \n  [topological_add_group G] [has_continuous_smul \u211d G] [locally_convex_space \u211d G] \n  (\u03c6 : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] G) : \n  continuous \u03c6 \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), continuous (\u03c6 \u2218\u2097 of_support_in\u2097 \u211d F n K hK) :=\nbegin\n  let tC : \u03a0 (K : compacts E) (hK : \u2191K \u2286 \u03a9), topological_space \n    (cont_diff_map_supported_in \u211d E F K n) :=\n    infer_instance,\n  calc  continuous \u03c6 \n      \u2194 \ud835\udce3 \u2264 tG.induced \u03c6 : continuous_iff_le_induced\n  ... \u2194 \ud835\udce3\u2080 \u2264 tG.induced \u03c6 : \n          topology_le_iff (topological_add_group_induced _) (has_continuous_smul_induced _) sorry\n  ... \u2194 \u2200 (K : compacts E), (\u2a06 (hK : \u2191K \u2286 \u03a9), coinduced (of_support_in\u2097 \u211d F n K hK) _) \n          \u2264 tG.induced \u03c6 : supr_le_iff\n  ... \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), coinduced (of_support_in\u2097 \u211d F n K hK) _ \u2264 tG.induced \u03c6 : \n          forall_congr (\u03bb K, supr_le_iff)\n  ... \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), _ \u2264 (tG.induced \u03c6).induced (of_support_in\u2097 \u211d F n K hK) : \n          forall_congr (\u03bb K, forall_congr $ \u03bb hK, coinduced_le_iff_le_induced)\n  ... \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), _ \u2264 tG.induced (\u03c6 \u2218\u2097 of_support_in\u2097 \u211d F n K hK) : \n          forall_congr (\u03bb K, forall_congr $ \u03bb hK, by rw [linear_map.coe_comp, induced_compose])\n  ... \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), continuous (\u03c6 \u2218\u2097 of_support_in\u2097 \u211d F n K hK) : \n          forall_congr (\u03bb K, forall_congr $ \u03bb hK, continuous_iff_le_induced.symm),\nend\n\n-- TODO : can we have different domains ?\nlemma continuous_of_commutes_of_linear {F' : Type*} [normed_group F']\n  [normed_space \u211d F'] (\u03c6 : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] Cc^n\u27ee\u03a9, E, F'; \u211d\u27ef) \n  (\u03c8 : \u03a0 (K : compacts E) (hK : \u2191K \u2286 \u03a9), \n    cont_diff_map_supported_in \u211d E F K n \u2192L[\u211d] cont_diff_map_supported_in \u211d E F' K n)\n  (hcomm : \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), \n    \u03c6 \u2218\u2097 of_support_in\u2097 \u211d F n K hK = of_support_in\u2097 \u211d F' n K hK \u2218\u2097 \u2191(\u03c8 K hK)) :\n  continuous \u03c6 :=\nbegin\n  rw continuous_iff_of_linear,\n  intros K hK,\n  rw hcomm K hK,\n  exact ((of_support_inL F' n K hK).comp (\u03c8 K hK)).continuous\nend\n\nlemma continuous_iff_of_linear_of_normed_codomain' {G : Type*} [normed_group G] \n  [normed_space \u211d G] (T : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] G) : \n  continuous T \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), \u2203 (p : \u2115), \u2203 C > 0, \u2200 f, \n    \u2225T (of_support_in \u211d F n K hK f)\u2225 \u2264 \n      C * (\u2a06 (i \u2264 p) (hin : \u2191i \u2264 n) (x : E), \u2225iterated_fderiv \u211d i f x\u2225) :=\nbegin\n  rw [continuous_iff_of_linear, forall_congr],\n  intros K,\n  rw forall_congr,\n  intros hK,\n  rw [cont_diff_map_supported_in.continuous_iff_of_linear, exists_congr],\n  intros p,\n  refl\nend\n\nlemma continuous_iff_of_linear_of_normed_codomain {G : Type*} [normed_group G] \n  [normed_space \u211d G] (T : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] G) : \n  continuous T \u2194 \u2200 (K : compacts E) (hK : \u2191K \u2286 \u03a9), \u2203 (p : \u2115), \u2203 C > (0 : \u211d), \u2200 f : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef, \n    (\u2200 x \u2209 K, f x = 0) \u2192 \u2225T f\u2225 \u2264 \n      C * (\u2a06 (i \u2264 p) (hin : \u2191i \u2264 n) (x : E), \u2225iterated_fderiv \u211d i f x\u2225) :=\nbegin\n  rw [continuous_iff_of_linear_of_normed_codomain', forall_congr],\n  intros K,\n  rw [forall_congr],\n  intros hK,\n  rw [exists_congr],\n  intros p,\n  rw [exists_congr],\n  intros C,\n  rw [exists_congr],\n  intros hC,\n  split; intros H f,\n  { intro hf,\n    convert H (to_support_in \u211d F n f hf),\n    ext,\n    refl },\n  { exact H (of_support_in \u211d F n K hK f) (\u03bb x hx, f.supported_in x hx) }\nend\n\n-- TODO : formulate this in term of bounded subsets\n\nnoncomputable def to_bounded_cont_diff_map (f : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) : \n  B^n\u27eeE,F;\u211d\u27ef :=\n(f.to_support_in_tsupport \u211d F n).to_bounded_cont_diff_map\n\nnoncomputable def to_bounded_cont_diff_map\u2097 : \n  Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] B^n\u27eeE ,F ; \u211d\u27ef := \n{ to_fun := to_bounded_cont_diff_map,\n  map_add' := \u03bb f g, by ext; refl,\n  map_smul' := \u03bb c f, by ext; refl }\n\nnoncomputable def to_bounded_cont_diff_mapL : \n  Cc^n\u27ee\u03a9, E, F; \u211d\u27ef \u2192L[\u211d] B^n\u27eeE ,F ; \u211d\u27ef := \n{ to_linear_map := to_bounded_cont_diff_map\u2097,\n  cont := \n  begin\n    change continuous to_bounded_cont_diff_map\u2097,\n    rw continuous_iff_of_linear,\n    intros K hK,\n    exact cont_diff_map_supported_in.to_bounded_cont_diff_mapL.continuous\n  end }\n\nlemma mem_\u2112p (f : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : \u211d\u22650\u221e) (\u03bc : measure E) [fact (1 \u2264 p)]\n  [is_finite_measure_on_compacts \u03bc] : mem_\u2112p f p \u03bc :=\nf.continuous.mem_\u2112p_of_has_compact_support f.has_compact_support p \u03bc\n\nlemma integrable (f : Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) \n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (\u03bc : measure E)\n  [is_finite_measure_on_compacts \u03bc] : integrable f \u03bc :=\nmem_\u2112p_one_iff_integrable.mp (f.mem_\u2112p 1 \u03bc)\n\nvariable (n)\n\nnoncomputable def to_Lp\u2097\n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : \u211d\u22650\u221e) (\u03bc : measure E) [fact (1 \u2264 p)]\n  [is_finite_measure_on_compacts \u03bc] : \n  (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) \u2192\u2097[\u211d] (Lp F p \u03bc) :=\n{ to_fun := \u03bb f, (f.mem_\u2112p p \u03bc).to_Lp f,\n  map_add' := \u03bb f g, (f.mem_\u2112p p \u03bc).to_Lp_add (g.mem_\u2112p p \u03bc),\n  map_smul' := \u03bb c f, (f.mem_\u2112p p \u03bc).to_Lp_const_smul c }  \n\nnoncomputable def to_Lp\n  {m : measurable_space E} [opens_measurable_space E] [measurable_space F] \n  [second_countable_topology F] [borel_space F] (p : \u211d\u22650\u221e) (\u03bc : measure E) [fact (1 \u2264 p)]\n  [is_finite_measure_on_compacts \u03bc] : \n  (Cc^n\u27ee\u03a9, E, F; \u211d\u27ef) \u2192L[\u211d] (Lp F p \u03bc) :=\n{ to_linear_map := to_Lp\u2097 n p \u03bc,\n  cont := \n  begin\n    change continuous (to_Lp\u2097 n p \u03bc),\n    rw continuous_iff_of_linear,\n    intros K hK,\n    exact (cont_diff_map_supported_in.to_Lp n p \u03bc).continuous,\n  end } \n\nvariable {n}\n\nsection infinity\n\nlemma differentiable (f : Cc^\u22a4\u27ee\u03a9, E, F; \u211d\u27ef) : differentiable \u211d f := \nf.cont_diff.differentiable le_top\n\nprotected noncomputable def fderiv (f : Cc^\u22a4\u27ee\u03a9, E, F; \u211d\u27ef) : Cc^\u22a4\u27ee\u03a9, E, E \u2192L[\u211d] F; \u211d\u27ef := \nof_support_in \u211d (E \u2192L[\u211d] F) \u22a4 f.tsupport f.tsupport_subset (f.to_support_in_tsupport \u211d F \u22a4).fderiv\n\n@[simp] lemma fderiv_apply (f : Cc^\u22a4\u27ee\u03a9, E, F; \u211d\u27ef) (x : E) : f.fderiv x = fderiv \u211d f x := rfl\n\nprotected noncomputable def fderiv\u2097 : Cc^\u22a4\u27ee\u03a9, E, F; \u211d\u27ef \u2192\u2097[\u211d] Cc^\u22a4\u27ee\u03a9, E, E \u2192L[\u211d] F; \u211d\u27ef := \n{ to_fun := test_function.fderiv,\n  map_add' := \u03bb f g,\n  begin\n    ext x : 1,\n    exact fderiv_add f.differentiable.differentiable_at\n      g.differentiable.differentiable_at,\n  end,\n  map_smul' := \u03bb a f,\n  begin\n    ext x : 1,\n    exact fderiv_const_smul f.differentiable.differentiable_at _\n  end }\n\nprotected noncomputable def fderivL : Cc^\u22a4\u27ee\u03a9, E, F; \u211d\u27ef \u2192L[\u211d] Cc^\u22a4\u27ee\u03a9, E, E \u2192L[\u211d] F; \u211d\u27ef := \n{ to_linear_map := test_function.fderiv\u2097,\n  cont := \n  begin\n    change continuous test_function.fderiv\u2097,\n    exact continuous_of_commutes_of_linear _ \n      (\u03bb K hK, cont_diff_map_supported_in.fderivL) (\u03bb K hK, rfl)\n  end }\n\nend infinity\n\nend real\n\nend test_function", "meta": {"author": "ADedecker", "repo": "distributions", "sha": "b4d124142788db55cf781184aff03bcc46aa2b10", "save_path": "github-repos/lean/ADedecker-distributions", "path": "github-repos/lean/ADedecker-distributions/distributions-b4d124142788db55cf781184aff03bcc46aa2b10/src/spaces/test_function.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5039061705290806, "lm_q1q2_score": 0.3652666109886312}}
{"text": "import Huber_ring.basic\n\nlocal attribute [instance] discrete_top_ring\n\n/-- Chambert-Loir's lemma: a discrete ring is Huber. -/\nlemma discrete_Huber_ring {A : Type*} [comm_ring A] [topological_space A] [discrete_topology A] :\n  Huber_ring A :=\n\u27e8\u27e8A, by assumption, by assumption, by apply_instance,\n           \u27e8{ emb := open_embedding_id,\n              J := \u22a5,\n              fin := submodule.fg_bot,\n              top := is_bot_adic_iff.mpr \u2039_\u203a,\n              .. algebra.id A }\u27e9\u27e9\u27e9\n", "meta": {"author": "leanprover-community", "repo": "lean-perfectoid-spaces", "sha": "95a6520ce578b30a80b4c36e36ab2d559a842690", "save_path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces", "path": "github-repos/lean/leanprover-community-lean-perfectoid-spaces/lean-perfectoid-spaces-95a6520ce578b30a80b4c36e36ab2d559a842690/src/examples/discrete.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7248702880639791, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.36526661098863117}}
{"text": "import  .groupk\nopen category_theory\nopen category_theory.limits\nopen category_theory.category\nuniverses v u\nopen Product_stuff\nnamespace GROUP_OBJ\nstructure group_obj (C : Type u)[\ud835\udc9e : category.{v} C][has_binary_products.{v} C][has_terminal.{v} C] :=\n(X : C)\n(\u03bc : X \u2a2f X \u27f6 X)\n(inv : X \u27f6 X)\n(\u03b5 :  T C \u27f6 X)\n(hyp_one_mul  :  (T X | \ud835\udfd9 X) \u226b (\u03b5 \u2297 \ud835\udfd9 X) \u226b  \u03bc  = \ud835\udfd9 X)\n(hyp_mul_one  :  (\ud835\udfd9 X | T X) \u226b ( \ud835\udfd9 X \u2297 \u03b5) \u226b \u03bc  = \ud835\udfd9 X)\n(hyp_mul_inv  :  (\ud835\udfd9 X | inv) \u226b  \u03bc = (T X) \u226b \u03b5 )\n(hyp_assoc    :  (\u03bc \u2297 \ud835\udfd9 X) \u226b (\u03bc) = (prod.associator X X X).hom \u226b (\ud835\udfd9 X \u2297 \u03bc)  \u226b \u03bc )   -- (a *b) * c = (a * (b * c))\n\n\n#print \"ger\"\nstructure  Rr (C : Type u)[\ud835\udc9e : category.{v} C][has_binary_products.{v} C][has_terminal.{v} C] := \n#print notation\nopen lem \n#print notation\n-- structure group_obj (C : Type u)[\ud835\udc9e : category.{v} C][has_binary_products.{v} C][has_terminal.{v} C] :=\n-- (X            :  C)\n-- (\u03bc            :  X \u2a2f X \u27f6 X)\n-- (inv          :  X \u27f6 X) \n-- (\u03b5            :  T C \u27f6 X)\n-- (hyp_one_mul  :  (T X | \ud835\udfd9 X) \u226b (\u03b5 \u2297 \ud835\udfd9 X) \u226b  \u03bc  = \ud835\udfd9 X) \n-- (hyp_mul_one  :  (\ud835\udfd9 X | T X) \u226b ( \ud835\udfd9 X \u2297 \u03b5) \u226b \u03bc  = \ud835\udfd9 X)\n-- (hyp_mul_inv  :  (\ud835\udfd9 X | inv) \u226b  \u03bc = (T X) \u226b \u03b5 )   \n-- (hyp_assoc    :  (\u03bc \u2297 \ud835\udfd9 X) \u226b (\u03bc) = (prod.associator X X X).hom \u226b (\ud835\udfd9 X \u2297 \u03bc)  \u226b \u03bc )   -- (a *b) * c = (a * (b * c))\n\n-- variables {C : Type u}\n-- variables [\ud835\udc9e : category.{v} C]\n-- variables  [has_binary_products.{v} C][has_terminal.{v} C]\n-- include \ud835\udc9e \n-- instance coee : has_coe (group_obj C) C := \u27e8\u03bb F, F.X\u27e9 \n-- variables (G : group_obj C)\n-- #eval 2+2 \n-- /-\n-- First Goal : make a instance of group on the point Hom (Y, G)  = G(Y) \n-- -/\n\n-- --  Idea  Fix R : We have (\u0393 \u00d7 \u0393 )(R ) \u2243  \u0393 (R) \u00d7 \u0393 (R) : Let g1 g2 \u2208 \u0393 (R)\n-- --   we get \u03c6  \u2208 (\u0393 \u00d7 \u0393) R. Next : \n-- --  \u03b5 : \u0393 \u00d7 \u0393  \u2192 \u0393 give  \u03b2  : (\u0393 \u00d7 \u0393) R \u2192 \u0393 R via Yoneda.map  finaly : \u03b2 \u03c6 is ok !  \n-- --   \n-- --\n-- -- \n\n-- def one   (R : C) : R[G.X] :=  \n-- begin \n--      exact (terminal.from R \u226b G.\u03b5),\n-- end\n\n-- def mul (R : C) : R[G.X] \u2192 R[G.X] \u2192 R[G.X] :=  \u03bb g1 g2, \n-- begin \n--      let \u03c6 := ( g1 | g2),\n--      -- let \u03b3 := (prod.mk g1 g2 : (yoneda.obj G.X).obj (op R) \u00d7 (yoneda.obj G.X).obj (op R)), -- \u00d7 versus \u2a2f  \n--      -- let \u03b8 :=  (Yoneda_preserve_product R G.X G.X ).inv,\n--      let \u03b2 := (R< (G.\u03bc) > : R[G.X \u2a2f G.X] \u27f6 R[G.X]),\n--      exact \u03b2 \u03c6,\n-- end\n-- variables (R : C)\n-- include R\n-- instance yoneda_mul : has_mul (R[(G : C)]) := \u27e8mul G R \u27e9 \n-- instance yoneda_one : has_one (R[(G :C)]) := \u27e8one G R \u27e9\n-- lemma mul_comp (a b : R [(G : C)] ) : a * b = (R < G.\u03bc >) (a | b) := rfl -- priority R < g.\u03bc > (a | b) not ()\n-- lemma one_comp :  (1 : (R[(G : C)])) = terminal.from R \u226b G.\u03b5 := rfl\n-- #print group \n-- -- group.mul : \u03a0 {\u03b1 : Type u} [c : group \u03b1], \u03b1 \u2192 \u03b1 \u2192 \u03b1\n-- -- group.mul_assoc : \u2200 {\u03b1 : Type u} [c : group \u03b1] (a b c_1 : \u03b1), a * b * c_1 = a * (b * c_1)\n-- -- group.one : \u03a0 (\u03b1 : Type u) [c : group \u03b1], \u03b1\n-- -- group.one_mul : \u2200 {\u03b1 : Type u} [c : group \u03b1] (a : \u03b1), 1 * a = a\n-- -- group.mul_one : \u2200 {\u03b1 : Type u} [c : group \u03b1] (a : \u03b1), a * 1 = a\n-- -- group.inv : \u03a0 {\u03b1 : Type u} [c : group \u03b1], \u03b1 \u2192 \u03b1\n-- -- group.mul_left_inv : \u2200 {\u03b1 : Type u} [c : group \u03b1] (a : \u03b1), a\u207b\u00b9 * a = 1\n-- -- lemma pre_des (R: C) : (R < T G.X> | \ud835\udfd9 (R[G.X])) \u226b (R < G.\u03b5 > \u2297 \ud835\udfd9 (R[G.X])) =  (( R < T G.X>  \u226b (R < G.\u03b5>)) | \ud835\udfd9 (R[G.X])) := \n-- -- begin exact destruction (R < T G.X>) (R < G.\u03b5 >), end\n-- def one_mul' (a : R[(G : C)]) :  1 * a = a := begin\n-- sorry,\n--      -- rw mul_comp,rw one_comp,\n--      -- --  (hyp_one_mul  :  (T X | \ud835\udfd9 X) \u226b (\u03b5 \u2297 \ud835\udfd9 X) \u226b  \u03bc  = \ud835\udfd9 X) \n--      -- have V : (R <(T G.X | \ud835\udfd9 G.X)>) \u226b (R<(G.\u03b5 \u2297 \ud835\udfd9 G.X)>) \u226b  (R<G.\u03bc>)  = (R<\ud835\udfd9 G.X>),\n--      --      rw \u2190 yoneda_sugar.composition,rw \u2190 yoneda_sugar.composition,\n--      --      rw G.hyp_one_mul,\n--      -- rw yoneda_sugar_prod_inv at V,rw \u2190 assoc at V,\n--      -- rw yoneda_sugar.otimes at V, \n--      -- have hyp : (((R < T G.X> | R < \ud835\udfd9 G.X>) \u226b (yoneda_sugar.prod R (T C) G.X).inv) \u226b\n--      --     (yoneda_sugar.prod R (T C) G.X).hom \u226b\n--      --       (R < G.\u03b5> \u2297 R < \ud835\udfd9 G.X>) \u226b (yoneda_sugar.prod R G.X G.X).inv) \u226b\n--      --  (R < G.\u03bc>) = (((R < T G.X> | R < \ud835\udfd9 G.X>) \u226b ((yoneda_sugar.prod R (T C) G.X).inv) \u226b\n--      --     (yoneda_sugar.prod R (T C) G.X).hom) \u226b\n--      --       (R < G.\u03b5> \u2297 R < \ud835\udfd9 G.X>) \u226b (yoneda_sugar.prod R G.X G.X).inv) \u226b\n--      --  (R < G.\u03bc>), \n--      --      simp,\n--      -- rw yoneda_sugar.prod.inv_hom at hyp,rw hyp at V, \n--      -- -- rw yoneda_sugar.id at V,have V' : (\ud835\udfd9 (R[(G : C)])) a = a, exact rfl,\n--      -- -- erw \u2190  V at V', rw \u2190 V', \n--      -- have fact_2 : ((R < T G.X> | R < \ud835\udfd9 G.X>) \u226b \ud835\udfd9 (R[T C] \u2a2f R[G.X])) = (R < T G.X> | R < \ud835\udfd9 G.X>), \n--      --      simp,\n--      -- rw fact_2 at V,\n--      -- have fact_3 : ((R < T G.X> | R < \ud835\udfd9 G.X>) \u226b (R < G.\u03b5> \u2297 R < \ud835\udfd9 G.X>) \u226b (yoneda_sugar.prod R G.X G.X).inv) \u226b\n--      --  (R < G.\u03bc>) = (((R < T G.X> | R < \ud835\udfd9 G.X>) \u226b (R < G.\u03b5> \u2297 R < \ud835\udfd9 G.X>)) \u226b (yoneda_sugar.prod R G.X G.X).inv) \u226b\n--      --  (R < G.\u03bc>), sorry,\n--      --  rw yoneda_sugar.id at fact_3,\n--      -- rw pre_des R at fact_3, \n--      -- scott_and_kevin_ultimate_tatic --   :D\n--      -- -- rw destruction(R < T G.X>) (R < G.\u03b5>) at fact_3,\n--      -- sorry, -- tooooooooo difficult for the moment !!!! \n-- end\n end GROUP_OBJ", "meta": {"author": "Or7ando", "repo": "lean", "sha": "d41169cf4e416a0d42092fb6bdc14131cee9dd15", "save_path": "github-repos/lean/Or7ando-lean", "path": "github-repos/lean/Or7ando-lean/lean-d41169cf4e416a0d42092fb6bdc14131cee9dd15/.github/workflows/geo/src/group_objet/G.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5273165233795672, "lm_q1q2_score": 0.36524156918455064}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h1 : \u03b1 \u2260 0) (h2 : \u03b1 \u2209 set.range (\u03bb n, n\u207b\u00b9 : \u2124 \u2192 \u211d)) :\n\u2200 y \u2208 Icc 0 1, \u2203 (x : \u2124) (\u03b5 > 0), \u03b5 \u2264 y \u2227 \u03b5 + x * \u03b1 \u2208 Icc 0 1 :=\nbegin\n  assume y h3 : y \u2208 Icc 0 1,\n\n  obtain \u27e8n, h4\u27e9 : \u2203 n : \u2124, y \u2264 n, from exists_lt_of_le y,\n  obtain \u27e8m, h5\u27e9 : \u2203 m : \u2124, y \u2265 m, from exists_lt_of_lt y,\n\n  let N1 := n + 1,\n  let N2 := m - 1,\n\n  have h6 : \u2203 N : \u2124, (N1 \u2264 N) \u2227 (N \u2264 N2), from exists_lt_of_le N1,\n\n  cases h6 with N h7,\n\n  have h8 : (N1 \u2264 N) \u2227 (N \u2264 N2), from h7,\n\n  have h9 : y < N + 1, from by auto [lt_iff_le_and_ne, le_add_right, h7, h4],\n  have h10 : N < y + 1, from by auto [add_lt_add_iff_right, h7, h5],\n\n  have h11 : y \u2264 N, from by auto [le_of_lt, h9],\n  have h12 : N \u2264 y, from by auto [le_of_lt, h10],\n\n  let \u03b5 := (y - N) * \u03b1,\n\n  have h13 : \u03b5 > 0, from by auto [lt_of_le_of_lt, sub_pos.mpr, sub_nonneg.mpr, h11],\n\n  have h14 : \u03b5 + N * \u03b1 = y * \u03b1, from by auto [mul_sub_right_distrib, mul_self_cancel h1, one_mul],\n  have h15 : \u03b5 + N * \u03b1 = (N + 1) * \u03b1 - \u03b1, from by auto [add_comm, mul_comm, h14, add_mul, mul_add, mul_comm, add_comm],\n  have h16 : \u03b5 + N * \u03b1 = (N + 1) * \u03b1 - 1, from by auto [h15, mul_one],\n\n  have h17 : \u03b5 + N * \u03b1 = (N + 1) * \u03b1 - 1, from by auto [add_comm, mul_comm, h14, add_mul, mul_add, mul_comm, add_comm],\n  have h18 : \u03b5 + N * \u03b1 = (N + 1) * \u03b1 - 1, from by auto [h17, mul_one],\n\n  have h19 : \u03b5 + N * \u03b1 \u2208 Icc 0 1, from by auto [Icc_subset_right, h9, h18, mul_nonneg.mpr, add_nonneg.mpr, mul_nonneg.mpr, h11],\n\n  use N,\n  use \u03b5,\n  use h13,\n  use h19,\nend\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (ra : \u03b1 \u2209 \u211a) : {n : \u2115 // n \u2265 1} \u2192 \u2203! i : \u2124, \u2200 j : \u2124, (i - j : \u211d) \u2260 0 \u2192 (n : \u211d) * i \u2260 n * j :=\nbegin\n  assume (n : \u2115) (h1 : n \u2265 1),\n  have h2 : \u2200 i j : \u2124, (i - j : \u211d) \u2260 0 \u2192 (n : \u211d) * i \u2260 n * j, from by auto [mul_left_cancel],\n  use n,\n  show \u2200 (j : \u2124), (n - j : \u211d) \u2260 0 \u2192 (n : \u211d) * n \u2260 n * j, from by auto [h2, sub_eq_iff_eq_add],\n  have h3 : \u2200 i j : \u2124, (i - j : \u211d) \u2260 0 \u2192 (n : \u211d) * i \u2260 n * j, from by auto [h2],\n  assume (i : \u2124) (h4 : \u2200 j : \u2124, (i - j : \u211d) \u2260 0 \u2192 (n : \u211d) * i \u2260 n * j),\n  assume (j : \u2124) (h5 : (i - j : \u211d) \u2260 0),\n  have h6 : (n : \u211d) * i \u2260 n * j, from by auto [h3, h5],\n  show (n : \u211d) * i \u2260 n * j, from by auto [h4, h5, h6],\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=orbit_density (\u03b1 : \u211d) : \u03b1 \u2209 \u211a \u2192 \u2200 (\u03b5 : \u211d) (y : \u211d) (h : \u03b5 > 0), \u2203 n : \u2124, |y - n * \u03b1| < \u03b5 :=\nbegin\n  assume h1 (\u03b5 : \u211d) (y : \u211d) (h2 : \u03b5 > 0),\n  have h3 : \u2200 n : \u2124, \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h4 : \u2200 n : \u2124, \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h5 : \u2200 n : \u2124, \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h6 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h7 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h8 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h9 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h10 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h11 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h12 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h13 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h14 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h15 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h16 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n  have h17 : \u2203 x : \u211d, x \u2208 (set.range (\u03bb (n : \u2124), n * \u03b1)) \u2227 ((abs (x - y)) < \u03b5), from by auto [set.subset_iff] using [abs_sub_lt_iff] using [mul_sub_left_distrib, lt_sub_iff_add_lt, abs_add, abs_mul, abs_nonneg, le_of_lt],\n\n  cases h17 with x h18,\n  cases h18 with h19 h20,\n  have h21 : \u2203 n, x = n * \u03b1, from by auto [set.mem_range] using [h19],\n  cases h21 with n h22,\n  have h23 : |y - n * \u03b1| < \u03b5, from by auto [h22, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg, abs_of_nonneg] using [h20, abs_of_nonneg],\n  use n,\n  show |y - n * \u03b1| < \u03b5, from by auto [h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_irrational_orbit (\u03b1 : \u211d) : irrat \u03b1 \u2192 \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 (i : \u2124), i > N \u2192 |((\u03b1 * i) % 1) - 0| < \u03b5 :=\nbegin\n  assume h1 : irrat \u03b1,\n  assume (h2 : \u03b5 > 0),\n  let S : set \u211d := {((n : \u2124) : \u211d) : \u211d | \u2203 (i : \u2124), n = (\u03b1 * i) % 1},\n  have h3 : \u2200 (i j : \u2124), i \u2260 j \u2192 ((\u03b1 * i) % 1) \u2260 ((\u03b1 * j) % 1), \n  from by auto [irrat.def, abs_add_lt_iff, abs_mul_lt_iff, abs_sub_lt_iff, one_mul, add_mul, sub_mul, mul_sub, mul_add, add_sub, mul_comm, sub_add_cancel, add_sub_cancel, mul_assoc, mul_one, sub_self, add_self] using [linarith],\n\n  have h4 : \u2200 (i : \u2124), \u2203 (x : \u211d), x \u2208 S, from by auto [exists.intro ((\u03b1 * i) % 1), set.mem_set_of_eq],\n  have h5 : \u2200 (x : \u211d), \u2203 (i : \u2124), x = ((\u03b1 * i) % 1), from by auto [abs_lt_iff, abs_add_lt_iff, abs_mul_lt_iff, abs_sub_lt_iff, one_mul, add_mul, sub_mul, mul_sub, mul_add, add_sub, mul_comm, sub_add_cancel, add_sub_cancel, mul_assoc, mul_one, sub_self, add_self] using [linarith],\n\n  have h6 : S.nonempty, from by auto [h4],\n  have h7 : \u2200 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 y \u2208 S, from by auto [h3, h5],\n  have h8 : \u2203 (y : \u211d), y \u2208 S \u2227 y \u2208 S, from by auto [h6, h7],\n  have h9 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h6, h7],\n  have h10 : \u2203 (y : \u211d), \u2203 (z : \u211d), y \u2260 z \u2227 y \u2208 S \u2227 z \u2208 S, from by auto [h8, h7],\n  have h11 : S.finite, from by auto [set.finite_iff_card_lt_omega, h9, h10],\n  have h12 : S.finite, from by auto [h11],\n  have h13 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h14 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h10],\n\n  --have h15 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h16 : \u2203 (x : \u211d), x \u2208 S \u2227 x \u2208 S, from by auto [h8],\n  have h17 : \u2203 (x : \u211d), x \u2208 S, from by auto [h16],\n  have h18 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h19 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h17, h18],\n  have h20 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h21 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h22 : \u2203 (x : \u211d), x \u2208 S \u2227 x \u2208 S, from by auto [h8],\n  have h23 : \u2203 (x : \u211d), x \u2208 S, from by auto [h22],\n  have h24 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h21, h7],\n  have h25 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h23, h24],\n  have h26 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h27 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h28 : \u2203 (x : \u211d), x \u2208 S \u2227 x \u2208 S, from by auto [h8],\n  have h29 : \u2203 (x : \u211d), x \u2208 S, from by auto [h28],\n  have h30 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h27, h7],\n  have h31 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h29, h30],\n  have h32 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h33 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h34 : \u2203 (x : \u211d), x \u2208 S \u2227 x \u2208 S, from by auto [h8],\n  have h35 : \u2203 (x : \u211d), x \u2208 S, from by auto [h34],\n  have h36 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h33, h7],\n  have h37 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h35, h36],\n  have h38 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h39 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h14, h7],\n  have h40 : \u2203 (x : \u211d), x \u2208 S \u2227 x \u2208 S, from by auto [h8],\n  have h41 : \u2203 (x : \u211d), x \u2208 S, from by auto [h40],\n  have h42 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from by auto [h39, h7],\n  have h43 : S.infinite, from by auto [set.infinite_iff_nonempty_of_inhabited_of_not_finite, h41, h42],\n  have h44 : \u2203 (x : \u211d), x \u2208 S, from by auto [h4],\n  have h45 : \u2203 (x : \u211d), \u2203 (y : \u211d), x \u2260 y \u2227 x \u2208 S \u2227 y \u2208 S, from\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit {x : \u211d} (hx : \u00ac(\u2203 n : \u2115, x = n)) : \n  let orbit : \u2124 \u2192 \u211d := \u03bb (n : \u2124), n*x in\n  let orbit_frac_part : \u2124 \u2192 \u211d := \u03bb (n : \u2124), n*x - \u230an*x\u230b in\n  let orbit_frac_part_set : set \u211d := { n*x - \u230an*x\u230b | n : \u2124 } in\n  let orbit_frac_part_set_0_1 : set \u211d := orbit_frac_part_set \u2229 Icc 0 1 in\n  let orbit_frac_part_set_0_1_dense : Prop := \u2200 x : \u211d, \u2203 y : \u211d, y \u2208 orbit_frac_part_set_0_1 \u2227 |x - y| < 1 in\n  orbit_frac_part_set_0_1_dense :=\nbegin\n  assume orbit orbit_frac_part orbit_frac_part_set orbit_frac_part_set_0_1 orbit_frac_part_set_0_1_dense,\n\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2260 orbit_frac_part j,\n  from by auto [orbit_frac_part, hx, eq_of_mul_eq_mul_left],\n\n  have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2260 orbit j,\n  from by auto [orbit, hx, eq_of_mul_eq_mul_left],\n\n  have h3 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2209 orbit_frac_part_set_0_1 \u2194 orbit_frac_part j \u2209 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h1],\n\n  have h4 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2209 orbit_frac_part_set_0_1 \u2194 orbit j \u2209 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h2],\n\n  have h5 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2209 orbit_frac_part_set \u2194 orbit_frac_part j \u2209 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h6 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2209 orbit_frac_part_set \u2194 orbit j \u2209 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h2],\n\n  have h7 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set \u2194 orbit_frac_part j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h8 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set \u2194 orbit j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h2],\n\n  have h9 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set_0_1 \u2194 orbit_frac_part j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h1],\n\n  have h10 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set_0_1 \u2194 orbit j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h2],\n\n  have h11 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set \u2194 orbit_frac_part j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h12 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set \u2194 orbit j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h2],\n\n  have h13 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2209 orbit_frac_part_set_0_1 \u2194 orbit_frac_part j \u2209 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h1],\n\n  have h14 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2209 orbit_frac_part_set_0_1 \u2194 orbit j \u2209 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h2],\n\n  have h15 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set \u2194 orbit_frac_part j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h16 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set \u2194 orbit j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h2],\n\n  have h17 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set_0_1 \u2194 orbit_frac_part j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h1],\n\n  have h18 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set_0_1 \u2194 orbit j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h2],\n\n  have h19 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set \u2194 orbit_frac_part j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h20 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set \u2194 orbit j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h2],\n\n  have h21 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set_0_1 \u2194 orbit_frac_part j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h1],\n\n  have h22 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set_0_1 \u2194 orbit j \u2208 orbit_frac_part_set_0_1,\n  from by auto [orbit_frac_part_set_0_1, h2],\n\n  have h23 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit_frac_part i \u2208 orbit_frac_part_set \u2194 orbit_frac_part j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_part_set, h1],\n\n  have h24 : \u2200 (i j : \u2124), i \u2260 j \u2192 orbit i \u2208 orbit_frac_part_set \u2194 orbit j \u2208 orbit_frac_part_set,\n  from by auto [orbit_frac_\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) : \nlet irrational : \u211d \u2192 Prop := \u03bb x, \u00ac x \u2208 set.range (\u03bb (n : \u2115), n : \u211d) in\nirrational \u03b1 \u2192 \nlet frac_part : \u211d \u2192 \u211d := \u03bb x, x - (x.to_int : \u211d) in\nlet S : set \u211d := \u03bb x, \u2203 i : \u2124, x = frac_part (i * \u03b1) in\nlet is_dense : set \u211d \u2192 Prop := \u03bb x, \u2200 y : \u211d, \u2203 z : \u211d, z \u2208 x \u2227 y < z \u2227 z < y + 1 in\nis_dense S :=\nbegin\n  assume (h1 : irrational \u03b1),\n  assume (h3 : \u2200 (y : \u211d), \u2203 (z : \u211d), z \u2208 S \u2227 y < z \u2227 z < y + 1),\n\n  show \u2200 (x : \u211d), \u2203 (z : \u211d), z \u2208 S \u2227 x < z \u2227 z < x + 1,\n  from by auto [h1, h3] using [exists_unique.unique, exists_unique.exists, exists_unique.not_exists, exists_unique.not_exists_left, exists_unique.not_exists_right, exists_unique.ne, exists_unique.not_mem_iff, exists_unique.mem_iff],\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac \u2203 r : \u211a, \u03b1 = r) : \n\u2200 x \u2208 set.range (\u03bb n, (n : \u2115) * \u03b1), \u2203 y \u2208 set.range (\u03bb n, (n : \u2115) * \u03b1), y \u2260 x \u2227 abs (y - x) < 1 :=\nbegin\n  assume (x : \u211d) (hx : x \u2208 set.range (\u03bb n, (n : \u2115) * \u03b1)), \n\n  have hx1 : \u2203 n, x = (n : \u2115) * \u03b1, from set.mem_range.1 hx,\n  cases hx1 with n hn,\n  subst x,\n\n  have hn1 : \u2203! r : \u211a, r * \u03b1 = (n : \u2115) * \u03b1, from by auto using [exists_unique.exists, exists_unique.unique, eq_of_mul_eq_mul_left],\n  have hn2 : \u2203! r : \u211a, r * \u03b1 = (n + 1) * \u03b1, from by auto using [exists_unique.exists, exists_unique.unique, eq_of_mul_eq_mul_left],\n\n  have h1 : \u00ac (\u2203 (r : \u211a), r * \u03b1 = (n : \u2115) * \u03b1 \u2227 (\u2203 (r : \u211a), r * \u03b1 = (n + 1) * \u03b1)),\n  from by auto [hn1, hn2, classical.not_forall, classical.not_exists, h\u03b1, eq_of_mul_eq_mul_left],\n\n  have h2 : \u2200 (r : \u211a), r * \u03b1 \u2260 (n : \u2115) * \u03b1 \u2228 r * \u03b1 \u2260 (n + 1) * \u03b1,\n  from by auto [h1, classical.not_and_iff_not_or_not, classical.not_forall, classical.not_exists],\n\n  have h3 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 > (n + 1) * \u03b1,\n  from by auto using [h2, eq_of_mul_eq_mul_left],\n\n  have h4 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1,\n  from by auto [lt_or_gt],\n\n  have h5 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1 \u2228 (n : \u2115) * \u03b1 = r * \u03b1,\n  from by auto [lt_or_gt, eq_or_lt, lt_or_eq_of_le],\n\n  have h6 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1 \u2228 ((n : \u2115) * \u03b1 = r * \u03b1),\n  from by auto [h5, h2, eq_of_mul_eq_mul_left],\n\n  have h7 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1 \u2228 ((n : \u2115) * \u03b1 = r * \u03b1),\n  from by auto [h6, h3, not_or_distrib],\n\n  have h8 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1 \u2228 (n : \u2115) * \u03b1 = r * \u03b1,\n  from by auto [h7, eq_of_mul_eq_mul_left],\n\n  have h9 : \u2200 (r : \u211a), r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 < r * \u03b1 \u2228 (n : \u2115) * \u03b1 = r * \u03b1,\n  from by auto [h8, h4],\n\n  have h10 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 (n : \u2115) * \u03b1 = r * \u03b1,\n  from by auto [h9, lt_or_gt],\n\n  have h11 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h10, eq_of_mul_eq_mul_left],\n\n  have h12 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h11],\n\n  have h13 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h12],\n\n  have h14 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h13],\n\n  have h15 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h14],\n\n  have h16 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h15],\n\n  have h17 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h16],\n\n  have h18 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h17],\n\n  have h19 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h18],\n\n  have h20 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h19],\n\n  have h21 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h20],\n\n  have h22 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h21],\n\n  have h23 : \u2200 (r : \u211a), (n : \u2115) * \u03b1 < r * \u03b1 \u2228 r * \u03b1 < (n : \u2115) * \u03b1 \u2228 r * \u03b1 = (n : \u2115) * \u03b1,\n  from by auto [h22],\n\n  have h24 : \u2200 (r : \u211a),\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_irrational_orbit (\u03b1 : \u211d) (h : \u03b1 \u2209 \u211a) : \u2200 x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1), \u2203 y \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1), |x - y| < 1 :=\nbegin\n  assume x hx,\n  have h1 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 set.Ico 0 1 (i * \u03b1) \u2260 set.Ico 0 1 (j * \u03b1), from by auto [not_iff_comm, mem_Ico] using [h, mul_mem_Ico],\n  have h2 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 set.Ico 0 1 (i * \u03b1) \u2229 set.Ico 0 1 (j * \u03b1) = \u2205, from by auto [set.inter_eq_empty_of_disjoint h1],\n  have h3 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [not_iff_comm, ne.def, mem_Ico] using [h, mul_mem_Ico],\n  have h4 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2209 set.Ico 0 1 (j : \u211d) * \u03b1, from by auto [not_iff_comm, mem_Ico] using [h, mul_mem_Ico],\n  have h5 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2209 set.Ico 0 1 (j : \u211d) * \u03b1, from by auto [not_iff_comm, mem_Ico] using [h, mul_mem_Ico],\n  have h6 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 set.Ico 0 1 ((j : \u211d) * \u03b1) = \u2205, from by auto [set.inter_eq_empty_of_disjoint h4],\n  have h7 : \u2200 (i : \u2124) (j : \u2124), i \u2260 j \u2192 set.Ico 0 1 ((j : \u211d) * \u03b1) \u2229 set.Ico 0 1 ((i : \u211d) * \u03b1) = \u2205, from by auto [set.inter_eq_empty_of_disjoint h5],\n  have h8 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((-i : \u211d) * \u03b1)) = \u2205, from by auto [h7, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h7],\n  have h9 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((-i : \u211d) * \u03b1)) = \u2205, from by auto [h7, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h7],\n  have h10 : \u2200 i : \u2124, set.Ico 0 1 ((-i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h8, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h9],\n  have h11 : \u2200 i : \u2124, set.Ico 0 1 ((-i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h8, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h9],\n  have h12 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((-i : \u211d) * \u03b1)) = \u2205, from by auto [h10, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h11],\n  have h13 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((-i : \u211d) * \u03b1)) = \u2205, from by auto [h10, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h11],\n  have h14 : \u2200 i : \u2124, set.Ico 0 1 ((-i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h12, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h13],\n  have h15 : \u2200 i : \u2124, set.Ico 0 1 ((-i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h12, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h13],\n\n  have h16 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 (-(i : \u211d) * \u03b1)) = \u2205, from by auto [h14, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h15],\n  have h17 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 (-(i : \u211d) * \u03b1)) = \u2205, from by auto [h14, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h15],\n  have h18 : \u2200 i : \u2124, set.Ico 0 1 (-(i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h16, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h17],\n  have h19 : \u2200 i : \u2124, set.Ico 0 1 (-(i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h16, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h17],\n  have h20 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 (-(i : \u211d) * \u03b1)) = \u2205, from by auto [h18, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h19],\n  have h21 : \u2200 i : \u2124, set.Ico 0 1 ((i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 (-(i : \u211d) * \u03b1)) = \u2205, from by auto [h18, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h19],\n  have h22 : \u2200 i : \u2124, set.Ico 0 1 (-(i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i : \u211d) * \u03b1)) = \u2205, from by auto [h20, neg_neg_eq_of_pos, set.inter_eq_empty_of_disjoint h21],\n  have h23 : \u2200 i : \u2124, set.Ico 0 1 (-(i : \u211d) * \u03b1) \u2229 (set.Ico 0 1 ((i\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  assume (h7 : \u03b5 > 0),\n\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by auto [h8, h10, h5, h9],\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by auto [h11] using [linarith],\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/lean_proof_auto-4_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3652415558021249}}
{"text": "import category_theory.preadditive.yoneda\n\nopen opposite\n\nnamespace category_theory\n\nuniverses v u\nvariables (A : Type u) [category.{v} A] [preadditive A]\n\nlemma is_iso_of_is_iso_preadditive_yoneda_map_app {X Y : A}\n  (f : X \u27f6 Y) [\u2200 W : A, is_iso ((preadditive_yoneda.map f).app (op W))] :\n  is_iso f :=\nbegin\n  let e := (preadditive_yoneda.map f).app (op Y),\n  let g := (preadditive_yoneda.map f).app (op X),\n  use inv e (\ud835\udfd9 _),\n  split,\n  { apply_fun g,\n    swap,\n    { intros i j h,\n      apply_fun inv g at h,\n      simpa only [\u2190 comp_apply, is_iso.hom_inv_id] using h },\n    dsimp [g],\n    simp only [category.id_comp, category.assoc],\n    change f \u226b e _ = _,\n    simp only [\u2190 comp_apply, is_iso.hom_inv_id],\n    simp },\n  { change e (inv e (\ud835\udfd9 _)) = _,\n    simp only [\u2190 comp_apply, is_iso.inv_hom_id],\n    simpa },\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/preadditive_yoneda.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7662936430859597, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3651999570342197}}
{"text": "\nimport util.category\nimport util.predicate\nimport temporal_logic\n\nopen predicate\n\nuniverse variables u u' v'\n\nnamespace unitb\n\nvariable {\u03b1 : Type u}\n\nclass lifted_pred (cat : pred' \u03b1 \u2192 pred' \u03b1 \u2192 Sort u')\nextends category cat :=\n  (imp : \u2200 p q, p \u27f9 q \u2192 cat p q)\n  (imp_self_eq_ident : \u2200 {p : pred' \u03b1}, imp p p (by refl) = ident)\n  (imp_comp_imp_eq_imp_trans : \u2200 {p q r : pred' \u03b1} (h\u2080 : p \u27f9 q) (h\u2081 : q \u27f9 r),\n     (imp q r h\u2081 <<< imp p q h\u2080) = imp p r (entails_trans h\u2080 h\u2081))\n\nclass finite_disjunctive (cat : pred' \u03b1 \u2192 pred' \u03b1 \u2192 Sort u')\nextends lifted_pred cat :=\n  (disj : \u2200 {p q r : pred' \u03b1}, cat p r \u2192 cat q r \u2192 cat (p \u22c1 q) r)\n  (comp_over_disj_right : \u2200 {p q r r' : pred' \u03b1}\n                            (Pp : cat p r)\n                            (Pq : cat q r)\n                            (Pr : cat r r'),\n             (Pr <<< disj Pp Pq) = disj (Pr <<< Pp) (Pr <<< Pq))\n  (select_left_disj  : \u2200 {p q r : pred' \u03b1}\n                            (Pp : cat p r)\n                            (Pq : cat q r),\n             (disj Pp Pq <<< imp p _ (p_or_intro_left _ _)) = Pp)\n  (disj_imp_imp : \u2200 {p q r : pred' \u03b1}\n                            (Pp : p \u27f9 r)\n                            (Pq : q \u27f9 r),\n             disj (imp _ _ Pp) (imp _ _ Pq) =\n             imp (p \u22c1 q) r (p_or_entails_of_entails Pp Pq))\n  (disj_flip : \u2200 {p q r : pred' \u03b1}\n                            (P\u2080 : cat p r) (P\u2081 : cat q r),\n             disj P\u2080 P\u2081 = (disj P\u2081 P\u2080 <<< imp _ _ (by rw [p_or_comm])) )\n\nclass disjunctive (cat : pred' \u03b1 \u2192 pred' \u03b1 \u2192 Sort u')\nextends finite_disjunctive cat :=\n  (disj' : \u2200 {t : Type u} (p : t \u2192 pred' \u03b1) r : pred' \u03b1,\n      (\u2200 x, cat (p x) r) \u2192 cat (\u2203\u2203 x, p x) r)\n  (disj :=\n   begin\n     intros p q r X Y,\n     let f := \u03bb x : ulift.{u} bool, if x.down then p else q,\n     have H : p \u22c1 q = \u2203\u2203 x, f x,\n     { apply mutual_entails,\n       apply p_or_entails_of_entails,\n       { apply @p_exists_intro _ _ f \u27e8 tt \u27e9, },\n       { apply @p_exists_intro _ _ f \u27e8 ff \u27e9, },\n       apply p_exists_elim,\n       intro x, cases x, cases down,\n       { apply p_or_intro_right },\n       { apply p_or_intro_left }, },\n     rw H, clear H,\n     apply disj',\n     apply ulift.rec,\n     apply bool.rec ; revert f\n     ; simp\n     ; assumption\n   end )\n\nexport lifted_pred (imp_comp_imp_eq_imp_trans imp_self_eq_ident)\nexport finite_disjunctive (disj comp_over_disj_right select_left_disj\n                           disj_imp_imp disj_flip)\nexport disjunctive (disj')\n\nsection lemmas\n\nparameters {\u03c3 : Type u}\nparameters (cat : pred' \u03c3 \u2192 pred' \u03c3 \u2192 Sort v')\n@[reducible]\nprivate def pred := pred' \u03c3\n\nlocal infix ` \u2907 `:25 := cat\n   -- M-x insert-char RET right*arrow\n   -- code point: 0x2907\nparameters {t : Type u}\n\nlemma disj.select_left_disj' [finite_disjunctive cat]\n  {p p' q r : pred}\n  (H : p' \u27f9 p)\n  (Pp : cat p r)\n  (Pq : cat q r)\n: (disj \u03c3 Pp Pq <<< lifted_pred.imp _ p' (p \u22c1 q)\n         (entails_trans H $ p_or_intro_left _ _)) =\n  (Pp <<< lifted_pred.imp _ p' p H) :=\nbegin\n  have h\u2080 : p' \u27f9 p \u22c1 q,\n  { transitivity p, assumption, apply p_or_intro_left },\n  have h\u2081 := p_or_intro_left p q,\n  have H' : lifted_pred.imp cat p' (p \u22c1 q) h\u2080 =\n            (lifted_pred.imp cat p (p \u22c1 q) h\u2081 <<< lifted_pred.imp cat p' p H),\n  { rw imp_comp_imp_eq_imp_trans },\n  rw [H',semigroupoid.assoc,select_left_disj],\nend\n\ndef select_right_disj [finite_disjunctive cat] {p q r : pred}\n  (Pp : cat p r)\n  (Pq : cat q r)\n: (disj _ Pp Pq <<< lifted_pred.imp cat q _ (p_or_intro_right _ _)) = Pq :=\nby rw [disj_flip,\u2190 category.assoc,imp_comp_imp_eq_imp_trans,select_left_disj]\n\ndef monotonicity [lifted_pred cat]\n  {p p' q q'}\n  (hp : p' \u27f9 p)\n  (hq : q  \u27f9 q')\n  (hcat : cat p q)\n: cat p' q' :=\nbegin\n  apply has_comp.comp,\n  apply lifted_pred.imp cat _ _ hq,\n  apply has_comp.comp _ hcat,\n  apply lifted_pred.imp cat _ _ hp,\nend\n\ndef antimono_left [lifted_pred cat]\n  (p) {p' q}\n  (h : p' \u27f9 p)\n: cat p q \u2192 cat p' q :=\nmonotonicity h (by refl)\n\ndef lifted_pred.weaken [lifted_pred cat]\n  {p q}\n  (h : p \u27f9 q)\n: cat p q :=\nlifted_pred.imp cat p q h\n\ndef lifted_pred.mono_right [lifted_pred cat]\n  {p} (q) {q'}\n  (h : q  \u27f9 q')\n: cat p q \u2192 cat p q' :=\nmonotonicity (by refl) h\n\ndef disj_rng [category cat] [disjunctive cat]\n  {p : t \u2192 pred} {q : pred} {r : t \u2192 Prop}\n  (h : \u2200 i, r i \u2192 p i \u2907 q)\n: (\u2203\u2203 i, r i \u22c0 p i) \u2907 q :=\nbegin\n  have h' : (\u2203\u2203 (i : t), \u2191(r i) \u22c0 p i) =\n              (\u2203\u2203 (i : { x : t // r x }), p i),\n  { funext x,\n    split,\n    { intro h, cases h with j h,\n      exact \u27e8\u27e8j, h^.left\u27e9, h^.right\u27e9 },\n    { intro h\u2080, cases h\u2080 with j h\u2080, cases j with j h\u2081 h\u2082,\n      exact \u27e8j,h\u2081,h\u2080\u27e9, } },\n  rw h',\n  apply disj',\n  intro i,\n  apply h,\n  apply i^.property,\nend\n\ndef gen_disj [finite_disjunctive cat]\n  {p\u2080 p\u2081 q\u2080 q\u2081 : pred' \u03c3}\n  (P\u2080 : p\u2080 \u2907 q\u2080)\n  (P\u2081 : p\u2081 \u2907 q\u2081)\n: p\u2080 \u22c1 p\u2081 \u2907 q\u2080 \u22c1 q\u2081 :=\nbegin\n  apply disj,\n  { apply lifted_pred.mono_right cat _ _ P\u2080,\n    apply p_or_intro_left },\n  { apply lifted_pred.mono_right cat _ _ P\u2081,\n    apply p_or_intro_right },\nend\n\ndef gen_disj' [disjunctive cat]\n  {t : Type u} {p q : t \u2192 pred' \u03c3}\n  (P : \u2200 x, p x \u2907 q x)\n: (\u2203\u2203 x, p x) \u2907 (\u2203\u2203 x, q x) :=\nbegin\n  have P' : \u03a0 (x : t), cat (p x) (\u2203\u2203 (x : t), q x),\n  { intro x,\n    apply lifted_pred.mono_right cat _ _ (P x),\n    apply p_exists_intro },\n  apply disj' p _ P',\nend\n\ndef cancellation'\n  [lifted_pred cat] [finite_disjunctive cat]\n  {p : pred' \u03c3} (q : pred' \u03c3)\n  {r b : pred' \u03c3}\n  (P\u2080 : p \u2907 q \u22c1 b)\n  (P\u2081 : q \u2907 r)\n: p \u2907 r \u22c1 b :=\nbegin\n  apply has_comp.comp _ _ P\u2080,\n  apply gen_disj _ P\u2081 (ident _),\nend\n\ndef cancellation\n  [lifted_pred cat] [finite_disjunctive cat]\n  {p : pred' \u03c3} (q : pred' \u03c3)\n  {r b : pred' \u03c3}\n  (P\u2080 : p \u2907 q \u22c1 b)\n  (P\u2081 : q \u2907 r \u22c1 b)\n: p \u2907 r \u22c1 b :=\nbegin\n  apply has_comp.comp _ _ P\u2080,\n  apply finite_disjunctive.disj _ P\u2081,\n  apply lifted_pred.imp,\n  apply p_or_intro_right,\nend\n\n-- set_option pp.implicit true\n\nlemma cancellation_assoc\n  [finite_disjunctive cat]\n  {p q : pred' \u03c3}\n  {r r' b : pred' \u03c3}\n  (P\u2080 : p \u2907 q \u22c1 b)\n  (P\u2081 : q \u2907 r \u22c1 b)\n  (P\u2082 : r \u2907 r' \u22c1 b)\n:   cancellation _ (cancellation _ P\u2080 P\u2081) P\u2082\n  = cancellation _ P\u2080 (cancellation _ P\u2081 P\u2082)  :=\nbegin\n  unfold cancellation,\n  rw [semigroupoid.assoc, comp_over_disj_right, select_right_disj],\nend\n\ntheorem induction [disjunctive cat]\n  {\u03b2 : Type u}\n  [has_well_founded \u03b2]\n  (V : var \u03c3 \u03b2)\n  {p q : pred}\n  (P : \u2200 v : \u03b2, p \u22c0 V \u2243 v  \u2907  p \u22c0 V \u227a\u227a v \u22c1 q)\n: p \u2907 q :=\nbegin\n  have P' : (\u2203\u2203 v : \u03b2, p \u22c0 V \u2243 v)  \u2907 q,\n  { apply disjunctive.disj', intro i,\n    let PP := \u03bb v : \u03b2, p \u22c0 V \u2243 v  \u2907  q,\n    change PP i,\n    apply @well_founded.recursion _ _ (has_well_founded.wf _) PP,\n    intros j IH,\n    change _ \u2907 _,\n    apply lifted_pred.mono_right cat (q \u22c1 q),\n    { simp [p_or_self] },\n    apply cancellation' _ (p \u22c0 V \u227a\u227a j) (P _),\n    have h' : (p \u22c0 V \u227a\u227a j) = (\u2203\u2203v : \u03b2, \u2191(v << j) \u22c0 (p \u22c0 V \u2243 v)),\n    { funext x, split,\n      { intros H\u2080, cases H\u2080 with H\u2080 H\u2081,\n        existsi V.apply x, TL_simp,\n        simp at H\u2081,\n        split ; assumption, },\n      { intro h, apply exists.elim h,\n        intros s h', cases h' with h\u2080 h\u2081, cases h\u2081,\n        TL_simp [function.comp] at right h\u2080 \u22a2, subst s,\n        split ; assumption,  }, },\n    simp [h'], clear h',\n    -- apply @disj_rng _ _ \u03b2 _ _  (lt j) ,\n    apply disj_rng,\n    apply IH, },\n  { have h : (\u2203\u2203 (v : \u03b2), p \u22c0 V \u2243 v) = p,\n    { funext x, TL_simp [function.comp] },\n    rw h at P',\n    apply P' }\nend\n\nend lemmas\n\nend unitb\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/category/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3650730576745188}}
{"text": "/-\nCopyright (c) 2021 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn, Joachim Breitner\n-/\nimport algebra.free_monoid\nimport group_theory.congruence\nimport group_theory.is_free_group\nimport group_theory.subgroup.pointwise\nimport data.list.chain\nimport set_theory.cardinal\n/-!\n# The free product of groups or monoids\n\nGiven an `\u03b9`-indexed family `M` of monoids, we define their free product (categorical coproduct)\n`free_product M`. When `\u03b9` and all `M i` have decidable equality, the free product bijects with the\ntype `word M` of reduced words. This bijection is constructed by defining an action of\n`free_product M` on `word M`.\n\nWhen `M i` are all groups, `free_product M` is also a group (and the coproduct in the category of\ngroups).\n\n## Main definitions\n\n- `free_product M`: the free product, defined as a quotient of a free monoid.\n- `free_product.of {i} : M i \u2192* free_product M`.\n- `free_product.lift : (\u03a0 {i}, M i \u2192* N) \u2243 (free_product M \u2192* N)`: the universal property.\n- `free_product.word M`: the type of reduced words.\n- `free_product.word.equiv M : free_product M \u2243 word M`.\n- `free_product.neword M i j`: an inductive description of non-empty words with first letter from\n  `M i` and last letter from `M j`, together with an API (`singleton`, `append`, `head`, `tail`,\n  `to_word`, `prod`, `inv`). Used in the proof of the Ping-Pong-lemma.\n- `free_product.lift_injective_of_ping_pong`: The Ping-Pong-lemma, proving injectivity of the\n  `lift`. See the documentation of that theorem for more information.\n\n## Remarks\n\nThere are many answers to the question \"what is the free product of a family `M` of monoids?\", and\nthey are all equivalent but not obviously equivalent. We provide two answers. The first, almost\ntautological answer is given by `free_product M`, which is a quotient of the type of words in the\nalphabet `\u03a3 i, M i`. It's straightforward to define and easy to prove its universal property. But\nthis answer is not completely satisfactory, because it's difficult to tell when two elements\n`x y : free_product M` are distinct since `free_product M` is defined as a quotient.\n\nThe second, maximally efficient answer is given by `word M`. An element of `word M` is a word in the\nalphabet `\u03a3 i, M i`, where the letter `\u27e8i, 1\u27e9` doesn't occur and no adjacent letters share an index\n`i`. Since we only work with reduced words, there is no need for quotienting, and it is easy to tell\nwhen two elements are distinct. However it's not obvious that this is even a monoid!\n\nWe prove that every element of `free_product M` can be represented by a unique reduced word, i.e.\n`free_product M` and `word M` are equivalent types. This means that `word M` can be given a monoid\nstructure, and it lets us tell when two elements of `free_product M` are distinct.\n\nThere is also a completely tautological, maximally inefficient answer given by\n`algebra.category.Mon.colimits`. Whereas `free_product M` at least ensures that (any instance of)\nassociativity holds by reflexivity, in this answer associativity holds because of quotienting. Yet\nanother answer, which is constructively more satisfying, could be obtained by showing that\n`free_product.rel` is confluent.\n\n## References\n\n[van der Waerden, *Free products of groups*][MR25465]\n\n-/\n\nvariables {\u03b9 : Type*} (M : \u03a0 i : \u03b9, Type*) [\u03a0 i, monoid (M i)]\n\n/-- A relation on the free monoid on alphabet `\u03a3 i, M i`, relating `\u27e8i, 1\u27e9` with `1` and\n`\u27e8i, x\u27e9 * \u27e8i, y\u27e9` with `\u27e8i, x * y\u27e9`. -/\ninductive free_product.rel : free_monoid (\u03a3 i, M i) \u2192 free_monoid (\u03a3 i, M i) \u2192 Prop\n| of_one (i : \u03b9) : free_product.rel (free_monoid.of \u27e8i, 1\u27e9) 1\n| of_mul {i : \u03b9} (x y : M i) : free_product.rel (free_monoid.of \u27e8i, x\u27e9 * free_monoid.of \u27e8i, y\u27e9)\n  (free_monoid.of \u27e8i, x * y\u27e9)\n\n/-- The free product (categorical coproduct) of an indexed family of monoids. -/\n@[derive [monoid, inhabited]]\ndef free_product : Type* := (con_gen (free_product.rel M)).quotient\n\nnamespace free_product\n\n/-- The type of reduced words. A reduced word cannot contain a letter `1`, and no two adjacent\nletters can come from the same summand. -/\n@[ext] structure word :=\n(to_list : list (\u03a3 i, M i))\n(ne_one : \u2200 l \u2208 to_list, sigma.snd l \u2260 1)\n(chain_ne : to_list.chain' (\u03bb l l', sigma.fst l \u2260 sigma.fst l'))\n\nvariable {M}\n\n/-- The inclusion of a summand into the free product. -/\ndef of {i : \u03b9} : M i \u2192* free_product M :=\n{ to_fun   := \u03bb x, con.mk' _ (free_monoid.of $ sigma.mk i x),\n  map_one' := (con.eq _).mpr (con_gen.rel.of _ _ (free_product.rel.of_one i)),\n  map_mul' := \u03bb x y, eq.symm $ (con.eq _).mpr (con_gen.rel.of _ _ (free_product.rel.of_mul x y)) }\n\nlemma of_apply {i} (m : M i) : of m = con.mk' _ (free_monoid.of $ sigma.mk i m) := rfl\n\nvariables {N : Type*} [monoid N]\n\n/-- See note [partially-applied ext lemmas]. -/\n@[ext] lemma ext_hom (f g : free_product M \u2192* N) (h : \u2200 i, f.comp (of : M i \u2192* _) = g.comp of) :\n  f = g :=\n(monoid_hom.cancel_right con.mk'_surjective).mp $ free_monoid.hom_eq $ \u03bb \u27e8i, x\u27e9,\n  by rw [monoid_hom.comp_apply, monoid_hom.comp_apply, \u2190of_apply,\n    \u2190monoid_hom.comp_apply, \u2190monoid_hom.comp_apply, h]\n\n/-- A map out of the free product corresponds to a family of maps out of the summands. This is the\nuniversal property of the free product, charaterizing it as a categorical coproduct. -/\n@[simps symm_apply]\ndef lift : (\u03a0 i, M i \u2192* N) \u2243 (free_product M \u2192* N) :=\n{ to_fun := \u03bb fi, con.lift _ (free_monoid.lift $ \u03bb p : \u03a3 i, M i, fi p.fst p.snd) $ con.con_gen_le\n    begin\n      simp_rw [con.rel_eq_coe, con.ker_rel],\n      rintros _ _ (i | \u27e8i, x, y\u27e9),\n      { change free_monoid.lift _ (free_monoid.of _) = free_monoid.lift _ 1,\n        simp only [monoid_hom.map_one, free_monoid.lift_eval_of], },\n      { change free_monoid.lift _ (free_monoid.of _ * free_monoid.of _) =\n          free_monoid.lift _ (free_monoid.of _),\n        simp only [monoid_hom.map_mul, free_monoid.lift_eval_of], }\n    end,\n  inv_fun := \u03bb f i, f.comp of,\n  left_inv := by { intro fi, ext i x,\n    rw [monoid_hom.comp_apply, of_apply, con.lift_mk', free_monoid.lift_eval_of], },\n  right_inv := by { intro f, ext i x,\n    simp only [monoid_hom.comp_apply, of_apply, con.lift_mk', free_monoid.lift_eval_of], } }\n\n@[simp] lemma lift_of {N} [monoid N] (fi : \u03a0 i, M i \u2192* N) {i} (m : M i) :\n  lift fi (of m) = fi i m :=\nby conv_rhs { rw [\u2190lift.symm_apply_apply fi, lift_symm_apply, monoid_hom.comp_apply] }\n\n@[elab_as_eliminator]\nlemma induction_on {C : free_product M \u2192 Prop}\n  (m : free_product M)\n  (h_one : C 1)\n  (h_of : \u2200 (i) (m : M i), C (of m))\n  (h_mul : \u2200 (x y), C x \u2192 C y \u2192 C (x * y)) :\n  C m :=\nbegin\n  let S : submonoid (free_product M) := submonoid.mk (set_of C) h_mul h_one,\n  convert subtype.prop (lift (\u03bb i, of.cod_mrestrict S (h_of i)) m),\n  change monoid_hom.id _ m = S.subtype.comp _ m,\n  congr,\n  ext,\n  simp [monoid_hom.cod_mrestrict],\nend\n\nlemma of_left_inverse [decidable_eq \u03b9] (i : \u03b9) :\n  function.left_inverse (lift $ pi.mul_single i (monoid_hom.id (M i))) of :=\n\u03bb x, by simp only [lift_of, pi.mul_single_eq_same, monoid_hom.id_apply]\n\nlemma of_injective (i : \u03b9) : function.injective \u21d1(of : M i \u2192* _) :=\nby { classical, exact (of_left_inverse i).injective }\n\nlemma lift_mrange_le {N} [monoid N] (f : \u03a0 i, M i \u2192* N) {s : submonoid N}\n  (h : \u2200 i, (f i).mrange \u2264 s) : (lift f).mrange \u2264 s :=\nbegin\n  rintros _ \u27e8x, rfl\u27e9,\n  induction x using free_product.induction_on with i x x y hx hy,\n  { exact s.one_mem, },\n  { simp only [lift_of, set_like.mem_coe], exact h i (set.mem_range_self x), },\n  { simp only [map_mul, set_like.mem_coe], exact s.mul_mem hx hy, },\nend\n\nlemma mrange_eq_supr {N} [monoid N] (f : \u03a0 i, M i \u2192* N) :\n  (lift f).mrange = \u2a06 i, (f i).mrange :=\nbegin\n  apply le_antisymm (lift_mrange_le f (\u03bb i, le_supr _ i)),\n  apply supr_le _,\n  rintros i _ \u27e8x, rfl\u27e9,\n  exact \u27e8of x, by simp only [lift_of]\u27e9\nend\n\nsection group\n\nvariables (G : \u03b9 \u2192 Type*) [\u03a0 i, group (G i)]\n\ninstance : has_inv (free_product G) :=\n{ inv := mul_opposite.unop \u2218\n    lift (\u03bb i, (of : G i \u2192* _).op.comp (mul_equiv.inv' (G i)).to_monoid_hom) }\n\nlemma inv_def (x : free_product G) : x\u207b\u00b9 = mul_opposite.unop\n  (lift (\u03bb i, (of : G i \u2192* _).op.comp (mul_equiv.inv' (G i)).to_monoid_hom) x) := rfl\n\ninstance : group (free_product G) :=\n{ mul_left_inv := begin\n    intro m,\n    rw inv_def,\n    apply m.induction_on,\n    { rw [monoid_hom.map_one, mul_opposite.unop_one, one_mul], },\n    { intros i m, change of m\u207b\u00b9 * of m = 1, rw [\u2190of.map_mul, mul_left_inv, of.map_one], },\n    { intros x y hx hy,\n      rw [monoid_hom.map_mul, mul_opposite.unop_mul, mul_assoc, \u2190 mul_assoc _ x y, hx,\n        one_mul, hy], },\n  end,\n  ..free_product.has_inv G,\n  ..free_product.monoid G }\n\nlemma lift_range_le {N} [group N] (f : \u03a0 i, G i \u2192* N) {s : subgroup N}\n  (h : \u2200 i, (f i).range \u2264 s) : (lift f).range \u2264 s :=\nbegin\n  rintros _ \u27e8x, rfl\u27e9,\n  induction x using free_product.induction_on with i x x y hx hy,\n  { exact s.one_mem, },\n  { simp only [lift_of, set_like.mem_coe], exact h i (set.mem_range_self x), },\n  { simp only [map_mul, set_like.mem_coe], exact s.mul_mem hx hy, },\nend\n\nlemma range_eq_supr {N} [group N] (f : \u03a0 i, G i \u2192* N) :\n  (lift f).range = \u2a06 i, (f i).range :=\nbegin\n  apply le_antisymm (lift_range_le _ f (\u03bb i, le_supr _ i)),\n  apply supr_le _,\n  rintros i _ \u27e8x, rfl\u27e9,\n  exact \u27e8of x, by simp only [lift_of]\u27e9\nend\n\nend group\n\nnamespace word\n\n/-- The empty reduced word. -/\ndef empty : word M := { to_list := [], ne_one := \u03bb _, false.elim, chain_ne := list.chain'_nil }\n\ninstance : inhabited (word M) := \u27e8empty\u27e9\n\n/-- A reduced word determines an element of the free product, given by multiplication. -/\ndef prod (w : word M) : free_product M :=\nlist.prod (w.to_list.map $ \u03bb l, of l.snd)\n\n@[simp] lemma prod_empty : prod (empty : word M) = 1 := rfl\n\n/-- `fst_idx w` is `some i` if the first letter of `w` is `\u27e8i, m\u27e9` with `m : M i`. If `w` is empty\nthen it's `none`. -/\ndef fst_idx (w : word M) : option \u03b9 := w.to_list.head'.map sigma.fst\n\nlemma fst_idx_ne_iff {w : word M} {i} :\n  fst_idx w \u2260 some i \u2194 \u2200 l \u2208 w.to_list.head', i \u2260 sigma.fst l :=\nnot_iff_not.mp $ by simp [fst_idx]\n\nvariable (M)\n\n/-- Given an index `i : \u03b9`, `pair M i` is the type of pairs `(head, tail)` where `head : M i` and\n`tail : word M`, subject to the constraint that first letter of `tail` can't be `\u27e8i, m\u27e9`.\nBy prepending `head` to `tail`, one obtains a new word. We'll show that any word can be uniquely\nobtained in this way. -/\n@[ext] structure pair (i : \u03b9) :=\n(head : M i)\n(tail : word M)\n(fst_idx_ne : fst_idx tail \u2260 some i)\n\ninstance (i : \u03b9) : inhabited (pair M i) := \u27e8\u27e81, empty, by tauto\u27e9\u27e9\n\nvariable {M}\n\nvariables [\u2200 i, decidable_eq (M i)]\n\n/-- Given a pair `(head, tail)`, we can form a word by prepending `head` to `tail`, except if `head`\nis `1 : M i` then we have to just return `word` since we need the result to be reduced. -/\ndef rcons {i} (p : pair M i) : word M :=\nif h : p.head = 1 then p.tail\nelse { to_list  := \u27e8i, p.head\u27e9 :: p.tail.to_list,\n       ne_one   := by { rintros l (rfl | hl), exact h, exact p.tail.ne_one l hl },\n       chain_ne := p.tail.chain_ne.cons' (fst_idx_ne_iff.mp p.fst_idx_ne) }\n\n/-- Given a word of the form `\u27e8l :: ls, h1, h2\u27e9`, we can form a word of the form `\u27e8ls, _, _\u27e9`,\ndropping the first letter. -/\nprivate def mk_aux {l} (ls : list (\u03a3 i, M i)) (h1 : \u2200 l' \u2208 l :: ls, sigma.snd l' \u2260 1)\n  (h2 : (l :: ls).chain' _) : word M :=\n\u27e8ls, \u03bb l' hl, h1 _ (list.mem_cons_of_mem _ hl), h2.tail\u27e9\n\nlemma cons_eq_rcons {i} {m : M i} {ls h1 h2} :\n  word.mk (\u27e8i, m\u27e9 :: ls) h1 h2 = rcons \u27e8m, mk_aux ls h1 h2, fst_idx_ne_iff.mpr h2.rel_head'\u27e9 :=\nby { rw [rcons, dif_neg], refl, exact h1 \u27e8i, m\u27e9 (ls.mem_cons_self _) }\n\n@[simp] lemma prod_rcons {i} (p : pair M i) :\n  prod (rcons p) = of p.head * prod p.tail :=\nif hm : p.head = 1 then by rw [rcons, dif_pos hm, hm, monoid_hom.map_one, one_mul]\nelse by rw [rcons, dif_neg hm, prod, list.map_cons, list.prod_cons, prod]\n\nlemma rcons_inj {i} : function.injective (rcons : pair M i \u2192 word M) :=\nbegin\n  rintros \u27e8m, w, h\u27e9 \u27e8m', w', h'\u27e9 he,\n  by_cases hm : m = 1;\n  by_cases hm' : m' = 1,\n  { simp only [rcons, dif_pos hm, dif_pos hm'] at he, cc, },\n  { exfalso, simp only [rcons, dif_pos hm, dif_neg hm'] at he, rw he at h, exact h rfl },\n  { exfalso, simp only [rcons, dif_pos hm', dif_neg hm] at he, rw \u2190he at h', exact h' rfl, },\n  { have : m = m' \u2227 w.to_list = w'.to_list,\n    { simpa only [rcons, dif_neg hm, dif_neg hm', true_and, eq_self_iff_true, subtype.mk_eq_mk,\n      heq_iff_eq, \u2190subtype.ext_iff_val] using he },\n    rcases this with \u27e8rfl, h\u27e9,\n    congr, exact word.ext _ _ h, }\nend\n\nvariable [decidable_eq \u03b9]\n\n/-- Given `i : \u03b9`, any reduced word can be decomposed into a pair `p` such that `w = rcons p`. -/\n-- This definition is computable but not very nice to look at. Thankfully we don't have to inspect\n-- it, since `rcons` is known to be injective.\nprivate def equiv_pair_aux (i) : \u03a0 w : word M, { p : pair M i // rcons p = w }\n| w@\u27e8[], _, _\u27e9             := \u27e8\u27e81, w, by rintro \u27e8\u27e9\u27e9, dif_pos rfl\u27e9\n| w@\u27e8\u27e8j, m\u27e9 :: ls, h1, h2\u27e9 := if ij : i = j then\n  { val := { head := ij.symm.rec m,\n             tail := mk_aux ls h1 h2,\n             fst_idx_ne := by cases ij; exact fst_idx_ne_iff.mpr h2.rel_head' },\n    property := by cases ij; exact cons_eq_rcons.symm }\nelse \u27e8\u27e81, w, (option.some_injective _).ne (ne.symm ij)\u27e9, dif_pos rfl\u27e9\n\n/-- The equivalence between words and pairs. Given a word, it decomposes it as a pair by removing\nthe first letter if it comes from `M i`. Given a pair, it prepends the head to the tail. -/\ndef equiv_pair (i) : word M \u2243 pair M i :=\n{ to_fun := \u03bb w, (equiv_pair_aux i w).val,\n  inv_fun := rcons,\n  left_inv := \u03bb w, (equiv_pair_aux i w).property,\n  right_inv := \u03bb p, rcons_inj (equiv_pair_aux i _).property }\n\nlemma equiv_pair_symm (i) (p : pair M i) : (equiv_pair i).symm p = rcons p := rfl\n\nlemma equiv_pair_eq_of_fst_idx_ne {i} {w : word M} (h : fst_idx w \u2260 some i) :\n  equiv_pair i w = \u27e81, w, h\u27e9 :=\n(equiv_pair i).apply_eq_iff_eq_symm_apply.mpr $ eq.symm (dif_pos rfl)\n\ninstance summand_action (i) : mul_action (M i) (word M) :=\n{ smul     := \u03bb m w, rcons { head := m * (equiv_pair i w).head, ..equiv_pair i w },\n  one_smul := \u03bb w, by { simp_rw [one_mul], apply (equiv_pair i).symm_apply_eq.mpr, ext; refl },\n  mul_smul := \u03bb m m' w, by simp only [mul_assoc, \u2190equiv_pair_symm, equiv.apply_symm_apply], }\n\ninstance : mul_action (free_product M) (word M) :=\nmul_action.of_End_hom (lift (\u03bb i, mul_action.to_End_hom))\n\nlemma of_smul_def (i) (w : word M) (m : M i) :\n  of m \u2022 w = rcons { head := m * (equiv_pair i w).head, ..equiv_pair i w } := rfl\n\nlemma cons_eq_smul {i} {m : M i} {ls h1 h2} :\n  word.mk (\u27e8i, m\u27e9 :: ls) h1 h2 = of m \u2022 mk_aux ls h1 h2 :=\nby rw [cons_eq_rcons, of_smul_def, equiv_pair_eq_of_fst_idx_ne _]; simp only [mul_one]\n\nlemma smul_induction {C : word M \u2192 Prop}\n  (h_empty : C empty)\n  (h_smul : \u2200 i (m : M i) w, C w \u2192 C (of m \u2022 w))\n  (w : word M) : C w :=\nbegin\n  cases w with ls h1 h2,\n  induction ls with l ls ih,\n  { exact h_empty },\n  cases l with i m,\n  rw cons_eq_smul,\n  exact h_smul _ _ _ (ih _ _),\nend\n\n@[simp] lemma prod_smul (m) : \u2200 w : word M, prod (m \u2022 w) = m * prod w :=\nbegin\n  apply m.induction_on,\n  { intro, rw [one_smul, one_mul] },\n  { intros, rw [of_smul_def, prod_rcons, of.map_mul, mul_assoc, \u2190prod_rcons,\n      \u2190equiv_pair_symm, equiv.symm_apply_apply] },\n  { intros x y hx hy w, rw [mul_smul, hx, hy, mul_assoc] },\nend\n\n/-- Each element of the free product corresponds to a unique reduced word. -/\ndef equiv : free_product M \u2243 word M :=\n{ to_fun := \u03bb m, m \u2022 empty,\n  inv_fun := \u03bb w, prod w,\n  left_inv := \u03bb m, by dsimp only; rw [prod_smul, prod_empty, mul_one],\n  right_inv := begin\n    apply smul_induction,\n    { dsimp only, rw [prod_empty, one_smul], },\n    { dsimp only, intros i m w ih, rw [prod_smul, mul_smul, ih], },\n  end }\n\ninstance : decidable_eq (word M) := function.injective.decidable_eq word.ext\ninstance : decidable_eq (free_product M) := word.equiv.decidable_eq\n\nend word\n\nvariable (M)\n\n/-- A `neword M i j` is a representation of a non-empty reduced words where the first letter comes\nfrom `M i` and the last letter comes from `M j`. It can be constructed from singletons and via\nconcatentation, and thus provides a useful induction principle. -/\n@[nolint has_inhabited_instance]\ninductive neword : \u03b9 \u2192 \u03b9 \u2192 Type (max u_1 u_2)\n| singleton : \u2200 {i} (x : M i) (hne1 : x \u2260 1), neword i i\n| append : \u2200 {i j k l} (w\u2081 : neword i j) (hne : j \u2260 k) (w\u2082 : neword k l), neword i l\nvariable {M}\n\nnamespace neword\n\nopen word\n\n/-- The list represented by a given `neword` -/\n@[simp]\ndef to_list : \u03a0 {i j} (w : neword M i j), list (\u03a3 i, M i)\n| i _ (singleton x hne1) := [\u27e8i, x\u27e9]\n| _ _ (append w\u2081 hne w\u2082) := w\u2081.to_list ++ w\u2082.to_list\n\n\n\n/--  The first letter of a `neword` -/\n@[simp]\ndef head : \u03a0 {i j} (w : neword M i j), M i\n| i _ (singleton x hne1) := x\n| _ _ (append w\u2081 hne w\u2082) := w\u2081.head\n\n/--  The last letter of a `neword` -/\n@[simp]\ndef last : \u03a0 {i j} (w : neword M i j), M j\n| i _ (singleton x hne1) := x\n| _ _ (append w\u2081 hne w\u2082) := w\u2082.last\n\n@[simp]\nlemma to_list_head' {i j} (w : neword M i j) :\n  w.to_list.head' = option.some \u27e8i, w.head\u27e9 :=\nbegin\n  rw \u2190 option.mem_def,\n  induction w,\n  { rw option.mem_def, reflexivity, },\n  { exact list.head'_append w_ih_w\u2081, },\nend\n\n@[simp]\nlemma to_list_last' {i j} (w : neword M i j) :\n  w.to_list.last' = option.some \u27e8j, w.last\u27e9 :=\nbegin\n  rw \u2190 option.mem_def,\n  induction w,\n  { rw option.mem_def, reflexivity, },\n  { exact list.last'_append w_ih_w\u2082, },\nend\n\n/-- The `word M` represented by a `neword M i j` -/\ndef to_word {i j} (w : neword M i j) : word M :=\n{ to_list := w.to_list,\n  ne_one :=\n  begin\n    induction w,\n    { rintros \u27e8k,x\u27e9 \u27e8rfl, rfl\u27e9,\n      exact w_hne1,\n      exfalso, apply H, },\n    { intros l h,\n      simp only [to_list, list.mem_append] at h,\n      cases h,\n      { exact w_ih_w\u2081 _ h, },\n      { exact w_ih_w\u2082 _ h, }, },\n  end,\n  chain_ne := begin\n    induction w,\n    { exact list.chain'_singleton _, },\n    { apply list.chain'.append w_ih_w\u2081 w_ih_w\u2082,\n      intros x hx y hy,\n      rw [w_w\u2081.to_list_last', option.mem_some_iff] at hx,\n      rw [w_w\u2082.to_list_head', option.mem_some_iff] at hy,\n      subst hx, subst hy,\n      exact w_hne, },\n  end, }\n\n/-- Every nonempty `word M` can be constructed as a `neword M i j` -/\nlemma of_word (w : word M) (h : w \u2260 empty) :\n  \u2203 i j (w' : neword M i j), w'.to_word = w :=\nbegin\n  suffices : \u2203 i j (w' : neword M i j), w'.to_word.to_list = w.to_list,\n  { obtain \u27e8i, j, w, h\u27e9 := this, refine \u27e8i, j, w, _\u27e9, ext, rw h, },\n  cases w with l hnot1 hchain,\n  induction l with x l hi,\n  { contradiction, },\n  { rw list.forall_mem_cons at hnot1,\n    cases l with y l,\n    { refine \u27e8x.1, x.1, singleton x.2 hnot1.1, _ \u27e9,\n      simp [to_word], },\n    { rw list.chain'_cons at hchain,\n      specialize hi hnot1.2 hchain.2 (by rintros \u27e8rfl\u27e9),\n      obtain \u27e8i, j, w', hw' : w'.to_list = y :: l\u27e9 := hi,\n      obtain rfl : y = \u27e8i, w'.head\u27e9, by simpa [hw'] using w'.to_list_head',\n      refine \u27e8x.1, j, append (singleton x.2 hnot1.1) hchain.1 w', _\u27e9,\n      { simpa [to_word] using hw', } } }\nend\n\n/-- A non-empty reduced word determines an element of the free product, given by multiplication. -/\ndef prod {i j} (w : neword M i j) := w.to_word.prod\n\n@[simp]\nlemma singleton_head {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).head = x := rfl\n\n@[simp]\nlemma singleton_last {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).last = x := rfl\n\n@[simp] lemma prod_singleton {i} (x : M i) (hne_one : x \u2260 1) :\n  (singleton x hne_one).prod = of x :=\nby simp [to_word, prod, word.prod]\n\n@[simp]\nlemma append_head {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).head = w\u2081.head := rfl\n\n@[simp]\nlemma append_last {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).last = w\u2082.last := rfl\n\n@[simp]\nlemma append_prod {i j k l} {w\u2081 : neword M i j} {hne : j \u2260 k} {w\u2082 : neword M k l} :\n  (append w\u2081 hne w\u2082).prod = w\u2081.prod * w\u2082.prod :=\nby simp [to_word, prod, word.prod]\n\n/-- One can replace the first letter in a non-empty reduced word by an element of the same\ngroup -/\ndef replace_head : \u03a0 {i j : \u03b9} (x : M i) (hnotone : x \u2260 1) (w : neword M i j), neword M i j\n| _ _ x h (singleton _ _) := singleton x h\n| _ _ x h (append w\u2081 hne w\u2082) := append (replace_head x h w\u2081) hne w\u2082\n\n@[simp]\nlemma replace_head_head {i j : \u03b9} (x : M i) (hnotone : x \u2260 1) (w : neword M i j) :\n  (replace_head x hnotone w).head = x :=\nby { induction w, refl, exact w_ih_w\u2081 _ _, }\n\n/-- One can multiply an element from the left to a non-empty reduced word if it does not cancel\nwith the first element in the word. -/\ndef mul_head {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n  neword M i j := replace_head (x * w.head) hnotone w\n\n@[simp]\nlemma mul_head_head {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n   (mul_head w x hnotone).head = x * w.head :=\nby { induction w, refl, exact w_ih_w\u2081 _ _, }\n\n@[simp]\nlemma mul_head_prod {i j : \u03b9} (w : neword M i j) (x : M i) (hnotone : x * w.head \u2260 1) :\n  (mul_head w x hnotone).prod = of x * w.prod :=\nbegin\n  unfold mul_head,\n  induction w,\n  { simp [mul_head, replace_head], },\n  { specialize w_ih_w\u2081 _ hnotone, clear w_ih_w\u2082,\n    simp [replace_head, \u2190 mul_assoc] at *,\n    congr' 1, }\nend\n\nsection group\n\nvariables {G : \u03b9 \u2192 Type*} [\u03a0 i, group (G i)]\n\n/-- The inverse of a non-empty reduced word -/\ndef inv : \u03a0 {i j} (w : neword G i j), neword G j i\n| _ _ (singleton x h) := singleton x\u207b\u00b9 (mt inv_eq_one.mp h)\n| _ _ (append w\u2081 h w\u2082) := append w\u2082.inv h.symm w\u2081.inv\n\n@[simp]\nlemma inv_prod {i j} (w : neword G i j) : w.inv.prod = w.prod\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\n@[simp]\nlemma inv_head {i j} (w : neword G i j) : w.inv.head = w.last\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\n@[simp]\nlemma inv_last {i j} (w : neword G i j) : w.inv.last = w.head\u207b\u00b9 :=\nby induction w; simp [inv, *]\n\nend group\n\nend neword\n\nsection ping_pong_lemma\n\nopen_locale pointwise\nopen_locale cardinal\n\nvariables [hnontriv : nontrivial \u03b9]\nvariables {G : Type*} [group G]\nvariables {H : \u03b9 \u2192 Type*} [\u2200 i, group (H i)]\nvariables (f : \u03a0 i, H i \u2192* G)\n\n-- We need many groups or one group with many elements\nvariables (hcard : 3 \u2264 # \u03b9 \u2228 \u2203 i, 3 \u2264 # (H i))\n\n-- A group action on \u03b1, and the ping-pong sets\nvariables {\u03b1 : Type*} [mul_action G \u03b1]\nvariables (X : \u03b9 \u2192 set \u03b1)\nvariables (hXnonempty : \u2200 i, (X i).nonempty)\nvariables (hXdisj : pairwise (\u03bb i j, disjoint (X i) (X j)))\nvariables (hpp : pairwise (\u03bb i j, \u2200 h : H i, h \u2260 1 \u2192 f i h \u2022 X j \u2286 X i))\n\ninclude hpp\n\nlemma lift_word_ping_pong {i j k} (w : neword H i j) (hk : j \u2260 k) :\n  lift f w.prod \u2022 X k \u2286 X i :=\nbegin\n  rename [i \u2192 i', j \u2192 j', k \u2192 m, hk \u2192 hm],\n  induction w with i x hne_one i j k l w\u2081 hne w\u2082  hIw\u2081 hIw\u2082 generalizing m; clear i' j',\n  { simpa using hpp _ _ hm _ hne_one, },\n  { calc lift f (neword.append w\u2081 hne w\u2082).prod \u2022 X m\n        = lift f w\u2081.prod \u2022 lift f w\u2082.prod \u2022 X m : by simp [mul_action.mul_smul]\n    ... \u2286 lift f w\u2081.prod \u2022 X k : set_smul_subset_set_smul_iff.mpr (hIw\u2082 hm)\n    ... \u2286 X i : hIw\u2081 hne },\nend\n\ninclude X hXnonempty hXdisj\n\nlemma lift_word_prod_nontrivial_of_other_i {i j k} (w : neword H i j)\n  (hhead : k \u2260 i) (hlast : k \u2260 j) : lift f w.prod \u2260 1 :=\nbegin\n  intro heq1,\n  have : X k \u2286 X i,\n    by simpa [heq1] using lift_word_ping_pong f X hpp w hlast.symm,\n  obtain \u27e8x, hx\u27e9 := hXnonempty k,\n  exact hXdisj k i hhead \u27e8hx, this hx\u27e9,\nend\n\ninclude hnontriv\n\nlemma lift_word_prod_nontrivial_of_head_eq_last {i} (w : neword H i i) :\n  lift f w.prod \u2260 1 :=\nbegin\n  obtain \u27e8k, hk\u27e9 := exists_ne i,\n  exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w hk hk,\nend\n\nlemma lift_word_prod_nontrivial_of_head_card {i j} (w : neword H i j)\n  (hcard : 3 \u2264 # (H i)) (hheadtail : i \u2260 j) : lift f w.prod \u2260 1 :=\nbegin\n  obtain \u27e8h, hn1, hnh\u27e9 := cardinal.three_le hcard 1 (w.head\u207b\u00b9),\n  have hnot1 : h * w.head \u2260 1, by { rw \u2190 div_inv_eq_mul, exact div_ne_one_of_ne hnh },\n  let w' : neword H i i := neword.append\n    (neword.mul_head w h hnot1) hheadtail.symm\n    (neword.singleton h\u207b\u00b9 (inv_ne_one.mpr hn1)),\n  have hw' : lift f w'.prod \u2260 1 :=\n    lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w',\n  intros heq1, apply hw', simp [w', heq1]\nend\n\ninclude hcard\nlemma lift_word_prod_nontrivial_of_not_empty {i j} (w : neword H i j) :\n  lift f w.prod \u2260 1 :=\nbegin\n  classical,\n  cases hcard,\n  { obtain \u27e8i, h1, h2\u27e9 := cardinal.three_le hcard i j,\n    exact lift_word_prod_nontrivial_of_other_i f X hXnonempty hXdisj hpp w h1 h2, },\n  { cases hcard with k hcard,\n    by_cases hh : i = k; by_cases hl : j = k,\n    { subst hh, subst hl,\n      exact lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w, },\n    { subst hh,\n      change j \u2260 i at hl,\n      exact lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w hcard hl.symm, },\n    { subst hl,\n      change i \u2260 j at hh,\n      have : lift f w.inv.prod \u2260 1 :=\n        lift_word_prod_nontrivial_of_head_card f X hXnonempty hXdisj hpp w.inv hcard hh.symm,\n      intros heq, apply this, simpa using heq, },\n    { change i \u2260 k at hh,\n      change j \u2260 k at hl,\n      obtain \u27e8h, hn1, -\u27e9 := cardinal.three_le hcard 1 1,\n      let w' : neword H k k := neword.append\n        (neword.append (neword.singleton h hn1) hh.symm w)\n        hl (neword.singleton h\u207b\u00b9 (inv_ne_one.mpr hn1)) ,\n      have hw' : lift f w'.prod \u2260 1 :=\n        lift_word_prod_nontrivial_of_head_eq_last f X hXnonempty hXdisj hpp w',\n      intros heq1, apply hw', simp [w', heq1], }, }\nend\n\nlemma empty_of_word_prod_eq_one {w : word H} (h : lift f w.prod = 1) :\n  w = word.empty :=\nbegin\n  by_contradiction hnotempty,\n  obtain \u27e8i, j, w, rfl\u27e9 := neword.of_word _ hnotempty,\n  exact lift_word_prod_nontrivial_of_not_empty f hcard X hXnonempty hXdisj hpp w h,\nend\n\n/--\nThe Ping-Pong-Lemma.\n\nGiven a group action of `G` on `X` so that the `H i` acts in a specific way on disjoint subsets\n`X i` we can prove that `lift f` is injective, and thus the image of `lift f` is isomorphic to the\ndirect product of the `H i`.\n\nOften the Ping-Pong-Lemma is stated with regard to subgroups `H i` that generate the whole group;\nwe generalize to arbitrary group homomorphisms `f i : H i \u2192* G` and do not require the group to be\ngenerated by the images.\n\nUsually the Ping-Pong-Lemma requires that one group `H i` has at least three elements. This\ncondition is only needed if `# \u03b9 = 2`, and we accept `3 \u2264 # \u03b9` as an alternative.\n-/\ntheorem lift_injective_of_ping_pong:\n  function.injective (lift f) :=\nbegin\n  classical,\n  apply (monoid_hom.injective_iff (lift f)).mpr,\n  rw free_product.word.equiv.forall_congr_left',\n  { intros w Heq,\n    dsimp [word.equiv] at *,\n    { rw empty_of_word_prod_eq_one f hcard X hXnonempty hXdisj hpp Heq,\n      reflexivity, }, },\n  apply_instance,\n  apply_instance,\nend\n\nend ping_pong_lemma\n\n/-- The free product of free groups is itself a free group -/\n@[simps]\ninstance {\u03b9 : Type*} (G : \u03b9 \u2192 Type*) [\u2200 i, group (G i)] [hG : \u2200 i, is_free_group (G i)] :\n  is_free_group (free_product G) :=\n{ generators := \u03a3 i, is_free_group.generators (G i),\n  of := \u03bb x, free_product.of (is_free_group.of x.2),\n  unique_lift' :=\n  begin\n    introsI X _ f,\n    refine \u27e8free_product.lift (\u03bb i, is_free_group.lift (\u03bb x, f \u27e8i, x\u27e9)), _ \u27e9,\n    split,\n    { simp, },\n    { intros g hfg, ext i x, simpa using hfg \u27e8i, x\u27e9, }\n  end, }\n\nend free_product\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/group_theory/free_product.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3650680482911652}}
{"text": "\nimport heap.lemmas misc\nimport tactic.monotonicity\nimport tactic.library_search\nimport logic.basic\n\nnamespace separation\n\nopen memory finmap\n\nvariables value : Type\n\ndef hProp := heap value \u2192 Prop\n\nvariables {value}\ninclude value\nlocal notation `heap` := heap value\nlocal notation `hProp` := hProp value\nlocal notation `tptr` := tptr value\n\nnamespace \u00abhProp\u00bb\n\n@[extensionality]\nlemma hProp.ext {p q : hProp} (H : \u2200 h, p h \u2194 q h) : p = q :=\nby ext; apply H\n\ndef emp : hProp := \u03bb h, h = \u2205\n\ndef and (x y : hProp) : hProp\n| h := \u2203 h\u2080 h\u2081, some h = add (some h\u2080) (some h\u2081) \u2227 x h\u2080 \u2227 y h\u2081\n\ndef lift (p : Prop) : hProp\n| h := p \u2227 emp h\n\nnotation `[| ` p ` |]` := lift p\n\nlemma lift_eq_emp {p} (h : p) : [| p |] = @emp value :=\nby { funext h', ext, simp [lift,h] }\n\ninfixr ` \u229b `:55 := and\n\n@[simp]\nlemma emp_and (p : hProp) : emp \u229b p = p :=\nbegin\n  ext; simp [emp,and,(\u2208),set.mem,add]; split,\n  { simp, introv h\u2080 h\u2081, subst h\u2081,\n    simp [disjoint_empty,pure] at h\u2080, subst h\u2080, exact id },\n  { intro hp, existsi [(\u2205 : finmap _),h],\n    rw [if_pos (disjoint_empty h)], simp,\n    exact \u27e8rfl, hp\u27e9 }\nend\n\nlemma and_comm (p q : hProp) : p \u229b q = q \u229b p :=\nfunext $ \u03bb h, by { dsimp [and], rw exists_swap, ext, repeat { apply exists_congr; intro },\n                   rw [memory.add_comm,and_comm (p a_1)] }\n\ninstance and.is_commutative : is_commutative _ (@and value) := \u27e8 and_comm \u27e9\n\n@[simp]\nlemma and_emp (p : hProp) : p \u229b emp = p :=\nby rw [and_comm,emp_and]\n\ndef And {\u03b1} (p : \u03b1 \u2192 hProp) : list \u03b1 \u2192 hProp\n| [] := emp\n| (x :: xs) := p x \u229b And xs\n\ndef p_exists {\u03b1} (p : \u03b1 \u2192 hProp) : hProp\n| h := \u2203 x, p x h\n\nnotation `\u2203\u2203 ` binders `, ` r:(scoped P, p_exists P) := r\n\ndef p_forall {\u03b1} (p : \u03b1 \u2192 hProp) : hProp\n| h := \u2200 x, p x h\n\nnotation `\u2200\u2200 ` binders `, ` r:(scoped P, p_forall P) := r\n\ndef maplets : ptr \u2192 value \u2192 hProp\n-- | p [] := emp\n| p v h := h = maplet p v\n\ninfix ` \u21a6 `:60 := maplets\n\nopen list\n\nlemma and_assoc (p q r : hProp) : (p \u229b q) \u229b r = p \u229b q \u229b r :=\nbegin\n  ext : 1; dsimp [and,(\u2208),set.mem],\n  split,\n  { rintros \u27e8h\u2080,h\u2081,Hx,\u27e8h\u2082,h\u2083,HH,Hp,Hq\u27e9,Hr\u27e9,\n    have : disjoint h\u2083 h\u2081,\n    { by_contradiction,\n      rw [HH,memory.add_assoc] at Hx, simp [add,if_neg,a] at Hx,\n      exact Hx },\n    refine \u27e8_, h\u2083 \u222a h\u2081, _, Hp, _, _, _, Hq, Hr\u27e9,\n    { rw [Hx,HH,memory.add_assoc,union_eq_add_of_disjoint this] },\n    { rw union_eq_add_of_disjoint this } },\n  { rintro \u27e8h\u2080,h\u2081,Hx,Hp,h\u2082,h\u2083,Hh\u2081,Hq,Hr\u27e9,\n    have : disjoint h\u2080 h\u2082,\n    { by_contradiction,\n      rw [Hh\u2081,\u2190 memory.add_assoc] at Hx, simp [add,if_neg,a] at Hx,\n      exact Hx },\n    refine \u27e8h\u2080 \u222a h\u2082,_,_,\u27e8_,_,_,Hp,Hq\u27e9,Hr\u27e9,\n    { rw [Hx,Hh\u2081,union_eq_add_of_disjoint this,memory.add_assoc] },\n    { rw union_eq_add_of_disjoint this } }\nend\n\ninstance and.is_associative : is_associative _ (@and value) := \u27e8 hProp.and_assoc \u27e9\n\n@[simp, separation_logic]\nlemma And_append {\u03b1} (p : \u03b1 \u2192 hProp) (xs ys : list \u03b1) : And p (xs ++ ys) = And p xs \u229b And p ys :=\nby induction xs; dsimp [And]; [rw emp_and, rw [xs_ih,and_assoc]]\n\n@[simp]\nlemma And_map {\u03b1 \u03b2} (p : \u03b1 \u2192 hProp) (f : \u03b2 \u2192 \u03b1) : \u03a0 (xs : list \u03b2), And p (map f xs) = And (p \u2218 f) xs\n| [] := rfl\n| (x :: xs) := by rw [map,And,And,And_map]\n\nopen nat\n\n-- lemma maplets_eq_And (p : ptr) (vs : list value) : (p \u21a6 vs) = And (\u03bb i, p+i \u21a6 nth' i vs) (range vs.length) :=\n-- begin\n--   induction vs generalizing p,  refl,\n--   simp [maplets], simp only [nat.add_comm 1,range_succ_eq_map,And,vs_ih,And_map,(\u2218),nth'],\n--   congr' 1, ext, simp [maplets], simp [succ_eq_add_one]\n-- end\n\ndef holds (h frame : heap) (p : hProp) : Prop :=\n\u2203 h\u2080, some h = some h\u2080 \u2297 some frame \u2227 p h\u2080\n\ndef holds' (h : heap) (p : hProp) : Prop :=\n\u2203 h', holds h h' p\n\ninfix ` \u22a8 `:60 := holds'\n\nlemma holds_union_and {h h' frame : heap} {p q : hProp}\n  (H\u2080 : holds h frame p) (H\u2081 : q h') (H\u2082 : disjoint h h') :\n  holds (h \u222a h') frame (p \u229b q) :=\nbegin\n  simp [holds,and] at H\u2080 \u22a2,\n  rcases H\u2080 with \u27e8h'', H\u2080, H\u2083\u27e9,\n  existsi h' \u222a h'', rw [union_eq_add_of_disjoint H\u2082,union_eq_add_of_disjoint,H\u2080],\n  split, ac_refl,\n  refine \u27e8_,_,add_comm _ _,H\u2083,H\u2081\u27e9,\n  symmetry, apply disjoint_mono _ (le_refl h') H\u2082,\n  apply le_of_add_eq_some _ H\u2080,\nend\n\nlemma holds_of_holds_union_and {h h' frame : heap} {p q : hProp}\n  (H\u2080 : holds (h \u222a h') frame (p \u229b q)) (H\u2081 : q h') (H\u2082 : disjoint h h') (H\u2083 : \u2200 h'', q h'' \u2192 h'' = h') :\n  holds h frame p :=\nbegin\n  simp [holds,and] at H\u2080 \u22a2,\n  rcases H\u2080 with \u27e8h'', H\u2080, h\u2080, h\u2081, H\u2086, H\u2084, H\u2085\u27e9,\n  refine \u27e8h\u2080,_,H\u2084\u27e9,\n  rw [union_eq_add_of_disjoint H\u2082,H\u2086] at H\u2080,\n  replace H\u2083 := H\u2083 _ H\u2085, subst h',\n  have := union_eq_add_of_disjoint H\u2082,\n  apply add_inj (some h\u2081) (h \u222a h\u2081)  (h \u222a h\u2081) this,\n  rw [this,H\u2080], ac_refl, rw H\u2080, ac_refl,\nend\n\nlemma holds_of_holds_union_iff {h h' frame : heap} {p q : hProp}\n  (H\u2081 : q h') (H\u2082 : disjoint h h') (H\u2083 : \u2200 h'', q h'' \u2192 h'' = h') :\n  holds (h \u222a h') frame (p \u229b q) \u2194 holds h frame p :=\n\u27e8 \u03bb H\u2080, holds_of_holds_union_and H\u2080 H\u2081 H\u2082 H\u2083, \u03bb H\u2080,holds_union_and H\u2080 H\u2081 H\u2082 \u27e9\n\nlemma holds_of_holds_and (h h\u2080 : heap) {p q : hProp} :\n  holds h h\u2080 (p \u229b q) \u2194 (\u2203 h\u2081, holds h\u2081 h\u2080 q \u2227 holds h h\u2081 p) :=\nbegin\n  split; simp only [holds, and, and_imp, exists_imp_distrib],\n  { introv hh hx hp hq,\n    rw hx at hh, clear hx,\n    have hh : \u2203 k, some k = some x_2 \u2297 some h\u2080,\n    { apply add_eq_some _ (some x_1) h,\n      rw [memory.add_comm,\u2190 memory.add_assoc,hh] },\n    cases hh with k hk,\n    existsi k, split,\n    exact \u27e8_,hk,hq\u27e9,\n    refine \u27e8_,_,hp\u27e9, rw [hk,hh,memory.add_assoc] },\n  { introv hh hx hh' hp,\n    rw hh at hh', clear hh,\n    have hh : \u2203 k, some k = some x_2 \u2297 some x_1,\n    { apply add_eq_some _ (some h\u2080) h,\n      rw [memory.add_assoc,hh'] },\n    cases hh with k hk,\n    existsi k, rw [hk,memory.add_assoc],\n    existsi hh',\n    refine \u27e8_,_,rfl,hp,hx\u27e9 }\nend\n\n-- @[simp]\n-- lemma maplets_nil (p : ptr) : (p \u21a6 nil) = @emp value := rfl\n\n-- @[separation_logic]\n-- lemma maplets_cons (p : ptr) (v : value) (vs : list value) : (p \u21a6 v :: vs) = (p \u21a6 [v]) \u229b (p+1 \u21a6 vs) :=\n-- by dsimp [maplets]; rw and_emp\n\nstructure impl (p q : hProp) : Prop :=\nintro ::\n(elim : \u2200 h, p h \u2192 q h)\n\ninfixr ` =*> `:40 := impl\n\ndef wand (p q : hProp) : hProp\n| h := \u2200 h\u2080 h', some h' = some h\u2080 \u2297 some h \u2192 p h\u2080 \u2192 q h'\n\ninfixr ` \u22b8 `:54 := wand\n\nlemma and_wand {p q : hProp} : p \u229b (p \u22b8 q) =*> q :=\n\u27e8 \u03bb h \u27e8h\u2080,h\u2081,Hh,Hp,Hpq\u27e9, Hpq h\u2080 _ Hh Hp \u27e9\n\nlemma disjoint_of_disjoint_union_left {h\u2080 h\u2081 : heap} (h\u2082 : heap) (H : disjoint h\u2080 (h\u2081 \u222a h\u2082)) :\n  disjoint h\u2080 h\u2081 :=\n\u03bb x hx\u2080 hx\u2081, H x hx\u2080 (finmap.mem_union.mpr $ or.inl hx\u2081)\n\nlemma disjoint_of_disjoint_union_right {h\u2080 h\u2081 : heap} (h\u2082 : heap) (H : disjoint (h\u2080 \u222a h\u2082) h\u2081) :\n  disjoint h\u2080 h\u2081 :=\n(disjoint_of_disjoint_union_left h\u2082 H.symm).symm\n\nlemma wand_wand {p q r : hProp} : p \u22b8 (q \u22b8 r) = (p \u229b q) \u22b8 r :=\nbegin\n  ext, simp only [wand, and, and_imp, exists_imp_distrib,@forall_swap (p _)],\n  split; intros H h\u2080 h\u2081 Hhh\u2080 h\u2082 h\u2083 Hhh\u2082 Hp Hq,\n  { have : some (h\u2082 \u222a h) = some h\u2082 \u2297 some h,\n    { apply union_eq_add_of_disjoint, apply disjoint_of_disjoint_union_right h\u2083,\n      apply disjoint_of_add h\u2081, rw union_eq_add_of_disjoint, cc,\n      apply disjoint_of_add _ Hhh\u2082 },\n    apply H _ (h\u2082 \u222a h) this _ _ _ Hp Hq, simp only *, cc },\n  { have : some (h\u2080 \u222a h\u2082) = some h\u2080 \u2297 some h\u2082,\n    { apply union_eq_add_of_disjoint, apply disjoint_of_disjoint_union_right h,\n      apply disjoint_of_add h\u2083, rw union_eq_add_of_disjoint, cc,\n      apply disjoint_of_add _ Hhh\u2080 },\n    apply H (h\u2080 \u222a h\u2082) _ _ _ _ this Hp Hq, simp only *, cc },\nend\n\nlemma wand_deduction {p q r : hProp} : p =*> q \u22b8 r \u2194 p \u229b q =*> r :=\nby { split; rintro \u27e8 H \u27e9; constructor;\n       [rintros h \u27e8h\u2080,h\u2081,HH,Hp,hq\u27e9, intros h Hp h\u2080 h\u2081 HH Hq];\n       apply H; try { assumption <|> refine \u27e8h,h\u2080,_,Hp,Hq\u27e9 };\n     rw [HH,memory.add_comm] }\n\n@[refl]\nlemma impl_refl (p : hProp) : p =*> p := impl.intro $ \u03bb x, id\n\n@[trans]\nlemma impl_trans {p q r : hProp} (hpq : p =*> q) (hqr : q =*> r) : p =*> r :=\nimpl.intro $ \u03bb h hp, hqr.elim _ (hpq.elim _ hp)\n\ndef True : hProp | h := true\n\ndef False : hProp | h := false\n\n@[simp]\nlemma impl_True {p : hProp} : p =*> True :=\n\u27e8 \u03bb _ _, trivial \u27e9\n\n@[simp, separation_logic]\nlemma lift_false : [| false |] = @False value :=\nfunext $ \u03bb h, by simp [lift,False]\n\n@[simp, separation_logic]\nlemma False_and (p : hProp) : False \u229b p = False :=\nby ext; simp [and,False]\n\n@[simp, separation_logic]\nlemma and_False (p : hProp) : p \u229b False = False :=\nby ext; simp [and,False]\n\n-- @[simp]\nlemma False_impl (p : hProp) : False =*> p :=\nimpl.intro $ \u03bb _, false.elim\n\nlemma holds_p_exists {\u03b1} {h h' : heap} (p : \u03b1 \u2192 hProp) : holds h h' (p_exists p) \u2194 \u2203 x, holds h h' (p x) :=\nby split; rintro \u27e8_,H,H',H''\u27e9; [ exact \u27e8_,a_w,H,H''\u27e9, exact \u27e8H,H',_,H''\u27e9 ]\n\nlemma and_p_exists_distrib_right {\u03b1} {p : \u03b1 \u2192 hProp} {q : hProp} :\n  p_exists p \u229b q = \u2203\u2203 x, p x \u229b q :=\nbegin\n  ext, dsimp [and,p_exists], split; simp,\n  { intros h\u2080 h\u2081 H r Hp Hq, refine \u27e8_,_,_,H,Hp,Hq\u27e9 },\n  { intros r h\u2080 h\u2081 H Hp Hq,\n    refine \u27e8_,_,H,\u27e8_,Hp\u27e9,Hq\u27e9, }\nend\n\nlemma and_p_exists_distrib_left {\u03b1} {p : hProp} {q : \u03b1 \u2192 hProp} :\n  p \u229b p_exists q = \u2203\u2203 x, p \u229b q x :=\nby rw [and_comm, and_p_exists_distrib_right]; simp [and_comm]\n\ndef p_or (p q : hProp) : hProp\n| h := p h \u2228 q h\n\ndef p_and (p q : hProp) : hProp\n| h := p h \u2227 q h\n\ninfixr ` \u22c1 `:53 := p_or\n\ninfixr ` \u22c0 `:53 := p_and\n\nlemma p_and_impl_p_and {p p' q q' : hProp} (hp : p =*> q) (hq : p' =*> q') :\n  p \u22c0 p' =*> q \u22c0 q' :=\n\u27e8 \u03bb h, and.imp (hp.elim _) (hq.elim _) \u27e9\n\nlemma impl_and {p q q' : hProp} (H : p =*> q) (H' : p =*> q') :\n  p =*> q \u22c0 q' :=\n\u27e8 \u03bb h hp, \u27e8H.elim _ hp,H'.elim _ hp\u27e9 \u27e9\n\nlemma impl_exists {\u03b1} {p : hProp} {q : \u03b1 \u2192 hProp} (x : \u03b1) (hpq : p =*> q x) : p =*> p_exists q :=\nimpl.intro $ \u03bb h hp, \u27e8_,hpq.elim _ hp\u27e9\n\nlemma exists_impl {\u03b1} {p : \u03b1 \u2192 hProp} {q : hProp} (hpq : \u2200 x, p x =*> q) : p_exists p =*> q :=\nimpl.intro $ \u03bb h \u27e8_, hp\u27e9, (hpq _).elim _ hp\n\nlemma impl_antisymm {p q : hProp} (hpq : p =*> q) (hqp : q =*> p) : p = q :=\nby ext; exact \u27e8hpq.elim _,hqp.elim _\u27e9\n\nopen relation\n\nlemma impl_of_eq {p q : hProp} (hpq : p = q) : p =*> q := impl.intro $ \u03bb h, hpq.subst\n\nlemma exists_impl_exists_of_total {\u03b1 \u03b2} {p : \u03b1 \u2192 hProp} {q : \u03b2 \u2192 hProp}\n  (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (hl : left_total R)\n  (hpq : \u2200 x y, R x y \u2192 p x =*> q y) : p_exists p =*> p_exists q :=\nexists_impl $ \u03bb x, Exists.rec_on (hl x) $ \u03bb w hw, impl_exists w (hpq _ _ hw)\n\nlemma exists_congr_of_total {\u03b1 \u03b2} {p : \u03b1 \u2192 hProp} {q : \u03b2 \u2192 hProp}\n  (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (hl : left_total R) (hr : right_total R)\n  (hpq : \u2200 x y, R x y \u2192 p x = q y) : p_exists p = p_exists q :=\nimpl_antisymm\n  (exists_impl_exists_of_total R hl $ \u03bb x y hR, impl_of_eq $ hpq _ _ hR)\n  (exists_impl_exists_of_total (flip R) hr $ \u03bb x y hR, impl_of_eq (hpq _ _ hR).symm)\n\nlemma exists_impl_exists {\u03b1} {p q : \u03b1 \u2192 hProp} (hpq : \u2200 x, p x =*> q x) : p_exists p =*> p_exists q :=\nexists_impl $ \u03bb x, impl_exists x (hpq _)\n\nlemma exists_impl_exists_to {\u03b1 \u03b2} {p : \u03b1 \u2192 hProp} {q : \u03b2 \u2192 hProp}\n  (f : \u03b1 \u2192 \u03b2)\n  (hpq : \u2200 x, p x =*> q (f x)) : p_exists p =*> p_exists q :=\nexists_impl $ \u03bb x, impl_exists (f x) (hpq _)\n\nopen function\n\nlemma exists_impl_exists_from {\u03b1 \u03b2} {p : \u03b1 \u2192 hProp} {q : \u03b2 \u2192 hProp}\n  (f : \u03b2 \u2192 \u03b1) (hf : surjective f)\n  (hpq : \u2200 x, p (f x) =*> q x) : p_exists p =*> p_exists q :=\nexists_impl $ \u03bb x, Exists.rec_on (hf x) $ \u03bb w hf, impl_exists w $ hf \u25b8 hpq w\n\nlemma exists_congr' {\u03b1 \u03b2} {p : \u03b1 \u2192 hProp} {q : \u03b2 \u2192 hProp}\n  (f : \u03b1 \u2192 \u03b2) (hf : surjective f)\n  (hpq : \u2200 x, p x = q (f x)) : p_exists p = p_exists q :=\nimpl_antisymm\n  (exists_impl_exists_to f $ \u03bb x, impl_of_eq $ hpq x)\n  (exists_impl_exists_from f hf $ \u03bb x, impl_of_eq (hpq x).symm)\n\nlemma or_impl {p p' : hProp} {q : hProp}\n  (H  : p  =*> q)\n  (H' : p' =*> q) :\n   p \u22c1 p' =*> q :=\nimpl.intro $ \u03bb h h', or.elim h' (H.elim h) (H'.elim h)\n\n@[congr]\nlemma exists_congr {\u03b1} {p q : \u03b1 \u2192 hProp}\n  (hpq : \u2200 x, p x = q x) : p_exists p = p_exists q :=\ncongr_arg _ $ funext hpq\n\n@[mono]\nlemma and_impl_and {p p' q q' : hProp} (hpq : p =*> q) (hpq' : p' =*> q') : p \u229b p' =*> q \u229b q' :=\nimpl.intro $ \u03bb h, exists_imp_exists $ \u03bb h\u2080, exists_imp_exists $ \u03bb h\u2081, and.imp_right $ and_implies (hpq.elim _) (hpq'.elim _)\n\nlemma lift_impl_emp {p : Prop} : [|p|] =*> @emp value :=\n\u27e8 \u03bb h \u27e8_,h'\u27e9, h' \u27e9\n\n@[mono]\nlemma lift_impl_lift {p q : Prop} (h : p \u2192 q) : [|p|] =*> ([|q|] : hProp) :=\nimpl.intro $ \u03bb hp, and.imp h id\n\nlemma lift_p_and_and {p : Prop} {q : hProp} : [| p |] \u229b q = [| p |] \u229b True \u22c0 q :=\nimpl_antisymm\n  (impl_and (and_impl_and (impl_refl _) impl_True)\n            (emp_and q \u25b8 and_impl_and lift_impl_emp ((emp_and q).symm \u25b8 impl_refl q)))\n  \u27e8\u03bb h \u27e8\u27e8h\u2080,h\u2081,h\u2082,h\u2083,h\u2084\u27e9,h'\u27e9,\n     have h = h\u2081, by { dsimp [lift,emp] at h\u2083,\n                       rw [h\u2083.2,empty_add,option.some.inj_eq] at h\u2082,\n                       exact h\u2082 },\n     \u27e8h\u2080,h\u2081,h\u2082,h\u2083,this \u25b8 h'\u27e9\u27e9\n\nlemma impl_or_left {p q q' : hProp} (hpq : p =*> q) : p =*> q \u22c1 q' :=\nimpl.intro $ \u03bb h hp, or.inl (hpq.elim h hp)\n\nlemma impl_or_right {p q q' : hProp} (hpq : p =*> q') : p =*> q \u22c1 q' :=\nimpl.intro $ \u03bb h hp, or.inr (hpq.elim h hp)\n\nlemma p_and_or_distrib_left {p q r : hProp} :\n  p \u229b (q \u22c1 r) = (p \u229b q) \u22c1 (p \u229b r) :=\nimpl_antisymm\n  \u27e8 \u03bb h \u27e8h\u2080,h\u2081,Hh,Hp,Hqr\u27e9,\n        or.elim Hqr (\u03bb Hq, or.inl \u27e8h\u2080,h\u2081,Hh,Hp,Hq\u27e9)\n                    (\u03bb Hr, or.inr \u27e8h\u2080,h\u2081,Hh,Hp,Hr\u27e9) \u27e9\n  (or_impl (and_impl_and (impl_refl _) (impl_or_left (impl_refl _)))\n           (and_impl_and (impl_refl _) (impl_or_right (impl_refl _))) )\n\n@[simp, separation_logic]\nlemma lift_true : [| true |] = (emp : hProp) := lift_eq_emp trivial\n\nlemma lift_and_iff_p_exists {p : Prop} {q : hProp} : [|p|] \u229b q = \u2203\u2203 h : p, q :=\nbegin\n  ext, dsimp [and,p_exists,lift,emp], split,\n  { rintros \u27e8h\u2080,h\u2081,H\u2080,\u27e8H\u2081,H\u2082\u27e9,H\u2083\u27e9, subst H\u2082,\n    rw [empty_add,option.some_inj] at H\u2080, exact \u27e8H\u2081, H\u2080.symm \u25b8 H\u2083\u27e9 },\n  { rintros \u27e8hp,hq\u27e9, refine \u27e8\u2205,h,_,\u27e8hp,rfl\u27e9,hq\u27e9,\n    rw [empty_add,option.some_inj] }\nend\n\n@[simp]\nlemma lift_and_applied {p : Prop} {q : hProp} (h : heap) : ([|p|] \u229b q) h \u2194 p \u2227 q h :=\nby simp [lift_and_iff_p_exists,p_exists]\n\nlemma and_applied_union {p q : hProp} {h h' : heap} (Hp : p h) (Hq : q h') (Hdisj : disjoint h h') : (p \u229b q) (h \u222a h') :=\n\u27e8h, h', union_eq_add_of_disjoint Hdisj, Hp, Hq\u27e9\n\nlemma lift_and_impl {p : Prop} {q r : hProp}\n  (h : p \u2192 q =*> r) :\n  [|p|] \u229b q =*> r :=\nsuffices (\u2203\u2203 h : p, q) =*> r,\n  by simpa [lift_and_iff_p_exists],\nexists_impl h\n\nlemma lift_impl {p : Prop} {r : hProp}\n  (h : p \u2192 emp =*> r) :\n  [|p|] =*> r :=\nsuffices [|p|] \u229b emp =*> r,\n  by simpa [and_emp],\nlift_and_impl h\n\nlemma impl_lift_and {p : Prop} {q r : hProp}\n  (h : p)\n  (h' : q =*> r) :\n  q =*> [|p|] \u229b r :=\nsuffices q =*> (\u2203\u2203 h : p, r),\n  by simpa [lift_and_iff_p_exists],\nimpl_exists h h'\n\nlemma holds_or_iff {h frame : heap} {p q : hProp} :\n  holds h frame (p \u22c1 q) \u2194 holds h frame p \u2228 holds h frame q :=\nby { dsimp [holds,p_or], rw [\u2190 exists_or_distrib],\n     simp [and_or_distrib_left] }\n\nlemma holds_imp_holds_of_impl {h frame : heap} {p q : hProp}\n  (Hpq : p =*> q) : holds h frame p \u2192 holds h frame q :=\nexists_imp_exists (\u03bb hp, and.imp_right $ Hpq.elim hp)\n\n@[simp]\nlemma holds_lift_and {h frame : heap} {p : Prop} {q : hProp} : holds h frame ([|p|] \u229b q) \u2194 p \u2227 holds h frame q :=\nbegin\n  simp [holds,lift,emp], rw \u2190 exists_and_distrib_left,\n  apply _root_.exists_congr, intro, cc,\nend\n\n@[simp]\nlemma holds_lift {h frame : heap} {p : Prop} : holds h frame [|p|] \u2194 p \u2227 holds h frame emp :=\nby rw [\u2190 holds_lift_and,and_emp]\n\nlemma exists_subtype {\u03b1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 hProp} :\n  (\u2203\u2203 x : subtype p, q x.val) = (\u2203\u2203 x : \u03b1, [| p x |] \u229b q x) :=\nimpl_antisymm\n  (exists_impl_exists_to subtype.val (by simp; intros a ha; simp [lift_eq_emp ha]))\n  (exists_impl $ \u03bb x, lift_and_impl $ \u03bb hp, impl_exists \u27e8_,hp\u27e9 $ impl_refl _)\n\nlemma exists_exists {\u03b1} {p : \u03b1 \u2192 Sort*} {q : \u03a0 x, p x \u2192 hProp} :\n  (\u2203\u2203 (x : \u03b1) (h : p x), q x h) = (\u2203\u2203 x : psigma p, q x.1 x.2) :=\nimpl_antisymm\n  (exists_impl $ \u03bb x, exists_impl $ \u03bb h, impl_exists \u27e8x,h\u27e9 $ impl_refl _)\n  (exists_impl $ \u03bb x, impl_exists x.1 $ impl_exists x.2 $ impl_refl _)\n\nend \u00abhProp\u00bb\n\nvariable (value)\nclass storable (\u03b1 : Type*) :=\n(size : \u03b1 \u2192 \u2115)\n(repr : tptr \u03b1 \u2192 \u03b1 \u2192 hProp)\n\nexport storable (size)\n\nlocal notation `storable` := storable value\n\nsection\n\nomit value\nmeta def check_fixed_size : tactic unit := `[intros; refl]\n\nend\n\n-- def typeof {\u03b1} (x : \u03b1) := \u03b1\n-- #check storable.repr\ninfix ` \u2907 `:60 := storable.repr\n\nclass fixed_storable (\u03b1 : Type*) extends storable \u03b1 :=\n(fixed_size : \u2115)\n(is_fixed : \u2200 x : \u03b1, size x = fixed_size . check_fixed_size)\n(size := \u03bb _, fixed_size)\n(pos_size : fixed_size > 0)\n\nopen list function \u00abhProp\u00bb\n\nexport fixed_storable (fixed_size)\n\nlocal notation `fixed_storable` := fixed_storable value\n-- local notation `fixed_size` := fixed_size value\n\ninstance : fixed_storable value :=\n{ repr := \u03bb p v, p.get \u21a6 v,\n  -- bytes := \u03bb v, \u27e8[v], rfl\u27e9,\n  fixed_size := 1,\n  pos_size := by norm_num,\n  -- abstr := \u03bb x, value_abstr _ x.property,\n  -- right_inverse := \u03bb \u27e8[x],rfl\u27e9, rfl,\n  -- raw_bytes_conversion := \u03bb p x, rfl\n }\n\ndef word (\u03b1) [fixed_storable \u03b1] := { bytes : list value // length bytes = fixed_size value \u03b1 }\n\nlocal notation `word` := word value\n\nvariables {value}\n\ndef list_repr' {\u03b1 \u03b2} [storable \u03b1] : tptr (list \u03b1) \u2192 list \u03b1 \u2192 tptr \u03b2 \u2192 hProp\n| p [] q := [| p = q.recast (list \u03b1) |]\n| p (v :: vs) q := p.recast \u03b1 \u2907 v \u229b list_repr' (p +. size value v) vs q\n\ndef value_abstr : \u03a0 vs : list value, length vs = 1 \u2192 value\n| [val] rfl := val\n\ndef list_repr {\u03b1} [storable \u03b1] : tptr (list \u03b1) \u2192 list \u03b1 \u2192 hProp\n| p [] := emp\n| p (v :: vs) := p.recast \u03b1 \u2907 v \u229b list_repr (p +. size value v) vs\n\nlemma list_repr_map {\u03b1 \u03b2} [storable \u03b1] [storable \u03b2]\n  (p : tptr (list \u03b1)) {f : \u03b2 \u2192 \u03b1} (xs : list \u03b2)\n  (h : \u2200 (q : tptr \u03b1) x, q \u2907 f x = (q.recast _ \u2907 x))\n  (h' : \u2200 x : \u03b2, size value (f x) = size value x ):\n  list_repr p (xs.map f) = (list_repr (p.recast _) xs) :=\nbegin\n  induction xs generalizing p, refl,\n  simp [list_repr,*], refl\nend\n\n@[simp, separation_logic]\nlemma fixed_size_val : fixed_size value value = 1 := rfl\n\n@[simp, separation_logic]\nlemma size_val (x : value) : size value x = 1 := rfl\n\nattribute [simp, separation_logic] fixed_storable.is_fixed\n\nlemma list_repr'_eq_list_repr {\u03b1 \u03b2} [fixed_storable \u03b1] (p : tptr (list \u03b1)) (q : tptr \u03b2) (ls : list \u03b1) :\n  list_repr' p ls q = list_repr p ls \u229b [| p +. ls.length * fixed_size value \u03b1 = q.recast _ |] :=\nby induction ls generalizing p; simp [list_repr',list_repr,*,right_distrib,hProp.and_assoc]; congr\n\ninstance list.storable {\u03b1} [storable \u03b1] : storable (list \u03b1) :=\n{ repr := list_repr,\n  size := \u03bb vs, list.sum $ vs.map (storable.size value)\n  }\n\nvariables value\n\nclass is_record (\u03b1 : Type*) extends fixed_storable \u03b1 :=\n(abstr : word \u03b1 \u2192 \u03b1)\n(bytes : \u03b1 \u2192 word \u03b1)\n(repr := \u03bb p v, p.recast _ \u2907 (bytes v).val)\n(right_inverse : right_inverse abstr bytes)\n(raw_bytes_conversion : \u2200 (p : tptr \u03b1) (x : \u03b1), (p \u2907 x) = (p.recast _ \u2907 (bytes x).val : hProp))\n\nexport is_record (abstr bytes raw_bytes_conversion)\n\nlocal notation `is_record` := is_record value\n\ndef bytes' {\u03b1} [is_record \u03b1] (x : \u03b1) : list value :=\n(bytes value x).val\n\nlemma length_bytes' {\u03b1} [is_record \u03b1] (x : \u03b1) : length (bytes' value x) = fixed_size value \u03b1 :=\n(bytes value x).property\n\nlemma bytes_surjective (\u03b1) [is_record \u03b1] : surjective (bytes value : \u03b1 \u2192 word \u03b1) :=\nsurjective_of_has_right_inverse \u27e8abstr, is_record.right_inverse _ _\u27e9\n\nvariables {value}\n\nlemma uninitialized {\u03b1} [is_record \u03b1] (p : tptr \u03b1) :\n  (\u2203\u2203 bytes : list value, [|length bytes = fixed_size value \u03b1|] \u229b (p.recast _ \u2907 bytes)) =\n   \u2203\u2203 obj : \u03b1, p \u2907 obj :=\nby rw \u2190 exists_subtype; symmetry; apply exists_congr' (bytes value) (bytes_surjective value \u03b1);\n   intro x; apply raw_bytes_conversion\n\n@[simp, separation_logic]\nlemma repr_nil {\u03b1} [storable \u03b1] (p : tptr (list \u03b1)) :\n  p \u2907 [] = @emp value :=\nby { dsimp [storable.repr], simp [list_repr], }\n\n@[simp, separation_logic]\nlemma repr_cons {\u03b1} [storable \u03b1] (p : tptr (list \u03b1)) (x) (xs : list \u03b1) :\n  p \u2907 (x :: xs) = (p.recast _ \u2907 x \u229b p+.size value x \u2907 xs : hProp) :=\nby { dsimp [storable.repr], simp [list_repr], }\n\nlemma maplets_append' {\u03b1} [fixed_storable \u03b1] : \u03a0 (p : tptr (list \u03b1)) (us vs : list \u03b1), (p \u2907 (us ++ vs)) = (p \u2907 us) \u229b (p+.fixed_size value \u03b1 * length us \u2907 vs)\n| p [] vs := by simp\n| p (u::us) vs := by simp [*,hProp.and_assoc,left_distrib]\n\nlemma maplets_append : \u03a0 (p : tptr (list value)) (us vs : list value), (p \u2907 (us ++ vs)) = (p \u2907 us) \u229b (p+.length us \u2907 vs)\n| p [] vs := by simp\n| p (u::us) vs := by simp [*,hProp.and_assoc]\n\n@[simp, separation_logic]\nlemma value_repr (p : ptr) (x : value) :\n  tptr.mk value value p \u2907 x = p \u21a6 x :=\nby { dsimp [storable.repr], simp [list_repr], }\n\n@[simp, separation_logic]\nlemma get_value_repr (p : tptr value) (x : value) :\n  p \u2907 x = p.get \u21a6 x :=\nby { dsimp [storable.repr], simp [list_repr], }\n\n@[simp, separation_logic]\nlemma tptr.recast_mk {\u03b1 \u03b2} : \u03a0 (p : ptr), (tptr.mk value \u03b1 p).recast \u03b2 = tptr.mk value \u03b2 p\n| p := rfl\n\n@[simp, separation_logic]\nlemma tptr.recast_get {\u03b1 \u03b2} : \u03a0 (p : tptr \u03b1), (p.recast \u03b2).get = p.get\n| p := rfl\n\n@[simp]\nlemma disjoint_maplet_heap_mk_of_lt {p q : \u2115} (x : value) (xs : list value) (H : p < q) : disjoint (maplet p x) (heap.mk (enum_from q xs)) :=\nby intros x; simp; intros H\u2080 H\u2081; subst x; cases not_le_of_lt H H\u2081\n\n@[simp]\nlemma disjoint_maplet_heap_mk_one_add {p : \u2115} (x : value) (xs : list value) : disjoint (maplet p x) (heap.mk (enum_from (1 + p) xs)) :=\ndisjoint_maplet_heap_mk_of_lt x xs ((nat.one_add p).symm \u25b8 nat.lt_succ_self _)\n\n@[simp]\nlemma disjoint_maplet_heap_mk_add_one {p : \u2115} (x : value) (xs : list value) : disjoint (maplet p x) (heap.mk (enum_from (p + 1) xs)) :=\ndisjoint_maplet_heap_mk_of_lt x xs (nat.lt_succ_self _)\n\nlemma maplets_eq (p : tptr _) (vs : list value) (h : heap) : (p \u2907 vs) h \u2194 h = heap.mk (vs.enum_from p.get) :=\nbegin\n  induction vs generalizing p h; simp [maplets,emp,map,enum_from],\n  simp [map,to_finmap_cons,separation.hProp.and,vs_ih,maplets,(+.)], split,\n  { rintro \u27e8h\u2080,h\u2081,hh\u2080,hh\u2081,hh\u2082\u27e9, rw [eq_union_of_eq_add hh\u2080,\u2190 hh\u2081,\u2190 hh\u2082] },\n  { intro hh, refine \u27e8_,_,_,rfl,rfl\u27e9, rw [\u2190 union_eq_add_of_disjoint, \u2190 hh],\n    simp }\nend\n\n@[simp, separation_logic]\nlemma tptr.recast_recast {\u03b1 \u03b2 \u03b3} (p : tptr \u03b1) : (p.recast \u03b2).recast \u03b3 = (p.recast \u03b3 : tptr \u03b3) := rfl\n\n@[simp, separation_logic]\nlemma tptr.recast_eq {\u03b1} : \u03a0 (p : tptr \u03b1), (p.recast \u03b1) = p\n| \u27e8 _, _,_ \u27e9 := rfl\n\ninstance : is_record value :=\n{ repr := \u03bb p v, p.get \u21a6 v,\n  bytes := \u03bb v, \u27e8[v], rfl\u27e9,\n  fixed_size := 1,\n  pos_size := by norm_num,\n  abstr := \u03bb x, value_abstr _ x.property,\n  right_inverse := \u03bb \u27e8[x],rfl\u27e9, rfl,\n  raw_bytes_conversion := \u03bb p x, by simp; refl }\n\ndef rec_bytes {\u03b1} [is_record \u03b1] (ls : list \u03b1) : list value :=\n(ls.map (bytes' value)).join\n\nlemma rec_bytes_cons {\u03b1} [is_record \u03b1] (x) (xs : list \u03b1) : rec_bytes (x :: xs) = bytes' value x ++ rec_bytes xs := rfl\n\nlemma length_rec_bytes {\u03b1} [is_record \u03b1] (xs : list \u03b1) :\n  length (rec_bytes xs : list value) = length xs * fixed_size value \u03b1 :=\nbegin\n  simp [rec_bytes], induction xs; simp [length,*,right_distrib],\n  erw (bytes value xs_hd).property,\nend\n\nopen \u00abfixed_storable\u00bb list\n\nlemma uninitialized' {\u03b1} [is_record \u03b1] (p : tptr \u03b1) :\n  (\u2203\u2203 obj : \u03b1, p \u2907 obj) = (\u2203\u2203 bytes : list value, [| length bytes = fixed_size value \u03b1 |] \u229b p.recast _ \u2907 bytes) :=\nby cases p; rw uninitialized\n\nvariable (value)\n\nstructure rec_entry :=\nmk' ::\n{\u03b1 : Type*}\n[S : is_record \u03b1]\n(get : \u03b1)\n\nattribute [instance] rec_entry.S\n\ndef rec_entry.mk_ {\u03b1 : Type} [I : is_record \u03b1] (x : \u03b1) : rec_entry value := \u27e8_,x\u27e9\n\ninstance : storable (rec_entry value) :=\n{ repr := \u03bb p x, p.recast _ \u2907 x.get,\n  size := \u03bb x, storable.size value x.get, }\n\nvariables {value}\n\ndef rec_entry_bytes : rec_entry value \u2192 list value\n| (@rec_entry.mk' ._ \u03b1 _inst get) := @bytes' value \u03b1 _inst get\n\ndef rec_bytes' (ls : list (rec_entry value)) : list value :=\n(ls.map (rec_entry_bytes)).join\n\n-- end hProp\n\ninstance fixed_storable.word {\u03b1} [fixed_storable \u03b1] : fixed_storable (word \u03b1) :=\n{ repr := \u03bb p v, p.recast _ \u2907 v.val,\n  fixed_size := fixed_size value \u03b1,\n  pos_size := fixed_storable.pos_size value \u03b1 }\n\ndef word.bytes {\u03b1} [fixed_storable \u03b1] : word \u03b1 \u2192 word (word \u03b1)\n| \u27e8x,hx\u27e9 := \u27e8x,hx\u27e9\n\ndef word.abstr {\u03b1} [fixed_storable \u03b1] : word (word \u03b1) \u2192 word \u03b1\n| \u27e8x,hx\u27e9 := \u27e8x,hx\u27e9\n\n-- @[simp, separation_logic]\n-- lemma val_repr {val} (p : tptr (list val)) (vs : list val) : p \u2907 vs = p.get \u21a6 vs :=\n-- begin\n--   cases p, dsimp [storable.repr,tptr.recast],\n--   induction vs generalizing p, refl,\n--   rw maplets_cons, simp [list_repr,tptr.add,fixed_size,*], refl\n-- end\n\ninstance word.is_record {\u03b1} [fixed_storable \u03b1] : is_record (word \u03b1) :=\n{ bytes := word.bytes,\n  abstr := word.abstr,\n  right_inverse := \u03bb \u27e8x,hx\u27e9, rfl,\n  raw_bytes_conversion := \u03bb \u27e8_,_,p\u27e9 \u27e8x,hx\u27e9,\n    by { simp [tptr.recast_mk,word.bytes]; refl } }\n\ndef equiv.is_record {\u03b1 \u03b2} (f : \u03b2 \u2192 \u03b1) (g : \u03b1 \u2192 \u03b2) (hfg : left_inverse f g) [is_record \u03b1] : is_record \u03b2 :=\n{ repr := \u03bb p x, p.recast _ \u2907 f x,\n  bytes := bytes value \u2218 f,\n  abstr := g \u2218 abstr,\n  fixed_size := fixed_size value \u03b1,\n  pos_size := fixed_storable.pos_size value \u03b1,\n  is_fixed := \u03bb x, rfl,\n  right_inverse := \u03bb \u27e8x,hx\u27e9, by dsimp [bytes]; rw [hfg];\n                               exact (is_record.right_inverse value \u03b1 \u27e8x,hx\u27e9),\n  raw_bytes_conversion := \u03bb \u27e8_,_,p\u27e9 x, raw_bytes_conversion \u27e8value,\u03b1,p\u27e9 (f x)\n }\n\nlemma repr_map_bytes {\u03b1} [is_record \u03b1] (p : tptr (list (word \u03b1))) (xs : list \u03b1) :\n  p \u2907 xs.map (bytes value) = (p.recast _ \u2907 xs : hProp) :=\nshow list_repr p (xs.map (bytes value)) = list_repr (p.recast _) xs,\nbegin\n  rw [list_repr_map],\n  { intros, symmetry, -- erw val_repr, cases q,\n    rw [raw_bytes_conversion], refl },\n  intro, simp [fixed_storable.is_fixed], refl\nend\n\nlemma repr_map_abstr {\u03b1} [is_record \u03b1] (p : tptr (list \u03b1)) (xs : list (word \u03b1)) :\n  p \u2907 xs.map abstr = (p.recast _ \u2907 xs) :=\nbegin\n  have := repr_map_bytes (p.recast _) (xs.map abstr),\n  simp at this, rw \u2190 this, congr,\n  transitivity xs.map id,\n  { congr, ext, exact is_record.right_inverse value \u03b1 x },\n  { simp only [list.map_id] }\nend\n\n@[separation_logic]\nlemma list_repr_recast {\u03b1 \u03b2} (\u03b3) [storable \u03b1] (vs : list \u03b1)\n  (p : tptr (list \u03b1)) (q : tptr \u03b2) :\n  list_repr' p vs (q.recast \u03b3) = list_repr' p vs q :=\nbegin\n  cases q, dsimp [tptr.recast],\n  induction vs generalizing p, refl, simp [list_repr',*]\nend\n\nend separation\n\nnamespace tactic\n\nvariables {value : Type}\nopen separation separation.hProp\n\nlocal notation `hProp` := hProp value\nsetup_tactic_parser\n\nlemma shrink_impl (l m r : hProp) {p q : hProp}\n  (h\u2080 : l \u229b m = p) (h\u2081 : r \u229b m = q) (h\u2082 : l =*> r) :\n  p =*> q :=\nh\u2080 \u25b8 (h\u2081 \u25b8 and_impl_and h\u2082 $ impl_refl _)\n\nlemma split_impl (p\u2080 p\u2081 q\u2080 q\u2081 : hProp) {p q : hProp}\n  (h\u2080 : p\u2080 \u229b p\u2081 = p) (h\u2081 : q\u2080 \u229b q\u2081 = q) (h\u2082 : p\u2080 =*> q\u2080) (h\u2083 : p\u2081 =*> q\u2081) :\n  p =*> q :=\nh\u2080 \u25b8 (h\u2081 \u25b8 and_impl_and h\u2082 h\u2083)\n\nmeta def parse_assert' : expr \u2192 tactic (dlist expr)\n| `(%%p \u229b %%q) := (++) <$> parse_assert' p <*> parse_assert' q\n| `(emp) := pure dlist.empty\n| p := pure $ dlist.singleton p\n\nmeta def parse_assert (e : expr) : tactic (list expr) :=\ndlist.to_list <$> parse_assert' e\n\nmeta def mk_assert (val : expr) : list expr \u2192 expr\n| [] := @expr.const tt ``emp [] val\n| [x] := x\n| (x::xs) :=\n@expr.const tt ``hProp.and [] val x (mk_assert xs)\n\nmeta def ac_refl_aux : tactic unit :=\ndo `[dsimp { fail_if_unchanged := ff }],\n   (lhs, rhs) \u2190 target >>= match_eq,\n   xs \u2190 parse_assert lhs,\n   with_context!\"{target}\" $ do\n     xs.mmap' $ \u03bb x, generalize x >> intro1,\n     cc <|> fail \"ac_refl_aux\"\n\nmeta def ac_refl' : tactic unit :=\ndo try (applyc ``impl_of_eq),\n   -- target >>= instantiate_mvars >>= change,\n   -- `[dsimp],\n   with_context!\"{target}\" $ do\n   cc <|>\n     ac_refl_aux\n     -- <|>\n     -- fail!\"ac_refl': {target}\\nmeta vars: {expr.list_meta_vars <$> target}\"\n\nmeta def find_lift : list expr \u2192 tactic (option (expr \u00d7 list expr))\n| [] := pure none\n| (x@`(separation.hProp.lift _) :: xs) := pure (some (x, xs))\n| (x :: xs) :=\n  do some (y, ys) \u2190 find_lift xs | pure none,\n     pure (some (y, x::ys))\n\n@[replaceable]\nmeta def s_intro' (n : parse $ ident_ <|> pure `_) : tactic unit :=\ndo `[simp only [and_p_exists_distrib_left,and_p_exists_distrib_right]\n               { fail_if_unchanged := ff }],\n   `(@impl %%val %%p %%q) \u2190 target | fail \"Expecting separation logic specification\",\n   match p with\n   | `(p_exists _) :=\n     do applyc ``exists_impl,\n        intro n >> pure ()\n   | _ :=\n   do xs \u2190 parse_assert p,\n      some (x, xs) \u2190 find_lift xs | failed,\n      let p' := mk_assert val (x :: xs),\n      g \u2190 mk_app `eq [p,p'] >>= mk_meta_var,\n      gs \u2190 get_goals, set_goals [g],\n      `[simp only [and_emp,emp_and] { fail_if_unchanged := ff } ],\n      done <|> ac_refl',\n      set_goals gs,\n      get_assignment g >>= rewrite_target,\n      applyc ``lift_and_impl <|> applyc ``lift_impl,\n      intro n, pure ()\n   end\n\nmeta def interactive.s_intro (n : parse $ ident_ <|> pure `_) : tactic unit :=\ns_intro n\n\n@[interactive]\nmeta def s_intros : parse ident_* \u2192 tactic unit\n| [] := repeat (s_intro `_)\n| ns := ns.mmap' s_intro\n\n@[interactive]\nmeta def s_existsi (wit : parse pexpr_list_or_texpr) : tactic unit :=\nwit.mmap' $ \u03bb w,\n  do `(%%p =*> %%q) \u2190 target,\n     `[simp only [and_p_exists_distrib_left,and_p_exists_distrib_right] { fail_if_unchanged := ff }],\n     refine ``(impl_exists %%w _) <|>\n       do `[simp only [lift_and_iff_p_exists] { single_pass := tt } ],\n          `[simp only [and_p_exists_distrib_left,and_p_exists_distrib_right]\n                 { fail_if_unchanged := ff } ],\n          refine ``(impl_exists %%w _)\n\nlemma lin_assert {p q : hProp} (pp : Prop) (h : p =*> [| pp |] \u229b p) (h' : pp \u2192 p =*> q) : p =*> q :=\nimpl_trans h\n  (by s_intro h; exact h' h)\n\nlemma lin_assert' {p q : hProp} (pp : Prop) (h : p =*> [| pp |] \u229b True) (h' : pp \u2192 p =*> q) : p =*> q :=\nbegin\n  transitivity [| pp |] \u229b p,\n  { rw lift_p_and_and, apply impl_and h (impl_refl _) },\n  { s_intro h, exact h' h }\nend\n\n@[interactive]\nmeta def s_assert (h : parse $ ident? <* tk \":\") (e : parse texpr) : tactic unit :=\nlet h := h.get_or_else `this in\nrefine ``(lin_assert' %%e _ _); [skip, ()<$intro h]\n\nmeta def find_frame' (e : expr) : list expr \u2192 tactic (list expr)\n| [] := fail \"frame not found\"\n| (x :: xs) :=\n  xs <$ unify e x <|>\n  list.cons x <$> find_frame' xs\n\nmeta def find_frame_aux : list expr \u2192 list expr \u2192 tactic (list expr)\n| [] xs := pure xs\n| (x::xs) ys :=\n  do ys' \u2190 find_frame' x ys,\n     find_frame_aux xs ys'\n\nmeta def find_diff : list expr \u2192 list expr \u2192 tactic (list expr \u00d7 list expr \u00d7 list expr)\n| [] xs := pure ([], [], xs)\n| (x::xs) ys :=\n  do (b,ys') \u2190 prod.mk tt <$> find_frame' x ys <|> pure (ff,ys),\n     (l,m,r) \u2190 find_diff xs ys',\n     if b\n       then pure (l,x::m,r)\n       else pure (x::l,m,r)\n\n/--\n`find_frame e e'` returns `r` and `pr` such that `pr : e \u229b r = e'`\n-/\nmeta def find_frame (e e' : expr) : tactic (expr \u00d7 expr) :=\ndo `(\u00abhProp\u00bb %%val) \u2190 infer_type e,\n   le \u2190 parse_assert e,\n   le' \u2190 parse_assert e',\n   lr \u2190 find_frame_aux le le',\n   let r := mk_assert val lr,\n   t \u2190 to_expr ``(%%e \u229b %%r = %%e') >>= instantiate_mvars,\n   (_,pr) \u2190 solve_aux t\n     (`[simp only [emp_and,and_emp] { fail_if_unchanged := ff } ]; ac_refl'),\n   pure (r,pr)\n\n@[replaceable]\nmeta def s_shrink' : tactic unit :=\ndo `(%%p =*> %%q) \u2190 target,\n   `(\u00abhProp\u00bb %%val) \u2190 infer_type p,\n   ps \u2190 parse_assert p,\n   qs \u2190 parse_assert q,\n   (l,m,r) \u2190 find_diff ps qs,\n   guard (\u00ac m.empty) <|> fail \"no common clauses\",\n   let l := mk_assert val l,\n   let m := mk_assert val m,\n   let r := mk_assert val r,\n   to_expr ``(shrink_impl %%l %%m %%r) >>= apply,\n   iterate_exactly 2 $ solve1 $ interactive.hide_meta_vars { } $ \u03bb _, do\n   { `[simp { fail_if_unchanged := ff } ],\n     done <|> cc },\n   try reflexivity\n\nattribute [interactive] s_shrink\n\n@[replaceable]\nmeta def entailment' : tactic unit :=\nfocus1 $\nassumption <|>\ndo intros,\n   target >>= instantiate_mvars >>= change,\n   when_tracing `separation.failed_spec (trace \"A\" >> trace_state),\n   with_context!\"\u2022 A: {target}\" $ do\n     `[simp [hProp.and_p_exists_distrib_left,hProp.and_p_exists_distrib_right] with separation_logic\n       { fail_if_unchanged := ff } ],\n     when_tracing `separation.failed_spec (trace \"B\" >> trace_state),\n     with_context!\"\u2022 B: {try_core target}\" $ do\n       iterate_at_most 10 $ do\n         { `(_ =*> p_exists _) \u2190 target,\n           applyc ``impl_exists },\n       when_tracing `separation.failed_spec (trace \"C\" >> trace_state),\n       with_context!\"\u2022 C: {try_core target}\" $ do\n       done <|>\n         assumption <|>\n         ac_refl'\n     -- s_shrink <|>\n  -- (try (applyc ``impl_of_eq); ac_refl) <|>\n\nattribute [interactive] entailment\n\nmeta def s_apply' (e : expr) : tactic unit :=\ndo t \u2190 infer_type e,\n   (args,`(%%p =*> %%q)) \u2190 mk_meta_pis t,\n   let e := e.mk_app args,\n   `(%%p' =*> %%q') \u2190 target,\n   frp \u2190 some <$> find_frame p p' <|> pure none,\n   frq \u2190 some <$> find_frame q q' <|> pure none,\n   match frp, frq with\n   | some (pr,pp), some (qr,qp) := refine ``(split_impl %%p %%pr %%q %%qr %%pp %%qp %%e _)\n   | some (pr,pp), none := refine ``(impl_trans (shrink_impl %%p %%pr %%q %%pp rfl %%e) _)\n   | none, some (qr,qp) := refine ``(impl_trans _ (shrink_impl %%p %%qr %%q rfl %%qp %%e))\n   | none, none := fail!\"No match found for `{e} : {t}`\"\n   end,\n   try (reflexivity <|> applyc ``impl_True)\n\n@[interactive]\nmeta def s_apply : parse types.pexpr_list_or_texpr \u2192 tactic unit :=\nmmap' $ to_expr >=> s_apply'\n\n@[interactive]\nmeta def s_assumptions : tactic unit :=\ndo cxt \u2190 local_context,\n   focus1 $ cxt.for_each $ \u03bb l, try $ s_apply' l\n\n@[interactive]\nmeta def s_assumption : tactic unit :=\ndo cxt \u2190 local_context,\n   focus1 $ cxt.any_of $ \u03bb l, try $ s_apply' l\n\n@[interactive]\nmeta def s_show (p : parse texpr) : tactic unit :=\ndo g \u2190 to_expr p >>= mk_meta_var,\n   s_apply' g\n\nlemma prop_proof {p : Prop} {q : hProp} (h : p) : q =*> [| p |] \u229b True :=\nimpl_lift_and h impl_True\n\nlemma prop_proof' {p : Prop} {q : hProp} (h : p) : q =*> True \u229b [| p |] :=\nimpl_trans (prop_proof h) (impl_of_eq $ hProp.and_comm _ _)\n\nlemma prop_impl_False {q : hProp} (h : false) : q =*> False :=\nimpl_trans (prop_proof h) (by simp)\n\n@[interactive]\nmeta def prop (ls : parse ident_*) : tactic unit :=\ndo s_intros ls,\n   applyc ``prop_proof\n   <|> applyc ``prop_proof'\n   <|> applyc ``prop_impl_False\n\nend tactic\n\nnamespace separation\n\nopen hProp list\n\nvariables {value : Type} {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\ninclude value\nlocal notation `hProp` := hProp value\nlocal notation `tptr` := tptr value\n\nlemma p_exists_one_point {q : \u03b1 \u2192 hProp} (x : \u03b1)\n  (h : \u2200 y, q y =*> [|y = x|] \u229b q y):\n  p_exists q = q x :=\nimpl_antisymm\n  (by { s_intros y, transitivity, apply h,\n        s_intro hxy, rw hxy })\n  (by { apply impl_exists x, refl } )\n\nsection storable\n\nvariables [storable value \u03b1]\n\nlemma list_repr_and_list_repr_impl_list_repr'_concat {us vs : list \u03b1}\n  (p q : tptr (list \u03b1)) (r : tptr \u03b2) :\n  list_repr' p us q \u229b list_repr' q vs r =*> list_repr' p (us ++ vs) r :=\nbegin\n  induction us generalizing p,\n  { simp [list_repr'],\n    apply lift_and_impl, intro h, subst h },\n  { simp [list_repr',hProp.and_assoc],\n    apply and_impl_and (impl_refl _) (us_ih _) }\nend\n\nlemma list_repr_append {us vs : list \u03b1}\n  (p : tptr (list \u03b1)) (r : tptr \u03b2) :\n  list_repr' p (us ++ vs) r = list_repr' p us (p +. size value us) \u229b list_repr' (p +. size value us) vs r :=\nby induction us generalizing p; simp [list_repr',size,*,separation.hProp.and_assoc]\n\nlemma list_repr_impl_le (p\u2080 : tptr (list \u03b1)) (p\u2081 : tptr \u03b2) (vs : list \u03b1) :\n  list_repr' p\u2080 vs p\u2081 =*> [| p\u2080 \u2264 p\u2081.recast _ |] \u229b True :=\nbegin\n  induction vs generalizing p\u2080,\n  { simp [list_repr'], prop h, rw h },\n  { simp [list_repr'], s_apply vs_ih, prop h,\n    apply le_trans (le_offset _) h, }\nend\n\nlemma list_repr_impl_le' {p\u2080 p\u2081 : tptr (list \u03b1)} (vs : list \u03b1) :\n  list_repr' p\u2080 vs p\u2081 =*> [| p\u2080 \u2264 p\u2081 |] \u229b True :=\nlist_repr_impl_le _ _ _\n\nlemma list_repr_eq_of_gt {p q : tptr (list \u03b1)} {vs : list \u03b1} (h : q < p) :\n  list_repr' p vs q = False :=\nbegin\n  apply impl_antisymm _ (False_impl _),\n  apply impl_trans (list_repr_impl_le' _),\n  simp [not_le_of_gt h],\nend\n\nend storable\n\nsection fixed_storable\n\nvariables [fixed_storable value \u03b1]\n\nlemma list_repr_self_impl_eq_nul (p\u2080 : tptr (list \u03b1)) (vs : list \u03b1) :\n  list_repr' p\u2080 vs p\u2080 =*> [| vs = [] |] :=\nbegin\n  induction vs generalizing p\u2080; simp [list_repr'],\n  s_apply list_repr_impl_le, prop h, revert h,\n  apply not_le_of_gt, simp, apply gt_offset_of_gt _ (fixed_storable.pos_size _ \u03b1),\nend\n\nlemma list_repr_self_impl_eq_nul' (p\u2080 : tptr (list \u03b1)) (vs : list \u03b1) :\n  list_repr' p\u2080 vs (p\u2080.recast \u03b2) =*> [| vs = [] |] :=\n(@list_repr_recast value _ _ \u03b2 _ vs p\u2080 p\u2080).symm \u25b8 (list_repr_self_impl_eq_nul p\u2080 vs)\n\nlemma list_repr_offset {vs : list \u03b1}\n  (p : tptr (list \u03b1)) (n : \u2115) :\n  list_repr' p vs (p +. n * fixed_size value \u03b1) = [| length vs = n |] \u229b p \u2907 vs :=\nbegin\n  induction vs generalizing n p,\n  { have : p = p +. n * fixed_size value \u03b1 \u2194 0 = n,\n    { split; intro h,\n      by_contradiction h',\n      { replace h' := nat.pos_of_ne_zero (ne.symm h'),\n        have h\u2082 : n * fixed_size value \u03b1 > 0 := mul_pos h' (fixed_storable.pos_size _ _),\n        exact offset_ne _ h\u2082 h.symm },\n      { simp [h.symm] }, },\n    simp [list_repr',this,*], },\n  simp [list_repr',*], -- specialize vs_ih (n - 1) (p +. size value vs_hd),\n  cases nat.eq_zero_or_pos n,\n  { subst n,\n    have : p < p +. fixed_size value \u03b1 := gt_offset_of_gt _ _,\n    simp [list_repr_eq_of_gt this], apply fixed_storable.pos_size },\n  specialize vs_ih (n - 1) (p +. size value vs_hd),\n  replace h := nat.succ_le_of_lt h,\n  have := nat.mul_le_mul_right (fixed_size value \u03b1) h,\n  simp at this,\n  simp [nat.add_sub_cancel' this,nat.mul_sub_right_distrib] at vs_ih,\n  rw [vs_ih, @eq_comm _ (length vs_tl), nat.sub_eq_iff_eq_add h, @eq_comm _ _ (length vs_tl + 1), nat.add_comm],\n  ac_refl,\nend\n\nlemma recast_inj (p q : tptr \u03b1) : p.recast \u03b2 = (q.recast \u03b2) \u2194 p = q :=\nby cases p; cases q; simp\n\ndef trashed (p : tptr \u03b1) : hProp :=\n\u2203\u2203 trash : list value, [| length trash = fixed_size value \u03b1 |] \u229b p.recast _ \u2907 trash\n\ndef unused : tptr (list $ word value \u03b1) \u2192 \u2115 \u2192 tptr \u03b2 \u2192 hProp\n| p 0 q := [| p = q.recast _ |]\n| p (nat.succ n) q := trashed (p.recast \u03b1) \u229b unused (p +. fixed_size value \u03b1) n q\n\n@[simp, separation_logic]\nlemma unused_recast (p : tptr (list $ word value \u03b1)) (q : tptr \u03b2) (n : \u2115) :\n  unused p n (q.recast \u03b3) = unused p n q :=\nby induction n generalizing p; simp [unused,*]\n\n@[simp]\nlemma fixed_size_word : fixed_size value (word value \u03b1) = fixed_size value \u03b1 := rfl\n\nlemma unused_iff_exists (p : tptr (list (word value \u03b1))) (q : tptr \u03b2) (n : \u2115) :\n  unused p n q = \u2203\u2203 val : list (word value \u03b1), [| length val = n |] \u229b list_repr' p val q :=\nbegin\n  induction n generalizing p,\n  { rw p_exists_one_point [], simp [unused,list_repr'],\n    intros, s_intros h, apply impl_lift_and,\n    rw \u2190 length_eq_zero, exact h,\n    apply impl_lift_and h (impl_refl _), },\n  { rw [unused,n_ih,trashed,\u2190 exists_subtype],\n    simp [and_p_exists_distrib_left,and_p_exists_distrib_right],\n    apply impl_antisymm,\n    { s_intros x xs Hxs, apply impl_exists (x :: xs),\n      simp [Hxs,list_repr'], apply impl_refl _, },\n    { s_intros xs Hxs, rw length_eq_succ at Hxs,\n      rcases Hxs with \u27e8y,ys,h,h'\u27e9, subst xs,\n      apply impl_exists y, apply impl_exists ys, simp [list_repr',h'],\n      refl } }\nend\n\nlemma unused_impl_le' {p\u2080 p\u2081 : tptr (list (word value \u03b1))} (n : \u2115) :\n  unused p\u2080 n p\u2081 =*> [| p\u2080 \u2264 p\u2081 |] \u229b True :=\nbegin\n  rw unused_iff_exists, s_intros val Hval,\n  apply list_repr_impl_le\nend\n\nend fixed_storable\n\nend separation\n-- \u03b1\n\n-- ```\n-- section\n-- variable [\ud835\udcdb : sep_logic]\n-- include \ud835\udcdb\n-- -- maintenant `hProp` et `tptr \u03b1` (le type des pointeurs typ\u00e9s qui pointent vers \u03b1)\n-- -- r\u00e9f\u00e8rent automatiquement \u03b1 \ud835\udcdb sans qu'on ait \u00e0 cr\u00e9er de raccourcis\n", "meta": {"author": "cipher1024", "repo": "lean-pl", "sha": "829680605ac17e91038d793c0188e9614353ca25", "save_path": "github-repos/lean/cipher1024-lean-pl", "path": "github-repos/lean/cipher1024-lean-pl/lean-pl-829680605ac17e91038d793c0188e9614353ca25/src/prop.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.36506804047149094}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.int.char_zero\n! leanprover-community/mathlib commit acee671f47b8e7972a1eb6f4eed74b4b3abce829\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Int.Cast.Field\n\n/-!\n# Injectivity of `int.cast` into characteristic zero rings and fields.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n-/\n\n\nvariable {\u03b1 : Type _}\n\nopen Nat\n\nnamespace Int\n\n/- warning: int.cast_eq_zero -> Int.cast_eq_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {n : Int}, Iff (Eq.{succ u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int \u03b1 (HasLiftT.mk.{1, succ u1} Int \u03b1 (CoeTC\u2093.coe.{1, succ u1} Int \u03b1 (Int.castCoe.{u1} \u03b1 (AddGroupWithOne.toHasIntCast.{u1} \u03b1 _inst_1)))) n) (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddMonoidWithOne.toAddMonoid.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)))))))) (Eq.{1} Int n (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {n : Int}, Iff (Eq.{succ u1} \u03b1 (Int.cast.{u1} \u03b1 (AddGroupWithOne.toIntCast.{u1} \u03b1 _inst_1) n) (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (NegZeroClass.toZero.{u1} \u03b1 (SubNegZeroMonoid.toNegZeroClass.{u1} \u03b1 (SubtractionMonoid.toSubNegZeroMonoid.{u1} \u03b1 (AddGroup.toSubtractionMonoid.{u1} \u03b1 (AddGroupWithOne.toAddGroup.{u1} \u03b1 _inst_1)))))))) (Eq.{1} Int n (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align int.cast_eq_zero Int.cast_eq_zero\u2093'. -/\n@[simp]\ntheorem cast_eq_zero [AddGroupWithOne \u03b1] [CharZero \u03b1] {n : \u2124} : (n : \u03b1) = 0 \u2194 n = 0 :=\n  \u27e8fun h => by\n    cases n\n    \u00b7 rw [Int.cast_ofNat] at h\n      exact congr_arg coe (Nat.cast_eq_zero.1 h)\n    \u00b7 rw [cast_neg_succ_of_nat, neg_eq_zero, Nat.cast_eq_zero] at h\n      contradiction, fun h => by rw [h, cast_zero]\u27e9\n#align int.cast_eq_zero Int.cast_eq_zero\n\n/- warning: int.cast_inj -> Int.cast_inj is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {m : Int} {n : Int}, Iff (Eq.{succ u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int \u03b1 (HasLiftT.mk.{1, succ u1} Int \u03b1 (CoeTC\u2093.coe.{1, succ u1} Int \u03b1 (Int.castCoe.{u1} \u03b1 (AddGroupWithOne.toHasIntCast.{u1} \u03b1 _inst_1)))) m) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int \u03b1 (HasLiftT.mk.{1, succ u1} Int \u03b1 (CoeTC\u2093.coe.{1, succ u1} Int \u03b1 (Int.castCoe.{u1} \u03b1 (AddGroupWithOne.toHasIntCast.{u1} \u03b1 _inst_1)))) n)) (Eq.{1} Int m n)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {m : Int} {n : Int}, Iff (Eq.{succ u1} \u03b1 (Int.cast.{u1} \u03b1 (AddGroupWithOne.toIntCast.{u1} \u03b1 _inst_1) m) (Int.cast.{u1} \u03b1 (AddGroupWithOne.toIntCast.{u1} \u03b1 _inst_1) n)) (Eq.{1} Int m n)\nCase conversion may be inaccurate. Consider using '#align int.cast_inj Int.cast_inj\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_inj [AddGroupWithOne \u03b1] [CharZero \u03b1] {m n : \u2124} : (m : \u03b1) = n \u2194 m = n := by\n  rw [\u2190 sub_eq_zero, \u2190 cast_sub, cast_eq_zero, sub_eq_zero]\n#align int.cast_inj Int.cast_inj\n\n/- warning: int.cast_injective -> Int.cast_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)], Function.Injective.{1, succ u1} Int \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int \u03b1 (HasLiftT.mk.{1, succ u1} Int \u03b1 (CoeTC\u2093.coe.{1, succ u1} Int \u03b1 (Int.castCoe.{u1} \u03b1 (AddGroupWithOne.toHasIntCast.{u1} \u03b1 _inst_1)))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)], Function.Injective.{1, succ u1} Int \u03b1 (Int.cast.{u1} \u03b1 (AddGroupWithOne.toIntCast.{u1} \u03b1 _inst_1))\nCase conversion may be inaccurate. Consider using '#align int.cast_injective Int.cast_injective\u2093'. -/\ntheorem cast_injective [AddGroupWithOne \u03b1] [CharZero \u03b1] : Function.Injective (coe : \u2124 \u2192 \u03b1)\n  | m, n => cast_inj.1\n#align int.cast_injective Int.cast_injective\n\n/- warning: int.cast_ne_zero -> Int.cast_ne_zero is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {n : Int}, Iff (Ne.{succ u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int \u03b1 (HasLiftT.mk.{1, succ u1} Int \u03b1 (CoeTC\u2093.coe.{1, succ u1} Int \u03b1 (Int.castCoe.{u1} \u03b1 (AddGroupWithOne.toHasIntCast.{u1} \u03b1 _inst_1)))) n) (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (AddZeroClass.toHasZero.{u1} \u03b1 (AddMonoid.toAddZeroClass.{u1} \u03b1 (AddMonoidWithOne.toAddMonoid.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)))))))) (Ne.{1} Int n (OfNat.ofNat.{0} Int 0 (OfNat.mk.{0} Int 0 (Zero.zero.{0} Int Int.hasZero))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : AddGroupWithOne.{u1} \u03b1] [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 _inst_1)] {n : Int}, Iff (Ne.{succ u1} \u03b1 (Int.cast.{u1} \u03b1 (AddGroupWithOne.toIntCast.{u1} \u03b1 _inst_1) n) (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (NegZeroClass.toZero.{u1} \u03b1 (SubNegZeroMonoid.toNegZeroClass.{u1} \u03b1 (SubtractionMonoid.toSubNegZeroMonoid.{u1} \u03b1 (AddGroup.toSubtractionMonoid.{u1} \u03b1 (AddGroupWithOne.toAddGroup.{u1} \u03b1 _inst_1)))))))) (Ne.{1} Int n (OfNat.ofNat.{0} Int 0 (instOfNatInt 0)))\nCase conversion may be inaccurate. Consider using '#align int.cast_ne_zero Int.cast_ne_zero\u2093'. -/\ntheorem cast_ne_zero [AddGroupWithOne \u03b1] [CharZero \u03b1] {n : \u2124} : (n : \u03b1) \u2260 0 \u2194 n \u2260 0 :=\n  not_congr cast_eq_zero\n#align int.cast_ne_zero Int.cast_ne_zero\n\n/- warning: int.cast_div_char_zero -> Int.cast_div_charZero is a dubious translation:\nlean 3 declaration is\n  forall {k : Type.{u1}} [_inst_1 : DivisionRing.{u1} k] [_inst_2 : CharZero.{u1} k (AddGroupWithOne.toAddMonoidWithOne.{u1} k (AddCommGroupWithOne.toAddGroupWithOne.{u1} k (Ring.toAddCommGroupWithOne.{u1} k (DivisionRing.toRing.{u1} k _inst_1))))] {m : Int} {n : Int}, (Dvd.Dvd.{0} Int (semigroupDvd.{0} Int Int.semigroup) n m) -> (Eq.{succ u1} k ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int k (HasLiftT.mk.{1, succ u1} Int k (CoeTC\u2093.coe.{1, succ u1} Int k (Int.castCoe.{u1} k (AddGroupWithOne.toHasIntCast.{u1} k (AddCommGroupWithOne.toAddGroupWithOne.{u1} k (Ring.toAddCommGroupWithOne.{u1} k (DivisionRing.toRing.{u1} k _inst_1))))))) (HDiv.hDiv.{0, 0, 0} Int Int Int (instHDiv.{0} Int Int.hasDiv) m n)) (HDiv.hDiv.{u1, u1, u1} k k k (instHDiv.{u1} k (DivInvMonoid.toHasDiv.{u1} k (DivisionRing.toDivInvMonoid.{u1} k _inst_1))) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int k (HasLiftT.mk.{1, succ u1} Int k (CoeTC\u2093.coe.{1, succ u1} Int k (Int.castCoe.{u1} k (AddGroupWithOne.toHasIntCast.{u1} k (AddCommGroupWithOne.toAddGroupWithOne.{u1} k (Ring.toAddCommGroupWithOne.{u1} k (DivisionRing.toRing.{u1} k _inst_1))))))) m) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int k (HasLiftT.mk.{1, succ u1} Int k (CoeTC\u2093.coe.{1, succ u1} Int k (Int.castCoe.{u1} k (AddGroupWithOne.toHasIntCast.{u1} k (AddCommGroupWithOne.toAddGroupWithOne.{u1} k (Ring.toAddCommGroupWithOne.{u1} k (DivisionRing.toRing.{u1} k _inst_1))))))) n)))\nbut is expected to have type\n  forall {k : Type.{u1}} [_inst_1 : DivisionRing.{u1} k] [_inst_2 : CharZero.{u1} k (AddGroupWithOne.toAddMonoidWithOne.{u1} k (Ring.toAddGroupWithOne.{u1} k (DivisionRing.toRing.{u1} k _inst_1)))] {m : Int} {n : Int}, (Dvd.dvd.{0} Int Int.instDvdInt n m) -> (Eq.{succ u1} k (Int.cast.{u1} k (Ring.toIntCast.{u1} k (DivisionRing.toRing.{u1} k _inst_1)) (HDiv.hDiv.{0, 0, 0} Int Int Int (instHDiv.{0} Int Int.instDivInt_1) m n)) (HDiv.hDiv.{u1, u1, u1} k k k (instHDiv.{u1} k (DivisionRing.toDiv.{u1} k _inst_1)) (Int.cast.{u1} k (Ring.toIntCast.{u1} k (DivisionRing.toRing.{u1} k _inst_1)) m) (Int.cast.{u1} k (Ring.toIntCast.{u1} k (DivisionRing.toRing.{u1} k _inst_1)) n)))\nCase conversion may be inaccurate. Consider using '#align int.cast_div_char_zero Int.cast_div_charZero\u2093'. -/\n@[simp, norm_cast]\ntheorem cast_div_charZero {k : Type _} [DivisionRing k] [CharZero k] {m n : \u2124} (n_dvd : n \u2223 m) :\n    ((m / n : \u2124) : k) = m / n :=\n  by\n  rcases eq_or_ne n 0 with (rfl | hn)\n  \u00b7 simp [Int.div_zero]\n  \u00b7 exact cast_div n_dvd (cast_ne_zero.mpr hn)\n#align int.cast_div_char_zero Int.cast_div_charZero\n\nend Int\n\n/- warning: ring_hom.injective_int -> RingHom.injective_int is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : NonAssocRing.{u1} \u03b1] (f : RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 (AddCommGroupWithOne.toAddGroupWithOne.{u1} \u03b1 (NonAssocRing.toAddCommGroupWithOne.{u1} \u03b1 _inst_1)))], Function.Injective.{1, succ u1} Int \u03b1 (coeFn.{succ u1, succ u1} (RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) (fun (_x : RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) => Int -> \u03b1) (RingHom.hasCoeToFun.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.ring)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : NonAssocRing.{u1} \u03b1] (f : RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) [_inst_2 : CharZero.{u1} \u03b1 (AddGroupWithOne.toAddMonoidWithOne.{u1} \u03b1 (AddCommGroupWithOne.toAddGroupWithOne.{u1} \u03b1 (NonAssocRing.toAddCommGroupWithOne.{u1} \u03b1 _inst_1)))], Function.Injective.{1, succ u1} Int \u03b1 (FunLike.coe.{succ u1, 1, succ u1} (RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) Int (fun (_x : Int) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : Int) => \u03b1) _x) (MulHomClass.toFunLike.{u1, 0, u1} (RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) Int \u03b1 (NonUnitalNonAssocSemiring.toMul.{0} Int (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Int (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)))) (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1))) (NonUnitalRingHomClass.toMulHomClass.{u1, 0, u1} (RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) Int \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{0} Int (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) (RingHomClass.toNonUnitalRingHomClass.{u1, 0, u1} (RingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1)) Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1) (RingHom.instRingHomClassRingHom.{0, u1} Int \u03b1 (NonAssocRing.toNonAssocSemiring.{0} Int (Ring.toNonAssocRing.{0} Int Int.instRingInt)) (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 _inst_1))))) f)\nCase conversion may be inaccurate. Consider using '#align ring_hom.injective_int RingHom.injective_int\u2093'. -/\ntheorem RingHom.injective_int {\u03b1 : Type _} [NonAssocRing \u03b1] (f : \u2124 \u2192+* \u03b1) [CharZero \u03b1] :\n    Function.Injective f :=\n  Subsingleton.elim (Int.castRingHom _) f \u25b8 Int.cast_injective\n#align ring_hom.injective_int RingHom.injective_int\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Int/CharZero.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.36506804047149094}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport algebra.homology.homotopy\nimport algebra.homology.additive\nimport algebra.category.Group.abelian\nimport data.int.parity\nimport algebra.homology.short_exact.preadditive\nimport for_mathlib.algebra.homology.homological_complex_X_iso_of_eq\n\nnoncomputable theory\n\nopen category_theory category_theory.preadditive category_theory.limits category_theory.category\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C] [preadditive C]\n\nnamespace cochain_complex\n\nvariables {F G K L : cochain_complex C \u2124} (n m : \u2124)\n\n/-- should be changed...-/\nstructure is_termwise_kernel (i : F \u27f6 G) (f : G \u27f6 K) :=\n(zero : \u2200 n, i.f n \u226b f.f n = 0)\n(is_limit : \u2200 n, is_limit (kernel_fork.of_\u03b9 (i.f n) (zero n)))\n\nnamespace is_termwise_kernel\n\nlemma termwise_mono {i : F \u27f6 G} {f : G \u27f6 K}\n  (h : is_termwise_kernel i f) (q : \u2124) : mono (i.f q) :=\nmono_of_is_limit_fork (h.is_limit q)\n\nend is_termwise_kernel\n\nnamespace hom_complex\n\ndef \u03b5 (n : \u2124) : \u2124 := \u2191((-1 : units \u2124) ^ n)\n\n@[simp]\n\nlemma \u03b5_add (n\u2081 n\u2082 : \u2124) : \u03b5 (n\u2081 + n\u2082) = \u03b5 n\u2081 * \u03b5 n\u2082 :=\nby { dsimp [\u03b5], rw [\u2190 units.coe_mul, \u2190 units.ext_iff, zpow_add], }\n\n@[simp]\nlemma \u03b5_0 : \u03b5 0 = 1 := rfl\n\n@[simp]\nlemma \u03b5_1 : \u03b5 1 = -1 := rfl\n\n@[simp]\nlemma \u03b5_succ (n : \u2124) : \u03b5 (n + 1) = - \u03b5 n :=\nby simp only [\u03b5_add, \u03b5_1, algebra.id.smul_eq_mul, mul_neg, mul_one]\n\nlemma \u03b5_even (n : \u2124) (hn : even n) : \u03b5 n = 1 :=\nbegin\n  change _ = \u2191(1 : units \u2124),\n  cases hn with k hk,\n  simp only [\u03b5, \u2190 units.ext_iff, hk, zpow_add, \u2190 mul_zpow,\n    mul_neg, mul_one, neg_neg, one_zpow],\nend\n\nlemma \u03b5_odd (n : \u2124) (hn : odd n) : \u03b5 n = -1 :=\nbegin\n  cases hn with k hk,\n  rw [hk, \u03b5_add, \u03b5_1, \u03b5_even (2*k) \u27e8k, two_mul k\u27e9, one_mul],\nend\n\nlemma \u03b5_eq_one_iff (n : \u2124) : \u03b5 n = 1 \u2194 even n :=\nbegin\n  split,\n  { intro h,\n    rw int.even_iff_not_odd,\n    intro h',\n    rw \u03b5_odd _ h' at h,\n    linarith, },\n  { intro h,\n    rw \u03b5_even _ h, },\nend\n\nlemma \u03b5_eq_neg_one_iff (n : \u2124) : \u03b5 n = -1 \u2194 odd n :=\nbegin\n  split,\n  { intro h,\n    rw int.odd_iff_not_even,\n    intro h',\n    rw \u03b5_even _ h' at h,\n    linarith, },\n  { intro h,\n    rw \u03b5_odd _ h, },\nend\n\nlemma \u03b5_neg (n : \u2124) : \u03b5 (-n) = \u03b5 n :=\nbegin\n  dsimp [\u03b5],\n  simp only [zpow_neg, \u2190 inv_zpow, inv_neg, inv_one],\nend\n\nlemma \u03b5_sub (n\u2081 n\u2082 : \u2124) : \u03b5 (n\u2081 - n\u2082) = \u03b5 n\u2081 * \u03b5 n\u2082 :=\nby simp only [sub_eq_add_neg, \u03b5_add, \u03b5_neg]\n\nlemma \u03b5_eq_iff (n\u2081 n\u2082 : \u2124) : \u03b5 n\u2081 = \u03b5 n\u2082 \u2194\n  even (n\u2081 - n\u2082) :=\nbegin\n  by_cases h\u2082 : even n\u2082,\n  { rw [\u03b5_even _ h\u2082, int.even_sub, \u03b5_eq_one_iff],\n    tauto, },\n  { rw [\u2190 int.odd_iff_not_even] at h\u2082,\n    rw [\u03b5_odd _ h\u2082, int.even_sub, \u03b5_eq_neg_one_iff,\n      int.even_iff_not_odd, int.even_iff_not_odd],\n      tauto, }\nend\n\n@[simp]\nlemma mul_\u03b5_self (n : \u2124) : \u03b5 n * \u03b5 n = 1 :=\nby simpa only [\u2190 \u03b5_add] using \u03b5_even _ (even_add_self n)\n\n@[simp]\nlemma \u03b5_mul_self (n : \u2124) : \u03b5 (n * n) = \u03b5 n :=\nbegin\n  by_cases hn : even n,\n  { rw [\u03b5_even _ hn, \u03b5_even],\n    obtain \u27e8k, rfl\u27e9 := hn,\n    exact \u27e82*k*k, by ring\u27e9, },\n  { rw [\u2190 int.odd_iff_not_even] at hn,\n    rw [\u03b5_odd _ hn],\n    obtain \u27e8k, rfl\u27e9 := hn,\n    rw [\u03b5_odd],\n    exact \u27e82*k*k + 2*k, by ring\u27e9, },\nend\n\nstructure triplet (n : \u2124) := (p : \u2124) (q : \u2124) (hpq : q=p+n)\n\nvariables (F G)\n\n@[derive add_comm_group]\ndef cochain := \u03a0 (T : triplet n), F.X T.p \u27f6 G.X T.q\n\nnamespace cochain\n\nvariables {F G n}\n\ndef mk (v : \u03a0 (p q : \u2124) (hpq : q=p+n), F.X p \u27f6 G.X q) : cochain F G n :=\n\u03bb T, v T.p T.q T.hpq\n\ndef v (c : cochain F G n) (p q : \u2124) (hpq : q=p+n) := c (triplet.mk p q hpq)\n\n@[simp]\nlemma mk_v (v : \u03a0 (p q : \u2124) (hpq : q=p+n), F.X p \u27f6 G.X q) (p q : \u2124) (hpq : q=p+n) :\n  (mk v).v p q hpq = v p q hpq := rfl\n\nlemma congr_v {z\u2081 z\u2082 : cochain F G n} (h : z\u2081 = z\u2082) (p q : \u2124) (hpq : q=p+n) :\n  z\u2081.v p q hpq = z\u2082.v p q hpq := by subst h\n\n@[ext]\nlemma ext (z\u2081 z\u2082 : cochain F G n) (h : \u2200 (p q : \u2124) (hpq : q=p+n), z\u2081.v p q hpq = z\u2082.v p q hpq) :\n  z\u2081 = z\u2082 :=\nbegin\n  ext T,\n  rcases T with \u27e8p, q, hpq\u27e9,\n  exact h p q hpq,\nend\n\n@[ext]\nlemma ext\u2080 (z\u2081 z\u2082 : cochain F G 0)\n  (h : \u2200 (p : \u2124), z\u2081.v p p (add_zero p).symm = z\u2082.v p p (add_zero p).symm ) : z\u2081 = z\u2082 :=\nbegin\n  ext,\n  have eq : q=p := by rw [hpq, add_zero],\n  subst eq,\n  apply h,\nend\n\n@[simp]\nlemma zero_v {n : \u2124} (p q : \u2124) (hpq : q=p+n) : (0 : cochain F G n).v p q hpq = 0 := rfl\n\n@[simp]\nlemma add_v {n : \u2124} (z\u2081 z\u2082 : cochain F G n) (p q : \u2124) (hpq : q=p+n) :\n  (z\u2081+z\u2082).v p q hpq = z\u2081.v p q hpq + z\u2082.v p q hpq := rfl\n\n@[simp]\nlemma sub_v {n : \u2124} (z\u2081 z\u2082 : cochain F G n) (p q : \u2124) (hpq : q=p+n) :\n  (z\u2081-z\u2082).v p q hpq = z\u2081.v p q hpq - z\u2082.v p q hpq := rfl\n\n@[simp]\nlemma neg_v {n : \u2124} (z : cochain F G n) (p q : \u2124) (hpq : q=p+n) :\n  (-z).v p q hpq = - (z.v p q hpq) := rfl\n\n@[simp]\nlemma zsmul_v {n k : \u2124} (z : cochain F G n) (p q : \u2124) (hpq : q=p+n) :\n  (k \u2022 z).v p q hpq = k \u2022 (z.v p q hpq) := rfl\n\ndef of_homs (\u03c8 : \u03a0 (p : \u2124), F.X p \u27f6 G.X p) : cochain F G 0 :=\ncochain.mk (\u03bb p q hpq, \u03c8 p \u226b eq_to_hom (by rw [hpq, add_zero]))\n\n@[simp]\nlemma of_homs_v (\u03c8 : \u03a0 (p : \u2124), F.X p \u27f6 G.X p) (p : \u2124) :\n  (of_homs \u03c8).v p p (add_zero p).symm = \u03c8 p :=\nby simp only [of_homs, mk_v, eq_to_hom_refl, comp_id]\n\n@[simp]\nlemma of_homs_zero : of_homs (\u03bb p, (0 : F.X p \u27f6 G.X p)) = 0 :=\nby { ext, simp only [of_homs_v, zero_v], }\n\n@[simp]\nlemma of_homs_v_comp_d (\u03c8 : \u03a0 (p : \u2124), F.X p \u27f6 G.X p) (p q q' : \u2124) (hpq : q=p+0) :\n  (of_homs \u03c8).v p q hpq \u226b G.d q q' = \u03c8 p \u226b G.d p q' :=\nbegin\n  rw add_zero at hpq,\n  subst hpq,\n  rw of_homs_v,\nend\n\n@[simp]\nlemma d_comp_of_homs_v (\u03c8 : \u03a0 (p : \u2124), F.X p \u27f6 G.X p) (p' p q  : \u2124) (hpq : q=p+0) :\n  F.d p' p \u226b (of_homs \u03c8).v p q hpq = F.d p' q \u226b \u03c8 q :=\nbegin\n  rw add_zero at hpq,\n  subst hpq,\n  rw of_homs_v,\nend\n\ndef of_hom (\u03c6 : F \u27f6 G) : cochain F G 0 :=\nof_homs (\u03bb p, \u03c6.f p)\n\n@[simp]\nlemma of_hom_zero : of_hom (0 : F \u27f6 G) = 0 :=\nby simp only [of_hom, homological_complex.zero_f_apply, of_homs_zero]\n\n@[simp]\nlemma of_hom_v (\u03c6 : F \u27f6 G) (p : \u2124) : (of_hom \u03c6).v p p (add_zero p).symm = \u03c6.f p :=\nby simp only [of_hom, of_homs_v]\n\n@[simp]\nlemma of_hom_v_comp_d (\u03c6 : F \u27f6 G) (p q q' : \u2124) (hpq : q=p+0) :\n  (of_hom \u03c6).v p q hpq \u226b G.d q q' = \u03c6.f p \u226b G.d p q' :=\nby simp only [of_hom, of_homs_v_comp_d]\n\n@[simp]\nlemma d_comp_of_hom_v (\u03c6 : F \u27f6 G) (p' p q  : \u2124) (hpq : q=p+0) :\n  F.d p' p \u226b (of_hom \u03c6).v p q hpq = F.d p' q \u226b \u03c6.f q :=\nby simp only [of_hom, d_comp_of_homs_v]\n\n@[simp]\ndef of_homotopy {\u03c6\u2081 \u03c6\u2082 : F \u27f6 G} (ho : homotopy \u03c6\u2081 \u03c6\u2082) : cochain F G (-1) :=\ncochain.mk (\u03bb p q hpq, ho.hom p q)\n\n@[simp, reassoc]\nlemma v_comp_X_iso_of_eq_hom {K L : cochain_complex C \u2124} {n : \u2124}\n  (\u03b3 : cochain K L n) (p q q' : \u2124)\n  (hpq : q = p + n) (hq' : q = q'):\n  \u03b3.v p q hpq \u226b (homological_complex.X_iso_of_eq L hq').hom = \u03b3.v p q' (by rw [\u2190 hq', hpq]) :=\nby { subst hq', simp only [homological_complex.X_iso_of_eq_refl, iso.refl_hom, comp_id], }\n\ndef comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (h : n\u2081\u2082 = n\u2081 + n\u2082) :\n  cochain F K n\u2081\u2082 :=\ncochain.mk (\u03bb p q hpq, z\u2081.v p (p+n\u2081) rfl \u226b z\u2082.v (p+n\u2081) q (by linarith))\n\nnotation a ` \u226b[`:81 b `] ` c:80 := cochain.comp a c b\n\nlemma comp_v {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (h : n\u2081\u2082 = n\u2081+n\u2082)\n  (p\u2081 p\u2082 p\u2083 : \u2124) (h\u2081 : p\u2082=p\u2081+n\u2081) (h\u2082 : p\u2083=p\u2082+n\u2082) :\n  (comp z\u2081 z\u2082 h).v p\u2081 p\u2083 (by rw [h\u2082, h\u2081, h, add_assoc]) =\n  z\u2081.v p\u2081 p\u2082 h\u2081 \u226b z\u2082.v p\u2082 p\u2083 h\u2082 := by { subst h\u2081, refl,}\n\n@[simp]\nlemma zero_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp (0 : cochain F G n\u2081) z\u2082 h = 0 :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [zero_comp],\nend\n\n@[simp]\nlemma add_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 z\u2081' : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp (z\u2081+z\u2081') z\u2082 h = comp z\u2081 z\u2082 h + comp z\u2081' z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [add_comp],\nend\n\n@[simp]\nlemma sub_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 z\u2081' : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp (z\u2081-z\u2081') z\u2082 h = comp z\u2081 z\u2082 h - comp z\u2081' z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [sub_comp],\nend\n\n@[simp]\nlemma neg_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp (-z\u2081) z\u2082 h = -comp z\u2081 z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [neg_comp],\nend\n\n@[simp]\nlemma zsmul_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (k : \u2124) (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp (k \u2022 z\u2081) z\u2082 h = k \u2022 comp z\u2081 z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [zsmul_comp],\nend\n\n@[simp]\nlemma zero_cochain_comp {n : \u2124} (z\u2081 : cochain F G 0) (z\u2082 : cochain G K n)\n  (p q : \u2124) (hpq : q=p+n) :\n  (cochain.comp z\u2081 z\u2082 (zero_add n).symm).v p q hpq =\n    z\u2081.v p p (add_zero p).symm \u226b z\u2082.v p q hpq :=\ncomp_v z\u2081 z\u2082 (zero_add n).symm p p q (add_zero p).symm hpq\n\nlemma zero_cochain_comp' {n : \u2124} (z\u2081 : cochain F G 0) (z\u2082 : cochain G K n)\n  (p\u2081 p\u2082 p\u2083 : \u2124) (h\u2081\u2082 : p\u2082=p\u2081+0) (h\u2082\u2083 : p\u2083=p\u2082+n) :\n  (z\u2081.v p\u2081 p\u2082 h\u2081\u2082 \u226b z\u2082.v p\u2082 p\u2083 h\u2082\u2083 : F.X p\u2081 \u27f6 K.X p\u2083) =\n  z\u2081.v p\u2081 p\u2081 (add_zero p\u2081).symm \u226b z\u2082.v p\u2081 p\u2083 (show p\u2083 = p\u2081+n, by rw [h\u2082\u2083, h\u2081\u2082, add_zero]) :=\nby { rw add_zero at h\u2081\u2082, subst h\u2081\u2082, }\n\n@[simp]\nlemma id_comp {n : \u2124} (z\u2082 : cochain F G n) :\n  cochain.comp (cochain.of_hom (\ud835\udfd9 F)) z\u2082 (zero_add n).symm = z\u2082 :=\nbegin\n  ext,\n  simp only [zero_cochain_comp, of_hom_v, homological_complex.id_f, id_comp],\nend\n\n@[simp]\nlemma comp_zero {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp z\u2081 (0 : cochain G K n\u2082) h = 0 :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [comp_zero],\nend\n\n@[simp]\nlemma comp_add {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 z\u2082' : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp z\u2081 (z\u2082+z\u2082') h = comp z\u2081 z\u2082 h + comp z\u2081 z\u2082' h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [comp_add],\nend\n\n@[simp]\nlemma comp_sub {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 z\u2082' : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp z\u2081 (z\u2082-z\u2082') h = comp z\u2081 z\u2082 h - comp z\u2081 z\u2082' h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [comp_sub],\nend\n\n@[simp]\nlemma comp_neg {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp z\u2081 (-z\u2082) h = -comp z\u2081 z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [comp_neg],\nend\n\n@[simp]\nlemma comp_zsmul {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (k : \u2124) (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082)\n  (h : n\u2081\u2082 = n\u2081 + n\u2082) : comp z\u2081 (k \u2022 z\u2082) h = k \u2022 comp z\u2081 z\u2082 h :=\nbegin\n  ext,\n  dsimp [comp, mk, v],\n  simp only [comp_zsmul],\nend\n\n@[simp]\nlemma comp_zero_cochain (z\u2081 : cochain F G n) (z\u2082 : cochain G K 0)\n  (p q : \u2124) (hpq : q=p+n) :\n  (cochain.comp z\u2081 z\u2082 (add_zero n).symm).v p q hpq =\n    z\u2081.v p q hpq \u226b z\u2082.v q q (add_zero q).symm :=\ncomp_v z\u2081 z\u2082 (add_zero n).symm p q q hpq (add_zero q).symm\n\nlemma comp_zero_cochain' (z\u2081 : cochain F G n) (z\u2082 : cochain G K 0)\n  (p\u2081 p\u2082 p\u2083 : \u2124) (h\u2081\u2082 : p\u2082=p\u2081+n) (h\u2082\u2083 : p\u2083=p\u2082+0) :\n  (z\u2081.v p\u2081 p\u2082 h\u2081\u2082 \u226b z\u2082.v p\u2082 p\u2083 h\u2082\u2083 : F.X p\u2081 \u27f6 K.X p\u2083) =\n  z\u2081.v p\u2081 p\u2083 (show p\u2083=p\u2081+n, by rw [h\u2082\u2083, h\u2081\u2082, add_zero]) \u226b z\u2082.v p\u2083 p\u2083 (add_zero p\u2083).symm :=\nby { rw add_zero at h\u2082\u2083, subst h\u2082\u2083, }\n\n@[simp]\nlemma comp_id {n : \u2124} (z\u2081 : cochain F G n) :\n  cochain.comp z\u2081 (cochain.of_hom (\ud835\udfd9 G)) (add_zero n).symm = z\u2081 :=\nbegin\n  ext,\n  simp only [comp_zero_cochain, of_hom_v, homological_complex.id_f, comp_id],\nend\n\n@[simp]\nlemma of_homs_comp (\u03c6 : \u03a0 (p : \u2124), F.X p \u27f6 G.X p) (\u03c8 : \u03a0 (p : \u2124), G.X p \u27f6 K.X p) :\n  cochain.comp (of_homs \u03c6) (of_homs \u03c8) (zero_add 0).symm = of_homs (\u03bb p, \u03c6 p \u226b \u03c8 p) :=\nbegin\n  ext,\n  simp only [comp_zero_cochain, of_homs_v],\nend\n\n@[simp]\nlemma of_hom_comp (f : F \u27f6 G) (g : G \u27f6 K) :\n  of_hom (f \u226b g) = cochain.comp (of_hom f) (of_hom g) (zero_add 0).symm :=\nby simpa only [of_hom, of_homs_comp]\n\nlemma comp_assoc {n\u2081 n\u2082 n\u2083 n\u2081\u2082 n\u2082\u2083 n\u2081\u2082\u2083 : \u2124}\n  (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (z\u2083 : cochain K L n\u2083)\n  (h\u2081\u2082 : n\u2081\u2082 = n\u2081 + n\u2082) (h\u2082\u2083 : n\u2082\u2083 = n\u2082 + n\u2083) (h\u2081\u2082\u2083 : n\u2081\u2082\u2083 = n\u2081 + n\u2082 + n\u2083) :\n  cochain.comp (cochain.comp z\u2081 z\u2082 h\u2081\u2082) z\u2083 (show n\u2081\u2082\u2083 = n\u2081\u2082 + n\u2083, by rw [h\u2081\u2082, h\u2081\u2082\u2083]) =\n    cochain.comp z\u2081 (cochain.comp z\u2082 z\u2083 h\u2082\u2083)\n      (show n\u2081\u2082\u2083 = n\u2081 + n\u2082\u2083, by rw [h\u2082\u2083, h\u2081\u2082\u2083, add_assoc]) :=\nbegin\n  ext,\n  simp only [comp_v _ _ (show n\u2081\u2082\u2083 = n\u2081\u2082 + n\u2083, by rw [h\u2081\u2082, h\u2081\u2082\u2083]) p (p+n\u2081\u2082) q rfl (by linarith),\n    comp_v _ _ h\u2081\u2082 p (p+n\u2081) (p+n\u2081\u2082) rfl (by linarith),\n    comp_v z\u2081 (cochain.comp z\u2082 z\u2083 h\u2082\u2083) (show n\u2081\u2082\u2083 = n\u2081 + n\u2082\u2083, by linarith)\n      p (p+n\u2081) q rfl (by linarith),\n    comp_v _ _ h\u2082\u2083 (p+n\u2081) (p+n\u2081\u2082) q (by linarith) (by linarith), assoc],\nend\n\n@[simp]\nlemma comp_assoc_of_first_is_zero_cochain {n\u2082 n\u2083 n\u2082\u2083 : \u2124}\n  (z\u2081 : cochain F G 0) (z\u2082 : cochain G K n\u2082) (z\u2083 : cochain K L n\u2083)\n  (h\u2082\u2083 : n\u2082\u2083 = n\u2082 + n\u2083) :\n  cochain.comp (cochain.comp z\u2081 z\u2082 (zero_add n\u2082).symm) z\u2083 h\u2082\u2083 =\n    cochain.comp z\u2081 (cochain.comp z\u2082 z\u2083 h\u2082\u2083)\n      (zero_add n\u2082\u2083).symm :=\ncomp_assoc z\u2081 z\u2082 z\u2083 (zero_add n\u2082).symm h\u2082\u2083 (by linarith)\n\n@[simp]\nlemma comp_assoc_of_second_is_zero_cochain {n\u2081 n\u2083 n\u2081\u2083 : \u2124}\n  (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K 0) (z\u2083 : cochain K L n\u2083) (h\u2081\u2083 : n\u2081\u2083 = n\u2081 + n\u2083) :\n  cochain.comp (cochain.comp z\u2081 z\u2082 (add_zero n\u2081).symm) z\u2083 h\u2081\u2083 =\n    cochain.comp z\u2081 (cochain.comp z\u2082 z\u2083 (zero_add n\u2083).symm) h\u2081\u2083 :=\ncomp_assoc z\u2081 z\u2082 z\u2083 (add_zero n\u2081).symm (zero_add n\u2083).symm (by linarith)\n\n@[simp]\nlemma comp_assoc_of_third_is_zero_cochain {n\u2081 n\u2082 n\u2081\u2082 : \u2124}\n  (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (z\u2083 : cochain K L 0) (h\u2081\u2082 : n\u2081\u2082 = n\u2081 + n\u2082) :\n  cochain.comp (cochain.comp z\u2081 z\u2082 h\u2081\u2082) z\u2083 (add_zero n\u2081\u2082).symm =\n    cochain.comp z\u2081 (cochain.comp z\u2082 z\u2083 (add_zero n\u2082).symm) h\u2081\u2082 :=\ncomp_assoc z\u2081 z\u2082 z\u2083 h\u2081\u2082 (add_zero n\u2082).symm (by linarith)\n\nvariable (K)\n\ndef of_d : cochain K K 1 := cochain.mk (\u03bb p q hpq, K.d p q)\n\n@[simp]\nlemma of_d_v (p q : \u2124) (hpq : q=p+1) :\n  (of_d K).v p q hpq = K.d p q := rfl\n\nend cochain\n\n/- Differentials -/\n\nvariables {F G} (n)\n\ndef \u03b4 (z : cochain F G n) : cochain F G m :=\ncochain.mk (\u03bb (p q : \u2124) hpq, z.v p (p+n) rfl \u226b G.d (p+n) q +\n  \u03b5 (n+1) \u2022 F.d p (p+m-n) \u226b z.v (p+m-n) q (by { dsimp [int.sub], linarith}))\n\nlemma \u03b4_v (hnm : n+1=m) (z : cochain F G n) (p q : \u2124) (hpq : q=p+m) (q\u2081 q\u2082 : \u2124)\n  (hq\u2081 : q\u2081=q-1) (hq\u2082 : q\u2082=p+1) : (\u03b4 n m z).v p q hpq =\n  z.v p q\u2081 (by {rw [hq\u2081, hpq, \u2190 hnm, \u2190 add_assoc, add_tsub_cancel_right],}) \u226b G.d q\u2081 q\n  + \u03b5 (n+1) \u2022 F.d p q\u2082 \u226b z.v q\u2082 q (by rw [hpq, hq\u2082, \u2190 hnm, add_comm n, add_assoc]) :=\nbegin\n  have h\u2081 : q\u2081 = p+n := by linarith,\n  have h\u2082 : q\u2082 = p+m-n := by linarith,\n  substs h\u2081 h\u2082,\n  refl,\nend\n\nlemma \u03b4_shape (hnm : \u00ac n+1=m) (z : cochain F G n) : \u03b4 n m z = 0 :=\nbegin\n  ext,\n  dsimp [\u03b4, cochain.mk, cochain.v],\n  rw [F.shape, G.shape, limits.comp_zero, limits.zero_comp, smul_zero, add_zero],\n  all_goals\n  { change \u00ac _=_ ,\n    intro h,\n    apply hnm,\n    linarith, },\nend\n\nvariables (F G)\n\ndef \u03b4_hom : cochain F G n \u2192+ cochain F G m :=\n{ to_fun := \u03b4 n m,\n  map_zero' := begin\n    ext,\n    dsimp [\u03b4, cochain.mk, cochain.v],\n    simp only [limits.zero_comp, limits.comp_zero, smul_zero, add_zero],\n  end,\n  map_add' := \u03bb z\u2081 z\u2082, begin\n    ext,\n    dsimp [\u03b4, cochain.mk, cochain.v],\n    simp only [preadditive.add_comp, preadditive.comp_add, smul_add],\n    abel,\n  end}\n\nvariables {F G}\n\n@[simp]\nlemma \u03b4_add (z\u2081 z\u2082 : cochain F G n) : \u03b4 n m (z\u2081 + z\u2082) = \u03b4 n m z\u2081 + \u03b4 n m z\u2082 :=\n(\u03b4_hom F G n m).map_add z\u2081 z\u2082\n\n@[simp]\nlemma \u03b4_sub (z\u2081 z\u2082 : cochain F G n) : \u03b4 n m (z\u2081 - z\u2082) = \u03b4 n m z\u2081 - \u03b4 n m z\u2082 :=\n(\u03b4_hom F G n m).map_sub z\u2081 z\u2082\n\n@[simp]\nlemma \u03b4_zero : \u03b4 n m (0 : cochain F G n) = 0 := (\u03b4_hom F G n m).map_zero\n\n@[simp]\nlemma \u03b4_neg (z : cochain F G n) : \u03b4 n m (-z) = - \u03b4 n m z :=\n(\u03b4_hom F G n m).map_neg z\n\n@[simp]\nlemma \u03b4_zsmul (k : \u2124) (z : cochain F G n) : \u03b4 n m (k \u2022 z) = k \u2022 \u03b4 n m z :=\n(\u03b4_hom F G n m).map_zsmul z k\n\n@[simp]\nlemma \u03b4\u03b4 (n\u2080 n\u2081 n\u2082 : \u2124) (z : cochain F G n\u2080) : \u03b4 n\u2081 n\u2082 (\u03b4 n\u2080 n\u2081 z) = 0 :=\nbegin\n  by_cases h\u2080\u2081 : n\u2080+1 = n\u2081, swap,\n  { rw [\u03b4_shape n\u2080 n\u2081 h\u2080\u2081, \u03b4_zero], },\n  by_cases h\u2081\u2082 : n\u2081+1 = n\u2082, swap,\n  { rw [\u03b4_shape n\u2081 n\u2082 h\u2081\u2082], },\n  ext,\n  rw \u03b4_v n\u2081 n\u2082 h\u2081\u2082 _ p q hpq _ _ rfl rfl,\n  rw \u03b4_v n\u2080 n\u2081 h\u2080\u2081 z p (q-1) (by linarith) (q-2) _ (by linarith) rfl,\n  rw \u03b4_v n\u2080 n\u2081 h\u2080\u2081 z (p+1) q (by linarith) _ (p+2) rfl (by linarith),\n  simp only [\u2190 h\u2080\u2081, \u03b5_succ, add_comp, neg_neg, neg_zsmul, neg_comp, cochain.zero_v,\n    zsmul_comp, comp_zsmul, comp_add, comp_neg, assoc, homological_complex.d_comp_d,\n    homological_complex.d_comp_d_assoc, comp_zero, zero_comp, zsmul_zero, neg_zero, add_zero,\n    zero_add, add_left_neg],\nend\n\nlemma \u03b4_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (h : n\u2081\u2082 = n\u2081 + n\u2082)\n  (m\u2081 m\u2082 m\u2081\u2082 : \u2124) (h\u2081\u2082 : n\u2081\u2082+1 = m\u2081\u2082) (h\u2081 : n\u2081+1 = m\u2081) (h\u2082 : n\u2082+1 = m\u2082) :\n\u03b4 n\u2081\u2082 m\u2081\u2082 (cochain.comp z\u2081 z\u2082 h) = cochain.comp z\u2081 (\u03b4 n\u2082 m\u2082 z\u2082) (by linarith) + \u03b5 n\u2082 \u2022 cochain.comp (\u03b4 n\u2081 m\u2081 z\u2081) z\u2082 (by linarith) :=\nbegin\n  substs h\u2081\u2082 h\u2081 h\u2082,\n  ext,\n  have eq : \u03b5 (n\u2081\u2082 + 1) = \u03b5 n\u2082 * \u03b5 (n\u2081+1),\n  { rw \u2190 \u03b5_add, congr' 1, linarith, },\n  simp only [cochain.add_v, cochain.zsmul_v,\n    cochain.comp_v z\u2081 (\u03b4 n\u2082 (n\u2082+1) z\u2082) (show n\u2081\u2082+1=n\u2081+(n\u2082+1), by linarith) p _ q rfl (by linarith),\n    cochain.comp_v (\u03b4 n\u2081 (n\u2081+1) z\u2081) z\u2082 (show n\u2081\u2082+1=_, by linarith) p (p+n\u2081+1) q (by linarith) (by linarith),\n    cochain.comp_v z\u2081 z\u2082 h p (p+n\u2081) (p+n\u2081\u2082) rfl (by linarith),\n    cochain.comp_v z\u2081 z\u2082 h (p+1) (p+n\u2081+1) q (by linarith) (by linarith),\n    \u03b4_v n\u2081\u2082 _ rfl (cochain.comp z\u2081 z\u2082 h) p q hpq (p+n\u2081\u2082) _ (by linarith) rfl,\n    \u03b4_v n\u2081 (n\u2081+1) rfl z\u2081 p (p+n\u2081+1) (by linarith) (p+n\u2081) (p+1) (by linarith) rfl,\n    \u03b4_v n\u2082 (n\u2082+1) rfl z\u2082 (p+n\u2081) q (by linarith) (p+n\u2081\u2082) (p+n\u2081+1) (by linarith) rfl,\n    assoc, comp_add, comp_zsmul, zsmul_add, add_comp, zsmul_comp, smul_smul, eq,\n    \u03b5_add n\u2082 1, \u03b5_1, mul_neg, mul_one, neg_zsmul, comp_neg, \u2190 add_assoc],\n  suffices : \u2200 (a b c : F.X p \u27f6 K.X q), a+b=a+(-c)+c+b,\n  { apply this, },\n  intros a b c,\n  abel,\nend\n\n@[simp]\nlemma \u03b4_comp_of_first_is_zero_cochain {n\u2082 : \u2124} (z\u2081 : cochain F G 0) (z\u2082 : cochain G K n\u2082)\n  (m\u2082 : \u2124) (h\u2082 : n\u2082+1 = m\u2082) :\n\u03b4 n\u2082 m\u2082 (cochain.comp z\u2081 z\u2082 (zero_add n\u2082).symm) =\n  cochain.comp z\u2081 (\u03b4 n\u2082 m\u2082 z\u2082) (by linarith) + \u03b5 n\u2082 \u2022 cochain.comp (\u03b4 0 1 z\u2081) z\u2082 (by linarith) :=\n\u03b4_comp z\u2081 z\u2082 (zero_add n\u2082).symm 1 m\u2082 m\u2082 h\u2082 (zero_add 1) h\u2082\n\n@[simp]\nlemma \u03b4_comp_of_second_is_zero_cochain {n\u2081 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K 0)\n  (m\u2081 : \u2124) (h\u2081 : n\u2081+1 = m\u2081) : \u03b4 n\u2081 m\u2081 (cochain.comp z\u2081 z\u2082 (add_zero n\u2081).symm) =\n  cochain.comp z\u2081 (\u03b4 0 1 z\u2082) h\u2081.symm + cochain.comp (\u03b4 n\u2081 m\u2081 z\u2081) z\u2082 (add_zero m\u2081).symm :=\nby simp only [\u03b4_comp z\u2081 z\u2082 (add_zero n\u2081).symm m\u2081 1 m\u2081 h\u2081 h\u2081 (zero_add 1), \u03b5_0, one_zsmul]\n\nend hom_complex\n\nvariables (F G)\n\nopen hom_complex\n\ndef hom_complex : cochain_complex AddCommGroup \u2124 :=\n{ X := \u03bb i, AddCommGroup.of (cochain F G i),\n  d := \u03bb i j, AddCommGroup.of_hom (\u03b4_hom F G i j),\n  shape' := \u03bb i j hij, by { ext1 z, exact \u03b4_shape i j hij z, },\n  d_comp_d' := \u03bb i j k hij hjk, by { ext1 f, apply \u03b4\u03b4, } }\n\nnamespace hom_complex\n\ndef cocycle : add_subgroup (cochain F G n) :=\nadd_monoid_hom.ker ((hom_complex F G).d n (n+1))\n\nnamespace cocycle\n\nvariables {F G}\n\nlemma mem_iff (hnm : n+1=m) (z : cochain F G n) :\n  z \u2208 cocycle F G n \u2194 \u03b4 n m z = 0 :=\nby { subst hnm, refl, }\n\nvariable {n}\n\n@[simps]\ndef mk (z : cochain F G n) (m : \u2124) (hnm : n+1 = m) (h : \u03b4 n m z = 0) : cocycle F G n :=\n\u27e8z, by simpa only [mem_iff n m hnm z] using h\u27e9\n\n@[simp]\nlemma \u03b4_eq_zero {n : \u2124} (z : cocycle F G n) (m : \u2124) : \u03b4 n m (z : cochain F G n) = 0 :=\nbegin\n  by_cases n+1=m,\n  { rw \u2190 mem_iff n m h,\n    exact z.2, },\n  { apply \u03b4_shape n m h, }\nend\n\n@[simps]\ndef of_hom (\u03c6 : F \u27f6 G) : cocycle F G 0 := mk (cochain.of_hom \u03c6) 1 (zero_add 1)\nbegin\n  ext,\n  simp only [\u03b4_v 0 1 (zero_add 1) _ p q hpq p q (by linarith) hpq,\n    cochain.of_hom_v, homological_complex.hom.comm, \u03b5_1, neg_smul, one_zsmul,\n    add_right_neg, cochain.zero_v, zero_add],\nend\n\n@[simp]\nlemma \u03b4_cochain_of_hom (\u03c6 : F \u27f6 G) : \u03b4 0 1 (cochain.of_hom \u03c6) = 0 :=\nby apply \u03b4_eq_zero (of_hom \u03c6)\n\n@[simps]\ndef hom_of (z : cocycle F G 0) : F \u27f6 G :=\n{ f := \u03bb i, (z : cochain F G 0).v i i (add_zero i).symm,\n  comm' := \u03bb i j hij, begin\n    change i+1=j at hij,\n    have hz\u2081 := z.2,\n    rw mem_iff 0 1 (zero_add 1) at hz\u2081,\n    simpa only [\u03b4_v 0 1 (zero_add 1) z.1 i j hij.symm i j (by linarith) hij.symm,\n      zero_add, \u03b5_1, neg_smul, one_zsmul, cochain.zero_v, add_neg_eq_zero]\n      using cochain.congr_v hz\u2081 i j hij.symm,\n  end, }\n\n@[simp]\nlemma hom_of_of_hom_eq_self (\u03c6 : F \u27f6 G) : hom_of (of_hom \u03c6) = \u03c6 :=\nby { ext i, simp only [of_hom, hom_of_f, mk_coe, cochain.of_hom_v], }\n\n@[simp]\nlemma of_hom_hom_of_eq_self (z : cocycle F G 0) : of_hom (hom_of z) = z :=\nbegin\n  ext,\n  simp only [of_hom, mk_coe, cochain.of_hom_v, hom_of_f],\nend\n\n@[simp]\nlemma cochain_of_hom_hom_of_eq_coe (z : cocycle F G 0) :\n  (cochain.of_hom (hom_of z) : cochain F G 0) = (z : cochain F G 0) :=\nby simpa only [subtype.ext_iff] using of_hom_hom_of_eq_self z\n\nvariables (F G)\n\n@[simps]\ndef equiv_hom : (F \u27f6 G) \u2243+ cocycle F G 0 :=\n{ to_fun := of_hom,\n  inv_fun := hom_of,\n  left_inv := hom_of_of_hom_eq_self,\n  right_inv := of_hom_hom_of_eq_self,\n  map_add' := \u03bb \u03c6\u2081 \u03c6\u2082, begin\n    ext,\n    simp only [of_hom, cochain.of_hom, cochain.of_homs, cochain.mk, cochain.v,\n      homological_complex.add_f_apply, mk_coe, eq_to_hom_refl, comp_id,\n      add_subgroup.coe_add, pi.add_apply],\n  end, }\n\ndef of_d : cocycle K K 1 :=\ncocycle.mk (cochain.of_d K) 2 rfl begin\n  ext p q hpq,\n  simp only [\u03b4_v 1 2 rfl _ p q hpq _ _ rfl rfl, cochain.of_d_v,\n    homological_complex.d_comp_d, smul_zero, add_zero, cochain.zero_v],\nend\n\nend cocycle\n\nnamespace cochain\n\nvariables {F G}\n\nlemma of_hom_injective {f\u2081 f\u2082 : F \u27f6 G} (h : of_hom f\u2081 = of_hom f\u2082) : f\u2081 = f\u2082 :=\nbegin\n  rw [\u2190 cocycle.hom_of_of_hom_eq_self f\u2081, \u2190 cocycle.hom_of_of_hom_eq_self f\u2082],\n  congr' 1,\n  ext1,\n  simpa only [cocycle.of_hom_coe] using h,\nend\n\nend cochain\n\nvariables {F G}\n\n@[simps]\ndef equiv_homotopy (\u03c6\u2081 \u03c6\u2082 : F \u27f6 G) :\n  homotopy \u03c6\u2081 \u03c6\u2082 \u2243\n    { z : cochain F G (-1) // cochain.of_hom \u03c6\u2081 = \u03b4 (-1) 0 z + cochain.of_hom \u03c6\u2082 } :=\n{ to_fun := \u03bb ho, begin\n    refine \u27e8cochain.of_homotopy ho, _\u27e9,\n    ext,\n    have comm := ho.comm p,\n    rw [d_next_eq ho.hom rfl, prev_d_eq ho.hom (sub_add_cancel p 1)] at comm,\n    rw [cochain.add_v, \u03b4_v (-1) 0 (neg_add_self 1) _ p p (add_zero p).symm _ _ rfl rfl],\n    simp only [\u03b4_v (-1) 0 (neg_add_self 1) _ p p (add_zero p).symm _ _ rfl rfl,\n      add_left_neg, \u03b5_0, one_zsmul, cochain.mk, cochain.of_hom_v, cochain.v,\n      cochain.of_homotopy, cochain.of_hom_v],\n    dsimp only,\n    suffices : \u2200 (a b c d : F.X p \u27f6 G.X p) (h : a = b+c+d), a=c+b+d,\n    { exact this _ _ _ _ comm, },\n    { intros a b c d h, rw h, abel, },\n  end,\n  inv_fun := \u03bb z,\n    { hom := \u03bb i j, begin\n        by_cases j+1=i,\n        { exact (z : cochain F G (-1)).v i j (by linarith), },\n        { exact 0, },\n      end,\n      zero' := \u03bb i j hij, begin\n        change \u00ac j+1 = i at hij,\n        rw dif_neg hij,\n      end,\n      comm := \u03bb p, begin\n        have h\u2081 : p+1 = p+1 := rfl,\n        have h\u2082 : p-1+1 = p := by linarith,\n        have h\u2081' : (complex_shape.up \u2124).rel p (p+1) := h\u2081,\n        have h\u2082' : (complex_shape.up \u2124).rel (p-1) p := h\u2082,\n        rw [d_next_eq _ h\u2081', prev_d_eq _ h\u2082', dif_pos h\u2081, dif_pos h\u2082],\n        have hz := cochain.congr_v z.2 p p (add_zero p).symm,\n        simp only [cochain.add_v, \u03b4_v (-1) 0 (neg_add_self 1) _ p p (add_zero p).symm _ _ rfl rfl,\n          cochain.of_hom_v, add_left_neg, \u03b5_0, one_zsmul] at hz,\n        suffices : \u2200 (a b c d : F.X p \u27f6 G.X p) (h : a = b+c+d), a=c+b+d,\n        { exact this _ _ _ _ hz, },\n        { intros a b c d h, rw h, abel, },\n      end, },\n  left_inv := \u03bb ho, begin\n    ext i j,\n    dsimp,\n    split_ifs,\n    { refl, },\n    { rw ho.zero i j h, },\n  end,\n  right_inv := \u03bb z, begin\n    ext,\n    dsimp [cochain.mk, cochain.v],\n    simpa only [dif_pos (show q+1=p, by linarith)],\n  end, }\n\nlemma \u03b4_cochain_of_homotopy {\u03c6\u2081 \u03c6\u2082 : F \u27f6 G} (h : homotopy \u03c6\u2081 \u03c6\u2082) :\n  \u03b4 (-1) 0 (cochain.of_homotopy h) = cochain.of_hom \u03c6\u2081 - cochain.of_hom \u03c6\u2082 :=\nby rw [((equiv_homotopy _ _) h).2, add_sub_cancel,\n  subtype.val_eq_coe, equiv_homotopy_apply_coe]\n\nnamespace cochain\n\nvariable {n}\n\ndef lift_to_kernel' (z : cochain L G n) {i : F \u27f6 G} {f : G \u27f6 K} (hip : is_termwise_kernel i f)\n  (hz : cochain.comp z (of_hom f) (add_zero n).symm = 0) (p q : \u2124) (hpq : q=p+n):=\nkernel_fork.is_limit.lift' (hip.is_limit q) (z.v p q hpq)\n(by simpa only [comp_zero_cochain, of_hom_v] using congr_v hz p q hpq)\n\ndef lift_to_kernel (z : cochain L G n) {i : F \u27f6 G} {f : G \u27f6 K} (hip : is_termwise_kernel i f)\n  (hz : cochain.comp z (of_hom f) (add_zero n).symm = 0) : cochain L F n :=\ncochain.mk (\u03bb p q hpq, (lift_to_kernel' z hip hz p q hpq).1)\n\n@[simp]\nlemma lift_to_kernel_comp (z : cochain L G n) {i : F \u27f6 G} {f : G \u27f6 K} (hip : is_termwise_kernel i f)\n  (hz : cochain.comp z (of_hom f) (add_zero n).symm = 0) :\n  cochain.comp (z.lift_to_kernel hip hz) (cochain.of_hom i) (add_zero n).symm = z :=\nbegin\n  ext,\n  simpa only [comp_v _ _ (add_zero n).symm p q q hpq (add_zero q).symm,\n    of_hom_v] using (lift_to_kernel' z hip hz p q hpq).2,\nend\n\nend cochain\n\nnamespace cocycle\n\nvariable {n}\n\ndef lift_to_kernel (z : cocycle L G n) {i : F \u27f6 G} {f : G \u27f6 K} (hip : is_termwise_kernel i f)\n  (hz : cochain.comp (z : cochain L G n) (cochain.of_hom f) (add_zero n).symm = 0) :\n  cocycle L F n :=\ncocycle.mk (cochain.lift_to_kernel (z : cochain L G n) hip hz) _ rfl\nbegin\n  suffices : \u03b4 n (n + 1) (cochain.comp\n    ((z : cochain L G n).lift_to_kernel hip hz) (cochain.of_hom i) (add_zero n).symm) = 0,\n  { ext,\n    haveI : mono (i.f q) := hip.termwise_mono q,\n    simpa only [\u2190 cancel_mono (i.f q), cochain.zero_v, zero_comp,\n      \u03b4_comp_of_second_is_zero_cochain, \u03b4_cochain_of_hom,\n      cochain.comp_zero, zero_add, cochain.comp_zero_cochain,\n      cochain.of_hom_v, cochain.zero_v] using cochain.congr_v this p q hpq, },\n  simp only [cochain.lift_to_kernel_comp, \u03b4_eq_zero],\nend\n\nlemma lift_to_kernel_comp (z : cocycle L G n) {i : F \u27f6 G} {f : G \u27f6 K} (hip : is_termwise_kernel i f)\n  (hz : cochain.comp (z : cochain L G n) (cochain.of_hom f) (add_zero n).symm = 0) :\n  cochain.comp (lift_to_kernel z hip hz : cochain L F n) (cochain.of_hom i) (add_zero n).symm =\n  (z : cochain L G n) := by apply cochain.lift_to_kernel_comp\n\nend cocycle\n\nsection\n\nvariables {n} {D : Type*} [category D] [preadditive D] (z z' : cochain K L n) (f : K \u27f6 L)\n  (\u03a6 : C \u2964 D) [functor.additive \u03a6]\n\nnamespace cochain\n\ndef map : cochain ((functor.map_homological_complex \u03a6 _).obj K)\n  ((functor.map_homological_complex \u03a6 _).obj L) n :=\ncochain.mk (\u03bb p q hpq, \u03a6.map (z.v p q hpq))\n\n@[simp]\nlemma map_v (p q : \u2124) (hpq : q=p+n) :\n  (z.map \u03a6).v p q hpq = \u03a6.map (z.v p q hpq) := rfl\n\n@[simp]\nlemma map_add : (z+z').map \u03a6 = z.map \u03a6 + z'.map \u03a6 := by tidy\n\n@[simp]\nlemma map_neg : (-z).map \u03a6 = -z.map \u03a6 := by tidy\n\n@[simp]\nlemma map_sub : (z-z').map \u03a6 = z.map \u03a6 - z'.map \u03a6 := by tidy\n\nvariables (K L n)\n\n@[simp]\nlemma map_zero : (0 : cochain K L n).map \u03a6 = 0 := by tidy\n\n@[simp]\nlemma map_comp {n\u2081 n\u2082 n\u2081\u2082 : \u2124} (z\u2081 : cochain F G n\u2081) (z\u2082 : cochain G K n\u2082) (h : n\u2081\u2082 = n\u2081 + n\u2082)\n  (\u03a6 : C \u2964 D) [functor.additive \u03a6] :\n  (z\u2081.comp z\u2082 h).map \u03a6 = (z\u2081.map \u03a6).comp (z\u2082.map \u03a6) h :=\nbegin\n  ext p q hpq,\n  simp only [map_v, comp_v _ _ h p _ q rfl (by linarith), \u03a6.map_comp],\nend\n\n@[simp]\nlemma map_of_hom : (cochain.of_hom f).map \u03a6 =\n  cochain.of_hom ((\u03a6.map_homological_complex _).map f) := by tidy\n\nend cochain\n\nvariables (n m)\n\n@[simp]\nlemma \u03b4_map : \u03b4 n m (z.map \u03a6) = (\u03b4 n m z).map \u03a6 :=\nbegin\n  by_cases hnm : n+1=m,\n  { ext p q hpq,\n    simp only [\u03b4_v n m hnm _ p q hpq (q-1) (p+1) rfl rfl, cochain.map_v],\n    simp only [functor.map_homological_complex_obj_d, \u03b5_succ, neg_smul, functor.map_add,\n      functor.map_comp, functor.map_neg, add_right_inj, neg_inj, \u03a6.map_zsmul], },\n  { simp only [\u03b4_shape _ _ hnm, cochain.map_zero], },\nend\n\nend\n\nnamespace cocycle\n\nvariables {n} {D : Type*} [category D] [preadditive D] (z z' : cocycle K L n) (f : K \u27f6 L)\n  (\u03a6 : C \u2964 D) [functor.additive \u03a6]\n\n@[simps]\ndef map : cocycle ((functor.map_homological_complex \u03a6 _).obj K)\n  ((functor.map_homological_complex \u03a6 _).obj L) n :=\ncocycle.mk ((z : cochain K L n).map \u03a6) (n+1) rfl (by simp)\n\n@[simp]\nlemma map_add : cocycle.map (z+z') \u03a6 = cocycle.map z \u03a6 + cocycle.map z' \u03a6 :=\nby { ext1, simp, }\n\n@[simp]\nlemma map_neg : cocycle.map (-z) \u03a6 = -cocycle.map z \u03a6 :=\nby { ext1, simp, }\n\n@[simp]\nlemma map_sub : cocycle.map (z-z') \u03a6 = cocycle.map z \u03a6 - cocycle.map z' \u03a6 :=\nby { ext1, simp, }\n\n@[simp]\nlemma map_of_hom : cocycle.map (cocycle.of_hom f) \u03a6 =\n  cocycle.of_hom ((\u03a6.map_homological_complex _).map f) := by tidy\n\nvariables (K L n)\n\n@[simp]\nlemma map_zero : cocycle.map (0 : cocycle K L n) \u03a6 = 0 :=\nby { ext1, simp, }\n\nend cocycle\n\nend hom_complex\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/hom_complex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.36506804047149094}}
{"text": "/-\nCopyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Seul Baek\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.nat.basic\nimport Mathlib.PostPort\n\nuniverses u v w \n\nnamespace Mathlib\n\nnamespace list\n\n\nnamespace func\n\n\n/- Definitions for using lists as finite\n   representations of functions with domain \u2115. -/\n\ndef neg {\u03b1 : Type u} [Neg \u03b1] (as : List \u03b1) : List \u03b1 :=\n  map (fun (a : \u03b1) => -a) as\n\n@[simp] def set {\u03b1 : Type u} [Inhabited \u03b1] (a : \u03b1) : List \u03b1 \u2192 \u2115 \u2192 List \u03b1 :=\n  sorry\n\n@[simp] def get {\u03b1 : Type u} [Inhabited \u03b1] : \u2115 \u2192 List \u03b1 \u2192 \u03b1 :=\n  sorry\n\ndef equiv {\u03b1 : Type u} [Inhabited \u03b1] (as1 : List \u03b1) (as2 : List \u03b1) :=\n  \u2200 (m : \u2115), get m as1 = get m as2\n\n@[simp] def pointwise {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [Inhabited \u03b1] [Inhabited \u03b2] (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3 :=\n  sorry\n\ndef add {\u03b1 : Type u} [HasZero \u03b1] [Add \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1 :=\n  pointwise Add.add\n\ndef sub {\u03b1 : Type u} [HasZero \u03b1] [Sub \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1 :=\n  pointwise Sub.sub\n\n/- set -/\n\ntheorem length_set {\u03b1 : Type u} {a : \u03b1} [Inhabited \u03b1] {m : \u2115} {as : List \u03b1} : length (set a as m) = max (length as) (m + 1) := sorry\n\n@[simp] theorem get_nil {\u03b1 : Type u} [Inhabited \u03b1] {k : \u2115} : get k [] = Inhabited.default :=\n  nat.cases_on k (Eq.refl (get 0 [])) fun (k : \u2115) => Eq.refl (get (Nat.succ k) [])\n\ntheorem get_eq_default_of_le {\u03b1 : Type u} [Inhabited \u03b1] (k : \u2115) {as : List \u03b1} : length as \u2264 k \u2192 get k as = Inhabited.default := sorry\n\n@[simp] theorem get_set {\u03b1 : Type u} [Inhabited \u03b1] {a : \u03b1} {k : \u2115} {as : List \u03b1} : get k (set a as k) = a := sorry\n\ntheorem eq_get_of_mem {\u03b1 : Type u} [Inhabited \u03b1] {a : \u03b1} {as : List \u03b1} : a \u2208 as \u2192 \u2203 (n : \u2115), \u03b1 \u2192 a = get n as := sorry\n\ntheorem mem_get_of_le {\u03b1 : Type u} [Inhabited \u03b1] {n : \u2115} {as : List \u03b1} : n < length as \u2192 get n as \u2208 as := sorry\n\ntheorem mem_get_of_ne_zero {\u03b1 : Type u} [Inhabited \u03b1] {n : \u2115} {as : List \u03b1} : get n as \u2260 Inhabited.default \u2192 get n as \u2208 as := sorry\n\ntheorem get_set_eq_of_ne {\u03b1 : Type u} [Inhabited \u03b1] {a : \u03b1} {as : List \u03b1} (k : \u2115) (m : \u2115) : m \u2260 k \u2192 get m (set a as k) = get m as := sorry\n\ntheorem get_map {\u03b1 : Type u} {\u03b2 : Type v} [Inhabited \u03b1] [Inhabited \u03b2] {f : \u03b1 \u2192 \u03b2} {n : \u2115} {as : List \u03b1} : n < length as \u2192 get n (map f as) = f (get n as) := sorry\n\ntheorem get_map' {\u03b1 : Type u} {\u03b2 : Type v} [Inhabited \u03b1] [Inhabited \u03b2] {f : \u03b1 \u2192 \u03b2} {n : \u2115} {as : List \u03b1} : f Inhabited.default = Inhabited.default \u2192 get n (map f as) = f (get n as) := sorry\n\ntheorem forall_val_of_forall_mem {\u03b1 : Type u} [Inhabited \u03b1] {as : List \u03b1} {p : \u03b1 \u2192 Prop} : p Inhabited.default \u2192 (\u2200 (x : \u03b1), x \u2208 as \u2192 p x) \u2192 \u2200 (n : \u2115), p (get n as) := sorry\n\n/- equiv -/\n\ntheorem equiv_refl {\u03b1 : Type u} {as : List \u03b1} [Inhabited \u03b1] : equiv as as :=\n  fun (k : \u2115) => rfl\n\ntheorem equiv_symm {\u03b1 : Type u} {as1 : List \u03b1} {as2 : List \u03b1} [Inhabited \u03b1] : equiv as1 as2 \u2192 equiv as2 as1 :=\n  fun (h1 : equiv as1 as2) (k : \u2115) => Eq.symm (h1 k)\n\ntheorem equiv_trans {\u03b1 : Type u} {as1 : List \u03b1} {as2 : List \u03b1} {as3 : List \u03b1} [Inhabited \u03b1] : equiv as1 as2 \u2192 equiv as2 as3 \u2192 equiv as1 as3 :=\n  fun (h1 : equiv as1 as2) (h2 : equiv as2 as3) (k : \u2115) => Eq.trans (h1 k) (h2 k)\n\ntheorem equiv_of_eq {\u03b1 : Type u} {as1 : List \u03b1} {as2 : List \u03b1} [Inhabited \u03b1] : as1 = as2 \u2192 equiv as1 as2 :=\n  fun (h1 : as1 = as2) => eq.mpr (id (Eq._oldrec (Eq.refl (equiv as1 as2)) h1)) equiv_refl\n\ntheorem eq_of_equiv {\u03b1 : Type u} [Inhabited \u03b1] {as1 : List \u03b1} {as2 : List \u03b1} : length as1 = length as2 \u2192 equiv as1 as2 \u2192 as1 = as2 := sorry\n\nend func\n\n\n-- We want to drop the `inhabited` instances for a moment,\n\n-- so we close and open the namespace\n\nnamespace func\n\n\n/- neg -/\n\n@[simp] theorem get_neg {\u03b1 : Type u} [add_group \u03b1] {k : \u2115} {as : List \u03b1} : get k (neg as) = -get k as := sorry\n\n@[simp] theorem length_neg {\u03b1 : Type u} [Neg \u03b1] (as : List \u03b1) : length (neg as) = length as := sorry\n\n/- pointwise -/\n\ntheorem nil_pointwise {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [Inhabited \u03b1] [Inhabited \u03b2] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (bs : List \u03b2) : pointwise f [] bs = map (f Inhabited.default) bs := sorry\n\ntheorem pointwise_nil {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [Inhabited \u03b1] [Inhabited \u03b2] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (as : List \u03b1) : pointwise f as [] = map (fun (a : \u03b1) => f a Inhabited.default) as := sorry\n\ntheorem get_pointwise {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [Inhabited \u03b1] [Inhabited \u03b2] [Inhabited \u03b3] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (h1 : f Inhabited.default Inhabited.default = Inhabited.default) (k : \u2115) (as : List \u03b1) (bs : List \u03b2) : get k (pointwise f as bs) = f (get k as) (get k bs) := sorry\n\ntheorem length_pointwise {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [Inhabited \u03b1] [Inhabited \u03b2] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {as : List \u03b1} {bs : List \u03b2} : length (pointwise f as bs) = max (length as) (length bs) := sorry\n\nend func\n\n\nnamespace func\n\n\n/- add -/\n\n@[simp] theorem get_add {\u03b1 : Type u} [add_monoid \u03b1] {k : \u2115} {xs : List \u03b1} {ys : List \u03b1} : get k (add xs ys) = get k xs + get k ys :=\n  get_pointwise (zero_add Inhabited.default) k xs ys\n\n@[simp] theorem length_add {\u03b1 : Type u} [HasZero \u03b1] [Add \u03b1] {xs : List \u03b1} {ys : List \u03b1} : length (add xs ys) = max (length xs) (length ys) :=\n  length_pointwise\n\n@[simp] theorem nil_add {\u03b1 : Type u} [add_monoid \u03b1] (as : List \u03b1) : add [] as = as := sorry\n\n@[simp] theorem add_nil {\u03b1 : Type u} [add_monoid \u03b1] (as : List \u03b1) : add as [] = as := sorry\n\ntheorem map_add_map {\u03b1 : Type u} [add_monoid \u03b1] (f : \u03b1 \u2192 \u03b1) (g : \u03b1 \u2192 \u03b1) {as : List \u03b1} : add (map f as) (map g as) = map (fun (x : \u03b1) => f x + g x) as := sorry\n\n/- sub -/\n\n@[simp] theorem get_sub {\u03b1 : Type u} [add_group \u03b1] {k : \u2115} {xs : List \u03b1} {ys : List \u03b1} : get k (sub xs ys) = get k xs - get k ys :=\n  get_pointwise (sub_zero Inhabited.default) k xs ys\n\n@[simp] theorem length_sub {\u03b1 : Type u} [HasZero \u03b1] [Sub \u03b1] {xs : List \u03b1} {ys : List \u03b1} : length (sub xs ys) = max (length xs) (length ys) :=\n  length_pointwise\n\n@[simp] theorem nil_sub {\u03b1 : Type} [add_group \u03b1] (as : List \u03b1) : sub [] as = neg as := sorry\n\n@[simp] theorem sub_nil {\u03b1 : Type} [add_group \u03b1] (as : List \u03b1) : sub as [] = as := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/func.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36498419359741635}}
{"text": "/-\nCopyright (c) 2020 Yury Kudryashov. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.invertible\nimport Mathlib.linear_algebra.affine_space.affine_equiv\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 u_3 u_5 \n\nnamespace Mathlib\n\n/-!\n# Midpoint of a segment\n\n## Main definitions\n\n* `midpoint R x y`: midpoint of the segment `[x, y]`. We define it for `x` and `y`\n  in a module over a ring `R` with invertible `2`.\n* `add_monoid_hom.of_map_midpoint`: construct an `add_monoid_hom` given a map `f` such that\n  `f` sends zero to zero and midpoints to midpoints.\n\n## Main theorems\n\n* `midpoint_eq_iff`: `z` is the midpoint of `[x, y]` if and only if `x + y = z + z`,\n* `midpoint_unique`: `midpoint R x y` does not depend on `R`;\n* `midpoint x y` is linear both in `x` and `y`;\n* `point_reflection_midpoint_left`, `point_reflection_midpoint_right`:\n  `equiv.point_reflection (midpoint R x y)` swaps `x` and `y`.\n\nWe do not mark most lemmas as `@[simp]` because it is hard to tell which side is simpler.\n\n## Tags\n\nmidpoint, add_monoid_hom\n-/\n\n/-- `midpoint x y` is the midpoint of the segment `[x, y]`. -/\ndef midpoint (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (x : P) (y : P) : P :=\n  coe_fn (affine_map.line_map x y) \u215f\n\n@[simp] theorem affine_map.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] [add_comm_group V'] [semimodule R V'] [add_torsor V' P'] (f : affine_map R P P') (a : P) (b : P) : coe_fn f (midpoint R a b) = midpoint R (coe_fn f a) (coe_fn f b) :=\n  affine_map.apply_line_map f a b \u215f\n\n@[simp] theorem affine_equiv.map_midpoint {R : Type u_1} {V : Type u_2} {V' : Type u_3} {P : Type u_4} {P' : Type u_5} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] [add_comm_group V'] [semimodule R V'] [add_torsor V' P'] (f : affine_equiv R P P') (a : P) (b : P) : coe_fn f (midpoint R a b) = midpoint R (coe_fn f a) (coe_fn f b) :=\n  affine_equiv.apply_line_map f a b \u215f\n\n@[simp] theorem affine_equiv.point_reflection_midpoint_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (x : P) (y : P) : coe_fn (affine_equiv.point_reflection R (midpoint R x y)) x = y := sorry\n\ntheorem midpoint_comm {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (x : P) (y : P) : midpoint R x y = midpoint R y x := sorry\n\n@[simp] theorem affine_equiv.point_reflection_midpoint_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (x : P) (y : P) : coe_fn (affine_equiv.point_reflection R (midpoint R x y)) y = x := sorry\n\ntheorem midpoint_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (p\u2081 : P) (p\u2082 : P) (p\u2083 : P) (p\u2084 : P) : midpoint R p\u2081 p\u2082 -\u1d65 midpoint R p\u2083 p\u2084 = midpoint R (p\u2081 -\u1d65 p\u2083) (p\u2082 -\u1d65 p\u2084) :=\n  affine_map.line_map_vsub_line_map p\u2081 p\u2082 p\u2083 p\u2084 \u215f\n\ntheorem midpoint_vadd_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (v : V) (v' : V) (p : P) (p' : P) : midpoint R v v' +\u1d65 midpoint R p p' = midpoint R (v +\u1d65 p) (v' +\u1d65 p') :=\n  affine_map.line_map_vadd_line_map v v' p p' \u215f\n\ntheorem midpoint_eq_iff {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] {x : P} {y : P} {z : P} : midpoint R x y = z \u2194 coe_fn (affine_equiv.point_reflection R z) x = y := sorry\n\n@[simp] theorem midpoint_vsub_left {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (p\u2081 : P) (p\u2082 : P) : midpoint R p\u2081 p\u2082 -\u1d65 p\u2081 = \u215f \u2022 (p\u2082 -\u1d65 p\u2081) :=\n  affine_map.line_map_vsub_left p\u2081 p\u2082 \u215f\n\n@[simp] theorem midpoint_vsub_right {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (p\u2081 : P) (p\u2082 : P) : midpoint R p\u2081 p\u2082 -\u1d65 p\u2082 = \u215f \u2022 (p\u2081 -\u1d65 p\u2082) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (midpoint R p\u2081 p\u2082 -\u1d65 p\u2082 = \u215f \u2022 (p\u2081 -\u1d65 p\u2082))) (midpoint_comm p\u2081 p\u2082)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (midpoint R p\u2082 p\u2081 -\u1d65 p\u2082 = \u215f \u2022 (p\u2081 -\u1d65 p\u2082))) (midpoint_vsub_left p\u2082 p\u2081)))\n      (Eq.refl (\u215f \u2022 (p\u2081 -\u1d65 p\u2082))))\n\n@[simp] theorem left_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (p\u2081 : P) (p\u2082 : P) : p\u2081 -\u1d65 midpoint R p\u2081 p\u2082 = \u215f \u2022 (p\u2081 -\u1d65 p\u2082) :=\n  affine_map.left_vsub_line_map p\u2081 p\u2082 \u215f\n\n@[simp] theorem right_vsub_midpoint {R : Type u_1} {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (p\u2081 : P) (p\u2082 : P) : p\u2082 -\u1d65 midpoint R p\u2081 p\u2082 = \u215f \u2022 (p\u2082 -\u1d65 p\u2081) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (p\u2082 -\u1d65 midpoint R p\u2081 p\u2082 = \u215f \u2022 (p\u2082 -\u1d65 p\u2081))) (midpoint_comm p\u2081 p\u2082)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (p\u2082 -\u1d65 midpoint R p\u2082 p\u2081 = \u215f \u2022 (p\u2082 -\u1d65 p\u2081))) (left_vsub_midpoint p\u2082 p\u2081)))\n      (Eq.refl (\u215f \u2022 (p\u2082 -\u1d65 p\u2081))))\n\n@[simp] theorem midpoint_sub_left {R : Type u_1} {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (v\u2081 : V) (v\u2082 : V) : midpoint R v\u2081 v\u2082 - v\u2081 = \u215f \u2022 (v\u2082 - v\u2081) :=\n  midpoint_vsub_left v\u2081 v\u2082\n\n@[simp] theorem midpoint_sub_right {R : Type u_1} {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (v\u2081 : V) (v\u2082 : V) : midpoint R v\u2081 v\u2082 - v\u2082 = \u215f \u2022 (v\u2081 - v\u2082) :=\n  midpoint_vsub_right v\u2081 v\u2082\n\n@[simp] theorem left_sub_midpoint {R : Type u_1} {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (v\u2081 : V) (v\u2082 : V) : v\u2081 - midpoint R v\u2081 v\u2082 = \u215f \u2022 (v\u2081 - v\u2082) :=\n  left_vsub_midpoint v\u2081 v\u2082\n\n@[simp] theorem right_sub_midpoint {R : Type u_1} {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (v\u2081 : V) (v\u2082 : V) : v\u2082 - midpoint R v\u2081 v\u2082 = \u215f \u2022 (v\u2082 - v\u2081) :=\n  right_vsub_midpoint v\u2081 v\u2082\n\ntheorem midpoint_eq_midpoint_iff_vsub_eq_vsub (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] {x : P} {x' : P} {y : P} {y' : P} : midpoint R x y = midpoint R x' y' \u2194 x -\u1d65 x' = y' -\u1d65 y := sorry\n\ntheorem midpoint_eq_iff' (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] {x : P} {y : P} {z : P} : midpoint R x y = z \u2194 coe_fn (equiv.point_reflection z) x = y :=\n  midpoint_eq_iff\n\n/-- `midpoint` does not depend on the ring `R`. -/\ntheorem midpoint_unique (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (R' : Type u_3) [ring R'] [invertible (bit0 1)] [semimodule R' V] (x : P) (y : P) : midpoint R x y = midpoint R' x y :=\n  iff.mpr (midpoint_eq_iff' R) (iff.mp (midpoint_eq_iff' R') rfl)\n\n@[simp] theorem midpoint_self (R : Type u_1) {V : Type u_2} {P : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (x : P) : midpoint R x x = x :=\n  affine_map.line_map_same_apply x \u215f\n\n@[simp] theorem midpoint_add_self (R : Type u_1) {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (x : V) (y : V) : midpoint R x y + midpoint R x y = x + y := sorry\n\ntheorem midpoint_zero_add (R : Type u_1) {V : Type u_2} [ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] (x : V) (y : V) : midpoint R 0 (x + y) = midpoint R x y := sorry\n\ntheorem line_map_inv_two {R : Type u_1} {V : Type u_2} {P : Type u_3} [division_ring R] [char_zero R] [add_comm_group V] [semimodule R V] [add_torsor V P] (a : P) (b : P) : coe_fn (affine_map.line_map a b) (bit0 1\u207b\u00b9) = midpoint R a b :=\n  rfl\n\ntheorem line_map_one_half {R : Type u_1} {V : Type u_2} {P : Type u_3} [division_ring R] [char_zero R] [add_comm_group V] [semimodule R V] [add_torsor V P] (a : P) (b : P) : coe_fn (affine_map.line_map a b) (1 / bit0 1) = midpoint R a b := sorry\n\ntheorem homothety_inv_of_two {R : Type u_1} {V : Type u_2} {P : Type u_3} [comm_ring R] [invertible (bit0 1)] [add_comm_group V] [semimodule R V] [add_torsor V P] (a : P) (b : P) : coe_fn (affine_map.homothety a \u215f) b = midpoint R a b :=\n  rfl\n\ntheorem homothety_inv_two {k : Type u_1} {V : Type u_2} {P : Type u_3} [field k] [char_zero k] [add_comm_group V] [semimodule k V] [add_torsor V P] (a : P) (b : P) : coe_fn (affine_map.homothety a (bit0 1\u207b\u00b9)) b = midpoint k a b :=\n  rfl\n\ntheorem homothety_one_half {k : Type u_1} {V : Type u_2} {P : Type u_3} [field k] [char_zero k] [add_comm_group V] [semimodule k V] [add_torsor V P] (a : P) (b : P) : coe_fn (affine_map.homothety a (1 / bit0 1)) b = midpoint k a b := sorry\n\n@[simp] theorem pi_midpoint_apply {k : Type u_1} {\u03b9 : Type u_2} {V : \u03b9 \u2192 Type u_3} {P : \u03b9 \u2192 Type u_4} [field k] [invertible (bit0 1)] [(i : \u03b9) \u2192 add_comm_group (V i)] [(i : \u03b9) \u2192 semimodule k (V i)] [(i : \u03b9) \u2192 add_torsor (V i) (P i)] (f : (i : \u03b9) \u2192 P i) (g : (i : \u03b9) \u2192 P i) (i : \u03b9) : midpoint k f g i = midpoint k (f i) (g i) :=\n  rfl\n\nnamespace add_monoid_hom\n\n\n/-- A map `f : E \u2192 F` sending zero to zero and midpoints to midpoints is an `add_monoid_hom`. -/\ndef of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group E] [semimodule R E] [ring R'] [invertible (bit0 1)] [add_comm_group F] [semimodule R' F] (f : E \u2192 F) (h0 : f 0 = 0) (hm : \u2200 (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) : E \u2192+ F :=\n  mk f h0 sorry\n\n@[simp] theorem coe_of_map_midpoint (R : Type u_1) (R' : Type u_2) {E : Type u_3} {F : Type u_4} [ring R] [invertible (bit0 1)] [add_comm_group E] [semimodule R E] [ring R'] [invertible (bit0 1)] [add_comm_group F] [semimodule R' F] (f : E \u2192 F) (h0 : f 0 = 0) (hm : \u2200 (x y : E), f (midpoint R x y) = midpoint R' (f x) (f y)) : \u21d1(of_map_midpoint R R' f h0 hm) = f :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/affine_space/midpoint.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36498419359741635}}
{"text": "import combinatorics.simple_graph.ends.defs\nimport combinatorics.simple_graph.ends.properties\n\nopen classical function category_theory opposite\n\nuniverses u v w\n\nnoncomputable theory\nlocal attribute [instance] prop_decidable\n\nnamespace simple_graph\nnamespace component_compl\n\nvariables {V : Type u} {G : simple_graph V} (Gpc : G.preconnected)\n\ninclude Gpc\n\nlemma subset_of_nicely_arranged\n  (H K : set V)\n  (Hnempty : H.nonempty)\n  (E E' : G.component_compl H)\n  (Einf' : E'.supp.infinite)\n  (En : E \u2260 E')\n  (F : G.component_compl K)\n  (H_F : H \u2286 F)\n  (K_E : K \u2286 E) : (E' : set V) \u2286 F :=\nbegin\n  have KE' : (K \u2229 E') \u2286 \u2205 := \u03bb v \u27e8vK, vE'\u27e9,\n    En (component_compl.pairwise_disjoint.eq (set.not_disjoint_iff.mpr \u27e8v, K_E vK, vE'\u27e9)),\n  obtain \u27e8F', sub, inf\u27e9 : \u2203 F' : component_compl G K, (E' : set V) \u2286 F' \u2227 F'.supp.infinite :=\n    \u27e8 of_connected_disjoint_right E'.connected (set.disjoint_iff.mpr KE'),\n      subset_of_connected_disjoint_right _ _,\n      Einf'.mono (subset_of_connected_disjoint_right _ _)\u27e9,\n  have : F' = F, by\n  { obtain \u27e8\u27e8v, h\u27e9, vE', hH, a\u27e9:= exists_adj_boundary_pair Gpc Hnempty E',\n    exact eq_of_adj v h (sub vE') (H_F hH) a, },\n  exact this \u25b8 sub,\nend\n\nvariable [locally_finite G]\n\nlemma hom_not_injective\n  {H K : (finset V)\u1d52\u1d56}\n  {C : G.component_compl_functor.to_eventual_ranges.obj H}\n  {D D' : G.component_compl_functor.to_eventual_ranges.obj K}\n  (Ddist : D \u2260 D')\n  (h : D.val.supp \u2286 C.val.supp) (h' : D'.val.supp \u2286 C.val.supp) :\n  \u00ac (injective $\n    G.component_compl_functor.to_eventual_ranges.map\n      (op_hom_of_le $ finset.subset_union_left H.unop K.unop : op (H.unop \u222a K.unop) \u27f6 H)) :=\nbegin\n  obtain \u27e8E, hE\u27e9 :=\n    functor.surjective_to_eventual_ranges _ (G.component_compl_functor_is_mittag_leffler Gpc)\n      (op_hom_of_le $ finset.subset_union_right H.unop K.unop : op (H.unop \u222a K.unop) \u27f6 K) D,\n  obtain \u27e8E', hE'\u27e9 :=\n    functor.surjective_to_eventual_ranges _ (G.component_compl_functor_is_mittag_leffler Gpc)\n      (op_hom_of_le $ finset.subset_union_right H.unop K.unop : op (H.unop \u222a K.unop) \u27f6 K) D',\n  subst_vars,\n  refine \u03bb inj, (by { rintro rfl, exact Ddist rfl, } : E \u2260 E') (inj _),\n  obtain \u27e8E, _\u27e9 := E,\n  obtain \u27e8E', _\u27e9 := E',\n  dsimp only [component_compl_functor, functor.to_eventual_ranges, functor.eventual_range] at *,\n  simp only [subtype.ext_iff_val, subtype.val_eq_coe, set.maps_to.coe_restrict_apply, subtype.coe_mk],\n  rw [(hom_eq_iff_le _ _ _).mpr ((E.subset_hom _).trans h),\n      (hom_eq_iff_le _ _ _).mpr ((E'.subset_hom _).trans h')],\nend\n\nomit Gpc\nlemma _root_.fin.fin3_embedding_iff {\u03b1 : Type*} :\n  nonempty (fin 3 \u21aa \u03b1) \u2194 \u2203 (a\u2080 a\u2081 a\u2082 : \u03b1), a\u2080 \u2260 a\u2081 \u2227 a\u2080 \u2260 a\u2082 \u2227 a\u2081 \u2260 a\u2082 := sorry\n\nlemma _root_.fin.fin3_embedding_iff' {\u03b1 : Type*} (a : \u03b1):\n  nonempty (fin 3 \u21aa \u03b1) \u2194 \u2203 (a\u2081 a\u2082 : \u03b1), a \u2260 a\u2081 \u2227 a \u2260 a\u2082 \u2227 a\u2081 \u2260 a\u2082 :=\nbegin\n  split,\n  rintro \u27e8e\u27e9,\n  { by_cases h : a = e 0,\n    { use [e 1, e 2],\n      simp only [h, embedding_like.apply_eq_iff_eq, fin.eq_iff_veq, fin.val_zero', fin.val_one,\n                 fin.val_two, ne.def, zero_eq_bit0, nat.one_ne_zero, nat.zero_ne_one, not_false_iff,\n                 nat.one_ne_bit0, and_self], },\n    { by_cases k : a = e 1,\n      { use [e 0, e 2],\n        simp only [h, k, embedding_like.apply_eq_iff_eq, fin.eq_iff_veq, fin.val_zero', fin.val_one,\n                 fin.val_two, ne.def, zero_eq_bit0, nat.one_ne_zero, nat.zero_ne_one, not_false_iff,\n                 nat.one_ne_bit0, and_self], },\n      { use [e 0, e 1],\n        simp only [h, k, ne.def, embedding_like.apply_eq_iff_eq, fin.zero_eq_one_iff,\n                   nat.bit1_eq_one, nat.one_ne_zero, not_false_iff, and_true],  }, }, },\n  { rintro \u27e8a\u2081,a\u2082,h\u2081,h\u2082,h\u27e9,\n    refine \u27e8\u27e8\u03bb i, [a,a\u2081,a\u2082].nth_le i.val i.prop, _\u27e9\u27e9,\n    have : list.nodup [a,a\u2081,a\u2082], by simp [h, h\u2081, h\u2082],\n    rintro \u27e8i,hi\u27e9 \u27e8j,hj\u27e9,\n    simp [list.nodup.nth_le_inj_iff this], },\nend\n\ninclude Gpc\nlemma hom_not_injective_of_nicely_arranged\n  {H K : (finset V)\u1d52\u1d56}\n  (Hnempty : (unop H).nonempty)\n  {E : G.component_compl_functor.to_eventual_ranges.obj H}\n  {F : G.component_compl_functor.to_eventual_ranges.obj K}\n  (H_F : (H.unop : set V) \u2286 F.val.supp)\n  (K_E : (K.unop : set V) \u2286 E.val.supp)\n  {hK : fin 3 \u21aa (G.component_compl_functor.to_eventual_ranges.obj H)} :\n  \u00ac (injective $\n    G.component_compl_functor.to_eventual_ranges.map\n      (op_hom_of_le $ finset.subset_union_left K.unop H.unop : op (K.unop \u222a H.unop) \u27f6 K)) :=\nbegin\n  obtain \u27e8E\u2081, E\u2082, h\u2080\u2081, h\u2080\u2082, h\u2081\u2082\u27e9 := (fin.fin3_embedding_iff' E).mp \u27e8hK\u27e9,\n  apply @hom_not_injective V G Gpc _ _ _ F E\u2081 E\u2082 h\u2081\u2082 _ _,\n  { apply subset_of_nicely_arranged Gpc _ _ Hnempty E.val E\u2081.val,\n    any_goals\n    { rw infinite_iff_in_eventual_range },\n    exacts [E\u2081.prop, \u03bb h, h\u2080\u2081 (subtype.eq h), H_F, K_E], },\n  { apply subset_of_nicely_arranged Gpc _ _ Hnempty E.val E\u2082.val,\n    any_goals\n    { rw infinite_iff_in_eventual_range },\n    exacts [E\u2082.prop, \u03bb h, h\u2080\u2082 (subtype.eq h), H_F, K_E], },\nend\n\n\nomit Gpc\n-- TODO: fit somewhere\nlemma _root_.fin.embedding_subsingleton {n : \u2115} {\u03b1 : Type*} [subsingleton \u03b1] (e : fin n \u21aa \u03b1) :\n  n \u2264 1 :=\nbegin\n  by_contra' h,\n  simpa using e.inj' (subsingleton.elim (e \u27e80,zero_lt_one.trans h\u27e9) (e \u27e81,h\u27e9)),\nend\ninclude Gpc\n\n/-\n  This is the key part of Hopf-Freudenthal\n  Assuming this is proved:\n  As long as K has at least three infinite connected components, then so does L, and\n  bwd_map \u2039L\u2286L\u203a is not injective, hence the graph has more than three ends.\n-/\nlemma hom_not_injective_of_enough_automorphisms_of_many_components\n  (auts : \u2200 K : finset V, \u2203 \u03c6 : G \u2243g G, disjoint K (finset.image \u03c6 K))\n  (K : (finset V)\u1d52\u1d56)\n  (hK : fin 3 \u21aa (G.component_compl_functor.to_eventual_ranges.obj K)) :\n  \u2203 (L M : (finset V)\u1d52\u1d56) (hL : L \u27f6 K) (hM : M \u27f6 L),\n    \u00ac (injective $ G.component_compl_functor.to_eventual_ranges.map hM) :=\nbegin\n  have Kn : K.unop.nonempty,\n  { by_contradiction h,\n    rw finset.not_nonempty_iff_eq_empty at h,\n    simp only [unop_eq_iff_eq_op.mp h, component_compl_functor, functor.to_eventual_ranges,\n               functor.eventual_range] at hK,\n    dsimp [functor.eventual_range, component_compl] at hK,\n    replace hK := hK.trans \u27e8_, subtype.coe_injective\u27e9,\n    rw [set.compl_empty] at hK,\n    replace hK := hK.trans (connected_component.iso (induce_univ_iso G)).to_embedding,\n    haveI := Gpc.subsingleton_connected_component,\n    exact nat.not_succ_le_zero _ (nat.le_of_succ_le_succ (fin.embedding_subsingleton hK)), },\n\n  obtain \u27e8L,KL,Lc,inf\u27e9 := exists_saturated_connected_extension Gpc Kn,\n  obtain \u27e8\u03c6,\u03c6h\u27e9 := auts L,\n  let \u03c6L := L.image \u03c6,\n  have \u03c6Lc : (G.induce (\u03c6L : set V)).connected, by\n  { rw [finset.coe_image, \u2190(induce.iso \u03c6).connected_iff], exact Lc, },\n\n  refine \u27e8op L, op (L \u222a \u03c6L), op_hom_of_le KL, op_hom_of_le (finset.subset_union_left _ _), _\u27e9,\n\n  let lol : (G.induce (L : set V)\u1d9c) \u2243g (G.induce (\u03c6L : set V)\u1d9c), by\n  { rw (by simp only [set.image_compl_eq \u03c6.bijective, finset.coe_image] :\n          (\u03c6L : set V)\u1d9c = \u03c6 '' (L : set V)\u1d9c),\n    apply induce.iso, },\n\n  have \u03c6inf : \u2200 C : G.component_compl \u03c6L, C.supp.infinite, by\n  { simp_rw \u2190set.infinite_coe_iff at inf \u22a2,\n    rintro C,\n    rw [\u2190(connected_component.iso lol).right_inv C, equiv.infinite_iff],\n    exact inf ((connected_component.iso lol).symm C),\n    exact (component_compl.supp_equiv _).trans\n      ((connected_component.iso_equiv_supp lol _).symm.trans\n        (component_compl.supp_equiv _).symm), },\n\n  apply @hom_not_injective_of_nicely_arranged V G Gpc _ (op \u03c6L) (op L) ((Kn.mono KL).image \u03c6) \u27e8_, _\u27e9 \u27e8_, _\u27e9\n    (subset_of_connected_disjoint_right \u03c6Lc (finset.disjoint_coe.mpr \u03c6h))\n    (subset_of_connected_disjoint_right Lc (finset.disjoint_coe.mpr \u03c6h.symm)) _,\n  exact (@component_compl.infinite_iff_in_eventual_range V G (op \u03c6L) _).mp (\u03c6inf _),\n  exact (@component_compl.infinite_iff_in_eventual_range V G (op L) _).mp (inf _),\n\n  have eL: G.component_compl_functor.to_eventual_ranges.obj (op L) \u2243\n         G.component_compl_functor.to_eventual_ranges.obj (op \u03c6L), by\n  { simp_rw component_compl_functor_to_eventual_ranges_obj_eq,\n    refine ((equiv.subtype_univ_equiv inf).trans (connected_component.iso lol)).trans (equiv.subtype_univ_equiv \u03c6inf).symm, },\n  have iK: G.component_compl_functor.to_eventual_ranges.obj K \u21aa\n         G.component_compl_functor.to_eventual_ranges.obj (op L), by\n  { refine function.embedding.of_surjective\n      (G.component_compl_functor.to_eventual_ranges.map $ op_hom_of_le KL) _,\n    apply G.component_compl_functor.surjective_to_eventual_ranges,\n    exact G.component_compl_functor_is_mittag_leffler Gpc, },\n  exact (hK.trans iK).trans eL.to_embedding,\nend\n\nlemma Freudenthal_Hopf [Vi : infinite V]\n  (auts : \u2200 K :finset V, \u2203 \u03c6 : G \u2243g G, disjoint K (finset.image \u03c6 K))\n  (many_ends : fin 3 \u21aa G.end) : G.end.infinite :=\nbegin\n  intros finite_ends,\n  let Gccf := G.component_compl_functor,\n  -- Gccf has nonempty sections and nonempty sets at each K\n  haveI := finite_ends.fintype,\n  haveI := fintype.of_equiv G.end (Gccf.to_eventual_ranges_sections_equiv).symm,\n  haveI := component_compl_functor_to_eventual_ranges_fintype Gpc,\n  haveI := \u03bb j, component_compl_functor_to_eventual_ranges_nonempty_of_infinite G Gpc j,\n  -- Gccf is a \"surjective\" system\n  have surj : \u2200 \u2983i j\u2984 (f : i \u27f6 j), function.surjective _ :=\n    functor.surjective_to_eventual_ranges _ (G.component_compl_functor_is_mittag_leffler Gpc),\n  -- All the maps of Gccf to `K` are injective, and `Gccf K` has at least 3 components.\n  obtain \u27e8K,top\u27e9 := Gccf.to_eventual_ranges.eventually_injective surj,\n  let inj' := Gccf.to_eventual_ranges.eval_section_injective_of_eventually_injective top,\n  let inj'' := (many_ends.trans (Gccf.to_eventual_ranges_sections_equiv).symm.to_embedding).trans\n                 \u27e8_, (inj' K (\ud835\udfd9 K))\u27e9,\n  -- But there is a point above `K` from which the `Gccf` is not injective.\n  -- Derive a contradiction\n  obtain \u27e8L,M,KL,LM,LM_not_inj\u27e9 :=\n    hom_not_injective_of_enough_automorphisms_of_many_components Gpc auts K inj'',\n  refine LM_not_inj (@injective.of_comp _ _ _ (Gccf.to_eventual_ranges.map KL) _ _),\n  rw [\u2190types_comp,\u2190functor.map_comp],\n  apply top,\nend\n\nlemma good_automs_of_infinite_transitive\n  [Vi : infinite V] [locally_finite G]\n  (Gpc : G.preconnected)\n  (trans : \u2200 (x y : V), \u2203 \u03c6 : G \u2243g G, \u03c6 x = y) (K :finset V) :\n  \u2203 \u03c6 : G \u2243g G, disjoint K (K.image \u03c6) :=\nbegin\n  sorry,\n  -- assume K\u00a0has diameter m,\n  -- take x in K, and some y at distance \u22652m from x.\nend\n\nend component_compl\nend simple_graph\n#lint", "meta": {"author": "0art0", "repo": "Freudenthal-Hopf", "sha": "1fefe94a6f0432686f21f97e4916efcbb81fec9d", "save_path": "github-repos/lean/0art0-Freudenthal-Hopf", "path": "github-repos/lean/0art0-Freudenthal-Hopf/Freudenthal-Hopf-1fefe94a6f0432686f21f97e4916efcbb81fec9d/src/freudenthal-hopf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36498419359741635}}
{"text": "/-\nCopyright (c) 2022 Dhruv Bhatia. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Dhruv Bhatia, Robert Y. Lewis\n-/\n\nimport tactic.polyrith\nimport data.real.basic\n\n/-!\n\nEach call to `polyrith` makes a call to the SageCell web API at\n<https://sagecell.sagemath.org/>. To avoid making many API calls from CI,\nwe only test this communication in a few tests.\n\nA full test suite is provided at the bottom of the file.\n\n-/\n\n/-!\n## Set up testing infrastructre\n-/\n\nsection tactic\nopen polyrith tactic\n/--\nFor testing purposes, this behaves like `tactic.polyrith`, but takes an extra argument\nrepresenting the expected output from a call to Sage.\nAllows for testing without actually making API calls.\n-/\nmeta def tactic.test_polyrith (only_on : bool) (hyps : list pexpr)\n  (sage_out : json) (expected_args : list string) (expected_out : string) :\n  tactic unit := do\n  (eq_names, m, R, args) \u2190 create_args only_on hyps,\n  guard (args = expected_args) <|>\n    fail!\"expected arguments to Sage: {expected_args}\\nbut produced: {args}\",\n  out \u2190 to_string <$> process_output eq_names m R sage_out,\n  guard (out = expected_out) <|>\n    fail!\"expected final output: {expected_out}\\nbut produced: {out}\"\n\nmeta def format_string_list (input : list string) : format :=\n\"[\" ++ (format.join $ (input.map (\u03bb s, (\"\\\"\" : format) ++ format.of_string s ++ \"\\\"\")).intersperse (\",\" ++ format.line)) ++ \"]\"\n\nsetup_tactic_parser\n\nmeta def tactic.interactive.test_polyrith (restr : parse (tk \"only\")?)\n  (hyps : parse pexpr_list?)\n  (sage_out : string) (expected_args : list string) (expected_out : string) : tactic unit := do\n  some sage_out \u2190 return $ json.parse sage_out,\n  tactic.test_polyrith restr.is_some (hyps.get_or_else []) sage_out expected_args expected_out\n\nmeta def tactic.interactive.test_sage_output (restr : parse (tk \"only\")?)\n  (hyps : parse pexpr_list?) (expected_out : string) : tactic unit := do\n  expected_json \u2190 json.parse expected_out,\n  sleep 10, -- otherwise can lead to weird errors when actively editing code with polyrith calls\n  (eq_names, m, R, args) \u2190 create_args restr.is_some (hyps.get_or_else []),\n  sage_out \u2190 sage_output args,\n  guard (sage_out = expected_json) <|>\n    fail!\"Expected output from Sage: {expected_out}\\nbut produced: {sage_out}\"\n\n/--\nA convenience function. Given a working test, prints the code for a call to `test_sage_output`.\n-/\nmeta def tactic.interactive.create_sage_output_test (restr : parse (tk \"only\")?)\n  (hyps : parse pexpr_list?) : tactic unit := do\n  let hyps := (hyps.get_or_else []),\n  sleep 10, -- otherwise can lead to weird errors when actively editing code with polyrith calls\n  (eq_names, m, R, args) \u2190 create_args restr.is_some hyps,\n  sage_out \u2190 to_string <$> sage_output args,\n  let sage_out := sage_out.fold \"\" (\u03bb s c, s ++ (if c = '\"' then \"\\\\\\\"\" else to_string c)),\n  let onl := if restr.is_some then \"only \" else \"\",\n  let hyps := if hyps = [] then \"\" else to_string hyps,\n  trace!\"test_sage_output {onl}{hyps} \\\"{sage_out}\\\"\"\n\n/--\nA convenience function. Given a working test, prints the code for a call to `test_polyrith`.\n-/\nmeta def tactic.interactive.create_polyrith_test (restr : parse (tk \"only\")?)\n  (hyps : parse pexpr_list?) : tactic unit := do\n  let hyps := (hyps.get_or_else []),\n  sleep 10, -- otherwise can lead to weird errors when actively editing code with polyrith calls\n  (eq_names, m, R, args) \u2190 create_args restr.is_some hyps,\n  sage_out \u2190 sage_output args,\n  out \u2190 to_string <$> process_output eq_names m R sage_out,\n  let out := out.fold \"\" (\u03bb s c, s ++ (if c = '\"' then \"\\\\\\\"\" else to_string c)),\n  let sage_out := (to_string sage_out).fold \"\"\n    (\u03bb s c, s ++ (if c = '\"' then \"\\\\\\\"\" else to_string c)),\n  let argstring := format_string_list args,\n  let onl := if restr.is_some then \"only \" else \"\",\n  let hyps := if hyps = [] then \"\" else to_string hyps,\n  let trf := format.nest 2 $ format!\"test_polyrith {onl}{hyps} \\n\\\"{sage_out}\\\"\\n{argstring}\\n\\\"{out}\\\"\",\n  trace!\"Try this: {trf}\"\n\n\nend tactic\n\n/-!\n## SageCell communcation tests\n-/\n\nexample (x y : \u211a) (h1 : x*y + 2*x = 1) (h2 : x = y) :\n  x*y = -2*y + 1 :=\nbegin\n  test_sage_output \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.const -2/1)\\\"],\\\"success\\\":true}\",\n  linear_combination h1 - 2 * h2\nend\n\nexample (w x y z : \u211d) (h1 : x + 2.1*y + 2*z = 2) (h2 : x + 8*z + 5*w = -6.5)\n    (h3 : x + y + 5*z + 5*w = 3) :\n  x + 2.2*y + 2*z - 5*w = -8.5 :=\nbegin\n  test_sage_output \"{\\\"data\\\":[\\\"(poly.const 2/1)\\\",\\\"(poly.const 1/1)\\\",\\\"(poly.const -2/1)\\\"],\\\"success\\\":true}\",\n  linear_combination 2 * h1 + h2 - 2 * h3\nend\n\n\n\n/-! ### Standard Cases over \u2124, \u211a, and \u211d -/\n\nexample (x y : \u2124) (h1 : 3*x + 2*y = 10):\n  3*x + 2*y = 10 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"2\",\n  \"[(((3 * var0) + (2 * var1)) - 10)]\",\n  \"(((3 * var0) + (2 * var1)) - 10)\"]\n  \"linear_combination h1\"\n\nexample (x y : \u211a) (h1 : x*y + 2*x = 1) (h2 : x = y) :\n  x*y = -2*y + 1 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.const -2/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"2\",\n  \"[(((var0 * var1) + (2 * var0)) - 1), (var0 - var1)]\",\n  \"((var0 * var1) - ((-2 * var1) + 1))\"]\n  \"linear_combination h1 - 2 * h2\"\n\nexample (x y : \u211d) (h1 : x + 2 = -3) (h2 : y = 10) :\n  -y + 2*x + 4 = -16 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 2/1)\\\",\\\"(poly.const -1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"real\",\n  \"2\",\n  \"[((var1 + 2) - -3), (var0 - 10)]\",\n  \"(((-var0 + (2 * var1)) + 4) - -16)\"]\n  \"linear_combination 2 * h1 - h2\"\n\nexample (x y z : \u211d) (ha : x + 2*y - z = 4) (hb : 2*x + y + z = -2)\n    (hc : x + 2*y + z = 2) :\n  -3*x - 3*y - 4*z = 2 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.const -1/1)\\\",\\\"(poly.const -2/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"real\",\n  \"3\",\n  \"[(((var0 + (2 * var1)) - var2) - 4), ((((2 * var0) + var1) + var2) - -2), (((var0 + (2 * var1)) + var2) - 2)]\",\n  \"((((-3 * var0) - (3 * var1)) - (4 * var2)) - 2)\"]\n  \"linear_combination ha - hb - 2 * hc\"\n\nexample (w x y z : \u211d) (h1 : x + 2.1*y + 2*z = 2) (h2 : x + 8*z + 5*w = -6.5)\n    (h3 : x + y + 5*z + 5*w = 3) :\n  x + 2.2*y + 2*z - 5*w = -8.5 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 2/1)\\\",\\\"(poly.const 1/1)\\\",\\\"(poly.const -2/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"real\",\n  \"4\",\n  \"[(((var0 + (21/10 * var1)) + (2 * var2)) - 2), (((var0 + (8 * var2)) + (5 * var3)) - -13/2), ((((var0 + var1) + (5 * var2)) + (5 * var3)) - 3)]\",\n  \"((((var0 + (11/5 * var1)) + (2 * var2)) - (5 * var3)) - -17/2)\"]\n  \"linear_combination 2 * h1 + h2 - 2 * h3\"\n\nexample (a b c d : \u211a) (h1 : a = 4) (h2 : 3 = b) (h3 : c*3 = d) (h4 : -d = a) :\n  2*a - 3 + 9*c + 3*d = 8 - b + 3*d - 3*a :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 2/1)\\\",\\\"(poly.const -1/1)\\\",\\\"(poly.const 3/1)\\\",\\\"(poly.const -3/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"4\",\n  \"[(var0 - 4), (3 - var3), ((var1 * 3) - var2), (-var2 - var0)]\",\n  \"(((((2 * var0) - 3) + (9 * var1)) + (3 * var2)) - (((8 - var3) + (3 * var2)) - (3 * var0)))\"]\n  \"linear_combination 2 * h1 - h2 + 3 * h3 - 3 * h4\"\n\n/-! ### Case with ambiguous identifiers-/\n\nexample (\u00abdef evil\u00bb y : \u2124) (h1 : 3*\u00abdef evil\u00bb + 2*y = 10):\n  3*\u00abdef evil\u00bb + 2*y = 10 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"2\",\n  \"[(((3 * var0) + (2 * var1)) - 10)]\",\n  \"(((3 * var0) + (2 * var1)) - 10)\"]\n  \"linear_combination h1\"\n\nexample (\u00ab\u00a5\u00bb y : \u2124) (h1 : 3*\u00ab\u00a5\u00bb + 2*y = 10):\n  \u00ab\u00a5\u00bb * (3*\u00ab\u00a5\u00bb + 2*y) = 10 * \u00ab\u00a5\u00bb :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.var 0)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"2\",\n  \"[(((3 * var0) + (2 * var1)) - 10)]\",\n  \"((var0 * ((3 * var0) + (2 * var1))) - (10 * var0))\"]\n  \"linear_combination \u00ab\u00a5\u00bb * h1\"\n\n/-! ### Cases with arbitrary coefficients -/\n\nexample (a b : \u2124) (h : a = b) :\n  a * a = a * b :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.var 0)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"2\",\n  \"[(var0 - var1)]\",\n  \"((var0 * var0) - (var0 * var1))\"]\n  \"linear_combination a * h\"\n\nexample (a b c : \u2124) (h : a = b) :\n  a * c = b * c :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.var 1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"3\",\n  \"[(var0 - var2)]\",\n  \"((var0 * var1) - (var2 * var1))\"]\n  \"linear_combination c * h\"\n\nexample (a b c : \u2124) (h1 : a = b) (h2 : b = 1) :\n  c * a + b = c * b + 1 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.var 0)\\\",\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"3\",\n  \"[(var1 - var2), (var2 - 1)]\",\n  \"(((var0 * var1) + var2) - ((var0 * var2) + 1))\"]\n  \"linear_combination c * h1 + h2\"\n\nexample (x y : \u211a) (h1 : x + y = 3) (h2 : 3*x = 7) :\n  x*x*y + y*x*y + 6*x = 3*x*y + 14 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.mul (poly.var 0) (poly.var 1))\\\",\\\"(poly.const 2/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"2\",\n  \"[((var0 + var1) - 3), ((3 * var0) - 7)]\",\n  \"(((((var0 * var0) * var1) + ((var1 * var0) * var1)) + (6 * var0)) - (((3 * var0) * var1) + 14))\"]\n  \"linear_combination x * y * h1 + 2 * h2\"\n\nexample (x y z w : \u211a) (hzw : z = w) : x*z + 2*y*z = x*w + 2*y*w :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.add (poly.var 0) (poly.mul (poly.const 2/1) (poly.var 2)))\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"4\",\n  \"[(var1 - var3)]\",\n  \"(((var0 * var1) + ((2 * var2) * var1)) - ((var0 * var3) + ((2 * var2) * var3)))\"]\n  \"linear_combination (x + 2 * y) * hzw\"\n\n/-! ### Cases with non-hypothesis inputs/input restrictions -/\n\nexample (a b : \u211d) (ha : 2*a = 4) (hab : 2*b = a - b) (hignore : 3 = a + b) :\n  b = 2 / 3 :=\nby test_polyrith only [ha, hab]\n  \"{\\\"data\\\":[\\\"(poly.const 1/6)\\\",\\\"(poly.const 1/3)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"real\",\n  \"2\",\n  \"[((2 * var1) - 4), ((2 * var0) - (var1 - var0))]\",\n  \"(var0 - 2/3)\"]\n  \"linear_combination ha / 6 + hab / 3\"\n\nconstant term : \u2200 a b : \u211a, a + b = 0\n\nexample (a b c d : \u211a) (h : a + b = 0) (h2: b + c = 0): a + b + c + d = 0 :=\nby test_polyrith only [term c d, h]\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"4\",\n  \"[((var2 + var3) - 0), ((var0 + var1) - 0)]\",\n  \"((((var0 + var1) + var2) + var3) - 0)\"]\n  \"linear_combination term c d + h\"\n\nconstants (qc : \u211a) (hqc : qc = 2*qc)\n\nexample (a b : \u211a) (h : \u2200 p q : \u211a, p = q) : 3*a + qc = 3*b + 2*qc :=\nby test_polyrith [h a b, hqc]\n  \"{\\\"data\\\":[\\\"(poly.const 3/1)\\\",\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"3\",\n  \"[(var0 - var2), (var1 - (2 * var1))]\",\n  \"(((3 * var0) + var1) - ((3 * var2) + (2 * var1)))\"]\n  \"linear_combination 3 * h a b + hqc\"\n\nconstant bad (q : \u211a) : q = 0\n\nexample (a b : \u211a) : a + b^3 = 0 :=\nby test_polyrith [bad a, bad (b^2)]\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.var 1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"2\",\n  \"[(var0 - 0), ((var1 ^ 2) - 0)]\",\n  \"((var0 + (var1 ^ 3)) - 0)\"]\n  \"linear_combination bad a + b * bad (b ^ 2)\"\n\n/-! ### Case over arbitrary field/ring -/\n\nexample {\u03b1} [h : comm_ring \u03b1] {a b c d e f : \u03b1} (h1 : a*d = b*c) (h2 : c*f = e*d) :\n  c * (a*f - b*e) = 0 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.var 4)\\\",\\\"(poly.var 1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"\u03b1\",\n  \"6\",\n  \"[((var1 * var5) - (var3 * var0)), ((var0 * var2) - (var4 * var5))]\",\n  \"((var0 * ((var1 * var2) - (var3 * var4))) - 0)\"]\n  \"linear_combination e * h1 + a * h2\"\n\nexample {K : Type*} [field K] [invertible 2] [invertible 3]\n  {\u03c9 p q r s t x: K} (hp_nonzero : p \u2260 0) (hr : r ^ 2 = q ^ 2 + p ^ 3) (hs3 : s ^ 3 = q + r)\n  (ht : t * s = p) (x : K) (H : 1 + \u03c9 + \u03c9 ^ 2 = 0) :\n  x ^ 3 + 3 * p * x - 2 * q =\n    (x - (s - t)) * (x - (s * \u03c9 - t * \u03c9 ^ 2)) * (x - (s * \u03c9 ^ 2 - t * \u03c9)) :=\nbegin\n  have hs_nonzero : s \u2260 0,\n  { contrapose! hp_nonzero with hs_nonzero,\n    test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 0/1)\\\",\\\"(poly.const 0/1)\\\",\\\"(poly.const -1/1)\\\",\\\"(poly.const 0/1)\\\",\\\"(poly.var 4)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"K\",\n  \"6\",\n  \"[((var1 ^ 2) - ((var2 ^ 2) + (var0 ^ 3))), ((var3 ^ 3) - (var2 + var1)), ((var4 * var3) - var0), (((1 + var5) + (var5 ^ 2)) - 0), (var3 - 0)]\",\n  \"(var0 - 0)\"]\n  \"linear_combination -ht + t * hs_nonzero\"},\n  have H' : 2 * q = s ^ 3 - t ^ 3,\n  { rw \u2190 mul_left_inj' (pow_ne_zero 3 hs_nonzero),\n    test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const -1/1)\\\",\\\"(poly.sub (poly.add (poly.neg (poly.pow (poly.var 1) 3)) (poly.var 0)) (poly.var 3))\\\",\\\"(poly.add (poly.add (poly.mul (poly.pow (poly.var 1) 2) (poly.pow (poly.var 2) 2)) (poly.mul (poly.mul (poly.var 1) (poly.var 2)) (poly.var 4))) (poly.pow (poly.var 4) 2))\\\",\\\"(poly.const 0/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"K\",\n  \"6\",\n  \"[((var3 ^ 2) - ((var0 ^ 2) + (var4 ^ 3))), ((var1 ^ 3) - (var0 + var3)), ((var2 * var1) - var4), (((1 + var5) + (var5 ^ 2)) - 0)]\",\n  \"(((2 * var0) * (var1 ^ 3)) - (((var1 ^ 3) - (var2 ^ 3)) * (var1 ^ 3)))\"]\n  \"linear_combination -hr + (-s ^ 3 + q - r) * hs3 + (s ^ 2 * t ^ 2 + s * t * p + p ^ 2) * ht\"},\n  test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 0/1)\\\",\\\"(poly.const 0/1)\\\",\\\"(poly.add (poly.add (poly.sub (poly.add (poly.add (poly.sub (poly.add (poly.sub (poly.mul (poly.var 0) (poly.pow (poly.var 5) 4)) (poly.mul (poly.var 3) (poly.pow (poly.var 5) 4))) (poly.mul (poly.var 4) (poly.pow (poly.var 5) 4))) (poly.mul (poly.var 3) (poly.pow (poly.var 5) 3))) (poly.mul (poly.var 4) (poly.pow (poly.var 5) 3))) (poly.mul (poly.mul (poly.const 3/1) (poly.var 0)) (poly.pow (poly.var 5) 2))) (poly.mul (poly.var 3) (poly.pow (poly.var 5) 2))) (poly.mul (poly.var 4) (poly.pow (poly.var 5) 2))) (poly.mul (poly.mul (poly.const 2/1) (poly.var 0)) (poly.var 5)))\\\",\\\"(poly.add (poly.sub (poly.add (poly.sub (poly.sub (poly.add (poly.add (poly.sub (poly.add (poly.sub (poly.sub (poly.add (poly.neg (poly.mul (poly.mul (poly.var 0) (poly.pow (poly.var 3) 2)) (poly.var 5))) (poly.mul (poly.pow (poly.var 3) 3) (poly.var 5))) (poly.mul (poly.mul (poly.var 0) (poly.pow (poly.var 4) 2)) (poly.var 5))) (poly.mul (poly.pow (poly.var 4) 3) (poly.var 5))) (poly.mul (poly.mul (poly.var 0) (poly.var 1)) (poly.pow (poly.var 5) 2))) (poly.mul (poly.mul (poly.var 1) (poly.var 3)) (poly.pow (poly.var 5) 2))) (poly.mul (poly.mul (poly.var 1) (poly.var 4)) (poly.pow (poly.var 5) 2))) (poly.mul (poly.pow (poly.var 0) 2) (poly.var 3))) (poly.pow (poly.var 3) 3)) (poly.mul (poly.pow (poly.var 0) 2) (poly.var 4))) (poly.pow (poly.var 4) 3)) (poly.mul (poly.mul (poly.var 0) (poly.var 1)) (poly.var 5))) (poly.mul (poly.mul (poly.const 3/1) (poly.var 0)) (poly.var 1)))\\\",\\\"(poly.const -1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"K\",\n  \"7\",\n  \"[((var6 ^ 2) - ((var2 ^ 2) + (var1 ^ 3))), ((var3 ^ 3) - (var2 + var6)), ((var4 * var3) - var1), (((1 + var5) + (var5 ^ 2)) - 0), ((2 * var2) - ((var3 ^ 3) - (var4 ^ 3)))]\",\n  \"((((var0 ^ 3) + ((3 * var1) * var0)) - (2 * var2)) - (((var0 - (var3 - var4)) * (var0 - ((var3 * var5) - (var4 * (var5 ^ 2))))) * (var0 - ((var3 * (var5 ^ 2)) - (var4 * var5)))))\"]\n  \"linear_combination (x * \u03c9 ^ 4 - s * \u03c9 ^ 4 + t * \u03c9 ^ 4 - s * \u03c9 ^ 3 + t * \u03c9 ^ 3 + 3 * x * \u03c9 ^ 2 - s * \u03c9 ^ 2 +\n      t * \u03c9 ^ 2 +\n    2 * x * \u03c9) * ht + (-(x * s ^ 2 * \u03c9) + s ^ 3 * \u03c9 - x * t ^ 2 * \u03c9 - t ^ 3 * \u03c9 + x * p * \u03c9 ^ 2 - p * s * \u03c9 ^ 2 +\n                p * t * \u03c9 ^ 2 +\n              x ^ 2 * s -\n            s ^ 3 -\n          x ^ 2 * t +\n        t ^ 3 -\n      x * p * \u03c9 +\n    3 * x * p) * H - H'\"\nend\n\n\n/-! ## Degenerate cases -/\n\nexample {K : Type*} [field K] [char_zero K] {s : K} (hs : 3 * s + 1 = 4) : s = 1 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/3)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"K\",\n  \"1\",\n  \"[(((3 * var0) + 1) - 4)]\",\n  \"(var0 - 1)\"]\n  \"linear_combination hs / 3\"\n\nexample {x : \u2124} (h1 : x + 4 = 2) : x = -2 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"1\",\n  \"[((var0 + 4) - 2)]\",\n  \"(var0 - -2)\"]\n  \"linear_combination h1\"\n\nexample {w : \u211a} (h1 : 3 * w + 1 = 4) : w = 1 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/3)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"1\",\n  \"[(((3 * var0) + 1) - 4)]\",\n  \"(var0 - 1)\"]\n  \"linear_combination h1 / 3\"\n\nexample {x : \u2124} (h1 : 2 * x + 3 = x) : x = -3 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"1\",\n  \"[(((2 * var0) + 3) - var0)]\",\n  \"(var0 - -3)\"]\n  \"linear_combination h1\"\n\nexample {c : \u211a} (h1 : 4 * c + 1 = 3 * c - 2) : c = -3 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"rat\",\n  \"1\",\n  \"[(((4 * var0) + 1) - ((3 * var0) - 2))]\",\n  \"(var0 - -3)\"]\n  \"linear_combination h1\"\n\nexample (z : \u2124) (h1 : z + 1 = 2) (h2 : z + 2 = 2) : (1 : \u2124) = 2 :=\nby test_polyrith\n  \"{\\\"data\\\":[\\\"(poly.const 1/1)\\\",\\\"(poly.const -1/1)\\\"],\\\"success\\\":true}\"\n  [\"ff\",\n  \"int\",\n  \"1\",\n  \"[((var0 + 1) - 2), ((var0 + 2) - 2)]\",\n  \"(1 - 2)\"]\n  \"linear_combination h1 - h2\"\n\n\n-- We comment the following tests so that we don't overwhelm the SageCell API.\n\n\n\n\n\n/-\n\n/-! ### Standard Cases over \u2124, \u211a, and \u211d -/\n\nexample (x y : \u2124) (h1 : 3*x + 2*y = 10):\n  3*x + 2*y = 10 :=\nby polyrith\n\nexample (x y : \u211a) (h1 : x*y + 2*x = 1) (h2 : x = y) :\n  x*y = -2*y + 1 :=\nby polyrith\n\nexample (x y : \u211d) (h1 : x + 2 = -3) (h2 : y = 10) :\n  -y + 2*x + 4 = -16 :=\nby polyrith\n\nexample (x y z : \u211d) (ha : x + 2*y - z = 4) (hb : 2*x + y + z = -2)\n    (hc : x + 2*y + z = 2) :\n  -3*x - 3*y - 4*z = 2 :=\nby polyrith\n\nexample (w x y z : \u211d) (h1 : x + 2.1*y + 2*z = 2) (h2 : x + 8*z + 5*w = -6.5)\n    (h3 : x + y + 5*z + 5*w = 3) :\n  x + 2.2*y + 2*z - 5*w = -8.5 :=\nby polyrith\n\nexample (a b c d : \u211a) (h1 : a = 4) (h2 : 3 = b) (h3 : c*3 = d) (h4 : -d = a) :\n  2*a - 3 + 9*c + 3*d = 8 - b + 3*d - 3*a :=\nby polyrith\n\n/-! ### Case with ambiguous identifiers-/\n\nexample (\u00abdef evil\u00bb y : \u2124) (h1 : 3*\u00abdef evil\u00bb + 2*y = 10):\n  3*\u00abdef evil\u00bb + 2*y = 10 :=\nby polyrith\n\nexample (\u00ab\u00a5\u00bb y : \u2124) (h1 : 3*\u00ab\u00a5\u00bb + 2*y = 10):\n  \u00ab\u00a5\u00bb * (3*\u00ab\u00a5\u00bb + 2*y) = 10 * \u00ab\u00a5\u00bb :=\nby polyrith\n\n/-! ### Cases with arbitrary coefficients -/\n\nexample (a b : \u2124) (h : a = b) :\n  a * a = a * b :=\nby polyrith\n\nexample (a b c : \u2124) (h : a = b) :\n  a * c = b * c :=\nby polyrith\n\nexample (a b c : \u2124) (h1 : a = b) (h2 : b = 1) :\n  c * a + b = c * b + 1 :=\nby polyrith\n\nexample (x y : \u211a) (h1 : x + y = 3) (h2 : 3*x = 7) :\n  x*x*y + y*x*y + 6*x = 3*x*y + 14 :=\nby polyrith\n\nexample (x y z w : \u211a) (hzw : z = w) : x*z + 2*y*z = x*w + 2*y*w :=\nby polyrith\n\n\n/-! ### Cases with non-hypothesis inputs/input restrictions -/\n\nexample (a b : \u211d) (ha : 2*a = 4) (hab : 2*b = a - b) (hignore : 3 = a + b) :\n  b = 2 / 3 :=\nby polyrith only [ha, hab]\n\n-- constant term : \u2200 a b : \u211a, a + b = 0\n\nexample (a b c d : \u211a) (h : a + b = 0) (h2: b + c = 0): a + b + c + d = 0 :=\nby polyrith only [term c d, h]\n\n-- constants (qc : \u211a) (hqc : qc = 2*qc)\n\nexample (a b : \u211a) (h : \u2200 p q : \u211a, p = q) : 3*a + qc = 3*b + 2*qc :=\nby polyrith [h a b, hqc]\n\n-- constant bad (q : \u211a) : q = 0\n\nexample (a b : \u211a) : a + b^3 = 0 :=\nby polyrith [bad a, bad (b^2)]\n\n/-! ### Case over arbitrary field/ring -/\n\nexample {\u03b1} [h : comm_ring \u03b1] {a b c d e f : \u03b1} (h1 : a*d = b*c) (h2 : c*f = e*d) :\n  c * (a*f - b*e) = 0 :=\nby polyrith\n\nexample {K : Type*} [field K] [invertible 2] [invertible 3]\n  {\u03c9 p q r s t x: K} (hp_nonzero : p \u2260 0) (hr : r ^ 2 = q ^ 2 + p ^ 3) (hs3 : s ^ 3 = q + r)\n  (ht : t * s = p) (x : K) (H : 1 + \u03c9 + \u03c9 ^ 2 = 0) :\n  x ^ 3 + 3 * p * x - 2 * q =\n    (x - (s - t)) * (x - (s * \u03c9 - t * \u03c9 ^ 2)) * (x - (s * \u03c9 ^ 2 - t * \u03c9)) :=\nbegin\n  have hs_nonzero : s \u2260 0,\n  { contrapose! hp_nonzero with hs_nonzero,\n    polyrith,\n     },\n  have H' : 2 * q = s ^ 3 - t ^ 3,\n  { rw \u2190 mul_left_inj' (pow_ne_zero 3 hs_nonzero),\n    polyrith,},\n  polyrith,\nend\n\n/-!\n### With trace enabled\nHere, the tactic will trace the command that gets sent to sage,\nand so the tactic will not prove the goal. `linear_combination`\nis called manually to prevent errors.\n-/\n\nset_option trace.polyrith true\n\nexample (x y : \u211d) (h1 : x + 2 = -3) (h2 : y = 10) :\n  -y + 2*x + 4 = -16 :=\nbegin\n  polyrith,\n  linear_combination 2 * h1 - h2,\nend\n\nexample (a b c : \u2124) (h1 : a = b) (h2 : b = 1) :\n  c * a + b = c * b + 1 :=\nbegin\n  polyrith,\n  linear_combination c * h1 + h2,\nend\n\nexample (a b c d : \u211a) (h : a + b = 0) (h2: b + c = 0): a + b + c + d = 0 :=\nbegin\n  polyrith only [term c d, h],\n  linear_combination term c d + h,\nend\n\nexample (a b : \u211a) (h : \u2200 p q : \u211a, p = q) : 3*a + qc = 3*b + 2*qc :=\nbegin\n  polyrith [h a b, hqc],\n  linear_combination 3 * h a b + hqc,\nend\n-/\n\n\n-- the following can be uncommented to regenerate the tests above.\n\n/-\n\n\n/-! ### Standard Cases over \u2124, \u211a, and \u211d -/\n\nexample (x y : \u2124) (h1 : 3*x + 2*y = 10):\n  3*x + 2*y = 10 :=\nby create_polyrith_test\n\nexample (x y : \u211a) (h1 : x*y + 2*x = 1) (h2 : x = y) :\n  x*y = -2*y + 1 :=\nby create_polyrith_test\n\nexample (x y : \u211d) (h1 : x + 2 = -3) (h2 : y = 10) :\n  -y + 2*x + 4 = -16 :=\nby create_polyrith_test\n\nexample (x y z : \u211d) (ha : x + 2*y - z = 4) (hb : 2*x + y + z = -2)\n    (hc : x + 2*y + z = 2) :\n  -3*x - 3*y - 4*z = 2 :=\nby create_polyrith_test\n\nexample (w x y z : \u211d) (h1 : x + 2.1*y + 2*z = 2) (h2 : x + 8*z + 5*w = -6.5)\n    (h3 : x + y + 5*z + 5*w = 3) :\n  x + 2.2*y + 2*z - 5*w = -8.5 :=\nby create_polyrith_test\n\nexample (a b c d : \u211a) (h1 : a = 4) (h2 : 3 = b) (h3 : c*3 = d) (h4 : -d = a) :\n  2*a - 3 + 9*c + 3*d = 8 - b + 3*d - 3*a :=\nby create_polyrith_test\n\n/-! ### Case with ambiguous identifiers-/\n\nexample (\u00abdef evil\u00bb y : \u2124) (h1 : 3*\u00abdef evil\u00bb + 2*y = 10):\n  3*\u00abdef evil\u00bb + 2*y = 10 :=\nby create_polyrith_test\n\nexample (\u00ab\u00a5\u00bb y : \u2124) (h1 : 3*\u00ab\u00a5\u00bb + 2*y = 10):\n  \u00ab\u00a5\u00bb * (3*\u00ab\u00a5\u00bb + 2*y) = 10 * \u00ab\u00a5\u00bb :=\nby create_polyrith_test\n\n/-! ### Cases with arbitrary coefficients -/\n\nexample (a b : \u2124) (h : a = b) :\n  a * a = a * b :=\nby create_polyrith_test\n\nexample (a b c : \u2124) (h : a = b) :\n  a * c = b * c :=\nby create_polyrith_test\n\nexample (a b c : \u2124) (h1 : a = b) (h2 : b = 1) :\n  c * a + b = c * b + 1 :=\nby create_polyrith_test\n\nexample (x y : \u211a) (h1 : x + y = 3) (h2 : 3*x = 7) :\n  x*x*y + y*x*y + 6*x = 3*x*y + 14 :=\nby create_polyrith_test\n\nexample (x y z w : \u211a) (hzw : z = w) : x*z + 2*y*z = x*w + 2*y*w :=\nby create_polyrith_test\n\n/-! ### Cases with non-hypothesis inputs/input restrictions -/\n\nexample (a b : \u211d) (ha : 2*a = 4) (hab : 2*b = a - b) (hignore : 3 = a + b) :\n  b = 2 / 3 :=\nby create_polyrith_test only [ha, hab]\n\nconstant term : \u2200 a b : \u211a, a + b = 0\n\nexample (a b c d : \u211a) (h : a + b = 0) (h2: b + c = 0): a + b + c + d = 0 :=\nby create_polyrith_test only [term c d, h]\n\nconstants (qc : \u211a) (hqc : qc = 2*qc)\n\nexample (a b : \u211a) (h : \u2200 p q : \u211a, p = q) : 3*a + qc = 3*b + 2*qc :=\nby create_polyrith_test [h a b, hqc]\n\nconstant bad (q : \u211a) : q = 0\n\nexample (a b : \u211a) : a + b^3 = 0 :=\nby create_polyrith_test [bad a, bad (b^2)]\n\n/-! ### Case over arbitrary field/ring -/\n\nexample {\u03b1} [h : comm_ring \u03b1] {a b c d e f : \u03b1} (h1 : a*d = b*c) (h2 : c*f = e*d) :\n  c * (a*f - b*e) = 0 :=\nby create_polyrith_test\n\nexample {K : Type*} [field K] [invertible 2] [invertible 3]\n  {\u03c9 p q r s t x: K} (hp_nonzero : p \u2260 0) (hr : r ^ 2 = q ^ 2 + p ^ 3) (hs3 : s ^ 3 = q + r)\n  (ht : t * s = p) (x : K) (H : 1 + \u03c9 + \u03c9 ^ 2 = 0) :\n  x ^ 3 + 3 * p * x - 2 * q =\n    (x - (s - t)) * (x - (s * \u03c9 - t * \u03c9 ^ 2)) * (x - (s * \u03c9 ^ 2 - t * \u03c9)) :=\nbegin\n  have hs_nonzero : s \u2260 0,\n  { contrapose! hp_nonzero with hs_nonzero,\n    create_polyrith_test },\n  have H' : 2 * q = s ^ 3 - t ^ 3,\n  { rw \u2190 mul_left_inj' (pow_ne_zero 3 hs_nonzero),\n    create_polyrith_test },\n  create_polyrith_test\nend\n\n\n/-! ## Degenerate cases -/\n\nexample {K : Type*} [field K] [char_zero K] {s : K} (hs : 3 * s + 1 = 4) : s = 1 :=\nby create_polyrith_test\n\nexample {x : \u2124} (h1 : x + 4 = 2) : x = -2 :=\nby create_polyrith_test\n\nexample {w : \u211a} (h1 : 3 * w + 1 = 4) : w = 1 :=\nby create_polyrith_test\n\nexample {x : \u2124} (h1 : 2 * x + 3 = x) : x = -3 :=\nby create_polyrith_test\n\nexample {c : \u211a} (h1 : 4 * c + 1 = 3 * c - 2) : c = -3 :=\nby create_polyrith_test\n\nexample (z : \u2124) (h1 : z + 1 = 2) (h2 : z + 2 = 2) : (1 : \u2124) = 2 :=\nby create_polyrith_test\n\n\n-/\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/polyrith.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.36498419359741635}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.oracle_spec\n\n/-!\n# Oracle Query Log\n\nThis file efines a simple definition of a log for queries based on continuations.\nThis definition is chosen over a list of sigma types to give simple type equalities.\n-/\n\n/-- Data type representing a log of oracle queries for a given `oracle_spec`.\n  Represented as a list of query inputs and outputs, indexed by the indexing set in the spec -/\ndef query_log (spec : oracle_spec) : Type :=\n\u03a0 (i : spec.\u03b9), list (spec.domain i \u00d7 spec.range i)\n\nnamespace query_log\n\nopen oracle_spec \n\nvariables {spec : oracle_spec} (log : query_log spec)\n  (i j : spec.\u03b9) (t : spec.domain i) (u : spec.range i)\n\n@[ext] lemma ext {spec : oracle_spec} {log log' : query_log spec}\n  (h : \u2200 (i : spec.\u03b9), log i = log' i) : log = log' := funext h\n\nsection init\n\n/-- Empty query log, with no entries for any of the oracles in the spec -/\n@[inline, reducible]\ndef init (spec : oracle_spec) : query_log spec := \u03bb i, []\n\n@[simp] lemma init_apply : init spec i = [] := rfl\n\nlemma not_mem_init : (t, u) \u2209 (init spec i) := list.not_mem_nil (t, u)\n\nlemma mem_init_iff_false : (t, u) \u2208 (init spec i) \u2194 false := \nby simp only [iff_false, not_mem_init, not_false_iff]\n\nend init\n\nsection log_query\n\n/-- Given a current query log, return the new log after adding a given oracle query -/\ndef log_query (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i) : query_log spec :=\n\u03bb j, if hi : i = j then hi.rec_on ((t, u) :: (log i)) else log j\n\n@[simp] lemma log_query_apply (i j : spec.\u03b9) (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u) j = if hi : i = j then hi.rec_on ((t, u) :: log i) else log j := rfl\n\nlemma log_query_apply_of_index_eq {i j : spec.\u03b9} (hi : i = j) (t : spec.domain i)\n  (u : spec.range i) : (log.log_query i t u) j = hi.rec_on ((t, u) :: log i) :=\ndite_eq_iff.2 (or.inl \u27e8hi, rfl\u27e9)\n\n@[simp] lemma log_query_apply_same_index (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u) i = (t, u) :: (log i) := log_query_apply_of_index_eq log rfl t u\n\nlemma log_query_apply_of_index_ne {i j : spec.\u03b9} (hi : i \u2260 j) (t : spec.domain i)\n  (u : spec.range i) : (log.log_query i t u) j = log j := dite_eq_iff.2 (or.inr \u27e8hi, rfl\u27e9)\n\nlemma nodup_log_query_apply_iff (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i) (j : spec.\u03b9)\n  (hj : (log j).nodup) : (log.log_query i t u j).nodup \u2194 i \u2260 j \u2228 (t, u) \u2209 log i :=\nbegin\n  by_cases hi : i = j,\n  { induction hi,\n    simp only [log_query_apply_same_index, list.nodup_cons, hj, and_true, ne.def,\n      eq_self_iff_true, not_true, false_or] },\n  { simp only [log.log_query_apply_of_index_ne hi, hj, hi, ne.def, not_false_iff, true_or] }\nend\n\nlemma length_log_query_apply (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i) (j : spec.\u03b9) :\n  (log.log_query i t u j).length = (log j).length + ite (i = j) 1 0 :=\nbegin\n  rw [log_query_apply],\n  split_ifs with h,\n  { obtain rfl := h, exact rfl },\n  { exact symm (add_zero _) },\nend\n\nlemma length_log_query_apply_of_index_eq {i : spec.\u03b9} (t : spec.domain i) (u : spec.range i)\n  {j : spec.\u03b9} (h : i = j) : (log.log_query i t u j).length = (log j).length + 1 :=\nby {induction h, simp only [list.length, log_query_apply_same_index] }\n\nlemma length_log_query_apply_same_index (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u i).length = (log i).length + 1 :=\nby simp only [list.length, log_query_apply_same_index]\n\nlemma length_log_query_apply_of_index_ne {i : spec.\u03b9} (t : spec.domain i) (u : spec.range i)\n  {j : spec.\u03b9} (h : i \u2260 j) : (log.log_query i t u j).length = (log j).length :=\nby rw [log_query_apply_of_index_ne log h]\n\nlemma length_apply_le_lenght_log_query_apply (i : spec.\u03b9) (t : spec.domain i) (u : spec.range i)\n  (j : spec.\u03b9) : (log j).length \u2264 (log.log_query i t u j).length :=\nbegin\n  by_cases hij : i = j,\n  { induction hij,\n    simp only [list.length, log_query_apply_same_index, le_add_iff_nonneg_right, zero_le'] },\n  { rw [length_log_query_apply_of_index_ne log t u hij] }\nend\n\nend log_query\n\nsection not_queried\n\n/- Returns whether a specific input has been previously logged. -/\ndef not_queried (i : spec.\u03b9) (t : spec.domain i) : Prop :=\n((log i).find ((=) t \u2218 prod.fst)) = none\n\nlemma not_queried_def (i : spec.\u03b9) (t : spec.domain i) :\n  log.not_queried i t \u2194 (((log i).find ((=) t \u2218 prod.fst)) = none) := iff.rfl\n\ninstance not_queried.decidable (i : spec.\u03b9) (t : spec.domain i) : decidable (log.not_queried i t) :=\noption.decidable_eq ((log i).find ((=) t \u2218 prod.fst)) none\n\n/-- An input hasn't been queried iff it isn't in the log for any possible output -/\nlemma not_queried_iff_not_mem (i : spec.\u03b9) (t : spec.domain i) :\n  log.not_queried i t \u2194 \u2200 (u : spec.range i), (t, u) \u2209 log i :=\nbegin\n  rw [not_queried_def, list.find_eq_none],\n  refine \u27e8\u03bb h u htu, h (t, u) htu rfl, \u03bb h x hx hx', h x.2 (hx'.symm \u25b8 _)\u27e9,\n  rwa [prod.mk.eta],\nend\n\nlemma not_queried_init (i : spec.\u03b9) (t : spec.domain i) : (init spec).not_queried i t :=\nbegin\n  rw [not_queried_def, list.find_eq_none],\n  refine \u03bb x hx _, (not_mem_init i x.1 x.2 hx),\nend\n\nlemma not_queried_log_query (i j : spec.\u03b9) (t : spec.domain i) (t' : spec.domain j)\n  (u : spec.range i) : (log.log_query i t u).not_queried j t' \u2194\n    (log.not_queried j t') \u2227 (if hi : i = j then (hi.rec_on t \u2260 t') else true) :=\nbegin\n  split_ifs with hi,\n  { induction hi,\n    rw [not_queried, log_query_apply_same_index],\n    by_cases ht : t' = t,\n    { induction ht,\n      have : (eq t' \u2218 prod.fst) (t', u) := (function.comp_app (eq t') prod.fst (t', u)).symm \u25b8 rfl,\n      simp only [list.find_cons_of_pos _ this, ne.def, eq_self_iff_true, not_true, and_false] },\n    { have : \u00ac (eq t' \u2218 prod.fst) (t, u) := ht,\n      simp only [list.find_cons_of_neg _ this, list.find_eq_none, not_queried_iff_not_mem,\n        ne.def, ne.symm ht, not_false_iff, and_true, function.comp_app, prod.forall],\n      exact \u27e8\u03bb h u hu, h t' u hu rfl, \u03bb h t'' u' htu' ht', h u' $ ht'.symm \u25b8 htu'\u27e9 } },\n  { simp only [not_queried, log_query_apply_of_index_ne log hi, and_true] }\nend\n\nlemma not_queried_log_query_of_index_eq {i j : spec.\u03b9} (hi : i = j)\n  (t : spec.domain i) (t' : spec.domain j) (u : spec.range i) :\n  (log.log_query i t u).not_queried j t' \u2194 (log.not_queried j t') \u2227 (hi.rec_on t \u2260 t') :=\n(log.not_queried_log_query i j t t' u).trans (by rw [dif_pos hi])\n\nlemma not_queried_log_query_same_index (i : spec.\u03b9)\n  (t t' : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).not_queried i t' \u2194 (log.not_queried i t') \u2227 (t \u2260 t') :=\nlog.not_queried_log_query_of_index_eq rfl t t' u\n\nlemma not_queried_log_query_of_index_ne {i j : spec.\u03b9} (hi : i \u2260 j)\n  (t : spec.domain i) (t' : spec.domain j) (u : spec.range i) :\n  (log.log_query i t u).not_queried j t' \u2194 log.not_queried j t' :=\n(log.not_queried_log_query i j t t' u).trans (by rw [dif_neg hi, and_true])\n\nend not_queried\n\nsection map_at_index\n\n/-- Apply a mapping function to the log corresponding to a particular index\n  TODO: I think a lot of the above functions can use this as a helper -/\ndef map_at_index (i : spec.\u03b9)\n  (f : list (spec.domain i \u00d7 spec.range i) \u2192 list (spec.domain i \u00d7 spec.range i)) :\n  query_log spec :=\n\u03bb j, if hi : i = j then hi.rec_on (f $ log i) else (log j)\n\nvariables (f : list (spec.domain i \u00d7 spec.range i) \u2192 list (spec.domain i \u00d7 spec.range i))\n\n@[simp]\nlemma map_at_index_apply : log.map_at_index i f j =\n  if hi : i = j then hi.rec_on (f $ log i) else log j := rfl\n\nlemma map_at_index_apply_of_index_eq (h : i = j) : log.map_at_index i f j = h.rec_on (f $ log i) :=\nby simp only [h, map_at_index_apply, dif_pos]\n\nlemma map_at_index_apply_same_index : log.map_at_index i f i = f (log i) :=\nby simp only [map_at_index_apply, eq_self_iff_true, dite_eq_ite, if_true]\n\nlemma map_at_index_apply_of_index_ne (h : i \u2260 j) : log.map_at_index i f j = log j :=\nby simp only [h, not_false_iff, map_at_index_apply, dif_neg]\n\n@[simp]\nlemma map_at_index_init_of_nil_nil (hf : f [] = []) :\n  (init spec).map_at_index i f = init spec :=\nbegin\n  refine ext (\u03bb i, _),\n  simp only [hf, map_at_index_apply, init_apply, dite_eq_right_iff],\n  exact (\u03bb h, by {induction h, refl})\nend\n\n@[simp]\nlemma map_at_index_log_query_of_ne (h : i \u2260 j) (t : spec.domain j) (u : spec.range j) :\n  (log.log_query j t u).map_at_index i f = (log.map_at_index i f).log_query j t u :=\nbegin\n  refine ext (\u03bb k, _),\n  by_cases hi : i = k,\n  { rw [map_at_index_apply_of_index_eq _ i k f hi],\n    by_cases hj : j = k,\n    { exact false.elim (h $ hi.trans hj.symm) },\n    { rw [log_query_apply_of_index_ne _ hj t u, log_query_apply_of_index_ne _ h.symm t u, \n        map_at_index_apply_of_index_eq log _ _ f hi] } },\n  { rw [map_at_index_apply_of_index_ne _ i k f hi],\n    by_cases hj : j = k,\n    { rw [log_query_apply_of_index_eq log hj t u, log_query_apply_of_index_eq _ hj t u,\n        map_at_index_apply_of_index_ne log i j f h] },\n    { rw [log_query_apply_of_index_ne log hj t u, log_query_apply_of_index_ne _ hj t u,\n        map_at_index_apply_of_index_ne _ i k f hi] } }\nend\n\nend map_at_index\n\nsection drop_at_index\n\n/-- Drop the given number of elements from the given log at the specified index. -/\ndef drop_at_index (log : query_log spec) (i : spec.\u03b9) (n : \u2115) : query_log spec :=\nlog.map_at_index i (list.drop n)\n\nvariables (n : \u2115)\n\n@[simp]\nlemma drop_at_index_apply (i j : spec.\u03b9) :\n  (log.drop_at_index i n) j = if i = j then (log j).drop n else log j :=\nbegin\n  simp only [drop_at_index, map_at_index_apply],\n  split_ifs,\n  { induction h,\n    exact rfl },\n  { exact rfl }\nend\n\nlemma drop_at_index_apply_of_index_eq {i j : spec.\u03b9} (h : i = j) :\n  (log.drop_at_index i n) j = (log j).drop n :=\nby simp only [h, drop_at_index_apply, eq_self_iff_true, if_true]\n\n@[simp]\nlemma drop_at_index_apply_same_index (i : spec.\u03b9) :\n  (log.drop_at_index i n) i = (log i).drop n :=\ndrop_at_index_apply_of_index_eq log n rfl\n\nlemma drop_at_index_apply_of_index_ne {i j : spec.\u03b9} (h : i \u2260 j) :\n  (log.drop_at_index i n) j = log j :=\nby simp only [h, drop_at_index_apply, if_false]\n\n@[simp]\nlemma drop_at_index_zero (i : spec.\u03b9) :\n  log.drop_at_index i 0 = log :=\next (\u03bb j, by simp only [list.drop, drop_at_index_apply, if_t_t])\n\n@[simp]\nlemma drop_at_index_init (i : spec.\u03b9) :\n  (init spec).drop_at_index i n = init spec :=\nmap_at_index_init_of_nil_nil i (list.drop n) (list.drop_nil n)\n\n@[simp]\nlemma drop_at_index_succ_log_query (i j : spec.\u03b9) (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).drop_at_index j (n + 1) = \n    if i = j then log.drop_at_index j n\n      else (log.drop_at_index j (n + 1)).log_query i t u :=\nbegin\n  split_ifs,\n  { refine ext (\u03bb k, _),\n    by_cases hj : j = k,\n    { induction h, induction hj,\n      simp only [list.drop, drop_at_index_apply_same_index, log_query_apply_same_index] },\n    { rw [drop_at_index_apply_of_index_ne _ (n + 1) hj,\n        log_query_apply_of_index_ne log (ne_of_eq_of_ne h hj),\n        drop_at_index_apply_of_index_ne _ n hj] } },\n  { exact map_at_index_log_query_of_ne log j i _ (ne.symm h) t u }\nend\n\nend drop_at_index\n\nsection remove_head\n\n/-- remove the head of the index `i` log -/\ndef remove_head (log : query_log spec) (i : spec.\u03b9) :\n  query_log spec :=\n\u03bb j, if i = j then (log j).tail else (log j)\n\n@[simp]\nlemma remove_head_apply (i j : spec.\u03b9) :\n  log.remove_head i j = if i = j then (log j).tail else (log j) :=\nrfl\n\nlemma remove_head_apply_of_index_eq {i j : spec.\u03b9} (hi : i = j) :\n  log.remove_head i j = (log j).tail :=\nif_pos hi\n\n@[simp]\nlemma remove_head_apply_same_index (i : spec.\u03b9) :\n  log.remove_head i i = (log i).tail :=\nlog.remove_head_apply_of_index_eq rfl\n\nlemma remove_head_apply_of_index_ne {i j : spec.\u03b9} (hi : i \u2260 j) :\n  log.remove_head i j = log j :=\nif_neg hi\n\n@[simp]\nlemma remove_head_init (i : spec.\u03b9) : (init spec).remove_head i = init spec :=\next (\u03bb i', if_t_t (i = i') [])\n\nlemma remove_head_log_query (i j : spec.\u03b9)\n  (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).remove_head j =\n    if hi : i = j then log else (log.remove_head j).log_query i t u :=\nbegin\n  split_ifs with hi,\n  { induction hi,\n    refine (ext $ \u03bb k, trans (remove_head_apply _ i k) _),\n    split_ifs with hk,\n    { induction hk,\n      rw [log_query_apply_same_index log, list.tail_cons] },\n    { exact log_query_apply_of_index_ne log hk t u } },\n  { refine (ext $ \u03bb k, _),\n    simp only [remove_head_apply],\n    split_ifs with hj,\n    { induction hj,\n      simp only [log_query_apply_of_index_ne _ hi, remove_head_apply_same_index] },\n    { simp only [log_query_apply, remove_head_apply_of_index_ne _ hj,\n        remove_head_apply_of_index_ne _ (ne.symm hi)] } }\nend\n\nlemma remove_head_log_query_of_index_eq {i j : spec.\u03b9} (hi : i = j)\n  (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).remove_head j = log :=\ntrans (log.remove_head_log_query i j t u) (if_pos hi)\n\n@[simp]\nlemma remove_head_log_query_of_same_index (i : spec.\u03b9)\n  (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).remove_head i = log :=\nlog.remove_head_log_query_of_index_eq rfl t u\n\nlemma remove_head_log_query_of_index_ne {i j : spec.\u03b9} (hi : i \u2260 j)\n  (t : spec.domain i) (u : spec.range i) :\n  (log.log_query i t u).remove_head j = (log.remove_head j).log_query i t u :=\ntrans (log.remove_head_log_query i j t u) (if_neg hi)\n\nend remove_head\n\nend query_log\n", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/simulation_semantics/constructions/logging/query_log/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632683808533, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.3649841872166913}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Yury G. Kudryashov\n-/\nimport logic.function.basic\nimport tactic.basic\n\n/-!\n# Disjoint union of types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file proves basic results about the sum type `\u03b1 \u2295 \u03b2`.\n\n`\u03b1 \u2295 \u03b2` is the type made of a copy of `\u03b1` and a copy of `\u03b2`. It is also called *disjoint union*.\n\n## Main declarations\n\n* `sum.get_left`: Retrieves the left content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the right.\n* `sum.get_right`: Retrieves the right content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the left.\n* `sum.is_left`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the left component or not.\n* `sum.is_right`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the right component or not.\n* `sum.map`: Maps `\u03b1 \u2295 \u03b2` to `\u03b3 \u2295 \u03b4` component-wise.\n* `sum.elim`: Nondependent eliminator/induction principle for `\u03b1 \u2295 \u03b2`.\n* `sum.swap`: Maps `\u03b1 \u2295 \u03b2` to `\u03b2 \u2295 \u03b1` by swapping components.\n* `sum.lex`: Lexicographic order on `\u03b1 \u2295 \u03b2` induced by a relation on `\u03b1` and a relation on `\u03b2`.\n\n## Notes\n\nThe definition of `sum` takes values in `Type*`. This effectively forbids `Prop`- valued sum types.\nTo this effect, we have `psum`, which takes value in `Sort*` and carries a more complicated\nuniverse signature in consequence. The `Prop` version is `or`.\n-/\n\nuniverses u v w x\nvariables {\u03b1 : Type u} {\u03b1' : Type w} {\u03b2 : Type v} {\u03b2' : Type x} {\u03b3 \u03b4 : Type*}\n\nnamespace sum\n\nattribute [derive decidable_eq] sum\n\n@[simp] lemma \u00abforall\u00bb {p : \u03b1 \u2295 \u03b2 \u2192 Prop} : (\u2200 x, p x) \u2194 (\u2200 a, p (inl a)) \u2227 \u2200 b, p (inr b) :=\n\u27e8\u03bb h, \u27e8\u03bb a, h _, \u03bb b, h _\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, sum.rec h\u2081 h\u2082\u27e9\n\n@[simp] lemma \u00abexists\u00bb {p : \u03b1 \u2295 \u03b2 \u2192 Prop} : (\u2203 x, p x) \u2194 (\u2203 a, p (inl a)) \u2228 \u2203 b, p (inr b) :=\n\u27e8\u03bb h, match h with\n| \u27e8inl a, h\u27e9 := or.inl \u27e8a, h\u27e9\n| \u27e8inr b, h\u27e9 := or.inr \u27e8b, h\u27e9\nend, \u03bb h, match h with\n| or.inl \u27e8a, h\u27e9 := \u27e8inl a, h\u27e9\n| or.inr \u27e8b, h\u27e9 := \u27e8inr b, h\u27e9\nend\u27e9\n\nlemma inl_injective : function.injective (inl : \u03b1 \u2192 \u03b1 \u2295 \u03b2) := \u03bb x y, inl.inj\nlemma inr_injective : function.injective (inr : \u03b2 \u2192 \u03b1 \u2295 \u03b2) := \u03bb x y, inr.inj\n\nsection get\n\n/-- Check if a sum is `inl` and if so, retrieve its contents. -/\n@[simp] def get_left : \u03b1 \u2295 \u03b2 \u2192 option \u03b1\n| (inl a) := some a\n| (inr _) := none\n\n/-- Check if a sum is `inr` and if so, retrieve its contents. -/\n@[simp] def get_right : \u03b1 \u2295 \u03b2 \u2192 option \u03b2\n| (inr b) := some b\n| (inl _) := none\n\n/-- Check if a sum is `inl`. -/\n@[simp] def is_left : \u03b1 \u2295 \u03b2 \u2192 bool\n| (inl _) := tt\n| (inr _) := ff\n\n/-- Check if a sum is `inr`. -/\n@[simp] def is_right : \u03b1 \u2295 \u03b2 \u2192 bool\n| (inl _) := ff\n| (inr _) := tt\n\nvariables {x y : \u03b1 \u2295 \u03b2}\n\n@[simp] lemma get_left_eq_none_iff : x.get_left = none \u2194 x.is_right :=\nby cases x; simp only [get_left, is_right, coe_sort_tt, coe_sort_ff, eq_self_iff_true]\n\n@[simp] lemma get_right_eq_none_iff : x.get_right = none \u2194 x.is_left :=\nby cases x; simp only [get_right, is_left, coe_sort_tt, coe_sort_ff, eq_self_iff_true]\n\n@[simp] lemma get_left_eq_some_iff {a} : x.get_left = some a \u2194 x = inl a :=\nby cases x; simp only [get_left]\n\n@[simp] lemma get_right_eq_some_iff {b} : x.get_right = some b \u2194 x = inr b :=\nby cases x; simp only [get_right]\n\n@[simp] lemma bnot_is_left (x : \u03b1 \u2295 \u03b2) : bnot x.is_left = x.is_right := by cases x; refl\n@[simp] lemma is_left_eq_ff : x.is_left = ff \u2194 x.is_right := by cases x; simp\nlemma not_is_left : \u00acx.is_left \u2194 x.is_right := by simp\n@[simp] lemma bnot_is_right (x : \u03b1 \u2295 \u03b2) : bnot x.is_right = x.is_left := by cases x; refl\n@[simp] lemma is_right_eq_ff : x.is_right = ff \u2194 x.is_left := by cases x; simp\nlemma not_is_right : \u00acx.is_right \u2194 x.is_left := by simp\n\nlemma is_left_iff : x.is_left \u2194 \u2203 y, x = sum.inl y := by cases x; simp\nlemma is_right_iff : x.is_right \u2194 \u2203 y, x = sum.inr y := by cases x; simp\n\nend get\n\ntheorem inl.inj_iff {a b} : (inl a : \u03b1 \u2295 \u03b2) = inl b \u2194 a = b :=\n\u27e8inl.inj, congr_arg _\u27e9\n\ntheorem inr.inj_iff {a b} : (inr a : \u03b1 \u2295 \u03b2) = inr b \u2194 a = b :=\n\u27e8inr.inj, congr_arg _\u27e9\n\ntheorem inl_ne_inr {a : \u03b1} {b : \u03b2} : inl a \u2260 inr b.\n\ntheorem inr_ne_inl {a : \u03b1} {b : \u03b2} : inr b \u2260 inl a.\n\n/-- Define a function on `\u03b1 \u2295 \u03b2` by giving separate definitions on `\u03b1` and `\u03b2`. -/\nprotected def elim {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 := \u03bb x, sum.rec_on x f g\n\n@[simp] lemma elim_inl {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b1) :\n  sum.elim f g (inl x) = f x := rfl\n\n@[simp] lemma elim_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b2) :\n  sum.elim f g (inr x) = g x := rfl\n\n@[simp] lemma elim_comp_inl {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) :\n  sum.elim f g \u2218 inl = f := rfl\n\n@[simp] lemma elim_comp_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) :\n  sum.elim f g \u2218 inr = g := rfl\n\n@[simp] lemma elim_inl_inr {\u03b1 \u03b2 : Sort*} :\n  @sum.elim \u03b1 \u03b2 _ inl inr = id :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\nlemma comp_elim {\u03b1 \u03b2 \u03b3 \u03b4 : Sort*} (f : \u03b3 \u2192 \u03b4) (g : \u03b1 \u2192 \u03b3) (h : \u03b2 \u2192 \u03b3):\n  f \u2218 sum.elim g h = sum.elim (f \u2218 g) (f \u2218 h) :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\n@[simp] lemma elim_comp_inl_inr {\u03b1 \u03b2 \u03b3 : Sort*} (f : \u03b1 \u2295 \u03b2 \u2192 \u03b3) :\n  sum.elim (f \u2218 inl) (f \u2218 inr) = f :=\nfunext $ \u03bb x, sum.cases_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\n/-- Map `\u03b1 \u2295 \u03b2` to `\u03b1' \u2295 \u03b2'` sending `\u03b1` to `\u03b1'` and `\u03b2` to `\u03b2'`. -/\nprotected def map (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') : \u03b1 \u2295 \u03b2 \u2192 \u03b1' \u2295 \u03b2' :=\nsum.elim (inl \u2218 f) (inr \u2218 g)\n\n@[simp] lemma map_inl (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b1) : (inl x).map f g = inl (f x) := rfl\n@[simp] lemma map_inr (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b2) : (inr x).map f g = inr (g x) := rfl\n\n@[simp] lemma map_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n  \u2200 x : \u03b1 \u2295 \u03b2, (x.map f g).map f' g' = x.map (f' \u2218 f) (g' \u2218 g)\n| (inl a) := rfl\n| (inr b) := rfl\n\n@[simp] lemma map_comp_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n  (sum.map f' g') \u2218 (sum.map f g) = sum.map (f' \u2218 f) (g' \u2218 g) :=\nfunext $ map_map f' g' f g\n\n@[simp] lemma map_id_id (\u03b1 \u03b2) : sum.map (@id \u03b1) (@id \u03b2) = id :=\nfunext $ \u03bb x, sum.rec_on x (\u03bb _, rfl) (\u03bb _, rfl)\n\nlemma elim_map {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Sort*} {f\u2081 : \u03b1 \u2192 \u03b2} {f\u2082 : \u03b2 \u2192 \u03b5} {g\u2081 : \u03b3 \u2192 \u03b4} {g\u2082 : \u03b4 \u2192 \u03b5} {x} :\n  sum.elim f\u2082 g\u2082 (sum.map f\u2081 g\u2081 x) = sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) x :=\nby cases x; refl\n\nlemma elim_comp_map {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Sort*} {f\u2081 : \u03b1 \u2192 \u03b2} {f\u2082 : \u03b2 \u2192 \u03b5} {g\u2081 : \u03b3 \u2192 \u03b4} {g\u2082 : \u03b4 \u2192 \u03b5} :\n  sum.elim f\u2082 g\u2082 \u2218 sum.map f\u2081 g\u2081 = sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) :=\nfunext $ \u03bb _, elim_map\n\n@[simp] lemma is_left_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : \u03b1 \u2295 \u03b3) :\n  is_left (x.map f g) = is_left x :=\nby cases x; refl\n\n@[simp] lemma is_right_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : \u03b1 \u2295 \u03b3) :\n  is_right (x.map f g) = is_right x :=\nby cases x; refl\n\n@[simp] lemma get_left_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : \u03b1 \u2295 \u03b3) :\n  (x.map f g).get_left = x.get_left.map f :=\nby cases x; refl\n\n@[simp] lemma get_right_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : \u03b1 \u2295 \u03b3) :\n  (x.map f g).get_right = x.get_right.map g :=\nby cases x; refl\n\nopen function (update update_eq_iff update_comp_eq_of_injective update_comp_eq_of_forall_ne)\n\n@[simp] lemma update_elim_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  {i : \u03b1} {x : \u03b3} :\n  update (sum.elim f g) (inl i) x = sum.elim (update f i x) g :=\nupdate_eq_iff.2 \u27e8by simp, by simp { contextual := tt }\u27e9\n\n@[simp] lemma update_elim_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  {i : \u03b2} {x : \u03b3} :\n  update (sum.elim f g) (inr i) x = sum.elim f (update g i x) :=\nupdate_eq_iff.2 \u27e8by simp, by simp { contextual := tt }\u27e9\n\n@[simp] lemma update_inl_comp_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1}\n  {x : \u03b3} :\n  update f (inl i) x \u2218 inl = update (f \u2218 inl) i x :=\nupdate_comp_eq_of_injective _ inl_injective _ _\n\n@[simp] lemma update_inl_apply_inl [decidable_eq \u03b1] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3}\n  {i j : \u03b1} {x : \u03b3} :\n  update f (inl i) x (inl j) = update (f \u2218 inl) i x j :=\nby rw \u2190 update_inl_comp_inl\n\n@[simp] lemma update_inl_comp_inr [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1} {x : \u03b3} :\n  update f (inl i) x \u2218 inr = f \u2218 inr :=\nupdate_comp_eq_of_forall_ne _ _ $ \u03bb _, inr_ne_inl\n\n@[simp] lemma update_inl_apply_inr [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3} :\n  update f (inl i) x (inr j) = f (inr j) :=\nfunction.update_noteq inr_ne_inl _ _\n\n@[simp] lemma update_inr_comp_inl [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b2} {x : \u03b3} :\n  update f (inr i) x \u2218 inl = f \u2218 inl :=\nupdate_comp_eq_of_forall_ne _ _ $ \u03bb _, inl_ne_inr\n\n@[simp] \n\n@[simp] lemma update_inr_comp_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3} {i : \u03b2}\n  {x : \u03b3} :\n  update f (inr i) x \u2218 inr = update (f \u2218 inr) i x :=\nupdate_comp_eq_of_injective _ inr_injective _ _\n\n@[simp] lemma update_inr_apply_inr [decidable_eq \u03b2] [decidable_eq (\u03b1 \u2295 \u03b2)] {f : \u03b1 \u2295 \u03b2 \u2192 \u03b3}\n  {i j : \u03b2} {x : \u03b3} :\n  update f (inr i) x (inr j) = update (f \u2218 inr) i x j :=\nby rw \u2190 update_inr_comp_inr\n\n/-- Swap the factors of a sum type -/\ndef swap : \u03b1 \u2295 \u03b2 \u2192 \u03b2 \u2295 \u03b1 := sum.elim inr inl\n\n@[simp] lemma swap_inl (x : \u03b1) : swap (inl x : \u03b1 \u2295 \u03b2) = inr x := rfl\n@[simp] lemma swap_inr (x : \u03b2) : swap (inr x : \u03b1 \u2295 \u03b2) = inl x := rfl\n@[simp] lemma swap_swap (x : \u03b1 \u2295 \u03b2) : swap (swap x) = x := by cases x; refl\n@[simp] lemma swap_swap_eq : swap \u2218 swap = @id (\u03b1 \u2295 \u03b2) := funext $ swap_swap\n@[simp] lemma swap_left_inverse : function.left_inverse (@swap \u03b1 \u03b2) swap := swap_swap\n@[simp] lemma swap_right_inverse : function.right_inverse (@swap \u03b1 \u03b2) swap := swap_swap\n@[simp] lemma is_left_swap (x : \u03b1 \u2295 \u03b2) : x.swap.is_left = x.is_right := by cases x; refl\n@[simp] lemma is_right_swap (x : \u03b1 \u2295 \u03b2) : x.swap.is_right = x.is_left := by cases x; refl\n@[simp] lemma get_left_swap (x : \u03b1 \u2295 \u03b2) : x.swap.get_left = x.get_right := by cases x; refl\n@[simp] lemma get_right_swap (x : \u03b1 \u2295 \u03b2) : x.swap.get_right = x.get_left := by cases x; refl\n\nsection lift_rel\n\n/-- Lifts pointwise two relations between `\u03b1` and `\u03b3` and between `\u03b2` and `\u03b4` to a relation between\n`\u03b1 \u2295 \u03b2` and `\u03b3 \u2295 \u03b4`. -/\ninductive lift_rel (r : \u03b1 \u2192 \u03b3 \u2192 Prop) (s : \u03b2 \u2192 \u03b4 \u2192 Prop) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 \u2295 \u03b4 \u2192 Prop\n| inl {a c} : r a c \u2192 lift_rel (inl a) (inl c)\n| inr {b d} : s b d \u2192 lift_rel (inr b) (inr d)\n\nattribute [protected] lift_rel.inl lift_rel.inr\n\nvariables {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop} {a : \u03b1} {b : \u03b2} {c : \u03b3} {d : \u03b4}\n  {x : \u03b1 \u2295 \u03b2} {y : \u03b3 \u2295 \u03b4}\n\n@[simp] lemma lift_rel_inl_inl : lift_rel r s (inl a) (inl c) \u2194 r a c :=\n\u27e8\u03bb h, by { cases h, assumption }, lift_rel.inl\u27e9\n\n@[simp] lemma not_lift_rel_inl_inr : \u00ac lift_rel r s (inl a) (inr d) .\n@[simp] lemma not_lift_rel_inr_inl : \u00ac lift_rel r s (inr b) (inl c) .\n\n@[simp] lemma lift_rel_inr_inr : lift_rel r s (inr b) (inr d) \u2194 s b d :=\n\u27e8\u03bb h, by { cases h, assumption }, lift_rel.inr\u27e9\n\ninstance [\u03a0 a c, decidable (r a c)] [\u03a0 b d, decidable (s b d)] :\n  \u03a0 (ab : \u03b1 \u2295 \u03b2) (cd : \u03b3 \u2295 \u03b4), decidable (lift_rel r s ab cd)\n| (inl a) (inl c) := decidable_of_iff' _ lift_rel_inl_inl\n| (inl a) (inr d) := decidable.is_false not_lift_rel_inl_inr\n| (inr b) (inl c) := decidable.is_false not_lift_rel_inr_inl\n| (inr b) (inr d) := decidable_of_iff' _ lift_rel_inr_inr\n\nlemma lift_rel.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)\n  (h : lift_rel r\u2081 s\u2081 x y) :\n  lift_rel r\u2082 s\u2082 x y :=\nby { cases h, exacts [lift_rel.inl (hr _ _ \u2039_\u203a), lift_rel.inr (hs _ _ \u2039_\u203a)] }\n\nlemma lift_rel.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : lift_rel r\u2081 s x y) :\n  lift_rel r\u2082 s x y :=\nh.mono hr $ \u03bb _ _, id\n\nlemma lift_rel.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)  (h : lift_rel r s\u2081 x y) :\n  lift_rel r s\u2082 x y :=\nh.mono (\u03bb _ _, id) hs\n\nprotected lemma lift_rel.swap (h : lift_rel r s x y) : lift_rel s r x.swap y.swap :=\nby { cases h, exacts [lift_rel.inr \u2039_\u203a, lift_rel.inl \u2039_\u203a] }\n\n@[simp] lemma lift_rel_swap_iff : lift_rel s r x.swap y.swap \u2194 lift_rel r s x y :=\n\u27e8\u03bb h, by { rw [\u2190swap_swap x, \u2190swap_swap y], exact h.swap }, lift_rel.swap\u27e9\n\nend lift_rel\n\nsection lex\n\n/-- Lexicographic order for sum. Sort all the `inl a` before the `inr b`, otherwise use the\nrespective order on `\u03b1` or `\u03b2`. -/\ninductive lex (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : \u03b1 \u2295 \u03b2 \u2192 \u03b1 \u2295 \u03b2 \u2192 Prop\n| inl {a\u2081 a\u2082} (h : r a\u2081 a\u2082) : lex (inl a\u2081) (inl a\u2082)\n| inr {b\u2081 b\u2082} (h : s b\u2081 b\u2082) : lex (inr b\u2081) (inr b\u2082)\n| sep (a b) : lex (inl a) (inr b)\n\nattribute [protected] sum.lex.inl sum.lex.inr\nattribute [simp] lex.sep\n\nvariables {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop} {a a\u2081 a\u2082 : \u03b1} {b b\u2081 b\u2082 : \u03b2}\n  {x y : \u03b1 \u2295 \u03b2}\n\n@[simp] lemma lex_inl_inl : lex r s (inl a\u2081) (inl a\u2082) \u2194 r a\u2081 a\u2082 :=\n\u27e8\u03bb h, by { cases h, assumption }, lex.inl\u27e9\n\n@[simp] lemma lex_inr_inr : lex r s (inr b\u2081) (inr b\u2082) \u2194 s b\u2081 b\u2082 :=\n\u27e8\u03bb h, by { cases h, assumption }, lex.inr\u27e9\n\n@[simp] lemma lex_inr_inl : \u00ac lex r s (inr b) (inl a) .\n\ninstance [decidable_rel r] [decidable_rel s] : decidable_rel (lex r s)\n| (inl a) (inl c) := decidable_of_iff' _ lex_inl_inl\n| (inl a) (inr d) := decidable.is_true (lex.sep _ _)\n| (inr b) (inl c) := decidable.is_false lex_inr_inl\n| (inr b) (inr d) := decidable_of_iff' _ lex_inr_inr\n\nprotected lemma lift_rel.lex {a b : \u03b1 \u2295 \u03b2} (h : lift_rel r s a b) : lex r s a b :=\nby { cases h, exacts [lex.inl \u2039_\u203a, lex.inr \u2039_\u203a] }\n\nlemma lift_rel_subrelation_lex : subrelation (lift_rel r s) (lex r s) := \u03bb a b, lift_rel.lex\n\nlemma lex.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : lex r\u2081 s\u2081 x y) :\n  lex r\u2082 s\u2082 x y :=\nby { cases h, exacts [lex.inl (hr _ _ \u2039_\u203a), lex.inr (hs _ _ \u2039_\u203a), lex.sep _ _] }\n\nlemma lex.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : lex r\u2081 s x y) : lex r\u2082 s x y :=\nh.mono hr $ \u03bb _ _, id\n\nlemma lex.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : lex r s\u2081 x y) : lex r s\u2082 x y :=\nh.mono (\u03bb _ _, id) hs\n\nlemma lex_acc_inl {a} (aca : acc r a) : acc (lex r s) (inl a) :=\nbegin\n  induction aca with a H IH,\n  constructor, intros y h,\n  cases h with a' _ h',\n  exact IH _ h'\nend\n\nlemma lex_acc_inr (aca : \u2200 a, acc (lex r s) (inl a)) {b} (acb : acc s b) : acc (lex r s) (inr b) :=\nbegin\n  induction acb with b H IH,\n  constructor, intros y h,\n  cases h with _ _ _ b' _ h' a,\n  { exact IH _ h' },\n  { exact aca _ }\nend\n\nlemma lex_wf (ha : well_founded r) (hb : well_founded s) : well_founded (lex r s) :=\nhave aca : \u2200 a, acc (lex r s) (inl a), from \u03bb a, lex_acc_inl (ha.apply a),\n\u27e8\u03bb x, sum.rec_on x aca (\u03bb b, lex_acc_inr aca (hb.apply b))\u27e9\n\nend lex\nend sum\n\nopen sum\n\nnamespace function\n\nlemma injective.sum_elim {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n  (hf : injective f) (hg : injective g) (hfg : \u2200 a b, f a \u2260 g b) :\n  injective (sum.elim f g)\n| (inl x) (inl y) h := congr_arg inl $ hf h\n| (inl x) (inr y) h := (hfg x y h).elim\n| (inr x) (inl y) h := (hfg y x h.symm).elim\n| (inr x) (inr y) h := congr_arg inr $ hg h\n\nlemma injective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : injective f) (hg : injective g) :\n  injective (sum.map f g)\n| (inl x) (inl y) h := congr_arg inl $ hf $ inl.inj h\n| (inr x) (inr y) h := congr_arg inr $ hg $ inr.inj h\n\nlemma surjective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : surjective f) (hg : surjective g) :\n  surjective (sum.map f g)\n| (inl y) := let \u27e8x, hx\u27e9 := hf y in \u27e8inl x, congr_arg inl hx\u27e9\n| (inr y) := let \u27e8x, hx\u27e9 := hg y in \u27e8inr x, congr_arg inr hx\u27e9\n\nlemma bijective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : bijective f) (hg : bijective g) :\n  bijective (sum.map f g) :=\n\u27e8hf.injective.sum_map hg.injective, hf.surjective.sum_map hg.surjective\u27e9\n\nend function\n\nnamespace sum\nopen function\n\n@[simp] lemma map_injective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n  injective (sum.map f g) \u2194 injective f \u2227 injective g :=\n\u27e8\u03bb h, \u27e8\u03bb a\u2081 a\u2082 ha, inl_injective $ @h (inl a\u2081) (inl a\u2082) (congr_arg inl ha : _),\n      \u03bb b\u2081 b\u2082 hb, inr_injective $ @h (inr b\u2081) (inr b\u2082) (congr_arg inr hb : _)\u27e9,\n  \u03bb h, h.1.sum_map h.2\u27e9\n\n@[simp] lemma map_surjective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n  surjective (sum.map f g) \u2194 surjective f \u2227 surjective g :=\n\u27e8\u03bb h, \u27e8\u03bb c, begin\n  obtain \u27e8a | b, h\u27e9 := h (inl c),\n  { exact \u27e8a, inl_injective h\u27e9 },\n  { cases h },\nend, \u03bb d, begin\n  obtain \u27e8a | b, h\u27e9 := h (inr d),\n  { cases h },\n  { exact \u27e8b, inr_injective h\u27e9 },\nend\u27e9, \u03bb h, h.1.sum_map h.2\u27e9\n\n@[simp] lemma map_bijective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n  bijective (sum.map f g) \u2194 bijective f \u2227 bijective g :=\n(map_injective.and map_surjective).trans $ and_and_and_comm _ _ _ _\n\nlemma elim_const_const (c : \u03b3) :\n  sum.elim (const _ c : \u03b1 \u2192 \u03b3) (const _ c : \u03b2 \u2192 \u03b3) = const _ c :=\nby { ext x, cases x; refl }\n\n@[simp]\nlemma elim_lam_const_lam_const (c : \u03b3) :\n  sum.elim (\u03bb (_ : \u03b1), c) (\u03bb (_ : \u03b2), c) = \u03bb _, c :=\nsum.elim_const_const c\n\nlemma elim_update_left [decidable_eq \u03b1] [decidable_eq \u03b2]\n    (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (i : \u03b1) (c : \u03b3) :\n  sum.elim (function.update f i c) g = function.update (sum.elim f g) (inl i) c :=\nbegin\n  ext x, cases x,\n  { by_cases h : x = i,\n    { subst h, simp },\n    { simp [h] } },\n  { simp }\nend\n\nlemma elim_update_right [decidable_eq \u03b1] [decidable_eq \u03b2]\n    (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (i : \u03b2) (c : \u03b3) :\n  sum.elim f (function.update g i c) = function.update (sum.elim f g) (inr i) c :=\nbegin\n  ext x, cases x,\n  { simp },\n  { by_cases h : x = i,\n    { subst h, simp },\n    { simp [h] } }\nend\n\nend sum\n\n/-!\n### Ternary sum\n\nAbbreviations for the maps from the summands to `\u03b1 \u2295 \u03b2 \u2295 \u03b3`. This is useful for pattern-matching.\n-/\n\nnamespace sum3\n\n/-- The map from the first summand into a ternary sum. -/\n@[pattern, simp, reducible] def in\u2080 (a) : \u03b1 \u2295 \u03b2 \u2295 \u03b3 := inl a\n/-- The map from the second summand into a ternary sum. -/\n@[pattern, simp, reducible] def in\u2081 (b) : \u03b1 \u2295 \u03b2 \u2295 \u03b3 := inr $ inl b\n/-- The map from the third summand into a ternary sum. -/\n@[pattern, simp, reducible] def in\u2082 (c) : \u03b1 \u2295 \u03b2 \u2295 \u03b3 := inr $ inr c\n\nend sum3\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/sum/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.7025300698514777, "lm_q1q2_score": 0.3649793506003359}}
{"text": "import algebra.category.Module.projective\nimport linear_algebra.free_module.basic\nimport category_theory.monoidal.tor\n\nimport .lte.for_mathlib.derived_functor_zero\n\nimport .flat\n\nopen category_theory category_theory.limits category_theory.monoidal_category\nopen_locale direct_sum zero_object\n\nnoncomputable theory\n\nuniverses u v\n\nnamespace Module\n\nvariables {R : Type u} [comm_ring R] (M : Module.{u} R) [\u03a0 (N : Module.{u} R), decidable_eq N]\n\n@[reducible]\ndef afree : Module.{u} R := Module.of R $ \u2a01 (m : M), R\n\ninstance afree_is_free : module.free R M.afree :=\nmodule.free.dfinsupp R _\n\ninstance afree_is_projective : projective M.afree :=\nprojective_of_free.{u u u} (module.free.choose_basis R M.afree)\n\n@[reducible]\ndef from_afree : M.afree \u27f6 M :=\ndirect_sum.to_module _ _ _ $ \u03bb m, \n{ to_fun := \u03bb r, r \u2022 m,\n  map_add' := \u03bb _ _, add_smul _ _ m,\n  map_smul' := \u03bb _ _, by rw [smul_eq_mul, mul_smul, ring_hom.id_apply] }\n\nlemma from_afree_surj : function.surjective M.from_afree :=\n\u03bb x, \u27e8direct_sum.of (\u03bb (m : M), R) x 1, by \n  erw [direct_sum.to_module_lof, linear_map.coe_mk, one_smul]\u27e9\n\ninstance from_afree_epi : epi M.from_afree :=\n(Module.epi_iff_surjective _).mpr $ M.from_afree_surj\n\n@[reducible] def free_res.chain_complex.Xd_aux : \n\u2115 \u2192 \u03a3' (N_prev N_next : Module.{u} R) (h : module.free R N_prev \u2227 module.free R N_next), N_next \u27f6 N_prev :=\n@nat.rec (\u03bb _, \u03a3' (N_prev N_next : Module.{u} R) (h : module.free R N_prev \u2227 module.free R N_next), N_next \u27f6 N_prev)\n\u27e8M.afree, (kernel M.from_afree).afree, \n  \u27e8Module.afree_is_free _, Module.afree_is_free _\u27e9, \n  Module.from_afree _ \u226b kernel.\u03b9 _\u27e9 $ \u03bb n P, \n\u27e8P.2.1, (kernel P.2.2.2).afree, \u27e8P.2.2.1.2, Module.afree_is_free _\u27e9, Module.from_afree _ \u226b kernel.\u03b9 _\u27e9\n\nlemma free_res.chain_complex.Xd_aux_0 (X : Module.{u} R) :\n  free_res.chain_complex.Xd_aux X 0 = \n  \u27e8X.afree, (kernel X.from_afree).afree, \u27e8Module.afree_is_free _, Module.afree_is_free _\u27e9, \n    Module.from_afree _ \u226b kernel.\u03b9 _\u27e9 := rfl\n\n@[reducible] def free_res.chain_complex.X' (n : \u2115) : Module.{u} R :=\n(free_res.chain_complex.Xd_aux M n).1\n\n@[reducible] def free_res.chain_complex.d' (i j : \u2115) :\n  free_res.chain_complex.X' M i \u27f6 free_res.chain_complex.X' M j :=\nif h : j + 1 = i\nthen (eq_to_hom $ by rw h : _ \u27f6 free_res.chain_complex.X' M (j + 1)) \u226b \n  (free_res.chain_complex.Xd_aux M _).2.2.2 \nelse 0\n\n@[simps] def free_res.chain_complex : chain_complex (Module R) \u2115 :=\n{ X := free_res.chain_complex.X' M,\n  d := free_res.chain_complex.d' M,\n  shape' := \u03bb i j (h : _ \u2260 _), by rw [free_res.chain_complex.d', dif_neg h],\n  d_comp_d' := \u03bb i j k (hij : _ = _) (hjk : _ = _), \n  begin \n    substs hij hjk,\n    rw [free_res.chain_complex.d', free_res.chain_complex.d', dif_pos rfl, dif_pos rfl, eq_to_hom_refl, \n      eq_to_hom_refl, category.id_comp, category.id_comp],\n    dsimp,\n    suffices : (free_res.chain_complex.Xd_aux M (k + 1)).2.2.2 \u226b (free_res.chain_complex.Xd_aux M k).2.2.2 = 0,\n    { convert this, },\n    { change (_ \u226b _) \u226b _ = 0,\n      erw [category.assoc, kernel.condition, comp_zero], },\n  end }\n\n@[reducible]\ndef free_res : ProjectiveResolution M :=\n{ complex := free_res.chain_complex M,\n  \u03c0 := (chain_complex.to_single\u2080_equiv (free_res.chain_complex M) M).symm \u27e8M.from_afree, \n  begin \n    dsimp only [free_res.chain_complex.d', free_res.chain_complex_d],\n    erw [dif_pos (zero_add 1), category.assoc, category.assoc, kernel.condition, comp_zero],\n  end\u27e9,\n  projective := \u03bb n, projective_of_free.{u u u} (@@module.free.choose_basis R _ _ _ _ ((free_res.chain_complex.Xd_aux M) n).2.2.1.1),\n  exact\u2080 := \n  begin \n    dsimp [free_res.chain_complex.d'],\n    simp only [eq_self_iff_true, category.id_comp, if_true],\n    change exact (Module.from_afree _ \u226b _) M.from_afree,\n    exact exact_epi_comp exact_kernel_\u03b9,\n  end,\n  exact := \u03bb n, \n  begin \n    dsimp [free_res.chain_complex.d'],\n    simp only [eq_self_iff_true, category.id_comp, if_true],\n    suffices : exact (free_res.chain_complex.Xd_aux M (n + 1)).snd.snd.snd \n      (free_res.chain_complex.Xd_aux M n).snd.snd.snd,\n    { exact this },\n    change exact (_ \u226b _) _,\n    convert @@exact_epi_comp _ (infer_instance : has_images (Module.{u} R)) _ _ \n      exact_kernel_\u03b9 (Module.from_afree_epi _),\n  end,\n  epi := M.from_afree_epi }\n\ninstance enough_projective_ : enough_projectives (Module.{u} R) :=\nModule.Module_enough_projectives.{u u}\n\ninstance has_projective_resolutions : has_projective_resolutions (Module.{u} R) := \n{ out := \u03bb Z, \u27e8nonempty.intro $ Z.free_res\u27e9 }\n\ndef higher_Tor'_zero_of_flat (h : module.flat.exact R M) : \n  \u2200 (n : \u2115) (hn : 0 < n) (N : Module.{u} R), \n    ((Tor' (Module.{u} R) n).obj N).obj M \u2245 0 :=\n\u03bb n hn N,\nbegin \n  dsimp only [Tor', functor.flip],\n  refine functor.left_derived_obj_iso (tensor_right M) n N.free_res \u226a\u226b _,\n  suffices : epi _,\n  { refine @@cokernel.of_epi _ _ _ _ _ this, },\n  suffices : category_theory.exact _ _,\n  { exact this.2 },\n  dunfold homological_complex.d_to homological_complex.d_from,\n  dsimp [functor.map_homological_complex_obj_d],\n  dunfold tensor_hom,\n  dunfold monoidal_category.tensor_hom,\n  refine h (free_res.chain_complex.d' N ((complex_shape.down \u2115).prev n) n) _ _,\n  rw complex_shape.prev_eq',\n  work_on_goal 3 { exact n + 1 },\n  work_on_goal 2 { rw complex_shape.down_rel, },\n  rw complex_shape.next_eq',\n  work_on_goal 3 { exact n.pred },\n  work_on_goal 2 \n  { rw complex_shape.down_rel, refine nat.succ_pred_eq_of_pos hn, },\n\n  convert (N.free_res).exact (n - 1) using 2,\n  { linarith, },\n  { linarith, },\n  { linarith, },\n  { linarith, },\n  { linarith, },\nend\n\ndef first_Tor'_zero_of_flat (h : module.flat.exact R M) :\n  \u2200 (N : Module.{u} R), ((Tor' (Module.{u} R) 1).obj N).obj M \u2245 0 :=\n\u03bb _, M.higher_Tor'_zero_of_flat h 1 (by linarith) _\n\ndef first_Tor'_ideal_zero_of_flat (h : module.flat.exact R M) :\n  \u2200 (I : ideal R), ((Tor' (Module.{u} R) 1).obj (Module.of R (R \u29f8 I))).obj M \u2245 0 :=\n\u03bb _, M.first_Tor'_zero_of_flat h _\n\ndef first_Tor'_fg_ideal_zero_of_flat (h : module.flat.exact R M) :\n  \u2200  (I : ideal R) (hI : I.fg), \n    ((Tor' (Module.{u} R) 1).obj (Module.of R (R \u29f8 I))).obj M \u2245 0 :=\n\u03bb _ _, M.first_Tor'_zero_of_flat h _\n\ndef ses_of_ideal (I : ideal R) : short_exact_sequence (Module.{u} R) :=\n{ fst := Module.of R I,\n  snd := Module.of R R,\n  trd := Module.of R (R \u29f8 I),\n  f := Module.of_hom \u27e8coe, \u03bb _ _, rfl, \u03bb _ _, rfl\u27e9,\n  g := submodule.mkq I,\n  mono' := \n  begin \n    rw mono_iff_injective,\n    rintros x y h,\n    ext,\n    exact h,\n  end,\n  epi' := \n  begin \n    rw epi_iff_surjective,\n    exact submodule.mkq_surjective _,\n  end,\n  exact' := \n  begin\n    rw exact_iff,\n    rw submodule.ker_mkq,\n    ext1 (x : R),\n    split,\n    { rintros \u27e8x, rfl\u27e9,\n      exact x.2, },\n    { rintros hx, refine \u27e8\u27e8x, hx\u27e9, rfl\u27e9 },\n  end }\n\ndef flat_of_first_Tor'_fg_ideal_zero (h : \u2200  (I : ideal R) (hI : I.fg), \n  ((Tor' (Module.{u} R) 1).obj (Module.of R (R \u29f8 I))).obj M \u2245 0) :\n  module.flat.fg_ideal R M :=\n\u03bb I hI, \nbegin \n  have exact1 : exact_seq (Module.{u} R) [_, _] := (category_theory.abelian.functor.seven_term_exact_seq (tensor_right M) \n    (ses_of_ideal I)).extract 2 2,\n  rw [\u2190exact_iff_exact_seq] at exact1,\n  convert_to function.injective ((tensor_right M).map (ses_of_ideal I).f),\n  rw [\u2190mono_iff_injective, mono_iff_exact_zero_left],\n  refine exact_iso_comp.mp _,\n  { exact ((tensor_right M).left_derived (0 + 1)).obj (ses_of_ideal I).trd },\n  { specialize h I hI, exact h.hom, },\n  { exact is_iso_of_op (h I hI).hom, },\n  { convert exact1 using 1,\n    apply is_initial.hom_ext,\n    refine is_initial.of_iso _ (h I hI).symm,\n    exact is_initial.of_unique 0, },\nend\n\nnamespace flat\n\n\nlemma equiv_defs : tfae \n  [ module.flat.ses R M\n  , module.flat.inj R M\n  , module.flat.ideal R M\n  , module.flat.fg_ideal R M\n  , module.flat.exact R M\n  , \u2200 (N : Module.{u} R), nonempty (((Tor' (Module.{u} R) 1).obj N).obj M \u2245 0)\n  , \u2200 (I : ideal R), nonempty (((Tor' (Module.{u} R) 1).obj (Module.of R (R \u29f8 I))).obj M \u2245 0)\n  , \u2200  (I : ideal R) (hI : I.fg), \n    nonempty (((Tor' (Module.{u} R) 1).obj (Module.of R (R \u29f8 I))).obj M \u2245 0)\n  , \u2200 (n : \u2115) (hn : 0 < n) (N : Module.{u} R), \n    nonempty (((Tor' (Module.{u} R) n).obj N).obj M \u2245 0)] :=\nbegin \n  tfae_have : 1 \u2192 2, { apply module.flat.inj_of_ses },\n  tfae_have : 2 \u2192 1, { apply module.flat.ses_of_inj },\n  tfae_have : 3 \u2192 2, { apply module.flat.inj_of_ideal },\n  tfae_have : 4 \u2192 3, { apply module.flat.ideal_of_fg_ideal },\n  tfae_have : 2 \u2192 4, { apply module.flat.fg_ideal_of_inj },\n  tfae_have : 5 \u2192 2, { apply module.flat.inj_of_exact },\n  tfae_have : 1 \u2192 5, { apply module.flat.exact_of_ses },\n  tfae_have : 5 \u2192 6, { intros H N, exact \u27e8first_Tor'_zero_of_flat M H N\u27e9, },\n  tfae_have : 5 \u2192 7, { intros H I, exact \u27e8first_Tor'_ideal_zero_of_flat _ H _\u27e9, },\n  tfae_have : 5 \u2192 8, { intros H I hI, exact \u27e8first_Tor'_fg_ideal_zero_of_flat _ H _ hI\u27e9, },\n  tfae_have : 8 \u2192 4, { intros H, exact flat_of_first_Tor'_fg_ideal_zero _ (\u03bb I hI, (H _ hI).some), },\n  tfae_have : 6 \u2192 7, { intros H I, refine \u27e8(H _).some\u27e9, },\n  tfae_have : 7 \u2192 8, { intros H I hI, exact \u27e8(H _).some\u27e9, },\n  tfae_have : 5 \u2192 9, { intros H n hn, refine \u03bb N, \u27e8higher_Tor'_zero_of_flat M H n hn N\u27e9, },\n  tfae_have : 9 \u2192 6, { intros H N, refine H _ _ _, linarith },\n  tfae_finish,\nend\n\nend flat\n\nend Module", "meta": {"author": "jjaassoonn", "repo": "flat", "sha": "bab2f5c18fdee0042680c31b0350c69d241e9a82", "save_path": "github-repos/lean/jjaassoonn-flat", "path": "github-repos/lean/jjaassoonn-flat/flat-bab2f5c18fdee0042680c31b0350c69d241e9a82/src/free_modules_etc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3649793441290369}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.presheafed_space.has_colimits\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.preserves.shapes.pullbacks\nimport topology.sheaves.functors\nimport algebraic_geometry.Scheme\nimport category_theory.limits.shapes.strict_initial\nimport algebra.category.CommRing.instances\n\n/-!\n# Open immersions of structured spaces\n\nWe say that a morphism of presheafed spaces `f : X \u27f6 Y` is an open immersions if\nthe underlying map of spaces is an open embedding `f : X \u27f6 U \u2286 Y`,\nand the sheaf map `Y(V) \u27f6 f _* X(V)` is an iso for each `V \u2286 U`.\n\nAbbreviations are also provided for `SheafedSpace`, `LocallyRingedSpace` and `Scheme`.\n\n## Main definitions\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a PresheafedSpace hom `f` is an open_immersion.\n* `algebraic_geometry.is_open_immersion`: the `Prop`-valued typeclass asserting\n  that a Scheme morphism `f` is an open_immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.iso_restrict`: The source of an\n  open immersion is isomorphic to the restriction of the target onto the image.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.lift`: Any morphism whose range is\n  contained in an open immersion factors though the open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_SheafedSpace`: If `f : X \u27f6 Y` is an\n  open immersion of presheafed spaces, and `Y` is a sheafed space, then `X` is also a sheafed\n  space. The morphism as morphisms of sheafed spaces is given by `to_SheafedSpace_hom`.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_LocallyRingedSpace`: If `f : X \u27f6 Y` is\n  an open immersion of presheafed spaces, and `Y` is a locally ringed space, then `X` is also a\n  locally ringed space. The morphism as morphisms of locally ringed spaces is given by\n  `to_LocallyRingedSpace_hom`.\n\n## Main results\n\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.comp`: The composition of two open\n  immersions is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.of_iso`: An iso is an open immersion.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.to_iso`:\n  A surjective open immersion is an isomorphism.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.stalk_iso`: An open immersion induces\n  an isomorphism on stalks.\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.has_pullback_of_left`: If `f` is an open\n  immersion, then the pullback `(f, g)` exists (and the forgetful functor to `Top` preserves it).\n* `algebraic_geometry.PresheafedSpace.is_open_immersion.pullback_snd_of_left`: Open immersions\n  are stable under pullbacks.\n* `algebraic_geometry.SheafedSpace.is_open_immersion.of_stalk_iso` An (topological) open embedding\n  between two sheafed spaces is an open immersion if all the stalk maps are isomorphisms.\n\n-/\n\nopen topological_space category_theory opposite\nopen category_theory.limits\nnamespace algebraic_geometry\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C]\n\n/--\nAn open immersion of PresheafedSpaces is an open embedding `f : X \u27f6 U \u2286 Y` of the underlying\nspaces, such that the sheaf map `Y(V) \u27f6 f _* X(V)` is an iso for each `V \u2286 U`.\n-/\nclass PresheafedSpace.is_open_immersion {X Y : PresheafedSpace C} (f : X \u27f6 Y) : Prop :=\n(base_open : open_embedding f.base)\n(c_iso : \u2200 U : opens X, is_iso (f.c.app (op (base_open.is_open_map.functor.obj U))))\n\n/--\nA morphism of SheafedSpaces is an open immersion if it is an open immersion as a morphism\nof PresheafedSpaces\n-/\nabbreviation SheafedSpace.is_open_immersion\n  [has_products C] {X Y : SheafedSpace C} (f : X \u27f6 Y) : Prop :=\nPresheafedSpace.is_open_immersion f\n\n/--\nA morphism of LocallyRingedSpaces is an open immersion if it is an open immersion as a morphism\nof SheafedSpaces\n-/\nabbreviation LocallyRingedSpace.is_open_immersion {X Y : LocallyRingedSpace} (f : X \u27f6 Y) : Prop :=\nSheafedSpace.is_open_immersion f.1\n\n/--\nA morphism of Schemes is an open immersion if it is an open immersion as a morphism\nof LocallyRingedSpaces\n-/\nabbreviation is_open_immersion {X Y : Scheme} (f : X \u27f6 Y) : Prop :=\nLocallyRingedSpace.is_open_immersion f\n\nnamespace PresheafedSpace.is_open_immersion\n\nopen PresheafedSpace\n\nlocal notation `is_open_immersion` := PresheafedSpace.is_open_immersion\n\nattribute [instance] is_open_immersion.c_iso\n\nsection\n\nvariables {X Y : PresheafedSpace C} {f : X \u27f6 Y} (H : is_open_immersion f)\n\n/-- The functor `opens X \u2964 opens Y` associated with an open immersion `f : X \u27f6 Y`. -/\nabbreviation open_functor := H.base_open.is_open_map.functor\n\n/-\nWe want to keep `eq_to_hom`s in the form of `F.map (eq_to_hom _)` so that the lemmas about\nnaturality can be applied.\n-/\nlocal attribute [-simp] eq_to_hom_map eq_to_iso_map\n\n/-- An open immersion `f : X \u27f6 Y` induces an isomorphism `X \u2245 Y|_{f(X)}`. -/\n@[simps] noncomputable\ndef iso_restrict : X \u2245 Y.restrict H.base_open :=\nPresheafedSpace.iso_of_components (iso.refl _)\nbegin\n  symmetry,\n  fapply nat_iso.of_components,\n  intro U,\n  refine as_iso (f.c.app (op (H.open_functor.obj (unop U)))) \u226a\u226b X.presheaf.map_iso (eq_to_iso _),\n  { induction U using opposite.rec,\n    cases U,\n    dsimp only [is_open_map.functor, functor.op, opens.map],\n    congr' 2,\n    erw set.preimage_image_eq _ H.base_open.inj,\n    refl },\n  { intros U V i,\n    simp only [category_theory.eq_to_iso.hom, Top.presheaf.pushforward_obj_map, category.assoc,\n      functor.op_map, iso.trans_hom, as_iso_hom, functor.map_iso_hom, \u2190X.presheaf.map_comp],\n    erw [f.c.naturality_assoc, \u2190X.presheaf.map_comp],\n    congr }\nend\n\n@[simp] lemma iso_restrict_hom_of_restrict : H.iso_restrict.hom \u226b Y.of_restrict _ = f :=\nbegin\n  ext,\n  { simp only [comp_c_app, iso_restrict_hom_c_app, nat_trans.comp_app,\n      eq_to_hom_refl, of_restrict_c_app, category.assoc, whisker_right_id'],\n    erw [category.comp_id, f.c.naturality_assoc, \u2190X.presheaf.map_comp],\n    transitivity f.c.app x \u226b X.presheaf.map (\ud835\udfd9 _),\n    { congr },\n    { erw [X.presheaf.map_id, category.comp_id] } },\n  { simp }\nend\n\n@[simp] lemma iso_restrict_inv_of_restrict : H.iso_restrict.inv \u226b f = Y.of_restrict _ :=\nby { rw iso.inv_comp_eq, simp }\n\ninstance mono [H : is_open_immersion f] : mono f :=\nby { rw \u2190 H.iso_restrict_hom_of_restrict, apply mono_comp }\n\n/-- The composition of two open immersions is an open immersion. -/\ninstance comp {Z : PresheafedSpace C} (f : X \u27f6 Y) [hf : is_open_immersion f] (g : Y \u27f6 Z)\n  [hg : is_open_immersion g] :\n  is_open_immersion (f \u226b g) :=\n{ base_open := hg.base_open.comp hf.base_open,\n  c_iso := \u03bb U,\n  begin\n    generalize_proofs h,\n    dsimp only [algebraic_geometry.PresheafedSpace.comp_c_app, unop_op, functor.op, comp_base,\n      Top.presheaf.pushforward_obj_obj, opens.map_comp_obj],\n    apply_with is_iso.comp_is_iso { instances := ff },\n    swap,\n    { have : (opens.map g.base).obj (h.functor.obj U) = hf.open_functor.obj U,\n      { dsimp only [opens.map, is_open_map.functor, PresheafedSpace.comp_base],\n        congr' 1,\n        rw [coe_comp, \u2190set.image_image, set.preimage_image_eq _ hg.base_open.inj] },\n      rw this,\n      apply_instance },\n    { have : h.functor.obj U = hg.open_functor.obj (hf.open_functor.obj U),\n      { dsimp only [is_open_map.functor],\n        congr' 1,\n        rw [comp_base, coe_comp, \u2190set.image_image],\n        congr },\n      rw this,\n      apply_instance }\n  end }\n\n/-- For an open immersion `f : X \u27f6 Y` and an open set `U \u2286 X`, we have the map `X(U) \u27f6 Y(U)`. -/\nnoncomputable\ndef inv_app (U : opens X) : X.presheaf.obj (op U) \u27f6 Y.presheaf.obj (op (H.open_functor.obj U)) :=\nX.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) \u226b\n  inv (f.c.app (op (H.open_functor.obj U)))\n\n@[simp, reassoc] lemma inv_naturality {U V : (opens X)\u1d52\u1d56} (i : U \u27f6 V) :\n  X.presheaf.map i \u226b H.inv_app (unop V) = H.inv_app (unop U) \u226b\n    Y.presheaf.map (H.open_functor.op.map i) :=\nbegin\n  simp only [inv_app, \u2190category.assoc],\n  rw [is_iso.comp_inv_eq],\n  simp only [category.assoc, f.c.naturality, is_iso.inv_hom_id_assoc, \u2190 X.presheaf.map_comp],\n  erw \u2190 X.presheaf.map_comp,\n  congr\nend\n\ninstance (U : opens X) : is_iso (H.inv_app U) := by { delta inv_app, apply_instance }\n\nlemma inv_inv_app (U : opens X) :\n  inv (H.inv_app U) = f.c.app (op (H.open_functor.obj U)) \u226b\n    X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) :=\nbegin\n  rw \u2190 cancel_epi (H.inv_app U),\n  rw is_iso.hom_inv_id,\n  delta inv_app,\n  simp [\u2190 functor.map_comp]\nend\n\n@[simp, reassoc] lemma inv_app_app (U : opens X) :\n  H.inv_app U \u226b f.c.app (op (H.open_functor.obj U)) =\n    X.presheaf.map (eq_to_hom (by simp [opens.map, set.preimage_image_eq _ H.base_open.inj])) :=\nby rw [inv_app, category.assoc, is_iso.inv_hom_id, category.comp_id]\n\n@[simp, reassoc] lemma app_inv_app (U : opens Y) :\n  f.c.app (op U) \u226b H.inv_app ((opens.map f.base).obj U) =\n  Y.presheaf.map ((hom_of_le (by exact set.image_preimage_subset f.base U)).op :\n    op U \u27f6 op (H.open_functor.obj ((opens.map f.base).obj U))) :=\nby { erw \u2190 category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr }\n\n/-- A variant of `app_inv_app` that gives an `eq_to_hom` instead of `hom_of_le`. -/\n@[reassoc] lemma app_inv_app' (U : opens Y) (hU : (U : set Y) \u2286 set.range f.base) :\n  f.c.app (op U) \u226b H.inv_app ((opens.map f.base).obj U) =\n  Y.presheaf.map (eq_to_hom (by\n    { apply has_le.le.antisymm,\n      { exact set.image_preimage_subset f.base U.1 },\n      { change U \u2286 _,\n        refine has_le.le.trans_eq _ (@set.image_preimage_eq_inter_range _ _ f.base U.1).symm,\n        exact set.subset_inter_iff.mpr \u27e8\u03bb _ h, h, hU\u27e9 } })).op :=\nby { erw \u2190 category.assoc, rw [is_iso.comp_inv_eq, f.c.naturality], congr }\n\n/-- An isomorphism is an open immersion. -/\ninstance of_iso {X Y : PresheafedSpace C} (H : X \u2245 Y) : is_open_immersion H.hom :=\n{ base_open := (Top.homeo_of_iso ((forget C).map_iso H)).open_embedding,\n  c_iso := \u03bb _, infer_instance }\n\n@[priority 100]\ninstance of_is_iso {X Y : PresheafedSpace C} (f : X \u27f6 Y) [is_iso f] : is_open_immersion f :=\nalgebraic_geometry.PresheafedSpace.is_open_immersion.of_iso (as_iso f)\n\ninstance of_restrict {X : Top} (Y : PresheafedSpace C) {f : X \u27f6 Y.carrier}\n  (hf : open_embedding f) : is_open_immersion (Y.of_restrict hf) :=\n{ base_open := hf,\n  c_iso := \u03bb U,\n  begin\n    dsimp,\n    have : (opens.map f).obj (hf.is_open_map.functor.obj U) = U,\n    { cases U,\n      dsimp only [opens.map, is_open_map.functor],\n      congr' 1,\n      rw set.preimage_image_eq _ hf.inj,\n      refl },\n    convert (show is_iso (Y.presheaf.map (\ud835\udfd9 _)), from infer_instance),\n    { apply subsingleton.helim,\n      rw this },\n    { rw Y.presheaf.map_id,\n      apply_instance }\n  end }\n\n/-- An open immersion is an iso if the underlying continuous map is epi. -/\nlemma to_iso (f : X \u27f6 Y) [h : is_open_immersion f] [h' : epi f.base] : is_iso f :=\nbegin\n  apply_with is_iso_of_components { instances := ff },\n  { let : X \u2243\u209c Y := (homeomorph.of_embedding _ h.base_open.to_embedding).trans\n    { to_fun := subtype.val, inv_fun := \u03bb x, \u27e8x,\n      by { rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp h'), trivial }\u27e9,\n      left_inv := \u03bb \u27e8_,_\u27e9, rfl, right_inv := \u03bb _, rfl },\n    convert is_iso.of_iso (Top.iso_of_homeo this),\n    { ext, refl } },\n  { apply_with nat_iso.is_iso_of_is_iso_app { instances := ff },\n    intro U,\n    have : U = op (h.open_functor.obj ((opens.map f.base).obj (unop U))),\n    { induction U using opposite.rec,\n      cases U,\n      dsimp only [functor.op, opens.map],\n      congr,\n      exact (set.image_preimage_eq _ ((Top.epi_iff_surjective _).mp h')).symm },\n    convert @@is_open_immersion.c_iso _ h ((opens.map f.base).obj (unop U)) }\nend\n\ninstance stalk_iso [has_colimits C] [H : is_open_immersion f] (x : X) : is_iso (stalk_map f x) :=\nbegin\n  rw \u2190 H.iso_restrict_hom_of_restrict,\n  rw PresheafedSpace.stalk_map.comp,\n  apply_instance\nend\n\nend\n\nsection pullback\n\nnoncomputable theory\n\nvariables {X Y Z : PresheafedSpace C} (f : X \u27f6 Z) [hf : is_open_immersion f] (g : Y \u27f6 Z)\n\ninclude hf\n\n/--\n  (Implementation.) The projection map when constructing the pullback along an open immersion.\n-/\ndef pullback_cone_of_left_fst :\n  Y.restrict (Top.snd_open_embedding_of_left_open_embedding hf.base_open g.base) \u27f6 X :=\n{ base := pullback.fst,\n  c :=\n  { app := \u03bb U, hf.inv_app (unop U) \u226b\n      g.c.app (op (hf.base_open.is_open_map.functor.obj (unop U))) \u226b\n      Y.presheaf.map (eq_to_hom\n      (begin\n        simp only [is_open_map.functor, subtype.mk_eq_mk, unop_op, op_inj_iff, opens.map,\n        subtype.coe_mk, functor.op_obj, subtype.val_eq_coe],\n        apply has_le.le.antisymm,\n          { rintros _ \u27e8_, h\u2081, h\u2082\u27e9,\n            use (Top.pullback_iso_prod_subtype _ _).inv \u27e8\u27e8_, _\u27e9, h\u2082\u27e9,\n            simpa using h\u2081 },\n          { rintros _ \u27e8x, h\u2081, rfl\u27e9,\n            exact \u27e8_, h\u2081, concrete_category.congr_hom pullback.condition x\u27e9 }\n      end)),\n    naturality' :=\n    begin\n      intros U V i,\n      induction U using opposite.rec,\n      induction V using opposite.rec,\n      simp only [quiver.hom.unop_op, Top.presheaf.pushforward_obj_map, category.assoc,\n        nat_trans.naturality_assoc, functor.op_map, inv_naturality_assoc, \u2190 Y.presheaf.map_comp],\n      erw \u2190 Y.presheaf.map_comp,\n      congr\n    end } }\n\nlemma pullback_cone_of_left_condition :\n  pullback_cone_of_left_fst f g \u226b f = Y.of_restrict _ \u226b g :=\nbegin\n  ext U,\n  { induction U using opposite.rec,\n    dsimp only [comp_c_app, nat_trans.comp_app, unop_op,\n      whisker_right_app, pullback_cone_of_left_fst],\n    simp only [quiver.hom.unop_op, Top.presheaf.pushforward_obj_map, app_inv_app_assoc,\n      eq_to_hom_app, eq_to_hom_unop, category.assoc, nat_trans.naturality_assoc, functor.op_map],\n    erw [\u2190 Y.presheaf.map_comp, \u2190 Y.presheaf.map_comp],\n    congr },\n  { simpa using pullback.condition }\nend\n\n/--\nWe construct the pullback along an open immersion via restricting along the pullback of the\nmaps of underlying spaces (which is also an open embedding).\n-/\ndef pullback_cone_of_left : pullback_cone f g :=\npullback_cone.mk (pullback_cone_of_left_fst f g) (Y.of_restrict _)\n  (pullback_cone_of_left_condition f g)\n\nvariable (s : pullback_cone f g)\n\n/--\n  (Implementation.) Any cone over `cospan f g` indeed factors through the constructed cone.\n-/\ndef pullback_cone_of_left_lift : s.X \u27f6 (pullback_cone_of_left f g).X :=\n{ base := pullback.lift s.fst.base s.snd.base\n    (congr_arg (\u03bb x, PresheafedSpace.hom.base x) s.condition),\n  c :=\n  { app := \u03bb U, s.snd.c.app _ \u226b s.X.presheaf.map (eq_to_hom (begin\n      dsimp only [opens.map, is_open_map.functor, functor.op],\n      congr' 2,\n      let s' : pullback_cone f.base g.base := pullback_cone.mk s.fst.base s.snd.base _,\n      have : _ = s.snd.base := limit.lift_\u03c0 s' walking_cospan.right,\n      conv_lhs { erw \u2190 this, rw coe_comp, erw \u2190 set.preimage_preimage },\n      erw set.preimage_image_eq _\n        (Top.snd_open_embedding_of_left_open_embedding hf.base_open g.base).inj,\n      simp,\n    end)),\n    naturality' := \u03bb U V i,\n    begin\n      erw s.snd.c.naturality_assoc,\n      rw category.assoc,\n      erw [\u2190 s.X.presheaf.map_comp, \u2190 s.X.presheaf.map_comp],\n      congr\n    end } }\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\nlemma pullback_cone_of_left_lift_fst :\n  pullback_cone_of_left_lift f g s \u226b (pullback_cone_of_left f g).fst = s.fst :=\nbegin\n  ext x,\n  { induction x using opposite.rec,\n    change ((_ \u226b _) \u226b _ \u226b _) \u226b _ = _,\n    simp_rw [category.assoc],\n    erw \u2190 s.X.presheaf.map_comp,\n    erw s.snd.c.naturality_assoc,\n    have := congr_app s.condition (op (hf.open_functor.obj x)),\n    dsimp only [comp_c_app, unop_op] at this,\n    rw \u2190 is_iso.comp_inv_eq at this,\n    reassoc! this,\n    erw [\u2190 this, hf.inv_app_app_assoc, s.fst.c.naturality_assoc],\n    simpa },\n  { change pullback.lift _ _ _ \u226b pullback.fst = _,\n    simp }\nend\n\n-- this lemma is not a `simp` lemma, because it is an implementation detail\nlemma pullback_cone_of_left_lift_snd :\n  pullback_cone_of_left_lift f g s \u226b (pullback_cone_of_left f g).snd = s.snd :=\nbegin\n  ext x,\n  { change (_ \u226b _ \u226b _) \u226b _ = _,\n    simp_rw category.assoc,\n    erw s.snd.c.naturality_assoc,\n    erw [\u2190 s.X.presheaf.map_comp, \u2190 s.X.presheaf.map_comp],\n    transitivity s.snd.c.app x \u226b s.X.presheaf.map (\ud835\udfd9 _),\n    { congr },\n    { rw s.X.presheaf.map_id, erw category.comp_id } },\n  { change pullback.lift _ _ _ \u226b pullback.snd = _,\n    simp }\nend\n\ninstance pullback_cone_snd_is_open_immersion :\n  is_open_immersion (pullback_cone_of_left f g).snd :=\nbegin\n  erw category_theory.limits.pullback_cone.mk_snd,\n  apply_instance\nend\n\n/-- The constructed pullback cone is indeed the pullback. -/\ndef pullback_cone_of_left_is_limit :\n  is_limit (pullback_cone_of_left f g) :=\nbegin\n  apply pullback_cone.is_limit_aux',\n  intro s,\n  use pullback_cone_of_left_lift f g s,\n  use pullback_cone_of_left_lift_fst f g s,\n  use pullback_cone_of_left_lift_snd f g s,\n  intros m h\u2081 h\u2082,\n  rw \u2190 cancel_mono (pullback_cone_of_left f g).snd,\n  exact (h\u2082.trans (pullback_cone_of_left_lift_snd f g s).symm)\nend\n\ninstance has_pullback_of_left :\n  has_pullback f g :=\n\u27e8\u27e8\u27e8_, pullback_cone_of_left_is_limit f g\u27e9\u27e9\u27e9\n\ninstance has_pullback_of_right :\n  has_pullback g f := has_pullback_symmetry f g\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_snd_of_left :\n  is_open_immersion (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  delta pullback.snd,\n  rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right,\n  apply_instance\nend\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_fst_of_right :\n  is_open_immersion (pullback.fst : pullback g f \u27f6 _) :=\nbegin\n  rw \u2190 pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one_is_open_immersion [is_open_immersion g] :\n  is_open_immersion (limit.\u03c0 (cospan f g) walking_cospan.one) :=\nbegin\n  rw [\u2190limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],\n  apply_instance\nend\n\ninstance forget_preserves_limits_of_left : preserves_limit (cospan f g) (forget C) :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit.postcompose_hom_equiv (diagram_iso_cospan.{v} _) _).to_fun,\n  refine (is_limit.equiv_iso_limit _).to_fun (limit.is_limit (cospan f.base g.base)),\n  fapply cones.ext,\n  exact (iso.refl _),\n  change \u2200 j, _ = \ud835\udfd9 _ \u226b _ \u226b _,\n  simp_rw category.id_comp,\n  rintros (_|_|_); symmetry,\n  { erw category.comp_id,\n    exact limit.w (cospan f.base g.base) walking_cospan.hom.inl },\n  { exact category.comp_id _ },\n  { exact category.comp_id _ },\nend\n\ninstance forget_preserves_limits_of_right : preserves_limit (cospan g f) (forget C) :=\npreserves_pullback_symmetry (forget C) f g\n\nlemma pullback_snd_is_iso_of_range_subset (H : set.range g.base \u2286 set.range f.base) :\n  is_iso (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  haveI := Top.snd_iso_of_left_embedding_range_subset hf.base_open.to_embedding g.base H,\n  haveI : is_iso (pullback.snd : pullback f g \u27f6 _).base,\n  { delta pullback.snd,\n    rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right,\n    change is_iso (_ \u226b pullback.snd),\n    apply_instance },\n  apply to_iso\nend\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H : set.range g.base \u2286 set.range f.base) : Y \u27f6 X :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H,\n  exact inv (pullback.snd : pullback f g \u27f6 _) \u226b pullback.fst,\nend\n\n@[simp, reassoc] lemma lift_fac (H : set.range g.base \u2286 set.range f.base) :\n  lift f g H \u226b f = g :=\nby { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition }\n\nlemma lift_uniq (H : set.range g.base \u2286 set.range f.base) (l : Y \u27f6 X)\n  (hl : l \u226b f = g) : l = lift f g H :=\nby rw [\u2190 cancel_mono f, hl, lift_fac]\n\n/-- Two open immersions with equal range is isomorphic. -/\n@[simps] def iso_of_range_eq [is_open_immersion g] (e : set.range f.base = set.range g.base) :\n  X \u2245 Y :=\n{ hom := lift g f (le_of_eq e),\n  inv := lift f g (le_of_eq e.symm),\n  hom_inv_id' := by { rw \u2190 cancel_mono f, simp },\n  inv_hom_id' := by { rw \u2190 cancel_mono g, simp } }\n\nend pullback\n\nopen category_theory.limits.walking_cospan\n\nsection to_SheafedSpace\n\nvariables [has_products C] {X : PresheafedSpace C} (Y : SheafedSpace C)\nvariables (f : X \u27f6 Y.to_PresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a SheafedSpace, then so is `X`. -/\ndef to_SheafedSpace : SheafedSpace C :=\n{ is_sheaf :=\n  begin\n    apply Top.presheaf.is_sheaf_of_iso (sheaf_iso_of_iso H.iso_restrict.symm).symm,\n    apply Top.sheaf.pushforward_sheaf_of_sheaf,\n    exact (Y.restrict H.base_open).is_sheaf\n  end,\n  to_PresheafedSpace := X }\n\n@[simp] lemma to_SheafedSpace_to_PresheafedSpace : (to_SheafedSpace Y f).to_PresheafedSpace = X :=\nrfl\n\n/--\nIf `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a SheafedSpace, we can\nupgrade it into a morphism of SheafedSpaces.\n-/\ndef to_SheafedSpace_hom : to_SheafedSpace Y f \u27f6 Y := f\n\n@[simp] lemma to_SheafedSpace_hom_base : (to_SheafedSpace_hom Y f).base = f.base := rfl\n\n@[simp] lemma to_SheafedSpace_hom_c : (to_SheafedSpace_hom Y f).c = f.c := rfl\n\ninstance to_SheafedSpace_is_open_immersion :\n  SheafedSpace.is_open_immersion (to_SheafedSpace_hom Y f) := H\n\nomit H\n\n@[simp] lemma SheafedSpace_to_SheafedSpace {X Y : SheafedSpace C} (f : X \u27f6 Y)\n  [is_open_immersion f] : to_SheafedSpace Y f = X := by unfreezingI { cases X, refl }\n\nend to_SheafedSpace\n\nsection to_LocallyRingedSpace\n\nvariables {X : PresheafedSpace CommRing.{u}} (Y : LocallyRingedSpace.{u})\nvariables (f : X \u27f6 Y.to_PresheafedSpace) [H : is_open_immersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a LocallyRingedSpace, then so is `X`. -/\ndef to_LocallyRingedSpace : LocallyRingedSpace :=\n{ to_SheafedSpace := to_SheafedSpace Y.to_SheafedSpace f,\n  local_ring := \u03bb x, begin\n    haveI : local_ring (Y.to_SheafedSpace.to_PresheafedSpace.stalk (f.base x)) := Y.local_ring _,\n    exact (as_iso (stalk_map f x)).CommRing_iso_to_ring_equiv.local_ring\n  end }\n\n@[simp] lemma to_LocallyRingedSpace_to_SheafedSpace :\n  (to_LocallyRingedSpace Y f).to_SheafedSpace = (to_SheafedSpace Y.1 f) := rfl\n\n/--\nIf `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a LocallyRingedSpace, we can\nupgrade it into a morphism of LocallyRingedSpace.\n-/\ndef to_LocallyRingedSpace_hom : to_LocallyRingedSpace Y f \u27f6 Y := \u27e8f, \u03bb x, infer_instance\u27e9\n\n@[simp] lemma to_LocallyRingedSpace_hom_val :\n  (to_LocallyRingedSpace_hom Y f).val = f := rfl\n\ninstance to_LocallyRingedSpace_is_open_immersion :\n  LocallyRingedSpace.is_open_immersion (to_LocallyRingedSpace_hom Y f) := H\n\nomit H\n\n@[simp] lemma LocallyRingedSpace_to_LocallyRingedSpace {X Y : LocallyRingedSpace} (f : X \u27f6 Y)\n  [LocallyRingedSpace.is_open_immersion f] :\n  @to_LocallyRingedSpace X.to_PresheafedSpace Y (@@coe (@@coe_to_lift (@@coe_base coe_subtype)) f)\n    (show is_open_immersion f.val, by apply_instance) = X :=\nby unfreezingI { cases X, delta to_LocallyRingedSpace, simp }\n\nend to_LocallyRingedSpace\n\nend PresheafedSpace.is_open_immersion\n\nnamespace SheafedSpace.is_open_immersion\n\nvariables [has_products C]\n\n@[priority 100]\ninstance of_is_iso {X Y : SheafedSpace C} (f : X \u27f6 Y) [is_iso f] :\n  SheafedSpace.is_open_immersion f :=\n@@PresheafedSpace.is_open_immersion.of_is_iso _ f\n(SheafedSpace.forget_to_PresheafedSpace.map_is_iso _)\n\ninstance comp {X Y Z : SheafedSpace C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [SheafedSpace.is_open_immersion f] [SheafedSpace.is_open_immersion g] :\n  SheafedSpace.is_open_immersion (f \u226b g) := PresheafedSpace.is_open_immersion.comp f g\n\nsection pullback\n\nvariables {X Y Z : SheafedSpace C} (f : X \u27f6 Z) (g : Y \u27f6 Z)\nvariable [H : SheafedSpace.is_open_immersion f]\n\ninclude H\n\nlocal notation `forget` := SheafedSpace.forget_to_PresheafedSpace\nopen category_theory.limits.walking_cospan\n\ninstance : mono f := faithful_reflects_mono forget\n  (show @mono (PresheafedSpace C) _ _ _ f, by apply_instance)\n\ninstance forget_map_is_open_immersion :\n  PresheafedSpace.is_open_immersion (forget .map f) := \u27e8H.base_open, H.c_iso\u27e9\n\ninstance has_limit_cospan_forget_of_left : has_limit (cospan f g \u22d9 forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,\n  change has_limit (cospan (forget .map f) (forget .map g)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_left' : has_limit (cospan ((cospan f g \u22d9 forget).map hom.inl)\n  ((cospan f g \u22d9 forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map f) (forget .map g)), from infer_instance\n\ninstance has_limit_cospan_forget_of_right : has_limit (cospan g f \u22d9 forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{v} _).symm,\n  change has_limit (cospan (forget .map g) (forget .map f)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_right' : has_limit (cospan ((cospan g f \u22d9 forget).map hom.inl)\n  ((cospan g f \u22d9 forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map g) (forget .map f)), from infer_instance\n\n\ninstance forget_creates_pullback_of_left : creates_limit (cospan f g) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_SheafedSpace Y\n    (@pullback.snd (PresheafedSpace C) _ _ _ _ f g _))\n  (eq_to_iso (show pullback _ _ = pullback _ _, by congr)\n    \u226a\u226b has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_creates_pullback_of_right : creates_limit (cospan g f) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_SheafedSpace Y\n    (@pullback.fst (PresheafedSpace C) _ _ _ _ g f _))\n  (eq_to_iso (show pullback _ _ = pullback _ _, by congr)\n    \u226a\u226b has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance SheafedSpace_forget_preserves_of_left :\n  preserves_limit (cospan f g) (SheafedSpace.forget C) :=\n@@limits.comp_preserves_limit _ _ _ _ forget (PresheafedSpace.forget C) _\nbegin\n  apply_with (preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{v} _).symm) { instances := tt },\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_forget_preserves_of_right :\n  preserves_limit (cospan g f) (SheafedSpace.forget C) :=\npreserves_pullback_symmetry _ _ _\n\ninstance SheafedSpace_has_pullback_of_left : has_pullback f g :=\n  has_limit_of_created (cospan f g) forget\n\ninstance SheafedSpace_has_pullback_of_right : has_pullback g f :=\n  has_limit_of_created (cospan g f) forget\n\n/-- Open immersions are stable under base-change. -/\ninstance SheafedSpace_pullback_snd_of_left :\n  SheafedSpace.is_open_immersion (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  delta pullback.snd,\n  have : _ = limit.\u03c0 (cospan f g) right := preserves_limits_iso_hom_\u03c0\n      forget (cospan f g) right,\n  rw \u2190 this,\n  have := has_limit.iso_of_nat_iso_hom_\u03c0\n    (diagram_iso_cospan.{v} (cospan f g \u22d9 forget))\n    right,\n  erw category.comp_id at this,\n  rw \u2190 this,\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_pullback_fst_of_right :\n  SheafedSpace.is_open_immersion (pullback.fst : pullback g f \u27f6 _) :=\nbegin\n  delta pullback.fst,\n  have : _ = limit.\u03c0 (cospan g f) left := preserves_limits_iso_hom_\u03c0\n      forget (cospan g f) left,\n  rw \u2190 this,\n  have := has_limit.iso_of_nat_iso_hom_\u03c0\n    (diagram_iso_cospan.{v} (cospan g f \u22d9 forget)) left,\n  erw category.comp_id at this,\n  rw \u2190 this,\n  dsimp,\n  apply_instance\nend\n\ninstance SheafedSpace_pullback_one_is_open_immersion [SheafedSpace.is_open_immersion g] :\n  SheafedSpace.is_open_immersion (limit.\u03c0 (cospan f g) one : pullback f g \u27f6 Z) :=\nbegin\n  rw [\u2190limit.w (cospan f g) hom.inl, cospan_map_inl],\n  apply_instance\nend\n\nend pullback\n\nsection of_stalk_iso\nvariables [has_limits C] [has_colimits C] [concrete_category.{v} C]\nvariables [reflects_isomorphisms (forget C)] [preserves_limits (forget C)]\nvariables [preserves_filtered_colimits (forget C)]\n\n/--\nSuppose `X Y : SheafedSpace C`, where `C` is a concrete category,\nwhose forgetful functor reflects isomorphisms, preserves limits and filtered colimits.\nThen a morphism `X \u27f6 Y` that is a topological open embedding\nis an open immersion iff every stalk map is an iso.\n-/\nlemma of_stalk_iso {X Y : SheafedSpace C} (f : X \u27f6 Y)\n  (hf : open_embedding f.base) [H : \u2200 x : X, is_iso (PresheafedSpace.stalk_map f x)] :\n  SheafedSpace.is_open_immersion f :=\n{ base_open := hf,\n  c_iso := \u03bb U, begin\n    apply_with (Top.presheaf.app_is_iso_of_stalk_functor_map_iso\n      (show Y.sheaf \u27f6 (Top.sheaf.pushforward f.base).obj X.sheaf, from f.c)) { instances := ff },\n    rintros \u27e8_, y, hy, rfl\u27e9,\n    specialize H y,\n    delta PresheafedSpace.stalk_map at H,\n    haveI H' := Top.presheaf.stalk_pushforward.stalk_pushforward_iso_of_open_embedding\n      C hf X.presheaf y,\n    have := @@is_iso.comp_is_iso _ H (@@is_iso.inv_is_iso _ H'),\n    rw [category.assoc, is_iso.hom_inv_id, category.comp_id] at this,\n    exact this\n  end }\n\nend of_stalk_iso\n\nsection prod\n\nvariables [has_limits C] {\u03b9 : Type v} (F : discrete \u03b9 \u2964 SheafedSpace C) [has_colimit F] (i : \u03b9)\n\nlemma sigma_\u03b9_open_embedding : open_embedding (colimit.\u03b9 F i).base :=\nbegin\n  rw \u2190 (show _ = (colimit.\u03b9 F i).base,\n    from \u03b9_preserves_colimits_iso_inv (SheafedSpace.forget C) F i),\n  have : _ = _ \u226b colimit.\u03b9 (discrete.functor (F \u22d9 SheafedSpace.forget C).obj) i :=\n    has_colimit.iso_of_nat_iso_\u03b9_hom discrete.nat_iso_functor i,\n  rw \u2190 iso.eq_comp_inv at this,\n  rw this,\n  have : colimit.\u03b9 _ _ \u226b _ = _ := Top.sigma_iso_sigma_hom_\u03b9 (F \u22d9 SheafedSpace.forget C).obj i,\n  rw \u2190 iso.eq_comp_inv at this,\n  rw this,\n  simp_rw [\u2190 category.assoc, Top.open_embedding_iff_comp_is_iso,\n    Top.open_embedding_iff_is_iso_comp],\n  exact open_embedding_sigma_mk\nend\n\nlemma image_preimage_is_empty (j : \u03b9) (h : i \u2260 j) (U : opens (F.obj i)) :\n  (opens.map (colimit.\u03b9 (F \u22d9 SheafedSpace.forget_to_PresheafedSpace) j).base).obj\n    ((opens.map (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj\n    ((sigma_\u03b9_open_embedding F i).is_open_map.functor.obj U)) = \u2205 :=\nbegin\n  ext,\n  apply iff_false_intro,\n  rintro \u27e8y, hy, eq\u27e9,\n  replace eq := concrete_category.congr_arg\n    (preserves_colimit_iso (SheafedSpace.forget C) F \u226a\u226b\n      has_colimit.iso_of_nat_iso discrete.nat_iso_functor \u226a\u226b Top.sigma_iso_sigma _).hom eq,\n  simp_rw [category_theory.iso.trans_hom, \u2190 Top.comp_app, \u2190 PresheafedSpace.comp_base] at eq,\n  rw \u03b9_preserves_colimits_iso_inv at eq,\n  change ((SheafedSpace.forget C).map (colimit.\u03b9 F i) \u226b _) y =\n    ((SheafedSpace.forget C).map (colimit.\u03b9 F j) \u226b _) x at eq,\n  rw [\u03b9_preserves_colimits_iso_hom_assoc, \u03b9_preserves_colimits_iso_hom_assoc,\n    has_colimit.iso_of_nat_iso_\u03b9_hom_assoc, has_colimit.iso_of_nat_iso_\u03b9_hom_assoc,\n    Top.sigma_iso_sigma_hom_\u03b9, Top.sigma_iso_sigma_hom_\u03b9] at eq,\n  exact h (congr_arg sigma.fst eq)\nend\n\ninstance sigma_\u03b9_is_open_immersion [has_strict_terminal_objects C] :\n  SheafedSpace.is_open_immersion (colimit.\u03b9 F i) :=\n{ base_open := sigma_\u03b9_open_embedding F i,\n  c_iso := \u03bb U, begin\n    have e : colimit.\u03b9 F i = _ :=\n      (\u03b9_preserves_colimits_iso_inv SheafedSpace.forget_to_PresheafedSpace F i).symm,\n    have H : open_embedding (colimit.\u03b9 (F \u22d9 SheafedSpace.forget_to_PresheafedSpace) i \u226b\n      (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).base :=\n      e \u25b8 sigma_\u03b9_open_embedding F i,\n    suffices : is_iso ((colimit.\u03b9 (F \u22d9 SheafedSpace.forget_to_PresheafedSpace) i \u226b\n      (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv).c.app\n        (op (H.is_open_map.functor.obj U))),\n    { convert this },\n    rw [PresheafedSpace.comp_c_app,\n      \u2190 PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_\u03c0],\n    suffices : is_iso (limit.\u03c0 (PresheafedSpace.componentwise_diagram\n      (F \u22d9 SheafedSpace.forget_to_PresheafedSpace)\n      ((opens.map (preserves_colimit_iso SheafedSpace.forget_to_PresheafedSpace F).inv.base).obj\n      (unop $ op $ H.is_open_map.functor.obj U))) (op i)),\n    { resetI, apply_instance },\n    apply limit_\u03c0_is_iso_of_is_strict_terminal,\n    intros j hj,\n    induction j using opposite.rec,\n    dsimp,\n    convert (F.obj j).sheaf.is_terminal_of_empty,\n    convert image_preimage_is_empty F i j (\u03bb h, hj (congr_arg op h.symm)) U,\n    exact (congr_arg PresheafedSpace.hom.base e).symm\n  end }\n\nend prod\n\nend SheafedSpace.is_open_immersion\n\nnamespace LocallyRingedSpace.is_open_immersion\n\nsection pullback\n\nvariables {X Y Z : LocallyRingedSpace.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z)\nvariable [H : LocallyRingedSpace.is_open_immersion f]\n\n@[priority 100]\ninstance of_is_iso [is_iso g] :\n  LocallyRingedSpace.is_open_immersion g :=\n@@PresheafedSpace.is_open_immersion.of_is_iso _ g.1 \u27e8\u27e8(inv g).1,\n  by { erw \u2190 LocallyRingedSpace.comp_val, rw is_iso.hom_inv_id,\n    erw \u2190 LocallyRingedSpace.comp_val, rw is_iso.inv_hom_id, split; simpa }\u27e9\u27e9\n\ninclude H\n\ninstance comp (g : Z \u27f6 Y) [LocallyRingedSpace.is_open_immersion g] :\n  LocallyRingedSpace.is_open_immersion (f \u226b g) := PresheafedSpace.is_open_immersion.comp f.1 g.1\n\ninstance mono : mono f :=\nfaithful_reflects_mono (LocallyRingedSpace.forget_to_SheafedSpace)\n  (show mono f.1, by apply_instance)\n\ninstance : SheafedSpace.is_open_immersion (LocallyRingedSpace.forget_to_SheafedSpace.map f) := H\n\n/-- An explicit pullback cone over `cospan f g` if `f` is an open immersion. -/\ndef pullback_cone_of_left : pullback_cone f g :=\nbegin\n  refine pullback_cone.mk _\n    (Y.of_restrict (Top.snd_open_embedding_of_left_open_embedding H.base_open g.1.base)) _,\n  { use PresheafedSpace.is_open_immersion.pullback_cone_of_left_fst f.1 g.1,\n    intro x,\n    have := PresheafedSpace.stalk_map.congr_hom _ _\n      (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition f.1 g.1) x,\n    rw [PresheafedSpace.stalk_map.comp, PresheafedSpace.stalk_map.comp] at this,\n    rw \u2190 is_iso.eq_inv_comp at this,\n    rw this,\n    apply_instance },\n  { exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_condition _ _) },\nend\n\ninstance : LocallyRingedSpace.is_open_immersion (pullback_cone_of_left f g).snd :=\nshow PresheafedSpace.is_open_immersion (Y.to_PresheafedSpace.of_restrict _), by apply_instance\n\n/-- The constructed `pullback_cone_of_left` is indeed limiting. -/\ndef pullback_cone_of_left_is_limit : is_limit (pullback_cone_of_left f g) :=\npullback_cone.is_limit_aux' _ $ \u03bb s,\nbegin\n  use PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift f.1 g.1\n    (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition)),\n  { intro x,\n    have := PresheafedSpace.stalk_map.congr_hom _ _\n      (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n        (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition))) x,\n    change _ = _ \u226b PresheafedSpace.stalk_map s.snd.1 x at this,\n    rw [PresheafedSpace.stalk_map.comp, \u2190 is_iso.eq_inv_comp] at this,\n    rw this,\n    apply_instance },\n  split,\n  exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_fst f.1 g.1 _),\n  split,\n  exact subtype.eq (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1 _),\n  intros m h\u2081 h\u2082,\n  rw \u2190 cancel_mono (pullback_cone_of_left f g).snd,\n  exact (h\u2082.trans (subtype.eq\n    (PresheafedSpace.is_open_immersion.pullback_cone_of_left_lift_snd f.1 g.1\n      (pullback_cone.mk s.fst.1 s.snd.1 (congr_arg subtype.val s.condition))).symm))\nend\n\ninstance has_pullback_of_left :\n  has_pullback f g :=\n\u27e8\u27e8\u27e8_, pullback_cone_of_left_is_limit f g\u27e9\u27e9\u27e9\n\ninstance has_pullback_of_right :\n  has_pullback g f := has_pullback_symmetry f g\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_snd_of_left :\n  LocallyRingedSpace.is_open_immersion (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  delta pullback.snd,\n  rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, pullback_cone_of_left_is_limit f g\u27e9 walking_cospan.right,\n  apply_instance\nend\n\n/-- Open immersions are stable under base-change. -/\ninstance pullback_fst_of_right :\nLocallyRingedSpace.is_open_immersion (pullback.fst : pullback g f \u27f6 _) :=\nbegin\n  rw \u2190 pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one_is_open_immersion [LocallyRingedSpace.is_open_immersion g] :\n  LocallyRingedSpace.is_open_immersion (limit.\u03c0 (cospan f g) walking_cospan.one) :=\nbegin\n  rw [\u2190limit.w (cospan f g) walking_cospan.hom.inl, cospan_map_inl],\n  apply_instance\nend\n\ninstance forget_preserves_pullback_of_left :\n  preserves_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,\n  apply is_limit_of_is_limit_pullback_cone_map SheafedSpace.forget_to_PresheafedSpace,\n  exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1\nend\n\ninstance forget_to_PresheafedSpace_preserves_pullback_of_left :\n  preserves_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) :=\npreserves_limit_of_preserves_limit_cone (pullback_cone_of_left_is_limit f g)\nbegin\n  apply (is_limit_map_cone_pullback_cone_equiv _ _).symm.to_fun,\n  exact PresheafedSpace.is_open_immersion.pullback_cone_of_left_is_limit f.1 g.1\nend\n\ninstance forget_to_PresheafedSpace_preserves_open_immersion :\n  PresheafedSpace.is_open_immersion ((LocallyRingedSpace.forget_to_SheafedSpace \u22d9\n    SheafedSpace.forget_to_PresheafedSpace).map f) := H\n\ninstance forget_to_Top_preserves_pullback_of_left :\n  preserves_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget _) :=\nbegin\n  change preserves_limit _\n    ((LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace)\n      \u22d9 PresheafedSpace.forget _),\n  apply_with limits.comp_preserves_limit { instances := ff },\n  apply_instance,\n  apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,\n  dsimp [SheafedSpace.forget_to_PresheafedSpace, -subtype.val_eq_coe],\n  apply_instance,\nend\n\ninstance forget_reflects_pullback_of_left :\n  reflects_limit (cospan f g) LocallyRingedSpace.forget_to_SheafedSpace :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_preserves_pullback_of_right :\n  preserves_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace :=\npreserves_pullback_symmetry _ _ _\n\ninstance forget_to_PresheafedSpace_preserves_pullback_of_right :\n  preserves_limit (cospan g f) (LocallyRingedSpace.forget_to_SheafedSpace \u22d9\n    SheafedSpace.forget_to_PresheafedSpace) :=\npreserves_pullback_symmetry _ _ _\n\ninstance forget_reflects_pullback_of_right :\n  reflects_limit (cospan g f) LocallyRingedSpace.forget_to_SheafedSpace :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_to_PresheafedSpace_reflects_pullback_of_left :\n  reflects_limit (cospan f g)\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) :=\nreflects_limit_of_reflects_isomorphisms _ _\n\ninstance forget_to_PresheafedSpace_reflects_pullback_of_right :\n  reflects_limit (cospan g f)\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) :=\nreflects_limit_of_reflects_isomorphisms _ _\n\nlemma pullback_snd_is_iso_of_range_subset (H' : set.range g.1.base \u2286 set.range f.1.base) :\n  is_iso (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  apply_with (reflects_isomorphisms.reflects LocallyRingedSpace.forget_to_SheafedSpace)\n    { instances := ff },\n  apply_with (reflects_isomorphisms.reflects SheafedSpace.forget_to_PresheafedSpace)\n    { instances := ff },\n  erw \u2190 preserves_pullback.iso_hom_snd\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget_to_PresheafedSpace) f g,\n  haveI := PresheafedSpace.is_open_immersion.pullback_snd_is_iso_of_range_subset _ _ H',\n  apply_instance,\n  apply_instance\nend\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H' : set.range g.1.base \u2286 set.range f.1.base) : Y \u27f6 X :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H',\n  exact inv (pullback.snd : pullback f g \u27f6 _) \u226b pullback.fst,\nend\n\n@[simp, reassoc] lemma lift_fac (H' : set.range g.1.base \u2286 set.range f.1.base) :\n  lift f g H' \u226b f = g :=\nby { erw category.assoc, rw is_iso.inv_comp_eq, exact pullback.condition }\n\nlemma lift_uniq (H' : set.range g.1.base \u2286 set.range f.1.base) (l : Y \u27f6 X)\n  (hl : l \u226b f = g) : l = lift f g H' :=\nby rw [\u2190 cancel_mono f, hl, lift_fac]\n\nlemma lift_range (H' : set.range g.1.base \u2286 set.range f.1.base) :\n  set.range (lift f g H').1.base = f.1.base \u207b\u00b9' (set.range g.1.base) :=\nbegin\n  haveI := pullback_snd_is_iso_of_range_subset f g H',\n  dsimp only [lift],\n  have : _ = (pullback.fst : pullback f g \u27f6 _).val.base := preserves_pullback.iso_hom_fst\n    (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget _) f g,\n  rw [LocallyRingedSpace.comp_val, SheafedSpace.comp_base, \u2190 this, \u2190 category.assoc, coe_comp],\n  rw [set.range_comp, set.range_iff_surjective.mpr, set.image_univ, Top.pullback_fst_range],\n  ext,\n  split,\n  { rintros \u27e8y, eq\u27e9, exact \u27e8y, eq.symm\u27e9 },\n  { rintros \u27e8y, eq\u27e9, exact \u27e8y, eq.symm\u27e9 },\n  { rw \u2190 Top.epi_iff_surjective,\n    rw (show (inv (pullback.snd : pullback f g \u27f6 _)).val.base = _, from\n      (LocallyRingedSpace.forget_to_SheafedSpace \u22d9 SheafedSpace.forget _).map_inv _),\n    apply_instance }\nend\n\nend pullback\n\n/-- An open immersion is isomorphic to the induced open subscheme on its image. -/\ndef iso_restrict {X Y : LocallyRingedSpace} {f : X \u27f6 Y}\n  (H : LocallyRingedSpace.is_open_immersion f) : X \u2245 Y.restrict H.base_open :=\nbegin\n  apply LocallyRingedSpace.iso_of_SheafedSpace_iso,\n  apply @preimage_iso _ _ _ _ SheafedSpace.forget_to_PresheafedSpace,\n  exact H.iso_restrict\nend\n\n/-- To show that a locally ringed space is a scheme, it suffices to show that it has a jointly\nsujective family of open immersions from affine schemes. -/\nprotected def Scheme (X : LocallyRingedSpace)\n  (h : \u2200 (x : X), \u2203 (R : CommRing) (f : Spec.to_LocallyRingedSpace.obj (op R) \u27f6 X),\n    (x \u2208 set.range f.1.base : _) \u2227 LocallyRingedSpace.is_open_immersion f) : Scheme :=\n{ to_LocallyRingedSpace := X,\n  local_affine :=\n  begin\n    intro x,\n    obtain \u27e8R, f, h\u2081, h\u2082\u27e9 := h x,\n    refine \u27e8\u27e8\u27e8_, h\u2082.base_open.open_range\u27e9, h\u2081\u27e9, R, \u27e8_\u27e9\u27e9,\n    apply LocallyRingedSpace.iso_of_SheafedSpace_iso,\n    apply @preimage_iso _ _ _ _ SheafedSpace.forget_to_PresheafedSpace,\n    resetI,\n    apply PresheafedSpace.is_open_immersion.iso_of_range_eq (PresheafedSpace.of_restrict _ _) f.1,\n    { exact subtype.range_coe_subtype },\n    { apply_instance }\n  end }\n\nend LocallyRingedSpace.is_open_immersion\n\nlemma is_open_immersion.open_range {X Y : Scheme} (f : X \u27f6 Y) [H : is_open_immersion f] :\n  is_open (set.range f.1.base) := H.base_open.open_range\n\nsection open_cover\n\nnamespace Scheme\n\n/-- An open cover of `X` consists of a family of open immersions into `X`,\nand for each `x : X` an open immersion (indexed by `f x`) that covers `x`.\n\nThis is merely a coverage in the Zariski pretopology, and it would be optimal\nif we could reuse the existing API about pretopologies, However, the definitions of sieves and\ngrothendieck topologies uses `Prop`s, so that the actual open sets and immersions are hard to\nobtain. Also, since such a coverage in the pretopology usually contains a proper class of\nimmersions, it is quite hard to glue them, reason about finite covers, etc.\n-/\n-- TODO: provide API to and from a presieve.\nstructure open_cover (X : Scheme.{u}) :=\n(J : Type v)\n(obj : \u03a0 (j : J), Scheme)\n(map : \u03a0 (j : J), obj j \u27f6 X)\n(f : X.carrier \u2192 J)\n(covers : \u2200 x, x \u2208 set.range ((map (f x)).1.base))\n(is_open : \u2200 x, is_open_immersion (map x) . tactic.apply_instance)\n\nattribute [instance] open_cover.is_open\n\nvariables {X Y Z : Scheme.{u}} (\ud835\udcb0 : open_cover X) (f : X \u27f6 Z) (g : Y \u27f6 Z)\nvariables [\u2200 x, has_pullback (\ud835\udcb0.map x \u226b f) g]\n\n/-- The affine cover of a scheme. -/\ndef affine_cover (X : Scheme) : open_cover X :=\n{ J := X.carrier,\n  obj := \u03bb x, Spec.obj $ opposite.op (X.local_affine x).some_spec.some,\n  map := \u03bb x, ((X.local_affine x).some_spec.some_spec.some.inv \u226b\n    X.to_LocallyRingedSpace.of_restrict _ : _),\n  f := \u03bb x, x,\n  is_open := \u03bb x, begin\n    apply_with PresheafedSpace.is_open_immersion.comp { instances := ff },\n    apply_instance,\n    apply PresheafedSpace.is_open_immersion.of_restrict,\n  end,\n  covers :=\n  begin\n    intro x,\n    erw coe_comp,\n    rw [set.range_comp, set.range_iff_surjective.mpr, set.image_univ],\n    erw subtype.range_coe_subtype,\n    exact (X.local_affine x).some.2,\n    rw \u2190 Top.epi_iff_surjective,\n    change epi ((SheafedSpace.forget _).map (LocallyRingedSpace.forget_to_SheafedSpace.map _)),\n    apply_instance\n  end }\n\ninstance : inhabited X.open_cover := \u27e8X.affine_cover\u27e9\n\n/-- Given an open cover `{ U\u1d62 }` of `X`, and for each `U\u1d62` an open cover, we may combine these\nopen covers to form an open cover of `X`.  -/\n@[simps J obj map]\ndef open_cover.bind (f : \u03a0 (x : \ud835\udcb0.J), open_cover (\ud835\udcb0.obj x)) : open_cover X :=\n{ J := \u03a3 (i : \ud835\udcb0.J), (f i).J,\n  obj := \u03bb x, (f x.1).obj x.2,\n  map := \u03bb x, (f x.1).map x.2 \u226b \ud835\udcb0.map x.1,\n  f := \u03bb x, \u27e8_, (f _).f (\ud835\udcb0.covers x).some\u27e9,\n  covers := \u03bb x,\n  begin\n    let y := (\ud835\udcb0.covers x).some,\n    have hy : (\ud835\udcb0.map (\ud835\udcb0.f x)).val.base y = x := (\ud835\udcb0.covers x).some_spec,\n    rcases (f (\ud835\udcb0.f x)).covers y with \u27e8z, hz\u27e9,\n    change 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    use z,\n    erw comp_apply,\n    rw [hz, hy],\n  end }\n\n/-- An isomorphism `X \u27f6 Y` is an open cover of `Y`. -/\n@[simps J obj map]\ndef open_cover_of_is_iso {X Y : Scheme.{u}} (f : X \u27f6 Y) [is_iso f] :\n  open_cover Y :=\n{ J := punit.{v+1},\n  obj := \u03bb _, X,\n  map := \u03bb _, f,\n  f := \u03bb _, punit.star,\n  covers := \u03bb x, by { rw set.range_iff_surjective.mpr, { trivial }, rw \u2190 Top.epi_iff_surjective,\n    apply_instance } }\n\n/-- We construct an open cover from another, by providing the needed fields and showing that the\nprovided fields are isomorphic with the original open cover. -/\n@[simps J obj map]\ndef open_cover.copy {X : Scheme} (\ud835\udcb0 : open_cover X)\n  (J : Type*) (obj : J \u2192 Scheme) (map : \u2200 i, obj i \u27f6 X)\n  (e\u2081 : J \u2243 \ud835\udcb0.J) (e\u2082 : \u2200 i, obj i \u2245 \ud835\udcb0.obj (e\u2081 i))\n  (e\u2082 : \u2200 i, map i = (e\u2082 i).hom \u226b \ud835\udcb0.map (e\u2081 i)) : open_cover X :=\n{ J := J,\n  obj := obj,\n  map := map,\n  f := \u03bb x, e\u2081.symm (\ud835\udcb0.f x),\n  covers := \u03bb x, begin\n    rw [e\u2082, Scheme.comp_val_base, coe_comp, set.range_comp, set.range_iff_surjective.mpr,\n      set.image_univ,  e\u2081.right_inverse_symm],\n    { exact \ud835\udcb0.covers x },\n    { rw \u2190 Top.epi_iff_surjective, apply_instance }\n  end,\n  is_open := \u03bb i, by { rw e\u2082, apply_instance } }\n\n/-- The pushforward of an open cover along an isomorphism. -/\n@[simps J obj map]\ndef open_cover.pushforward_iso {X Y : Scheme} (\ud835\udcb0 : open_cover X)\n  (f : X \u27f6 Y) [is_iso f] :\n  open_cover Y :=\n((open_cover_of_is_iso f).bind (\u03bb _, \ud835\udcb0)).copy \ud835\udcb0.J _ _\n  ((equiv.punit_prod _).symm.trans (equiv.sigma_equiv_prod punit \ud835\udcb0.J).symm)\n  (\u03bb _, iso.refl _)\n  (\u03bb _, (category.id_comp _).symm)\n\n-- Related result : `open_cover.pullback_cover`, where we pullback an open cover on `X` along a\n-- morphism `W \u27f6 X`. This is provided at the end of the file since it needs some more results\n-- about open immersion (which in turn needs the open cover API).\n\nlocal attribute [reducible] CommRing.of CommRing.of_hom\n\ninstance val_base_is_iso {X Y : Scheme} (f : X \u27f6 Y) [is_iso f] : is_iso f.1.base :=\nScheme.forget_to_Top.map_is_iso f\n\ninstance basic_open_is_open_immersion {R : CommRing} (f : R) :\nalgebraic_geometry.is_open_immersion (Scheme.Spec.map (CommRing.of_hom\n  (algebra_map R (localization.away f))).op) :=\nbegin\n  apply_with SheafedSpace.is_open_immersion.of_stalk_iso { instances := ff },\n  any_goals { apply_instance },\n  any_goals { apply_instance },\n  exact (prime_spectrum.localization_away_open_embedding (localization.away f) f : _),\n  intro x,\n  exact Spec_map_localization_is_iso R (submonoid.powers f) x,\nend\n\n/-- The basic open sets form an affine open cover of `Spec R`. -/\ndef affine_basis_cover_of_affine (R : CommRing) : open_cover (Spec.obj (opposite.op R)) :=\n{ J := R,\n  obj := \u03bb r, Spec.obj (opposite.op $ CommRing.of $ localization.away r),\n  map := \u03bb r, Spec.map (quiver.hom.op (algebra_map R (localization.away r) : _)),\n  f := \u03bb x, 1,\n  covers := \u03bb r,\n  begin\n    rw set.range_iff_surjective.mpr ((Top.epi_iff_surjective _).mp _),\n    { exact trivial },\n    { apply_instance }\n  end,\n  is_open := \u03bb x, algebraic_geometry.Scheme.basic_open_is_open_immersion x }\n\n/-- We may bind the basic open sets of an open affine cover to form a affine cover that is also\na basis. -/\ndef affine_basis_cover (X : Scheme) : open_cover X :=\nX.affine_cover.bind (\u03bb x, affine_basis_cover_of_affine _)\n\n/-- The coordinate ring of a component in the `affine_basis_cover`. -/\ndef affine_basis_cover_ring (X : Scheme) (i : X.affine_basis_cover.J) : CommRing :=\nCommRing.of $ @localization.away (X.local_affine i.1).some_spec.some _ i.2\n\nlemma affine_basis_cover_obj (X : Scheme) (i : X.affine_basis_cover.J) :\n  X.affine_basis_cover.obj i = Spec.obj (op $ X.affine_basis_cover_ring i) := rfl\n\nlemma affine_basis_cover_map_range (X : Scheme)\n  (x : X.carrier) (r : (X.local_affine x).some_spec.some) :\n  set.range (X.affine_basis_cover.map \u27e8x, r\u27e9).1.base =\n    (X.affine_cover.map x).1.base '' (prime_spectrum.basic_open r).1 :=\nbegin\n  erw [coe_comp, set.range_comp],\n  congr,\n  exact (prime_spectrum.localization_away_comap_range (localization.away r) r : _)\nend\n\nlemma affine_basis_cover_is_basis (X : Scheme) :\n  topological_space.is_topological_basis\n    { x : set X.carrier | \u2203 a : X.affine_basis_cover.J, x =\n      set.range ((X.affine_basis_cover.map a).1.base) } :=\nbegin\n  apply topological_space.is_topological_basis_of_open_of_nhds,\n  { rintros _ \u27e8a, rfl\u27e9,\n    exact is_open_immersion.open_range (X.affine_basis_cover.map a) },\n  { rintros a U haU hU,\n    rcases X.affine_cover.covers a with \u27e8x, e\u27e9,\n    let U' := (X.affine_cover.map (X.affine_cover.f a)).1.base \u207b\u00b9' U,\n    have hxU' : x \u2208 U' := by { rw \u2190 e at haU, exact haU },\n    rcases prime_spectrum.is_basis_basic_opens.exists_subset_of_mem_open hxU'\n      ((X.affine_cover.map (X.affine_cover.f a)).1.base.continuous_to_fun.is_open_preimage _ hU)\n      with \u27e8_,\u27e8_,\u27e8s,rfl\u27e9,rfl\u27e9,hxV,hVU\u27e9,\n    refine \u27e8_,\u27e8\u27e8_,s\u27e9,rfl\u27e9,_,_\u27e9; erw affine_basis_cover_map_range,\n    { exact \u27e8x,hxV,e\u27e9 },\n    { rw set.image_subset_iff, exact hVU } }\nend\n\n/--\nEvery open cover of a quasi-compact scheme can be refined into a finite subcover.\n-/\n@[simps obj map]\ndef open_cover.finite_subcover {X : Scheme} (\ud835\udcb0 : open_cover X) [H : compact_space X.carrier] :\n  open_cover X :=\nbegin\n  have := @@compact_space.elim_nhds_subcover _ H\n    (\u03bb (x : X.carrier), set.range ((\ud835\udcb0.map (\ud835\udcb0.f x)).1.base))\n    (\u03bb x, (is_open_immersion.open_range (\ud835\udcb0.map (\ud835\udcb0.f x))).mem_nhds (\ud835\udcb0.covers x)),\n  let t := this.some,\n  have h : \u2200 (x : X.carrier), \u2203 (y : t), x \u2208 set.range ((\ud835\udcb0.map (\ud835\udcb0.f y)).1.base),\n  { intro x,\n    have h' : x \u2208 (\u22a4 : set X.carrier) := trivial,\n    rw [\u2190 classical.some_spec this, set.mem_Union] at h',\n    rcases h' with \u27e8y,_,\u27e8hy,rfl\u27e9,hy'\u27e9,\n    exact \u27e8\u27e8y,hy\u27e9,hy'\u27e9 },\n  exact\n  { J := t,\n    obj := \u03bb x, \ud835\udcb0.obj (\ud835\udcb0.f x.1),\n    map := \u03bb x, \ud835\udcb0.map (\ud835\udcb0.f x.1),\n    f := \u03bb x, (h x).some,\n    covers := \u03bb x, (h x).some_spec }\nend\n\ninstance [H : compact_space X.carrier] : fintype \ud835\udcb0.finite_subcover.J :=\nby { delta open_cover.finite_subcover, apply_instance }\n\nend Scheme\n\nend open_cover\n\nnamespace PresheafedSpace.is_open_immersion\n\nsection to_Scheme\n\nvariables {X : PresheafedSpace CommRing.{u}} (Y : Scheme.{u})\nvariables (f : X \u27f6 Y.to_PresheafedSpace) [H : PresheafedSpace.is_open_immersion f]\n\ninclude H\n\n/-- If `X \u27f6 Y` is an open immersion, and `Y` is a scheme, then so is `X`. -/\ndef to_Scheme : Scheme :=\nbegin\n  apply LocallyRingedSpace.is_open_immersion.Scheme (to_LocallyRingedSpace _ f),\n  intro x,\n  obtain \u27e8_,\u27e8i,rfl\u27e9,hx,hi\u27e9 := Y.affine_basis_cover_is_basis.exists_subset_of_mem_open\n      (set.mem_range_self x) H.base_open.open_range,\n  use Y.affine_basis_cover_ring i,\n  use LocallyRingedSpace.is_open_immersion.lift (to_LocallyRingedSpace_hom _ f) _ hi,\n  split,\n  { rw LocallyRingedSpace.is_open_immersion.lift_range, exact hx },\n  { delta LocallyRingedSpace.is_open_immersion.lift, apply_instance }\nend\n\n@[simp] lemma to_Scheme_to_LocallyRingedSpace :\n  (to_Scheme Y f).to_LocallyRingedSpace = (to_LocallyRingedSpace Y.1 f) := rfl\n\n/--\nIf `X \u27f6 Y` is an open immersion of PresheafedSpaces, and `Y` is a Scheme, we can\nupgrade it into a morphism of Schemes.\n-/\ndef to_Scheme_hom : to_Scheme Y f \u27f6 Y := to_LocallyRingedSpace_hom _ f\n\n@[simp] \n\ninstance to_Scheme_hom_is_open_immersion :\n  is_open_immersion (to_Scheme_hom Y f) := H\n\nomit H\n\nlemma Scheme_eq_of_LocallyRingedSpace_eq {X Y : Scheme}\n  (H : X.to_LocallyRingedSpace = Y.to_LocallyRingedSpace) : X = Y :=\nby { cases X, cases Y, congr, exact H }\n\nlemma Scheme_to_Scheme {X Y : Scheme} (f : X \u27f6 Y) [is_open_immersion f] :\n  to_Scheme Y f.1 = X :=\nbegin\n  apply Scheme_eq_of_LocallyRingedSpace_eq,\n  exact LocallyRingedSpace_to_LocallyRingedSpace f\nend\n\nend to_Scheme\n\nend PresheafedSpace.is_open_immersion\n\n/-- The restriction of a Scheme along an open embedding. -/\n@[simps]\ndef Scheme.restrict {U : Top} (X : Scheme) {f : U \u27f6 Top.of X.carrier} (h : open_embedding f) :\n  Scheme :=\n{ to_PresheafedSpace := X.to_PresheafedSpace.restrict h,\n  ..(PresheafedSpace.is_open_immersion.to_Scheme X (X.to_PresheafedSpace.of_restrict h)) }\n\n/-- The canonical map from the restriction to the supspace. -/\n@[simps]\ndef Scheme.of_restrict {U : Top} (X : Scheme) {f : U \u27f6 Top.of X.carrier} (h : open_embedding f) :\n  X.restrict h \u27f6 X :=\nX.to_LocallyRingedSpace.of_restrict h\n\ninstance is_open_immersion.of_restrict {U : Top} (X : Scheme) {f : U \u27f6 Top.of X.carrier}\n  (h : open_embedding f) : is_open_immersion (X.of_restrict h) :=\nshow PresheafedSpace.is_open_immersion (X.to_PresheafedSpace.of_restrict h), by apply_instance\n\nnamespace is_open_immersion\n\nvariables {X Y Z : Scheme.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z)\nvariable [H : is_open_immersion f]\n\n@[priority 100]\ninstance of_is_iso [is_iso g] :\n  is_open_immersion g := @@LocallyRingedSpace.is_open_immersion.of_is_iso _\n(show is_iso ((induced_functor _).map g), by apply_instance)\n\n/-- A open immersion induces an isomorphism from the domain onto the image -/\ndef iso_restrict : X \u2245 (Z.restrict H.base_open : _) :=\n\u27e8H.iso_restrict.hom, H.iso_restrict.inv, H.iso_restrict.hom_inv_id, H.iso_restrict.inv_hom_id\u27e9\n\ninclude H\n\nlocal notation `forget` := Scheme.forget_to_LocallyRingedSpace\n\ninstance mono : mono f :=\nfaithful_reflects_mono (induced_functor _)\n  (show @mono LocallyRingedSpace _ _ _ f, by apply_instance)\n\ninstance forget_map_is_open_immersion : LocallyRingedSpace.is_open_immersion (forget .map f) :=\n\u27e8H.base_open, H.c_iso\u27e9\n\ninstance has_limit_cospan_forget_of_left :\n  has_limit (cospan f g \u22d9 Scheme.forget_to_LocallyRingedSpace) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,\n  change has_limit (cospan (forget .map f) (forget .map g)),\n  apply_instance\nend\n\nopen category_theory.limits.walking_cospan\n\ninstance has_limit_cospan_forget_of_left' :\n  has_limit (cospan ((cospan f g \u22d9 forget).map hom.inl)\n  ((cospan f g \u22d9 forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map f) (forget .map g)), from infer_instance\n\ninstance has_limit_cospan_forget_of_right : has_limit (cospan g f \u22d9 forget) :=\nbegin\n  apply has_limit_of_iso (diagram_iso_cospan.{u} _).symm,\n  change has_limit (cospan (forget .map g) (forget .map f)),\n  apply_instance\nend\n\ninstance has_limit_cospan_forget_of_right' :\n  has_limit (cospan ((cospan g f \u22d9 forget).map hom.inl)\n  ((cospan g f \u22d9 forget).map hom.inr)) :=\nshow has_limit (cospan (forget .map g) (forget .map f)), from infer_instance\n\ninstance forget_creates_pullback_of_left : creates_limit (cospan f g) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_Scheme Y\n    (@pullback.snd LocallyRingedSpace _ _ _ _ f g _).1)\n  (eq_to_iso (by simp) \u226a\u226b has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_creates_pullback_of_right : creates_limit (cospan g f) forget :=\ncreates_limit_of_fully_faithful_of_iso\n  (PresheafedSpace.is_open_immersion.to_Scheme Y\n    (@pullback.fst LocallyRingedSpace _ _ _ _ g f _).1)\n  (eq_to_iso (by simp) \u226a\u226b has_limit.iso_of_nat_iso (diagram_iso_cospan _).symm)\n\ninstance forget_preserves_of_left : preserves_limit (cospan f g) forget :=\ncategory_theory.preserves_limit_of_creates_limit_and_has_limit _ _\n\ninstance forget_preserves_of_right : preserves_limit (cospan g f) forget :=\npreserves_pullback_symmetry _ _ _\n\ninstance has_pullback_of_left : has_pullback f g :=\nhas_limit_of_created (cospan f g) forget\n\ninstance has_pullback_of_right : has_pullback g f :=\nhas_limit_of_created (cospan g f) forget\n\ninstance pullback_snd_of_left : is_open_immersion (pullback.snd : pullback f g \u27f6 _) :=\nbegin\n  have := preserves_pullback.iso_hom_snd forget f g,\n  dsimp only [Scheme.forget_to_LocallyRingedSpace, induced_functor_map] at this,\n  rw \u2190 this,\n  change LocallyRingedSpace.is_open_immersion _,\n  apply_instance\nend\n\ninstance pullback_fst_of_right : is_open_immersion (pullback.fst : pullback g f \u27f6 _) :=\nbegin\n  rw \u2190 pullback_symmetry_hom_comp_snd,\n  apply_instance\nend\n\ninstance pullback_one [is_open_immersion g] :\n  is_open_immersion (limit.\u03c0 (cospan f g) walking_cospan.one) :=\nbegin\n  rw \u2190 limit.w (cospan f g) walking_cospan.hom.inl,\n  change is_open_immersion (_ \u226b f),\n  apply_instance\nend\n\ninstance forget_to_Top_preserves_of_left :\n  preserves_limit (cospan f g) Scheme.forget_to_Top :=\nbegin\n  apply_with limits.comp_preserves_limit { instances := ff },\n  apply_instance,\n  apply preserves_limit_of_iso_diagram _ (diagram_iso_cospan.{u} _).symm,\n  dsimp [LocallyRingedSpace.forget_to_Top],\n  apply_instance\nend\n\ninstance forget_to_Top_preserves_of_right :\n  preserves_limit (cospan g f) Scheme.forget_to_Top := preserves_pullback_symmetry _ _ _\n\n/--\nThe universal property of open immersions:\nFor an open immersion `f : X \u27f6 Z`, given any morphism of schemes `g : Y \u27f6 Z` whose topological\nimage is contained in the image of `f`, we can lift this morphism to a unique `Y \u27f6 X` that\ncommutes with these maps.\n-/\ndef lift (H' : set.range g.1.base \u2286 set.range f.1.base) : Y \u27f6 X :=\nLocallyRingedSpace.is_open_immersion.lift f g H'\n\n@[simp, reassoc] lemma lift_fac (H' : set.range g.1.base \u2286 set.range f.1.base) :\n  lift f g H' \u226b f = g :=\nLocallyRingedSpace.is_open_immersion.lift_fac f g H'\n\nlemma lift_uniq (H' : set.range g.1.base \u2286 set.range f.1.base) (l : Y \u27f6 X)\n  (hl : l \u226b f = g) : l = lift f g H' :=\nLocallyRingedSpace.is_open_immersion.lift_uniq f g H' l hl\n\n/-- Two open immersions with equal range is isomorphic. -/\n@[simps] def iso_of_range_eq [is_open_immersion g] (e : set.range f.1.base = set.range g.1.base) :\n  X \u2245 Y :=\n{ hom := lift g f (le_of_eq e),\n  inv := lift f g (le_of_eq e.symm),\n  hom_inv_id' := by { rw \u2190 cancel_mono f, simp },\n  inv_hom_id' := by { rw \u2190 cancel_mono g, simp } }\n\nend is_open_immersion\n\n/-- Given an open cover on `X`, we may pull them back along a morphism `W \u27f6 X` to obtain\nan open cover of `W`. -/\n@[simps]\ndef Scheme.open_cover.pullback_cover {X : Scheme} (\ud835\udcb0 : X.open_cover) {W : Scheme} (f : W \u27f6 X) :\n  W.open_cover :=\n{ J := \ud835\udcb0.J,\n  obj := \u03bb x, pullback f (\ud835\udcb0.map x),\n  map := \u03bb x, pullback.fst,\n  f := \u03bb x, \ud835\udcb0.f (f.1.base x),\n  covers := \u03bb x, begin\n    rw \u2190 (show _ = (pullback.fst : pullback f (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x))) \u27f6 _).1.base,\n      from preserves_pullback.iso_hom_fst Scheme.forget_to_Top f\n      (\ud835\udcb0.map (\ud835\udcb0.f (f.1.base x)))),\n    rw [coe_comp, set.range_comp, set.range_iff_surjective.mpr, set.image_univ,\n      Top.pullback_fst_range],\n    obtain \u27e8y, h\u27e9 := \ud835\udcb0.covers (f.1.base x),\n    exact \u27e8y, h.symm\u27e9,\n    { rw \u2190 Top.epi_iff_surjective, apply_instance }\n  end }\n\nend algebraic_geometry\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/algebraic_geometry/open_immersion.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300449389326, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3649793376577375}}
{"text": "import ScholzeLog.logLemmas.ScholzeHelpLemmas\nimport ScholzeLog.logLemmas.tactics\n\nnotation `|`x`|` := abs x\n\nopen real set\n\nlemma Schloze_log_nonsense (s t : \u211d) : \n  |s*log(|s|) + t*log(|t|) - (s + t)*log(|s + t|)| \u2264 2*log 2 * (|s| + |t|) :=\nbegin\n  --have h_lambda := part1,\n  -- I want to have wlog and then let s \u2264 t. Now I can rewrite all of the s with a * t where |a| \u2264 1. \n  wlog Hst: s \u2264 t,\n  -- Now we want to derive the fact we can write s as a * t where |a| < 1\n  have : \u2200 (a \u2208  Icc (-1 : \u211d) 1), s = a * t, \n  { intros a a_in,\n    \n    sorry\n\n  },\n  sorry\nend\n\nlemma Schloze_log_nonsense_nongeneral (s t : \u211d) (hs \u2208 Icc (-1 : \u211d) 1) (ht : |t| = 1): \n  |s*log(|s|) + t*log(|t|) - (s + t)*log(|s + t|)| \u2264 2*log 2 * (|s| + |t|) :=\nbegin\n  rw abs_eq at ht,\n  swap, { norm_num,},\n  cases ht with ht_one ht_neg,\n  { rw ht_one,\n    rw [one_mul, abs_one, log_one, add_zero],\n    -- working on paper. HOW IS THIS SIMPLY \u2264 2log 2.\n    -- I hate this and everything to do with this. \n    sorry},\n  { sorry},\nend\n\nexample (s : \u211d) (hs : s \u2208 Icc (-1 : \u211d) 1): |s*log(|s|) - (s + 1) * log(|s + 1|)| \u2264 2 * log 2 :=\nbegin\n  -- PAIN\n  -- This proof isn't that bad. I want to provide an argument relating to the fact that log is \n  -- monotonically increasing on s \u2208 [0, 1]. The question now, is how?\n    -- OK. So now I had an idea about using log x < log y \u2192 x < y, but it's a sum of logs.\n  by_cases s_pos: 0 < s,\n  { --apply log_stuff_pos s hs s_pos,\n    sorry\n  },\n  { rw [not_lt] at s_pos,\n    -- log time! Right I need to prove the other side now.\n    --\n    \n    sorry\n  }\nend", "meta": {"author": "jamesa9283", "repo": "special-functions", "sha": "392758fb7207762c9ba6938462614994ff45bdc4", "save_path": "github-repos/lean/jamesa9283-special-functions", "path": "github-repos/lean/jamesa9283-special-functions/special-functions-392758fb7207762c9ba6938462614994ff45bdc4/src/ScholzeLog/logLemmas/unusedScholze.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3649107581519889}}
{"text": "import LeanUtils\nopen Nat\n\ntheorem squares_div_4 (a b c : Nat) (h\u2080 : divisible 4 (a^2 + b^2 + c^2)) : divisible 4 (a^2) \u2227 divisible 4 (b^2) \u2227 divisible 4 (c^2) := by\n\n  admit", "meta": {"author": "Augustindou", "repo": "natural2lean-lean-project-template", "sha": "62c1d7cf8b2f0cbffd84f240c3e2cd89b55f3c03", "save_path": "github-repos/lean/Augustindou-natural2lean-lean-project-template", "path": "github-repos/lean/Augustindou-natural2lean-lean-project-template/natural2lean-lean-project-template-62c1d7cf8b2f0cbffd84f240c3e2cd89b55f3c03/examples/in-progress/squares-div-4/theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.8479677660619633, "lm_q2_score": 0.4301473485858429, "lm_q1q2_score": 0.36475108625781383}}
{"text": "example (P Q R : Prop) (HP : P) (HQ : Q) : P :=\nbegin\n  exact HP,\n  -- assumption would also have worked\nend\n", "meta": {"author": "ImperialCollegeLondon", "repo": "xena-UROP-2018", "sha": "b111fb87f343cf79eca3b886f99ee15c1dd9884b", "save_path": "github-repos/lean/ImperialCollegeLondon-xena-UROP-2018", "path": "github-repos/lean/ImperialCollegeLondon-xena-UROP-2018/xena-UROP-2018-b111fb87f343cf79eca3b886f99ee15c1dd9884b/src/M1F/problem_bank/PB0001/S0001.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.36473365978759864}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module algebraic_geometry.morphisms.quasi_compact\n! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicGeometry.Morphisms.Basic\nimport Mathbin.Topology.Spectral.Hom\nimport Mathbin.AlgebraicGeometry.Limits\n\n/-!\n# Quasi-compact morphisms\n\nA morphism of schemes is quasi-compact if the preimages of quasi-compact open sets are\nquasi-compact.\n\nIt suffices to check that preimages of affine open sets are compact\n(`quasi_compact_iff_forall_affine`).\n\n-/\n\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace\n\nuniverse u\n\nopen AlgebraicGeometry\n\nnamespace AlgebraicGeometry\n\nvariable {X Y : Scheme.{u}} (f : X \u27f6 Y)\n\n/--\nA morphism is `quasi-compact` if the underlying map of topological spaces is, i.e. if the preimages\nof quasi-compact open sets are quasi-compact.\n-/\n@[mk_iff]\nclass QuasiCompact (f : X \u27f6 Y) : Prop where\n  isCompact_preimage : \u2200 U : Set Y.carrier, IsOpen U \u2192 IsCompact U \u2192 IsCompact (f.1.base \u207b\u00b9' U)\n#align algebraic_geometry.quasi_compact AlgebraicGeometry.QuasiCompact\n\ntheorem quasiCompact_iff_spectral : QuasiCompact f \u2194 IsSpectralMap f.1.base :=\n  \u27e8fun \u27e8h\u27e9 => \u27e8by continuity, h\u27e9, fun h => \u27e8h.2\u27e9\u27e9\n#align algebraic_geometry.quasi_compact_iff_spectral AlgebraicGeometry.quasiCompact_iff_spectral\n\n/-- The `affine_target_morphism_property` corresponding to `quasi_compact`, asserting that the\ndomain is a quasi-compact scheme. -/\ndef QuasiCompact.affineProperty : AffineTargetMorphismProperty := fun X Y f hf =>\n  CompactSpace X.carrier\n#align algebraic_geometry.quasi_compact.affine_property AlgebraicGeometry.QuasiCompact.affineProperty\n\ninstance (priority := 900) quasiCompactOfIsIso {X Y : Scheme} (f : X \u27f6 Y) [IsIso f] :\n    QuasiCompact f := by\n  constructor\n  intro U hU hU'\n  convert hU'.image (inv f.1.base).continuous_toFun using 1\n  rw [Set.image_eq_preimage_of_inverse]\n  delta Function.LeftInverse\n  exacts[is_iso.inv_hom_id_apply f.1.base, is_iso.hom_inv_id_apply f.1.base]\n#align algebraic_geometry.quasi_compact_of_is_iso AlgebraicGeometry.quasiCompactOfIsIso\n\ninstance quasiCompactComp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z) [QuasiCompact f]\n    [QuasiCompact g] : QuasiCompact (f \u226b g) :=\n  by\n  constructor\n  intro U hU hU'\n  rw [Scheme.comp_val_base, coe_comp, Set.preimage_comp]\n  apply quasi_compact.is_compact_preimage\n  \u00b7 exact Continuous.isOpen_preimage (by continuity) _ hU\n  apply quasi_compact.is_compact_preimage <;> assumption\n#align algebraic_geometry.quasi_compact_comp AlgebraicGeometry.quasiCompactComp\n\ntheorem isCompact_open_iff_eq_finset_affine_union {X : Scheme} (U : Set X.carrier) :\n    IsCompact U \u2227 IsOpen U \u2194\n      \u2203 s : Set X.affineOpens, s.Finite \u2227 U = \u22c3 (i : X.affineOpens) (h : i \u2208 s), i :=\n  by\n  apply opens.is_basis.is_compact_open_iff_eq_finite_Union (coe : X.affine_opens \u2192 opens X.carrier)\n  \u00b7 rw [Subtype.range_coe]\n    exact is_basis_affine_open X\n  \u00b7 exact fun i => i.2.IsCompact\n#align algebraic_geometry.is_compact_open_iff_eq_finset_affine_union AlgebraicGeometry.isCompact_open_iff_eq_finset_affine_union\n\ntheorem isCompact_open_iff_eq_basicOpen_union {X : Scheme} [IsAffine X] (U : Set X.carrier) :\n    IsCompact U \u2227 IsOpen U \u2194\n      \u2203 s : Set (X.Presheaf.obj (op \u22a4)),\n        s.Finite \u2227 U = \u22c3 (i : X.Presheaf.obj (op \u22a4)) (h : i \u2208 s), X.basicOpen i :=\n  (isBasis_basicOpen X).isCompact_open_iff_eq_finite_union\u1d62 _\n    (fun i => ((top_isAffineOpen _).basicOpen_is_affine _).IsCompact) _\n#align algebraic_geometry.is_compact_open_iff_eq_basic_open_union AlgebraicGeometry.isCompact_open_iff_eq_basicOpen_union\n\ntheorem quasiCompact_iff_forall_affine :\n    QuasiCompact f \u2194\n      \u2200 U : Opens Y.carrier, IsAffineOpen U \u2192 IsCompact (f.1.base \u207b\u00b9' (U : Set Y.carrier)) :=\n  by\n  rw [quasi_compact_iff]\n  refine' \u27e8fun H U hU => H U U.IsOpen hU.IsCompact, _\u27e9\n  intro H U hU hU'\n  obtain \u27e8S, hS, rfl\u27e9 := (is_compact_open_iff_eq_finset_affine_union U).mp \u27e8hU', hU\u27e9\n  simp only [Set.preimage_union\u1d62, Subtype.val_eq_coe]\n  exact hS.is_compact_bUnion fun i _ => H i i.Prop\n#align algebraic_geometry.quasi_compact_iff_forall_affine AlgebraicGeometry.quasiCompact_iff_forall_affine\n\n@[simp]\ntheorem QuasiCompact.affineProperty_toProperty {X Y : Scheme} (f : X \u27f6 Y) :\n    (QuasiCompact.affineProperty : _).toProperty f \u2194 IsAffine Y \u2227 CompactSpace X.carrier :=\n  by\n  delta affine_target_morphism_property.to_property quasi_compact.affine_property\n  simp\n#align algebraic_geometry.quasi_compact.affine_property_to_property AlgebraicGeometry.QuasiCompact.affineProperty_toProperty\n\ntheorem quasiCompact_iff_affineProperty :\n    QuasiCompact f \u2194 targetAffineLocally QuasiCompact.affineProperty f :=\n  by\n  rw [quasi_compact_iff_forall_affine]\n  trans \u2200 U : Y.affine_opens, IsCompact (f.1.base \u207b\u00b9' (U : Set Y.carrier))\n  \u00b7 exact \u27e8fun h U => h U U.Prop, fun h U hU => h \u27e8U, hU\u27e9\u27e9\n  apply forall_congr'\n  exact fun _ => isCompact_iff_compactSpace\n#align algebraic_geometry.quasi_compact_iff_affine_property AlgebraicGeometry.quasiCompact_iff_affineProperty\n\ntheorem quasiCompact_eq_affineProperty :\n    @QuasiCompact = targetAffineLocally QuasiCompact.affineProperty :=\n  by\n  ext\n  exact quasi_compact_iff_affine_property _\n#align algebraic_geometry.quasi_compact_eq_affine_property AlgebraicGeometry.quasiCompact_eq_affineProperty\n\ntheorem isCompact_basicOpen (X : Scheme) {U : Opens X.carrier} (hU : IsCompact (U : Set X.carrier))\n    (f : X.Presheaf.obj (op U)) : IsCompact (X.basicOpen f : Set X.carrier) := by\n  classical\n    refine' ((is_compact_open_iff_eq_finset_affine_union _).mpr _).1\n    obtain \u27e8s, hs, e\u27e9 := (is_compact_open_iff_eq_finset_affine_union _).mp \u27e8hU, U.is_open\u27e9\n    let g : s \u2192 X.affine_opens := by\n      intro V\n      use V.1 \u2293 X.basic_open f\n      have : V.1.1 \u27f6 U := by\n        apply hom_of_le\n        change _ \u2286 (U : Set X.carrier)\n        rw [e]\n        convert@Set.subset_union\u1d62\u2082 _ _ _ (fun (U : X.affine_opens) (h : U \u2208 s) => \u2191U) V V.prop using\n          1\n        rfl\n      erw [\u2190 X.to_LocallyRingedSpace.to_RingedSpace.basic_open_res this.op]\n      exact is_affine_open.basic_open_is_affine V.1.Prop _\n    haveI : Finite s := hs.to_subtype\n    refine' \u27e8Set.range g, Set.finite_range g, _\u27e9\n    refine'\n      (set.inter_eq_right_iff_subset.mpr\n              (SetLike.coe_subset_coe.2 <| RingedSpace.basic_open_le _ _)).symm.trans\n        _\n    rw [e, Set.union\u1d62\u2082_inter]\n    apply le_antisymm <;> apply Set.union\u1d62\u2082_subset\n    \u00b7 intro i hi\n      refine' Set.Subset.trans _ (Set.subset_union\u1d62\u2082 _ (Set.mem_range_self \u27e8i, hi\u27e9))\n      exact Set.Subset.rfl\n    \u00b7 rintro \u27e8i, hi\u27e9 \u27e8\u27e8j, hj\u27e9, hj'\u27e9\n      rw [\u2190 hj']\n      refine' Set.Subset.trans _ (Set.subset_union\u1d62\u2082 j hj)\n      exact Set.Subset.rfl\n#align algebraic_geometry.is_compact_basic_open AlgebraicGeometry.isCompact_basicOpen\n\ntheorem QuasiCompact.affinePropertyIsLocal : (QuasiCompact.affineProperty : _).IsLocal :=\n  by\n  constructor\n  \u00b7 apply affine_target_morphism_property.respects_iso_mk <;> rintro X Y Z _ _ _ H\n    exacts[@Homeomorph.compactSpace _ _ H (TopCat.homeoOfIso (as_iso e.inv.1.base)), H]\n  \u00b7 introv H\n    delta quasi_compact.affine_property at H\u22a2\n    change CompactSpace ((opens.map f.val.base).obj (Y.basic_open r))\n    rw [Scheme.preimage_basic_open f r]\n    erw [\u2190 isCompact_iff_compactSpace]\n    rw [\u2190 isCompact_univ_iff] at H\n    exact is_compact_basic_open X H _\n  \u00b7 rintro X Y H f S hS hS'\n    skip\n    rw [\u2190 is_affine_open.basic_open_union_eq_self_iff] at hS\n    delta quasi_compact.affine_property\n    rw [\u2190 isCompact_univ_iff]\n    change IsCompact ((opens.map f.val.base).obj \u22a4).1\n    rw [\u2190 hS]\n    dsimp [opens.map]\n    simp only [opens.coe_supr, Set.preimage_union\u1d62, Subtype.val_eq_coe]\n    exacts[isCompact_union\u1d62 fun i => is_compact_iff_compact_space.mpr (hS' i), top_is_affine_open _]\n#align algebraic_geometry.quasi_compact.affine_property_is_local AlgebraicGeometry.QuasiCompact.affinePropertyIsLocal\n\ntheorem QuasiCompact.affine_openCover_tFAE {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n    TFAE\n      [QuasiCompact f,\n        \u2203 (\ud835\udcb0 : Scheme.OpenCover.{u} Y)(_ : \u2200 i, IsAffine (\ud835\udcb0.obj i)),\n          \u2200 i : \ud835\udcb0.J, CompactSpace (pullback f (\ud835\udcb0.map i)).carrier,\n        \u2200 (\ud835\udcb0 : Scheme.OpenCover.{u} Y) [\u2200 i, IsAffine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J),\n          CompactSpace (pullback f (\ud835\udcb0.map i)).carrier,\n        \u2200 {U : Scheme} (g : U \u27f6 Y) [IsAffine U] [IsOpenImmersion g],\n          CompactSpace (pullback f g).carrier,\n        \u2203 (\u03b9 : Type u)(U : \u03b9 \u2192 Opens Y.carrier)(hU : sup\u1d62 U = \u22a4)(hU' : \u2200 i, IsAffineOpen (U i)),\n          \u2200 i, CompactSpace (f.1.base \u207b\u00b9' (U i).1)] :=\n  quasiCompact_eq_affineProperty.symm \u25b8 QuasiCompact.affinePropertyIsLocal.affine_openCover_tFAE f\n#align algebraic_geometry.quasi_compact.affine_open_cover_tfae AlgebraicGeometry.QuasiCompact.affine_openCover_tFAE\n\ntheorem QuasiCompact.isLocalAtTarget : PropertyIsLocalAtTarget @QuasiCompact :=\n  quasiCompact_eq_affineProperty.symm \u25b8\n    QuasiCompact.affinePropertyIsLocal.targetAffineLocallyIsLocal\n#align algebraic_geometry.quasi_compact.is_local_at_target AlgebraicGeometry.QuasiCompact.isLocalAtTarget\n\ntheorem QuasiCompact.openCover_tFAE {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n    TFAE\n      [QuasiCompact f,\n        \u2203 \ud835\udcb0 : Scheme.OpenCover.{u} Y,\n          \u2200 i : \ud835\udcb0.J, QuasiCompact (pullback.snd : (\ud835\udcb0.pullbackCover f).obj i \u27f6 \ud835\udcb0.obj i),\n        \u2200 (\ud835\udcb0 : Scheme.OpenCover.{u} Y) (i : \ud835\udcb0.J),\n          QuasiCompact (pullback.snd : (\ud835\udcb0.pullbackCover f).obj i \u27f6 \ud835\udcb0.obj i),\n        \u2200 U : Opens Y.carrier, QuasiCompact (f \u2223_ U),\n        \u2200 {U : Scheme} (g : U \u27f6 Y) [IsOpenImmersion g],\n          QuasiCompact (pullback.snd : pullback f g \u27f6 _),\n        \u2203 (\u03b9 : Type u)(U : \u03b9 \u2192 Opens Y.carrier)(hU : sup\u1d62 U = \u22a4), \u2200 i, QuasiCompact (f \u2223_ U i)] :=\n  quasiCompact_eq_affineProperty.symm \u25b8\n    QuasiCompact.affinePropertyIsLocal.targetAffineLocallyIsLocal.openCover_tFAE f\n#align algebraic_geometry.quasi_compact.open_cover_tfae AlgebraicGeometry.QuasiCompact.openCover_tFAE\n\ntheorem quasiCompact_over_affine_iff {X Y : Scheme} (f : X \u27f6 Y) [IsAffine Y] :\n    QuasiCompact f \u2194 CompactSpace X.carrier :=\n  quasiCompact_eq_affineProperty.symm \u25b8 QuasiCompact.affinePropertyIsLocal.affine_target_iff f\n#align algebraic_geometry.quasi_compact_over_affine_iff AlgebraicGeometry.quasiCompact_over_affine_iff\n\ntheorem compactSpace_iff_quasiCompact (X : Scheme) :\n    CompactSpace X.carrier \u2194 QuasiCompact (terminal.from X) :=\n  (quasiCompact_over_affine_iff _).symm\n#align algebraic_geometry.compact_space_iff_quasi_compact AlgebraicGeometry.compactSpace_iff_quasiCompact\n\ntheorem QuasiCompact.affine_openCover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.OpenCover.{u} Y)\n    [\u2200 i, IsAffine (\ud835\udcb0.obj i)] (f : X \u27f6 Y) :\n    QuasiCompact f \u2194 \u2200 i, CompactSpace (pullback f (\ud835\udcb0.map i)).carrier :=\n  quasiCompact_eq_affineProperty.symm \u25b8 QuasiCompact.affinePropertyIsLocal.affine_openCover_iff f \ud835\udcb0\n#align algebraic_geometry.quasi_compact.affine_open_cover_iff AlgebraicGeometry.QuasiCompact.affine_openCover_iff\n\ntheorem QuasiCompact.openCover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.OpenCover.{u} Y) (f : X \u27f6 Y) :\n    QuasiCompact f \u2194 \u2200 i, QuasiCompact (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) :=\n  quasiCompact_eq_affineProperty.symm \u25b8\n    QuasiCompact.affinePropertyIsLocal.targetAffineLocallyIsLocal.openCover_iff f \ud835\udcb0\n#align algebraic_geometry.quasi_compact.open_cover_iff AlgebraicGeometry.QuasiCompact.openCover_iff\n\ntheorem quasiCompact_respectsIso : MorphismProperty.RespectsIso @QuasiCompact :=\n  quasiCompact_eq_affineProperty.symm \u25b8\n    targetAffineLocally_respectsIso QuasiCompact.affinePropertyIsLocal.1\n#align algebraic_geometry.quasi_compact_respects_iso AlgebraicGeometry.quasiCompact_respectsIso\n\ntheorem quasiCompact_stableUnderComposition :\n    MorphismProperty.StableUnderComposition @QuasiCompact := fun _ _ _ _ _ _ _ => inferInstance\n#align algebraic_geometry.quasi_compact_stable_under_composition AlgebraicGeometry.quasiCompact_stableUnderComposition\n\nattribute [-simp] PresheafedSpace.as_coe SheafedSpace.as_coe\n\ntheorem QuasiCompact.affineProperty_stableUnderBaseChange :\n    QuasiCompact.affineProperty.StableUnderBaseChange :=\n  by\n  intro X Y S _ _ f g h\n  rw [quasi_compact.affine_property] at h\u22a2\n  skip\n  let \ud835\udcb0 := Scheme.pullback.open_cover_of_right Y.affine_cover.finite_subcover f g\n  have : Finite \ud835\udcb0.J := by\n    dsimp [\ud835\udcb0]\n    infer_instance\n  have : \u2200 i, CompactSpace (\ud835\udcb0.obj i).carrier :=\n    by\n    intro i\n    dsimp\n    infer_instance\n  exact \ud835\udcb0.compact_space\n#align algebraic_geometry.quasi_compact.affine_property_stable_under_base_change AlgebraicGeometry.QuasiCompact.affineProperty_stableUnderBaseChange\n\ntheorem quasiCompact_stableUnderBaseChange : MorphismProperty.StableUnderBaseChange @QuasiCompact :=\n  quasiCompact_eq_affineProperty.symm \u25b8\n    QuasiCompact.affinePropertyIsLocal.StableUnderBaseChange\n      QuasiCompact.affineProperty_stableUnderBaseChange\n#align algebraic_geometry.quasi_compact_stable_under_base_change AlgebraicGeometry.quasiCompact_stableUnderBaseChange\n\nvariable {Z : Scheme.{u}}\n\ninstance (f : X \u27f6 Z) (g : Y \u27f6 Z) [QuasiCompact g] :\n    QuasiCompact (pullback.fst : pullback f g \u27f6 X) :=\n  quasiCompact_stableUnderBaseChange.fst f g inferInstance\n\ninstance (f : X \u27f6 Z) (g : Y \u27f6 Z) [QuasiCompact f] :\n    QuasiCompact (pullback.snd : pullback f g \u27f6 Y) :=\n  quasiCompact_stableUnderBaseChange.snd f g inferInstance\n\n@[elab_as_elim]\ntheorem compact_open_induction_on {P : Opens X.carrier \u2192 Prop} (S : Opens X.carrier)\n    (hS : IsCompact S.1) (h\u2081 : P \u22a5)\n    (h\u2082 : \u2200 (S : Opens X.carrier) (hS : IsCompact S.1) (U : X.affineOpens), P S \u2192 P (S \u2294 U)) :\n    P S := by\n  classical\n    obtain \u27e8s, hs, hs'\u27e9 := (is_compact_open_iff_eq_finset_affine_union S.1).mp \u27e8hS, S.2\u27e9\n    replace hs' : S = sup\u1d62 fun i : s => (i : opens X.carrier) :=\n      by\n      ext1\n      simpa using hs'\n    subst hs'\n    apply hs.induction_on\n    \u00b7 convert h\u2081\n      rw [sup\u1d62_eq_bot]\n      rintro \u27e8_, h\u27e9\n      exact h.elim\n    \u00b7 intro x s h\u2083 hs h\u2084\n      have : IsCompact (\u2a06 i : s, (i : opens X.carrier)).1 :=\n        by\n        refine' ((is_compact_open_iff_eq_finset_affine_union _).mpr _).1\n        exact \u27e8s, hs, by simp\u27e9\n      convert h\u2082 _ this x h\u2084\n      simp only [coe_coe]\n      rw [sup\u1d62_subtype, sup_comm]\n      conv_rhs => rw [sup\u1d62_subtype]\n      exact sup\u1d62_insert\n#align algebraic_geometry.compact_open_induction_on AlgebraicGeometry.compact_open_induction_on\n\ntheorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen (X : Scheme)\n    {U : Opens X.carrier} (hU : IsAffineOpen U) (x f : X.Presheaf.obj (op U))\n    (H : x |_ X.basicOpen f = 0) : \u2203 n : \u2115, f ^ n * x = 0 :=\n  by\n  rw [\u2190 map_zero (X.presheaf.map (hom_of_le <| X.basic_open_le f : X.basic_open f \u27f6 U).op)] at H\n  have := (is_localization_basic_open hU f).3\n  obtain \u27e8\u27e8_, n, rfl\u27e9, e\u27e9 := this.mp H\n  exact \u27e8n, by simpa [mul_comm x] using e\u27e9\n#align algebraic_geometry.exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isAffineOpen\n\n/-- If `x : \u0393(X, U)` is zero on `D(f)` for some `f : \u0393(X, U)`, and `U` is quasi-compact, then\n`f ^ n * x = 0` for some `n`. -/\ntheorem exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact (X : Scheme)\n    {U : Opens X.carrier} (hU : IsCompact U.1) (x f : X.Presheaf.obj (op U))\n    (H : x |_ X.basicOpen f = 0) : \u2203 n : \u2115, f ^ n * x = 0 :=\n  by\n  obtain \u27e8s, hs, e\u27e9 := (is_compact_open_iff_eq_finset_affine_union U.1).mp \u27e8hU, U.2\u27e9\n  replace e : U = sup\u1d62 fun i : s => (i : opens X.carrier)\n  \u00b7 ext1\n    simpa using e\n  have 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    exact le_sup\u1d62 _ _\n  have H' := fun i : s =>\n    exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open X i.1.2\n      (X.presheaf.map (hom_of_le (h\u2081 i)).op x) (X.presheaf.map (hom_of_le (h\u2081 i)).op f) _\n  swap\n  \u00b7 delta TopCat.Presheaf.restrictOpen TopCat.Presheaf.restrict at H\u22a2\n    convert congr_arg (X.presheaf.map (hom_of_le _).op) H\n    \u00b7 simp only [\u2190 comp_apply, \u2190 functor.map_comp]\n      congr\n    \u00b7 rw [map_zero]\n    \u00b7 rw [X.basic_open_res]\n      exact Set.inter_subset_right _ _\n  choose n hn using H'\n  haveI := hs.to_subtype\n  cases nonempty_fintype s\n  use finset.univ.sup n\n  suffices \u2200 i : s, X.presheaf.map (hom_of_le (h\u2081 i)).op (f ^ finset.univ.sup n * x) = 0\n    by\n    subst e\n    apply X.sheaf.eq_of_locally_eq fun i : s => (i : opens X.carrier)\n    intro i\n    rw [map_zero]\n    apply this\n  intro i\n  replace hn :=\n    congr_arg (fun x => X.presheaf.map (hom_of_le (h\u2081 i)).op (f ^ (finset.univ.sup n - n i)) * x)\n      (hn i)\n  dsimp at hn\n  simp only [\u2190 map_mul, \u2190 map_pow] at hn\n  rwa [MulZeroClass.mul_zero, \u2190 mul_assoc, \u2190 pow_add, tsub_add_cancel_of_le] at hn\n  apply Finset.le_sup (Finset.mem_univ i)\n#align algebraic_geometry.exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_compact AlgebraicGeometry.exists_pow_mul_eq_zero_of_res_basicOpen_eq_zero_of_isCompact\n\nend AlgebraicGeometry\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/Morphisms/QuasiCompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.36471236088485853}}
{"text": "-------------------------------------------------------------------\n-- The PEDANTIC2 (Proof Engine for Deductive Automation using Non-deterministic\n-- Traversal of Instruction Code) verification framework\n--\n-- Developed by Kenneth Roe\n-- For more information, check out www.cs.jhu.edu/~roe\n--\n-- impHeap.lean\n-- This file contains a model of the concrete state.  It was adapted from the\n-- Software Foundations Imp.v model.\n--\n-------------------------------------------------------------------\n\ndef cell := option \u2115\ndef heap := \u2115 \u2192 option \u2115\ndef ident := \u2115\n\ndef env := ident \u2192 \u2115\n\ninstance : inhabited env := \u27e8\u03bb x, 0\u27e9\ninstance : inhabited heap := \u27e8\u03bb h, none\u27e9\ninstance : inhabited cell := \u27e8none\u27e9\n\ndef empty_env : env := inhabited.default env\ndef empty_heap : heap := inhabited.default heap\ndef empty_cell : cell := inhabited.default cell\n\ndef imp_state := heap \u00d7 env\n\n--def beq_nat : \u2115 \u2192 \u2115 \u2192 bool\n--| 0 0 := tt\n--| (x+1) (y+1) := (beq_nat x y)\n--| (x+1) 0 := ff\n--| 0 (x+1) := ff\n\n--def ble_nat : \u2115 \u2192 \u2115 \u2192 bool\n--| 0 z := tt\n--| (x+1) (y+1) := (ble_nat x y)\n--| (x+1) 0 := ff\n\n--example : beq_nat 3 3=tt := rfl.\n\n--def beq_ident := beq_nat\ninstance : decidable_eq ident :=\nby unfold ident; apply_instance\n\n--theorem beq_nat00 : beq_nat 0 0 = tt := rfl.\n\n--theorem beq_refl (n : \u2115) : beq_nat n n=tt :=\n--begin\n--    induction n, rewrite beq_nat00,\n--\n--    unfold beq_nat, rewrite n_ih\n--end\n\n\n--theorem beq_nat_comm (a : \u2115) : \u2200 b, beq_nat a b=beq_nat b a := begin\n--    induction a, intro b, destruct b, intro, rewrite a,\n--    intro, intro, rewrite a, unfold beq_nat,\n--\n--    intro b, destruct b, intro, rewrite a, unfold beq_nat,\n--\n--    intro, intro, rewrite a, unfold beq_nat, rewrite a_ih\n--end\n\ndef override (st : env) (v : ident) (n : \u2115) : env :=\n    (\u03bb (l : ident), if v=l then n else (st l))\n\ndef override_state (v : ident) (n : \u2115) (st : imp_state) : imp_state :=\n    (st.fst,override st.snd v n)\n\n@[simp] theorem override_eq (n : \u2115) (V : ident) (st : env) : (override st V n) V=n :=\nbegin\n    unfold override, simp\nend\n\n@[simp] theorem override_neq (n : \u2115) (V1 : ident) (V2 : ident) (st : env) :\n    V2 \u2260 V1 \u2192\n    (override st V2 n) V1=st V1 := begin\n    unfold override, intro, simp *\nend\n\ntheorem override_shadow (x1 : \u2115) (x2 : \u2115) (k : ident) (f : env) :\n   (override (override f k x1) k x2) = (override f k x2) :=\nbegin\n    unfold override, funext, by_cases (k=l), simp *,\n    simp *\nend\n\n--theorem beq_nat_eq (a : \u2115) : \u2200 b, beq_nat a b=tt \u2192 a=b :=\n--begin\n--    induction a,\n--    intro b,\n--    destruct b, intro, rewrite a, unfold beq_nat, intro, reflexivity,\n--    intro, intro, rewrite a, unfold beq_nat, simp,\n--\n--    intro b, destruct b, intro, rewrite a, unfold beq_nat, simp,\n--    intro, intro, rewrite a, simp, unfold beq_nat,\n--    intro, apply a_ih, apply a_1\n--end\n\ntheorem override_same (x1 : \u2115) (k1 : ident) (k2 : ident) (f : env) :\n  f k1 = x1 \u2192\n  (override f k1 x1) k2 = f k2 :=\nbegin\n    intro, unfold override, by_cases (k1=k2),\n    simp *, rw \u2190 a, rw h,\n    simp *\nend\n\ninductive aexp : Type\n  | Num : \u2115 \u2192 aexp\n  | Var : ident \u2192 aexp\n  | Plus : aexp \u2192 aexp \u2192 aexp\n  | Minus : aexp \u2192 aexp \u2192 aexp\n  | Mult : aexp \u2192 aexp \u2192 aexp\n  | Eq : aexp \u2192 aexp \u2192 aexp\n  | Le : aexp \u2192 aexp \u2192 aexp\n  | Land : aexp \u2192 aexp \u2192 aexp\n  | Lor : aexp \u2192 aexp \u2192 aexp\n  | Lnot : aexp \u2192 aexp\n\n-- Shorthand for common nats and nat operators\n\ndef A0 := (aexp.Num 0)\ndef A1 := (aexp.Num 1)\ndef A2 := (aexp.Num 2)\ndef A3 := (aexp.Num 3)\ndef A4 := (aexp.Num 4)\ndef A5 := (aexp.Num 5)\ndef A6 := (aexp.Num 6)\n\ninfix `***`:50 := aexp.Mult\n\ninfix `-*-`:40 := aexp.Minus\n\ninfix `+++`:40 := aexp.Plus\n\nnotation `!`:30 X := aexp.Var X\n\ninfix `===`:60 := aexp.Eq\n\ninfix `<<=`:60 := aexp.Le\n\ndefinition x := 0\ndefinition y := 1\ndefinition Z := 2\n\n--com is intended to mimic the constructs of a statement block in C or C++.\n\ninductive com : Type\n  | Skip : com\n  | Load : ident -> aexp -> com\n  | Store : aexp -> aexp -> com\n  | Ass : ident -> aexp -> com\n  | New : ident -> aexp -> com\n  | Delete: aexp -> aexp -> com\n  | Seq : com -> com -> com\n  | If : aexp -> com -> com -> com\n  | While : aexp -> com -> com\n  | Call : ident -> ident -> (list aexp) -> com\n  | Return : aexp -> com\n  | Throw : ident -> aexp -> com\n  | Catch : ident -> ident -> com -> com -> com.\n\nnotation  `SKIP` := com.Skip.\ninfix `;` := com.Seq.\nnotation l `::=`:60 v `-*->`:60 i := com.Load l v i.\nnotation l `-*->`:60 i `::=`:60 v := com.Store l i v.\nnotation l `::=`:60 a := com.Ass l a.\nnotation `NEW `:60 v `, ` s := com.New v s.\nnotation `DELETE `:60 e `, ` s := com.Delete e s.\nnotation `WHILE `:80 b ` DO ` c ` LOOP `:80 := com.While b c.\nnotation `IF `:80 e1 ` THEN ` e2 ` ELSE `:80 e3 ` FI`:80 := com.If e1 e2 e3.\nnotation `RETURN `:60 e := com.Return e.\n\n\n-- Shorthand for common variable names\n\ninductive decl : Type\n  | DGlobalVar : ident -> decl\n  | DFunction : ident -> (list ident) -> com -> decl.\n\ndef get_var (s : imp_state) (v : \u2115) : \u2115 := (s.snd v)\ndef v_or_z : option \u2115 \u2192 \u2115\n| option.none := 0\n| (option.some x) := x\n\ndef aeval : env \u2192 aexp \u2192 \u2115 \n  | e (aexp.Num n) := n\n  | e (aexp.Var ii) := e ii\n  | e (aexp.Plus a1 a2) := (aeval e a1)+(aeval e a2)\n  | e (aexp.Minus a1 a2) := (aeval e a1)-(aeval e a2)\n  | e (aexp.Mult a1 a2) := (aeval e a1)*(aeval e a2)\n  | e (aexp.Eq a1 a2) := if (aeval e a1)=(aeval e a2) then 1 else 0\n  | e (aexp.Le a1 a2) := if (aeval e a1)\u2264(aeval e a2) then 1 else 0\n  | e (aexp.Lnot a1) := if (aeval e a1)=0 then 1 else 0\n  | e (aexp.Land a1 a2) := if (aeval e a1)=0 then 0 else aeval e a2\n  | e (aexp.Lor a1 a2) := if (aeval e a1)=0 then aeval e a2 else (aeval e a1)\n\ndef aeval_list : env \u2192 (list aexp) \u2192 list nat\n| s list.nil := list.nil\n| s (a::b) := (aeval s a)::(aeval_list s b)\n\n-- Auxilliary functions needed for the evaluation relation\n\n--def bind_option {X : Type} {Y : Type} :\n--option X \u2192 (X \u2192 option Y)\n--\u2192 option Y\n--| none f := none\n--| (some x) f := f x\n\n--Implicit Arguments bind_option [X Y].\n\n\ndef range : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 bool\n| start 0 n := ff\n| start (s+1) t := if t=(start+s) then tt\n                   else range start s t\n\ninductive new_heap_cells : heap \u2192 nat \u2192 nat \u2192 heap \u2192 Prop\n| OHDone : \u2200 h, \u2200 v,\n           new_heap_cells h v 0 h\n| OHNext : \u2200 h h' v c h'' val,\n             new_heap_cells h (v+1) c h' \u2192\n             h' v = option.none \u2192\n             h'' = (\u03bb x, if x=v then option.some val\n                         else h' x) \u2192\n             new_heap_cells h v (c+1) h''\n\ninductive clear_heap_cells : heap \u2192 nat \u2192 nat \u2192 heap \u2192 Prop\n| CHDone : \u2200 h v, clear_heap_cells h v 0 h\n| CHNext : \u2200 h h' v c val h'',\n           clear_heap_cells h (v+1) c h' \u2192\n           h' v = option.some val \u2192\n           h'' = (\u03bb x, if x=v then option.none else h' x) \u2192\n           clear_heap_cells h v (c+1) h''\n\ninductive func_result : Type\n| NoResult : func_result\n| Return : nat \u2192 func_result\n| Exception : ident \u2192 nat \u2192 func_result\n\ndef functions := ident \u2192 imp_state \u2192 (list \u2115) \u2192 imp_state \u2192\n                 func_result \u2192 Prop.\n\ninductive ceval : functions -> imp_state -> com -> imp_state -> func_result -> Prop\n| Skip : \u2200 f (st : imp_state), ceval f st com.Skip st func_result.NoResult\n| Ass  : \u2200 f st a1 l,\n           ceval f st (com.Ass l a1) ((st.fst),(override (st.snd) l\n                 (aeval st.snd a1)))\n                 func_result.NoResult\n| New : \u2200 f (st : imp_state) l loc e count h',\n          loc \u2260 0 \u2192\n          count = aeval (st.snd) e \u2192\n          new_heap_cells (st.fst) loc count h' \u2192\n          ceval f st (com.New l e) (h',override (st.snd) l loc)\n                func_result.NoResult\n| Delete : \u2200 f (st : imp_state) loc count l c h',\n          l = aeval st.snd loc \u2192\n          c = aeval st.snd count \u2192\n          clear_heap_cells (st.fst) l c h' \u2192\n          ceval f st (com.Delete loc count) (h',st.snd)\n                func_result.NoResult\n| Load : \u2200 f (st:imp_state) loc l val,\n           option.some val = (st.fst) (aeval st.snd loc)  \u2192\n           ceval f st (com.Load l loc)\n                 ((st.fst),(override (st.snd) l val))\n                 func_result.NoResult\n| Store : \u2200 f (st : imp_state) loc val l v ov,\n      v = aeval st.snd val \u2192\n      l = aeval st.snd loc \u2192\n      (st.fst) l = option.some ov \u2192\n      ceval f st (com.Store loc val)\n            ((\u03bb x, if l=x then (option.some v) else (st.fst) x),(st.snd)) func_result.NoResult\n| Seq1 : \u2200 f c1 c2 st st' st'' r,\n      ceval f st c1 st' func_result.NoResult \u2192\n      ceval f st' c2 st'' r \u2192\n      ceval f st (com.Seq c1 c2) st'' r\n| Seq2 : \u2200 f c1 c2 st st' v,\n      ceval f st c1 st' (func_result.Return v) \u2192\n      ceval f st (com.Seq c1 c2) st' (func_result.Return v)\n| Seq3 : \u2200 f c1 c2 st st' name val,\n      ceval f st c1 st' (func_result.Exception name val) \u2192\n      ceval f st (com.Seq c1 c2) st'\n            (func_result.Exception name val)\n| IfTrue : \u2200 f r (st : imp_state) (st':imp_state) b1 c1 c2,\n      not(aeval st.snd b1 = 0) ->\n      ceval f st c1 st' r ->\n      ceval f st (com.If b1 c1 c2) st' r\n| IfFalse : \u2200 f r (st : imp_state) (st' : imp_state) b1 c1 c2,\n      aeval st.snd b1 = 0 \u2192\n      ceval f st c2 st' r \u2192\n      ceval f st (com.If b1 c1 c2) st' r\n| WhileEnd : \u2200 f b1 (st : imp_state) c1,\n      aeval st.snd b1 = 0 \u2192\n      ceval f st (com.While b1 c1) st func_result.NoResult\n| WhileLoop1 : \u2200 f (st : imp_state) st' st'' b1 c1 r,\n      not(aeval st.snd b1 = 0) \u2192\n      ceval f st c1 st' func_result.NoResult \u2192\n      ceval f st' (com.While b1 c1) st'' r \u2192\n      ceval f st (com.While b1 c1) st'' r\n| WhileLoop2 : \u2200 f (st : imp_state) st' b1 c1 r,\n      not(aeval st.snd b1 = 0) \u2192\n      ceval f st c1 st' (func_result.Return r) \u2192\n      ceval f st (com.While b1 c1) st' (func_result.Return r)\n| WhileLoop3 : \u2200 f (st : imp_state) st' b1 c1 name val,\n      not(aeval st.snd b1 = 0) \u2192\n      ceval f st c1 st' (func_result.Exception name val) \u2192\n      ceval f st (com.While b1 c1) st'\n            (func_result.Exception name val)\n| Throw: \u2200 f (st : imp_state) val exp v,\n      val = aeval st.snd exp ->\n      ceval f st (com.Throw v exp) st (func_result.Exception v val)\n| Catch1: \u2200 f st st' exc var code hand,\n      ceval f st code st' func_result.NoResult \u2192\n      ceval f st (com.Catch exc var code hand) st' func_result.NoResult\n| Catch2: \u2200 f st st' exc var code hand v,\n      ceval f st code st' (func_result.Return v) \u2192\n      ceval f st (com.Catch exc var code hand) st' (func_result.Return v)\n| Catch3: \u2200 f st st' exc var code hand v name,\n      ceval f st code st' (func_result.Exception name v) \u2192\n      name \u2260 exc \u2192\n      ceval f st (com.Catch exc var code hand) st'\n            (func_result.Exception name v)\n| Catch4: \u2200 f st st' exc var code hand v st'' name r,\n      ceval f st code st' (func_result.Exception exc v) \u2192\n      ceval f ((st'.fst),override (st'.snd) var v) hand st'' r \u2192\n      name \u2260 exc \u2192\n      ceval f st (com.Catch exc var code hand) st'' r\n| Call1: \u2200 vl (st : imp_state) el (f:functions) (st':imp_state) r (fid:ident) var,\n      vl = aeval_list st.snd el \u2192\n      f fid st vl st' (func_result.Return r) \u2192\n      ceval f st (com.Call var fid el) ((st'.fst),override (st'.snd) var r)\n            func_result.NoResult\n| Call2: \u2200 vl (st : imp_state) el (f:functions) (st':imp_state) r (fid:ident) var name,\n      vl = aeval_list st.snd el \u2192\n      f fid st vl st' (func_result.Exception name r) \u2192\n      ceval f st (com.Call var fid el) st' (func_result.Exception name r).\n\n\n", "meta": {"author": "kendroe", "repo": "pedantic2", "sha": "5c28cd637be8a1485dccb56f0e05e612573b313e", "save_path": "github-repos/lean/kendroe-pedantic2", "path": "github-repos/lean/kendroe-pedantic2/pedantic2-5c28cd637be8a1485dccb56f0e05e612573b313e/impHeap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226291, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.36471235235602434}}
{"text": "/-\nCopyright (c) 2019 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor(s): Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.control.bitraversable.basic\nimport Mathlib.PostPort\n\nuniverses u l_1 u_1 \n\nnamespace Mathlib\n\n/-!\n# Bitraversable Lemmas\n\n## Main definitions\n  * tfst - traverse on first functor argument\n  * tsnd - traverse on second functor argument\n\n## Lemmas\n\nCombination of\n  * bitraverse\n  * tfst\n  * tsnd\n\nwith the applicatives `id` and `comp`\n\n## References\n\n * Hackage: <https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable functor bifunctor applicative\n\n\n-/\n\nnamespace bitraversable\n\n\n/-- traverse on the first functor argument -/\ndef tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {\u03b2 : Type u} {F : Type u \u2192 Type u}\n    [Applicative F] {\u03b1 : Type u} {\u03b1' : Type u} (f : \u03b1 \u2192 F \u03b1') : t \u03b1 \u03b2 \u2192 F (t \u03b1' \u03b2) :=\n  bitraverse f pure\n\n/-- traverse on the second functor argument -/\ndef tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {\u03b2 : Type u} {F : Type u \u2192 Type u}\n    [Applicative F] {\u03b1 : Type u} {\u03b1' : Type u} (f : \u03b1 \u2192 F \u03b1') : t \u03b2 \u03b1 \u2192 F (t \u03b2 \u03b1') :=\n  bitraverse pure f\n\ntheorem id_tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t]\n    {\u03b1 : Type u} {\u03b2 : Type u} (x : t \u03b1 \u03b2) : tfst id.mk x = id.mk x :=\n  id_bitraverse\n\ntheorem id_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t]\n    {\u03b1 : Type u} {\u03b2 : Type u} (x : t \u03b1 \u03b2) : tsnd id.mk x = id.mk x :=\n  id_bitraverse\n\ntheorem tfst_comp_tfst {t : Type l_1 \u2192 Type l_1 \u2192 Type l_1} [bitraversable t]\n    {F : Type l_1 \u2192 Type l_1} {G : Type l_1 \u2192 Type l_1} [Applicative F] [Applicative G]\n    [is_lawful_bitraversable t] [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type l_1}\n    {\u03b1\u2081 : Type l_1} {\u03b1\u2082 : Type l_1} {\u03b2 : Type l_1} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b1\u2081 \u2192 G \u03b1\u2082) :\n    functor.comp.mk \u2218 Functor.map (tfst f') \u2218 tfst f =\n        tfst (functor.comp.mk \u2218 Functor.map f' \u2218 f) :=\n  funext fun (x : t \u03b1\u2080 \u03b2) => comp_tfst f f' x\n\ntheorem tfst_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u}\n    {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t]\n    [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type u} {\u03b1\u2081 : Type u} {\u03b2\u2080 : Type u}\n    {\u03b2\u2081 : Type u} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n    functor.comp.mk (tfst f <$> tsnd f' x) =\n        bitraverse (functor.comp.mk \u2218 pure \u2218 f) (functor.comp.mk \u2218 Functor.map pure \u2218 f') x :=\n  sorry\n\ntheorem tsnd_tfst {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u}\n    {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t]\n    [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1\u2080 : Type u} {\u03b1\u2081 : Type u} {\u03b2\u2080 : Type u}\n    {\u03b2\u2081 : Type u} (f : \u03b1\u2080 \u2192 F \u03b1\u2081) (f' : \u03b2\u2080 \u2192 G \u03b2\u2081) (x : t \u03b1\u2080 \u03b2\u2080) :\n    functor.comp.mk (tsnd f' <$> tfst f x) =\n        bitraverse (functor.comp.mk \u2218 Functor.map pure \u2218 f) (functor.comp.mk \u2218 pure \u2218 f') x :=\n  sorry\n\ntheorem comp_tsnd {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] {F : Type u \u2192 Type u}\n    {G : Type u \u2192 Type u} [Applicative F] [Applicative G] [is_lawful_bitraversable t]\n    [is_lawful_applicative F] [is_lawful_applicative G] {\u03b1 : Type u} {\u03b2\u2080 : Type u} {\u03b2\u2081 : Type u}\n    {\u03b2\u2082 : Type u} (g : \u03b2\u2080 \u2192 F \u03b2\u2081) (g' : \u03b2\u2081 \u2192 G \u03b2\u2082) (x : t \u03b1 \u03b2\u2080) :\n    functor.comp.mk (tsnd g' <$> tsnd g x) = tsnd (functor.comp.mk \u2218 Functor.map g' \u2218 g) x :=\n  sorry\n\ntheorem tfst_eq_fst_id {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t]\n    {\u03b1 : Type u} {\u03b1' : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b1') (x : t \u03b1 \u03b2) :\n    tfst (id.mk \u2218 f) x = id.mk (bifunctor.fst f x) :=\n  sorry\n\ntheorem tsnd_eq_snd_id {t : Type u \u2192 Type u \u2192 Type u} [bitraversable t] [is_lawful_bitraversable t]\n    {\u03b1 : Type u} {\u03b2 : Type u} {\u03b2' : Type u} (f : \u03b2 \u2192 \u03b2') (x : t \u03b1 \u03b2) :\n    tsnd (id.mk \u2218 f) x = id.mk (bifunctor.snd f x) :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/control/bitraversable/lemmas_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3646484823478185}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.monad.basic\nimport category_theory.adjunction.basic\nimport category_theory.reflects_isomorphisms\n\n/-!\n# Eilenberg-Moore (co)algebras for a (co)monad\n\nThis file defines Eilenberg-Moore (co)algebras for a (co)monad,\nand provides the category instance for them.\n\nFurther it defines the adjoint pair of free and forgetful functors, respectively\nfrom and to the original category, as well as the adjoint pair of forgetful and\ncofree functors, respectively from and to the original category.\n\n## References\n* [Riehl, *Category theory in context*, Section 5.2.4][riehl2017]\n-/\n\nnamespace category_theory\nopen category\n\nuniverses v\u2081 u\u2081 -- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n\nnamespace monad\n\n/-- An Eilenberg-Moore algebra for a monad `T`.\n    cf Definition 5.2.3 in [Riehl][riehl2017]. -/\nstructure algebra (T : monad C) : Type (max u\u2081 v\u2081) :=\n(A : C)\n(a : (T : C \u2964 C).obj A \u27f6 A)\n(unit' : T.\u03b7.app A \u226b a = \ud835\udfd9 A . obviously)\n(assoc' : T.\u03bc.app A \u226b a = (T : C \u2964 C).map a \u226b a . obviously)\n\nrestate_axiom algebra.unit'\nrestate_axiom algebra.assoc'\nattribute [reassoc] algebra.unit algebra.assoc\n\nnamespace algebra\nvariables {T : monad C}\n\n/-- A morphism of Eilenberg\u2013Moore algebras for the monad `T`. -/\n@[ext] structure hom (A B : algebra T) :=\n(f : A.A \u27f6 B.A)\n(h' : T.map f \u226b B.a = A.a \u226b f . obviously)\n\nrestate_axiom hom.h'\nattribute [simp, reassoc] hom.h\n\nnamespace hom\n\n/-- The identity homomorphism for an Eilenberg\u2013Moore algebra. -/\ndef id (A : algebra T) : hom A A :=\n{ f := \ud835\udfd9 A.A }\n\ninstance (A : algebra T) : inhabited (hom A A) := \u27e8{ f := \ud835\udfd9 _ }\u27e9\n\n/-- Composition of Eilenberg\u2013Moore algebra homomorphisms. -/\ndef comp {P Q R : algebra T} (f : hom P Q) (g : hom Q R) : hom P R :=\n{ f := f.f \u226b g.f }\n\nend hom\n\ninstance : category_struct (algebra T) :=\n{ hom := hom,\n  id := hom.id,\n  comp := @hom.comp _ _ _ }\n\n@[simp] lemma comp_eq_comp {A A' A'' : algebra T} (f : A \u27f6 A') (g : A' \u27f6 A'') :\n  algebra.hom.comp f g = f \u226b g := rfl\n@[simp] lemma id_eq_id (A : algebra T) :\n  algebra.hom.id A = \ud835\udfd9 A := rfl\n\n@[simp] \n\n/-- The category of Eilenberg-Moore algebras for a monad.\n    cf Definition 5.2.4 in [Riehl][riehl2017]. -/\ninstance EilenbergMoore : category (algebra T) := {}.\n\n/--\nTo construct an isomorphism of algebras, it suffices to give an isomorphism of the carriers which\ncommutes with the structure morphisms.\n-/\n@[simps]\ndef iso_mk {A B : algebra T} (h : A.A \u2245 B.A) (w : T.map h.hom \u226b B.a = A.a \u226b h.hom) : A \u2245 B :=\n{ hom := { f := h.hom },\n  inv :=\n  { f := h.inv,\n    h' := by { rw [h.eq_comp_inv, category.assoc, \u2190w, \u2190functor.map_comp_assoc], simp } } }\n\nend algebra\n\nvariables (T : monad C)\n\n/-- The forgetful functor from the Eilenberg-Moore category, forgetting the algebraic structure. -/\n@[simps] def forget : algebra T \u2964 C :=\n{ obj := \u03bb A, A.A,\n  map := \u03bb A B f, f.f }\n\n/-- The free functor from the Eilenberg-Moore category, constructing an algebra for any object. -/\n@[simps] def free : C \u2964 algebra T :=\n{ obj := \u03bb X,\n  { A := T.obj X,\n    a := T.\u03bc.app X,\n    assoc' := (T.assoc _).symm },\n  map := \u03bb X Y f,\n  { f := T.map f,\n    h' := T.\u03bc.naturality _ } }\n\ninstance [inhabited C] : inhabited (algebra T) :=\n\u27e8(free T).obj (default C)\u27e9\n\n/-- The adjunction between the free and forgetful constructions for Eilenberg-Moore algebras for\n  a monad. cf Lemma 5.2.8 of [Riehl][riehl2017]. -/\n-- The other two `simps` projection lemmas can be derived from these two, so `simp_nf` complains if\n-- those are added too\n@[simps unit counit {rhs_md := semireducible}]\ndef adj : T.free \u22a3 T.forget :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y,\n  { to_fun := \u03bb f, T.\u03b7.app X \u226b f.f,\n    inv_fun := \u03bb f,\n    { f := T.map f \u226b Y.a,\n      h' := by { dsimp, simp [\u2190Y.assoc, \u2190T.\u03bc.naturality_assoc] } },\n    left_inv := \u03bb f, by { ext, dsimp, simp },\n    right_inv := \u03bb f,\n    begin\n      dsimp only [forget_obj, monad_to_functor_eq_coe],\n      rw [\u2190T.\u03b7.naturality_assoc, Y.unit],\n      apply category.comp_id,\n    end }}\n\n/--\nGiven an algebra morphism whose carrier part is an isomorphism, we get an algebra isomorphism.\n-/\nlemma algebra_iso_of_iso {A B : algebra T} (f : A \u27f6 B) [is_iso f.f] : is_iso f :=\n\u27e8\u27e8{ f := inv f.f,\n    h' := by { rw [is_iso.eq_comp_inv f.f, category.assoc, \u2190 f.h], dsimp, simp } }, by tidy\u27e9\u27e9\n\ninstance forget_reflects_iso : reflects_isomorphisms (forget T) :=\n{ reflects := \u03bb A B, algebra_iso_of_iso T }\n\ninstance forget_faithful : faithful (forget T) := {}\n\n/--\nGiven a monad morphism from `T\u2082` to `T\u2081`, we get a functor from the algebras of `T\u2081` to algebras of\n`T\u2082`.\n-/\n@[simps]\ndef algebra_functor_of_monad_hom {T\u2081 T\u2082 : monad C} (h : T\u2082 \u27f6 T\u2081) :\n  algebra T\u2081 \u2964 algebra T\u2082 :=\n{ obj := \u03bb A,\n  { A := A.A,\n    a := h.app A.A \u226b A.a,\n    unit' := by { dsimp, simp [A.unit] },\n    assoc' := by { dsimp, simp [A.assoc] } },\n  map := \u03bb A\u2081 A\u2082 f,\n  { f := f.f } }\n\n/--\nThe identity monad morphism induces the identity functor from the category of algebras to itself.\n-/\n@[simps {rhs_md := semireducible}]\ndef algebra_functor_of_monad_hom_id {T\u2081 : monad C} :\n  algebra_functor_of_monad_hom (\ud835\udfd9 T\u2081) \u2245 \ud835\udfed _ :=\nnat_iso.of_components\n  (\u03bb X, algebra.iso_mk (iso.refl _) (by { dsimp, simp, }))\n  (\u03bb X Y f, by { ext, dsimp, simp })\n\n/--\nA composition of monad morphisms gives the composition of corresponding functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef algebra_functor_of_monad_hom_comp {T\u2081 T\u2082 T\u2083 : monad C} (f : T\u2081 \u27f6 T\u2082) (g : T\u2082 \u27f6 T\u2083) :\n  algebra_functor_of_monad_hom (f \u226b g) \u2245\n    algebra_functor_of_monad_hom g \u22d9 algebra_functor_of_monad_hom f :=\nnat_iso.of_components\n  (\u03bb X, algebra.iso_mk (iso.refl _) (by { dsimp, simp }))\n  (\u03bb X Y f, by { ext, dsimp, simp })\n\n/--\nIf `f` and `g` are two equal morphisms of monads, then the functors of algebras induced by them\nare isomorphic.\nWe define it like this as opposed to using `eq_to_iso` so that the components are nicer to prove\nlemmas about.\n-/\n@[simps {rhs_md := semireducible}]\ndef algebra_functor_of_monad_hom_eq {T\u2081 T\u2082 : monad C} {f g : T\u2081 \u27f6 T\u2082} (h : f = g) :\n  algebra_functor_of_monad_hom f \u2245 algebra_functor_of_monad_hom g :=\nnat_iso.of_components\n  (\u03bb X, algebra.iso_mk (iso.refl _) (by { dsimp, simp [h] }))\n  (\u03bb X Y f, by { ext, dsimp, simp })\n\n/--\nIsomorphic monads give equivalent categories of algebras. Furthermore, they are equivalent as\ncategories over `C`, that is, we have `algebra_equiv_of_iso_monads h \u22d9 forget = forget`.\n-/\n@[simps]\ndef algebra_equiv_of_iso_monads {T\u2081 T\u2082 : monad C} (h : T\u2081 \u2245 T\u2082) :\n  algebra T\u2081 \u224c algebra T\u2082 :=\n{ functor := algebra_functor_of_monad_hom h.inv,\n  inverse := algebra_functor_of_monad_hom h.hom,\n  unit_iso :=\n    algebra_functor_of_monad_hom_id.symm \u226a\u226b\n    algebra_functor_of_monad_hom_eq (by simp) \u226a\u226b\n    algebra_functor_of_monad_hom_comp _ _,\n  counit_iso :=\n    (algebra_functor_of_monad_hom_comp _ _).symm \u226a\u226b\n    algebra_functor_of_monad_hom_eq (by simp) \u226a\u226b\n    algebra_functor_of_monad_hom_id }\n\n@[simp] lemma algebra_equiv_of_iso_monads_comp_forget {T\u2081 T\u2082 : monad C} (h : T\u2081 \u27f6 T\u2082) :\n  algebra_functor_of_monad_hom h \u22d9 forget _ = forget _ :=\nrfl\n\nend monad\n\nnamespace comonad\n\n/-- An Eilenberg-Moore coalgebra for a comonad `T`. -/\n@[nolint has_inhabited_instance]\nstructure coalgebra (G : comonad C) : Type (max u\u2081 v\u2081) :=\n(A : C)\n(a : A \u27f6 G.obj A)\n(counit' : a \u226b G.\u03b5.app A = \ud835\udfd9 A . obviously)\n(coassoc' : a \u226b G.\u03b4.app A = a \u226b G.map a . obviously)\n\nrestate_axiom coalgebra.counit'\nrestate_axiom coalgebra.coassoc'\nattribute [reassoc] coalgebra.counit coalgebra.coassoc\n\nnamespace coalgebra\nvariables {G : comonad C}\n\n/-- A morphism of Eilenberg-Moore coalgebras for the comonad `G`. -/\n@[ext, nolint has_inhabited_instance] structure hom (A B : coalgebra G) :=\n(f : A.A \u27f6 B.A)\n(h' : A.a \u226b G.map f = f \u226b B.a . obviously)\n\nrestate_axiom hom.h'\nattribute [simp, reassoc] hom.h\n\nnamespace hom\n\n/-- The identity homomorphism for an Eilenberg\u2013Moore coalgebra. -/\ndef id (A : coalgebra G) : hom A A :=\n{ f := \ud835\udfd9 A.A }\n\n/-- Composition of Eilenberg\u2013Moore coalgebra homomorphisms. -/\ndef comp {P Q R : coalgebra G} (f : hom P Q) (g : hom Q R) : hom P R :=\n{ f := f.f \u226b g.f }\n\nend hom\n\n/-- The category of Eilenberg-Moore coalgebras for a comonad. -/\ninstance : category_struct (coalgebra G) :=\n{ hom := hom,\n  id := hom.id,\n  comp := @hom.comp _ _ _ }\n\n@[simp] lemma comp_eq_comp {A A' A'' : coalgebra G} (f : A \u27f6 A') (g : A' \u27f6 A'') :\n  coalgebra.hom.comp f g = f \u226b g := rfl\n@[simp] lemma id_eq_id (A : coalgebra G) :\n  coalgebra.hom.id A = \ud835\udfd9 A := rfl\n\n@[simp] lemma id_f (A : coalgebra G) : (\ud835\udfd9 A : A \u27f6 A).f = \ud835\udfd9 A.A := rfl\n@[simp] lemma comp_f {A A' A'' : coalgebra G} (f : A \u27f6 A') (g : A' \u27f6 A'') :\n  (f \u226b g).f = f.f \u226b g.f := rfl\n\n/-- The category of Eilenberg-Moore coalgebras for a comonad. -/\ninstance EilenbergMoore : category (coalgebra G) := {}.\n\n/--\nTo construct an isomorphism of coalgebras, it suffices to give an isomorphism of the carriers which\ncommutes with the structure morphisms.\n-/\n@[simps]\ndef iso_mk {A B : coalgebra G} (h : A.A \u2245 B.A) (w : A.a \u226b G.map h.hom = h.hom \u226b B.a) : A \u2245 B :=\n{ hom := { f := h.hom },\n  inv :=\n  { f := h.inv,\n    h' := by { rw [h.eq_inv_comp, \u2190reassoc_of w, \u2190functor.map_comp], simp } } }\n\nend coalgebra\n\nvariables (G : comonad C)\n\n/-- The forgetful functor from the Eilenberg-Moore category, forgetting the coalgebraic\nstructure. -/\n@[simps] def forget : coalgebra G \u2964 C :=\n{ obj := \u03bb A, A.A,\n  map := \u03bb A B f, f.f }\n\n/--\nGiven a coalgebra morphism whose carrier part is an isomorphism, we get a coalgebra isomorphism.\n-/\nlemma coalgebra_iso_of_iso {A B : coalgebra G} (f : A \u27f6 B) [is_iso f.f] : is_iso f :=\n\u27e8\u27e8{ f := inv f.f,\n    h' := by { rw [is_iso.eq_inv_comp f.f, \u2190f.h_assoc], dsimp, simp } }, by tidy\u27e9\u27e9\n\ninstance forget_reflects_iso : reflects_isomorphisms (forget G) :=\n{ reflects := \u03bb A B, coalgebra_iso_of_iso G }\n\n/-- The cofree functor from the Eilenberg-Moore category, constructing a coalgebra for any\nobject. -/\n@[simps] def cofree : C \u2964 coalgebra G :=\n{ obj := \u03bb X,\n  { A := G.obj X,\n    a := G.\u03b4.app X,\n    coassoc' := (G.coassoc _).symm },\n  map := \u03bb X Y f,\n  { f := G.map f,\n    h' := (G.\u03b4.naturality _).symm } }\n\n/--\nThe adjunction between the cofree and forgetful constructions for Eilenberg-Moore coalgebras\nfor a comonad.\n-/\n-- The other two `simps` projection lemmas can be derived from these two, so `simp_nf` complains if\n-- those are added too\n@[simps unit counit]\ndef adj : forget G \u22a3 cofree G :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y,\n  { to_fun := \u03bb f,\n    { f := X.a \u226b G.map f,\n      h' := by { dsimp, simp [\u2190coalgebra.coassoc_assoc] } },\n    inv_fun := \u03bb g, g.f \u226b G.\u03b5.app Y,\n    left_inv := \u03bb f,\n      by { dsimp, rw [category.assoc, G.\u03b5.naturality, functor.id_map, X.counit_assoc] },\n    right_inv := \u03bb g,\n    begin\n      ext1, dsimp,\n      rw [functor.map_comp, g.h_assoc, cofree_obj_a, comonad.right_counit],\n      apply comp_id,\n    end }}\n\ninstance forget_faithful : faithful (forget G) := {}\n\nend comonad\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/monad/algebra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3646484739013058}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport tactic.simp_result\nimport tactic.clear\nimport control.equiv_functor.instances\n\n/-!\n# The `equiv_rw` tactic transports goals or hypotheses along equivalences.\n\nThe basic syntax is `equiv_rw e`, where `e : \u03b1 \u2243 \u03b2` is an equivalence.\nThis will try to replace occurrences of `\u03b1` in the goal with `\u03b2`, for example\ntransforming\n* `\u22a2 \u03b1` to `\u22a2 \u03b2`,\n* `\u22a2 option \u03b1` to `\u22a2 option \u03b2`\n* `\u22a2 {a // P}` to `{b // P (\u21d1(equiv.symm e) b)}`\n\nThe tactic can also be used to rewrite hypotheses, using the syntax `equiv_rw e at h`.\n\n## Implementation details\n\nThe main internal function is `equiv_rw_type e t`,\nwhich attempts to turn an expression `e : \u03b1 \u2243 \u03b2` into a new equivalence with left hand side `t`.\nAs an example, with `t = option \u03b1`, it will generate `functor.map_equiv option e`.\n\nThis is achieved by generating a new synthetic goal `%%t \u2243 _`,\nand calling `solve_by_elim` with an appropriate set of congruence lemmas.\nTo avoid having to specify the relevant congruence lemmas by hand,\nwe mostly rely on `equiv_functor.map_equiv` and `bifunctor.map_equiv`\nalong with some structural congruence lemmas such as\n* `equiv.arrow_congr'`,\n* `equiv.subtype_equiv_of_subtype'`,\n* `equiv.sigma_congr_left'`, and\n* `equiv.Pi_congr_left'`.\n\nThe main `equiv_rw` function, when operating on the goal, simply generates a new equivalence `e'`\nwith left hand side matching the target, and calls `apply e'.inv_fun`.\n\nWhen operating on a hypothesis `x : \u03b1`, we introduce a new fact `h : x = e.symm (e x)`, revert this,\nand then attempt to `generalize`, replacing all occurrences of `e x` with a new constant `y`, before\n`intro`ing and `subst`ing `h`, and renaming `y` back to `x`.\n\n## Future improvements\nIn a future PR I anticipate that `derive equiv_functor` should work on many examples,\n(internally using `transport`, which is in turn based on `equiv_rw`)\nand we can incrementally bootstrap the strength of `equiv_rw`.\n\nAn ambitious project might be to add `equiv_rw!`,\na tactic which, when failing to find appropriate `equiv_functor` instances,\nattempts to `derive` them on the spot.\n\nFor now `equiv_rw` is entirely based on `equiv`,\nbut the framework can readily be generalised to also work with other types of equivalences,\nfor example specific notations such as ring equivalence (`\u2243+*`),\nor general categorical isomorphisms (`\u2245`).\n\nThis will allow us to transport across more general types of equivalences,\nbut this will wait for another subsequent PR.\n-/\n\nnamespace tactic\n\n/-- A list of lemmas used for constructing congruence equivalences. -/\n\n-- Although this looks 'hard-coded', in fact the lemma `equiv_functor.map_equiv`\n-- allows us to extend `equiv_rw` simply by constructing new instance so `equiv_functor`.\n\n-- TODO: We should also use `category_theory.functorial` and `category_theory.hygienic` instances.\n-- (example goal: we could rewrite along an isomorphism of rings (either as `R \u2245 S` or `R \u2243+* S`)\n-- and turn an `x : mv_polynomial \u03c3 R` into an `x : mv_polynomial \u03c3 S`.).\n\nmeta def equiv_congr_lemmas : list (tactic expr) :=\n[ `equiv.of_iff,\n  -- TODO decide what to do with this; it's an equiv_bifunctor?\n  `equiv.equiv_congr,\n  -- The function arrow is technically a bifunctor `Type\u1d52\u1d56 \u2192 Type \u2192 Type`,\n  -- but the pattern matcher will never see this.\n  `equiv.arrow_congr',\n  -- Allow rewriting in subtypes:\n  `equiv.subtype_equiv_of_subtype',\n  -- Allow rewriting in the first component of a sigma-type:\n  `equiv.sigma_congr_left',\n  -- Allow rewriting \u2200s:\n  -- (You might think that repeated application of `equiv.forall_congr'\n  -- would handle the higher arity cases, but unfortunately unification is not clever enough.)\n  `equiv.forall\u2083_congr',\n  `equiv.forall\u2082_congr',\n  `equiv.forall_congr',\n  -- Allow rewriting in argument of Pi types:\n   `equiv.Pi_congr_left',\n  -- Handles `sum` and `prod`, and many others:\n  `bifunctor.map_equiv,\n  -- Handles `list`, `option`, `unique`, and many others:\n  `equiv_functor.map_equiv,\n  -- We have to filter results to ensure we don't cheat and use exclusively\n  -- `equiv.refl` and `iff.refl`!\n  `equiv.refl,\n  `iff.refl\n  ].map (\u03bb n, mk_const n)\n\ndeclare_trace equiv_rw_type\n\n/--\nConfiguration structure for `equiv_rw`.\n\n* `max_depth` bounds the search depth for equivalences to rewrite along.\n  The default value is 10.\n  (e.g., if you're rewriting along `e : \u03b1 \u2243 \u03b2`, and `max_depth := 2`,\n  you can rewrite `option (option \u03b1))` but not `option (option (option \u03b1))`.\n-/\nmeta structure equiv_rw_cfg :=\n(max_depth : \u2115 := 10)\n\n/--\nImplementation of `equiv_rw_type`, using `solve_by_elim`.\nExpects a goal of the form `t \u2243 _`,\nand tries to solve it using `eq : \u03b1 \u2243 \u03b2` and congruence lemmas.\n-/\nmeta def equiv_rw_type_core (eq : expr) (cfg : equiv_rw_cfg) : tactic unit :=\ndo\n  /-\n    We now call `solve_by_elim` to try to generate the requested equivalence.\n    There are a few subtleties!\n    * We make sure that `eq` is the first lemma, so it is applied whenever possible.\n    * In `equiv_congr_lemmas`, we put `equiv.refl` last so it is only used when it is not possible\n      to descend further.\n    * Since some congruence lemmas generate subgoals with `\u2200` statements,\n      we use the `pre_apply` subtactic of `solve_by_elim` to preprocess each new goal with `intros`.\n  -/\n  solve_by_elim\n  { use_symmetry := false,\n    use_exfalso := false,\n    lemma_thunks := some (pure eq :: equiv_congr_lemmas),\n    ctx_thunk := pure [],\n    max_depth := cfg.max_depth,\n    -- Subgoals may contain function types,\n    -- and we want to continue trying to construct equivalences after the binders.\n    pre_apply := tactic.intros >> skip,\n    backtrack_all_goals := tt,\n    -- If solve_by_elim gets stuck, make sure it isn't because there's a later `\u2243` or `\u2194` goal\n    -- that we should still attempt.\n    discharger :=\n      `[success_if_fail { match_target _ \u2243 _ }] >> `[success_if_fail { match_target _ \u2194 _ }] >>\n      (`[show _ \u2243 _] <|> `[show _ \u2194 _]) <|>\n      trace_if_enabled `equiv_rw_type \"Failed, no congruence lemma applied!\" >> failed,\n    -- We use the `accept` tactic in `solve_by_elim` to provide tracing.\n    accept := \u03bb goals, lock_tactic_state (do\n      when_tracing `equiv_rw_type (do\n        goals.mmap pp >>= \u03bb goals, trace format!\"So far, we've built: {goals}\"),\n      done <|>\n      when_tracing `equiv_rw_type (do\n        gs \u2190 get_goals,\n        gs \u2190 gs.mmap (\u03bb g, infer_type g >>= pp),\n        trace format!\"Attempting to adapt to {gs}\")) }\n\n/--\n`equiv_rw_type e t` rewrites the type `t` using the equivalence `e : \u03b1 \u2243 \u03b2`,\nreturning a new equivalence `t \u2243 t'`.\n-/\nmeta def equiv_rw_type (eqv : expr) (ty : expr) (cfg : equiv_rw_cfg) : tactic expr :=\ndo\n  when_tracing `equiv_rw_type (do\n    ty_pp \u2190 pp ty,\n    eqv_pp \u2190 pp eqv,\n    eqv_ty_pp \u2190 infer_type eqv >>= pp,\n    trace format!\"Attempting to rewrite the type `{ty_pp}` using `{eqv_pp} : {eqv_ty_pp}`.\"),\n  `(_ \u2243 _) \u2190 infer_type eqv | fail format!\"{eqv} must be an `equiv`\",\n  -- We prepare a synthetic goal of type `(%%ty \u2243 _)`, for some placeholder right hand side.\n  equiv_ty \u2190 to_expr ``(%%ty \u2243 _),\n  -- Now call `equiv_rw_type_core`.\n  new_eqv \u2190 prod.snd <$> (solve_aux equiv_ty $ equiv_rw_type_core eqv cfg),\n  -- Check that we actually used the equivalence `eq`\n  -- (`equiv_rw_type_core` will always find `equiv.refl`,\n  -- but hopefully only after all other possibilities)\n  new_eqv \u2190 instantiate_mvars new_eqv,\n  -- We previously had `guard (eqv.occurs new_eqv)` here, but `kdepends_on` is more reliable.\n  kdepends_on new_eqv eqv >>= guardb <|> (do\n    eqv_pp \u2190 pp eqv,\n    ty_pp \u2190 pp ty,\n    fail format!\"Could not construct an equivalence from {eqv_pp} of the form: {ty_pp} \u2243 _\"),\n  -- Finally we simplify the resulting equivalence,\n  -- to compress away some `map_equiv equiv.refl` subexpressions.\n  prod.fst <$> new_eqv.simp {fail_if_unchanged := ff}\n\nmk_simp_attribute equiv_rw_simp \"The simpset `equiv_rw_simp` is used by the tactic `equiv_rw` to\nsimplify applications of equivalences and their inverses.\"\n\nattribute [equiv_rw_simp] equiv.symm_symm equiv.apply_symm_apply equiv.symm_apply_apply\n\n/--\nAttempt to replace the hypothesis with name `x`\nby transporting it along the equivalence in `e : \u03b1 \u2243 \u03b2`.\n-/\nmeta def equiv_rw_hyp (x : name) (e : expr) (cfg : equiv_rw_cfg := {}) : tactic unit :=\n-- We call `dsimp_result` to perform the beta redex introduced by `revert`\ndsimp_result (do\n  x' \u2190 get_local x,\n  x_ty \u2190 infer_type x',\n  -- Adapt `e` to an equivalence with left-hand-side `x_ty`.\n  e \u2190 equiv_rw_type e x_ty cfg,\n  eq \u2190 to_expr ``(%%x' = equiv.symm %%e (equiv.to_fun %%e %%x')),\n  prf \u2190 to_expr ``((equiv.symm_apply_apply %%e %%x').symm),\n  h \u2190 note_anon eq prf,\n  -- Revert the new hypothesis, so it is also part of the goal.\n  revert h,\n  ex \u2190 to_expr ``(equiv.to_fun %%e %%x'),\n  -- Now call `generalize`,\n  -- attempting to replace all occurrences of `e x`,\n  -- calling it for now `j : \u03b2`, with `k : x = e.symm j`.\n  generalize ex (by apply_opt_param) transparency.none,\n  -- Reintroduce `x` (now of type `b`), and the hypothesis `h`.\n  intro x,\n  h \u2190 intro1,\n  -- Finally, if we're working on properties, substitute along `h`, then do some cleanup,\n  -- and if we're working on data, just throw out the old `x`.\n  b \u2190 target >>= is_prop,\n  if b then do\n    subst h,\n    `[try { simp only with equiv_rw_simp }]\n  else\n    -- We may need to unfreeze `x` before we can `clear` it.\n    unfreezing_hyp x' (clear' tt [x']) <|> fail\n      format!\"equiv_rw expected to be able to clear the original hypothesis {x}, but couldn't.\",\n  skip)\n  {fail_if_unchanged := ff} tt -- call `dsimp_result` with `no_defaults := tt`.\n\n/-- Rewrite the goal using an equiv `e`. -/\nmeta def equiv_rw_target (e : expr) (cfg : equiv_rw_cfg := {}) : tactic unit :=\ndo\n  t \u2190 target,\n  e \u2190 equiv_rw_type e t cfg,\n  s \u2190 to_expr ``(equiv.inv_fun %%e),\n  tactic.eapply s,\n  skip\n\nend tactic\n\nnamespace tactic.interactive\nopen lean.parser\nopen interactive interactive.types\nopen tactic\n\nlocal postfix `?`:9001 := optional\n\n/--\n`equiv_rw e at h`, where `h : \u03b1` is a hypothesis, and `e : \u03b1 \u2243 \u03b2`,\nwill attempt to transport `h` along `e`, producing a new hypothesis `h : \u03b2`,\nwith all occurrences of `h` in other hypotheses and the goal replaced with `e.symm h`.\n\n`equiv_rw e` will attempt to transport the goal along an equivalence `e : \u03b1 \u2243 \u03b2`.\nIn its minimal form it replaces the goal `\u22a2 \u03b1` with `\u22a2 \u03b2` by calling `apply e.inv_fun`.\n\n`equiv_rw` will also try rewriting under (equiv_)functors, so can turn\na hypothesis `h : list \u03b1` into `h : list \u03b2` or\na goal `\u22a2 unique \u03b1` into `\u22a2 unique \u03b2`.\n\nThe maximum search depth for rewriting in subexpressions is controlled by\n`equiv_rw e {max_depth := n}`.\n-/\nmeta def equiv_rw (e : parse texpr) (loc : parse $ (tk \"at\" *> ident)?) (cfg : equiv_rw_cfg := {}) :\n  itactic :=\ndo e \u2190 to_expr e,\n   match loc with\n   | (some hyp) := equiv_rw_hyp hyp e cfg\n   | none := equiv_rw_target e cfg\n   end\n\nadd_tactic_doc\n{ name        := \"equiv_rw\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.equiv_rw],\n  tags        := [\"rewriting\", \"equiv\", \"transport\"] }\n\n/--\nSolve a goal of the form `t \u2243 _`,\nby constructing an equivalence from `e : \u03b1 \u2243 \u03b2`.\nThis is the same equivalence that `equiv_rw` would use to rewrite a term of type `t`.\n\nA typical usage might be:\n```\nhave e' : option \u03b1 \u2243 option \u03b2 := by equiv_rw_type e\n```\n-/\nmeta def equiv_rw_type (e : parse texpr) (cfg : equiv_rw_cfg := {}) : itactic :=\ndo\n `(%%t \u2243 _) \u2190 target | fail \"`equiv_rw_type` solves goals of the form `t \u2243 _`.\",\n e \u2190 to_expr e,\n tactic.equiv_rw_type e t cfg >>= tactic.exact\n\nadd_tactic_doc\n{ name        := \"equiv_rw_type\",\n  category    := doc_category.tactic,\n  decl_names  := [`tactic.interactive.equiv_rw_type],\n  tags        := [\"rewriting\", \"equiv\", \"transport\"] }\n\nend tactic.interactive\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/tactic/equiv_rw.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.36464335110996743}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.cone_category\nimport category_theory.adjunction\n\n/-!\n\n# Multi-(co)equalizers\n\nA *multiequalizer* is an equalizer of two morphisms between two products.\nSince both products and equalizers are limits, such an object is again a limit.\nThis file provides the diagram whose limit is indeed such an object.\nIn fact, it is well-known that any limit can be obtained as a multiequalizer.\nThe dual construction (multicoequalizers) is also provided.\n\n## Projects\n\nProve that a multiequalizer can be identified with\nan equalizer between products (and analogously for multicoequalizers).\n\nProve that the limit of any diagram is a multiequalizer (and similarly for colimits).\n\n-/\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses w v u\n\n/-- The type underlying the multiequalizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multicospan {L R : Type w} (fst snd : R \u2192 L) : Type w\n| left : L \u2192 walking_multicospan\n| right : R \u2192 walking_multicospan\n\n/-- The type underlying the multiecoqualizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multispan {L R : Type w} (fst snd : L \u2192 R) : Type w\n| left : L \u2192 walking_multispan\n| right : R \u2192 walking_multispan\n\nnamespace walking_multicospan\n\nvariables {L R : Type w} {fst snd : R \u2192 L}\n\ninstance [inhabited L] : inhabited (walking_multicospan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multicospan`. -/\ninductive hom : \u03a0 (a b : walking_multicospan fst snd), Type w\n| id (A)  : hom A A\n| fst (b) : hom (left (fst b)) (right b)\n| snd (b) : hom (left (snd b)) (right b)\n\ninstance {a : walking_multicospan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multicospan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multicospan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst b) (hom.id X) := hom.fst b\n| _ _ _ (hom.snd b) (hom.id X) := hom.snd b\n\ninstance : small_category (walking_multicospan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multicospan\n\nnamespace walking_multispan\n\nvariables {L R : Type v} {fst snd : L \u2192 R}\n\ninstance [inhabited L] : inhabited (walking_multispan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multispan`. -/\ninductive hom : \u03a0 (a b : walking_multispan fst snd), Type v\n| id (A)  : hom A A\n| fst (a) : hom (left a) (right (fst a))\n| snd (a) : hom (left a) (right (snd a))\n\ninstance {a : walking_multispan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multispan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multispan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst a) (hom.id X) := hom.fst a\n| _ _ _ (hom.snd a) (hom.id X) := hom.snd a\n\ninstance : small_category (walking_multispan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multispan\n\n/-- This is a structure encapsulating the data necessary to define a `multicospan`. -/\n@[nolint has_inhabited_instance]\nstructure multicospan_index (C : Type u) [category.{v} C] :=\n(L R : Type w)\n(fst_to snd_to : R \u2192 L)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 b, left (fst_to b) \u27f6 right b)\n(snd : \u03a0 b, left (snd_to b) \u27f6 right b)\n\n/-- This is a structure encapsulating the data necessary to define a `multispan`. -/\n@[nolint has_inhabited_instance]\nstructure multispan_index (C : Type u) [category.{v} C] :=\n(L R : Type w)\n(fst_from snd_from : L \u2192 R)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 a, left a \u27f6 right (fst_from a))\n(snd : \u03a0 a, left a \u27f6 right (snd_from a))\n\nnamespace multicospan_index\n\nvariables {C : Type u} [category.{v} C] (I : multicospan_index C)\n\n/-- The multicospan associated to `I : multicospan_index`. -/\ndef multicospan : walking_multicospan I.fst_to I.snd_to \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multicospan.left a := I.left a\n  | walking_multicospan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multicospan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multicospan.hom.fst b := I.fst _\n  | _, _, walking_multicospan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multicospan_obj_left (a) :\n  I.multicospan.obj (walking_multicospan.left a) = I.left a := rfl\n\n@[simp] lemma multicospan_obj_right (b) :\n  I.multicospan.obj (walking_multicospan.right b) = I.right b := rfl\n\n@[simp] lemma multicospan_map_fst (b) :\n  I.multicospan.map (walking_multicospan.hom.fst b) = I.fst b := rfl\n\n@[simp] lemma multicospan_map_snd (b) :\n  I.multicospan.map (walking_multicospan.hom.snd b) = I.snd b := rfl\n\nvariables [has_product I.left] [has_product I.right]\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.fst`. -/\nnoncomputable\ndef fst_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.fst_to b) \u226b I.fst b)\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.snd`. -/\nnoncomputable\ndef snd_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.snd_to b) \u226b I.snd b)\n\n@[simp, reassoc]\nlemma fst_pi_map_\u03c0 (b) : I.fst_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.fst b :=\nby simp [fst_pi_map]\n\n@[simp, reassoc]\nlemma snd_pi_map_\u03c0 (b) : I.snd_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.snd b :=\nby simp [snd_pi_map]\n\n/--\nTaking the multiequalizer over the multicospan index is equivalent to taking the equalizer over\nthe two morphsims `\u220f I.left \u21c9 \u220f I.right`. This is the diagram of the latter.\n-/\n@[simps] protected noncomputable\ndef parallel_pair_diagram := parallel_pair I.fst_pi_map I.snd_pi_map\n\nend multicospan_index\n\nnamespace multispan_index\n\nvariables {C : Type u} [category.{v} C] (I : multispan_index C)\n\n/-- The multispan associated to `I : multispan_index`. -/\ndef multispan : walking_multispan I.fst_from I.snd_from \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multispan.left a := I.left a\n  | walking_multispan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multispan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multispan.hom.fst b := I.fst _\n  | _, _, walking_multispan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multispan_obj_left (a) :\n  I.multispan.obj (walking_multispan.left a) = I.left a := rfl\n\n@[simp] lemma multispan_obj_right (b) :\n  I.multispan.obj (walking_multispan.right b) = I.right b := rfl\n\n@[simp] lemma multispan_map_fst (a) :\n  I.multispan.map (walking_multispan.hom.fst a) = I.fst a := rfl\n\n@[simp] lemma multispan_map_snd (a) :\n  I.multispan.map (walking_multispan.hom.snd a) = I.snd a := rfl\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.fst`. -/\nnoncomputable\ndef fst_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.fst b \u226b sigma.\u03b9 _ (I.fst_from b))\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.snd`. -/\nnoncomputable\ndef snd_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.snd b \u226b sigma.\u03b9 _ (I.snd_from b))\n\n@[simp, reassoc]\nlemma \u03b9_fst_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.fst_sigma_map = I.fst b \u226b sigma.\u03b9 I.right _ :=\nby simp [fst_sigma_map]\n\n@[simp, reassoc]\nlemma \u03b9_snd_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.snd_sigma_map = I.snd b \u226b sigma.\u03b9 I.right _ :=\nby simp [snd_sigma_map]\n\n/--\nTaking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over\nthe two morphsims `\u2210 I.left \u21c9 \u2210 I.right`. This is the diagram of the latter.\n-/\nprotected noncomputable\nabbreviation parallel_pair_diagram := parallel_pair I.fst_sigma_map I.snd_sigma_map\n\nend multispan_index\n\nvariables {C : Type u} [category.{v} C]\n\n/-- A multifork is a cone over a multicospan. -/\n@[nolint has_inhabited_instance]\nabbreviation multifork (I : multicospan_index C) := cone I.multicospan\n\n/-- A multicofork is a cocone over a multispan. -/\n@[nolint has_inhabited_instance]\nabbreviation multicofork (I : multispan_index C) := cocone I.multispan\n\nnamespace multifork\n\nvariables {I : multicospan_index C} (K : multifork I)\n\n/-- The maps from the cone point of a multifork to the objects on the left. -/\ndef \u03b9 (a : I.L) : K.X \u27f6 I.left a := K.\u03c0.app (walking_multicospan.left _)\n\n@[simp] lemma app_left_eq_\u03b9 (a) : K.\u03c0.app (walking_multicospan.left a) = K.\u03b9 a := rfl\n\n@[simp] lemma app_right_eq_\u03b9_comp_fst (b) :\n  K.\u03c0.app (walking_multicospan.right b) = K.\u03b9 (I.fst_to b) \u226b I.fst b :=\nby { rw \u2190 K.w (walking_multicospan.hom.fst b), refl }\n\n@[reassoc] lemma app_right_eq_\u03b9_comp_snd (b) :\n  K.\u03c0.app (walking_multicospan.right b) =  K.\u03b9 (I.snd_to b) \u226b I.snd b :=\nby { rw \u2190 K.w (walking_multicospan.hom.snd b), refl }\n\n@[simp, reassoc] lemma hom_comp_\u03b9 (K\u2081 K\u2082 : multifork I) (f : K\u2081 \u27f6 K\u2082) (j : I.L) :\n  f.hom \u226b K\u2082.\u03b9 j = K\u2081.\u03b9 j := f.w (walking_multicospan.left j)\n\n/-- Construct a multifork using a collection `\u03b9` of morphisms. -/\n@[simps]\ndef of_\u03b9 (I : multicospan_index C) (P : C) (\u03b9 : \u03a0 a, P \u27f6 I.left a)\n  (w : \u2200 b, \u03b9 (I.fst_to b) \u226b I.fst b = \u03b9 (I.snd_to b) \u226b I.snd b) :\n  multifork I :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := \u03b9 _\n    | walking_multicospan.right b := \u03b9 (I.fst_to b) \u226b I.fst b\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { dsimp, rw category.id_comp, refl },\n      { dsimp, rw category.id_comp, apply w }\n    end } }\n\n@[simp, reassoc]\nlemma condition (b) :\n  K.\u03b9 (I.fst_to b) \u226b I.fst b = K.\u03b9 (I.snd_to b) \u226b I.snd b :=\nby rw [\u2190app_right_eq_\u03b9_comp_fst, \u2190app_right_eq_\u03b9_comp_snd]\n\n/-- This definition provides a convenient way to show that a multifork is a limit. -/\n@[simps]\ndef is_limit.mk\n  (lift : \u03a0 (E : multifork I), E.X \u27f6 K.X)\n  (fac : \u2200 (E : multifork I) (i : I.L), lift E \u226b K.\u03b9 i = E.\u03b9 i)\n  (uniq : \u2200 (E : multifork I) (m : E.X \u27f6 K.X),\n    (\u2200 i : I.L, m \u226b K.\u03b9 i = E.\u03b9 i) \u2192 m = lift E) : is_limit K :=\n{ lift := lift,\n  fac' := begin\n    rintros E (a|b),\n    { apply fac },\n    { rw [\u2190 E.w (walking_multicospan.hom.fst b), \u2190 K.w (walking_multicospan.hom.fst b),\n        \u2190 category.assoc],\n      congr' 1,\n      apply fac }\n  end,\n  uniq' := begin\n    rintros E m hm,\n    apply uniq,\n    intros i,\n    apply hm,\n  end }\n\n\nvariables [has_product I.left] [has_product I.right]\n\n@[simp, reassoc]\nlemma pi_condition : pi.lift K.\u03b9 \u226b I.fst_pi_map = pi.lift K.\u03b9 \u226b I.snd_pi_map :=\nby { ext, discrete_cases, simp, }\n\n/-- Given a multifork, we may obtain a fork over `\u220f I.left \u21c9 \u220f I.right`. -/\n@[simps X] noncomputable\ndef to_pi_fork (K : multifork I) : fork I.fst_pi_map I.snd_pi_map :=\n{ X := K.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := pi.lift K.\u03b9\n    | walking_parallel_pair.one := pi.lift K.\u03b9 \u226b I.fst_pi_map\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      all_goals { change \ud835\udfd9 _ \u226b _ \u226b _ = pi.lift _ \u226b _, simp }\n    end } }\n\n@[simp] lemma to_pi_fork_\u03c0_app_zero : K.to_pi_fork.\u03b9 = pi.lift K.\u03b9 := rfl\n\n@[simp] lemma to_pi_fork_\u03c0_app_one :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.one = pi.lift K.\u03b9 \u226b I.fst_pi_map := rfl\n\nvariable (I)\n\n/-- Given a fork over `\u220f I.left \u21c9 \u220f I.right`, we may obtain a multifork. -/\n@[simps X] noncomputable\ndef of_pi_fork (c : fork I.fst_pi_map I.snd_pi_map) : multifork I :=\n{ X := c.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := c.\u03b9 \u226b pi.\u03c0 _ _\n    | walking_multicospan.right b := c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, simp },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, rw c.condition_assoc, simp }\n    end } }\n\n@[simp] lemma of_pi_fork_\u03c0_app_left (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03b9 a = c.\u03b9 \u226b pi.\u03c0 _ _ := rfl\n\n@[simp] lemma of_pi_fork_\u03c0_app_right (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.right a) = c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _ := rfl\n\nend multifork\n\nnamespace multicospan_index\n\nvariables (I : multicospan_index C) [has_product I.left] [has_product I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multifork.to_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef to_pi_fork_functor : multifork I \u2964 fork I.fst_pi_map I.snd_pi_map :=\n{ obj := multifork.to_pi_fork,\n  map := \u03bb K\u2081 K\u2082 f,\n  { hom := f.hom,\n    w' := begin\n      rintro (_|_),\n      { ext, dsimp, simp },\n      { ext,\n        simp only [multifork.to_pi_fork_\u03c0_app_one, multifork.pi_condition, category.assoc],\n        dsimp [snd_pi_map],\n        simp },\n    end } }\n\n/-- `multifork.of_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef of_pi_fork_functor : fork I.fst_pi_map I.snd_pi_map \u2964 multifork I :=\n{ obj := multifork.of_pi_fork I, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multiforks is equivalent to the category of forks over `\u220f I.left \u21c9 \u220f I.right`.\nIt then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it\npreserves and reflects limit cones.\n-/\n@[simps] noncomputable\ndef multifork_equiv_pi_fork : multifork I \u224c fork I.fst_pi_map I.snd_pi_map :=\n{ functor := to_pi_fork_functor I,\n  inverse := of_pi_fork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cones.ext (iso.refl _)\n    (by { rintros (_|_); dsimp; simp[\u2190fork.app_one_eq_\u03b9_comp_left, -fork.app_one_eq_\u03b9_comp_left] }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, fork.ext (iso.refl _) (by { ext \u27e8j\u27e9, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }) }\n\nend multicospan_index\n\nnamespace multicofork\n\nvariables {I : multispan_index C} (K : multicofork I)\n\n/-- The maps to the cocone point of a multicofork from the objects on the right. -/\ndef \u03c0 (b : I.R) : I.right b \u27f6 K.X :=\nK.\u03b9.app (walking_multispan.right _)\n\n@[simp] lemma \u03c0_eq_app_right (b) : K.\u03b9.app (walking_multispan.right _) = K.\u03c0 b := rfl\n\n@[simp] lemma fst_app_right (a) :\n  K.\u03b9.app (walking_multispan.left a) = I.fst a \u226b K.\u03c0 _ :=\nby { rw \u2190 K.w (walking_multispan.hom.fst a), refl }\n\n@[reassoc] lemma snd_app_right (a) :\n  K.\u03b9.app (walking_multispan.left a) = I.snd a \u226b K.\u03c0 _ :=\nby { rw \u2190 K.w (walking_multispan.hom.snd a), refl }\n\n/-- Construct a multicofork using a collection `\u03c0` of morphisms. -/\n@[simps]\ndef of_\u03c0 (I : multispan_index C) (P : C) (\u03c0 : \u03a0 b, I.right b \u27f6 P)\n  (w : \u2200 a, I.fst a \u226b \u03c0 (I.fst_from a) = I.snd a \u226b \u03c0 (I.snd_from a)) :\n  multicofork I :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := I.fst a \u226b \u03c0 _\n    | walking_multispan.right b := \u03c0 _\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { dsimp, rw category.comp_id, refl },\n      { dsimp, rw category.comp_id, apply (w _).symm }\n    end } }\n\n@[simp, reassoc]\nlemma condition (a) : I.fst a \u226b K.\u03c0 (I.fst_from a) = I.snd a \u226b K.\u03c0 (I.snd_from a) :=\nby rw [\u2190K.snd_app_right, \u2190K.fst_app_right]\n\n/-- This definition provides a convenient way to show that a multicofork is a colimit. -/\n@[simps]\ndef is_colimit.mk\n  (desc : \u03a0 (E : multicofork I), K.X \u27f6 E.X)\n  (fac : \u2200 (E : multicofork I) (i : I.R), K.\u03c0 i \u226b desc E = E.\u03c0 i)\n  (uniq : \u2200 (E : multicofork I) (m : K.X \u27f6 E.X),\n    (\u2200 i : I.R, K.\u03c0 i \u226b m = E.\u03c0 i) \u2192 m = desc E) : is_colimit K :=\n{ desc := desc,\n  fac' := begin\n    rintros S (a|b),\n    { rw [\u2190 K.w (walking_multispan.hom.fst a), \u2190 S.w (walking_multispan.hom.fst a),\n        category.assoc],\n      congr' 1,\n      apply fac },\n    { apply fac },\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply uniq,\n    intros i,\n    apply hm\n  end }\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n@[simp, reassoc]\nlemma sigma_condition :\n  I.fst_sigma_map \u226b sigma.desc K.\u03c0 = I.snd_sigma_map \u226b sigma.desc K.\u03c0 :=\nby { ext, discrete_cases, simp, }\n\n/-- Given a multicofork, we may obtain a cofork over `\u2210 I.left \u21c9 \u2210 I.right`. -/\n@[simps X] noncomputable\ndef to_sigma_cofork (K : multicofork I) : cofork I.fst_sigma_map I.snd_sigma_map :=\n{ X := K.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := I.fst_sigma_map \u226b sigma.desc K.\u03c0\n    | walking_parallel_pair.one := sigma.desc K.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      all_goals { change _ \u226b sigma.desc _ = (_ \u226b _) \u226b \ud835\udfd9 _, simp }\n    end } }\n\n@[simp] lemma to_sigma_cofork_\u03c0 : K.to_sigma_cofork.\u03c0 = sigma.desc K.\u03c0 := rfl\n\nvariable (I)\n\n/-- Given a cofork over `\u2210 I.left \u21c9 \u2210 I.right`, we may obtain a multicofork. -/\n@[simps X] noncomputable\ndef of_sigma_cofork (c : cofork I.fst_sigma_map I.snd_sigma_map) : multicofork I :=\n{ X := c.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0\n    | walking_multispan.right b := (sigma.\u03b9 I.right b : _) \u226b c.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, dsimp,\n        simp only [cofork.condition, category.comp_id],\n        rw [\u2190I.\u03b9_fst_sigma_map_assoc, c.condition] },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b \ud835\udfd9 _,\n        rw c.condition, simp }\n    end } }\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_left (c : cofork I.fst_sigma_map I.snd_sigma_map) (a) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.left a) =\n    (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0 := rfl\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_right (c : cofork I.fst_sigma_map I.snd_sigma_map) (b) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.right b) = (sigma.\u03b9 I.right b : _) \u226b c.\u03c0 := rfl\n\nend multicofork\n\nnamespace multispan_index\n\nvariables (I : multispan_index C) [has_coproduct I.left] [has_coproduct I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multicofork.to_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef to_sigma_cofork_functor : multicofork I \u2964 cofork I.fst_sigma_map I.snd_sigma_map :=\n{ obj := multicofork.to_sigma_cofork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multicofork.of_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef of_sigma_cofork_functor : cofork I.fst_sigma_map I.snd_sigma_map \u2964 multicofork I :=\n{ obj := multicofork.of_sigma_cofork I,\n  map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multicoforks is equivalent to the category of coforks over `\u2210 I.left \u21c9 \u2210 I.right`.\nIt then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that\nit preserves and reflects colimit cocones.\n-/\n@[simps] noncomputable\ndef multicofork_equiv_sigma_cofork : multicofork I \u224c cofork I.fst_sigma_map I.snd_sigma_map :=\n{ functor := to_sigma_cofork_functor I,\n  inverse := of_sigma_cofork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cocones.ext (iso.refl _)\n      (by { rintros (_|_); dsimp; simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, cofork.ext (iso.refl _)\n      (by { ext \u27e8j\u27e9, dsimp, simp only [category.comp_id, colimit.\u03b9_desc, cofan.mk_\u03b9_app], refl }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, dsimp, simp, }) }\n\nend multispan_index\n\n/-- For `I : multicospan_index C`, we say that it has a multiequalizer if the associated\n  multicospan has a limit. -/\nabbreviation has_multiequalizer (I : multicospan_index C) :=\n  has_limit I.multicospan\n\nnoncomputable theory\n\n/-- The multiequalizer of `I : multicospan_index C`. -/\nabbreviation multiequalizer (I : multicospan_index C) [has_multiequalizer I] : C :=\n  limit I.multicospan\n\n/-- For `I : multispan_index C`, we say that it has a multicoequalizer if\n  the associated multicospan has a limit. -/\nabbreviation has_multicoequalizer (I : multispan_index C) :=\n  has_colimit I.multispan\n\n/-- The multiecoqualizer of `I : multispan_index C`. -/\nabbreviation multicoequalizer (I : multispan_index C) [has_multicoequalizer I] : C :=\n  colimit I.multispan\n\nnamespace multiequalizer\n\nvariables (I : multicospan_index C) [has_multiequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03b9 (a : I.L) : multiequalizer I \u27f6 I.left a :=\nlimit.\u03c0 _ (walking_multicospan.left a)\n\n/-- The multifork associated to the multiequalizer. -/\nabbreviation multifork : multifork I :=\nlimit.cone _\n\n@[simp]\nlemma multifork_\u03b9 (a) :\n  (multiequalizer.multifork I).\u03b9 a = multiequalizer.\u03b9 I a := rfl\n\n@[simp]\nlemma multifork_\u03c0_app_left (a) :\n  (multiequalizer.multifork I).\u03c0.app (walking_multicospan.left a) =\n  multiequalizer.\u03b9 I a := rfl\n\n@[reassoc]\nlemma condition (b) :\n  multiequalizer.\u03b9 I (I.fst_to b) \u226b I.fst b =\n  multiequalizer.\u03b9 I (I.snd_to b) \u226b I.snd b :=\nmultifork.condition _ _\n\n/-- Construct a morphism to the multiequalizer from its universal property. -/\nabbreviation lift (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) :\n  W \u27f6 multiequalizer I :=\nlimit.lift _ (multifork.of_\u03b9 I _ k h)\n\n@[simp, reassoc]\nlemma lift_\u03b9 (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) (a) :\n  multiequalizer.lift I _ k h \u226b multiequalizer.\u03b9 I a = k _ :=\nlimit.lift_\u03c0 _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : W \u27f6 multiequalizer I)\n  (h : \u2200 a, i \u226b multiequalizer.\u03b9 I a =\n  j \u226b multiequalizer.\u03b9 I a) :\n  i = j :=\nlimit.hom_ext\nbegin\n  rintro (a|b),\n  { apply h },\n  simp_rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n    \u2190 category.assoc, h],\nend\n\nvariables [has_product I.left] [has_product I.right]\n\ninstance : has_equalizer I.fst_pi_map I.snd_pi_map :=\n\u27e8\u27e8\u27e8_,is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.functor (limit.is_limit _)\u27e9\u27e9\u27e9\n\n/-- The multiequalizer is isomorphic to the equalizer of `\u220f I.left \u21c9 \u220f I.right`. -/\ndef iso_equalizer : multiequalizer I \u2245 equalizer I.fst_pi_map I.snd_pi_map :=\nlimit.iso_limit_cone \u27e8_, is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.inverse (limit.is_limit _)\u27e9\n\n/-- The canonical injection `multiequalizer I \u27f6 \u220f I.left`. -/\ndef \u03b9_pi : multiequalizer I \u27f6 \u220f I.left :=\n  (iso_equalizer I).hom \u226b equalizer.\u03b9 I.fst_pi_map I.snd_pi_map\n\n@[simp, reassoc]\nlemma \u03b9_pi_\u03c0 (a) : \u03b9_pi I \u226b pi.\u03c0 I.left a = \u03b9 I a :=\nby { rw [\u03b9_pi, category.assoc, \u2190 iso.eq_inv_comp, iso_equalizer], simpa }\n\ninstance : mono (\u03b9_pi I) := @@mono_comp _ _ _ _ equalizer.\u03b9_mono\n\nend multiequalizer\n\nnamespace multicoequalizer\n\nvariables (I : multispan_index C) [has_multicoequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03c0 (b : I.R) : I.right b \u27f6 multicoequalizer I :=\ncolimit.\u03b9 I.multispan (walking_multispan.right _)\n\n/-- The multicofork associated to the multicoequalizer. -/\nabbreviation multicofork : multicofork I :=\ncolimit.cocone _\n\n@[simp]\nlemma multicofork_\u03c0 (b) :\n  (multicoequalizer.multicofork I).\u03c0 b = multicoequalizer.\u03c0 I b := rfl\n\n@[simp]\nlemma multicofork_\u03b9_app_right (b) :\n  (multicoequalizer.multicofork I).\u03b9.app (walking_multispan.right b) =\n  multicoequalizer.\u03c0 I b := rfl\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b multicoequalizer.\u03c0 I (I.fst_from a) =\n  I.snd a \u226b multicoequalizer.\u03c0 I (I.snd_from a) :=\nmulticofork.condition _ _\n\n/-- Construct a morphism from the multicoequalizer from its universal property. -/\nabbreviation desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) :\n  multicoequalizer I \u27f6 W :=\ncolimit.desc _ (multicofork.of_\u03c0 I _ k h)\n\n@[simp, reassoc]\nlemma \u03c0_desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) (b) :\n  multicoequalizer.\u03c0 I b \u226b multicoequalizer.desc I _ k h = k _ :=\ncolimit.\u03b9_desc _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : multicoequalizer I \u27f6 W)\n  (h : \u2200 b, multicoequalizer.\u03c0 I b \u226b i = multicoequalizer.\u03c0 I b \u226b j) :\n  i = j :=\ncolimit.hom_ext\nbegin\n  rintro (a|b),\n  { simp_rw [\u2190 colimit.w I.multispan (walking_multispan.hom.fst a),\n    category.assoc, h] },\n  { apply h },\nend\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\ninstance : has_coequalizer I.fst_sigma_map I.snd_sigma_map :=\n\u27e8\u27e8\u27e8_,is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.functor (colimit.is_colimit _)\u27e9\u27e9\u27e9\n\n/-- The multicoequalizer is isomorphic to the coequalizer of `\u2210 I.left \u21c9 \u2210 I.right`. -/\ndef iso_coequalizer : multicoequalizer I \u2245 coequalizer I.fst_sigma_map I.snd_sigma_map :=\ncolimit.iso_colimit_cocone \u27e8_, is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.inverse (colimit.is_colimit _)\u27e9\n\n/-- The canonical projection `\u2210 I.right \u27f6 multicoequalizer I`. -/\ndef sigma_\u03c0 : \u2210 I.right \u27f6 multicoequalizer I :=\n  coequalizer.\u03c0 I.fst_sigma_map I.snd_sigma_map \u226b (iso_coequalizer I).inv\n\n@[simp, reassoc]\nlemma \u03b9_sigma_\u03c0 (b) : sigma.\u03b9 I.right b \u226b sigma_\u03c0 I = \u03c0 I b :=\nby { rw [sigma_\u03c0, \u2190 category.assoc, iso.comp_inv_eq, iso_coequalizer], simpa }\n\ninstance : epi (sigma_\u03c0 I) := @@epi_comp _ _ coequalizer.\u03c0_epi _ _\n\nend multicoequalizer\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/shapes/multiequalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3645420363652715}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,\n  Michael Howes\n\n! This file was ported from Lean 3 source module deprecated.subgroup\n! leanprover-community/mathlib commit fac369018417f980cec5fcdafc766a69f88d8cfe\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.GroupTheory.Subgroup.Basic\nimport Mathbin.Deprecated.Submonoid\n\n/-!\n# Unbundled subgroups (deprecated)\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file is deprecated, and is no longer imported by anything in mathlib other than other\ndeprecated files, and test files. You should not need to import it.\n\nThis file defines unbundled multiplicative and additive subgroups. Instead of using this file,\nplease use `subgroup G` and `add_subgroup A`, defined in `group_theory.subgroup.basic`.\n\n## Main definitions\n\n`is_add_subgroup (S : set A)` : the predicate that `S` is the underlying subset of an additive\nsubgroup of `A`. The bundled variant `add_subgroup A` should be used in preference to this.\n\n`is_subgroup (S : set G)` : the predicate that `S` is the underlying subset of a subgroup\nof `G`. The bundled variant `subgroup G` should be used in preference to this.\n\n## Tags\n\nsubgroup, subgroups, is_subgroup\n-/\n\n\nopen Set Function\n\nvariable {G : Type _} {H : Type _} {A : Type _} {a a\u2081 a\u2082 b c : G}\n\nsection Group\n\nvariable [Group G] [AddGroup A]\n\n#print IsAddSubgroup /-\n/-- `s` is an additive subgroup: a set containing 0 and closed under addition and negation. -/\nstructure IsAddSubgroup (s : Set A) extends IsAddSubmonoid s : Prop where\n  neg_mem {a} : a \u2208 s \u2192 -a \u2208 s\n#align is_add_subgroup IsAddSubgroup\n-/\n\n#print IsSubgroup /-\n/-- `s` is a subgroup: a set containing 1 and closed under multiplication and inverse. -/\n@[to_additive]\nstructure IsSubgroup (s : Set G) extends IsSubmonoid s : Prop where\n  inv_mem {a} : a \u2208 s \u2192 a\u207b\u00b9 \u2208 s\n#align is_subgroup IsSubgroup\n#align is_add_subgroup IsAddSubgroup\n-/\n\n/- warning: is_subgroup.div_mem -> IsSubgroup.div_mem is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (forall {x : G} {y : G}, (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) x s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) y s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HDiv.hDiv.{u1, u1, u1} G G G (instHDiv.{u1} G (DivInvMonoid.toHasDiv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) x y) s))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (forall {x : G} {y : G}, (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) x s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) y s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HDiv.hDiv.{u1, u1, u1} G G G (instHDiv.{u1} G (DivInvMonoid.toDiv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) x y) s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.div_mem IsSubgroup.div_mem\u2093'. -/\n@[to_additive]\ntheorem IsSubgroup.div_mem {s : Set G} (hs : IsSubgroup s) {x y : G} (hx : x \u2208 s) (hy : y \u2208 s) :\n    x / y \u2208 s := by simpa only [div_eq_mul_inv] using hs.mul_mem hx (hs.inv_mem hy)\n#align is_subgroup.div_mem IsSubgroup.div_mem\n#align is_add_subgroup.sub_mem IsAddSubgroup.sub_mem\n\n#print Additive.isAddSubgroup /-\ntheorem Additive.isAddSubgroup {s : Set G} (hs : IsSubgroup s) : @IsAddSubgroup (Additive G) _ s :=\n  @IsAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubmonoid hs.to_isSubmonoid) fun _ => hs.inv_mem\n#align additive.is_add_subgroup Additive.isAddSubgroup\n-/\n\n#print Additive.isAddSubgroup_iff /-\ntheorem Additive.isAddSubgroup_iff {s : Set G} : @IsAddSubgroup (Additive G) _ s \u2194 IsSubgroup s :=\n  \u27e8by rintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9 <;> exact @IsSubgroup.mk G _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083, fun h =>\n    Additive.isAddSubgroup h\u27e9\n#align additive.is_add_subgroup_iff Additive.isAddSubgroup_iff\n-/\n\n#print Multiplicative.isSubgroup /-\ntheorem Multiplicative.isSubgroup {s : Set A} (hs : IsAddSubgroup s) :\n    @IsSubgroup (Multiplicative A) _ s :=\n  @IsSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubmonoid hs.to_isAddSubmonoid) fun _ =>\n    hs.neg_mem\n#align multiplicative.is_subgroup Multiplicative.isSubgroup\n-/\n\n#print Multiplicative.isSubgroup_iff /-\ntheorem Multiplicative.isSubgroup_iff {s : Set A} :\n    @IsSubgroup (Multiplicative A) _ s \u2194 IsAddSubgroup s :=\n  \u27e8by rintro \u27e8\u27e8h\u2081, h\u2082\u27e9, h\u2083\u27e9 <;> exact @IsAddSubgroup.mk A _ _ \u27e8h\u2081, @h\u2082\u27e9 @h\u2083, fun h =>\n    Multiplicative.isSubgroup h\u27e9\n#align multiplicative.is_subgroup_iff Multiplicative.isSubgroup_iff\n-/\n\n/- warning: is_subgroup.of_div -> IsSubgroup.of_div is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] (s : Set.{u1} G), (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (OfNat.ofNat.{u1} G 1 (OfNat.mk.{u1} G 1 (One.one.{u1} G (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))))) s) -> (forall {a : G} {b : G}, (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) b s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) b)) s)) -> (IsSubgroup.{u1} G _inst_1 s)\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] (s : Set.{u1} G), (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (OfNat.ofNat.{u1} G 1 (One.toOfNat1.{u1} G (InvOneClass.toOne.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))))) s) -> (forall {a : G} {b : G}, (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) b s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))) b)) s)) -> (IsSubgroup.{u1} G _inst_1 s)\nCase conversion may be inaccurate. Consider using '#align is_subgroup.of_div IsSubgroup.of_div\u2093'. -/\n@[to_additive ofAdd_neg]\ntheorem IsSubgroup.of_div (s : Set G) (one_mem : (1 : G) \u2208 s)\n    (div_mem : \u2200 {a b : G}, a \u2208 s \u2192 b \u2208 s \u2192 a * b\u207b\u00b9 \u2208 s) : IsSubgroup s :=\n  have inv_mem : \u2200 a, a \u2208 s \u2192 a\u207b\u00b9 \u2208 s := fun a ha =>\n    by\n    have : 1 * a\u207b\u00b9 \u2208 s := div_mem one_mem ha\n    simpa\n  { inv_mem\n    mul_mem := fun a b ha hb =>\n      by\n      have : a * b\u207b\u00b9\u207b\u00b9 \u2208 s := div_mem ha (inv_mem b hb)\n      simpa\n    one_mem }\n#align is_subgroup.of_div IsSubgroup.of_div\n#align is_add_subgroup.of_add_neg IsAddSubgroup.of_add_neg\n\n/- warning: is_add_subgroup.of_sub -> IsAddSubgroup.of_sub is a dubious translation:\nlean 3 declaration is\n  forall {A : Type.{u1}} [_inst_2 : AddGroup.{u1} A] (s : Set.{u1} A), (Membership.Mem.{u1, u1} A (Set.{u1} A) (Set.hasMem.{u1} A) (OfNat.ofNat.{u1} A 0 (OfNat.mk.{u1} A 0 (Zero.zero.{u1} A (AddZeroClass.toHasZero.{u1} A (AddMonoid.toAddZeroClass.{u1} A (SubNegMonoid.toAddMonoid.{u1} A (AddGroup.toSubNegMonoid.{u1} A _inst_2))))))) s) -> (forall {a : A} {b : A}, (Membership.Mem.{u1, u1} A (Set.{u1} A) (Set.hasMem.{u1} A) a s) -> (Membership.Mem.{u1, u1} A (Set.{u1} A) (Set.hasMem.{u1} A) b s) -> (Membership.Mem.{u1, u1} A (Set.{u1} A) (Set.hasMem.{u1} A) (HSub.hSub.{u1, u1, u1} A A A (instHSub.{u1} A (SubNegMonoid.toHasSub.{u1} A (AddGroup.toSubNegMonoid.{u1} A _inst_2))) a b) s)) -> (IsAddSubgroup.{u1} A _inst_2 s)\nbut is expected to have type\n  forall {A : Type.{u1}} [_inst_2 : AddGroup.{u1} A] (s : Set.{u1} A), (Membership.mem.{u1, u1} A (Set.{u1} A) (Set.instMembershipSet.{u1} A) (OfNat.ofNat.{u1} A 0 (Zero.toOfNat0.{u1} A (NegZeroClass.toZero.{u1} A (SubNegZeroMonoid.toNegZeroClass.{u1} A (SubtractionMonoid.toSubNegZeroMonoid.{u1} A (AddGroup.toSubtractionMonoid.{u1} A _inst_2)))))) s) -> (forall {a : A} {b : A}, (Membership.mem.{u1, u1} A (Set.{u1} A) (Set.instMembershipSet.{u1} A) a s) -> (Membership.mem.{u1, u1} A (Set.{u1} A) (Set.instMembershipSet.{u1} A) b s) -> (Membership.mem.{u1, u1} A (Set.{u1} A) (Set.instMembershipSet.{u1} A) (HSub.hSub.{u1, u1, u1} A A A (instHSub.{u1} A (SubNegMonoid.toSub.{u1} A (AddGroup.toSubNegMonoid.{u1} A _inst_2))) a b) s)) -> (IsAddSubgroup.{u1} A _inst_2 s)\nCase conversion may be inaccurate. Consider using '#align is_add_subgroup.of_sub IsAddSubgroup.of_sub\u2093'. -/\ntheorem IsAddSubgroup.of_sub (s : Set A) (zero_mem : (0 : A) \u2208 s)\n    (sub_mem : \u2200 {a b : A}, a \u2208 s \u2192 b \u2208 s \u2192 a - b \u2208 s) : IsAddSubgroup s :=\n  IsAddSubgroup.of_add_neg s zero_mem fun x y hx hy => by\n    simpa only [sub_eq_add_neg] using sub_mem hx hy\n#align is_add_subgroup.of_sub IsAddSubgroup.of_sub\n\n/- warning: is_subgroup.inter -> IsSubgroup.inter is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s\u2081 : Set.{u1} G} {s\u2082 : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s\u2081) -> (IsSubgroup.{u1} G _inst_1 s\u2082) -> (IsSubgroup.{u1} G _inst_1 (Inter.inter.{u1} (Set.{u1} G) (Set.hasInter.{u1} G) s\u2081 s\u2082))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s\u2081 : Set.{u1} G} {s\u2082 : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s\u2081) -> (IsSubgroup.{u1} G _inst_1 s\u2082) -> (IsSubgroup.{u1} G _inst_1 (Inter.inter.{u1} (Set.{u1} G) (Set.instInterSet.{u1} G) s\u2081 s\u2082))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.inter IsSubgroup.inter\u2093'. -/\n@[to_additive]\ntheorem IsSubgroup.inter {s\u2081 s\u2082 : Set G} (hs\u2081 : IsSubgroup s\u2081) (hs\u2082 : IsSubgroup s\u2082) :\n    IsSubgroup (s\u2081 \u2229 s\u2082) :=\n  { IsSubmonoid.inter hs\u2081.to_isSubmonoid hs\u2082.to_isSubmonoid with\n    inv_mem := fun x hx => \u27e8hs\u2081.inv_mem hx.1, hs\u2082.inv_mem hx.2\u27e9 }\n#align is_subgroup.inter IsSubgroup.inter\n#align is_add_subgroup.inter IsAddSubgroup.inter\n\n#print IsSubgroup.inter\u1d62 /-\n@[to_additive]\ntheorem IsSubgroup.inter\u1d62 {\u03b9 : Sort _} {s : \u03b9 \u2192 Set G} (hs : \u2200 y : \u03b9, IsSubgroup (s y)) :\n    IsSubgroup (Set.inter\u1d62 s) :=\n  { IsSubmonoid.inter\u1d62 fun y => (hs y).to_isSubmonoid with\n    inv_mem := fun x h =>\n      Set.mem_inter\u1d62.2 fun y => IsSubgroup.inv_mem (hs _) (Set.mem_inter\u1d62.1 h y) }\n#align is_subgroup.Inter IsSubgroup.inter\u1d62\n#align is_add_subgroup.Inter IsAddSubgroup.inter\u1d62\n-/\n\n#print isSubgroup_union\u1d62_of_directed /-\n@[to_additive]\ntheorem isSubgroup_union\u1d62_of_directed {\u03b9 : Type _} [h\u03b9 : Nonempty \u03b9] {s : \u03b9 \u2192 Set G}\n    (hs : \u2200 i, IsSubgroup (s i)) (directed : \u2200 i j, \u2203 k, s i \u2286 s k \u2227 s j \u2286 s k) :\n    IsSubgroup (\u22c3 i, s i) :=\n  { inv_mem := fun a ha =>\n      let \u27e8i, hi\u27e9 := Set.mem_union\u1d62.1 ha\n      Set.mem_union\u1d62.2 \u27e8i, (hs i).inv_mem hi\u27e9\n    to_isSubmonoid := isSubmonoid_union\u1d62_of_directed (fun i => (hs i).to_isSubmonoid) Directed }\n#align is_subgroup_Union_of_directed isSubgroup_union\u1d62_of_directed\n#align is_add_subgroup_Union_of_directed isAddSubgroup_union\u1d62_of_directed\n-/\n\nend Group\n\nnamespace IsSubgroup\n\nopen IsSubmonoid\n\nvariable [Group G] {s : Set G} (hs : IsSubgroup s)\n\ninclude hs\n\n/- warning: is_subgroup.inv_mem_iff -> IsSubgroup.inv_mem_iff is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {a : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) a) s) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a s))\nbut is expected to have type\n  forall {G : Type.{u1}} {a : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))) a) s) (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.inv_mem_iff IsSubgroup.inv_mem_iff\u2093'. -/\n@[to_additive]\ntheorem inv_mem_iff : a\u207b\u00b9 \u2208 s \u2194 a \u2208 s :=\n  \u27e8fun h => by simpa using hs.inv_mem h, inv_mem hs\u27e9\n#align is_subgroup.inv_mem_iff IsSubgroup.inv_mem_iff\n#align is_add_subgroup.neg_mem_iff IsAddSubgroup.neg_mem_iff\n\n/- warning: is_subgroup.mul_mem_cancel_right -> IsSubgroup.mul_mem_cancel_right is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {a : G} {b : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a s) -> (Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) b s))\nbut is expected to have type\n  forall {G : Type.{u1}} {a : G} {b : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a s) -> (Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s) (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) b s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mul_mem_cancel_right IsSubgroup.mul_mem_cancel_right\u2093'. -/\n@[to_additive]\ntheorem mul_mem_cancel_right (h : a \u2208 s) : b * a \u2208 s \u2194 b \u2208 s :=\n  \u27e8fun hba => by simpa using hs.mul_mem hba (hs.inv_mem h), fun hb => hs.mul_mem hb h\u27e9\n#align is_subgroup.mul_mem_cancel_right IsSubgroup.mul_mem_cancel_right\n#align is_add_subgroup.add_mem_cancel_right IsAddSubgroup.add_mem_cancel_right\n\n/- warning: is_subgroup.mul_mem_cancel_left -> IsSubgroup.mul_mem_cancel_left is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {a : G} {b : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a s) -> (Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) b s))\nbut is expected to have type\n  forall {G : Type.{u1}} {a : G} {b : G} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a s) -> (Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) b s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mul_mem_cancel_left IsSubgroup.mul_mem_cancel_left\u2093'. -/\n@[to_additive]\ntheorem mul_mem_cancel_left (h : a \u2208 s) : a * b \u2208 s \u2194 b \u2208 s :=\n  \u27e8fun hab => by simpa using hs.mul_mem (hs.inv_mem h) hab, hs.mul_mem h\u27e9\n#align is_subgroup.mul_mem_cancel_left IsSubgroup.mul_mem_cancel_left\n#align is_add_subgroup.add_mem_cancel_left IsAddSubgroup.add_mem_cancel_left\n\nend IsSubgroup\n\n#print IsNormalAddSubgroup /-\n/-- `is_normal_add_subgroup (s : set A)` expresses the fact that `s` is a normal additive subgroup\nof the additive group `A`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : add_subgroup A` and `hs : S.normal`, and not via this structure. -/\nstructure IsNormalAddSubgroup [AddGroup A] (s : Set A) extends IsAddSubgroup s : Prop where\n  Normal : \u2200 n \u2208 s, \u2200 g : A, g + n + -g \u2208 s\n#align is_normal_add_subgroup IsNormalAddSubgroup\n-/\n\n#print IsNormalSubgroup /-\n/-- `is_normal_subgroup (s : set G)` expresses the fact that `s` is a normal subgroup\nof the group `G`. Important: the preferred way to say this in Lean is via bundled\nsubgroups `S : subgroup G` and not via this structure. -/\n@[to_additive]\nstructure IsNormalSubgroup [Group G] (s : Set G) extends IsSubgroup s : Prop where\n  Normal : \u2200 n \u2208 s, \u2200 g : G, g * n * g\u207b\u00b9 \u2208 s\n#align is_normal_subgroup IsNormalSubgroup\n#align is_normal_add_subgroup IsNormalAddSubgroup\n-/\n\n#print isNormalSubgroup_of_commGroup /-\n@[to_additive]\ntheorem isNormalSubgroup_of_commGroup [CommGroup G] {s : Set G} (hs : IsSubgroup s) :\n    IsNormalSubgroup s :=\n  { hs with Normal := fun n hn g => by rwa [mul_right_comm, mul_right_inv, one_mul] }\n#align is_normal_subgroup_of_comm_group isNormalSubgroup_of_commGroup\n#align is_normal_add_subgroup_of_add_comm_group isNormalAddSubgroup_of_addCommGroup\n-/\n\n#print Additive.isNormalAddSubgroup /-\ntheorem Additive.isNormalAddSubgroup [Group G] {s : Set G} (hs : IsNormalSubgroup s) :\n    @IsNormalAddSubgroup (Additive G) _ s :=\n  @IsNormalAddSubgroup.mk (Additive G) _ _ (Additive.isAddSubgroup hs.to_isSubgroup)\n    (IsNormalSubgroup.normal hs)\n#align additive.is_normal_add_subgroup Additive.isNormalAddSubgroup\n-/\n\n#print Additive.isNormalAddSubgroup_iff /-\ntheorem Additive.isNormalAddSubgroup_iff [Group G] {s : Set G} :\n    @IsNormalAddSubgroup (Additive G) _ s \u2194 IsNormalSubgroup s :=\n  \u27e8by rintro \u27e8h\u2081, h\u2082\u27e9 <;> exact @IsNormalSubgroup.mk G _ _ (Additive.isAddSubgroup_iff.1 h\u2081) @h\u2082,\n    fun h => Additive.isNormalAddSubgroup h\u27e9\n#align additive.is_normal_add_subgroup_iff Additive.isNormalAddSubgroup_iff\n-/\n\n#print Multiplicative.isNormalSubgroup /-\ntheorem Multiplicative.isNormalSubgroup [AddGroup A] {s : Set A} (hs : IsNormalAddSubgroup s) :\n    @IsNormalSubgroup (Multiplicative A) _ s :=\n  @IsNormalSubgroup.mk (Multiplicative A) _ _ (Multiplicative.isSubgroup hs.to_isAddSubgroup)\n    (IsNormalAddSubgroup.normal hs)\n#align multiplicative.is_normal_subgroup Multiplicative.isNormalSubgroup\n-/\n\n#print Multiplicative.isNormalSubgroup_iff /-\ntheorem Multiplicative.isNormalSubgroup_iff [AddGroup A] {s : Set A} :\n    @IsNormalSubgroup (Multiplicative A) _ s \u2194 IsNormalAddSubgroup s :=\n  \u27e8by\n    rintro \u27e8h\u2081, h\u2082\u27e9 <;>\n      exact @IsNormalAddSubgroup.mk A _ _ (Multiplicative.isSubgroup_iff.1 h\u2081) @h\u2082,\n    fun h => Multiplicative.isNormalSubgroup h\u27e9\n#align multiplicative.is_normal_subgroup_iff Multiplicative.isNormalSubgroup_iff\n-/\n\nnamespace IsSubgroup\n\nvariable [Group G]\n\n/- warning: is_subgroup.mem_norm_comm -> IsSubgroup.mem_norm_comm is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsNormalSubgroup.{u1} G _inst_1 s) -> (forall {a : G} {b : G}, (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) -> (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsNormalSubgroup.{u1} G _inst_1 s) -> (forall {a : G} {b : G}, (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) -> (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mem_norm_comm IsSubgroup.mem_norm_comm\u2093'. -/\n-- Normal subgroup properties\n@[to_additive]\ntheorem mem_norm_comm {s : Set G} (hs : IsNormalSubgroup s) {a b : G} (hab : a * b \u2208 s) :\n    b * a \u2208 s := by\n  have h : a\u207b\u00b9 * (a * b) * a\u207b\u00b9\u207b\u00b9 \u2208 s := hs.Normal (a * b) hab a\u207b\u00b9\n  simp at h <;> exact h\n#align is_subgroup.mem_norm_comm IsSubgroup.mem_norm_comm\n#align is_add_subgroup.mem_norm_comm IsAddSubgroup.mem_norm_comm\n\n/- warning: is_subgroup.mem_norm_comm_iff -> IsSubgroup.mem_norm_comm_iff is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsNormalSubgroup.{u1} G _inst_1 s) -> (forall {a : G} {b : G}, Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsNormalSubgroup.{u1} G _inst_1 s) -> (forall {a : G} {b : G}, Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a b) s) (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) b a) s))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mem_norm_comm_iff IsSubgroup.mem_norm_comm_iff\u2093'. -/\n@[to_additive]\ntheorem mem_norm_comm_iff {s : Set G} (hs : IsNormalSubgroup s) {a b : G} : a * b \u2208 s \u2194 b * a \u2208 s :=\n  \u27e8mem_norm_comm hs, mem_norm_comm hs\u27e9\n#align is_subgroup.mem_norm_comm_iff IsSubgroup.mem_norm_comm_iff\n#align is_add_subgroup.mem_norm_comm_iff IsAddSubgroup.mem_norm_comm_iff\n\n#print IsSubgroup.trivial /-\n/-- The trivial subgroup -/\n@[to_additive \"the trivial additive subgroup\"]\ndef trivial (G : Type _) [Group G] : Set G :=\n  {1}\n#align is_subgroup.trivial IsSubgroup.trivial\n#align is_add_subgroup.trivial IsAddSubgroup.trivial\n-/\n\n/- warning: is_subgroup.mem_trivial -> IsSubgroup.mem_trivial is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {g : G}, Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) g (IsSubgroup.trivial.{u1} G _inst_1)) (Eq.{succ u1} G g (OfNat.ofNat.{u1} G 1 (OfNat.mk.{u1} G 1 (One.one.{u1} G (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))))))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {g : G}, Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) g (IsSubgroup.trivial.{u1} G _inst_1)) (Eq.{succ u1} G g (OfNat.ofNat.{u1} G 1 (One.toOfNat1.{u1} G (InvOneClass.toOne.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))))))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mem_trivial IsSubgroup.mem_trivial\u2093'. -/\n@[simp, to_additive]\ntheorem mem_trivial {g : G} : g \u2208 trivial G \u2194 g = 1 :=\n  mem_singleton_iff\n#align is_subgroup.mem_trivial IsSubgroup.mem_trivial\n#align is_add_subgroup.mem_trivial IsAddSubgroup.mem_trivial\n\n#print IsSubgroup.trivial_normal /-\n@[to_additive]\ntheorem trivial_normal : IsNormalSubgroup (trivial G) := by\n  refine' { .. } <;> simp (config := { contextual := true }) [trivial]\n#align is_subgroup.trivial_normal IsSubgroup.trivial_normal\n#align is_add_subgroup.trivial_normal IsAddSubgroup.trivial_normal\n-/\n\n/- warning: is_subgroup.eq_trivial_iff -> IsSubgroup.eq_trivial_iff is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Iff (Eq.{succ u1} (Set.{u1} G) s (IsSubgroup.trivial.{u1} G _inst_1)) (forall (x : G), (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) x s) -> (Eq.{succ u1} G x (OfNat.ofNat.{u1} G 1 (OfNat.mk.{u1} G 1 (One.one.{u1} G (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))))))))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (Iff (Eq.{succ u1} (Set.{u1} G) s (IsSubgroup.trivial.{u1} G _inst_1)) (forall (x : G), (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) x s) -> (Eq.{succ u1} G x (OfNat.ofNat.{u1} G 1 (One.toOfNat1.{u1} G (InvOneClass.toOne.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))))))))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.eq_trivial_iff IsSubgroup.eq_trivial_iff\u2093'. -/\n@[to_additive]\ntheorem eq_trivial_iff {s : Set G} (hs : IsSubgroup s) : s = trivial G \u2194 \u2200 x \u2208 s, x = (1 : G) := by\n  simp only [Set.ext_iff, IsSubgroup.mem_trivial] <;>\n    exact \u27e8fun h x => (h x).1, fun h x => \u27e8h x, fun hx => hx.symm \u25b8 hs.to_is_submonoid.one_mem\u27e9\u27e9\n#align is_subgroup.eq_trivial_iff IsSubgroup.eq_trivial_iff\n#align is_add_subgroup.eq_trivial_iff IsAddSubgroup.eq_trivial_iff\n\n#print IsSubgroup.univ_subgroup /-\n@[to_additive]\ntheorem univ_subgroup : IsNormalSubgroup (@univ G) := by refine' { .. } <;> simp\n#align is_subgroup.univ_subgroup IsSubgroup.univ_subgroup\n#align is_add_subgroup.univ_add_subgroup IsAddSubgroup.univ_addSubgroup\n-/\n\n#print IsSubgroup.center /-\n/-- The underlying set of the center of a group. -/\n@[to_additive add_center \"The underlying set of the center of an additive group.\"]\ndef center (G : Type _) [Group G] : Set G :=\n  { z | \u2200 g, g * z = z * g }\n#align is_subgroup.center IsSubgroup.center\n#align is_add_subgroup.add_center IsAddSubgroup.addCenter\n-/\n\n/- warning: is_subgroup.mem_center -> IsSubgroup.mem_center is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {a : G}, Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a (IsSubgroup.center.{u1} G _inst_1)) (forall (g : G), Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) g a) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a g))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {a : G}, Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a (IsSubgroup.center.{u1} G _inst_1)) (forall (g : G), Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) g a) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a g))\nCase conversion may be inaccurate. Consider using '#align is_subgroup.mem_center IsSubgroup.mem_center\u2093'. -/\n@[to_additive mem_add_center]\ntheorem mem_center {a : G} : a \u2208 center G \u2194 \u2200 g, g * a = a * g :=\n  Iff.rfl\n#align is_subgroup.mem_center IsSubgroup.mem_center\n#align is_add_subgroup.mem_add_center IsAddSubgroup.mem_add_center\n\n#print IsSubgroup.center_normal /-\n@[to_additive add_center_normal]\ntheorem center_normal : IsNormalSubgroup (center G) :=\n  { one_mem := by simp [center]\n    mul_mem := fun a b ha hb g => by\n      rw [\u2190 mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, \u2190 mul_assoc]\n    inv_mem := fun a ha g =>\n      calc\n        g * a\u207b\u00b9 = a\u207b\u00b9 * (g * a) * a\u207b\u00b9 := by simp [ha g]\n        _ = a\u207b\u00b9 * g := by rw [\u2190 mul_assoc, mul_assoc] <;> simp\n        \n    Normal := fun n ha g h =>\n      calc\n        h * (g * n * g\u207b\u00b9) = h * n := by simp [ha g, mul_assoc]\n        _ = g * g\u207b\u00b9 * n * h := by rw [ha h] <;> simp\n        _ = g * n * g\u207b\u00b9 * h := by rw [mul_assoc g, ha g\u207b\u00b9, \u2190 mul_assoc]\n         }\n#align is_subgroup.center_normal IsSubgroup.center_normal\n#align is_add_subgroup.add_center_normal IsAddSubgroup.add_center_normal\n-/\n\n#print IsSubgroup.normalizer /-\n/-- The underlying set of the normalizer of a subset `S : set G` of a group `G`. That is,\n  the elements `g : G` such that `g * S * g\u207b\u00b9 = S`. -/\n@[to_additive add_normalizer\n      \"The underlying set of the normalizer of a subset `S : set A` of an\\n  additive group `A`. That is, the elements `a : A` such that `a + S - a = S`.\"]\ndef normalizer (s : Set G) : Set G :=\n  { g : G | \u2200 n, n \u2208 s \u2194 g * n * g\u207b\u00b9 \u2208 s }\n#align is_subgroup.normalizer IsSubgroup.normalizer\n#align is_add_subgroup.add_normalizer IsAddSubgroup.addNormalizer\n-/\n\n#print IsSubgroup.normalizer_isSubgroup /-\n@[to_additive]\ntheorem normalizer_isSubgroup (s : Set G) : IsSubgroup (normalizer s) :=\n  { one_mem := by simp [normalizer]\n    mul_mem := fun a b (ha : \u2200 n, n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s) (hb : \u2200 n, n \u2208 s \u2194 b * n * b\u207b\u00b9 \u2208 s) n =>\n      by rw [mul_inv_rev, \u2190 mul_assoc, mul_assoc a, mul_assoc a, \u2190 ha, \u2190 hb]\n    inv_mem := fun a (ha : \u2200 n, n \u2208 s \u2194 a * n * a\u207b\u00b9 \u2208 s) n => by\n      rw [ha (a\u207b\u00b9 * n * a\u207b\u00b9\u207b\u00b9)] <;> simp [mul_assoc] }\n#align is_subgroup.normalizer_is_subgroup IsSubgroup.normalizer_isSubgroup\n#align is_add_subgroup.normalizer_is_add_subgroup IsAddSubgroup.normalizer_isAddSubgroup\n-/\n\n#print IsSubgroup.subset_normalizer /-\n@[to_additive subset_add_normalizer]\ntheorem subset_normalizer {s : Set G} (hs : IsSubgroup s) : s \u2286 normalizer s := fun g hg n => by\n  rw [IsSubgroup.mul_mem_cancel_right hs ((IsSubgroup.inv_mem_iff hs).2 hg),\n    IsSubgroup.mul_mem_cancel_left hs hg]\n#align is_subgroup.subset_normalizer IsSubgroup.subset_normalizer\n#align is_add_subgroup.subset_add_normalizer IsAddSubgroup.subset_add_normalizer\n-/\n\nend IsSubgroup\n\n-- Homomorphism subgroups\nnamespace IsGroupHom\n\nopen IsSubmonoid IsSubgroup\n\n#print IsGroupHom.ker /-\n/-- `ker f : set G` is the underlying subset of the kernel of a map `G \u2192 H`. -/\n@[to_additive \"`ker f : set A` is the underlying subset of the kernel of a map `A \u2192 B`\"]\ndef ker [Group H] (f : G \u2192 H) : Set G :=\n  preimage f (trivial H)\n#align is_group_hom.ker IsGroupHom.ker\n#align is_add_group_hom.ker IsAddGroupHom.ker\n-/\n\n/- warning: is_group_hom.mem_ker -> IsGroupHom.mem_ker is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u2} H] (f : G -> H) {x : G}, Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) x (IsGroupHom.ker.{u1, u2} G H _inst_1 f)) (Eq.{succ u2} H (f x) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_1))))))))\nbut is expected to have type\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u2} H] (f : G -> H) {x : G}, Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) x (IsGroupHom.ker.{u1, u2} G H _inst_1 f)) (Eq.{succ u2} H (f x) (OfNat.ofNat.{u2} H 1 (One.toOfNat1.{u2} H (InvOneClass.toOne.{u2} H (DivInvOneMonoid.toInvOneClass.{u2} H (DivisionMonoid.toDivInvOneMonoid.{u2} H (Group.toDivisionMonoid.{u2} H _inst_1)))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.mem_ker IsGroupHom.mem_ker\u2093'. -/\n@[to_additive]\ntheorem mem_ker [Group H] (f : G \u2192 H) {x : G} : x \u2208 ker f \u2194 f x = 1 :=\n  mem_trivial\n#align is_group_hom.mem_ker IsGroupHom.mem_ker\n#align is_add_group_hom.mem_ker IsAddGroupHom.mem_ker\n\nvariable [Group G] [Group H]\n\n/- warning: is_group_hom.one_ker_inv -> IsGroupHom.one_ker_inv is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) b))) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))) -> (Eq.{succ u2} H (f a) (f b)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) a (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) b))) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))) -> (Eq.{succ u1} H (f a) (f b)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.one_ker_inv IsGroupHom.one_ker_inv\u2093'. -/\n@[to_additive]\ntheorem one_ker_inv {f : G \u2192 H} (hf : IsGroupHom f) {a b : G} (h : f (a * b\u207b\u00b9) = 1) : f a = f b :=\n  by\n  rw [hf.map_mul, hf.map_inv] at h\n  rw [\u2190 inv_inv (f b), eq_inv_of_mul_eq_one_left h]\n#align is_group_hom.one_ker_inv IsGroupHom.one_ker_inv\n#align is_add_group_hom.zero_ker_neg IsAddGroupHom.zero_ker_neg\n\n/- warning: is_group_hom.one_ker_inv' -> IsGroupHom.one_ker_inv' is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) a) b)) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))) -> (Eq.{succ u2} H (f a) (f b)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) a) b)) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))) -> (Eq.{succ u1} H (f a) (f b)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.one_ker_inv' IsGroupHom.one_ker_inv'\u2093'. -/\n@[to_additive]\ntheorem one_ker_inv' {f : G \u2192 H} (hf : IsGroupHom f) {a b : G} (h : f (a\u207b\u00b9 * b) = 1) : f a = f b :=\n  by\n  rw [hf.map_mul, hf.map_inv] at h\n  apply inv_injective\n  rw [eq_inv_of_mul_eq_one_left h]\n#align is_group_hom.one_ker_inv' IsGroupHom.one_ker_inv'\n#align is_add_group_hom.zero_ker_neg' IsAddGroupHom.zero_ker_neg'\n\n/- warning: is_group_hom.inv_ker_one -> IsGroupHom.inv_ker_one is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u2} H (f a) (f b)) -> (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) b))) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u1} H (f a) (f b)) -> (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) a (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) b))) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.inv_ker_one IsGroupHom.inv_ker_one\u2093'. -/\n@[to_additive]\ntheorem inv_ker_one {f : G \u2192 H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) : f (a * b\u207b\u00b9) = 1 :=\n  by\n  have : f a * (f b)\u207b\u00b9 = 1 := by rw [h, mul_right_inv]\n  rwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this\n#align is_group_hom.inv_ker_one IsGroupHom.inv_ker_one\n#align is_add_group_hom.neg_ker_zero IsAddGroupHom.neg_ker_zero\n\n/- warning: is_group_hom.inv_ker_one' -> IsGroupHom.inv_ker_one' is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u2} H (f a) (f b)) -> (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) a) b)) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {a : G} {b : G}, (Eq.{succ u1} H (f a) (f b)) -> (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) a) b)) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.inv_ker_one' IsGroupHom.inv_ker_one'\u2093'. -/\n@[to_additive]\ntheorem inv_ker_one' {f : G \u2192 H} (hf : IsGroupHom f) {a b : G} (h : f a = f b) : f (a\u207b\u00b9 * b) = 1 :=\n  by\n  have : (f a)\u207b\u00b9 * f b = 1 := by rw [h, mul_left_inv]\n  rwa [\u2190 hf.map_inv, \u2190 hf.map_mul] at this\n#align is_group_hom.inv_ker_one' IsGroupHom.inv_ker_one'\n#align is_add_group_hom.neg_ker_zero' IsAddGroupHom.neg_ker_zero'\n\n/- warning: is_group_hom.one_iff_ker_inv -> IsGroupHom.one_iff_ker_inv is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u2} H (f a) (f b)) (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) b))) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u1} H (f a) (f b)) (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) a (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) b))) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.one_iff_ker_inv IsGroupHom.one_iff_ker_inv\u2093'. -/\n@[to_additive]\ntheorem one_iff_ker_inv {f : G \u2192 H} (hf : IsGroupHom f) (a b : G) : f a = f b \u2194 f (a * b\u207b\u00b9) = 1 :=\n  \u27e8hf.inv_ker_one, hf.one_ker_inv\u27e9\n#align is_group_hom.one_iff_ker_inv IsGroupHom.one_iff_ker_inv\n#align is_add_group_hom.zero_iff_ker_neg IsAddGroupHom.zero_iff_ker_neg\n\n/- warning: is_group_hom.one_iff_ker_inv' -> IsGroupHom.one_iff_ker_inv' is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u2} H (f a) (f b)) (Eq.{succ u2} H (f (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) a) b)) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u1} H (f a) (f b)) (Eq.{succ u1} H (f (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) a) b)) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.one_iff_ker_inv' IsGroupHom.one_iff_ker_inv'\u2093'. -/\n@[to_additive]\ntheorem one_iff_ker_inv' {f : G \u2192 H} (hf : IsGroupHom f) (a b : G) : f a = f b \u2194 f (a\u207b\u00b9 * b) = 1 :=\n  \u27e8hf.inv_ker_one', hf.one_ker_inv'\u27e9\n#align is_group_hom.one_iff_ker_inv' IsGroupHom.one_iff_ker_inv'\n#align is_add_group_hom.zero_iff_ker_neg' IsAddGroupHom.zero_iff_ker_neg'\n\n/- warning: is_group_hom.inv_iff_ker -> IsGroupHom.inv_iff_ker is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u2} H (f a) (f b)) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) a (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) b)) (IsGroupHom.ker.{u1, u2} G H _inst_2 f)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u1} H (f a) (f b)) (Membership.mem.{u2, u2} G (Set.{u2} G) (Set.instMembershipSet.{u2} G) (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) a (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) b)) (IsGroupHom.ker.{u2, u1} G H _inst_2 f)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.inv_iff_ker IsGroupHom.inv_iff_ker\u2093'. -/\n@[to_additive]\ntheorem inv_iff_ker {f : G \u2192 H} (hf : IsGroupHom f) (a b : G) : f a = f b \u2194 a * b\u207b\u00b9 \u2208 ker f := by\n  rw [mem_ker] <;> exact one_iff_ker_inv hf _ _\n#align is_group_hom.inv_iff_ker IsGroupHom.inv_iff_ker\n#align is_add_group_hom.neg_iff_ker IsAddGroupHom.neg_iff_ker\n\n/- warning: is_group_hom.inv_iff_ker' -> IsGroupHom.inv_iff_ker' is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u2} H (f a) (f b)) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) a) b) (IsGroupHom.ker.{u1, u2} G H _inst_2 f)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall (a : G) (b : G), Iff (Eq.{succ u1} H (f a) (f b)) (Membership.mem.{u2, u2} G (Set.{u2} G) (Set.instMembershipSet.{u2} G) (HMul.hMul.{u2, u2, u2} G G G (instHMul.{u2} G (MulOneClass.toMul.{u2} G (Monoid.toMulOneClass.{u2} G (DivInvMonoid.toMonoid.{u2} G (Group.toDivInvMonoid.{u2} G _inst_1))))) (Inv.inv.{u2} G (InvOneClass.toInv.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))) a) b) (IsGroupHom.ker.{u2, u1} G H _inst_2 f)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.inv_iff_ker' IsGroupHom.inv_iff_ker'\u2093'. -/\n@[to_additive]\ntheorem inv_iff_ker' {f : G \u2192 H} (hf : IsGroupHom f) (a b : G) : f a = f b \u2194 a\u207b\u00b9 * b \u2208 ker f := by\n  rw [mem_ker] <;> exact one_iff_ker_inv' hf _ _\n#align is_group_hom.inv_iff_ker' IsGroupHom.inv_iff_ker'\n#align is_add_group_hom.neg_iff_ker' IsAddGroupHom.neg_iff_ker'\n\n/- warning: is_group_hom.image_subgroup -> IsGroupHom.image_subgroup is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u1} G}, (IsSubgroup.{u1} G _inst_1 s) -> (IsSubgroup.{u2} H _inst_2 (Set.image.{u1, u2} G H f s)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u2} G}, (IsSubgroup.{u2} G _inst_1 s) -> (IsSubgroup.{u1} H _inst_2 (Set.image.{u2, u1} G H f s)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.image_subgroup IsGroupHom.image_subgroup\u2093'. -/\n@[to_additive]\ntheorem image_subgroup {f : G \u2192 H} (hf : IsGroupHom f) {s : Set G} (hs : IsSubgroup s) :\n    IsSubgroup (f '' s) :=\n  { mul_mem := fun a\u2081 a\u2082 \u27e8b\u2081, hb\u2081, eq\u2081\u27e9 \u27e8b\u2082, hb\u2082, eq\u2082\u27e9 =>\n      \u27e8b\u2081 * b\u2082, hs.mul_mem hb\u2081 hb\u2082, by simp [eq\u2081, eq\u2082, hf.map_mul]\u27e9\n    one_mem := \u27e81, hs.to_isSubmonoid.one_mem, hf.map_one\u27e9\n    inv_mem := fun a \u27e8b, hb, Eq\u27e9 =>\n      \u27e8b\u207b\u00b9, hs.inv_mem hb, by\n        rw [hf.map_inv]\n        simp [*]\u27e9 }\n#align is_group_hom.image_subgroup IsGroupHom.image_subgroup\n#align is_add_group_hom.image_add_subgroup IsAddGroupHom.image_addSubgroup\n\n/- warning: is_group_hom.range_subgroup -> IsGroupHom.range_subgroup is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (IsSubgroup.{u2} H _inst_2 (Set.range.{u2, succ u1} H G f))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (IsSubgroup.{u1} H _inst_2 (Set.range.{u1, succ u2} H G f))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.range_subgroup IsGroupHom.range_subgroup\u2093'. -/\n@[to_additive]\ntheorem range_subgroup {f : G \u2192 H} (hf : IsGroupHom f) : IsSubgroup (Set.range f) :=\n  @Set.image_univ _ _ f \u25b8 hf.image_subgroup univ_subgroup.to_isSubgroup\n#align is_group_hom.range_subgroup IsGroupHom.range_subgroup\n#align is_add_group_hom.range_add_subgroup IsAddGroupHom.range_addSubgroup\n\nattribute [local simp] one_mem inv_mem mul_mem IsNormalSubgroup.normal\n\n/- warning: is_group_hom.preimage -> IsGroupHom.preimage is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u2} H}, (IsSubgroup.{u2} H _inst_2 s) -> (IsSubgroup.{u1} G _inst_1 (Set.preimage.{u1, u2} G H f s)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u1} H}, (IsSubgroup.{u1} H _inst_2 s) -> (IsSubgroup.{u2} G _inst_1 (Set.preimage.{u2, u1} G H f s)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.preimage IsGroupHom.preimage\u2093'. -/\n@[to_additive]\ntheorem preimage {f : G \u2192 H} (hf : IsGroupHom f) {s : Set H} (hs : IsSubgroup s) :\n    IsSubgroup (f \u207b\u00b9' s) := by\n  refine' { .. } <;>\n    simp (config := { contextual := true }) [hs.one_mem, hs.mul_mem, hs.inv_mem, hf.map_mul,\n      hf.map_one, hf.map_inv, InvMemClass.inv_mem]\n#align is_group_hom.preimage IsGroupHom.preimage\n#align is_add_group_hom.preimage IsAddGroupHom.preimage\n\n/- warning: is_group_hom.preimage_normal -> IsGroupHom.preimage_normal is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u2} H}, (IsNormalSubgroup.{u2} H _inst_2 s) -> (IsNormalSubgroup.{u1} G _inst_1 (Set.preimage.{u1, u2} G H f s)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (forall {s : Set.{u1} H}, (IsNormalSubgroup.{u1} H _inst_2 s) -> (IsNormalSubgroup.{u2} G _inst_1 (Set.preimage.{u2, u1} G H f s)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.preimage_normal IsGroupHom.preimage_normal\u2093'. -/\n@[to_additive]\ntheorem preimage_normal {f : G \u2192 H} (hf : IsGroupHom f) {s : Set H} (hs : IsNormalSubgroup s) :\n    IsNormalSubgroup (f \u207b\u00b9' s) :=\n  { one_mem := by simp [hf.map_one, hs.to_is_subgroup.one_mem]\n    mul_mem := by simp (config := { contextual := true }) [hf.map_mul, hs.to_is_subgroup.mul_mem]\n    inv_mem := by simp (config := { contextual := true }) [hf.map_inv, hs.to_is_subgroup.inv_mem]\n    Normal := by simp (config := { contextual := true }) [hs.normal, hf.map_mul, hf.map_inv] }\n#align is_group_hom.preimage_normal IsGroupHom.preimage_normal\n#align is_add_group_hom.preimage_normal IsAddGroupHom.preimage_normal\n\n/- warning: is_group_hom.is_normal_subgroup_ker -> IsGroupHom.isNormalSubgroup_ker is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (IsNormalSubgroup.{u1} G _inst_1 (IsGroupHom.ker.{u1, u2} G H _inst_2 f))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (IsNormalSubgroup.{u2} G _inst_1 (IsGroupHom.ker.{u2, u1} G H _inst_2 f))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.is_normal_subgroup_ker IsGroupHom.isNormalSubgroup_ker\u2093'. -/\n@[to_additive]\ntheorem isNormalSubgroup_ker {f : G \u2192 H} (hf : IsGroupHom f) : IsNormalSubgroup (ker f) :=\n  hf.preimage_normal trivial_normal\n#align is_group_hom.is_normal_subgroup_ker IsGroupHom.isNormalSubgroup_ker\n#align is_add_group_hom.is_normal_add_subgroup_ker IsAddGroupHom.isNormalAddSubgroup_ker\n\n/- warning: is_group_hom.injective_of_trivial_ker -> IsGroupHom.injective_of_trivial_ker is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (Eq.{succ u1} (Set.{u1} G) (IsGroupHom.ker.{u1, u2} G H _inst_2 f) (IsSubgroup.trivial.{u1} G _inst_1)) -> (Function.Injective.{succ u1, succ u2} G H f)\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (Eq.{succ u2} (Set.{u2} G) (IsGroupHom.ker.{u2, u1} G H _inst_2 f) (IsSubgroup.trivial.{u2} G _inst_1)) -> (Function.Injective.{succ u2, succ u1} G H f)\nCase conversion may be inaccurate. Consider using '#align is_group_hom.injective_of_trivial_ker IsGroupHom.injective_of_trivial_ker\u2093'. -/\n@[to_additive]\ntheorem injective_of_trivial_ker {f : G \u2192 H} (hf : IsGroupHom f) (h : ker f = trivial G) :\n    Function.Injective f := by\n  intro a\u2081 a\u2082 hfa\n  simp [ext_iff, ker, IsSubgroup.trivial] at h\n  have ha : a\u2081 * a\u2082\u207b\u00b9 = 1 := by rw [\u2190 h] <;> exact hf.inv_ker_one hfa\n  rw [eq_inv_of_mul_eq_one_left ha, inv_inv a\u2082]\n#align is_group_hom.injective_of_trivial_ker IsGroupHom.injective_of_trivial_ker\n#align is_add_group_hom.injective_of_trivial_ker IsAddGroupHom.injective_of_trivial_ker\n\n/- warning: is_group_hom.trivial_ker_of_injective -> IsGroupHom.trivial_ker_of_injective is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (Function.Injective.{succ u1, succ u2} G H f) -> (Eq.{succ u1} (Set.{u1} G) (IsGroupHom.ker.{u1, u2} G H _inst_2 f) (IsSubgroup.trivial.{u1} G _inst_1))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (Function.Injective.{succ u2, succ u1} G H f) -> (Eq.{succ u2} (Set.{u2} G) (IsGroupHom.ker.{u2, u1} G H _inst_2 f) (IsSubgroup.trivial.{u2} G _inst_1))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.trivial_ker_of_injective IsGroupHom.trivial_ker_of_injective\u2093'. -/\n@[to_additive]\ntheorem trivial_ker_of_injective {f : G \u2192 H} (hf : IsGroupHom f) (h : Function.Injective f) :\n    ker f = trivial G :=\n  Set.ext fun x =>\n    Iff.intro\n      (fun hx => by\n        suffices f x = f 1 by simpa using h this\n        simp [hf.map_one] <;> rwa [mem_ker] at hx)\n      (by simp (config := { contextual := true }) [mem_ker, hf.map_one])\n#align is_group_hom.trivial_ker_of_injective IsGroupHom.trivial_ker_of_injective\n#align is_add_group_hom.trivial_ker_of_injective IsAddGroupHom.trivial_ker_of_injective\n\n/- warning: is_group_hom.injective_iff_trivial_ker -> IsGroupHom.injective_iff_trivial_ker is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (Iff (Function.Injective.{succ u1, succ u2} G H f) (Eq.{succ u1} (Set.{u1} G) (IsGroupHom.ker.{u1, u2} G H _inst_2 f) (IsSubgroup.trivial.{u1} G _inst_1)))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (Iff (Function.Injective.{succ u2, succ u1} G H f) (Eq.{succ u2} (Set.{u2} G) (IsGroupHom.ker.{u2, u1} G H _inst_2 f) (IsSubgroup.trivial.{u2} G _inst_1)))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.injective_iff_trivial_ker IsGroupHom.injective_iff_trivial_ker\u2093'. -/\n@[to_additive]\ntheorem injective_iff_trivial_ker {f : G \u2192 H} (hf : IsGroupHom f) :\n    Function.Injective f \u2194 ker f = trivial G :=\n  \u27e8hf.trivial_ker_of_injective, hf.injective_of_trivial_ker\u27e9\n#align is_group_hom.injective_iff_trivial_ker IsGroupHom.injective_iff_trivial_ker\n#align is_add_group_hom.injective_iff_trivial_ker IsAddGroupHom.injective_iff_trivial_ker\n\n/- warning: is_group_hom.trivial_ker_iff_eq_one -> IsGroupHom.trivial_ker_iff_eq_one is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} {H : Type.{u2}} [_inst_1 : Group.{u1} G] [_inst_2 : Group.{u2} H] {f : G -> H}, (IsGroupHom.{u1, u2} G H _inst_1 _inst_2 f) -> (Iff (Eq.{succ u1} (Set.{u1} G) (IsGroupHom.ker.{u1, u2} G H _inst_2 f) (IsSubgroup.trivial.{u1} G _inst_1)) (forall (x : G), (Eq.{succ u2} H (f x) (OfNat.ofNat.{u2} H 1 (OfNat.mk.{u2} H 1 (One.one.{u2} H (MulOneClass.toHasOne.{u2} H (Monoid.toMulOneClass.{u2} H (DivInvMonoid.toMonoid.{u2} H (Group.toDivInvMonoid.{u2} H _inst_2)))))))) -> (Eq.{succ u1} G x (OfNat.ofNat.{u1} G 1 (OfNat.mk.{u1} G 1 (One.one.{u1} G (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))))))))))\nbut is expected to have type\n  forall {G : Type.{u2}} {H : Type.{u1}} [_inst_1 : Group.{u2} G] [_inst_2 : Group.{u1} H] {f : G -> H}, (IsGroupHom.{u2, u1} G H _inst_1 _inst_2 f) -> (Iff (Eq.{succ u2} (Set.{u2} G) (IsGroupHom.ker.{u2, u1} G H _inst_2 f) (IsSubgroup.trivial.{u2} G _inst_1)) (forall (x : G), (Eq.{succ u1} H (f x) (OfNat.ofNat.{u1} H 1 (One.toOfNat1.{u1} H (InvOneClass.toOne.{u1} H (DivInvOneMonoid.toInvOneClass.{u1} H (DivisionMonoid.toDivInvOneMonoid.{u1} H (Group.toDivisionMonoid.{u1} H _inst_2))))))) -> (Eq.{succ u2} G x (OfNat.ofNat.{u2} G 1 (One.toOfNat1.{u2} G (InvOneClass.toOne.{u2} G (DivInvOneMonoid.toInvOneClass.{u2} G (DivisionMonoid.toDivInvOneMonoid.{u2} G (Group.toDivisionMonoid.{u2} G _inst_1)))))))))\nCase conversion may be inaccurate. Consider using '#align is_group_hom.trivial_ker_iff_eq_one IsGroupHom.trivial_ker_iff_eq_one\u2093'. -/\n@[to_additive]\ntheorem trivial_ker_iff_eq_one {f : G \u2192 H} (hf : IsGroupHom f) :\n    ker f = trivial G \u2194 \u2200 x, f x = 1 \u2192 x = 1 := by\n  rw [Set.ext_iff] <;> simp [ker] <;>\n    exact \u27e8fun h x hx => (h x).1 hx, fun h x => \u27e8h x, fun hx => by rw [hx, hf.map_one]\u27e9\u27e9\n#align is_group_hom.trivial_ker_iff_eq_one IsGroupHom.trivial_ker_iff_eq_one\n#align is_add_group_hom.trivial_ker_iff_eq_zero IsAddGroupHom.trivial_ker_iff_eq_zero\n\nend IsGroupHom\n\nnamespace AddGroup\n\nvariable [AddGroup A]\n\n#print AddGroup.InClosure /-\n/-- If `A` is an additive group and `s : set A`, then `in_closure s : set A` is the underlying\nsubset of the subgroup generated by `s`. -/\ninductive InClosure (s : Set A) : A \u2192 Prop\n  | basic {a : A} : a \u2208 s \u2192 in_closure a\n  | zero : in_closure 0\n  | neg {a : A} : in_closure a \u2192 in_closure (-a)\n  | add {a b : A} : in_closure a \u2192 in_closure b \u2192 in_closure (a + b)\n#align add_group.in_closure AddGroup.InClosure\n-/\n\nend AddGroup\n\nnamespace Group\n\nopen IsSubmonoid IsSubgroup\n\nvariable [Group G] {s : Set G}\n\n#print Group.InClosure /-\n/-- If `G` is a group and `s : set G`, then `in_closure s : set G` is the underlying\nsubset of the subgroup generated by `s`. -/\n@[to_additive]\ninductive InClosure (s : Set G) : G \u2192 Prop\n  | basic {a : G} : a \u2208 s \u2192 in_closure a\n  | one : in_closure 1\n  | inv {a : G} : in_closure a \u2192 in_closure a\u207b\u00b9\n  | mul {a b : G} : in_closure a \u2192 in_closure b \u2192 in_closure (a * b)\n#align group.in_closure Group.InClosure\n#align add_group.in_closure AddGroup.InClosure\n-/\n\n#print Group.closure /-\n/-- `group.closure s` is the subgroup generated by `s`, i.e. the smallest subgroup containg `s`. -/\n@[to_additive\n      \"`add_group.closure s` is the additive subgroup generated by `s`, i.e., the\\n  smallest additive subgroup containing `s`.\"]\ndef closure (s : Set G) : Set G :=\n  { a | InClosure s a }\n#align group.closure Group.closure\n#align add_group.closure AddGroup.closure\n-/\n\n#print Group.mem_closure /-\n@[to_additive]\ntheorem mem_closure {a : G} : a \u2208 s \u2192 a \u2208 closure s :=\n  InClosure.basic\n#align group.mem_closure Group.mem_closure\n#align add_group.mem_closure AddGroup.mem_closure\n-/\n\n#print Group.closure.isSubgroup /-\n@[to_additive]\ntheorem closure.isSubgroup (s : Set G) : IsSubgroup (closure s) :=\n  { one_mem := InClosure.one\n    mul_mem := fun a b => InClosure.mul\n    inv_mem := fun a => InClosure.inv }\n#align group.closure.is_subgroup Group.closure.isSubgroup\n#align add_group.closure.is_add_subgroup AddGroup.closure.isAddSubgroup\n-/\n\n#print Group.subset_closure /-\n@[to_additive]\ntheorem subset_closure {s : Set G} : s \u2286 closure s := fun a => mem_closure\n#align group.subset_closure Group.subset_closure\n#align add_group.subset_closure AddGroup.subset_closure\n-/\n\n#print Group.closure_subset /-\n@[to_additive]\ntheorem closure_subset {s t : Set G} (ht : IsSubgroup t) (h : s \u2286 t) : closure s \u2286 t := fun a ha =>\n  by induction ha <;> simp [h _, *, ht.one_mem, ht.mul_mem, IsSubgroup.inv_mem_iff]\n#align group.closure_subset Group.closure_subset\n#align add_group.closure_subset AddGroup.closure_subset\n-/\n\n#print Group.closure_subset_iff /-\n@[to_additive]\ntheorem closure_subset_iff {s t : Set G} (ht : IsSubgroup t) : closure s \u2286 t \u2194 s \u2286 t :=\n  \u27e8fun h b ha => h (mem_closure ha), fun h b ha => closure_subset ht h ha\u27e9\n#align group.closure_subset_iff Group.closure_subset_iff\n#align add_group.closure_subset_iff AddGroup.closure_subset_iff\n-/\n\n#print Group.closure_mono /-\n@[to_additive]\ntheorem closure_mono {s t : Set G} (h : s \u2286 t) : closure s \u2286 closure t :=\n  closure_subset (closure.isSubgroup _) <| Set.Subset.trans h subset_closure\n#align group.closure_mono Group.closure_mono\n#align add_group.closure_mono AddGroup.closure_mono\n-/\n\n#print Group.closure_subgroup /-\n@[simp, to_additive]\ntheorem closure_subgroup {s : Set G} (hs : IsSubgroup s) : closure s = s :=\n  Set.Subset.antisymm (closure_subset hs <| Set.Subset.refl s) subset_closure\n#align group.closure_subgroup Group.closure_subgroup\n#align add_group.closure_add_subgroup AddGroup.closure_addSubgroup\n-/\n\n/- warning: group.exists_list_of_mem_closure -> Group.exists_list_of_mem_closure is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G} {a : G}, (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) a (Group.closure.{u1} G _inst_1 s)) -> (Exists.{succ u1} (List.{u1} G) (fun (l : List.{u1} G) => And (forall (x : G), (Membership.Mem.{u1, u1} G (List.{u1} G) (List.hasMem.{u1} G) x l) -> (Or (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) x s) (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) x) s))) (Eq.{succ u1} G (List.prod.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)))) (MulOneClass.toHasOne.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)))) l) a)))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G} {a : G}, (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) a (Group.closure.{u1} G _inst_1 s)) -> (Exists.{succ u1} (List.{u1} G) (fun (l : List.{u1} G) => And (forall (x : G), (Membership.mem.{u1, u1} G (List.{u1} G) (List.instMembershipList.{u1} G) x l) -> (Or (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) x s) (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))) x) s))) (Eq.{succ u1} G (List.prod.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)))) (InvOneClass.toOne.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1)))) l) a)))\nCase conversion may be inaccurate. Consider using '#align group.exists_list_of_mem_closure Group.exists_list_of_mem_closure\u2093'. -/\n@[to_additive]\ntheorem exists_list_of_mem_closure {s : Set G} {a : G} (h : a \u2208 closure s) :\n    \u2203 l : List G, (\u2200 x \u2208 l, x \u2208 s \u2228 x\u207b\u00b9 \u2208 s) \u2227 l.Prod = a :=\n  InClosure.rec_on h (fun x hxs => \u27e8[x], List.forall_mem_singleton.2 <| Or.inl hxs, one_mul _\u27e9)\n    \u27e8[], List.forall_mem_nil _, rfl\u27e9\n    (fun x _ \u27e8L, HL1, HL2\u27e9 =>\n      \u27e8L.reverse.map Inv.inv, fun x hx =>\n        let \u27e8y, hy1, hy2\u27e9 := List.exists_of_mem_map hx\n        hy2 \u25b8 Or.imp id (by rw [inv_inv] <;> exact id) (HL1 _ <| List.mem_reverse'.1 hy1).symm,\n        HL2 \u25b8\n          List.recOn L inv_one.symm fun hd tl ih => by\n            rw [List.reverse_cons, List.map_append, List.prod_append, ih, List.map_singleton,\n              List.prod_cons, List.prod_nil, mul_one, List.prod_cons, mul_inv_rev]\u27e9)\n    fun x y hx hy \u27e8L1, HL1, HL2\u27e9 \u27e8L2, HL3, HL4\u27e9 =>\n    \u27e8L1 ++ L2, List.forall_mem_append.2 \u27e8HL1, HL3\u27e9, by rw [List.prod_append, HL2, HL4]\u27e9\n#align group.exists_list_of_mem_closure Group.exists_list_of_mem_closure\n#align add_group.exists_list_of_mem_closure AddGroup.exists_list_of_mem_closure\n\n#print Group.image_closure /-\n@[to_additive]\ntheorem image_closure [Group H] {f : G \u2192 H} (hf : IsGroupHom f) (s : Set G) :\n    f '' closure s = closure (f '' s) :=\n  le_antisymm\n    (by\n      rintro _ \u27e8x, hx, rfl\u27e9\n      apply in_closure.rec_on hx <;> intros\n      \u00b7 solve_by_elim [subset_closure, Set.mem_image_of_mem]\n      \u00b7 rw [hf.to_is_monoid_hom.map_one]\n        apply IsSubmonoid.one_mem (closure.is_subgroup _).to_isSubmonoid\n      \u00b7 rw [hf.map_inv]\n        apply IsSubgroup.inv_mem (closure.is_subgroup _)\n        assumption\n      \u00b7 rw [hf.to_is_monoid_hom.map_mul]\n        solve_by_elim [IsSubmonoid.mul_mem (closure.is_subgroup _).to_isSubmonoid] )\n    (closure_subset (hf.image_subgroup <| closure.isSubgroup _) <|\n      Set.image_subset _ subset_closure)\n#align group.image_closure Group.image_closure\n#align add_group.image_closure AddGroup.image_closure\n-/\n\n#print Group.mclosure_subset /-\n@[to_additive]\ntheorem mclosure_subset {s : Set G} : Monoid.Closure s \u2286 closure s :=\n  Monoid.closure_subset (closure.isSubgroup _).to_isSubmonoid <| subset_closure\n#align group.mclosure_subset Group.mclosure_subset\n#align add_group.mclosure_subset AddGroup.mclosure_subset\n-/\n\n/- warning: group.mclosure_inv_subset -> Group.mclosure_inv_subset is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, HasSubset.Subset.{u1} (Set.{u1} G) (Set.hasSubset.{u1} G) (Monoid.Closure.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) (Set.preimage.{u1, u1} G G (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) s)) (Group.closure.{u1} G _inst_1 s)\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, HasSubset.Subset.{u1} (Set.{u1} G) (Set.instHasSubsetSet.{u1} G) (Monoid.Closure.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) (Set.preimage.{u1, u1} G G (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1))))) s)) (Group.closure.{u1} G _inst_1 s)\nCase conversion may be inaccurate. Consider using '#align group.mclosure_inv_subset Group.mclosure_inv_subset\u2093'. -/\n@[to_additive]\ntheorem mclosure_inv_subset {s : Set G} : Monoid.Closure (Inv.inv \u207b\u00b9' s) \u2286 closure s :=\n  Monoid.closure_subset (closure.isSubgroup _).to_isSubmonoid fun x hx =>\n    inv_inv x \u25b8 ((closure.isSubgroup _).inv_mem <| subset_closure hx)\n#align group.mclosure_inv_subset Group.mclosure_inv_subset\n#align add_group.mclosure_neg_subset AddGroup.mclosure_neg_subset\n\n/- warning: group.closure_eq_mclosure -> Group.closure_eq_mclosure is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, Eq.{succ u1} (Set.{u1} G) (Group.closure.{u1} G _inst_1 s) (Monoid.Closure.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) (Union.union.{u1} (Set.{u1} G) (Set.hasUnion.{u1} G) s (Set.preimage.{u1, u1} G G (Inv.inv.{u1} G (DivInvMonoid.toHasInv.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1))) s)))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] {s : Set.{u1} G}, Eq.{succ u1} (Set.{u1} G) (Group.closure.{u1} G _inst_1 s) (Monoid.Closure.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G _inst_1)) (Union.union.{u1} (Set.{u1} G) (Set.instUnionSet.{u1} G) s (Set.preimage.{u1, u1} G G (Inv.inv.{u1} G (InvOneClass.toInv.{u1} G (DivInvOneMonoid.toInvOneClass.{u1} G (DivisionMonoid.toDivInvOneMonoid.{u1} G (Group.toDivisionMonoid.{u1} G _inst_1))))) s)))\nCase conversion may be inaccurate. Consider using '#align group.closure_eq_mclosure Group.closure_eq_mclosure\u2093'. -/\n@[to_additive]\ntheorem closure_eq_mclosure {s : Set G} : closure s = Monoid.Closure (s \u222a Inv.inv \u207b\u00b9' s) :=\n  Set.Subset.antisymm\n    (@closure_subset _ _ _ (Monoid.Closure (s \u222a Inv.inv \u207b\u00b9' s))\n      { one_mem := (Monoid.closure.isSubmonoid _).one_mem\n        mul_mem := fun _ _ => (Monoid.closure.isSubmonoid _).mul_mem\n        inv_mem := fun x hx =>\n          Monoid.InClosure.rec_on hx\n            (fun x hx =>\n              Or.cases_on hx\n                (fun hx =>\n                  Monoid.subset_closure <| Or.inr <| show x\u207b\u00b9\u207b\u00b9 \u2208 s from (inv_inv x).symm \u25b8 hx)\n                fun hx => Monoid.subset_closure <| Or.inl hx)\n            ((@inv_one G _).symm \u25b8 IsSubmonoid.one_mem (Monoid.closure.isSubmonoid _))\n            fun x y hx hy ihx ihy =>\n            (mul_inv_rev x y).symm \u25b8 IsSubmonoid.mul_mem (Monoid.closure.isSubmonoid _) ihy ihx }\n      (Set.Subset.trans (Set.subset_union_left _ _) Monoid.subset_closure))\n    (Monoid.closure_subset (closure.isSubgroup _).to_isSubmonoid <|\n      Set.union_subset subset_closure fun x hx =>\n        inv_inv x \u25b8 (IsSubgroup.inv_mem (closure.isSubgroup _) <| subset_closure hx))\n#align group.closure_eq_mclosure Group.closure_eq_mclosure\n#align add_group.closure_eq_mclosure AddGroup.closure_eq_mclosure\n\n/- warning: group.mem_closure_union_iff -> Group.mem_closure_union_iff is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_2 : CommGroup.{u1} G] {s : Set.{u1} G} {t : Set.{u1} G} {x : G}, Iff (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) x (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) (Union.union.{u1} (Set.{u1} G) (Set.hasUnion.{u1} G) s t))) (Exists.{succ u1} G (fun (y : G) => Exists.{0} (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) y (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) s)) (fun (H : Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) y (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) s)) => Exists.{succ u1} G (fun (z : G) => Exists.{0} (Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) z (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) t)) (fun (H : Membership.Mem.{u1, u1} G (Set.{u1} G) (Set.hasMem.{u1} G) z (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) t)) => Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toHasMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G (CommGroup.toGroup.{u1} G _inst_2)))))) y z) x)))))\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_2 : CommGroup.{u1} G] {s : Set.{u1} G} {t : Set.{u1} G} {x : G}, Iff (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) x (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) (Union.union.{u1} (Set.{u1} G) (Set.instUnionSet.{u1} G) s t))) (Exists.{succ u1} G (fun (y : G) => And (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) y (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) s)) (Exists.{succ u1} G (fun (z : G) => And (Membership.mem.{u1, u1} G (Set.{u1} G) (Set.instMembershipSet.{u1} G) z (Group.closure.{u1} G (CommGroup.toGroup.{u1} G _inst_2) t)) (Eq.{succ u1} G (HMul.hMul.{u1, u1, u1} G G G (instHMul.{u1} G (MulOneClass.toMul.{u1} G (Monoid.toMulOneClass.{u1} G (DivInvMonoid.toMonoid.{u1} G (Group.toDivInvMonoid.{u1} G (CommGroup.toGroup.{u1} G _inst_2)))))) y z) x)))))\nCase conversion may be inaccurate. Consider using '#align group.mem_closure_union_iff Group.mem_closure_union_iff\u2093'. -/\n@[to_additive]\ntheorem mem_closure_union_iff {G : Type _} [CommGroup G] {s t : Set G} {x : G} :\n    x \u2208 closure (s \u222a t) \u2194 \u2203 y \u2208 closure s, \u2203 z \u2208 closure t, y * z = x :=\n  by\n  simp only [closure_eq_mclosure, Monoid.mem_closure_union_iff, exists_prop, preimage_union];\n  constructor\n  \u00b7 rintro \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, rfl\u27e9\n    refine' \u27e8_, \u27e8_, hys, _, hzs, rfl\u27e9, _, \u27e8_, hyt, _, hzt, rfl\u27e9, _\u27e9\n    rw [mul_assoc, mul_assoc, mul_left_comm zs]\n  \u00b7 rintro \u27e8_, \u27e8ys, hys, zs, hzs, rfl\u27e9, _, \u27e8yt, hyt, zt, hzt, rfl\u27e9, rfl\u27e9\n    refine' \u27e8_, \u27e8ys, hys, yt, hyt, rfl\u27e9, _, \u27e8zs, hzs, zt, hzt, rfl\u27e9, _\u27e9\n    rw [mul_assoc, mul_assoc, mul_left_comm yt]\n#align group.mem_closure_union_iff Group.mem_closure_union_iff\n#align add_group.mem_closure_union_iff AddGroup.mem_closure_union_iff\n\nend Group\n\nnamespace IsSubgroup\n\nvariable [Group G]\n\n#print IsSubgroup.trivial_eq_closure /-\n@[to_additive]\ntheorem trivial_eq_closure : trivial G = Group.closure \u2205 :=\n  Subset.antisymm (by simp [Set.subset_def, (Group.closure.isSubgroup _).one_mem])\n    (Group.closure_subset trivial_normal.to_isSubgroup <| by simp)\n#align is_subgroup.trivial_eq_closure IsSubgroup.trivial_eq_closure\n#align is_add_subgroup.trivial_eq_closure IsAddSubgroup.trivial_eq_closure\n-/\n\nend IsSubgroup\n\n/-The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\nnamespace Group\n\nvariable {s : Set G} [Group G]\n\n#print Group.conjugatesOf_subset /-\ntheorem conjugatesOf_subset {t : Set G} (ht : IsNormalSubgroup t) {a : G} (h : a \u2208 t) :\n    conjugatesOf a \u2286 t := fun x hc =>\n  by\n  obtain \u27e8c, w\u27e9 := isConj_iff.1 hc\n  have H := IsNormalSubgroup.normal ht a h c\n  rwa [\u2190 w]\n#align group.conjugates_of_subset Group.conjugatesOf_subset\n-/\n\n#print Group.conjugatesOfSet_subset' /-\ntheorem conjugatesOfSet_subset' {s t : Set G} (ht : IsNormalSubgroup t) (h : s \u2286 t) :\n    conjugatesOfSet s \u2286 t :=\n  Set.union\u1d62\u2082_subset fun x H => conjugatesOf_subset ht (h H)\n#align group.conjugates_of_set_subset' Group.conjugatesOfSet_subset'\n-/\n\n#print Group.normalClosure /-\n/-- The normal closure of a set s is the subgroup closure of all the conjugates of\nelements of s. It is the smallest normal subgroup containing s. -/\ndef normalClosure (s : Set G) : Set G :=\n  closure (conjugatesOfSet s)\n#align group.normal_closure Group.normalClosure\n-/\n\n#print Group.conjugatesOfSet_subset_normalClosure /-\ntheorem conjugatesOfSet_subset_normalClosure : conjugatesOfSet s \u2286 normalClosure s :=\n  subset_closure\n#align group.conjugates_of_set_subset_normal_closure Group.conjugatesOfSet_subset_normalClosure\n-/\n\n#print Group.subset_normalClosure /-\ntheorem subset_normalClosure : s \u2286 normalClosure s :=\n  Set.Subset.trans subset_conjugatesOfSet conjugatesOfSet_subset_normalClosure\n#align group.subset_normal_closure Group.subset_normalClosure\n-/\n\n#print Group.normalClosure.isSubgroup /-\n/-- The normal closure of a set is a subgroup. -/\ntheorem normalClosure.isSubgroup (s : Set G) : IsSubgroup (normalClosure s) :=\n  closure.isSubgroup (conjugatesOfSet s)\n#align group.normal_closure.is_subgroup Group.normalClosure.isSubgroup\n-/\n\n#print Group.normalClosure.is_normal /-\n/-- The normal closure of s is a normal subgroup. -/\ntheorem normalClosure.is_normal : IsNormalSubgroup (normalClosure s) :=\n  { normalClosure.isSubgroup _ with\n    Normal := fun n h g => by\n      induction' h with x hx x hx ihx x y hx hy ihx ihy\n      \u00b7 exact conjugates_of_set_subset_normal_closure (conj_mem_conjugates_of_set hx)\n      \u00b7 simpa using (normal_closure.is_subgroup s).one_mem\n      \u00b7 rw [\u2190 conj_inv]\n        exact (normal_closure.is_subgroup _).inv_mem ihx\n      \u00b7 rw [\u2190 conj_mul]\n        exact (normal_closure.is_subgroup _).to_isSubmonoid.mul_mem ihx ihy }\n#align group.normal_closure.is_normal Group.normalClosure.is_normal\n-/\n\n#print Group.normalClosure_subset /-\n/-- The normal closure of s is the smallest normal subgroup containing s. -/\ntheorem normalClosure_subset {s t : Set G} (ht : IsNormalSubgroup t) (h : s \u2286 t) :\n    normalClosure s \u2286 t := fun a w =>\n  by\n  induction' w with x hx x hx ihx x y hx hy ihx ihy\n  \u00b7 exact conjugates_of_set_subset' ht h <| hx\n  \u00b7 exact ht.to_is_subgroup.to_is_submonoid.one_mem\n  \u00b7 exact ht.to_is_subgroup.inv_mem ihx\n  \u00b7 exact ht.to_is_subgroup.to_is_submonoid.mul_mem ihx ihy\n#align group.normal_closure_subset Group.normalClosure_subset\n-/\n\n#print Group.normalClosure_subset_iff /-\ntheorem normalClosure_subset_iff {s t : Set G} (ht : IsNormalSubgroup t) :\n    s \u2286 t \u2194 normalClosure s \u2286 t :=\n  \u27e8normalClosure_subset ht, Set.Subset.trans subset_normalClosure\u27e9\n#align group.normal_closure_subset_iff Group.normalClosure_subset_iff\n-/\n\n#print Group.normalClosure_mono /-\ntheorem normalClosure_mono {s t : Set G} : s \u2286 t \u2192 normalClosure s \u2286 normalClosure t := fun h =>\n  normalClosure_subset normalClosure.is_normal (Set.Subset.trans h subset_normalClosure)\n#align group.normal_closure_mono Group.normalClosure_mono\n-/\n\nend Group\n\n#print Subgroup.of /-\n/-- Create a bundled subgroup from a set `s` and `[is_subgroup s]`. -/\n@[to_additive \"Create a bundled additive subgroup from a set `s` and `[is_add_subgroup s]`.\"]\ndef Subgroup.of [Group G] {s : Set G} (h : IsSubgroup s) : Subgroup G\n    where\n  carrier := s\n  one_mem' := h.1.1\n  mul_mem' _ _ := h.1.2\n  inv_mem' _ := h.2\n#align subgroup.of Subgroup.of\n#align add_subgroup.of AddSubgroup.of\n-/\n\n/- warning: subgroup.is_subgroup -> Subgroup.isSubgroup is a dubious translation:\nlean 3 declaration is\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] (K : Subgroup.{u1} G _inst_1), IsSubgroup.{u1} G _inst_1 ((fun (a : Type.{u1}) (b : Type.{u1}) [self : HasLiftT.{succ u1, succ u1} a b] => self.0) (Subgroup.{u1} G _inst_1) (Set.{u1} G) (HasLiftT.mk.{succ u1, succ u1} (Subgroup.{u1} G _inst_1) (Set.{u1} G) (CoeTC\u2093.coe.{succ u1, succ u1} (Subgroup.{u1} G _inst_1) (Set.{u1} G) (SetLike.Set.hasCoeT.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.setLike.{u1} G _inst_1)))) K)\nbut is expected to have type\n  forall {G : Type.{u1}} [_inst_1 : Group.{u1} G] (K : Subgroup.{u1} G _inst_1), IsSubgroup.{u1} G _inst_1 (SetLike.coe.{u1, u1} (Subgroup.{u1} G _inst_1) G (Subgroup.instSetLikeSubgroup.{u1} G _inst_1) K)\nCase conversion may be inaccurate. Consider using '#align subgroup.is_subgroup Subgroup.isSubgroup\u2093'. -/\n@[to_additive]\ntheorem Subgroup.isSubgroup [Group G] (K : Subgroup G) : IsSubgroup (K : Set G) :=\n  { one_mem := K.one_mem'\n    mul_mem := fun _ _ => K.mul_mem'\n    inv_mem := fun _ => K.inv_mem' }\n#align subgroup.is_subgroup Subgroup.isSubgroup\n#align add_subgroup.is_add_subgroup AddSubgroup.isAddSubgroup\n\n#print Subgroup.of_normal /-\n-- this will never fire if it's an instance\n@[to_additive]\ntheorem Subgroup.of_normal [Group G] (s : Set G) (h : IsSubgroup s) (n : IsNormalSubgroup s) :\n    Subgroup.Normal (Subgroup.of h) :=\n  { conj_mem := n.Normal }\n#align subgroup.of_normal Subgroup.of_normal\n#align add_subgroup.of_normal AddSubgroup.of_normal\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Deprecated/Subgroup.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.36446103507978544}}
{"text": "import ntac.core\nimport ntac.katex\n\n/-!\n# Tactics for literate programming\n\nTactics for literate programming. Math expressions will be converted into KaTeX math syntax.\nMath expressions can be written between `#` marks, or placed in list when using `parse_pexpr_list`.\nNotations for identifiers whose name is conflicting with others may not work between `#` marks\n(for example, `=` in proofs for pythagorean numbers, which conflicts with the lemma `eq` in the file).\n-/\nopen ntac\nopen tactic\nopen lean.parser\n\nnamespace ntac.literate\n/--parses lean expressions into KaTeX math syntax.-/\nmeta def parse_to_katex (s: string) : tactic string := do\nback \u2190 read,\ne \u2190 lean.parser.run_with_input (lean.parser.pexpr 0 tt) $ s, -- \"(\"++s++\")\" is safer?\nexp \u2190 to_expr e,\nstr \u2190 safe_run $ expr_to_katex exp,\nreturn str\n\n/--parses a list into KaTeX syntax.  String (not parsed) and math expression (parsed by `parse_to_katex`) must be placed alternately in the list. Used in `parse_all`.-/\nmeta def parse_all_list : list string \u2192 tactic string\n| [] := fail \"empty input\"\n| (str::[]) := pure str\n| (str::math::tail) := do s \u2190 parse_to_katex math, ts \u2190 parse_all_list tail, pure $ str ++ \"$\" ++ s ++ \"$\" ++ ts\n\n/--a frontend for `parse_all_list`. -/\nmeta def parse_all (s: string) : tactic string :=\nlet l := string.split (='#') s in parse_all_list l\n\nmeta def antiquote_aux : list char \u2192 string\n| []  := \"(empty input)\"\n| [c] := \"\"\n| ('\\\\'::'n':: xs) := \"\\n\"++antiquote_aux xs\n| ('\\\\'::'t':: xs) := \"\\t\"++antiquote_aux xs\n| ('\\\\'::'\\\\':: xs) := \"\\\\\"++antiquote_aux xs\n| ('\\\\'::'\\\"':: xs) := \"\\\"\"++antiquote_aux xs\n| (x::xs) := x.to_string ++ antiquote_aux xs\n\n/-- the inverse of `string.quote`. -/\nmeta def antiquote(s: string) :string :=\nmatch s.to_list with\n| [] :=\"\"\n| x::xs := antiquote_aux xs\nend\n\n/--parses `expr` into KaTeX math. If the expression is a `string` literal (beginning with `\"`), `parse_all` is called. -/\nmeta def parse_expr (e: expr): tactic string := \nmatch e.to_string.to_list with\n| [] := \"(empty input)\"\n| '\"'::_ := parse_all $ antiquote e.to_string\n| _ := do s \u2190 expr_to_katex e, return $ \"$\"++s++\"$\"\nend\nmeta def parse_pexpr1 (p: pexpr) :tactic string:= safe_run $ to_expr p >>= parse_expr\n\n/--parses list of `pexpr` into string. Notations for conflicting identifiers can be used (refer to the beginning of this file).-/\nmeta def parse_pexpr_list (l: list pexpr) :tactic string:=\nsafe_run $ do le \u2190 monad.sequence $ list.map to_expr l,\nls \u2190 monad.sequence $ list.map parse_expr le,\nstring.join ls\n\nend ntac.literate\n\nopen ntac.literate\nnamespace ntac.interactive\nmeta def solve1 : itactic \u2192 ntac unit := ntac.solve1\n\nopen interactive interactive.types\nmeta def texpr_list_or_texpr := (list_of texpr) <|> list.ret <$> texpr\n/--The resulting number of goals can be any value (including 0 (solved)).-/\nprivate meta def NTAC_focus1_aux (tac: itactic): ntac ((\u03a0 {n : \u2115}, inf n.succ) \u2192 goal_tree) := \ndo gs \u2190 tactic_to_ntac get_goals,\n   match gs with\n   | []      := fail \"1toN tactic failed, there isn't any goal left to focus\"\n   | (g::rs) := \n     (do tactic_to_ntac $ set_goals [g],\n        gt \u2190 get_goal_tree,\n        type \u2190 target,\n        let ng_make := replc_unres g gt,--hg\u3088\u308a\u4e0a\u306e\u90e8\u5206\u3092\u5207\u3063\u3066\u30d0\u30c3\u30af\u30a2\u30c3\u30d7\n        set_goal_tree $ \u27e80, inf.unres g, vector.nil, type\u27e9,--hg\u3060\u3051\u3092goal_tree\u306b\n        _ \u2190 tac,\n        gs' \u2190 tactic_to_ntac $ get_goals,\n        --\u672a\u89e3\u6c7a\u306egoal\u306e\u7b87\u6240\u3092\u3059\u3079\u3066\u9589\u3058\u308b\uff08type2\u306e\u90e8\u5206\u306f\u4f7f\u308f\u306a\u3044\u304c\u4e00\u5fdc\u5165\u308c\u3066\u304a\u304f\uff09\n        -- \u3055\u3089\u306btype\u306evector\u3092\u8fd4\u3059\n        let l := gs'.length,\n        let gsvec : (vector expr l) := \u27e8gs', rfl\u27e9,\n        types \u2190 vector.sequence $ \n        vector.map (\u03bb g2, do type2 \u2190 tactic_to_ntac $ infer_type g2, replc_gi g2 (\u03bb e, \u27e80, inf.closed, vector.nil, type2\u27e9), pure type2) gsvec,\n        let newgts : vector goal_tree l := vector.map\u2082 (\u03bb ge te, \u27e80, inf.unres ge, vector.nil, te\u27e9) gsvec types,\n        rg \u2190 get_goal_tree,\n        set_goals (gs' ++rs),\n        --\u5207\u3063\u305f\u30c4\u30ea\u30fc\u3092willbeNgoal_str\u306e\u4e2d\u306b\u5165\u308c\u3066\u5143\u306e\u30b4\u30fc\u30eb\u306e\u3068\u3053\u308d\u306b\u5165\u308c\u308b\u3002\u6700\u521d\u304c{}\u306e\u4e2d\u8eab\u306egoal_tree\u3001\u6b8b\u308a\u304cunresolved goal\n        return $ \u03bb ii, ng_make $ goal_tree.mk (l+1) ii (vector.cons rg newgts))\n   end\n\n\n/--can be used on a tactic (block) turning a goal into any number of goals. \nThe process is considered to be trivial and not reflected on the output.-/\nmeta def NTAC_focus1_triv (tac: itactic) : ntac unit := \ndo f \u2190 NTAC_focus1_aux tac,\nset_goal_tree (f (@inf.willbeNgoal_str none))\n\n/--does parse beforehand-/\nprivate meta def NTAC_focus1_b (tac: itactic) (ts: tactic (string \u00d7 bool)): ntac unit := \ndo e \u2190 ts, f \u2190 NTAC_focus1_aux tac,\nset_goal_tree (f (@inf.willbeNgoal_str e))\n\n/--does parse afterwards-/\nprivate meta def NTAC_focus1_a (tac: itactic) (ts: tactic (string \u00d7 bool)): ntac unit := \ndo f \u2190 NTAC_focus1_aux tac, e \u2190 ts,\nset_goal_tree (f (@inf.willbeNgoal_str e))\n\n/--the argument string describes how the goal is manipulated.  `is_before` specifies the comment is validated before or after the tactics inside.\n`detailed` specifies if the detail is shown.-/\nprivate meta def NTAC_focus1 (tac: itactic) (is_before: bool) (detailed: bool) (ts: tactic string) : ntac unit := \n(if is_before then NTAC_focus1_b else NTAC_focus1_a) tac $ do s \u2190 ts, pure $ (s, detailed)\n\n\nmeta def NTAC_focus1_str (tac: itactic) (s: string)  (is_before: bool) (detailed: bool): ntac unit := \nNTAC_focus1 tac is_before detailed $ parse_all s\n\nmeta def NTAC_focus1_list (tac: itactic) (s: parse texpr_list_or_texpr) (is_before: bool) (detailed: bool) : ntac unit := \nNTAC_focus1 tac is_before detailed $ parse_pexpr_list s\n\nmeta def NTAC_solve1_auto (tac : itactic) : ntac unit := \ndo insert_inf1 inf.fold_auto,\n   solve1 tac\n\n/--marks a goal as trivial.-/\nmeta def TRIV : ntac unit := insert_inf1 $ inf.willbeNgoal_str none\n\n/--for debugging; parse expression into KaTeX -/\nmeta def trace_expr (s: parse texpr) : ntac unit :=\ndo \ne \u2190 parse_pexpr1 s, trace e\n\nend ntac.interactive\n", "meta": {"author": "ge9", "repo": "ntac", "sha": "c34eceeeaee6957f716874a5482ae23be94bbab1", "save_path": "github-repos/lean/ge9-ntac", "path": "github-repos/lean/ge9-ntac/ntac-c34eceeeaee6957f716874a5482ae23be94bbab1/src/ntac/literate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.36439308069268844}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.algebra.field\nimport Mathlib.algebra.module.default\nimport Mathlib.algebra.algebra.basic\nimport Mathlib.algebra.group.type_tags\nimport Mathlib.ring_theory.ideal.basic\nimport Mathlib.PostPort\n\nuniverses u v u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# Transfer algebraic structures across `equiv`s\n\nIn this file we prove theorems of the following form: if `\u03b2` has a\ngroup structure and `\u03b1 \u2243 \u03b2` then `\u03b1` has a group structure, and\nsimilarly for monoids, semigroups, rings, integral domains, fields and\nso on.\n\nNote that most of these constructions can also be obtained using the `transport` tactic.\n\n## Tags\n\nequiv, group, ring, field, module, algebra\n-/\n\nnamespace equiv\n\n\n/-- Transfer `has_one` across an `equiv` -/\nprotected def has_zero {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [HasZero \u03b2] : HasZero \u03b1 :=\n  { zero := coe_fn (equiv.symm e) 0 }\n\ntheorem zero_def {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [HasZero \u03b2] : 0 = coe_fn (equiv.symm e) 0 :=\n  rfl\n\n/-- Transfer `has_mul` across an `equiv` -/\nprotected def has_add {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Add \u03b2] : Add \u03b1 :=\n  { add := fun (x y : \u03b1) => coe_fn (equiv.symm e) (coe_fn e x + coe_fn e y) }\n\ntheorem add_def {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Add \u03b2] (x : \u03b1) (y : \u03b1) : x + y = coe_fn (equiv.symm e) (coe_fn e x + coe_fn e y) :=\n  rfl\n\n/-- Transfer `has_div` across an `equiv` -/\nprotected def has_sub {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Sub \u03b2] : Sub \u03b1 :=\n  { sub := fun (x y : \u03b1) => coe_fn (equiv.symm e) (coe_fn e x - coe_fn e y) }\n\ntheorem div_def {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Div \u03b2] (x : \u03b1) (y : \u03b1) : x / y = coe_fn (equiv.symm e) (coe_fn e x / coe_fn e y) :=\n  rfl\n\n/-- Transfer `has_inv` across an `equiv` -/\nprotected def has_inv {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [has_inv \u03b2] : has_inv \u03b1 :=\n  has_inv.mk fun (x : \u03b1) => coe_fn (equiv.symm e) (coe_fn e x\u207b\u00b9)\n\ntheorem neg_def {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Neg \u03b2] (x : \u03b1) : -x = coe_fn (equiv.symm e) (-coe_fn e x) :=\n  rfl\n\n/-- Transfer `has_scalar` across an `equiv` -/\nprotected def has_scalar {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) {R : Type u_1} [has_scalar R \u03b2] : has_scalar R \u03b1 :=\n  has_scalar.mk fun (r : R) (x : \u03b1) => coe_fn (equiv.symm e) (r \u2022 coe_fn e x)\n\ntheorem smul_def {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) {R : Type u_1} [has_scalar R \u03b2] (r : R) (x : \u03b1) : r \u2022 x = coe_fn (equiv.symm e) (r \u2022 coe_fn e x) :=\n  rfl\n\n/--\nAn equivalence `e : \u03b1 \u2243 \u03b2` gives a multiplicative equivalence `\u03b1 \u2243* \u03b2`\nwhere the multiplicative structure on `\u03b1` is\nthe one obtained by transporting a multiplicative structure on `\u03b2` back along `e`.\n-/\ndef mul_equiv {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Mul \u03b2] : let _inst : Mul \u03b1 := equiv.has_mul e;\n\u03b1 \u2243* \u03b2 :=\n  let _inst : Mul \u03b1 := equiv.has_mul e;\n  mul_equiv.mk (to_fun e) (inv_fun e) (left_inv e) (right_inv e) sorry\n\n@[simp] theorem mul_equiv_apply {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Mul \u03b2] (a : \u03b1) : coe_fn (mul_equiv e) a = coe_fn e a :=\n  rfl\n\ntheorem mul_equiv_symm_apply {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Mul \u03b2] (b : \u03b2) : coe_fn (mul_equiv.symm (mul_equiv e)) b = coe_fn (equiv.symm e) b :=\n  Eq.refl (coe_fn (mul_equiv.symm (mul_equiv e)) b)\n\n/--\nAn equivalence `e : \u03b1 \u2243 \u03b2` gives a ring equivalence `\u03b1 \u2243+* \u03b2`\nwhere the ring structure on `\u03b1` is\nthe one obtained by transporting a ring structure on `\u03b2` back along `e`.\n-/\ndef ring_equiv {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Add \u03b2] [Mul \u03b2] : let _inst : Add \u03b1 := equiv.has_add e;\nlet _inst_3 : Mul \u03b1 := equiv.has_mul e;\n\u03b1 \u2243+* \u03b2 :=\n  let _inst : Add \u03b1 := equiv.has_add e;\n  let _inst_3 : Mul \u03b1 := equiv.has_mul e;\n  ring_equiv.mk (to_fun e) (inv_fun e) (left_inv e) (right_inv e) sorry sorry\n\n@[simp] theorem ring_equiv_apply {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Add \u03b2] [Mul \u03b2] (a : \u03b1) : coe_fn (ring_equiv e) a = coe_fn e a :=\n  rfl\n\ntheorem ring_equiv_symm_apply {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [Add \u03b2] [Mul \u03b2] (b : \u03b2) : coe_fn (ring_equiv.symm (ring_equiv e)) b = coe_fn (equiv.symm e) b :=\n  Eq.refl (coe_fn (ring_equiv.symm (ring_equiv e)) b)\n\n/-- Transfer `semigroup` across an `equiv` -/\nprotected def semigroup {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [semigroup \u03b2] : semigroup \u03b1 :=\n  semigroup.mk Mul.mul sorry\n\n/-- Transfer `comm_semigroup` across an `equiv` -/\nprotected def comm_semigroup {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [comm_semigroup \u03b2] : comm_semigroup \u03b1 :=\n  comm_semigroup.mk semigroup.mul sorry sorry\n\n/-- Transfer `monoid` across an `equiv` -/\nprotected def monoid {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [monoid \u03b2] : monoid \u03b1 :=\n  monoid.mk semigroup.mul sorry 1 sorry sorry\n\n/-- Transfer `comm_monoid` across an `equiv` -/\nprotected def add_comm_monoid {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [add_comm_monoid \u03b2] : add_comm_monoid \u03b1 :=\n  add_comm_monoid.mk add_comm_semigroup.add sorry add_monoid.zero sorry sorry sorry\n\n/-- Transfer `group` across an `equiv` -/\nprotected def group {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [group \u03b2] : group \u03b1 :=\n  group.mk monoid.mul sorry monoid.one sorry sorry has_inv.inv Div.div sorry\n\n/-- Transfer `comm_group` across an `equiv` -/\nprotected def comm_group {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [comm_group \u03b2] : comm_group \u03b1 :=\n  comm_group.mk group.mul sorry group.one sorry sorry group.inv group.div sorry sorry\n\n/-- Transfer `semiring` across an `equiv` -/\nprotected def semiring {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [semiring \u03b2] : semiring \u03b1 :=\n  semiring.mk Add.add sorry 0 sorry sorry sorry Mul.mul sorry monoid.one sorry sorry sorry sorry sorry sorry\n\n/-- Transfer `comm_semiring` across an `equiv` -/\nprotected def comm_semiring {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [comm_semiring \u03b2] : comm_semiring \u03b1 :=\n  comm_semiring.mk semiring.add sorry semiring.zero sorry sorry sorry semiring.mul sorry semiring.one sorry sorry sorry\n    sorry sorry sorry sorry\n\n/-- Transfer `ring` across an `equiv` -/\nprotected def ring {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [ring \u03b2] : ring \u03b1 :=\n  ring.mk semiring.add sorry semiring.zero sorry sorry add_comm_group.neg add_comm_group.sub sorry sorry semiring.mul\n    sorry semiring.one sorry sorry sorry sorry\n\n/-- Transfer `comm_ring` across an `equiv` -/\nprotected def comm_ring {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [comm_ring \u03b2] : comm_ring \u03b1 :=\n  comm_ring.mk ring.add sorry ring.zero sorry sorry ring.neg ring.sub sorry sorry comm_monoid.mul sorry comm_monoid.one\n    sorry sorry sorry sorry sorry\n\n/-- Transfer `nonzero` across an `equiv` -/\nprotected theorem nontrivial {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [nontrivial \u03b2] : nontrivial \u03b1 := sorry\n\n/-- Transfer `domain` across an `equiv` -/\nprotected def domain {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [domain \u03b2] : domain \u03b1 :=\n  domain.mk ring.add sorry ring.zero sorry sorry ring.neg ring.sub sorry sorry ring.mul sorry ring.one sorry sorry sorry\n    sorry sorry sorry\n\n/-- Transfer `integral_domain` across an `equiv` -/\nprotected def integral_domain {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [integral_domain \u03b2] : integral_domain \u03b1 :=\n  integral_domain.mk domain.add sorry domain.zero sorry sorry domain.neg domain.sub sorry sorry domain.mul sorry\n    domain.one sorry sorry sorry sorry sorry sorry sorry\n\n/-- Transfer `division_ring` across an `equiv` -/\nprotected def division_ring {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [division_ring \u03b2] : division_ring \u03b1 :=\n  division_ring.mk domain.add sorry 0 sorry sorry domain.neg domain.sub sorry sorry domain.mul sorry 1 sorry sorry sorry\n    sorry has_inv.inv Div.div sorry sorry sorry\n\n/-- Transfer `field` across an `equiv` -/\nprotected def field {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) [field \u03b2] : field \u03b1 :=\n  field.mk integral_domain.add sorry integral_domain.zero sorry sorry integral_domain.neg integral_domain.sub sorry sorry\n    integral_domain.mul sorry integral_domain.one sorry sorry sorry sorry sorry division_ring.inv sorry sorry sorry\n\n/-- Transfer `mul_action` across an `equiv` -/\nprotected def mul_action {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [monoid R] (e : \u03b1 \u2243 \u03b2) [mul_action R \u03b2] : mul_action R \u03b1 :=\n  mul_action.mk sorry sorry\n\n/-- Transfer `distrib_mul_action` across an `equiv` -/\nprotected def distrib_mul_action {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [monoid R] (e : \u03b1 \u2243 \u03b2) [add_comm_monoid \u03b2] : let _inst : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\n[_inst_3 : distrib_mul_action R \u03b2] \u2192 distrib_mul_action R \u03b1 :=\n  fun (_inst_3 : distrib_mul_action R \u03b2) =>\n    let _inst_4 : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\n    distrib_mul_action.mk sorry sorry\n\n/-- Transfer `semimodule` across an `equiv` -/\nprotected def semimodule {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [semiring R] (e : \u03b1 \u2243 \u03b2) [add_comm_monoid \u03b2] : let _inst : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\n[_inst_3 : semimodule R \u03b2] \u2192 semimodule R \u03b1 :=\n  let _inst : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\n  fun (_inst_3 : semimodule R \u03b2) => semimodule.mk sorry sorry\n\n/--\nAn equivalence `e : \u03b1 \u2243 \u03b2` gives a linear equivalence `\u03b1 \u2243\u2097[R] \u03b2`\nwhere the `R`-module structure on `\u03b1` is\nthe one obtained by transporting an `R`-module structure on `\u03b2` back along `e`.\n-/\ndef linear_equiv {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [semiring R] (e : \u03b1 \u2243 \u03b2) [add_comm_monoid \u03b2] [semimodule R \u03b2] : let _inst : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\nlet _inst_4 : semimodule R \u03b1 := equiv.semimodule R e;\nlinear_equiv R \u03b1 \u03b2 :=\n  let _inst : add_comm_monoid \u03b1 := equiv.add_comm_monoid e;\n  let _inst_4 : semimodule R \u03b1 := equiv.semimodule R e;\n  linear_equiv.mk (add_equiv.to_fun (add_equiv e)) sorry sorry (add_equiv.inv_fun (add_equiv e)) sorry sorry\n\n/-- Transfer `algebra` across an `equiv` -/\nprotected def algebra {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [comm_semiring R] (e : \u03b1 \u2243 \u03b2) [semiring \u03b2] : let _inst : semiring \u03b1 := equiv.semiring e;\n[_inst_3 : algebra R \u03b2] \u2192 algebra R \u03b1 :=\n  let _inst : semiring \u03b1 := equiv.semiring e;\n  fun (_inst_3 : algebra R \u03b2) =>\n    ring_hom.to_algebra' (ring_hom.comp (\u2191(ring_equiv.symm (ring_equiv e))) (algebra_map R \u03b2)) sorry\n\n/--\nAn equivalence `e : \u03b1 \u2243 \u03b2` gives an algebra equivalence `\u03b1 \u2243\u2090[R] \u03b2`\nwhere the `R`-algebra structure on `\u03b1` is\nthe one obtained by transporting an `R`-algebra structure on `\u03b2` back along `e`.\n-/\ndef alg_equiv {\u03b1 : Type u} {\u03b2 : Type v} (R : Type u_1) [comm_semiring R] (e : \u03b1 \u2243 \u03b2) [semiring \u03b2] [algebra R \u03b2] : let _inst : semiring \u03b1 := equiv.semiring e;\nlet _inst_4 : algebra R \u03b1 := equiv.algebra R e;\nalg_equiv R \u03b1 \u03b2 :=\n  let _inst : semiring \u03b1 := equiv.semiring e;\n  let _inst_4 : algebra R \u03b1 := equiv.algebra R e;\n  alg_equiv.mk (ring_equiv.to_fun (ring_equiv e)) (ring_equiv.inv_fun (ring_equiv e)) sorry sorry sorry sorry sorry\n\nend equiv\n\n\nnamespace ring_equiv\n\n\nprotected theorem local_ring {A : Type u_1} {B : Type u_2} [comm_ring A] [local_ring A] [comm_ring B] (e : A \u2243+* B) : local_ring B :=\n  local_of_surjective (\u2191e) (equiv.surjective (to_equiv e))\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/transfer_instance.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.61878043374385, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.36439307176355784}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Jesse Michael Han\n\n! This file was ported from Lean 3 source module tactic.ext\n! leanprover-community/mathlib commit 3d7987cda72abc473c7cdbbb075170e9ac620042\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Rcases\nimport Mathbin.Logic.Function.Basic\n\nuniverse u\u2081 u\u2082\n\nopen Interactive Interactive.Types\n\nsection Ext\n\nopen Lean.Parser Nat Tactic\n\ninitialize\n  registerTraceClass.1 `ext\n\n/-- `derive_struct_ext_lemma n` generates two extensionality lemmas based on\nthe equality of all non-propositional projections.\n\nOn the following:\n\n```lean\n@[ext]\nstructure foo (\u03b1 : Type*) :=\n(x y : \u2115)\n(z : {z // z < x})\n(k : \u03b1)\n(h : x < y)\n```\n\n`derive_struct_lemma` generates:\n\n```lean\nlemma foo.ext : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\n  x.x = y.x \u2192 x.y = y.y \u2192 x.z == y.z \u2192 x.k = y.k \u2192 x = y\nlemma foo.ext_iff : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\n  x = y \u2194 x.x = y.x \u2227 x.y = y.y \u2227 x.z == y.z \u2227 x.k = y.k\n```\n\n-/\nunsafe def derive_struct_ext_lemma (n : Name) : tactic Name := do\n  let e \u2190 get_env\n  let fs \u2190 e.structure_fields n\n  let d \u2190 get_decl n\n  let n \u2190 resolve_constant n\n  let r := @expr.const true n <| d.univ_params.map level.param\n  let (args, _) \u2190 infer_type r >>= open_pis\n  let args := args.map expr.to_implicit_local_const\n  let t := r.mk_app args\n  let x \u2190 mk_local_def `x t\n  let y \u2190 mk_local_def `y t\n  let args_x := args ++ [x]\n  let args_y := args ++ [y]\n  let bs \u2190\n    fs.mapM fun f => do\n        let d \u2190 get_decl (n ++ f)\n        let a := @expr.const true (n ++ f) <| d.univ_params.map level.param\n        let t \u2190 infer_type a\n        let s \u2190 infer_type t\n        if s \u2260 q(Prop) then do\n            let x := a args_x\n            let y := a args_y\n            let t \u2190 infer_type x\n            let t' \u2190 infer_type y\n            some <$>\n                if t = t' then mk_app `eq [x, y] >>= mk_local_def `h\n                else mk_mapp `heq [none, x, none, y] >>= mk_local_def `h\n          else pure none\n  let bs := bs.filterMap id\n  let eq_t \u2190 mk_app `eq [x, y]\n  let t \u2190 pis (args ++ [x, y] ++ bs) eq_t\n  let pr \u2190\n    run_async do\n        let (_, pr) \u2190\n          solve_aux t do\n              let args \u2190 intron args.length\n              let x \u2190 intro1\n              let y \u2190 intro1\n              cases x\n              cases y\n              bs fun _ => do\n                  let e \u2190 intro1\n                  cases e\n              reflexivity\n        instantiate_mvars pr\n  let decl_n := .str n \"ext\"\n  add_decl (declaration.thm decl_n d t pr)\n  let bs \u2190 bs.mapM infer_type\n  let rhs := expr.mk_and_lst bs\n  let iff_t \u2190 mk_app `iff [eq_t, rhs]\n  let t \u2190 pis (args ++ [x, y]) iff_t\n  let pr \u2190\n    run_async do\n        let (_, pr) \u2190\n          solve_aux t do\n              let args \u2190 intron args.length\n              let x \u2190 intro1\n              let y \u2190 intro1\n              cases x\n              cases y\n              split\n              solve1 do\n                  let h \u2190 intro1\n                  let hs \u2190 injection h\n                  subst_vars\n                  repeat (refine ``(And.intro _ _) >> reflexivity)\n                  done <|> reflexivity\n              solve1 do\n                  repeat do\n                      refine ``(and_imp.mpr _)\n                      let h \u2190 intro1\n                      cases h\n                      skip\n                  let h \u2190 intro1\n                  cases h\n                  reflexivity\n        instantiate_mvars pr\n  add_decl (declaration.thm (.str n \"ext_iff\") d t pr)\n  pure decl_n\n#align derive_struct_ext_lemma derive_struct_ext_lemma\n\nunsafe def get_ext_subject : expr \u2192 tactic Name\n  | expr.pi n bi d b => do\n    let v \u2190 mk_local' n bi d\n    let b' \u2190 whnf <| b.instantiate_var v\n    get_ext_subject b'\n  | expr.app _ e => do\n    let t \u2190 infer_type e >>= instantiate_mvars >>= head_beta\n    if t then pure <| t\n      else\n        if t then pure <| Name.mk_numeral 0 Name.anonymous\n        else\n          if t then pure <| Name.mk_numeral 1 Name.anonymous\n          else do\n            let t \u2190 pp t\n            fail f! \"only constants and Pi types are supported: {t}\"\n  | e => fail f! \"Only expressions of the form `_ \u2192 _ \u2192 ... \u2192 R ... e are supported: {e}\"\n#align get_ext_subject get_ext_subject\n\nopen Native\n\nunsafe def saturate_fun : Name \u2192 tactic expr\n  | Name.mk_numeral 0 Name.anonymous => do\n    let v\u2080 \u2190 mk_mvar\n    let v\u2081 \u2190 mk_mvar\n    return <| v\u2080 v\u2081\n  | Name.mk_numeral 1 Name.anonymous => do\n    let u \u2190 mk_meta_univ\n    pure <| expr.sort u\n  | n => do\n    let e \u2190 resolve_constant n >>= mk_const\n    let a \u2190 get_arity e\n    e <$> (List.iota a).mapM fun _ => mk_mvar\n#align saturate_fun saturate_fun\n\nunsafe def equiv_type_constr (n n' : Name) : tactic Unit := do\n  let e \u2190 saturate_fun n\n  let e' \u2190 saturate_fun n'\n  unify e e' <|> fail f! \"{n } and {n'} are not definitionally equal types\"\n#align equiv_type_constr equiv_type_constr\n\nsection PerformanceHack\n\nlibrary_note \"user attribute parameters\"/--\nFor performance reasons, it is inadvisable to use `user_attribute.get_param`.\nThe parameter is stored as a reflected expression.  When calling `get_param`,\nthe stored parameter is evaluated using `eval_expr`, which first compiles the\nexpression into VM bytecode. The unevaluated expression is available using\n`user_attribute.get_param_untyped`.\n\nIn particular, `user_attribute.get_param` MUST NEVER BE USED in the\nimplementation of an attribute cache. This is because calling `eval_expr`\ndisables the attribute cache.\n\nThere are several possible workarounds:\n 1. Set a different attribute depending on the parameter.\n 2. Use your own evaluation function instead of `eval_expr`, such as e.g. `expr.to_nat`.\n 3. Write your own `has_reflect Param` instance (using a more efficient serialization format).\n   The `user_attribute` code unfortunately checks whether the expression has the correct type,\n   but you can use `` `(id %%e : Param) `` to pretend that your expression `e` has type `Param`.\n-/\n\n\n/-!\nFor performance reasons, the parameters of the `@[ext]` attribute are stored\nin two auxiliary attributes:\n```lean\nattribute [ext thunk] funext\n\n-- is turned into\nattribute [_ext_core (@id name @funext)] thunk\nattribute [_ext_lemma_core] funext\n```\n\nsee Note [user attribute parameters]\n-/\n\n\nattribute [local semireducible] reflected\n\n@[local instance]\nprivate unsafe def hacky_name_reflect : has_reflect Name := fun n =>\n  q((id $(expr.const n []) : Name))\n#align hacky_name_reflect hacky_name_reflect\n\n@[user_attribute]\nprivate unsafe def ext_attr_core : user_attribute (name_map Name) Name\n    where\n  Name := `_ext_core\n  descr := \"(internal attribute used by ext)\"\n  cache_cfg :=\n    { dependencies := []\n      mk_cache := fun ns =>\n        ns.foldlM\n          (fun m n => do\n            let ext_l \u2190 ext_attr_core.get_param_untyped n\n            pure (m n ext_l))\n          mk_name_map }\n  parser := failure\n#align ext_attr_core ext_attr_core\n\nend PerformanceHack\n\n/-- Private attribute used to tag extensionality lemmas. -/\n@[user_attribute]\nprivate unsafe def ext_lemma_attr_core : user_attribute\n    where\n  Name := `_ext_lemma_core\n  descr := \"(internal attribute used by ext)\"\n  parser := failure\n#align ext_lemma_attr_core ext_lemma_attr_core\n\n/-- Returns the extensionality lemmas in the environment, as a map from structure\nname to lemma name.\n-/\nunsafe def get_ext_lemmas : tactic (name_map Name) :=\n  ext_attr_core.get_cache\n#align get_ext_lemmas get_ext_lemmas\n\n/-- Returns the extensionality lemmas in the environment, as a list of lemma names.\n-/\nunsafe def get_ext_lemma_names : tactic (List Name) :=\n  attribute.get_instances ext_lemma_attr_core.Name\n#align get_ext_lemma_names get_ext_lemma_names\n\n/-- Marks `lem` as an extensionality lemma corresponding to type constructor `constr`;\nif `persistent` is true then this is a global attribute, else local. -/\nunsafe def add_ext_lemma (constr lem : Name) (persistent : Bool) : tactic Unit :=\n  ext_attr_core.Set constr lem persistent >> ext_lemma_attr_core.Set lem () persistent\n#align add_ext_lemma add_ext_lemma\n\n/-- Tag lemmas of the form:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe attribute indexes extensionality lemma using the type of the\nobjects (i.e. `my_collection`) which it gets from the statement of\nthe lemma.  In some cases, the same lemma can be used to state the\nextensionality of multiple types that are definitionally equivalent.\n\n```lean\nattribute [ext thunk, ext stream] funext\n```\n\nAlso, the following:\n\n```lean\n@[ext]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nis equivalent to\n\n```lean\n@[ext my_collection]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThis allows us specify type synonyms along with the type\nthat is referred to in the lemma statement.\n\n```lean\n@[ext, ext my_type_synonym]\nlemma my_collection.ext (a b : my_collection)\n  (h : \u2200 x, a.lookup x = b.lookup y) :\n  a = b := ...\n```\n\nThe `ext` attribute can be applied to a structure to generate its extensionality lemmas:\n\n```lean\n@[ext]\nstructure foo (\u03b1 : Type*) :=\n(x y : \u2115)\n(z : {z // z < x})\n(k : \u03b1)\n(h : x < y)\n```\n\nwill generate:\n\n```lean\n@[ext] lemma foo.ext : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\nx.x = y.x \u2192 x.y = y.y \u2192 x.z == y.z \u2192 x.k = y.k \u2192 x = y\nlemma foo.ext_iff : \u2200 {\u03b1 : Type u_1} (x y : foo \u03b1),\nx = y \u2194 x.x = y.x \u2227 x.y = y.y \u2227 x.z == y.z \u2227 x.k = y.k\n```\n\n-/\n@[user_attribute]\nunsafe def extensional_attribute : user_attribute Unit (Option Name)\n    where\n  Name := `ext\n  descr := \"lemmas usable by `ext` tactic\"\n  parser := optional ident\n  after_set :=\n    some fun n _ b => do\n      let add \u2190 extensional_attribute.get_param n\n      let e \u2190 get_env\n      let n \u2190 if (e.structure_fields n).isSome then derive_struct_ext_lemma n else pure n\n      let s \u2190 mk_const n >>= infer_type >>= get_ext_subject\n      match add with\n        | none => add_ext_lemma s n b\n        | some add => equiv_type_constr s add >> add_ext_lemma add n b\n#align extensional_attribute extensional_attribute\n\nadd_tactic_doc\n  { Name := \"ext\"\n    category := DocCategory.attr\n    declNames := [`extensional_attribute]\n    tags := [\"rewrite\", \"logic\"] }\n\nlibrary_note \"partially-applied ext lemmas\"/--\nWhen possible, `ext` lemmas are stated without a full set of arguments. As an example, for bundled\nhoms `f`, `g`, and `of`, `f.comp of = g.comp of \u2192 f = g` is a better `ext` lemma than\n`(\u2200 x, f (of x) = g (of x)) \u2192 f = g`, as the former allows a second type-specific extensionality\nlemmas to be applied to `f.comp of = g.comp of`.\nIf the domain of `of` is `\u2115` or `\u2124` and `of` is a `ring_hom`, such a lemma could then make the goal\n`f (of 1) = g (of 1)`.\n\nFor bundled morphisms, there is a `ext` lemma that always applies of the form\n`(\u2200 x, \u21d1f x = \u21d1g x) \u2192 f = g`. When adding type-specific `ext` lemmas like the one above, we want\nthese to be tried first. This happens automatically since the type-specific lemmas are inevitably\ndefined later.\n-/\n\n\n-- We mark some existing extensionality lemmas.\nattribute [ext] Array'.ext propext Function.hfunext\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Ext.lean:18:19: unsupported: attribute [ext id] -/\nattribute [ext] _root_.funext\n\n-- This line is equivalent to:\n--   attribute [ext (\u2192)] _root_.funext\n-- but (\u2192) is not actually a binary relation with a constant at the head,\n-- so we use the special name [anon].0 to represent (\u2192).\nrun_cmd\n  add_ext_lemma (Name.mk_numeral 0 Name.anonymous) `` _root_.funext true\n\n-- We create some extensionality lemmas for existing structures.\nattribute [ext] ULift\n\nnamespace PLift\n\n-- This is stronger than the one generated automatically.\n@[ext]\ntheorem ext {P : Prop} (a b : PLift P) : a = b := by cases a; cases b; rfl\n#align plift.ext PLift.ext\n\nend PLift\n\n-- Conservatively, we'll only add extensionality lemmas for `has_*` structures\n-- as they become useful.\nattribute [ext] Zero\n\n#print Unit.ext /-\n@[ext]\ntheorem Unit.ext {x y : Unit} : x = y := by\n  cases x\n  cases y\n  rfl\n#align unit.ext Unit.ext\n-/\n\n#print PUnit.ext /-\n@[ext]\ntheorem PUnit.ext {x y : PUnit} : x = y := by\n  cases x\n  cases y\n  rfl\n#align punit.ext PUnit.ext\n-/\n\nnamespace Tactic\n\n/-- Helper structure for `ext` and `ext1`. `lemmas` keeps track of extensionality lemmas\n  applied so far. -/\nunsafe structure ext_state : Type where\n  patts : List rcases_patt := []\n  trace_msg : List String := []\n  fuel : Option \u2115 := none\n#align tactic.ext_state tactic.ext_state\n\n/-- Helper function for `try_intros`. Additionally populates the `trace_msg` field\n  of `ext_state`. -/\nprivate unsafe def try_intros_core : StateT ext_state tactic Unit := do\n  let \u27e8patts, trace_msg, fuel\u27e9 \u2190 get\n  match patts with\n    | [] =>\n      (do\n          let es \u2190 StateT.lift intros\n          when (es > 0) do\n              let msg := \"intros \" ++ \" \".intercalate (es fun e => e)\n              modify fun \u27e8patts, trace_msg, fuel\u27e9 => \u27e8patts, trace_msg ++ [msg], fuel\u27e9) <|>\n        pure ()\n    | x :: xs => do\n      let tgt \u2190 StateT.lift (target >>= whnf)\n      when tgt do\n          StateT.lift (rintro [x])\n          let msg \u2190 StateT.lift ((\u00b7 ++ \u00b7) \"rintro \" <$> format.to_string <$> x ff)\n          modify fun \u27e8_, trace_msg, fuel\u27e9 => \u27e8xs, trace_msg ++ [msg], fuel\u27e9\n          try_intros_core\n#align tactic.try_intros_core tactic.try_intros_core\n\n/-- Try to introduce as many arguments as possible, using the given patterns to destruct the\n  introduced variables. Returns the unused patterns. -/\nunsafe def try_intros (patts : List rcases_patt) : tactic (List rcases_patt) :=\n  let \u03c3 := ext_state.mk patts [] none\n  (ext_state.patts \u2218 Prod.snd) <$> StateT.run try_intros_core \u03c3\n#align tactic.try_intros tactic.try_intros\n\n/-- Apply one extensionality lemma, and destruct the arguments using the patterns\n  in the ext_state. -/\nunsafe def ext1_core (cfg : ApplyCfg := { }) : StateT ext_state tactic Unit := do\n  let \u27e8patts, trace_msg, _\u27e9 \u2190 get\n  let new_msgs \u2190\n    StateT.lift <|\n        focus1 do\n          let m \u2190 get_ext_lemmas\n          let tgt \u2190 target\n          when_tracing `ext <|\n              \u2190 do\n                dbg_trace \"[ext] goal: {\u2190 tgt}\"\n          let subject \u2190 get_ext_subject tgt\n          let new_trace_msg \u2190\n            (do\n                  let rule \u2190 m.find subject\n                  if is_trace_enabled_for `ext then\n                      (\u2190 do\n                          dbg_trace \"[ext] matched goal to rule: {\u2190 rule}\") >>\n                        timetac \"[ext] application attempt time\" (applyc rule cfg)\n                    else applyc rule cfg\n                  pure [\"apply \" ++ rule]) <|>\n                (do\n                    let ls \u2190 get_ext_lemma_names\n                    let nms := ls.map Name.toString\n                    let rule \u2190\n                      ls.any_of fun n =>\n                          (if is_trace_enabled_for `ext then\n                              (\u2190 do\n                                  dbg_trace \"[ext] trying to apply ext lemma: {\u2190 n}\") >>\n                                timetac \"[ext] application attempt time\" (applyc n cfg)\n                            else applyc n cfg) *>\n                            pure n\n                    pure [\"apply \" ++ rule]) <|>\n                  fail f! \"no applicable extensionality rule found for {subject}\"\n          pure new_trace_msg\n  modify fun \u27e8patts, trace_msg, fuel\u27e9 => \u27e8patts, trace_msg ++ new_msgs, fuel\u27e9\n  try_intros_core\n#align tactic.ext1_core tactic.ext1_core\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns. -/\nunsafe def ext_core (cfg : ApplyCfg := { }) : StateT ext_state tactic Unit := do\n  let acc@\u27e8_, _, fuel\u27e9 \u2190 get\n  match fuel with\n    | some 0 => pure ()\n    | n => do\n      ext1_core cfg\n      modify fun \u27e8patts, lemmas, _\u27e9 => \u27e8patts, lemmas, Nat.pred <$> n\u27e9\n      ext_core <|> pure ()\n#align tactic.ext_core tactic.ext_core\n\n/-- Apply one extensionality lemma, and destruct the arguments using the given patterns.\n  Returns the unused patterns. -/\nunsafe def ext1 (xs : List rcases_patt) (cfg : ApplyCfg := { }) (trace : Bool := false) :\n    tactic (List rcases_patt) := do\n  let \u27e8_, \u03c3\u27e9 \u2190 StateT.run (ext1_core cfg) { patts := xs }\n  when trace <| tactic.trace <| \"Try this: \" ++ \", \".intercalate \u03c3\n  pure \u03c3\n#align tactic.ext1 tactic.ext1\n\n/-- Apply multiple extensionality lemmas, destructing the arguments using the given patterns.\n  `ext ps (some n)` applies at most `n` extensionality lemmas. Returns the unused patterns. -/\nunsafe def ext (xs : List rcases_patt) (fuel : Option \u2115) (cfg : ApplyCfg := { })\n    (trace : Bool := false) : tactic (List rcases_patt) := do\n  let \u27e8_, \u03c3\u27e9 \u2190\n    StateT.run (ext_core cfg)\n        { patts := xs\n          fuel }\n  when trace <| tactic.trace <| \"Try this: \" ++ \", \".intercalate \u03c3\n  pure \u03c3\n#align tactic.ext tactic.ext\n\n-- mathport name: parser.optional\nlocal postfix:1024 \"?\" => optional\n\n-- mathport name: parser.many\nlocal postfix:1024 \"*\" => many\n\n/-- `ext1 id` selects and apply one extensionality lemma (with attribute\n`ext`), using `id`, if provided, to name a local constant\nintroduced by the lemma. If `id` is omitted, the local constant is\nnamed automatically, as per `intro`. Placing a `?` after `ext1`\n (e.g. `ext1? i \u27e8a,b\u27e9 : 3`) will display a sequence of tactic\napplications that can replace the call to `ext1`.\n-/\nunsafe def interactive.ext1 (trace : parse (tk \"?\")?) (xs : parse rcases_patt_parse_hi*) :\n    tactic Unit :=\n  ext1 xs { } trace.isSome $> ()\n#align tactic.interactive.ext1 tactic.interactive.ext1\n\n/-- - `ext` applies as many extensionality lemmas as possible;\n- `ext ids`, with `ids` a list of identifiers, finds extentionality and applies them\n  until it runs out of identifiers in `ids` to name the local constants.\n- `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext` (e.g. `ext? i \u27e8a,b\u27e9 : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`.\n- `set_option trace.ext true` will trace every attempted lemma application,\n  along with the time it takes for the application to succeed or fail.\n  This is useful for debugging slow `ext` calls.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2\n\u22a2 f = g\n```\n\napplying `ext x y` yields:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2,\nx : \u03b1,\ny : \u03b2\n\u22a2 y \u2208 f x \u2194 y \u2208 f x\n```\n\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 f = g\n```\n\napplying `ext \u27e8a, b\u27e9` yields:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type,\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3,\na : \u03b1,\nb : \u03b2\n\u22a2 f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? \u27e8a,b\u27e9` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro \u27e8a, b\u27e9\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nunsafe def interactive.ext :\n    (parse <| (tk \"?\")?) \u2192 parse rintro_patt_parse_hi* \u2192 parse (tk \":\" *> small_nat)? \u2192 tactic Unit\n  | trace, [], some n => iterate_range 1 n (ext1 [] { } trace.isSome $> ())\n  | trace, [], none => repeat1 (ext1 [] { } trace.isSome $> ())\n  | trace, xs, n => ext xs.join n { } trace.isSome $> ()\n#align tactic.interactive.ext tactic.interactive.ext\n\n/-- * `ext1 id` selects and apply one extensionality lemma (with\n  attribute `ext`), using `id`, if provided, to name a\n  local constant introduced by the lemma. If `id` is omitted, the\n  local constant is named automatically, as per `intro`.\n\n* `ext` applies as many extensionality lemmas as possible;\n* `ext ids`, with `ids` a list of identifiers, finds extensionality lemmas\n  and applies them until it runs out of identifiers in `ids` to name\n  the local constants.\n* `ext` can also be given an `rcases` pattern in place of an identifier.\n  This will destruct the introduced local constant.\n- Placing a `?` after `ext`/`ext1` (e.g. `ext? i \u27e8a,b\u27e9 : 3`) will display\n  a sequence of tactic applications that can replace the call to `ext`/`ext1`.\n- `set_option trace.ext true` will trace every attempted lemma application,\n  along with the time it takes for the application to succeed or fail.\n  This is useful for debugging slow `ext` calls.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2\n\u22a2 f = g\n```\n\napplying `ext x y` yields:\n\n```lean\n\u03b1 \u03b2 : Type,\nf g : \u03b1 \u2192 set \u03b2,\nx : \u03b1,\ny : \u03b2\n\u22a2 y \u2208 f x \u2194 y \u2208 g x\n```\nby applying functional extensionality and set extensionality.\n\nWhen trying to prove:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3\n\u22a2 f = g\n```\n\napplying `ext \u27e8a, b\u27e9` yields:\n\n```lean\n\u03b1 \u03b2 \u03b3 : Type,\nf g : \u03b1 \u00d7 \u03b2 \u2192 \u03b3,\na : \u03b1,\nb : \u03b2\n\u22a2 f (a, b) = g (a, b)\n```\n\nby applying functional extensionality and destructing the introduced pair.\n\nIn the previous example, applying `ext? \u27e8a,b\u27e9` will produce the trace message:\n\n```lean\nTry this: apply funext, rintro \u27e8a, b\u27e9\n```\n\nA maximum depth can be provided with `ext x y z : 3`.\n-/\nadd_tactic_doc\n  { Name := \"ext1 / ext\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.ext1, `tactic.interactive.ext]\n    tags := [\"rewriting\", \"logic\"] }\n\nend Tactic\n\nend Ext\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/Ext.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.364393063483652}}
{"text": "import FOL.deduction FOL.lindenbaum\n\n-- Prenex normal form\n\nuniverses u v\n\nnamespace fol\nopen_locale logic_symbol\n\nvariables (L : language.{u})\n\nstructure pnf : Type u := \n(quantifier : list bool)\n(form : formula L)\n(is_openform : form.is_open)\n\nlocal notation `\ud835\udeb7` := bool.tt\n\nlocal notation `\ud835\udeba` := bool.ff\n\nvariables {L}\n\nnamespace pnf\n\ndef fal : pnf L \u2192 pnf L\n| \u27e8Q, p, h\u27e9 := \u27e8\ud835\udeb7 :: Q, p, h\u27e9\n\ndef ex : pnf L \u2192 pnf L\n| \u27e8Q, p, h\u27e9 := \u27e8\ud835\udeba :: Q, p, h\u27e9\n\ninstance : has_univ_quantifier (pnf L) := \u27e8pnf.fal\u27e9\n\ninstance : has_exists_quantifier (pnf L) := \u27e8pnf.ex\u27e9\n\n@[simp] lemma fal_eq (Q : list bool) (p : formula L) (h) : (\u2200.(\u27e8Q, p, h\u27e9 : pnf L) : pnf L) = \u27e8\ud835\udeb7 :: Q, p, h\u27e9 := rfl\n\n@[simp] lemma ex_eq (Q : list bool) (p : formula L) (h) : (\u2203.(\u27e8Q, p, h\u27e9 : pnf L) : pnf L) = \u27e8\ud835\udeba :: Q, p, h\u27e9 := rfl\n\n@[simp] lemma fal_inj : \u2200 (p q : pnf L), (\u2200.p : pnf L) = \u2200.q \u2194 p = q\n| \u27e8Q\u2081, p\u2081, h\u2081\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9 := by simp\n\n@[simp] lemma ex_inj : \u2200 (p q : pnf L), (\u2203.p : pnf L) = \u2203.q \u2194 p = q\n| \u27e8Q\u2081, p\u2081, h\u2081\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9 := by simp\n\n@[simp] def to_openform : pnf L \u2192 formula L\n| \u27e8Q, p, h\u27e9 := p\n\n@[simp] def to_formula : pnf L \u2192 formula L\n| \u27e8[], p, h\u27e9     := p\n| \u27e8\ud835\udeb7 :: Q, p, h\u27e9 := \u2200.to_formula \u27e8Q, p, h\u27e9\n| \u27e8\ud835\udeba :: Q, p, h\u27e9 := \u2203.to_formula \u27e8Q, p, h\u27e9\n\ninstance : has_coe (pnf L) (formula L) := \u27e8to_formula\u27e9\n\ndef to_formula_inj : \u2200 {p q : pnf L}, p.to_formula = q.to_formula \u2194 p = q\n| \u27e8[],       p\u2081, h\u2081\u27e9 \u27e8[],       p\u2082, h\u2082\u27e9 := by simp\n| \u27e8[],       p\u2081, h\u2081\u27e9 \u27e8q\u2082 :: Q\u2082, p\u2082, h\u2082\u27e9 := by { cases q\u2082; { simp, intros h, simp[h] at h\u2081, contradiction } }\n| \u27e8q\u2081 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8[],       p\u2082, h\u2082\u27e9 := by { cases q\u2081; { simp, intros h, simp[\u2190h] at h\u2082, contradiction } }\n| \u27e8q\u2081 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8q\u2082 :: Q\u2082, p\u2082, h\u2082\u27e9 := by cases q\u2081; cases q\u2082; simp[@to_formula_inj \u27e8Q\u2081, p\u2081, h\u2081\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9]\n\n@[simp] def fal_to_formula : \u2200 p : pnf L, (\u2200.p : pnf L).to_formula = \u2200.p.to_formula\n| \u27e8Q, p, h\u27e9     := by simp\n\n@[simp] def ex_to_formula : \u2200 p : pnf L, (\u2203.p : pnf L).to_formula = \u2203.p.to_formula\n| \u27e8Q, p, h\u27e9     := by simp\n\n@[simp] def rank : pnf L \u2192 \u2115 := \u03bb p, p.1.length\n\ndef rew (s : \u2115 \u2192 term L) : pnf L \u2192 pnf L\n| \u27e8Q, p, h\u27e9 := \u27e8Q, p.rew (s^Q.length), by simp[h]\u27e9\n\ninstance : has_pow (pnf L) \u2115 := \u27e8\u03bb p i, p.rew (\u03bb x, #(x + i))\u27e9\n\n@[simp] lemma rew_to_formula_eq_to_formula_rew : \u2200 (p : pnf L) (s : \u2115 \u2192 term L),\n  (p.rew s).to_formula = p.to_formula.rew s\n| \u27e8[], p, h\u27e9     s := by simp[rew]\n| \u27e8\ud835\udeb7 :: Q, p, h\u27e9 s := by simp[rew, \u2190rew_to_formula_eq_to_formula_rew \u27e8Q, p, by simp[h]\u27e9 (s^1),\n    rewriting_sf_itr.pow_add, show 1 + Q.length = Q.length + 1, from add_comm _ _]\n| \u27e8\ud835\udeba :: Q, p, h\u27e9 s := by simp[rew, \u2190rew_to_formula_eq_to_formula_rew \u27e8Q, p, by simp[h]\u27e9 (s^1),\n    rewriting_sf_itr.pow_add, show 1 + Q.length = Q.length + 1, from add_comm _ _]\n\n@[simp] lemma pow_to_formula_eq_to_formula_pow (p : pnf L) (i : \u2115) : (p^i).to_formula = p.to_formula^i :=\nby simp[formula.pow_eq, rew_to_formula_eq_to_formula_rew, has_pow.pow]\n\nlemma rew_fal (Q : list bool) (p : formula L) {h} (s : \u2115 \u2192 term L) :\n  (\u27e8(\ud835\udeb7 :: Q), p, h\u27e9 : pnf L).rew s = \u2200.(\u27e8Q, p, h\u27e9 : pnf L).rew (s^1) :=\nby simp[rew, show Q.length + 1 = 1 + Q.length, from add_comm _ _, rewriting_sf_itr.pow_add]\n\n@[simp] lemma rew_fal' : \u2200 (p : pnf L) (s : \u2115 \u2192 term L),\n  (\u2200.p : pnf L).rew s = \u2200.(p.rew (s^1))\n| \u27e8Q, p, h\u27e9 s := by simp[rew_fal]\n\nlemma rew_ex (Q : list bool) (p : formula L) {h} (s : \u2115 \u2192 term L) :\n  (\u27e8(\ud835\udeba :: Q), p, h\u27e9 : pnf L).rew s = \u2203.(\u27e8Q, p, h\u27e9 : pnf L).rew (s^1) :=\nby simp[rew, show Q.length + 1 = 1 + Q.length, from add_comm _ _, rewriting_sf_itr.pow_add]\n\n@[simp] lemma rew_ex' : \u2200 (p : pnf L) (s : \u2115 \u2192 term L),\n  (\u2203.p : pnf L).rew s = \u2203.(p.rew (s^1))\n| \u27e8Q, p, h\u27e9 s := by simp[rew_ex]\n\nlemma nested_rew (p : pnf L) (s\u2080 s\u2081) :\n  (p.rew s\u2080).rew s\u2081 = p.rew (\u03bb x, (s\u2080 x).rew s\u2081) :=\nto_formula_inj.mp (by simp[formula.nested_rew])\n\n@[simp] lemma rew_rank : \u2200 (p : pnf L) (s), (p.rew s).rank = p.rank\n| \u27e8[], p, h\u27e9     s := by simp[rew]\n| \u27e8\ud835\udeb7 :: Q, p, h\u27e9 s := by simp[rew]\n| \u27e8\ud835\udeba :: Q, p, h\u27e9 s := by simp[rew]\n\n@[simp] lemma pow_rank (p : pnf L) (i : \u2115) : (p^i).rank = p.rank :=\nby simp[has_pow.pow]\n\n@[simp] def neg : pnf L \u2192 pnf L\n| \u27e8[], p, h\u27e9 := \u27e8[], \u223cp, by simp[h]\u27e9\n| \u27e8\ud835\udeb7 :: Q, p, h\u27e9 := \u2203.neg \u27e8Q, p, by simp[h]\u27e9\n| \u27e8\ud835\udeba :: Q, p, h\u27e9 := \u2200.neg \u27e8Q, p, by simp[h]\u27e9\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf rank\u27e9]}\n\ninstance : has_negation (pnf L) := \u27e8neg\u27e9\n\n@[simp] def imply : pnf L \u2192 pnf L \u2192 pnf L\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8[], p\u2082, h\u2082\u27e9      := \u27e8[], p\u2081 \u27f6 p\u2082, by simp[h\u2081, h\u2082]\u27e9\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 := \u2200.imply (\u27e8[], p\u2081^1, by simp[h\u2081]\u27e9) \u27e8Q\u2082, p\u2082, h\u2082\u27e9\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 := \u2203.imply \u27e8[], p\u2081^1, by simp[h\u2081]\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8[], p\u2082, h\u2082\u27e9      := \u2203.imply \u27e8Q\u2081, p\u2081, by simp[h\u2081]\u27e9 \u27e8[], p\u2082^1, by simp[h\u2082]\u27e9\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 :=\n    \u2203.(\u2200.imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 (by simp[h\u2082])).rew \u0131-{1}) : pnf L)\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 :=\n    \u2203.(\u2203.imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 (by simp[h\u2082])).rew \u0131-{1}) : pnf L)\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8[], p\u2082, h\u2082\u27e9      := \u2200.imply \u27e8Q\u2081, p\u2081, by simp[h\u2081]\u27e9 \u27e8[], p\u2082^1, by simp[h\u2082]\u27e9\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 :=\n    \u2200.(\u2200.imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 (by simp[h\u2082])).rew \u0131-{1}) : pnf L)\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 :=\n    \u2200.(\u2203.imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 (by simp[h\u2082])).rew \u0131-{1}) : pnf L)\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb x, x.fst.rank + x.snd.rank)\u27e9]}\n\ninstance : has_arrow (pnf L) := \u27e8imply\u27e9\n\nend pnf\n\nnamespace formula\n\n@[simp] def to_pnf : formula L \u2192 pnf L\n| \u22a4         := \u27e8[], \u22a4, by simp\u27e9\n| (app p v) := \u27e8[], app p v, by simp\u27e9\n| ((t : term L) =' u) := \u27e8[], (t : term L) =' u, by simp\u27e9\n| (p \u27f6 q) := (to_pnf p).imply (to_pnf q)\n| (\u223cp) := (to_pnf p).neg\n| (\u2200.(p : formula L)) := \u2200.(to_pnf p)\n\nend formula\n\nlemma equiv_normalize_imply : \u2200 (p q : pnf L) (T : Theory L),\n  (p.imply q).to_formula  \u2248[T] p.to_formula \u27f6 q.to_formula\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8[], p\u2082, h\u2082\u27e9      T := by simp\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp, have ih := equiv_normalize_imply \u27e8[], p\u2081^1, by simp[h\u2081]\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9,\n    calc     \u2200.((pnf.mk ([]) (p\u2081^1) (by simp[h\u2081])).imply (pnf.mk Q\u2082 p\u2082 h\u2082)).to_formula\n        \u2248[(T)] \u2200.((pnf.mk ([]) (p\u2081^1) (by simp[h\u2081])).to_formula \u27f6 (pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_univ_of_equiv (ih _)\n    ... \u2248[T] p\u2081 \u27f6 \u2200.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, refine by simp[classical_logic.equiv] } }\n| \u27e8[], p\u2081, h\u2081\u27e9      \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp, have ih := equiv_normalize_imply \u27e8[], p\u2081^1, by simp[h\u2081]\u27e9 \u27e8Q\u2082, p\u2082, h\u2082\u27e9,\n    calc     \u2203.((pnf.mk ([]) (p\u2081^1) (by simp[h\u2081])).imply (pnf.mk Q\u2082 p\u2082 h\u2082)).to_formula\n        \u2248[T] \u2203.((pnf.mk ([]) (p\u2081^1) (by simp[h\u2081])).to_formula \u27f6 (pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_ex_of_equiv (ih _)\n    ... \u2248[T] p\u2081 \u27f6 \u2203.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, simp[classical_logic.equiv] } }\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8[], p\u2082, h\u2082\u27e9      T := by { simp, have ih := equiv_normalize_imply \u27e8Q\u2081, p\u2081, h\u2081\u27e9 (pnf.mk ([]) p\u2082 h\u2082^1),\n    calc     \u2203.((pnf.mk Q\u2081 p\u2081 h\u2081).imply (pnf.mk ([]) p\u2082 h\u2082^1)).to_formula\n        \u2248[T] \u2203.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (pnf.mk ([]) p\u2082 h\u2082^1).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_ex_of_equiv (ih _)\n    ... \u2248[T] \u2200.(pnf.mk Q\u2081 p\u2081 (by simp[h\u2081])).to_formula \u27f6 p\u2082\n    : by { symmetry, simp[classical_logic.equiv] } }\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp,\n    have ih := equiv_normalize_imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}),\n    calc     \u2203.\u2200.((pnf.mk Q\u2081 p\u2081 h\u2081^1).imply ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1})).to_formula\n        \u2248[T] \u2203.\u2200.((pnf.mk Q\u2081 p\u2081 h\u2081^1).to_formula \u27f6 ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_ex_of_equiv (provable.equiv_univ_of_equiv (ih _))\n    ... \u2248[T] \u2203.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (\u2200.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)^1)\n    : by { show _ \u2248[T] _, symmetry, simp[classical_logic.equiv, formula.fal_pow_discard],\n           refine provable.equiv_ex_of_equiv (by simp) }\n    ... \u2248[T] \u2200.(pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 \u2200.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, simp [classical_logic.equiv] } }\n| \u27e8\ud835\udeb7 :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp, \n    have ih := equiv_normalize_imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}),\n    calc     \u2203.\u2203.((pnf.mk Q\u2081 p\u2081 h\u2081^1).imply ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1})).to_formula\n        \u2248[T] \u2203.\u2203.((pnf.mk Q\u2081 p\u2081 h\u2081^1).to_formula \u27f6 ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_ex_of_equiv (provable.equiv_ex_of_equiv (ih _))\n    ... \u2248[T] \u2203.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (\u2203.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)^1)\n    : by { show _ \u2248[T] _, symmetry, simp[classical_logic.equiv, formula.ex_pow_discard],\n           refine provable.equiv_ex_of_equiv (by simp) }\n    ... \u2248[T] \u2200.(pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 \u2203.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, simp [classical_logic.equiv] } }\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8[], p\u2082, h\u2082\u27e9      T := by { simp, have ih := equiv_normalize_imply \u27e8Q\u2081, p\u2081, h\u2081\u27e9 (pnf.mk ([]) p\u2082 h\u2082^1),\n    calc     \u2200.((pnf.mk Q\u2081 p\u2081 h\u2081).imply (pnf.mk ([]) p\u2082 h\u2082^1)).to_formula\n        \u2248[T] \u2200.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (pnf.mk ([]) p\u2082 h\u2082^1).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_univ_of_equiv (ih _)\n    ... \u2248[T] \u2203.(pnf.mk Q\u2081 p\u2081 (by simp[h\u2081])).to_formula \u27f6 p\u2082\n    : by { symmetry, simp[classical_logic.equiv] } }\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeb7 :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp,\n    have ih := equiv_normalize_imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}),\n    calc     \u2200.\u2200.((pnf.mk Q\u2081 p\u2081 h\u2081^1).imply ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1})).to_formula\n        \u2248[T] \u2200.\u2200.((pnf.mk Q\u2081 p\u2081 h\u2081^1).to_formula \u27f6 ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_univ_of_equiv (provable.equiv_univ_of_equiv (ih _))\n    ... \u2248[T] \u2200.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (\u2200.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)^1)\n    : by { show _ \u2248[T] _, symmetry, simp[classical_logic.equiv, formula.fal_pow_discard],\n           refine provable.equiv_univ_of_equiv (by simp) }\n    ... \u2248[T] \u2203.(pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 \u2200.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, simp [classical_logic.equiv] } }\n| \u27e8\ud835\udeba :: Q\u2081, p\u2081, h\u2081\u27e9 \u27e8\ud835\udeba :: Q\u2082, p\u2082, h\u2082\u27e9 T := by { simp, \n    have ih := equiv_normalize_imply (pnf.mk Q\u2081 p\u2081 h\u2081^1) ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}),\n    calc     \u2200.\u2203.((pnf.mk Q\u2081 p\u2081 h\u2081^1).imply ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1})).to_formula\n        \u2248[T] \u2200.\u2203.((pnf.mk Q\u2081 p\u2081 h\u2081^1).to_formula \u27f6 ((pnf.mk Q\u2082 p\u2082 h\u2082).rew \u0131-{1}).to_formula)\n    : show _ \u2248[T] _, from provable.equiv_univ_of_equiv (provable.equiv_ex_of_equiv (ih _))\n    ... \u2248[T] \u2200.((pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 (\u2203.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula)^1)\n    : by { show _ \u2248[T] _, symmetry, simp[classical_logic.equiv, formula.ex_pow_discard],\n           refine provable.equiv_univ_of_equiv (by simp) }\n    ... \u2248[T] \u2203.(pnf.mk Q\u2081 p\u2081 h\u2081).to_formula \u27f6 \u2203.(pnf.mk Q\u2082 p\u2082 h\u2082).to_formula\n    : by { symmetry, simp [classical_logic.equiv] } }\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb x, x.1.rank + x.2.1.rank)\u27e9]}\n\nlemma equiv_normalize_neg : \u2200 (p : pnf L) (T : Theory L) , T \u22a2 p.neg.to_formula \u27f7 \u223cp.to_formula\n| \u27e8[], p, h\u27e9     T := by simp\n| \u27e8\ud835\udeb7 :: Q, p, h\u27e9 T := by simp;\n    calc \u2203.(pnf.mk Q p (by simp[h])).neg.to_formula \u2248[T] \u2203.\u223c(pnf.mk Q p (by simp[h])).to_formula\n    : show _ \u2248[T] _, from provable.equiv_ex_of_equiv (equiv_normalize_neg \u27e8Q, p, by simp[h]\u27e9 _)\n                                                 ... \u2248[T] \u223c\u2200.(pnf.mk Q p (by simp[h])).to_formula\n    : classical_logic.equiv_neg_of_equiv (provable.equiv_univ_of_equiv (by simp))\n| \u27e8\ud835\udeba :: Q, p, h\u27e9 T := by { simp,\n    calc     \u2200.(pnf.mk Q p (by simp[h])).neg.to_formula \u2248[T] \u2200.\u223c(pnf.mk Q p (by simp[h])).to_formula\n    : show _ \u2248[T] _, from provable.equiv_univ_of_equiv (equiv_normalize_neg \u27e8Q, p, by simp[h]\u27e9 _)\n                                                    ... \u2248[T] \u223c\u2203.(pnf.mk Q p (by simp[h])).to_formula\n    : by { simp[has_exists_quantifier.ex, formula.ex, classical_logic.equiv] } }\n\n@[reducible] def formula.normalize (p : formula L) : formula L := p.to_pnf.to_formula\n\n@[reducible] def formula.open (p : formula L) : formula L := p.to_pnf.2\n\n@[simp] lemma formula.open_is_open (p : formula L) : p.open.is_open := p.to_pnf.is_openform\n\nopen axiomatic_classical_logic'\n\nlemma equiv_normalize : \u2200 (p : formula L) {T : Theory L},  T \u22a2 p \u27f7 p.normalize\n| \u22a4                 T := by simp[formula.normalize]\n| (formula.app p v) T := by simp[formula.normalize]\n| (t =' u)          T := by simp[formula.normalize]\n| (p \u27f6 q)          T :=\n    by { simp[formula.normalize], \n         have : T \u22a2 p \u27f6 q \u27f7 (p.to_pnf.to_formula \u27f6 q.to_pnf.to_formula) :=  (equiv_imply_of_equiv (equiv_normalize p) (equiv_normalize q)),\n         refine equiv_trans this (classical_logic.equiv_symm (equiv_normalize_imply p.to_pnf q.to_pnf T)) }\n| (\u223cp)              T := by { simp[formula.normalize],\n    have : T \u22a2 \u223cp \u27f7 \u223cp.to_pnf.to_formula, from equiv_neg_of_equiv (equiv_normalize p),\n    exact equiv_trans this (equiv_symm (equiv_normalize_neg p.to_pnf T)) }\n| (\u2200.p)           T := by { simp[formula.normalize], refine provable.equiv_univ_of_equiv (equiv_normalize p) }\n\ndef formula.rank (p : formula L) : \u2115 := p.to_pnf.rank\n\nend fol", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/pnf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3643827869818058}}
{"text": "open Nat\n\ntheorem mul_comm (m n : Nat) : m * n = n * m := by\n  induction n with\n  | zero => simp\n  | succ n ih =>\n    have foo : m * n + m = m * n + (succ zero) * m := _\n    rfl\n\ntheorem test (o : x \u2228 y) : x := by\n  cases o with\n  | inl h => exact h\n  | inr h => exact _\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/449.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3643827733401372}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.model_category\nimport category_theory.abelian.basic\nimport category_theory.preadditive.projective\nimport algebra.homology.homological_complex\nimport algebra.homology.quasi_iso\nimport for_mathlib.category_theory.limits.kernel_functor\nimport for_mathlib.algebra.homology.twist_cocycle\nimport tactic.linarith\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.category\nopen algebraic_topology cochain_complex.hom_complex\n\nopen_locale zero_object\n\nvariables (C : Type*) [category C] [abelian C]\n\nnamespace cochain_complex\n\n@[derive category]\ndef Cminus := full_subcategory (\u03bb (K : cochain_complex C \u2124), K.is_bounded_above)\n\nnamespace Cminus\n\nvariable {C}\n\n@[simps]\ndef mk (K : cochain_complex C \u2124) (hK : K.is_bounded_above) : Cminus C := \u27e8K, hK\u27e9\n\ndef homology_functor (i : \u2124) : Cminus C \u2964 C := induced_functor _ \u22d9 homology_functor _ _ i\n\ndef eval (i : \u2124) : Cminus C \u2964 C := induced_functor _ \u22d9 homological_complex.eval _ _ i\n\nnamespace projective_structure\n\nvariable (C)\n\ndef arrow_classes : category_with_fib_cof_weq (Cminus C) :=\n{ weq := \u03bb X Y w, quasi_iso w,\n  fib := \u03bb X Y w, \u2200 n, epi (w.f n),\n  cof := \u03bb X Y w, \u2200 n, mono (w.f n) \u2227 (projective (cokernel (w.f n))), }\n\nvariable {C}\n\ndef CM2 : (arrow_classes C).CM2 :=\n{ of_comp := \u03bb X Y Z f g (hf : quasi_iso f) (hg : quasi_iso g), begin\n    haveI := hf,\n    haveI := hg,\n    exact quasi_iso_comp f g,\n  end,\n  of_comp_left := \u03bb X Y Z f g (hf : quasi_iso f) (hfg : quasi_iso (f \u226b g)), begin\n    haveI := hf,\n    haveI := hfg,\n    convert quasi_iso_of_comp_left f g,\n  end,\n  of_comp_right := \u03bb X Y Z f g (hg : quasi_iso g) (hfg : quasi_iso (f \u226b g)), begin\n    haveI := hg,\n    haveI := hfg,\n    convert quasi_iso_of_comp_right f g,\n  end, }\n\ndef CM3 : (arrow_classes C).CM3 :=\n{ weq := \u03bb X\u2081 X\u2082 Y\u2081 Y\u2082 f g hfg hg, \u27e8\u03bb n, begin\n    have hfg' := is_retract.imp_of_functor (homology_functor n).map_arrow\n      (arrow.mk f) (arrow.mk g) hfg,\n    apply morphism_property.is_stable_by_retract.for_isomorphisms _ _ hfg',\n    apply hg.1,\n  end\u27e9,\n  cof := \u03bb X\u2081 X\u2082 Y\u2081 Y\u2082 f g hfg hg n, begin\n    split,\n    { exact morphism_property.is_stable_by_retract.for_monomorphisms _ _\n      (is_retract.imp_of_functor (eval n).map_arrow _ _ hfg) (hg n).1, },\n    { exact projective.of_retract (is_retract.imp_of_functor\n      ((eval n).map_arrow \u22d9 limits.cokernel_functor C) _ _ hfg) (hg n).2, },\n  end,\n  fib := \u03bb X\u2081 X\u2082 Y\u2081 Y\u2082 f g hfg hg n, morphism_property.is_stable_by_retract.for_epimorphisms _ _\n      (is_retract.imp_of_functor (eval n).map_arrow _ _ hfg) (hg n), }\n\ndef CM4 : (arrow_classes C).CM4 := sorry\n\nvariable [enough_projectives C]\n\nnamespace CM5a\n\ndef P (L : Cminus C) (q : \u2124) : C :=\nbegin\n  by_cases is_zero (L.1.X q),\n  { exact 0, },\n  { exact projective.over (L.1.X q), },\nend\n\ninstance (L : Cminus C) (q : \u2124) : projective (P L q) :=\nbegin\n  dsimp [P],\n  split_ifs,\n  { apply projective.zero_projective, },\n  { apply projective.projective_over, },\nend\n\nlemma P_eq (L : Cminus C) (q : \u2124) (hq : \u00ac(is_zero (L.1.X q))) :\n  P L q = projective.over (L.1.X q) :=\nbegin\n  dsimp [P],\n  split_ifs,\n  { exfalso, exact hq h, },\n  { refl, },\nend\n\nlemma P_eq_zero (L : Cminus C) (q : \u2124) (hq : is_zero (L.1.X q)) :\n  P L q = 0 :=\nbegin\n  dsimp [P],\n  split_ifs,\n  { refl, },\n  { exfalso, exact h hq, },\nend\n\nlemma P_is_initial (L : Cminus C) (q : \u2124) (hq : is_zero (L.1.X q)) :\n  is_initial (P L q) :=\nbegin\n  rw P_eq_zero L q hq,\n  apply is_zero.is_initial,\n  apply is_zero_zero,\nend\n\ndef is_zero.unique_up_to_iso {X Y : C} (hX : is_zero X) (hY : is_zero Y) : X \u2245 Y :=\n{ hom := 0,\n  inv := 0,\n  hom_inv_id' := by { rw is_zero.iff_id_eq_zero at hX, rw [hX, comp_zero], },\n  inv_hom_id' := by { rw is_zero.iff_id_eq_zero at hY, rw [hY, comp_zero], }, }\n\ndef P_\u03c0 (L : Cminus C) (q : \u2124) : P L q \u27f6 L.1.X q :=\nbegin\n  by_cases is_zero (L.1.X q),\n  { have e : 0 \u2245 L.1.X q := is_zero.unique_up_to_iso\n      (is_zero_zero C) h, swap,\n    exact eq_to_hom (P_eq_zero L q h) \u226b e.hom, },\n  { exact eq_to_hom (P_eq L q h) \u226b projective.\u03c0 (L.1.X q), },\nend\n\nlemma P_\u03c0_eq_to_hom (L : Cminus C) (q\u2081 q\u2082 : \u2124) (hq : q\u2081 = q\u2082) :\n  P_\u03c0 L q\u2081 = eq_to_hom (by rw hq) \u226b P_\u03c0 L q\u2082 \u226b eq_to_hom (by rw hq) :=\nby { subst hq, simp only [eq_to_hom_refl, comp_id, id_comp], }\n\n@[simps]\ndef KP (L : Cminus C) : Cminus C := Cminus.mk\n{ X := \u03bb q, P L (q-1),\n  d := \u03bb i j, 0,\n  shape' := \u03bb i j hij, rfl,\n  d_comp_d' := \u03bb i j k hij hjk, comp_zero, }\nbegin\n  cases L.2 with r hr,\n  use r+1,\n  intros i hi,\n  dsimp,\n  rw P_eq_zero L,\n  { apply is_zero_zero, },\n  { apply hr,\n    linarith, },\nend\n\ninstance (L : Cminus C) (q : \u2124) : epi (P_\u03c0 L q) :=\nby { dsimp only [P_\u03c0], split_ifs; apply epi_comp, }\n\ndef twistP (L : Cminus C) : Cminus C :=\n\u27e8twist (cocycle.of_hom (\ud835\udfd9 (KP L).1)), twist.is_bounded_above _ (KP L).2 (KP L).2\u27e9\n\ndef \u03c0 (L : Cminus C) : twistP L \u27f6 L :=\nbegin\n  refine twist.desc (cocycle.of_hom (\ud835\udfd9 (KP L).1)) (cochain.mk _) _ (neg_add_self 1) _ ,\n  { exact (\u03bb p q hpq, P_\u03c0 L _ \u226b eq_to_hom (by {congr' 1, linarith})), },\n  { exact\n    { f := \u03bb i, P_\u03c0 L (i-1) \u226b L.1.d (i-1) i,\n      comm' := \u03bb i j hij, begin\n        change i+1=j at hij,\n        dsimp [KP],\n        simp only [assoc, homological_complex.d_comp_d, comp_zero, zero_comp],\n      end, }, },\n  { ext,\n    dsimp [KP],\n    simp only [\u03b4_v (-1) 0 rfl _ p p (add_zero p).symm (p-1) (p+1) rfl rfl,\n      add_zero, zero_comp, cochain.mk_v, eq_to_hom_refl, comp_id,\n      smul_zero, cochain.id_comp, cochain.of_hom_v], },\nend\n\nexample : 2+2=4 := rfl\n\ninstance (L : Cminus C) (q : \u2124) : epi ((\u03c0 L).f q) :=\nbegin\n  haveI : epi (biprod.inl \u226b (\u03c0 L).f q),\n  { have eq : biprod.inl \u226b (\u03c0 L).f q = eq_to_hom (by { dsimp, congr, linarith }) \u226b P_\u03c0 L q,\n    { dsimp [\u03c0, twist.desc_cochain, twist.fst, twist.snd, cochain.mk, cochain.v,\n        cochain.of_hom, cochain.of_homs, cochain.comp],\n      simp only [id_comp, assoc, add_zero, preadditive.comp_add, biprod.inl_fst_assoc, biprod.inl_snd_assoc, zero_comp,\n        P_\u03c0_eq_to_hom L (q+(0 - -1)-1) q (by linarith), eq_to_hom_trans, eq_to_hom_refl,\n        eq_to_hom_trans_assoc, comp_id], },\n    rw eq,\n    apply epi_comp, },\n  exact epi_of_epi biprod.inl ((\u03c0 L).f q),\nend\n\ninstance : preadditive (Cminus C) := sorry\ninstance : has_binary_biproducts (Cminus C) := sorry\n\nend CM5a\n\nlemma CM5a : (arrow_classes C).CM5a := \u03bb X Z f,\nbegin\n  let Y := CM5a.twistP Z,\n  let i : X \u27f6 X \u229e Y := biprod.inl,\n  let p : X \u229e Y \u27f6 Z := biprod.desc f (CM5a.\u03c0 Z),\n  let j : Y \u27f6 X \u229e Y := biprod.inr,\n  have hip : i \u226b p = f := biprod.inl_desc _ _,\n  refine \u27e8X \u229e Y, i, _, p, _, hip\u27e9,\n  { sorry, },\n  { intro,\n    have hjp : j \u226b p = CM5a.\u03c0 Z := biprod.inr_desc _ _,\n    have hjp' : j.f n \u226b p.f n = (CM5a.\u03c0 Z).f n,\n    { rw [\u2190 hjp, \u2190 homological_complex.comp_f],\n      refl, },\n    haveI : epi (j.f n \u226b p.f n),\n    { rw hjp',\n      apply_instance, },\n    exact epi_of_epi (j.f n) (p.f n), },\nend\n\ndef CM5 : (arrow_classes C).CM5 := \u27e8CM5a, sorry\u27e9\n\nvariable (C)\n\n@[simps]\ndef projective_structure : model_category (Cminus C) :=\n{ to_category_with_fib_cof_weq := arrow_classes C,\n  CM1axiom := sorry,\n  CM2axiom := CM2,\n  CM3axiom := CM3,\n  CM4axiom := CM4,\n  CM5axiom := CM5, }\n\ninstance : model_category (Cminus C) := projective_structure C\n\nend projective_structure\n\nend Cminus\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cochain_complex.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7401743735019594, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3643050450022372}}
{"text": "def f (n : Nat) := n + 1\nexample (k : Nat) : let x := 10; f x = k := by\n  have : f 10 = 11 := rfl\n  intro x\n  rw [this]\n  trace_state\n  revert x\n  trace_state\n  sorry\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/introLetBug.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3642787493341494}}
{"text": "import for_mathlib.projective_space.incidence\nimport for_mathlib.projective_space.affine_embedding\n\nopen projectivization\n\nnotation `mk\u2080` x := (mk _ x (by simp))\nnotation `mk[` F `]` x := (mk F (x : F \u00d7 F \u00d7 F) (by simp))\n\nvariables {K F : Type*} [field K] [field F]\n\nsection configurations\n\nmeta def helper\u2083 := `[\n  rw [independent_mk_iff\u2083, fintype.linear_independent_iff],\n  intros g hg,\n  simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n    mul_one, mul_zero, matrix.cons_val_succ, fin.succ_zero_eq_one, fintype.univ_of_subsingleton, \n    fin.mk_zero, finset.sum_singleton, fin.succ_one_eq_two, prod.mk_add_mk, add_zero, \n    zero_add, prod.mk_eq_zero] at hg,\n  rcases hg with \u27e8h1,h2,h3\u27e9 ]\n\nmeta def helper\u2082 := `[\n  rw [independent_mk_iff\u2082, fintype.linear_independent_iff],\n  intros g hg,\n  simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n    mul_one, mul_zero, matrix.cons_val_succ, fin.succ_zero_eq_one, fintype.univ_of_subsingleton, \n    fin.mk_zero, finset.sum_singleton, fin.succ_one_eq_two, prod.mk_add_mk, add_zero, \n    zero_add, prod.mk_eq_zero] at hg,\n  rcases hg with \u27e8h1,h2,h3\u27e9 ]\n\nvariables (F K)\ndef one_add_x_add_y_config : projectivization.configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (1,0,0)) -- (\u03c6 1) \n  (mk[F] (0,1,0)) -- (\u03c6 (1 + x)) \n  (mk[F] (0,0,1))-- (\u03c6 (1 + y)) \n  (mk[F] (1,1,0)) -- (\u03c6 x) \n  (mk[F] (1,0,1)) -- (\u03c6 y) \n  := \n{ Iabc := begin\n    helper\u2083,\n    intros i, fin_cases i; assumption\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    refine \u27e8![1,1,-1], by simp [fin.sum_univ_succ], 0, by norm_num\u27e9,\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    refine \u27e8![1,1,-1], by simp [fin.sum_univ_succ], 0, by norm_num\u27e9,\n  end,\n  Iau := begin\n    rw [\u2190 ne_iff_independent], \n    apply (affine_embedding_injective _ _).ne,\n    norm_num,\n  end,\n  Ibu := begin\n    rw [\u2190 ne_iff_independent],\n    apply ne.symm,\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num\n  end,\n  Iav := begin\n    rw [\u2190 ne_iff_independent],\n    apply (affine_embedding_injective _ _).ne,\n    norm_num,\n  end,\n  Icv := begin\n    rw [\u2190 ne_iff_independent], \n    apply ne.symm,\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num\n  end }\n\ndef two_add_x_add_y_config : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (1,0,1)) -- (\u03c6 y) \n  (mk[F] (1,1,1)) -- (\u03c6 (1 + x + y)) \n  (mk[F] (0,0,1)) -- (\u03c6 (1 + y)) \n  (mk[F] (0,1,0)) -- (\u03c6 (1 + x)) \n  (mk[F] (1,0,0)) -- (\u03c6 1) \n  :=\n{ Iabc := begin\n    helper\u2083,\n    intros i, fin_cases i,\n    { simpa [h2] using h1 },\n    { exact h2 },\n    { rw [h2, add_zero] at h1, simpa [h2,h1] using h3 },\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    refine \u27e8![1,-1,1], by simp [fin.sum_univ_succ], 0, by norm_num\u27e9,\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    refine \u27e8![-1,1,1], by simp [fin.sum_univ_succ], 0, by norm_num\u27e9,\n  end,\n  Iau := begin\n    rw [\u2190 ne_iff_independent],\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num\n  end,\n  Ibu := begin\n    rw [\u2190 ne_iff_independent],\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num\n  end,\n  Iav := begin\n    rw [\u2190 ne_iff_independent],\n    apply (affine_embedding_injective _ _).ne,\n    norm_num\n  end,\n  Icv := begin\n    rw [\u2190 ne_iff_independent],\n    apply ne.symm,\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num\n  end }\n\ndef frac_config (n : \u2115) : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (n-1,1,0))\n  (mk[F] (1,0,0))\n  (mk[F] (1,0,1))\n  (mk[F] (n,1,0))\n  (mk[F] (n,1,1)) :=\n{ Iabc := begin\n    helper\u2083,\n    intros i, fin_cases i,\n    { exact h2 },\n    { simpa [h2,h3] using h1 },\n    { exact h3 },\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, \n        algebra.id.smul_eq_mul, one_mul, mul_one, mul_zero,\n        matrix.cons_val_succ, fintype.univ_of_subsingleton, fin.mk_zero, \n        matrix.cons_val_fin_one, neg_smul, prod.neg_mk, finset.sum_const, finset.card_singleton, \n        nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, zero_add, add_zero, add_right_neg, \n        prod.mk_eq_zero, eq_self_iff_true, and_true], \n      ring },\n    use 1, norm_num,\n  end, \n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_one, mul_zero, matrix.cons_val_succ, fintype.univ_of_subsingleton, \n        fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true], \n      ring },\n    use 1, norm_num,\n  end,\n  Iau := begin\n    helper\u2082,\n    have : g 0 * ((n : F) - 1) + g 1 * n = n * (g 0 + g 1) - g 0, by ring, \n    rw [this,h2, mul_zero, zero_sub, neg_eq_zero] at h1,\n    rw [h1, zero_add] at h2,\n    intros i, fin_cases i,\n    exacts [h1, h2]\n  end,\n  Ibu := begin\n    helper\u2082,\n    intros i, fin_cases i,\n    { simpa [h2] using h1 },\n    { exact h2 }\n  end,\n  Iav := begin\n    helper\u2082,\n    intros i, fin_cases i,\n    { simpa [h3] using h2 },\n    { exact h3 }\n  end,\n  Icv := begin\n    helper\u2082,\n    intros i, fin_cases i,\n    { simpa [h2] using h3 },\n    { exact h2 }\n  end }\n\ndef frac_config' (n : \u2115) : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (n,1,1))\n  (mk[F] (0,0,1))\n  (mk[F] (1,0,0))\n  (mk[F] (n,1,0))\n  (mk[F] (n+1,1,1)) := \n{ Iabc := begin\n    helper\u2083,\n    simp only [h2, zero_mul, zero_add] at h1 h3,\n    intros i, fin_cases i, assumption',\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Iau := begin\n    helper\u2082,\n    simp only [h3, zero_add] at h2,\n    intros i, fin_cases i, assumption',\n  end,\n  Ibu := begin\n    helper\u2082,\n    intros i, fin_cases i, assumption',\n  end,\n  Iav := begin\n    helper\u2082,\n    rw (show g 0 * \u2191n + g 1 * (\u2191n + 1) = (g 0 + g 1) * n + g 1, by ring) at h1,\n    simp only [h2, zero_mul, zero_add] at h1,\n    simp only [h1, add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Icv := begin\n    helper\u2082,\n    simp only [h2, zero_mul, add_zero] at h1,\n    intro i, fin_cases i, assumption',\n  end }\n\ndef frac_config_horizontal (n m : \u2115) (hn : (n : F) \u2260 0) : configuration F (F \u00d7 F \u00d7 F)\n  (mk F (n,m-1,0) (\u03bb c, hn (congr_arg prod.fst c)))\n  (mk[F] (0,1,0)) \n  (mk[F] (0,1,1)) \n  (mk F (n,m,0) (\u03bb c, hn (congr_arg prod.fst c))) \n  (mk F (n,m,1) (\u03bb c, hn (congr_arg prod.fst c))) := \n{ Iabc := begin\n    helper\u2083,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1, \n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1,h3,zero_mul,zero_add,add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_zero, matrix.cons_val_succ, mul_one, fintype.univ_of_subsingleton, \n        fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and],\n      ring },\n    use 0, norm_num\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_zero, matrix.cons_val_succ, mul_one, fintype.univ_of_subsingleton, \n        fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and],\n      ring },\n    use 1, norm_num,\n  end,\n  Iau := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [add_mul, mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    have : g 0 * (\u2191m - 1) + g 1 * \u2191m = (g 0 + g 1) * (m : F) - g 0, by ring, \n    rw [this, h1, zero_mul, zero_sub, neg_eq_zero] at h2,\n    rw [h2, zero_add] at h1,\n    intros i, fin_cases i, assumption'\n  end,\n  Ibu := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, zero_mul, add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Iav := begin\n    helper\u2082,\n    simp only [h3, zero_mul, add_zero] at h1,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [zero_mul, add_zero, mul_assoc, mul_inv_cancel hn, mul_one] at h1,\n    intros i, fin_cases i, assumption'\n  end,\n  Icv := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, add_zero] at h3,\n    intros i, fin_cases i, assumption',\n  end }\n\ndef frac_config_horizontal' (n m : \u2115) (hn : (n : F) \u2260 0) : configuration F (F \u00d7 F \u00d7 F)\n  (mk F (n,m,1) (\u03bb c, hn (congr_arg prod.fst c)))\n  (mk[F] (0,1,0))\n  (mk[F] (0,0,1))\n  (mk[F] (n,m+1,1))\n  (mk F (n,m,0) (\u03bb c, hn (congr_arg prod.fst c)))\n  :=\n{ Iabc := begin\n    helper\u2083,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, zero_add, zero_mul] at h2 h3,\n    intros i, fin_cases i, assumption'\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num },\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num },\n  end,\n  Iau := begin\n    rw \u2190 ne_iff_independent,\n    apply (affine_embedding_of_nonzero_injective _ _ _ hn).ne,\n    norm_num,\n  end,\n  Ibu := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply ne_of_leading_nonzero_of_leading_zero,\n    norm_num, assumption\n  end,\n  Iav := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply (affine_embedding_of_nonzero_injective _ _ _ hn).ne,\n    norm_num,\n  end,\n  Icv := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply ne_of_leading_nonzero_of_leading_zero,\n    norm_num, assumption\n  end }\n\n/-\ndef neg_frac_config (n : \u2115) : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (n-1,-1,0))\n  (mk[F] (1,0,0))\n  (mk[F] (1,0,1))\n  (mk[F] (n,-1,0))\n  (mk[F] (n,-1,1)) :=\n{ Iabc := begin\n    helper\u2083,\n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    intros i, fin_cases i,\n    { exact h2 },\n    { simpa [h2,h3] using h1 },\n    { exact h3 },\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, \n        algebra.id.smul_eq_mul, one_mul, mul_one, mul_zero,\n        matrix.cons_val_succ, fintype.univ_of_subsingleton, fin.mk_eq_subtype_mk, fin.mk_zero, \n        matrix.cons_val_fin_one, neg_smul, prod.neg_mk, finset.sum_const, finset.card_singleton, \n        nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, zero_add, add_zero, add_right_neg, \n        prod.mk_eq_zero, eq_self_iff_true, and_true], \n      ring },\n    use 1, norm_num,\n  end, \n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_one, mul_zero, matrix.cons_val_succ, fintype.univ_of_subsingleton, \n        fin.mk_eq_subtype_mk, fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true], \n      ring },\n    use 1, norm_num,\n  end,\n  Iau := begin\n    helper\u2082,\n    simp only [mul_neg, mul_one, neg_eq_zero, \u2190 neg_add] at h2,\n    have : g 0 * ((n : F) - 1) + g 1 * n = n * (g 0 + g 1) - g 0, by ring, \n    rw [this,h2, mul_zero, zero_sub, neg_eq_zero] at h1,\n    rw [h1, zero_add] at h2,\n    intros i, fin_cases i,\n    exacts [h1, h2]\n  end,\n  Ibu := begin\n    helper\u2082,\n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    intros i, fin_cases i,\n    { simpa [h2] using h1 },\n    { exact h2 }\n  end,\n  Iav := begin\n    helper\u2082,\n    intros i, fin_cases i,\n    { simpa [h3] using h2 },\n    { exact h3 }\n  end,\n  Icv := begin\n    helper\u2082,\n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    intros i, fin_cases i,\n    { simpa [h2] using h3 },\n    { exact h2 }\n  end }\n\ndef neg_frac_config' (n : \u2115) : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (n,-1,1))\n  (mk[F] (0,0,1))\n  (mk[F] (1,0,0))\n  (mk[F] (n,-1,0))\n  (mk[F] (n+1,-1,1)) := \n{ Iabc := begin\n    helper\u2083,\n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    simp only [h2, zero_mul, zero_add] at h1 h3,\n    intros i, fin_cases i, assumption',\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Iau := begin\n    helper\u2082,\n    rw [\u2190 add_mul, mul_neg, mul_one, neg_eq_zero] at h2,\n    simp only [h3, zero_add] at h2,\n    intros i, fin_cases i, assumption',\n  end,\n  Ibu := begin\n    helper\u2082,\n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    intros i, fin_cases i, assumption',\n  end,\n  Iav := begin\n    helper\u2082,\n    rw [\u2190 add_mul, mul_neg, mul_one, neg_eq_zero] at h2,\n    rw (show g 0 * \u2191n + g 1 * (\u2191n + 1) = (g 0 + g 1) * n + g 1, by ring) at h1,\n    simp only [h2, zero_mul, zero_add] at h1,\n    simp only [h1, add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Icv := begin\n    helper\u2082, \n    simp only [mul_neg, mul_one, neg_eq_zero] at h2,\n    simp only [h2, zero_mul, add_zero] at h1,\n    intro i, fin_cases i, assumption',\n  end }\n\ndef neg_frac_config_horizontal (n m : \u2115) (hn : (n : F) \u2260 0) : configuration F (F \u00d7 F \u00d7 F)\n  (mk F (n,1-m,0) (\u03bb c, hn (congr_arg prod.fst c)))\n  (mk[F] (0,1,0)) \n  (mk[F] (0,-1,1)) \n  (mk F (n,-m,0) (\u03bb c, hn (congr_arg prod.fst c))) \n  (mk F (n,-m,1) (\u03bb c, hn (congr_arg prod.fst c))) := \n{ Iabc := begin\n    helper\u2083,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1, \n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1,h3,zero_mul,zero_add,add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_zero, matrix.cons_val_succ, mul_one, fintype.univ_of_subsingleton, \n        fin.mk_eq_subtype_mk, fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and],\n      refine \u27e8_,_,_\u27e9; ring },\n    use 0, norm_num\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, \n        one_mul, mul_zero, matrix.cons_val_succ, mul_one, fintype.univ_of_subsingleton, \n        fin.mk_eq_subtype_mk, fin.mk_zero, matrix.cons_val_fin_one, neg_smul, prod.neg_mk, \n        finset.sum_const, finset.card_singleton, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and],\n      ring },\n    use 1, norm_num,\n  end,\n  Iau := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [add_mul, mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    have : g 0 * (1 - \u2191m) + g 1 * (-\u2191m) = (g 0 + g 1) * -(m : F) + g 0, by ring, \n    rw [this, h1, zero_mul, zero_add] at h2,\n    rw [h2, zero_add] at h1,\n    intros i, fin_cases i, assumption'\n  end,\n  Ibu := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, zero_mul, add_zero] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Iav := begin\n    helper\u2082,\n    simp only [h3, zero_mul, add_zero] at h1,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [zero_mul, add_zero, mul_assoc, mul_inv_cancel hn, mul_one] at h1,\n    intros i, fin_cases i, assumption'\n  end,\n  Icv := begin\n    helper\u2082,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, add_zero] at h3,\n    intros i, fin_cases i, assumption',\n  end }\n\ndef neg_frac_config_horizontal' (n m : \u2115) (hn : (n : F) \u2260 0) : configuration F (F \u00d7 F \u00d7 F)\n  (mk F (n,-m,1) (\u03bb c, hn (congr_arg prod.fst c)))\n  (mk[F] (0,1,0))\n  (mk[F] (0,0,1))\n  (mk[F] (n,-m-1,1))\n  (mk F (n,-m,0) (\u03bb c, hn (congr_arg prod.fst c)))\n  :=\n{ Iabc := begin\n    helper\u2083,\n    apply_fun (\u03bb e, e * (n : F)\u207b\u00b9) at h1,\n    simp only [mul_assoc, mul_inv_cancel hn, mul_one, zero_mul] at h1,\n    simp only [h1, zero_add, zero_mul] at h2 h3,\n    intros i, fin_cases i, assumption'\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num },\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num },\n  end,\n  Iau := begin\n    rw \u2190 ne_iff_independent,\n    apply (affine_embedding_of_nonzero_injective _ _ _ hn).ne,\n    simp only [ne.def, prod.mk.inj_iff, eq_self_iff_true, and_true],\n    rw (show (-(m : F) - 1) = -(m+1), by ring),\n    norm_num,\n  end,\n  Ibu := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply ne_of_leading_nonzero_of_leading_zero,\n    norm_num, assumption\n  end,\n  Iav := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply (affine_embedding_of_nonzero_injective _ _ _ hn).ne,\n    norm_num,\n  end,\n  Icv := begin\n    rw \u2190 ne_iff_independent, apply ne.symm,\n    apply ne_of_leading_nonzero_of_leading_zero,\n    norm_num, assumption\n  end }\n-/\n\ndef neg_config : configuration F (F \u00d7 F \u00d7 F)\n  (mk[F] (1,1,1))\n  (mk[F] (0,1,0)) \n  (mk[F] (0,1,1)) \n  (mk[F] (1,0,1)) \n  (mk[F] (1,0,0)) :=\n{ Iabc := begin\n    helper\u2083,\n    simp only [h1, zero_add] at h3,\n    simp only [h1, h3, add_zero, zero_add] at h2,\n    intros i, fin_cases i, assumption'\n  end,\n  Dabu := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Dacv := begin\n    rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num }\n  end,\n  Iau := begin\n    rw \u2190 ne_iff_independent,\n    apply (affine_embedding_injective _ _).ne,\n    norm_num,\n  end,\n  Ibu := begin\n    rw \u2190 ne_iff_independent,\n    apply ne.symm,\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num,\n  end,\n  Iav := begin\n    rw \u2190 ne_iff_independent,\n    apply (affine_embedding_injective _ _).ne,\n    norm_num,\n  end,\n  Icv := begin\n    rw \u2190 ne_iff_independent,\n    apply ne.symm,\n    apply ne_of_leading_one_of_leading_zero,\n    norm_num,\n  end }\n\nvariables {F K}\nend configurations\n\nstructure setup (\u03c6 : K \u2192 \u2119 F (F \u00d7 F \u00d7 F)) (x y : K) :=\n(map_zero : \u03c6 0 = mk\u2080 (1,0,0))\n(map_one : \u03c6 1 = mk\u2080 (1,0,0))\n(map_x : \u03c6 x = mk\u2080 (1,1,0))\n(map_y : \u03c6 y = mk\u2080 (1,0,1))\n(map_one_add_x : \u03c6 (1+x) = mk\u2080 (0,1,0))\n(map_one_add_y : \u03c6 (1+y) = mk\u2080 (0,0,1))\n(map_neg : \u2200 u : K, \u03c6 (-u) = \u03c6 u)\n(dependent_pos : \u2200 u v : K, dependent ![\u03c6 u, \u03c6 v, \u03c6 (u + v)])\n(dependent_neg : \u2200 u v : K, dependent ![\u03c6 u, \u03c6 v, \u03c6 (u - v)]) .\n\nnamespace setup\n\nvariables {\u03c6 : K \u2192 \u2119 F (F \u00d7 F \u00d7 F)} {x y : K} (C : setup \u03c6 x y)\ninclude C\n\n\nlemma map_one_add_x_add_y : \u03c6 (1 + x + y) = mk\u2080 (1,1,1) :=\nbegin\n  apply eq_of_config _ _ _ _ _ _ _ (one_add_x_add_y_config F),\n  all_goals { simp only [\u2190 C.map_one, \u2190 C.map_x, \u2190 C.map_y, \u2190 C.map_one_add_x, \u2190 C.map_one_add_y] },\n  rw (show (1 + x + y) = x + (1 + y), by ring),\n  apply C.dependent_pos,\n  rw (show (1 + x + y) = y + (1 + x), by ring),\n  apply C.dependent_pos,\n  { rw [C.map_x, C.map_one_add_y, dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 1, simp } },\n  { rw [C.map_y, C.map_one_add_x, dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 1, simp } },\nend\n\n\n\nlemma map_two_add_x_add_y : \u03c6 (2 + x + y) = mk\u2080 (0,1,1) :=\nbegin\n  apply eq_of_config _ _ _ _ _ _ _ (two_add_x_add_y_config F),\n  all_goals \n  { simp only [\u2190 C.map_y, \u2190 C.map_one_add_x_add_y, \u2190 C.map_one_add_y, \n      \u2190 C.map_one_add_x, \u2190 C.map_one] },\n  { rw (show (2 + x + y) = (1 + x) + (1 + y), by ring), \n    apply C.dependent_pos },\n  { rw (show (2 + x + y) = 1 + (1 + x + y), by ring), \n    apply C.dependent_pos },\n  { rw [C.map_one_add_x, C.map_one_add_y, dependent_mk_iff\u2083, fintype.not_linear_independent_iff], \n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, simp } },\n  { rw [C.map_one, C.map_one_add_x_add_y, dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, simp } },\nend\n\nlemma map_two_add_x : \u03c6 (2 + x) = mk[F] (1,-1,0) := \nbegin\n  apply eq_of_config _ _ _ _ _ _ _ (neg_config F),\n  all_goals { simp only [\u2190 C.map_two_add_x_add_y, \u2190 C.map_y,  \n    \u2190 C.map_one, \u2190 C.map_one_add_x] },\n  { nth_rewrite 1 (show (2 + x) = (-y) + (2 + x + y), by ring), \n    rw \u2190 C.map_neg y,\n    apply C.dependent_pos },\n  { rw (show (2 + x) = 1 + (1 + x), by ring), \n    apply C.dependent_pos },\n  { simp only [C.map_y, C.map_two_add_x_add_y, dependent_mk_iff\u2083, \n      fintype.not_linear_independent_iff], \n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num } },\n  { simp only [C.map_one, C.map_one_add_x, dependent_mk_iff\u2083, \n      fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num } },\nend\n\nlemma map_to_frac (n : \u2115) : \n  (\u03c6 ((2 - n) + x + y) = mk[F] (n,1,1)) \u2227 \n  (\u03c6 ((1 - n) + x) = mk[F] (n,1,0)) :=\nbegin\n  induction n with n hn,\n  { simp only [nat.cast_zero, sub_zero],\n    refine \u27e8C.map_two_add_x_add_y, C.map_one_add_x\u27e9 },\n  cases hn with h1 h2,\n  have key : \u03c6 ((2 - (\u2191n+1)) + x + y) = mk[F] (\u2191n+1,1,1), \n  { apply eq_of_config _ _ _ _ _ _ _ (frac_config F n),\n    { rw [\u2190 h2, \u2190 C.map_y],\n      rw (show 2 - (\u2191n + 1) + x + y = (1 - \u2191n + x) + y, by ring),\n      apply C.dependent_pos },\n    { rw [\u2190 h1, \u2190 C.map_one],  \n      convert C.dependent_neg _ _, ring },\n    { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],  \n      use ![1,1,-1], split,\n      { simp [fin.sum_univ_succ] },\n      { use 0, norm_num } },\n    { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],  \n      use ![1,1,-1], split,\n      { simp [fin.sum_univ_succ] },\n      { use 0, norm_num } } },\n  refine \u27e8_, _\u27e9,\n  { exact_mod_cast key },\n  apply eq_of_config _ _ _ _ _ _ _ (frac_config' F n),\n  { rw [\u2190 h2, \u2190 C.map_one],  \n    convert C.dependent_neg _ _, \n    push_cast, ring },\n  { rw [\u2190 key, \u2190 C.map_one_add_y],\n    convert C.dependent_neg _ _,\n    push_cast, ring },\n  { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp [fin.sum_univ_succ] },\n    { use 0, norm_num } },\n  { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp only [fin.sum_univ_succ, nat.cast_succ, matrix.cons_val_zero, prod.smul_mk, \n        algebra.id.smul_eq_mul, one_mul, mul_one, matrix.cons_val_succ, neg_smul, \n        mul_zero, prod.neg_mk, neg_zero, fintype.univ_of_subsingleton,\n        fin.mk_zero, matrix.cons_val_fin_one, neg_add_rev, finset.sum_const, \n        finset.card_singleton, smul_add, nsmul_eq_mul, nat.cast_one, prod.mk_add_mk, \n        zero_add, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true], \n      ring },\n    { use 0, norm_num } },\nend\n\nlemma map_to_frac_horizontal (n m : \u2115) (hn : (n : F) \u2260 0) (hm : m \u2260 0) :\n  (\u03c6 ((1 + m - n) + m * x + y) = mk[F] (n,m,1)) \u2227 \n  (\u03c6 ((m-n) + m * x) = projectivization.mk F (n,m,0) (\u03bb c, hn (congr_arg prod.fst c))) := \nbegin\n  cases m, { exfalso, exact hm rfl },\n  induction m with m H,\n  { convert (C.map_to_frac n) using 3, \n    { push_cast, ring },\n    { exact_mod_cast rfl },\n    { push_cast, ring },\n    { exact_mod_cast rfl } },\n  specialize H (m.succ_ne_zero),\n  have key : \n    \u03c6 (1 + m + 2 - n + (m + 2) * x + y) = \n    projectivization.mk F (n, m + 1 + 1, 1) (\u03bb c, hn (congr_arg prod.fst c)), \n  { apply eq_of_config _ _ _ _ _ _ _ (frac_config_horizontal F n (m + 1) hn),\n    { rw [\u2190 H.2, \u2190 C.map_two_add_x_add_y],\n      convert C.dependent_pos _ _ using 5,\n      push_cast, ring },\n    { erw [\u2190 H.1, \u2190 C.map_one_add_x],\n      convert C.dependent_pos _ _ using 5,\n      push_cast, ring },\n    { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],  \n      use ![1,1,-1], split,\n      { simp only [fin.sum_univ_succ, nat.cast_add, nat.cast_one, matrix.cons_val_zero, \n          prod.smul_mk, algebra.id.smul_eq_mul, one_mul, mul_zero, matrix.cons_val_succ, \n          mul_one, fintype.univ_of_subsingleton, fin.mk_zero,\n          matrix.cons_val_fin_one, neg_smul, prod.neg_mk, neg_add_rev, finset.sum_const, \n          finset.card_singleton, nsmul_eq_mul, prod.mk_add_mk, zero_add, add_right_neg, \n          prod.mk_eq_zero, eq_self_iff_true, and_true, true_and], \n        ring },\n      { use 0, norm_num } },\n    { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n      use ![1,1,-1], split,\n      { simp only [fin.sum_univ_succ, nat.cast_add, nat.cast_one, matrix.cons_val_zero, \n          prod.smul_mk, algebra.id.smul_eq_mul, one_mul, mul_zero, matrix.cons_val_succ, \n          mul_one, fintype.univ_of_subsingleton, fin.mk_zero,\n          matrix.cons_val_fin_one, neg_smul, prod.neg_mk, neg_add_rev, finset.sum_const, \n          finset.card_singleton, nsmul_eq_mul, prod.mk_add_mk, zero_add, add_right_neg, \n          prod.mk_eq_zero, eq_self_iff_true, and_true, true_and], \n        ring },\n      { use 0, norm_num } } },\n  split, { exact_mod_cast key },\n  push_cast at H,\n  apply eq_of_config _ _ _ _ _ _ _ (frac_config_horizontal' F n (m + 1) hn),\n  { push_cast, rw [\u2190 key, \u2190 C.map_one_add_y],\n    convert C.dependent_neg _ _ using 5, ring },\n  { push_cast, rw [\u2190 H.2, \u2190 C.map_one_add_x],\n    convert C.dependent_pos _ _ using 5, ring },\n  { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,-1,-1], split,\n    { simp only [fin.sum_univ_succ, nat.cast_add, nat.cast_one, nat.cast_succ, \n        matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, one_mul, \n        mul_one, matrix.cons_val_succ, neg_smul, mul_zero, prod.neg_mk, neg_zero,\n        fintype.univ_of_subsingleton, fin.mk_zero, \n        matrix.cons_val_fin_one, neg_add_rev, finset.sum_const, finset.card_singleton, \n        nsmul_eq_mul, mul_neg, smul_add, smul_zero, prod.mk_add_mk, zero_add, add_zero, \n        add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and], \n      simp only [nat.cast_zero, zero_add, one_mul, prod.mk_add_mk, add_zero, \n        add_right_neg, prod.mk_eq_zero, eq_self_iff_true,\n        and_true, true_and], \n      ring },\n    { use 0, norm_num } },\n  { rw [dependent_mk_iff\u2083, fintype.not_linear_independent_iff],\n    use ![1,1,-1], split,\n    { simp only [fin.sum_univ_succ, nat.cast_add, nat.cast_one, nat.cast_succ, \n        matrix.cons_val_zero, prod.smul_mk, algebra.id.smul_eq_mul, one_mul, \n        mul_zero, matrix.cons_val_succ, mul_one, fintype.univ_of_subsingleton,\n        fin.mk_zero, matrix.cons_val_fin_one, neg_smul, \n        prod.neg_mk, neg_add_rev, neg_zero, finset.sum_const, finset.card_singleton, \n        nsmul_eq_mul, mul_neg, smul_add, smul_zero, prod.mk_add_mk, zero_add,\n        add_neg_cancel_left, add_zero, add_right_neg, prod.mk_eq_zero, eq_self_iff_true, \n        and_true, true_and, ne.def],\n      simp only [nat.cast_zero, zero_add, one_mul, prod.mk_add_mk, add_neg_cancel_left, \n        add_right_neg, prod.mk_eq_zero, eq_self_iff_true, and_true, true_and],\n      ring },\n    { use 0, norm_num } },\nend\n\n/-! \n\nNow we use the symmetry of the situation.\n\n-/\n\nomit C\n\nvariable (F)\n\n@[simps]\ndef negate_snd_equiv : (F \u00d7 F \u00d7 F) \u2243\u2097[F] (F \u00d7 F \u00d7 F) := \n{ to_fun := \u03bb a, \u27e8a.1, -a.2.1, a.2.2\u27e9,\n  map_add' := by { intros x y, ext; simp; ring },\n  map_smul' := by { intros r x, ext; dsimp; simp; ring },\n  inv_fun := \u03bb a, \u27e8a.1, -a.2.1, a.2.2\u27e9,\n  left_inv := by { intros x, ext; simp },\n  right_inv := by { intros x, ext; simp } }\n\n@[simp]\nlemma negate_snd_equiv_negate_snd_equiv_apply (x : F \u00d7 F \u00d7 F) : \n  negate_snd_equiv F (negate_snd_equiv F x) = x :=\nbegin\n  cases x, dsimp, simp,\nend\n\n@[simp]\nlemma negate_snd_equiv_comp_negate_snd_equiv : \n  (negate_snd_equiv F).to_linear_map.comp (negate_snd_equiv F).to_linear_map = linear_map.id := \nby { apply fun_like.ext, dsimp, simp }\n\nvariable {F}\ninclude C\n\ndef negate_snd : setup (map (negate_snd_equiv F).to_linear_map (negate_snd_equiv F).injective \u2218 \u03c6)\n  (-2 - x) y :=\n{ map_zero := begin\n    dsimp,\n    rw [C.map_zero, map_mk_eq_mk],\n    dsimp, simp only [neg_zero], \n  end,\n  map_one := begin\n    dsimp,\n    rw [C.map_one, map_mk_eq_mk],\n    dsimp, simp only [neg_zero], \n  end,\n  map_x := begin\n    dsimp,\n    rw [(show -2 - x = -(2+x), by ring), C.map_neg, C.map_two_add_x, map_mk_eq_mk],\n    dsimp, simp only [neg_neg],\n  end,\n  map_y := begin\n    dsimp,\n    rw [C.map_y, map_mk_eq_mk],\n    dsimp, simp only [neg_zero], \n  end,\n  map_one_add_x := begin\n    dsimp,\n    rw [(show 1 + (-2 - x) = -(1 + x), by ring), C.map_neg, C.map_one_add_x, map_mk_eq_mk],\n    dsimp, rw mk_eq_mk_iff, use (-1), norm_num,\n  end,\n  map_one_add_y := begin\n    dsimp, rw [C.map_one_add_y, map_mk_eq_mk],\n    dsimp, simp only [neg_zero],\n  end,\n  map_neg := begin\n    intros u, dsimp, rw [C.map_neg],\n  end,\n  dependent_pos := begin\n    intros u v,\n    rw \u2190 dependent_iff_dependent\u2083, apply C.dependent_pos,\n  end,\n  dependent_neg := begin\n    intros u v,\n    rw \u2190 dependent_iff_dependent\u2083, apply C.dependent_neg,\n  end }\n\ntheorem main_theorem (m : \u2124) (n : \u2115) (h : ((n : F), (m : F), (0 : F)) \u2260 0) : \n  \u2203 (t : K), \u03c6 t = projectivization.mk F ((n : F), (m : F), 0) h :=\nbegin\n  induction m,\n  { cases m, \n    { have H : ((n : F), (0 : F), (0 : F)) \u2260 0, by simpa using h,\n      simp only [nat.nat_zero_eq_zero, int.of_nat_eq_coe, int.coe_nat_zero, int.cast_zero],\n      use 1, symmetry, rw [C.map_one, mk_eq_mk_iff],\n      simp only [ne.def, prod.mk_eq_zero, eq_self_iff_true, and_true] at H,\n      use (is_unit.mk0 _ H).unit, simp [units.smul_def] },\n    simp only [int.of_nat_eq_coe, int.coe_nat_succ, int.cast_add, int.cast_coe_nat, int.cast_one],\n    by_cases hn : (n : F) = 0,\n    { use 1 + x, \n      have H : (m : F) + 1 \u2260 0, by simpa [hn] using h,\n      simp only [hn], rw C.map_one_add_x, symmetry, rw mk_eq_mk_iff,\n      use (is_unit.mk0 ((m : F) + 1) H).unit, simp [units.smul_def] },\n    { have := (C.map_to_frac_horizontal n (m + 1) hn (by norm_num)).2, \n      push_cast at this,\n      refine \u27e8_, this\u27e9 } },\n  { let H := h, --push_cast at H, \n    by_cases hn : (n : F) = 0, \n    { simp only [hn, int.cast_neg_succ_of_nat, neg_add_rev, ne.def, prod.mk_eq_zero, \n        eq_self_iff_true, and_true, true_and] at H \u22a2,  \n      use 1 + x,\n      symmetry, rw [C.map_one_add_x, mk_eq_mk_iff],\n      use (is_unit.mk0 _ H).unit, simp [units.smul_def] },\n    { have := (C.negate_snd.map_to_frac_horizontal n (m + 1) hn (by norm_num)).2,\n      push_cast at this \u22a2,\n      let E := negate_snd_equiv F,\n      apply_fun (map E.to_linear_map E.injective) at this,\n      dsimp [map_mk_eq_mk] at this,\n      change ((map E.to_linear_map E.injective) \u2218 (map E.to_linear_map E.injective)) _ = _ at this, \n      rw \u2190 map_comp at this,\n      conv_lhs at this \n      { congr, erw negate_snd_equiv_comp_negate_snd_equiv },\n      erw map_id at this, dsimp [id] at this,\n      refine \u27e8_, by exact_mod_cast this\u27e9, } }\nend\n\n\n--lemma map_to_frac_horizontal (n m : \u2115) (hn : (n : F) \u2260 0) (hm : m \u2260 0) :\n--  (\u03c6 ((1 + m - n) + m * x + y) = mk[F] (n,m,1)) \u2227 \ntheorem main_theorem_char (m : \u2115) (hm : m \u2260 0) : \n  (\u03c6 ((m-1) + m * x) = mk[F] (1,m,0)) := \nby convert (map_to_frac_horizontal C 1 m _ hm).2; norm_num\n\nend setup", "meta": {"author": "adamtopaz", "repo": "lean-acl-pairs", "sha": "6ac31d86ca2739b6c18d3f05b7007e720f66299f", "save_path": "github-repos/lean/adamtopaz-lean-acl-pairs", "path": "github-repos/lean/adamtopaz-lean-acl-pairs/lean-acl-pairs-6ac31d86ca2739b6c18d3f05b7007e720f66299f/src/main_induction_step.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.36422578108397413}}
{"text": "/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nnamespace Std\nuniverse u v w w'\n\nnamespace PersistentHashMap\n\ninductive Entry (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : Type w) where\n  | entry (key : \u03b1) (val : \u03b2) : Entry \u03b1 \u03b2 \u03c3\n  | ref   (node : \u03c3) : Entry \u03b1 \u03b2 \u03c3\n  | null  : Entry \u03b1 \u03b2 \u03c3\n\ninstance {\u03b1 \u03b2 \u03c3} : Inhabited (Entry \u03b1 \u03b2 \u03c3) := \u27e8Entry.null\u27e9\n\ninductive Node (\u03b1 : Type u) (\u03b2 : Type v) : Type (max u v) where\n  | entries   (es : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : Node \u03b1 \u03b2\n  | collision (ks : Array \u03b1) (vs : Array \u03b2) (h : ks.size = vs.size) : Node \u03b1 \u03b2\n\ninstance {\u03b1 \u03b2} : Inhabited (Node \u03b1 \u03b2) := \u27e8Node.entries #[]\u27e9\n\nabbrev shift         : USize  := 5\nabbrev branching     : USize  := USize.ofNat (2 ^ shift.toNat)\nabbrev maxDepth      : USize  := 7\nabbrev maxCollisions : Nat    := 4\n\ndef mkEmptyEntriesArray {\u03b1 \u03b2} : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2)) :=\n  (Array.mkArray PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)\n\nend PersistentHashMap\n\nstructure PersistentHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] where\n  root    : PersistentHashMap.Node \u03b1 \u03b2 := PersistentHashMap.Node.entries PersistentHashMap.mkEmptyEntriesArray\n  size    : Nat                        := 0\n\nabbrev PHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] := PersistentHashMap \u03b1 \u03b2\n\nnamespace PersistentHashMap\nvariable {\u03b1 : Type u} {\u03b2 : Type v}\n\ndef empty [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 := {}\n\ndef isEmpty [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) : Bool :=\n  m.size == 0\n\ninstance [BEq \u03b1] [Hashable \u03b1] : Inhabited (PersistentHashMap \u03b1 \u03b2) := \u27e8{}\u27e9\n\ndef mkEmptyEntries {\u03b1 \u03b2} : Node \u03b1 \u03b2 :=\n  Node.entries mkEmptyEntriesArray\n\nabbrev mul2Shift (i : USize) (shift : USize) : USize := i.shiftLeft shift\nabbrev div2Shift (i : USize) (shift : USize) : USize := i.shiftRight shift\nabbrev mod2Shift (i : USize) (shift : USize) : USize := USize.land i ((USize.shiftLeft 1 shift) - 1)\n\ninductive IsCollisionNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (keys : Array \u03b1) (vals : Array \u03b2) (h : keys.size = vals.size) : IsCollisionNode (Node.collision keys vals h)\n\nabbrev CollisionNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsCollisionNode n }\n\ninductive IsEntriesNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (entries : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : IsEntriesNode (Node.entries entries)\n\nabbrev EntriesNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsEntriesNode n }\n\nprivate theorem size_set {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (i : Fin ks.size) (j : Fin vs.size) (k : \u03b1) (v : \u03b2)\n                           : (ks.set i k).size = (vs.set j v).size := by\n  simp [h]\n\nprivate theorem size_push {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (k : \u03b1) (v : \u03b2) : (ks.push k).size = (vs.push v).size := by\n  simp [h]\n\npartial def insertAtCollisionNodeAux [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 Nat \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2\n  | n@\u27e8Node.collision keys vals heq, _\u27e9, i, k, v =>\n    if h : i < keys.size then\n      let idx : Fin keys.size := \u27e8i, h\u27e9;\n      let k' := keys.get idx;\n      if k == k' then\n         let j : Fin vals.size := \u27e8i, by rw [\u2190heq]; assumption\u27e9\n         \u27e8Node.collision (keys.set idx k) (vals.set j v) (size_set heq idx j k v), IsCollisionNode.mk _ _ _\u27e9\n      else insertAtCollisionNodeAux n (i+1) k v\n    else\n      \u27e8Node.collision (keys.push k) (vals.push v) (size_push heq k v), IsCollisionNode.mk _ _ _\u27e9\n  | \u27e8Node.entries _, h\u27e9, _, _, _ => False.elim (nomatch h)\n\ndef insertAtCollisionNode [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2 :=\n  fun n k v => insertAtCollisionNodeAux n 0 k v\n\ndef getCollisionNodeSize : CollisionNode \u03b1 \u03b2 \u2192 Nat\n  | \u27e8Node.collision keys _ _, _\u27e9 => keys.size\n  | \u27e8Node.entries _, h\u27e9          => False.elim (nomatch h)\n\ndef mkCollisionNode (k\u2081 : \u03b1) (v\u2081 : \u03b2) (k\u2082 : \u03b1) (v\u2082 : \u03b2) : Node \u03b1 \u03b2 :=\n  let ks : Array \u03b1 := Array.mkEmpty maxCollisions\n  let ks := (ks.push k\u2081).push k\u2082\n  let vs : Array \u03b2 := Array.mkEmpty maxCollisions\n  let vs := (vs.push v\u2081).push v\u2082\n  Node.collision ks vs rfl\n\npartial def insertAux [BEq \u03b1] [Hashable \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 USize \u2192 \u03b1 \u2192 \u03b2 \u2192 Node \u03b1 \u03b2\n  | Node.collision keys vals heq, _, depth, k, v =>\n    let newNode := insertAtCollisionNode \u27e8Node.collision keys vals heq, IsCollisionNode.mk _ _ _\u27e9 k v\n    if depth >= maxDepth || getCollisionNodeSize newNode < maxCollisions then newNode.val\n    else match newNode with\n      | \u27e8Node.entries _, h\u27e9 => False.elim (nomatch h)\n      | \u27e8Node.collision keys vals heq, _\u27e9 =>\n        let rec traverse (i : Nat) (entries : Node \u03b1 \u03b2) : Node \u03b1 \u03b2 :=\n          if h : i < keys.size then\n            let k := keys.get \u27e8i, h\u27e9\n            let v := vals.get \u27e8i, heq \u25b8 h\u27e9\n            let h := hash k |>.toUSize\n            let h := div2Shift h (shift * (depth - 1))\n            traverse (i+1) (insertAux entries h depth k v)\n          else\n            entries\n        traverse 0 mkEmptyEntries\n  | Node.entries entries, h, depth, k, v =>\n    let j     := (mod2Shift h shift).toNat\n    Node.entries $ entries.modify j fun entry =>\n      match entry with\n      | Entry.null        => Entry.entry k v\n      | Entry.ref node    => Entry.ref $ insertAux node (div2Shift h shift) (depth+1) k v\n      | Entry.entry k' v' =>\n        if k == k' then Entry.entry k v\n        else Entry.ref $ mkCollisionNode k' v' k v\n\ndef insert [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k, v => { root := insertAux n (hash k |>.toUSize) 1 k v, size := sz + 1 }\n\npartial def findAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option \u03b2 :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then some (vals.get \u27e8i, by rw [\u2190heq]; assumption\u27e9)\n    else findAtAux keys vals heq (i+1) k\n  else none\n\npartial def findAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option \u03b2\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some v else none\n  | Node.collision keys vals heq, _, k => findAtAux keys vals heq 0 k\n\ndef find? [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option \u03b2\n  | { root := n, .. }, k => findAux n (hash k |>.toUSize) k\n\n@[inline] def getOp [BEq \u03b1] [Hashable \u03b1] (self : PersistentHashMap \u03b1 \u03b2) (idx : \u03b1) : Option \u03b2 :=\n  self.find? idx\n\n@[inline] def findD [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) (b\u2080 : \u03b2) : \u03b2 :=\n  (m.find? a).getD b\u2080\n\n@[inline] def find! [BEq \u03b1] [Hashable \u03b1] [Inhabited \u03b2] (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) : \u03b2 :=\n  match m.find? a with\n  | some b => b\n  | none   => panic! \"key is not in the map\"\n\npartial def findEntryAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then some (k', vals.get \u27e8i, by rw [\u2190heq]; assumption\u27e9)\n    else findEntryAtAux keys vals heq (i+1) k\n  else none\n\npartial def findEntryAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findEntryAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some (k', v) else none\n  | Node.collision keys vals heq, _, k => findEntryAtAux keys vals heq 0 k\n\ndef findEntry? [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | { root := n, .. }, k => findEntryAux n (hash k |>.toUSize) k\n\npartial def containsAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Bool :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then true\n    else containsAtAux keys vals heq (i+1) k\n  else false\n\npartial def containsAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Bool\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => false\n    | Entry.ref node   => containsAux node (div2Shift h shift) k\n    | Entry.entry k' v => k == k'\n  | Node.collision keys vals heq, _, k => containsAtAux keys vals heq 0 k\n\ndef contains [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Bool\n  | { root := n, .. }, k => containsAux n (hash k |>.toUSize) k\n\npartial def isUnaryEntries (a : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) (i : Nat) (acc : Option (\u03b1 \u00d7 \u03b2)) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < a.size then\n    match a.get \u27e8i, h\u27e9 with\n    | Entry.null      => isUnaryEntries a (i+1) acc\n    | Entry.ref _     => none\n    | Entry.entry k v =>\n      match acc with\n      | none   => isUnaryEntries a (i+1) (some (k, v))\n      | some _ => none\n  else acc\n\ndef isUnaryNode : Node \u03b1 \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries         => isUnaryEntries entries 0 none\n  | Node.collision keys vals heq =>\n    if h : 1 = keys.size then\n      have : 0 < keys.size := by rw [\u2190h]; decide\n      some (keys.get \u27e80, this\u27e9, vals.get \u27e80, by rw [\u2190heq]; assumption\u27e9)\n    else\n      none\n\npartial def eraseAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Node \u03b1 \u03b2 \u00d7 Bool\n  | n@(Node.collision keys vals heq), _, k =>\n    match keys.indexOf? k with\n    | some idx =>\n      let \u27e8keys', keq\u27e9 := keys.eraseIdx' idx\n      let \u27e8vals', veq\u27e9 := vals.eraseIdx' (Eq.ndrec idx heq)\n      have : keys.size - 1 = vals.size - 1 := by rw [heq]\n      (Node.collision keys' vals' (keq.trans (this.trans veq.symm)), true)\n    | none     => (n, false)\n  | n@(Node.entries entries), h, k =>\n    let j       := (mod2Shift h shift).toNat\n    let entry   := entries.get! j\n    match entry with\n    | Entry.null       => (n, false)\n    | Entry.entry k' v =>\n      if k == k' then (Node.entries (entries.set! j Entry.null), true) else (n, false)\n    | Entry.ref node   =>\n      let entries := entries.set! j Entry.null\n      let (newNode, deleted) := eraseAux node (div2Shift h shift) k\n      if !deleted then (n, false)\n      else match isUnaryNode newNode with\n        | none        => (Node.entries (entries.set! j (Entry.ref newNode)), true)\n        | some (k, v) => (Node.entries (entries.set! j (Entry.entry k v)), true)\n\ndef erase [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k =>\n    let h := hash k |>.toUSize\n    let (n, del) := eraseAux n h k\n    { root := n, size := if del then sz - 1 else sz }\n\nsection\nvariable {m : Type w \u2192 Type w'} [Monad m]\nvariable {\u03c3 : Type w}\n\n@[specialize] partial def foldlMAux (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) : Node \u03b1 \u03b2 \u2192 \u03c3 \u2192 m \u03c3\n  | Node.collision keys vals heq, acc =>\n    let rec traverse (i : Nat) (acc : \u03c3) : m \u03c3 := do\n      if h : i < keys.size then\n        let k := keys.get \u27e8i, h\u27e9\n        let v := vals.get \u27e8i, heq \u25b8 h\u27e9\n        traverse (i+1) (\u2190 f acc k v)\n      else\n        pure acc\n    traverse 0 acc\n  | Node.entries entries, acc => entries.foldlM (fun acc entry =>\n    match entry with\n    | Entry.null      => pure acc\n    | Entry.entry k v => f acc k v\n    | Entry.ref node  => foldlMAux f node acc)\n    acc\n\n@[specialize] def foldlM [BEq \u03b1] [Hashable \u03b1] (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) (init : \u03c3) : m \u03c3 :=\n  foldlMAux f map.root init\n\n@[specialize] def forM [BEq \u03b1] [Hashable \u03b1] (map : PersistentHashMap \u03b1 \u03b2) (f : \u03b1 \u2192 \u03b2 \u2192 m PUnit) : m PUnit :=\n  map.foldlM (fun _ => f) \u27e8\u27e9\n\n@[specialize] def foldl [BEq \u03b1] [Hashable \u03b1] (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03c3) (init : \u03c3) : \u03c3 :=\n  Id.run $ map.foldlM f init\nend\n\ndef toList [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) : List (\u03b1 \u00d7 \u03b2) :=\n  m.foldl (init := []) fun ps k v => (k, v) :: ps\n\nstructure Stats where\n  numNodes      : Nat := 0\n  numNull       : Nat := 0\n  numCollisions : Nat := 0\n  maxDepth      : Nat := 0\n\npartial def collectStats : Node \u03b1 \u03b2 \u2192 Stats \u2192 Nat \u2192 Stats\n  | Node.collision keys _ _, stats, depth =>\n    { stats with\n      numNodes      := stats.numNodes + 1,\n      numCollisions := stats.numCollisions + keys.size - 1,\n      maxDepth      := Nat.max stats.maxDepth depth }\n  | Node.entries entries, stats, depth =>\n    let stats :=\n      { stats with\n        numNodes      := stats.numNodes + 1,\n        maxDepth      := Nat.max stats.maxDepth depth }\n    entries.foldl (fun stats entry =>\n      match entry with\n      | Entry.null      => { stats with numNull := stats.numNull + 1 }\n      | Entry.ref node  => collectStats node stats (depth + 1)\n      | Entry.entry _ _ => stats)\n      stats\n\ndef stats [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) : Stats :=\n  collectStats m.root {} 1\n\ndef Stats.toString (s : Stats) : String :=\n  s!\"\\{ nodes := {s.numNodes}, null := {s.numNull}, collisions := {s.numCollisions}, depth := {s.maxDepth}}\"\n\ninstance : ToString Stats := \u27e8Stats.toString\u27e9\n\nend PersistentHashMap\nend Std\n", "meta": {"author": "JLimperg", "repo": "lean4-aesop", "sha": "5c4b9a3e05c32f69a4357c3047c274f4b94f9c71", "save_path": "github-repos/lean/JLimperg-lean4-aesop", "path": "github-repos/lean/JLimperg-lean4-aesop/lean4-aesop-5c4b9a3e05c32f69a4357c3047c274f4b94f9c71/stage0/src/Std/Data/PersistentHashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592642, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.36422577404497003}}
{"text": "import logic.funrel\nimport logic.finite\nimport logic.idecidable\nimport .basic .bool .decfree\n\nnamespace binary_module\n\nlocal attribute [instance] binary_abelian\n\n--- Type for matrices whose entries are indexed in subtypes of `finord`. This is defined so that `matrix p q` is isomorphic to the type of homomorphisms from the free `binary_module` generated by `subtype q` to that by `subtype p`.\ndefinition submatrix {m n : \u2115} (p : finord m \u2192 Prop) (q : finord n \u2192 Prop) : Type _ :=\n  {a : finord m \u2192 subtype q \u2192 bool // \u2200 i y, a i y \u2260 0 \u2192 p i}\n\n\nnamespace submatrix\n\nvariables {m n : \u2115} {p : finord m \u2192 Prop} {q : finord n \u2192 Prop}\n\n--- In each column, 'non-zero' is a decidable condition.\ninstance col_decidable_neq_zero (a : submatrix p q) (y : subtype q) : decidable_pred (\u03bb i, p i \u2227 a.val i y \u2260 0) :=\n  begin\n    intros i; dsimp *,\n    cases bool.decidable_eq (a.val i y) 0 with haiy haiy,\n    case is_false {\n      exact is_true \u27e8a.property _ _ haiy, haiy\u27e9,\n    },\n    case is_true {\n      apply is_false,\n      intros h,\n      exact h.right haiy\n    }\n  end\n\n--- The 'support' of each column; i.e. the `exhaustive_list` of non-zero entries.\ndefinition col_support (a : submatrix p q) (y : subtype q) : exhaustive_list {x : subtype p // a.val x.val y \u2260 0} :=\n  exhaustive_list.translate\n    (@bijection.subtype_uncurry _ p (\u03bb i, a.val i y \u2260 0)).inv_is_bijective\n    ((finord.exhaustive_list m).restrict (\u03bb i, p i \u2227 a.val i y \u2260 0))\n\n--- In each column of `submatrix`, only finitely many entries are non-zero.\nlemma col_finite (a : submatrix p q) (y : subtype q) : is_finite {x : subtype p // a.val x.val y \u2260 0} :=\n  is_finite.of_exhaustive_list (a.col_support y)\n\n--- Realize matrices as maps into free modules generated by the row indices (aka. Kleisli arrows).\ndefinition to_fun (a : submatrix p q) : subtype q \u2192 finsupp_bits (subtype p) :=\n  \u03bb y, subtype.mk (\u03bb (x : subtype p), a.val x.val y) $ a.col_finite y\n\n--- For every term `finsupp_bits (subtype p)`, its support is internally-decidable in the super type.\nlemma support_idec {\u03b1 : Type _} (f : \u03b1 \u2192 finsupp_bits (subtype p)) (a : \u03b1): \u2200 (i : finord m), idecidable (\u2203 (h : p i), (f a).val \u27e8i,h\u27e9 \u2260 0) :=\n  \u03bb i, @is_finite.idec_in_super _ _ p (\u03bb x, (f a).val x \u2260 0) (f a).property i\n\nnamespace unsafe\n\n--- Representation matrix\nnoncomputable definition mk_mat (f : subtype q \u2192 finsupp_bits (subtype p)) : submatrix p q :=\n{\n  val := \u03bb i y,\n    decidable.cases_on\n      (@unsafe.decidable_of_idecidable _ (support_idec f y i))\n      (\u03bb _, ff) (\u03bb _, tt),\n  property :=\n    begin\n      intros i y h,\n      cases @whether _ (support_idec f y i) with hpi hnpi,\n      case or.inl { cases hpi with hpi _, exact hpi },\n      case or.inr {\n        rw [@unsafe.dec_of_idec_neg _ (support_idec f y i) hnpi] at h,\n        dsimp * at h,\n        exfalso; exact h rfl\n      }\n    end\n}\n\n--- The representation matrix of a function defined from a matrix is exactly the original one.\nlemma mat_of_to_fun (a : submatrix p q) : mk_mat a.to_fun = a :=\n  begin\n    apply subtype.eq,\n    dsimp [mk_mat],\n    funext,\n    cases @whether _ (support_idec a.to_fun y i) with hpi hnpi,\n    case or.inl {\n      rw [@unsafe.dec_of_idec_pos _ (support_idec _ _ _) hpi],\n      dsimp *,\n      symmetry,\n      cases hpi with hpi hai,\n      dsimp [to_fun] at hai,\n      exact (neq_ff_iff _).mp hai,\n    },\n    case or.inr {\n      rw [@unsafe.dec_of_idec_neg _ (support_idec _ _ _) hnpi],\n      dsimp *,\n      symmetry,\n      dsimp [to_fun] at hnpi,\n      cases haiy: a.val i y,\n      case ff { refl },\n      case tt {\n        exfalso; apply hnpi; clear hnpi,\n        existsi a.property i y ((neq_ff_iff _).mpr haiy),\n        rw [haiy],\n        intro h; injection h,\n      }\n    }\n  end\n\n--- The function defined from a representation matrix is exactly the original one.\nlemma to_fun_of_mat (f : subtype q \u2192 finsupp_bits (subtype p)) : (mk_mat f).to_fun = f :=\n  begin\n    funext y,\n    apply subtype.eq,\n    funext x,\n    dsimp [mk_mat, to_fun],\n    cases @whether _ (support_idec f y x.val) with hp hnp,\n    case or.inl {\n      rw [@unsafe.dec_of_idec_pos _ (support_idec _ _ _) hp],\n      dsimp *,\n      symmetry,\n      cases hp with hpx hfy,\n      rw [subtype.eta x hpx] at hfy,\n      exact (neq_ff_iff _).mp hfy\n    },\n    case or.inr {\n      rw [@unsafe.dec_of_idec_neg _ (support_idec _ _ _) hnp],\n      dsimp *,\n      symmetry,\n      cases hfy: (f y).val x,\n      case ff { refl },\n      case tt {\n        exfalso; apply hnp; clear hnp,\n        existsi x.property,\n        rw [subtype.eta x x.property],\n        exact (neq_ff_iff _).mpr hfy,\n      }\n    }\n  end\n\nend unsafe\n\nend submatrix\n\nend binary_module\n", "meta": {"author": "Junology", "repo": "groth-lean", "sha": "5aa1ba624cd0f5145f63fa86130f99b85bbbcac2", "save_path": "github-repos/lean/Junology-groth-lean", "path": "github-repos/lean/Junology-groth-lean/groth-lean-5aa1ba624cd0f5145f63fa86130f99b85bbbcac2/src/algebra/binary_module/submatrix.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.36419216738929155}}
{"text": "/-\n## Literal tensors\n\nThis file provides utilities to validate `TensorElem` objects (literal tensors)\nwith shape inference and shape/type verification. It defines a `TensorLiteral`\nstructure which extends `TensorElem` with invariants and can be extracted into\na concrete `builtin.tensor` value.\n-/\n\nimport MLIR.AST\nimport MLIR.Semantics.Types\nimport MLIR.Util.List\nimport MLIR.Util.KDTensor\n\nopen MLIR.AST\n\n/-\n### Decidable equality\n-/\n\nmutual\ndef TensorElem.eq (e\u2081 e\u2082: TensorElem): Decidable (e\u2081 = e\u2082) := by\n  cases e\u2081 <;> cases e\u2082\n  <;> try (simp; exact inferInstance)\n  <;> try apply isFalse TensorElem.noConfusion\n\n  case float.float f\u2081 f\u2082 =>\n    -- FIXME: We shouldn't have DecidableEq on floats o(x_x)o\n    exact if f\u2081 == f\u2082 then isTrue sorry else isFalse sorry\n\n  case nested.nested l\u2081 l\u2082 =>\n    match eqList l\u2081 l\u2082 with\n    | isTrue h => exact isTrue $ by rw [h]\n    | isFalse h => exact isFalse fun h' => by cases h'; cases h rfl\n\nprivate def TensorElem.eqList (l\u2081 l\u2082: List TensorElem): Decidable (l\u2081 = l\u2082) :=\n  match l\u2081, l\u2082 with\n  | [], [] => isTrue rfl\n  | e\u2081::l\u2081, e\u2082::l\u2082 =>\n      match eq e\u2081 e\u2082, eqList l\u2081 l\u2082 with\n      | isTrue h\u03c4, isTrue hl => isTrue $ by rw [h\u03c4,hl]\n      | isFalse h\u03c4, _ => isFalse fun h => by cases h; cases h\u03c4 rfl\n      | _, isFalse hl => isFalse fun h => by cases h; cases hl rfl\n  | [], _::_ => isFalse List.noConfusion\n  | _::_, [] => isFalse List.noConfusion\nend\n termination_by\n   TensorElem.eqList l\u2081 l\u2082  => sizeOf l\u2081\n   TensorElem.eq e\u2081 _ => sizeOf e\u2081\n\n\ninstance: DecidableEq TensorElem :=\n  TensorElem.eq\n\n\n/-\n### Shape inference\n\nThis section defines shape verification and shape inference for TensorElem\n(tensor literals), *excluding the case of uniform tensor literals*. The shape\ninference is proven correct, and the `flatten` method is defined that exports\nthe tensor literal to a flat array suitable for use in a `RankedTensor`.\n\n`RankedTensor` provides the functions that actually turn tensor literals into\nranked tensors and properly handle uniform tensor literals.\n\nTODO: Integrate TensorElem invariants into the verifier\n-/\n\nnamespace MLIR.AST.TensorElem\n\n-- Check whether a tensor literal matches a concrete shape\ndef hasShape: TensorElem \u2192 List Nat \u2192 Bool\n  | TensorElem.empty, _ =>\n      false\n  | TensorElem.int _, [] =>\n      true\n  | TensorElem.int _, _::_ =>\n      false\n  | TensorElem.bool _, [] =>\n      true\n  | TensorElem.bool _, _::_ =>\n      false\n  | TensorElem.float _, [] =>\n      true\n  | TensorElem.float _, _::_ =>\n      false\n  | TensorElem.nested l, rank::size =>\n      l.length = rank \u2227 l.all (hasShape . size)\n  | TensorElem.nested _, [] =>\n      false\n\n-- Check whether a tensor literal has a particular data type\ndef hasType: TensorElem \u2192 MLIRTy \u2192 Bool\n  | TensorElem.int n, .int sgn sz =>\n      FinInt.isInBounds sgn sz n\n  | TensorElem.bool _, .i1 =>\n      true\n  | TensorElem.float _, .float _ =>\n      true\n  | TensorElem.nested [], \u03c4 =>\n      true\n  | TensorElem.nested (e::l), \u03c4 =>\n      e.hasType \u03c4 \u2227 (TensorElem.nested l).hasType \u03c4\n  | _, _ =>\n      false\n\ntheorem hasType_list_1 {l} {\u03c4: MLIRTy}:\n    hasType (.nested l) \u03c4 \u2192 l.all (hasType . \u03c4) := by\n  induction l; simp\n  case cons e l ih' =>\n    simp [hasType, List.all_cons]\n    intro h' h\n    constructor <;> simp\n    assumption\n    simp[h, h']\n    sorry\n\n\ntheorem hasType_list_2 {l} {\u03c4: MLIRTy}:\n    l.all (hasType . \u03c4) \u2192 hasType (.nested l) \u03c4 := by\n  induction l; simp [hasType]\n  case cons e l ih =>\n    simp [hasType, List.all_cons]\n    intro h2 h\n    simp [h2]\n    apply ih\n    simp[h]\n    assumption\n\ndef mapWithType {\u03c4: MLIRTy} l (f: (e: TensorElem) \u2192 (h: e.hasType \u03c4) \u2192 \u03b1)\n    (h: hasType (TensorElem.nested l) \u03c4): List \u03b1 :=\n  match l, h with\n  | [], h =>\n      []\n  | e::l, h =>\n      let h\u2081 := (by simp [hasType] at h; apply h.1)\n      let h\u2082 := (by simp [hasType] at h; apply h.2)\n      f e h\u2081 :: mapWithType l f h\u2082\n\n\n-- Shape inference function; this determines the unique shape that we allow a\n-- non-uniform tensor can have (`hasShape` is more liberal with empty lists,\n-- but the MLIR compiler is not)\ndef inferredShape: TensorElem \u2192 Option (List Nat)\n  | TensorElem.empty =>\n      none\n  | TensorElem.int _ =>\n      some []\n  | TensorElem.bool _ =>\n      some []\n  | TensorElem.float _ =>\n      some []\n  | TensorElem.nested [] =>\n      some [0]\n  | TensorElem.nested [e] => do\n      1 :: (\u2190 inferredShape e)\n  | TensorElem.nested (e::l) => do\n      let s1 \u2190 inferredShape e\n      let s2 \u2190 inferredShape (.nested l)\n      match s2 with\n      | [] => none /- impossible -/\n      | head :: tail => if s1 = tail then some ((head+1) :: tail) else none\n\n-- First let's prove the list case equivalent to a more readable form\n\ntheorem inferredShape_cons: \u2200 head tail s_head s_tail,\n    inferredShape (.nested tail) = some (s_head :: s_tail) \u2192\n    inferredShape head = some s_tail \u2192\n    inferredShape (.nested (head :: tail)) =\n      some ((s_head+1) :: s_tail) := by\n  intros head tail s_head s_tail H1 H2\n  cases tail\n  . simp [inferredShape, H2, bind, Option.bind] at *; simp[H1.1];\n  . simp [inferredShape, H2, bind, Option.bind, H1]\n\ntheorem inferredShape_cons_inv: \u2200 {head mid tail s_head s_tail},\n    inferredShape (.nested (head::mid::tail)) = some (s_head::s_tail) \u2192\n    s_head > 0 \u2227\n    inferredShape head = some s_tail \u2227\n    inferredShape (.nested (mid::tail)) = some ((s_head-1) :: s_tail) := by\n  intros head mid tail s_head s_tail\n  simp [inferredShape, bind, Option.bind]\n  cases inferredShape head <;> simp\n  case some head_shape =>\n  cases inferredShape (.nested (mid :: tail)) <;> simp\n  case some tail_shape =>\n  cases tail_shape <;> simp\n  case cons s_head' s_tail' =>\n  apply dite (head_shape = s_tail')\n  . intros Heq; rw [Heq]; simp\n    intros H H'; rw [\u2190H, Nat.add_sub_self_right, \u2190H']\n    exact \u27e8by simp_arith, rfl, rfl, rfl\u27e9\n  . intros Hne; simp [Hne]\n\ntheorem inferredShape_list {l head tail}:\n    inferredShape (.nested l) = some (head::tail) \u2192\n    head = l.length \u2227 l.all (inferredShape . = some tail) := by\n  revert head tail; induction l <;> simp\n  case nil =>\n    intros head tile H; simp [inferredShape, List.all_nil] at *; simp [H.1]\n  case cons head tail ih =>\n    intros s_head s_tail H\n    cases tail\n    case nil =>\n      simp [inferredShape, List.all_one]\n      simp [inferredShape, bind, Option.bind, List.all_one] at H\n      split at H; trivial; simp at H\n      case h_2 _ s_mid H' =>\n        simp [\u2190H.1, H', H.2]\n    case cons mid tail =>\n      let H' := inferredShape_cons_inv H\n      specialize (ih H'.2.2)\n      constructor\n      . have helper: forall {n m}, n > 0 \u2192 n - 1 = m \u2192 n = m + 1 := by\n          sorry\n        simp [helper H'.1 ih.1]\n      . sorry\n\ntheorem inferredShape_list_to_cons {l s}:\n    inferredShape (.nested l) = some s \u2192\n    \u2203 tail, s = l.length :: tail := by\n  cases s <;> simp [inferredShape]\n  case nil =>\n    cases l <;> simp [inferredShape]\n    case cons head tail =>\n      cases tail <;> simp [inferredShape]\n      case nil =>\n        cases inferredShape head <;> simp [bind, Option.bind]\n      case cons mid tail =>\n        cases inferredShape head <;> simp [bind, Option.bind]\n        cases inferredShape (.nested (mid :: tail)) <;> simp [Option.bind]\n        case some.some s1 s2 =>\n          cases s2 <;> simp\n          case cons s2_head s2_tail =>\n            apply dite (s1 = s2_tail) <;> intros H <;> simp [H]\n  case cons s_head s_tail =>\n    intro H\n    let H' := inferredShape_list H\n    exact H'.1\n\n-- We can now show that the shape inference function is correct\n\ntheorem hasShape_inferredShape:\n  \u2200 (e: TensorElem) (shape: List Nat),\n    e.inferredShape = some shape \u2192 e.hasShape shape := by\n  intro e\n  -- Cannot use the [induction] tactic because TensorElem is a nested inductive\n  -- and the tactic only supports recursors with a single motive\n  apply @TensorElem.recOn\n    (motive_1 := fun e =>\n      \u2200s, e.inferredShape = some s \u2192 e.hasShape s)\n    (motive_2 := fun l =>\n      \u2200s, l.all (TensorElem.inferredShape . = some s) \u2192\n        l.all (TensorElem.hasShape . s))\n  case int =>\n    intros _ s H; cases s <;> simp [inferredShape, hasShape] at *\n  case bool =>\n    intros _ s H; cases s <;> simp [inferredShape, hasShape] at *\n  case float =>\n    intros _ s H; cases s <;> simp [inferredShape, hasShape] at *\n  case nested =>\n    intros l motive_2 s H\n    let H' := inferredShape_list_to_cons H\n    cases H'; case intro s_tail Hs =>\n      rw [Hs]; rw [Hs] at H; clear Hs H'\n      let H' := inferredShape_list H\n      simp [hasShape, motive_2 _ H'.2]\n  case empty =>\n    simp [inferredShape]\n  case nil =>\n    intros s H; simp [List.all_nil]\n  case cons =>\n    intros head tail motive_1 ih s H; simp [List.all_cons] at *\n    simp [motive_1 _ H.1, ih _ H.2]\n    sorry\n\nend MLIR.AST.TensorElem\n\n\n/-\n### Tools for generation of ranked tensors\n-/\n\n@[inline]\ndef DimList := List Dimension\n\nnamespace DimList\n\nderiving instance DecidableEq for DimList\n\ndef str (D: DimList): String :=\n  \"x\".intercalate <| D.map fun\n    | .Known n => toString n\n    | .Unknown => \"?\"\n\ninstance: ToString DimList where\n  toString := str\n\n@[simp]\ndef shapeRefines: List Nat \u2192 DimList \u2192 Bool\n  | [], [] => true\n  | size::shape, .Unknown::dim => shapeRefines shape dim\n  | size::shape, .Known d::dim => size = d && shapeRefines shape dim\n  | (_::_), [] => false\n  | [], (_::_) => false\n\n@[simp]\ndef prod: DimList \u2192 Nat\n  | [] => 1\n  | .Known n :: D => n * prod D\n  | .Unknown :: _ => 0\n\n@[simp]\ndef project: DimList \u2192 List Nat\n  | [] => []\n  | .Known n :: D => n :: project D\n  | .Unknown :: D => project D\n\n@[simp]\ndef known: DimList \u2192 Bool\n  | [] => true\n  | .Known n :: D => known D\n  | .Unknown :: _ => false\n\n@[simp]\ndef defaultRefinement: DimList \u2192 List Nat\n  | [] => []\n  | .Known n :: D => n :: defaultRefinement D\n  | .Unknown :: D => 0 :: defaultRefinement D\n\ntheorem dim_lift_refines (S: List Nat):\n    shapeRefines S (S.map Dimension.Known) := by\n  induction S <;> simp; assumption\n\ntheorem dim_known_project_refines {D: DimList}:\n    D.known \u2192 shapeRefines D.project D := by\n  intros h <;> induction D <;> simp\n  case cons head tail ih =>\n    cases head <;> simp at *; apply (ih h)\n\ntheorem dim_known_refines_inv {D: DimList} {S: List Nat}:\n    D.known \u2192 shapeRefines S D \u2192 D = S.map Dimension.Known := by\n  intros Hknown; revert S; induction D <;> intros S Hrefines\n  case nil =>\n    cases S; simp [List.map]; simp at Hrefines\n  case cons head tail ih =>\n    cases S; simp at Hrefines\n    simp [List.map]; cases head <;> simp at *\n    rw [Hrefines.1, \u2190ih Hknown]; apply Hrefines.2\n\ntheorem dim_known_project_eq {D: DimList}:\n    D.known \u2192 shapeRefines S D \u2192 D.project = S := by\n  intros Hknown Hrefines\n  rw [dim_known_refines_inv Hknown Hrefines]\n  clear D Hknown Hrefines\n  induction S <;> simp; assumption\n\ntheorem dim_known_prod_refines {D: DimList}:\n    D.known \u2192 shapeRefines S D \u2192 shapeProd S = D.prod := by\n  intros Hknown; revert S; induction D <;> intros S Hrefines <;> simp\n  case nil =>\n    cases S; simp; simp at Hrefines\n  case cons head tail ih =>\n    cases S; simp at Hrefines\n    cases head <;> simp at *\n    rw [\u2190Hrefines.1, \u2190ih Hknown Hrefines.2]\n\ntheorem dim_known_prod (D: DimList):\n    D.known \u2192 shapeProd D.project = D.prod :=\n  fun Hknown =>\n    dim_known_prod_refines Hknown (dim_known_project_refines Hknown)\n\ntheorem defaultRefinement_refines (D: DimList):\n    shapeRefines D.defaultRefinement D := by\n  induction D <;> simp\n  case cons head _ ih =>\n    cases head <;> simp <;> apply ih\n\nend DimList\n\nnamespace MLIR.AST.TensorElem\n\ndef flatten {\u03c4: MLIRTy} (e: TensorElem) (h: e.hasType \u03c4): List \u03c4.eval :=\n  match e, \u03c4 with\n  | TensorElem.int i, .int sgn sz =>\n      [FinInt.ofInt sz i]\n  | TensorElem.bool b, .int sgn sz =>\n      [FinInt.ofInt sz (if b then 1 else 0)]\n  | TensorElem.float f, .float _ =>\n      [f]\n  | TensorElem.nested [], _ =>\n      []\n  | TensorElem.nested (e::l), \u03c4 =>\n      let h\u2081 := (by simp [hasType] at h; apply h.1)\n      let h\u2082 := (by simp [hasType] at h; apply h.2)\n      flatten e h\u2081 ++ flatten (TensorElem.nested l) h\u2082\n  | _, _ =>\n      [] /- impossible -/\n\n-- Once again, we prove a more friendly version of the list case first\n\ntheorem flatten_list {\u03c4: MLIRTy} (l: List TensorElem) (h: hasType (.nested l) \u03c4):\n    flatten (.nested l) h = (mapWithType l flatten h).join := by\n  revert h\n  induction l <;> intros h\n  case nil =>\n    simp [flatten, mapWithType, List.join]\n  case cons _ _ ih =>\n    simp [flatten, mapWithType, List.join, ih]\n\n/- LONG PROOF -/\ntheorem flatten_size {\u03c4: MLIRTy} (e: TensorElem) (shape: List Nat):\n    e.hasShape shape \u2192 (h: e.hasType \u03c4) \u2192 (e.flatten h).length = shapeProd shape := by\n  revert shape\n  apply @TensorElem.recOn\n    (motive_1 := fun e =>\n      \u2200s, e.hasShape s \u2192 (h: e.hasType \u03c4) \u2192 (e.flatten h).length = shapeProd s)\n    (motive_2 := fun l =>\n      \u2200s, l.all (TensorElem.hasShape . s) \u2192 (h: l.all (hasType . \u03c4)) \u2192\n        (mapWithType l flatten (hasType_list_2 h)).join.length = l.length * shapeProd s)\n    <;> simp <;> clear e\n  case int =>\n    intros i s Hshape Htype;\n    cases \u03c4 <;> simp [hasType] at Htype\n    cases s <;> simp [flatten, hasShape] at *\n  case float =>\n    intros i s Hshape Htype;\n    cases \u03c4 <;> simp [hasType] at Htype\n    cases s <;> simp [flatten, hasShape] at *\n  case bool =>\n    constructor;\n    intros s;\n    intros Hshape;\n    intros Htype;\n    cases \u03c4 <;> simp [hasType] at Htype\n    cases s <;> simp [flatten, hasShape] at *\n    sorry\n  case nested =>\n    intros l motive_2 s Hshape Htype\n    cases s <;> simp [hasShape] at Hshape\n    case cons s_head s_tail =>\n    simp [TensorElem.flatten_list, shapeProd, List.foldr]\n    simp [motive_2 s_tail Hshape.2 (hasType_list_1 Htype)]\n    simp [shapeProd, Nat.mul_comm, Hshape.1]\n  case empty =>\n    intros s Hshape Htype\n    simp [hasType] at Htype\n  /- tag not found??\n  case nil =>\n    intros _ Htype\n    simp [mapWithType, List.join]\n    sorry\n  -/\n  case cons =>\n    intros head tail motive_1 IH2 s Hshape1 Hshape2 Htype1\n    simp [List.map, List.join]\n    rw [Nat.add_comm]\n    simp [Nat.succ_eq_add_one, Nat.right_distrib]\n    simp [List.all_cons] at Hshape1\n    simp [List.all_cons] at Htype1\n    sorry\n    /-\n    simp [IH2 s Hshape2 Htype1]\n    rw [motive_1 s Hshape1 _]\n    sorry\n    -/\n\ninductive rankCompatibleWith (e: TensorElem) (D: DimList): MLIRTy \u2192 Type :=\n  | UniformInt (i: Int) (sgn: Signedness) (sz: Nat):\n      FinInt.isInBounds sgn sz i \u2192\n      e = TensorElem.int i \u2192\n      e.rankCompatibleWith D (.int sgn sz)\n  -- TODO: Only allow .Signless, once the dependent matching bug is fixed\n  | UniformBool (b: Bool) (sgn: Signedness):\n      e = TensorElem.bool b \u2192\n      e.rankCompatibleWith D (.int sgn 1)\n  | UniformFloat (f: Float) bitsize:\n      -- TODO: Check range of uniform tensor value\n      e = TensorElem.float f \u2192\n      e.rankCompatibleWith D (.float bitsize)\n  | HasShape s \u03c4:\n      e.hasShape s \u2192\n      D.shapeRefines s \u2192\n      e.rankCompatibleWith D \u03c4\n\nend MLIR.AST.TensorElem\n\n\n/-\n### `TensorLiteral` type\n\nThe `TensorLiteral` bundles a `TensorElem` with all the elements required for\nthe generation of a flat value array and (later) a `RankedTensor`.\n-/\n\nstructure TensorLiteral (D: DimList) (\u03c4: MLIRTy) where\n  elem: TensorElem\n  h_type: elem.hasType \u03c4\n  h_rank: elem.rankCompatibleWith D \u03c4\n\ndef TensorLiteral.ofTensorElemInferred (elem: TensorElem) (\u03c4: MLIRTy):\n    Option ((D: DimList) \u00d7 TensorLiteral D \u03c4) :=\n  if h_type: elem.hasType \u03c4 then\n    match h: elem.inferredShape with\n    | some shape =>\n        let h := TensorElem.hasShape_inferredShape _ _ h\n        -- No dimension specified: use the inferred shape\n        some \u27e8shape.map Dimension.Known, {\n          elem := elem,\n          h_type := h_type,\n          h_rank := .HasShape _ _ h (DimList.dim_lift_refines _) }\u27e9\n    | none => none\n  else none\n\ndef TensorLiteral.ofTensorElem (elem: TensorElem) (D: DimList) (\u03c4: MLIRTy):\n    Option (TensorLiteral D \u03c4) :=\n  if h_type: elem.hasType \u03c4 then\n    match h: elem.inferredShape with\n    | some shape =>\n        let h := TensorElem.hasShape_inferredShape _ _ h\n        -- Dimension is specified and matching: use it\n        if h': D.shapeRefines shape then\n          some {\n            elem := elem,\n            h_type := h_type,\n            h_rank := .HasShape _ _ h h' }\n        -- Dimension is not specified, but tensor is uniform: do uniform\n        else match h': elem, \u03c4 with\n        | TensorElem.int i, .int sgn sz =>\n            if h'': FinInt.isInBounds sgn sz i then\n              some {\n                elem := elem,\n                h_type := by simp [h', h_type],\n                h_rank := .UniformInt _ _ _ h'' h' }\n            else\n              none\n        | TensorElem.bool _, .int sgn 1 =>\n            some {\n              elem := elem,\n              h_type := by simp [h', h_type],\n              h_rank := .UniformBool _ sgn h' }\n        | TensorElem.float _, .float _ =>\n            some {\n              elem := elem,\n              h_type := by simp [h', h_type],\n              h_rank := .UniformFloat _ _ h' }\n        -- Otherwise: no\n        | _, _ => none\n    | none => none\n  else none\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Semantics/TensorElem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.364176935448373}}
{"text": "example : \u03b1 \u2192 \u03b1 := by\n                  --^ $/lean/plainGoal\n                   --^ $/lean/plainGoal\n  intro a\n--^ $/lean/plainGoal\n --^ $/lean/plainGoal\n --v $/lean/plainGoal\n  focus\n    apply a\n\nexample : \u03b1 \u2192 \u03b1 := by\n                  --^ $/lean/plainGoal\n\nexample : 0 + n = n := by\n  induction n with\n  | zero => simp; simp\n       --^ $/lean/plainGoal\n  | succ\n   --^ $/lean/plainGoal\n\nexample : \u03b1 \u2192 \u03b1 := by\n  intro a; apply a\n       --^ $/lean/plainGoal\n        --^ $/lean/plainGoal\n         --^ $/lean/plainGoal\n\nexample (h1 : n = m) (h2 : m = 0) : 0 = n := by\n  rw [h1, h2]\n --^ $/lean/plainGoal\n       --^ $/lean/plainGoal\n           --^ $/lean/plainGoal\n\nexample : 0 + n = n := by\n  induction n\n  focus\n --^ $/lean/plainGoal\n    rfl\n-- TODO: goal state after dedent\n\nexample : 0 + n = n := by\n  induction n\n --^ $/lean/plainGoal\n\nexample : 0 + n = n := by\n  cases n\n --^ $/lean/plainGoal\n\nexample : \u2200 a b : Nat, a = b := by\n  intro a b\n --^ $/lean/plainGoal\n\nexample : \u03b1 \u2192 \u03b1 := (by\n                  --^ $/lean/plainGoal\n\nexample (p : \u03b1 \u2192 Prop) (a b : \u03b1) [DecidablePred p] (h : \u2200 {p} [DecidablePred p], p a \u2192 p b) : p b := by\n  apply h _\n --^ $/lean/plainGoal\n -- should not display solved goal `\u22a2 DecidablePred p`\n\nexample : True \u2227 False := by\n  constructor\n  { constructor }\n --^ $/lean/plainGoal\n  { }\n --^ $/lean/plainGoal\n\nexample : True \u2227 False := by\n  constructor\n  \u00b7 constructor\n --^ $/lean/plainGoal\n  \u00b7\n --^ $/lean/plainGoal\n\ntheorem left_distrib (t a b : Nat) : t * (a + b) = t * a + t * b := by\n  induction b\n  next => simp\n  next =>\n    rw [Nat.add_succ]\n    repeat (rw [Nat.mul_succ])\n                           --^ $/lean/plainGoal\n\nexample (as bs cs : List \u03b1) : (as ++ bs) ++ cs = as ++ (bs ++ cs) := by\n  induction as <;> skip <;> (try rename_i h; simp[h]) <;> rfl\n                                                   --^ $/lean/plainGoal\n                                                    --^ $/lean/plainGoal\n\nexample : True := (by exact True.intro)\n                                    --^ $/lean/plainGoal\n\nexample : True := (by exact True.intro )\n                                     --^ $/lean/plainGoal\n\nexample : True \u2227 False := by\n  \u00b7 constructor; constructor\n              --^ $/lean/plainGoal\n\nexample : True = True := by\n  conv =>\n      --^ $/lean/plainGoal\n    whnf\n  --^ $/lean/plainGoal\n  --\n--^ $/lean/plainGoal\n\nexample : True := by\n  have : True := by\n    -- type here\n  --^ $/lean/plainGoal\n-- no `this` here either, but seems okay\n--^ $/lean/plainGoal\n\nexample : True := by\n  have : True := by\n    -- type here\n  --^ $/lean/plainGoal\n  apply this\n--^ $/lean/plainGoal\n-- note: no output here at all because of parse error\n\nexample : False := by\n-- EOF test\n--^ $/lean/plainGoal\n\nexample (hp : p) (hq : q) : p \u2227 q := by\n  suffices q \u2227 p by\n               --^ $/lean/plainGoal\n\nexample (hp : p) (hq : q) : p \u2227 q :=\n  show id (p \u2227 q) by\n                --^ $/lean/plainGoal\n\nexample : True \u2227 False := by\n  constructor\n  \u00b7 --\n --^ $/lean/plainGoal\n  --^ $/lean/plainGoal\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/interactive/plainGoal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.3641769316258149}}
{"text": "/-\nCopyright (c) 2020 Oliver Nash. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Oliver Nash\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.abel\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\nnamespace tactic\n\n\nnamespace interactive\n\n\n/-- A tactic for simplifying identities in not-necessarily-commutative rings.\n\nAn example:\n```lean\nexample {R : Type*} [ring R] (a b c : R) : a * (b + c + c - b) = 2*a*c :=\nby noncomm_ring\n```\n-/\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/noncomm_ring_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3639877244529887}}
{"text": "/-\nCopyright (c) 2018 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,\nAmelia Livingston, Yury Kudryashov\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.group_theory.submonoid.basic\nimport Mathlib.data.equiv.mul_add\nimport Mathlib.algebra.group.prod\nimport Mathlib.algebra.group.inj_surj\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Operations on `submonoid`s\n\nIn this file we define various operations on `submonoid`s and `monoid_hom`s.\n\n## Main definitions\n\n### Conversion between multiplicative and additive definitions\n\n* `submonoid.to_add_submonoid`, `submonoid.of_add_submonoid`, `add_submonoid.to_submonoid`,\n  `add_submonoid.of_submonoid`: convert between multiplicative and additive submonoids of `M`,\n  `multiplicative M`, and `additive M`.\n* `submonoid.add_submonoid_equiv`: equivalence between `submonoid M`\n  and `add_submonoid (additive M)`.\n\n### (Commutative) monoid structure on a submonoid\n\n* `submonoid.to_monoid`, `submonoid.to_comm_monoid`: a submonoid inherits a (commutative) monoid\n  structure.\n\n### Operations on submonoids\n\n* `submonoid.comap`: preimage of a submonoid under a monoid homomorphism as a submonoid of the\n  domain;\n* `submonoid.map`: image of a submonoid under a monoid homomorphism as a submonoid of the codomain;\n* `submonoid.prod`: product of two submonoids `s : submonoid M` and `t : submonoid N` as a submonoid\n  of `M \u00d7 N`;\n\n### Monoid homomorphisms between submonoid\n\n* `submonoid.subtype`: embedding of a submonoid into the ambient monoid.\n* `submonoid.inclusion`: given two submonoids `S`, `T` such that `S \u2264 T`, `S.inclusion T` is the\n  inclusion of `S` into `T` as a monoid homomorphism;\n* `mul_equiv.submonoid_congr`: converts a proof of `S = T` into a monoid isomorphism between `S`\n  and `T`.\n* `submonoid.prod_equiv`: monoid isomorphism between `s.prod t` and `s \u00d7 t`;\n\n### Operations on `monoid_hom`s\n\n* `monoid_hom.mrange`: range of a monoid homomorphism as a submonoid of the codomain;\n* `monoid_hom.mrestrict`: restrict a monoid homomorphism to a submonoid;\n* `monoid_hom.cod_mrestrict`: restrict the codomain of a monoid homomorphism to a submonoid;\n* `monoid_hom.mrange_restrict`: restrict a monoid homomorphism to its range;\n\n## Tags\n\nsubmonoid, range, product, map, comap\n-/\n\n/-!\n### Conversion to/from `additive`/`multiplicative`\n-/\n\n/-- Map from submonoids of monoid `M` to `add_submonoid`s of `additive M`. -/\ndef submonoid.to_add_submonoid {M : Type u_1} [monoid M] (S : submonoid M) :\n    add_submonoid (additive M) :=\n  add_submonoid.mk (submonoid.carrier S) (submonoid.one_mem' S) (submonoid.mul_mem' S)\n\n/-- Map from `add_submonoid`s of `additive M` to submonoids of `M`. -/\ndef submonoid.of_add_submonoid {M : Type u_1} [monoid M] (S : add_submonoid (additive M)) :\n    submonoid M :=\n  submonoid.mk (add_submonoid.carrier S) sorry sorry\n\n/-- Map from `add_submonoid`s of `add_monoid M` to submonoids of `multiplicative M`. -/\ndef add_submonoid.to_submonoid {M : Type u_1} [add_monoid M] (S : add_submonoid M) :\n    submonoid (multiplicative M) :=\n  submonoid.mk (add_submonoid.carrier S) (add_submonoid.zero_mem' S) (add_submonoid.add_mem' S)\n\n/-- Map from submonoids of `multiplicative M` to `add_submonoid`s of `add_monoid M`. -/\ndef add_submonoid.of_submonoid {M : Type u_1} [add_monoid M] (S : submonoid (multiplicative M)) :\n    add_submonoid M :=\n  add_submonoid.mk (submonoid.carrier S) sorry sorry\n\n/-- Submonoids of monoid `M` are isomorphic to additive submonoids of `additive M`. -/\ndef submonoid.add_submonoid_equiv (M : Type u_1) [monoid M] :\n    submonoid M \u2243 add_submonoid (additive M) :=\n  equiv.mk submonoid.to_add_submonoid submonoid.of_add_submonoid sorry sorry\n\nnamespace submonoid\n\n\n/-!\n### `comap` and `map`\n-/\n\n/-- The preimage of a submonoid along a monoid homomorphism is a submonoid. -/\ndef comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) (S : submonoid N) :\n    submonoid M :=\n  mk (\u21d1f \u207b\u00b9' \u2191S) sorry sorry\n\n@[simp] theorem coe_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid N)\n    (f : M \u2192* N) : \u2191(comap f S) = \u21d1f \u207b\u00b9' \u2191S :=\n  rfl\n\n@[simp] theorem mem_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {S : submonoid N}\n    {f : M \u2192* N} {x : M} : x \u2208 comap f S \u2194 coe_fn f x \u2208 S :=\n  iff.rfl\n\ntheorem Mathlib.add_submonoid.comap_comap {M : Type u_1} {N : Type u_2} {P : Type u_3}\n    [add_monoid M] [add_monoid N] [add_monoid P] (S : add_submonoid P) (g : N \u2192+ P) (f : M \u2192+ N) :\n    add_submonoid.comap f (add_submonoid.comap g S) =\n        add_submonoid.comap (add_monoid_hom.comp g f) S :=\n  rfl\n\n@[simp] theorem Mathlib.add_submonoid.comap_id {P : Type u_3} [add_monoid P] (S : add_submonoid P) :\n    add_submonoid.comap (add_monoid_hom.id P) S = S :=\n  sorry\n\n/-- The image of a submonoid along a monoid homomorphism is a submonoid. -/\ndef map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) (S : submonoid M) :\n    submonoid N :=\n  mk (\u21d1f '' \u2191S) sorry sorry\n\n@[simp] theorem coe_map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N)\n    (S : submonoid M) : \u2191(map f S) = \u21d1f '' \u2191S :=\n  rfl\n\n@[simp] theorem mem_map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    {S : submonoid M} {y : N} : y \u2208 map f S \u2194 \u2203 (x : M), \u2203 (H : x \u2208 S), coe_fn f x = y :=\n  set.mem_image_iff_bex\n\ntheorem Mathlib.add_submonoid.mem_map_of_mem {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] (S : add_submonoid M) (f : M \u2192+ N) (x : \u21a5S) :\n    coe_fn f \u2191x \u2208 add_submonoid.map f S :=\n  set.mem_image_of_mem (\u21d1f) (subtype.property x)\n\ntheorem map_map {M : Type u_1} {N : Type u_2} {P : Type u_3} [monoid M] [monoid N] [monoid P]\n    (S : submonoid M) (g : N \u2192* P) (f : M \u2192* N) : map g (map f S) = map (monoid_hom.comp g f) S :=\n  ext' (set.image_image (fun (a : N) => coe_fn g a) (fun (a : M) => coe_fn f a) \u2191S)\n\ntheorem map_le_iff_le_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    {S : submonoid M} {T : submonoid N} : map f S \u2264 T \u2194 S \u2264 comap f T :=\n  set.image_subset_iff\n\ntheorem Mathlib.add_submonoid.gc_map_comap {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] (f : M \u2192+ N) : galois_connection (add_submonoid.map f) (add_submonoid.comap f) :=\n  fun (S : add_submonoid M) (T : add_submonoid N) => add_submonoid.map_le_iff_le_comap\n\ntheorem map_le_of_le_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid M)\n    {T : submonoid N} {f : M \u2192* N} : S \u2264 comap f T \u2192 map f S \u2264 T :=\n  galois_connection.l_le (gc_map_comap f)\n\ntheorem le_comap_of_map_le {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid M)\n    {T : submonoid N} {f : M \u2192* N} : map f S \u2264 T \u2192 S \u2264 comap f T :=\n  galois_connection.le_u (gc_map_comap f)\n\ntheorem le_comap_map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid M)\n    {f : M \u2192* N} : S \u2264 comap f (map f S) :=\n  galois_connection.le_u_l (gc_map_comap f) S\n\ntheorem map_comap_le {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {S : submonoid N}\n    {f : M \u2192* N} : map f (comap f S) \u2264 S :=\n  galois_connection.l_u_le (gc_map_comap f) S\n\ntheorem monotone_map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N} :\n    monotone (map f) :=\n  galois_connection.monotone_l (gc_map_comap f)\n\ntheorem Mathlib.add_submonoid.monotone_comap {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] {f : M \u2192+ N} : monotone (add_submonoid.comap f) :=\n  galois_connection.monotone_u (add_submonoid.gc_map_comap f)\n\n@[simp] theorem map_comap_map {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid M)\n    {f : M \u2192* N} : map f (comap f (map f S)) = map f S :=\n  congr_fun (galois_connection.l_u_l_eq_l (gc_map_comap f)) S\n\n@[simp] theorem comap_map_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {S : submonoid N} {f : M \u2192* N} : comap f (map f (comap f S)) = comap f S :=\n  congr_fun (galois_connection.u_l_u_eq_u (gc_map_comap f)) S\n\ntheorem map_sup {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (S : submonoid M)\n    (T : submonoid M) (f : M \u2192* N) : map f (S \u2294 T) = map f S \u2294 map f T :=\n  galois_connection.l_sup (gc_map_comap f)\n\ntheorem map_supr {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {\u03b9 : Sort u_3} (f : M \u2192* N)\n    (s : \u03b9 \u2192 submonoid M) : map f (supr s) = supr fun (i : \u03b9) => map f (s i) :=\n  galois_connection.l_supr (gc_map_comap f)\n\ntheorem Mathlib.add_submonoid.comap_inf {M : Type u_1} {N : Type u_2} [add_monoid M] [add_monoid N]\n    (S : add_submonoid N) (T : add_submonoid N) (f : M \u2192+ N) :\n    add_submonoid.comap f (S \u2293 T) = add_submonoid.comap f S \u2293 add_submonoid.comap f T :=\n  galois_connection.u_inf (add_submonoid.gc_map_comap f)\n\ntheorem Mathlib.add_submonoid.comap_infi {M : Type u_1} {N : Type u_2} [add_monoid M] [add_monoid N]\n    {\u03b9 : Sort u_3} (f : M \u2192+ N) (s : \u03b9 \u2192 add_submonoid N) :\n    add_submonoid.comap f (infi s) = infi fun (i : \u03b9) => add_submonoid.comap f (s i) :=\n  galois_connection.u_infi (add_submonoid.gc_map_comap f)\n\n@[simp] theorem map_bot {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) :\n    map f \u22a5 = \u22a5 :=\n  galois_connection.l_bot (gc_map_comap f)\n\n@[simp] theorem comap_top {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) :\n    comap f \u22a4 = \u22a4 :=\n  galois_connection.u_top (gc_map_comap f)\n\n@[simp] theorem Mathlib.add_submonoid.map_id {M : Type u_1} [add_monoid M] (S : add_submonoid M) :\n    add_submonoid.map (add_monoid_hom.id M) S = S :=\n  sorry\n\n/-- `map f` and `comap f` form a `galois_coinsertion` when `f` is injective. -/\ndef gci_map_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) : galois_coinsertion (map f) (comap f) :=\n  galois_connection.to_galois_coinsertion (gc_map_comap f) sorry\n\ntheorem comap_map_eq_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) (S : submonoid M) : comap f (map f S) = S :=\n  galois_coinsertion.u_l_eq (gci_map_comap hf) S\n\ntheorem comap_surjective_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.injective \u21d1f) : function.surjective (comap f) :=\n  galois_coinsertion.u_surjective (gci_map_comap hf)\n\ntheorem map_injective_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) : function.injective (map f) :=\n  galois_coinsertion.l_injective (gci_map_comap hf)\n\ntheorem comap_inf_map_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) (S : submonoid M) (T : submonoid M) :\n    comap f (map f S \u2293 map f T) = S \u2293 T :=\n  galois_coinsertion.u_inf_l (gci_map_comap hf) S T\n\ntheorem comap_infi_map_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {\u03b9 : Type u_4} {f : M \u2192* N} (hf : function.injective \u21d1f) (S : \u03b9 \u2192 submonoid M) :\n    comap f (infi fun (i : \u03b9) => map f (S i)) = infi S :=\n  galois_coinsertion.u_infi_l (gci_map_comap hf) fun (i : \u03b9) => S i\n\ntheorem comap_sup_map_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) (S : submonoid M) (T : submonoid M) :\n    comap f (map f S \u2294 map f T) = S \u2294 T :=\n  galois_coinsertion.u_sup_l (gci_map_comap hf) S T\n\ntheorem comap_supr_map_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {\u03b9 : Type u_4} {f : M \u2192* N} (hf : function.injective \u21d1f) (S : \u03b9 \u2192 submonoid M) :\n    comap f (supr fun (i : \u03b9) => map f (S i)) = supr S :=\n  galois_coinsertion.u_supr_l (gci_map_comap hf) fun (i : \u03b9) => S i\n\ntheorem map_le_map_iff_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.injective \u21d1f) {S : submonoid M} {T : submonoid M} : map f S \u2264 map f T \u2194 S \u2264 T :=\n  galois_coinsertion.l_le_l_iff (gci_map_comap hf)\n\ntheorem map_strict_mono_of_injective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.injective \u21d1f) : strict_mono (map f) :=\n  galois_coinsertion.strict_mono_l (gci_map_comap hf)\n\n/-- `map f` and `comap f` form a `galois_insertion` when `f` is surjective. -/\ndef gi_map_comap {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.surjective \u21d1f) : galois_insertion (map f) (comap f) :=\n  galois_connection.to_galois_insertion (gc_map_comap f) sorry\n\ntheorem map_comap_eq_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.surjective \u21d1f) (S : submonoid N) : map f (comap f S) = S :=\n  galois_insertion.l_u_eq (gi_map_comap hf) S\n\ntheorem map_surjective_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.surjective \u21d1f) : function.surjective (map f) :=\n  galois_insertion.l_surjective (gi_map_comap hf)\n\ntheorem comap_injective_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.surjective \u21d1f) : function.injective (comap f) :=\n  galois_insertion.u_injective (gi_map_comap hf)\n\ntheorem map_inf_comap_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.surjective \u21d1f) (S : submonoid N) (T : submonoid N) :\n    map f (comap f S \u2293 comap f T) = S \u2293 T :=\n  galois_insertion.l_inf_u (gi_map_comap hf) S T\n\ntheorem map_infi_comap_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {\u03b9 : Type u_4} {f : M \u2192* N} (hf : function.surjective \u21d1f) (S : \u03b9 \u2192 submonoid N) :\n    map f (infi fun (i : \u03b9) => comap f (S i)) = infi S :=\n  galois_insertion.l_infi_u (gi_map_comap hf) fun (i : \u03b9) => S i\n\ntheorem map_sup_comap_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    (hf : function.surjective \u21d1f) (S : submonoid N) (T : submonoid N) :\n    map f (comap f S \u2294 comap f T) = S \u2294 T :=\n  galois_insertion.l_sup_u (gi_map_comap hf) S T\n\ntheorem map_supr_comap_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {\u03b9 : Type u_4} {f : M \u2192* N} (hf : function.surjective \u21d1f) (S : \u03b9 \u2192 submonoid N) :\n    map f (supr fun (i : \u03b9) => comap f (S i)) = supr S :=\n  galois_insertion.l_supr_u (gi_map_comap hf) fun (i : \u03b9) => S i\n\ntheorem comap_le_comap_iff_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.surjective \u21d1f) {S : submonoid N} {T : submonoid N} :\n    comap f S \u2264 comap f T \u2194 S \u2264 T :=\n  galois_insertion.u_le_u_iff (gi_map_comap hf)\n\ntheorem comap_strict_mono_of_surjective {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    {f : M \u2192* N} (hf : function.surjective \u21d1f) : strict_mono (comap f) :=\n  galois_insertion.strict_mono_u (gi_map_comap hf)\n\n/-- A submonoid of a monoid inherits a multiplication. -/\nprotected instance Mathlib.add_submonoid.has_add {M : Type u_1} [add_monoid M]\n    (S : add_submonoid M) : Add \u21a5S :=\n  { add := fun (a b : \u21a5S) => { val := subtype.val a + subtype.val b, property := sorry } }\n\n/-- A submonoid of a monoid inherits a 1. -/\nprotected instance has_one {M : Type u_1} [monoid M] (S : submonoid M) : HasOne \u21a5S :=\n  { one := { val := 1, property := one_mem S } }\n\n@[simp] theorem coe_mul {M : Type u_1} [monoid M] (S : submonoid M) (x : \u21a5S) (y : \u21a5S) :\n    \u2191(x * y) = \u2191x * \u2191y :=\n  rfl\n\n@[simp] theorem Mathlib.add_submonoid.coe_zero {M : Type u_1} [add_monoid M] (S : add_submonoid M) :\n    \u21910 = 0 :=\n  rfl\n\n/-- A submonoid of a monoid inherits a monoid structure. -/\nprotected instance to_monoid {M : Type u_1} [monoid M] (S : submonoid M) : monoid \u21a5S :=\n  function.injective.monoid coe (coe_injective S) sorry sorry\n\n/-- A submonoid of a `comm_monoid` is a `comm_monoid`. -/\nprotected instance to_comm_monoid {M : Type u_1} [comm_monoid M] (S : submonoid M) :\n    comm_monoid \u21a5S :=\n  function.injective.comm_monoid coe sorry sorry sorry\n\n/-- The natural monoid hom from a submonoid of monoid `M` to `M`. -/\ndef Mathlib.add_submonoid.subtype {M : Type u_1} [add_monoid M] (S : add_submonoid M) : \u21a5S \u2192+ M :=\n  add_monoid_hom.mk coe sorry sorry\n\n@[simp] theorem Mathlib.add_submonoid.coe_subtype {M : Type u_1} [add_monoid M]\n    (S : add_submonoid M) : \u21d1(add_submonoid.subtype S) = coe :=\n  rfl\n\n/-- An induction principle on elements of the type `submonoid.closure s`.\nIf `p` holds for `1` and all elements of `s`, and is preserved under multiplication, then `p`\nholds for all elements of the closure of `s`.\n\nThe difference with `submonoid.closure_induction` is that this acts on the subtype.\n-/\ntheorem Mathlib.add_submonoid.closure_induction' {M : Type u_1} [add_monoid M] (s : set M)\n    {p : \u21a5(add_submonoid.closure s) \u2192 Prop}\n    (Hs : \u2200 (x : M) (h : x \u2208 s), p { val := x, property := add_submonoid.subset_closure h })\n    (H1 : p 0) (Hmul : \u2200 (x y : \u21a5(add_submonoid.closure s)), p x \u2192 p y \u2192 p (x + y))\n    (x : \u21a5(add_submonoid.closure s)) : p x :=\n  sorry\n\n/-- Given `submonoid`s `s`, `t` of monoids `M`, `N` respectively, `s \u00d7 t` as a submonoid\nof `M \u00d7 N`. -/\ndef prod {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid M) (t : submonoid N) :\n    submonoid (M \u00d7 N) :=\n  mk (set.prod \u2191s \u2191t) sorry sorry\n\ntheorem coe_prod {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid M)\n    (t : submonoid N) : \u2191(prod s t) = set.prod \u2191s \u2191t :=\n  rfl\n\ntheorem Mathlib.add_submonoid.mem_prod {M : Type u_1} {N : Type u_2} [add_monoid M] [add_monoid N]\n    {s : add_submonoid M} {t : add_submonoid N} {p : M \u00d7 N} :\n    p \u2208 add_submonoid.prod s t \u2194 prod.fst p \u2208 s \u2227 prod.snd p \u2208 t :=\n  iff.rfl\n\ntheorem prod_mono {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {s\u2081 : submonoid M}\n    {s\u2082 : submonoid M} {t\u2081 : submonoid N} {t\u2082 : submonoid N} (hs : s\u2081 \u2264 s\u2082) (ht : t\u2081 \u2264 t\u2082) :\n    prod s\u2081 t\u2081 \u2264 prod s\u2082 t\u2082 :=\n  set.prod_mono hs ht\n\ntheorem prod_top {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid M) :\n    prod s \u22a4 = comap (monoid_hom.fst M N) s :=\n  sorry\n\ntheorem top_prod {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid N) :\n    prod \u22a4 s = comap (monoid_hom.snd M N) s :=\n  sorry\n\n@[simp] theorem top_prod_top {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] : prod \u22a4 \u22a4 = \u22a4 :=\n  Eq.trans (top_prod \u22a4) (comap_top (monoid_hom.snd M N))\n\ntheorem bot_prod_bot {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] : prod \u22a5 \u22a5 = \u22a5 := sorry\n\n/-- The product of submonoids is isomorphic to their product as monoids. -/\ndef prod_equiv {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid M)\n    (t : submonoid N) : \u21a5(prod s t) \u2243* \u21a5s \u00d7 \u21a5t :=\n  mul_equiv.mk (equiv.to_fun (equiv.set.prod \u2191s \u2191t)) (equiv.inv_fun (equiv.set.prod \u2191s \u2191t)) sorry\n    sorry sorry\n\ntheorem map_inl {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid M) :\n    map (monoid_hom.inl M N) s = prod s \u22a5 :=\n  sorry\n\ntheorem map_inr {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (s : submonoid N) :\n    map (monoid_hom.inr M N) s = prod \u22a5 s :=\n  sorry\n\n@[simp] theorem prod_bot_sup_bot_prod {M : Type u_1} {N : Type u_2} [monoid M] [monoid N]\n    (s : submonoid M) (t : submonoid N) : prod s \u22a5 \u2294 prod \u22a5 t = prod s t :=\n  sorry\n\nend submonoid\n\n\nnamespace monoid_hom\n\n\n/-- The range of a monoid homomorphism is a submonoid. -/\ndef mrange {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) : submonoid N :=\n  submonoid.map f \u22a4\n\n@[simp] theorem coe_mrange {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) :\n    \u2191(mrange f) = set.range \u21d1f :=\n  set.image_univ\n\n@[simp] theorem mem_mrange {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] {f : M \u2192* N}\n    {y : N} : y \u2208 mrange f \u2194 \u2203 (x : M), coe_fn f x = y :=\n  sorry\n\ntheorem map_mrange {M : Type u_1} {N : Type u_2} {P : Type u_3} [monoid M] [monoid N] [monoid P]\n    (g : N \u2192* P) (f : M \u2192* N) : submonoid.map g (mrange f) = mrange (comp g f) :=\n  submonoid.map_map \u22a4 g f\n\ntheorem Mathlib.add_monoid_hom.mrange_top_iff_surjective {M : Type u_1} [add_monoid M]\n    {N : Type u_2} [add_monoid N] {f : M \u2192+ N} :\n    add_monoid_hom.mrange f = \u22a4 \u2194 function.surjective \u21d1f :=\n  sorry\n\n/-- The range of a surjective monoid hom is the whole of the codomain. -/\ntheorem Mathlib.add_monoid_hom.mrange_top_of_surjective {M : Type u_1} [add_monoid M] {N : Type u_2}\n    [add_monoid N] (f : M \u2192+ N) (hf : function.surjective \u21d1f) : add_monoid_hom.mrange f = \u22a4 :=\n  iff.mpr add_monoid_hom.mrange_top_iff_surjective hf\n\ntheorem Mathlib.add_monoid_hom.mrange_eq_map {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] (f : M \u2192+ N) : add_monoid_hom.mrange f = add_submonoid.map f \u22a4 :=\n  rfl\n\ntheorem mclosure_preimage_le {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N)\n    (s : set N) : submonoid.closure (\u21d1f \u207b\u00b9' s) \u2264 submonoid.comap f (submonoid.closure s) :=\n  iff.mpr submonoid.closure_le\n    fun (x : M) (hx : x \u2208 \u21d1f \u207b\u00b9' s) =>\n      iff.mpr submonoid.mem_coe (iff.mpr submonoid.mem_comap (submonoid.subset_closure hx))\n\n/-- The image under a monoid hom of the submonoid generated by a set equals the submonoid generated\n    by the image of the set. -/\ntheorem Mathlib.add_monoid_hom.map_mclosure {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] (f : M \u2192+ N) (s : set M) :\n    add_submonoid.map f (add_submonoid.closure s) = add_submonoid.closure (\u21d1f '' s) :=\n  sorry\n\n/-- Restriction of a monoid hom to a submonoid of the domain. -/\ndef mrestrict {M : Type u_1} [monoid M] {N : Type u_2} [monoid N] (f : M \u2192* N) (S : submonoid M) :\n    \u21a5S \u2192* N :=\n  comp f (submonoid.subtype S)\n\n@[simp] theorem Mathlib.add_monoid_hom.mrestrict_apply {M : Type u_1} [add_monoid M]\n    (S : add_submonoid M) {N : Type u_2} [add_monoid N] (f : M \u2192+ N) (x : \u21a5S) :\n    coe_fn (add_monoid_hom.mrestrict f S) x = coe_fn f \u2191x :=\n  rfl\n\n/-- Restriction of a monoid hom to a submonoid of the codomain. -/\ndef cod_mrestrict {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] (f : M \u2192* N) (S : submonoid N)\n    (h : \u2200 (x : M), coe_fn f x \u2208 S) : M \u2192* \u21a5S :=\n  mk (fun (n : M) => { val := coe_fn f n, property := h n }) sorry sorry\n\n/-- Restriction of a monoid hom to its range interpreted as a submonoid. -/\ndef mrange_restrict {M : Type u_1} [monoid M] {N : Type u_2} [monoid N] (f : M \u2192* N) :\n    M \u2192* \u21a5(mrange f) :=\n  cod_mrestrict f (mrange f) sorry\n\n@[simp] theorem Mathlib.add_monoid_hom.coe_mrange_restrict {M : Type u_1} [add_monoid M]\n    {N : Type u_2} [add_monoid N] (f : M \u2192+ N) (x : M) :\n    \u2191(coe_fn (add_monoid_hom.mrange_restrict f) x) = coe_fn f x :=\n  rfl\n\nend monoid_hom\n\n\nnamespace submonoid\n\n\ntheorem Mathlib.add_submonoid.mrange_inl {M : Type u_1} {N : Type u_2} [add_monoid M]\n    [add_monoid N] : add_monoid_hom.mrange (add_monoid_hom.inl M N) = add_submonoid.prod \u22a4 \u22a5 :=\n  add_submonoid.map_inl \u22a4\n\ntheorem mrange_inr {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.inr M N) = prod \u22a5 \u22a4 :=\n  map_inr \u22a4\n\ntheorem mrange_inl' {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.inl M N) = comap (monoid_hom.snd M N) \u22a5 :=\n  Eq.trans mrange_inl (top_prod \u22a5)\n\ntheorem mrange_inr' {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.inr M N) = comap (monoid_hom.fst M N) \u22a5 :=\n  Eq.trans mrange_inr (prod_top \u22a5)\n\n@[simp] theorem mrange_fst {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.fst M N) = \u22a4 :=\n  monoid_hom.mrange_top_of_surjective (monoid_hom.fst M N) prod.fst_surjective\n\n@[simp] theorem mrange_snd {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.snd M N) = \u22a4 :=\n  monoid_hom.mrange_top_of_surjective (monoid_hom.snd M N) prod.snd_surjective\n\n@[simp] theorem mrange_inl_sup_mrange_inr {M : Type u_1} {N : Type u_2} [monoid M] [monoid N] :\n    monoid_hom.mrange (monoid_hom.inl M N) \u2294 monoid_hom.mrange (monoid_hom.inr M N) = \u22a4 :=\n  sorry\n\n/-- The monoid hom associated to an inclusion of submonoids. -/\ndef inclusion {M : Type u_1} [monoid M] {S : submonoid M} {T : submonoid M} (h : S \u2264 T) :\n    \u21a5S \u2192* \u21a5T :=\n  monoid_hom.cod_mrestrict (subtype S) T sorry\n\n@[simp] theorem range_subtype {M : Type u_1} [monoid M] (s : submonoid M) :\n    monoid_hom.mrange (subtype s) = s :=\n  ext' (Eq.trans (monoid_hom.coe_mrange (subtype s)) subtype.range_coe)\n\ntheorem eq_bot_iff_forall {M : Type u_1} [monoid M] (S : submonoid M) :\n    S = \u22a5 \u2194 \u2200 (x : M), x \u2208 S \u2192 x = 1 :=\n  sorry\n\ntheorem Mathlib.add_submonoid.nontrivial_iff_exists_ne_zero {M : Type u_1} [add_monoid M]\n    (S : add_submonoid M) : nontrivial \u21a5S \u2194 \u2203 (x : M), \u2203 (H : x \u2208 S), x \u2260 0 :=\n  sorry\n\n/-- A submonoid is either the trivial submonoid or nontrivial. -/\ntheorem bot_or_nontrivial {M : Type u_1} [monoid M] (S : submonoid M) : S = \u22a5 \u2228 nontrivial \u21a5S :=\n  sorry\n\n/-- A submonoid is either the trivial submonoid or contains a nonzero element. -/\ntheorem Mathlib.add_submonoid.bot_or_exists_ne_zero {M : Type u_1} [add_monoid M]\n    (S : add_submonoid M) : S = \u22a5 \u2228 \u2203 (x : M), \u2203 (H : x \u2208 S), x \u2260 0 :=\n  sorry\n\nend submonoid\n\n\nnamespace mul_equiv\n\n\n/-- Makes the identity isomorphism from a proof that two submonoids of a multiplicative\n    monoid are equal. -/\ndef Mathlib.add_equiv.add_submonoid_congr {M : Type u_1} [add_monoid M] {S : add_submonoid M}\n    {T : add_submonoid M} (h : S = T) : \u21a5S \u2243+ \u21a5T :=\n  add_equiv.mk (equiv.to_fun (equiv.set_congr sorry)) (equiv.inv_fun (equiv.set_congr sorry)) sorry\n    sorry sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/group_theory/submonoid/operations_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3639679193499405}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.adjunction.basic\nimport category_theory.conj\nimport category_theory.yoneda\n\n/-!\n# Adjoints of fully faithful functors\n\nA left adjoint is fully faithful, if and only if the unit is an isomorphism\n(and similarly for right adjoints and the counit).\n\n`adjunction.restrict_fully_faithful` shows that an adjunction can be restricted along fully faithful\ninclusions.\n\n## Future work\n\nThe statements from Riehl 4.5.13 for adjoints which are either full, or faithful.\n-/\n\nopen category_theory\n\nnamespace category_theory\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category\nopen opposite\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D]\nvariables {L : C \u2964 D} {R : D \u2964 C} (h : L \u22a3 R)\n\n/--\nIf the left adjoint is fully faithful, then the unit is an isomorphism.\n\nSee\n* Lemma 4.5.13 from [Riehl][riehl2017]\n* https://math.stackexchange.com/a/2727177\n* https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)\n-/\ninstance unit_is_iso_of_L_fully_faithful [full L] [faithful L] : is_iso (adjunction.unit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.unit h) $ \u03bb X,\n@yoneda.is_iso _ _ _ _ ((adjunction.unit h).app X)\n\u27e8\u27e8{ app := \u03bb Y f, L.preimage ((h.hom_equiv (unop Y) (L.obj X)).symm f) },\n  \u27e8begin\n    ext x f, dsimp,\n    apply L.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_counit, preimage_comp, preimage_map, category.assoc],\n    rw \u2190h.unit_naturality,\n    simp,\n  end\u27e9\u27e9\u27e9\n\n/--\nIf the right adjoint is fully faithful, then the counit is an isomorphism.\n\nSee https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)\n-/\ninstance counit_is_iso_of_R_fully_faithful [full R] [faithful R] : is_iso (adjunction.counit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.counit h) $ \u03bb X,\n@is_iso_of_op _ _ _ _ _ $\n@coyoneda.is_iso _ _ _ _ ((adjunction.counit h).app X).op\n\u27e8\u27e8{ app := \u03bb Y f, R.preimage ((h.hom_equiv (R.obj X) Y) f) },\n  \u27e8begin\n    ext x f, dsimp,\n    apply R.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_unit, preimage_comp, preimage_map],\n    rw \u2190h.counit_naturality,\n    simp,\n  end\u27e9\u27e9\u27e9\n\n/-- If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given\nby L whiskered with the counit. -/\n@[simp]\nlemma inv_map_unit {X : C} [is_iso (h.unit.app X)] :\n  inv (L.map (h.unit.app X)) = h.counit.app (L.obj X) :=\nis_iso.inv_eq_of_hom_inv_id h.left_triangle_components\n\n/-- If the unit is an isomorphism, bundle one has an isomorphism `L \u22d9 R \u22d9 L \u2245 L`. -/\n@[simps]\nnoncomputable def whisker_left_L_counit_iso_of_is_iso_unit [is_iso h.unit] :\n  L \u22d9 R \u22d9 L \u2245 L :=\n(L.associator R L).symm \u226a\u226b iso_whisker_right (as_iso h.unit).symm L \u226a\u226b functor.left_unitor _\n\n/-- If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given\nby R whiskered with the unit. -/\n@[simp]\nlemma inv_counit_map {X : D} [is_iso (h.counit.app X)] :\n  inv (R.map (h.counit.app X)) = h.unit.app (R.obj X) :=\nis_iso.inv_eq_of_inv_hom_id h.right_triangle_components\n\n/-- If the counit of an is an isomorphism, one has an isomorphism `(R \u22d9 L \u22d9 R) \u2245 R`. -/\n@[simps]\nnoncomputable def whisker_left_R_unit_iso_of_is_iso_counit [is_iso h.counit] :\n  (R \u22d9 L \u22d9 R) \u2245 R :=\n(R.associator L R).symm \u226a\u226b iso_whisker_right (as_iso h.counit) R \u226a\u226b functor.left_unitor _\n\n/-- If the unit is an isomorphism, then the left adjoint is full-/\nnoncomputable\ndef L_full_of_unit_is_iso [is_iso h.unit] : full L :=\n{ preimage := \u03bb X Y f, (h.hom_equiv X (L.obj Y) f) \u226b inv (h.unit.app Y) }\n\n/-- If the unit is an isomorphism, then the left adjoint is faithful-/\nlemma L_faithful_of_unit_is_iso [is_iso h.unit] : faithful L :=\n{ map_injective' := \u03bb X Y f g H,\n  begin\n    rw \u2190(h.hom_equiv X (L.obj Y)).apply_eq_iff_eq at H,\n    simpa using H =\u226b inv (h.unit.app Y),\n  end }\n\n/-- If the counit is an isomorphism, then the right adjoint is full-/\nnoncomputable\ndef R_full_of_counit_is_iso [is_iso h.counit] : full R :=\n{ preimage := \u03bb X Y f, inv (h.counit.app X) \u226b (h.hom_equiv (R.obj X) Y).symm f }\n\n/-- If the counit is an isomorphism, then the right adjoint is faithful-/\nlemma R_faithful_of_counit_is_iso [is_iso h.counit] : faithful R :=\n{ map_injective' := \u03bb X Y f g H,\n  begin\n    rw \u2190(h.hom_equiv (R.obj X) Y).symm.apply_eq_iff_eq at H,\n    simpa using inv (h.counit.app X) \u226b= H,\n  end }\n\ninstance whisker_left_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_left L h.counit) :=\nbegin\n  have := h.left_triangle,\n  rw \u2190is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_right h.counit R) :=\nbegin\n  have := h.right_triangle,\n  rw \u2190is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_left_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_left R h.unit) :=\nbegin\n  have := h.right_triangle,\n  rw \u2190is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_right h.unit L) :=\nbegin\n  have := h.left_triangle,\n  rw \u2190is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\n-- TODO also do the statements from Riehl 4.5.13 for full and faithful separately?\n\nuniverses v\u2083 v\u2084 u\u2083 u\u2084\n\nvariables {C' : Type u\u2083} [category.{v\u2083} C']\nvariables {D' : Type u\u2084} [category.{v\u2084} D']\n\n-- TODO: This needs some lemmas describing the produced adjunction, probably in terms of `adj`,\n-- `iC` and `iD`.\n/--\nIf `C` is a full subcategory of `C'` and `D` is a full subcategory of `D'`, then we can restrict\nan adjunction `L' \u22a3 R'` where `L' : C' \u2964 D'` and `R' : D' \u2964 C'` to `C` and `D`.\nThe construction here is slightly more general, in that `C` is required only to have a full and\nfaithful \"inclusion\" functor `iC : C \u2964 C'` (and similarly `iD : D \u2964 D'`) which commute (up to\nnatural isomorphism) with the proposed restrictions.\n-/\ndef adjunction.restrict_fully_faithful (iC : C \u2964 C') (iD : D \u2964 D') {L' : C' \u2964 D'} {R' : D' \u2964 C'}\n  (adj : L' \u22a3 R') {L : C \u2964 D} {R : D \u2964 C} (comm1 : iC \u22d9 L' \u2245 L \u22d9 iD) (comm2 : iD \u22d9 R' \u2245 R \u22d9 iC)\n  [full iC] [faithful iC] [full iD] [faithful iD] :\n  L \u22a3 R :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y,\n  calc (L.obj X \u27f6 Y) \u2243 (iD.obj (L.obj X) \u27f6 iD.obj Y) : equiv_of_fully_faithful iD\n       ... \u2243 (L'.obj (iC.obj X) \u27f6 iD.obj Y) : iso.hom_congr (comm1.symm.app X) (iso.refl _)\n       ... \u2243 (iC.obj X \u27f6 R'.obj (iD.obj Y)) : adj.hom_equiv _ _\n       ... \u2243 (iC.obj X \u27f6 iC.obj (R.obj Y)) : iso.hom_congr (iso.refl _) (comm2.app Y)\n       ... \u2243 (X \u27f6 R.obj Y) : (equiv_of_fully_faithful iC).symm,\n  hom_equiv_naturality_left_symm' := \u03bb X' X Y f g,\n  begin\n    apply iD.map_injective,\n    simpa using (comm1.inv.naturality_assoc f _).symm,\n  end,\n  hom_equiv_naturality_right' := \u03bb X Y' Y f g,\n  begin\n    apply iC.map_injective,\n    suffices : R'.map (iD.map g) \u226b comm2.hom.app Y = comm2.hom.app Y' \u226b iC.map (R.map g),\n      simp [this],\n    apply comm2.hom.naturality g,\n  end }\n\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/adjunction/fully_faithful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.36391681657188596}}
{"text": "\nimport unitb.logic\nimport unitb.scheduling\n\nimport util.logic\n\nimport temporal_logic\n\nuniverse variables u u'\nnamespace nondet\n\nsection nondet\n\nopen predicate\n\nparameter \u03b1 : Type\n\n@[reducible]\nprivate def pred := \u03b1 \u2192 Prop\n\nstructure event : Type :=\n  (coarse_sch : pred)\n  (fine_sch : pred)\n  (step : \u2200 s, coarse_sch s \u2192 fine_sch s \u2192 pred)\n  (fis : \u2200 s CS FS, \u2203 s', step s CS FS s')\n\nparameter {\u03b1}\n\ndef event.guard (e : event) : pred :=\ne.coarse_sch \u22c0 e.fine_sch\n\nparameter \u03b1\n\nstructure program : Type 2 :=\n  (lbl : Type)\n  (lbl_is_sched : scheduling.sched lbl)\n  (first : \u03b1 \u2192 Prop)\n  (first_fis : \u2203 s, first s)\n  (event' : lbl \u2192 event)\n\ninstance {p : program} : scheduling.sched p.lbl :=\np.lbl_is_sched\n\nparameter {\u03b1}\n\ndef event.step_of (e : event) (\u03c3 \u03c3' : \u03b1) : Prop :=\n\u2203 Hc Hf, e.step \u03c3 Hc Hf \u03c3'\n\ndef skip : event :=\n{ coarse_sch := True\n, fine_sch := True\n, step := \u03bb s _ _ s', s = s'\n, fis := assume s _ _, \u27e8s,rfl\u27e9 }\n\ndef program.event (s : program) : option s.lbl \u2192 event\n  | none := skip\n  | (some e) := s^.event' e\n\ndef program.init (s : program) (p : pred) : Prop :=\ns^.first \u27f9 p\n\ndef program.guard  (s : program) (e : option s.lbl) : \u03b1 \u2192 Prop :=\n(s^.event e)^.coarse_sch \u22c0 (s^.event e)^.fine_sch\n\ndef program.guard_none_holds (s : program) (\u03c3 : \u03b1)\n: s.guard none \u03c3 :=\n\u27e8trivial,trivial\u27e9\n\ndef program.coarse_sch_of (s : program) (act : option s.lbl) : \u03b1 \u2192 Prop :=\n(s.event act).coarse_sch\n\n@[simp]\nlemma program.coarse_sch_of_none (s : program)\n: s.coarse_sch_of none = True :=\nby refl\n\n@[simp]\nlemma program.coarse_sch_of_some (s : program) (e : s.lbl)\n: s.coarse_sch_of (some e) = (s.event' e).coarse_sch :=\nby refl\n\ndef program.fine_sch_of (s : program) (act : option s.lbl) : \u03b1 \u2192 Prop :=\n(s.event act).fine_sch\n\n@[simp]\nlemma program.fine_sch_of_none (s : program)\n: s.fine_sch_of none = True :=\nby refl\n\ndef program.take_step (s : program)\n: \u2200 (e : option s.lbl) (\u03c3 : \u03b1), s^.coarse_sch_of e \u03c3 \u2192 s^.fine_sch_of e \u03c3 \u2192 \u03b1 \u2192 Prop\n  | none \u03c3 _ _ := \u03bb \u03c3', \u03c3 = \u03c3'\n  | (some e) \u03c3 Hc Hf := (s^.event e)^.step \u03c3 Hc Hf\n\ndef program.step_of (s : program) (act : option s.lbl) : \u03b1 \u2192 \u03b1 \u2192 Prop :=\n(s.event act).step_of\n\n@[simp]\nlemma program.step_of_none (s : program)\n: s.step_of none = eq :=\nbegin\n  apply funext, intro \u03c3,\n  apply funext, intro \u03c3',\n  dunfold program.step_of program.event skip event.step_of,\n  dsimp [event.coarse_sch,event.fine_sch],\n  simp [exists_true],\nend\n\ndef is_step (s : program) (\u03c3 \u03c3' : \u03b1) : Prop :=\n\u2203 ev, s.step_of ev \u03c3 \u03c3'\n\nopen temporal\n\nlemma step_of_none  (s : program) : s.step_of none = eq :=\nbegin\n  dunfold program.step_of program.event skip,\n  apply funext, intro \u03c3,\n  apply funext, intro \u03c3',\n  dunfold event.step_of,\n  dunfold event.coarse_sch event.fine_sch event.step,\n  dunfold True lifted\u2080,\n  simp [exists_true],\nend\n\nlemma is_step_exists_event  (s : program)\n : temporal.action (is_step s) = (\u27e6 eq \u27e7 \u22c1 \u2203\u2203 ev : s.lbl, \u27e6 (s.event' ev).step_of \u27e7) :=\nbegin\n  simp [exists_action,or_action],\n  apply congr_arg,\n  apply funext, intro \u03c3,\n  apply funext, intro \u03c3',\n  unfold is_step,\n  simp [exists_option],\n  rw or_congr,\n  { simp [step_of_none] },\n  { apply exists_congr, intro e,\n    refl }\nend\n\nlemma is_step_exists_event'  (s : program)\n : temporal.action (is_step s) = (\u2203\u2203 ev : option s.lbl, \u27e6 s.step_of ev \u27e7) :=\nbegin\n  simp [exists_action,or_action],\n  apply congr_arg,\n  apply funext, intro \u03c3,\n  apply funext, intro \u03c3',\n  unfold is_step,\nend\n\nlemma is_step_inst' (s : program) (ev : option s.lbl)\n: \u27e6 s.step_of ev \u27e7 \u27f9 \u27e6 is_step s \u27e7 :=\nbegin\n  rw is_step_exists_event',\n  intros \u03c4, simp,\n  apply exists.intro ev,\nend\n\ndef pair  (\u03c3 \u03c3' : \u03b1) : stream \u03b1\n  | 0 := \u03c3\n  | (nat.succ i) := \u03c3'\n\nlemma is_step_inst (s : program) (ev : option s.lbl) (\u03c3 \u03c3' : \u03b1)\n  (h : s.step_of ev \u03c3 \u03c3')\n: is_step s \u03c3 \u03c3' :=\nbegin\n  change \u27e6 is_step s \u27e7 (pair \u03c3 \u03c3'),\n  apply is_step_inst' _ ev,\n  apply h\nend\n\nlocal attribute [instance] classical.prop_decidable\n\nnoncomputable def program.object_mch (p : program)\n: scheduling.unitb.target_mch (option p.lbl) :=\n{ \u03c3 := \u03b1\n, s\u2080 := classical.some p.first_fis\n, req := \u03bb s, { l | p.guard l s }\n, req_nemp := assume x,\n  begin\n    apply @set.ne_empty_of_mem _ _ none,\n    simp [mem_set_of], exact \u27e8trivial,trivial\u27e9,\n  end\n, next := \u03bb l s (h : p.guard l s),\n                 classical.some ((p.event l).fis s h.left h.right)\n                  }\n\ninstance : unitb.has_safety program :=\n  { \u03c3 := \u03b1\n  , step := is_step }\n\nstructure program.falsify (s : program) (act : option s.lbl) (p q : pred' \u03b1) : Prop :=\n  (enable : q \u27f9 s^.coarse_sch_of act)\n  (schedule : p \u27f9 s^.fine_sch_of act)\n  (negate' : \u2983 \u2022q \u27f6 \u27e6 s^.step_of act \u27e7 \u27f6 \u2299-\u2022q \u2984)\n\ndef program.transient (s : program) (p q : pred' \u03b1) : Prop :=\n\u2203 (act : option s.lbl), s.falsify act p q\n\nopen temporal has_mem scheduling.unitb function\n\nlemma object_mch_action_eq_step_of (s : program) (e : option s.lbl)\n: \u2022mem e \u2218 s.object_mch.req \u22c0 action (s.object_mch.action e) \u27f9 action (s.step_of e) :=\nbegin\n  rw [init_eq_action,action_and_action],\n  apply action_entails_action,\n  intro \u03c3, intro \u03c3',\n  dunfold comp program.object_mch,\n  unfold  target_mch.action target_mch.next target_mch.req,\n  simp [mem_set_of],\n  intros h\u2080 h\u2081 h\u2082,\n  cases h\u2081 with P h\u2081,\n  rw [h\u2082],\n  dunfold program.step_of event.step_of program.object_mch._proof_3,\n  existsi h\u2080.left, existsi h\u2080.right,\n  apply classical.some_spec,\nend\n\nlemma mem_object_req_eq_csch_and_fsch (s : program) (e : option s.lbl)\n: mem e \u2218 s.object_mch.req = s.coarse_sch_of e \u22c0 s.fine_sch_of e :=\nrfl\n\nlemma program.falsify.negate\n   {s : program} {act : option s.lbl} {p q : pred}\n:  s.falsify act p q\n\u2192  \u2022q \u22c0 \u27e6 s^.step_of act \u27e7 \u27f9 \u25c7-\u2022q :=\nbegin\n  intros h\u2080 \u03c4 h\u2081,\n  have h\u2082 := h\u2080.negate' _ h\u2081.left h\u2081.right,\n  unfold eventually p_not init,\n  existsi 1,\n  apply h\u2082,\nend\n\n\nsection theorems\n\nvariable (s : program)\n\nopen program\nopen event\n\ntheorem program.transient_false {p : pred}\n: transient s p False :=\nbegin\n  dunfold program.transient,\n  existsi none,\n  apply falsify.mk,\n  { intros \u03c3 h, apply trivial },\n  { intros \u03c4 i,\n    apply trivial, },\n  { intros \u03c3 h\u2080 h\u2081, cases h\u2080 },\nend\n\ndef program.transient_antimono (s : program) {p q p' q' : pred}\n  (hp : p' \u27f9 p)\n  (hq : q' \u27f9 q)\n: s.transient p q \u2192 s.transient p' q' :=\nbegin\n  dunfold transient,\n  apply exists_imp_exists,\n  intros e h',\n  apply falsify.mk,\n  { apply entails_trans _ hq h'.enable, },\n  { apply entails_trans _ hp h'.schedule, },\n  { have hp' := init_entails_init hp,\n    have hq' := init_entails_init hq,\n    apply' ew_imp_ew _ h'.negate',\n    refine p_imp_entails_p_imp hq' _,\n    refine p_imp_entails_p_imp_right _,\n    apply' next_imp_next _ ,\n    apply' p_not_entails_p_not_right hq', }\nend\n\nend theorems\n\ninstance prog_is_system : unitb.system program :=\n{ \u03c3 := \u03b1\n, transient := program.transient\n, step  := is_step\n, init  := program.init\n, transient_false := \u03bb s p, program.transient_false s\n, transient_antimono := program.transient_antimono }\n\nsection soundness\n\ndef fair' (s : program) (e : option s.lbl) (\u03c4 : stream \u03b1) : Prop :=\n(\u25c7\u25fb\u2022 s^.coarse_sch_of e) \u03c4 \u2192\n(\u25fb\u25c7\u2022s.fine_sch_of e) \u03c4 \u2192\n(\u25fb\u25c7\u27e6 s.step_of e \u27e7) \u03c4\n\nstructure program.ex (s : program) (\u03c4 : stream \u03b1) : Prop :=\n    (init : s^.first (\u03c4 0))\n    (safety : unitb.saf_ex s \u03c4)\n    (liveness : \u2200 e, fair' s e \u03c4)\n\nopen program\n\nvariables {s : program} {p q : pred' \u03b1}\nvariables (\u03c4 : stream \u03b1)\n\nlemma transient.semantics'\n  (h : \u2200 e, fair' s e \u03c4)\n  (T\u2080 : s.transient p q)\n: (\u25fb\u25c7\u2022p) \u03c4 \u2192 (\u25fb\u25c7-\u2022q) \u03c4 :=\nbegin\n  cases (temporal.em' (\u2022q) \u03c4) with h_q ev_nq,\n  { dunfold nondet.program.transient at T\u2080,\n    cases T\u2080 with ev T\u2080,\n    have Hc : (\u25c7\u25fb\u2022s.coarse_sch_of ev) \u03c4,\n    { apply stable_entails_stable' _ _ h_q,\n      apply T\u2080.enable },\n    intro Hp,\n    have Hf : (\u25fb\u25c7\u2022s.fine_sch_of ev) \u03c4,\n    { apply inf_often_entails_inf_often' _ _ Hp,\n      apply T\u2080.schedule, },\n    have live := h ev Hc Hf,\n    have act := coincidence h_q (h ev Hc Hf),\n    rw [\u2190 eventually_eventually],\n    apply inf_often_entails_inf_often _ _ act,\n    apply entails_imp_entails_left _ T\u2080.negate,\n    refl, },\n  { intro, apply ev_nq },\nend\n\nvariables (h : ex s \u03c4)\ninclude h\n\nlemma init_sem\n  (I\u2080 : init s p)\n: (\u2022p) \u03c4 :=\nbegin\n  unfold temporal.init,\n  apply I\u2080,\n  apply h.init,\nend\n\nlemma transient.semantics\n  (T\u2080 : s.transient p q)\n: (\u25fb\u25c7\u2022p) \u03c4 \u2192 (\u25fb\u25c7-\u2022q) \u03c4 :=\nnondet.transient.semantics' \u03c4 h.liveness T\u2080\n\nend soundness\n\nopen scheduling nat list\n\nnoncomputable def program.first_state (s : program) := (classical.some s.first_fis)\n\nopen unitb has_mem\n\nlemma program.witness (s : program)\n: \u2203 (\u03c4 : stream \u03b1), s.ex \u03c4 :=\nbegin\n  apply exists_imp_exists _ (sched.sched_str s.object_mch),\n  intros \u03c4 h,\n  apply ex.mk,\n  { rw h.init,\n    apply classical.some_spec },\n  { unfold saf_ex,\n    revert_p h.valid, monotonicity,\n    rw p_exists_entails_eq_p_forall_entails,\n    intro l,\n    intros \u03c4 h,\n    apply is_step_inst' _ l,\n    apply object_mch_action_eq_step_of,\n    revert_p h, refl },\n  { have h' := h.fair, revert h',\n    intros_mono e,\n    intros Hsch Hc Hf,\n    apply inf_often_entails_inf_often,\n    apply (object_mch_action_eq_step_of _ _),\n    apply Hsch,\n    rw mem_object_req_eq_csch_and_fsch,\n    apply coincidence Hc Hf },\nend\n\n-- instance {\u03b1} [sched lbl] : system_sem (program lbl) :=\ninstance : unitb.system_sem program :=\n  { (_ : unitb.system program) with\n    ex := program.ex\n  , safety := @program.ex.safety _\n  , inhabited := program.witness\n  , init_sem := @init_sem\n  , transient_sem := @transient.semantics }\n\nopen unitb classical\n\ndef unless_except (s : program) (p q : pred' \u03b1) (evts : set event) : Prop :=\nunless' s p q (\u03bb \u03c3 \u03c3', \u2203 e : event, e \u2208 evts \u2227 e.step_of \u03c3 \u03c3')\n\nlemma unless_except_imp_unless {F : program} {p q : pred' \u03b1} (exp : set event)\n  (S : unless F p q)\n: unless_except F p q exp :=\nbegin\n  intros s s' STEP Hexcp,\n  apply S _ _ STEP,\nend\n\ntheorem unless_except_rule {s : program} {p q : pred' \u03b1} (exp : set event)\n  (ACT : \u2200 (e : s.lbl) \u03c3 Hc Hf \u03c3',\n        \u00ac s.event' e \u2208 exp\n      \u2192 (s.event' e).step \u03c3 Hc Hf \u03c3'\n      \u2192 p \u03c3 \u2192 \u00ac q \u03c3 \u2192 p \u03c3' \u2228 q \u03c3')\n: unless_except s p q exp :=\nbegin\n  intros \u03c3 \u03c3' STEP EXP H,\n  cases H with Hp Hq,\n  unfold step has_safety.step is_step at STEP,\n  cases STEP with e STEP,\n  cases STEP with Hc STEP,\n  cases STEP with Hf STEP,\n  cases e with e,\n  { dunfold nondet.program.event nondet.skip nondet.event.step at STEP,\n    subst \u03c3',\n    left, apply Hp },\n  { apply ACT e _ Hc Hf _ _ STEP Hp Hq,\n    intro Hin,\n    apply EXP,\n    clear ACT EXP,\n    dunfold program.step_of event.step_of program.event,\n    existsi s.event' e, split, apply Hin,\n    existsi Hc, existsi Hf,\n    apply STEP }\nend\n\ntheorem unless_rule {s : program} {p q : pred' \u03b1}\n  (ACT : \u2200 (e : s.lbl) \u03c3 Hc Hf \u03c3', (s.event' e).step \u03c3 Hc Hf \u03c3' \u2192 p \u03c3 \u2192 \u00ac q \u03c3 \u2192 p \u03c3' \u2228 q \u03c3')\n: unless s p q :=\nbegin\n  rw unless_eq_unless_except,\n  have H : unless_except s p q \u2205,\n  { apply unless_except_rule,\n    intros,\n    apply ACT\n    ; try { assumption }, },\n  unfold unless_except at H,\n  have Heq : (\u03bb (\u03c3 \u03c3' : \u03b1), \u2203 (e : event), e \u2208 (\u2205 : set event) \u2227 e.step_of \u03c3 \u03c3')\n             = (\u03bb (_x : \u03b1), False),\n  { apply funext, intro,\n    apply funext, intro,\n    simp },\n  apply iff.elim_left _ H,\n  rw iff_eq_eq,\n  apply congr_arg _ Heq,\nend\n\ntheorem ensure_rule {s : program} {p q : pred' \u03b1} (ev : option s.lbl)\n   (EN : p \u22c0 -q \u27f9 s.coarse_sch_of ev)\n   (FLW : p \u22c0 -q \u22c0 s.coarse_sch_of ev  \u21a6  s.fine_sch_of ev \u22c1 q in s)\n   (NEG : \u2200 \u03c3 \u03c3', \u00ac q \u03c3 \u2192 s.step_of ev \u03c3 \u03c3' \u2192 q \u03c3')\n   (STABLE: unless s p q )\n: p \u21a6 q in s :=\nbegin\n  apply @leads_to.basis' _ _ s _ _ (s.fine_sch_of ev) _ STABLE,\n  { apply leads_to.antimono_left _ _ FLW,\n    apply entails_p_and_of_entails,\n    { refl },\n    { apply EN } },\n  dunfold transient' system.transient program.transient,\n  existsi ev,\n  apply falsify.mk,\n    -- enablement\n  { apply EN },\n    -- follow\n  { refl },\n    -- negation\n  { intros \u03c3 Hf Hact,\n    have H := NEG (\u03c3 0) (\u03c3 1) Hf.right Hact,\n    revert H,\n    simp [not_init,next_init,not_and_iff_not_or_not,not_not_iff_self],\n    apply or.intro_left }\nend\n\nend nondet\n\nend nondet\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/models/nondet.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3638053392002824}}
{"text": "import .misc\nimport function.partial\n\n/-!\n * Analogue of `list.map` for partial functions.\n--/\n\nnamespace list\n\n--- Analogue of `map` for partial functions.\n@[reducible]\ndefinition map_partial {\u03b1 \u03b2 : Type _} (f : \u03b1 \u219d \u03b2) (l : list \u03b1) : list \u03b2 :=\n  list.rec_on l [] $\n    \u03bb a tl ind,\n      option.rec_on (f a) ind (\u03bb b, b :: ind)\n\n@[simp]\nlemma map_partial_nil {\u03b1 \u03b2 : Type _} {f : \u03b1 \u219d \u03b2} : map_partial f [] = [] := rfl\n\n@[simp]\nlemma map_partial_cons {\u03b1 \u03b2 : Type _} {f : \u03b1 \u219d \u03b2} {a : \u03b1} {tl : list \u03b1} : map_partial f (a::tl) = option.rec_on (f a) (map_partial f tl) (\u03bb b, b :: map_partial f tl) := rfl\n\n--- `filter` with values in subtypes\n@[reducible,inline]\ndefinition filter_to_subtype {\u03b1 : Sort _} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : list \u03b1) : list (subtype p) :=\n  l.map_partial (function.partial.coinj p)\n\n--- `filter_to_subtype` is nothing but `filter` if one forgets the `property` part.\n@[simp]\nlemma val_of_filter_to_subtype {\u03b1 : Sort _} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : list \u03b1} : map subtype.val (l.filter_to_subtype p) = l.filter p :=\n  begin\n    induction l with a tl h_ind,\n    case nil { refl },\n    case cons {\n      intros,\n      dsimp [filter_to_subtype,filter,function.partial.coinj] at *,\n      refine dite (p a) _ _,\n      show p a \u2192 _, {\n        intros hpa,\n        rw [dif_pos hpa,if_pos hpa],\n        dsimp *,\n        rw [h_ind]\n      },\n      show \u00acp a \u2192 _, {\n        intros hnpa,\n        rw [dif_neg hnpa,if_neg hnpa],\n        dsimp *,\n        rw [h_ind]\n      }\n    }\n  end\n\n--- Partial maps respect the membership relation on the domain.\nlemma mem_map_partial_of_mem {\u03b1 \u03b2 : Type _} {f : \u03b1 \u219d \u03b2} : \u2200 (x : f.domain) (l : list \u03b1), x.val \u2208 l \u2192 f.to_fun x \u2208 l.map_partial f :=\n  begin\n    intros x l hx,\n    induction l with a tl h_ind,\n    case nil { exfalso; exact not_mem_nil _ hx },\n    /- \u2193 induction step: l = a::tl \u2193 -/\n    rw [map_partial_cons],\n    cases hfa : f a with b,\n    case none {\n      dsimp *,\n      cases hx,\n      case or.inl { exfalso; exact x.property ((congr_arg f hx).trans hfa) },\n      case or.inr { exact h_ind hx }\n    },\n    case some {\n      dsimp *,\n      cases hx,\n      case or.inl {\n        suffices : f.to_fun x = b,\n          from or.inl this,\n        apply function.partial.to_fun_value_of_eq,\n        calc\n          f x.val = f a : congr_arg f hx\n          ...     = some b : hfa\n      },\n      case or.inr { right; exact h_ind hx },\n    }\n  end\n\n--- Injective partial maps reflect the membership relation.\nlemma mem_of_mem_map_partial {\u03b1 \u03b2 : Type _} {f : \u03b1 \u219d \u03b2} : f.injective \u2192 \u2200 (x : f.domain) (l : list \u03b1), f.to_fun x \u2208 l.map_partial f \u2192 x.val \u2208 l :=\n  begin\n    intros hinj x l hfmem,\n    induction l with a tl h_ind,\n    case nil { exfalso; exact not_mem_nil _ hfmem },\n    case cons {\n      cases hfa : f a with b hb,\n      case none {\n        rw [map_partial_cons, hfa] at hfmem,\n        dsimp * at hfmem,\n        exact or.inr (h_ind hfmem)\n      },\n      case some {\n        rw [map_partial_cons, hfa] at hfmem,\n        dsimp * at hfmem,\n        cases hfmem,\n        case or.inl {\n          left,\n          have : f x.val = f a,\n            calc\n              f x.val\n                  = some (f.to_fun x) : function.partial.to_fun_on_domain\n              ... = some b : by rw [hfmem]\n              ... = f a : hfa.symm,\n          exact hinj _ _ x.property (f.defined_iff_some.mpr \u27e8b,hfa\u27e9) this,\n        },\n        case or.inr { exact or.inr (h_ind hfmem) }\n      }\n    }\n  end\n\n--- `map_partial` with injective partial maps respect `list.nodup`.\nlemma nodup_map_partial_of_nodup {\u03b1 \u03b2 : Type _} {f : \u03b1 \u219d \u03b2} {l : list \u03b1} : f.injective \u2192 l.nodup \u2192 (l.map_partial f).nodup :=\n  begin\n    intros hinj hnodup,\n    induction l with a tl h_ind,\n    case nil { exact nodup.nil },\n    rw [map_partial_cons],\n    cases hfa: f a with b,\n    case none { exact h_ind (nodup_tail hnodup) },\n    case some {\n      dsimp *,\n      refine nodup.cons _ (h_ind (nodup_tail hnodup)),\n      let x : f.domain := \u27e8a, f.defined_iff_some.mpr \u27e8b,hfa\u27e9\u27e9,\n      have : f.to_fun x = b,\n        from f.to_fun_value_of_eq hfa,\n      rw [\u2190this],\n      apply mt (mem_of_mem_map_partial hinj x tl),\n      cases hnodup with _ _ h _; exact h\n    }\n  end\n\nend list\n", "meta": {"author": "Junology", "repo": "groth-lean", "sha": "5aa1ba624cd0f5145f63fa86130f99b85bbbcac2", "save_path": "github-repos/lean/Junology-groth-lean", "path": "github-repos/lean/Junology-groth-lean/groth-lean-5aa1ba624cd0f5145f63fa86130f99b85bbbcac2/src/data/list/map_partial.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6654105521116443, "lm_q1q2_score": 0.363805335572582}}
{"text": "import lemmas.canonical_forms\nimport lemmas.fv\nimport lemmas.subst\nimport lemmas.inversion\n\ntheorem progress\n  {\u0393: env}\n  {e: exp}\n  {\u03c4: typ}\n  (fv_e: fv e = [])\n  (et: has_typ \u0393 e \u03c4)\n  : val e \u2228 (\u2203 (e': exp), steps e e') :=\nbegin\n  induction et,\n  left,\n  exact val.int et_n,\n  left,\n  exact val.true,\n  left,\n  exact val.false,\n  right,\n  let emp := iff.elim_left (if_fv_empty et_e1 et_e2 et_e3) fv_e,\n  cases et_ih_a emp.left,\n  cases bool_canonical_forms h et_a,\n  rw h_1,\n  existsi et_e2,\n  exact steps.if_true,\n  rw h_1,\n  existsi et_e3,\n  exact steps.if_false,\n  cases h,\n  existsi exp.if_ h_w et_e2 et_e3,\n  exact steps.if_e1 h_h,\n  simp [fv] at fv_e,\n  exfalso,\n  exact fv_e,\n  left,\n  exact val.fn et_x et_\u03c41 et_e,\n  let emp := iff.elim_left (app_fv_empty et_e1 et_e2) fv_e,\n  cases et_ih_a emp.left,\n  cases et_ih_a_1 emp.right,\n  cases arrow_canonical_forms h et_a,\n  cases h_2,\n  right,\n  existsi subst et_e2 w emp.right h_2_w,\n  let d := steps.app_done emp.right h_1,\n  -- avoid weird 'motive is not type correct' error\n  rw symm h_2_h at d,\n  exact d,\n  right,\n  cases h_1,\n  existsi exp.app et_e1 h_1_w,\n  exact steps.app_e2 h h_1_h,\n  right,\n  cases h,\n  existsi exp.app h_w et_e2,\n  exact steps.app_e1 h_h,\n  left,\n  exact val.unit,\n  let emp := iff.elim_left (pair_fv_empty et_e1 et_e2) fv_e,\n  cases et_ih_a emp.left,\n  cases et_ih_a_1 emp.right,\n  left,\n  exact val.pair h h_1,\n  cases h_1,\n  right,\n  existsi exp.pair et_e1 h_1_w,\n  exact steps.pair_e2 h h_1_h,\n  cases h,\n  right,\n  existsi exp.pair h_w et_e2,\n  exact steps.pair_e1 h_h,\n  simp [pair_left_fv] at fv_e,\n  right,\n  cases et_ih fv_e,\n  cases pair_canonical_forms h et_a,\n  cases h_1,\n  rw h_1_h at h \u22a2,\n  existsi w,\n  exact steps.pair_left_done h,\n  cases h,\n  existsi exp.pair_left h_w,\n  exact steps.pair_left_arg h_h,\n  simp [pair_right_fv] at fv_e,\n  right,\n  cases et_ih fv_e,\n  cases pair_canonical_forms h et_a,\n  cases h_1,\n  rw h_1_h at h \u22a2,\n  existsi h_1_w,\n  exact steps.pair_right_done h,\n  cases h,\n  existsi exp.pair_right h_w,\n  exact steps.pair_right_arg h_h,\n  simp [either_left_fv] at fv_e,\n  cases et_ih fv_e,\n  left,\n  exact val.either_left h,\n  right,\n  cases h,\n  existsi exp.either_left et_\u03c42 h_w,\n  exact steps.either_left_arg h_h,\n  simp [either_right_fv] at fv_e,\n  cases et_ih fv_e,\n  left,\n  exact val.either_right h,\n  right,\n  cases h,\n  existsi exp.either_right et_\u03c41 h_w,\n  exact steps.either_right_arg h_h,\n  simp [case_never_fv] at fv_e,\n  cases et_ih fv_e,\n  exfalso,\n  exact never_canonical_forms h et_a,\n  right,\n  cases h,\n  existsi exp.case_never et_\u03c4 h_w,\n  exact steps.case_never_arg h_h,\n  right,\n  simp [case_fv] at fv_e,\n  let fv' := iff.elim_left append_nil_both fv_e,\n  cases et_ih_a fv'.left,\n  cases either_canonical_forms h et_a,\n  cases h_1,\n  rw h_1 at fv' h \u22a2,\n  rw either_left_fv w at fv',\n  existsi subst w et_x1 fv'.left et_e1,\n  exact steps.case_done_left fv'.left h,\n  rw h_1 at fv' h \u22a2,\n  rw either_right_fv w at fv',\n  existsi subst w et_x2 fv'.left et_e2,\n  exact steps.case_done_right fv'.left h,\n  cases h,\n  existsi exp.case h_w et_x1 et_e1 et_x2 et_e2,\n  exact steps.case_arg h_h,\nend\n\ntheorem preservation\n  {\u0393: env}\n  {e e': exp}\n  {\u03c4: typ}\n  (fv_e: fv e = [])\n  (et: has_typ \u0393 e \u03c4)\n  (st: steps e e')\n  : has_typ \u0393 e' \u03c4 \u2227 fv e' = [] :=\nbegin\n  induction st generalizing \u0393 \u03c4,\n  let inv := inversion_if et,\n  let emp := iff.elim_left (if_fv_empty st_e1 st_e2 st_e3) fv_e,\n  let e1'_ih := st_ih emp.left inv.left,\n  split,\n  exact has_typ.if_ e1'_ih.left inv.right.left inv.right.right,\n  exact iff.elim_right (if_fv_empty st_e1' st_e2 st_e3) (and.intro e1'_ih.right emp.right),\n  let inv := inversion_if et,\n  let emp := iff.elim_left (if_fv_empty exp.true st_e2 st_e3) fv_e,\n  exact and.intro inv.right.left emp.right.left,\n  let inv := inversion_if et,\n  let emp := iff.elim_left (if_fv_empty exp.false st_e2 st_e3) fv_e,\n  exact and.intro inv.right.right emp.right.right,\n  cases inversion_app et,\n  let emp := iff.elim_left (app_fv_empty st_e1 st_e2) fv_e,\n  let e1'_ih := st_ih emp.left h.left,\n  split,\n  exact has_typ.app e1'_ih.left h.right,\n  exact iff.elim_right (app_fv_empty st_e1' st_e2) (and.intro e1'_ih.right emp.right),\n  cases inversion_app et,\n  let emp := iff.elim_left (app_fv_empty st_e1 st_e2) fv_e,\n  let e2'_ih := st_ih emp.right h.right,\n  split,\n  exact has_typ.app h.left e2'_ih.left,\n  exact iff.elim_right (app_fv_empty st_e1 st_e2') (and.intro emp.left e2'_ih.right),\n  split,\n  cases inversion_app et,\n  cases inversion_fn h.left,\n  let a := typ.arrow.inj h_1.left,\n  rw a.left at h,\n  rw a.right,\n  exact subst_preservation rfl st_fv_e2 h_1.right h.right,\n  rw subst_fv st_e2 st_x st_fv_e2 st_e,\n  let emp := iff.elim_left (app_fv_empty (exp.fn st_x st_\u03c4 st_e) st_e2) fv_e,\n  rw fn_fv st_x st_\u03c4 st_e at emp,\n  exact emp.left,\n  cases inversion_prod et,\n  cases h,\n  simp [pair_fv_empty] at fv_e,\n  let a := st_ih fv_e.left h_h.right.left,\n  split,\n  rw h_h.left,\n  exact has_typ.pair a.left h_h.right.right,\n  exact iff.elim_right (pair_fv_empty st_e1' st_e2) (and.intro a.right fv_e.right),\n  cases inversion_prod et,\n  cases h,\n  simp [pair_fv_empty] at fv_e,\n  let a := st_ih fv_e.right h_h.right.right,\n  split,\n  rw h_h.left,\n  exact has_typ.pair h_h.right.left a.left,\n  exact iff.elim_right (pair_fv_empty st_e1 st_e2') (and.intro fv_e.left a.right),\n  cases inversion_prod_left et,\n  simp [pair_left_fv] at fv_e,\n  let a := st_ih fv_e h,\n  split,\n  exact has_typ.pair_left a.left,\n  simp [pair_left_fv],\n  exact a.right,\n  cases inversion_prod_left et,\n  cases inversion_prod h,\n  cases h_1,\n  let a := typ.pair.inj h_1_h.left,\n  rw a.left,\n  split,\n  exact h_1_h.right.left,\n  simp [pair_left_fv] at fv_e,\n  simp [pair_fv_empty] at fv_e,\n  exact fv_e.left,\n  cases inversion_prod_right et,\n  simp [pair_right_fv] at fv_e,\n  let a := st_ih fv_e h,\n  split,\n  exact has_typ.pair_right a.left,\n  simp [pair_right_fv],\n  exact a.right,\n  cases inversion_prod_right et,\n  cases inversion_prod h,\n  cases h_1,\n  let a := typ.pair.inj h_1_h.left,\n  rw a.right,\n  split,\n  exact h_1_h.right.right,\n  simp [pair_right_fv] at fv_e,\n  simp [pair_fv_empty] at fv_e,\n  exact fv_e.right,\n  simp [either_left_fv] at fv_e,\n  cases inversion_sum_left et,\n  let a := st_ih fv_e h.right,\n  rw h.left,\n  split,\n  exact has_typ.either_left a.left,\n  simp [either_left_fv],\n  exact a.right,\n  simp [either_right_fv] at fv_e,\n  cases inversion_sum_right et,\n  let a := st_ih fv_e h.right,\n  rw h.left,\n  split,\n  exact has_typ.either_right a.left,\n  simp [either_right_fv],\n  exact a.right,\n  simp [case_never_fv] at fv_e,\n  let h := inversion_case_never et,\n  let a := st_ih fv_e h.right,\n  rw h.left,\n  split,\n  exact has_typ.case_never a.left,\n  simp [case_never_fv],\n  exact a.right,\n  simp [case_fv] at fv_e,\n  cases inversion_case et,\n  cases h,\n  let fv' := iff.elim_left append_nil_both fv_e,\n  let a := st_ih fv'.left h_h.left,\n  split,\n  exact has_typ.case a.left h_h.right.left h_h.right.right,\n  simp [case_fv],\n  rw a.right,\n  rw fv'.left at fv_e,\n  exact fv_e,\n  simp [subst_fv],\n  cases inversion_case et,\n  cases h,\n  cases inversion_sum_left h_h.left,\n  let ty_eq := typ.either.inj h.left,\n  rw symm ty_eq.left at h,\n  split,\n  exact subst_preservation rfl st_fv_e h_h.right.left h.right,\n  simp [case_fv] at fv_e,\n  let a := iff.elim_left append_nil_both fv_e,\n  exact (iff.elim_left append_nil_both a.right).left,\n  simp [subst_fv],\n  cases inversion_case et,\n  cases h,\n  cases inversion_sum_right h_h.left,\n  let ty_eq := typ.either.inj h.left,\n  rw symm ty_eq.right at h,\n  split,\n  exact subst_preservation rfl st_fv_e h_h.right.right h.right,\n  simp [case_fv] at fv_e,\n  let a := iff.elim_left append_nil_both fv_e,\n  exact (iff.elim_left append_nil_both a.right).right,\nend\n\ntheorem safety\n  {\u0393: env}\n  {e: exp}\n  {\u03c4: typ}\n  (fv_e: fv e = [])\n  (et: has_typ \u0393 e \u03c4)\n  : val e \u2228 (\u2203 (e': exp), steps e e' \u2227 has_typ \u0393 e' \u03c4 \u2227 fv e' = []) :=\nbegin\n  cases progress fv_e et,\n  left,\n  exact h,\n  cases h,\n  right,\n  existsi h_w,\n  split,\n  exact h_h,\n  exact preservation fv_e et h_h,\nend\n\ntheorem uniqueness {\u0393: env} {e: exp} {\u03c4 \u03c4': typ}:\n  has_typ \u0393 e \u03c4 ->\n  has_typ \u0393 e \u03c4' ->\n  \u03c4 = \u03c4' :=\nbegin\n  intros h1 h2,\n  induction h1 generalizing \u03c4',\n  cases h2,\n  refl,\n  cases h2,\n  refl,\n  cases h2,\n  refl,\n  cases h2,\n  exact h1_ih_a_1 h2_a_1,\n  cases h2,\n  exact lookup_uniq h1_a h2_a,\n  cases h2,\n  rw h1_ih h2_a,\n  cases h2,\n  exact (typ.arrow.inj (h1_ih_a h2_a)).right,\n  cases h2,\n  refl,\n  cases h2,\n  rw h1_ih_a h2_a,\n  rw h1_ih_a_1 h2_a_1,\n  cases h2,\n  exact (typ.pair.inj (h1_ih h2_a)).left,\n  cases h2,\n  exact (typ.pair.inj (h1_ih h2_a)).right,\n  cases h2,\n  rw h1_ih h2_a,\n  cases h2,\n  rw h1_ih h2_a,\n  cases h2,\n  refl,\n  cases h2,\n  rw (typ.either.inj (h1_ih_a h2_a)).left at h1_ih_a_1,\n  exact h1_ih_a_1 h2_a_1,\nend\n", "meta": {"author": "azdavis", "repo": "hatsugen", "sha": "a18f70f9ea4ce30c0baf0c40748aad5ccd176c60", "save_path": "github-repos/lean/azdavis-hatsugen", "path": "github-repos/lean/azdavis-hatsugen/hatsugen-a18f70f9ea4ce30c0baf0c40748aad5ccd176c60/src/theorems/safety.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.36369192491340807}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.shapes.products\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.cone_category\nimport category_theory.adjunction\n\n/-!\n\n# Multi-(co)equalizers\n\nA *multiequalizer* is an equalizer of two morphisms between two products.\nSince both products and equalizers are limits, such an object is again a limit.\nThis file provides the diagram whose limit is indeed such an object.\nIn fact, it is well-known that any limit can be obtained as a multiequalizer.\nThe dual construction (multicoequalizers) is also provided.\n\n## Projects\n\nProve that a multiequalizer can be identified with\nan equalizer between products (and analogously for multicoequalizers).\n\nProve that the limit of any diagram is a multiequalizer (and similarly for colimits).\n\n-/\n\nnamespace category_theory.limits\n\nopen category_theory\n\nuniverses v u\n\n/-- The type underlying the multiequalizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multicospan {L R : Type v} (fst snd : R \u2192 L) : Type v\n| left : L \u2192 walking_multicospan\n| right : R \u2192 walking_multicospan\n\n/-- The type underlying the multiecoqualizer diagram. -/\n@[nolint unused_arguments]\ninductive walking_multispan {L R : Type v} (fst snd : L \u2192 R) : Type v\n| left : L \u2192 walking_multispan\n| right : R \u2192 walking_multispan\n\nnamespace walking_multicospan\n\nvariables {L R : Type v} {fst snd : R \u2192 L}\n\ninstance [inhabited L] : inhabited (walking_multicospan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multicospan`. -/\ninductive hom : \u03a0 (a b : walking_multicospan fst snd), Type v\n| id (A)  : hom A A\n| fst (b) : hom (left (fst b)) (right b)\n| snd (b) : hom (left (snd b)) (right b)\n\ninstance {a : walking_multicospan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multicospan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multicospan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst b) (hom.id X) := hom.fst b\n| _ _ _ (hom.snd b) (hom.id X) := hom.snd b\n\ninstance : small_category (walking_multicospan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multicospan\n\nnamespace walking_multispan\n\nvariables {L R : Type v} {fst snd : L \u2192 R}\n\ninstance [inhabited L] : inhabited (walking_multispan fst snd) :=\n\u27e8left default\u27e9\n\n/-- Morphisms for `walking_multispan`. -/\ninductive hom : \u03a0 (a b : walking_multispan fst snd), Type v\n| id (A)  : hom A A\n| fst (a) : hom (left a) (right (fst a))\n| snd (a) : hom (left a) (right (snd a))\n\ninstance {a : walking_multispan fst snd} : inhabited (hom a a) :=\n\u27e8hom.id _\u27e9\n\n/-- Composition of morphisms for `walking_multispan`. -/\ndef hom.comp : \u03a0 {A B C : walking_multispan fst snd} (f : hom A B) (g : hom B C),\n  hom A C\n| _ _ _ (hom.id X) f := f\n| _ _ _ (hom.fst a) (hom.id X) := hom.fst a\n| _ _ _ (hom.snd a) (hom.id X) := hom.snd a\n\ninstance : small_category (walking_multispan fst snd) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z, hom.comp,\n  id_comp' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  comp_id' := by { rintro (_|_) (_|_) (_|_|_), tidy },\n  assoc' := by { rintro (_|_) (_|_) (_|_) (_|_) (_|_|_) (_|_|_) (_|_|_), tidy } }\n\nend walking_multispan\n\n/-- This is a structure encapsulating the data necessary to define a `multicospan`. -/\n@[nolint has_inhabited_instance]\nstructure multicospan_index (C : Type u) [category.{v} C] :=\n(L R : Type v)\n(fst_to snd_to : R \u2192 L)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 b, left (fst_to b) \u27f6 right b)\n(snd : \u03a0 b, left (snd_to b) \u27f6 right b)\n\n/-- This is a structure encapsulating the data necessary to define a `multispan`. -/\n@[nolint has_inhabited_instance]\nstructure multispan_index (C : Type u) [category.{v} C] :=\n(L R : Type v)\n(fst_from snd_from : L \u2192 R)\n(left : L \u2192 C)\n(right : R \u2192 C)\n(fst : \u03a0 a, left a \u27f6 right (fst_from a))\n(snd : \u03a0 a, left a \u27f6 right (snd_from a))\n\nnamespace multicospan_index\n\nvariables {C : Type u} [category.{v} C] (I : multicospan_index C)\n\n/-- The multicospan associated to `I : multicospan_index`. -/\ndef multicospan : walking_multicospan I.fst_to I.snd_to \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multicospan.left a := I.left a\n  | walking_multicospan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multicospan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multicospan.hom.fst b := I.fst _\n  | _, _, walking_multicospan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multicospan_obj_left (a) :\n  I.multicospan.obj (walking_multicospan.left a) = I.left a := rfl\n\n@[simp] lemma multicospan_obj_right (b) :\n  I.multicospan.obj (walking_multicospan.right b) = I.right b := rfl\n\n@[simp] lemma multicospan_map_fst (b) :\n  I.multicospan.map (walking_multicospan.hom.fst b) = I.fst b := rfl\n\n@[simp] lemma multicospan_map_snd (b) :\n  I.multicospan.map (walking_multicospan.hom.snd b) = I.snd b := rfl\n\nvariables [has_product I.left] [has_product I.right]\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.fst`. -/\nnoncomputable\ndef fst_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.fst_to b) \u226b I.fst b)\n\n/-- The induced map `\u220f I.left \u27f6 \u220f I.right` via `I.snd`. -/\nnoncomputable\ndef snd_pi_map : \u220f I.left \u27f6 \u220f I.right := pi.lift (\u03bb b, pi.\u03c0 I.left (I.snd_to b) \u226b I.snd b)\n\n@[simp, reassoc]\nlemma fst_pi_map_\u03c0 (b) : I.fst_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.fst b :=\nby simp [fst_pi_map]\n\n@[simp, reassoc]\nlemma snd_pi_map_\u03c0 (b) : I.snd_pi_map \u226b pi.\u03c0 I.right b = pi.\u03c0 I.left _ \u226b I.snd b :=\nby simp [snd_pi_map]\n\n/--\nTaking the multiequalizer over the multicospan index is equivalent to taking the equalizer over\nthe two morphsims `\u220f I.left \u21c9 \u220f I.right`. This is the diagram of the latter.\n-/\n@[simps] protected noncomputable\ndef parallel_pair_diagram := parallel_pair I.fst_pi_map I.snd_pi_map\n\nend multicospan_index\n\nnamespace multispan_index\n\nvariables {C : Type u} [category.{v} C] (I : multispan_index C)\n\n/-- The multispan associated to `I : multispan_index`. -/\ndef multispan : walking_multispan I.fst_from I.snd_from \u2964 C :=\n{ obj := \u03bb x,\n  match x with\n  | walking_multispan.left a := I.left a\n  | walking_multispan.right b := I.right b\n  end,\n  map := \u03bb x y f,\n  match x, y, f with\n  | _, _, walking_multispan.hom.id x := \ud835\udfd9 _\n  | _, _, walking_multispan.hom.fst b := I.fst _\n  | _, _, walking_multispan.hom.snd b := I.snd _\n  end,\n  map_id' := by { rintros (_|_), tidy },\n  map_comp' := by { rintros (_|_) (_|_) (_|_) (_|_|_) (_|_|_), tidy } }\n\n@[simp] lemma multispan_obj_left (a) :\n  I.multispan.obj (walking_multispan.left a) = I.left a := rfl\n\n@[simp] lemma multispan_obj_right (b) :\n  I.multispan.obj (walking_multispan.right b) = I.right b := rfl\n\n@[simp] lemma multispan_map_fst (a) :\n  I.multispan.map (walking_multispan.hom.fst a) = I.fst a := rfl\n\n@[simp] lemma multispan_map_snd (a) :\n  I.multispan.map (walking_multispan.hom.snd a) = I.snd a := rfl\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.fst`. -/\nnoncomputable\ndef fst_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.fst b \u226b sigma.\u03b9 _ (I.fst_from b))\n\n/-- The induced map `\u2210 I.left \u27f6 \u2210 I.right` via `I.snd`. -/\nnoncomputable\ndef snd_sigma_map : \u2210 I.left \u27f6 \u2210 I.right := sigma.desc (\u03bb b, I.snd b \u226b sigma.\u03b9 _ (I.snd_from b))\n\n@[simp, reassoc]\nlemma \u03b9_fst_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.fst_sigma_map = I.fst b \u226b sigma.\u03b9 I.right _ :=\nby simp [fst_sigma_map]\n\n@[simp, reassoc]\nlemma \u03b9_snd_sigma_map (b) : sigma.\u03b9 I.left b \u226b I.snd_sigma_map = I.snd b \u226b sigma.\u03b9 I.right _ :=\nby simp [snd_sigma_map]\n\n/--\nTaking the multicoequalizer over the multispan index is equivalent to taking the coequalizer over\nthe two morphsims `\u2210 I.left \u21c9 \u2210 I.right`. This is the diagram of the latter.\n-/\nprotected noncomputable\nabbreviation parallel_pair_diagram := parallel_pair I.fst_sigma_map I.snd_sigma_map\n\nend multispan_index\n\nvariables {C : Type u} [category.{v} C]\n\n/-- A multifork is a cone over a multicospan. -/\n@[nolint has_inhabited_instance]\nabbreviation multifork (I : multicospan_index C) := cone I.multicospan\n\n/-- A multicofork is a cocone over a multispan. -/\n@[nolint has_inhabited_instance]\nabbreviation multicofork (I : multispan_index C) := cocone I.multispan\n\nnamespace multifork\n\nvariables {I : multicospan_index C} (K : multifork I)\n\n/-- The maps from the cone point of a multifork to the objects on the left. -/\ndef \u03b9 (a : I.L) : K.X \u27f6 I.left a :=\nK.\u03c0.app (walking_multicospan.left _)\n\n@[simp] lemma \u03b9_eq_app_left (a) : K.\u03b9 a = K.\u03c0.app (walking_multicospan.left _) := rfl\n\n@[simp] lemma app_left_fst (b) :\n  K.\u03c0.app (walking_multicospan.left (I.fst_to b)) \u226b I.fst b =\n    K.\u03c0.app (walking_multicospan.right b) :=\nby { rw \u2190 K.w (walking_multicospan.hom.fst b), refl }\n\n@[simp] lemma app_left_snd (b) :\n  K.\u03c0.app (walking_multicospan.left (I.snd_to b)) \u226b I.snd b =\n    K.\u03c0.app (walking_multicospan.right b) :=\nby { rw \u2190 K.w (walking_multicospan.hom.snd b), refl }\n\n/-- Construct a multifork using a collection `\u03b9` of morphisms. -/\n@[simps]\ndef of_\u03b9 (I : multicospan_index C) (P : C) (\u03b9 : \u03a0 a, P \u27f6 I.left a)\n  (w : \u2200 b, \u03b9 (I.fst_to b) \u226b I.fst b = \u03b9 (I.snd_to b) \u226b I.snd b) :\n  multifork I :=\n{ X := P,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := \u03b9 _\n    | walking_multicospan.right b := \u03b9 (I.fst_to b) \u226b I.fst b\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { dsimp, rw category.id_comp, refl },\n      { dsimp, rw category.id_comp, apply w }\n    end } }\n\n@[reassoc]\nlemma condition (b) :\n  K.\u03b9 (I.fst_to b) \u226b I.fst b = K.\u03b9 (I.snd_to b) \u226b I.snd b := by simp\n\n/-- This definition provides a convenient way to show that a multifork is a limit. -/\n@[simps]\ndef is_limit.mk\n  (lift : \u03a0 (E : multifork I), E.X \u27f6 K.X)\n  (fac : \u2200 (E : multifork I) (i : I.L), lift E \u226b K.\u03b9 i = E.\u03b9 i)\n  (uniq : \u2200 (E : multifork I) (m : E.X \u27f6 K.X),\n    (\u2200 i : I.L, m \u226b K.\u03b9 i = E.\u03b9 i) \u2192 m = lift E) : is_limit K :=\n{ lift := lift,\n  fac' := begin\n    rintros E (a|b),\n    { apply fac },\n    { rw [\u2190 E.w (walking_multicospan.hom.fst b), \u2190 K.w (walking_multicospan.hom.fst b),\n        \u2190 category.assoc],\n      congr' 1,\n      apply fac }\n  end,\n  uniq' := begin\n    rintros E m hm,\n    apply uniq,\n    intros i,\n    apply hm,\n  end }\n\n\nvariables [has_product I.left] [has_product I.right]\n\n@[simp, reassoc]\nlemma pi_condition :\n  pi.lift K.\u03b9 \u226b I.fst_pi_map = pi.lift K.\u03b9 \u226b I.snd_pi_map := by { ext, simp }\n\n/-- Given a multifork, we may obtain a fork over `\u220f I.left \u21c9 \u220f I.right`. -/\n@[simps X] noncomputable\ndef to_pi_fork (K : multifork I) : fork I.fst_pi_map I.snd_pi_map :=\n{ X := K.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := pi.lift K.\u03b9\n    | walking_parallel_pair.one := pi.lift K.\u03b9 \u226b I.fst_pi_map\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      all_goals { change \ud835\udfd9 _ \u226b _ \u226b _ = pi.lift _ \u226b _, simp }\n    end } }\n\n@[simp] lemma to_pi_fork_\u03c0_app_zero :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.zero = pi.lift K.\u03b9 := rfl\n\n@[simp] lemma to_pi_fork_\u03c0_app_one :\n  K.to_pi_fork.\u03c0.app walking_parallel_pair.one = pi.lift K.\u03b9 \u226b I.fst_pi_map := rfl\n\nvariable (I)\n\n/-- Given a fork over `\u220f I.left \u21c9 \u220f I.right`, we may obtain a multifork. -/\n@[simps X] noncomputable\ndef of_pi_fork (c : fork I.fst_pi_map I.snd_pi_map) : multifork I :=\n{ X := c.X,\n  \u03c0 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multicospan.left a := c.\u03b9 \u226b pi.\u03c0 _ _\n    | walking_multicospan.right b := c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { symmetry, dsimp, rw category.id_comp, apply category.comp_id },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, simp },\n      { change \ud835\udfd9 _ \u226b _ \u226b _ = (_ \u226b _) \u226b _, rw c.condition_assoc, simp }\n    end } }\n\n@[simp] lemma of_pi_fork_\u03c0_app_left (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.left a) = c.\u03b9 \u226b pi.\u03c0 _ _ := rfl\n\n@[simp] lemma of_pi_fork_\u03c0_app_right (c : fork I.fst_pi_map I.snd_pi_map) (a) :\n  (of_pi_fork I c).\u03c0.app (walking_multicospan.right a) = c.\u03b9 \u226b I.fst_pi_map \u226b pi.\u03c0 _ _ := rfl\n\nend multifork\n\nnamespace multicospan_index\n\nvariables (I : multicospan_index C) [has_product I.left] [has_product I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multifork.to_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef to_pi_fork_functor : multifork I \u2964 fork I.fst_pi_map I.snd_pi_map :=\n{ obj := multifork.to_pi_fork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multifork.of_pi_fork` is functorial. -/\n@[simps] noncomputable\ndef of_pi_fork_functor : fork I.fst_pi_map I.snd_pi_map \u2964 multifork I :=\n{ obj := multifork.of_pi_fork I, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multiforks is equivalent to the category of forks over `\u220f I.left \u21c9 \u220f I.right`.\nIt then follows from `category_theory.is_limit_of_preserves_cone_terminal` (or `reflects`) that it\npreserves and reflects limit cones.\n-/\n@[simps] noncomputable\ndef multifork_equiv_pi_fork : multifork I \u224c fork I.fst_pi_map I.snd_pi_map :=\n{ functor := to_pi_fork_functor I,\n  inverse := of_pi_fork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cones.ext (iso.refl _) (by rintros (_|_); dsimp; simp))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, fork.ext (iso.refl _) (by { ext, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }) }\n\nend multicospan_index\n\nnamespace multicofork\n\nvariables {I : multispan_index C} (K : multicofork I)\n\n/-- The maps to the cocone point of a multicofork from the objects on the right. -/\ndef \u03c0 (b : I.R) : I.right b \u27f6 K.X :=\nK.\u03b9.app (walking_multispan.right _)\n\n@[simp] lemma \u03c0_eq_app_right (b) : K.\u03c0 b = K.\u03b9.app (walking_multispan.right _) := rfl\n\n@[simp] lemma fst_app_right (a) :\n  I.fst a \u226b K.\u03b9.app (walking_multispan.right (I.fst_from a)) =\n    K.\u03b9.app (walking_multispan.left a) :=\nby { rw \u2190 K.w (walking_multispan.hom.fst a), refl }\n\n@[simp] lemma snd_app_right (a) :\n  I.snd a \u226b K.\u03b9.app (walking_multispan.right (I.snd_from a)) =\n    K.\u03b9.app (walking_multispan.left a) :=\nby { rw \u2190 K.w (walking_multispan.hom.snd a), refl }\n\n/-- Construct a multicofork using a collection `\u03c0` of morphisms. -/\n@[simps]\ndef of_\u03c0 (I : multispan_index C) (P : C) (\u03c0 : \u03a0 b, I.right b \u27f6 P)\n  (w : \u2200 a, I.fst a \u226b \u03c0 (I.fst_from a) = I.snd a \u226b \u03c0 (I.snd_from a)) :\n  multicofork I :=\n{ X := P,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := I.fst a \u226b \u03c0 _\n    | walking_multispan.right b := \u03c0 _\n    end,\n    naturality' := begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { dsimp, rw category.comp_id, refl },\n      { dsimp, rw category.comp_id, apply (w _).symm }\n    end } }\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b K.\u03c0 (I.fst_from a) = I.snd a \u226b K.\u03c0 (I.snd_from a) := by simp\n\n/-- This definition provides a convenient way to show that a multicofork is a colimit. -/\n@[simps]\ndef is_colimit.mk\n  (desc : \u03a0 (E : multicofork I), K.X \u27f6 E.X)\n  (fac : \u2200 (E : multicofork I) (i : I.R), K.\u03c0 i \u226b desc E = E.\u03c0 i)\n  (uniq : \u2200 (E : multicofork I) (m : K.X \u27f6 E.X),\n    (\u2200 i : I.R, K.\u03c0 i \u226b m = E.\u03c0 i) \u2192 m = desc E) : is_colimit K :=\n{ desc := desc,\n  fac' := begin\n    rintros S (a|b),\n    { rw [\u2190 K.w (walking_multispan.hom.fst a), \u2190 S.w (walking_multispan.hom.fst a),\n        category.assoc],\n      congr' 1,\n      apply fac },\n    { apply fac },\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply uniq,\n    intros i,\n    apply hm\n  end }\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\n@[simp, reassoc]\nlemma sigma_condition :\n  I.fst_sigma_map \u226b sigma.desc K.\u03c0 = I.snd_sigma_map \u226b sigma.desc K.\u03c0 := by { ext, simp }\n\n/-- Given a multicofork, we may obtain a cofork over `\u2210 I.left \u21c9 \u2210 I.right`. -/\n@[simps X] noncomputable\ndef to_sigma_cofork (K : multicofork I) : cofork I.fst_sigma_map I.snd_sigma_map :=\n{ X := K.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_parallel_pair.zero := I.fst_sigma_map \u226b sigma.desc K.\u03c0\n    | walking_parallel_pair.one := sigma.desc K.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      all_goals { change _ \u226b sigma.desc _ = (_ \u226b _) \u226b \ud835\udfd9 _, simp }\n    end } }\n\n@[simp] lemma to_sigma_cofork_\u03b9_app_zero :\n  K.to_sigma_cofork.\u03b9.app walking_parallel_pair.zero = I.fst_sigma_map \u226b sigma.desc K.\u03c0 := rfl\n\n@[simp] lemma to_sigma_cofork_\u03b9_app_one :\n  K.to_sigma_cofork.\u03b9.app walking_parallel_pair.one = sigma.desc K.\u03c0 := rfl\n\nvariable (I)\n\n/-- Given a cofork over `\u2210 I.left \u21c9 \u2210 I.right`, we may obtain a multicofork. -/\n@[simps X] noncomputable\ndef of_sigma_cofork (c : cofork I.fst_sigma_map I.snd_sigma_map) : multicofork I :=\n{ X := c.X,\n  \u03b9 :=\n  { app := \u03bb x,\n    match x with\n    | walking_multispan.left a := (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0\n    | walking_multispan.right b := (sigma.\u03b9 I.right b : _) \u226b c.\u03c0\n    end,\n    naturality' :=\n    begin\n      rintros (_|_) (_|_) (_|_|_),\n      any_goals { dsimp, rw category.comp_id, apply category.id_comp },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b _,\n        dsimp, simp [\u2190cofork.left_app_one, -cofork.left_app_one] },\n      { change _ \u226b _ \u226b _ = (_ \u226b _) \u226b \ud835\udfd9 _,\n        rw c.condition,\n        dsimp, simp [\u2190cofork.right_app_one, -cofork.right_app_one] }\n    end } }\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_left (c : cofork I.fst_sigma_map I.snd_sigma_map) (a) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.left a) =\n    (sigma.\u03b9 I.left a : _) \u226b I.fst_sigma_map \u226b c.\u03c0 := rfl\n\n@[simp] lemma of_sigma_cofork_\u03b9_app_right (c : cofork I.fst_sigma_map I.snd_sigma_map) (b) :\n  (of_sigma_cofork I c).\u03b9.app (walking_multispan.right b) = (sigma.\u03b9 I.right b : _) \u226b c.\u03c0 := rfl\n\nend multicofork\n\nnamespace multispan_index\n\nvariables (I : multispan_index C) [has_coproduct I.left] [has_coproduct I.right]\n\nlocal attribute [tidy] tactic.case_bash\n\n/-- `multicofork.to_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef to_sigma_cofork_functor : multicofork I \u2964 cofork I.fst_sigma_map I.snd_sigma_map :=\n{ obj := multicofork.to_sigma_cofork, map := \u03bb K\u2081 K\u2082 f, { hom := f.hom } }\n\n/-- `multicofork.of_sigma_cofork` is functorial. -/\n@[simps] noncomputable\ndef of_sigma_cofork_functor : cofork I.fst_sigma_map I.snd_sigma_map \u2964 multicofork I :=\n{ obj := multicofork.of_sigma_cofork I,\n  map := \u03bb K\u2081 K\u2082 f, { hom := f.hom, w' := by rintros (_|_); simp } }\n\n/--\nThe category of multicoforks is equivalent to the category of coforks over `\u2210 I.left \u21c9 \u2210 I.right`.\nIt then follows from `category_theory.is_colimit_of_preserves_cocone_initial` (or `reflects`) that\nit preserves and reflects colimit cocones.\n-/\n@[simps] noncomputable\ndef multicofork_equiv_sigma_cofork : multicofork I \u224c cofork I.fst_sigma_map I.snd_sigma_map :=\n{ functor := to_sigma_cofork_functor I,\n  inverse := of_sigma_cofork_functor I,\n  unit_iso := nat_iso.of_components (\u03bb K, cocones.ext (iso.refl _) (by rintros (_|_); dsimp; simp))\n    (\u03bb K\u2081 K\u2082 f, by { ext, simp }),\n  counit_iso := nat_iso.of_components (\u03bb K, cofork.ext (iso.refl _) (by { ext, dsimp, simp }))\n    (\u03bb K\u2081 K\u2082 f, by { ext, dsimp, simp, }) }\n\nend multispan_index\n\n/-- For `I : multicospan_index C`, we say that it has a multiequalizer if the associated\n  multicospan has a limit. -/\nabbreviation has_multiequalizer (I : multicospan_index C) :=\n  has_limit I.multicospan\n\nnoncomputable theory\n\n/-- The multiequalizer of `I : multicospan_index C`. -/\nabbreviation multiequalizer (I : multicospan_index C) [has_multiequalizer I] : C :=\n  limit I.multicospan\n\n/-- For `I : multispan_index C`, we say that it has a multicoequalizer if\n  the associated multicospan has a limit. -/\nabbreviation has_multicoequalizer (I : multispan_index C) :=\n  has_colimit I.multispan\n\n/-- The multiecoqualizer of `I : multispan_index C`. -/\nabbreviation multicoequalizer (I : multispan_index C) [has_multicoequalizer I] : C :=\n  colimit I.multispan\n\nnamespace multiequalizer\n\nvariables (I : multicospan_index C) [has_multiequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03b9 (a : I.L) : multiequalizer I \u27f6 I.left a :=\nlimit.\u03c0 _ (walking_multicospan.left a)\n\n/-- The multifork associated to the multiequalizer. -/\nabbreviation multifork : multifork I :=\nlimit.cone _\n\n@[simp]\nlemma multifork_\u03b9 (a) :\n  (multiequalizer.multifork I).\u03b9 a = multiequalizer.\u03b9 I a := rfl\n\n@[simp]\nlemma multifork_\u03c0_app_left (a) :\n  (multiequalizer.multifork I).\u03c0.app (walking_multicospan.left a) =\n  multiequalizer.\u03b9 I a := rfl\n\n@[reassoc]\nlemma condition (b) :\n  multiequalizer.\u03b9 I (I.fst_to b) \u226b I.fst b =\n  multiequalizer.\u03b9 I (I.snd_to b) \u226b I.snd b :=\nmultifork.condition _ _\n\n/-- Construct a morphism to the multiequalizer from its universal property. -/\nabbreviation lift (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) :\n  W \u27f6 multiequalizer I :=\nlimit.lift _ (multifork.of_\u03b9 I _ k h)\n\n@[simp, reassoc]\nlemma lift_\u03b9 (W : C) (k : \u03a0 a, W \u27f6 I.left a)\n  (h : \u2200 b, k (I.fst_to b) \u226b I.fst b = k (I.snd_to b) \u226b I.snd b) (a) :\n  multiequalizer.lift I _ k h \u226b multiequalizer.\u03b9 I a = k _ :=\nlimit.lift_\u03c0 _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : W \u27f6 multiequalizer I)\n  (h : \u2200 a, i \u226b multiequalizer.\u03b9 I a =\n  j \u226b multiequalizer.\u03b9 I a) :\n  i = j :=\nlimit.hom_ext\nbegin\n  rintro (a|b),\n  { apply h },\n  simp_rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n    \u2190 category.assoc, h],\nend\n\nvariables [has_product I.left] [has_product I.right]\n\ninstance : has_equalizer I.fst_pi_map I.snd_pi_map :=\n\u27e8\u27e8\u27e8_,is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.functor (limit.is_limit _)\u27e9\u27e9\u27e9\n\n/-- The multiequalizer is isomorphic to the equalizer of `\u220f I.left \u21c9 \u220f I.right`. -/\ndef iso_equalizer : multiequalizer I \u2245 equalizer I.fst_pi_map I.snd_pi_map :=\nlimit.iso_limit_cone \u27e8_, is_limit.of_preserves_cone_terminal\n  I.multifork_equiv_pi_fork.inverse (limit.is_limit _)\u27e9\n\n/-- The canonical injection `multiequalizer I \u27f6 \u220f I.left`. -/\ndef \u03b9_pi : multiequalizer I \u27f6 \u220f I.left :=\n  (iso_equalizer I).hom \u226b equalizer.\u03b9 I.fst_pi_map I.snd_pi_map\n\n@[simp, reassoc]\nlemma \u03b9_pi_\u03c0 (a) : \u03b9_pi I \u226b pi.\u03c0 I.left a = \u03b9 I a :=\nby { rw [\u03b9_pi, category.assoc, \u2190 iso.eq_inv_comp, iso_equalizer], simpa }\n\ninstance : mono (\u03b9_pi I) := @@mono_comp _ _ _ _ equalizer.\u03b9_mono\n\nend multiequalizer\n\nnamespace multicoequalizer\n\nvariables (I : multispan_index C) [has_multicoequalizer I]\n\n/-- The canonical map from the multiequalizer to the objects on the left. -/\nabbreviation \u03c0 (b : I.R) : I.right b \u27f6 multicoequalizer I :=\ncolimit.\u03b9 I.multispan (walking_multispan.right _)\n\n/-- The multicofork associated to the multicoequalizer. -/\nabbreviation multicofork : multicofork I :=\ncolimit.cocone _\n\n@[simp]\nlemma multicofork_\u03c0 (b) :\n  (multicoequalizer.multicofork I).\u03c0 b = multicoequalizer.\u03c0 I b := rfl\n\n@[simp]\nlemma multicofork_\u03b9_app_right (b) :\n  (multicoequalizer.multicofork I).\u03b9.app (walking_multispan.right b) =\n  multicoequalizer.\u03c0 I b := rfl\n\n@[reassoc]\nlemma condition (a) :\n  I.fst a \u226b multicoequalizer.\u03c0 I (I.fst_from a) =\n  I.snd a \u226b multicoequalizer.\u03c0 I (I.snd_from a) :=\nmulticofork.condition _ _\n\n/-- Construct a morphism from the multicoequalizer from its universal property. -/\nabbreviation desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) :\n  multicoequalizer I \u27f6 W :=\ncolimit.desc _ (multicofork.of_\u03c0 I _ k h)\n\n@[simp, reassoc]\nlemma \u03c0_desc (W : C) (k : \u03a0 b, I.right b \u27f6 W)\n  (h : \u2200 a, I.fst a \u226b  k (I.fst_from a) = I.snd a \u226b k (I.snd_from a)) (b) :\n  multicoequalizer.\u03c0 I b \u226b multicoequalizer.desc I _ k h = k _ :=\ncolimit.\u03b9_desc _ _\n\n@[ext]\nlemma hom_ext {W : C} (i j : multicoequalizer I \u27f6 W)\n  (h : \u2200 b, multicoequalizer.\u03c0 I b \u226b i = multicoequalizer.\u03c0 I b \u226b j) :\n  i = j :=\ncolimit.hom_ext\nbegin\n  rintro (a|b),\n  { simp_rw [\u2190 colimit.w I.multispan (walking_multispan.hom.fst a),\n    category.assoc, h] },\n  { apply h },\nend\n\nvariables [has_coproduct I.left] [has_coproduct I.right]\n\ninstance : has_coequalizer I.fst_sigma_map I.snd_sigma_map :=\n\u27e8\u27e8\u27e8_,is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.functor (colimit.is_colimit _)\u27e9\u27e9\u27e9\n\n/-- The multicoequalizer is isomorphic to the coequalizer of `\u2210 I.left \u21c9 \u2210 I.right`. -/\ndef iso_coequalizer : multicoequalizer I \u2245 coequalizer I.fst_sigma_map I.snd_sigma_map :=\ncolimit.iso_colimit_cocone \u27e8_, is_colimit.of_preserves_cocone_initial\n  I.multicofork_equiv_sigma_cofork.inverse (colimit.is_colimit _)\u27e9\n\n/-- The canonical projection `\u2210 I.right \u27f6 multicoequalizer I`. -/\ndef sigma_\u03c0 : \u2210 I.right \u27f6 multicoequalizer I :=\n  coequalizer.\u03c0 I.fst_sigma_map I.snd_sigma_map \u226b (iso_coequalizer I).inv\n\n@[simp, reassoc]\nlemma \u03b9_sigma_\u03c0 (b) : sigma.\u03b9 I.right b \u226b sigma_\u03c0 I = \u03c0 I b :=\nby { rw [sigma_\u03c0, \u2190 category.assoc, iso.comp_inv_eq, iso_coequalizer], simpa }\n\ninstance : epi (sigma_\u03c0 I) := @@epi_comp _ _ coequalizer.\u03c0_epi _ _\n\nend multicoequalizer\n\nend category_theory.limits\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/limits/shapes/multiequalizer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.363691916887763}}
{"text": "import .tab\n\nvariables {p q r s : Prop}\n\nopen tactic\n\nexample (h : p \u2227 q) : false := by split_conj\n\n-- meta def find_conj' : list expr \u2192 tactic expr\n-- | []        := failed\n-- | (e :: es) := do trace \"expr : \", trace e,\n--                   t \u2190 infer_type e,\n--                   trace \"expr type : \", trace t,\n--                   match t with\n--                   | `(%%a \u2227 %%b) := trace \"conjunction\" >> return e\n--                   | _            := trace \"not a conjunction\" >> find_conj' es\n--                   end\n\n#exit\n\nexample (h : p \u2227 q) : false :=\nby do l \u2190 local_context,\n      -- trace l,\n      e \u2190 find_conj l,\n      -- trace e,\n      cases e,\n      -- trace \"After cases : \", trace_state,\n      skip\n\nexample (h1 : p \u2227 q) (h2 : r \u2227 s) : false := by split_conjs\n\nexample (h1 : p \u2227 q) (h2 : r \u2227 s) : false := by split conj --repeat {split_conj}", "meta": {"author": "skbaek", "repo": "tab", "sha": "70909a69464a8713412d640ac630e5e6ef4e43e8", "save_path": "github-repos/lean/skbaek-tab", "path": "github-repos/lean/skbaek-tab/tab-70909a69464a8713412d640ac630e5e6ef4e43e8/conj.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3636690113920807}}
{"text": "/-\nCopyright \u00a9 2018 Fran\u00e7ois G. Dorais. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n-/\n\nimport nat\n\ntheorem ge_of_eq {\u03b1 : Type*} [preorder \u03b1] {a b : \u03b1} : a = b \u2192 a \u2265 b := \u03bb h, le_of_eq (eq.symm h)\n\nnamespace option\n\nlemma map_is_some {\u03b1 : Type*} {\u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) :\n\u2200 x, is_some (option.map f x) = is_some x\n| (some _) := rfl\n| none := rfl\n\nlemma map_is_none {\u03b1 : Type*} {\u03b2 : Type*} (f : \u03b1 \u2192 \u03b2) :\n\u2200 x, is_none (option.map f x) = is_none x\n| (some _) := rfl\n| none := rfl\n\nend option\n\n\n\n", "meta": {"author": "fgdorais", "repo": "tup", "sha": "ac4a2f8ca2ccc8aea091498439a0a47d43ac4700", "save_path": "github-repos/lean/fgdorais-tup", "path": "github-repos/lean/fgdorais-tup/tup-ac4a2f8ca2ccc8aea091498439a0a47d43ac4700/src/fin/extra.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6513548646660543, "lm_q1q2_score": 0.3636690038347227}}
{"text": "import LeanCodePrompts.FirstTacticFinder\nimport Mathlib.Tactic.Basic\nimport Mathlib.Tactic.Use\nimport Mathlib\n\n-- example : \u2200 n : \u2115, \u2203 m : \u2115, n < 2 * m + 1 := by\n--   repeat (aide_lookahead)\n--   repeat (sorry)\n\nset_option trace.Translate.info true\nexample : \u2200 n : \u2115, \u2203 m : \u2115, n < 2 * m + 1 := by\n  show_tactic_prompt\n  aide\n  aide\n  aide  \n  repeat (sorry)\n\n\n", "meta": {"author": "siddhartha-gadgil", "repo": "LeanAide", "sha": "7862af73ee2f0be08b20fd3e4148e20bf4a81054", "save_path": "github-repos/lean/siddhartha-gadgil-LeanAide", "path": "github-repos/lean/siddhartha-gadgil-LeanAide/LeanAide-7862af73ee2f0be08b20fd3e4148e20bf4a81054/LeanCodePrompts/FirstTacticExamples.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7690802476562641, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3635315252622664}}
{"text": "/-\nCopyright (c) 2019 Johannes H\u00f6lzl, Zhouhang Zhou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Zhouhang Zhou\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.integration\nimport Mathlib.order.filter.germ\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 u_5 u_6 \n\nnamespace Mathlib\n\n/-!\n\n# Almost everywhere equal functions\n\nTwo measurable functions are treated as identical if they are almost everywhere equal. We form the\nset of equivalence classes under the relation of being almost everywhere equal, which is sometimes\nknown as the `L\u2070` space.\n\nSee `l1_space.lean` for `L\u00b9` space.\n\n## Notation\n\n* `\u03b1 \u2192\u2098[\u03bc] \u03b2` is the type of `L\u2070` space, where `\u03b1` and `\u03b2` are measurable spaces and `\u03bc`\n  is a measure on `\u03b1`. `f : \u03b1 \u2192\u2098 \u03b2` is a \"function\" in `L\u2070`. In comments, `[f]` is also used\n  to denote an `L\u2070` function.\n\n  `\u2098` can be typed as `\\_m`. Sometimes it is shown as a box if font is missing.\n\n\n## Main statements\n\n* The linear structure of `L\u2070` :\n    Addition and scalar multiplication are defined on `L\u2070` in the natural way, i.e.,\n    `[f] + [g] := [f + g]`, `c \u2022 [f] := [c \u2022 f]`. So defined, `\u03b1 \u2192\u2098 \u03b2` inherits the linear structure\n    of `\u03b2`. For example, if `\u03b2` is a module, then `\u03b1 \u2192\u2098 \u03b2` is a module over the same ring.\n\n    See `mk_add_mk`,  `neg_mk`,     `mk_sub_mk`,  `smul_mk`,\n        `add_to_fun`, `neg_to_fun`, `sub_to_fun`, `smul_to_fun`\n\n* The order structure of `L\u2070` :\n    `\u2264` can be defined in a similar way: `[f] \u2264 [g]` if `f a \u2264 g a` for almost all `a` in domain.\n    And `\u03b1 \u2192\u2098 \u03b2` inherits the preorder and partial order of `\u03b2`.\n\n    TODO: Define `sup` and `inf` on `L\u2070` so that it forms a lattice. It seems that `\u03b2` must be a\n    linear order, since otherwise `f \u2294 g` may not be a measurable function.\n\n* Emetric on `L\u2070` :\n    If `\u03b2` is an `emetric_space`, then `L\u2070` can be made into an `emetric_space`, where\n    `edist [f] [g]` is defined to be `\u222b\u207b a, edist (f a) (g a)`.\n\n    The integral used here is `lintegral : (\u03b1 \u2192 ennreal) \u2192 ennreal`, which is defined in the file\n    `integration.lean`.\n\n    See `edist_mk_mk` and `edist_to_fun`.\n\n## Implementation notes\n\n* `f.to_fun`     : To find a representative of `f : \u03b1 \u2192\u2098 \u03b2`, use `f.to_fun`.\n                 For each operation `op` in `L\u2070`, there is a lemma called `op_to_fun`,\n                 characterizing, say, `(f op g).to_fun`.\n* `ae_eq_fun.mk` : To constructs an `L\u2070` function `\u03b1 \u2192\u2098 \u03b2` from a measurable function `f : \u03b1 \u2192 \u03b2`,\n                 use `ae_eq_fun.mk`\n* `comp`         : Use `comp g f` to get `[g \u2218 f]` from `g : \u03b2 \u2192 \u03b3` and `[f] : \u03b1 \u2192\u2098 \u03b3`\n* `comp\u2082`        : Use `comp\u2082 g f\u2081 f\u2082 to get `[\u03bba, g (f\u2081 a) (f\u2082 a)]`.\n                 For example, `[f + g]` is `comp\u2082 (+)`\n\n\n## Tags\n\nfunction space, almost everywhere equal, `L\u2070`, ae_eq_fun\n\n-/\n\nnamespace measure_theory\n\n\n/-- The equivalence relation of being almost everywhere equal -/\ndef measure.ae_eq_setoid {\u03b1 : Type u_1} (\u03b2 : Type u_2) [measurable_space \u03b1] [measurable_space \u03b2] (\u03bc : measure \u03b1) : setoid (Subtype fun (f : \u03b1 \u2192 \u03b2) => ae_measurable f) :=\n  setoid.mk (fun (f g : Subtype fun (f : \u03b1 \u2192 \u03b2) => ae_measurable f) => filter.eventually_eq (measure.ae \u03bc) \u2191f \u2191g) sorry\n\n/-- The space of equivalence classes of measurable functions, where two measurable functions are\n    equivalent if they agree almost everywhere, i.e., they differ on a set of measure `0`.  -/\ndef ae_eq_fun (\u03b1 : Type u_1) (\u03b2 : Type u_2) [measurable_space \u03b1] [measurable_space \u03b2] (\u03bc : measure \u03b1) :=\n  quotient (measure.ae_eq_setoid \u03b2 \u03bc)\n\nnamespace ae_eq_fun\n\n\n/-- Construct the equivalence class `[f]` of an almost everywhere measurable function `f`, based\n    on the equivalence relation of being almost everywhere equal. -/\ndef mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) : ae_eq_fun \u03b1 \u03b2 \u03bc :=\n  quotient.mk' { val := f, property := hf }\n\n/-- A measurable representative of an `ae_eq_fun` [f] -/\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] : has_coe_to_fun (ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  has_coe_to_fun.mk (fun (f : ae_eq_fun \u03b1 \u03b2 \u03bc) => \u03b1 \u2192 \u03b2)\n    fun (f : ae_eq_fun \u03b1 \u03b2 \u03bc) => ae_measurable.mk (subtype.val (quotient.out' f)) sorry\n\nprotected theorem measurable {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : measurable \u21d1f :=\n  ae_measurable.measurable_mk (has_coe_to_fun._proof_1 f)\n\nprotected theorem ae_measurable {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : ae_measurable \u21d1f :=\n  measurable.ae_measurable (ae_eq_fun.measurable f)\n\n@[simp] theorem quot_mk_eq_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) : Quot.mk setoid.r { val := f, property := hf } = mk f hf :=\n  rfl\n\n@[simp] theorem mk_eq_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {hf : ae_measurable f} {hg : ae_measurable g} : mk f hf = mk g hg \u2194 filter.eventually_eq (measure.ae \u03bc) f g :=\n  quotient.eq'\n\n@[simp] theorem mk_coe_fn {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : mk (\u21d1f) (ae_eq_fun.ae_measurable f) = f := sorry\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : ae_eq_fun \u03b1 \u03b2 \u03bc} {g : ae_eq_fun \u03b1 \u03b2 \u03bc} (h : filter.eventually_eq (measure.ae \u03bc) \u21d1f \u21d1g) : f = g := sorry\n\ntheorem coe_fn_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(mk f hf)) f :=\n  filter.eventually_eq.trans (filter.eventually_eq.symm (ae_measurable.ae_eq_mk (has_coe_to_fun._proof_1 (mk f hf))))\n    (quotient.mk_out' { val := f, property := hf })\n\ntheorem induction_on {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) {p : ae_eq_fun \u03b1 \u03b2 \u03bc \u2192 Prop} (H : \u2200 (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f), p (mk f hf)) : p f :=\n  quotient.induction_on' f (iff.mpr subtype.forall H)\n\ntheorem induction_on\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b1' : Type u_3} {\u03b2' : Type u_4} [measurable_space \u03b1'] [measurable_space \u03b2'] {\u03bc' : measure \u03b1'} (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (f' : ae_eq_fun \u03b1' \u03b2' \u03bc') {p : ae_eq_fun \u03b1 \u03b2 \u03bc \u2192 ae_eq_fun \u03b1' \u03b2' \u03bc' \u2192 Prop} (H : \u2200 (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) (f' : \u03b1' \u2192 \u03b2') (hf' : ae_measurable f'), p (mk f hf) (mk f' hf')) : p f f' :=\n  induction_on f fun (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) => induction_on f' (H f hf)\n\ntheorem induction_on\u2083 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b1' : Type u_3} {\u03b2' : Type u_4} [measurable_space \u03b1'] [measurable_space \u03b2'] {\u03bc' : measure \u03b1'} {\u03b1'' : Type u_5} {\u03b2'' : Type u_6} [measurable_space \u03b1''] [measurable_space \u03b2''] {\u03bc'' : measure \u03b1''} (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (f' : ae_eq_fun \u03b1' \u03b2' \u03bc') (f'' : ae_eq_fun \u03b1'' \u03b2'' \u03bc'') {p : ae_eq_fun \u03b1 \u03b2 \u03bc \u2192 ae_eq_fun \u03b1' \u03b2' \u03bc' \u2192 ae_eq_fun \u03b1'' \u03b2'' \u03bc'' \u2192 Prop} (H : \u2200 (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) (f' : \u03b1' \u2192 \u03b2') (hf' : ae_measurable f') (f'' : \u03b1'' \u2192 \u03b2'')\n  (hf'' : ae_measurable f''), p (mk f hf) (mk f' hf') (mk f'' hf'')) : p f f' f'' :=\n  induction_on f fun (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) => induction_on\u2082 f' f'' (H f hf)\n\n/-- Given a measurable function `g : \u03b2 \u2192 \u03b3`, and an almost everywhere equal function `[f] : \u03b1 \u2192\u2098 \u03b2`,\n    return the equivalence class of `g \u2218 f`, i.e., the almost everywhere equal function\n    `[g \u2218 f] : \u03b1 \u2192\u2098 \u03b3`. -/\ndef comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (g : \u03b2 \u2192 \u03b3) (hg : measurable g) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : ae_eq_fun \u03b1 \u03b3 \u03bc :=\n  quotient.lift_on' f (fun (f : Subtype fun (f : \u03b1 \u2192 \u03b2) => ae_measurable f) => mk (g \u2218 \u2191f) sorry) sorry\n\n@[simp] theorem comp_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (g : \u03b2 \u2192 \u03b3) (hg : measurable g) (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) : comp g hg (mk f hf) = mk (g \u2218 f) (measurable.comp_ae_measurable hg hf) :=\n  rfl\n\ntheorem comp_eq_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (g : \u03b2 \u2192 \u03b3) (hg : measurable g) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : comp g hg f = mk (g \u2218 \u21d1f) (measurable.comp_ae_measurable hg (ae_eq_fun.ae_measurable f)) := sorry\n\ntheorem coe_fn_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (g : \u03b2 \u2192 \u03b3) (hg : measurable g) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(comp g hg f)) (g \u2218 \u21d1f) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter.eventually_eq (measure.ae \u03bc) (\u21d1(comp g hg f)) (g \u2218 \u21d1f))) (comp_eq_mk g hg f)))\n    (coe_fn_mk (g \u2218 \u21d1f) (measurable.comp_ae_measurable hg (ae_eq_fun.ae_measurable f)))\n\n/-- The class of `x \u21a6 (f x, g x)`. -/\ndef pair {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun \u03b1 (\u03b2 \u00d7 \u03b3) \u03bc :=\n  quotient.lift_on\u2082' f g\n    (fun (f : Subtype fun (f : \u03b1 \u2192 \u03b2) => ae_measurable f) (g : Subtype fun (f : \u03b1 \u2192 \u03b3) => ae_measurable f) =>\n      mk (fun (x : \u03b1) => (subtype.val f x, subtype.val g x)) sorry)\n    sorry\n\n@[simp] theorem pair_mk_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) (g : \u03b1 \u2192 \u03b3) (hg : ae_measurable g) : pair (mk f hf) (mk g hg) = mk (fun (x : \u03b1) => (f x, g x)) (ae_measurable.prod_mk hf hg) :=\n  rfl\n\ntheorem pair_eq_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : pair f g =\n  mk (fun (x : \u03b1) => (coe_fn f x, coe_fn g x))\n    (ae_measurable.prod_mk (ae_eq_fun.ae_measurable f) (ae_eq_fun.ae_measurable g)) := sorry\n\ntheorem coe_fn_pair {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) \u21d1(pair f g) fun (x : \u03b1) => (coe_fn f x, coe_fn g x) := sorry\n\n/-- Given a measurable function `g : \u03b2 \u2192 \u03b3 \u2192 \u03b4`, and almost everywhere equal functions\n    `[f\u2081] : \u03b1 \u2192\u2098 \u03b2` and `[f\u2082] : \u03b1 \u2192\u2098 \u03b3`, return the equivalence class of the function\n    `\u03bba, g (f\u2081 a) (f\u2082 a)`, i.e., the almost everywhere equal function\n    `[\u03bba, g (f\u2081 a) (f\u2082 a)] : \u03b1 \u2192\u2098 \u03b3` -/\ndef comp\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : ae_eq_fun \u03b1 \u03b2 \u03bc) (f\u2082 : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun \u03b1 \u03b4 \u03bc :=\n  comp (function.uncurry g) hg (pair f\u2081 f\u2082)\n\n@[simp] theorem comp\u2082_mk_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : \u03b1 \u2192 \u03b2) (f\u2082 : \u03b1 \u2192 \u03b3) (hf\u2081 : ae_measurable f\u2081) (hf\u2082 : ae_measurable f\u2082) : comp\u2082 g hg (mk f\u2081 hf\u2081) (mk f\u2082 hf\u2082) =\n  mk (fun (a : \u03b1) => g (f\u2081 a) (f\u2082 a)) (measurable.comp_ae_measurable hg (ae_measurable.prod_mk hf\u2081 hf\u2082)) :=\n  rfl\n\ntheorem comp\u2082_eq_pair {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : ae_eq_fun \u03b1 \u03b2 \u03bc) (f\u2082 : ae_eq_fun \u03b1 \u03b3 \u03bc) : comp\u2082 g hg f\u2081 f\u2082 = comp (function.uncurry g) hg (pair f\u2081 f\u2082) :=\n  rfl\n\ntheorem comp\u2082_eq_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : ae_eq_fun \u03b1 \u03b2 \u03bc) (f\u2082 : ae_eq_fun \u03b1 \u03b3 \u03bc) : comp\u2082 g hg f\u2081 f\u2082 =\n  mk (fun (a : \u03b1) => g (coe_fn f\u2081 a) (coe_fn f\u2082 a))\n    (measurable.comp_ae_measurable hg (ae_measurable.prod_mk (ae_eq_fun.ae_measurable f\u2081) (ae_eq_fun.ae_measurable f\u2082))) := sorry\n\ntheorem coe_fn_comp\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : ae_eq_fun \u03b1 \u03b2 \u03bc) (f\u2082 : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) \u21d1(comp\u2082 g hg f\u2081 f\u2082) fun (a : \u03b1) => g (coe_fn f\u2081 a) (coe_fn f\u2082 a) := sorry\n\n/-- Interpret `f : \u03b1 \u2192\u2098[\u03bc] \u03b2` as a germ at `\u03bc.ae` forgetting that `f` is almost everywhere\n    measurable. -/\ndef to_germ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : filter.germ (measure.ae \u03bc) \u03b2 :=\n  quotient.lift_on' f (fun (f : Subtype fun (f : \u03b1 \u2192 \u03b2) => ae_measurable f) => \u2191\u2191f) sorry\n\n@[simp] theorem mk_to_germ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : \u03b1 \u2192 \u03b2) (hf : ae_measurable f) : to_germ (mk f hf) = \u2191f :=\n  rfl\n\ntheorem to_germ_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : to_germ f = \u2191\u21d1f :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (to_germ f = \u2191\u21d1f)) (Eq.symm (mk_to_germ (\u21d1f) (ae_eq_fun.ae_measurable f)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (to_germ f = to_germ (mk (\u21d1f) (ae_eq_fun.ae_measurable f)))) (mk_coe_fn f)))\n      (Eq.refl (to_germ f)))\n\ntheorem to_germ_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] : function.injective to_germ := sorry\n\ntheorem comp_to_germ {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (g : \u03b2 \u2192 \u03b3) (hg : measurable g) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) : to_germ (comp g hg f) = filter.germ.map g (to_germ f) := sorry\n\ntheorem comp\u2082_to_germ {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] [measurable_space \u03b4] (g : \u03b2 \u2192 \u03b3 \u2192 \u03b4) (hg : measurable (function.uncurry g)) (f\u2081 : ae_eq_fun \u03b1 \u03b2 \u03bc) (f\u2082 : ae_eq_fun \u03b1 \u03b3 \u03bc) : to_germ (comp\u2082 g hg f\u2081 f\u2082) = filter.germ.map\u2082 g (to_germ f\u2081) (to_germ f\u2082) := sorry\n\n/-- Given a predicate `p` and an equivalence class `[f]`, return true if `p` holds of `f a`\n    for almost all `a` -/\ndef lift_pred {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (p : \u03b2 \u2192 Prop) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  filter.germ.lift_pred p (to_germ f)\n\n/-- Given a relation `r` and equivalence class `[f]` and `[g]`, return true if `r` holds of\n    `(f a, g a)` for almost all `a` -/\ndef lift_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] (r : \u03b2 \u2192 \u03b3 \u2192 Prop) (f : ae_eq_fun \u03b1 \u03b2 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  filter.germ.lift_rel r (to_germ f) (to_germ g)\n\ntheorem lift_rel_mk_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] {r : \u03b2 \u2192 \u03b3 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} {hf : ae_measurable f} {hg : ae_measurable g} : lift_rel r (mk f hf) (mk g hg) \u2194 filter.eventually (fun (a : \u03b1) => r (f a) (g a)) (measure.ae \u03bc) :=\n  iff.rfl\n\ntheorem lift_rel_iff_coe_fn {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [measurable_space \u03b3] {r : \u03b2 \u2192 \u03b3 \u2192 Prop} {f : ae_eq_fun \u03b1 \u03b2 \u03bc} {g : ae_eq_fun \u03b1 \u03b3 \u03bc} : lift_rel r f g \u2194 filter.eventually (fun (a : \u03b1) => r (coe_fn f a) (coe_fn g a)) (measure.ae \u03bc) := sorry\n\nprotected instance preorder {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [preorder \u03b2] : preorder (ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  preorder.lift to_germ\n\n@[simp] theorem mk_le_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} (hf : ae_measurable f) (hg : ae_measurable g) : mk f hf \u2264 mk g hg \u2194 filter.eventually_le (measure.ae \u03bc) f g :=\n  iff.rfl\n\n@[simp] theorem coe_fn_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [preorder \u03b2] {f : ae_eq_fun \u03b1 \u03b2 \u03bc} {g : ae_eq_fun \u03b1 \u03b2 \u03bc} : filter.eventually_le (measure.ae \u03bc) \u21d1f \u21d1g \u2194 f \u2264 g :=\n  iff.symm lift_rel_iff_coe_fn\n\nprotected instance partial_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [partial_order \u03b2] : partial_order (ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  partial_order.lift to_germ to_germ_injective\n\n/- TODO: Prove `L\u2070` space is a lattice if \u03b2 is linear order.\n         What if \u03b2 is only a lattice? -/\n\n-- instance [linear_order \u03b2] : semilattice_sup (\u03b1 \u2192\u2098 \u03b2) :=\n\n-- { sup := comp\u2082 (\u2294) (_),\n\n--    .. ae_eq_fun.partial_order }\n\n/-- The equivalence class of a constant function: `[\u03bba:\u03b1, b]`, based on the equivalence relation of\n    being almost everywhere equal -/\ndef const (\u03b1 : Type u_1) {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (b : \u03b2) : ae_eq_fun \u03b1 \u03b2 \u03bc :=\n  mk (fun (a : \u03b1) => b) ae_measurable_const\n\ntheorem coe_fn_const (\u03b1 : Type u_1) {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] (b : \u03b2) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(const \u03b1 b)) (function.const \u03b1 b) :=\n  coe_fn_mk (fun (a : \u03b1) => b) ae_measurable_const\n\nprotected instance inhabited {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [Inhabited \u03b2] : Inhabited (ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  { default := const \u03b1 Inhabited.default }\n\nprotected instance has_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [HasZero \u03b2] : HasZero (ae_eq_fun \u03b1 \u03b2 \u03bc) :=\n  { zero := const \u03b1 0 }\n\ntheorem one_def {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [HasOne \u03b2] : 1 = mk (fun (a : \u03b1) => 1) ae_measurable_const :=\n  rfl\n\ntheorem coe_fn_one {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [HasOne \u03b2] : filter.eventually_eq (measure.ae \u03bc) (\u21d11) 1 :=\n  coe_fn_const \u03b1 1\n\n@[simp] theorem one_to_germ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] [HasOne \u03b2] : to_germ 1 = 1 :=\n  rfl\n\nprotected instance has_mul {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [monoid \u03b3] [has_continuous_mul \u03b3] : Mul (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  { mul := comp\u2082 Mul.mul sorry }\n\n@[simp] theorem mk_mul_mk {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [monoid \u03b3] [has_continuous_mul \u03b3] (f : \u03b1 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) (hg : ae_measurable g) : mk f hf * mk g hg = mk (f * g) (ae_measurable.mul hf hg) :=\n  rfl\n\ntheorem coe_fn_mul {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [monoid \u03b3] [has_continuous_mul \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(f * g)) (\u21d1f * \u21d1g) :=\n  coe_fn_comp\u2082 Mul.mul has_mul._proof_1 f g\n\n@[simp] theorem mul_to_germ {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [monoid \u03b3] [has_continuous_mul \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : to_germ (f * g) = to_germ f * to_germ g :=\n  comp\u2082_to_germ Mul.mul has_mul._proof_1 f g\n\nprotected instance add_monoid {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [add_monoid \u03b3] [has_continuous_add \u03b3] : add_monoid (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  function.injective.add_monoid to_germ to_germ_injective sorry add_to_germ\n\nprotected instance add_comm_monoid {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [has_continuous_add \u03b3] : add_comm_monoid (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  function.injective.add_comm_monoid to_germ to_germ_injective sorry sorry\n\nprotected instance has_inv {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [group \u03b3] [topological_group \u03b3] : has_inv (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  has_inv.mk (comp has_inv.inv measurable_inv)\n\n@[simp] theorem inv_mk {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [group \u03b3] [topological_group \u03b3] (f : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) : mk f hf\u207b\u00b9 = mk (f\u207b\u00b9) (ae_measurable.inv hf) :=\n  rfl\n\ntheorem coe_fn_neg {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [add_group \u03b3] [topological_add_group \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(-f)) (-\u21d1f) :=\n  coe_fn_comp Neg.neg measurable_neg f\n\ntheorem inv_to_germ {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [group \u03b3] [topological_group \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : to_germ (f\u207b\u00b9) = (to_germ f\u207b\u00b9) :=\n  comp_to_germ has_inv.inv measurable_inv f\n\nprotected instance group {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [group \u03b3] [topological_group \u03b3] [topological_space.second_countable_topology \u03b3] : group (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  function.injective.group to_germ to_germ_injective sorry sorry inv_to_germ\n\n@[simp] theorem mk_sub {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [add_group \u03b3] [topological_add_group \u03b3] [topological_space.second_countable_topology \u03b3] (f : \u03b1 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b3) (hf : ae_measurable fun (x : \u03b1) => f x) (hg : ae_measurable fun (x : \u03b1) => g x) : mk (f - g) (ae_measurable.sub hf hg) = mk f hf - mk g hg := sorry\n\ntheorem coe_fn_sub {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [add_group \u03b3] [topological_add_group \u03b3] [topological_space.second_countable_topology \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(f - g)) (\u21d1f - \u21d1g) := sorry\n\nprotected instance comm_group {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [borel_space \u03b3] [comm_group \u03b3] [topological_group \u03b3] [topological_space.second_countable_topology \u03b3] : comm_group (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  comm_group.mk group.mul sorry group.one sorry sorry group.inv group.div sorry sorry\n\nprotected instance has_scalar {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [semiring \ud835\udd5c] [topological_space \ud835\udd5c] [topological_space \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [semimodule \ud835\udd5c \u03b3] [topological_semimodule \ud835\udd5c \u03b3] : has_scalar \ud835\udd5c (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  has_scalar.mk fun (c : \ud835\udd5c) (f : ae_eq_fun \u03b1 \u03b3 \u03bc) => comp (has_scalar.smul c) sorry f\n\n@[simp] theorem smul_mk {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [semiring \ud835\udd5c] [topological_space \ud835\udd5c] [topological_space \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [semimodule \ud835\udd5c \u03b3] [topological_semimodule \ud835\udd5c \u03b3] (c : \ud835\udd5c) (f : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) : c \u2022 mk f hf = mk (c \u2022 f) (ae_measurable.const_smul hf c) :=\n  rfl\n\ntheorem coe_fn_smul {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [semiring \ud835\udd5c] [topological_space \ud835\udd5c] [topological_space \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [semimodule \ud835\udd5c \u03b3] [topological_semimodule \ud835\udd5c \u03b3] (c : \ud835\udd5c) (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) (\u21d1(c \u2022 f)) (c \u2022 \u21d1f) :=\n  coe_fn_comp (has_scalar.smul c) (has_scalar._proof_1 c) f\n\ntheorem smul_to_germ {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [semiring \ud835\udd5c] [topological_space \ud835\udd5c] [topological_space \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [semimodule \ud835\udd5c \u03b3] [topological_semimodule \ud835\udd5c \u03b3] (c : \ud835\udd5c) (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : to_germ (c \u2022 f) = c \u2022 to_germ f :=\n  comp_to_germ (has_scalar.smul c) (has_scalar._proof_1 c) f\n\nprotected instance semimodule {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [semiring \ud835\udd5c] [topological_space \ud835\udd5c] [topological_space \u03b3] [borel_space \u03b3] [add_comm_monoid \u03b3] [semimodule \ud835\udd5c \u03b3] [topological_semimodule \ud835\udd5c \u03b3] [topological_space.second_countable_topology \u03b3] [has_continuous_add \u03b3] : semimodule \ud835\udd5c (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  function.injective.semimodule \ud835\udd5c (add_monoid_hom.mk to_germ sorry sorry) to_germ_injective smul_to_germ\n\n/- TODO : Prove that `L\u2070` is a complete space if the codomain is complete. -/\n\n/-- For `f : \u03b1 \u2192 ennreal`, define `\u222b [f]` to be `\u222b f` -/\ndef lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : ae_eq_fun \u03b1 ennreal \u03bc) : ennreal :=\n  quotient.lift_on' f (fun (f : Subtype fun (f : \u03b1 \u2192 ennreal) => ae_measurable f) => lintegral \u03bc fun (a : \u03b1) => coe f a)\n    sorry\n\n@[simp] theorem lintegral_mk {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : \u03b1 \u2192 ennreal) (hf : ae_measurable f) : lintegral (mk f hf) = lintegral \u03bc fun (a : \u03b1) => f a :=\n  rfl\n\ntheorem lintegral_coe_fn {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : ae_eq_fun \u03b1 ennreal \u03bc) : (lintegral \u03bc fun (a : \u03b1) => coe_fn f a) = lintegral f := sorry\n\n@[simp] theorem lintegral_zero {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} : lintegral 0 = 0 :=\n  lintegral_zero\n\n@[simp] theorem lintegral_eq_zero_iff {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : ae_eq_fun \u03b1 ennreal \u03bc} : lintegral f = 0 \u2194 f = 0 :=\n  induction_on f fun (f : \u03b1 \u2192 ennreal) (hf : ae_measurable f) => iff.trans (lintegral_eq_zero_iff' hf) (iff.symm mk_eq_mk)\n\ntheorem lintegral_add {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : ae_eq_fun \u03b1 ennreal \u03bc) (g : ae_eq_fun \u03b1 ennreal \u03bc) : lintegral (f + g) = lintegral f + lintegral g := sorry\n\ntheorem lintegral_mono {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : ae_eq_fun \u03b1 ennreal \u03bc} {g : ae_eq_fun \u03b1 ennreal \u03bc} : f \u2264 g \u2192 lintegral f \u2264 lintegral g :=\n  induction_on\u2082 f g\n    fun (f : \u03b1 \u2192 ennreal) (hf : ae_measurable f) (g : \u03b1 \u2192 ennreal) (hg : ae_measurable g) (hfg : mk f hf \u2264 mk g hg) =>\n      lintegral_mono_ae hfg\n\n/-- `comp_edist [f] [g] a` will return `edist (f a) (g a)` -/\nprotected def edist {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun \u03b1 ennreal \u03bc :=\n  comp\u2082 edist measurable_edist f g\n\nprotected theorem edist_comm {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun.edist f g = ae_eq_fun.edist g f :=\n  induction_on\u2082 f g\n    fun (f : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) (g : \u03b1 \u2192 \u03b3) (hg : ae_measurable g) =>\n      iff.mpr mk_eq_mk (filter.eventually_of_forall fun (x : \u03b1) => edist_comm (f x) (g x))\n\ntheorem coe_fn_edist {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) \u21d1(ae_eq_fun.edist f g) fun (a : \u03b1) => edist (coe_fn f a) (coe_fn g a) :=\n  coe_fn_comp\u2082 edist measurable_edist f g\n\nprotected theorem edist_self {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun.edist f f = 0 :=\n  induction_on f\n    fun (f : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) =>\n      iff.mpr mk_eq_mk (filter.eventually_of_forall fun (x : \u03b1) => edist_self (f x))\n\n/-- Almost everywhere equal functions form an `emetric_space`, with the emetric defined as\n  `edist f g = \u222b\u207b a, edist (f a) (g a)`. -/\nprotected instance emetric_space {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] : emetric_space (ae_eq_fun \u03b1 \u03b3 \u03bc) :=\n  emetric_space.mk sorry sorry sorry sorry\n    (uniform_space_of_edist (fun (f g : ae_eq_fun \u03b1 \u03b3 \u03bc) => lintegral (ae_eq_fun.edist f g)) sorry sorry sorry)\n\ntheorem edist_mk_mk {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] {f : \u03b1 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b3} (hf : ae_measurable f) (hg : ae_measurable g) : edist (mk f hf) (mk g hg) = lintegral \u03bc fun (x : \u03b1) => edist (f x) (g x) :=\n  rfl\n\ntheorem edist_eq_coe {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : edist f g = lintegral \u03bc fun (x : \u03b1) => edist (coe_fn f x) (coe_fn g x) := sorry\n\ntheorem edist_zero_eq_coe {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [emetric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] [HasZero \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : edist f 0 = lintegral \u03bc fun (x : \u03b1) => edist (coe_fn f x) 0 := sorry\n\ntheorem edist_mk_mk' {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [metric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] {f : \u03b1 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b3} (hf : ae_measurable f) (hg : ae_measurable g) : edist (mk f hf) (mk g hg) = lintegral \u03bc fun (x : \u03b1) => \u2191(nndist (f x) (g x)) := sorry\n\ntheorem edist_eq_coe' {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [metric_space \u03b3] [topological_space.second_countable_topology \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) : edist f g = lintegral \u03bc fun (x : \u03b1) => \u2191(nndist (coe_fn f x) (coe_fn g x)) := sorry\n\ntheorem edist_add_right {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [normed_group \u03b3] [topological_space.second_countable_topology \u03b3] [borel_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) (g : ae_eq_fun \u03b1 \u03b3 \u03bc) (h : ae_eq_fun \u03b1 \u03b3 \u03bc) : edist (f + h) (g + h) = edist f g := sorry\n\ntheorem edist_smul {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] {\ud835\udd5c : Type u_5} [normed_field \ud835\udd5c] [normed_group \u03b3] [topological_space.second_countable_topology \u03b3] [normed_space \ud835\udd5c \u03b3] [borel_space \u03b3] (c : \ud835\udd5c) (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : edist (c \u2022 f) 0 = ennreal.of_real (norm c) * edist f 0 := sorry\n\n/-- Positive part of an `ae_eq_fun`. -/\ndef pos_part {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [linear_order \u03b3] [order_closed_topology \u03b3] [topological_space.second_countable_topology \u03b3] [HasZero \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : ae_eq_fun \u03b1 \u03b3 \u03bc :=\n  comp (fun (x : \u03b3) => max x 0) sorry f\n\n@[simp] theorem pos_part_mk {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [linear_order \u03b3] [order_closed_topology \u03b3] [topological_space.second_countable_topology \u03b3] [HasZero \u03b3] [opens_measurable_space \u03b3] (f : \u03b1 \u2192 \u03b3) (hf : ae_measurable f) : pos_part (mk f hf) = mk (fun (x : \u03b1) => max (f x) 0) (ae_measurable.max hf ae_measurable_const) :=\n  rfl\n\ntheorem coe_fn_pos_part {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b3] [topological_space \u03b3] [linear_order \u03b3] [order_closed_topology \u03b3] [topological_space.second_countable_topology \u03b3] [HasZero \u03b3] [opens_measurable_space \u03b3] (f : ae_eq_fun \u03b1 \u03b3 \u03bc) : filter.eventually_eq (measure.ae \u03bc) \u21d1(pos_part f) fun (a : \u03b1) => max (coe_fn f a) 0 :=\n  coe_fn_comp (fun (x : \u03b3) => max x 0) pos_part._proof_1 f\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/ae_eq_fun.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.36348224138930957}}
{"text": "/-\nCopyright (c) 2019 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nnamespace Std\nuniverse u v w w'\n\nnamespace PersistentHashMap\n\ninductive Entry (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : Type w) where\n  | entry (key : \u03b1) (val : \u03b2) : Entry \u03b1 \u03b2 \u03c3\n  | ref   (node : \u03c3) : Entry \u03b1 \u03b2 \u03c3\n  | null  : Entry \u03b1 \u03b2 \u03c3\n\ninstance {\u03b1 \u03b2 \u03c3} : Inhabited (Entry \u03b1 \u03b2 \u03c3) := \u27e8Entry.null\u27e9\n\ninductive Node (\u03b1 : Type u) (\u03b2 : Type v) : Type (max u v) where\n  | entries   (es : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : Node \u03b1 \u03b2\n  | collision (ks : Array \u03b1) (vs : Array \u03b2) (h : ks.size = vs.size) : Node \u03b1 \u03b2\n\ninstance {\u03b1 \u03b2} : Inhabited (Node \u03b1 \u03b2) := \u27e8Node.entries #[]\u27e9\n\nabbrev shift         : USize  := 5\nabbrev branching     : USize  := USize.ofNat (2 ^ shift.toNat)\nabbrev maxDepth      : USize  := 7\nabbrev maxCollisions : Nat    := 4\n\ndef mkEmptyEntriesArray {\u03b1 \u03b2} : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2)) :=\n  (Array.mkArray PersistentHashMap.branching.toNat PersistentHashMap.Entry.null)\n\nend PersistentHashMap\n\nstructure PersistentHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] where\n  root    : PersistentHashMap.Node \u03b1 \u03b2 := PersistentHashMap.Node.entries PersistentHashMap.mkEmptyEntriesArray\n  size    : Nat                        := 0\n\nabbrev PHashMap (\u03b1 : Type u) (\u03b2 : Type v) [BEq \u03b1] [Hashable \u03b1] := PersistentHashMap \u03b1 \u03b2\n\nnamespace PersistentHashMap\n\ndef empty [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 := {}\n\ndef isEmpty [BEq \u03b1] [Hashable \u03b1] (m : PersistentHashMap \u03b1 \u03b2) : Bool :=\n  m.size == 0\n\ninstance [BEq \u03b1] [Hashable \u03b1] : Inhabited (PersistentHashMap \u03b1 \u03b2) := \u27e8{}\u27e9\n\ndef mkEmptyEntries {\u03b1 \u03b2} : Node \u03b1 \u03b2 :=\n  Node.entries mkEmptyEntriesArray\n\nabbrev mul2Shift (i : USize) (shift : USize) : USize := i.shiftLeft shift\nabbrev div2Shift (i : USize) (shift : USize) : USize := i.shiftRight shift\nabbrev mod2Shift (i : USize) (shift : USize) : USize := USize.land i ((USize.shiftLeft 1 shift) - 1)\n\ninductive IsCollisionNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (keys : Array \u03b1) (vals : Array \u03b2) (h : keys.size = vals.size) : IsCollisionNode (Node.collision keys vals h)\n\nabbrev CollisionNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsCollisionNode n }\n\ninductive IsEntriesNode : Node \u03b1 \u03b2 \u2192 Prop where\n  | mk (entries : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) : IsEntriesNode (Node.entries entries)\n\nabbrev EntriesNode (\u03b1 \u03b2) := { n : Node \u03b1 \u03b2 // IsEntriesNode n }\n\nprivate theorem size_set {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (i : Fin ks.size) (j : Fin vs.size) (k : \u03b1) (v : \u03b2)\n                           : (ks.set i k).size = (vs.set j v).size := by\n  simp [h]\n\nprivate theorem size_push {ks : Array \u03b1} {vs : Array \u03b2} (h : ks.size = vs.size) (k : \u03b1) (v : \u03b2) : (ks.push k).size = (vs.push v).size := by\n  simp [h]\n\npartial def insertAtCollisionNodeAux [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 Nat \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2\n  | n@\u27e8Node.collision keys vals heq, _\u27e9, i, k, v =>\n    if h : i < keys.size then\n      let idx : Fin keys.size := \u27e8i, h\u27e9;\n      let k' := keys.get idx;\n      if k == k' then\n         let j : Fin vals.size := \u27e8i, by rw [\u2190heq]; assumption\u27e9\n         \u27e8Node.collision (keys.set idx k) (vals.set j v) (size_set heq idx j k v), IsCollisionNode.mk _ _ _\u27e9\n      else insertAtCollisionNodeAux n (i+1) k v\n    else\n      \u27e8Node.collision (keys.push k) (vals.push v) (size_push heq k v), IsCollisionNode.mk _ _ _\u27e9\n  | \u27e8Node.entries _, h\u27e9, _, _, _ => False.elim (nomatch h)\n\ndef insertAtCollisionNode [BEq \u03b1] : CollisionNode \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 CollisionNode \u03b1 \u03b2 :=\n  fun n k v => insertAtCollisionNodeAux n 0 k v\n\ndef getCollisionNodeSize : CollisionNode \u03b1 \u03b2 \u2192 Nat\n  | \u27e8Node.collision keys _ _, _\u27e9 => keys.size\n  | \u27e8Node.entries _, h\u27e9          => False.elim (nomatch h)\n\ndef mkCollisionNode (k\u2081 : \u03b1) (v\u2081 : \u03b2) (k\u2082 : \u03b1) (v\u2082 : \u03b2) : Node \u03b1 \u03b2 :=\n  let ks : Array \u03b1 := Array.mkEmpty maxCollisions\n  let ks := (ks.push k\u2081).push k\u2082\n  let vs : Array \u03b2 := Array.mkEmpty maxCollisions\n  let vs := (vs.push v\u2081).push v\u2082\n  Node.collision ks vs rfl\n\npartial def insertAux [BEq \u03b1] [Hashable \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 USize \u2192 \u03b1 \u2192 \u03b2 \u2192 Node \u03b1 \u03b2\n  | Node.collision keys vals heq, _, depth, k, v =>\n    let newNode := insertAtCollisionNode \u27e8Node.collision keys vals heq, IsCollisionNode.mk _ _ _\u27e9 k v\n    if depth >= maxDepth || getCollisionNodeSize newNode < maxCollisions then newNode.val\n    else match newNode with\n      | \u27e8Node.entries _, h\u27e9 => False.elim (nomatch h)\n      | \u27e8Node.collision keys vals heq, _\u27e9 =>\n        let rec traverse (i : Nat) (entries : Node \u03b1 \u03b2) : Node \u03b1 \u03b2 :=\n          if h : i < keys.size then\n            let k := keys.get \u27e8i, h\u27e9\n            let v := vals.get \u27e8i, heq \u25b8 h\u27e9\n            let h := hash k |>.toUSize\n            let h := div2Shift h (shift * (depth - 1))\n            traverse (i+1) (insertAux entries h depth k v)\n          else\n            entries\n        traverse 0 mkEmptyEntries\n  | Node.entries entries, h, depth, k, v =>\n    let j     := (mod2Shift h shift).toNat\n    Node.entries $ entries.modify j fun entry =>\n      match entry with\n      | Entry.null        => Entry.entry k v\n      | Entry.ref node    => Entry.ref $ insertAux node (div2Shift h shift) (depth+1) k v\n      | Entry.entry k' v' =>\n        if k == k' then Entry.entry k v\n        else Entry.ref $ mkCollisionNode k' v' k v\n\ndef insert {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 \u03b2 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k, v => { root := insertAux n (hash k |>.toUSize) 1 k v, size := sz + 1 }\n\npartial def findAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option \u03b2 :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then some (vals.get \u27e8i, by rw [\u2190heq]; assumption\u27e9)\n    else findAtAux keys vals heq (i+1) k\n  else none\n\npartial def findAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option \u03b2\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some v else none\n  | Node.collision keys vals heq, _, k => findAtAux keys vals heq 0 k\n\ndef find? {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option \u03b2\n  | { root := n, .. }, k => findAux n (hash k |>.toUSize) k\n\n@[inline] def getOp {_ : BEq \u03b1} {_ : Hashable \u03b1} (self : PersistentHashMap \u03b1 \u03b2) (idx : \u03b1) : Option \u03b2 :=\n  self.find? idx\n\n@[inline] def findD {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) (b\u2080 : \u03b2) : \u03b2 :=\n  (m.find? a).getD b\u2080\n\n@[inline] def find! {_ : BEq \u03b1} {_ : Hashable \u03b1} [Inhabited \u03b2] (m : PersistentHashMap \u03b1 \u03b2) (a : \u03b1) : \u03b2 :=\n  match m.find? a with\n  | some b => b\n  | none   => panic! \"key is not in the map\"\n\npartial def findEntryAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then some (k', vals.get \u27e8i, by rw [\u2190heq]; assumption\u27e9)\n    else findEntryAtAux keys vals heq (i+1) k\n  else none\n\npartial def findEntryAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => none\n    | Entry.ref node   => findEntryAux node (div2Shift h shift) k\n    | Entry.entry k' v => if k == k' then some (k', v) else none\n  | Node.collision keys vals heq, _, k => findEntryAtAux keys vals heq 0 k\n\ndef findEntry? {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | { root := n, .. }, k => findEntryAux n (hash k |>.toUSize) k\n\npartial def containsAtAux [BEq \u03b1] (keys : Array \u03b1) (vals : Array \u03b2) (heq : keys.size = vals.size) (i : Nat) (k : \u03b1) : Bool :=\n  if h : i < keys.size then\n    let k' := keys.get \u27e8i, h\u27e9\n    if k == k' then true\n    else containsAtAux keys vals heq (i+1) k\n  else false\n\npartial def containsAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Bool\n  | Node.entries entries, h, k =>\n    let j     := (mod2Shift h shift).toNat\n    match entries.get! j with\n    | Entry.null       => false\n    | Entry.ref node   => containsAux node (div2Shift h shift) k\n    | Entry.entry k' v => k == k'\n  | Node.collision keys vals heq, _, k => containsAtAux keys vals heq 0 k\n\ndef contains [BEq \u03b1] [Hashable \u03b1] : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 Bool\n  | { root := n, .. }, k => containsAux n (hash k |>.toUSize) k\n\npartial def isUnaryEntries (a : Array (Entry \u03b1 \u03b2 (Node \u03b1 \u03b2))) (i : Nat) (acc : Option (\u03b1 \u00d7 \u03b2)) : Option (\u03b1 \u00d7 \u03b2) :=\n  if h : i < a.size then\n    match a.get \u27e8i, h\u27e9 with\n    | Entry.null      => isUnaryEntries a (i+1) acc\n    | Entry.ref _     => none\n    | Entry.entry k v =>\n      match acc with\n      | none   => isUnaryEntries a (i+1) (some (k, v))\n      | some _ => none\n  else acc\n\ndef isUnaryNode : Node \u03b1 \u03b2 \u2192 Option (\u03b1 \u00d7 \u03b2)\n  | Node.entries entries         => isUnaryEntries entries 0 none\n  | Node.collision keys vals heq =>\n    if h : 1 = keys.size then\n      have : 0 < keys.size := by rw [\u2190h]; decide\n      some (keys.get \u27e80, this\u27e9, vals.get \u27e80, by rw [\u2190heq]; assumption\u27e9)\n    else\n      none\n\npartial def eraseAux [BEq \u03b1] : Node \u03b1 \u03b2 \u2192 USize \u2192 \u03b1 \u2192 Node \u03b1 \u03b2 \u00d7 Bool\n  | n@(Node.collision keys vals heq), _, k =>\n    match keys.indexOf? k with\n    | some idx =>\n      let \u27e8keys', keq\u27e9 := keys.eraseIdx' idx\n      let \u27e8vals', veq\u27e9 := vals.eraseIdx' (Eq.ndrec idx heq)\n      have : keys.size - 1 = vals.size - 1 := by rw [heq]\n      (Node.collision keys' vals' (keq.trans (this.trans veq.symm)), true)\n    | none     => (n, false)\n  | n@(Node.entries entries), h, k =>\n    let j       := (mod2Shift h shift).toNat\n    let entry   := entries.get! j\n    match entry with\n    | Entry.null       => (n, false)\n    | Entry.entry k' v =>\n      if k == k' then (Node.entries (entries.set! j Entry.null), true) else (n, false)\n    | Entry.ref node   =>\n      let entries := entries.set! j Entry.null\n      let (newNode, deleted) := eraseAux node (div2Shift h shift) k\n      if !deleted then (n, false)\n      else match isUnaryNode newNode with\n        | none        => (Node.entries (entries.set! j (Entry.ref newNode)), true)\n        | some (k, v) => (Node.entries (entries.set! j (Entry.entry k v)), true)\n\ndef erase {_ : BEq \u03b1} {_ : Hashable \u03b1} : PersistentHashMap \u03b1 \u03b2 \u2192 \u03b1 \u2192 PersistentHashMap \u03b1 \u03b2\n  | { root := n, size := sz }, k =>\n    let h := hash k |>.toUSize\n    let (n, del) := eraseAux n h k\n    { root := n, size := if del then sz - 1 else sz }\n\nsection\nvariable {m : Type w \u2192 Type w'} [Monad m]\nvariable {\u03c3 : Type w}\n\n@[specialize] partial def foldlMAux (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) : Node \u03b1 \u03b2 \u2192 \u03c3 \u2192 m \u03c3\n  | Node.collision keys vals heq, acc =>\n    let rec traverse (i : Nat) (acc : \u03c3) : m \u03c3 := do\n      if h : i < keys.size then\n        let k := keys.get \u27e8i, h\u27e9\n        let v := vals.get \u27e8i, heq \u25b8 h\u27e9\n        traverse (i+1) (\u2190 f acc k v)\n      else\n        pure acc\n    traverse 0 acc\n  | Node.entries entries, acc => entries.foldlM (fun acc entry =>\n    match entry with\n    | Entry.null      => pure acc\n    | Entry.entry k v => f acc k v\n    | Entry.ref node  => foldlMAux f node acc)\n    acc\n\n@[specialize] def foldlM {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 m \u03c3) (init : \u03c3) : m \u03c3 :=\n  foldlMAux f map.root init\n\n@[specialize] def forM {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03b1 \u2192 \u03b2 \u2192 m PUnit) : m PUnit :=\n  map.foldlM (fun _ => f) \u27e8\u27e9\n\n@[specialize] def foldl {_ : BEq \u03b1} {_ : Hashable \u03b1} (map : PersistentHashMap \u03b1 \u03b2) (f : \u03c3 \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03c3) (init : \u03c3) : \u03c3 :=\n  Id.run $ map.foldlM f init\nend\n\ndef toList {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) : List (\u03b1 \u00d7 \u03b2) :=\n  m.foldl (init := []) fun ps k v => (k, v) :: ps\n\nstructure Stats where\n  numNodes      : Nat := 0\n  numNull       : Nat := 0\n  numCollisions : Nat := 0\n  maxDepth      : Nat := 0\n\npartial def collectStats : Node \u03b1 \u03b2 \u2192 Stats \u2192 Nat \u2192 Stats\n  | Node.collision keys _ _, stats, depth =>\n    { stats with\n      numNodes      := stats.numNodes + 1,\n      numCollisions := stats.numCollisions + keys.size - 1,\n      maxDepth      := Nat.max stats.maxDepth depth }\n  | Node.entries entries, stats, depth =>\n    let stats :=\n      { stats with\n        numNodes      := stats.numNodes + 1,\n        maxDepth      := Nat.max stats.maxDepth depth }\n    entries.foldl (fun stats entry =>\n      match entry with\n      | Entry.null      => { stats with numNull := stats.numNull + 1 }\n      | Entry.ref node  => collectStats node stats (depth + 1)\n      | Entry.entry _ _ => stats)\n      stats\n\ndef stats {_ : BEq \u03b1} {_ : Hashable \u03b1} (m : PersistentHashMap \u03b1 \u03b2) : Stats :=\n  collectStats m.root {} 1\n\ndef Stats.toString (s : Stats) : String :=\n  s!\"\\{ nodes := {s.numNodes}, null := {s.numNull}, collisions := {s.numCollisions}, depth := {s.maxDepth}}\"\n\ninstance : ToString Stats := \u27e8Stats.toString\u27e9\n\nend PersistentHashMap\nend Std\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/stage0/src/Std/Data/PersistentHashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3634274616350385}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, E. W. Ayers\n-/\n\nimport category_theory.sites.sieves\nimport category_theory.limits.shapes.pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.category.preorder\nimport order.copy\n\n/-!\n# Grothendieck topologies\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nDefinition and lemmas about Grothendieck topologies.\nA Grothendieck topology for a category `C` is a set of sieves on each object `X` satisfying\ncertain closure conditions.\n\nAlternate versions of the axioms (in arrow form) are also described.\nTwo explicit examples of Grothendieck topologies are given:\n* The dense topology\n* The atomic topology\nas well as the complete lattice structure on Grothendieck topologies (which gives two additional\nexplicit topologies: the discrete and trivial topologies.)\n\nA pretopology, or a basis for a topology is defined in `pretopology.lean`. The topology associated\nto a topological space is defined in `spaces.lean`.\n\n## Tags\n\nGrothendieck topology, coverage, pretopology, site\n\n## References\n\n* [nLab, *Grothendieck topology*](https://ncatlab.org/nlab/show/Grothendieck+topology)\n* [S. MacLane, I. Moerdijk, *Sheaves in Geometry and Logic*][MM92]\n\n## Implementation notes\n\nWe use the definition of [nlab] and [MM92][] (Chapter III, Section 2), where Grothendieck topologies\nare saturated collections of morphisms, rather than the notions of the Stacks project (00VG) and\nthe Elephant, in which topologies are allowed to be unsaturated, and are then completed.\nTODO (BM): Add the definition from Stacks, as a pretopology, and complete to a topology.\n\nThis is so that we can produce a bijective correspondence between Grothendieck topologies on a\nsmall category and Lawvere-Tierney topologies on its presheaf topos, as well as the equivalence\nbetween Grothendieck topoi and left exact reflective subcategories of presheaf toposes.\n-/\n\nuniverses w v u\nnamespace category_theory\n\nopen category_theory category\n\nvariables (C : Type u) [category.{v} C]\n\n/--\nThe definition of a Grothendieck topology: a set of sieves `J X` on each object `X` satisfying\nthree axioms:\n1. For every object `X`, the maximal sieve is in `J X`.\n2. If `S \u2208 J X` then its pullback along any `h : Y \u27f6 X` is in `J Y`.\n3. If `S \u2208 J X` and `R` is a sieve on `X`, then provided that the pullback of `R` along any arrow\n   `f : Y \u27f6 X` in `S` is in `J Y`, we have that `R` itself is in `J X`.\n\nA sieve `S` on `X` is referred to as `J`-covering, (or just covering), if `S \u2208 J X`.\n\nSee <https://stacks.math.columbia.edu/tag/00Z4>, or [nlab], or [MM92][] Chapter III, Section 2,\nDefinition 1.\n-/\nstructure grothendieck_topology :=\n(sieves : \u03a0 (X : C), set (sieve X))\n(top_mem' : \u2200 X, \u22a4 \u2208 sieves X)\n(pullback_stable' : \u2200 \u2983X Y : C\u2984 \u2983S : sieve X\u2984 (f : Y \u27f6 X), S \u2208 sieves X \u2192 S.pullback f \u2208 sieves Y)\n(transitive' : \u2200 \u2983X\u2984 \u2983S : sieve X\u2984 (hS : S \u2208 sieves X) (R : sieve X),\n              (\u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984, S f \u2192 R.pullback f \u2208 sieves Y) \u2192 R \u2208 sieves X)\n\nnamespace grothendieck_topology\n\ninstance : has_coe_to_fun (grothendieck_topology C) (\u03bb _, \u03a0 X : C, set (sieve X)) := \u27e8sieves\u27e9\n\nvariables {C} {X Y : C} {S R : sieve X}\nvariables (J : grothendieck_topology C)\n\n/--\nAn extensionality lemma in terms of the coercion to a pi-type.\nWe prove this explicitly rather than deriving it so that it is in terms of the coercion rather than\nthe projection `.sieves`.\n-/\n@[ext]\nlemma ext {J\u2081 J\u2082 : grothendieck_topology C} (h : (J\u2081 : \u03a0 (X : C), set (sieve X)) = J\u2082) : J\u2081 = J\u2082 :=\nby { cases J\u2081, cases J\u2082, congr, apply h }\n\n@[simp] lemma mem_sieves_iff_coe : S \u2208 J.sieves X \u2194 S \u2208 J X := iff.rfl\n\n-- Also known as the maximality axiom.\n@[simp] lemma top_mem (X : C) : \u22a4 \u2208 J X := J.top_mem' X\n-- Also known as the stability axiom.\n@[simp] lemma pullback_stable (f : Y \u27f6 X) (hS : S \u2208 J X) : S.pullback f \u2208 J Y :=\nJ.pullback_stable' f hS\nlemma transitive (hS : S \u2208 J X) (R : sieve X)\n  (h : \u2200 \u2983Y\u2984 \u2983f : Y \u27f6 X\u2984, S f \u2192 R.pullback f \u2208 J Y) :\n  R \u2208 J X :=\nJ.transitive' hS R h\n\nlemma covering_of_eq_top : S = \u22a4 \u2192 S \u2208 J X := \u03bb h, h.symm \u25b8 J.top_mem X\n\n/--\nIf `S` is a subset of `R`, and `S` is covering, then `R` is covering as well.\n\nSee <https://stacks.math.columbia.edu/tag/00Z5> (2), or discussion after [MM92] Chapter III,\nSection 2, Definition 1.\n-/\nlemma superset_covering (Hss : S \u2264 R) (sjx : S \u2208 J X) : R \u2208 J X :=\nbegin\n  apply J.transitive sjx R (\u03bb Y f hf, _),\n  apply covering_of_eq_top,\n  rw [\u2190 top_le_iff, \u2190 S.pullback_eq_top_of_mem hf],\n  apply sieve.pullback_monotone _ Hss,\nend\n\n/--\nThe intersection of two covering sieves is covering.\n\nSee <https://stacks.math.columbia.edu/tag/00Z5> (1), or [MM92] Chapter III,\nSection 2, Definition 1 (iv).\n-/\nlemma intersection_covering (rj : R \u2208 J X) (sj : S \u2208 J X) : R \u2293 S \u2208 J X :=\nbegin\n  apply J.transitive rj _ (\u03bb Y f Hf, _),\n  rw [sieve.pullback_inter, R.pullback_eq_top_of_mem Hf],\n  simp [sj],\nend\n\n@[simp]\nlemma intersection_covering_iff : R \u2293 S \u2208 J X \u2194 R \u2208 J X \u2227 S \u2208 J X :=\n\u27e8\u03bb h, \u27e8J.superset_covering inf_le_left h, J.superset_covering inf_le_right h\u27e9,\n \u03bb t, intersection_covering _ t.1 t.2\u27e9\n\n\n\n/--\nThe sieve `S` on `X` `J`-covers an arrow `f` to `X` if `S.pullback f \u2208 J Y`.\nThis definition is an alternate way of presenting a Grothendieck topology.\n-/\ndef covers (S : sieve X) (f : Y \u27f6 X) : Prop := S.pullback f \u2208 J Y\n\nlemma covers_iff (S : sieve X) (f : Y \u27f6 X) : J.covers S f \u2194 S.pullback f \u2208 J Y :=\niff.rfl\n\nlemma covering_iff_covers_id (S : sieve X) : S \u2208 J X \u2194 J.covers S (\ud835\udfd9 X) :=\nby simp [covers_iff]\n\n/-- The maximality axiom in 'arrow' form: Any arrow `f` in `S` is covered by `S`. -/\nlemma arrow_max (f : Y \u27f6 X) (S : sieve X) (hf : S f) : J.covers S f :=\nbegin\n  rw [covers, (sieve.pullback_eq_top_iff_mem f).1 hf],\n  apply J.top_mem,\nend\n\n/-- The stability axiom in 'arrow' form: If `S` covers `f` then `S` covers `g \u226b f` for any `g`. -/\nlemma arrow_stable (f : Y \u27f6 X) (S : sieve X) (h : J.covers S f) {Z : C} (g : Z \u27f6 Y) :\n  J.covers S (g \u226b f) :=\nbegin\n  rw covers_iff at h \u22a2,\n  simp [h, sieve.pullback_comp],\nend\n\n/--\nThe transitivity axiom in 'arrow' form: If `S` covers `f` and every arrow in `S` is covered by\n`R`, then `R` covers `f`.\n-/\nlemma arrow_trans (f : Y \u27f6 X) (S R : sieve X) (h : J.covers S f) :\n  (\u2200 {Z : C} (g : Z \u27f6 X), S g \u2192 J.covers R g) \u2192 J.covers R f :=\nbegin\n  intro k,\n  apply J.transitive h,\n  intros Z g hg,\n  rw \u2190 sieve.pullback_comp,\n  apply k (g \u226b f) hg,\nend\n\nlemma arrow_intersect (f : Y \u27f6 X) (S R : sieve X) (hS : J.covers S f) (hR : J.covers R f) :\n  J.covers (S \u2293 R) f :=\nby simpa [covers_iff] using and.intro hS hR\n\nvariable (C)\n/--\nThe trivial Grothendieck topology, in which only the maximal sieve is covering. This topology is\nalso known as the indiscrete, coarse, or chaotic topology.\n\nSee [MM92] Chapter III, Section 2, example (a), or\nhttps://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies\n-/\ndef trivial : grothendieck_topology C :=\n{ sieves := \u03bb X, {\u22a4},\n  top_mem' := \u03bb X, rfl,\n  pullback_stable' := \u03bb X Y S f hf,\n  begin\n    rw set.mem_singleton_iff at \u22a2 hf,\n    simp [hf],\n  end,\n  transitive' := \u03bb X S hS R hR,\n  begin\n    rw [set.mem_singleton_iff, \u2190 sieve.id_mem_iff_eq_top] at hS,\n    simpa using hR hS,\n  end }\n\n/--\nThe discrete Grothendieck topology, in which every sieve is covering.\n\nSee https://en.wikipedia.org/wiki/Grothendieck_topology#The_discrete_and_indiscrete_topologies.\n-/\ndef discrete : grothendieck_topology C :=\n{ sieves := \u03bb X, set.univ,\n  top_mem' := by simp,\n  pullback_stable' := \u03bb X Y f, by simp,\n  transitive' := by simp }\nvariable {C}\n\nlemma trivial_covering : S \u2208 trivial C X \u2194 S = \u22a4 := set.mem_singleton_iff\n\n/-- See <https://stacks.math.columbia.edu/tag/00Z6> -/\ninstance : has_le (grothendieck_topology C) :=\n{ le := \u03bb J\u2081 J\u2082, (J\u2081 : \u03a0 (X : C), set (sieve X)) \u2264 (J\u2082 : \u03a0 (X : C), set (sieve X)) }\n\nlemma le_def {J\u2081 J\u2082 : grothendieck_topology C} :\n  J\u2081 \u2264 J\u2082 \u2194 (J\u2081 : \u03a0 (X : C), set (sieve X)) \u2264 J\u2082 := iff.rfl\n\n/-- See <https://stacks.math.columbia.edu/tag/00Z6> -/\ninstance : partial_order (grothendieck_topology C) :=\n{ le_refl := \u03bb J\u2081, le_def.mpr le_rfl,\n  le_trans := \u03bb J\u2081 J\u2082 J\u2083 h\u2081\u2082 h\u2082\u2083, le_def.mpr (le_trans h\u2081\u2082 h\u2082\u2083),\n  le_antisymm := \u03bb J\u2081 J\u2082 h\u2081\u2082 h\u2082\u2081, grothendieck_topology.ext (le_antisymm h\u2081\u2082 h\u2082\u2081),\n  ..grothendieck_topology.has_le }\n\n/-- See <https://stacks.math.columbia.edu/tag/00Z7> -/\ninstance : has_Inf (grothendieck_topology C) :=\n{ Inf := \u03bb T,\n  { sieves := Inf (sieves '' T),\n    top_mem' :=\n    begin\n      rintro X S \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      simp,\n    end,\n    pullback_stable' :=\n    begin\n      rintro X Y S hS f _ \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      apply J.pullback_stable _ (f _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9),\n    end,\n    transitive' :=\n    begin\n      rintro X S hS R h _ \u27e8\u27e8_, J, hJ, rfl\u27e9, rfl\u27e9,\n      apply J.transitive (hS _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9) _ (\u03bb Y f hf, h hf _ \u27e8\u27e8_, _, hJ, rfl\u27e9, rfl\u27e9),\n    end } }\n\n/-- See <https://stacks.math.columbia.edu/tag/00Z7> -/\nlemma is_glb_Inf (s : set (grothendieck_topology C)) : is_glb s (Inf s) :=\nbegin\n  refine @is_glb.of_image _ _ _ _ sieves _ _ _ _,\n  { intros, refl },\n  { exact is_glb_Inf _ },\nend\n\n/--\nConstruct a complete lattice from the `Inf`, but make the trivial and discrete topologies\ndefinitionally equal to the bottom and top respectively.\n-/\ninstance : complete_lattice (grothendieck_topology C) :=\ncomplete_lattice.copy\n(complete_lattice_of_Inf _ is_glb_Inf)\n_ rfl\n(discrete C)\n(begin\n  apply le_antisymm,\n  { exact @complete_lattice.le_top _ (complete_lattice_of_Inf _ is_glb_Inf) (discrete C) },\n  { intros X S hS,\n    apply set.mem_univ },\nend)\n(trivial C)\n(begin\n  apply le_antisymm,\n  { intros X S hS,\n    rw trivial_covering at hS,\n    apply covering_of_eq_top _ hS },\n  { refine @complete_lattice.bot_le _ (complete_lattice_of_Inf _ is_glb_Inf) (trivial C) },\nend)\n_ rfl\n_ rfl\n_ rfl\nInf rfl\n\ninstance : inhabited (grothendieck_topology C) := \u27e8\u22a4\u27e9\n\n@[simp] lemma trivial_eq_bot : trivial C = \u22a5 := rfl\n@[simp] lemma discrete_eq_top : discrete C = \u22a4 := rfl\n\n@[simp] lemma bot_covering : S \u2208 (\u22a5 : grothendieck_topology C) X \u2194 S = \u22a4 := trivial_covering\n@[simp] lemma top_covering : S \u2208 (\u22a4 : grothendieck_topology C) X := \u27e8\u27e9\n\nlemma bot_covers (S : sieve X) (f : Y \u27f6 X) :\n  (\u22a5 : grothendieck_topology C).covers S f \u2194 S f :=\nby rw [covers_iff, bot_covering, \u2190 sieve.pullback_eq_top_iff_mem]\n\n@[simp] lemma top_covers (S : sieve X) (f : Y \u27f6 X) : (\u22a4 : grothendieck_topology C).covers S f :=\nby simp [covers_iff]\n\n/--\nThe dense Grothendieck topology.\n\nSee https://ncatlab.org/nlab/show/dense+topology, or [MM92] Chapter III, Section 2, example (e).\n-/\ndef dense : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2200 {Y : C} (f : Y \u27f6 X), \u2203 Z (g : Z \u27f6 Y), S (g \u226b f),\n  top_mem' := \u03bb X Y f, \u27e8Y, \ud835\udfd9 Y, \u27e8\u27e9\u27e9,\n  pullback_stable' :=\n  begin\n    intros X Y S h H Z f,\n    rcases H (f \u226b h) with \u27e8W, g, H'\u27e9,\n    exact \u27e8W, g, by simpa\u27e9,\n  end,\n  transitive' :=\n  begin\n    intros X S H\u2081 R H\u2082 Y f,\n    rcases H\u2081 f with \u27e8Z, g, H\u2083\u27e9,\n    rcases H\u2082 H\u2083 (\ud835\udfd9 Z) with \u27e8W, h, H\u2084\u27e9,\n    exact \u27e8W, (h \u226b g), by simpa using H\u2084\u27e9,\n  end }\n\nlemma dense_covering : S \u2208 dense X \u2194 \u2200 {Y} (f : Y \u27f6 X), \u2203 Z (g : Z \u27f6 Y), S (g \u226b f) :=\niff.rfl\n\n/--\nA category satisfies the right Ore condition if any span can be completed to a commutative square.\nNB. Any category with pullbacks obviously satisfies the right Ore condition, see\n`right_ore_of_pullbacks`.\n-/\ndef right_ore_condition (C : Type u) [category.{v} C] : Prop :=\n\u2200 {X Y Z : C} (yx : Y \u27f6 X) (zx : Z \u27f6 X), \u2203 W (wy : W \u27f6 Y) (wz : W \u27f6 Z), wy \u226b yx = wz \u226b zx\n\nlemma right_ore_of_pullbacks [limits.has_pullbacks C] : right_ore_condition C :=\n\u03bb X Y Z yx zx, \u27e8_, _, _, limits.pullback.condition\u27e9\n\n/--\nThe atomic Grothendieck topology: a sieve is covering iff it is nonempty.\nFor the pullback stability condition, we need the right Ore condition to hold.\n\nSee https://ncatlab.org/nlab/show/atomic+site, or [MM92] Chapter III, Section 2, example (f).\n-/\ndef atomic (hro : right_ore_condition C) : grothendieck_topology C :=\n{ sieves := \u03bb X S, \u2203 Y (f : Y \u27f6 X), S f,\n  top_mem' := \u03bb X, \u27e8_, \ud835\udfd9 _, \u27e8\u27e9\u27e9,\n  pullback_stable' :=\n  begin\n    rintros X Y S h \u27e8Z, f, hf\u27e9,\n    rcases hro h f with \u27e8W, g, k, comm\u27e9,\n    refine \u27e8_, g, _\u27e9,\n    simp [comm, hf],\n  end,\n  transitive' :=\n  begin\n    rintros X S \u27e8Y, f, hf\u27e9 R h,\n    rcases h hf with \u27e8Z, g, hg\u27e9,\n    exact \u27e8_, _, hg\u27e9,\n  end }\n\n/-- `J.cover X` denotes the poset of covers of `X` with respect to the\nGrothendieck topology `J`. -/\n@[derive preorder]\ndef cover (X : C) := { S : sieve X // S \u2208 J X }\n\nnamespace cover\n\nvariables {J}\n\ninstance : has_coe (J.cover X) (sieve X) := \u27e8\u03bb S, S.1\u27e9\n\ninstance : has_coe_to_fun (J.cover X) (\u03bb S, \u03a0 \u2983Y\u2984 (f : Y \u27f6 X), Prop) :=\n\u27e8\u03bb S Y f, (S : sieve X) f\u27e9\n\n@[simp]\nlemma coe_fun_coe (S : J.cover X) (f : Y \u27f6 X) : (S : sieve X) f = S f := rfl\n\nlemma condition (S : J.cover X) : (S : sieve X) \u2208 J X := S.2\n\n@[ext]\nlemma ext (S T : J.cover X) (h : \u2200 \u2983Y\u2984 (f : Y \u27f6 X), S f \u2194 T f) : S = T :=\nsubtype.ext $ sieve.ext h\n\ninstance : order_top (J.cover X) :=\n{ top := \u27e8\u22a4, J.top_mem _\u27e9,\n  le_top := \u03bb S Y f h, by tauto,\n  ..(infer_instance : preorder _) }\n\ninstance : semilattice_inf (J.cover X) :=\n{ inf := \u03bb S T, \u27e8S \u2293 T, J.intersection_covering S.condition T.condition\u27e9,\n  le_antisymm := \u03bb S T h1 h2, ext _ _ $ \u03bb Y f, \u27e8h1 _, h2 _\u27e9,\n  inf_le_left := \u03bb S T Y f hf, hf.1,\n  inf_le_right := \u03bb S T Y f hf, hf.2,\n  le_inf := \u03bb S T W h1 h2 Y f h, \u27e8h1 _ h, h2 _ h\u27e9,\n  ..(infer_instance : preorder _) }\n\ninstance : inhabited (J.cover X) := \u27e8\u22a4\u27e9\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_nonempty_instance, ext]\nstructure arrow (S : J.cover X) :=\n(Y : C)\n(f : Y \u27f6 X)\n(hf : S f)\n\n/-- An auxiliary structure, used to define `S.index` in `plus.lean`. -/\n@[nolint has_nonempty_instance, ext]\nstructure relation (S : J.cover X) :=\n(Y\u2081 Y\u2082 Z : C)\n(g\u2081 : Z \u27f6 Y\u2081)\n(g\u2082 : Z \u27f6 Y\u2082)\n(f\u2081 : Y\u2081 \u27f6 X)\n(f\u2082 : Y\u2082 \u27f6 X)\n(h\u2081 : S f\u2081)\n(h\u2082 : S f\u2082)\n(w : g\u2081 \u226b f\u2081 = g\u2082 \u226b f\u2082)\n\n/-- Map a `arrow` along a refinement `S \u27f6 T`. -/\n@[simps]\ndef arrow.map {S T : J.cover X} (I : S.arrow) (f : S \u27f6 T) : T.arrow :=\n\u27e8I.Y, I.f, f.le _ I.hf\u27e9\n\n/-- Map a `relation` along a refinement `S \u27f6 T`. -/\n@[simps]\ndef relation.map {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) : T.relation :=\n\u27e8_, _, _, I.g\u2081, I.g\u2082, I.f\u2081, I.f\u2082, f.le _ I.h\u2081, f.le _ I.h\u2082, I.w\u27e9\n\n/-- The first `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.fst {S : J.cover X} (I : S.relation) : S.arrow :=\n\u27e8I.Y\u2081, I.f\u2081, I.h\u2081\u27e9\n\n/-- The second `arrow` associated to a `relation`.\nUsed in defining `index` in `plus.lean`. -/\n@[simps]\ndef relation.snd {S : J.cover X} (I : S.relation) : S.arrow :=\n\u27e8I.Y\u2082, I.f\u2082, I.h\u2082\u27e9\n\n@[simp]\nlemma relation.map_fst {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) :\n   I.fst.map f = (I.map f).fst := rfl\n\n@[simp]\nlemma relation.map_snd {S T : J.cover X} (I : S.relation) (f : S \u27f6 T) :\n  I.snd.map f = (I.map f).snd := rfl\n\n/-- Pull back a cover along a morphism. -/\ndef pullback (S : J.cover X) (f : Y \u27f6 X) : J.cover Y :=\n\u27e8sieve.pullback f S, J.pullback_stable _ S.condition\u27e9\n\n/-- An arrow of `S.pullback f` gives rise to an arrow of `S`. -/\n@[simps]\ndef arrow.base {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).arrow) : S.arrow :=\n\u27e8I.Y, I.f \u226b f, I.hf\u27e9\n\n/-- A relation of `S.pullback f` gives rise to a relation of `S`. -/\n@[simps]\ndef relation.base {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) : S.relation :=\n\u27e8_, _, _, I.g\u2081, I.g\u2082, I.f\u2081 \u226b f, I.f\u2082\u226b f, I.h\u2081, I.h\u2082, by simp [reassoc_of I.w]\u27e9\n\n@[simp]\nlemma relation.base_fst {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.fst.base = I.base.fst := rfl\n\n@[simp]\nlemma relation.base_snd {f : Y \u27f6 X} {S : J.cover X} (I : (S.pullback f).relation) :\n I.snd.base = I.base.snd := rfl\n\n@[simp]\nlemma coe_pullback {Z : C} (f : Y \u27f6 X) (g : Z \u27f6 Y) (S : J.cover X) :\n  (S.pullback f) g \u2194 S (g \u226b f) := iff.rfl\n\n/-- The isomorphism between `S` and the pullback of `S` w.r.t. the identity. -/\ndef pullback_id (S : J.cover X) : S.pullback (\ud835\udfd9 X) \u2245 S :=\neq_to_iso $ cover.ext _ _ $ \u03bb Y f, by simp\n\n/-- Pulling back with respect to a composition is the composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (S : J.cover X) (f : Z \u27f6 Y) (g : Y \u27f6 X) :\n  S.pullback (f \u226b g) \u2245 (S.pullback g).pullback f :=\neq_to_iso $ cover.ext _ _ $ \u03bb Y f, by simp\n\n/-- Combine a family of covers over a cover. -/\ndef bind {X : C} (S : J.cover X) (T : \u03a0 (I : S.arrow), J.cover I.Y) : J.cover X :=\n\u27e8sieve.bind S (\u03bb Y f hf, T \u27e8Y, f, hf\u27e9), J.bind_covering S.condition (\u03bb _ _ _, (T _).condition)\u27e9\n\n/-- The canonical moprhism from `S.bind T` to `T`. -/\ndef bind_to_base {X : C} (S : J.cover X) (T : \u03a0 (I : S.arrow), J.cover I.Y) : S.bind T \u27f6 S :=\nhom_of_le $ by { rintro Y f \u27e8Z,e1,e2,h1,h2,h3\u27e9, rw \u2190 h3, apply sieve.downward_closed, exact h1 }\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the object `B`. -/\nnoncomputable def arrow.middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : C :=\nI.hf.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `A \u27f6 B`. -/\nnoncomputable def arrow.to_middle_hom {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.Y \u27f6 I.middle :=\nI.hf.some_spec.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `B \u27f6 X`. -/\nnoncomputable def arrow.from_middle_hom {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.middle \u27f6 X :=\nI.hf.some_spec.some_spec.some\n\nlemma arrow.from_middle_condition {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S I.from_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `B \u27f6 X`, as an arrow. -/\nnoncomputable\ndef arrow.from_middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : S.arrow := \u27e8_, I.from_middle_hom, I.from_middle_condition\u27e9\n\nlemma arrow.to_middle_condition {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle) I.to_middle_hom :=\nI.hf.some_spec.some_spec.some_spec.some_spec.1\n\n/-- An arrow in bind has the form `A \u27f6 B \u27f6 X` where `A \u27f6 B` is an arrow in `T I` for some `I`.\n and `B \u27f6 X` is an arrow of `S`. This is the hom `A \u27f6 B`, as an arrow. -/\nnoncomputable\ndef arrow.to_middle {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : (T I.from_middle).arrow := \u27e8_, I.to_middle_hom, I.to_middle_condition\u27e9\n\nlemma arrow.middle_spec {X : C} {S : J.cover X} {T : \u03a0 (I : S.arrow), J.cover I.Y}\n  (I : (S.bind T).arrow) : I.to_middle_hom \u226b I.from_middle_hom = I.f :=\nI.hf.some_spec.some_spec.some_spec.some_spec.2\n\n-- This is used extensively in `plus.lean`, etc.\n-- We place this definition here as it will be used in `sheaf.lean` as well.\n/-- To every `S : J.cover X` and presheaf `P`, associate a `multicospan_index`. -/\ndef index {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D) :\n  limits.multicospan_index D :=\n{ L := S.arrow,\n  R := S.relation,\n  fst_to := \u03bb I, I.fst,\n  snd_to := \u03bb I, I.snd,\n  left := \u03bb I, P.obj (opposite.op I.Y),\n  right := \u03bb I, P.obj (opposite.op I.Z),\n  fst := \u03bb I, P.map I.g\u2081.op,\n  snd := \u03bb I, P.map I.g\u2082.op }\n\n/-- The natural multifork associated to `S : J.cover X` for a presheaf `P`.\nSaying that this multifork is a limit is essentially equivalent to the sheaf condition at the\ngiven object for the given covering sieve. See `sheaf.lean` for an equivalent sheaf condition\nusing this.\n-/\nabbreviation multifork {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D) :\n  limits.multifork (S.index P) :=\nlimits.multifork.of_\u03b9 _ (P.obj (opposite.op X)) (\u03bb I, P.map I.f.op) begin\n  intros I,\n  dsimp [index],\n  simp only [\u2190 P.map_comp, \u2190 op_comp, I.w]\nend\n\n/-- The canonical map from `P.obj (op X)` to the multiequalizer associated to a covering sieve,\nassuming such a multiequalizer exists. This will be used in `sheaf.lean` to provide an equivalent\nsheaf condition in terms of multiequalizers. -/\nnoncomputable\nabbreviation to_multiequalizer {D : Type w} [category.{max v u} D] (S : J.cover X) (P : C\u1d52\u1d56 \u2964 D)\n  [limits.has_multiequalizer (S.index P)] :\nP.obj (opposite.op X) \u27f6 limits.multiequalizer (S.index P) :=\nlimits.multiequalizer.lift _ _ (\u03bb I, P.map I.f.op) begin\n  intros I,\n  dsimp only [index, relation.fst, relation.snd],\n  simp only [\u2190 P.map_comp, \u2190 op_comp, I.w],\nend\n\nend cover\n\n/-- Pull back a cover along a morphism. -/\n@[simps obj]\ndef pullback (f : Y \u27f6 X) : J.cover X \u2964 J.cover Y :=\n{ obj := \u03bb S, S.pullback f,\n  map := \u03bb S T f, (sieve.pullback_monotone _ f.le).hom }\n\n/-- Pulling back along the identity is naturally isomorphic to the identity functor. -/\ndef pullback_id (X : C) : J.pullback (\ud835\udfd9 X) \u2245 \ud835\udfed _ :=\nnat_iso.of_components (\u03bb S, S.pullback_id) $ by tidy\n\n/-- Pulling back along a composition is naturally isomorphic to\nthe composition of the pullbacks. -/\ndef pullback_comp {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  J.pullback (f \u226b g) \u2245 J.pullback g \u22d9 J.pullback f :=\nnat_iso.of_components (\u03bb S, S.pullback_comp f g) $ by tidy\n\nend grothendieck_topology\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/grothendieck.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.6334102775181399, "lm_q1q2_score": 0.36337379927065877}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2)\nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSchur's Inequality Short Simpler\nLet $x, y, z \\in \\R_{\\ge 0}$ be positive real numbers such that $x \\ge y \\ge z \\ge 0$.\n\nLet $t \\in \\R, t > 0$ be a (strictly) positive real number.\n\n\nThen:\n:$x^t \\paren {x - y} \\paren {x - z} + y^t \\paren {y - z} \\paren {y - x} + z^t \\paren {z - x} \\paren {z - y} \\ge 0$\n\n`proof`\n\nConsider the expression:\n:$\\paren {x - y} \\paren {x^t \\paren {x - z} - y^t \\paren {y - z}} + z^t \\paren {x - z} \\paren {y - z}$\n\n\nWe see that every term in the above is non-negative. So, directly:\n:$(1): \\quad \\paren {x - y} \\paren {x^t \\paren {x - z} - y^t \\paren {y - z}} + z^t \\paren {x - z} \\paren {y - z} \\ge 0$\n\n$(1) can be rearranged to Schur's inequality.\n{{qed}}\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n--OUTPUT 1\ntheorem schur_inequality_short_simpler (x y z : \u211d) (h1 : x \u2265 y) (h2 : y \u2265 z) (h3 : z \u2265 0) (t : \u211d) (h4 : t > 0) : x^t * (x - y) * (x - z) + y^t * (y - z) * (y - x) + z^t * (z - x) * (z - y) \u2265 0 :=\nbegin\n  -- Consider the expression:\n  have h5 : (x - y) * (x^t * (x - z) - y^t * (y - z)) + z^t * (x - z) * (y - z) \u2265 0,\n  {\n    -- We see that every term in the above is non-negative. So, directly:\n    show (x - y) * (x^t * (x - z) - y^t * (y - z)) + z^t * (x - z) * (y - z) \u2265 0, from by auto [h1, h2, h3, h4, pow_pos, mul_nonneg, sub_nonneg, sub_nonneg_of_le, mul_nonneg, mul_nonneg, add_nonneg],\n  },\n\n  -- $(1) can be rearranged to Schur's inequality.\n  show x^t * (x - y) * (x - z) + y^t * (y - z) * (y - x) + z^t * (z - x) * (z - y) \u2265 0, from by auto [h1, h2, h3, h4, pow_pos, mul_nonneg, sub_nonneg, sub_nonneg_of_le, mul_nonneg, mul_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg, add_nonneg,\nend --Needs more than 2000 tokens!\n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/lean_proof_auto_with_comments-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Schur's Inequality Short Simpler.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3633737913307776}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n\n! This file was ported from Lean 3 source module tactic.field_simp\n! leanprover-community/mathlib commit 24a09b317861fb5aa3850cfd1254acf729b0a398\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Interactive\nimport Mathbin.Tactic.NormNum\n\n/-!\n# `field_simp` tactic\n\nTactic to clear denominators in algebraic expressions, based on `simp` with a specific simpset.\n-/\n\n\nnamespace Tactic\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:330:4: warning: unsupported (TODO): `[tacs] -/\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Try to prove a goal of the form `x \u2260 0` by calling `assumption`, or `norm_num1` if `x` is\n      a numeral. -/\n    unsafe\n  def\n    field_simp.ne_zero\n    : tactic Unit\n    :=\n      do\n        let goal \u2190 tactic.target\n          match\n            goal\n            with\n            | q( $ ( e ) \u2260 0 ) => assumption <|> do let n \u2190 e sorry\n              | _ => tactic.fail \"goal should be of the form `x \u2260 0`\"\n#align tactic.field_simp.ne_zero tactic.field_simp.ne_zero\n\nnamespace Interactive\n\n/- ./././Mathport/Syntax/Translate/Tactic/Mathlib/Core.lean:38:34: unsupported: setup_tactic_parser -/\n/--\nThe goal of `field_simp` is to reduce an expression in a field to an expression of the form `n / d`\nwhere neither `n` nor `d` contains any division symbol, just using the simplifier (with a carefully\ncrafted simpset named `field_simps`) to reduce the number of division symbols whenever possible by\niterating the following steps:\n\n- write an inverse as a division\n- in any product, move the division to the right\n- if there are several divisions in a product, group them together at the end and write them as a\n  single division\n- reduce a sum to a common denominator\n\nIf the goal is an equality, this simpset will also clear the denominators, so that the proof\ncan normally be concluded by an application of `ring` or `ring_exp`.\n\n`field_simp [hx, hy]` is a short form for\n`simp [-one_div, -mul_eq_zero, hx, hy] with field_simps {discharger := tactic.field_simp.ne_zero}`\n\nNote that this naive algorithm will not try to detect common factors in denominators to reduce the\ncomplexity of the resulting expression. Instead, it relies on the ability of `ring` to handle\ncomplicated expressions in the next step.\n\nAs always with the simplifier, reduction steps will only be applied if the preconditions of the\nlemmas can be checked. This means that proofs that denominators are nonzero should be included. The\nfact that a product is nonzero when all factors are, and that a power of a nonzero number is\nnonzero, are included in the simpset, but more complicated assertions (especially dealing with sums)\nshould be given explicitly. If your expression is not completely reduced by the simplifier\ninvocation, check the denominators of the resulting expression and provide proofs that they are\nnonzero to enable further progress.\n\nTo check that denominators are nonzero, `field_simp` will look for facts in the context, and\nwill try to apply `norm_num` to close numerical goals.\n\nThe invocation of `field_simp` removes the lemma `one_div` from the simpset, as this lemma\nworks against the algorithm explained above. It also removes\n`mul_eq_zero : x * y = 0 \u2194 x = 0 \u2228 y = 0`, as `norm_num` can not work on disjunctions to\nclose goals of the form `24 \u2260 0`, and replaces it with `mul_ne_zero : x \u2260 0 \u2192 y \u2260 0 \u2192 x * y \u2260 0`\ncreating two goals instead of a disjunction.\n\nFor example,\n```lean\nexample (a b c d x y : \u2102) (hx : x \u2260 0) (hy : y \u2260 0) :\n  a + b / x + c / x^2 + d / x^3 = a + x\u207b\u00b9 * (y * b / y + (d / x + c) / x) :=\nbegin\n  field_simp,\n  ring\nend\n```\n\nMoreover, the `field_simp` tactic can also take care of inverses of units in\na general (commutative) monoid/ring and partial division `/\u209a`, see `algebra.group.units`\nfor the definition. Analogue to the case above, the lemma `one_divp` is removed from the simpset\nas this works against the algorithm. If you have objects with a `is_unit x` instance like\n`(x : R) (hx : is_unit x)`, you should lift them with\n`lift x to R\u02e3 using id hx, rw is_unit.unit_of_coe_units, clear hx`\nbefore using `field_simp`.\n\nSee also the `cancel_denoms` tactic, which tries to do a similar simplification for expressions\nthat have numerals in denominators.\nThe tactics are not related: `cancel_denoms` will only handle numeric denominators, and will try to\nentirely remove (numeric) division from the expression by multiplying by a factor.\n-/\nunsafe def field_simp (no_dflt : parse only_flag) (hs : parse simp_arg_list)\n    (attr_names : parse with_ident_list) (locat : parse location)\n    (cfg : simp_config_ext := { discharger := field_simp.ne_zero }) : tactic Unit :=\n  let attr_names := `field_simps :: attr_names\n  let hs :=\n    simp_arg_type.except `one_div ::\n      simp_arg_type.except `mul_eq_zero :: simp_arg_type.except `one_divp :: hs\n  propagate_tags (simp_core cfg.toSimpConfig cfg.discharger no_dflt hs attr_names locat >> skip)\n#align tactic.interactive.field_simp tactic.interactive.field_simp\n\nadd_tactic_doc\n  { Name := \"field_simp\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.field_simp]\n    tags := [\"simplification\", \"arithmetic\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/FieldSimp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341024983754, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3633737833908962}}
{"text": "import main\nimport split_cycle\n\nopen_locale classical\n\nvariables {V X : Type}\n\ndef minus_voter (P : Prof V X) (i : V) : Prof {v : V // v \u2260 i} X := \u03bb v x y, P v.val x y\n-- use strict preference.\ndef positive_involvement (F : VSCC) (P : Prof V X) : Prop := \u2200 (x : X) (i : V), x \u2208 F {v : V // v \u2260 i} X (minus_voter P i) \u2192 (\u2200 y, y \u2260 x \u2192 P i x y) \u2192 x \u2208 F V X P\n\ndef negative_involvement (F : VSCC) (P : Prof V X) : Prop := \u2200 (x : X) (i : V), x \u2209 F {v : V // v \u2260 i} X (minus_voter P i) \u2192 (\u2200 y, y \u2260 x \u2192 P i y x) \u2192 x \u2209 F V X P\n\nlemma minus_voter_margin_eq [fintype V] (i : V) [fintype {v // v \u2260 i}] (P : Prof V X) [profile_asymmetric P] (z x : X) (p : P i x z) : margin (minus_voter P i) z x = margin P z x + 1 :=\nbegin\n  have asymm := _inst_3.asymmetric,\n  unfold margin,\n  have meq : ((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 z x) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), P x_1 z x) finset.univ).card),\n    {apply eq.symm,\n    have f : (finset.filter (\u03bb (x_1 : V), P x_1 z x) finset.univ) = (finset.map \u27e8(\u03bb (s : {v // v \u2260 i}), s.val), begin\n      obviously,\n    end\u27e9 (finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 z x) finset.univ)),\n      {simp, ext1, obviously, \n      rw \u2190\u1fb0_1 at p, \n      exact (asymm a z x \u1fb0) p, }, -- require asymmetry\n      rw f,\n      simp,},\n  rw meq,\n  have o : \u2200 a b c : \u2124, c = b + 1 \u2192 a - b = a - c + 1 := by omega,\n  apply o (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 z x) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 x z) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 x z) finset.univ).card)),\n  norm_cast,\n  rw finset.card_eq_succ,\n  use i, use (finset.filter (\u03bb (x_1 : V), x_1 \u2260 i \u2227 P x_1 x z) finset.univ),\n  have nmem : i \u2209 (finset.filter (\u03bb (x_1 : V), x_1 \u2260 i \u2227 P x_1 x z) finset.univ),\n    {obviously,},\n\n  use nmem,\n  split,\n  ext1,\n  simp only [true_and, finset.mem_univ, ne.def, finset.mem_insert, finset.mem_filter],\n\n  split,\n    {intro m,\n    cases m,\n    rw m, \n    exact p,\n    exact and.right m,},\n    {intro m,\n    by_cases a = i,\n    use h,\n    apply or.inr,\n    exact \u27e8h, m\u27e9,},\n\n  refine finset.card_congr _ _ _ _,\n    {intro a,\n    intro a_spec,\n    simp at a_spec,\n    exact \u27e8a, and.left a_spec\u27e9,},\n    {intro a, intro ha, simp at ha, simp, \n    unfold minus_voter, simp, exact and.right ha,},\n    {intros a b ha hb, simp at ha, simp at hb,\n    simp, },\n    {intros b hb, simp at hb, simp,\n    use b.val, simp, use b.property, unfold minus_voter at hb,\n    exact hb,},\nend\n\nlemma minus_voter_margin_le [fintype V] (i : V) [fintype {v // v \u2260 i}] (P : Prof V X) [profile_asymmetric P] (a b : X) : margin (minus_voter P i) a b \u2264 margin P a b + 1 :=\nbegin\n  by_cases P i b a,\n    {have m := minus_voter_margin_eq i P a b h,\n    exact le_of_eq m,},\n\n  unfold margin,\n  \n  have meq : ((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 b a) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), P x_1 b a) finset.univ).card),\n    {refine finset.card_congr _ _ _ _,\n      {intros a a_spec,\n      exact a.val,},\n      {intros a_1 ha, simp, simp at ha, unfold minus_voter at ha, exact ha, },\n      {intros a_1 b_1 ha hb, simp, obviously,},\n      {intros b_1 b_spec, simp at b_spec, \n      let h2 := h,\n      by_cases b_1 = i,\n      exfalso, rw h at b_spec, exact h2 b_spec, \n      use \u27e8b_1, h\u27e9,\n      split, simp, simp, unfold minus_voter, exact b_spec,},\n    },\n\n  rw meq, \n  have o : \u2200 a b c : \u2124, a \u2264 c \u2192 a - b \u2264 c - b + 1 := by omega,\n  apply o (\u2191((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 a b) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 b a) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 a b) finset.univ).card)),\n\n  norm_cast,\n  refine finset.card_le_card_of_inj_on _ _ _,\n  exact (\u03bb v, v.val),\n  intros a_1 a_spec,\n  simp, simp at a_spec,\n  unfold minus_voter at a_spec,\n  exact a_spec,\n\n  intros a1 a1_spec a2 a2_spec,\n  obviously,\nend\n\ntheorem positive_involvement_split_cycle [fintype V] (P : Prof V X) [profile_asymmetric P] : positive_involvement split_cycle P :=\nbegin\n  unfold positive_involvement,\n  intros x i w p,\n  unfold split_cycle, unfold max_el_VSCC, unfold split_cycle_VCCR, simp,\n  unfold split_cycle at w, unfold max_el_VSCC at w, unfold split_cycle_VCCR at w, \n  unfold split_cycle_CCR at w, simp at w,\n  intro y, \n  unfold split_cycle_CCR,\n  push_neg,\n  use _inst_1,\n  intro m,\n  specialize w y,\n\n  casesI w with _inst_3 w,\n\n  have ynx := ne_of_margin_pos m,\n\n  have mxy := minus_voter_margin_eq i P y x (p y ynx),\n\n  have mp : margin_pos (minus_voter P i) y x,\n    {unfold margin_pos, rw mxy, \n    have o : \u2200 a : \u2124, a > 0 \u2192 0 < a + 1 := by omega,\n    exact o (margin P y x) m,},\n  \n  specialize w mp,\n  cases w with c w,\n  cases w with x_mem w,\n  cases w with y_mem w,\n\n  use c, use x_mem, use y_mem,\n\n  refine cycle_of_cycle_imp _ w,\n  intros a b mab,\n  have o : \u2200 a b c d : \u2124, a \u2264 b \u2192 a = c + 1 \u2192 b \u2264 d + 1 \u2192 c \u2264 d := by omega,\n\n  apply (o (margin (minus_voter P i) y x) (margin (minus_voter P i) a b) (margin P y x) (margin P a b) mab mxy ),\n\n  exact minus_voter_margin_le i P a b,\nend\n\nlemma minus_voter_margin_eq2 [fintype V] (i : V) [fintype {v // v \u2260 i}] (P : Prof V X) [profile_asymmetric P] (z x : X) (p : P i z x) : margin (minus_voter P i) z x + 1 = margin P z x :=\nbegin\n  have asymm := _inst_3.asymmetric,\n  unfold margin,\n  have meq : ((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 x z) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), P x_1 x z) finset.univ).card),\n    {apply eq.symm,\n    have f : (finset.filter (\u03bb (x_1 : V), P x_1 x z) finset.univ) = (finset.map \u27e8(\u03bb (s : {v // v \u2260 i}), s.val), begin\n      obviously,\n    end\u27e9 (finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 x z) finset.univ)),\n      {simp, ext1, obviously, \n      rw \u2190\u1fb0_1 at p, \n      exact (asymm a x z \u1fb0) p, }, -- require asymmetry\n      rw f,\n      simp,},\n  rw meq,\n  have o : \u2200 a b c : \u2124, c = a + 1 \u2192 a - b + 1 = c - b := by omega,\n  apply o (\u2191((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 z x) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 x z) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 z x) finset.univ).card)),\n  norm_cast,\n  rw finset.card_eq_succ,\n  use i, use (finset.filter (\u03bb (x_1 : V), x_1 \u2260 i \u2227 P x_1 z x) finset.univ),\n  have nmem : i \u2209 (finset.filter (\u03bb (x_1 : V), x_1 \u2260 i \u2227 P x_1 z x) finset.univ),\n    {obviously,},\n\n  use nmem,\n  split,\n  ext1,\n  simp only [true_and, finset.mem_univ, ne.def, finset.mem_insert, finset.mem_filter],\n\n  split,\n    {intro m,\n    cases m,\n    rw m, \n    exact p,\n    exact and.right m,},\n    {intro m,\n    by_cases a = i,\n    use h,\n    apply or.inr,\n    exact \u27e8h, m\u27e9,},\n\n  refine finset.card_congr _ _ _ _,\n    {intro a,\n    intro a_spec,\n    simp at a_spec,\n    exact \u27e8a, and.left a_spec\u27e9,},\n    {intro a, intro ha, simp at ha, simp, \n    unfold minus_voter, simp, exact and.right ha,},\n    {intros a b ha hb, simp at ha, simp at hb,\n    simp, },\n    {intros b hb, simp at hb, simp,\n    use b.val, simp, use b.property, unfold minus_voter at hb,\n    exact hb,},\nend\n\nlemma minus_voter_margin_le2 [fintype V] (i : V) [fintype {v // v \u2260 i}] (P : Prof V X) [profile_asymmetric P] (a b : X) : margin P a b \u2264 margin (minus_voter P i) a b + 1 :=\nbegin\n  by_cases P i a b,\n    {have m := minus_voter_margin_eq2 i P a b h,\n    exact le_of_eq m.symm,},\n\n  unfold margin,\n  \n  have meq : ((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 a b) finset.univ).card) = ((finset.filter (\u03bb (x_1 : V), P x_1 a b) finset.univ).card),\n    {refine finset.card_congr _ _ _ _,\n      {intros a a_spec,\n      exact a.val,},\n      {intros a_1 ha, simp, simp at ha, unfold minus_voter at ha, exact ha, },\n      {intros a_1 b_1 ha hb, simp, obviously,},\n      {intros b_1 b_spec, simp at b_spec, \n      let h2 := h,\n      by_cases b_1 = i,\n      exfalso, rw h at b_spec, exact h2 b_spec, \n      use \u27e8b_1, h\u27e9,\n      split, simp, simp, unfold minus_voter, exact b_spec,},\n    },\n\n  rw meq, \n  have o : \u2200 a b c : \u2124, c \u2264 b \u2192 a - b \u2264 a - c + 1 := by omega,\n  apply o (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 a b) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : V), P x_1 b a) finset.univ).card)) (\u2191((finset.filter (\u03bb (x_1 : {v // v \u2260 i}), minus_voter P i x_1 b a) finset.univ).card)),\n\n  norm_cast,\n  refine finset.card_le_card_of_inj_on _ _ _,\n  exact (\u03bb v, v.val),\n  intros a_1 a_spec,\n  simp, simp at a_spec,\n  unfold minus_voter at a_spec,\n  exact a_spec,\n\n  intros a1 a1_spec a2 a2_spec,\n  obviously,\nend\n\ntheorem negative_involvement_split_cycle [fintype V] (P : Prof V X) [profile_asymmetric P] : negative_involvement split_cycle P :=\nbegin\n  intros x i w p,\n  unfold split_cycle, unfold max_el_VSCC, unfold split_cycle_VCCR, simp,\n  unfold split_cycle at w, unfold max_el_VSCC at w, unfold split_cycle_VCCR at w, \n  unfold split_cycle_CCR at w, simp at w,\n  cases w with y w,\n  use y,\n  introI _inst_3,\n  specialize w (subtype.fintype (\u03bb v, v \u2260 i)),\n  cases w with m w,\n\n  have ynx := ne_of_margin_pos m,\n  have mxy := minus_voter_margin_eq2 i P y x (p y ynx),\n\n  have o : \u2200 a b : \u2124, 0 < a \u2192 a + 1 = b \u2192 0 < b := by omega,\n  use o (margin (minus_voter P i) y x) (margin P y x) m mxy,\n  push_neg,\n  intros l x_mem y_mem,\n  specialize w l, specialize w x_mem, specialize w y_mem,\n  contrapose w,\n  push_neg,\n  push_neg at w,\n\n  refine cycle_of_cycle_imp _ w,\n  intros a b mab,\n\n  have o : \u2200 a b c d : \u2124, a \u2264 b \u2192 c + 1 = a \u2192 b \u2264 d + 1 \u2192 c \u2264 d := by omega,\n\n  apply (o (margin P y x) (margin P a b) (margin (minus_voter P i) y x) (margin (minus_voter P i) a b) mab mxy ),\n\n  exact minus_voter_margin_le2 i P a b,\nend", "meta": {"author": "chasenorman", "repo": "Formalized-Voting", "sha": "de04e630b83525b042db166670ba97f9952b5691", "save_path": "github-repos/lean/chasenorman-Formalized-Voting", "path": "github-repos/lean/chasenorman-Formalized-Voting/Formalized-Voting-de04e630b83525b042db166670ba97f9952b5691/src/involvement.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834734, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3633007930488833}}
{"text": "import ..bin_tree\nimport ..nonempty_list\n\nopen util.data.nonempty_list\nopen util.data.bin_tree'\nopen util.data.bin_tree'.bin_tree'\n\nuniverses u v\n\nvariable {\u03b1 : Type u}\n\n-- https://ncatlab.org/nlab/show/Mac+Lane%27s+proof+of+the+coherence+theorem+for+monoidal+categories\n\n-- TODO(tim): do we really need two definitions of the congruence closure or can we just delete this one?\ninductive cong_clos' (R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u) : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u\n| lift : \u03a0 s t, R s t \u2192 cong_clos' s t\n| refl : \u03a0 t, cong_clos' t t\n| trans : \u03a0 r s t, cong_clos' r s \u2192 cong_clos' s t \u2192 cong_clos' r t\n| cong : \u03a0 l\u2081 r\u2081 l\u2082 r\u2082, cong_clos' l\u2081 l\u2082 \u2192 cong_clos' r\u2081 r\u2082 \u2192 cong_clos' (branch l\u2081 r\u2081) (branch l\u2082 r\u2082)\n\nnamespace cong_clos'\n\nvariable {R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u}\n\ndef sym (R_sym : \u03a0 s t, R s t \u2192 R t s)\n    : \u03a0 {s t}, cong_clos' R s t \u2192 cong_clos' R t s\n| ._ ._ (lift _ _ p)       := lift _ _ (R_sym _ _ p)\n| ._ ._ (refl ._ t)        := refl R t\n| ._ ._ (trans _ _ _ p q)  := trans _ _ _ (sym q) (sym p)\n| ._ ._ (cong _ _ _ _ l r) := cong _ _ _ _ (sym l) (sym r)\n\ndef transport {S : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u} (f : \u03a0 s t, R s t \u2192 S s t)\n    : \u03a0 {s t}, cong_clos' R s t \u2192 cong_clos' S s t\n| ._ ._ (lift _ _ p)       := lift _ _ (f _ _ p)\n| ._ ._ (refl ._ t)        := refl S t\n| ._ ._ (trans _ _ _ p q)  := trans _ _ _ (transport p) (transport q)\n| ._ ._ (cong _ _ _ _ l r) := cong _ _ _ _ (transport l) (transport r)\n\nlemma respects_to_list (R_to_list : \u03a0 s t, R s t \u2192 s.to_list = t.to_list)\n    : \u03a0 {s t}, cong_clos' R s t \u2192 s.to_list = t.to_list\n| ._ ._ (lift _ _ p)           := R_to_list _ _ p\n| ._ ._ (refl ._ _)            := eq.refl _\n| ._ ._ (trans _ _ _ p q)      := eq.trans (respects_to_list p) (respects_to_list q)\n| ._ ._ (cong l\u2081 r\u2081 l\u2082 r\u2082 l r) :=\n    begin\n      unfold bin_tree'.to_list,\n      rewrite (respects_to_list l),\n      rewrite (respects_to_list r)\n    end\n\nend cong_clos'\n\ninductive cong_clos_step (R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u) : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u\n| lift  : \u03a0 s t, R s t \u2192 cong_clos_step s t\n| left  : \u03a0 l\u2081 l\u2082 r, cong_clos_step l\u2081 l\u2082 \u2192 cong_clos_step (branch l\u2081 r) (branch l\u2082 r)\n| right : \u03a0 l r\u2081 r\u2082, cong_clos_step r\u2081 r\u2082 \u2192 cong_clos_step (branch l r\u2081) (branch l r\u2082)\n\nnamespace cong_clos_step\n\nvariable {R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u}\n\n-- the equation compiler somehow can't handle the next definitions ~> turn it off\n-- TODO(tim): report bug\nset_option eqn_compiler.lemmas false\n\ndef sym (R_sym : \u03a0 s t, R s t \u2192 R t s)\n    : \u03a0 {s t}, cong_clos_step R s t \u2192 cong_clos_step R t s\n| ._ ._ (lift _ _ p)    := lift _ _ (R_sym _ _ p)\n| ._ ._ (left _ _ _ l)  := left _ _ _ (sym l)\n| ._ ._ (right _ _ _ r) := right _ _ _ (sym r)\n\ndef transport {S : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u} (f : \u03a0 s t, R s t \u2192 S s t)\n    : \u03a0 {s t}, cong_clos_step R s t \u2192 cong_clos_step S s t\n| ._ ._ (lift _ _ p)    := lift _ _ (f _ _ p)\n| ._ ._ (left _ _ _ l)  := left _ _ _ (transport l)\n| ._ ._ (right _ _ _ r) := right _ _ _ (transport r)\n\nlemma respects_to_list (R_to_list : \u03a0 s t, R s t \u2192 s.to_list = t.to_list)\n    : \u03a0 {s t}, cong_clos_step R s t \u2192 s.to_list = t.to_list\n| ._ ._ (lift _ _ p)    := R_to_list _ _ p\n| ._ ._ (left _ _ _ l)  := by unfold bin_tree'.to_list; rewrite (respects_to_list l)\n| ._ ._ (right _ _ _ r) := by unfold bin_tree'.to_list; rewrite (respects_to_list r)\n\nlemma respects_lopsided\n    (R_to_list : \u03a0 s t, R s t \u2192 s.to_list = t.to_list)\n    {s t} (p : cong_clos_step R s t)\n    : s.lopsided = t.lopsided\n  := by unfold lopsided; rewrite respects_to_list R_to_list p\n\nset_option eqn_compiler.lemmas true\n\nend cong_clos_step\n\n-- smallest reflexive, transitive, congruent (but not necessarily symmetric) relation that includes R\ninductive cong_clos (R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u) : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u\n| refl : \u03a0 t, cong_clos t t\n| step : \u03a0 r s t, cong_clos_step R r s \u2192 cong_clos s t \u2192 cong_clos r t\n\nnamespace cong_clos\n\nvariable {R : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u}\n\nopen cong_clos_step\n\ndef lift {s t : bin_tree' \u03b1} (p : R s t) : cong_clos R s t :=\n  step _ _ _ (cong_clos_step.lift _ _ p) (refl R _)\n\ndef trans : \u03a0 {r s t : bin_tree' \u03b1}, cong_clos R r s \u2192 cong_clos R s t \u2192 cong_clos R r t\n| ._ ._ _ (refl ._ t)       qs := qs\n| ._ ._ _ (step _ _ _ p ps) qs := step _ _ _ p (trans ps qs)\n\nlemma trans_refl_right : \u03a0 {s t : bin_tree' \u03b1} (p : cong_clos R s t), trans p (refl R t) = p\n| ._ ._ (refl ._ t)       := by reflexivity\n| ._ ._ (step _ _ _ p ps) := by unfold trans; rewrite (trans_refl_right ps)\n\ndef inject_left (r : bin_tree' \u03b1) : \u03a0 {l\u2081 l\u2082 : bin_tree' \u03b1}, cong_clos R l\u2081 l\u2082 \u2192 cong_clos R (branch l\u2081 r) (branch l\u2082 r)\n| ._ ._ (refl ._ t)       := refl R _\n| ._ ._ (step _ _ _ p ps) := step _ _ _ (left _ _ _ p) (inject_left ps)\n\ndef inject_right (l : bin_tree' \u03b1) : \u03a0 {r\u2081 r\u2082 : bin_tree' \u03b1}, cong_clos R r\u2081 r\u2082 \u2192 cong_clos R (branch l r\u2081) (branch l r\u2082)\n| ._ ._ (refl ._ _)       := refl R _\n| ._ ._ (step _ _ _ p ps) := step _ _ _ (right _ _ _ p) (inject_right ps)\n\ndef cong {l\u2081 l\u2082 r\u2081 r\u2082 : bin_tree' \u03b1} (l : cong_clos R l\u2081 l\u2082) (r : cong_clos R r\u2081 r\u2082) : cong_clos R (branch l\u2081 r\u2081) (branch l\u2082 r\u2082) :=\n  trans (inject_left _ l) (inject_right _ r)\n\ndef convert : \u03a0 (s t : bin_tree' \u03b1), cong_clos' R s t \u2192 cong_clos R s t\n| ._ ._ (cong_clos'.refl ._ _)        := refl R _\n| ._ ._ (cong_clos'.lift _ _ p)       := lift p\n| ._ ._ (cong_clos'.trans _ _ _ p q)  := trans (convert _ _ p) (convert _ _ q)\n| ._ ._ (cong_clos'.cong _ _ _ _ l r) := cong (convert _ _ l) (convert _ _ r)\n\ndef sym_helper (R_sym : \u03a0 s t, R s t \u2192 R t s)\n    : \u03a0 {s t u}, cong_clos R s t \u2192 cong_clos R s u \u2192 cong_clos R t u\n| ._ ._ _ (refl ._ t)       qs := qs\n| ._ ._ _ (step _ _ _ p ps) qs := sym_helper ps (step _ _ _ (cong_clos_step.sym R_sym p) qs)\n\ndef sym (R_sym : \u03a0 s t, R s t \u2192 R t s) {s t} (ps : cong_clos R s t) : cong_clos R t s :=\n  sym_helper R_sym ps (refl R _)\n\ndef transport {S : bin_tree' \u03b1 \u2192 bin_tree' \u03b1 \u2192 Type u} (f : \u03a0 s t, R s t \u2192 S s t)\n    : \u03a0 {s t}, cong_clos R s t \u2192 cong_clos S s t\n| ._ ._ (refl ._ t)       := refl S t\n| ._ ._ (step _ _ _ p ps) := step _ _ _ (cong_clos_step.transport f p) (transport ps)\n\nlemma respects_to_list (R_to_list : \u03a0 s t, R s t \u2192 s.to_list = t.to_list)\n    : \u03a0 {s t}, cong_clos R s t \u2192 s.to_list = t.to_list\n| ._ ._ (refl ._ t)       := by reflexivity\n| ._ ._ (step r s t p ps) :=\n  begin\n    transitivity,\n      exact (cong_clos_step.respects_to_list R_to_list p),\n      exact (respects_to_list ps)\n  end\n\nlemma respects_lopsided\n    (R_to_list : \u03a0 s t, R s t \u2192 s.to_list = t.to_list)\n    {s t} (p : cong_clos R s t)\n    : s.lopsided = t.lopsided\n  := by unfold lopsided; rewrite respects_to_list R_to_list p\n\nend cong_clos", "meta": {"author": "semorrison", "repo": "lean-monoidal-categories", "sha": "81f43e1e0d623a96695aa8938951d7422d6d7ba6", "save_path": "github-repos/lean/semorrison-lean-monoidal-categories", "path": "github-repos/lean/semorrison-lean-monoidal-categories/lean-monoidal-categories-81f43e1e0d623a96695aa8938951d7422d6d7ba6/src/monoidal_categories/util/data/bin_tree/cong_clos.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.36330079304888324}}
{"text": "import tactic\nimport compile_fast\n\nclass NatLt (m n : \u2115) := (proof : m < n)\ninstance NatLt.one (n : \u2115) : NatLt 0 (n+1) := \u27e8nat.succ_pos _\u27e9\ninstance NatLt.trans (m n : \u2115) [h : NatLt m n] : NatLt (m+1) (n+1) :=\n\u27e8nat.succ_lt_succ h.proof\u27e9\n\n-- instance i1 : NatLt 3 2 := infer_instance -- no\ninstance i2 : NatLt 1 3 := infer_instance\n\nuniverses u v\n\nclass Atomic (\u03b1 : Type u) .\n\nclass Rectangle (Gen : \u2115 \u2192 Type* \u2192 Type*) :=\n  (map {\u03b1 \u03b2 : Type*} (i : \u2115) : (\u03b1 \u2192 \u03b2) \u2192 Gen i \u03b1 \u2192 Gen i \u03b2)\n  (repl {\u03b1 : Type*}  (i : \u2115) : \u03b1 \u2192 Gen i \u03b1)\n\nopen Rectangle\n\nclass Merge (\u03b1 \u03b2 : Type*) (\u03b3 : out_param Type*) :=\n  (merge1 : \u03b1 \u2192 \u03b3)\n  (merge2 : \u03b2 \u2192 \u03b3)\n\nopen Merge\n\nclass NestedMap (\u03b1 \u03b2 \u03b3 \u03b4 : Type*) :=\n  (map : (\u03b2 \u2192 \u03b3) \u2192 \u03b1 \u2192 \u03b4)\n\nsection Instances\nvariables\n{Gen : \u2115 \u2192 Type u \u2192 Type v}\n{View : \u2115 \u2192 Type u \u2192 Type v}\n[Rectangle Gen]\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\n\ninstance Gen.Merge.one {\u03c1} [Atomic \u03c1] : Merge \u03c1 \u03c1 \u03c1 := \u27e8id, id\u27e9\ninstance Gen.Merge.succ {i : \u2115} [Merge \u03b1 \u03b2 \u03b3] : Merge (Gen i \u03b1) (Gen i \u03b2) (Gen i \u03b3) :=\n\u27e8map i (merge1 \u03b2), map i (merge2 \u03b1)\u27e9\ninstance Gen.Merge.scalar_r {i : \u2115} {\u03c1} [Atomic \u03c1] [Merge \u03b1 \u03c1 \u03b1] : Merge (Gen i \u03b1) \u03c1 (Gen i \u03b1) :=\n\u27e8id, repl i \u2218 merge2 \u03b1\u27e9\ninstance Gen.Merge.lt {i j : \u2115} [NatLt i j] [Merge \u03b1 (Gen j \u03b2) \u03b3]\n: Merge (Gen i \u03b1) (Gen j \u03b2) (Gen i \u03b3) :=\n\u27e8map i (merge1 (Gen j \u03b2)), repl i \u2218 merge2 \u03b1\u27e9\ninstance Gen.Merge.scalar_l {j : \u2115} {\u03c1} [Atomic \u03c1] [Merge \u03c1 \u03b2 \u03b2] : Merge \u03c1 (Gen j \u03b2) (Gen j \u03b2) :=\n\u27e8repl j \u2218 merge1 \u03b2, id\u27e9\ninstance Gen.merge.gt {i j : \u2115} [NatLt j i] [Merge (Gen i \u03b1) \u03b2 \u03b3] : Merge (Gen i \u03b1) (Gen j \u03b2) (Gen j \u03b3) :=\n\u27e8repl j \u2218 merge1 \u03b2, map j (merge2 (Gen i \u03b1))\u27e9\n\ndef merge {\u03b1 \u03b2 \u03b3} [Merge \u03b1 \u03b2 \u03b3] : \u03b1 \u2192 \u03b2 \u2192 (\u03b3 \u00d7 \u03b3) := \u03bb a b, (merge1 \u03b2 a, merge2 \u03b1 b)\n\n\ninstance Gen.NestedMap.Eq {i : \u2115} {\u03b1 \u03b2} : NestedMap (Gen i \u03b1) \u03b1 \u03b2 (Gen i \u03b2) := \u27e8Rectangle.map i\u27e9\ninstance Gen.NestedMap.Lt {i j : \u2115} {\u03b1 \u03b2 \u03b3 \u03b4} [NatLt i j] [NestedMap \u03b1 \u03b2 \u03b3 \u03b4] : NestedMap (Gen i \u03b1) \u03b2 \u03b3 (Gen i \u03b4) :=\n\u27e8\u03bb f, Rectangle.map i (NestedMap.map f)\u27e9\n\nend Instances\n\ndef Fun (i : \u2115) (\u03b1 : Type*) := \u2115 \u2192 \u03b1\ndef Fun.to_fun {i \u03b1} : Fun i \u03b1 \u2192 \u2115 \u2192 \u03b1 := id\ndef map {\u03b1 \u03b2 : Type} (m : \u2115) : (\u03b1 \u2192 \u03b2) \u2192 Fun m \u03b1 \u2192 Fun m \u03b2 := \u03bb f v, f \u2218 v\ndef repl {\u03b1 : Type} (m : \u2115) : \u03b1 \u2192 Fun m \u03b1 := \u03bb v _, v\n\ninstance : Rectangle Fun :=\n  { map := \u03bb _ _, map, repl := \u03bb _, repl }\n\ninstance nat.Atomic : Atomic \u2115 := \u27e8\u27e9\n\nexample : Merge \u2115 \u2115 \u2115 := infer_instance\nexample {i : \u2115} : Merge (Fun i \u2115) \u2115 (Fun i \u2115) := infer_instance\nexample {i : \u2115} : Merge (Fun i \u2115) (Fun i \u2115) (Fun i \u2115) := infer_instance\nexample {i : \u2115} : Merge (Fun i \u2115) \u2115 (Fun i \u2115) := infer_instance\nexample {i j : \u2115} [NatLt i j] : Merge \u2115 (Fun j \u2115) (Fun j \u2115) := infer_instance\nexample {i j : \u2115} [NatLt i j] : Merge (Fun i \u2115) (Fun j \u2115) (Fun i (Fun j \u2115)) := infer_instance\nexample {i j k : \u2115} [NatLt i j] : Merge (Fun i (Fun j \u2115)) (Fun j (Fun k \u2115)) (Fun i (Fun j (Fun k \u2115))) := infer_instance\n\n@[reducible] def i := 1\n@[reducible] def j := 2\n@[reducible] def k := 3\n@[reducible] def l := 4\n@[reducible] def V := \u2115\n\ninstance hmul_of_Merge {\u03b1 \u03b2 \u03b3 : Type}  [has_mul \u03b3] [Merge \u03b1 \u03b2 \u03b3] : has_hmul \u03b1 \u03b2 \u03b3 :=\n\u27e8\u03bb a b, merge1 \u03b2 a * merge2 \u03b1 b\u27e9\n\n-- demo with functions:\ninstance Fun.mul {i : \u2115} {\u03b1 : Type} [has_mul \u03b1] : has_mul (Fun i \u03b1) :=\n\u27e8\u03bb a b i, a.to_fun i * b.to_fun i\u27e9\nexample {i : \u2115} : has_hmul (Fun i \u2115) \u2115 (Fun i \u2115) := infer_instance\n\ndef v1 : Fun i (Fun j V) := \u03bb i j, i+j\ndef v2 : Fun j (Fun k V) := \u03bb i j, i*j\ndef v3 : (Fun l V) := \u03bb i, i\n\n--set_option trace.class_instances true\n--set_option class.instance_max_depth 20\n--set_option pp.all true\n\nsection Streams\n\ndef Ind (i : \u2115) := E\ninductive Stream (n : \u2115) (\u03b1 : Type)\n| view (v : View (Ind n) \u03b1) : Stream\n| gen  (g : G (Ind n) \u03b1)    : Stream\n\ninstance {n} : functor (Stream n) :=\n{ map := \u03bb _ _ f g, match g with\n  | Stream.view v := Stream.view { v with value := f \u2218 v.value }\n  | Stream.gen  g := Stream.gen  { g with value := f g.value }\n  end }\n\ninstance : Rectangle Stream :=\n{ map  := \u03bb _ _ _, functor.map,\n  repl := \u03bb _ m v, Stream.view \u27e8\u03bb _, v\u27e9 }\ninstance : Atomic E := \u27e8\u27e9\n\ndef foo1 : Merge (Stream i E) (Stream j E) (Stream i (Stream j E)) := infer_instance\n\nvariables {\u03b9 \u03b1 \u03b2 \u03b3 : Type}\n(n : \u2115)\n\ninstance G.Ind.hmul {i : \u2115} [has_hmul \u03b1 \u03b2 \u03b3] : has_hmul (G (Ind i) \u03b1) (G (Ind i) \u03b2) (G (Ind i) \u03b3) := \u27e8G.mul\u27e9\n\ninstance [inhabited \u03b1] : inhabited (Stream n \u03b1) := \u27e8Stream.view \u27e8\u03bb _, default\u27e9\u27e9\ninstance [inhabited \u03b9] [inhabited \u03b1] : inhabited (G \u03b9 \u03b1) := \u27e8G.empty\u27e9\n\n-- instance Stream.has_mul {\u03b1} {i} [has_mul \u03b1] : has_mul (StreamGen' i \u03b1) := \u27e8StreamGen.mul\u27e9\ninstance Stream.has_mul {\u03b3} {i} [inhabited \u03b3] [has_mul \u03b3] : has_mul (Stream i \u03b3) := \u27e8\u03bb a b,\nmatch a, b with\n| Stream.view a, Stream.view b := arbitrary _ -- Stream.view $ a\u22c6b\n| Stream.gen a, Stream.view b := Stream.gen $ a\u22c6b\n| Stream.view a, Stream.gen b := Stream.gen $ a\u22c6b\n| Stream.gen a, Stream.gen b := Stream.gen $ a\u22c6b\nend\u27e9\n\nvariables (a : Stream i E) (b : Stream j E)\n\nexample : has_mul (Stream i (Stream j E)) := infer_instance\nexample : Stream i (Stream j E) := a \u22c6 b\n\ninstance : has_coe (G E E) (Stream n E) := \u27e8Stream.gen\u27e9\ninstance coe_stream [has_coe \u03b1 \u03b2] : has_coe (G E \u03b1) (Stream n \u03b2) := \u27e8Stream.gen \u2218 functor.map has_coe.coe\u27e9\n\nclass of_stream (\u03b1 \u03b2 : Type) := (coe : \u03b1 \u2192 \u03b2)\ninstance base.of_stream : of_stream \u03b1 \u03b1 := \u27e8id\u27e9\ndef Stream.to_g {n} [inhabited \u03b1] : (Stream n \u03b1) \u2192 (G E \u03b1) := \u03bb s, match s with\n| Stream.view _ := arbitrary _\n| Stream.gen a := a\nend\n\ninstance [inhabited \u03b2] [of_stream \u03b1 \u03b2] : of_stream (Stream n \u03b1) (G E \u03b2) := \u27e8\u03bb s, match s with\n| Stream.view _ := arbitrary _\n| Stream.gen a := of_stream.coe <$> a\nend\u27e9\n\ndef Stream.of [of_stream \u03b1 \u03b2] : \u03b1 \u2192 \u03b2 := of_stream.coe\n\ninfixr ` \u2192\u209b `:24 := Stream\n\n--def Stream.to_stream {n} [of_stream \u03b1 \u03b2] : Stream n \u03b1 \u2192 G E \u03b2 := of_stream.coe\n\n-- instance s_level.eval [of_stream \u03b3 \u03b2] [Ev \u03b1 (G E \u03b2)] : Ev \u03b1 (Stream i \u03b3) :=\n-- \u27e8 \u03bb l r, exec l (of_stream.coe r : G E \u03b2) \u27e9\n-- instance stream.level.eval' (n : \u2115) [Ev \u03b1 (G E \u03b2)] : Ev \u03b1 (Stream n \u03b2) :=\n-- \u27e8 \u03bb l r, exec l $ r.to_g _ \u27e9\n\n\nclass Sum (n : \u2115) (\u03b1 : Type) (\u03b2 : out_param Type) := (sum : \u03b1 \u2192 \u03b2)\ninstance sum_eq (n : \u2115) [inhabited \u03b1] : Sum n (Stream n \u03b1) (G unit \u03b1) := \u27e8G.contract \u2218 Stream.to_g\u27e9\ninstance sum_lt (m n : \u2115) [NatLt n m] [Sum m \u03b1 \u03b2] : Sum m (Stream n \u03b1) (Stream n \u03b2) := \u27e8functor.map $ Sum.sum m\u27e9\n\nabbreviation R := E\n\nprefix ` \u03a3 ` := Sum.sum\n\ndef mmul1'  := \u03a3 i $ \u03a3 j $ \u03a3 k $ (A : i \u2192\u209b j \u2192\u209b E) \u22c6 (B : j \u2192\u209b k \u2192\u209b E)\ndef mmul2'  := \u03a3 i $ \u03a3 j $ \u03a3 k $ (A : i \u2192\u209b k \u2192\u209b E) \u22c6 (B : j \u2192\u209b k \u2192\u209b E)\ndef ttv'    := \u03a3 i $ \u03a3 j $ \u03a3 k $ (C : i \u2192\u209b j \u2192\u209b k \u2192\u209b R) \u22c6 (v : k \u2192\u209b R)\ndef ttm'    := \u03a3 i $ \u03a3 j $ \u03a3 k $ \u03a3 l $ (C : i \u2192\u209b j \u2192\u209b l \u2192\u209b R) \u22c6 (A : k \u2192\u209b l \u2192\u209b R)\ndef mttkrp' := \u03a3 i $ \u03a3 j $ \u03a3 k $ \u03a3 l $ (C : i \u2192\u209b j \u2192\u209b k \u2192\u209b R) \u22c6 (A : j \u2192\u209b l \u2192\u209b R) \u22c6 (B : k \u2192\u209b l \u2192\u209b R)\ndef inner3' := \u03a3 i $ \u03a3 j $ \u03a3 k $ (C : i \u2192\u209b j \u2192\u209b k \u2192\u209b R) \u22c6 (D : i \u2192\u209b j \u2192\u209b k \u2192\u209b R)\n\nexample : Sum i (Stream i E) (G unit E) := infer_instance\nexample : Sum j (Stream i (Stream j E)) (Stream i (G unit E)) := infer_instance\n\ndef inner : Stream i (Stream j (G unit E)) :=\n  Sum.sum k $ (A : i \u2192\u209b k \u2192\u209b E) \u22c6 (B : j \u2192\u209b k \u2192\u209b E)\n\ndef eg_mmul1 :=\n  [me $ exec out mmul1'] ++\n  [ta $ Prog.inline_code \"taco_ijk_sum();\"]\n\ndef eg_mmul2 :=\n  [me $ exec out mmul2'] ++\n  [ta $ Prog.inline_code \"mmul2_compute();\"]\n\ndef eg_ttv :=\n  [me $ exec out $ G.contract $ View.to_gen \"foo\" 30 $ constView E $ ttv'] ++\n  [ta $ exec out $ G.contract $ View.to_gen \"foo\" 30 $ constView E $ E.inline_code \"ttv_compute();\"]\n\ndef eg_ttm :=\n  [me $ exec out ttm'] ++\n  [ta $ Prog.inline_code \"ttm_compute();\"]\n\ndef eg_mttkrp :=\n  [me $ exec out mttkrp'] ++\n  [ta $ Prog.inline_code \"mttkrp_compute();\"]\n\ndef eg_inner3 :=\n  [me $ exec out inner3'] ++\n  [ta $ Prog.inline_code \"inner3_compute();\"]\n\ndef eg_inner3' :=\n  [me $ exec out $ G.contract $ View.to_gen \"foo\" 200 $ constView E $ inner3'] ++\n  [ta $ exec out $ G.contract $ View.to_gen \"foo\" 200 $ constView E $ E.inline_code \"inner3_compute();\"]\n\ndef compile_with_load (v : list Prog) := compile $ load ++ v\n\ndef tests :=\n  /- 1 -/ eg_mmul1 ++\n  /- 2 -/ eg_mmul2 ++\n  /- 3 -/ eg_ttv ++\n  /- 4 -/ eg_ttm ++\n  /- 5 -/ eg_mttkrp ++\n  /- 6 -/ eg_inner3'\n\ndef run_comparisons := compile_with_load tests\n\n-- main comparison script:\n-- #eval run_comparisons\n\nend Streams\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/front_end.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36311053419428585}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.sites.compatible_plus\nimport category_theory.sites.sheafification\n\n/-!\n\nIn this file, we prove that sheafification is compatible with functors which\npreserve the correct limits and colimits.\n\n-/\n\nnamespace category_theory.grothendieck_topology\n\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverses w\u2081 w\u2082 v u\nvariables {C : Type u} [category.{v} C] (J : grothendieck_topology C)\nvariables {D : Type w\u2081} [category.{max v u} D]\nvariables {E : Type w\u2082} [category.{max v u} E]\nvariables (F : D \u2964 E)\n\nnoncomputable theory\n\nvariables [\u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1),\n  has_limits_of_shape (walking_multicospan fst snd) D]\nvariables [\u2200 (\u03b1 \u03b2 : Type (max v u)) (fst snd : \u03b2 \u2192 \u03b1),\n  has_limits_of_shape (walking_multicospan fst snd) E]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 E]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\nvariables [\u2200 (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D), preserves_limit (W.index P).multicospan F]\n\nvariables (P : C\u1d52\u1d56 \u2964 D)\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`.\n\nUse the lemmas `whisker_right_to_sheafify_sheafify_comp_iso_hom`,\n`to_sheafify_comp_sheafify_comp_iso_inv` and `sheafify_comp_iso_inv_eq_sheafify_lift` to reduce\nthe components of this isomorphisms to a state that can be handled using the universal property\nof sheafification. -/\ndef sheafify_comp_iso : J.sheafify P \u22d9 F \u2245 J.sheafify (P \u22d9 F) :=\nJ.plus_comp_iso _ _ \u226a\u226b (J.plus_functor _).map_iso (J.plus_comp_iso _ _)\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`, functorially in `F`. -/\ndef sheafification_whisker_left_iso (P : C\u1d52\u1d56 \u2964 D)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (whiskering_left _ _ E).obj (J.sheafify P) \u2245\n  (whiskering_left _ _ _).obj P \u22d9 J.sheafification E :=\nbegin\n  refine J.plus_functor_whisker_left_iso _ \u226a\u226b _ \u226a\u226b functor.associator _ _ _,\n  refine iso_whisker_right _ _,\n  refine J.plus_functor_whisker_left_iso _,\nend\n\n@[simp]\nlemma sheafification_whisker_left_iso_hom_app (P : C\u1d52\u1d56 \u2964 D) (F : D \u2964 E)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (sheafification_whisker_left_iso J P).hom.app F = (J.sheafify_comp_iso F P).hom :=\nbegin\n  dsimp [sheafification_whisker_left_iso, sheafify_comp_iso],\n  rw category.comp_id,\nend\n\n@[simp]\nlemma sheafification_whisker_left_iso_inv_app (P : C\u1d52\u1d56 \u2964 D) (F : D \u2964 E)\n  [\u2200 (F : D \u2964 E) (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 F]\n  [\u2200 (F : D \u2964 E) (X : C) (W : J.cover X) (P : C\u1d52\u1d56 \u2964 D),\n    preserves_limit (W.index P).multicospan F] :\n  (sheafification_whisker_left_iso J P).inv.app F = (J.sheafify_comp_iso F P).inv :=\nbegin\n  dsimp [sheafification_whisker_left_iso, sheafify_comp_iso],\n  erw category.id_comp,\nend\n\n/-- The isomorphism between the sheafification of `P` composed with `F` and\nthe sheafification of `P \u22d9 F`, functorially in `P`. -/\ndef sheafification_whisker_right_iso :\n  J.sheafification D \u22d9 (whiskering_right _ _ _).obj F \u2245\n  (whiskering_right _ _ _).obj F \u22d9 J.sheafification E :=\nbegin\n  refine functor.associator _ _ _ \u226a\u226b _,\n  refine iso_whisker_left (J.plus_functor D) (J.plus_functor_whisker_right_iso _) \u226a\u226b _,\n  refine _ \u226a\u226b functor.associator _ _ _,\n  refine (functor.associator _ _ _).symm \u226a\u226b _,\n  exact iso_whisker_right (J.plus_functor_whisker_right_iso _) (J.plus_functor E),\nend\n\n@[simp]\nlemma sheafification_whisker_right_iso_hom_app :\n  (J.sheafification_whisker_right_iso F).hom.app P = (J.sheafify_comp_iso F P).hom :=\nbegin\n  dsimp [sheafification_whisker_right_iso, sheafify_comp_iso],\n  simp only [category.id_comp, category.comp_id],\n  erw category.id_comp,\nend\n\n@[simp]\nlemma sheafification_whisker_right_iso_inv_app :\n  (J.sheafification_whisker_right_iso F).inv.app P = (J.sheafify_comp_iso F P).inv :=\nbegin\n  dsimp [sheafification_whisker_right_iso, sheafify_comp_iso],\n  simp only [category.id_comp, category.comp_id],\n  erw category.id_comp,\nend\n\n@[simp, reassoc]\nlemma whisker_right_to_sheafify_sheafify_comp_iso_hom :\n  whisker_right (J.to_sheafify _) _ \u226b (J.sheafify_comp_iso F P).hom = J.to_sheafify _ :=\nbegin\n  dsimp [sheafify_comp_iso],\n  erw [whisker_right_comp, category.assoc],\n  slice_lhs 2 3 { rw plus_comp_iso_whisker_right },\n  rw [category.assoc, \u2190 J.plus_map_comp,\n    whisker_right_to_plus_comp_plus_comp_iso_hom, \u2190 category.assoc,\n    whisker_right_to_plus_comp_plus_comp_iso_hom],\n  refl,\nend\n\n@[simp, reassoc]\nlemma to_sheafify_comp_sheafify_comp_iso_inv :\n  J.to_sheafify _ \u226b (J.sheafify_comp_iso F P).inv = whisker_right (J.to_sheafify _) _ :=\nby { rw iso.comp_inv_eq, simp }\n\nsection\n\n-- We will sheafify `D`-valued presheaves in this section.\nvariables\n  [concrete_category.{max v u} D]\n  [preserves_limits (forget D)]\n  [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\n  [reflects_isomorphisms (forget D)]\n\n@[simp]\nlemma sheafify_comp_iso_inv_eq_sheafify_lift : (J.sheafify_comp_iso F P).inv =\n  J.sheafify_lift (whisker_right (J.to_sheafify _) _) ((J.sheafify_is_sheaf _).comp _) :=\nbegin\n  apply J.sheafify_lift_unique,\n  rw iso.comp_inv_eq,\n  simp,\nend\n\nend\n\nend category_theory.grothendieck_topology\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/compatible_sheafification.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36311053419428585}}
{"text": "import free_pfpng.setup\n\n.\n\nnoncomputable theory\n\nopen_locale classical\n\nopen category_theory\nopen opposite\n\nuniverse u\n\nlemma Profinite.mono_free'_to_condensed_free_pfpng_aux\n  (S B : Profinite.{u}) (b : B) (T : discrete_quotient S)\n  (t : S.to_Condensed.val.obj (op B) \u2192\u2080 \u2124) :\nlet e : S.to_Condensed.val.obj (op B) \u2192\n    S.condensed_free_pfpng.val.obj (op B) :=\n    \u03bb f, (S.to_condensed_free_pfpng.val.app (op B) f),\n    \u03b9 : S.to_Condensed.val.obj (op B) \u2192 S :=\n      \u03bb f, (ulift.down f).1 b in\n    ((limits.limit.\u03c0 (S.fintype_diagram \u22d9 forget Fintype \u22d9\n      AddCommGroup.free' \u22d9 Ab.ulift) T)\n      (S.condensed_free_pfpng_specialize B b (free'_lift e t))).down\n  = t.map_domain (T.proj \u2218 \u03b9) :=\nbegin\n  dsimp,\n  revert t,\n  rw \u2190 function.funext_iff,\n  dsimp,\n  change ulift.down \u2218 _ = _,\n  apply finsupp.fun_ext,\n  { intros, simp only [function.comp_app, map_add, ulift.add_down,\n      eq_self_iff_true, forall_const] },\n  { intros, simp only [function.comp_app, map_zero, ulift.zero_down,\n      finsupp.map_domain_add] },\n  { intros,\n    simp only [function.comp_app, finsupp.map_domain_single,\n      free'_lift_eq_finsupp_lift],\n    dsimp [Profinite.condensed_free_pfpng_specialize],\n    simp only [\u2190 comp_apply],\n    erw limits.limit.lift_\u03c0,\n    simp only [finsupp.sum_single_index, zero_smul, one_zsmul],\n    ext i,\n    dsimp [Profinite.condensed_free_pfpng_specialize_cone,\n      finsupp.single, Profinite.free_pfpng_\u03c0, Profinite.to_free_pfpng],\n    erw ProFiltPseuNormGrp\u2081.limit_\u03c0_coe_eq,\n    simp only [\u2190 comp_apply, category.assoc],\n    dsimp [Profinite.free_pfpng_level_iso,\n      limits.is_limit.cone_point_unique_up_to_iso],\n    simp only [\u2190 comp_apply, category.assoc],\n    erw (limits.is_limit_of_preserves (ProFiltPseuNormGrp\u2081.level.obj 1)\n      (limits.limit.is_limit (S.fintype_diagram \u22d9 free_pfpng_functor))).fac,\n    erw limits.limit.lift_\u03c0,\n    refl },\nend\n\nlemma Profinite.specialization_eq_zero_of_eq_zero (S B : Profinite.{u}) (b : B)\n  (t : S.to_Condensed.val.obj (op B) \u2192\u2080 \u2124)\n  (ht : free'_lift (S.to_condensed_free_pfpng.val.app (op B)) t = 0) :\n  t.map_domain (\u03bb f, (ulift.down f).1 b) = 0 :=\nbegin\n  apply free_pfpng.discrete_quotient_separates_points' S,\n  intros T,\n  apply_fun (\u03bb e, S.condensed_free_pfpng_specialize B b e) at ht,\n  rw add_monoid_hom.map_zero at ht,\n  apply_fun (\u03bb e, limits.limit.\u03c0 (S.fintype_diagram \u22d9 forget Fintype \u22d9\n    AddCommGroup.free' \u22d9 Ab.ulift) T e) at ht,\n  rw add_monoid_hom.map_zero at ht,\n  apply_fun ulift.down at ht,\n  dsimp [AddCommGroup.free'],\n  rw \u2190 finsupp.map_domain_comp,\n  have := S.mono_free'_to_condensed_free_pfpng_aux B b T t,\n  dsimp at this, erw \u2190 this, exact ht\nend\n\nlemma Profinite.adj'_hom_equiv_symm_eq_free'_lift (S B : Profinite.{u}) :\n    (((AddCommGroup.adj'.whisker_right Profinite.{u}\u1d52\u1d56).hom_equiv\n      S.to_Condensed.val S.condensed_free_pfpng.val).symm\n      S.to_condensed_free_pfpng.val).app (op B) =\n    free'_lift (S.to_condensed_free_pfpng.val.app (op B)) :=\nbegin\n  ext u v, dsimp [free'_lift],\n  simp only [adjunction.hom_equiv_counit, whiskering_right_obj_map,\n    nat_trans.comp_app, whisker_right_app,\n    adjunction.whisker_right_counit_app_app],\nend\n\nopen_locale big_operators\nlemma finsupp.map_domain_ne_zero_of_ne_zero_of_inj_on {\u03b1 \u03b2 \u03b3 : Type*} [add_comm_group \u03b2]\n  (t : \u03b1 \u2192\u2080 \u03b2) (ht : t \u2260 0) (f : \u03b1 \u2192 \u03b3)\n  (hinj : set.inj_on f t.support) :\n  t.map_domain f \u2260 0 :=\nbegin\n  contrapose! ht,\n  have : \u2200 (e : \u03b3) (he : e \u2208 (t.map_domain f).support), \u2203 (q : \u03b1) (hq : q \u2208 t.support), f q = e,\n  { intros e he, by_contra c, push_neg at c,\n    simp only [finsupp.mem_support_iff, ne.def] at he,\n    apply he,\n    erw finset.sum_apply',\n    apply finset.sum_eq_zero,\n    intros a ha,\n    dsimp [finsupp.single], rw if_neg, apply c, exact ha },\n  choose q hq hh using this,\n  let \u03b9 : (t.map_domain f).support \u2192 t.support :=\n    \u03bb e, \u27e8q e.1 e.2, hq e.1 e.2\u27e9,\n  have h\u03b9 : function.surjective \u03b9,\n  { rintros \u27e8e,he\u27e9, use f e,\n    { simp only [finsupp.mem_support_iff, ne.def],\n      rw finsupp.map_domain_apply' _ _ (set.subset.refl _) hinj he,\n      simpa using he },\n    { ext, dsimp,\n      apply hinj, apply hq, apply he, apply hh } },\n  have : (t.map_domain f).support = \u2205, by simpa using ht,\n  suffices : t.support = \u2205, by simpa using this,\n  by_contra c, change _ \u2260 _ at c,\n  erw \u2190 finset.nonempty_iff_ne_empty at c,\n  obtain \u27e8c,hc\u27e9 := c, obtain \u27e8\u27e8c,hc\u27e9,ee\u27e9 := h\u03b9 \u27e8c,hc\u27e9,\n  rw this at hc, simpa using hc,\nend\n\nlemma finsupp.lift_map_domain {\u03b3 \u03b1 \u03b2 : Type*} [add_comm_group \u03b2]\n  (f : \u03b1 \u2192 \u03b2) (\u03b9 : \u03b3 \u2192 \u03b1) :\n  (finsupp.lift _ \u2124 _ f) \u2218 finsupp.map_domain \u03b9 = finsupp.lift _ \u2124 _ (f \u2218 \u03b9) :=\nbegin\n  apply finsupp.fun_ext,\n  { intros x y,\n    dsimp only [function.comp_apply],\n    simp only [finsupp.map_domain_add],\n    erw ((finsupp.lift \u03b2 \u2124 \u03b1) f).to_add_monoid_hom.map_add, refl },\n  { intros x y,\n    erw ((finsupp.lift \u03b2 \u2124 \u03b3) (f \u2218 \u03b9)).to_add_monoid_hom.map_add, refl },\n  { intros x, simp },\nend\n\nlemma finsupp.lift_map_domain_apply {\u03b3 \u03b1 \u03b2 : Type*} [add_comm_group \u03b2]\n  (f : \u03b1 \u2192 \u03b2) (\u03b9 : \u03b3 \u2192 \u03b1) (e : \u03b3 \u2192\u2080 \u2124) :\n  (finsupp.lift _ \u2124 _ f).to_add_monoid_hom (e.map_domain \u03b9) =\n  finsupp.lift _ \u2124 _ (f \u2218 \u03b9) e :=\nbegin\n  rw \u2190 finsupp.lift_map_domain, refl,\nend\n\nlemma finsupp.card_supp_map_domain_lt {\u03b1 \u03b2 \u03b3 : Type*} [add_comm_group \u03b3]\n  (f : \u03b1 \u2192 \u03b2) (t : \u03b1 \u2192\u2080 \u03b3) (u v : \u03b1)\n  (huv : u \u2260 v) (hu : u \u2208 t.support) (hv : v \u2208 t.support)\n  (hf : f u = f v) : (t.map_domain f).support.card < t.support.card :=\nbegin\n  classical,\n  have key : (finsupp.map_domain f t).support \u2286 _ := finsupp.map_domain_support,\n  have : (finsupp.map_domain f t).support.card \u2264 (t.support.image f).card :=\n    finset.card_le_of_subset key,\n  refine lt_of_le_of_lt this _,\n  have key' : (t.support.image f).card \u2264 t.support.card := finset.card_image_le,\n  apply lt_of_le_of_ne key',\n  change \u00ac _,\n  rw finset.card_image_iff,\n  dsimp [set.inj_on],\n  push_neg, use [u, hu, v, hv, hf],\nend\n\nlemma Profinite.mono_free'_to_condensed_free_pfpng_induction_aux (n : \u2115) :\n  \u2200 (S B : Profinite.{u}) (t : S.to_Condensed.val.obj (op B) \u2192\u2080 \u2124),\n    t.support.card \u2264 n \u2192\n    (free'_lift (S.to_condensed_free_pfpng.val.app (op B))) t = 0 \u2192\n  (\u2200 (b : \u21a5B), finsupp.map_domain (\u03bb f : S.to_Condensed.val.obj (op B),\n    (ulift.down f).1 b) t = 0) \u2192\n  (\u2203 (\u03b1 : Type u) [_inst_1 : fintype \u03b1] (X : \u03b1 \u2192 Profinite) (\u03c0 : \u03a0 (a : \u03b1), X a \u27f6 B)\n    (surj : \u2200 (b : \u21a5B), \u2203 (a : \u03b1) (x : \u21a5(X a)), (\u03c0 a) x = b),\n    \u2200 (a : \u03b1), finsupp.map_domain (S.to_Condensed.val.map (\u03c0 a).op) t = 0) :=\nbegin\n  /-\n  TODO: This proof is very slow. It would be better to pull out a few\n  of the `have` statements into separate lemmas to (hopefully)\n  speed this up.\n  -/\n  induction n,\n  case nat.zero\n  { intros S B t ht, simp at ht, rw ht, intros h1 h2,\n    use [punit, infer_instance, \u03bb _, B, \u03bb _, \ud835\udfd9 _],\n    split, { intros b, use [punit.star, b], refl },\n    { intros _, rw finsupp.map_domain_zero, } },\n  case nat.succ : n hn\n  { intros S B t ht1 ht2 H,\n    by_cases ht1' : t.support.card = n+1, swap,\n    { apply hn, exact nat.le_of_lt_succ (nat.lt_of_le_and_ne ht1 ht1'),\n      assumption' },\n    clear ht1,\n    let F := t.support,\n    let e : F \u2192 (B \u27f6 S) := \u03bb f, f.1.1,\n    obtain \u27e8Q,h1,h2,ee,-\u27e9 : \u2203 (\u03b1 : Type u) (h\u03b11 : fintype \u03b1)\n      (h\u03b12 : linear_order \u03b1) (ee : \u03b1 \u2243 F), true,\n    { refine \u27e8ulift (fin (fintype.card F)), infer_instance,\n        is_well_order.linear_order well_ordering_rel,\n        equiv.ulift.trans (fintype.equiv_fin _).symm, trivial\u27e9, },\n    resetI,\n    let E\u2080 := { a : Q \u00d7 Q | a.1 < a.2 },\n    let X\u2080 : E\u2080 \u2192 Profinite.{u} := \u03bb i, Profinite.equalizer (e (ee i.1.1)) (e (ee i.1.2)),\n    let \u03c0\u2080 : \u03a0 (i : E\u2080), X\u2080 i \u27f6 B := \u03bb i, Profinite.equalizer.\u03b9 _ _,\n\n    have surj\u2080 : \u2200 (b : B), \u2203 (e\u2080 : E\u2080) (x : X\u2080 e\u2080), \u03c0\u2080 _ x = b,\n    { intro b, specialize H b,\n      contrapose! H,\n      have key : \u2200 (i j : Q) (h : i < j), e (ee i) b \u2260 e (ee j) b,\n      { intros i j h, specialize H \u27e8\u27e8i,j\u27e9, h\u27e9, intro c,\n        specialize H, dsimp [X\u2080] at H, specialize H \u27e8b, c\u27e9,\n        apply H, refl },\n      apply finsupp.map_domain_ne_zero_of_ne_zero_of_inj_on,\n      { intro c, rw c at ht1', simpa using ht1' },\n      { intros x hx y hy hxy, dsimp at hxy,\n        let i : Q := ee.symm \u27e8x,hx\u27e9,\n        let j : Q := ee.symm \u27e8y,hy\u27e9,\n        rcases lt_trichotomy i j with (hhh|hhh|hhh),\n        { specialize key i j hhh, contrapose hxy, convert key,\n          { dsimp [i], rw ee.apply_symm_apply, refl },\n          { dsimp [j], rw ee.apply_symm_apply, refl } },\n        { apply_fun (\u03bb q, (ee q).1) at hhh, dsimp [i,j] at hhh,\n          simp_rw ee.apply_symm_apply at hhh, exact hhh },\n        { specialize key j i hhh, contrapose hxy, convert key.symm,\n          { dsimp [j], rw ee.apply_symm_apply, refl },\n          { dsimp [i], rw ee.apply_symm_apply, refl } } } },\n\n    let f\u2080 : \u03a0 (i : E\u2080), S.to_Condensed.val.obj (op B) \u2192 S.to_Condensed.val.obj (op (X\u2080 i)) :=\n      \u03bb i, S.to_Condensed.val.map (\u03c0\u2080 i).op,\n\n    let t\u2080 : \u03a0 (i : E\u2080), S.to_Condensed.val.obj (op (X\u2080 i)) \u2192\u2080 \u2124 :=\n      \u03bb i, t.map_domain (f\u2080 i),\n\n    have card\u2080 : \u2200 (i : E\u2080), (t\u2080 i).support.card \u2264 n,\n    { intros i, suffices : (t\u2080 i).support.card < n + 1,\n        by exact nat.lt_succ_iff.mp this,\n      rw \u2190 ht1',\n      fapply finsupp.card_supp_map_domain_lt,\n      refine (ee i.1.1).1,\n      refine (ee i.1.2).1,\n      { change \u00ac _,\n        erw \u2190 subtype.ext_iff,\n        apply ee.injective.ne,\n        apply ne_of_lt,\n        exact i.2 },\n      refine (ee i.1.1).2,\n      refine (ee i.1.2).2,\n      { dsimp [f\u2080, \u03c0\u2080, Profinite.to_Condensed], ext1, dsimp,\n        -- missing Profinite.equalizer.condition\n        ext t, exact t.2 } },\n\n    have lift\u2080 : \u2200 (i : E\u2080), free'_lift (S.to_condensed_free_pfpng.val.app (op (X\u2080 i))) (t\u2080 i) = 0,\n    { intros i, rw free'_lift_eq_finsupp_lift, dsimp only [t\u2080, f\u2080],\n      apply_fun (\u03bb q, S.condensed_free_pfpng.val.map (\u03c0\u2080 i).op q) at ht2,\n      rw [add_monoid_hom.map_zero, free'_lift_eq_finsupp_lift] at ht2,\n      convert ht2,\n      rw finsupp.lift_map_domain_apply,\n      dsimp [finsupp.lift],\n      rw (S.condensed_free_pfpng.val.map (\u03c0\u2080 i).op).map_finsupp_sum,\n      refl },\n\n    have map\u2080 : \u2200 (i : E\u2080) (b : \u21a5(X\u2080 i)),\n        finsupp.map_domain\n          (\u03bb (f : S.to_Condensed.val.obj (op (X\u2080 i))), f.down.to_fun b) (t\u2080 i) = 0,\n    { intros i b, dsimp [t\u2080], rw \u2190 finsupp.map_domain_comp,\n      exact H (\u03c0\u2080 i b) },\n\n    have key := \u03bb i, hn S (X\u2080 i) (t\u2080 i) (card\u2080 i) (lift\u2080 i) (map\u2080 i),\n\n    choose A hA X\u2081 \u03c0\u2081 surj\u2081 key using key, resetI,\n\n    let E := \u03a3 (e : E\u2080), A e,\n    let X : E \u2192 Profinite.{u} := \u03bb i, X\u2081 i.1 i.2,\n    let \u03c0 : \u03a0 (e : E), X e \u27f6 B := \u03bb e, \u03c0\u2081 e.1 e.2 \u226b \u03c0\u2080 e.1,\n\n    use [E, infer_instance, X, \u03c0], split,\n\n    { intros b,\n      obtain \u27e8e\u2080,x,hx\u27e9 := surj\u2080 b,\n      obtain \u27e8i,q,hq\u27e9 := surj\u2081 e\u2080 x,\n      use [\u27e8e\u2080,i\u27e9,q], dsimp [\u03c0], rw [hq, hx] },\n    { intros a,\n      dsimp [\u03c0], rw functor.map_comp,\n      erw finsupp.map_domain_comp,\n      apply key } },\nend\n\ninstance Profinite.mono_free'_to_condensed_free_pfpng\n  (S : Profinite.{u}) : mono S.free'_to_condensed_free_pfpng :=\nbegin\n  apply presheaf_to_Condensed_Ab_map_mono_of_exists, intros B t ht,\n  let e : S.to_Condensed.val.obj (op B) \u2192\n    S.condensed_free_pfpng.val.obj (op B) :=\n    \u03bb f, (S.to_condensed_free_pfpng.val.app (op B) f),\n  dsimp at t ht,\n  replace ht : free'_lift e t = 0, by rwa \u2190 S.adj'_hom_equiv_symm_eq_free'_lift,\n  let \u03b9 : \u03a0 b : B, S.to_Condensed.val.obj (op B) \u2192 S :=\n    \u03bb b f, (ulift.down f).1 b,\n  have aux : \u2200 b : B, t.map_domain (\u03b9 b) = 0 :=\n    \u03bb b, S.specialization_eq_zero_of_eq_zero B b t ht,\n  dsimp,\n  apply Profinite.mono_free'_to_condensed_free_pfpng_induction_aux,\n  refl,\n  assumption',\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/free_pfpng/mono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623216, "lm_q2_score": 0.5698526514141572, "lm_q1q2_score": 0.36301369512078757}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n\n! This file was ported from Lean 3 source module category_theory.limits.constructions.finite_products_of_binary_products\n! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts\nimport Mathbin.CategoryTheory.Limits.Preserves.Shapes.Products\nimport Mathbin.CategoryTheory.Limits.Shapes.BinaryProducts\nimport Mathbin.CategoryTheory.Limits.Shapes.FiniteProducts\nimport Mathbin.Logic.Equiv.Fin\n\n/-!\n# Constructing finite products from binary products and terminal.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIf a category has binary products and a terminal object then it has finite products.\nIf a functor preserves binary products and the terminal object then it preserves finite products.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\n\nuniverse v v' u u'\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Limits\n\nnamespace CategoryTheory\n\nvariable {J : Type v} [SmallCategory J]\n\nvariable {C : Type u} [Category.{v} C]\n\nvariable {D : Type u'} [Category.{v'} D]\n\n/- warning: category_theory.extend_fan -> CategoryTheory.extendFan is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} {f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> C} (c\u2081 : CategoryTheory.Limits.Fan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))), (CategoryTheory.Limits.BinaryFan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) -> (CategoryTheory.Limits.Fan.{0, u1, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) C _inst_2 f)\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} {f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> C} (c\u2081 : CategoryTheory.Limits.Fan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))), (CategoryTheory.Limits.BinaryFan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) -> (CategoryTheory.Limits.Fan.{0, u1, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) C _inst_2 f)\nCase conversion may be inaccurate. Consider using '#align category_theory.extend_fan CategoryTheory.extendFan\u2093'. -/\n/--\nGiven `n+1` objects of `C`, a fan for the last `n` with point `c\u2081.X` and a binary fan on `c\u2081.X` and\n`f 0`, we can build a fan for all `n+1`.\n\nIn `extend_fan_is_limit` we show that if the two given fans are limits, then this fan is also a\nlimit.\n-/\n@[simps (config := { rhsMd := semireducible })]\ndef extendFan {n : \u2115} {f : Fin (n + 1) \u2192 C} (c\u2081 : Fan fun i : Fin n => f i.succ)\n    (c\u2082 : BinaryFan (f 0) c\u2081.pt) : Fan f :=\n  Fan.mk c\u2082.pt\n    (by\n      refine' Fin.cases _ _\n      \u00b7 apply c\u2082.fst\n      \u00b7 intro i\n        apply c\u2082.snd \u226b c\u2081.\u03c0.app \u27e8i\u27e9)\n#align category_theory.extend_fan CategoryTheory.extendFan\n\n/- warning: category_theory.extend_fan_is_limit -> CategoryTheory.extendFanIsLimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} (f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> C) {c\u2081 : CategoryTheory.Limits.Fan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))} {c\u2082 : CategoryTheory.Limits.BinaryFan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)}, (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_2 (CategoryTheory.Limits.pair.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) c\u2082) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (CategoryTheory.discreteCategory.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) f) (CategoryTheory.extendFan.{u1, u2} C _inst_2 n f c\u2081 c\u2082))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} (f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> C) {c\u2081 : CategoryTheory.Limits.Fan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))} {c\u2082 : CategoryTheory.Limits.BinaryFan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)}, (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_2 (CategoryTheory.Limits.pair.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) c\u2082) -> (CategoryTheory.Limits.IsLimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (CategoryTheory.discreteCategory.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) f) (CategoryTheory.extendFan.{u1, u2} C _inst_2 n f c\u2081 c\u2082))\nCase conversion may be inaccurate. Consider using '#align category_theory.extend_fan_is_limit CategoryTheory.extendFanIsLimit\u2093'. -/\n/-- Show that if the two given fans in `extend_fan` are limits, then the constructed fan is also a\nlimit.\n-/\ndef extendFanIsLimit {n : \u2115} (f : Fin (n + 1) \u2192 C) {c\u2081 : Fan fun i : Fin n => f i.succ}\n    {c\u2082 : BinaryFan (f 0) c\u2081.pt} (t\u2081 : IsLimit c\u2081) (t\u2082 : IsLimit c\u2082) : IsLimit (extendFan c\u2081 c\u2082)\n    where\n  lift s := by\n    apply (binary_fan.is_limit.lift' t\u2082 (s.\u03c0.app \u27e80\u27e9) _).1\n    apply t\u2081.lift \u27e8_, discrete.nat_trans fun \u27e8i\u27e9 => s.\u03c0.app \u27e8i.succ\u27e9\u27e9\n  fac := fun s \u27e8j\u27e9 => by\n    apply Fin.inductionOn j\n    \u00b7 apply (binary_fan.is_limit.lift' t\u2082 _ _).2.1\n    \u00b7 rintro i -\n      dsimp only [extend_fan_\u03c0_app]\n      rw [Fin.cases_succ, \u2190 assoc, (binary_fan.is_limit.lift' t\u2082 _ _).2.2, t\u2081.fac]\n      rfl\n  uniq s m w := by\n    apply binary_fan.is_limit.hom_ext t\u2082\n    \u00b7 rw [(binary_fan.is_limit.lift' t\u2082 _ _).2.1]\n      apply w \u27e80\u27e9\n    \u00b7 rw [(binary_fan.is_limit.lift' t\u2082 _ _).2.2]\n      apply t\u2081.uniq \u27e8_, _\u27e9\n      rintro \u27e8j\u27e9\n      rw [assoc]\n      dsimp only [discrete.nat_trans_app, extend_fan_is_limit._match_1]\n      rw [\u2190 w \u27e8j.succ\u27e9]\n      dsimp only [extend_fan_\u03c0_app]\n      rw [Fin.cases_succ]\n#align category_theory.extend_fan_is_limit CategoryTheory.extendFanIsLimit\n\nsection\n\nvariable [HasBinaryProducts C] [HasTerminal C]\n\n/-- If `C` has a terminal object and binary products, then it has a product for objects indexed by\n`fin n`.\nThis is a helper lemma for `has_finite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate theorem has_product_fin : \u2200 (n : \u2115) (f : Fin n \u2192 C), HasProduct f\n  | 0 => fun f =>\n    by\n    letI : has_limits_of_shape (discrete (Fin 0)) C :=\n      has_limits_of_shape_of_equivalence (Discrete.equivalence.{0} fin_zero_equiv'.symm)\n    infer_instance\n  | n + 1 => fun f => by\n    haveI := has_product_fin n\n    apply has_limit.mk \u27e8_, extend_fan_is_limit f (limit.is_limit _) (limit.is_limit _)\u27e9\n#align category_theory.has_product_fin category_theory.has_product_fin\n\n#print CategoryTheory.hasFiniteProducts_of_has_binary_and_terminal /-\n/-- If `C` has a terminal object and binary products, then it has finite products. -/\ntheorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C :=\n  by\n  refine' \u27e8fun n => \u27e8fun K => _\u27e9\u27e9\n  letI := has_product_fin n fun n => K.obj \u27e8n\u27e9\n  let this : (discrete.functor fun n => K.obj \u27e8n\u27e9) \u2245 K := discrete.nat_iso fun \u27e8i\u27e9 => iso.refl _\n  apply has_limit_of_iso this\n#align category_theory.has_finite_products_of_has_binary_and_terminal CategoryTheory.hasFiniteProducts_of_has_binary_and_terminal\n-/\n\nend\n\nsection Preserves\n\nvariable (F : C \u2964 D)\n\nvariable [PreservesLimitsOfShape (Discrete WalkingPair) F]\n\nvariable [PreservesLimitsOfShape (Discrete.{0} PEmpty) F]\n\nvariable [HasFiniteProducts.{v} C]\n\n/- warning: category_theory.preserves_fin_of_preserves_binary_and_terminal -> CategoryTheory.preservesFinOfPreservesBinaryAndTerminal\u2093 is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_2 D _inst_3) [_inst_4 : CategoryTheory.Limits.PreservesLimitsOfShape.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) F] [_inst_5 : CategoryTheory.Limits.PreservesLimitsOfShape.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} PEmpty.{1}) (CategoryTheory.discreteCategory.{0} PEmpty.{1}) F] [_inst_6 : CategoryTheory.Limits.HasFiniteProducts.{u1, u3} C _inst_2] (n : Nat) (f : (Fin n) -> C), CategoryTheory.Limits.PreservesLimit.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) (CategoryTheory.Discrete.functor.{u1, 0, u3} C _inst_2 (Fin n) f) F\nbut is expected to have type\n  forall {C : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] {D : Type.{u2}} [_inst_3 : CategoryTheory.Category.{u4, u2} D] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_4 : CategoryTheory.Limits.PreservesLimitsOfShape.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) F] [_inst_5 : CategoryTheory.Limits.PreservesLimitsOfShape.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} PEmpty.{1}) (CategoryTheory.discreteCategory.{0} PEmpty.{1}) F] [_inst_6 : CategoryTheory.Limits.HasFiniteProducts.{u3, u1} C _inst_2] (n : Nat) (f : (Fin n) -> C), CategoryTheory.Limits.PreservesLimit.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) (CategoryTheory.Discrete.functor.{u3, 0, u1} C _inst_2 (Fin n) f) F\nCase conversion may be inaccurate. Consider using '#align category_theory.preserves_fin_of_preserves_binary_and_terminal CategoryTheory.preservesFinOfPreservesBinaryAndTerminal\u2093\u2093'. -/\n/-- If `F` preserves the terminal object and binary products, then it preserves products indexed by\n`fin n` for any `n`.\n-/\nnoncomputable def preservesFinOfPreservesBinaryAndTerminal :\n    \u2200 (n : \u2115) (f : Fin n \u2192 C), PreservesLimit (Discrete.functor f) F\n  | 0 => fun f =>\n    by\n    letI : preserves_limits_of_shape (discrete (Fin 0)) F :=\n      preservesLimitsOfShapeOfEquiv.{0, 0} (discrete.equivalence fin_zero_equiv'.symm) _\n    infer_instance\n  | n + 1 => by\n    haveI := preserves_fin_of_preserves_binary_and_terminal n\n    intro f\n    refine'\n      preserves_limit_of_preserves_limit_cone\n        (extend_fan_is_limit f (limit.is_limit _) (limit.is_limit _)) _\n    apply (is_limit_map_cone_fan_mk_equiv _ _ _).symm _\n    let this :=\n      extend_fan_is_limit (fun i => F.obj (f i)) (is_limit_of_has_product_of_preserves_limit F _)\n        (is_limit_of_has_binary_product_of_preserves_limit F _ _)\n    refine' is_limit.of_iso_limit this _\n    apply cones.ext _ _\n    apply iso.refl _\n    rintro \u27e8j\u27e9\n    apply Fin.inductionOn j\n    \u00b7 apply (category.id_comp _).symm\n    \u00b7 rintro i -\n      dsimp only [extend_fan_\u03c0_app, iso.refl_hom, fan.mk_\u03c0_app]\n      rw [Fin.cases_succ, Fin.cases_succ]\n      change F.map _ \u226b _ = \ud835\udfd9 _ \u226b _\n      rw [id_comp, \u2190 F.map_comp]\n      rfl\n#align category_theory.preserves_fin_of_preserves_binary_and_terminal CategoryTheory.preservesFinOfPreservesBinaryAndTerminal\u2093\n\n#print CategoryTheory.preservesShapeFinOfPreservesBinaryAndTerminal /-\n/-- If `F` preserves the terminal object and binary products, then it preserves limits of shape\n`discrete (fin n)`.\n-/\ndef preservesShapeFinOfPreservesBinaryAndTerminal (n : \u2115) :\n    PreservesLimitsOfShape (Discrete (Fin n)) F\n    where PreservesLimit K :=\n    by\n    let this : (discrete.functor fun n => K.obj \u27e8n\u27e9) \u2245 K := discrete.nat_iso fun \u27e8i\u27e9 => iso.refl _\n    haveI := preserves_fin_of_preserves_binary_and_terminal F n fun n => K.obj \u27e8n\u27e9\n    apply preserves_limit_of_iso_diagram F this\n#align category_theory.preserves_shape_fin_of_preserves_binary_and_terminal CategoryTheory.preservesShapeFinOfPreservesBinaryAndTerminal\n-/\n\n#print CategoryTheory.preservesFiniteProductsOfPreservesBinaryAndTerminal /-\n/-- If `F` preserves the terminal object and binary products then it preserves finite products. -/\ndef preservesFiniteProductsOfPreservesBinaryAndTerminal (J : Type) [Fintype J] :\n    PreservesLimitsOfShape (Discrete J) F := by\n  classical\n    let e := Fintype.equivFin J\n    haveI := preserves_shape_fin_of_preserves_binary_and_terminal F (Fintype.card J)\n    apply preservesLimitsOfShapeOfEquiv.{0, 0} (discrete.equivalence e).symm\n#align category_theory.preserves_finite_products_of_preserves_binary_and_terminal CategoryTheory.preservesFiniteProductsOfPreservesBinaryAndTerminal\n-/\n\nend Preserves\n\n/- warning: category_theory.extend_cofan -> CategoryTheory.extendCofan is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} {f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> C} (c\u2081 : CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))), (CategoryTheory.Limits.BinaryCofan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) -> (CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) C _inst_2 f)\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} {f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> C} (c\u2081 : CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))), (CategoryTheory.Limits.BinaryCofan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) -> (CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) C _inst_2 f)\nCase conversion may be inaccurate. Consider using '#align category_theory.extend_cofan CategoryTheory.extendCofan\u2093'. -/\n/-- Given `n+1` objects of `C`, a cofan for the last `n` with point `c\u2081.X`\nand a binary cofan on `c\u2081.X` and `f 0`, we can build a cofan for all `n+1`.\n\nIn `extend_cofan_is_colimit` we show that if the two given cofans are colimits,\nthen this cofan is also a colimit.\n-/\n@[simps (config := { rhsMd := semireducible })]\ndef extendCofan {n : \u2115} {f : Fin (n + 1) \u2192 C} (c\u2081 : Cofan fun i : Fin n => f i.succ)\n    (c\u2082 : BinaryCofan (f 0) c\u2081.pt) : Cofan f :=\n  Cofan.mk c\u2082.pt\n    (by\n      refine' Fin.cases _ _\n      \u00b7 apply c\u2082.inl\n      \u00b7 intro i\n        apply c\u2081.\u03b9.app \u27e8i\u27e9 \u226b c\u2082.inr)\n#align category_theory.extend_cofan CategoryTheory.extendCofan\n\n/- warning: category_theory.extend_cofan_is_colimit -> CategoryTheory.extendCofanIsColimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} (f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) -> C) {c\u2081 : CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))} {c\u2082 : CategoryTheory.Limits.BinaryCofan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)}, (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_2 (CategoryTheory.Limits.pair.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (OfNat.mk.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) 0 (Zero.zero.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne))) (Fin.hasZeroOfNeZero (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (One.one.{0} Nat Nat.hasOne)) (NeZero.succ n)))))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) c\u2082) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) (CategoryTheory.discreteCategory.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne)))))) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) n (OfNat.ofNat.{0} Nat 1 (OfNat.mk.{0} Nat 1 (One.one.{0} Nat Nat.hasOne))))) f) (CategoryTheory.extendCofan.{u1, u2} C _inst_2 n f c\u2081 c\u2082))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_2 : CategoryTheory.Category.{u1, u2} C] {n : Nat} (f : (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) -> C) {c\u2081 : CategoryTheory.Limits.Cofan.{0, u1, u2} (Fin n) C _inst_2 (fun (i : Fin n) => f (Fin.succ n i))} {c\u2082 : CategoryTheory.Limits.BinaryCofan.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)}, (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) C _inst_2 (CategoryTheory.Limits.pair.{u1, u2} C _inst_2 (f (OfNat.ofNat.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) 0 (Fin.instOfNatFin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))) 0 (NeZero.succ n)))) (CategoryTheory.Limits.Cocone.pt.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin n) (fun (i : Fin n) => f (Fin.succ n i))) c\u2081)) c\u2082) -> (CategoryTheory.Limits.IsColimit.{0, u1, 0, u2} (CategoryTheory.Discrete.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) (CategoryTheory.discreteCategory.{0} (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1))))) C _inst_2 (CategoryTheory.Discrete.functor.{u1, 0, u2} C _inst_2 (Fin (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) n (OfNat.ofNat.{0} Nat 1 (instOfNatNat 1)))) f) (CategoryTheory.extendCofan.{u1, u2} C _inst_2 n f c\u2081 c\u2082))\nCase conversion may be inaccurate. Consider using '#align category_theory.extend_cofan_is_colimit CategoryTheory.extendCofanIsColimit\u2093'. -/\n/-- Show that if the two given cofans in `extend_cofan` are colimits,\nthen the constructed cofan is also a colimit.\n-/\ndef extendCofanIsColimit {n : \u2115} (f : Fin (n + 1) \u2192 C) {c\u2081 : Cofan fun i : Fin n => f i.succ}\n    {c\u2082 : BinaryCofan (f 0) c\u2081.pt} (t\u2081 : IsColimit c\u2081) (t\u2082 : IsColimit c\u2082) :\n    IsColimit (extendCofan c\u2081 c\u2082)\n    where\n  desc s := by\n    apply (binary_cofan.is_colimit.desc' t\u2082 (s.\u03b9.app \u27e80\u27e9) _).1\n    apply t\u2081.desc \u27e8_, discrete.nat_trans fun i => s.\u03b9.app \u27e8i.as.succ\u27e9\u27e9\n  fac s := by\n    rintro \u27e8j\u27e9\n    apply Fin.inductionOn j\n    \u00b7 apply (binary_cofan.is_colimit.desc' t\u2082 _ _).2.1\n    \u00b7 rintro i -\n      dsimp only [extend_cofan_\u03b9_app]\n      rw [Fin.cases_succ, assoc, (binary_cofan.is_colimit.desc' t\u2082 _ _).2.2, t\u2081.fac]\n      rfl\n  uniq s m w := by\n    apply binary_cofan.is_colimit.hom_ext t\u2082\n    \u00b7 rw [(binary_cofan.is_colimit.desc' t\u2082 _ _).2.1]\n      apply w \u27e80\u27e9\n    \u00b7 rw [(binary_cofan.is_colimit.desc' t\u2082 _ _).2.2]\n      apply t\u2081.uniq \u27e8_, _\u27e9\n      rintro \u27e8j\u27e9\n      dsimp only [discrete.nat_trans_app]\n      rw [\u2190 w \u27e8j.succ\u27e9]\n      dsimp only [extend_cofan_\u03b9_app]\n      rw [Fin.cases_succ, assoc]\n#align category_theory.extend_cofan_is_colimit CategoryTheory.extendCofanIsColimit\n\nsection\n\nvariable [HasBinaryCoproducts C] [HasInitial C]\n\n/--\nIf `C` has an initial object and binary coproducts, then it has a coproduct for objects indexed by\n`fin n`.\nThis is a helper lemma for `has_cofinite_products_of_has_binary_and_terminal`, which is more general\nthan this.\n-/\nprivate theorem has_coproduct_fin : \u2200 (n : \u2115) (f : Fin n \u2192 C), HasCoproduct f\n  | 0 => fun f =>\n    by\n    letI : has_colimits_of_shape (discrete (Fin 0)) C :=\n      has_colimits_of_shape_of_equivalence (Discrete.equivalence.{0} fin_zero_equiv'.symm)\n    infer_instance\n  | n + 1 => fun f => by\n    haveI := has_coproduct_fin n\n    apply\n      has_colimit.mk \u27e8_, extend_cofan_is_colimit f (colimit.is_colimit _) (colimit.is_colimit _)\u27e9\n#align category_theory.has_coproduct_fin category_theory.has_coproduct_fin\n\n#print CategoryTheory.hasFiniteCoproducts_of_has_binary_and_initial /-\n/-- If `C` has an initial object and binary coproducts, then it has finite coproducts. -/\ntheorem hasFiniteCoproducts_of_has_binary_and_initial : HasFiniteCoproducts C :=\n  by\n  refine' \u27e8fun n => \u27e8fun K => _\u27e9\u27e9\n  letI := has_coproduct_fin n fun n => K.obj \u27e8n\u27e9\n  let this : K \u2245 discrete.functor fun n => K.obj \u27e8n\u27e9 := discrete.nat_iso fun \u27e8i\u27e9 => iso.refl _\n  apply has_colimit_of_iso this\n#align category_theory.has_finite_coproducts_of_has_binary_and_initial CategoryTheory.hasFiniteCoproducts_of_has_binary_and_initial\n-/\n\nend\n\nsection Preserves\n\nvariable (F : C \u2964 D)\n\nvariable [PreservesColimitsOfShape (Discrete WalkingPair) F]\n\nvariable [PreservesColimitsOfShape (Discrete.{0} PEmpty) F]\n\nvariable [HasFiniteCoproducts.{v} C]\n\n/- warning: category_theory.preserves_fin_of_preserves_binary_and_initial -> CategoryTheory.preservesFinOfPreservesBinaryAndInitial\u2093 is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_2 : CategoryTheory.Category.{u1, u3} C] {D : Type.{u4}} [_inst_3 : CategoryTheory.Category.{u2, u4} D] (F : CategoryTheory.Functor.{u1, u2, u3, u4} C _inst_2 D _inst_3) [_inst_4 : CategoryTheory.Limits.PreservesColimitsOfShape.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) F] [_inst_5 : CategoryTheory.Limits.PreservesColimitsOfShape.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} PEmpty.{1}) (CategoryTheory.discreteCategory.{0} PEmpty.{1}) F] [_inst_6 : CategoryTheory.Limits.HasFiniteCoproducts.{u1, u3} C _inst_2] (n : Nat) (f : (Fin n) -> C), CategoryTheory.Limits.PreservesColimit.{0, 0, u1, u2, u3, u4} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) (CategoryTheory.Discrete.functor.{u1, 0, u3} C _inst_2 (Fin n) f) F\nbut is expected to have type\n  forall {C : Type.{u1}} [_inst_2 : CategoryTheory.Category.{u3, u1} C] {D : Type.{u2}} [_inst_3 : CategoryTheory.Category.{u4, u2} D] (F : CategoryTheory.Functor.{u3, u4, u1, u2} C _inst_2 D _inst_3) [_inst_4 : CategoryTheory.Limits.PreservesColimitsOfShape.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} CategoryTheory.Limits.WalkingPair) (CategoryTheory.discreteCategory.{0} CategoryTheory.Limits.WalkingPair) F] [_inst_5 : CategoryTheory.Limits.PreservesColimitsOfShape.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} PEmpty.{1}) (CategoryTheory.discreteCategory.{0} PEmpty.{1}) F] [_inst_6 : CategoryTheory.Limits.HasFiniteCoproducts.{u3, u1} C _inst_2] (n : Nat) (f : (Fin n) -> C), CategoryTheory.Limits.PreservesColimit.{0, 0, u3, u4, u1, u2} C _inst_2 D _inst_3 (CategoryTheory.Discrete.{0} (Fin n)) (CategoryTheory.discreteCategory.{0} (Fin n)) (CategoryTheory.Discrete.functor.{u3, 0, u1} C _inst_2 (Fin n) f) F\nCase conversion may be inaccurate. Consider using '#align category_theory.preserves_fin_of_preserves_binary_and_initial CategoryTheory.preservesFinOfPreservesBinaryAndInitial\u2093\u2093'. -/\n/-- If `F` preserves the initial object and binary coproducts, then it preserves products indexed by\n`fin n` for any `n`.\n-/\nnoncomputable def preservesFinOfPreservesBinaryAndInitial :\n    \u2200 (n : \u2115) (f : Fin n \u2192 C), PreservesColimit (Discrete.functor f) F\n  | 0 => fun f =>\n    by\n    letI : preserves_colimits_of_shape (discrete (Fin 0)) F :=\n      preservesColimitsOfShapeOfEquiv.{0, 0} (discrete.equivalence fin_zero_equiv'.symm) _\n    infer_instance\n  | n + 1 => by\n    haveI := preserves_fin_of_preserves_binary_and_initial n\n    intro f\n    refine'\n      preserves_colimit_of_preserves_colimit_cocone\n        (extend_cofan_is_colimit f (colimit.is_colimit _) (colimit.is_colimit _)) _\n    apply (is_colimit_map_cocone_cofan_mk_equiv _ _ _).symm _\n    let this :=\n      extend_cofan_is_colimit (fun i => F.obj (f i))\n        (is_colimit_of_has_coproduct_of_preserves_colimit F _)\n        (is_colimit_of_has_binary_coproduct_of_preserves_colimit F _ _)\n    refine' is_colimit.of_iso_colimit this _\n    apply cocones.ext _ _\n    apply iso.refl _\n    rintro \u27e8j\u27e9\n    apply Fin.inductionOn j\n    \u00b7 apply category.comp_id\n    \u00b7 rintro i -\n      dsimp only [extend_cofan_\u03b9_app, iso.refl_hom, cofan.mk_\u03b9_app]\n      rw [Fin.cases_succ, Fin.cases_succ]\n      erw [comp_id, \u2190 F.map_comp]\n      rfl\n#align category_theory.preserves_fin_of_preserves_binary_and_initial CategoryTheory.preservesFinOfPreservesBinaryAndInitial\u2093\n\n#print CategoryTheory.preservesShapeFinOfPreservesBinaryAndInitial /-\n/-- If `F` preserves the initial object and binary coproducts, then it preserves colimits of shape\n`discrete (fin n)`.\n-/\ndef preservesShapeFinOfPreservesBinaryAndInitial (n : \u2115) :\n    PreservesColimitsOfShape (Discrete (Fin n)) F\n    where PreservesColimit K :=\n    by\n    let this : (discrete.functor fun n => K.obj \u27e8n\u27e9) \u2245 K := discrete.nat_iso fun \u27e8i\u27e9 => iso.refl _\n    haveI := preserves_fin_of_preserves_binary_and_initial F n fun n => K.obj \u27e8n\u27e9\n    apply preserves_colimit_of_iso_diagram F this\n#align category_theory.preserves_shape_fin_of_preserves_binary_and_initial CategoryTheory.preservesShapeFinOfPreservesBinaryAndInitial\n-/\n\n#print CategoryTheory.preservesFiniteCoproductsOfPreservesBinaryAndInitial /-\n/-- If `F` preserves the initial object and binary coproducts then it preserves finite products. -/\ndef preservesFiniteCoproductsOfPreservesBinaryAndInitial (J : Type) [Fintype J] :\n    PreservesColimitsOfShape (Discrete J) F := by\n  classical\n    let e := Fintype.equivFin J\n    haveI := preserves_shape_fin_of_preserves_binary_and_initial F (Fintype.card J)\n    apply preservesColimitsOfShapeOfEquiv.{0, 0} (discrete.equivalence e).symm\n#align category_theory.preserves_finite_coproducts_of_preserves_binary_and_initial CategoryTheory.preservesFiniteCoproductsOfPreservesBinaryAndInitial\n-/\n\nend Preserves\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141572, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.36301369119395055}}
{"text": "import data.complex.basic\nimport number_theory.divisors\n\ntheorem AMC_2021_A_25 (N:\u2115)(f:\u2115 \u2192 \u211d)(h:\u2200(n:\u2115), f n = (finset.card (nat.divisors n))/(n^(1/3)))(hN:\u2200(n:\u2115),n\u2260N \u2192 (f N > f n)):\nN=2520\n:=\nbegin\n  sorry\nend", "meta": {"author": "ahayat16", "repo": "lean_exos", "sha": "682f2552d5b04a8c8eb9e4ab15f875a91b03845c", "save_path": "github-repos/lean/ahayat16-lean_exos", "path": "github-repos/lean/ahayat16-lean_exos/lean_exos-682f2552d5b04a8c8eb9e4ab15f875a91b03845c/AMC_2021_A_25.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.36287659852054643}}
{"text": "import Smt\nimport Smt.Data.BitVec\n\ntheorem append_eq_shl_or_2 (x y : BitVec 2)\n    : x ++ y = (x.zeroExtend 2 <<< 2) ||| y.zeroExtend 2 := by\n  smt\n  sorry\n\ntheorem append_eq_shl_or_3 (x y : BitVec 3)\n    : x ++ y = (x.zeroExtend 3 <<< 3) ||| y.zeroExtend 3 := by\n  smt\n  sorry\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/BitVec/Shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7431680086124811, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3628765985205464}}
{"text": "import model_theory.substructures\nimport model_theory.bundled\n\n\n--OUTPUT 1\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  have h1 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {exact h},\n  have h2 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], \u00ac(@fintype.card m mfin : \u2115) < n, from by {\n    assume (n : \u2115), \n    have h3 : \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from h1 n,\n    use classical.some h3,\n    have h4 : n \u2264 classical.some_spec (h1 n), from classical.some_spec (h1 n),\n    rw lt_iff_not_ge at h4,\n    exact h4,\n    },\n  have h3 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], \u00ac(@fintype.card m mfin : \u2115) < n, from by {exact h2},\n  let \u0393 : L.Theory := F.union (range (\u03bb n, \u2203' (@fintype.card \u2115 n) \u2227 \u2227' (\u03bb i : fin n, \u00ac(i = 0)))),\n  have h4 : F.consistent \u0393 := by {exact \u0393.compact},\n  have h5 : \u2203 (M : F.Model), (\u2200 (m : F.Model) [mfin : fintype m], \u00ac(@fintype.card m mfin : \u2115) < @fintype.card M M), from by {\n    have h6 : F.nonempty_models \u0393, from by { apply F.nonempty_of_consistent h4 },\n    use classical.some h6,\n    have h15 : F.consistent (F.union (range (\u03bb (n : \u2115), \u2203' (@fintype.card \u2115 n) \u2227 \u2227' (\u03bb (i : fin n), \u00ac(i = 0))))), from by {exact \u0393.compact},\n    have h16 : \u2203 (m : F.Model) [mfin : fintype m], F.consistent (F.union (range (\u03bb (n : \u2115), \u2203' (@fintype.card \u2115 n) \u2227 \u2227' (\u03bb (i : fin n), \u00ac(i = 0)))))\n      \u2227 \u2200 (m' : F.Model) [mfin' : fintype m'], (\u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0))) \u2286 M, from by {\n      have h17 : \u2200 (m' : F.Model) [mfin' : fintype m'], (\u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0))) \u2286 M, from by {\n        assume (m' : F.Model) [mfin' : fintype m'],\n        have h18 : (\u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0))) \u2286 M, from by {\n          have h19 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)) \u2286 M, from by {\n            have h20 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)) \u2286 M, from by {\n              have h31 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)) \u2286 M, from by {\n                have h21 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')), from by {\n                  have h22 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')), from by {\n                    have h23 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')), from by {\n                      rw \u2190 fintype.card_le,\n                      exact \u27e8(@fintype.card m' mfin')+1,nat.succ_le_succ (@fintype.card m' mfin')\u27e9,\n                    },\n                    exact \u27e8h23\u27e9,\n                  },\n                  have h24 : fintype.fin (@fintype.card \u2115 (@fintype.card m' mfin')), from by {\n                    exact fintype.mk (@fintype.card \u2115 (@fintype.card m' mfin')) (\u03bb (n : \u2115), \u27e8n,\u27e8@fintype.card m' mfin'\u27e9\u27e9)\n                    \u27e8\u03bb (n : \u2115), @fintype.card m' mfin'\u27e9\n                    \u27e8\u03bb (n m : \u2115), @fintype.card m' mfin'\u27e9,  \n                  },\n                  exact \u27e8h22,h24\u27e9,\n                },\n                have h25 : \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)), from by {\n                  have h26 : \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)), from by {\n                    have h27 : \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)), from by {\n                      have h28 : \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)), from by {\n                        have h29 : \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)), from by {\n                          simp,\n                          exact \u27e8\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)\u27e9,\n                        },\n                        exact \u27e8h29\u27e9,\n                      },\n                      exact \u27e8h28\u27e9,\n                    },\n                    exact \u27e8h26\u27e9,\n                  },\n                  exact \u27e8h25\u27e9,\n                },\n                have h30 : \u2203' (@fintype.card \u2115 (@fintype.card m' mfin')) \u2227 \u2227' (\u03bb (i : fin (@fintype.card m' mfin')), \u00ac(i = 0)) \u2286 M, from by {\n                  exact \u27e8h21,h25\u27e9,\n                },\n                exact \u27e8h30\u27e9,\n              },\n              exact \u27e8h31\u27e9,\n            },\n            exact \u27e8h20\u27e9,\n          },\n          exact \u27e8h19\u27e9,\n        },\n        exact \u27e8h17\u27e9,\n      },\n      exact \u27e8h16\u27e9,\n    },\n    have h10 : \u2200 n : \u2115, n \u2264 @fintype.card M M, from by {\n      assume (n : \u2115),\n      have h11 : n \u2264 @fintype.card M M, from by {\n        have h12 : n \u2264 @fintype.card M M, from by {\n          have h13 : n \u2264 @fintype.card M M, from\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  -- Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size.\n  assume (F : L.Theory) (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin),\n\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let \u0393 : L.Theory := F,\n  let A (n : \u2115) : L.Formula := F.exists (\u03bb x, L.nne x (list.range n)),\n\n  -- Then $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n  have hA_iff : \u2200 (n : \u2115) (A : F.Model) ,F.models A A n \u2194 fintype.card A \u2265 n, from by {\n    assume (n : \u2115) (A : F.Model),\n    split,\n    {\n      --\u22a2 (A \u22a8 A n ) \u2192 (card A \u2265 n)\n      assume (h1 : F.models A A n),\n      -- $A$ has at least $n$ elements\n      have h2 : fintype.card A \u2265 n, from by {\n        --\u22a2 (\u2203 m, fintype A \u2192 card A \u2265 n)\n        have h3 : \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from exists.intro A (and.intro (by apply_instance) (by {rw nat.le_of_lt,apply nat.lt_succ_self})),\n        show fintype.card A \u2265 n, from by {apply exists.elim h3, assume m, cases m with mfin, apply nat.le_of_lt (mt fintype.card_le_one_iff.mp mfin.right),},\n      },\n      show fintype.card A \u2265 n, from h2,\n    },\n    {\n      --\u22a2 (card A \u2265 n) \u2192 (A \u22a8 A n )\n      assume (h1 : fintype.card A \u2265 n),\n\n      -- Let $[a_1, ..., a_n]$ be the range of $A$\n      let (A_var : fin A.\u03c3) := list.range n.succ,\n      let (A_list : list A.\u03c3) := \u03bb z, z.val,\n      let A_range := list.range A.\u03c3,\n      have hA_var : fintype A_var, from by { apply_instance },\n      have hA_list : A_list A_var = \u03bb (z : fin A.\u03c3), z.val, from by simp[A_list],\n      have hA_range : A_range = \u03bb (z : fin A.\u03c3), z.val, from by simp [A_range],\n      have hA_range_eq : A_list A_var = A_range, from by {rw [hA_var,hA_list,hA_range],},\n      have hA_range_eq_n : A_list A_var = A_range \u2227 fintype.card A_var = n.succ, from by {split,simp [hA_range_eq],rw fintype.card_range,},\n\n      -- By Finite Union Property of Finite Models, there are $a_1, ..., a_n$ such that: $a_1 \\ne a_2 \\land a_1 \\ne a_3 \\land \\ldots \\land a_{n - 1} \\ne a_n$\n      have hA_ne : \u2203 (a :  F.Model.\u03c3 \u2192 fin A_var), \n        (\u2200 (i j : fin A_var), i \u2260 j \u2192 F.Model.\u03c6 a i \u2260 F.Model.\u03c6 a j \u2227 F.Model.\u03c6 a i \u2208 A.\u03c3) \u2227\n        \u2200 (i j h2 : fin A_var) (h1 : F.Model.\u03c6 a i \u2208 A.\u03c3),\n          F.Model.\u03c6 a j \u2208 A.\u03c3 \u2192 \n          F.Model.\u03c6 a j = F.Model.\u03c6 a i, from by apply F.finite_union_property,\n      -- Let: $a_1, ..., a_n$ be a model of $A$\n      have hA_models_A : F.models A A n, from by {\n        apply exists.elim hA_ne,\n        --\u22a2 \u2203 m, fintype A \u2192 card A \u2265 n\n        have h4 : \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {apply exists.intro A (by {split, apply_instance, rw nat.le_of_lt,show n < n.succ, from nat.lt_succ_self n,}),},\n        --\u22a2 exists.elim h4\n        assume (m : F.Model) (mfin : fintype m),\n        --\u22a2 \u2203 m \u2208 list.range A.\u03c3, fintype A \u2192 card A \u2265 n\n        have h5 : \u2203 (m : F.Model.\u03c3 \u2192 fin A.\u03c3), m \u2208 list.range A.\u03c3 \u2227 fintype A \u2192 @fintype.card (F.Model.\u03c3 \u2192 fin A.\u03c3) m \u2286 n, from\n          exists.intro (F.Model.\u03c6 \u2218 (list.range n.succ).val)\n          (and.intro (by {rw hA_range_eq,}) (by {rw fintype.card_range,rw nat.le_iff_lt_or_eq,rw nat.le_iff_lt_or_eq,rw nat.lt_succ_iff,rw nat.le_iff_lt_or_eq,rw nat.le_iff_lt_or_eq,rw nat.lt_succ_iff,apply mt fintype.card_le_one_iff.mp,apply and.right,apply and.left,})),\n\n        --\u22a2 (exists.elim h5)\n        assume (m : F.Model.\u03c3 \u2192 fin A_var) (h5_1 : m \u2208 list.range A.\u03c3) (h5_2 : fintype A \u2192 @fintype.card (F.Model.\u03c3 \u2192 fin A_var) m \u2286 n),\n        --\u22a2 (\u2200 x, x \u2208 list.range A.\u03c3 \u2192 @fintype.card (F.Model.\u03c3 \u2192 fin A_var) x \u2264 @fintype.card (F.Model.\u03c3 \u2192 fin A_var) m)\n        have h6 : \u2200 x, x \u2208 list.range A.\u03c3 \u2192 fintype.card x \u2264 fintype.card m, from\n          (list.forall_mem_range).mp (by {apply set.forall_le_iff.mp h5_2,}),\n        --\u22a2 (A \u22a8 A n )\n        show F.models A A n, from\n          F.models_of_mem_models (and.left hA_ne) hA_range_eq_n.right (list.mem_range A.\u03c3) h1,\n      },\n\n      -- Hence $A$ has at least $n$ elements.\n      have h2 : fintype.card A \u2265 n, from by {apply hA_iff,from hA_models_A},\n      show fintype.card A \u2265 n, from h2,\n    },\n  },\n\n  -- Take:\n  -- $$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$  \n  let \u0393 : L.Theory := by {apply set.union, apply F, apply classical.some (exists_eq_fintype.mp h),},\n\n  -- Since $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n  have h_models : \u2200 (\u0393 : set L.Formula) (h_finite : finite \u0393), \u2203 (A : F.Model), F.satisfiable A \u0393, from by {\n    assume (\u0393 : set L.Formula\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  rcases h 1 with \u27e8M,mfinM,h\u27e9,\n\n  have h2 : 1 \u2264 @fintype.card M mfinM, from h,\n  have h3 : @fintype.card M mfinM = 1, from le_antisymm h h2,\n  -- this is a contradiction\n  have h4 : fintype M, from card_le_one.elim2 h3,\n  have h5 : \u00ac infinite M, from not_infinite_iff_finite.mpr h4,\n  contradiction,\nend\n\n--OUTPUT 4\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  let \u03c8_n := \u03bb n : \u2115, list_to_set L.predicates $\n    ((list.range n).pmap (\u03bb i j, (L.symb \u2218 L.fns) i \u2260 (L.symb \u2218 L.fns) j)).map @nonempty.elim,\n  let \u03c8 := (\u222a) (range \u03c8_n),\n  have h1 : \u2200 n : \u2115, \u2200 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin \u2192 \u22a8' \u03c8_n n m, from\n    assume n : \u2115, assume (m : F.Model) [mfin : fintype m] (hmn : n \u2264 @fintype.card m mfin), have\n      h\u2080 : \u2203 (i : fin m.n) (j : fin m.n), \u00ac(i = j), from fintype.exists_ne mfin,\n    have h\u2081 : \u2203 (i : fin m.n) (j : fin m.n) (hneij : i \u2260 j),\n      from fintype.exists_ne_of_card_eq_succ hmn,\n    have h\u2082 : \u2200 i, \u2200 j, \u2200 hneij, (\u03bb (i : fin n) (j : fin n), (L.symb \u2218 L.fns) i \u2260 (L.symb \u2218 L.fns) j) i j, from\n      assume i : fin m.n, assume j : fin m.n, assume hneij : i \u2260 j,\n      have h\u2083 : \u2200 f : \u2115 \u2192 L.predicates, \u2200 i j : fin (m.n), (f i \u2260 f j), from assume f : \u2115 \u2192 L.predicates,\n      assume i : fin m.n, assume j : fin m.n, assume hfneij : f i \u2260 f j, have h\u2084 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y,\n        L.predicates.injective ((L.symb \u2218 L.fns) : fin m.n \u2192 L.predicates) hfneij, from \n        assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y, have h\u2085 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y,\n          L.predicates.injective L.symb hfneij, from assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y,\n          L.symb.injective L.symb hfneij, have h\u2086 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from\n          assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y, have h\u2087 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y,\n            L.predicates.injective (L.symb \u2218 L.fns) hfneij, from assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y,\n            L.predicates.injective L.symb hfneij, have h\u2088 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y,\n          from assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y, L.fns.injective hfneij, have h\u2089 :\n            \u2200 x : fin m.n, \u2200 y : fin m.n, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from assume x : fin m.n, assume y : fin m.n,\n            assume hneij : x \u2260 y, have h\u2081\u2080 : \u2200 i : fin m.n, \u2200 j : fin m.n, \u2200 hneij : i \u2260 j, L.symb (L.fns i) \u2260 L.symb (L.fns j),\n            from assume i : fin m.n, assume j : fin m.n, assume hneij : i \u2260 j, have h\u2081\u2081 : \u2200 i : fin m.n, \u2200 j : fin m.n, \u2200 hneij : i \u2260 j,\n              L.symb (L.fns i) \u2260 L.symb (L.fns j), from assume i : fin m.n, assume j : fin m.n, assume hneij : i \u2260 j,\n              L.predicates.injective L.symb hfneij, have h\u2081\u2082 : \u2200 i : fin m.n, \u2200 j : fin m.n, \u2200 hneij : i \u2260 j,\n              L.symb (L.fns i) \u2260 L.symb (L.fns j), from h\u2081\u2081 i j hneij, have h\u2081\u2083 : \u2200 i : fin m.n, \u2200 j : fin m.n, \u2200 hneij : i \u2260 j,\n              L.fns i \u2260 L.fns j, from by apply h\u2089, h\u2081\u2083 x y hneij, have h\u2081\u2084 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y,\n            from by rw [\u2190 fin.cast_up, \u2190 fin.cast_up, h\u2089], have h\u2081\u2085 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from h\u2081\u2084,\n            have h\u2081\u2086 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y,\n              have h\u2081\u2087 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from by apply h\u2081\u2084, have h\u2081\u2088 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y,\n                L.fns x \u2260 L.fns y, from h\u2081\u2087 x y hneij, have h\u2081\u2089 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from assume x : \u2115, assume y : \u2115,\n                assume hneij : x \u2260 y, have h\u2082\u2080 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from have h\u2082\u2081 : \u2200 x : \u2115,\n                  \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from assume x : \u2115, assume y : \u2115, assume hneij : x \u2260 y,\n                  have h\u2082\u2082 : \u2200 x : \u2115, \u2200 y : \u2115, \u2200 hneij : x \u2260 y, L.fns x \u2260 L.fns y, from by apply h\u2081\u2084, have h\u2082\u2083 :\n                  \u2200 x : \u2115, \u2200 y : \u2115,\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  -- let F be a set of first-order formulas which has finite models of arbitrarily large size\n  assume F : L.Theory,\n  assume h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin,\n\n  -- let A_n be the formula $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n  let A_n :=  exists (x1 : F.struct.\u03c3) $ exists (x2 : F.struct.\u03c3) $ exists (x3 : F.struct.\u03c3)\n    (n - 3) \u27e8x2 \u2260 x3, by apply nat.le_pred a\u27e9\n    (\u2203 x_1 \u2203 x_2 \u2203 x_3 (n - 3) (\u27e8x_2 \u2260 x_3, by apply nat.le_pred a\u27e9) \n    (\u2200 x_1 \u2200 x_2 \u2200 x_3 (n - 3) (\u27e8x_2 \u2260 x_3, by apply nat.le_pred a\u27e9) \n    \u2203 x_n (n - 1) (\u27e8x_1 \u2260 x_n, by apply nat.le_pred a\u27e9)), \n\n  -- Gamma is the union of F and the above formula for all values of n\n  let Gamma := F.set \u222a (\u222a i : \u2115, A_n i)\n\n  -- We know that every finite subset of Gamma is satisfiable.\n  have h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from\n    assume (n : \u2115), h n,\n\n  -- The compactness theorem tells us Gamma is satisfiable in some model\n  have h3 : \u2203 (M : F.Model), M \u22a8 Gamma, from by apply compactness Gamma,\n\n  -- this model must be infinite since for each value of n, the above formula has a solution\n  have h4 : \u2203 (M : F.Model), infinite M, from \u27e8classical.some h3, assume x : M,\n    classical.some_spec h3 \u27e8x, assume \u27e8w,(h\u2081 : w \u2208 F.set) | (h\u2082 : w \u2208 A_n)\u27e9,\n      begin\n        -- w is either in F or one of the n formulas\n        assume h\u2081 | h\u2082,\n        {\n          -- w is in F\n          assume h\u2081,\n          have h5 : F \u22a8 w, from \u27e8w,assume h\u2081,h\u2081, h\u2081\u27e9,\n          have h6 : M \u22a8 w, from classical.some_spec h3 \u27e8w, h5\u27e9,\n          exact h6 x,\n        },\n        {\n          -- w is in the n formulas\n          assume h\u2082,\n\n          -- w is a formula of the form $\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n          have h7 : \u2203 (x_1 : F.struct.\u03c3) \u2203 (x_2 : F.struct.\u03c3) \u2203 (x_3 : F.struct.\u03c3) (n - 3) \n            \u27e8x_2 \u2260 x_3, by apply nat.le_pred a\u27e9 \u2203 x_n (n - 1) \u27e8x_1 \u2260 x_n, by apply nat.le_pred a\u27e9,\n            from classical.some_spec h3 \u27e8w,h\u2082\u27e9,\n\n          have h8 : \u2203 (x_1 : F.struct.\u03c3) \u2203 (x_2 : F.struct.\u03c3) \u2203 (x_3 : F.struct.\u03c3) (n - 3) \n            (\u27e8x_2 \u2260 x_3, by apply nat.le_pred a\u27e9) \u2203 x_n (n - 1) \u27e8x_1 \u2260 x_n, by apply nat.le_pred a\u27e9,\n            from h7,\n\n          have h9 : \u2203 (x_1 : F.struct.\u03c3) \u2203 (x_2 : F.struct.\u03c3) \u2203 (x_3 : F.struct.\u03c3) (n - 3) \n            (\u27e8x_2 \u2260 x_3, by apply nat.le_pred a\u27e9) \u2203 x_n (n - 1) \u27e8x_1 \u2260 x_n, by apply nat.le_pred a\u27e9,\n            from h8,\n\n          -- set x to be the value of x_n\n          have h10 : x \u2260 F.struct.\u03c3 (n - 3), from classical.some_spec h9.right.right.right.right,\n          have h11 : x \u2260 F.struct.\u03c3 (n - 2), from classical.some_spec h9.right.right.right,\n          have h12 : x \u2260 F.struct.\u03c3 (n - 1), from classical.some_spec h9.right,\n\n          -- x_n is an element of the model, so x is an element of the model\n          have h13 : x \u2208 M, from classical.some_spec h9.left.right.right.left.right,\n\n          -- the model is infinite since it has at least n elements\n          have h14 : infinite M, from classical.some_spec h3.right n,\n          show \u2203 x : M, true, from \u27e8x,true.intro\u27e9,\n        },\n      end,\n      classical.some_spec h3 \u27e8w,h\u2081\u27e9\u27e9\u27e9,\n  exact h4\nend\n\n--OUTPUT 6\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  -- Let $F$ be a set of first-order formulas which has finite models of arbitrarily large size.\n  assume (L : first_order.language) (F : L.Theory) (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin),\n  -- Then $F$ has an infinite model.\n\n  -- For each $n$, let $\\mathbf A_n$ be the formula:\n  let (An : L.formula) (n : \u2115) := \n  begin\n    have : \u2203 x1 : L.sort, \u2203 x2 : L.sort, \u22c0 i : fin 2, x1 \u2260 x2, from \n      \u27e8L.variable (L.sort 1) 0, L.variable (L.sort 1) 1, assume i : fin 2, begin cases i, refl, end\u27e9,\n    let x : L.variable (L.sort 1) := (this).1,\n    let y : L.variable (L.sort 1) := (this).2,\n    use L.exists' (L.variable (L.sort 1) 0) (L.exists' (L.variable (L.sort 1) 1) (L.forall' \u2115 2 (\u03bb i, L.ne x y))),\n  end,\n  -- Then $A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n  have hAn : \u2200 n : \u2115, (F.Model) \u2192 Prop := \u03bb n, (\u03bb m, \u2203 (l : list (L.sort m)), card (set.range (@list.to_finset m l)) = n),\n  have hAn1 : \u2200 n : \u2115, (F.Model) \u2192 Prop := \u03bb n, (\u03bb m, \u2203 (l : list (L.sort m)), card (set.range (@list.to_finset m l)) \u2260 n),\n\n  -- Take:\n  let \u0393 : list L.formula := F.axioms ++ (list.univ (\u03bb n : \u2115, An n)),\n  -- Since $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n  have h\u0393_sat : \u2200 \u0393' : list L.formula, (\u0393'.length \u2264 \u0393.length) \u2192 F.satisfiable \u0393', from assume \u0393' : list L.formula,\n    assume h : \u0393'.length \u2264 \u0393.length,\n    show F.satisfiable \u0393', from begin\n      -- Let $n$ be the number of formulas in $\\Gamma'$\n      let n : \u2115 := \u0393'.length,\n      -- Since $F$ has models of arbitrarily large size, \n      have h1 : \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from (hF _),\n      -- Let $m$ be a model of $F$ with more than $n$ elements\n      let m : F.Model := classical.some h1.1,\n      have h1' : fintype m, from (classical.some_spec h1.1).2,\n      have h1'' : n \u2264 @fintype.card m h1', from (classical.some_spec h1.1).1,\n\n      -- then \u0393' is satisfiable in $m$\n      have h2 : @F.satisfiable m h1' \u0393', from begin\n        show @F.satisfiable m h1' \u0393', from by {\n          -- $F$ is satisfiable in $m$\n          have h3 : @F.satisfiable m h1' F.axioms, from by apply @F.satisfiable_of_satisfiable _ h1' F.axioms,\n          -- and $A_n$ is satisfiable in $m$\n          have h4 : @F.satisfiable m h1' (An (n + 1)), from by {\n            have h5 : m \u22a8 (An (n + 1)), from begin\n              -- $\\Gamma'$ has $n$ formulas\n              have h6 : \u0393'.length = n, from by simp,\n              have h7 : exists s : L.sort m, \u2203 l : list (L.sort m), card (set.range (list.to_finset s l)) = n, from begin\n                -- Let $n$ be the number of formulas in $\\Gamma'$\n                let n : \u2115 := \u0393'.length,\n                -- Since $F$ has models of arbitrarily large size, \n                have h1 : \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from (hF _),\n                -- Let $m$ be a model of $F$ with more than $n$ elements\n                let m : F.Model := classical.some h1.1,\n                have h1' : fintype m, from (classical.some_spec h1.1).2,\n                have h1'' : n \u2264 @fintype.card m h1', from (classical.some_spec h1.1).1,\n                let s : L.sort m := L.sort 1,\n                have h5 : \u2200 {n : \u2115} {\u03b1 : Type n} [fintype \u03b1] (l : list (L.sort \u03b1)), card (set.range (list.to_finset s l)) = n \u2192 \u2203 l' : list (L.sort \u03b1), card (set.range (list.to_finset s l')) = n + 1, from assume n \u03b1 fintype_\u03b1 l h,\n                  by {\n                    assume h,\n                    have h' : \u2200 (l : list (L.sort \u03b1)), \u2203 s', fintype s' \u2227 s' \u2260 (L.sort \u03b1), from assume l,\n                      by {\n                        have h' : \u2203 (n : \u2115) (m : n), \u2203 (\u03b1' : L.sort \u03b1) (a : n), \u2203 (\u03b2 : Type n), \n                          \u2203 (f : L.sort \u03b1 \u2192 L.sort \u03b1 \u2192 Type n), \u2200 (a' : n), \u2203 (b : Type n), \u2203 (g : L.sort \u03b1 \u2192 L.sort \u03b1 \u2192 b), \n                          \u2203 (h : b \u2192 b \u2192 Type n), f a' a' = h (g a' a') (g a' a'),\n                          from begin\n                            have h' : @fintype.card (L.sort (L.sort \u03b1)) _ \u2260 0, from by {\n                              have h' : 0 \u2264 @fintype.card (L.sort (L.sort \u03b1)) _, from by {\n                                apply (set.card_pos_iff_ne_empty).mp,\n                                have h' : (\u2203 (n : \u2115) (m : n), \u2203 (\u03b1' : L.sort \u03b1) (a : n), \u2203 (\u03b2 : Type n), \u2203 (f : \u03b1' \u2192 \u03b1' \u2192 \u03b2), \u2200 (a' : n), \u2203 (b : Type n), \u2203 (g : \u03b1' \u2192 \u03b1' \u2192 b), \u2203 (h : b \u2192 b \u2192 Type n), f a' a' = h (g a' a') (g a' a')), from by {\n                                  use 1,\n                                  use 0,\n                                  use (L.variable (L.sort \u03b1) 0),\n                                  use 0,\n                                  use bool,\n                                  use eq,\n                                  assume a' : 1,\n                                  use bool,\n                                  use eq,\n                                  use eq.refl ff,\n                                },\n                                exact ne_empty_of_mem h',\n                              },\n                              have h' : 0 < @fintype.card (L.sort (L.sort \u03b1)) _, from begin\n                                apply nat.lt_of_le_of_ne,\n                                exact h',\n                                assume h',\n                                rw h',\n                                exact (dec_trivial : 0 \u2260 1),\n                              end,\n                              have h' : { l : list (L.sort (L.sort \u03b1)) // card (set.range (list.to_finset (L.sort (L.sort \u03b1))\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  sorry\nend\n\n--OUTPUT 8\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  have h1 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by exact h,\n  have h2 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h3 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h4 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h5 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h6 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h7 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h8 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h9 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h10 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h11 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h12 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h13 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h14 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h15 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h16 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h17 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h18 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h19 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h20 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h21 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right,\n  },\n  have h22 : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin, from by {\n    assume n : \u2115,\n    cases (h1 n) with m hm,\n    use m,\n    use hm.left,\n    exact hm.right\nend --Needs more than 2000 tokens!\n\n--OUTPUT 9\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nbegin\n  -- There is a formula consisting of a set of first-order formulas:\n  -- \u2200 i, exists x1,x2,\u2026,xn such that {x1!=x2 && x1!=x3 && x2!=x3 && x2!=x4 && x3!=x4 && \u2026 && x(n-1)!=xn}\n  -- This formula is true iff there are at least n elements\n  have hAi : \u2200 (n : \u2115), \u2203 Ai : F.Formula, @F.formula.satisfies L F Ai (finset.univ) (\u03bb x, \u27e8x, \u27e8n,finset.singleton_eq_singleton.mp (by {exact finset.mem_univ x})\u27e9\u27e9) = ff,\n  {\n    assume n : \u2115,\n    use (finset.univ.bexists $ \u03bb (x1 : F.Model) (h1 : \u2203 h1 : fintype x1, n \u2264 @fintype.card x1 h1),\n            finset.univ.bexists $ \u03bb (x2 : F.Model) (h2 : \u2203 h2 : fintype x2, n \u2264 @fintype.card x2 h2),\n              finset.univ.bexists $ \u03bb (x3 : F.Model) (h3 : \u2203 h3 : fintype x3, n \u2264 @fintype.card x3 h3),\n                (x1 \u2260 x2) \u2227 (x1 \u2260 x3) \u2227 (x2 \u2260 x3) ),\n    assume x : F.Model,\n    have h1 : \u2200 x : F.Model, \u2203 (h1 : fintype x), n \u2264 @fintype.card x h1, from assume x : F.Model, h n,\n    assume hx : @F.formula.satisfies L F (finset.univ.bexists $ \u03bb (x1 : F.Model) (h1 : \u2203 h1 : fintype x1, n \u2264 @fintype.card x1 h1),\n            finset.univ.bexists $ \u03bb (x2 : F.Model) (h2 : \u2203 h2 : fintype x2, n \u2264 @fintype.card x2 h2),\n              finset.univ.bexists $ \u03bb (x3 : F.Model) (h3 : \u2203 h3 : fintype x3, n \u2264 @fintype.card x3 h3),\n                (x1 \u2260 x2) \u2227 (x1 \u2260 x3) \u2227 (x2 \u2260 x3) ) (finset.univ) (\u03bb x, \u27e8x, \u27e8n,finset.singleton_eq_singleton.mp (by {exact finset.mem_univ x})\u27e9\u27e9) = ff,\n    have h2 : n \u2264 @fintype.card x \u27e8x, by {exact finset.mem_univ x}\u27e9, from @F.formula.satisfies_of_mem L F \n      (finset.univ.bexists $ \u03bb (x1 : F.Model) (h1 : \u2203 h1 : fintype x1, n \u2264 @fintype.card x1 h1),\n            finset.univ.bexists $ \u03bb (x2 : F.Model) (h2 : \u2203 h2 : fintype x2, n \u2264 @fintype.card x2 h2),\n              finset.univ.bexists $ \u03bb (x3 : F.Model) (h3 : \u2203 h3 : fintype x3, n \u2264 @fintype.card x3 h3),\n                (x1 \u2260 x2) \u2227 (x1 \u2260 x3) \u2227 (x2 \u2260 x3) ) (finset.univ) (\u03bb x, \u27e8x, \u27e8n,finset.singleton_eq_singleton.mp (by {exact finset.mem_univ x})\u27e9\u27e9) (ff)\n      hx \u27e8x, by {exact finset.mem_univ x}\u27e9,\n    show \u2203 (hx : fintype x), n \u2264 @fintype.card x hx, from \u27e8x, h2\u27e9,\n  },\n\n  -- \u2200 i, exists Ai, false if Ai is true\n  -- This is a contradiction to compactness: \n  -- there exists Ai and false if Ai is true for all i\n  have h1 : \u2203 (Ai : \u2115 \u2192 F.Formula), (\u2200 (n : \u2115), \u2203 (x : F.Model), F.formula.satisfies Ai n x = ff) \n      \u2227 (\u2200 (n : \u2115), @F.formula.satisfies L F (Ai n) (finset.univ) (\u03bb x, \u27e8x, \u27e8n,finset.singleton_eq_singleton.mp (by {exact finset.mem_univ x})\u27e9\u27e9) = ff),\n  {\n    use hAi,\n    obviously,\n  },\n\n  -- F has infinite models\n  exact F.model.infinite_of_satisfies_unbounded_formula h1,\nend\n\n--OUTPUT 10\ntheorem overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M := \nbegin\n  -- Define the formula $\\mathbf A_n$\n  def \u0393 : F.Theory :=\n    F.union (\u222a (range (\u03bb n, (\u2203 (x\u2081 : M), \u2203 (x\u2082 : M), \n    \u2203 (a : \u03a0 {i : \u2115}, fin i \u2192 M), (\u2203 (b : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (c : \u03a0 {i : \u2115}, fin i \u2192 M), (\u2203 (d : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (e : \u03a0 {i : \u2115}, fin i \u2192 M),\n    (\u2203 (f : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (g : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (h : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (p : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (q : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (r : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (s : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (t : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (u : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (v : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (w : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (z : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (y : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03b1 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03b2 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03c7 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03c8 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03c9 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u2205 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03b4 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    \u2203 (\u03b5 : \u03a0 {i : \u2115}, fin i \u2192 M),\n    neq L x\u2081 x\u2082 \u2227 neq L x\u2081 x\u2083 \u2227 neq L x\u2081 x\u2084 \u2227 neq L x\u2081 x\u2085 \u2227 neq L x\u2081 x\u2086 \u2227 neq L x\u2081 x\u2087 \u2227 neq L x\u2081 x\u2088 \u2227 neq L x\u2081 x\u2089 \u2227 neq L x\u2081 x\u2081\u2080 \u2227 neq L x\u2081 x\u2081\u2081 \u2227 neq L x\u2081 x\u2081\u2082 \u2227 neq L x\u2081 x\u2081\u2083 \u2227 neq L x\u2081 x\u2081\u2084 \u2227 neq L x\u2081 x\u2081\u2085 \u2227 neq L x\u2081 x\u2081\u2086 \u2227 neq L x\u2081 x\u2081\u2087 \u2227 neq L x\u2081 x\u2081\u2088 \u2227 neq L x\u2081 x\u2081\u2089 \u2227 neq L x\u2081 x\u2082\u2080 \u2227 neq L x\u2081 x\u2082\u2081 \u2227 neq L x\u2081 x\u2082\u2082 \u2227 neq L x\u2081 x\u2082\u2083 \u2227 neq L x\u2081 x\u2082\u2084 \u2227 neq L x\u2081 x\u2082\u2085 \u2227 neq L x\u2081 x\u2082\u2086 \u2227 neq L x\u2081 x\u2082\u2087 \u2227 neq L x\u2081 x\u2082\u2088 \u2227 neq L x\u2081 x\u2082\u2089 \u2227 neq L x\u2081 x\u2083\u2080 \u2227 neq L x\u2081 x\u2083\u2081 \u2227 neq L x\u2081 x\u2083\u2082 \u2227 neq L x\u2081 x\u2083\u2083 \u2227 neq L x\u2081 x\u2083\u2084 \u2227 neq L x\u2081 x\u2083\u2085 \u2227 neq L x\u2081 x\u2083\u2086 \u2227 neq L x\u2081 x\u2083\u2087 \u2227 neq L x\u2081 x\u2083\u2088 \u2227 neq L x\u2081 x\u2083\u2089 \u2227 neq L x\u2081 x\u2084\u2080 \u2227 neq L x\u2081 x\u2084\u2081 \u2227 neq L x\u2081 a\u2080 \u2227 neq L x\u2081 b\u2080 \u2227 neq L x\u2081 c\u2080 \u2227 neq L x\u2081 d\u2080 \u2227 neq L x\u2081 e\u2080 \u2227 neq L x\u2081 f\u2080 \u2227 neq L x\u2081 g\u2080 \u2227 neq L x\u2081 h\u2080 \u2227 neq L x\u2081 p\u2080 \u2227 neq L x\u2081 q\u2080 \u2227 neq L x\u2081 r\u2080 \u2227 neq L x\u2081 s\u2080 \u2227 neq L x\u2081 t\u2080 \u2227 neq L x\u2081 u\u2080 \u2227 neq L x\u2081 v\u2080 \u2227 neq L x\u2081 w\u2080 \u2227 neq L x\u2081 z\u2080 \u2227 neq L x\u2081 y\u2080 \u2227 neq L x\u2081 \u03b1\u2080 \u2227 neq L x\u2081 \u03b2\u2080 \u2227 neq L x\u2081 \u03c7\u2080 \u2227 neq L x\u2081 \u03c8\u2080 \u2227 neq L x\u2081 \u03c9\u2080 \u2227 neq L x\u2081 \u2205\u2080 \u2227 neq L x\u2081 \u03b4\u2080 \u2227 neq L x\u2081 \u03b5\u2080 \u2227 neq L x\u2082 x\u2083 \u2227 neq L x\u2082 x\u2084 \u2227 neq L x\u2082 x\u2085 \u2227 neq L x\u2082 x\u2086 \u2227 neq L x\u2082 x\u2087 \u2227 neq L x\u2082 x\u2088 \u2227 neq L x\u2082 x\u2089 \u2227 neq L x\u2082 x\u2081\u2080 \u2227 neq L x\u2082 x\u2081\u2081 \u2227 neq L x\u2082 x\u2081\u2082 \u2227 neq L x\u2082 x\u2081\u2083 \u2227 neq L x\u2082 x\u2081\u2084 \u2227 neq L x\u2082 x\u2081\u2085 \u2227 neq L x\u2082 x\u2081\u2086 \u2227 neq L x\u2082 x\u2081\u2087 \u2227 neq L x\u2082 x\u2081\u2088 \u2227 neq L x\u2082 x\u2081\u2089 \u2227 neq L x\u2082 x\u2082\u2080 \u2227 neq L x\u2082 x\u2082\u2081 \u2227 neq L x\u2082 x\u2082\u2082 \u2227 neq L x\u2082 x\u2082\u2083 \u2227 neq L x\u2082 x\u2082\u2084 \u2227 neq L x\u2082 x\u2082\u2085 \u2227 neq L x\u2082 x\u2082\u2086 \u2227 neq L x\u2082 x\u2082\u2087 \u2227 neq L x\u2082 x\u2082\u2088 \u2227 neq L x\u2082 x\u2082\u2089 \u2227 neq L x\u2082 x\u2083\u2080\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nOverflow theorem\nLet $F$ be a set of first-order formulas which has finite models of arbitrarily large size. Then $F$ has an infinite model.\n`proof`\nFor each $n$, let $\\mathbf A_n$ be the formula:\n\n$\\exists x_1 \\exists x_2 \\ldots \\exists x_n: \\{x_1 \\ne x_2 \\land x_1 \\ne x_3 \\land \\ldots \\land x_{n - 1} \\ne x_n\\}$\n\nThen $\\mathbf A_i$ is true in a structure $\\AA$ iff $\\AA$ has at least $n$ elements.\n\nTake:\n$$ \\Gamma := F \\cup \\bigcup_{i \\mathop = 1}^\\infty A_i $$\n\nSince $F$ has models of arbitrarily large size, every finite subset of $\\Gamma$ is satisfiable.\n\nFrom the Compactness Theorem, $\\Gamma$ is satisfiable in some model $\\mathbf{M}$.\n\nBut since $\\mathbf{M} \\models A_i$ for each $i$, $\\mathbf{M}$ must be infinite.\n\nSo $F$ has an infinite model.\n\nQED\n-/\ntheorem  overflow {L : first_order.language} {F : L.Theory} (h : \u2200 n : \u2115, \u2203 (m : F.Model) [mfin : fintype m], n \u2264 @fintype.card m mfin) : \u2203 (M : F.Model), infinite M :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_with_comments-3_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872243177518, "lm_q2_score": 0.4843800842769844, "lm_q1q2_score": 0.36284293284584496}}
{"text": "import data.complex.basic\nimport algebra.field\n\ntheorem cpge_group_3_a {G1 : Type*} [group G1] {G2 : Type*} [group G2] \n                      (H : (subgroup G1)) (f : (G1 \u2192* G2) ) : is_subgroup (f '' H) := sorry\n", "meta": {"author": "ahayat16", "repo": "lean_exos", "sha": "682f2552d5b04a8c8eb9e4ab15f875a91b03845c", "save_path": "github-repos/lean/ahayat16-lean_exos", "path": "github-repos/lean/ahayat16-lean_exos/lean_exos-682f2552d5b04a8c8eb9e4ab15f875a91b03845c/src_icannos_totilas/groupes/cpge_groupe_3_a.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.737158174177441, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3628205074797402}}
{"text": "import .lib .integers .floats .maps .errors\n\nnamespace ast\nopen integers maps errors floats\n\n/- * Syntactic elements -/\n\n/- Identifiers (names of local variables, of global symbols and functions,\n  etc) are represented by the type [positive] of positive integers. -/\n\ndef ident := pos_num\n\ninstance pos_num_eq : decidable_eq pos_num := by tactic.mk_dec_eq_instance\ninstance ident_eq : decidable_eq ident := by tactic.mk_dec_eq_instance\n\n/- The intermediate languages are weakly typed, using the following types: -/\n\ninductive typ : Type\n| Tint                /- 32-bit integers or pointers -/\n| Tfloat              /- 64-bit double-precision floats -/\n| Tlong               /- 64-bit integers -/\n| Tsingle             /- 32-bit single-precision floats -/\n| Tany32              /- any 32-bit value -/\n| Tany64              /- any 64-bit value, i.e. any value -/\n\ndef typ.Tptr : typ := if archi.ptr64 then typ.Tlong else typ.Tint\nopen typ\n\ninstance typ_eq : decidable_eq typ := by tactic.mk_dec_eq_instance\n\ndef typesize : typ \u2192 \u2124\n| Tint    := 4\n| Tfloat  := 8\n| Tlong   := 8\n| Tsingle := 4\n| Tany32  := 4\n| Tany64  := 8\n\nlemma typesize_pos (ty) : typesize ty > 0 :=\nby cases ty; exact dec_trivial\n\nlemma typesize_Tptr : typesize Tptr = if archi.ptr64 then 8 else 4 :=\nby delta Tptr; cases archi.ptr64; refl\n\n/- All values of size 32 bits are also of type [Tany32].  All values\n  are of type [Tany64].  This corresponds to the following subtyping\n  relation over types. -/\n\ndef subtype : typ \u2192 typ \u2192 bool\n| Tint    Tint    := tt\n| Tlong   Tlong   := tt\n| Tfloat  Tfloat  := tt\n| Tsingle Tsingle := tt\n| Tint    Tany32  := tt\n| Tsingle Tany32  := tt\n| Tany32  Tany32  := tt\n| _       Tany64  := tt\n| _       _       := ff\n\ndef subtype_list : list typ \u2192 list typ \u2192 bool\n| [] [] := tt\n| (ty1::tys1) (ty2::tys2) := subtype ty1 ty2 && subtype_list tys1 tys2\n| _ _ := ff\n\n/- Additionally, function definitions and function calls are annotated\n  by function signatures indicating:\n- the number and types of arguments;\n- the type of the returned value, if any;\n- additional information on which calling convention to use.\n\nThese signatures are used in particular to determine appropriate\ncalling conventions for the function. -/\n\nstructure calling_convention : Type := mkcallconv ::\n(cc_vararg : bool)                      /- variable-arity function -/\n(cc_unproto : bool)                     /- old-style unprototyped function -/\n(cc_structret : bool)                   /- function returning a struct  -/\n\ninstance calling_convention_eq : decidable_eq calling_convention := by tactic.mk_dec_eq_instance\n\ndef cc_default : calling_convention :=\n{ cc_vararg := false, cc_unproto := false, cc_structret := false }\n\nstructure signature : Type :=\n(sig_args : list typ)\n(sig_res : option typ)\n(sig_cc : calling_convention)\n\ndef proj_sig_res (s : signature) : typ :=\ns.sig_res.get_or_else Tint\n\ninstance signature_eq : decidable_eq signature := by tactic.mk_dec_eq_instance\n\ndef signature_main : signature :=\n{ sig_args := [], sig_res := some Tint, sig_cc := cc_default }\n\n/- Memory accesses (load and store instructions) are annotated by\n  a ``memory chunk'' indicating the type, size and signedness of the\n  chunk of memory being accessed. -/\n\ninductive memory_chunk : Type\n| Mint8signed     /- 8-bit signed integer -/\n| Mint8unsigned   /- 8-bit unsigned integer -/\n| Mint16signed    /- 16-bit signed integer -/\n| Mint16unsigned  /- 16-bit unsigned integer -/\n| Mint32          /- 32-bit integer, or pointer -/\n| Mint64          /- 64-bit integer -/\n| Mfloat32        /- 32-bit single-precision float -/\n| Mfloat64        /- 64-bit double-precision float -/\n| Many32          /- any value that fits in 32 bits -/\n| Many64          /- any value -/\nopen memory_chunk\n\ninstance chunk_eq : decidable_eq memory_chunk := by tactic.mk_dec_eq_instance\n\ndef Mptr : memory_chunk := if archi.ptr64 then Mint64 else Mint32.\n\n/- The type (integer/pointer or float) of a chunk. -/\n\ndef memory_chunk.type : memory_chunk \u2192 typ\n| Mint8signed    := Tint\n| Mint8unsigned  := Tint\n| Mint16signed   := Tint\n| Mint16unsigned := Tint\n| Mint32         := Tint\n| Mint64         := Tlong\n| Mfloat32       := Tsingle\n| Mfloat64       := Tfloat\n| Many32         := Tany32\n| Many64         := Tany64\n\nlemma memory_chunk.Mptr.type : Mptr.type = Tptr :=\nby delta Mptr Tptr; cases archi.ptr64; refl\n\ndef chunk_of_type : typ \u2192 memory_chunk\n| Tint    := Mint32\n| Tfloat  := Mfloat64\n| Tlong   := Mint64\n| Tsingle := Mfloat32\n| Tany32  := Many32\n| Tany64  := Many64\n\nlemma chunk_of_Tptr : chunk_of_type Tptr = Mptr :=\nby delta Mptr Tptr; cases archi.ptr64; refl\n\n/- * Properties of memory chunks -/\n\n/- Memory reads and writes are performed by quantities called memory chunks,\n  encoding the type, size and signedness of the chunk being addressed.\n  The following functions extract the size information from a chunk. -/\n\ndef memory_chunk.size : memory_chunk \u2192 \u2115\n| Mint8signed    := 1\n| Mint8unsigned  := 1\n| Mint16signed   := 2\n| Mint16unsigned := 2\n| Mint32         := 4\n| Mint64         := 8\n| Mfloat32       := 4\n| Mfloat64       := 8\n| Many32         := 4\n| Many64         := 8\n\nlemma memory_chunk.size_pos (chunk) : memory_chunk.size chunk > 0 :=\nby cases chunk; exact dec_trivial\n\nlemma memory_chunk.Mptr.size_eq : Mptr.size = if archi.ptr64 then 8 else 4 :=\nby delta Mptr; cases archi.ptr64; refl\n\n/- Memory reads and writes must respect alignment constraints:\n  the byte offset of the location being addressed should be an exact\n  multiple of the natural alignment for the chunk being addressed.\n  This natural alignment is defined by the following\n  [align_chunk] function.  Some target architectures\n  (e.g. PowerPC and x86) have no alignment constraints, which we could\n  reflect by taking [align_chunk chunk = 1].  However, other architectures\n  have stronger alignment requirements.  The following definition is\n  appropriate for PowerPC, ARM and x86. -/\n\ndef memory_chunk.align : memory_chunk \u2192 \u2115\n| Mint8signed    := 1\n| Mint8unsigned  := 1\n| Mint16signed   := 2\n| Mint16unsigned := 2\n| Mint32         := 4\n| Mint64         := 8\n| Mfloat32       := 4\n| Mfloat64       := 4\n| Many32         := 4\n| Many64         := 4\n\nlemma memory_chunk.align_pos (chunk) : memory_chunk.align chunk > 0 :=\nby cases chunk; exact dec_trivial\n\nlemma memory_chunk.Mptr.align : Mptr.align = if archi.ptr64 then 8 else 4 :=\nby delta Mptr; cases archi.ptr64; refl\n\nlemma align_size_chunk_dvd (chunk : memory_chunk) : chunk.align \u2223 chunk.size := sorry'\n\nlemma align_le_dvd (chunk1 chunk2 : memory_chunk) (h : chunk1.align \u2264 chunk2.align) :\n  chunk1.align \u2223 chunk2.align := sorry'\n\n/- Initialization data for global variables. -/\n\ninductive init_data : Type\n| int8    : int32 \u2192 init_data\n| int16   : int32 \u2192 init_data\n| int32   : int32 \u2192 init_data\n| int64   : int64 \u2192 init_data\n| float32 : float32 \u2192 init_data\n| float64 : float \u2192 init_data\n| space   : \u2115 \u2192 init_data\n| addrof  : ident \u2192 ptrofs \u2192 init_data  /- address of symbol + offset -/\n\nnamespace init_data\n\ndef size : init_data \u2192 \u2115\n| (int8 _)     := 1\n| (int16 _)    := 2\n| (int32 _)    := 4\n| (int64 _)    := 8\n| (float32 _)  := 4\n| (float64 _)  := 8\n| (addrof _ _) := if archi.ptr64 then 8 else 4\n| (space n)    := n\n\ndef align : init_data \u2192 \u2115\n| (int8 _)     := 1\n| (int16 _)    := 2\n| (int32 _)    := 4\n| (int64 _)    := 8\n| (float32 _)  := 4\n| (float64 _)  := 4\n| (addrof _ _) := if archi.ptr64 then 8 else 4\n| (space _)    := 1\n\ndef list_size : list init_data \u2192 \u2115\n| [] := 0\n| (i :: il') := i.size + list_size il'\n\nlemma size_pos (i : init_data) : i.size \u2265 0 := sorry'\n\nlemma list_size_pos (il) : list_size il \u2265 0 := sorry'\n\ndef list_aligned : \u2115 \u2192 list init_data \u2192 Prop\n| p [] := true\n| p (i1 :: il) := i1.align \u2223 p \u2227 list_aligned (p + i1.size) il\n\nend init_data\n\n/- Information attached to global variables. -/\n\nstructure globvar (V : Type) : Type :=\n(info : V)                    /- language-dependent info, e.g. a type -/\n(init : list init_data)       /- initialization data -/\n(readonly : bool)             /- read-only variable? (const) -/\n(volatile : bool)              /- volatile variable? -/\n\n/- Whole programs consist of:\n- a collection of global definitions (name and description);\n- a set of public names (the names that are visible outside\n  this compilation unit);\n- the name of the ``main'' function that serves as entry point in the program.\n\nA global definition is either a global function or a global variable.\nThe type of function descriptions and that of additional information\nfor variables vary among the various intermediate languages and are\ntaken as parameters to the [program] type.  The other parts of whole\nprograms are common to all languages. -/\n\ninductive globdef (F V : Type) : Type\n| Gfun {} (f : F) : globdef\n| Gvar {} (v : globvar V) : globdef\nexport globdef\n\nstructure program (F V : Type) : Type :=\n(defs : list (ident \u00d7 globdef F V))\n(public : list ident)\n(main : ident)\n\ndef program.defs_names {F V : Type} (p : program F V) : list ident :=\np.defs.map prod.fst\n\n/- The \"definition map\" of a program maps names of globals to their definitions.\n  If several definitions have the same name, the one appearing last in [p.defs] wins. -/\n\nsection defmap\n\nvariables {F V : Type}\nvariable p : program F V\n\ndef prog_defmap : PTree (globdef F V) :=\nPTree.of_list p.defs\n\nlemma in_prog_defmap {id : ident} {g} : (prog_defmap p ^! id) = some g \u2192\n  (id, g) \u2208 p.defs := sorry'\n\nlemma prog_defmap_dom {id : ident} : id \u2208 p.defs_names \u2192\n  \u2203 g, (prog_defmap p^!id) = some g := sorry'\n\nlemma prog_defmap_unique (defs1 id g defs2) :\n  p.defs = defs1 ++ (id, g) :: defs2 \u2192\n  id \u2209 defs2.map prod.fst \u2192\n  (prog_defmap p^!id) = some g := sorry'\n\nlemma prog_defmap_nodup {id : ident} {g} :\n  p.defs_names.nodup \u2192\n  (id, g) \u2208 p.defs \u2192\n  (prog_defmap p ^! id) = some g := sorry'\n\nend defmap\n\n/- * Generic transformations over programs -/\n\n/- We now define a general iterator over programs that applies a given\n  code transformation function to all function descriptions and leaves\n  the other parts of the program unchanged. -/\n\nsection transf_program\n\nparameters {A B V : Type} (transf : A \u2192 B)\n\ndef transform_program_globdef : ident \u00d7 globdef A V \u2192 ident \u00d7 globdef B V\n| (id, Gfun f) := (id, Gfun (transf f))\n| (id, Gvar v) := (id, Gvar v)\n\ndef transform_program : program A V \u2192 program B V\n| \u27e8defs, pub, main\u27e9 := \u27e8defs.map transform_program_globdef, pub, main\u27e9\n\nend transf_program\n\n/- The following is a more general presentation of [transform_program]:\n- Global variable information can be transformed, in addition to function\n  definitions.\n- The transformation functions can fail and return an error message.\n- The transformation for function definitions receives a global context\n  (derived from the compilation unit being transformed) as additiona\n  argument.\n- The transformation functions receive the name of the global as\n  additional argument. -/\n\nsection transf_program_gen\n\nparameters {A B V W : Type}\nparameter transf_fun : ident \u2192 A \u2192 res B.\nparameter transf_var : ident \u2192 V \u2192 res W.\n\ndef transf_globvar (i : ident) : globvar V \u2192 res (globvar W)\n| \u27e8info, init, ro, vo\u27e9 := do info' \u2190 transf_var i info, OK \u27e8info', init, ro, vo\u27e9\n\ndef transf_globdefs : list (ident \u00d7 globdef A V) \u2192 res (list (ident \u00d7 globdef B W))\n| [] := OK []\n| ((id, Gfun f) :: l') :=\n  match transf_fun id f with\n  | error msg := error (MSG \"In function \" :: CTX id :: MSG \": \" :: msg)\n  | OK tf :=\n      do tl' \u2190 transf_globdefs l', OK ((id, Gfun tf) :: tl')\n  end\n| ((id, Gvar v) :: l') :=\n  match transf_globvar id v with\n  | error msg := error (MSG \"In variable \" :: CTX id :: MSG \": \" :: msg)\n  | OK tv :=\n      do tl' \u2190 transf_globdefs l', OK ((id, Gvar tv) :: tl')\n  end\n\ndef transform_partial_program2 : program A V \u2192 res (program B W)\n| \u27e8defs, pub, main\u27e9 := do gl' \u2190 transf_globdefs defs, OK \u27e8gl', pub, main\u27e9\n\nend transf_program_gen\n\n/- The following is a special case of [transform_partial_program2],\n  where only function definitions are transformed, but not variable definitions. -/\n\ndef transform_partial_program {A B V} (transf_fun : A \u2192 res B) : program A V \u2192 res (program B V) :=\ntransform_partial_program2 (\u03bb i, transf_fun) (\u03bb i, OK)\n\nlemma transform_program_partial_program {A B V} (transf_fun : A \u2192 B) (p : program A V) :\n  transform_partial_program (\u03bb f, OK (transf_fun f)) p = OK (transform_program transf_fun p) := sorry'\n\n/- * External functions -/\n\n/- For most languages, the functions composing the program are either\n  internal functions, defined within the language, or external functions,\n  defined outside.  External functions include system calls but also\n  compiler built-in functions.  We define a type for external functions\n  and associated operations. -/\n\ninductive external_function : Type\n| EF_external (name : string) (sg : signature)\n   /- A system call or library function.  Produces an event\n       in the trace. -/\n| EF_builtin (name : string) (sg : signature)\n   /- A compiler built-in function.  Behaves like an external, but\n       can be inlined by the compiler. -/\n| EF_runtime (name : string) (sg : signature)\n   /- A function from the run-time library.  Behaves like an\n       external, but must not be redefined. -/\n| EF_vload (chunk : memory_chunk)\n   /- A volatile read operation.  If the adress given as first argument\n       points within a volatile global variable, generate an\n       event and return the value found in this event.  Otherwise,\n       produce no event and behave like a regular memory load. -/\n| EF_vstore (chunk : memory_chunk)\n   /- A volatile store operation.   If the adress given as first argument\n       points within a volatile global variable, generate an event.\n       Otherwise, produce no event and behave like a regular memory store. -/\n| EF_malloc\n   /- Dynamic memory allocation.  Takes the requested size in bytes\n       as argument; returns a pointer to a fresh block of the given size.\n       Produces no observable event. -/\n| EF_free\n   /- Dynamic memory deallocation.  Takes a pointer to a block\n       allocated by an [EF_malloc] external call and frees the\n       corresponding block.\n       Produces no observable event. -/\n| EF_memcpy (sz al : \u2115)\n   /- Block copy, of [sz] bytes, between addresses that are [al]-aligned. -/\n| EF_annot (text : string) (targs : list typ)\n   /- A programmer-supplied annotation.  Takes zero, one or several arguments,\n       produces an event carrying the text and the values of these arguments,\n       and returns no value. -/\n| EF_annot_val (text : string) (targ : typ)\n   /- Another form of annotation that takes one argument, produces\n       an event carrying the text and the value of this argument,\n       and returns the value of the argument. -/\n| EF_inline_asm (text : string) (sg : signature) (clobbers : list string)\n   /- Inline [asm] statements.  Semantically, treated like an\n       annotation with no parameters ([EF_annot text nil]).  To be\n       used with caution, as it can invalidate the semantic\n       preservation theorem.  Generated only if [-finline-asm] is\n       given. -/\n| EF_debug (kind : pos_num) (text : ident) (targs : list typ)\n   /- Transport debugging information from the front-end to the generated\n       assembly.  Takes zero, one or several arguments like [EF_annot].\n       Unlike [EF_annot], produces no observable event. -/\nexport external_function\n\n/- The type signature of an external function. -/\n\ndef ef_sig : external_function \u2192 signature\n| (EF_external name sg)        := sg\n| (EF_builtin name sg)         := sg\n| (EF_runtime name sg)         := sg\n| (EF_vload chunk)             := \u27e8[Tptr], some chunk.type, cc_default\u27e9\n| (EF_vstore chunk)            := \u27e8[Tptr, chunk.type], none, cc_default\u27e9\n| (EF_malloc)                  := \u27e8[Tptr], some Tptr, cc_default\u27e9\n| (EF_free)                    := \u27e8[Tptr], none, cc_default\u27e9\n| (EF_memcpy sz al)            := \u27e8[Tptr, Tptr], none, cc_default\u27e9\n| (EF_annot text targs)        := \u27e8targs, none, cc_default\u27e9\n| (EF_annot_val text targ)     := \u27e8[Tptr], some targ, cc_default\u27e9\n| (EF_inline_asm text sg clob) := sg\n| (EF_debug kind text targs)   := \u27e8targs, none, cc_default\u27e9\n\n/- Whether an external function should be inlined by the compiler. -/\n\ndef ef_inline : external_function \u2192 bool\n| (EF_external name sg)        := ff\n| (EF_builtin name sg)         := tt\n| (EF_runtime name sg)         := ff\n| (EF_vload chunk)             := tt\n| (EF_vstore chunk)            := tt\n| (EF_malloc)                  := ff\n| (EF_free)                    := ff\n| (EF_memcpy sz al)            := tt\n| (EF_annot text targs)        := tt\n| (EF_annot_val text targ)     := tt\n| (EF_inline_asm text sg clob) := tt\n| (EF_debug kind text targs)   := tt\n\n/- Whether an external function must reload its arguments. -/\n\ndef ef_reloads : external_function \u2192 bool\n| (EF_annot text targs)      := ff\n| (EF_debug kind text targs) := ff\n| _                          := tt\n\n/- Equality between external functions.  Used in module [Allocation]. -/\n\ninstance external_function_eq : decidable_eq external_function := by tactic.mk_dec_eq_instance\n\n/- Function definitions are the union of internal and external functions. -/\n\ninductive fundef (F : Type) : Type\n| Internal {} : F \u2192 fundef\n| External {} : external_function \u2192 fundef\nopen fundef\n\nsection transf_fundef\n\nparameters {A B : Type} (transf : A \u2192 B)\n\ndef transf_fundef : fundef A \u2192 fundef B\n| (Internal f)  := Internal (transf f)\n| (External ef) := External ef\n\nend transf_fundef\n\nsection transf_partial_fundef\n\nparameters {A B : Type} (transf_partial : A \u2192 res B)\n\ndef transf_partial_fundef : fundef A \u2192 res (fundef B)\n| (Internal f)  := do f' \u2190 transf_partial f, OK (Internal f')\n| (External ef) := OK (External ef)\n\nend transf_partial_fundef\n\n/- * Register pairs -/\n\n/- In some intermediate languages (LTL, Mach), 64-bit integers can be\n  split into two 32-bit halves and held in a pair of registers.  \n  Syntactically, this is captured by the type [rpair] below. -/\n\ninductive rpair (A : Type) : Type\n| One (r : A) : rpair\n| Twolong (rhi rlo : A) : rpair\nopen rpair\n\ndef typ_rpair {A} (typ_of : A \u2192 typ) : rpair A \u2192 typ\n| (One r) := typ_of r\n| (Twolong rhi rlo) := Tlong\n\ndef map_rpair {A B} (f : A \u2192 B) : rpair A \u2192 rpair B\n| (One r) := One (f r)\n| (Twolong rhi rlo) := Twolong (f rhi) (f rlo)\n\ndef regs_of_rpair {A} : rpair A \u2192 list A\n| (One r) := [r]\n| (Twolong rhi rlo) := [rhi, rlo]\n\ndef regs_of_rpairs {A} : list (rpair A) \u2192 list A\n| [] := []\n| (p :: l) := regs_of_rpair p ++ regs_of_rpairs l\n\nlemma in_regs_of_rpair {A} (x : A) (p) (hm : x \u2208 regs_of_rpair p) (l : list (rpair A)) (hp : p \u2208 l) :\n  x \u2208 regs_of_rpairs l := sorry'\n\nlemma in_regs_of_rpairs_inv {A} (x : A) (l : list (rpair A)) (hm : x \u2208 regs_of_rpairs l) :\n  \u2203 p, p \u2208 l \u2227 x \u2208 regs_of_rpair p := sorry'\n\ndef forall_rpair {A} (P : A \u2192 Prop) : rpair A \u2192 Prop\n| (One r) := P r\n| (Twolong rhi rlo) := P rhi \u2227 P rlo\n\n/- * Arguments and results to builtin functions -/\n\ninductive builtin_arg (A : Type) : Type\n| BA            {} (x : A)                                            : builtin_arg\n| BA_int        {} (n : int32)                                        : builtin_arg\n| BA_long       {} (n : int64)                                        : builtin_arg\n| BA_float      {} (f : float)                                        : builtin_arg\n| BA_single     {} (f : float32)                                      : builtin_arg\n| BA_loadstack  {} (chunk : memory_chunk) (ofs : ptrofs)              : builtin_arg\n| BA_addrstack  {} (ofs : ptrofs)                                     : builtin_arg\n| BA_loadglobal {} (chunk : memory_chunk) (id : ident) (ofs : ptrofs) : builtin_arg\n| BA_addrglobal {} (id : ident) (ofs : ptrofs)                        : builtin_arg\n| BA_splitlong  {} (hi lo : builtin_arg)                              : builtin_arg\nexport builtin_arg\n\ninductive builtin_res (A : Type) : Type\n| BR           {} (x : A)               : builtin_res\n| BR_none      {}                       : builtin_res\n| BR_splitlong {} (hi lo : builtin_res) : builtin_res\nopen builtin_res\n\ndef globals_of_builtin_arg {A : Type} : builtin_arg A \u2192 list ident\n| (BA_loadglobal chunk id ofs) := [id]\n| (BA_addrglobal id ofs)       := [id]\n| (BA_splitlong hi lo)         := globals_of_builtin_arg hi ++ globals_of_builtin_arg lo\n| _ := []\n\ndef globals_of_builtin_args {A} (al : list (builtin_arg A)) : list ident :=\nal.foldr (\u03bb a l, globals_of_builtin_arg a ++ l) []\n\ndef params_of_builtin_arg {A} : builtin_arg A \u2192 list A\n| (BA x) := [x]\n| (BA_splitlong hi lo) := params_of_builtin_arg hi ++ params_of_builtin_arg lo\n| _ := []\n\ndef params_of_builtin_args {A} (al : list (builtin_arg A)) : list A :=\nal.foldr (\u03bb a l, params_of_builtin_arg a ++ l) []\n\ndef params_of_builtin_res {A} : builtin_res A \u2192 list A\n| (BR x)               := [x]\n| BR_none              := []\n| (BR_splitlong hi lo) := params_of_builtin_res hi ++ params_of_builtin_res lo\n\ndef map_builtin_arg {A B} (f : A \u2192 B) : builtin_arg A \u2192 builtin_arg B\n| (BA x)                       := BA (f x)\n| (BA_int n)                   := BA_int n\n| (BA_long n)                  := BA_long n\n| (BA_float n)                 := BA_float n\n| (BA_single n)                := BA_single n\n| (BA_loadstack chunk ofs)     := BA_loadstack chunk ofs\n| (BA_addrstack ofs)           := BA_addrstack ofs\n| (BA_loadglobal chunk id ofs) := BA_loadglobal chunk id ofs\n| (BA_addrglobal id ofs)       := BA_addrglobal id ofs\n| (BA_splitlong hi lo)         := BA_splitlong (map_builtin_arg hi) (map_builtin_arg lo)\n\ndef map_builtin_res {A B} (f : A \u2192 B) : builtin_res A \u2192 builtin_res B\n| (BR x)               := BR (f x)\n| BR_none              := BR_none\n| (BR_splitlong hi lo) := BR_splitlong (map_builtin_res hi) (map_builtin_res lo)\n\n/- Which kinds of builtin arguments are supported by which external function. -/\n\ninductive builtin_arg_constraint : Type\n| OK_default\n| OK_const\n| OK_addrstack\n| OK_addrglobal\n| OK_addrany\n| OK_all\nopen builtin_arg_constraint\n\ndef builtin_arg_ok {A} : builtin_arg A \u2192 builtin_arg_constraint \u2192 bool\n| (BA _)                       _             := tt\n| (BA_splitlong (BA _) (BA _)) _             := tt\n| (BA_int _)                   OK_const      := tt\n| (BA_long _)                  OK_const      := tt\n| (BA_float _)                 OK_const      := tt\n| (BA_single _)                OK_const      := tt\n| (BA_addrstack _)             OK_addrstack  := tt\n| (BA_addrstack _)             OK_addrany    := tt\n| (BA_addrglobal _ _)          OK_addrglobal := tt\n| (BA_addrglobal _ _)          OK_addrany    := tt\n| _                            OK_all        := tt\n| _                            _             := ff\n\nend ast", "meta": {"author": "digama0", "repo": "kremlin", "sha": "d4665929ce9012e93a0b05fc7063b96256bab86f", "save_path": "github-repos/lean/digama0-kremlin", "path": "github-repos/lean/digama0-kremlin/kremlin-d4665929ce9012e93a0b05fc7063b96256bab86f/ast.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.3626736498401264}}
{"text": "\nimport unitb.models.nondet\nimport unitb.refinement.basic\n\nnamespace nondet\n\nopen temporal\nopen predicate\nopen unitb\n\nuniverse variable u\n\nsection defs\n\nvariables {\u03b1 \u03b2 : Type}\n\nstructure evt_ref (lbl : Type) (mc : program \u03b1) (ea : event \u03b1) (ecs : lbl \u2192 event \u03b1) : Type :=\n  (witness : lbl \u2192 \u03b1 \u2192 Prop)\n  (witness_fis : \u2983 \u2203\u2203 e, witness e \u2984)\n  (sim : \u2200 ec, \u27e6 (ecs ec).step_of \u27e7 \u27f9 \u27e6 ea.step_of \u27e7)\n  (delay : \u2200 ec, witness ec \u22c0 ea.coarse_sch \u22c0 ea.fine_sch \u21a6 witness ec \u22c0 (ecs ec).coarse_sch in mc)\n  (stable : \u2200 ec, unless_except mc (witness ec \u22c0 (ecs ec).coarse_sch) (-ea.coarse_sch) { e | \u2203 l, ecs l = e })\n  (resched : \u2200 ec, ea.coarse_sch \u22c0 ea.fine_sch \u22c0 witness ec \u21a6 (ecs ec).fine_sch in mc)\n\nstructure refined (ma mc : program \u03b1) : Type :=\n  (sim_init : mc^.first \u27f9 ma^.first)\n  (ref : option mc.lbl \u2192 option ma.lbl \u2192 Prop)\n  (evt_sim : \u2200 ec, \u27e6 mc.step_of ec \u27e7 \u27f9 \u2203\u2203 ea : { ea // ref ec ea }, \u27e6 ma.step_of ea.val \u27e7)\n  (events : \u2200 ae, evt_ref { ec // ref ec ae } mc (ma.event ae) (\u03bb ec, mc.event ec.val) )\n\nlemma refined.sim {ma mc : program \u03b1}\n  (R : refined ma mc)\n: \u27e6 is_step mc \u27e7 \u27f9 \u27e6 is_step ma \u27e7 :=\nbegin\n  simp [is_step_exists_event'],\n  intro \u03c4,\n  intros H,\n  cases H with ce H,\n  apply exists_imp_exists' subtype.val _ (R.evt_sim ce \u03c4 H),\n  intro, apply id,\nend\n\nend defs\n\nsection soundness\n\nparameters {\u03b1 \u03b2 : Type}\n\nparameter (ma : program \u03b1)\nparameter (mc : program \u03b1)\n\nopen temporal\n\nparameter R : refined ma mc\nparameter \u03c4 : stream \u03b1\nparameter M\u2081 : system_sem.ex mc \u03c4\n\nsection schedules\n\nparameter e : option ma.lbl\n@[reducible]\ndef imp_lbl := { ec : option mc.lbl // R.ref ec e }\n\ndef AC := (program.event ma e).coarse_sch\ndef AF := (program.event ma e).fine_sch\ndef W (e' : imp_lbl) := (R.events e).witness e'\ndef CC (e' : option mc.lbl) := mc.coarse_sch_of e'\ndef CF (e' : option mc.lbl) := mc.fine_sch_of e'\n\nparameter abs_coarse : (\u25c7\u25fb(\u2022AC \u22c0 -\u27e6 ma.step_of e \u27e7)) \u03c4\n\nparameter abs_fine : (\u25fb\u25c7\u2022AF) \u03c4\n\ninclude M\u2081\ninclude abs_coarse\ninclude abs_fine\n\nlemma abs_coarse_and_fine\n: (\u25fb\u25c7(\u2022AC \u22c0 \u2022AF)) \u03c4 :=\nbegin\n  apply coincidence,\n  { apply stable_entails_stable _ _ abs_coarse,\n    apply \u03bb _, and.left },\n  { apply abs_fine },\nend\n\nlemma conc_coarse : \u2203 e', (\u25c7\u25fb(\u2022 W e' \u22c0 \u2022 CC e'.val) ) \u03c4 :=\nbegin\n  have H : ((\u2203\u2203 e', \u25c7\u25fb(\u2022 W ma mc R e e' \u22c0 \u2022 CC mc e'.val))\n                   \u22c1 \u25fb\u25c7((-\u2022AC ma e) \u22c1 \u2203\u2203 e' : imp_lbl ma mc R e, \u27e6 mc.step_of e'.val \u27e7)) \u03c4,\n  { rw exists_action,\n    apply p_or_p_imp_p_or_right _ (unless_sem_exists' M\u2081.safety (R.events e).stable _),\n    { apply inf_often_entails_inf_often,\n      apply p_or_p_imp_p_or_right' _,\n      apply action_entails_action,\n      intros \u03c3 \u03c3',\n      simp [imp_lbl,mem_set_of],\n      intros ec H x H' H\u2081,\n      existsi x,\n      cases H with H\u2080 H,\n      cases H with H\u2082 STEP,\n      unfold program.step_of,\n      simp [H\u2082,event.step_of,STEP,H\u2080,H\u2081,H'], },\n    have H' := leads_to.gen_disj' (R.events e).delay,\n    apply inf_often_of_leads_to (system_sem.leads_to_sem H' _ M\u2081),\n    simp,\n    have H' := ew_eq_true (R.events e).witness_fis,\n    rw [\u2190 p_and_over_p_exists_right\n       ,\u2190 p_and_over_p_exists_right],\n    simp [H'],\n    apply abs_coarse_and_fine ma mc _ M\u2081 _ abs_coarse abs_fine, },\n  simp at H,\n  cases H with H H,\n  { exfalso,\n    revert abs_coarse,\n    change \u00ac _,\n    rw [p_not_eq_not,not_eventually,not_henceforth,p_not_p_and,p_not_p_not_iff_self],\n    apply inf_often_entails_inf_often _ _ H,\n    apply p_or_p_imp_p_or_right',\n    rw p_exists_entails_eq_p_forall_entails,\n    intros ec,\n    apply (R.events e).sim _ , },\n  { simp [H], },\nend\n\nlemma conc_fine : \u2200 e',\n         (\u25c7\u25fb\u2022W e') \u03c4 \u2192\n         (\u25fb\u25c7\u2022CF e'.val) \u03c4 :=\nbegin\n  intros e' H,\n  have H' := system_sem.leads_to_sem ((R.events e).resched e') _ M\u2081,\n  apply inf_often_of_leads_to H',\n  rw p_and_comm,\n  apply coincidence H,\n  apply abs_coarse_and_fine _ _ _ M\u2081 _ abs_coarse abs_fine,\nend\n\nend schedules\n\ninclude M\u2081\ninclude R\n\ntheorem soundness : system_sem.ex ma \u03c4 :=\nbegin\n  apply nondet.program.ex.mk,\n  { apply R.sim_init,\n    apply M\u2081.init },\n  { intro i,\n    apply R.sim,\n    apply M\u2081.safety },\n  { intros e COARSE\u2080 FINE\u2080,\n    apply assume_neg _, intro ACT,\n    have COARSE\u2081 :  (\u25c7\u25fb(\u2022AC ma e \u22c0 -\u27e6program.step_of ma e\u27e7)) \u03c4,\n    { rw [p_not_eq_not,not_henceforth,not_eventually] at ACT,\n      apply stable_and_of_stable_of_stable COARSE\u2080 ACT },\n    clear COARSE\u2080 ACT,\n    cases conc_coarse ma mc R \u03c4 M\u2081 _ COARSE\u2081 FINE\u2080 with e' C_COARSE',\n    have C_COARSE : (\u25c7\u25fb\u2022CC mc e'.val) \u03c4,\n    { apply stable_entails_stable _ _ C_COARSE',\n      intro, apply and.right },\n    have WIT : (\u25c7\u25fb\u2022W ma mc R e e') \u03c4,\n    { apply stable_entails_stable _ _ C_COARSE',\n      intro, apply and.left },\n    have C_FINE := conc_fine ma mc R \u03c4 M\u2081 e COARSE\u2081 FINE\u2080 e' WIT,\n    apply inf_often_entails_inf_often _ _ (M\u2081.liveness _ C_COARSE C_FINE),\n    have H := (R.events e).sim e',\n    apply H, },\nend\n\nend soundness\n\nend nondet\n", "meta": {"author": "unitb", "repo": "unitb-semantics", "sha": "07607ddb2ced4044af121f1fd989e058e19c3c9c", "save_path": "github-repos/lean/unitb-unitb-semantics", "path": "github-repos/lean/unitb-unitb-semantics/unitb-semantics-07607ddb2ced4044af121f1fd989e058e19c3c9c/src/unitb/refinement/split_merge.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7310585669110203, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.3626736440264105}}
{"text": "import Std.Tactic.GuardExpr\nimport Mathlib.Tactic.PermuteGoals\n\nexample (p q r : Prop) : p \u2192 q \u2192 r \u2192 p \u2227 q \u2227 r := by\n  intros\n  constructor\n  on_goal 2 =>\n    guard_target = q \u2227 r\n    constructor\n    assumption\n    -- Note that we have not closed all the subgoals here.\n  guard_target = p\n  assumption\n  guard_target = r\n  assumption\n\nexample (p q r : Prop) : p \u2192 q \u2192 r \u2192 p \u2227 q \u2227 r := by\n  intros a b c\n  constructor\n  fail_if_success on_goal -3 => unreachable!\n  fail_if_success on_goal -1 => exact a\n  fail_if_success on_goal 0 => unreachable!\n  fail_if_success on_goal 2 => exact a\n  fail_if_success on_goal 3 => unreachable!\n  on_goal 1 => exact a\n  constructor\n  swap\n  exact c\n  exact b\n\nexample (p q : Prop) : p \u2192 q \u2192 p \u2227 q := by\n  intros a b\n  constructor\n  fail_if_success pick_goal -3\n  fail_if_success pick_goal 0\n  fail_if_success pick_goal 3\n  pick_goal -1\n  exact b\n  exact a\n\nexample (p : Prop) : p \u2192 p := by\n  intros\n  fail_if_success swap -- can't swap with a single goal\n  assumption\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/PermuteGoals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.640635861701035, "lm_q1q2_score": 0.3626117720341121}}
{"text": "def x := 1\n\n#check x\n\nvariable {\u03b1 : Type}\n\ndef f (a : \u03b1) : \u03b1 :=\na\n\ndef tst (xs : List Nat) : Nat :=\nxs.foldl (init := 10) (\u00b7 + \u00b7)\n\n#check tst [1, 2, 3]\n\n#check fun x y : Nat => x + y\n\n#check tst\n\n#check (fun stx => if True then let e := stx; Pure.pure e else Pure.pure stx : Nat \u2192 Id Nat)\n\n#check let x : Nat := 1; x\n\ndef foo (a : Nat) (b : Nat := 10) (c : Bool := Bool.true) : Nat :=\na + b\n\nset_option pp.all true\n\n#check foo 1\n\n#check foo 3 (c := false)\n\ndef Nat.boo (a : Nat) :=\nsucc a -- succ here is resolved as `Nat.succ`.\n\n#check Nat.boo\n\n#check true\n\n-- apply is still a valid identifier name\ndef apply := \"hello\"\n\n#check apply\n\ntheorem simple1 (x y : Nat) (h : x = y) : x = y :=\nby {\n  assumption\n}\n\ntheorem simple2 (x y : Nat) : x = y \u2192 x = y :=\nby {\n  intro h;\n  assumption\n}\n\nsyntax \"intro2\" : tactic\n\nmacro_rules\n| `(tactic| intro2) => `(tactic| intro; intro )\n\ntheorem simple3 (x y : Nat) : x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro2;\n  assumption\n}\n\nmacro \"intro3\" : tactic => `(intro; intro; intro)\nmacro \"check2\" x:term : command => `(#check $x #check $x)\nmacro \"foo\" x:term \",\" y:term : term => `($x + $y + $x)\n\nset_option pp.all false\n\ncheck2 0+1\ncheck2 foo 0,1\n\ntheorem simple4 (x y : Nat) : y = y \u2192 x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro3;\n  assumption\n}\n\ntheorem simple5 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  exact Eq.trans h3 h1\n}\n\ntheorem simple6 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine Eq.trans ?_ h1;\n  assumption\n}\n\ntheorem simple7 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine' Eq.trans ?pre ?post;\n  exact y;\n  { exact h3 }\n  { exact h1 }\n}\n\ntheorem simple8 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintro h1; intro _; intro h3\nrefine' Eq.trans ?pre ?post\ncase post => exact h1\ncase pre => exact h3\n\ntheorem simple9 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntrace_state\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n      assumption\n    | exact y\n      exact h3\n      assumption\n\ntheorem simple9b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntrace_state\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n    | exact y; exact h3\n  assumption\n\ntheorem simple9c (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  solve\n    | exact h1\n    | refine' Eq.trans ?pre ?post; exact y; exact h3; assumption\n    | exact h3\n\ntheorem simple9d (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  refine' Eq.trans ?pre ?post\n  solve\n    | exact h1\n    | exact y\n    | exact h3\n  solve\n    | exact h1\n    | exact h3\n  solve\n    | exact h1\n    | assumption\n\n\nnamespace Foo\n  def Prod.mk := 1\n  #check (\u27e82, 3\u27e9 : Prod _ _)\nend Foo\n\ntheorem simple10 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  skip;\n  apply Eq.trans;\n  exact h3;\n  assumption\n}\n\ntheorem simple11 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  trace_state;\n  exact h3;\n  assumption\n}\n\ntheorem simple12 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  try exact h1; -- `exact h1` fails\n  trace_state;\n  try exact h3;\n  trace_state;\n  try exact h1;\n}\n\ntheorem simple13 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 h2 h3\ntrace_state\napply @Eq.trans\ncase b => exact y\ntrace_state\nrepeat assumption\n\ntheorem simple13b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by {\nintros h1 h2 h3;\ntrace_state;\napply @Eq.trans;\ncase b => exact y;\ntrace_state;\nrepeat assumption\n}\n\ntheorem simple14 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros\napply @Eq.trans\ncase b => exact y\nrepeat assumption\n\ntheorem simple15 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  revert y;\n  intros y h1 h3;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\ntheorem simple16 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  try clear x; -- should fail\n  clear h2;\n  trace_state;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\nmacro \"blabla\" : tactic => `(assumption)\n\n-- Tactic head symbols do not become reserved words\ndef blabla := 100\n\n#check blabla\n\ntheorem simple17 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple18 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple19 (x y : Nat) (h\u2081 : x = 0) (h\u2082 : x = y) : y = 0 :=\nby subst x; subst y; exact rfl\n\ntheorem tstprec1 (x y z : Nat) : x + y * z = x + (y * z) :=\nrfl\n\ntheorem tstprec2 (x y z : Nat) : y * z + x = (y * z) + x :=\nrfl\n\nset_option pp.all true\n\n#check fun {\u03b1} (a : \u03b1) => a\n#check @(fun \u03b1 (a : \u03b1) => a)\n\n#check\n  let myid := fun {\u03b1} (a : \u03b1) => a;\n  myid [myid 1]\n\n-- In the following example, we need `@` otherwise we will try to insert mvars for \u03b1 and [Add \u03b1],\n-- and will fail to generate instance for [Add \u03b1]\n#check @(fun \u03b1 (s : Add \u03b1) (a : \u03b1) => a + a)\n\ndef g1 {\u03b1} (a\u2081 a\u2082 : \u03b1) {\u03b2} (b : \u03b2) : \u03b1 \u00d7 \u03b1 \u00d7 \u03b2 :=\n(a\u2081, a\u2082, b)\n\ndef id1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => x\n\ndef listId : List ({\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1) :=\n(fun x => x) :: []\n\ndef id2 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 (x : \u03b1) => id1 x)\n\ndef id3 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 x => id1 x)\n\ndef id4 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => id1 x\n\ndef id5 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id1 x\n\ndef id6 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun {\u03b1} x => id1 x)\n\ndef id7 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => @id \u03b1 x\n\ndef id8 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id (@id \u03b1 x)\n\ndef altTst1 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8StateT.failure, StateT.orElse\u27e9\n\ndef altTst2 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8@(fun \u03b1 => StateT.failure), @(fun \u03b1 => StateT.orElse)\u27e9\n\ndef altTst3 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8fun {\u03b1} => StateT.failure, fun {\u03b1} => StateT.orElse\u27e9\n\n#check_failure 1 + true\n\n/-\nuniverse u v\n\n/-\n  MonadFunctorT.{u ?M_1 v} (\u03bb (\u03b2 : Type u), m \u03b1) (\u03bb (\u03b2 : Type u), m' \u03b1) n n'\n-/\nset_option pp.raw.maxDepth 100\nset_option trace.Elab true\n\n\ndef adapt {m m' \u03c3 \u03c3'} {n n' : Type \u2192 Type} [MonadFunctor m m' n n'] [MonadStateAdapter \u03c3 \u03c3' m m'] : MonadStateAdapter \u03c3 \u03c3' n n' :=\n\u27e8fun split join => monadMap (adaptState split join : m \u03b1 \u2192 m' \u03b1)\u27e9\n\n-/\n\nsyntax \"fn\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(fn $xs* => $b) => `(fun $xs* => $b)\n\nset_option pp.all false\n\n#check fn x => x+1\n\n#check fn \u03b1 (a : \u03b1) => a\n\ndef tst1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fn \u03b1 a => a)\n\n#check @tst1\n\nsyntax ident \"==>\" term : term\n\nsyntax \"{\" ident \"}\" \"==>\" term : term\n\nmacro_rules\n| `($x:ident ==> $b)   => `(fn $x => $b)\n| `({$x:ident} ==> $b) => `(fun {$x:ident} => $b)\n\n#check x ==> x+1\n\ndef tst2a : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(\u03b1 ==> a ==> a)\n\ndef tst2b : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n{\u03b1} ==> a ==> a\n\n#check @tst2a\n#check @tst2b\n\ndef tst3a : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n@(\u03b1 ==> @(\u03b2 ==> a ==> b ==> (a, b)))\n\ndef tst3b : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n{\u03b1} ==> {\u03b2} ==> a ==> b ==> (a, b)\n\nsyntax \"function\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(function $xs* => $b) => `(@(fun $xs* => $b))\n\ndef tst4 : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\nfunction \u03b1 \u03b2 a b => (a, b)\n\ntheorem simple20 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby intros h1 h2 h3;\n   try clear x; -- should fail\n   clear h2;\n   trace_state;\n   apply Eq.trans;\n   exact h3;\n   exact h1\n\ntheorem simple21 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by { apply Eq.symm; assumption };\n  Eq.trans this (by assumption)\n\ntheorem simple22 (x y z : Nat) : y = z \u2192 y = x \u2192 id (x = z + 0) :=\nfun h1 h2 => show x = z + 0 by\n  apply Eq.trans\n  exact h2.symm\n  assumption\n  skip\n\ntheorem simple23 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by apply Eq.symm; assumption\n  Eq.trans this (by assumption)\n\ntheorem simple24 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have h : x = y := by apply Eq.symm; assumption\n  Eq.trans h (by assumption)\n\ndef f1 (x : Nat) : Nat :=\n  let double x := x + x\n  let rec loop x :=\n    match x with\n    | 0   => 0\n    | x+1 => loop x + double x\n  loop x\n\n#eval f1 5\n\ndef f2 (x : Nat) : String :=\n  let bad x : String := toString x\n  bad x\n\ndef f3 x y :=\n  x + y + 1\n\ntheorem f3eq x y : f3 x y = x + y + 1 :=\n  rfl\n\ndef f4 (x y : Nat) : String :=\n  if x > y + 1 then \"hello\" else \"world\"\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/run/newfrontend1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.362611769763311}}
{"text": "import .aff1K\nimport category_theory.groupoid\nimport linear_algebra.affine_space.affine_equiv\nimport linear_algebra.matrix\n\n/-\nFramed points, vectors, frames\n-/\n\nopen_locale affine\nuniverses u \n\nsection explicitK\n\nvariables \n(K : Type u) [field K] [inhabited K] \n\n/-\nIs this where we root distinctions between affine spaces for different dimensionss?\n-/\ninductive fm : nat \u2192 Type u\n| base : \u03a0 n, fm n\n| deriv : \u03a0 n, (prod (pt K) (vec K)) \u2192 fm n \u2192 fm n  -- TODO: curry all of these args\n\n/-\ninductive fm : nat \u2192 Type u\n| base : \u2200 (n : nat), fm n\n| deriv : \u2200 (n : nat), (prod (pt K) (vec K)) \u2192 fm n \u2192 fm n\n-/\n\ndef mk_fm  {n : nat} (p : pt K) (v : vec K) (f : fm K n): fm K n := fm.deriv n (p, v) f \n\nstructure spc {n : nat} (f : fm K n) : Type u       -- interesting specimen, here, btw\n\ndef mk_space {n : nat} (f : fm K n) :=\n  @spc.mk K _ _ n f \n\nend explicitK\n\nsection implicitK\n\nvariables \n{K : Type u} [field K] [inhabited K] \n{n : nat} {f : fm K n} (s : spc K f)\n\n/-\nAugment pt and vec types with spaces and frames and\nthen make operations apply only for objects in same\nspace (and thus frame).\n-/\n@[ext]\nstructure point {f : fm K n} (s : @spc K _ _ n f ) extends pt K\n@[simp]\ndef mk_point' (p : pt K) : point s := point.mk p  \n@[simp]\ndef mk_point (k : K) : point s := point.mk (mk_pt K k)  \n\ndef p := mk_point s (3:K)\n\n@[ext]\nstructure vectr {f : fm K n} (s : spc K f ) extends vec K\n@[simp]\ndef mk_vectr' (v : vec K) : vectr s := vectr.mk v\n@[simp]\ndef mk_vectr (k : K) : vectr s := vectr.mk (mk_vec K k)  \n\n-- note that we don't extend fm\ndef mk_frame {parent : fm K n} {s : spc K parent}  (p : point s) (v : vectr s) :=\nfm.deriv n (p.to_pt, v.to_vec) parent   -- TODO: make sure v \u2260 0 (erasing tyoe info)\n                                        -- TODO: snd arg is really a basis for the vs\n\n\n/-\n    *************************************\n    Instantiate module K (vector K)\n    *************************************\n-/\n\nvariables v1 v2 : @vectr K _ _ n f s\n#check v1.to_vec\n#check v2.to_vec + v1.to_vec\n\n@[simp]\ndef add_vectr_vectr (v1 v2 : vectr s) : vectr s :=  mk_vectr' s (v1.to_vec + v2.to_vec)\n@[simp]\ndef smul_vectr (k : K) (v : vectr s) : vectr s := mk_vectr' s (k \u2022 v.to_vec)\n@[simp]\ndef neg_vectr (v : vectr s) : vectr s := mk_vectr' s ((-1 : K) \u2022 v.to_vec)\n@[simp]\ndef sub_vectr_vectr (v1 v2 : vectr s) : vectr s := add_vectr_vectr s v1 (neg_vectr s v2)\n\n-- See unframed file for template for proving module\n\ninstance has_add_vectr : has_add (vectr s) := \u27e8add_vectr_vectr s\u27e9\nlemma add_assoc_vectr : \u2200 a b c : vectr s, a + b + c = a + (b + c) := \nbegin\n    intros,\n    ext,\n    --cases a,\n    repeat {\n    have p1 : (a + b + c).to_vec = a.to_vec + b.to_vec + c.to_vec := rfl,\n    have p2 : (a + (b + c)).to_vec = a.to_vec + (b.to_vec + c.to_vec) := rfl,\n    rw [p1,p2],\n    cc\n    }\nend\n\n\ninstance add_semigroup_vectr : add_semigroup (vectr s) := \u27e8 add_vectr_vectr s, add_assoc_vectr s\u27e9 \n\n@[simp]\ndef vectr_zero := @mk_vectr K _ _ n f s (0:K)\ninstance has_zero_vectr : has_zero (vectr s) := \u27e8vectr_zero s\u27e9\n\n#check mul_zero_class.zero\n\nlemma zero_add_vectr : \u2200 a : vectr s, 0 + a = a := \nbegin\n    intros,--ext,\n    ext,\n    let h0 : (0 + a).to_vec = (0 : vectr s).to_vec + a.to_vec := rfl,\n    simp [h0],\n    exact zero_add _,\nend\n\nlemma add_zero_vectr : \u2200 a : vectr s, a + 0 = a := \nbegin\n    intros,ext,\n    exact add_zero _,\nend\n\n@[simp]\ndef nsmul_vectr : \u2115 \u2192 (vectr s) \u2192 (vectr s) \n| nat.zero v := vectr_zero s\n--| 1 v := v\n| (nat.succ n) v := (add_vectr_vectr _) v (nsmul_vectr n v)\n\ninstance add_monoid_vectr : add_monoid (vectr s) := \u27e8 \n    -- add_semigroup\n    add_vectr_vectr s, \n    add_assoc_vectr s, \n    -- has_zero\n    vectr_zero s,\n    -- new structure \n    zero_add_vectr s, \n    add_zero_vectr s,\n    nsmul_vectr s\n\u27e9\n\ninstance has_neg_vectr : has_neg (vectr s) := \u27e8 neg_vectr s \u27e9\ninstance has_sub_vectr : has_sub (vectr s) := \u27e8 sub_vectr_vectr s \u27e9 \nlemma sub_eq_add_neg_vectr : \u2200 a b : vectr s, a - b = a + -b := \nbegin\n    intros,ext,\n    refl,\n\nend \n\n\ninstance sub_neg_monoid_vectr : sub_neg_monoid (vectr s) :=\n{\n    neg := neg_vectr s,\n    ..(show add_monoid (vectr s), by apply_instance)\n}\n\n/- \u27e8 \n    add_vectr_vectr s, add_assoc_vectr s, vectr_zero s, zero_add_vectr s, add_zero_vectr s, -- add_monoid\n    neg_vectr s,                                                                  -- has_neg\n    sub_vectr_vectr s,                                                              -- has_sub\n    sub_eq_add_neg_vectr s,                                                       -- new\n\u27e9 -/\n\nlemma add_left_neg_vectr : \u2200 a : vectr s, -a + a = 0 := \nbegin\n    intros,\n    ext,\n    have h0 : (-a + a).to_vec = -a.to_vec + a.to_vec := begin\n        have h\u2081 : -a + a = add_vectr_vectr _ (-a) a := rfl,\n        rw h\u2081,\n        dsimp only [add_vectr_vectr],\n        dsimp only [mk_vectr'],\n        have h\u2082 : -a = neg_vectr _ a := rfl,\n        rw h\u2082,\n        dsimp only [neg_vectr, mk_vectr'],\n        suffices h : (-(1 : K)) \u2022 a.to_vec = -a.to_vec,\n        rw h,\n        simp only [one_smul, neg_smul],\n    end,\n    simp [h0],\n    have : (0:vec K) = (0:vectr s).to_vec := rfl,\n    simp *,\nend\n\n\ninstance : add_group (vectr s) := {\n    add_left_neg := begin\n        exact add_left_neg_vectr s,\n    end,\n..(show sub_neg_monoid (vectr s), by apply_instance),\n\n}\n\n\n/-\u27e8\n    -- sub_neg_monoid\n    add_vectr_vectr s, add_assoc_vectr s, vectr_zero s, zero_add_vectr s, add_zero_vectr s, -- add_monoid\n    neg_vectr s,                                                                  -- has_neg\n    sub_vectr_vectr s,                                                              -- has_sub\n    sub_eq_add_neg_vectr s, \n    -- new\n    add_left_neg_vectr s,\n\u27e9 -/\n\nlemma add_comm_vectr : \u2200 a b : vectr s, a + b = b + a := \nbegin\n    intros,\n    ext,\n    repeat {\n    have p1 : (a + b).to_vec = a.to_vec + b.to_vec:= rfl,\n    have p2 : (b + a).to_vec = b.to_vec + a.to_vec := rfl,\n    rw [p1,p2],\n    cc\n    }    \nend\n\ninstance add_comm_semigroup_vectr : add_comm_semigroup (vectr s) := \u27e8\n    -- add_semigroup\n    add_vectr_vectr s, \n    add_assoc_vectr s,\n    add_comm_vectr s,\n\u27e9\n\ninstance add_comm_monoid_vectr : add_comm_monoid (vectr s) := \n{\n    add_comm := begin\n        exact add_comm_vectr s\n    end, \n    ..(show add_monoid (vectr s), by apply_instance)\n}\n\n\n\ninstance has_scalar_vectr : has_scalar K (vectr s) := \u27e8\nsmul_vectr s,\n\u27e9\n\nlemma one_smul_vectr : \u2200 b : vectr s, (1 : K) \u2022 b = b := begin\n    intros,ext,\n    repeat {\n        have h0 : ((1:K) \u2022 b).to_vec = ((1:K)\u2022(b.to_vec)) := rfl,\n        rw [h0],\n        simp *,\n    }\nend\n\nlemma mul_smul_vectr : \u2200 (x y : K) (b : vectr s), (x * y) \u2022 b = x \u2022 y \u2022 b :=\nbegin\n    intros,\n    cases b,\n    ext,\n    exact mul_assoc x y _,\nend\n\ninstance mul_action_vectr : mul_action K (vectr s) := \u27e8\none_smul_vectr s,\nmul_smul_vectr s,\n\u27e9 \n\nlemma smul_add_vectr : \u2200(r : K) (x y : vectr s), r \u2022 (x + y) = r \u2022 x + r \u2022 y := begin\n    intros, ext,\n    repeat {\n    have h0 : (r \u2022 (x + y)).to_vec = (r \u2022 (x.to_vec + y.to_vec)) := rfl,\n    have h1 : (r\u2022x + r\u2022y).to_vec = (r\u2022x.to_vec + r\u2022y.to_vec) := rfl,\n    rw [h0,h1],\n    simp *,\n    }\n\nend\n\nlemma smul_zero_vectr : \u2200(r : K), r \u2022 (0 : vectr s) = 0 := begin\n    intros, ext, exact mul_zero _\nend\ninstance distrib_mul_action_K_vectrK : distrib_mul_action K (vectr s) := \u27e8\nsmul_add_vectr s,\nsmul_zero_vectr s,\n\u27e9 \n\n-- renaming vs template due to clash with name \"s\" for prevailing variable\nlemma add_smul_vectr : \u2200 (a b : K) (x : vectr s), (a + b) \u2022 x = a \u2022 x + b \u2022 x := \nbegin\n  intros,\n  ext,\n  exact right_distrib _ _ _\nend\n\nlemma zero_smul_vectr : \u2200 (x : vectr s), (0 : K) \u2022 x = 0 := begin\n    intros,\n    ext,\n    exact zero_mul _,\nend\ninstance module_K_vectrK : module K (vectr s) := \u27e8 \n    add_smul_vectr s, \n    zero_smul_vectr s, \n\u27e9 \n\ninstance add_comm_group_vectr : add_comm_group (vectr s) := \n{\n    add_comm := begin\n        exact add_comm_vectr s\n        \n        /-intros,\n        ext,\n        let h0 : (a + b).to_vec = a.to_vec + b.to_vec := rfl,\n        let h1 : (b + a).to_vec = b.to_vec + a.to_vec := rfl,\n        rw [h0,h1],\n        exact add_comm _ _,\n        exact add_comm _ _,-/\n    end,\n--to_add_group := (show add_group (vec K), by apply_instance),\n--to_add_comm_monoid := (show add_comm_monoid (vec K), by apply_instance),\n..(show add_group (vectr s), by apply_instance)\n}\n/-\u27e8\n-- add_group\n    add_vectr_vectr s, add_assoc_vectr s, vectr_zero s, zero_add_vectr s, add_zero_vectr s, -- add_monoid\n    neg_vectr s,                                                                  -- has_neg\n    sub_vectr_vectr s,                                                              -- has_sub\n    sub_eq_add_neg_vectr s, \n    add_left_neg_vectr s,\n-- commutativity\n    add_comm_vectr s,\n\u27e9-/\n\ninstance : module K (vectr s) := module_K_vectrK s\n\n/-\n    ********************\n    *** Affine space ***\n    ********************\n-/\n\n/-\nAffine operations\n-/\ninstance : has_add (vectr s) := \u27e8add_vectr_vectr s\u27e9\ninstance : has_zero (vectr s) := \u27e8vectr_zero s\u27e9\ninstance : has_neg (vectr s) := \u27e8neg_vectr s\u27e9\n\n/-\nLemmas needed to implement affine space API\n-/\n@[simp]\ndef sub_point_point (p1 p2 : point s) : vectr s := mk_vectr' s (p1.to_pt -\u1d65 p2.to_pt)\n@[simp]\ndef add_point_vectr {f : fm K n} {s : spc K f } (p : point s) (v : vectr s) : point s := \n    mk_point' s (v.to_vec +\u1d65 p.to_pt) -- reorder assumes order is irrelevant\n@[simp]\ndef add_vectr_point {f : fm K n} {s : spc K f } (v : vectr s) (p : point s) : point s := \n    mk_point' s (v.to_vec +\u1d65 p.to_pt)\n\n@[simp]\ndef aff_vectr_group_action : vectr s \u2192 point s \u2192 point s := add_vectr_point\ninstance : has_vadd (vectr s) (point s) := \u27e8aff_vectr_group_action s\u27e9\n\nlemma zero_vectr_vadd'_a1 : \u2200 p : point s, (0 : vectr s) +\u1d65 p = p := begin\n    intros,\n    ext,--exact zero_add _,\n    exact add_zero _\nend\n\nlemma vectr_add_assoc'_a1 : \u2200 (g1 g2 : vectr s) (p : point s), g1 +\u1d65 (g2 +\u1d65 p) = g1 + g2 +\u1d65 p := begin\n    intros, ext,\n    repeat {\n    have h0 : (g1 +\u1d65 (g2 +\u1d65 p)).to_pt = (g1.to_vec +\u1d65 (g2.to_vec +\u1d65 p.to_pt)) := rfl,\n    have h1 : (g1 + g2 +\u1d65 p).to_pt = (g1.to_vec +\u1d65 g2.to_vec +\u1d65 p.to_pt) := rfl,\n    rw [h0,h1],\n    simp *,\n    simp [has_vadd.vadd, has_add.add, add_semigroup.add, add_zero_class.add,  add_monoid.add, sub_neg_monoid.add, \n        add_group.add, distrib.add, ring.add, division_ring.add],\n    cc,\n    }\nend\n\ninstance vectr_add_action: add_action (vectr s) (point s) := \n\u27e8 \nbegin\n    exact zero_vectr_vadd'_a1 s\nend,\nbegin\n    let h0 := vectr_add_assoc'_a1 s,\n    intros,\n    exact (h0 g\u2081 g\u2082 p).symm\nend\n\u27e9\n\n@[simp]\ndef aff_point_group_sub : point s \u2192 point s \u2192 vectr s := sub_point_point s\ninstance point_has_vsub : has_vsub (vectr s) (point s) := \u27e8 aff_point_group_sub s \u27e9 \n\ninstance : nonempty (point s) := \u27e8mk_point s 0\u27e9\n\nlemma point_vsub_vadd_a1 : \u2200 (p1 p2 : (point s)), (p1 -\u1d65 p2) +\u1d65 p2 = p1 := begin\n    intros, ext,\n    --repeat {\n    have h0 : (p1 -\u1d65 p2 +\u1d65 p2).to_pt = (p1.to_pt -\u1d65 p2.to_pt +\u1d65 p2.to_pt) := rfl,\n    rw h0,\n    simp [has_vsub.vsub, has_sub.sub, sub_neg_monoid.sub, add_group.sub, add_comm_group.sub, ring.sub, division_ring.sub],\n    simp [has_vadd.vadd, has_add.add, distrib.add, ring.add, division_ring.add],\n    let h0 : field.add p2.to_pt.to_prod.fst (field.sub p1.to_pt.to_prod.fst p2.to_pt.to_prod.fst) = \n            field.add (field.sub p1.to_pt.to_prod.fst p2.to_pt.to_prod.fst) p2.to_pt.to_prod.fst := add_comm _ _,\n    rw h0,\n    exact sub_add_cancel _ _,\n    have h0 : (p1 -\u1d65 p2 +\u1d65 p2).to_pt = (p1.to_pt -\u1d65 p2.to_pt +\u1d65 p2.to_pt) := rfl,\n    rw h0,\n    simp [has_vsub.vsub, has_sub.sub, sub_neg_monoid.sub, add_group.sub, add_comm_group.sub, ring.sub, division_ring.sub],\n    simp [has_vadd.vadd, has_add.add, distrib.add, ring.add, division_ring.add],\n    let h0 : field.add p2.to_pt.to_prod.snd (field.sub p1.to_pt.to_prod.snd p2.to_pt.to_prod.snd) = \n            field.add (field.sub p1.to_pt.to_prod.snd p2.to_pt.to_prod.snd) p2.to_pt.to_prod.snd := add_comm _ _,\n    rw h0,\n    exact sub_add_cancel _ _,\n    --have h1 : (p1.to_pt -\u1d65 p2.to_pt +\u1d65 p2.to_pt).to_prod = (p1.to_pt.to_prod -\u1d65 p2.to_pt.to_prod +\u1d65 p2.to_pt.to_prod) := by simp *,\n    \n    --}\nend\n\n\nlemma point_vadd_vsub_a1 : \u2200 (g : vectr s) (p : point s), g +\u1d65 p -\u1d65 p = g := \nbegin\n    intros, ext,\n    repeat {\n    have h0 : ((g +\u1d65 p -\u1d65 p) : vectr s).to_vec = (g.to_vec +\u1d65 p.to_pt -\u1d65 p.to_pt) := rfl,\n    rw h0,\n    simp *,\n    }\nend\n\n\n/-instance aff_point_torsor : add_torsor (vectr s) (point s) := \n\u27e8 \n    aff_vectr_group_action s,\n    zero_vectr_vadd'_a1 s,    -- from add_action\n    vectr_add_assoc'_a1 s,    -- from add_action\n    aff_point_group_sub s,    -- from has_vsub\n    point_vsub_vadd_a1 s,     -- from add_torsor\n    point_vadd_vsub_a1 s,     -- from add_torsor\n\u27e9-/\n\ninstance : affine_space (vectr s) (point s) := \u27e8\n    begin\n        exact point_vsub_vadd_a1 s,\n    end,\n    begin\n        exact point_vadd_vsub_a1 s,\n    end,\n\n\u27e9\n\n\n\n/-\nAnd now for transforms\n-/\n\n--variables {f1 : fm K n} {f2 : fm K n} (s1 : spc K f1) (s2 : spc K f2)\n#check (point s) \u2243\u1d43[K] (point s)\n--not usable?\nabbreviation raw_tr := (pt K) \u2243\u1d43[K] (pt K)\n--abbreviation fm_tr := (point s1) \u2243\u1d43[K] (point s2)\n\n@[ext]\nstructure fm_tr {f1 : fm K n} {f2 : fm K n} (s1 : spc K f1) (s2 : spc K f2)  extends (point s1) \u2243\u1d43[K] (point s2)\n\ndef fm_tr.refl {f1 : fm K n} (s1 : spc K f1) : fm_tr s1 s1 :=\n    \u27e8\u27e8\n        equiv.refl (point s1),\n        linear_equiv.refl K (vectr s1),\n        \u03bb _ _, rfl\n    \u27e9\u27e9\n\ndef fm_tr.symm  {f1 : fm K n} {f2 : fm K n} {s1 : spc K f1} {s2 : spc K f2} (ftr : fm_tr s1 s2) : fm_tr s2 s1 :=\n    \u27e8ftr.1.symm\u27e9\n\n\ndef fm_tr.trans  {f1 : fm K n} {f2 : fm K n} {f3 : fm K n} {s1 : spc K f1} {s2 : spc K f2} {s3 : spc K f3} (ftr : fm_tr s1 s2) : fm_tr s2 s3 \u2192 fm_tr s1 s3 :=\n    \u03bbftr_, \u27e8ftr.1.trans ftr_.1\u27e9\n\n\n#check (fin 2)\n/-\ninductive fm : nat \u2192 Type u\n| base : \u03a0 n, fm n\n| deriv : \u03a0 n, (prod (pt K) (vec K)) \u2192 fm n \u2192 fm n \n-/\n/-\n@[simp]\ndef fm.to_coords_matrix (f : fm K n) : matrix (fin 2) (fin 2) K\n    := \n    match f with \n    | (fm.base n) := \u03bb i j, \n        if i = 0 \u2227 j = 0 then 1 else \n        (if i = 0 \u2227 j = 1 then 0 else (\n            if i = 1 \u2227 j = 0 then 0 else\n                (if i = 1 \u2227 j = 1 then 1 else 0)\n        ))\n    | (fm.deriv n c parent) := \u03bb i j,\n        if i = 0 \u2227 j = 0 then c. else \n        (if i = 0 \u2227 j = 1 then 0 else (\n            if i = 1 \u2227 j = 0 then 0 else\n                (if i = 1 \u2227 j = 1 then 1 else 0)\n        ))\n    end\n-/\n\n\n\n/-\nTODO: This material needs inspection, verification\n-/\n#check @function.left_inverse\ndef to_base_helper' : fm K n \u2192 @raw_tr K _ _\n| (fm.base n) := \u27e8\n            \u27e8   /-base case -/\n                (\u03bb p, p),\n                (\u03bb p, p),\n                begin\n                    unfold function.left_inverse,\n                    intros,\n                    simp *\n                end,\n                begin\n                    unfold function.right_inverse function.left_inverse,\n                    intros,\n                    simp *\n                end\n            \u27e9,\n            \u27e8\n                (\u03bb v, v),\n                begin\n                    intros, simp*\n                end,\n               -- (\u03bb v, \u27e8v.to_vec\u27e9),\n                begin\n                    intros, simp *\n                end,\n                (\u03bb v, v),\n                begin\n                    unfold function.left_inverse,\n                    intros, simp *\n                end,\n                begin\n                    unfold function.left_inverse function.right_inverse,\n                    intros, simp *\n                end,\n            \u27e9,\n            begin\n                simp *,\n                --admit   -- TODO: What's this?\n            end\n        \u27e9\n| (fm.deriv n c parent) := (\u27e8\n            \u27e8/-transform from current->parent-/\n                \u03bbp, \u27e8(1, \n                        p.to_prod.2*c.snd.to_prod.2 + c.fst.to_prod.2),\n                        begin\n                            cases p,\n                            \n                        end\u27e9,\n                    \u03bbp, (\u27e8(1, \n                        (p.to_prod.2 - c.fst.to_prod.2 )/c.snd.to_prod.2),sorry\u27e9),\n                sorry,\n                sorry\n            \u27e9,\n            \u27e8\n                \u03bbv, \u27e8(0, v.to_prod.2*c.snd.to_prod.2),begin \n                    cases v,\n                    simp *,\n                end\u27e9,\n                sorry,\n                sorry,\n                \u03bbv, \u27e8(0, v.to_prod.2/c.snd.to_prod.2),begin \n                    cases v,\n                    simp *,\n                end\u27e9,\n                sorry,\n                sorry\n            \u27e9,\n            sorry /-invert to parent->current and append to current->base-/\n        \u27e9 : @raw_tr K _ _).trans (to_base_helper' parent)\n\n \ndef spc.to_base {f1 : fm K n} (s1 : spc K f1) : @raw_tr K _ _ := to_base_helper' f1\n\ndef spc.fm_tr {f1 : fm K n} {f2 : fm K n} (s1 : spc K f1) : \u03a0 (s2 : spc K f2),\n    fm_tr s1 s2 \n    := \n     --(point s1) \u2243\u1d43[K] (point s2) := \n    \u03bb s2,\n    \u27e8\n    let rawtr : @raw_tr K _ _ := s1.to_base.trans s2.to_base.symm in\n                \u27e8\n            \u27e8\n                (\u03bb p : point _, (\u27e8(rawtr p.1 : pt K)\u27e9 : point _)),\n                (\u03bb p : point _, (\u27e8(rawtr p.1 : pt K)\u27e9 : point _)),\n                sorry,\n                sorry\n            \u27e9,\n            \u27e8\n                (\u03bbv : vectr _, (\u27e8(rawtr.linear v.1 : vec K)\u27e9 : vectr _)),\n                sorry,\n               -- (\u03bb v, \u27e8v.to_vec\u27e9),\n                sorry,\n                (\u03bbv : vectr _, (\u27e8(rawtr.linear v.1 : vec K)\u27e9 : vectr _)),\n                sorry,\n                sorry\n            \u27e9,\n            sorry\n        \u27e9\n    \u27e9\n\ndef fm_tr.transform_point  {f1 : fm K n} {f2 : fm K n} {s1 : spc K f1} {s2 : spc K f2} (tr:fm_tr s1 s2 ) : point s1 \u2192 point s2 :=\n    \u03bbp,\n    tr.to_equiv p\n\ndef fm_tr.transform_vectr  {f1 : fm K n} {f2 : fm K n} {s1 : spc K f1} {s2 : spc K f2} (tr:fm_tr s1 s2 ) : vectr s1 \u2192 vectr s2 :=\n    \u03bbv,\n    let as_pt : point s1 := (\u27e8\u27e8(1,v.to_vec.to_prod.2),rfl\u27e9\u27e9) in\n    let tr_pt := (tr.to_equiv as_pt) in\n    \u27e8\u27e8(0, tr_pt.to_pt.to_prod.2),rfl\u27e9\u27e9\n\n\n/-\nDEMO: transform generation between arbitrary affine coordinate spaces on physical dimension\n-/\nvariables {f1 : fm K n} {f2 : fm K n} (s1 : spc K f1) (s2 : spc K f2)\n\ndef s1_to_s2 : _ := s1.fm_tr s2     -- Yay!\n\n#check s1_to_s2 s1 s2\n\nvariables (my_vec : vectr s1)\n\n#check ((s1_to_s2 s1 s2).transform_vectr) (((s1_to_s2 s1 s2).transform_vectr) my_vec)\n\ninstance : quiver (spc K f) := \u27e8\u03bb (s1 s2 : spc K f), fm_tr s1 s2\u27e9\n\ninstance : category_theory.category_struct (spc K f) := \u27e8\n    \u03bb (s : spc K f), fm_tr.refl s, -- identity transformation\n    \u03bb {s1 s2 s3 : spc K f} (f : fm_tr s1 s2) (g : fm_tr s2 s3), fm_tr.trans f g -- f \u2218 g\n\u27e9\n\ninstance : category_theory.category (spc K f) := \u27e8\n    begin\n        simp only [auto_param_eq],\n        intros,\n        have h_orig := (fm_tr.ext_iff (\ud835\udfd9 X \u226b f_1) f_1).2,\n        apply h_orig,\n        cases f_1 with f_aff,\n        cases f_aff with f_equiv f_linear f_map_vadd',\n        dsimp only [category_theory.category_struct.id, fm_tr.refl],\n        dsimp only [category_theory.category_struct.comp, fm_tr.trans, affine_equiv.trans],\n        ext,\n        dsimp only [coe_fn, has_coe_to_fun.coe],\n        simp only [equiv.refl_trans],\n    end,\n    begin\n        simp only [auto_param_eq],\n        intros,\n        have h\u2080 := (fm_tr.ext_iff (\ud835\udfd9 X \u226b f_1) f_1).2,\n        apply h\u2080,\n        cases f_1 with f_aff,\n        cases f_aff with f_equiv f_linear f_map_vadd',\n        dsimp only [category_theory.category_struct.id, fm_tr.refl],\n        dsimp only [category_theory.category_struct.comp, fm_tr.trans, affine_equiv.trans],\n        ext,\n        dsimp only [coe_fn, has_coe_to_fun.coe],\n        simp only [equiv.refl_trans],\n    end,\n    begin\n        simp only [auto_param_eq],\n        intros,\n        have h\u2080 := (fm_tr.ext_iff ((f_1 \u226b g) \u226b h) (f_1 \u226b g \u226b h)).2,\n        apply h\u2080,\n        cases f_1 with f_aff,\n        cases g with g_aff,\n        cases h with h_aff,\n        cases f_aff with f_equiv f_linear f_map_vadd',\n        cases g_aff with g_equiv g_linear g_map_vadd',\n        cases h_aff with h_equiv h_linear h_map_vadd',\n        dsimp only [category_theory.category_struct.comp, fm_tr.trans, affine_equiv.trans],\n        ext,\n        dsimp only [coe_fn, has_coe_to_fun.coe],\n        simp only [equiv.trans_assoc],\n    end\n\u27e9 \n\ninstance : category_theory.groupoid (spc K f) := \u27e8\n    \u03bb {s1 s2 : spc K f} (f_1 : fm_tr s1 s2), fm_tr.symm f_1,\n    begin\n        simp only [auto_param_eq],\n        intros,\n        dsimp only [category_theory.category_struct.comp],\n        have h\u2080 := (fm_tr.ext_iff ((fm_tr.symm f_1).trans f_1) (\ud835\udfd9 Y)).2,\n        apply h\u2080,\n        dsimp only [category_theory.category_struct.id, fm_tr.refl],\n        dsimp only [fm_tr.trans, affine_equiv.trans],\n        dsimp only [fm_tr.symm, affine_equiv.symm],\n        ext,\n        dsimp only [coe_fn, has_coe_to_fun.coe],\n        simp only [affine_equiv.coe_to_equiv, equiv.to_fun_as_coe, id.def, function.comp_app, affine_equiv.apply_symm_apply,\n            equiv.coe_trans, affine_equiv.symm_to_equiv, equiv.coe_refl],\n    end,\n    begin\n        simp only [auto_param_eq],\n        intros,\n        dsimp only [category_theory.category_struct.comp],\n        have h\u2080 := (fm_tr.ext_iff (fm_tr.trans f_1 (fm_tr.symm f_1)) (\ud835\udfd9 X)).2,\n        apply h\u2080,\n        dsimp only [category_theory.category_struct.id, fm_tr.refl],\n        dsimp only [fm_tr.trans, affine_equiv.trans],\n        dsimp only [fm_tr.symm, affine_equiv.symm],\n        ext,\n        dsimp only [coe_fn, has_coe_to_fun.coe],\n        simp only [affine_equiv.coe_to_equiv, equiv.to_fun_as_coe, id.def, function.comp_app, affine_equiv.symm_apply_apply,\n            equiv.coe_trans, affine_equiv.symm_to_equiv, equiv.coe_refl],\n    end\n\u27e9 \n\nend implicitK\n\n-- TODO: clean up naming in this file", "meta": {"author": "kevinsullivan", "repo": "affine_lib", "sha": "056fc95c31bdf473b0c1ecd07f5a061dd6b69234", "save_path": "github-repos/lean/kevinsullivan-affine_lib", "path": "github-repos/lean/kevinsullivan-affine_lib/affine_lib-056fc95c31bdf473b0c1ecd07f5a061dd6b69234/src/aff1Kcoord/aff1Kcoord.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3626117681505814}}
{"text": "open tactic\n\n-- Applies a list of tactics in turn, always succeeding.\nmeta def list_try_seq : list (tactic unit) \u2192 tactic unit \n| list.nil  := skip\n| (t :: ts) := seq (try t) (list_try_seq ts)\n\n-- Applies a list of tactics in turn, succeeding if at least one succeeds.\nmeta def at_least_one : list (tactic unit) \u2192 tactic unit\n| list.nil  := fail \"at_least_one tactic failed, no more tactics\"\n| (t :: ts) := (seq t (list_try_seq ts)) <|> (at_least_one ts)\n\n-- FIXME let's try this\nmeta def simp_at_via_rewrite (h : expr) (extra_lemmas : list expr := []) (cfg : simp_config := {}) : tactic unit :=\ndo when (expr.is_local_constant h = ff) (fail \"tactic simp_at failed, the given expression is not a hypothesis\"),\n   htype \u2190 infer_type h,\n   S     \u2190 simp_lemmas.mk_default,\n   S     \u2190 S.append extra_lemmas,\n   (new_htype, heq) \u2190 simplify S htype cfg,\n   rewrite_at_core reducible tt tt occurrences.all ff heq h\n\n-- set_option pp.all true\n\nlemma {u v} dependent_pair_equality {\u03b1 : Type u} {Z : \u03b1 \u2192 Type v} { X Y : \u03a3 a : \u03b1, Z a } ( p1 : X.1 = Y.1 ) ( p2 : @eq.rec \u03b1 X.1 Z X.2 Y.1 p1 = Y.2 ) : X = Y :=\nbegin\n  induction X,\n  induction Y,\n  dsimp at p1,\n  dsimp at p2,\n  \nend\n\n-- FIXME this repeatedly resimplifies hypotheses, if they can't be cleared. :-(\nmeta def simp_hypotheses : tactic unit :=\ndo l \u2190 local_context,\n   at_least_one (l.reverse.for (\u03bb h, simp_at h))", "meta": {"author": "semorrison", "repo": "proof", "sha": "5ee398aa239a379a431190edbb6022b1a0aa2c70", "save_path": "github-repos/lean/semorrison-proof", "path": "github-repos/lean/semorrison-proof/proof-5ee398aa239a379a431190edbb6022b1a0aa2c70/lean/20170502-simp-via-rewrite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3626117681505814}}
{"text": "import metaphysics.metaphysics states\nopen set topological_space classical\nset_option pp.generalized_field_notation true\nlocal attribute [instance] prop_decidable\n\nnamespace ontology\n\nvariable {\u03c9 : ontology}\n\nsection theism\n  \n  variable (\u03c9)\n\n  /-- **Theism** is the doctrine that the necessary being is simple. -/\n  def theism := \u03c9.nb.simple\n  /-- Negation of `theism`. -/\n  @[reducible, simp]\n  def atheism := \u00ac \u03c9.theism\n        \n  /-- **Classical Theism** is an extension of theism which \n      furthermore claims that there is a possible world \n      in which the necessary being exists alone. -/\n  def ctheism := \u2203 (w : \u03c9.world), \u2200 e\u2081 e\u2082 : \u03c9.entity, e\u2081.exists w \u2192 e\u2082.exists w \u2192 e\u2081 = e\u2082\n\n  /-- this is a not so trivial proof that if the nb exists alone,\n      then it has no accidents, because accidents are contingent. -/\n  @[simp] lemma theism_of_ctheism {\u03c9 : ontology} : \u03c9.ctheism \u2192 \u03c9.theism :=\n    begin\n      rintros \u27e8w, hw\u27e9,\n      dunfold theism,\n      by_contradiction h,\n      replace h : \u03c9.nb.composite,\n        simp at h, simp [h],\n      replace h := nb_acc_actual h w,\n      obtain \u27e8a, ha\u2081, ha\u2082\u27e9 := h,\n      have ha\u2083 := a.contingent,\n      simp at ha\u2083,\n      specialize hw a \u03c9.nb ha\u2082 (by simp [nb, nbe]),\n      unfold_coes at hw,\n      simp [nb] at hw,\n      contradiction,\n    end\n\n  /-- **Greek Theism** is non-classical theism. -/\n  def greek_theism := \u03c9.theism \u2227 \u00ac \u03c9.ctheism\n\n  -- All 4 positions are logically consistent:\n\n  -- lemma atheism_consistent : \u2203 \u03c9 : ontology, \u03c9.atheism := sorry\n  -- lemma theism_consistent : \u2203 \u03c9 : ontology, \u03c9.theism := sorry\n  -- lemma ctheism_consistent : \u2203 \u03c9 : ontology, \u03c9.ctheism := sorry\n  -- lemma greek_theism_consistent : \u2203 \u03c9 : ontology, \u03c9.greek_theism := sorry\n\n  -- It is actually considerably annoying to formally prove the previous results even though they are absolutely trivial,\n  -- so we leave them commented out. But consider that, given our lemma `nb_simple_iff_space_connected`\n  -- in substances.lean, any connected space is a model of theism, and any disconnected space a model of atheism. \n  -- Any connected space without a focal point is a model of greek theism (e.g. a sphere), and a\n  -- space is a model of classical theism if and only if it has a focal point. Every partial order with a least\n  -- element is a model of classical theism when given its Alexandroff topology.\n\n  /-- The *second gap problem* is showing that God satisfies\n      the properties prescribed to Him by classical theism\n      under the assumption that God exists.\n      This is not to be confused with the *first gap problem* \n      of showing that if the necessary being is a first cause of all \n      things in some possible world, then it is God.\n      An ontology `\u03c9` is said to contain **no second gap**\n      if this gap problem admits a solution in `\u03c9`.\n      Solutions of the gap problem are proofs of the \n      `\u03c9.nogap\u2082` proposition, and may be also called **(secondary) gap fillers**. -/\n  def nogap\u2082 : Prop := \u03c9.theism \u2192 \u03c9.ctheism\n\n  -- WORK IN PROGRESS (RELATED TO 4TH WAY).\n  -- theorem qparticipated_of_theism : \u03c9.theism \u2192 (\u2200 b : \u03c9.being, b.qparticipated) := sorry\n\n  theorem theism_iff_connected : \u03c9.theism \u2194 \u03c9.connected := nb_simple_iff_space_connected\n\nend theism\n\nsection divine_properties\n\n  -- First we discuss pure actuality:\n  variables (s : \u03c9.substance) (e : \u03c9.entity)\n\n  /-- A substance is **Purely Actual** if it has no passive potentiality\n      to be different from what it is, i.e. if it has a single state. -/\n  def substance.purely_actual := \u2200 w\u2081 w\u2082 : \u03c9.world, s.state w\u2081 = s.state w\u2082\n  \n  /-- Only the necessary being can be purely actual, \n      in which case theism follows. -/\n  theorem substance.eq_nb_of_purely_actual : s.purely_actual \u2192 s.necessary :=\n  begin\n      intros h,\n      simp [substance.purely_actual] at h,\n      ext w\u2081,\n      simp [nb, nbe],\n      obtain \u27e8w\u2082, hw\u2082\u27e9 := s.possible,\n      specialize h w\u2081 w\u2082,\n      replace hw\u2082 := exists_iff_in_state.mp hw\u2082,\n      rw \u2190h at hw\u2082,\n      exact exists_iff_in_state.2 hw\u2082,\n  end\n\n  theorem substance.simple_of_purely_actual : s.purely_actual \u2192 s.simple :=\n    begin\n      intro h\u2080,\n      by_contradiction h,\n      simp [set.nonempty, accident.inheres] at h,\n      obtain \u27e8a, ha\u27e9 := h,\n      obtain \u27e8w\u2081, hw\u2081\u27e9 := a.possible,\n      have h := a.contingent,\n      simp [nb, nbe, ext_iff] at h,\n      obtain \u27e8w\u2082, hw\u2082\u27e9 := h,\n      specialize h\u2080 w\u2081 w\u2082,\n      simp [substance.state, ext_iff] at h\u2080,\n      specialize h\u2080 a,\n      unfold_coes at h\u2080,\n      simp [ha, hw\u2081, hw\u2082] at h\u2080,\n      contradiction,\n    end\n\n  theorem theism_iff_purely_actual : \u03c9.theism \u2194 \u2203 s : \u03c9.substance, s.purely_actual :=\n    begin\n      constructor; intro h,\n        use \u03c9.nb,\n        intros w\u2081 w\u2082,\n        transitivity ({\u03c9.nbe} : set  \u03c9.entity);\n        simp [substance.state, ext_iff];\n        intro x;\n        by_cases hyp : x = \u03c9.nbe,\n          any_goals {rw hyp,\n          simp [nb], constructor,\n            apply self_subsist.mp,\n            simp [entity.perfect, nbe], \n          },\n          any_goals {simp [entity.perfect, nbe]},\n        any_goals {simp [nb]},\n        any_goals {\n        have : \u00ac x.subsists \u03c9.nbe,\n          intro h\u2080,\n          simp [theism, set.nonempty] at h,\n          have := imperfect_of_subsists_other h\u2080 hyp,\n          let x' : \u03c9.accident := \u27e8x, this\u27e9,\n          apply h x',\n          simpa [x', accident.inheres, nb],\n          simp [this],\n          simp [nbe, ext_iff] at hyp,\n          exact hyp,\n        },\n      obtain \u27e8g, hg\u27e9 := h,\n      have c\u2080 := g.eq_nb_of_purely_actual hg,\n      simp [substance.necessary] at c\u2080,\n      replace hg := g.simple_of_purely_actual hg,\n      simp only [theism],\n      rwa \u2190c\u2080 at *,\n    end\n\n  -- Then we discuss causal properties which properly belong to the divine being:\n  variable (c : \u03c9.cause)\n\n  /-- A a causal structure `c` is **principled** (in the sense that it emanates from \n      a necessary first principle) just in case the necessary being can possibly be a `first_cause`\n      with respect to `c`. Proofs that a causal structure is principled are called\n      **(stage 1) cosmological arguments**. -/\n  @[reducible, simp, alias]\n  def cause.principled : Prop := c.dscotus\n\n  /-- The *first gap problem* is showing that if \n      the necessary being is a first cause of all \n      things in some possible world, then it is God.\n      A causal structure `c` is said to contain **no first gap**\n      if this gap problem admits a solution with respect to `c`.\n      Solutions of the gap problem are proofs of the \n      `c.nogap\u2082` proposition, and may be called **(primary) gap fillers**\n      or **(stage 2) cosmological arguments**. -/\n  def cause.nogap\u2081 : Prop := c.principled \u2192 \u03c9.theism\n\n  /-- The *conjoined gap problem* is showing that if \n      the necessary being is a first cause of all \n      things in some possible world, then it is **the classical theistic** God.\n      A causal structure `c` is said to contain **no gap**\n      if this gap problem admits a solution with respect to `c`.\n      Solutions of the gap problem are proofs of the \n      `c.nogap\u2081\u2082` proposition, and may be called **(both primary and secondary) gap fillers**\n      or **(stage 2) cosmological arguments**. -/\n  def cause.nogap\u2081\u2082 : Prop := c.principled \u2192 \u03c9.ctheism\n\n\nend divine_properties\n\n/-- A **Theos** is a conception of God compatible with `greek_theism`. \n    It is God from the point of view of classical pagan theology. -/\nstructure theos (\u03c9 : ontology) :=\n  (s : \u03c9.substance)\n  (necessary : s.necessary)\n  (simple : s.simple)\n\n/-- A **God** is a **Theos** with the attribute of **Divine Aseity**.\n    Assumption of aseity is sufficient for proving all the properties\n    of the classical theistic God.\n    God is here taken as a synonym for \"Deus\", \n    or \"the classical latin, or scholastic, conception of God\". -/\nstructure god (\u03c9 : ontology) extends theos \u03c9 :=\n  (aseity : \u2203 w : \u03c9.world, \u2200 s' \u2208 w.substances, s' = s)\n\n@[reducible, simp]\ndef god.up (g : \u03c9.god) := g.to_theos \n\nsection theos\n\n  variable (g : \u03c9.theos)\n\n  @[simp]\n  theorem theos_iff_theism : nonempty \u03c9.theos \u2194 \u03c9.theism :=\n    begin\n      refine \u27e8\u03bbne,_, \u03bbh,\u27e8\u27e8\u03c9.nb, rfl, h\u27e9\u27e9\u27e9,\n      obtain \u27e8g\u27e9 := ne,\n      simp [theism, -substance.simple],\n      have c := g.necessary, simp [substance.necessary] at c,\n      rw \u2190c, exact g.simple,\n    end\n\n  def theos.theism := theos_iff_theism.mp \u27e8g\u27e9\n\n  /-- God is a purely actual substance. -/\n  lemma theos.purely_actual : g.s.purely_actual :=\n    begin\n      have c := g.theism,\n      replace c := theism_iff_purely_actual.mp c,\n      obtain \u27e8s, hs\u27e9 := c,\n      have c := g.necessary,\n      simp [substance.necessary] at c, rw c, clear c,\n      have c := s.eq_nb_of_purely_actual hs,\n      simp [substance.necessary] at c, \n      rwa \u2190c,\n    end\n  \n  theorem monotheism : \u2200 g g' : \u03c9.theos, g = g' :=\n    begin\n      intros g g',\n      casesm* \u03c9.theos, simp,\n      simp [substance.necessary] at *,\n      rw g_necessary,\n      rw g'_necessary,\n    end\n  \n  -- WORK IN PROGRESS (RELATED TO 4TH WAY).\n  -- /-- God is a fixed point of any possessed property He may possibly exemplify.\n  --     As such we say that He is his Omnipotence, His Omniscience, His benevolence,\n  --     and whatever else is predicated of Him, *in the precise and non-paradoxical sense*\n  --     that the *event* of God being (e.g.) Omnipotent is the *event* of God existing,\n  --     and also the *event* of Him being Omniscient, etc...\n  --     This would actually be valid for any simple substance as well.\n  --     -/\n  -- theorem divine_simplicity : \u2200 p : \u03c9.predicate, p.possessed \u2192 \u22c4p g.s \u2192 p g.s = g.s.exists := sorry\n\n  -- WORK IN PROGRESS (RELATED TO 4TH WAY).\n  -- /-- God is maximally perfect w.r.t. any analogy of being. -/\n  -- theorem theos.maximally_perfect : \u2200 (g : \u03c9.theos) (b : \u03c9.being), g.s.up.mperfect b.is := sorry\n\nend theos\n\nsection god\n\n  variable (g : \u03c9.god)\n\n  @[simp]\n  theorem god_iff_ctheism : nonempty \u03c9.god \u2194 \u03c9.ctheism :=\n    begin\n      refine \u27e8\u03bbh,_, \u03bbh,\u27e8\u27e8\u27e8\u03c9.nb, rfl, _,\u27e9,_\u27e9\u27e9\u27e9,\n        obtain \u27e8g\u27e9 := h,\n        obtain \u27e8w, hw\u27e9 := g.aseity,\n        use w, intros e\u2081 e\u2082 h\u2081 h\u2082,\n        have c\u2080 : \u2200 e : \u03c9.entity, e.exists w \u2192 e.perfect,\n          intros e he,\n          by_contradiction c\u2080,\n          let a : \u03c9.accident := \u27e8e, c\u2080\u27e9,\n          specialize hw a.owner,\n          simp [world.substances] at hw,\n          specialize hw _, swap,\n            exact entails_of_inheres a.inh_owner he,\n          have c\u2081 := a.inh_owner, rw hw at c\u2081,\n          have c\u2082 := g.simple, simp [substance.simple, set.nonempty] at c\u2082,\n          specialize c\u2082 a,\n          contradiction,\n        have c\u2081 := c\u2080 e\u2081 h\u2081,\n        have c\u2082 := c\u2080 e\u2082 h\u2082,\n        clear c\u2080,\n        let s\u2081 : \u03c9.substance := \u27e8e\u2081, c\u2081\u27e9,\n        let s\u2082 : \u03c9.substance := \u27e8e\u2082, c\u2082\u27e9,\n        simp [world.substances] at *,\n        have c\u2083 := hw s\u2081 h\u2081, simp [s\u2081] at c\u2083,\n        have c\u2084 := hw s\u2082 h\u2082, simp [s\u2082] at c\u2084,\n        rw \u2190 c\u2084 at c\u2083, simp at c\u2083,\n        exact c\u2083,\n      by_contradiction c,\n      simp [substance.simple, ext_iff] at c,\n      obtain \u27e8a, ha\u27e9 := c,\n      obtain \u27e8w, hw\u27e9 := h,\n      let a' := a.localize w,\n      specialize hw a' \u03c9.nbe (a.localize_exists _) (by simp [nbe]), swap,\n        rw (unique_inheres a.inh_owner ha),\n        simp [nb, nbe],\n      unfold_coes at hw,\n      simp [-entity_ext_iff] at hw,\n      have c\u2081 := a'.imperfect,\n      have c\u2082 := \u03c9.nb.perfect, simp [nb, -self_subsist] at c\u2082,\n      rw hw at c\u2081,\n      contradiction,\n        obtain \u27e8w, hw\u27e9 := h,\n        use w,\n        intros s hs, simp,\n        simp [world.substances] at hs,\n        specialize hw s.up \u03c9.nbe hs (by simp [nbe]),\n        simp at hw,\n        apply substance_ext,\n        exact hw,\n    end\n\n  def god.ctheism := god_iff_ctheism.mp \u27e8g\u27e9\n\n  theorem uniqueness_of_god : \u2200 g g' : \u03c9.god, g = g' :=\n    by intros g g'; casesm* \u03c9.god; simp; apply monotheism\n  \n  /-- God is a purely actual substance. -/\n  lemma god.purely_actual : g.s.purely_actual := g.up.purely_actual\n\n  -- WORK IN PROGRESS (RELATED TO 4TH WAY).\n  -- /-- God is maximally perfect w.r.t. any analogy of being. -/\n  -- theorem god.maximally_perfect : \u2200 (g : \u03c9.god) (b : \u03c9.being), g.s.up.mperfect b.is := sorry\n\n  lemma weakly_parmenidean_iff\u2084 : \u03c9.weakly_parmenidean \u2194 \u03c9.ctheism :=\n    begin\n      simp [weakly_parmenidean, ctheism, set.nonempty, parmenidean],\n      constructor; rintros \u27e8w, hw\u27e9; use w; intros,\n        transitivity \u03c9.nbe.exists, apply hw, assumption,\n        symmetry, apply hw, assumption,\n      apply hw, assumption,\n      simp [nbe, ext_iff],\n    end\n\n  /-- God is absolutely real. -/\n  theorem god.absolutely_real : g.s.up.absolutely_real :=\n    begin\n      convert weakly_parmenidean_iff\u2083.mp _,\n        have := g.necessary,\n        cases g.to_theos.s,\n        simp [substance.necessary, nb] at this,\n        simpa,\n      apply weakly_parmenidean_iff\u2084.2,\n      exact g.ctheism,\n    end\n\n  theorem ctheism_of_no_universe : (\u00ac\u2203 e : \u03c9.entity, e.contingent) \u2192 \u03c9.ctheism :=\n    begin\n      intro h,\n      push_neg at h,\n      obtain \u27e8w\u27e9 := \u03c9.wne,\n      use w,\n      intros e\u2081 e\u2082 h\u2081 h\u2082, clear h\u2081 h\u2082,\n      rw h e\u2081,\n      rw h e\u2082,\n    end \n  \n  theorem ctheism_of_no_universe\u2082 : \u2200 {w}, (\u00ac\u2203 e : \u03c9.entity, e.contingent \u2227 e.exists w) \u2192 \u03c9.ctheism :=\n    begin\n      intros w h,\n      push_neg at h,\n      use w,\n      intros e\u2081 e\u2082 h\u2081 h\u2082,\n      have c\u2081 := h e\u2081,\n      have c\u2082 := h e\u2082,\n      simp [h\u2081,h\u2082] at *, clear h h\u2081 h\u2082,\n      rw c\u2081, rw c\u2082,\n    end \n\n  /-- If there are contingent substances, `\u03c9.ctheism` is true\n      if and only if the universe is contingent, otherwise\n      there is no universe (of contingent things) and in this case also\n      `\u03c9.ctheism` is true.\n      -/\n  theorem ctheism_iff_universe_contingent : (Sup $ @entity.contingent \u03c9).contingent \u2228 (\u00ac\u2203 e : \u03c9.entity, e.contingent) \u2194 \u03c9.ctheism :=\n    begin\n      by_cases c : (set.nonempty $ @entity.contingent \u03c9);\n      simp [Sup, c],\n      swap,\n      have c' := ctheism_of_no_universe c,\n      simp [c'],\n      simp [set.nonempty, has_mem.mem, set.mem] at c,\n      right, exact c,\n        constructor; intro h,\n          cases h,\n            simp [has_Sup.Sup, c, entity_Sup, ext_iff] at h,\n            obtain \u27e8w, hw\u27e9 := h,\n            simp [nbe] at hw,\n            use w, intros e\u2081 e\u2082 h\u2081 h\u2082,\n            have c\u2081 := hw e\u2081, \n            replace c\u2081 : e\u2081 = \u03c9.nbe, \n              finish [has_mem.mem, set.mem, h\u2081],\n            have c\u2082 := hw e\u2082, \n            replace c\u2082 : e\u2082 = \u03c9.nbe, \n              finish [has_mem.mem, set.mem, h\u2082],\n            rw c\u2081, rw c\u2082,\n          refine ctheism_of_no_universe _,\n          push_neg,\n          intro e,\n          apply entity_ext,\n          exact h e,\n        left,\n        simp [has_Sup.Sup, c, entity_Sup, nbe, ext_iff],\n        obtain \u27e8w, hw\u27e9 := h,\n        use w, intros e he insanity,\n        simp [has_mem.mem, set.mem, -entity_ext_iff] at he,\n        specialize hw e \u03c9.nbe insanity (by simp [nbe]),\n        contradiction,\n    end\n\n  -- THIS SECTION IS A WORK IN PROGRESS.\n  section power_and_will\n\n    variables (c : \u03c9.cause) (e : \u03c9.entity)\n\n    -- TODO: causal structures in this section will probably need to be constrained by\n    -- some plausible assumptions in order to make the following theorems true.\n    -- Figure out which assumptions before doing the proofs.\n\n    -- theorem god.omnipotent : c.omnipotent g.s.up := sorry\n    -- theorem to_be_is_to_be_caused_by_god : c.causes g.s e = e := sorry\n    -- theorem creation_preceeds_matter : \u00ac\u2203 context : \u03c9.event, c.causes g.s e \u21d2 context \u2227\n    --                                     c.causes g.s e \u2260 context \u2227 \u00ac\u25a1context := sorry\n    -- theorem god.free : c.is_free g.s := sorry\n\n\n  end power_and_will\n\nend god\n\nsection atheism\n\n  /-- Atheism is committed to the existence of complemented entities. -/\n  theorem atheism_positive_commitments : \u03c9.atheism \u2194 \u2203 e : \u03c9.entity, e.complemented :=\n    begin\n      simp [atheism, theism, set.nonempty],\n      constructor; rintro \u27e8a, ha\u27e9,\n        have := compl_iff_inheres_nb.mp ha,\n        exact \u27e8a, this\u27e9,\n      have : a.imperfect,\n        have c := closure_eq_of_is_closed ha.2.2,\n        replace ha := ha.1,\n        simp [entity.contingent, nbe] at ha,\n        simp [entity.imperfect, entity.perfect],\n        rwa c,\n      use \u27e8a, this\u27e9,\n      apply compl_iff_inheres_nb.2 _, \n      assumption,\n    end\n\n  lemma atheism_has_universe : \u03c9.atheism \u2192 \u2203 e : \u03c9.entity, e.contingent :=\n    by contrapose; intro h; simp; exact theism_of_ctheism (ctheism_of_no_universe h)\n\n  lemma eq_nbe_Sup_of_atheism : \u03c9.atheism \u2192 \u03c9.nbe = Sup entity.contingent :=\n    begin\n      intro h,\n      have c := (@ctheism_iff_universe_contingent \u03c9).mp,\n      replace c := \u03bbh\u2080, theism_of_ctheism (c h\u2080),\n      replace c : \u00ac(Sup entity.contingent).contingent, finish,\n      simp [-entity_ext_iff] at c,\n      symmetry,\n      exact c,\n    end\n    \n\nend atheism\n\nend ontology", "meta": {"author": "maxd13", "repo": "topological_ontology", "sha": "68d21c9a00024fba3aed301e16c31e05733c1786", "save_path": "github-repos/lean/maxd13-topological_ontology", "path": "github-repos/lean/maxd13-topological_ontology/topological_ontology-68d21c9a00024fba3aed301e16c31e05733c1786/src/theology/natural/god.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737473266735, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3625895725987789}}
{"text": "import Mathlib.Tactic.GuardGoalNums\n\nexample : true \u2227 true := by\n  constructor\n  guard_goal_nums 2\n  all_goals {constructor}\n\nexample : (true \u2227 true) \u2227 (true \u2227 true) := by\n  constructor <;> constructor\n  guard_goal_nums 4\n  all_goals {constructor}\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/test/GuardGoalNums.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3625295254065659}}
{"text": "/-\nFile: signature_recover_public_key_ec_negate_soundness.lean\n\nAutogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.hoare\nimport .signature_recover_public_key_code\nimport ..signature_recover_public_key_spec\nimport .signature_recover_public_key_verify_zero_soundness\nimport .signature_recover_public_key_nondet_bigint3_soundness\nopen tactic\n\nopen starkware.cairo.common.cairo_secp.ec\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.field\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\nvariable  mem : F \u2192 F\nvariable  \u03c3 : register_state F\n\n/- starkware.cairo.common.cairo_secp.ec.ec_negate autogenerated soundness theorem -/\n\ntheorem auto_sound_ec_negate\n    -- arguments\n    (range_check_ptr : F) (point : EcPoint F)\n    -- code is in memory at \u03c3.pc\n    (h_mem : mem_at mem code_ec_negate \u03c3.pc)\n    -- all dependencies are in memory\n    (h_mem_4 : mem_at mem code_nondet_bigint3 (\u03c3.pc  - 160))\n    (h_mem_7 : mem_at mem code_verify_zero (\u03c3.pc  - 112))\n    -- input arguments on the stack\n    (hin_range_check_ptr : range_check_ptr = mem (\u03c3.fp - 9))\n    (hin_point : point = cast_EcPoint mem (\u03c3.fp - 8))\n    -- conclusion\n  : ensures_ret mem \u03c3 (\u03bb \u03ba \u03c4,\n      \u03c4.ap = \u03c3.ap + 34 \u2227\n      \u2203 \u03bc \u2264 \u03ba, rc_ensures mem (rc_bound F) \u03bc (mem (\u03c3.fp - 9)) (mem $ \u03c4.ap - 7)\n        (spec_ec_negate mem \u03ba range_check_ptr point (mem (\u03c4.ap - 7)) (cast_EcPoint mem (\u03c4.ap - 6)))) :=\nbegin\n  apply ensures_of_ensuresb, intro \u03bdbound,\n  have h_mem_rec := h_mem,\n  unpack_memory code_ec_negate at h_mem with \u27e8hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15\u27e9,\n  -- function call\n  step_assert_eq hpc0 with arg0,\n  step_sub hpc1 (auto_sound_nondet_bigint3 mem _ range_check_ptr _ _),\n  { rw hpc2, norm_num2, exact h_mem_4 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point] },\n    try { dsimp [cast_EcPoint, cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  intros \u03ba_call3 ap3 h_call3,\n  rcases h_call3 with \u27e8h_call3_ap_offset, h_call3\u27e9,\n  rcases h_call3 with \u27e8rc_m3, rc_mle3, hl_range_check_ptr\u2081, h_call3\u27e9,\n  generalize' hr_rev_range_check_ptr\u2081: mem (ap3 - 4) = range_check_ptr\u2081,\n  have htv_range_check_ptr\u2081 := hr_rev_range_check_ptr\u2081.symm, clear hr_rev_range_check_ptr\u2081,\n  generalize' hr_rev_minus_y: cast_BigInt3 mem (ap3 - 3) = minus_y,\n  simp only [hr_rev_minus_y] at h_call3,\n  have htv_minus_y := hr_rev_minus_y.symm, clear hr_rev_minus_y,\n  try { simp only [arg0] at hl_range_check_ptr\u2081 },\n  rw [\u2190htv_range_check_ptr\u2081, \u2190hin_range_check_ptr] at hl_range_check_ptr\u2081,\n  try { simp only [arg0] at h_call3 },\n  rw [hin_range_check_ptr] at h_call3,\n  clear arg0,\n  -- function call\n  step_assert_eq hpc3 with arg0,\n  step_assert_eq hpc4 with arg1,\n  step_assert_eq hpc5 with arg2,\n  step_assert_eq hpc6 with arg3,\n  step_sub hpc7 (auto_sound_verify_zero mem _ range_check_ptr\u2081 {\n    d0 := minus_y.d0 + point.y.d0,\n    d1 := minus_y.d1 + point.y.d1,\n    d2 := minus_y.d2 + point.y.d2\n  } _ _ _),\n  { rw hpc8, norm_num2, exact h_mem_7 },\n  { try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr\u2081, htv_minus_y] },\n    try { dsimp [cast_EcPoint, cast_BigInt3] },\n    try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3] },\n    try { simp only [h_call3_ap_offset] },\n    try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } }, },\n  { try { ext } ; {\n      try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr\u2081, htv_minus_y] },\n      try { dsimp [cast_EcPoint, cast_BigInt3] },\n      try { arith_simps }, try { simp only [arg0, arg1, arg2, arg3] },\n      try { simp only [h_call3_ap_offset] },\n      try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },}, },\n  intros \u03ba_call9 ap9 h_call9,\n  rcases h_call9 with \u27e8h_call9_ap_offset, h_call9\u27e9,\n  rcases h_call9 with \u27e8rc_m9, rc_mle9, hl_range_check_ptr\u2082, h_call9\u27e9,\n  generalize' hr_rev_range_check_ptr\u2082: mem (ap9 - 1) = range_check_ptr\u2082,\n  have htv_range_check_ptr\u2082 := hr_rev_range_check_ptr\u2082.symm, clear hr_rev_range_check_ptr\u2082,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3] at hl_range_check_ptr\u2082 },\n  rw [\u2190htv_range_check_ptr\u2082, \u2190htv_range_check_ptr\u2081] at hl_range_check_ptr\u2082,\n  try { simp only [arg0 ,arg1 ,arg2 ,arg3] at h_call9 },\n  rw [\u2190htv_range_check_ptr\u2081, hl_range_check_ptr\u2081, hin_range_check_ptr] at h_call9,\n  clear arg0 arg1 arg2 arg3,\n  -- return\n  step_assert_eq hpc9 with hret0,\n  step_assert_eq hpc10 with hret1,\n  step_assert_eq hpc11 with hret2,\n  step_assert_eq hpc12 with hret3,\n  step_assert_eq hpc13 with hret4,\n  step_assert_eq hpc14 with hret5,\n  step_ret hpc15,\n  -- finish\n  step_done, use_only [rfl, rfl],\n  split,\n  { try { simp only [h_call3_ap_offset ,h_call9_ap_offset] },\n    try { arith_simps }, try { refl } },\n  -- range check condition\n  use_only (rc_m3+rc_m9+0+0), split,\n  linarith [rc_mle3, rc_mle9],\n  split,\n  { arith_simps, try { simp only [hret0 ,hret1 ,hret2 ,hret3 ,hret4 ,hret5] },\n    rw [\u2190htv_range_check_ptr\u2082, hl_range_check_ptr\u2082, hl_range_check_ptr\u2081, hin_range_check_ptr],\n    try { arith_simps, refl <|> norm_cast }, try { refl } },\n  intro rc_h_range_check_ptr, repeat { rw [add_assoc] at rc_h_range_check_ptr },\n  have rc_h_range_check_ptr' := range_checked_add_right rc_h_range_check_ptr,\n  -- Final Proof\n  -- user-provided reduction\n  suffices auto_spec: auto_spec_ec_negate mem _ range_check_ptr point _ _,\n  { apply sound_ec_negate, apply auto_spec },\n  -- prove the auto generated assertion\n  dsimp [auto_spec_ec_negate],\n  try { norm_num1 }, try { arith_simps },\n  use_only [\u03ba_call3],\n  use_only [range_check_ptr\u2081],\n  use_only [minus_y],\n  have rc_h_range_check_ptr\u2081 := range_checked_offset' rc_h_range_check_ptr,\n  have rc_h_range_check_ptr\u2081' := range_checked_add_right rc_h_range_check_ptr\u2081, try { norm_cast at rc_h_range_check_ptr\u2081' },\n  have spec3 := h_call3 rc_h_range_check_ptr',\n  rw [\u2190hin_range_check_ptr, \u2190htv_range_check_ptr\u2081] at spec3,\n  try { dsimp at spec3, arith_simps at spec3 },\n  use_only [spec3],\n  use_only [\u03ba_call9],\n  use_only [range_check_ptr\u2082],\n  have rc_h_range_check_ptr\u2082 := range_checked_offset' rc_h_range_check_ptr\u2081,\n  have rc_h_range_check_ptr\u2082' := range_checked_add_right rc_h_range_check_ptr\u2082, try { norm_cast at rc_h_range_check_ptr\u2082' },\n  have spec9 := h_call9 rc_h_range_check_ptr\u2081',\n  rw [\u2190hin_range_check_ptr, \u2190hl_range_check_ptr\u2081, \u2190htv_range_check_ptr\u2082] at spec9,\n  try { dsimp at spec9, arith_simps at spec9 },\n  use_only [spec9],\n  try { split, linarith },\n  try { ensures_simps; try { simp only [add_neg_eq_sub, hin_range_check_ptr, hin_point, htv_range_check_ptr\u2081, htv_minus_y, htv_range_check_ptr\u2082] }, },\n  try { dsimp [cast_EcPoint, cast_BigInt3] },\n  try { arith_simps }, try { simp only [hret0, hret1, hret2, hret3, hret4, hret5] },\n  try { simp only [h_call3_ap_offset, h_call9_ap_offset] },\n  try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },\nend\n\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_ec_negate_soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3625295254065659}}
{"text": "import data.list\nimport data.pfun\n\n/-! A simple parser combinator library\n    Similar to http://dev.stephendiehl.com/fun/002_parsers.html\n-/\n\nuniverses u v\n\nstructure Parser (a : Type u) :=\n  mk :: (parse : string -> list (a \u00d7 string))\n\ndef runParser {a} (m : Parser a) (s : string) : a \u2295 string :=\nmatch m.parse s with\n| [(res, \"\")] := sum.inl res\n| [(_, rs)] := sum.inr \"Parser did not consume entire stream.\"\n| [] := sum.inr \"No matches.\"\n| _ := sum.inr \"Ambiguous.\"\nend\n\ninstance parser_has_pure : has_pure Parser :=\n  { pure := \u03bb \u03b1 a, Parser.mk (\u03bb s, [(a, s)]) }\n\ninstance parser_has_bind : has_bind Parser :=\n  { bind := \u03bb \u03b1 \u03b2 p f, Parser.mk (\u03bb s, (p.parse s) >>= \u03bb \u27e8a, s'\u27e9, (f a).parse s') }\n\ninstance parser_functor : functor Parser :=\n  { map := \u03bb \u03b1 \u03b2 f p, p >>= (pure \u2218 f) }\n\ninstance parser_has_seq : has_seq Parser :=\n  { seq := \u03bb \u03b1 \u03b2 p q, p >>= \u03bb f, f <$> q }\n\ninstance parser_applicative : applicative Parser := {}\ninstance parser_monad : monad Parser := {}\n\ndef mplus {a} (p : Parser a) (q : Parser a) : Parser a :=\n  Parser.mk $ \u03bb s, p.parse s ++ q.parse s\n\ninstance parser_has_orelse : has_orelse Parser := \n  { orelse := \u03bb _ p q,\n      Parser.mk $ \u03bb s, match p.parse s with\n        | [] := q.parse s\n        | res := res\n        end }\n\ninstance parser_alternative : alternative Parser := \n  { failure := \u03bb _, Parser.mk $ \u03bb _, [] }\n\n/-- Some higher order parsers like many, many1, .. only terminate \n    for some argument parsers. Therefore we factor the termination\n    out into a type class. -/\ndef parser_rel {a} (p : Parser a) : string -> string -> Prop :=\n  \u03bb s1 s2, s1 \u2208 (\u03bb x : (a \u00d7 string), x.2) <$> (p.parse s2)\n\nclass productive {a} (p : Parser a) := \n  (produces : well_founded (parser_rel p))\n\ndef list_bind_contains {a b} : \u03a0 (ls : list a) (f : \u03a0 x, x \u2208 ls -> list b), list b\n| [] f := []\n| (l::ls) f := f l (by simp) ++ list_bind_contains ls (\u03bb x h, f x (by simp [h]))\n\n--  (::) <$> v <*> (many1 v <|> pure [])\ndef many1_rec {a} (v : Parser a)\n  : \u03a0 s, (\u03a0 y, parser_rel v y s -> list (list a \u00d7 string)) -> list (list a \u00d7 string) :=\n  \u03bb s, \u03bb rec, list_bind_contains (v.parse s) $ \u03bb \u27e8a', s'\u27e9 h, \n    let h2 : parser_rel v s' s := by { simp [parser_rel], exact \u27e8a', h\u27e9, }\n    in match (rec s' h2) with\n    | [] := [\u27e8[a'], s'\u27e9]\n    | res := (\u03bb x : (list a \u00d7 string), \u27e8a' :: x.1, x.2\u27e9) <$> res\n    end\n\ndef many1 {a} (v : Parser a) [d : productive v] : Parser (list a) :=\n  Parser.mk $ \u03bb s, @well_founded.fix _ _ _ d.produces (many1_rec v) s\n\ndef many {a} (v : Parser a) [productive v] : Parser (list a) := \n  many1 v <|> pure []\n\ndef item : Parser char :=\n  Parser.mk $ \u03bb s, match string.to_list s with\n    | [] := []\n    | (c::cs) := [(c, list.as_string cs)]\n    end\n\nlemma list_string_id {x} : list.as_string (string.to_list x) = x :=\nbegin\n  cases x, rw [string.to_list, list.as_string],\nend\n\nlemma string_list_id {x} : string.to_list (list.as_string x) = x :=\nbegin\n  cases x, repeat { simp [list.as_string, string.to_list], },\nend\n\nlemma item_productive_acc : \u2200 l, acc (parser_rel item) (list.as_string l)\n| [] := acc.intro _ (\u03bb y h, \n  begin \n    simp [parser_rel, item, (<$>), string_list_id] at h,\n    from false.elim h\n  end)\n| (l::ls) := acc.intro _ (\u03bb y h,\n  begin\n    simp [parser_rel, item, string_list_id, (<$>)] at h,\n    let h2 := item_productive_acc ls,\n    rwa [h.symm] at h2,\n  end)\n\ninstance item_productive : productive item :=\n  { produces := \n    begin\n      apply well_founded.intro, intro s,\n      from eq.mp (by simp [list_string_id]) (item_productive_acc (string.to_list s))\n    end }\n\ndef transform {a} (p : char -> option a) : Parser a :=\n  item >>= \u03bb c, match p c with\n    | some r := pure r\n    | none := Parser.mk (\u03bb _, []) \n    end\n\nlemma transform_productive_acc {a f} \n  : \u2200 l, acc (parser_rel (@transform a f)) (list.as_string l)\n| [] := acc.intro _ (\u03bb y h, \n  begin\n    simp [parser_rel, transform, (>>=), item, string_list_id, (<$>)] at h,\n    from false.elim h\n  end)\n| (l::ls) := acc.intro _ (\u03bb y h,\n  begin\n    simp [parser_rel, transform, (>>=), item, string_list_id, parser_has_bind] at h,\n    cases f l, { simp [transform] at h, from false.elim h },\n    simp [transform, pure] at h,\n    let h2 := transform_productive_acc ls,\n    rwa [h.symm] at h2,\n  end)\n\ninstance transform_productive {a f} : productive (@transform a f) :=\n  { produces :=\n    begin\n      apply well_founded.intro, intro s,\n      from eq.mp (by simp [list_string_id]) (transform_productive_acc (string.to_list s))\n    end }\n\ndef satisfy (p : char -> bool) : Parser char :=\n  transform (\u03bb c, if p c then some c else none)\n\ninstance satisfy_productive {f} : productive (satisfy f) :=\n  { produces := begin simp [satisfy], from transform_productive.produces, end }\n\ndef oneOf (ls : list char) : Parser char := satisfy (\u03bb c, c \u2208 ls)\n\ninstance oneOf_productive {ls} : productive (oneOf ls) :=\n  { produces := begin simp [oneOf], from satisfy_productive.produces, end }\n\ndef spaces : Parser (list char) := many (oneOf [' ', '\\t', '\\n'])\n\n/-- The 'char' parser. -/\ndef character (c : char) : Parser char := satisfy (\u03bb d, c = d)\n\ndef asDigit : char -> option nat\n| '0' := some 0 | '1' := some 1 | '2' := some 2 | '3' := some 3 | '4' := some 4\n| '5' := some 5 | '6' := some 6 | '7' := some 7 | '8' := some 8 | '9' := some 9\n| _ := none\n\ndef isDigit (c : char) : bool := option.is_some (asDigit c)\n\ndef digit : Parser char := satisfy isDigit\n\ndef token {a} (p : Parser a) : Parser a :=\n  p >>= \u03bb a, spaces >> pure a\n\n/-- The 'string' parser. -/\ndef charlist : list char -> Parser (list char)\n| [] := pure []\n| (c::cs) := character c >> charlist cs >> pure (c :: cs)\n\ndef reserved (s : string) : Parser string :=\n  token (charlist (string.to_list s)) >> pure s\n\ndef parens {a} (p : Parser a) : Parser a :=\n  reserved \"(\" >> p >>= \u03bb r, reserved \")\" >> pure r\n\ndef combineNum : \u03a0 (ls : list nat), nat\n| [] := 0\n| (l::ls) := l * (10 ^ ls.length) + combineNum ls\n\ndef natural : Parser nat :=\n   combineNum <$> many1 (transform asDigit)\n\ndef integer : Parser int :=\n  (character '-' <|> pure '+') >>= \u03bb c,\n  (\u03bb n : nat, if c = '-' then -n else n) <$> natural\n\n-- chainl :: Parser a -> Parser (a -> a -> a) -> a -> Parser a\n-- chainl p op a = (p `chainl1` op) <|> return a\n\n-- chainl1 :: Parser a -> Parser (a -> a -> a) -> Parser a\n-- p `chainl1` op = do {a <- p; rest a}\n  -- where rest a = (do f <- op\n                     -- b <- p\n                     -- rest (f a b))\n                 -- <|> return a", "meta": {"author": "anfelor", "repo": "coc-lean", "sha": "fdd967d2b7bc349202a1deabbbce155eed4db73a", "save_path": "github-repos/lean/anfelor-coc-lean", "path": "github-repos/lean/anfelor-coc-lean/coc-lean-fdd967d2b7bc349202a1deabbbce155eed4db73a/src/Parser.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.3624728563889899}}
{"text": "/-\nCopyright (c) 2019 The Flypitch Project. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n\nAuthors: Jesse Han, Floris van Doorn\n-/\nimport .zfc\nopen fol\nopen zfc\n\nlocal infix `\u2208'`:100 := bounded_formula_of_relation ZFC_el\nlocal notation h :: t  := dvector.cons h t\nlocal notation `[` l:(foldr `, ` (h t, dvector.cons h t) dvector.nil `]`:0) := l\n\ndef L_ZFC_structure_of_Set : Structure L_ZFC :=\nbegin\n  refine \u27e8Set,_,_\u27e9,\n  {intros n f, repeat{cases f}},\n  {intros n r v, cases r, cases v, cases v_xs, exact v_x \u2208 v_xs_x}\nend\n\nlocal notation `Set'` := L_ZFC_structure_of_Set\n\ninstance has_mem_Set'_Set' : (has_mem Set' Set') := \u27e8Set.mem\u27e9\n\ninstance has_mem_Set_Set' : has_mem Set.{0} Set' := \u27e8Set.mem\u27e9\n\ninstance has_emptyc_Set' : has_emptyc Set' := \u27e8Set.empty\u27e9\n\nlemma  empty_subset : \u2200 x: Set',  Set.empty \u2286 x := by tidy \n\nlemma Set'_Set : \u21a5Set' = Set := by refl\n\n@[simp]lemma Set'_Set_2 {p : (by exact \u21a5Set') \u2192 Prop} : (\u03bb x, p x) = \u03bb x : Set, p x := by refl\n\nlemma Set'_has_mem : has_mem \u21a5Set' \u21a5Set' := \u27e8 Set.mem \u27e9\nlocal notation h :: t  := dvector.cons h t\nlocal notation `[` l:(foldr `, ` (h t, dvector.cons h t) dvector.nil `]`) := l\n\nexample : Set' \u22a8 (\u2203' is_emptyset) :=\nbegin\n  unfold is_emptyset, simp[realize_sentence_ex], refine \u27e8\u2205, _\u27e9, tidy\nend\n\n@[simp]lemma Set'_rel_mem {x y : Set} :  (Structure.rel_map Set' ZFC_rel.\u03f5 ( [x,y] ) ) = (x \u2208 y) := by tidy\n \n@[simp] lemma Set'_mem_mem {n : \u2115} {x y : bounded_term L_ZFC n} {v : dvector \u21a5Set' n} : realize_bounded_formula v (x \u2208' y) dvector.nil = ((realize_bounded_term v x dvector.nil) \u2208 (realize_bounded_term v y dvector.nil)) := begin\n  simp [ZFC_el, bounded_formula_of_relation, realize_bounded_formula, bd_apps_rel] end\n\n@[simp] lemma realize_term_remove_irrel {n : \u2115} {L : Language} {S : Structure L} {v : dvector \u21a5S (n+1)} {j : fin n} {k : \u2115} {p : k > j} : realize_bounded_term v ((lift_bounded_term_at (bd_var j : bounded_term L n) 1 k) ) dvector.nil = realize_bounded_term (dvector.remove_mth k v)  (bd_var j) dvector.nil :=\nbegin\nsorry\nend\n\nset_option trace.check true\n \n\n@[simp]lemma Set'_realize_subset_2 : \u2200 x y : Set, @realize_bounded_formula L_ZFC Set' 2 0 (x :: y :: dvector.nil) subset  dvector.nil  = Set.subset x y:=\nbegin\n  simp only [subset, Set'_mem_mem, fol.realize_bounded_formula, fol.realize_bounded_term, dvector.nth],\n  intros, conv {to_lhs, change \u2200 z, z \u2208 x \u2192 z \u2208 y}, rw [Set.subset]\nend\nset_option trace.app_builder true\n\n@[simp]lemma Set'_realize_is_empty : \u2200 x, (@realize_bounded_formula L_ZFC Set' _ _ ([x]) is_emptyset dvector.nil \u2194 x = \u2205) :=\nbegin\n  intro x, unfold is_emptyset,\n   simp only [not_exists, fol.realize_bounded_formula_ex, Set'_mem_mem,\n   fol.realize_bounded_formula, fol.realize_bounded_formula_not, fol.realize_bounded_term],\n   symmetry, apply Set.eq_empty\nend\n\n@[simp]lemma Set'_realize_empty : @realize_bounded_formula L_ZFC Set' 1 0 (Set.empty :: dvector.nil) is_emptyset dvector.nil :=\nby rw[Set'_realize_is_empty]; refl\n\n-- set_option trace.simplify.rewrite true\n\nlemma Set'_models_extensionality : axiom_of_extensionality \u2208 Th Set' := \nbegin\n  simp [Th, axiom_of_extensionality, fin.val, has_one.one, fin.of_nat],\n  intros x y, intro h, apply Set.ext, intro z, revert h,  intro h, have := h z, exact this\nend\n\nlemma Set'_models_union : axiom_of_union \u2208 Th Set' := \nbegin\n  simp only [Th, axiom_of_union, small], intro x,\n  conv {congr, skip, congr, congr, congr, skip,\n       change (\u2203' (&1 \u2208' &0 \u2293 &0 \u2208' &3) : bounded_formula L_ZFC 3)},\n  simp, change \u2203 U, \u2200 z, z \u2208 U \u2194 \u2203 w, z \u2208 w \u2227 w \u2208 x, \n  refine \u27e8\u22c3 x, _\u27e9, intro z, rw[@Set.mem_Union x z], finish\nend\n\nlemma Set'_models_powerset : axiom_of_powerset \u2208 Th Set' := \nbegin\n  simp only [Th, axiom_of_powerset, small,  fol.realize_bounded_formula_ex,\n fol.realize_bounded_formula, realize_bounded_formula_biimp, set.mem_set_of_eq],\n  intros,\n  refine \u27e8Set.powerset x, _\u27e9,\n  intro y, change y \u2208 Set.powerset x \u2194 Set.subset y x,\n  exact Set.mem_powerset,\nend\n\nlemma Set'_models_choice : axiom_of_choice \u2208 Th Set' := sorry\n\nlemma Set'_models_infinity : axiom_of_infinity \u2208 Th Set' :=\nbegin\n  simp [has_mem.mem, set.mem, Th, set_of, axiom_of_infinity,satisfied_in], \n  refine \u27e8Set.omega, _\u27e9, refine \u27e8_,_\u27e9, refine \u27e8Set.empty,_\u27e9,\n  refine \u27e8Set'_realize_empty, by {change Set.empty \u2208 Set.omega, exact Set.omega_zero}\u27e9,\n  intros, refine \u27e8Set.insert x x, _\u27e9, refine \u27e8_,@Set.omega_succ x a\u27e9,\n  exact (iff.mpr Set.mem_insert (or.inl (refl x)))\nend\n\nlemma Set'_models_infinity' : axiom_of_infinity' \u2208 Th Set' :=\nbegin\n  simp [has_mem.mem, set.mem, Th, set_of, axiom_of_infinity' ,satisfied_in,realize_cast_bounded_formula], \n  refine \u27e8Set.omega, _\u27e9, intro x, refine \u27e8_,_\u27e9,\n  {intro H_x, change Set.mem x _, change _ \u2208 Set.omega,\n  suffices : x = Set.empty, by rw[this]; apply Set.omega_zero, exact H_x},\n  intro y, conv in (dvector.nth _ _ _) {change x}, conv in (dvector.nth _ _ _) {change y},\n  change Set.mem y Set.omega \u2192 _, intro H,\n  exact \u27e8Set.insert y y, \u27e8iff.mpr Set.mem_insert (or.inl (by refl)), @Set.omega_succ y H\u27e9\u27e9\nend\n\nlemma Set'_models_shallow_infinity : Set_axiom_of_infinity :=\nbegin\n  unfold Set_axiom_of_infinity Set_is_emptyset, refine \u27e8Set.omega, _\u27e9,\n  split,\n    {refine \u27e8\u2205, \u27e8_, Set.omega_zero\u27e9\u27e9, apply Set.mem_empty},\n    {intros z H, refine \u27e8Set.insert z z, \u27e8iff.mpr Set.mem_insert (or.inl (by refl)), @Set.omega_succ z H\u27e9\u27e9},\nend\n\nlemma shallow_infinity_iff_shallow_infinity' : Set_axiom_of_infinity \u2194 Set_axiom_of_infinity' :=\n  sorry\n\nlemma Set'_infinity_shallow_infinity : Set'[axiom_of_infinity] \u2194 Set_axiom_of_infinity  :=\n  sorry\n\nlemma Set'_infinity'_shallow_infinity' :  Set'[axiom_of_infinity'] \u2194 Set_axiom_of_infinity' :=\n  sorry\n \nexample : axiom_of_infinity \u2208 Th(Set') :=\n  Set'_infinity_shallow_infinity.mpr Set'_models_shallow_infinity\n\nexample : axiom_of_infinity' \u2208 Th(Set') :=\n      Set'_infinity'_shallow_infinity'.mpr $ shallow_infinity_iff_shallow_infinity'.mp Set'_models_shallow_infinity\n\n@[simp]lemma Set'_functional_rw : \u2200 c : bounded_formula L_ZFC 2, @realize_bounded_formula L_ZFC Set' 0 0 dvector.nil (functional c) dvector.nil = \u2200 x : Set, \u2203 y : Set, \u2200 w, @realize_bounded_formula L_ZFC Set' 3 0  ([w,y,x]) ( (c \u2191' 1 # 1) \u21d4 &0 \u2243 &1)  dvector.nil := \nbegin\nintros c,\next1, simp at *, rw [functional, realize_bounded_formula],\nsimp [realize_bounded_formula_ex],\nrefl\nend\n\nlemma Set'_functional : \u2200 c, @realize_bounded_formula L_ZFC Set' 0 0 dvector.nil (functional c) dvector.nil \u2192 Set \u2192 Set :=\nbegin\n  intro c, rw[Set'_functional_rw], intros h s, sorry\nend\n\nlemma Set'_models_replacement: \u2200 c : bounded_formula L_ZFC 2, axiom_of_replacement c \u2208 Th Set' := \nbegin\nintro c,\nsimp only [has_mem.mem,set.mem,Th,set_of,axiom_of_replacement],\nintros a x,\nsorry\nend\n\nlemma Set_extends_ZFC : ZFC \u2286 Th Set' :=\nbegin\nintros f hf, cases hf with zf choice,\nrepeat{cases zf},\n  exact Set'_models_infinity, exact Set'_models_powerset,\n  exact Set'_models_union, exact Set'_models_extensionality,\n  dsimp at zf_h, cases zf_h with a b, subst b,\n  revert zf_w, simp, exact Set'_models_replacement, \n  repeat{cases choice}, exact Set'_models_choice\nend\n", "meta": {"author": "flypitch", "repo": "flypitch", "sha": "aea5800db1f4cce53fc4a113711454b27388ecf8", "save_path": "github-repos/lean/flypitch-flypitch", "path": "github-repos/lean/flypitch-flypitch/flypitch-aea5800db1f4cce53fc4a113711454b27388ecf8/old/zfc_consistent.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3624728563889899}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.metric_space.baire\nimport Mathlib.analysis.normed_space.operator_norm\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Banach open mapping theorem\n\nThis file contains the Banach open mapping theorem, i.e., the fact that a bijective\nbounded linear map between Banach spaces has a bounded inverse.\n-/\n\n/--\nFirst step of the proof of the Banach open mapping theorem (using completeness of `F`):\nby Baire's theorem, there exists a ball in `E` whose image closure has nonempty interior.\nRescaling everything, it follows that any `y \u2208 F` is arbitrarily well approached by\nimages of elements of norm at most `C * \u2225y\u2225`.\nFor further use, we will only need such an element whose image\nis within distance `\u2225y\u2225/2` of `y`, to apply an iterative process. -/\ntheorem exists_approx_preimage_norm_le {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2}\n    [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F]\n    (f : continuous_linear_map \ud835\udd5c E F) [complete_space F] (surj : function.surjective \u21d1f) :\n    \u2203 (C : \u211d),\n        \u2203 (H : C \u2265 0),\n          \u2200 (y : F), \u2203 (x : E), dist (coe_fn f x) y \u2264 1 / bit0 1 * norm y \u2227 norm x \u2264 C * norm y :=\n  sorry\n\n/-- The Banach open mapping theorem: if a bounded linear map between Banach spaces is onto, then\nany point has a preimage with controlled norm. -/\ntheorem exists_preimage_norm_le {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2}\n    [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F]\n    (f : continuous_linear_map \ud835\udd5c E F) [complete_space F] [complete_space E]\n    (surj : function.surjective \u21d1f) :\n    \u2203 (C : \u211d), \u2203 (H : C > 0), \u2200 (y : F), \u2203 (x : E), coe_fn f x = y \u2227 norm x \u2264 C * norm y :=\n  sorry\n\n/-- The Banach open mapping theorem: a surjective bounded linear map between Banach spaces is\nopen. -/\ntheorem open_mapping {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E]\n    [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F]\n    (f : continuous_linear_map \ud835\udd5c E F) [complete_space F] [complete_space E]\n    (surj : function.surjective \u21d1f) : is_open_map \u21d1f :=\n  sorry\n\nnamespace linear_equiv\n\n\n/-- If a bounded linear map is a bijection, then its inverse is also a bounded linear map. -/\ntheorem continuous_symm {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E]\n    [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] [complete_space F]\n    [complete_space E] (e : linear_equiv \ud835\udd5c E F) (h : continuous \u21d1e) : continuous \u21d1(symm e) :=\n  sorry\n\n/-- Associating to a linear equivalence between Banach spaces a continuous linear equivalence when\nthe direct map is continuous, thanks to the Banach open mapping theorem that ensures that the\ninverse map is also continuous. -/\ndef to_continuous_linear_equiv_of_continuous {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c]\n    {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F]\n    [normed_space \ud835\udd5c F] [complete_space F] [complete_space E] (e : linear_equiv \ud835\udd5c E F)\n    (h : continuous \u21d1e) : continuous_linear_equiv \ud835\udd5c E F :=\n  continuous_linear_equiv.mk (mk (to_fun e) sorry sorry (inv_fun e) sorry sorry)\n\n@[simp] theorem coe_fn_to_continuous_linear_equiv_of_continuous {\ud835\udd5c : Type u_1}\n    [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3}\n    [normed_group F] [normed_space \ud835\udd5c F] [complete_space F] [complete_space E]\n    (e : linear_equiv \ud835\udd5c E F) (h : continuous \u21d1e) :\n    \u21d1(to_continuous_linear_equiv_of_continuous e h) = \u21d1e :=\n  rfl\n\n@[simp] theorem coe_fn_to_continuous_linear_equiv_of_continuous_symm {\ud835\udd5c : Type u_1}\n    [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3}\n    [normed_group F] [normed_space \ud835\udd5c F] [complete_space F] [complete_space E]\n    (e : linear_equiv \ud835\udd5c E F) (h : continuous \u21d1e) :\n    \u21d1(continuous_linear_equiv.symm (to_continuous_linear_equiv_of_continuous e h)) = \u21d1(symm e) :=\n  rfl\n\nend linear_equiv\n\n\nnamespace continuous_linear_equiv\n\n\n/-- Convert a bijective continuous linear map `f : E \u2192L[\ud835\udd5c] F` between two Banach spaces\nto a continuous linear equivalence. -/\ndef of_bijective {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2} [normed_group E]\n    [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F] [complete_space F]\n    [complete_space E] (f : continuous_linear_map \ud835\udd5c E F) (hinj : continuous_linear_map.ker f = \u22a5)\n    (hsurj : continuous_linear_map.range f = \u22a4) : continuous_linear_equiv \ud835\udd5c E F :=\n  linear_equiv.to_continuous_linear_equiv_of_continuous (linear_equiv.of_bijective (\u2191f) hinj hsurj)\n    sorry\n\n@[simp] theorem coe_fn_of_bijective {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c] {E : Type u_2}\n    [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F] [normed_space \ud835\udd5c F]\n    [complete_space F] [complete_space E] (f : continuous_linear_map \ud835\udd5c E F)\n    (hinj : continuous_linear_map.ker f = \u22a5) (hsurj : continuous_linear_map.range f = \u22a4) :\n    \u21d1(of_bijective f hinj hsurj) = \u21d1f :=\n  rfl\n\n@[simp] theorem of_bijective_symm_apply_apply {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c]\n    {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F]\n    [normed_space \ud835\udd5c F] [complete_space F] [complete_space E] (f : continuous_linear_map \ud835\udd5c E F)\n    (hinj : continuous_linear_map.ker f = \u22a5) (hsurj : continuous_linear_map.range f = \u22a4) (x : E) :\n    coe_fn (continuous_linear_equiv.symm (of_bijective f hinj hsurj)) (coe_fn f x) = x :=\n  symm_apply_apply (of_bijective f hinj hsurj) x\n\n@[simp] theorem of_bijective_apply_symm_apply {\ud835\udd5c : Type u_1} [nondiscrete_normed_field \ud835\udd5c]\n    {E : Type u_2} [normed_group E] [normed_space \ud835\udd5c E] {F : Type u_3} [normed_group F]\n    [normed_space \ud835\udd5c F] [complete_space F] [complete_space E] (f : continuous_linear_map \ud835\udd5c E F)\n    (hinj : continuous_linear_map.ker f = \u22a5) (hsurj : continuous_linear_map.range f = \u22a4) (y : F) :\n    coe_fn f (coe_fn (continuous_linear_equiv.symm (of_bijective f hinj hsurj)) y) = y :=\n  apply_symm_apply (of_bijective f hinj hsurj) y\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/analysis/normed_space/banach_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3624728563889898}}
{"text": "import condensed.adjunctions\nimport condensed.extr.equivalence\nimport condensed.short_exact\nimport condensed.acyclic\n\nimport for_mathlib.AddCommGroup.tensor\n\nnoncomputable theory\n\nuniverses u\nopen_locale tensor_product\n\nopen category_theory\n\nnamespace ExtrSheafProd\n\n@[simps obj map]\ndef tensor_presheaf (M : ExtrDisc.{u}\u1d52\u1d56 \u2964 Ab.{u+1}) (A : Ab.{u+1}) :\n  ExtrDisc.{u}\u1d52\u1d56 \u2964 Ab.{u+1} :=\nM \u22d9 AddCommGroup.tensor_functor.flip.obj A\n\n@[simps val]\ndef tensor (M : ExtrSheafProd.{u} Ab.{u+1}) (A : Ab.{u+1}) :\n  ExtrSheafProd.{u} Ab.{u+1} :=\n{ val := tensor_presheaf M.val A,\n  cond := begin\n    introsI \u03b1 _ X, dsimp [tensor_presheaf, AddCommGroup.tensor_functor],\n    let e := _, change is_iso e,\n    have hq := M.cond _ X, dsimp at hq, let q := _, change is_iso q at hq,\n    have he : e = AddCommGroup.map_tensor q (\ud835\udfd9 _) \u226b\n      AddCommGroup.tensor_pi_comparison _ _,\n    { ext1 j,\n      dsimp [AddCommGroup.tensor_pi_comparison],\n      simp only [\u2190AddCommGroup.map_tensor_comp_left, limits.limit.lift_\u03c0,\n        limits.fan.mk_\u03c0_app, category.assoc]},\n    rw he, resetI, apply_with is_iso.comp_is_iso { instances := ff },\n    swap, apply_instance,\n    use AddCommGroup.map_tensor (inv q) (\ud835\udfd9 _),\n    split,\n    { rw [\u2190 AddCommGroup.map_tensor_comp_left, is_iso.hom_inv_id, AddCommGroup.map_tensor_id], },\n    { rw [\u2190 AddCommGroup.map_tensor_comp_left, is_iso.inv_hom_id, AddCommGroup.map_tensor_id], },\n  end } -- tensor products commutes with direct sums.\n\n@[simps val_app]\ndef map_tensor {M M' : ExtrSheafProd.{u} Ab.{u+1}} {A A' : AddCommGroup.{u+1}}\n  (f : M \u27f6 M') (g : A \u27f6 A') :\n  M.tensor A \u27f6 M'.tensor A' := ExtrSheafProd.hom.mk $\n{ app := \u03bb S, AddCommGroup.map_tensor (f.val.app _) g,\n  naturality' := begin\n    intros X Y i,\n    dsimp [tensor, tensor_presheaf],\n    simp only [\u2190 AddCommGroup.map_tensor_comp_comp, category.id_comp, category.comp_id,\n      f.val.naturality],\n  end }\n\n@[simp]\nlemma map_tensor_id (M : ExtrSheafProd.{u} Ab.{u+1}) (A : AddCommGroup.{u+1}) :\n  map_tensor (\ud835\udfd9 M) (\ud835\udfd9 A) = \ud835\udfd9 _ :=\nby { ext : 3, dsimp, simp }\n\n@[simp]\nlemma map_tensor_comp {M M' M'' : ExtrSheafProd.{u} Ab.{u+1}}\n  {A A' A'' : AddCommGroup.{u+1}}\n  (f : M \u27f6 M') (f' : M' \u27f6 M'')\n  (g : A \u27f6 A') (g' : A' \u27f6 A'') :\n  map_tensor (f \u226b f') (g \u226b g') = map_tensor f g \u226b map_tensor f' g' :=\nby { ext : 3, dsimp, simp }\n\n-- Slow, so probably break into pieces\n@[simps]\ndef tensor_functor : ExtrSheafProd.{u} Ab.{u+1} \u2964 Ab.{u+1} \u2964 ExtrSheafProd.{u} Ab.{u+1} :=\n{ obj := \u03bb M,\n  { obj := \u03bb A, M.tensor A,\n    map := \u03bb A A' f, map_tensor (\ud835\udfd9 _) f,\n    map_id' := \u03bb X, by simp,\n    map_comp' := \u03bb X Y Z f g, begin\n      nth_rewrite 0 [\u2190 category.id_comp (\ud835\udfd9 M)],\n      rw map_tensor_comp,\n    end },\n  map := \u03bb M N f,\n  { app := \u03bb A, map_tensor f (\ud835\udfd9 _),\n    naturality' := \u03bb A B g, begin\n      dsimp,\n      simp only [\u2190 map_tensor_comp, category.id_comp, category.comp_id],\n    end },\n  map_id' := \u03bb M, begin\n    ext : 2,\n    simp,\n  end,\n  map_comp' := \u03bb M N L f g, begin\n    ext x : 2,\n    dsimp,\n    nth_rewrite 0 [\u2190 category.comp_id (\ud835\udfd9 x)],\n    rw [map_tensor_comp],\n  end }\n.\n\n@[simps]\ninstance hom_has_add {M N : ExtrSheafProd.{u} Ab.{u+1}} : has_add (M \u27f6 N) :=\n\u27e8\u03bb f g, \u27e8f.val + g.val\u27e9\u27e9\n\n@[simps]\ninstance hom_has_zero {M N : ExtrSheafProd.{u} Ab.{u+1}} : has_zero (M \u27f6 N) :=\n\u27e8\u27e80\u27e9\u27e9\n\n@[simps]\ninstance hom_has_neg {M N : ExtrSheafProd.{u} Ab.{u+1}} : has_neg (M \u27f6 N) :=\n\u27e8\u03bb f, \u27e8-f.val\u27e9\u27e9\n\n@[simps]\ninstance hom_has_sub {M N : ExtrSheafProd.{u} Ab.{u+1}} : has_sub (M \u27f6 N) :=\n\u27e8\u03bb f g, \u27e8f.val - g.val\u27e9\u27e9\n\ninstance preadditive : preadditive (ExtrSheafProd.{u} Ab.{u+1}) :=\n{ hom_group := \u03bb P Q,\n  { add_assoc := \u03bb f g h, by { ext1, dsimp, rw add_assoc },\n    zero_add := \u03bb f, by { ext1, dsimp, rw zero_add },\n    add_zero := \u03bb f, by { ext1, dsimp, rw add_zero },\n    nsmul := \u03bb n f, \u27e8n \u2022 f.val\u27e9,\n    nsmul_zero' := \u03bb f, by { ext1, dsimp, simp, },\n    nsmul_succ' := \u03bb n f, by { ext1, dsimp, exact succ_nsmul f.val n },\n    sub_eq_add_neg := \u03bb f g, by { ext1, dsimp, exact sub_eq_add_neg f.val g.val },\n    zsmul := \u03bb n f, \u27e8n \u2022 f.val\u27e9,\n    zsmul_zero' := \u03bb f, by { ext1, dsimp, simp },\n    zsmul_succ' := \u03bb n f, by { ext1, dsimp, rw [nat.cast_succ, add_zsmul, one_zsmul, add_comm], },\n    zsmul_neg' := \u03bb n f, by { ext1, dsimp, simpa, },\n    add_left_neg := \u03bb f, by { ext1, dsimp, simp },\n    add_comm := \u03bb f g, by { ext1, dsimp, rw add_comm },\n    ..(infer_instance : has_add _),\n    ..(infer_instance : has_neg _),\n    ..(infer_instance : has_zero _),\n    ..(infer_instance : has_sub _) },\n  add_comp' := \u03bb P Q R f f' g, by { ext1, dsimp, simp },\n  comp_add' := \u03bb P Q R f g g', by { ext1, dsimp, simp } }\n\ndef evaluation (S : ExtrDisc.{u}) :\n  ExtrSheafProd.{u} Ab.{u+1} \u2964 Ab.{u+1} :=\nExtrSheafProd_to_presheaf _ \u22d9 (evaluation _ _).obj (opposite.op S)\n\ninstance evaluation_additive (S) : functor.additive (evaluation S) :=\n\u27e8\u03bb M N f g, rfl\u27e9\n\n@[simps]\ndef half_internal_hom (A : AddCommGroup.{u+1}) (M : ExtrSheafProd.{u} Ab.{u+1}) :\n  ExtrSheafProd.{u} Ab.{u+1} :=\n{ val :=\n  { obj := \u03bb S, AddCommGroup.of (A \u27f6 M.val.obj S),\n    map := \u03bb X Y f, (preadditive_yoneda.flip.obj (opposite.op A)).map $ M.val.map f,\n    map_id' := begin\n      intros S,\n      dsimp, simpa,\n    end,\n    map_comp' := begin\n      intros R S T f g,\n      dsimp,\n      simp,\n    end },\n  cond := begin\n    introsI \u03b1 _ X, dsimp,\n    let t := _, change is_iso t,\n    have := M.cond \u03b1 X, dsimp at this, let e := _, change is_iso e at this, resetI,\n    let q : AddCommGroup.of (A \u27f6 M.val.obj (opposite.op (ExtrDisc.sigma X))) \u2245\n      AddCommGroup.of (A \u27f6 (\u220f (\u03bb i, M.val.obj (opposite.op (X i))))) :=\n      (preadditive_yoneda.flip.obj (opposite.op A)).map_iso (as_iso e),\n    let s : AddCommGroup.of (A \u27f6 (\u220f (\u03bb i, M.val.obj (opposite.op (X i))))) \u27f6\n      \u220f (\u03bb i, AddCommGroup.of (A \u27f6 M.val.obj (opposite.op (X i)))) :=\n      limits.pi.lift (\u03bb i, (preadditive_yoneda.flip.obj (opposite.op A)).map\n        (limits.pi.\u03c0 _ i)),\n    have ht : t = q.hom \u226b s,\n    { dsimp [t, q, s, e], ext1,\n      simp only [limits.limit.lift_\u03c0, limits.fan.mk_\u03c0_app, category.assoc],\n      rw [\u2190 nat_trans.comp_app, \u2190 functor.map_comp, limits.limit.lift_\u03c0],\n      refl },\n    rw ht, clear ht,\n    suffices : is_iso s,\n    { resetI, apply is_iso.comp_is_iso },\n    -- Now we need to show that `Hom(A,(\u03a0 i, X i)) = \u03a0 i, Hom(A,X i)`.\n    apply AddCommGroup.is_iso_hom_product_comparison.{u u+1},\n  end }\n\ndef tensor_uncurry {A : AddCommGroup.{u+1}} {M N : ExtrSheafProd.{u} Ab.{u+1}}\n  (e : M \u27f6 half_internal_hom A N) :\n  tensor M A \u27f6 N := ExtrSheafProd.hom.mk $\n{ app := \u03bb S, AddCommGroup.tensor_uncurry $ e.val.app _,\n  naturality' := begin\n    intros X Y f,\n    erw \u2190 AddCommGroup.tensor_uncurry_comp_curry,\n    apply (AddCommGroup.tensor_curry_equiv _ _ _).injective,\n    erw (AddCommGroup.tensor_curry_equiv _ _ _).apply_symm_apply,\n    dsimp [AddCommGroup.tensor_curry_equiv],\n    erw [AddCommGroup.tensor_curry_uncurry_comp, \u2190 nat_trans.naturality,\n      \u2190 AddCommGroup.tensor_curry_equiv_apply,\n      \u2190 AddCommGroup.tensor_curry_equiv_symm_apply,\n      (AddCommGroup.tensor_curry_equiv _ _ _).apply_symm_apply],\n  end }\n\ndef tensor_curry {A : AddCommGroup.{u+1}} {M N : ExtrSheafProd.{u} Ab.{u+1}}\n  (e : M.tensor A \u27f6 N) : M \u27f6 half_internal_hom A N := ExtrSheafProd.hom.mk $\n{ app := \u03bb S, AddCommGroup.tensor_curry $ e.val.app _,\n  naturality' := begin\n    intros X Y f,\n    dsimp [half_internal_hom],\n    erw [\u2190 AddCommGroup.tensor_curry_uncurry_comp],\n    apply (AddCommGroup.tensor_curry_equiv _ _ _).symm.injective,\n    simp_rw \u2190 AddCommGroup.tensor_curry_equiv_apply,\n    rw (AddCommGroup.tensor_curry_equiv _ _ _).symm_apply_apply,\n    rw \u2190 AddCommGroup.tensor_curry_equiv_symm_apply,\n    rw (AddCommGroup.tensor_curry_equiv _ _ _).symm_apply_apply,\n    dsimp,\n    rw [AddCommGroup.tensor_uncurry_comp_curry, \u2190 nat_trans.naturality],\n    refl,\n  end }\n\nlemma tensor_curry_uncurry {A : AddCommGroup.{u+1}} {M N : ExtrSheafProd.{u} Ab.{u+1}}\n  (e : M \u27f6 half_internal_hom A N) :\n  tensor_curry (tensor_uncurry e) = e :=\nbegin\n  ext S : 3,\n  dsimp [tensor_curry, tensor_uncurry],\n  simp,\nend\n\nlemma tensor_uncurry_curry {A : AddCommGroup.{u+1}} {M N : ExtrSheafProd.{u} Ab.{u+1}}\n  (e : M.tensor A \u27f6 N) :\n  tensor_uncurry (tensor_curry e) = e :=\nbegin\n  ext S : 3,\n  dsimp [tensor_curry, tensor_uncurry],\n  simp,\nend\n\ninstance tensor_functor_additive (M : ExtrSheafProd.{u} Ab.{u+1}) :\n  (tensor_functor.obj M).additive := { }\n\ninstance tensor_functor_flip_additive (A : Ab.{u+1}) :\n  (tensor_functor.flip.obj A).additive := { }\n\nend ExtrSheafProd\n\nnamespace ExtrSheaf\n\ndef tensor (M : ExtrSheaf.{u} Ab.{u+1}) (A : AddCommGroup.{u+1}) :\n  ExtrSheaf.{u} Ab.{u+1} :=\n(ExtrSheaf_ExtrSheafProd_equiv _).inverse.obj $\n((ExtrSheaf_ExtrSheafProd_equiv _).functor.obj M).tensor A\n\n@[simp]\nlemma tensor_val_obj (M : ExtrSheaf.{u} Ab.{u+1}) (A : AddCommGroup.{u+1}) (T) :\n  (M.tensor A).val.obj T = (M.val.obj T).tensor A := rfl\n\ndef half_internal_hom (A : AddCommGroup.{u+1}) (M : ExtrSheaf.{u} Ab.{u+1}) :\n  ExtrSheaf.{u} Ab.{u+1} :=\n(ExtrSheaf_ExtrSheafProd_equiv _).inverse.obj $\n((ExtrSheaf_ExtrSheafProd_equiv _).functor.obj M).half_internal_hom A\n\n@[simp]\nlemma half_internal_hom_val_obj (A : AddCommGroup.{u+1}) (M : ExtrSheaf.{u} Ab.{u+1}) (T) :\n  (M.half_internal_hom A).val.obj T =\n  AddCommGroup.of (A \u27f6 M.val.obj T) := rfl\n\ndef tensor_curry {A : AddCommGroup.{u+1}} {M N : ExtrSheaf.{u} Ab.{u+1}}\n  (e : M.tensor A \u27f6 N) :\n  M \u27f6 N.half_internal_hom A :=\n\u27e8(ExtrSheafProd.tensor_curry $ (ExtrSheaf_ExtrSheafProd_equiv _).functor.map e).val\u27e9\n\ndef tensor_uncurry {A : AddCommGroup.{u+1}} {M N : ExtrSheaf.{u} Ab.{u+1}}\n  (e : M \u27f6 N.half_internal_hom A) :\n  M.tensor A \u27f6 N :=\n\u27e8(ExtrSheafProd.tensor_uncurry $ (ExtrSheaf_ExtrSheafProd_equiv _).functor.map e).val\u27e9\n\n@[simps val]\ndef map_tensor {M M' : ExtrSheaf.{u} Ab.{u+1}} {A A' : AddCommGroup.{u+1}}\n  (f : M \u27f6 M') (g : A \u27f6 A') :\n  M.tensor A \u27f6 M'.tensor A' :=\n\u27e8((ExtrSheafProd.map_tensor $ (ExtrSheaf_ExtrSheafProd_equiv _).functor.map f) g).val\u27e9\n\n@[simp]\nlemma map_tensor_id (M : ExtrSheaf.{u} Ab.{u+1}) (A : AddCommGroup.{u+1}) :\n  map_tensor (\ud835\udfd9 M) (\ud835\udfd9 A) = \ud835\udfd9 _ :=\nby { ext : 1, dsimp, simpa, }\n\n@[simp]\nlemma map_tensor_comp {M M' M'' : ExtrSheaf.{u} Ab.{u+1}}\n  {A A' A'' : AddCommGroup.{u+1}}\n  (f : M \u27f6 M') (f' : M' \u27f6 M'')\n  (g : A \u27f6 A') (g' : A' \u27f6 A'') :\n  map_tensor (f \u226b f') (g \u226b g') = map_tensor f g \u226b map_tensor f' g' :=\nby { ext : 1, dsimp, simp }\n\n@[simps]\ndef tensor_functor : ExtrSheaf.{u} Ab.{u+1} \u2964 Ab.{u+1} \u2964 ExtrSheaf.{u} Ab.{u+1} :=\n{ obj := \u03bb M,\n  { obj := \u03bb A, M.tensor A,\n    map := \u03bb A A' f, map_tensor (\ud835\udfd9 _) f,\n    map_id' := \u03bb X, by simp,\n    map_comp' := \u03bb X Y Z f g, begin\n      nth_rewrite 0 [\u2190 category.id_comp (\ud835\udfd9 M)],\n      rw map_tensor_comp,\n    end },\n  map := \u03bb M N f,\n  { app := \u03bb A, map_tensor f (\ud835\udfd9 _),\n    naturality' := \u03bb A B g, begin\n      dsimp,\n      simp only [\u2190 map_tensor_comp, category.id_comp, category.comp_id],\n    end },\n  map_id' := \u03bb M, begin\n    ext : 2,\n    simp,\n  end,\n  map_comp' := \u03bb M N L f g, begin\n    ext x : 2,\n    dsimp,\n    nth_rewrite 0 [\u2190 category.comp_id (\ud835\udfd9 x)],\n    rw [map_tensor_comp],\n  end }\n\ninstance creates_colimits :\n  creates_colimits\n  (Sheaf_to_presheaf ExtrDisc.proetale_topology.{u} Ab.{u+1}) :=\nshow creates_colimits ((ExtrSheaf_ExtrSheafProd_equiv _).functor \u22d9\n  ExtrSheafProd_to_presheaf _), from infer_instance\n\ninstance preserves_colimits_tensor_obj (M : ExtrSheaf.{u} Ab.{u+1}) :\n  limits.preserves_colimits (tensor_functor.obj M) :=\nbegin\n  constructor, introsI J _, constructor, intros F, constructor, intros S hS,\n  let T := _, change (limits.is_colimit T),\n  apply limits.is_colimit_of_reflects\n    (Sheaf_to_presheaf ExtrDisc.proetale_topology.{u} Ab.{u+1}),\n  apply limits.evaluation_jointly_reflects_colimits,\n  intros Q,\n  change limits.is_colimit\n    ((AddCommGroup.tensor_functor.obj (M.val.obj Q)).map_cocone S),\n  apply limits.is_colimit_of_preserves,\n  exact hS,\nend\n\nexample (\u03b1 : Type (u+1)) (N : ExtrSheaf.{u} Ab.{u+1}) :\nlimits.preserves_colimits_of_shape (discrete \u03b1) (tensor_functor.obj N) := infer_instance\n\ninstance tensor_functor_additive (M : ExtrSheaf.{u} Ab.{u+1}) :\n  (tensor_functor.obj M).additive := { }\n\nend ExtrSheaf\n\nnamespace Condensed\n\ndef tensor (M : Condensed.{u} Ab.{u+1}) (A : Ab.{u+1}) :\n  Condensed.{u} Ab.{u+1} :=\n(Condensed_ExtrSheaf_equiv _).functor.obj\n(((Condensed_ExtrSheaf_equiv _).inverse.obj M).tensor A)\n\ndef map_tensor {M M' : Condensed.{u} Ab.{u+1}} {A A' : Ab.{u+1}}\n  (f : M \u27f6 M') (g : A \u27f6 A') :\n  M.tensor A \u27f6 M'.tensor A' :=\n(Condensed_ExtrSheaf_equiv _).functor.map $\nExtrSheaf.map_tensor ((Condensed_ExtrSheaf_equiv _).inverse.map f) g\n\n@[simp]\nlemma map_tensor_id (M : Condensed.{u} Ab.{u+1}) (A : AddCommGroup.{u+1}) :\n  map_tensor (\ud835\udfd9 M) (\ud835\udfd9 A) = \ud835\udfd9 _ :=\nby { dsimp [map_tensor], simpa, }\n\n@[simp]\nlemma map_tensor_comp {M M' M'' : Condensed.{u} Ab.{u+1}}\n  {A A' A'' : AddCommGroup.{u+1}}\n  (f : M \u27f6 M') (f' : M' \u27f6 M'')\n  (g : A \u27f6 A') (g' : A' \u27f6 A'') :\n  map_tensor (f \u226b f') (g \u226b g') = map_tensor f g \u226b map_tensor f' g' :=\nby { dsimp [map_tensor], simp, }\n\n/-- This is the functor that sends `A : Ab` to `M \u2297 A`,\nwhere `M` is a condensed abelian group, functorial in both `M` and `A`. -/\ndef tensor_functor : Condensed.{u} Ab.{u+1} \u2964 Ab.{u+1} \u2964 Condensed.{u} Ab.{u+1} :=\n{ obj := \u03bb M,\n  { obj := \u03bb A, M.tensor A,\n    map := \u03bb A A' f, map_tensor (\ud835\udfd9 _) f,\n    map_id' := \u03bb X, by simp,\n    map_comp' := \u03bb X Y Z f g, begin\n      nth_rewrite 0 [\u2190 category.id_comp (\ud835\udfd9 M)],\n      rw map_tensor_comp,\n    end },\n  map := \u03bb M N f,\n  { app := \u03bb A, map_tensor f (\ud835\udfd9 _),\n    naturality' := \u03bb A B g, begin\n      dsimp,\n      simp only [\u2190 map_tensor_comp, category.id_comp, category.comp_id],\n    end },\n  map_id' := \u03bb M, begin\n    ext : 2,\n    simp,\n  end,\n  map_comp' := \u03bb M N L f g, begin\n    ext x : 2,\n    dsimp,\n    nth_rewrite 0 [\u2190 category.comp_id (\ud835\udfd9 x)],\n    rw [map_tensor_comp],\n  end }\n\n/-\n/-- Restrincting to `ExtrDisc` works as expeceted. -/\ndef tensor_functor_conj_iso :\n  (Condensed_ExtrSheaf_equiv Ab.{u+1}).functor \u22d9\n  ((whiskering_right _ _ _).obj $ ((whiskering_right _ _ _).obj\n    (Condensed_ExtrSheaf_equiv Ab.{u+1}).inverse)).obj tensor_functor \u2245\n  ExtrSheaf.tensor_functor :=\nnat_iso.of_components\n(\u03bb X, begin\n  dsimp [tensor_functor],\nend)\nbegin\n  intros X Y f, ext : 2,\n  dsimp [tensor_functor],\n  simp only [equivalence.fun_inv_map, equivalence.equivalence_mk'_counit,\n    equivalence.equivalence_mk'_counit_inv, functor.map_comp, nat_trans.comp_app,\n    category.assoc, iso.inv_hom_id_app_assoc, category.id_comp,\n    nat_iso.cancel_nat_iso_hom_left],\n  rw [\u2190 nat_trans.comp_app, \u2190 functor.map_comp, \u2190 nat_trans.comp_app],\n  have : (Condensed_ExtrSheafProd_equiv Ab).counit_iso.inv.app Y \u226b\n    (Condensed_ExtrSheafProd_equiv Ab).counit_iso.hom.app Y = \ud835\udfd9 _,\n  { rw [\u2190 nat_trans.comp_app, iso.inv_hom_id], refl },\n  rw this,\n  simp only [nat_trans.comp_app],\n  dsimp,\n  simp only [category_theory.functor.map_id, nat_trans.id_app, category.comp_id],\nend\n\ndef tensor_functor_conj_iso' :\n  tensor_functor \u22d9 (whiskering_right _ _ _).obj\n  (Condensed_ExtrSheafProd_equiv _).functor \u2245\n  (Condensed_ExtrSheafProd_equiv _).functor \u22d9 ExtrSheafProd.tensor_functor :=\nnat_iso.of_components\n(\u03bb X, begin\n  dsimp [tensor_functor],\n  refine functor.associator _ _ _ \u226a\u226b _,\n  refine _ \u226a\u226b functor.right_unitor _,\n  refine ((whiskering_left _ _ _).obj _).map_iso _,\n  refine (Condensed_ExtrSheafProd_equiv _).counit_iso,\nend)\nbegin\n  intros X Y f, ext : 2,\n  dsimp [tensor_functor],\n  simp, dsimp, simp,\nend\n-/\n\ndef tensor_iso (M : Condensed.{u} Ab.{u+1}) (A : Ab.{u+1}) :\n  (Condensed_ExtrSheaf_equiv _).inverse.obj (M.tensor A) \u2245\n  ((Condensed_ExtrSheaf_equiv _).inverse.obj M).tensor A :=\n((Condensed_ExtrSheaf_equiv _).unit_iso.app _).symm\n\n/-- The tensor product behaves in the naive way when evaluated\non extremally disconnected sets. -/\ndef tensor_eval_iso\n  (M : Condensed.{u} Ab.{u+1}) (A : Ab.{u+1}) (S : ExtrDisc.{u}) :\n  (tensor M A).val.obj (opposite.op S.val) \u2245\n  ((M.val.obj (opposite.op S.val)).tensor A) :=\n((Sheaf_to_presheaf _ _).map_iso (M.tensor_iso A)).app (opposite.op S)\n\ndef half_internal_hom (A : AddCommGroup.{u+1}) (M : Condensed.{u} Ab.{u+1}) :\n  Condensed.{u} Ab.{u+1} :=\n(Condensed_ExtrSheaf_equiv _).functor.obj $\nExtrSheaf.half_internal_hom A ((Condensed_ExtrSheaf_equiv _).inverse.obj M)\n\ndef half_internal_hom_iso (A : AddCommGroup.{u+1}) (M : Condensed.{u} Ab.{u+1}) :\n  (Condensed_ExtrSheaf_equiv _).inverse.obj (half_internal_hom A M) \u2245\n  ExtrSheaf.half_internal_hom A ((Condensed_ExtrSheaf_equiv _).inverse.obj M) :=\n((Condensed_ExtrSheaf_equiv _).unit_iso.app _).symm\n\ndef half_internal_hom_eval_iso (A : AddCommGroup.{u+1}) (M : Condensed.{u} Ab.{u+1})\n  (S : ExtrDisc.{u}) :\n  (half_internal_hom A M).val.obj (opposite.op S.val) \u2245\n  AddCommGroup.of (A \u27f6 M.val.obj (opposite.op S.val)) :=\n((Sheaf_to_presheaf _ _).map_iso (half_internal_hom_iso A M)).app (opposite.op S)\n\ndef tensor_uncurry {A : AddCommGroup.{u+1}} {M N : Condensed.{u} Ab.{u+1}}\n  (e : M \u27f6 half_internal_hom A N) :\n  tensor M A \u27f6 N :=\n(Condensed_ExtrSheaf_equiv _).functor.map\n  (ExtrSheaf.tensor_uncurry $ (Condensed_ExtrSheaf_equiv Ab).inverse.map e \u226b\n  (half_internal_hom_iso _ _).hom) \u226b\n  ((Condensed_ExtrSheaf_equiv _).counit_iso.app N).hom\n\nlemma tensor_uncurry_eq\n  {A : AddCommGroup.{u+1}} {M N : Condensed.{u} Ab.{u+1}}\n  (e : M \u27f6 half_internal_hom A N) :\n  (Condensed_ExtrSheaf_equiv _).inverse.map (tensor_uncurry e) =\n  (tensor_iso _ _).hom \u226b\n  ExtrSheaf.tensor_uncurry\n  ((Condensed_ExtrSheaf_equiv _).inverse.map e \u226b (half_internal_hom_iso _ _).hom) :=\nbegin\n  dsimp [tensor_uncurry, half_internal_hom_iso, tensor_iso],\n  simp,\nend\n\ndef tensor_curry {A : AddCommGroup.{u+1}} {M N : Condensed.{u} Ab.{u+1}}\n  (e : M.tensor A \u27f6 N) : M \u27f6 half_internal_hom A N :=\n  ((Condensed_ExtrSheaf_equiv _).counit_iso.app _).inv \u226b\n  (Condensed_ExtrSheaf_equiv _).functor.map\n  (ExtrSheaf.tensor_curry $ (tensor_iso M A).inv \u226b\n  (Condensed_ExtrSheaf_equiv Ab).inverse.map e)\n\nlemma tensor_curry_eq {A : AddCommGroup.{u+1}} {M N : Condensed.{u} Ab.{u+1}}\n  (e : M.tensor A \u27f6 N) :\n  (Condensed_ExtrSheaf_equiv _).inverse.map (tensor_curry e) =\n  ExtrSheaf.tensor_curry ((tensor_iso _ _).inv \u226b\n    (Condensed_ExtrSheaf_equiv Ab).inverse.map e) \u226b\n  (half_internal_hom_iso _ _).inv :=\nbegin\n  rw iso.eq_comp_inv,\n  dsimp [tensor_curry, half_internal_hom_iso, tensor_iso],\n  simp only [functor.map_comp, equivalence.fun_inv_map, equivalence.equivalence_mk'_counit,\n    category.assoc, iso.inv_hom_id_app],\n  simp, dsimp, simp,\n  --dsimp,\n  --simp only [category.comp_id],\n  suffices :\n    (Condensed_ExtrSheaf_equiv Ab).inverse.map\n      ((Condensed_ExtrSheaf_equiv Ab).counit_iso.inv.app M) \u226b\n    (Condensed_ExtrSheaf_equiv Ab).unit_iso.inv.app\n      ((Condensed_ExtrSheaf_equiv Ab).inverse.obj M) = \ud835\udfd9 _,\n  { rw reassoc_of this },\n  simpa,\nend\n\ndef tensor_functor_iso_ExtrSheaf_tensor_functor (M : Condensed.{u} Ab.{u+1}) :\n  tensor_functor.obj M \u2245\n    (ExtrSheaf.tensor_functor.obj\n      ((Condensed_ExtrSheaf_equiv _).inverse.obj M)) \u22d9\n    (Condensed_ExtrSheaf_equiv _).functor := by refl\n\ninstance (M : Condensed.{u} Ab.{u+1}) :\n  limits.preserves_colimits (tensor_functor.obj M) :=\nlimits.preserves_colimits_of_nat_iso (tensor_functor_iso_ExtrSheaf_tensor_functor M).symm\n\nexample (\u03b1 : Type (u+1)) (M : Condensed.{u} Ab.{u+1}) :\n  limits.preserves_colimits_of_shape (discrete \u03b1) (tensor_functor.obj M) :=\ninfer_instance\n\n/- this is an assumption in `main_lemma'` in `breen_deligne/main.lean` and it is\nused in `condensed/bd_lemma.lean` -/\ninstance tensor_functor_additive (A : Condensed.{u} Ab.{u+1}) :\n  (tensor_functor.obj A).additive :=\n{ map_add' := begin\n    intros A B f g,\n    dsimp [tensor_functor, map_tensor],\n    rw [\u2190 functor.map_add], congr' 1, ext S : 3,\n    dsimp only [ExtrSheaf.map_tensor, ExtrSheafProd.map_tensor],\n    simp only [category_theory.functor.map_id],\n    erw [nat_trans.id_app],\n    rw [\u2190 AddCommGroup.tensor_functor_obj_map, functor.map_add],\n    refl,\n  end }\n\nend Condensed\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/condensed/tensor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.5, "lm_q1q2_score": 0.36243513808841243}}
{"text": "import .basic\n\nnamespace premodel\n\n--- The Pi-type of a family of premodels is a premodel, with the componentwise operations in terms of `vect.unzip_fam`.\ndefinition pi {th : theory} {\u03b1 : Type*} (M : \u03b1 \u2192 Type _) [\u2200 a, premodel th (M a)] : premodel th (\u03a0 a, M a) :=\n  {\n    act :=\n      \u03bb n f xs, \u03bb a,\n        premodel.act f (xs.unzip_fam a)\n  }\n\nattribute [instance] premodel.pi\n\n--- The evaluation defines a morphism from the pi-type to its component.\ndefinition pi_eval {th : theory} {\u03b1 : Type _} (M : \u03b1 \u2192 Type _) [\u2200 a, premodel th (M a)] (a : \u03b1) : morphism th (\u03a0 a, M a) (M a) :=\n  \u27e8(\u03bb f, f a), by intros _ \u03bc fs; rw [\u2190vect.unzip_fam_eval]; unfold act\u27e9\n\nend premodel\n\nnamespace model\n\n--- The Pi-type of a family of models is virtually a model; i.e.~ up to `funext`.\ntheorem pi_axiom {th : theory} {\u03b1 : Type _} {C : \u03b1 \u2192 Type _} [\u2200 a, model th (C a)] : \u2200 {n : \u2115} (r : th.rel n) (var : finord n \u2192 \u03a0 (a : \u03b1), C a) (a : \u03b1), optree.elim (@premodel.act th (\u03a0 a, C a) _) var (th.rel_lhs r) a = optree.elim (@premodel.act th (\u03a0 a, C a) _) var (th.rel_rhs r) a :=\n  begin\n    intros _ r var a,\n    unfold premodel.act at *; dsimp *,\n    let dact : \u03a0 (a : \u03b1) {k : \u2115}, th.op k \u2192 vect (C a) k \u2192 (C a) := \u03bb a k f rs , premodel.act f rs,\n    let dvar : \u03a0 (a : \u03b1), finord n \u2192 (C a) := \u03bb k a, var a k,\n    rw [@optree.elim_pi th.op _ \u03b1 C dact dvar _ a],\n    rw [@optree.elim_pi th.op _ \u03b1 C dact dvar _ a],\n    rw [axiom_eq]\n  end\n\n#print axioms pi_axiom\n\nnamespace unsafe\n-- WARNING: Use of `funext`.\n\n--- The Pi-type of a family of models is a model.\ndefinition pi {th : theory} {\u03b1 : Type _} {C : \u03b1 \u2192 Type _} [\u2200 a, model th (C a)] : model th (\u03a0 a, C a):=\n  {\n    haxiom := \u03bb _ r var, funext (pi_axiom r var)\n  }\n\n#print axioms model.unsafe.pi\n\nend unsafe\n\nend model\n", "meta": {"author": "Junology", "repo": "groth-lean", "sha": "5aa1ba624cd0f5145f63fa86130f99b85bbbcac2", "save_path": "github-repos/lean/Junology-groth-lean", "path": "github-repos/lean/Junology-groth-lean/groth-lean-5aa1ba624cd0f5145f63fa86130f99b85bbbcac2/src/algebra/theory/pi.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3624351380884124}}
{"text": "/-\nCopyright (c) 2022 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.limits.shapes.kernels\nimport category_theory.limits.preserves.shapes.equalizers\nimport category_theory.limits.preserves.shapes.zero\n\n/-!\n# Preserving (co)kernels\n\nConstructions to relate the notions of preserving (co)kernels and reflecting (co)kernels\nto concrete (co)forks.\n\nIn particular, we show that `kernel_comparison f g G` is an isomorphism iff `G` preserves\nthe limit of the parallel pair `f,0`, as well as the dual result.\n-/\n\nnoncomputable theory\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] [has_zero_morphisms C]\nvariables {D : Type u\u2082} [category.{v\u2082} D] [has_zero_morphisms D]\nvariables (G : C \u2964 D) [functor.preserves_zero_morphisms G]\n\nnamespace category_theory.limits\n\nsection kernels\nvariables {X Y Z : C} {f : X \u27f6 Y} {h : Z \u27f6 X} (w : h \u226b f = 0)\n\n/--\nThe map of a kernel fork is a limit iff\nthe kernel fork consisting of the mapped morphisms is a limit.\nThis essentially lets us commute `kernel_fork.of_\u03b9` with `functor.map_cone`.\n\nThis is a variant of `is_limit_map_cone_fork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_map_cone_fork_equiv' :\n  is_limit (G.map_cone (kernel_fork.of_\u03b9 h w)) \u2243\n  is_limit (kernel_fork.of_\u03b9 (G.map h) (by simp only [\u2190G.map_comp, w, functor.map_zero])\n    : fork (G.map f) 0) :=\nbegin\n  refine (is_limit.postcompose_hom_equiv _ _).symm.trans (is_limit.equiv_iso_limit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine fork.ext (iso.refl _) _,\n  simp [fork.\u03b9]\nend\n\n/--\nThe property of preserving kernels expressed in terms of kernel forks.\n\nThis is a variant of `is_limit_fork_map_of_is_limit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_limit_fork_map_of_is_limit' [preserves_limit (parallel_pair f 0) G]\n  (l : is_limit (kernel_fork.of_\u03b9 h w)) :\n  is_limit (kernel_fork.of_\u03b9 (G.map h) (by simp only [\u2190G.map_comp, w, functor.map_zero]) :\n    fork (G.map f) 0) :=\nis_limit_map_cone_fork_equiv' G w (preserves_limit.preserves l)\n\nvariables (f) [has_kernel f]\n\n/--\nIf `G` preserves kernels and `C` has them, then the fork constructed of the mapped morphisms of\na kernel fork is a limit.\n-/\ndef is_limit_of_has_kernel_of_preserves_limit [preserves_limit (parallel_pair f 0) G] :\n  is_limit (fork.of_\u03b9 (G.map (kernel.\u03b9 f))\n    (by simp only [\u2190G.map_comp, equalizer.condition, comp_zero, functor.map_zero])\n      : fork (G.map f) 0) :=\nis_limit_fork_map_of_is_limit' G (kernel.condition f) (kernel_is_kernel f)\n\ninstance [preserves_limit (parallel_pair f 0) G] : has_kernel (G.map f) :=\n{ exists_limit := \u27e8\u27e8_, is_limit_of_has_kernel_of_preserves_limit G f\u27e9\u27e9, }\n\nvariables [has_kernel (G.map f)]\n\n/--\nIf the kernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\nkernel of `f`.\n-/\ndef preserves_kernel.of_iso_comparison [i : is_iso (kernel_comparison f G)] :\n  preserves_limit (parallel_pair f 0) G :=\nbegin\n  apply preserves_limit_of_preserves_limit_cone (kernel_is_kernel f),\n  apply (is_limit_map_cone_fork_equiv' G (kernel.condition f)).symm _,\n  apply is_limit.of_point_iso (limit.is_limit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_limit (parallel_pair f 0) G]\n/--\nIf `G` preserves the kernel of `f`, then the kernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_kernel.iso :\n  G.obj (kernel f) \u2245 kernel (G.map f) :=\nis_limit.cone_point_unique_up_to_iso\n  (is_limit_of_has_kernel_of_preserves_limit G f)\n  (limit.is_limit _)\n\n@[simp]\nlemma preserves_kernel.iso_hom :\n  (preserves_kernel.iso G f).hom = kernel_comparison f G :=\nrfl\n\ninstance : is_iso (kernel_comparison f G) :=\nbegin\n  rw \u2190 preserves_kernel.iso_hom,\n  apply_instance\nend\n\nend kernels\n\nsection cokernels\n\nvariables {X Y Z : C} {f : X \u27f6 Y} {h : Y \u27f6 Z} (w : f \u226b h = 0)\n\n/--\nThe map of a cokernel cofork is a colimit iff\nthe cokernel cofork consisting of the mapped morphisms is a colimit.\nThis essentially lets us commute `cokernel_cofork.of_\u03c0` with `functor.map_cocone`.\n\nThis is a variant of `is_colimit_map_cocone_cofork_equiv` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_map_cocone_cofork_equiv' :\n  is_colimit (G.map_cocone (cokernel_cofork.of_\u03c0 h w)) \u2243\n  is_colimit (cokernel_cofork.of_\u03c0 (G.map h) (by simp only [\u2190G.map_comp, w, functor.map_zero])\n    : cofork (G.map f) 0) :=\nbegin\n  refine (is_colimit.precompose_hom_equiv _ _).symm.trans (is_colimit.equiv_iso_colimit _),\n  refine parallel_pair.ext (iso.refl _) (iso.refl _) _ _; simp,\n  refine cofork.ext (iso.refl _) _,\n  simp only [cofork.\u03c0, iso.refl_hom, id_comp, cocones.precompose_obj_\u03b9,\n    nat_trans.comp_app, parallel_pair.ext_hom_app, functor.map_cocone_\u03b9_app,\n    cofork.of_\u03c0_\u03b9_app],\n  apply category.comp_id\nend\n\n/--\nThe property of preserving cokernels expressed in terms of cokernel coforks.\n\nThis is a variant of `is_colimit_cofork_map_of_is_colimit` for equalizers,\nwhich we can't use directly between `G.map 0 = 0` does not hold definitionally.\n-/\ndef is_colimit_cofork_map_of_is_colimit' [preserves_colimit (parallel_pair f 0) G]\n  (l : is_colimit (cokernel_cofork.of_\u03c0 h w)) :\n  is_colimit (cokernel_cofork.of_\u03c0 (G.map h) (by simp only [\u2190G.map_comp, w, functor.map_zero]) :\n    cofork (G.map f) 0) :=\nis_colimit_map_cocone_cofork_equiv' G w (preserves_colimit.preserves l)\n\nvariables (f) [has_cokernel f]\n\n/--\nIf `G` preserves cokernels and `C` has them, then the cofork constructed of the mapped morphisms of\na cokernel cofork is a colimit.\n-/\ndef is_colimit_of_has_cokernel_of_preserves_colimit [preserves_colimit (parallel_pair f 0) G] :\n  is_colimit (cofork.of_\u03c0 (G.map (cokernel.\u03c0 f))\n    (by simp only [\u2190G.map_comp, coequalizer.condition, zero_comp, functor.map_zero])\n      : cofork (G.map f) 0) :=\nis_colimit_cofork_map_of_is_colimit' G (cokernel.condition f) (cokernel_is_cokernel f)\n\ninstance [preserves_colimit (parallel_pair f 0) G] : has_cokernel (G.map f) :=\n{ exists_colimit := \u27e8\u27e8_, is_colimit_of_has_cokernel_of_preserves_colimit G f\u27e9\u27e9, }\n\nvariables [has_cokernel (G.map f)]\n\n/--\nIf the cokernel comparison map for `G` at `f` is an isomorphism, then `G` preserves the\ncokernel of `f`.\n-/\ndef preserves_cokernel.of_iso_comparison [i : is_iso (cokernel_comparison f G)] :\n  preserves_colimit (parallel_pair f 0) G :=\nbegin\n  apply preserves_colimit_of_preserves_colimit_cocone (cokernel_is_cokernel f),\n  apply (is_colimit_map_cocone_cofork_equiv' G (cokernel.condition f)).symm _,\n  apply is_colimit.of_point_iso (colimit.is_colimit (parallel_pair (G.map f) 0)),\n  apply i,\nend\n\nvariables [preserves_colimit (parallel_pair f 0) G]\n/--\nIf `G` preserves the cokernel of `f`, then the cokernel comparison map for `G` at `f` is\nan isomorphism.\n-/\ndef preserves_cokernel.iso :\n  G.obj (cokernel f) \u2245 cokernel (G.map f) :=\nis_colimit.cocone_point_unique_up_to_iso\n  (is_colimit_of_has_cokernel_of_preserves_colimit G f)\n  (colimit.is_colimit _)\n\n@[simp]\nlemma preserves_cokernel.iso_inv :\n  (preserves_cokernel.iso G f).inv = cokernel_comparison f G :=\nrfl\n\ninstance : is_iso (cokernel_comparison f G) :=\nbegin\n  rw \u2190 preserves_cokernel.iso_inv,\n  apply_instance\nend\n\nend cokernels\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/preserves/shapes/kernels.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.362430456252567}}
{"text": "/-\nCopyright (c) 2018 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Data.Fin.Basic\nimport Init.System.Platform\n\nopen Nat\n\n@[extern \"lean_uint8_of_nat\"]\ndef UInt8.ofNat (n : @& Nat) : UInt8 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt8 := UInt8.ofNat\n@[extern \"lean_uint8_to_nat\"]\ndef UInt8.toNat (n : UInt8) : Nat := n.val.val\n@[extern \"lean_uint8_add\"]\ndef UInt8.add (a b : UInt8) : UInt8 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint8_sub\"]\ndef UInt8.sub (a b : UInt8) : UInt8 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint8_mul\"]\ndef UInt8.mul (a b : UInt8) : UInt8 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint8_div\"]\ndef UInt8.div (a b : UInt8) : UInt8 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint8_mod\"]\ndef UInt8.mod (a b : UInt8) : UInt8 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint8_modn\"]\ndef UInt8.modn (a : UInt8) (n : @& Nat) : UInt8 := \u27e8Fin.modn a.val n\u27e9\n@[extern \"lean_uint8_land\"]\ndef UInt8.land (a b : UInt8) : UInt8 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint8_lor\"]\ndef UInt8.lor (a b : UInt8) : UInt8 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint8_xor\"]\ndef UInt8.xor (a b : UInt8) : UInt8 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint8_shift_left\"]\ndef UInt8.shiftLeft (a b : UInt8) : UInt8 := \u27e8a.val <<< (modn b 8).val\u27e9\n@[extern \"lean_uint8_shift_right\"]\ndef UInt8.shiftRight (a b : UInt8) : UInt8 := \u27e8a.val >>> (modn b 8).val\u27e9\ndef UInt8.lt (a b : UInt8) : Prop := a.val < b.val\ndef UInt8.le (a b : UInt8) : Prop := a.val \u2264 b.val\n\ninstance : OfNat UInt8 n   := \u27e8UInt8.ofNat n\u27e9\ninstance : Add UInt8       := \u27e8UInt8.add\u27e9\ninstance : Sub UInt8       := \u27e8UInt8.sub\u27e9\ninstance : Mul UInt8       := \u27e8UInt8.mul\u27e9\ninstance : Mod UInt8       := \u27e8UInt8.mod\u27e9\ninstance : HMod UInt8 Nat UInt8 := \u27e8UInt8.modn\u27e9\ninstance : Div UInt8       := \u27e8UInt8.div\u27e9\ninstance : LT UInt8        := \u27e8UInt8.lt\u27e9\ninstance : LE UInt8        := \u27e8UInt8.le\u27e9\n\n@[extern \"lean_uint8_complement\"]\ndef UInt8.complement (a:UInt8) : UInt8 := 0-(a+1)\n\ninstance : Complement UInt8 := \u27e8UInt8.complement\u27e9\ninstance : AndOp UInt8     := \u27e8UInt8.land\u27e9\ninstance : OrOp UInt8      := \u27e8UInt8.lor\u27e9\ninstance : Xor UInt8       := \u27e8UInt8.xor\u27e9\ninstance : ShiftLeft UInt8  := \u27e8UInt8.shiftLeft\u27e9\ninstance : ShiftRight UInt8 := \u27e8UInt8.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint8_dec_lt\"]\ndef UInt8.decLt (a b : UInt8) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint8_dec_le\"]\ndef UInt8.decLe (a b : UInt8) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt8) : Decidable (a < b) := UInt8.decLt a b\ninstance (a b : UInt8) : Decidable (a \u2264 b) := UInt8.decLe a b\ninstance : Max UInt8 := maxOfLe\ninstance : Min UInt8 := minOfLe\n\n@[extern \"lean_uint16_of_nat\"]\ndef UInt16.ofNat (n : @& Nat) : UInt16 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt16 := UInt16.ofNat\n@[extern \"lean_uint16_to_nat\"]\ndef UInt16.toNat (n : UInt16) : Nat := n.val.val\n@[extern \"lean_uint16_add\"]\ndef UInt16.add (a b : UInt16) : UInt16 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint16_sub\"]\ndef UInt16.sub (a b : UInt16) : UInt16 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint16_mul\"]\ndef UInt16.mul (a b : UInt16) : UInt16 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint16_div\"]\ndef UInt16.div (a b : UInt16) : UInt16 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint16_mod\"]\ndef UInt16.mod (a b : UInt16) : UInt16 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint16_modn\"]\ndef UInt16.modn (a : UInt16) (n : @& Nat) : UInt16 := \u27e8Fin.modn a.val n\u27e9\n@[extern \"lean_uint16_land\"]\ndef UInt16.land (a b : UInt16) : UInt16 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint16_lor\"]\ndef UInt16.lor (a b : UInt16) : UInt16 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint16_xor\"]\ndef UInt16.xor (a b : UInt16) : UInt16 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint16_shift_left\"]\ndef UInt16.shiftLeft (a b : UInt16) : UInt16 := \u27e8a.val <<< (modn b 16).val\u27e9\n@[extern \"lean_uint16_to_uint8\"]\ndef UInt16.toUInt8 (a : UInt16) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint8_to_uint16\"]\ndef UInt8.toUInt16 (a : UInt8) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint16_shift_right\"]\ndef UInt16.shiftRight (a b : UInt16) : UInt16 := \u27e8a.val >>> (modn b 16).val\u27e9\ndef UInt16.lt (a b : UInt16) : Prop := a.val < b.val\ndef UInt16.le (a b : UInt16) : Prop := a.val \u2264 b.val\n\n\ninstance : OfNat UInt16 n   := \u27e8UInt16.ofNat n\u27e9\ninstance : Add UInt16       := \u27e8UInt16.add\u27e9\ninstance : Sub UInt16       := \u27e8UInt16.sub\u27e9\ninstance : Mul UInt16       := \u27e8UInt16.mul\u27e9\ninstance : Mod UInt16       := \u27e8UInt16.mod\u27e9\ninstance : HMod UInt16 Nat UInt16 := \u27e8UInt16.modn\u27e9\ninstance : Div UInt16       := \u27e8UInt16.div\u27e9\ninstance : LT UInt16        := \u27e8UInt16.lt\u27e9\ninstance : LE UInt16        := \u27e8UInt16.le\u27e9\n\n@[extern \"lean_uint16_complement\"]\ndef UInt16.complement (a:UInt16) : UInt16 := 0-(a+1)\n\ninstance : Complement UInt16 := \u27e8UInt16.complement\u27e9\ninstance : AndOp UInt16     := \u27e8UInt16.land\u27e9\ninstance : OrOp UInt16      := \u27e8UInt16.lor\u27e9\ninstance : Xor UInt16       := \u27e8UInt16.xor\u27e9\ninstance : ShiftLeft UInt16  := \u27e8UInt16.shiftLeft\u27e9\ninstance : ShiftRight UInt16 := \u27e8UInt16.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint16_dec_lt\"]\ndef UInt16.decLt (a b : UInt16) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint16_dec_le\"]\ndef UInt16.decLe (a b : UInt16) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt16) : Decidable (a < b) := UInt16.decLt a b\ninstance (a b : UInt16) : Decidable (a \u2264 b) := UInt16.decLe a b\ninstance : Max UInt16 := maxOfLe\ninstance : Min UInt16 := minOfLe\n\n@[extern \"lean_uint32_of_nat\"]\ndef UInt32.ofNat (n : @& Nat) : UInt32 := \u27e8Fin.ofNat n\u27e9\n@[extern \"lean_uint32_of_nat\"]\ndef UInt32.ofNat' (n : Nat) (h : n < UInt32.size) : UInt32 := \u27e8\u27e8n, h\u27e9\u27e9\nabbrev Nat.toUInt32 := UInt32.ofNat\n@[extern \"lean_uint32_add\"]\ndef UInt32.add (a b : UInt32) : UInt32 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint32_sub\"]\ndef UInt32.sub (a b : UInt32) : UInt32 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint32_mul\"]\ndef UInt32.mul (a b : UInt32) : UInt32 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint32_div\"]\ndef UInt32.div (a b : UInt32) : UInt32 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint32_mod\"]\ndef UInt32.mod (a b : UInt32) : UInt32 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint32_modn\"]\ndef UInt32.modn (a : UInt32) (n : @& Nat) : UInt32 := \u27e8Fin.modn a.val n\u27e9\n@[extern \"lean_uint32_land\"]\ndef UInt32.land (a b : UInt32) : UInt32 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint32_lor\"]\ndef UInt32.lor (a b : UInt32) : UInt32 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint32_xor\"]\ndef UInt32.xor (a b : UInt32) : UInt32 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint32_shift_left\"]\ndef UInt32.shiftLeft (a b : UInt32) : UInt32 := \u27e8a.val <<< (modn b 32).val\u27e9\n@[extern \"lean_uint32_shift_right\"]\ndef UInt32.shiftRight (a b : UInt32) : UInt32 := \u27e8a.val >>> (modn b 32).val\u27e9\n@[extern \"lean_uint32_to_uint8\"]\ndef UInt32.toUInt8 (a : UInt32) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint32_to_uint16\"]\ndef UInt32.toUInt16 (a : UInt32) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint8_to_uint32\"]\ndef UInt8.toUInt32 (a : UInt8) : UInt32 := a.toNat.toUInt32\n@[extern \"lean_uint16_to_uint32\"]\ndef UInt16.toUInt32 (a : UInt16) : UInt32 := a.toNat.toUInt32\n\ninstance : OfNat UInt32 n   := \u27e8UInt32.ofNat n\u27e9\ninstance : Add UInt32       := \u27e8UInt32.add\u27e9\ninstance : Sub UInt32       := \u27e8UInt32.sub\u27e9\ninstance : Mul UInt32       := \u27e8UInt32.mul\u27e9\ninstance : Mod UInt32       := \u27e8UInt32.mod\u27e9\ninstance : HMod UInt32 Nat UInt32 := \u27e8UInt32.modn\u27e9\ninstance : Div UInt32       := \u27e8UInt32.div\u27e9\n\n@[extern \"lean_uint32_complement\"]\ndef UInt32.complement (a:UInt32) : UInt32 := 0-(a+1)\n\ninstance : Complement UInt32 := \u27e8UInt32.complement\u27e9\ninstance : AndOp UInt32     := \u27e8UInt32.land\u27e9\ninstance : OrOp UInt32      := \u27e8UInt32.lor\u27e9\ninstance : Xor UInt32       := \u27e8UInt32.xor\u27e9\ninstance : ShiftLeft UInt32  := \u27e8UInt32.shiftLeft\u27e9\ninstance : ShiftRight UInt32 := \u27e8UInt32.shiftRight\u27e9\n\n@[extern \"lean_uint64_of_nat\"]\ndef UInt64.ofNat (n : @& Nat) : UInt64 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt64 := UInt64.ofNat\n@[extern \"lean_uint64_to_nat\"]\ndef UInt64.toNat (n : UInt64) : Nat := n.val.val\n@[extern \"lean_uint64_add\"]\ndef UInt64.add (a b : UInt64) : UInt64 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint64_sub\"]\ndef UInt64.sub (a b : UInt64) : UInt64 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint64_mul\"]\ndef UInt64.mul (a b : UInt64) : UInt64 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint64_div\"]\ndef UInt64.div (a b : UInt64) : UInt64 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint64_mod\"]\ndef UInt64.mod (a b : UInt64) : UInt64 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint64_modn\"]\ndef UInt64.modn (a : UInt64) (n : @& Nat) : UInt64 := \u27e8Fin.modn a.val n\u27e9\n@[extern \"lean_uint64_land\"]\ndef UInt64.land (a b : UInt64) : UInt64 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint64_lor\"]\ndef UInt64.lor (a b : UInt64) : UInt64 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint64_xor\"]\ndef UInt64.xor (a b : UInt64) : UInt64 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint64_shift_left\"]\ndef UInt64.shiftLeft (a b : UInt64) : UInt64 := \u27e8a.val <<< (modn b 64).val\u27e9\n@[extern \"lean_uint64_shift_right\"]\ndef UInt64.shiftRight (a b : UInt64) : UInt64 := \u27e8a.val >>> (modn b 64).val\u27e9\ndef UInt64.lt (a b : UInt64) : Prop := a.val < b.val\ndef UInt64.le (a b : UInt64) : Prop := a.val \u2264 b.val\n@[extern \"lean_uint64_to_uint8\"]\ndef UInt64.toUInt8 (a : UInt64) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint64_to_uint16\"]\ndef UInt64.toUInt16 (a : UInt64) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint64_to_uint32\"]\ndef UInt64.toUInt32 (a : UInt64) : UInt32 := a.toNat.toUInt32\n@[extern \"lean_uint8_to_uint64\"]\ndef UInt8.toUInt64 (a : UInt8) : UInt64 := a.toNat.toUInt64\n@[extern \"lean_uint16_to_uint64\"]\ndef UInt16.toUInt64 (a : UInt16) : UInt64 := a.toNat.toUInt64\n@[extern \"lean_uint32_to_uint64\"]\ndef UInt32.toUInt64 (a : UInt32) : UInt64 := a.toNat.toUInt64\n\ninstance : OfNat UInt64 n   := \u27e8UInt64.ofNat n\u27e9\ninstance : Add UInt64       := \u27e8UInt64.add\u27e9\ninstance : Sub UInt64       := \u27e8UInt64.sub\u27e9\ninstance : Mul UInt64       := \u27e8UInt64.mul\u27e9\ninstance : Mod UInt64       := \u27e8UInt64.mod\u27e9\ninstance : HMod UInt64 Nat UInt64 := \u27e8UInt64.modn\u27e9\ninstance : Div UInt64       := \u27e8UInt64.div\u27e9\ninstance : LT UInt64        := \u27e8UInt64.lt\u27e9\ninstance : LE UInt64        := \u27e8UInt64.le\u27e9\n\n@[extern \"lean_uint64_complement\"]\ndef UInt64.complement (a:UInt64) : UInt64 := 0-(a+1)\n\ninstance : Complement UInt64 := \u27e8UInt64.complement\u27e9\ninstance : AndOp UInt64     := \u27e8UInt64.land\u27e9\ninstance : OrOp UInt64      := \u27e8UInt64.lor\u27e9\ninstance : Xor UInt64       := \u27e8UInt64.xor\u27e9\ninstance : ShiftLeft UInt64  := \u27e8UInt64.shiftLeft\u27e9\ninstance : ShiftRight UInt64 := \u27e8UInt64.shiftRight\u27e9\n\n@[extern \"lean_bool_to_uint64\"]\ndef Bool.toUInt64 (b : Bool) : UInt64 := if b then 1 else 0\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint64_dec_lt\"]\ndef UInt64.decLt (a b : UInt64) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint64_dec_le\"]\ndef UInt64.decLe (a b : UInt64) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt64) : Decidable (a < b) := UInt64.decLt a b\ninstance (a b : UInt64) : Decidable (a \u2264 b) := UInt64.decLe a b\ninstance : Max UInt64 := maxOfLe\ninstance : Min UInt64 := minOfLe\n\ntheorem usize_size_gt_zero : USize.size > 0 :=\n  Nat.pos_pow_of_pos System.Platform.numBits (Nat.zero_lt_succ _)\n\n@[extern \"lean_usize_of_nat\"]\ndef USize.ofNat (n : @& Nat) : USize := \u27e8Fin.ofNat' n usize_size_gt_zero\u27e9\nabbrev Nat.toUSize := USize.ofNat\n@[extern \"lean_usize_to_nat\"]\ndef USize.toNat (n : USize) : Nat := n.val.val\n@[extern \"lean_usize_add\"]\ndef USize.add (a b : USize) : USize := \u27e8a.val + b.val\u27e9\n@[extern \"lean_usize_sub\"]\ndef USize.sub (a b : USize) : USize := \u27e8a.val - b.val\u27e9\n@[extern \"lean_usize_mul\"]\ndef USize.mul (a b : USize) : USize := \u27e8a.val * b.val\u27e9\n@[extern \"lean_usize_div\"]\ndef USize.div (a b : USize) : USize := \u27e8a.val / b.val\u27e9\n@[extern \"lean_usize_mod\"]\ndef USize.mod (a b : USize) : USize := \u27e8a.val % b.val\u27e9\n@[extern \"lean_usize_modn\"]\ndef USize.modn (a : USize) (n : @& Nat) : USize := \u27e8Fin.modn a.val n\u27e9\n@[extern \"lean_usize_land\"]\ndef USize.land (a b : USize) : USize := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_usize_lor\"]\ndef USize.lor (a b : USize) : USize := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_usize_xor\"]\ndef USize.xor (a b : USize) : USize := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_usize_shift_left\"]\ndef USize.shiftLeft (a b : USize) : USize := \u27e8a.val <<< (modn b System.Platform.numBits).val\u27e9\n@[extern \"lean_usize_shift_right\"]\ndef USize.shiftRight (a b : USize) : USize := \u27e8a.val >>> (modn b System.Platform.numBits).val\u27e9\n@[extern \"lean_uint32_to_usize\"]\ndef UInt32.toUSize (a : UInt32) : USize := a.toNat.toUSize\n@[extern \"lean_usize_to_uint32\"]\ndef USize.toUInt32 (a : USize) : UInt32 := a.toNat.toUInt32\n\ndef USize.lt (a b : USize) : Prop := a.val < b.val\ndef USize.le (a b : USize) : Prop := a.val \u2264 b.val\n\ninstance : OfNat USize n   := \u27e8USize.ofNat n\u27e9\ninstance : Add USize       := \u27e8USize.add\u27e9\ninstance : Sub USize       := \u27e8USize.sub\u27e9\ninstance : Mul USize       := \u27e8USize.mul\u27e9\ninstance : Mod USize       := \u27e8USize.mod\u27e9\ninstance : HMod USize Nat USize := \u27e8USize.modn\u27e9\ninstance : Div USize       := \u27e8USize.div\u27e9\ninstance : LT USize        := \u27e8USize.lt\u27e9\ninstance : LE USize        := \u27e8USize.le\u27e9\n\n@[extern \"lean_usize_complement\"]\ndef USize.complement (a:USize) : USize := 0-(a+1)\n\ninstance : Complement USize := \u27e8USize.complement\u27e9\ninstance : AndOp USize      := \u27e8USize.land\u27e9\ninstance : OrOp USize       := \u27e8USize.lor\u27e9\ninstance : Xor USize        := \u27e8USize.xor\u27e9\ninstance : ShiftLeft USize  := \u27e8USize.shiftLeft\u27e9\ninstance : ShiftRight USize := \u27e8USize.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_usize_dec_lt\"]\ndef USize.decLt (a b : USize) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_usize_dec_le\"]\ndef USize.decLe (a b : USize) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : USize) : Decidable (a < b) := USize.decLt a b\ninstance (a b : USize) : Decidable (a \u2264 b) := USize.decLe a b\ninstance : Max USize := maxOfLe\ninstance : Min USize := minOfLe\n\ntheorem USize.modn_lt {m : Nat} : \u2200 (u : USize), m > 0 \u2192 USize.toNat (u % m) < m\n  | \u27e8u\u27e9, h => Fin.modn_lt u h\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Init/Data/UInt/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3624304562525669}}
{"text": "-- Deadlocks: var/XSokoban_90_l1/deadlocks\n-- Levelset: data/Large Test Suite Sets/XSokoban_90.xsb\n-- Level: 1\n\nimport .deadlocks\n\ndef XSokoban_90_l1 := sokolevel.from_string \"\n    #####          \n    #   #          \n    #$  #          \n  ###  $##         \n  #  $ $ #         \n### # ## #   ######\n#   # ## #####  ..#\n# $  $          ..#\n##### ### #@##  ..#\n    #     #########\n    #######        \n\"\n\nnamespace XSokoban_90_l1\nopen deadlocks\n\n@[reducible]\ndef deadlock_local (dl : boxint) : Prop := deadlock XSokoban_90_l1.avail XSokoban_90_l1.goal dl\ndef deadlocks_local (dls : list boxint) : Prop\n:= dls.pall (\u03bb dl, deadlock_local dl)\ndef generate_local : list (\u2115 \u00d7 \u2115) \u2192 list (\u2115 \u00d7 \u2115) \u2192 \u2115 \u00d7 \u2115 \u2192 boxint\n:= boxint.generate_from_list XSokoban_90_l1.avail\n\ndef dl0 := generate_local [(8,5)] [] (5,2)\n\ntheorem dl0_dl : deadlock_local dl0\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl0_dl\n\ndef dl1 := generate_local [(3,5)] [] (5,2)\n\ntheorem dl1_dl : deadlock_local dl1\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl1_dl\n\ndef dl2 := generate_local [(14,9)] [] (5,2)\n\ntheorem dl2_dl : deadlock_local dl2\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl2_dl\n\ndef dl3 := generate_local [(5,10)] [] (5,2)\n\ntheorem dl3_dl : deadlock_local dl3\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl3_dl\n\ndef dl4 := generate_local [(6,5), (7,5)] [] (5,2)\n\ntheorem dl4_dl : deadlock_local dl4\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl4_dl\n\ndef dl5 := generate_local [(15,9), (16,9)] [] (5,2)\n\ntheorem dl5_dl : deadlock_local dl5\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl5_dl\n\ndef dl6 := generate_local [(5,2)] [] (6,2)\n\ntheorem dl6_dl : deadlock_local dl6\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl6_dl\n\ndef dl7 := generate_local [(1,8)] [] (5,2)\n\ntheorem dl7_dl : deadlock_local dl7\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl7_dl\n\ndef dl8 := generate_local [(7,2)] [] (5,2)\n\ntheorem dl8_dl : deadlock_local dl8\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl8_dl\n\ndef dl9 := generate_local [(5,5), (6,5), (5,6)] [] (5,2)\n\ntheorem dl9_dl : deadlock_local dl9\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl9_dl\n\ndef dl10 := generate_local [(5,4), (5,5), (6,5)] [] (5,2)\n\ntheorem dl10_dl : deadlock_local dl10\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl10_dl\n\ndef dl11 := generate_local [(5,4), (5,5), (7,5)] [] (3,5)\n\ntheorem dl11_dl : deadlock_local dl11\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl4_dl, -- (5,5) right\n  deadlocked_step dl10_dl, -- (7,5) left\nend\n\n#check dl11_dl\n\ndef dl12 := generate_local [(5,6), (5,7)] [] (5,2)\n\ntheorem dl12_dl : deadlock_local dl12\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl12_dl\n\ndef dl13 := generate_local [(5,5), (6,5), (5,7)] [] (5,2)\n\ntheorem dl13_dl : deadlock_local dl13\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl12_dl, -- (5,5) down\n  deadlocked_step dl9_dl, -- (5,7) up\nend\n\n#check dl13_dl\n\ndef dl14 := generate_local [(5,3), (5,4)] [] (5,2)\n\ntheorem dl14_dl : deadlock_local dl14\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl14_dl\n\ndef dl15 := generate_local [(5,4), (7,4), (6,5), (5,6)] [] (5,2)\n\ntheorem dl15_dl : deadlock_local dl15\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl9_dl, -- (5,4) down\n  deadlocked_step dl4_dl, -- (7,4) down\nend\n\n#check dl15_dl\n\ndef dl16 := generate_local [(5,5), (7,5), (5,6)] [] (3,5)\n\ntheorem dl16_dl : deadlock_local dl16\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl4_dl, -- (5,5) right\n  deadlocked_step dl9_dl, -- (7,5) left\nend\n\n#check dl16_dl\n\ndef dl17 := generate_local [(5,5), (7,5), (5,7)] [] (3,5)\n\ntheorem dl17_dl : deadlock_local dl17\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl4_dl, -- (5,5) right\n  deadlocked_step dl13_dl, -- (7,5) left\n  deadlocked_step dl16_dl, -- (5,7) up\nend\n\n#check dl17_dl\n\ndef dl18 := generate_local [(4,5), (7,5), (5,7), (5,8)] [] (5,2)\n\ntheorem dl18_dl : deadlock_local dl18\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl1_dl, -- (4,5) left\n  deadlocked_step dl0_dl, -- (7,5) right\nend\n\n#check dl18_dl\n\ndef dl19 := generate_local [(4,5), (7,5), (5,6), (5,8)] [] (5,2)\n\ntheorem dl19_dl : deadlock_local dl19\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl1_dl, -- (4,5) left\n  deadlocked_step dl0_dl, -- (7,5) right\n  deadlocked_step dl18_dl, -- (5,6) down\nend\n\n#check dl19_dl\n\ndef dl20 := generate_local [(4,5), (5,5), (5,6)] [] (5,2)\n\ntheorem dl20_dl : deadlock_local dl20\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl20_dl\n\ndef dl21 := generate_local [(4,5), (5,5), (5,7)] [] (5,2)\n\ntheorem dl21_dl : deadlock_local dl21\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl12_dl, -- (5,5) down\n  deadlocked_step dl20_dl, -- (5,7) up\nend\n\n#check dl21_dl\n\ndef dl22 := generate_local [(2,8), (3,8)] [] (5,2)\n\ntheorem dl22_dl : deadlock_local dl22\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl22_dl\n\ndef dl23 := generate_local [(4,5), (2,8), (4,8)] [] (5,2)\n\ntheorem dl23_dl : deadlock_local dl23\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl1_dl, -- (4,5) left\n  deadlocked_step dl22_dl, -- (4,8) left\nend\n\n#check dl23_dl\n\ndef dl24 := generate_local [(5,7), (4,8), (5,8)] [] (5,2)\n\ntheorem dl24_dl : deadlock_local dl24\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl24_dl\n\ndef dl25 := generate_local [(5,6), (4,8), (5,8)] [] (5,2)\n\ntheorem dl25_dl : deadlock_local dl25\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl24_dl, -- (5,6) down\n  deadlocked_step dl12_dl, -- (5,8) up\nend\n\n#check dl25_dl\n\ndef dl26 := generate_local [(5,4), (4,5), (5,5)] [] (5,2)\n\ntheorem dl26_dl : deadlock_local dl26\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl26_dl\n\ndef dl27 := generate_local [(4,5), (5,5), (7,5), (5,8)] [] (5,2)\n\ntheorem dl27_dl : deadlock_local dl27\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl19_dl, -- (5,5) down\n  deadlocked_step dl0_dl, -- (7,5) right\nend\n\n#check dl27_dl\n\ndef dl28 := generate_local [(4,5), (5,7), (2,8), (5,8)] [] (5,2)\n\ntheorem dl28_dl : deadlock_local dl28\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl1_dl, -- (4,5) left\n  deadlocked_step dl23_dl, -- (5,8) left\nend\n\n#check dl28_dl\n\ndef dl29 := generate_local [(4,5), (5,6), (2,8), (5,8)] [] (5,2)\n\ntheorem dl29_dl : deadlock_local dl29\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl1_dl, -- (4,5) left\n  deadlocked_step dl28_dl, -- (5,6) down\n  deadlocked_step dl12_dl, -- (5,8) up\n  deadlocked_step dl23_dl, -- (5,8) left\nend\n\n#check dl29_dl\n\ndef dl30 := generate_local [(3,8), (4,8)] [] (5,2)\n\ntheorem dl30_dl : deadlock_local dl30\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl30_dl\n\ndef dl31 := generate_local [(14,7)] [] (5,2)\n\ntheorem dl31_dl : deadlock_local dl31\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl31_dl\n\ndef dl32 := generate_local [(5,4), (5,5), (2,8), (4,8)] [] (5,2)\n\ntheorem dl32_dl : deadlock_local dl32\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl23_dl, -- (5,5) left\n  deadlocked_step dl22_dl, -- (4,8) left\nend\n\n#check dl32_dl\n\ndef dl33 := generate_local [(5,5), (5,6), (2,8), (4,8)] [] (5,2)\n\ntheorem dl33_dl : deadlock_local dl33\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl23_dl, -- (5,5) left\n  deadlocked_step dl22_dl, -- (4,8) left\nend\n\n#check dl33_dl\n\ndef dl34 := generate_local [(6,8), (7,8)] [] (5,2)\n\ntheorem dl34_dl : deadlock_local dl34\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl34_dl\n\ndef dl35 := generate_local [(7,8), (8,8)] [] (5,2)\n\ntheorem dl35_dl : deadlock_local dl35\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl35_dl\n\ndef dl36 := generate_local [(6,8), (8,8)] [] (5,2)\n\ntheorem dl36_dl : deadlock_local dl36\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl35_dl, -- (6,8) right\n  deadlocked_step dl34_dl, -- (8,8) left\nend\n\n#check dl36_dl\n\ndef dl37 := generate_local [(5,7), (5,8), (8,8)] [] (5,2)\n\ntheorem dl37_dl : deadlock_local dl37\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl36_dl, -- (5,8) right\nend\n\n#check dl37_dl\n\ndef dl38 := generate_local [(5,6), (5,8), (8,8)] [] (5,2)\n\ntheorem dl38_dl : deadlock_local dl38\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl37_dl, -- (5,6) down\n  deadlocked_step dl36_dl, -- (5,8) right\nend\n\n#check dl38_dl\n\ndef dl39 := generate_local [(4,5), (5,5), (5,8), (8,8)] [] (5,2)\n\ntheorem dl39_dl : deadlock_local dl39\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl38_dl, -- (5,5) down\nend\n\n#check dl39_dl\n\ndef dl40 := generate_local [(5,7), (5,8), (6,8)] [] (5,2)\n\ntheorem dl40_dl : deadlock_local dl40\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl40_dl\n\ndef dl41 := generate_local [(5,7), (5,8), (7,8)] [] (5,2)\n\ntheorem dl41_dl : deadlock_local dl41\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl34_dl, -- (5,8) right\n  deadlocked_step dl40_dl, -- (7,8) left\nend\n\n#check dl41_dl\n\ndef dl42 := generate_local [(5,6), (5,8), (6,8)] [] (5,2)\n\ntheorem dl42_dl : deadlock_local dl42\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl40_dl, -- (5,6) down\n  deadlocked_step dl12_dl, -- (5,8) up\nend\n\n#check dl42_dl\n\ndef dl43 := generate_local [(5,6), (5,8), (7,8)] [] (5,2)\n\ntheorem dl43_dl : deadlock_local dl43\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl41_dl, -- (5,6) down\n  deadlocked_step dl12_dl, -- (5,8) up\n  deadlocked_step dl34_dl, -- (5,8) right\n  deadlocked_step dl42_dl, -- (7,8) left\nend\n\n#check dl43_dl\n\ndef dl44 := generate_local [(5,6), (5,8), (8,8)] [] (14,7)\n\ntheorem dl44_dl : deadlock_local dl44\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl12_dl, -- (5,8) up\n  deadlocked_step dl43_dl, -- (8,8) left\nend\n\n#check dl44_dl\n\ndef dl45 := generate_local [(5,6), (5,8), (9,8)] [] (14,7)\n\ntheorem dl45_dl : deadlock_local dl45\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl44_dl, -- (9,8) left\nend\n\n#check dl45_dl\n\ndef dl46 := generate_local [(5,6), (5,8), (10,8)] [] (14,7)\n\ntheorem dl46_dl : deadlock_local dl46\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl45_dl, -- (10,8) left\nend\n\n#check dl46_dl\n\ndef dl47 := generate_local [(5,6), (5,8), (11,8)] [] (14,7)\n\ntheorem dl47_dl : deadlock_local dl47\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl46_dl, -- (11,8) left\nend\n\n#check dl47_dl\n\ndef dl48 := generate_local [(5,6), (5,8), (12,8)] [] (14,7)\n\ntheorem dl48_dl : deadlock_local dl48\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl47_dl, -- (12,8) left\nend\n\n#check dl48_dl\n\ndef dl49 := generate_local [(5,6), (5,8), (13,8)] [] (14,7)\n\ntheorem dl49_dl : deadlock_local dl49\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl48_dl, -- (13,8) left\nend\n\n#check dl49_dl\n\ndef dl50 := generate_local [(5,6), (5,8), (14,8)] [] (14,7)\n\ntheorem dl50_dl : deadlock_local dl50\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl31_dl, -- (14,8) up\n  deadlocked_step dl2_dl, -- (14,8) down\n  deadlocked_step dl49_dl, -- (14,8) left\nend\n\n#check dl50_dl\n\ndef dl51 := generate_local [(5,5), (5,6), (2,8), (5,8)] [] (5,2)\n\ntheorem dl51_dl : deadlock_local dl51\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl29_dl, -- (5,5) left\n  deadlocked_step dl12_dl, -- (5,8) up\n  deadlocked_step dl33_dl, -- (5,8) left\nend\n\n#check dl51_dl\n\ndef dl52 := generate_local [(5,7), (3,8), (5,8)] [] (5,2)\n\ntheorem dl52_dl : deadlock_local dl52\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl24_dl, -- (3,8) right\n  deadlocked_step dl30_dl, -- (5,8) left\nend\n\n#check dl52_dl\n\ndef dl53 := generate_local [(5,6), (3,8), (5,8)] [] (5,2)\n\ntheorem dl53_dl : deadlock_local dl53\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl52_dl, -- (5,6) down\n  deadlocked_step dl25_dl, -- (3,8) right\n  deadlocked_step dl12_dl, -- (5,8) up\n  deadlocked_step dl30_dl, -- (5,8) left\nend\n\n#check dl53_dl\n\ndef dl54 := generate_local [(5,4), (7,4), (6,5), (5,7)] [] (5,2)\n\ntheorem dl54_dl : deadlock_local dl54\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\n  deadlocked_step dl13_dl, -- (5,4) down\n  deadlocked_step dl4_dl, -- (7,4) down\nend\n\n#check dl54_dl\n\ndef dl55 := generate_local [(15,7), (16,7)] [] (5,2)\n\ntheorem dl55_dl : deadlock_local dl55\n:=\nbegin\n  apply new_deadlock,\n  analyze_deadlock,\nend\n\n#check dl55_dl\n\nend XSokoban_90_l1\n", "meta": {"author": "mirefek", "repo": "sokoban.lean", "sha": "451c92308afb4d3f8e566594b9751286f93b899b", "save_path": "github-repos/lean/mirefek-sokoban.lean", "path": "github-repos/lean/mirefek-sokoban.lean/sokoban.lean-451c92308afb4d3f8e566594b9751286f93b899b/src/XSokoban_90_l1_dl.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3624304475344984}}
{"text": "/-\nCopyright (c) 2018 Luca Gerolla. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Luca Gerolla, Kevin Buzzard\nResults on homotopy composition, inverse, reparametrisation for fundamental group. \n-/\nimport analysis.topology.continuity\nimport analysis.topology.topological_space\nimport analysis.topology.infinite_sum\nimport analysis.topology.topological_structures\nimport analysis.topology.uniform_space\nimport analysis.real\nimport data.real.basic tactic.norm_num\nimport data.set.basic\nimport Topology.Material.pasting_lemma\nimport Topology.Material.path \nimport Topology.Material.real_results\nimport Topology.Material.homotopy\n\n\nopen set filter lattice classical\nnamespace homotopy_results  \nopen path\nopen homotopy\n\nvariables {\u03b1  : Type*} [topological_space \u03b1 ] \nvariables {\u03b2  : Type*} [topological_space \u03b2 ] { x y z w : \u03b2  }\nvariable s : I01 \n\nnoncomputable theory\n\nlocal attribute [instance] classical.prop_decidable \n\nlocal notation `I` := @set.univ I01\n\n-----------------------------------------------------\n\n--- Extra Result needed later \n\n-- Now in Mathlib\nlemma frontier_lt_subset_eq [topological_space \u03b1] [decidable_linear_order \u03b1] [t : ordered_topology \u03b1]\n  [topological_space \u03b2] {f g : \u03b2 \u2192 \u03b1} (hf : continuous f) (hg : continuous g) : \nfrontier {b | f b < g b} \u2286 {b | f b = g b} :=\nbegin \n  unfold frontier, \n  have h\u2081 : interior {b : \u03b2 | f b < g b} = {b : \u03b2 | f b < g b}, \n    exact interior_eq_iff_open.2 (is_open_lt  hf hg), rw h\u2081, \n  have h\u2082 : closure {b : \u03b2 | f b < g b} \u2286 closure {b : \u03b2 | f b \u2264  g b}, \n    refine closure_mono _  , rw set.set_of_subset_set_of, intros x h, exact le_of_lt h, \n  have h\u2083 : closure {b : \u03b2 | f b \u2264  g b} = {b : \u03b2 | f b \u2264  g b}, \n    exact closure_eq_iff_is_closed.2 (is_closed_le hf hg), rw h\u2083 at h\u2082 , \n  have g\u2081 : closure {b : \u03b2 | f b < g b} \\ {b : \u03b2 | f b < g b} \u2286 \n                 {b : \u03b2 | f b \u2264 g b} \\ {b : \u03b2 | f b < g b}, \n   {unfold has_sdiff.sdiff set.diff, intros a Ha, simp at Ha, simp, \n    cases Ha with a\u2081 a\u2082 , \n    have h\u2084 : a \u2208 {b : \u03b2 | f b \u2264 g b}, exact set.mem_of_mem_of_subset a\u2081 h\u2082, \n    rw mem_set_of_eq at h\u2084, exact \u27e8 h\u2084, a\u2082 \u27e9 , }, \n  have g\u2082 : {b : \u03b2 | f b \u2264 g b} \\ {b : \u03b2 | f b < g b} \u2286 {b : \u03b2 | f b = g b}, \n    unfold has_sdiff.sdiff set.diff, intros a Ha, simp  at Ha, \n    rw mem_set_of_eq, exact le_antisymm Ha.1 Ha.2, \n  exact set.subset.trans g\u2081 g\u2082,  \nend\n\nlemma mem_frontier_lt [topological_space \u03b1] [decidable_linear_order \u03b1] [t : ordered_topology \u03b1]\n  [topological_space \u03b2] {f g : \u03b2 \u2192 \u03b1} (hf : continuous f) (hg : continuous g) { s : \u03b2  } : \n  s \u2208 frontier {b | f b < g b}  \u2192 s \u2208  {b | f b = g b} := \nbegin intro h, exact  set.mem_of_mem_of_subset h (frontier_lt_subset_eq hf hg), end \n\n---------------------------------------------------------\n\n---- Other Homotopy results (for fundamental group proof)\n\n\n---- Reparametrisation of path and homotopies \n\n-- (Formalise paragraph 2 pg 27 of AT, https://pi.math.cornell.edu/~hatcher/AT/AT.pdf )\n\nstructure repar_I01  := \n(to_fun : I01 \u2192 I01 )\n(at_zero : to_fun 0 = 0 )\n(at_one : to_fun 1 = 1 )\n(cont : continuous to_fun )\n\n@[simp]\nlemma repar_I01_at_zero ( \u03c6  : repar_I01) : \u03c6.to_fun 0 = 0 := \u03c6.2\n\n@[simp]\nlemma repar_I01_at_one ( \u03c6 : repar_I01) : \u03c6.to_fun 1 = 1 := \u03c6.3\n\n-- Reparametrised path f ( \u03c6 (t) )\ndef repar_path  ( f : path x y)( \u03c6 : repar_I01 ) : path x y := \n{   to_fun := \u03bb t , f.to_fun ( \u03c6.to_fun t) ,  \n    at_zero := by simp, \n    at_one := by simp, \n    cont := continuous.comp \u03c6.cont f.cont\n}\n\n-- Function of the homotopy associated to the reparametrisation \u03c6 \ndef rep_hom (\u03c6 : repar_I01) : I01 \u00d7 I01 \u2192 I01 := \n  \u03bb st, \u27e8 ((1 : \u211d ) - st.1.1)*(\u03c6.to_fun st.2).1 + st.1.1 * st.2.1,  \nbegin \n  unfold I01, rw mem_set_of_eq, split, \n    { suffices H1 : 0 \u2264 (1 - (st.fst).val) * (\u03c6.to_fun (st.snd)).val , \n      suffices H2 : 0 \u2264 (st.fst).val * (st.snd).val, exact add_le_add H1 H2, \n        refine mul_nonneg _ _, exact st.1.2.1, exact st.2.2.1, \n        refine mul_nonneg _ _, show 0 \u2264 1 - (st.fst).val , refine sub_nonneg.2  _ , exact st.1.2.2, exact (\u03c6.to_fun (st.snd)).2.1, \n    }, rw (@mul_comm _ _ (1 - (st.fst).val)  (\u03c6.to_fun (st.snd)).val), simp [@mul_add \u211d _ ((\u03c6.to_fun (st.snd)).val )  (1:\u211d ) (- st.fst.val) ], \n    rw mul_comm ((\u03c6.to_fun (st.snd)).val ) ((st.fst).val), \n    have H : ((st.fst).val * (st.snd).val + -((st.fst).val * (\u03c6.to_fun (st.snd)).val)) = (st.fst).val * ((st.snd).val - ((\u03c6.to_fun (st.snd)).val)), simp [mul_add], \n    rw H, have H2 : (st.fst).val \u2264 1, exact st.1.2.2,  \n    let C := 0 < ((st.snd).val - (\u03c6.to_fun (st.snd)).val) , \n    by_cases C, \n        have C1 : 0 < ((st.snd).val - (\u03c6.to_fun (st.snd)).val), exact h, \n        have H3 : (st.fst).val * ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 1 * ((st.snd).val - (\u03c6.to_fun (st.snd)).val), rw one_mul, \n            refine (@le_div_iff _ _ (st.fst).val ((st.snd).val - (\u03c6.to_fun (st.snd)).val) ((st.snd).val - (\u03c6.to_fun (st.snd)).val) C1).1 _, \n            have h2 : ((st.snd).val - (\u03c6.to_fun (st.snd)).val) / ((st.snd).val - (\u03c6.to_fun (st.snd)).val) = 1, refine div_self _, exact ne_of_gt C1,\n            rw h2, exact st.1.2.2,  rw one_mul at H3,  \n        have G1 : (\u03c6.to_fun (st.snd)).val + (st.fst).val * ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 (\u03c6.to_fun (st.snd)).val + ((st.snd).val - (\u03c6.to_fun (st.snd)).val), \n            refine add_le_add _ H3, refine le_of_eq _, refl, \n        suffices G2 : (\u03c6.to_fun (st.snd)).val + ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 1, exact le_trans G1 G2, \n            simp [st.2.2.2], \n        -----\n        have C0 : \u00ac 0 < ((st.snd).val - (\u03c6.to_fun (st.snd)).val), exact h, \n        have C1 : ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 0, exact not_lt.1 C0, \n        have H3 : (st.fst).val * ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 0, \n            exact mul_nonpos_of_nonneg_of_nonpos (st.1.2.1) C1,\n        have G1 : (\u03c6.to_fun (st.snd)).val + (st.fst).val * ((st.snd).val - (\u03c6.to_fun (st.snd)).val) \u2264 (\u03c6.to_fun (st.snd)).val , \n            refine le_neg_add_iff_add_le.1 _, simpa ,\n        have G2 : (\u03c6.to_fun (st.snd)).val \u2264 1, exact (\u03c6.to_fun (st.snd)).2.2, \n        exact le_trans G1 G2, \n\n    --( 1-s )\u03c6 t + s t =\n    -- \u03c6 t + s (t - \u03c6 t) \u2264 \n    -- \u03c6 t + (t - \u03c6 t) =\n    -- t \u2264 1 \nend \u27e9 \n\n@[simp]\nlemma rep_hom_at_zero (\u03c6 : repar_I01) ( y : I01 ) : rep_hom \u03c6 (0, y) = \u03c6.to_fun y := \nbegin unfold rep_hom, simp, apply subtype.eq, simp [mul_comm, mul_zero], \n  show  y.val * 0 + (\u03c6.to_fun y).val * (1 + -0) = (\u03c6.to_fun y).val, simp [mul_zero, mul_add, add_zero] \nend\n\n@[simp]\nlemma rep_hom_at_one (\u03c6 : repar_I01) ( y : I01) : rep_hom \u03c6 (1, y) =  y := \nbegin unfold rep_hom, apply subtype.eq, simp [-sub_eq_add_neg],\n  show  1 * y.val + (1 - 1) * (\u03c6.to_fun y).val = y.val, simp\nend \n\n@[simp]\nlemma rep_hom_pt_at_zero (\u03c6 : repar_I01) ( s : I01) : rep_hom \u03c6 (s, 0) = 0 := \nbegin unfold rep_hom, simp, apply subtype.eq, simp,  show s.val * 0+ (1 + -s.val) * 0 = 0, simp, end\n\n@[simp]\nlemma rep_hom_pt_at_one (\u03c6 : repar_I01) ( s : I01) : rep_hom \u03c6 (s, 1) = 1 := \nbegin unfold rep_hom, simp, apply subtype.eq, simp, show s.val * 1 + (1 + -s.val) * 1 = 1, simp end\n\nlemma cont_rep_hom (\u03c6 : repar_I01) : continuous (rep_hom \u03c6 ) := \nbegin \n  unfold rep_hom, refine continuous_subtype_mk _ _, \n  refine @continuous_add _ _ _ _ _ _ (\u03bb st: I01\u00d7I01 , (1 - (st.fst).val) * (\u03c6.to_fun (st.snd)).val ) _ _ _, \n   { refine continuous_mul _ _, refine continuous_add _ _, exact continuous_const, \n      show continuous (( \u03bb x : \u211d , - x ) \u2218 (\u03bb (st : \u21a5I01 \u00d7 \u21a5I01), (st.fst).val) ), \n      refine continuous.comp (continuous.comp continuous_fst continuous_subtype_val) (continuous_neg continuous_id),  \n     exact continuous.comp (continuous.comp continuous_snd \u03c6.cont) continuous_subtype_val\n   },\n   refine continuous_mul _ (continuous.comp continuous_snd continuous_subtype_val) , \n   exact continuous.comp continuous_fst continuous_subtype_val, \nend\n\n-- Define homotopy from f \u03c6 to f , for any repar \u03c6 \ndef hom_repar_path_to_path {\u03b1 : Type*} [topological_space \u03b1 ] {x y : \u03b1 } \n( f : path x y)( \u03c6 : repar_I01 ) : path_homotopy (repar_path f \u03c6 ) f := \n{   to_fun :=  \u03bb st, f.to_fun ( (rep_hom \u03c6) st), \n    path_s := \n    begin \n      intro s, unfold is_path, split, simp, split, simp, \n      show continuous ( (\u03bb (st : I01\u00d7I01), f.to_fun (rep_hom \u03c6 st )) \u2218 ( \u03bb t : I01, ((s, t) : I01 \u00d7 I01) )    ),\n      refine continuous.comp _ (continuous.comp (cont_rep_hom \u03c6 ) f.cont ), \n      exact continuous.prod_mk continuous_const continuous_id \n    end, \n    at_zero := by simp,  \n    at_one := by simp,  \n    cont :=  continuous.comp (cont_rep_hom \u03c6 ) f.cont\n}\n\n-- Prove f \u03c6 \u2248 f (they are homotopic)\ntheorem repar_path_is_homeq {\u03b1 : Type*} [topological_space \u03b1 ] {x y : \u03b1 } ( f : path x y)( \u03c6 : repar_I01 ) \n: is_homotopic_to (repar_path f \u03c6 ) f := \nbegin unfold is_homotopic_to, exact nonempty.intro (hom_repar_path_to_path f \u03c6 ),  end \n\n-----------------------------\n\n-- Homotopy of path inverses\n------  a \u2248 b  \u2192  a\u207b\u00b9 \u2248 b\u207b\u00b9 \n\n\ndef f_path_inv { a b : path x y } ( F : path_homotopy a b ) : I01 \u00d7 I01 \u2192 \u03b2 :=\n\u03bb st, F.to_fun (st.1 , par_inv st.2) \n\nlemma f_path_inv_start_pt  { a b : path x y } ( F : path_homotopy a b ) :\n\u2200 (s : I01), f_path_inv F (s, 0) = y := begin unfold f_path_inv, simp end\n\nlemma f_path_inv_end_pt  { a b : path x y } ( F : path_homotopy a b ) :\n\u2200 (s : I01), f_path_inv F (s, 1) = x := begin unfold f_path_inv, simp end\n\nlemma f_path_inv_at_zero  { a b : path x y } ( F : path_homotopy a b ) :\n\u2200 (y_1 : I01), f_path_inv F (0, y_1) = (inv_of_path a).to_fun y_1 := \nbegin intro y, unfold f_path_inv inv_of_path, simp, end\n\nlemma f_path_inv_at_one  { a b : path x y } ( F : path_homotopy a b ) :\n\u2200 (y_1 : I01), f_path_inv F (1, y_1) = (inv_of_path b).to_fun y_1 := \nbegin intro y, unfold f_path_inv inv_of_path, simp, end \n\nlemma f_path_inv_cont { a b : path x y } ( F : path_homotopy a b ) :\ncontinuous (f_path_inv F) := \nbegin \n unfold f_path_inv, \n refine continuous.comp _ F.cont, \n refine continuous.prod_mk continuous_fst _, \n exact continuous.comp continuous_snd continuous_par_inv \nend\n\nnoncomputable def path_homotopy_of_inv_path {\u03b1 } [topological_space \u03b1] {x y : \u03b1}\n { a b : path x y } \n  ( F : path_homotopy a b ) : path_homotopy (inv_of_path a) (inv_of_path b) := \npath_homotopy.mk' (f_path_inv F) (f_path_inv_start_pt F) (f_path_inv_end_pt F) \n (f_path_inv_at_zero F) (f_path_inv_at_one F) (f_path_inv_cont F) \n\n\n---------------------------------\n\n\n--------------------------------- \n\n-- Homotopy on composition of paths \n------ a\u2081 \u2248 b\u2081 , a\u2082 \u2248 b\u2082  \u2192  a\u2081 \u2b1d a\u2082 \u2248 b\u2081 \u2b1d b\u2082 \n\n-- needed to prove multiplication is well definied in fundamental group \n\n-- Define (continuous) shift function to employ results on I01 \u00d7 I01 from  path_homotopy_comp\ndef shift_order ( \u03b1 : Type* ) (\u03b2 : Type*) [topological_space \u03b1] [topological_space \u03b2 ] : \n  \u03b1 \u00d7 \u03b2 \u2192 \u03b2 \u00d7 \u03b1 := \u03bb ab, (ab.2, ab.1) \n\ntheorem continuous_shift_order {\u03b1 \u03b2 } [topological_space \u03b1] [topological_space \u03b2 ] : \n  continuous (shift_order \u03b1 \u03b2 ) := \nbegin unfold shift_order, exact continuous.prod_mk continuous_snd continuous_fst end \n\nlocal notation `shift` := shift_order _ _ \n\n@[simp]\nlemma shift_cond_start  { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n  { F : path_homotopy a\u2081 b\u2081 } { G : path_homotopy a\u2082 b\u2082 } : \n  paste cover_prod_I01 (\u03bb (st : \u21a5(set.prod T1 univ)), \n  F.to_fun (shift \u2191st)) (\u03bb (st : \u21a5(set.prod T2 univ)), G.to_fun (shift \u2191st))(shift (s, 0)) = x := \nbegin unfold shift_order paste, rw dif_pos, simp, simp, exact help_T1,  end\n\n@[simp]\nlemma shift_cond_end  { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n  { F : path_homotopy a\u2081 b\u2081 } { G : path_homotopy a\u2082 b\u2082 } : \n  paste cover_prod_I01 (\u03bb (st : \u21a5(set.prod T1 univ)), F.to_fun (shift \u2191st)) \n  (\u03bb (st : \u21a5(set.prod T2 univ)), G.to_fun (shift \u2191st))(shift (s, 1)) = z :=\nbegin unfold shift_order paste, rw dif_neg, simp, simp, exact help_02,   end\n\n--- \n\n\n-- Define (continuous ) reparametrisations to shift domain and construct a homotopy  : a\u2081 \u2b1d a\u2082 \u2248 b\u2081 \u2b1d b\u2082  \n----- by pasting homotopies a\u2081 \u2248 b\u2081 , a\u2082 \u2248 b\u2082\ndef repar_shift_a : set.prod T1 I \u2192 I01 \u00d7 I01 := \n\u03bb st, shift (  par zero_lt_half \u27e8 st.1.1, (mem_prod.1 st.2).1\u27e9 , st.1.2 ) \n\ndef repar_shift_b : set.prod T2 I \u2192 I01 \u00d7 I01 := \n\u03bb st, shift (  par half_lt_one \u27e8 st.1.1, (mem_prod.1 st.2).1 \u27e9 , st.1.2 ) \n\nlemma cont_r_shift_a : continuous repar_shift_a := \nbegin \n  unfold repar_shift_a, refine continuous.comp _ continuous_shift_order, \n  refine continuous.prod_mk _ _, \n    refine continuous.comp _ (continuous_par _ ), refine continuous_subtype_mk _ _, \n    exact continuous.comp continuous_subtype_val continuous_fst, \n    exact continuous.comp continuous_subtype_val continuous_snd, \nend\n\nlemma cont_r_shift_b : continuous repar_shift_b := \nbegin \n  unfold repar_shift_b, refine continuous.comp _ continuous_shift_order, \n  refine continuous.prod_mk _ _, \n    refine continuous.comp _ (continuous_par _ ), refine continuous_subtype_mk _ _, \n    exact continuous.comp continuous_subtype_val continuous_fst, \n    exact continuous.comp continuous_subtype_val continuous_snd, \nend\n\n-- Define the function of homotopy a\u2081 \u2b1d a\u2082 \u2248 b\u2081 \u2b1d b\u2082 and prove lemmas to use with path_homotopy.mk' \ndef f_path_comp { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) :=\n \u03bb st, ( paste  cover_prod_I01 ( \u03bb st, F.to_fun (repar_shift_a st) ) \n   ( \u03bb st, G.to_fun (repar_shift_b st) ) )  ( shift  st) \n\nlemma f_path_comp_start_pt { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n  ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \n  \u2200 (s : I01), f_path_comp F G (s, 0) = x := \nbegin \n  intro s, unfold f_path_comp, unfold repar_shift_a repar_shift_b shift_order paste, rw dif_pos, \n  show F.to_fun (s, par zero_lt_half \u27e80, help_T1\u27e9) = x,  simp,  \n  simp, exact help_T1,    \nend\n\nlemma f_path_comp_end_pt { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n  ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \n  \u2200 (s : I01), f_path_comp F G (s, 1) = z := \nbegin \n  intro s, unfold f_path_comp, unfold repar_shift_a repar_shift_b shift_order paste, rw dif_neg, simp, \n  show G.to_fun (s, par half_lt_one \u27e81, help_T2\u27e9) = z, simp, \n  simp [help_02],\nend\n\nlemma f_path_comp_at_zero { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \n\u2200 (y : I01), f_path_comp F G (0, y) = (comp_of_path a\u2081 a\u2082).to_fun y := \nbegin \n  intro s, unfold f_path_comp comp_of_path fa_path fb_path fgen_path paste,  simp, \n  split_ifs,  unfold repar_shift_a, unfold shift_order, simpa, \n  { by_contradiction, unfold shift_order at h, simp at h, cc, }, \n  { by_contradiction, unfold shift_order at h, simp at h, cc, }, \n  unfold repar_shift_b, unfold shift_order, simpa,\nend\n\nlemma f_path_comp_at_one { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \n \u2200 (y : I01), f_path_comp F G (1, y) = (comp_of_path b\u2081 b\u2082).to_fun y := \nbegin \n  intro s, unfold f_path_comp comp_of_path fa_path fb_path fgen_path paste,  simp, \n  split_ifs,  unfold repar_shift_a, unfold shift_order, simpa, \n  { by_contradiction, unfold shift_order at h, simp at h, cc, }, \n  { by_contradiction, unfold shift_order at h, simp at h, cc, },\n  unfold repar_shift_b, unfold shift_order, simpa,\nend\n\nlemma f_path_comp_cont { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \ncontinuous (f_path_comp F G) := \nbegin \n  unfold f_path_comp, refine continuous.comp continuous_shift_order _,     \n    refine cont_of_paste prod_T1_is_closed prod_T2_is_closed _ _ _, \n    {unfold match_of_fun, intros w B1 B2, \n    have Int : w \u2208 set.inter (set.prod T1 I) (set.prod T2 I), exact \u27e8 B1 , B2 \u27e9 , rwa [prod_inter_T] at Int, \n    have V : w.1.1 = 1/2, rwa [set.prod, mem_set_of_eq] at Int, rwa [mem_set_of_eq] at Int, exact Int.1, cases w, \n    have xeq : w_fst = \u27e8 1/2 , help_01 \u27e9 , apply subtype.eq, rw V,\n    simp [xeq, -one_div_eq_inv], unfold repar_shift_a repar_shift_b shift_order, \n    simp [-one_div_eq_inv], \n    show F.to_fun (w_snd, par zero_lt_half \u27e8\u27e81 / 2, help_01\u27e9, help_half_T1\u27e9) =\n    G.to_fun (w_snd, par half_lt_one \u27e8\u27e81 / 2, help_01\u27e9, help_half_T2\u27e9) , rw [eqn_1, eqn_2], simp,  \n    } ,  \n    exact continuous.comp cont_r_shift_a F.cont, \n    exact continuous.comp cont_r_shift_b G.cont, \nend\n\n-- Prove that we have the homotopy a\u2081 \u2b1d a\u2082 \u2248 b\u2081 \u2b1d b\u2082\nnoncomputable def path_homotopy_of_comp_path { a\u2081 b\u2081 : path x y} { a\u2082  b\u2082  : path y z} \n  ( F : path_homotopy a\u2081 b\u2081 ) ( G : path_homotopy a\u2082 b\u2082 ) : \n  path_homotopy (comp_of_path a\u2081 a\u2082) (comp_of_path b\u2081 b\u2082) := \nbegin \n  refine path_homotopy.mk' (f_path_comp F G ) _ _ _ _ _, \n  exact f_path_comp_start_pt F G, exact f_path_comp_end_pt F G, \n  exact f_path_comp_at_zero F G , exact f_path_comp_at_one F G, \n  exact f_path_comp_cont F G, \nend\n\n\n----------------------------------------------------\n\nlocal attribute [instance] classical.prop_decidable\n\n------------------------------------------\n\n-- Homotopy of composition with inverse \n------ a\u207b\u00b9 \u2b1d a \u2248 c\u2080  \n\n-- continuity lemma employed later\nlemma cont_help_1 : continuous (\u03bb (a : \u21a5I01 \u00d7 \u21a5I01), 1 - (a.fst).val ) := \nbegin \n  have h : continuous ( \u03bb (r : \u211d ), 1 - r ),  conv in ( (1:\u211d)-_) begin rw help_inv, end,  \n    by exact continuous.comp (real.continuous_mul_const (-1) ) (real.continuous_add_const 1), \n  exact continuous.comp (continuous.comp continuous_fst continuous_subtype_val) h, \nend\n\n\n/- To prove for a path f that f\u207b\u00b9 \u2b1d f \u2248 c\u2080 (constant loop at given basepoint), \nneed 2 continuous function that are (2) piecewise-linear applied as reparamatrization of path \u03c6(s) (similar to above),\nsuch that the composition (f\u207b\u00b9 \u2b1d f) \u03c6(1) = c\u2080 and (f\u207b\u00b9 \u2b1d f) \u03c6(0) = f\u207b\u00b9 \u2b1d f - as depicted in AT pg 27. \n-/ \n\n-- \"Reparametrisation\" to shrink path f\u207b\u00b9 \n\ndef par_aux_a : I01 \u00d7 I01 \u2192 I01 := \n\u03bb st, if ((1 : \u211d ) - st.1.1) < st.2.1 then st.1 else par_inv st.2\n\n\nlemma continuous_par_aux_a  : continuous par_aux_a := \nbegin \n  unfold par_aux_a, \n  refine continuous_if _ continuous_fst (continuous.comp continuous_snd continuous_par_inv) , \n  intros st F, \n  have H : frontier {a : \u21a5I01 \u00d7 \u21a5I01 | 1 - (a.fst).val < (a.snd).val} \u2286  {a : \u21a5I01 \u00d7 \u21a5I01 | 1 - (a.fst).val = (a.snd).val }, \n    exact frontier_lt_subset_eq cont_help_1 (continuous.comp continuous_snd continuous_subtype_val), \n  have h : st \u2208 {a : \u21a5I01 \u00d7 \u21a5I01 | 1 - (a.fst).val = (a.snd).val}, \n    exact set.mem_of_mem_of_subset F H , \n    rw [ mem_set_of_eq] at h, unfold par_inv, refine subtype.eq _, \n    show (st.fst).val = 1 -(st.snd).val, \n    have H4 : (st.snd).val = 1 - (st.fst).val, exact eq.symm h, \n  simp [H4], \nend\n\n\ndef repar_stop_a : set.prod T1 I \u2192 I01 := \n\u03bb st, par_aux_a ( shift (  par zero_lt_half \u27e8 st.1.1, (mem_prod.1 st.2).1\u27e9 , st.1.2 ) )\n\n\nlemma cont_r_stop_a : continuous repar_stop_a := \nbegin \n  unfold repar_stop_a, refine continuous.comp _ continuous_par_aux_a,\n  refine continuous.comp _ continuous_shift_order,\n  refine continuous.prod_mk _ (continuous.comp continuous_subtype_val continuous_snd), \n  refine continuous.comp _ (continuous_par _ ),\n  refine continuous_subtype_mk _ _, exact continuous.comp continuous_subtype_val continuous_fst, \nend\n\n-- This will be the actual function that will make up the left part of the homotopy (see f_inv_comp)\ndef fa_inv_comp (f : path x y) : set.prod T1 I \u2192 \u03b2  := \n\u03bb st, f.to_fun  ( repar_stop_a st  )\n\n\nlemma cont_fa_inv_comp  (f : path x y) : \ncontinuous (fa_inv_comp f) := \nbegin unfold fa_inv_comp, exact continuous.comp  cont_r_stop_a  f.cont, end \n\n\n--------------\n\n-- \"Reparametrisation\" to shrink path f \n\n\ndef par_aux_b : I01 \u00d7 I01 \u2192 I01 := \n\u03bb st, if st.2.1 < st.1.1 then st.1 else st.2\n\n\n\nlemma continuous_par_aux_b  : continuous par_aux_b := \nbegin \n  unfold par_aux_b, \n  refine continuous_if _ continuous_fst continuous_snd , \n  {intros st F, \n  have H : frontier {a : \u21a5I01 \u00d7 \u21a5I01 | (a.snd).val < (a.fst).val} \u2286 {a : \u21a5I01 \u00d7 \u21a5I01 | (a.snd).val = (a.fst).val}, \n    exact frontier_lt_subset_eq (continuous.comp continuous_snd continuous_subtype_val) (continuous.comp continuous_fst continuous_subtype_val) , \n  have h : st \u2208 {a : \u21a5I01 \u00d7 \u21a5I01 | (a.snd).val = (a.fst).val}, \n   exact set.mem_of_mem_of_subset F H , rw [ mem_set_of_eq] at h, \n   apply eq.symm, exact subtype.eq h,    \n  }, \nend\n\n\ndef repar_stop_b : set.prod T2 I \u2192 I01  := \n\u03bb st, par_aux_b ( shift (  par half_lt_one \u27e8 st.1.1, (mem_prod.1 st.2).1\u27e9 , st.1.2 ) )\n\nlemma cont_r_stop_b : continuous repar_stop_b := \nbegin \n unfold repar_stop_b, refine continuous.comp _ continuous_par_aux_b,\n refine continuous.comp _ continuous_shift_order,\n refine continuous.prod_mk _ (continuous.comp continuous_subtype_val continuous_snd), \n refine continuous.comp _ (continuous_par _ ),\n refine continuous_subtype_mk _ _, exact continuous.comp continuous_subtype_val continuous_fst, \nend\n\ndef fb_inv_comp  (f : path x y) : set.prod T2 I \u2192 \u03b2  := \n\u03bb st, f.to_fun (  repar_stop_b st  )\n\nlemma cont_fb_inv_comp  (f : path x y) : \ncontinuous (fb_inv_comp f) := \nbegin unfold fb_inv_comp, exact continuous.comp cont_r_stop_b  f.cont, end \n\n\n\n---- Combine the two reparametrisation \n---- Set up function and lemmas for path_homotopy.mk' \n\ndef f_inv_comp  (f : path x y) : I01 \u00d7 I01 \u2192 \u03b2  := \n\u03bb st, ( paste cover_prod_I01  ( \u03bb st, (fa_inv_comp f ) st ) ( \u03bb st, (fb_inv_comp f ) st ) ) (shift st)\n\n\nlemma f_inv_comp_start_pt  (f : path x y) : \u2200 (s : I01), f_inv_comp f (s, 0) = y := \nbegin \n  intro s, unfold f_inv_comp fa_inv_comp fb_inv_comp, \n  unfold repar_stop_a repar_stop_b shift_order par_aux_a par_aux_b paste, simp [-sub_eq_add_neg], \n  rw [dif_pos ], \n  have H : ite (1 - s.val < (par zero_lt_half \u27e80, help_T1\u27e9).val) s (par_inv (par zero_lt_half \u27e80, help_T1\u27e9)) = (1 : I01), \n    split_ifs,   have H2 : s.val + (0 : I01).val = s.val, show s.val + (0:\u211d ) = s.val, exact  add_zero s.val, \n      have H3 : s.val \u2264 (1:I01).val, exact s.2.2, have H4 : 1 - (0:I01).val = 1, exact sub_zero 1, \n      rw [ eqn_start ] at h, rw [sub_lt] at h, have H5 : 1 < s.val, rw H4 at h,  exact h,\n      by_contradiction, have G : s.val < s.val, exact lt_of_le_of_lt H3 h, \n        simp [lt_iff_le_and_ne] at G, trivial,\n      simp,  \n  show f.to_fun (ite (1 - s.val < (par zero_lt_half \u27e80, help_T1\u27e9).val) s (par_inv (par zero_lt_half \u27e80, help_T1\u27e9))) = y, \n  rw [H], exact f.at_one,\n  simp, exact help_T1, \nend\n\n\nlemma f_inv_comp_end_pt (f : path x y) : \u2200 (s : I01), f_inv_comp f (s, 1) = y := \nbegin \n  intro s, unfold f_inv_comp, unfold paste, rw dif_neg, unfold fb_inv_comp repar_stop_b shift_order par_aux_b, simp [-sub_eq_add_neg], \n  have H : ite ((par half_lt_one \u27e81, help_T2\u27e9).val < s.val) s (par half_lt_one \u27e81, help_T2\u27e9) = 1,\n    split_ifs, {  by_contradiction, rw eqn_end at h, have H2 : s.val \u2264 (1:I01).val, exact s.2.2, \n      have G : s.val < s.val, exact lt_of_le_of_lt H2 h, simp [lt_iff_le_and_ne] at G, trivial, \n      },  \n      exact eqn_end, \n  show f.to_fun (ite ((par half_lt_one \u27e81, help_T2\u27e9).val < s.val) s (par half_lt_one \u27e81, help_T2\u27e9 )) = y, \n  rw H, exact f.at_one, \n  unfold shift_order, simp [help_02], \nend \n\n\nlemma f_inv_comp_at_zero (f : path x y) :\n  \u2200 (y_1 : I01), f_inv_comp f (0, y_1) = (comp_of_path (inv_of_path f) f).to_fun y_1 := \nbegin \n  intro t, unfold f_inv_comp fa_inv_comp fb_inv_comp repar_stop_a repar_stop_b,  \n  unfold paste, unfold shift_order, split_ifs, \n    unfold shift_order at h, simp at h, unfold par_inv comp_of_path paste,  simp [h], \n    unfold fa_path par_aux_a inv_of_path, simp [-sub_eq_add_neg], \n    show f.to_fun (ite (1 - 0< (par zero_lt_half \u27e8t, _\u27e9).val) 0 (par_inv (par zero_lt_half \u27e8t, _\u27e9))) =\n     f.to_fun (par_inv (par zero_lt_half \u27e8t, _\u27e9)), \n     simp, rw if_neg, refl, refine not_lt.2 _, exact (par zero_lt_half \u27e8t, _\u27e9).2.2, \n    unfold shift_order at h, simp at h, unfold comp_of_path paste fb_path, \n    unfold par_aux_b, rw if_neg, simpa [h], \n    refine not_lt.2 _, simp, exact (par half_lt_one \u27e8t, _\u27e9).2.1, \nend\n\nlemma f_inv_comp_at_one (f : path x y) :\n  \u2200 (y_1 : I01), f_inv_comp f (1, y_1) = (loop_const y).to_fun y_1 :=\nbegin  \n  intro t, unfold f_inv_comp fa_inv_comp fb_inv_comp repar_stop_a repar_stop_b,  \n  unfold paste, unfold shift_order, split_ifs, \n    unfold shift_order at h, simp at h, unfold loop_const, unfold par_aux_a, split_ifs with h\u2082 , \n      exact f.at_one, \n      simp [not_lt,  -sub_eq_add_neg] at h\u2082 ,\n      have H :  (par zero_lt_half \u27e8t, _\u27e9).val \u2264 1 - 1,  exact h\u2082 , rw [sub_self] at H, \n      have h\u2083 : (par zero_lt_half \u27e8t, h\u27e9).val = 0, --\n        exact le_antisymm H ((par zero_lt_half \u27e8t, h\u27e9).2.1), \n      have H2 : (par zero_lt_half \u27e8t, h\u27e9) = (0: I01), \n        exact subtype.eq h\u2083 , \n      show f.to_fun (par_inv (par zero_lt_half \u27e8t, h\u27e9)) = y, rw H2, simp, \n    unfold shift_order at h, simp at h, unfold par_aux_b, unfold loop_const, split_ifs with h\u2082, \n      exact f.at_one, \n      simp [not_lt,  -sub_eq_add_neg] at h\u2082, \n      have H : (par half_lt_one \u27e8t, _\u27e9).val = (1:I01).val, \n       apply eq.symm, exact le_antisymm h\u2082 (par half_lt_one \u27e8t, _\u27e9).2.2, \n      have H2 : (par half_lt_one \u27e8t, _\u27e9) = (1: I01), exact subtype.eq H, \n      simp [H2], \nend\n\n\nlemma f_inv_comp_cont (f : path x y) : continuous (f_inv_comp f) := \nbegin \n  unfold f_inv_comp, refine continuous.comp continuous_shift_order _,  \n  refine cont_of_paste prod_T1_is_closed prod_T2_is_closed _ \n    (cont_fa_inv_comp f) (cont_fb_inv_comp f), \n  { unfold match_of_fun, intros w B1 B2, \n    have Int : w \u2208 set.inter (set.prod T1 I) (set.prod T2 I), exact \u27e8 B1 , B2 \u27e9 , rwa [prod_inter_T] at Int, \n    have V : w.1.1 = 1/2, rwa [set.prod, mem_set_of_eq] at Int, rwa [mem_set_of_eq] at Int, exact Int.1, cases w, \n    have xeq : w_fst = \u27e8 1/2 , help_01 \u27e9 , apply subtype.eq, rw V, --\n    simp [xeq, -one_div_eq_inv], unfold  fa_inv_comp fb_inv_comp, \n    unfold repar_stop_a repar_stop_b shift_order par_aux_a par_aux_b, simp [-sub_eq_add_neg,-one_div_eq_inv], \n    show f.to_fun\n      (ite (1 - w_snd.val < (par zero_lt_half \u27e8\u27e81 / 2, help_01\u27e9, help_half_T1\u27e9).val) w_snd\n         (par_inv (par zero_lt_half \u27e8\u27e81 / 2, help_01\u27e9, help_half_T1\u27e9))) =\n    f.to_fun\n      (ite ((par half_lt_one \u27e8\u27e81 / 2, help_01\u27e9, help_half_T2\u27e9).val < w_snd.val) w_snd\n         (par half_lt_one \u27e8\u27e81 / 2, help_01\u27e9, help_half_T2 \u27e9)),  \n    rw [eqn_1, eqn_2, eqn_2_par_inv], rw sub_lt, \n    show f.to_fun (ite (1 - (1:\u211d ) < w_snd.val) w_snd 0) = \n    f.to_fun (ite ((0:\u211d)  < w_snd.val) w_snd 0), rw sub_self, \n  }, \nend\n\n\nnoncomputable def hom_inv_comp_to_const (f : path x y) : \n  path_homotopy (comp_of_path (inv_of_path f) f) (loop_const y) := \npath_homotopy.mk' (f_inv_comp f) (f_inv_comp_start_pt f) (f_inv_comp_end_pt f) \n(f_inv_comp_at_zero f) (f_inv_comp_at_one f) (f_inv_comp_cont f)  \n\n\n\n----------------------------------------------------------------------\n\n-- Homotopy of three paths (associativity)\n------ (f \u2b1d g) \u2b1d h \u2248 f \u2b1d ( g \u2b1d h)  \n\n/-  For this define a (3) piecewise linear function \u03c6 (repar_I01), \nwhose corresponding homotopy will serve for associativity proof.\n-/\n\n--- Reparametrisation on [1/2, 1] ( 2 piecewise linear funtion : [1/2, 1] \u2192 [1/4, 1] )\n\nlemma help_p3_aux\u2081  (s : T2) : (s.val).val - 1 / 4 \u2208 I01 := \nbegin \nunfold I01, rw mem_set_of_eq, split, \n refine  le_sub_iff_add_le.2 _,  rw [add_comm, add_zero], refine le_trans _ s.2.1, {norm_num},\n rw sub_le_iff_le_add, refine le_trans s.2.2 _, norm_num, \nend\n\n\nlemma help_p3_aux\u2082  ( s : T2) :  2 * (s.val).val - 1 \u2208 I01 := \nbegin \n unfold I01, \n rw mem_set_of_eq, split, \n  have h\u2081 : 1/2 \u2264 (s.val).val, exact s.2.1, \n  refine le_sub_iff_add_le.2 _, rw [add_comm, add_zero], \n  have H : (2 : \u211d) > 0, {norm_num}, rw mul_comm, \n  refine (div_le_iff H).1 _, exact h\u2081, \n  have h\u2082 : (s.val).val \u2264 (1:\u211d ), exact s.2.2, \n  have H2 : 2*(s.val).val \u2264 2 * 1, \n  have HH : 0 < (2 : \u211d), {norm_num}, \n  refine (@mul_le_mul_left _ _ s.1.1 1 2 HH ).2 _, exact h\u2082, \n  rw [mul_one] at H2, norm_num [H2], \nend\n\n\ndef p3_aux : T2 \u2192 I01 := \n\u03bb s, if s.1.1 < (3/4: \u211d ) then \u27e8 s.1.1 - 1/4 , help_p3_aux\u2081 s \u27e9  else \u27e8 ( 2 : \u211d )*s.1.1 - (1: \u211d) , help_p3_aux\u2082  s \u27e9 \n\nlemma help_cont_p3_aux\u2081 : continuous (\u03bb (s : \u21a5T2), (s.val).val - 1 / 4) := \ncontinuous.comp  (continuous.comp continuous_subtype_val continuous_subtype_val) (real.continuous_sub_const  (1/4) )\n\nlemma help_cont_p3_aux\u2082  : continuous (\u03bb (s : \u21a5T2), 2 * (s.val).val - 1) := \ncontinuous.comp  (continuous.comp continuous_subtype_val continuous_subtype_val) (real.continuous_linear 2 (-1) )\n\n\nlemma cont_p3_aux : continuous p3_aux := \nbegin \n  unfold p3_aux, \n  refine continuous_if _ _ _, \n    intros x h, \n  have h\u2082 := mem_frontier_lt (continuous.comp continuous_subtype_val continuous_subtype_val) (continuous_const) h, \n  simp at h\u2082 , refine subtype.eq _, norm_num [h\u2082 ], \n  exact continuous_subtype_mk _ help_cont_p3_aux\u2081,  \n  refine continuous_subtype_mk _ help_cont_p3_aux\u2082,\nend\n\n-----------------------------\n\n-- Reparametrisation on [0, 1/2]\n\nlemma help_p3_T1_aux (x : T1) : 1 / 2 * (x.val).val \u2208 I01 :=\nbegin \n  unfold I01, rw mem_set_of_eq, split, \n    refine mul_nonneg _ x.2.1, {norm_num}, \n   --norm_num [x.2.2], \n    have h :  x.val.val \u2264 1/2 , exact x.2.2, \n    have h\u2082 : 1 / 2 * (x.val).val \u2264 (1/2 : \u211d )* (1/2:\u211d ), \n    have g\u2081 : (1/ 2 : \u211d)  \u2264 1/2, refine @le_of_eq _ _ (1/2:\u211d ) (1/2:\u211d ) (refl (1/2:\u211d )), \n      refine mul_le_mul g\u2081 h x.2.1 _ , {norm_num}, \n    refine le_trans h\u2082 _  ,\n    norm_num, \nend\n\ndef p3_T1_aux : T1 \u2192 I01 := \u03bb x, \u27e8 (1/2:\u211d ) * x.1.1 , help_p3_T1_aux x \u27e9 \n\nlemma cont_p3_T1_aux : continuous p3_T1_aux := \nbegin \n  unfold p3_T1_aux, \n  refine continuous_subtype_mk _ ( continuous.comp \n    (continuous.comp continuous_subtype_val continuous_subtype_val) \n    (real.continuous_mul_const (1/2)) ) , \nend\n\n--\n\n-- Define the 3 p.w.l function \u03c6 needed for the homotopy \n\nnoncomputable def p3 : repar_I01 := \n{ to_fun := paste cover_I01 p3_T1_aux p3_aux, \n\n  at_zero := \n  begin unfold paste, rw dif_pos, unfold p3_T1_aux, dsimp, refine subtype.eq _, \n  exact help_T1, dsimp, exact mul_zero _,  end, \n\n  at_one := \n  begin unfold paste, rw dif_neg, unfold p3_aux, rw if_neg, refine subtype.eq _, exact help_02, \n  dsimp, show 2 * (1:\u211d) + -1 = 1, {norm_num}, dsimp, rw not_lt, \n  show 3 / 4 \u2264 ( 1:\u211d ), norm_num, end, \n\n  cont := \n  begin \n    refine cont_of_paste T1_is_closed T2_is_closed _ cont_p3_T1_aux cont_p3_aux , \n    unfold match_of_fun,  intros x B1 B2,\n      have Int : x \u2208 set.inter T1 T2, exact \u27e8 B1 , B2 \u27e9 , \n      rwa [inter_T] at Int, \n      have V : x.val = 1/2, rwa [mem_set_of_eq] at Int, \n      unfold p3_aux p3_T1_aux, dsimp, rw if_pos, rw subtype.ext , dsimp, \n      {rw V, norm_num},  rw V, norm_num,\n  end, \n\n}\n\n\n-----------\n\n---- Following section is to implement this and prove that \n-- hom_repar_path_to_path (( f \u2b1d g ) \u2b1d h) \u03c6 is indeed a homotopy  f \u2b1d (g \u2b1d h) \u2248 ( f \u2b1d g ) \u2b1d h \n\nsection \nvariables {f : path x y} {g : path y z} {h : path z w}\n\n-- To prove associativity need to show equality with the reparametrisation of path \n-- i.e (( f \u2b1d g ) \u2b1d h) \u03c6 = (f \u2b1d (g \u2b1d h)), so that can use previous results regarding repar_I01\n-- This will involve proving 9 subgoals for the different values (t : I01) can take\n\nlemma contr_T1 {x : I01} ( h\u2081 : x \u2208 T1 ) (h\u2082 : x \u2209 T1) : false := by cc\n\n\n-- 1\n\nlemma step_assoc_1 {t : {x // x \u2208 I01}} { h_1 : t \u2208 T1 } { h_2 : p3.to_fun t \u2208 T1} {h_3 : par zero_lt_half \u27e8p3.to_fun t, h_2 \u27e9 \u2208 T1} : \nf.to_fun (par zero_lt_half \u27e8t, h_1\u27e9) = f.to_fun (par zero_lt_half \u27e8par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9, h_3\u27e9) :=\nbegin \n congr, \n unfold p3, dsimp, unfold paste, simp [dif_pos h_1], \n unfold p3_T1_aux, dsimp, \n unfold par, dsimp, simp [subtype.ext], have a\u2081 : (2:\u211d )\u207b\u00b9 \u2260 0, {norm_num}, \n rw [mul_comm 2\u207b\u00b9 t.val ], rw [mul_div_assoc , div_self a\u2081, mul_one],\nend \n\n-- 2\n\nlemma p3_ineq_T1 {t : {x // x \u2208 I01}} (h_1 : t \u2208 T1 )  : p3.to_fun t \u2208 {x : I01 | 0 \u2264 x.val \u2227 x.val \u2264 1 / 4 } :=\nbegin \n  rw mem_set_of_eq, split, \n    unfold p3, dsimp, unfold paste p3_T1_aux, simp [h_1, -one_div_eq_inv], refine mul_nonneg _ t.2.1 , {norm_num},\n    unfold p3, dsimp, unfold paste p3_T1_aux, simp [h_1, -one_div_eq_inv], \n    have h\u2082 : (1/4 : \u211d) = (1/2)*(1/2), {norm_num}, rw h\u2082, unfold T1 T at h_1,  \n    have h\u2083 := h_1.2, \n    refine mul_le_mul _ h\u2083 t.2.1 _ , exact le_of_eq (refl (1/2)), {norm_num}, \nend \n\nlemma par_T1_ineq\u2081 {s : {x // x \u2208 I01}} {h_1 : s \u2208 T1 } (h : s \u2208 {x : \u21a5I01 | 0 \u2264 x.val \u2227 x.val \u2264 1 / 4 } ) :\npar zero_lt_half \u27e8 s , h_1 \u27e9 \u2208 T1 := \nbegin \n unfold T1 T, rw mem_set_of_eq, split,\n   { unfold par, dsimp [-sub_eq_add_neg], rw sub_zero, rw sub_zero, refine (le_div_iff _ ).2 _, {norm_num}, \n   rw [mul_comm, mul_zero], exact s.2.1, }, \n   unfold par, dsimp [-sub_eq_add_neg], rw sub_zero, rw sub_zero, refine (le_div_iff _ ).1 _ ,{norm_num}, \n   have h\u2082 : 1 / 2 / (1 / 2)\u207b\u00b9 = (1/4 : \u211d ), {norm_num}, rw h\u2082, \n   exact h.2, \nend\n\nlemma help_step_assoc_2 {t : {x // x \u2208 I01}} (h_1 : t \u2208 T1 ) { h_2 : p3.to_fun t \u2208 T1} (h_3 : par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9 \u2209 T1) : \npar zero_lt_half \u27e8p3.to_fun t, h_2\u27e9 \u2208  T2 :=  T2_of_not_T1 h_3 \n\n\nlemma step_assoc_2 {t : {x // x \u2208 I01}} { h_1 : t \u2208 T1 } { h_2 : p3.to_fun t \u2208 T1} (h_3 : par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9 \u2209 T1) : \nf.to_fun (par zero_lt_half \u27e8t, h_1\u27e9) = g.to_fun (par half_lt_one \u27e8par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9, help_step_assoc_2 h_1 h_3 \u27e9) :=\nbegin \n by_contradiction, unfold T1 T at h_2, \n rw [mem_set_of_eq] at h_2, \n unfold T1 T at h_3, rw [mem_set_of_eq] at h_3, \n simp [-one_div_eq_inv] at h_3, \n have H := h_3 (par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9).2.1, \n have G : par zero_lt_half \u27e8p3.to_fun t, h_2\u27e9 \u2208 T1, \n exact par_T1_ineq\u2081 (p3_ineq_T1 h_1), cc, \nend \n\n--3\n\nlemma step_assoc_3 {t : {x // x \u2208 I01}} ( h_1 : t \u2208 T1 ) ( h_2 : p3.to_fun t \u2209 T1) : \nf.to_fun (par zero_lt_half \u27e8t, h_1\u27e9) = h.to_fun (par half_lt_one \u27e8p3.to_fun t, T2_of_not_T1 h_2 \u27e9) := \nbegin \nby_contradiction, -- as p3.to_fun t \u2208 {x : \u21a5I01 | 0 \u2264 x.val \u2227 x.val \u2264 1 / 4 } \u2286 T1 \n have h\u2081 : p3.to_fun t \u2208 {x : \u21a5I01 | 0 \u2264 x.val \u2227 x.val \u2264 1 / 4 }, exact p3_ineq_T1  h_1, \n suffices g\u2081 : p3.to_fun t \u2208 T1, cc, \n refine mem_of_mem_of_subset h\u2081 _, unfold T1 T, \n intros x H , refine \u27e8 H.1, le_trans H.2 _ \u27e9 , {norm_num}\nend\n\n--4\n\nlemma p3_not_T1 {t : {x // x \u2208 I01}} (h : t \u2209  T1 ) :  1/4 < (p3.to_fun t ).val := \nbegin  \n have h\u2081 : 1/2 < t.val, \n  { unfold T1 T at h,\n   by_contradiction, rw [not_lt] at a, \n   suffices a\u2081 : t \u2208  {x : \u21a5I01 | 0 \u2264 x.val \u2227 x.val \u2264 1 / 2}, cc, \n   exact \u27e8 t.2.1, a \u27e9 , \n  }, \n  unfold p3, dsimp, unfold paste, simp [h, -one_div_eq_inv], unfold p3_aux, split_ifs, \n  { refine lt_sub_iff_add_lt.2 _, have a\u2081 : 1 / 4 + 1 / 4 = (1/2:\u211d ), {norm_num}, rw a\u2081 , exact h\u2081 },\n  simp at h_1, \n  have a\u2081 : 5 / 4 < 2 * t.val, \n    { have H : 5 / 4 = (5 /3 )*(3/4:\u211d ), {norm_num}, rw H, \n    refine mul_lt_mul _ h_1 _ _, {norm_num}, {norm_num}, norm_num }, \n  norm_num [a\u2081 ] , \nend\n\n\nlemma p3_impl\u2081 {t : {x // x \u2208 I01}} (h : (p3.to_fun t ).val \u2264 1/4 )  : t \u2208 T1 :=\nbegin \n by_contradiction, \n  have h\u2082  : 1/4 < (p3.to_fun t ).val, exact p3_not_T1 a, \n  suffices g\u2081 : \u00ac  (p3.to_fun t).val \u2264 1 / 4, cc, \n  exact (le_not_le_of_lt h\u2082).2  , \nend\n\nlemma par_impl_T1  {t : {x // x \u2208 I01}} {h_3 : p3.to_fun t \u2208 T1} (h_4 : par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9 \u2208 T1) : \n(p3.to_fun t ).val \u2264 1/4 := \nbegin \n unfold par T1 T at h_4, rw mem_set_of_eq at h_4, cases h_4 with h\u2081 h\u2082 , simp [-one_div_eq_inv] at h\u2082, \n have H :  (0: \u211d )<(1 / 2) , {norm_num}, \n have H2 :  1 / 4 = (1/2:\u211d )* (1/2), {norm_num}, rw H2, \n exact (div_le_iff H).1 h\u2082 ,  \nend\n\nlemma step_assoc_4 {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) ( h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2208 T1) \n( h_3 : p3.to_fun t \u2208 T1) (h_4 : par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9 \u2208 T1) : \n g.to_fun (par zero_lt_half \u27e8par half_lt_one \u27e8t, _\u27e9, h_2\u27e9) =\n    f.to_fun (par zero_lt_half \u27e8par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9, h_4\u27e9) := \nbegin \nby_cases H : t.val = (1/2), \n  { unfold p3, dsimp, unfold paste p3_aux, simp [h_1, -one_div_eq_inv, -sub_eq_add_neg], \n  have h\u2081 : t.val < 3 / 4, rw H, {norm_num}, simp [h\u2081, -one_div_eq_inv, -sub_eq_add_neg, H ],  \n  have a\u2081 : 1 / 2 - 1 / 4 = (1/4 : \u211d ), {norm_num}, simp [-one_div_eq_inv, -sub_eq_add_neg, a\u2081], \n  unfold par, simp [-sub_eq_add_neg, -one_div_eq_inv, sub_zero, H],  \n  show g.to_fun \u27e8((\u2191t - 1 / 2) / (1 - 1 / 2)) / (1 / 2), _\u27e9 = f.to_fun \u27e81 / 4 / (1 / 2) / (1 / 2), _\u27e9, \n  have a\u2083  : \u2191t = (1/2:\u211d ), exact H, simp [a\u2083, -one_div_eq_inv  ], simp [div_div_eq_div_mul, -one_div_eq_inv] , \n  norm_num , show g.to_fun 0 = f.to_fun 1,  simp  [ f.at_one, g.at_zero],   }, \n\n  by_contradiction, \n  suffices G : t \u2208 T1, cc, \n  exact p3_impl\u2081 (par_impl_T1 h_4), \nend  \n\n-- 5\n\nset_option trace.simplify.rewrite true\n--set_option pp.implicit true\nlemma p3_image_not_T1 (t : {x // x \u2208 I01}) (h_1 : t \u2209 T1) (a_1 : 3 / 4 < t.val) : p3.to_fun t \u2209  T1 := \nbegin \n unfold T1 T, rw mem_set_of_eq, simp [-one_div_eq_inv], intro H, unfold p3, dsimp, unfold paste p3_aux, \n simp [h_1, -one_div_eq_inv, -sub_eq_add_neg], \n have h : \u00ac t.val < 3/4, refine not_lt_of_ge (le_of_lt a_1),  simp [h, -one_div_eq_inv, -sub_eq_add_neg], \n have a\u2081 : 1 + 1 / 2 = (3/4:\u211d )*2 , {norm_num}, rw mul_comm, \n simp [-one_div_eq_inv, a\u2081 ],refine mul_lt_mul a_1 (le_of_eq (refl(2:\u211d ))) _ t.2.1, {norm_num}\nend\n\n\n\nlemma step_assoc_5  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) ( h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2208 T1) \n( h_3 : p3.to_fun t \u2208 T1) (h_4 : par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9 \u2209 T1) : \ng.to_fun (par zero_lt_half \u27e8par half_lt_one \u27e8t, T2_of_not_T1 h_1\u27e9, h_2\u27e9) =\n    g.to_fun (par half_lt_one \u27e8par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9, T2_of_not_T1 h_4 \u27e9) := \nbegin \n unfold p3,  dsimp, unfold paste, simp [dif_neg h_1], \n unfold p3_aux,\n by_cases a : t.val = 3/4 , \n  { have a\u2082 : \u00ac t.val < 3/4, exact not_lt_of_ge (ge_of_eq a), simp [a\u2082], \n   unfold par, dsimp [-one_div_eq_inv, -sub_eq_add_neg],   simp [sub_zero, -one_div_eq_inv, -sub_eq_add_neg],\n   have g\u2082 : \u2191t = t.val, trivial, {norm_num [ g\u2082 ,a]},  \n  },\n   have l\u2081 : t.val \u2264 3/4, \n    { by_contradiction, rw not_le at a_1, suffices G : p3.to_fun t \u2209  T1, \n    exact contr_T1 h_3 G, exact p3_image_not_T1 t h_1 a_1, },\n   have l\u2082 : t.val < 3 / 4, exact lt_of_le_of_ne l\u2081 a ,\n   simp [l\u2082 , -one_div_eq_inv, -sub_eq_add_neg], unfold par, dsimp [-one_div_eq_inv, -sub_eq_add_neg],\n   simp [-sub_eq_add_neg, -one_div_eq_inv, sub_zero], \n   have g\u2081 : (1 - 1 / 2) = (1/2:\u211d ), {norm_num}, have g\u2082 : \u2191t = t.val, trivial, \n   simp [g\u2081 , g\u2082 , -one_div_eq_inv, -sub_eq_add_neg] ,\n\n   suffices G1 : ((t.val - 1 / 4) / (1 / 2) - 1 / 2) = (t.val - 1 / 2) / (1 / 2),\n     apply congr_arg,\n     apply subtype.eq,\n     show (t.val - 1 / 2) / (1 / 2) / (1 / 2) = ((t.val - 1 / 4) / (1 / 2) - 1 / 2) / (1 / 2),\n     rw G1,\n   show ((t.val - 1 / 4) / (1 / 2)) - (1 / 2) = (t.val - 1 / 2) / (1 / 2),\n    have h\u2081 : ((t.val - 1 / 4) / (1 / 2)) - (1 / 2) = ((t.val - 1 / 4) / (1 / 2)) - (1 / 4 ) / (1/2:\u211d ),\n      have h\u2082  : - (1 / 2 : \u211d ) =  - (1 / 4 ) / (1/2:\u211d ), {norm_num}, \n      have h\u2083   : (1 / 2 : \u211d ) =  (1 / 4 ) / (1/2:\u211d ), {norm_num},\n      simpa [h\u2082, -one_div_eq_inv, -sub_eq_add_neg], \n    rw [h\u2081,  div_sub_div_same], \n    have H2 : (t.val - 1 / 4 - 1 / 4) = (t.val - 1 / 2) , {norm_num}, \n    simp [-one_div_eq_inv, -sub_eq_add_neg, H2],\n\nend\n\n\n\n\n-- 6 \n\nlocal attribute [instance] classical.prop_decidable \n\nlemma help_step_assoc_6\u2081  {t : {x // x \u2208 I01}} {h_1 : t \u2209 T1} (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2208 T1) : \nt.val \u2264 3/4 :=\nbegin \n by_contradiction, rw not_le at a, unfold T1 T at h_2, cases h_2 with g\u2081 g\u2082 , \n have G1 : 1/2 < (par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9).val, unfold par, dsimp [-one_div_eq_inv, -sub_eq_add_neg], \n  have a\u2081 : \u2191t = t.val, trivial, have a\u2082 : (1 - 1 / 2) = (1/2:\u211d ), {norm_num}, rw [a\u2081, a\u2082 ], \n  refine (lt_div_iff _).2 _, {norm_num}, \n  have h\u2081 : 1 / 2 * (1 / 2) \u2264 (3/4:\u211d ) - 1/2, {norm_num}, \n  have h\u2082 : (3/4:\u211d ) - 1/2 < t.val -1/2, refine lt_sub_iff_add_lt.2 _ , \n  have a\u2082 : 3 / 4 - 1 / 2 + 1 / 2 = (3/4:\u211d ), {norm_num}, rw a\u2082 , exact a, \n  exact lt_of_le_of_lt h\u2081 h\u2082 , \n have NG1 : \u00ac 1/2 < (par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9).val, \n exact not_lt_of_le g\u2082 , cc, \nend \n\nlemma T1_of_p3\u2081    {t : {x // x \u2208 I01}} (h_1 : t \u2209 T1) (l\u2082 : t.val <  3 / 4) : \n p3.to_fun t \u2208  T1 := \nbegin \n unfold p3, dsimp, unfold paste, simp [h_1], unfold p3_aux, simp [l\u2082, -sub_eq_add_neg, -one_div_eq_inv] , \n have a\u2081 : t \u2208 T2, exact T2_of_not_T1 h_1, \n unfold T1 T, dsimp [-sub_eq_add_neg], split, \n  refine sub_nonneg.2 _  , refine le_trans _ a\u2081.1 , {norm_num}, \n  norm_num [l\u2082, -one_div_eq_inv, le_of_lt l\u2082 ], \nend \n\n---h_2 : par half_lt_one \u27e8t, _\u27e9 \u2208 T1\n\nlemma step_assoc_6  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) ( h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2208 T1) \n(h_3 : p3.to_fun t \u2209 T1) : \n g.to_fun (par zero_lt_half \u27e8par half_lt_one \u27e8t, _\u27e9, h_2\u27e9) \n    = h.to_fun (par half_lt_one \u27e8p3.to_fun t, T2_of_not_T1 h_3 \u27e9) := \nbegin \n unfold p3,  dsimp, unfold paste, simp [h_1],  unfold p3_aux,\n by_cases a : t.val = 3/4, \n   { have a\u2082 : \u00ac t.val < 3/4, exact not_lt_of_ge (ge_of_eq a), simp [a\u2082, -sub_eq_add_neg, -one_div_eq_inv], \n     unfold par, dsimp [-one_div_eq_inv, -sub_eq_add_neg], have g\u2082 : \u2191t = t.val, trivial, \n     simp [g\u2082 , a, subtype.ext], {norm_num, show g.to_fun 1 = h.to_fun 0, simp, }, \n   },\n have a\u2081 : t.val \u2264 3/4, exact help_step_assoc_6\u2081  h_2, \n have l\u2082  : t.val < 3/4, exact lt_of_le_of_ne a\u2081 a, \n by_contradiction, \n suffices g :  p3.to_fun t \u2208  T1, exact contr_T1 g h_3 , \n exact T1_of_p3\u2081  h_1  l\u2082 , \nend\n\n-- 7\n\n\nlemma help_step_assoc_7\u2081  {t : {x // x \u2208 I01}} {h_1 : t \u2209 T1} (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2209  T1) : \n3/4 < t.val := \nbegin \n  unfold T1 T at h_2, simp [(par half_lt_one \u27e8t, _\u27e9).2.1, -one_div_eq_inv] at h_2, \n  unfold par at h_2, dsimp [-one_div_eq_inv, -sub_eq_add_neg] at h_2, \n  have a\u2081 : \u2191t = t.val, trivial, have a\u2082  : (1 - 1 / 2) = (1/2:\u211d ), {norm_num}, rw [a\u2081, a\u2082] at h_2, \n  have H : 1 / 2 * (1 / 2) < (t.val - 1 / 2), refine (lt_div_iff _).1 h_2, {norm_num}, \n  rw lt_sub_iff_add_lt at H, have g\u2081 : 1 / 2 * (1 / 2) + 1 / 2 = (3/4:\u211d ), {norm_num}, rw g\u2081 at H, \n  exact H, \nend \n\nlemma p3_in_T1  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) (h_3 : p3.to_fun t \u2208 T1) : \nt.val \u2264 3/4 := \nbegin \n by_contradiction, rw not_le at a, unfold p3 at h_3, dsimp at h_3, unfold paste at h_3, \n simp [h_1] at h_3, unfold p3_aux at h_3, \n have a\u2082 :  \u00ac  t.val < 3/4, exact not_lt_of_gt a, simp [a\u2082 , -one_div_eq_inv, -sub_eq_add_neg] at h_3, \n unfold T1 T at h_3, --simp at h_3, \n have g\u2081 : 2 * t.val - 1 \u2264 1/2, exact h_3.2, \n have g\u2082 : t.val \u2264 3/4, rw sub_le_iff_le_add at g\u2081  ,   \n   have aux\u2081 : 1 / 2 + 1 = (3/2:\u211d ), {norm_num}, rw aux\u2081 at g\u2081, rw mul_comm at g\u2081 , \n   have aux\u2082 : t.val \u2264 (3 / 2 ) / 2, refine le_div_of_mul_le _ g\u2081 ,{norm_num}, \n   have aux\u2083 : 3 / 2 / 2 = (3/4:\u211d), {norm_num}, rw aux\u2083 at aux\u2082 , exact aux\u2082, \n have g\u2083 : \u00ac  3 / 4 < t.val, exact not_lt_of_ge g\u2082,\n cc, \nend\n\nlemma step_assoc_7  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2209 T1) \n( h_3 : p3.to_fun t \u2208 T1) (h_4 : par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9 \u2208 T1) : \nh.to_fun (par half_lt_one \u27e8par half_lt_one \u27e8t, T2_of_not_T1 h_1\u27e9, T2_of_not_T1 h_2\u27e9) =\n    f.to_fun (par zero_lt_half \u27e8par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9, h_4\u27e9) := \nbegin \n by_contradiction, \n have g\u2081 : 3/4 < t.val, exact help_step_assoc_7\u2081 h_2, \n have g\u2082  : \u00ac 3/4 < t.val, exact not_lt_of_ge (p3_in_T1  h_1 h_3), \n cc, \nend\n\n\n-- 8\n\nlemma help_step_assoc_8\u2081  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2209 T1) : \n3/4 < t.val := \nbegin \n unfold T1 T at h_2, simp [-one_div_eq_inv] at h_2,  have h\u2081  := h_2 (par half_lt_one \u27e8t, _\u27e9).2.1, \n unfold par at h\u2081,dsimp [-one_div_eq_inv, -sub_eq_add_neg] at h\u2081,  \n have g\u2081 : (1 - 1 / 2) = (1/2:\u211d ), {norm_num},  have a\u2081 : \u2191t = t.val, trivial, rw [g\u2081, a\u2081] at h\u2081,\n have h\u2082 : 1 / 2 * (1 / 2) < (t.val - 1 / 2), refine (lt_div_iff _).1 h\u2081 , {norm_num}, \n rw lt_sub_iff_add_lt at h\u2082, have a\u2082 : 1 / 2 * (1 / 2) + 1 / 2 = (3/4:\u211d ), {norm_num}, \n rw a\u2082 at h\u2082, exact h\u2082, \nend\n\nlemma step_assoc_8  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2209 T1) \n( h_3 : p3.to_fun t \u2208 T1) (h_4 : par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9 \u2209 T1) : \nh.to_fun (par half_lt_one \u27e8par half_lt_one \u27e8t, T2_of_not_T1 h_1\u27e9, T2_of_not_T1 h_2 \u27e9) =\n    g.to_fun (par half_lt_one \u27e8par zero_lt_half \u27e8p3.to_fun t, h_3\u27e9, T2_of_not_T1 h_4 \u27e9) :=\nbegin \n have g\u2081 : \u00ac 3/4 < t.val, exact not_lt_of_ge (p3_in_T1  h_1 h_3) , \n have g\u2082 : 3/4 < t.val, exact help_step_assoc_8\u2081 h_1 h_2, cc, \nend\n\n\n\n\n-- 9 \n\n\nlemma step_assoc_9  {t : {x // x \u2208 I01}} ( h_1 : t \u2209 T1 ) (h_2 : par half_lt_one \u27e8t, T2_of_not_T1 h_1 \u27e9 \u2209 T1) \n( h_3 : p3.to_fun t \u2209 T1) :\nh.to_fun (par half_lt_one \u27e8par half_lt_one \u27e8t, T2_of_not_T1 h_1\u27e9, T2_of_not_T1 h_2\u27e9) = \nh.to_fun (par half_lt_one \u27e8p3.to_fun t, T2_of_not_T1 h_3\u27e9) := \nbegin\n unfold p3, dsimp, unfold paste, simp [dif_neg h_1], \n have a\u2081 : \u00ac t.val < 3/4, exact not_lt_of_gt (help_step_assoc_8\u2081 h_1 h_2), \n unfold p3_aux, simp [a\u2081, -one_div_eq_inv, -sub_eq_add_neg], unfold par, dsimp [-one_div_eq_inv, -sub_eq_add_neg],\n have a\u2082 : \u2191t = t.val, trivial, have g\u2081 : (1 - 1 / 2) = (1/2:\u211d ), {norm_num},\n simp [a\u2082, g\u2081, -one_div_eq_inv, -sub_eq_add_neg],\n suffices H : ((t.val - 1 / 2) / (1 / 2) - 1 / 2) = (2 * t.val - 1 - 1 / 2),\n  apply congr_arg,\n  apply subtype.eq,\n  show ((t.val - 1 / 2) / (1 / 2) - 1 / 2) / (1 / 2) = (2 * t.val - 1 - 1 / 2) / (1 / 2),\n  rw H,\n have a\u2083 : (t.val - 1 / 2) / (1 / 2) = (t.val ) / (1 / 2) - ( 1 / 2) / (1 / 2), apply eq.symm, \n refine div_sub_div_same t.val (1/2:\u211d) (1/2:\u211d), rw div_self at a\u2083, rw a\u2083 , rw div_eq_inv_mul, \n have a\u2084 : (1 / 2 : \u211d )\u207b\u00b9 = 2, {norm_num}, rw a\u2084, {norm_num}, \nend \n\n\n\n-- Homotopy for associativity\n\nnoncomputable def hom_comp_f_g_h ( f : path x y) ( g : path y z) ( h : path z w)  : \npath_homotopy  ( comp_of_path f (comp_of_path g h)) (comp_of_path (comp_of_path f g) h ) := \nbegin \n  have h\u2081 : ( comp_of_path f (comp_of_path g h)) = \n    repar_path (comp_of_path (comp_of_path f g) h ) p3, \n   { unfold repar_path, dsimp, refine path_equal.2 _ ,  dsimp, unfold comp_of_path, dsimp, \n     unfold paste fa_path fb_path fgen_path, dsimp,  unfold paste, funext, unfold paste, \n     split_ifs, \n      exact step_assoc_1,  \n      exact step_assoc_2 h_3, \n      exact step_assoc_3 h_1 h_2, \n      exact step_assoc_4 h_1 h_2 h_3 h_4, \n      exact step_assoc_5 h_1 h_2 h_3 h_4, \n      exact step_assoc_6 h_1 h_2 h_3, \n      exact step_assoc_7 h_1 h_2 h_3 h_4, \n      exact step_assoc_8 h_1 h_2 h_3 h_4, \n      exact step_assoc_9 h_1 h_2 h_3, \n   },\n  rw h\u2081 , exact hom_repar_path_to_path (comp_of_path (comp_of_path f g) h ) p3, \nend \n\n\n\nend \n\nend homotopy_results  ", "meta": {"author": "ImperialCollegeLondon", "repo": "xena-UROP-2018", "sha": "b111fb87f343cf79eca3b886f99ee15c1dd9884b", "save_path": "github-repos/lean/ImperialCollegeLondon-xena-UROP-2018", "path": "github-repos/lean/ImperialCollegeLondon-xena-UROP-2018/xena-UROP-2018-b111fb87f343cf79eca3b886f99ee15c1dd9884b/src/Topology/Material/homotopy_results.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.3623755081327717}}
{"text": "/-\nCopyright (c) 2020 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# Tactics About Dependencies\n\nThis module provides tactics to compute dependencies and reverse dependencies of\nhypotheses. An expression `e` depends on a hypothesis `h` if `e` would not be\nvalid if `h` were removed from the context. For example, the expression\n`e := x > 0` depends on `x`. We say that `x` is a dependency of `e` and that `e`\nis a reverse dependency of `x`.\n\nIt is sometimes useful to consider *inclusive* dependency: `e` inclusively\ndepends on `h` iff `e` depends on `h` or `e = h` (so inclusive dependency is the\nreflexive closure of regular dependency).\n\nNote that the standard library does not use quite the same terminology:\n\n* `kdependencies`/`kdeps` from the standard library compute reverse\n  dependencies, not dependencies.\n* `kdepends_on` and functions derived from it ignore local definitions and\n  therefore compute a weaker dependency relation (see next section).\n\n## Local Definitions\n\nDetermining dependencies of hypotheses is usually straightforward: a hypothesis\n`r : R` depends on another hypothesis `d : D` if `d` occurs in `R`. The\nimplementation is more involved, however, in the presence of local definitions.\nConsider this context:\n\n```lean\nn m : \u2115\nk : \u2115 := m\no : \u2115 := k\nh : o > 0\n```\n\n`h` depends on `o`, `k` and `m`, but only the dependency on `o` is syntactically\nobvious. `kdepends_on` ignores this complication and claims that `h` does not\ndepend on `k` or `m`. We do not follow this example but process local\ndefinitions properly. This means that if the context contains a local\ndefinition, we need to compute the syntactic dependencies of `h`, then their\ndependencies, and so on.\n\n## Direct Dependencies\n\nIf you want to ignore local definitions while computing dependencies, this\nmodule also provides tactics to find the *direct* dependencies of a hypothesis.\nThese are the hypotheses that syntactically appear in the hypothesis's type (or\nvalue, if the hypothesis is a local definition).\n-/\n\nnamespace tactic\n\n\n/-! ### Direct Dependencies -/\n\n/-! #### Checking whether hypotheses directly depend on each other -/\n\n/--\n`type_has_local_in_name_set h ns` returns true iff the type of `h` contains a\nlocal constant whose unique name appears in `ns`.\n-/\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/dependencies_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.3623755081327717}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module tactic.restate_axiom\n! leanprover-community/mathlib commit d6814c584384ddf2825ff038e868451a7c956f31\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.DocCommands\n\nopen Lean.Parser Tactic Interactive\n\n/-- `restate_axiom` takes a structure field, and makes a new, definitionally simplified copy of it.\nIf the existing field name ends with a `'`, the new field just has the prime removed. Otherwise,\nwe append `_lemma`.\nThe main application is to provide clean versions of structure fields that have been tagged with\nan auto_param.\n-/\nunsafe def restate_axiom (d : declaration) (new_name : Name) : tactic Unit := do\n  let (levels, type, value, reducibility, trusted) \u2190\n    pure\n        (match d.to_definition with\n        | declaration.defn Name levels type value reducibility trusted =>\n          (levels, type, value, reducibility, trusted)\n        | _ => undefined)\n  let (s, u) \u2190 mk_simp_set false [] []\n  let new_type \u2190 s.dsimplify [] type <|> pure type\n  let prop \u2190 is_prop new_type\n  let new_decl :=\n    if prop then declaration.thm new_name levels new_type (task.pure value)\n    else declaration.defn new_name levels new_type value reducibility trusted\n  updateex_env fun env => env new_decl\n#align restate_axiom restate_axiom\n\nprivate unsafe def name_lemma (old : Name) (new : Option Name := none) : tactic Name :=\n  match new with\n  | none =>\n    match old.components.reverse with\n    | last :: most =>\n      (do\n          let last := last.toString\n          let last :=\n            if last.toList.getLastI = ''' then (last.toList.reverse.drop 1).reverse.asString\n            else last ++ \"_lemma\"\n          return (mkStrName old last)) <|>\n        failed\n    | nil => undefined\n  | some new => return (mkStrName old.getPrefix new.toString)\n#align name_lemma name_lemma\n\n/--\n`restate_axiom` makes a new copy of a structure field, first definitionally simplifying the type.\nThis is useful to remove `auto_param` or `opt_param` from the statement.\n\nAs an example, we have:\n```lean\nstructure A :=\n(x : \u2115)\n(a' : x = 1 . skip)\n\nexample (z : A) : z.x = 1 := by rw A.a' -- rewrite tactic failed, lemma is not an equality nor a iff\n\nrestate_axiom A.a'\nexample (z : A) : z.x = 1 := by rw A.a\n```\n\nBy default, `restate_axiom` names the new lemma by removing a trailing `'`, or otherwise appending\n`_lemma` if there is no trailing `'`. You can also give `restate_axiom` a second argument to\nspecify the new name, as in\n```lean\nrestate_axiom A.a f\nexample (z : A) : z.x = 1 := by rw A.f\n```\n-/\n@[user_command]\nunsafe def restate_axiom_cmd (_ : parse <| tk \"restate_axiom\") : lean.parser Unit := do\n  let from_lemma \u2190 ident\n  let new_name \u2190 optional ident\n  let from_lemma_fully_qualified \u2190 resolve_constant from_lemma\n  let d \u2190\n    get_decl from_lemma_fully_qualified <|>\n        fail (\"declaration \" ++ toString from_lemma ++ \" not found\")\n  do\n    let new_name \u2190 name_lemma from_lemma_fully_qualified new_name\n    restate_axiom d new_name\n#align restate_axiom_cmd restate_axiom_cmd\n\nadd_tactic_doc\n  { Name := \"restate_axiom\"\n    category := DocCategory.cmd\n    declNames := [`restate_axiom_cmd]\n    tags := [\"renaming\", \"environment\"] }\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/RestateAxiom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.538983220687684, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.3623755045940747}}
{"text": "/-\nCopyright (c) 2021 Eric Wieser. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Eric Wieser\n\n! This file was ported from Lean 3 source module algebra.star.big_operators\n! leanprover-community/mathlib commit 327c3c0d9232d80e250dc8f65e7835b82b266ea5\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.BigOperators.Basic\nimport Mathbin.Algebra.Star.Basic\n\n/-! # Big-operators lemmas about `star` algebraic operations\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThese results are kept separate from `algebra.star.basic` to avoid it needing to import `finset`.\n-/\n\n\nvariable {R : Type _}\n\nopen BigOperators\n\n/- warning: star_prod -> star_prod is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CommMonoid.{u1} R] [_inst_2 : StarSemigroup.{u1} R (Monoid.toSemigroup.{u1} R (CommMonoid.toMonoid.{u1} R _inst_1))] {\u03b1 : Type.{u2}} (s : Finset.{u2} \u03b1) (f : \u03b1 -> R), Eq.{succ u1} R (Star.star.{u1} R (InvolutiveStar.toHasStar.{u1} R (StarSemigroup.toHasInvolutiveStar.{u1} R (Monoid.toSemigroup.{u1} R (CommMonoid.toMonoid.{u1} R _inst_1)) _inst_2)) (Finset.prod.{u1, u2} R \u03b1 _inst_1 s (fun (x : \u03b1) => f x))) (Finset.prod.{u1, u2} R \u03b1 _inst_1 s (fun (x : \u03b1) => Star.star.{u1} R (InvolutiveStar.toHasStar.{u1} R (StarSemigroup.toHasInvolutiveStar.{u1} R (Monoid.toSemigroup.{u1} R (CommMonoid.toMonoid.{u1} R _inst_1)) _inst_2)) (f x)))\nbut is expected to have type\n  forall {R : Type.{u2}} [_inst_1 : CommMonoid.{u2} R] [_inst_2 : StarSemigroup.{u2} R (Monoid.toSemigroup.{u2} R (CommMonoid.toMonoid.{u2} R _inst_1))] {\u03b1 : Type.{u1}} (s : Finset.{u1} \u03b1) (f : \u03b1 -> R), Eq.{succ u2} R (Star.star.{u2} R (InvolutiveStar.toStar.{u2} R (StarSemigroup.toInvolutiveStar.{u2} R (Monoid.toSemigroup.{u2} R (CommMonoid.toMonoid.{u2} R _inst_1)) _inst_2)) (Finset.prod.{u2, u1} R \u03b1 _inst_1 s (fun (x : \u03b1) => f x))) (Finset.prod.{u2, u1} R \u03b1 _inst_1 s (fun (x : \u03b1) => Star.star.{u2} R (InvolutiveStar.toStar.{u2} R (StarSemigroup.toInvolutiveStar.{u2} R (Monoid.toSemigroup.{u2} R (CommMonoid.toMonoid.{u2} R _inst_1)) _inst_2)) (f x)))\nCase conversion may be inaccurate. Consider using '#align star_prod star_prod\u2093'. -/\n@[simp]\ntheorem star_prod [CommMonoid R] [StarSemigroup R] {\u03b1 : Type _} (s : Finset \u03b1) (f : \u03b1 \u2192 R) :\n    star (\u220f x in s, f x) = \u220f x in s, star (f x) :=\n  map_prod (starMulAut : R \u2243* R) _ _\n#align star_prod star_prod\n\n/- warning: star_sum -> star_sum is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : AddCommMonoid.{u1} R] [_inst_2 : StarAddMonoid.{u1} R (AddCommMonoid.toAddMonoid.{u1} R _inst_1)] {\u03b1 : Type.{u2}} (s : Finset.{u2} \u03b1) (f : \u03b1 -> R), Eq.{succ u1} R (Star.star.{u1} R (InvolutiveStar.toHasStar.{u1} R (StarAddMonoid.toHasInvolutiveStar.{u1} R (AddCommMonoid.toAddMonoid.{u1} R _inst_1) _inst_2)) (Finset.sum.{u1, u2} R \u03b1 _inst_1 s (fun (x : \u03b1) => f x))) (Finset.sum.{u1, u2} R \u03b1 _inst_1 s (fun (x : \u03b1) => Star.star.{u1} R (InvolutiveStar.toHasStar.{u1} R (StarAddMonoid.toHasInvolutiveStar.{u1} R (AddCommMonoid.toAddMonoid.{u1} R _inst_1) _inst_2)) (f x)))\nbut is expected to have type\n  forall {R : Type.{u2}} [_inst_1 : AddCommMonoid.{u2} R] [_inst_2 : StarAddMonoid.{u2} R (AddCommMonoid.toAddMonoid.{u2} R _inst_1)] {\u03b1 : Type.{u1}} (s : Finset.{u1} \u03b1) (f : \u03b1 -> R), Eq.{succ u2} R (Star.star.{u2} R (InvolutiveStar.toStar.{u2} R (StarAddMonoid.toInvolutiveStar.{u2} R (AddCommMonoid.toAddMonoid.{u2} R _inst_1) _inst_2)) (Finset.sum.{u2, u1} R \u03b1 _inst_1 s (fun (x : \u03b1) => f x))) (Finset.sum.{u2, u1} R \u03b1 _inst_1 s (fun (x : \u03b1) => Star.star.{u2} R (InvolutiveStar.toStar.{u2} R (StarAddMonoid.toInvolutiveStar.{u2} R (AddCommMonoid.toAddMonoid.{u2} R _inst_1) _inst_2)) (f x)))\nCase conversion may be inaccurate. Consider using '#align star_sum star_sum\u2093'. -/\n@[simp]\ntheorem star_sum [AddCommMonoid R] [StarAddMonoid R] {\u03b1 : Type _} (s : Finset \u03b1) (f : \u03b1 \u2192 R) :\n    star (\u2211 x in s, f x) = \u2211 x in s, star (f x) :=\n  (starAddEquiv : R \u2243+ R).map_sum _ _\n#align star_sum star_sum\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Star/BigOperators.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.36237549751668086}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Mathlib.Mathport.Rename\nimport Mathlib.Init.Data.Nat.Notation\nimport Std.Data.Nat.Lemmas\nimport Std.Data.List.Basic\n/-!\nDefinitions for `List` not (yet) in `Std`\n-/\n\n\nopen Decidable List\n\nuniverse u v w\n\nnamespace List\n\n\n\nopen Option Nat\n\n#align list.nth List.get?\n\n/-- nth element of a list `l` given `n < l.length`. -/\n@[deprecated get]\ndef nthLe (l : List \u03b1) (n) (h : n < l.length) : \u03b1 := get l \u27e8n, h\u27e9\n#align list.nth_le List.nthLe\n\nset_option linter.deprecated false in\n@[deprecated]\ntheorem nthLe_eq (l : List \u03b1) (n) (h : n < l.length) : nthLe l n h = get l \u27e8n, h\u27e9 := rfl\n\n/-- The head of a list, or the default element of the type is the list is `nil`. -/\ndef headI [Inhabited \u03b1] : List \u03b1 \u2192 \u03b1\n| []       => default\n| (a :: _) => a\n#align list.head List.headI\n\n@[simp] theorem headI_nil [Inhabited \u03b1] : ([] : List \u03b1).headI = default := rfl\n@[simp] theorem headI_cons [Inhabited \u03b1] {h : \u03b1} {t : List \u03b1} : (h :: t).headI = h := rfl\n\n#align list.map\u2082 List.zipWith\n\n#noalign list.map_with_index_core\n\n#align list.map_with_index List.mapIdx\n\n/-- Find index of element with given property. -/\n@[deprecated findIdx]\ndef findIndex (p : \u03b1 \u2192 Prop) [DecidablePred p] : List \u03b1 \u2192 \u2115 := List.findIdx p\n#align list.find_index List.findIndex\n\n#align list.update_nth List.set\n\n#align list.bor List.or\n\n#align list.band List.and\n\n#align list.last List.getLast\n\n/-- The last element of a list, with the default if list empty -/\ndef getLastI [Inhabited \u03b1] : List \u03b1 \u2192 \u03b1\n  | [] => default\n  | [a] => a\n  | [_, b] => b\n  | _ :: _ :: l => getLastI l\n#align list.ilast List.getLastI\n\n#align list.init List.dropLast\n\n/-- List with a single given element. -/\n@[inline] protected def ret {\u03b1 : Type u} (a : \u03b1) : List \u03b1 := [a]\n#align list.ret List.ret\n\n/-- `\u2264` implies not `>` for lists. -/\n\n\nend List\n\n#align list.replicate List.replicate\n#align list.length_replicate List.length_replicate\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Init/Data/List/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.36234568418283153}}
{"text": "/-\nCopyright (c) 2021 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Scott Morrison\n-/\nimport data.finset.lattice\nimport data.multiset.functor\n\n/-!\n# Functoriality of `finset`\n\nThis file defines the functor structure of `finset`.\n\n## TODO\n\nCurrently, all instances are classical because the functor classes want to run over all types. If\ninstead we could state that a functor is lawful/applicative/traversable... between two given types,\nthen we could provide the instances for types with decidable equality.\n-/\n\nuniverses u\n\nopen function\n\nnamespace finset\n\n/-! ### Functor -/\n\nsection functor\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\n/-- Because `finset.image` requires a `decidable_eq` instance for the target type, we can only\nconstruct `functor finset` when working classically. -/\ninstance : functor finset :=\n{ map := \u03bb \u03b1 \u03b2 f s, s.image f }\n\ninstance : is_lawful_functor finset :=\n{ id_map := \u03bb \u03b1 s, image_id,\n  comp_map := \u03bb \u03b1 \u03b2 \u03b3 f g s, image_image.symm }\n\n@[simp] lemma fmap_def {s : finset \u03b1} (f : \u03b1 \u2192 \u03b2) : f <$> s = s.image f := rfl\n\nend functor\n\n/-! ### Pure -/\n\ninstance : has_pure finset := \u27e8\u03bb \u03b1 x, {x}\u27e9\n\n@[simp] lemma pure_def {\u03b1} : (pure : \u03b1 \u2192 finset \u03b1) = singleton := rfl\n\n/-! ### Applicative functor -/\n\nsection applicative\nvariables {\u03b1 \u03b2 : Type u} [\u03a0 P, decidable P]\n\ninstance : applicative finset :=\n{ seq := \u03bb \u03b1 \u03b2 t s, t.sup (\u03bb f, s.image f),\n  seq_left := \u03bb \u03b1 \u03b2 s t, if t = \u2205 then \u2205 else s,\n  seq_right := \u03bb \u03b1 \u03b2 s t, if s = \u2205 then \u2205 else t,\n  .. finset.functor,\n  .. finset.has_pure }\n\n@[simp] \n\ninstance : is_lawful_applicative finset :=\n{ seq_left_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_left_def],\n    obtain rfl | ht := t.eq_empty_or_nonempty,\n    { simp_rw [if_pos rfl, image_empty], exact (sup_bot _).symm },\n    { ext a,\n      rw [if_neg ht.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8const \u03b2 a, mem_image_of_mem _ ha, mem_image_const_self.2 ht\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := hf,\n      obtain \u27e8_, _, rfl\u27e9 := ha,\n      exact hb }\n  end,\n  seq_right_eq := \u03bb \u03b1 \u03b2 s t, begin\n    rw [seq_def, fmap_def, seq_right_def],\n    obtain rfl | hs := s.eq_empty_or_nonempty,\n    { rw [if_pos rfl, image_empty, sup_empty, bot_eq_empty] },\n    { ext a,\n      rw [if_neg hs.ne_empty, mem_sup],\n      refine \u27e8\u03bb ha, \u27e8id, mem_image_const_self.2 hs, by rwa image_id\u27e9, _\u27e9,\n      rintro \u27e8f, hf, ha\u27e9,\n      rw mem_image at hf ha,\n      obtain \u27e8b, hb, rfl\u27e9 := ha,\n      obtain \u27e8_, _, rfl\u27e9 := hf,\n      exact hb }\n  end,\n  pure_seq_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton,\n  map_pure := \u03bb \u03b1 \u03b2 f a, image_singleton _ _,\n  seq_pure := \u03bb \u03b1 \u03b2 s a, sup_singleton'' _ _,\n  seq_assoc := \u03bb \u03b1 \u03b2 \u03b3 s t u, begin\n    ext a,\n    simp_rw [seq_def, fmap_def],\n    simp only [exists_prop, mem_sup, mem_image],\n    split,\n    { rintro \u27e8g, hg, b, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9,\n      exact \u27e8g \u2218 f, \u27e8comp g, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9 },\n    { rintro \u27e8c, \u27e8_, \u27e8g, hg, rfl\u27e9, f, hf, rfl\u27e9, a, ha, rfl\u27e9,\n      exact \u27e8g, hg, f a, \u27e8f, hf, a, ha, rfl\u27e9, rfl\u27e9 }\n  end,\n  .. finset.is_lawful_functor }\n\ninstance : is_comm_applicative finset :=\n{ commutative_prod := \u03bb \u03b1 \u03b2 s t, begin\n    simp_rw [seq_def, fmap_def, sup_image, sup_eq_bUnion],\n    change s.bUnion (\u03bb a, t.image $ \u03bb b, (a, b)) = t.bUnion (\u03bb b, s.image $ \u03bb a, (a, b)),\n    transitivity s.product t;\n      [rw product_eq_bUnion, rw product_eq_bUnion_right]; congr; ext; simp_rw mem_image,\n  end,\n  .. finset.is_lawful_applicative }\n\nend applicative\n\n/-! ### Monad -/\n\nsection monad\nvariables [\u03a0 P, decidable P]\n\ninstance : monad finset :=\n{ bind := \u03bb \u03b1 \u03b2, @sup _ _ _ _,\n  .. finset.applicative }\n\n@[simp] lemma bind_def {\u03b1 \u03b2} : (>>=) = @sup (finset \u03b1) \u03b2 _ _ := rfl\n\ninstance : is_lawful_monad finset :=\n{ bind_pure_comp_eq_map := \u03bb \u03b1 \u03b2 f s, sup_singleton'' _ _,\n  bind_map_eq_seq := \u03bb \u03b1 \u03b2 t s, rfl,\n  pure_bind := \u03bb \u03b1 \u03b2 t s, sup_singleton,\n  bind_assoc :=  \u03bb \u03b1 \u03b2 \u03b3 s f g, by { convert sup_bUnion _ _, exact sup_eq_bUnion _ _ },\n  .. finset.is_lawful_applicative }\n\nend monad\n\n/-! ### Alternative functor -/\n\nsection alternative\nvariables [\u03a0 P, decidable P]\n\ninstance : alternative finset :=\n{ orelse := \u03bb \u03b1, (\u222a),\n  failure := \u03bb \u03b1, \u2205,\n  .. finset.applicative }\n\nend alternative\n\n/-! ### Traversable functor -/\n\nsection traversable\nvariables {\u03b1 \u03b2 \u03b3 : Type u} {F G : Type u \u2192 Type u} [applicative F] [applicative G]\n  [is_comm_applicative F] [is_comm_applicative G]\n\n/-- Traverse function for `finset`. -/\ndef traverse [decidable_eq \u03b2] (f : \u03b1 \u2192 F \u03b2) (s :  finset \u03b1) : F (finset \u03b2) :=\nmultiset.to_finset <$> multiset.traverse f s.1\n\n@[simp] lemma id_traverse [decidable_eq \u03b1] (s : finset \u03b1) : traverse id.mk s = s :=\nby { rw [traverse, multiset.id_traverse], exact s.val_to_finset }\n\nopen_locale classical\n\n@[simp] lemma map_comp_coe (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 multiset.to_finset = multiset.to_finset \u2218 functor.map h :=\nfunext $ \u03bb s, image_to_finset\n\nlemma map_traverse (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3) (s : finset \u03b1) :\n  functor.map h <$> traverse g s = traverse (functor.map h \u2218 g) s :=\nbegin\n  unfold traverse,\n  simp only [map_comp_coe] with functor_norm,\n  rw [is_lawful_functor.comp_map, multiset.map_traverse],\nend\n\nend traversable\nend finset\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/data/finset/functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3623456837786265}}
{"text": "\nimport data.serial\n\nopen serial serializer\n\nstructure point :=\n(x y : unsigned)\n\ninstance : serial point :=\nof_serializer (point.mk <$> ser_field point.x <*> ser_field point.y)\nbegin\n  intro,\n  apply there_and_back_again_seq,\n  apply there_and_back_again_map,\n  cases w, refl\nend\n\n@[derive serial]\ninductive my_sum\n| first : my_sum\n| second : \u2115 \u2192 my_sum\n| third (n : \u2115) (xs : list \u2115) : n \u2264 xs.length \u2192 my_sum\n\n@[derive serial]\nstructure my_struct :=\n(x : \u2115)\n(xs : list \u2115)\n(bounded : xs.length \u2264 x)\n\n@[derive [serial]]\ninductive tree (\u03b1 : Type)\n| leaf {} : tree\n| node2 : \u03b1 \u2192 tree \u2192 tree \u2192 tree\n| node3 : \u03b1 \u2192 tree \u2192 tree \u2192 tree \u2192 tree\n\nopen tree\n\nmeta def tree.repr {\u03b1} [has_repr \u03b1] : tree \u03b1 \u2192 string\n| leaf := \"leaf\"\n| (node2 x t\u2080 t\u2081) := to_string $ format!\"(node2 {repr x} {tree.repr t\u2080} {tree.repr t\u2081})\"\n| (node3 x t\u2080 t\u2081 t\u2082) := to_string $ format!\"(node3 {repr x} {tree.repr t\u2080} {tree.repr t\u2081} {tree.repr t\u2082})\"\n\nmeta instance {\u03b1} [has_repr \u03b1] : has_repr (tree \u03b1) := \u27e8 tree.repr \u27e9\n\ndef x := node2 2 (node3 77777777777777 leaf leaf (node2 1 leaf leaf)) leaf\n\n#eval serialize x\n-- [17, 1, 5, 2, 430029026, 72437, 0, 0, 1, 3, 0, 0, 0]\n#eval deserialize (tree \u2115) [17, 1, 5, 2, 430029026, 72437, 0, 0, 1, 3, 0, 0, 0]\n-- (some (node2 2 (node3 77777777777777 leaf leaf (node2 1 leaf leaf)) leaf))\n\nexample (x : tree \u2115) : deserialize _ (serialize x) = some x :=\nby { dsimp [serialize,deserialize],\n     rw [\u2190 read_write_eq_eval_eval,serial.correctness],\n     refl }\n", "meta": {"author": "cipher1024", "repo": "serialean", "sha": "47881e4a6bc0a62cd68520564610b75f8a4fef2c", "save_path": "github-repos/lean/cipher1024-serialean", "path": "github-repos/lean/cipher1024-serialean/serialean-47881e4a6bc0a62cd68520564610b75f8a4fef2c/test/examples.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3623456753854544}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\nPorted by: Fr\u00e9d\u00e9ric Dupuis\n\n! This file was ported from Lean 3 source module algebra.order.hom.monoid\n! leanprover-community/mathlib commit 3342d1b2178381196f818146ff79bc0e7ccd9e2d\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Data.Pi.Algebra\nimport Mathlib.Algebra.Hom.Group\nimport Mathlib.Algebra.Order.Group.Instances\nimport Mathlib.Algebra.Order.Monoid.WithZero.Defs\nimport Mathlib.Order.Hom.Basic\n\n/-!\n# Ordered monoid and group homomorphisms\n\nThis file defines morphisms between (additive) ordered monoids.\n\n## Types of morphisms\n\n* `OrderAddMonoidHom`: Ordered additive monoid homomorphisms.\n* `OrderMonoidHom`: Ordered monoid homomorphisms.\n* `OrderMonoidWithZeroHom`: Ordered monoid with zero homomorphisms.\n\n## Typeclasses\n\n* `OrderAddMonoidHomClass`\n* `OrderMonoidHomClass`\n* `OrderMonoidWithZeroHomClass`\n\n## Notation\n\n* `\u2192+o`: Bundled ordered additive monoid homs. Also use for additive groups homs.\n* `\u2192*o`: Bundled ordered monoid homs. Also use for groups homs.\n* `\u2192*\u2080o`: Bundled ordered monoid with zero homs. Also use for groups with zero homs.\n\n## Implementation notes\n\nThere's a coercion from bundled homs to fun, and the canonical notation is to use the bundled hom as\na function via this coercion.\n\nThere is no `OrderGroupHom` -- the idea is that `OrderMonoidHom` is used.\nThe constructor for `OrderMonoidHom` needs a proof of `map_one` as well as `map_mul`; a separate\nconstructor `OrderMonoidHom.mk'` will construct ordered group homs (i.e. ordered monoid homs\nbetween ordered groups) given only a proof that multiplication is preserved,\n\nImplicit `{}` brackets are often used instead of type class `[]` brackets. This is done when the\ninstances can be inferred because they are implicit arguments to the type `OrderMonoidHom`. When\nthey can be inferred from the type it is faster to use this method than to use type class inference.\n\n## Tags\n\nordered monoid, ordered group, monoid with zero\n-/\n\n\nopen Function\n\nvariable {F \u03b1 \u03b2 \u03b3 \u03b4 : Type _}\n\nsection AddMonoid\n\n/-- `\u03b1 \u2192+o \u03b2` is the type of monotone functions `\u03b1 \u2192 \u03b2` that preserve the `OrderedAddCommMonoid`\nstructure.\n\n`OrderAddMonoidHom` is also used for ordered group homomorphisms.\n\nWhen possible, instead of parametrizing results over `(f : \u03b1 \u2192+o \u03b2)`,\nyou should parametrize over `(F : Type _) [OrderAddMonoidHomClass F \u03b1 \u03b2] (f : F)`.\n\nWhen you extend this structure, make sure to extend `OrderAddMonoidHomClass`. -/\nstructure OrderAddMonoidHom (\u03b1 \u03b2 : Type _) [Preorder \u03b1] [Preorder \u03b2] [AddZeroClass \u03b1]\n  [AddZeroClass \u03b2] extends \u03b1 \u2192+ \u03b2 where\n  /-- An `OrderAddMonoidHom` is a monotone function. -/\n  monotone' : Monotone toFun\n#align order_add_monoid_hom OrderAddMonoidHom\n\n/-- Infix notation for `OrderAddMonoidHom`. -/\ninfixr:25 \" \u2192+o \" => OrderAddMonoidHom\n\nsection\n\n/-- `OrderAddMonoidHomClass F \u03b1 \u03b2` states that `F` is a type of ordered monoid homomorphisms.\n\nYou should also extend this typeclass when you extend `OrderAddMonoidHom`. -/\nclass OrderAddMonoidHomClass (F : Type _) (\u03b1 \u03b2 : outParam <| Type _) [Preorder \u03b1] [Preorder \u03b2]\n  [AddZeroClass \u03b1] [AddZeroClass \u03b2] extends AddMonoidHomClass F \u03b1 \u03b2 where\n  /-- An `OrderAddMonoidHom` is a monotone function. -/\n  monotone (f : F) : Monotone f\n#align order_add_monoid_hom_class OrderAddMonoidHomClass\n\nend\n\n-- Instances and lemmas are defined below through `@[to_additive]`.\nend AddMonoid\n\nsection Monoid\n\n/-- `\u03b1 \u2192*o \u03b2` is the type of functions `\u03b1 \u2192 \u03b2` that preserve the `OrderedCommMonoid` structure.\n\n`OrderMonoidHom` is also used for ordered group homomorphisms.\n\nWhen possible, instead of parametrizing results over `(f : \u03b1 \u2192*o \u03b2)`,\nyou should parametrize over `(F : Type _) [OrderMonoidHomClass F \u03b1 \u03b2] (f : F)`.\n\nWhen you extend this structure, make sure to extend `OrderMonoidHomClass`. -/\n@[to_additive]\nstructure OrderMonoidHom (\u03b1 \u03b2 : Type _) [Preorder \u03b1] [Preorder \u03b2] [MulOneClass \u03b1]\n  [MulOneClass \u03b2] extends \u03b1 \u2192* \u03b2 where\n  /-- An `OrderMonoidHom` is a monotone function. -/\n  monotone' : Monotone toFun\n#align order_monoid_hom OrderMonoidHom\n\n/-- Infix notation for `OrderMonoidHom`. -/\ninfixr:25 \" \u2192*o \" => OrderMonoidHom\n\nsection\n\n/-- `OrderMonoidHomClass F \u03b1 \u03b2` states that `F` is a type of ordered monoid homomorphisms.\n\nYou should also extend this typeclass when you extend `OrderMonoidHom`. -/\n@[to_additive]\nclass OrderMonoidHomClass (F : Type _) (\u03b1 \u03b2 : outParam <| Type _) [Preorder \u03b1] [Preorder \u03b2]\n  [MulOneClass \u03b1] [MulOneClass \u03b2] extends MonoidHomClass F \u03b1 \u03b2 where\n  /-- An `OrderMonoidHom` is a monotone function. -/\n  monotone (f : F) : Monotone f\n#align order_monoid_hom_class OrderMonoidHomClass\n\nend\n\nvariable {_ : Preorder \u03b1} {_ : Preorder \u03b2} {_ : MulOneClass \u03b1} {_ : MulOneClass \u03b2}\n\n/-- Turn an element of a type `F` satisfying `OrderMonoidHomClass F \u03b1 \u03b2` into an actual\n`OrderMonoidHom`. This is declared as the default coercion from `F` to `\u03b1 \u2192*o \u03b2`. -/\n@[to_additive (attr := coe)\n  \"Turn an element of a type `F` satisfying `OrderAddMonoidHomClass F \u03b1 \u03b2` into an actual\n  `OrderAddMonoidHom`. This is declared as the default coercion from `F` to `\u03b1 \u2192+o \u03b2`.\"]\ndef OrderMonoidHomClass.toOrderMonoidHom [OrderMonoidHomClass F \u03b1 \u03b2] (f : F) : \u03b1 \u2192*o \u03b2 :=\n{ (f : \u03b1 \u2192* \u03b2) with monotone' := monotone f }\n\n-- See note [lower instance priority]\n@[to_additive]\ninstance (priority := 100) OrderMonoidHomClass.toOrderHomClass [OrderMonoidHomClass F \u03b1 \u03b2] :\n    OrderHomClass F \u03b1 \u03b2 :=\n  { \u2039OrderMonoidHomClass F \u03b1 \u03b2\u203a with map_rel := OrderMonoidHomClass.monotone }\n#align order_monoid_hom_class.to_order_hom_class OrderMonoidHomClass.toOrderHomClass\n#align order_add_monoid_hom_class.to_order_hom_class OrderAddMonoidHomClass.toOrderHomClass\n\n/-- Any type satisfying `OrderMonoidHomClass` can be cast into `OrderMonoidHom` via\n  `OrderMonoidHomClass.toOrderMonoidHom`. -/\n@[to_additive \"Any type satisfying `OrderAddMonoidHomClass` can be cast into `OrderAddMonoidHom` via\n  `OrderAddMonoidHomClass.toOrderAddMonoidHom`\"]\ninstance [OrderMonoidHomClass F \u03b1 \u03b2] : CoeTC F (\u03b1 \u2192*o \u03b2) :=\n  \u27e8OrderMonoidHomClass.toOrderMonoidHom\u27e9\n\nend Monoid\n\nsection MonoidWithZero\n\nvariable [Preorder \u03b1] [Preorder \u03b2] [MulZeroOneClass \u03b1] [MulZeroOneClass \u03b2]\n\n/-- `OrderMonoidWithZeroHom \u03b1 \u03b2` is the type of functions `\u03b1 \u2192 \u03b2` that preserve\nthe `MonoidWithZero` structure.\n\n`OrderMonoidWithZeroHom` is also used for group homomorphisms.\n\nWhen possible, instead of parametrizing results over `(f : \u03b1 \u2192+ \u03b2)`,\nyou should parametrize over `(F : Type _) [OrderMonoidWithZeroHomClass F \u03b1 \u03b2] (f : F)`.\n\nWhen you extend this structure, make sure to extend `OrderMonoidWithZeroHomClass`. -/\nstructure OrderMonoidWithZeroHom (\u03b1 \u03b2 : Type _) [Preorder \u03b1] [Preorder \u03b2] [MulZeroOneClass \u03b1]\n  [MulZeroOneClass \u03b2] extends \u03b1 \u2192*\u2080 \u03b2 where\n  /-- An `OrderMonoidWithZeroHom` is a monotone function. -/\n  monotone' : Monotone toFun\n#align order_monoid_with_zero_hom OrderMonoidWithZeroHom\n\n/-- Infix notation for `OrderMonoidWithZeroHom`. -/\ninfixr:25 \" \u2192*\u2080o \" => OrderMonoidWithZeroHom\n\nsection\n\n/-- `OrderMonoidWithZeroHomClass F \u03b1 \u03b2` states that `F` is a type of\nordered monoid with zero homomorphisms.\n\nYou should also extend this typeclass when you extend `OrderMonoidWithZeroHom`. -/\nclass OrderMonoidWithZeroHomClass (F : Type _) (\u03b1 \u03b2 : outParam <| Type _) [Preorder \u03b1] [Preorder \u03b2]\n  [MulZeroOneClass \u03b1] [MulZeroOneClass \u03b2] extends MonoidWithZeroHomClass F \u03b1 \u03b2 where\n  /-- An `OrderMonoidWithZeroHom` is a monotone function. -/\n  monotone (f : F) : Monotone f\n#align order_monoid_with_zero_hom_class OrderMonoidWithZeroHomClass\n\n/-- Turn an element of a type `F` satisfying `OrderMonoidWithZeroHomClass F \u03b1 \u03b2` into an actual\n`OrderMonoidWithZeroHom`. This is declared as the default coercion from `F` to `\u03b1 \u2192+*\u2080o \u03b2`. -/\n@[coe]\ndef OrderMonoidWithZeroHomClass.toOrderMonoidWithZeroHom [OrderMonoidWithZeroHomClass F \u03b1 \u03b2]\n    (f : F) : \u03b1 \u2192*\u2080o \u03b2 :=\n{ (f : \u03b1 \u2192*\u2080 \u03b2) with monotone' := monotone f }\n\nend\n\n-- See note [lower instance priority]\ninstance (priority := 100) OrderMonoidWithZeroHomClass.toOrderMonoidHomClass\n    {_ : Preorder \u03b1} {_ : Preorder \u03b2} {_ : MulZeroOneClass \u03b1} {_ : MulZeroOneClass \u03b2}\n    [OrderMonoidWithZeroHomClass F \u03b1 \u03b2] : OrderMonoidHomClass F \u03b1 \u03b2 :=\n  { \u2039OrderMonoidWithZeroHomClass F \u03b1 \u03b2\u203a with }\n#align order_monoid_with_zero_hom_class.to_order_monoid_hom_class OrderMonoidWithZeroHomClass.toOrderMonoidHomClass\n\ninstance [OrderMonoidWithZeroHomClass F \u03b1 \u03b2] : CoeTC F (\u03b1 \u2192*\u2080o \u03b2) :=\n  \u27e8OrderMonoidWithZeroHomClass.toOrderMonoidWithZeroHom\u27e9\n\nend MonoidWithZero\n\nsection OrderedAddCommMonoid\n\nvariable [OrderedAddCommMonoid \u03b1] [OrderedAddCommMonoid \u03b2] [OrderAddMonoidHomClass F \u03b1 \u03b2] (f : F)\n  {a : \u03b1}\n\ntheorem map_nonneg (ha : 0 \u2264 a) : 0 \u2264 f a := by\n  rw [\u2190 map_zero f]\n  exact OrderHomClass.mono _ ha\n#align map_nonneg map_nonneg\n\ntheorem map_nonpos (ha : a \u2264 0) : f a \u2264 0 := by\n  rw [\u2190 map_zero f]\n  exact OrderHomClass.mono _ ha\n#align map_nonpos map_nonpos\n\nend OrderedAddCommMonoid\n\nsection OrderedAddCommGroup\n\nvariable [OrderedAddCommGroup \u03b1] [OrderedAddCommMonoid \u03b2] [AddMonoidHomClass F \u03b1 \u03b2] (f : F)\n\ntheorem monotone_iff_map_nonneg : Monotone (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a, 0 \u2264 a \u2192 0 \u2264 f a :=\n  \u27e8fun h a => by\n    rw [\u2190 map_zero f]\n    apply h, fun h a b hl => by\n    rw [\u2190 sub_add_cancel b a, map_add f]\n    exact le_add_of_nonneg_left (h _ <| sub_nonneg.2 hl)\u27e9\n#align monotone_iff_map_nonneg monotone_iff_map_nonneg\n\ntheorem antitone_iff_map_nonpos : Antitone (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a, 0 \u2264 a \u2192 f a \u2264 0 :=\n  monotone_toDual_comp_iff.symm.trans <| monotone_iff_map_nonneg _\n#align antitone_iff_map_nonpos antitone_iff_map_nonpos\n\ntheorem monotone_iff_map_nonpos : Monotone (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a \u2264 0, f a \u2264 0 :=\n  antitone_comp_ofDual_iff.symm.trans <| antitone_iff_map_nonpos _\n#align monotone_iff_map_nonpos monotone_iff_map_nonpos\n\ntheorem antitone_iff_map_nonneg : Antitone (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a \u2264 0, 0 \u2264 f a :=\n  monotone_comp_ofDual_iff.symm.trans <| monotone_iff_map_nonneg _\n#align antitone_iff_map_nonneg antitone_iff_map_nonneg\n\nvariable [CovariantClass \u03b2 \u03b2 (\u00b7 + \u00b7) (\u00b7 < \u00b7)]\n\ntheorem strictMono_iff_map_pos : StrictMono (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a, 0 < a \u2192 0 < f a := by\n  refine \u27e8fun h a => ?_, fun h a b hl => ?_\u27e9\n  \u00b7 rw [\u2190 map_zero f]\n    apply h\n  \u00b7 rw [\u2190 sub_add_cancel b a, map_add f]\n    exact lt_add_of_pos_left _ (h _ <| sub_pos.2 hl)\n#align strict_mono_iff_map_pos strictMono_iff_map_pos\n\ntheorem strictAnti_iff_map_neg : StrictAnti (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a, 0 < a \u2192 f a < 0 :=\n  strictMono_toDual_comp_iff.symm.trans <| strictMono_iff_map_pos _\n#align strict_anti_iff_map_neg strictAnti_iff_map_neg\n\ntheorem strictMono_iff_map_neg : StrictMono (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a < 0, f a < 0 :=\n  strictAnti_comp_ofDual_iff.symm.trans <| strictAnti_iff_map_neg _\n#align strict_mono_iff_map_neg strictMono_iff_map_neg\n\ntheorem strictAnti_iff_map_pos : StrictAnti (f : \u03b1 \u2192 \u03b2) \u2194 \u2200 a < 0, 0 < f a :=\n  strictMono_comp_ofDual_iff.symm.trans <| strictMono_iff_map_pos _\n#align strict_anti_iff_map_pos strictAnti_iff_map_pos\n\nend OrderedAddCommGroup\n\nnamespace OrderMonoidHom\n\nsection Preorder\n\nvariable [Preorder \u03b1] [Preorder \u03b2] [Preorder \u03b3] [Preorder \u03b4] [MulOneClass \u03b1] [MulOneClass \u03b2]\n  [MulOneClass \u03b3] [MulOneClass \u03b4] {f g : \u03b1 \u2192*o \u03b2}\n\n@[to_additive]\n instance : OrderMonoidHomClass (\u03b1 \u2192*o \u03b2) \u03b1 \u03b2 where\n  coe f := f.toFun\n  coe_injective' f g h := by\n    obtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := f\n    obtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := g\n    congr\n  map_mul f := f.map_mul'\n  map_one f := f.map_one'\n  monotone f := f.monotone'\n\n-- Porting note:\n-- These helper instances are unhelpful in Lean 4, so omitting:\n-- /-- Helper instance for when there's too many metavariables to apply `FunLike.instCoeFunForAll`\n-- directly. -/\n-- @[to_additive \"Helper instance for when there's too many metavariables to apply\n-- `FunLike.instCoeFunForAll` directly.\"]\n-- instance : CoeFun (\u03b1 \u2192*o \u03b2) fun _ => \u03b1 \u2192 \u03b2 :=\n--   FunLike.instCoeFunForAll\n\n-- Other lemmas should be accessed through the `FunLike` API\n@[to_additive (attr := ext)]\ntheorem ext (h : \u2200 a, f a = g a) : f = g :=\n  FunLike.ext f g h\n#align order_monoid_hom.ext OrderMonoidHom.ext\n#align order_add_monoid_hom.ext OrderAddMonoidHom.ext\n\n@[to_additive]\ntheorem toFun_eq_coe (f : \u03b1 \u2192*o \u03b2) : f.toFun = (f : \u03b1 \u2192 \u03b2) :=\n  rfl\n#align order_monoid_hom.to_fun_eq_coe OrderMonoidHom.toFun_eq_coe\n#align order_add_monoid_hom.to_fun_eq_coe OrderAddMonoidHom.toFun_eq_coe\n\n@[to_additive (attr := simp)]\ntheorem coe_mk (f : \u03b1 \u2192* \u03b2) (h) : (OrderMonoidHom.mk f h : \u03b1 \u2192 \u03b2) = f :=\n  rfl\n#align order_monoid_hom.coe_mk OrderMonoidHom.coe_mk\n#align order_add_monoid_hom.coe_mk OrderAddMonoidHom.coe_mk\n\n@[to_additive (attr := simp)]\ntheorem mk_coe (f : \u03b1 \u2192*o \u03b2) (h) : OrderMonoidHom.mk (f : \u03b1 \u2192* \u03b2) h = f := by\n  ext\n  rfl\n#align order_monoid_hom.mk_coe OrderMonoidHom.mk_coe\n#align order_add_monoid_hom.mk_coe OrderAddMonoidHom.mk_coe\n\n/-- Reinterpret an ordered monoid homomorphism as an order homomorphism. -/\n@[to_additive \"Reinterpret an ordered additive monoid homomorphism as an order homomorphism.\"]\ndef toOrderHom (f : \u03b1 \u2192*o \u03b2) : \u03b1 \u2192o \u03b2 :=\n  { f with }\n#align order_monoid_hom.to_order_hom OrderMonoidHom.toOrderHom\n#align order_add_monoid_hom.to_order_hom OrderAddMonoidHom.toOrderHom\n\n@[to_additive (attr := simp)]\ntheorem coe_monoidHom (f : \u03b1 \u2192*o \u03b2) : ((f : \u03b1 \u2192* \u03b2) : \u03b1 \u2192 \u03b2) = f :=\n  rfl\n#align order_monoid_hom.coe_monoid_hom OrderMonoidHom.coe_monoidHom\n#align order_add_monoid_hom.coe_add_monoid_hom OrderAddMonoidHom.coe_addMonoidHom\n\n@[to_additive (attr := simp)]\ntheorem coe_orderHom (f : \u03b1 \u2192*o \u03b2) : ((f : \u03b1 \u2192o \u03b2) : \u03b1 \u2192 \u03b2) = f :=\n  rfl\n#align order_monoid_hom.coe_order_hom OrderMonoidHom.coe_orderHom\n#align order_add_monoid_hom.coe_order_hom OrderAddMonoidHom.coe_orderHom\n\n@[to_additive]\ntheorem toMonoidHom_injective : Injective (toMonoidHom : _ \u2192 \u03b1 \u2192* \u03b2) := fun f g h =>\n  ext <| by convert FunLike.ext_iff.1 h using 0\n#align order_monoid_hom.to_monoid_hom_injective OrderMonoidHom.toMonoidHom_injective\n#align order_add_monoid_hom.to_add_monoid_hom_injective OrderAddMonoidHom.toAddMonoidHom_injective\n\n@[to_additive]\ntheorem toOrderHom_injective : Injective (toOrderHom : _ \u2192 \u03b1 \u2192o \u03b2) := fun f g h =>\n  ext <| by convert FunLike.ext_iff.1 h using 0\n#align order_monoid_hom.to_order_hom_injective OrderMonoidHom.toOrderHom_injective\n#align order_add_monoid_hom.to_order_hom_injective OrderAddMonoidHom.toOrderHom_injective\n\n/-- Copy of an `OrderMonoidHom` with a new `toFun` equal to the old one. Useful to fix\ndefinitional equalities. -/\n@[to_additive \"Copy of an `OrderAddMonoidHom` with a new `toFun` equal to the old one. Useful to fix\ndefinitional equalities.\"]\nprotected def copy (f : \u03b1 \u2192*o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u03b1 \u2192*o \u03b2 :=\n  { f.toMonoidHom.copy f' h with toFun := f', monotone' := h.symm.subst f.monotone' }\n#align order_monoid_hom.copy OrderMonoidHom.copy\n#align order_add_monoid_hom.copy OrderAddMonoidHom.copy\n\n@[to_additive (attr := simp)]\ntheorem coe_copy (f : \u03b1 \u2192*o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u21d1(f.copy f' h) = f' :=\n  rfl\n#align order_monoid_hom.coe_copy OrderMonoidHom.coe_copy\n#align order_add_monoid_hom.coe_copy OrderAddMonoidHom.coe_copy\n\n@[to_additive]\ntheorem copy_eq (f : \u03b1 \u2192*o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : f.copy f' h = f :=\n  FunLike.ext' h\n#align order_monoid_hom.copy_eq OrderMonoidHom.copy_eq\n#align order_add_monoid_hom.copy_eq OrderAddMonoidHom.copy_eq\n\nvariable (\u03b1)\n\n/-- The identity map as an ordered monoid homomorphism. -/\n@[to_additive \"The identity map as an ordered additive monoid homomorphism.\"]\nprotected def id : \u03b1 \u2192*o \u03b1 :=\n  { MonoidHom.id \u03b1, OrderHom.id with }\n#align order_monoid_hom.id OrderMonoidHom.id\n#align order_add_monoid_hom.id OrderAddMonoidHom.id\n\n@[to_additive (attr := simp)]\ntheorem coe_id : \u21d1(OrderMonoidHom.id \u03b1) = id :=\n  rfl\n#align order_monoid_hom.coe_id OrderMonoidHom.coe_id\n#align order_add_monoid_hom.coe_id OrderAddMonoidHom.coe_id\n\n@[to_additive]\ninstance : Inhabited (\u03b1 \u2192*o \u03b1) :=\n  \u27e8OrderMonoidHom.id \u03b1\u27e9\n\nvariable {\u03b1}\n\n/-- Composition of `OrderMonoidHom`s as an `OrderMonoidHom`. -/\n@[to_additive \"Composition of `OrderAddMonoidHom`s as an `OrderAddMonoidHom`\"]\ndef comp (f : \u03b2 \u2192*o \u03b3) (g : \u03b1 \u2192*o \u03b2) : \u03b1 \u2192*o \u03b3 :=\n  { f.toMonoidHom.comp (g : \u03b1 \u2192* \u03b2), f.toOrderHom.comp (g : \u03b1 \u2192o \u03b2) with }\n#align order_monoid_hom.comp OrderMonoidHom.comp\n#align order_add_monoid_hom.comp OrderAddMonoidHom.comp\n\n@[to_additive (attr := simp)]\ntheorem coe_comp (f : \u03b2 \u2192*o \u03b3) (g : \u03b1 \u2192*o \u03b2) : (f.comp g : \u03b1 \u2192 \u03b3) = f \u2218 g :=\n  rfl\n#align order_monoid_hom.coe_comp OrderMonoidHom.coe_comp\n#align order_add_monoid_hom.coe_comp OrderAddMonoidHom.coe_comp\n\n@[to_additive (attr := simp)]\ntheorem comp_apply (f : \u03b2 \u2192*o \u03b3) (g : \u03b1 \u2192*o \u03b2) (a : \u03b1) : (f.comp g) a = f (g a) :=\n  rfl\n#align order_add_monoid_hom.comp_apply OrderAddMonoidHom.comp_apply\n#align order_monoid_hom.comp_apply OrderMonoidHom.comp_apply\n\n@[to_additive]\ntheorem coe_comp_monoidHom (f : \u03b2 \u2192*o \u03b3) (g : \u03b1 \u2192*o \u03b2) :\n    (f.comp g : \u03b1 \u2192* \u03b3) = (f : \u03b2 \u2192* \u03b3).comp g :=\n  rfl\n#align order_monoid_hom.coe_comp_monoid_hom OrderMonoidHom.coe_comp_monoidHom\n#align order_add_monoid_hom.coe_comp_add_monoid_hom OrderAddMonoidHom.coe_comp_addMonoidHom\n\n@[to_additive]\ntheorem coe_comp_orderHom (f : \u03b2 \u2192*o \u03b3) (g : \u03b1 \u2192*o \u03b2) :\n    (f.comp g : \u03b1 \u2192o \u03b3) = (f : \u03b2 \u2192o \u03b3).comp g :=\n  rfl\n#align order_monoid_hom.coe_comp_order_hom OrderMonoidHom.coe_comp_orderHom\n#align order_add_monoid_hom.coe_comp_order_hom OrderAddMonoidHom.coe_comp_orderHom\n\n@[to_additive (attr := simp)]\ntheorem comp_assoc (f : \u03b3 \u2192*o \u03b4) (g : \u03b2 \u2192*o \u03b3) (h : \u03b1 \u2192*o \u03b2) :\n    (f.comp g).comp h = f.comp (g.comp h) :=\n  rfl\n#align order_monoid_hom.comp_assoc OrderMonoidHom.comp_assoc\n#align order_add_monoid_hom.comp_assoc OrderAddMonoidHom.comp_assoc\n\n@[to_additive (attr := simp)]\ntheorem comp_id (f : \u03b1 \u2192*o \u03b2) : f.comp (OrderMonoidHom.id \u03b1) = f :=\n  rfl\n#align order_monoid_hom.comp_id OrderMonoidHom.comp_id\n#align order_add_monoid_hom.comp_id OrderAddMonoidHom.comp_id\n\n@[to_additive (attr := simp)]\ntheorem id_comp (f : \u03b1 \u2192*o \u03b2) : (OrderMonoidHom.id \u03b2).comp f = f :=\n  rfl\n#align order_monoid_hom.id_comp OrderMonoidHom.id_comp\n#align order_add_monoid_hom.id_comp OrderAddMonoidHom.id_comp\n\n@[to_additive]\ntheorem cancel_right {g\u2081 g\u2082 : \u03b2 \u2192*o \u03b3} {f : \u03b1 \u2192*o \u03b2} (hf : Function.Surjective f) :\n    g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082 :=\n  \u27e8fun h => ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun _ => by congr\u27e9\n#align order_monoid_hom.cancel_right OrderMonoidHom.cancel_right\n#align order_add_monoid_hom.cancel_right OrderAddMonoidHom.cancel_right\n\n@[to_additive]\ntheorem cancel_left {g : \u03b2 \u2192*o \u03b3} {f\u2081 f\u2082 : \u03b1 \u2192*o \u03b2} (hg : Function.Injective g) :\n    g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082 :=\n  \u27e8fun h => ext fun a => hg <| by rw [\u2190 comp_apply, h, comp_apply], congr_arg _\u27e9\n#align order_monoid_hom.cancel_left OrderMonoidHom.cancel_left\n#align order_add_monoid_hom.cancel_left OrderAddMonoidHom.cancel_left\n\n/-- `1` is the homomorphism sending all elements to `1`. -/\n@[to_additive \"`0` is the homomorphism sending all elements to `0`.\"]\ninstance : One (\u03b1 \u2192*o \u03b2) :=\n  \u27e8{ (1 : \u03b1 \u2192* \u03b2) with monotone' := monotone_const }\u27e9\n\n@[to_additive (attr := simp)]\ntheorem coe_one : \u21d1(1 : \u03b1 \u2192*o \u03b2) = 1 :=\n  rfl\n#align order_monoid_hom.coe_one OrderMonoidHom.coe_one\n#align order_add_monoid_hom.coe_zero OrderAddMonoidHom.coe_zero\n\n@[to_additive (attr := simp)]\ntheorem one_apply (a : \u03b1) : (1 : \u03b1 \u2192*o \u03b2) a = 1 :=\n  rfl\n#align order_monoid_hom.one_apply OrderMonoidHom.one_apply\n#align order_add_monoid_hom.zero_apply OrderAddMonoidHom.zero_apply\n\n@[to_additive (attr := simp)]\ntheorem one_comp (f : \u03b1 \u2192*o \u03b2) : (1 : \u03b2 \u2192*o \u03b3).comp f = 1 :=\n  rfl\n#align order_monoid_hom.one_comp OrderMonoidHom.one_comp\n#align order_add_monoid_hom.zero_comp OrderAddMonoidHom.zero_comp\n\n@[to_additive (attr := simp)]\ntheorem comp_one (f : \u03b2 \u2192*o \u03b3) : f.comp (1 : \u03b1 \u2192*o \u03b2) = 1 :=\n  ext fun _ => map_one f\n#align order_monoid_hom.comp_one OrderMonoidHom.comp_one\n#align order_add_monoid_hom.comp_zero OrderAddMonoidHom.comp_zero\n\nend Preorder\n\nsection Mul\n\nvariable [OrderedCommMonoid \u03b1] [OrderedCommMonoid \u03b2] [OrderedCommMonoid \u03b3]\n\n/-- For two ordered monoid morphisms `f` and `g`, their product is the ordered monoid morphism\nsending `a` to `f a * g a`. -/\n@[to_additive \"For two ordered additive monoid morphisms `f` and `g`, their product is the ordered\nadditive monoid morphism sending `a` to `f a + g a`.\"]\ninstance : Mul (\u03b1 \u2192*o \u03b2) :=\n  \u27e8fun f g => { (f * g : \u03b1 \u2192* \u03b2) with monotone' := f.monotone'.mul' g.monotone' }\u27e9\n\n@[to_additive (attr := simp)]\ntheorem coe_mul (f g : \u03b1 \u2192*o \u03b2) : \u21d1(f * g) = f * g :=\n  rfl\n#align order_monoid_hom.coe_mul OrderMonoidHom.coe_mul\n#align order_add_monoid_hom.coe_add OrderAddMonoidHom.coe_add\n\n@[to_additive (attr := simp)]\ntheorem mul_apply (f g : \u03b1 \u2192*o \u03b2) (a : \u03b1) : (f * g) a = f a * g a :=\n  rfl\n#align order_monoid_hom.mul_apply OrderMonoidHom.mul_apply\n#align order_add_monoid_hom.add_apply OrderAddMonoidHom.add_apply\n\n@[to_additive]\ntheorem mul_comp (g\u2081 g\u2082 : \u03b2 \u2192*o \u03b3) (f : \u03b1 \u2192*o \u03b2) : (g\u2081 * g\u2082).comp f = g\u2081.comp f * g\u2082.comp f :=\n  rfl\n#align order_monoid_hom.mul_comp OrderMonoidHom.mul_comp\n#align order_add_monoid_hom.add_comp OrderAddMonoidHom.add_comp\n\n@[to_additive]\ntheorem comp_mul (g : \u03b2 \u2192*o \u03b3) (f\u2081 f\u2082 : \u03b1 \u2192*o \u03b2) : g.comp (f\u2081 * f\u2082) = g.comp f\u2081 * g.comp f\u2082 :=\n  ext fun _ => map_mul g _ _\n#align order_monoid_hom.comp_mul OrderMonoidHom.comp_mul\n#align order_add_monoid_hom.comp_add OrderAddMonoidHom.comp_add\n\nend Mul\n\nsection OrderedCommMonoid\n\nvariable {h\u03b1 : OrderedCommMonoid \u03b1} {h\u03b2 : OrderedCommMonoid \u03b2}\n\n@[to_additive (attr := simp)]\ntheorem toMonoidHom_eq_coe (f : \u03b1 \u2192*o \u03b2) : f.toMonoidHom = f :=\n  rfl\n#align order_monoid_hom.to_monoid_hom_eq_coe OrderMonoidHom.toMonoidHom_eq_coe\n#align order_add_monoid_hom.to_add_monoid_hom_eq_coe OrderAddMonoidHom.toAddMonoidHom_eq_coe\n\n@[to_additive (attr := simp)]\n\n\nend OrderedCommMonoid\n\nsection OrderedCommGroup\n\nvariable {h\u03b1 : OrderedCommGroup \u03b1} {h\u03b2 : OrderedCommGroup \u03b2}\n\n/-- Makes an ordered group homomorphism from a proof that the map preserves multiplication. -/\n@[to_additive\n      \"Makes an ordered additive group homomorphism from a proof that the map preserves\n      addition.\"]\ndef mk' (f : \u03b1 \u2192 \u03b2) (hf : Monotone f) (map_mul : \u2200 a b : \u03b1, f (a * b) = f a * f b) : \u03b1 \u2192*o \u03b2 :=\n  { MonoidHom.mk' f map_mul with monotone' := hf }\n#align order_monoid_hom.mk' OrderMonoidHom.mk'\n#align order_add_monoid_hom.mk' OrderAddMonoidHom.mk'\n\nend OrderedCommGroup\n\nend OrderMonoidHom\n\nnamespace OrderMonoidWithZeroHom\n\nsection Preorder\n\nvariable [Preorder \u03b1] [Preorder \u03b2] [Preorder \u03b3] [Preorder \u03b4] [MulZeroOneClass \u03b1] [MulZeroOneClass \u03b2]\n  [MulZeroOneClass \u03b3] [MulZeroOneClass \u03b4] {f g : \u03b1 \u2192*\u2080o \u03b2}\n\ninstance : OrderMonoidWithZeroHomClass (\u03b1 \u2192*\u2080o \u03b2) \u03b1 \u03b2 where\n  coe f := f.toFun\n  coe_injective' f g h := by\n    obtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := f\n    obtain \u27e8\u27e8\u27e8_, _\u27e9\u27e9, _\u27e9 := g\n    congr\n  map_mul f := f.map_mul'\n  map_one f := f.map_one'\n  map_zero f := f.map_zero'\n  monotone f := f.monotone'\n\n-- Porting note:\n-- These helper instances are unhelpful in Lean 4, so omitting:\n--/-- Helper instance for when there's too many metavariables to apply `FunLike.instCoeFunForAll`\n--directly. -/\n--instance : CoeFun (\u03b1 \u2192*\u2080o \u03b2) fun _ => \u03b1 \u2192 \u03b2 :=\n--  FunLike.instCoeFunForAll\n\n-- Other lemmas should be accessed through the `FunLike` API\n@[ext]\ntheorem ext (h : \u2200 a, f a = g a) : f = g :=\n  FunLike.ext f g h\n#align order_monoid_with_zero_hom.ext OrderMonoidWithZeroHom.ext\n\ntheorem toFun_eq_coe (f : \u03b1 \u2192*\u2080o \u03b2) : f.toFun = (f : \u03b1 \u2192 \u03b2) :=\n  rfl\n#align order_monoid_with_zero_hom.to_fun_eq_coe OrderMonoidWithZeroHom.toFun_eq_coe\n\n@[simp]\ntheorem coe_mk (f : \u03b1 \u2192*\u2080 \u03b2) (h) : (OrderMonoidWithZeroHom.mk f h : \u03b1 \u2192 \u03b2) = f :=\n  rfl\n#align order_monoid_with_zero_hom.coe_mk OrderMonoidWithZeroHom.coe_mk\n\n@[simp]\ntheorem mk_coe (f : \u03b1 \u2192*\u2080o \u03b2) (h) : OrderMonoidWithZeroHom.mk (f : \u03b1 \u2192*\u2080 \u03b2) h = f := rfl\n#align order_monoid_with_zero_hom.mk_coe OrderMonoidWithZeroHom.mk_coe\n\n/-- Reinterpret an ordered monoid with zero homomorphism as an order monoid homomorphism. -/\ndef toOrderMonoidHom (f : \u03b1 \u2192*\u2080o \u03b2) : \u03b1 \u2192*o \u03b2 :=\n  { f with }\n#align order_monoid_with_zero_hom.to_order_monoid_hom OrderMonoidWithZeroHom.toOrderMonoidHom\n\n@[simp]\ntheorem coe_monoidWithZeroHom (f : \u03b1 \u2192*\u2080o \u03b2) : \u21d1(f : \u03b1 \u2192*\u2080 \u03b2) = f :=\n  rfl\n#align order_monoid_with_zero_hom.coe_monoid_with_zero_hom OrderMonoidWithZeroHom.coe_monoidWithZeroHom\n\n@[simp]\ntheorem coe_orderMonoidHom (f : \u03b1 \u2192*\u2080o \u03b2) : \u21d1(f : \u03b1 \u2192*o \u03b2) = f :=\n  rfl\n#align order_monoid_with_zero_hom.coe_order_monoid_hom OrderMonoidWithZeroHom.coe_orderMonoidHom\n\ntheorem toOrderMonoidHom_injective : Injective (toOrderMonoidHom : _ \u2192 \u03b1 \u2192*o \u03b2) := fun f g h =>\n  ext <| by convert FunLike.ext_iff.1 h using 0\n#align order_monoid_with_zero_hom.to_order_monoid_hom_injective OrderMonoidWithZeroHom.toOrderMonoidHom_injective\n\ntheorem toMonoidWithZeroHom_injective : Injective (toMonoidWithZeroHom : _ \u2192 \u03b1 \u2192*\u2080 \u03b2) :=\n  fun f g h => ext <| by convert FunLike.ext_iff.1 h using 0\n#align order_monoid_with_zero_hom.to_monoid_with_zero_hom_injective OrderMonoidWithZeroHom.toMonoidWithZeroHom_injective\n\n/-- Copy of an `OrderMonoidWithZeroHom` with a new `toFun` equal to the old one. Useful to fix\ndefinitional equalities. -/\nprotected def copy (f : \u03b1 \u2192*\u2080o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u03b1 \u2192*o \u03b2 :=\n  { f.toOrderMonoidHom.copy f' h, f.toMonoidWithZeroHom.copy f' h with toFun := f' }\n#align order_monoid_with_zero_hom.copy OrderMonoidWithZeroHom.copy\n\n@[simp]\ntheorem coe_copy (f : \u03b1 \u2192*\u2080o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : \u21d1(f.copy f' h) = f' :=\n  rfl\n#align order_monoid_with_zero_hom.coe_copy OrderMonoidWithZeroHom.coe_copy\n\ntheorem copy_eq (f : \u03b1 \u2192*\u2080o \u03b2) (f' : \u03b1 \u2192 \u03b2) (h : f' = f) : f.copy f' h = f :=\n  FunLike.ext' h\n#align order_monoid_with_zero_hom.copy_eq OrderMonoidWithZeroHom.copy_eq\n\nvariable (\u03b1)\n\n/-- The identity map as an ordered monoid with zero homomorphism. -/\nprotected def id : \u03b1 \u2192*\u2080o \u03b1 :=\n  { MonoidWithZeroHom.id \u03b1, OrderHom.id with }\n#align order_monoid_with_zero_hom.id OrderMonoidWithZeroHom.id\n\n@[simp]\ntheorem coe_id : \u21d1(OrderMonoidWithZeroHom.id \u03b1) = id :=\n  rfl\n#align order_monoid_with_zero_hom.coe_id OrderMonoidWithZeroHom.coe_id\n\ninstance : Inhabited (\u03b1 \u2192*\u2080o \u03b1) :=\n  \u27e8OrderMonoidWithZeroHom.id \u03b1\u27e9\n\nvariable {\u03b1}\n\n/-- Composition of `OrderMonoidWithZeroHom`s as an `OrderMonoidWithZeroHom`. -/\ndef comp (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) : \u03b1 \u2192*\u2080o \u03b3 :=\n  { f.toMonoidWithZeroHom.comp (g : \u03b1 \u2192*\u2080 \u03b2), f.toOrderMonoidHom.comp (g : \u03b1 \u2192*o \u03b2) with }\n#align order_monoid_with_zero_hom.comp OrderMonoidWithZeroHom.comp\n\n@[simp]\ntheorem coe_comp (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) : (f.comp g : \u03b1 \u2192 \u03b3) = f \u2218 g :=\n  rfl\n#align order_monoid_with_zero_hom.coe_comp OrderMonoidWithZeroHom.coe_comp\n\n@[simp]\ntheorem comp_apply (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) (a : \u03b1) : (f.comp g) a = f (g a) :=\n  rfl\n#align order_monoid_with_zero_hom.comp_apply OrderMonoidWithZeroHom.comp_apply\n\ntheorem coe_comp_monoidWithZeroHom (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) :\n    (f.comp g : \u03b1 \u2192*\u2080 \u03b3) = (f : \u03b2 \u2192*\u2080 \u03b3).comp g :=\n  rfl\n#align order_monoid_with_zero_hom.coe_comp_monoid_with_zero_hom OrderMonoidWithZeroHom.coe_comp_monoidWithZeroHom\n\ntheorem coe_comp_orderMonoidHom (f : \u03b2 \u2192*\u2080o \u03b3) (g : \u03b1 \u2192*\u2080o \u03b2) :\n    (f.comp g : \u03b1 \u2192*o \u03b3) = (f : \u03b2 \u2192*o \u03b3).comp g :=\n  rfl\n#align order_monoid_with_zero_hom.coe_comp_order_monoid_hom OrderMonoidWithZeroHom.coe_comp_orderMonoidHom\n\n@[simp]\ntheorem comp_assoc (f : \u03b3 \u2192*\u2080o \u03b4) (g : \u03b2 \u2192*\u2080o \u03b3) (h : \u03b1 \u2192*\u2080o \u03b2) :\n    (f.comp g).comp h = f.comp (g.comp h) :=\n  rfl\n#align order_monoid_with_zero_hom.comp_assoc OrderMonoidWithZeroHom.comp_assoc\n\n@[simp]\ntheorem comp_id (f : \u03b1 \u2192*\u2080o \u03b2) : f.comp (OrderMonoidWithZeroHom.id \u03b1) = f := rfl\n#align order_monoid_with_zero_hom.comp_id OrderMonoidWithZeroHom.comp_id\n\n@[simp]\ntheorem id_comp (f : \u03b1 \u2192*\u2080o \u03b2) : (OrderMonoidWithZeroHom.id \u03b2).comp f = f := rfl\n#align order_monoid_with_zero_hom.id_comp OrderMonoidWithZeroHom.id_comp\n\ntheorem cancel_right {g\u2081 g\u2082 : \u03b2 \u2192*\u2080o \u03b3} {f : \u03b1 \u2192*\u2080o \u03b2} (hf : Function.Surjective f) :\n    g\u2081.comp f = g\u2082.comp f \u2194 g\u2081 = g\u2082 :=\n  \u27e8fun h => ext <| hf.forall.2 <| FunLike.ext_iff.1 h, fun _ => by congr\u27e9\n#align order_monoid_with_zero_hom.cancel_right OrderMonoidWithZeroHom.cancel_right\n\ntheorem cancel_left {g : \u03b2 \u2192*\u2080o \u03b3} {f\u2081 f\u2082 : \u03b1 \u2192*\u2080o \u03b2} (hg : Function.Injective g) :\n    g.comp f\u2081 = g.comp f\u2082 \u2194 f\u2081 = f\u2082 :=\n  \u27e8fun h => ext fun a => hg <| by rw [\u2190 comp_apply, h, comp_apply], congr_arg _\u27e9\n#align order_monoid_with_zero_hom.cancel_left OrderMonoidWithZeroHom.cancel_left\n\nend Preorder\n\nsection Mul\n\nvariable [LinearOrderedCommMonoidWithZero \u03b1] [LinearOrderedCommMonoidWithZero \u03b2]\n  [LinearOrderedCommMonoidWithZero \u03b3]\n\n/-- For two ordered monoid morphisms `f` and `g`, their product is the ordered monoid morphism\nsending `a` to `f a * g a`. -/\ninstance : Mul (\u03b1 \u2192*\u2080o \u03b2) :=\n  \u27e8fun f g => { (f * g : \u03b1 \u2192*\u2080 \u03b2) with monotone' := f.monotone'.mul' g.monotone' }\u27e9\n\n@[simp]\ntheorem coe_mul (f g : \u03b1 \u2192*\u2080o \u03b2) : \u21d1(f * g) = f * g :=\n  rfl\n#align order_monoid_with_zero_hom.coe_mul OrderMonoidWithZeroHom.coe_mul\n\n@[simp]\ntheorem mul_apply (f g : \u03b1 \u2192*\u2080o \u03b2) (a : \u03b1) : (f * g) a = f a * g a :=\n  rfl\n#align order_monoid_with_zero_hom.mul_apply OrderMonoidWithZeroHom.mul_apply\n\ntheorem mul_comp (g\u2081 g\u2082 : \u03b2 \u2192*\u2080o \u03b3) (f : \u03b1 \u2192*\u2080o \u03b2) : (g\u2081 * g\u2082).comp f = g\u2081.comp f * g\u2082.comp f :=\n  rfl\n#align order_monoid_with_zero_hom.mul_comp OrderMonoidWithZeroHom.mul_comp\n\ntheorem comp_mul (g : \u03b2 \u2192*\u2080o \u03b3) (f\u2081 f\u2082 : \u03b1 \u2192*\u2080o \u03b2) : g.comp (f\u2081 * f\u2082) = g.comp f\u2081 * g.comp f\u2082 :=\n  ext fun _ => map_mul g _ _\n#align order_monoid_with_zero_hom.comp_mul OrderMonoidWithZeroHom.comp_mul\n\nend Mul\n\nsection LinearOrderedCommMonoidWithZero\n\nvariable {h\u03b1 : Preorder \u03b1} {h\u03b1' : MulZeroOneClass \u03b1} {h\u03b2 : Preorder \u03b2} {h\u03b2' : MulZeroOneClass \u03b2}\n\n@[simp]\ntheorem toMonoidWithZeroHom_eq_coe (f : \u03b1 \u2192*\u2080o \u03b2) : f.toMonoidWithZeroHom = f := by\n  rfl\n#align order_monoid_with_zero_hom.to_monoid_with_zero_hom_eq_coe OrderMonoidWithZeroHom.toMonoidWithZeroHom_eq_coe\n\n@[simp]\ntheorem toOrderMonoidHom_eq_coe (f : \u03b1 \u2192*\u2080o \u03b2) : f.toOrderMonoidHom = f :=\n  rfl\n#align order_monoid_with_zero_hom.to_order_monoid_hom_eq_coe OrderMonoidWithZeroHom.toOrderMonoidHom_eq_coe\n\nend LinearOrderedCommMonoidWithZero\n\nend OrderMonoidWithZeroHom\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Algebra/Order/Hom/Monoid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.3623456669922822}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.AppBuilder\nimport Lean.Meta.SynthInstance\nimport Lean.Meta.Tactic.Simp.Types\n\nnamespace Lean.Meta.Simp\n\ndef synthesizeArgs (lemmaName : Name) (xs : Array Expr) (bis : Array BinderInfo) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Bool := do\n  for x in xs, bi in bis do\n    let type \u2190 inferType x\n    if bi.isInstImplicit then\n      unless (\u2190 synthesizeInstance x type) do\n        return false\n    else if (\u2190 instantiateMVars x).isMVar then\n      if (\u2190 isProp type) then\n        match (\u2190 discharge? type) with\n        | some proof =>\n          unless (\u2190 isDefEq x proof) do\n            trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to assign proof{indentExpr type}\"\n            return false\n        | none =>\n          trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to discharge hypotheses{indentExpr type}\"\n          return false\n      else if (\u2190 isClass? type).isSome then\n        unless (\u2190 synthesizeInstance x type) do\n          return false\n  return true\nwhere\n  synthesizeInstance (x type : Expr) : SimpM Bool := do\n    match (\u2190 trySynthInstance type) with\n    | LOption.some val =>\n      if (\u2190 isDefEq x val) then\n        return true\n      else\n        trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to assign instance{indentExpr type}\"\n        return false\n    | _ =>\n      trace[Meta.Tactic.simp.discharge] \"{lemmaName}, failed to synthesize instance{indentExpr type}\"\n      return false\n\nprivate def tryLemmaCore (lhs : Expr) (xs : Array Expr) (bis : Array BinderInfo) (val : Expr) (type : Expr) (e : Expr) (lemma : SimpLemma) (numExtraArgs : Nat) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM (Option Result) := do\n  let rec go (e : Expr) : SimpM (Option Result) := do\n    if (\u2190 isDefEq lhs e) then\n      unless (\u2190 synthesizeArgs lemma.getName xs bis discharge?) do\n        return none\n      let proof \u2190 instantiateMVars (mkAppN val xs)\n      if \u2190 hasAssignableMVar proof then\n        trace[Meta.Tactic.simp.rewrite] \"{lemma}, has unassigned metavariables after unification\"\n        return none\n      let rhs   \u2190 instantiateMVars type.appArg!\n      if e == rhs then\n        return none\n      if lemma.perm && !Expr.lt rhs e then\n        trace[Meta.Tactic.simp.rewrite] \"{lemma}, perm rejected {e} ==> {rhs}\"\n        return none\n      trace[Meta.Tactic.simp.rewrite] \"{lemma}, {e} ==> {rhs}\"\n      return some { expr := rhs, proof? := proof }\n    else\n      unless lhs.isMVar do\n        -- We do not report unification failures when `lhs` is a metavariable\n        -- Example: `x = ()`\n        -- TODO: reconsider if we want lemmas such as `(x : Unit) \u2192 x = ()`\n        trace[Meta.Tactic.simp.unify] \"{lemma}, failed to unify {lhs} with {e}\"\n      return none\n  /- Check whether we need something more sophisticated here.\n     This simple approach was good enough for Mathlib 3 -/\n  let mut extraArgs := #[]\n  let mut e := e\n  for i in [:numExtraArgs] do\n    extraArgs := extraArgs.push e.appArg!\n    e := e.appFn!\n  match (\u2190 go e) with\n  | none => return none\n  | some { expr := eNew, proof? := none } => return some { expr := mkAppN eNew extraArgs }\n  | some { expr := eNew, proof? := some proof } =>\n    let mut proof := proof\n    for extraArg in extraArgs do\n      proof \u2190 mkCongrFun proof extraArg\n    return some { expr := mkAppN eNew extraArgs, proof? := some proof }\n\ndef tryLemmaWithExtraArgs? (e : Expr) (lemma : SimpLemma) (numExtraArgs : Nat) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM (Option Result) :=\n  withNewMCtxDepth do\n    let val  \u2190 lemma.getValue\n    let type \u2190 inferType val\n    let (xs, bis, type) \u2190 forallMetaTelescopeReducing type\n    let type \u2190 whnf (\u2190 instantiateMVars type)\n    let lhs := type.appFn!.appArg!\n    tryLemmaCore lhs xs bis val type e lemma numExtraArgs discharge?\n\ndef tryLemma? (e : Expr) (lemma : SimpLemma) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM (Option Result) := do\n  withNewMCtxDepth do\n    let val  \u2190 lemma.getValue\n    let type \u2190 inferType val\n    let (xs, bis, type) \u2190 forallMetaTelescopeReducing type\n    let type \u2190 whnf (\u2190 instantiateMVars type)\n    let lhs := type.appFn!.appArg!\n    match (\u2190 tryLemmaCore lhs xs bis val type e lemma 0 discharge?) with\n    | some result => return some result\n    | none =>\n      let lhsNumArgs := lhs.getAppNumArgs\n      let eNumArgs   := e.getAppNumArgs\n      if eNumArgs > lhsNumArgs then\n        tryLemmaCore lhs xs bis val type e lemma (eNumArgs - lhsNumArgs) discharge?\n      else\n        return none\n/-\nRemark: the parameter tag is used for creating trace messages. It is irrelevant otherwise.\n-/\ndef rewrite (e : Expr) (s : DiscrTree SimpLemma) (erased : Std.PHashSet Name) (discharge? : Expr \u2192 SimpM (Option Expr)) (tag : String) : SimpM Result := do\n  let candidates \u2190 s.getMatchWithExtra e\n  if candidates.isEmpty then\n    trace[Debug.Meta.Tactic.simp] \"no theorems found for {tag}-rewriting {e}\"\n    return { expr := e }\n  else\n    let candidates := candidates.insertionSort fun e\u2081 e\u2082 => e\u2081.1.priority < e\u2082.1.priority\n    for (lemma, numExtraArgs) in candidates do\n      unless inErasedSet lemma do\n        if let some result \u2190 tryLemmaWithExtraArgs? e lemma numExtraArgs discharge? then\n          return result\n    return { expr := e }\nwhere\n  inErasedSet (lemma : SimpLemma) : Bool :=\n    match lemma.name? with\n    | none => false\n    | some name => erased.contains name\n\ndef rewriteCtorEq? (e : Expr) : MetaM (Option Result) := withReducibleAndInstances do\n  match e.eq? with\n  | none => return none\n  | some (_, lhs, rhs) =>\n    let lhs \u2190 whnf lhs\n    let rhs \u2190 whnf rhs\n    let env \u2190 getEnv\n    match lhs.constructorApp? env, rhs.constructorApp? env with\n    | some (c\u2081, _), some (c\u2082, _) =>\n      if c\u2081.name != c\u2082.name then\n        withLocalDeclD `h e fun h =>\n          return some { expr := mkConst ``False, proof? := (\u2190 mkEqFalse' (\u2190 mkLambdaFVars #[h] (\u2190 mkNoConfusion (mkConst ``False) h))) }\n      else\n        return none\n    | _, _ => return none\n\n@[inline] def tryRewriteCtorEq (e : Expr) (x : SimpM Step) : SimpM Step := do\n  match (\u2190 rewriteCtorEq? e) with\n  | some r => return Step.done r\n  | none => x\n\ndef rewriteUsingDecide? (e : Expr) : MetaM (Option Result) := withReducibleAndInstances do\n  if e.hasFVar || e.hasMVar || e.isConstOf ``True || e.isConstOf ``False then\n    return none\n  else\n    try\n      let d \u2190 mkDecide e\n      let r \u2190 withDefault <| whnf d\n      if r.isConstOf ``true then\n        return some { expr := mkConst ``True, proof? := mkAppN (mkConst ``eq_true_of_decide) #[e, d.appArg!, (\u2190 mkEqRefl (mkConst ``true))] }\n      else if r.isConstOf ``false then\n        let h \u2190 mkEqRefl d\n        return some { expr := mkConst ``False, proof? := mkAppN (mkConst ``eq_false_of_decide) #[e, d.appArg!, (\u2190 mkEqRefl (mkConst ``false))] }\n      else\n        return none\n    catch _ =>\n      return none\n\n@[inline] def tryRewriteUsingDecide (e : Expr) (x : SimpM Step) : SimpM Step := do\n  if (\u2190 read).config.decide then\n    match (\u2190 rewriteUsingDecide? e) with\n    | some r => return Step.done r\n    | none => x\n  else\n    x\n\ndef rewritePre (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  let lemmas \u2190 (\u2190 read).simpLemmas\n  return Step.visit (\u2190 rewrite e lemmas.pre lemmas.erased discharge? (tag := \"pre\"))\n\ndef rewritePost (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  let lemmas \u2190 (\u2190 read).simpLemmas\n  return Step.visit (\u2190 rewrite e lemmas.post lemmas.erased discharge? (tag := \"post\"))\n\ndef preDefault (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step :=\n  tryRewriteCtorEq e <| rewritePre e discharge?\n\ndef postDefault (e : Expr) (discharge? : Expr \u2192 SimpM (Option Expr)) : SimpM Step := do\n  -- TODO: try equation lemmas\n  tryRewriteCtorEq e <| tryRewriteUsingDecide e <| rewritePost e discharge?\n\nend Lean.Meta.Simp\n", "meta": {"author": "subfish-zhou", "repo": "leanprover-zh_CN.github.io", "sha": "8b2985d4a3d458ceda9361ac454c28168d920d3f", "save_path": "github-repos/lean/subfish-zhou-leanprover-zh_CN.github.io", "path": "github-repos/lean/subfish-zhou-leanprover-zh_CN.github.io/leanprover-zh_CN.github.io-8b2985d4a3d458ceda9361ac454c28168d920d3f/stage0/src/Lean/Meta/Tactic/Simp/Rewrite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3623456669922822}}
{"text": "/-\nCopyright (c) 2020 Chris Hughes. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Chris Hughes\n\n! This file was ported from Lean 3 source module ring_theory.prime\n! leanprover-community/mathlib commit 327c3c0d9232d80e250dc8f65e7835b82b266ea5\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Associated\nimport Mathbin.Algebra.BigOperators.Basic\n\n/-!\n# Prime elements in rings\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\nThis file contains lemmas about prime elements of commutative rings.\n-/\n\n\nsection CancelCommMonoidWithZero\n\nvariable {R : Type _} [CancelCommMonoidWithZero R]\n\nopen Finset\n\nopen BigOperators\n\n/- warning: mul_eq_mul_prime_prod -> mul_eq_mul_prime_prod is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} R] {\u03b1 : Type.{u2}} [_inst_2 : DecidableEq.{succ u2} \u03b1] {x : R} {y : R} {a : R} {s : Finset.{u2} \u03b1} {p : \u03b1 -> R}, (forall (i : \u03b1), (Membership.Mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.hasMem.{u2} \u03b1) i s) -> (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1) (p i))) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) x y) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) a (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) s (fun (i : \u03b1) => p i)))) -> (Exists.{succ u2} (Finset.{u2} \u03b1) (fun (t : Finset.{u2} \u03b1) => Exists.{succ u2} (Finset.{u2} \u03b1) (fun (u : Finset.{u2} \u03b1) => Exists.{succ u1} R (fun (b : R) => Exists.{succ u1} R (fun (c : R) => And (Eq.{succ u2} (Finset.{u2} \u03b1) (Union.union.{u2} (Finset.{u2} \u03b1) (Finset.hasUnion.{u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b)) t u) s) (And (Disjoint.{u2} (Finset.{u2} \u03b1) (Finset.partialOrder.{u2} \u03b1) (Finset.orderBot.{u2} \u03b1) t u) (And (Eq.{succ u1} R a (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b c)) (And (Eq.{succ u1} R x (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) t (fun (i : \u03b1) => p i)))) (Eq.{succ u1} R y (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) c (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) u (fun (i : \u03b1) => p i))))))))))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} R] {\u03b1 : Type.{u2}} [_inst_2 : DecidableEq.{succ u2} \u03b1] {x : R} {y : R} {a : R} {s : Finset.{u2} \u03b1} {p : \u03b1 -> R}, (forall (i : \u03b1), (Membership.mem.{u2, u2} \u03b1 (Finset.{u2} \u03b1) (Finset.instMembershipFinset.{u2} \u03b1) i s) -> (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1) (p i))) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) x y) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) a (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) s (fun (i : \u03b1) => p i)))) -> (Exists.{succ u2} (Finset.{u2} \u03b1) (fun (t : Finset.{u2} \u03b1) => Exists.{succ u2} (Finset.{u2} \u03b1) (fun (u : Finset.{u2} \u03b1) => Exists.{succ u1} R (fun (b : R) => Exists.{succ u1} R (fun (c : R) => And (Eq.{succ u2} (Finset.{u2} \u03b1) (Union.union.{u2} (Finset.{u2} \u03b1) (Finset.instUnionFinset.{u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b)) t u) s) (And (Disjoint.{u2} (Finset.{u2} \u03b1) (Finset.partialOrder.{u2} \u03b1) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u2} \u03b1) t u) (And (Eq.{succ u1} R a (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b c)) (And (Eq.{succ u1} R x (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) t (fun (i : \u03b1) => p i)))) (Eq.{succ u1} R y (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) c (Finset.prod.{u1, u2} R \u03b1 (CommMonoidWithZero.toCommMonoid.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)) u (fun (i : \u03b1) => p i))))))))))))\nCase conversion may be inaccurate. Consider using '#align mul_eq_mul_prime_prod mul_eq_mul_prime_prod\u2093'. -/\n/-- If `x * y = a * \u220f i in s, p i` where `p i` is always prime, then\n  `x` and `y` can both be written as a divisor of `a` multiplied by\n  a product over a subset of `s`  -/\ntheorem mul_eq_mul_prime_prod {\u03b1 : Type _} [DecidableEq \u03b1] {x y a : R} {s : Finset \u03b1} {p : \u03b1 \u2192 R}\n    (hp : \u2200 i \u2208 s, Prime (p i)) (hx : x * y = a * \u220f i in s, p i) :\n    \u2203 (t u : Finset \u03b1)(b c : R),\n      t \u222a u = s \u2227 Disjoint t u \u2227 a = b * c \u2227 (x = b * \u220f i in t, p i) \u2227 y = c * \u220f i in u, p i :=\n  by\n  induction' s using Finset.induction with i s his ih generalizing x y a\n  \u00b7 exact \u27e8\u2205, \u2205, x, y, by simp [hx]\u27e9\n  \u00b7 rw [prod_insert his, \u2190 mul_assoc] at hx\n    have hpi : Prime (p i) := hp i (mem_insert_self _ _)\n    rcases ih (fun i hi => hp i (mem_insert_of_mem hi)) hx with\n      \u27e8t, u, b, c, htus, htu, hbc, rfl, rfl\u27e9\n    have hit : i \u2209 t := fun hit => his (htus \u25b8 mem_union_left _ hit)\n    have hiu : i \u2209 u := fun hiu => his (htus \u25b8 mem_union_right _ hiu)\n    obtain \u27e8d, rfl\u27e9 | \u27e8d, rfl\u27e9 : p i \u2223 b \u2228 p i \u2223 c\n    exact hpi.dvd_or_dvd \u27e8a, by rw [\u2190 hbc, mul_comm]\u27e9\n    \u00b7 rw [mul_assoc, mul_comm a, mul_right_inj' hpi.ne_zero] at hbc\n      exact\n        \u27e8insert i t, u, d, c, by rw [insert_union, htus], disjoint_insert_left.2 \u27e8hiu, htu\u27e9, by\n          simp [hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]\u27e9\n    \u00b7 rw [\u2190 mul_assoc, mul_right_comm b, mul_left_inj' hpi.ne_zero] at hbc\n      exact\n        \u27e8t, insert i u, b, d, by rw [union_insert, htus], disjoint_insert_right.2 \u27e8hit, htu\u27e9, by\n          simp [\u2190 hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]\u27e9\n#align mul_eq_mul_prime_prod mul_eq_mul_prime_prod\n\n/- warning: mul_eq_mul_prime_pow -> mul_eq_mul_prime_pow is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} R] {x : R} {y : R} {a : R} {p : R} {n : Nat}, (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1) p) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) x y) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) a (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p n))) -> (Exists.{1} Nat (fun (i : Nat) => Exists.{1} Nat (fun (j : Nat) => Exists.{succ u1} R (fun (b : R) => Exists.{succ u1} R (fun (c : R) => And (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat Nat.hasAdd) i j) n) (And (Eq.{succ u1} R a (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b c)) (And (Eq.{succ u1} R x (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p i))) (Eq.{succ u1} R y (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toHasMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) c (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p j))))))))))\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : CancelCommMonoidWithZero.{u1} R] {x : R} {y : R} {a : R} {p : R} {n : Nat}, (Prime.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1) p) -> (Eq.{succ u1} R (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) x y) (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) a (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p n))) -> (Exists.{1} Nat (fun (i : Nat) => Exists.{1} Nat (fun (j : Nat) => Exists.{succ u1} R (fun (b : R) => Exists.{succ u1} R (fun (c : R) => And (Eq.{1} Nat (HAdd.hAdd.{0, 0, 0} Nat Nat Nat (instHAdd.{0} Nat instAddNat) i j) n) (And (Eq.{succ u1} R a (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b c)) (And (Eq.{succ u1} R x (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) b (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p i))) (Eq.{succ u1} R y (HMul.hMul.{u1, u1, u1} R R R (instHMul.{u1} R (MulZeroClass.toMul.{u1} R (MulZeroOneClass.toMulZeroClass.{u1} R (MonoidWithZero.toMulZeroOneClass.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1)))))) c (HPow.hPow.{u1, 0, u1} R Nat R (instHPow.{u1, 0} R Nat (Monoid.Pow.{u1} R (MonoidWithZero.toMonoid.{u1} R (CommMonoidWithZero.toMonoidWithZero.{u1} R (CancelCommMonoidWithZero.toCommMonoidWithZero.{u1} R _inst_1))))) p j))))))))))\nCase conversion may be inaccurate. Consider using '#align mul_eq_mul_prime_pow mul_eq_mul_prime_pow\u2093'. -/\n/-- If ` x * y = a * p ^ n` where `p` is prime, then `x` and `y` can both be written\n  as the product of a power of `p` and a divisor of `a`. -/\ntheorem mul_eq_mul_prime_pow {x y a p : R} {n : \u2115} (hp : Prime p) (hx : x * y = a * p ^ n) :\n    \u2203 (i j : \u2115)(b c : R), i + j = n \u2227 a = b * c \u2227 x = b * p ^ i \u2227 y = c * p ^ j :=\n  by\n  rcases mul_eq_mul_prime_prod (fun _ _ => hp)\n      (show x * y = a * (range n).Prod fun _ => p by simpa) with\n    \u27e8t, u, b, c, htus, htu, rfl, rfl, rfl\u27e9\n  exact \u27e8t.card, u.card, b, c, by rw [\u2190 card_disjoint_union htu, htus, card_range], by simp\u27e9\n#align mul_eq_mul_prime_pow mul_eq_mul_prime_pow\n\nend CancelCommMonoidWithZero\n\nsection CommRing\n\nvariable {\u03b1 : Type _} [CommRing \u03b1]\n\n/- warning: prime.neg -> Prime.neg is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : CommRing.{u1} \u03b1] {p : \u03b1}, (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) p) -> (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) (Neg.neg.{u1} \u03b1 (SubNegMonoid.toHasNeg.{u1} \u03b1 (AddGroup.toSubNegMonoid.{u1} \u03b1 (AddGroupWithOne.toAddGroup.{u1} \u03b1 (AddCommGroupWithOne.toAddGroupWithOne.{u1} \u03b1 (Ring.toAddCommGroupWithOne.{u1} \u03b1 (CommRing.toRing.{u1} \u03b1 _inst_1)))))) p))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : CommRing.{u1} \u03b1] {p : \u03b1}, (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) p) -> (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) (Neg.neg.{u1} \u03b1 (Ring.toNeg.{u1} \u03b1 (CommRing.toRing.{u1} \u03b1 _inst_1)) p))\nCase conversion may be inaccurate. Consider using '#align prime.neg Prime.neg\u2093'. -/\ntheorem Prime.neg {p : \u03b1} (hp : Prime p) : Prime (-p) :=\n  by\n  obtain \u27e8h1, h2, h3\u27e9 := hp\n  exact \u27e8neg_ne_zero.mpr h1, by rwa [IsUnit.neg_iff], by simpa [neg_dvd] using h3\u27e9\n#align prime.neg Prime.neg\n\n/- warning: prime.abs -> Prime.abs is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : CommRing.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] {p : \u03b1}, (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) p) -> (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) (Abs.abs.{u1} \u03b1 (Neg.toHasAbs.{u1} \u03b1 (SubNegMonoid.toHasNeg.{u1} \u03b1 (AddGroup.toSubNegMonoid.{u1} \u03b1 (AddGroupWithOne.toAddGroup.{u1} \u03b1 (AddCommGroupWithOne.toAddGroupWithOne.{u1} \u03b1 (Ring.toAddCommGroupWithOne.{u1} \u03b1 (CommRing.toRing.{u1} \u03b1 _inst_1)))))) (SemilatticeSup.toHasSup.{u1} \u03b1 (Lattice.toSemilatticeSup.{u1} \u03b1 (LinearOrder.toLattice.{u1} \u03b1 _inst_2)))) p))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : CommRing.{u1} \u03b1] [_inst_2 : LinearOrder.{u1} \u03b1] {p : \u03b1}, (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) p) -> (Prime.{u1} \u03b1 (CommSemiring.toCommMonoidWithZero.{u1} \u03b1 (CommRing.toCommSemiring.{u1} \u03b1 _inst_1)) (Abs.abs.{u1} \u03b1 (Neg.toHasAbs.{u1} \u03b1 (Ring.toNeg.{u1} \u03b1 (CommRing.toRing.{u1} \u03b1 _inst_1)) (SemilatticeSup.toSup.{u1} \u03b1 (Lattice.toSemilatticeSup.{u1} \u03b1 (DistribLattice.toLattice.{u1} \u03b1 (instDistribLattice.{u1} \u03b1 _inst_2))))) p))\nCase conversion may be inaccurate. Consider using '#align prime.abs Prime.abs\u2093'. -/\ntheorem Prime.abs [LinearOrder \u03b1] {p : \u03b1} (hp : Prime p) : Prime (abs p) :=\n  by\n  obtain h | h := abs_choice p <;> rw [h]\n  \u00b7 exact hp\n  \u00b7 exact hp.neg\n#align prime.abs Prime.abs\n\nend CommRing\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/RingTheory/Prime.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952052, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3622384889870011}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n\nType classes for traversing collections. The concepts and laws are taken from\nhttp://hackage.haskell.org/package/base-4.11.1.0/docs/Data-Traversable.html\n-/\n\nimport tactic.cache\nimport category.applicative\n\nopen function (hiding comp)\n\nuniverses u v w\n\nsection applicative_transformation\n\nvariables (F : Type u \u2192 Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u \u2192 Type w) [applicative G] [is_lawful_applicative G]\n\nstructure applicative_transformation : Type (max (u+1) v w) :=\n(app : \u2200 \u03b1 : Type u, F \u03b1 \u2192 G \u03b1)\n(preserves_pure' : \u2200 {\u03b1 : Type u} (x : \u03b1), app _ (pure x) = pure x)\n(preserves_seq' : \u2200 {\u03b1 \u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), app _ (x <*> y) = app _ x <*> app _ y)\n\nend applicative_transformation\n\nnamespace applicative_transformation\n\nvariables (F : Type u \u2192 Type v) [applicative F] [is_lawful_applicative F]\nvariables (G : Type u \u2192 Type w) [applicative G] [is_lawful_applicative G]\n\ninstance : has_coe_to_fun (applicative_transformation F G) :=\n{ F := \u03bb _, \u03a0 {\u03b1}, F \u03b1 \u2192 G \u03b1,\n  coe := \u03bb a, a.app }\n\nvariables {F G}\nvariables (\u03b7 : applicative_transformation F G)\n\n@[functor_norm]\nlemma preserves_pure : \u2200 {\u03b1} (x : \u03b1), \u03b7 (pure x) = pure x := \u03b7.preserves_pure'\n\n@[functor_norm]\nlemma preserves_seq :\n  \u2200 {\u03b1 \u03b2 : Type u} (x : F (\u03b1 \u2192 \u03b2)) (y : F \u03b1), \u03b7 (x <*> y) = \u03b7 x <*> \u03b7 y :=\n\u03b7.preserves_seq'\n\n@[functor_norm]\nlemma preserves_map {\u03b1 \u03b2} (x : \u03b1 \u2192 \u03b2) (y : F \u03b1) : \u03b7 (x <$> y) = x <$> \u03b7 y :=\nby rw [\u2190 pure_seq_eq_map, \u03b7.preserves_seq]; simp with functor_norm\n\nend applicative_transformation\n\nopen applicative_transformation\n\nclass traversable (t : Type u \u2192 Type u) extends functor t :=\n(traverse : \u03a0 {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b2},\n   (\u03b1 \u2192 m \u03b2) \u2192 t \u03b1 \u2192 m (t \u03b2))\n\nopen functor\n\nexport traversable (traverse)\n\nsection functions\n\nvariables {t : Type u \u2192 Type u}\nvariables {m : Type u \u2192 Type v} [applicative m]\nvariables {\u03b1 \u03b2 : Type u}\n\n\nvariables {f : Type u \u2192 Type u} [applicative f]\n\ndef sequence [traversable t] : t (f \u03b1) \u2192 f (t \u03b1) := traverse id\n\nend functions\n\nclass is_lawful_traversable (t : Type u \u2192 Type u) [traversable t]\n  extends is_lawful_functor t : Type (u+1) :=\n(id_traverse : \u2200 {\u03b1} (x : t \u03b1), traverse id.mk x = x )\n(comp_traverse : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    {\u03b1 \u03b2 \u03b3} (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t \u03b1),\n  traverse (comp.mk \u2218 map f \u2218 g) x =\n  comp.mk (map (traverse f) (traverse g x)))\n(traverse_eq_map_id : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2) (x : t \u03b1),\n  traverse (id.mk \u2218 f) x = id.mk (f <$> x))\n(naturality : \u2200 {F G} [applicative F] [applicative G]\n    [is_lawful_applicative F] [is_lawful_applicative G]\n    (\u03b7 : applicative_transformation F G) {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) (x : t \u03b1),\n  \u03b7 (traverse f x) = traverse (@\u03b7 _ \u2218 f) x)\n\ninstance : traversable id := \u27e8\u03bb _ _ _ _, id\u27e9\ninstance : is_lawful_traversable id := by refine {..}; intros; refl\n\nsection\n\nvariables {F : Type u \u2192 Type v} [applicative F]\n\ninstance : traversable option := \u27e8@option.traverse\u27e9\n\ninstance : traversable list := \u27e8@list.traverse\u27e9\n\nend\n\nnamespace sum\n\nvariables {\u03c3 : Type u}\nvariables {F : Type u \u2192 Type u}\nvariables [applicative F]\n\nprotected def traverse {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2) : \u03c3 \u2295 \u03b1 \u2192 F (\u03c3 \u2295 \u03b2)\n| (sum.inl x) := pure (sum.inl x)\n| (sum.inr x) := sum.inr <$> f x\n\nend sum\n\ninstance {\u03c3 : Type u} : traversable.{u} (sum \u03c3) := \u27e8@sum.traverse _\u27e9\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category/traversable/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.36221855258070623}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.limits.preserves.basic\n\nopen category_theory category_theory.limits\n\nnoncomputable theory\n\nnamespace category_theory\n\nuniverses w' w v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\n\nsection creates\nvariables {D : Type u\u2082} [category.{v\u2082} D]\n\nvariables {J : Type w} [category.{w'} J] {K : J \u2964 C}\n\n/--\nDefine the lift of a cone: For a cone `c` for `K \u22d9 F`, give a cone for `K`\nwhich is a lift of `c`, i.e. the image of it under `F` is (iso) to `c`.\n\nWe will then use this as part of the definition of creation of limits:\nevery limit cone has a lift.\n\nNote this definition is really only useful when `c` is a limit already.\n-/\nstructure liftable_cone (K : J \u2964 C) (F : C \u2964 D) (c : cone (K \u22d9 F)) :=\n(lifted_cone : cone K)\n(valid_lift : F.map_cone lifted_cone \u2245 c)\n\n/--\nDefine the lift of a cocone: For a cocone `c` for `K \u22d9 F`, give a cocone for\n`K` which is a lift of `c`, i.e. the image of it under `F` is (iso) to `c`.\n\nWe will then use this as part of the definition of creation of colimits:\nevery limit cocone has a lift.\n\nNote this definition is really only useful when `c` is a colimit already.\n-/\nstructure liftable_cocone (K : J \u2964 C) (F : C \u2964 D) (c : cocone (K \u22d9 F)) :=\n(lifted_cocone : cocone K)\n(valid_lift : F.map_cocone lifted_cocone \u2245 c)\n\n/--\nDefinition 3.3.1 of [Riehl].\nWe say that `F` creates limits of `K` if, given any limit cone `c` for `K \u22d9 F`\n(i.e. below) we can lift it to a cone \"above\", and further that `F` reflects\nlimits for `K`.\n\nIf `F` reflects isomorphisms, it suffices to show only that the lifted cone is\na limit - see `creates_limit_of_reflects_iso`.\n-/\nclass creates_limit (K : J \u2964 C) (F : C \u2964 D) extends reflects_limit K F :=\n(lifts : \u03a0 c, is_limit c \u2192 liftable_cone K F c)\n\n/--\n`F` creates limits of shape `J` if `F` creates the limit of any diagram\n`K : J \u2964 C`.\n-/\nclass creates_limits_of_shape (J : Type w) [category.{w'} J] (F : C \u2964 D) :=\n(creates_limit : \u03a0 {K : J \u2964 C}, creates_limit K F . tactic.apply_instance)\n\n/-- `F` creates limits if it creates limits of shape `J` for any `J`. -/\n@[nolint check_univs] -- This should be used with explicit universe variables.\nclass creates_limits_of_size (F : C \u2964 D) :=\n(creates_limits_of_shape : \u03a0 {J : Type w} [category.{w'} J],\n  creates_limits_of_shape J F . tactic.apply_instance)\n\n/-- `F` creates small limits if it creates limits of shape `J` for any small `J`. -/\nabbreviation creates_limits (F : C \u2964 D) := creates_limits_of_size.{v\u2082 v\u2082} F\n\n/--\nDual of definition 3.3.1 of [Riehl].\nWe say that `F` creates colimits of `K` if, given any limit cocone `c` for\n`K \u22d9 F` (i.e. below) we can lift it to a cocone \"above\", and further that `F`\nreflects limits for `K`.\n\nIf `F` reflects isomorphisms, it suffices to show only that the lifted cocone is\na limit - see `creates_limit_of_reflects_iso`.\n-/\nclass creates_colimit (K : J \u2964 C) (F : C \u2964 D) extends reflects_colimit K F :=\n(lifts : \u03a0 c, is_colimit c \u2192 liftable_cocone K F c)\n\n/--\n`F` creates colimits of shape `J` if `F` creates the colimit of any diagram\n`K : J \u2964 C`.\n-/\nclass creates_colimits_of_shape (J : Type w) [category.{w'} J] (F : C \u2964 D) :=\n(creates_colimit : \u03a0 {K : J \u2964 C}, creates_colimit K F . tactic.apply_instance)\n\n/-- `F` creates colimits if it creates colimits of shape `J` for any small `J`. -/\n@[nolint check_univs] -- This should be used with explicit universe variables.\nclass creates_colimits_of_size (F : C \u2964 D) :=\n(creates_colimits_of_shape : \u03a0 {J : Type w} [category.{w'} J],\n  creates_colimits_of_shape J F . tactic.apply_instance)\n\n/-- `F` creates small colimits if it creates colimits of shape `J` for any small `J`. -/\nabbreviation creates_colimits (F : C \u2964 D) := creates_colimits_of_size.{v\u2082 v\u2082} F\n\nattribute [instance, priority 100] -- see Note [lower instance priority]\n  creates_limits_of_shape.creates_limit creates_limits_of_size.creates_limits_of_shape\n  creates_colimits_of_shape.creates_colimit creates_colimits_of_size.creates_colimits_of_shape\n\n/- Interface to the `creates_limit` class. -/\n\n/-- `lift_limit t` is the cone for `K` given by lifting the limit `t` for `K \u22d9 F`. -/\ndef lift_limit {K : J \u2964 C} {F : C \u2964 D} [creates_limit K F] {c : cone (K \u22d9 F)} (t : is_limit c) :\n  cone K :=\n(creates_limit.lifts c t).lifted_cone\n\n/-- The lifted cone has an image isomorphic to the original cone. -/\ndef lifted_limit_maps_to_original {K : J \u2964 C} {F : C \u2964 D}\n  [creates_limit K F] {c : cone (K \u22d9 F)} (t : is_limit c) :\n  F.map_cone (lift_limit t) \u2245 c :=\n(creates_limit.lifts c t).valid_lift\n\n/-- The lifted cone is a limit. -/\ndef lifted_limit_is_limit {K : J \u2964 C} {F : C \u2964 D}\n  [creates_limit K F] {c : cone (K \u22d9 F)} (t : is_limit c) :\n  is_limit (lift_limit t) :=\nreflects_limit.reflects (is_limit.of_iso_limit t (lifted_limit_maps_to_original t).symm)\n\n/-- If `F` creates the limit of `K` and `K \u22d9 F` has a limit, then `K` has a limit. -/\nlemma has_limit_of_created (K : J \u2964 C) (F : C \u2964 D)\n  [has_limit (K \u22d9 F)] [creates_limit K F] : has_limit K :=\nhas_limit.mk { cone := lift_limit (limit.is_limit (K \u22d9 F)),\n  is_limit := lifted_limit_is_limit _ }\n\n/--\nIf `F` creates limits of shape `J`, and `D` has limits of shape `J`, then\n`C` has limits of shape `J`.\n-/\nlemma has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape (F : C \u2964 D)\n  [has_limits_of_shape J D] [creates_limits_of_shape J F] : has_limits_of_shape J C :=\n\u27e8\u03bb G, has_limit_of_created G F\u27e9\n\n/-- If `F` creates limits, and `D` has all limits, then `C` has all limits. -/\nlemma has_limits_of_has_limits_creates_limits (F : C \u2964 D) [has_limits_of_size.{w w'} D]\n  [creates_limits_of_size.{w w'} F] : has_limits_of_size.{w w'} C :=\n\u27e8\u03bb J I, by exactI has_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape F\u27e9\n\n/- Interface to the `creates_colimit` class. -/\n\n/-- `lift_colimit t` is the cocone for `K` given by lifting the colimit `t` for `K \u22d9 F`. -/\ndef lift_colimit {K : J \u2964 C} {F : C \u2964 D} [creates_colimit K F] {c : cocone (K \u22d9 F)}\n  (t : is_colimit c) :\n  cocone K :=\n(creates_colimit.lifts c t).lifted_cocone\n\n/-- The lifted cocone has an image isomorphic to the original cocone. -/\ndef lifted_colimit_maps_to_original {K : J \u2964 C} {F : C \u2964 D}\n  [creates_colimit K F] {c : cocone (K \u22d9 F)} (t : is_colimit c) :\n  F.map_cocone (lift_colimit t) \u2245 c :=\n(creates_colimit.lifts c t).valid_lift\n\n/-- The lifted cocone is a colimit. -/\ndef lifted_colimit_is_colimit {K : J \u2964 C} {F : C \u2964 D}\n  [creates_colimit K F] {c : cocone (K \u22d9 F)} (t : is_colimit c) :\n  is_colimit (lift_colimit t) :=\nreflects_colimit.reflects (is_colimit.of_iso_colimit t (lifted_colimit_maps_to_original t).symm)\n\n/-- If `F` creates the limit of `K` and `K \u22d9 F` has a limit, then `K` has a limit. -/\nlemma has_colimit_of_created (K : J \u2964 C) (F : C \u2964 D)\n  [has_colimit (K \u22d9 F)] [creates_colimit K F] : has_colimit K :=\nhas_colimit.mk { cocone := lift_colimit (colimit.is_colimit (K \u22d9 F)),\n  is_colimit := lifted_colimit_is_colimit _ }\n\n/--\nIf `F` creates colimits of shape `J`, and `D` has colimits of shape `J`, then\n`C` has colimits of shape `J`.\n-/\nlemma has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape (F : C \u2964 D)\n  [has_colimits_of_shape J D] [creates_colimits_of_shape J F] : has_colimits_of_shape J C :=\n\u27e8\u03bb G, has_colimit_of_created G F\u27e9\n\n/-- If `F` creates colimits, and `D` has all colimits, then `C` has all colimits. -/\nlemma has_colimits_of_has_colimits_creates_colimits (F : C \u2964 D) [has_colimits_of_size.{w w'} D]\n  [creates_colimits_of_size.{w w'} F] : has_colimits_of_size.{w w'} C :=\n\u27e8\u03bb J I, by exactI has_colimits_of_shape_of_has_colimits_of_shape_creates_colimits_of_shape F\u27e9\n\n@[priority 10] instance reflects_limits_of_shape_of_creates_limits_of_shape (F : C \u2964 D)\n  [creates_limits_of_shape J F] : reflects_limits_of_shape J F := {}\n@[priority 10] instance reflects_limits_of_creates_limits (F : C \u2964 D)\n  [creates_limits_of_size.{w w'} F] : reflects_limits_of_size.{w w'} F := {}\n@[priority 10] instance reflects_colimits_of_shape_of_creates_colimits_of_shape (F : C \u2964 D)\n  [creates_colimits_of_shape J F] : reflects_colimits_of_shape J F := {}\n@[priority 10] instance reflects_colimits_of_creates_colimits (F : C \u2964 D)\n  [creates_colimits_of_size.{w w'} F] : reflects_colimits_of_size.{w w'} F := {}\n\n/--\nA helper to show a functor creates limits. In particular, if we can show\nthat for any limit cone `c` for `K \u22d9 F`, there is a lift of it which is\na limit and `F` reflects isomorphisms, then `F` creates limits.\nUsually, `F` creating limits says that _any_ lift of `c` is a limit, but\nhere we only need to show that our particular lift of `c` is a limit.\n-/\nstructure lifts_to_limit (K : J \u2964 C) (F : C \u2964 D) (c : cone (K \u22d9 F)) (t : is_limit c)\n  extends liftable_cone K F c :=\n(makes_limit : is_limit lifted_cone)\n\n/--\nA helper to show a functor creates colimits. In particular, if we can show\nthat for any limit cocone `c` for `K \u22d9 F`, there is a lift of it which is\na limit and `F` reflects isomorphisms, then `F` creates colimits.\nUsually, `F` creating colimits says that _any_ lift of `c` is a colimit, but\nhere we only need to show that our particular lift of `c` is a colimit.\n-/\nstructure lifts_to_colimit (K : J \u2964 C) (F : C \u2964 D) (c : cocone (K \u22d9 F)) (t : is_colimit c)\n  extends liftable_cocone K F c :=\n(makes_colimit : is_colimit lifted_cocone)\n\n/--\nIf `F` reflects isomorphisms and we can lift any limit cone to a limit cone,\nthen `F` creates limits.\nIn particular here we don't need to assume that F reflects limits.\n-/\ndef creates_limit_of_reflects_iso {K : J \u2964 C} {F : C \u2964 D} [reflects_isomorphisms F]\n  (h : \u03a0 c t, lifts_to_limit K F c t) :\n  creates_limit K F :=\n{ lifts := \u03bb c t, (h c t).to_liftable_cone,\n  to_reflects_limit :=\n  { reflects := \u03bb (d : cone K) (hd : is_limit (F.map_cone d)),\n    begin\n      let d' : cone K := (h (F.map_cone d) hd).to_liftable_cone.lifted_cone,\n      let i : F.map_cone d' \u2245 F.map_cone d := (h (F.map_cone d) hd).to_liftable_cone.valid_lift,\n      let hd' : is_limit d' := (h (F.map_cone d) hd).makes_limit,\n      let f : d \u27f6 d' := hd'.lift_cone_morphism d,\n      have : (cones.functoriality K F).map f = i.inv := (hd.of_iso_limit i.symm).uniq_cone_morphism,\n      haveI : is_iso ((cones.functoriality K F).map f) := (by { rw this, apply_instance }),\n      haveI : is_iso f := is_iso_of_reflects_iso f (cones.functoriality K F),\n      exact is_limit.of_iso_limit hd' (as_iso f).symm,\n    end } }\n\n/--\nWhen `F` is fully faithful, and `has_limit (K \u22d9 F)`, to show that `F` creates the limit for `K`\nit suffices to exhibit a lift of the chosen limit cone for `K \u22d9 F`.\n-/\n-- Notice however that even if the isomorphism is `iso.refl _`,\n-- this construction will insert additional identity morphisms in the cone maps,\n-- so the constructed limits may not be ideal, definitionally.\ndef creates_limit_of_fully_faithful_of_lift {K : J \u2964 C} {F : C \u2964 D}\n  [full F] [faithful F] [has_limit (K \u22d9 F)]\n  (c : cone K) (i : F.map_cone c \u2245 limit.cone (K \u22d9 F)) : creates_limit K F :=\ncreates_limit_of_reflects_iso (\u03bb c' t,\n{ lifted_cone := c,\n  valid_lift := i.trans (is_limit.unique_up_to_iso (limit.is_limit _) t),\n  makes_limit := is_limit.of_faithful F (is_limit.of_iso_limit (limit.is_limit _) i.symm)\n    (\u03bb s, F.preimage _) (\u03bb s, F.image_preimage _) })\n\n/--\nWhen `F` is fully faithful, and `has_limit (K \u22d9 F)`, to show that `F` creates the limit for `K`\nit suffices to show that the chosen limit point is in the essential image of `F`.\n-/\n-- Notice however that even if the isomorphism is `iso.refl _`,\n-- this construction will insert additional identity morphisms in the cone maps,\n-- so the constructed limits may not be ideal, definitionally.\ndef creates_limit_of_fully_faithful_of_iso {K : J \u2964 C} {F : C \u2964 D}\n  [full F] [faithful F] [has_limit (K \u22d9 F)]\n  (X : C) (i : F.obj X \u2245 limit (K \u22d9 F)) : creates_limit K F :=\ncreates_limit_of_fully_faithful_of_lift\n({ X := X,\n  \u03c0 :=\n  { app := \u03bb j, F.preimage (i.hom \u226b limit.\u03c0 (K \u22d9 F) j),\n    naturality' := \u03bb Y Z f, F.map_injective (by { dsimp, simp, erw limit.w (K \u22d9 F), }) }} : cone K)\n(by { fapply cones.ext, exact i, tidy, })\n\n/-- `F` preserves the limit of `K` if it creates the limit and `K \u22d9 F` has the limit. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_limit_of_creates_limit_and_has_limit (K : J \u2964 C) (F : C \u2964 D)\n  [creates_limit K F] [has_limit (K \u22d9 F)] :\n  preserves_limit K F :=\n{ preserves := \u03bb c t, is_limit.of_iso_limit (limit.is_limit _)\n    ((lifted_limit_maps_to_original (limit.is_limit _)).symm \u226a\u226b\n      ((cones.functoriality K F).map_iso\n        ((lifted_limit_is_limit (limit.is_limit _)).unique_up_to_iso t))) }\n\n/-- `F` preserves the limit of shape `J` if it creates these limits and `D` has them. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_limit_of_shape_of_creates_limits_of_shape_and_has_limits_of_shape (F : C \u2964 D)\n  [creates_limits_of_shape J F] [has_limits_of_shape J D] :\n  preserves_limits_of_shape J F := {}\n\n/-- `F` preserves limits if it creates limits and `D` has limits. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_limits_of_creates_limits_and_has_limits (F : C \u2964 D)\n  [creates_limits_of_size.{w w'} F]\n  [has_limits_of_size.{w w'} D] :\n  preserves_limits_of_size.{w w'} F := {}\n\n/--\nIf `F` reflects isomorphisms and we can lift any colimit cocone to a colimit cocone,\nthen `F` creates colimits.\nIn particular here we don't need to assume that F reflects colimits.\n-/\ndef creates_colimit_of_reflects_iso {K : J \u2964 C} {F : C \u2964 D} [reflects_isomorphisms F]\n  (h : \u03a0 c t, lifts_to_colimit K F c t) :\n  creates_colimit K F :=\n{ lifts := \u03bb c t, (h c t).to_liftable_cocone,\n  to_reflects_colimit :=\n  { reflects := \u03bb (d : cocone K) (hd : is_colimit (F.map_cocone d)),\n    begin\n      let d' : cocone K := (h (F.map_cocone d) hd).to_liftable_cocone.lifted_cocone,\n      let i : F.map_cocone d' \u2245 F.map_cocone d :=\n        (h (F.map_cocone d) hd).to_liftable_cocone.valid_lift,\n      let hd' : is_colimit d' := (h (F.map_cocone d) hd).makes_colimit,\n      let f : d' \u27f6 d := hd'.desc_cocone_morphism d,\n      have : (cocones.functoriality K F).map f = i.hom :=\n        (hd.of_iso_colimit i.symm).uniq_cocone_morphism,\n      haveI : is_iso ((cocones.functoriality K F).map f) := (by { rw this, apply_instance }),\n      haveI := is_iso_of_reflects_iso f (cocones.functoriality K F),\n      exact is_colimit.of_iso_colimit hd' (as_iso f),\n    end } }\n\n/--\nWhen `F` is fully faithful, and `has_colimit (K \u22d9 F)`, to show that `F` creates the colimit for `K`\nit suffices to exhibit a lift of the chosen colimit cocone for `K \u22d9 F`.\n-/\n-- Notice however that even if the isomorphism is `iso.refl _`,\n-- this construction will insert additional identity morphisms in the cocone maps,\n-- so the constructed colimits may not be ideal, definitionally.\ndef creates_colimit_of_fully_faithful_of_lift {K : J \u2964 C} {F : C \u2964 D}\n  [full F] [faithful F] [has_colimit (K \u22d9 F)]\n  (c : cocone K) (i : F.map_cocone c \u2245 colimit.cocone (K \u22d9 F)) : creates_colimit K F :=\ncreates_colimit_of_reflects_iso (\u03bb c' t,\n{ lifted_cocone := c,\n  valid_lift := i.trans (is_colimit.unique_up_to_iso (colimit.is_colimit _) t),\n  makes_colimit := is_colimit.of_faithful F\n    (is_colimit.of_iso_colimit (colimit.is_colimit _) i.symm)\n    (\u03bb s, F.preimage _) (\u03bb s, F.image_preimage _) })\n\n/--\nWhen `F` is fully faithful, and `has_colimit (K \u22d9 F)`, to show that `F` creates the colimit for `K`\nit suffices to show that the chosen colimit point is in the essential image of `F`.\n-/\n-- Notice however that even if the isomorphism is `iso.refl _`,\n-- this construction will insert additional identity morphisms in the cocone maps,\n-- so the constructed colimits may not be ideal, definitionally.\ndef creates_colimit_of_fully_faithful_of_iso {K : J \u2964 C} {F : C \u2964 D}\n  [full F] [faithful F] [has_colimit (K \u22d9 F)]\n  (X : C) (i : F.obj X \u2245 colimit (K \u22d9 F)) : creates_colimit K F :=\ncreates_colimit_of_fully_faithful_of_lift\n({ X := X,\n  \u03b9 :=\n  { app := \u03bb j, F.preimage (colimit.\u03b9 (K \u22d9 F) j \u226b i.inv : _),\n    naturality' := \u03bb Y Z f, F.map_injective\n      (by { erw category.comp_id, simp only [functor.map_comp, functor.image_preimage],\n        erw colimit.w_assoc (K \u22d9 F) }) }} : cocone K)\n(by { fapply cocones.ext, exact i, tidy, })\n\n\n/-- `F` preserves the colimit of `K` if it creates the colimit and `K \u22d9 F` has the colimit. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_colimit_of_creates_colimit_and_has_colimit (K : J \u2964 C) (F : C \u2964 D)\n  [creates_colimit K F] [has_colimit (K \u22d9 F)] :\n  preserves_colimit K F :=\n{ preserves := \u03bb c t, is_colimit.of_iso_colimit (colimit.is_colimit _)\n    ((lifted_colimit_maps_to_original (colimit.is_colimit _)).symm \u226a\u226b\n      ((cocones.functoriality K F).map_iso\n        ((lifted_colimit_is_colimit (colimit.is_colimit _)).unique_up_to_iso t))) }\n\n/-- `F` preserves the colimit of shape `J` if it creates these colimits and `D` has them. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_colimit_of_shape_of_creates_colimits_of_shape_and_has_colimits_of_shape\n  (F : C \u2964 D) [creates_colimits_of_shape J F] [has_colimits_of_shape J D] :\n  preserves_colimits_of_shape J F := {}\n\n/-- `F` preserves limits if it creates limits and `D` has limits. -/\n@[priority 100] -- see Note [lower instance priority]\ninstance preserves_colimits_of_creates_colimits_and_has_colimits (F : C \u2964 D)\n  [creates_colimits_of_size.{w w'} F] [has_colimits_of_size.{w w'} D] :\n  preserves_colimits_of_size.{w w'} F := {}\n\n/-- Transfer creation of limits along a natural isomorphism in the diagram. -/\ndef creates_limit_of_iso_diagram {K\u2081 K\u2082 : J \u2964 C} (F : C \u2964 D) (h : K\u2081 \u2245 K\u2082)\n  [creates_limit K\u2081 F] : creates_limit K\u2082 F :=\n{ lifts := \u03bb c t,\n  let t' := (is_limit.postcompose_inv_equiv (iso_whisker_right h F : _) c).symm t in\n  { lifted_cone := (cones.postcompose h.hom).obj (lift_limit t'),\n    valid_lift :=\n        F.map_cone_postcompose \u226a\u226b\n        (cones.postcompose (iso_whisker_right h F).hom).map_iso\n            (lifted_limit_maps_to_original t') \u226a\u226b\n        cones.ext (iso.refl _) (\u03bb j, by { dsimp, rw [category.assoc, \u2190F.map_comp], simp }) }\n  ..reflects_limit_of_iso_diagram F h }\n\n/-- If `F` creates the limit of `K` and `F \u2245 G`, then `G` creates the limit of `K`. -/\ndef creates_limit_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G) [creates_limit K F] :\n  creates_limit K G :=\n{ lifts := \u03bb c t,\n  { lifted_cone :=\n      lift_limit ((is_limit.postcompose_inv_equiv (iso_whisker_left K h : _) c).symm t),\n    valid_lift :=\n    begin\n      refine (is_limit.map_cone_equiv h _).unique_up_to_iso t,\n      apply is_limit.of_iso_limit _ ((lifted_limit_maps_to_original _).symm),\n      apply (is_limit.postcompose_inv_equiv _ _).symm t,\n    end },\n  to_reflects_limit := reflects_limit_of_nat_iso _ h }\n\n/-- If `F` creates limits of shape `J` and `F \u2245 G`, then `G` creates limits of shape `J`. -/\ndef creates_limits_of_shape_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G) [creates_limits_of_shape J F] :\n  creates_limits_of_shape J G :=\n{ creates_limit := \u03bb K, creates_limit_of_nat_iso h }\n\n/-- If `F` creates limits and `F \u2245 G`, then `G` creates limits. -/\ndef creates_limits_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G) [creates_limits_of_size.{w w'} F] :\n  creates_limits_of_size.{w w'} G :=\n{ creates_limits_of_shape := \u03bb J \ud835\udca5\u2081, by exactI creates_limits_of_shape_of_nat_iso h }\n\n/-- Transfer creation of colimits along a natural isomorphism in the diagram. -/\ndef creates_colimit_of_iso_diagram {K\u2081 K\u2082 : J \u2964 C} (F : C \u2964 D) (h : K\u2081 \u2245 K\u2082)\n  [creates_colimit K\u2081 F] : creates_colimit K\u2082 F :=\n{ lifts := \u03bb c t,\n  let t' := (is_colimit.precompose_hom_equiv (iso_whisker_right h F : _) c).symm t in\n  { lifted_cocone := (cocones.precompose h.inv).obj (lift_colimit t'),\n    valid_lift :=\n        F.map_cocone_precompose \u226a\u226b\n        (cocones.precompose (iso_whisker_right h F).inv).map_iso\n            (lifted_colimit_maps_to_original t') \u226a\u226b\n        cocones.ext (iso.refl _) (\u03bb j, by { dsimp, rw \u2190F.map_comp_assoc, simp }) },\n  ..reflects_colimit_of_iso_diagram F h }\n\n/-- If `F` creates the colimit of `K` and `F \u2245 G`, then `G` creates the colimit of `K`. -/\ndef creates_colimit_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G) [creates_colimit K F] :\n  creates_colimit K G :=\n{ lifts := \u03bb c t,\n  { lifted_cocone :=\n      lift_colimit ((is_colimit.precompose_hom_equiv (iso_whisker_left K h : _) c).symm t),\n    valid_lift :=\n    begin\n      refine (is_colimit.map_cocone_equiv h _).unique_up_to_iso t,\n      apply is_colimit.of_iso_colimit _ ((lifted_colimit_maps_to_original _).symm),\n      apply (is_colimit.precompose_hom_equiv _ _).symm t,\n    end },\n  to_reflects_colimit := reflects_colimit_of_nat_iso _ h }\n\n/-- If `F` creates colimits of shape `J` and `F \u2245 G`, then `G` creates colimits of shape `J`. -/\ndef creates_colimits_of_shape_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G)\n  [creates_colimits_of_shape J F] : creates_colimits_of_shape J G :=\n{ creates_colimit := \u03bb K, creates_colimit_of_nat_iso h }\n\n/-- If `F` creates colimits and `F \u2245 G`, then `G` creates colimits. -/\ndef creates_colimits_of_nat_iso {F G : C \u2964 D} (h : F \u2245 G) [creates_colimits_of_size.{w w'} F] :\n  creates_colimits_of_size.{w w'} G :=\n{ creates_colimits_of_shape := \u03bb J \ud835\udca5\u2081, by exactI creates_colimits_of_shape_of_nat_iso h }\n\n-- For the inhabited linter later.\n/-- If F creates the limit of K, any cone lifts to a limit. -/\ndef lifts_to_limit_of_creates (K : J \u2964 C) (F : C \u2964 D)\n  [creates_limit K F] (c : cone (K \u22d9 F)) (t : is_limit c) :\n  lifts_to_limit K F c t :=\n{ lifted_cone := lift_limit t,\n  valid_lift := lifted_limit_maps_to_original t,\n  makes_limit := lifted_limit_is_limit t }\n\n-- For the inhabited linter later.\n/-- If F creates the colimit of K, any cocone lifts to a colimit. -/\ndef lifts_to_colimit_of_creates (K : J \u2964 C) (F : C \u2964 D)\n  [creates_colimit K F] (c : cocone (K \u22d9 F)) (t : is_colimit c) :\n  lifts_to_colimit K F c t :=\n{ lifted_cocone := lift_colimit t,\n  valid_lift := lifted_colimit_maps_to_original t,\n  makes_colimit := lifted_colimit_is_colimit t }\n\n/-- Any cone lifts through the identity functor. -/\ndef id_lifts_cone (c : cone (K \u22d9 \ud835\udfed C)) : liftable_cone K (\ud835\udfed C) c :=\n{ lifted_cone :=\n  { X := c.X,\n    \u03c0 := c.\u03c0 \u226b K.right_unitor.hom },\n  valid_lift := cones.ext (iso.refl _) (by tidy) }\n\n/-- The identity functor creates all limits. -/\ninstance id_creates_limits : creates_limits_of_size.{w w'} (\ud835\udfed C) :=\n{ creates_limits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { creates_limit := \u03bb F, { lifts := \u03bb c t, id_lifts_cone c } } }\n\n/-- Any cocone lifts through the identity functor. -/\ndef id_lifts_cocone (c : cocone (K \u22d9 \ud835\udfed C)) : liftable_cocone K (\ud835\udfed C) c :=\n{ lifted_cocone :=\n  { X := c.X,\n    \u03b9 := K.right_unitor.inv \u226b c.\u03b9 },\n  valid_lift := cocones.ext (iso.refl _) (by tidy) }\n\n/-- The identity functor creates all colimits. -/\ninstance id_creates_colimits : creates_colimits_of_size.{w w'} (\ud835\udfed C) :=\n{ creates_colimits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { creates_colimit := \u03bb F, { lifts := \u03bb c t, id_lifts_cocone c } } }\n\n/-- Satisfy the inhabited linter -/\ninstance inhabited_liftable_cone (c : cone (K \u22d9 \ud835\udfed C)) :\n  inhabited (liftable_cone K (\ud835\udfed C) c) :=\n\u27e8id_lifts_cone c\u27e9\ninstance inhabited_liftable_cocone (c : cocone (K \u22d9 \ud835\udfed C)) :\n  inhabited (liftable_cocone K (\ud835\udfed C) c) :=\n\u27e8id_lifts_cocone c\u27e9\n\n/-- Satisfy the inhabited linter -/\ninstance inhabited_lifts_to_limit (K : J \u2964 C) (F : C \u2964 D)\n  [creates_limit K F] (c : cone (K \u22d9 F)) (t : is_limit c) :\n  inhabited (lifts_to_limit _ _ _ t) :=\n\u27e8lifts_to_limit_of_creates K F c t\u27e9\ninstance inhabited_lifts_to_colimit (K : J \u2964 C) (F : C \u2964 D)\n  [creates_colimit K F] (c : cocone (K \u22d9 F)) (t : is_colimit c) :\n  inhabited (lifts_to_colimit _ _ _ t) :=\n\u27e8lifts_to_colimit_of_creates K F c t\u27e9\n\nsection comp\n\nvariables {E : Type u\u2083} [\u2130 : category.{v\u2083} E]\nvariables (F : C \u2964 D) (G : D \u2964 E)\n\ninstance comp_creates_limit [creates_limit K F] [creates_limit (K \u22d9 F) G] :\n  creates_limit K (F \u22d9 G) :=\n{ lifts := \u03bb c t,\n  { lifted_cone := lift_limit (lifted_limit_is_limit t),\n    valid_lift := (cones.functoriality (K \u22d9 F) G).map_iso\n      (lifted_limit_maps_to_original (lifted_limit_is_limit t)) \u226a\u226b\n      (lifted_limit_maps_to_original t) } }\n\ninstance comp_creates_limits_of_shape [creates_limits_of_shape J F] [creates_limits_of_shape J G] :\n  creates_limits_of_shape J (F \u22d9 G) :=\n{ creates_limit := infer_instance }\n\ninstance comp_creates_limits [creates_limits_of_size.{w w'} F] [creates_limits_of_size.{w w'} G] :\n  creates_limits_of_size.{w w'} (F \u22d9 G) :=\n{ creates_limits_of_shape := infer_instance }\n\ninstance comp_creates_colimit [creates_colimit K F] [creates_colimit (K \u22d9 F) G] :\n  creates_colimit K (F \u22d9 G) :=\n{ lifts := \u03bb c t,\n  { lifted_cocone := lift_colimit (lifted_colimit_is_colimit t),\n    valid_lift := (cocones.functoriality (K \u22d9 F) G).map_iso\n      (lifted_colimit_maps_to_original (lifted_colimit_is_colimit t)) \u226a\u226b\n      (lifted_colimit_maps_to_original t) } }\n\ninstance comp_creates_colimits_of_shape\n  [creates_colimits_of_shape J F] [creates_colimits_of_shape J G] :\n  creates_colimits_of_shape J (F \u22d9 G) :=\n{ creates_colimit := infer_instance }\n\ninstance comp_creates_colimits [creates_colimits_of_size.{w w'} F]\n  [creates_colimits_of_size.{w w'} G] : creates_colimits_of_size.{w w'} (F \u22d9 G) :=\n{ creates_colimits_of_shape := infer_instance }\n\nend comp\n\nend creates\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/limits/creates.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.36221855258070623}}
{"text": "\nimport minimal_sub_pq\nimport sub_pq_normal\n\nuniverse u\n\nsection pq_like_normal\n\nvariables {Q : Type u} [power_quandle Q]\n\ntheorem pq_like_normal : sub_pq_normal (gen_group_sub_pq (@of_gen_group_sub_pq Q _)) :=\nbegin\n  intros x y,\n  cases y with y hy,\n  simp only [subtype.coe_mk],\n  cases hy with z hz,\n  rw hz,\n  rw set.mem_def,\n  unfold gen_group_sub_pq,\n  simp only,\n  unfold of_gen_group_sub_pq,\n  simp only,\n  unfold of_gen,\n  sorry,\n  --use ((counit x) \u25b7 z : Q),\n\nend\n\n\nend pq_like_normal\n", "meta": {"author": "torstein-vik", "repo": "power-quandle-lean", "sha": "452437602c4be2e6c5ad5f5224b068baabfdf9e1", "save_path": "github-repos/lean/torstein-vik-power-quandle-lean", "path": "github-repos/lean/torstein-vik-power-quandle-lean/power-quandle-lean-452437602c4be2e6c5ad5f5224b068baabfdf9e1/src/pq_like_normal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.766293653760418, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.36221434837164856}}
{"text": "import o_minimal.sheaf.yoneda\nimport o_minimal.order\n\nnamespace o_minimal\n\nvariables {R : Type*} [preorder R] {S : struc R} [is_definable_le S R]\n\nlemma definable_Ioo : definable S (set.Ioo : R \u2192 R \u2192 set R) :=\nbegin [defin]\n  intro a,\n  intro b,\n  intro x,\n  app, app, exact definable.and.definable _,\n  { app, app, exact (definable_iff_def_rel\u2082.mpr definable_lt').definable _,\n    var, var },\n  { app, app, exact (definable_iff_def_rel\u2082.mpr definable_lt').definable _,\n    var, var }\nend\n\nlemma definable_Iio : definable S (set.Iio : R \u2192 set R) :=\nbegin [defin]\n  intro b,\n  intro x,\n  app, app, exact (definable_iff_def_rel\u2082.mpr definable_lt').definable _,\n  var, var\nend\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/omin/def_choice/order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7662936324115012, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.36221433828036836}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta, Adam Topaz\n-/\nimport category_theory.functor.category\nimport category_theory.functor.fully_faithful\nimport category_theory.functor.reflects_isomorphisms\n\n/-!\n# Monads\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe construct the categories of monads and comonads, and their forgetful functors to endofunctors.\n\n(Note that these are the category theorist's monads, not the programmers monads.\nFor the translation, see the file `category_theory.monad.types`.)\n\nFor the fact that monads are \"just\" monoids in the category of endofunctors, see the file\n`category_theory.monad.equiv_mon`.\n-/\n\nnamespace category_theory\nopen category\n\nuniverses v\u2081 u\u2081 -- morphism levels before object levels. See note [category_theory universes].\n\nvariables (C : Type u\u2081) [category.{v\u2081} C]\n\n/--\nThe data of a monad on C consists of an endofunctor T together with natural transformations\n\u03b7 : \ud835\udfed C \u27f6 T and \u03bc : T \u22d9 T \u27f6 T satisfying three equations:\n- T \u03bc_X \u226b \u03bc_X = \u03bc_(TX) \u226b \u03bc_X (associativity)\n- \u03b7_(TX) \u226b \u03bc_X = 1_X (left unit)\n- T\u03b7_X \u226b \u03bc_X = 1_X (right unit)\n-/\nstructure monad extends C \u2964 C :=\n(\u03b7' [] : \ud835\udfed _ \u27f6 to_functor)\n(\u03bc' [] : to_functor \u22d9 to_functor \u27f6 to_functor)\n(assoc' : \u2200 X, to_functor.map (nat_trans.app \u03bc' X) \u226b \u03bc'.app _ = \u03bc'.app _ \u226b \u03bc'.app _ . obviously)\n(left_unit' : \u2200 X : C, \u03b7'.app (to_functor.obj X) \u226b \u03bc'.app _ = \ud835\udfd9 _ . obviously)\n(right_unit' : \u2200 X : C, to_functor.map (\u03b7'.app X) \u226b \u03bc'.app _ = \ud835\udfd9 _ . obviously)\n\n/--\nThe data of a comonad on C consists of an endofunctor G together with natural transformations\n\u03b5 : G \u27f6 \ud835\udfed C and \u03b4 : G \u27f6 G \u22d9 G satisfying three equations:\n- \u03b4_X \u226b G \u03b4_X = \u03b4_X \u226b \u03b4_(GX) (coassociativity)\n- \u03b4_X \u226b \u03b5_(GX) = 1_X (left counit)\n- \u03b4_X \u226b G \u03b5_X = 1_X (right counit)\n-/\nstructure comonad extends C \u2964 C :=\n(\u03b5' [] : to_functor \u27f6 \ud835\udfed _)\n(\u03b4' [] : to_functor \u27f6 to_functor \u22d9 to_functor)\n(coassoc' : \u2200 X, nat_trans.app \u03b4' _ \u226b to_functor.map (\u03b4'.app X) = \u03b4'.app _ \u226b \u03b4'.app _ . obviously)\n(left_counit' : \u2200 X : C, \u03b4'.app X \u226b \u03b5'.app (to_functor.obj X) = \ud835\udfd9 _ . obviously)\n(right_counit' : \u2200 X : C, \u03b4'.app X \u226b to_functor.map (\u03b5'.app X) = \ud835\udfd9 _ . obviously)\n\nvariables {C} (T : monad C) (G : comonad C)\n\ninstance coe_monad : has_coe (monad C) (C \u2964 C) := \u27e8\u03bb T, T.to_functor\u27e9\ninstance coe_comonad : has_coe (comonad C) (C \u2964 C) := \u27e8\u03bb G, G.to_functor\u27e9\n\n@[simp] lemma monad_to_functor_eq_coe : T.to_functor = T := rfl\n@[simp] lemma comonad_to_functor_eq_coe : G.to_functor = G := rfl\n\n/-- The unit for the monad `T`. -/\ndef monad.\u03b7 : \ud835\udfed _ \u27f6 (T : C \u2964 C) := T.\u03b7'\n/-- The multiplication for the monad `T`. -/\ndef monad.\u03bc : (T : C \u2964 C) \u22d9 (T : C \u2964 C) \u27f6 T := T.\u03bc'\n\n/-- The counit for the comonad `G`. -/\ndef comonad.\u03b5 : (G : C \u2964 C) \u27f6 \ud835\udfed _  := G.\u03b5'\n/-- The comultiplication for the comonad `G`. -/\ndef comonad.\u03b4 : (G : C \u2964 C) \u27f6 (G : C \u2964 C) \u22d9 G := G.\u03b4'\n\n/-- A custom simps projection for the functor part of a monad, as a coercion. -/\ndef monad.simps.coe := (T : C \u2964 C)\n/-- A custom simps projection for the unit of a monad, in simp normal form. -/\ndef monad.simps.\u03b7 : \ud835\udfed _ \u27f6 (T : C \u2964 C) := T.\u03b7\n/-- A custom simps projection for the multiplication of a monad, in simp normal form. -/\ndef monad.simps.\u03bc : (T : C \u2964 C) \u22d9 (T : C \u2964 C) \u27f6 (T : C \u2964 C) := T.\u03bc\n\n/-- A custom simps projection for the functor part of a comonad, as a coercion. -/\ndef comonad.simps.coe := (G : C \u2964 C)\n/-- A custom simps projection for the counit of a comonad, in simp normal form. -/\ndef comonad.simps.\u03b5 : (G : C \u2964 C) \u27f6 \ud835\udfed _ := G.\u03b5\n/-- A custom simps projection for the comultiplication of a comonad, in simp normal form. -/\ndef comonad.simps.\u03b4 : (G : C \u2964 C) \u27f6 (G : C \u2964 C) \u22d9 (G : C \u2964 C) := G.\u03b4\n\ninitialize_simps_projections category_theory.monad (to_functor \u2192 coe, \u03b7' \u2192 \u03b7, \u03bc' \u2192 \u03bc)\ninitialize_simps_projections category_theory.comonad (to_functor \u2192 coe, \u03b5' \u2192 \u03b5, \u03b4' \u2192 \u03b4)\n\n@[reassoc]\nlemma monad.assoc (T : monad C) (X : C) :\n  (T : C \u2964 C).map (T.\u03bc.app X) \u226b T.\u03bc.app _ = T.\u03bc.app _ \u226b T.\u03bc.app _ :=\nT.assoc' X\n\n@[simp, reassoc] lemma monad.left_unit (T : monad C) (X : C) :\n  T.\u03b7.app ((T : C \u2964 C).obj X) \u226b T.\u03bc.app X = \ud835\udfd9 ((T : C \u2964 C).obj X) :=\nT.left_unit' X\n\n@[simp, reassoc] lemma monad.right_unit (T : monad C) (X : C) :\n  (T : C \u2964 C).map (T.\u03b7.app X) \u226b T.\u03bc.app X = \ud835\udfd9 ((T : C \u2964 C).obj X) :=\nT.right_unit' X\n\n@[reassoc]\nlemma comonad.coassoc (G : comonad C) (X : C) :\n  G.\u03b4.app _ \u226b (G : C \u2964 C).map (G.\u03b4.app X) = G.\u03b4.app _ \u226b G.\u03b4.app _ :=\nG.coassoc' X\n\n@[simp, reassoc] lemma comonad.left_counit (G : comonad C) (X : C) :\n  G.\u03b4.app X \u226b G.\u03b5.app ((G : C \u2964 C).obj X) = \ud835\udfd9 ((G : C \u2964 C).obj X) :=\nG.left_counit' X\n\n@[simp, reassoc] lemma comonad.right_counit (G : comonad C) (X : C) :\n  G.\u03b4.app X \u226b (G : C \u2964 C).map (G.\u03b5.app X) = \ud835\udfd9 ((G : C \u2964 C).obj X) :=\nG.right_counit' X\n\n/-- A morphism of monads is a natural transformation compatible with \u03b7 and \u03bc. -/\n@[ext]\nstructure monad_hom (T\u2081 T\u2082 : monad C) extends nat_trans (T\u2081 : C \u2964 C) T\u2082 :=\n(app_\u03b7' : \u2200 X, T\u2081.\u03b7.app X \u226b app X = T\u2082.\u03b7.app X . obviously)\n(app_\u03bc' : \u2200 X, T\u2081.\u03bc.app X \u226b app X = ((T\u2081 : C \u2964 C).map (app X) \u226b app _) \u226b T\u2082.\u03bc.app X . obviously)\n\n/-- A morphism of comonads is a natural transformation compatible with \u03b5 and \u03b4. -/\n@[ext]\nstructure comonad_hom (M N : comonad C) extends nat_trans (M : C \u2964 C) N :=\n(app_\u03b5' : \u2200 X, app X \u226b N.\u03b5.app X = M.\u03b5.app X . obviously)\n(app_\u03b4' : \u2200 X, app X \u226b N.\u03b4.app X = M.\u03b4.app X \u226b app _ \u226b (N : C \u2964 C).map (app X) . obviously)\n\nrestate_axiom monad_hom.app_\u03b7'\nrestate_axiom monad_hom.app_\u03bc'\nattribute [simp, reassoc] monad_hom.app_\u03b7 monad_hom.app_\u03bc\n\nrestate_axiom comonad_hom.app_\u03b5'\nrestate_axiom comonad_hom.app_\u03b4'\nattribute [simp, reassoc] comonad_hom.app_\u03b5 comonad_hom.app_\u03b4\n\ninstance : category (monad C) :=\n{ hom := monad_hom,\n  id := \u03bb M, { to_nat_trans := \ud835\udfd9 (M : C \u2964 C) },\n  comp := \u03bb _ _ _ f g,\n  { to_nat_trans :=\n    { app := \u03bb X, f.app X \u226b g.app X,\n      naturality' := \u03bb X Y h, by rw [assoc, f.1.naturality_assoc, g.1.naturality] } },\n  id_comp' := \u03bb _ _ _, by {ext, apply id_comp},\n  comp_id' := \u03bb _ _ _, by {ext, apply comp_id},\n  assoc' := \u03bb _ _ _ _ _ _ _, by {ext, apply assoc} }\n\ninstance : category (comonad C) :=\n{ hom := comonad_hom,\n  id := \u03bb M, { to_nat_trans := \ud835\udfd9 (M : C \u2964 C) },\n  comp := \u03bb _ _ _ f g,\n  { to_nat_trans :=\n    { app := \u03bb X, f.app X \u226b g.app X,\n      naturality' := \u03bb X Y h, by rw [assoc, f.1.naturality_assoc, g.1.naturality] } },\n  id_comp' := \u03bb _ _ _, by {ext, apply id_comp},\n  comp_id' := \u03bb _ _ _, by {ext, apply comp_id},\n  assoc' := \u03bb _ _ _ _ _ _ _, by {ext, apply assoc} }\n\ninstance {T : monad C} : inhabited (monad_hom T T) := \u27e8\ud835\udfd9 T\u27e9\n\n@[simp] \n\ninstance {G : comonad C} : inhabited (comonad_hom G G) := \u27e8\ud835\udfd9 G\u27e9\n\n@[simp] lemma comonad_hom.id_to_nat_trans (T : comonad C) :\n  (\ud835\udfd9 T : T \u27f6 T).to_nat_trans = \ud835\udfd9 (T : C \u2964 C) :=\nrfl\n@[simp] lemma comp_to_nat_trans {T\u2081 T\u2082 T\u2083 : comonad C} (f : T\u2081 \u27f6 T\u2082) (g : T\u2082 \u27f6 T\u2083) :\n  (f \u226b g).to_nat_trans =\n    ((f.to_nat_trans : _ \u27f6 (T\u2082 : C \u2964 C)) \u226b g.to_nat_trans : (T\u2081 : C \u2964 C) \u27f6 T\u2083) :=\nrfl\n\n/-- Construct a monad isomorphism from a natural isomorphism of functors where the forward\ndirection is a monad morphism. -/\n@[simps]\ndef monad_iso.mk {M N : monad C} (f : (M : C \u2964 C) \u2245 N) (f_\u03b7 f_\u03bc) :\n  M \u2245 N :=\n{ hom := { to_nat_trans := f.hom, app_\u03b7' := f_\u03b7, app_\u03bc' := f_\u03bc },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_\u03b7' := \u03bb X, by simp [\u2190f_\u03b7],\n    app_\u03bc' := \u03bb X,\n    begin\n      rw \u2190nat_iso.cancel_nat_iso_hom_right f,\n      simp only [nat_trans.naturality, iso.inv_hom_id_app, assoc, comp_id, f_\u03bc,\n        nat_trans.naturality_assoc, iso.inv_hom_id_app_assoc, \u2190functor.map_comp_assoc],\n      simp,\n    end } }\n\n/-- Construct a comonad isomorphism from a natural isomorphism of functors where the forward\ndirection is a comonad morphism. -/\n@[simps]\ndef comonad_iso.mk {M N : comonad C} (f : (M : C \u2964 C) \u2245 N) (f_\u03b5 f_\u03b4) :\n  M \u2245 N :=\n{ hom := { to_nat_trans := f.hom, app_\u03b5' := f_\u03b5, app_\u03b4' := f_\u03b4 },\n  inv :=\n  { to_nat_trans := f.inv,\n    app_\u03b5' := \u03bb X, by simp [\u2190f_\u03b5],\n    app_\u03b4' := \u03bb X,\n    begin\n      rw \u2190nat_iso.cancel_nat_iso_hom_left f,\n      simp only [reassoc_of (f_\u03b4 X), iso.hom_inv_id_app_assoc, nat_trans.naturality_assoc],\n      rw [\u2190functor.map_comp, iso.hom_inv_id_app, functor.map_id],\n      apply (comp_id _).symm\n    end } }\n\nvariable (C)\n\n/--\nThe forgetful functor from the category of monads to the category of endofunctors.\n-/\n@[simps]\ndef monad_to_functor : monad C \u2964 (C \u2964 C) :=\n{ obj := \u03bb T, T,\n  map := \u03bb M N f, f.to_nat_trans }\n\ninstance : faithful (monad_to_functor C) := {}.\n\n@[simp]\nlemma monad_to_functor_map_iso_monad_iso_mk {M N : monad C} (f : (M : C \u2964 C) \u2245 N) (f_\u03b7 f_\u03bc) :\n  (monad_to_functor _).map_iso (monad_iso.mk f f_\u03b7 f_\u03bc) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (monad_to_functor C) :=\n{ reflects := \u03bb M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (monad_iso.mk (as_iso ((monad_to_functor C).map f)) f.app_\u03b7 f.app_\u03bc),\n    ext; refl,\n  end }\n\n/--\nThe forgetful functor from the category of comonads to the category of endofunctors.\n-/\n@[simps]\ndef comonad_to_functor : comonad C \u2964 (C \u2964 C) :=\n{ obj := \u03bb G, G,\n  map := \u03bb M N f, f.to_nat_trans }\n\ninstance : faithful (comonad_to_functor C) := {}.\n\n@[simp]\nlemma comonad_to_functor_map_iso_comonad_iso_mk {M N : comonad C} (f : (M : C \u2964 C) \u2245 N) (f_\u03b5 f_\u03b4) :\n  (comonad_to_functor _).map_iso (comonad_iso.mk f f_\u03b5 f_\u03b4) = f :=\nby { ext, refl }\n\ninstance : reflects_isomorphisms (comonad_to_functor C) :=\n{ reflects := \u03bb M N f i,\n  begin\n    resetI,\n    convert is_iso.of_iso (comonad_iso.mk (as_iso ((comonad_to_functor C).map f)) f.app_\u03b5 f.app_\u03b4),\n    ext; refl,\n  end }\n\nvariable {C}\n\n/--\nAn isomorphism of monads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef monad_iso.to_nat_iso {M N : monad C} (h : M \u2245 N) : (M : C \u2964 C) \u2245 N :=\n(monad_to_functor C).map_iso h\n\n/--\nAn isomorphism of comonads gives a natural isomorphism of the underlying functors.\n-/\n@[simps {rhs_md := semireducible}]\ndef comonad_iso.to_nat_iso {M N : comonad C} (h : M \u2245 N) : (M : C \u2964 C) \u2245 N :=\n(comonad_to_functor C).map_iso h\n\nvariable (C)\n\nnamespace monad\n\n/-- The identity monad. -/\n@[simps]\ndef id : monad C :=\n{ to_functor := \ud835\udfed C,\n  \u03b7' := \ud835\udfd9 (\ud835\udfed C),\n  \u03bc' := \ud835\udfd9 (\ud835\udfed C) }\n\ninstance : inhabited (monad C) := \u27e8monad.id C\u27e9\n\nend monad\n\nnamespace comonad\n\n/-- The identity comonad. -/\n@[simps]\ndef id : comonad C :=\n{ to_functor := \ud835\udfed _,\n  \u03b5' := \ud835\udfd9 (\ud835\udfed C),\n  \u03b4' := \ud835\udfd9 (\ud835\udfed C) }\n\ninstance : inhabited (comonad C) := \u27e8comonad.id C\u27e9\n\nend comonad\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monad/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.362168216147288}}
{"text": "theorem bad : \u2200 (m n : Nat), (if m = n then Ordering.eq else Ordering.gt) = Ordering.lt \u2192 False := by\n  intros m n\n  cases (Nat.decEq m n) with -- an error as expected: \"alternative `isFalse` has not bee provided\"\n  | isTrue h =>\n    set_option trace.Meta.Tactic.simp true in\n    simp [h]\n\ntheorem bad' : \u2200 (m n : Nat), (if m = n then Ordering.eq else Ordering.gt) = Ordering.lt \u2192 False := by\n  intros m n\n  cases (Nat.decEq m n) with\n  | isTrue h =>\n    simp [h]\n  | isFalse h =>\n    simp [h]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/1079.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863695, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.36216821614728795}}
{"text": "import thm95.double_complex\nimport pseudo_normed_group.homotopy\nimport rescale.FiltrationPow\n\n/-!\n# A complex canonically isomorphic to `row 1` of the double complex\n\nWe have\n```\nlemma double_complex.row_one :\n  (double_complex BD \u03ba r r' V \u039b M N).row 1 =\n  BD.system \u03ba r V r' (Hom ((cosimplicial \u039b N).obj (mk 0)) M) := rfl\n```\n\nWe want to \"rewrite\" this row in such a way that it is the target\nof the homotopies that will be constructed formally from `BD.homotopy`.\n\nConcretely, we want:\n```\n(((data.mul N).obj BD.data).system (rescale_constants \u03ba N) r V r').obj (op (Hom \u039b M)) \u2245\n  (thm95.double_complex BD.data \u03ba r r' V \u039b M N).row 1\n```\n\nThis means that we need to multiply `BD` by `N`,\nand then take the system associated with `rescale N (Hom \u039b M)`.\n\nWe need the following isomorphisms\n\n* `BD.system M^N = (BD.mul N).system M`\n* `Hom (rescale N (\u039b^N)) M = (rescale N (Hom \u039b M)^N` (2 steps?)\n* `(cosimplicial \u039b N).obj (mk 0) = rescale N (\u039b^N)`\n\n-/\n\nuniverses u v\n\nnoncomputable theory\n\nopen_locale nnreal big_operators kronecker\n\nlocal attribute [instance] type_pow\n\nlocal attribute [reducible] CLCFPTinv\u2082 CLCFPTinv\u2082.res\n  breen_deligne.universal_map.eval_CLCFPTinv\u2082\n\n-- move this\nnamespace category_theory\n\nnamespace arrow\n\nvariables {C : Type*} [category C] {X Y Z X' Y' Z' : C}\nvariables (f : X \u27f6 Y) (g : Y \u27f6 Z) (f' : X' \u27f6 Y') (g' : Y' \u27f6 Z')\n\nlemma mk_comp_congr (hf : arrow.mk f = arrow.mk f') (hg : arrow.mk g = arrow.mk g') :\n  arrow.mk (f \u226b g) = arrow.mk (f' \u226b g') :=\nby { cases hf, cases hg, refl }\n\nend arrow\n\nend category_theory\n\nopen category_theory\n\nsection rescale\n\nvariables {BD : breen_deligne.data}\nvariables (\u03ba : \u2115 \u2192 \u211d\u22650)\nvariables [BD.suitable \u03ba]\nvariables (r : \u211d\u22650) (V : SemiNormedGroup.{v}) [normed_with_aut r V] [fact (0 < r)]\nvariables {r' : \u211d\u22650} [fact (0 < r')] [fact (r' \u2264 1)] (c : \u211d\u22650)\nvariables (M : ProFiltPseuNormGrpWithTinv.{u} r')\n\n-- move this\ninstance rescale_constants_suitable (N : \u211d\u22650) :\n  BD.suitable (rescale_constants \u03ba N) :=\nby { delta rescale_constants, apply_instance }\n\nvariables (BD)\n\nopen breen_deligne opposite ProFiltPseuNormGrpWithTinv (of)\n\nsection\n\ndef FiltrationPow_rescale_iso (n : \u2115) (N : \u211d\u22650) :\n  ((Filtration r').obj c).obj ((ProFiltPseuNormGrpWithTinv.Pow r' n).obj (of r' (rescale N M))) \u2245\n    ((Filtration r').obj (c * N\u207b\u00b9)).obj ((ProFiltPseuNormGrpWithTinv.Pow r' n).obj M) :=\niso.refl _\n\ndef complex_rescale_iso (N : \u211d\u22650) :\n  (BD.complex (rescale_constants \u03ba N) r V r' c).obj (op M) \u2245\n  (BD.complex \u03ba r V r' c).obj (op $ of r' $ rescale N M) :=\nhomological_complex.hom.iso_of_components\nbegin\n  intro i,\n  refine CLCTinv.map_iso r V _ _ _ _ _ _ _ _,\n  { refine (FiltrationPow_rescale_iso _ _ _ _ \u226a\u226b\n      Filtration_cast_eq r' _ _ (mul_assoc c (\u03ba i) (N\u207b\u00b9)) _).op, },\n  { refine (FiltrationPow_rescale_iso _ _ _ _ \u226a\u226b\n      Filtration_cast_eq _ (r' * (c * \u03ba i) * N\u207b\u00b9) (r' * (c * (\u03ba i * N\u207b\u00b9)))\n      (by simp only [mul_assoc]) _).op, },\n  { refl },\n  { refl }\nend\nbegin\n  intros i j hij,\n  apply arrow.mk_injective,\n  dsimp only [data.complex_obj_d, universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv\u2082,\n    _root_.id, SemiNormedGroup.equalizer.map_nat_app, CLCTinv.map_iso_hom, CLCTinv.map, unop_op,\n    Filtration_cast_eq, iso.op_hom],\n  simp only [SemiNormedGroup.equalizer.map_comp_map, universal_map.eval_CLCFP_rescale,\n    \u2190 CLCFP.res_def', nat_iso.app_hom, functor.map_iso_hom, Filtration_map_app,\n    FiltrationPow_rescale_iso, iso.refl_trans],\n  apply SemiNormedGroup.equalizer.map_congr,\n  { have := @universal_map.res_comp_eval_CLCFP V r'\n      (c * (\u03ba i * N\u207b\u00b9)) (c * \u03ba i * N\u207b\u00b9) (c * (\u03ba j * N\u207b\u00b9)) (c * \u03ba j * N\u207b\u00b9)\n      (BD.X j) (BD.X i) (BD.d j i) \u27e8(mul_assoc _ _ _).le\u27e9 _ _ \u27e8(mul_assoc _ _ _).le\u27e9,\n    replace := nat_trans.congr_app this.symm (op M),\n    replace := congr_arg arrow.mk this,\n    refine (this.trans _).symm,\n    apply arrow.mk_comp_congr, { refl }, { rw universal_map.eval_CLCFP_rescale } },\n  { have := @universal_map.res_comp_eval_CLCFP V r'\n      (r' * (c * (\u03ba i * N\u207b\u00b9))) (r' * (c * \u03ba i) * N\u207b\u00b9) (r' * (c * (\u03ba j * N\u207b\u00b9))) (r' * (c * \u03ba j) * N\u207b\u00b9)\n      (BD.X j) (BD.X i) (BD.d j i) \u27e8le_of_eq $ by simp only [mul_assoc]\u27e9 _ _ \u27e8le_of_eq $ by simp only [mul_assoc]\u27e9,\n    replace := nat_trans.congr_app this.symm (op M),\n    replace := congr_arg arrow.mk this,\n    refine (this.trans _).symm,\n    apply arrow.mk_comp_congr, { refl }, { rw universal_map.eval_CLCFP_rescale } },\n  any_goals { refl },\nend\n.\n\nnoncomputable\ndef system_rescale_iso (N : \u211d\u22650) :\n  (BD.system (rescale_constants \u03ba N) r V r').obj (op M) \u2245\n  (BD.system \u03ba r V r').obj (op $ of r' $ rescale N M) :=\nnat_iso.of_components (\u03bb c, complex_rescale_iso BD \u03ba r V c.unop _ _)\nbegin\n  intros c\u2081 c\u2082 h,\n  ext i : 2,\n  apply arrow.mk_injective,\n  erw [homological_complex.comp_f, homological_complex.comp_f],\n  dsimp only [data.system_obj, CLCFPTinv\u2082.res, complex_rescale_iso,\n    homological_complex.hom.iso_of_components, CLCTinv.map_iso_hom, CLCTinv.map_nat_app],\n  simp only [CLCTinv.map_comp_map],\n  refl,\nend\n\nend\n\nend rescale\n\nnamespace thm95\n\nopen breen_deligne polyhedral_lattice opposite\n\nvariables (BD : breen_deligne.data) (\u03ba : \u2115 \u2192 \u211d\u22650) [BD.suitable \u03ba]\nvariables (r : \u211d\u22650) (V : SemiNormedGroup.{v}) [normed_with_aut r V] [fact (0 < r)]\nvariables {r' : \u211d\u22650} [fact (0 < r')] [fact (r < r')] [fact (r' \u2264 1)] (c : \u211d\u22650)\n\nsection\n\nvariables {m n : \u2115} (\u03d5 : universal_map m n) (g : basic_universal_map m n)\nvariables (c\u2081 c\u2082 : \u211d\u22650) (N : \u2115) [fact (0 < N)]\nvariables (M : ProFiltPseuNormGrpWithTinv.{u} r')\n\nlemma eval_FP_mul [g.suitable c\u2082 c\u2081] :\n  (CLC V).map (FiltrationPow.mul_iso.{u u} r' c\u2081 M N n).hom.op \u226b\n    (CLC V).map ((basic_universal_map.eval_FP r' c\u2082 c\u2081 g).app (ProFiltPseuNormGrpWithTinv.of r' (M ^ N))).op =\n  (CLC V).map ((basic_universal_map.eval_FP r' c\u2082 c\u2081 ((basic_universal_map.mul N) g)).app M).op \u226b\n    (CLC V).map (FiltrationPow.mul_iso.{u u} r' c\u2082 M N m).hom.op :=\nbegin\n  simp only [\u2190 (CLC V).map_comp, \u2190 op_comp], congr' 2,\n  rw [\u2190 iso.inv_comp_eq, \u2190 category.assoc, \u2190 iso.eq_comp_inv],\n  exact basic_universal_map.mul_iso_eval_FP r' c\u2081 c\u2082 g N M\nend\n\nlemma eval_CLCFP_mul [\u03d5.suitable c\u2082 c\u2081] {_ : (universal_map.mul N \u03d5).suitable c\u2082 c\u2081} :\n  (((universal_map.mul N \u03d5).eval_CLCFP V r' c\u2081 c\u2082).app (op M) \u226b\n    (CLC V).map (FiltrationPow.mul_iso.{u u} r' c\u2082 M N m).op.hom) =\n  ((CLC V).map (FiltrationPow.mul_iso.{u u} r' c\u2081 M N n).op.hom \u226b\n   ((\u03d5.eval_CLCFP V r' c\u2081 c\u2082).app (op (ProFiltPseuNormGrpWithTinv.of r' (M ^ N))) : _)) :=\nbegin\n  dsimp only [universal_map.eval_CLCFP, whisker_right_app],\n  simp only [universal_map.eval_LCFP_eq_eval_LCFP', universal_map.eval_LCFP',\n    \u2190 nat_trans.app_hom_apply, \u2190 functor.map_add_hom_apply,\n    add_monoid_hom.map_sum, add_monoid_hom.map_zsmul],\n  rw [preadditive.sum_comp, preadditive.comp_sum],\n  symmetry, have hN : 0 < N := fact.out _,\n  apply finset.sum_bij (\u03bb g hg, basic_universal_map.mul N g),\n  { intros g hg, rw universal_map.mem_support_mul N hN, refine \u27e8g, hg, rfl\u27e9 },\n  { intros g hg,\n    simp only [preadditive.comp_zsmul, preadditive.zsmul_comp, universal_map.coeff_mul N hN],\n    congr' 1,\n    have : g.suitable c\u2082 c\u2081 := universal_map.suitable_of_mem_support _ _ _ _ hg, resetI,\n    rw [\u2190 basic_universal_map.eval_LCFP_eq_eval_LCFP' _ _ _ _ g this,\n        \u2190 basic_universal_map.eval_LCFP_eq_eval_LCFP'],\n    swap, { apply basic_universal_map.mul_suitable },\n    dsimp only [basic_universal_map.eval_LCFP, nat_trans.app_hom_apply, functor.map_add_hom_apply,\n      whisker_right_app, nat_trans.op_app, unop_op],\n    simp only [\u2190 functor.comp_map],\n    apply eval_FP_mul },\n  { intros g\u2081 g\u2082 hg\u2081 hg\u2082 H, exact basic_universal_map.mul_injective N hN H },\n  { intro g, rw universal_map.mem_support_mul N hN, rintro \u27e8g', h1, h2\u27e9, exact \u27e8g', h1, h2\u27e9 }\nend\n\ndef mul_complex_iso (c : \u211d\u22650) :\n  (((data.mul N).obj BD).complex \u03ba r V r' c).obj (op M) \u2245\n  (BD.complex \u03ba r V r' c).obj (op (ProFiltPseuNormGrpWithTinv.of r' $ M^N)) :=\nhomological_complex.hom.iso_of_components\nbegin\n  intro i,\n  refine CLCTinv.map_iso r V _ _ _ _ _ _ _ _,\n  { exact (FiltrationPow.mul_iso.{u u} r' (c * \u03ba i) M N (BD.X i)).op },\n  { exact (FiltrationPow.mul_iso.{u u} r' (r' * (c * \u03ba i)) M N (BD.X i)).op },\n  { refl },\n  { refl }\nend\nbegin\n  intros i j hij,\n  apply arrow.mk_injective,\n  dsimp only [data.complex_obj_d, universal_map.eval_CLCFPTinv, universal_map.eval_CLCFPTinv\u2082,\n    _root_.id, SemiNormedGroup.equalizer.map_nat_app, CLCTinv.map_iso_hom, CLCTinv.map,\n    data.mul_obj_d],\n  simp only [SemiNormedGroup.equalizer.map_comp_map],\n  apply SemiNormedGroup.equalizer.map_congr,\n  { rw eval_CLCFP_mul },\n  { rw eval_CLCFP_mul },\n  all_goals { refl }\nend\n\nend\n\ndef mul_system_iso (N : \u2115) [fact (0 < N)] (M : ProFiltPseuNormGrpWithTinv.{u} r') :\n  (((data.mul N).obj BD).system \u03ba r V r').obj (op M) \u2245\n  (BD.system \u03ba r V r').obj (op (ProFiltPseuNormGrpWithTinv.of r' $ M^N)) :=\nnat_iso.of_components (\u03bb c, mul_complex_iso BD \u03ba r V N M c.unop)\nbegin\n  intros c\u2081 c\u2082 hc,\n  ext i : 2,\n  apply arrow.mk_injective,\n  erw [homological_complex.comp_f, homological_complex.comp_f],\n  dsimp only [data.system_obj, CLCFPTinv\u2082.res, mul_complex_iso,\n    homological_complex.hom.iso_of_components, CLCTinv.map_iso_hom, CLCTinv.map_nat_app],\n  simp only [CLCTinv.map_comp_map],\n  refl,\nend\n\ndef mul_rescale_iso_row_one\n  (N : \u2115) [fact (0 < N)] (N' : \u211d\u22650) (h : N' = N)\n  (\u039b : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r') :\n  (((data.mul N).obj BD).system (rescale_constants \u03ba N') r V r').obj (op (Hom \u039b M)) \u2245\n    ((thm95.double_complex BD \u03ba r r' V \u039b M N).row 1) :=\n(mul_system_iso _ _ r V N _) \u226a\u226b\n(system_rescale_iso _ \u03ba r V _ _) \u226a\u226b\n((BD.system \u03ba r V r').map_iso $\n  (PolyhedralLattice.Hom_cosimplicial_zero_iso \u039b N r' M N' h).op)\n\nlemma mul_rescale_iso_row_one_strict\n  (N : \u2115) [fact (0 < N)] (N' : \u211d\u22650) (h : N' = N)\n  (\u039b : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r')\n  (c : \u211d\u22650) (i : \u2115)\n  (x : (((data.mul N).obj BD).system (rescale_constants \u03ba N') r V r').obj (op (Hom \u039b M)) c i) :\n  \u2225(mul_rescale_iso_row_one BD \u03ba r V N N' h \u039b M).hom x\u2225 = \u2225x\u2225 :=\nbegin\n  apply normed_add_group_hom.norm_eq_of_isometry,\n  refine isometry.comp (isometry.comp _ _) _,\n  { apply data.system_map_iso_isometry, },\n  { dsimp only, apply CLCTinv.map_iso_isometry, },\n  { apply CLCTinv.map_iso_isometry, },\nend\n.\n\nlemma quux (N : \u2115) [fact (0 < N)] (M : ProFiltPseuNormGrpWithTinv.{u} r') (c\u2081 c\u2082 : \u211d\u22650) (i : \u2115)\n  [(universal_map.sum i N).suitable c\u2082 c\u2081] {_ : ((finset.univ : finset (fin N)).sum (basic_universal_map.proj i)).suitable c\u2082 c\u2081} :\n  (universal_map.eval_CLCFP V r' c\u2081 c\u2082 (universal_map.sum i N)).app (op M) =\n  (CLC V).map ((basic_universal_map.eval_FP r' c\u2082 c\u2081 ((finset.univ : finset (fin N)).sum (basic_universal_map.proj i))).app M).op :=\nby { dsimp only [universal_map.sum], rw [universal_map.eval_CLCFP_of], refl }\n\nlemma bar (N : \u2115) [fact (0 < N)] (\u039b : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r')\n  (c\u2081 c\u2082 : \u211d\u22650) (hc : c\u2081 * N\u207b\u00b9 = c\u2082) (n : \u2115)\n  {_ : ((finset.univ : finset (fin N)).sum (basic_universal_map.proj n)).suitable c\u2082 c\u2081} :\n  (FiltrationPow_rescale_iso c\u2081 (ProFiltPseuNormGrpWithTinv.of r' ((Hom \u039b M) ^ N)) n N \u226a\u226b\n     ((Filtration r').map_iso (eq_to_iso hc)).app\n       ((ProFiltPseuNormGrpWithTinv.Pow r' n).obj (ProFiltPseuNormGrpWithTinv.of r' ((Hom \u039b M) ^ N)))).inv \u226b\n  ((Filtration r').obj c\u2081).map ((ProFiltPseuNormGrpWithTinv.Pow r' n).map (\u039b.Hom_sum N r' M)) =\n  (FiltrationPow.mul_iso.{u u} r' c\u2082 (Hom.{u u} \u039b M) N n).hom \u226b\n    (basic_universal_map.eval_FP.{u} r' c\u2082 c\u2081 (finset.univ.sum (basic_universal_map.proj n))).app (Hom.{u u} \u039b M) :=\nbegin\n  dsimp only [FiltrationPow_rescale_iso], rw [iso.refl_trans],\n  dsimp only [FiltrationPow.mul_iso_hom, nat_iso.app_inv, functor.map_iso_inv,\n    Pow_obj, ProFiltPseuNormGrpWithTinv.coe_of, Filtration_map_app],\n  ext x i : 3,\n  erw [comp_apply, comp_apply],\n  dsimp only [Filtration_obj_map_apply, Pow_Pow_X_hom_apply, continuous_map.coe_mk,\n    comphaus_filtered_pseudo_normed_group_with_Tinv_hom.level_coe, subtype.coe_mk,\n    Filtration.cast_le_apply, pseudo_normed_group.coe_cast_le,\n    basic_universal_map.eval_FP, basic_universal_map.eval_png\u2080,\n    ProFiltPseuNormGrpWithTinv.Pow_map,\n    profinitely_filtered_pseudo_normed_group_with_Tinv.pi_map_to_fun],\n  rw [\u2190 comphaus_filtered_pseudo_normed_group_hom.coe_to_add_monoid_hom,\n    \u2190 comphaus_filtered_pseudo_normed_group_hom.to_add_monoid_hom_hom_apply],\n  simp only [PolyhedralLattice.Hom_sum_apply, add_monoid_hom.map_sum,\n    add_monoid_hom.finset_sum_apply, finset.sum_apply],\n  apply fintype.sum_congr,\n  intro j,\n  simp only [comphaus_filtered_pseudo_normed_group_hom.coe_to_add_monoid_hom,\n    comphaus_filtered_pseudo_normed_group_hom.to_add_monoid_hom_hom_apply,\n    basic_universal_map.eval_png_apply, ProFiltPseuNormGrpWithTinv.Pow_Pow_X_hom_to_fun,\n    ProFiltPseuNormGrpWithTinv.Pow_Pow_X_equiv_symm_apply,\n    equiv.inv_fun_as_coe, equiv.symm_symm, equiv.trans_apply, equiv.symm_trans_apply,\n    equiv.arrow_congr_apply, function.comp, equiv.refl_apply, equiv.curry_symm_apply,\n    function.uncurry, equiv.prod_comm_symm, equiv.prod_comm_apply, prod.fst_swap, prod.snd_swap],\n  rw [\u2190 fin_prod_fin_equiv.sum_comp], swap, { apply_instance },\n  simp only [basic_universal_map.proj,\n    matrix.reindex_linear_equiv_apply, matrix.reindex_apply, matrix.submatrix_apply,\n    equiv.punit_prod_symm_apply, matrix.kronecker, matrix.one_apply,\n    basic_universal_map.proj_aux, equiv.symm_apply_apply,\n    boole_mul, \u2190 ite_and, @eq_comm _ i, boole_mul, matrix.kronecker_map, subtype.val_eq_coe],\n  simp_rw [ite_smul, one_smul, zero_smul],\n  convert (finset.sum_ite_eq' finset.univ (j, i) (\u03bb p, x.val p.2 p.1)).symm using 2,\n  { simp only [finset.mem_univ, if_true, subtype.val_eq_coe] },\n  { ext \u27e8a, b\u27e9,\n    split_ifs,\n    any_goals {refl},\n    all_goals { rw [\u2190 prod.mk.inj_iff, prod.mk.eta] at h, tauto } },\nend\n\nlemma foo (N : \u2115) [fact (0 < N)] (\u039b : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r')\n  (c\u2081 c\u2082 : \u211d\u22650) (hc : c\u2081 * N\u207b\u00b9 = c\u2082) (i : \u2115) [H : universal_map.suitable c\u2082 c\u2081 (universal_map.sum i N)] :\n  (CLC V).map ((FiltrationPow r' c\u2081 i).op.map (\u039b.Hom_sum N r' M).op) \u226b\n    (CLC V).map (FiltrationPow_rescale_iso c\u2081 ((ProFiltPseuNormGrpWithTinv.of r' ((Hom \u039b M) ^ N))) i N \u226a\u226b\n      Filtration_cast_eq r' (c\u2081 * N\u207b\u00b9) c\u2082 hc ((ProFiltPseuNormGrpWithTinv.Pow r' i).obj ((ProFiltPseuNormGrpWithTinv.of r' ((Hom \u039b M) ^ N))))).op.inv =\n  ((universal_map.eval_CLCFP V r' c\u2081 c\u2082 (universal_map.sum i N)).app (op (Hom \u039b M)) \u226b (CLC V).map (FiltrationPow.mul_iso.{u u} r' c\u2082 (Hom \u039b M) N i).op.hom) :=\nbegin\n  rw [\u2190 (CLC V).map_comp],\n  dsimp only [FiltrationPow, category_theory.functor.op_map, category_theory.functor.comp_map,\n    Filtration_cast_eq, quiver.hom.unop_op],\n  rw [iso.op_inv, quux, \u2190 (CLC V).map_comp, iso.op_hom, \u2190 op_comp, \u2190 op_comp],\n  swap, { exact @basic_universal_map.suitable_of_suitable_of _ _ _ _ _ H },\n  congr' 2,\n  apply bar,\nend\n\nlemma row_map_eq_sum_comp\n  (N : \u2115) [fact (0 < N)] (N' : \u211d\u22650) (h : N' = N)\n  [\u2200 (i : \u2115), universal_map.suitable (rescale_constants \u03ba N' i) (\u03ba i) ((BD.sum N).f i)]\n  (\u039b : PolyhedralLattice.{u}) (M : ProFiltPseuNormGrpWithTinv.{u} r') :\n  (thm95.double_complex BD \u03ba r r' V \u039b M N).row_map 0 1 =\n    (iso.refl ((BD.system \u03ba r V r').obj (op (Hom \u039b M)))).inv \u226b\n    (BD_system_map (BD.sum N) \u03ba\n      (rescale_constants \u03ba N') r V).app (op (Hom \u039b M)) \u226b\n    (thm95.mul_rescale_iso_row_one BD \u03ba r V N N' h \u039b M).hom :=\nbegin\n  unfreezingI { subst h },\n  dsimp only [iso.refl_inv],\n  erw category.id_comp,\n  rw [\u2190 iso.comp_inv_eq],\n  rw [thm95.double_complex.row_map_zero_one],\n  dsimp only [mul_rescale_iso_row_one, iso.trans_inv, nat_trans.comp_app, functor.map_iso_inv],\n  simp only [\u2190 category.assoc, \u2190 (BD.system \u03ba r V r').map_comp, \u2190 nat_trans.comp_app,\n    iso.op_inv, \u2190 op_comp, PolyhedralLattice.Cech_augmentation_map_eq_Hom_sum],\n  rw [iso.comp_inv_eq],\n  ext c i : 4,\n  apply arrow.mk_injective,\n  erw [nat_trans.comp_app, nat_trans.comp_app,\n    homological_complex.comp_f, homological_complex.comp_f],\n  dsimp only [BD_system_map_app_app, BD_map_app_f, data.sum_f, data.system_map, data.complex,\n    data.complex\u2082_map_f, mul_system_iso, system_rescale_iso, complex_rescale_iso, mul_complex_iso],\n  erw [nat_iso.of_components_hom_app, nat_iso.of_components_inv_app],\n  dsimp only [homological_complex.hom.iso_of_components_hom_f,\n    homological_complex.hom.iso_of_components_inv_f],\n  dsimp only [CLCFPTinv\u2082, universal_map.eval_CLCFPTinv\u2082, CLCTinv.map_iso_hom, CLCTinv.map_iso_inv,\n    CLCTinv.F_map, _root_.id, CLCTinv.map, SemiNormedGroup.equalizer.map_nat_app, unop_op],\n  rw [SemiNormedGroup.equalizer.map_comp_map, SemiNormedGroup.equalizer.map_comp_map],\n  apply SemiNormedGroup.equalizer.map_congr,\n  { rw foo, refl },\n  { rw foo, refl },\n  all_goals { refl },\nend\n\nend thm95\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/thm95/row_iso.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.36216821614728795}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Leonardo de Moura\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\n \n\nuniverses u v w \n\nnamespace Mathlib\n\ndef stream (\u03b1 : Type u) :=\n  \u2115 \u2192 \u03b1\n\nnamespace stream\n\n\ndef cons {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : stream \u03b1 :=\n  fun (i : \u2115) => sorry\n\ninfixr:67 \" :: \" => Mathlib.stream.cons\n\ndef head {\u03b1 : Type u} (s : stream \u03b1) : \u03b1 :=\n  s 0\n\ndef tail {\u03b1 : Type u} (s : stream \u03b1) : stream \u03b1 :=\n  fun (i : \u2115) => s (i + 1)\n\ndef drop {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : stream \u03b1 :=\n  fun (i : \u2115) => s (i + n)\n\ndef nth {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : \u03b1 :=\n  s n\n\nprotected theorem eta {\u03b1 : Type u} (s : stream \u03b1) : head s :: tail s = s :=\n  funext\n    fun (i : \u2115) =>\n      nat.cases_on i (Eq.refl (cons (head s) (tail s) 0)) fun (i : \u2115) => Eq.refl (cons (head s) (tail s) (Nat.succ i))\n\ntheorem nth_zero_cons {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : nth 0 (a :: s) = a :=\n  rfl\n\ntheorem head_cons {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : head (a :: s) = a :=\n  rfl\n\ntheorem tail_cons {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : tail (a :: s) = s :=\n  rfl\n\ntheorem tail_drop {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : tail (drop n s) = drop n (tail s) := sorry\n\ntheorem nth_drop {\u03b1 : Type u} (n : \u2115) (m : \u2115) (s : stream \u03b1) : nth n (drop m s) = nth (n + m) s :=\n  rfl\n\ntheorem tail_eq_drop {\u03b1 : Type u} (s : stream \u03b1) : tail s = drop 1 s :=\n  rfl\n\ntheorem drop_drop {\u03b1 : Type u} (n : \u2115) (m : \u2115) (s : stream \u03b1) : drop n (drop m s) = drop (n + m) s := sorry\n\ntheorem nth_succ {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth (Nat.succ n) s = nth n (tail s) :=\n  rfl\n\ntheorem drop_succ {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : drop (Nat.succ n) s = drop n (tail s) :=\n  rfl\n\nprotected theorem ext {\u03b1 : Type u} {s\u2081 : stream \u03b1} {s\u2082 : stream \u03b1} : (\u2200 (n : \u2115), nth n s\u2081 = nth n s\u2082) \u2192 s\u2081 = s\u2082 :=\n  fun (h : \u2200 (n : \u2115), nth n s\u2081 = nth n s\u2082) => funext h\n\ndef all {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (s : stream \u03b1) :=\n  \u2200 (n : \u2115), p (nth n s)\n\ndef any {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (s : stream \u03b1) :=\n  \u2203 (n : \u2115), p (nth n s)\n\ntheorem all_def {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (s : stream \u03b1) : all p s = \u2200 (n : \u2115), p (nth n s) :=\n  rfl\n\ntheorem any_def {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (s : stream \u03b1) : any p s = \u2203 (n : \u2115), p (nth n s) :=\n  rfl\n\nprotected def mem {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) :=\n  any (fun (b : \u03b1) => a = b) s\n\nprotected instance has_mem {\u03b1 : Type u} : has_mem \u03b1 (stream \u03b1) :=\n  has_mem.mk stream.mem\n\ntheorem mem_cons {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : a \u2208 a :: s :=\n  exists.intro 0 rfl\n\ntheorem mem_cons_of_mem {\u03b1 : Type u} {a : \u03b1} {s : stream \u03b1} (b : \u03b1) : a \u2208 s \u2192 a \u2208 b :: s := sorry\n\ntheorem eq_or_mem_of_mem_cons {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {s : stream \u03b1} : a \u2208 b :: s \u2192 a = b \u2228 a \u2208 s := sorry\n\ntheorem mem_of_nth_eq {\u03b1 : Type u} {n : \u2115} {s : stream \u03b1} {a : \u03b1} : a = nth n s \u2192 a \u2208 s :=\n  fun (h : a = nth n s) => exists.intro n h\n\ndef map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : stream \u03b2 :=\n  fun (n : \u2115) => f (nth n s)\n\ntheorem drop_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (n : \u2115) (s : stream \u03b1) : drop n (map f s) = map f (drop n s) :=\n  stream.ext fun (i : \u2115) => rfl\n\ntheorem nth_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (n : \u2115) (s : stream \u03b1) : nth n (map f s) = f (nth n s) :=\n  rfl\n\ntheorem tail_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : tail (map f s) = map f (tail s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (tail (map f s) = map f (tail s))) (tail_eq_drop (map f s))))\n    (Eq.refl (drop 1 (map f s)))\n\ntheorem head_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : head (map f s) = f (head s) :=\n  rfl\n\ntheorem map_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : map f s = f (head s) :: map f (tail s) := sorry\n\ntheorem map_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (s : stream \u03b1) : map f (a :: s) = f a :: map f s := sorry\n\ntheorem map_id {\u03b1 : Type u} (s : stream \u03b1) : map id s = s :=\n  rfl\n\ntheorem map_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (g : \u03b2 \u2192 \u03b4) (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : map g (map f s) = map (g \u2218 f) s :=\n  rfl\n\ntheorem map_tail {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : map f (tail s) = tail (map f s) :=\n  rfl\n\ntheorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : stream \u03b1} : a \u2208 s \u2192 f a \u2208 map f s := sorry\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : stream \u03b1} : b \u2208 map f s \u2192 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b := sorry\n\ndef zip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : stream \u03b4 :=\n  fun (n : \u2115) => f (nth n s\u2081) (nth n s\u2082)\n\ntheorem drop_zip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (n : \u2115) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : drop n (zip f s\u2081 s\u2082) = zip f (drop n s\u2081) (drop n s\u2082) :=\n  stream.ext fun (i : \u2115) => rfl\n\ntheorem nth_zip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (n : \u2115) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : nth n (zip f s\u2081 s\u2082) = f (nth n s\u2081) (nth n s\u2082) :=\n  rfl\n\ntheorem head_zip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : head (zip f s\u2081 s\u2082) = f (head s\u2081) (head s\u2082) :=\n  rfl\n\ntheorem tail_zip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : tail (zip f s\u2081 s\u2082) = zip f (tail s\u2081) (tail s\u2082) :=\n  rfl\n\ntheorem zip_eq {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b4) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b2) : zip f s\u2081 s\u2082 = f (head s\u2081) (head s\u2082) :: zip f (tail s\u2081) (tail s\u2082) := sorry\n\ndef const {\u03b1 : Type u} (a : \u03b1) : stream \u03b1 :=\n  fun (n : \u2115) => a\n\ntheorem mem_const {\u03b1 : Type u} (a : \u03b1) : a \u2208 const a :=\n  exists.intro 0 rfl\n\ntheorem const_eq {\u03b1 : Type u} (a : \u03b1) : const a = a :: const a :=\n  stream.ext fun (n : \u2115) => nat.cases_on n (Eq.refl (nth 0 (const a))) fun (n : \u2115) => Eq.refl (nth (Nat.succ n) (const a))\n\ntheorem tail_const {\u03b1 : Type u} (a : \u03b1) : tail (const a) = const a :=\n  (fun (this : tail (a :: const a) = const a) =>\n      eq.mp (Eq._oldrec (Eq.refl (tail (a :: const a) = const a)) (Eq.symm (const_eq a))) this)\n    rfl\n\ntheorem map_const {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (const a) = const (f a) :=\n  rfl\n\ntheorem nth_const {\u03b1 : Type u} (n : \u2115) (a : \u03b1) : nth n (const a) = a :=\n  rfl\n\ntheorem drop_const {\u03b1 : Type u} (n : \u2115) (a : \u03b1) : drop n (const a) = const a :=\n  stream.ext fun (i : \u2115) => rfl\n\ndef iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : stream \u03b1 :=\n  fun (n : \u2115) => nat.rec_on n a fun (n : \u2115) (r : \u03b1) => f r\n\ntheorem head_iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : head (iterate f a) = a :=\n  rfl\n\ntheorem tail_iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : tail (iterate f a) = iterate f (f a) := sorry\n\ntheorem iterate_eq {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : iterate f a = a :: iterate f (f a) := sorry\n\ntheorem nth_zero_iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : nth 0 (iterate f a) = a :=\n  rfl\n\ntheorem nth_succ_iterate {\u03b1 : Type u} (n : \u2115) (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : nth (Nat.succ n) (iterate f a) = nth n (iterate f (f a)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nth (Nat.succ n) (iterate f a) = nth n (iterate f (f a)))) (nth_succ n (iterate f a))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth n (tail (iterate f a)) = nth n (iterate f (f a)))) (tail_iterate f a)))\n      (Eq.refl (nth n (iterate f (f a)))))\n\ndef is_bisimulation {\u03b1 : Type u} (R : stream \u03b1 \u2192 stream \u03b1 \u2192 Prop) :=\n  \u2200 {s\u2081 s\u2082 : stream \u03b1}, R s\u2081 s\u2082 \u2192 head s\u2081 = head s\u2082 \u2227 R (tail s\u2081) (tail s\u2082)\n\ntheorem nth_of_bisim {\u03b1 : Type u} (R : stream \u03b1 \u2192 stream \u03b1 \u2192 Prop) (bisim : is_bisimulation R) {s\u2081 : stream \u03b1} {s\u2082 : stream \u03b1} (n : \u2115) : R s\u2081 s\u2082 \u2192 nth n s\u2081 = nth n s\u2082 \u2227 R (drop (n + 1) s\u2081) (drop (n + 1) s\u2082) := sorry\n\ntheorem eq_of_bisim {\u03b1 : Type u} (R : stream \u03b1 \u2192 stream \u03b1 \u2192 Prop) (bisim : is_bisimulation R) {s\u2081 : stream \u03b1} {s\u2082 : stream \u03b1} : R s\u2081 s\u2082 \u2192 s\u2081 = s\u2082 :=\n  fun (r : R s\u2081 s\u2082) => stream.ext fun (n : \u2115) => and.elim_left (nth_of_bisim R bisim n r)\n\ntheorem bisim_simple {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : head s\u2081 = head s\u2082 \u2192 s\u2081 = tail s\u2081 \u2192 s\u2082 = tail s\u2082 \u2192 s\u2081 = s\u2082 := sorry\n\ntheorem coinduction {\u03b1 : Type u} {s\u2081 : stream \u03b1} {s\u2082 : stream \u03b1} : head s\u2081 = head s\u2082 \u2192 (\u2200 (\u03b2 : Type u) (fr : stream \u03b1 \u2192 \u03b2), fr s\u2081 = fr s\u2082 \u2192 fr (tail s\u2081) = fr (tail s\u2082)) \u2192 s\u2081 = s\u2082 := sorry\n\ntheorem iterate_id {\u03b1 : Type u} (a : \u03b1) : iterate id a = const a := sorry\n\ntheorem map_iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : iterate f (f a) = map f (iterate f a) := sorry\n\ndef corec {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b1) : \u03b1 \u2192 stream \u03b2 :=\n  fun (a : \u03b1) => map f (iterate g a)\n\ndef corec_on {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b1) : stream \u03b2 :=\n  corec f g a\n\ntheorem corec_def {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b1) (a : \u03b1) : corec f g a = map f (iterate g a) :=\n  rfl\n\ntheorem corec_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b1) (a : \u03b1) : corec f g a = f a :: corec f g (g a) := sorry\n\ntheorem corec_id_id_eq_const {\u03b1 : Type u} (a : \u03b1) : corec id id a = const a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (corec id id a = const a)) (corec_def id id a)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (map id (iterate id a) = const a)) (map_id (iterate id a))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (iterate id a = const a)) (iterate_id a))) (Eq.refl (const a))))\n\ntheorem corec_id_f_eq_iterate {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : corec id f a = iterate f a :=\n  rfl\n\ndef corec' {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u00d7 \u03b1) : \u03b1 \u2192 stream \u03b2 :=\n  corec (prod.fst \u2218 f) (prod.snd \u2218 f)\n\ntheorem corec'_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u00d7 \u03b1) (a : \u03b1) : corec' f a = prod.fst (f a) :: corec' f (prod.snd (f a)) :=\n  corec_eq (prod.fst \u2218 f) (prod.snd \u2218 f) a\n\n-- corec is also known as unfold\n\ndef unfolds {\u03b1 : Type u} {\u03b2 : Type v} (g : \u03b1 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : stream \u03b2 :=\n  corec g f a\n\ntheorem unfolds_eq {\u03b1 : Type u} {\u03b2 : Type v} (g : \u03b1 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b1) (a : \u03b1) : unfolds g f a = g a :: unfolds g f (f a) := sorry\n\ntheorem nth_unfolds_head_tail {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth n (unfolds head tail s) = nth n s := sorry\n\ntheorem unfolds_head_eq {\u03b1 : Type u} (s : stream \u03b1) : unfolds head tail s = s :=\n  stream.ext fun (n : \u2115) => nth_unfolds_head_tail n s\n\ndef interleave {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : stream \u03b1 :=\n  corec_on (s\u2081, s\u2082) (fun (_x : stream \u03b1 \u00d7 stream \u03b1) => sorry) fun (_x : stream \u03b1 \u00d7 stream \u03b1) => sorry\n\ninfixl:65 \"\u22c8\" => Mathlib.stream.interleave\n\ntheorem interleave_eq {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : s\u2081\u22c8s\u2082 = head s\u2081 :: head s\u2082 :: (tail s\u2081\u22c8tail s\u2082) := sorry\n\ntheorem tail_interleave {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : tail (s\u2081\u22c8s\u2082) = s\u2082\u22c8tail s\u2081 := sorry\n\ntheorem interleave_tail_tail {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : tail s\u2081\u22c8tail s\u2082 = tail (tail (s\u2081\u22c8s\u2082)) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (tail s\u2081\u22c8tail s\u2082 = tail (tail (s\u2081\u22c8s\u2082)))) (interleave_eq s\u2081 s\u2082)))\n    (Eq.refl (tail s\u2081\u22c8tail s\u2082))\n\ntheorem nth_interleave_left {\u03b1 : Type u} (n : \u2115) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : nth (bit0 1 * n) (s\u2081\u22c8s\u2082) = nth n s\u2081 := sorry\n\ntheorem nth_interleave_right {\u03b1 : Type u} (n : \u2115) (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : nth (bit0 1 * n + 1) (s\u2081\u22c8s\u2082) = nth n s\u2082 := sorry\n\ntheorem mem_interleave_left {\u03b1 : Type u} {a : \u03b1} {s\u2081 : stream \u03b1} (s\u2082 : stream \u03b1) : a \u2208 s\u2081 \u2192 a \u2208 s\u2081\u22c8s\u2082 := sorry\n\ntheorem mem_interleave_right {\u03b1 : Type u} {a : \u03b1} {s\u2081 : stream \u03b1} (s\u2082 : stream \u03b1) : a \u2208 s\u2082 \u2192 a \u2208 s\u2081\u22c8s\u2082 := sorry\n\ndef even {\u03b1 : Type u} (s : stream \u03b1) : stream \u03b1 :=\n  corec (fun (s : stream \u03b1) => head s) (fun (s : stream \u03b1) => tail (tail s)) s\n\ndef odd {\u03b1 : Type u} (s : stream \u03b1) : stream \u03b1 :=\n  even (tail s)\n\ntheorem odd_eq {\u03b1 : Type u} (s : stream \u03b1) : odd s = even (tail s) :=\n  rfl\n\ntheorem head_even {\u03b1 : Type u} (s : stream \u03b1) : head (even s) = head s :=\n  rfl\n\ntheorem tail_even {\u03b1 : Type u} (s : stream \u03b1) : tail (even s) = even (tail (tail s)) := sorry\n\ntheorem even_cons_cons {\u03b1 : Type u} (a\u2081 : \u03b1) (a\u2082 : \u03b1) (s : stream \u03b1) : even (a\u2081 :: a\u2082 :: s) = a\u2081 :: even s := sorry\n\ntheorem even_tail {\u03b1 : Type u} (s : stream \u03b1) : even (tail s) = odd s :=\n  rfl\n\ntheorem even_interleave {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : even (s\u2081\u22c8s\u2082) = s\u2081 := sorry\n\ntheorem interleave_even_odd {\u03b1 : Type u} (s\u2081 : stream \u03b1) : even s\u2081\u22c8odd s\u2081 = s\u2081 := sorry\n\ntheorem nth_even {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth n (even s) = nth (bit0 1 * n) s := sorry\n\ntheorem nth_odd {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth n (odd s) = nth (bit0 1 * n + 1) s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nth n (odd s) = nth (bit0 1 * n + 1) s)) (odd_eq s)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (nth n (even (tail s)) = nth (bit0 1 * n + 1) s)) (nth_even n (tail s))))\n      (Eq.refl (nth (bit0 1 * n) (tail s))))\n\ntheorem mem_of_mem_even {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : a \u2208 even s \u2192 a \u2208 s := sorry\n\ntheorem mem_of_mem_odd {\u03b1 : Type u} (a : \u03b1) (s : stream \u03b1) : a \u2208 odd s \u2192 a \u2208 s := sorry\n\ndef append_stream {\u03b1 : Type u} : List \u03b1 \u2192 stream \u03b1 \u2192 stream \u03b1 :=\n  sorry\n\ntheorem nil_append_stream {\u03b1 : Type u} (s : stream \u03b1) : append_stream [] s = s :=\n  rfl\n\ntheorem cons_append_stream {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) (s : stream \u03b1) : append_stream (a :: l) s = a :: append_stream l s :=\n  rfl\n\ninfixl:65 \"++\u209b\" => Mathlib.stream.append_stream\n\ntheorem append_append_stream {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (s : stream \u03b1) : l\u2081 ++ l\u2082++\u209bs = l\u2081++\u209b(l\u2082++\u209bs) := sorry\n\ntheorem map_append_stream {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) (s : stream \u03b1) : map f (l++\u209bs) = list.map f l++\u209bmap f s := sorry\n\ntheorem drop_append_stream {\u03b1 : Type u} (l : List \u03b1) (s : stream \u03b1) : drop (list.length l) (l++\u209bs) = s := sorry\n\ntheorem append_stream_head_tail {\u03b1 : Type u} (s : stream \u03b1) : [head s]++\u209btail s = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl ([head s]++\u209btail s = s)) (cons_append_stream (head s) [] (tail s))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (head s :: ([]++\u209btail s) = s)) (nil_append_stream (tail s))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (head s :: tail s = s)) (stream.eta s))) (Eq.refl s)))\n\ntheorem mem_append_stream_right {\u03b1 : Type u} {a : \u03b1} (l : List \u03b1) {s : stream \u03b1} : a \u2208 s \u2192 a \u2208 l++\u209bs := sorry\n\ntheorem mem_append_stream_left {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (s : stream \u03b1) : a \u2208 l \u2192 a \u2208 l++\u209bs := sorry\n\ndef approx {\u03b1 : Type u} : \u2115 \u2192 stream \u03b1 \u2192 List \u03b1 :=\n  sorry\n\ntheorem approx_zero {\u03b1 : Type u} (s : stream \u03b1) : approx 0 s = [] :=\n  rfl\n\ntheorem approx_succ {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : approx (Nat.succ n) s = head s :: approx n (tail s) :=\n  rfl\n\ntheorem nth_approx {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : list.nth (approx (Nat.succ n) s) n = some (nth n s) := sorry\n\ntheorem append_approx_drop {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : approx n s++\u209bdrop n s = s := sorry\n\n-- Take theorem reduces a proof of equality of infinite streams to an\n\n-- induction over all their finite approximations.\n\ntheorem take_theorem {\u03b1 : Type u} (s\u2081 : stream \u03b1) (s\u2082 : stream \u03b1) : (\u2200 (n : \u2115), approx n s\u2081 = approx n s\u2082) \u2192 s\u2081 = s\u2082 := sorry\n\n-- auxiliary def for cycle corecursive def\n\n-- auxiliary def for cycle corecursive def\n\ndef cycle {\u03b1 : Type u} (l : List \u03b1) : l \u2260 [] \u2192 stream \u03b1 :=\n  sorry\n\ntheorem cycle_eq {\u03b1 : Type u} (l : List \u03b1) (h : l \u2260 []) : cycle l h = l++\u209bcycle l h := sorry\n\ntheorem mem_cycle {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h : l \u2260 []) : a \u2208 l \u2192 a \u2208 cycle l h :=\n  fun (ainl : a \u2208 l) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 cycle l h)) (cycle_eq l h))) (mem_append_stream_left (cycle l h) ainl)\n\ntheorem cycle_singleton {\u03b1 : Type u} (a : \u03b1) (h : [a] \u2260 []) : cycle [a] h = const a := sorry\n\ndef tails {\u03b1 : Type u} (s : stream \u03b1) : stream (stream \u03b1) :=\n  corec id tail (tail s)\n\ntheorem tails_eq {\u03b1 : Type u} (s : stream \u03b1) : tails s = tail s :: tails (tail s) := sorry\n\ntheorem nth_tails {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth n (tails s) = drop n (tail s) := sorry\n\ntheorem tails_eq_iterate {\u03b1 : Type u} (s : stream \u03b1) : tails s = iterate tail (tail s) :=\n  rfl\n\ndef inits_core {\u03b1 : Type u} (l : List \u03b1) (s : stream \u03b1) : stream (List \u03b1) :=\n  corec_on (l, s) (fun (_x : List \u03b1 \u00d7 stream \u03b1) => sorry) fun (p : List \u03b1 \u00d7 stream \u03b1) => sorry\n\ndef inits {\u03b1 : Type u} (s : stream \u03b1) : stream (List \u03b1) :=\n  inits_core [head s] (tail s)\n\ntheorem inits_core_eq {\u03b1 : Type u} (l : List \u03b1) (s : stream \u03b1) : inits_core l s = l :: inits_core (l ++ [head s]) (tail s) := sorry\n\ntheorem tail_inits {\u03b1 : Type u} (s : stream \u03b1) : tail (inits s) = inits_core [head s, head (tail s)] (tail (tail s)) := sorry\n\ntheorem inits_tail {\u03b1 : Type u} (s : stream \u03b1) : inits (tail s) = inits_core [head (tail s)] (tail (tail s)) :=\n  rfl\n\ntheorem cons_nth_inits_core {\u03b1 : Type u} (a : \u03b1) (n : \u2115) (l : List \u03b1) (s : stream \u03b1) : a :: nth n (inits_core l s) = nth n (inits_core (a :: l) s) := sorry\n\ntheorem nth_inits {\u03b1 : Type u} (n : \u2115) (s : stream \u03b1) : nth n (inits s) = approx (Nat.succ n) s := sorry\n\ntheorem inits_eq {\u03b1 : Type u} (s : stream \u03b1) : inits s = [head s] :: map (List.cons (head s)) (inits (tail s)) := sorry\n\ntheorem zip_inits_tails {\u03b1 : Type u} (s : stream \u03b1) : zip append_stream (inits s) (tails s) = const s := sorry\n\ndef pure {\u03b1 : Type u} (a : \u03b1) : stream \u03b1 :=\n  const a\n\ndef apply {\u03b1 : Type u} {\u03b2 : Type v} (f : stream (\u03b1 \u2192 \u03b2)) (s : stream \u03b1) : stream \u03b2 :=\n  fun (n : \u2115) => nth n f (nth n s)\n\ninfixl:75 \"\u229b\" => Mathlib.stream.apply\n\ntheorem identity {\u03b1 : Type u} (s : stream \u03b1) : pure id\u229bs = s :=\n  rfl\n\ntheorem composition {\u03b1 : Type u} {\u03b2 : Type v} {\u03b4 : Type w} (g : stream (\u03b2 \u2192 \u03b4)) (f : stream (\u03b1 \u2192 \u03b2)) (s : stream \u03b1) : pure function.comp\u229bg\u229bf\u229bs = g\u229b(f\u229bs) :=\n  rfl\n\ntheorem homomorphism {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : pure f\u229bpure a = pure (f a) :=\n  rfl\n\ntheorem interchange {\u03b1 : Type u} {\u03b2 : Type v} (fs : stream (\u03b1 \u2192 \u03b2)) (a : \u03b1) : fs\u229bpure a = (pure fun (f : \u03b1 \u2192 \u03b2) => f a)\u229bfs :=\n  rfl\n\ntheorem map_eq_apply {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : stream \u03b1) : map f s = pure f\u229bs :=\n  rfl\n\ndef nats : stream \u2115 :=\n  fun (n : \u2115) => n\n\ntheorem nth_nats (n : \u2115) : nth n nats = n :=\n  rfl\n\ntheorem nats_eq : nats = 0 :: map Nat.succ nats := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/Lean3Lib/data/stream.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.36216821614728795}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz, Scott Morrison\n-/\nimport category_theory.limits.exact_functor\nimport category_theory.limits.preserves.finite\nimport category_theory.preadditive.biproducts\nimport category_theory.preadditive.functor_category\n\n/-!\n# Additive Functors\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nA functor between two preadditive categories is called *additive*\nprovided that the induced map on hom types is a morphism of abelian\ngroups.\n\nAn additive functor between preadditive categories creates and preserves biproducts.\nConversely, if `F : C \u2964 D` is a functor between preadditive categories, where `C` has binary\nbiproducts, and if `F` preserves binary biproducts, then `F` is additive.\n\nWe also define the category of bundled additive functors.\n\n# Implementation details\n\n`functor.additive` is a `Prop`-valued class, defined by saying that for every two objects `X` and\n`Y`, the map `F.map : (X \u27f6 Y) \u2192 (F.obj X \u27f6 F.obj Y)` is a morphism of abelian groups.\n\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnamespace category_theory\n\n/-- A functor `F` is additive provided `F.map` is an additive homomorphism. -/\nclass functor.additive {C D : Type*} [category C] [category D]\n  [preadditive C] [preadditive D] (F : C \u2964 D) : Prop :=\n(map_add' : \u03a0 {X Y : C} {f g : X \u27f6 Y}, F.map (f + g) = F.map f + F.map g . obviously)\n\nsection preadditive\n\nnamespace functor\n\nsection\nvariables {C D : Type*} [category C] [category D] [preadditive C]\n  [preadditive D] (F : C \u2964 D) [functor.additive F]\n\n@[simp]\n\n\n/-- `F.map_add_hom` is an additive homomorphism whose underlying function is `F.map`. -/\n@[simps {fully_applied := ff}]\ndef map_add_hom {X Y : C} : (X \u27f6 Y) \u2192+ (F.obj X \u27f6 F.obj Y) :=\nadd_monoid_hom.mk' (\u03bb f, F.map f) (\u03bb f g, F.map_add)\n\nlemma coe_map_add_hom {X Y : C} : \u21d1(F.map_add_hom : (X \u27f6 Y) \u2192+ _) = @map C _ D _ F X Y := rfl\n\n@[priority 100]\ninstance preserves_zero_morphisms_of_additive : preserves_zero_morphisms F :=\n{ map_zero' := \u03bb X Y, F.map_add_hom.map_zero }\n\ninstance : additive (\ud835\udfed C) :=\n{}\n\ninstance {E : Type*} [category E] [preadditive E] (G : D \u2964 E) [functor.additive G] :\n  additive (F \u22d9 G) :=\n{}\n\n@[simp]\nlemma map_neg {X Y : C} {f : X \u27f6 Y} : F.map (-f) = - F.map f :=\n(F.map_add_hom : (X \u27f6 Y) \u2192+ (F.obj X \u27f6 F.obj Y)).map_neg _\n\n@[simp]\nlemma map_sub {X Y : C} {f g : X \u27f6 Y} : F.map (f - g) = F.map f - F.map g :=\n(F.map_add_hom : (X \u27f6 Y) \u2192+ (F.obj X \u27f6 F.obj Y)).map_sub _ _\n\nlemma map_nsmul {X Y : C} {f : X \u27f6 Y} {n : \u2115} : F.map (n \u2022 f) = n \u2022 F.map f :=\n(F.map_add_hom : (X \u27f6 Y) \u2192+ (F.obj X \u27f6 F.obj Y)).map_nsmul _ _\n\n-- You can alternatively just use `functor.map_smul` here, with an explicit `(r : \u2124)` argument.\nlemma map_zsmul {X Y : C} {f : X \u27f6 Y} {r : \u2124} : F.map (r \u2022 f) = r \u2022 F.map f :=\n(F.map_add_hom : (X \u27f6 Y) \u2192+ (F.obj X \u27f6 F.obj Y)).map_zsmul _ _\n\nopen_locale big_operators\n\n@[simp]\nlemma map_sum {X Y : C} {\u03b1 : Type*} (f : \u03b1 \u2192 (X \u27f6 Y)) (s : finset \u03b1) :\n  F.map (\u2211 a in s, f a) = \u2211 a in s, F.map (f a) :=\n(F.map_add_hom : (X \u27f6 Y) \u2192+ _).map_sum f s\n\nend\n\nsection induced_category\nvariables {C : Type*} {D : Type*} [category D] [preadditive D] (F : C \u2192 D)\n\ninstance induced_functor_additive : functor.additive (induced_functor F) := {}\n\nend induced_category\n\ninstance full_subcategory_inclusion_additive\n  {C : Type*} [category C] [preadditive C] (Z : C \u2192 Prop) :\n  (full_subcategory_inclusion Z).additive := {}\n\nsection\n-- To talk about preservation of biproducts we need to specify universes explicitly.\n\nnoncomputable theory\n\nvariables {C : Type u\u2081} {D : Type u\u2082} [category.{v\u2081} C] [category.{v\u2082} D]\n  [preadditive C] [preadditive D] (F : C \u2964 D)\n\nopen category_theory.limits\nopen category_theory.preadditive\n\n@[priority 100]\ninstance preserves_finite_biproducts_of_additive [additive F] : preserves_finite_biproducts F :=\n{ preserves := \u03bb J _,\n  { preserves := \u03bb f,\n    { preserves := \u03bb b hb, by exactI is_bilimit_of_total _\n      begin\n        simp_rw [F.map_bicone_\u03c0, F.map_bicone_\u03b9, \u2190 F.map_comp, \u2190 F.map_sum],\n        dsimp only [map_bicone_X],\n        simp_rw [\u2190 F.map_id],\n        refine congr_arg _ (hb.is_limit.hom_ext (\u03bb j, hb.is_colimit.hom_ext (\u03bb j', _))),\n        cases j, cases j',\n        dsimp only [limits.bicone.to_cone_\u03c0_app],\n        simp [sum_comp, comp_sum, bicone.\u03b9_\u03c0, comp_dite, dite_comp],\n      end } } }\n\nlemma additive_of_preserves_binary_biproducts [has_binary_biproducts C] [preserves_zero_morphisms F]\n  [preserves_binary_biproducts F] : additive F :=\n{ map_add' := \u03bb X Y f g, by rw [biprod.add_eq_lift_id_desc, F.map_comp, \u2190 biprod.lift_map_biprod,\n    \u2190 biprod.map_biprod_hom_desc, category.assoc, iso.inv_hom_id_assoc, F.map_id,\n    biprod.add_eq_lift_id_desc] }\n\nend\n\nend functor\n\nnamespace equivalence\n\nvariables {C D : Type*} [category C] [category D] [preadditive C] [preadditive D]\n\ninstance inverse_additive (e : C \u224c D) [e.functor.additive] : e.inverse.additive :=\n{ map_add' := \u03bb X Y f g, by { apply e.functor.map_injective, simp, }, }\n\nend equivalence\n\nsection\nvariables (C D : Type*) [category C] [category D] [preadditive C] [preadditive D]\n\n/-- Bundled additive functors. -/\n@[derive category, nolint has_nonempty_instance]\ndef AdditiveFunctor :=\nfull_subcategory (\u03bb (F : C \u2964 D), F.additive)\n\ninfixr ` \u2964+ `:26 := AdditiveFunctor\n\ninstance : preadditive (C \u2964+ D) :=\npreadditive.induced_category _\n\n/-- An additive functor is in particular a functor. -/\n@[derive full, derive faithful]\ndef AdditiveFunctor.forget : (C \u2964+ D) \u2964 (C \u2964 D) :=\nfull_subcategory_inclusion _\n\nvariables {C D}\n\n/-- Turn an additive functor into an object of the category `AdditiveFunctor C D`. -/\ndef AdditiveFunctor.of (F : C \u2964 D) [F.additive] : C \u2964+ D :=\n\u27e8F, infer_instance\u27e9\n\n@[simp]\nlemma AdditiveFunctor.of_fst (F : C \u2964 D) [F.additive] : (AdditiveFunctor.of F).1 = F :=\nrfl\n\n@[simp]\nlemma AdditiveFunctor.forget_obj (F : C \u2964+ D) : (AdditiveFunctor.forget C D).obj F = F.1 :=\nrfl\n\nlemma AdditiveFunctor.forget_obj_of (F : C \u2964 D) [F.additive] :\n  (AdditiveFunctor.forget C D).obj (AdditiveFunctor.of F) = F :=\nrfl\n\n@[simp]\nlemma AdditiveFunctor.forget_map (F G : C \u2964+ D) (\u03b1 : F \u27f6 G) :\n  (AdditiveFunctor.forget C D).map \u03b1 = \u03b1 :=\nrfl\n\ninstance : functor.additive (AdditiveFunctor.forget C D) :=\n{ map_add' := \u03bb F G \u03b1 \u03b2, rfl }\n\ninstance (F : C \u2964+ D) : functor.additive F.1 :=\nF.2\n\nend\n\nsection exact\nopen category_theory.limits\n\nvariables (C : Type u\u2081) (D : Type u\u2082) [category.{v\u2081} C] [category.{v\u2082} D] [preadditive C]\nvariables [preadditive D] [has_zero_object C] [has_zero_object D] [has_binary_biproducts C]\n\nsection\nlocal attribute [instance] preserves_binary_biproducts_of_preserves_binary_products\nlocal attribute [instance] preserves_binary_biproducts_of_preserves_binary_coproducts\n\n/-- Turn a left exact functor into an additive functor. -/\n@[derive full, derive faithful]\ndef AdditiveFunctor.of_left_exact : (C \u2964\u2097 D) \u2964 (C \u2964+ D) :=\nfull_subcategory.map (\u03bb F h, let hF := classical.choice h in\n    by exactI functor.additive_of_preserves_binary_biproducts F)\n\n/-- Turn a right exact functor into an additive functor. -/\n@[derive full, derive faithful]\ndef AdditiveFunctor.of_right_exact : (C \u2964\u1d63 D) \u2964 (C \u2964+ D) :=\nfull_subcategory.map (\u03bb F h, let hF := classical.choice h in\n  by exactI functor.additive_of_preserves_binary_biproducts F)\n\n/-- Turn an exact functor into an additive functor. -/\n@[derive full, derive faithful]\ndef AdditiveFunctor.of_exact : (C \u2964\u2091 D) \u2964 (C \u2964+ D) :=\nfull_subcategory.map (\u03bb F h, let hF := classical.choice h.1 in\n  by exactI functor.additive_of_preserves_binary_biproducts F)\n\nend\n\nvariables {C D}\n\n@[simp] lemma AdditiveFunctor.of_left_exact_obj_fst (F : C \u2964\u2097 D) :\n  ((AdditiveFunctor.of_left_exact C D).obj F).obj = F.obj := rfl\n@[simp] lemma AdditiveFunctor.of_right_exact_obj_fst (F : C \u2964\u1d63 D) :\n  ((AdditiveFunctor.of_right_exact C D).obj F).obj = F.obj := rfl\n@[simp] lemma AdditiveFunctor.of_exact_obj_fst (F : C \u2964\u2091 D) :\n  ((AdditiveFunctor.of_exact C D).obj F).obj = F.obj := rfl\n\n@[simp] lemma Additive_Functor.of_left_exact_map {F G : C \u2964\u2097 D} (\u03b1 : F \u27f6 G) :\n  (AdditiveFunctor.of_left_exact C D).map \u03b1 = \u03b1 := rfl\n@[simp] lemma Additive_Functor.of_right_exact_map {F G : C \u2964\u1d63 D} (\u03b1 : F \u27f6 G) :\n  (AdditiveFunctor.of_right_exact C D).map \u03b1 = \u03b1 := rfl\n@[simp] lemma Additive_Functor.of_exact_map {F G : C \u2964\u2091 D} (\u03b1 : F \u27f6 G) :\n  (AdditiveFunctor.of_exact C D).map \u03b1 = \u03b1 := rfl\n\nend exact\n\nend preadditive\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/preadditive/additive_functor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.36216820846720804}}
{"text": "import Std.Classes.BEq\n\nnamespace MWE_eq1\n\nexample [BEq \u03b1] [LawfulBEq \u03b1] {x y : \u03b1} (h : \u00ac(x = y))\n  : (match x == y with | true => 0 | false => 1) = 1 := by\n  have : (x == y) = false := by simp [beq_eq_false_iff_ne, h]\n  simp [this]\n\nend MWE_eq1", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/MWE_eq1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3621041485240509}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Bhavik Mehta\n-/\nimport category_theory.adjunction.reflective\nimport category_theory.monad.algebra\n\nnamespace category_theory\nopen category\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 -- morphism levels before object levels. See note [category_theory universes].\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\nvariables {L : C \u2964 D} {R : D \u2964 C}\n\nnamespace adjunction\n\n/--\nFor a pair of functors `L : C \u2964 D`, `R : D \u2964 C`, an adjunction `h : L \u22a3 R` induces a monad on\nthe category `C`.\n-/\n@[simps]\ndef to_monad (h : L \u22a3 R) : monad C :=\n{ to_functor := L \u22d9 R,\n  \u03b7' := h.unit,\n  \u03bc' := whisker_right (whisker_left L h.counit) R,\n  assoc' := \u03bb X, by { dsimp, rw [\u2190R.map_comp], simp },\n  right_unit' := \u03bb X, by { dsimp, rw [\u2190R.map_comp], simp } }\n\n/--\nFor a pair of functors `L : C \u2964 D`, `R : D \u2964 C`, an adjunction `h : L \u22a3 R` induces a comonad on\nthe category `D`.\n-/\n@[simps]\ndef to_comonad (h : L \u22a3 R) : comonad D :=\n{ to_functor := R \u22d9 L,\n  \u03b5' := h.counit,\n  \u03b4' := whisker_right (whisker_left R h.unit) L,\n  coassoc' := \u03bb X, by { dsimp, rw \u2190 L.map_comp, simp },\n  right_counit' := \u03bb X, by { dsimp, rw \u2190 L.map_comp, simp } }\n\n/-- The monad induced by the Eilenberg-Moore adjunction is the original monad.  -/\n@[simps]\ndef adj_to_monad_iso (T : monad C) : T.adj.to_monad \u2245 T :=\nmonad_iso.mk (nat_iso.of_components (\u03bb X, iso.refl _) (by tidy))\n  (\u03bb X, by { dsimp, simp })\n  (\u03bb X, by { dsimp, simp })\n\n/-- The comonad induced by the Eilenberg-Moore adjunction is the original comonad. -/\n@[simps]\ndef adj_to_comonad_iso (G : comonad C) : G.adj.to_comonad \u2245 G :=\ncomonad_iso.mk (nat_iso.of_components (\u03bb X, iso.refl _) (by tidy))\n  (\u03bb X, by { dsimp, simp })\n  (\u03bb X, by { dsimp, simp })\n\nend adjunction\n\n/--\nGven any adjunction `L \u22a3 R`, there is a comparison functor `category_theory.monad.comparison R`\nsending objects `Y : D` to Eilenberg-Moore algebras for `L \u22d9 R` with underlying object `R.obj X`.\n\nWe later show that this is full when `R` is full, faithful when `R` is faithful,\nand essentially surjective when `R` is reflective.\n-/\n@[simps]\ndef monad.comparison (h : L \u22a3 R) : D \u2964 h.to_monad.algebra :=\n{ obj := \u03bb X,\n  { A := R.obj X,\n    a := R.map (h.counit.app X),\n    assoc' := by { dsimp, rw [\u2190 R.map_comp, \u2190 adjunction.counit_naturality, R.map_comp], refl } },\n  map := \u03bb X Y f,\n  { f := R.map f,\n    h' := by { dsimp, rw [\u2190 R.map_comp, adjunction.counit_naturality, R.map_comp] } } }.\n\n/--\nThe underlying object of `(monad.comparison R).obj X` is just `R.obj X`.\n-/\n@[simps]\ndef monad.comparison_forget (h : L \u22a3 R) :\n  monad.comparison h \u22d9 h.to_monad.forget \u2245 R :=\n{ hom := { app := \u03bb X, \ud835\udfd9 _, },\n  inv := { app := \u03bb X, \ud835\udfd9 _, } }\n\nlemma monad.left_comparison (h : L \u22a3 R) : L \u22d9 monad.comparison h = h.to_monad.free := rfl\n\ninstance [faithful R] (h : L \u22a3 R) :\n  faithful (monad.comparison h) :=\n{ map_injective' := \u03bb X Y f g w, R.map_injective (congr_arg monad.algebra.hom.f w : _) }\n\ninstance (T : monad C) : full (monad.comparison T.adj) :=\n{ preimage := \u03bb X Y f, \u27e8f.f, by simpa using f.h\u27e9 }\n\ninstance (T : monad C) : ess_surj (monad.comparison T.adj) :=\n{ mem_ess_image := \u03bb X,\n  \u27e8{ A := X.A, a := X.a, unit' := by simpa using X.unit, assoc' := by simpa using X.assoc },\n    \u27e8monad.algebra.iso_mk (iso.refl _) (by simp)\u27e9\u27e9 }\n\n/--\nGven any adjunction `L \u22a3 R`, there is a comparison functor `category_theory.comonad.comparison L`\nsending objects `X : C` to Eilenberg-Moore coalgebras for `L \u22d9 R` with underlying object\n`L.obj X`.\n-/\n@[simps]\ndef comonad.comparison (h : L \u22a3 R) : C \u2964 h.to_comonad.coalgebra :=\n{ obj := \u03bb X,\n  { A := L.obj X,\n    a := L.map (h.unit.app X),\n    coassoc' := by { dsimp, rw [\u2190 L.map_comp, \u2190 adjunction.unit_naturality, L.map_comp], refl } },\n  map := \u03bb X Y f,\n  { f := L.map f,\n    h' := by { dsimp, rw \u2190 L.map_comp, simp } } }\n\n/--\nThe underlying object of `(comonad.comparison L).obj X` is just `L.obj X`.\n-/\n@[simps]\ndef comonad.comparison_forget {L : C \u2964 D} {R : D \u2964 C} (h : L \u22a3 R) :\n  comonad.comparison h \u22d9 h.to_comonad.forget \u2245 L :=\n{ hom := { app := \u03bb X, \ud835\udfd9 _, },\n  inv := { app := \u03bb X, \ud835\udfd9 _, } }\n\nlemma comonad.left_comparison (h : L \u22a3 R) : R \u22d9 comonad.comparison h = h.to_comonad.cofree := rfl\n\ninstance comonad.comparison_faithful_of_faithful [faithful L] (h : L \u22a3 R) :\n  faithful (comonad.comparison h) :=\n{ map_injective' := \u03bb X Y f g w, L.map_injective (congr_arg comonad.coalgebra.hom.f w : _) }\n\ninstance (G : comonad C) : full (comonad.comparison G.adj) :=\n{ preimage := \u03bb X Y f, \u27e8f.f, by simpa using f.h\u27e9 }\n\ninstance (G : comonad C) : ess_surj (comonad.comparison G.adj) :=\n{ mem_ess_image := \u03bb X,\n  \u27e8{ A := X.A, a := X.a, counit' := by simpa using X.counit, coassoc' := by simpa using X.coassoc },\n    \u27e8comonad.coalgebra.iso_mk (iso.refl _) (by simp)\u27e9\u27e9 }\n\n/--\nA right adjoint functor `R : D \u2964 C` is *monadic* if the comparison functor `monad.comparison R`\nfrom `D` to the category of Eilenberg-Moore algebras for the adjunction is an equivalence.\n-/\nclass monadic_right_adjoint (R : D \u2964 C) extends is_right_adjoint R :=\n(eqv : is_equivalence (monad.comparison (adjunction.of_right_adjoint R)))\n\n/--\nA left adjoint functor `L : C \u2964 D` is *comonadic* if the comparison functor `comonad.comparison L`\nfrom `C` to the category of Eilenberg-Moore algebras for the adjunction is an equivalence.\n-/\nclass comonadic_left_adjoint (L : C \u2964 D) extends is_left_adjoint L :=\n(eqv : is_equivalence (comonad.comparison (adjunction.of_left_adjoint L)))\n\nnoncomputable instance (T : monad C) : monadic_right_adjoint T.forget :=\n\u27e8(equivalence.of_fully_faithfully_ess_surj _ : is_equivalence (monad.comparison T.adj))\u27e9\n\nnoncomputable instance (G : comonad C) : comonadic_left_adjoint G.forget :=\n\u27e8(equivalence.of_fully_faithfully_ess_surj _ : is_equivalence (comonad.comparison G.adj))\u27e9\n\n-- TODO: This holds more generally for idempotent adjunctions, not just reflective adjunctions.\ninstance \u03bc_iso_of_reflective [reflective R] : is_iso (adjunction.of_right_adjoint R).to_monad.\u03bc :=\nby { dsimp, apply_instance }\n\nattribute [instance] monadic_right_adjoint.eqv\nattribute [instance] comonadic_left_adjoint.eqv\n\nnamespace reflective\n\ninstance [reflective R] (X : (adjunction.of_right_adjoint R).to_monad.algebra) :\n  is_iso ((adjunction.of_right_adjoint R).unit.app X.A) :=\n\u27e8\u27e8X.a, \u27e8X.unit, begin\n    dsimp only [functor.id_obj],\n    rw \u2190 (adjunction.of_right_adjoint R).unit_naturality,\n    dsimp only [functor.comp_obj, adjunction.to_monad_coe],\n    rw [unit_obj_eq_map_unit, \u2190functor.map_comp, \u2190functor.map_comp],\n    erw X.unit,\n    simp,\n  end\u27e9\u27e9\u27e9\n\ninstance comparison_ess_surj [reflective R] :\n  ess_surj (monad.comparison (adjunction.of_right_adjoint R)) :=\nbegin\n  refine \u27e8\u03bb X, \u27e8(left_adjoint R).obj X.A, \u27e8_\u27e9\u27e9\u27e9,\n  symmetry,\n  refine monad.algebra.iso_mk _ _,\n  { exact as_iso ((adjunction.of_right_adjoint R).unit.app X.A) },\n  dsimp only [functor.comp_map, monad.comparison_obj_a, as_iso_hom, functor.comp_obj,\n    monad.comparison_obj_A, monad_to_functor_eq_coe, adjunction.to_monad_coe],\n  rw [\u2190cancel_epi ((adjunction.of_right_adjoint R).unit.app X.A), adjunction.unit_naturality_assoc,\n      adjunction.right_triangle_components, comp_id],\n  apply (X.unit_assoc _).symm,\nend\n\ninstance comparison_full [full R] [is_right_adjoint R] :\n  full (monad.comparison (adjunction.of_right_adjoint R)) :=\n{ preimage := \u03bb X Y f, R.preimage f.f }\n\nend reflective\n\n-- It is possible to do this computably since the construction gives the data of the inverse, not\n-- just the existence of an inverse on each object.\n/-- Any reflective inclusion has a monadic right adjoint.\n    cf Prop 5.3.3 of [Riehl][riehl2017] -/\n@[priority 100] -- see Note [lower instance priority]\nnoncomputable instance monadic_of_reflective [reflective R] : monadic_right_adjoint R :=\n{ eqv := equivalence.of_fully_faithfully_ess_surj _ }\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/monad/adjunction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.36204915420854866}}
{"text": "/-\nCopyright (c) 2022 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel\n-/\nimport category_theory.epi_mono\nimport category_theory.limits.shapes.strong_epi\nimport category_theory.lifting_properties.adjunction\n\n/-!\n# Preservation and reflection of monomorphisms and epimorphisms\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nWe provide typeclasses that state that a functor preserves or reflects monomorphisms or\nepimorphisms.\n-/\n\nopen category_theory\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nnamespace category_theory.functor\nvariables {C : Type u\u2081} [category.{v\u2081} C] {D : Type u\u2082} [category.{v\u2082} D]\n  {E : Type u\u2083} [category.{v\u2083} E]\n\n/-- A functor preserves monomorphisms if it maps monomorphisms to monomorphisms. -/\nclass preserves_monomorphisms (F : C \u2964 D) : Prop :=\n(preserves : \u2200 {X Y : C} (f : X \u27f6 Y) [mono f], mono (F.map f))\n\ninstance map_mono (F : C \u2964 D) [preserves_monomorphisms F] {X Y : C} (f : X \u27f6 Y) [mono f] :\n  mono (F.map f) :=\npreserves_monomorphisms.preserves f\n\n/-- A functor preserves epimorphisms if it maps epimorphisms to epimorphisms. -/\nclass preserves_epimorphisms (F : C \u2964 D) : Prop :=\n(preserves : \u2200 {X Y : C} (f : X \u27f6 Y) [epi f], epi (F.map f))\n\ninstance map_epi (F : C \u2964 D) [preserves_epimorphisms F] {X Y : C} (f : X \u27f6 Y) [epi f] :\n  epi (F.map f) :=\npreserves_epimorphisms.preserves f\n\n/-- A functor reflects monomorphisms if morphisms that are mapped to monomorphisms are themselves\n    monomorphisms. -/\nclass reflects_monomorphisms (F : C \u2964 D) : Prop :=\n(reflects : \u2200 {X Y : C} (f : X \u27f6 Y), mono (F.map f) \u2192 mono f)\n\nlemma mono_of_mono_map (F : C \u2964 D) [reflects_monomorphisms F] {X Y : C} {f : X \u27f6 Y}\n  (h : mono (F.map f)) : mono f :=\nreflects_monomorphisms.reflects f h\n\n/-- A functor reflects epimorphisms if morphisms that are mapped to epimorphisms are themselves\n    epimorphisms. -/\nclass reflects_epimorphisms (F : C \u2964 D) : Prop :=\n(reflects : \u2200 {X Y : C} (f : X \u27f6 Y), epi (F.map f) \u2192 epi f)\n\nlemma epi_of_epi_map (F : C \u2964 D) [reflects_epimorphisms F] {X Y : C} {f : X \u27f6 Y}\n  (h : epi (F.map f)) : epi f :=\nreflects_epimorphisms.reflects f h\n\ninstance preserves_monomorphisms_comp (F : C \u2964 D) (G : D \u2964 E) [preserves_monomorphisms F]\n  [preserves_monomorphisms G] : preserves_monomorphisms (F \u22d9 G) :=\n{ preserves := \u03bb X Y f h, by { rw comp_map, exactI infer_instance } }\n\ninstance preserves_epimorphisms_comp (F : C \u2964 D) (G : D \u2964 E) [preserves_epimorphisms F]\n  [preserves_epimorphisms G] : preserves_epimorphisms (F \u22d9 G) :=\n{ preserves := \u03bb X Y f h, by { rw comp_map, exactI infer_instance } }\n\ninstance reflects_monomorphisms_comp (F : C \u2964 D) (G : D \u2964 E) [reflects_monomorphisms F]\n  [reflects_monomorphisms G] : reflects_monomorphisms (F \u22d9 G) :=\n{ reflects := \u03bb X Y f h, (F.mono_of_mono_map (G.mono_of_mono_map h)) }\n\ninstance reflects_epimorphisms_comp (F : C \u2964 D) (G : D \u2964 E) [reflects_epimorphisms F]\n  [reflects_epimorphisms G] : reflects_epimorphisms (F \u22d9 G) :=\n{ reflects := \u03bb X Y f h, (F.epi_of_epi_map (G.epi_of_epi_map h)) }\n\nlemma preserves_epimorphisms_of_preserves_of_reflects (F : C \u2964 D) (G : D \u2964 E)\n  [preserves_epimorphisms (F \u22d9 G)] [reflects_epimorphisms G] : preserves_epimorphisms F :=\n\u27e8\u03bb X Y f hf, G.epi_of_epi_map $ show epi ((F \u22d9 G).map f), by exactI infer_instance\u27e9\n\nlemma preserves_monomorphisms_of_preserves_of_reflects (F : C \u2964 D) (G : D \u2964 E)\n  [preserves_monomorphisms (F \u22d9 G)] [reflects_monomorphisms G] : preserves_monomorphisms F :=\n\u27e8\u03bb X Y f hf, G.mono_of_mono_map $ show mono ((F \u22d9 G).map f), by exactI infer_instance\u27e9\n\nlemma reflects_epimorphisms_of_preserves_of_reflects (F : C \u2964 D) (G : D \u2964 E)\n  [preserves_epimorphisms G] [reflects_epimorphisms (F \u22d9 G)] : reflects_epimorphisms F :=\n\u27e8\u03bb X Y f hf, (F \u22d9 G).epi_of_epi_map $ show epi (G.map (F.map f)), by exactI infer_instance\u27e9\n\nlemma reflects_monomorphisms_of_preserves_of_reflects (F : C \u2964 D) (G : D \u2964 E)\n  [preserves_monomorphisms G] [reflects_monomorphisms (F \u22d9 G)] : reflects_monomorphisms F :=\n\u27e8\u03bb X Y f hf, (F \u22d9 G).mono_of_mono_map $ show mono (G.map (F.map f)), by exactI infer_instance\u27e9\n\nlemma preserves_monomorphisms.of_iso {F G : C \u2964 D} [preserves_monomorphisms F] (\u03b1 : F \u2245 G) :\n  preserves_monomorphisms G :=\n{ preserves := \u03bb X Y f h,\n  begin\n    haveI : mono (F.map f \u226b (\u03b1.app Y).hom) := by exactI mono_comp _ _,\n    convert (mono_comp _ _ : mono ((\u03b1.app X).inv \u226b F.map f \u226b (\u03b1.app Y).hom)),\n    rw [iso.eq_inv_comp, iso.app_hom, iso.app_hom, nat_trans.naturality]\n  end }\n\nlemma preserves_monomorphisms.iso_iff {F G : C \u2964 D} (\u03b1 : F \u2245 G) :\n  preserves_monomorphisms F \u2194 preserves_monomorphisms G :=\n\u27e8\u03bb h, by exactI preserves_monomorphisms.of_iso \u03b1,\n \u03bb h, by exactI preserves_monomorphisms.of_iso \u03b1.symm\u27e9\n\nlemma preserves_epimorphisms.of_iso {F G : C \u2964 D} [preserves_epimorphisms F] (\u03b1 : F \u2245 G) :\n  preserves_epimorphisms G :=\n{ preserves := \u03bb X Y f h,\n  begin\n    haveI : epi (F.map f \u226b (\u03b1.app Y).hom) := by exactI epi_comp _ _,\n    convert (epi_comp _ _ : epi ((\u03b1.app X).inv \u226b F.map f \u226b (\u03b1.app Y).hom)),\n    rw [iso.eq_inv_comp, iso.app_hom, iso.app_hom, nat_trans.naturality]\n  end }\n\nlemma preserves_epimorphisms.iso_iff {F G : C \u2964 D} (\u03b1 : F \u2245 G) :\n  preserves_epimorphisms F \u2194 preserves_epimorphisms G :=\n\u27e8\u03bb h, by exactI preserves_epimorphisms.of_iso \u03b1,\n \u03bb h, by exactI preserves_epimorphisms.of_iso \u03b1.symm\u27e9\n\nlemma reflects_monomorphisms.of_iso {F G : C \u2964 D} [reflects_monomorphisms F] (\u03b1 : F \u2245 G) :\n  reflects_monomorphisms G :=\n{ reflects := \u03bb X Y f h,\n  begin\n    apply F.mono_of_mono_map,\n    haveI : mono (G.map f \u226b (\u03b1.app Y).inv) := by exactI mono_comp _ _,\n    convert (mono_comp _ _ : mono ((\u03b1.app X).hom \u226b G.map f \u226b (\u03b1.app Y).inv)),\n    rw [\u2190 category.assoc, iso.eq_comp_inv, iso.app_hom, iso.app_hom, nat_trans.naturality]\n  end }\n\nlemma reflects_monomorphisms.iso_iff {F G : C \u2964 D} (\u03b1 : F \u2245 G) :\n  reflects_monomorphisms F \u2194 reflects_monomorphisms G :=\n\u27e8\u03bb h, by exactI reflects_monomorphisms.of_iso \u03b1,\n \u03bb h, by exactI reflects_monomorphisms.of_iso \u03b1.symm\u27e9\n\nlemma reflects_epimorphisms.of_iso {F G : C \u2964 D} [reflects_epimorphisms F] (\u03b1 : F \u2245 G) :\n  reflects_epimorphisms G :=\n{ reflects := \u03bb X Y f h,\n  begin\n    apply F.epi_of_epi_map,\n    haveI : epi (G.map f \u226b (\u03b1.app Y).inv) := by exactI epi_comp _ _,\n    convert (epi_comp _ _ : epi ((\u03b1.app X).hom \u226b G.map f \u226b (\u03b1.app Y).inv)),\n    rw [\u2190 category.assoc, iso.eq_comp_inv, iso.app_hom, iso.app_hom, nat_trans.naturality]\n  end }\n\nlemma reflects_epimorphisms.iso_iff {F G : C \u2964 D} (\u03b1 : F \u2245 G) :\n  reflects_epimorphisms F \u2194 reflects_epimorphisms G :=\n\u27e8\u03bb h, by exactI reflects_epimorphisms.of_iso \u03b1, \u03bb h, by exactI reflects_epimorphisms.of_iso \u03b1.symm\u27e9\n\nlemma preserves_epimorphsisms_of_adjunction {F : C \u2964 D} {G : D \u2964 C} (adj : F \u22a3 G) :\n preserves_epimorphisms F :=\n{ preserves := \u03bb X Y f hf,\n  \u27e8begin\n    introsI Z g h H,\n    replace H := congr_arg (adj.hom_equiv X Z) H,\n    rwa [adj.hom_equiv_naturality_left, adj.hom_equiv_naturality_left, cancel_epi,\n      equiv.apply_eq_iff_eq] at H\n  end\u27e9 }\n\n@[priority 100]\ninstance preserves_epimorphisms_of_is_left_adjoint (F : C \u2964 D) [is_left_adjoint F] :\n  preserves_epimorphisms F :=\npreserves_epimorphsisms_of_adjunction (adjunction.of_left_adjoint F)\n\n\n\n@[priority 100]\ninstance preserves_monomorphisms_of_is_right_adjoint (F : C \u2964 D) [is_right_adjoint F] :\n  preserves_monomorphisms F :=\npreserves_monomorphisms_of_adjunction (adjunction.of_right_adjoint F)\n\n@[priority 100]\ninstance reflects_monomorphisms_of_faithful (F : C \u2964 D) [faithful F] : reflects_monomorphisms F :=\n{ reflects := \u03bb X Y f hf, \u27e8\u03bb Z g h hgh, by exactI F.map_injective ((cancel_mono (F.map f)).1\n    (by rw [\u2190 F.map_comp, hgh, F.map_comp]))\u27e9 }\n\n@[priority 100]\ninstance reflects_epimorphisms_of_faithful (F : C \u2964 D) [faithful F] : reflects_epimorphisms F :=\n{ reflects := \u03bb X Y f hf, \u27e8\u03bb Z g h hgh, by exactI F.map_injective ((cancel_epi (F.map f)).1\n    (by rw [\u2190 F.map_comp, hgh, F.map_comp]))\u27e9 }\n\nsection\n\nvariables (F : C \u2964 D) {X Y : C} (f : X \u27f6 Y)\n\n/-- If `F` is a fully faithful functor, split epimorphisms are preserved and reflected by `F`. -/\ndef split_epi_equiv [full F] [faithful F] : split_epi f \u2243 split_epi (F.map f) :=\n{ to_fun := \u03bb f, f.map F,\n  inv_fun := \u03bb s, begin\n    refine \u27e8F.preimage s.section_, _\u27e9,\n    apply F.map_injective,\n    simp only [map_comp, image_preimage, map_id],\n    apply split_epi.id,\n  end,\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\n@[simp]\nlemma is_split_epi_iff [full F] [faithful F] : is_split_epi (F.map f) \u2194 is_split_epi f :=\nbegin\n  split,\n  { intro h, exact is_split_epi.mk' ((split_epi_equiv F f).inv_fun h.exists_split_epi.some), },\n  { intro h, exact is_split_epi.mk' ((split_epi_equiv F f).to_fun h.exists_split_epi.some), },\nend\n\n/-- If `F` is a fully faithful functor, split monomorphisms are preserved and reflected by `F`. -/\ndef split_mono_equiv [full F] [faithful F] : split_mono f \u2243 split_mono (F.map f) :=\n{ to_fun := \u03bb f, f.map F,\n  inv_fun := \u03bb s, begin\n    refine \u27e8F.preimage s.retraction, _\u27e9,\n    apply F.map_injective,\n    simp only [map_comp, image_preimage, map_id],\n    apply split_mono.id,\n  end,\n  left_inv := by tidy,\n  right_inv := by tidy, }\n\n@[simp]\nlemma is_split_mono_iff [full F] [faithful F] : is_split_mono (F.map f) \u2194 is_split_mono f :=\nbegin\n  split,\n  { intro h, exact is_split_mono.mk' ((split_mono_equiv F f).inv_fun h.exists_split_mono.some), },\n  { intro h, exact is_split_mono.mk' ((split_mono_equiv F f).to_fun h.exists_split_mono.some), },\nend\n\n@[simp]\nlemma epi_map_iff_epi [hF\u2081 : preserves_epimorphisms F] [hF\u2082 : reflects_epimorphisms F] :\n  epi (F.map f) \u2194 epi f :=\nbegin\n  split,\n  { exact F.epi_of_epi_map, },\n  { introI h,\n    exact F.map_epi f, },\nend\n\n@[simp]\nlemma mono_map_iff_mono [hF\u2081 : preserves_monomorphisms F] [hF\u2082 : reflects_monomorphisms F] :\n  mono (F.map f) \u2194 mono f :=\nbegin\n  split,\n  { exact F.mono_of_mono_map, },\n  { introI h,\n    exact F.map_mono f, },\nend\n\n/-- If `F : C \u2964 D` is an equivalence of categories and `C` is a `split_epi_category`,\nthen `D` also is. -/\ndef split_epi_category_imp_of_is_equivalence [is_equivalence F] [split_epi_category C] :\n  split_epi_category D :=\n\u27e8\u03bb X Y f, begin\n  introI,\n  rw \u2190 F.inv.is_split_epi_iff f,\n  apply is_split_epi_of_epi,\nend\u27e9\n\nend\n\nend category_theory.functor\n\nnamespace category_theory.adjunction\n\nvariables {C D : Type*} [category C] [category D] {F : C \u2964 D} {F' : D \u2964 C} {A B : C}\n\nlemma strong_epi_map_of_strong_epi (adj : F \u22a3 F') (f : A \u27f6 B)\n  [h\u2081 : F'.preserves_monomorphisms] [h\u2082 : F.preserves_epimorphisms] [strong_epi f] :\n  strong_epi (F.map f) :=\n\u27e8infer_instance, \u03bb X Y Z, by { introI, rw adj.has_lifting_property_iff, apply_instance, }\u27e9\n\ninstance strong_epi_map_of_is_equivalence [is_equivalence F] (f : A \u27f6 B) [h : strong_epi f] :\n  strong_epi (F.map f) :=\nF.as_equivalence.to_adjunction.strong_epi_map_of_strong_epi f\n\nend category_theory.adjunction\n\nnamespace category_theory.functor\n\nvariables {C D : Type*} [category C] [category D] {F : C \u2964 D} {A B : C} (f : A \u27f6 B)\n\n@[simp]\nlemma strong_epi_map_iff_strong_epi_of_is_equivalence [is_equivalence F] :\n  strong_epi (F.map f) \u2194 strong_epi f  :=\nbegin\n  split,\n  { introI,\n    have e : arrow.mk f \u2245 arrow.mk (F.inv.map (F.map f)) :=\n      arrow.iso_of_nat_iso F.as_equivalence.unit_iso (arrow.mk f),\n    rw strong_epi.iff_of_arrow_iso e,\n    apply_instance, },\n  { introI,\n    apply_instance, },\nend\n\nend category_theory.functor\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/functor/epi_mono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.36204913775525654}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport topology.sheaves.presheaf\n\n/-!\n# Presheafed spaces\n\nIntroduces the category of topological spaces equipped with a presheaf (taking values in an\narbitrary target category `C`.)\n\nWe further describe how to apply functors and natural transformations to the values of the\npresheaves.\n-/\n\nuniverses v u\n\nopen category_theory\nopen Top\nopen topological_space\nopen opposite\nopen category_theory.category category_theory.functor\n\nvariables (C : Type u) [category.{v} C]\n\nlocal attribute [tidy] tactic.op_induction'\n\nnamespace algebraic_geometry\n\n/-- A `PresheafedSpace C` is a topological space equipped with a presheaf of `C`s. -/\nstructure PresheafedSpace :=\n(carrier : Top)\n(presheaf : carrier.presheaf C)\n\nvariables {C}\n\nnamespace PresheafedSpace\n\nattribute [protected] presheaf\n\ninstance coe_carrier : has_coe (PresheafedSpace C) Top :=\n{ coe := \u03bb X, X.carrier }\n\n@[simp] lemma as_coe (X : PresheafedSpace C) : X.carrier = (X : Top.{v}) := rfl\n@[simp] lemma mk_coe (carrier) (presheaf) : (({ carrier := carrier, presheaf := presheaf } :\n  PresheafedSpace.{v} C) : Top.{v}) = carrier := rfl\n\ninstance (X : PresheafedSpace.{v} C) : topological_space X := X.carrier.str\n\n/-- The constant presheaf on `X` with value `Z`. -/\ndef const (X : Top) (Z : C) : PresheafedSpace C :=\n{ carrier := X,\n  presheaf :=\n  { obj := \u03bb U, Z,\n    map := \u03bb U V f, \ud835\udfd9 Z, } }\n\ninstance [inhabited C] : inhabited (PresheafedSpace C) := \u27e8const (Top.of pempty) (default C)\u27e9\n\n/-- A morphism between presheafed spaces `X` and `Y` consists of a continuous map\n    `f` between the underlying topological spaces, and a (notice contravariant!) map\n    from the presheaf on `Y` to the pushforward of the presheaf on `X` via `f`. -/\nstructure hom (X Y : PresheafedSpace C) :=\n(base : (X : Top.{v}) \u27f6 (Y : Top.{v}))\n(c : Y.presheaf \u27f6 base _* X.presheaf)\n\n@[ext] lemma ext {X Y : PresheafedSpace C} (\u03b1 \u03b2 : hom X Y)\n  (w : \u03b1.base = \u03b2.base)\n  (h : \u03b1.c \u226b (whisker_right (nat_trans.op (opens.map_iso _ _ w).inv) X.presheaf) = \u03b2.c) :\n  \u03b1 = \u03b2 :=\nbegin\n  cases \u03b1, cases \u03b2,\n  dsimp [presheaf.pushforward_obj] at *,\n  tidy, -- TODO including `injections` would make tidy work earlier.\nend\n.\n\n/-- The identity morphism of a `PresheafedSpace`. -/\ndef id (X : PresheafedSpace C) : hom X X :=\n{ base := \ud835\udfd9 (X : Top.{v}),\n  c := (functor.left_unitor _).inv \u226b whisker_right (nat_trans.op (opens.map_id X.carrier).hom) _ }\n\ninstance hom_inhabited (X : PresheafedSpace C) : inhabited (hom X X) := \u27e8id X\u27e9\n\n/-- Composition of morphisms of `PresheafedSpace`s. -/\ndef comp {X Y Z : PresheafedSpace C} (\u03b1 : hom X Y) (\u03b2 : hom Y Z) : hom X Z :=\n{ base := \u03b1.base \u226b \u03b2.base,\n  c := \u03b2.c \u226b (whisker_left (opens.map \u03b2.base).op \u03b1.c) \u226b (Top.presheaf.pushforward.comp _ _ _).inv }\n\nvariables (C)\n\nsection\nlocal attribute [simp] id comp\n\n/- The proofs below can be done by `tidy`, but it is too slow,\n   and we don't have a tactic caching mechanism. -/\n/-- The category of PresheafedSpaces. Morphisms are pairs, a continuous map and a presheaf map\n    from the presheaf on the target to the pushforward of the presheaf on the source. -/\ninstance category_of_PresheafedSpaces : category (PresheafedSpace C) :=\n{ hom := hom,\n  id := id,\n  comp := \u03bb X Y Z f g, comp f g,\n  id_comp' := \u03bb X Y f,\n  begin\n    ext1, swap,\n    { dsimp, simp only [id_comp] },  -- See note [dsimp, simp].\n    { ext U, op_induction, cases U,\n      dsimp,\n      simp only [presheaf.pushforward.comp_inv_app, opens.map_iso_inv_app],\n      dsimp,\n      simp only [comp_id, comp_id, map_id], },\n  end,\n  comp_id' := \u03bb X Y f,\n  begin\n    ext1, swap,\n    { dsimp, simp only [comp_id] },\n    { ext U, op_induction, cases U,\n      dsimp,\n      simp only [presheaf.pushforward.comp_inv_app, opens.map_iso_inv_app],\n      dsimp,\n      simp only [id_comp, comp_id, map_id], }\n  end,\n  assoc' := \u03bb W X Y Z f g h,\n  begin\n     ext1, swap,\n     refl,\n     { ext U, op_induction, cases U,\n       dsimp,\n       simp only [assoc, presheaf.pushforward.comp_inv_app, opens.map_iso_inv_app],\n       dsimp,\n       simp only [comp_id, id_comp, map_id], }\n  end }\n\nend\n\nvariables {C}\n\n@[simp] lemma id_base (X : PresheafedSpace C) :\n  ((\ud835\udfd9 X) : X \u27f6 X).base = (\ud835\udfd9 (X : Top.{v})) := rfl\n\nlemma id_c (X : PresheafedSpace C) :\n  ((\ud835\udfd9 X) : X \u27f6 X).c =\n  (functor.left_unitor _).inv \u226b whisker_right (nat_trans.op (opens.map_id X.carrier).hom) _ := rfl\n\n@[simp] lemma id_c_app (X : PresheafedSpace C) (U) :\n  ((\ud835\udfd9 X) : X \u27f6 X).c.app U = eq_to_hom (by { op_induction U, cases U, refl }) :=\nby { op_induction U, cases U, simp only [id_c], dsimp, simp, }\n\n@[simp] lemma comp_base {X Y Z : PresheafedSpace C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  (f \u226b g).base = f.base \u226b g.base := rfl\n\n@[simp] lemma comp_c_app {X Y Z : PresheafedSpace C} (\u03b1 : X \u27f6 Y) (\u03b2 : Y \u27f6 Z) (U) :\n  (\u03b1 \u226b \u03b2).c.app U = (\u03b2.c).app U \u226b (\u03b1.c).app (op ((opens.map (\u03b2.base)).obj (unop U))) \u226b\n    (Top.presheaf.pushforward.comp _ _ _).inv.app U := rfl\n\nlemma congr_app {X Y : PresheafedSpace C} {\u03b1 \u03b2 : X \u27f6 Y} (h : \u03b1 = \u03b2) (U) :\n  \u03b1.c.app U = \u03b2.c.app U \u226b X.presheaf.map (eq_to_hom (by subst h)) :=\nby { subst h, dsimp, simp, }\n\nsection\nvariables (C)\n\n/-- The forgetful functor from `PresheafedSpace` to `Top`. -/\n@[simps]\ndef forget : PresheafedSpace C \u2964 Top :=\n{ obj := \u03bb X, (X : Top.{v}),\n  map := \u03bb X Y f, f.base }\n\nend\n\n/--\nThe restriction of a presheafed space along an open embedding into the space.\n-/\n@[simps]\ndef restrict {U : Top} (X : PresheafedSpace C)\n  (f : U \u27f6 (X : Top.{v})) (h : open_embedding f) : PresheafedSpace C :=\n{ carrier := U,\n  presheaf := h.is_open_map.functor.op \u22d9 X.presheaf }\n\n/--\nThe map from the restriction of a presheafed space.\n-/\n@[simps]\ndef of_restrict (U : Top) (X : PresheafedSpace C)\n  (f : U \u27f6 (X : Top.{v})) (h : open_embedding f) :\n  X.restrict f h \u27f6 X :=\n{ base := f,\n  c := { app := \u03bb V, X.presheaf.map $\n      ((h.is_open_map.adjunction.hom_equiv _ _).symm (\ud835\udfd9 $ (opens.map f).obj $ unop V)).op,\n    naturality':= \u03bb U V f, show _ = _ \u226b X.presheaf.map _,\n      by { rw [\u2190 map_comp, \u2190 map_comp], refl } } }\n\n/--\nThe map to the restriction of a presheafed space along the canonical inclusion from the top\nsubspace.\n-/\n@[simps]\ndef to_restrict_top (X : PresheafedSpace C) :\n  X \u27f6 X.restrict (opens.inclusion \u22a4) (opens.open_embedding \u22a4) :=\n{ base := \u27e8\u03bb x, \u27e8x, trivial\u27e9, continuous_def.2 $ \u03bb U \u27e8S, hS, hSU\u27e9, hSU \u25b8 hS\u27e9,\n  c := { app := \u03bb U, X.presheaf.map $ (hom_of_le $ \u03bb x hxU, \u27e8\u27e8x, trivial\u27e9, hxU, rfl\u27e9 :\n      (opens.map (\u27e8\u03bb x, \u27e8x, trivial\u27e9, continuous_def.2 $ \u03bb U \u27e8S, hS, hSU\u27e9, hSU \u25b8 hS\u27e9 :\n          X.1 \u27f6 (opens.to_Top X.1).obj \u22a4)).obj (unop U) \u27f6\n        (opens.open_embedding \u22a4).is_open_map.functor.obj (unop U)).op,\n    naturality':= \u03bb U V f, show X.presheaf.map _ \u226b _ = _ \u226b X.presheaf.map _,\n      by { rw [\u2190 map_comp, \u2190 map_comp], refl } } }\n\n/--\nThe isomorphism from the restriction to the top subspace.\n-/\n@[simps]\ndef restrict_top_iso (X : PresheafedSpace C) :\n  X.restrict (opens.inclusion \u22a4) (opens.open_embedding \u22a4) \u2245 X :=\n{ hom := X.of_restrict _ _ _,\n  inv := X.to_restrict_top,\n  hom_inv_id' := ext _ _ (concrete_category.hom_ext _ _ $ \u03bb \u27e8x, _\u27e9, rfl) $\n    nat_trans.ext _ _ $ funext $ \u03bb U, by { op_induction U,\n      dsimp only [nat_trans.comp_app, comp_c_app, to_restrict_top, of_restrict,\n          whisker_right_app, comp_base, nat_trans.op_app, opens.map_iso_inv_app],\n      erw [presheaf.pushforward.comp_inv_app, comp_id, \u2190 X.presheaf.map_comp,\n          \u2190 X.presheaf.map_comp, id_c_app],\n      exact X.presheaf.map_id _ },\n  inv_hom_id' := ext _ _ rfl $ nat_trans.ext _ _ $ funext $ \u03bb U, by { op_induction U,\n    dsimp only [nat_trans.comp_app, comp_c_app, of_restrict, to_restrict_top,\n        whisker_right_app, comp_base, nat_trans.op_app, opens.map_iso_inv_app],\n    erw [\u2190 X.presheaf.map_comp, \u2190 X.presheaf.map_comp, \u2190 X.presheaf.map_comp, id_c_app],\n    convert eq_to_hom_map X.presheaf _,\n    erw [op_obj, id_base, opens.map_id_obj], refl } }\n\n/--\nThe global sections, notated Gamma.\n-/\n@[simps]\ndef \u0393 : (PresheafedSpace C)\u1d52\u1d56 \u2964 C :=\n{ obj := \u03bb X, (unop X).presheaf.obj (op \u22a4),\n  map := \u03bb X Y f, f.unop.c.app (op \u22a4) \u226b (unop Y).presheaf.map (opens.le_map_top _ _).op,\n  map_id' := \u03bb X, begin\n    op_induction X,\n    erw [unop_id_op, id_c_app, eq_to_hom_refl, id_comp],\n    exact X.presheaf.map_id _\n  end,\n  map_comp' := \u03bb X Y Z f g, begin\n    rw [unop_comp, comp_c_app],\n    simp_rw category.assoc,\n    erw [nat_trans.naturality_assoc, presheaf.pushforward.comp_inv_app, id_comp,\n        category_theory.functor.comp_map, \u2190 map_comp],\n    refl\n  end }\n\nlemma \u0393_obj_op (X : PresheafedSpace C) : \u0393.obj (op X) = X.presheaf.obj (op \u22a4) := rfl\n\nlemma \u0393_map_op {X Y : PresheafedSpace C} (f : X \u27f6 Y) :\n  \u0393.map f.op = f.c.app (op \u22a4) \u226b X.presheaf.map (opens.le_map_top _ _).op := rfl\n\nend PresheafedSpace\n\nend algebraic_geometry\n\nopen algebraic_geometry algebraic_geometry.PresheafedSpace\n\nvariables {C}\n\nnamespace category_theory\n\nvariables {D : Type u} [category.{v} D]\n\nlocal attribute [simp] presheaf.pushforward_obj\n\nnamespace functor\n\n/-- We can apply a functor `F : C \u2964 D` to the values of the presheaf in any `PresheafedSpace C`,\n    giving a functor `PresheafedSpace C \u2964 PresheafedSpace D` -/\ndef map_presheaf (F : C \u2964 D) : PresheafedSpace C \u2964 PresheafedSpace D :=\n{ obj := \u03bb X, { carrier := X.carrier, presheaf := X.presheaf \u22d9 F },\n  map := \u03bb X Y f, { base := f.base, c := whisker_right f.c F }, }\n\n@[simp] lemma map_presheaf_obj_X (F : C \u2964 D) (X : PresheafedSpace C) :\n  ((F.map_presheaf.obj X) : Top.{v}) = (X : Top.{v}) := rfl\n@[simp] lemma map_presheaf_obj_presheaf (F : C \u2964 D) (X : PresheafedSpace C) :\n  (F.map_presheaf.obj X).presheaf = X.presheaf \u22d9 F := rfl\n@[simp] lemma map_presheaf_map_f (F : C \u2964 D) {X Y : PresheafedSpace C} (f : X \u27f6 Y) :\n  (F.map_presheaf.map f).base = f.base := rfl\n@[simp] lemma map_presheaf_map_c (F : C \u2964 D) {X Y : PresheafedSpace C} (f : X \u27f6 Y) :\n  (F.map_presheaf.map f).c = whisker_right f.c F := rfl\n\nend functor\n\nnamespace nat_trans\n\n/--\nA natural transformation induces a natural transformation between the `map_presheaf` functors.\n-/\ndef on_presheaf {F G : C \u2964 D} (\u03b1 : F \u27f6 G) : G.map_presheaf \u27f6 F.map_presheaf :=\n{ app := \u03bb X,\n  { base := \ud835\udfd9 _,\n    c := whisker_left X.presheaf \u03b1 \u226b (functor.left_unitor _).inv \u226b\n           whisker_right (nat_trans.op (opens.map_id X.carrier).hom) _ }, }\n\n-- TODO Assemble the last two constructions into a functor\n--   `(C \u2964 D) \u2964 (PresheafedSpace C \u2964 PresheafedSpace D)`\nend nat_trans\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/algebraic_geometry/presheafed_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6187804196836383, "lm_q1q2_score": 0.36204913775525654}}
{"text": "example : \u2200 a b c : \u03b1, a = b \u2192 b = c \u2192 a = c :=\n  fun a b c h1 h2 =>\n    match b, h1 with\n    | _, Eq.refl a => h2\n\nexample : \u2200 a b c : \u03b1, a = b \u2192 b = c \u2192 a = c :=\n  fun a b c h1 h2 =>\n    match h1 with\n    | Eq.refl a => h2\n\nexample : \u2200 a b c : \u03b1, a = b \u2192 b = c \u2192 a = c :=\n  fun a b c (Eq.refl a) => fun h2 => h2\n\nexample : \u2200 a b c : \u03b1, a = b \u2192 b = c \u2192 a = c :=\n  fun a b c (Eq.refl a) h2 => h2\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/matchGenIssue.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.36183753302803023}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.seq.seq\nimport Mathlib.Lean3Lib.data.dlist\nimport Mathlib.PostPort\n\nuniverses u_1 u v w \n\nnamespace Mathlib\n\n/-\ncoinductive wseq (\u03b1 : Type u) : Type u\n| nil : wseq \u03b1\n| cons : \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1\n| think : wseq \u03b1 \u2192 wseq \u03b1\n-/\n\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef wseq (\u03b1 : Type u_1) :=\n  seq (Option \u03b1)\n\nnamespace wseq\n\n\n/-- Turn a sequence into a weak sequence -/\ndef of_seq {\u03b1 : Type u} : seq \u03b1 \u2192 wseq \u03b1 :=\n  Functor.map some\n\n/-- Turn a list into a weak sequence -/\ndef of_list {\u03b1 : Type u} (l : List \u03b1) : wseq \u03b1 :=\n  of_seq \u2191l\n\n/-- Turn a stream into a weak sequence -/\ndef of_stream {\u03b1 : Type u} (l : stream \u03b1) : wseq \u03b1 :=\n  of_seq \u2191l\n\nprotected instance coe_seq {\u03b1 : Type u} : has_coe (seq \u03b1) (wseq \u03b1) :=\n  has_coe.mk of_seq\n\nprotected instance coe_list {\u03b1 : Type u} : has_coe (List \u03b1) (wseq \u03b1) :=\n  has_coe.mk of_list\n\nprotected instance coe_stream {\u03b1 : Type u} : has_coe (stream \u03b1) (wseq \u03b1) :=\n  has_coe.mk of_stream\n\n/-- The empty weak sequence -/\ndef nil {\u03b1 : Type u} : wseq \u03b1 :=\n  seq.nil\n\nprotected instance inhabited {\u03b1 : Type u} : Inhabited (wseq \u03b1) :=\n  { default := nil }\n\n/-- Prepend an element to a weak sequence -/\ndef cons {\u03b1 : Type u} (a : \u03b1) : wseq \u03b1 \u2192 wseq \u03b1 :=\n  seq.cons (some a)\n\n/-- Compute for one tick, without producing any elements -/\ndef think {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 :=\n  seq.cons none\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct {\u03b1 : Type u} : wseq \u03b1 \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  computation.corec fun (s : wseq \u03b1) => sorry\n\ndef cases_on {\u03b1 : Type u} {C : wseq \u03b1 \u2192 Sort v} (s : wseq \u03b1) (h1 : C nil) (h2 : (x : \u03b1) \u2192 (s : wseq \u03b1) \u2192 C (cons x s)) (h3 : (s : wseq \u03b1) \u2192 C (think s)) : C s :=\n  seq.cases_on s h1 fun (o : Option \u03b1) => option.cases_on o h3 h2\n\nprotected def mem {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) :=\n  seq.mem (some a) s\n\nprotected instance has_mem {\u03b1 : Type u} : has_mem \u03b1 (wseq \u03b1) :=\n  has_mem.mk wseq.mem\n\ntheorem not_mem_nil {\u03b1 : Type u} (a : \u03b1) : \u00aca \u2208 nil :=\n  seq.not_mem_nil \u2191a\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head {\u03b1 : Type u} (s : wseq \u03b1) : computation (Option \u03b1) :=\n  computation.map (Functor.map prod.fst) (destruct s)\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten {\u03b1 : Type u} : computation (wseq \u03b1) \u2192 wseq \u03b1 :=\n  seq.corec fun (c : computation (wseq \u03b1)) => sorry\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail {\u03b1 : Type u} (s : wseq \u03b1) : wseq \u03b1 :=\n  flatten ((fun (o : Option (\u03b1 \u00d7 wseq \u03b1)) => option.rec_on o nil prod.snd) <$> destruct s)\n\n/-- drop the first `n` elements from `s`. -/\n@[simp] def drop {\u03b1 : Type u} (s : wseq \u03b1) : \u2115 \u2192 wseq \u03b1 :=\n  sorry\n\n/-- Get the nth element of `s`. -/\ndef nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : computation (Option \u03b1) :=\n  head (drop s n)\n\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef to_list {\u03b1 : Type u} (s : wseq \u03b1) : computation (List \u03b1) :=\n  computation.corec (fun (_x : List \u03b1 \u00d7 wseq \u03b1) => sorry) ([], s)\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length {\u03b1 : Type u} (s : wseq \u03b1) : computation \u2115 :=\n  computation.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (0, s)\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\ndef is_finite {\u03b1 : Type u} (s : wseq \u03b1) :=\n  computation.terminates (to_list s)\n\nprotected instance to_list_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : is_finite s] : computation.terminates (to_list s) :=\n  h\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get {\u03b1 : Type u} (s : wseq \u03b1) [is_finite s] : List \u03b1 :=\n  computation.get (to_list s)\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\ndef productive {\u03b1 : Type u} (s : wseq \u03b1) :=\n  \u2200 (n : \u2115), computation.terminates (nth s n)\n\nprotected instance nth_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : productive s] (n : \u2115) : computation.terminates (nth s n) :=\n  h\n\nprotected instance head_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : productive s] : computation.terminates (head s) :=\n  h 0\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef update_nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) (a : \u03b1) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n + 1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef remove_nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n + 1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filter_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 :=\n  seq.corec fun (s : wseq \u03b1) => sorry\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] : wseq \u03b1 \u2192 wseq \u03b1 :=\n  filter_map fun (a : \u03b1) => ite (p a) (some a) none\n\n-- example of infinite list manipulations\n\n/-- Get the first element of `s` satisfying `p`. -/\ndef find {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation (Option \u03b1) :=\n  head (filter p s)\n\n/-- Zip a function over two weak sequences -/\ndef zip_with {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (s1 : wseq \u03b1) (s2 : wseq \u03b2) : wseq \u03b3 :=\n  seq.corec (fun (_x : wseq \u03b1 \u00d7 wseq \u03b2) => sorry) (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip {\u03b1 : Type u} {\u03b2 : Type v} : wseq \u03b1 \u2192 wseq \u03b2 \u2192 wseq (\u03b1 \u00d7 \u03b2) :=\n  zip_with Prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef find_indexes {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : wseq \u2115 :=\n  filter_map (fun (_x : \u03b1 \u00d7 \u2115) => sorry) (zip s \u2191stream.nats)\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef find_index {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation \u2115 :=\n  (fun (o : Option \u2115) => option.get_or_else o 0) <$> head (find_indexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef index_of {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 computation \u2115 :=\n  find_index (Eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexes_of {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 wseq \u2115 :=\n  find_indexes (Eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union {\u03b1 : Type u} (s1 : wseq \u03b1) (s2 : wseq \u03b1) : wseq \u03b1 :=\n  seq.corec (fun (_x : wseq \u03b1 \u00d7 wseq \u03b1) => sorry) (s1, s2)\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef is_empty {\u03b1 : Type u} (s : wseq \u03b1) : computation Bool :=\n  computation.map option.is_none (head s)\n\n/-- Calculate one step of computation -/\ndef compute {\u03b1 : Type u} (s : wseq \u03b1) : wseq \u03b1 :=\n  sorry\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef split_at {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : computation (List \u03b1 \u00d7 wseq \u03b1) :=\n  computation.corec (fun (_x : \u2115 \u00d7 List \u03b1 \u00d7 wseq \u03b1) => sorry) (n, [], s)\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any {\u03b1 : Type u} (s : wseq \u03b1) (p : \u03b1 \u2192 Bool) : computation Bool :=\n  computation.corec (fun (s : wseq \u03b1) => sorry) s\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all {\u03b1 : Type u} (s : wseq \u03b1) (p : \u03b1 \u2192 Bool) : computation Bool :=\n  computation.corec (fun (s : wseq \u03b1) => sorry) s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (s : wseq \u03b2) : wseq \u03b1 :=\n  cons a (seq.corec (fun (_x : \u03b1 \u00d7 wseq \u03b2) => sorry) (a, s))\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits {\u03b1 : Type u} (s : wseq \u03b1) : wseq (List \u03b1) :=\n  cons [] (seq.corec (fun (_x : dlist \u03b1 \u00d7 wseq \u03b1) => sorry) (dlist.empty, s))\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : List \u03b1 :=\n  list.filter_map id (seq.take n s)\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1 :=\n  seq.append\n\n/-- Map a function over a weak sequence -/\ndef map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 :=\n  seq.map (option.map f)\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join {\u03b1 : Type u} (S : wseq (wseq \u03b1)) : wseq \u03b1 :=\n  seq.join ((fun (o : Option (wseq \u03b1)) => sorry) <$> S)\n\n/-- Monadic bind operator for weak sequences -/\ndef bind {\u03b1 : Type u} {\u03b2 : Type v} (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2) : wseq \u03b2 :=\n  join (map f s)\n\n@[simp] def lift_rel_o {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop) : Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Option (\u03b2 \u00d7 wseq \u03b2) \u2192 Prop :=\n  sorry\n\ntheorem lift_rel_o.imp {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {S : \u03b1 \u2192 \u03b2 \u2192 Prop} {C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} {D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} (H1 : \u2200 (a : \u03b1) (b : \u03b2), R a b \u2192 S a b) (H2 : \u2200 (s : wseq \u03b1) (t : wseq \u03b2), C s t \u2192 D s t) {o : Option (\u03b1 \u00d7 wseq \u03b1)} {p : Option (\u03b2 \u00d7 wseq \u03b2)} : lift_rel_o R C o p \u2192 lift_rel_o S D o p := sorry\n\ntheorem lift_rel_o.imp_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} {D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} (H : \u2200 (s : wseq \u03b1) (t : wseq \u03b2), C s t \u2192 D s t) {o : Option (\u03b1 \u00d7 wseq \u03b1)} {p : Option (\u03b2 \u00d7 wseq \u03b2)} : lift_rel_o R C o p \u2192 lift_rel_o R D o p :=\n  lift_rel_o.imp (fun (_x : \u03b1) (_x_1 : \u03b2) => id) H\n\n@[simp] def bisim_o {\u03b1 : Type u} (R : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop) : Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Prop :=\n  lift_rel_o Eq R\n\ntheorem bisim_o.imp {\u03b1 : Type u} {R : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop} {S : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop} (H : \u2200 (s t : wseq \u03b1), R s t \u2192 S s t) {o : Option (\u03b1 \u00d7 wseq \u03b1)} {p : Option (\u03b1 \u00d7 wseq \u03b1)} : bisim_o R o p \u2192 bisim_o S o p :=\n  lift_rel_o.imp_right Eq H\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef lift_rel {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1) (t : wseq \u03b2) :=\n  \u2203 (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop),\n    C s t \u2227 \u2200 {s : wseq \u03b1} {t : wseq \u03b2}, C s t \u2192 computation.lift_rel (lift_rel_o R C) (destruct s) (destruct t)\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef equiv {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop :=\n  lift_rel Eq\n\ntheorem lift_rel_destruct {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} : lift_rel R s t \u2192 computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) := sorry\n\ntheorem lift_rel_destruct_iff {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} : lift_rel R s t \u2194 computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) := sorry\n\ninfixl:50 \" ~ \" => Mathlib.wseq.equiv\n\ntheorem destruct_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2192 computation.lift_rel (bisim_o equiv) (destruct s) (destruct t) :=\n  lift_rel_destruct\n\ntheorem destruct_congr_iff {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2194 computation.lift_rel (bisim_o equiv) (destruct s) (destruct t) :=\n  lift_rel_destruct_iff\n\ntheorem lift_rel.refl {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : reflexive R) : reflexive (lift_rel R) := sorry\n\ntheorem lift_rel_o.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop) : function.swap (lift_rel_o R C) = lift_rel_o (function.swap R) (function.swap C) := sorry\n\ntheorem lift_rel.swap_lem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s1 : wseq \u03b1} {s2 : wseq \u03b2} (h : lift_rel R s1 s2) : lift_rel (function.swap R) s2 s1 := sorry\n\ntheorem lift_rel.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) : function.swap (lift_rel R) = lift_rel (function.swap R) :=\n  funext fun (x : wseq \u03b2) => funext fun (y : wseq \u03b1) => propext { mp := lift_rel.swap_lem, mpr := lift_rel.swap_lem }\n\ntheorem lift_rel.symm {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : symmetric R) : symmetric (lift_rel R) := sorry\n\ntheorem lift_rel.trans {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : transitive R) : transitive (lift_rel R) := sorry\n\ntheorem lift_rel.equiv {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : equivalence R \u2192 equivalence (lift_rel R) := sorry\n\ntheorem equiv.refl {\u03b1 : Type u} (s : wseq \u03b1) : s ~ s :=\n  lift_rel.refl Eq Eq.refl\n\ntheorem equiv.symm {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2192 t ~ s :=\n  lift_rel.symm Eq Eq.symm\n\ntheorem equiv.trans {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} {u : wseq \u03b1} : s ~ t \u2192 t ~ u \u2192 s ~ u :=\n  lift_rel.trans Eq Eq.trans\n\ntheorem equiv.equivalence {\u03b1 : Type u} : equivalence equiv :=\n  { left := equiv.refl, right := { left := equiv.symm, right := equiv.trans } }\n\n@[simp] theorem destruct_nil {\u03b1 : Type u} : destruct nil = computation.return none :=\n  computation.destruct_eq_ret rfl\n\n@[simp] theorem destruct_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : destruct (cons a s) = computation.return (some (a, s)) := sorry\n\n@[simp] theorem destruct_think {\u03b1 : Type u} (s : wseq \u03b1) : destruct (think s) = computation.think (destruct s) := sorry\n\n@[simp] theorem seq_destruct_nil {\u03b1 : Type u} : seq.destruct nil = none :=\n  seq.destruct_nil\n\n@[simp] theorem seq_destruct_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : seq.destruct (cons a s) = some (some a, s) :=\n  seq.destruct_cons (some a) s\n\n@[simp] theorem seq_destruct_think {\u03b1 : Type u} (s : wseq \u03b1) : seq.destruct (think s) = some (none, s) :=\n  seq.destruct_cons none s\n\n@[simp] theorem head_nil {\u03b1 : Type u} : head nil = computation.return none := sorry\n\n@[simp] theorem head_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : head (cons a s) = computation.return (some a) := sorry\n\n@[simp] theorem head_think {\u03b1 : Type u} (s : wseq \u03b1) : head (think s) = computation.think (head s) := sorry\n\n@[simp] theorem flatten_ret {\u03b1 : Type u} (s : wseq \u03b1) : flatten (computation.return s) = s := sorry\n\n@[simp] theorem flatten_think {\u03b1 : Type u} (c : computation (wseq \u03b1)) : flatten (computation.think c) = think (flatten c) := sorry\n\n@[simp] theorem destruct_flatten {\u03b1 : Type u} (c : computation (wseq \u03b1)) : destruct (flatten c) = c >>= destruct := sorry\n\ntheorem head_terminates_iff {\u03b1 : Type u} (s : wseq \u03b1) : computation.terminates (head s) \u2194 computation.terminates (destruct s) :=\n  computation.terminates_map_iff (Functor.map prod.fst) (destruct s)\n\n@[simp] theorem tail_nil {\u03b1 : Type u} : tail nil = nil := sorry\n\n@[simp] theorem tail_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : tail (cons a s) = s := sorry\n\n@[simp] theorem tail_think {\u03b1 : Type u} (s : wseq \u03b1) : tail (think s) = think (tail s) := sorry\n\n@[simp] theorem dropn_nil {\u03b1 : Type u} (n : \u2115) : drop nil n = nil := sorry\n\n@[simp] theorem dropn_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) (n : \u2115) : drop (cons a s) (n + 1) = drop s n := sorry\n\n@[simp] theorem dropn_think {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : drop (think s) n = think (drop s n) := sorry\n\ntheorem dropn_add {\u03b1 : Type u} (s : wseq \u03b1) (m : \u2115) (n : \u2115) : drop s (m + n) = drop (drop s m) n := sorry\n\ntheorem dropn_tail {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : drop (tail s) n = drop s (n + 1) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (drop (tail s) n = drop s (n + 1))) (add_comm n 1))) (Eq.symm (dropn_add s 1 n))\n\ntheorem nth_add {\u03b1 : Type u} (s : wseq \u03b1) (m : \u2115) (n : \u2115) : nth s (m + n) = nth (drop s m) n :=\n  congr_arg head (dropn_add s m n)\n\ntheorem nth_tail {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : nth (tail s) n = nth s (n + 1) :=\n  congr_arg head (dropn_tail s n)\n\n@[simp] theorem join_nil {\u03b1 : Type u} : join nil = nil :=\n  seq.join_nil\n\n@[simp] theorem join_think {\u03b1 : Type u} (S : wseq (wseq \u03b1)) : join (think S) = think (join S) := sorry\n\n@[simp] theorem join_cons {\u03b1 : Type u} (s : wseq \u03b1) (S : wseq (wseq \u03b1)) : join (cons s S) = think (append s (join S)) := sorry\n\n@[simp] theorem nil_append {\u03b1 : Type u} (s : wseq \u03b1) : append nil s = s :=\n  seq.nil_append s\n\n@[simp] theorem cons_append {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) (t : wseq \u03b1) : append (cons a s) t = cons a (append s t) :=\n  seq.cons_append (some a) s t\n\n@[simp] theorem think_append {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) : append (think s) t = think (append s t) :=\n  seq.cons_append none s t\n\n@[simp] theorem append_nil {\u03b1 : Type u} (s : wseq \u03b1) : append s nil = s :=\n  seq.append_nil s\n\n@[simp] theorem append_assoc {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) (u : wseq \u03b1) : append (append s t) u = append s (append t u) :=\n  seq.append_assoc s t u\n\n@[simp] def tail.aux {\u03b1 : Type u} : Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem destruct_tail {\u03b1 : Type u} (s : wseq \u03b1) : destruct (tail s) = destruct s >>= tail.aux := sorry\n\n@[simp] def drop.aux {\u03b1 : Type u} : \u2115 \u2192 Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem drop.aux_none {\u03b1 : Type u} (n : \u2115) : drop.aux n none = computation.return none := sorry\n\ntheorem destruct_dropn {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : destruct (drop s n) = destruct s >>= drop.aux n := sorry\n\ntheorem head_terminates_of_head_tail_terminates {\u03b1 : Type u} (s : wseq \u03b1) [T : computation.terminates (head (tail s))] : computation.terminates (head s) := sorry\n\ntheorem destruct_some_of_destruct_tail_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1 \u00d7 wseq \u03b1} (h : some a \u2208 destruct (tail s)) : \u2203 (a' : \u03b1 \u00d7 wseq \u03b1), some a' \u2208 destruct s := sorry\n\ntheorem head_some_of_head_tail_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : some a \u2208 head (tail s)) : \u2203 (a' : \u03b1), some a' \u2208 head s := sorry\n\ntheorem head_some_of_nth_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} (h : some a \u2208 nth s n) : \u2203 (a' : \u03b1), some a' \u2208 head s := sorry\n\nprotected instance productive_tail {\u03b1 : Type u} (s : wseq \u03b1) [productive s] : productive (tail s) :=\n  fun (n : \u2115) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (computation.terminates (nth (tail s) n))) (nth_tail s n)))\n      (wseq.nth_terminates s (n + 1))\n\nprotected instance productive_dropn {\u03b1 : Type u} (s : wseq \u03b1) [productive s] (n : \u2115) : productive (drop s n) :=\n  fun (m : \u2115) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (computation.terminates (nth (drop s n) m))) (Eq.symm (nth_add s n m))))\n      (wseq.nth_terminates s (n + m))\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef to_seq {\u03b1 : Type u} (s : wseq \u03b1) [productive s] : seq \u03b1 :=\n  { val := fun (n : \u2115) => computation.get (nth s n), property := sorry }\n\ntheorem nth_terminates_le {\u03b1 : Type u} {s : wseq \u03b1} {m : \u2115} {n : \u2115} (h : m \u2264 n) : computation.terminates (nth s n) \u2192 computation.terminates (nth s m) := sorry\n\ntheorem head_terminates_of_nth_terminates {\u03b1 : Type u} {s : wseq \u03b1} {n : \u2115} : computation.terminates (nth s n) \u2192 computation.terminates (head s) :=\n  nth_terminates_le (nat.zero_le n)\n\ntheorem destruct_terminates_of_nth_terminates {\u03b1 : Type u} {s : wseq \u03b1} {n : \u2115} (T : computation.terminates (nth s n)) : computation.terminates (destruct s) :=\n  iff.mp (head_terminates_iff s) (head_terminates_of_nth_terminates T)\n\ntheorem mem_rec_on {\u03b1 : Type u} {C : wseq \u03b1 \u2192 Prop} {a : \u03b1} {s : wseq \u03b1} (M : a \u2208 s) (h1 : \u2200 (b : \u03b1) (s' : wseq \u03b1), a = b \u2228 C s' \u2192 C (cons b s')) (h2 : \u2200 (s : wseq \u03b1), C s \u2192 C (think s)) : C s := sorry\n\n@[simp] theorem mem_think {\u03b1 : Type u} (s : wseq \u03b1) (a : \u03b1) : a \u2208 think s \u2194 a \u2208 s := sorry\n\ntheorem eq_or_mem_iff_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {a' : \u03b1} {s' : wseq \u03b1} : some (a', s') \u2208 destruct s \u2192 (a \u2208 s \u2194 a = a' \u2228 a \u2208 s') := sorry\n\n@[simp] theorem mem_cons_iff {\u03b1 : Type u} (s : wseq \u03b1) (b : \u03b1) {a : \u03b1} : a \u2208 cons b s \u2194 a = b \u2228 a \u2208 s := sorry\n\ntheorem mem_cons_of_mem {\u03b1 : Type u} {s : wseq \u03b1} (b : \u03b1) {a : \u03b1} (h : a \u2208 s) : a \u2208 cons b s :=\n  iff.mpr (mem_cons_iff s b) (Or.inr h)\n\ntheorem mem_cons {\u03b1 : Type u} (s : wseq \u03b1) (a : \u03b1) : a \u2208 cons a s :=\n  iff.mpr (mem_cons_iff s a) (Or.inl rfl)\n\ntheorem mem_of_mem_tail {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} : a \u2208 tail s \u2192 a \u2208 s := sorry\n\ntheorem mem_of_mem_dropn {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} : a \u2208 drop s n \u2192 a \u2208 s := sorry\n\ntheorem nth_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} : some a \u2208 nth s n \u2192 a \u2208 s := sorry\n\ntheorem exists_nth_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) : \u2203 (n : \u2115), some a \u2208 nth s n := sorry\n\ntheorem exists_dropn_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) : \u2203 (n : \u2115), \u2203 (s' : wseq \u03b1), some (a, s') \u2208 destruct (drop s n) := sorry\n\ntheorem lift_rel_dropn_destruct {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} (H : lift_rel R s t) (n : \u2115) : computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct (drop s n)) (destruct (drop t n)) := sorry\n\ntheorem exists_of_lift_rel_left {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} (H : lift_rel R s t) {a : \u03b1} (h : a \u2208 s) : Exists fun {b : \u03b2} => b \u2208 t \u2227 R a b := sorry\n\ntheorem exists_of_lift_rel_right {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} (H : lift_rel R s t) {b : \u03b2} (h : b \u2208 t) : Exists fun {a : \u03b1} => a \u2208 s \u2227 R a b :=\n  exists_of_lift_rel_left\n    (eq.mp (Eq._oldrec (Eq.refl (lift_rel R s t)) (Eq.symm (lift_rel.swap fun (x : \u03b2) (y : \u03b1) => R y x))) H) h\n\ntheorem head_terminates_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) : computation.terminates (head s) := sorry\n\ntheorem of_mem_append {\u03b1 : Type u} {s\u2081 : wseq \u03b1} {s\u2082 : wseq \u03b1} {a : \u03b1} : a \u2208 append s\u2081 s\u2082 \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\n  seq.of_mem_append\n\ntheorem mem_append_left {\u03b1 : Type u} {s\u2081 : wseq \u03b1} {s\u2082 : wseq \u03b1} {a : \u03b1} : a \u2208 s\u2081 \u2192 a \u2208 append s\u2081 s\u2082 :=\n  seq.mem_append_left\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : wseq \u03b1} : b \u2208 map f s \u2192 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b := sorry\n\n@[simp] theorem lift_rel_nil {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) : lift_rel R nil nil := sorry\n\n@[simp] theorem lift_rel_cons {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1) (b : \u03b2) (s : wseq \u03b1) (t : wseq \u03b2) : lift_rel R (cons a s) (cons b t) \u2194 R a b \u2227 lift_rel R s t := sorry\n\n@[simp] theorem lift_rel_think_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1) (t : wseq \u03b2) : lift_rel R (think s) t \u2194 lift_rel R s t := sorry\n\n@[simp] theorem lift_rel_think_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1) (t : wseq \u03b2) : lift_rel R s (think t) \u2194 lift_rel R s t := sorry\n\ntheorem cons_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (a : \u03b1) (h : s ~ t) : cons a s ~ cons a t := sorry\n\ntheorem think_equiv {\u03b1 : Type u} (s : wseq \u03b1) : think s ~ s :=\n  eq.mpr (id (congr_fun (congr_fun equiv.equations._eqn_1 (think s)) s))\n    (eq.mpr (id (propext (lift_rel_think_left Eq s s))) (equiv.refl s))\n\ntheorem think_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (a : \u03b1) (h : s ~ t) : think s ~ think t :=\n  eq.mpr (id (congr_fun (congr_fun equiv.equations._eqn_1 (think s)) (think t)))\n    (eq.mpr (id (Eq.trans (propext (lift_rel_think_right Eq (think s) t)) (propext (lift_rel_think_left Eq s t)))) h)\n\ntheorem head_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2192 head s ~ head t := sorry\n\ntheorem flatten_equiv {\u03b1 : Type u} {c : computation (wseq \u03b1)} {s : wseq \u03b1} (h : s \u2208 c) : flatten c ~ s := sorry\n\ntheorem lift_rel_flatten {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {c1 : computation (wseq \u03b1)} {c2 : computation (wseq \u03b2)} (h : computation.lift_rel (lift_rel R) c1 c2) : lift_rel R (flatten c1) (flatten c2) := sorry\n\ntheorem flatten_congr {\u03b1 : Type u} {c1 : computation (wseq \u03b1)} {c2 : computation (wseq \u03b1)} : computation.lift_rel equiv c1 c2 \u2192 flatten c1 ~ flatten c2 :=\n  lift_rel_flatten\n\ntheorem tail_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) : tail s ~ tail t := sorry\n\ntheorem dropn_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (n : \u2115) : drop s n ~ drop t n := sorry\n\ntheorem nth_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (n : \u2115) : nth s n ~ nth t n :=\n  head_congr (dropn_congr h n)\n\ntheorem mem_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (a : \u03b1) : a \u2208 s \u2194 a \u2208 t := sorry\n\ntheorem productive_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) : productive s \u2194 productive t :=\n  forall_congr fun (n : \u2115) => computation.terminates_congr (nth_congr h n)\n\ntheorem equiv.ext {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : \u2200 (n : \u2115), nth s n ~ nth t n) : s ~ t := sorry\n\ntheorem length_eq_map {\u03b1 : Type u} (s : wseq \u03b1) : length s = computation.map list.length (to_list s) := sorry\n\n@[simp] theorem of_list_nil {\u03b1 : Type u} : of_list [] = nil :=\n  rfl\n\n@[simp] theorem of_list_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : of_list (a :: l) = cons a (of_list l) := sorry\n\n@[simp] theorem to_list'_nil {\u03b1 : Type u} (l : List \u03b1) : computation.corec to_list._match_2 (l, nil) = computation.return (list.reverse l) :=\n  computation.destruct_eq_ret rfl\n\n@[simp] theorem to_list'_cons {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) (a : \u03b1) : computation.corec to_list._match_2 (l, cons a s) = computation.think (computation.corec to_list._match_2 (a :: l, s)) := sorry\n\n@[simp] theorem to_list'_think {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) : computation.corec to_list._match_2 (l, think s) = computation.think (computation.corec to_list._match_2 (l, s)) := sorry\n\ntheorem to_list'_map {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) : computation.corec to_list._match_2 (l, s) = append (list.reverse l) <$> to_list s := sorry\n\n@[simp] theorem to_list_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : to_list (cons a s) = computation.think (List.cons a <$> to_list s) := sorry\n\n@[simp] theorem to_list_nil {\u03b1 : Type u} : to_list nil = computation.return [] :=\n  computation.destruct_eq_ret rfl\n\ntheorem to_list_of_list {\u03b1 : Type u} (l : List \u03b1) : l \u2208 to_list (of_list l) := sorry\n\n@[simp] theorem destruct_of_seq {\u03b1 : Type u} (s : seq \u03b1) : destruct (of_seq s) = computation.return (option.map (fun (a : \u03b1) => (a, of_seq (seq.tail s))) (seq.head s)) := sorry\n\n@[simp] theorem head_of_seq {\u03b1 : Type u} (s : seq \u03b1) : head (of_seq s) = computation.return (seq.head s) := sorry\n\n@[simp] theorem tail_of_seq {\u03b1 : Type u} (s : seq \u03b1) : tail (of_seq s) = of_seq (seq.tail s) := sorry\n\n@[simp] theorem dropn_of_seq {\u03b1 : Type u} (s : seq \u03b1) (n : \u2115) : drop (of_seq s) n = of_seq (seq.drop s n) := sorry\n\ntheorem nth_of_seq {\u03b1 : Type u} (s : seq \u03b1) (n : \u2115) : nth (of_seq s) n = computation.return (seq.nth s n) := sorry\n\nprotected instance productive_of_seq {\u03b1 : Type u} (s : seq \u03b1) : productive (of_seq s) :=\n  fun (n : \u2115) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (computation.terminates (nth (of_seq s) n))) (nth_of_seq s n)))\n      (computation.ret_terminates (seq.nth s n))\n\ntheorem to_seq_of_seq {\u03b1 : Type u} (s : seq \u03b1) : to_seq (of_seq s) = s := sorry\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret {\u03b1 : Type u} (a : \u03b1) : wseq \u03b1 :=\n  of_list [a]\n\n@[simp] theorem map_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : map f nil = nil :=\n  rfl\n\n@[simp] theorem map_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (s : wseq \u03b1) : map f (cons a s) = cons (f a) (map f s) :=\n  seq.map_cons (option.map f) (some a) s\n\n@[simp] theorem map_think {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) : map f (think s) = think (map f s) :=\n  seq.map_cons (option.map f) none s\n\n@[simp] theorem map_id {\u03b1 : Type u} (s : wseq \u03b1) : map id s = s := sorry\n\n@[simp] theorem map_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (ret a) = ret (f a) := sorry\n\n@[simp] theorem map_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) (t : wseq \u03b1) : map f (append s t) = append (map f s) (map f t) :=\n  seq.map_append (option.map f) s t\n\ntheorem map_comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (s : wseq \u03b1) : map (g \u2218 f) s = map g (map f s) := sorry\n\ntheorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : wseq \u03b1} : a \u2208 s \u2192 f a \u2208 map f s :=\n  seq.mem_map (option.map f)\n\n-- The converse is not true without additional assumptions\n\ntheorem exists_of_mem_join {\u03b1 : Type u} {a : \u03b1} {S : wseq (wseq \u03b1)} : a \u2208 join S \u2192 \u2203 (s : wseq \u03b1), s \u2208 S \u2227 a \u2208 s := sorry\n\ntheorem exists_of_mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : wseq \u03b1} {f : \u03b1 \u2192 wseq \u03b2} {b : \u03b2} (h : b \u2208 bind s f) : \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a := sorry\n\ntheorem destruct_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) : destruct (map f s) = computation.map (option.map (prod.map f (map f))) (destruct s) := sorry\n\ntheorem lift_rel_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b2} {f1 : \u03b1 \u2192 \u03b3} {f2 : \u03b2 \u2192 \u03b4} (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)) : lift_rel S (map f1 s1) (map f2 s2) := sorry\n\ntheorem map_congr {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) : map f s ~ map f t :=\n  lift_rel_map Eq Eq h fun (_x _x_1 : \u03b1) => congr_arg fun (_x : \u03b1) => f _x\n\n@[simp] def destruct_append.aux {\u03b1 : Type u} (t : wseq \u03b1) : Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem destruct_append {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) : destruct (append s t) = computation.bind (destruct s) (destruct_append.aux t) := sorry\n\n@[simp] def destruct_join.aux {\u03b1 : Type u} : Option (wseq \u03b1 \u00d7 wseq (wseq \u03b1)) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem destruct_join {\u03b1 : Type u} (S : wseq (wseq \u03b1)) : destruct (join S) = computation.bind (destruct S) destruct_join.aux := sorry\n\ntheorem lift_rel_append {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b1} {t1 : wseq \u03b2} {t2 : wseq \u03b2} (h1 : lift_rel R s1 t1) (h2 : lift_rel R s2 t2) : lift_rel R (append s1 s2) (append t1 t2) := sorry\n\ntheorem lift_rel_join.lem {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S : wseq (wseq \u03b1)} {T : wseq (wseq \u03b2)} {U : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} (ST : lift_rel (lift_rel R) S T) (HU : \u2200 (s1 : wseq \u03b1) (s2 : wseq \u03b2),\n  (\u2203 (s : wseq \u03b1),\n      \u2203 (t : wseq \u03b2),\n        \u2203 (S : wseq (wseq \u03b1)),\n          \u2203 (T : wseq (wseq \u03b2)),\n            s1 = append s (join S) \u2227 s2 = append t (join T) \u2227 lift_rel R s t \u2227 lift_rel (lift_rel R) S T) \u2192\n    U s1 s2) {a : Option (\u03b1 \u00d7 wseq \u03b1)} (ma : a \u2208 destruct (join S)) : Exists fun {b : Option (\u03b2 \u00d7 wseq \u03b2)} => b \u2208 destruct (join T) \u2227 lift_rel_o R U a b := sorry\n\ntheorem lift_rel_join {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S : wseq (wseq \u03b1)} {T : wseq (wseq \u03b2)} (h : lift_rel (lift_rel R) S T) : lift_rel R (join S) (join T) := sorry\n\ntheorem join_congr {\u03b1 : Type u} {S : wseq (wseq \u03b1)} {T : wseq (wseq \u03b1)} (h : lift_rel equiv S T) : join S ~ join T :=\n  lift_rel_join Eq h\n\ntheorem lift_rel_bind {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b2} {f1 : \u03b1 \u2192 wseq \u03b3} {f2 : \u03b2 \u2192 wseq \u03b4} (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 lift_rel S (f1 a) (f2 b)) : lift_rel S (bind s1 f1) (bind s2 f2) :=\n  lift_rel_join S (lift_rel_map R (lift_rel S) h1 h2)\n\ntheorem bind_congr {\u03b1 : Type u} {\u03b2 : Type v} {s1 : wseq \u03b1} {s2 : wseq \u03b1} {f1 : \u03b1 \u2192 wseq \u03b2} {f2 : \u03b1 \u2192 wseq \u03b2} (h1 : s1 ~ s2) (h2 : \u2200 (a : \u03b1), f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=\n  lift_rel_bind Eq Eq h1\n    fun (a b : \u03b1) (h : a = b) => eq.mpr (id (Eq._oldrec (Eq.refl (lift_rel Eq (f1 a) (f2 b))) h)) (h2 b)\n\n@[simp] theorem join_ret {\u03b1 : Type u} (s : wseq \u03b1) : join (ret s) ~ s := sorry\n\n@[simp] theorem join_map_ret {\u03b1 : Type u} (s : wseq \u03b1) : join (map ret s) ~ s := sorry\n\n@[simp] theorem join_append {\u03b1 : Type u} (S : wseq (wseq \u03b1)) (T : wseq (wseq \u03b1)) : join (append S T) ~ append (join S) (join T) := sorry\n\n@[simp] theorem bind_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) : bind s (ret \u2218 f) ~ map f s :=\n  id (eq.mpr (id (Eq._oldrec (Eq.refl (join (map (ret \u2218 f) s) ~ map f s)) (map_comp f ret s))) (join_map_ret (map f s)))\n\n@[simp] theorem ret_bind {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (f : \u03b1 \u2192 wseq \u03b2) : bind (ret a) f ~ f a := sorry\n\n@[simp] theorem map_join {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (S : wseq (wseq \u03b1)) : map f (join S) = join (map (map f) S) := sorry\n\n@[simp] theorem join_join {\u03b1 : Type u} (SS : wseq (wseq (wseq \u03b1))) : join (join SS) ~ join (map join SS) := sorry\n\n@[simp] theorem bind_assoc {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2) (g : \u03b2 \u2192 wseq \u03b3) : bind (bind s f) g ~ bind s fun (x : \u03b1) => bind (f x) g := sorry\n\nprotected instance monad : Monad wseq :=\n  { toApplicative :=\n      { toFunctor := { map := map, mapConst := fun (\u03b1 \u03b2 : Type u_1) => map \u2218 function.const \u03b2 },\n        toPure := { pure := ret },\n        toSeq := { seq := fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) => bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : wseq \u03b1) (b : wseq \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) => bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x) \u03b2 \u03b1\n                  (map (function.const \u03b2) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : wseq \u03b1) (b : wseq \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) => bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x) \u03b2 \u03b2\n                  (map (function.const \u03b1 id) a) b } },\n    toBind := { bind := bind } }\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/seq/wseq.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6224593312018545, "lm_q1q2_score": 0.36183753302803023}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  -- Let $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), \n  from by auto [int.fract_eq_iff, h\u03b1_irrat, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_eq_iff, int.fract_\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : \u03b5 > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 \u03b5 h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 \u03b5 h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by auto [h8, h10, h5, h9],\n\n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by auto [h11] using [linarith],\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-4_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7718435083355187, "lm_q2_score": 0.4687906266262437, "lm_q1q2_score": 0.36183300193000617}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Mario Carneiro\n-/\nimport topology.metric_space.basic\nimport topology.algebra.uniform_group\nimport topology.algebra.uniform_mul_action\nimport topology.algebra.ring.basic\nimport topology.algebra.star\nimport topology.algebra.order.field\nimport ring_theory.subring.basic\nimport group_theory.archimedean\nimport algebra.order.group.bounds\nimport algebra.periodic\nimport topology.instances.int\n\n/-!\n# Topological properties of \u211d\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nnoncomputable theory\nopen classical filter int metric set topological_space\nopen_locale classical topology filter uniformity interval\n\nuniverses u v w\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w}\n\ninstance : noncompact_space \u211d := int.closed_embedding_coe_real.noncompact_space\n\ntheorem real.uniform_continuous_add : uniform_continuous (\u03bbp : \u211d \u00d7 \u211d, p.1 + p.2) :=\nmetric.uniform_continuous_iff.2 $ \u03bb \u03b5 \u03b50,\nlet \u27e8\u03b4, \u03b40, H\u03b4\u27e9 := rat_add_continuous_lemma abs \u03b50 in\n\u27e8\u03b4, \u03b40, \u03bb a b h, let \u27e8h\u2081, h\u2082\u27e9 := max_lt_iff.1 h in H\u03b4 h\u2081 h\u2082\u27e9\n\n\ntheorem real.uniform_continuous_neg : uniform_continuous (@has_neg.neg \u211d _) :=\nmetric.uniform_continuous_iff.2 $ \u03bb \u03b5 \u03b50, \u27e8_, \u03b50, \u03bb a b h,\n  by rw dist_comm at h; simpa [real.dist_eq] using h\u27e9\n\ninstance : has_continuous_star \u211d := \u27e8continuous_id\u27e9\n\ninstance : uniform_add_group \u211d :=\nuniform_add_group.mk' real.uniform_continuous_add real.uniform_continuous_neg\n\n -- short-circuit type class inference\ninstance : topological_add_group \u211d := by apply_instance\n\ninstance : proper_space \u211d :=\n{ is_compact_closed_ball := \u03bbx r, by { rw real.closed_ball_eq_Icc, apply is_compact_Icc } }\n\ninstance : second_countable_topology \u211d := second_countable_of_proper\n\nlemma real.is_topological_basis_Ioo_rat :\n  @is_topological_basis \u211d _ (\u22c3(a b : \u211a) (h : a < b), {Ioo a b}) :=\nis_topological_basis_of_open_of_nhds\n  (by simp [is_open_Ioo] {contextual:=tt})\n  (assume a v hav hv,\n    let \u27e8l, u, \u27e8hl, hu\u27e9, h\u27e9 := mem_nhds_iff_exists_Ioo_subset.mp (is_open.mem_nhds hv hav),\n        \u27e8q, hlq, hqa\u27e9 := exists_rat_btwn hl,\n        \u27e8p, hap, hpu\u27e9 := exists_rat_btwn hu in\n    \u27e8Ioo q p,\n      by { simp only [mem_Union], exact \u27e8q, p, rat.cast_lt.1 $ hqa.trans hap, rfl\u27e9 },\n      \u27e8hqa, hap\u27e9, assume a' \u27e8hqa', ha'p\u27e9, h \u27e8hlq.trans hqa', ha'p.trans hpu\u27e9\u27e9)\n\n@[simp] lemma real.cocompact_eq : cocompact \u211d = at_bot \u2294 at_top :=\nby simp only [\u2190 comap_dist_right_at_top_eq_cocompact (0 : \u211d), real.dist_eq, sub_zero,\n  comap_abs_at_top]\n\n/- TODO(Mario): Prove that these are uniform isomorphisms instead of uniform embeddings\nlemma uniform_embedding_add_rat {r : \u211a} : uniform_embedding (\u03bbp:\u211a, p + r) :=\n_\n\nlemma uniform_embedding_mul_rat {q : \u211a} (hq : q \u2260 0) : uniform_embedding ((*) q) :=\n_ -/\n\nlemma real.mem_closure_iff {s : set \u211d} {x : \u211d} :\n  x \u2208 closure s \u2194 \u2200 \u03b5 > 0, \u2203 y \u2208 s, |y - x| < \u03b5 :=\nby simp [mem_closure_iff_nhds_basis nhds_basis_ball, real.dist_eq]\n\nlemma real.uniform_continuous_inv (s : set \u211d) {r : \u211d} (r0 : 0 < r) (H : \u2200 x \u2208 s, r \u2264 |x|) :\n  uniform_continuous (\u03bbp:s, p.1\u207b\u00b9) :=\nmetric.uniform_continuous_iff.2 $ \u03bb \u03b5 \u03b50,\nlet \u27e8\u03b4, \u03b40, H\u03b4\u27e9 := rat_inv_continuous_lemma abs \u03b50 r0 in\n\u27e8\u03b4, \u03b40, \u03bb a b h, H\u03b4 (H _ a.2) (H _ b.2) h\u27e9\n\nlemma real.uniform_continuous_abs : uniform_continuous (abs : \u211d \u2192 \u211d) :=\nmetric.uniform_continuous_iff.2 $ \u03bb \u03b5 \u03b50,\n  \u27e8\u03b5, \u03b50, \u03bb a b, lt_of_le_of_lt (abs_abs_sub_abs_le_abs_sub _ _)\u27e9\n\nlemma real.tendsto_inv {r : \u211d} (r0 : r \u2260 0) : tendsto (\u03bbq, q\u207b\u00b9) (\ud835\udcdd r) (\ud835\udcdd r\u207b\u00b9) :=\nby rw \u2190 abs_pos at r0; exact\ntendsto_of_uniform_continuous_subtype\n  (real.uniform_continuous_inv {x | |r| / 2 < |x|} (half_pos r0) (\u03bb x h, le_of_lt h))\n  (is_open.mem_nhds ((is_open_lt' (|r| / 2)).preimage continuous_abs) (half_lt_self r0))\n\nlemma real.continuous_inv : continuous (\u03bba:{r:\u211d // r \u2260 0}, a.val\u207b\u00b9) :=\ncontinuous_iff_continuous_at.mpr $ assume \u27e8r, hr\u27e9,\n  tendsto.comp (real.tendsto_inv hr) (continuous_iff_continuous_at.mp continuous_subtype_val _)\n\nlemma real.continuous.inv [topological_space \u03b1] {f : \u03b1 \u2192 \u211d} (h : \u2200a, f a \u2260 0) (hf : continuous f) :\n  continuous (\u03bba, (f a)\u207b\u00b9) :=\nshow continuous ((has_inv.inv \u2218 @subtype.val \u211d (\u03bbr, r \u2260 0)) \u2218 \u03bba, \u27e8f a, h a\u27e9),\n  from real.continuous_inv.comp (hf.subtype_mk _)\n\nlemma real.uniform_continuous_const_mul {x : \u211d} : uniform_continuous ((*) x) :=\nuniform_continuous_const_smul x\n\nlemma real.uniform_continuous_mul (s : set (\u211d \u00d7 \u211d))\n  {r\u2081 r\u2082 : \u211d} (H : \u2200 x \u2208 s, |(x : \u211d \u00d7 \u211d).1| < r\u2081 \u2227 |x.2| < r\u2082) :\n  uniform_continuous (\u03bbp:s, p.1.1 * p.1.2) :=\nmetric.uniform_continuous_iff.2 $ \u03bb \u03b5 \u03b50,\nlet \u27e8\u03b4, \u03b40, H\u03b4\u27e9 := rat_mul_continuous_lemma abs \u03b50 in\n\u27e8\u03b4, \u03b40, \u03bb a b h,\n  let \u27e8h\u2081, h\u2082\u27e9 := max_lt_iff.1 h in H\u03b4 (H _ a.2).1 (H _ b.2).2 h\u2081 h\u2082\u27e9\n\nprotected lemma real.continuous_mul : continuous (\u03bbp : \u211d \u00d7 \u211d, p.1 * p.2) :=\ncontinuous_iff_continuous_at.2 $ \u03bb \u27e8a\u2081, a\u2082\u27e9,\ntendsto_of_uniform_continuous_subtype\n  (real.uniform_continuous_mul\n    ({x | |x| < |a\u2081| + 1} \u00d7\u02e2 {x | |x| < |a\u2082| + 1})\n    (\u03bb x, id))\n  (is_open.mem_nhds\n    (((is_open_gt' (|a\u2081| + 1)).preimage continuous_abs).prod\n      ((is_open_gt' (|a\u2082| + 1)).preimage continuous_abs ))\n    \u27e8lt_add_one (|a\u2081|), lt_add_one (|a\u2082|)\u27e9)\n\ninstance : topological_ring \u211d :=\n{ continuous_mul := real.continuous_mul, ..real.topological_add_group }\n\ninstance : complete_space \u211d :=\nbegin\n  apply complete_of_cauchy_seq_tendsto,\n  intros u hu,\n  let c : cau_seq \u211d abs := \u27e8u, metric.cauchy_seq_iff'.1 hu\u27e9,\n  refine \u27e8c.lim, \u03bb s h, _\u27e9,\n  rcases metric.mem_nhds_iff.1 h with \u27e8\u03b5, \u03b50, h\u03b5\u27e9,\n  have := c.equiv_lim \u03b5 \u03b50,\n  simp only [mem_map, mem_at_top_sets, mem_set_of_eq],\n  refine this.imp (\u03bb N hN n hn, h\u03b5 (hN n hn))\nend\n\nlemma real.totally_bounded_ball (x \u03b5 : \u211d) : totally_bounded (ball x \u03b5) :=\nby rw real.ball_eq_Ioo; apply totally_bounded_Ioo\n\nsection\n\nlemma closure_of_rat_image_lt {q : \u211a} : closure ((coe:\u211a \u2192 \u211d) '' {x | q < x}) = {r | \u2191q \u2264 r} :=\nsubset.antisymm\n  ((is_closed_ge' _).closure_subset_iff.2\n    (image_subset_iff.2 $ \u03bb p h, le_of_lt $ (@rat.cast_lt \u211d _ _ _).2 h)) $\n\u03bb x hx, mem_closure_iff_nhds.2 $ \u03bb t ht,\nlet \u27e8\u03b5, \u03b50, h\u03b5\u27e9 := metric.mem_nhds_iff.1 ht in\nlet \u27e8p, h\u2081, h\u2082\u27e9 := exists_rat_btwn ((lt_add_iff_pos_right x).2 \u03b50) in\n\u27e8_, h\u03b5 (show abs _ < _,\n    by rwa [abs_of_nonneg (le_of_lt $ sub_pos.2 h\u2081), sub_lt_iff_lt_add']),\n  p, rat.cast_lt.1 (@lt_of_le_of_lt \u211d _ _ _ _ hx h\u2081), rfl\u27e9\n\n/- TODO(Mario): Put these back only if needed later\nlemma closure_of_rat_image_le_eq {q : \u211a} : closure ((coe:\u211a \u2192 \u211d) '' {x | q \u2264 x}) = {r | \u2191q \u2264 r} :=\n_\n\nlemma closure_of_rat_image_le_le_eq {a b : \u211a} (hab : a \u2264 b) :\n  closure (of_rat '' {q:\u211a | a \u2264 q \u2227 q \u2264 b}) = {r:\u211d | of_rat a \u2264 r \u2227 r \u2264 of_rat b} :=\n_-/\n\nlemma real.bounded_iff_bdd_below_bdd_above {s : set \u211d} : bounded s \u2194 bdd_below s \u2227 bdd_above s :=\n\u27e8begin\n  assume bdd,\n  rcases (bounded_iff_subset_ball 0).1 bdd with \u27e8r, hr\u27e9, -- hr : s \u2286 closed_ball 0 r\n  rw real.closed_ball_eq_Icc at hr, -- hr : s \u2286 Icc (0 - r) (0 + r)\n  exact \u27e8bdd_below_Icc.mono hr, bdd_above_Icc.mono hr\u27e9\nend,\n\u03bb h, bounded_of_bdd_above_of_bdd_below h.2 h.1\u27e9\n\nlemma real.subset_Icc_Inf_Sup_of_bounded {s : set \u211d} (h : bounded s) :\n  s \u2286 Icc (Inf s) (Sup s) :=\nsubset_Icc_cInf_cSup (real.bounded_iff_bdd_below_bdd_above.1 h).1\n  (real.bounded_iff_bdd_below_bdd_above.1 h).2\n\nend\n\nsection periodic\n\nnamespace function\n\nlemma periodic.compact_of_continuous' [topological_space \u03b1] {f : \u211d \u2192 \u03b1} {c : \u211d}\n  (hp : periodic f c) (hc : 0 < c) (hf : continuous f) :\n  is_compact (range f) :=\nbegin\n  convert is_compact_Icc.image hf,\n  ext x,\n  refine \u27e8_, mem_range_of_mem_image f (Icc 0 c)\u27e9,\n  rintros \u27e8y, h1\u27e9,\n  obtain \u27e8z, hz, h2\u27e9 := hp.exists_mem_Ico\u2080 hc y,\n  exact \u27e8z, mem_Icc_of_Ico hz, h2.symm.trans h1\u27e9,\nend\n\n/-- A continuous, periodic function has compact range. -/\nlemma periodic.compact_of_continuous [topological_space \u03b1] {f : \u211d \u2192 \u03b1} {c : \u211d}\n  (hp : periodic f c) (hc : c \u2260 0) (hf : continuous f) :\n  is_compact (range f) :=\nbegin\n  cases lt_or_gt_of_ne hc with hneg hpos,\n  exacts [hp.neg.compact_of_continuous' (neg_pos.mpr hneg) hf, hp.compact_of_continuous' hpos hf],\nend\n\n/-- A continuous, periodic function is bounded. -/\nlemma periodic.bounded_of_continuous [pseudo_metric_space \u03b1] {f : \u211d \u2192 \u03b1} {c : \u211d}\n  (hp : periodic f c) (hc : c \u2260 0) (hf : continuous f) :\n  bounded (range f) :=\n(hp.compact_of_continuous hc hf).bounded\n\nend function\n\nend periodic\n\nsection subgroups\n\nnamespace int\nopen metric\n\n/-- Under the coercion from `\u2124` to `\u211d`, inverse images of compact sets are finite. -/\nlemma tendsto_coe_cofinite : tendsto (coe : \u2124 \u2192 \u211d) cofinite (cocompact \u211d) :=\nbegin\n  refine tendsto_cocompact_of_tendsto_dist_comp_at_top (0 : \u211d) _,\n  simp only [filter.tendsto_at_top, eventually_cofinite, not_le, \u2190 mem_ball],\n  change \u2200 r : \u211d, (coe \u207b\u00b9' (ball (0 : \u211d) r)).finite,\n  simp [real.ball_eq_Ioo, set.finite_Ioo],\nend\n\n/-- For nonzero `a`, the \"multiples of `a`\" map `zmultiples_hom` from `\u2124` to `\u211d` is discrete, i.e.\ninverse images of compact sets are finite. -/\nlemma tendsto_zmultiples_hom_cofinite {a : \u211d} (ha : a \u2260 0) :\n  tendsto (zmultiples_hom \u211d a) cofinite (cocompact \u211d) :=\nbegin\n  convert (tendsto_cocompact_mul_right\u2080 ha).comp int.tendsto_coe_cofinite,\n  ext n,\n  simp,\nend\n\nend int\n\nnamespace add_subgroup\n\n/-- The subgroup \"multiples of `a`\" (`zmultiples a`) is a discrete subgroup of `\u211d`, i.e. its\nintersection with compact sets is finite. -/\nlemma tendsto_zmultiples_subtype_cofinite (a : \u211d) :\n  tendsto (zmultiples a).subtype cofinite (cocompact \u211d) :=\nbegin\n  rcases eq_or_ne a 0 with rfl | ha,\n  { rw add_subgroup.zmultiples_zero_eq_bot,\n    intros K hK,\n    rw [filter.mem_map, mem_cofinite],\n    apply set.to_finite },\n  intros K hK,\n  have H := int.tendsto_zmultiples_hom_cofinite ha hK,\n  simp only [filter.mem_map, mem_cofinite, \u2190 preimage_compl] at \u22a2 H,\n  rw [\u2190 (zmultiples_hom \u211d a).range_restrict_surjective.image_preimage\n    ((zmultiples a).subtype \u207b\u00b9' K\u1d9c), \u2190 preimage_comp, \u2190 add_monoid_hom.coe_comp_range_restrict],\n  exact finite.image _ H,\nend\n\nend add_subgroup\n\n/-- Given a nontrivial subgroup `G \u2286 \u211d`, if `G \u2229 \u211d_{>0}` has no minimum then `G` is dense. -/\nlemma real.subgroup_dense_of_no_min {G : add_subgroup \u211d} {g\u2080 : \u211d} (g\u2080_in : g\u2080 \u2208 G) (g\u2080_ne : g\u2080 \u2260 0)\n  (H' : \u00ac \u2203 a : \u211d, is_least {g : \u211d | g \u2208 G \u2227 0 < g} a) :\n  dense (G : set \u211d) :=\nbegin\n  let G_pos := {g : \u211d | g \u2208 G \u2227 0 < g},\n  push_neg at H',\n  intros x,\n  suffices : \u2200 \u03b5 > (0 : \u211d), \u2203 g \u2208 G, |x - g| < \u03b5,\n    by simpa only [real.mem_closure_iff, abs_sub_comm],\n  intros \u03b5 \u03b5_pos,\n  obtain \u27e8g\u2081, g\u2081_in, g\u2081_pos\u27e9 : \u2203 g\u2081 : \u211d, g\u2081 \u2208 G \u2227 0 < g\u2081,\n  { cases lt_or_gt_of_ne g\u2080_ne with Hg\u2080 Hg\u2080,\n    { exact \u27e8-g\u2080, G.neg_mem g\u2080_in, neg_pos.mpr Hg\u2080\u27e9 },\n    { exact \u27e8g\u2080, g\u2080_in, Hg\u2080\u27e9 } },\n  obtain \u27e8a, ha\u27e9 : \u2203 a, is_glb G_pos a :=\n    \u27e8Inf G_pos, is_glb_cInf \u27e8g\u2081, g\u2081_in, g\u2081_pos\u27e9 \u27e80, \u03bb _ hx, le_of_lt hx.2\u27e9\u27e9,\n  have a_notin : a \u2209 G_pos,\n  { intros H,\n    exact H' a \u27e8H, ha.1\u27e9 },\n  obtain \u27e8g\u2082, g\u2082_in, g\u2082_pos, g\u2082_lt\u27e9 : \u2203 g\u2082 : \u211d, g\u2082 \u2208 G \u2227 0 < g\u2082 \u2227 g\u2082 < \u03b5,\n  { obtain \u27e8b, hb, hb', hb''\u27e9 := ha.exists_between_self_add' a_notin \u03b5_pos,\n    obtain \u27e8c, hc, hc', hc''\u27e9 := ha.exists_between_self_add' a_notin (sub_pos.2 hb'),\n    refine \u27e8b - c, G.sub_mem hb.1 hc.1, _, _\u27e9 ;\n    linarith },\n  refine \u27e8floor (x/g\u2082) * g\u2082, _, _\u27e9,\n  { exact add_subgroup.int_mul_mem _ g\u2082_in },\n  { rw abs_of_nonneg (sub_floor_div_mul_nonneg x g\u2082_pos),\n    linarith [sub_floor_div_mul_lt x g\u2082_pos] }\nend\n\n/-- Subgroups of `\u211d` are either dense or cyclic. See `real.subgroup_dense_of_no_min` and\n`subgroup_cyclic_of_min` for more precise statements. -/\nlemma real.subgroup_dense_or_cyclic (G : add_subgroup \u211d) :\n  dense (G : set \u211d) \u2228 \u2203 a : \u211d, G = add_subgroup.closure {a} :=\nbegin\n  cases add_subgroup.bot_or_exists_ne_zero G with H H,\n  { right,\n    use 0,\n    rw [H, add_subgroup.closure_singleton_zero] },\n  { let G_pos := {g : \u211d | g \u2208 G \u2227 0 < g},\n    by_cases H' : \u2203 a, is_least G_pos a,\n    { right,\n      rcases H' with \u27e8a, ha\u27e9,\n      exact \u27e8a, add_subgroup.cyclic_of_min ha\u27e9 },\n    { left,\n      rcases H with \u27e8g\u2080, g\u2080_in, g\u2080_ne\u27e9,\n      exact real.subgroup_dense_of_no_min g\u2080_in g\u2080_ne H' } }\nend\n\nend subgroups\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/topology/instances/real.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.36171248182902893}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n--Prove that Overflow  in Lean mathlib\n \n--OUTPUT 1\ntheorem overflow_thm : \u2200 n m p : \u2124, (n < m) \u2227 (m < p) \u2192 (n < p) :=\nbegin\n  intros n m p h,\n  cases h with nm mp,\n  apply lt_of_lt_of_le,\n  -- This proof is done using 3 different tactics in one block\n  { apply nm },\n  { apply le_trans mp nm }\n\n--OUTPUT 2\ntheorem Overflow:\n\u2200(n:\u2115), \u2203(y:\u2115), (2 * y) \u2265 n\n-> \u2203(x:\u2115), (2 * x) < n\n-> n = 0\n:=\nbegin\n\nintro n,\nintro EX1,\nintro EX2,\ncases EX2 with x EX2,\ncases EX1 with y EX1,\n\nhave HYP1: 2 * y \u2265 (2 * x) + 1,\nhave HYP2: 2 * x < 2 * y,\n\nlinarith,\n\nhave HYP3: \u00ac(2 * x + 1 = 2 * y),\n\nintro CONTRA,\n\nhave LHS: 2 * y = 2 * x + 1,\nrw CONTRA,\n\nhave RHS: 2 * x + 1 \u2223 2 * y,\nexistsi (2 * y) / (2 * x + 1),\nsimp,\nnorm_num,\n\nhave HYP4: \u2203(w:\u2115), 2 * x + 1 = 2 * w,\napply eq_two_factors,\nassumption,\n\nhave HYP5: 2 * x + 1 = 2 * (2 * x + 1),\nexistsi (2 * x + 1),\nsimp,\nnorm_num,\n\nhave HYP6: 2 \u2223 (2 * x + 1),\nexistsi (2 * x + 1),\nsimp,\nnorm_num,\n\nhave HYP7: 2 \u2223 (2 * y),\nexistsi (2 * y) / 2,\nsimp,\nnorm_num,\n\nhave RHS: (2 * x + 1) = (2 * y),\nrw LHS,\n\nhave HYP8: 2 \u2264 2 * x + 1,\nlinarith,\n\nhave HYP9: 2 \u2264 2 * y,\nlinarith,\n\napply div_eq_of_eq_mul,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\nassumption,\n\nhave HYP10: 2 * y \u2265 n,\nlinarith,\n\nhave HYP11: n = 0,\nlinarith,\nassumption,\n\nassumption,\nassumption,\nassumption,\nassumption,\n\n\n--OUTPUT 3\ntheorem overflow_proof {a b c : \u2124} (h : a * b = c) (bound: \u2124) (bound_pos: 0 < bound) : c \u2264 a * (b - bound) \u2227 c \u2265 a * (b + bound) :=\nbegin\n    rw \u2190 h,\n    -- From the assumption, we know that a * b = c.\n    -- So we can just rewrite the concatenation and\n    -- start to prove it.\n    have h1 : a * (b - bound) \u2264 a * b + -(a * bound), {\n        apply add_le_add_right,\n        apply mul_le_mul_of_ge_left h_1; \n        -- Here, we use the hypothesis 'h_1' which\n        -- means b - bound >= 0, given by the \n        -- tactic 'cases' above.\n        all_goals {exact bound_pos},\n        -- And we can also use all_goals to \n        -- examine all the goals generated, in\n        -- this case, we get a new subgoal which\n        -- claims that a >= 0. We can use the\n        -- hypothesis 'bound_pos' which is given\n        -- by the parameter of the function.\n    },\n    have h2 : a * (b - bound) \u2264 a * b + -(a * bound), {\n        linarith,\n        -- The tactic 'linarith' will\n        -- automatically prove this\n        -- goal for us.\n    },\n    have h3 : a * b + -(a * bound) \u2264 c, {\n        rw \u2190 h,\n        -- rewrite the equation again\n        linarith,\n    },\n    -- Now, we can just use the transitivity of\n    -- the relation '<=' to prove the first goal\n    -- which is c <= a * (b - bound).\n    have h4 : c \u2264 a * (b - bound), {\n        transitivity,\n        {\n            exact h1,\n        },\n        {\n            exact h2,\n        },\n    },\n    -- Now let's prove the second goal c >= a * (b + bound).\n    have h5 : a * b + -(a * bound) \u2264 a * (b + bound), {\n        linarith,\n    },\n    have h6 : c \u2264 a * (b + bound), {\n        transitivity,\n        {\n            exact h3,\n        },\n        {\n            exact h5,\n        },\n    },\n    split,\n    {\n        exact h6,\n    },\n    {\n        -- To prove the second goal, we need to \n        -- prove the commutativity of the relation \n        -- '<=' firstly.\n        have h7 : a * (b + bound) \u2264 a * (b - bound), {\n            have h8 : b + bound \u2264 b - bound, {\n                linarith,\n            },\n            apply mul_le_mul_of_ge_left h8; \n            -- Here, we use the hypothesis 'h8' which\n            -- means b + bound <= b - bound, given by the \n            -- tactic 'linarith' above.\n            all_goals {exact bound_pos},\n            -- And we can also use all_goals to \n            -- examine all the goals generated, in\n            -- this case, we get a new subgoal which\n            -- claims that a >= 0. We can use the\n            -- hypothesis 'bound_pos' which is given\n            -- by the parameter of the function.\n        },\n        -- Now, we just need to use the transitivity \n        -- of the relation '<=' to prove the second \n        -- goal.\n        have h9 : c \u2265 a * (b + bound), {\n            transitivity,\n            {\n                exact h4,\n            },\n            {\n                exact h7,\n            },\n        },\n        exact h9,\n    }\n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/Zero-shot-theorem-names/Zero-Shot-prompts_temperature_0.8_max_tokens_2000_n_3/clean_files/Overflow theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3617124818290289}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn\n-/\nimport category_theory.natural_transformation\nimport category_theory.isomorphism\n\n/-!\n# The category of functors and natural transformations between two fixed categories.\n\nWe provide the category instance on `C \u2964 D`, with morphisms the natural transformations.\n\n## Universes\n\nIf `C` and `D` are both small categories at the same universe level,\nthis is another small category at that level.\nHowever if `C` and `D` are both large categories at the same universe level,\nthis is a small category at the next higher level.\n-/\n\nnamespace category_theory\n\n-- declare the `v`'s first; see `category_theory.category` for an explanation\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nopen nat_trans category category_theory.functor\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2082) [category.{v\u2082} D]\n\nlocal attribute [simp] vcomp_app\n/--\n`functor.category C D` gives the category structure on functors and natural transformations\nbetween categories `C` and `D`.\n\nNotice that if `C` and `D` are both small categories at the same universe level,\nthis is another small category at that level.\nHowever if `C` and `D` are both large categories at the same universe level,\nthis is a small category at the next higher level.\n-/\ninstance functor.category : category.{(max u\u2081 v\u2082)} (C \u2964 D) :=\n{ hom     := \u03bb F G, nat_trans F G,\n  id      := \u03bb F, nat_trans.id F,\n  comp    := \u03bb _ _ _ \u03b1 \u03b2, vcomp \u03b1 \u03b2 }\n\nvariables {C D} {E : Type u\u2083} [category.{v\u2083} E]\nvariables {F G H I : C \u2964 D}\n\nnamespace nat_trans\n\n@[simp] lemma vcomp_eq_comp (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) : vcomp \u03b1 \u03b2 = \u03b1 \u226b \u03b2 := rfl\n\nlemma vcomp_app' (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) (X : C) :\n  (\u03b1 \u226b \u03b2).app X = (\u03b1.app X) \u226b (\u03b2.app X) := rfl\n\nlemma congr_app {\u03b1 \u03b2 : F \u27f6 G} (h : \u03b1 = \u03b2) (X : C) : \u03b1.app X = \u03b2.app X := by rw h\n@[simp] lemma id_app (F : C \u2964 D) (X : C) : (\ud835\udfd9 F : F \u27f6 F).app X = \ud835\udfd9 (F.obj X) := rfl\n@[simp] lemma comp_app {F G H : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) (X : C) :\n  (\u03b1 \u226b \u03b2).app X = \u03b1.app X \u226b \u03b2.app X := rfl\n\nlemma app_naturality {F G : C \u2964 (D \u2964 E)} (T : F \u27f6 G) (X : C) {Y Z : D} (f : Y \u27f6 Z) :\n  ((F.obj X).map f) \u226b ((T.app X).app Z) = ((T.app X).app Y) \u226b ((G.obj X).map f) :=\n(T.app X).naturality f\n\nlemma naturality_app {F G : C \u2964 (D \u2964 E)} (T : F \u27f6 G) (Z : D) {X Y : C} (f : X \u27f6 Y) :\n  ((F.map f).app Z) \u226b ((T.app Y).app Z) = ((T.app X).app Z) \u226b ((G.map f).app Z) :=\ncongr_fun (congr_arg app (T.naturality f)) Z\n\n/-- A natural transformation is a monomorphism if each component is. -/\nlemma mono_app_of_mono (\u03b1 : F \u27f6 G) [\u2200 (X : C), mono (\u03b1.app X)] : mono \u03b1 :=\n\u27e8\u03bb H g h eq, by { ext X, rw [\u2190cancel_mono (\u03b1.app X), \u2190comp_app, eq, comp_app] }\u27e9\n\n/-- A natural transformation is an epimorphism if each component is. -/\nlemma epi_app_of_epi (\u03b1 : F \u27f6 G) [\u2200 (X : C), epi (\u03b1.app X)] : epi \u03b1 :=\n\u27e8\u03bb H g h eq, by { ext X, rw [\u2190cancel_epi (\u03b1.app X), \u2190comp_app, eq, comp_app] }\u27e9\n\n/-- `hcomp \u03b1 \u03b2` is the horizontal composition of natural transformations. -/\ndef hcomp {H I : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) : (F \u22d9 H) \u27f6 (G \u22d9 I) :=\n{ app         := \u03bb X : C, (\u03b2.app (F.obj X)) \u226b (I.map (\u03b1.app X)),\n  naturality' := \u03bb X Y f,\n  begin\n    rw [functor.comp_map, functor.comp_map, \u2190assoc, naturality, assoc,\n        \u2190map_comp I, naturality, map_comp, assoc]\n  end }\n\ninfix ` \u25eb `:80 := hcomp\n\n@[simp] lemma hcomp_app {H I : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (X : C) :\n  (\u03b1 \u25eb \u03b2).app X = (\u03b2.app (F.obj X)) \u226b (I.map (\u03b1.app X)) := rfl\n\n@[simp] lemma hcomp_id_app {H : D \u2964 E} (\u03b1 : F \u27f6 G) (X : C) : (\u03b1 \u25eb \ud835\udfd9 H).app X = H.map (\u03b1.app X) :=\n  by {dsimp, simp} -- See note [dsimp, simp].\n\nlemma id_hcomp_app {H : E \u2964 C} (\u03b1 : F \u27f6 G) (X : E) : (\ud835\udfd9 H \u25eb \u03b1).app X = \u03b1.app _ := by simp\n\n-- Note that we don't yet prove a `hcomp_assoc` lemma here: even stating it is painful, because we\n-- need to use associativity of functor composition. (It's true without the explicit associator,\n-- because functor composition is definitionally associative,\n-- but relying on the definitional equality causes bad problems with elaboration later.)\n\nlemma exchange {I J K : D \u2964 E} (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H)\n  (\u03b3 : I \u27f6 J) (\u03b4 : J \u27f6 K) : (\u03b1 \u226b \u03b2) \u25eb (\u03b3 \u226b \u03b4) = (\u03b1 \u25eb \u03b3) \u226b (\u03b2 \u25eb \u03b4) :=\nby ext; simp\n\nend nat_trans\nopen nat_trans\nnamespace functor\n\n/-- Flip the arguments of a bifunctor. See also `currying.lean`. -/\nprotected def flip (F : C \u2964 (D \u2964 E)) : D \u2964 (C \u2964 E) :=\n{ obj := \u03bb k,\n  { obj := \u03bb j, (F.obj j).obj k,\n    map := \u03bb j j' f, (F.map f).app k,\n    map_id' := \u03bb X, begin rw category_theory.functor.map_id, refl end,\n    map_comp' := \u03bb X Y Z f g, by rw [map_comp, \u2190comp_app] },\n  map := \u03bb c c' f,\n  { app := \u03bb j, (F.obj j).map f } }.\n\n@[simp] lemma flip_obj_obj (F : C \u2964 (D \u2964 E)) (c) (d) : (F.flip.obj d).obj c = (F.obj c).obj d := rfl\n@[simp] lemma flip_obj_map (F : C \u2964 (D \u2964 E)) {c c' : C} (f : c \u27f6 c') (d : D) :\n  (F.flip.obj d).map f = (F.map f).app d := rfl\n@[simp] lemma flip_map_app (F : C \u2964 (D \u2964 E)) {d d' : D} (f : d \u27f6 d') (c : C) :\n  (F.flip.map f).app c = (F.obj c).map f := rfl\n\nend functor\n\n@[simp, reassoc] lemma map_hom_inv_app (F : C \u2964 D \u2964 E) {X Y : C} (e : X \u2245 Y) (Z : D) :\n  (F.map e.hom).app Z \u226b (F.map e.inv).app Z = \ud835\udfd9 _ :=\nby simp [\u2190 nat_trans.comp_app, \u2190 functor.map_comp]\n\n@[simp, reassoc] lemma map_inv_hom_app (F : C \u2964 D \u2964 E) {X Y : C} (e : X \u2245 Y) (Z : D) :\n  (F.map e.inv).app Z \u226b (F.map e.hom).app Z = \ud835\udfd9 _ :=\nby simp [\u2190 nat_trans.comp_app, \u2190 functor.map_comp]\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/functor_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512488, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3616832395577797}}
{"text": "import .diffeomorph\n\nopen function set\nopen_locale topological_space\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\n{E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n{E' : Type*} [normed_group E'] [normed_space \ud835\udd5c E']\n{F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n{F' : Type*} [normed_group F'] [normed_space \ud835\udd5c F']\n{H : Type*} [topological_space H]\n{H' : Type*} [topological_space H']\n{G : Type*} [topological_space G]\n{G' : Type*} [topological_space G']\n\n/-- local diffeomorphisms, defined on open subsets of the space -/\n@[nolint has_inhabited_instance]\nstructure local_diffeomorph\n(I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n(M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\nextends local_homeomorph M M' :=\n(smooth_to_fun      : smooth_in_charts_on I I' to_fun source)\n(smooth_inv_fun     : smooth_in_charts_on I' I inv_fun target)\n\n/-- A diffomorphism induces a local diffeomorphism on the whole space -/\ndef diffeomorph.to_local_diffeomorph\n(I : model_with_corners \ud835\udd5c E H) (I' : model_with_corners \ud835\udd5c E' H')\n(M : Type*) [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n(M' : Type*) [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n(e : diffeomorph I I' M M') :\n  local_diffeomorph I I' M M' :=\n{ smooth_to_fun  := by { simp only [smooth_in_charts_on], erw times_cont_mdiff_in_charts_on_univ, exact e.smooth_to_fun },\n  smooth_inv_fun := by { simp only [smooth_in_charts_on], erw times_cont_mdiff_in_charts_on_univ, exact e.smooth_inv_fun },\n  ..e.to_homeomorph.to_local_homeomorph }\n\nnamespace local_diffeomorph\n\nvariables {I : model_with_corners \ud835\udd5c E H} {I' : model_with_corners \ud835\udd5c E' H'}\n{J : model_with_corners \ud835\udd5c F G} {J' : model_with_corners \ud835\udd5c F' G'}\n{M : Type*} [topological_space M] [charted_space H M] [smooth_manifold_with_corners I M]\n{M' : Type*} [topological_space M'] [charted_space H' M'] [smooth_manifold_with_corners I' M']\n{N : Type*} [topological_space N] [charted_space G N] [smooth_manifold_with_corners J N]\n{N' : Type*} [topological_space N'] [charted_space G' N'] [smooth_manifold_with_corners J' N']\n(e : local_diffeomorph I I' M M') (e' : local_diffeomorph J J' N N')\n\ninstance : has_coe (local_diffeomorph I I' M M') (local_homeomorph M M') := \u27e8local_diffeomorph.to_local_homeomorph\u27e9\ninstance : has_coe_to_fun (local_diffeomorph I I' M M') := \u27e8_, \u03bb e, e.to_local_equiv.to_fun\u27e9\n\n/-- The inverse of a local homeomorphism -/\nprotected def symm : local_diffeomorph I' I M' M :=\n{ smooth_to_fun      := e.smooth_inv_fun,\n  smooth_inv_fun     := e.smooth_to_fun,\n  ..e.to_local_homeomorph.symm }\n\nprotected lemma smooth_in_charts_on : smooth_in_charts_on I I' e e.source := e.smooth_to_fun\n\nlemma smooth_in_charts_on_symm : smooth_in_charts_on I' I e.symm e.target := e.smooth_inv_fun\n\n/- Register a few simp lemmas to make sure that `simp` puts the application of a local\ndiffeomorphism in its normal form, i.e., in terms of its coercion to a function. -/\n\n@[simp, mfld_simps] lemma to_fun_eq_coe (e : local_diffeomorph I I' M M') : e.to_fun = e := rfl\n\n@[simp, mfld_simps] lemma inv_fun_eq_coe (e : local_diffeomorph I I' M M') : e.inv_fun = e.symm := rfl\n\n@[simp, mfld_simps] lemma coe_coe : (e.to_local_equiv : M \u2192 M') = e := rfl\n\n@[simp, mfld_simps] lemma coe_coe_symm : (e.to_local_equiv.symm : M' \u2192 M) = e.symm := rfl\n\n@[simp, mfld_simps] lemma map_source {x : M} (h : x \u2208 e.source) : e x \u2208 e.target :=\ne.map_source' h\n\n@[simp, mfld_simps] lemma map_target {x : M'} (h : x \u2208 e.target) : e.symm x \u2208 e.source :=\ne.map_target' h\n\n@[simp, mfld_simps] lemma left_inv {x : M} (h : x \u2208 e.source) : e.symm (e x) = x :=\ne.left_inv' h\n\n@[simp, mfld_simps] lemma right_inv {x : M'} (h : x \u2208 e.target) : e (e.symm x) = x :=\ne.right_inv' h\n\nlemma eq_of_local_equiv_eq {e e' : local_diffeomorph I I' M M'}\n  (h : e.to_local_equiv = e'.to_local_equiv) : e = e' :=\nbegin\n  cases e, cases e',\n  dsimp at *,\n  induction h,\n  refl\nend\n\nlemma eventually_left_inverse (e : local_diffeomorph I I' M M') {x} (hx : x \u2208 e.source) :\n  \u2200\u1da0 y in \ud835\udcdd x, e.symm (e y) = y :=\nfilter.eventually.mono (mem_nhds_sets e.open_source hx) e.left_inv'\n\nend local_diffeomorph", "meta": {"author": "Nicknamen", "repo": "lie_group", "sha": "e0d5c4f859654e3dea092702f1320c3c72a49983", "save_path": "github-repos/lean/Nicknamen-lie_group", "path": "github-repos/lean/Nicknamen-lie_group/lie_group-e0d5c4f859654e3dea092702f1320c3c72a49983/src/local_diffeomorph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521105, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3616832319650137}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem two_colorable_bipartite {G : Type*} [graph G] : two_colorable G \u2194 bipartite G :=\nbegin\n  have h1 : two_colorable G \u2192 bipartite G, from sorry,\n  have h2 : bipartite G \u2192 two_colorable G, from sorry,\n  show two_colorable G \u2194 bipartite G, from sorry,\nend\n\n--OUTPUT 2\ntheorem two_colorable_iff_bipartite {G : Type*} [graph G] : two_colorable G \u2194 bipartite G :=\nbegin\n  sorry,\nend\n\n--OUTPUT 3\ntheorem bipartite_graph_two_colorable {G : Type*} [graph G] : is_bipartite G \u2194 is_two_colorable G :=\nbegin\n  show is_bipartite G \u2194 is_two_colorable G, from sorry,\nend\n\n--OUTPUT 4\ntheorem bipartite_two_colorable (G : Type*) [graph G] : two_colorable G \u2194 bipartite G :=\nbegin\n  have h1 : two_colorable G \u2192 bipartite G, from by {\n    assume h1 : two_colorable G,\n    have h2 : \u2203 A B : set G, \u2200 a : G, a \u2208 A \u2228 a \u2208 B \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac(a b)) \u2227 (\u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac(a b)) \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 (a b)), from sorry,\n    show bipartite G, from sorry,\n  },\n  have h2 : bipartite G \u2192 two_colorable G, from by {\n    assume h2 : bipartite G,\n    have h3 : \u2203 A B : set G, \u2200 a : G, a \u2208 A \u2228 a \u2208 B \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac(a b)) \u2227 (\u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac(a b)) \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 (a b)), from sorry,\n    show two_colorable G, from sorry,\n  },\n  show two_colorable G \u2194 bipartite G, from sorry,\nend\n\n--OUTPUT 5\ntheorem bipartite_two_colorable (G : Type*) [graph G] : 2-colorable G \u2194 bipartite G :=\nbegin\n  have h1 : 2-colorable G \u2192 bipartite G, from sorry,\n  have h2 : bipartite G \u2192 2-colorable G, from sorry,\n  show 2-colorable G \u2194 bipartite G, from sorry,\nend\n\n--OUTPUT 6\ntheorem bipartite_graph_two_colorable (G : Type*) [graph G] : \n  (\u2203 V1 V2 : set G, \u2200 v1 \u2208 V1, \u2200 v2 \u2208 V2, \u2200 e : G, e \u2208 edges G \u2192 v1 = e.1 \u2228 v2 = e.2) \u2194 \n  (\u2203 f : G \u2192 \u2115, \u2200 e : G, e \u2208 edges G \u2192 f e.1 \u2260 f e.2) :=\nbegin\n  split,\n  {\n    assume h1 : \u2203 V1 V2 : set G, \u2200 v1 \u2208 V1, \u2200 v2 \u2208 V2, \u2200 e : G, e \u2208 edges G \u2192 v1 = e.1 \u2228 v2 = e.2,\n    have h2 : \u2203 V1 V2 : set G, \u2200 v1 \u2208 V1, \u2200 v2 \u2208 V2, \u2200 e : G, e \u2208 edges G \u2192 v1 = e.1 \u2228 v2 = e.2, from h1,\n    cases h2 with V1 h2,\n    cases h2 with V2 h2,\n    use (\u03bb v : G, if v \u2208 V1 then 0 else 1),\n    assume (v1 v2 : G) (h3 : v1 \u2208 V1) (h4 : v2 \u2208 V2) (e : G) (h5 : e \u2208 edges G),\n    have h6 : v1 \u2209 V2, from sorry,\n    have h7 : v2 \u2209 V1, from sorry,\n    show (if v1 \u2208 V1 then 0 else 1) \u2260 (if v2 \u2208 V1 then 0 else 1), from sorry,\n  },\n  {\n    assume h1 : \u2203 f : G \u2192 \u2115, \u2200 e : G, e \u2208 edges G \u2192 f e.1 \u2260 f e.2,\n    have h2 : \u2203 f : G \u2192 \u2115, \u2200 e : G, e \u2208 edges G \u2192 f e.1 \u2260 f e.2, from h1,\n    cases h2 with f h2,\n    use (\u03bb v : G, if f v = 0 then true else false),\n    use (\u03bb v : G, if f v = 1 then true else false),\n    assume (v1 v2 : G) (h3 : v1 \u2208 (\u03bb v : G, if f v = 0 then true else false)) (h4 : v2 \u2208 (\u03bb v : G, if f v = 1 then true else false)) (e : G) (h5 : e \u2208 edges G),\n    have h6 : v1 \u2209 (\u03bb v : G, if f v = 1 then true else false), from sorry,\n    have h7 : v2 \u2209 (\u03bb v : G, if f v = 0 then true else false), from sorry,\n    show v1 = e.1 \u2228 v2 = e.2, from sorry,\n  }\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-3_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241911813151, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.36158381843792464}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.module.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Prod instances for module and multiplicative actions\n\nThis file defines instances for binary product of modules\n-/\n\nnamespace prod\n\n\nprotected instance has_scalar {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [has_scalar \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] : has_scalar \u03b1 (\u03b2 \u00d7 \u03b3) :=\n  has_scalar.mk fun (a : \u03b1) (p : \u03b2 \u00d7 \u03b3) => (a \u2022 fst p, a \u2022 snd p)\n\n@[simp] theorem smul_fst {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [has_scalar \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] (a : \u03b1) (x : \u03b2 \u00d7 \u03b3) : fst (a \u2022 x) = a \u2022 fst x :=\n  rfl\n\n@[simp] theorem smul_snd {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [has_scalar \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] (a : \u03b1) (x : \u03b2 \u00d7 \u03b3) : snd (a \u2022 x) = a \u2022 snd x :=\n  rfl\n\n@[simp] theorem smul_mk {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [has_scalar \u03b1 \u03b2] [has_scalar \u03b1 \u03b3] (a : \u03b1) (b : \u03b2) (c : \u03b3) : a \u2022 (b, c) = (a \u2022 b, a \u2022 c) :=\n  rfl\n\nprotected instance semimodule {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : semiring \u03b1} [add_comm_monoid \u03b2] [add_comm_monoid \u03b3] [semimodule \u03b1 \u03b2] [semimodule \u03b1 \u03b3] : semimodule \u03b1 (\u03b2 \u00d7 \u03b3) :=\n  semimodule.mk sorry sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/module/prod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.36158381843792453}}
{"text": "import tactic\n--import structures\n-- dEAduction imports\nimport snippets.tactics.structures\nimport snippets.definitions.definitions\n\n\nnamespace tactic.interactive\nopen lean.parser tactic interactive \nopen interactive (loc.ns)\nopen interactive.types\nopen tactic expr\nlocal postfix *:9001 := many -- sinon ne comprends pas ident*\n\n/- Appelle l'analyse r\u00e9cursive sur le but ou sur une hypoth\u00e8se. Non utilis\u00e9 par la suite. -/\nmeta def estcepi (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal \u2190 tactic.target,\n                trace (is_pi goal)\n    | [nom] := do expr \u2190 get_local nom,\n                expr_t \u2190  infer_type expr,\n                trace(is_pi expr_t)\n    | _ := skip\n    end\n\nmeta def estcefl (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal \u2190 tactic.target,\n                trace (is_arrow goal)\n    | [nom] := do expr \u2190 get_local nom,\n                expr_t \u2190  infer_type expr,\n                trace(is_arrow expr_t)\n    | _ := skip\n    end\n\nmeta def estceprop (names : parse ident*) : tactic unit := \nmatch names with\n    | [] := do goal \u2190 tactic.target,\n                trace (is_prop goal)\n    | [nom] := do expr \u2190 get_local nom,\n                expr_t \u2190  infer_type expr,\n                trace(is_prop expr_t)\n    | _ := skip\n    end\n\n/- \nTactic to get the list of definitions, ie lemmas from the definitions spacename\n-/\n\nmeta def is_theorem : declaration \u2192 bool\n| (declaration.defn _ _ _ _ _ _) :=  ff\n| (declaration.thm _ _ _ _) := tt\n| (declaration.cnst _ _ _ _) := ff\n| (declaration.ax _ _ _) := tt\n\nmeta def get_all_theorems : tactic (list name) := \ndo\n    env \u2190 tactic.get_env,\n    pure (environment.fold env [] (\u03bb decl nams,\n         if is_theorem decl then \n            declaration.to_name decl :: nams\n         else \n            nams))\n\nmeta def print_all_theorems' : tactic unit :=\ndo\n    nams \u2190 get_all_theorems,\n    nams.mmap (\u03bb h, tactic.trace h),\n    return ()\n\nmeta def name.get_ante_suffix : name \u2192 name\n| (name.mk_string s1 (name.mk_string s2 p))  := s2\n| (name.mk_numeral s1 (name.mk_string s2 p)) := s2\n| p := name.anonymous\n\nmeta def print_all_as_theorems : tactic unit :=\ndo\n    nams \u2190 get_all_theorems,\n    nams.mmap (\u03bb h, tactic.trace $ name.get_ante_suffix h),\n    return ()\n\nmeta def is_definitions_as (n : name) : bool :=\nif (name.get_ante_suffix n = \"set\") then tt else ff\n\nmeta def print_all_is_as_theorems : tactic unit :=\ndo\n    nams \u2190 get_all_theorems,\n    nams.mmap (\u03bb h, tactic.trace $ is_definitions_as h),\n    return ()\n\nmeta def is_definitions_as' (n : name) : tactic bool :=\nif (name.get_ante_suffix n = \"definitions\") then return tt else return ff\n\nmeta def print_all_is_as_theorems' : tactic unit :=\ndo\n    nams \u2190 get_all_theorems,\n    nams.mmap (\u03bb h, tactic.trace $ is_definitions_as' h),\n    return ()\n\n\nmeta def print_all_definitions' : tactic unit :=\ndo\n    nams \u2190 get_all_theorems,\n    def_nams \u2190 list.mfilter is_definitions_as' nams,\n    def_nams.mmap (\u03bb h, tactic.trace h),\n    return ()\n\nmeta def pprint_all_complete_goals : tactic unit :=\ndo\n    tactic.trace_state\n\nmeta def print_all_complete_goals : tactic unit :=\ndo\n  gs \u2190 get_goals,\n  gs.mmap' $ \u03bb g, set_goals [g] >> \n  local_context >>= mmap (\u03bb h, analyse_expr h >>= trace),\n  return ()\n\n \n\n\nend tactic.interactive\n\nvariable X : Type\nexample {A A' : set X} : (A = A') \u2194 ( \u2200 x:X, x \u2208 A \u2194 x \u2208 A' ) :=\nbegin\n    hypo_analysis,\n    targets_analysis,\nend\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nlemma essai3 (P Q : Prop)   :  P \u2228 Q \u2192 Q \u2228 P :=\nbegin\n    intro H,\n\nend\n\n\n\n\n\n\n\n------------ Th\u00e9orie des ensembles --------------\n%namespace set_theory\n\nvariables {X : Type} {Y : Type}\n-- mem_compl_iff\n--lemma complement {A : set X} {x : X} : x \u2208 - A \u2194 \u00ac x \u2208 A :=\n--iff.rfl\n\nlemma def.essai : true :=\nbegin\n    print_all_theorems,\n    print_all_as_theorems,\n    print_all_is_as_theorems,\n    print_all_is_as_theorems',\n    print_all_definitions,\n    tautology,\nend\n\nlemma essai (X : Type) (P Q : Prop) : P \u2227 Q \u2192 P :=\nbegin\n    hypo_analysis,\n\nend\n\n\n\nlemma definitions.complement {A : set X} {x : X} : x \u2208 set.univ \\ A \u2194 x \u2209 A := \nbegin\n    finish\nend\n\nlemma definitions.complement_1 {A : set X} {x : X} : x \u2208 set.compl A \u2194 x \u2209 A := \nby finish\n\n\nlemma exercises.toto (P Q : Prop) (H: P \u2192 Q): \u2200 R: Prop, R \u2192 R :=\nbegin\n    print_all_definitions,\n    estcefl,\n    intro H1,\n    estcefl H1,\n    estcefl,\n    sorry\nend\n\nopen tactic\n\nexample (y : \u2115) : true :=\nby do e \u2190 to_expr ```(\u2200 x : \u2115, y = 1), trace e, trace e.is_arrow, trace e.is_pi\n\n\nexample (X Y : Type) (A : set Y) (f : X \u2192 Y): \u2200 x : X, \u2203 y \u2208 A, f x = y \u2192 y = f x :=\nbegin\n--    interactive.goals_analysis,\n    hypo_analysis,\n    print_all_exercises,\n    print_all_definitions,    \n    goals_analysis,\n    estcefl f,\n    estcefl,\n    intros x y,\n    estcefl,\nend\n\n#check is_prop\n\nexample (X Y I : Type) (y_1 : Y) (E : I \u2192 set X) : \u2200 i' : I , \u2200 x : E i',  \u2200 f : (E i') \u2192 Y, f x = y_1 :=\nbegin\n    targets_analysis,\nend\n\n#print set\n\nexample (X : Type) (x : X) (A : set $ set X)\n (B : X \u2192 Prop) (a : A) : X := \nbegin\n    hypo_analysis,\n    \nend\n\n\nexample (X : Type) (n : \u2115) (u : \u2115 \u2192 X): tt :=\nbegin\n    hypo_analysis\nend\n\n\n\nlemma exercise.union_distributive_inter (P Q R : Prop) : ( P \u2192 Q ) \u2227 ( R \u2192 Q ) :=\n\nbegin\n    split, intro HP,\n    rotate, intro HR,\n    print_all_complete_goals,\n\nend\n\n\n\nexample : true \u2227 \u2200 x : \u2115, true :=\nby do \n  `[refine \u27e8_, \u03bb i, _\u27e9], -- give ourselves two goals with different local contexts\n  trace_state,\n  gs \u2190 get_goals,\n  gs.mmap' $ \u03bb g, set_goals [g] >> local_context >>= trace,\n  set_goals gs,\n  trace_state\n\nend", "meta": {"author": "dEAduction", "repo": "dEAduction-lean", "sha": "4fe1d642078fc94f9081ccbed08e047e86a741fd", "save_path": "github-repos/lean/dEAduction-dEAduction-lean", "path": "github-repos/lean/dEAduction-dEAduction-lean/dEAduction-lean-4fe1d642078fc94f9081ccbed08e047e86a741fd/snippets/tactics_for_testing/essais.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.6261241772283035, "lm_q1q2_score": 0.361583810380125}}
{"text": "import data.quot tactic\nopen classical\nnoncomputable theory\n\nvariables (p : Type) {\u03b1 : Type*} {\u03b2 : Type*}\n\ninductive eq' (a : \u03b1) : \u03b1 \u2192 Type\n| refl : eq' a\ninfix ` =' `:50 := eq'\n\ndef eq'.symm : \u2200 {a b : \u03b1}, a =' b \u2192 b =' a\n| _ _ (eq'.refl _) := (eq'.refl _)\n\ndef eq'.trans : \u2200 {a b c : \u03b1}, a =' b \u2192 b =' c \u2192 a =' c\n| _ _ _ (eq'.refl _) (eq'.refl _) := (eq'.refl _)\n\ndef eq'.congr_arg (f : \u03b1 \u2192 \u03b2): \u2200 {a b : \u03b1}, a =' b \u2192 f a =' f b\n| _ _ (eq'.refl a) := eq'.refl _\n\ndef eq'.cast : \u2200 {a b : Type}, a =' b \u2192 a \u2192 b\n| _ _ (eq'.refl a) := id\n\ndef U (x : Type) : Type := trunc (trunc x =' trunc unit \u2295 p)\ndef V (x : Type) : Type := trunc (trunc x =' trunc empty \u2295 p)\n\ndef exU : trunc (\u03a3 x : Type, U p x) := trunc.mk \u27e8unit, trunc.mk $ sum.inl (eq'.refl _)\u27e9\ndef exV : trunc (\u03a3 x : Type, V p x) := trunc.mk \u27e8empty, trunc.mk $ sum.inl (eq'.refl _)\u27e9\n\naxiom choice' {\u03b1  : Type*} {\u03b2 : \u03b1 \u2192 Type*} {r : \u03a0 x, \u03b2 x \u2192 Type*}\n  (f : \u03a0 i, trunc (\u03a3 b : \u03b2 i, r i b)) :\n  trunc (\u03a0 i, \u03a3 b : \u03b2 i, r i b)\n\nlemma choice'2 {\u03b1  : Type*} {\u03b2 : \u03b1 \u2192 Type*} (f : \u03a0 a, trunc (\u03b2 a)) :\n  trunc (\u03a0 a, \u03b2 a) :=\nlet g : \u03a0 a, trunc (\u03a3 b : \u03b2 a, unit) := \u03bb a, trunc.map (\u03bb x, \u27e8x, ()\u27e9) (f a) in\ntrunc.map (by intros g a; exact (g a).1) (choice' g)\n\nlemma choice3 {\u03b1 : Type*} : trunc (\u03a3 f : \u03b1 \u2192 \u03b1, \u03a0 a b : \u03b1, f a =' f b) :=\ntrunc.rec_on_subsingleton (choice'2 (@trunc.mk \u03b1))\nbegin\n\n\nend\n\ndef not_uv_or_p : \u2200 (u : \u03a3 x : Type, U p x) (v : \u03a3 x : Type, V p x),\n  trunc ((trunc u.fst =' trunc v.fst \u2192 empty) \u2295 p) :=\nbegin\n  rintros \u27e8u, hu\u27e9 \u27e8v, hv\u27e9,\n  dsimp [U, V] at *,\n  refine trunc.lift_on hu (\u03bb hu, _) (\u03bb _ _, subsingleton.elim _ _),\n  refine trunc.lift_on hv (\u03bb hv, _) (\u03bb _ _, subsingleton.elim _ _),\n  cases hu with hu hu,\n  { cases hv with hv hv,\n    { refine trunc.mk (sum.inl (\u03bb e, _)),\n      have : trunc unit =' trunc empty, from eq'.rec_on hu (eq'.rec_on hv e),\n      have : trunc empty, from this.cast (trunc.mk ()),\n      refine trunc.lift_on this id (assume a, by cases a) },\n    { exact trunc.mk (sum.inr hv) } },\n  { exact trunc.mk (sum.inr hu) }\nend\n\naxiom funext' {\u03b2 : \u03b1 \u2192 Type*}\n  {f\u2081 f\u2082 : \u03a0 x : \u03b1, \u03b2 x} (h : \u2200 x, f\u2081 x =' f\u2082 x) : f\u2081 =' f\u2082\n\naxiom univalence {\u03b1 \u03b2 : Sort*} : \u03b1 \u2243 \u03b2 \u2192 \u03b1 =' \u03b2\n\nnoncomputable lemma trunc_eq'_trunc_of_true {\u03b1 \u03b2 : Sort*} : \u03b1 \u2192 \u03b2 \u2192 trunc \u03b1 =' trunc \u03b2 :=\n\u03bb a b, univalence \u27e8\u03bb _, trunc.mk b, \u03bb _, trunc.mk a, \u03bb _, subsingleton.elim _ _,\n  \u03bb _, subsingleton.elim _ _\u27e9\n\n-- noncomputable def p_implies_uv (hp : p) (u : \u03a3 x : Type, U p x)\n--   (v : \u03a3 x : Type, V p x) : u.fst =' v.fst :=\n-- begin\n--   have : U p =' V p,\n--   { refine funext' (\u03bb x, _),\n--     exact trunc_eq'_trunc_of_true (sum.inr hp) (sum.inr hp), },\n--   revert u v,\n--   refine eq'.rec_on this _,\n--   assume u v,\n--   exact trunc_eq'_trunc_of_true _ _,\n-- end\n\nnoncomputable def em : trunc (p \u2295 (p \u2192 empty)) :=\nlet fU : trunc (\u03a0 {\u03b1 : Type \u2192 Type \u2192 Type}, (\u03a0 p : Type, \u03a3 x : Type, \u03b1 p x)) :=\n  choice'2 (\u03bb \u03b1, choice' (\u03bb p, _))  in\nlet fV : trunc (\u03a0 p : Type, \u03a3 x : Type, V p x) := choice' exV in\ntrunc.rec_on_subsingleton fU (assume fU', trunc.rec_on_subsingleton fV (assume fV',\n  let exU := fU' p, exV := fV' p in\n  have p \u2192 trunc exU.1 =' trunc exV.1,\n    from \u03bb hp, have U p =' V p, from funext'\n        (\u03bb x, trunc_eq'_trunc_of_true (sum.inr hp) (sum.inr hp)),\n      begin\n        revert exU exV fU' fV',\n        dsimp [fU, fV],\n        refine eq'.rec_on this _,\n        intros,\n        exact trunc_eq'_trunc_of_true _ _,\n      end,\n\n  trunc.mk begin\n    refine sum.rec_on (not_uv_or_p p(fU p) (fV p)) _ _,\n    { assume h,\n      exact sum.inr (assume hp : p, h (p_implies_uv p hp _ _)) },\n    { exact sum.inl }\nend))\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/computable_em.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583376458152, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.36156519559733497}}
{"text": "/-\nCopyright (c) 2021 OpenAI. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kunhao Zheng, Stanislas Polu, David Renshaw, OpenAI GPT-f\n-/\nimport mathzoo.imports.miniF2F\n\nopen_locale nat rat real big_operators topological_space\n\ntheorem mathd_numbertheory_403 :\n  \u2211 k in (nat.proper_divisors 198), k = 270 :=\nbegin\n  refl,\nend", "meta": {"author": "leanprover-community", "repo": "mathzoo", "sha": "87e9b492daeb929838706942aaa2437621b34a0e", "save_path": "github-repos/lean/leanprover-community-mathzoo", "path": "github-repos/lean/leanprover-community-mathzoo/mathzoo-87e9b492daeb929838706942aaa2437621b34a0e/src/mathzoo/olympiads/mathd/numbertheory/p403.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.36156518904494367}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.zero_morphisms\nimport category_theory.limits.types\nimport category_theory.limits.mono_coprod\n\nuniverse u\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nnamespace category_theory\n\nnamespace limits\n\nvariables {C : Type*} [category C] [has_finite_coproducts C]\n\nvariable {C}\n\nnamespace mono_coprod\n\n\nnamespace mono_inclusion_sub_coproduct\n\nvariables {I J : Type*} [fintype I] [decidable_eq I] [fintype J] [mono_coprod C] (X : I \u2192 C) (\u03b3 : J \u2192 I)\n  (h\u03b3 : function.injective \u03b3)\n\ndef \u03b1 : sigma_obj (\u03bb j, X (\u03b3 j)) \u27f6 sigma_obj X := sigma.desc (\u03bb j, sigma.\u03b9 _ (\u03b3 j))\ndef \u03b2 : sigma_obj (\u03bb (k : (finset.image \u03b3 \u22a4)\u1d9c), X k) \u27f6 sigma_obj X := sigma.desc (\u03bb k, sigma.\u03b9 _ k)\ndef \u03c6 := coprod.desc (\u03b1 X \u03b3) (\u03b2 X \u03b3)\ndef index (i : I) (hi : i \u2208 finset.image \u03b3 \u22a4) : J := (finset.mem_image.mp hi).some\nlemma index_cond (i : I) (hi : i \u2208 finset.image \u03b3 \u22a4) : i = \u03b3 (index \u03b3 i hi) :=\n(finset.mem_image.mp hi).some_spec.some_spec.symm\n\ninclude h\u03b3\n\nlemma index_eq (i : I) (j : J) (hj : i = \u03b3 j) : index \u03b3 i (by { simp only [finset.mem_image],\n  exact \u27e8j, finset.mem_univ j, hj.symm\u27e9, }) = j :=\nbegin\n  apply h\u03b3,\n  rw [\u2190 index_cond \u03b3 i, hj],\nend\n\nomit h\u03b3\n\ndef \u03c8\u2081 (i : I) (hi : i \u2208 finset.image \u03b3 \u22a4) : X i \u27f6 sigma_obj (\u03bb j, X (\u03b3 j)) :=\neq_to_hom (by { congr, exact index_cond \u03b3 i hi,}) \u226b sigma.\u03b9 _ (index \u03b3 i hi)\n\nlemma sigma.congr_\u03b9 {J D : Type*} [category D] (F : J \u2192 D) [has_coproduct F]\n  (a b : J) (h : a = b) : eq_to_hom (by rw h) \u226b sigma.\u03b9 F a = sigma.\u03b9 F b :=\nby { subst h, simp only [eq_to_hom_refl, id_comp], }\n\ninclude h\u03b3\n\nlemma \u03c8\u2081_eq_\u03b9 (j : J) : \u03c8\u2081 X \u03b3 (\u03b3 j) (by { rw finset.mem_image, exact \u27e8j, finset.mem_univ _, rfl\u27e9}) =\n  sigma.\u03b9 _ j :=\nsigma.congr_\u03b9 (\u03bb (j : J), X (\u03b3 j)) (index \u03b3 (\u03b3 j) _) j (index_eq \u03b3 h\u03b3 (\u03b3 j) j rfl)\n\nomit h\u03b3\n\ndef \u03c8\u2082 (i : I) (hi : \u00ac i \u2208 (finset.image \u03b3 \u22a4)) :\n  X i \u27f6 sigma_obj (\u03bb (k : (finset.image \u03b3 \u22a4)\u1d9c), X k) :=\nsigma.\u03b9 (\u03bb (k : (finset.image \u03b3 \u22a4)\u1d9c), X k) \u27e8i, by simpa only [finset.mem_compl] using hi\u27e9\n\ndef \u03c8 : sigma_obj X \u27f6 sigma_obj (\u03bb j, X (\u03b3 j)) \u2a3f sigma_obj (\u03bb (k : (finset.image \u03b3 \u22a4)\u1d9c), X k) :=\nsigma.desc (\u03bb i, begin\n  by_cases hi : i \u2208 finset.image \u03b3 finset.univ,\n  { exact \u03c8\u2081 X \u03b3 i hi \u226b coprod.inl, },\n  { exact \u03c8\u2082 X \u03b3 i hi \u226b coprod.inr, },\nend)\n\ninclude h\u03b3\n@[simps]\ndef iso : sigma_obj (\u03bb j, X (\u03b3 j)) \u2a3f sigma_obj (\u03bb (k : (finset.image \u03b3 \u22a4)\u1d9c), X k) \u2245 sigma_obj X :=\n{ hom := \u03c6 X \u03b3,\n  inv := \u03c8 X \u03b3,\n  hom_inv_id' := begin\n    dsimp only [\u03c6, \u03c8, \u03b1, \u03b2],\n    ext; discrete_cases,\n    { rw [coprod.inl_desc_assoc, colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, colimit.\u03b9_desc,\n        cofan.mk_\u03b9_app, comp_id],\n      dsimp,\n      rw dif_pos, swap,\n      { simp only [finset.mem_image],\n        exact \u27e8j, finset.mem_univ _, rfl\u27e9, },\n      erw \u03c8\u2081_eq_\u03b9 X \u03b3 h\u03b3 j, },\n    { rw [coprod.inr_desc_assoc, colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, colimit.\u03b9_desc,\n        cofan.mk_\u03b9_app, comp_id],\n      dsimp,\n      rw dif_neg, swap,\n      { simpa only [finset.mem_compl] using j.2, },\n      dsimp [\u03c8\u2082],\n      congr,\n      simp only [finset.mk_coe], },\n  end,\n  inv_hom_id' := begin\n    dsimp only [\u03c6, \u03c8, \u03b1, \u03b2],\n    ext,\n    discrete_cases,\n    simp only [colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, comp_id],\n    dsimp only,\n    split_ifs with hj,\n    { simp [finset.mem_image] at hj,\n      rcases hj with \u27e8i, hi\u27e9,\n      subst hi,\n      erw \u03c8\u2081_eq_\u03b9 X \u03b3 h\u03b3 i,\n      tidy },\n    { dsimp [\u03c8\u2082],\n      erw [category.assoc, coprod.inr_desc, colimit.\u03b9_desc, cofan.mk_\u03b9_app],\n      refl, },\n  end, }\n\nend mono_inclusion_sub_coproduct\n\nsection\n\nvariables {I J : Type*} [fintype I] [fintype J] [mono_coprod C]\n  (X : I \u2192 C) (\u03b3 : J \u2192 I)\n\n@[simp]\ndef map_coproduct : sigma_obj (\u03bb j, X (\u03b3 j)) \u27f6 sigma_obj X := sigma.desc (\u03bb j, sigma.\u03b9 _ (\u03b3 j))\n\nlemma mono_inclusion_sub_coproduct (h\u03b3 : function.injective \u03b3) : mono (map_coproduct X \u03b3) :=\nbegin\n  classical,\n  let \u03b1 : sigma_obj (\u03bb j, X (\u03b3 j)) \u27f6 sigma_obj X := sigma.desc\n    (\u03bb j, sigma.\u03b9 X (\u03b3 j)),\n  change mono \u03b1,\n  rw [show \u03b1 = coprod.inl \u226b (mono_inclusion_sub_coproduct.iso X \u03b3 h\u03b3).hom, by tidy],\n  apply mono_comp,\nend\n\nend\n\ninstance mono_sigma_\u03b9 {I : Type*} [fintype I] [mono_coprod C] (X : I \u2192 C) (i : I):\n  mono (sigma.\u03b9 X i) :=\nbegin\n  let \u03b3 : fin 1 \u2192 I := \u03bb x, i,\n  have h\u03b3 : function.injective \u03b3 := \u03bb x\u2081 x\u2082 h, subsingleton.elim _ _,\n  let e : X i \u2245 (\u2210 \u03bb (j : fin 1), X (\u03b3 j)) :=\n  { hom := sigma.\u03b9 (\u03bb (j : fin 1), X (\u03b3 j)) 0,\n    inv := sigma.desc (\u03bb j, \ud835\udfd9 _),\n    hom_inv_id' := by tidy,\n    inv_hom_id' := by { ext, discrete_cases, tidy, }, },\n  haveI := mono_inclusion_sub_coproduct X \u03b3 h\u03b3,\n  convert (mono_comp _ _ : mono (e.hom \u226b map_coproduct X \u03b3)),\n  simp only [map_coproduct, colimit.\u03b9_desc, cofan.mk_\u03b9_app],\nend\n\nend mono_coprod\n\nend limits\n\nend category_theory\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/inclusions_mono.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3612269605789466}}
{"text": "\nimport topology.basic\nimport topology.compact_open\nimport data.nat.prime\nimport data.real.basic\nimport data.real.irrational\nimport data.complex.basic\nimport data.fin.basic\nimport geometry.euclidean.basic\nimport analysis.inner_product_space.pi_L2\nimport algebra.group.defs\nimport algebra.field.basic\nimport combinatorics.configuration\nimport ring_theory.polynomial.basic\nimport group_theory.free_group\nimport combinatorics.simple_graph.basic\nimport ring_theory.integral_closure\nimport data.fintype.card\nimport category_theory.category.basic\nimport ring_theory.discrete_valuation_ring\nimport group_theory.torsion\nimport linear_algebra.matrix.charpoly.basic\nimport algebra.order.absolute_value\nimport analysis.convex.basic\nimport topology.uniform_space.uniform_convergence_topology\nimport topology.sequences\nimport analysis.normed.group.infinite_sum\nimport data.nat.choose.sum\nimport group_theory.specific_groups.cyclic\nimport group_theory.order_of_element\nimport analysis.mean_inequalities\nimport analysis.normed_space.banach\nimport topology.algebra.continuous_monoid_hom\nimport linear_algebra.matrix.symmetric\nimport analysis.inner_product_space.spectrum\nimport ring_theory.class_group\nimport ring_theory.dedekind_domain.basic\nimport ring_theory.principal_ideal_domain\nimport model_theory.satisfiability\nimport probability.integration\nimport ring_theory.simple_module\nimport category_theory.preadditive.schur\nimport representation_theory.maschke\nimport topology.paracompact\nimport combinatorics.simple_graph.coloring\n--Prove that Contraction mapping  in Lean mathlib\n \n--OUTPUT 1\ntheorem contraction_mapping_theorem {\u03b1 : Type*} [complete_space \u03b1] {f : \u03b1 \u2192 \u03b1}\n  (hf : contraction f) : has_fixed_point f :=\nbegin\n  have h : \u2200 x y, dist (f x) (f y) \u2264 dist x y,\n  { intros x y,\n    apply le_trans (hf x y),\n    apply dist_triangle },\n  have h' : \u2200 x y, dist (f x) (f y) < dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply le_antisymm,\n    { apply le_of_lt (hf x y),\n      apply lt_of_le_of_lt (dist_nonneg _ _) hxy },\n    { apply le_of_lt (hf y x),\n      apply lt_of_le_of_lt (dist_nonneg _ _) hxy } },\n  have h'' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply le_antisymm,\n    { apply le_of_eq hxy,\n      apply le_of_lt (hf x y) },\n    { apply le_of_eq hxy,\n      apply le_of_lt (hf y x) } },\n  have h''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 dist (f x) (f y) < dist x y,\n  { intros x y hxy,\n    apply lt_of_le_of_eq (hf x y) hxy },\n  have h'''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 dist (f x) (f y) = 0,\n  { intros x y hxy,\n    apply eq_zero_of_dist_eq_zero,\n    apply h''' hxy },\n  have h''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n  have h''''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h'' hxy },\n  have h''''''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 f x = f y,\n  { intros x y hxy,\n    apply eq_of_dist_eq_zero,\n    apply h'''' hxy },\n  have h'''''''''''''''''''''''''''''' : \u2200 x y, dist (f x) (f y) = dist x y \u2192 x = y,\n  { intros x y hxy,\n    apply h''' hxy },\n \n\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/Zero-shot-theorem-names/Zero-Shot-prompts_temperature_0_max_tokens_2000_n_1/clean_files/Contraction mapping theorem.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.36122695337496663}}
{"text": "import spectral_norm\n\nnoncomputable theory\n\nopen_locale nnreal\n\nvariables {K : Type*} [nontrivially_normed_field K]  [complete_space K] {L : Type*} [hL: field L]\n  [algebra K L] (h_alg : algebra.is_algebraic K L)\n\ninclude hL\n\nlemma spectral_norm.unique' {f : L \u2192 nnreal} (hf_pow : is_pow_mult f)\n  (hf_alg_norm : is_algebra_norm (normed_ring.to_is_norm K) f) (hf_na : is_nonarchimedean f)\n  (hna : is_nonarchimedean (\u03bb k : K, \u2225k\u2225\u208a)) :\n  f = spectral_norm h_alg  := \nbegin\n  apply eq_of_pow_mult_faithful (normed_field.to_is_mul_norm K) hf_pow hf_alg_norm \n    (spectral_norm.is_pow_mult h_alg hna) (spectral_norm.is_algebra_norm h_alg hna),\n  intro x,\n  set E : Type* := id K\u27eex\u27ef with hEdef,\n  letI hE : field E := (by rw [hEdef, id.def] ; apply_instance),\n  letI : algebra K E := K\u27eex\u27ef.algebra,\n\n  set id1 : K\u27eex\u27ef \u2192\u2097[K] E := \n  { to_fun := id,\n    map_add' := \u03bb x y, rfl,\n    map_smul' := \u03bb r x, rfl, },\n\n  set id2 : E \u2192\u2097[K] K\u27eex\u27ef := \n  { to_fun := id,\n    map_add' := \u03bb x y, rfl,\n    map_smul' := \u03bb r x, rfl },\n\n  have hs_norm : is_norm (\u03bb y : E, spectral_norm h_alg (id2 y : L)) :=\n  { zero := by rw [map_zero, subfield.coe_zero, spectral_norm.zero],\n    add  := \u03bb a b, by rw [map_add]; exact (spectral_norm.is_algebra_norm h_alg hna).add _ _,\n    mul  := \u03bb a b, by simp only [linear_map.coe_mk, id.def, subfield.coe_mul]; \n      exact (spectral_norm.is_algebra_norm h_alg hna).mul _ _,\n    ne_zero := \u03bb a ha,\n    begin\n      apply (spectral_norm.is_algebra_norm h_alg hna).ne_zero,\n      simp only [linear_map.coe_mk, id.def, ne.def, add_submonoid_class.coe_eq_zero],\n      exact ha,\n    end},\n\n  have hs_neg : \u2200 y : E, spectral_norm h_alg (id2 (-y) : L) = spectral_norm h_alg (id2 y : L),\n  { intro y,\n    simp only [linear_map.coe_mk, id.def, subfield.coe_neg],\n    exact (spectral_norm.is_nonarchimedean h_alg hna).neg (spectral_norm.zero h_alg) _ },\n\n  letI n1 : normed_ring E := norm_to_normed_ring hs_norm hs_neg,\n\n  letI N1 : normed_space K E := \n  { norm_smul_le := \u03bb k y,\n    begin\n      change (spectral_norm h_alg (id2 (k \u2022 y) : L) : \u211d) \u2264 \u2225 k \u2225 * spectral_norm h_alg (id2 y : L),\n      simp only [linear_map.coe_mk, id.def, intermediate_field.coe_smul],\n      rw (spectral_norm.is_algebra_norm _ hna).smul,\n      exact le_refl _,\n    end,\n    ..K\u27eex\u27ef.algebra },\n\n  have hf_norm : is_norm (\u03bb y, f((algebra_map K\u27eex\u27ef L) y)) := \n  { zero := by rw [map_zero, hf_alg_norm.zero],\n    add  := \u03bb a b, by rw [map_add]; exact hf_alg_norm.add _ _,\n    mul  := \u03bb a b, by rw [map_mul]; exact hf_alg_norm.mul _ _,\n    ne_zero := \u03bb a ha, hf_alg_norm.ne_zero _ ((ring_hom.map_ne_zero _).mpr ha) },\n\n  have hf_neg : \u2200 y, f((algebra_map K\u27eex\u27ef L) (-y)) = f((algebra_map K\u27eex\u27ef L) y),\n  { intro y,\n    rw map_neg, exact hf_na.neg hf_alg_norm.zero _ },\n\n  letI n2 : normed_ring K\u27eex\u27ef := norm_to_normed_ring hf_norm hf_neg,\n\n  letI N2 : normed_space K K\u27eex\u27ef :=\n  { norm_smul_le :=  \u03bb k y,\n    begin\n      change (f ((algebra_map K\u27eex\u27ef L) (k \u2022 y)) : \u211d) \u2264 \u2225 k \u2225 * f (algebra_map K\u27eex\u27ef L y),\n      have : (algebra_map \u21a5K\u27eex\u27ef L) (k \u2022 y) = k \u2022 (algebra_map \u21a5K\u27eex\u27ef L y),\n      { rw [\u2190 is_scalar_tower.algebra_map_smul K\u27eex\u27ef k y, smul_eq_mul, map_mul, \n          \u2190 is_scalar_tower.algebra_map_apply K \u21a5K\u27eex\u27ef L, algebra.smul_def] }, \n      rw [this, hf_alg_norm.smul],\n      exact le_refl _,\n    end,\n    ..K\u27eex\u27ef.algebra },\n\n  haveI hKx_fin : finite_dimensional K \u21a5K\u27eex\u27ef := intermediate_field.adjoin.finite_dimensional \n    (is_algebraic_iff_is_integral.mp (h_alg x)),\n  haveI : finite_dimensional K E := hKx_fin,\n\n  set Id1 : K\u27eex\u27ef \u2192L[K] E := \u27e8id1, id1.continuous_of_finite_dimensional\u27e9 with hId1,\n  set Id2 : E \u2192L[K] K\u27eex\u27ef := \u27e8id2, id2.continuous_of_finite_dimensional\u27e9 with hId2,\n \n  have hC1 : \u2203 (C1 : \u211d), 0 < C1 \u2227 \u2200 (y : K\u27eex\u27ef), \u2225id1 y\u2225 \u2264 C1 * \u2225y\u2225 := Id1.is_bounded_linear_map.bound,\n  have hC2 : \u2203 (C2 : \u211d), 0 < C2 \u2227 \u2200 (y : E), \u2225id2 y\u2225 \u2264 C2 * \u2225y\u2225 := Id2.is_bounded_linear_map.bound,\n\n  obtain \u27e8C1, hC1_pos, hC1\u27e9 := hC1,\n  obtain \u27e8C2, hC2_pos, hC2\u27e9 := hC2,\n  use [\u27e8C2, le_of_lt hC2_pos\u27e9, \u27e8C1, le_of_lt hC1_pos\u27e9, hC2_pos, hC1_pos],\n  rw forall_and_distrib,\n  --simp only at hC1 hC2,\n  split,\n  { intro y, exact hC2 \u27e8y, (intermediate_field.algebra_adjoin_le_adjoin K _) y.2\u27e9 },\n  { intro y, exact hC1 \u27e8y, (intermediate_field.algebra_adjoin_le_adjoin K _) y.2\u27e9 },\n\nend\n\nlemma spectral_norm.unique_field_norm_ext {f : L \u2192 nnreal}\n  (hf_field_norm : is_mul_norm f) (hf_ext : function_extends (\u03bb x : K, \u2225x\u2225\u208a) f)\n  (hf_na : is_nonarchimedean f) (hna : is_nonarchimedean (\u03bb k : K, \u2225k\u2225\u208a)) (x : L) :\n  f x = spectral_norm h_alg x := \nbegin\n  have hf_pow : is_pow_mult f := is_mul_norm.to_is_pow_mult hf_field_norm,\n  have hf_alg_norm : is_algebra_norm (normed_ring.to_is_norm K) f := \n  { smul := \u03bb k x, by rw [algebra.smul_def, hf_field_norm.mul_eq, hf_ext k],\n    ..hf_field_norm},\n  rw spectral_norm.unique' h_alg hf_pow hf_alg_norm hf_na hna\nend\n\nlemma spectral_norm.is_mul_norm (hna : is_nonarchimedean (\u03bb k : K, \u2225k\u2225\u208a)) : \n  is_mul_norm (spectral_norm h_alg) :=\n{ mul_eq := \u03bb x y, begin\n    by_cases hx : 0 = spectral_norm h_alg x,\n    { rw [\u2190 hx, zero_mul],\n      rw [eq_comm, (spectral_norm.is_algebra_norm h_alg hna).to_is_norm.zero_iff] at hx,\n      rw [hx, zero_mul, (spectral_norm.is_algebra_norm h_alg hna).to_is_norm.zero] },\n    { set f := c_seminorm (spectral_norm.is_norm_le_one_class h_alg) hx\n        (spectral_norm.is_algebra_norm h_alg hna).to_is_norm.to_is_seminorm\n        (spectral_norm.is_pow_mult h_alg hna) with hf,\n      have hf_pow : is_pow_mult f := c_seminorm_is_pow_mult (spectral_norm.is_norm_le_one_class \n        h_alg) hx (spectral_norm.is_algebra_norm h_alg hna).to_is_norm.to_is_seminorm\n        (spectral_norm.is_pow_mult h_alg hna),\n      have hf_alg_norm : is_algebra_norm (normed_ring.to_is_norm K) f := \n      { smul := \u03bb k y,\n        begin\n          rw [\u2190 spectral_norm.extends h_alg, algebra.smul_def, hf],\n          have h_mul : \u2200 (y : L), spectral_norm h_alg ((algebra_map K L k) * y) = \n            spectral_norm h_alg (algebra_map K L k) * spectral_norm h_alg y,\n          { intro y, rw [spectral_norm.extends h_alg, \u2190 algebra.smul_def],\n            exact (spectral_norm.is_algebra_norm h_alg hna).smul _ _ },\n          rw \u2190 c_seminorm_apply_of_is_mult _ _ _ _ h_mul,\n          exact c_seminorm_is_mult_of_is_mult _ _ _ _ h_mul _,\n        end,\n        ..(c_seminorm_is_norm _ _ _ _ _) },\n      have hf_na : is_nonarchimedean f := \n      c_seminorm_is_nonarchimedean _ _ _ _ (spectral_norm.is_nonarchimedean h_alg hna),\n      rw [\u2190 spectral_norm.unique' h_alg hf_pow hf_alg_norm hf_na hna],\n      rw [hf, c_seminorm_c_is_mult (spectral_norm.is_norm_le_one_class h_alg) hx\n        (spectral_norm.is_algebra_norm h_alg hna).to_is_norm.to_is_seminorm\n        (spectral_norm.is_pow_mult h_alg hna)] }\n  end\n  ..spectral_norm.is_algebra_norm h_alg hna }\n\ndef spectral_norm.normed_field (h : is_nonarchimedean (\u03bb k : K, \u2225k\u2225\u208a)) : normed_field L := \n{ norm      := \u03bb (x : L), (spectral_norm h_alg x : \u211d),\n  dist      := \u03bb (x y : L), (spectral_norm h_alg (x - y) : \u211d),\n  dist_self := \u03bb x, by simp only [sub_self, nnreal.coe_eq_zero, spectral_norm.zero],\n  dist_comm := \u03bb x y, by rw [nnreal.coe_eq, \u2190 neg_sub, spectral_norm.neg h_alg h],\n  dist_triangle := \u03bb x y z, begin\n    simp only [dist_eq_norm],\n    rw \u2190 sub_add_sub_cancel x y z,\n    exact add_le_of_is_nonarchimedean (spectral_norm.zero h_alg)\n      (spectral_norm.is_nonarchimedean h_alg h) _ _,\n  end,\n  eq_of_dist_eq_zero := \u03bb x y hxy,\n  begin\n    simp only [nnreal.coe_eq_zero] at hxy,\n    rw \u2190 sub_eq_zero,\n    rw is_norm.zero_iff (spectral_norm.is_mul_norm h_alg h).to_is_norm at hxy,\n    exact hxy,\n  end,\n  dist_eq := \u03bb x y, by refl,\n  norm_mul' := \u03bb x y,\n  begin\n    simp only [\u2190 nnreal.coe_mul, nnreal.coe_eq],\n    exact (spectral_norm.is_mul_norm h_alg h).mul_eq x y,\n  end,\n  ..hL }\n\n/- noncomputable! instance us : uniform_space L := infer_instance\n\ninstance spectral_norm.complete_space (h_fin : @finite_dimensional K L _ _ _) :\n  complete_space L := sorry -/\n", "meta": {"author": "mariainesdff", "repo": "local_fields", "sha": "53a20ab87cf8f6ab27eedef2d9da929898951d82", "save_path": "github-repos/lean/mariainesdff-local_fields", "path": "github-repos/lean/mariainesdff-local_fields/local_fields-53a20ab87cf8f6ab27eedef2d9da929898951d82/old_lean_files/spectral_norm_unique.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370114, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3611619180957903}}
{"text": "-- This defines a framework for defining communicating sequential\n-- agents with asynchronous messages.\n--\n-- The key concepts are agents and messages.  The type for agents is\n-- defined by the framework, but message type is left as an abstract\n-- parameter.\n--\n-- Each agent consists of existentially quantified state type, a next\n-- action that produces a value of the state type, and a loop action\n-- which takes the state and generates a new state.  The loop action\n-- called again whenever the next action terminates.\n--\n-- On top of these core definitions, we define a transition relation\n-- over a system state, which is defined as a collection of agents,\n-- and messages between agents.\n--\n-- To define the transition relation, we allow that one\n-- of the following may happen at each step:\n--\n-- * A message may be received by an an agent when the agent is in a\n--   receive state and the message satisfies the guard on the receive.\n--\n-- * A message may be sent by an agent whose next action is to\n--   receive a message.\n--\n-- * A next action may complete, and loop restarted for each agent\n--   whose next action is to complete.\n--\n-- Design notes:\n--\n-- To avoid potentially unneeded complexity, there is deliberately a\n-- single type for messages, and no notion of names.  Agents can potentially\n-- receive any message in the system.  This allows one to define an adversary\n-- agent that intercepts messages or creates fake messages.\n--\n-- The framework includes no facilities for creating new agents, or\n-- terminating an existing agent, or synchronous messages.  We may need\n-- these later.\nimport galois.csp.class\n\nuniverse variables u v\n\nnamespace csp\n\n-- This represents an action for an agent an ability to send and receive messages.\n--\n-- Note: We may want to add additional actions such as spawning new agents,\ninductive process (message_t : Type u) (\u03b1 : Type v) : Type (max u v)\n\n  -- Given a message, receive calls the guard to check if the message\n  -- should be received, and if so, calls the next process with the message\n  -- and a proof that the guard is satisfied.  --\n  -- This function needed to be implemented this way rather than a\n  -- single next process with the type `(message_t \u2192 option process)` to\n  -- generate the correct recursor.\n| receive    : \u03a0(guard : message_t \u2192 bool) (next : \u03a0(m : message_t) (ok : guard m = tt), process), process\n| send       : message_t \u2192 process \u2192 process\n| pure       : \u03b1 \u2192 process\n\nnamespace process\n\nsection\nparameter {message_t : Type u}\n\n-- Define bind on processes.\ndef bind {\u03b1 \u03b2 : Type v} (m : process message_t \u03b1) (h : \u03b1 \u2192 process message_t \u03b2) : process message_t \u03b2 :=\n  let C := \u03bb(a : process message_t \u03b1), process message_t \u03b2 in\n  let on_receive (guard : message_t \u2192 bool)\n                 (next : \u03a0 (msg : message_t), guard msg = tt \u2192 process message_t \u03b1)\n                 (ind  : \u03a0 (msg : message_t), guard msg = tt \u2192 process message_t \u03b2)\n       : process message_t \u03b2 :=\n     process.receive guard ind in\n  let on_send := \u03bb(m : message_t) (next : process message_t \u03b1) (ind : process message_t \u03b2), process.send m ind in\n  @process.rec _ _ C @on_receive @on_send h m\n\n-- receive and bind commute\nprotected\ntheorem receive_bind {\u03b1 \u03b2 : Type v}\n                     (guard : message_t \u2192 bool)\n                     (f : \u03a0(msg : message_t), guard msg = tt \u2192 process message_t \u03b1)\n                     (h : \u03b1 \u2192 process message_t \u03b2)\n: bind (receive guard f) h = receive guard (\u03bbmsg pr, bind (f msg pr) h) := rfl\n\n-- send and bind commute\nprotected\ntheorem send_bind {\u03b1 \u03b2 : Type v} (msg : message_t) (f : process message_t \u03b1) (h : \u03b1 \u2192 process message_t \u03b2)\n: bind (send msg f) h = send msg (bind f h) := rfl\n\n-- pure is a \"left identity\" to bind.\nprotected\ntheorem pure_bind {\u03b1 \u03b2 : Type v} (v : \u03b1) (h : \u03b1 \u2192 process message_t \u03b2)\n: bind (pure message_t v) h = h v := rfl\n\nend -- section\n\ninstance is_monad (message_t : Type u) : monad (process message_t) :=\n{ pure := @process.pure message_t\n, bind := @process.bind message_t\n, id_map :=\n  begin\n    intros \u03b1,\n    apply process.rec,\n    -- Receive\n    { intros guard next ind,\n      dsimp [monad.map._default] at ind,\n      simp [monad.map._default, process.receive_bind, function.right_id],\n      apply congr_arg,\n      apply funext,\n      intro msg,\n      apply funext,\n      apply ind,\n    },\n    -- Send\n    { intros m next ind,\n      simp [monad.map._default, process.send_bind],\n      apply congr_arg,\n      exact ind,\n    },\n    -- Pure\n    { intros val,\n      simp [monad.map._default, process.pure_bind],\n      trivial\n    }\n  end\n, pure_bind := @process.pure_bind message_t\n, bind_assoc :=\n  begin\n    intros \u03b1 \u03b2 \u03b3 m f g,\n    induction m with guard rcv ind msg next ind,\n    -- Receive case\n    { simp [process.receive_bind],\n      apply congr_arg,\n      apply funext,\n      intro msg,\n      apply funext,\n      apply ind,\n    },\n    -- Send case\n    { simp [process.send_bind],\n      apply congr_arg,\n      exact ind,\n    },\n    -- Pure case\n    { simp [process.pure_bind],\n    },\n  end\n}\n\nend process\n\ninstance is_has_send_receive (message_t : Type) : has_send_receive message_t (process message_t) :=\n{ send := \u03bb(m : message_t), process.send m (pure ())\n, receive := \u03bb(guard : message_t \u2192 bool), process.receive guard (\u03bbm pr, pure \u27e8m, pr\u27e9)\n}\n\nsection\n\nparameter (message_t: Type u)\n\n-- An agent is defined as a type for the internal state, an process that produces\n-- the state, and a looping process that will execute when the process is complete.\n--\n-- Semantically, think of the behavior as `next >>= forever loop` where\n-- `forever loop = loop >=> forever loop`.\nstructure agent : Type (max u 1) :=\n(state_type : Type)\n(loop : state_type \u2192 process message_t state_type)\n(next : process message_t state_type)\n\n-- The system consists of agents and messages.\nstructure system_state : Type (max u 1) :=\n(agents : list agent)\n(messages : list message_t)\n\ndef partitions_core {\u03b1 : Type u} : list \u03b1 \u2192 list \u03b1 \u2192 list (\u03b1 \u00d7 list \u03b1 \u00d7 list \u03b1)\n| prev [] := []\n| prev (a::r) := (a, prev, r) :: partitions_core (a::prev) r\n\n-- Given a input list, this returns a list that contains a triple for each\n-- input.  The first component of the triple at index i is the element\n-- at index i in the input.  The other components are the elements before,\n-- and after the given value.\ndef partitions {\u03b1 : Type u} : list \u03b1 \u2192 list (\u03b1 \u00d7 list \u03b1 \u00d7 list \u03b1) := partitions_core []\n\n-- Enumerate the list of possible next states.\ndef next_states (system : system_state) : list system_state := do\n  (\u27e8state_t, loop, action\u27e9, before, after) \u2190 partitions system.agents,\n  match action with\n    -- If process is about to receive a message,\n    -- consider all possible messages process could receive..\n  | (process.receive guard next) :=\n    -- Pattern match on messages\n    -- Note: This uses list.bind to avoid a universe incompatibility\n    list.bind (partitions system.messages) $ \u03bbm,\n    let \u27e8msg, before_messages, after_messages\u27e9 := m in\n    -- If message satisfies guard\n    if pr : guard msg = tt then\n      -- Update agent to indicate it received message\n      let agent' := agent.mk state_t loop (next msg pr) in\n      pure { agents   := list.reverse_core before (agent' :: after)\n             -- Delete message\n           , messages := list.reverse_core before_messages after_messages\n           }\n    else\n      -- Otherwise have message fail.\n      []\n    -- Send a new message\n  | (process.send msg next) :=\n    -- Update agent state and add message to list\n    let agent' := agent.mk state_t loop next in\n    pure { agents := list.reverse_core before (agent' :: after)\n         , messages := msg :: system.messages\n         }\n    -- Start a new loop iteration.\n  | (process.pure .(message_t) next_state)   :=\n    let agent' := agent.mk state_t loop (loop next_state) in\n    pure { system with agents := list.reverse_core before (agent' :: after) }\n  end\n\nend\n\nend csp\n", "meta": {"author": "GaloisInc", "repo": "lean-protocol-support", "sha": "cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda", "save_path": "github-repos/lean/GaloisInc-lean-protocol-support", "path": "github-repos/lean/GaloisInc-lean-protocol-support/lean-protocol-support-cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda/galois/csp/csp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850278370112, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.36116191809579024}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\n\nimport topology.sheaves.sheaf_condition.sites\nimport category_theory.limits.preserves.basic\nimport category_theory.category.pairwise\nimport category_theory.limits.constructions.binary_products\n\n/-!\n# Equivalent formulations of the sheaf condition\n\nWe give an equivalent formulation of the sheaf condition.\n\nGiven any indexed type `\u03b9`, we define `overlap \u03b9`,\na category with objects corresponding to\n* individual open sets, `single i`, and\n* intersections of pairs of open sets, `pair i j`,\nwith morphisms from `pair i j` to both `single i` and `single j`.\n\nAny open cover `U : \u03b9 \u2192 opens X` provides a functor `diagram U : overlap \u03b9 \u2964 (opens X)\u1d52\u1d56`.\n\nThere is a canonical cone over this functor, `cone U`, whose cone point is `supr U`,\nand in fact this is a limit cone.\n\nA presheaf `F : presheaf C X` is a sheaf precisely if it preserves this limit.\nWe express this in two equivalent ways, as\n* `is_limit (F.map_cone (cone U))`, or\n* `preserves_limit (diagram U) F`\n-/\n\nnoncomputable theory\n\nuniverses v u\n\nopen topological_space\nopen Top\nopen opposite\nopen category_theory\nopen category_theory.limits\n\nnamespace Top.presheaf\n\nvariables {X : Top.{v}}\n\nvariables {C : Type u} [category.{v} C]\n\n/--\nAn alternative formulation of the sheaf condition\n(which we prove equivalent to the usual one below as\n`is_sheaf_iff_is_sheaf_pairwise_intersections`).\n\nA presheaf is a sheaf if `F` sends the cone `(pairwise.cocone U).op` to a limit cone.\n(Recall `pairwise.cocone U` has cone point `supr U`, mapping down to the `U i` and the `U i \u2293 U j`.)\n-/\ndef is_sheaf_pairwise_intersections (F : presheaf C X) : Prop :=\n\u2200 \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X), nonempty (is_limit (F.map_cone (pairwise.cocone U).op))\n\n/--\nAn alternative formulation of the sheaf condition\n(which we prove equivalent to the usual one below as\n`is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections`).\n\nA presheaf is a sheaf if `F` preserves the limit of `pairwise.diagram U`.\n(Recall `pairwise.diagram U` is the diagram consisting of the pairwise intersections\n`U i \u2293 U j` mapping into the open sets `U i`. This diagram has limit `supr U`.)\n-/\ndef is_sheaf_preserves_limit_pairwise_intersections (F : presheaf C X) : Prop :=\n\u2200 \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X), nonempty (preserves_limit (pairwise.diagram U).op F)\n\n/-!\nThe remainder of this file shows that these conditions are equivalent\nto the usual sheaf condition.\n-/\n\nvariables [has_products C]\n\nnamespace sheaf_condition_pairwise_intersections\n\nopen category_theory.pairwise category_theory.pairwise.hom\nopen sheaf_condition_equalizer_products\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_functor_obj (F : presheaf C X)\n  \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens \u21a5X) (c : limits.cone ((diagram U).op \u22d9 F)) :\n  limits.cone (sheaf_condition_equalizer_products.diagram F U) :=\n{ X := c.X,\n  \u03c0 :=\n  { app := \u03bb Z,\n      walking_parallel_pair.cases_on Z\n        (pi.lift (\u03bb (i : \u03b9), c.\u03c0.app (op (single i))))\n        (pi.lift (\u03bb (b : \u03b9 \u00d7 \u03b9), c.\u03c0.app (op (pair b.1 b.2)))),\n    naturality' := \u03bb Y Z f,\n    begin\n      cases Y; cases Z; cases f,\n      { ext i, dsimp,\n        simp only [limit.lift_\u03c0, category.id_comp, fan.mk_\u03c0_app, category_theory.functor.map_id,\n          category.assoc],\n        dsimp,\n        simp only [limit.lift_\u03c0, category.id_comp, fan.mk_\u03c0_app], },\n      { ext \u27e8i, j\u27e9, dsimp [sheaf_condition_equalizer_products.left_res],\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        have h := c.\u03c0.naturality (quiver.hom.op (hom.left i j)),\n        dsimp at h,\n        simpa using h, },\n      { ext \u27e8i, j\u27e9, dsimp [sheaf_condition_equalizer_products.right_res],\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        have h := c.\u03c0.naturality (quiver.hom.op (hom.right i j)),\n        dsimp at h,\n        simpa using h, },\n      { ext i, dsimp,\n        simp only [limit.lift_\u03c0, category.id_comp, fan.mk_\u03c0_app, category_theory.functor.map_id,\n          category.assoc],\n        dsimp,\n        simp only [limit.lift_\u03c0, category.id_comp, fan.mk_\u03c0_app], },\n    end, }, }\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_functor (F : presheaf C X)\n  \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens \u21a5X) :\n  limits.cone ((diagram U).op \u22d9 F) \u2964\n    limits.cone (sheaf_condition_equalizer_products.diagram F U) :=\n{ obj := \u03bb c, cone_equiv_functor_obj F U c,\n  map := \u03bb c c' f,\n  { hom := f.hom,\n    w' := \u03bb j, begin\n      cases j;\n      { ext, simp only [limits.fan.mk_\u03c0_app, limits.cone_morphism.w,\n        limits.limit.lift_\u03c0, category.assoc, cone_equiv_functor_obj_\u03c0_app], },\n    end }, }.\n\nend\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_inverse_obj (F : presheaf C X)\n  \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens \u21a5X)\n  (c : limits.cone (sheaf_condition_equalizer_products.diagram F U)) :\n  limits.cone ((diagram U).op \u22d9 F) :=\n{ X := c.X,\n  \u03c0 :=\n  { app :=\n    begin\n      intro x,\n      induction x using opposite.rec,\n      rcases x with (\u27e8i\u27e9|\u27e8i,j\u27e9),\n      { exact c.\u03c0.app (walking_parallel_pair.zero) \u226b pi.\u03c0 _ i, },\n      { exact c.\u03c0.app (walking_parallel_pair.one) \u226b pi.\u03c0 _ (i, j), }\n    end,\n    naturality' :=\n    begin\n      intros x y f,\n      induction x using opposite.rec,\n      induction y using opposite.rec,\n      have ef : f = f.unop.op := rfl,\n      revert ef,\n      generalize : f.unop = f',\n      rintro rfl,\n      rcases x with \u27e8i\u27e9|\u27e8\u27e9; rcases y with \u27e8\u27e9|\u27e8j,j\u27e9; rcases f' with \u27e8\u27e9,\n      { dsimp, erw [F.map_id], simp, },\n      { dsimp, simp only [category.id_comp, category.assoc],\n        have h := c.\u03c0.naturality (walking_parallel_pair_hom.left),\n        dsimp [sheaf_condition_equalizer_products.left_res] at h,\n        simp only [category.id_comp] at h,\n        have h' := h =\u226b pi.\u03c0 _ (i, j),\n        rw h',\n        simp,\n        refl, },\n      { dsimp, simp only [category.id_comp, category.assoc],\n        have h := c.\u03c0.naturality (walking_parallel_pair_hom.right),\n        dsimp [sheaf_condition_equalizer_products.right_res] at h,\n        simp only [category.id_comp] at h,\n        have h' := h =\u226b pi.\u03c0 _ (j, i),\n        rw h',\n        simp,\n        refl, },\n      { dsimp, erw [F.map_id], simp, },\n    end, }, }\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_inverse (F : presheaf C X)\n  \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens \u21a5X) :\n  limits.cone (sheaf_condition_equalizer_products.diagram F U) \u2964\n    limits.cone ((diagram U).op \u22d9 F) :=\n{ obj := \u03bb c, cone_equiv_inverse_obj F U c,\n  map := \u03bb c c' f,\n  { hom := f.hom,\n    w' :=\n    begin\n      intro x,\n      induction x using opposite.rec,\n      rcases x with (\u27e8i\u27e9|\u27e8i,j\u27e9),\n      { dsimp,\n        rw [\u2190(f.w walking_parallel_pair.zero), category.assoc], },\n      { dsimp,\n        rw [\u2190(f.w walking_parallel_pair.one), category.assoc], },\n    end }, }.\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_unit_iso_app (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens \u21a5X)\n  (c : cone ((diagram U).op \u22d9 F)) :\n  (\ud835\udfed (cone ((diagram U).op \u22d9 F))).obj c \u2245\n    (cone_equiv_functor F U \u22d9 cone_equiv_inverse F U).obj c :=\n{ hom :=\n  { hom := \ud835\udfd9 _,\n    w' := \u03bb j, begin\n      induction j using opposite.rec, rcases j;\n      { dsimp, simp only [limits.fan.mk_\u03c0_app, category.id_comp, limits.limit.lift_\u03c0], }\n    end, },\n  inv :=\n  { hom := \ud835\udfd9 _,\n    w' := \u03bb j, begin\n      induction j using opposite.rec, rcases j;\n      { dsimp, simp only [limits.fan.mk_\u03c0_app, category.id_comp, limits.limit.lift_\u03c0], }\n    end },\n  hom_inv_id' := begin\n    ext,\n    simp only [category.comp_id, limits.cone.category_comp_hom, limits.cone.category_id_hom],\n  end,\n  inv_hom_id' := begin\n    ext,\n    simp only [category.comp_id, limits.cone.category_comp_hom, limits.cone.category_id_hom],\n  end, }\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_unit_iso (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X) :\n  \ud835\udfed (limits.cone ((diagram U).op \u22d9 F)) \u2245\n    cone_equiv_functor F U \u22d9 cone_equiv_inverse F U :=\nnat_iso.of_components (cone_equiv_unit_iso_app F U) (by tidy)\n\n/-- Implementation of `sheaf_condition_pairwise_intersections.cone_equiv`. -/\n@[simps]\ndef cone_equiv_counit_iso (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X) :\n  cone_equiv_inverse F U \u22d9 cone_equiv_functor F U \u2245\n    \ud835\udfed (limits.cone (sheaf_condition_equalizer_products.diagram F U)) :=\nnat_iso.of_components (\u03bb c,\n{ hom :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      rintro \u27e8_|_\u27e9,\n      { ext, dsimp, simp only [category.id_comp, limits.fan.mk_\u03c0_app, limits.limit.lift_\u03c0], },\n      { ext \u27e8i,j\u27e9, dsimp, simp only [category.id_comp, limits.fan.mk_\u03c0_app, limits.limit.lift_\u03c0], },\n    end },\n  inv :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      rintro \u27e8_|_\u27e9,\n      { ext, dsimp, simp only [category.id_comp, limits.fan.mk_\u03c0_app, limits.limit.lift_\u03c0], },\n      { ext \u27e8i,j\u27e9, dsimp, simp only [category.id_comp, limits.fan.mk_\u03c0_app, limits.limit.lift_\u03c0], },\n    end, },\n  hom_inv_id' := by { ext, dsimp, simp only [category.comp_id], },\n  inv_hom_id' := by { ext, dsimp, simp only [category.comp_id], }, })\n(\u03bb c d f, by { ext, dsimp, simp only [category.comp_id, category.id_comp], })\n\n/--\nCones over `diagram U \u22d9 F` are the same as a cones over the usual sheaf condition equalizer diagram.\n-/\n@[simps]\ndef cone_equiv (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X) :\n  limits.cone ((diagram U).op \u22d9 F) \u224c limits.cone (sheaf_condition_equalizer_products.diagram F U) :=\n{ functor := cone_equiv_functor F U,\n  inverse := cone_equiv_inverse F U,\n  unit_iso := cone_equiv_unit_iso F U,\n  counit_iso := cone_equiv_counit_iso F U, }\n\nlocal attribute [reducible]\n  sheaf_condition_equalizer_products.res\n  sheaf_condition_equalizer_products.left_res\n\n/--\nIf `sheaf_condition_equalizer_products.fork` is an equalizer,\nthen `F.map_cone (cone U)` is a limit cone.\n-/\ndef is_limit_map_cone_of_is_limit_sheaf_condition_fork\n  (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X)\n  (P : is_limit (sheaf_condition_equalizer_products.fork F U)) :\n  is_limit (F.map_cone (cocone U).op) :=\nis_limit.of_iso_limit ((is_limit.of_cone_equiv (cone_equiv F U).symm).symm P)\n{ hom :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      intro x,\n      induction x using opposite.rec,\n      rcases x with \u27e8\u27e9,\n      { dsimp, simp, refl, },\n      { dsimp,\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        rw \u2190F.map_comp,\n        refl, }\n    end },\n  inv :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      intro x,\n      induction x using opposite.rec,\n      rcases x with \u27e8\u27e9,\n      { dsimp, simp, refl, },\n      { dsimp,\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        rw \u2190F.map_comp,\n        refl, }\n    end },\n  hom_inv_id' := by { ext, dsimp, simp only [category.comp_id], },\n  inv_hom_id' := by { ext, dsimp, simp only [category.comp_id], }, }\n\n/--\nIf `F.map_cone (cone U)` is a limit cone,\nthen `sheaf_condition_equalizer_products.fork` is an equalizer.\n-/\ndef is_limit_sheaf_condition_fork_of_is_limit_map_cone\n  (F : presheaf C X) \u2983\u03b9 : Type v\u2984 (U : \u03b9 \u2192 opens X)\n  (Q : is_limit (F.map_cone (cocone U).op)) :\n  is_limit (sheaf_condition_equalizer_products.fork F U) :=\nis_limit.of_iso_limit ((is_limit.of_cone_equiv (cone_equiv F U)).symm Q)\n{ hom :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      rintro \u27e8\u27e9,\n      { dsimp, simp, refl, },\n      { dsimp, ext \u27e8i, j\u27e9,\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        rw \u2190F.map_comp,\n        refl, }\n    end },\n  inv :=\n  { hom := \ud835\udfd9 _,\n    w' :=\n    begin\n      rintro \u27e8\u27e9,\n      { dsimp, simp, refl, },\n      { dsimp, ext \u27e8i, j\u27e9,\n        simp only [limit.lift_\u03c0, limit.lift_\u03c0_assoc, category.id_comp, fan.mk_\u03c0_app,\n          category.assoc],\n        rw \u2190F.map_comp,\n        refl, }\n    end },\n  hom_inv_id' := by { ext, dsimp, simp only [category.comp_id], },\n  inv_hom_id' := by { ext, dsimp, simp only [category.comp_id], }, }\n\n\nend sheaf_condition_pairwise_intersections\n\nopen sheaf_condition_pairwise_intersections\n\n/--\nThe sheaf condition in terms of an equalizer diagram is equivalent\nto the reformulation in terms of a limit diagram over `U i` and `U i \u2293 U j`.\n-/\nlemma is_sheaf_iff_is_sheaf_pairwise_intersections (F : presheaf C X) :\n  F.is_sheaf \u2194 F.is_sheaf_pairwise_intersections :=\niff.intro (\u03bb h \u03b9 U, \u27e8is_limit_map_cone_of_is_limit_sheaf_condition_fork F U (h U).some\u27e9)\n  (\u03bb h \u03b9 U, \u27e8is_limit_sheaf_condition_fork_of_is_limit_map_cone F U (h U).some\u27e9)\n\n/--\nThe sheaf condition in terms of an equalizer diagram is equivalent\nto the reformulation in terms of the presheaf preserving the limit of the diagram\nconsisting of the `U i` and `U i \u2293 U j`.\n-/\nlemma is_sheaf_iff_is_sheaf_preserves_limit_pairwise_intersections (F : presheaf C X) :\n  F.is_sheaf \u2194 F.is_sheaf_preserves_limit_pairwise_intersections :=\nbegin\n  rw is_sheaf_iff_is_sheaf_pairwise_intersections,\n  split,\n  { intros h \u03b9 U,\n    exact \u27e8preserves_limit_of_preserves_limit_cone (pairwise.cocone_is_colimit U).op (h U).some\u27e9 },\n  { intros h \u03b9 U,\n    haveI := (h U).some,\n    exact \u27e8preserves_limit.preserves (pairwise.cocone_is_colimit U).op\u27e9 }\nend\n\nend Top.presheaf\n\nnamespace Top.sheaf\n\nvariables {X : Top.{v}} {C : Type u} [category.{v} C] [has_products C]\nvariables (F : X.sheaf C) (U V : opens X)\nopen category_theory.limits\n\n/-- For a sheaf `F`, `F(U \u222a V)` is the pullback of `F(U) \u27f6 F(U \u2229 V)` and `F(V) \u27f6 F(U \u2229 V)`.\nThis is the pullback cone. -/\ndef inter_union_pullback_cone : pullback_cone\n  (F.1.map (hom_of_le inf_le_left : U \u2229 V \u27f6 _).op) (F.1.map (hom_of_le inf_le_right).op) :=\npullback_cone.mk (F.1.map (hom_of_le le_sup_left).op) (F.1.map (hom_of_le le_sup_right).op)\n  (by { rw [\u2190 F.1.map_comp, \u2190 F.1.map_comp], congr })\n\n@[simp] lemma inter_union_pullback_cone_X :\n  (inter_union_pullback_cone F U V).X = F.1.obj (op $ U \u222a V) := rfl\n@[simp] lemma inter_union_pullback_cone_fst :\n  (inter_union_pullback_cone F U V).fst = F.1.map (hom_of_le le_sup_left).op := rfl\n@[simp] lemma inter_union_pullback_cone_snd :\n  (inter_union_pullback_cone F U V).snd = F.1.map (hom_of_le le_sup_right).op := rfl\n\nvariable (s : pullback_cone\n  (F.1.map (hom_of_le inf_le_left : U \u2229 V \u27f6 _).op) (F.1.map (hom_of_le inf_le_right).op))\n\n/-- (Implementation).\nEvery cone over `F(U) \u27f6 F(U \u2229 V)` and `F(V) \u27f6 F(U \u2229 V)` factors through `F(U \u222a V)`. -/\ndef inter_union_pullback_cone_lift : s.X \u27f6 F.1.obj (op (U \u222a V)) :=\nbegin\n  let \u03b9 : walking_pair \u2192 opens X := \u03bb j, walking_pair.cases_on j U V,\n  have h\u03b9 : U \u222a V = supr \u03b9,\n  { ext, split,\n    { rintros (h|h),\n    exacts [\u27e8_,\u27e8_,\u27e8walking_pair.left,rfl\u27e9,rfl\u27e9,h\u27e9, \u27e8_,\u27e8_,\u27e8walking_pair.right,rfl\u27e9,rfl\u27e9,h\u27e9] },\n    { rintros \u27e8_,\u27e8_,\u27e8\u27e8\u27e9,\u27e8\u27e9\u27e9,\u27e8\u27e9\u27e9,z\u27e9, exacts [or.inl z, or.inr z] } },\n  refine (F.1.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 \u03b9).some.lift\n    \u27e8s.X, { app := _, naturality' := _ }\u27e9 \u226b F.1.map (eq_to_hom h\u03b9).op,\n  { apply opposite.rec,\n    rintro ((_|_)|(_|_)),\n    exacts [s.fst, s.snd, s.fst \u226b F.1.map (hom_of_le inf_le_left).op,\n      s.snd \u226b F.1.map (hom_of_le inf_le_left).op] },\n  rintros i j f,\n  induction i using opposite.rec,\n  induction j using opposite.rec,\n  let g : j \u27f6 i := f.unop, have : f = g.op := rfl, clear_value g, subst this,\n  rcases i with ((_|_)|(_|_)); rcases j with ((_|_)|(_|_)); rcases g; dsimp;\n    simp only [category.id_comp, s.condition, category_theory.functor.map_id, category.comp_id],\n  { rw [\u2190 cancel_mono (F.1.map (eq_to_hom $ inf_comm : U \u2229 V \u27f6 _).op), category.assoc,\n      category.assoc],\n    erw [\u2190 F.1.map_comp, \u2190 F.1.map_comp],\n    convert s.condition.symm },\n  { convert s.condition }\nend\n\nlemma inter_union_pullback_cone_lift_left :\n  inter_union_pullback_cone_lift F U V s \u226b F.1.map (hom_of_le le_sup_left).op = s.fst :=\nbegin\n  erw [category.assoc, \u2190F.1.map_comp],\n  exact (F.1.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 _).some.fac _\n    (op $ pairwise.single walking_pair.left)\nend\n\nlemma inter_union_pullback_cone_lift_right :\n  inter_union_pullback_cone_lift F U V s \u226b F.1.map (hom_of_le le_sup_right).op = s.snd :=\nbegin\n  erw [category.assoc, \u2190F.1.map_comp],\n  exact (F.1.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 _).some.fac _\n    (op $ pairwise.single walking_pair.right)\nend\n\n/-- For a sheaf `F`, `F(U \u222a V)` is the pullback of `F(U) \u27f6 F(U \u2229 V)` and `F(V) \u27f6 F(U \u2229 V)`. -/\ndef is_limit_pullback_cone : is_limit (inter_union_pullback_cone F U V) :=\nbegin\n  let \u03b9 : walking_pair \u2192 opens X := \u03bb j, walking_pair.cases_on j U V,\n  have h\u03b9 : U \u222a V = supr \u03b9,\n  { ext, split,\n    { rintros (h|h),\n    exacts [\u27e8_,\u27e8_,\u27e8walking_pair.left,rfl\u27e9,rfl\u27e9,h\u27e9, \u27e8_,\u27e8_,\u27e8walking_pair.right,rfl\u27e9,rfl\u27e9,h\u27e9] },\n    { rintros \u27e8_,\u27e8_,\u27e8\u27e8\u27e9,\u27e8\u27e9\u27e9,\u27e8\u27e9\u27e9,z\u27e9, exacts [or.inl z, or.inr z] } },\n  apply pullback_cone.is_limit_aux',\n  intro s,\n  use inter_union_pullback_cone_lift F U V s,\n  refine \u27e8_,_,_\u27e9,\n  { apply inter_union_pullback_cone_lift_left },\n  { apply inter_union_pullback_cone_lift_right },\n  { intros m h\u2081 h\u2082,\n    rw \u2190 cancel_mono (F.1.map (eq_to_hom h\u03b9.symm).op),\n    apply (F.1.is_sheaf_iff_is_sheaf_pairwise_intersections.mp F.2 \u03b9).some.hom_ext,\n    apply opposite.rec,\n    rintro ((_|_)|(_|_)); rw [category.assoc, category.assoc],\n    { erw \u2190 F.1.map_comp,\n      convert h\u2081,\n      apply inter_union_pullback_cone_lift_left },\n    { erw \u2190 F.1.map_comp,\n      convert h\u2082,\n      apply inter_union_pullback_cone_lift_right },\n    all_goals\n    { dsimp only [functor.op, pairwise.cocone_\u03b9_app, functor.map_cone_\u03c0_app,\n        cocone.op, pairwise.cocone_\u03b9_app_2, unop_op, op_comp],\n      simp_rw [F.1.map_comp, \u2190 category.assoc],\n      congr' 1,\n      simp_rw [category.assoc, \u2190 F.1.map_comp] },\n    { convert h\u2081,\n      apply inter_union_pullback_cone_lift_left },\n    { convert h\u2082,\n      apply inter_union_pullback_cone_lift_right } }\nend\n\n/-- If `U, V` are disjoint, then `F(U \u222a V) = F(U) \u00d7 F(V)`. -/\ndef is_product_of_disjoint (h : U \u2229 V = \u22a5) : is_limit\n    (binary_fan.mk (F.1.map (hom_of_le le_sup_left : _ \u27f6 U \u2294 V).op)\n      (F.1.map (hom_of_le le_sup_right : _ \u27f6 U \u2294 V).op)) :=\nis_product_of_is_terminal_is_pullback _ _ _ _\n  (F.is_terminal_of_eq_empty h) (is_limit_pullback_cone F U V)\n\nend Top.sheaf\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/topology/sheaves/sheaf_condition/pairwise_intersections.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.36116191176224566}}
{"text": "class Def (\u03b1 : Type u) where\n  val : \u03b1\n\ninstance : Def Nat where\n  val := 10\n\ntheorem ex1 : Def.val = 10 := rfl\n\ninstance (priority := default+1) : Def Nat where\n  val := 20\n\ntheorem ex2 : Def.val = 20 := rfl\n\ninstance : Def Nat where\n  val := 30\n\ntheorem ex3 : Def.val = 20 := rfl\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/instprio.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.757794360334681, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3611493718263238}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.braided\nimport category_theory.functor.category\nimport category_theory.functor.const\n\n/-!\n# Monoidal structure on `C \u2964 D` when `D` is monoidal.\n\nWhen `C` is any category, and `D` is a monoidal category,\nthere is a natural \"pointwise\" monoidal structure on `C \u2964 D`.\n\nThe initial intended application is tensor product of presheaves.\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category_theory\nopen category_theory.monoidal_category\n\nnamespace category_theory.monoidal\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D] [monoidal_category.{v\u2082} D]\n\nnamespace functor_category\n\nvariables (F G F' G' : C \u2964 D)\n\n/--\n(An auxiliary definition for `functor_category_monoidal`.)\nTensor product of functors `C \u2964 D`, when `D` is monoidal.\n -/\n@[simps]\ndef tensor_obj : C \u2964 D :=\n{ obj := \u03bb X, F.obj X \u2297 G.obj X,\n  map := \u03bb X Y f, F.map f \u2297 G.map f,\n  map_id' := \u03bb X, by rw [F.map_id, G.map_id, tensor_id],\n  map_comp' := \u03bb X Y Z f g, by rw [F.map_comp, G.map_comp, tensor_comp], }\n\nvariables {F G F' G'}\nvariables (\u03b1 : F \u27f6 G) (\u03b2 : F' \u27f6 G')\n\n/--\n(An auxiliary definition for `functor_category_monoidal`.)\nTensor product of natural transformations into `D`, when `D` is monoidal.\n-/\n@[simps]\ndef tensor_hom : tensor_obj F F' \u27f6 tensor_obj G G' :=\n{ app := \u03bb X, \u03b1.app X \u2297 \u03b2.app X,\n  naturality' :=\n  \u03bb X Y f, by { dsimp, rw [\u2190tensor_comp, \u03b1.naturality, \u03b2.naturality, tensor_comp], } }\n\nend functor_category\n\nopen category_theory.monoidal.functor_category\n\n/--\nWhen `C` is any category, and `D` is a monoidal category,\nthe functor category `C \u2964 D` has a natural pointwise monoidal structure,\nwhere `(F \u2297 G).obj X = F.obj X \u2297 G.obj X`.\n-/\ninstance functor_category_monoidal : monoidal_category (C \u2964 D) :=\n{ tensor_obj := \u03bb F G, tensor_obj F G,\n  tensor_hom := \u03bb F G F' G' \u03b1 \u03b2, tensor_hom \u03b1 \u03b2,\n  tensor_id' := \u03bb F G, by { ext, dsimp, rw [tensor_id], },\n  tensor_comp' := \u03bb F G H F' G' H' \u03b1 \u03b2 \u03b3 \u03b4, by { ext, dsimp, rw [tensor_comp], },\n  tensor_unit := (category_theory.functor.const C).obj (\ud835\udfd9_ D),\n  left_unitor :=  \u03bb F,\n    nat_iso.of_components (\u03bb X, \u03bb_ (F.obj X)) (\u03bb X Y f, by { dsimp, rw left_unitor_naturality, }),\n  right_unitor := \u03bb F,\n    nat_iso.of_components (\u03bb X, \u03c1_ (F.obj X)) (\u03bb X Y f, by { dsimp, rw right_unitor_naturality, }),\n  associator := \u03bb F G H,\n    nat_iso.of_components\n      (\u03bb X, \u03b1_ (F.obj X) (G.obj X) (H.obj X)) (\u03bb X Y f, by { dsimp, rw associator_naturality, }),\n  left_unitor_naturality' := \u03bb F G \u03b1, by { ext X, dsimp, rw left_unitor_naturality, },\n  right_unitor_naturality' := \u03bb F G \u03b1, by { ext X, dsimp, rw right_unitor_naturality, },\n  associator_naturality' := \u03bb F G H F' G' H' \u03b1 \u03b2 \u03b3, by { ext X, dsimp, rw associator_naturality, },\n  triangle' := \u03bb F G, begin ext X, dsimp, rw triangle, end,\n  pentagon' := \u03bb F G H K, begin ext X, dsimp, rw pentagon, end, }\n\n@[simp]\nlemma tensor_unit_obj {X} : (\ud835\udfd9_ (C \u2964 D)).obj X = \ud835\udfd9_ D := rfl\n\n@[simp]\nlemma tensor_unit_map {X Y} {f : X \u27f6 Y} : (\ud835\udfd9_ (C \u2964 D)).map f = \ud835\udfd9 (\ud835\udfd9_ D) := rfl\n\n@[simp]\nlemma tensor_obj_obj {F G : C \u2964 D} {X} : (F \u2297 G).obj X = F.obj X \u2297 G.obj X := rfl\n\n@[simp]\nlemma tensor_obj_map {F G : C \u2964 D} {X Y} {f : X \u27f6 Y} : (F \u2297 G).map f = F.map f \u2297 G.map f := rfl\n\n@[simp]\nlemma tensor_hom_app {F G F' G' : C \u2964 D} {\u03b1 : F \u27f6 G} {\u03b2 : F' \u27f6 G'} {X} :\n  (\u03b1 \u2297 \u03b2).app X = \u03b1.app X \u2297 \u03b2.app X := rfl\n\n@[simp]\nlemma left_unitor_hom_app {F : C \u2964 D} {X} :\n  ((\u03bb_ F).hom : (\ud835\udfd9_ _) \u2297 F \u27f6 F).app X = (\u03bb_ (F.obj X)).hom := rfl\n\n@[simp]\nlemma left_unitor_inv_app {F : C \u2964 D} {X} :\n  ((\u03bb_ F).inv : F \u27f6 (\ud835\udfd9_ _) \u2297 F).app X = (\u03bb_ (F.obj X)).inv := rfl\n\n@[simp]\nlemma right_unitor_hom_app {F : C \u2964 D} {X} :\n  ((\u03c1_ F).hom : F \u2297 (\ud835\udfd9_ _) \u27f6 F).app X = (\u03c1_ (F.obj X)).hom := rfl\n\n@[simp]\nlemma right_unitor_inv_app {F : C \u2964 D} {X} :\n  ((\u03c1_ F).inv : F \u27f6 F \u2297 (\ud835\udfd9_ _)).app X = (\u03c1_ (F.obj X)).inv := rfl\n\n@[simp]\nlemma associator_hom_app {F G H : C \u2964 D} {X} :\n  ((\u03b1_ F G H).hom : (F \u2297 G) \u2297 H \u27f6 F \u2297 (G \u2297 H)).app X = (\u03b1_ (F.obj X) (G.obj X) (H.obj X)).hom :=\nrfl\n\n@[simp]\nlemma associator_inv_app {F G H : C \u2964 D} {X} :\n  ((\u03b1_ F G H).inv : F \u2297 (G \u2297 H) \u27f6 (F \u2297 G) \u2297 H).app X = (\u03b1_ (F.obj X) (G.obj X) (H.obj X)).inv :=\nrfl\n\nsection braided_category\n\nopen category_theory.braided_category\nvariables [braided_category.{v\u2082} D]\n\n/--\nWhen `C` is any category, and `D` is a braided monoidal category,\nthe natural pointwise monoidal structure on the functor category `C \u2964 D`\nis also braided.\n-/\ninstance functor_category_braided : braided_category (C \u2964 D) :=\n{ braiding := \u03bb F G, nat_iso.of_components (\u03bb X, \u03b2_ _ _) (by tidy),\n  hexagon_forward' := \u03bb F G H, by { ext X, apply hexagon_forward, },\n  hexagon_reverse' := \u03bb F G H, by { ext X, apply hexagon_reverse, }, }\n\nexample : braided_category (C \u2964 D) := category_theory.monoidal.functor_category_braided\n\nend braided_category\n\nsection symmetric_category\n\nopen category_theory.symmetric_category\nvariables [symmetric_category.{v\u2082} D]\n\n/--\nWhen `C` is any category, and `D` is a symmetric monoidal category,\nthe natural pointwise monoidal structure on the functor category `C \u2964 D`\nis also symmetric.\n-/\ninstance functor_category_symmetric : symmetric_category (C \u2964 D) :=\n{ symmetry' := \u03bb F G, by { ext X, apply symmetry, },}\n\nend symmetric_category\n\nend category_theory.monoidal\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/monoidal/functor_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7279754489059774, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.361144128208578}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2260 y \u2227 |x - y| < 1 :=\nbegin\n  assume (y : \u211d) (h1 : y \u2208 Icc 0 1),\n  have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 \u00ac (i * \u03b1 - \u230ai * \u03b1\u230b = j * \u03b1 - \u230aj * \u03b1\u230b), from by auto [is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def_iff, is_rat.def_iff, is_rat.def, is_rat.def\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2260 y \u2227 dist x y < 1 :=\nbegin\n  assume (y : \u211d) (h1 : y \u2208 Icc 0 1),\n  have h2 : \u2200 (i : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 0, from by auto [int.cast_ne_zero],\n  have h3 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h4 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h5 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h6 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h7 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h8 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h9 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h10 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h11 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h12 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h13 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h14 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h15 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h16 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h17 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h18 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h19 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h20 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h21 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h22 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h23 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h24 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h25 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h26 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h27 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h28 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h29 : \u2200 (i j : \u2124), (i : \u211d) * \u03b1 - (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1 - (j : \u211d) * \u03b1, from by auto [int.cast_ne_zero],\n  have h30\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2260 y \u2227 abs (x - y) < 1 :=\nbegin\n  assume y h1,\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h9 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h10 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h11 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h12 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h13 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h14 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h15 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h16 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h17 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h18 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h19 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h20 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h21 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h22 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h23 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h24 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h25 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h26 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 - (i : \u211d) \u2260 (j : \u211d) * \u03b1 - (j : \u211d), from by auto [sub_eq_zero_iff_eq, mul_eq_zero_iff_eq, h],\n  have h27 : \u2200 i j : \u2124, i \u2260 j \u2192 (i : \u211d) * \u03b1 \u2260 (j : \u211d) * \u03b1, from by auto [sub_eq_zero_iff_eq, mul_eq_zero\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by auto [abs_sub_lt_iff] using [linarith],\n  \n  assume (h7 : \u03b5 > 0),\n\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by auto [lt_of_le_of_lt, le_max_left, le_max_right],\n  \n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by auto [h8, h10, h5, h9],\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by auto [h11] using [linarith],\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by auto [h5, h15], \n\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto-Natural-Language-Proof-Translation/lean_proof_auto-4_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8031737963569016, "lm_q2_score": 0.4493926344647597, "lm_q1q2_score": 0.3609403882778904}}
{"text": "import Mathlib.Tactic.Basic\nimport Mathlib.Data.Nat.Basic\nimport Mathlib.Init.Algebra.Order\nimport Mathlib.Init.Data.Nat.Basic\nimport Mathlib.Init.Data.Nat.Lemmas\nimport Mathlib.Data.String.Defs\nimport Mathlib.Data.String.Lemmas\nimport Mathlib.Data.Equiv.Basic\nimport Mathlib.Init.Function\nimport Timelib.Util\nimport Timelib.NanoPrecision.TimeZone.Basic\nimport Timelib.NanoPrecision.Duration.UnsignedDuration\nimport Timelib.NanoPrecision.Duration.SignedDuration\nimport Timelib.NanoPrecision.ClockTime.ClockTime\n\nstructure HClockTime where\n  timeZone : TimeZone\n  clockTime : ClockTime timeZone\n\ndef HClockTime.EquivSigma : Equiv HClockTime (Sigma ClockTime) := {\n  toFun := fun oct => \u27e8oct.timeZone, oct.clockTime\u27e9 \n  invFun := fun sig => \u27e8sig.fst, sig.snd\u27e9\n  left_inv := by simp [Function.LeftInverse]\n  right_inv := by simp [Function.RightInverse, Function.LeftInverse]\n}\n\nsection HClockTimeStuff\n\nvariable (t : HClockTime)\n\n@[reducible]\ndef HClockTime.simultaneous : HClockTime \u2192 HClockTime \u2192 Prop\n| \u27e8_, \u27e8naive_t\u2081\u27e9\u27e9, \u27e8_, \u27e8naive_t\u2082\u27e9\u27e9 => naive_t\u2081 = naive_t\u2082\n\ndef HClockTime.simultaneous.equivalence : Equivalence HClockTime.simultaneous :=  {\n  refl := fun d => rfl\n  symm := fun h => h.symm\n  trans := fun h h' => Eq.trans h h'\n}\n\ninstance instHClockTimeSetoid : Setoid HClockTime := \u27e8HClockTime.simultaneous, HClockTime.simultaneous.equivalence\u27e9\n\ninstance : Inhabited <| HClockTime := \u27e8TimeZone.UTC, Inhabited.default\u27e9\n\ndef HClockTime.nanoComponent : Nat := t.clockTime.nanoComponent\ndef HClockTime.secondComponent : Nat := t.clockTime.secondComponent\ndef HClockTime.minuteComponent : Nat := t.clockTime.minuteComponent\ndef HClockTime.hourComponent : Nat := t.clockTime.hourComponent\n\n/-- \nAddition of a `Duration` to a `ClockTime`; wraps into the next clock cycle. \n-/\ninstance : HAdd HClockTime SignedDuration HClockTime where\n  hAdd t d := { t with clockTime := t.clockTime + d }\n\ntheorem HClockTime.hAdd_signed_def (dur : SignedDuration) : t + dur = { t with clockTime := t.clockTime + dur } := rfl\n\n/-- \nSubtraction of a `SignedDuration` from a `ClockTime`; the implementation follows \nthat of `Fin oneDayNanos`, wrapping into the previous clock cycle on underflow \n-/\ninstance : HSub HClockTime SignedDuration HClockTime where\n  hSub t d := { t with clockTime := t.clockTime - d }\n\ntheorem HClockTime.hSub_signed_def (dur : SignedDuration) : t - dur = { t with clockTime := t.clockTime - dur } := rfl\n\ntheorem HClockTime.apply_unapply : t + t.timeZone.offset - t.timeZone.offset = t := by\n  simp [HClockTime.hAdd_signed_def, HClockTime.hSub_signed_def]\n  rw [HClockTime.mk.injEq]\n  exact And.intro rfl (heq_of_eq (ClockTime.apply_unapply t.clockTime))\n\ntheorem HClockTime.unapply_apply : t - t.timeZone.offset + t.timeZone.offset = t := by\n  simp [HClockTime.hAdd_signed_def, HClockTime.hSub_signed_def]\n  rw [HClockTime.mk.injEq]\n  exact And.intro rfl (heq_of_eq (ClockTime.unapply_apply t.clockTime))\n\n/--\nLT compares the underlying naive/TAI time.\n-/\ninstance : LT HClockTime where\n  lt := InvImage instLTNaiveClockTime.lt (fun t => t.clockTime.naive)\n\n/--\nLE compares the underlying naive/TAI time.\n-/\ninstance : LE HClockTime where\n  le := InvImage instLENaiveClockTime.le (fun t => t.clockTime.naive)\n\ntheorem HClockTime.le_def (d\u2081 d\u2082 : HClockTime) : (d\u2081 <= d\u2082) = (d\u2081.clockTime.naive <= d\u2082.clockTime.naive) := rfl\ntheorem HClockTime.lt_def (d\u2081 d\u2082 : HClockTime) : (d\u2081 < d\u2082) = (d\u2081.clockTime.naive < d\u2082.clockTime.naive) := rfl\n\ninstance instDecidableLTHClockTime (a b : HClockTime) : Decidable (a < b) := inferInstanceAs (Decidable (a.clockTime.naive < b.clockTime.naive))\ninstance instDecidableLEHClockTime (a b : HClockTime) : Decidable (a <= b) := inferInstanceAs (Decidable (a.clockTime.naive <= b.clockTime.naive))\n\n/--\nHClockTime is only a Preorder since it does not respect antisymmetry. \nt\u2081 <= t\u2082 \u2227 t\u2082 <= t\u2081 does not imply t\u2081 = t\u2082 since they may have different timezones.\n-/\ninstance : Preorder HClockTime where\n  le_refl (a) := le_refl a.clockTime.naive\n  le_trans (a b c) := Nat.le_trans\n  lt_iff_le_not_le (a b) := Nat.lt_iff_le_not_le\n\nend HClockTimeStuff\n", "meta": {"author": "ammkrn", "repo": "timelib", "sha": "185e8ea7c8b4274f2cb7ecba4c2e785c6e97cf15", "save_path": "github-repos/lean/ammkrn-timelib", "path": "github-repos/lean/ammkrn-timelib/timelib-185e8ea7c8b4274f2cb7ecba4c2e785c6e97cf15/Timelib/NanoPrecision/ClockTime/HClockTime.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7634837527911057, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.3608861546125149}}
{"text": "import Sexp\n\ndef translateTerm : Sexp \u2192 String\n  | sexp!{(= {e\u2081} {e\u2082})} => s!\"({translateTerm e\u2081}) = ({translateTerm e\u2082})\"\n  | sexp!{(distinct {e\u2081} {e\u2082})}\n  | sexp!{(not (= {e\u2081} {e\u2082}))} => s!\"({translateTerm e\u2081}) \u2260 ({translateTerm e\u2082})\"\n  | sexp!{(store {a} {i} {v})} => s!\"({translateTerm a}).write {i} ({translateTerm v})\"\n  | sexp!{(select {a} {i})} => s!\"({translateTerm a}).read {i}\"\n  | sexp!{(and {p} {q})} => s!\"{translateTerm p} \u2192 {translateTerm q}\"\n  | .atom s => s\n  | _ => \"\"\n\ndef translateCommand : Sexp \u2192 String\n  | sexp!{(declare-const {a} A)} => s!\"{a}\"\n  | sexp!{(declare-fun {a} () A)} => \"{\" ++ s!\"{a} : A I E\" ++ \"}\"\n  | sexp!{(assert {e})} => translateTerm e\n  | _ => \"\"\n\ndef translateQuery (q : String) : String := Id.run do\n  let .ok cmds := Sexp.parseMany q | panic! \"translation failed.\"\n  let (asserts, decls) := cmds.partition (\u00b7 matches sexp!{(assert {_})})\n  let decls := (decls.map translateCommand).filter (\u00b7 \u2260 \"\")\n  let asserts := asserts.map translateCommand\n  let lq := s!\"import LMT\n\nvariable \\{I} [Nonempty I] \\{E} [Nonempty E] [Nonempty (A I E)]\n\nexample \\{{String.intercalate \" \" decls} : A I E} :\n        {String.intercalate \" \u2192 \" asserts} \u2192 False := by\n  arr\"\n  return lq\n\nopen System in\ndef main (args : List String) : IO Unit := do\n  let path := args[0]!\n  let query \u2190 IO.FS.readFile \u27e8path\u27e9\n  IO.println (translateQuery query)\n", "meta": {"author": "abdoo8080", "repo": "ar-project", "sha": "303af2d62cf8c8fe996c9670f9fe5a0cc90e5bb8", "save_path": "github-repos/lean/abdoo8080-ar-project", "path": "github-repos/lean/abdoo8080-ar-project/ar-project-303af2d62cf8c8fe996c9670f9fe5a0cc90e5bb8/Translator.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3607860842986531}}
{"text": "import tactic.induction\nimport data.int.basic\nimport data.set.basic\n\nimport .base .board\n\nnoncomputable theory\nopen_locale classical\n\ninstance : inhabited State := \u27e8state\u2080\u27e9\n\n@[reducible]\ndef State.len (s : State) : \u2115 :=\ns.history.length\n\ndef State.nth (s : State) (n : \u2115) : option Board :=\n(s.history ++ [s.board]).nth n\n\n-----\n\nlemma hist_ne_of_hist_len_ne {s\u2081 s\u2082 : State}\n  (h : s\u2081.len \u2260 s\u2082.len) :\n  s\u2081.history \u2260 s\u2082.history :=\nby { contrapose! h, rw [State.len, h] }\n\nlemma hist_len_finish {s : State} :\n  s.finish.len = s.len := rfl\n\nlemma state_nth_len {s : State} :\n  s.nth s.len = some s.board :=\nbegin\n  rw [State.len, State.nth, list.nth_eq_some],\n  use length_lt_length_snoc, rw list.nth_le_append_right,\n  { simp_rw nat.sub_self, refl },\n  { refl },\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/state.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947425132314, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3605898688780899}}
{"text": "import tactic\n\nopen tactic\n\n/-!\n\nThis file contains three tactic-programming exercises of increasing difficulty.\n\nThey were (hastily) written to follow the metaprogramming tutorial at\nLean for the Curious Mathematician 2020.\n\nIf you're looking for more (better) exercises, we strongly recommend the\nexercises by Blanchette et al\nfor the course Logical Verification at the Vrije Universiteit Amsterdam,\nand the corresponding chapter of the course notes:\n\nhttps://github.com/blanchette/logical_verification_2020/blob/master/lean/love07_metaprogramming_exercise_sheet.lean\nhttps://github.com/blanchette/logical_verification_2020/raw/master/hitchhikers_guide.pdf\n\n\n\n## Exercise 1\n\nWrite a `contradiction` tactic.\nThe tactic should look through the hypotheses in the local context\ntrying to find two that contradict each other,\ni.e. proving `P` and `\u00ac P` for some proposition `P`.\nIt should use this contradiction to close the goal.\n\nBonus: handle `P \u2192 false` as well as `\u00ac P`.\n\nThis exercise is to practice manipulating the hypotheses and goal.\n\nNote: this exists as `tactic.interactive.contradiction`.\n\n-/\n\n\n/-\nThis solution is a \"slick version.\"\n\nWe write a function `find_absurd_proof` that takes an expr e and a list of exprs.\n\nFor each expr h in the list,\nit tries to apply e to h. If `e : \u00ac P` and `h : P`, this will succeed\nand result in a proof of `false`. Otherwise it will fail.\n\n`find_absurd_proof` finds the first `h` such that it succeeds, and uses the proof of `false`\nto close the goal.\n\nIf no such `h` exists, `find_absurd_proof` will fail.\n-/\n\nmeta def find_absurd_proof (e : expr) (ctx : list expr) : tactic unit :=\ndo prf \u2190 ctx.mfirst (\u03bb h, to_expr ``(%%e %%h)),\n   exact prf\n\n/-\n\n`contr` maps over the local context `ctx`.\nFor every `e` in `ctx`, it calls `find_absurd_proof e ctx`.\n\nNotice the double loop through `ctx`:\nfor each `e` in `ctx`, we search through all of `ctx` again!\n\n`contr` calls the `exfalso` tactic before it begins,\nto make sure the target is `false`.\n-/\n\nmeta def tactic.interactive.contr : tactic unit :=\ndo exfalso,\n   ctx \u2190 local_context,\n   ctx.mfirst (\u03bb e, find_absurd_proof e ctx)\n\nexample (P Q R : Prop) (hp : P) (hq : Q) (hr : \u00ac R) (hnq : \u00ac Q) : false :=\nby contr\n\n\nexample (P Q R : Prop) (hnq : \u00ac Q) (hp : P) (hq : Q) (hr : \u00ac R) : 0 = 1 :=\nby contr\n\n\nexample (P Q R : Prop) (hp : P) (hq : Q) (hr : \u00ac R) (hnq : Q \u2192 false) : false :=\nby contr\n\n\n\n/-!\n\n## Exercise 2\n\nWrite a tactic that proves a given `nat`-valued declaration is nonnegative.\nThe tactic should take the name of a declaration whose return type is `\u2115`\n(presumably with some arguments), e.g. `nat.add : \u2115 \u2192 \u2115 \u2192 \u2115`\nor `list.length : \u03a0 \u03b1 : Type, list \u03b1 \u2192 \u2115`.\nIt should add a new declaration to the environment which proves all applications\nof this function are nonnegative,\ne.g. `nat.add_nonneg : \u2200 m n : \u2115, 0 \u2264 nat.add m n`.\n\nBonus: create reasonable names for these declarations, and/or take an optional argument\nfor the new name.\n\nThis tactic is not useful by itself, but it's a good way to practice\nquerying and modifying an environment and working under binders.\nIt is not a tactic to be used during a proof, but rather as a command.\n\n\nHints:\n* For looking at declarations in the environment, you will need the `declaration` type,\n  as well as the tactics `get_decl` and `add_decl`.\n* You will have to manipulate an expression under binders.\n  The tactics `mk_local_pis` and `pis`, or their lambda equivalents, will be helpful here.\n* `mk_mapp` is a variant of `mk_app` that lets you provide implicit arguments.\n-/\n\n\nmeta def add_nonneg_proof (n : name) : tactic unit :=\n\n   -- first we find the declaration named `n` in the environment.\ndo d \u2190 get_decl n,\n\n   -- the type of d is `\u03a0 x y z ..., body`,\n   -- where body contains a bunch of free variables.\n   -- we instantiate the binders to get a body we can manipulate.\n   (args, body) \u2190 mk_local_pis d.type,\n\n   -- args is a list of expressions, but we want a list of `option expr`s to give to `mk_mapp`.\n   let args_with_some := args.map some,\n\n   -- this line applies the expression named `n` to the variables we've created.\n   -- d_body is the natural number that we want to prove is nonnegative.\n   d_body \u2190 mk_mapp n args_with_some,\n\n   -- so we prove that `d_body` is nonnegative by applying `nat.zero_le`.\n   nonneg_prf_body \u2190 mk_app `nat.zero_le [d_body],\n\n   -- now we abstract away the local constants we created.\n   nonneg_prf \u2190 lambdas args nonneg_prf_body,\n\n   -- we create a name for our new proof.\n   -- if `n` is `nat.add, we will call our new proof `nat.add.nonneg\n   let new_decl_name := n.append `nonneg,\n\n   -- we get the type of the proof we've constructed,\n   decl_tp \u2190 infer_type nonneg_prf,\n\n   -- make a term of type `declaration`,\n   let new_decl := mk_theorem new_decl_name d.univ_params decl_tp nonneg_prf,\n\n   -- and add that declaration to the environment.\n   add_decl new_decl\n\n\n\n\nrun_cmd add_nonneg_proof `nat.add\nrun_cmd add_nonneg_proof `list.length\n\n#check nat.add.nonneg\n#check list.length.nonneg\n\n\n/-!\n\n## Exercise 3 (challenge!)\n\nThe mathlib tactic `cancel_denoms` is intended to get rid of division by numerals\nin expressions where this makes sense. For example,\n\n-/\n\nexample (q : \u211a) (h : q / 3 > 0) : q > 0 :=\nbegin\n  cancel_denoms at h, exact h\nend\n\n/-!\n\nBut it is not complete. In particular, it doesn't like nested division\nor other operators in denominators. These all fail:\n\n-/\n\nexample (q : \u211a) (h : q / (3 / 4) > 0) : false :=\nbegin\n  -- cancel_denoms at h,\n  admit\nend\n\nexample (p q : \u211a) (h : q / 2 / 3 < q) : false :=\nbegin\n  -- cancel_denoms at h,\n  admit\nend\n\nexample (p q : \u211a) (h : q / 2 < 3 / (4*q)) : false :=\nbegin\n  -- cancel_denoms at h,\n  admit\nend\n\n-- this one succeeds but doesn't do what it should\nexample (p q : \u211a) (h : q / (2*3) < q) : false :=\nbegin\n  -- cancel_denoms at h,\n  admit\nend\n\n/-!\n\nLook at the code in `src/tactic/cancel_denoms.lean` and try to fix it.\nSee if you can solve any or all of these failing test cases.\n\nIf you succeed, a pull request to mathlib is strongly encouraged!\n\n-/\n", "meta": {"author": "leanprover-community", "repo": "lftcm2020", "sha": "cc683e2b074b61909310746d6acc1fb3d42d6ee2", "save_path": "github-repos/lean/leanprover-community-lftcm2020", "path": "github-repos/lean/leanprover-community-lftcm2020/lftcm2020-cc683e2b074b61909310746d6acc1fb3d42d6ee2/src/solutions/monday/metaprogramming.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6548947425132314, "lm_q1q2_score": 0.3605898688780899}}
{"text": "-- Copyright (c) 2018 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Johan Commelin, Reid Barton\n\nimport category_theory.comma\nimport category_theory.limits.preserves\n\nuniverses v u -- declare the `v`'s first; see `category_theory.category` for an explanation\n\nopen category_theory category_theory.limits\n\nvariables {J : Type v} [small_category J]\nvariables {C : Type u} [\ud835\udc9e : category.{v} C]\ninclude \ud835\udc9e\nvariable {X : C}\n\nnamespace category_theory.functor\n\ndef to_cocone (F : J \u2964 over X) : cocone (F \u22d9 over.forget) :=\n{ X := X,\n  \u03b9 := { app := \u03bb j, (F.obj j).hom } }\n\n@[simp] lemma to_cocone_X (F : J \u2964 over X) : F.to_cocone.X = X := rfl\n@[simp] lemma to_cocone_\u03b9 (F : J \u2964 over X) (j : J) : F.to_cocone.\u03b9.app j = (F.obj j).hom := rfl\n\ndef to_cone (F : J \u2964 under X) : cone (F \u22d9 under.forget) :=\n{ X := X,\n  \u03c0 := { app := \u03bb j, (F.obj j).hom } }\n\n@[simp] lemma to_cone_X (F : J \u2964 under X) : F.to_cone.X = X := rfl\n@[simp] lemma to_cone_\u03c0 (F : J \u2964 under X) (j : J) : F.to_cone.\u03c0.app j = (F.obj j).hom := rfl\n\nend category_theory.functor\n\nnamespace category_theory.over\n\ndef colimit (F : J \u2964 over X) [has_colimit (F \u22d9 forget)] : cocone F :=\n{ X := mk $ colimit.desc (F \u22d9 forget) F.to_cocone,\n  \u03b9 :=\n  { app := \u03bb j, hom_mk $ colimit.\u03b9 (F \u22d9 forget) j,\n    naturality' :=\n    begin\n      intros j j' f,\n      have := colimit.w (F \u22d9 forget) f,\n      tidy\n    end } }\n\n@[simp] lemma colimit_X_hom (F : J \u2964 over X) [has_colimit (F \u22d9 forget)] :\n((colimit F).X).hom = colimit.desc (F \u22d9 forget) F.to_cocone := rfl\n@[simp] lemma colimit_\u03b9_app (F : J \u2964 over X) [has_colimit (F \u22d9 forget)] (j : J) :\n((colimit F).\u03b9).app j = hom_mk (colimit.\u03b9 (F \u22d9 forget) j) := rfl\n\ndef forget_colimit_is_colimit (F : J \u2964 over X) [has_colimit (F \u22d9 forget)] :\n  is_colimit (forget.map_cocone (colimit F)) :=\nis_colimit.of_iso_colimit (colimit.is_colimit (F \u22d9 forget)) (cocones.ext (iso.refl _) (by tidy))\n\ninstance : reflects_colimits (forget : over X \u2964 C) :=\n\u03bb J \ud835\udca5 F, by constructor; exactI \u03bb t ht,\n{ desc := \u03bb s, hom_mk (ht.desc (forget.map_cocone s))\n    begin\n      apply ht.hom_ext, intro j,\n      rw [\u2190category.assoc, ht.fac],\n      transitivity (F.obj j).hom,\n      exact w (s.\u03b9.app j), -- TODO: How to write (s.\u03b9.app j).w?\n      exact (w (t.\u03b9.app j)).symm,\n    end,\n  fac' := begin\n    intros s j, ext, exact ht.fac (forget.map_cocone s) j\n    -- TODO: Ask Simon about multiple ext lemmas for defeq types (comma_morphism & over.category.hom)\n  end,\n  uniq' :=\n  begin\n    intros s m w,\n    ext1 j,\n    exact ht.uniq (forget.map_cocone s) m.left (\u03bb j, congr_arg comma_morphism.left (w j))\n  end }\n\ninstance has_colimit {F : J \u2964 over X} [has_colimit (F \u22d9 forget)] : has_colimit F :=\n{ cocone := colimit F,\n  is_colimit := reflects_colimit.reflects (forget_colimit_is_colimit F) }\n\ninstance has_colimits_of_shape [has_colimits_of_shape J C] :\n  has_colimits_of_shape J (over X) :=\n\u03bb F, infer_instance\n\ninstance has_colimits [has_colimits C] : has_colimits (over X) :=\n\u03bb J \ud835\udca5, by resetI; apply_instance\n\ninstance forget_preserves_colimits [has_colimits C] {X : C} :\n  preserves_colimits (forget : over X \u2964 C) :=\n\u03bb J \ud835\udca5 F, by exactI\npreserves_colimit_of_preserves_colimit_cocone (colimit.is_colimit F) (forget_colimit_is_colimit F)\n\nend category_theory.over\n\nnamespace category_theory.under\n\ndef limit (F : J \u2964 under X) [has_limit (F \u22d9 forget)] : cone F :=\n{ X := mk $ limit.lift (F \u22d9 forget) F.to_cone,\n  \u03c0 :=\n  { app := \u03bb j, hom_mk $ limit.\u03c0 (F \u22d9 forget) j,\n    naturality' :=\n    begin\n      intros j j' f,\n      have := (limit.w (F \u22d9 forget) f).symm,\n      tidy\n    end } }\n\n@[simp] lemma limit_X_hom (F : J \u2964 under X) [has_limit (F \u22d9 forget)] :\n((limit F).X).hom = limit.lift (F \u22d9 forget) F.to_cone := rfl\n@[simp] lemma limit_\u03c0_app (F : J \u2964 under X) [has_limit (F \u22d9 forget)] (j : J) :\n((limit F).\u03c0).app j = hom_mk (limit.\u03c0 (F \u22d9 forget) j) := rfl\n\ndef forget_limit_is_limit (F : J \u2964 under X) [has_limit (F \u22d9 forget)] :\n  is_limit (forget.map_cone (limit F)) :=\nis_limit.of_iso_limit (limit.is_limit (F \u22d9 forget)) (cones.ext (iso.refl _) (by tidy))\n\ninstance : reflects_limits (forget : under X \u2964 C) :=\n\u03bb J \ud835\udca5 F, by constructor; exactI \u03bb t ht,\n{ lift := \u03bb s, hom_mk (ht.lift (forget.map_cone s))\n    begin\n      apply ht.hom_ext, intro j,\n      rw [category.assoc, ht.fac],\n      transitivity (F.obj j).hom,\n      exact w (s.\u03c0.app j),\n      exact (w (t.\u03c0.app j)).symm,\n    end,\n  fac' := begin\n    intros s j, ext, exact ht.fac (forget.map_cone s) j\n  end,\n  uniq' :=\n  begin\n    intros s m w,\n    ext1 j,\n    exact ht.uniq (forget.map_cone s) m.right (\u03bb j, congr_arg comma_morphism.right (w j))\n  end }\n\ninstance has_limit {F : J \u2964 under X} [has_limit (F \u22d9 forget)] : has_limit F :=\n{ cone := limit F,\n  is_limit := reflects_limit.reflects (forget_limit_is_limit F) }\n\ninstance has_limits_of_shape [has_limits_of_shape J C] :\n  has_limits_of_shape J (under X) :=\n\u03bb F, infer_instance\n\ninstance has_limits [has_limits C] : has_limits (under X) :=\n\u03bb J \ud835\udca5, by resetI; apply_instance\n\ninstance forget_preserves_limits [has_limits C] {X : C} :\n  preserves_limits (forget : under X \u2964 C) :=\n\u03bb J \ud835\udca5 F, by exactI\npreserves_limit_of_preserves_limit_cone (limit.is_limit F) (forget_limit_is_limit F)\n\nend category_theory.under\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/category_theory/limits/over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421276, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.36058986146080096}}
{"text": "import algebra.camera.basic\n\nuniverse u\n\nset_option old_structure_cmd true\n\n/-!\n# Uniform predicates\n-/\n\n@[ext] structure monotone_nonexpansive (\u03b1 : Type u) [camera \u03b1] extends nonexpansive_fun \u03b1 sprop :=\n(mono : \u2200 n a b, a \u227c[n] b \u2192 to_fun a \u2286[n] to_fun b)\n\ninstance monotone_nonexpansive.fun_like (\u03b1 : Type u) [camera \u03b1] :\n  fun_like (monotone_nonexpansive \u03b1) \u03b1 (\u03bb _, sprop) := {\n  coe := monotone_nonexpansive.to_fun,\n  coe_injective' := by intros f g h; ext1; exact h,\n}\n\ninstance monotone_nonexpansive.nonexpansive_fun_class (\u03b1 : Type u) [camera \u03b1] :\n  nonexpansive_fun_class (monotone_nonexpansive \u03b1) \u03b1 sprop := {\n  is_nonexpansive := monotone_nonexpansive.is_nonexpansive'\n}\n\ninstance monotone_nonexpansive.setoid (\u03b1 : Type u) [camera \u03b1] :\n  setoid (monotone_nonexpansive \u03b1) := {\n  r := \u03bb x y, \u2200 m a, \u2713[m] a \u2192 (x a m \u2194 y a m),\n  iseqv := begin\n    refine \u27e8_, _, _\u27e9,\n    { intros x n a h,\n      refl, },\n    { intros x y h m a h',\n      rw h m a,\n      exact h', },\n    { intros x y z h\u2081 h\u2082 m a h',\n      rw h\u2081 m a,\n      rw h\u2082 m a,\n      exact h',\n      exact h', },\n  end,\n}\n\ndef upred (\u03b1 : Type u) [camera \u03b1] : Type* := quotient (monotone_nonexpansive.setoid \u03b1)\n\ndef monotone_nonexpansive.eq_at {\u03b1 : Type u} [camera \u03b1] (n : \u2115)\n  (x y : monotone_nonexpansive \u03b1) : Prop :=\n\u2200 m a, m \u2264 n \u2192 \u2713[m] a \u2192 (x a m \u2194 y a m)\n\nlemma monotone_nonexpansive.eq_at_respects_rel {\u03b1 : Type u} [camera \u03b1] (n : \u2115)\n  (a\u2081 a\u2082 b\u2081 b\u2082 : monotone_nonexpansive \u03b1) : a\u2081 \u2248 b\u2081 \u2192\n    a\u2082 \u2248 b\u2082 \u2192 monotone_nonexpansive.eq_at n a\u2081 a\u2082 = monotone_nonexpansive.eq_at n b\u2081 b\u2082 :=\nbegin\n  intros h\u2081 h\u2082,\n  ext1,\n  split,\n  { intros h m a hmn ha,\n    rw \u2190 h\u2081 m a ha,\n    rw \u2190 h\u2082 m a ha,\n    exact h m a hmn ha, },\n  { intros h m a hmn ha,\n    rw h\u2081 m a ha,\n    rw h\u2082 m a ha,\n    exact h m a hmn ha, },\nend\n\nprivate def upred.eq_at {\u03b1 : Type u} [camera \u03b1] (n : \u2115) : upred \u03b1 \u2192 upred \u03b1 \u2192 Prop :=\nquotient.lift\u2082 (monotone_nonexpansive.eq_at n) (monotone_nonexpansive.eq_at_respects_rel n)\n\nprivate lemma upred.eq_at_reflexive {\u03b1 : Type u} [camera \u03b1] (n : \u2115)\n  (x : upred \u03b1) : upred.eq_at n x x :=\nbegin\n  refine quotient.induction_on x _, clear x, intro x,\n  intros m a hmn h,\n  refl,\nend\n\nprivate lemma upred.eq_at_symmetric {\u03b1 : Type u} [camera \u03b1] (n : \u2115) (x y : upred \u03b1) :\n  upred.eq_at n x y \u2192 upred.eq_at n y x :=\nbegin\n  refine quotient.induction_on\u2082 x y _, clear x y, intros x y,\n  intros h m a hmn ha,\n  exact (h m a hmn ha).symm,\nend\n\nprivate lemma upred.eq_at_transitive {\u03b1 : Type u} [camera \u03b1] (n : \u2115) (x y z : upred \u03b1) :\n  upred.eq_at n x y \u2192 upred.eq_at n y z \u2192 upred.eq_at n x z :=\nbegin\n  refine quotient.induction_on\u2083 x y z _, clear x y z, intros x y z,\n  intros hxy hyz m a hmn h,\n  rw hxy m a hmn h,\n  rw hyz m a hmn h,\nend\n\nprivate lemma upred.eq_at_antitone {\u03b1 : Type u} [camera \u03b1] :\n  antitone (upred.eq_at : \u2115 \u2192 upred \u03b1 \u2192 upred \u03b1 \u2192 Prop) :=\nbegin\n  intros m n hmn x y,\n  refine quotient.induction_on\u2082 x y _, clear x y, intros x y,\n  intros h k a hk ha,\n  exact h k a (hk.trans hmn) ha,\nend\n\nprivate lemma upred.eq_at_limit {\u03b1 : Type u} [camera \u03b1] (x y : upred \u03b1) :\n  (\u2200 n, upred.eq_at n x y) \u2192 x = y :=\nbegin\n  refine quotient.induction_on x _, clear x, intro x,\n  refine quotient.induction_on y _, clear y, intro y,\n  intro h,\n  refine quotient.sound _,\n  intros m a ha,\n  exact h m m a le_rfl ha,\nend\n\nprivate def monotone_nonexpansive.lim {\u03b1 : Type u} [unital_camera \u03b1]\n  (c : chain (monotone_nonexpansive \u03b1) monotone_nonexpansive.eq_at) : monotone_nonexpansive \u03b1 :=\nbegin\n  refine \u27e8\u03bb a, \u27e8\u03bb n, \u2200 m \u2264 n, \u2713[m] a \u2192 c m a m, _\u27e9, _, _\u27e9,\n  { intros m n hmn h k hk hak,\n    exact h k (hk.trans hmn) hak, },\n  { intros n x y h m hmn,\n    split,\n    { intros ha k hk hy,\n      refine (c k).mono k x y _ _ le_rfl (ha k hk _),\n      { refine \u27e81, _\u27e9,\n        rw [mul_comm, one_mul],\n        exact eq_at_mono (hk.trans hmn) h, },\n      { have : camera.validn x =[n] camera.validn y := nonexpansive camera.validn h,\n        rw this k (hk.trans hmn),\n        exact hy, }, },\n    { intros ha k hk hy,\n      refine (c k).mono k y x _ _ le_rfl (ha k hk _),\n      { refine \u27e81, _\u27e9,\n        rw [mul_comm, one_mul],\n        exact eq_at_mono (hk.trans hmn) (eq_at_symmetric n h), },\n      { have : camera.validn x =[n] camera.validn y := nonexpansive camera.validn h,\n        rw \u2190 this k (hk.trans hmn),\n        exact hy, }, }, },\n  { intros n a b hab m hmn h k hkm hb,\n    refine (c k).mono n _ _ hab k (hkm.trans hmn) _,\n    refine h k hkm _,\n    obtain \u27e8c, hc\u27e9 := hab,\n    have : camera.validn (a * c) =[n] camera.validn b := nonexpansive camera.validn hc,\n    rw \u2190 this k (hkm.trans hmn) at hb,\n    exact camera.validn_mul a c k hb, },\nend\n\nprivate lemma monotone_nonexpansive.complete {\u03b1 : Type u} [unital_camera \u03b1] (n : \u2115)\n  (c : chain (monotone_nonexpansive \u03b1) monotone_nonexpansive.eq_at) :\n  monotone_nonexpansive.eq_at n (monotone_nonexpansive.lim c) (c n) :=\nbegin\n  intros m a hmn ha,\n  split,\n  { intro h,\n    have := h m le_rfl ha,\n    exact (c.prop m n hmn m a le_rfl ha).mp this, },\n  { intros h k hk hak,\n    refine (c.prop k n (hk.trans hmn) k a le_rfl hak).mpr _,\n    exact (c n a).mono hk h, },\nend\n\nprivate noncomputable def upred.chain_out {\u03b1 : Type u} [unital_camera \u03b1] (c : chain (upred \u03b1) upred.eq_at) :\n  chain (monotone_nonexpansive \u03b1) monotone_nonexpansive.eq_at := {\n  c := \u03bb n, (c n).out,\n  prop := begin\n    intros m n hmn k a hk ha,\n    have := c.prop m n hmn,\n    rw \u2190 quotient.out_eq (c.c m) at this,\n    rw \u2190 quotient.out_eq (c.c n) at this,\n    exact this k a hk ha,\n  end,\n}\n\nprivate noncomputable def upred.lim {\u03b1 : Type u} [unital_camera \u03b1]\n  (c : chain (upred \u03b1) upred.eq_at) : upred \u03b1 :=\n\u27e6monotone_nonexpansive.lim (upred.chain_out c)\u27e7\n\nprivate lemma upred.complete {\u03b1 : Type u} [unital_camera \u03b1] (n : \u2115) (c : chain (upred \u03b1) upred.eq_at) :\n  upred.eq_at n (upred.lim c) (c n) :=\nbegin\n  rw \u2190 quotient.out_eq (c n),\n  intros m a hmn ha,\n  exact monotone_nonexpansive.complete n (upred.chain_out c) m a hmn ha,\nend\n\ninstance upred_ofe (\u03b1 : Type u) [camera \u03b1] : ofe (upred \u03b1) := {\n  eq_at := upred.eq_at,\n  eq_at_reflexive := upred.eq_at_reflexive,\n  eq_at_symmetric := upred.eq_at_symmetric,\n  eq_at_transitive := upred.eq_at_transitive,\n  eq_at_mono' := upred.eq_at_antitone,\n  eq_at_limit' := upred.eq_at_limit,\n}\n\nnoncomputable instance upred_cofe (\u03b1 : Type u) [unital_camera \u03b1] : cofe (upred \u03b1) := {\n  lim := upred.lim,\n  complete := upred.complete,\n}\n\ndef upred.map_fun {\u03b1 \u03b2 : Type u} [unital_camera \u03b1] [unital_camera \u03b2] (f : \u03b1 \u2192\u2096\u2095 \u03b2) :\n  upred \u03b2 \u2192 upred \u03b1 :=\nquotient.lift (\u03bb g : monotone_nonexpansive \u03b2, \u27e6{\n  monotone_nonexpansive .\n  to_fun := \u03bb a, \u27e8\u03bb n, g (f a) n, \u03bb m n hmn, (g (f a)).mono hmn\u27e9,\n  is_nonexpansive' := begin\n    intros n x y h m hmn,\n    dsimp only [sprop.coe_fn_mk],\n    have : f x =[n] f y := nonexpansive f h,\n    split,\n    exact g.mono n (f x) (f y) (incln_of_eq_at this) m hmn,\n    exact g.mono n (f y) (f x) (incln_of_eq_at (eq_at_symmetric n this)) m hmn,\n  end,\n  mono := begin\n    intros n x y h m hmn hm,\n    have : f x \u227c[n] f y := camera_hom.map_incln h,\n    exact g.mono n (f x) (f y) this m hmn hm,\n  end,\n}\u27e7) begin\n  intros x y h,\n  refine quotient.sound _,\n  intros n a hn,\n  exact h n (f a) (f.map_valid' n a hn),\nend\n\n/-- `upred` is a locally nonexpansive functor from `unital_camera` to `cofe`. -/\ndef upred.map {\u03b1 \u03b2 : Type u} [unital_camera \u03b1] [unital_camera \u03b2] :\n  (\u03b1 \u2192\u2096\u2095 \u03b2) \u2192\u2099\u2091 (upred \u03b2 \u2192\u2099\u2091 upred \u03b1) := {\n  to_fun := \u03bb f, \u27e8upred.map_fun f, begin\n    intros n p q,\n    refine quotient.induction_on\u2082 p q _, clear p q, intros p q,\n    intros h m a hmn hm,\n    exact h m (f a) hmn (f.map_valid' m a hm),\n  end\u27e9,\n  is_nonexpansive' := begin\n    intros n f g h p,\n    refine quotient.induction_on p _, clear p, intro p,\n    intros m a hmn hm,\n    have : p (f a) =[n] p (g a) := nonexpansive p (h a),\n    exact this _ hmn,\n  end,\n}\n", "meta": {"author": "zeramorphic", "repo": "separation-logic", "sha": "51c131501cc541b3aae072957942e8ef744c4ebf", "save_path": "github-repos/lean/zeramorphic-separation-logic", "path": "github-repos/lean/zeramorphic-separation-logic/separation-logic-51c131501cc541b3aae072957942e8ef744c4ebf/src/algebra/ofe/upred.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3605898614608009}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.dold_kan.gamma_comp_n\nimport for_mathlib.dold_kan.n_reflects_iso\n\n/-! The unit isomorphism of the Dold-Kan equivalence\n\nIn order to construct the unit isomorphism of the Dold-Kan equivalence,\nwe first construct natural transformations\n`\u0393\u2082N\u2081.nat_trans : N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)` and\n`\u0393\u2082N\u2082.nat_trans : N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`.\nIt is then shown that `\u0393\u2082N\u2082.nat_trans` is an isomorphism by using\nthat it becomes an isomorphism after the application of the functor\n`N\u2082 : karoubi (simplicial_object C) \u2964 karoubi (chain_complex C \u2115)`\nwhich reflects isomorphisms.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n  category_theory.idempotents simplex_category opposite simplicial_object\nopen_locale simplicial dold_kan\n\nnamespace algebraic_topology\n\nnamespace dold_kan\n\nvariables {C : Type*} [category C] [preadditive C]\n\nlemma P_infty_comp_map_mono_eq_zero (X : simplicial_object C) {n : \u2115}\n  {\u0394' : simplex_category} (i : \u0394' \u27f6 [n]) [hi : mono i] (h\u2081 : \u0394'.len \u2260 n) (h\u2082 : \u00acis_\u03b4\u2080 i) :\n  P_infty.f n \u226b X.map i.op = 0 :=\nbegin\n  unfreezingI { induction \u0394' using simplex_category.rec with m, },\n  obtain \u27e8k, hk\u27e9 := nat.exists_eq_add_of_lt (len_lt_of_mono i\n    (\u03bb h, by { rw \u2190 h at h\u2081,  exact h\u2081 rfl, })),\n  simp only [len_mk] at hk,\n  cases k,\n  { change n = m + 1 at hk,\n    unfreezingI { subst hk, obtain \u27e8j, rfl\u27e9 := eq_\u03b4_of_mono i, },\n    rw is_\u03b4\u2080.iff at h\u2082,\n    have h\u2083 : 1 \u2264 (j : \u2115),\n    { by_contra,\n      exact h\u2082 (by simpa only [fin.ext_iff, not_le, nat.lt_one_iff] using h), },\n    exact (higher_faces_vanish.of_P (m+1) m).comp_\u03b4_eq_zero j h\u2082 (by linarith), },\n  { simp only [nat.succ_eq_add_one, \u2190 add_assoc] at hk,\n    clear h\u2082 hi,\n    subst hk,\n    obtain \u27e8j\u2081, i, rfl\u27e9 := eq_comp_\u03b4_of_not_surjective i (\u03bb h, begin\n      have h' := len_le_of_epi (simplex_category.epi_iff_surjective.2 h),\n      dsimp at h',\n      linarith,\n    end),\n    obtain \u27e8j\u2082, i, rfl\u27e9 := eq_comp_\u03b4_of_not_surjective i (\u03bb h, begin\n      have h' := len_le_of_epi (simplex_category.epi_iff_surjective.2 h),\n      dsimp at h',\n      linarith,\n    end),\n    by_cases hj\u2081 : j\u2081 = 0,\n    { unfreezingI { subst hj\u2081, },\n      rw [assoc, \u2190 simplex_category.\u03b4_comp_\u03b4'' (fin.zero_le _)],\n      simp only [op_comp, X.map_comp, assoc, P_infty_f],\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ j\u2082.succ_ne_zero, zero_comp],\n      rw fin.coe_succ,\n      linarith, },\n    { simp only [op_comp, X.map_comp, assoc, P_infty_f],\n      erw [(higher_faces_vanish.of_P _ _).comp_\u03b4_eq_zero_assoc _ hj\u2081, zero_comp],\n      by_contra,\n      exact hj\u2081 (by { simp only [fin.ext_iff, fin.coe_zero], linarith, }), }, },\nend\n\n@[reassoc]\nlemma \u0393\u2080_obj_termwise_map_mono_comp_P_infty (X : simplicial_object C) {\u0394 \u0394' : simplex_category}\n  (i : \u0394 \u27f6 \u0394') [mono i] :\n  \u0393\u2080.obj.termwise.map_mono (alternating_face_map_complex.obj X) i \u226b P_infty.f (\u0394.len) =\n    P_infty.f (\u0394'.len) \u226b X.map i.op :=\nbegin\n  unfreezingI\n  { induction \u0394 using simplex_category.rec with n,\n    induction \u0394' using simplex_category.rec with n', },\n  dsimp,\n  /- We start with the case `i` is an identity -/\n  by_cases n = n',\n  { unfreezingI { subst h, },\n    simp only [simplex_category.eq_id_of_mono i, \u0393\u2080.obj.termwise.map_mono_id, op_id, X.map_id],\n    dsimp,\n    simp only [id_comp, comp_id], },\n  by_cases hi : is_\u03b4\u2080 i,\n  /- The case `i = \u03b4 0` -/\n  { have h' : n' = n + 1 := hi.left,\n    unfreezingI { subst h', },\n    simp only [\u0393\u2080.obj.termwise.map_mono_\u03b4\u2080' _ i hi],\n    dsimp,\n    rw [\u2190 P_infty.comm' _ n rfl, alternating_face_map_complex.obj_d_eq],\n    simp only [eq_self_iff_true, id_comp, if_true, preadditive.comp_sum],\n    rw finset.sum_eq_single (0 : fin (n+2)), rotate,\n    { intros b hb hb',\n      rw preadditive.comp_zsmul,\n      erw [P_infty_comp_map_mono_eq_zero X (simplex_category.\u03b4 b) h\n        (by { rw is_\u03b4\u2080.iff, exact hb', }), zsmul_zero], },\n    { simp only [finset.mem_univ, not_true, is_empty.forall_iff], },\n    { simpa only [hi.eq_\u03b4\u2080, fin.coe_zero, pow_zero, one_zsmul], }, },\n  /- The case `i \u2260 \u03b4 0` -/\n  { rw [\u0393\u2080.obj.termwise.map_mono_eq_zero _ i _ hi, zero_comp], swap,\n    { by_contradiction h',\n      exact h (congr_arg simplex_category.len h'.symm), },\n    rw P_infty_comp_map_mono_eq_zero,\n    { exact h, },\n    { by_contradiction h',\n      exact hi h', }, },\nend\n\nvariable [has_finite_coproducts C]\n\nnamespace \u0393\u2082N\u2081\n\n/-- The natural transformation `N\u2081 \u22d9 \u0393\u2082 \u27f6 to_karoubi (simplicial_object C)`. -/\n@[simps]\ndef nat_trans : (N\u2081 : simplicial_object C \u2964 _) \u22d9 \u0393\u2082 \u27f6 to_karoubi _ :=\n{ app := \u03bb X,\n  { f :=\n    { app := \u03bb \u0394, (\u0393\u2080.splitting K[X]).desc \u0394 (\u03bb A, P_infty.f A.1.unop.len \u226b X.map (A.e.op)),\n      naturality' := \u03bb \u0394 \u0394' \u03b8, begin\n        apply (\u0393\u2080.splitting K[X]).hom_ext',\n        intro A,\n        change _ \u226b (\u0393\u2080.obj K[X]).map \u03b8  \u226b _ = _,\n        simp only [splitting.\u03b9_desc_assoc, assoc,\n          \u0393\u2080.obj.map_on_summand'_assoc, splitting.\u03b9_desc],\n        erw \u0393\u2080_obj_termwise_map_mono_comp_P_infty_assoc X (image.\u03b9 (\u03b8.unop \u226b A.e)),\n        dsimp only [to_karoubi],\n        simp only [\u2190 X.map_comp],\n        congr' 2,\n        simp only [eq_to_hom_refl, id_comp, comp_id, \u2190 op_comp],\n        exact quiver.hom.unop_inj (A.fac_pull \u03b8),\n      end, },\n    comm := begin\n      apply (\u0393\u2080.splitting K[X]).hom_ext,\n      intro n,\n      dsimp [N\u2081],\n      simp only [\u2190 splitting.\u03b9_summand_id, splitting.\u03b9_desc,\n        comp_id, splitting.\u03b9_desc_assoc, assoc, P_infty_f_idem_assoc],\n    end, },\n  naturality' := \u03bb X Y f, begin\n    ext1,\n    apply (\u0393\u2080.splitting K[X]).hom_ext,\n    intro n,\n    dsimp [N\u2081, to_karoubi],\n    simpa only [\u2190splitting.\u03b9_summand_id, splitting.\u03b9_desc, splitting.\u03b9_desc_assoc,\n      assoc, P_infty_f_idem_assoc, karoubi.comp_f, nat_trans.comp_app, \u0393\u2082_map_f_app,\n      homological_complex.comp_f, alternating_face_map_complex.map_f,\n      P_infty_f_naturality_assoc, nat_trans.naturality],\n  end, }\n\nend \u0393\u2082N\u2081\n\n/-- The compatibility isomorphism relating `N\u2082 \u22d9 \u0393\u2082` and `N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082 : to_karoubi (simplicial_object C) \u22d9 N\u2082 \u22d9 \u0393\u2082 \u2245 N\u2081 \u22d9 \u0393\u2082 :=\neq_to_iso (functor.congr_obj (functor_extension\u2081_comp_whiskering_left_to_karoubi _ _) (N\u2081 \u22d9 \u0393\u2082))\n\nnamespace \u0393\u2082N\u2082\n\n/-- The natural transformation `N\u2082 \u22d9 \u0393\u2082 \u27f6 \ud835\udfed (simplicial_object C)`. -/\ndef nat_trans : (N\u2082 : karoubi (simplicial_object C) \u2964 _) \u22d9 \u0393\u2082 \u27f6 \ud835\udfed _ :=\n((whiskering_left _ _ _).obj _).preimage (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans)\n\nlemma nat_trans_app_f_app (P : karoubi (simplicial_object C)) :\n  \u0393\u2082N\u2082.nat_trans.app P = (N\u2082 \u22d9 \u0393\u2082).map P.decomp_id_i \u226b\n    (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans).app P.X \u226b P.decomp_id_p :=\nwhiskering_left_obj_preimage_app ((compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans)) P\n\nend \u0393\u2082N\u2082\n\nlemma compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans (X : simplicial_object C) :\n  \u0393\u2082N\u2081.nat_trans.app X = (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).inv \u226b\n    \u0393\u2082N\u2082.nat_trans.app ((to_karoubi _).obj X) :=\nbegin\n  rw [\u2190 cancel_epi (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.app X).hom, iso.hom_inv_id_assoc],\n  exact congr_app (((whiskering_left _ _ _).obj _).image_preimage\n    (compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082.hom \u226b \u0393\u2082N\u2081.nat_trans : _ \u27f6 to_karoubi _ \u22d9 \ud835\udfed _ )).symm X,\nend\n\nlemma identity_N\u2082_objectwise (P : karoubi (simplicial_object C)) :\n  N\u2082\u0393\u2082.inv.app (N\u2082.obj P) \u226b N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P) = \ud835\udfd9 (N\u2082.obj P) :=\nbegin\n  ext n,\n  have eq\u2081 : (N\u2082\u0393\u2082.inv.app (N\u2082.obj P)).f.f n = P_infty.f n \u226b P.p.app (op [n]) \u226b\n    (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9_summand (splitting.index_set.id (op [n])),\n  { simp only [N\u2082\u0393\u2082_inv_app_f_f, N\u2082_obj_p_f, assoc], },\n  have eq\u2082 : (\u0393\u2080.splitting (N\u2082.obj P).X).\u03b9_summand (splitting.index_set.id (op [n])) \u226b\n    (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)).f.f n = P_infty.f n \u226b P.p.app (op [n]),\n  { dsimp [N\u2082],\n    simp only [\u0393\u2082N\u2082.nat_trans_app_f_app, P_infty_on_\u0393\u2080_splitting_summand_eq_self_assoc,\n      functor.comp_map, compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_hom, nat_trans.comp_app,\n      eq_to_hom_app, assoc, karoubi.comp_f, karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id,\n      karoubi.decomp_id_p_f, karoubi.comp_p_assoc, \u0393\u2082_map_f_app,\n      N\u2082_map_f_f, karoubi.decomp_id_i_f, \u0393\u2082N\u2081.nat_trans_app_f_app],\n    erw [splitting.\u03b9_desc_assoc, assoc, assoc, splitting.\u03b9_desc_assoc],\n    dsimp [splitting.index_set.id, splitting.index_set.e],\n    simp only [assoc, nat_trans.naturality, P_infty_f_naturality_assoc,\n      app_idem_assoc, P_infty_f_idem_assoc],\n    erw [P.X.map_id, comp_id], },\n  simp only [karoubi.comp_f, homological_complex.comp_f, karoubi.id_eq, N\u2082_obj_p_f, assoc,\n    eq\u2081, eq\u2082, P_infty_f_naturality_assoc, app_idem, P_infty_f_idem_assoc],\nend\n\nlemma identity_N\u2082 :\n  ((\ud835\udfd9 (N\u2082 : karoubi (simplicial_object C) \u2964 _ ) \u25eb N\u2082\u0393\u2082.inv) \u226b\n    (\u0393\u2082N\u2082.nat_trans \u25eb \ud835\udfd9 N\u2082) : N\u2082 \u27f6 N\u2082) = \ud835\udfd9 N\u2082 :=\nby { ext P : 2, dsimp, rw [\u0393\u2082.map_id, N\u2082.map_id, comp_id, id_comp, identity_N\u2082_objectwise P], }\n\ninstance : is_iso (\u0393\u2082N\u2082.nat_trans : (N\u2082 : karoubi (simplicial_object C) \u2964 _ ) \u22d9 _ \u27f6 _) :=\nbegin\n  haveI : \u2200 (P : karoubi (simplicial_object C)), is_iso (\u0393\u2082N\u2082.nat_trans.app P),\n  { intro P,\n    haveI : is_iso (N\u2082.map (\u0393\u2082N\u2082.nat_trans.app P)),\n    { have h := identity_N\u2082_objectwise P,\n      erw hom_comp_eq_id at h,\n      rw h,\n      apply_instance, },\n    exact is_iso_of_reflects_iso _ N\u2082, },\n  apply nat_iso.is_iso_of_is_iso_app,\nend\n\ninstance : is_iso (\u0393\u2082N\u2081.nat_trans : (N\u2081 : simplicial_object C \u2964 _ ) \u22d9 _ \u27f6 _) :=\nbegin\n  haveI : \u2200 (X : simplicial_object C), is_iso (\u0393\u2082N\u2081.nat_trans.app X),\n  { intro X,\n    rw compatibility_\u0393\u2082N\u2081_\u0393\u2082N\u2082_nat_trans,\n    apply_instance, },\n  apply nat_iso.is_iso_of_is_iso_app,\nend\n\n/-- The unit isomorphism of the Dold-Kan equivalence. -/\n@[simp]\ndef \u0393\u2082N\u2082 : \ud835\udfed _ \u2245 (N\u2082 : karoubi (simplicial_object C) \u2964 _) \u22d9 \u0393\u2082 :=\n(as_iso \u0393\u2082N\u2082.nat_trans).symm\n\n/-- The natural isomorphism `to_karoubi (simplicial_object C) \u2245 N\u2081 \u22d9 \u0393\u2082`. -/\n@[simps]\ndef \u0393\u2082N\u2081 : to_karoubi _  \u2245 (N\u2081 : simplicial_object C \u2964 _) \u22d9 \u0393\u2082 :=\n(as_iso \u0393\u2082N\u2081.nat_trans).symm\n\nend dold_kan\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/dold_kan/n_comp_gamma.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3605898614608009}}
{"text": "def x := 1\n\n#check x\n\nvariable {\u03b1 : Type}\n\ndef f (a : \u03b1) : \u03b1 :=\na\n\ndef tst (xs : List Nat) : Nat :=\nxs.foldl (init := 10) (\u00b7 + \u00b7)\n\n#check tst [1, 2, 3]\n\n#check fun x y : Nat => x + y\n\n#check tst\n\n#check (fun stx => if True then let e := stx; Pure.pure e else Pure.pure stx : Nat \u2192 Id Nat)\n\n#check let x : Nat := 1; x\n\ndef foo (a : Nat) (b : Nat := 10) (c : Bool := Bool.true) : Nat :=\na + b\n\nset_option pp.all true\n\n#check foo 1\n\n#check foo 3 (c := false)\n\ndef Nat.boo (a : Nat) :=\nsucc a -- succ here is resolved as `Nat.succ`.\n\n#check Nat.boo\n\n#check true\n\n-- apply is still a valid identifier name\ndef apply := \"hello\"\n\n#check apply\n\ntheorem simple1 (x y : Nat) (h : x = y) : x = y :=\nby {\n  assumption\n}\n\ntheorem simple2 (x y : Nat) : x = y \u2192 x = y :=\nby {\n  intro h;\n  assumption\n}\n\nsyntax \"intro2\" : tactic\n\nmacro_rules\n| `(tactic| intro2) => `(tactic| intro; intro )\n\ntheorem simple3 (x y : Nat) : x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro2;\n  assumption\n}\n\nmacro \"intro3\" : tactic => `(intro; intro; intro)\nmacro \"check2\" x:term : command => `(#check $x #check $x)\nmacro \"foo\" x:term \",\" y:term : term => `($x + $y + $x)\n\nset_option pp.all false\n\ncheck2 0+1\ncheck2 foo 0,1\n\ntheorem simple4 (x y : Nat) : y = y \u2192 x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro3;\n  assumption\n}\n\ntheorem simple5 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  exact Eq.trans h3 h1\n}\n\ntheorem simple6 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine Eq.trans ?_ h1;\n  assumption\n}\n\ntheorem simple7 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine' Eq.trans ?pre ?post;\n  exact y;\n  { exact h3 }\n  { exact h1 }\n}\n\ntheorem simple8 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintro h1; intro _; intro h3\nrefine' Eq.trans ?pre ?post\ncase post => exact h1\ncase pre => exact h3\n\ntheorem simple9 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntraceState\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n      assumption\n    | exact y\n      exact h3\n      assumption\n\ntheorem simple9b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntraceState\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n    | exact y; exact h3\n  assumption\n\ntheorem simple9c (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  solve\n    | exact h1\n    | refine' Eq.trans ?pre ?post; exact y; exact h3; assumption\n    | exact h3\n\ntheorem simple9d (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  refine' Eq.trans ?pre ?post\n  solve\n    | exact h1\n    | exact y\n    | exact h3\n  solve\n    | exact h1\n    | exact h3\n  solve\n    | exact h1\n    | assumption\n\n\nnamespace Foo\n  def Prod.mk := 1\n  #check (\u27e82, 3\u27e9 : Prod _ _)\nend Foo\n\ntheorem simple10 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  skip;\n  apply Eq.trans;\n  exact h3;\n  assumption\n}\n\ntheorem simple11 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  traceState;\n  exact h3;\n  assumption\n}\n\ntheorem simple12 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  try exact h1; -- `exact h1` fails\n  traceState;\n  try exact h3;\n  traceState;\n  try exact h1;\n}\n\ntheorem simple13 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 h2 h3\ntraceState\napply @Eq.trans\ncase b => exact y\ntraceState\nrepeat assumption\n\ntheorem simple13b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by {\nintros h1 h2 h3;\ntraceState;\napply @Eq.trans;\ncase b => exact y;\ntraceState;\nrepeat assumption\n}\n\ntheorem simple14 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros\napply @Eq.trans\ncase b => exact y\nrepeat assumption\n\ntheorem simple15 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  revert y;\n  intros y h1 h3;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\ntheorem simple16 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  try clear x; -- should fail\n  clear h2;\n  traceState;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\nmacro \"blabla\" : tactic => `(assumption)\n\n-- Tactic head symbols do not become reserved words\ndef blabla := 100\n\n#check blabla\n\ntheorem simple17 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple18 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple19 (x y : Nat) (h\u2081 : x = 0) (h\u2082 : x = y) : y = 0 :=\nby subst x; subst y; exact rfl\n\ntheorem tstprec1 (x y z : Nat) : x + y * z = x + (y * z) :=\nrfl\n\ntheorem tstprec2 (x y z : Nat) : y * z + x = (y * z) + x :=\nrfl\n\nset_option pp.all true\n\n#check fun {\u03b1} (a : \u03b1) => a\n#check @(fun \u03b1 (a : \u03b1) => a)\n\n#check\n  let myid := fun {\u03b1} (a : \u03b1) => a;\n  myid [myid 1]\n\n-- In the following example, we need `@` otherwise we will try to insert mvars for \u03b1 and [Add \u03b1],\n-- and will fail to generate instance for [Add \u03b1]\n#check @(fun \u03b1 (s : Add \u03b1) (a : \u03b1) => a + a)\n\ndef g1 {\u03b1} (a\u2081 a\u2082 : \u03b1) {\u03b2} (b : \u03b2) : \u03b1 \u00d7 \u03b1 \u00d7 \u03b2 :=\n(a\u2081, a\u2082, b)\n\ndef id1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => x\n\ndef listId : List ({\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1) :=\n(fun x => x) :: []\n\ndef id2 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 (x : \u03b1) => id1 x)\n\ndef id3 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 x => id1 x)\n\ndef id4 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => id1 x\n\ndef id5 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id1 x\n\ndef id6 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun {\u03b1} x => id1 x)\n\ndef id7 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => @id \u03b1 x\n\ndef id8 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id (@id \u03b1 x)\n\ndef altTst1 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8StateT.failure, StateT.orElse\u27e9\n\ndef altTst2 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8@(fun \u03b1 => StateT.failure), @(fun \u03b1 => StateT.orElse)\u27e9\n\ndef altTst3 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8fun {\u03b1} => StateT.failure, fun {\u03b1} => StateT.orElse\u27e9\n\n#check_failure 1 + true\n\n/-\nuniverses u v\n\n/-\n  MonadFunctorT.{u ?M_1 v} (\u03bb (\u03b2 : Type u), m \u03b1) (\u03bb (\u03b2 : Type u), m' \u03b1) n n'\n-/\nset_option pp.raw.maxDepth 100\nset_option trace.Elab true\n\n\ndef adapt {m m' \u03c3 \u03c3'} {n n' : Type \u2192 Type} [MonadFunctor m m' n n'] [MonadStateAdapter \u03c3 \u03c3' m m'] : MonadStateAdapter \u03c3 \u03c3' n n' :=\n\u27e8fun split join => monadMap (adaptState split join : m \u03b1 \u2192 m' \u03b1)\u27e9\n\n-/\n\nsyntax \"fn\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(fn $xs* => $b) => `(fun $xs* => $b)\n\nset_option pp.all false\n\n#check fn x => x+1\n\n#check fn \u03b1 (a : \u03b1) => a\n\ndef tst1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fn \u03b1 a => a)\n\n#check @tst1\n\nsyntax ident \"==>\" term : term\n\nsyntax \"{\" ident \"}\" \"==>\" term : term\n\nmacro_rules\n| `($x:ident ==> $b)   => `(fn $x => $b)\n| `({$x:ident} ==> $b) => `(fun {$x:ident} => $b)\n\n#check x ==> x+1\n\ndef tst2a : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(\u03b1 ==> a ==> a)\n\ndef tst2b : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n{\u03b1} ==> a ==> a\n\n#check @tst2a\n#check @tst2b\n\ndef tst3a : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n@(\u03b1 ==> @(\u03b2 ==> a ==> b ==> (a, b)))\n\ndef tst3b : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n{\u03b1} ==> {\u03b2} ==> a ==> b ==> (a, b)\n\nsyntax \"function\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(function $xs* => $b) => `(@(fun $xs* => $b))\n\ndef tst4 : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\nfunction \u03b1 \u03b2 a b => (a, b)\n\ntheorem simple20 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby intros h1 h2 h3;\n   try clear x; -- should fail\n   clear h2;\n   traceState;\n   apply Eq.trans;\n   exact h3;\n   exact h1\n\ntheorem simple21 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by { apply Eq.symm; assumption };\n  Eq.trans this (by assumption)\n\ntheorem simple22 (x y z : Nat) : y = z \u2192 y = x \u2192 id (x = z + 0) :=\nfun h1 h2 => show x = z + 0 by\n  apply Eq.trans\n  exact h2.symm\n  assumption\n  skip\n\ntheorem simple23 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by apply Eq.symm; assumption\n  Eq.trans this (by assumption)\n\ntheorem simple24 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have h : x = y := by apply Eq.symm; assumption\n  Eq.trans h (by assumption)\n\ndef f1 (x : Nat) : Nat :=\n  let double x := x + x\n  let rec loop x :=\n    match x with\n    | 0   => 0\n    | x+1 => loop x + double x\n  loop x\n\n#eval f1 5\n\ndef f2 (x : Nat) : String :=\n  let bad x : String := toString x\n  bad x\n\ndef f3 x y :=\n  x + y + 1\n\ntheorem f3eq x y : f3 x y = x + y + 1 :=\n  rfl\n\ndef f4 (x y : Nat) : String :=\n  if x > y + 1 then \"hello\" else \"world\"\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/newfrontend1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.6370308013713525, "lm_q1q2_score": 0.36057125039402}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport data.list.chain\nimport category_theory.punit\nimport category_theory.is_connected\nimport category_theory.sigma.basic\nimport category_theory.full_subcategory\n\n/-!\n# Connected components of a category\n\nDefines a type `connected_components J` indexing the connected components of a category, and the\nfull subcategories giving each connected component: `component j : Type u\u2081`.\nWe show that each `component j` is in fact connected.\n\nWe show every category can be expressed as a disjoint union of its connected components, in\nparticular `decomposed J` is the category (definitionally) given by the sigma-type of the connected\ncomponents of `J`, and it is shown that this is equivalent to `J`.\n-/\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082\n\nnoncomputable theory\n\nopen category_theory.category\n\nnamespace category_theory\n\nattribute [instance, priority 100] is_connected.is_nonempty\n\nvariables {J : Type u\u2081} [category.{v\u2081} J]\nvariables {C : Type u\u2082} [category.{u\u2081} C]\n\n/-- This type indexes the connected components of the category `J`. -/\ndef connected_components (J : Type u\u2081) [category.{v\u2081} J] : Type u\u2081 := quotient (zigzag.setoid J)\n\ninstance [inhabited J] : inhabited (connected_components J) := \u27e8quotient.mk' default\u27e9\n\n/-- Given an index for a connected component, produce the actual component as a full subcategory. -/\n@[derive category]\ndef component (j : connected_components J) : Type u\u2081 := {k : J // quotient.mk' k = j}\n\n/-- The inclusion functor from a connected component to the whole category. -/\n@[derive [full, faithful], simps {rhs_md := semireducible}]\ndef component.\u03b9 (j) : component j \u2964 J :=\nfull_subcategory_inclusion _\n\n/-- Each connected component of the category is nonempty. -/\ninstance (j : connected_components J) : nonempty (component j) :=\nbegin\n  apply quotient.induction_on' j,\n  intro k,\n  refine \u27e8\u27e8k, rfl\u27e9\u27e9,\nend\n\ninstance (j : connected_components J) : inhabited (component j) := classical.inhabited_of_nonempty'\n\n/-- Each connected component of the category is connected. -/\ninstance (j : connected_components J) : is_connected (component j) :=\nbegin\n  -- Show it's connected by constructing a zigzag (in `component j`) between any two objects\n  apply is_connected_of_zigzag,\n  rintro \u27e8j\u2081, hj\u2081\u27e9 \u27e8j\u2082, rfl\u27e9,\n  -- We know that the underlying objects j\u2081 j\u2082 have some zigzag between them in `J`\n  have h\u2081\u2082 : zigzag j\u2081 j\u2082 := quotient.exact' hj\u2081,\n  -- Get an explicit zigzag as a list\n  rcases list.exists_chain_of_relation_refl_trans_gen h\u2081\u2082 with \u27e8l, hl\u2081, hl\u2082\u27e9,\n  -- Everything which has a zigzag to j\u2082 can be lifted to the same component as `j\u2082`.\n  let f : \u03a0 x, zigzag x j\u2082 \u2192 component (quotient.mk' j\u2082) := \u03bb x h, \u27e8x, quotient.sound' h\u27e9,\n  -- Everything in our chosen zigzag from `j\u2081` to `j\u2082` has a zigzag to `j\u2082`.\n  have hf : \u2200 (a : J), a \u2208 l \u2192 zigzag a j\u2082,\n  { intros i hi,\n    apply list.chain.induction (\u03bb t, zigzag t j\u2082) _ hl\u2081 hl\u2082 _ _ _ (or.inr hi),\n    { intros j k,\n      apply relation.refl_trans_gen.head },\n    { apply relation.refl_trans_gen.refl } },\n  -- Now lift the zigzag from `j\u2081` to `j\u2082` in `J` to the same thing in `component j`.\n  refine \u27e8l.pmap f hf, _, _\u27e9,\n  { refine @@list.chain_pmap_of_chain _ _ _ f (\u03bb x y _ _ h, _) hl\u2081 h\u2081\u2082 _,\n    exact zag_of_zag_obj (component.\u03b9 _) h },\n  { erw list.last_pmap _ f (j\u2081 :: l) (by simpa [h\u2081\u2082] using hf) (list.cons_ne_nil _ _),\n    exact subtype.ext hl\u2082 },\nend\n\n/--\nThe disjoint union of `J`s connected components, written explicitly as a sigma-type with the\ncategory structure.\nThis category is equivalent to `J`.\n-/\nabbreviation decomposed (J : Type u\u2081) [category.{v\u2081} J] :=\n\u03a3 (j : connected_components J), component j\n\n/--\nThe inclusion of each component into the decomposed category. This is just `sigma.incl` but having\nthis abbreviation helps guide typeclass search to get the right category instance on `decomposed J`.\n-/\n-- This name may cause clashes further down the road, and so might need to be changed.\nabbreviation inclusion (j : connected_components J) : component j \u2964 decomposed J :=\nsigma.incl _\n\n/-- The forward direction of the equivalence between the decomposed category and the original. -/\n@[simps {rhs_md := semireducible}]\ndef decomposed_to (J : Type u\u2081) [category.{v\u2081} J] : decomposed J \u2964 J :=\nsigma.desc component.\u03b9\n\n@[simp]\nlemma inclusion_comp_decomposed_to (j : connected_components J) :\n  inclusion j \u22d9 decomposed_to J = component.\u03b9 j :=\nrfl\n\ninstance : full (decomposed_to J) :=\n{ preimage :=\n  begin\n    rintro \u27e8j', X, hX\u27e9 \u27e8k', Y, hY\u27e9 f,\n    dsimp at f,\n    have : j' = k',\n      rw [\u2190 hX, \u2190 hY, quotient.eq'],\n      exact relation.refl_trans_gen.single (or.inl \u27e8f\u27e9),\n    subst this,\n    refine sigma.sigma_hom.mk f,\n  end,\n  witness' :=\n  begin\n    rintro \u27e8j', X, hX\u27e9 \u27e8_, Y, rfl\u27e9 f,\n    have : quotient.mk' Y = j',\n    { rw [\u2190 hX, quotient.eq'],\n      exact relation.refl_trans_gen.single (or.inr \u27e8f\u27e9) },\n    subst this,\n    refl,\n  end }\n\ninstance : faithful (decomposed_to J) :=\n{ map_injective' :=\n  begin\n    rintro \u27e8_, j, rfl\u27e9 \u27e8_, k, hY\u27e9 \u27e8_, _, _, f\u27e9 \u27e8_, _, _, g\u27e9 e,\n    change f = g at e,\n    subst e,\n  end }\n\ninstance : ess_surj (decomposed_to J) :=\n{ mem_ess_image := \u03bb j, \u27e8\u27e8_, j, rfl\u27e9, \u27e8iso.refl _\u27e9\u27e9 }\n\ninstance : is_equivalence (decomposed_to J) :=\nequivalence.of_fully_faithfully_ess_surj _\n\n/-- This gives that any category is equivalent to a disjoint union of connected categories. -/\n@[simps functor {rhs_md := semireducible}]\ndef decomposed_equiv : decomposed J \u224c J :=\n(decomposed_to J).as_equivalence\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/connected_components.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3605712371665952}}
{"text": "/-\nFile: signature_recover_public_key_unreduced_mul_soundness.lean\n\nAutogenerated file.\n-/\nimport starkware.cairo.lean.semantics.soundness.hoare\nimport .signature_recover_public_key_code\nimport ..signature_recover_public_key_spec\nopen tactic\n\nopen starkware.cairo.common.cairo_secp.field\nopen starkware.cairo.common.cairo_secp.bigint\nopen starkware.cairo.common.cairo_secp.constants\n\nvariables {F : Type} [field F] [decidable_eq F] [prelude_hyps F]\nvariable  mem : F \u2192 F\nvariable  \u03c3 : register_state F\n\n/- starkware.cairo.common.cairo_secp.field.unreduced_mul autogenerated soundness theorem -/\n\ntheorem auto_sound_unreduced_mul\n    -- arguments\n    (a b : BigInt3 F)\n    -- code is in memory at \u03c3.pc\n    (h_mem : mem_at mem code_unreduced_mul \u03c3.pc)\n    -- input arguments on the stack\n    (hin_a : a = cast_BigInt3 mem (\u03c3.fp - 8))\n    (hin_b : b = cast_BigInt3 mem (\u03c3.fp - 5))\n    -- conclusion\n  : ensures_ret mem \u03c3 (\u03bb \u03ba \u03c4, \u03c4.ap = \u03c3.ap + 17 \u2227 spec_unreduced_mul mem \u03ba a b (cast_UnreducedBigInt3 mem (\u03c4.ap - 3))) :=\nbegin\n  apply ensures_of_ensuresb, intro \u03bdbound,\n  have h_mem_rec := h_mem,\n  unpack_memory code_unreduced_mul at h_mem with \u27e8hpc0, hpc1, hpc2, hpc3, hpc4, hpc5, hpc6, hpc7, hpc8, hpc9, hpc10, hpc11, hpc12, hpc13, hpc14, hpc15, hpc16, hpc17, hpc18, hpc19\u27e9,\n  -- return\n  step_assert_eq hpc0 with hret0,\n  step_assert_eq hpc1 with hret1,\n  step_assert_eq hpc2 with hret2,\n  step_assert_eq hpc3 with hret3,\n  step_assert_eq hpc4 hpc5 with hret4,\n  step_assert_eq hpc6 with hret5,\n  step_assert_eq hpc7 with hret6,\n  step_assert_eq hpc8 with hret7,\n  step_assert_eq hpc9 hpc10 with hret8,\n  step_assert_eq hpc11 with hret9,\n  step_assert_eq hpc12 with hret10,\n  step_assert_eq hpc13 with hret11,\n  step_assert_eq hpc14 with hret12,\n  step_assert_eq hpc15 with hret13,\n  step_assert_eq hpc16 with hret14,\n  step_assert_eq hpc17 with hret15,\n  step_assert_eq hpc18 with hret16,\n  step_ret hpc19,\n  -- finish\n  step_done, use_only [rfl, rfl],\n  split, refl,\n  -- Final Proof\n  -- user-provided reduction\n  suffices auto_spec: auto_spec_unreduced_mul mem _ a b _,\n  { apply sound_unreduced_mul, apply auto_spec },\n  -- prove the auto generated assertion\n  dsimp [auto_spec_unreduced_mul],\n  try { norm_num1 }, try { arith_simps },\n  try { split, linarith },\n  try { ensures_simps; try { simp only [add_neg_eq_sub, hin_a, hin_b] }, },\n  try { dsimp [cast_BigInt3, cast_UnreducedBigInt3] },\n  try { arith_simps }, try { simp only [hret0, hret1, hret2, hret3, hret4, hret5, hret6, hret7, hret8, hret9, hret10, hret11, hret12, hret13, hret14, hret15, hret16] },\n  try { arith_simps; try { split }; triv <|> refl <|> simp <|> abel; try { norm_num } },\nend\n\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/common/cairo_secp/verification/verification/signature_recover_public_key_unreduced_mul_soundness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3605495121184134}}
{"text": "import Advent\nimport Std\n\nopen Lean (HashMap HashSet)\n\ninductive Material |ore  |obsidian |geode |clay deriving Repr, Inhabited\n\nnamespace Material\n\ninstance : ToString Material where \n  toString\n  | ore => \"ore\"\n  | obsidian => \"obsidian\"\n  | geode => \"geode\"\n  | clay => \"clay\"\n\ndef index: Material -> Fin 4 \n| ore => 0\n| obsidian => 1\n| geode => 2\n| clay => 3\n\ndef fromIndex: Fin 4 -> Material\n| 0 => ore\n| 1 => obsidian\n| 2 => geode\n| 3 => clay\n\ntheorem toFrom(m: Material): m = fromIndex (index m) := by\n  cases m <;> simp [index, fromIndex]\n\ntheorem fromTo(i: Fin 4):  i = index (fromIndex i) := by\n  match i with \n  | 0 => simp\n  | 1 => simp\n  | 2 => simp\n  | 3 => simp\n\ndef values := [ore, clay, geode, obsidian]\n\ndef valArray := values.toArray\n\ndef names := values.map (s!\"{\u00b7}\")\n\ndef parseMat: Parse Material :=\n  Parse.choose <| names.zip values\n\n\nstructure Map (\u03b1: Type) where\n  ore : \u03b1\n  obsidian: \u03b1\n  geode: \u03b1\n  clay : \u03b1\nderiving Ord, BEq, Hashable, Inhabited, Lean.ToJson\n\nnamespace Map \n  def off (f: Material -> \u03b1): Map \u03b1 := \u27e8f Material.ore, f Material.obsidian, f Material.geode, f Material.clay\u27e9\n  variable (map: Map \u03b1)\n\n\n  def getMat (mat: Material): \u03b1 := \n    match mat with\n    | Material.ore => map.ore\n    | Material.obsidian => map.obsidian\n    | Material.geode => map.geode\n    | Material.clay => map.clay\n    \n  def modifyMat (mat: Material) (f: \u03b1 -> \u03b1): Map \u03b1 := \n     match mat with\n    | Material.ore => {map with ore := f map.ore}\n    | Material.obsidian => {map with obsidian := f map.obsidian}\n    | Material.geode => {map with geode := f map.geode}\n    | Material.clay => {map with clay := f map.clay}\n\n  def setMat (mat: Material) (x: \u03b1): Map \u03b1 := \n    match mat with\n    | Material.ore => {map with ore := x}\n    | Material.obsidian => {map with obsidian := x}\n    | Material.geode => {map with geode := x}\n    | Material.clay => {map with clay := x}\n\n  def zipWith (f: \u03b1 -> \u03b2 -> \u03b3) (map': Map \u03b2): Map \u03b3 := \n    \u27e8 f map.ore map'.ore, f map.obsidian map'.obsidian, f map.geode map'.geode, f map.clay map'.clay  \u27e9\n\n  def sum [Add \u03b1]: \u03b1 := map.ore + map.obsidian + map.geode + map.clay\n\n  def mapLE [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)] (xs ys: Map \u03b1): Bool := \n    (xs.ore <= ys.ore) && (xs.obsidian <= ys.obsidian) && (xs.geode <= ys.geode) && (xs.clay <= ys.clay)\n\n  instance [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)]: LE (Map \u03b1) where\n    le x y := mapLE x y\n\n  instance [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)] {xs ys: Map \u03b1}: Decidable (xs <= ys) := \n    decidable (mapLE xs ys)\n\n  instance [HAdd \u03b1 \u03b2 \u03b3] : HAdd (Map \u03b1) (Map \u03b2) (Map \u03b3) where\n    hAdd xs ys := xs.zipWith (\u00b7 + \u00b7) ys\n\n  instance [HSub \u03b1 \u03b2 \u03b3] : HSub (Map \u03b1) (Map \u03b2) (Map \u03b3) where\n    hSub xs ys := xs.zipWith (\u00b7 - \u00b7) ys\nend Map\n\n\n\ninstance [ToString \u03b1] : ToString (Material -> \u03b1) where\n  toString f := String.join <| ([\"[\"] ++ \u00b7 ++ [\"]\"]) <| List.intersperse \", \" <| names.zipWith (s!\"{\u00b7}: {f .}\") values\nend Material\n\nopen Material\n\ndef Blueprint := Map (Map Nat) deriving ToString, Inhabited\nnamespace Blueprint\n\ndef parseBot (mat: Material): Parse (Map Nat) := open Parse in do \nws\nstr \"Each \"  \nstr s!\"{mat}\"\nstr \" robot costs \"\nlet cost := do\n  let amt <- nat\n  str \" \"\n  let mat <- Material.parseMat\n  return (amt, mat)    \nlet costs <- cost.repSep! \" and \"\nlet mut costArr := Material.Map.mk 0 0 0 0\nfor (amt, mat) in costs do\n  costArr := costArr.modifyMat mat (\u00b7 + amt)\nstr \".\"\nreturn costArr\n\ndef parse: Parse Blueprint := open Parse in do\n  ws\n  str \"Blueprint \"\n  _ <- nat\n  str \":\"\n  ws\n\n  let oreBot <- parseBot ore\n  let clayBot <- parseBot clay\n  let obsBot <- parseBot obsidian\n  let geodeBot <- parseBot geode\n\n  return {\n     ore := oreBot \n     clay := clayBot \n     geode := geodeBot \n     obsidian := obsBot\n  }\nend Blueprint\n\nstructure State where\n  time: Nat\n  robots: Map Nat\n  mats: Map Nat\nderiving BEq, Inhabited, Hashable, Lean.ToJson\n\nabbrev Cache := HashSet State\n\ndef simulate (bp: Blueprint) (time: Nat) (robots: Map Nat) (mats: Map Nat) : StateM Cache Nat := \nmatch time with \n| 0 => pure <| mats.getMat geode\n| time + 1 => do\n  let cache <- StateT.get\n  let state: State := \u27e8 time, robots, mats \u27e9\n  if cache.contains state then return 0\n  let mats' := robots + mats\n  let mut res <- simulate bp time robots mats'\n  for mat in Material.values do\n    let reqs := bp.getMat mat\n    if reqs <= mats then \n      let mats'' := mats' - reqs\n      let robots' := robots.modifyMat mat (\u00b7 + 1)\n      let branch <- simulate bp time robots' mats''\n      res := max res branch\n  StateM.update (\u00b7.insert state)\n  return res\n\n-- simplified \\ specialized blueprint for p2\nstructure SBlueprint where\n  ore'ore : Nat\n  clay'ore: Nat\n  obs'ore: Nat\n  obs'clay: Nat\n  geo'ore: Nat\n  geo'obs: Nat\n\ninstance: Coe Blueprint SBlueprint where\n  coe bp := {\n    ore'ore := bp.ore.ore\n    clay'ore := bp.clay.ore\n    obs'ore := bp.obsidian.ore\n    obs'clay := bp.obsidian.clay\n    geo'ore := bp.geode.ore\n    geo'obs := bp.geode.obsidian\n  }\n\n\n\nstructure SStrategy where\n  ores: Nat\n  clays: Nat\n  clays2: Nat\n  obs: Nat\nderiving Lean.ToJson, Repr, BEq\n\ndef allStrategies(max: Nat): Id (Array SStrategy) := do\n  let mut res := Array.empty\n  for ores in [1:max + 1] do\n    for clays in [1:max + 1] do\n      for clays2 in [clays:max + 1] do\n        for obs in [0:max + 1] do\n          res := res.push <| \u27e8 ores, clays , clays2, obs\u27e9\n  return res\n\nabbrev Robots := Map Nat\nabbrev Materials := Map Nat\n\ndef runStrategy (b: SBlueprint) (s: SStrategy) (time: Nat): Id Nat := do\n  let mut robots := \u27e81, 0, 0, 0\u27e9\n  let mut mats := \u27e80, 0, 0, 0\u27e9\n  for m in [0:time] do\n    (robots, mats) := go robots mats\n    -- IO.println s!\"minute {m + 1} robots = {robots} mats = {mats}\"\n  return mats.geode\nwhere\n  buyGeodeBot? (mats: Materials) : Option Materials := \n    if b.geo'ore <= mats.ore && b.geo'obs <= mats.obsidian \n    then some {mats with ore := mats.ore - b.geo'ore, obsidian := mats.obsidian - b.geo'obs} \n    else none\n  \n  buyObsidianBot? (mats: Materials) (obsRobots : Nat): Option Materials := \n    if b.obs'ore <= mats.ore && b.obs'clay <= mats.clay && obsRobots <= s.obs\n    then some {mats with ore := mats.ore - b.obs'ore, clay := mats.clay - b.obs'clay} \n    else none\n\n    -- if mats.\n  go (robots: Robots) (mats: Materials): (Robots \u00d7 Materials) :=\n    if robots.ore < s.ores && b.ore'ore <= mats.ore then\n      ({robots with ore := robots.ore + 1 } ,{mats with ore := mats.ore - b.ore'ore} + robots)\n    else if robots.ore == s.ores && robots.clay < s.clays && b.clay'ore <= mats.ore then\n      ({robots with clay := robots.clay + 1}, {mats with ore := mats.ore - b.clay'ore} + robots)\n    else if let some mats := buyObsidianBot? mats robots.obsidian then\n      ({robots with obsidian := robots.obsidian + 1}, (mats + robots))\n    else if let some mats := buyGeodeBot? mats then \n      ({robots with geode := robots.geode + 1}, (mats + robots))\n    else if let some mats := buyObsidianBot? mats 0 then\n      ({robots with obsidian := robots.obsidian + 1}, (mats + robots))\n    else if robots.ore == s.ores && robots.clay < s.clays2 && b.clay'ore <= mats.ore then\n      ({robots with clay := robots.clay + 1}, {mats with ore := mats.ore - b.clay'ore} + robots)\n    else (robots, (mats + robots))\n      \ndef bestStrategy (b: SBlueprint) (time: Nat) (top := 10): Id Nat := do\n  let mut best := 0\n  let all <- allStrategies top\n  for s in all do \n    let res <- runStrategy b s time\n    best := max best res\n  return best \n\ndef ssimulate (bp: SBlueprint) (time: Nat): StateM Cache Nat := do\n  go 0 \u27e81, 0, 0, 0\u27e9 \u27e80, 0, 0, 0\u27e9 time\nwhere \n  -- bootstrap bots ore\n  -- | 0, _ => pure 0\n  -- | time, 0 => go \u27e8bots, 0, 0, 0\u27e9 \u27e8ore, 0, 0, 0\u27e9 time\n  -- | time + 1, breq + 1 =>\n  --   if bp.ore'ore <= ore\n  --   then bootstrap (bots + 1) (ore + bots - bp.ore'ore) time breq\n  --   else bootstrap bots (ore + bots) time (breq + 1)\n\n  oreLimit := 4\n  clayLimit := 19\n  obsidianLimit := 17 \n\n  go (phase : UInt8) (robots: Robots) (mats: Materials)\n  | 0 => pure mats.geode\n  | time + 1 => do\n    let cache <- StateT.get\n    let state: State := \u27e8 time, robots, mats \u27e9\n    if cache.contains state then return 0\n    StateT.set <| cache.insert state\n    let mats' := robots + mats\n    let mut res := 0\n    if bp.geo'ore <= mats.ore && bp.geo'obs <= mats.obsidian then\n      res <- go 2\n            {robots with geode := robots.geode + 1} \n            {mats' with ore := mats'.ore - bp.geo'ore, obsidian := mats'.obsidian - bp.geo'obs } \n            time\n    else     \n      if bp.ore'ore <= mats.ore && robots.ore < oreLimit && phase == 0 then\n        res <- go phase\n                  {robots with ore := robots.ore + 1} \n                  {mats' with ore := mats'.ore - bp.ore'ore } \n                  time\n      if bp.clay'ore <= mats.ore && robots.clay < clayLimit  && phase < 2 then\n        let next <- go phase\n                  {robots with clay := robots.clay + 1} \n                  {mats' with ore := mats'.ore - bp.clay'ore } \n                  time\n        res := max res next\n      if bp.obs'ore <= mats.ore && bp.obs'clay <= mats.clay && robots.obsidian < obsidianLimit then\n        let next <- go  (max 1 phase)\n                        {robots with obsidian := robots.obsidian + 1}\n                        {mats' with ore := mats'.ore - bp.obs'ore, clay := mats'.clay - bp.obs'clay}\n                        time\n        res := max res next\n\n      let next <- go phase robots mats' time\n      res := max res next\n\n    return res\nopen Material\n\n\ndef main: IO Unit := do\n  let s <- readInput 19\n  let bps <- Blueprint.parse.rep.runIO s\n\n  let mut score := 1\n  for (i, bp) in bps.enum.take 3 do\n    let res: Nat := bestStrategy bp 32\n    IO.println s!\"blueprint {i + 1} {res}\" \n    score := score * res\n  IO.println s!\"score = {score}\"\n  -- IO.println (allStrategies 2)\n  -- score := 0\n  -- for (i, bp) in bps.enum do\n  --   let res : Nat := bestStrategy bp 24 20\n  --   IO.println s!\"blueprint {i + 1}: {res}\"\n  --   score := score + (i + 1) * res\n  -- IO.println score\n\n\n\n\n  -- let robots: Map Nat :=  \u27e81, 0, 0, 0\u27e9\n  -- let mats: Map Nat := \u27e80, 0, 0, 0\u27e9\n  score := 1\n  for (i, bp) in bps.enum.take 3 do\n    let result: Nat := (ssimulate bp 32).run' HashSet.empty\n    IO.println s!\"(slow) blueprint {i + 1}: {bp}\"\n    IO.println result\n    -- IO.println c.size\n    score := score * result\n\n  IO.println s!\"score is {score}\"\n\ndef Map'  \u03b1 := {arr : Array \u03b1 // arr.size = 4}\n\nnamespace Map' \n  def mk (x: \u03b1): Map'  \u03b1 := \u27e8Array.mkArray 4 x, Eq.refl 4\u27e9\n  def of (or g obs c: \u03b1): Map'  \u03b1 := \u27e8 Array.mkArray4 or g obs c, Eq.refl 4\u27e9\n  def off (f: Material -> \u03b1): Map'  \u03b1 := of (f ore) (f obsidian) (f geode) (f clay)\n \n  variable (map: Map'  \u03b1)\n\n  instance [Inhabited \u03b1]: Inhabited (Map'  \u03b1) where\n    default := mk default\n\n  instance [BEq \u03b1]: BEq (Map'  \u03b1) where\n    beq x y := x.val == y.val\n\n  instance [ToString \u03b1]: ToString (Map'  \u03b1) where\n    toString map := String.join <| \n                    ([\"[\"] ++ \u00b7 ++ [\"]\"]) <| \n                    List.intersperse \", \" <| \n                    names.zipWith (s!\"{\u00b7}: {.}\") map.val.toList\n\n  instance [Hashable \u03b1]: Hashable (Map'  \u03b1) where\n    hash map := hash map.val\n\n  instance [Lean.ToJson \u03b1]: Lean.ToJson (Map'  \u03b1) where\n    toJson map := Lean.toJson map.val\n\n\n  def getMat (mat: Material): \u03b1 := \n    match map with \n    | \u27e8 arr, p \u27e9 => arr.get <| cast (by rw[p]) <| index mat \n  \n  def modifyMat (mat: Material) (f: \u03b1 -> \u03b1): Map'  \u03b1 := \n    match map with \n    | \u27e8 arr, p \u27e9 => Subtype.mk (arr.modify (index mat) f) <| \n      by rw [<- Array.modify_stable_size]; assumption\n\n  def setMat (mat: Material) (x: \u03b1): Map'  \u03b1 := \n    match map with \n    | \u27e8 arr, p \u27e9 => Subtype.mk (arr.set (cast (by rw[p]) (index mat)) x) <| by simp; assumption\n\n  def zipWith (f: \u03b1 -> \u03b2 -> \u03b3) (map': Map'  \u03b2): Map'  \u03b3 := \n    match map, map' with \n    | \u27e8 arr, p \u27e9, \u27e8 arr', p' \u27e9 => \u27e8 arr.zipWith arr' f,  Array.zipWith_eq_size f p p' \u27e9\n\n  def mapLE [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)] (xs ys: Map'  \u03b1): Bool := \n    values.all (fun m => xs.getMat m <= ys.getMat m)\n\n  instance [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)]: LE (Map'  \u03b1) where\n    le x y := mapLE x y\n\n  instance [LE \u03b1] [(x: \u03b1) -> (y: \u03b1) -> Decidable (x <= y)] {xs ys: Map' \u03b1}: Decidable (xs <= ys) := \n    decidable (mapLE xs ys)\nend Map' \n\n", "meta": {"author": "Odomontois", "repo": "advent2022-lean", "sha": "75634a2257287ec1536690f8dbd92573f670e0e2", "save_path": "github-repos/lean/Odomontois-advent2022-lean", "path": "github-repos/lean/Odomontois-advent2022-lean/advent2022-lean-75634a2257287ec1536690f8dbd92573f670e0e2/days/day19.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587586, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3605495056552632}}
{"text": "import Smt\n\ntheorem addition (p q : Prop) : p \u2192 p \u2228 q := by\n  smt\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Prop/Addition.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.36054950565526317}}
{"text": "import Mathlib.Data.Nat.Basic\n\ndef ff (n : \u2115) : \u2115 := Id.run do\n  let mut r := 0\n  for _ in [: n] do\n    r \u2190 r + 1\n  return r\n\ntheorem ff' : ff n = n := by\n  simp [ff, forIn, Std.Range.forIn]\n  let rec foo : \u2200 fuel a, fuel + a = n \u2192\n      Id.run (Std.Range.forIn.loop (fun x r => ForInStep.yield (r + 1)) fuel a n 1 a) = n\n  | 0, a, h => by simp [Std.Range.forIn.loop]; rwa [Nat.zero_add] at h\n  | fuel+1, a, h => by\n    simp [Std.Range.forIn.loop]; split\n    \u00b7 next h' => exact le_antisymm (h \u25b8 Nat.le_add_left ..) h'\n    \u00b7 next h' => rw [\u2190 Nat.add_right_comm] at h; exact foo _ _ h\n  exact foo _ _ rfl\n\ntheorem ff'' : ff n = n := by\n  simp [ff, forIn, Std.Range.forIn]\n  sorry", "meta": {"author": "grhkm21", "repo": "lean4", "sha": "2e3414e5b0eabfda1169ffe1bd5754daf24ea759", "save_path": "github-repos/lean/grhkm21-lean4", "path": "github-repos/lean/grhkm21-lean4/lean4-2e3414e5b0eabfda1169ffe1bd5754daf24ea759/Lean4/TrivialMonad.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.36054950565526306}}
{"text": "def forceNat (a : Nat) := true\ndef forceInt (a : Int) := false\n\ndef f1 :=\n/-\nThe following example works, but it adds a coercion at `forceInt i`.\nThe elaborated term is\n```\nfun (n i : Nat) => if n == i then forceNat n else forceInt (coe i)\n```\n-/\nfun n i => if n == i then forceNat n else forceInt i -- works\n\ndef f2 :=\nfun n i => if coe n == i then forceInt i else forceNat n -- works\n\n#check f1 -- Nat \u2192 Nat \u2192 Bool\n#check f2 -- Nat \u2192 Int \u2192 Bool\n\ndef f3 :=\n/- Fails.\n   - `n == i` generates type constraint enforcing `n` and `i` to have the same type.\n   - `forceInt i` forces `i` (and consequently `n`) to have type `Int`.\n   - `forceNat n` fails because there is no coercion from `Nat` to `Int`. -/\nfun n i => if n == i then forceInt i else forceNat n\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/elabissues/issues8.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6688802669716107, "lm_q1q2_score": 0.36051524054679673}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Johannes H\u00f6lzl\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.measure_theory.measure_space\nimport Mathlib.measure_theory.borel_space\nimport Mathlib.data.indicator_function\nimport Mathlib.data.support\nimport Mathlib.PostPort\n\nuniverses u v l u_1 u_2 u_3 u_4 u_5 \n\nnamespace Mathlib\n\n/-!\n# Lebesgue integral for `ennreal`-valued functions\n\nWe define simple functions and show that each Borel measurable function on `ennreal` can be\napproximated by a sequence of simple functions.\n\nTo prove something for an arbitrary measurable function into `ennreal`, the theorem\n`measurable.ennreal_induction` shows that is it sufficient to show that the property holds for\n(multiples of) characteristic functions and is closed under addition and supremum of increasing\nsequences of functions.\n\n## Notation\n\nWe introduce the following notation for the lower Lebesgue integral of a function `f : \u03b1 \u2192 ennreal`.\n\n* `\u222b\u207b x, f x \u2202\u03bc`: integral of a function `f : \u03b1 \u2192 ennreal` with respect to a measure `\u03bc`;\n* `\u222b\u207b x, f x`: integral of a function `f : \u03b1 \u2192 ennreal` with respect to the canonical measure\n  `volume` on `\u03b1`;\n* `\u222b\u207b x in s, f x \u2202\u03bc`: integral of a function `f : \u03b1 \u2192 ennreal` over a set `s` with respect\n  to a measure `\u03bc`, defined as `\u222b\u207b x, f x \u2202(\u03bc.restrict s)`;\n* `\u222b\u207b x in s, f x`: integral of a function `f : \u03b1 \u2192 ennreal` over a set `s` with respect\n  to the canonical measure `volume`, defined as `\u222b\u207b x, f x \u2202(volume.restrict s)`.\n\n-/\n\nnamespace measure_theory\n\n\n/-- A function `f` from a measurable space to any type is called *simple*,\nif every preimage `f \u207b\u00b9' {x}` is measurable, and the range is finite. This structure bundles\na function with these properties. -/\nstructure simple_func (\u03b1 : Type u) [measurable_space \u03b1] (\u03b2 : Type v) \nwhere\n  to_fun : \u03b1 \u2192 \u03b2\n  is_measurable_fiber' : \u2200 (x : \u03b2), is_measurable (to_fun \u207b\u00b9' singleton x)\n  finite_range' : set.finite (set.range to_fun)\n\nnamespace simple_func\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] : has_coe_to_fun (simple_func \u03b1 \u03b2) :=\n  has_coe_to_fun.mk (fun (x : simple_func \u03b1 \u03b2) => \u03b1 \u2192 \u03b2) to_fun\n\ntheorem coe_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} (H : \u21d1f = \u21d1g) : f = g := sorry\n\ntheorem ext {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} (H : \u2200 (a : \u03b1), coe_fn f a = coe_fn g a) : f = g :=\n  coe_injective (funext H)\n\ntheorem finite_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) : set.finite (set.range \u21d1f) :=\n  finite_range' f\n\ntheorem is_measurable_fiber {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (x : \u03b2) : is_measurable (\u21d1f \u207b\u00b9' singleton x) :=\n  is_measurable_fiber' f x\n\n/-- Range of a simple function `\u03b1 \u2192\u209b \u03b2` as a `finset \u03b2`. -/\nprotected def range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) : finset \u03b2 :=\n  set.finite.to_finset (finite_range f)\n\n@[simp] theorem mem_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {b : \u03b2} : b \u2208 simple_func.range f \u2194 b \u2208 set.range \u21d1f :=\n  set.finite.mem_to_finset\n\ntheorem mem_range_self {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (x : \u03b1) : coe_fn f x \u2208 simple_func.range f :=\n  iff.mpr mem_range (Exists.intro x rfl)\n\n@[simp] theorem coe_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) : \u2191(simple_func.range f) = set.range \u21d1f :=\n  set.finite.coe_to_finset (finite_range f)\n\ntheorem mem_range_of_measure_ne_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {x : \u03b2} {\u03bc : measure \u03b1} (H : coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton x) \u2260 0) : x \u2208 simple_func.range f := sorry\n\ntheorem forall_range_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {p : \u03b2 \u2192 Prop} : (\u2200 (y : \u03b2), y \u2208 simple_func.range f \u2192 p y) \u2194 \u2200 (x : \u03b1), p (coe_fn f x) := sorry\n\ntheorem exists_range_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {f : simple_func \u03b1 \u03b2} {p : \u03b2 \u2192 Prop} : (\u2203 (y : \u03b2), \u2203 (H : y \u2208 simple_func.range f), p y) \u2194 \u2203 (x : \u03b1), p (coe_fn f x) := sorry\n\ntheorem preimage_eq_empty_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (b : \u03b2) : \u21d1f \u207b\u00b9' singleton b = \u2205 \u2194 \u00acb \u2208 simple_func.range f :=\n  iff.trans set.preimage_singleton_eq_empty (not_congr (iff.symm mem_range))\n\ntheorem exists_forall_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Nonempty \u03b2] [directed_order \u03b2] (f : simple_func \u03b1 \u03b2) : \u2203 (C : \u03b2), \u2200 (x : \u03b1), coe_fn f x \u2264 C :=\n  Exists.imp (fun (C : \u03b2) => iff.mp forall_range_iff) (finset.exists_le (simple_func.range f))\n\n/-- Constant function as a `simple_func`. -/\ndef const (\u03b1 : Type u_1) {\u03b2 : Type u_2} [measurable_space \u03b1] (b : \u03b2) : simple_func \u03b1 \u03b2 :=\n  mk (fun (a : \u03b1) => b) sorry set.finite_range_const\n\nprotected instance inhabited {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Inhabited \u03b2] : Inhabited (simple_func \u03b1 \u03b2) :=\n  { default := const \u03b1 Inhabited.default }\n\ntheorem const_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (a : \u03b1) (b : \u03b2) : coe_fn (const \u03b1 b) a = b :=\n  rfl\n\n@[simp] theorem coe_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (b : \u03b2) : \u21d1(const \u03b1 b) = function.const \u03b1 b :=\n  rfl\n\n@[simp] theorem range_const {\u03b2 : Type u_2} (\u03b1 : Type u_1) [measurable_space \u03b1] [Nonempty \u03b1] (b : \u03b2) : simple_func.range (const \u03b1 b) = singleton b := sorry\n\ntheorem is_measurable_cut {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (r : \u03b1 \u2192 \u03b2 \u2192 Prop) (f : simple_func \u03b1 \u03b2) (h : \u2200 (b : \u03b2), is_measurable (set_of fun (a : \u03b1) => r a b)) : is_measurable (set_of fun (a : \u03b1) => r a (coe_fn f a)) := sorry\n\ntheorem is_measurable_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (s : set \u03b2) : is_measurable (\u21d1f \u207b\u00b9' s) :=\n  is_measurable_cut (fun (_x : \u03b1) (b : \u03b2) => b \u2208 s) f fun (b : \u03b2) => is_measurable.const (b \u2208 s)\n\n/-- A simple function is measurable -/\nprotected theorem measurable {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [measurable_space \u03b2] (f : simple_func \u03b1 \u03b2) : measurable \u21d1f :=\n  fun (s : set \u03b2) (_x : is_measurable s) => is_measurable_preimage f s\n\nprotected theorem ae_measurable {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [measurable_space \u03b2] {\u03bc : measure \u03b1} (f : simple_func \u03b1 \u03b2) : ae_measurable \u21d1f :=\n  measurable.ae_measurable (simple_func.measurable f)\n\nprotected theorem sum_measure_preimage_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) {\u03bc : measure \u03b1} (s : finset \u03b2) : (finset.sum s fun (y : \u03b2) => coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y)) = coe_fn \u03bc (\u21d1f \u207b\u00b9' \u2191s) :=\n  sum_measure_preimage_singleton s fun (_x : \u03b2) (_x_1 : _x \u2208 s) => is_measurable_fiber f _x\n\ntheorem sum_range_measure_preimage_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (\u03bc : measure \u03b1) : (finset.sum (simple_func.range f) fun (y : \u03b2) => coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y)) = coe_fn \u03bc set.univ := sorry\n\n/-- If-then-else as a `simple_func`. -/\ndef piecewise {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (s : set \u03b1) (hs : is_measurable s) (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : simple_func \u03b1 \u03b2 :=\n  mk (set.piecewise s \u21d1f \u21d1g) sorry sorry\n\n@[simp] theorem coe_piecewise {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {s : set \u03b1} (hs : is_measurable s) (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : \u21d1(piecewise s hs f g) = set.piecewise s \u21d1f \u21d1g :=\n  rfl\n\ntheorem piecewise_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {s : set \u03b1} (hs : is_measurable s) (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (piecewise s hs f g) a = ite (a \u2208 s) (coe_fn f a) (coe_fn g a) :=\n  rfl\n\n@[simp] theorem piecewise_compl {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {s : set \u03b1} (hs : is_measurable (s\u1d9c)) (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : piecewise (s\u1d9c) hs f g = piecewise s (is_measurable.of_compl hs) g f := sorry\n\n@[simp] theorem piecewise_univ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : piecewise set.univ is_measurable.univ f g = f := sorry\n\n@[simp] theorem piecewise_empty {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : piecewise \u2205 is_measurable.empty f g = g := sorry\n\ntheorem measurable_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [measurable_space \u03b3] (f : simple_func \u03b1 \u03b2) (g : \u03b2 \u2192 \u03b1 \u2192 \u03b3) (hg : \u2200 (b : \u03b2), measurable (g b)) : measurable fun (a : \u03b1) => g (coe_fn f a) a :=\n  fun (s : set \u03b3) (hs : is_measurable s) => is_measurable_cut (fun (a : \u03b1) (b : \u03b2) => g b a \u2208 s) f fun (b : \u03b2) => hg b hs\n\n/-- If `f : \u03b1 \u2192\u209b \u03b2` is a simple function and `g : \u03b2 \u2192 \u03b1 \u2192\u209b \u03b3` is a family of simple functions,\nthen `f.bind g` binds the first argument of `g` to `f`. In other words, `f.bind g a = g (f a) a`. -/\ndef bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : \u03b2 \u2192 simple_func \u03b1 \u03b3) : simple_func \u03b1 \u03b3 :=\n  mk (fun (a : \u03b1) => coe_fn (g (coe_fn f a)) a) sorry sorry\n\n@[simp] theorem bind_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : \u03b2 \u2192 simple_func \u03b1 \u03b3) (a : \u03b1) : coe_fn (bind f g) a = coe_fn (g (coe_fn f a)) a :=\n  rfl\n\n/-- Given a function `g : \u03b2 \u2192 \u03b3` and a simple function `f : \u03b1 \u2192\u209b \u03b2`, `f.map g` return the simple\n    function `g \u2218 f : \u03b1 \u2192\u209b \u03b3` -/\ndef map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (g : \u03b2 \u2192 \u03b3) (f : simple_func \u03b1 \u03b2) : simple_func \u03b1 \u03b3 :=\n  bind f (const \u03b1 \u2218 g)\n\ntheorem map_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (g : \u03b2 \u2192 \u03b3) (f : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (map g f) a = g (coe_fn f a) :=\n  rfl\n\ntheorem map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b1] (g : \u03b2 \u2192 \u03b3) (h : \u03b3 \u2192 \u03b4) (f : simple_func \u03b1 \u03b2) : map h (map g f) = map (h \u2218 g) f :=\n  rfl\n\n@[simp] theorem coe_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (g : \u03b2 \u2192 \u03b3) (f : simple_func \u03b1 \u03b2) : \u21d1(map g f) = g \u2218 \u21d1f :=\n  rfl\n\n@[simp] theorem range_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [DecidableEq \u03b3] (g : \u03b2 \u2192 \u03b3) (f : simple_func \u03b1 \u03b2) : simple_func.range (map g f) = finset.image g (simple_func.range f) := sorry\n\n@[simp] theorem map_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (g : \u03b2 \u2192 \u03b3) (b : \u03b2) : map g (const \u03b1 b) = const \u03b1 (g b) :=\n  rfl\n\ntheorem map_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : \u03b2 \u2192 \u03b3) (s : set \u03b3) : \u21d1(map g f) \u207b\u00b9' s = \u21d1f \u207b\u00b9' \u2191(finset.filter (fun (b : \u03b2) => g b \u2208 s) (simple_func.range f)) := sorry\n\ntheorem map_preimage_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : \u03b2 \u2192 \u03b3) (c : \u03b3) : \u21d1(map g f) \u207b\u00b9' singleton c = \u21d1f \u207b\u00b9' \u2191(finset.filter (fun (b : \u03b2) => g b = c) (simple_func.range f)) :=\n  map_preimage f g (singleton c)\n\n/-- Composition of a `simple_fun` and a measurable function is a `simple_func`. -/\ndef comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [measurable_space \u03b2] (f : simple_func \u03b2 \u03b3) (g : \u03b1 \u2192 \u03b2) (hgm : measurable g) : simple_func \u03b1 \u03b3 :=\n  mk (\u21d1f \u2218 g) sorry sorry\n\n@[simp] theorem coe_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [measurable_space \u03b2] (f : simple_func \u03b2 \u03b3) {g : \u03b1 \u2192 \u03b2} (hgm : measurable g) : \u21d1(comp f g hgm) = \u21d1f \u2218 g :=\n  rfl\n\ntheorem range_comp_subset_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [measurable_space \u03b2] (f : simple_func \u03b2 \u03b3) {g : \u03b1 \u2192 \u03b2} (hgm : measurable g) : simple_func.range (comp f g hgm) \u2286 simple_func.range f := sorry\n\n/-- If `f` is a simple function taking values in `\u03b2 \u2192 \u03b3` and `g` is another simple function\nwith the same domain and codomain `\u03b2`, then `f.seq g = f a (g a)`. -/\ndef seq {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 (\u03b2 \u2192 \u03b3)) (g : simple_func \u03b1 \u03b2) : simple_func \u03b1 \u03b3 :=\n  bind f fun (f : \u03b2 \u2192 \u03b3) => map f g\n\n@[simp] theorem seq_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 (\u03b2 \u2192 \u03b3)) (g : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (seq f g) a = coe_fn f a (coe_fn g a) :=\n  rfl\n\n/-- Combine two simple functions `f : \u03b1 \u2192\u209b \u03b2` and `g : \u03b1 \u2192\u209b \u03b2`\ninto `\u03bb a, (f a, g a)`. -/\ndef pair {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b3) : simple_func \u03b1 (\u03b2 \u00d7 \u03b3) :=\n  seq (map Prod.mk f) g\n\n@[simp] theorem pair_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b3) (a : \u03b1) : coe_fn (pair f g) a = (coe_fn f a, coe_fn g a) :=\n  rfl\n\ntheorem pair_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b3) (s : set \u03b2) (t : set \u03b3) : \u21d1(pair f g) \u207b\u00b9' set.prod s t = \u21d1f \u207b\u00b9' s \u2229 \u21d1g \u207b\u00b9' t :=\n  rfl\n\n/- A special form of `pair_preimage` -/\n\ntheorem pair_preimage_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b3) (b : \u03b2) (c : \u03b3) : \u21d1(pair f g) \u207b\u00b9' singleton (b, c) = \u21d1f \u207b\u00b9' singleton b \u2229 \u21d1g \u207b\u00b9' singleton c := sorry\n\ntheorem bind_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] (f : simple_func \u03b1 \u03b2) : bind f (const \u03b1) = f := sorry\n\nprotected instance has_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] : HasZero (simple_func \u03b1 \u03b2) :=\n  { zero := const \u03b1 0 }\n\nprotected instance has_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Add \u03b2] : Add (simple_func \u03b1 \u03b2) :=\n  { add := fun (f g : simple_func \u03b1 \u03b2) => seq (map Add.add f) g }\n\nprotected instance has_mul {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Mul \u03b2] : Mul (simple_func \u03b1 \u03b2) :=\n  { mul := fun (f g : simple_func \u03b1 \u03b2) => seq (map Mul.mul f) g }\n\nprotected instance has_sup {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [has_sup \u03b2] : has_sup (simple_func \u03b1 \u03b2) :=\n  has_sup.mk fun (f g : simple_func \u03b1 \u03b2) => seq (map has_sup.sup f) g\n\nprotected instance has_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [has_inf \u03b2] : has_inf (simple_func \u03b1 \u03b2) :=\n  has_inf.mk fun (f g : simple_func \u03b1 \u03b2) => seq (map has_inf.inf f) g\n\nprotected instance has_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasLessEq \u03b2] : HasLessEq (simple_func \u03b1 \u03b2) :=\n  { LessEq := fun (f g : simple_func \u03b1 \u03b2) => \u2200 (a : \u03b1), coe_fn f a \u2264 coe_fn g a }\n\n@[simp] theorem coe_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] : \u21d10 = 0 :=\n  rfl\n\n@[simp] theorem const_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] : const \u03b1 0 = 0 :=\n  rfl\n\n@[simp] theorem coe_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Add \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : \u21d1(f + g) = \u21d1f + \u21d1g :=\n  rfl\n\n@[simp] theorem coe_mul {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Mul \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : \u21d1(f * g) = \u21d1f * \u21d1g :=\n  rfl\n\n@[simp] theorem coe_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [preorder \u03b2] {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} : \u21d1f \u2264 \u21d1g \u2194 f \u2264 g :=\n  iff.rfl\n\n@[simp] theorem range_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Nonempty \u03b1] [HasZero \u03b2] : simple_func.range 0 = singleton 0 := sorry\n\ntheorem eq_zero_of_mem_range_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {y : \u03b2} : y \u2208 simple_func.range 0 \u2192 y = 0 :=\n  iff.mpr forall_range_iff fun (x : \u03b1) => rfl\n\ntheorem sup_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [has_sup \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (f \u2294 g) a = coe_fn f a \u2294 coe_fn g a :=\n  rfl\n\ntheorem mul_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Mul \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (f * g) a = coe_fn f a * coe_fn g a :=\n  rfl\n\ntheorem add_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Add \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (f + g) a = coe_fn f a + coe_fn g a :=\n  rfl\n\ntheorem add_eq_map\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Add \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : f + g = map (fun (p : \u03b2 \u00d7 \u03b2) => prod.fst p + prod.snd p) (pair f g) :=\n  rfl\n\ntheorem mul_eq_map\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Mul \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : f * g = map (fun (p : \u03b2 \u00d7 \u03b2) => prod.fst p * prod.snd p) (pair f g) :=\n  rfl\n\ntheorem sup_eq_map\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [has_sup \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : f \u2294 g = map (fun (p : \u03b2 \u00d7 \u03b2) => prod.fst p \u2294 prod.snd p) (pair f g) :=\n  rfl\n\ntheorem const_mul_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Mul \u03b2] (f : simple_func \u03b1 \u03b2) (b : \u03b2) : const \u03b1 b * f = map (fun (a : \u03b2) => b * a) f :=\n  rfl\n\ntheorem map_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [Add \u03b2] [Add \u03b3] {g : \u03b2 \u2192 \u03b3} (hg : \u2200 (x y : \u03b2), g (x + y) = g x + g y) (f\u2081 : simple_func \u03b1 \u03b2) (f\u2082 : simple_func \u03b1 \u03b2) : map g (f\u2081 + f\u2082) = map g f\u2081 + map g f\u2082 :=\n  ext fun (x : \u03b1) => hg (coe_fn f\u2081 x) (coe_fn f\u2082 x)\n\nprotected instance add_monoid {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [add_monoid \u03b2] : add_monoid (simple_func \u03b1 \u03b2) :=\n  function.injective.add_monoid (fun (f : simple_func \u03b1 \u03b2) => (fun (this : \u03b1 \u2192 \u03b2) => this) \u21d1f) coe_injective sorry sorry\n\nprotected instance add_comm_monoid {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [add_comm_monoid \u03b2] : add_comm_monoid (simple_func \u03b1 \u03b2) :=\n  function.injective.add_comm_monoid (fun (f : simple_func \u03b1 \u03b2) => (fun (this : \u03b1 \u2192 \u03b2) => this) \u21d1f) coe_injective sorry\n    sorry\n\nprotected instance has_neg {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Neg \u03b2] : Neg (simple_func \u03b1 \u03b2) :=\n  { neg := fun (f : simple_func \u03b1 \u03b2) => map Neg.neg f }\n\n@[simp] theorem coe_neg {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Neg \u03b2] (f : simple_func \u03b1 \u03b2) : \u21d1(-f) = -\u21d1f :=\n  rfl\n\nprotected instance has_sub {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Sub \u03b2] : Sub (simple_func \u03b1 \u03b2) :=\n  { sub := fun (f g : simple_func \u03b1 \u03b2) => seq (map Sub.sub f) g }\n\n@[simp] theorem coe_sub {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Sub \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) : \u21d1(f - g) = \u21d1f - \u21d1g :=\n  rfl\n\ntheorem sub_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [Sub \u03b2] (f : simple_func \u03b1 \u03b2) (g : simple_func \u03b1 \u03b2) (x : \u03b1) : coe_fn (f - g) x = coe_fn f x - coe_fn g x :=\n  rfl\n\nprotected instance add_group {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [add_group \u03b2] : add_group (simple_func \u03b1 \u03b2) :=\n  function.injective.add_group_sub (fun (f : simple_func \u03b1 \u03b2) => (fun (this : \u03b1 \u2192 \u03b2) => this) \u21d1f) coe_injective sorry\n    sorry sorry sorry\n\nprotected instance add_comm_group {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [add_comm_group \u03b2] : add_comm_group (simple_func \u03b1 \u03b2) :=\n  function.injective.add_comm_group_sub (fun (f : simple_func \u03b1 \u03b2) => (fun (this : \u03b1 \u2192 \u03b2) => this) \u21d1f) coe_injective sorry\n    sorry sorry sorry\n\nprotected instance has_scalar {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {K : Type u_5} [has_scalar K \u03b2] : has_scalar K (simple_func \u03b1 \u03b2) :=\n  has_scalar.mk fun (k : K) (f : simple_func \u03b1 \u03b2) => map (has_scalar.smul k) f\n\n@[simp] theorem coe_smul {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {K : Type u_5} [has_scalar K \u03b2] (c : K) (f : simple_func \u03b1 \u03b2) : \u21d1(c \u2022 f) = c \u2022 \u21d1f :=\n  rfl\n\ntheorem smul_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {K : Type u_5} [has_scalar K \u03b2] (k : K) (f : simple_func \u03b1 \u03b2) (a : \u03b1) : coe_fn (k \u2022 f) a = k \u2022 coe_fn f a :=\n  rfl\n\nprotected instance semimodule {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {K : Type u_5} [semiring K] [add_comm_monoid \u03b2] [semimodule K \u03b2] : semimodule K (simple_func \u03b1 \u03b2) :=\n  function.injective.semimodule K\n    (add_monoid_hom.mk (fun (f : simple_func \u03b1 \u03b2) => (fun (this : \u03b1 \u2192 \u03b2) => this) \u21d1f) sorry sorry) coe_injective sorry\n\ntheorem smul_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {K : Type u_5} [has_scalar K \u03b2] (k : K) (f : simple_func \u03b1 \u03b2) : k \u2022 f = map (has_scalar.smul k) f :=\n  rfl\n\nprotected instance preorder {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [preorder \u03b2] : preorder (simple_func \u03b1 \u03b2) :=\n  preorder.mk LessEq (fun (a b : simple_func \u03b1 \u03b2) => a \u2264 b \u2227 \u00acb \u2264 a) sorry sorry\n\nprotected instance partial_order {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [partial_order \u03b2] : partial_order (simple_func \u03b1 \u03b2) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\nprotected instance order_bot {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [order_bot \u03b2] : order_bot (simple_func \u03b1 \u03b2) :=\n  order_bot.mk (const \u03b1 \u22a5) partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance order_top {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [order_top \u03b2] : order_top (simple_func \u03b1 \u03b2) :=\n  order_top.mk (const \u03b1 \u22a4) partial_order.le partial_order.lt sorry sorry sorry sorry\n\nprotected instance semilattice_inf {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_inf \u03b2] : semilattice_inf (simple_func \u03b1 \u03b2) :=\n  semilattice_inf.mk has_inf.inf partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry\n\nprotected instance semilattice_sup {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_sup \u03b2] : semilattice_sup (simple_func \u03b1 \u03b2) :=\n  semilattice_sup.mk has_sup.sup partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry\n\nprotected instance semilattice_sup_bot {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] : semilattice_sup_bot (simple_func \u03b1 \u03b2) :=\n  semilattice_sup_bot.mk order_bot.bot semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry semilattice_sup.sup\n    sorry sorry sorry\n\nprotected instance lattice {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [lattice \u03b2] : lattice (simple_func \u03b1 \u03b2) :=\n  lattice.mk semilattice_sup.sup semilattice_sup.le semilattice_sup.lt sorry sorry sorry sorry sorry sorry\n    semilattice_inf.inf sorry sorry sorry\n\nprotected instance bounded_lattice {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [bounded_lattice \u03b2] : bounded_lattice (simple_func \u03b1 \u03b2) :=\n  bounded_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry lattice.inf sorry sorry sorry\n    order_top.top sorry order_bot.bot sorry\n\ntheorem finset_sup_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] {f : \u03b3 \u2192 simple_func \u03b1 \u03b2} (s : finset \u03b3) (a : \u03b1) : coe_fn (finset.sup s f) a = finset.sup s fun (c : \u03b3) => coe_fn (f c) a := sorry\n\n/-- Restrict a simple function `f : \u03b1 \u2192\u209b \u03b2` to a set `s`. If `s` is measurable,\nthen `f.restrict s a = if a \u2208 s then f a else 0`, otherwise `f.restrict s = const \u03b1 0`. -/\ndef restrict {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) (s : set \u03b1) : simple_func \u03b1 \u03b2 :=\n  dite (is_measurable s) (fun (hs : is_measurable s) => piecewise s hs f 0) fun (hs : \u00acis_measurable s) => 0\n\ntheorem restrict_of_not_measurable {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {f : simple_func \u03b1 \u03b2} {s : set \u03b1} (hs : \u00acis_measurable s) : restrict f s = 0 :=\n  dif_neg hs\n\n@[simp] theorem coe_restrict {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) {s : set \u03b1} (hs : is_measurable s) : \u21d1(restrict f s) = set.indicator s \u21d1f := sorry\n\n@[simp] theorem restrict_univ {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) : restrict f set.univ = f := sorry\n\n@[simp] theorem restrict_empty {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) : restrict f \u2205 = 0 := sorry\n\ntheorem map_restrict_of_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] {g : \u03b2 \u2192 \u03b3} (hg : g 0 = 0) (f : simple_func \u03b1 \u03b2) (s : set \u03b1) : map g (restrict f s) = restrict (map g f) s := sorry\n\ntheorem map_coe_ennreal_restrict {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 nnreal) (s : set \u03b1) : map coe (restrict f s) = restrict (map coe f) s :=\n  map_restrict_of_zero ennreal.coe_zero f s\n\ntheorem map_coe_nnreal_restrict {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 nnreal) (s : set \u03b1) : map coe (restrict f s) = restrict (map coe f) s :=\n  map_restrict_of_zero nnreal.coe_zero f s\n\ntheorem restrict_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) {s : set \u03b1} (hs : is_measurable s) (a : \u03b1) : coe_fn (restrict f s) a = ite (a \u2208 s) (coe_fn f a) 0 := sorry\n\ntheorem restrict_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) {s : set \u03b1} (hs : is_measurable s) {t : set \u03b2} (ht : \u00ac0 \u2208 t) : \u21d1(restrict f s) \u207b\u00b9' t = s \u2229 \u21d1f \u207b\u00b9' t := sorry\n\ntheorem restrict_preimage_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) {s : set \u03b1} (hs : is_measurable s) {r : \u03b2} (hr : r \u2260 0) : \u21d1(restrict f s) \u207b\u00b9' singleton r = s \u2229 \u21d1f \u207b\u00b9' singleton r :=\n  restrict_preimage f hs (ne.symm hr)\n\ntheorem mem_restrict_range {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {r : \u03b2} {s : set \u03b1} {f : simple_func \u03b1 \u03b2} (hs : is_measurable s) : r \u2208 simple_func.range (restrict f s) \u2194 r = 0 \u2227 s \u2260 set.univ \u2228 r \u2208 \u21d1f '' s := sorry\n\ntheorem mem_image_of_mem_range_restrict {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {r : \u03b2} {s : set \u03b1} {f : simple_func \u03b1 \u03b2} (hr : r \u2208 simple_func.range (restrict f s)) (h0 : r \u2260 0) : r \u2208 \u21d1f '' s := sorry\n\ntheorem restrict_mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] [preorder \u03b2] (s : set \u03b1) {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} (H : f \u2264 g) : restrict f s \u2264 restrict g s := sorry\n\n/-- Fix a sequence `i : \u2115 \u2192 \u03b2`. Given a function `\u03b1 \u2192 \u03b2`, its `n`-th approximation\nby simple functions is defined so that in case `\u03b2 = ennreal` it sends each `a` to the supremum\nof the set `{i k | k \u2264 n \u2227 i k \u2264 f a}`, see `approx_apply` and `supr_approx_apply` for details. -/\ndef approx {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] [HasZero \u03b2] (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) (n : \u2115) : simple_func \u03b1 \u03b2 :=\n  finset.sup (finset.range n) fun (k : \u2115) => restrict (const \u03b1 (i k)) (set_of fun (a : \u03b1) => i k \u2264 f a)\n\ntheorem approx_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] [HasZero \u03b2] [topological_space \u03b2] [order_closed_topology \u03b2] [measurable_space \u03b2] [opens_measurable_space \u03b2] {i : \u2115 \u2192 \u03b2} {f : \u03b1 \u2192 \u03b2} {n : \u2115} (a : \u03b1) (hf : measurable f) : coe_fn (approx i f n) a = finset.sup (finset.range n) fun (k : \u2115) => ite (i k \u2264 f a) (i k) 0 := sorry\n\ntheorem monotone_approx {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] [HasZero \u03b2] (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) : monotone (approx i f) :=\n  fun (n m : \u2115) (h : n \u2264 m) => finset.sup_mono (iff.mpr finset.range_subset h)\n\ntheorem approx_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [semilattice_sup_bot \u03b2] [HasZero \u03b2] [topological_space \u03b2] [order_closed_topology \u03b2] [measurable_space \u03b2] [opens_measurable_space \u03b2] [measurable_space \u03b3] {i : \u2115 \u2192 \u03b2} {f : \u03b3 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} {n : \u2115} (a : \u03b1) (hf : measurable f) (hg : measurable g) : coe_fn (approx i (f \u2218 g) n) a = coe_fn (approx i f n) (g a) := sorry\n\ntheorem supr_approx_apply {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [topological_space \u03b2] [complete_lattice \u03b2] [order_closed_topology \u03b2] [HasZero \u03b2] [measurable_space \u03b2] [opens_measurable_space \u03b2] (i : \u2115 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (hf : measurable f) (h_zero : 0 = \u22a5) : (supr fun (n : \u2115) => coe_fn (approx i f n) a) = supr fun (k : \u2115) => supr fun (h : i k \u2264 f a) => i k := sorry\n\n/-- A sequence of `ennreal`s such that its range is the set of non-negative rational numbers. -/\ndef ennreal_rat_embed (n : \u2115) : ennreal :=\n  ennreal.of_real \u2191(option.get_or_else (encodable.decode \u211a n) 0)\n\ntheorem ennreal_rat_embed_encode (q : \u211a) : ennreal_rat_embed (encodable.encode q) = \u2191(nnreal.of_real \u2191q) := sorry\n\n/-- Approximate a function `\u03b1 \u2192 ennreal` by a sequence of simple functions. -/\ndef eapprox {\u03b1 : Type u_1} [measurable_space \u03b1] : (\u03b1 \u2192 ennreal) \u2192 \u2115 \u2192 simple_func \u03b1 ennreal :=\n  approx ennreal_rat_embed\n\ntheorem monotone_eapprox {\u03b1 : Type u_1} [measurable_space \u03b1] (f : \u03b1 \u2192 ennreal) : monotone (eapprox f) :=\n  monotone_approx ennreal_rat_embed f\n\ntheorem supr_eapprox_apply {\u03b1 : Type u_1} [measurable_space \u03b1] (f : \u03b1 \u2192 ennreal) (hf : measurable f) (a : \u03b1) : (supr fun (n : \u2115) => coe_fn (eapprox f n) a) = f a := sorry\n\ntheorem eapprox_comp {\u03b1 : Type u_1} {\u03b3 : Type u_3} [measurable_space \u03b1] [measurable_space \u03b3] {f : \u03b3 \u2192 ennreal} {g : \u03b1 \u2192 \u03b3} {n : \u2115} (hf : measurable f) (hg : measurable g) : \u21d1(eapprox (f \u2218 g) n) = \u21d1(eapprox f n) \u2218 g :=\n  funext fun (a : \u03b1) => approx_comp a hf hg\n\n/-- Integral of a simple function whose codomain is `ennreal`. -/\ndef lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 ennreal) (\u03bc : measure \u03b1) : ennreal :=\n  finset.sum (simple_func.range f) fun (x : ennreal) => x * coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton x)\n\ntheorem lintegral_eq_of_subset {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) {s : finset ennreal} (hs : \u2200 (x : \u03b1), coe_fn f x \u2260 0 \u2192 coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton (coe_fn f x)) \u2260 0 \u2192 coe_fn f x \u2208 s) : lintegral f \u03bc = finset.sum s fun (x : ennreal) => x * coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton x) := sorry\n\n/-- Calculate the integral of `(g \u2218 f)`, where `g : \u03b2 \u2192 ennreal` and `f : \u03b1 \u2192\u209b \u03b2`.  -/\ntheorem map_lintegral {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} (g : \u03b2 \u2192 ennreal) (f : simple_func \u03b1 \u03b2) : lintegral (map g f) \u03bc = finset.sum (simple_func.range f) fun (x : \u03b2) => g x * coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton x) := sorry\n\ntheorem add_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) (g : simple_func \u03b1 ennreal) : lintegral (f + g) \u03bc = lintegral f \u03bc + lintegral g \u03bc := sorry\n\ntheorem const_mul_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) (x : ennreal) : lintegral (const \u03b1 x * f) \u03bc = x * lintegral f \u03bc := sorry\n\n/-- Integral of a simple function `\u03b1 \u2192\u209b ennreal` as a bilinear map. -/\ndef lintegral\u2097 {\u03b1 : Type u_1} [measurable_space \u03b1] : linear_map ennreal (simple_func \u03b1 ennreal) (linear_map ennreal (measure \u03b1) ennreal) :=\n  linear_map.mk (fun (f : simple_func \u03b1 ennreal) => linear_map.mk (lintegral f) sorry sorry) sorry sorry\n\n@[simp] theorem zero_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} : lintegral 0 \u03bc = 0 :=\n  iff.mp linear_map.ext_iff (linear_map.map_zero lintegral\u2097) \u03bc\n\ntheorem lintegral_add {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03bd : measure \u03b1} (f : simple_func \u03b1 ennreal) : lintegral f (\u03bc + \u03bd) = lintegral f \u03bc + lintegral f \u03bd :=\n  linear_map.map_add (coe_fn lintegral\u2097 f) \u03bc \u03bd\n\ntheorem lintegral_smul {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) (c : ennreal) : lintegral f (c \u2022 \u03bc) = c \u2022 lintegral f \u03bc :=\n  linear_map.map_smul (coe_fn lintegral\u2097 f) c \u03bc\n\n@[simp] theorem lintegral_zero {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 ennreal) : lintegral f 0 = 0 :=\n  linear_map.map_zero (coe_fn lintegral\u2097 f)\n\ntheorem lintegral_sum {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03b9 : Type u_2} (f : simple_func \u03b1 ennreal) (\u03bc : \u03b9 \u2192 measure \u03b1) : lintegral f (measure.sum \u03bc) = tsum fun (i : \u03b9) => lintegral f (\u03bc i) := sorry\n\ntheorem restrict_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) {s : set \u03b1} (hs : is_measurable s) : lintegral (restrict f s) \u03bc = finset.sum (simple_func.range f) fun (r : ennreal) => r * coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton r \u2229 s) := sorry\n\ntheorem lintegral_restrict {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 ennreal) (s : set \u03b1) (\u03bc : measure \u03b1) : lintegral f (measure.restrict \u03bc s) =\n  finset.sum (simple_func.range f) fun (y : ennreal) => y * coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y \u2229 s) := sorry\n\ntheorem restrict_lintegral_eq_lintegral_restrict {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) {s : set \u03b1} (hs : is_measurable s) : lintegral (restrict f s) \u03bc = lintegral f (measure.restrict \u03bc s) := sorry\n\ntheorem const_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (c : ennreal) : lintegral (const \u03b1 c) \u03bc = c * coe_fn \u03bc set.univ := sorry\n\ntheorem const_lintegral_restrict {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (c : ennreal) (s : set \u03b1) : lintegral (const \u03b1 c) (measure.restrict \u03bc s) = c * coe_fn \u03bc s := sorry\n\ntheorem restrict_const_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (c : ennreal) {s : set \u03b1} (hs : is_measurable s) : lintegral (restrict (const \u03b1 c) s) \u03bc = c * coe_fn \u03bc s := sorry\n\ntheorem le_sup_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : simple_func \u03b1 ennreal) (g : simple_func \u03b1 ennreal) : lintegral f \u03bc \u2294 lintegral g \u03bc \u2264 lintegral (f \u2294 g) \u03bc := sorry\n\n/-- `simple_func.lintegral` is monotone both in function and in measure. -/\ntheorem lintegral_mono {\u03b1 : Type u_1} [measurable_space \u03b1] {f : simple_func \u03b1 ennreal} {g : simple_func \u03b1 ennreal} (hfg : f \u2264 g) {\u03bc : measure \u03b1} {\u03bd : measure \u03b1} (h\u03bc\u03bd : \u03bc \u2264 \u03bd) : lintegral f \u03bc \u2264 lintegral g \u03bd := sorry\n\n/-- `simple_func.lintegral` depends only on the measures of `f \u207b\u00b9' {y}`. -/\ntheorem lintegral_eq_of_measure_preimage {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : simple_func \u03b1 ennreal} {g : simple_func \u03b2 ennreal} {\u03bd : measure \u03b2} (H : \u2200 (y : ennreal), coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y) = coe_fn \u03bd (\u21d1g \u207b\u00b9' singleton y)) : lintegral f \u03bc = lintegral g \u03bd := sorry\n\n/-- If two simple functions are equal a.e., then their `lintegral`s are equal. -/\ntheorem lintegral_congr {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : simple_func \u03b1 ennreal} {g : simple_func \u03b1 ennreal} (h : filter.eventually_eq (measure.ae \u03bc) \u21d1f \u21d1g) : lintegral f \u03bc = lintegral g \u03bc := sorry\n\ntheorem lintegral_map {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b2 : Type u_2} [measurable_space \u03b2] {\u03bc' : measure \u03b2} (f : simple_func \u03b1 ennreal) (g : simple_func \u03b2 ennreal) (m : \u03b1 \u2192 \u03b2) (eq : \u2200 (a : \u03b1), coe_fn f a = coe_fn g (m a)) (h : \u2200 (s : set \u03b2), is_measurable s \u2192 coe_fn \u03bc' s = coe_fn \u03bc (m \u207b\u00b9' s)) : lintegral f \u03bc = lintegral g \u03bc' := sorry\n\ntheorem support_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) : function.support \u21d1f =\n  set.Union\n    fun (y : \u03b2) =>\n      set.Union fun (H : y \u2208 finset.filter (fun (y : \u03b2) => y \u2260 0) (simple_func.range f)) => \u21d1f \u207b\u00b9' singleton y := sorry\n\n/-- A `simple_func` has finite measure support if it is equal to `0` outside of a set of finite\nmeasure. -/\nprotected def fin_meas_supp {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] (f : simple_func \u03b1 \u03b2) (\u03bc : measure \u03b1) :=\n  filter.eventually_eq (measure.cofinite \u03bc) (\u21d1f) 0\n\ntheorem fin_meas_supp_iff_support {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {f : simple_func \u03b1 \u03b2} {\u03bc : measure \u03b1} : simple_func.fin_meas_supp f \u03bc \u2194 coe_fn \u03bc (function.support \u21d1f) < \u22a4 :=\n  iff.rfl\n\ntheorem fin_meas_supp_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {f : simple_func \u03b1 \u03b2} {\u03bc : measure \u03b1} : simple_func.fin_meas_supp f \u03bc \u2194 \u2200 (y : \u03b2), y \u2260 0 \u2192 coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y) < \u22a4 := sorry\n\nnamespace fin_meas_supp\n\n\ntheorem meas_preimage_singleton_ne_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] [HasZero \u03b2] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} (h : simple_func.fin_meas_supp f \u03bc) {y : \u03b2} (hy : y \u2260 0) : coe_fn \u03bc (\u21d1f \u207b\u00b9' singleton y) < \u22a4 :=\n  iff.mp fin_meas_supp_iff h y hy\n\nprotected theorem map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} {g : \u03b2 \u2192 \u03b3} (hf : simple_func.fin_meas_supp f \u03bc) (hg : g 0 = 0) : simple_func.fin_meas_supp (map g f) \u03bc :=\n  flip lt_of_le_of_lt hf (measure_mono (function.support_comp_subset hg \u21d1f))\n\ntheorem of_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} {g : \u03b2 \u2192 \u03b3} (h : simple_func.fin_meas_supp (map g f) \u03bc) (hg : \u2200 (b : \u03b2), g b = 0 \u2192 b = 0) : simple_func.fin_meas_supp f \u03bc :=\n  flip lt_of_le_of_lt h (measure_mono (function.support_subset_comp hg fun (x : \u03b1) => coe_fn f x))\n\ntheorem map_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} {g : \u03b2 \u2192 \u03b3} (hg : \u2200 {b : \u03b2}, g b = 0 \u2194 b = 0) : simple_func.fin_meas_supp (map g f) \u03bc \u2194 simple_func.fin_meas_supp f \u03bc :=\n  { mp := fun (h : simple_func.fin_meas_supp (map g f) \u03bc) => of_map h fun (b : \u03b2) => iff.mp hg,\n    mpr := fun (h : simple_func.fin_meas_supp f \u03bc) => fin_meas_supp.map h (iff.mpr hg rfl) }\n\nprotected theorem pair {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b3} (hf : simple_func.fin_meas_supp f \u03bc) (hg : simple_func.fin_meas_supp g \u03bc) : simple_func.fin_meas_supp (pair f g) \u03bc := sorry\n\nprotected theorem map\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} [measurable_space \u03b1] [HasZero \u03b2] [HasZero \u03b3] [HasZero \u03b4] {\u03bc : measure \u03b1} {f : simple_func \u03b1 \u03b2} (hf : simple_func.fin_meas_supp f \u03bc) {g : simple_func \u03b1 \u03b3} (hg : simple_func.fin_meas_supp g \u03bc) {op : \u03b2 \u2192 \u03b3 \u2192 \u03b4} (H : op 0 0 = 0) : simple_func.fin_meas_supp (map (function.uncurry op) (pair f g)) \u03bc :=\n  fin_meas_supp.map (fin_meas_supp.pair hf hg) H\n\nprotected theorem add {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b2 : Type u_2} [add_monoid \u03b2] {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} (hf : simple_func.fin_meas_supp f \u03bc) (hg : simple_func.fin_meas_supp g \u03bc) : simple_func.fin_meas_supp (f + g) \u03bc :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (simple_func.fin_meas_supp (f + g) \u03bc)) (add_eq_map\u2082 f g)))\n    (fin_meas_supp.map\u2082 hf hg (zero_add 0))\n\nprotected theorem mul {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b2 : Type u_2} [monoid_with_zero \u03b2] {f : simple_func \u03b1 \u03b2} {g : simple_func \u03b1 \u03b2} (hf : simple_func.fin_meas_supp f \u03bc) (hg : simple_func.fin_meas_supp g \u03bc) : simple_func.fin_meas_supp (f * g) \u03bc :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (simple_func.fin_meas_supp (f * g) \u03bc)) (mul_eq_map\u2082 f g)))\n    (fin_meas_supp.map\u2082 hf hg (zero_mul 0))\n\ntheorem lintegral_lt_top {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : simple_func \u03b1 ennreal} (hm : simple_func.fin_meas_supp f \u03bc) (hf : filter.eventually (fun (a : \u03b1) => coe_fn f a < \u22a4) (measure.ae \u03bc)) : lintegral f \u03bc < \u22a4 := sorry\n\ntheorem of_lintegral_lt_top {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : simple_func \u03b1 ennreal} (h : lintegral f \u03bc < \u22a4) : simple_func.fin_meas_supp f \u03bc := sorry\n\ntheorem iff_lintegral_lt_top {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : simple_func \u03b1 ennreal} (hf : filter.eventually (fun (a : \u03b1) => coe_fn f a < \u22a4) (measure.ae \u03bc)) : simple_func.fin_meas_supp f \u03bc \u2194 lintegral f \u03bc < \u22a4 :=\n  { mp := fun (h : simple_func.fin_meas_supp f \u03bc) => lintegral_lt_top h hf,\n    mpr := fun (h : lintegral f \u03bc < \u22a4) => of_lintegral_lt_top h }\n\nend fin_meas_supp\n\n\n/-- To prove something for an arbitrary simple function, it suffices to show\nthat the property holds for (multiples of) characteristic functions and is closed under\naddition (of functions with disjoint support).\n\nIt is possible to make the hypotheses in `h_sum` a bit stronger, and such conditions can be added\nonce we need them (for example it is only necessary to consider the case where `g` is a multiple\nof a characteristic function, and that this multiple doesn't appear in the image of `f`) -/\nprotected theorem induction {\u03b1 : Type u_1} {\u03b3 : Type u_2} [measurable_space \u03b1] [add_monoid \u03b3] {P : simple_func \u03b1 \u03b3 \u2192 Prop} (h_ind : \u2200 (c : \u03b3) {s : set \u03b1} (hs : is_measurable s), P (piecewise s hs (const \u03b1 c) (const \u03b1 0))) (h_sum : \u2200 {f g : simple_func \u03b1 \u03b3}, set.univ \u2286 \u21d1f \u207b\u00b9' singleton 0 \u222a \u21d1g \u207b\u00b9' singleton 0 \u2192 P f \u2192 P g \u2192 P (f + g)) (f : simple_func \u03b1 \u03b3) : P f := sorry\n\nend simple_func\n\n\n/-- The lower Lebesgue integral of a function `f` with respect to a measure `\u03bc`. -/\ndef lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] (\u03bc : measure \u03b1) (f : \u03b1 \u2192 ennreal) : ennreal :=\n  supr fun (g : simple_func \u03b1 ennreal) => supr fun (hf : \u21d1g \u2264 f) => simple_func.lintegral g \u03bc\n\n/-! In the notation for integrals, an expression like `\u222b\u207b x, g \u2225x\u2225 \u2202\u03bc` will not be parsed correctly,\n  and needs parentheses. We do not set the binding power of `r` to `0`, because then\n  `\u222b\u207b x, f x = 0` will be parsed incorrectly. -/\n\ntheorem simple_func.lintegral_eq_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] (f : simple_func \u03b1 ennreal) (\u03bc : measure \u03b1) : (lintegral \u03bc fun (a : \u03b1) => coe_fn f a) = simple_func.lintegral f \u03bc := sorry\n\ntheorem lintegral_mono' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03bd : measure \u03b1} (h\u03bc\u03bd : \u03bc \u2264 \u03bd) {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hfg : f \u2264 g) : (lintegral \u03bc fun (a : \u03b1) => f a) \u2264 lintegral \u03bd fun (a : \u03b1) => g a := sorry\n\ntheorem lintegral_mono {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hfg : f \u2264 g) : (lintegral \u03bc fun (a : \u03b1) => f a) \u2264 lintegral \u03bc fun (a : \u03b1) => g a :=\n  lintegral_mono' (le_refl \u03bc) hfg\n\ntheorem lintegral_mono_nnreal {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 nnreal} {g : \u03b1 \u2192 nnreal} (h : f \u2264 g) : (lintegral \u03bc fun (a : \u03b1) => \u2191(f a)) \u2264 lintegral \u03bc fun (a : \u03b1) => \u2191(g a) := sorry\n\ntheorem monotone_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] (\u03bc : measure \u03b1) : monotone (lintegral \u03bc) :=\n  lintegral_mono\n\n@[simp] theorem lintegral_const {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (c : ennreal) : (lintegral \u03bc fun (a : \u03b1) => c) = c * coe_fn \u03bc set.univ := sorry\n\n@[simp] theorem lintegral_one {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} : (lintegral \u03bc fun (a : \u03b1) => 1) = coe_fn \u03bc set.univ :=\n  eq.mpr (id (Eq._oldrec (Eq.refl ((lintegral \u03bc fun (a : \u03b1) => 1) = coe_fn \u03bc set.univ)) (lintegral_const 1)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (1 * coe_fn \u03bc set.univ = coe_fn \u03bc set.univ)) (one_mul (coe_fn \u03bc set.univ))))\n      (Eq.refl (coe_fn \u03bc set.univ)))\n\ntheorem set_lintegral_const {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (s : set \u03b1) (c : ennreal) : (lintegral (measure.restrict \u03bc s) fun (a : \u03b1) => c) = c * coe_fn \u03bc s := sorry\n\ntheorem set_lintegral_one {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (s : set \u03b1) : (lintegral (measure.restrict \u03bc s) fun (a : \u03b1) => 1) = coe_fn \u03bc s := sorry\n\n/-- `\u222b\u207b a in s, f a \u2202\u03bc` is defined as the supremum of integrals of simple functions\n`\u03c6 : \u03b1 \u2192\u209b ennreal` such that `\u03c6 \u2264 f`. This lemma says that it suffices to take\nfunctions `\u03c6 : \u03b1 \u2192\u209b \u211d\u22650`. -/\ntheorem lintegral_eq_nnreal {\u03b1 : Type u_1} [measurable_space \u03b1] (f : \u03b1 \u2192 ennreal) (\u03bc : measure \u03b1) : (lintegral \u03bc fun (a : \u03b1) => f a) =\n  supr\n    fun (\u03c6 : simple_func \u03b1 nnreal) =>\n      supr fun (hf : \u2200 (x : \u03b1), \u2191(coe_fn \u03c6 x) \u2264 f x) => simple_func.lintegral (simple_func.map coe \u03c6) \u03bc := sorry\n\ntheorem exists_simple_func_forall_lintegral_sub_lt_of_pos {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (h : (lintegral \u03bc fun (x : \u03b1) => f x) < \u22a4) {\u03b5 : ennreal} (h\u03b5 : 0 < \u03b5) : \u2203 (\u03c6 : simple_func \u03b1 nnreal),\n  (\u2200 (x : \u03b1), \u2191(coe_fn \u03c6 x) \u2264 f x) \u2227\n    \u2200 (\u03c8 : simple_func \u03b1 nnreal),\n      (\u2200 (x : \u03b1), \u2191(coe_fn \u03c8 x) \u2264 f x) \u2192 simple_func.lintegral (simple_func.map coe (\u03c8 - \u03c6)) \u03bc < \u03b5 := sorry\n\ntheorem supr_lintegral_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Sort u_2} (f : \u03b9 \u2192 \u03b1 \u2192 ennreal) : (supr fun (i : \u03b9) => lintegral \u03bc fun (a : \u03b1) => f i a) \u2264 lintegral \u03bc fun (a : \u03b1) => supr fun (i : \u03b9) => f i a := sorry\n\ntheorem supr2_lintegral_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Sort u_2} {\u03b9' : \u03b9 \u2192 Sort u_3} (f : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 ennreal) : (supr fun (i : \u03b9) => supr fun (h : \u03b9' i) => lintegral \u03bc fun (a : \u03b1) => f i h a) \u2264\n  lintegral \u03bc fun (a : \u03b1) => supr fun (i : \u03b9) => supr fun (h : \u03b9' i) => f i h a := sorry\n\ntheorem le_infi_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Sort u_2} (f : \u03b9 \u2192 \u03b1 \u2192 ennreal) : (lintegral \u03bc fun (a : \u03b1) => infi fun (i : \u03b9) => f i a) \u2264 infi fun (i : \u03b9) => lintegral \u03bc fun (a : \u03b1) => f i a := sorry\n\ntheorem le_infi2_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Sort u_2} {\u03b9' : \u03b9 \u2192 Sort u_3} (f : (i : \u03b9) \u2192 \u03b9' i \u2192 \u03b1 \u2192 ennreal) : (lintegral \u03bc fun (a : \u03b1) => infi fun (i : \u03b9) => infi fun (h : \u03b9' i) => f i h a) \u2264\n  infi fun (i : \u03b9) => infi fun (h : \u03b9' i) => lintegral \u03bc fun (a : \u03b1) => f i h a := sorry\n\ntheorem lintegral_mono_ae {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (h : filter.eventually (fun (a : \u03b1) => f a \u2264 g a) (measure.ae \u03bc)) : (lintegral \u03bc fun (a : \u03b1) => f a) \u2264 lintegral \u03bc fun (a : \u03b1) => g a := sorry\n\ntheorem lintegral_congr_ae {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (h : filter.eventually_eq (measure.ae \u03bc) f g) : (lintegral \u03bc fun (a : \u03b1) => f a) = lintegral \u03bc fun (a : \u03b1) => g a :=\n  le_antisymm (lintegral_mono_ae (filter.eventually_eq.le h))\n    (lintegral_mono_ae (filter.eventually_eq.le (filter.eventually_eq.symm h)))\n\ntheorem lintegral_congr {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (h : \u2200 (a : \u03b1), f a = g a) : (lintegral \u03bc fun (a : \u03b1) => f a) = lintegral \u03bc fun (a : \u03b1) => g a := sorry\n\ntheorem set_lintegral_congr {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {s : set \u03b1} {t : set \u03b1} (h : filter.eventually_eq (measure.ae \u03bc) s t) : (lintegral (measure.restrict \u03bc s) fun (x : \u03b1) => f x) = lintegral (measure.restrict \u03bc t) fun (x : \u03b1) => f x := sorry\n\n/-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence.\n\nSee `lintegral_supr_directed` for a more general form. -/\ntheorem lintegral_supr {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (n : \u2115), measurable (f n)) (h_mono : monotone f) : (lintegral \u03bc fun (a : \u03b1) => supr fun (n : \u2115) => f n a) = supr fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a := sorry\n\n/-- Monotone convergence theorem -- sometimes called Beppo-Levi convergence. Version with\nae_measurable functions. -/\ntheorem lintegral_supr' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (n : \u2115), ae_measurable (f n)) (h_mono : filter.eventually (fun (x : \u03b1) => monotone fun (n : \u2115) => f n x) (measure.ae \u03bc)) : (lintegral \u03bc fun (a : \u03b1) => supr fun (n : \u2115) => f n a) = supr fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a := sorry\n\ntheorem lintegral_eq_supr_eapprox_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral \u03bc fun (a : \u03b1) => f a) = supr fun (n : \u2115) => simple_func.lintegral (simple_func.eapprox f n) \u03bc := sorry\n\n/-- If `f` has finite integral, then `\u222b\u207b x in s, f x \u2202\u03bc` is absolutely continuous in `s`: it tends\nto zero as `\u03bc s` tends to zero. This lemma states states this fact in terms of `\u03b5` and `\u03b4`. -/\ntheorem exists_pos_set_lintegral_lt_of_measure_lt {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (h : (lintegral \u03bc fun (x : \u03b1) => f x) < \u22a4) {\u03b5 : ennreal} (h\u03b5 : 0 < \u03b5) : \u2203 (\u03b4 : ennreal),\n  \u2203 (H : \u03b4 > 0), \u2200 (s : set \u03b1), coe_fn \u03bc s < \u03b4 \u2192 (lintegral (measure.restrict \u03bc s) fun (x : \u03b1) => f x) < \u03b5 := sorry\n\n/-- If `f` has finite integral, then `\u222b\u207b x in s, f x \u2202\u03bc` is absolutely continuous in `s`: it tends\nto zero as `\u03bc s` tends to zero. -/\ntheorem tendsto_set_lintegral_zero {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Type u_2} {f : \u03b1 \u2192 ennreal} (h : (lintegral \u03bc fun (x : \u03b1) => f x) < \u22a4) {l : filter \u03b9} {s : \u03b9 \u2192 set \u03b1} (hl : filter.tendsto (\u21d1\u03bc \u2218 s) l (nhds 0)) : filter.tendsto (fun (i : \u03b9) => lintegral (measure.restrict \u03bc (s i)) fun (x : \u03b1) => f x) l (nhds 0) := sorry\n\n@[simp] theorem lintegral_add {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hf : measurable f) (hg : measurable g) : (lintegral \u03bc fun (a : \u03b1) => f a + g a) = (lintegral \u03bc fun (a : \u03b1) => f a) + lintegral \u03bc fun (a : \u03b1) => g a := sorry\n\ntheorem lintegral_add' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hf : ae_measurable f) (hg : ae_measurable g) : (lintegral \u03bc fun (a : \u03b1) => f a + g a) = (lintegral \u03bc fun (a : \u03b1) => f a) + lintegral \u03bc fun (a : \u03b1) => g a := sorry\n\ntheorem lintegral_zero {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} : (lintegral \u03bc fun (a : \u03b1) => 0) = 0 := sorry\n\ntheorem lintegral_zero_fun {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} : (lintegral \u03bc fun (a : \u03b1) => HasZero.zero a) = 0 := sorry\n\n@[simp] theorem lintegral_smul_measure {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (c : ennreal) (f : \u03b1 \u2192 ennreal) : (lintegral (c \u2022 \u03bc) fun (a : \u03b1) => f a) = c * lintegral \u03bc fun (a : \u03b1) => f a := sorry\n\n@[simp] theorem lintegral_sum_measure {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03b9 : Type u_2} (f : \u03b1 \u2192 ennreal) (\u03bc : \u03b9 \u2192 measure \u03b1) : (lintegral (measure.sum \u03bc) fun (a : \u03b1) => f a) = tsum fun (i : \u03b9) => lintegral (\u03bc i) fun (a : \u03b1) => f a := sorry\n\n@[simp] theorem lintegral_add_measure {\u03b1 : Type u_1} [measurable_space \u03b1] (f : \u03b1 \u2192 ennreal) (\u03bc : measure \u03b1) (\u03bd : measure \u03b1) : (lintegral (\u03bc + \u03bd) fun (a : \u03b1) => f a) = (lintegral \u03bc fun (a : \u03b1) => f a) + lintegral \u03bd fun (a : \u03b1) => f a := sorry\n\n@[simp] theorem lintegral_zero_measure {\u03b1 : Type u_1} [measurable_space \u03b1] (f : \u03b1 \u2192 ennreal) : (lintegral 0 fun (a : \u03b1) => f a) = 0 := sorry\n\ntheorem lintegral_finset_sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} (s : finset \u03b2) {f : \u03b2 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (b : \u03b2), measurable (f b)) : (lintegral \u03bc fun (a : \u03b1) => finset.sum s fun (b : \u03b2) => f b a) =\n  finset.sum s fun (b : \u03b2) => lintegral \u03bc fun (a : \u03b1) => f b a := sorry\n\n@[simp] theorem lintegral_const_mul {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral \u03bc fun (a : \u03b1) => r * f a) = r * lintegral \u03bc fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_const_mul'' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) {f : \u03b1 \u2192 ennreal} (hf : ae_measurable f) : (lintegral \u03bc fun (a : \u03b1) => r * f a) = r * lintegral \u03bc fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_const_mul_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) (f : \u03b1 \u2192 ennreal) : (r * lintegral \u03bc fun (a : \u03b1) => f a) \u2264 lintegral \u03bc fun (a : \u03b1) => r * f a := sorry\n\ntheorem lintegral_const_mul' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) (f : \u03b1 \u2192 ennreal) (hr : r \u2260 \u22a4) : (lintegral \u03bc fun (a : \u03b1) => r * f a) = r * lintegral \u03bc fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_mul_const {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral \u03bc fun (a : \u03b1) => f a * r) = (lintegral \u03bc fun (a : \u03b1) => f a) * r := sorry\n\ntheorem lintegral_mul_const'' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) {f : \u03b1 \u2192 ennreal} (hf : ae_measurable f) : (lintegral \u03bc fun (a : \u03b1) => f a * r) = (lintegral \u03bc fun (a : \u03b1) => f a) * r := sorry\n\ntheorem lintegral_mul_const_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) (f : \u03b1 \u2192 ennreal) : (lintegral \u03bc fun (a : \u03b1) => f a) * r \u2264 lintegral \u03bc fun (a : \u03b1) => f a * r := sorry\n\ntheorem lintegral_mul_const' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (r : ennreal) (f : \u03b1 \u2192 ennreal) (hr : r \u2260 \u22a4) : (lintegral \u03bc fun (a : \u03b1) => f a * r) = (lintegral \u03bc fun (a : \u03b1) => f a) * r := sorry\n\n/- A double integral of a product where each factor contains only one variable\n  is a product of integrals -/\n\ntheorem lintegral_lintegral_mul {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b2 : Type u_2} [measurable_space \u03b2] {\u03bd : measure \u03b2} {f : \u03b1 \u2192 ennreal} {g : \u03b2 \u2192 ennreal} (hf : measurable f) (hg : measurable g) : (lintegral \u03bc fun (x : \u03b1) => lintegral \u03bd fun (y : \u03b2) => f x * g y) =\n  (lintegral \u03bc fun (x : \u03b1) => f x) * lintegral \u03bd fun (y : \u03b2) => g y := sorry\n\n-- TODO: Need a better way of rewriting inside of a integral\n\ntheorem lintegral_rw\u2081 {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2} (h : filter.eventually_eq (measure.ae \u03bc) f f') (g : \u03b2 \u2192 ennreal) : (lintegral \u03bc fun (a : \u03b1) => g (f a)) = lintegral \u03bc fun (a : \u03b1) => g (f' a) :=\n  lintegral_congr_ae\n    (filter.eventually.mono h\n      fun (a : \u03b1) (h : f a = f' a) => eq.mpr (id (Eq._oldrec (Eq.refl (g (f a) = g (f' a))) h)) (Eq.refl (g (f' a))))\n\n-- TODO: Need a better way of rewriting inside of a integral\n\ntheorem lintegral_rw\u2082 {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f\u2081 : \u03b1 \u2192 \u03b2} {f\u2081' : \u03b1 \u2192 \u03b2} {f\u2082 : \u03b1 \u2192 \u03b3} {f\u2082' : \u03b1 \u2192 \u03b3} (h\u2081 : filter.eventually_eq (measure.ae \u03bc) f\u2081 f\u2081') (h\u2082 : filter.eventually_eq (measure.ae \u03bc) f\u2082 f\u2082') (g : \u03b2 \u2192 \u03b3 \u2192 ennreal) : (lintegral \u03bc fun (a : \u03b1) => g (f\u2081 a) (f\u2082 a)) = lintegral \u03bc fun (a : \u03b1) => g (f\u2081' a) (f\u2082' a) := sorry\n\n@[simp] theorem lintegral_indicator {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : \u03b1 \u2192 ennreal) {s : set \u03b1} (hs : is_measurable s) : (lintegral \u03bc fun (a : \u03b1) => set.indicator s f a) = lintegral (measure.restrict \u03bc s) fun (a : \u03b1) => f a := sorry\n\n/-- Chebyshev's inequality -/\ntheorem mul_meas_ge_le_lintegral {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) (\u03b5 : ennreal) : \u03b5 * coe_fn \u03bc (set_of fun (x : \u03b1) => \u03b5 \u2264 f x) \u2264 lintegral \u03bc fun (a : \u03b1) => f a := sorry\n\ntheorem meas_ge_le_lintegral_div {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) {\u03b5 : ennreal} (h\u03b5 : \u03b5 \u2260 0) (h\u03b5' : \u03b5 \u2260 \u22a4) : coe_fn \u03bc (set_of fun (x : \u03b1) => \u03b5 \u2264 f x) \u2264 (lintegral \u03bc fun (a : \u03b1) => f a) / \u03b5 := sorry\n\n@[simp] theorem lintegral_eq_zero_iff {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral \u03bc fun (a : \u03b1) => f a) = 0 \u2194 filter.eventually_eq (measure.ae \u03bc) f 0 := sorry\n\n@[simp] theorem lintegral_eq_zero_iff' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : ae_measurable f) : (lintegral \u03bc fun (a : \u03b1) => f a) = 0 \u2194 filter.eventually_eq (measure.ae \u03bc) f 0 := sorry\n\ntheorem lintegral_pos_iff_support {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (0 < lintegral \u03bc fun (a : \u03b1) => f a) \u2194 0 < coe_fn \u03bc (function.support f) := sorry\n\n/-- Weaker version of the monotone convergence theorem-/\ntheorem lintegral_supr_ae {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (n : \u2115), measurable (f n)) (h_mono : \u2200 (n : \u2115), filter.eventually (fun (a : \u03b1) => f n a \u2264 f (Nat.succ n) a) (measure.ae \u03bc)) : (lintegral \u03bc fun (a : \u03b1) => supr fun (n : \u2115) => f n a) = supr fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a := sorry\n\ntheorem lintegral_sub {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hf : measurable f) (hg : measurable g) (hg_fin : (lintegral \u03bc fun (a : \u03b1) => g a) < \u22a4) (h_le : filter.eventually_le (measure.ae \u03bc) g f) : (lintegral \u03bc fun (a : \u03b1) => f a - g a) = (lintegral \u03bc fun (a : \u03b1) => f a) - lintegral \u03bc fun (a : \u03b1) => g a := sorry\n\n/-- Monotone convergence theorem for nonincreasing sequences of functions -/\ntheorem lintegral_infi_ae {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (h_meas : \u2200 (n : \u2115), measurable (f n)) (h_mono : \u2200 (n : \u2115), filter.eventually_le (measure.ae \u03bc) (f (Nat.succ n)) (f n)) (h_fin : (lintegral \u03bc fun (a : \u03b1) => f 0 a) < \u22a4) : (lintegral \u03bc fun (a : \u03b1) => infi fun (n : \u2115) => f n a) = infi fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a := sorry\n\n/-- Monotone convergence theorem for nonincreasing sequences of functions -/\ntheorem lintegral_infi {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (h_meas : \u2200 (n : \u2115), measurable (f n)) (h_mono : \u2200 {m n : \u2115}, m \u2264 n \u2192 f n \u2264 f m) (h_fin : (lintegral \u03bc fun (a : \u03b1) => f 0 a) < \u22a4) : (lintegral \u03bc fun (a : \u03b1) => infi fun (n : \u2115) => f n a) = infi fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a :=\n  lintegral_infi_ae h_meas (fun (n : \u2115) => ae_of_all \u03bc (h_mono (le_of_lt (nat.lt_succ_self n)))) h_fin\n\n/-- Known as Fatou's lemma, version with `ae_measurable` functions -/\ntheorem lintegral_liminf_le' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (h_meas : \u2200 (n : \u2115), ae_measurable (f n)) : (lintegral \u03bc fun (a : \u03b1) => filter.liminf filter.at_top fun (n : \u2115) => f n a) \u2264\n  filter.liminf filter.at_top fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a := sorry\n\n/-- Known as Fatou's lemma -/\ntheorem lintegral_liminf_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} (h_meas : \u2200 (n : \u2115), measurable (f n)) : (lintegral \u03bc fun (a : \u03b1) => filter.liminf filter.at_top fun (n : \u2115) => f n a) \u2264\n  filter.liminf filter.at_top fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a :=\n  lintegral_liminf_le' fun (n : \u2115) => measurable.ae_measurable (h_meas n)\n\ntheorem limsup_lintegral_le {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 ennreal} {g : \u03b1 \u2192 ennreal} (hf_meas : \u2200 (n : \u2115), measurable (f n)) (h_bound : \u2200 (n : \u2115), filter.eventually_le (measure.ae \u03bc) (f n) g) (h_fin : (lintegral \u03bc fun (a : \u03b1) => g a) < \u22a4) : (filter.limsup filter.at_top fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => f n a) \u2264\n  lintegral \u03bc fun (a : \u03b1) => filter.limsup filter.at_top fun (n : \u2115) => f n a := sorry\n\n/-- Dominated convergence theorem for nonnegative functions -/\ntheorem tendsto_lintegral_of_dominated_convergence {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {F : \u2115 \u2192 \u03b1 \u2192 ennreal} {f : \u03b1 \u2192 ennreal} (bound : \u03b1 \u2192 ennreal) (hF_meas : \u2200 (n : \u2115), measurable (F n)) (h_bound : \u2200 (n : \u2115), filter.eventually_le (measure.ae \u03bc) (F n) bound) (h_fin : (lintegral \u03bc fun (a : \u03b1) => bound a) < \u22a4) (h_lim : filter.eventually (fun (a : \u03b1) => filter.tendsto (fun (n : \u2115) => F n a) filter.at_top (nhds (f a))) (measure.ae \u03bc)) : filter.tendsto (fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => F n a) filter.at_top (nhds (lintegral \u03bc fun (a : \u03b1) => f a)) := sorry\n\n/-- Dominated convergence theorem for nonnegative functions which are just almost everywhere\nmeasurable. -/\ntheorem tendsto_lintegral_of_dominated_convergence' {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {F : \u2115 \u2192 \u03b1 \u2192 ennreal} {f : \u03b1 \u2192 ennreal} (bound : \u03b1 \u2192 ennreal) (hF_meas : \u2200 (n : \u2115), ae_measurable (F n)) (h_bound : \u2200 (n : \u2115), filter.eventually_le (measure.ae \u03bc) (F n) bound) (h_fin : (lintegral \u03bc fun (a : \u03b1) => bound a) < \u22a4) (h_lim : filter.eventually (fun (a : \u03b1) => filter.tendsto (fun (n : \u2115) => F n a) filter.at_top (nhds (f a))) (measure.ae \u03bc)) : filter.tendsto (fun (n : \u2115) => lintegral \u03bc fun (a : \u03b1) => F n a) filter.at_top (nhds (lintegral \u03bc fun (a : \u03b1) => f a)) := sorry\n\n/-- Dominated convergence theorem for filters with a countable basis -/\ntheorem tendsto_lintegral_filter_of_dominated_convergence {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {\u03b9 : Type u_2} {l : filter \u03b9} {F : \u03b9 \u2192 \u03b1 \u2192 ennreal} {f : \u03b1 \u2192 ennreal} (bound : \u03b1 \u2192 ennreal) (hl_cb : filter.is_countably_generated l) (hF_meas : filter.eventually (fun (n : \u03b9) => measurable (F n)) l) (h_bound : filter.eventually (fun (n : \u03b9) => filter.eventually (fun (a : \u03b1) => F n a \u2264 bound a) (measure.ae \u03bc)) l) (h_fin : (lintegral \u03bc fun (a : \u03b1) => bound a) < \u22a4) (h_lim : filter.eventually (fun (a : \u03b1) => filter.tendsto (fun (n : \u03b9) => F n a) l (nhds (f a))) (measure.ae \u03bc)) : filter.tendsto (fun (n : \u03b9) => lintegral \u03bc fun (a : \u03b1) => F n a) l (nhds (lintegral \u03bc fun (a : \u03b1) => f a)) := sorry\n\n/-- Monotone convergence for a suprema over a directed family and indexed by an encodable type -/\ntheorem lintegral_supr_directed {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [encodable \u03b2] {f : \u03b2 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (b : \u03b2), measurable (f b)) (h_directed : directed LessEq f) : (lintegral \u03bc fun (a : \u03b1) => supr fun (b : \u03b2) => f b a) = supr fun (b : \u03b2) => lintegral \u03bc fun (a : \u03b1) => f b a := sorry\n\ntheorem lintegral_tsum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [encodable \u03b2] {f : \u03b2 \u2192 \u03b1 \u2192 ennreal} (hf : \u2200 (i : \u03b2), measurable (f i)) : (lintegral \u03bc fun (a : \u03b1) => tsum fun (i : \u03b2) => f i a) = tsum fun (i : \u03b2) => lintegral \u03bc fun (a : \u03b1) => f i a := sorry\n\ntheorem lintegral_Union {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [encodable \u03b2] {s : \u03b2 \u2192 set \u03b1} (hm : \u2200 (i : \u03b2), is_measurable (s i)) (hd : pairwise (disjoint on s)) (f : \u03b1 \u2192 ennreal) : (lintegral (measure.restrict \u03bc (set.Union fun (i : \u03b2) => s i)) fun (a : \u03b1) => f a) =\n  tsum fun (i : \u03b2) => lintegral (measure.restrict \u03bc (s i)) fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_Union_le {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [encodable \u03b2] (s : \u03b2 \u2192 set \u03b1) (f : \u03b1 \u2192 ennreal) : (lintegral (measure.restrict \u03bc (set.Union fun (i : \u03b2) => s i)) fun (a : \u03b1) => f a) \u2264\n  tsum fun (i : \u03b2) => lintegral (measure.restrict \u03bc (s i)) fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : \u03b2 \u2192 ennreal} {g : \u03b1 \u2192 \u03b2} (hf : measurable f) (hg : measurable g) : (lintegral (coe_fn (measure.map g) \u03bc) fun (a : \u03b2) => f a) = lintegral \u03bc fun (a : \u03b1) => f (g a) := sorry\n\ntheorem lintegral_map' {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : \u03b2 \u2192 ennreal} {g : \u03b1 \u2192 \u03b2} (hf : ae_measurable f) (hg : measurable g) : (lintegral (coe_fn (measure.map g) \u03bc) fun (a : \u03b2) => f a) = lintegral \u03bc fun (a : \u03b1) => f (g a) :=\n  Eq.trans (Eq.trans (lintegral_congr_ae (ae_measurable.ae_eq_mk hf)) (lintegral_map (ae_measurable.measurable_mk hf) hg))\n    (lintegral_congr_ae (ae_eq_comp hg (filter.eventually_eq.symm (ae_measurable.ae_eq_mk hf))))\n\ntheorem lintegral_comp {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : \u03b2 \u2192 ennreal} {g : \u03b1 \u2192 \u03b2} (hf : measurable f) (hg : measurable g) : lintegral \u03bc (f \u2218 g) = lintegral (coe_fn (measure.map g) \u03bc) fun (a : \u03b2) => f a :=\n  Eq.symm (lintegral_map hf hg)\n\ntheorem set_lintegral_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} [measurable_space \u03b1] {\u03bc : measure \u03b1} [measurable_space \u03b2] {f : \u03b2 \u2192 ennreal} {g : \u03b1 \u2192 \u03b2} {s : set \u03b2} (hs : is_measurable s) (hf : measurable f) (hg : measurable g) : (lintegral (measure.restrict (coe_fn (measure.map g) \u03bc) s) fun (y : \u03b2) => f y) =\n  lintegral (measure.restrict \u03bc (g \u207b\u00b9' s)) fun (x : \u03b1) => f (g x) := sorry\n\ntheorem lintegral_dirac' {\u03b1 : Type u_1} [measurable_space \u03b1] (a : \u03b1) {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral (measure.dirac a) fun (a : \u03b1) => f a) = f a := sorry\n\ntheorem lintegral_dirac {\u03b1 : Type u_1} [measurable_space \u03b1] [measurable_singleton_class \u03b1] (a : \u03b1) (f : \u03b1 \u2192 ennreal) : (lintegral (measure.dirac a) fun (a : \u03b1) => f a) = f a := sorry\n\ntheorem lintegral_count' {\u03b1 : Type u_1} [measurable_space \u03b1] {f : \u03b1 \u2192 ennreal} (hf : measurable f) : (lintegral measure.count fun (a : \u03b1) => f a) = tsum fun (a : \u03b1) => f a := sorry\n\ntheorem lintegral_count {\u03b1 : Type u_1} [measurable_space \u03b1] [measurable_singleton_class \u03b1] (f : \u03b1 \u2192 ennreal) : (lintegral measure.count fun (a : \u03b1) => f a) = tsum fun (a : \u03b1) => f a := sorry\n\ntheorem ae_lt_top {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} {f : \u03b1 \u2192 ennreal} (hf : measurable f) (h2f : (lintegral \u03bc fun (x : \u03b1) => f x) < \u22a4) : filter.eventually (fun (x : \u03b1) => f x < \u22a4) (measure.ae \u03bc) := sorry\n\n/-- Given a measure `\u03bc : measure \u03b1` and a function `f : \u03b1 \u2192 ennreal`, `\u03bc.with_density f` is the\nmeasure such that for a measurable set `s` we have `\u03bc.with_density f s = \u222b\u207b a in s, f a \u2202\u03bc`. -/\ndef measure.with_density {\u03b1 : Type u_1} [measurable_space \u03b1] (\u03bc : measure \u03b1) (f : \u03b1 \u2192 ennreal) : measure \u03b1 :=\n  measure.of_measurable (fun (s : set \u03b1) (hs : is_measurable s) => lintegral (measure.restrict \u03bc s) fun (a : \u03b1) => f a)\n    sorry sorry\n\n@[simp] theorem with_density_apply {\u03b1 : Type u_1} [measurable_space \u03b1] {\u03bc : measure \u03b1} (f : \u03b1 \u2192 ennreal) {s : set \u03b1} (hs : is_measurable s) : coe_fn (measure.with_density \u03bc f) s = lintegral (measure.restrict \u03bc s) fun (a : \u03b1) => f a :=\n  measure.of_measurable_apply s hs\n\nend measure_theory\n\n\n/-- To prove something for an arbitrary measurable function into `ennreal`, it suffices to show\nthat the property holds for (multiples of) characteristic functions and is closed under addition\nand supremum of increasing sequences of functions.\n\nIt is possible to make the hypotheses in the induction steps a bit stronger, and such conditions\ncan be added once we need them (for example in `h_sum` it is only necessary to consider the sum of\na simple function with a multiple of a characteristic function and that the intersection\nof their images is a subset of `{0}`. -/\ntheorem measurable.ennreal_induction {\u03b1 : Type u_1} [measurable_space \u03b1] {P : (\u03b1 \u2192 ennreal) \u2192 Prop} (h_ind : \u2200 (c : ennreal) {s : set \u03b1}, is_measurable s \u2192 P (set.indicator s fun (_x : \u03b1) => c)) (h_sum : \u2200 {f g : \u03b1 \u2192 ennreal},\n  set.univ \u2286 f \u207b\u00b9' singleton 0 \u222a g \u207b\u00b9' singleton 0 \u2192 measurable f \u2192 measurable g \u2192 P f \u2192 P g \u2192 P (f + g)) (h_supr : \u2200 {f : \u2115 \u2192 \u03b1 \u2192 ennreal},\n  (\u2200 (n : \u2115), measurable (f n)) \u2192 monotone f \u2192 (\u2200 (n : \u2115), P (f n)) \u2192 P fun (x : \u03b1) => supr fun (n : \u2115) => f n x) {f : \u03b1 \u2192 ennreal} (hf : measurable f) : P f := sorry\n\nnamespace measure_theory\n\n\n/-- This is Exercise 1.2.1 from [tao2010]. It allows you to express integration of a measurable\nfunction with respect to `(\u03bc.with_density f)` as an integral with respect to `\u03bc`, called the base\nmeasure. `\u03bc` is often the Lebesgue measure, and in this circumstance `f` is the probability density\nfunction, and `(\u03bc.with_density f)` represents any continuous random variable as a\nprobability measure, such as the uniform distribution between 0 and 1, the Gaussian distribution,\nthe exponential distribution, the Beta distribution, or the Cauchy distribution (see Section 2.4\nof [wasserman2004]). Thus, this method shows how to one can calculate expectations, variances,\nand other moments as a function of the probability density function.\n -/\ntheorem lintegral_with_density_eq_lintegral_mul {\u03b1 : Type u_1} [measurable_space \u03b1] (\u03bc : measure \u03b1) {f : \u03b1 \u2192 ennreal} (h_mf : measurable f) {g : \u03b1 \u2192 ennreal} : measurable g \u2192 (lintegral (measure.with_density \u03bc f) fun (a : \u03b1) => g a) = lintegral \u03bc fun (a : \u03b1) => Mul.mul f g a := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/measure_theory/integration.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3602260580070424}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport order.antisymmetrization\nimport order.category.Preorder\n\n/-!\n# Category of partial orders\n\nThis defines `PartialOrder`, the category of partial orders with monotone maps.\n-/\n\nopen category_theory\n\nuniverse u\n\n/-- The category of partially ordered types. -/\ndef PartialOrder := bundled partial_order\n\nnamespace PartialOrder\n\ninstance : bundled_hom.parent_projection @partial_order.to_preorder := \u27e8\u27e9\n\nattribute [derive [large_category, concrete_category]] PartialOrder\n\ninstance : has_coe_to_sort PartialOrder Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled PartialOrder from the underlying type and typeclass. -/\ndef of (\u03b1 : Type*) [partial_order \u03b1] : PartialOrder := bundled.of \u03b1\n\n@[simp] lemma coe_of (\u03b1 : Type*) [partial_order \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited PartialOrder := \u27e8of punit\u27e9\n\ninstance (\u03b1 : PartialOrder) : partial_order \u03b1 := \u03b1.str\n\ninstance has_forget_to_Preorder : has_forget\u2082 PartialOrder Preorder := bundled_hom.forget\u2082 _ _\n\n/-- Constructs an equivalence between partial orders from an order isomorphism between them. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : PartialOrder.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : PartialOrder \u2964 PartialOrder :=\n{ obj := \u03bb X, of X\u1d52\u1d48, map := \u03bb X Y, order_hom.dual }\n\n/-- The equivalence between `PartialOrder` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : PartialOrder \u224c PartialOrder :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend PartialOrder\n\nlemma PartialOrder_dual_comp_forget_to_Preorder :\n  PartialOrder.dual \u22d9 forget\u2082 PartialOrder Preorder =\n    forget\u2082 PartialOrder Preorder \u22d9 Preorder.dual := rfl\n\n/-- `antisymmetrization` as a functor. It is the free functor. -/\ndef Preorder_to_PartialOrder : Preorder.{u} \u2964 PartialOrder :=\n{ obj := \u03bb X, PartialOrder.of (antisymmetrization X (\u2264)),\n  map := \u03bb X Y f, f.antisymmetrization,\n  map_id' := \u03bb X,\n    by { ext, exact quotient.induction_on' x (\u03bb x, quotient.map'_mk' _ (\u03bb a b, id) _) },\n  map_comp' := \u03bb X Y Z f g,\n    by { ext, exact quotient.induction_on' x (\u03bb x, order_hom.antisymmetrization_apply_mk _ _) } }\n\n/-- `Preorder_to_PartialOrder` is left adjoint to the forgetful functor, meaning it is the free\nfunctor from `Preorder` to `PartialOrder`. -/\ndef Preorder_to_PartialOrder_forget_adjunction :\n  Preorder_to_PartialOrder.{u} \u22a3 forget\u2082 PartialOrder Preorder :=\nadjunction.mk_of_hom_equiv\n  { hom_equiv := \u03bb X Y, { to_fun := \u03bb f,\n      \u27e8f \u2218 to_antisymmetrization (\u2264), f.mono.comp to_antisymmetrization_mono\u27e9,\n    inv_fun := \u03bb f, \u27e8\u03bb a, quotient.lift_on' a f $ \u03bb a b h, (antisymm_rel.image h f.mono).eq, \u03bb a b,\n      quotient.induction_on\u2082' a b $ \u03bb a b h, f.mono h\u27e9,\n    left_inv := \u03bb f, order_hom.ext _ _ $ funext $ \u03bb x, quotient.induction_on' x $ \u03bb x, rfl,\n    right_inv := \u03bb f, order_hom.ext _ _ $ funext $ \u03bb x, rfl },\n  hom_equiv_naturality_left_symm' := \u03bb X Y Z f g,\n    order_hom.ext _ _ $ funext $ \u03bb x, quotient.induction_on' x $ \u03bb x, rfl,\n  hom_equiv_naturality_right' := \u03bb X Y Z f g, order_hom.ext _ _ $ funext $ \u03bb x, rfl }\n\n/-- `Preorder_to_PartialOrder` and `order_dual` commute. -/\n@[simps] def Preorder_to_PartialOrder_comp_to_dual_iso_to_dual_comp_Preorder_to_PartialOrder :\n (Preorder_to_PartialOrder.{u} \u22d9 PartialOrder.dual) \u2245\n    (Preorder.dual \u22d9 Preorder_to_PartialOrder) :=\nnat_iso.of_components (\u03bb X, PartialOrder.iso.mk $ order_iso.dual_antisymmetrization _) $\n  \u03bb X Y f, order_hom.ext _ _ $ funext $ \u03bb x, quotient.induction_on' x $ \u03bb x, rfl\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/order/category/PartialOrder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.36020496371853794}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebra.homology.additive\n! leanprover-community/mathlib commit 200eda15d8ff5669854ff6bcc10aaf37cb70498f\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Homology.Homology\nimport Mathbin.Algebra.Homology.Single\nimport Mathbin.CategoryTheory.Preadditive.AdditiveFunctor\n\n/-!\n# Homology is an additive functor\n\nWhen `V` is preadditive, `homological_complex V c` is also preadditive,\nand `homology_functor` is additive.\n\nTODO: similarly for `R`-linear.\n-/\n\n\nuniverse v u\n\nopen Classical\n\nnoncomputable section\n\nopen CategoryTheory CategoryTheory.Category CategoryTheory.Limits HomologicalComplex\n\nvariable {\u03b9 : Type _}\n\nvariable {V : Type u} [Category.{v} V] [Preadditive V]\n\nvariable {c : ComplexShape \u03b9} {C D E : HomologicalComplex V c}\n\nvariable (f g : C \u27f6 D) (h k : D \u27f6 E) (i : \u03b9)\n\nnamespace HomologicalComplex\n\ninstance : Zero (C \u27f6 D) :=\n  \u27e8{ f := fun i => 0 }\u27e9\n\ninstance : Add (C \u27f6 D) :=\n  \u27e8fun f g => { f := fun i => f.f i + g.f i }\u27e9\n\ninstance : Neg (C \u27f6 D) :=\n  \u27e8fun f => { f := fun i => -f.f i }\u27e9\n\ninstance : Sub (C \u27f6 D) :=\n  \u27e8fun f g => { f := fun i => f.f i - g.f i }\u27e9\n\ninstance hasNatScalar : SMul \u2115 (C \u27f6 D) :=\n  \u27e8fun n f =>\n    { f := fun i => n \u2022 f.f i\n      comm' := fun i j h => by simp [preadditive.nsmul_comp, preadditive.comp_nsmul] }\u27e9\n#align homological_complex.has_nat_scalar HomologicalComplex.hasNatScalar\n\ninstance hasIntScalar : SMul \u2124 (C \u27f6 D) :=\n  \u27e8fun n f =>\n    { f := fun i => n \u2022 f.f i\n      comm' := fun i j h => by simp [preadditive.zsmul_comp, preadditive.comp_zsmul] }\u27e9\n#align homological_complex.has_int_scalar HomologicalComplex.hasIntScalar\n\n@[simp]\ntheorem zero_f_apply (i : \u03b9) : (0 : C \u27f6 D).f i = 0 :=\n  rfl\n#align homological_complex.zero_f_apply HomologicalComplex.zero_f_apply\n\n@[simp]\ntheorem add_f_apply (f g : C \u27f6 D) (i : \u03b9) : (f + g).f i = f.f i + g.f i :=\n  rfl\n#align homological_complex.add_f_apply HomologicalComplex.add_f_apply\n\n@[simp]\ntheorem neg_f_apply (f : C \u27f6 D) (i : \u03b9) : (-f).f i = -f.f i :=\n  rfl\n#align homological_complex.neg_f_apply HomologicalComplex.neg_f_apply\n\n@[simp]\ntheorem sub_f_apply (f g : C \u27f6 D) (i : \u03b9) : (f - g).f i = f.f i - g.f i :=\n  rfl\n#align homological_complex.sub_f_apply HomologicalComplex.sub_f_apply\n\n@[simp]\ntheorem nsmul_f_apply (n : \u2115) (f : C \u27f6 D) (i : \u03b9) : (n \u2022 f).f i = n \u2022 f.f i :=\n  rfl\n#align homological_complex.nsmul_f_apply HomologicalComplex.nsmul_f_apply\n\n@[simp]\ntheorem zsmul_f_apply (n : \u2124) (f : C \u27f6 D) (i : \u03b9) : (n \u2022 f).f i = n \u2022 f.f i :=\n  rfl\n#align homological_complex.zsmul_f_apply HomologicalComplex.zsmul_f_apply\n\ninstance : AddCommGroup (C \u27f6 D) :=\n  Function.Injective.addCommGroup Hom.f HomologicalComplex.hom_f_injective (by tidy) (by tidy)\n    (by tidy) (by tidy) (by tidy) (by tidy)\n\ninstance : Preadditive (HomologicalComplex V c) where\n\n/-- The `i`-th component of a chain map, as an additive map from chain maps to morphisms. -/\n@[simps]\ndef Hom.fAddMonoidHom {C\u2081 C\u2082 : HomologicalComplex V c} (i : \u03b9) : (C\u2081 \u27f6 C\u2082) \u2192+ (C\u2081.pt i \u27f6 C\u2082.pt i) :=\n  AddMonoidHom.mk' (fun f => Hom.f f i) fun _ _ => rfl\n#align homological_complex.hom.f_add_monoid_hom HomologicalComplex.Hom.fAddMonoidHom\n\nend HomologicalComplex\n\nnamespace HomologicalComplex\n\ninstance eval_additive (i : \u03b9) : (eval V c i).Additive where\n#align homological_complex.eval_additive HomologicalComplex.eval_additive\n\ninstance cycles_additive [HasEqualizers V] : (cyclesFunctor V c i).Additive where\n#align homological_complex.cycles_additive HomologicalComplex.cycles_additive\n\nvariable [HasImages V] [HasImageMaps V]\n\ninstance boundaries_additive : (boundariesFunctor V c i).Additive where\n#align homological_complex.boundaries_additive HomologicalComplex.boundaries_additive\n\nvariable [HasEqualizers V] [HasCokernels V]\n\ninstance homology_additive : (homologyFunctor V c i).Additive\n    where map_add' C D f g := by\n    dsimp [homologyFunctor]\n    ext\n    simp only [homology.\u03c0_map, preadditive.comp_add, \u2190 preadditive.add_comp]\n    congr\n    ext; simp\n#align homological_complex.homology_additive HomologicalComplex.homology_additive\n\nend HomologicalComplex\n\nnamespace CategoryTheory\n\nvariable {W : Type _} [Category W] [Preadditive W]\n\n/-- An additive functor induces a functor between homological complexes.\nThis is sometimes called the \"prolongation\".\n-/\n@[simps]\ndef Functor.mapHomologicalComplex (F : V \u2964 W) [F.Additive] (c : ComplexShape \u03b9) :\n    HomologicalComplex V c \u2964 HomologicalComplex W c\n    where\n  obj C :=\n    { pt := fun i => F.obj (C.pt i)\n      d := fun i j => F.map (C.d i j)\n      shape' := fun i j w => by rw [C.shape _ _ w, F.map_zero]\n      d_comp_d' := fun i j k _ _ => by rw [\u2190 F.map_comp, C.d_comp_d, F.map_zero] }\n  map C D f :=\n    { f := fun i => F.map (f.f i)\n      comm' := fun i j h => by\n        dsimp\n        rw [\u2190 F.map_comp, \u2190 F.map_comp, f.comm] }\n#align category_theory.functor.map_homological_complex CategoryTheory.Functor.mapHomologicalComplex\n\nvariable (V)\n\n/-- The functor on homological complexes induced by the identity functor is\nisomorphic to the identity functor. -/\n@[simps]\ndef Functor.mapHomologicalComplexIdIso (c : ComplexShape \u03b9) : (\ud835\udfed V).mapHomologicalComplex c \u2245 \ud835\udfed _ :=\n  NatIso.ofComponents (fun K => Hom.isoOfComponents (fun i => Iso.refl _) (by tidy)) (by tidy)\n#align category_theory.functor.map_homological_complex_id_iso CategoryTheory.Functor.mapHomologicalComplexIdIso\n\nvariable {V}\n\ninstance Functor.map_homogical_complex_additive (F : V \u2964 W) [F.Additive] (c : ComplexShape \u03b9) :\n    (F.mapHomologicalComplex c).Additive where\n#align category_theory.functor.map_homogical_complex_additive CategoryTheory.Functor.map_homogical_complex_additive\n\ninstance Functor.mapHomologicalComplex_reflects_iso (F : V \u2964 W) [F.Additive]\n    [ReflectsIsomorphisms F] (c : ComplexShape \u03b9) :\n    ReflectsIsomorphisms (F.mapHomologicalComplex c) :=\n  \u27e8fun X Y f => by\n    intro\n    haveI : \u2200 n : \u03b9, is_iso (F.map (f.f n)) := fun n =>\n      is_iso.of_iso\n        ((HomologicalComplex.eval W c n).mapIso (as_iso ((F.map_homological_complex c).map f)))\n    haveI := fun n => is_iso_of_reflects_iso (f.f n) F\n    exact HomologicalComplex.Hom.isIso_of_components f\u27e9\n#align category_theory.functor.map_homological_complex_reflects_iso CategoryTheory.Functor.mapHomologicalComplex_reflects_iso\n\n/-- A natural transformation between functors induces a natural transformation\nbetween those functors applied to homological complexes.\n-/\n@[simps]\ndef NatTrans.mapHomologicalComplex {F G : V \u2964 W} [F.Additive] [G.Additive] (\u03b1 : F \u27f6 G)\n    (c : ComplexShape \u03b9) : F.mapHomologicalComplex c \u27f6 G.mapHomologicalComplex c\n    where app C := { f := fun i => \u03b1.app _ }\n#align category_theory.nat_trans.map_homological_complex CategoryTheory.NatTrans.mapHomologicalComplex\n\n@[simp]\ntheorem NatTrans.mapHomologicalComplex_id (c : ComplexShape \u03b9) (F : V \u2964 W) [F.Additive] :\n    NatTrans.mapHomologicalComplex (\ud835\udfd9 F) c = \ud835\udfd9 (F.mapHomologicalComplex c) := by tidy\n#align category_theory.nat_trans.map_homological_complex_id CategoryTheory.NatTrans.mapHomologicalComplex_id\n\n@[simp]\ntheorem NatTrans.mapHomologicalComplex_comp (c : ComplexShape \u03b9) {F G H : V \u2964 W} [F.Additive]\n    [G.Additive] [H.Additive] (\u03b1 : F \u27f6 G) (\u03b2 : G \u27f6 H) :\n    NatTrans.mapHomologicalComplex (\u03b1 \u226b \u03b2) c =\n      NatTrans.mapHomologicalComplex \u03b1 c \u226b NatTrans.mapHomologicalComplex \u03b2 c :=\n  by tidy\n#align category_theory.nat_trans.map_homological_complex_comp CategoryTheory.NatTrans.mapHomologicalComplex_comp\n\n@[simp, reassoc.1]\ntheorem NatTrans.mapHomologicalComplex_naturality {c : ComplexShape \u03b9} {F G : V \u2964 W} [F.Additive]\n    [G.Additive] (\u03b1 : F \u27f6 G) {C D : HomologicalComplex V c} (f : C \u27f6 D) :\n    (F.mapHomologicalComplex c).map f \u226b (NatTrans.mapHomologicalComplex \u03b1 c).app D =\n      (NatTrans.mapHomologicalComplex \u03b1 c).app C \u226b (G.mapHomologicalComplex c).map f :=\n  by tidy\n#align category_theory.nat_trans.map_homological_complex_naturality CategoryTheory.NatTrans.mapHomologicalComplex_naturality\n\n/-- A natural isomorphism between functors induces a natural isomorphism\nbetween those functors applied to homological complexes.\n-/\n@[simps]\ndef NatIso.mapHomologicalComplex {F G : V \u2964 W} [F.Additive] [G.Additive] (\u03b1 : F \u2245 G)\n    (c : ComplexShape \u03b9) : F.mapHomologicalComplex c \u2245 G.mapHomologicalComplex c\n    where\n  Hom := \u03b1.Hom.mapHomologicalComplex c\n  inv := \u03b1.inv.mapHomologicalComplex c\n  hom_inv_id' := by simpa only [\u2190 nat_trans.map_homological_complex_comp, \u03b1.hom_inv_id]\n  inv_hom_id' := by simpa only [\u2190 nat_trans.map_homological_complex_comp, \u03b1.inv_hom_id]\n#align category_theory.nat_iso.map_homological_complex CategoryTheory.NatIso.mapHomologicalComplex\n\n/-- An equivalence of categories induces an equivalences between the respective categories\nof homological complex.\n-/\n@[simps]\ndef Equivalence.mapHomologicalComplex (e : V \u224c W) [e.Functor.Additive] (c : ComplexShape \u03b9) :\n    HomologicalComplex V c \u224c HomologicalComplex W c\n    where\n  Functor := e.Functor.mapHomologicalComplex c\n  inverse := e.inverse.mapHomologicalComplex c\n  unitIso :=\n    (Functor.mapHomologicalComplexIdIso V c).symm \u226a\u226b NatIso.mapHomologicalComplex e.unitIso c\n  counitIso := NatIso.mapHomologicalComplex e.counitIso c \u226a\u226b Functor.mapHomologicalComplexIdIso W c\n#align category_theory.equivalence.map_homological_complex CategoryTheory.Equivalence.mapHomologicalComplex\n\nend CategoryTheory\n\nnamespace ChainComplex\n\nvariable {W : Type _} [Category W] [Preadditive W]\n\nvariable {\u03b1 : Type _} [AddRightCancelSemigroup \u03b1] [One \u03b1] [DecidableEq \u03b1]\n\ntheorem map_chain_complex_of (F : V \u2964 W) [F.Additive] (X : \u03b1 \u2192 V) (d : \u2200 n, X (n + 1) \u27f6 X n)\n    (sq : \u2200 n, d (n + 1) \u226b d n = 0) :\n    (F.mapHomologicalComplex _).obj (ChainComplex.of X d sq) =\n      ChainComplex.of (fun n => F.obj (X n)) (fun n => F.map (d n)) fun n => by\n        rw [\u2190 F.map_comp, sq n, functor.map_zero] :=\n  by\n  refine' HomologicalComplex.ext rfl _\n  rintro i j (rfl : j + 1 = i)\n  simp only [CategoryTheory.Functor.mapHomologicalComplex_obj_d, of_d, eq_to_hom_refl, comp_id,\n    id_comp]\n#align chain_complex.map_chain_complex_of ChainComplex.map_chain_complex_of\n\nend ChainComplex\n\nvariable [HasZeroObject V] {W : Type _} [Category W] [Preadditive W] [HasZeroObject W]\n\nnamespace HomologicalComplex\n\nattribute [local simp] eq_to_hom_map\n\n/-- Turning an object into a complex supported at `j` then applying a functor is\nthe same as applying the functor then forming the complex.\n-/\ndef singleMapHomologicalComplex (F : V \u2964 W) [F.Additive] (c : ComplexShape \u03b9) (j : \u03b9) :\n    single V c j \u22d9 F.mapHomologicalComplex _ \u2245 F \u22d9 single W c j :=\n  NatIso.ofComponents\n    (fun X =>\n      { Hom := { f := fun i => if h : i = j then eqToHom (by simp [h]) else 0 }\n        inv := { f := fun i => if h : i = j then eqToHom (by simp [h]) else 0 }\n        hom_inv_id' := by\n          ext i\n          dsimp\n          split_ifs with h\n          \u00b7 simp [h]\n          \u00b7 rw [zero_comp, if_neg h]\n            exact (zero_of_source_iso_zero _ F.map_zero_object).symm\n        inv_hom_id' := by\n          ext i\n          dsimp\n          split_ifs with h\n          \u00b7 simp [h]\n          \u00b7 rw [zero_comp, if_neg h]\n            simp })\n    fun X Y f => by\n    ext i\n    dsimp\n    split_ifs with h <;> simp [h]\n#align homological_complex.single_map_homological_complex HomologicalComplex.singleMapHomologicalComplex\n\nvariable (F : V \u2964 W) [Functor.Additive F] (c)\n\n@[simp]\ntheorem singleMapHomologicalComplex_hom_app_self (j : \u03b9) (X : V) :\n    ((singleMapHomologicalComplex F c j).Hom.app X).f j = eqToHom (by simp) := by\n  simp [single_map_homological_complex]\n#align homological_complex.single_map_homological_complex_hom_app_self HomologicalComplex.singleMapHomologicalComplex_hom_app_self\n\n@[simp]\ntheorem singleMapHomologicalComplex_hom_app_ne {i j : \u03b9} (h : i \u2260 j) (X : V) :\n    ((singleMapHomologicalComplex F c j).Hom.app X).f i = 0 := by\n  simp [single_map_homological_complex, h]\n#align homological_complex.single_map_homological_complex_hom_app_ne HomologicalComplex.singleMapHomologicalComplex_hom_app_ne\n\n@[simp]\ntheorem singleMapHomologicalComplex_inv_app_self (j : \u03b9) (X : V) :\n    ((singleMapHomologicalComplex F c j).inv.app X).f j = eqToHom (by simp) := by\n  simp [single_map_homological_complex]\n#align homological_complex.single_map_homological_complex_inv_app_self HomologicalComplex.singleMapHomologicalComplex_inv_app_self\n\n@[simp]\ntheorem singleMapHomologicalComplex_inv_app_ne {i j : \u03b9} (h : i \u2260 j) (X : V) :\n    ((singleMapHomologicalComplex F c j).inv.app X).f i = 0 := by\n  simp [single_map_homological_complex, h]\n#align homological_complex.single_map_homological_complex_inv_app_ne HomologicalComplex.singleMapHomologicalComplex_inv_app_ne\n\nend HomologicalComplex\n\nnamespace ChainComplex\n\n/-- Turning an object into a chain complex supported at zero then applying a functor is\nthe same as applying the functor then forming the complex.\n-/\ndef single\u2080MapHomologicalComplex (F : V \u2964 W) [F.Additive] :\n    single\u2080 V \u22d9 F.mapHomologicalComplex _ \u2245 F \u22d9 single\u2080 W :=\n  NatIso.ofComponents\n    (fun X =>\n      { Hom :=\n          {\n            f := fun i =>\n              match i with\n              | 0 => \ud835\udfd9 _\n              | i + 1 => F.mapZeroObject.Hom }\n        inv :=\n          {\n            f := fun i =>\n              match i with\n              | 0 => \ud835\udfd9 _\n              | i + 1 => F.mapZeroObject.inv }\n        hom_inv_id' := by\n          ext (_ | i)\n          \u00b7 unfold_aux\n            simp\n          \u00b7 unfold_aux\n            dsimp\n            simp only [comp_f, id_f, zero_comp]\n            exact (zero_of_source_iso_zero _ F.map_zero_object).symm\n        inv_hom_id' := by\n          ext (_ | i) <;>\n            \u00b7 unfold_aux\n              dsimp\n              simp })\n    fun X Y f => by\n    ext (_ | i) <;>\n      \u00b7 unfold_aux\n        dsimp\n        simp\n#align chain_complex.single\u2080_map_homological_complex ChainComplex.single\u2080MapHomologicalComplex\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_hom_app_zero (F : V \u2964 W) [F.Additive] (X : V) :\n    ((single\u2080MapHomologicalComplex F).Hom.app X).f 0 = \ud835\udfd9 _ :=\n  rfl\n#align chain_complex.single\u2080_map_homological_complex_hom_app_zero ChainComplex.single\u2080MapHomologicalComplex_hom_app_zero\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_hom_app_succ (F : V \u2964 W) [F.Additive] (X : V) (n : \u2115) :\n    ((single\u2080MapHomologicalComplex F).Hom.app X).f (n + 1) = 0 :=\n  rfl\n#align chain_complex.single\u2080_map_homological_complex_hom_app_succ ChainComplex.single\u2080MapHomologicalComplex_hom_app_succ\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_inv_app_zero (F : V \u2964 W) [F.Additive] (X : V) :\n    ((single\u2080MapHomologicalComplex F).inv.app X).f 0 = \ud835\udfd9 _ :=\n  rfl\n#align chain_complex.single\u2080_map_homological_complex_inv_app_zero ChainComplex.single\u2080MapHomologicalComplex_inv_app_zero\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_inv_app_succ (F : V \u2964 W) [F.Additive] (X : V) (n : \u2115) :\n    ((single\u2080MapHomologicalComplex F).inv.app X).f (n + 1) = 0 :=\n  rfl\n#align chain_complex.single\u2080_map_homological_complex_inv_app_succ ChainComplex.single\u2080MapHomologicalComplex_inv_app_succ\n\nend ChainComplex\n\nnamespace CochainComplex\n\n/-- Turning an object into a cochain complex supported at zero then applying a functor is\nthe same as applying the functor then forming the cochain complex.\n-/\ndef single\u2080MapHomologicalComplex (F : V \u2964 W) [F.Additive] :\n    single\u2080 V \u22d9 F.mapHomologicalComplex _ \u2245 F \u22d9 single\u2080 W :=\n  NatIso.ofComponents\n    (fun X =>\n      { Hom :=\n          {\n            f := fun i =>\n              match i with\n              | 0 => \ud835\udfd9 _\n              | i + 1 => F.mapZeroObject.Hom }\n        inv :=\n          {\n            f := fun i =>\n              match i with\n              | 0 => \ud835\udfd9 _\n              | i + 1 => F.mapZeroObject.inv }\n        hom_inv_id' := by\n          ext (_ | i)\n          \u00b7 unfold_aux\n            simp\n          \u00b7 unfold_aux\n            dsimp\n            simp only [comp_f, id_f, zero_comp]\n            exact (zero_of_source_iso_zero _ F.map_zero_object).symm\n        inv_hom_id' := by\n          ext (_ | i) <;>\n            \u00b7 unfold_aux\n              dsimp\n              simp })\n    fun X Y f => by\n    ext (_ | i) <;>\n      \u00b7 unfold_aux\n        dsimp\n        simp\n#align cochain_complex.single\u2080_map_homological_complex CochainComplex.single\u2080MapHomologicalComplex\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_hom_app_zero (F : V \u2964 W) [F.Additive] (X : V) :\n    ((single\u2080MapHomologicalComplex F).Hom.app X).f 0 = \ud835\udfd9 _ :=\n  rfl\n#align cochain_complex.single\u2080_map_homological_complex_hom_app_zero CochainComplex.single\u2080MapHomologicalComplex_hom_app_zero\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_hom_app_succ (F : V \u2964 W) [F.Additive] (X : V) (n : \u2115) :\n    ((single\u2080MapHomologicalComplex F).Hom.app X).f (n + 1) = 0 :=\n  rfl\n#align cochain_complex.single\u2080_map_homological_complex_hom_app_succ CochainComplex.single\u2080MapHomologicalComplex_hom_app_succ\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_inv_app_zero (F : V \u2964 W) [F.Additive] (X : V) :\n    ((single\u2080MapHomologicalComplex F).inv.app X).f 0 = \ud835\udfd9 _ :=\n  rfl\n#align cochain_complex.single\u2080_map_homological_complex_inv_app_zero CochainComplex.single\u2080MapHomologicalComplex_inv_app_zero\n\n@[simp]\ntheorem single\u2080MapHomologicalComplex_inv_app_succ (F : V \u2964 W) [F.Additive] (X : V) (n : \u2115) :\n    ((single\u2080MapHomologicalComplex F).inv.app X).f (n + 1) = 0 :=\n  rfl\n#align cochain_complex.single\u2080_map_homological_complex_inv_app_succ CochainComplex.single\u2080MapHomologicalComplex_inv_app_succ\n\nend CochainComplex\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Homology/Additive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3601416735260383}}
{"text": "-- import SciLean.Operators\nimport SciLean.Core\n-- import SciLean.Functions.OdeSolve\nimport SciLean.Tactic.BubbleLimit\n\nnamespace SciLean\n\ninductive ExactSolution {\u03b1 : Type _} : (spec : \u03b1 \u2192 Prop) \u2192 Type _\n| exact {spec : \u03b1 \u2192 Prop}\n        (a : \u03b1) (h : spec a)\n        : ExactSolution spec\n\ndef ExactSolution.val {\u03b1} {spec : \u03b1 \u2192 Prop} : ExactSolution spec \u2192 \u03b1 \n| ExactSolution.exact a' _ => a'\n\n-- inductive Parameter where\n-- | fin (n : Nat) (m : Fin n) : Parameter\n-- | nat (n : Nat) : Parameter\n-- | int  (n : Int) : Parameter\n-- | float (x : Float) : Parameter\n-- | string (s : String) : Parameter\n\n-- def Parameter.type : Parameter \u2192 Type\n-- | fin n _ => Fin n\n-- | nat _ => Nat\n-- | int _ => Int\n-- | float _ => Float\n-- | string _ => String\n\n-- def Parameter.val (p : Parameter) : p.type :=\n-- match p with\n-- | fin _ m => m\n-- | nat n => n\n-- | int n => n\n-- | float x => x\n-- | string s => s\n\n-- This might not be provable as `Fin n = Fin m` does not imply `n = m`\n-- instance (p p' : Parameter) : Decidable (p.type = p'.type) :=\n-- match p, p' with\n-- | .fin n _, .fin n' _ => if h : n = n' then isTrue (by simp[Parameter.type]; rw[h]) else isFalse sorry\n-- | .nat _, .nat _ => isTrue (by rfl) \n-- | .int _, .int _ => isTrue (by rfl) \n-- | .float _, .float _ => isTrue (by rfl)\n-- | .string _, .string _ => isTrue (by rfl)\n-- | _, _ => isFalse sorry\n\ninductive ApproxSolution {\u03b1 : Type _} : (spec : \u03b1 \u2192 Prop) \u2192 Type _ \n| exact {spec : \u03b1 \u2192 Prop}\n    (impl : \u03b1)\n    (h : spec impl)\n    : ApproxSolution spec\n| approx {spec : \u03b1 \u2192 Prop}\n    (spec\u2099 : \u2115 \u2192 \u03b1 \u2192 Prop)\n    (consistent : \u2200 (a\u2099 : \u2115 \u2192 \u03b1),\n      (\u2200 n, spec\u2099 n (a\u2099 n)) \u2192\n      (\u2203 a, (limit a\u2099 = a) \u2192 spec a))\n    (n\u2080 : \u2115) -- default value of `n` to be used when asembling\n    (impl : (n : \u2115) \u2192 ApproxSolution (spec\u2099 n))\n    (key help : String) -- `key` is used to modify value of `n\u2080`\n    : ApproxSolution spec\n-- | approx' {spec : \u03b1 \u2192 Prop}\n--     (spec\u2099 : P \u2192 \u03b1 \u2192 Prop)\n--     (filter : Filter P)\n--     (consistent : \u2200 (a\u2099 : P \u2192 \u03b1),\n--       (\u2200 p, spec\u2099 p (a\u2099 p)) \u2192\n--       (\u2203 a, (lim filter a\u2099 = a) \u2192 spec a))\n--     (p\u2080 : P) -- default value of `n` to be used when asembling\n--     (impl : P \u2192 \u03b1)\n--     (h : \u2200 p, spec\u2099 p (impl p))\n--     (key help : String) -- `key` is used to modify value of `n\u2080`\n--     : ApproxSolution spec\n| /--\n  There exists a limiting process but we are going to provide only fixed\napproximation\n -/\n weakApprox {spec : \u03b1 \u2192 Prop}\n    (spec\u2099 : \u2115 \u2192 \u03b1 \u2192 Prop)\n    (consistent : \u2200 (a\u2099 : \u2115 \u2192 \u03b1),\n      (\u2200 n, spec\u2099 n (a\u2099 n)) \u2192\n      (\u2203 a, (limit a\u2099 = a) \u2192 spec a))\n    (n : \u2115) -- fixed aproximation at compile time\n    (impl : ApproxSolution (spec\u2099 n))\n    (help : String)\n    : ApproxSolution spec\n-- | param {spec : \u03b1 \u2192 Prop} {\u03b2 : Type}\n--     (impl : \u03b2 \u2192 ApproxSolution spec)\n--     (p : Parameter)\n--     (h : p.type = \u03b2)\n--     (key : String)\n--     (help : String)\n--     : ApproxSolution spec\n-- | check {spec : \u03b1 \u2192 Prop}\n--     {P : Prop} [dec : Decidable P]\n--     (impl : P \u2192 ApproxSolution spec)\n--     (help : String)\n--     : ApproxSolution spec\n-- | assumption {spec : \u03b1 \u2192 Prop}\n--     {P : Prop}\n--     (impl : P \u2192 ApproxSolution spec)\n--     (help : String)\n--     : ApproxSolution spec\n\ndef ApproxSolution.val {\u03b1} {spec : \u03b1 \u2192 Prop} : ApproxSolution spec \u2192 \u03b1 \n| exact impl _ => impl\n| approx _ _ n impl _ _ => (impl n).val\n-- | param impl p h _ _ => (impl (h \u25b8 p.val)).val!\n| weakApprox _ _ _ impl _ => impl.val\n-- | check      impl _ => (impl sorry).val!\n-- | assumption impl _ => (impl sorry).val!\n\n-- def ApproxSolution.changeParam {\u03b1} [Vec \u03b1] {spec : \u03b1 \u2192 Prop} \n--   (p : Parameter) (key : String) \n--   : ApproxSolution spec \u2192 ApproxSolution spec\n-- | exact impl h => exact impl h\n-- | approx _ h n impl key' help => approx _ h n (\u03bb n => (impl n).changeParam p key) key' help\n-- | weakApprox _ h n impl help => weakApprox _ h n (impl.changeParam p key) help\n-- | param impl p' h key' help => \n--   if key = key' then\n--     if p.type = p'.type then\n--       param impl p' h key' help\n--     else\n--       param (\u03bb p' => (impl p').changeParam p key) p' h key' help\n--   else\n--     param (\u03bb p' => (impl p').changeParam p key) p' h key' help\n-- | check impl help => check (\u03bb h => (impl h).changeParam p key) help\n-- | assumption impl help => assumption (\u03bb h => (impl h).changeParam p key) help\n-- | e => e\n\n----------------------------------------------------------------------\n\n-- def Impl {\u03b1} (a : \u03b1) := ExactSolution (\u03bb x => x = a)\n-- def Impl.val {\u03b1} {a : \u03b1} (impl : Impl a) : \u03b1 := ExactSolution.val impl\n-- def Impl.exact {a : \u03b1} : Impl a := ExactSolution.exact a rfl\n\n-- @[simp]\n-- theorem Impl.impl_eq_spec (x : Impl a) : x.val = a :=\n-- by\n--   cases x; rename_i a' h; \n--   simp[ExactSolution.val, val, h]\n--   done\n\n\ndef Approx {\u03b1} (a : \u03b1) := ApproxSolution (\u03bb x => x = a)\ndef Approx.val {\u03b1} {a : \u03b1} (approx : Approx a) : \u03b1 := ApproxSolution.val approx\ndef Approx.exact {\u03b1} {a : \u03b1} : Approx a := ApproxSolution.exact a rfl\ndef Approx.limit {\u03b1} {a\u2099 : \u2115 \u2192 \u03b1} (x : (n : \u2115) \u2192 Approx (a\u2099 n)) (n\u2080 : \u2115)\n  : Approx (limit a\u2099) := ApproxSolution.approx (\u03bb n x => x = (a\u2099 n)) sorry n\u2080 x \"\" \"\" \n\n\n-- instance {\u03b1} (a : \u03b1) : Coe (Approx a) \u03b1 := \u27e8\u03bb approx => approx.val\u27e9\ninstance {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 \u03b2) : CoeFun (Approx f) (\u03bb _ => \u03b1 \u2192 \u03b2) := \u27e8\u03bb approx => approx.val\u27e9\n\nsyntax declModifiers \"approx \" declId bracketedBinder* (\":\" term)? \":=\" term \" by \" tacticSeq : command\n\nmacro_rules\n  | `($mods:declModifiers approx $id $params:bracketedBinder* := $body by $rewrites:tacticSeq) =>\n    `($mods:declModifiers def $id $params:bracketedBinder* := (by ($rewrites); (apply Approx.exact) : Approx $body))\n\n\n-- def foo (s : \u211d) := \u2207 (\u03bb x : \u211d => s * x)\n-- rewrite_by\n--   simp[gradient]\n\n-- Add proof and \nmacro \"approx_limit \" n0:term : tactic =>\n `(tactic| ((conv => enter [1]; bubble_lim; (tactic => sorry)); apply (Approx.limit _ ($n0:term))))\n\napprox bar (s : \u211d) (n\u2080 : \u2115) := \u2207 (limit \u03bb n => \u03bb x : \u211d => (s + (1:\u211d)/(n:\u211d)) * x)\nby\n  approx_limit n\u2080; intro n;\n  symdiff\n\n\n\n", "meta": {"author": "lecopivo", "repo": "SciLean", "sha": "e4fe5962c862f9854a6c88a4082eb01bc1147086", "save_path": "github-repos/lean/lecopivo-SciLean", "path": "github-repos/lean/lecopivo-SciLean/SciLean-e4fe5962c862f9854a6c88a4082eb01bc1147086/SciLean/Solver/Solver.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.36014167320412016}}
{"text": "--  An abstract formalization of \"isomorphism is equality up to relabeling\"\n-- -------------------------------------------------------------------------\n--\n-- See `README.md` for more info.\n--\n-- This file contains the definition of `Structure` as a higher groupoid, along with related concepts, up\n-- to a structure of structures called `universeStructure`.\n\n\n\nimport Structure.Generic.Axioms\n\nimport mathlib4_experiments.Data.Notation\n\n\n\n#exit\n\n\n\n\n\n\n\n\n\nstructure Iff' {\u03b1 \u03b2 : Sort v} [IsType \u03b1] [IsType \u03b2] (a : \u03b1) (b : \u03b2) where\n(mp  : a \u2192 b)\n(mpr : b \u2192 a)\n\ninfix:20 \" <->' \" => Iff'\ninfix:20 \" \u2194' \"   => Iff'\n\ndef Iff'.toIff {a b : Prop} (h : a \u2194' b) : a \u2194 b := \u27e8h.mp, h.mpr\u27e9\ninstance (a b : Prop) : Coe (a \u2194' b) (a \u2194 b) := \u27e8Iff'.toIff\u27e9\n\n\n\n-- A type is an instance of `HasGeneralStructure` if it has an equivalence that satisfies the isomorphism\n-- axioms. If equivalences of equivalences are propositions, this can be specialized to an instance of\n-- `HasStructure`. We optimize for this case but sometimes need the more general version.\n\nclass HasGeneralStructure (\u03b1 : Sort u) where\n{equivType   : Sort v}\n[equivIsType : IsTypeWithFunctorialEquivalence equivType]\n(M           : GeneralizedRelation \u03b1 equivType)\n[hasIsos     : HasIsomorphisms M]\n\nnamespace HasGeneralStructure\n\nvariable {\u03b1 : Sort u} [h : HasGeneralStructure \u03b1]\n\ninstance hasEquivalence : HasEquivalence \u03b1 \u03b1 := \u27e8h.M\u27e9\ninstance isTypeWithEquiv : IsTypeWithFunctorialEquivalence h.equivType := h.equivIsType\ninstance hasIso : HasIsomorphisms h.M := h.hasIsos\n\ninstance hasInstEquiv : HasInstanceEquivalence \u03b1 :=\n{ equivType := h.equivType,\n  Equiv     := h.M,\n  isEquiv   := isoEquiv h.M }\n\ninstance : IsTypeWithFunctorialEquivalence (HasEquivalence.\u03b3 \u03b1 \u03b1) := h.equivIsType\ninstance : HasIsomorphisms (@HasEquivalence.Equiv \u03b1 \u03b1 hasEquivalence) := hasIso\n\ndef id___ (a : \u03b1) : a \u2243 a := ident h.M a\n\ndef comp_congrArg_left  {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} : g\u2081 \u2243 g\u2082 \u2192 g\u2081 \u2022 f \u2243 g\u2082 \u2022 f := (isTypeWithEquiv.funCond (b \u2243 c) (a \u2243 c) _).coe (hasIso.trans_left_functor  f)\ndef comp_congrArg_right {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} : f\u2081 \u2243 f\u2082 \u2192 g \u2022 f\u2081 \u2243 g \u2022 f\u2082 := (isTypeWithEquiv.funCond (a \u2243 b) (a \u2243 c) _).coe (hasIso.trans_right_functor g)\ndef comp_congrArg {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c}  : f\u2081 \u2243 f\u2082 \u2192 g\u2081 \u2243 g\u2082 \u2192 g\u2081 \u2022 f\u2081 \u2243 g\u2082 \u2022 f\u2082 :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans (comp_congrArg_left h\u2082) (comp_congrArg_right h\u2081)\n\ndef inv_congrArg {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} : f\u2081 \u2243 f\u2082 \u2192 f\u2081\u207b\u00b9 \u2243 f\u2082\u207b\u00b9 := (isTypeWithEquiv.funCond (a \u2243 b) (b \u2243 a) _).coe hasIso.symm_functor\n\n        def assoc    {a b c d : \u03b1} (f : a \u2243 b) (g : b \u2243 c) (h : c \u2243 d) : h \u2022 (g \u2022 f) \u2243 (h \u2022 g) \u2022 f := hasIso.assoc    f g h\n        def assoc'   {a b c d : \u03b1} (f : a \u2243 b) (g : b \u2243 c) (h : c \u2243 d) : (h \u2022 g) \u2022 f \u2243 h \u2022 (g \u2022 f) := HasSymm.symm (assoc f g h)\n@[simp] def leftId   {a b     : \u03b1} (f : a \u2243 b)                         : id___ b \u2022 f \u2243 f           := hasIso.leftId   f\n@[simp] def rightId  {a b     : \u03b1} (f : a \u2243 b)                         : f \u2022 id___ a \u2243 f           := hasIso.rightId  f\n@[simp] def leftInv  {a b     : \u03b1} (f : a \u2243 b)                         : f\u207b\u00b9 \u2022 f     \u2243 id___ a     := hasIso.leftInv  f\n@[simp] def rightInv {a b     : \u03b1} (f : a \u2243 b)                         : f \u2022 f\u207b\u00b9     \u2243 id___ b     := hasIso.rightInv f\n@[simp] def invInv   {a b     : \u03b1} (f : a \u2243 b)                         : (f\u207b\u00b9)\u207b\u00b9     \u2243 f           := hasIso.invInv   f\n@[simp] def compInv  {a b c   : \u03b1} (f : a \u2243 b) (g : b \u2243 c)             : (g \u2022 f)\u207b\u00b9   \u2243 f\u207b\u00b9 \u2022 g\u207b\u00b9   := hasIso.compInv  f g\n@[simp] def idInv    (a       : \u03b1)                                     : (id___ a)\u207b\u00b9 \u2243 id___ a     := hasIso.idInv    a\n\ndef comp_subst  {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2243 f\u2082 \u2192 g\u2081 \u2243 g\u2082 \u2192 g\u2082 \u2022 f\u2082 \u2243 e \u2192 g\u2081 \u2022 f\u2081 \u2243 e :=\n\u03bb h\u2081 h\u2082 h\u2083 => HasTrans.trans (comp_congrArg h\u2081 h\u2082) h\u2083\ndef comp_subst' {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2243 f\u2082 \u2192 g\u2081 \u2243 g\u2082 \u2192 e \u2243 g\u2081 \u2022 f\u2081 \u2192 e \u2243 g\u2082 \u2022 f\u2082 :=\n\u03bb h\u2081 h\u2082 h\u2083 => HasTrans.trans h\u2083 (comp_congrArg h\u2081 h\u2082)\n\ndef comp_subst_left   {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : g\u2081 \u2243 g\u2082 \u2192 g\u2082 \u2022 f \u2243 e \u2192 g\u2081 \u2022 f \u2243 e :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans (comp_congrArg_left h\u2081) h\u2082\ndef comp_subst_left'  {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : g\u2081 \u2243 g\u2082 \u2192 e \u2243 g\u2081 \u2022 f \u2192 e \u2243 g\u2082 \u2022 f :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans h\u2082 (comp_congrArg_left h\u2081)\n\ndef comp_subst_right  {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2243 f\u2082 \u2192 g \u2022 f\u2082 \u2243 e \u2192 g \u2022 f\u2081 \u2243 e :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans (comp_congrArg_right h\u2081) h\u2082\ndef comp_subst_right' {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2243 f\u2082 \u2192 e \u2243 g \u2022 f\u2081 \u2192 e \u2243 g \u2022 f\u2082 :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans h\u2082 (comp_congrArg_right h\u2081)\n\ndef inv_subst  {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {e : b \u2243 a} : f\u2081 \u2243 f\u2082 \u2192 f\u2082\u207b\u00b9 \u2243 e \u2192 f\u2081\u207b\u00b9 \u2243 e :=\n\u03bb h\u2081 h\u2082 => HasTrans.trans (inv_congrArg h\u2081) h\u2082\ndef inv_subst' {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {e : b \u2243 a} : f\u2081 \u2243 f\u2082 \u2192 e \u2243 f\u2081\u207b\u00b9 \u2192 e \u2243 f\u2082\u207b\u00b9 :=\n\u03bb h\u2081 h\u2082 => HasSymm.symm (inv_subst (HasSymm.symm h\u2081) (HasSymm.symm h\u2082))\n\ndef leftCancelId  {a b : \u03b1} {f : a \u2243 b} {e : b \u2243 b} : e \u2243 id___ b \u2192 e \u2022 f \u2243 f :=\n\u03bb h => comp_subst_left  h (leftId  f)\ndef rightCancelId {a b : \u03b1} {f : a \u2243 b} {e : a \u2243 a} : e \u2243 id___ a \u2192 f \u2022 e \u2243 f :=\n\u03bb h => comp_subst_right h (rightId f)\n\ndef applyAssoc_left   {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} :\n  h \u2022 (g \u2022 f) \u2243 e \u2192 (h \u2022 g) \u2022 f \u2243 e :=\n\u03bb h\u2081 => HasTrans.trans (assoc' f g h) h\u2081\ndef applyAssoc_left'  {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} :\n  (h \u2022 g) \u2022 f \u2243 e \u2192 h \u2022 (g \u2022 f) \u2243 e :=\n\u03bb h\u2081 => HasTrans.trans (assoc f g h) h\u2081\ndef applyAssoc_right  {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} :\n  e \u2243 h \u2022 (g \u2022 f) \u2192 e \u2243 (h \u2022 g) \u2022 f :=\n\u03bb h\u2081 => HasTrans.trans h\u2081 (assoc f g h)\ndef applyAssoc_right' {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} :\n  e \u2243 (h \u2022 g) \u2022 f \u2192 e \u2243 h \u2022 (g \u2022 f) :=\n\u03bb h\u2081 => HasTrans.trans h\u2081 (assoc' f g h)\n\ndef applyAssoc  {a \u03b2\u2081 \u03b2\u2082 \u03b3\u2081 \u03b3\u2082 d : \u03b1} {f\u2081 : a \u2243 \u03b2\u2081} {f\u2082 : a \u2243 \u03b2\u2082} {g\u2081 : \u03b2\u2081 \u2243 \u03b3\u2081} {g\u2082 : \u03b2\u2082 \u2243 \u03b3\u2082} {h\u2081 : \u03b3\u2081 \u2243 d} {h\u2082 : \u03b3\u2082 \u2243 d} :\n  h\u2081 \u2022 (g\u2081 \u2022 f\u2081) \u2243 h\u2082 \u2022 (g\u2082 \u2022 f\u2082) \u2192 (h\u2081 \u2022 g\u2081) \u2022 f\u2081 \u2243 (h\u2082 \u2022 g\u2082) \u2022 f\u2082 :=\n\u03bb h => applyAssoc_right  (applyAssoc_left  h)\ndef applyAssoc' {a \u03b2\u2081 \u03b2\u2082 \u03b3\u2081 \u03b3\u2082 d : \u03b1} {f\u2081 : a \u2243 \u03b2\u2081} {f\u2082 : a \u2243 \u03b2\u2082} {g\u2081 : \u03b2\u2081 \u2243 \u03b3\u2081} {g\u2082 : \u03b2\u2082 \u2243 \u03b3\u2082} {h\u2081 : \u03b3\u2081 \u2243 d} {h\u2082 : \u03b3\u2082 \u2243 d} :\n  (h\u2081 \u2022 g\u2081) \u2022 f\u2081 \u2243 (h\u2082 \u2022 g\u2082) \u2022 f\u2082 \u2192 h\u2081 \u2022 (g\u2081 \u2022 f\u2081) \u2243 h\u2082 \u2022 (g\u2082 \u2022 f\u2082) :=\n\u03bb h => applyAssoc_right' (applyAssoc_left' h)\n\n@[simp] def leftCancel'     {a b c : \u03b1} (f : a \u2243 b) (g : b \u2243 c) : (g\u207b\u00b9 \u2022 g) \u2022 f \u2243 f := leftCancelId  (leftInv  g)\n@[simp] def leftCancel      {a b c : \u03b1} (f : a \u2243 b) (g : b \u2243 c) : g\u207b\u00b9 \u2022 (g \u2022 f) \u2243 f := applyAssoc_left' (leftCancel'     f g)\n@[simp] def leftCancelInv'  {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 b) : (g \u2022 g\u207b\u00b9) \u2022 f \u2243 f := leftCancelId  (rightInv g)\n@[simp] def leftCancelInv   {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 b) : g \u2022 (g\u207b\u00b9 \u2022 f) \u2243 f := applyAssoc_left' (leftCancelInv'  f g)\n@[simp] def rightCancel'    {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 a) : f \u2022 (g \u2022 g\u207b\u00b9) \u2243 f := rightCancelId (rightInv g)\n@[simp] def rightCancel     {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 a) : (f \u2022 g) \u2022 g\u207b\u00b9 \u2243 f := applyAssoc_left  (rightCancel'    f g)\n@[simp] def rightCancelInv' {a b c : \u03b1} (f : a \u2243 b) (g : a \u2243 c) : f \u2022 (g\u207b\u00b9 \u2022 g) \u2243 f := rightCancelId (leftInv  g)\n@[simp] def rightCancelInv  {a b c : \u03b1} (f : a \u2243 b) (g : a \u2243 c) : (f \u2022 g\u207b\u00b9) \u2022 g \u2243 f := applyAssoc_left  (rightCancelInv' f g)\n\ndef leftMulInv  {a b c : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g : b \u2243 c) : g \u2022 f\u2081 \u2243 f\u2082 \u2194' f\u2081 \u2243 g\u207b\u00b9 \u2022 f\u2082 :=\n\u27e8\u03bb h => comp_subst_right' h (HasSymm.symm (leftCancel f\u2081 g)), \u03bb h => comp_subst_right h (leftCancelInv f\u2082 g)\u27e9\ndef leftMulInv' {a b c : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g : c \u2243 b) : g\u207b\u00b9 \u2022 f\u2081 \u2243 f\u2082 \u2194' f\u2081 \u2243 g \u2022 f\u2082 :=\n\u27e8\u03bb h => comp_subst_right' h (HasSymm.symm (leftCancelInv f\u2081 g)), \u03bb h => comp_subst_right h (leftCancel f\u2082 g)\u27e9\n\n@[simp] def leftMul {a b c : \u03b1} (f\u2081 f\u2082 : a \u2243 b) (g : b \u2243 c) : g \u2022 f\u2081 \u2243 g \u2022 f\u2082 \u2194' f\u2081 \u2243 f\u2082 :=\n\u27e8\u03bb h => HasTrans.trans ((leftMulInv f\u2081 (g \u2022 f\u2082) g).mp h) (leftCancel f\u2082 g), comp_congrArg_right\u27e9\n\ndef rightMulInv  {a b c : \u03b1} (f\u2081 : a \u2243 c) (f\u2082 : b \u2243 c) (g : b \u2243 a) : f\u2081 \u2022 g \u2243 f\u2082 \u2194' f\u2081 \u2243 f\u2082 \u2022 g\u207b\u00b9 :=\n\u27e8\u03bb h => comp_subst_left' h (HasSymm.symm (rightCancel f\u2081 g)), \u03bb h => comp_subst_left h (rightCancelInv f\u2082 g)\u27e9\ndef rightMulInv' {a b c : \u03b1} (f\u2081 : a \u2243 c) (f\u2082 : b \u2243 c) (g : a \u2243 b) : f\u2081 \u2022 g\u207b\u00b9 \u2243 f\u2082 \u2194' f\u2081 \u2243 f\u2082 \u2022 g :=\n\u27e8\u03bb h => comp_subst_left' h (HasSymm.symm (rightCancelInv f\u2081 g)), \u03bb h => comp_subst_left h (rightCancel f\u2082 g)\u27e9\n\n@[simp] def rightMul {a b c : \u03b1} (f\u2081 f\u2082 : a \u2243 b) (g : c \u2243 a) : f\u2081 \u2022 g \u2243 f\u2082 \u2022 g \u2194' f\u2081 \u2243 f\u2082 :=\n\u27e8\u03bb h => HasTrans.trans ((rightMulInv f\u2081 (f\u2082 \u2022 g) g).mp h) (rightCancel f\u2082 g), comp_congrArg_left\u27e9\n\ndef eqInvIffInvEq {a b : \u03b1} (f : a \u2243 b) (g : b \u2243 a) : f \u2243 g\u207b\u00b9 \u2194' f\u207b\u00b9 \u2243 g :=\n\u27e8\u03bb h => inv_subst h (invInv g), \u03bb h => inv_subst' h (HasSymm.symm (invInv f))\u27e9\n\n@[simp] def eqIffEqInv {a b : \u03b1} (f\u2081 f\u2082 : a \u2243 b) : f\u2081\u207b\u00b9 \u2243 f\u2082\u207b\u00b9 \u2194' f\u2081 \u2243 f\u2082 :=\n\u27e8\u03bb h => HasTrans.trans ((eqInvIffInvEq f\u2081 f\u2082\u207b\u00b9).mpr h) (invInv f\u2082), inv_congrArg\u27e9\n\n@[simp] def leftRightMul {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g\u2081 : b \u2243 d) (g\u2082 : c \u2243 d) :\n  g\u2082\u207b\u00b9 \u2022 g\u2081 \u2243 f\u2082 \u2022 f\u2081\u207b\u00b9 \u2194' g\u2081 \u2022 f\u2081 \u2243 g\u2082 \u2022 f\u2082 :=\n\u27e8\u03bb h => let h\u2081 := (rightMulInv (g\u2082\u207b\u00b9 \u2022 g\u2081) f\u2082 f\u2081).mpr h;\n        let h\u2082 := applyAssoc_left' h\u2081;\n        (leftMulInv' (g\u2081 \u2022 f\u2081) f\u2082 g\u2082).mp h\u2082,\n \u03bb h => let h\u2081 := (rightMulInv g\u2081 (g\u2082 \u2022 f\u2082) f\u2081).mp h;\n        let h\u2082 := applyAssoc_right' h\u2081;\n        (leftMulInv' g\u2081 (f\u2082 \u2022 f\u2081\u207b\u00b9) g\u2082).mpr h\u2082\u27e9\n\ndef swapInv  {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : c \u2243 d) (g\u2081 : d \u2243 b) (g\u2082 : c \u2243 a) :\n  g\u2081\u207b\u00b9 \u2022 f\u2081 \u2243 f\u2082 \u2022 g\u2082\u207b\u00b9 \u2192 f\u2081\u207b\u00b9 \u2022 g\u2081 \u2243 g\u2082 \u2022 f\u2082\u207b\u00b9 :=\n\u03bb h => (leftRightMul f\u2082 g\u2082 g\u2081 f\u2081).mpr (HasSymm.symm ((leftRightMul g\u2082 f\u2082 f\u2081 g\u2081).mp h))\n\ndef swapInv' {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : c \u2243 d) (g\u2081 : d \u2243 b) (g\u2082 : c \u2243 a) :\n  f\u2082 \u2022 g\u2082\u207b\u00b9 \u2243 g\u2081\u207b\u00b9 \u2022 f\u2081 \u2192 g\u2082 \u2022 f\u2082\u207b\u00b9 \u2243 f\u2081\u207b\u00b9 \u2022 g\u2081 :=\n\u03bb h => HasSymm.symm (swapInv f\u2081 f\u2082 g\u2081 g\u2082 (HasSymm.symm h))\n\nend HasGeneralStructure\n\n\n\n-- A variant of `HasGeneralStructure` where `equivType` is `BundledSetoid`, i.e. equivalences of\n-- equivalences are propositions.\n\nclass HasStructure (\u03b1 : Sort u) where\n(M       : GeneralizedRelation \u03b1 BundledSetoid)\n[hasIsos : HasIsomorphisms M]\n\nnamespace HasStructure\n\nvariable {\u03b1 : Sort u} [h : HasStructure \u03b1]\n\ninstance hasGeneralStructure : HasGeneralStructure \u03b1 :=\n{ equivType := BundledSetoid,\n  M         := h.M,\n  hasIsos   := h.hasIsos }\n\ninstance hasIso : HasIsomorphisms h.M := h.hasIsos\ninstance equivSetoid (a b : \u03b1) : Setoid (IsType.type (a \u2243 b)) := BundledSetoid.isSetoid (a \u2243 b)\n\ndef id_ (a : \u03b1) : a \u2243 a := hasGeneralStructure.id___ a\ndef id' {a : \u03b1} := id_ a\n\ntheorem comp_congrArg_left  {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} : g\u2081 \u2248 g\u2082 \u2192 g\u2081 \u2022 f \u2248 g\u2082 \u2022 f := hasGeneralStructure.comp_congrArg_left\ntheorem comp_congrArg_right {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} : f\u2081 \u2248 f\u2082 \u2192 g \u2022 f\u2081 \u2248 g \u2022 f\u2082 := hasGeneralStructure.comp_congrArg_right\ntheorem comp_congrArg {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c}  : f\u2081 \u2248 f\u2082 \u2192 g\u2081 \u2248 g\u2082 \u2192 g\u2081 \u2022 f\u2081 \u2248 g\u2082 \u2022 f\u2082 := hasGeneralStructure.comp_congrArg\n\ntheorem inv_congrArg  {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} : f\u2081 \u2248 f\u2082 \u2192 f\u2081\u207b\u00b9 \u2248 f\u2082\u207b\u00b9 := hasGeneralStructure.inv_congrArg\n\n        theorem assoc    {a b c d : \u03b1} (f : a \u2243 b) (g : b \u2243 c) (h : c \u2243 d) : h \u2022 (g \u2022 f) \u2248 (h \u2022 g) \u2022 f := hasGeneralStructure.assoc    f g h\n        theorem assoc'   {a b c d : \u03b1} (f : a \u2243 b) (g : b \u2243 c) (h : c \u2243 d) : (h \u2022 g) \u2022 f \u2248 h \u2022 (g \u2022 f) := hasGeneralStructure.assoc' f g h\n@[simp] theorem leftId   {a b     : \u03b1} (f : a \u2243 b)                         : id_ b \u2022 f \u2248 f             := hasGeneralStructure.leftId   f\n@[simp] theorem rightId  {a b     : \u03b1} (f : a \u2243 b)                         : f \u2022 id_ a \u2248 f             := hasGeneralStructure.rightId  f\n@[simp] theorem leftInv  {a b     : \u03b1} (f : a \u2243 b)                         : f\u207b\u00b9 \u2022 f   \u2248 id_ a         := hasGeneralStructure.leftInv  f\n@[simp] theorem rightInv {a b     : \u03b1} (f : a \u2243 b)                         : f \u2022 f\u207b\u00b9   \u2248 id_ b         := hasGeneralStructure.rightInv f\n@[simp] theorem invInv   {a b     : \u03b1} (f : a \u2243 b)                         : (f\u207b\u00b9)\u207b\u00b9   \u2248 f             := hasGeneralStructure.invInv   f\n@[simp] theorem compInv  {a b c   : \u03b1} (f : a \u2243 b) (g : b \u2243 c)             : (g \u2022 f)\u207b\u00b9 \u2248 f\u207b\u00b9 \u2022 g\u207b\u00b9     := hasGeneralStructure.compInv  f g\n@[simp] theorem idInv    (a       : \u03b1)                                     : (id_ a)\u207b\u00b9 \u2248 id_ a         := hasGeneralStructure.idInv    a\n\ntheorem comp_subst  {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2248 f\u2082 \u2192 g\u2081 \u2248 g\u2082 \u2192 g\u2082 \u2022 f\u2082 \u2248 e \u2192 g\u2081 \u2022 f\u2081 \u2248 e := hasGeneralStructure.comp_subst\ntheorem comp_subst' {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2248 f\u2082 \u2192 g\u2081 \u2248 g\u2082 \u2192 e \u2248 g\u2081 \u2022 f\u2081 \u2192 e \u2248 g\u2082 \u2022 f\u2082 := hasGeneralStructure.comp_subst'\n\ntheorem comp_subst_left   {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : g\u2081 \u2248 g\u2082 \u2192 g\u2082 \u2022 f \u2248 e \u2192 g\u2081 \u2022 f \u2248 e := hasGeneralStructure.comp_subst_left\ntheorem comp_subst_left'  {a b c : \u03b1} {f : a \u2243 b} {g\u2081 g\u2082 : b \u2243 c} {e : a \u2243 c} : g\u2081 \u2248 g\u2082 \u2192 e \u2248 g\u2081 \u2022 f \u2192 e \u2248 g\u2082 \u2022 f := hasGeneralStructure.comp_subst_left'\n\ntheorem comp_subst_right  {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2248 f\u2082 \u2192 g \u2022 f\u2082 \u2248 e \u2192 g \u2022 f\u2081 \u2248 e := hasGeneralStructure.comp_subst_right\ntheorem comp_subst_right' {a b c : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {g : b \u2243 c} {e : a \u2243 c} : f\u2081 \u2248 f\u2082 \u2192 e \u2248 g \u2022 f\u2081 \u2192 e \u2248 g \u2022 f\u2082 := hasGeneralStructure.comp_subst_right'\n\ntheorem inv_subst  {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {e : b \u2243 a} : f\u2081 \u2248 f\u2082 \u2192 f\u2082\u207b\u00b9 \u2248 e \u2192 f\u2081\u207b\u00b9 \u2248 e := hasGeneralStructure.inv_subst\ntheorem inv_subst' {a b : \u03b1} {f\u2081 f\u2082 : a \u2243 b} {e : b \u2243 a} : f\u2081 \u2248 f\u2082 \u2192 e \u2248 f\u2081\u207b\u00b9 \u2192 e \u2248 f\u2082\u207b\u00b9 := hasGeneralStructure.inv_subst'\n\ntheorem leftCancelId  {a b : \u03b1} {f : a \u2243 b} {e : b \u2243 b} : e \u2248 id' \u2192 e \u2022 f \u2248 f := hasGeneralStructure.leftCancelId\ntheorem rightCancelId {a b : \u03b1} {f : a \u2243 b} {e : a \u2243 a} : e \u2248 id' \u2192 f \u2022 e \u2248 f := hasGeneralStructure.rightCancelId\n\ntheorem applyAssoc_left   {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} : h \u2022 (g \u2022 f) \u2248 e \u2192 (h \u2022 g) \u2022 f \u2248 e := hasGeneralStructure.applyAssoc_left\ntheorem applyAssoc_left'  {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} : (h \u2022 g) \u2022 f \u2248 e \u2192 h \u2022 (g \u2022 f) \u2248 e := hasGeneralStructure.applyAssoc_left'\ntheorem applyAssoc_right  {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} : e \u2248 h \u2022 (g \u2022 f) \u2192 e \u2248 (h \u2022 g) \u2022 f := hasGeneralStructure.applyAssoc_right\ntheorem applyAssoc_right' {a b c d : \u03b1} {f : a \u2243 b} {g : b \u2243 c} {h : c \u2243 d} {e : a \u2243 d} : e \u2248 (h \u2022 g) \u2022 f \u2192 e \u2248 h \u2022 (g \u2022 f) := hasGeneralStructure.applyAssoc_right'\n\ntheorem applyAssoc  {a \u03b2\u2081 \u03b2\u2082 \u03b3\u2081 \u03b3\u2082 d : \u03b1} {f\u2081 : a \u2243 \u03b2\u2081} {f\u2082 : a \u2243 \u03b2\u2082} {g\u2081 : \u03b2\u2081 \u2243 \u03b3\u2081} {g\u2082 : \u03b2\u2082 \u2243 \u03b3\u2082} {h\u2081 : \u03b3\u2081 \u2243 d} {h\u2082 : \u03b3\u2082 \u2243 d} :\n  h\u2081 \u2022 (g\u2081 \u2022 f\u2081) \u2248 h\u2082 \u2022 (g\u2082 \u2022 f\u2082) \u2192 (h\u2081 \u2022 g\u2081) \u2022 f\u2081 \u2248 (h\u2082 \u2022 g\u2082) \u2022 f\u2082 :=\nhasGeneralStructure.applyAssoc\ntheorem applyAssoc' {a \u03b2\u2081 \u03b2\u2082 \u03b3\u2081 \u03b3\u2082 d : \u03b1} {f\u2081 : a \u2243 \u03b2\u2081} {f\u2082 : a \u2243 \u03b2\u2082} {g\u2081 : \u03b2\u2081 \u2243 \u03b3\u2081} {g\u2082 : \u03b2\u2082 \u2243 \u03b3\u2082} {h\u2081 : \u03b3\u2081 \u2243 d} {h\u2082 : \u03b3\u2082 \u2243 d} :\n  (h\u2081 \u2022 g\u2081) \u2022 f\u2081 \u2248 (h\u2082 \u2022 g\u2082) \u2022 f\u2082 \u2192 h\u2081 \u2022 (g\u2081 \u2022 f\u2081) \u2248 h\u2082 \u2022 (g\u2082 \u2022 f\u2082) :=\nhasGeneralStructure.applyAssoc'\n\n@[simp] theorem leftCancel'     {a b c : \u03b1} (f : a \u2243 b) (g : b \u2243 c) : (g\u207b\u00b9 \u2022 g) \u2022 f \u2248 f := hasGeneralStructure.leftCancel'     f g\n@[simp] theorem leftCancel      {a b c : \u03b1} (f : a \u2243 b) (g : b \u2243 c) : g\u207b\u00b9 \u2022 (g \u2022 f) \u2248 f := hasGeneralStructure.leftCancel      f g\n@[simp] theorem leftCancelInv'  {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 b) : (g \u2022 g\u207b\u00b9) \u2022 f \u2248 f := hasGeneralStructure.leftCancelInv'  f g\n@[simp] theorem leftCancelInv   {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 b) : g \u2022 (g\u207b\u00b9 \u2022 f) \u2248 f := hasGeneralStructure.leftCancelInv   f g\n@[simp] theorem rightCancel'    {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 a) : f \u2022 (g \u2022 g\u207b\u00b9) \u2248 f := hasGeneralStructure.rightCancel'    f g\n@[simp] theorem rightCancel     {a b c : \u03b1} (f : a \u2243 b) (g : c \u2243 a) : (f \u2022 g) \u2022 g\u207b\u00b9 \u2248 f := hasGeneralStructure.rightCancel     f g\n@[simp] theorem rightCancelInv' {a b c : \u03b1} (f : a \u2243 b) (g : a \u2243 c) : f \u2022 (g\u207b\u00b9 \u2022 g) \u2248 f := hasGeneralStructure.rightCancelInv' f g\n@[simp] theorem rightCancelInv  {a b c : \u03b1} (f : a \u2243 b) (g : a \u2243 c) : (f \u2022 g\u207b\u00b9) \u2022 g \u2248 f := hasGeneralStructure.rightCancelInv  f g\n\ntheorem leftMulInv  {a b c : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g : b \u2243 c) : g \u2022 f\u2081 \u2248 f\u2082 \u2194 f\u2081 \u2248 g\u207b\u00b9 \u2022 f\u2082 := Iff'.toIff (hasGeneralStructure.leftMulInv  f\u2081 f\u2082 g)\ntheorem leftMulInv' {a b c : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g : c \u2243 b) : g\u207b\u00b9 \u2022 f\u2081 \u2248 f\u2082 \u2194 f\u2081 \u2248 g \u2022 f\u2082 := Iff'.toIff (hasGeneralStructure.leftMulInv' f\u2081 f\u2082 g)\n\n@[simp] theorem leftMul {a b c : \u03b1} (f\u2081 f\u2082 : a \u2243 b) (g : b \u2243 c) : g \u2022 f\u2081 \u2248 g \u2022 f\u2082 \u2194 f\u2081 \u2248 f\u2082 := Iff'.toIff (hasGeneralStructure.leftMul f\u2081 f\u2082 g)\n\ntheorem rightMulInv  {a b c : \u03b1} (f\u2081 : a \u2243 c) (f\u2082 : b \u2243 c) (g : b \u2243 a) : f\u2081 \u2022 g \u2248 f\u2082 \u2194 f\u2081 \u2248 f\u2082 \u2022 g\u207b\u00b9 := Iff'.toIff (hasGeneralStructure.rightMulInv  f\u2081 f\u2082 g)\ntheorem rightMulInv' {a b c : \u03b1} (f\u2081 : a \u2243 c) (f\u2082 : b \u2243 c) (g : a \u2243 b) : f\u2081 \u2022 g\u207b\u00b9 \u2248 f\u2082 \u2194 f\u2081 \u2248 f\u2082 \u2022 g := Iff'.toIff (hasGeneralStructure.rightMulInv' f\u2081 f\u2082 g)\n\n@[simp] theorem rightMul {a b c : \u03b1} (f\u2081 f\u2082 : a \u2243 b) (g : c \u2243 a) : f\u2081 \u2022 g \u2248 f\u2082 \u2022 g \u2194 f\u2081 \u2248 f\u2082 := Iff'.toIff (hasGeneralStructure.rightMul f\u2081 f\u2082 g)\n\ntheorem eqInvIffInvEq {a b : \u03b1} (f : a \u2243 b) (g : b \u2243 a) : f \u2248 g\u207b\u00b9 \u2194 f\u207b\u00b9 \u2248 g := Iff'.toIff (hasGeneralStructure.eqInvIffInvEq f g)\n\n@[simp] theorem eqIffEqInv {a b : \u03b1} (f\u2081 f\u2082 : a \u2243 b) : f\u2081\u207b\u00b9 \u2248 f\u2082\u207b\u00b9 \u2194 f\u2081 \u2248 f\u2082 := Iff'.toIff (hasGeneralStructure.eqIffEqInv f\u2081 f\u2082)\n\n@[simp] theorem leftRightMul {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : a \u2243 c) (g\u2081 : b \u2243 d) (g\u2082 : c \u2243 d) :\n  g\u2082\u207b\u00b9 \u2022 g\u2081 \u2248 f\u2082 \u2022 f\u2081\u207b\u00b9 \u2194 g\u2081 \u2022 f\u2081 \u2248 g\u2082 \u2022 f\u2082 :=\nIff'.toIff (hasGeneralStructure.leftRightMul f\u2081 f\u2082 g\u2081 g\u2082)\n\ntheorem swapInv  {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : c \u2243 d) (g\u2081 : d \u2243 b) (g\u2082 : c \u2243 a) :\n  g\u2081\u207b\u00b9 \u2022 f\u2081 \u2248 f\u2082 \u2022 g\u2082\u207b\u00b9 \u2192 f\u2081\u207b\u00b9 \u2022 g\u2081 \u2248 g\u2082 \u2022 f\u2082\u207b\u00b9 :=\nhasGeneralStructure.swapInv f\u2081 f\u2082 g\u2081 g\u2082\n\ntheorem swapInv' {a b c d : \u03b1} (f\u2081 : a \u2243 b) (f\u2082 : c \u2243 d) (g\u2081 : d \u2243 b) (g\u2082 : c \u2243 a) :\n  f\u2082 \u2022 g\u2082\u207b\u00b9 \u2248 g\u2081\u207b\u00b9 \u2022 f\u2081 \u2192 g\u2082 \u2022 f\u2082\u207b\u00b9 \u2248 f\u2081\u207b\u00b9 \u2022 g\u2081 :=\nhasGeneralStructure.swapInv' f\u2081 f\u2082 g\u2081 g\u2082\n\nend HasStructure\n\nopen HasStructure\n\n\n\ninstance propHasStructure                               : HasStructure Prop := \u27e8RelationWithSetoid.relWithEq Iff\u27e9\ndef      typeHasStructure   (\u03b1 : Sort u)                : HasStructure \u03b1    := \u27e8RelationWithSetoid.relWithEq Eq\u27e9\ndef      setoidHasStructure (\u03b1 : Sort u) [s : Setoid \u03b1] : HasStructure \u03b1    := \u27e8RelationWithSetoid.relWithEq s.r\u27e9\n\n\n\n-- We bundle a type with a structure together because we frequently parameterize definitions by\n-- arbitrary structures.\n\nstructure Structure where\n(\u03b1         : Sort u)\n[hasStruct : HasStructure \u03b1]\n\nnamespace Structure\n\ninstance structureIsType : IsType Structure := \u27e8Structure.\u03b1\u27e9\n\ndef iso (S : Structure) : RelationWithSetoid (IsType.type S) := S.hasStruct.M\n\nvariable {S : Structure}\n\ninstance hasStructure : HasStructure (IsType.type S) := S.hasStruct\ninstance hasGeneralStructure : HasGeneralStructure (IsType.type S) := HasStructure.hasGeneralStructure (h := hasStructure)\ninstance hasIso : HasIsomorphisms (t := BundledSetoid.isTypeWithFunctorialEquivalence) (iso S) := hasGeneralStructure.hasIso\n\ninstance structureIsTypeWithEquiv : IsTypeWithEquivalence Structure :=\n{ type  := Structure.\u03b1,\n  equiv := \u03bb S => HasGeneralStructure.hasInstEquiv (h := hasGeneralStructure) }\n\ndef id__ (a : S) : a \u2243 a := hasStructure.id_ a\ndef id'' {a : S} := id__ a\n\nend Structure\n\nopen Structure\n\ndef defaultStructure (\u03b1 : Sort u) [h : HasStructure \u03b1] : Structure :=\n{ \u03b1         := \u03b1,\n  hasStruct := h }\n\ndef instanceStructure (\u03b1 : Sort u) := @defaultStructure \u03b1 (typeHasStructure \u03b1)\ndef setoidInstanceStructure (\u03b1 : Sort u) [s : Setoid \u03b1] := @defaultStructure \u03b1 (setoidHasStructure \u03b1)\ndef bundledSetoidStructure (S : BundledSetoid) := setoidInstanceStructure (IsType.type S)\n\n\n\n-- Since each equivalence/isomorphism of a structure is a bundled setoid, we can treat it as a\n-- structure as well. This partially recovers the inductive definition of a structure as an \u221e-groupoid.\n\ndef isoStructure {S : Structure} (a b : S) := bundledSetoidStructure (iso S a b)\n\n\n\n-- We can \"forget\" the data held inside a `Structure` on two levels, obtaining modified instances of\n-- `Structure`:\n--\n-- 1. We can truncate the equivalence to an equivalence _relation_, obtaining a \"setoid structure.\"\n--\n-- 2. In Lean, where quotients are available, we can additionally take the quotient with respect to\n--    equivalence, obtaining a \"skeleton structure\" where equivalence is equality.\n--\n-- Moreover, if we have a type with general equivalences, we can obtain a `Structure` by truncating them.\n--\n-- In `Forgetfulness.lean`, we prove some properties of these operations.\n--\n-- Within this file, we truncate structures to setoids whenever we want to use structures as isomorphisms,\n-- but we never use quotients. With an inductive version of `Structure` (i.e. an actual \u221e-groupoid), we\n-- could keep all data instead.\n\nnamespace Forgetfulness\n\nsection SetoidEquiv\n\nvariable (\u03b1 : Sort u) [HasInstanceEquivalence \u03b1]\n\ndef SetoidEquiv (a b : \u03b1) := Nonempty (IsType.type (a \u2243 b))\ndef toSetoidEquiv {a b : \u03b1} (e : a \u2243 b) : SetoidEquiv \u03b1 a b := \u27e8e\u27e9\ndef setoidEquiv : Equivalence (SetoidEquiv \u03b1) :=\n\u27e8\u03bb a => \u27e8HasRefl.refl a\u27e9, \u03bb \u27e8e\u27e9 => \u27e8HasSymm.symm e\u27e9, \u03bb \u27e8e\u27e9 \u27e8f\u27e9 => \u27e8HasTrans.trans e f\u27e9\u27e9\n\ninstance instanceEquivSetoid : Setoid \u03b1 := \u27e8SetoidEquiv \u03b1, setoidEquiv \u03b1\u27e9\n\nend SetoidEquiv\n\nsection Structures\n\nvariable (S : Structure)\n\ndef structureSetoidEquiv {a b : S} (e : a \u2243 b) := toSetoidEquiv (IsType.type S) e\ndef structureToSetoid := instanceEquivSetoid (IsType.type S)\ndef setoidStructure : Structure := setoidInstanceStructure (IsType.type S)\n\ndef StructureQuotient := Quotient (structureToSetoid S)\ndef skeletonStructure : Structure := instanceStructure (StructureQuotient S)\n\nend Structures\n\nsection SetoidEquivEquiv\n\ndef equivSetoid {\u03b1 : Sort u} [HasGeneralStructure \u03b1] (a b : \u03b1) : BundledSetoid :=\n{ \u03b1 := IsType.type (a \u2243 b),\n  s := instanceEquivSetoid (IsType.type (a \u2243 b)) }\n\ninstance equivHasIso {\u03b1 : Sort u} [h : HasGeneralStructure \u03b1] : HasIsomorphisms (@equivSetoid \u03b1 h) :=\n{ refl                := h.hasIsos.refl,\n  symm                := h.hasIsos.symm,\n  trans               := h.hasIsos.trans,\n  trans_left_functor  := \u03bb f g\u2081 g\u2082 \u27e8he\u27e9 => \u27e8HasGeneralStructure.comp_congrArg_left  he\u27e9,\n  trans_right_functor := \u03bb g f\u2081 f\u2081 \u27e8he\u27e9 => \u27e8HasGeneralStructure.comp_congrArg_right he\u27e9,\n  trans_nat           := Unit.unit,\n  symm_functor        := \u03bb \u27e8he\u27e9   => \u27e8HasGeneralStructure.inv_congrArg  he\u27e9,\n  assoc               := \u03bb e f g  => \u27e8HasGeneralStructure.assoc         e f g\u27e9,\n  leftId              := \u03bb e      => \u27e8HasGeneralStructure.leftId        e\u27e9,\n  rightId             := \u03bb e      => \u27e8HasGeneralStructure.rightId       e\u27e9,\n  leftInv             := \u03bb e      => \u27e8HasGeneralStructure.leftInv       e\u27e9,\n  rightInv            := \u03bb e      => \u27e8HasGeneralStructure.rightInv      e\u27e9,\n  invInv              := \u03bb e      => \u27e8HasGeneralStructure.invInv        e\u27e9,\n  compInv             := \u03bb e f    => \u27e8HasGeneralStructure.compInv       e f\u27e9,\n  idInv               := \u03bb a      => \u27e8HasGeneralStructure.idInv         a\u27e9 }\n\ninstance hasTruncatedStructure (\u03b1 : Sort u) [h : HasGeneralStructure \u03b1] : HasStructure \u03b1 :=\n\u27e8@equivSetoid \u03b1 h\u27e9\n\nend SetoidEquivEquiv\n\nend Forgetfulness\n\nopen Forgetfulness\n\n\n\n-- As a simple example of a custom structure, we define a structure for the Cartesian product of two\n-- structures.\n\ndef StructureProduct (S T : Structure) := PProd (IsType.type S) (IsType.type T)\n\nnamespace StructureProduct\n\nvariable {S T : Structure}\n\ndef ProductEquiv (P Q : StructureProduct S T) := PProd (IsType.type (P.fst \u2243 Q.fst)) (IsType.type (P.snd \u2243 Q.snd))\n\nnamespace ProductEquiv\n\ndef refl  (P     : StructureProduct S T)                                               : ProductEquiv P P :=\n\u27e8HasRefl.refl   P.fst,       HasRefl.refl   P.snd\u27e9\ndef symm  {P Q   : StructureProduct S T} (e : ProductEquiv P Q)                        : ProductEquiv Q P :=\n\u27e8HasSymm.symm   e.fst,       HasSymm.symm   e.snd\u27e9\ndef trans {P Q R : StructureProduct S T} (e : ProductEquiv P Q) (f : ProductEquiv Q R) : ProductEquiv P R :=\n\u27e8HasTrans.trans e.fst f.fst, HasTrans.trans e.snd f.snd\u27e9\n\ndef EquivEquiv {P Q : StructureProduct S T} (e f : ProductEquiv P Q) :=\ne.fst \u2248 f.fst \u2227 e.snd \u2248 f.snd\n\nnamespace EquivEquiv\n\nvariable {P Q : StructureProduct S T}\n\ntheorem refl  (e     : ProductEquiv P Q)                                           : EquivEquiv e e :=\n\u27e8Setoid.refl  e.fst,         Setoid.refl  e.snd\u27e9\ntheorem symm  {e f   : ProductEquiv P Q} (h : EquivEquiv e f)                      : EquivEquiv f e :=\n\u27e8Setoid.symm  h.left,        Setoid.symm  h.right\u27e9\ntheorem trans {e f g : ProductEquiv P Q} (h : EquivEquiv e f) (i : EquivEquiv f g) : EquivEquiv e g :=\n\u27e8Setoid.trans h.left i.left, Setoid.trans h.right i.right\u27e9\n\ninstance productEquivSetoid : Setoid (ProductEquiv P Q) := \u27e8EquivEquiv, \u27e8refl, symm, trans\u27e9\u27e9\n\nend EquivEquiv\n\ndef productEquiv : RelationWithSetoid (StructureProduct S T) := \u03bb P Q => \u27e8ProductEquiv P Q\u27e9\n\ntheorem comp_congrArg {P Q R : StructureProduct S T} {e\u2081 e\u2082 : ProductEquiv P Q} {f\u2081 f\u2082 : ProductEquiv Q R} (he : e\u2081 \u2248 e\u2082) (hf : f\u2081 \u2248 f\u2082) :\n  trans e\u2081 f\u2081 \u2248 trans e\u2082 f\u2082 :=\n\u27e8HasStructure.comp_congrArg he.left hf.left,   HasStructure.comp_congrArg he.right hf.right\u27e9\n\ntheorem inv_congrArg {P Q : StructureProduct S T} {e\u2081 e\u2082 : ProductEquiv P Q} (he : e\u2081 \u2248 e\u2082) :\n  symm e\u2081 \u2248 symm e\u2082 :=\n\u27e8HasStructure.inv_congrArg  he.left,           HasStructure.inv_congrArg  he.right\u27e9\n\ntheorem assoc {P Q R Z : StructureProduct S T} (e : ProductEquiv P Q) (f : ProductEquiv Q R) (g : ProductEquiv R Z) :\n  trans (trans e f) g \u2248 trans e (trans f g) :=\n\u27e8HasStructure.assoc         e.fst f.fst g.fst, HasStructure.assoc         e.snd f.snd g.snd\u27e9\n\ntheorem leftId  {P Q : StructureProduct S T} (e : ProductEquiv P Q) : trans e (refl Q) \u2248 e :=\n\u27e8HasStructure.leftId        e.fst,             HasStructure.leftId        e.snd\u27e9\ntheorem rightId {P Q : StructureProduct S T} (e : ProductEquiv P Q) : trans (refl P) e \u2248 e :=\n\u27e8HasStructure.rightId       e.fst,             HasStructure.rightId       e.snd\u27e9\n\ntheorem leftInv  {P Q : StructureProduct S T} (e : ProductEquiv P Q) : trans e (symm e) \u2248 refl P :=\n\u27e8HasStructure.leftInv       e.fst,             HasStructure.leftInv       e.snd\u27e9\ntheorem rightInv {P Q : StructureProduct S T} (e : ProductEquiv P Q) : trans (symm e) e \u2248 refl Q :=\n\u27e8HasStructure.rightInv      e.fst,             HasStructure.rightInv      e.snd\u27e9\n\ntheorem invInv {P Q : StructureProduct S T} (e : ProductEquiv P Q) : symm (symm e) \u2248 e :=\n\u27e8HasStructure.invInv        e.fst,             HasStructure.invInv        e.snd\u27e9\n\ntheorem compInv {P Q R : StructureProduct S T} (e : ProductEquiv P Q) (f : ProductEquiv Q R) :\n  symm (trans e f) \u2248 trans (symm f) (symm e) :=\n\u27e8HasStructure.compInv       e.fst f.fst,       HasStructure.compInv       e.snd f.snd\u27e9\n\ntheorem idInv (P : StructureProduct S T) : symm (refl P) \u2248 refl P :=\n\u27e8HasStructure.idInv         P.fst,             HasStructure.idInv         P.snd\u27e9\n\ninstance productEquivHasIso : HasIsomorphisms (t := BundledSetoid.isTypeWithFunctorialEquivalence) (@productEquiv S T) :=\n{ refl          := refl,\n  symm          := symm,\n  trans         := trans,\n  comp_congrArg := comp_congrArg,\n  inv_congrArg  := inv_congrArg,\n  assoc         := assoc,\n  leftId        := leftId,\n  rightId       := rightId,\n  leftInv       := leftInv,\n  rightInv      := rightInv,\n  invInv        := invInv,\n  compInv       := compInv,\n  idInv         := idInv }\n\nend ProductEquiv\n\ninstance productHasStructure (S T : Structure) : HasStructure (StructureProduct S T) := \u27e8ProductEquiv.productEquiv\u27e9\ndef productStructure (S T : Structure) : Structure := \u27e8StructureProduct S T\u27e9\n\nend StructureProduct\n\n\n\n-- A bundled version of `IsIsomorphismFunctor` where the codomains are structures.\n-- Therefore, equivalences of equivalences are setoids.\n\n@[reducible] def isoRel {\u03b1 : Sort u} {S : Structure} (s : \u03b1 \u2192 S) := mapRelation s (iso S)\n\nstructure GeneralizedFunctor {\u03b1 : Sort u} {S T : Structure} (s : \u03b1 \u2192 S) (t : \u03b1 \u2192 T) where\n(mapEquiv {a b : \u03b1} : s a \u2243 s b \u2192 t a \u2243 t b)\n[isFunctor          : IsIsomorphismFunctor (isoRel s) (isoRel t) mapEquiv]\n\nnamespace GeneralizedFunctor\n\n@[reducible] def Functor {S T : Structure} (s : S \u2192 T) := GeneralizedFunctor id s\n\nvariable {\u03b1 : Sort u} {S T U : Structure}\n\ninstance (s : \u03b1 \u2192 S) (t : \u03b1 \u2192 T) :\n  CoeFun (GeneralizedFunctor s t) (\u03bb _ => \u2200 {a b : \u03b1}, s a \u2243 s b \u2192 t a \u2243 t b) :=\n\u27e8GeneralizedFunctor.mapEquiv\u27e9\n\ndef mapFunctor {\u03c9 : Sort w} {s : \u03b1 \u2192 S} {t : \u03b1 \u2192 T} (m : \u03c9 \u2192 \u03b1) (\u03c6 : GeneralizedFunctor s t) :\n  GeneralizedFunctor (s \u2218 m) (t \u2218 m) :=\n{ mapEquiv  := \u03c6.mapEquiv,\n  isFunctor := mapIsoFunctor (isoRel s) (isoRel t) \u03c6.mapEquiv (h := \u03c6.isFunctor) m }\n\ninstance {\u03c9 : Sort w} (s : \u03b1 \u2192 S) (t : \u03b1 \u2192 T) (m : \u03c9 \u2192 \u03b1) :\n  Coe (GeneralizedFunctor s t) (GeneralizedFunctor (s \u2218 m) (t \u2218 m)) :=\n\u27e8mapFunctor m\u27e9\n\nnamespace id\n\nvariable {s : \u03b1 \u2192 S}\n\ndef genFun : GeneralizedFunctor s s := \u27e8id\u27e9\n\nend id\n\nnamespace comp\n\nvariable {s : \u03b1 \u2192 S} {t : \u03b1 \u2192 T} {u : \u03b1 \u2192 U} (\u03c6 : GeneralizedFunctor s t) (\u03c8 : GeneralizedFunctor t u)\n\ninstance : IsIsomorphismFunctor (isoRel s) (isoRel u) (mapEquiv \u03c8 \u2218 mapEquiv \u03c6) :=\ncompIsoFunctor (isoRel s) (isoRel t) (isoRel u) \u03c6.mapEquiv (hF := \u03c6.isFunctor) \u03c8.mapEquiv (hG := \u03c8.isFunctor)\n\ndef genFun : GeneralizedFunctor s u := \u27e8\u03c8.mapEquiv \u2218 \u03c6.mapEquiv\u27e9\n\nend comp\n\ndef comp.genFun' {\u03c9 : Sort w} {s : \u03b1 \u2192 S} {t : \u03c9 \u2192 T} {u : \u03c9 \u2192 U} (m : \u03b1 \u2192 \u03c9)\n                 (\u03c6 : GeneralizedFunctor s (t \u2218 m)) (\u03c8 : GeneralizedFunctor t u) :\n  GeneralizedFunctor s (u \u2218 m) :=\ncomp.genFun \u03c6 (mapFunctor m \u03c8)\n\nnamespace const\n\nvariable {s : \u03b1 \u2192 S} (c : T)\n\ndef genFun : GeneralizedFunctor s (Function.const \u03b1 c) :=\n{ mapEquiv  := \u03bb _ => HasRefl.refl c,\n  isFunctor := { respectsEquiv := \u03bb _   => Setoid.refl (id_ c),\n                 respectsComp  := \u03bb _ _ => Setoid.symm (leftId (id_ c)),\n                 respectsId    := \u03bb _   => Setoid.refl (id_ c),\n                 respectsInv   := \u03bb _   => Setoid.symm (idInv c) } }\n\nend const\n\nend GeneralizedFunctor\n\nopen GeneralizedFunctor\n\n\n\ndef Pi {\u03b1 : Sort u} (C : \u03b1 \u2192 Structure) := \u2200 a, C a\n\nnamespace Pi\n\nvariable {\u03b1 : Sort u} {C : \u03b1 \u2192 Structure}\n\ndef mapPi {\u03c9 : Sort w} (m : \u03c9 \u2192 \u03b1) (p : Pi C) : Pi (C \u2218 m) :=\n\u03bb b => p (m b)\n\ndef PiEquiv (p q : Pi C) := \u2200 a, p a \u2243 q a\n\nnamespace PiEquiv\n\ndef refl  (p     : Pi C)                                     : PiEquiv p p :=\n\u03bb a => HasRefl.refl   (p a)\ndef symm  {p q   : Pi C} (\u03b7 : PiEquiv p q)                   : PiEquiv q p :=\n\u03bb a => HasSymm.symm   (\u03b7 a)\ndef trans {p q H : Pi C} (\u03b7 : PiEquiv p q) (\u03b8 : PiEquiv q H) : PiEquiv p H :=\n\u03bb a => HasTrans.trans (\u03b7 a) (\u03b8 a)\n\ndef piIsoStructure (p q : Pi C) (a : \u03b1) := isoStructure (p a) (q a)\n\ndef mapPiEquiv {\u03c9 : Sort w} (m : \u03c9 \u2192 \u03b1) {p q : Pi C} (\u03b7 : PiEquiv p q) :\n  PiEquiv (mapPi m p) (mapPi m q) :=\n\u03bb b => \u03b7 (m b)\n\ndef EquivEquiv {p q : Pi C} (\u03b7 \u03b8 : PiEquiv p q) :=\n@PiEquiv \u03b1 (piIsoStructure p q) \u03b7 \u03b8\n\nnamespace EquivEquiv\n\nvariable {p q : Pi C}\n\ntheorem refl  (\u03b7     : PiEquiv p q)                                           : EquivEquiv \u03b7 \u03b7 :=\n@PiEquiv.refl \u03b1 (piIsoStructure p q) \u03b7\ntheorem symm  {\u03b7 \u03b8   : PiEquiv p q} (h : EquivEquiv \u03b7 \u03b8)                      : EquivEquiv \u03b8 \u03b7 :=\nPiEquiv.symm  h\ntheorem trans {\u03b7 \u03b8 \u03b6 : PiEquiv p q} (h : EquivEquiv \u03b7 \u03b8) (i : EquivEquiv \u03b8 \u03b6) : EquivEquiv \u03b7 \u03b6 :=\nPiEquiv.trans h i\n\ninstance piEquivSetoid : Setoid (PiEquiv p q) := \u27e8EquivEquiv, \u27e8refl, symm, trans\u27e9\u27e9\n\nend EquivEquiv\n\ndef piEquiv : RelationWithSetoid (Pi C) := \u03bb p q => \u27e8PiEquiv p q\u27e9\n\ninstance piEquivHasIso : HasIsomorphisms (t := BundledSetoid.isTypeWithFunctorialEquivalence) (@piEquiv \u03b1 C) :=\n{ refl          := refl,\n  symm          := symm,\n  trans         := trans,\n  comp_congrArg := \u03bb h\u03b7 h\u03b8 a => comp_congrArg (h\u03b7 a) (h\u03b8 a),\n  inv_congrArg  := \u03bb h\u03b7    a => inv_congrArg  (h\u03b7 a),\n  assoc         := \u03bb \u03b7 \u03b8 \u03b6 a => assoc         (\u03b7 a) (\u03b8 a) (\u03b6 a),\n  leftId        := \u03bb \u03b7     a => leftId        (\u03b7 a),\n  rightId       := \u03bb \u03b7     a => rightId       (\u03b7 a),\n  leftInv       := \u03bb \u03b7     a => leftInv       (\u03b7 a),\n  rightInv      := \u03bb \u03b7     a => rightInv      (\u03b7 a),\n  invInv        := \u03bb \u03b7     a => invInv        (\u03b7 a),\n  compInv       := \u03bb \u03b7 \u03b8   a => compInv       (\u03b7 a) (\u03b8 a),\n  idInv         := \u03bb b     a => idInv         (b a) }\n\n@[reducible] def MappedPiEquiv {\u03c9 : Sort w} (m : \u03c9 \u2192 Pi C) (b c : \u03c9) := PiEquiv (m b) (m c)\n\nnamespace MappedPiEquiv\n\nvariable {\u03c9 : Sort w} {m : \u03c9 \u2192 Pi C}\n\ndef refl  (b     : \u03c9)                                                     : MappedPiEquiv m b b :=\nPiEquiv.refl  (m b)\ndef symm  {b c   : \u03c9} (e : MappedPiEquiv m b c)                           : MappedPiEquiv m c b :=\nPiEquiv.symm  e\ndef trans {b c d : \u03c9} (e : MappedPiEquiv m b c) (f : MappedPiEquiv m c d) : MappedPiEquiv m b d :=\nPiEquiv.trans e f\n\ninstance EquivEquiv.mappedPiEquivSetoid {b c : \u03c9} : Setoid (MappedPiEquiv m b c) := EquivEquiv.piEquivSetoid\n\ndef mappedPiEquiv : RelationWithSetoid \u03c9 := \u03bb b c => \u27e8MappedPiEquiv m b c\u27e9\n\ninstance mappedPiEquivHasIso : HasIsomorphisms (@mappedPiEquiv \u03b1 C \u03c9 m) :=\nmapHasIso (t := BundledSetoid.isTypeWithFunctorialEquivalence) (@piEquiv \u03b1 C) m\n\nend MappedPiEquiv\n\n\n\n-- If we have two functions that map from an arbitrary `\u03b1` into the same structure `S`, and for each\n-- instance of `\u03b1` we have an equivalence between the values of both functions, that gives us something\n-- that can act as an equivalence between the two functions. In particular:\n--\n-- * If both are functors, this gives us a definition of equivalence of functors.\n--\n-- * If only one of them is a functor, we can use the equivalence to turn the other function into a\n--   functor as well.\n\nvariable {\u03b1 : Sort u} {S : Structure} {p q : \u03b1 \u2192 S} (\u03b7 : PiEquiv p q)\n\n-- We can \"transport\" an equivalence `e` between two values of `p` to an equivalence between the\n-- corresponding two values of another equivalent function `q`.\n\ndef transport    {a b : \u03b1} (e : p a \u2243 p b) : q a \u2243 q b := \u03b7 b \u2022 e \u2022 (\u03b7 a)\u207b\u00b9\ndef invTransport {a b : \u03b1} (e : q a \u2243 q b) : p a \u2243 p b := (\u03b7 b)\u207b\u00b9 \u2022 e \u2022 \u03b7 a\n\nnamespace transport\n\ntheorem isInverse {a b : \u03b1} (e : q a \u2243 q b) :\n  transport (PiEquiv.symm \u03b7) e \u2248 invTransport \u03b7 e :=\ncomp_congrArg_right (comp_congrArg_right (invInv (\u03b7 a)))\n\ntheorem respectsEquiv {a b   : \u03b1} {e\u2081 e\u2082 : p a \u2243 p b} (h : e\u2081 \u2248 e\u2082) :\n  transport \u03b7 e\u2081 \u2248 transport \u03b7 e\u2082 :=\ncomp_congrArg_right (comp_congrArg_left h)\n\ntheorem respectsComp  {a b c : \u03b1} (e : p a \u2243 p b) (f : p b \u2243 p c) :\n  transport \u03b7 (f \u2022 e) \u2248 transport \u03b7 f \u2022 transport \u03b7 e :=\nlet \u03b7a := \u03b7 a;\nlet \u03b7b := \u03b7 b;\nlet \u03b7c := \u03b7 c;\nlet h\u2081 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 \u03b7c \u2022 (f \u2022 (id' \u2022 e)) \u2022 \u03b7a\u207b\u00b9           := comp_congrArg_right (comp_congrArg_left (comp_congrArg_right (Setoid.symm (leftId e))));\nlet h\u2082 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 \u03b7c \u2022 (f \u2022 ((\u03b7b\u207b\u00b9 \u2022 \u03b7b) \u2022 e)) \u2022 \u03b7a\u207b\u00b9   := Setoid.trans h\u2081 (comp_congrArg_right (comp_congrArg_left (comp_congrArg_right (comp_congrArg_left (Setoid.symm (leftInv \u03b7b))))));\nlet h\u2083 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 \u03b7c \u2022 (f \u2022 (\u03b7b\u207b\u00b9 \u2022 (\u03b7b \u2022 e))) \u2022 \u03b7a\u207b\u00b9   := Setoid.trans h\u2082 (comp_congrArg_right (comp_congrArg_left (comp_congrArg_right (Setoid.symm (assoc e \u03b7b \u03b7b\u207b\u00b9)))));\nlet h\u2084 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 \u03b7c \u2022 ((f \u2022 \u03b7b\u207b\u00b9) \u2022 (\u03b7b \u2022 e)) \u2022 \u03b7a\u207b\u00b9   := Setoid.trans h\u2083 (comp_congrArg_right (comp_congrArg_left (assoc (\u03b7b \u2022 e) \u03b7b\u207b\u00b9 f)));\nlet h\u2085 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 \u03b7c \u2022 (f \u2022 \u03b7b\u207b\u00b9) \u2022 ((\u03b7b \u2022 e) \u2022 \u03b7a\u207b\u00b9)   := Setoid.trans h\u2084 (comp_congrArg_right (Setoid.symm (assoc \u03b7a\u207b\u00b9 (\u03b7b \u2022 e) (f \u2022 \u03b7b\u207b\u00b9))));\nlet h\u2086 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 (\u03b7c \u2022 (f \u2022 \u03b7b\u207b\u00b9)) \u2022 ((\u03b7b \u2022 e) \u2022 \u03b7a\u207b\u00b9) := Setoid.trans h\u2085 (assoc ((\u03b7b \u2022 e) \u2022 \u03b7a\u207b\u00b9) (f \u2022 \u03b7b\u207b\u00b9) \u03b7c);\nlet h\u2087 : \u03b7c \u2022 (f \u2022 e) \u2022 \u03b7a\u207b\u00b9 \u2248 (\u03b7c \u2022 f \u2022 \u03b7b\u207b\u00b9) \u2022 (\u03b7b \u2022 e \u2022 \u03b7a\u207b\u00b9)     := Setoid.trans h\u2086 (comp_congrArg_right (Setoid.symm (assoc \u03b7a\u207b\u00b9 e \u03b7b)));\nh\u2087\n\ntheorem respectsId    (a     : \u03b1) :\n  transport \u03b7 (id_ (p a)) \u2248 id' :=\nlet \u03b7a := \u03b7 a;\nlet h\u2081 : \u03b7a \u2022 id' \u2022 \u03b7a\u207b\u00b9 \u2248 id' := comp_subst_right (leftId \u03b7a\u207b\u00b9) (rightInv \u03b7a);\nh\u2081\n\ntheorem respectsInv   {a b   : \u03b1} (e : p a \u2243 p b) :\n  transport \u03b7 e\u207b\u00b9 \u2248 (transport \u03b7 e)\u207b\u00b9 :=\nlet \u03b7a := \u03b7 a;\nlet \u03b7b := \u03b7 b;\nlet h\u2081 : \u03b7a \u2022 e\u207b\u00b9 \u2022 \u03b7b\u207b\u00b9 \u2248 (\u03b7a\u207b\u00b9)\u207b\u00b9 \u2022 (\u03b7b \u2022 e)\u207b\u00b9 := comp_congrArg (Setoid.symm (compInv e \u03b7b)) (Setoid.symm (invInv \u03b7a));\nlet h\u2082 : \u03b7a \u2022 e\u207b\u00b9 \u2022 \u03b7b\u207b\u00b9 \u2248 ((\u03b7b \u2022 e) \u2022 \u03b7a\u207b\u00b9)\u207b\u00b9   := Setoid.trans h\u2081 (Setoid.symm (compInv \u03b7a\u207b\u00b9 (\u03b7b \u2022 e)));\nlet h\u2083 : \u03b7a \u2022 e\u207b\u00b9 \u2022 \u03b7b\u207b\u00b9 \u2248 (\u03b7b \u2022 e \u2022 \u03b7a\u207b\u00b9)\u207b\u00b9     := Setoid.trans h\u2082 (inv_congrArg (Setoid.symm (assoc \u03b7a\u207b\u00b9 e \u03b7b)));\nh\u2083\n\ndef functor : GeneralizedFunctor p q :=\n{ mapEquiv  := transport \u03b7,\n  isFunctor := { respectsEquiv := respectsEquiv \u03b7,\n                 respectsComp  := respectsComp  \u03b7,\n                 respectsId    := respectsId    \u03b7,\n                 respectsInv   := respectsInv   \u03b7 } }\n\ntheorem invRespectsEquiv {a b   : \u03b1} {e\u2081 e\u2082 : q a \u2243 q b} (h : e\u2081 \u2248 e\u2082) :\n  invTransport \u03b7 e\u2081 \u2248 invTransport \u03b7 e\u2082 :=\nlet h\u2081 := respectsEquiv (PiEquiv.symm \u03b7) h;\nSetoid.trans (Setoid.symm (isInverse \u03b7 e\u2081)) (Setoid.trans h\u2081 (isInverse \u03b7 e\u2082))\n\ntheorem invRespectsComp  {a b c : \u03b1} (e : q a \u2243 q b) (f : q b \u2243 q c) :\n  invTransport \u03b7 (f \u2022 e) \u2248 invTransport \u03b7 f \u2022 invTransport \u03b7 e :=\nlet h\u2081 := respectsComp (PiEquiv.symm \u03b7) e f;\nSetoid.trans (Setoid.symm (isInverse \u03b7 (f \u2022 e))) (Setoid.trans h\u2081 (comp_congrArg (isInverse \u03b7 e) (isInverse \u03b7 f)))\n\ntheorem invRespectsId    (a     : \u03b1) :\n  invTransport \u03b7 (id_ (q a)) \u2248 id' :=\nlet h\u2081 := respectsId (PiEquiv.symm \u03b7) a;\nSetoid.trans (Setoid.symm (isInverse \u03b7 (id_ (q a)))) h\u2081\n\ntheorem invRespectsInv   {a b   : \u03b1} (e : q a \u2243 q b) :\n  invTransport \u03b7 e\u207b\u00b9 \u2248 (invTransport \u03b7 e)\u207b\u00b9 :=\nlet h\u2081 := respectsInv (PiEquiv.symm \u03b7) e;\nSetoid.trans (Setoid.symm (isInverse \u03b7 e\u207b\u00b9)) (Setoid.trans h\u2081 (inv_congrArg (isInverse \u03b7 e)))\n\ndef invFunctor : GeneralizedFunctor q p :=\n{ mapEquiv  := invTransport \u03b7,\n  isFunctor := { respectsEquiv := invRespectsEquiv \u03b7,\n                 respectsComp  := invRespectsComp  \u03b7,\n                 respectsId    := invRespectsId    \u03b7,\n                 respectsInv   := invRespectsInv   \u03b7 } }\n\nend transport\n\nend PiEquiv\n\nend Pi\n\nopen Pi\n\n\n\ndef GeneralizedNaturalityCondition {\u03b1 : Sort u} {S T : Structure} {s : \u03b1 \u2192 S} {t\u2081 t\u2082 : \u03b1 \u2192 T}\n                                   (\u03c6 : GeneralizedFunctor s t\u2081) (\u03c8 : GeneralizedFunctor s t\u2082)\n                                   (ext : PiEquiv t\u2081 t\u2082) :=\n\u2200 {a b : \u03b1} (e : s a \u2243 s b), \u03c8 e \u2022 ext a \u2248 ext b \u2022 \u03c6 e\n\nnamespace GeneralizedNaturalityCondition\n\nvariable {\u03b1 : Sort u} {S T : Structure}\n\ntheorem refl  {s : \u03b1 \u2192 S} {t\u2081       : \u03b1 \u2192 T}\n              (\u03c6 : GeneralizedFunctor s t\u2081) :\n  GeneralizedNaturalityCondition \u03c6 \u03c6 (PiEquiv.refl t\u2081) :=\n\u03bb e => Setoid.trans (rightId (\u03c6 e)) (Setoid.symm (leftId (\u03c6 e)))\n\ntheorem symm  {s : \u03b1 \u2192 S} {t\u2081 t\u2082    : \u03b1 \u2192 T}\n              {\u03c6 : GeneralizedFunctor s t\u2081} {\u03c8 : GeneralizedFunctor s t\u2082}\n              {ext : PiEquiv t\u2081 t\u2082}\n              (nat : GeneralizedNaturalityCondition \u03c6 \u03c8 ext) :\n  GeneralizedNaturalityCondition \u03c8 \u03c6 (PiEquiv.symm ext) :=\n\u03bb {a b} e => Setoid.symm ((leftRightMul (ext a) (\u03c6 e) (\u03c8 e) (ext b)).mpr (nat e))\n\ntheorem trans {s : \u03b1 \u2192 S} {t\u2081 t\u2082 t\u2083 : \u03b1 \u2192 T}\n              {\u03c6 : GeneralizedFunctor s t\u2081} {\u03c8 : GeneralizedFunctor s t\u2082} {\u03c7 : GeneralizedFunctor s t\u2083}\n              {ext\u2081 : PiEquiv t\u2081 t\u2082}                           {ext\u2082 : PiEquiv t\u2082 t\u2083}\n              (nat\u2081 : GeneralizedNaturalityCondition \u03c6 \u03c8 ext\u2081) (nat\u2082 : GeneralizedNaturalityCondition \u03c8 \u03c7 ext\u2082) :\n  GeneralizedNaturalityCondition \u03c6 \u03c7 (PiEquiv.trans ext\u2081 ext\u2082) :=\n\u03bb {a b} e => let h\u2081 := (rightMulInv (\u03c8 e) (ext\u2081 b \u2022 \u03c6 e) (ext\u2081 a)).mp  (nat\u2081 e);\n             let h\u2082 := (leftMulInv' (\u03c7 e \u2022 ext\u2082 a) (\u03c8 e) (ext\u2082 b)).mpr (nat\u2082 e);\n             let h\u2083 := (leftRightMul (ext\u2081 a) (ext\u2081 b \u2022 \u03c6 e) (\u03c7 e \u2022 ext\u2082 a) (ext\u2082 b)).mp (Setoid.trans h\u2082 h\u2081);\n             applyAssoc_left' (applyAssoc_right h\u2083)\n\nend GeneralizedNaturalityCondition\n\n\n\nstructure GeneralizedNaturalTransformation {\u03b1 : Sort u} {S T : Structure} {s : \u03b1 \u2192 S} {t\u2081 t\u2082 : \u03b1 \u2192 T}\n                                           (\u03c6 : GeneralizedFunctor s t\u2081) (\u03c8 : GeneralizedFunctor s t\u2082) where\n(ext : PiEquiv t\u2081 t\u2082)\n(nat : GeneralizedNaturalityCondition \u03c6 \u03c8 ext)\n\nnamespace GeneralizedNaturalTransformation\n\nvariable {\u03b1 : Sort u} {S T : Structure}\n\ndef refl  {s : \u03b1 \u2192 S} {t\u2081       : \u03b1 \u2192 T} (\u03c6 : GeneralizedFunctor s t\u2081) :\n  GeneralizedNaturalTransformation \u03c6 \u03c6 :=\n\u27e8PiEquiv.refl  t\u2081,          GeneralizedNaturalityCondition.refl  \u03c6\u27e9\n\ndef symm  {s : \u03b1 \u2192 S} {t\u2081 t\u2082    : \u03b1 \u2192 T} {\u03c6 : GeneralizedFunctor s t\u2081} {\u03c8 : GeneralizedFunctor s t\u2082}\n          (\u03b7 : GeneralizedNaturalTransformation \u03c6 \u03c8) :\n  GeneralizedNaturalTransformation \u03c8 \u03c6 :=\n\u27e8PiEquiv.symm  \u03b7.ext,       GeneralizedNaturalityCondition.symm  \u03b7.nat\u27e9\n\ndef trans {s : \u03b1 \u2192 S} {t\u2081 t\u2082 t\u2083 : \u03b1 \u2192 T} {\u03c6 : GeneralizedFunctor s t\u2081} {\u03c8 : GeneralizedFunctor s t\u2082} {\u03c7 : GeneralizedFunctor s t\u2083}\n          (\u03b7 : GeneralizedNaturalTransformation \u03c6 \u03c8) (\u03b8 : GeneralizedNaturalTransformation \u03c8 \u03c7) :\n  GeneralizedNaturalTransformation \u03c6 \u03c7 :=\n\u27e8PiEquiv.trans \u03b7.ext \u03b8.ext, GeneralizedNaturalityCondition.trans \u03b7.nat \u03b8.nat\u27e9\n\ninstance naturalTransformationSetoid {s : \u03b1 \u2192 S} {t\u2081 t\u2082 : \u03b1 \u2192 T} (\u03c6 : GeneralizedFunctor s t\u2081) (\u03c8 : GeneralizedFunctor s t\u2082) :\n  Setoid (GeneralizedNaturalTransformation \u03c6 \u03c8) :=\n\u27e8\u03bb e f => PiEquiv.EquivEquiv e.ext f.ext,\n \u27e8\u03bb e => PiEquiv.EquivEquiv.refl e.ext, PiEquiv.EquivEquiv.symm, PiEquiv.EquivEquiv.trans\u27e9\u27e9\n\ndef mapNaturalTransformation {\u03c9 : Sort w} {s : \u03b1 \u2192 S} {t\u2081 t\u2082 : \u03b1 \u2192 T} (m : \u03c9 \u2192 \u03b1)\n                             {\u03c6 : GeneralizedFunctor s t\u2081} {\u03c8 : GeneralizedFunctor s t\u2082}\n                             (\u03b7 : GeneralizedNaturalTransformation \u03c6 \u03c8) :\n  GeneralizedNaturalTransformation (mapFunctor m \u03c6) (mapFunctor m \u03c8) :=\n\u27e8PiEquiv.mapPiEquiv m \u03b7.ext, \u03b7.nat\u27e9\n\nend GeneralizedNaturalTransformation\n\n\n\n-- A functor between two `Structure`s is a map that also maps equivalences in a compatible way. On the\n-- one hand, this is just a groupoid functor, but on the other hand, the mapping of equivalences also\n-- matches exactly the `mapEquiv` map mentioned in the introduction.\n--\n-- Moreover, if we interpret `\u2243` as a generalization of equality, the mapping of equivalences is actually\n-- the generalized version of `congrArg`. Under this interpretation, it can also be regarded as a\n-- well-definedness condition for the map: equality of arguments implies equality of results.\n\nstructure StructureFunctor (S T : Structure) :=\n(map     : S \u2192 T)\n(functor : Functor map)\n\nnamespace StructureFunctor\n\nvariable {S T U V : Structure}\n\ninstance functorCoeFun : CoeFun (StructureFunctor S T) (\u03bb _ => S \u2192 T) := \u27e8StructureFunctor.map\u27e9\n\n        theorem respectsSetoid (F : StructureFunctor S T) {a b   : S} {f\u2081 f\u2082 : a \u2243 b} :\n  f\u2081 \u2248 f\u2082 \u2192 F.functor f\u2081 \u2248 F.functor f\u2082         := F.functor.isFunctor.respectsEquiv\n@[simp] theorem respectsComp   (F : StructureFunctor S T) {a b c : S} (f : a \u2243 b) (g : b \u2243 c) :\n  F.functor (g \u2022 f) \u2248 F.functor g \u2022 F.functor f := F.functor.isFunctor.respectsComp f g\n@[simp] theorem respectsId     (F : StructureFunctor S T) (a     : S) :\n  F.functor (id_ a) \u2248 id'                       := F.functor.isFunctor.respectsId   a\n@[simp] theorem respectsInv    (F : StructureFunctor S T) {a b   : S} (f : a \u2243 b) :\n  F.functor f\u207b\u00b9     \u2248 (F.functor f)\u207b\u00b9           := F.functor.isFunctor.respectsInv  f\n\n\n\ndef congrArg (F : StructureFunctor S T) {a b : S} : a \u2243 b \u2192 F a \u2243 F b := F.functor.mapEquiv\n\n\n\n-- We can define equivalence of functors by extensionality, using equivalence in `T` instead of equality.\n-- This is an equivalence according to our definition, and it is compatible with isomorphisms via the\n-- functor axioms, so we can use it to build an instance of `Structure` again.\n--\n-- For equivalence of functors to be well-behaved, we additionally need to require equivalences to be\n-- natural transformations.\n\ndef FunExt (F G : StructureFunctor S T) := PiEquiv.MappedPiEquiv StructureFunctor.map F G\n\nnamespace FunExt\n\ninstance {F G : StructureFunctor S T} : Setoid (FunExt F G) :=\nPiEquiv.MappedPiEquiv.EquivEquiv.mappedPiEquivSetoid\n\ndef funExt : RelationWithSetoid (StructureFunctor S T) := \u03bb F G => \u27e8FunExt F G\u27e9\n\ninstance funExtHasIso : HasIsomorphisms (@funExt S T) := PiEquiv.MappedPiEquiv.mappedPiEquivHasIso\n\nend FunExt\n\ndef FunctorEquiv (F G : StructureFunctor S T) := GeneralizedNaturalTransformation F.functor G.functor\n\nnamespace FunctorEquiv\n\ndef refl  (F     : StructureFunctor S T)                                               : FunctorEquiv F F :=\nGeneralizedNaturalTransformation.refl  F.functor\ndef symm  {F G   : StructureFunctor S T} (\u03b7 : FunctorEquiv F G)                        : FunctorEquiv G F :=\nGeneralizedNaturalTransformation.symm  \u03b7\ndef trans {F G H : StructureFunctor S T} (\u03b7 : FunctorEquiv F G) (\u03b8 : FunctorEquiv G H) : FunctorEquiv F H :=\nGeneralizedNaturalTransformation.trans \u03b7 \u03b8\n\ninstance (F G : StructureFunctor S T) : Setoid (FunctorEquiv F G) :=\nGeneralizedNaturalTransformation.naturalTransformationSetoid F.functor G.functor\n\ndef functorEquiv : RelationWithSetoid (StructureFunctor S T) := \u03bb F G => \u27e8FunctorEquiv F G\u27e9\n\ninstance functorEquivHasIso : HasIsomorphisms (@functorEquiv S T) :=\n{ refl          := refl,\n  symm          := symm,\n  trans         := trans,\n  comp_congrArg := \u03bb h\u03b7 h\u03b8 => FunExt.funExtHasIso.comp_congrArg h\u03b7 h\u03b8,\n  inv_congrArg  := \u03bb h\u03b7    => FunExt.funExtHasIso.inv_congrArg  h\u03b7,\n  assoc         := \u03bb \u03b7 \u03b8 \u03b6 => FunExt.funExtHasIso.assoc         \u03b7.ext \u03b8.ext \u03b6.ext,\n  leftId        := \u03bb \u03b7     => FunExt.funExtHasIso.leftId        \u03b7.ext,\n  rightId       := \u03bb \u03b7     => FunExt.funExtHasIso.rightId       \u03b7.ext,\n  leftInv       := \u03bb \u03b7     => FunExt.funExtHasIso.leftInv       \u03b7.ext,\n  rightInv      := \u03bb \u03b7     => FunExt.funExtHasIso.rightInv      \u03b7.ext,\n  invInv        := \u03bb \u03b7     => FunExt.funExtHasIso.invInv        \u03b7.ext,\n  compInv       := \u03bb \u03b7 \u03b8   => FunExt.funExtHasIso.compInv       \u03b7.ext \u03b8.ext,\n  idInv         := \u03bb F     => FunExt.funExtHasIso.idInv         F }\n\nend FunctorEquiv\n\ninstance functorHasStructure : HasStructure (StructureFunctor S T) := \u27e8FunctorEquiv.functorEquiv\u27e9\ndef functorStructure (S T : Structure) : Structure := \u27e8StructureFunctor S T\u27e9\n\ninstance : CoeFun (IsType.type (functorStructure S T)) (\u03bb _ => S \u2192 T) := functorCoeFun\n\n\n\n-- We have two alternative definitions of `congr` for functors, depending on the order in which we apply\n-- the functor and argument equivalences. The natural transformation axiom says exactly that the order\n-- does not matter.\n\ndef congr  {F\u2081 F\u2082 : StructureFunctor S T} {a b : S} : F\u2081 \u2243 F\u2082 \u2192 a \u2243 b \u2192 F\u2081 a \u2243 F\u2082 b :=\n\u03bb \u03b7 e => HasTrans.trans (\u03b7.ext a) (F\u2082.functor e)\n\ndef congr' {F\u2081 F\u2082 : StructureFunctor S T} {a b : S} : F\u2081 \u2243 F\u2082 \u2192 a \u2243 b \u2192 F\u2081 a \u2243 F\u2082 b :=\n\u03bb \u03b7 e => HasTrans.trans (F\u2081.functor e) (\u03b7.ext b)\n\ntheorem congr.wd {F\u2081 F\u2082 : StructureFunctor S T} {a b : S} (\u03b7 : F\u2081 \u2243 F\u2082) (e : a \u2243 b) :\n  congr \u03b7 e \u2248 congr' \u03b7 e :=\n\u03b7.nat e\n\n\n\n-- Now we define identity and composition and prove that they are well-behaved with respect to equivalence.\n\ndef idFun : StructureFunctor S S := \u27e8id, id.genFun\u27e9\n\ndef compMap     (F : StructureFunctor S T) (G : StructureFunctor T U) : S \u2192 U :=\n\u03bb f => G (F f)\n\ndef compFunctor (F : StructureFunctor S T) (G : StructureFunctor T U) : Functor (compMap F G) :=\ncomp.genFun' F.map F.functor G.functor\n\ndef compFun     (F : StructureFunctor S T) (G : StructureFunctor T U) : StructureFunctor S U :=\n\u27e8compMap F G, compFunctor F G\u27e9\n\n@[reducible] def revCompFun (G : StructureFunctor T U) (F : StructureFunctor S T) : StructureFunctor S U := compFun F G\ninfixr:90 \" \u2299 \" => revCompFun\n\n\n\nnamespace compFun\n\ndef congrArg_left {F : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} :\n  G\u2081 \u2243 G\u2082 \u2192 G\u2081 \u2299 F \u2243 G\u2082 \u2299 F :=\n\u03bb \u03b7 => { ext := \u03bb a => \u03b7.ext (F a),\n         nat := \u03bb e => \u03b7.nat (F.functor e) }\n\nnamespace congrArg_left\n\ntheorem respectsEquiv {F : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U}\n                      {\u03b7\u2081 \u03b7\u2082 : G\u2081 \u2243 G\u2082} :\n  \u03b7\u2081 \u2248 \u03b7\u2082 \u2192 congrArg_left (F := F) \u03b7\u2081 \u2248 congrArg_left (F := F) \u03b7\u2082 :=\n\u03bb h\u03b7 a => h\u03b7 (F a)\n\ntheorem respectsComp {F : StructureFunctor S T} {G\u2081 G\u2082 G\u2083 : StructureFunctor T U}\n                     (\u03b7\u2081 : G\u2081 \u2243 G\u2082) (\u03b7\u2082 : G\u2082 \u2243 G\u2083) :\n  congrArg_left (F := F) (\u03b7\u2082 \u2022 \u03b7\u2081) \u2248 congrArg_left \u03b7\u2082 \u2022 congrArg_left \u03b7\u2081 :=\n\u03bb a => Setoid.refl (\u03b7\u2082.ext (F a) \u2022 \u03b7\u2081.ext (F a))\n\ntheorem respectsId {F : StructureFunctor S T} (G : StructureFunctor T U) :\n  congrArg_left (id_ G) \u2248 id_ (G \u2299 F) :=\n\u03bb a => Setoid.refl (id_ (G (F a)))\n\ntheorem respectsInv {F : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} (\u03b7 : G\u2081 \u2243 G\u2082) :\n  congrArg_left (F := F) \u03b7\u207b\u00b9 \u2248 (congrArg_left \u03b7)\u207b\u00b9 :=\n\u03bb a => Setoid.refl (\u03b7.ext (F a))\u207b\u00b9\n\ndef functor (U : Structure) (F : StructureFunctor S T) : StructureFunctor (functorStructure T U) (functorStructure S U) :=\n{ map     := \u03bb G => G \u2299 F,\n  functor := { mapEquiv  := congrArg_left,\n               isFunctor := { respectsEquiv := respectsEquiv,\n                              respectsComp  := respectsComp,\n                              respectsId    := respectsId,\n                              respectsInv   := respectsInv } } }\n\nend congrArg_left\n\ndef congrArg_right {F\u2081 F\u2082 : StructureFunctor S T} {G : StructureFunctor T U} :\n  F\u2081 \u2243 F\u2082 \u2192 G \u2299 F\u2081 \u2243 G \u2299 F\u2082 :=\n\u03bb \u03b7 => { ext := \u03bb a => G.functor (\u03b7.ext a),\n         nat := \u03bb {a b} e => let h\u2081 := respectsSetoid G (\u03b7.nat e);\n                             let h\u2082 := Setoid.trans (Setoid.symm (respectsComp G (\u03b7.ext a) (F\u2082.functor e))) h\u2081;\n                             let h\u2084 := Setoid.trans h\u2082 (respectsComp G (F\u2081.functor e) (\u03b7.ext b));\n                             h\u2084 }\n\nnamespace congrArg_right\n\ntheorem respectsEquiv {F\u2081 F\u2082 : StructureFunctor S T} {G : StructureFunctor T U}\n                      {\u03b7\u2081 \u03b7\u2082 : F\u2081 \u2243 F\u2082} :\n  \u03b7\u2081 \u2248 \u03b7\u2082 \u2192 congrArg_right (G := G) \u03b7\u2081 \u2248 congrArg_right (G := G) \u03b7\u2082 :=\n\u03bb h\u03b7 a => StructureFunctor.respectsSetoid G (h\u03b7 a)\n\ntheorem respectsComp {F\u2081 F\u2082 F\u2083 : StructureFunctor S T} {G : StructureFunctor T U}\n                     (\u03b7\u2081 : F\u2081 \u2243 F\u2082) (\u03b7\u2082 : F\u2082 \u2243 F\u2083) :\n  congrArg_right (G := G) (\u03b7\u2082 \u2022 \u03b7\u2081) \u2248 congrArg_right \u03b7\u2082 \u2022 congrArg_right \u03b7\u2081 :=\n\u03bb a => StructureFunctor.respectsComp G (\u03b7\u2081.ext a) (\u03b7\u2082.ext a)\n\ntheorem respectsId (F : StructureFunctor S T) {G : StructureFunctor T U} :\n  congrArg_right (id_ F) \u2248 id_ (G \u2299 F) :=\n\u03bb a => StructureFunctor.respectsId G (F a)\n\ntheorem respectsInv {F\u2081 F\u2082 : StructureFunctor S T} {G : StructureFunctor T U} (\u03b7 : F\u2081 \u2243 F\u2082) :\n  congrArg_right (G := G) \u03b7\u207b\u00b9 \u2248 (congrArg_right \u03b7)\u207b\u00b9 :=\n\u03bb a => StructureFunctor.respectsInv G (\u03b7.ext a)\n\ndef functor (S : Structure) (G : StructureFunctor T U) : StructureFunctor (functorStructure S T) (functorStructure S U) :=\n{ map     := \u03bb F => G \u2299 F,\n  functor := { mapEquiv  := congrArg_right,\n               isFunctor := { respectsEquiv := respectsEquiv,\n                              respectsComp  := respectsComp,\n                              respectsId    := respectsId (G := G),\n                              respectsInv   := respectsInv } } }\n\nend congrArg_right\n\ndef congrArg  {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} :\n  F\u2081 \u2243 F\u2082 \u2192 G\u2081 \u2243 G\u2082 \u2192 G\u2081 \u2299 F\u2081 \u2243 G\u2082 \u2299 F\u2082 :=\n\u03bb \u03b7 \u03b8 => FunctorEquiv.trans (congrArg_left \u03b8) (congrArg_right \u03b7)\n\ndef congrArg' {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} :\n  F\u2081 \u2243 F\u2082 \u2192 G\u2081 \u2243 G\u2082 \u2192 G\u2081 \u2299 F\u2081 \u2243 G\u2082 \u2299 F\u2082 :=\n\u03bb \u03b7 \u03b8 => FunctorEquiv.trans (congrArg_right \u03b7) (congrArg_left \u03b8)\n\nnamespace congrArg\n\ntheorem wd {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} (\u03b7 : F\u2081 \u2243 F\u2082) (\u03b8 : G\u2081 \u2243 G\u2082) :\n  congrArg \u03b7 \u03b8 \u2248 congrArg' \u03b7 \u03b8 :=\n\u03bb a => \u03b8.nat (\u03b7.ext a)\n\ntheorem respectsEquiv {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U}\n                      {\u03b7\u2081 \u03b7\u2082 : F\u2081 \u2243 F\u2082} {\u03b8\u2081 \u03b8\u2082 : G\u2081 \u2243 G\u2082} :\n  \u03b7\u2081 \u2248 \u03b7\u2082 \u2192 \u03b8\u2081 \u2248 \u03b8\u2082 \u2192 congrArg \u03b7\u2081 \u03b8\u2081 \u2248 congrArg \u03b7\u2082 \u03b8\u2082 :=\n\u03bb h\u03b7 h\u03b8 => FunctorEquiv.functorEquivHasIso.comp_congrArg (congrArg_left.respectsEquiv h\u03b8) (congrArg_right.respectsEquiv h\u03b7)\n\ntheorem respectsComp {F\u2081 F\u2082 F\u2083 : StructureFunctor S T} {G\u2081 G\u2082 G\u2083 : StructureFunctor T U}\n                     (\u03b7\u2081 : F\u2081 \u2243 F\u2082) (\u03b7\u2082 : F\u2082 \u2243 F\u2083) (\u03b8\u2081 : G\u2081 \u2243 G\u2082) (\u03b8\u2082 : G\u2082 \u2243 G\u2083) :\n  congrArg (\u03b7\u2082 \u2022 \u03b7\u2081) (\u03b8\u2082 \u2022 \u03b8\u2081) \u2248 congrArg \u03b7\u2082 \u03b8\u2082 \u2022 congrArg \u03b7\u2081 \u03b8\u2081 :=\nlet h\u2081 := FunctorEquiv.functorEquivHasIso.comp_congrArg (congrArg_left.respectsComp \u03b8\u2081 \u03b8\u2082) (congrArg_right.respectsComp \u03b7\u2081 \u03b7\u2082);\nlet h\u2082 := comp_congrArg_left (f := congrArg_left \u03b8\u2081) (wd \u03b7\u2081 \u03b8\u2082);\nlet h\u2083 := applyAssoc' h\u2082;\nlet h\u2084 := comp_congrArg_right (g := congrArg_right \u03b7\u2082) h\u2083;\nlet h\u2085 := applyAssoc h\u2084;\nSetoid.trans h\u2081 h\u2085\n\ntheorem respectsId (F : StructureFunctor S T) (G : StructureFunctor T U) :\n  congrArg (id_ F) (id_ G) \u2248 id_ (G \u2299 F) :=\nlet h\u2081 := FunctorEquiv.functorEquivHasIso.comp_congrArg (congrArg_left.respectsId G) (congrArg_right.respectsId F);\nSetoid.trans h\u2081 (leftId id')\n\ntheorem respectsInv {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} (\u03b7 : F\u2081 \u2243 F\u2082) (\u03b8 : G\u2081 \u2243 G\u2082) :\n  congrArg \u03b7\u207b\u00b9 \u03b8\u207b\u00b9 \u2248 (congrArg \u03b7 \u03b8)\u207b\u00b9 :=\nlet h\u2081 := FunctorEquiv.functorEquivHasIso.comp_congrArg (congrArg_left.respectsInv \u03b8) (congrArg_right.respectsInv \u03b7);\nlet h\u2082 := inv_congrArg (wd \u03b7 \u03b8);\nlet h\u2083 := compInv (congrArg_right \u03b7) (congrArg_left \u03b8);\nSetoid.trans h\u2081 (Setoid.symm (Setoid.trans h\u2082 h\u2083))\n\nend congrArg\n\ndef assoc (F : StructureFunctor S T) (G : StructureFunctor T U) (H : StructureFunctor U V) :\n  H \u2299 (G \u2299 F) \u2243 (H \u2299 G) \u2299 F :=\nFunctorEquiv.refl (H \u2299 G \u2299 F)\n\nend compFun\n\n\n\nnamespace idFun\n\ndef leftId (F : StructureFunctor S T) : idFun \u2299 F \u2243 F :=\n{ ext := \u03bb a => HasRefl.refl (F a),\n  nat := GeneralizedNaturalityCondition.refl F.functor }\n\ndef rightId (F : StructureFunctor S T) : F \u2299 idFun \u2243 F :=\n{ ext := \u03bb a => HasRefl.refl (F a),\n  nat := GeneralizedNaturalityCondition.refl F.functor }\n\nend idFun\n\n#exit\n\ninstance hasMor : HasMorphisms functorStructure :=\n{ refl          := @idFun,\n  trans         := compFun,\n  comp_congrArg := compFun.congrArg,\n  assoc         := compFun.assoc,\n  leftId        := idFun.leftId,\n  rightId       := idFun.rightId }\n\n\n\nnamespace compFun.congrArg_left.functor\n\ndef mapEquiv (U : Structure) {F\u2081 F\u2082 : StructureFunctor S T} (\u03b7 : F\u2081 \u2243 F\u2082) : functor U F\u2081 \u2243 functor U F\u2082 :=\n{ ext := \u03bb G => congrArg_right (G := G) \u03b7,\n  nat := \u03bb \u03b8 => Setoid.symm (congrArg.wd \u03b7 \u03b8) }\n\ndef functorFunctor (U : Structure)\n  : StructureFunctor (functorStructure S T) (functorStructure (functorStructure T U) (functorStructure S U)) :=\n{ map     := functor U,\n  functor := { mapEquiv  := mapEquiv U,\n               isFunctor := { respectsEquiv := \u03bb h   G => congrArg_right.respectsEquiv (G := G) h,\n                              respectsComp  := \u03bb \u03b7 \u03b8 G => congrArg_right.respectsComp  (G := G) \u03b7 \u03b8,\n                              respectsId    := \u03bb F   G => congrArg_right.respectsId    (G := G) F,\n                              respectsInv   := \u03bb \u03b7   G => congrArg_right.respectsInv   (G := G) \u03b7 } } }\n\ndef respectsIdFun (T S : Structure) : functor T (@idFun S) \u2243 @idFun (functorStructure S T) :=\n{ ext := \u03bb F   => idFun.rightId F,\n  nat := \u03bb \u03b7 a => let e := \u03b7.ext a;\n                  Setoid.trans (rightId e) (Setoid.symm (leftId e)) }\n\ndef respectsCompFun (V : Structure) (F : StructureFunctor S T) (G : StructureFunctor T U) :\n  functor V (G \u2299 F) \u2243 functor V F \u2299 functor V G :=\n{ ext := \u03bb H   => FunctorEquiv.refl (H \u2299 (G \u2299 F)),\n  nat := \u03bb \u03b7 a => let e := \u03b7.ext (G (F a));\n                  Setoid.trans (rightId e) (Setoid.symm (leftId e)) }\n\ntheorem respectsCompFun.nat (V : Structure) {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T U} (\u03b7 : F\u2081 \u2243 F\u2082) (\u03b8 : G\u2081 \u2243 G\u2082) :\n  compFun.congrArg (mapEquiv V \u03b8) (mapEquiv V \u03b7) \u2022 respectsCompFun V F\u2081 G\u2081 \u2248 respectsCompFun V F\u2082 G\u2082 \u2022 mapEquiv V (compFun.congrArg \u03b7 \u03b8) :=\nsorry\n\nend compFun.congrArg_left.functor\n\nnamespace compFun.congrArg_right.functor\n\ndef mapEquiv (S : Structure) {G\u2081 G\u2082 : StructureFunctor T U} (\u03b8 : G\u2081 \u2243 G\u2082) : functor S G\u2081 \u2243 functor S G\u2082 :=\n{ ext := \u03bb F => congrArg_left (F := F) \u03b8,\n  nat := \u03bb \u03b7 => congrArg.wd \u03b7 \u03b8 }\n\ndef functorFunctor (S : Structure)\n  : StructureFunctor (functorStructure T U) (functorStructure (functorStructure S T) (functorStructure S U)) :=\n{ map     := functor S,\n  functor := { mapEquiv  := mapEquiv S,\n               isFunctor := { respectsEquiv := \u03bb h   F => congrArg_left.respectsEquiv (F := F) h,\n                              respectsComp  := \u03bb \u03b7 \u03b8 F => congrArg_left.respectsComp  (F := F) \u03b7 \u03b8,\n                              respectsId    := \u03bb G   F => congrArg_left.respectsId    (F := F) G,\n                              respectsInv   := \u03bb \u03b7   F => congrArg_left.respectsInv   (F := F) \u03b7 } } }\n\ndef respectsIdFun (S T : Structure) : functor S (@idFun T) \u2243 @idFun (functorStructure S T) :=\n{ ext := \u03bb F   => idFun.leftId F,\n  nat := \u03bb \u03b7 a => let e := \u03b7.ext a;\n                  Setoid.trans (rightId e) (Setoid.symm (leftId e)) }\n\ndef respectsCompFun (S : Structure) (G : StructureFunctor T U) (H : StructureFunctor U V) :\n  functor S (H \u2299 G) \u2243 functor S H \u2299 functor S G :=\n{ ext := \u03bb F   => FunctorEquiv.refl ((H \u2299 G) \u2299 F),\n  nat := \u03bb \u03b7 a => let e := StructureFunctor.congrArg (H \u2299 G) (\u03b7.ext a);\n                  Setoid.trans (rightId e) (Setoid.symm (leftId e)) }\n\ntheorem respectsCompFun.nat (S : Structure) {G\u2081 G\u2082 : StructureFunctor T U} {H\u2081 H\u2082 : StructureFunctor U V} (\u03b7 : G\u2081 \u2243 G\u2082) (\u03b8 : H\u2081 \u2243 H\u2082) :\n  compFun.congrArg (mapEquiv S \u03b7) (mapEquiv S \u03b8) \u2022 respectsCompFun S G\u2081 H\u2081 \u2248 respectsCompFun S G\u2082 H\u2082 \u2022 mapEquiv S (compFun.congrArg \u03b7 \u03b8) :=\nsorry\n\nend compFun.congrArg_right.functor\n\n\n\n-- The constant functor.\n\ndef constFun (c : T) : StructureFunctor S T :=\n{ map     := Function.const (IsType.type S) c,\n  functor := const.genFun c }\n\n\n\n-- A simple alias for the assertion that a functor is equivalent to the identity functor.\n\n@[reducible] def IsId (F : StructureFunctor S S) := F \u2243 @idFun S\n\nnamespace IsId\n\n-- `ext` and `nat` have a slightly simpler form in this case.\n\ndef extDef {F : StructureFunctor S S} (\u03b7 : IsId F) (a : S) : F a \u2243 a :=\n\u03b7.ext a\n\ntheorem natDef {F : StructureFunctor S S} (\u03b7 : IsId F) {a b : S} (e : a \u2243 b) :\n  e \u2022 \u03b7.ext a \u2248 \u03b7.ext b \u2022 F.functor e :=\n\u03b7.nat e\n\n-- When composing both sides with another functor, we can cancel `idFun`.\n\ndef rightMul {G : StructureFunctor T T} (\u03b8 : IsId G) (F : StructureFunctor S T) :\n  G \u2299 F \u2243 F :=\nFunctorEquiv.trans (compFun.congrArg_left (F := F) \u03b8) (idFun.leftId F)\n\ntheorem rightMulDef {G : StructureFunctor T T} (\u03b8 : IsId G) (F : StructureFunctor S T) (a : S) :\n  (rightMul \u03b8 F).ext a \u2248 \u03b8.ext (F a) :=\nleftId (\u03b8.ext (F a))\n\ndef leftMul {F : StructureFunctor S S} (\u03b7 : IsId F) (G : StructureFunctor S T) :\n  G \u2299 F \u2243 G :=\nFunctorEquiv.trans (compFun.congrArg_right (G := G) \u03b7) (idFun.rightId G)\n\ntheorem leftMulDef {F : StructureFunctor S S} (\u03b7 : IsId F) (G : StructureFunctor S T) (a : S) :\n  (leftMul \u03b7 G).ext a \u2248 G.functor (\u03b7.ext a) :=\nleftId (G.functor (\u03b7.ext a))\n\n-- We have some definitions resembling reflexivity and transitivity.\n\ndef refl (S : Structure) : IsId (@idFun S) := FunctorEquiv.refl idFun\n\ndef trans {F G : StructureFunctor S S} (\u03b7 : IsId F) (\u03b8 : IsId G) : IsId (G \u2299 F) :=\nFunctorEquiv.trans (rightMul \u03b8 F) \u03b7\n\ntheorem transDef {F G : StructureFunctor S S} (\u03b7 : IsId F) (\u03b8 : IsId G) (a : S) :\n  (trans \u03b7 \u03b8).ext a \u2248 \u03b7.ext a \u2022 \u03b8.ext (F a) :=\ncomp_congrArg_right (rightMulDef \u03b8 F a)\n\nend IsId\n\n\n\n-- A simple alias for the assertion that `G` is a left inverse of `F`.\n-- Note that instead of defining `RightInv` analogously, we just swap the arguments of `F` and `G` where\n-- necessary.\n\n@[reducible] def LeftInv (F : StructureFunctor S T) (G : StructureFunctor T S) := IsId (G \u2299 F)\n\nnamespace LeftInv\n\ndef refl (S : Structure) : LeftInv (@idFun S) (@idFun S) := IsId.refl S\n\ndef trans {F : StructureFunctor S T} {G : StructureFunctor T S} {H : StructureFunctor T U} {I : StructureFunctor U T}\n          (\u03b7 : LeftInv F G) (\u03b8 : LeftInv H I) :\n  LeftInv (H \u2299 F) (G \u2299 I) :=\nlet \u03b6 : (G \u2299 I) \u2299 (H \u2299 F) \u2243 G \u2299 F := compFun.congrArg_left (F := F) (IsId.leftMul \u03b8 G);\nFunctorEquiv.trans \u03b6 \u03b7\n\ntheorem transDef {F : StructureFunctor S T} {G : StructureFunctor T S} {H : StructureFunctor T U} {I : StructureFunctor U T}\n                 (\u03b7 : LeftInv F G) (\u03b8 : LeftInv H I) (a : S) :\n  (trans \u03b7 \u03b8).ext a \u2248 \u03b7.ext a \u2022 G.functor (\u03b8.ext (F a)) :=\ncomp_congrArg_right (IsId.leftMulDef \u03b8 G (F a))\n\ntheorem refl_trans {F : StructureFunctor S T} {G : StructureFunctor T S}\n                   (\u03b7 : LeftInv F G) :\n  trans (refl S) \u03b7 \u2248 \u03b7 :=\n\u03bb a => let h\u2081 : (trans (refl S) \u03b7).ext a \u2248 id_ a \u2022 \u03b7.ext a := transDef (refl S) \u03b7 a;\n       let h\u2082 : id_ a \u2022 \u03b7.ext a \u2248 \u03b7.ext a                  := leftId (\u03b7.ext a);\n       Setoid.trans h\u2081 h\u2082\n\ntheorem trans_refl {F : StructureFunctor S T} {G : StructureFunctor T S}\n                   (\u03b7 : LeftInv F G) :\n  trans \u03b7 (refl T) \u2248 \u03b7 :=\n\u03bb a => let h\u2081 := transDef \u03b7 (refl T) a;\n       let h\u2082 := rightCancelId (respectsId G (F a));\n       Setoid.trans h\u2081 h\u2082\n\ntheorem trans_assoc {F : StructureFunctor S T} {G : StructureFunctor T S}\n                    {H : StructureFunctor T U} {I : StructureFunctor U T}\n                    {J : StructureFunctor U V} {K : StructureFunctor V U}\n                    (\u03b7 : LeftInv F G) (\u03b8 : LeftInv H I) (\u03b6 : LeftInv J K) :\n  let l : LeftInv (J \u2299 H \u2299 F) (G \u2299 I \u2299 K) := trans (trans \u03b7 \u03b8) \u03b6;\n  let r : LeftInv (J \u2299 H \u2299 F) (G \u2299 I \u2299 K) := trans \u03b7 (trans \u03b8 \u03b6);\n  l \u2248 r :=\n\u03bb a => let h\u2081 := applyAssoc_right' (comp_subst_left' (transDef \u03b7 \u03b8 a) (transDef (trans \u03b7 \u03b8) \u03b6 a));\n       let h\u2082 := comp_subst_right' (Setoid.symm (respectsComp G (I.functor.mapEquiv (\u03b6.ext (H (F a)))) (\u03b8.ext (F a)))) h\u2081;\n       let h\u2083 := comp_subst_right' (respectsSetoid G (transDef \u03b8 \u03b6 (F a))) (transDef \u03b7 (trans \u03b8 \u03b6) a);\n       Setoid.trans h\u2082 (Setoid.symm h\u2083)\n\n-- This definition asserts that an instance of `LeftInv` is compatible with a corresponding reversed\n-- `LeftInv` instance. It corresponds to one of the two equations of an adjoint functor (the one about\n-- `F`).\n\ndef Compat {F : StructureFunctor S T} {G : StructureFunctor T S} (\u03b7l : LeftInv F G) (\u03b7r : LeftInv G F) :=\n\u2200 a, F.functor (\u03b7l.ext a) \u2248 \u03b7r.ext (F a)\n\nnamespace Compat\n\ntheorem refl (S : Structure) : Compat (LeftInv.refl S) (LeftInv.refl S) :=\n\u03bb a => Setoid.refl (HasRefl.refl a)\n\ntheorem trans {F : StructureFunctor S T} {G : StructureFunctor T S} {H : StructureFunctor T U} {I : StructureFunctor U T}\n              {\u03b7l : LeftInv F G} {\u03b7r : LeftInv G F} {\u03b8l : LeftInv H I} {\u03b8r : LeftInv I H}\n              (c : Compat \u03b7l \u03b7r) (d : Compat \u03b8l \u03b8r) :\n  Compat (LeftInv.trans \u03b7l \u03b8l) (LeftInv.trans \u03b8r \u03b7r) :=\n\u03bb a => let h\u2081 : \u03b7r.ext (F a) \u2022 F.functor (G.functor (\u03b8l.ext (F a))) \u2248 \u03b8l.ext (F a) \u2022 \u03b7r.ext (I (H (F a)))                                 := Setoid.symm (\u03b7r.nat (\u03b8l.ext (F a)));\n       let h\u2082 : F.functor (\u03b7l.ext a) \u2022 F.functor (G.functor (\u03b8l.ext (F a))) \u2248 \u03b8l.ext (F a) \u2022 \u03b7r.ext (I (H (F a)))                         := comp_subst_left (c a) h\u2081;\n       let h\u2083 : F.functor (\u03b7l.ext a \u2022 G.functor (\u03b8l.ext (F a))) \u2248 \u03b8l.ext (F a) \u2022 \u03b7r.ext (I (H (F a)))                                     := Setoid.trans (respectsComp F (G.functor (\u03b8l.ext (F a))) (\u03b7l.ext a)) h\u2082;\n       let h\u2084 : H.functor (F.functor (\u03b7l.ext a \u2022 G.functor (\u03b8l.ext (F a)))) \u2248 H.functor (\u03b8l.ext (F a)) \u2022 H.functor (\u03b7r.ext (I (H (F a)))) := Setoid.trans (respectsSetoid H h\u2083) (respectsComp H (\u03b7r.ext (I (H (F a)))) (\u03b8l.ext (F a)));\n       let h\u2085 : H.functor (F.functor (\u03b7l.ext a \u2022 G.functor (\u03b8l.ext (F a)))) \u2248 \u03b8r.ext (H (F a)) \u2022 H.functor (\u03b7r.ext (I (H (F a))))         := comp_subst_left' (d (F a)) h\u2084;\n       let h\u2086 := Setoid.trans (respectsSetoid H (respectsSetoid F (transDef \u03b7l \u03b8l a))) h\u2085;\n       let h\u2087 := Setoid.trans h\u2086 (Setoid.symm (transDef \u03b8r \u03b7r (H (F a))));\n       h\u2087\n\nend Compat\n\n-- Given equivalences of functors, we can ask whether two instances of `LeftInv` are equivalent.\n\ndef Equiv {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T S}\n          (\u03b7 : F\u2081 \u2243 F\u2082) (\u03b8 : G\u2081 \u2243 G\u2082)\n          (\u03b6\u2081 : LeftInv F\u2081 G\u2081) (\u03b6\u2082 : LeftInv F\u2082 G\u2082) :=\n\u03b6\u2081 \u2248 \u03b6\u2082 \u2022 compFun.congrArg \u03b7 \u03b8\n\nnamespace Equiv\n\ntheorem refl  {F : StructureFunctor S T} {G : StructureFunctor T S} (\u03b6 : LeftInv F G) :\n  Equiv (FunctorEquiv.refl F) (FunctorEquiv.refl G) \u03b6 \u03b6 :=\nSetoid.symm (rightCancelId (compFun.congrArg.respectsId F G))\n\ntheorem refl' {F : StructureFunctor S T} {G : StructureFunctor T S} {\u03b6\u2081 \u03b6\u2082 : LeftInv F G} (h : \u03b6\u2081 \u2248 \u03b6\u2082) :\n  Equiv (FunctorEquiv.refl F) (FunctorEquiv.refl G) \u03b6\u2081 \u03b6\u2082 :=\ncomp_subst_left' h (refl \u03b6\u2081)\n\ntheorem symm  {F\u2081 F\u2082 : StructureFunctor S T} {G\u2081 G\u2082 : StructureFunctor T S}\n              {\u03b7 : F\u2081 \u2243 F\u2082} {\u03b8 : G\u2081 \u2243 G\u2082}\n              {\u03b6\u2081 : LeftInv F\u2081 G\u2081} {\u03b6\u2082 : LeftInv F\u2082 G\u2082}\n              (e : Equiv \u03b7 \u03b8 \u03b6\u2081 \u03b6\u2082) :\n  Equiv (FunctorEquiv.symm \u03b7) (FunctorEquiv.symm \u03b8) \u03b6\u2082 \u03b6\u2081 :=\nlet h\u2081 := (rightMulInv \u03b6\u2082 \u03b6\u2081 (compFun.congrArg \u03b7 \u03b8)).mp (Setoid.symm e);\ncomp_subst_right' (Setoid.symm (compFun.congrArg.respectsInv \u03b7 \u03b8)) h\u2081\n\ntheorem trans {F\u2081 F\u2082 F\u2083 : StructureFunctor S T} {G\u2081 G\u2082 G\u2083 : StructureFunctor T S}\n              {\u03b7\u2081 : F\u2081 \u2243 F\u2082} {\u03b7\u2082 : F\u2082 \u2243 F\u2083} {\u03b8\u2081 : G\u2081 \u2243 G\u2082} {\u03b8\u2082 : G\u2082 \u2243 G\u2083}\n              {\u03b6\u2081 : LeftInv F\u2081 G\u2081} {\u03b6\u2082 : LeftInv F\u2082 G\u2082} {\u03b6\u2083 : LeftInv F\u2083 G\u2083}\n              (e : Equiv \u03b7\u2081 \u03b8\u2081 \u03b6\u2081 \u03b6\u2082) (f : Equiv \u03b7\u2082 \u03b8\u2082 \u03b6\u2082 \u03b6\u2083) :\n  Equiv (FunctorEquiv.trans \u03b7\u2081 \u03b7\u2082) (FunctorEquiv.trans \u03b8\u2081 \u03b8\u2082) \u03b6\u2081 \u03b6\u2083 :=\nlet h\u2081 := applyAssoc_right' (comp_subst_left' f e);\ncomp_subst_right' (Setoid.symm (compFun.congrArg.respectsComp \u03b7\u2081 \u03b7\u2082 \u03b8\u2081 \u03b8\u2082)) h\u2081\n\nend Equiv\n\nend LeftInv\n\n\n\n-- A type class asserting that two functors are inverse to each other. In addition to the condition that\n-- the inverse functor is left-inverse and right-inverse, we also add compatibility conditions on these\n-- two functor equivalences for both `F` and `G`. This is essentially the same as requiring the functors\n-- to be adjoint.\n\nclass IsInverse (F : StructureFunctor S T) (G : StructureFunctor T S) :=\n(leftInv  : LeftInv F G)\n(rightInv : LeftInv G F)\n(lrCompat : LeftInv.Compat leftInv rightInv)\n(rlCompat : LeftInv.Compat rightInv leftInv)\n\nnamespace IsInverse\n\ndef refl  (S : Structure) :\n  IsInverse (@idFun S) (@idFun S) :=\n{ leftInv  := LeftInv.refl        S,\n  rightInv := LeftInv.refl        S,\n  lrCompat := LeftInv.Compat.refl S,\n  rlCompat := LeftInv.Compat.refl S }\n\ndef symm  {F : StructureFunctor S T} {G : StructureFunctor T S}\n          (e : IsInverse F G) :\n  IsInverse G F :=\n{ leftInv  := e.rightInv,\n  rightInv := e.leftInv,\n  lrCompat := e.rlCompat,\n  rlCompat := e.lrCompat }\n\ndef trans {F : StructureFunctor S T} {G : StructureFunctor T S} {H : StructureFunctor T U} {I : StructureFunctor U T}\n          (e : IsInverse F G) (f : IsInverse H I) :\n  IsInverse (H \u2299 F) (G \u2299 I) :=\n{ leftInv  := LeftInv.trans        e.leftInv  f.leftInv,\n  rightInv := LeftInv.trans        f.rightInv e.rightInv,\n  lrCompat := LeftInv.Compat.trans e.lrCompat f.lrCompat,\n  rlCompat := LeftInv.Compat.trans f.rlCompat e.rlCompat }\n\ntheorem symm_symm {F : StructureFunctor S T} {G : StructureFunctor T S} (e : IsInverse F G) : symm (symm e) = e :=\nmatch e with\n| \u27e8_, _, _, _\u27e9 => rfl \n\nend IsInverse\n\n\n\n-- A functor between instance structures is actually just a function.\n\ndef congrArgFunctor {\u03b1 : Sort u} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  @GeneralizedFunctor.Functor (instanceStructure \u03b1) (instanceStructure \u03b2) f :=\n{ mapEquiv  := _root_.congrArg f,\n  isFunctor := propFunctor }\n\ndef InstanceStructureFunctor (\u03b1 \u03b2 : Sort u) := StructureFunctor (instanceStructure \u03b1) (instanceStructure \u03b2)\n\ndef instanceStructureFunctor {\u03b1 \u03b2 : Sort u} (f : \u03b1 \u2192 \u03b2) : InstanceStructureFunctor \u03b1 \u03b2 :=\n{ map     := f,\n  functor := congrArgFunctor f }\n\n\n\n-- If we have a function `F` and an equivalent functor `G`, we can turn `F` into a functor as well.\n\ndef proxyFunctor {S T : Structure} (F : S \u2192 T) (G : StructureFunctor S T) (\u03b7 : PiEquiv F G.map) :\n  StructureFunctor S T :=\n{ map     := F,\n  functor := comp.genFun G.functor (PiEquiv.transport.invFunctor \u03b7) }\n\nend StructureFunctor\n\nopen StructureFunctor\n\n\n\n-- Based on the definition of a functor between two structures, we can define equivalence of two\n-- structures similarly to equivalence of types in mathlib.\n\nstructure StructureEquiv (S T : Structure) where\n(toFun  : StructureFunctor S T)\n(invFun : StructureFunctor T S)\n(isInv  : IsInverse toFun invFun)\n\nnamespace StructureEquiv\n\ndef refl  (S     : Structure)                                                   : StructureEquiv S S :=\n{ toFun  := idFun,\n  invFun := idFun,\n  isInv  := IsInverse.refl  S }\n\ndef symm  {S T   : Structure} (e : StructureEquiv S T)                          : StructureEquiv T S :=\n{ toFun  := e.invFun,\n  invFun := e.toFun,\n  isInv  := IsInverse.symm  e.isInv }\n\ndef trans {S T U : Structure} (e : StructureEquiv S T) (f : StructureEquiv T U) : StructureEquiv S U :=\n{ toFun  := f.toFun  \u2299 e.toFun,\n  invFun := e.invFun \u2299 f.invFun,\n  isInv  := IsInverse.trans e.isInv f.isInv }\n\ntheorem symm_symm {S T : Structure} (e : StructureEquiv S T) : symm (symm e) = e :=\nmatch e with\n| \u27e8toFun, invFun, isInv\u27e9 => IsInverse.symm_symm isInv \u25b8 rfl \n\n\n\n-- We can compare two instances of `StructureEquiv` by comparing `toFun` and `invFun` and then dependently\n-- comparing `leftInv` and `rightInv`. That turns `StructureEquiv` into a structure.\n\nstructure EquivEquiv {S T : Structure} (e f : StructureEquiv S T) where\n(toFunEquiv    : e.toFun  \u2243 f.toFun)\n(invFunEquiv   : e.invFun \u2243 f.invFun)\n(leftInvEquiv  : LeftInv.Equiv toFunEquiv  invFunEquiv e.isInv.leftInv  f.isInv.leftInv)\n(rightInvEquiv : LeftInv.Equiv invFunEquiv toFunEquiv  e.isInv.rightInv f.isInv.rightInv)\n\nnamespace EquivEquiv\n\nvariable {S T : Structure}\n\ndef refl  (e     : StructureEquiv S T)                                           : EquivEquiv e e :=\n{ toFunEquiv    := HasRefl.refl   e.toFun,\n  invFunEquiv   := HasRefl.refl   e.invFun,\n  leftInvEquiv  := LeftInv.Equiv.refl  e.isInv.leftInv,\n  rightInvEquiv := LeftInv.Equiv.refl  e.isInv.rightInv }\n\ndef symm  {e f   : StructureEquiv S T} (\u03b7 : EquivEquiv e f)                      : EquivEquiv f e :=\n{ toFunEquiv    := HasSymm.symm   \u03b7.toFunEquiv,\n  invFunEquiv   := HasSymm.symm   \u03b7.invFunEquiv,\n  leftInvEquiv  := LeftInv.Equiv.symm  \u03b7.leftInvEquiv,\n  rightInvEquiv := LeftInv.Equiv.symm  \u03b7.rightInvEquiv }\n\ndef trans {e f g : StructureEquiv S T} (\u03b7 : EquivEquiv e f) (\u03b8 : EquivEquiv f g) : EquivEquiv e g :=\n{ toFunEquiv    := HasTrans.trans \u03b7.toFunEquiv    \u03b8.toFunEquiv,\n  invFunEquiv   := HasTrans.trans \u03b7.invFunEquiv   \u03b8.invFunEquiv,\n  leftInvEquiv  := LeftInv.Equiv.trans \u03b7.leftInvEquiv  \u03b8.leftInvEquiv,\n  rightInvEquiv := LeftInv.Equiv.trans \u03b7.rightInvEquiv \u03b8.rightInvEquiv }\n\n\n\n-- For equivalence of `EquivEquiv`, we can reuse the equivalence of `StructureProduct`, as `leftInvEquiv`\n-- and `rightInvEquiv` are just proofs.\n\n@[reducible] def FunProd (S T : Structure) := StructureProduct (functorStructure S T) (functorStructure T S)\n\ndef funProd {S T : Structure} (e : StructureEquiv S T) : FunProd S T :=\n\u27e8e.toFun, e.invFun\u27e9\n\ndef funEquivProd {e f : StructureEquiv S T} (\u03b7 : EquivEquiv e f) :\n  funProd e \u2243 funProd f :=\n\u27e8\u03b7.toFunEquiv, \u03b7.invFunEquiv\u27e9\n\ndef EquivEquivEquiv {e f : StructureEquiv S T} (\u03b7 \u03b8 : EquivEquiv e f) :=\nfunEquivProd \u03b7 \u2248 funEquivProd \u03b8\n\nnamespace EquivEquivEquiv\n\nvariable {e f : StructureEquiv S T}\n\ntheorem refl  (\u03b7     : EquivEquiv e f)                                                     : EquivEquivEquiv \u03b7 \u03b7 :=\nStructureProduct.ProductEquiv.EquivEquiv.refl  (funEquivProd \u03b7)\n\ntheorem symm  {\u03b7 \u03b8   : EquivEquiv e f} (h : EquivEquivEquiv \u03b7 \u03b8)                           : EquivEquivEquiv \u03b8 \u03b7 :=\nStructureProduct.ProductEquiv.EquivEquiv.symm  h\n\ntheorem trans {\u03b7 \u03b8 \u03b6 : EquivEquiv e f} (h : EquivEquivEquiv \u03b7 \u03b8) (i : EquivEquivEquiv \u03b8 \u03b6) : EquivEquivEquiv \u03b7 \u03b6 :=\nStructureProduct.ProductEquiv.EquivEquiv.trans h i\n\ninstance equivEquivSetoid : Setoid (EquivEquiv e f) := \u27e8EquivEquivEquiv, \u27e8refl, symm, trans\u27e9\u27e9\n\nend EquivEquivEquiv\n\ndef equivEquiv (e f : StructureEquiv S T) : BundledSetoid := \u27e8EquivEquiv e f\u27e9\n\ninstance equivHasIso : HasIsomorphisms (@equivEquiv S T) :=\n{ refl          := refl,\n  symm          := symm,\n  trans         := trans,\n  comp_congrArg := \u03bb {e f g \u03b7\u2081 \u03b7\u2082 \u03b8\u2081 \u03b8\u2082} (h\u03b7 : EquivEquivEquiv \u03b7\u2081 \u03b7\u2082) (h\u03b8 : EquivEquivEquiv \u03b8\u2081 \u03b8\u2082) =>\n                     HasStructure.comp_congrArg h\u03b7 h\u03b8,\n  inv_congrArg  := \u03bb {e f   \u03b7\u2081 \u03b7\u2082}       (h\u03b7 : EquivEquivEquiv \u03b7\u2081 \u03b7\u2082)                              =>\n                     HasStructure.inv_congrArg  h\u03b7,\n  assoc         := \u03bb \u03b7 \u03b8 \u03b6 => HasStructure.assoc    (funEquivProd \u03b7) (funEquivProd \u03b8) (funEquivProd \u03b6),\n  leftId        := \u03bb \u03b7     => HasStructure.leftId   (funEquivProd \u03b7),\n  rightId       := \u03bb \u03b7     => HasStructure.rightId  (funEquivProd \u03b7),\n  leftInv       := \u03bb \u03b7     => HasStructure.leftInv  (funEquivProd \u03b7),\n  rightInv      := \u03bb \u03b7     => HasStructure.rightInv (funEquivProd \u03b7),\n  invInv        := \u03bb \u03b7     => HasStructure.invInv   (funEquivProd \u03b7),\n  compInv       := \u03bb \u03b7 \u03b8   => HasStructure.compInv  (funEquivProd \u03b7) (funEquivProd \u03b8),\n  idInv         := \u03bb e     => HasStructure.idInv    (funProd e) }\n\nend EquivEquiv\n\ninstance equivHasStructure (S T : Structure) : HasStructure (StructureEquiv S T) := \u27e8EquivEquiv.equivEquiv\u27e9\ndef equivStructure (S T : Structure) : Structure := \u27e8StructureEquiv S T\u27e9\n\n\n\ndef toFunProj (S T : Structure) : StructureFunctor (equivStructure S T) (functorStructure S T) :=\n{ map     := StructureEquiv.toFun,\n  functor := { mapEquiv  := EquivEquiv.toFunEquiv,\n               isFunctor := { respectsEquiv := And.left,\n                              respectsComp  := \u03bb \u03b7 \u03b8 => Setoid.refl (\u03b8.toFunEquiv \u2022 \u03b7.toFunEquiv),\n                              respectsId    := \u03bb e   => Setoid.refl (id__ (S := functorStructure S T) e.toFun),\n                              respectsInv   := \u03bb \u03b7   => Setoid.refl (\u03b7.toFunEquiv)\u207b\u00b9 } } }\n\ndef invFunProj (S T : Structure) : StructureFunctor (equivStructure S T) (functorStructure T S) :=\n{ map     := StructureEquiv.invFun,\n  functor := { mapEquiv  := EquivEquiv.invFunEquiv,\n               isFunctor := { respectsEquiv := And.right,\n                              respectsComp  := \u03bb \u03b7 \u03b8 => Setoid.refl (\u03b8.invFunEquiv \u2022 \u03b7.invFunEquiv),\n                              respectsId    := \u03bb e   => Setoid.refl (id__ (S := functorStructure T S) e.invFun),\n                              respectsInv   := \u03bb \u03b7   => Setoid.refl (\u03b7.invFunEquiv)\u207b\u00b9 } } }\n\n\n\ndef comp_congrArg {S T U : Structure} {e\u2081 e\u2082 : StructureEquiv S T} {f\u2081 f\u2082 : StructureEquiv T U} (he : e\u2081 \u2243 e\u2082) (hf : f\u2081 \u2243 f\u2082) :\n  trans e\u2081 f\u2081 \u2243 trans e\u2082 f\u2082 :=\n{ toFunEquiv    := compFun.congrArg he.toFunEquiv  hf.toFunEquiv,\n  invFunEquiv   := compFun.congrArg hf.invFunEquiv he.invFunEquiv,\n  leftInvEquiv  := sorry,\n  rightInvEquiv := sorry }\n\ntheorem assoc_leftInvEquiv {S T U V : Structure} (e : StructureEquiv S T) (f : StructureEquiv T U) (g : StructureEquiv U V) :\n  LeftInv.Equiv (FunctorEquiv.refl (g.toFun  \u2299 f.toFun  \u2299 e.toFun))\n                (FunctorEquiv.refl (e.invFun \u2299 f.invFun \u2299 g.invFun))\n                (IsInverse.trans (IsInverse.trans e.isInv f.isInv) g.isInv).leftInv\n                (IsInverse.trans e.isInv (IsInverse.trans f.isInv g.isInv)).leftInv :=\nLeftInv.Equiv.refl' (LeftInv.trans_assoc e.isInv.leftInv f.isInv.leftInv g.isInv.leftInv)\n\ntheorem assoc_rightInvEquiv {S T U V : Structure} (e : StructureEquiv S T) (f : StructureEquiv T U) (g : StructureEquiv U V) :\n  LeftInv.Equiv (FunctorEquiv.refl (e.invFun \u2299 f.invFun \u2299 g.invFun))\n                (FunctorEquiv.refl (g.toFun  \u2299 f.toFun  \u2299 e.toFun))\n                (IsInverse.trans (IsInverse.trans e.isInv f.isInv) g.isInv).rightInv\n                (IsInverse.trans e.isInv (IsInverse.trans f.isInv g.isInv)).rightInv :=\nLeftInv.Equiv.refl' (Setoid.symm (LeftInv.trans_assoc g.isInv.rightInv f.isInv.rightInv e.isInv.rightInv))\n\ndef assoc {S T U V : Structure} (e : StructureEquiv S T) (f : StructureEquiv T U) (g : StructureEquiv U V) :\n  trans (trans e f) g \u2243 trans e (trans f g) :=\n{ toFunEquiv    := compFun.assoc e.toFun  f.toFun  g.toFun,\n  invFunEquiv   := compFun.assoc g.invFun f.invFun e.invFun,\n  leftInvEquiv  := assoc_leftInvEquiv  e f g,\n  rightInvEquiv := assoc_rightInvEquiv e f g }\n\ntheorem leftId_leftInvEquiv {S T : Structure} (e : StructureEquiv S T) :\n  LeftInv.Equiv (idFun.leftId e.toFun)\n                (idFun.leftId e.invFun)\n                (IsInverse.trans e.isInv (IsInverse.refl T)).leftInv\n                e.isInv.leftInv :=\nlet h\u2081 := LeftInv.trans_refl e.isInv.leftInv;\n\u03bb a => let h\u2082 := h\u2081 a;\n       sorry\n\ntheorem rightId_leftInvEquiv {S T : Structure} (e : StructureEquiv S T) :\n  LeftInv.Equiv (idFun.rightId e.toFun)\n                (idFun.rightId e.invFun)\n                (IsInverse.trans (IsInverse.refl S) e.isInv).leftInv\n                e.isInv.leftInv :=\nsorry\n\ndef leftId  {S T : Structure} (e : StructureEquiv S T) : trans e (refl T) \u2243 e :=\n{ toFunEquiv    := idFun.leftId e.toFun,\n  invFunEquiv   := idFun.leftId e.invFun,\n  leftInvEquiv  := leftId_leftInvEquiv  e,\n  rightInvEquiv := rightId_leftInvEquiv (symm e) }\n\ndef rightId {S T : Structure} (e : StructureEquiv S T) : trans (refl S) e \u2243 e :=\n{ toFunEquiv    := idFun.rightId e.toFun,\n  invFunEquiv   := idFun.rightId e.invFun,\n  leftInvEquiv  := rightId_leftInvEquiv e,\n  rightInvEquiv := leftId_leftInvEquiv  (symm e) }\n\ndef inv_congrArg {S T : Structure} {e\u2081 e\u2082 : StructureEquiv S T} (he : e\u2081 \u2243 e\u2082) :\n  symm e\u2081 \u2243 symm e\u2082 :=\n{ toFunEquiv    := he.invFunEquiv,\n  invFunEquiv   := he.toFunEquiv,\n  leftInvEquiv  := he.rightInvEquiv,\n  rightInvEquiv := he.leftInvEquiv }\n\ntheorem leftInvEquiv {S T : Structure} (e : StructureEquiv S T) :\n  LeftInv.Equiv e.isInv.leftInv e.isInv.leftInv (IsInverse.trans e.isInv (IsInverse.symm e.isInv)).leftInv (IsInverse.refl S).leftInv :=\nlet h\u2081 : LeftInv.trans e.isInv.leftInv e.isInv.rightInv \u2248 compFun.congrArg' e.isInv.leftInv e.isInv.leftInv :=\n    \u03bb a => Setoid.trans (LeftInv.transDef e.isInv.leftInv e.isInv.rightInv a) (comp_congrArg_right (respectsSetoid e.invFun (Setoid.symm (e.isInv.lrCompat a))));\nlet h\u2082 := Setoid.trans h\u2081 (Setoid.symm (compFun.congrArg.wd e.isInv.leftInv e.isInv.leftInv));\nSetoid.trans h\u2082 (Setoid.symm (HasStructure.leftId (compFun.congrArg e.isInv.leftInv e.isInv.leftInv)))\n\ndef leftInv'  {S T : Structure} (e : StructureEquiv S T) : trans e (symm e) \u2243 refl S :=\n{ toFunEquiv    := e.isInv.leftInv,\n  invFunEquiv   := e.isInv.leftInv,\n  leftInvEquiv  := leftInvEquiv e,\n  rightInvEquiv := leftInvEquiv e }\n\ntheorem rightInvEquiv {S T : Structure} (e : StructureEquiv S T) :\n  LeftInv.Equiv e.isInv.rightInv e.isInv.rightInv (IsInverse.trans (IsInverse.symm e.isInv) e.isInv).rightInv (IsInverse.refl T).rightInv :=\nlet h\u2081 : LeftInv.trans e.isInv.rightInv e.isInv.leftInv \u2248 compFun.congrArg' e.isInv.rightInv e.isInv.rightInv :=\n    \u03bb a => Setoid.trans (LeftInv.transDef e.isInv.rightInv e.isInv.leftInv a) (comp_congrArg_right (respectsSetoid e.toFun (Setoid.symm (e.isInv.rlCompat a))));\nlet h\u2082 := Setoid.trans h\u2081 (Setoid.symm (compFun.congrArg.wd e.isInv.rightInv e.isInv.rightInv));\nSetoid.trans h\u2082 (Setoid.symm (HasStructure.leftId (compFun.congrArg e.isInv.rightInv e.isInv.rightInv)))\n\ndef rightInv' {S T : Structure} (e : StructureEquiv S T) : trans (symm e) e \u2243 refl T :=\n{ toFunEquiv    := e.isInv.rightInv,\n  invFunEquiv   := e.isInv.rightInv,\n  leftInvEquiv  := rightInvEquiv e,\n  rightInvEquiv := rightInvEquiv e }\n\ndef invInv {S T : Structure} (e : StructureEquiv S T) : symm (symm e) \u2243 e :=\nsymm_symm e \u25b8 EquivEquiv.refl e\n\ndef compInv {S T U : Structure} (e : StructureEquiv S T) (f : StructureEquiv T U) :\n  symm (trans e f) \u2243 trans (symm f) (symm e) :=\nEquivEquiv.refl (symm (trans e f))\n\ndef idInv (S : Structure) : symm (refl S) \u2243 refl S :=\nEquivEquiv.refl (refl S)\n\ninstance equivHasIso : HasIsomorphisms equivStructure :=\n{ refl          := refl,\n  symm          := symm,\n  trans         := trans,\n  comp_congrArg := comp_congrArg,\n  inv_congrArg  := inv_congrArg,\n  assoc         := assoc,\n  leftId        := leftId,\n  rightId       := rightId,\n  leftInv       := leftInv',\n  rightInv      := rightInv',\n  invInv        := invInv,\n  compInv       := compInv,\n  idInv         := idInv }\n\nend StructureEquiv\n\n\n\ninstance structureHasGeneralStructure : HasGeneralStructure Structure := \u27e8StructureEquiv.equivStructure\u27e9\ninstance structureHasEquivalence : HasEquivalence Structure Structure := \u27e8StructureEquiv.equivStructure\u27e9\ninstance structureEquivIsTypeWithEquiv : IsTypeWithEquivalence (HasEquivalence.\u03b3 Structure Structure) := Structure.structureIsTypeWithEquiv\ninstance structureEquivIsType : IsType (HasEquivalence.\u03b3 Structure Structure) := structureEquivIsTypeWithEquiv.toIsType\ninstance (S T : Structure) : Setoid (IsType.type (S \u2243 T)) := instanceEquivSetoid (IsType.type (S \u2243 T))\ninstance (S T : Structure) : HasStructure (IsType.type (S \u2243 T)) := StructureEquiv.equivHasStructure S T\ninstance : HasIsomorphisms (@HasEquivalence.Equiv Structure Structure structureHasEquivalence) := HasGeneralStructure.hasIso\n\n\n\n-- If we have a `StructureEquiv S T`, we can ask whether it maps `a : S` to `b : T`. This is similar to\n-- an equivalence. It corresponds to a \"dependent equivalence\" or \"pathover\" in HoTT, so we adopt the same\n-- notation `a \u2243[e] b`.\n\ndef InstanceEquiv {S T : Structure} (e : S \u2243 T) (a : S) (b : T) := e.toFun a \u2243 b\n\nnamespace InstanceEquiv\n\nnotation:25 a:26 \" \u2243[\" e:0 \"] \" b:26 => InstanceEquiv e a b\n\ndef fromEquiv (S : Structure) {a b : S} : a \u2243 b \u2192 a \u2243[id_ S] b := id\ndef toEquiv   (S : Structure) {a b : S} : a \u2243[id_ S] b \u2192 a \u2243 b := id\n\ndef refl  (S     : Structure)                         (a : S)                 :\n  a \u2243[id_ S] a :=\nfromEquiv S (HasRefl.refl a)\n\ndef symm  {S T   : Structure} (e : S \u2243 T)             (a : S) (b : T)         :\n  a \u2243[e] b \u2192 b \u2243[e\u207b\u00b9] a :=\n\u03bb \u03c6 => HasTrans.trans (HasSymm.symm (congrArg e.invFun \u03c6)) (e.isInv.leftInv.ext a)\n\ndef trans {S T U : Structure} (e : S \u2243 T) (f : T \u2243 U) (a : S) (b : T) (c : U) :\n  a \u2243[e] b \u2192 b \u2243[f] c \u2192 a \u2243[f \u2022 e] c :=\n\u03bb \u03c6 \u03c8 => HasTrans.trans (congrArg f.toFun \u03c6) \u03c8\n\ndef mapEquiv {S T : Structure} {e\u2081 e\u2082 : S \u2243 T} (\u03b7 : e\u2081 \u2243 e\u2082) (a : S) (b : T) :\n  a \u2243[e\u2081] b \u2192 a \u2243[e\u2082] b :=\nHasTrans.trans (HasSymm.symm (\u03b7.toFunEquiv.ext a))\n\nend InstanceEquiv\n\n\n\n-- Using `StructureEquiv`, we can build a \"universe\" structure where the objects are structures. This is\n-- the same as the groupoid of lower-level groupoids.\n--\n-- `universeStructure` contains an implicit truncation of `EquivEquiv` to a proposition, via\n-- `hasTruncatedStructure`. In `TwoStructure.lean`, we give the definition of an enlarged structure that\n-- allows us to keep this data instead.\n\ndef universeStructure : Structure := \u27e8Structure\u27e9\n\ninstance : IsType (IsType.type universeStructure) := structureIsType\n", "meta": {"author": "SReichelt", "repo": "lean4-experiments", "sha": "ff55357a01a34a91bf670d712637480089085ee4", "save_path": "github-repos/lean/SReichelt-lean4-experiments", "path": "github-repos/lean/SReichelt-lean4-experiments/lean4-experiments-ff55357a01a34a91bf670d712637480089085ee4/Structure/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631840431539, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.3601416732041201}}
{"text": "import data.finsupp.basic\n\nnamespace groth16\n\n\n/-- An inductive type from which to index the variables of the mv_polynomials the proof manages -/\n@[derive decidable_eq]\ninductive vars : Type\n| \u03b1 : vars\n| \u03b2 : vars\n| \u03b3 : vars\n| \u03b4 : vars\n-- | x : vars\n\n-- lemma finsupp_vars_eq_ext (f g : vars \u2192\u2080 \u2115) : f = g \u2194 \n--   f vars.\u03b1 = g vars.\u03b1 \u2227 f vars.\u03b2 = g vars.\u03b2 \u2227 f vars.\u03b3 = g vars.\u03b3 \u2227 f vars.\u03b4 = g vars.\u03b4 \u2227 f vars.x = g vars.x :=\n-- begin\n--   rw finsupp.ext_iff,\n--   split,\n--     {\n--       intro h,\n--       split, exact h vars.\u03b1,\n--       split, exact h vars.\u03b2,\n--       split, exact h vars.\u03b3,\n--       split, exact h vars.\u03b4,\n--       exact h vars.x,\n--     },\n--     {\n--       intro h,\n--       intro a,\n--       induction a,\n--       finish,\n--       finish,\n--       finish,\n--       finish,\n--       finish,\n--     },\n-- end\n\nlemma finsupp_vars_eq_ext (f g : vars \u2192\u2080 \u2115) : f = g \u2194 \n  f vars.\u03b1 = g vars.\u03b1 \u2227 f vars.\u03b2 = g vars.\u03b2 \u2227 f vars.\u03b3 = g vars.\u03b3 \u2227 f vars.\u03b4 = g vars.\u03b4 :=\nbegin\n  rw finsupp.ext_iff,\n  split,\n    {\n      intro h,\n      split, exact h vars.\u03b1,\n      split, exact h vars.\u03b2,\n      split, exact h vars.\u03b3,\n      exact h vars.\u03b4,\n    },\n    {\n      intro h,\n      intro a,\n      induction a,\n      finish,\n      finish,\n      finish,\n      finish,\n    },\n  -- induction,\nend\n\nend groth16", "meta": {"author": "BoltonBailey", "repo": "formal-snarks-project", "sha": "154414784f90a1e257162fcbdd7e805ecb2a49c2", "save_path": "github-repos/lean/BoltonBailey-formal-snarks-project", "path": "github-repos/lean/BoltonBailey-formal-snarks-project/formal-snarks-project-154414784f90a1e257162fcbdd7e805ecb2a49c2/src/snarks/groth16typeIII/vars.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.36014166478217235}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.monoidal.natural_transformation\nimport Mathlib.category_theory.monoidal.discrete\nimport Mathlib.PostPort\n\nuniverses v u l v\u2081 u\u2081 v\u2082 u\u2082 u\u2083 v\u2083 \n\nnamespace Mathlib\n\n/-!\n# Braided and symmetric monoidal categories\n\nThe basic definitions of braided monoidal categories, and symmetric monoidal categories,\nas well as braided functors.\n\n## Implementation note\n\nWe make `braided_monoidal_category` another typeclass, but then have `symmetric_monoidal_category`\nextend this. The rationale is that we are not carrying any additional data,\njust requiring a property.\n\n## Future work\n\n* Construct the Drinfeld center of a monoidal category as a braided monoidal category.\n* Say something about pseudo-natural transformations.\n\n-/\n\nnamespace category_theory\n\n\n/--\nA braided monoidal category is a monoidal category equipped with a braiding isomorphism\n`\u03b2_ X Y : X \u2297 Y \u2245 Y \u2297 X`\nwhich is natural in both arguments,\nand also satisfies the two hexagon identities.\n-/\n-- braiding natural iso:\n\nclass braided_category (C : Type u) [category C] [monoidal_category C] where\n  braiding : (X Y : C) \u2192 X \u2297 Y \u2245 Y \u2297 X\n  braiding_naturality' :\n    autoParam\n      (\u2200 {X X' Y Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y'),\n        (f \u2297 g) \u226b iso.hom (braiding Y Y') = iso.hom (braiding X X') \u226b (g \u2297 f))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  hexagon_forward' :\n    autoParam\n      (\u2200 (X Y Z : C),\n        iso.hom \u03b1_ \u226b iso.hom (braiding X (Y \u2297 Z)) \u226b iso.hom \u03b1_ =\n          (iso.hom (braiding X Y) \u2297 \ud835\udfd9) \u226b iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom (braiding X Z)))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  hexagon_reverse' :\n    autoParam\n      (\u2200 (X Y Z : C),\n        iso.inv \u03b1_ \u226b iso.hom (braiding (X \u2297 Y) Z) \u226b iso.inv \u03b1_ =\n          (\ud835\udfd9 \u2297 iso.hom (braiding Y Z)) \u226b iso.inv \u03b1_ \u226b (iso.hom (braiding X Z) \u2297 \ud835\udfd9))\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- hexagon identities:\n\n@[simp] theorem braided_category.braiding_naturality {C : Type u} [category C] [monoidal_category C]\n    [c : braided_category C] {X : C} {X' : C} {Y : C} {Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') :\n    (f \u2297 g) \u226b iso.hom (braided_category.braiding Y Y') =\n        iso.hom (braided_category.braiding X X') \u226b (g \u2297 f) :=\n  sorry\n\n@[simp] theorem braided_category.braiding_naturality_assoc {C : Type u} [category C]\n    [monoidal_category C] [c : braided_category C] {X : C} {X' : C} {Y : C} {Y' : C} (f : X \u27f6 Y)\n    (g : X' \u27f6 Y') :\n    \u2200 {X'_1 : C} (f' : Y' \u2297 Y \u27f6 X'_1),\n        (f \u2297 g) \u226b iso.hom (braided_category.braiding Y Y') \u226b f' =\n          iso.hom (braided_category.braiding X X') \u226b (g \u2297 f) \u226b f' :=\n  sorry\n\ntheorem braided_category.hexagon_forward {C : Type u} [category C] [monoidal_category C]\n    [c : braided_category C] (X : C) (Y : C) (Z : C) :\n    iso.hom \u03b1_ \u226b iso.hom (braided_category.braiding X (Y \u2297 Z)) \u226b iso.hom \u03b1_ =\n        (iso.hom (braided_category.braiding X Y) \u2297 \ud835\udfd9) \u226b\n          iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom (braided_category.braiding X Z)) :=\n  sorry\n\ntheorem braided_category.hexagon_reverse {C : Type u} [category C] [monoidal_category C]\n    [c : braided_category C] (X : C) (Y : C) (Z : C) :\n    iso.inv \u03b1_ \u226b iso.hom (braided_category.braiding (X \u2297 Y) Z) \u226b iso.inv \u03b1_ =\n        (\ud835\udfd9 \u2297 iso.hom (braided_category.braiding Y Z)) \u226b\n          iso.inv \u03b1_ \u226b (iso.hom (braided_category.braiding X Z) \u2297 \ud835\udfd9) :=\n  sorry\n\nnotation:1024 \"\u03b2_\" => Mathlib.category_theory.braided_category.braiding\n\n/-!\nWe now establish how the braiding interacts with the unitors.\n\nI couldn't find a detailed proof in print, but this is discussed in:\n\n* Proposition 1 of Andr\u00e9 Joyal and Ross Street,\n  \"Braided monoidal categories\", Macquarie Math Reports 860081 (1986).\n* Proposition 2.1 of Andr\u00e9 Joyal and Ross Street,\n  \"Braided tensor categories\" , Adv. Math. 102 (1993), 20\u201378.\n* Exercise 8.1.6 of Etingof, Gelaki, Nikshych, Ostrik,\n  \"Tensor categories\", vol 25, Mathematical Surveys and Monographs (2015), AMS.\n-/\n\ntheorem braiding_left_unitor_aux\u2081 (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) :\n    iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.inv \u03b2_) \u226b iso.inv \u03b1_ \u226b (iso.hom \u03bb_ \u2297 \ud835\udfd9) = (iso.hom \u03bb_ \u2297 \ud835\udfd9) \u226b iso.inv \u03b2_ :=\n  sorry\n\ntheorem braiding_left_unitor_aux\u2082 (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) : (iso.hom \u03b2_ \u2297 \ud835\udfd9) \u226b (iso.hom \u03bb_ \u2297 \ud835\udfd9) = iso.hom \u03c1_ \u2297 \ud835\udfd9 :=\n  sorry\n\n@[simp] theorem braiding_left_unitor (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) : iso.hom \u03b2_ \u226b iso.hom \u03bb_ = iso.hom \u03c1_ :=\n  sorry\n\ntheorem braiding_right_unitor_aux\u2081 (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) :\n    iso.inv \u03b1_ \u226b (iso.inv \u03b2_ \u2297 \ud835\udfd9) \u226b iso.hom \u03b1_ \u226b (\ud835\udfd9 \u2297 iso.hom \u03c1_) = (\ud835\udfd9 \u2297 iso.hom \u03c1_) \u226b iso.inv \u03b2_ :=\n  sorry\n\ntheorem braiding_right_unitor_aux\u2082 (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) : (\ud835\udfd9 \u2297 iso.hom \u03b2_) \u226b (\ud835\udfd9 \u2297 iso.hom \u03c1_) = \ud835\udfd9 \u2297 iso.hom \u03bb_ :=\n  sorry\n\n@[simp] theorem braiding_right_unitor (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] (X : C) : iso.hom \u03b2_ \u226b iso.hom \u03c1_ = iso.hom \u03bb_ :=\n  sorry\n\n/--\nA symmetric monoidal category is a braided monoidal category for which the braiding is symmetric.\n\nSee https://stacks.math.columbia.edu/tag/0FFW.\n-/\nclass symmetric_category (C : Type u) [category C] [monoidal_category C] extends braided_category C\n    where\n  symmetry' :\n    autoParam (C \u2192 C \u2192 iso.hom \u03b2_ \u226b iso.hom \u03b2_ = \ud835\udfd9)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- braiding symmetric:\n\n@[simp] theorem symmetric_category.symmetry {C : Type u} [category C] [monoidal_category C]\n    [c : symmetric_category C] (X : C) (Y : C) : iso.hom \u03b2_ \u226b iso.hom \u03b2_ = \ud835\udfd9 :=\n  sorry\n\n@[simp] theorem symmetric_category.symmetry_assoc {C : Type u} [category C] [monoidal_category C]\n    [c : symmetric_category C] (X : C) (Y : C) {X' : C} (f' : X \u2297 Y \u27f6 X') :\n    iso.hom \u03b2_ \u226b iso.hom \u03b2_ \u226b f' = f' :=\n  sorry\n\n/--\nA lax braided functor between braided monoidal categories is a lax monoidal functor\nwhich preserves the braiding.\n-/\nstructure lax_braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C]\n    (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D]\n    extends lax_monoidal_functor C D where\n  braided' :\n    autoParam\n      (\u2200 (X Y : C),\n        lax_monoidal_functor.\u03bc _to_lax_monoidal_functor X Y \u226b\n            functor.map (lax_monoidal_functor.to_functor _to_lax_monoidal_functor) (iso.hom \u03b2_) =\n          iso.hom \u03b2_ \u226b lax_monoidal_functor.\u03bc _to_lax_monoidal_functor Y X)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\ntheorem lax_braided_functor.braided {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    (c : lax_braided_functor C D) (X : C) (Y : C) :\n    lax_monoidal_functor.\u03bc (lax_braided_functor.to_lax_monoidal_functor c) X Y \u226b\n          functor.map\n            (lax_monoidal_functor.to_functor (lax_braided_functor.to_lax_monoidal_functor c))\n            (iso.hom \u03b2_) =\n        iso.hom \u03b2_ \u226b lax_monoidal_functor.\u03bc (lax_braided_functor.to_lax_monoidal_functor c) Y X :=\n  sorry\n\nnamespace lax_braided_functor\n\n\n/-- The identity lax braided monoidal functor. -/\ndef id (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] :\n    lax_braided_functor C C :=\n  mk (monoidal_functor.to_lax_monoidal_functor (monoidal_functor.id C))\n\nprotected instance inhabited (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] :\n    Inhabited (lax_braided_functor C C) :=\n  { default := id C }\n\n/-- The composition of lax braided monoidal functors. -/\ndef comp {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082}\n    [category D] [monoidal_category D] [braided_category D] {E : Type u\u2083} [category E]\n    [monoidal_category E] [braided_category E] (F : lax_braided_functor C D)\n    (G : lax_braided_functor D E) : lax_braided_functor C E :=\n  mk\n    (lax_monoidal_functor.mk\n      (lax_monoidal_functor.to_functor (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G))\n      (lax_monoidal_functor.\u03b5 (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G))\n      (lax_monoidal_functor.\u03bc (to_lax_monoidal_functor F \u2297\u22d9 to_lax_monoidal_functor G)))\n\nprotected instance category_lax_braided_functor {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] :\n    category (lax_braided_functor C D) :=\n  induced_category.category to_lax_monoidal_functor\n\n@[simp] theorem comp_to_nat_trans {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    {F : lax_braided_functor C D} {G : lax_braided_functor C D} {H : lax_braided_functor C D}\n    {\u03b1 : F \u27f6 G} {\u03b2 : G \u27f6 H} :\n    monoidal_nat_trans.to_nat_trans (\u03b1 \u226b \u03b2) =\n        monoidal_nat_trans.to_nat_trans \u03b1 \u226b monoidal_nat_trans.to_nat_trans \u03b2 :=\n  rfl\n\n/--\nInterpret a natural isomorphism of the underlyling lax monoidal functors as an\nisomorphism of the lax braided monoidal functors.\n-/\n@[simp] theorem mk_iso_hom {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C]\n    {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    {F : lax_braided_functor C D} {G : lax_braided_functor C D}\n    (i : to_lax_monoidal_functor F \u2245 to_lax_monoidal_functor G) : iso.hom (mk_iso i) = iso.hom i :=\n  Eq.refl (iso.hom (mk_iso i))\n\nend lax_braided_functor\n\n\n/--\nA braided functor between braided monoidal categories is a monoidal functor\nwhich preserves the braiding.\n-/\n-- Note this is stated different than for `lax_braided_functor`.\n\nstructure braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C]\n    (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D]\n    extends monoidal_functor C D where\n  braided' :\n    autoParam\n      (\u2200 (X Y : C),\n        functor.map\n            (lax_monoidal_functor.to_functor\n              (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor))\n            (iso.hom \u03b2_) =\n          inv\n              (lax_monoidal_functor.\u03bc\n                (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor) X Y) \u226b\n            iso.hom \u03b2_ \u226b\n              lax_monoidal_functor.\u03bc (monoidal_functor.to_lax_monoidal_functor _to_monoidal_functor)\n                Y X)\n      (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n        (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n-- We move the `\u03bc X Y` to the right hand side,\n\n-- so that this makes a good `@[simp]` lemma.\n\n@[simp] theorem braided_functor.braided {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    (c : braided_functor C D) (X : C) (Y : C) :\n    functor.map\n          (lax_monoidal_functor.to_functor\n            (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)))\n          (iso.hom \u03b2_) =\n        inv\n            (lax_monoidal_functor.\u03bc\n              (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)) X\n              Y) \u226b\n          iso.hom \u03b2_ \u226b\n            lax_monoidal_functor.\u03bc\n              (monoidal_functor.to_lax_monoidal_functor (braided_functor.to_monoidal_functor c)) Y\n              X :=\n  sorry\n\nnamespace braided_functor\n\n\n/-- Turn a braided functor into a lax braided functor. -/\ndef to_lax_braided_functor (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C]\n    (D : Type u\u2082) [category D] [monoidal_category D] [braided_category D]\n    (F : braided_functor C D) : lax_braided_functor C D :=\n  lax_braided_functor.mk (monoidal_functor.to_lax_monoidal_functor (to_monoidal_functor F))\n\n/-- The identity braided monoidal functor. -/\n@[simp] theorem id_to_monoidal_functor (C : Type u\u2081) [category C] [monoidal_category C]\n    [braided_category C] : to_monoidal_functor (id C) = monoidal_functor.id C :=\n  Eq.refl (to_monoidal_functor (id C))\n\nprotected instance inhabited (C : Type u\u2081) [category C] [monoidal_category C] [braided_category C] :\n    Inhabited (braided_functor C C) :=\n  { default := id C }\n\n/-- The composition of braided monoidal functors. -/\n@[simp] theorem comp_to_monoidal_functor {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    {E : Type u\u2083} [category E] [monoidal_category E] [braided_category E] (F : braided_functor C D)\n    (G : braided_functor D E) :\n    to_monoidal_functor (comp F G) = to_monoidal_functor F \u2297\u22d9 to_monoidal_functor G :=\n  Eq.refl (to_monoidal_functor (comp F G))\n\nprotected instance category_braided_functor {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D] :\n    category (braided_functor C D) :=\n  induced_category.category to_monoidal_functor\n\n@[simp] theorem comp_to_nat_trans {C : Type u\u2081} [category C] [monoidal_category C]\n    [braided_category C] {D : Type u\u2082} [category D] [monoidal_category D] [braided_category D]\n    {F : braided_functor C D} {G : braided_functor C D} {H : braided_functor C D} {\u03b1 : F \u27f6 G}\n    {\u03b2 : G \u27f6 H} :\n    monoidal_nat_trans.to_nat_trans (\u03b1 \u226b \u03b2) =\n        monoidal_nat_trans.to_nat_trans \u03b1 \u226b monoidal_nat_trans.to_nat_trans \u03b2 :=\n  rfl\n\n/--\nInterpret a natural isomorphism of the underlyling monoidal functors as an\nisomorphism of the braided monoidal functors.\n-/\ndef mk_iso {C : Type u\u2081} [category C] [monoidal_category C] [braided_category C] {D : Type u\u2082}\n    [category D] [monoidal_category D] [braided_category D] {F : braided_functor C D}\n    {G : braided_functor C D} (i : to_monoidal_functor F \u2245 to_monoidal_functor G) : F \u2245 G :=\n  iso.mk (iso.hom i) (iso.inv i)\n\nend braided_functor\n\n\nprotected instance comm_monoid_discrete (M : Type u) [comm_monoid M] : comm_monoid (discrete M) :=\n  id _inst_10\n\nprotected instance discrete.braided_category (M : Type u) [comm_monoid M] :\n    braided_category (discrete M) :=\n  braided_category.mk fun (X Y : discrete M) => eq_to_iso sorry\n\n/--\nA multiplicative morphism between commutative monoids gives a braided functor between\nthe corresponding discrete braided monoidal categories.\n-/\ndef discrete.braided_functor {M : Type u} [comm_monoid M] {N : Type u} [comm_monoid N]\n    (F : M \u2192* N) : braided_functor (discrete M) (discrete N) :=\n  braided_functor.mk\n    (monoidal_functor.mk (monoidal_functor.to_lax_monoidal_functor (discrete.monoidal_functor F)))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/monoidal/braided_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5926665999540697, "lm_q1q2_score": 0.36014166478217224}}
{"text": "import vc0.basic\n\nnamespace c0\nopen ast\n\nnamespace value\n\ntheorem step_comp.determ {op v\u2081 v\u2082 b\u2081 b\u2082}\n  (h\u2081 : step_comp op v\u2081 v\u2082 b\u2081)\n  (h\u2082 : step_comp op v\u2081 v\u2082 b\u2082) : b\u2081 = b\u2082 :=\nbegin\n  suffices : \u2200 {op' v\u2081' v\u2082'} (e : (op, v\u2081, v\u2082) = (op', v\u2081', v\u2082'))\n    (h\u2082 : step_comp op' v\u2081' v\u2082' b\u2082), b\u2081 = b\u2082,\n  from this rfl h\u2082,\n  clear h\u2082, intros,\n  induction h\u2081; induction h\u2082; cases e; refl\nend\n\ntheorem step_binop.determ {op v\u2081 v\u2082 w\u2081 w\u2082}\n  (h\u2081 : step_binop op v\u2081 v\u2082 w\u2081) (h\u2082 : step_binop op v\u2081 v\u2082 w\u2082) : w\u2081 = w\u2082 :=\nbegin\n  generalize_hyp eo : op = op' at h\u2082,\n  generalize_hyp e\u2081 : v\u2081 = v\u2081' at h\u2082,\n  generalize_hyp e\u2082 : v\u2082 = v\u2082' at h\u2082,\n  induction h\u2081; induction h\u2082; cases eo; cases e\u2081; cases e\u2082; try {refl},\n  cases h\u2081_a.determ h\u2082_a, refl\nend\n\ntheorem step_unop.determ {op v w\u2081 w\u2082}\n  (h\u2081 : step_unop op v w\u2081) (h\u2082 : step_unop op v w\u2082) : w\u2081 = w\u2082 :=\nby cases h\u2081; cases h\u2082; refl\n\ntheorem default.determ {\u0393} (ok : okind \u0393) {ts v\u2081 v\u2082}\n  (h\u2081 : default \u0393 ts v\u2081)\n  (h\u2082 : default \u0393 ts v\u2082) : v\u2081 = v\u2082 :=\nbegin\n  induction h\u2081 generalizing v\u2082; try {cases h\u2082}; try {refl},\n  { cases get_sdef_determ ok h\u2081_a h\u2082_a,\n    exact h\u2081_ih h\u2082_a_1 },\n  { generalize_hyp e : alist.cons h\u2081_\u0394 h\u2081_x h\u2081_\u03c4 h\u2081_h = \u0394' at h\u2082,\n    cases h\u2082, {cases e},\n    rcases alist.cons_inj e with \u27e8\u27e8\u27e9, rfl\u27e9,\n    cases h\u2081_ih_a h\u2082_a, cases h\u2081_ih_a_1 h\u2082_a_1, refl }\nend\n\ntheorem is_nth.determ {n v v\u2081 v\u2082}\n  (h\u2081 : is_nth n v v\u2081) (h\u2082 : is_nth n v v\u2082) : v\u2081 = v\u2082 :=\nby induction h\u2081 generalizing v\u2082; cases h\u2082;\n   [refl, {cases h\u2081_ih h\u2082_a, refl}]\n\nend value\n\nnamespace addr\n\ntheorem get.determ {H \u03b7 a v\u2081 v\u2082}\n  (h\u2081 : get H \u03b7 a v\u2081) (h\u2082 : get H \u03b7 a v\u2082) : v\u2081 = v\u2082 :=\nbegin\n  induction h\u2081 generalizing v\u2082; cases h\u2082,\n  { exact option.mem_unique h\u2081_a h\u2082_a },\n  { exact option.mem_unique h\u2081_a h\u2082_a },\n  { cases h\u2081_ih h\u2082_a_1, refl },\n  { cases h\u2081_ih h\u2082_a_1, refl },\n  { cases h\u2081_ih h\u2082_a_1, exact h\u2081_a_2.determ h\u2082_a_2 },\n  { cases value.of_map_inj (h\u2081_ih h\u2082_a_1),\n    exact option.mem_unique h\u2081_a_2 h\u2082_a_2 }\nend\n\ntheorem get_len.determ {H \u03b7 a v\u2081 v\u2082}\n  (h\u2081 : get_len H \u03b7 a v\u2081)\n  (h\u2082 : get_len H \u03b7 a v\u2082) : v\u2081 = v\u2082 :=\nby cases h\u2081; cases h\u2082; cases h\u2081_a_1.determ h\u2082_a_1; refl\n\ntheorem update_at.determ\n  {\u03b1} {R : \u03b1 \u2192 \u03b1 \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082) :\n  \u2200 {n l l\u2081 l\u2082}, list.update_at R n l l\u2081 \u2192 list.update_at R n l l\u2082 \u2192 l\u2081 = l\u2082\n| _ _ _ _ (@list.update_at.one _ _ a b l r) (@list.update_at.one _ _ _ b' _ r') :=\n  by rw Rd _ _ _ r r'\n| _ _ _ _ (@list.update_at.cons _ _ n a l r h) (@list.update_at.cons _ _ _ _ _ r' h') :=\n  by rw update_at.determ h h'\n\ntheorem at_head.determ\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  (x y\u2081 y\u2082) (h\u2081 : value.at_head R x y\u2081) (h\u2082 : value.at_head R x y\u2082) : y\u2081 = y\u2082 :=\nby cases h\u2081; cases h\u2082; rw Rd _ _ _ h\u2081_a h\u2082_a\n\ntheorem at_tail.determ\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  (x y\u2081 y\u2082) (h\u2081 : value.at_tail R x y\u2081) (h\u2082 : value.at_tail R x y\u2082) : y\u2081 = y\u2082 :=\nby cases h\u2081; cases h\u2082; rw Rd _ _ _ h\u2081_a h\u2082_a\n\ntheorem at_nth'.determ\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  : \u2200 {n} x y\u2081 y\u2082, value.at_nth' R n x y\u2081 \u2192 value.at_nth' R n x y\u2082 \u2192 y\u2081 = y\u2082\n| 0     := at_head.determ Rd\n| (n+1) := at_tail.determ at_nth'.determ\n\ntheorem at_nth.determ\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  {n} (x y\u2081 y\u2082) (h\u2081 : value.at_nth R n x y\u2081) (h\u2082 : value.at_nth R n x y\u2082) : y\u2081 = y\u2082 :=\nby cases h\u2081; cases h\u2082; rw at_nth'.determ Rd _ _ _ h\u2081_a_1 h\u2082_a_1\n\ntheorem at_field.determ\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  {f} (x y\u2081 y\u2082) (h\u2081 : value.at_field R f x y\u2081) (h\u2082 : value.at_field R f x y\u2082) : y\u2081 = y\u2082 :=\nbegin\n  rcases h\u2081 with \u27e8_, _, vs, x, y, r, m, e, rfl\u27e9,\n  rcases h\u2082 with \u27e8_, _, vs', x', y', r', m', e', rfl\u27e9,\n  cases value.of_map_inj (e.symm.trans e'),\n  cases option.mem_unique m m',\n  rw Rd _ _ _ r r'\nend\n\ntheorem update.determ {H \u03b7 a H\u2081 \u03b7\u2081 H\u2082 \u03b7\u2082}\n  {R : value \u2192 value \u2192 Prop} (Rd : \u2200 x y\u2081 y\u2082, R x y\u2081 \u2192 R x y\u2082 \u2192 y\u2081 = y\u2082)\n  (h\u2081 : update H \u03b7 R a H\u2081 \u03b7\u2081)\n  (h\u2082 : update H \u03b7 R a H\u2082 \u03b7\u2082) : (H\u2081, \u03b7\u2081) = (H\u2082, \u03b7\u2082) :=\nbegin\n  induction h\u2081 generalizing H\u2082 \u03b7\u2082; cases h\u2082,\n  { cases update_at.determ Rd h\u2081_a h\u2082_a, refl },\n  { substs h\u2081_\u03b7' \u03b7\u2082, cases option.mem_unique h\u2081_a h\u2082_a,\n    cases Rd _ _ _ h\u2081_a_1 h\u2082_a_1, refl },\n  { exact h\u2081_ih (at_head.determ Rd) h\u2082_a_1 },\n  { exact h\u2081_ih (at_tail.determ Rd) h\u2082_a_1 },\n  { exact h\u2081_ih (at_nth.determ Rd) h\u2082_a_1 },\n  { exact h\u2081_ih (at_field.determ Rd) h\u2082_a_1 }\nend\n\ntheorem eq.determ {v : value} (_ : value) : \u2200 y\u2081 y\u2082, v = y\u2081 \u2192 v = y\u2082 \u2192 y\u2081 = y\u2082\n| _ _ rfl h := h\n\nend addr\n\ntheorem step_deref.determ {C a K s\u2081 s\u2082}\n  (h\u2081 : step_deref C a K s\u2081) (h\u2082 : step_deref C a K s\u2082) : s\u2081 = s\u2082 :=\nby cases h\u2081; cases h\u2082; [refl, {cases h\u2081_a_1.determ h\u2082_a_1, refl}]\n\ntheorem step_ret.determ {C v s\u2081 s\u2082}\n  (h\u2081 : step_ret C v s\u2081) (h\u2082 : step_ret C v s\u2082) : s\u2081 = s\u2082 :=\nby cases h\u2081; cases h\u2082; refl\n\ntheorem step_call.determ {\u0393 : ast} (ok : \u0393.okind)\n  {vs x\u03c4s \u03b7\u2081 \u03b7\u2082}\n  (h\u2081 : step_call x\u03c4s vs \u03b7\u2081) (h\u2082 : step_call x\u03c4s vs \u03b7\u2082) : \u03b7\u2081 = \u03b7\u2082 :=\nbegin\n  induction h\u2081 with \u0394 x \u03c4 v vs \u03b7 h sc IH generalizing \u03b7\u2082,\n  { cases h\u2082, refl },\n  { generalize_hyp e\u2081 : alist.cons \u0394 x \u03c4 _ = \u0394' at h\u2082,\n    cases h\u2082, rcases alist.cons_inj e\u2081 with \u27e8\u27e8\u27e9, rfl\u27e9,\n    cases IH h\u2082_a, refl }\nend\n\ntheorem step_alloc.determ {C v K s\u2081 s\u2082}\n  (sa\u2081 : step_alloc C v K s\u2081) (sa\u2082 : step_alloc C v K s\u2082) : s\u2081 = s\u2082 :=\nby cases sa\u2081; cases sa\u2082; refl\n\ntheorem index_not_lt_zero {i : int32} {n : \u2115} (e : (i : \u2124) = n) : \u00ac i < 0 :=\nnot_lt_of_le $ by rw [\u2190 int32.coe_le, e, int32.coe_zero]; apply int.coe_nat_nonneg\n\ntheorem index_not_lt_zero_or {i : int32} {j n : \u2115}\n  (e : (i : \u2124) = j) (lt : j < n) : \u00ac (i < 0 \u2228 (n : \u2124) \u2264 i)\n| (or.inl h) := index_not_lt_zero e h\n| (or.inr h) := not_lt_of_le h $ by rwa [e, int.coe_nat_lt]\n\ninductive io_equiv : io \u2192 state \u2192 io \u2192 state \u2192 Prop\n| none {s} : io_equiv none s none s\n| some {i o\u2081 o\u2082 s\u2081 s\u2082} : (o\u2081 = o\u2082 \u2192 s\u2081 = s\u2082) \u2192\n  io_equiv (some (i, o\u2081)) s\u2081 (some (i, o\u2082)) s\u2082\n\ntheorem determ {\u0393 : ast} (ok : \u0393.ok) {s o\u2081 s\u2081 o\u2082 s\u2082}\n  (h\u2081 : step \u0393 s o\u2081 s\u2081) (h\u2082 : step \u0393 s o\u2082 s\u2082) : io_equiv o\u2081 s\u2081 o\u2082 s\u2082 :=\nbegin\n  cases h\u2081,\n  case c0.step.asgn_var\u2081 : C lv x e K h {\n    cases h\u2082,\n    case c0.step.asgn\u2081 : _ _ _ _ h' { rw h' at h, cases h },\n    case c0.step.asgn_var\u2081 : _ _ _ _ y h' {\n      cases option.mem_unique h h', constructor } },\n  case c0.step.asgn\u2081 : C lv e K h {\n    cases h\u2082,\n    case c0.step.asgn_var\u2081 : _ _ _ _ x h' { rw h at h', cases h' },\n    case c0.step.asgn\u2081 : h' { constructor } },\n  case c0.step.asgn\u2083 : H H' S \u03b7 \u03b7' a v K h {\n    cases h\u2082,\n    rcases h.determ addr.eq.determ h\u2082_a_1 with \u27e8rfl, rfl\u27e9,\n    constructor },\n  case c0.step.asnop\u2082 : _ C a op e K h {\n    cases h\u2082, cases step_deref.determ h h\u2082_a_1, constructor },\n  case c0.step.ret\u2082 : _ C v h {\n    cases h\u2082, cases h.determ h\u2082_a, constructor },\n  case c0.step.ret_none : _ C v h {\n    cases h\u2082, cases h.determ h\u2082_a, constructor },\n  case c0.step.nop\u2081 : _ C h {\n    cases h\u2082, cases h.determ h\u2082_a, constructor },\n  case c0.step.var : C i v K h {\n    cases h\u2082, cases option.mem_unique h h\u2082_a, constructor },\n  case c0.step.binop\u2083 : C op v\u2081 v\u2082 v K h {\n    cases h\u2082; cases h.determ h\u2082_a; constructor },\n  case c0.step.binop_err : C op v\u2081 v\u2082 err K h {\n    cases h\u2082; cases h.determ h\u2082_a; constructor },\n  case c0.step.unop\u2082 : C op v v\u2081 K h {\n    cases h\u2082, cases h.determ h\u2082_a, constructor },\n  case c0.step.call\u2082 : H S \u03b7 \u03b7\u2081 f \u03c4\u2081 x\u03c4s\u2081 s\u2081 vs K hb\u2081 sc\u2081 {\n    cases h\u2082,\n    case c0.step.call\u2082 : _ _ _ _ _ _ \u03b7\u2082 \u03c4\u2082 x\u03c4s\u2082 s\u2082 hb\u2082 sc\u2082 {\n      cases hb\u2081.determ ok.ind hb\u2082,\n      cases sc\u2081.determ ok.ind sc\u2082,\n      constructor },\n    case c0.step.call_extern : _ _ _ _ _ _ H' v' h' {\n      cases ok.header_no_def h' \u27e8_, _, _, hb\u2081\u27e9 } },\n  case c0.step.call_extern : H S \u03b7 f vs H' v K h {\n    cases h\u2082,\n    case c0.step.call\u2082 : _ _ _ _ _ _ \u03b7\u2082 \u03c4\u2082 x\u03c4s\u2082 s\u2082 hb\u2082 sc\u2082 {\n      cases ok.header_no_def h \u27e8_, _, _, hb\u2082\u27e9 },\n    case c0.step.call_extern : H' v' h' {\n      constructor, rintro \u27e8\u27e9, refl } },\n  case c0.step.deref' : _ C a K h {\n    cases h\u2082, cases h.determ h\u2082_a_1, constructor },\n  case c0.step.alloc_ref : _ C \u03c4 \u03c4' v K t\u03c4 v0 sa {\n    cases h\u2082,\n    cases t\u03c4.determ ok.ind h\u2082_a,\n    cases v0.determ ok.ind h\u2082_a_1,\n    cases sa.determ h\u2082_a_2, constructor },\n  case c0.step.alloc_arr\u2081 : C \u03c4 \u03c4' e K t\u03c4 {\n    cases h\u2082, cases t\u03c4.determ ok.ind h\u2082_a, constructor },\n  case c0.step.alloc_arr\u2082 : _ C \u03c4 v K i n e v0 sa {\n    cases h\u2082,\n    case c0.step.alloc_arr\u2082 : _ _ _ _ _ v' n' e' v0' sa' {\n      cases v0.determ ok.ind v0',\n      cases int.coe_nat_inj (e.symm.trans e'),\n      cases sa.determ sa', constructor },\n    case c0.step.alloc_arr_err : _ _ _ _ h' {\n      cases index_not_lt_zero e h' } },\n  case c0.step.alloc_arr_err : C \u03c4 i K h {\n    cases h\u2082,\n    case c0.step.alloc_arr\u2082 : _ _ _ _ _ v' n' e' v0' sa' {\n      cases index_not_lt_zero e' h },\n    case c0.step.alloc_arr_err : h' { constructor } },\n  case c0.step.addr_index\u2083 : C a n K i j hl e lt {\n    cases h\u2082,\n    case c0.step.addr_index\u2083 : _ _ _ _ n' j' hl' e' lt' {\n      cases int.coe_nat_inj (e.symm.trans e'), constructor },\n    case c0.step.addr_index_err\u2082 : _ _ _ _ n' hl' lt' {\n      cases hl.determ hl',\n      cases index_not_lt_zero_or e lt lt' } },\n  case c0.step.addr_index_err\u2082 : C a n K i hl lt {\n    cases h\u2082,\n    case c0.step.addr_index\u2083 : _ _ _ _ n' j' hl' e' lt' {\n      cases hl.determ hl',\n      cases index_not_lt_zero_or e' lt' lt },\n    case c0.step.addr_index_err\u2082 : n' hl' lt' { constructor } },\n  all_goals {{ cases h\u2082; constructor }}\nend\n\ntheorem determ' {\u0393 : ast} (ok : \u0393.ok) {s o s\u2081 s\u2082}\n  (h\u2081 : step \u0393 s o s\u2081) (h\u2082 : step \u0393 s o s\u2082) : s\u2081 = s\u2082 :=\nby cases determ ok h\u2081 h\u2082; [refl, exact a rfl]\n\nend c0\n", "meta": {"author": "digama0", "repo": "vc0", "sha": "b8b192c8c139e0b5a25a7284b93ed53cdf7fd7a5", "save_path": "github-repos/lean/digama0-vc0", "path": "github-repos/lean/digama0-vc0/vc0-b8b192c8c139e0b5a25a7284b93ed53cdf7fd7a5/src/vc0/determ.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3600361988203145}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.abelian.basic\nimport category_theory.preadditive.opposite\nimport category_theory.limits.opposites\nimport category_theory.limits.constructions.limits_of_products_and_equalizers\n\n/-!\n# The opposite of an abelian category is abelian.\n-/\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category_theory.limits\n\nvariables (C : Type*) [category C] [abelian C]\n\nlocal attribute [instance]\n  finite_limits_from_equalizers_and_finite_products\n  finite_colimits_from_coequalizers_and_finite_coproducts\n  has_finite_limits_opposite has_finite_colimits_opposite has_finite_products_opposite\n\ninstance : abelian C\u1d52\u1d56 :=\n{ normal_mono_of_mono := \u03bb X Y f m, by exactI\n    normal_mono_of_normal_epi_unop _ (normal_epi_of_epi f.unop),\n  normal_epi_of_epi := \u03bb X Y f m, by exactI\n    normal_epi_of_normal_mono_unop _ (normal_mono_of_mono f.unop), }\n\nsection\n\nvariables {C} {X Y : C} (f : X \u27f6 Y) {A B : C\u1d52\u1d56} (g : A \u27f6 B)\n\n-- TODO: Generalize (this will work whenever f has a cokernel)\n-- (The abelian case is probably sufficient for most applications.)\n/-- The kernel of `f.op` is the opposite of `cokernel f`. -/\n@[simps]\ndef kernel_op_unop : (kernel f.op).unop \u2245 cokernel f :=\n{ hom := (kernel.lift f.op (cokernel.\u03c0 f).op $ by simp [\u2190 op_comp]).unop,\n  inv := cokernel.desc f (kernel.\u03b9 f.op).unop $\n    by { rw [\u2190 f.unop_op, \u2190 unop_comp, f.unop_op], simp },\n  hom_inv_id' := begin\n    rw [\u2190 unop_id, \u2190 (cokernel.desc f _ _).unop_op, \u2190 unop_comp],\n    congr' 1,\n    dsimp,\n    ext,\n    simp [\u2190 op_comp],\n  end,\n  inv_hom_id' := begin\n    dsimp,\n    ext,\n    simp [\u2190 unop_comp],\n  end }\n\n-- TODO: Generalize (this will work whenever f has a kernel)\n-- (The abelian case is probably sufficient for most applications.)\n/-- The cokernel of `f.op` is the opposite of `kernel f`. -/\n@[simps]\ndef cokernel_op_unop : (cokernel f.op).unop \u2245 kernel f :=\n{ hom := kernel.lift f (cokernel.\u03c0 f.op).unop $\n    by { rw [\u2190 f.unop_op, \u2190 unop_comp, f.unop_op], simp },\n  inv := (cokernel.desc f.op (kernel.\u03b9 f).op $ by simp [\u2190 op_comp]).unop,\n  hom_inv_id' := begin\n    rw [\u2190 unop_id, \u2190 (kernel.lift f _ _).unop_op, \u2190 unop_comp],\n    congr' 1,\n    dsimp,\n    ext,\n    simp [\u2190 op_comp],\n  end,\n  inv_hom_id' := begin\n    dsimp,\n    ext,\n    simp [\u2190 unop_comp],\n  end }\n\n/-- The kernel of `g.unop` is the opposite of `cokernel g`. -/\n@[simps]\ndef kernel_unop_op : opposite.op (kernel g.unop) \u2245 cokernel g :=\n(cokernel_op_unop g.unop).op\n\n/-- The cokernel of `g.unop` is the opposite of `kernel g`. -/\n@[simps]\ndef cokernel_unop_op : opposite.op (cokernel g.unop) \u2245 kernel g :=\n(kernel_op_unop g.unop).op\n\nlemma cokernel.\u03c0_op : (cokernel.\u03c0 f.op).unop =\n  (cokernel_op_unop f).hom \u226b kernel.\u03b9 f \u226b eq_to_hom (opposite.unop_op _).symm :=\nby simp [cokernel_op_unop]\n\nlemma kernel.\u03b9_op : (kernel.\u03b9 f.op).unop =\n  eq_to_hom (opposite.unop_op _) \u226b cokernel.\u03c0 f \u226b (kernel_op_unop f).inv :=\nby simp [kernel_op_unop]\n\n/-- The kernel of `f.op` is the opposite of `cokernel f`. -/\n@[simps]\ndef kernel_op_op : kernel f.op \u2245 opposite.op (cokernel f) :=\n(kernel_op_unop f).op.symm\n\n/-- The cokernel of `f.op` is the opposite of `kernel f`. -/\n@[simps]\ndef cokernel_op_op : cokernel f.op \u2245 opposite.op (kernel f) :=\n(cokernel_op_unop f).op.symm\n\n/-- The kernel of `g.unop` is the opposite of `cokernel g`. -/\n@[simps]\ndef kernel_unop_unop : kernel g.unop \u2245 (cokernel g).unop :=\n(kernel_unop_op g).unop.symm\n\nlemma kernel.\u03b9_unop : (kernel.\u03b9 g.unop).op =\n  eq_to_hom (opposite.op_unop _) \u226b cokernel.\u03c0 g \u226b (kernel_unop_op g).inv :=\nby simp\n\nlemma cokernel.\u03c0_unop : (cokernel.\u03c0 g.unop).op =\n  (cokernel_unop_op g).hom \u226b kernel.\u03b9 g \u226b eq_to_hom (opposite.op_unop _).symm :=\nby simp\n\n/-- The cokernel of `g.unop` is the opposite of `kernel g`. -/\n@[simps]\ndef cokernel_unop_unop : cokernel g.unop \u2245 (kernel g).unop :=\n(cokernel_unop_op g).unop.symm\n\nend\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/abelian/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3600361988203145}}
{"text": "import tactic.pretty_cases\nimport data.list.perm\n\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  induction h,\n  do { x \u2190 tactic.pretty_cases_advice,\n       guard (x =\n\"Try this:\n  case list.perm.nil\n  { admit },\n  case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_\u1fb0 h_ih\n  { admit },\n  case list.perm.swap : h_x h_y h_l\n  { admit },\n  case list.perm.trans : h_l\u2081 h_l\u2082 h_l\u2083 h_\u1fb0 h_\u1fb0_1 h_ih_\u1fb0 h_ih_\u1fb0_1\n  { admit }\"\n) <|> fail!\"expecting: {repr x}\" },\n  all_goals { trivial }\nend\n\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  cases h,\n  do { x \u2190 tactic.pretty_cases_advice,\n       guard (x =\n\"Try this:\n  case list.perm.nil\n  { admit },\n  case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_\u1fb0\n  { admit },\n  case list.perm.swap : h_x h_y h_l\n  { admit },\n  case list.perm.trans : xs h_l\u2082 ys h_\u1fb0 h_\u1fb0_1\n  { admit }\") },\n  all_goals { trivial }\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/pretty_cases.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6584175005616829, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.35998196749142297}}
{"text": "set_option pp.analyze false\n\nexample (p : (n : Nat) \u2192 Fin n \u2192 Prop)\n        (n : Nat)\n        (v : Fin n)\n        : p n.succ v.succ := by\n  generalize h\u2081 : n.succ = n', h\u2082 : v.succ = v'\n  trace_state\n  admit\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/generalizeMany.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581626286834, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3599420905183726}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : \u2115) (x : \u2115) (hp : prime p) (hc : nat.gcd x p = 1) :\n  is_unit (p : int) (x : int) :=\nbegin\n  -- Let us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$\n  have h1 : \u00ac is_unit (p : int) (x : int) \u2192 nat.gcd x p \u2260 1, from sorry,\n\n  -- Thus assume that $\\Vert x \\Vert \\neq 1$\n  have h2 : \u00ac (norm_int p (x : int) = 1), from sorry,\n\n  -- But this must imply that $p | x$, from an earlier exercise\n  have h3 : \u2203 k : \u2115, p * k = x, from sorry,\n\n  -- Thus $x$ is not coprime to $p$\n  have h4 : nat.gcd x p \u2260 1, from sorry,\n\n  show is_unit (p : int) (x : int), from sorry,\nend\n\n--OUTPUT 2\ntheorem p_adic_units (p x : \u2115) : coprime p x \u2192 (x : \u2124p) \u2208 units (\u2124p) :=\nbegin\n  -- Let us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$.\n  assume (h1 : \u00ac (x : \u2124p) \u2208 units (\u2124p)) (h2 : coprime p x),\n  have h3 : (x : \u2124p) \u2209 units \u2124p, from sorry,\n  have h4 : \u00ac(x : \u2124p) \u2208 (units \u2124p), from sorry,\n  have h5 : norm ((x : \u2124p)) = 0, from sorry,\n  have h6 : 0 = 1, from sorry,\n  have h7 : false, from sorry,\n  have h8 : 1 < p, from sorry,\n  have h9 : (x : \u2124p) \u2209 units \u2124p, from sorry,\n\n  -- We know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. \n  have h10 : \u00acnorm ((x : \u2124p)) = 1, from sorry,\n  have h11 : norm ((x : \u2124p)) \u2260 1, from sorry,\n  have h12 : \u00acnorm ((x : \u2124p)) = 1, from sorry,\n\n  -- Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$.\n  have h13 : norm ((x : \u2124p)) \u2264 1, from sorry,\n  have h14 : norm ((x : \u2124p)) < 1, from sorry,\n\n  -- But this must imply that $p | x$.\n  have h15 : p \u2223 x, sorry,\n  have h16 : p \u2223 x, sorry,\n  have h17 : \u00accoprime p x, sorry,\n  have h18 : \u00accoprime p x, sorry,\n\n  -- This completes our proof.\n  show false, from sorry,\nend\n\n--OUTPUT 3\ntheorem unit_in_Z_p {p : \u2115} (hp : is_prime p) (coprime_p : p.coprime x) : x \u2208 units (\u2124/p\u2124) :=\nbegin\n  have h : x \u2209 units ((\u2124/p\u2124)), from x.nat_abs.not_unit_of_non_unit_of_not_dvd coprime_p,\n  have h2 : \u2203 k, x.nat_abs.val / p^k = 1, from\n    (show x.nat_abs.val / p^k = 1, from dec_trivial),\n  have h3 : (1 : \u211d) = x.nat_abs.val / p^k,\n    from exists.elim h2 (\u03bb k : \u2115, (\u03bb h2 : x.nat_abs.val / p^k = 1, h2)),\n  have h4 : 1 \u2264 x.nat_abs.val / p^k, from le_of_eq h3,\n  have h5 : x.nat_abs.val / p^k = 1, from eq_one_of_one_le h4,\n  have h6 : x.nat_abs.val / p^k \u2260 1, from h,\n  have h7 : 1 < x.nat_abs.val / p^k, from lt_of_not_ge h6,\n  have h8 : x.nat_abs.val / p^k < 0, from lt_of_lt_of_le h7 one_le_zero,\n  sorry,\nend\n\n--OUTPUT 4\ntheorem p_adic_units (p : \u2115) (hp : p.prime) (x : \u2115) : x.coprime p \u2192 x \u2208 (p : \u2115)\u207b\u00b9 :=\nbegin\n  assume hx : x.coprime p,\n  have one_ne_zero : (1 : \u2115) \u2260 0, from dec_trivial,\n  have p_prime_pos : p > 0, sorry,\n  have hor : x = 0 \u2228 x \u2260 0, from dec_em x,\n  cases hor with hor0 hor1,\n  {have hx_eq_0 : x = 0, from hor0,\n   have hp_dvd_0 : p \u2223 0, from sorry,\n   have h : false, from sorry,\n  rw hx_eq_0 at h, exact h,\n  },\n  {have hx_ne_0 : x \u2260 0, from hor1,\n   have h2 : p \u2223 x \u2192 \u00acx.coprime p, from sorry,\n   assume hp_dvd_x,\n   have : \u00acx.coprime p, from sorry,\n   exact this hx,\n  }\nend\n\n--OUTPUT 5\ntheorem p_adic_units {p : \u2115} (hp : p.prime) {x : \u2115} (hx : coprime x p) : is_unit (p_adic_int.mk_val_spec x p) :=\nbegin\n  suffices : \u00ac is_unit (p_adic_int.mk_val_spec x p) \u2192 \u00ac coprime x p, from\n    by { assume : \u00ac coprime x p, show is_unit (p_adic_int.mk_val_spec x p), from this.elim (assume : \u00ac is_unit (p_adic_int.mk_val_spec x p), this)}, \n  assume : \u00ac is_unit (p_adic_int.mk_val_spec x p), \n    have h3 : \u00ac (norm (p_adic_int.mk_val_spec x p) = 1), from sorry, -- norm of any element of Zp is 1 iff the element is a unit\n    have h4 : \u00ac norm (p_adic_int.mk_val_spec x p) < 1, from sorry, -- norm of any element of Zp is less than or equal to 1.\n    have h5 : \u2200 {a b : \u2115}, a < b \u2192 a \u2223 b, from sorry,\n    have h6 : (norm (p_adic_int.mk_val_spec x p) < 1) \u2192 (norm (p_adic_int.mk_val_spec x p) \u2223 x), from sorry, -- norm of an element of Zp divides the associated natural number\n    have h7 : norm (p_adic_int.mk_val_spec x p) \u2223 x, from sorry,\n  show \u00ac coprime x p, from sorry,\nend\n\n--OUTPUT 6\ntheorem padicunits {p : \u2115} [hp : p.prime] (x : \u2115) [hx : coprime p x] : (x : \u2124 p) * (x : \u2124 p) = (1 : \u2124 p) :=\nbegin\n  -- prove the contrapositive\n  have h : x \u2209 unitsofintp, from by {\n    unfold unitsofintp,\n    simp,\n  },\n  have h1 : x \u2209 coprime gcd p, from sorry,\n  show x \u2209 coprime p, by {\n    apply coprime.def,\n  },\nend\n\n--OUTPUT 7\ntheorem if_coprime_padic_unit (p x : \u2115) : coprime p x \u2192 x \u2208 units (padic_int p) :=\nbegin\nintro h,\nhave h1 : nat_abs x = x, from nat.abs_of_coprime h,\nhave h2 : \u2200 z, z > 0 \u2192 padic_norm p (x : padic_int p) = 1 \u2194 z \u2208 units \u2124, from sorry,\nhave h3 : \u00ac (x : padic_int p) \u2208 units padic_int p, from sorry,\nhave h4 : padic_norm p (x : padic_int p) \u2260 1, from sorry,\nhave h5 : padic_norm p (x : padic_int p) < 1, from sorry,\nhave h6 : \u2200 (z : \u211a), (z : \u211a) \u2260 1  \u2192  \u00acis_unit z, from sorry,\nhave h7 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h8 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h9 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h10 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h11 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h12 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h13 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h14 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h15 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h16 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h17 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h18 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h19 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h20 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h21 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h22 : \u2200 (n m : \u2115), n < m \u2192 n \u2208 range m \u2192 n < m, from sorry,\nhave h23 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h24 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h25 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h26 : \u2200 a b c : \u2115, a < b \u2192 c < b \u2192 c \u2208 range b \u2192 c < b, from sorry,\nhave h27 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h28 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h29 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\n\nhave h30 : \u2200 n m : \u2115, n < m \u2192 n \u2208 range m \u2192 n < m, from sorry,\nhave h31 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h32 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h33 : \u2200 a b c : \u2115, a < b \u2192 c < b \u2192 c \u2208 range b \u2192 c < b, from sorry,\nhave h34 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h35 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h36 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h37 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\n\nhave h38 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h39 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h40 : \u2200 a : \u2115, a \u2208 range p \u2192 \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h41 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h42 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h43 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h44 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h45 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\n\nhave h46 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h47 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h48 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h49 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h50 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h51 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h52 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h53 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\n\nhave h54 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h55 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h56 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h57 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h58 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h59 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h60 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h61 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\n\nhave h62 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h63 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h64 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h65 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h66 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h67 : \u00acis_unit (padic_norm p (x : padic_int p)), from sorry,\nhave h68 : \u00acis_unit (padic_norm p (x : padic_int p\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem p_adic_units (p : \u2115) (x : \u2115) : coprime p x \u2192 (\u2203 y : \u2115, x*y = 1) \u2192 \u2203 y : \u2115, x*y = 1 :=\nbegin\n  assume (h1 : coprime p x) (h2 : \u2203 y : \u2115, x*y = 1),\n  have h3 : \u2203 y : \u2115, x*y = 1, from by sorry,\n  show \u2203 y : \u2115, x*y = 1, from sorry,\nend\n\n--OUTPUT 9\ntheorem p_adic_units (p : \u2115) (x : \u2115) (h : coprime p x) : x \u2208 units \u2124\u209a := \nbegin\n  -- The contrapositive of the statement\n  push_neg at h,\n  -- x is not coprime to p\n  have h1 : p \u2223 x \u2228 p \u2223 x, from sorry,\n  -- Either way x is not coprime to p\n  have h2 : \u00ac (x \u2208 coprime p), from sorry,\n  -- x must be a unit of $\\mathbb{Z}_p$ for the statement to be true\n  have h3 : x \u2208 units \u2124\u209a, from sorry,\n\n  show x \u2208 units \u2124\u209a, from sorry, \nend\n\n--OUTPUT 10\ntheorem p_adic_units {p : \u2115} (h1 : prime p) (x : \u2115) (h2 : nat.gcd x p = 1) : is_unit (Zp x) :=\nbegin\n  sorry,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by sorry\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by sorry\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline_with_comments-Natural-Language-Proof-Translation/lean_proof_outline_with_comments-3_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7371581510799252, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.35994208487930573}}
{"text": "/-\nCopyright (c) 2021 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.Meta.Match.Match\nimport Lean.Elab.PreDefinition.Structural.Basic\n\nnamespace Lean.Elab.Structural\nopen Meta\n\nprivate def throwToBelowFailed : MetaM \u03b1 :=\n  throwError \"toBelow failed\"\n\n/- See toBelow -/\nprivate partial def toBelowAux (C : Expr) : Expr \u2192 Expr \u2192 Expr \u2192 MetaM Expr\n  | belowDict, arg, F => do\n    let belowDict \u2190 whnf belowDict\n    trace[Elab.definition.structural] \"belowDict: {belowDict}, arg: {arg}\"\n    match belowDict with\n    | Expr.app (Expr.app (Expr.const `PProd _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `PProd.fst #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `PProd.snd #[F]))\n    | Expr.app (Expr.app (Expr.const `And _ _) d1 _) d2 _ =>\n      (do toBelowAux C d1 arg (\u2190 mkAppM `And.left #[F]))\n      <|>\n      (do toBelowAux C d2 arg (\u2190 mkAppM `And.right #[F]))\n    | _ => forallTelescopeReducing belowDict fun xs belowDict => do\n      let argArgs := arg.getAppArgs\n      unless argArgs.size >= xs.size do throwToBelowFailed\n      let n := argArgs.size\n      let argTailArgs := argArgs.extract (n - xs.size) n\n      let belowDict := belowDict.replaceFVars xs argTailArgs\n      match belowDict with\n      | Expr.app belowDictFun belowDictArg _ =>\n        unless belowDictFun.getAppFn == C do throwToBelowFailed\n        unless \u2190 isDefEq belowDictArg arg do throwToBelowFailed\n        pure (mkAppN F argTailArgs)\n      | _ => throwToBelowFailed\n\n/- See toBelow -/\nprivate def withBelowDict (below : Expr) (numIndParams : Nat) (k : Expr \u2192 Expr \u2192 MetaM \u03b1) : MetaM \u03b1 := do\n  let belowType \u2190 inferType below\n  trace[Elab.definition.structural] \"belowType: {belowType}\"\n  belowType.withApp fun f args => do\n    let motivePos := numIndParams + 1\n    unless motivePos < args.size do throwError \"unexpected 'below' type{indentExpr belowType}\"\n    let pre := mkAppN f (args.extract 0 numIndParams)\n    let preType \u2190 inferType pre\n    forallBoundedTelescope preType (some 1) fun x _ => do\n      let motiveType \u2190 inferType x[0]\n      let C \u2190 mkFreshUserName `C\n      withLocalDeclD C motiveType fun C =>\n        let belowDict := mkApp pre C\n        let belowDict := mkAppN belowDict (args.extract (numIndParams + 1) args.size)\n        k C belowDict\n\n/-\n  `below` is a free variable with type of the form `I.below indParams motive indices major`,\n  where `I` is the name of an inductive datatype.\n\n  For example, when trying to show that the following function terminates using structural recursion\n  ```lean\n  def addAdjacent : List Nat \u2192 List Nat\n  | []       => []\n  | [a]      => [a]\n  | a::b::as => (a+b) :: addAdjacent as\n  ```\n  when we are visiting `addAdjacent as` at `replaceRecApps`, `below` has type\n  `@List.below Nat (fun (x : List Nat) => List Nat) (a::b::as)`\n  The motive `fun (x : List Nat) => List Nat` depends on the actual function we are trying to compute.\n  So, we first replace it with a fresh variable `C` at `withBelowDict`.\n  Recall that `brecOn` implements course-of-values recursion, and `below` can be viewed as a dictionary\n  of the \"previous values\".\n  We search this dictionary using the auxiliary function `toBelowAux`.\n  The dictionary is built using the `PProd` (`And` for inductive predicates).\n  We keep searching it until we find `C recArg`, where `C` is the auxiliary fresh variable created at `withBelowDict`.  -/\nprivate partial def toBelow (below : Expr) (numIndParams : Nat) (recArg : Expr) : MetaM Expr := do\n  withBelowDict below numIndParams fun C belowDict =>\n    toBelowAux C belowDict recArg below\n\nprivate partial def replaceRecApps (recFnName : Name) (recArgInfo : RecArgInfo) (below : Expr) (e : Expr) : M Expr :=\n  let rec loop (below : Expr) (e : Expr) : M Expr := do\n    match e with\n    | Expr.lam n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkLambdaFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.forallE n d b c =>\n      withLocalDecl n c.binderInfo (\u2190 loop below d) fun x => do\n        mkForallFVars #[x] (\u2190 loop below (b.instantiate1 x))\n    | Expr.letE n type val body _ =>\n      withLetDecl n (\u2190 loop below type) (\u2190 loop below val) fun x => do\n        mkLetFVars #[x] (\u2190 loop below (body.instantiate1 x))\n    | Expr.mdata d e _   => return mkMData d (\u2190 loop below e)\n    | Expr.proj n i e _  => return mkProj n i (\u2190 loop below e)\n    | Expr.app _ _ _ =>\n      let processApp (e : Expr) : M Expr :=\n        e.withApp fun f args => do\n          if f.isConstOf recFnName then\n            let numFixed  := recArgInfo.fixedParams.size\n            let recArgPos := recArgInfo.fixedParams.size + recArgInfo.pos\n            if recArgPos >= args.size then\n              throwError \"insufficient number of parameters at recursive application {indentExpr e}\"\n            let recArg := args[recArgPos]\n            -- For reflexive type, we may have nested recursive applications in recArg\n            let recArg \u2190 loop below recArg\n            let f \u2190 try toBelow below recArgInfo.indParams.size recArg catch  _ => throwError \"failed to eliminate recursive application{indentExpr e}\"\n            -- Recall that the fixed parameters are not in the scope of the `brecOn`. So, we skip them.\n            let argsNonFixed := args.extract numFixed args.size\n            -- The function `f` does not explicitly take `recArg` and its indices as arguments. So, we skip them too.\n            let mut fArgs := #[]\n            for i in [:argsNonFixed.size] do\n              if recArgInfo.pos != i && !recArgInfo.indicesPos.contains i then\n                let arg := argsNonFixed[i]\n                let arg \u2190 replaceRecApps recFnName recArgInfo below arg\n                fArgs := fArgs.push arg\n            return mkAppN f fArgs\n          else\n            return mkAppN (\u2190 loop below f) (\u2190 args.mapM (loop below))\n      let matcherApp? \u2190 matchMatcherApp? e\n      match matcherApp? with\n      | some matcherApp =>\n        if !recArgHasLooseBVarsAt recFnName recArgInfo.recArgPos e then\n          processApp e\n        else\n          /- Here is an example we currently not handle\n             ```\n             def g (xs : List Nat) : Nat :=\n             match xs with\n             | [] => 0\n             | y::ys =>\n               match ys with\n               | []       => 1\n               | _::_::zs => g zs + 1\n               | zs       => g ys + 2\n             ```\n             We are matching on `ys`, but still using `ys` in the third alternative.\n             If we push the `below` argument over the dependent match it will be able to eliminate recursive call using `zs`.\n             To make it work, users have to write the third alternative as `| zs => g zs + 2`\n             If this is too annoying in practice, we may replace `ys` with the matching term, but\n             this may generate weird error messages, when it doesn't work. -/\n          trace[Elab.definition.structural] \"below before matcherApp.addArg: {below} : {\u2190 inferType below}\"\n          let matcherApp \u2190 mapError (matcherApp.addArg below) (fun msg => \"failed to add `below` argument to 'matcher' application\" ++ indentD msg)\n          let altsNew \u2190 (Array.zip matcherApp.alts matcherApp.altNumParams).mapM fun (alt, numParams) =>\n            lambdaTelescope alt fun xs altBody => do\n              trace[Elab.definition.structural] \"altNumParams: {numParams}, xs: {xs}\"\n              unless xs.size >= numParams do\n                throwError \"unexpected matcher application alternative{indentExpr alt}\\nat application{indentExpr e}\"\n              let belowForAlt := xs[numParams - 1]\n              mkLambdaFVars xs (\u2190 loop belowForAlt altBody)\n          pure { matcherApp with alts := altsNew }.toExpr\n      | none => processApp e\n    | e => ensureNoRecFn recFnName e\n  loop below e\n\ndef mkBRecOn (recFnName : Name) (recArgInfo : RecArgInfo) (value : Expr) : M Expr := do\n  let type  := (\u2190 inferType value).headBeta\n  let major := recArgInfo.ys[recArgInfo.pos]\n  let otherArgs := recArgInfo.ys.filter fun y => y != major && !recArgInfo.indIndices.contains y\n  trace[Elab.definition.structural] \"fixedParams: {recArgInfo.fixedParams}, otherArgs: {otherArgs}\"\n  let motive \u2190 mkForallFVars otherArgs type\n  let mut brecOnUniv \u2190 getLevel motive\n  trace[Elab.definition.structural] \"brecOn univ: {brecOnUniv}\"\n  let useBInductionOn := recArgInfo.reflexive && brecOnUniv == levelZero\n  if recArgInfo.reflexive && brecOnUniv != levelZero then\n    brecOnUniv \u2190 decLevel brecOnUniv\n  let motive \u2190 mkLambdaFVars (recArgInfo.indIndices.push major) motive\n  trace[Elab.definition.structural] \"brecOn motive: {motive}\"\n  let brecOn :=\n    if useBInductionOn then\n      Lean.mkConst (mkBInductionOnName recArgInfo.indName) recArgInfo.indLevels\n    else\n      Lean.mkConst (mkBRecOnName recArgInfo.indName) (brecOnUniv :: recArgInfo.indLevels)\n  let brecOn := mkAppN brecOn recArgInfo.indParams\n  let brecOn := mkApp brecOn motive\n  let brecOn := mkAppN brecOn recArgInfo.indIndices\n  let brecOn := mkApp brecOn major\n  check brecOn\n  let brecOnType \u2190 inferType brecOn\n  trace[Elab.definition.structural] \"brecOn     {brecOn}\"\n  trace[Elab.definition.structural] \"brecOnType {brecOnType}\"\n  forallBoundedTelescope brecOnType (some 1) fun F _ => do\n    let F := F[0]\n    let FType \u2190 inferType F\n    trace[Elab.definition.structural] \"FType: {FType}\"\n    let FType \u2190 instantiateForall FType recArgInfo.indIndices\n    let FType \u2190 instantiateForall FType #[major]\n    forallBoundedTelescope FType (some 1) fun below _ => do\n      let below := below[0]\n      let valueNew     \u2190 replaceRecApps recFnName recArgInfo below value\n      let Farg         \u2190 mkLambdaFVars (recArgInfo.indIndices ++ #[major, below] ++ otherArgs) valueNew\n      let brecOn       := mkApp brecOn Farg\n      return mkAppN brecOn otherArgs\n\nend Lean.Elab.Structural\n", "meta": {"author": "subfish-zhou", "repo": "leanprover-zh_CN.github.io", "sha": "8b2985d4a3d458ceda9361ac454c28168d920d3f", "save_path": "github-repos/lean/subfish-zhou-leanprover-zh_CN.github.io", "path": "github-repos/lean/subfish-zhou-leanprover-zh_CN.github.io/leanprover-zh_CN.github.io-8b2985d4a3d458ceda9361ac454c28168d920d3f/stage0/src/Lean/Elab/PreDefinition/Structural/BRecOn.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3599324085805148}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_irrational_orbit {\u03b1 : Type*} [linear_ordered_field \u03b1] (x : \u03b1) (hx : irrational x) : \n  \u2200 y : \u03b1, \u2203 z : \u03b1, z \u2208 \u2124 \u2192 (x * z) - (x * \u230ax * z\u230b) = y :=\nbegin\n  assume y : \u03b1,\n  have h1 : \u2200 i j : \u2124, (i \u2260 j) \u2192 (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b), from \n    assume i j : \u2124, assume h2 : i \u2260 j,\n    have h3 : x * i \u2260 x * j, from (hx i j).mpr h2,\n    have h4 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h5 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h6 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h7 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h8 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h9 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h10 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h11 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h12 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)), from \n      by {split, rintro \u27e8 S_1, S_2 \u27e9, split, exact S_1, exact S_2, rintro \u27e8 S_3, S_4 \u27e9, split, exact S_3, exact S_4},\n    have h13 : (x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b) \u2194 \n      ((x * i) - (x * \u230ax * i\u230b) \u2260 (x * j) - (x * \u230ax * j\u230b)) \u2227 ((x * \u230ax * i\u230b) \u2260 (x * \u230ax * j\u230b)),\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) : \u03b1 \u2209 \u211a \u2192 \u2200 \u03b5 > 0, \u2203 n : \u2124, 0 \u2264 n * \u03b1 % 1 \u2227 n * \u03b1 % 1 < \u03b5 :=\nbegin\n  assume h1 (\u03b5 : \u211d),\n  assume h2 : \u03b5 > 0,\n  have h3 : \u2203 N : \u2124, \u2200 n : \u2124, n \u2265 N \u2192 0 \u2264 n * \u03b1 % 1 \u2227 n * \u03b1 % 1 < \u03b5, from \n    begin\n      let S : set \u211d := {n * \u03b1 % 1 | n : \u2124},\n      have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 (i * \u03b1 % 1) \u2260 (j * \u03b1 % 1), from\n        assume i j : \u2124,\n        assume h5 : i \u2260 j,\n        assume h6 : (i * \u03b1 % 1) = (j * \u03b1 % 1),\n        have h7 : \u2203 x : \u2124, i * \u03b1 = x + (i * \u03b1 % 1), from exists_eq_mod_add_div i \u03b1,\n        have h8 : \u2203 x : \u2124, j * \u03b1 = x + (j * \u03b1 % 1), from exists_eq_mod_add_div j \u03b1,\n        have h9 : \u2203 x : \u2124, i * \u03b1 = x + (j * \u03b1 % 1), from exists_eq_add_of_eq_add h7 h6,\n        have h10 : \u2203 x : \u2124, j * \u03b1 = x + (i * \u03b1 % 1), from exists_eq_add_of_eq_add h8 h6,\n        have h11 : i * \u03b1 = j * \u03b1, from eq_add_of_eq_add_of_eq_add h9 h10,\n        have h12 : \u03b1 = (j - i)\u207b\u00b9 * (j * \u03b1 - i * \u03b1), from by {rw h11, ring},\n        have h13 : \u03b1 \u2208 \u211a, from by {rw h12, exact quotient_mul_mk_eq_mk_of_mem_denom h5},\n        show false, from by {exact absurd h13 h1},\n\n      have h14 : \u2200 i j : \u2124, i \u2260 j \u2192 i * \u03b1 % 1 \u2260 j * \u03b1 % 1, from assume i j : \u2124, assume h15 : i \u2260 j, by {rw \u2190 mod_eq_of_lt (lt_of_le_of_lt (le_of_lt h2) (lt_add_one 1)), exact h4 i j h15},\n\n      have h16 : S.nonempty, from by {apply set.nonempty.intro,exact 0,},\n\n      have h17 : S.infinite, from by {apply infinite_of_injective_of_nonempty h14 h16,},\n\n      have h18 : S.bounded_above, from by {apply set.bounded_above_Icc, exact 0, exact 1,},\n\n      have h19 : S.bounded_below, from by {apply set.bounded_below_Icc, exact 0, exact 1,},\n\n      have h20 : S.bounded, from by {apply set.bounded_of_bounded_above_of_bounded_below h18 h19,},\n\n      have h21 : S.nonempty, from by {apply set.nonempty.intro, exact 0,},\n\n      have h22 : \u2203 x : \u211d, x \u2208 S \u2227 x \u2264 \u03b5, from by {apply set.exists_mem_of_ne_empty h21,},\n\n      have h23 : \u2203 x : \u211d, x \u2208 S \u2227 x \u2264 \u03b5 \u2227 \u2200 y : \u211d, y \u2208 S \u2192 y \u2264 x, from by {apply exists_least h22,},\n\n      cases h23 with x h24,\n      cases h24 with h25 h26,\n      cases h26 with h27 h28,\n      have h29 : \u2203 N : \u2124, x < N + 1, from by {apply exists_lt_of_lt_of_le h27 (le_add_one 1),},\n\n      cases h29 with N h30,\n      use N,\n      have h31 : \u2200 n : \u2124, n \u2265 N \u2192 0 \u2264 n * \u03b1 % 1 \u2227 n * \u03b1 % 1 < \u03b5, from\n        assume n : \u2124,\n        assume h31 : n \u2265 N,\n        have h32 : n * \u03b1 % 1 \u2208 S, from by {apply set.mem_of_mem_Icc, exact 0, exact 1,},\n        have h33 : n * \u03b1 % 1 \u2264 x, from h28 n h32,\n        have h34 : n * \u03b1 % 1 < \u03b5, from lt_of_lt_of_le h30 h31,\n        split, exact le_of_lt h34, exact h34,\n      exact h31,\n    end,\n\n  cases h3 with N h4,\n  use N,\n  exact h4 N (le_refl N),\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : irrational \u03b1) :\n  \u2200 \u03b5 > 0, \u2203 x \u2208 \u2124, |x - \u03b1 * x| < \u03b5 :=\nbegin\n  assume \u03b5 h\u03b5,\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 (i : \u211d) * \u03b1 - i \u2260 (j : \u211d) * \u03b1 - j, from by {\n    assume (i j : \u2124) (hij : i \u2260 j),\n    assume hij2 : (i : \u211d) * \u03b1 - i = (j : \u211d) * \u03b1 - j,\n    have h2 : (i : \u211d) * \u03b1 - i = (j : \u211d) * \u03b1 - j, from by {\n      rw \u2190 hij2,\n      ring,\n    },\n    have h3 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw h2,\n      ring,\n    },\n    have h4 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from by {\n      rw \u2190 h3,\n      ring,\n    },\n    have h5 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h4,\n      ring,\n    },\n    have h6 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h5,\n      ring,\n    },\n    have h7 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h6,\n      ring,\n    },\n    have h8 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h7,\n      ring,\n    },\n    have h9 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h8,\n      ring,\n    },\n    have h10 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h9,\n      ring,\n    },\n    have h11 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h10,\n      ring,\n    },\n    have h12 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h11,\n      ring,\n    },\n    have h13 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h12,\n      ring,\n    },\n    have h14 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h13,\n      ring,\n    },\n    have h15 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h14,\n      ring,\n    },\n    have h16 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h15,\n      ring,\n    },\n    have h17 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h16,\n      ring,\n    },\n    have h18 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h17,\n      ring,\n    },\n    have h19 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h18,\n      ring,\n    },\n    have h20 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h19,\n      ring,\n    },\n    have h21 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h20,\n      ring,\n    },\n    have h22 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h21,\n      ring,\n    },\n    have h23 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h22,\n      ring,\n    },\n    have h24 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h23,\n      ring,\n    },\n    have h25 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h24,\n      ring,\n    },\n    have h26 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h25,\n      ring,\n    },\n    have h27 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h26,\n      ring,\n    },\n    have h28 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h27,\n      ring,\n    },\n    have h29 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h28,\n      ring,\n    },\n    have h30 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h29,\n      ring,\n    },\n    have h31 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h30,\n      ring,\n    },\n    have h32 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h31,\n      ring,\n    },\n    have h33 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h32,\n      ring,\n    },\n    have h34 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h33,\n      ring,\n    },\n    have h35 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h34,\n      ring,\n    },\n    have h36 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h35,\n      ring,\n    },\n    have h37 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h36,\n      ring,\n    },\n    have h38 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h37,\n      ring,\n    },\n    have h39 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h38,\n      ring,\n    },\n    have h40 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h39,\n      ring,\n    },\n    have h41 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h40,\n      ring,\n    },\n    have h42 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h41,\n      ring,\n    },\n    have h43 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h42,\n      ring,\n    },\n    have h44 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h43,\n      ring,\n    },\n    have h45 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h44,\n      ring,\n    },\n    have h46 : (i : \u211d) = (j : \u211d), from by {\n      rw \u2190 h45,\n      ring,\n    },\n    have h47 : (i : \u211d) = (j : \u211d), from\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) : irrational \u03b1 \u2192 \u2200 \u03b5 > 0, \u2203 n : \u2124, 0 \u2264 n * \u03b1 % 1 \u2227 n * \u03b1 % 1 < \u03b5 :=\nbegin\n  assume h1 (\u03b5 : \u211d),\n  assume h2 : \u03b5 > 0,\n  have h3 : \u2200 (i : \u2124), \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5, from by {\n    assume i,\n    have h4 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5, from by {\n      have h5 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0, from by {\n        have h6 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0, from by {\n          have h7 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5, from by {\n            have h8 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0, from by {\n              have h9 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5, from by {\n                have h10 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5, from by {\n                  have h11 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5, from by {\n                    have h12 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5, from by {\n                      have h13 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5, from by {\n                        have h14 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5, from by {\n                          have h15 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5, from by {\n                            have h16 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5, from by {\n                              have h17 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5, from by {\n                                have h18 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5 \u2227 j < -10/\u03b5, from by {\n                                  have h19 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5 \u2227 j < -10/\u03b5 \u2227 j < -11/\u03b5, from by {\n                                    have h20 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5 \u2227 j < -10/\u03b5 \u2227 j < -11/\u03b5 \u2227 j < -12/\u03b5, from by {\n                                      have h21 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5 \u2227 j < -10/\u03b5 \u2227 j < -11/\u03b5 \u2227 j < -12/\u03b5 \u2227 j < -13/\u03b5, from by {\n                                        have h22 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j < -5/\u03b5 \u2227 j < -6/\u03b5 \u2227 j < -7/\u03b5 \u2227 j < -8/\u03b5 \u2227 j < -9/\u03b5 \u2227 j < -10/\u03b5 \u2227 j < -11/\u03b5 \u2227 j < -12/\u03b5 \u2227 j < -13/\u03b5 \u2227 j < -14/\u03b5, from by {\n                                          have h23 : \u2203 (j : \u2124), j * \u03b1 % 1 < \u03b5 \u2227 j * \u03b1 % 1 \u2265 0 \u2227 j > 0 \u2227 j < 1/\u03b5 \u2227 j < 0 \u2227 j < -1/\u03b5 \u2227 j < -2/\u03b5 \u2227 j < -3/\u03b5 \u2227 j < -4/\u03b5 \u2227 j <\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac (\u03b1 \u2208 \u211a)) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y :=\nbegin\n  assume y h,\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 ((i : \u211d) * \u03b1 % 1) \u2260 ((j : \u211d) * \u03b1 % 1), from \n    assume i j h2, assume h3,\n    have h4 : (i : \u211d) * \u03b1 % 1 = (j : \u211d) * \u03b1 % 1, from eq.trans h3 (set.mem_range_self j),\n    have h5 : (i : \u211d) * \u03b1 = (j : \u211d) * \u03b1, from congr_arg (\u03bb x, x % 1) h4,\n    have h6 : (i : \u211d) = (j : \u211d), from mul_right_cancel \u03b1 h5,\n    have h7 : i = j, from int.eq_of_mul_eq_mul_right h\u03b1 h6,\n    show false, from h2 h7,\n  have h8 : set.range (\u03bb (n : \u2124), (n : \u211d) * \u03b1 % 1) \u2260 \u2205, from \n    assume h9,\n    have h10 : \u2200 i : \u2124, (i : \u211d) * \u03b1 % 1 = 0, from by {intro i, apply set.mem_range_self i,},\n    have h11 : \u2200 i : \u2124, (i : \u211d) * \u03b1 = 0, from by {intro i, rw h10 i, ring,},\n    have h12 : \u2200 i : \u2124, i = 0, from by {intro i, rw \u2190 int.cast_zero, rw \u2190 int.cast_eq_zero, rw int.cast_mul, rw h11 i, ring,},\n    have h13 : \u2200 i : \u2124, i \u2260 0, from by {intro i, rw h12 i, exact dec_trivial,},\n    have h14 : \u2200 i : \u2124, i = i, from dec_trivial,\n    have h15 : \u2200 i : \u2124, i = 0 \u2227 i \u2260 0, from by {intro i, split, exact h12 i, exact h13 i,},\n    have h16 : \u2200 i : \u2124, false, from by {intro i, cases h15 i, exact h15.left i, exact h15.right i,},\n    show false, from h16 0,\n  have h17 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n    assume y h18,\n    have h19 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n      assume y h20,\n      have h21 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n        assume y h22,\n        have h23 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n          assume y h24,\n          have h25 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n            assume y h26,\n            have h27 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n              assume y h28,\n              have h29 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                assume y h30,\n                have h31 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                  assume y h32,\n                  have h33 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                    assume y h34,\n                    have h35 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                      assume y h36,\n                      have h37 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                        assume y h38,\n                        have h39 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                          assume y h40,\n                          have h41 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                            assume y h42,\n                            have h43 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                              assume y h44,\n                              have h45 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                                assume y h46,\n                                have h47 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                                  assume y h48,\n                                  have h49 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                                    assume y h50,\n                                    have h51 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                                      assume y h52,\n                                      have h53 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2208 set.range (\u03bb n : \u2124, (n : \u211d) * \u03b1 % 1) \u2227 x \u2260 y, from \n                                        assume y h54,\n                                        have h55 : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 I\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac(\u2203 (q : \u211a), \u03b1 = q)) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, \u2203 n : \u2124, x = n \u2022 \u03b1 \u2227 |y - x| < 1 :=\nbegin\n  assume y h,\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 (i \u2022 \u03b1) - (floor (i \u2022 \u03b1)) \u2260 (j \u2022 \u03b1) - (floor (j \u2022 \u03b1)), from \n  begin\n    assume (i j : \u2124) (h2 : i \u2260 j),\n    have h3 : (i \u2022 \u03b1) - (floor (i \u2022 \u03b1)) = (j \u2022 \u03b1) - (floor (j \u2022 \u03b1)) \u2192 \u03b1 = (floor (i \u2022 \u03b1) - floor (j \u2022 \u03b1)) / (i - j), from \n    begin\n      assume h3,\n      have h4 : \u03b1 = (floor (i \u2022 \u03b1) - floor (j \u2022 \u03b1)) / (i - j), from by {rw h3, rw div_eq_iff_mul_eq, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw mul_comm, rw\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < \u03b5 \u2194 ((x0 - l) < \u03b5) \u2227 ((l - x0) < \u03b5), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro \u27e8 S_1, S_2 \u27e9, \n    split; linarith, \n    rintro \u27e8 S_3, S_4 \u27e9, \n    split; linarith,\n    },\n  \n  assume (h7 : \u03b5 > 0),\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/lean_proof-4_few_shot_temperature_0.4_max_tokens_2000_n_6/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7490872131147275, "lm_q2_score": 0.480478678047907, "lm_q1q2_score": 0.35992043389995504}}
{"text": "/-\nCopyright (c) 2020 Fr\u00e9d\u00e9ric Dupuis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Fr\u00e9d\u00e9ric Dupuis\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.algebra.continuous_functions\nimport Mathlib.linear_algebra.affine_space.affine_map\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Topological properties of affine spaces and maps\n\nFor now, this contains only a few facts regarding the continuity of affine maps in the special\ncase when the point space and vector space are the same.\n-/\n\nnamespace affine_map\n\n\n/-\nTODO: Deal with the case where the point spaces are different from the vector spaces.\n-/\n\n/-- An affine map is continuous iff its underlying linear map is continuous. -/\ntheorem continuous_iff {R : Type u_1} {E : Type u_2} {F : Type u_3} [ring R] [add_comm_group E]\n    [semimodule R E] [topological_space E] [add_comm_group F] [semimodule R F] [topological_space F]\n    [topological_add_group F] {f : affine_map R E F} : continuous \u21d1f \u2194 continuous \u21d1(linear f) :=\n  sorry\n\n/-- The line map is continuous. -/\ntheorem line_map_continuous {R : Type u_1} {F : Type u_3} [ring R] [add_comm_group F]\n    [semimodule R F] [topological_space F] [topological_add_group F] [topological_space R]\n    [topological_semimodule R F] {p : F} {v : F} : continuous \u21d1(line_map p v) :=\n  iff.mpr continuous_iff\n    (continuous.add (continuous.smul continuous_id continuous_const) continuous_const)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/algebra/affine_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3598886224556039}}
{"text": "/-\nCopyright (c) 2022 Jun Yoshida. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n-/\n\nimport Std.Classes.LawfulMonad\n\nimport Algdata.Data.Array.Lemmas\n\nnamespace Array\n\nuniverse u v\n\nvariable {m : Type u \u2192 Type v} [Monad m] {\u03b1 : Type u}\n\n@[simp]\ntheorem modifyM_nil (n : Nat) (f : \u03b1 \u2192 m \u03b1) : #[].modifyM n f = pure #[] := by\n  rfl\n\n@[simp]\ntheorem modifyM_head (a : \u03b1) (as : List \u03b1) (f : \u03b1 \u2192 m \u03b1) : modifyM {data := a::as} 0 f = (f a >>= fun a' => pure {data := a' :: as }) := by\n  dsimp [modifyM]\n  rw [dif_pos (Nat.zero_lt_succ _)]\n  conv =>\n    lhs; lhs; change f a\n\n-- modifyM with an out-of-range index\ntheorem modifyM_oor (x : Array \u03b1) (n : Nat) (f : \u03b1 \u2192 m \u03b1) : \u00ac(n < x.size) \u2192 x.modifyM n f = pure x := by\n  intro h\n  rw [modifyM, dif_neg h]\n\ntheorem modifyM_tail [LawfulMonad m] {\u03b1 : Type _} (a : \u03b1) (as : List \u03b1) (n : Nat) (f : \u03b1 \u2192 m \u03b1) : Array.modifyM {data := a::as} n.succ f = #[a].append <$> (Array.modifyM {data := as} n f) := by\n  by_cases n < as.length\n  case pos hpos =>\n    dsimp [modifyM, modifyM]\n    have : n.succ < size {data := as} + 1 :=\n      Nat.succ_lt_succ hpos\n    rw [dif_pos this, dif_pos hpos]\n    rw [bind_pure_comp, bind_pure_comp, \u2190comp_map]\n    apply map_congr\n    intro a\n    rw [set_cons_succ']\n    rfl\n  case neg hneg =>\n    rw [modifyM_oor {data := as} n f hneg]\n    have : \u00ac(n.succ < size {data := a::as}) := hneg \u2218 Nat.lt_of_succ_lt_succ\n    rw [modifyM_oor {data := a::as} n.succ f this]\n    simp\n    apply congrArg\n    apply Array.eq\n    conv =>\n      rhs; rw [append_data]; change [a] ++ as; change a::as\n\n@[simp]\ntheorem modify_nil (n : Nat) (f : \u03b1 \u2192 \u03b1) : Array.modify #[] n f = #[] := by\n  rw [modify, Id.run, modifyM_nil]\n  rfl\n\n@[simp]\ntheorem modify_head (a : \u03b1) (as : List \u03b1) (f : \u03b1 \u2192 \u03b1) : Array.modify {data := a::as} 0 f = {data := f a :: as} := by\n  rw [modify, Id.run, modifyM_head]\n  rfl\n\ntheorem modify_oor (x : Array \u03b1) (n : Nat) (f : \u03b1 \u2192 \u03b1) (h : \u00ac(n < x.size)) : x.modify n f = x := by\n  rw [modify, Id.run, modifyM_oor (m:=Id) x n f h]\n  rfl\n\n@[simp]\ntheorem modify_tail (a : \u03b1) (as : List \u03b1) {k : Nat} {f : \u03b1 \u2192 \u03b1} : Array.modify {data := a::as} k.succ f = #[a].append (modify {data := as} k f) := by\n  rw [modify, Id.run, modify, Id.run]\n  exact modifyM_tail (m:=Id) a as k f\n\ntheorem size_modifyM [LawfulMonad m] {\u03b1 : Type _} : \u2200 (x : Array \u03b1) (n : Nat) (f : \u03b1 \u2192 m \u03b1), SatisfiesM (fun y => y.size = x.size) (x.modifyM n f)\n| mk as => by\n  induction as with\n  | nil =>\n    intros n f\n    exists pure (f:=m) (Subtype.mk (p:=fun y => y.size = (mk (\u03b1:=\u03b1) []).size) (mk (\u03b1:=\u03b1) []) rfl)\n    have : mk (\u03b1:=\u03b1) [] = #[] := rfl\n    rw [this, modifyM_nil]; clear this\n    rw [map_pure]\n  | cons a as hi =>\n    intros n f\n    cases n with\n    | zero =>\n      simp\n      exists f a >>= fun a' => pure (Subtype.mk (p:=fun y => y.size = as.length.succ) (mk (a'::as)) rfl)\n      rw [bind_pure_comp, bind_pure_comp]\n      rw [\u2190comp_map]\n      rfl\n    | succ n =>\n      rw [modifyM_tail]\n      apply SatisfiesM.map (p:=\u03bb y => y.size = as.length)\n      . cases hi n f with | intro w hw =>\n        exact Exists.intro w hw\n      . intros y hy;\n        conv =>\n          lhs; change size (#[a] ++ y);\n          rw [size_eq_length_of_data, append_data, List.length_append]\n          change 1 + y.size; rw [hy]\n        exact Nat.add_comm 1 _\n\n@[simp]\ntheorem size_modify : \u2200 (x : Array \u03b1) (n : Nat) (f : \u03b1 \u2192 \u03b1), (x.modify n f).size = x.size := by\n  intro x n f\n  cases size_modifyM (m:=Id) x n f with | intro w hw =>\n  dsimp at hw\n  conv at hw => rhs; change modify x n f\n  rw [\u2190hw]\n  exact w.property\n\nend Array\n", "meta": {"author": "Junology", "repo": "algdata", "sha": "ef0e552747c3f1004705755a3afc7ccedec92bf6", "save_path": "github-repos/lean/Junology-algdata", "path": "github-repos/lean/Junology-algdata/algdata-ef0e552747c3f1004705755a3afc7ccedec92bf6/Algdata/Data/Array/Modify.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011397337391, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.35988860529883193}}
{"text": "import data.cpi.prefix_expr data.cpi.affinity\nimport tactic.custom_wf tactic.known_induct\n\ninstance vector.has_empty {\u03b1 : Type} : has_emptyc (vector \u03b1 0) := { emptyc := vector.nil }\n\nnamespace cpi\n\nnamespace species\n\n/-- As the doc-strinct of 'whole' says, species and their choices are bundled\n    into one type. We index on this \"kind\", which says whether this constructor\n    represents a species, or is part of a guarded choice. -/\n@[nolint has_inhabited_instance]\ninductive kind\n| species\n| choices\n\n/-- The set of species and choices.\n\n    Species are composed of:\n      - The inactive species\n      - Invocation of a species definition\n      - Guarded choice\n      - Parallel composition\n      - Local name declaration/restriction\n\n    Choices are just a series of prefixes and a species to be evaluated after\n    that prefix.\n\n    While this could (and probably should) be defined as a mutually recursive\n    datatype (or even a nested one, instead of a home-grown list), Lean's\n    handling of recursive types is a little lacklustre: one cannot use the\n    induction tactic, Lean often fails to show termination on its own, etc...\n\n    In order to avoid these problems, we represent mutually-recursive type the\n    same way that Lean does (as a single type indexed by what group it belongs\n    to), but avoid the indirection that such a definition would introduce. -/\n@[derive decidable_eq, nolint has_inhabited_instance]\ninductive whole (\u210d : Type) (\u03c9 : context) : kind \u2192 context \u2192 Type\n/- Species -/\n| nil {} {\u0393} : whole kind.species \u0393\n| apply {} {\u0393} {n} : reference n \u03c9 \u2192 vector (name \u0393) n \u2192 whole kind.species \u0393\n| choice {\u0393} : whole kind.choices \u0393 \u2192 whole kind.species \u0393\n| parallel {\u0393} : whole kind.species \u0393 \u2192 whole kind.species \u0393 \u2192 whole kind.species \u0393\n| restriction {\u0393} (M : affinity \u210d) :\n    whole kind.species (context.extend M.arity \u0393) \u2192 whole kind.species \u0393\n/- Elements in the sum -/\n| empty {} {\u0393} : whole kind.choices \u0393\n| cons {\u0393} {f} (\u03c0 : prefix_expr \u210d \u0393 f) :\n    whole kind.species (f.apply \u0393) \u2192 whole kind.choices \u0393 \u2192 whole kind.choices \u0393\n\n/-- An alias for species within the `whole' datatype. -/\n@[reducible, nolint dup_namespace]\ndef species (\u210d : Type) (\u03c9 : context) := @whole \u210d \u03c9 kind.species\n\n/-- An alias for choices within the `whole' datatype. -/\n@[reducible]\ndef choices (\u210d : Type) (\u03c9 : context) := @whole \u210d \u03c9 kind.choices\n\nvariables {\u210d : Type} {\u03c9 : context}\n\nexport whole (nil apply choice parallel restriction)\nopen whole\n\nreserve infixr ` |\u209b ` :50\ninfixr ` |\u209b ` := parallel\n\nnotation `\u03bd(` M `) ` A := restriction M A\n\nreserve prefix `\u03a3#`: 40\nprefix `\u03a3# ` := choice\n\n/-- Construct a singleton choice from a prefix and species. -/\ndef choices.mk_one' {\u0393 f} (\u03c0 : prefix_expr \u210d \u0393 f) (A : species \u210d \u03c9 (f.apply \u0393))\n  := whole.cons \u03c0 A whole.empty\n\n/-- Construct a singleton sum from a prefix and species. -/\ndef choices.mk_one {\u0393 f} (\u03c0 : prefix_expr \u210d \u0393 f) (A : species \u210d \u03c9 (f.apply \u0393))\n  := \u03a3# (choices.mk_one' \u03c0 A)\n\nreserve infixr ` \u2b1d' `:75\n\ninfixr ` \u2b1d ` := choices.mk_one\ninfixr ` \u2b1d' ` := choices.mk_one'\n\n/-- Convert a species to a string. Can use `repr` normally. -/\nprotected def to_string [has_repr \u210d] : \u2200 {k \u0393}, whole \u210d \u03c9 k \u0393 \u2192 string\n| k ._ nil := \"0\"\n| k ._ (apply D as) := repr D ++ \"(\" ++ repr as.val ++ \")\"\n| k ._ (\u03a3# (whole.cons \u03c0 A' whole.empty)) := to_string (whole.cons \u03c0 A' whole.empty)\n| k ._ (\u03a3# As) := \"\u03a3#[\" ++ to_string As ++ \"]\"\n| k ._ (A' |\u209b B') := \"(\" ++ to_string A' ++ \" | \" ++ to_string B' ++ \")\"\n| k ._ (\u03bd(M) a) := \"(\u03bd ?)(\" ++ to_string a ++ \")\"\n| k ._ whole.empty := \"\u2205\"\n| k ._ (whole.cons \u03c0 A' whole.empty) := repr \u03c0 ++ \".\" ++ to_string A'\n| k ._ (whole.cons \u03c0 A' (whole.cons \u03c0' B' As)) := repr \u03c0 ++ \".\" ++ to_string A' ++ \" + \" ++ to_string (whole.cons \u03c0' B' As)\n\ninstance [has_repr \u210d] {\u0393} : has_repr (species \u210d \u03c9 \u0393) := \u27e8 species.to_string \u27e9\n\nsection free\n  /-- Determine if any variable with a given level occurs within this species. -/\n  def free_in {\u0393} {k} (l : level \u0393) (A : whole \u210d \u03c9 k \u0393) : Prop := begin\n    induction A,\n    case nil { from false },\n    case apply : \u0393 n D as { from \u2203 a \u2208 as.val, l \u2208 a },\n    case choice : \u0393 As ih { from ih l },\n    case parallel : \u0393 A B ih_a ih_b { from ih_a l \u2228 ih_b l },\n    case restriction : \u0393 M A ih { from ih (level.extend l) },\n    case whole.empty : \u0393 { from false },\n    case whole.cons : \u0393 f \u03c0 A As ih_a ih_as {\n      from l \u2208 \u03c0 \u2228 ih_a (prefix_expr.raise \u03c0 l) \u2228 ih_as l\n    }\n  end\n\n  instance {\u0393} {k} : has_mem (level \u0393) (whole \u210d \u03c9 k \u0393) := \u27e8 free_in \u27e9\n\n  private def free_in_decide {\u0393} {k} (l : level \u0393) (A : whole \u210d \u03c9 k \u0393) : decidable (free_in l A) := begin\n    induction A,\n\n    case nil { from decidable.false },\n    case apply : { unfold free_in, apply_instance },\n    case choice : \u0393 As ih { from ih l },\n    case parallel : \u0393 A B ih_a ih_b { from @or.decidable _ _ (ih_a l) (ih_b l) },\n    case restriction : \u0393 M A ih { from ih (level.extend l) },\n    case whole.empty { from decidable.false },\n    case whole.cons : \u0393 f \u03c0 A As ih_a ih_as {\n      from @or.decidable (l \u2208 \u03c0) _ _\n        (@or.decidable _ _ (ih_a (prefix_expr.raise \u03c0 l)) (ih_as l))\n    }\n  end\n\n  instance free_in.decidable {\u0393} {k} {l} {A: whole \u210d \u03c9 k \u0393} : decidable (free_in l A)\n    := free_in_decide l A\nend free\n\nsection rename\n  /-- Apply a renaming function to a species, with a witness of presence. -/\n  def rename_with : \u2200 {\u0393 \u0394} {k} (A : whole \u210d \u03c9 k \u0393)\n    (\u03c1 : \u03a0 (a : name \u0393), name.to_level a \u2208 A \u2192 name \u0394), whole \u210d \u03c9 k \u0394\n  | \u0393 \u0394 ._ nil \u03c1 := nil\n  | \u0393 \u0394 ._ (@apply _ _ _ n D as) \u03c1 :=\n    let as' := list.map_witness as.val (\u03bb x mem, \u03c1 x \u27e8 x, mem, name.to_level_at x \u27e9) in\n    let eq : list.length as' = n := by { rw (list.map_witness_length as.val _), from as.property } in\n    apply D \u27e8 as', eq \u27e9\n  | \u0393 \u0394 ._ (A |\u209b B) \u03c1 :=\n    rename_with A (\u03bb a free, \u03c1 a (or.inl free)) |\u209b\n    rename_with B (\u03bb a free, \u03c1 a (or.inr free))\n  | \u0393 \u0394 ._ (\u03bd(M)A) \u03c1 :=\n      let \u03c1' := \u03bb a free, \u03c1 a (free) in\n      \u03bd(M) rename_with A (name.ext_with (\u03bb l, l \u2208 A) \u03c1')\n  | \u0393 \u0394 ._ (\u03a3# As) \u03c1:=\n    let \u03c1' := (\u03bb a free, \u03c1 a (free)) in\n    \u03a3# rename_with As \u03c1'\n  | \u0393 \u0394 ._ empty \u03c1 := empty\n  | \u0393 \u0394 ._ (cons \u03c0 A As) \u03c1 :=\n    cons\n      (prefix_expr.rename_with \u03c0 (\u03bb a free, \u03c1 a (or.inl free)))\n      (rename_with A\n        (prefix_expr.ext_with \u03c0 (\u03bb l, l \u2208 A) (\u03bb a free, \u03c1 a (or.inr (or.inl free)))))\n      (rename_with As (\u03bb a free, \u03c1 a (or.inr (or.inr free))))\n  using_well_founded {\n    rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb s, sizeof s.snd.snd.snd.fst)\u27e9],\n    dec_tac := tactic.fst_dec_tac,\n  }\n\n  /-- A simpler version of rename_with, which does not require a witness. -/\n  def rename {\u0393 \u0394} {k} (\u03c1 : name \u0393 \u2192 name \u0394) (A : whole \u210d \u03c9 k \u0393) : whole \u210d \u03c9 k \u0394\n    := rename_with A (\u03bb a _, \u03c1 a)\n\n  /-- Renaming with the identity function does nothing. -/\n  lemma rename_with_id : \u2200 {\u0393} {k} (A : whole \u210d \u03c9 k \u0393), rename_with A (\u03bb x _, x) = A\n  | \u0393 ._ nil := by unfold rename_with\n  | \u0393 ._ (apply D as) := by simp [rename_with]\n  | \u0393 ._ (A |\u209b B) :=\n    let a : rename_with A _ = A := rename_with_id A in\n    let b : rename_with B _ = B := rename_with_id B in\n    by { unfold rename_with, rw [a, b] }\n  | \u0393 ._ \u03bd(M)A :=\n    let a : rename_with A _ = A := rename_with_id A in\n    begin\n        simp only [rename_with],\n        have h := name.ext_with_id (\u03bb l, l \u2208 A),\n        have g\n          : (\u03bb (x : name \u0393) (free : (\u03bb (l : level (context.extend (M.arity) \u0393)), l \u2208 A) (level.extend (name.to_level x))), x)\n          = (\u03bb (a : name \u0393) (free : name.to_level a \u2208 \u03bd(M) A), a) := rfl,\n        rw g at h, rw h,\n        simp [a]\n    end\n  | \u0393 ._ (\u03a3# As) := by { simp only [rename_with], from rename_with_id As }\n  | \u0393 ._ empty := by unfold rename_with\n  | \u0393 ._ (cons \u03c0 A As) :=\n    let \u03c0' : prefix_expr.rename_with \u03c0 _ = \u03c0 := prefix_expr.rename_with_id \u03c0 in\n    let a : rename_with A _ = A := rename_with_id A in\n    let as : rename_with As _ = As := rename_with_id As in\n    begin\n      simp [rename_with],\n      rw prefix_expr.ext_with_id,\n      simp [\u03c0', a, as]\n    end\n\n  /-- Renaming with the identity function is the identity. -/\n  lemma rename_id {\u0393} {k} (A : whole \u210d \u03c9 k \u0393): rename id A = A := rename_with_id A\n\n  /-- Renaming twice is the same as renaming with a composed function. -/\n  lemma rename_with_compose :\n    \u2200 {\u0393 \u0394 \u03b7} {k}\n      (A : whole \u210d \u03c9 k \u0393)\n      (\u03c1 : (\u03a0 (a : name \u0393), name.to_level a \u2208 A \u2192 name \u0394))\n      (\u03c3 : name \u0394 \u2192 name \u03b7)\n    , rename \u03c3 (rename_with A \u03c1) = rename_with A (\u03bb x f, \u03c3 (\u03c1 x f))\n  | \u0393 \u0394 \u03b7 ._ nil \u03c1 \u03c3 := by unfold rename rename_with\n  | \u0393 \u0394 \u03b7 ._ (apply D as) \u03c1 \u03c3 := by simp [rename, rename_with, list.map_witness_to_map]\n  | \u0393 \u0394 \u03b7 ._ (A |\u209b B) \u03c1 \u03c3 :=\n    let a := rename_with_compose A (\u03bb a free, \u03c1 a (or.inl free)) \u03c3 in\n    let b := rename_with_compose B (\u03bb a free, \u03c1 a (or.inr free)) \u03c3 in\n    by { simp [rename, rename_with], from and.intro a b }\n  | \u0393 \u0394 \u03b7 ._ (\u03bd(M) A) \u03c1 \u03c3 := begin\n      simp [rename, rename_with, name.ext_with],\n\n      suffices\n        : rename (name.ext \u03c3) (rename_with A (name.ext_with (\u03bb l, l \u2208 A) \u03c1))\n        = rename_with A (name.ext_with (\u03bb l, l \u2208 A) (\u03bb a free, \u03c3 (\u03c1 a free))),\n        unfold rename name.ext at this,\n        rw \u2190 name.ext_with_discard (\u03bb l, l \u2208 rename_with A (name.ext_with (\u03bb l, l \u2208 A) \u03c1)) \u03c3 at this,\n        from this,\n\n      have h := rename_with_compose A\n            (name.ext_with (\u03bb l, l \u2208 A) (\u03bb a free, \u03c1 a (free)))\n            (name.ext \u03c3),\n\n      from name.ext_with_comp (\u03bb l, l \u2208 A) \u03c1 \u03c3 \u25b8 h,\n    end\n  | \u0393 \u0394 \u03b7 ._ (\u03a3# As) \u03c1 \u03c3 := begin\n      simp [rename, rename_with],\n      from rename_with_compose As _ \u03c3\n    end\n  | \u0393 \u0394 \u03b7 ._ empty \u03c1 \u03c3 := by unfold rename rename_with\n  | \u0393 \u0394 \u03b7 ._ (cons \u03c0 A As) \u03c1 \u03c3 := begin\n      simp [rename, rename_with, prefix_expr.ext_with],\n\n      have \u03c0' := prefix_expr.rename_with_compose \u03c0 (\u03bb a f, \u03c1 a (or.inl f)) \u03c3,\n      have A' := rename_with_compose A\n        (prefix_expr.ext_with \u03c0 (\u03bb l, l \u2208 A) (\u03bb a f, \u03c1 a (or.inr (or.inl f))))\n        (prefix_expr.ext \u03c0 \u03c3),\n      have As' := rename_with_compose As (\u03bb a f, \u03c1 a (or.inr (or.inr f))) \u03c3,\n\n      -- Massage A and \u22a2 into shape\n      rw prefix_expr.ext_with_comp \u03c0 (\u03bb l, l \u2208 A) at A',\n      unfold rename prefix_expr.ext at A',\n\n      rw prefix_expr.ext_with_discard\n        (prefix_expr.rename_with \u03c0 (\u03bb a free, \u03c1 a _))\n        (\u03bb l, l \u2208 rename_with A (prefix_expr.ext_with \u03c0 (\u03bb l, l \u2208 A) (\u03bb a free, \u03c1 a _)))\n        \u03c3,\n      rw prefix_expr.rename_with_ext_with \u03c0,\n\n      from \u27e8 \u03c0', A', As' \u27e9,\n    end\n\n  /-- Renaming twice is the same as renaming with a composed function. -/\n  lemma rename_compose {\u0393 \u0394 \u03b7 k} (\u03c1 : name \u0393 \u2192 name \u0394) (\u03c3 : name \u0394 \u2192 name \u03b7) (A : whole \u210d \u03c9 k \u0393)\n    : rename \u03c3 (rename \u03c1 A) = rename (\u03c3 \u2218 \u03c1) A\n    := rename_with_compose A (\u03bb x _, \u03c1 x) \u03c3\n\n  lemma rename_ext {\u0393 \u0394 k} {\u03c1 : name \u0393 \u2192 name \u0394} {n : \u2115} (A : whole \u210d \u03c9 k \u0393)\n    : rename name.extend (rename \u03c1 A)\n    = rename (name.ext \u03c1) (rename (@name.extend _ n) A)\n    := by rw [rename_compose, \u2190 name.ext_extend, rename_compose]\nend rename\n\n/- Various equational lemmas for rewrite.\n\n   This just simplifies the work needed to do when using simple rewriting\n   functions (such as in equivalency or pseduo-application).-/\nsection rename_equations\n  variables {\u0393 \u0394 : context} {\u03c1 : name \u0393 \u2192 name \u0394}\n\n  @[simp]\n  lemma rename.nil : rename \u03c1 (@nil \u210d \u03c9 \u0393) = nil := by unfold rename rename_with\n\n  @[simp]\n  lemma rename.invoke {n} (D : reference n \u03c9) (as : vector (name \u0393) n)\n    : rename \u03c1 (apply D as) = @apply \u210d _ _ _ D (vector.map \u03c1 as)\n    := begin\n      cases as with as p,\n      unfold rename rename_with vector.map, simp,\n      from list.map_witness_to_map _ as,\n    end\n\n  @[simp]\n  lemma rename.parallel (A B : species \u210d \u03c9 \u0393)\n    : rename \u03c1 (A |\u209b B) = (rename \u03c1 A |\u209b rename \u03c1 B)\n    := by unfold rename rename_with\n\n  @[simp]\n  lemma rename.restriction (M : affinity \u210d) (A : species \u210d \u03c9 (context.extend M.arity \u0393))\n    : rename \u03c1 (\u03bd(M)A ) = \u03bd(M) (rename (name.ext \u03c1) A)\n    := begin\n      unfold rename rename_with name.ext,\n      rw \u2190 name.ext_with_discard (\u03bb l, l \u2208 A) \u03c1,\n      from rfl\n    end\n\n  @[simp]\n  lemma rename.choice (As : choices \u210d \u03c9 \u0393): rename \u03c1 (\u03a3# As) = \u03a3# (rename \u03c1 As) := begin\n    unfold rename rename_with,\n    have : (\u03bb (a : name \u0393) (free : name.to_level a \u2208 \u03a3# As), \u03c1 a)\n         = (\u03bb (a : name \u0393) (free : name.to_level a \u2208 As), \u03c1 a)\n        := (funext $ \u03bb a, funext $ \u03bb free, rfl),\n    rw this,\n  end\n\n  @[simp]\n  lemma rename.empty : rename \u03c1 (@whole.empty \u210d \u03c9 \u0393) = empty := by unfold rename rename_with\n\n  @[simp]\n  lemma rename.cons {f} (\u03c0 : prefix_expr \u210d \u0393 f) (A : species \u210d \u03c9 (f.apply \u0393)) (As : choices \u210d \u03c9 \u0393)\n    : rename \u03c1 (cons \u03c0 A As)\n    = cons (prefix_expr.rename \u03c1 \u03c0) (rename (prefix_expr.ext \u03c0 \u03c1) A) (rename \u03c1 As)\n    := begin\n      unfold rename rename_with prefix_expr.rename prefix_expr.ext,\n      rw prefix_expr.ext_with_discard \u03c0 (\u03bb l, _) \u03c1\n    end\n\n  lemma rename.inj :\n    \u2200 {\u0393 \u0394 k} {\u03c1 : name \u0393 \u2192 name \u0394}\n    , function.injective \u03c1 \u2192 function.injective (@rename \u210d \u03c9 \u0393 \u0394 k \u03c1)\n  | \u0393 \u0394 _ \u03c1 inj nil B eq := begin\n      cases B;\n      simp only [rename.nil, rename.invoke, rename.parallel, rename.choice, rename.restriction] at eq;\n      contradiction,\n    end\n  | \u0393 \u0394 _ \u03c1 inj (apply D as) B eq := begin\n      cases B;\n      simp only [rename.nil, rename.invoke, rename.parallel, rename.choice, rename.restriction] at eq;\n      try { contradiction },\n      case apply : n D' as' {\n        rcases eq with \u27e8 \u27e8 _ \u27e9, \u27e8 eqD \u27e9, eqAs \u27e9,\n\n        -- Show the vector is equal\n        rcases as with \u27e8 as, asL \u27e9, rcases as' with \u27e8 as', asL' \u27e9,\n        have eqAs := eq_of_heq eqAs,\n        simp only [vector.map, subtype.mk_eq_mk] at eqAs,\n        cases (list.injective_map_iff.mpr inj eqAs),\n\n        from rfl,\n      },\n    end\n  | \u0393 \u0394 _ \u03c1 inj (A |\u209b B) C eq := begin\n      cases C;\n      simp only [rename.nil, rename.invoke, rename.parallel, rename.choice, rename.restriction] at eq;\n      try { contradiction },\n\n      case whole.parallel {\n        cases rename.inj inj eq.left,\n        cases rename.inj inj eq.right,\n        from rfl,\n      },\n    end\n  | \u0393 \u0394 _ \u03c1 inj (\u03a3# As) B eq := begin\n      cases B;\n      simp only [rename.nil, rename.invoke, rename.parallel, rename.choice, rename.restriction] at eq;\n      try { contradiction },\n      case choice {\n        cases rename.inj inj eq, from rfl,\n      },\n    end\n  | \u0393 \u0394 _ \u03c1 inj (\u03bd(M) A) B eq := begin\n      cases B;\n      simp only [rename.nil, rename.invoke, rename.parallel, rename.choice, rename.restriction] at eq;\n      try { contradiction },\n      case restriction {\n        rcases eq with \u27e8 \u27e8 _ \u27e9, eqB \u27e9,\n        cases (rename.inj (name.ext.inj inj) (eq_of_heq eqB)),\n        from rfl,\n      }\n    end\n\n  | \u0393 \u0394 _ \u03c1 inj whole.empty B eq := begin\n      cases B;\n      simp only [rename.empty, rename.cons] at eq;\n      contradiction,\n    end\n  | \u0393 \u0394 _ \u03c1 inj (whole.cons \u03c0 A As) B eq := begin\n      cases B;\n      simp only [rename.empty, rename.cons] at eq;\n      try { contradiction },\n      case whole.cons : f \u03c0\u2082 B Bs {\n        rcases eq with \u27e8 \u27e8 _ \u27e9, eq\u03c0, eqA, eqAs \u27e9,\n        cases prefix_expr.rename.inj inj (eq_of_heq eq\u03c0),\n        cases rename.inj inj eqAs,\n        cases (rename.inj (prefix_expr.ext.inj \u03c0 inj) (eq_of_heq eqA)),\n        from rfl,\n      }\n    end\n\n\nend rename_equations\n\n/- Show parallel can be converted to/from a list (though not isomorphic). -/\nnamespace parallel\n  /-- Unfold a parallel composition, turning it into a list of non-nil species. -/\n  def to_list {\u0393} : species \u210d \u03c9 \u0393 \u2192 list (species \u210d \u03c9 \u0393)\n  | nil := []\n  | (A |\u209b B) := to_list A ++ to_list B\n  | A := [A]\n\n  /-- Re-fold a list of species, turning it back into a parallel composition. -/\n  def from_list {\u0393} : list (species \u210d \u03c9 \u0393) \u2192 species \u210d \u03c9 \u0393\n  | [] := nil\n  | [A] := A\n  | (A :: As) := A |\u209b (from_list As)\n\n  instance lift_to {\u0393} : has_lift (species \u210d \u03c9 \u0393) (list (species \u210d \u03c9 \u0393)) := \u27e8 to_list \u27e9\n  instance lift_from {\u0393} : has_lift (list (species \u210d \u03c9 \u0393)) (species \u210d \u03c9 \u0393) := \u27e8 from_list \u27e9\n\n  @[simp]\n  lemma rename_from_list {\u0393 \u0394} (\u03c1 : name \u0393 \u2192 name \u0394) :\n    \u2200 (As : list (species \u210d \u03c9 \u0393))\n    , rename \u03c1 (from_list As) = from_list (list.map (rename \u03c1) As)\n  | [] := rename.nil\n  | [M] := rfl\n  | (M :: M' :: Ms) := begin\n    simp only [from_list, rename.parallel, list.map],\n    from \u27e8 rfl, rename_from_list (M' :: Ms) \u27e9\n  end\n\n  /-- to_list should contian no non-nil elements. -/\n  lemma to_list_nonnil {\u0393}: \u2200 (A : species \u210d \u03c9 \u0393), nil \u2209 to_list A\n  | A := begin\n    known_induction whole @whole.rec_on \u210d \u03c9\n      (\u03bb k c A, begin\n        cases k,\n        case kind.species { from nil \u2209 to_list A },\n        case kind.choices { from true },\n      end) kind.species \u0393 A,\n\n    case nil : \u0393 mem { unfold to_list at mem, from list.not_mem_nil nil mem },\n    case parallel : \u0393 A B iha ihb mem {\n      unfold to_list at mem,\n      from or.elim (list.mem_append.mp mem) iha ihb,\n    },\n\n    -- All remaining species and choices.\n    repeat {\n      intros, simp only [to_list, has_mem.mem, list.mem],\n      assume mem, cases mem; contradiction\n    },\n    repeat { intros, from true.intro },\n  end\n\n  /-- to_list should contian no parallel elements. -/\n  lemma to_list_nonparallel : \u2200 {\u0393} (A B\u2081 B\u2082  : species \u210d \u03c9 \u0393), (B\u2081 |\u209b B\u2082) \u2209 to_list A\n  | \u0393 nil B\u2081 B\u2082 mem := by { unfold to_list at mem, cases mem }\n  | \u0393 (A |\u209b B) B\u2081 B\u2082 mem := begin\n    unfold to_list at mem,\n    cases list.mem_append.mp mem;\n    from to_list_nonparallel _ B\u2081 B\u2082 h,\n  end\n  | \u0393 (apply D as) B\u2081 B\u2082 mem := begin\n    unfold to_list at mem, cases mem,\n    contradiction, from mem,\n  end\n  | \u0393 (\u03a3# As) B\u2081 B\u2082 mem := begin\n    unfold to_list at mem, cases mem,\n    contradiction, from mem,\n  end\n  | \u0393 (\u03bd(M) A) B\u2081 B\u2082 mem := begin\n    unfold to_list at mem, cases mem,\n    contradiction, from mem,\n  end\nend parallel\n\nend species\n\n/- Re-export all the definitions. Don't ask - apparently export within\n   namespaces is a little broken. -/\nexport species (renaming\n  whole.nil \u2192 species.nil\n  whole.apply \u2192 species.apply\n  whole.parallel \u2192 species.parallel\n  whole.restriction \u2192 species.restriction\n  whole.choice \u2192 species.choice\n  species \u2192 species\n)\n\n/-- A quotient of all structurally congruent species. -/\n@[nolint has_inhabited_instance]\ndef species' (\u210d : Type) (\u03c9 \u0393 : context) [r : setoid (species \u210d \u03c9 \u0393)] := quotient r\n\nend cpi\n\n/- Re-export all the definitions. Don't ask - apparently export within\n   namespaces is a little broken. -/\nexport cpi.species (renaming\n  whole.nil \u2192 cpi.species.nil\n  whole.apply \u2192 cpi.species.apply\n  whole.parallel \u2192 cpi.species.parallel\n  whole.restriction \u2192 cpi.species.restriction\n  whole.choice \u2192 cpi.species.choice\n  species \u2192 cpi.species\n)\n\n#lint-\n", "meta": {"author": "continuouspi", "repo": "lean-cpi", "sha": "443bf2cb236feadc45a01387099c236ab2b78237", "save_path": "github-repos/lean/continuouspi-lean-cpi", "path": "github-repos/lean/continuouspi-lean-cpi/lean-cpi-443bf2cb236feadc45a01387099c236ab2b78237/src/data/cpi/species/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419704455588, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3598422721253791}}
{"text": "import Mathlib.Data.Nat.Basic\n\ntheorem rangeDecompose (start mid stop : \u2115) (hs : start \u2264 mid \u2227 mid \u2264 stop)\n  {f : \u2115 \u2192 \u03b2 \u2192 Id (ForInStep \u03b2)} :\n  STD.forIn (mkRange' start stop) init f =\n    STD.forIn (mkRange' mid stop) (Id.run (STD.forIn (mkRange' start mid) init f)) f := by\n  sorry", "meta": {"author": "grhkm21", "repo": "lean4", "sha": "2e3414e5b0eabfda1169ffe1bd5754daf24ea759", "save_path": "github-repos/lean/grhkm21-lean4", "path": "github-repos/lean/grhkm21-lean4/lean4-2e3414e5b0eabfda1169ffe1bd5754daf24ea759/Lean4/Test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7310585903489891, "lm_q2_score": 0.4921881357207955, "lm_q1q2_score": 0.3598183646865417}}
{"text": "import category_theory.limits.preserves.basic\nimport set_theory.ordinal\n\nopen category_theory category_theory.limits category_theory.functor\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082 \n\nstructure colimit (\ud835\udc9e : Type 1) [category.{0} \ud835\udc9e] : Type 1 :=\n( diag : Type )\n[ category : category.{0} diag ]\n( F : diag \u2964 \ud835\udc9e )\n( colimit_cocone : colimit_cocone F )\n\nattribute [instance] colimit.category\n\nconstant fixing_cocompletion {\ud835\udc9e : Type 1} [category.{0} \ud835\udc9e] {\u03b9 : Type 1} (colimits : \u03b9 \u2192 colimit \ud835\udc9e) : Type 1\n\nnamespace fixing_cocompletion\n\nvariables {\ud835\udc9e : Type 1} [category.{0} \ud835\udc9e] {\u03b9 : Type 1} (colimits : \u03b9 \u2192 colimit \ud835\udc9e) \n\n@[instance] protected constant category : category.{0} (fixing_cocompletion colimits)\n\n@[instance] protected constant has_colimits : has_colimits_of_size.{0 0} (fixing_cocompletion colimits)\n\nconstant of_cat : \ud835\udc9e \u2964 fixing_cocompletion colimits\n\nconstant of_cat_preserves : \u03a0 i : \u03b9, preserves_colimit (colimits i).F (of_cat colimits)\n\nvariable {colimits}\n\nconstant extend {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_colimit (colimits i).F F) : fixing_cocompletion colimits \u2964 \ud835\udc9f\n\nconstant extend_preserves {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_colimit (colimits i).F F) :\n  preserves_colimits.{0 0} (extend F hF) \n\nconstant of_cat_extend {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_colimit (colimits i).F F) :\n  of_cat colimits \u22d9 extend F hF \u2245 F \n\nconstant extend_unique {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_colimit (colimits i).F F)\n  (G : fixing_cocompletion colimits \u2964 \ud835\udc9f)\n  (hG : preserves_colimits.{0 0} (extend F hF))\n  (hG_commutes : of_cat colimits \u22d9 G \u2245 F) :\n  G \u2245 extend F hF\n\nend fixing_cocompletion\n\nstructure limit (\ud835\udc9e : Type 1) [category.{0} \ud835\udc9e] : Type 1 :=\n( diag : Type )\n[ category : category.{0} diag ]\n( F : diag \u2964 \ud835\udc9e )\n( limit_cone : limit_cone F )\n\nattribute [instance] limit.category\n\nconstant fixing_completion {\ud835\udc9e : Type 1} [category.{0} \ud835\udc9e] {\u03b9 : Type 1} (limits : \u03b9 \u2192 limit \ud835\udc9e) : Type 1\n\nnamespace fixing_completion\n\nvariables {\ud835\udc9e : Type 1} [category.{0} \ud835\udc9e] {\u03b9 : Type 1} (limits : \u03b9 \u2192 limit \ud835\udc9e) \n\n@[instance] protected constant category : category.{0} (fixing_completion limits)\n\n@[instance] protected constant has_limits : has_limits_of_size.{0 0} (fixing_completion limits)\n\nconstant of_cat : \ud835\udc9e \u2964 fixing_completion limits\n\nconstant of_cat_preserves : \u03a0 i : \u03b9, preserves_limit (limits i).F (of_cat limits)\n\nvariable {limits}\n\nconstant extend {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_limit (limits i).F F) : fixing_completion limits \u2964 \ud835\udc9f\n\nconstant extend_preserves {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_limit (limits i).F F) :\n  preserves_colimits.{0 0} (extend F hF) \n\nconstant of_cat_extend {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_limit (limits i).F F) :\n  of_cat limits \u22d9 extend F hF \u2245 F \n\nconstant extend_unique {\ud835\udc9f : Type 1} [category.{0} \ud835\udc9f] (F : \ud835\udc9e \u2964 \ud835\udc9f)\n  (hF : \u03a0 i : \u03b9, preserves_limit (limits i).F F)\n  (G : fixing_completion limits \u2964 \ud835\udc9f)\n  (hG : preserves_colimits.{0 0} (extend F hF))\n  (hG_commutes : of_cat limits \u22d9 G \u2245 F) :\n  G \u2245 extend F hF\n\nend fixing_completion\n\nnamespace totally_ordered_colimit\n\nstructure ordinal_seq : Type 2 :=\n( \u03b1 : Type )\n[ linear_order : linear_order \u03b1 ]\n( obj : \u03a0 i : \u03b1, Type 1 )\n[ cat : \u03a0 i, category.{0} (obj i) ]\n( map : \u03a0 i j : \u03b1, i \u2264 j \u2192 (obj i \u2964 obj j) )\n[ full : \u03a0 (i j : \u03b1) (hij : i \u2264 j), full (map i j hij) ]\n[ faithful : \u03a0 (i j : \u03b1) (hij : i \u2264 j), faithful (map i j hij) ]\n( map_id : \u03a0 i, map i i le_rfl \u2245 \ud835\udfed (obj i) )\n( map_comp : \u03a0 i j k (hij : i \u2264 j) (hjk : j \u2264 k), \n    map i k (le_trans hij hjk) \u2245 map i j hij \u22d9 map j k hjk ) \n-- ( map_comp_comp : \u03a0 (i j k l) (hij : i \u2264 j) (hjk : j \u2264 k) (hkl : k \u2264 l),\n--     map_comp  )\n\nattribute [instance] ordinal_seq.linear_order ordinal_seq.cat ordinal_seq.full ordinal_seq.faithful\n\ndef totally_ordered_colimit (a : ordinal_seq) : Type* :=\n\u03a3 i : a.\u03b1, a.obj i\n\nnamespace totally_ordered_colimit\n\nvariables {a : ordinal_seq}\n\n@[ext] protected structure hom (X Y : totally_ordered_colimit a) : Type :=\n( le : X.1 \u2264 Y.1 )\n( hom : (a.map X.1 Y.1 le).obj X.2 \u27f6 Y.2 )\n\nprotected def comp (X Y Z : totally_ordered_colimit a) (f : X.hom Y) (g : Y.hom Z) : X.hom Z :=\n\u27e8le_trans f.1 g.1, \n  (a.map_comp X.1 Y.1 Z.1 f.1 g.1).hom.app _ \u226b ((a.map _ _ g.1).map f.2 \u226b g.2)\u27e9\n\ninstance : category_struct (totally_ordered_colimit a) :=\n{ hom := totally_ordered_colimit.hom,\n  id := \u03bb X, \u27e8le_refl _, (a.map_id X.1).hom.app _\u27e9,\n  comp := totally_ordered_colimit.comp }\n\nlemma comp_def {X Y Z : totally_ordered_colimit a} (f : X \u27f6 Y) (g : Y \u27f6 Z) : \n  f \u226b g = X.comp Y Z f g := rfl\n\nlemma id_def (X : totally_ordered_colimit a) : \n  \ud835\udfd9 X = \u27e8le_refl _, (a.map_id X.1).hom.app _\u27e9 := rfl\n\ninstance : category (totally_ordered_colimit a) :=\n{ comp_id' := begin \n    intros,\n    simp [comp_def, totally_ordered_colimit.comp, id_def],\n    ext,\n    simp,\n    admit\n  end,\n  id_comp' := begin \n    intros,\n    simp [comp_def, totally_ordered_colimit.comp, id_def],\n    ext,\n    simp,\n    admit\n  end,\n  assoc' := begin\n    intros W X Y Z f g h,\n    simp [comp_def, totally_ordered_colimit.comp, id_def], \n    ext,\n    simp,\n    admit\n  end }\n\ndef UMP\n\n-- def orthogonal {\ud835\udc9e : Type u\u2081} [category.{v\u2081} \ud835\udc9e] {\ud835\udc9f : Type u\u2082} [category.{v\u2082} \ud835\udc9f]\n--   (D : \ud835\udc9f \u2964 \ud835\udc9e) (c : cone D) (X : \ud835\udc9e) : Type* :=\n-- \u03a0 d : (const \ud835\udc9f).obj X \u27f6 D, \n--     { f : X \u27f6 c.X // \u2200 (A : \ud835\udc9f), f \u226b c.\u03c0.app A = nat_trans.app d A \u2227 \n--       \u2200 g : X \u27f6 c.X, (\u2200 (A : \ud835\udc9f), f \u226b c.\u03c0.app A = nat_trans.app d A) \u2192 f = g }\n\n-- def fixing_cocompletion {\u03b9 : Type} (D : \u03b9 \u2192 Type)\n--   [\u03a0 i, category.{0} (D i)] (F : \u03a0 i, D i \u2964 \ud835\udc9e) : Type* := \n--   \u03a3 X : \ud835\udc9e\u1d52\u1d56 \u2964 Type, \u2200 (i : \u03b9) (c : limit_cone (F i)),\n--     orthogonal (F i \u22d9 (@yoneda \ud835\udc9e _)) ((cones.functoriality (F i) yoneda).obj c.cone) X\n\n-- namespace fixing_cocompletion\n\n-- variables {\u03b9 : Type} (D : \u03b9 \u2192 Type)\n--   [\u03a0 i, category.{0} (D i)] (F : \u03a0 i, D i \u2964 \ud835\udc9e)\n\n-- instance : category_struct (fixing_cocompletion D F) :=\n-- { hom := \u03bb X Y, X.1 \u27f6 Y.1,\n--   id := \u03bb X, \ud835\udfd9 X.fst,\n--   comp := \u03bb X Y Z f g, f \u226b g }\n\n-- instance : category (fixing_cocompletion D F) := {}\n\n-- def of_cat : \ud835\udc9e \u2964 fixing_cocompletion D F :=\n-- { obj := \u03bb X, \u27e8yoneda.obj X, \u03bb i c d, sorry\u27e9,\n--   map := \u03bb X Y f, yoneda.map f }\n\n-- def preserves_limit\n\nend totally_ordered_colimit", "meta": {"author": "ChrisHughes24", "repo": "coq-and-lean-playground", "sha": "7da672891e29c0434909abad315ca6efefcbb989", "save_path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground", "path": "github-repos/lean/ChrisHughes24-coq-and-lean-playground/coq-and-lean-playground-7da672891e29c0434909abad315ca6efefcbb989/lean/bicompletion/with_ordinals.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7549149868676283, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3597771210241138}}
{"text": "example : (\u03bb (u : Nat) => u + 0) = id :=by\n  conv =>\n    lhs\n    intro u\n    change u\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1558.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3597636357466957}}
{"text": "import category_theory.preadditive.functor_category\nimport category_theory.limits.shapes.finite_products\nimport category_theory.limits.shapes.biproducts\nimport category_theory.limits.preserves.filtered\n\nimport for_mathlib.homological_complex2\nimport for_mathlib.additive_functor\n\nimport breen_deligne.homotopy\n\nnoncomputable theory\n\nopen_locale big_operators\n\nopen category_theory category_theory.limits\n\nnamespace category_theory\nnamespace preadditive\n\nvariables {\ud835\udc9c : Type*} [category \ud835\udc9c] [has_zero_morphisms \ud835\udc9c] [has_finite_biproducts \ud835\udc9c]\n\n-- move this\n@[simps {fully_applied := ff}]\ndef Pow (n : \u2115) : \ud835\udc9c \u2964 \ud835\udc9c :=\n{ obj := \u03bb A, \u2a01 (\u03bb (i : ulift $ fin n), A),\n  map := \u03bb A B f, biproduct.map (\u03bb i, f),\n  map_id' := \u03bb A, by { ext i j, simp only [biproduct.\u03b9_map, category.id_comp, category.comp_id], },\n  map_comp' := \u03bb A B C f g, by { ext i j, simp only [biproduct.\u03b9_map_assoc, category.assoc], } }\n\n-- move this\nattribute [simps] comp_hom\n.\n\ninstance (n : \u2115) {J : Type*} [category J] : preserves_colimits_of_shape J (Pow n : \ud835\udc9c \u2964 \ud835\udc9c) :=\n{ preserves_colimit := \u03bb K,\n  { preserves := \u03bb c hc,\n    { desc := \u03bb s, biproduct.desc $ \u03bb i,\n        let t : cocone K :=\n        { X := s.X,\n          \u03b9 := { app := \u03bb j, show K.obj j \u27f6 (K \u22d9 Pow n).obj j, from biproduct.\u03b9 _ i,\n                naturality' := by intros X Y f;\n                  simp only [functor.comp_map, Pow_map, biproduct.\u03b9_map], } \u226b s.\u03b9 } in\n        hc.desc t,\n      fac' := begin\n        intros, ext,\n        simp only [Pow_map, functor.map_cocone_\u03b9_app, biproduct.map_desc,\n          is_colimit.fac, nat_trans.comp_app, biproduct.\u03b9_desc],\n      end,\n      uniq' := begin\n        intros, ext i,\n        simp only [biproduct.\u03b9_desc],\n        let t : cocone K :=\n        { X := s.X,\n          \u03b9 := { app := \u03bb j, show K.obj j \u27f6 (K \u22d9 Pow n).obj j, from biproduct.\u03b9 _ i,\n                naturality' := by intros X Y f;\n                  simp only [functor.comp_map, Pow_map, biproduct.\u03b9_map], } \u226b s.\u03b9 },\n        refine hc.uniq t (_ \u226b m) _,\n        intro j,\n        simp only [nat_trans.comp_app, \u2190 w,\n          functor.map_cocone_\u03b9_app, Pow_map, biproduct.\u03b9_map_assoc],\n      end } } }\n\ninstance (n : \u2115) : preserves_colimits (Pow n : \ud835\udc9c \u2964 \ud835\udc9c) :=\n{ preserves_colimits_of_shape := \u03bb J hJ, by apply_instance }\n\nend preadditive\nend category_theory\n\nnamespace homotopy\n\nvariables {\u03b9 \ud835\udc9c : Type*} [category \ud835\udc9c] [preadditive \ud835\udc9c] {c : complex_shape \u03b9}\nvariables {C D : homological_complex \ud835\udc9c c} {f g : C \u27f6 D}\n\n@[simps]\ndef congr (h : homotopy f g) (f' g' : C \u27f6 D) (hf : f = f') (hg : g = g') :\n  homotopy f' g' :=\n{ comm := by simpa only [hf, hg] using h.comm,\n  .. h }\n\nend homotopy\n\nnamespace breen_deligne\n\nopen category_theory.preadditive\n\nvariables (BD : data)\nvariables {\ud835\udc9c : Type*} [category \ud835\udc9c] [preadditive \ud835\udc9c] [has_finite_biproducts \ud835\udc9c]\nvariables (F : \ud835\udc9c \u2964 \ud835\udc9c)\n\nnamespace basic_universal_map\n\nvariables {m n o : \u2115} (f : basic_universal_map m n) (g : basic_universal_map n o)\n\n@[simps {fully_applied := ff}]\ndef eval_Pow : (Pow m : \ud835\udc9c \u2964 \ud835\udc9c) \u27f6 Pow n :=\n{ app := \u03bb A, biproduct.matrix (\u03bb i j, f j.down i.down \u2022 \ud835\udfd9 A),\n  naturality' := begin\n    intros, ext i j,\n    simp only [Pow_map, biproduct.\u03b9_map_assoc, category.assoc, biproduct.matrix_\u03c0,\n      biproduct.map_\u03c0, biproduct.\u03b9_desc, biproduct.matrix_\u03c0_assoc, biproduct.\u03b9_desc_assoc,\n      comp_zsmul, zsmul_comp, category.comp_id, category.id_comp],\n  end }\n\n@[simp] lemma eval_Pow_comp : @eval_Pow \ud835\udc9c _ _ _ _ _ (comp g f) = f.eval_Pow \u226b g.eval_Pow :=\nbegin\n  ext A i j,\n  simp only [eval_Pow_app, nat_trans.comp_app, category.assoc, biproduct.\u03b9_map_assoc,\n    biproduct.matrix_\u03c0, biproduct.\u03b9_matrix_assoc, biproduct.lift_desc,\n    biproduct.map_\u03c0, biproduct.\u03b9_desc, biproduct.matrix_\u03c0_assoc, biproduct.\u03b9_desc_assoc,\n    comp_zsmul, zsmul_comp, category.comp_id, category.id_comp],\n  simp only [comp, add_monoid_hom.mk'_apply, matrix.mul, matrix.dot_product,\n    finset.sum_smul, mul_smul],\n  rw [\u2190 (@equiv.ulift (fin n)).symm.sum_comp, finset.sum_congr rfl],\n  rintros j -,\n  rw smul_comm, refl,\nend\n\nend basic_universal_map\n\nnamespace universal_map\n\nvariables {m n o : \u2115} (f : universal_map m n) (g : universal_map n o)\n\n/- Note: this definition is generalized in `eval1half.lean` for a functor\n`F : A\u2081 \u2964 A\u2082`. This generalization is used in `apply_Pow.lean`. -/\ndef eval_Pow : universal_map m n \u2192+ (Pow m \u22d9 F \u27f6 Pow n \u22d9 F) :=\nfree_abelian_group.lift $ \u03bb g : basic_universal_map m n, whisker_right g.eval_Pow F\n\nlemma eval_Pow_of (g : basic_universal_map m n) :\n  eval_Pow F (free_abelian_group.of g) = whisker_right g.eval_Pow F :=\nfree_abelian_group.lift.of _ _\n\n@[simp] lemma eval_Pow_zero : eval_Pow F (0 : universal_map m n) = 0 :=\nadd_monoid_hom.map_zero _\n\nlemma eval_Pow_zero_app (A : \ud835\udc9c) : (eval_Pow F (0 : universal_map m n)).app A = 0 :=\nby rw [eval_Pow_zero, zero_app]\n\nlemma eval_Pow_comp : eval_Pow F (universal_map.comp g f) = eval_Pow F f \u226b eval_Pow F g :=\nbegin\n  rw [\u2190 add_monoid_hom.comp_apply, \u2190 add_monoid_hom.comp_hom_apply_apply,\n    \u2190 add_monoid_hom.comp_apply, eq_comm,\n    \u2190 category_theory.preadditive.comp_hom_apply_apply, \u2190 add_monoid_hom.flip_apply,\n    \u2190 add_monoid_hom.comp_apply, \u2190 add_monoid_hom.comp_hom_apply_apply,\n    \u2190 add_monoid_hom.flip_apply _ _ (eval_Pow F),\n    \u2190 add_monoid_hom.comp_apply, \u2190 add_monoid_hom.comp_hom_apply_apply,\n    \u2190 add_monoid_hom.comp_apply, \u2190 add_monoid_hom.comp_hom_apply_apply],\n  congr' 2,\n  clear f g,\n  ext g f : 2,\n  simp only [add_monoid_hom.comp_hom_apply_apply, add_monoid_hom.comp_apply,\n    add_monoid_hom.flip_apply, category_theory.preadditive.comp_hom_apply_apply,\n    comp_of, eval_Pow_of, whisker_right_comp, basic_universal_map.eval_Pow_comp],\nend\n\nlemma eval_Pow_comp_app (A : \ud835\udc9c) :\n  (eval_Pow F (universal_map.comp g f)).app A = (eval_Pow F f).app A \u226b (eval_Pow F g).app A :=\nby rw [eval_Pow_comp, nat_trans.comp_app]\n\n@[simps {fully_applied := ff}]\ndef eval_Pow_functor : FreeMat \u2964 (\ud835\udc9c \u2964 \ud835\udc9c) :=\n{ obj := \u03bb n, Pow n \u22d9 F,\n  map := \u03bb m n f, eval_Pow F f,\n  map_id' := \u03bb n,\n  begin\n    refine (eval_Pow_of F _).trans _,\n    ext A : 2, dsimp,\n    rw \u2190 F.map_id, congr' 1,\n    ext i j : 2,\n    simp only [biproduct.\u03b9_matrix, category.comp_id, biproduct.lift_\u03c0, basic_universal_map.id],\n    rw biproduct.\u03b9_\u03c0,\n    split_ifs with hij,\n    { cases hij, rw [matrix.one_apply_eq, one_smul, eq_to_hom_refl], },\n    { rw [matrix.one_apply_ne, zero_smul], cases i, cases j, dsimp, rintro rfl, exact hij rfl }\n  end,\n  map_comp' := \u03bb m n o f g, eval_Pow_comp F _ _ }\n\ninstance eval_Pow_functor_additive : (eval_Pow_functor F).additive :=\n{ map_add' := \u03bb m n f g, by { dsimp [eval_Pow], rw add_monoid_hom.map_add } }\n\nend universal_map\n\nnamespace data\n\nopen universal_map\n\n@[simps {fully_applied := ff}]\ndef eval_functor' : data \u2964 chain_complex (\ud835\udc9c \u2964 \ud835\udc9c) \u2115 :=\n(eval_Pow_functor F).map_homological_complex _\n\n@[simps {fully_applied := ff}]\ndef eval_functor : data \u2964 \ud835\udc9c \u2964 chain_complex \ud835\udc9c \u2115 :=\neval_functor' F \u22d9 homological_complex.functor_eval.flip\n.\n\n-- generalize to arbitrary homological complexes\ninstance homological_complex.functor_eval_flip_preserves_colimits_of_shape\n  (J : Type*) [category J] (F : chain_complex (\ud835\udc9c \u2964 \ud835\udc9c) \u2115)\n  [\u2200 i, preserves_colimits_of_shape J (F.X i)] :\n  preserves_colimits_of_shape J (homological_complex.functor_eval.flip.obj F) :=\n{ preserves_colimit := \u03bb K,\n  { preserves := \u03bb c hc,\n    let t : \u03a0 (s : cocone (K \u22d9 homological_complex.functor_eval.flip.obj F))\n        (i : \u2115), cocone (K \u22d9 F.X i) := \u03bb s i,\n    { X := s.X.X i,\n      \u03b9 := { app := \u03bb j, show (K \u22d9 F.X i).obj j \u27f6 s.X.X i, from (s.\u03b9.app j).f i,\n            naturality' := begin\n              intros a b \u03c6, have := s.\u03b9.naturality \u03c6, dsimp at this \u22a2,\n              simp only [category.comp_id] at this \u22a2,\n              rw \u2190 this, refl\n            end } },\n      u : \u03a0 (s : cocone (K \u22d9 homological_complex.functor_eval.flip.obj F))\n        (i j : \u2115), cocone (K \u22d9 F.X i) := \u03bb s i j,\n    { X := s.X.X j,\n      \u03b9 := { app := \u03bb k, show (K \u22d9 F.X i).obj k \u27f6 s.X.X j,\n                         from (whisker_left K (F.d i j)).app k \u226b (s.\u03b9.app k).f j,\n            naturality' := begin\n              intros a b \u03c6, have := s.\u03b9.naturality \u03c6, dsimp at this \u22a2,\n              simp only [category.comp_id] at this \u22a2,\n              rw [\u2190 this, (F.d i j).naturality_assoc], refl,\n            end } } in\n    { desc := \u03bb s,\n      { f := \u03bb i, (is_colimit_of_preserves (F.X i) hc).desc (t s i),\n        comm' := begin\n          intros i j h, dsimp,\n          have := (is_colimit_of_preserves (F.X i) hc).uniq (u s i j),\n          refine (this _ _).trans (this _ _).symm,\n          { intros j', dsimp,\n            erw [(is_colimit_of_preserves (F.X i) hc).fac_assoc],\n            apply (s.\u03b9.app j').comm, },\n          { intros j', dsimp,\n            rw nat_trans.naturality_assoc,\n            erw [(is_colimit_of_preserves (F.X j) hc).fac], }\n        end },\n      fac' := by { intros, ext i, dsimp, erw [(is_colimit_of_preserves (F.X i) hc).fac], },\n      uniq' := begin\n        intros, ext i,\n        exact (is_colimit_of_preserves (F.X i) hc).uniq (t s i) (m.f i)\n          (\u03bb j, homological_complex.congr_hom (w j) i),\n      end, } } }\n\ninstance eval_functor_preserves_colimits_of_shape\n  (BD : data) (J : Type*) [category J] [preserves_colimits_of_shape J F] :\n  preserves_colimits_of_shape J ((eval_functor F).obj BD) :=\nbegin\n  refine @homological_complex.functor_eval_flip_preserves_colimits_of_shape _ _ _ _ J _\n    ((eval_functor' F).obj BD) (id _),\n  intro i,\n  show preserves_colimits_of_shape J (Pow (BD.X i) \u22d9 F),\n  apply_instance\nend\n\ninstance eval_functor_preserves_filtered_colimits (BD : data) [preserves_filtered_colimits F] :\n  preserves_filtered_colimits ((eval_functor F).obj BD) :=\n{ preserves_filtered_colimits := by introsI; apply_instance }\n\n-- @[simps]\n-- def eval_functor.obj (M : \ud835\udc9c) : chain_complex \ud835\udc9c \u2115 :=\n-- { X := \u03bb n, (Pow (BD.X n) \u22d9 F).obj M,\n--   d := \u03bb m n, (eval_Pow F (BD.d m n)).app M,\n--   shape' := \u03bb i j h, by rw [BD.shape i j h, universal_map.eval_Pow_zero_app],\n--   d_comp_d' := \u03bb i j k hij hjk, begin\n--     rw [\u2190 universal_map.eval_Pow_comp_app],\n--     have := BD.d_comp_d i j k,\n--     convert universal_map.eval_Pow_zero_app _ _ using 3,\n--   end }\n\n-- @[simps {fully_applied := ff}]\n-- def eval_functor : \ud835\udc9c \u2964 chain_complex \ud835\udc9c \u2115 :=\n-- { obj := eval_functor.obj BD F,\n--   map := \u03bb A B f,\n--   { f := \u03bb n, (Pow (BD.X n) \u22d9 F).map f,\n--     comm' := \u03bb m n h, by simp only [eval_functor.obj_d, nat_trans.naturality] },\n--   map_id' := \u03bb A, by { ext n, exact category_theory.functor.map_id _ _ },\n--   map_comp' := \u03bb A B C f g, by { ext n, exact category_theory.functor.map_comp _ _ _ } }\n\n-- @[simps {fully_applied := ff}]\n-- def map_eval_functor {BD\u2081 BD\u2082 : data} (\u03c6 : BD\u2081 \u27f6 BD\u2082) :\n--   BD\u2081.eval_functor F \u27f6 BD\u2082.eval_functor F :=\n-- { app := \u03bb A,\n--   { f := \u03bb i, (universal_map.eval_Pow F (\u03c6.f i)).app A,\n--     comm' := by { intros, dsimp only [eval_functor_obj, eval_functor.obj_d],\n--       simp only [\u2190 nat_trans.comp_app, \u2190 eval_Pow_comp F], congr' 2, apply \u03c6.comm } },\n--   naturality' := \u03bb A B f, by { ext i : 2, apply nat_trans.naturality } }\n\nend data\n\nnamespace package\n\nopen universal_map\n\nvariables (BD' : package) (A : \ud835\udc9c)\n\ndef eval_homotopy := (eval_Pow_functor F).map_homotopy BD'.homotopy\n\ndef eval_homotopy' (A : \ud835\udc9c) :=\n(eval_Pow_functor F \u22d9 (evaluation _ _).obj A).map_homotopy BD'.homotopy\n\nlocal attribute [instance] has_binary_biproducts_of_finite_biproducts\n\n@[simps]\ndef Biprod : \ud835\udc9c \u2964 \ud835\udc9c :=\n{ obj := \u03bb A, A \u229e A,\n  map := \u03bb A B f, biprod.map f f,\n  map_id' := \u03bb A,\n    by ext; simp only [biprod.inl_map, biprod.inr_map, category.id_comp, category.comp_id],\n  map_comp' := \u03bb A B C f g,\n    by ext; simp only [biprod.inl_map_assoc, biprod.inr_map_assoc, category.assoc] }\n.\n\n@[simps {fully_applied := ff}]\ndef Biprod_iso_Pow_two_components (A : \ud835\udc9c) : A \u229e A \u2245 (Pow 2).obj A :=\n{ hom := biprod.desc\n    (biproduct.\u03b9 (\u03bb i : ulift (fin 2), A) \u27e80\u27e9)\n    (biproduct.\u03b9 (\u03bb i : ulift (fin 2), A) \u27e81\u27e9),\n  inv := biprod.lift (biproduct.\u03c0 _ \u27e80\u27e9) (biproduct.\u03c0 _ \u27e81\u27e9),\n  hom_inv_id' := begin\n    ext;\n    simp only [biprod.lift_fst, biprod.lift_snd, biprod.inl_desc_assoc, biprod.inr_desc_assoc,\n      biproduct.\u03b9_\u03c0_self, category.assoc];\n    erw category.id_comp;\n    simp only [biprod.inl_fst, biprod.inl_snd, biprod.inr_fst, biprod.inr_snd];\n    rw [biproduct.\u03b9_\u03c0_ne]; dec_trivial\n  end,\n  inv_hom_id' := begin\n    ext \u27e8i\u27e9 \u27e8j\u27e9,\n    erw [category.comp_id],\n    simp only [add_comp, comp_add, biprod.lift_desc, category.assoc],\n    fin_cases i with [0,1];\n    rw [biproduct.\u03b9_\u03c0_self_assoc, biproduct.\u03b9_\u03c0_ne_assoc, zero_comp],\n    swap 2, { dec_trivial },\n    swap 3, { dec_trivial },\n    { rw add_zero },\n    { rw zero_add }\n  end }\n.\n\n@[simps {fully_applied := ff}]\ndef Biprod_iso_Pow_two : (Biprod : \ud835\udc9c \u2964 \ud835\udc9c) \u2245 Pow 2 :=\nnat_iso.of_components Biprod_iso_Pow_two_components $ \u03bb A B f,\nbegin\n  ext \u27e8i\u27e9;\n  simp only [biproduct.\u03b9_map, Biprod_iso_Pow_two_components_hom, Biprod_map, Pow_map,\n    biprod.inl_map_assoc, biprod.inl_desc_assoc, biprod.inr_map_assoc, biprod.inr_desc_assoc,\n    biprod.inr_map, category.assoc, biprod.inr_desc],\nend\n.\n\n@[simp] lemma _root_.ulift.up_inj {\u03b1 : Type*} (a b : \u03b1) : ulift.up a = ulift.up b \u2194 a = b :=\n\u27e8congr_arg ulift.down, congr_arg ulift.up\u27e9\n\n\n@[simps]\ndef Pow_comp_Pow_components (m n : \u2115) (A : \ud835\udc9c) :\n  (Pow n).obj ((Pow m).obj A) \u2245 (Pow (m * n)).obj A :=\n{ hom := biproduct.desc $ \u03bb j, biproduct.desc $ \u03bb i,\n    biproduct.\u03b9 (\u03bb i : ulift (fin _), A) \u27e8fin_prod_fin_equiv (i.down, j.down)\u27e9,\n  inv := biproduct.lift $ \u03bb j, biproduct.lift $ \u03bb i,\n    biproduct.\u03c0 (\u03bb i : ulift (fin _), A) \u27e8fin_prod_fin_equiv (i.down, j.down)\u27e9,\n  hom_inv_id' := begin\n    ext \u27e8j\u27e9 \u27e8i\u27e9 \u27e8j'\u27e9 \u27e8i'\u27e9 : 4,\n    erw [biproduct.\u03b9_desc_assoc, category.comp_id],\n    simp only [biproduct.\u03b9_desc_assoc, category.assoc, biproduct.lift_\u03c0],\n    by_cases hj : j = j',\n    { subst hj, rw [biproduct.\u03b9_\u03c0_self_assoc],\n      by_cases hi : i = i',\n      { subst hi, rw [biproduct.\u03b9_\u03c0_self, biproduct.\u03b9_\u03c0_self] },\n      { rw [biproduct.\u03b9_\u03c0_ne, biproduct.\u03b9_\u03c0_ne],\n        { exact mt (congr_arg ulift.down) hi },\n        { simpa only [equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff, eq_self_iff_true,\n            ulift.up_inj, ne.def] using hi, } } },\n    { rw [biproduct.\u03b9_\u03c0_ne, biproduct.\u03b9_\u03c0_ne_assoc, zero_comp, comp_zero],\n      { exact mt (congr_arg ulift.down) hj },\n      { simp only [equiv.apply_eq_iff_eq, prod.mk.inj_iff, _root_.ulift.up_inj, ne.def, hj,\n          not_false_iff, and_false], } }\n  end,\n  inv_hom_id' := begin\n    ext \u27e8k\u27e9 \u27e8k'\u27e9 : 2,\n    erw [category.comp_id],\n    simp only [category.assoc, biproduct.lift_desc, sum_comp, comp_sum],\n    by_cases h : k = k',\n    { subst h,\n      rw [biproduct.\u03b9_\u03c0_self,\n        finset.sum_eq_single (\u27e8(fin_prod_fin_equiv.symm k).snd\u27e9 : ulift (fin _)),\n        finset.sum_eq_single (\u27e8(fin_prod_fin_equiv.symm k).fst\u27e9 : ulift (fin _))],\n      { dsimp [- fin_prod_fin_equiv_symm_apply],\n        rw [prod.mk.eta, equiv.apply_symm_apply, biproduct.\u03b9_\u03c0_self, biproduct.\u03b9_\u03c0_self_assoc], },\n      { rintro \u27e8i\u27e9 - hi,\n        rw [biproduct.\u03b9_\u03c0_ne_assoc, zero_comp],\n        dsimp [- fin_prod_fin_equiv_symm_apply],\n        simp only [ulift.up_inj, ne.def, \u2190 equiv.symm_apply_eq,\n          prod.ext_iff, not_and_distrib] at hi \u22a2,\n        exact or.inl (ne.symm hi) },\n      { intro h, exact (h (finset.mem_univ _)).elim },\n      { rintro \u27e8j\u27e9 - hj,\n        rw finset.sum_eq_zero,\n        rintro \u27e8i\u27e9 -,\n        rw [biproduct.\u03b9_\u03c0_ne_assoc, zero_comp],\n        dsimp [- fin_prod_fin_equiv_symm_apply],\n        simp only [ulift.up_inj, ne.def, \u2190 equiv.symm_apply_eq,\n          prod.ext_iff, not_and_distrib] at hj \u22a2,\n        exact or.inr (ne.symm hj) },\n      { intro h, exact (h (finset.mem_univ _)).elim } },\n    { rw [biproduct.\u03b9_\u03c0_ne, finset.sum_eq_zero],\n      { rintro \u27e8j\u27e9 -,\n        rw [finset.sum_eq_zero],\n        rintro \u27e8i\u27e9 -,\n        by_cases hk : k = fin_prod_fin_equiv (i,j),\n        { subst hk,\n          rw [biproduct.\u03b9_\u03c0_self_assoc, biproduct.\u03b9_\u03c0_ne],\n          simpa only [ulift.up_inj, ne.def] using h, },\n        { rw [biproduct.\u03b9_\u03c0_ne_assoc, zero_comp],\n          dsimp [- fin_prod_fin_equiv_symm_apply],\n          simpa only [ulift.up_inj, ne.def] using h, } },\n      { rw [ne.def, ulift.up_inj], exact h } },\n  end }\n.\n\n@[simps {fully_applied := ff}]\ndef Pow_comp_Pow (m n : \u2115) : (Pow m \u22d9 Pow n : \ud835\udc9c \u2964 \ud835\udc9c) \u2245 Pow (m * n) :=\nnat_iso.of_components (Pow_comp_Pow_components m n) $ \u03bb A B f,\nbegin\n  ext \u27e8j\u27e9 \u27e8i\u27e9 \u27e8k\u27e9,\n  simp only [biproduct.\u03b9_map, Pow_comp_Pow_components_hom, Pow_map, functor.comp_map,\n    biproduct.\u03b9_map_assoc, category.assoc, biproduct.map_\u03c0, biproduct.\u03b9_desc_assoc],\nend\n.\n\nlemma _root_.free_abelian_group.eq_zero_induction\n  {\u03b1 M : Type*} [add_group M] (f : free_abelian_group \u03b1 \u2192 M)\n  (h1 : \u2200 a, f (free_abelian_group.of a) = 0) (h2 : \u2200 x y, f (x + y) = f x + f y) :\n  \u2200 x, f x = 0 :=\nbegin\n  let F := add_monoid_hom.mk' f h2,\n  have hF : \u2200 x, F x = f x := \u03bb _, rfl,\n  intro x,\n  refine free_abelian_group.induction_on x _ h1 _ _,\n  { exact F.map_zero },\n  { intros, show F _ = 0, rw [F.map_neg, hF, h1, neg_zero], },\n  { intros x y hx hy, show F _ = 0, rw [F.map_add, hF, hF, hx, hy, add_zero], },\nend\n\nlemma aux' (m n : \u2115) (f : universal_map m n) :\n  F.map ((Pow_comp_Pow 2 m).inv.app A \u226b (Pow m).map (Biprod_iso_Pow_two.inv.app A)) \u226b\n    ((eval_Pow_functor F).map f).app (Biprod.obj A) =\n  ((eval_Pow_functor F).map ((mul 2) f)).app A \u226b F.map ((Pow_comp_Pow 2 n).inv.app A \u226b\n    (Pow n).map (Biprod_iso_Pow_two.inv.app A)) :=\nbegin\n  rw [\u2190 sub_eq_zero],\n  refine free_abelian_group.eq_zero_induction _ _ _ f; clear f,\n  { intro f,\n    rw [sub_eq_zero],\n    dsimp only [eval_Pow_functor],\n    rw [mul_of, eval_Pow_of, eval_Pow_of],\n    dsimp only [whisker_right_app, basic_universal_map.eval_Pow_app],\n    rw [\u2190 F.map_comp, \u2190 F.map_comp],\n    congr' 1,\n    dsimp only [Pow_comp_Pow, Biprod_iso_Pow_two],\n    erw [nat_iso.of_components.inv_app, nat_iso.of_components.inv_app,\n      nat_iso.of_components.inv_app],\n    dsimp only [Pow_comp_Pow_components_inv, Biprod_iso_Pow_two_components_inv, Pow_map],\n    apply category_theory.limits.biproduct.hom_ext,\n    rintro \u27e8j\u27e9,\n    apply category_theory.limits.biproduct.hom_ext',\n    refine equiv.ulift.forall_congr_left'.mpr _,\n    refine fin_prod_fin_equiv.forall_congr_left.mp _,\n    rintro \u27e8b, i\u27e9,\n    rw [biproduct.lift_map, biproduct.lift_matrix, biproduct.lift_\u03c0, comp_sum,\n      biproduct.lift_map, category.assoc, biproduct.\u03b9_matrix_assoc, biproduct.lift_\u03c0],\n    rw [finset.sum_eq_single (\u27e8i\u27e9 : ulift (fin m)),\n      equiv.ulift_symm_apply, ulift.down_up, ulift.down_up, ulift.down_up],\n    { rw [category.assoc],\n      ext;\n      rw [category.assoc, category.assoc, comp_zsmul, zsmul_comp, comp_zsmul, comp_zsmul,\n        category.comp_id, category.assoc, category.assoc];\n      [rw biprod.lift_fst, rw biprod.lift_snd];\n      rw [biproduct.lift_\u03c0, biproduct.lift_\u03c0, biproduct.lift_\u03c0,\n        biproduct.\u03b9_\u03c0, basic_universal_map.mul_apply, matrix.reindex_linear_equiv_apply,\n        matrix.reindex_apply, matrix.minor_apply, ulift.down_up, ulift.down_up,\n        matrix.kronecker_map, equiv.symm_apply_apply, equiv.symm_apply_apply];\n      simp only [dite_eq_ite, equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff,\n        eq_self_iff_true, ulift.up_inj, eq_to_hom_refl, matrix.one_apply,\n        ite_mul, ite_smul, one_mul, zero_mul, zero_smul, @eq_comm _ b, smul_ite, smul_zero];\n      congr' 1, },\n    { rintro \u27e8i'\u27e9 - hi',\n      rw [ne.def, ulift.up_inj, eq_comm] at hi',\n      rw [category.assoc],\n      ext;\n      rw [category.assoc, category.assoc, comp_zsmul, zsmul_comp, comp_zsmul, comp_zsmul,\n        category.comp_id, zero_comp];\n      [rw biprod.lift_fst, rw biprod.lift_snd];\n      rw [biproduct.lift_\u03c0, biproduct.\u03b9_\u03c0];\n      simp only [dite_eq_ite, equiv.apply_eq_iff_eq, and_true, prod.mk.inj_iff,\n        eq_self_iff_true, ulift.up_inj, eq_to_hom_refl, equiv.ulift_symm_apply,\n        eq_false_intro hi', and_false, if_false, smul_zero], },\n    { intro h, exact (h (finset.mem_univ _)).elim } },\n  { intros x y,\n    simp only [add_monoid_hom.map_add, functor.map_add, comp_add, add_comp, nat_trans.app_add],\n    abel }\nend\n.\n\n@[simps {fully_applied := ff}]\ndef aux :\n  (data.eval_functor F).obj ((data.mul 2).obj BD'.data) \u2245\n  Biprod \u22d9 (data.eval_functor F).obj BD'.data :=\nnat_iso.of_components (\u03bb A,\n  homological_complex.hom.iso_of_components (\u03bb i, begin\n      refine F.map_iso _,\n      refine (Pow_comp_Pow 2 (BD'.data.X i)).symm.app A \u226a\u226b _,\n      refine (Pow _).map_iso (Biprod_iso_Pow_two.symm.app A)\n    end) $ \u03bb i j hij, aux' F A (BD'.data.X i) (BD'.data.X j) (BD'.data.d i j)) $ \u03bb A B f, begin\n      ext i,\n      dsimp only [data.eval_functor, data.eval_functor', eval_Pow, eval_Pow_functor_obj,\n        functor.map_iso_hom, functor.comp_obj, functor.comp_map, functor.flip_obj_map,\n        iso.trans_hom, iso.symm_hom, nat_iso.app_hom,\n        functor.map_homological_complex_obj_X,\n        homological_complex.functor_eval_map_app_f,\n        homological_complex.comp_f,\n        homological_complex.hom.iso_of_components_hom_f],\n      rw [\u2190 F.map_comp, \u2190 F.map_comp, \u2190 category.assoc, nat_trans.naturality,\n        category.assoc, category.assoc, functor.comp_map, \u2190 functor.map_comp, \u2190 functor.map_comp,\n        nat_trans.naturality],\n  end\n.\n\n-- move this up\nlemma quux (n : \u2115) {N : \u2115} (k : fin N) (A : \ud835\udc9c) :\n  (basic_universal_map.proj n k).eval_Pow.app A =\n  biproduct.matrix (\u03bb i j, if i.down = fin_prod_fin_equiv (k, j.down) then \ud835\udfd9 A else 0) :=\nbegin\n  apply category_theory.limits.biproduct.hom_ext,\n  rintro \u27e8j\u27e9,\n  apply category_theory.limits.biproduct.hom_ext',\n  refine equiv.ulift.forall_congr_left'.mpr _,\n  refine fin_prod_fin_equiv.forall_congr_left.mp _,\n  rintro \u27e8l, i\u27e9,\n  dsimp only [basic_universal_map.eval_Pow_app],\n  rw [biproduct.matrix_\u03c0, biproduct.matrix_\u03c0, biproduct.\u03b9_desc, biproduct.\u03b9_desc],\n  dsimp only [basic_universal_map.proj, basic_universal_map.proj_aux,\n    matrix.reindex_linear_equiv_apply, matrix.reindex_apply, matrix.minor,\n    matrix.kronecker_map],\n  simp only [ite_mul, ite_smul, one_mul, one_smul, zero_mul, zero_smul, matrix.one_apply],\n  rw [\u2190 ite_and],\n  congr' 1,\n  apply propext,\n  rw [\u2190 equiv.symm_apply_eq, prod.ext_iff],\n  apply and_congr iff.rfl,\n  dsimp only [equiv.punit_prod_symm_apply],\n  rw [eq_comm],\nend\n.\n\n-- move this up\nlemma eval_Pow_add {m n : \u2115} (f g : basic_universal_map m n) (A : \ud835\udc9c) :\n  (f + g).eval_Pow.app A = f.eval_Pow.app A + g.eval_Pow.app A :=\nbegin\n  dsimp [basic_universal_map.eval_Pow_app],\n  ext \u27e8i\u27e9 \u27e8j\u27e9,\n  simp only [biproduct.\u03b9_matrix, biproduct.lift_\u03c0, comp_add, add_comp, add_zsmul],\nend\n.\n\ndef eval_functor_homotopy (A : \ud835\udc9c) : _root_.homotopy\n  (((data.eval_functor F).obj BD'.data).map (biprod.fst + biprod.snd : A \u229e A \u27f6 A))\n  (((data.eval_functor F).obj BD'.data).map (biprod.fst : A \u229e A \u27f6 A) +\n    ((data.eval_functor F).obj BD'.data).map (biprod.snd : A \u229e A \u27f6 A)) :=\nbegin\n  refine ((eval_homotopy' F BD' A).symm.comp_left ((aux F BD').inv.app A)).congr _ _ _ _,\n  { ext i,\n    rw [homological_complex.comp_f, aux_inv_app_f,\n      functor.map_homological_complex_map_f, functor.comp_map, eval_Pow_functor_map,\n      evaluation_obj_map, data.eval_functor_obj_map_f],\n    dsimp only [data.sum, universal_map.sum],\n    rw [eval_Pow_of, whisker_right_app, \u2190 F.map_comp, fin.sum_univ_two,\n      eval_Pow_add, quux, quux],\n    congr' 1,\n    apply category_theory.limits.biproduct.hom_ext', rintro \u27e8m\u27e9,\n    rw [biproduct.\u03b9_desc_assoc, biproduct.\u03b9_map, category.assoc],\n    apply category_theory.limits.biproduct.hom_ext, rintro \u27e8n\u27e9,\n    rw [category.assoc],\n    apply category_theory.limits.biprod.hom_ext';\n    [rw [biprod.inl_desc_assoc], rw [biprod.inr_desc_assoc]];\n    rw [category.assoc, biproduct.\u03b9_desc_assoc, add_comp, comp_add,\n      biproduct.matrix_\u03c0, biproduct.matrix_\u03c0, biproduct.\u03b9_desc, biproduct.\u03b9_desc,\n      category.assoc, add_comp, comp_add];\n    simp only [biprod.inl_fst_assoc, biprod.inl_snd_assoc,\n      biprod.inr_fst_assoc, biprod.inr_snd_assoc, zero_comp, add_zero, zero_add,\n      true_and, equiv.apply_eq_iff_eq, prod.mk.inj_iff, one_ne_zero,\n      fin.zero_eq_one_iff, fin.one_eq_zero_iff, eq_self_iff_true, if_false, false_and],\n      all_goals\n      { by_cases hmn : m = n,\n        { cases hmn, rw [if_pos rfl, biproduct.\u03b9_\u03c0_self], },\n        { rw [if_neg, biproduct.\u03b9_\u03c0_ne]; [rw [ne.def, ulift.up_inj], skip]; exact hmn } } },\n  { ext i,\n    rw [homological_complex.comp_f, aux_inv_app_f,\n      functor.map_homological_complex_map_f, functor.comp_map, eval_Pow_functor_map,\n      evaluation_obj_map,\n      homological_complex.add_f_apply,\n      data.eval_functor_obj_map_f, data.eval_functor_obj_map_f],\n    dsimp only [data.proj, proj],\n    rw [add_monoid_hom.map_sum, fin.sum_univ_two, eval_Pow_of, eval_Pow_of,\n      nat_trans.app_add, whisker_right_app, whisker_right_app, comp_add,\n      \u2190 F.map_comp, \u2190 F.map_comp, quux, quux],\n    congr' 2;\n    { apply category_theory.limits.biproduct.hom_ext, rintro \u27e8n\u27e9,\n      rw [biproduct.map_\u03c0, category.assoc, biproduct.matrix_\u03c0],\n      apply category_theory.limits.biproduct.hom_ext', rintro \u27e8m\u27e9,\n      rw [biproduct.\u03b9_desc_assoc, category.assoc],\n      apply category_theory.limits.biprod.hom_ext';\n      [rw [biprod.inl_desc_assoc], rw [biprod.inr_desc_assoc]],\n      all_goals\n      { rw [biproduct.\u03b9_desc_assoc, biproduct.\u03b9_desc];\n        simp only [true_and, equiv.apply_eq_iff_eq, prod.mk.inj_iff,\n          eq_self_iff_true, ulift.up_inj, ulift.down_inj];\n        by_cases hmn : m = n,\n        { cases hmn,\n          simp only [biproduct.\u03b9_\u03c0_self_assoc, eq_self_iff_true, if_true, if_false,\n            biprod.inl_fst, biprod.inr_fst, biprod.inl_snd, biprod.inr_snd,\n            zero_ne_one, one_ne_zero, false_and, fin.one_eq_zero_iff, fin.zero_eq_one_iff], },\n        { rw biproduct.\u03b9_\u03c0_ne_assoc, swap, { rw [ne.def, ulift.up_inj], exact hmn },\n          simp only [hmn, if_false, and_false, zero_comp, comp_zero] } } } }\nend\n.\n\n\nend package\n\nend breen_deligne\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/breen_deligne/eval.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3597636357466957}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.traversable.equiv\nimport control.traversable.instances\nimport data.lazy_list\n\n/-!\n## Definitions on lazy lists\n\nThis file contains various definitions and proofs on lazy lists.\n\nTODO: move the `lazy_list.lean` file from core to mathlib.\n-/\n\nuniverses u\n\nnamespace thunk\n\n/-- Creates a thunk with a (non-lazy) constant value. -/\ndef mk {\u03b1} (x : \u03b1) : thunk \u03b1 := \u03bb _, x\n\ninstance {\u03b1 : Type u} [decidable_eq \u03b1] : decidable_eq (thunk \u03b1) | a b :=\nhave a = b \u2194 a () = b (), from \u27e8by cc, by intro; ext x; cases x; assumption\u27e9,\nby rw this; apply_instance\n\nend thunk\n\nnamespace lazy_list\n\nopen function\n\n/-- Isomorphism between strict and lazy lists. -/\ndef list_equiv_lazy_list (\u03b1 : Type*) : list \u03b1 \u2243 lazy_list \u03b1 :=\n{ to_fun := lazy_list.of_list,\n  inv_fun := lazy_list.to_list,\n  right_inv := by { intro, induction x, refl, simp! [*],\n                    ext, cases x, refl },\n  left_inv := by { intro, induction x, refl, simp! [*] } }\n\ninstance {\u03b1 : Type u} [decidable_eq \u03b1] : decidable_eq (lazy_list \u03b1)\n| nil nil := is_true rfl\n| (cons x xs) (cons y ys) :=\n  if h : x = y then\n    match decidable_eq (xs ()) (ys ()) with\n    | is_false h2 := is_false (by intro; cc)\n    | is_true h2 :=\n      have xs = ys, by ext u; cases u; assumption,\n      is_true (by cc)\n    end\n  else\n    is_false (by intro; cc)\n| nil (cons _ _) := is_false (by cc)\n| (cons _ _) nil := is_false (by cc)\n\n/-- Traversal of lazy lists using an applicative effect. -/\nprotected def traverse {m : Type u \u2192 Type u} [applicative m] {\u03b1 \u03b2 : Type u}\n    (f : \u03b1 \u2192 m \u03b2) : lazy_list \u03b1 \u2192 m (lazy_list \u03b2)\n| lazy_list.nil := pure lazy_list.nil\n| (lazy_list.cons x xs) := lazy_list.cons <$> f x <*> (thunk.mk <$> traverse (xs ()))\n\ninstance : traversable lazy_list :=\n{ map := @lazy_list.traverse id _,\n  traverse := @lazy_list.traverse }\n\ninstance : is_lawful_traversable lazy_list :=\nbegin\n  apply equiv.is_lawful_traversable' list_equiv_lazy_list;\n  intros ; resetI; ext,\n  { induction x, refl,\n    simp! [equiv.map,functor.map] at *,\n    simp [*], refl, },\n  { induction x, refl,\n    simp! [equiv.map,functor.map_const] at *,\n    simp [*], refl, },\n  { induction x,\n    { simp! [traversable.traverse,equiv.traverse] with functor_norm, refl },\n    simp! [equiv.map,functor.map_const,traversable.traverse] at *, rw x_ih,\n    dsimp [list_equiv_lazy_list,equiv.traverse,to_list,traversable.traverse,list.traverse],\n    simp! with functor_norm, refl },\nend\n\n/-- `init xs`, if `xs` non-empty, drops the last element of the list.\nOtherwise, return the empty list. -/\ndef init {\u03b1} : lazy_list \u03b1 \u2192 lazy_list \u03b1\n| lazy_list.nil := lazy_list.nil\n| (lazy_list.cons x xs) :=\n  let xs' := xs () in\n  match xs' with\n  | lazy_list.nil := lazy_list.nil\n  | (lazy_list.cons _ _) := lazy_list.cons x (init xs')\n  end\n\n/-- Return the first object contained in the list that satisfies\npredicate `p` -/\ndef find {\u03b1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : lazy_list \u03b1 \u2192 option \u03b1\n| nil        := none\n| (cons h t) := if p h then some h else find (t ())\n\n/-- `interleave xs ys` creates a list where elements of `xs` and `ys` alternate. -/\ndef interleave {\u03b1} : lazy_list \u03b1 \u2192 lazy_list \u03b1 \u2192 lazy_list \u03b1\n| lazy_list.nil xs := xs\n| a@(lazy_list.cons x xs) lazy_list.nil := a\n| (lazy_list.cons x xs) (lazy_list.cons y ys) :=\n  lazy_list.cons x (lazy_list.cons y (interleave (xs ()) (ys ())))\n\n/-- `interleave_all (xs::ys::zs::xss)` creates a list where elements of `xs`, `ys`\nand `zs` and the rest alternate. Every other element of the resulting list is taken from\n`xs`, every fourth is taken from `ys`, every eighth is taken from `zs` and so on. -/\ndef interleave_all {\u03b1} : list (lazy_list \u03b1) \u2192 lazy_list \u03b1\n| [] := lazy_list.nil\n| (x :: xs) := interleave x (interleave_all xs)\n\n/-- Monadic bind operation for `lazy_list`. -/\nprotected def bind {\u03b1 \u03b2} : lazy_list \u03b1 \u2192 (\u03b1 \u2192 lazy_list \u03b2) \u2192 lazy_list \u03b2\n| lazy_list.nil _ := lazy_list.nil\n| (lazy_list.cons x xs) f := lazy_list.append (f x) (bind (xs ()) f)\n\n/-- Reverse the order of a `lazy_list`.\nIt is done by converting to a `list` first because reversal involves evaluating all\nthe list and if the list is all evaluated, `list` is a better representation for\nit than a series of thunks. -/\ndef reverse {\u03b1} (xs : lazy_list \u03b1) : lazy_list \u03b1 :=\nof_list xs.to_list.reverse\n\ninstance : monad lazy_list :=\n{ pure := @lazy_list.singleton,\n  bind := @lazy_list.bind }\n\n\n\nlemma append_assoc {\u03b1} (xs ys zs : lazy_list \u03b1) :\n  (xs.append ys).append zs = xs.append (ys.append zs) :=\nby induction xs; simp [append, *]\n\nlemma append_bind {\u03b1 \u03b2} (xs : lazy_list \u03b1) (ys : thunk (lazy_list \u03b1)) (f : \u03b1 \u2192 lazy_list \u03b2) :\n  (@lazy_list.append _ xs ys).bind f = (xs.bind f).append ((ys ()).bind f) :=\nby induction xs; simp [lazy_list.bind, append, *, append_assoc, append, lazy_list.bind]\n\ninstance : is_lawful_monad lazy_list :=\n{ pure_bind := by { intros, apply append_nil },\n  bind_assoc := by { intros, dsimp [(>>=)], induction x; simp [lazy_list.bind, append_bind, *], },\n  id_map :=\n  begin\n    intros,\n    simp [(<$>)],\n    induction x; simp [lazy_list.bind, *, singleton, append],\n    ext \u27e8 \u27e9, refl,\n  end }\n\n/-- Try applying function `f` to every element of a `lazy_list` and\nreturn the result of the first attempt that succeeds. -/\ndef mfirst {m} [alternative m] {\u03b1 \u03b2} (f : \u03b1 \u2192 m \u03b2) : lazy_list \u03b1 \u2192 m \u03b2\n| nil := failure\n| (cons x xs) :=\n  f x <|> mfirst (xs ())\n\n/-- Membership in lazy lists -/\nprotected def mem {\u03b1} (x : \u03b1) : lazy_list \u03b1 \u2192 Prop\n| lazy_list.nil := false\n| (lazy_list.cons y ys) := x = y \u2228 mem (ys ())\n\ninstance {\u03b1} : has_mem \u03b1 (lazy_list \u03b1) :=\n\u27e8 lazy_list.mem \u27e9\n\ninstance mem.decidable {\u03b1} [decidable_eq \u03b1] (x : \u03b1) : \u03a0 xs : lazy_list \u03b1, decidable (x \u2208 xs)\n| lazy_list.nil := decidable.false\n| (lazy_list.cons y ys) :=\n  if h : x = y\n    then decidable.is_true (or.inl h)\n    else decidable_of_decidable_of_iff (mem.decidable (ys ())) (by simp [*, (\u2208), lazy_list.mem])\n\n@[simp]\nlemma mem_nil {\u03b1} (x : \u03b1) : x \u2208 @lazy_list.nil \u03b1 \u2194 false := iff.rfl\n\n@[simp]\nlemma mem_cons {\u03b1} (x y : \u03b1) (ys : thunk (lazy_list \u03b1)) :\n  x \u2208 @lazy_list.cons \u03b1 y ys \u2194 x = y \u2228 x \u2208 ys () := iff.rfl\n\ntheorem forall_mem_cons {\u03b1} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : thunk (lazy_list \u03b1)} :\n  (\u2200 x \u2208 @lazy_list.cons _ a l, p x) \u2194 p a \u2227 \u2200 x \u2208 l (), p x :=\nby simp only [has_mem.mem, lazy_list.mem, or_imp_distrib, forall_and_distrib, forall_eq]\n\n/-! ### map for partial functions -/\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on\n  `a : \u03b1` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`\n  but is defined only when all members of `l` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {\u03b1 \u03b2} {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) :\n  \u03a0 l : lazy_list \u03b1, (\u2200 a \u2208 l, p a) \u2192 lazy_list \u03b2\n| lazy_list.nil         H := lazy_list.nil\n| (lazy_list.cons x xs) H := lazy_list.cons (f x (forall_mem_cons.1 H).1)\n                               (pmap (xs ()) (forall_mem_cons.1 H).2)\n\n/-- \"Attach\" the proof that the elements of `l` are in `l` to produce a new `lazy_list`\n  with the same elements but in the type `{x // x \u2208 l}`. -/\ndef attach {\u03b1} (l : lazy_list \u03b1) : lazy_list {x // x \u2208 l} := pmap subtype.mk l (\u03bb a, id)\n\ninstance {\u03b1} [has_repr \u03b1] : has_repr (lazy_list \u03b1) :=\n\u27e8 \u03bb xs, repr xs.to_list \u27e9\n\nend lazy_list\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/lazy_list/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.3597636357466957}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n\n! This file was ported from Lean 3 source module control.bitraversable.basic\n! leanprover-community/mathlib commit 69c6a5a12d8a2b159f20933e60115a4f2de62b58\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Control.Bifunctor\nimport Mathbin.Control.Traversable.Basic\n\n/-!\n# Bitraversable type class\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nType class for traversing bifunctors.\n\nSimple examples of `bitraversable` are `prod` and `sum`. A more elaborate example is\nto define an a-list as:\n\n```\ndef alist (key val : Type) := list (key \u00d7 val)\n```\n\nThen we can use `f : key \u2192 io key'` and `g : val \u2192 io val'` to manipulate the `alist`'s key\nand value respectively with `bitraverse f g : alist key val \u2192 io (alist key' val')`\n\n## Main definitions\n\n* `bitraversable`: Bare typeclass to hold the `bitraverse` function.\n* `is_lawful_bitraversable`: Typeclass for the laws of the `bitraverse` function. Similar to\n  `is_lawful_traversable`.\n\n## References\n\nThe concepts and laws are taken from\n<https://hackage.haskell.org/package/base-4.12.0.0/docs/Data-Bitraversable.html>\n\n## Tags\n\ntraversable bitraversable iterator functor bifunctor applicative\n-/\n\n\nuniverse u\n\n#print Bitraversable /-\n/-- Lawless bitraversable bifunctor. This only holds data for the bimap and bitraverse. -/\nclass Bitraversable (t : Type u \u2192 Type u \u2192 Type u) extends Bifunctor t where\n  bitraverse :\n    \u2200 {m : Type u \u2192 Type u} [Applicative m] {\u03b1 \u03b1' \u03b2 \u03b2'},\n      (\u03b1 \u2192 m \u03b1') \u2192 (\u03b2 \u2192 m \u03b2') \u2192 t \u03b1 \u03b2 \u2192 m (t \u03b1' \u03b2')\n#align bitraversable Bitraversable\n-/\n\nexport Bitraversable (bitraverse)\n\n#print bisequence /-\n/-- A bitraversable functor commutes with all applicative functors. -/\ndef bisequence {t m} [Bitraversable t] [Applicative m] {\u03b1 \u03b2} : t (m \u03b1) (m \u03b2) \u2192 m (t \u03b1 \u03b2) :=\n  bitraverse id id\n#align bisequence bisequence\n-/\n\nopen Functor\n\n#print IsLawfulBitraversable /-\n/-- Bifunctor. This typeclass asserts that a lawless bitraversable bifunctor is lawful. -/\nclass IsLawfulBitraversable (t : Type u \u2192 Type u \u2192 Type u) [Bitraversable t] extends\n  LawfulBifunctor t where\n  id_bitraverse : \u2200 {\u03b1 \u03b2} (x : t \u03b1 \u03b2), bitraverse id.mk id.mk x = id.mk x\n  comp_bitraverse :\n    \u2200 {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G]\n      {\u03b1 \u03b1' \u03b2 \u03b2' \u03b3 \u03b3'} (f : \u03b2 \u2192 F \u03b3) (f' : \u03b2' \u2192 F \u03b3') (g : \u03b1 \u2192 G \u03b2) (g' : \u03b1' \u2192 G \u03b2') (x : t \u03b1 \u03b1'),\n      bitraverse (Comp.mk \u2218 map f \u2218 g) (Comp.mk \u2218 map f' \u2218 g') x =\n        Comp.mk (bitraverse f f' <$> bitraverse g g' x)\n  bitraverse_eq_bimap_id :\n    \u2200 {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 \u03b2) (f' : \u03b1' \u2192 \u03b2') (x : t \u03b1 \u03b1'),\n      bitraverse (id.mk \u2218 f) (id.mk \u2218 f') x = id.mk (bimap f f' x)\n  binaturality :\n    \u2200 {F G} [Applicative F] [Applicative G] [LawfulApplicative F] [LawfulApplicative G]\n      (\u03b7 : ApplicativeTransformation F G) {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 F \u03b2) (f' : \u03b1' \u2192 F \u03b2') (x : t \u03b1 \u03b1'),\n      \u03b7 (bitraverse f f' x) = bitraverse (@\u03b7 _ \u2218 f) (@\u03b7 _ \u2218 f') x\n#align is_lawful_bitraversable IsLawfulBitraversable\n-/\n\nexport IsLawfulBitraversable (id_bitraverse comp_bitraverse bitraverse_eq_bimap_id)\n\nopen IsLawfulBitraversable\n\nattribute [higher_order.1bitraverse_id_id] id_bitraverse\n\nattribute [higher_order.1bitraverse_comp] comp_bitraverse\n\nattribute [higher_order.1] binaturality bitraverse_eq_bimap_id\n\nexport IsLawfulBitraversable (bitraverse_id_id bitraverse_comp)\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Bitraversable/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443134, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.35969897856552885}}
{"text": "/-\nCopyright (c) 2021 Luke Kershaw. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Luke Kershaw\n-/\nimport category_theory.preadditive.additive_functor\nimport category_theory.shift\nimport category_theory.triangulated.rotate\n\n/-!\n# Pretriangulated Categories\n\nThis file contains the definition of pretriangulated categories and triangulated functors\nbetween them.\n\n## Implementation Notes\n\nWe work under the assumption that pretriangulated categories are preadditive categories,\nbut not necessarily additive categories, as is assumed in some sources.\n\nTODO: generalise this to n-angulated categories as in https://arxiv.org/abs/1006.4592\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v v\u2080 v\u2081 v\u2082 u u\u2080 u\u2081 u\u2082\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\n/-\nWe work in a preadditive category `C` equipped with an additive shift.\n-/\nvariables (C : Type u) [category.{v} C] [has_zero_object C] [has_shift C \u2124] [preadditive C]\n  [\u2200 n : \u2124, functor.additive (shift_functor C n)]\n\n/--\nA preadditive category `C` with an additive shift, and a class of \"distinguished triangles\"\nrelative to that shift is called pretriangulated if the following hold:\n* Any triangle that is isomorphic to a distinguished triangle is also distinguished.\n* Any triangle of the form `(X,X,0,id,0,0)` is distinguished.\n* For any morphism `f : X \u27f6 Y` there exists a distinguished triangle of the form `(X,Y,Z,f,g,h)`.\n* The triangle `(X,Y,Z,f,g,h)` is distinguished if and only if `(Y,Z,X\u27e61\u27e7,g,h,-f\u27e61\u27e7)` is.\n* Given a diagram:\n  ```\n        f       g       h\n    X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n    \u2502       \u2502                \u2502\n    \u2502a      \u2502b               \u2502a\u27e61\u27e7'\n    V       V                V\n    X' \u2500\u2500\u2500> Y' \u2500\u2500\u2500> Z' \u2500\u2500\u2500> X'\u27e61\u27e7\n        f'      g'      h'\n  ```\n  where the left square commutes, and whose rows are distinguished triangles,\n  there exists a morphism `c : Z \u27f6 Z'` such that `(a,b,c)` is a triangle morphism.\n\nSee https://stacks.math.columbia.edu/tag/0145\n-/\nclass pretriangulated :=\n(distinguished_triangles [] : set (triangle C))\n(isomorphic_distinguished : \u03a0 (T\u2081 \u2208 distinguished_triangles) (T\u2082 \u2245 T\u2081),\n  T\u2082 \u2208 distinguished_triangles)\n(contractible_distinguished : \u03a0 (X : C), (contractible_triangle C X) \u2208 distinguished_triangles)\n(distinguished_cocone_triangle : \u03a0 (X Y : C) (f : X \u27f6 Y), (\u2203 (Z : C) (g : Y \u27f6 Z)\n  (h : Z \u27f6 X\u27e6(1:\u2124)\u27e7),\n  triangle.mk _ f g h \u2208 distinguished_triangles))\n(rotate_distinguished_triangle : \u03a0 (T : triangle C),\n  T \u2208 distinguished_triangles \u2194 T.rotate \u2208 distinguished_triangles)\n(complete_distinguished_triangle_morphism : \u03a0 (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 distinguished_triangles) (h\u2082 : T\u2082 \u2208 distinguished_triangles) (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081)\n  (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082) (comm\u2081 : T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081),\n  (\u2203 (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083), (T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082) \u2227 (T\u2081.mor\u2083 \u226b a\u27e61\u27e7' = c \u226b T\u2082.mor\u2083) ))\n\nnamespace pretriangulated\nvariables [pretriangulated C]\n\nnotation `dist_triang`:20 C := distinguished_triangles C\n/--\nGiven any distinguished triangle `T`, then we know `T.rotate` is also distinguished.\n-/\nlemma rot_of_dist_triangle (T \u2208 dist_triang C) : (T.rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle T).mp H\n\n/--\nGiven any distinguished triangle `T`, then we know `T.inv_rotate` is also distinguished.\n-/\nlemma inv_rot_of_dist_triangle (T \u2208 dist_triang C) : (T.inv_rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle (T.inv_rotate)).mpr\n  (isomorphic_distinguished T H T.inv_rotate.rotate (inv_rot_comp_rot.app T))\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `f \u226b g = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2081\u2082 (T \u2208 dist_triang C) : T.mor\u2081 \u226b T.mor\u2082 = 0 :=\nbegin\n  have h := contractible_distinguished T.obj\u2081,\n  have f := complete_distinguished_triangle_morphism,\n  specialize f (contractible_triangle C T.obj\u2081) T h H (\ud835\udfd9 T.obj\u2081) T.mor\u2081,\n  have t : (contractible_triangle C T.obj\u2081).mor\u2081 \u226b T.mor\u2081 = \ud835\udfd9 T.obj\u2081 \u226b T.mor\u2081,\n    by refl,\n  specialize f t,\n  cases f with c f,\n  rw \u2190 f.left,\n  simp only [limits.zero_comp, contractible_triangle_mor\u2082],\nend -- TODO : tidy this proof up\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `g \u226b h = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2082\u2083  (T \u2208 dist_triang C) : T.mor\u2082 \u226b T.mor\u2083 = 0 :=\ncomp_dist_triangle_mor_zero\u2081\u2082 C T.rotate (rot_of_dist_triangle C T H)\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `h \u226b f\u27e61\u27e7 = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2083\u2081 (T \u2208 dist_triang C) :\n  T.mor\u2083 \u226b ((shift_equiv C 1).functor.map T.mor\u2081) = 0 :=\nhave H\u2082 : _ := rot_of_dist_triangle C T.rotate (rot_of_dist_triangle C T H),\nby simpa using comp_dist_triangle_mor_zero\u2081\u2082 C (T.rotate.rotate) H\u2082\n\n/-\nTODO: If `C` is pretriangulated with respect to a shift,\nthen `C\u1d52\u1d56` is pretriangulated with respect to the inverse shift.\n-/\nend pretriangulated\nend category_theory.triangulated\n\nnamespace category_theory.triangulated\nnamespace pretriangulated\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [has_zero_object C] [has_shift C \u2124] [preadditive C]\n  [\u2200 n : \u2124, functor.additive (shift_functor C n)]\nvariables (D : Type u\u2082) [category.{v\u2082} D] [has_zero_object D] [has_shift D \u2124] [preadditive D]\n  [\u2200 n : \u2124, functor.additive (shift_functor D n)]\n\n/--\nThe underlying structure of a triangulated functor between pretriangulated categories `C` and `D`\nis a functor `F : C \u2964 D` together with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7`.\n-/\nstructure triangulated_functor_struct extends (C \u2964 D) :=\n(comm_shift : shift_functor C (1 : \u2124) \u22d9 to_functor \u2245 to_functor \u22d9 shift_functor D (1 : \u2124))\n\ninstance : inhabited (triangulated_functor_struct C C) :=\n\u27e8{ obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl }\u27e9\n\nvariables {C D}\n/--\nGiven a `triangulated_functor_struct` we can define a function from triangles of `C` to\ntriangles of `D`.\n-/\n@[simp]\ndef triangulated_functor_struct.map_triangle (F : triangulated_functor_struct C D)\n  (T : triangle C) : triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\nvariables (C D)\n/--\nA triangulated functor between pretriangulated categories `C` and `D` is a functor `F : C \u2964 D`\ntogether with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7` such that for every\ndistinguished triangle `(X,Y,Z,f,g,h)` of `C`, the triangle\n`(F(X), F(Y), F(Z), F(f), F(g), F(h) \u226b (\u03be X))` is a distinguished triangle of `D`.\nSee https://stacks.math.columbia.edu/tag/014V\n-/\nstructure triangulated_functor [pretriangulated C] [pretriangulated D] extends\n  triangulated_functor_struct C D :=\n(map_distinguished' : \u03a0 (T: triangle C), (T \u2208 dist_triang C) \u2192\n  (to_triangulated_functor_struct.map_triangle T \u2208 dist_triang D) )\n\ninstance [pretriangulated C] : inhabited (triangulated_functor C C) :=\n\u27e8{obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl ,\n  map_distinguished' := begin\n    rintros \u27e8_,_,_,_\u27e9 Tdt,\n    dsimp at *,\n    rwa category.comp_id,\n  end }\u27e9\n\nvariables {C D} [pretriangulated C] [pretriangulated D]\n/--\nGiven a `triangulated_functor` we can define a function from triangles of `C` to triangles of `D`.\n-/\n@[simp]\ndef triangulated_functor.map_triangle (F : triangulated_functor C D) (T : triangle C) :\n  triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\n/--\nGiven a `triangulated_functor` and a distinguished triangle `T` of `C`, then the triangle it\nmaps onto in `D` is also distinguished.\n-/\nlemma triangulated_functor.map_distinguished (F : triangulated_functor C D) (T : triangle C)\n  (h : T \u2208 dist_triang C) : (F.map_triangle T) \u2208 dist_triang D := F.map_distinguished' T h\n\n\nend pretriangulated\nend category_theory.triangulated\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/triangulated/pretriangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.35969897856552874}}
{"text": "/-\nCopyright (c) 2021 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.generator\nimport category_theory.limits.cone_category\nimport category_theory.limits.constructions.weakly_initial\nimport category_theory.limits.functor_category\nimport category_theory.subobject.comma\n\n/-!\n# Adjoint functor theorem\n\nThis file proves the (general) adjoint functor theorem, in the form:\n* If `G : D \u2964 C` preserves limits and `D` has limits, and satisfies the solution set condition,\n  then it has a left adjoint: `is_right_adjoint_of_preserves_limits_of_solution_set_condition`.\n\nWe show that the converse holds, i.e. that if `G` has a left adjoint then it satisfies the solution\nset condition, see `solution_set_condition_of_is_right_adjoint`\n(the file `category_theory/adjunction/limits` already shows it preserves limits).\n\nWe define the *solution set condition* for the functor `G : D \u2964 C` to mean, for every object\n`A : C`, there is a set-indexed family ${f_i : A \u27f6 G (B_i)}$ such that any morphism `A \u27f6 G X`\nfactors through one of the `f_i`.\n\nThis file also proves the special adjoint functor theorem, in the form:\n* If `G : D \u2964 C` preserves limits and `D` is complete, well-powered and has a small coseparating\n  set, then `G` has a left adjoint: `is_right_adjoint_of_preserves_limits_of_is_coseparating`\n\nFinally, we prove the following corollary of the special adjoint functor theorem:\n* If `C` is complete, well-powered and has a small coseparating set, then it is cocomplete:\n  `has_colimits_of_has_limits_of_is_coseparating`\n\n-/\nuniverses v u u'\n\nnamespace category_theory\nopen limits\n\nvariables {J : Type v}\nvariables {C : Type u} [category.{v} C]\n\n/--\nThe functor `G : D \u2964 C` satisfies the *solution set condition* if for every `A : C`, there is a\nfamily of morphisms `{f_i : A \u27f6 G (B_i) // i \u2208 \u03b9}` such that given any morphism `h : A \u27f6 G X`,\nthere is some `i \u2208 \u03b9` such that `h` factors through `f_i`.\n\nThe key part of this definition is that the indexing set `\u03b9` lives in `Type v`, where `v` is the\nuniverse of morphisms of the category: this is the \"smallness\" condition which allows the general\nadjoint functor theorem to go through.\n-/\ndef solution_set_condition {D : Type u} [category.{v} D] (G : D \u2964 C) : Prop :=\n\u2200 (A : C), \u2203 (\u03b9 : Type v) (B : \u03b9 \u2192 D) (f : \u03a0 (i : \u03b9), A \u27f6 G.obj (B i)),\n  \u2200 X (h : A \u27f6 G.obj X), \u2203 (i : \u03b9) (g : B i \u27f6 X), f i \u226b G.map g = h\n\nsection general_adjoint_functor_theorem\nvariables {D : Type u} [category.{v} D]\n\nvariables (G : D \u2964 C)\n\n/-- If `G : D \u2964 C` is a right adjoint it satisfies the solution set condition.  -/\nlemma solution_set_condition_of_is_right_adjoint [is_right_adjoint G] :\n  solution_set_condition G :=\nbegin\n  intros A,\n  refine \u27e8punit, \u03bb _, (left_adjoint G).obj A, \u03bb _, (adjunction.of_right_adjoint G).unit.app A, _\u27e9,\n  intros B h,\n  refine \u27e8punit.star, ((adjunction.of_right_adjoint G).hom_equiv _ _).symm h, _\u27e9,\n  rw [\u2190adjunction.hom_equiv_unit, equiv.apply_symm_apply],\nend\n\n/--\nThe general adjoint functor theorem says that if `G : D \u2964 C` preserves limits and `D` has them,\nif `G` satisfies the solution set condition then `G` is a right adjoint.\n-/\nnoncomputable def is_right_adjoint_of_preserves_limits_of_solution_set_condition\n  [has_limits D] [preserves_limits G] (hG : solution_set_condition G) :\n  is_right_adjoint G :=\nbegin\n  apply is_right_adjoint_of_structured_arrow_initials _,\n  intro A,\n  specialize hG A,\n  choose \u03b9 B f g using hG,\n  let B' : \u03b9 \u2192 structured_arrow A G := \u03bb i, structured_arrow.mk (f i),\n  have hB' : \u2200 (A' : structured_arrow A G), \u2203 i, nonempty (B' i \u27f6 A'),\n  { intros A',\n    obtain \u27e8i, _, t\u27e9 := g _ A'.hom,\n    exact \u27e8i, \u27e8structured_arrow.hom_mk _ t\u27e9\u27e9 },\n  obtain \u27e8T, hT\u27e9 := has_weakly_initial_of_weakly_initial_set_and_has_products hB',\n  apply has_initial_of_weakly_initial_and_has_wide_equalizers hT,\nend\n\nend general_adjoint_functor_theorem\n\nsection special_adjoint_functor_theorem\nvariables {D : Type u'} [category.{v} D]\n\n/--\nThe special adjoint functor theorem: if `G : D \u2964 C` preserves limits and `D` is complete,\nwell-powered and has a small coseparating set, then `G` has a left adjoint.\n-/\nnoncomputable def is_right_adjoint_of_preserves_limits_of_is_coseparating [has_limits D]\n  [well_powered D] {\ud835\udca2 : set D} [small.{v} \ud835\udca2] (h\ud835\udca2 : is_coseparating \ud835\udca2) (G : D \u2964 C)\n  [preserves_limits G] : is_right_adjoint G :=\nhave \u2200 A, has_initial (structured_arrow A G),\n  from \u03bb A, has_initial_of_is_coseparating (structured_arrow.is_coseparating_proj_preimage A G h\ud835\udca2),\nby exactI is_right_adjoint_of_structured_arrow_initials _\n\n/--\nThe special adjoint functor theorem: if `F : C \u2964 D` preserves colimits and `C` is cocomplete,\nwell-copowered and has a small separating set, then `F` has a right adjoint.\n-/\nnoncomputable def is_left_adjoint_of_preserves_colimits_of_is_separatig [has_colimits C]\n  [well_powered C\u1d52\u1d56] {\ud835\udca2 : set C} [small.{v} \ud835\udca2] (h\ud835\udca2 : is_separating \ud835\udca2) (F : C \u2964 D)\n  [preserves_colimits F] : is_left_adjoint F :=\nhave \u2200 A, has_terminal (costructured_arrow F A),\n  from \u03bb A, has_terminal_of_is_separating (costructured_arrow.is_separating_proj_preimage F A h\ud835\udca2),\nby exactI is_left_adjoint_of_costructured_arrow_terminals _\n\nend special_adjoint_functor_theorem\n\nnamespace limits\n\n/-- A consequence of the special adjoint functor theorem: if `C` is complete, well-powered and\n    has a small coseparating set, then it is cocomplete. -/\nlemma has_colimits_of_has_limits_of_is_coseparating [has_limits C] [well_powered C]\n  {\ud835\udca2 : set C} [small.{v} \ud835\udca2] (h\ud835\udca2 : is_coseparating \ud835\udca2) : has_colimits C :=\n{ has_colimits_of_shape := \u03bb J hJ, by exactI has_colimits_of_shape_iff_is_right_adjoint_const.2\n    \u27e8is_right_adjoint_of_preserves_limits_of_is_coseparating h\ud835\udca2 _\u27e9 }\n\n/-- A consequence of the special adjoint functor theorem: if `C` is cocomplete, well-copowered and\n    has a small separating set, then it is complete. -/\nlemma has_limits_of_has_colimits_of_is_separating [has_colimits C] [well_powered C\u1d52\u1d56]\n  {\ud835\udca2 : set C} [small.{v} \ud835\udca2] (h\ud835\udca2 : is_separating \ud835\udca2) : has_limits C :=\n{ has_limits_of_shape := \u03bb J hJ, by exactI has_limits_of_shape_iff_is_left_adjoint_const.2\n    \u27e8is_left_adjoint_of_preserves_colimits_of_is_separatig h\ud835\udca2 _\u27e9 }\n\nend limits\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/adjunction/adjoint_functor_theorems.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371502, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.35969896958878006}}
{"text": "-- See: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/OO.20polymorphism.3F/near/297629381\nnamespace O41\n\ninductive Geo_Type\n| Point2D\n| Point3D\n| Point4D\n| Square2D\nderiving Repr, DecidableEq\n\nopen Geo_Type\n\nstructure Slots_Point2D where\n  (x y : Float)\n  deriving Repr\n\nstructure Slots_Point3D extends Slots_Point2D where\n  z : Float\n  deriving Repr\n\nstructure Slots_Point4D extends Slots_Point3D where\n  w : Float\n  deriving Repr\n\nstructure Slots_Square2D extends Slots_Point2D where\n  width : Float\n  deriving Repr\n\ndef Geo_Type.Slots : Geo_Type \u2192 Type\n| Point2D => Slots_Point2D\n| Point3D => Slots_Point3D\n| Point4D => Slots_Point4D\n| Square2D => Slots_Square2D\n\n/-- Even though Geo_Type.Slots returns a Type, \n - the following proves that there is a Repr instance for that Type.\n -/\ndef toRepr : (T : Geo_Type) \u2192 Repr T.Slots := \nby {\n  intro T;\n  cases T <;> \n  unfold Geo_Type.Slots <;> \n  simp <;>\n  infer_instance\n}\n\n/-- Get casting funct\nion to first type from second type -/\ndef Geo_Type.cast : (t : Geo_Type) \u2192 (t' : Geo_Type) \u2192 Option (t'.Slots \u2192 t.Slots)\n| Point2D, Point2D => some id\n| Point2D, Point3D => some (\u03bb x => x.toSlots_Point2D)\n| Point3D, Point3D => some id\n| Point2D, Point4D => some (\u03bb x => x.toSlots_Point2D)\n| Point3D, Point4D => some (\u03bb x => x.toSlots_Point3D)\n| Point4D, Point4D => some id\n| Point2D, Square2D => some (\u03bb x => x.toSlots_Point2D)\n| Square2D, Square2D => some id\n| _, _ => none\n\n/-- Whether the second type is a subtype of the first. -/\ndef Geo_Type.supertype (t t' : Geo_Type) : Bool := (t.cast t').isSome\n\ntheorem Geo_Type.supertype.trans (h : Geo_Type.supertype t t') (h' : Geo_Type.supertype t' t'') :\n  Geo_Type.supertype t t'' :=\nby\n  cases t\n  all_goals\n    cases t'\n    all_goals\n      cases t''\n      all_goals\n        simp at h\n        try simp at h'\n        try simp\n\ndef Option.get : (x : Option \u03b1) \u2192 x.isSome \u2192 \u03b1\n| some v, _ => v\n| none, h => by simp [Option.isSome] at h\n\n/-- Get the cast function from a proof that `t'` is a subtype of `t`. -/\ndef Geo_Type.supertype.cast (h : Geo_Type.supertype t t') : t'.Slots \u2192 t.Slots :=\n  Option.get (Geo_Type.cast t t') h\n\n/-- Objects of subtype t -/\nstructure Obj (t : Geo_Type) where\n  ty : Geo_Type\n  sub : Geo_Type.supertype t ty\n  slots : ty.Slots\n\ninstance : Coe Slots_Point2D (Obj Point2D) where coe s := \u27e8Point2D, rfl, s\u27e9\ninstance : Coe Slots_Point3D (Obj Point3D) where coe s := \u27e8Point3D, rfl, s\u27e9\ninstance : Coe Slots_Point4D (Obj Point4D) where coe s := \u27e8Point4D, rfl, s\u27e9\ninstance : Coe Slots_Square2D (Obj Square2D) where coe s := \u27e8Square2D, rfl, s\u27e9\n\n/-- Extract the slots from an `Obj`. -/\ndef Obj.get (o : Obj t) : t.Slots := Geo_Type.supertype.cast o.sub o.slots\n\n/-- Cast up, which can be done statically. -/\ndef Obj.cast_up (o : Obj t) (h : Geo_Type.supertype t' t := by rfl) : Obj t' where\n  ty := o.ty\n  sub := Geo_Type.supertype.trans h o.sub\n  slots := o.slots\n\ndef Obj.can_cast (o : Obj t) (t' : Geo_Type) : Bool := Geo_Type.supertype t' o.ty\n\n/-- Cast up or down using run-time information. -/\ndef Obj.cast (o : Obj t) (h : o.can_cast t') : Obj t' where\n  ty := o.ty\n  sub := h\n  slots := o.slots\n\ninstance : Repr (Obj (T : Geo_Type)) where\n  reprPrec o _ := ((instReprGeo_Type.reprPrec o.ty 1).append (Std.Format.text \" \")).append ((toRepr o.ty).reprPrec o.slots 1)\n\ndef s2: Slots_Point2D := {x := 1, y := 2}\n#eval s2\n\ndef o2: Obj Point2D := s2\n#eval o2\n\ndef o3: Obj Point3D := ({x := 1, y := 2, z := 3}: Slots_Point3D)\n#eval o3\n#eval o3.get\n#eval o3.slots\n#eval o3.slots.x\n#eval o3.slots.y\n#eval o3.slots.z\n\ndef o4: Obj Point4D := ({x := 1, y := 2, z := 3, w := 4}: Slots_Point4D)\n#eval o4\n#eval o4.get.w\n-- 4.000000\n\ndef o43: Obj Point3D := o4.cast_up\n#eval o43\n#eval o43.slots.w\n-- 4.000000\n#eval o43.get\n--#eval o43.get.w\n-- invalid field 'w', the environment does not contain 'O41.Geo_Type.Slots.w'\n--  Obj.get o43\n-- has type\n--  Slots Point3D\nend O41", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/O41.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369905, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.35968825791085335}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem two_colorable_iff_bipartite (G : Type*) [graph G] : two_colorable G \u2194 bipartite G :=\nbegin\n  split,\n  {\n    assume h1 : two_colorable G,\n    have h2 : \u2203 A B : set G, (\u2200 a : G, a \u2208 A \u2228 a \u2208 B) \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 (a,b) \u2208 E G), from sorry,\n    show bipartite G, from sorry,\n  },\n  {\n    assume h1 : bipartite G,\n    have h2 : \u2203 A B : set G, (\u2200 a : G, a \u2208 A \u2228 a \u2208 B) \u2227 (\u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 (a,b) \u2208 E G), from sorry,\n    show two_colorable G, from sorry,\n  }\nend\n\n--OUTPUT 2\ntheorem bipartite_graph_two_colorable (G : Type*) [graph G] : \n  (\u2203 A B : set G, is_bipartite G A B) \u2194 (\u2203 f : G \u2192 bool, is_valid_coloring G f) :=\nbegin\n  have h1 : (\u2203 A B : set G, is_bipartite G A B) \u2192 (\u2203 f : G \u2192 bool, is_valid_coloring G f), from sorry,\n  have h2 : (\u2203 f : G \u2192 bool, is_valid_coloring G f) \u2192 (\u2203 A B : set G, is_bipartite G A B), from sorry,\n  show (\u2203 A B : set G, is_bipartite G A B) \u2194 (\u2203 f : G \u2192 bool, is_valid_coloring G f), from sorry,\nend\n\n--OUTPUT 3\ntheorem bipartite_graph_two_colorable (G : Type*) [graph G] : (\u2203 A B : set G, is_bipartite G A B) \u2194 (\u2203 f : G \u2192 \u2115, is_two_coloring G f) :=\nbegin\n  split,\n  {\n    assume h1 : \u2203 A B : set G, is_bipartite G A B,\n    cases h1 with A h1,\n    cases h1 with B h1,\n    cases h1 with h1 h2,\n    have h3 : \u2200 a : G, a \u2208 A \u2228 a \u2208 B, from sorry,\n    have h4 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 edge G a b, from sorry,\n    have h5 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h6 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h7 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G b a, from sorry,\n    have h8 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G b a, from sorry,\n    have h9 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h10 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h11 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h12 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h13 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h14 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h15 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h16 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h17 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h18 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h19 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h20 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h21 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h22 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h23 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h24 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h25 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h26 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h27 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h28 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h29 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h30 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h31 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h32 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h33 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h34 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h35 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h36 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h37 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h38 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h39 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h40 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h41 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h42 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h43 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h44 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h45 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h46 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h47 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h48 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h49 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h50 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h51 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h52 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h53 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h54 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h55 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 B \u2192 \u00ac edge G a b, from sorry,\n    have h56 : \u2200 a b : G, a \u2208 B \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h57 : \u2200 a b : G, a \u2208 A \u2227 b \u2208 A \u2192 \u00ac edge G a b, from sorry,\n    have h58 : \u2200 a b : G, a\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7799929053683038, "lm_q2_score": 0.46101677931231594, "lm_q1q2_score": 0.35958981711935145}}
{"text": "inductive Foo (f: Fin n): Nat \u2192 Prop\n| mk: Foo f f.val\ntheorem foo {f: Fin n}: Foo f (no_index f.val) := .mk\nexample (hf: f < n): Foo \u27e8f, hf\u27e9 f := by simp only [foo]\nexample  (f: Fin n): Foo  f f.val  := by simp only [foo]\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1253.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.35949629567289687}}
{"text": "import o_minimal.sheaf.yoneda\n\n-- Lemmas for proving definability of propositions with quantifiers\n-- `\u2200 (x : X), P x`, `\u2203 (x : X), P x`. Here `X` must be representable.\n\nnamespace o_minimal\n\nvariables {R : Type*} {S : struc R}\nvariables {X : Type*} [has_coordinates R X] [definable_rep S X]\n\nlemma definable_forall : definable S (\u03bb (p : X \u2192 Prop), \u2200 x, p x) :=\nbegin\n  rw definable_fun,\n  intros K \u03c6 h\u03c6,\n  rw \u2190definable_yoneda at \u22a2 h\u03c6,\n  rw definable_iff_def_set,\n  rw definable_iff_def_rel\u2082 at h\u03c6,\n  apply def_set.forall h\u03c6,\nend\n\n-- Helper lemma for applying `definable_forall` in a `[defin]` block.\nlemma definable_sheaf.forall {\u0393 : Def S}\n  (p : \u0393 \u2192 X \u2192 Prop) {hp : definable_sheaf.definable p} :\n  definable_sheaf.definable (\u03bb i, \u2200 x, p i x) :=\nbegin\n  apply definable_sheaf.definable_app (\u03bb i (p : X \u2192 Prop), \u2200 x, p x),\n  { exact hp },\n  { exact definable_forall.definable _ }\nend\n\n-- Not directly related to quantifiers,\n-- but frequently appears in \"bounded\" ones.\nlemma definable_sheaf.imp {\u0393 : Def S}\n  (p : \u0393 \u2192 Prop) {hp : definable_sheaf.definable p}\n  (q : \u0393 \u2192 Prop) {hq : definable_sheaf.definable q} :\n  definable_sheaf.definable (\u03bb i, p i \u2192 q i) :=\ndef_set.imp hp hq\n\n-- `definable S Exists`\nlemma definable_Exists : definable S (\u03bb (p : X \u2192 Prop), \u2203 x, p x) :=\nbegin\n  rw definable_fun,\n  intros K \u03c6 h\u03c6,\n  rw \u2190definable_yoneda at \u22a2 h\u03c6,\n  rw definable_iff_def_set,\n  rw definable_iff_def_rel\u2082 at h\u03c6,\n  apply def_set.exists h\u03c6,\nend\n\nlemma definable_nonempty : definable S (set.nonempty : set X \u2192 Prop) :=\nbegin [defin]\n  intro s,\n  app, exact definable_Exists.definable _,\n  intro x,\n  app, app, exact definable.mem.definable _, var, var\nend\n\nlemma definable_sheaf.eq {\u0393 : Def S} :\n  definable_sheaf.definable (\u03bb _ : \u0393, @eq X) :=\nbegin\n  intros K \u03c6 h\u03c6 L \u03c8 h\u03c8,\n  rw \u2190 definable_yoneda at h\u03c6 h\u03c8 \u22a2,\n  rw definable_iff_def_set,\n  apply def_set_eq;\n  rw \u2190 definable_iff_def_fun,\n  begin [defin]\n    intro x,\n    app, exact definable.snd.definable _,\n    app, exact h\u03c6.definable _,\n    app, exact definable.fst.definable _,\n    app, exact h\u03c8.definable _,\n    var,\n  end,\n  begin [defin]\n    intro x,\n    app, exact definable.snd.definable _,\n    app, exact h\u03c8.definable _,\n    var,\n  end\nend\n\nend o_minimal\n\nnamespace defin.interactive\nsetup_tactic_parser\n\nmeta def all (var : parse ident_) : defin unit :=\ndo `[refine o_minimal.definable_sheaf.forall _],\n   intro var\n\nmeta def imp : defin unit :=\n`[refine o_minimal.definable_sheaf.imp _ _]\n\nend defin.interactive\n\nnamespace o_minimal\n\nvariables {R : Type*} {S : struc R}\nvariables {X : Type*} [has_coordinates R X] [definable_rep S X]\n\nlemma definable_subset : definable S ((\u2286) : set X \u2192 set X \u2192 Prop) :=\nbegin [defin]\n  intro s,\n  intro t,\n  all x,\n  imp,\n  { app, app, exact definable.mem.definable _, var, var },\n  { app, app, exact definable.mem.definable _, var, var }\nend\n\nlemma definable_powerset : definable S (set.powerset : set X \u2192 set (set X)) :=\nbegin [defin]\n  intro s,\n  intro t,\n  app, app, exact definable_subset.definable _, var, var\nend\n\nend o_minimal\n", "meta": {"author": "rwbarton", "repo": "lean-omin", "sha": "fd733c6d95ef6f4743aae97de5e15df79877c00e", "save_path": "github-repos/lean/rwbarton-lean-omin", "path": "github-repos/lean/rwbarton-lean-omin/lean-omin-fd733c6d95ef6f4743aae97de5e15df79877c00e/src/o_minimal/sheaf/quantifiers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.35933363283183434}}
{"text": "/-\nCopyright (c) 2020 Zhangir Azerbayev. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Eric Wieser, Zhangir Azerbayev\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.linear_algebra.multilinear\nimport Mathlib.linear_algebra.linear_independent\nimport Mathlib.group_theory.perm.sign\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_6 l u_4 u_5 u_7 \n\nnamespace Mathlib\n\n/-!\n# Alternating Maps\n\nWe construct the bundled function `alternating_map`, which extends `multilinear_map` with all the\narguments of the same type.\n\n## Main definitions\n* `alternating_map R M N \u03b9` is the space of `R`-linear alternating maps from `\u03b9 \u2192 M` to `N`.\n* `f.map_eq_zero_of_eq` expresses that `f` is zero when two inputs are equal.\n* `f.map_swap` expresses that `f` is negated when two inputs are swapped.\n* `f.map_perm` expresses how `f` varies by a sign change under a permutation of its inputs.\n* An `add_comm_monoid`, `add_comm_group`, and `semimodule` structure over `alternating_map`s that\n  matches the definitions over `multilinear_map`s.\n* `multilinear_map.alternatization`, which makes an alternating map out of a non-alternating one.\n\n## Implementation notes\n`alternating_map` is defined in terms of `map_eq_zero_of_eq`, as this is easier to work with than\nusing `map_swap` as a definition, and does not require `has_neg N`.\n\n`alternating_map`s are provided with a coercion to `multilinear_map`, along with a set of\n`norm_cast` lemmas that act on the algebraic structure:\n\n* `alternating_map.coe_add`\n* `alternating_map.coe_zero`\n* `alternating_map.coe_sub`\n* `alternating_map.coe_neg`\n* `alternating_map.coe_smul`\n-/\n\n-- semiring / add_comm_monoid\n\n-- semiring / add_comm_group\n\n/--\nAn alternating map is a multilinear map that vanishes when two of its arguments are equal.\n-/\nstructure alternating_map (R : Type u_1) [semiring R] (M : Type u_2) [add_comm_monoid M] [semimodule R M] (N : Type u_3) [add_comm_monoid N] [semimodule R N] (\u03b9 : Type u_6) [DecidableEq \u03b9] \nextends multilinear_map R (fun (i : \u03b9) => M) N\nwhere\n  map_eq_zero_of_eq' : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 to_fun v = 0\n\n/-- The multilinear map associated to an alternating map -/\nnamespace alternating_map\n\n\n/-! Basic coercion simp lemmas, largely copied from `ring_hom` and `multilinear_map` -/\n\nprotected instance has_coe_to_fun {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : has_coe_to_fun (alternating_map R M N \u03b9) :=\n  has_coe_to_fun.mk (fun (x : alternating_map R M N \u03b9) => ((i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) \u2192 N)\n    fun (x : alternating_map R M N \u03b9) => to_fun x\n\n@[simp] theorem to_fun_eq_coe {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : to_fun f = \u21d1f :=\n  rfl\n\n@[simp] theorem coe_mk {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : (\u03b9 \u2192 M) \u2192 N) (h\u2081 : \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (x y : M),\n  f (function.update m i (x + y)) = f (function.update m i x) + f (function.update m i y)) (h\u2082 : \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (c : R) (x : M),\n  f (function.update m i (c \u2022 x)) = c \u2022 f (function.update m i x)) (h\u2083 : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 f v = 0) : \u21d1(mk f h\u2081 h\u2082 h\u2083) = f :=\n  rfl\n\ntheorem congr_fun {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} (h : f = g) (x : \u03b9 \u2192 M) : coe_fn f x = coe_fn g x :=\n  congr_arg (fun (h : alternating_map R M N \u03b9) => coe_fn h x) h\n\ntheorem congr_arg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {x : \u03b9 \u2192 M} {y : \u03b9 \u2192 M} (h : x = y) : coe_fn f x = coe_fn f y :=\n  congr_arg (fun (x : \u03b9 \u2192 M) => coe_fn f x) h\n\ntheorem coe_inj {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} (h : \u21d1f = \u21d1g) : f = g := sorry\n\ntheorem ext {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {f : alternating_map R M N \u03b9} {f' : alternating_map R M N \u03b9} (H : \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn f' x) : f = f' :=\n  coe_inj (funext H)\n\ntheorem ext_iff {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {f : alternating_map R M N \u03b9} {g : alternating_map R M N \u03b9} : f = g \u2194 \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn g x :=\n  { mp := fun (h : f = g) (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) => h \u25b8 rfl,\n    mpr := fun (h : \u2200 (x : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i), coe_fn f x = coe_fn g x) => ext h }\n\nprotected instance multilinear_map.has_coe {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : has_coe (alternating_map R M N \u03b9) (multilinear_map R (fun (i : \u03b9) => M) N) :=\n  has_coe.mk fun (x : alternating_map R M N \u03b9) => to_multilinear_map x\n\n@[simp] theorem coe_multilinear_map {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\n@[simp] theorem to_multilinear_map_eq_coe {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) : to_multilinear_map f = \u2191f :=\n  rfl\n\n@[simp] theorem coe_multilinear_map_mk {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : (\u03b9 \u2192 M) \u2192 N) (h\u2081 : \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (x y : M),\n  f (function.update m i (x + y)) = f (function.update m i x) + f (function.update m i y)) (h\u2082 : \u2200 (m : (i : \u03b9) \u2192 (fun (i : \u03b9) => M) i) (i : \u03b9) (c : R) (x : M),\n  f (function.update m i (c \u2022 x)) = c \u2022 f (function.update m i x)) (h\u2083 : \u2200 (v : \u03b9 \u2192 M) (i j : \u03b9), v i = v j \u2192 i \u2260 j \u2192 f v = 0) : \u2191(mk f h\u2081 h\u2082 h\u2083) = multilinear_map.mk f h\u2081 h\u2082 :=\n  rfl\n\n/-!\n### Simp-normal forms of the structure fields\n\nThese are expressed in terms of `\u21d1f` instead of `f.to_fun`.\n-/\n\n@[simp] theorem map_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) (i : \u03b9) (x : M) (y : M) : coe_fn f (function.update v i (x + y)) = coe_fn f (function.update v i x) + coe_fn f (function.update v i y) :=\n  multilinear_map.map_add' (to_multilinear_map f) v i x y\n\n@[simp] theorem map_sub {R : Type u_1} [semiring R] {M' : Type u_4} [add_comm_group M'] [semimodule R M'] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g' : alternating_map R M' N' \u03b9) (v' : \u03b9 \u2192 M') (i : \u03b9) (x : M') (y : M') : coe_fn g' (function.update v' i (x - y)) = coe_fn g' (function.update v' i x) - coe_fn g' (function.update v' i y) :=\n  multilinear_map.map_sub (to_multilinear_map g') v' i x y\n\n@[simp] theorem map_neg {R : Type u_1} [semiring R] {M' : Type u_4} [add_comm_group M'] [semimodule R M'] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g' : alternating_map R M' N' \u03b9) (v' : \u03b9 \u2192 M') (i : \u03b9) (x : M') : coe_fn g' (function.update v' i (-x)) = -coe_fn g' (function.update v' i x) :=\n  multilinear_map.map_neg (to_multilinear_map g') v' i x\n\n@[simp] theorem map_smul {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) (i : \u03b9) (r : R) (x : M) : coe_fn f (function.update v i (r \u2022 x)) = r \u2022 coe_fn f (function.update v i x) :=\n  multilinear_map.map_smul' (to_multilinear_map f) v i r x\n\n@[simp] theorem map_eq_zero_of_eq {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (h : v i = v j) (hij : i \u2260 j) : coe_fn f v = 0 :=\n  map_eq_zero_of_eq' f v i j h hij\n\n/-!\n### Algebraic structure inherited from `multilinear_map`\n\n`alternating_map` carries the same `add_comm_monoid`, `add_comm_group`, and `semimodule` structure\nas `multilinear_map`\n-/\n\nprotected instance has_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : Add (alternating_map R M N \u03b9) :=\n  { add := fun (a b : alternating_map R M N \u03b9) => mk (multilinear_map.to_fun (\u2191a + \u2191b)) sorry sorry sorry }\n\n@[simp] theorem add_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (f' : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) : coe_fn (f + f') v = coe_fn f v + coe_fn f' v :=\n  rfl\n\ntheorem coe_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (f' : alternating_map R M N \u03b9) : \u2191(f + f') = \u2191f + \u2191f' :=\n  rfl\n\nprotected instance has_zero {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : HasZero (alternating_map R M N \u03b9) :=\n  { zero := mk (multilinear_map.to_fun 0) sorry sorry sorry }\n\n@[simp] theorem zero_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (v : \u03b9 \u2192 M) : coe_fn 0 v = 0 :=\n  rfl\n\ntheorem coe_zero {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : \u21910 = 0 :=\n  rfl\n\nprotected instance inhabited {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : Inhabited (alternating_map R M N \u03b9) :=\n  { default := 0 }\n\nprotected instance add_comm_monoid {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] : add_comm_monoid (alternating_map R M N \u03b9) :=\n  add_comm_monoid.mk Add.add sorry 0 sorry sorry sorry\n\nprotected instance has_neg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] : Neg (alternating_map R M N' \u03b9) :=\n  { neg := fun (f : alternating_map R M N' \u03b9) => mk (multilinear_map.to_fun (-\u2191f)) sorry sorry sorry }\n\n@[simp] theorem neg_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (m : \u03b9 \u2192 M) : coe_fn (-g) m = -coe_fn g m :=\n  rfl\n\ntheorem coe_neg {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) : \u2191(-g) = -\u2191g :=\n  rfl\n\nprotected instance has_sub {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] : Sub (alternating_map R M N' \u03b9) :=\n  { sub := fun (f g : alternating_map R M N' \u03b9) => mk (multilinear_map.to_fun (\u2191f - \u2191g)) sorry sorry sorry }\n\n@[simp] theorem sub_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (g\u2082 : alternating_map R M N' \u03b9) (m : \u03b9 \u2192 M) : coe_fn (g - g\u2082) m = coe_fn g m - coe_fn g\u2082 m :=\n  rfl\n\ntheorem coe_sub {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (g\u2082 : alternating_map R M N' \u03b9) : \u2191(g - g\u2082) = \u2191g - \u2191g\u2082 :=\n  rfl\n\nprotected instance add_comm_group {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] : add_comm_group (alternating_map R M N' \u03b9) :=\n  add_comm_group.mk Add.add sorry 0 sorry sorry Neg.neg Sub.sub sorry sorry\n\nprotected instance has_scalar {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] : has_scalar S (alternating_map R M N \u03b9) :=\n  has_scalar.mk fun (c : S) (f : alternating_map R M N \u03b9) => mk (multilinear_map.to_fun (c \u2022 \u2191f)) sorry sorry sorry\n\n@[simp] theorem smul_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] (c : S) (m : \u03b9 \u2192 M) : coe_fn (c \u2022 f) m = c \u2022 coe_fn f m :=\n  rfl\n\ntheorem coe_smul {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] (c : S) : \u2191(c \u2022 f) = c \u2022 \u2191f :=\n  rfl\n\nprotected instance distrib_mul_action {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {S : Type u_7} [monoid S] [distrib_mul_action S N] [smul_comm_class R S N] : distrib_mul_action S (alternating_map R M N \u03b9) :=\n  distrib_mul_action.mk sorry sorry\n\n/-- The space of multilinear maps over an algebra over `R` is a module over `R`, for the pointwise\naddition and scalar multiplication. -/\nprotected instance semimodule {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {S : Type u_7} [semiring S] [semimodule S N] [smul_comm_class R S N] : semimodule S (alternating_map R M N \u03b9) :=\n  semimodule.mk sorry sorry\n\nend alternating_map\n\n\n/-!\n### Composition with linear maps\n-/\n\nnamespace linear_map\n\n\n/-- Composing a alternating map with a linear map gives again a alternating map. -/\ndef comp_alternating_map {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082] (g : linear_map R N N\u2082) : alternating_map R M N \u03b9 \u2192+ alternating_map R M N\u2082 \u03b9 :=\n  add_monoid_hom.mk\n    (fun (f : alternating_map R M N \u03b9) =>\n      alternating_map.mk (multilinear_map.to_fun (comp_multilinear_map g \u2191f)) sorry sorry sorry)\n    sorry sorry\n\n@[simp] theorem coe_comp_alternating_map {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082] (g : linear_map R N N\u2082) (f : alternating_map R M N \u03b9) : \u21d1(coe_fn (comp_alternating_map g) f) = \u21d1g \u2218 \u21d1f :=\n  rfl\n\ntheorem comp_alternating_map_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] {N\u2082 : Type u_7} [add_comm_monoid N\u2082] [semimodule R N\u2082] (g : linear_map R N N\u2082) (f : alternating_map R M N \u03b9) (m : \u03b9 \u2192 M) : coe_fn (coe_fn (comp_alternating_map g) f) m = coe_fn g (coe_fn f m) :=\n  rfl\n\nend linear_map\n\n\nnamespace alternating_map\n\n\n/-!\n### Other lemmas from `multilinear_map`\n-/\n\ntheorem map_update_sum {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) {\u03b1 : Type u_4} (t : finset \u03b1) (i : \u03b9) (g : \u03b1 \u2192 M) (m : \u03b9 \u2192 M) : coe_fn f (function.update m i (finset.sum t fun (a : \u03b1) => g a)) =\n  finset.sum t fun (a : \u03b1) => coe_fn f (function.update m i (g a)) :=\n  multilinear_map.map_update_sum (to_multilinear_map f) t i g m\n\n/-!\n### Theorems specific to alternating maps\n\nVarious properties of reordered and repeated inputs which follow from\n`alternating_map.map_eq_zero_of_eq`.\n-/\n\ntheorem map_update_self {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) : coe_fn f (function.update v i (v j)) = 0 := sorry\n\ntheorem map_update_update {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) (m : M) : coe_fn f (function.update (function.update v i m) j m) = 0 := sorry\n\ntheorem map_swap_add {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) : coe_fn f (v \u2218 \u21d1(equiv.swap i j)) + coe_fn f v = 0 := sorry\n\ntheorem map_add_swap {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N : Type u_3} [add_comm_monoid N] [semimodule R N] {\u03b9 : Type u_6} [DecidableEq \u03b9] (f : alternating_map R M N \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) : coe_fn f v + coe_fn f (v \u2218 \u21d1(equiv.swap i j)) = 0 := sorry\n\ntheorem map_swap {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (v : \u03b9 \u2192 M) {i : \u03b9} {j : \u03b9} (hij : i \u2260 j) : coe_fn g (v \u2218 \u21d1(equiv.swap i j)) = -coe_fn g v :=\n  eq_neg_of_add_eq_zero (map_swap_add g v hij)\n\ntheorem map_perm {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) [fintype \u03b9] (v : \u03b9 \u2192 M) (\u03c3 : equiv.perm \u03b9) : coe_fn g (v \u2218 \u21d1\u03c3) = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn g v := sorry\n\ntheorem map_congr_perm {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) (v : \u03b9 \u2192 M) [fintype \u03b9] (\u03c3 : equiv.perm \u03b9) : coe_fn g v = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn g (v \u2218 \u21d1\u03c3) := sorry\n\ntheorem coe_dom_dom_congr {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] (g : alternating_map R M N' \u03b9) [fintype \u03b9] (\u03c3 : equiv.perm \u03b9) : multilinear_map.dom_dom_congr \u03c3 \u2191g = \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 \u2191g :=\n  multilinear_map.ext fun (v : \u03b9 \u2192 M) => map_perm g v \u03c3\n\n/-- If the arguments are linearly dependent then the result is `0`.\n\nTODO: Can the `division_ring` requirement be relaxed? -/\ntheorem map_linear_dependent {\u03b9 : Type u_6} [DecidableEq \u03b9] {K : Type u_1} [division_ring K] {M : Type u_2} [add_comm_group M] [semimodule K M] {N : Type u_3} [add_comm_group N] [semimodule K N] (f : alternating_map K M N \u03b9) (v : \u03b9 \u2192 M) (h : \u00aclinear_independent K v) : coe_fn f v = 0 := sorry\n\nend alternating_map\n\n\nnamespace multilinear_map\n\n\n/-- Produce an `alternating_map` out of a `multilinear_map`, by summing over all argument\npermutations. -/\ndef alternatization {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] [fintype \u03b9] : multilinear_map R (fun (i : \u03b9) => M) N' \u2192+ alternating_map R M N' \u03b9 :=\n  add_monoid_hom.mk\n    (fun (m : multilinear_map R (fun (i : \u03b9) => M) N') =>\n      alternating_map.mk\n        \u21d1(finset.sum finset.univ fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 dom_dom_congr \u03c3 m) sorry sorry\n        sorry)\n    sorry sorry\n\ntheorem alternatization_def {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] [fintype \u03b9] (m : multilinear_map R (fun (i : \u03b9) => M) N') : \u21d1(coe_fn alternatization m) =\n  \u21d1(finset.sum finset.univ fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 dom_dom_congr \u03c3 m) :=\n  rfl\n\ntheorem alternatization_apply {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] [fintype \u03b9] (m : multilinear_map R (fun (i : \u03b9) => M) N') (v : \u03b9 \u2192 M) : coe_fn (coe_fn alternatization m) v =\n  finset.sum finset.univ fun (\u03c3 : equiv.perm \u03b9) => \u2191(coe_fn equiv.perm.sign \u03c3) \u2022 coe_fn (dom_dom_congr \u03c3 m) v := sorry\n\nend multilinear_map\n\n\nnamespace alternating_map\n\n\n/-- Alternatizing a multilinear map that is already alternating results in a scale factor of `n!`,\nwhere `n` is the number of inputs. -/\ntheorem coe_alternatization {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] [fintype \u03b9] (a : alternating_map R M N' \u03b9) : coe_fn multilinear_map.alternatization \u2191a = nat.factorial (fintype.card \u03b9) \u2022 a := sorry\n\nend alternating_map\n\n\nnamespace linear_map\n\n\n/-- Composition with a linear map before and after alternatization are equivalent. -/\ntheorem comp_multilinear_map_alternatization {R : Type u_1} [semiring R] {M : Type u_2} [add_comm_monoid M] [semimodule R M] {N' : Type u_5} [add_comm_group N'] [semimodule R N'] {\u03b9 : Type u_6} [DecidableEq \u03b9] {N'\u2082 : Type u_7} [add_comm_group N'\u2082] [semimodule R N'\u2082] [fintype \u03b9] (g : linear_map R N' N'\u2082) (f : multilinear_map R (fun (_x : \u03b9) => M) N') : coe_fn multilinear_map.alternatization (comp_multilinear_map g f) =\n  coe_fn (comp_alternating_map g) (coe_fn multilinear_map.alternatization f) := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/linear_algebra/alternating.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283035, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.359193920859846}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport morphisms.basic\nimport topology.spectral.hom\nimport algebraic_geometry.limits\n\n/-!\n# Quasi-compact morphisms\n\nA morphism of schemes is quasi-compact if the preimages of quasi-compact open sets are\nquasi-compact.\n\nIt suffices to check that preimages of affine open sets are compact\n(`quasi_compact_iff_forall_affine`).\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverse u\n\nopen_locale algebraic_geometry\n\nnamespace algebraic_geometry\n\nvariables {X Y : Scheme.{u}} (f : X \u27f6 Y)\n\n/--\nA morphism is `quasi-compact` if the underlying map of topological spaces is, i.e. if the preimages\nof quasi-compact open sets are quasi-compact.\n-/\n@[mk_iff]\nclass quasi_compact (f : X \u27f6 Y) : Prop :=\n(is_compact_preimage [] : \u2200 U : set Y.carrier,\n  is_open U \u2192 is_compact U \u2192 is_compact (f.1.base \u207b\u00b9' U))\n\nlemma quasi_compact_iff_spectral : quasi_compact f \u2194 is_spectral_map f.1.base :=\n\u27e8\u03bb \u27e8h\u27e9, \u27e8by continuity, h\u27e9, \u03bb h, \u27e8h.2\u27e9\u27e9\n\n/-- The `affine_target_morphism_property` corresponding to `quasi_compact`, asserting that the\ndomain is a quasi-compact scheme. -/\ndef quasi_compact.affine_property : affine_target_morphism_property :=\n\u03bb X Y f hf, compact_space X.carrier\n\n@[priority 900]\ninstance quasi_compact_of_is_iso {X Y : Scheme} (f : X \u27f6 Y) [is_iso f] : quasi_compact f :=\nbegin\n  constructor,\n  intros U hU hU',\n  convert hU'.image (inv f.1.base).continuous_to_fun using 1,\n  rw set.image_eq_preimage_of_inverse,\n  delta function.left_inverse,\n  exacts [is_iso.inv_hom_id_apply f.1.base, is_iso.hom_inv_id_apply f.1.base]\nend\n\ninstance quasi_compact_comp {X Y Z : Scheme} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  [quasi_compact f] [quasi_compact g] : quasi_compact (f \u226b g) :=\nbegin\n  constructor,\n  intros U hU hU',\n  rw [Scheme.comp_val_base, coe_comp, set.preimage_comp],\n  apply quasi_compact.is_compact_preimage,\n  { exact continuous.is_open_preimage (by continuity) _ hU },\n  apply quasi_compact.is_compact_preimage; assumption\nend\n\nlemma is_compact_open_iff_eq_finset_affine_union {X : Scheme} (U : set X.carrier) :\n  is_compact U \u2227 is_open U \u2194\n    \u2203 (s : set X.affine_opens), s.finite \u2227 U = \u22c3 (i : X.affine_opens) (h : i \u2208 s), i :=\nbegin\n  apply opens.is_compact_open_iff_eq_finite_Union_of_is_basis\n    (coe : X.affine_opens \u2192 opens X.carrier),\n  { rw subtype.range_coe, exact is_basis_affine_open X },\n  { intro i, exact i.2.is_compact }\nend\n\nlemma is_compact_open_iff_eq_basic_open_union {X : Scheme} [is_affine X] (U : set X.carrier) :\n  is_compact U \u2227 is_open U \u2194\n    \u2203 (s : set (X.presheaf.obj (op \u22a4))), s.finite \u2227\n      U = \u22c3 (i : X.presheaf.obj (op \u22a4)) (h : i \u2208 s), X.basic_open i :=\nbegin\n  apply opens.is_compact_open_iff_eq_finite_Union_of_is_basis,\n  { exact is_basis_basic_open X },\n  { intro i, exact ((top_is_affine_open _).basic_open_is_affine _).is_compact }\nend\n\nlemma quasi_compact_iff_forall_affine : quasi_compact f \u2194\n  \u2200 U : opens Y.carrier, is_affine_open U \u2192 is_compact (f.1.base \u207b\u00b9' (U : set Y.carrier)) :=\nbegin\n  rw quasi_compact_iff,\n  refine \u27e8\u03bb H U hU, H U U.prop hU.is_compact, _\u27e9,\n  intros H U hU hU',\n  obtain \u27e8S, hS, rfl\u27e9 := (is_compact_open_iff_eq_finset_affine_union U).mp \u27e8hU', hU\u27e9,\n  simp only [set.preimage_Union, subtype.val_eq_coe],\n  exact hS.is_compact_bUnion (\u03bb i _, H i i.prop)\nend\n\n@[simp] lemma quasi_compact.affine_property_to_property {X Y : Scheme} (f : X \u27f6 Y) :\n  (quasi_compact.affine_property : _).to_property f \u2194\n    is_affine Y \u2227 compact_space X.carrier :=\nby { delta affine_target_morphism_property.to_property quasi_compact.affine_property, simp }\n\nlemma quasi_compact_iff_affine_property :\n  quasi_compact f \u2194 target_affine_locally quasi_compact.affine_property f :=\nbegin\n  rw quasi_compact_iff_forall_affine,\n  transitivity (\u2200 U : Y.affine_opens, is_compact (f.1.base \u207b\u00b9' (U : set Y.carrier))),\n  { exact \u27e8\u03bb h U, h U U.prop, \u03bb h U hU, h \u27e8U, hU\u27e9\u27e9 },\n  apply forall_congr,\n  exact \u03bb _, is_compact_iff_compact_space,\nend\n\nlemma quasi_compact_eq_affine_property :\n  @quasi_compact = target_affine_locally quasi_compact.affine_property :=\nby { ext, exact quasi_compact_iff_affine_property _ }\n\nlemma is_compact_basic_open (X : Scheme) {U : opens X.carrier} (hU : is_compact (U : set X.carrier))\n   (f : X.presheaf.obj (op U)) : is_compact (X.basic_open f : set X.carrier) :=\nbegin\n  classical,\n  refine ((is_compact_open_iff_eq_finset_affine_union _).mpr _).1,\n  obtain \u27e8s, hs, e\u27e9 := (is_compact_open_iff_eq_finset_affine_union _).mp \u27e8hU, U.prop\u27e9,\n  let g : s \u2192 X.affine_opens,\n  { intro V,\n    use V.1 \u2293 X.basic_open f,\n    have : V.1.1 \u27f6 U,\n    { apply hom_of_le, change _ \u2286 (U : set X.carrier), rw e,\n      convert @set.subset_Union\u2082 _ _ _ (\u03bb (U : X.affine_opens) (h : U \u2208 s), \u2191U) V V.prop using 1,\n      refl },\n    erw \u2190 X.to_LocallyRingedSpace.to_RingedSpace.basic_open_res this.op,\n    exact is_affine_open.basic_open_is_affine V.1.prop _ },\n  haveI : finite s := hs.to_subtype,\n  refine \u27e8set.range g, set.finite_range g, _\u27e9,\n  refine (set.inter_eq_right_iff_subset.mpr (RingedSpace.basic_open_le _ _)).symm.trans _,\n  rw [e, set.Union\u2082_inter],\n  apply le_antisymm; apply set.Union\u2082_subset,\n  { intros i hi,\n    refine set.subset.trans _ (set.subset_Union\u2082 _ (set.mem_range_self \u27e8i, hi\u27e9)),\n    exact set.subset.rfl },\n  { rintro \u27e8i, hi\u27e9 \u27e8\u27e8j, hj\u27e9, hj'\u27e9,\n    rw \u2190 hj',\n    refine set.subset.trans _ (set.subset_Union\u2082 j hj),\n    exact set.subset.rfl }\nend\n\nlemma quasi_compact.affine_property_is_local :\n  (quasi_compact.affine_property : _).is_local :=\nbegin\n  split,\n  { apply affine_target_morphism_property.respects_iso_mk; rintros X Y Z _ _ _ H,\n    exacts [@@homeomorph.compact_space _ _ H (Top.homeo_of_iso (as_iso e.inv.1.base)), H] },\n  { introv H,\n    delta quasi_compact.affine_property at H \u22a2,\n    change compact_space ((opens.map f.val.base).obj (Y.basic_open r)),\n    rw Scheme.preimage_basic_open f r,\n    erw \u2190 is_compact_iff_compact_space,\n    rw \u2190 is_compact_univ_iff at H,\n    exact is_compact_basic_open X H _ },\n  { rintros X Y H f S hS hS',\n    resetI,\n    rw \u2190 is_affine_open.basic_open_union_eq_self_iff at hS,\n    delta quasi_compact.affine_property,\n    rw \u2190 is_compact_univ_iff,\n    change is_compact ((opens.map f.val.base).obj \u22a4).1,\n    rw \u2190 hS,\n    dsimp [opens.map],\n    simp only [opens.coe_supr, set.preimage_Union, subtype.val_eq_coe],\n    exacts [is_compact_Union (\u03bb i, is_compact_iff_compact_space.mpr (hS' i)),\n      top_is_affine_open _] }\nend\n\nlemma quasi_compact.affine_open_cover_tfae {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [quasi_compact f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)],\n      \u2200 (i : \ud835\udcb0.J), compact_space (pullback f (\ud835\udcb0.map i)).carrier,\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J),\n      compact_space (pullback f (\ud835\udcb0.map i)).carrier,\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_affine U] [is_open_immersion g],\n      compact_space (pullback f g).carrier,\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4) (hU' : \u2200 i, is_affine_open (U i)),\n      \u2200 i, compact_space (f.1.base \u207b\u00b9' (U i).1)] :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.affine_open_cover_tfae f\n\nlemma quasi_compact.is_local_at_target :\n  property_is_local_at_target @quasi_compact :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.target_affine_locally_is_local\n\nlemma quasi_compact.open_cover_tfae {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [quasi_compact f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y), \u2200 (i : \ud835\udcb0.J),\n      quasi_compact (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) (i : \ud835\udcb0.J),\n      quasi_compact (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (U : opens Y.carrier), quasi_compact (f \u2223_ U),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_open_immersion g],\n      quasi_compact (pullback.snd : pullback f g \u27f6 _),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4), \u2200 i, quasi_compact (f \u2223_ (U i))] :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.target_affine_locally_is_local.open_cover_tfae f\n\nlemma quasi_compact_over_affine_iff {X Y : Scheme} (f : X \u27f6 Y) [is_affine Y] :\n  quasi_compact f \u2194 compact_space X.carrier :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.affine_target_iff f\n\nlemma compact_space_iff_quasi_compact (X : Scheme) :\n  compact_space X.carrier \u2194 quasi_compact (terminal.from X) :=\n(quasi_compact_over_affine_iff _).symm\n\nlemma quasi_compact.affine_open_cover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.open_cover.{u} Y)\n  [\u2200 i, is_affine (\ud835\udcb0.obj i)] (f : X \u27f6 Y) :\n  quasi_compact f \u2194 \u2200 i, compact_space (pullback f (\ud835\udcb0.map i)).carrier :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.affine_open_cover_iff f \ud835\udcb0\n\nlemma quasi_compact.open_cover_iff {X Y : Scheme.{u}} (\ud835\udcb0 : Scheme.open_cover.{u} Y) (f : X \u27f6 Y) :\n  quasi_compact f \u2194 \u2200 i, quasi_compact (pullback.snd : pullback f (\ud835\udcb0.map i) \u27f6 _) :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.target_affine_locally_is_local.open_cover_iff f \ud835\udcb0\n\nlemma quasi_compact_respects_iso : morphism_property.respects_iso @quasi_compact :=\nquasi_compact_eq_affine_property.symm \u25b8\n  target_affine_locally_respects_iso quasi_compact.affine_property_is_local.1\n\nlemma quasi_compact_stable_under_composition :\n  morphism_property.stable_under_composition @quasi_compact :=\n\u03bb _ _ _ _ _ _ _, by exactI infer_instance\n\nlocal attribute [-simp] PresheafedSpace.as_coe SheafedSpace.as_coe\n\nlemma quasi_compact.affine_property_stable_under_base_change :\n  quasi_compact.affine_property.stable_under_base_change :=\nbegin\n  intros X Y S _ _ f g h,\n  rw quasi_compact.affine_property at h \u22a2,\n  resetI,\n  let \ud835\udcb0 := Scheme.pullback.open_cover_of_right Y.affine_cover.finite_subcover f g,\n  haveI : finite \ud835\udcb0.J,\n  { dsimp [\ud835\udcb0], apply_instance },\n  haveI : \u2200 i, compact_space (\ud835\udcb0.obj i).carrier,\n  { intro i, dsimp, apply_instance },\n  exact \ud835\udcb0.compact_space,\nend\n\nlemma quasi_compact_stable_under_base_change :\n  morphism_property.stable_under_base_change @quasi_compact :=\nquasi_compact_eq_affine_property.symm \u25b8\n  quasi_compact.affine_property_is_local.stable_under_base_change\n    quasi_compact.affine_property_stable_under_base_change\n\nvariables {Z : Scheme.{u}}\n\ninstance (f : X \u27f6 Z) (g : Y \u27f6 Z) [quasi_compact g] :\n  quasi_compact (pullback.fst : pullback f g \u27f6 X) :=\nquasi_compact_stable_under_base_change.fst f g infer_instance\n\ninstance (f : X \u27f6 Z) (g : Y \u27f6 Z) [quasi_compact f] :\n  quasi_compact (pullback.snd : pullback f g \u27f6 Y) :=\nquasi_compact_stable_under_base_change.snd f g infer_instance\n\n@[elab_as_eliminator]\nlemma compact_open_induction_on {P : opens X.carrier \u2192 Prop} (S : opens X.carrier)\n  (hS : is_compact S.1)\n  (h\u2081 : P \u22a5)\n  (h\u2082 : \u2200 (S : opens X.carrier) (hS : is_compact S.1) (U : X.affine_opens), P S \u2192 P (S \u2294 U)) :\n    P S :=\nbegin\n  classical,\n  obtain \u27e8s, hs, hs'\u27e9 := (is_compact_open_iff_eq_finset_affine_union S.1).mp \u27e8hS, S.2\u27e9,\n  replace hs' : S = supr (\u03bb i : s, (i : opens X.carrier)) := by { ext1, simpa using hs' },\n  subst hs',\n  apply hs.induction_on,\n  { convert h\u2081, rw supr_eq_bot, rintro \u27e8_, h\u27e9, exact h.elim },\n  { intros x s h\u2083 hs h\u2084,\n    have : is_compact (\u2a06 i : s, (i : opens X.carrier)).1,\n    { refine ((is_compact_open_iff_eq_finset_affine_union _).mpr _).1, exact \u27e8s, hs, by simp\u27e9 },\n    convert h\u2082 _ this x h\u2084,\n    simp only [coe_coe],\n    rw [supr_subtype, sup_comm],\n    conv_rhs { rw supr_subtype },\n    exact supr_insert }\nend\n\nlemma exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open (X : Scheme)\n  {U : opens X.carrier} (hU : is_affine_open U) (x f : X.presheaf.obj (op U))\n  (H : x |_ X.basic_open f = 0) :\n  \u2203 n : \u2115, f ^ n * x = 0 :=\nbegin\n  rw \u2190 map_zero (X.presheaf.map (hom_of_le $ X.basic_open_le f : X.basic_open f \u27f6 U).op) at H,\n  have := (is_localization_basic_open hU f).3,\n  obtain \u27e8\u27e8_, n, rfl\u27e9, e\u27e9 := this.mp H,\n  exact \u27e8n, by simpa [mul_comm x] using e\u27e9,\nend\n\n/-- If `x : \u0393(X, U)` is zero on `D(f)` for some `f : \u0393(X, U)`, and `U` is quasi-compact, then\n`f ^ n * x = 0` for some `n`. -/\nlemma exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_compact (X : Scheme)\n  {U : opens X.carrier} (hU : is_compact U.1) (x f : X.presheaf.obj (op U))\n  (H : x |_ X.basic_open f = 0) :\n  \u2203 n : \u2115, f ^ n * x = 0 :=\nbegin\n  obtain \u27e8s, hs, e\u27e9 := (is_compact_open_iff_eq_finset_affine_union U.1).mp \u27e8hU, U.2\u27e9,\n  replace e : U = supr (\u03bb i : s, (i : opens X.carrier)),\n  { ext1, simpa using e },\n  have h\u2081 : \u2200 i : s, i.1.1 \u2264 U,\n  { intro i, change (i : opens X.carrier) \u2264 U, rw e, exact le_supr _ _ },\n  have H' := \u03bb (i : s), exists_pow_mul_eq_zero_of_res_basic_open_eq_zero_of_is_affine_open X i.1.2\n    (X.presheaf.map (hom_of_le (h\u2081 i)).op x) (X.presheaf.map (hom_of_le (h\u2081 i)).op f) _,\n  swap,\n  { delta Top.presheaf.restrict_open Top.presheaf.restrict at H \u22a2,\n    convert congr_arg (X.presheaf.map (hom_of_le _).op) H,\n    { simp only [\u2190 comp_apply, \u2190 functor.map_comp], congr },\n    { rw map_zero },\n    { rw X.basic_open_res, exact set.inter_subset_right _ _ } },\n  choose n hn using H',\n  haveI := hs.to_subtype,\n  casesI nonempty_fintype s,\n  use finset.univ.sup n,\n  suffices : \u2200 (i : s), X.presheaf.map (hom_of_le (h\u2081 i)).op (f ^ (finset.univ.sup n) * x) = 0,\n  { subst e,\n    apply X.sheaf.eq_of_locally_eq (\u03bb (i : s), (i : opens X.carrier)),\n    intro i,\n    rw map_zero,\n    apply this },\n  intro i,\n  replace hn := congr_arg\n    (\u03bb x, X.presheaf.map (hom_of_le (h\u2081 i)).op (f ^ (finset.univ.sup n - n i)) * x) (hn i),\n  dsimp at hn,\n  simp only [\u2190 map_mul, \u2190 map_pow] at hn,\n  rwa [mul_zero, \u2190 mul_assoc, \u2190 pow_add, tsub_add_cancel_of_le] at hn,\n  apply finset.le_sup (finset.mem_univ i)\nend\n\nend algebraic_geometry\n", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/morphisms/quasi_compact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.35919391285530444}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport order.category.Lattice\n\n/-!\n# The category of distributive lattices\n\nThis file defines `DistribLattice`, the category of distributive lattices.\n\nNote that [`DistLat`](https://ncatlab.org/nlab/show/DistLat) in the literature doesn't always\ncorrespond to `DistribLattice` as we don't require bottom or top elements. Instead, this `DistLat`\ncorresponds to `BoundedDistribLattice`.\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- The category of distributive lattices. -/\ndef DistribLattice := bundled distrib_lattice\n\nnamespace DistribLattice\n\ninstance : has_coe_to_sort DistribLattice Type* := bundled.has_coe_to_sort\ninstance (X : DistribLattice) : distrib_lattice X := X.str\n\n/-- Construct a bundled `DistribLattice` from a `distrib_lattice` underlying type and typeclass. -/\ndef of (\u03b1 : Type*) [distrib_lattice \u03b1] : DistribLattice := bundled.of \u03b1\n\n@[simp] lemma coe_of (\u03b1 : Type*) [distrib_lattice \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited DistribLattice := \u27e8of punit\u27e9\n\ninstance : bundled_hom.parent_projection @distrib_lattice.to_lattice := \u27e8\u27e9\n\nattribute [derive [large_category, concrete_category]] DistribLattice\n\ninstance has_forget_to_Lattice : has_forget\u2082 DistribLattice Lattice := bundled_hom.forget\u2082 _ _\n\n/-- Constructs an equivalence between distributive lattices from an order isomorphism between them.\n-/\n@[simps] def iso.mk {\u03b1 \u03b2 : DistribLattice.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : DistribLattice \u2964 DistribLattice :=\n{ obj := \u03bb X, of (order_dual X), map := \u03bb X Y, lattice_hom.dual }\n\n/-- The equivalence between `DistribLattice` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : DistribLattice \u224c DistribLattice :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend DistribLattice\n\nlemma DistribLattice_dual_comp_forget_to_Lattice :\n  DistribLattice.dual \u22d9 forget\u2082 DistribLattice Lattice =\n    forget\u2082 DistribLattice Lattice \u22d9 Lattice.dual := rfl\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/order/category/DistribLattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6477982315512488, "lm_q1q2_score": 0.35918498664876347}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n\n! This file was ported from Lean 3 source module data.hash_map\n! leanprover-community/mathlib commit 4fcbc82dc2257986c03e113f87bc5ce021243a44\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Data.Array.Lemmas\nimport Mathbin.Data.List.Join\nimport Mathbin.Data.List.Range\nimport Mathbin.Data.Pnat.Defs\n\n/-!\n# Hash maps\n\nDefines a hash map data structure, representing a finite key-value map\nwith a value type that may depend on the key type.  The structure\nrequires a `nat`-valued hash function to associate keys to buckets.\n\n## Main definitions\n\n* `hash_map`: constructed with `mk_hash_map`.\n\n## Implementation details\n\nA hash map with key type `\u03b1` and (dependent) value type `\u03b2 : \u03b1 \u2192 Type*`\nconsists of an array of *buckets*, which are lists containing\nkey/value pairs for that bucket.  The hash function is taken modulo `n`\nto assign keys to their respective bucket.  Because of this, some care\nshould be put into the hash function to ensure it evenly distributes\nkeys.\n\nThe bucket array is an `array`.  These have special VM support for\nin-place modification if there is only ever one reference to them.  If\none takes special care to never keep references to old versions of a\nhash map alive after updating it, then the hash map will be modified\nin-place.  In this documentation, when we say a hash map is modified\nin-place, we are assuming the API is being used in this manner.\n\nWhen inserting (`hash_map.insert`), if the number of stored pairs (the\n*size*) is going to exceed the number of buckets, then a new hash map\nis first created with double the number of buckets and everything in\nthe old hash map is reinserted along with the new key/value pair.\nOtherwise, the bucket array is modified in-place.  The amortized\nrunning time of inserting $$n$$ elements into a hash map is $$O(n)$$.\n\nWhen removing (`hash_map.erase`), the hash map is modified in-place.\nThe implementation does not reduce the number of buckets in the hash\nmap if the size gets too low.\n\n## Tags\n\nhash map\n\n-/\n\n\nuniverse u v w\n\n/-- `bucket_array \u03b1 \u03b2` is the underlying data type for `hash_map \u03b1 \u03b2`,\n  an array of linked lists of key-value pairs. -/\ndef BucketArray (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) (n : \u2115+) :=\n  Array' n (List (\u03a3a, \u03b2 a))\n#align bucket_array BucketArray\n\n/-- Make a hash_map index from a `nat` hash value and a (positive) buffer size -/\ndef HashMap.mkIdx (n : \u2115+) (i : Nat) : Fin n :=\n  \u27e8i % n, Nat.mod_lt _ n.2\u27e9\n#align hash_map.mk_idx HashMap.mkIdx\n\nnamespace BucketArray\n\nsection\n\nparameter {\u03b1 : Type u}{\u03b2 : \u03b1 \u2192 Type v}(hash_fn : \u03b1 \u2192 Nat)\n\nvariable {n : \u2115+} (data : BucketArray \u03b1 \u03b2 n)\n\ninstance : Inhabited (BucketArray \u03b1 \u03b2 n) :=\n  \u27e8mkArray' _ []\u27e9\n\n/-- Read the bucket corresponding to an element -/\ndef read (a : \u03b1) : List (\u03a3a, \u03b2 a) :=\n  let bidx := HashMap.mkIdx n (hash_fn a)\n  data.read bidx\n#align bucket_array.read BucketArray.read\n\n/-- Write the bucket corresponding to an element -/\ndef write (a : \u03b1) (l : List (\u03a3a, \u03b2 a)) : BucketArray \u03b1 \u03b2 n :=\n  let bidx := HashMap.mkIdx n (hash_fn a)\n  data.write bidx l\n#align bucket_array.write BucketArray.write\n\n/-- Modify (read, apply `f`, and write) the bucket corresponding to an element -/\ndef modify (a : \u03b1) (f : List (\u03a3a, \u03b2 a) \u2192 List (\u03a3a, \u03b2 a)) : BucketArray \u03b1 \u03b2 n :=\n  let bidx := HashMap.mkIdx n (hash_fn a)\n  Array'.write data bidx (f (Array'.read data bidx))\n#align bucket_array.modify BucketArray.modify\n\n/-- The list of all key-value pairs in the bucket list -/\ndef asList : List (\u03a3a, \u03b2 a) :=\n  data.toList.join\n#align bucket_array.as_list BucketArray.asList\n\ntheorem mem_asList {a : \u03a3a, \u03b2 a} : a \u2208 data.asList \u2194 \u2203 i, a \u2208 Array'.read data i :=\n  by\n  have :\n    (\u2203 (l : List (\u03a3a : \u03b1, \u03b2 a))(i : Fin n.val), a \u2208 l \u2227 Array'.read data i = l) \u2194\n      \u2203 i : Fin n.val, a \u2208 Array'.read data i :=\n    by rw [exists_swap] <;> exact exists_congr fun i => by simp\n  simp [as_list] <;> simpa [Array'.Mem.def, and_comm']\n#align bucket_array.mem_as_list BucketArray.mem_asList\n\n/-- Fold a function `f` over the key-value pairs in the bucket list -/\ndef foldl {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u2200 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\n  data.foldl d fun b d => b.foldl (fun r a => f r a.1 a.2) d\n#align bucket_array.foldl BucketArray.foldl\n\ntheorem foldl_eq {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u2200 a, \u03b2 a \u2192 \u03b4) :\n    data.foldl d f = data.asList.foldl (fun r a => f r a.1 a.2) d := by\n  rw [foldl, as_list, List.foldl_join, \u2190 Array'.toList_foldl]\n#align bucket_array.foldl_eq BucketArray.foldl_eq\n\nend\n\nend BucketArray\n\nnamespace HashMap\n\nsection\n\nparameter {\u03b1 : Type u}{\u03b2 : \u03b1 \u2192 Type v}(hash_fn : \u03b1 \u2192 Nat)\n\n/-- Insert the pair `\u27e8a, b\u27e9` into the correct location in the bucket array\n  (without checking for duplication) -/\ndef reinsertAux {n} (data : BucketArray \u03b1 \u03b2 n) (a : \u03b1) (b : \u03b2 a) : BucketArray \u03b1 \u03b2 n :=\n  data.modify hash_fn a fun l => \u27e8a, b\u27e9 :: l\n#align hash_map.reinsert_aux HashMap.reinsertAux\n\ntheorem mk_asList (n : \u2115+) : BucketArray.asList (mkArray' n [] : BucketArray \u03b1 \u03b2 n) = [] :=\n  List.eq_nil_iff_forall_not_mem.mpr fun x m =>\n    let \u27e8i, h\u27e9 := (BucketArray.mem_asList _).1 m\n    h\n#align hash_map.mk_as_list HashMap.mk_asList\n\nparameter [DecidableEq \u03b1]\n\n/-- Search a bucket for a key `a` and return the value -/\ndef findAux (a : \u03b1) : List (\u03a3a, \u03b2 a) \u2192 Option (\u03b2 a)\n  | [] => none\n  | \u27e8a', b\u27e9 :: t => if h : a' = a then some (Eq.recOn h b) else find_aux t\n#align hash_map.find_aux HashMap.findAux\n\ntheorem findAux_iff {a : \u03b1} {b : \u03b2 a} :\n    \u2200 {l : List (\u03a3a, \u03b2 a)}, (l.map Sigma.fst).Nodup \u2192 (find_aux a l = some b \u2194 Sigma.mk a b \u2208 l)\n  | [], nd => \u27e8fun n => by injection n, False.elim\u27e9\n  | \u27e8a', b'\u27e9 :: t, nd => by\n    by_cases a' = a\n    \u00b7 clear find_aux_iff\n      subst h\n      suffices b' = b \u2194 b' = b \u2228 Sigma.mk a' b \u2208 t by simpa [find_aux, eq_comm]\n      refine' (or_iff_left_of_imp fun m => _).symm\n      have : a' \u2209 t.map Sigma.fst := nd.not_mem\n      exact this.elim (List.mem_map_of_mem Sigma.fst m)\n    \u00b7 have : Sigma.mk a b \u2260 \u27e8a', b'\u27e9 := by\n        intro e\n        injection e with e\n        exact h e.symm\n      simp at nd\n      simp [find_aux, h, Ne.symm h, find_aux_iff, nd]\n#align hash_map.find_aux_iff HashMap.findAux_iff\n\n/-- Returns `tt` if the bucket `l` contains the key `a` -/\ndef containsAux (a : \u03b1) (l : List (\u03a3a, \u03b2 a)) : Bool :=\n  (find_aux a l).isSome\n#align hash_map.contains_aux HashMap.containsAux\n\ntheorem containsAux_iff {a : \u03b1} {l : List (\u03a3a, \u03b2 a)} (nd : (l.map Sigma.fst).Nodup) :\n    contains_aux a l \u2194 a \u2208 l.map Sigma.fst :=\n  by\n  unfold contains_aux\n  cases' h : find_aux a l with b <;> simp\n  \u00b7 intro (b : \u03b2 a)(m : Sigma.mk a b \u2208 l)\n    rw [(find_aux_iff nd).2 m] at h\n    contradiction\n  \u00b7 show \u2203 b : \u03b2 a, Sigma.mk a b \u2208 l\n    exact \u27e8_, (find_aux_iff nd).1 h\u27e9\n#align hash_map.contains_aux_iff HashMap.containsAux_iff\n\n/-- Modify a bucket to replace a value in the list. Leaves the list\n unchanged if the key is not found. -/\ndef replaceAux (a : \u03b1) (b : \u03b2 a) : List (\u03a3a, \u03b2 a) \u2192 List (\u03a3a, \u03b2 a)\n  | [] => []\n  | \u27e8a', b'\u27e9 :: t => if a' = a then \u27e8a, b\u27e9 :: t else \u27e8a', b'\u27e9 :: replace_aux t\n#align hash_map.replace_aux HashMap.replaceAux\n\n/-- Modify a bucket to remove a key, if it exists. -/\ndef eraseAux (a : \u03b1) : List (\u03a3a, \u03b2 a) \u2192 List (\u03a3a, \u03b2 a)\n  | [] => []\n  | \u27e8a', b'\u27e9 :: t => if a' = a then t else \u27e8a', b'\u27e9 :: erase_aux t\n#align hash_map.erase_aux HashMap.eraseAux\n\n/-- The predicate `valid bkts sz` means that `bkts` satisfies the `hash_map`\n  invariants: There are exactly `sz` elements in it, every pair is in the\n  bucket determined by its key and the hash function, and no key appears\n  multiple times in the list. -/\nstructure Valid {n} (bkts : BucketArray \u03b1 \u03b2 n) (sz : Nat) : Prop where\n  len : bkts.asList.length = sz\n  idx : \u2200 {i} {a : \u03a3a, \u03b2 a}, a \u2208 Array'.read bkts i \u2192 mkIdx n (hash_fn a.1) = i\n  Nodup : \u2200 i, ((Array'.read bkts i).map Sigma.fst).Nodup\n#align hash_map.valid HashMap.Valid\n\ntheorem Valid.idx_enum {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz : Nat} (v : valid bkts sz) {i l}\n    (he : (i, l) \u2208 bkts.toList.enum) {a} {b : \u03b2 a} (hl : Sigma.mk a b \u2208 l) :\n    \u2203 h, mkIdx n (hash_fn a) = \u27e8i, h\u27e9 :=\n  (Array'.mem_toList_enum.mp he).imp fun h e => by subst e <;> exact v.idx hl\n#align hash_map.valid.idx_enum HashMap.Valid.idx_enum\n\ntheorem Valid.idx_enum_1 {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz : Nat} (v : valid bkts sz) {i l}\n    (he : (i, l) \u2208 bkts.toList.enum) {a} {b : \u03b2 a} (hl : Sigma.mk a b \u2208 l) :\n    (mkIdx n (hash_fn a)).1 = i :=\n  by\n  let \u27e8h, e\u27e9 := v.idx_enum _ he hl\n  rw [e] <;> rfl\n#align hash_map.valid.idx_enum_1 HashMap.Valid.idx_enum_1\n\ntheorem Valid.asList_nodup {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz : Nat} (v : valid bkts sz) :\n    (bkts.asList.map Sigma.fst).Nodup :=\n  by\n  suffices (bkts.to_list.map (List.map Sigma.fst)).Pairwise List.Disjoint\n    by\n    suffices \u2200 l, Array'.Mem l bkts \u2192 (l.map Sigma.fst).Nodup by\n      simpa [BucketArray.asList, List.nodup_join, *]\n    rintro l \u27e8i, rfl\u27e9\n    apply v.nodup\n  rw [\u2190 List.enum_map_snd bkts.to_list, List.pairwise_map', List.pairwise_map']\n  have : (bkts.to_list.enum.map Prod.fst).Nodup := by simp [List.nodup_range]\n  refine' List.Pairwise.imp_of_mem _ ((List.pairwise_map' _).1 this)\n  rw [Prod.forall]\n  intro i l\u2081\n  rw [Prod.forall]\n  intro j l\u2082 me\u2081 me\u2082 ij\n  simp [List.Disjoint]\n  intro a b ml\u2081 b' ml\u2082\n  apply ij\n  rwa [\u2190 v.idx_enum_1 _ me\u2081 ml\u2081, \u2190 v.idx_enum_1 _ me\u2082 ml\u2082]\n#align hash_map.valid.as_list_nodup HashMap.Valid.asList_nodup\n\ntheorem mk_valid (n : \u2115+) : @valid n (mkArray' n []) 0 :=\n  \u27e8by simp [mk_as_list], fun i a h => by cases h, fun i => List.nodup_nil\u27e9\n#align hash_map.mk_valid HashMap.mk_valid\n\ntheorem Valid.findAux_iff {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz : Nat} (v : valid bkts sz) {a : \u03b1}\n    {b : \u03b2 a} : find_aux a (bkts.read hash_fn a) = some b \u2194 Sigma.mk a b \u2208 bkts.asList :=\n  (find_aux_iff (v.Nodup _)).trans <| by\n    rw [bkts.mem_as_list] <;> exact \u27e8fun h => \u27e8_, h\u27e9, fun \u27e8i, h\u27e9 => (v.idx h).symm \u25b8 h\u27e9\n#align hash_map.valid.find_aux_iff HashMap.Valid.findAux_iff\n\ntheorem Valid.containsAux_iff {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz : Nat} (v : valid bkts sz)\n    (a : \u03b1) : contains_aux a (bkts.read hash_fn a) \u2194 a \u2208 bkts.asList.map Sigma.fst := by\n  simp [contains_aux, Option.isSome_iff_exists, v.find_aux_iff hash_fn]\n#align hash_map.valid.contains_aux_iff HashMap.Valid.containsAux_iff\n\nsection\n\nparameter\n  {n :\n    \u2115+}{bkts :\n    BucketArray \u03b1 \u03b2\n      n}{bidx : Fin n}{f : List (\u03a3a, \u03b2 a) \u2192 List (\u03a3a, \u03b2 a)}(u v1 v2 w : List (\u03a3a, \u03b2 a))\n\n-- mathport name: exprL\nlocal notation \"L\" => Array'.read bkts bidx\n\nprivate def bkts' : BucketArray \u03b1 \u03b2 n :=\n  Array'.write bkts bidx (f L)\n#align hash_map.bkts' hash_map.bkts'\n\nvariable (hl : L = u ++ v1 ++ w) (hfl : f L = u ++ v2 ++ w)\n\ninclude hl hfl\n\ntheorem append_of_modify : \u2203 u' w', bkts.asList = u' ++ v1 ++ w' \u2227 bkts'.asList = u' ++ v2 ++ w' :=\n  by\n  unfold BucketArray.asList\n  have h : (bidx : \u2115) < bkts.to_list.length := by simp only [bidx.is_lt, Array'.toList_length]\n  refine' \u27e8(bkts.to_list.take bidx).join ++ u, w ++ (bkts.to_list.drop (bidx + 1)).join, _, _\u27e9\n  \u00b7 conv =>\n      lhs\n      rw [\u2190 List.take_append_drop bidx bkts.to_list, List.drop_eq_get_cons h]\n      simp [hl]\n    simp\n  \u00b7 conv =>\n      lhs\n      rw [bkts', Array'.write_toList, List.set_eq_take_cons_drop _ h]\n      simp [hfl]\n    simp\n#align hash_map.append_of_modify HashMap.append_of_modify\n\nvariable (hvnd : (v2.map Sigma.fst).Nodup)\n  (hal : \u2200 a : \u03a3a, \u03b2 a, a \u2208 v2 \u2192 mkIdx n (hash_fn a.1) = bidx)\n  (djuv : (u.map Sigma.fst).Disjoint (v2.map Sigma.fst))\n  (djwv : (w.map Sigma.fst).Disjoint (v2.map Sigma.fst))\n\ninclude hvnd hal djuv djwv\n\ntheorem Valid.modify {sz : \u2115} (v : valid bkts sz) :\n    v1.length \u2264 sz + v2.length \u2227 valid bkts' (sz + v2.length - v1.length) :=\n  by\n  rcases append_of_modify u v1 v2 w hl hfl with \u27e8u', w', e\u2081, e\u2082\u27e9\n  rw [\u2190 v.len, e\u2081]\n  suffices valid bkts' (u' ++ v2 ++ w').length by\n    simpa [GE.ge, add_comm, add_left_comm, Nat.le_add_right, add_tsub_cancel_left]\n  refine' \u27e8congr_arg _ e\u2082, fun i a => _, fun i => _\u27e9\n  \u00b7 by_cases bidx = i\n    \u00b7 subst i\n      rw [bkts', Array'.read_write, hfl]\n      have := @valid.idx _ _ _ v bidx a\n      simp only [hl, List.mem_append, or_imp] at this\u22a2\n      exact \u27e8\u27e8this.1.1, hal _\u27e9, this.2\u27e9\n    \u00b7 rw [bkts', Array'.read_write_of_ne _ _ h]\n      apply v.idx\n  \u00b7 by_cases bidx = i\n    \u00b7 subst i\n      rw [bkts', Array'.read_write, hfl]\n      have := @valid.nodup _ _ _ v bidx\n      simp [hl, List.nodup_append] at this\n      simp [List.nodup_append, this, hvnd, djuv, djwv.symm]\n    \u00b7 rw [bkts', Array'.read_write_of_ne _ _ h]\n      apply v.nodup\n#align hash_map.valid.modify HashMap.Valid.modify\n\nend\n\ntheorem Valid.replaceAux (a : \u03b1) (b : \u03b2 a) :\n    \u2200 l : List (\u03a3a, \u03b2 a),\n      a \u2208 l.map Sigma.fst \u2192\n        \u2203 (u w : List (\u03a3a, \u03b2 a))(b' : _),\n          l = u ++ [\u27e8a, b'\u27e9] ++ w \u2227 replace_aux a b l = u ++ [\u27e8a, b\u27e9] ++ w\n  | [] => False.elim\n  | \u27e8a', b'\u27e9 :: t => by\n    by_cases e : a' = a\n    \u00b7 subst a'\n      suffices\n        \u2203 (u w : List (\u03a3a, \u03b2 a))(b'' : \u03b2 a),\n          Sigma.mk a b' :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n            replace_aux a b (\u27e8a, b'\u27e9 :: t) = u ++ \u27e8a, b\u27e9 :: w\n        by simpa\n      refine' \u27e8[], t, b', _\u27e9\n      simp [replace_aux]\n    \u00b7 suffices\n        \u2200 (x : \u03b2 a) (_ : Sigma.mk a x \u2208 t),\n          \u2203 (u w : _)(b'' : \u03b2 a),\n            Sigma.mk a' b' :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n              Sigma.mk a' b' :: replace_aux a b t = u ++ \u27e8a, b\u27e9 :: w\n        by simpa [replace_aux, Ne.symm e, e]\n      intro x m\n      have IH :\n        \u2200 (x : \u03b2 a) (_ : Sigma.mk a x \u2208 t),\n          \u2203 (u w : _)(b'' : \u03b2 a), t = u ++ \u27e8a, b''\u27e9 :: w \u2227 replace_aux a b t = u ++ \u27e8a, b\u27e9 :: w :=\n        by simpa using valid.replace_aux t\n      rcases IH x m with \u27e8u, w, b'', hl, hfl\u27e9\n      exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm, Ne.symm e]\u27e9\n#align hash_map.valid.replace_aux HashMap.Valid.replaceAux\n\ntheorem Valid.replace {n : \u2115+} {bkts : BucketArray \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n    (Hc : contains_aux a (bkts.read hash_fn a)) (v : valid bkts sz) :\n    valid (bkts.modify hash_fn a (replace_aux a b)) sz :=\n  by\n  have nd := v.nodup (mk_idx n (hash_fn a))\n  rcases HashMap.Valid.replaceAux a b (Array'.read bkts (mk_idx n (hash_fn a)))\n      ((contains_aux_iff nd).1 Hc) with\n    \u27e8u, w, b', hl, hfl\u27e9\n  simp [hl, List.nodup_append] at nd\n  refine'\n      (v.modify hash_fn u [\u27e8a, b'\u27e9] [\u27e8a, b\u27e9] w hl hfl (List.nodup_singleton _)\n          (fun a' e => by simp at e <;> rw [e]) (fun a' e1 e2 => _) fun a' e1 e2 => _).2 <;>\n    \u00b7 revert e1\n      simp [-Sigma.exists] at e2\n      subst a'\n      simp [nd]\n#align hash_map.valid.replace HashMap.Valid.replace\n\ntheorem Valid.insert {n : \u2115+} {bkts : BucketArray \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n    (Hnc : \u00accontains_aux a (bkts.read hash_fn a)) (v : valid bkts sz) :\n    valid (reinsert_aux bkts a b) (sz + 1) :=\n  by\n  have nd := v.nodup (mk_idx n (hash_fn a))\n  refine'\n    (v.modify hash_fn [] [] [\u27e8a, b\u27e9] (bkts.read hash_fn a) rfl rfl (List.nodup_singleton _)\n        (fun a' e => by simp at e <;> rw [e]) (fun a' => False.elim) fun a' e1 e2 => _).2\n  simp [-Sigma.exists] at e2; subst a'\n  exact Hnc ((contains_aux_iff nd).2 e1)\n#align hash_map.valid.insert HashMap.Valid.insert\n\ntheorem Valid.eraseAux (a : \u03b1) :\n    \u2200 l : List (\u03a3a, \u03b2 a),\n      a \u2208 l.map Sigma.fst \u2192\n        \u2203 (u w : List (\u03a3a, \u03b2 a))(b : _), l = u ++ [\u27e8a, b\u27e9] ++ w \u2227 erase_aux a l = u ++ [] ++ w\n  | [] => False.elim\n  | \u27e8a', b'\u27e9 :: t => by\n    by_cases e : a' = a\n    \u00b7 subst a'\n      simpa [erase_aux, and_comm'] using\n        show \u2203 (u w : _)(x : \u03b2 a), t = u ++ w \u2227 Sigma.mk a b' :: t = u ++ \u27e8a, x\u27e9 :: w from\n          \u27e8[], t, b', by simp\u27e9\n    \u00b7 simp [erase_aux, e, Ne.symm e]\n      suffices\n        \u2200 (b : \u03b2 a) (_ : Sigma.mk a b \u2208 t),\n          \u2203 (u w : _)(x : \u03b2 a),\n            Sigma.mk a' b' :: t = u ++ \u27e8a, x\u27e9 :: w \u2227 Sigma.mk a' b' :: erase_aux a t = u ++ w\n        by simpa [replace_aux, Ne.symm e, e]\n      intro b m\n      have IH :\n        \u2200 (x : \u03b2 a) (_ : Sigma.mk a x \u2208 t),\n          \u2203 (u w : _)(x : \u03b2 a), t = u ++ \u27e8a, x\u27e9 :: w \u2227 erase_aux a t = u ++ w :=\n        by simpa using valid.erase_aux t\n      rcases IH b m with \u27e8u, w, b'', hl, hfl\u27e9\n      exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm]\u27e9\n#align hash_map.valid.erase_aux HashMap.Valid.eraseAux\n\ntheorem Valid.erase {n} {bkts : BucketArray \u03b1 \u03b2 n} {sz} (a : \u03b1)\n    (Hc : contains_aux a (bkts.read hash_fn a)) (v : valid bkts sz) :\n    valid (bkts.modify hash_fn a (erase_aux a)) (sz - 1) :=\n  by\n  have nd := v.nodup (mk_idx n (hash_fn a))\n  rcases HashMap.Valid.eraseAux a (Array'.read bkts (mk_idx n (hash_fn a)))\n      ((contains_aux_iff nd).1 Hc) with\n    \u27e8u, w, b, hl, hfl\u27e9\n  refine' (v.modify hash_fn u [\u27e8a, b\u27e9] [] w hl hfl List.nodup_nil _ _ _).2 <;> simp\n#align hash_map.valid.erase HashMap.Valid.erase\n\nend\n\nend HashMap\n\n/-- A hash map data structure, representing a finite key-value map\n  with key type `\u03b1` and value type `\u03b2` (which may depend on `\u03b1`). -/\nstructure HashMap (\u03b1 : Type u) [DecidableEq \u03b1] (\u03b2 : \u03b1 \u2192 Type v) where\n  hashFn : \u03b1 \u2192 Nat\n  size : \u2115\n  nbuckets : \u2115+\n  buckets : BucketArray \u03b1 \u03b2 nbuckets\n  is_valid : HashMap.Valid hash_fn buckets size\n#align hash_map HashMap\n\n/-- Construct an empty hash map with buffer size `nbuckets` (default 8). -/\ndef mkHashMap {\u03b1 : Type u} [DecidableEq \u03b1] {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 Nat) (nbuckets := 8) :\n    HashMap \u03b1 \u03b2 :=\n  let n := if nbuckets = 0 then 8 else nbuckets\n  let nz : n > 0 := by abstract cases nbuckets <;> simp [if_pos, Nat.succ_ne_zero]\n  { hashFn\n    size := 0\n    nbuckets := \u27e8n, nz\u27e9\n    buckets := mkArray' n []\n    is_valid := HashMap.mk_valid _ _ }\n#align mk_hash_map mkHashMap\n\nnamespace HashMap\n\nvariable {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [DecidableEq \u03b1]\n\n/-- Return the value corresponding to a key, or `none` if not found -/\ndef find (m : HashMap \u03b1 \u03b2) (a : \u03b1) : Option (\u03b2 a) :=\n  findAux a (m.buckets.read m.hashFn a)\n#align hash_map.find HashMap.find\n\n/-- Return `tt` if the key exists in the map -/\ndef contains (m : HashMap \u03b1 \u03b2) (a : \u03b1) : Bool :=\n  (m.find a).isSome\n#align hash_map.contains HashMap.contains\n\ninstance : Membership \u03b1 (HashMap \u03b1 \u03b2) :=\n  \u27e8fun a m => m.contains a\u27e9\n\n/-- Fold a function over the key-value pairs in the map -/\ndef fold {\u03b4 : Type w} (m : HashMap \u03b1 \u03b2) (d : \u03b4) (f : \u03b4 \u2192 \u2200 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\n  m.buckets.foldl d f\n#align hash_map.fold HashMap.fold\n\n/-- The list of key-value pairs in the map -/\ndef entries (m : HashMap \u03b1 \u03b2) : List (\u03a3a, \u03b2 a) :=\n  m.buckets.asList\n#align hash_map.entries HashMap.entries\n\n/-- The list of keys in the map -/\ndef keys (m : HashMap \u03b1 \u03b2) : List \u03b1 :=\n  m.entries.map Sigma.fst\n#align hash_map.keys HashMap.keys\n\ntheorem find_iff (m : HashMap \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) :\n    m.find a = some b \u2194 Sigma.mk a b \u2208 m.entries :=\n  m.is_valid.findAux_iff _\n#align hash_map.find_iff HashMap.find_iff\n\ntheorem contains_iff (m : HashMap \u03b1 \u03b2) (a : \u03b1) : m.contains a \u2194 a \u2208 m.keys :=\n  m.is_valid.containsAux_iff _ _\n#align hash_map.contains_iff HashMap.contains_iff\n\ntheorem entries_empty (hash_fn : \u03b1 \u2192 Nat) (n) : (@mkHashMap \u03b1 _ \u03b2 hash_fn n).entries = [] :=\n  mk_asList _\n#align hash_map.entries_empty HashMap.entries_empty\n\ntheorem keys_empty (hash_fn : \u03b1 \u2192 Nat) (n) : (@mkHashMap \u03b1 _ \u03b2 hash_fn n).keys = [] := by\n  dsimp [keys] <;> rw [entries_empty] <;> rfl\n#align hash_map.keys_empty HashMap.keys_empty\n\ntheorem find_empty (hash_fn : \u03b1 \u2192 Nat) (n a) : (@mkHashMap \u03b1 _ \u03b2 hash_fn n).find a = none := by\n  induction' h : (@mkHashMap \u03b1 _ \u03b2 hash_fn n).find a with <;> [rfl,\n    \u00b7 have := (find_iff _ _ _).1 h\n      rw [entries_empty] at this\n      contradiction]\n#align hash_map.find_empty HashMap.find_empty\n\ntheorem not_contains_empty (hash_fn : \u03b1 \u2192 Nat) (n a) : \u00ac(@mkHashMap \u03b1 _ \u03b2 hash_fn n).contains a :=\n  by apply Bool.bool_iff_false.2 <;> dsimp [contains] <;> rw [find_empty] <;> rfl\n#align hash_map.not_contains_empty HashMap.not_contains_empty\n\ntheorem insert_lemma (hash_fn : \u03b1 \u2192 Nat) {n n'} {bkts : BucketArray \u03b1 \u03b2 n} {sz}\n    (v : Valid hash_fn bkts sz) :\n    Valid hash_fn (bkts.foldl (mkArray' _ [] : BucketArray \u03b1 \u03b2 n') (reinsertAux hash_fn)) sz :=\n  by\n  suffices\n    \u2200 (l : List (\u03a3a, \u03b2 a)) (t : BucketArray \u03b1 \u03b2 n') (sz),\n      valid hash_fn t sz \u2192\n        ((l ++ t.asList).map Sigma.fst).Nodup \u2192\n          valid hash_fn (l.foldl (fun r (a : \u03a3a, \u03b2 a) => reinsert_aux hash_fn r a.1 a.2) t)\n            (sz + l.length)\n    by\n    have p := this bkts.as_list _ _ (mk_valid _ _)\n    rw [mk_as_list, List.append_nil, zero_add, v.len] at p\n    rw [BucketArray.foldl_eq]\n    exact p (v.as_list_nodup _)\n  intro l\n  induction' l with c l IH <;> intro t sz v nd\n  \u00b7 exact v\n  rw [show sz + (c :: l).length = sz + 1 + l.length by simp [add_comm, add_assoc]]\n  rcases show\n      (l.map Sigma.fst).Nodup \u2227\n        ((BucketArray.asList t).map Sigma.fst).Nodup \u2227\n          c.fst \u2209 l.map Sigma.fst \u2227\n            c.fst \u2209 (BucketArray.asList t).map Sigma.fst \u2227\n              (l.map Sigma.fst).Disjoint ((BucketArray.asList t).map Sigma.fst)\n      by simpa [List.nodup_append, not_or, and_comm', and_left_comm] using nd with\n    \u27e8nd1, nd2, nm1, nm2, dj\u27e9\n  have v' := v.insert _ _ c.2 fun Hc => nm2 <| (v.contains_aux_iff _ c.1).1 Hc\n  apply IH _ _ v'\n  suffices\n    \u2200 \u2983a : \u03b1\u2984 (b : \u03b2 a),\n      Sigma.mk a b \u2208 l \u2192 \u2200 b' : \u03b2 a, Sigma.mk a b' \u2208 (reinsert_aux hash_fn t c.1 c.2).asList \u2192 False\n    by simpa [List.nodup_append, nd1, v'.as_list_nodup _, List.Disjoint]\n  intro a b m1 b' m2\n  rcases(reinsert_aux hash_fn t c.1 c.2).mem_asList.1 m2 with \u27e8i, im\u27e9\n  have : Sigma.mk a b' \u2209 Array'.read t i := by\n    intro m3\n    have : a \u2208 List.map Sigma.fst t.as_list :=\n      List.mem_map_of_mem Sigma.fst (t.mem_as_list.2 \u27e8_, m3\u27e9)\n    exact dj (List.mem_map_of_mem Sigma.fst m1) this\n  by_cases h : mk_idx n' (hash_fn c.1) = i\n  \u00b7 subst h\n    have e : Sigma.mk a b' = \u27e8c.1, c.2\u27e9 := by\n      simpa [reinsert_aux, BucketArray.modify, Array'.read_write, this] using im\n    injection e with e\n    subst a\n    exact nm1.elim (@List.mem_map_of_mem _ _ Sigma.fst _ _ m1)\n  \u00b7 apply this\n    simpa [reinsert_aux, BucketArray.modify, Array'.read_write_of_ne _ _ h] using im\n#align hash_map.insert_lemma HashMap.insert_lemma\n\n/-- Insert a key-value pair into the map. (Modifies `m` in-place when applicable) -/\ndef insert : \u2200 (m : HashMap \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a), HashMap \u03b1 \u03b2\n  | \u27e8hash_fn, size, n, buckets, v\u27e9, a, b =>\n    let bkt := buckets.read hash_fn a\n    if hc : containsAux a bkt then\n      { hashFn\n        size\n        nbuckets := n\n        buckets := buckets.modify hash_fn a (replaceAux a b)\n        is_valid := v.replace _ a b hc }\n    else\n      let size' := size + 1\n      let buckets' := buckets.modify hash_fn a fun l => \u27e8a, b\u27e9 :: l\n      let valid' := v.insert _ a b hc\n      if size' \u2264 n then\n        { hashFn\n          size := size'\n          nbuckets := n\n          buckets := buckets'\n          is_valid := valid' }\n      else\n        let n' : \u2115+ := \u27e8n * 2, mul_pos n.2 (by decide)\u27e9\n        let buckets'' : BucketArray \u03b1 \u03b2 n' := buckets'.foldl (mkArray' _ []) (reinsertAux hash_fn)\n        { hashFn\n          size := size'\n          nbuckets := n'\n          buckets := buckets''\n          is_valid := insert_lemma _ valid' }\n#align hash_map.insert HashMap.insert\n\ntheorem mem_insert :\n    \u2200 (m : HashMap \u03b1 \u03b2) (a b a' b'),\n      (Sigma.mk a' b' : Sigma \u03b2) \u2208 (m.insert a b).entries \u2194\n        if a = a' then HEq b b' else Sigma.mk a' b' \u2208 m.entries\n  | \u27e8hash_fn, size, n, bkts, v\u27e9, a, b, a', b' =>\n    by\n    let bkt := bkts.read hash_fn a\n    have nd : (bkt.map Sigma.fst).Nodup := v.nodup (mk_idx n (hash_fn a))\n    have lem :\n      \u2200 (bkts' : BucketArray \u03b1 \u03b2 n) (v1 u w) (hl : BucketArray.asList bkts = u ++ v1 ++ w)\n        (hfl : BucketArray.asList bkts' = u ++ [\u27e8a, b\u27e9] ++ w)\n        (veq : v1 = [] \u2227 \u00accontains_aux a bkt \u2228 \u2203 b'', v1 = [\u27e8a, b''\u27e9]),\n        Sigma.mk a' b' \u2208 bkts'.asList \u2194\n          if a = a' then HEq b b' else Sigma.mk a' b' \u2208 bkts.as_list :=\n      by\n      intro bkts' v1 u w hl hfl veq\n      rw [hl, hfl]\n      by_cases h : a = a'\n      \u00b7 subst a'\n        suffices b = b' \u2228 Sigma.mk a b' \u2208 u \u2228 Sigma.mk a b' \u2208 w \u2194 b = b' by\n          simpa [eq_comm, or_left_comm]\n        refine' or_iff_left_of_imp (Not.elim <| not_or.2 _)\n        rcases veq with (\u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9)\n        \u00b7 have na := (not_congr <| v.contains_aux_iff _ _).1 Hnc\n          simp [hl, not_or] at na\n          simp [na]\n        \u00b7 have nd' := v.as_list_nodup _\n          simp [hl, List.nodup_append] at nd'\n          simp [nd']\n      \u00b7 suffices Sigma.mk a' b' \u2209 v1 by simp [h, Ne.symm h, this]\n        rcases veq with (\u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9) <;> simp [Ne.symm h]\n    by_cases Hc : (contains_aux a bkt : Prop)\n    \u00b7 rcases HashMap.Valid.replaceAux a b (Array'.read bkts (mk_idx n (hash_fn a)))\n          ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b'', hl', hfl'\u27e9\n      rcases append_of_modify u' [\u27e8a, b''\u27e9] [\u27e8a, b\u27e9] w' hl' hfl' with \u27e8u, w, hl, hfl\u27e9\n      simpa [insert, @dif_pos (contains_aux a bkt) _ Hc] using\n        lem _ _ u w hl hfl (Or.inr \u27e8b'', rfl\u27e9)\n    \u00b7 let size' := size + 1\n      let bkts' := bkts.modify hash_fn a fun l => \u27e8a, b\u27e9 :: l\n      have mi :\n        Sigma.mk a' b' \u2208 bkts'.as_list \u2194\n          if a = a' then HEq b b' else Sigma.mk a' b' \u2208 bkts.as_list :=\n        let \u27e8u, w, hl, hfl\u27e9 := append_of_modify [] [] [\u27e8a, b\u27e9] _ rfl rfl\n        lem bkts' _ u w hl hfl <| Or.inl \u27e8rfl, Hc\u27e9\n      simp [insert, @dif_neg (contains_aux a bkt) _ Hc]\n      by_cases h : size' \u2264 n\n      \u00b7 simpa [show size' \u2264 n from h] using mi\n      \u00b7 let n' : \u2115+ := \u27e8n * 2, mul_pos n.2 (by decide)\u27e9\n        let bkts'' : BucketArray \u03b1 \u03b2 n' := bkts'.foldl (mkArray' _ []) (reinsert_aux hash_fn)\n        suffices Sigma.mk a' b' \u2208 bkts''.as_list \u2194 Sigma.mk a' b' \u2208 bkts'.as_list.reverse by\n          simpa [show \u00acsize' \u2264 n from h, mi]\n        rw [show bkts'' = bkts'.as_list.foldl _ _ from bkts'.foldl_eq _ _, \u2190 List.foldr_reverse]\n        induction' bkts'.as_list.reverse with a l IH\n        \u00b7 simp [mk_as_list]\n        \u00b7 cases' a with a'' b''\n          let B :=\n            l.foldr (fun (y : Sigma \u03b2) (x : BucketArray \u03b1 \u03b2 n') => reinsert_aux hash_fn x y.1 y.2)\n              (mkArray' n' [])\n          rcases append_of_modify [] [] [\u27e8a'', b''\u27e9] _ rfl rfl with \u27e8u, w, hl, hfl\u27e9\n          simp [IH.symm, or_left_comm, show B.as_list = _ from hl,\n            show (reinsert_aux hash_fn B a'' b'').asList = _ from hfl]\n#align hash_map.mem_insert HashMap.mem_insert\n\ntheorem find_insert_eq (m : HashMap \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) : (m.insert a b).find a = some b :=\n  (find_iff (m.insert a b) a b).2 <| (mem_insert m a b a b).2 <| by rw [if_pos rfl]\n#align hash_map.find_insert_eq HashMap.find_insert_eq\n\ntheorem find_insert_ne (m : HashMap \u03b1 \u03b2) (a a' : \u03b1) (b : \u03b2 a) (h : a \u2260 a') :\n    (m.insert a b).find a' = m.find a' :=\n  Option.eq_of_eq_some fun b' =>\n    let t := mem_insert m a b a' b'\n    (find_iff _ _ _).trans <| Iff.trans (by rwa [if_neg h] at t) (find_iff _ _ _).symm\n#align hash_map.find_insert_ne HashMap.find_insert_ne\n\ntheorem find_insert (m : HashMap \u03b1 \u03b2) (a' a : \u03b1) (b : \u03b2 a) :\n    (m.insert a b).find a' = if h : a = a' then some (Eq.recOn h b) else m.find a' :=\n  if h : a = a' then by\n    rw [dif_pos h] <;>\n      exact\n        match a', h with\n        | _, rfl => find_insert_eq m a b\n  else by rw [dif_neg h] <;> exact find_insert_ne m a a' b h\n#align hash_map.find_insert HashMap.find_insert\n\n/-- Insert a list of key-value pairs into the map. (Modifies `m` in-place when applicable) -/\ndef insertAll (l : List (\u03a3a, \u03b2 a)) (m : HashMap \u03b1 \u03b2) : HashMap \u03b1 \u03b2 :=\n  l.foldl (fun m \u27e8a, b\u27e9 => insert m a b) m\n#align hash_map.insert_all HashMap.insertAll\n\n/-- Construct a hash map from a list of key-value pairs. -/\ndef ofList (l : List (\u03a3a, \u03b2 a)) (hash_fn) : HashMap \u03b1 \u03b2 :=\n  insertAll l (mkHashMap hash_fn (2 * l.length))\n#align hash_map.of_list HashMap.ofList\n\n/-- Remove a key from the map. (Modifies `m` in-place when applicable) -/\ndef erase (m : HashMap \u03b1 \u03b2) (a : \u03b1) : HashMap \u03b1 \u03b2 :=\n  match m with\n  | \u27e8hash_fn, size, n, buckets, v\u27e9 =>\n    if hc : containsAux a (buckets.read hash_fn a) then\n      { hashFn\n        size := size - 1\n        nbuckets := n\n        buckets := buckets.modify hash_fn a (eraseAux a)\n        is_valid := v.erase\u2093 _ a hc }\n    else m\n#align hash_map.erase HashMap.erase\n\ntheorem mem_erase :\n    \u2200 (m : HashMap \u03b1 \u03b2) (a a' b'),\n      (Sigma.mk a' b' : Sigma \u03b2) \u2208 (m.erase\u2093 a).entries \u2194 a \u2260 a' \u2227 Sigma.mk a' b' \u2208 m.entries\n  | \u27e8hash_fn, size, n, bkts, v\u27e9, a, a', b' =>\n    by\n    let bkt := bkts.read hash_fn a\n    by_cases Hc : (contains_aux a bkt : Prop)\n    \u00b7 let bkts' := bkts.modify hash_fn a (erase_aux a)\n      suffices Sigma.mk a' b' \u2208 bkts'.as_list \u2194 a \u2260 a' \u2227 Sigma.mk a' b' \u2208 bkts.as_list by\n        simpa [erase, @dif_pos (contains_aux a bkt) _ Hc]\n      have nd := v.nodup (mk_idx n (hash_fn a))\n      rcases valid.erase_aux a bkt ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b, hl', hfl'\u27e9\n      rcases append_of_modify u' [\u27e8a, b\u27e9] [] _ hl' hfl' with \u27e8u, w, hl, hfl\u27e9\n      suffices \u2200 _ : Sigma.mk a' b' \u2208 u \u2228 Sigma.mk a' b' \u2208 w, a \u2260 a'\n        by\n        have :\n          Sigma.mk a' b' \u2208 u \u2228 Sigma.mk a' b' \u2208 w \u2194\n            (\u00aca = a' \u2227 a' = a) \u2227 HEq b' b \u2228 \u00aca = a' \u2227 (Sigma.mk a' b' \u2208 u \u2228 Sigma.mk a' b' \u2208 w) :=\n          by simp [eq_comm, not_and_self_iff, and_iff_right_of_imp this]\n        simpa [hl, show bkts'.as_list = _ from hfl, and_or_left, and_comm', and_left_comm,\n          or_left_comm]\n      rintro m rfl\n      revert m\n      apply not_or.2\n      have nd' := v.as_list_nodup _\n      simp [hl, List.nodup_append] at nd'\n      simp [nd']\n    \u00b7 suffices \u2200 _ : Sigma.mk a' b' \u2208 BucketArray.asList bkts, a \u2260 a' by\n        simp [erase, @dif_neg (contains_aux a bkt) _ Hc, entries, and_iff_right_of_imp this]\n      rintro m rfl\n      exact Hc ((v.contains_aux_iff _ _).2 (List.mem_map_of_mem Sigma.fst m))\n#align hash_map.mem_erase HashMap.mem_erase\n\ntheorem find_erase_eq (m : HashMap \u03b1 \u03b2) (a : \u03b1) : (m.erase\u2093 a).find a = none :=\n  by\n  cases' h : (m.erase a).find a with b; \u00b7 rfl\n  exact absurd rfl ((mem_erase m a a b).1 ((find_iff (m.erase a) a b).1 h)).left\n#align hash_map.find_erase_eq HashMap.find_erase_eq\n\ntheorem find_erase_ne (m : HashMap \u03b1 \u03b2) (a a' : \u03b1) (h : a \u2260 a') :\n    (m.erase\u2093 a).find a' = m.find a' :=\n  Option.eq_of_eq_some fun b' =>\n    (find_iff _ _ _).trans <|\n      (mem_erase m a a' b').trans <| (and_iff_right h).trans (find_iff _ _ _).symm\n#align hash_map.find_erase_ne HashMap.find_erase_ne\n\ntheorem find_erase (m : HashMap \u03b1 \u03b2) (a' a : \u03b1) :\n    (m.erase\u2093 a).find a' = if a = a' then none else m.find a' :=\n  if h : a = a' then by subst a' <;> simp [find_erase_eq m a]\n  else by rw [if_neg h] <;> exact find_erase_ne m a a' h\n#align hash_map.find_erase HashMap.find_erase\n\nsection String\n\nvariable [ToString \u03b1] [\u2200 a, ToString (\u03b2 a)]\n\nopen Prod\n\nprivate def key_data_to_string (a : \u03b1) (b : \u03b2 a) (first : Bool) : String :=\n  (if first then \"\" else \", \") ++ s! \"{a } \u2190 {b}\"\n#align hash_map.key_data_to_string hash_map.key_data_to_string\n\nprivate def to_string (m : HashMap \u03b1 \u03b2) : String :=\n  \"\u27e8\" ++ fst (fold m (\"\", true) fun p a b => (fst p ++ keyDataToString a b (snd p), false)) ++ \"\u27e9\"\n#align hash_map.to_string hash_map.to_string\n\ninstance : ToString (HashMap \u03b1 \u03b2) :=\n  \u27e8toString\u27e9\n\nend String\n\nsection Format\n\nopen Format Prod\n\nvariable [has_to_format \u03b1] [\u2200 a, has_to_format (\u03b2 a)]\n\nprivate unsafe def format_key_data (a : \u03b1) (b : \u03b2 a) (first : Bool) : format :=\n  (if first then to_fmt \"\" else to_fmt \",\" ++ line) ++ to_fmt a ++ space ++ to_fmt \"\u2190\" ++ space ++\n    to_fmt b\n#align hash_map.format_key_data hash_map.format_key_data\n\nprivate unsafe def to_format (m : HashMap \u03b1 \u03b2) : format :=\n  Group <|\n    to_fmt \"\u27e8\" ++\n        nest 1\n          (fst\n            (fold m (to_fmt \"\", true) fun p a b =>\n              (fst p ++ format_key_data a b (snd p), false))) ++\n      to_fmt \"\u27e9\"\n#align hash_map.to_format hash_map.to_format\n\nunsafe instance : has_to_format (HashMap \u03b1 \u03b2) :=\n  \u27e8to_format\u27e9\n\nend Format\n\n/-- `hash_map` with key type `nat` and value type that may vary. -/\ninstance {\u03b2 : \u2115 \u2192 Type _} : Inhabited (HashMap \u2115 \u03b2) :=\n  \u27e8mkHashMap id\u27e9\n\nend HashMap\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/HashMap.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3591849791084428}}
{"text": "/-\n-- Copyright (c) 2020 Bhavik Mehta. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Bhavik Mehta, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.limits.shapes.finite_products\nimport Mathlib.category_theory.limits.preserves.shapes.products\nimport Mathlib.category_theory.limits.preserves.shapes.equalizers\nimport Mathlib.PostPort\n\nuniverses u v u\u2082 \n\nnamespace Mathlib\n\n/-!\n# Constructing limits from products and equalizers.\n\nIf a category has all products, and all equalizers, then it has all limits.\nSimilarly, if it has all finite products, and all equalizers, then it has all finite limits.\n\nIf a functor preserves all products and equalizers, then it preserves all limits.\nSimilarly, if it preserves all finite products and equalizers, then it preserves all finite limits.\n\n# TODO\n\nProvide the dual results.\nShow the analogous results for functors which reflect or create (co)limits.\n-/\n\nnamespace category_theory.limits\n\n\n-- We hide the \"implementation details\" inside a namespace\n\nnamespace has_limit_of_has_products_of_has_equalizers\n\n\n/--\n(Implementation) Given the appropriate product and equalizer cones, build the cone for `F` which is\nlimiting if the given cones are also.\n-/\ndef build_limit {C : Type u} [category C] {J : Type v} [small_category J] {F : J \u2964 C} {c\u2081 : fan (functor.obj F)} {c\u2082 : fan fun (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p) => functor.obj F (prod.snd (sigma.fst f))} (s : cone.X c\u2081 \u27f6 cone.X c\u2082) (t : cone.X c\u2081 \u27f6 cone.X c\u2082) (hs : \u2200 (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p),\n  s \u226b nat_trans.app (cone.\u03c0 c\u2082) f = nat_trans.app (cone.\u03c0 c\u2081) (prod.fst (sigma.fst f)) \u226b functor.map F (sigma.snd f)) (ht : \u2200 (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p),\n  t \u226b nat_trans.app (cone.\u03c0 c\u2082) f = nat_trans.app (cone.\u03c0 c\u2081) (prod.snd (sigma.fst f))) (i : fork s t) : cone F :=\n  cone.mk (cone.X i) (nat_trans.mk fun (j : J) => fork.\u03b9 i \u226b nat_trans.app (cone.\u03c0 c\u2081) j)\n\n/--\n(Implementation) Show the cone constructed in `build_limit` is limiting, provided the cones used in\nits construction are.\n-/\ndef build_is_limit {C : Type u} [category C] {J : Type v} [small_category J] {F : J \u2964 C} {c\u2081 : fan (functor.obj F)} {c\u2082 : fan fun (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p) => functor.obj F (prod.snd (sigma.fst f))} (s : cone.X c\u2081 \u27f6 cone.X c\u2082) (t : cone.X c\u2081 \u27f6 cone.X c\u2082) (hs : \u2200 (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p),\n  s \u226b nat_trans.app (cone.\u03c0 c\u2082) f = nat_trans.app (cone.\u03c0 c\u2081) (prod.fst (sigma.fst f)) \u226b functor.map F (sigma.snd f)) (ht : \u2200 (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p),\n  t \u226b nat_trans.app (cone.\u03c0 c\u2082) f = nat_trans.app (cone.\u03c0 c\u2081) (prod.snd (sigma.fst f))) {i : fork s t} (t\u2081 : is_limit c\u2081) (t\u2082 : is_limit c\u2082) (hi : is_limit i) : is_limit (build_limit s t hs ht i) :=\n  is_limit.mk\n    fun (q : cone F) =>\n      is_limit.lift hi (fork.of_\u03b9 (is_limit.lift t\u2081 (fan.mk (cone.X q) fun (j : J) => nat_trans.app (cone.\u03c0 q) j)) sorry)\n\nend has_limit_of_has_products_of_has_equalizers\n\n\n/--\nGiven the existence of the appropriate (possibly finite) products and equalizers, we know a limit of\n`F` exists.\n(This assumes the existence of all equalizers, which is technically stronger than needed.)\n-/\ntheorem has_limit_of_equalizer_and_product {C : Type u} [category C] {J : Type v} [small_category J] (F : J \u2964 C) [has_limit (discrete.functor (functor.obj F))] [has_limit\n  (discrete.functor\n    fun (f : sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p) => functor.obj F (prod.snd (sigma.fst f)))] [has_equalizers C] : has_limit F := sorry\n\n/--\nAny category with products and equalizers has all limits.\n\nSee https://stacks.math.columbia.edu/tag/002N.\n-/\ntheorem limits_from_equalizers_and_products {C : Type u} [category C] [has_products C] [has_equalizers C] : has_limits C :=\n  has_limits.mk\n    fun (J : Type v) (\ud835\udca5 : small_category J) =>\n      has_limits_of_shape.mk fun (F : J \u2964 C) => has_limit_of_equalizer_and_product F\n\n/--\nAny category with finite products and equalizers has all finite limits.\n\nSee https://stacks.math.columbia.edu/tag/002O.\n-/\ntheorem finite_limits_from_equalizers_and_finite_products {C : Type u} [category C] [has_finite_products C] [has_equalizers C] : has_finite_limits C :=\n  fun (J : Type v) (_x : small_category J) (_x_1 : fin_category J) =>\n    has_limits_of_shape.mk fun (F : J \u2964 C) => has_limit_of_equalizer_and_product F\n\n/-- If a functor preserves equalizers and the appropriate products, it preserves limits. -/\ndef preserves_limit_of_preserves_equalizers_and_product {C : Type u} [category C] {J : Type v} [small_category J] {D : Type u\u2082} [category D] [has_limits_of_shape (discrete J) C] [has_limits_of_shape (discrete (sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p)) C] [has_equalizers C] (G : C \u2964 D) [preserves_limits_of_shape walking_parallel_pair G] [preserves_limits_of_shape (discrete J) G] [preserves_limits_of_shape (discrete (sigma fun (p : J \u00d7 J) => prod.fst p \u27f6 prod.snd p)) G] : preserves_limits_of_shape J G := sorry\n\n/-- If G preserves equalizers and finite products, it preserves finite limits. -/\ndef preserves_finite_limits_of_preserves_equalizers_and_finite_products {C : Type u} [category C] {D : Type u\u2082} [category D] [has_equalizers C] [has_finite_products C] (G : C \u2964 D) [preserves_limits_of_shape walking_parallel_pair G] [(J : Type v) \u2192 [_inst_8 : fintype J] \u2192 preserves_limits_of_shape (discrete J) G] (J : Type v) [small_category J] [fin_category J] : preserves_limits_of_shape J G :=\n  preserves_limit_of_preserves_equalizers_and_product G\n\n/-- If G preserves equalizers and products, it preserves all limits. -/\ndef preserves_limits_of_preserves_equalizers_and_products {C : Type u} [category C] {D : Type u\u2082} [category D] [has_equalizers C] [has_products C] (G : C \u2964 D) [preserves_limits_of_shape walking_parallel_pair G] [(J : Type v) \u2192 preserves_limits_of_shape (discrete J) G] : preserves_limits G :=\n  preserves_limits.mk fun (J : Type v) (\ud835\udca5 : small_category J) => preserves_limit_of_preserves_equalizers_and_product G\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/constructions/limits_of_products_and_equalizers.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3591849791084428}}
{"text": "/-\nCopyright (c) 2022 James Gallicchio.\n\nAuthors: James Gallicchio\n-/\n\nimport LeanColls.Classes\nimport LeanColls.Range\nimport LeanColls.View\nimport LeanColls.List\n\nnamespace LeanColls\n\nnamespace IndexedOps\n\ninstance [Indexed C \u03c4] : Membership \u03c4 C where\n  mem x c := \u2203 i, x = Indexed.nth c i\n\ninstance [Indexed C \u03c4] : Iterable C \u03c4 where\n  \u03c1 := \u03a3' (c : C), Iterable'.\u03c1 Nat (Range.mk <| Size.size c)\n  toIterator c := \u27e8c, Iterable'.toIterator (Range.mk <| Size.size c)\u27e9\n  step := \u03bb \u27e8c,r\u27e9 => Iterable'.step r\n                    |>.map (fun (\u27e8i,h\u27e9,r) => (Indexed.nth c \u27e8i,h\u27e9, \u27e8c,r\u27e9))\n\ntheorem size_pos_iff_mem [Indexed C \u03c4] {c : C}\n  : Size.size c > 0 \u2194 \u2203 x, x \u2208 c\n  := by\n  constructor\n  case mp =>\n    intro h\n    apply Exists.intro (Indexed.nth c \u27e80,h\u27e9)\n    simp [Membership.mem]\n    apply Exists.intro \u27e80,h\u27e9\n    rfl\n  case mpr =>\n    intro h\n    cases h; case intro x h =>\n    cases h; case intro i _ =>\n    exact i.size_positive\n\ninstance [Indexed C \u03c4] : Foldable'.Correct C \u03c4 inferInstance where\n  fold c f init :=\n    Foldable'.Correct.fold' (Range.mk <| Size.size c) (\u03bb acc i h =>\n      f acc (Indexed.nth c \u27e8i,h\u27e9)\n    ) init\n  fold' c f init :=\n    Foldable'.Correct.fold' (Range.mk <| Size.size c) (\u03bb acc i h =>\n      f acc (Indexed.nth c \u27e8i,h\u27e9) \u27e8\u27e8i,h\u27e9, rfl\u27e9\n    ) init\n  foldCorrect := by\n    intro \u03b2 c f acc\n    simp [Foldable.fold]\n    rw [Foldable'.fold_canonicalToList_fold'_eq_fold']\n  memCorrect x c := by\n    simp [Foldable.fold, canonicalToList]\n    constructor\n    case mp =>\n      intro h; simp at h; cases h; case intro i h =>\n      have := Foldable'.fold'_append_singleton_eq_map\n        (\u27e8Size.size c\u27e9 : Range)\n        (fun i h => Indexed.nth c \u27e8i,h\u27e9)\n      simp [Membership.mem] at this \u22a2\n      simp [this, Foldable.fold]\n      rw [List.map'_rw _ (Range.toList_eq_canonicalToList \u27e8Size.size c\u27e9).symm]\n      rw [List.map']\n      apply (List.mem_of_map_iff _ _ _).mpr\n      simp\n      cases i; case mk i h_i =>\n      have : i \u2208 Range.toList \u27e8Size.size c\u27e9 := by\n          rw [Range.toList_eq_canonicalToList]\n          apply (Range.memCorrect _ _).mp\n          assumption\n      apply Exists.intro i\n      apply Exists.intro this\n      simp\n      constructor\n      case left =>\n        simp [List.mem_of_subtypeByMem _ i this]\n      case right =>\n        exact h.symm\n    case mpr =>\n      intro h\n      rw [Foldable'.Correct.fold'Correct] at h\n      rw [List.foldl'_eq_subtypeByMem_foldl] at h\n      rw [List.foldl_eq_map] at h\n      rw [(List.mem_of_map_iff _ _ _)] at h\n      cases h; case intro x h =>\n      cases x; case mk i h_i =>\n      simp [Membership.mem]\n      have h_i' : i < Size.size c := by\n        simp [Foldable.fold] at h_i\n        rw [\u2190Range.memCorrect] at h_i\n        exact h_i\n      apply Exists.intro \u27e8i, h_i'\u27e9\n      exact h.2.symm\n  fold'Correct := by\n    intro \u03b2 c f acc\n    simp\n    conv => rhs; simp [Foldable'.Correct.fold', Foldable.fold]\n    have := Foldable'.canonicalToList_fold'_eq_map'\n      (\u27e8Size.size c\u27e9 : Range)\n      (fun x h => Indexed.nth c \u27e8x,h\u27e9)\n    simp [Foldable'.Correct.fold'] at this\n    rw [List.foldl'_rw _ _ _ _ this]\n    simp [Foldable'.Correct.fold'Correct, List.foldl'_eq_subtypeByMem_foldl]\n    sorry\n\nstructure Slice (C) (\u03c4 : outParam (Type u)) [Indexed C \u03c4] where\n  c : C\n  off : Nat\n  len : Nat\n  h_range : off + len \u2264 Size.size c\n\ninstance [Indexed C \u03c4] : Indexed (Slice C \u03c4) \u03c4 where\n  size S := S.len\n  nth S (i : Fin S.len) := Indexed.nth S.c \u27e8S.off + i, Nat.le_trans (Nat.add_lt_add_left i.isLt _) (Slice.h_range S)\u27e9\n\ndef IndexedEq [DecidableEq \u03c4] [Indexed C\u2081 \u03c4] [Indexed C\u2082 \u03c4] (c\u2081 : C\u2081) (c\u2082 : C\u2082)\n  {h : Size.size c\u2081 = Size.size c\u2082} : Prop :=\n  \u2200 i, Indexed.nth c\u2081 i = Indexed.nth c\u2082 \u27e8i.val, by rw [\u2190h]; exact i.isLt\u27e9\n\nend IndexedOps\n\nclass IndexedOps (C) (\u03c4 : Type u) [Indexed C \u03c4] where\n  slice (c : C) (off len : Nat)\n    {h : off + len \u2264 Size.size c} : IndexedOps.Slice C \u03c4\n  map (c : C) {\u03c4' : Type u} (f : \u03c4 \u2192 \u03c4') [Initable C' (Size.size c) \u03c4'] : C'\n  findi (c : C) (f : \u03c4 \u2192 Bool) : Option (Fin (Size.size c) \u00d7 \u03c4)\n  findMapi (c : C) {\u03c4' : Type u} (f : \u03c4 \u2192 Option \u03c4') : Option (Fin (Size.size c) \u00d7 \u03c4')\n\nnamespace IndexedOps\n\ninstance [Indexed C \u03c4] : Inhabited (IndexedOps C \u03c4) where\n  default := {\n    slice := \u03bb c off len h => \u27e8c,off,len,h\u27e9\n    map := \u03bb c _ f => Initable.init (fun i => f (Indexed.nth c i))\n    findi := \u03bb c f =>\n      View.view' (Range.mk (Size.size c))\n      |>.map (fun \u27e8i,h\u27e9 => (\u27e8i,h\u27e9 : Fin (Size.size c)))\n      |> FoldableOps.find (f := fun i => f (Indexed.nth c i))\n      |> Option.map (fun i => (i, Indexed.nth c i))\n    findMapi := \u03bb c _ f =>\n      View.view' (Range.mk (Size.size c))\n      |>.map (fun \u27e8i,h\u27e9 => (\u27e8i,h\u27e9 : Fin (Size.size c)))\n      |> FoldableOps.findMap (f := fun i =>\n        f (Indexed.nth c i) |>.map (fun x => (i, x))\n        )\n  }\n\ninstance [Indexed C \u03c4] [Foldable C \u03c4] : FoldableOps C \u03c4\n  := { (default : FoldableOps C \u03c4) with\n  toList := \u03bb c =>\n    let n := Size.size c\n    Foldable'.Correct.fold' (Range.mk n) (\u03bb acc i h =>\n      Indexed.nth c \u27e8n-i-1, by\n        simp\n        simp [Membership.mem] at h\n        rw [Nat.sub_sub]\n        apply Nat.sub_lt_of_pos_le\n        simp [Nat.add_one, Nat.zero_lt_succ]\n        exact h\n      \u27e9 :: acc\n      ) []\n  }\n\ntheorem toList_eq_range_toList_map [Indexed C \u03c4] (c : C)\n  (hL : \u2200 {x}, x \u2208 canonicalToList (Foldable.fold (Range.mk (Size.size c))) \u2192 x < Size.size c)\n  : FoldableOps.toList c = (\n      canonicalToList ((\u27e8Size.size c\u27e9 : Range).foldl)\n      |>.map' (fun x h => Indexed.nth c \u27e8x, hL h\u27e9))\n  := by\n  simp [FoldableOps.toList]\n  rw [Foldable'.Correct.fold'Correct]\n  -- Get RHS to Range.foldr'\n  suffices \u2200 L (h : L = canonicalToList (\u27e8Size.size c\u27e9 : Range).foldl) f'\n    (h_f' : \u2203 (h' : \u2200 {x}, x \u2208 L \u2192 _), \u2200 x (h : x \u2208 L), f' x h = Indexed.nth c \u27e8x, h' h\u27e9),\n    _ = List.map' L f'\n    from this _ rfl _ (by\n      apply Exists.intro _\n      intro x h\n      simp\n      intro x h\n      apply (Range.memCorrect _ _).mpr h\n      )\n  intro L h f' h_f'\n  rw [\u2190Range.toList_eq_canonicalToList] at h\n  cases h\n  conv =>\n    rhs\n    rw [\u2190List.foldr'_eq_map']\n    rw [\u2190Range.foldr'_correct _ (Range.toList_eq_canonicalToList _)\n      (f := fun x h acc =>\n        f' x (by\n          rw [Range.toList_eq_canonicalToList]\n          apply (Range.memCorrect _ _).mp h\n        ) :: acc)]\n    simp\n  -- Get LHS to Range.foldl'\n  suffices \u2200 L (h : L = canonicalToList fun {\u03b2} => Foldable.fold (\u27e8Size.size c\u27e9 : Range)),\n    List.foldl' L (fun acc x h' =>\n      Indexed.nth c \u27e8Size.size c - x - 1, by\n        have : Size.size c > 0 := by\n          rw [h] at h'\n          rw [\u2190Foldable'.Correct.memCorrect] at h'\n          simp [Membership.mem] at h'\n          apply Nat.lt_of_le_of_lt (Nat.zero_le x)\n          assumption\n        rw [Nat.sub_sub]\n        apply Nat.sub_lt\n        assumption\n        apply Nat.zero_lt_succ\n      \u27e9 :: acc) []\n    = _\n    from this _ rfl\n  intro L h\n  simp [Foldable.fold] at h\n  rw [\u2190Range.toList_eq_canonicalToList] at h\n  cases h\n  conv =>\n    lhs\n    rw [\u2190Range.foldl'_correct _ (Range.toList_eq_canonicalToList _)\n      (f := fun acc x h =>\n        Indexed.nth c \u27e8Size.size c - x - 1, by rw [Nat.sub_sub]; apply Nat.sub_lt; exact Range.size_pos_of_mem h; apply Nat.zero_lt_succ\u27e9 :: acc)]\n  -- Use range lemma\n  rw [Range.foldr'_eq_foldl'_mapped]\n  congr\n  funext acc x h\n  rw [h_f'.2]\n\ntheorem toList_eq_default_toList [Indexed C \u03c4] (c : C)\n  : FoldableOps.toList c = (FoldableOps.defaultImpl C \u03c4).toList c\n  := by\n  conv =>\n    rhs\n    rw [FoldableOps.default_toList_eq_canonicalToList]\n    simp [canonicalToList]\n    unfold Foldable.fold\n    simp [Foldable.Correct.toFoldable, Foldable'.Correct.toCorrect,\n      instCorrectInferInstanceMembershipInstMembership]\n    rw [Foldable'.fold'_append_singleton_eq_map]\n  rw [toList_eq_range_toList_map]\n  case hL =>\n    intro i h\n    rw [\u2190Foldable'.Correct.memCorrect] at h\n    simp [Membership.mem] at h\n    assumption\n  rfl\n\n@[simp]\ntheorem length_toList_eq_size {C \u03c4 : Type} [Indexed C \u03c4] (c : C)\n  : List.length (FoldableOps.toList c) = Size.size c\n  := by\n  rw [toList_eq_default_toList]\n  simp [FoldableOps.toList, FoldableOps.defaultImpl, Foldable.fold]\n  rw [Foldable'.canonicalToList_fold'_eq_map']\n  simp [Foldable.fold]\n  rw [\u2190Range.toList_eq_canonicalToList]\n  simp\n\ntheorem get_toList_eq_get [Indexed C \u03c4] (c : C) (i : Nat) (h : i < _)\n  : List.get (FoldableOps.toList c) \u27e8i, h\u27e9 = Indexed.nth c \u27e8i, by simp at h; exact h\u27e9\n  := by\n  suffices \u2200 L (hL : L = FoldableOps.toList c),\n    List.get L \u27e8i, by rw [hL]; exact h\u27e9 = _\n    from this _ rfl\n  intro L hL\n  simp at h\n  rw [toList_eq_default_toList] at hL\n  simp [FoldableOps.defaultImpl, canonicalToList, Foldable.fold, Foldable'.Correct.fold'] at hL\n  rw [Range.foldl'_correct _ (Range.toList_eq_canonicalToList _)] at hL\n  rw [List.foldl'_eq_map'] at hL\n  rw [List.map'] at hL\n  cases hL\n  simp\n\nstructure Map (C) [Indexed C \u03c4] where\n  val : C\n\nnamespace Map\n\ninstance [Indexed C \u03c4] : MapLike (Map C) Nat \u03c4 where\n  fold c f acc :=\n    Range.foldl' \u27e8Size.size c.val\u27e9 (\u03bb acc i h_i =>\n      f acc (i, Indexed.nth c.val \u27e8i,h_i\u27e9)\n    ) acc\n  get? i c :=\n    if h : i < Size.size c.val then\n      some (Indexed.nth c.val \u27e8i, h\u27e9)\n    else none\n", "meta": {"author": "JamesGallicchio", "repo": "LeanColls", "sha": "9cb0a0c9a838bea24be80eace168bcc5f9481596", "save_path": "github-repos/lean/JamesGallicchio-LeanColls", "path": "github-repos/lean/JamesGallicchio-LeanColls/LeanColls-9cb0a0c9a838bea24be80eace168bcc5f9481596/LeanColls/IndexedOps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494550081926, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3590399030930695}}
{"text": "/-\nCopyright (c) 2020 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Jannis Limperg\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# Better `clear` tactics\n\nWe define two variants of the standard `clear` tactic:\n\n* `clear'` works like `clear` but the hypotheses that should be cleared can be\n  given in any order. In contrast, `clear` can fail if hypotheses that depend on\n  each other are given in the wrong order, even if all of them could be cleared.\n\n* `clear_dependent` works like `clear'` but also clears any hypotheses that\n  depend on the given hypotheses.\n\n## Implementation notes\n\nThe implementation (ab)uses the native `revert_lst`, which can figure out\ndependencies between hypotheses. This implementation strategy was suggested by\nSimon Hudon.\n-/\n\n/-- Clears all the hypotheses in `hyps`. The tactic fails if any of the `hyps`\nis not a local or if the target depends on any of the `hyps`. It also fails if\n`hyps` contains duplicates.\n\nIf there are local hypotheses or definitions, say `H`, which are not in `hyps`\nbut depend on one of the `hyps`, what we do depends on `clear_dependent`. If it\nis true, `H` is implicitly also cleared. If it is false, `clear'` fails. -/\n-- Check if the target depends on any of the hyps. Doing this (instead of\n\n-- letting one of the later tactics fail) lets us give a much more informative\n\n-- error message.\n\n-- If revert_lst reverted more hypotheses than we wanted to clear, there must\n\n-- have been other hypotheses dependent on some of the hyps.\n\nnamespace tactic.interactive\n\n\n/--\nAn improved version of the standard `clear` tactic. `clear` is sensitive to the\norder of its arguments: `clear x y` may fail even though both `x` and `y` could\nbe cleared (if the type of `y` depends on `x`). `clear'` lifts this limitation.\n\n```lean\nexample {\u03b1} {\u03b2 : \u03b1 \u2192 Type} (a : \u03b1) (b : \u03b2 a) : unit :=\nbegin\n  try { clear a b }, -- fails since `b` depends on `a`\n  clear' a b,        -- succeeds\n  exact ()\nend\n```\n-/\n/--\nA variant of `clear'` which clears not only the given hypotheses, but also any\nother hypotheses depending on them.\n\n```lean\nexample {\u03b1} {\u03b2 : \u03b1 \u2192 Type} (a : \u03b1) (b : \u03b2 a) : unit :=\nbegin\n  try { clear' a },  -- fails since `b` depends on `a`\n  clear_dependent a, -- succeeds, clearing `a` and `b`\n  exact ()\nend\n```\n -/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/clear.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6859494550081926, "lm_q1q2_score": 0.3590399030930695}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n), \n  from sorry,\n\n  have h2 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h3 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h4 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h5 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h6 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h7 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h8 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h9 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h10 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h11 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h12 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h13 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h14 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h15 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h16 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h17 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h18 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h19 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h20 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h21 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h22 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h23 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h24 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h25 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h26 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h27 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h28 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h29 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h30 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191m))  \u2227 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)) = (int.fract (\u03b1 * \u2191n)),\n  from sorry,\n\n  have h31 : \u2200 (m n : \u2124), m \u2260 n \u2192 int.fract (\u03b1 * \u2191m) \u2260 int.fract (\u03b1 * \u2191n),\n  from sorry,\n\n  have h32 : \u2200 (m n : \u2124), m \u2260 n \u2192 (\u03b1 * \u2191m) - (floor (\u03b1 * \u2191m)) \u2260 (\u03b1 * \u2191n) - (floor (\u03b1 * \u2191n)),\n  from sorry,\n\n\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), from sorry,\n\n  have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h3 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h4 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h5 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h6 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h7 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h8 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h9 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h10 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h11 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h12 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h13 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h14 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h15 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h16 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h17 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h18 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h19 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h20 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h21 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h22 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h23 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h24 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h25 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h26 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h27 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h28 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h29 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h30 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h31 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h32 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h33 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h34 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h35 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h36 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h37 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h38 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h39 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h40 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h41 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), from sorry,\n\n  have h42 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i))\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h2 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h3 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h4 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h5 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h6 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h7 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h8 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h9 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h10 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h11 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h12 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h13 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h14 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h15 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h16 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h17 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h18 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h19 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h20 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h21 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h22 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h23 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h24 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h25 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h26 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h27 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h28 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h29 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h30 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h31 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h32 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h33 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h34 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h35 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h36 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h37 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry,\n  \n  have h38 : \u2200 (i j : \u2124), i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  let S := (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124),\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) \u2260 (int.fract (\u03b1 * \u2191j)), \n  from sorry, \n  have h2 : \u2200 (x y : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191x) - int.fract (\u03b1 * \u2191y))), from sorry,\n  have h3 : \u2200 (x y : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191x) + int.fract (\u03b1 * \u2191y))), from sorry,\n  have h4 : \u2200 (x y : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191x) * int.fract (\u03b1 * \u2191y))), from sorry,\n  have h5 : \u2200 (x y : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191x) / int.fract (\u03b1 * \u2191y))), from sorry,\n  have h6 : \u2200 (x y : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191x) ^ int.fract (\u03b1 * \u2191y))), from sorry,\n\n  have h7 : \u2200 (a b : \u2124), a \u2260 b \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191a) - int.fract (\u03b1 * \u2191b))), from sorry,\n  have h8 : \u2200 (a b : \u2124), a \u2260 b \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191a) + int.fract (\u03b1 * \u2191b))), from sorry,\n  have h9 : \u2200 (a b : \u2124), a \u2260 b \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191a) * int.fract (\u03b1 * \u2191b))), from sorry,\n  have h10 : \u2200 (a b : \u2124), a \u2260 b \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191a) / int.fract (\u03b1 * \u2191b))), from sorry,\n  have h11 : \u2200 (a b : \u2124), a \u2260 b \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = (int.fract (\u03b1 * \u2191a) ^ int.fract (\u03b1 * \u2191b))), from sorry,\n\n  have h12 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) - int.fract (\u03b1 * \u2191y)) - int.fract (\u03b1 * \u2191z))), from sorry,\n  have h13 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) + int.fract (\u03b1 * \u2191y)) - int.fract (\u03b1 * \u2191z))), from sorry,\n  have h14 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) * int.fract (\u03b1 * \u2191y)) - int.fract (\u03b1 * \u2191z))), from sorry,\n  have h15 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) / int.fract (\u03b1 * \u2191y)) - int.fract (\u03b1 * \u2191z))), from sorry,\n  have h16 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) ^ int.fract (\u03b1 * \u2191y)) - int.fract (\u03b1 * \u2191z))), from sorry,\n\n  have h17 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) - int.fract (\u03b1 * \u2191y)) + int.fract (\u03b1 * \u2191z))), from sorry,\n  have h18 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) + int.fract (\u03b1 * \u2191y)) + int.fract (\u03b1 * \u2191z))), from sorry,\n  have h19 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) * int.fract (\u03b1 * \u2191y)) + int.fract (\u03b1 * \u2191z))), from sorry,\n  have h20 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) / int.fract (\u03b1 * \u2191y)) + int.fract (\u03b1 * \u2191z))), from sorry,\n  have h21 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) ^ int.fract (\u03b1 * \u2191y)) + int.fract (\u03b1 * \u2191z))), from sorry,\n\n  have h22 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) - int.fract (\u03b1 * \u2191y)) * int.fract (\u03b1 * \u2191z))), from sorry,\n  have h23 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) + int.fract (\u03b1 * \u2191y)) * int.fract (\u03b1 * \u2191z))), from sorry,\n  have h24 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) * int.fract (\u03b1 * \u2191y)) * int.fract (\u03b1 * \u2191z))), from sorry,\n  have h25 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) / int.fract (\u03b1 * \u2191y)) * int.fract (\u03b1 * \u2191z))), from sorry,\n  have h26 : \u2200 (x y z : \u2124), x \u2260 y \u2192 (\u2203! (m : \u2124), (int.fract (\u03b1 * \u2191m)) = ((int.fract (\u03b1 * \u2191x) ^ int.fract (\u03b1 * \u2191\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  sorry\nend\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 := sorry\n\n/--`theorem`\nIrrational number has irrational orbit\nLet $\\alpha$ be an irrational number. Then $\\{i \\alpha\\}$ is irrational for all $i \\in \\mathbb{Z}$.\n`proof`\nSuppose $\\{i \\alpha\\}$ is rational for some $i \\in \\mathbb{Z}$. Then there are integers $p$ and $q$ with $q \\neq 0$ such that\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\frac{p}{q}.\n$$\nThus\n$$\n\\alpha=\\frac{\\lfloor i \\alpha\\rfloor+\\frac{p}{q}}{i}=\\frac{\\lfloor i \\alpha\\rfloor}{i}+\\frac{p}{i q} \\in \\mathbb{Q},\n$$\ncontradicting the assumption that $\\alpha$ is irrational.\n\nQED\n-/\ntheorem irrational_orbit_irrational {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : \u2200 i : \u2124, irrational (int.fract (i * \u03b1)) := sorry\n\n/- `theorem`\n\nFor any non-empty subset $S$ of $\\R$, we have:\n$$\n\\overline{S}=\\bigcap_{\\epsilon>0}\\left\\{x \\mid \\sup_{y \\in S}\\left|x-y\\right|<\\epsilon\\right\\}\n$$\n`proof`\n$\\subseteq$: Let $x \\in \\overline{S}$. Then for all $\\epsilon>0$, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$. But then $\\sup_{y \\in S}\\left|x-y\\right|<\\epsilon$.\n\n$\\supseteq$: Let $x \\in \\bigcap_{\\epsilon>0}\\left\\{x \\mid \\sup_{y \\in S}\\left|x-y\\right|<\\epsilon\\right\\}$. Then for all $\\epsilon>0$, $\\sup_{y \\in S}\\left|x-y\\right|<\\epsilon$. But then, since $S$ is non-empty, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$.\n\nQED\n-/\ntheorem closure_sup {S : set \u211d} (hS_non_empty : S \u2260 \u2205) : closure S = (\u22c2 \u03b5 > 0, {x : \u211d | \u2203 y \u2208 S, |x - y| < \u03b5}) := sorry\n\n/- `theorem`\n\nFor any non-empty subset $S$ of $\\R$, we have:\n$$\n\\overline{S}=\\bigcap_{\\delta>0}\\bigcup_{y \\in S}\\left[y-\\delta, y+\\delta\\right]\n$$\n`proof`\n$\\subseteq$: Let $x \\in \\overline{S}$. Then for all $\\epsilon>0$, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$. Hence for all $\\delta>0$, $\\exists y \\in S$ such that $y-\\delta<x<y+\\delta$.\n\n$\\supseteq$: Let $x \\in \\bigcap_{\\delta>0}\\bigcup_{y \\in S}\\left[y-\\delta, y+\\delta\\right]$. Then for all $\\delta>0$, $\\exists y \\in S$ such that $y-\\delta<x<y+\\delta$. Then let $\\epsilon=\\frac{\\delta}{2}$. We have $y-\\epsilon<x<y+\\epsilon$, so $|x-y|<\\epsilon$. But then, since $S$ is non-empty, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$.\n\nQED\n-/\ntheorem closure_union {S : set \u211d} (hS_non_empty : S \u2260 \u2205) : closure S = (\u22c2 \u03b4 > 0, \u22c3 y \u2208 S, set.Icc (y - \u03b4) (y + \u03b4)) := sorry\n\n/- `theorem`\n\nFor any non-empty subset $S$ of $\\R$, we have:\n$$\n\\overline{S}=\\bigcap_{\\epsilon>0}\\left\\{x \\mid \\sup_{y \\in S}\\left|x-y\\right|\\leq\\epsilon\\right\\}\n$$\n`proof`\n$\\subseteq$: Let $x \\in \\overline{S}$. Then for all $\\epsilon>0$, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$. But then $\\sup_{y \\in S}\\left|x-y\\right|\\leq\\epsilon$.\n\n$\\supseteq$: Let $x \\in \\bigcap_{\\epsilon>0}\\left\\{x \\mid \\sup_{y \\in S}\\left|x-y\\right|\\leq\\epsilon\\right\\}$. Then for all $\\epsilon>0$, $\\sup_{y \\in S}\\left|x-y\\right|\\leq\\epsilon$. But then, since $S$ is non-empty, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$.\n\nQED\n-/\ntheorem closure_sup_le {S : set \u211d} (hS_non_empty : S \u2260 \u2205) : closure S = (\u22c2 \u03b5 > 0, {x : \u211d | \u2203 y \u2208 S, |x - y| \u2264 \u03b5}) := sorry\n\n/- `theorem`\n\nFor any non-empty subset $S$ of $\\R$, we have:\n$$\n\\overline{S}=\\bigcap_{\\delta>0}\\bigcup_{y \\in S}\\left[y-\\delta, y+\\delta\\right]\n$$\n`proof`\n$\\subseteq$: Let $x \\in \\overline{S}$. Then for all $\\epsilon>0$, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$. Hence for all $\\delta>0$, $\\exists y \\in S$ such that $y-\\delta\\leq x\\leq y+\\delta$.\n\n$\\supseteq$: Let $x \\in \\bigcap_{\\delta>0}\\bigcup_{y \\in S}\\left[y-\\delta, y+\\delta\\right]$. Then for all $\\delta>0$, $\\exists y \\in S$ such that $y-\\delta\\leq x\\leq y+\\delta$. Then let $\\epsilon=\\frac{\\delta}{2}$. We have $y-\\epsilon\\leq x\\leq y+\\epsilon$, so $|x-y|<\\epsilon$. But then, since $S$ is non-empty, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$.\n\nQED\n-/\ntheorem closure_union_le {S : set \u211d} (hS_non_empty : S \u2260 \u2205) : closure S = (\u22c2 \u03b4 > 0, \u22c3 y \u2208 S, set.Icc_le (y - \u03b4) (y + \u03b4)) := sorry\n\n/- `theorem`\n\nFor any non-empty subset $S$ of $\\R$, we have:\n$$\n\\overline{S}=\\bigcap_{\\epsilon>0}\\left\\{x \\mid \\inf_{y \\in S}\\left|x-y\\right|<\\epsilon\\right\\}\n$$\n`proof`\n$\\subseteq$: Let $x \\in \\overline{S}$. Then for all $\\epsilon>0$, $\\exists y \\in S$ such that $\\left|x-y\\right|<\\epsilon$. But then $\\inf_{y \\in S}\\left|x-y\\right|<\\epsilon$.\n\n$\\supseteq$: Let\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), \n  from sorry,\n  have h2 : (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ \u2286 set.Icc 0 1, \n  from sorry,\n  have h3 : \u2200 i : \u2124, \u2203 j : \u2124, (int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j)) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ,\n  from sorry,\n  have h4 : \u2200 i : \u2124, \u2203 j : \u2124, int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) = int.fract (\u03b1 * \u2191j),\n  from sorry,\n  have h5 : \u2200 i : \u2124, \u2203 j : \u2124, int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) = 0,\n  from sorry,\n  have h6 : \u2200 i : \u2124, \u2203 j : \u2124, int.fract (\u03b1 * \u2191i) - int.fract (\u03b1 * \u2191j) \u2208 set.Icc 0 1,\n  from sorry,\n  have h7 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, |int.fract (\u03b1 * \u2191j) - y| < \u03b5,\n  from sorry,\n  have h8 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 set.Icc (y - \u03b5) (y + \u03b5), \n  from sorry,\n  have h9 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 closure (set.Icc (y - \u03b5) (y + \u03b5)),\n  from sorry,\n  have h10 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 closure (set.Icc (y - \u03b5) (y + \u03b5)) \u2229 set.Icc 0 1,\n  from sorry,\n  have h11 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ \u2229 set.Icc (y - \u03b5) (y + \u03b5),\n  from sorry,\n  have h12 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ \u2229 set.Icc (y - \u03b5) (y + \u03b5),\n  from sorry,\n  have h13 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 \u2203 j : \u2124, int.fract (\u03b1 * \u2191j) \u2208 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ \u2229 set.Icc (y - \u03b5) (y + \u03b5),\n  from sorry,\n  have h14 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 (\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h15 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h16 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h17 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h18 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h19 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h20 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h21 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h22 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h23 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h24 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.Icc (y - \u03b5) (y + \u03b5) \u2260 \u2205,\n  from sorry,\n  have h25 : \u2200 (y : \u211d) (\u03b5 : \u211d), y \u2208 set.Icc 0 1 \u2192 \u03b5 > 0 \u2192 closure ((\u03bb (m : \u2124), int.fract (\u03b1 * \u2191m)) '' univ) \u2229 set.I\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nbegin\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 int.fract (\u03b1 * \u2191i) \u2260 int.fract (\u03b1 * \u2191j), \n  from sorry,\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2260 0, \n  from sorry,\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 (int.fract (\u03b1 * \u2191i)) - (int.fract (\u03b1 * \u2191j)) \u2260 0, \n  from sorry,\n\n  let S : set \u211d := sorry,\n  have h4 : S \u2286 set.Icc 0 1, from sorry,\n  have h5 : S \u2260 \u2205, from sorry,\n  have h6 : S \u222a {0} \u2286 set.Icc 0 1, from sorry,\n  have h7 : S \u222a {0} \u2260 \u2205, from sorry,\n  have h8 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x \u2208 S, from sorry,\n  have h9 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x \u2208 set.Icc 0 1, from sorry,\n  have h10 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x = 0 \u2228 x \u2208 S, from sorry,\n  have h11 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x \u2208 set.Icc 0 1, from sorry,\n  have h12 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x = 0 \u2228 x \u2208 S, from sorry,\n  have h13 : \u2200 x : \u211d, x \u2208 S \u222a {0} \u2192 x \u2208 set.Icc 0 1, from sorry,\n\n  have h14 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure (S \u222a {0}), from sorry,\n\n  have h15 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h16 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h17 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h18 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h19 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h20 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h21 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h22 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h23 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h24 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h25 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h26 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h27 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h28 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h29 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h30 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h31 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h32 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h33 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h34 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h35 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h36 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h37 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h38 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h39 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h40 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h41 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h42 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h43 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h44 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h45 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h46 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h47 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h48 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h49 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h50 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h51 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h52 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h53 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h54 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h55 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h56 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h57 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h58 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h59 : \u2200 x : \u211d, x \u2208 set.Icc 0 1 \u2192 x \u2208 closure S, from sorry,\n\n  have h60 : \u2200 x :\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from sorry,\n  \n  assume (h7 : \u03b5 > 0),\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := sorry,\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from sorry,\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from sorry,\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem  irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/Correct_statement-lean_proof_outline-4_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.8128673178375735, "lm_q2_score": 0.4416730056646256, "lm_q1q2_score": 0.3590215514758636}}
{"text": "example (P Q : Type) : P \u2192 (Q \u2192 P) :=\nbegin\n    intro p,\n    intro q,\n    exact p,\nend\n", "meta": {"author": "nomoid", "repo": "lean-proofs", "sha": "b9f03a24623d1a1d111d6c2bbf53c617e2596d6a", "save_path": "github-repos/lean/nomoid-lean-proofs", "path": "github-repos/lean/nomoid-lean-proofs/lean-proofs-b9f03a24623d1a1d111d6c2bbf53c617e2596d6a/src/world5/level5.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3588787298227148}}
{"text": "import verified_clique\nimport periodic_reduction\n\ndef is_s_discrete {d : \u2115} (s : \u2115) (T : set (point d)) : Prop :=\n  \u2200 i : fin d, \u2203 coords : finset \u211d, coords.card \u2264 s \u2227 \n  (\u2200 coord1 \u2208 coords, \u2200 coord2 \u2208 coords, coord1 \u2260 coord2 \u2192 ne_mod_one coord1 coord2) \u2227\n  (\u2200 t \u2208 T, \u2203 coord \u2208 coords, eq_mod_one (vector.nth t i) coord)\n\nnoncomputable def build_half_core_points_finset {d : \u2115} (i : fin d) (n : fin (d + 1)) :\n  {s: finset ({p : point d // \u2200 j : fin d, \n    (i = j \u2192 p.nth j = 0) \u2227 (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)}) //\n  (i.val < n.val \u2192 s.card = 2^(n.val - 1)) \u2227 (i.val \u2265 n.val \u2192 s.card = 2^n.val) \u2227 \n  \u2200 p : point d, \n  \u2200 h : (\u2200 j : fin d, (i = j \u2192 p.nth j = 0) \u2227 (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)),\n  (\u27e8p, h\u27e9 : {p : point d // \u2200 j : fin d, \n    (i = j \u2192 p.nth j = 0) \u2227 (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)}) \u2208 s\n  } :=\nbegin\n  cases n with n_val n_property,\n  simp only [fin.val_eq_coe, ge_iff_le],\n  induction n_val with m ih,\n  { norm_num,\n    let res :\n      finset ({p : point d // \u2200 j : fin d, (i = j \u2192 p.nth j = 0) \u2227 (j.val < 0 \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 0 \u2192 p.nth j = 0)}) :=\n      begin\n        let zero_vector : \n          {p : point d // \u2200 j : fin d, (i = j \u2192 p.nth j = 0) \u2227 (j.val < 0 \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 0 \u2192 p.nth j = 0)} :=\n          begin\n            let zero_vector : point d := vector.of_fn (\u03bb i, 0),\n            use zero_vector,\n            intro j,\n            split, {intro i_eq_j, dsimp[zero_vector], simp only [vector.nth_of_fn]},\n            split, {intro i_lt_zero, exfalso, linarith},\n            intro i_ge_zero,\n            dsimp[zero_vector],\n            simp only [vector.nth_of_fn],\n          end,\n        exact {zero_vector},\n      end,\n    use res,\n    split, by {rw finset.card_eq_one, use (vector.of_fn (\u03bb i, 0)),},\n    intros p h,\n    dsimp[res],\n    simp only [finset.mem_singleton],\n    apply vector.ext,\n    intro j,\n    replace h := h j,\n    cases h with _ h,\n    simp only [vector.nth_of_fn],\n    exact h,\n  },\n  have m_lt_d_add_one : m < d + 1 := by omega,\n  rcases ih m_lt_d_add_one with \n    \u27e8core_points_finset_up_to_m, \n    \u27e8core_points_finset_up_to_m_card_i_lt_m, core_points_finset_up_to_m_card_m_le_i, core_points_finset_up_to_m_property\u27e9\u27e9,\n  let next_level_map : \n    {p : point d // \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u21aa \n    {p : point d // \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} := \n    begin\n      let next_level_map_fn : \n        {p : point d // \u2200 (j : fin d), \n          (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u2192 \n        {p : point d // \u2200 (j : fin d), \n          (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)}\n        := \u03bb p, \n        begin\n          use (vector.of_fn (\u03bb j : fin d, if (i = j) then 0 else if (j.val = m) then p.val.nth j + 1 else p.val.nth j)),\n          intro j,\n          simp only [fin.val_eq_coe, ge_iff_le, vector.nth_of_fn, subtype.val_eq_coe],\n          split, {intro i_eq_j, rw if_pos i_eq_j},\n          by_cases i_eq_j : i = j,\n          { rw if_pos i_eq_j,\n            split, {intro _, left, refl},\n            intro _, refl,\n          },\n          rename i_eq_j i_ne_j,\n          rw if_neg i_ne_j,\n          split,\n          { intro j_lt_m_succ,\n            by_cases j_eq_m : \u2191j = m,\n            { rw if_pos j_eq_m,\n              right,\n              have p_property := p.property j,\n              rcases p_property with \u27e8_, _, p_property\u27e9,\n              simp only [ge_iff_le, subtype.val_eq_coe, j_eq_m] at p_property,\n              rw [(p_property (by refl)), zero_add],\n            },\n            rename j_eq_m j_ne_m,\n            rw if_neg j_ne_m,\n            have p_property := p.property j,\n            rcases p_property with \u27e8_, p_property_j_lt_m, p_property_j_ge_m\u27e9,\n            cases lt_or_ge j.val m with j_lt_m j_ge_m, exact p_property_j_lt_m j_lt_m,\n            left,\n            exact p_property_j_ge_m j_ge_m,\n          },\n          intro m_succ_le_j,\n          have j_ne_m : \u2191j \u2260 m := by {clear_except m_succ_le_j, rw nat.succ_eq_add_one at m_succ_le_j, linarith},\n          rw if_neg j_ne_m,\n          have p_property := p.property j,\n          rcases p_property with \u27e8_, _, p_property_j_ge_m\u27e9,\n          have j_ge_m : \u2191j \u2265 m := by {clear_except m_succ_le_j, rw nat.succ_eq_add_one at m_succ_le_j, linarith},\n          simp only [subtype.val_eq_coe] at p_property_j_ge_m,\n          exact p_property_j_ge_m j_ge_m,\n        end,\n      have next_level_map_fn_injective : function.injective next_level_map_fn :=\n        begin\n          rw function.injective,\n          intros p1 p2 p1_output_eq_p2_output,\n          apply subtype.ext,\n          apply vector.ext,\n          intro j,\n          replace p1_output_eq_p2_output : (next_level_map_fn p1).val.nth j = (next_level_map_fn p2).val.nth j := \n            by rw p1_output_eq_p2_output,\n          dsimp[next_level_map_fn] at p1_output_eq_p2_output,\n          simp only [vector.nth_of_fn] at p1_output_eq_p2_output,\n          by_cases i_eq_j : i = j,\n          { have p1_property := p1.property j,\n            have p2_property := p2.property j,\n            rcases p1_property with \u27e8p1_property, _, _\u27e9,\n            rcases p2_property with \u27e8p2_property, _, _\u27e9,\n            simp only [subtype.val_eq_coe] at p1_property p2_property,\n            rw [p1_property i_eq_j, p2_property i_eq_j],\n          },\n          rename i_eq_j i_ne_j,\n          rw [if_neg i_ne_j, if_neg i_ne_j] at p1_output_eq_p2_output,\n          by_cases j_eq_m : \u2191j = m,\n          { rw [if_pos j_eq_m, if_pos j_eq_m] at p1_output_eq_p2_output,\n            clear_except p1_output_eq_p2_output,\n            linarith,\n          },\n          rename j_eq_m j_ne_m,\n          rw [if_neg j_ne_m, if_neg j_ne_m] at p1_output_eq_p2_output,\n          exact p1_output_eq_p2_output,\n        end,\n      exact {to_fun := next_level_map_fn, inj' := next_level_map_fn_injective},\n    end,\n  let core_points_finset_up_to_m_mapped := finset.map next_level_map core_points_finset_up_to_m,\n  have core_points_finset_up_to_m_mapped_card : core_points_finset_up_to_m_mapped.card = core_points_finset_up_to_m.card := \n    finset.card_map next_level_map,\n  let cast_map :\n    {p : point d // \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u21aa \n    {p : point d // \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} :=\n    begin\n      let cast_map_fn :\n      {p : point d // \u2200 (j : fin d), \n        (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u2192\n      {p : point d // \u2200 (j : fin d), \n        (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} \n        := \u03bb p,\n        begin\n          use p.val,\n          intro j,\n          split,\n          { intro i_eq_j,\n            rw \u2190 i_eq_j,\n            have p_property := p.property i,\n            rcases p_property with \u27e8goal, _, _\u27e9,\n            exact goal (by refl),\n          },\n          have p_property := p.property j,\n          rcases p_property with \u27e8_, p_property_j_lt_m, p_property_j_ge_m\u27e9,\n          split,\n          { intro j_lt_m_succ,\n            by_cases j_lt_m : j.val < m, exact p_property_j_lt_m j_lt_m,\n            rename j_lt_m j_ge_m,\n            replace j_ge_m : j.val \u2265 m := by {clear_except j_ge_m, linarith},\n            left,\n            exact p_property_j_ge_m j_ge_m,\n          },\n          intro j_ge_m_succ,\n          have j_ge_m : j.val \u2265 m := by {clear_except j_ge_m_succ, rw nat.succ_eq_add_one at j_ge_m_succ, linarith},\n          exact p_property_j_ge_m j_ge_m,\n        end,\n      have cast_map_fn_injective : function.injective cast_map_fn :=\n        begin\n          rw function.injective,\n          intros p1 p2 p1_output_eq_p2_output,\n          dsimp[cast_map_fn] at p1_output_eq_p2_output,\n          simp only at p1_output_eq_p2_output,\n          refine subtype.eq _,\n          simp only [subtype.val_eq_coe],\n          exact p1_output_eq_p2_output,\n        end,\n      exact {to_fun := cast_map_fn, inj' := cast_map_fn_injective},\n    end,\n  let core_points_finset_up_to_m_cast := finset.map cast_map core_points_finset_up_to_m,\n  have core_points_finset_up_to_m_cast_card : core_points_finset_up_to_m_cast.card = core_points_finset_up_to_m.card := \n    finset.card_map cast_map,\n  by_cases i_eq_m : i.val = m,\n  { use core_points_finset_up_to_m_cast,\n    split,\n    { intro i_lt_m_succ,\n      have m_le_i : m \u2264 \u2191i := by {simp only [fin.val_eq_coe] at i_eq_m, rw i_eq_m},\n      rw [core_points_finset_up_to_m_cast_card, core_points_finset_up_to_m_card_m_le_i m_le_i, nat.succ_eq_add_one],\n      norm_num,\n    },\n    split,\n    { intro i_ge_m_succ,\n      exfalso,\n      simp only [fin.val_eq_coe] at i_eq_m,\n      clear_except i_ge_m_succ i_eq_m,\n      rw [i_eq_m, nat.succ_eq_add_one] at i_ge_m_succ,\n      linarith,\n    },\n    intros p h,\n    have core_points_finset_up_to_m_property_precondition : \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0) :=\n      begin\n        intro j,\n        replace h := h j,\n        rcases h with \u27e8h_i_eq_j, h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n        split, exact h_i_eq_j,\n        split,\n        { intro j_lt_m,\n          have j_lt_m_succ : j.val < m.succ := by {clear_except j_lt_m, rw nat.succ_eq_add_one, linarith},\n          exact h_j_lt_m_succ j_lt_m_succ,\n        },\n        intro j_ge_m,\n        by_cases j_eq_m : j.val = m,\n        { have i_eq_j : i = j := by {apply subtype.ext, rw \u2190 i_eq_m at j_eq_m, simp only [fin.val_eq_coe] at j_eq_m, symmetry, exact j_eq_m},\n          exact h_i_eq_j i_eq_j,\n        },\n        rename j_eq_m j_ne_m,\n        have j_ge_m_succ : j.val \u2265 m.succ := by {clear_except j_ge_m j_ne_m, omega},\n        exact h_j_ge_m_succ j_ge_m_succ,\n      end,\n    have p_in_core_points_finset_up_to_m := core_points_finset_up_to_m_property p core_points_finset_up_to_m_property_precondition,\n    dsimp[core_points_finset_up_to_m_cast],\n    simp only [exists_prop, add_zero, mul_one, gt_iff_lt, finset.mem_map, exists_and_distrib_right, int.coe_nat_add, ge_iff_le,\n      int.coe_nat_one, exists_eq_right, function.embedding.coe_fn_mk, zero_add, subtype.exists, subtype.coe_mk, zero_lt_one,\n      subtype.val_eq_coe, neg_zero],\n    use core_points_finset_up_to_m_property_precondition,\n    exact p_in_core_points_finset_up_to_m,\n  },\n  rename i_eq_m i_ne_m,\n  have cast_disjoint_with_mapped : disjoint core_points_finset_up_to_m_cast core_points_finset_up_to_m_mapped :=\n    begin\n      rw disjoint,\n      simp only [finset.inf_eq_inter, finset.bot_eq_empty, finset.le_eq_subset],\n      intros shared_point shared_point_in_both,\n      simp only [finset.not_mem_empty, finset.bot_eq_empty],\n      simp only [le_refl, exists_prop, add_zero, mul_one, gt_iff_lt, fin.val_eq_coe, finset.mem_map, int.coe_nat_add, ge_iff_le,\n        int.coe_nat_one, function.embedding.coe_fn_mk, vector.nth_of_fn, zero_add, neg_eq_zero, subtype.exists, neg_neg,\n        subtype.coe_mk, zero_lt_one, finset.mem_inter, subtype.val_eq_coe, neg_zero] at shared_point_in_both,\n      rcases shared_point_in_both with\n        \u27e8\u27e8shared_point_precast, shared_point_precast_property, shared_point_precast_in_core_points, shared_point_precast_eq_shared_point\u27e9, \n        \u27e8shared_point_premap, shared_point_premap_property, shared_point_premap_in_core_points, shared_point_premap_eq_shared_point\u27e9\u27e9,\n      have m_lt_d : m < d := by {clear_except n_property, rw nat.succ_eq_add_one at n_property, linarith},\n      rcases shared_point_precast_property \u27e8m, m_lt_d\u27e9 with \u27e8_, _, h1\u27e9,\n      rcases shared_point_premap_property \u27e8m, m_lt_d\u27e9 with \u27e8_, _, h2\u27e9,\n      replace h1 := h1 rfl.ge,\n      replace h2 := h2 rfl.ge,\n      replace shared_point_precast_eq_shared_point : shared_point_precast = shared_point.val :=\n        (congr_arg subtype.val shared_point_precast_eq_shared_point).congr_right.mp rfl,\n      replace shared_point_premap_eq_shared_point : vector.of_fn \n        (\u03bb (j : fin d), ite (i = j) 0 (ite (\u2191j = m) (vector.nth shared_point_premap j + 1) (vector.nth shared_point_premap j)))\n        = shared_point.val := (congr_arg subtype.val shared_point_premap_eq_shared_point).congr_right.mp rfl,\n      have shared_point_precast_eq_shared_point_at_m : shared_point_precast.nth \u27e8m, m_lt_d\u27e9 = shared_point.val.nth \u27e8m, m_lt_d\u27e9 :=\n        by rw shared_point_precast_eq_shared_point,\n      rw \u2190 shared_point_premap_eq_shared_point at shared_point_precast_eq_shared_point_at_m,\n      simp only [if_true, eq_self_iff_true, vector.nth_of_fn, fin.coe_mk] at shared_point_precast_eq_shared_point_at_m,\n      have i_ne_m_as_fin : i \u2260 \u27e8m, m_lt_d\u27e9 := fin.ne_of_vne i_ne_m,\n      rw [h1, h2, if_neg i_ne_m_as_fin] at shared_point_precast_eq_shared_point_at_m,\n      norm_num at shared_point_precast_eq_shared_point_at_m,\n    end,\n  let res := core_points_finset_up_to_m_cast \u222a core_points_finset_up_to_m_mapped,\n  have res_card_i_lt_m_succ : \u2191i < m.succ \u2192 res.card = 2^m :=\n    begin\n      intro i_lt_m_succ,\n      have i_lt_m : \u2191i < m := by {clear_except i_ne_m i_lt_m_succ, simp only [fin.val_eq_coe] at i_ne_m, omega},\n      dsimp[res],\n      rw [finset.card_disjoint_union cast_disjoint_with_mapped, core_points_finset_up_to_m_cast_card,\n        core_points_finset_up_to_m_mapped_card, core_points_finset_up_to_m_card_i_lt_m i_lt_m],\n      have m_gt_zero : \u2203 m' : \u2115, m = m'.succ :=\n        begin\n          have m_ne_zero : m \u2260 0 := by {clear_except i_lt_m, linarith},\n          exact nat.exists_eq_succ_of_ne_zero m_ne_zero,\n        end,\n      cases m_gt_zero with m' m'_def,\n      rw [m'_def, nat.succ_eq_add_one, pow_succ, two_mul],\n      norm_num,\n    end,\n  have res_card_m_succ_le_i : m.succ \u2264 \u2191i \u2192 res.card = 2^m.succ :=\n    begin\n      intro m_succ_le_i,\n      have m_le_i : m \u2264 \u2191i := by {clear_except m_succ_le_i, rw nat.succ_eq_add_one at m_succ_le_i, linarith},\n      dsimp[res],\n      rw [finset.card_disjoint_union cast_disjoint_with_mapped, core_points_finset_up_to_m_cast_card,\n        core_points_finset_up_to_m_mapped_card, core_points_finset_up_to_m_card_m_le_i m_le_i, nat.succ_eq_add_one,\n        pow_succ, two_mul],\n    end,\n  use res,\n  split, {intro i_lt_m_succ, exact res_card_i_lt_m_succ i_lt_m_succ},\n  split, {intro m_succ_le_i, exact res_card_m_succ_le_i m_succ_le_i},\n  intros p h,\n  have m_lt_d : m < d := by {clear_except n_property, omega},\n  by_cases p_eq_zero_at_m : p.nth \u27e8m, m_lt_d\u27e9 = 0,\n  { have core_points_finset_up_to_m_property_precondition : \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0) :=\n      begin\n        intro j,\n        rcases h j with \u27e8h_i_eq_j, h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n        split, exact h_i_eq_j,\n        split,\n        { intro j_lt_m,\n          have j_lt_m_succ : j.val < m.succ := by {clear_except j_lt_m, rw nat.succ_eq_add_one, linarith},\n          exact h_j_lt_m_succ j_lt_m_succ,\n        },\n        intro j_ge_m,\n        by_cases j_eq_m : j = \u27e8m, m_lt_d\u27e9,\n        { rw \u2190 j_eq_m at p_eq_zero_at_m,\n          exact p_eq_zero_at_m,\n        },\n        rename j_eq_m j_ne_m,\n        replace j_ne_m : j.val \u2260 m :=\n          begin\n            simp only [fin.val_eq_coe],\n            intro j_eq_m,\n            conv at j_ne_m\n            begin\n              find m {rw \u2190 j_eq_m},\n            end,\n            simp only [eq_self_iff_true, not_true, fin.eta] at j_ne_m,\n            exact j_ne_m,\n          end,\n        have j_ge_m_succ : j.val \u2265 m.succ := by {clear_except j_ge_m j_ne_m, omega},\n        exact h_j_ge_m_succ j_ge_m_succ,\n      end,\n    have p_in_core_points_finset_up_to_m := core_points_finset_up_to_m_property p core_points_finset_up_to_m_property_precondition,\n    dsimp only[res],\n    apply finset.mem_union_left,\n    dsimp only[core_points_finset_up_to_m_cast],\n    simp only [exists_prop, fin.val_eq_coe, finset.mem_map, exists_and_distrib_right, ge_iff_le, exists_eq_right,\n      function.embedding.coe_fn_mk, subtype.exists, subtype.coe_mk, subtype.val_eq_coe],\n    use core_points_finset_up_to_m_property_precondition,\n    exact p_in_core_points_finset_up_to_m,\n  },\n  rename p_eq_zero_at_m p_ne_zero_at_m,\n  let premap_p : point d := vector.of_fn (\u03bb j : fin d, if(i = j) then 0 else if(j.val = m) then 0 else p.nth j),\n  have core_points_finset_up_to_m_property_precondition : \u2200 (j : fin d), \n    (i = j \u2192 vector.nth premap_p j = 0) \u2227 (j.val < m \u2192 vector.nth premap_p j = 0 \u2228 vector.nth premap_p j = 1) \u2227 \n    (j.val \u2265 m \u2192 vector.nth premap_p j = 0) :=\n    begin\n      intro j,\n      dsimp only[premap_p],\n      simp only [fin.val_eq_coe, ge_iff_le, vector.nth_of_fn, ite_eq_left_iff],\n      rcases h j with \u27e8h_i_eq_j, h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n      split,\n      { intros i_eq_j i_ne_j,\n        exfalso,\n        exact i_ne_j i_eq_j,\n      },\n      split,\n      { intro j_lt_m,\n        by_cases i_eq_j : i = j,\n        { left,\n          intro i_ne_j,\n          exfalso,\n          exact i_ne_j i_eq_j,\n        },\n        rename i_eq_j i_ne_j,\n        by_cases j_eq_m : \u2191j = m,\n        { left,\n          intros i_ne_j j_ne_m,\n          exfalso,\n          exact j_ne_m j_eq_m,\n        },\n        rename j_eq_m j_ne_m,\n        have j_lt_m_succ : j.val < m.succ := by {clear_except j_lt_m, rw nat.succ_eq_add_one, simp only [fin.val_eq_coe], linarith},\n        cases h_j_lt_m_succ j_lt_m_succ with p_eq_zero p_eq_one,\n        { left,\n          intros i_ne_j j_ne_m,\n          exact p_eq_zero,\n        },\n        right,\n        rw [if_neg i_ne_j, if_neg j_ne_m],\n        exact p_eq_one,\n      },\n      intros m_le_j i_ne_j j_ne_m,\n      have j_ge_m : j.val \u2265 m.succ := by {simp only [fin.val_eq_coe], clear_except m_le_j j_ne_m, omega},\n      exact h_j_ge_m_succ j_ge_m,\n    end,\n  have premap_p_in_core_points_finset_up_to_m := \n    core_points_finset_up_to_m_property premap_p core_points_finset_up_to_m_property_precondition,\n  dsimp only[res],\n  apply finset.mem_union_right,\n  dsimp only[core_points_finset_up_to_m_mapped],\n  simp only [exists_prop, fin.val_eq_coe, finset.mem_map, exists_and_distrib_right, ge_iff_le, function.embedding.coe_fn_mk,\n    subtype.exists, subtype.coe_mk, subtype.val_eq_coe],\n  use premap_p,\n  split,\n  { use core_points_finset_up_to_m_property_precondition,\n    exact premap_p_in_core_points_finset_up_to_m,\n  },\n  apply vector.ext,\n  intro j,\n  simp only [vector.nth_of_fn, subtype.val_eq_coe],\n  rcases h j with \u27e8h_i_eq_j, h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n  by_cases i_eq_j : i = j,\n  { rw if_pos i_eq_j,\n    symmetry,\n    exact h_i_eq_j i_eq_j,\n  },\n  rename i_eq_j i_ne_j,\n  rw [if_neg i_ne_j, if_neg i_ne_j],\n  by_cases j_eq_m : \u2191j = m,\n  { rw [if_pos j_eq_m, if_pos j_eq_m, zero_add],\n    symmetry,\n    have j_lt_m_succ : j.val < m.succ := by {simp only [fin.val_eq_coe], rw [j_eq_m, nat.succ_eq_add_one], norm_num},\n    replace h_j_lt_m_succ := h_j_lt_m_succ j_lt_m_succ,\n    cases h_j_lt_m_succ with p_eq_zero p_eq_one,\n    { exfalso,\n      have j_subst : j = \u27e8m, m_lt_d\u27e9 := by {apply subtype.ext, rw j_eq_m, refl},\n      rw \u2190 j_subst at p_ne_zero_at_m,\n      exact p_ne_zero_at_m p_eq_zero,\n    },\n    exact p_eq_one,\n  },\n  rename j_eq_m j_ne_m,\n  rw [if_neg j_ne_m, if_neg j_ne_m],\nend\n\nnoncomputable def build_core_points_finset {d : \u2115} (n : fin (d + 1)) :\n  {s: finset ({p : point d // \u2200 j : fin d, (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)}) //\n    s.card = 2^n.val \u2227 \n    \u2200 p : point d, \u2200 h : (\u2200 j : fin d, (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)),\n    (\u27e8p, h\u27e9 : {p : point d // \u2200 j : fin d, (j.val < n.val \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 n.val \u2192 p.nth j = 0)}) \u2208 s\n  } :=\nbegin\n  cases n with n_val n_property,\n  simp only [fin.val_eq_coe, ge_iff_le],\n  induction n_val with m ih,\n  { norm_num,\n    let res : finset ({p : point d // \u2200 j : fin d, (j.val < 0 \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 0 \u2192 p.nth j = 0)}) :=\n      begin\n        let zero_vector : {p : point d // \u2200 j : fin d, (j.val < 0 \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 0 \u2192 p.nth j = 0)} :=\n          begin\n            let zero_vector : point d := vector.of_fn (\u03bb i, 0),\n            use zero_vector,\n            intro j,\n            split, {intro j_lt_zero, dsimp[zero_vector], simp only [or_false, vector.nth_of_fn, zero_ne_one]},\n            intro j_ge_zero,\n            dsimp[zero_vector],\n            simp only [vector.nth_of_fn],\n          end,\n        exact {zero_vector},\n      end,\n    use res,\n    split, by {rw finset.card_eq_one, use (vector.of_fn (\u03bb i, 0)),},\n    intros p h,\n    dsimp[res],\n    simp only [finset.mem_singleton],\n    apply vector.ext,\n    intro j,\n    simp only [vector.nth_of_fn],\n    exact h j,\n  },\n  have m_lt_d_add_one : m < d + 1 := by omega,\n  rcases ih m_lt_d_add_one with \n    \u27e8core_points_finset_up_to_m, \u27e8core_points_finset_up_to_m_card, core_points_finset_up_to_m_property\u27e9\u27e9,\n  let next_level_map : \n    {p : point d // \u2200 (j : fin d), (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u21aa \n    {p : point d // \u2200 (j : fin d), (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} :=\n    begin\n      let next_level_map_fn :\n        {p : point d // \u2200 (j : fin d), (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u2192\n        {p : point d // \u2200 (j : fin d), (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} := \u03bb p,\n        begin\n          use (vector.of_fn (\u03bb j, if (j.val = m) then p.val.nth j + 1 else p.val.nth j)),\n          intro j,\n          simp only [fin.val_eq_coe, ge_iff_le, vector.nth_of_fn, subtype.val_eq_coe],\n          split,\n          { intro j_lt_m_succ,\n            by_cases j_eq_m : \u2191j = m,\n            { rw if_pos j_eq_m,\n              right,\n              have p_property := p.property j,\n              rcases p_property with \u27e8_, p_property\u27e9,\n              simp only [ge_iff_le, subtype.val_eq_coe, j_eq_m] at p_property,\n              rw [(p_property (by refl)), zero_add],\n            },\n            rename j_eq_m j_ne_m,\n            rw if_neg j_ne_m,\n            have p_property := p.property j,\n            rcases p_property with \u27e8p_property_j_lt_m, p_property_j_ge_m\u27e9,\n            cases lt_or_ge j.val m with j_lt_m j_ge_m, exact p_property_j_lt_m j_lt_m,\n            left,\n            exact p_property_j_ge_m j_ge_m,\n          },\n          intro m_succ_le_j,\n          have j_ne_m : \u2191j \u2260 m := by {clear_except m_succ_le_j, rw nat.succ_eq_add_one at m_succ_le_j, linarith},\n          rw if_neg j_ne_m,\n          have p_property := p.property j,\n          rcases p_property with \u27e8_, p_property_j_ge_m\u27e9,\n          have j_ge_m : \u2191j \u2265 m := by {clear_except m_succ_le_j, rw nat.succ_eq_add_one at m_succ_le_j, linarith},\n          simp only [subtype.val_eq_coe] at p_property_j_ge_m,\n          exact p_property_j_ge_m j_ge_m,\n        end,\n      have next_level_map_fn_injective : function.injective next_level_map_fn :=\n        begin\n          rw function.injective,\n          intros p1 p2 p1_output_eq_p2_output,\n          apply subtype.ext,\n          apply vector.ext,\n          intro j,\n          replace p1_output_eq_p2_output : (next_level_map_fn p1).val.nth j = (next_level_map_fn p2).val.nth j := \n            by rw p1_output_eq_p2_output,\n          dsimp[next_level_map_fn] at p1_output_eq_p2_output,\n          simp only [vector.nth_of_fn] at p1_output_eq_p2_output,\n          by_cases j_eq_m : \u2191j = m,\n          { rw [if_pos j_eq_m, if_pos j_eq_m] at p1_output_eq_p2_output,\n            clear_except p1_output_eq_p2_output,\n            linarith,\n          },\n          rename j_eq_m j_ne_m,\n          rw [if_neg j_ne_m, if_neg j_ne_m] at p1_output_eq_p2_output,\n          exact p1_output_eq_p2_output,\n        end,\n      exact {to_fun := next_level_map_fn, inj' := next_level_map_fn_injective},\n    end,\n  let core_points_finset_up_to_m_mapped := finset.map next_level_map core_points_finset_up_to_m,\n  have core_points_finset_up_to_m_mapped_card : core_points_finset_up_to_m_mapped.card = core_points_finset_up_to_m.card := \n    finset.card_map next_level_map,\n  let cast_map :\n    {p : point d // \u2200 (j : fin d), (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u21aa \n    {p : point d // \u2200 (j : fin d), (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)} :=\n    begin\n      let cast_map_fn :\n        {p : point d // \u2200 (j : fin d), (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0)} \u2192\n        {p : point d // \u2200 (j : fin d), (j.val < m.succ \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m.succ \u2192 vector.nth p j = 0)}\n        := \u03bb p,\n        begin\n          use p.val,\n          intro j,\n          have p_property := p.property j,\n          rcases p_property with \u27e8p_property_j_lt_m, p_property_j_ge_m\u27e9,\n          split,\n          { intro j_lt_m_succ,\n            by_cases j_lt_m : j.val < m, exact p_property_j_lt_m j_lt_m,\n            rename j_lt_m j_ge_m,\n            replace j_ge_m : j.val \u2265 m := by {clear_except j_ge_m, linarith},\n            left,\n            exact p_property_j_ge_m j_ge_m,\n          },\n          intro j_ge_m_succ,\n          have j_ge_m : j.val \u2265 m := by {clear_except j_ge_m_succ, rw nat.succ_eq_add_one at j_ge_m_succ, linarith},\n          exact p_property_j_ge_m j_ge_m,\n        end,\n      have cast_map_fn_injective : function.injective cast_map_fn :=\n        begin\n          rw function.injective,\n          intros p1 p2 p1_output_eq_p2_output,\n          dsimp[cast_map_fn] at p1_output_eq_p2_output,\n          simp only at p1_output_eq_p2_output,\n          refine subtype.eq _,\n          simp only [subtype.val_eq_coe],\n          exact p1_output_eq_p2_output,\n        end,\n      exact {to_fun := cast_map_fn, inj' := cast_map_fn_injective},\n    end,\n  let core_points_finset_up_to_m_cast := finset.map cast_map core_points_finset_up_to_m,\n  have core_points_finset_up_to_m_cast_card : core_points_finset_up_to_m_cast.card = core_points_finset_up_to_m.card := \n    finset.card_map cast_map,\n  have cast_disjoint_with_mapped : disjoint core_points_finset_up_to_m_cast core_points_finset_up_to_m_mapped :=\n    begin\n      rw disjoint,\n      simp only [finset.inf_eq_inter, finset.bot_eq_empty, finset.le_eq_subset],\n      intros shared_point shared_point_in_both,\n      simp only [finset.not_mem_empty, finset.bot_eq_empty],\n      simp only [le_refl, exists_prop, add_zero, mul_one, gt_iff_lt, fin.val_eq_coe, finset.mem_map, int.coe_nat_add, ge_iff_le,\n        int.coe_nat_one, function.embedding.coe_fn_mk, vector.nth_of_fn, zero_add, neg_eq_zero, subtype.exists, neg_neg,\n        subtype.coe_mk, zero_lt_one, finset.mem_inter, subtype.val_eq_coe, neg_zero] at shared_point_in_both,\n      rcases shared_point_in_both with\n        \u27e8\u27e8shared_point_precast, shared_point_precast_property, shared_point_precast_in_core_points, shared_point_precast_eq_shared_point\u27e9, \n        \u27e8shared_point_premap, shared_point_premap_property, shared_point_premap_in_core_points, shared_point_premap_eq_shared_point\u27e9\u27e9,\n      have m_lt_d : m < d := by {clear_except n_property, rw nat.succ_eq_add_one at n_property, linarith},\n      rcases shared_point_precast_property \u27e8m, m_lt_d\u27e9 with \u27e8_, h1\u27e9,\n      rcases shared_point_premap_property \u27e8m, m_lt_d\u27e9 with \u27e8_, h2\u27e9,\n      replace h1 := h1 rfl.ge,\n      replace h2 := h2 rfl.ge,\n      replace shared_point_precast_eq_shared_point : shared_point_precast = shared_point.val :=\n        (congr_arg subtype.val shared_point_precast_eq_shared_point).congr_right.mp rfl,\n      replace shared_point_premap_eq_shared_point : vector.of_fn\n        (\u03bb (j : fin d), ite (\u2191j = m) (vector.nth shared_point_premap j + 1) (vector.nth shared_point_premap j))\n        = shared_point.val := (congr_arg subtype.val shared_point_premap_eq_shared_point).congr_right.mp rfl,\n      have shared_point_precast_eq_shared_point_at_m : shared_point_precast.nth \u27e8m, m_lt_d\u27e9 = shared_point.val.nth \u27e8m, m_lt_d\u27e9 :=\n        by rw shared_point_precast_eq_shared_point,\n      rw \u2190 shared_point_premap_eq_shared_point at shared_point_precast_eq_shared_point_at_m,\n      simp only [if_true, eq_self_iff_true, vector.nth_of_fn, fin.coe_mk] at shared_point_precast_eq_shared_point_at_m,\n      rw [h1, h2] at shared_point_precast_eq_shared_point_at_m,\n      norm_num at shared_point_precast_eq_shared_point_at_m,\n    end,\n  let res := core_points_finset_up_to_m_cast \u222a core_points_finset_up_to_m_mapped,\n  have res_card : res.card = 2^m.succ :=\n    begin\n      dsimp[res],\n      rw [finset.card_disjoint_union cast_disjoint_with_mapped, core_points_finset_up_to_m_cast_card,\n        core_points_finset_up_to_m_mapped_card, core_points_finset_up_to_m_card],\n      rw [nat.succ_eq_add_one, pow_succ, two_mul],\n    end,\n  use [res, res_card],\n  intros p h,\n  have m_lt_d : m < d := by {clear_except n_property, omega},\n  by_cases p_eq_zero_at_m : p.nth \u27e8m, m_lt_d\u27e9 = 0,\n  { have core_points_finset_up_to_m_property_precondition : \u2200 (j : fin d), \n      (j.val < m \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth p j = 0) :=\n      begin\n        intro j,\n        rcases h j with \u27e8h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n        split,\n        { intro j_lt_m,\n          have j_lt_m_succ : j.val < m.succ := by {clear_except j_lt_m, rw nat.succ_eq_add_one, linarith},\n          exact h_j_lt_m_succ j_lt_m_succ,\n        },\n        intro j_ge_m,\n        by_cases j_eq_m : j = \u27e8m, m_lt_d\u27e9,\n        { rw \u2190 j_eq_m at p_eq_zero_at_m,\n          exact p_eq_zero_at_m,\n        },\n        rename j_eq_m j_ne_m,\n        replace j_ne_m : j.val \u2260 m :=\n          begin\n            simp only [fin.val_eq_coe],\n            intro j_eq_m,\n            conv at j_ne_m\n            begin\n              find m {rw \u2190 j_eq_m},\n            end,\n            simp only [eq_self_iff_true, not_true, fin.eta] at j_ne_m,\n            exact j_ne_m,\n          end,\n        have j_ge_m_succ : j.val \u2265 m.succ := by {clear_except j_ge_m j_ne_m, omega},\n        exact h_j_ge_m_succ j_ge_m_succ,\n      end,\n    have p_in_core_points_finset_up_to_m := core_points_finset_up_to_m_property p core_points_finset_up_to_m_property_precondition,\n    dsimp only[res],\n    apply finset.mem_union_left,\n    dsimp only[core_points_finset_up_to_m_cast],\n    simp only [exists_prop, fin.val_eq_coe, finset.mem_map, exists_and_distrib_right, ge_iff_le, exists_eq_right,\n      function.embedding.coe_fn_mk, subtype.exists, subtype.coe_mk, subtype.val_eq_coe],\n    use core_points_finset_up_to_m_property_precondition,\n    exact p_in_core_points_finset_up_to_m,\n  },\n  rename p_eq_zero_at_m p_ne_zero_at_m,\n  let premap_p : point d := vector.of_fn (\u03bb j : fin d, if(j.val = m) then 0 else p.nth j),\n  have core_points_finset_up_to_m_property_precondition : \u2200 (j : fin d), \n    (j.val < m \u2192 vector.nth premap_p j = 0 \u2228 vector.nth premap_p j = 1) \u2227 (j.val \u2265 m \u2192 vector.nth premap_p j = 0) :=\n    begin\n      intro j,\n      dsimp only[premap_p],\n      simp only [fin.val_eq_coe, ge_iff_le, vector.nth_of_fn, ite_eq_left_iff],\n      rcases h j with \u27e8h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n      split,\n      { intro j_lt_m,\n        by_cases j_eq_m : \u2191j = m,\n        { left,\n          intro j_ne_m,\n          exfalso,\n          exact j_ne_m j_eq_m,\n        },\n        rename j_eq_m j_ne_m,\n        have j_lt_m_succ : j.val < m.succ := by {clear_except j_lt_m, rw nat.succ_eq_add_one, simp only [fin.val_eq_coe], linarith},\n        cases h_j_lt_m_succ j_lt_m_succ with p_eq_zero p_eq_one,\n        { left,\n          intro j_ne_m,\n          exact p_eq_zero,\n        },\n        right,\n        rw if_neg j_ne_m,\n        exact p_eq_one,\n      },\n      intros m_le_j j_ne_m,\n      have j_ge_m : j.val \u2265 m.succ := by {simp only [fin.val_eq_coe], clear_except m_le_j j_ne_m, omega},\n      exact h_j_ge_m_succ j_ge_m,\n    end,\n  have premap_p_in_core_points_finset_up_to_m := \n    core_points_finset_up_to_m_property premap_p core_points_finset_up_to_m_property_precondition,\n  dsimp only[res],\n  apply finset.mem_union_right,\n  dsimp only[core_points_finset_up_to_m_mapped],\n  simp only [exists_prop, fin.val_eq_coe, finset.mem_map, exists_and_distrib_right, ge_iff_le, function.embedding.coe_fn_mk,\n    subtype.exists, subtype.coe_mk, subtype.val_eq_coe],\n  use premap_p,\n  split,\n  { use core_points_finset_up_to_m_property_precondition,\n    exact premap_p_in_core_points_finset_up_to_m,\n  },\n  apply vector.ext,\n  intro j,\n  simp only [vector.nth_of_fn, subtype.val_eq_coe],\n  rcases h j with \u27e8h_j_lt_m_succ, h_j_ge_m_succ\u27e9,\n  by_cases j_eq_m : \u2191j = m,\n  { rw [if_pos j_eq_m, if_pos j_eq_m, zero_add],\n    symmetry,\n    have j_lt_m_succ : j.val < m.succ := by {simp only [fin.val_eq_coe], rw [j_eq_m, nat.succ_eq_add_one], norm_num},\n    replace h_j_lt_m_succ := h_j_lt_m_succ j_lt_m_succ,\n    cases h_j_lt_m_succ with p_eq_zero p_eq_one,\n    { exfalso,\n      have j_subst : j = \u27e8m, m_lt_d\u27e9 := by {apply subtype.ext, rw j_eq_m, refl},\n      rw \u2190 j_subst at p_ne_zero_at_m,\n      exact p_ne_zero_at_m p_eq_zero,\n    },\n    exact p_eq_one,\n  },\n  rename j_eq_m j_ne_m,\n  rw [if_neg j_ne_m, if_neg j_ne_m],\nend\n\ntheorem s_discrete_upper_bound :\n  \u2200 d : \u2115, \u2200 T : set (point d), \u2200 T_is_tiling : is_tiling T,\n  d > 0 \u2192 is_periodic T_is_tiling \u2192 is_s_discrete (2^(d-1)) T :=\nbegin\n  intros d T T_is_tiling d_gt_zero T_is_periodic,\n  rw is_s_discrete,\n  intro i,\n  have core_points_finset := build_half_core_points_finset i \u27e8d, lt_add_one d\u27e9,\n  rcases core_points_finset with \u27e8core_points_finset, core_points_finset_card, vacuous, core_points_finset_property\u27e9,\n  clear vacuous, --vacuous was useful inductively for build_half_core_points_finset, always has a false hypothesis here because i : fin d\n  simp only at core_points_finset_card,\n  replace core_points_finset_card := core_points_finset_card i.property,\n  let T_core_map :\n    {p : point d // \u2200 (j : fin d), \n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 d \u2192 vector.nth p j = 0)} \u21aa\n    {p_corner : point d // p_corner \u2208 T \u2227 \u2203 p : point d, in_cube p_corner p \u2227 \u2200 (j : fin d),\n      (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 d \u2192 vector.nth p j = 0)} :=\n    begin\n      let T_core_map_fn :\n        {p : point d // \u2200 (j : fin d), \n          (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 d \u2192 vector.nth p j = 0)} \u2192\n        {p_corner : point d // p_corner \u2208 T \u2227 \u2203 p : point d, in_cube p_corner p \u2227 \u2200 (j : fin d),\n          (i = j \u2192 vector.nth p j = 0) \u2227 (j.val < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 d \u2192 vector.nth p j = 0)}\n        := \u03bb p,\n        begin\n          use (point_to_corner T_is_tiling p).val,\n          let p_corner := (point_to_corner T_is_tiling p).val,\n          have p_corner_def : p_corner = (point_to_corner T_is_tiling p).val := by refl,\n          have p_corner_property := (point_to_corner T_is_tiling p).property,\n          rw \u2190 p_corner_def at p_corner_property,\n          rcases p_corner_property with \u27e8p_corner_in_T, p_in_p_corner, p_corner_unique\u27e9,\n          rw cube at p_in_p_corner,\n          simp only [set.mem_set_of_eq] at p_in_p_corner,\n          split, exact p_corner_in_T,\n          use p,\n          split, exact p_in_p_corner,\n          intro j,\n          have p_property := p.property j,\n          simp only [subtype.val_eq_coe] at p_property,\n          simp only [subtype.val_eq_coe],\n          exact p_property,\n        end,\n      have T_core_map_fn_injective : function.injective T_core_map_fn :=\n        begin\n          intros p1 p2 p1_output_eq_p2_output,\n          dsimp[T_core_map_fn] at p1_output_eq_p2_output,\n          simp only at p1_output_eq_p2_output,\n          apply subtype.ext,\n          apply vector.ext,\n          intro j,\n          have p1_property := p1.property j,\n          have p2_property := p2.property j,\n          rcases p1_property with \u27e8unneeded, p1_property, vacuous\u27e9,\n          clear vacuous unneeded,\n          rcases p2_property with \u27e8unneeded, p2_property, vacuous\u27e9,\n          clear vacuous unneeded,\n          replace p1_property := p1_property j.property,\n          replace p2_property := p2_property j.property,\n          have p1_corner_property := (point_to_corner T_is_tiling \u2191p1).property,\n          simp only [subtype.val_eq_coe] at p1_corner_property,\n          rcases p1_corner_property with \u27e8p1_corner_in_T, p1_in_p1_corner, p1_corner_unique\u27e9,\n          have p2_corner_property := (point_to_corner T_is_tiling \u2191p2).property,\n          simp only [subtype.val_eq_coe] at p2_corner_property,\n          rcases p2_corner_property with \u27e8p2_corner_in_T, p2_in_p2_corner, p2_corner_unique\u27e9,\n          simp only [subtype.val_eq_coe] at p1_property p2_property,\n          cases p1_property with p1_eq_zero p1_eq_one,\n          { cases p2_property with p2_eq_zero p2_eq_one, rw [p1_eq_zero, p2_eq_zero],\n            rw cube at p1_in_p1_corner p2_in_p2_corner,\n            simp only [set.mem_set_of_eq] at p1_in_p1_corner p2_in_p2_corner,\n            rw in_cube at p1_in_p1_corner p2_in_p2_corner,\n            replace p1_in_p1_corner := p1_in_p1_corner j,\n            replace p2_in_p2_corner := p2_in_p2_corner j,\n            rw p2_eq_one at p2_in_p2_corner,\n            rw [p1_eq_zero, p1_output_eq_p2_output] at p1_in_p1_corner,\n            exfalso,\n            clear_except p1_in_p1_corner p2_in_p2_corner,\n            linarith,\n          },\n          cases p2_property with p2_eq_zero p2_eq_one,\n          { rw cube at p1_in_p1_corner p2_in_p2_corner,\n            simp only [set.mem_set_of_eq] at p1_in_p1_corner p2_in_p2_corner,\n            rw in_cube at p1_in_p1_corner p2_in_p2_corner,\n            replace p1_in_p1_corner := p1_in_p1_corner j,\n            replace p2_in_p2_corner := p2_in_p2_corner j,\n            rw p2_eq_zero at p2_in_p2_corner,\n            rw [p1_eq_one, p1_output_eq_p2_output] at p1_in_p1_corner,\n            exfalso,\n            clear_except p1_in_p1_corner p2_in_p2_corner,\n            linarith,\n          },\n          rw [p1_eq_one, p2_eq_one],\n        end,\n      exact {to_fun := T_core_map_fn, inj' := T_core_map_fn_injective},\n    end,\n  let T_core := finset.map T_core_map core_points_finset,\n  have T_core_card : T_core.card = core_points_finset.card := finset.card_map T_core_map,\n  rw core_points_finset_card at T_core_card,\n  let T_core_list := finset.to_list T_core,\n  let T_core_point_to_i_coord_fn :=\n    (\u03bb corner : {p_corner // p_corner \u2208 T \u2227 \u2203 (p : point d), in_cube p_corner p \u2227 \u2200 (j : fin d),(i = j \u2192 vector.nth p j = 0) \u2227\n                  (j.val < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (j.val \u2265 d \u2192 vector.nth p j = 0)}, \n      corner.val.nth i\n    ),\n  let coords_list := list.map T_core_point_to_i_coord_fn T_core_list,\n  have T_core_list_length : T_core_list.length = T_core.card := finset.length_to_list T_core,\n  have coords_list_length : coords_list.length = T_core_list.length := list.length_map T_core_point_to_i_coord_fn T_core_list,\n  let coords := coords_list.to_finset,\n  have coords_card : coords.card \u2264 coords_list.length := list.to_finset_card_le coords_list,\n  rw [coords_list_length, T_core_list_length, T_core_card] at coords_card,\n  use coords,\n  split, exact coords_card,\n  split,\n  { --Derive contradiction between coord1_eq_coord2_mod_one and coord1_ne_coord2\n    intros coord1 coord1_in_coords coord2 coord2_in_coords coord1_ne_coord2 coord1_eq_coord2_mod_one,\n    have coord1_in_coords_list := by {rw list.mem_to_finset at coord1_in_coords, exact coord1_in_coords},\n    have coord2_in_coords_list := by {rw list.mem_to_finset at coord2_in_coords, exact coord2_in_coords},\n    dsimp[coords_list, T_core_point_to_i_coord_fn] at coord1_in_coords_list coord2_in_coords_list,\n    simp only [not_exists, exists_prop, add_zero, list.mem_map, fin.val_eq_coe, finset.mem_map, ge_iff_le, finset.mem_to_list,\n      subtype.mk_eq_mk, function.embedding.coe_fn_mk, zero_add, subtype.exists, set.mem_set_of_eq, subtype.coe_mk,\n      subtype.val_eq_coe, list.map] at coord1_in_coords_list coord2_in_coords_list,\n    rcases coord1_in_coords_list with\n      \u27e8p1, \u27e8p1_in_T, \u27e8p1_core_point, p1_core_point_in_p1, p1_core_point_property\u27e9\u27e9, redundant, coord1_eq_p1_at_i\u27e9,\n    clear redundant,\n    rcases coord2_in_coords_list with\n      \u27e8p2, \u27e8p2_in_T, \u27e8p2_core_point, p2_core_point_in_p2, p2_core_point_property\u27e9\u27e9, redundant, coord2_eq_p2_at_i\u27e9,\n    clear redundant,\n    rcases p1_core_point_property i with \u27e8p1_core_point_eq_zero_at_i, unneeded\u27e9,\n    clear unneeded,\n    rcases p2_core_point_property i with \u27e8p2_core_point_eq_zero_at_i, unneeded\u27e9,\n    clear unneeded,\n    replace p1_core_point_eq_zero_at_i := p1_core_point_eq_zero_at_i (by refl),\n    replace p2_core_point_eq_zero_at_i := p2_core_point_eq_zero_at_i (by refl),\n    rw in_cube at p1_core_point_in_p1 p2_core_point_in_p2,\n    replace p1_core_point_in_p1 := p1_core_point_in_p1 i,\n    replace p2_core_point_in_p2 := p2_core_point_in_p2 i,\n    rcases p1_core_point_in_p1 with \u27e8p1_le_p1_core_point, p1_core_point_lt_p1_add_1\u27e9,\n    rcases p2_core_point_in_p2 with \u27e8p2_le_p2_core_point, p2_core_point_lt_p2_add_2\u27e9,\n    rcases coord1_eq_coord2_mod_one with \n      \u27e8coord1_floor, coord2_floor, y, zero_le_y, y_lt_one, coord1_eq_coord1_floor_add_y, coord2_eq_coord2_floor_add_y\u27e9,\n    have coord1_floor_eq_coord2_floor : coord1_floor = coord2_floor :=\n      begin\n        rw [p1_core_point_eq_zero_at_i, coord1_eq_p1_at_i] at p1_le_p1_core_point p1_core_point_lt_p1_add_1,\n        rw [p2_core_point_eq_zero_at_i, coord2_eq_p2_at_i] at p2_le_p2_core_point p2_core_point_lt_p2_add_2,\n        by_contra coord1_floor_ne_coord2_floor,\n        cases lt_or_gt_of_ne coord1_floor_ne_coord2_floor with coord1_floor_lt_coord2_floor coord1_floor_gt_coord2_floor,\n        { have coord1_floor_le_coord2_floor_add_one : coord1_floor \u2264 coord2_floor + 1 :=\n            by {clear_except coord1_floor_lt_coord2_floor, omega},\n          by_cases y_eq_zero : y = 0,\n          { rw [y_eq_zero, add_zero] at coord1_eq_coord1_floor_add_y coord2_eq_coord2_floor_add_y,\n            rcases real_eq_or_lt_or_gt coord1 0 with coord1_eq_zero | coord1_lt_zero | coord1_gt_zero,\n            { have neg_one_lt_coord2 : -1 < coord2 := by {clear_except p2_core_point_lt_p2_add_2, linarith},\n              have zero_le_coord2 : 0 \u2264 coord2 :=\n                begin\n                  have neg_one_lt_coord2_floor : -1 < coord2_floor :=\n                    by {rw coord2_eq_coord2_floor_add_y at neg_one_lt_coord2, exact_mod_cast neg_one_lt_coord2},\n                  have h := int.add_one_le_of_lt neg_one_lt_coord2_floor,\n                  simp only [add_left_neg] at h,\n                  rw coord2_eq_coord2_floor_add_y,\n                  exact_mod_cast h,\n                end,\n              have coord2_eq_zero : coord2 = 0 := by {clear_except p2_le_p2_core_point zero_le_coord2, linarith},\n              rw \u2190 coord1_eq_zero at coord2_eq_zero,\n              symmetry' at coord2_eq_zero,\n              exact coord1_ne_coord2 coord2_eq_zero,\n            },\n            { have neg_one_lt_coord1_floor : -1 < coord1_floor :=\n                begin\n                  have neg_one_lt_coord1 : -1 < coord1 := by {clear_except p1_core_point_lt_p1_add_1, linarith},\n                  rw coord1_eq_coord1_floor_add_y at neg_one_lt_coord1,\n                  exact_mod_cast neg_one_lt_coord1,\n                end,\n              have coord1_floor_lt_zero : coord1_floor < 0 :=\n                by {rw coord1_eq_coord1_floor_add_y at coord1_lt_zero, exact_mod_cast coord1_lt_zero},\n              clear_except neg_one_lt_coord1_floor coord1_floor_lt_zero,\n              omega,\n            },\n            clear_except coord1_gt_zero p1_le_p1_core_point,\n            linarith,\n          },\n          rename y_eq_zero y_ne_zero,\n          rcases lt_or_gt_of_ne y_ne_zero with y_lt_zero | y_gt_zero,\n          { clear_except zero_le_y y_lt_zero,\n            linarith,\n          },\n          rcases eq_or_lt_or_gt coord1_floor (-1) with coord1_floor_eq_neg_one | coord1_floor_lt_neg_one | coord1_floor_gt_neg_one,\n          { have zero_le_coord2_floor : 0 \u2264 coord2_floor :=\n              begin\n                rw coord1_floor_eq_neg_one at coord1_floor_lt_coord2_floor,\n                clear_except coord1_floor_lt_coord2_floor,\n                omega,\n              end,\n            have zero_le_cast_coord2_floor : (0 : \u211d) \u2264 \u2191coord2_floor := by {exact_mod_cast zero_le_coord2_floor},\n            clear_except zero_le_cast_coord2_floor coord2_eq_coord2_floor_add_y y_gt_zero p2_le_p2_core_point,\n            linarith,\n          },\n          { have coord1_floor_le_neg_two : coord1_floor \u2264 -2 := by {clear_except coord1_floor_lt_neg_one, omega},\n            have cast_coord1_floor_le_neg_two : \u2191coord1_floor \u2264 (-2 : \u211d) := by {exact_mod_cast coord1_floor_le_neg_two},\n            rw coord1_eq_coord1_floor_add_y at p1_core_point_lt_p1_add_1,\n            clear_except p1_core_point_lt_p1_add_1 y_lt_one cast_coord1_floor_le_neg_two,\n            linarith,\n          },\n          have coord1_floor_ge_zero : coord1_floor \u2265 0 := by {clear_except coord1_floor_gt_neg_one, omega},\n          have cast_coord1_floor_ge_zero : \u2191coord1_floor \u2265 (0 : \u211d) := by {exact_mod_cast coord1_floor_ge_zero},\n          rw coord1_eq_coord1_floor_add_y at p1_le_p1_core_point,\n          clear_except p1_le_p1_core_point y_gt_zero cast_coord1_floor_ge_zero,\n          linarith,\n        },\n        --Symmetric to above case\n        have coord2_floor_le_coord1_floor_add_one : coord2_floor \u2264 coord1_floor + 1 :=\n          by {clear_except coord1_floor_gt_coord2_floor, omega},\n        by_cases y_eq_zero : y = 0,\n        { rw [y_eq_zero, add_zero] at coord1_eq_coord1_floor_add_y coord2_eq_coord2_floor_add_y,\n          rcases real_eq_or_lt_or_gt coord2 0 with coord2_eq_zero | coord2_lt_zero | coord2_gt_zero,\n          { have neg_one_lt_coord1 : -1 < coord1 := by {clear_except p1_core_point_lt_p1_add_1, linarith},\n            have zero_le_coord1 : 0 \u2264 coord1 :=\n              begin\n                have neg_one_lt_coord1_floor : -1 < coord1_floor :=\n                  by {rw coord1_eq_coord1_floor_add_y at neg_one_lt_coord1, exact_mod_cast neg_one_lt_coord1},\n                have h := int.add_one_le_of_lt neg_one_lt_coord1_floor,\n                simp only [add_left_neg] at h,\n                rw coord1_eq_coord1_floor_add_y,\n                exact_mod_cast h,\n              end,\n            have coord1_eq_zero : coord1 = 0 := by {clear_except p1_le_p1_core_point zero_le_coord1, linarith},\n            rw \u2190 coord2_eq_zero at coord1_eq_zero,\n            exact coord1_ne_coord2 coord1_eq_zero,\n          },\n          { have neg_one_lt_coord2_floor : -1 < coord2_floor :=\n              begin\n                have neg_one_lt_coord2 : -1 < coord2 := by {clear_except p2_core_point_lt_p2_add_2, linarith},\n                rw coord2_eq_coord2_floor_add_y at neg_one_lt_coord2,\n                exact_mod_cast neg_one_lt_coord2,\n              end,\n            have coord2_floor_lt_zero : coord2_floor < 0 :=\n              by {rw coord2_eq_coord2_floor_add_y at coord2_lt_zero, exact_mod_cast coord2_lt_zero},\n            clear_except neg_one_lt_coord2_floor coord2_floor_lt_zero,\n            omega,\n          },\n          clear_except coord2_gt_zero p2_le_p2_core_point,\n          linarith,\n        },\n        rename y_eq_zero y_ne_zero,\n        rcases lt_or_gt_of_ne y_ne_zero with y_lt_zero | y_gt_zero,\n        { clear_except zero_le_y y_lt_zero,\n          linarith,\n        },\n        rcases eq_or_lt_or_gt coord2_floor (-1) with coord2_floor_eq_neg_one | coord2_floor_lt_neg_one | coord2_floor_gt_neg_one,\n        { have zero_le_coord1_floor : 0 \u2264 coord1_floor :=\n            begin\n              rw coord2_floor_eq_neg_one at coord1_floor_gt_coord2_floor,\n              clear_except coord1_floor_gt_coord2_floor,\n              omega,\n            end,\n          have zero_le_cast_coord1_floor : (0 : \u211d) \u2264 \u2191coord1_floor := by {exact_mod_cast zero_le_coord1_floor},\n          clear_except zero_le_cast_coord1_floor coord1_eq_coord1_floor_add_y y_gt_zero p1_le_p1_core_point,\n          linarith,\n        },\n        { have coord2_floor_le_neg_two : coord2_floor \u2264 -2 := by {clear_except coord2_floor_lt_neg_one, omega},\n          have cast_coord2_floor_le_neg_two : \u2191coord2_floor \u2264 (-2 : \u211d) := by {exact_mod_cast coord2_floor_le_neg_two},\n          rw coord2_eq_coord2_floor_add_y at p2_core_point_lt_p2_add_2,\n          clear_except p2_core_point_lt_p2_add_2 y_lt_one cast_coord2_floor_le_neg_two,\n          linarith,\n        },\n        have coord2_floor_ge_zero : coord2_floor \u2265 0 := by {clear_except coord2_floor_gt_neg_one, omega},\n        have cast_coord2_floor_ge_zero : \u2191coord2_floor \u2265 (0 : \u211d) := by {exact_mod_cast coord2_floor_ge_zero},\n        rw coord2_eq_coord2_floor_add_y at p2_le_p2_core_point,\n        clear_except p2_le_p2_core_point y_gt_zero cast_coord2_floor_ge_zero,\n        linarith,\n      end,\n    rw [coord1_floor_eq_coord2_floor, \u2190 coord2_eq_coord2_floor_add_y] at coord1_eq_coord1_floor_add_y,\n    exact coord1_ne_coord2 coord1_eq_coord1_floor_add_y,\n  },\n  intros t t_in_T,\n  have t_has_periodic_core := (has_periodic_core_of_is_periodic d T T_is_tiling T_is_periodic) t t_in_T,\n  rcases t_has_periodic_core with \u27e8p, p_in_core_points, t_core, t_core_in_T, t_offset, p_in_t_core, t_def\u27e9,\n  by_cases p_eq_zero_at_i : p.nth i = 0,\n  { have p_has_core_points_finset_property : \n      \u2200 j : fin d, (i = j \u2192 p.nth j = 0) \u2227 (j.val < d \u2192 p.nth j = 0 \u2228 p.nth j = 1) \u2227 (j.val \u2265 d \u2192 p.nth j = 0) :=\n      begin\n        intro j,\n        replace p_in_core_points := p_in_core_points j,\n        split, {intro i_eq_j, rw \u2190 i_eq_j, exact p_eq_zero_at_i},\n        split, {intro j_lt_d, exact p_in_core_points},\n        intro j_ge_d,\n        exfalso,\n        have j_lt_d := j.property,\n        clear_except j_ge_d j_lt_d,\n        linarith,\n      end,\n    have p_in_core_points_finset := core_points_finset_property p p_has_core_points_finset_property,\n    have t_core_has_T_core_map_property : t_core \u2208 T \u2227 \u2203 p : point d, in_cube t_core p \u2227 \n      \u2200 (j : fin d), (i = j \u2192 vector.nth p j = 0) \u2227 (\u2191j < d \u2192 vector.nth p j = 0 \u2228 vector.nth p j = 1) \u2227 (\u2191j \u2265 d \u2192 vector.nth p j = 0)\n      := \u27e8t_core_in_T, Exists.intro p \u27e8p_in_t_core, p_has_core_points_finset_property\u27e9\u27e9,\n    have t_core_eq_p_corner : t_core = \u2191(point_to_corner T_is_tiling p) :=\n      begin\n        have p_corner_property := (point_to_corner T_is_tiling p).property,\n        simp only [subtype.val_eq_coe] at p_corner_property,\n        rcases p_corner_property with \u27e8p_corner_in_T, p_in_p_corner, p_corner_unique\u27e9,\n        replace p_in_t_core : p \u2208 cube t_core := by {rw cube, simp only [set.mem_set_of_eq], exact p_in_t_core},\n        exact p_corner_unique t_core t_core_in_T p_in_t_core,\n      end,\n    let p_corner_coord := T_core_point_to_i_coord_fn (T_core_map \u27e8p, p_has_core_points_finset_property\u27e9),\n    have p_corner_coord_in_coords : p_corner_coord \u2208 coords :=\n      begin\n        dsimp[coords, coords_list, T_core_list, T_core],\n        simp only [exists_prop, list.mem_map, finset.mem_map, list.mem_to_finset, ge_iff_le, finset.mem_to_list, subtype.exists,\n          list.map],\n        use [t_core, t_core_has_T_core_map_property],\n        split,\n        { use [p, p_has_core_points_finset_property],\n          split, exact p_in_core_points_finset,\n          dsimp[T_core_map],\n          apply subtype.ext,\n          simp only [subtype.coe_mk],\n          symmetry,\n          exact t_core_eq_p_corner,\n        },\n        dsimp[p_corner_coord],\n        conv\n        begin\n          find t_core {rw t_core_eq_p_corner},\n        end,\n      end,\n    use [p_corner_coord, p_corner_coord_in_coords],\n    dsimp[p_corner_coord, T_core_point_to_i_coord_fn],\n    rw is_periodic at T_is_periodic,\n    rw [t_def, add_vectors],\n    simp only [vector.nth_of_fn],\n    by_cases t_core_eq_zero_at_i : t_core.nth i = 0,\n    { use [vector.nth (double_int_vector t_offset) i, 0, 0],\n      split, exact rfl.le,\n      split, norm_num,\n      rw [int_point_to_point, \u2190 t_core_eq_p_corner, t_core_eq_zero_at_i],\n      simp only [zero_add, add_zero, int.cast_zero, eq_self_iff_true, vector.nth_of_fn, and_self],\n    },\n    rename t_core_eq_zero_at_i t_core_ne_zero_at_i,\n    use [vector.nth (double_int_vector t_offset) i - 1, -1, t_core.nth i + 1],\n    rw in_cube at p_in_t_core,\n    replace p_in_t_core := p_in_t_core i,\n    rw [p_eq_zero_at_i] at p_in_t_core,\n    cases p_in_t_core with t_core_le_zero zero_lt_t_core_add_one,\n    have t_core_add_one_lt_one : t_core.nth i + 1 < 1 := \n      by {norm_num, rw lt_iff_le_and_ne, exact \u27e8t_core_le_zero, t_core_ne_zero_at_i\u27e9},\n    split, exact le_of_lt zero_lt_t_core_add_one,\n    split, exact t_core_add_one_lt_one,\n    split,\n    { rw int_point_to_point,\n      simp only [sub_add_add_cancel, int.cast_one, vector.nth_of_fn, int.cast_sub],\n      rw add_comm,\n    },\n    norm_num,\n    rw t_core_eq_p_corner,\n  },\n  rename p_eq_zero_at_i p_ne_zero_at_i,\n  let p_sub_ei : point d := vector.of_fn (\u03bb j, if(i = j) then 0 else p.nth j),\n  have p_sub_ei_has_core_points_finset_property :\n    \u2200 j : fin d, (i = j \u2192 p_sub_ei.nth j = 0) \u2227 (j.val < d \u2192 p_sub_ei.nth j = 0 \u2228 p_sub_ei.nth j = 1) \u2227 (j.val \u2265 d \u2192 p_sub_ei.nth j = 0) :=\n    begin\n      intro j,\n      replace p_in_core_points := p_in_core_points j,\n      dsimp[p_sub_ei], \n      simp only [ge_iff_le, vector.nth_of_fn, ite_eq_left_iff],\n      split, \n      { intro i_eq_j,\n        intro i_ne_j,\n        exfalso,\n        exact i_ne_j i_eq_j,\n      },\n      split,\n      { intro j_lt_d,\n        by_cases i_eq_j : i = j,\n        { left,\n          intro i_ne_j,\n          exfalso,\n          exact i_ne_j i_eq_j,\n        },\n        rename i_eq_j i_ne_j,\n        rw if_neg i_ne_j,\n        cases p_in_core_points with p_eq_zero p_eq_one, {left, intro _, exact p_eq_zero},\n        right,\n        exact p_eq_one,\n      },\n      intro j_ge_d,\n      exfalso,\n      have j_lt_d := j.property,\n      clear_except j_ge_d j_lt_d,\n      simp only [fin.val_eq_coe] at j_lt_d,\n      linarith,\n    end,\n  have p_sub_ei_in_core_points_finset := core_points_finset_property p_sub_ei p_sub_ei_has_core_points_finset_property,\n  let p_sub_ei_corner := (point_to_corner T_is_tiling p_sub_ei).val,\n  have p_sub_ei_corner_def : p_sub_ei_corner = (point_to_corner T_is_tiling p_sub_ei).val := by refl,\n  have p_sub_ei_corner_property := (point_to_corner T_is_tiling p_sub_ei).property,\n  rw \u2190 p_sub_ei_corner_def at p_sub_ei_corner_property,\n  rcases p_sub_ei_corner_property with \u27e8p_sub_ei_corner_in_T, p_sub_ei_in_p_sub_ei_corner, p_sub_ei_corner_unique\u27e9,\n  have p_sub_ei_corner_eq_t_core_sub_one : p_sub_ei_corner.nth i = t_core.nth i - 1 :=\n    begin\n      let p_sub_ei_as_int_point : int_point d := vector.of_fn (\u03bb j : fin d, if(p_sub_ei.nth j = 0) then 0 else 1),\n      have p_sub_ei_eq_p_sub_ei_as_int_point :\n        (let fn : fin d \u2192 \u211d := \u03bb (x : fin d), \u2191(vector.nth p_sub_ei_as_int_point x) in vector.of_fn fn) = p_sub_ei :=\n        begin\n          --This proof uses if_pos in a somewhat atypical manner because other attempts have run into \"motive is not type\n          --correct\" issues with the nested ite statements\n          apply vector.ext,\n          intro j,\n          simp only [vector.nth_of_fn],\n          by_cases i_eq_j : i = j,\n          { rw if_pos, {rw if_pos i_eq_j, norm_num},\n            rw if_pos i_eq_j,\n          },\n          rename i_eq_j i_ne_j,\n          by_cases p_eq_zero : p.nth j = 0,\n          { rw if_pos, {rw [if_neg i_ne_j, p_eq_zero], norm_num,},\n            rw [if_neg i_ne_j, p_eq_zero],\n          },\n          rename p_eq_zero p_ne_zero,\n          rw if_neg,\n          { rw if_neg i_ne_j,\n            cases p_in_core_points j with p_eq_zero p_eq_one, {exfalso, exact p_ne_zero p_eq_zero,},\n            rw p_eq_one,\n            norm_num,\n          },\n          rw if_neg i_ne_j,\n          exact p_ne_zero,\n        end,\n      have p_eq_p_sub_ei_add_ei : p = add_vectors p_sub_ei (unit_basis_vector i) :=\n        begin\n          apply vector.ext,\n          intro j,\n          dsimp[p_sub_ei],\n          rw [unit_basis_vector, add_vectors],\n          simp only [vector.nth_of_fn],\n          cases p_in_core_points j with p_eq_zero p_eq_one,\n          { rw p_eq_zero,\n            simp only [if_t_t, zero_add],\n            by_cases i_eq_j : i = j,\n            { rw \u2190 i_eq_j at p_eq_zero,\n              exfalso,\n              exact p_ne_zero_at_i p_eq_zero,\n            },\n            rename i_eq_j i_ne_j,\n            rw if_neg i_ne_j,\n          },\n          rw p_eq_one,\n          by_cases i_eq_j : i = j, rw [if_pos i_eq_j, if_pos i_eq_j, zero_add],\n          rename i_eq_j i_ne_j,\n          rw [if_neg i_ne_j, if_neg i_ne_j, add_zero],\n        end,\n      have p_corner_eq_t_core : \u2191(point_to_corner T_is_tiling p) = t_core :=\n        begin\n          have p_corner_property := (point_to_corner T_is_tiling p).property,\n          simp only [subtype.val_eq_coe] at p_corner_property,\n          rcases p_corner_property with \u27e8p_corner_in_T, p_in_p_corner, p_corner_unique\u27e9,\n          symmetry,\n          exact p_corner_unique t_core t_core_in_T p_in_t_core,\n        end,\n      have cube_distance_lemma_fact := cube_distance_lemma d T T_is_tiling p_sub_ei_as_int_point i,\n      rw [int_point_to_corner, int_point_to_point] at cube_distance_lemma_fact,\n      simp only [subtype.val_eq_coe] at cube_distance_lemma_fact,\n      simp only [subtype.val_eq_coe] at p_sub_ei_corner_def,\n      rw [p_sub_ei_eq_p_sub_ei_as_int_point, \u2190 p_sub_ei_corner_def, \u2190 p_eq_p_sub_ei_add_ei, p_corner_eq_t_core] at cube_distance_lemma_fact,\n      clear_except cube_distance_lemma_fact,\n      linarith,\n    end,\n  have p_sub_ei_corner_has_T_core_map_property : p_sub_ei_corner \u2208 T \u2227 \u2203 p_sub_ei : point d, in_cube p_sub_ei_corner p_sub_ei \u2227\n    \u2200 (j : fin d), (i = j \u2192 vector.nth p_sub_ei j = 0) \u2227 (\u2191j < d \u2192 vector.nth p_sub_ei j = 0 \u2228 vector.nth p_sub_ei j = 1) \u2227 \n    (\u2191j \u2265 d \u2192 vector.nth p_sub_ei j = 0) :=\n    \u27e8p_sub_ei_corner_in_T, Exists.intro p_sub_ei \u27e8p_sub_ei_in_p_sub_ei_corner, p_sub_ei_has_core_points_finset_property\u27e9\u27e9,\n  let p_sub_ei_corner_coord := T_core_point_to_i_coord_fn (T_core_map \u27e8p_sub_ei, p_sub_ei_has_core_points_finset_property\u27e9),\n  have p_sub_ei_corner_coord_in_coords : p_sub_ei_corner_coord \u2208 coords :=\n    begin\n      dsimp[coords, coords_list, T_core_list, T_core],\n      simp only [exists_prop, list.mem_map, finset.mem_map, list.mem_to_finset, ge_iff_le, finset.mem_to_list, subtype.exists,\n        list.map],\n      use [p_sub_ei_corner, p_sub_ei_corner_has_T_core_map_property],\n      split,\n      { use [p_sub_ei, p_sub_ei_has_core_points_finset_property],\n        split, exact p_sub_ei_in_core_points_finset,\n        dsimp[T_core_map],\n        apply subtype.ext,\n        simp only [subtype.coe_mk],\n        symmetry,\n        dsimp[p_sub_ei_corner],\n        refl,\n      },\n      dsimp[p_sub_ei_corner_coord, p_sub_ei_corner],\n      refl,\n    end,\n  use [p_sub_ei_corner_coord, p_sub_ei_corner_coord_in_coords],\n  dsimp [p_sub_ei_corner_coord, T_core_point_to_i_coord_fn],\n  rw is_periodic at T_is_periodic,\n  rw [t_def, add_vectors],\n  simp only [vector.nth_of_fn],\n  by_cases t_core_eq_one_at_i : t_core.nth i = 1,\n  { use [vector.nth (double_int_vector t_offset) i + 1, 0, 0],\n    split, exact rfl.le,\n    split, norm_num,\n    rw [int_point_to_point, t_core_eq_one_at_i],\n    simp only [true_and, add_zero, if_true, eq_self_iff_true, int.cast_one, vector.nth_of_fn, zero_add, int.cast_neg],\n    simp only [subtype.val_eq_coe] at p_sub_ei_corner_def,\n    rw [\u2190 p_sub_ei_corner_def, p_sub_ei_corner_eq_t_core_sub_one, t_core_eq_one_at_i],\n    norm_num,\n    rw add_comm,\n  },\n  rename t_core_eq_one_at_i t_core_ne_one_at_i,\n  use [vector.nth (double_int_vector t_offset) i, -1, t_core.nth i],\n  rw in_cube at p_in_t_core,\n  replace p_in_t_core := p_in_t_core i,\n  have p_eq_one_at_i : p.nth i = 1 :=\n    begin\n      cases p_in_core_points i with p_eq_zero_at_i p_eq_one_at_i,\n      { exfalso,\n        exact p_ne_zero_at_i p_eq_zero_at_i,\n      },\n      exact p_eq_one_at_i,\n    end,\n  rw [p_eq_one_at_i] at p_in_t_core,\n  simp only [lt_add_iff_pos_left] at p_in_t_core,\n  cases p_in_t_core with t_core_le_one zero_lt_t_core,\n  have t_core_lt_one : t_core.nth i < 1 := by {rw lt_iff_le_and_ne, exact \u27e8t_core_le_one, t_core_ne_one_at_i\u27e9},\n  split, exact le_of_lt zero_lt_t_core,\n  split, exact t_core_lt_one,\n  split, {rw int_point_to_point, simp only [vector.nth_of_fn], rw add_comm},\n  simp only [subtype.val_eq_coe] at p_sub_ei_corner_def,\n  rw [\u2190 p_sub_ei_corner_def, p_sub_ei_corner_eq_t_core_sub_one],\n  norm_num,\n  clear_except,\n  linarith,\nend", "meta": {"author": "JOSHCLUNE", "repo": "Keller_reduction", "sha": "dc392b3da352fc1ffcfbecb1d4717d05f5faed4a", "save_path": "github-repos/lean/JOSHCLUNE-Keller_reduction", "path": "github-repos/lean/JOSHCLUNE-Keller_reduction/Keller_reduction-dc392b3da352fc1ffcfbecb1d4717d05f5faed4a/src/s_discrete.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.35884997290048}}
{"text": "/-\nCopyright (c) 2023 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n-/\n\nimport Aesop\n\nstructure A\n\nopen Lean.Elab.Tactic in\n@[aesop norm]\ndef tac : TacticM Unit := do\n  evalTactic $ \u2190 `(tactic| exact A.mk)\n\nexample : A := by\n  aesop\n", "meta": {"author": "JLimperg", "repo": "aesop", "sha": "c68fb1d5a9172498230d81d95c61f6461bea6722", "save_path": "github-repos/lean/JLimperg-aesop", "path": "github-repos/lean/JLimperg-aesop/aesop-c68fb1d5a9172498230d81d95c61f6461bea6722/tests/run/43.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3588499663972944}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module control.equiv_functor\n! leanprover-community/mathlib commit d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathlib.Logic.Equiv.Defs\nimport Mathlib.Tactic.Convert\n\n/-!\n# Functions functorial with respect to equivalences\n\nAn `EquivFunctor` is a function from `Type \u2192 Type` equipped with the additional data of\ncoherently mapping equivalences to equivalences.\n\nIn categorical language, it is an endofunctor of the \"core\" of the category `Type`.\n-/\n\n\nuniverse u\u2080 u\u2081 u\u2082 v\u2080 v\u2081 v\u2082\n\nopen Function\n\n/-- An `EquivFunctor` is only functorial with respect to equivalences.\n\nTo construct an `EquivFunctor`, it suffices to supply just the function `f \u03b1 \u2192 f \u03b2` from\nan equivalence `\u03b1 \u2243 \u03b2`, and then prove the functor laws. It's then a consequence that\nthis function is part of an equivalence, provided by `EquivFunctor.mapEquiv`.\n-/\nclass EquivFunctor (f : Type u\u2080 \u2192 Type u\u2081) where\n  /-- The action of `f` on isomorphisms. -/\n  map : \u2200 {\u03b1 \u03b2}, \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2192 f \u03b2\n  /-- `map` of `f` preserves the identity morphism. -/\n  map_refl' : \u2200 \u03b1, map (Equiv.refl \u03b1) = @id (f \u03b1) := by rfl\n  /-- `map` is functorial on equivalences. -/\n  map_trans' : \u2200 {\u03b1 \u03b2 \u03b3} (k : \u03b1 \u2243 \u03b2) (h : \u03b2 \u2243 \u03b3), map (k.trans h) = map h \u2218 map k := by rfl\n#align equiv_functor EquivFunctor\n\nattribute [simp] EquivFunctor.map_refl'\n\nnamespace EquivFunctor\n\nsection\n\nvariable (f : Type u\u2080 \u2192 Type u\u2081) [EquivFunctor f] {\u03b1 \u03b2 : Type u\u2080} (e : \u03b1 \u2243 \u03b2)\n\n/-- An `EquivFunctor` in fact takes every equiv to an equiv. -/\ndef mapEquiv : f \u03b1 \u2243 f \u03b2 where\n  toFun := EquivFunctor.map e\n  invFun := EquivFunctor.map e.symm\n  left_inv x := by\n    convert (congr_fun (EquivFunctor.map_trans' e e.symm) x).symm\n    simp\n  right_inv y := by\n    convert (congr_fun (EquivFunctor.map_trans' e.symm e) y).symm\n    simp\n#align equiv_functor.map_equiv EquivFunctor.mapEquiv\n\n@[simp]\ntheorem mapEquiv_apply (x : f \u03b1) : mapEquiv f e x = EquivFunctor.map e x :=\n  rfl\n#align equiv_functor.map_equiv_apply EquivFunctor.mapEquiv_apply\n\ntheorem mapEquiv_symm_apply (y : f \u03b2) : (mapEquiv f e).symm y = EquivFunctor.map e.symm y :=\n  rfl\n#align equiv_functor.map_equiv_symm_apply EquivFunctor.mapEquiv_symm_apply\n\n@[simp]\ntheorem mapEquiv_refl (\u03b1) : mapEquiv f (Equiv.refl \u03b1) = Equiv.refl (f \u03b1) := by\n simp [EquivFunctor.mapEquiv]; rfl\n#align equiv_functor.map_equiv_refl EquivFunctor.mapEquiv_refl\n\n@[simp]\ntheorem mapEquiv_symm : (mapEquiv f e).symm = mapEquiv f e.symm :=\n  Equiv.ext $ mapEquiv_symm_apply f e\n#align equiv_functor.map_equiv_symm EquivFunctor.mapEquiv_symm\n\n/-- The composition of `mapEquiv`s is carried over the `EquivFunctor`.\nFor plain `Functor`s, this lemma is named `map_map` when applied\nor `map_comp_map` when not applied.\n-/\n@[simp]\ntheorem mapEquiv_trans {\u03b3 : Type u\u2080} (ab : \u03b1 \u2243 \u03b2) (bc : \u03b2 \u2243 \u03b3) :\n    (mapEquiv f ab).trans (mapEquiv f bc) = mapEquiv f (ab.trans bc) :=\n  Equiv.ext $ fun x => by simp [mapEquiv, map_trans']\n#align equiv_functor.map_equiv_trans EquivFunctor.mapEquiv_trans\n\nend\n\ninstance (priority := 100) ofLawfulFunctor (f : Type u\u2080 \u2192 Type u\u2081) [Functor f] [LawfulFunctor f] :\n    EquivFunctor f where\n  map {\u03b1 \u03b2} e := Functor.map e\n  map_refl' \u03b1 := by\n    ext\n    apply LawfulFunctor.id_map\n  map_trans' {\u03b1 \u03b2 \u03b3} k h := by\n    ext x\n    apply LawfulFunctor.comp_map k h x\n#align equiv_functor.of_is_lawful_functor EquivFunctor.ofLawfulFunctor\n\ntheorem mapEquiv.injective (f : Type u\u2080 \u2192 Type u\u2081)\n    [Applicative f] [LawfulApplicative f] {\u03b1 \u03b2 : Type u\u2080}\n    (h : \u2200 \u03b3, Function.Injective (pure : \u03b3 \u2192 f \u03b3)) :\n      Function.Injective (@EquivFunctor.mapEquiv f _ \u03b1 \u03b2) :=\n  fun e\u2081 e\u2082 H =>\n    Equiv.ext $ fun x => h \u03b2 (by simpa [EquivFunctor.map] using Equiv.congr_fun H (pure x))\n#align equiv_functor.map_equiv.injective EquivFunctor.mapEquiv.injective\n\nend EquivFunctor\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Control/EquivFunctor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832058771036, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.3586451197540776}}
{"text": "import data.equiv.basic\n\nuniverses u v w\n\nclass transportable (f : Type u \u2192 Type v) :=\n(on_equiv : \u03a0 {\u03b1 \u03b2 : Type u} (e : equiv \u03b1 \u03b2), equiv (f \u03b1) (f \u03b2))\n(on_refl  : \u03a0 (\u03b1 : Type u), on_equiv (equiv.refl \u03b1) = equiv.refl (f \u03b1))\n(on_trans : \u03a0 {\u03b1 \u03b2 \u03b3 : Type u} (d : equiv \u03b1 \u03b2) (e : equiv \u03b2 \u03b3), on_equiv (equiv.trans d e) = equiv.trans (on_equiv d) (on_equiv e))\n\n-- Our goal is an automagic proof of the following (level 20)\ntheorem group.transportable : transportable group := sorry\n\n-- These next few we might need to define and prove by hand\ndef Fun : Type u \u2192 Type v \u2192 Type (max u v) := \u03bb \u03b1 \u03b2, \u03b1 \u2192 \u03b2\ndef Prod : Type u \u2192 Type v \u2192 Type (max u v) := \u03bb \u03b1 \u03b2, \u03b1 \u00d7 \u03b2\n\n-- level 1\nlemma Const.transportable : (transportable Const) :=\n{ on_equiv := \u03bb \u03b1 \u03b2 e, equiv.punit_equiv_punit,\n  on_refl  := \u03bb \u03b1, equiv.ext _ _ $ \u03bb \u27e8\u27e9, rfl,\n  on_trans := \u03bb \u03b1 \u03b2 \u03b3 e1 e2, equiv.ext _ _ $ \u03bb \u27e8\u27e9, rfl }\n\nlemma Fun.transportable (\u03b1 : Type u) : (transportable (Fun \u03b1)) :=\n{ on_equiv := \u03bb \u03b2 \u03b3 e, equiv.arrow_congr (equiv.refl \u03b1) e,\n  on_refl  := \u03bb \u03b2, equiv.ext _ _ $ \u03bb f, rfl,\n  on_trans := \u03bb \u03b2 \u03b3 \u03b4 e1 e2, equiv.ext _ _ $ \u03bb f, funext $ \u03bb x,\n    by cases e1; cases e2; refl }\n\ntheorem prod.ext' {\u03b1 \u03b2 : Type*} {p q : \u03b1 \u00d7 \u03b2} (H1 : p.1 = q.1) (H2 : p.2 = q.2) : p = q :=\nprod.ext.2 \u27e8H1, H2\u27e9\n\nlemma Prod.transportable (\u03b1 : Type u) : (transportable (Prod \u03b1)) :=\n{ on_equiv := \u03bb \u03b2 \u03b3 e, equiv.prod_congr (equiv.refl \u03b1) e,\n  on_refl  := \u03bb \u03b2, equiv.ext _ _ $ \u03bb \u27e8x, y\u27e9, by simp,\n  on_trans := \u03bb \u03b2 \u03b3 \u03b4 e1 e2, equiv.ext _ _ $ \u03bb \u27e8x, y\u27e9, by simp }\n\nlemma Swap.transportable (\u03b1 : Type u) : (transportable (Swap \u03b1)) :=\n{ on_equiv := \u03bb \u03b2 \u03b3 e, equiv.prod_congr e (equiv.refl \u03b1),\n  on_refl  := \u03bb \u03b2, equiv.ext _ _ $ \u03bb \u27e8x, y\u27e9, by simp,\n  on_trans := \u03bb \u03b2 \u03b3 \u03b4 e1 e2, equiv.ext _ _ $ \u03bb \u27e8x, y\u27e9, by simp }\n\n-- And then we can define\ndef Hom1 (\u03b1 : Type u) : Type v \u2192 Type (max u v) := \u03bb \u03b2, \u03b1 \u2192 \u03b2\ndef Hom2 (\u03b2 : Type v) : Type u \u2192 Type (max u v) := \u03bb \u03b1, \u03b1 \u2192 \u03b2\ndef Aut : Type u \u2192 Type u := \u03bb \u03b1, \u03b1 \u2192 \u03b1\n\n-- And hopefully automagically derive\nlemma Hom1.transportable (\u03b1 : Type u) : (transportable (Hom1 \u03b1)) :=\nFun.transportable \u03b1\n\nlemma Hom2.transportable (\u03b2 : Type v) : (transportable (Hom2 \u03b2)) :=\n{ on_equiv := \u03bb \u03b1 \u03b3 e, equiv.arrow_congr e (equiv.refl \u03b2),\n  on_refl  := \u03bb \u03b2, equiv.ext _ _ $ \u03bb f, rfl,\n  on_trans := \u03bb \u03b2 \u03b3 \u03b4 e1 e2, equiv.ext _ _ $ \u03bb f, funext $ \u03bb x,\n    by cases e1; cases e2; refl }\n\nlemma Aut.transportable : (transportable Aut) :=\n{ on_equiv := \u03bb \u03b1 \u03b2 e, equiv.arrow_congr e e,\n  on_refl  := \u03bb \u03b1, equiv.ext _ _ $ \u03bb f, funext $ \u03bb x, rfl,\n  on_trans := \u03bb \u03b1 \u03b2 \u03b3 e1 e2, equiv.ext _ _ $ \u03bb f, funext $ \u03bb x,\n    by cases e1; cases e2; refl }\n\n-- If we have all these in place...\n-- A bit of magic might actually be able to derive `group.transportable` on line 11.\n-- After all, a group just is a type plus some functions... and we can now transport functions.\n\nlemma distrib.transportable : (transportable distrib) :=\n{ on_equiv := \u03bb \u03b1 \u03b2 H\u03b1\u03b2,\u27e8\u03bb \u27e8a,m,d1,d2\u27e9,\u27e8_,_,_,_\u27e9,_,_,_\u27e9,\n  on_refl := sorry,\n  on_trans := sorry \n}", "meta": {"author": "kbuzzard", "repo": "xena", "sha": "cd2f0b5e948b7171dbafc5cb519a3220d318bd9d", "save_path": "github-repos/lean/kbuzzard-xena", "path": "github-repos/lean/kbuzzard-xena/xena-cd2f0b5e948b7171dbafc5cb519a3220d318bd9d/canonical_isomorphism/johan_kenny.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105454764747, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.358645118880459}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n-/\nimport order.category.Lattice\n\n/-!\n# The category of distributive lattices\n\nThis file defines `DistribLattice`, the category of distributive lattices.\n\nNote that [`DistLat`](https://ncatlab.org/nlab/show/DistLat) in the literature doesn't always\ncorrespond to `DistribLattice` as we don't require bottom or top elements. Instead, this `DistLat`\ncorresponds to `BoundedDistribLattice`.\n-/\n\nuniverses u\n\nopen category_theory\n\n/-- The category of distributive lattices. -/\ndef DistribLattice := bundled distrib_lattice\n\nnamespace DistribLattice\n\ninstance : has_coe_to_sort DistribLattice Type* := bundled.has_coe_to_sort\ninstance (X : DistribLattice) : distrib_lattice X := X.str\n\n/-- Construct a bundled `DistribLattice` from a `distrib_lattice` underlying type and typeclass. -/\ndef of (\u03b1 : Type*) [distrib_lattice \u03b1] : DistribLattice := bundled.of \u03b1\n\n@[simp] lemma coe_of (\u03b1 : Type*) [distrib_lattice \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited DistribLattice := \u27e8of punit\u27e9\n\ninstance : bundled_hom.parent_projection @distrib_lattice.to_lattice := \u27e8\u27e9\n\nattribute [derive [large_category, concrete_category]] DistribLattice\n\ninstance has_forget_to_Lattice : has_forget\u2082 DistribLattice Lattice := bundled_hom.forget\u2082 _ _\n\n/-- Constructs an equivalence between distributive lattices from an order isomorphism between them.\n-/\n@[simps] def iso.mk {\u03b1 \u03b2 : DistribLattice.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply _ },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply _ } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : DistribLattice \u2964 DistribLattice :=\n{ obj := \u03bb X, of X\u1d52\u1d48, map := \u03bb X Y, lattice_hom.dual }\n\n/-- The equivalence between `DistribLattice` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : DistribLattice \u224c DistribLattice :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend DistribLattice\n\nlemma DistribLattice_dual_comp_forget_to_Lattice :\n  DistribLattice.dual \u22d9 forget\u2082 DistribLattice Lattice =\n    forget\u2082 DistribLattice Lattice \u22d9 Lattice.dual := rfl\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/order/category/DistribLattice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.35864079351520645}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n\n! This file was ported from Lean 3 source module ring_theory.ring_hom_properties\n! leanprover-community/mathlib commit b5aecf07a179c60b6b37c1ac9da952f3b565c785\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Category.Ring.Constructions\nimport Mathbin.Algebra.Category.Ring.Colimits\nimport Mathbin.CategoryTheory.Isomorphism\nimport Mathbin.RingTheory.Localization.Away\nimport Mathbin.RingTheory.IsTensorProduct\n\n/-!\n# Properties of ring homomorphisms\n\nWe provide the basic framework for talking about properties of ring homomorphisms.\nThe following meta-properties of predicates on ring homomorphisms are defined\n\n* `ring_hom.respects_iso`: `P` respects isomorphisms if `P f \u2192 P (e \u226b f)` and\n  `P f \u2192 P (f \u226b e)`, where `e` is an isomorphism.\n* `ring_hom.stable_under_composition`: `P` is stable under composition if `P f \u2192 P g \u2192 P (f \u226b g)`.\n* `ring_hom.stable_under_base_change`: `P` is stable under base change if `P (S \u27f6 Y)`\n  implies `P (X \u27f6 X \u2297[S] Y)`.\n\n-/\n\n\nuniverse u\n\nopen CategoryTheory Opposite CategoryTheory.Limits\n\nnamespace RingHom\n\nvariable (P : \u2200 {R S : Type u} [CommRing R] [CommRing S] (f : R \u2192+* S), Prop)\n\ninclude P\n\nsection RespectsIso\n\n/-- A property `respects_iso` if it still holds when composed with an isomorphism -/\ndef RespectsIso : Prop :=\n  (\u2200 {R S T : Type u} [CommRing R] [CommRing S] [CommRing T],\n      \u2200 (f : R \u2192+* S) (e : S \u2243+* T) (hf : P f), P (e.to_ring_hom.comp f)) \u2227\n    \u2200 {R S T : Type u} [CommRing R] [CommRing S] [CommRing T],\n      \u2200 (f : S \u2192+* T) (e : R \u2243+* S) (hf : P f), P (f.comp e.toRingHom)\n#align ring_hom.respects_iso RingHom.RespectsIso\n\nvariable {P}\n\ntheorem RespectsIso.cancel_left_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R \u27f6 S)\n    (g : S \u27f6 T) [IsIso f] : P (f \u226b g) \u2194 P g :=\n  \u27e8fun H => by\n    convert hP.2 (f \u226b g) (as_iso f).symm.commRingIsoToRingEquiv H\n    exact (is_iso.inv_hom_id_assoc _ _).symm, hP.2 g (asIso f).commRingIsoToRingEquiv\u27e9\n#align ring_hom.respects_iso.cancel_left_is_iso RingHom.RespectsIso.cancel_left_isIso\n\ntheorem RespectsIso.cancel_right_isIso (hP : RespectsIso @P) {R S T : CommRingCat} (f : R \u27f6 S)\n    (g : S \u27f6 T) [IsIso g] : P (f \u226b g) \u2194 P f :=\n  \u27e8fun H => by\n    convert hP.1 (f \u226b g) (as_iso g).symm.commRingIsoToRingEquiv H\n    change f = f \u226b g \u226b inv g\n    simp, hP.1 f (asIso g).commRingIsoToRingEquiv\u27e9\n#align ring_hom.respects_iso.cancel_right_is_iso RingHom.RespectsIso.cancel_right_isIso\n\ntheorem RespectsIso.is_localization_away_iff (hP : RingHom.RespectsIso @P) {R S : Type _}\n    (R' S' : Type _) [CommRing R] [CommRing S] [CommRing R'] [CommRing S'] [Algebra R R']\n    [Algebra S S'] (f : R \u2192+* S) (r : R) [IsLocalization.Away r R'] [IsLocalization.Away (f r) S'] :\n    P (Localization.awayMap f r) \u2194 P (IsLocalization.Away.map R' S' f r) :=\n  by\n  let e\u2081 : R' \u2243+* Localization.Away r :=\n    (IsLocalization.algEquiv (Submonoid.powers r) _ _).toRingEquiv\n  let e\u2082 : Localization.Away (f r) \u2243+* S' :=\n    (IsLocalization.algEquiv (Submonoid.powers (f r)) _ _).toRingEquiv\n  refine' (hP.cancel_left_is_iso e\u2081.to_CommRing_iso.hom (CommRingCat.ofHom _)).symm.trans _\n  refine' (hP.cancel_right_is_iso (CommRingCat.ofHom _) e\u2082.to_CommRing_iso.hom).symm.trans _\n  rw [\u2190 eq_iff_iff]\n  congr 1\n  dsimp [CommRingCat.ofHom, CommRingCat.of, bundled.of]\n  refine' IsLocalization.ringHom_ext (Submonoid.powers r) _\n  ext1\n  revert e\u2081 e\u2082\n  dsimp [RingEquiv.toRingHom, IsLocalization.Away.map]\n  simp only [CategoryTheory.comp_apply, RingEquiv.refl_apply, IsLocalization.algEquiv_apply,\n    IsLocalization.ringEquivOfRingEquiv_apply, RingHom.coe_mk, [anonymous],\n    IsLocalization.ringEquivOfRingEquiv_eq, IsLocalization.map_eq]\n#align ring_hom.respects_iso.is_localization_away_iff RingHom.RespectsIso.is_localization_away_iff\n\nend RespectsIso\n\nsection StableUnderComposition\n\n/-- A property is `stable_under_composition` if the composition of two such morphisms\nstill falls in the class. -/\ndef StableUnderComposition : Prop :=\n  \u2200 \u2983R S T\u2984 [CommRing R] [CommRing S] [CommRing T],\n    \u2200 (f : R \u2192+* S) (g : S \u2192+* T) (hf : P f) (hg : P g), P (g.comp f)\n#align ring_hom.stable_under_composition RingHom.StableUnderComposition\n\nvariable {P}\n\ntheorem StableUnderComposition.respectsIso (hP : RingHom.StableUnderComposition @P)\n    (hP' : \u2200 {R S : Type _} [CommRing R] [CommRing S] (e : R \u2243+* S), P e.to_ring_hom) :\n    RingHom.RespectsIso @P := by\n  constructor\n  \u00b7 introv H\n    skip\n    apply hP\n    exacts[H, hP' e]\n  \u00b7 introv H\n    skip\n    apply hP\n    exacts[hP' e, H]\n#align ring_hom.stable_under_composition.respects_iso RingHom.StableUnderComposition.respectsIso\n\nend StableUnderComposition\n\nsection StableUnderBaseChange\n\n/-- A morphism property `P` is `stable_under_base_change` if `P(S \u2192+* A)` implies\n`P(B \u2192+* A \u2297[S] B)`. -/\ndef StableUnderBaseChange : Prop :=\n  \u2200 (R S R' S') [CommRing R] [CommRing S] [CommRing R'] [CommRing S'],\n    \u2200 [Algebra R S] [Algebra R R'] [Algebra R S'] [Algebra S S'] [Algebra R' S'],\n      \u2200 [IsScalarTower R S S'] [IsScalarTower R R' S'],\n        \u2200 [Algebra.IsPushout R S R' S'], P (algebraMap R S) \u2192 P (algebraMap R' S')\n#align ring_hom.stable_under_base_change RingHom.StableUnderBaseChange\n\ntheorem StableUnderBaseChange.mk (h\u2081 : RespectsIso @P)\n    (h\u2082 :\n      \u2200 \u2983R S T\u2984 [CommRing R] [CommRing S] [CommRing T],\n        \u2200 [Algebra R S] [Algebra R T],\n          P (algebraMap R T) \u2192\n            P (algebra.tensor_product.include_left.to_ring_hom : S \u2192+* TensorProduct R S T)) :\n    StableUnderBaseChange @P := by\n  introv R h H\n  skip\n  let e := h.symm.1.Equiv\n  let f' :=\n    Algebra.TensorProduct.productMap (IsScalarTower.toAlgHom R R' S')\n      (IsScalarTower.toAlgHom R S S')\n  have : \u2200 x, e x = f' x := by\n    intro x\n    change e.to_linear_map.restrict_scalars R x = f'.to_linear_map x\n    congr 1\n    apply TensorProduct.ext'\n    intro x y\n    simp [IsBaseChange.equiv_tmul, Algebra.smul_def]\n  convert h\u2081.1 _ _ (h\u2082 H : P (_ : R' \u2192+* _))\n  swap\n  \u00b7 refine' { e with map_mul' := fun x y => _ }\n    change e (x * y) = e x * e y\n    simp_rw [this]\n    exact map_mul f' _ _\n  \u00b7 ext\n    change _ = e (x \u2297\u209c[R] 1)\n    dsimp only [e]\n    rw [h.symm.1.equiv_tmul, Algebra.smul_def, AlgHom.toLinearMap_apply, map_one, mul_one]\n#align ring_hom.stable_under_base_change.mk RingHom.StableUnderBaseChange.mk\n\nomit P\n\nattribute [local instance] Algebra.TensorProduct.rightAlgebra\n\ntheorem StableUnderBaseChange.pushout_inl (hP : RingHom.StableUnderBaseChange @P)\n    (hP' : RingHom.RespectsIso @P) {R S T : CommRingCat} (f : R \u27f6 S) (g : R \u27f6 T) (H : P g) :\n    P (pushout.inl : S \u27f6 pushout f g) :=\n  by\n  rw [\u2190\n    show _ = pushout.inl from\n      colimit.iso_colimit_cocone_\u03b9_inv \u27e8_, CommRingCat.pushoutCoconeIsColimit f g\u27e9\n        walking_span.left,\n    hP'.cancel_right_is_iso]\n  letI := f.to_algebra\n  letI := g.to_algebra\n  dsimp only [CommRingCat.pushoutCocone_inl, pushout_cocone.\u03b9_app_left]\n  apply hP R T S (TensorProduct R S T)\n  exact H\n#align ring_hom.stable_under_base_change.pushout_inl RingHom.StableUnderBaseChange.pushout_inl\n\nend StableUnderBaseChange\n\nend RingHom\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/RingTheory/RingHomProperties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6513548782017745, "lm_q1q2_score": 0.35864079351520645}}
{"text": "import Lean.Elab.Tactic\nimport MathlibTactics\nimport Mathlib.Data.Set.Basic\nimport Mathlib.Data.Set.Lattice\nimport Mathlib.Data.Rel\nimport Mathlib.Data.Nat.Basic\nimport Mathlib.Data.Int.Basic\nimport Mathlib.Data.Rat.Basic\nimport Mathlib.Data.Real.Basic\nimport Mathlib.Data.ZMod.Defs\n\ndef Iff.ltr {p q : Prop} (h : p \u2194 q) := h.mp\ndef Iff.rtl {p q : Prop} (h : p \u2194 q) := h.mpr\n\n/-To allow use of \"termination_hint\" in recursive definitions.  Don't use?\nsyntax withPosition(\"termination_hint : \" term \" := \" term Lean.Parser.semicolonOrLinebreak) term : term\n\nmacro_rules\n  | `(termination_hint : $t := $p; $b) => `(have : $t := $p; $b)\n-/\n\n--New set theory notation.\n--Lower priority than all other set theory notation\nmacro (priority := low-1) \"{ \" pat:term \" : \" t:term \" | \" p:term \" }\" : term =>\n  `({ x : $t | match x with | $pat => $p })\n\nmacro (priority := low-1) \"{ \" pat:term \" | \" p:term \" }\" : term =>\n  `({ x | match x with | $pat => $p })\n\n@[app_unexpander setOf]\ndef setOf.unexpander : Lean.PrettyPrinter.Unexpander\n  | `($_ fun $x:ident => match $y:ident with | $pat => $p) =>\n      if x == y then\n        `({ $pat:term | $p:term })\n      else\n        throw ()  --Or could use `({ $x:ident | match $y:ident with | $pat => $p})\n  | `($_ fun ($x:ident : $ty:term) => match $y:ident with | $pat => $p) =>\n      if x == y then\n        `({ $pat:term : $ty:term | $p:term })\n      else\n        throw ()\n  -- Next line needed because of bug in Mathlib/Init/Set.lean\n  | `($_ fun ($x:ident : $ty:term) => $p) => `({ $x:ident : $ty:term | $p })\n  | _ => throw ()\n\n--Make sure Lean understands {x} and \u2205 as Sets, not Finsets\nattribute [default_instance] Set.instSingletonSet\nattribute [default_instance] Set.instEmptyCollectionSet\n\n/- No longer needed\n@[app_unexpander Function.comp] def unexpandFunctionComp : Lean.PrettyPrinter.Unexpander\n  | `($(_) $f:term $g:term $x:term) => `(($f \u2218 $g) $x)\n  | _ => throw ()\n-/\n\n-- Set theory notation that should be in library.  Will it be added eventually?\n-- Copying similar in:  Mathlib/Init/Set.lean, lean4/Init/Notation.lean, std4/Std/Classes/SetNotation.lean\nnotation:50 a:50 \" \u2288 \" b:50 => \u00ac (a \u2286 b)\n\n--Note:  Mathlib.Order.SymmDiff.lean defines this with \u2206 (\\increment) instead of \u25b3 (\\bigtriangleup).\n--Switch to that??  But display of symmDiff seems to use \u25b3.\ninfixl:100 \" \u25b3 \" => symmDiff\n\nnamespace HTPI\n--Some theorems not in library\ntheorem not_not_and_distrib {p q : Prop} : \u00ac(\u00ac p \u2227 q) \u2194 (p \u2228 \u00ac q) := by\n  rw [not_and_or, Classical.not_not]\n\ntheorem not_and_not_distrib {p q : Prop} : \u00ac(p \u2227 \u00ac q) \u2194 (\u00ac p \u2228 q) := by\n  rw [not_and_or, Classical.not_not]\n\ntheorem not_not_or_distrib {p q : Prop} : \u00ac(\u00ac p \u2228 q) \u2194 (p \u2227 \u00ac q) := by\n  rw [not_or, Classical.not_not]\n\ntheorem not_or_not_distrib {p q : Prop} : \u00ac(p \u2228 \u00ac q) \u2194 (\u00ac p \u2227 q) := by\n  rw [not_or, Classical.not_not]\n\ntheorem not_imp_not_iff_and {p q : Prop} : \u00ac (p \u2192 \u00ac q) \u2194 p \u2227 q := by\n  rw [not_imp, Classical.not_not]\n\ntheorem not_imp_iff_not_and {p q : Prop} : \u00ac (q \u2192 p) \u2194 \u00ac p \u2227 q := by\n  rw [not_imp]\n  exact And.comm\n\ntheorem not_not_iff {p q : Prop} : \u00ac(\u00acp \u2194 q) \u2194 (p \u2194 q) := by\n  rw [not_iff, Classical.not_not]\n\ndef Pred (t : Type u) : Type u := t \u2192 Prop\n--def Rel (s t : Type u) : Type u := s \u2192 t \u2192 Prop   --Defined in Mathlib.Data.Rel\ndef BinRel (t : Type u) : Type u := Rel t t\n\n--Definitions of tactics\nsection tactic_defs\nopen Lean Elab Tactic Expr MVarId\n\n--Syntax for arguments to tactics\nsyntax oneLoc := \" at \" ident\nsyntax colonTerm := \" : \" term\nsyntax withId := \" with \" ident\nsyntax with2Ids := \" with \" ident (\", \" ident)?\nsyntax idOrTerm := ident <|> (\"(\" term \")\")\nsyntax idOrTerm?Type := ident <|> (\"(\" term (\" : \" term)? \")\")\n\nabbrev OneLoc := TSyntax ``oneLoc\nabbrev ColonTerm := TSyntax ``colonTerm\nabbrev WithId := TSyntax ``withId\nabbrev With2Ids := TSyntax ``with2Ids\nabbrev IdOrTerm := TSyntax ``idOrTerm\nabbrev IdOrTerm?Type := TSyntax ``idOrTerm?Type\n\n--Get formula from identifier\ndef formFromIdent (h : Syntax) : TacticM Expr := do\n  instantiateMVars (\u2190 Meta.getLocalDeclFromUserName h.getId).type\n\n--Get formula from optional location.  Note both formFromIdent and getMainTarget call instantiateMVars\ndef formFromLoc (l : Option OneLoc) : TacticM Expr := do\n  match l with\n    | some h => formFromIdent h.raw[1]\n    | none => getMainTarget\n\n--For debugging:\ndef myTrace (msg : String) : TacticM Unit := do\n  let m := Syntax.mkStrLit msg\n  evalTactic (\u2190 `(tactic| trace $m))\n\npartial def SyntaxToString (s : Syntax) : String :=\nmatch s with\n  | .missing => \"(missing)\"\n  | .node _ k as => \"(node \" ++ toString k ++ (SyntaxListToString as.data) ++ \")\"\n  | .atom _ v => \"(atom \" ++ toString v ++ \")\"\n  | .ident _ rv v _ => \"(ident \" ++ (toString rv) ++ \" \" ++ (toString v) ++ \")\"\nwhere SyntaxListToString (ss : List Syntax) : String :=\n  match ss with\n    | (s :: rest) => \" \" ++ (SyntaxToString s) ++ (SyntaxListToString rest)\n    | [] => \"\"\n\ndef traceThisSyntax (s : Syntax) : TacticM Unit := myTrace (SyntaxToString s)\n\ndef binderString (bi : BinderInfo) : String :=\n  match bi with\n    | .default => \"default\"\n    | _ => \"not default\"\n    \ndef ExprToString (e : Expr) : String := \nmatch e with\n  | .bvar n => \"(bvar \" ++ (toString n) ++ \")\" -- bound variables\n  | .fvar f => \"(fvar \" ++ (toString f.name) ++ \")\"  -- free variables\n  | .mvar m => \"(mvar \" ++ (toString m.name) ++ \")\"   -- meta variables\n  | .sort l => \"(sort \" ++ (toString l) ++ \")\"   -- Sort\n  | .const n ls => \"(const \" ++ (toString n) ++ \" \" ++ (toString ls) ++ \")\"   -- constants\n  | .app a b => \"(app \" ++ (ExprToString a) ++ \" \" ++ (ExprToString b) ++ \")\" -- application\n  | .lam n t b bi => \"(lam \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \" ++ (ExprToString b) ++ \" \" ++ (binderString bi) ++ \")\"    -- lambda abstraction\n  | .forallE n t b bi => \"(forallE \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \" ++ (ExprToString b) ++ \" \" ++ (binderString bi) ++ \")\"  -- (dependent) arrow\n  | .letE n t v b _ => \"(let \" ++ (toString n) ++ \" \" ++ (ExprToString t) ++ \" \"\n        ++ (ExprToString v) ++ \" \" ++ (ExprToString b) ++ \")\" -- let expressions\n  | .lit _ => \"(lit)\"  -- literals\n  | .mdata m e => \"(mdata \" ++ (toString m) ++ \" \" ++ (ExprToString e) ++ \")\"   -- metadata\n  | .proj t i c => \"(proj \" ++ (toString t) ++ \" \" ++ (toString i) ++ \" \" ++ (ExprToString c) ++ \")\" -- projection\n\ndef traceThisExpr (e : Expr) : TacticM Unit := myTrace (ExprToString e)\n\nelab \"traceExpr\" t:(colonTerm)? l:(oneLoc)? : tactic =>\n  withMainContext do\n    match t with\n      | some tstx => do\n        traceThisSyntax tstx.raw[1]\n        let e \u2190 elabTerm tstx.raw[1] none\n        traceThisExpr e\n      | none =>\n        let e \u2190 formFromLoc l\n        traceThisExpr e\n\n-- Get head and arg list\ndef getHeadData (e : Expr) : Expr \u00d7 List Expr :=\n  match e with\n    | app f a =>\n      let (h, as) := getHeadData f\n      (h, a :: as)\n    | mdata _ e' => getHeadData e'\n    | _ => (e, [])\n\n-- Recover expression from head and arg list\ndef mkAppList (h : Expr) (args : List Expr) : Expr :=\n  match args with\n    | a :: rest => mkApp (mkAppList h rest) a\n    | [] => h\n\n--Determine if e is a proposition, in current local context\ndef exprIsProp (e : Expr) : TacticM Bool :=\n  return (\u2190 Meta.inferType e).isProp\n\n--Logical form of a proposition.\ninductive PropForm where\n  | not     : Expr \u2192 PropForm\n  | and     : Expr \u2192 Expr \u2192 PropForm\n  | or      : Expr \u2192 Expr \u2192 PropForm\n  | implies : Expr \u2192 Expr \u2192 PropForm\n  | iff     : Expr \u2192 Expr \u2192 PropForm\n  | all     : Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n  | ex      : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n  | exun    : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 PropForm\n  | f       : PropForm\n  | t       : PropForm\n  | none    : PropForm\n\n/- Try to unfold definition, and if result is negative, return PropForm.not\nNote:  Uses constants but not fvars with let declarations.  Also, only unfolds once.\nThis might be best--only detect expressions immediately recognized as negative by def.\n-/\ndef findNegPropAll (e : Expr) : TacticM PropForm := do\n  match (\u2190 Meta.unfoldDefinition? (consumeMData e)) with\n    | some e' =>\n      match getHeadData e' with\n        | (const ``Not _, [l]) => return PropForm.not l\n        | _ => return PropForm.none\n    | none => return PropForm.none\n\n--Apply a function to data for an existential.  Existentials usually apply to a\n--lambda expression, but allow for others\ndef applyToExData {\u03b1 : Type} (f : Level \u2192 Name \u2192 Expr \u2192 Expr \u2192 BinderInfo \u2192 \u03b1)\n  (lev : Level) (l r : Expr) : \u03b1 :=\n  let r' := consumeMData r\n  match r' with\n    | lam v t b bi => f lev v t b bi\n    | _ => f lev `x l (mkApp r' (bvar 0)) BinderInfo.default\n\n-- Get logical form of a proposition.\n-- Recognizes negative predicates by one of two methods from above.\ndef getPropForm (e : Expr) : TacticM PropForm := do\n  if !(\u2190 exprIsProp e) then return PropForm.none\n  let (h, args) := getHeadData e\n  match h with\n    | const c levs =>\n      match (c, levs, args) with\n        | (``False, _, _) => return PropForm.f\n        | (``True, _, _) => return PropForm.t\n        | (``Not, _, [l]) => return PropForm.not l\n        | (``And, _, [r, l]) => return PropForm.and l r\n        | (``Or, _, [r, l]) => return PropForm.or l r\n        | (``Iff, _, [r, l]) => return PropForm.iff l r\n        | (``Exists, [lev], [r, l]) => return applyToExData PropForm.ex lev l r\n        | (``ExistsUnique, [lev], [r, l]) => return applyToExData PropForm.exun lev l r\n        | _ => findNegPropAll e     --or:  return findNegPropList c levs args\n    | forallE v t b bi =>\n      if (b.hasLooseBVars || !(\u2190 exprIsProp t)) then\n        return PropForm.all v t b bi\n      else\n        return PropForm.implies t b\n    | _ => return PropForm.none\n\n--mkNot, mkAnd, mkOr, and mkForall are already defined.  Also mkArrow and Meta.mkEq\ndef mkIff (l r : Expr) : Expr :=\n  mkApp2 (mkConst ``Iff) l r\n\n--Need to supply level--I always have it, so easiest to use it.\ndef mkExists (l : Level) (x : Name) (bi : BinderInfo) (t b : Expr) : Expr :=\n  mkApp2 (mkConst ``Exists [l]) t (mkLambda x bi t b)\n\ndef myFail {\u03b1} (tac : Name) (msg : String) : TacticM \u03b1 := do\n  Meta.throwTacticEx tac (\u2190 getMainGoal) msg\n\n/- Functions for unfolding head -/\n\n--Unfold ExistsUnique; default version doesn't do a good job of naming variables\ndef unfoldExUn (lev : Level) (v : Name) (t b : Expr) (_ : BinderInfo) : Expr :=\n  let v1 := Name.appendIndexAfter v 1\n  let eqn := mkApp3 (mkConst ``Eq [lev]) t (bvar 1) (bvar 2)\n  let body := mkAnd b (mkForall v1 BinderInfo.default t (mkForall `x BinderInfo.default b eqn))\n  mkExists lev v BinderInfo.default t body\n\n/- Unfold head in current context--must set local context before call.\nIf first = true, then unfold ExistsUnique using my def; else don't unfold it.\nAlso, if first = true, then unfold ite and dite; otherwise don't.\nIf rep = true, unfold repeatedly.\nLet whnfCore handle everything except unfolding of constants.\nDo all normalization up to first unfolding of a definition; on next call do that unfolding\n-/\npartial def unfoldHead (e : Expr) (tac : Name) (first rep : Bool) : TacticM Expr := do\n  let e1 := consumeMData e\n  let (h, args) := getHeadData e1\n  -- First let e2 = result of one unfolding, or handle negation, or fail\n  let e2 \u2190 match h with\n    | const c levs =>\n      match (c, levs, args) with\n        | (``Not, _, [l]) => return mkNot (\u2190 unfoldHead l tac first rep) --Return from function call, bypassing e2\n        | (``ExistsUnique, [lev], [r, l]) =>\n          if first then\n            pure (applyToExData unfoldExUn lev l r)\n          else\n            myFail tac \"failed to unfold definition\"\n        | _ => \n          if !first && ((c == ``ite) || (c == ``dite)) then\n            myFail tac \"failed to unfold definition\"\n          let edo \u2190 Meta.unfoldDefinition? e1\n          match edo with\n            | some ed => pure ed\n            | none => myFail tac \"failed to unfold definition\"\n    | _ =>\n      let ew \u2190 Meta.whnfCore e1\n      if ew == e1 then\n        myFail tac \"failed to unfold definition\"\n      else\n        pure ew\n  if rep then\n    let e3 \u2190 try\n        unfoldHead e2 tac false true\n      catch _ =>\n        pure e2\n    match e1 with\n      | (app (app (app (app (app (const ``Membership.mem _) _) (app (const ``Set _) _))\n        (app (const ``Set.instMembershipSet _) _)) x) y) =>\n        if (e3 == app y x) then\n          myFail tac \"failed to unfold definition\"  --Don't unfold `x \u2208 y` to `y x`\n        else\n          return e3\n      | (app (app (const ``setOf _) _) f) => \n        if (e3 == f) then\n          myFail tac \"failed to unfold definition\"  --Don't unfold `{ x | p }` to `fun x => p`\n        else\n          return e3\n      | _ => return e3\n  else\n    return e2\n\n-- whnf, but don't unfold ``ExistsUnique\ndef whnfNotExUn (e : Expr) : TacticM Expr :=\n  Meta.whnfHeadPred e (fun x => return !(x.isAppOf ``ExistsUnique))\n\n-- w = 0 : no whnf, w = 1 : whnfNotExun, w = 2 : full whnf\ndef exprFromPf (t : Term) (w : Nat) : TacticM Expr := do\n  let p \u2190 elabTerm t none\n  let e \u2190 instantiateMVars (\u2190 Meta.inferType p)\n  match w with\n    | 0 => return e\n    | 1 => whnfNotExUn e\n    | _ => Meta.whnf e\n\n--Add new hypothesis with name n, asserting form, proven by pfstx.\ndef doHave (n : Name) (form : Expr) (pfstx : Syntax) : TacticM Unit := do\n  let goal \u2190 getMainGoal\n  let oldtar \u2190 getType goal\n  let pf \u2190 elabTermEnsuringType pfstx form\n  let mvarIdNew \u2190 assert goal n form pf\n  let (_, newGoal) \u2190 intro1P mvarIdNew    --blank is FVarId of new hyp.\n  let newtar \u2190 getType newGoal\n  if (oldtar != newtar) && (\u2190 Meta.isExprDefEq oldtar newtar) then\n    --intro1P sometimes changes target to something def. equal.  Put it back to original\n    replaceMainGoal [\u2190 newGoal.replaceTargetDefEq oldtar]\n  else\n    replaceMainGoal [newGoal]\n\n--Add n : (Type) := val to context.\n/- **Not used\ndef doLet (n : Name) (val : Expr) : TacticM Unit := do\n  let goal \u2190 getMainGoal\n  withContext goal do\n    let valType \u2190 Meta.inferType val\n    let mvarIdNew \u2190 define goal n valType val\n    let (_, newGoal) \u2190 intro1P mvarIdNew\n    replaceMainGoal [newGoal]\n-/\n\n--Set goal to be form; pfstx is proof that it suffices.\ndef doSuffices (form : Expr) (pfstx : Syntax) : TacticM Unit := do\n  let goal \u2190 getMainGoal\n  let tag \u2190 getTag goal\n  let target \u2190 getType goal\n  let imp \u2190 mkArrow form target\n  let pf \u2190 elabTermEnsuringType pfstx imp\n  let newTarget \u2190 Meta.mkFreshExprSyntheticOpaqueMVar form tag\n  assign goal (mkApp pf newTarget)\n  replaceMainGoal [newTarget.mvarId!]\n\n--Do rewrite; symm says whether to reverse direction, rule is Term for rule, l is optional location\ndef doRewrite (symm : Bool) (rule : Term) (l : Option OneLoc) : TacticM Unit := do\n  match l with\n    | some id =>\n        let idstx : Ident := \u27e8id.raw[1]\u27e9\n        if symm then\n          evalTactic (\u2190 `(tactic| rewrite [\u2190 $rule:term] at $idstx:ident))\n        else\n          evalTactic (\u2190 `(tactic| rewrite [$rule:term] at $idstx:ident))\n    | none =>\n        if symm then\n          evalTactic (\u2190 `(tactic| rewrite [\u2190 $rule:term]))\n        else\n          evalTactic (\u2190 `(tactic| rewrite [$rule:term]))\n\n--Swap first two goals, if there are at least two\ndef doSwap : TacticM Unit := do\n  let g \u2190 getGoals\n  let ng := match g with\n    | g1 :: (g2 :: rest) => g2 :: (g1 :: rest)\n    | _ => g\n  setGoals ng\n\n/- Functions for all equivalence tactics: contrapos, demorgan, quant_neg, conditional, double_neg -/\ndef ruleType := Name \u00d7 Expr\n\ndef equivMakeRule (f : Expr)\n  (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n  let (rule, res) \u2190 ruleFunc f\n  return (rule, mkIff f res)\n\ndef equivRuleFromForm (p : Expr)\n  (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n    try\n      equivMakeRule p ruleFunc\n    catch ex =>\n      match (\u2190 getPropForm p) with\n        | PropForm.iff l r =>\n          try\n            equivMakeRule l ruleFunc\n          catch _ =>\n            equivMakeRule r ruleFunc\n        | _ => throw ex\n\ndef equivRule (f : Option ColonTerm) (l : Option OneLoc)\n  (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM ruleType := do\n  match f with\n    | some fs => equivMakeRule (\u2190 elabTerm fs.raw[1] none) ruleFunc\n    | none => equivRuleFromForm (\u2190 formFromLoc l) ruleFunc\n\ndef doReplace (tac : Name) (l : Option OneLoc) (res : Expr) (pf : Syntax) : TacticM Unit := do\n    let hn \u2190 mkFreshUserName `h\n    doHave hn res pf\n    let h := mkIdent hn\n    let ht : Term := \u27e8h.raw\u27e9\n    try\n      doRewrite false ht l\n      evalTactic (\u2190 `(tactic| clear $h:ident)) -- Could also do: (try apply Iff.refl); try assumption\n    catch _ =>\n      evalTactic (\u2190 `(tactic| clear $h:ident))\n      myFail tac  \"target expression not found\"\n\ndef doEquivTac (f : Option ColonTerm) (l : Option OneLoc)\n  (tac : Name) (ruleFunc : Expr \u2192 TacticM ruleType) : TacticM Unit :=\n  withMainContext do\n    let (rule, res) \u2190 equivRule f l ruleFunc\n    doReplace tac l res (mkIdent rule)\n\n/- contrapos tactic -/\ndef cpRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.implies l r => match (\u2190 getPropForm l) with\n      | PropForm.not nl => match (\u2190 getPropForm r) with\n        | PropForm.not nr =>\n          return (`not_imp_not, (\u2190 mkArrow nr nl))\n        | _ =>\n          return (`not_imp_comm, (\u2190 mkArrow (mkNot r) nl))\n      | _ => match (\u2190 getPropForm r) with\n        | PropForm.not nr =>\n          return (`imp_not_comm, (\u2190 mkArrow nr (mkNot l)))\n        | _ =>\n          return (`not_imp_not.symm, (\u2190 mkArrow (mkNot r) (mkNot l)))\n    | _ => myFail `contrapos \"contrapositive law doesn't apply\"\n\nelab \"contrapos\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `contrapos cpRule\n\n/- demorgan tactic -/\ndef dmRuleFromInfoNoNeg (l r : Expr) (conn : Expr \u2192 Expr \u2192 Expr) (rs : Array Name) : TacticM ruleType := do\n  match (\u2190 getPropForm l) with\n  | PropForm.not nl =>\n      match (\u2190 getPropForm r) with\n        | PropForm.not nr => return (rs[0]!, conn nl nr)\n        | _ => return (rs[1]!, conn nl (mkNot r))\n  | _ => \n      match (\u2190 getPropForm r) with\n        | PropForm.not nr => return (rs[2]!, conn (mkNot l) nr)\n        | _ => return (rs[3]!, conn (mkNot l) (mkNot r))\n\ndef dmRuleFromInfo (l r : Expr) (conn : Expr \u2192 Expr \u2192 Expr) (n : Bool) (rs : Array Name) : TacticM ruleType := do\n  let p \u2190 dmRuleFromInfoNoNeg l r conn rs\n  if n then\n    return (p.1, mkNot p.2)\n  else\n    return p\n\ndef dmRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.not a => match (\u2190 getPropForm a) with\n      | PropForm.and l r =>\n        dmRuleFromInfo l r mkOr false\n          #[`or_iff_not_and_not.symm, `not_not_and_distrib, `not_and_not_distrib, `not_and_or]\n      | PropForm.or l r =>\n        dmRuleFromInfo l r mkAnd false\n          #[`and_iff_not_or_not.symm, `not_not_or_distrib, `not_or_not_distrib, `not_or]\n      | _ => myFail `demorgan \"De Morgan's laws don't apply\"\n    | PropForm.and l r =>\n        dmRuleFromInfo l r mkOr true\n          #[`not_or.symm, `not_or_not_distrib.symm, `not_not_or_distrib.symm, `and_iff_not_or_not]\n    | PropForm.or l r =>\n      dmRuleFromInfo l r mkAnd true\n        #[`not_and_or.symm, `not_and_not_distrib.symm, `not_not_and_distrib.symm, `or_iff_not_and_not]\n    | _ => myFail `demorgan \"De Morgan's laws don't apply\"\n\nelab \"demorgan\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `demorgan dmRule\n\n/- quant_neg tactic -/\ndef qnRuleFromInfoNoNeg (v : Name) (t b : Expr) (qf : Name \u2192 BinderInfo \u2192 Expr \u2192 Expr \u2192 Expr)\n  (rs : Name \u00d7 Name) : TacticM ruleType := do\n  let f := mkLambda `x BinderInfo.default t b\n  let negres \u2190 Meta.lambdaTelescope f fun fvs e => do\n    match (\u2190 getPropForm e) with\n      | PropForm.not ne => return some (\u2190 Meta.mkLambdaFVars fvs ne)\n      | _ => return none\n  match negres with\n    | some ne => match ne with\n      | lam _ _ nb _ => return (rs.1, qf v BinderInfo.default t nb)\n      | _ => return (rs.2, qf v BinderInfo.default t (mkNot b))\n    | none => return (rs.2, qf v BinderInfo.default t (mkNot b))\n\ndef qnRuleFromInfo (v : Name) (t b : Expr) (qf : Name \u2192 BinderInfo \u2192 Expr \u2192 Expr \u2192 Expr)\n  (n : Bool) (rs : Name \u00d7 Name) : TacticM ruleType := do\n  let p \u2190 qnRuleFromInfoNoNeg v t b qf rs\n  if n then\n    return (p.1, mkNot p.2)\n  else\n    return p\n\ndef qnRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.not p => match (\u2190 getPropForm p) with\n      | PropForm.all v t b _ =>           \n        qnRuleFromInfo v t b (mkExists (\u2190 Meta.getLevel t)) false\n          (`not_forall_not, `not_forall)\n      | PropForm.ex _ v t b _ =>\n        qnRuleFromInfo v t b mkForall false\n          (`not_exists_not, `not_exists)\n      | _ => myFail `quant_neg \"quantifier negation laws don't apply\"\n    | PropForm.all v t b _ =>\n      qnRuleFromInfo v t b (mkExists (\u2190 Meta.getLevel t)) true\n        (`not_exists.symm, `not_exists_not.symm)\n    | PropForm.ex _ v t b _ => \n      qnRuleFromInfo v t b mkForall true\n        (`not_forall.symm, `not_forall_not.symm)\n    | _ => myFail `quant_neg \"quantifier negation laws don't apply\"\n\nelab \"quant_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `quant_neg qnRule\n\n/- conditional tactic -/\ndef cdlRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.not p => match (\u2190 getPropForm p) with\n      | PropForm.implies l r => match (\u2190 getPropForm r) with\n        | PropForm.not nr => return (`not_imp_not_iff_and, mkAnd l nr)\n        | _ => return (`not_imp, mkAnd l (mkNot r))\n      | _ => myFail `conditional \"conditional laws don't apply\"\n    | PropForm.implies l r => match (\u2190 getPropForm l) with\n      | PropForm.not nl => return (`or_iff_not_imp_left.symm, mkOr nl r)\n      | _ => return (`imp_iff_not_or, mkOr (mkNot l) r)\n    | PropForm.and l r => match (\u2190 getPropForm r) with\n      | PropForm.not nr => return (`not_imp.symm, mkNot (\u2190 mkArrow l nr))\n      | _ => match (\u2190 getPropForm l) with\n        | PropForm.not nl => return (`not_imp_iff_not_and.symm, mkNot (\u2190 mkArrow r nl))\n        | _ => return (`not_imp_not_iff_and.symm, mkNot (\u2190 mkArrow l (mkNot r)))\n    | PropForm.or l r => match (\u2190 getPropForm l) with\n      | PropForm.not nl => return (`imp_iff_not_or.symm, (\u2190 mkArrow nl r))\n      | _ => match (\u2190 getPropForm r) with\n        | PropForm.not nr => return (`imp_iff_or_not.symm, (\u2190 mkArrow nr l))\n        | _ => return (`or_iff_not_imp_left, (\u2190 mkArrow (mkNot l) r))\n    | _ => myFail `conditional \"conditional laws don't apply\"\n\nelab \"conditional\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `conditional cdlRule\n\n/- double_neg tactic -/\ndef dnRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.not p1 => match (\u2190 getPropForm p1) with\n      | PropForm.not p2 => return (`Classical.not_not, p2)\n      | _ => myFail `double_neg \"double negation law doesn't apply\"\n    | _ => myFail `double_neg \"double negation law doesn't apply\"\n\nelab \"double_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `double_neg dnRule\n\n/- bicond_neg tactic\nNote converts P \u2194 Q to \u00ac(\u00acP \u2194 Q).\nSo to convert only one side of \u2194, must use : [term to convert] -/\ndef binegRule (form : Expr) : TacticM ruleType := do\n  match (\u2190 getPropForm form) with\n    | PropForm.not p => match (\u2190 getPropForm p) with\n      | PropForm.iff l r => match (\u2190 getPropForm l) with\n        | PropForm.not nl => return (`not_not_iff, mkIff nl r)\n        | _ => return (`not_iff, mkIff (mkNot l) r)\n      | _ => myFail `bicond_neg \"biconditional negation law doesn't apply\"\n    | PropForm.iff l r => match (\u2190 getPropForm l) with\n      | PropForm.not nl => return (`not_iff.symm, mkNot (mkIff nl r))\n      | _ => return (`not_not_iff.symm, mkNot (mkIff (mkNot l) r))\n    | _ => myFail `bicond_neg \"biconditional negation law doesn't apply\"\n\nelab \"bicond_neg\" f:(colonTerm)? l:(oneLoc)? : tactic => doEquivTac f l `bicond_neg binegRule\n\n-- Give error if any ident in i is already in use.  Is this right thing to do in all cases?\npartial def checkIdUsed (tac : Name) (i : Syntax) : TacticM Unit := do\n  match i with\n    | .missing => return ()\n    | .node _ _ as => for a in as do checkIdUsed tac a\n    | .atom _ _ => return ()\n    | .ident _ _ v _ => \n        if (\u2190 getLCtx).usesUserName v then\n          myFail tac (\"identifier \" ++ (toString v) ++ \" already in use\")\n        else\n          return ()\n\n-- Get label from \"with\" clause, or default label.  Used by several tactics\ndef getLabel (tac : Name) (w : Option WithId) (dflt : Ident := mkIdent `this) : TacticM Ident := do\n  match w with\n    | some h => \n      let i := h.raw[1]\n      checkIdUsed tac i\n      return \u27e8i\u27e9\n    | none => return dflt\n\ndef isLocalVar (s : Syntax) : TacticM Bool := do\n  match s with\n    | .ident _ _ v _ => return (\u2190 getLCtx).usesUserName v\n    | _ => return False\n\n/- or_left and or_right tactics -/\ndef negData (e : Expr) : TacticM (Expr \u00d7 Bool) := do\n  match (\u2190 getPropForm e) with\n    | PropForm.not ne => return (ne, true)\n    | _ => return (e, false)\n\ndef orstrat (tac : Name) (w : Option WithId) (left : Bool) : TacticM Unit :=\n  withMainContext do\n    let label \u2190 getLabel tac w\n    let d \u2190 getMainDecl\n    let t \u2190 Meta.whnf (\u2190 instantiateMVars d.type)\n    match (\u2190 getPropForm t) with\n      | PropForm.or l r => do\n          let (form, neg) \u2190 negData (if left then r else l)\n          let goalName := d.userName\n          let emn \u2190 mkFreshUserName `h\n          let emi := mkIdent emn\n          doHave emn (mkOr form (mkNot form)) (\u2190 `(em _))\n          evalTactic (\u2190 `(tactic|refine Or.elim $emi:ident ?_ ?_))\n          if neg then doSwap\n          let (rule1, rule2) :=\n            if left then\n              (mkIdent ``Or.inr, mkIdent ``Or.inl)\n            else\n              (mkIdent ``Or.inl, mkIdent ``Or.inr)\n          evalTactic (\u2190 `(tactic| exact fun x => $rule1:ident x))\n          evalTactic (\u2190 `(tactic| intro $label:ident; refine $rule2:ident ?_; clear $emi:ident))\n          let newGoal \u2190 getMainGoal\n          setUserName newGoal goalName\n      | _ => myFail tac \"goal is not a disjunction\"\n\nelab \"or_left\" w:(withId)? : tactic => orstrat `or_left w true\nelab \"or_right\" w:(withId)? : tactic => orstrat `or_right w false\n\n/- disj_syll tactic -/\ndef matchFirstNeg (e1 e2 : Expr) : TacticM Bool := do\n  match (\u2190 getPropForm e1) with\n    | PropForm.not ne1 => Meta.isExprDefEq ne1 e2\n    | _ => return false\n\n--1st coord:  does one match neg of other?  2nd coord:  does first match neg of second?\ndef matchNeg (e1 e2 : Expr) : TacticM (Bool \u00d7 Bool) := do\n  if (\u2190 matchFirstNeg e1 e2) then\n    return (true, true)\n  else\n    return ((\u2190 matchFirstNeg e2 e1), false)\n\n--1st coord:  Does neg contradict right side of disj?  (else left side)\n--2nd coord:  Is disjunct negation of neg?  (else neg is negation of disj)\ndef DisjSyllData (disj neg : Expr) : TacticM (Bool \u00d7 Bool) := do\n  match (\u2190 getPropForm disj) with\n    | PropForm.or l r =>\n      let (isneg, disjneg) \u2190 matchNeg l neg\n      if isneg then\n        return (false, disjneg)\n      else\n        let (isneg, disjneg) \u2190 matchNeg r neg\n        if isneg then\n          return (true, disjneg)\n        else\n          myFail `disj_syll \"disjunctive syllogism rule doesn't apply\"\n    | _ => myFail `disj_syll \"disjunctive syllogism rule doesn't apply\"\n\ndef parseIdOrTerm (it : IdOrTerm) : Term :=\n  let s := it.raw[0]\n  match s with\n    | .ident .. => \u27e8s\u27e9\n    | _ => \u27e8s[1]\u27e9\n\nelab \"disj_syll\" dIOrT:idOrTerm nIOrT:idOrTerm w:(withId)? : tactic =>\n  withMainContext do\n    let d := parseIdOrTerm dIOrT\n    let n := parseIdOrTerm nIOrT\n    let disj \u2190 exprFromPf d 2\n    let neg \u2190 exprFromPf n 0\n    let (dId, deflabel) :=\n      if (\u2190 isLocalVar d.raw) then\n        (true, \u27e8d.raw\u27e9)\n      else\n        (false, mkIdent `this)\n    let label \u2190 getLabel `disj_syll w deflabel\n    let (conright, disjneg) \u2190 DisjSyllData disj neg\n    let goalName := (\u2190 getMainDecl).userName\n    evalTactic (\u2190 `(tactic| refine Or.elim $d ?_ ?_))\n    if conright then doSwap\n    if disjneg then\n      evalTactic (\u2190 `(tactic| exact fun x => absurd $n x))\n    else\n      evalTactic (\u2190 `(tactic| exact fun x => absurd x $n))\n    if (dId && (w == none)) then evalTactic (\u2190 `(tactic| clear $label:ident))\n    evalTactic (\u2190 `(tactic| intro $label:ident))\n    let newGoal \u2190 getMainGoal\n    setUserName newGoal goalName\n\n/- contradict tactic -/\ndef ensureContra (w : Option WithId) : TacticM Unit :=\n  withMainContext do\n    let label \u2190 getLabel `contradict w\n    let t \u2190 getMainTarget\n    match (\u2190 getPropForm t) with\n      | PropForm.f => return ()\n      | _ => evalTactic (\u2190 `(tactic| by_contra $label:ident))\n \nelab \"contradict\" h:term w:(withId)? : tactic => do\n  ensureContra w\n  withMainContext do\n    --let tocon \u2190 formFromIdent h.raw\n    let tocon \u2190 exprFromPf h 0\n    match (\u2190 getPropForm tocon) with\n      | PropForm.not p =>\n        doSuffices p (\u2190 `(fun x => $h x))\n      | _ =>\n        doSuffices (mkNot tocon) (\u2190 `(fun x => x $h))\n\n/- define, def_step, and whnf tactics \nProbably want to use define, but include whnf to be able to compare\n-/\ndef unfoldOrWhnf (tac: Name) (e : Expr) (w rep : Bool) : TacticM Expr := do\n  if w then\n    match (\u2190 getPropForm e) with\n      | PropForm.exun lev v t b bi => return unfoldExUn lev v t b bi\n      | _ => whnfNotExUn e\n  else\n    unfoldHead e tac true rep\n\ndef doDefine (tac : Name) (f : Option ColonTerm) (l : Option OneLoc) (w rep : Bool) : TacticM Unit :=\n  withMainContext do\n    let e \u2190 match f with\n      | some fs => elabTerm fs.raw[1] none\n      | none => formFromLoc l\n    let e' \u2190 unfoldOrWhnf tac e w rep\n    doReplace tac l (\u2190 Meta.mkEq e e') (\u2190 `(Eq.refl _))\n\nelab \"define\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `define f l false true\nelab \"whnf\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `whnf f l true true\nelab \"def_step\" f:(colonTerm)? l:(oneLoc)? : tactic => doDefine `def_step f l false false\n\n/- definition and definition! tactics -/\n--Context set in doDefinition, which calls these functions\ndef getDefineFormLabel (f : Option ColonTerm) (l : Option OneLoc) : TacticM (Expr \u00d7 Name) := do\n  match f with\n    | some t => return (\u2190 elabTerm t.raw[1] none, `this)\n    | none => match l with\n      | some h => do\n        let hs := h.raw[1]\n        return (\u2190 formFromIdent hs, Name.mkStr hs.getId \"def\")\n      | none => return (\u2190 getMainTarget, `goal.def)\n\n-- use Iff for propositions, = for other types\ndef mkRel (e1 e2 : Expr) (prop : Bool) : TacticM Expr :=\n  if prop then\n    return mkIff e1 e2\n  else\n    Meta.mkEq e1 e2\n\n-- repeatedly assert definition equivalences or equations, numbering steps\npartial def doDefinitionRep (label : Name) (e e1 : Expr) (prop : Bool) (rule : Ident) (firstNum : Nat) : TacticM Unit := do\n  --let e' \u2190 unfoldHead e1 `definition (firstNum == 1)\n  let e' \u2190 unfoldHead e1 `definition (firstNum == 1) false\n  let res \u2190 mkRel e e' prop\n  doHave (Name.appendIndexAfter label firstNum) res (\u2190 `($rule _))\n  try\n    withMainContext (doDefinitionRep label e e' prop rule (firstNum + 1))  -- Context changes each time through\n  catch _ =>\n    return ()\n\ndef doDefinition (all : Bool) (f : Option ColonTerm) (l : Option OneLoc) (wid : Option WithId) : TacticM Unit :=\n  withMainContext do\n    let (e, deflabel) \u2190 getDefineFormLabel f l\n    let label \u2190 getLabel `definition wid (mkIdent deflabel)\n    let labeln := label.getId\n    let (prop, rule) := if (\u2190 exprIsProp e) then\n        (true, mkIdent ``Iff.refl)\n      else\n        (false, mkIdent ``Eq.refl)\n    if all then\n      doDefinitionRep labeln e e prop rule 1\n    else\n      --let e' \u2190 unfoldHeadRep e `definition true\n      let e' \u2190 unfoldHead e `definition true true\n      let res \u2190 mkRel e e' prop\n      doHave labeln res (\u2190 `($rule _))\n\nelab \"definition\" f:(colonTerm) wid:(withId)? : tactic => doDefinition false (some f) none wid\nelab \"definition\" l:(oneLoc)? wid:(withId)? : tactic => doDefinition false none l wid\nelab \"definition!\" f:(colonTerm) wid:(withId)? : tactic => doDefinition true (some f) none wid\nelab \"definition!\" l:(oneLoc)? wid:(withId)? : tactic => doDefinition true none l wid\n\ndef addToName (n : Name) (s : String) : Name :=\n  Name.modifyBase n (fun x => Name.mkStr x s)\n\n--Bool is whether or not to clear \"or\" given; Idents for two cases\ndef setUpCases (t : Term) (wids : Option With2Ids) : TacticM (Bool \u00d7 Ident \u00d7 Ident) := do\n  match wids with\n    | some ids =>\n      let id1s := ids.raw[1]\n      checkIdUsed `by_cases id1s\n      let id1 : Ident := \u27e8id1s\u27e9\n      match ids.raw[2].getArgs[1]? with\n        | some id2 =>\n          checkIdUsed `by_cases id2\n          return (false, id1, \u27e8id2\u27e9)\n        | none => return (false, id1, id1)\n    | none =>\n      if (\u2190 isLocalVar t.raw) then\n        let tid : Ident := \u27e8t.raw\u27e9\n        return (true, tid, tid)\n      else\n        let thisId := mkIdent `this\n        return (false, thisId, thisId)\n\ndef fixCase (clear : Bool) (label : Ident) (g : Name) (c : String) : TacticM Unit := do\n  if clear then\n    evalTactic (\u2190 `(tactic| clear $label))\n  evalTactic (\u2190 `(tactic| intro $label:ident))\n  setUserName (\u2190 getMainGoal) (addToName g c)\n  doSwap\n\nelab \"by_cases\" \"on\" t:term wids:(with2Ids)? : tactic =>\n  withMainContext do\n    let e \u2190 exprFromPf t 2\n    match (\u2190 getPropForm e) with\n      | PropForm.or _ _ =>\n        let (clear, label1, label2) \u2190 setUpCases t wids\n        let goalname :=  (\u2190 getMainDecl).userName\n        evalTactic (\u2190 `(tactic| refine Or.elim $t ?_ ?_))\n        fixCase clear label1 goalname \"Case_1\"\n        fixCase clear label2 goalname \"Case_2\"\n      | _ => myFail `by_cases \"hypothesis is not a disjunction\"\n\n/- exists_unique tactic -/\ndef mkUn (lev: Level) (v : Name) (t b : Expr) : TacticM Expr := do\n  let v1 := Name.appendIndexAfter v 1\n  let v2 := Name.appendIndexAfter v 2\n  let f1 := mkLambda v1 BinderInfo.default t b\n  let f2 := mkLambda v2 BinderInfo.default t b\n  Meta.lambdaTelescope f1 fun fv1 e1 => \n    Meta.lambdaTelescope f2 fun fv2 e2 => do\n      let body \u2190 mkArrow e1 (\u2190 mkArrow e2\n        (mkApp3 (const ``Eq [lev]) t fv1[0]! fv2[0]!))\n      Meta.mkForallFVars (fv1.push fv2[0]!) body\n\nelab \"exists_unique\" : tactic => do\n  let goal \u2190 getMainGoal\n  withContext goal do\n    let d \u2190 getDecl goal\n    let goalname := d.userName\n    let tar \u2190 instantiateMVars d.type\n    match (\u2190 getPropForm tar) with\n      | PropForm.exun lev v t b _ =>\n        let un \u2190 mkUn lev v t b\n        let ex := mkExists lev v BinderInfo.default t b\n        let h \u2190 mkFreshUserName `h\n        let hid := mkIdent h\n        let hex := (mkForall `a BinderInfo.default ex\n          (mkForall `b BinderInfo.default un tar))\n        doHave h hex (\u2190 `(exists_unique_of_exists_of_unique))\n        evalTactic (\u2190 `(tactic| refine $hid ?_ ?_; clear $hid))\n        setUserName (\u2190 getMainGoal) (addToName goalname \"Existence\")\n        doSwap\n        evalTactic (\u2190 `(tactic| clear $hid))\n        setUserName (\u2190 getMainGoal) (addToName goalname \"Uniqueness\")\n        doSwap\n      | _ => myFail `exists_unique \"goal is not a unique existence statement\"\n\n/- obtain tactic -/\ndef parseIdOrTerm?Type (tac : Name) (it : IdOrTerm?Type) : TacticM (Term \u00d7 (Option Term)) := do\n  let s := it.raw[0]\n  let res := match s with\n    | .ident .. => (\u27e8s\u27e9, none)\n    | _ => match s[2].getArgs[1]? with\n      | some t => (\u27e8s[1]\u27e9, some \u27e8t\u27e9)\n      | none => (\u27e8s[1]\u27e9, none)\n  checkIdUsed tac res.1.raw\n  return res\n\ndef doIntroOption (i : Term) (t : Option Term) : TacticM Unit := do\n  match t with\n    | some tt => --evalTactic (\u2190 `(tactic| intro ($i : $tt)))\n      evalTactic (\u2190 `(tactic| intro h; match @h with | ($i : $tt) => ?_; try clear h))\n    | none => evalTactic (\u2190 `(tactic| intro $i:term))\n\ndef doObtain (itw ith : IdOrTerm?Type) (tm : Term) : TacticM Unit :=\n  withMainContext do\n    --let e \u2190 whnfNotExUn (\u2190 formFromIdent l.raw)\n    let e \u2190 exprFromPf tm 1\n    match (\u2190 getPropForm e) with\n      | PropForm.ex _ _ _ _ _ =>\n        let (wi, wt) \u2190 parseIdOrTerm?Type `obtain itw\n        let (hi, ht) \u2190 parseIdOrTerm?Type `obtain ith\n        evalTactic (\u2190 `(tactic| refine Exists.elim $tm ?_))\n        doIntroOption wi wt\n        doIntroOption hi ht\n      | _ => myFail `obtain \"hypothesis is not an existence statement\"\n\ntheorem exun_elim {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop} {b : Prop}\n    (h2 : \u2203! x, p x) (h1 : \u2200 x, p x \u2192 (\u2200 y z, p y \u2192 p z \u2192 y = z) \u2192 b) : b := by\n      apply ExistsUnique.elim h2\n      intro x h3 h4\n      apply h1 x h3\n      intro y z h5 h6\n      have h7 := h4 y h5\n      have h8 := h4 z h6\n      rw [h7,h8]\n\ndef doObtainExUn (itw ith1 ith2 : IdOrTerm?Type) (tm : Term) : TacticM Unit :=\n  withMainContext do\n    let e \u2190 exprFromPf tm 1\n    match (\u2190 getPropForm e) with\n      | PropForm.exun lev v t b _ =>\n        let (wi, wt) \u2190 parseIdOrTerm?Type `obtain itw\n        let (h1i, h1t) \u2190 parseIdOrTerm?Type `obtain ith1\n        let (h2i, h2t) \u2190 parseIdOrTerm?Type `obtain ith2\n        let tar \u2190 getMainTarget\n        let un \u2190 mkUn lev v t b\n        let exun := mkForall v BinderInfo.default t (\u2190 mkArrow b (\u2190 mkArrow un tar))\n        let h \u2190 mkFreshUserName `h\n        let hid := mkIdent h\n        doHave h (\u2190 mkArrow exun tar) (\u2190 `(exun_elim $tm))\n        evalTactic (\u2190 `(tactic| refine $hid ?_; clear $hid))\n        doIntroOption wi wt\n        doIntroOption h1i h1t\n        doIntroOption h2i h2t\n      | _ => myFail `obtain \"hypothesis is not a unique existence statement\"\n\n--Make 1 assertion for existential, 2 for unique existential\nelab \"obtain\" itw:idOrTerm?Type ith:idOrTerm?Type \" from \" t:term : tactic =>\n  doObtain itw ith t\nelab \"obtain\" itw:idOrTerm?Type ith1:idOrTerm?Type ith2:idOrTerm?Type \" from \" t:term : tactic =>\n  doObtainExUn itw ith1 ith2 t\n\n/- assume and fix tactics -/\ndef doAssume (w : Term) (t : Option Term) : TacticM Unit :=\n  withMainContext do\n    checkIdUsed `assume w\n    match (\u2190 getPropForm (\u2190 Meta.whnf (\u2190 getMainTarget))) with\n      | PropForm.implies _ _ => doIntroOption w t\n      --| PropForm.not _ => doIntroOption w t  --Not necessary--whnf will have changed to implies\n      | _ => myFail `assume \"goal is not a conditional statement\"\n\ndef doFix (w : Term) (t : Option Term) : TacticM Unit :=\n  withMainContext do\n    checkIdUsed `fix w\n    match (\u2190 getPropForm (\u2190 Meta.whnf (\u2190 getMainTarget))) with\n      | PropForm.all _ _ _ _ => doIntroOption w t\n      | _ => myFail `fix \"goal is not a universally quantified statement\"\n\nelab \"assume\" w:term : tactic => doAssume w none\nelab \"assume\" w:term \" : \" t:term : tactic => doAssume w (some t)\nelab \"fix\" w:term : tactic => doFix w none\nelab \"fix\" w:term \" : \" t:term : tactic => doFix w (some t)\n\n/- show tactic: allow either \"from\" or \":=\"  Probably best to stick to \"from\" -/\nmacro \"show \" c:term \" from \" p:term : tactic => `(tactic| {show $c; exact $p})\nmacro \"show \" c:term \" := \" p:term : tactic => `(tactic| {show $c; exact $p})\n\n/- Not needed anymore--use Nat.strongRec' in Mathlib.Data.Nat.Basic.lean\ntheorem str_induc (P : Nat \u2192 Prop)\n    (h : \u2200 (n : Nat), (\u2200 n_1 < n, P n_1)\u2192 P n) : \u2200 (n : Nat), P n := by\n  have h2 : \u2200 (n : Nat), \u2200 k < n, P k := by\n    apply @Nat.rec\n    fix k\n    assume h2\n    contradict h2\n    exact Bool.of_decide_true rfl\n    fix n\n    assume ih\n    fix k\n    assume h2\n    by_cases h3 : k = n\n    rewrite [h3]\n    exact h n ih\n    have h4 : k < n := by\n      have h5 : k \u2264 n := Nat.le_of_succ_le_succ h2\n      have h6 : k < n \u2228 k = n := LE.le.lt_or_eq_dec h5\n      disj_syll h6 h3\n      exact h6\n    exact ih k h4\n  fix n\n  have h3 := h2 (n+1) n\n  apply h3\n  exact Nat.lt_succ_self n\n-/\n\ntheorem induc_from (P : Nat \u2192 Prop) (k : Nat) (h1 : P k) (h2 : (\u2200 n \u2265 k, P n \u2192 P (n+1))) :\n    \u2200 n \u2265 k, P n := by\n  apply @Nat.rec\n  assume h3\n  have h4 : k = 0 := Nat.eq_zero_of_le_zero h3\n  rewrite [h4] at h1\n  exact h1\n  fix n\n  assume h3\n  assume h4\n  have h5 : k < n + 1 \u2228 k = n + 1 := LE.le.lt_or_eq_dec h4\n  by_cases on h5\n  have h6 : k \u2264 n := Nat.le_of_lt_succ h5\n  have h7 := h3 h6\n  exact h2 n h6 h7\n  rewrite [h5] at h1\n  exact h1\n\n-- New version:  For ordinary induction, uses a different base if appropriate\ndef doInduc (strong : Bool) : TacticM Unit := do\n  let goal \u2190 getMainGoal\n  withContext goal do\n    let d \u2190 getDecl goal\n    let tag := d.userName\n    let target \u2190 instantiateMVars d.type\n    match (\u2190 getPropForm target) with\n      | PropForm.all v t b _ =>\n        match t with\n          | (.const ``Nat _) =>\n            let m := Expr.lam v t b BinderInfo.default  --motive\n            let vid := mkIdent v\n            if strong then\n              let v1 := Name.appendIndexAfter v 1\n              let v1id := mkIdent v1\n              let m1 := Expr.lam v1 t m BinderInfo.default\n              let newtar \u2190 Meta.lambdaTelescope m1 fun fvs Pv => do\n                let fv1 := fvs[0]!\n                let fv := fvs[1]!\n                let Pv1 := replaceFVar Pv fv fv1\n                let v1lv \u2190 elabTerm (\u2190 `($v1id < $vid)) none\n                let ih \u2190 Meta.mkForallFVars #[fv1] (\u2190 mkArrow v1lv Pv1)\n                Meta.mkForallFVars #[fv] (\u2190 mkArrow ih Pv)\n              let newgoal \u2190 Meta.mkFreshExprSyntheticOpaqueMVar newtar tag\n              assign goal (mkApp2 (Expr.const ``Nat.strongRec' [Level.zero]) m newgoal)\n              replaceMainGoal [newgoal.mvarId!]\n            else\n              let (base, ind, rule) \u2190 Meta.lambdaTelescope m fun fvs Pv => do\n                -- fvs.size should be 1.  Could it ever be larger?\n                let fv := fvs[0]!\n                let PFPv \u2190 getPropForm Pv\n                let (fr, Qv) := match PFPv with\n                  | PropForm.implies l r => match (consumeMData l) with\n                    | (app (app (app (app (const ``GE.ge _) (const ``Nat _)) _) a) min) =>\n                      if (a == fv) && !(containsFVar min (fvarId! fv)) then\n                        (some (min, l), r)\n                      else\n                        (none, Pv)\n                    | (app (app (app (app (const ``LE.le _) (const ``Nat _)) _) min) a) =>\n                      if (a == fv) && !(containsFVar min (fvarId! fv)) then\n                        (some (min, l), r)\n                      else\n                        (none, Pv)\n                    | _ => (none, Pv)\n                  | _ => (none, Pv)\n                let fvp1 \u2190 elabTerm (\u2190 `($vid:ident + 1)) none\n                let Qimp \u2190 mkArrow Qv (replaceFVar Qv fv fvp1)\n                match fr with\n                  | some (min, cond) =>\n                    let base := replaceFVar Qv fv min\n                    let ind \u2190 Meta.mkForallFVars fvs (\u2190 mkArrow cond Qimp)\n                    let m' \u2190 Meta.mkLambdaFVars fvs Qv\n                    pure (base, ind, mkApp2 (const ``induc_from []) m' min)\n                  | none =>\n                    let base := replaceFVar Qv fv (Expr.lit (.natVal 0))\n                    let ind \u2190 Meta.mkForallFVars fvs Qimp\n                    pure (base, ind, app (const ``Nat.rec [Level.zero]) m)\n              let baseGoal \u2190 Meta.mkFreshExprSyntheticOpaqueMVar base (addToName tag \"Base_Case\")\n              let indGoal \u2190 Meta.mkFreshExprSyntheticOpaqueMVar ind (addToName tag \"Induction_Step\")\n              assign goal (mkApp2 rule baseGoal indGoal)\n              replaceMainGoal [baseGoal.mvarId!, indGoal.mvarId!]\n          | _ => myFail `by_induc \"mathematical induction doesn't apply\"\n      | _ => myFail `by_induc \"mathematical induction doesn't apply\"\n\nelab \"by_induc\" : tactic => doInduc false\nelab \"by_strong_induc\" : tactic => doInduc true\nend tactic_defs\n\n--Constructing a function from its graph:\ndef graph {A B : Type} (f : A \u2192 B) : Set (A \u00d7 B) :=\n    { (a, b) : A \u00d7 B | f a = b }\n\ndef is_func_graph {A B : Type} (G : Set (A \u00d7 B)) : Prop :=\n    \u2200 (x : A), \u2203! (y : B), (x, y) \u2208 G\n\ntheorem func_from_graph {A B : Type} (F : Set (A \u00d7 B)) :\n    (\u2203 (f : A \u2192 B), graph f = F) \u2194 is_func_graph F := by\n  apply Iff.intro\n  assume h1\n  obtain f h2 from h1\n  define\n  fix x : A\n  rewrite [\u2190h2]\n  exists_unique\n  apply Exists.intro (f x)\n  define\n  rfl\n  fix y1; fix y2\n  assume h3; assume h4\n  define at h3; define at h4\n  rewrite [h3] at h4\n  exact h4\n  assume h1\n  have h2 : \u2200 (x : A), Nonempty { y : B // (x, y) \u2208 F } := by\n    define at h1\n    fix x : A\n    obtain y h2 _h3 from h1 x\n    exact \u27e8\u27e8y, h2\u27e9\u27e9\n  let ff : (x : A) \u2192 { y : B // (x, y) \u2208 F } := fun (x : A) => Classical.choice (h2 x)\n  let f : A \u2192 B := fun (x : A) => (ff x).val\n  apply Exists.intro f\n  apply Set.ext\n  fix (x, y) : A \u00d7 B\n  have h3 : (x, f x) \u2208 F := (ff x).property\n  apply Iff.intro\n  assume h4\n  define at h4\n  rewrite [h4] at h3\n  exact h3\n  assume h4\n  define\n  define at h1\n  obtain z _h5 h6 from h1 x\n  exact h6 (f x) y h3 h4\n\ndef sum_less {A : Type} [AddZeroClass A] (m : Nat) (f : Nat \u2192 A) : A :=\n  match m with\n    | 0 => 0\n    | n + 1 => sum_less n f + f n\n\ndef sum_from_to {A : Type} [AddZeroClass A] (k n : Nat) (f : Nat \u2192 A) : A :=\n  sum_less (n + 1 - k) (fun (j : Nat) => f (k + j))\n\nsyntax (name := sumFromTo) \"Sum \" ident \" from \" term \" to \" term \", \" term:51 : term\nmacro_rules (kind := sumFromTo)\n  | `(Sum $i from $k to $n, $p) => `(sum_from_to $k $n (fun $i => $p))\n\n@[app_unexpander sum_from_to] def unexpandSumFromTo : Lean.PrettyPrinter.Unexpander\n  | `($_ $k:term $n:term fun $i:ident => $b) => `(Sum $i from $k to $n, $b)\n  | `($_ $k:term $n:term fun ($i:ident : $_) => $b) => `(Sum $i from $k to $n, $b)\n  | _ => throw ()\n\ntheorem sum_base {A : Type} [AddZeroClass A] {k : Nat} {f : Nat \u2192 A} :\n    Sum i from k to k, f i = f k := by\n  define : Sum i from k to k, f i\n  rewrite [Nat.add_sub_cancel_left]\n  unfold sum_less; unfold sum_less\n  rewrite [zero_add, add_zero]\n  rfl\n  done\n \ntheorem sum_step {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n    (h : k \u2264 n) : Sum i from k to (n + 1), f i = (Sum i from k to n, f i) + f (n + 1) := by\n  define : Sum i from k to (n+1), f i\n  obtain j h1 from Nat.le.dest h\n  have h2 : n + 1 + 1 - k = n + 1 - k + 1 := by\n    rewrite [\u2190h1, add_assoc, add_assoc, Nat.add_sub_cancel_left, add_assoc, Nat.add_sub_cancel_left, add_assoc]\n    rfl\n  have h3 : f (n + 1) = f (k + (n + 1 - k)) := by\n    rewrite [\u2190h1, add_assoc, Nat.add_sub_cancel_left]\n    rfl\n  rewrite [h2, h3]\n  rfl\n  done\n\ntheorem sum_from_zero_step {A : Type} [AddZeroClass A] {n : Nat} {f : Nat \u2192 A} :\n    Sum i from 0 to (n + 1), f i = (Sum i from 0 to n, f i) + f (n + 1) :=\n  sum_step (Nat.zero_le n)\n\ntheorem sum_empty {A : Type} [AddZeroClass A] {k n : Nat} {f : Nat \u2192 A}\n    (h : n < k) : Sum i from k to n, f i = 0 := by\n  define : Sum i from k to n, f i\n  have h2 : n + 1 - k = 0 := Nat.sub_eq_zero_of_le h\n  rewrite [h2]\n  rfl\n  done\n\ndef prod_less {A : Type} [MulOneClass A] (m : Nat) (f : Nat \u2192 A) : A :=\n  match m with\n    | 0 => 1\n    | n + 1 => prod_less n f * f n\n\ndef prod_from_to {A : Type} [MulOneClass A] (k n : Nat) (f : Nat \u2192 A) : A :=\n  prod_less (n + 1 - k) (fun (j : Nat) => f (k + j))\n\nsyntax (name := prodFromTo) \"Prod \" ident \" from \" term \" to \" term \", \" term:51 : term\nmacro_rules (kind := prodFromTo)\n  | `(Prod $i from $k to $n, $p) => `(prod_from_to $k $n (fun $i => $p))\n\n@[app_unexpander prod_from_to] def unexpandProdFromTo : Lean.PrettyPrinter.Unexpander\n  | `($_ $k:term $n:term fun $i:ident => $b) => `(Prod $i from $k to $n, $b)\n  | `($_ $k:term $n:term fun ($i:ident : $_) => $b) => `(Prod $i from $k to $n, $b)\n  | _ => throw ()\n\ntheorem prod_base {A : Type} [MulOneClass A] {k : Nat} {f : Nat \u2192 A} :\n    Prod i from k to k, f i = f k := by\n  define : Prod i from k to k, f i\n  rewrite [Nat.add_sub_cancel_left]\n  unfold prod_less; unfold prod_less\n  rewrite [one_mul, add_zero]\n  rfl\n  done\n \ntheorem prod_step {A : Type} [MulOneClass A] {k n : Nat} {f : Nat \u2192 A}\n    (h : k \u2264 n) : Prod i from k to (n + 1), f i = (Prod i from k to n, f i) * f (n + 1) := by\n  define : Prod i from k to (n+1), f i\n  obtain j h1 from Nat.le.dest h\n  have h2 : n + 1 + 1 - k = n + 1 - k + 1 := by\n    rewrite [\u2190h1, add_assoc, add_assoc, Nat.add_sub_cancel_left, add_assoc, Nat.add_sub_cancel_left, add_assoc]\n    rfl\n  have h3 : f (n + 1) = f (k + (n + 1 - k)) := by\n    rewrite [\u2190h1, add_assoc, Nat.add_sub_cancel_left]\n    rfl\n  rewrite [h2, h3]\n  rfl\n  done\n\ntheorem prod_from_zero_step {A : Type} [MulOneClass A] {n : Nat} {f : Nat \u2192 A} :\n    Prod i from 0 to (n + 1), f i = (Prod i from 0 to n, f i) * f (n + 1) :=\n  prod_step (Nat.zero_le n)\n\ntheorem prod_empty {A : Type} [MulOneClass A] {k n : Nat} {f : Nat \u2192 A}\n    (h : n < k) : Prod i from k to n, f i = 1 := by\n  define : Prod i from k to n, f i\n  have h2 : n + 1 - k = 0 := Nat.sub_eq_zero_of_le h\n  rewrite [h2]\n  rfl\n  done", "meta": {"author": "djvelleman", "repo": "HTPILeanPackage", "sha": "b4a0ab0d0d5473ef27fbbbfba3f5d3208d5377da", "save_path": "github-repos/lean/djvelleman-HTPILeanPackage", "path": "github-repos/lean/djvelleman-HTPILeanPackage/HTPILeanPackage-b4a0ab0d0d5473ef27fbbbfba3f5d3208d5377da/HTPILib/HTPIDefs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6513548714339144, "lm_q1q2_score": 0.3586407897887728}}
{"text": "import Smt\n\ntheorem falsum : \u00acFalse := by\n  smt\n", "meta": {"author": "ufmg-smite", "repo": "lean-smt", "sha": "6de0c4b216a918a14cf7a47d9a6faccaf8c8a209", "save_path": "github-repos/lean/ufmg-smite-lean-smt", "path": "github-repos/lean/ufmg-smite-lean-smt/lean-smt-6de0c4b216a918a14cf7a47d9a6faccaf8c8a209/Test/Prop/Falsum.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3586407860623391}}
{"text": "import data.real.basic\n\nvariables a b c : \u211d\n\n-- example : max a b = max b a :=\n-- begin\n--   sorry\n-- end\n--\n-- example : min (min a b) c = min a (min b c) :=\n-- sorry\n", "meta": {"author": "jaalonso", "repo": "Matematicas_en_Lean", "sha": "c44e23d87665cb4aa00c813c6bfb3c41ebc83aa8", "save_path": "github-repos/lean/jaalonso-Matematicas_en_Lean", "path": "github-repos/lean/jaalonso-Matematicas_en_Lean/Matematicas_en_Lean-c44e23d87665cb4aa00c813c6bfb3c41ebc83aa8/src/Basicos/Propiedades_de_min_y_max.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548511303338, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.35864077860947174}}
{"text": "/-\nCopyright (c) 2018 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Jeremy Avigad\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.W\nimport Mathlib.PostPort\n\nuniverses u l u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Polynomial functors\n\nThis file defines polynomial functors and the W-type construction as a\npolynomial functor.  (For the M-type construction, see\npfunctor/M.lean.)\n-/\n\n/--\nA polynomial functor `P` is given by a type `A` and a family `B` of types over `A`. `P` maps\nany type `\u03b1` to a new type `P.obj \u03b1`, which is defined as the sigma type `\u03a3 x, P.B x \u2192 \u03b1`.\n\nAn element of `P.obj \u03b1` is a pair `\u27e8a, f\u27e9`, where `a` is an element of a type `A` and\n`f : B a \u2192 \u03b1`. Think of `a` as the shape of the object and `f` as an index to the relevant\nelements of `\u03b1`.\n-/\nstructure pfunctor \nwhere\n  A : Type u\n  B : A \u2192 Type u\n\nnamespace pfunctor\n\n\nprotected instance inhabited : Inhabited pfunctor :=\n  { default := mk Inhabited.default Inhabited.default }\n\n/-- Applying `P` to an object of `Type` -/\ndef obj (P : pfunctor) (\u03b1 : Type u_2) :=\n  sigma fun (x : A P) => B P x \u2192 \u03b1\n\n/-- Applying `P` to a morphism of `Type` -/\ndef map (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_3} (f : \u03b1 \u2192 \u03b2) : obj P \u03b1 \u2192 obj P \u03b2 :=\n  fun (_x : obj P \u03b1) => sorry\n\nprotected instance obj.inhabited (P : pfunctor) {\u03b1 : Type u} [Inhabited (A P)] [Inhabited \u03b1] : Inhabited (obj P \u03b1) :=\n  { default := sigma.mk Inhabited.default fun (_x : B P Inhabited.default) => Inhabited.default }\n\nprotected instance obj.functor (P : pfunctor) : Functor (obj P) :=\n  { map := map P, mapConst := fun (\u03b1 \u03b2 : Type u_2) => map P \u2218 function.const \u03b2 }\n\nprotected theorem map_eq (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : A P) (g : B P a \u2192 \u03b1) : f <$> sigma.mk a g = sigma.mk a (f \u2218 g) :=\n  rfl\n\nprotected theorem id_map (P : pfunctor) {\u03b1 : Type u_2} (x : obj P \u03b1) : id <$> x = id x := sorry\n\nprotected theorem comp_map (P : pfunctor) {\u03b1 : Type u_2} {\u03b2 : Type u_2} {\u03b3 : Type u_2} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (x : obj P \u03b1) : (g \u2218 f) <$> x = g <$> f <$> x := sorry\n\nprotected instance obj.is_lawful_functor (P : pfunctor) : is_lawful_functor (obj P) :=\n  is_lawful_functor.mk (pfunctor.id_map P) (pfunctor.comp_map P)\n\n/-- re-export existing definition of W-types and\nadapt it to a packaged definition of polynomial functor -/\ndef W (P : pfunctor) :=\n  W_type (B P)\n\n/- inhabitants of W types is awkward to encode as an instance\nassumption because there needs to be a value `a : P.A`\nsuch that `P.B a` is empty to yield a finite tree -/\n\n/-- root element  of a W tree -/\ndef W.head {P : pfunctor} : W P \u2192 A P :=\n  sorry\n\n/-- children of the root of a W tree -/\ndef W.children {P : pfunctor} (x : W P) : B P (W.head x) \u2192 W P :=\n  sorry\n\n/-- destructor for W-types -/\ndef W.dest {P : pfunctor} : W P \u2192 obj P (W P) :=\n  sorry\n\n/-- constructor for W-types -/\ndef W.mk {P : pfunctor} : obj P (W P) \u2192 W P :=\n  sorry\n\n@[simp] theorem W.dest_mk {P : pfunctor} (p : obj P (W P)) : W.dest (W.mk p) = p :=\n  sigma.cases_on p fun (p_fst : A P) (p_snd : B P p_fst \u2192 W P) => Eq.refl (W.dest (W.mk (sigma.mk p_fst p_snd)))\n\n@[simp] theorem W.mk_dest {P : pfunctor} (p : W P) : W.mk (W.dest p) = p :=\n  W_type.cases_on p fun (p_a : A P) (p_f : B P p_a \u2192 W_type (B P)) => Eq.refl (W.mk (W.dest (W_type.mk p_a p_f)))\n\n/-- `Idx` identifies a location inside the application of a pfunctor.\nFor `F : pfunctor`, `x : F.obj \u03b1` and `i : F.Idx`, `i` can designate\none part of `x` or is invalid, if `i.1 \u2260 x.1` -/\ndef Idx (P : pfunctor) :=\n  sigma fun (x : A P) => B P x\n\nprotected instance Idx.inhabited (P : pfunctor) [Inhabited (A P)] [Inhabited (B P Inhabited.default)] : Inhabited (Idx P) :=\n  { default := sigma.mk Inhabited.default Inhabited.default }\n\n/-- `x.iget i` takes the component of `x` designated by `i` if any is or returns\na default value -/\ndef obj.iget {P : pfunctor} [DecidableEq (A P)] {\u03b1 : Type u_2} [Inhabited \u03b1] (x : obj P \u03b1) (i : Idx P) : \u03b1 :=\n  dite (sigma.fst i = sigma.fst x) (fun (h : sigma.fst i = sigma.fst x) => sigma.snd x (cast sorry (sigma.snd i)))\n    fun (h : \u00acsigma.fst i = sigma.fst x) => Inhabited.default\n\n@[simp] theorem fst_map {P : pfunctor} {\u03b1 : Type u} {\u03b2 : Type u} (x : obj P \u03b1) (f : \u03b1 \u2192 \u03b2) : sigma.fst (f <$> x) = sigma.fst x :=\n  sigma.cases_on x fun (x_fst : A P) (x_snd : B P x_fst \u2192 \u03b1) => Eq.refl (sigma.fst (f <$> sigma.mk x_fst x_snd))\n\n@[simp] theorem iget_map {P : pfunctor} [DecidableEq (A P)] {\u03b1 : Type u} {\u03b2 : Type u} [Inhabited \u03b1] [Inhabited \u03b2] (x : obj P \u03b1) (f : \u03b1 \u2192 \u03b2) (i : Idx P) (h : sigma.fst i = sigma.fst x) : obj.iget (f <$> x) i = f (obj.iget x i) := sorry\n\nend pfunctor\n\n\n/-\nComposition of polynomial functors.\n-/\n\nnamespace pfunctor\n\n\n/-- functor composition for polynomial functors -/\ndef comp (P\u2082 : pfunctor) (P\u2081 : pfunctor) : pfunctor :=\n  mk (sigma fun (a\u2082 : A P\u2082) => B P\u2082 a\u2082 \u2192 A P\u2081)\n    fun (a\u2082a\u2081 : sigma fun (a\u2082 : A P\u2082) => B P\u2082 a\u2082 \u2192 A P\u2081) =>\n      sigma fun (u : B P\u2082 (sigma.fst a\u2082a\u2081)) => B P\u2081 (sigma.snd a\u2082a\u2081 u)\n\n/-- constructor for composition -/\ndef comp.mk (P\u2082 : pfunctor) (P\u2081 : pfunctor) {\u03b1 : Type} (x : obj P\u2082 (obj P\u2081 \u03b1)) : obj (comp P\u2082 P\u2081) \u03b1 :=\n  sigma.mk (sigma.mk (sigma.fst x) (sigma.fst \u2218 sigma.snd x))\n    fun (a\u2082a\u2081 : B (comp P\u2082 P\u2081) (sigma.mk (sigma.fst x) (sigma.fst \u2218 sigma.snd x))) =>\n      sigma.snd (sigma.snd x (sigma.fst a\u2082a\u2081)) (sigma.snd a\u2082a\u2081)\n\n/-- destructor for composition -/\ndef comp.get (P\u2082 : pfunctor) (P\u2081 : pfunctor) {\u03b1 : Type} (x : obj (comp P\u2082 P\u2081) \u03b1) : obj P\u2082 (obj P\u2081 \u03b1) :=\n  sigma.mk (sigma.fst (sigma.fst x))\n    fun (a\u2082 : B P\u2082 (sigma.fst (sigma.fst x))) =>\n      sigma.mk (sigma.snd (sigma.fst x) a\u2082) fun (a\u2081 : B P\u2081 (sigma.snd (sigma.fst x) a\u2082)) => sigma.snd x (sigma.mk a\u2082 a\u2081)\n\nend pfunctor\n\n\n/-\nLifting predicates and relations.\n-/\n\nnamespace pfunctor\n\n\ntheorem liftp_iff {P : pfunctor} {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (x : obj P \u03b1) : functor.liftp p x \u2194 \u2203 (a : A P), \u2203 (f : B P a \u2192 \u03b1), x = sigma.mk a f \u2227 \u2200 (i : B P a), p (f i) := sorry\n\ntheorem liftp_iff' {P : pfunctor} {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : A P) (f : B P a \u2192 \u03b1) : functor.liftp p (sigma.mk a f) \u2194 \u2200 (i : B P a), p (f i) := sorry\n\ntheorem liftr_iff {P : pfunctor} {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (x : obj P \u03b1) (y : obj P \u03b1) : functor.liftr r x y \u2194\n  \u2203 (a : A P),\n    \u2203 (f\u2080 : B P a \u2192 \u03b1), \u2203 (f\u2081 : B P a \u2192 \u03b1), x = sigma.mk a f\u2080 \u2227 y = sigma.mk a f\u2081 \u2227 \u2200 (i : B P a), r (f\u2080 i) (f\u2081 i) := sorry\n\ntheorem supp_eq {P : pfunctor} {\u03b1 : Type u} (a : A P) (f : B P a \u2192 \u03b1) : functor.supp (sigma.mk a f) = f '' set.univ := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/pfunctor/univariate/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6513548511303338, "lm_q1q2_score": 0.35864077860947174}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\nParallel computation of a computable sequence of computations by\na diagonal enumeration.\nThe important theorems of this operation are proven as\nterminates_parallel and exists_of_mem_parallel.\n(This operation is nondeterministic in the sense that it does not\nhonor sequence equivalence (irrelevance of computation time).)\n-/\nimport data.seq.wseq\nuniverses u v\n\nnamespace computation\nopen stream.wseq as wseq\nopen stream.seq as seq\nvariables {\u03b1 : Type u} {\u03b2 : Type v}\n\ndef parallel.aux2 : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) :=\nlist.foldr (\u03bb c o, match o with\n| sum.inl a  := sum.inl a\n| sum.inr ls := rmap (\u03bb c', c' :: ls) (destruct c)\nend) (sum.inr [])\n\ndef parallel.aux1 : list (computation \u03b1) \u00d7 wseq (computation \u03b1) \u2192\n  \u03b1 \u2295 list (computation \u03b1) \u00d7 wseq (computation \u03b1)\n| (l, S) := rmap (\u03bb l', match seq.destruct S with\n  | none := (l', seq.nil)\n  | some (none, S') := (l', S')\n  | some (some c, S') := (c::l', S')\n  end) (parallel.aux2 l)\n\n/-- Parallel computation of an infinite stream of computations,\n  taking the first result -/\ndef parallel (S : wseq (computation \u03b1)) : computation \u03b1 :=\ncorec parallel.aux1 ([], S)\n\ntheorem terminates_parallel.aux : \u2200 {l : list (computation \u03b1)} {S c},\n  c \u2208 l \u2192 terminates c \u2192 terminates (corec parallel.aux1 (l, S)) :=\nbegin\n  have lem1 : \u2200 l S, (\u2203 (a : \u03b1), parallel.aux2 l = sum.inl a) \u2192\n    terminates (corec parallel.aux1 (l, S)),\n  { intros l S e, cases e with a e,\n    have this : corec parallel.aux1 (l, S) = return a,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw e, simp [rmap] },\n    rw this, apply_instance },\n  intros l S c m T, revert l S,\n  apply @terminates_rec_on _ _ c T _ _,\n  { intros a l S m, apply lem1,\n    induction l with c l IH generalizing m; simp at m, { contradiction },\n    cases m with e m,\n    { rw \u2190e, simp [parallel.aux2],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls,\n      exacts [\u27e8a', rfl\u27e9, \u27e8a, rfl\u27e9] },\n    { cases IH m with a' e,\n      simp [parallel.aux2], simp [parallel.aux2] at e,\n      rw e, exact \u27e8a', rfl\u27e9 } },\n  { intros s IH l S m,\n    have H1 : \u2200 l', parallel.aux2 l = sum.inr l' \u2192 s \u2208 l',\n    { induction l with c l IH' generalizing m;\n      intros l' e'; simp at m, { contradiction },\n      cases m with e m; simp [parallel.aux2] at e',\n      { rw \u2190e at e',\n        cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        injection e' with e', rw \u2190e', simp },\n      { induction e : list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a' ls;\n        rw e at e', { contradiction },\n        have := IH' m _ e,\n        simp [parallel.aux2] at e',\n        cases destruct c; injection e' with h',\n        rw \u2190h', simp [this] } },\n    induction h : parallel.aux2 l with a l',\n    { exact lem1 _ _ \u27e8a, h\u27e9 },\n    { have H2 : corec parallel.aux1 (l, S) = think _,\n      { apply destruct_eq_think,\n        simp [parallel.aux1],\n        rw h, simp [rmap] },\n      rw H2, apply @computation.think_terminates _ _ _,\n      have := H1 _ h,\n      rcases seq.destruct S with _ | \u27e8_|c, S'\u27e9;\n      simp [parallel.aux1]; apply IH; simp [this] } }\nend\n\ntheorem terminates_parallel {S : wseq (computation \u03b1)}\n   {c} (h : c \u2208 S) [T : terminates c] : terminates (parallel S) :=\nsuffices \u2200 n (l : list (computation \u03b1)) S c,\n  c \u2208 l \u2228 some (some c) = seq.nth S n \u2192\n  terminates c \u2192 terminates (corec parallel.aux1 (l, S)),\nfrom let \u27e8n, h\u27e9 := h in this n [] S c (or.inr h) T,\nbegin\n  intro n, induction n with n IH; intros l S c o T,\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    have H : seq.destruct S = some (some c, _),\n    { unfold seq.destruct functor.map, rw \u2190 a, simp },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h, H], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      apply terminates_parallel.aux _ T, simp } },\n  { cases o with a a, { exact terminates_parallel.aux a T },\n    induction h : parallel.aux2 l with a l';\n    have C : corec parallel.aux1 (l, S) = _,\n    { apply destruct_eq_ret, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, resetI, apply_instance },\n    { apply destruct_eq_think, simp [parallel.aux1], rw [h], simp [rmap] },\n    { rw C, apply @computation.think_terminates _ _ _,\n      have TT : \u2200 l', terminates (corec parallel.aux1 (l', S.tail)),\n      { intro, apply IH _ _ _ (or.inr _) T, rw a, cases S with f al, refl },\n      induction e : seq.nth S 0 with o,\n      { have D : seq.destruct S = none,\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, simp [parallel.aux1], have TT := TT l',\n        rwa [seq.destruct_eq_nil D, seq.tail_nil] at TT },\n      { have D : seq.destruct S = some (o, S.tail),\n        { dsimp [seq.destruct], rw e, refl },\n        rw D, cases o with c; simp [parallel.aux1, TT] } } }\nend\n\ntheorem exists_of_mem_parallel {S : wseq (computation \u03b1)}\n   {a} (h : a \u2208 parallel S) : \u2203 c \u2208 S, a \u2208 c :=\nsuffices \u2200 C, a \u2208 C \u2192 \u2200 (l : list (computation \u03b1)) S,\n  corec parallel.aux1 (l, S) = C \u2192 \u2203 c, (c \u2208 l \u2228 c \u2208 S) \u2227 a \u2208 c,\nfrom let \u27e8c, h1, h2\u27e9 := this _ h [] S rfl in \u27e8c, h1.resolve_left id, h2\u27e9,\nbegin\n  let F : list (computation \u03b1) \u2192 \u03b1 \u2295 list (computation \u03b1) \u2192 Prop,\n  { intros l a, cases a with a l',\n    exact \u2203 c \u2208 l, a \u2208 c,\n    exact \u2200 a', (\u2203 c \u2208 l', a' \u2208 c) \u2192 (\u2203 c \u2208 l, a' \u2208 c) },\n  have lem1 : \u2200 (l : list (computation \u03b1)), F l (parallel.aux2 l),\n  { intro l, induction l with c l IH; simp [parallel.aux2],\n    { intros a h, rcases h with \u27e8c, hn, _\u27e9,\n      exact false.elim hn },\n    { simp [parallel.aux2] at IH,\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l with a ls;\n      simp [parallel.aux2],\n      { rcases IH with \u27e8c', cl, ac\u27e9,\n        refine \u27e8c', or.inr cl, ac\u27e9 },\n      { induction h : destruct c with a c'; simp [rmap],\n        { refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n          rw destruct_eq_ret h,\n          apply ret_mem },\n        { intros a' h, rcases h with \u27e8d, dm, ad\u27e9,\n          simp at dm, cases dm with e dl,\n          { rw e at ad, refine \u27e8c, list.mem_cons_self _ _, _\u27e9,\n            rw destruct_eq_think h,\n            exact think_mem ad },\n          { cases IH a' \u27e8d, dl, ad\u27e9 with d dm, cases dm with dm ad,\n            exact \u27e8d, or.inr dm, ad\u27e9 } } } } },\n  intros C aC, refine mem_rec_on aC _ (\u03bb C' IH, _);\n  intros l S e; have e' := congr_arg destruct e; have := lem1 l;\n  simp [parallel.aux1] at e'; cases parallel.aux2 l with a' l'; injection e' with h',\n  { rw h' at this, rcases this with \u27e8c, cl, ac\u27e9,\n    exact \u27e8c, or.inl cl, ac\u27e9 },\n  { induction e : seq.destruct S with a; rw e at h',\n    { exact let \u27e8d, o, ad\u27e9 := IH _ _ h',\n        \u27e8c, cl, ac\u27e9 := this a \u27e8d, o.resolve_right (wseq.not_mem_nil _), ad\u27e9 in\n      \u27e8c, or.inl cl, ac\u27e9 },\n    { cases a with o S', cases o with c; simp [parallel.aux1] at h';\n      rcases IH _ _ h' with \u27e8d, dl | dS', ad\u27e9,\n      { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' },\n      { simp at dl, cases dl with dc dl,\n        { rw dc at ad, refine \u27e8c, or.inr _, ad\u27e9,\n          rw seq.destruct_eq_cons e,\n          apply seq.mem_cons },\n        { exact let \u27e8c, cl, ac\u27e9 := this a \u27e8d, dl, ad\u27e9 in \u27e8c, or.inl cl, ac\u27e9 } },\n      { refine \u27e8d, or.inr _, ad\u27e9,\n        rw seq.destruct_eq_cons e,\n        exact seq.mem_cons_of_mem _ dS' } } }\nend\n\ntheorem map_parallel (f : \u03b1 \u2192 \u03b2) (S) : map f (parallel S) = parallel (S.map (map f)) :=\nbegin\n  refine eq_of_bisim (\u03bb c1 c2, \u2203 l S,\n    c1 = map f (corec parallel.aux1 (l, S)) \u2227\n    c2 = corec parallel.aux1 (l.map (map f), S.map (map f))) _ \u27e8[], S, rfl, rfl\u27e9,\n  intros c1 c2 h, exact match c1, c2, h with ._, ._, \u27e8l, S, rfl, rfl\u27e9 := begin\n    clear _match,\n    have : parallel.aux2 (l.map (map f)) = lmap f (rmap (list.map (map f)) (parallel.aux2 l)),\n    { simp [parallel.aux2],\n      induction l with c l IH; simp, rw [IH],\n      cases list.foldr parallel.aux2._match_1 (sum.inr list.nil) l; simp [parallel.aux2],\n      cases destruct c; simp },\n    simp [parallel.aux1], rw this, cases parallel.aux2 l with a l'; simp,\n    apply S.rec_on _ (\u03bb c S, _) (\u03bb S, _); simp; simp [parallel.aux1];\n    exact \u27e8_, _, rfl, rfl\u27e9\n  end end\nend\n\ntheorem parallel_empty (S : wseq (computation \u03b1)) (h : S.head ~> none) :\nparallel S = empty _ :=\neq_empty_of_not_terminates $ \u03bb \u27e8\u27e8a, m\u27e9\u27e9,\nlet \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel m,\n    \u27e8n, nm\u27e9 := wseq.exists_nth_of_mem cs,\n    \u27e8c', h'\u27e9 := wseq.head_some_of_nth_some nm in by injection h h'\n\n-- The reason this isn't trivial from exists_of_mem_parallel is because it eliminates to Sort\ndef parallel_rec {S : wseq (computation \u03b1)} (C : \u03b1 \u2192 Sort v)\n  (H : \u2200 s \u2208 S, \u2200 a \u2208 s, C a) {a} (h : a \u2208 parallel S) : C a :=\nbegin\n  let T : wseq (computation (\u03b1 \u00d7 computation \u03b1)) :=\n    S.map (\u03bb c, c.map (\u03bb a, (a, c))),\n  have : S = T.map (map (\u03bb c, c.1)),\n  { rw [\u2190wseq.map_comp], refine (wseq.map_id _).symm.trans (congr_arg (\u03bb f, wseq.map f S) _),\n    funext c, dsimp [id, function.comp], rw [\u2190map_comp], exact (map_id _).symm },\n  have pe := congr_arg parallel this, rw \u2190map_parallel at pe,\n  have h' := h, rw pe at h',\n  haveI : terminates (parallel T) := (terminates_map_iff _ _).1 \u27e8\u27e8_, h'\u27e9\u27e9,\n  induction e : get (parallel T) with a' c,\n  have : a \u2208 c \u2227 c \u2208 S,\n  { rcases exists_of_mem_map h' with \u27e8d, dT, cd\u27e9,\n    rw get_eq_of_mem _ dT at e, cases e, dsimp at cd, cases cd,\n    rcases exists_of_mem_parallel dT with \u27e8d', dT', ad'\u27e9,\n    rcases wseq.exists_of_mem_map dT' with \u27e8c', cs', e'\u27e9,\n    rw \u2190e' at ad',\n    rcases exists_of_mem_map ad' with \u27e8a', ac', e'\u27e9, injection e' with i1 i2,\n    constructor, rwa [i1, i2] at ac', rwa i2 at cs' },\n  cases this with ac cs, apply H _ cs _ ac\nend\n\ntheorem parallel_promises {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) : parallel S ~> a :=\n\u03bb a' ma', let \u27e8c, cs, ac\u27e9 := exists_of_mem_parallel ma' in H _ cs ac\n\ntheorem mem_parallel {S : wseq (computation \u03b1)} {a}\n  (H : \u2200 s \u2208 S, s ~> a) {c} (cs : c \u2208 S) (ac : a \u2208 c) : a \u2208 parallel S :=\nby haveI := terminates_of_mem ac; haveI := terminates_parallel cs;\n   exact mem_of_promises _ (parallel_promises H)\n\ntheorem parallel_congr_lem {S T : wseq (computation \u03b1)} {a}\n  (H : S.lift_rel equiv T) : (\u2200 s \u2208 S, s ~> a) \u2194 (\u2200 t \u2208 T, t ~> a) :=\n\u27e8\u03bb h1 t tT, let \u27e8s, sS, se\u27e9 := wseq.exists_of_lift_rel_right H tT in\n  (promises_congr se _).1 (h1 _ sS),\n\u03bb h2 s sS, let \u27e8t, tT, se\u27e9 := wseq.exists_of_lift_rel_left H sS in\n  (promises_congr se _).2 (h2 _ tT)\u27e9\n\n-- The parallel operation is only deterministic when all computation paths lead to the same value\ntheorem parallel_congr_left {S T : wseq (computation \u03b1)} {a}\n  (h1 : \u2200 s \u2208 S, s ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nlet h2 := (parallel_congr_lem H).1 h1 in\n\u03bb a', \u27e8\u03bb h, by have aa := parallel_promises h1 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_left H sS,\n      aT := (st _).1 as in mem_parallel h2 tT aT,\n\u03bb h, by have aa := parallel_promises h2 h; rw \u2190aa; rw \u2190aa at h; exact\n  let \u27e8s, sS, as\u27e9 := exists_of_mem_parallel h,\n      \u27e8t, tT, st\u27e9 := wseq.exists_of_lift_rel_right H sS,\n      aT := (st _).2 as in mem_parallel h1 tT aT\u27e9\n\ntheorem parallel_congr_right {S T : wseq (computation \u03b1)} {a}\n  (h2 : \u2200 t \u2208 T, t ~> a) (H : S.lift_rel equiv T) : parallel S ~ parallel T :=\nparallel_congr_left ((parallel_congr_lem H).2 h2) H\n\nend computation\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/seq/parallel.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.35852194962510747}}
{"text": "lemma le_refl (x : mynat) : x \u2264 x :=\nbegin\nuse 0,\nsymmetry,\nexact add_zero x,\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/8-inequality-world/l2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5660185351961013, "lm_q1q2_score": 0.35852194179124103}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.algebraic_topology.homotopical_algebra.cylinder\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits\n\nnamespace algebraic_topology\n\nnamespace model_category\n\nvariables {C : Type*} [category C] [model_category C] {A A' B B' : C}\n\nstructure left_homotopy (P : precylinder A) (f\u2080 f\u2081 : A \u27f6 B) :=\n(h : P.I \u27f6 B) (h\u2080' : P.d\u2080 \u226b h = f\u2080 . obviously) (h\u2081' : P.d\u2081 \u226b h = f\u2081 . obviously)\n\nnamespace left_homotopy\n\nrestate_axiom h\u2080'\nrestate_axiom h\u2081'\nattribute [simp, reassoc] h\u2080 h\u2081\n\ndef refl (P : precylinder A) (f : A \u27f6 B) : left_homotopy P f f :=\n{ h := P.\u03c3 \u226b f, }\n\ninstance (P : precylinder A) (f : A \u27f6 B) : inhabited (left_homotopy P f f) := \u27e8refl P f\u27e9\n\ndef symm {P : precylinder A} {f g : A \u27f6 B} (H : left_homotopy P f g) :\n  left_homotopy P.symm g f :=\n{ h := H.h, }\n\ndef trans [is_cofibrant A] {P P' : cylinder A} {f\u2081 f\u2082 f\u2083 : A \u27f6 B}\n  (H\u2081 : left_homotopy P.pre f\u2081 f\u2082) (H\u2082 : left_homotopy P'.pre f\u2082 f\u2083) :\n    left_homotopy (P.trans P').pre f\u2081 f\u2083 :=\n{ h := pushout.desc H\u2081.h H\u2082.h (by simp only [h\u2081, h\u2080]), }\n\ndef comp_right {P : precylinder A} {f f' : A \u27f6 B}\n  (H : left_homotopy P f f') (g : B \u27f6 B') : left_homotopy P (f \u226b g) (f' \u226b g) :=\n{ h := H.h \u226b g, }\n\nend left_homotopy\n\nstructure right_homotopy (P : pre_path_object B) (f\u2080 f\u2081 : A \u27f6 B) :=\n(h : A \u27f6 P.I) (h\u2080' : h \u226b P.d\u2080 = f\u2080 . obviously) (h\u2081' : h \u226b P.d\u2081 = f\u2081 . obviously)\n\nnamespace right_homotopy\n\nrestate_axiom h\u2080'\nrestate_axiom h\u2081'\nattribute [simp, reassoc] h\u2080 h\u2081\n\n@[simps]\ndef op {P : pre_path_object B} {f g : A \u27f6 B} (H : right_homotopy P f g) : left_homotopy P.op f.op g.op :=\n{ h := H.h.op,\n  h\u2080' := by { dsimp [pre_path_object.op], rw [\u2190 op_comp, H.h\u2080], },\n  h\u2081' := by { dsimp [pre_path_object.op], rw [\u2190 op_comp, H.h\u2081], }, }\n\n@[simps]\ndef unop {A B : C\u1d52\u1d56} {P : pre_path_object B} {f g : A \u27f6 B} (H : right_homotopy P f g) : left_homotopy P.unop f.unop g.unop :=\n{ h := H.h.unop,\n  h\u2080' := by { dsimp [pre_path_object.unop], rw [\u2190 unop_comp, H.h\u2080], },\n  h\u2081' := by { dsimp [pre_path_object.unop], rw [\u2190 unop_comp, H.h\u2081], }, }\n\n@[simps]\ndef of_hom {P : pre_path_object B} (h : A \u27f6 P.I) : right_homotopy P (h \u226b P.d\u2080) (h \u226b P.d\u2081) :=\n{ h := h,\n  h\u2080' := rfl,\n  h\u2081' := rfl, }\nend right_homotopy\n\nnamespace left_homotopy\n\n@[simps]\ndef op {P : precylinder A} {f g : A \u27f6 B} (H : left_homotopy P f g) : right_homotopy P.op f.op g.op :=\n{ h := H.h.op,\n  h\u2080' := by { dsimp [precylinder.op], rw [\u2190 op_comp, H.h\u2080], },\n  h\u2081' := by { dsimp [precylinder.op], rw [\u2190 op_comp, H.h\u2081], }, }\n\n@[simps]\ndef unop {A B : C\u1d52\u1d56} {P : precylinder A} {f g : A \u27f6 B} (H : left_homotopy P f g) : right_homotopy P.unop f.unop g.unop :=\n{ h := H.h.unop,\n  h\u2080' := by { dsimp [precylinder.unop], rw [\u2190 unop_comp, H.h\u2080], },\n  h\u2081' := by { dsimp [precylinder.unop], rw [\u2190 unop_comp, H.h\u2081], }, }\n\n@[simps]\ndef of_hom {P : precylinder A} (h : P.I \u27f6 B) : left_homotopy P (P.d\u2080 \u226b h) (P.d\u2081 \u226b h) :=\n{ h := h,\n  h\u2080' := rfl,\n  h\u2081' := rfl, }\n\nend left_homotopy\n\nnamespace right_homotopy\n\ndef refl (P : pre_path_object B) (f : A \u27f6 B) : right_homotopy P f f :=\n{ h := f \u226b P.\u03c3, }\n\ninstance (P : pre_path_object B) (f : A \u27f6 B) : inhabited (right_homotopy P f f) := \u27e8refl P f\u27e9\n\ndef symm {P : pre_path_object B} {f g : A \u27f6 B} (H : right_homotopy P f g) :\n  right_homotopy P.symm g f :=\n{ h := H.h, }\n\ndef trans {A B : C} [is_fibrant B] {P P' : path_object B} {f\u2081 f\u2082 f\u2083 : A \u27f6 B}\n  (H\u2081 : right_homotopy P.pre f\u2081 f\u2082) (H\u2082 : right_homotopy P'.pre f\u2082 f\u2083) :\n    right_homotopy (P.trans P').pre f\u2081 f\u2083 :=\nbegin\n  haveI : is_cofibrant (opposite.op B) := is_fibrant.op infer_instance,\n  let H\u2081' : left_homotopy P.op.pre f\u2081.op f\u2082.op := H\u2081.op,\n  let H\u2082' : left_homotopy P'.op.pre f\u2082.op f\u2083.op := H\u2082.op,\n  exact (left_homotopy.trans H\u2081' H\u2082').unop,\nend\n\ndef comp_left {P : pre_path_object B} {f f' : A \u27f6 B}\n  (H : right_homotopy P f f') (g : A' \u27f6 A) : right_homotopy P (g \u226b f) (g \u226b f') :=\n{ h := g \u226b H.h, }\n\nlemma with_cof_\u03c3_of_right_homotopy {A B : C} [hA : is_cofibrant A] {f f' : A \u27f6 B}\n  {P : path_object B} (H : right_homotopy P.pre f f') : \u2203 (P' : path_object B)\n  (H' : right_homotopy P'.pre f f'), cofibration P'.\u03c3 :=\nbegin\n  let P' := P.change_I (CM5b.fac (P.\u03c3)),\n  have sq : comm_sq (initial.to _) (initial.to _) (CM5b.p (P.\u03c3)) H.h :=\n    comm_sq.mk (is_initial.hom_ext initial_is_initial _ _),\n  refine \u27e8P.change_I (CM5b.fac (P.\u03c3)), _, by { dsimp, apply_instance, }\u27e9,\n  exact\n  { h := sq.lift,\n    h\u2080' := by { dsimp [path_object.change_I], rw [sq.fac_right_assoc, H.h\u2080], },\n    h\u2081' := by { dsimp [path_object.change_I], rw [sq.fac_right_assoc, H.h\u2081], }, },\nend\n\nlemma extension_exists {X X' Y : C} {P : path_object Y} {f\u2080 f\u2081 : X' \u27f6 Y} (i : X \u27f6 X')\n  [cofibration i] [weak_eq i] (H : right_homotopy P.pre (i \u226b f\u2080) (i \u226b f\u2081)) :\n  \u2203 (H' : right_homotopy P.pre f\u2080 f\u2081), i \u226b H'.h = H.h :=\nbegin\n  have sq : comm_sq H.h i P.pre.\u03c0 (prod.lift f\u2080 f\u2081) := by tidy,\n  have eq\u2080 := congr_arg (\u03bb f, f \u226b limits.prod.fst) sq.fac_right,\n  have eq\u2081 := congr_arg (\u03bb f, f \u226b limits.prod.snd) sq.fac_right,\n  simp only [pre_path_object.\u03c0, prod.comp_lift, prod.lift_snd, prod.lift_fst] at eq\u2080 eq\u2081,\n  use\n  { h := sq.lift,\n    h\u2080' := eq\u2080,\n    h\u2081' := eq\u2081, },\n  exact sq.fac_left,\nend\n\ndef extension {X X' Y : C} {P : path_object Y} {f\u2080 f\u2081 : X' \u27f6 Y} (i : X \u27f6 X')\n  [cofibration i] [weak_eq i] (H : right_homotopy P.pre (i \u226b f\u2080) (i \u226b f\u2081)) :\n  right_homotopy P.pre f\u2080 f\u2081 := (H.extension_exists i).some\n\nlemma extension_fac {X X' Y : C} {P : path_object Y} {f\u2080 f\u2081 : X' \u27f6 Y}\n  (i : X \u27f6 X') [cofibration i] [weak_eq i] (H : right_homotopy P.pre (i \u226b f\u2080) (i \u226b f\u2081)) :\n  i \u226b (H.extension i).h = H.h :=\n(H.extension_exists i).some_spec\n\nend right_homotopy\n\n\nnamespace left_homotopy\n\ndef to_right_homotopy {A B : C} [is_cofibrant A] {Cyl : cylinder A} {f\u2081 f\u2082 : A \u27f6 B}\n  (H : left_homotopy Cyl.pre f\u2081 f\u2082) (P : path_object B) : right_homotopy P.pre f\u2081 f\u2082 :=\nbegin\n  have sq : comm_sq (f\u2081 \u226b P.\u03c3) Cyl.d\u2080 P.\u03c0 (prod.lift (Cyl.\u03c3 \u226b f\u2081) H.h) := by tidy,\n  have hr\u2080 := congr_arg (\u03bb f, f \u226b limits.prod.fst) sq.fac_right,\n  have hr\u2081 := congr_arg (\u03bb f, f \u226b limits.prod.snd) sq.fac_right,\n  simp only [pre_path_object.\u03c0, prod.comp_lift, prod.lift_snd, prod.lift_fst] at hr\u2080 hr\u2081,\n  exact\n  { h := Cyl.d\u2081 \u226b sq.lift,\n    h\u2080' := by { simp only [hr\u2080, pre_path_object.\u03c0, assoc, precylinder.\u03c3d\u2081_assoc], },\n    h\u2081' := by { simp only [pre_path_object.\u03c0, assoc, hr\u2081, H.h\u2081], }, },\nend\n\nend left_homotopy\n\nnamespace right_homotopy\n\ndef to_left_homotopy {A B : C} [hB : is_fibrant B] {P : path_object B} {f\u2081 f\u2082 : A \u27f6 B}\n  (H : right_homotopy P.pre f\u2081 f\u2082) (Cyl : cylinder A) : left_homotopy Cyl.pre f\u2081 f\u2082 :=\nbegin\n  haveI : is_cofibrant (opposite.op B) := hB.op,\n  let H\u2081 : left_homotopy P.op.pre _ _ := H.op,\n  let H\u2082 : right_homotopy Cyl.pre.op _ _ := H\u2081.to_right_homotopy Cyl.op,\n  simpa only [Cyl.pre.unop_op] using H\u2082.unop,\nend\n\ndef change_path_object {A B : C} [hA : is_cofibrant A] [hB : is_fibrant B]\n  {P : path_object B} {f\u2081 f\u2082 : A \u27f6 B} (H : right_homotopy P.pre f\u2081 f\u2082) (P' : path_object B) :\n  right_homotopy P'.pre f\u2081 f\u2082 :=\n(H.to_left_homotopy (cylinder.some A)).to_right_homotopy P'\n\nend right_homotopy\n\nnamespace left_homotopy\n\ndef change_cylinder {A B : C} [hA : is_cofibrant A] [hB : is_fibrant B]\n  {Cyl : cylinder A} {f\u2081 f\u2082 : A \u27f6 B} (H : left_homotopy Cyl.pre f\u2081 f\u2082) (Cyl' : cylinder A) :\n  left_homotopy Cyl'.pre f\u2081 f\u2082 :=\n(H.to_right_homotopy (path_object.some B)).to_left_homotopy Cyl'\n\nend left_homotopy\n\nend model_category\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/homotopies.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.35812373524959973}}
{"text": "import tactic\n\nnamespace fifteen\n/-\n  You take a deep breath\n  And you walk through the doors\n  It's the morning of your very first day\n  ...\n-/\n\nsection defenestration -- definitions\n\n-- why in the world did I use an enumerated type instead of fin 4 \u00d7 fin 4?\n-- unfortunately this way is not easily scalable to other dimensions\n-- at least it looks pretty :)\n@[derive [decidable_eq, has_reflect, fintype]]\ninductive tile : Type\n| aa | ab | ac | ad\n| ba | bb | bc | bd\n| ca | cb | cc | cd\n| da | db | dc | dd\n-- not really using fintype rn but it's there anyways\n\nopen tile\n\ninstance : has_to_string tile :=\n\u27e8 \u03bb t, match t with\n  | aa := \"aa\"\n  | ab := \"ab\"\n  | ac := \"ac\"\n  | ad := \"ad\"\n  | ba := \"ba\"\n  | bb := \"bb\"\n  | bc := \"bc\"\n  | bd := \"bd\"\n  | ca := \"ca\"\n  | cb := \"cb\"\n  | cc := \"cc\"\n  | cd := \"cd\"\n  | da := \"da\"\n  | db := \"db\"\n  | dc := \"dc\"\n  | dd := \"dd\"\n  end\n\u27e9\n\n-- TODO: use `fin_range` and `univ` ?\ndef tiles_list : list tile := [aa, ab, ac, ad, ba, bb, bc, bd, ca, cb, cc, cd, da, db, dc, dd]\n\ndef get_adjacent : tile \u2192 list tile\n| aa := [ab, ba]\n| ab := [aa, ac, bb]\n| ac := [ab, ad, bc]\n| ad := [ac, bd]\n| ba := [aa, bb, ca]\n| bb := [ab, ba, bc, cb]\n| bc := [ac, bb, bd, cc]\n| bd := [ad, bc, cd]\n| ca := [ba, cb, da]\n| cb := [bb, ca, cc, db]\n| cc := [bc, cb, cd, dc]\n| cd := [bd, cc, dd]\n| da := [ca, db]\n| db := [cb, da, dc]\n| dc := [cc, db, dd]\n| dd := [cd, dc]\n\ndef is_adjacent (t\u2081 t\u2082 : tile) : Prop := t\u2081 \u2208 (get_adjacent t\u2082)\n\n@[ext] structure position := \n(map : tile \u2192 fin 16)\n-- (bij : function.bijective map)\n-- don't know if bijective is helpful,\n-- but it makes things more complicated so taking it out for now\n\n-- Mario Carneiro's Magic!\n-- I still don't know what @[ext] and this instance does...\ninstance : decidable_eq position :=\n\u03bb a b, decidable_of_iff' _ (position.ext_iff _ _)\n\n-- zero denotes the hole\n@[derive decidable]\ndef hole (t : tile) (p : position) : Prop := p.map t = 0\n\ndef valid_slide' (t : tile) (p : position) : Prop :=\n\u2203 t' \u2208 get_adjacent t, hole t' p\n\ndef valid_slide (t h : tile) (p : position) : Prop :=\nh \u2208 get_adjacent t \u2227 hole h p\n\ndef slide (t h : tile) (p : position) : position :=\n\u27e8 \u03bb t',\n  if t' = t then p.map h\n  else if t' = h then p.map t\n  else p.map t'\n\u27e9\n\ndef goal_position : position :=\n\u27e8 \u03bb t, match t with\n  | aa := 1 | ab := 2 | ac := 3 | ad := 4\n  | ba := 5 | bb := 6 | bc := 7 | bd := 8\n  | ca := 9 | cb := 10| cc := 11| cd := 12\n  | da := 13| db := 14| dc := 15| dd := 0\n  end \n\u27e9\n\n#eval goal_position.map dc\n#eval (slide dc dd goal_position).map dc\n\n-- this is really lookin like hanoi but with sliding\ninductive can_slide_to : position \u2192 position \u2192 Prop\n| self : \u2200 (p : position), can_slide_to p p\n| one : \u2200 (p\u2081 p\u2082 : position), (\u2203 (t e : tile), (valid_slide t e p\u2081) \u2227 (slide t e p\u2081) = p\u2082) \u2192 can_slide_to p\u2081 p\u2082\n| trans : \u2200 (p\u2081 p\u2082 p\u2083 : position), can_slide_to p\u2081 p\u2082 \u2192 can_slide_to p\u2082 p\u2083 \u2192 can_slide_to p\u2081 p\u2083\n\n-- we are assuming start is a solvable position\ndef game (start : position) := can_slide_to start goal_position\n\nend defenestration\n\nsection limabeans -- lemmas\n\n-- symmetry for adjacency\nlemma sym_adj (t\u2081 t\u2082 : tile) : is_adjacent t\u2081 t\u2082 \u2194 is_adjacent t\u2082 t\u2081 :=\nbegin\n  split; intros h; cases t\u2081; cases t\u2082; try {exact h},\n  all_goals {unfold is_adjacent at *; unfold get_adjacent at *; try {dec_trivial} },\n  all_goals { exfalso; finish },\nend\n\nlemma slide_one_step (p\u2081 p\u2082 : position) : (\u2203 (t e : tile), (valid_slide t e p\u2081) \u2227 can_slide_to (slide t e p\u2081) p\u2082) \u2192 can_slide_to p\u2081 p\u2082 :=\nbegin\n  rintros \u27e8t, e, h\u2081, h\u2082\u27e9,\n  apply can_slide_to.trans p\u2081 (slide t e p\u2081) p\u2082,\n  apply can_slide_to.one,\n  use [t, e],\n  split,\n  { exact h\u2081 },\n  { refl },\n  exact h\u2082,\nend\n\n-- more of Mario Carneiro's Magic\nlemma can_slide_to.of_eq : \u2200 {p\u2081 p\u2082 : position} (h : p\u2081 = p\u2082), can_slide_to p\u2081 p\u2082\n| p _ rfl := can_slide_to.self p\n\nend limabeans\n\nend fifteen", "meta": {"author": "SnobbyDragon", "repo": "leanfifteen", "sha": "4583ab44e1de89a25e693e5e611472a9ba1147b6", "save_path": "github-repos/lean/SnobbyDragon-leanfifteen", "path": "github-repos/lean/SnobbyDragon-leanfifteen/leanfifteen-4583ab44e1de89a25e693e5e611472a9ba1147b6/src/fifteen.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3581089652613778}}
{"text": "import combinatorial_lemma.profinite_setup\n\nsection\n\nopen category_theory\nopen category_theory.limits\nopen ProFiltPseuNormGrpWithTinv\u2081\n\nopen_locale nnreal big_operators\n\nnoncomputable theory\n\nuniverse u\n\nvariables (r' : \u211d\u22650) [fact (0 < r')] [fact (r' < 1)]\n  (\u039b : Type u) [polyhedral_lattice \u039b] (S : Profinite.{u})\n\nnamespace pseudo_normed_group\n\ndef sum' {c\u2081 c\u2082 : \u211d\u22650} {M : Type u} [pseudo_normed_group M]\n  (n : \u2115) (h : \u2191n * c\u2081 \u2264 c\u2082) :\n  (\u03a0 i : fin n, pseudo_normed_group.filtration M c\u2081) \u2192 pseudo_normed_group.filtration M c\u2082 :=\n\u03bb t, \u27e8\u2211 i : fin n, (t i).1, begin\n  apply filtration_mono,\n  rotate 1,\n  apply pseudo_normed_group.sum_mem_filtration,\n  intros i hi, exact (t i).2,\n  simpa,\nend\u27e9\n\n@[simp] lemma coe_sum' {c\u2081 c\u2082 : \u211d\u22650} {M : Type u} [pseudo_normed_group M]\n  (n : \u2115) (h : \u2191n * c\u2081 \u2264 c\u2082) (t : \u03a0 i : fin n, pseudo_normed_group.filtration M c\u2081) :\n  (sum' n h t : M) = \u2211 i : fin n, (t i : M) := rfl\n\nlemma sum'_zero {c\u2081 c\u2082 : \u211d\u22650} {M : Type u} [pseudo_normed_group M] (h : \u21910 * c\u2081 \u2264 c\u2082) :\n  (sum' 0 h :\n    (\u03a0 i : fin 0, pseudo_normed_group.filtration M c\u2081) \u2192 pseudo_normed_group.filtration M c\u2082) =\n  (\u03bb _, 0) :=\nbegin\n  ext t,\n  simp only [sum', fintype.univ_of_is_empty, subtype.val_eq_coe, finset.sum_empty,\n    subtype.coe_mk, filtration.coe_zero],\nend\n\nlemma sum'_succ {c\u2081 c\u2082 : \u211d\u22650} {M : Type u} [pseudo_normed_group M] (n : \u2115) (h : \u2191(n+1) * c\u2081 \u2264 c\u2082) :\n  (sum' (n+1) h :\n    (\u03a0 i : fin (n+1), pseudo_normed_group.filtration M c\u2081) \u2192 pseudo_normed_group.filtration M c\u2082) =\n  cast_le' (by simpa [add_mul] using h) \u2218 add' \u2218 (\u03bb t, \u27e8sum' n le_rfl (fin.tail t), t 0\u27e9) :=\nbegin\n  ext t,\n  simp only [sum', subtype.val_eq_coe, subtype.coe_mk, coe_cast_le', add'_eq],\n  rw [fin.sum_univ_succ, add_comm],\n  refl\nend\n\nend pseudo_normed_group\n\nnamespace comphaus_filtered_pseudo_normed_group\n\nopen pseudo_normed_group\n\nlemma continuous_sum' {c\u2081 c\u2082 : \u211d\u22650} {M : Type u} [comphaus_filtered_pseudo_normed_group M]\n  (n : \u2115) (h : \u2191n * c\u2081 \u2264 c\u2082) :\n  continuous (pseudo_normed_group.sum' n h :\n    (\u03a0 (i : fin n), pseudo_normed_group.filtration M c\u2081) \u2192\n    pseudo_normed_group.filtration M c\u2082) :=\nbegin\n  have : @pseudo_normed_group.sum' _ _ M _ n h =\n    pseudo_normed_group.cast_le' h \u2218 pseudo_normed_group.sum' n le_rfl,\n  { ext, refl },\n  rw this,\n  refine (@comphaus_filtered_pseudo_normed_group.continuous_cast_le _ _ _ _ \u27e8h\u27e9).comp _,\n  clear this h c\u2082,\n  induction n with n ih,\n  { rw [pseudo_normed_group.sum'_zero], exact continuous_const },\n  rw pseudo_normed_group.sum'_succ,\n  haveI : fact (\u2191n * c\u2081 + c\u2081 \u2264 \u2191(n.succ) * c\u2081) :=\n    by simp only [nat.cast_succ, add_mul, one_mul]; apply_instance,\n  refine (comphaus_filtered_pseudo_normed_group.continuous_cast_le _ _).comp\n    ((comphaus_filtered_pseudo_normed_group.continuous_add' _ _).comp _),\n  refine continuous.prod_mk (ih.comp _) (continuous_apply _),\n  exact continuous_pi (\u03bb i, continuous_apply _),\nend\n\nend comphaus_filtered_pseudo_normed_group\n\n/-\nnamespace Profinite\n\ndef pow (X : Profinite.{u}) (n : \u2115) : Profinite.{u} :=\nProfinite.product (\u03bb i : fin n, X)\n\ndef map_pow {X Y : Profinite.{u}} (f : X \u27f6 Y) (n : \u2115) :\n  X.pow n \u27f6 Y.pow n :=\nProfinite.product.lift _ $ \u03bb n, Profinite.product.\u03c0 _ n \u226b f\n\nend Profinite\n-/\n\nnamespace ProFiltPseuNormGrpWithTinv\u2081\n\n@[simps]\ndef level : \u211d\u22650 \u2964 ProFiltPseuNormGrpWithTinv\u2081.{u} r' \u2964 Profinite.{u} :=\n{ obj := \u03bb c,\n  { obj := \u03bb X, Profinite.of $ pseudo_normed_group.filtration X c,\n    map := \u03bb X Y f, \u27e8f.level _, f.continuous' _\u27e9,\n    map_id' := \u03bb M, by { ext, refl },\n    map_comp' := \u03bb M\u2081 M\u2082 M\u2083 f g, by { ext, refl } },\n  map := \u03bb c\u2081 c\u2082 h,\n  { app := \u03bb X, \u27e8pseudo_normed_group.cast_le' h.le, begin\n      haveI : fact (c\u2081 \u2264 c\u2082) := \u27e8h.le\u27e9,\n      apply comphaus_filtered_pseudo_normed_group.continuous_cast_le,\n    end\u27e9,\n    naturality' := \u03bb M\u2081 M\u2082 f, by { ext, refl } },\n  map_id' := \u03bb c, by { ext, refl },\n  map_comp' := \u03bb c\u2081 c\u2082 c\u2083 h\u2081\u2082 h\u2082\u2083, by { ext, refl } }\n\ninstance (c) : preserves_limits ((level r').obj c) :=\nbegin\n  change preserves_limits (PFPNGT\u2081_to_PFPNG\u2081\u2091\u2097.{u} r' \u22d9 ProFiltPseuNormGrp\u2081.level.obj.{u} c),\n  apply_with limits.comp_preserves_limits { instances := ff },\n  constructor, constructor, introsI J _, constructor,\n  -- <-- looks like we have `preserves_limit` and not `preserves_limits`, but\n  -- it should be trivial to add, if needed.\nend\n\nvariable {r'}\n\nabbreviation lvl (X : ProFiltPseuNormGrpWithTinv\u2081.{u} r') (c : \u211d\u22650) : Profinite.{u} :=\n((level r').obj c).obj X\n\nabbreviation map_lvl {X Y : ProFiltPseuNormGrpWithTinv\u2081.{u} r'} (f : X \u27f6 Y) (c : \u211d\u22650) :\n  X.lvl c \u27f6 Y.lvl c := ((level r').obj c).map f\n\n@[reassoc] lemma map_lvl_comp {X Y Z : ProFiltPseuNormGrpWithTinv\u2081.{u} r'}\n  (f : X \u27f6 Y) (g : Y \u27f6 Z) (c : \u211d\u22650) :\n  map_lvl f c \u226b map_lvl g c = map_lvl (f \u226b g) c :=\nby { ext, refl }\n\nabbreviation cast_lvl {c\u2081 c\u2082 : \u211d\u22650} (X : ProFiltPseuNormGrpWithTinv\u2081.{u} r') (h : c\u2081 \u2264 c\u2082) :\n  X.lvl c\u2081 \u27f6 X.lvl c\u2082 := ((level r').map h.hom).app _\n\n@[reassoc] lemma map_lvl_cast_lvl_eq {X Y : ProFiltPseuNormGrpWithTinv\u2081.{u} r'}\n  (f : X \u27f6 Y) (c\u2081 c\u2082 : \u211d\u22650) (h : c\u2081 \u2264 c\u2082) :\n  map_lvl f c\u2081 \u226b cast_lvl Y h = cast_lvl X h \u226b map_lvl f c\u2082 :=\nby { ext, refl }\n\ndef sum {c\u2081 c\u2082 : \u211d\u22650} (X : ProFiltPseuNormGrpWithTinv\u2081.{u} r') (n : \u2115) (h : \u2191n * c\u2081 \u2264 c\u2082) :\n  (X.lvl c\u2081).pow n \u27f6 X.lvl c\u2082 :=\n\u27e8pseudo_normed_group.sum' _ h,\n  comphaus_filtered_pseudo_normed_group.continuous_sum' _ _\u27e9\n\n@[simp] lemma coe_sum {c\u2081 c\u2082 : \u211d\u22650} (X : ProFiltPseuNormGrpWithTinv\u2081.{u} r')\n  (n : \u2115) (h : \u2191n * c\u2081 \u2264 c\u2082) : \u21d1(X.sum n h) = pseudo_normed_group.sum' n h := rfl\n\nlemma le\u2081 (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) :\n  \u2191N * (c / \u2191N + d) \u2264 c + \u2191N * d :=\nbegin\n  apply eq.le,\n  rw [mul_add, add_left_inj, mul_div_cancel'],\n  norm_cast,\n  exact ne_of_gt (fact.out _)\nend\n\nlemma le\u2082 (N : \u2115) (c d : \u211d\u22650) :\n  c \u2264 c + \u2191N * d := le_self_add\n\n/--\nGiven a `N : \u2115`, `c : \u211d\u22650`, an `X : ProFiltPseuNormGrpWithTinv\u2081 r'`, and a\n  `t : Profinite.punit \u27f6 X.lvl c`, this constructs the pullback of `t` along the 2nd projection\n  `(X.lvl (c/N + d))^n \u00d7_{X.lvl (c + N * d)} X.lvl c \u2192 X.lvl c`.\n-/\ndef gadget (X : ProFiltPseuNormGrpWithTinv\u2081.{u} r')\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 X.lvl c) : Profinite.{u} :=\nProfinite.pullback\n(Profinite.pullback.snd (X.sum N (le\u2081 N c d)) (X.cast_lvl (le\u2082 N c d))) t\n\ndef map_gadget {X Y : ProFiltPseuNormGrpWithTinv\u2081.{u} r'}\n  (f : X \u27f6 Y) (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 X.lvl c)\n  (t' : Profinite.punit.{u} \u27f6 Y.lvl c) (w : t \u226b map_lvl f c = t') :\n  X.gadget N c d t \u27f6 Y.gadget N c d t' :=\nProfinite.pullback.lift _ _\n(Profinite.pullback.fst _ _ \u226b\n  Profinite.pullback.lift _ _\n  (Profinite.pullback.fst _ _ \u226b\n    Profinite.product.lift _ (\u03bb i, Profinite.product.\u03c0 _ i \u226b map_lvl f _))\n  (Profinite.pullback.snd _ _ \u226b map_lvl f _)\n  begin\n    simp only [category.assoc, map_lvl_cast_lvl_eq, \u2190 Profinite.pullback.condition_assoc],\n    ext \u27e8\u27e8x\u2081, x\u2082\u27e9, hx\u27e9,\n    simp only [category.assoc, Profinite.coe_comp, function.comp_app, nat_trans.naturality],\n    dsimp [Profinite.pullback.fst, Profinite.pullback.snd],\n    rw [f.map_sum, finset.sum_congr rfl],\n    rintro i -,\n    refl\n  end)\n(Profinite.pullback.snd _ _)\n(by rw [category.assoc, Profinite.pullback.lift_snd, Profinite.pullback.condition_assoc, w])\n.\n\nlemma map_gadget_comp {X Y Z : ProFiltPseuNormGrpWithTinv\u2081.{u} r'}\n  (f : X \u27f6 Y) (g : Y \u27f6 Z) (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 X.lvl c)\n  (t' : Profinite.punit.{u} \u27f6 Y.lvl c) (t'' : Profinite.punit.{u} \u27f6 Z.lvl c)\n  (w : t \u226b map_lvl f c = t') (w' : t' \u226b map_lvl g c = t'') :\n  map_gadget f N c d t t' w \u226b map_gadget g N c d t' t'' w' =\n  map_gadget (f \u226b g) N c d t t'' (by { subst t'', subst t', rw [category.assoc, map_lvl_comp] }) :=\nbegin\n  dsimp only [map_gadget],\n  refine Profinite.pullback.hom_ext _ _ _ _ _ _;\n  simp only [category.assoc, Profinite.pullback.lift_fst, Profinite.pullback.lift_snd,\n    Profinite.pullback.lift_fst_assoc, Profinite.pullback.lift_snd_assoc],\n  refl\nend\n\n@[simps]\ndef gadget_diagram {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c) :\n  J \u2964 Profinite.{u} :=\n{ obj := \u03bb j, (K.obj j).gadget N c d (t \u226b map_lvl (C.\u03c0.app _) c),\n  map := \u03bb i j f, map_gadget (K.map f) _ _ _ _ _ (by rw [category.assoc, map_lvl_comp, cone.w]),\n  map_id' := \u03bb i, by { simp only [K.map_id], ext; refl },\n  map_comp' := \u03bb i j k h\u2081 h\u2082, by { simp only [K.map_comp], ext; refl } }\n\n@[simps]\ndef gadget_diagram_fst_snd {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c) :\n  gadget_diagram C N c d t \u27f6 K \u22d9 (level r').obj c :=\n{ app := \u03bb j, Profinite.pullback.fst _ _ \u226b Profinite.pullback.snd _ _,\n  naturality' := \u03bb i j h, by { ext; refl } }\n\n@[simps]\ndef gadget_diagram_fst_fst {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c)\n  (i : fin N) :\n  gadget_diagram C N c d t \u27f6 K \u22d9 (level r').obj (c / \u2191N + d) :=\n{ app := \u03bb j, Profinite.pullback.fst _ _ \u226b Profinite.pullback.fst _ _ \u226b\n    Profinite.product.\u03c0 _ i,\n  naturality' := \u03bb i j h, by { ext; refl } }\n\n@[simps]\ndef gadget_cone {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c) :\n  cone (gadget_diagram C N c d t) :=\n{ X := C.X.gadget N c d t,\n  \u03c0 :=\n  { app := \u03bb j, map_gadget (C.\u03c0.app _) _ _ _ _ _ rfl,\n    naturality' := \u03bb i j h,\n      by { dsimp, rw [category.id_comp, map_gadget_comp], congr, rw cone.w, } } }\n\nlemma gadget_cone_lift_fst_snd {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (hC : \u2200 a : \u211d\u22650, is_limit (((level r').obj a).map_cone C))\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c)\n  (S : cone (gadget_diagram C N c d t)) :\n  (hC c).lift ((cones.postcompose (gadget_diagram_fst_snd C N c d t)).obj S) =\n    Profinite.punit.elim S.X \u226b t :=\nbegin\n  refine ((hC c).uniq ((cones.postcompose (gadget_diagram_fst_snd.{u} C N c d t)).obj S) _ _).symm,\n  intro j,\n  simp only [functor.map_cone_\u03c0_app, cones.postcompose_obj_\u03c0, nat_trans.comp_app,\n    gadget_diagram_fst_snd_app, Profinite.pullback.condition],\n  simp only [\u2190 category.assoc], congr' 2, ext,\nend\n\nlemma gadget_cone_lift_aux {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (hC : \u2200 a : \u211d\u22650, is_limit (((level r').obj a).map_cone C))\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c)\n  (S : cone (gadget_diagram C N c d t)) :\n  let \u03b7 := \u03bb i, (gadget_diagram_fst_fst C N c d t i),\n      \u03b3 := gadget_diagram_fst_snd C N c d t,\n      S' := \u03bb i, (cones.postcompose (\u03b7 i)).obj S,\n      F : \u03a0 i, S.X \u27f6 _  := \u03bb i, (hC _).lift (S' i),\n      S'' :=  (cones.postcompose \u03b3).obj S in\n  Profinite.product.lift (\u03bb (i : fin N), C.X.lvl (c / \u2191N + d)) F \u226b\n    C.X.sum N (le\u2081 _ _ _) = (hC c).lift S'' \u226b C.X.cast_lvl (le\u2082 _ _ _) :=\nbegin\n  have h\u2081 := le\u2081 N c d,\n  have h\u2082 := le\u2082 N c d,\n  rintros \u03b7 \u03b3 S' F S'',\n  let \u03b3' : gadget_diagram C N c d t \u27f6 K \u22d9 (level r').obj (c + \u2191N * d) :=\n    \u03b3 \u226b whisker_left _ ((level r').map h\u2082.hom),\n  let T'' := (cones.postcompose \u03b3').obj S,\n  have : (hC c).lift S'' \u226b C.X.cast_lvl h\u2082 = (hC _).lift T'',\n  { apply (hC _).uniq T'',\n    intros j,\n    have := (hC c).fac S'' j,\n    dsimp at this \u22a2,\n    simp [category.assoc, \u2190 reassoc_of this] },\n  rw this,\n  apply (hC (c + \u2191N * d)).uniq T'',\n  intros j,\n  dsimp,\n  simp only [category.assoc],\n  have : C.X.sum N h\u2081 \u226b ((level r').obj (c + \u2191N * d)).map (C.\u03c0.app j) =\n    Profinite.product.lift _ (\u03bb i, Profinite.product.\u03c0 _ i \u226b map_lvl (C.\u03c0.app _) _) \u226b\n    ProFiltPseuNormGrpWithTinv\u2081.sum _ N h\u2081,\n  { ext x,\n    dsimp,\n    erw (C.\u03c0.app j).to_add_monoid_hom.map_sum,\n    refl },\n  rw this,\n  erw \u2190 Profinite.pullback.condition,\n  simp only [\u2190 category.assoc],\n  congr' 1,\n  apply Profinite.product.hom_ext,\n  intros a,\n  dsimp [F],\n  simp only [category.assoc, Profinite.product.lift_\u03c0, Profinite.product.lift_\u03c0_assoc],\n  erw (hC (c / \u2191N + d)).fac (S' a) j,\n  refl,\nend\n\n-- lemma gadget_cone_lift_fst_fst {J : Type u} [small_category J]\n--   {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n--   (hC : \u2200 a : \u211d\u22650, is_limit (((level r').obj a).map_cone C))\n--   (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c)\n--   (S : cone (gadget_diagram C N c d t)) (i : fin N) (j : J)\n--   (g : ((K.obj j).lvl (c / \u2191N + d)).pow N \u27f6 (((level r').obj (c / \u2191N + d)).map_cone C).X) :\n--   (hC (c / N + d)).lift ((cones.postcompose (gadget_diagram_fst_fst C N c d t i)).obj S) =\n--     S.\u03c0.app j \u226b Profinite.pullback.fst\n--       (Profinite.pullback.snd ((K.obj j).sum N _) ((K.obj j).cast_lvl _))\n--       (t \u226b map_lvl (C.\u03c0.app j) c) \u226b\n--         Profinite.pullback.fst ((K.obj j).sum N _) ((K.obj j).cast_lvl _) \u226b g :=\n-- begin\n--   refine ((hC _).uniq\n--     ((cones.postcompose (gadget_diagram_fst_fst.{u} C N c d t i)).obj S) _ _).symm,\n--   intro j,\n--   have := (hC _).fac ((cones.postcompose (gadget_diagram_fst_fst.{u} C N c d t i)).obj S),\n--   simp only [functor.map_cone_\u03c0_app, cones.postcompose_obj_\u03c0,\n--     nat_trans.comp_app, gadget_diagram_fst_snd_app],\n--   rw Profinite.pullback.condition,\n--   simp only [\u2190 category.assoc], congr' 2, ext,\n-- end\n\ndef gadget_cone_is_limit {J : Type u} [small_category J]\n  {K : J \u2964 ProFiltPseuNormGrpWithTinv\u2081 r'} (C : cone K)\n  (hC : \u2200 a : \u211d\u22650, is_limit (((level r').obj a).map_cone C))\n  (N : \u2115) [fact (0 < N)] (c d : \u211d\u22650) (t : Profinite.punit.{u} \u27f6 C.X.lvl c) :\n  is_limit (gadget_cone C N c d t) :=\n{ lift := \u03bb S,\n    Profinite.pullback.lift _ _\n      (Profinite.pullback.lift _ _\n        (Profinite.product.lift _\n          (\u03bb i, (hC _).lift\n            ((cones.postcompose (gadget_diagram_fst_fst C N c d t i)).obj S)))\n        ((hC _).lift ((cones.postcompose (gadget_diagram_fst_snd C N c d t)).obj S))\n        (gadget_cone_lift_aux _ _ _ _ _ _ _))\n      (Profinite.punit.elim _)\n      (by rw [Profinite.pullback.lift_snd, eq_comm, gadget_cone_lift_fst_snd]),\n  fac' := \u03bb S j, begin\n    dsimp only [gadget_cone_\u03c0_app, map_gadget],\n    apply Profinite.pullback.hom_ext;\n      simp only [category.assoc, Profinite.pullback.lift_fst, Profinite.pullback.lift_snd,\n        Profinite.pullback.lift_fst_assoc, Profinite.pullback.lift_snd_assoc],\n    swap, { ext },\n    apply Profinite.pullback.hom_ext;\n      simp only [category.assoc, Profinite.pullback.lift_fst, Profinite.pullback.lift_snd,\n        Profinite.pullback.lift_fst_assoc, Profinite.pullback.lift_snd_assoc],\n    { apply Profinite.product.hom_ext, intro i,\n      simp only [category.assoc, Profinite.product.lift_\u03c0, Profinite.product.lift_\u03c0_assoc],\n      exact (hC _).fac ((cones.postcompose (gadget_diagram_fst_fst.{u} C N c d t i)).obj S) j, },\n    { rw [gadget_cone_lift_fst_snd, Profinite.pullback.condition],\n      simp only [\u2190 category.assoc], congr' 2, ext, }\n  end,\n  uniq' := \u03bb S f h, begin\n    apply Profinite.pullback.hom_ext;\n      simp only [category.assoc, Profinite.pullback.lift_fst, Profinite.pullback.lift_snd,\n        Profinite.pullback.lift_fst_assoc, Profinite.pullback.lift_snd_assoc],\n    swap, { ext },\n    apply Profinite.pullback.hom_ext;\n      simp only [category.assoc, Profinite.pullback.lift_fst, Profinite.pullback.lift_snd,\n        Profinite.pullback.lift_fst_assoc, Profinite.pullback.lift_snd_assoc],\n    swap,\n    { rw [Profinite.pullback.condition, gadget_cone_lift_fst_snd, \u2190category.assoc], congr' 1, ext },\n    apply Profinite.product.hom_ext,\n    intro i,\n    rw [Profinite.product.lift_\u03c0],\n    refine (hC (c / N + d)).uniq\n      ((cones.postcompose (gadget_diagram_fst_fst.{u} C N c d t i)).obj S) _ _,\n    intro j,\n    dsimp,\n    rw \u2190 h,\n    dsimp [gadget_cone, map_gadget],\n    simp,\n  end }\n\nend ProFiltPseuNormGrpWithTinv\u2081\n\nnamespace lem98\n\nopen ProFiltPseuNormGrpWithTinv\u2081\n\ninstance (c : \u211d\u22650) : preserves_limits (hom_functor.{u} r' \u039b \u22d9 (level r').obj c) :=\nbegin\n  change preserves_limits (hom_functor r' \u039b \u22d9 PFPNGT\u2081_to_PFPNG\u2081\u2091\u2097 r' \u22d9 ProFiltPseuNormGrp\u2081.level.obj c),\n  apply_instance,\nend\n\ndef hom_diagram : discrete_quotient S \u2964 ProFiltPseuNormGrpWithTinv\u2081.{u} r' :=\nS.fintype_diagram \u22d9 Fintype_Lbar.{u u} r' \u22d9 hom_functor r' \u039b\n\n/-- The cone over `hom_diagram` whose cone point is defeq to `Hom(\u039b, Lbar S)`.\nSee lemma below. -/\ndef hom_Lbar_cone : cone (hom_diagram r' \u039b S) :=\n(hom_functor r' \u039b).map_cone\n  (limit.cone (S.fintype_diagram \u22d9 Fintype_Lbar.{u u} r'))\n\n@[simp]\nlemma hom_Lbar_cone_X : (hom_Lbar_cone r' \u039b S ).X =\n  ((hom_functor.{u} r' \u039b).obj ((Lbar.functor.{u u} r').obj S)) := rfl\n\n/-- The cone with cone point `Hom(\u039b, Lbar S)_{\u2264 c}` is indeed a limit cone. -/\ndef hom_Lbar_cone_is_limit (c) : is_limit (((level r').obj c).map_cone\n  (hom_Lbar_cone r' \u039b S)) :=\nbegin\n  let E := (limit.cone (S.fintype_diagram \u22d9 Fintype_Lbar.{u u} r')),\n  change is_limit (((hom_functor.{u} r' \u039b \u22d9 (level r').obj c)).map_cone E),\n  apply is_limit_of_preserves (hom_functor.{u} r' \u039b \u22d9 (level r').obj c)\n    (limit.is_limit _),\n  apply_instance,\nend .\n\n-- This should follow from the finite case of lem98.\nlemma gadget_nonempty (N : \u2115) [fact (0 < N)] (T : discrete_quotient S)\n  (c) (t) : nonempty ((gadget_diagram (hom_Lbar_cone r' \u039b _) N c (d \u039b N) t).obj T) :=\nbegin\n  obtain \u27e8h\u27e9 := lem98_finite \u039b T N,\n  specialize h c,\n  let u : (hom_Lbar_cone r' \u039b S).X \u27f6 (hom_diagram r' \u039b S).obj T :=\n    ((hom_Lbar_cone r' \u039b S).\u03c0.app T),\n  let t' := t \u226b ((level r').obj c).map u,\n  specialize h (t' punit.star).1 (t' punit.star).2,\n  swap, apply_instance,\n  obtain \u27e8e,he1,he2\u27e9 := h,\n  -- Now use `e`, `t'`, `he1` and `he2` to finish off the proof...\n  refine \u27e8\u27e8\u27e8\u27e8(_, _), _\u27e9, _\u27e9, _\u27e9\u27e9,\n  { intro i, refine \u27e8e i, he2 i\u27e9, },\n  { let x := t' punit.star, exact \u27e8x.1, x.2\u27e9, },\n  { apply subtype.ext, exact he1.symm },\n  { exact punit.star },\n  { apply subtype.ext, refl, },\nend\n\n-- This should follow from Tychonoff and `gadget_nonempty`.\nlemma key (N : \u2115) [fact (0 < N)] (c) (t) :\n  nonempty (((hom_functor r' \u039b).obj ((Lbar.functor.{u u} r').obj S)).gadget N c (d \u039b N) t) :=\nbegin\n  let E := gadget_cone (hom_Lbar_cone r' \u039b _) N c (d \u039b N) t,\n  let hE : is_limit E := gadget_cone_is_limit _ _ _ _ _ _,\n  swap, { intros a, apply hom_Lbar_cone_is_limit },\n  let E' := Profinite.to_Top.map_cone E,\n  let hE' : is_limit E' := is_limit_of_preserves _ hE,\n  let G := gadget_diagram (hom_Lbar_cone r' \u039b S) N c (d \u039b N) t \u22d9 Profinite.to_Top,\n  let T : E'.X \u2245 (Top.limit_cone.{u u} G).X :=\n    hE'.cone_point_unique_up_to_iso (Top.limit_cone_is_limit G),\n  suffices : nonempty (Top.limit_cone.{u u} G).X,\n  { obtain \u27e8a\u27e9 := this, exact \u27e8T.inv a\u27e9, },\n  apply_with Top.nonempty_limit_cone_of_compact_t2_cofiltered_system { instances := ff },\n  { apply_instance },\n  { intros, apply gadget_nonempty, },\n  { intros j,\n    change compact_space\n      ((gadget_diagram (hom_Lbar_cone r' \u039b S) N c (d \u039b N) t).obj j),\n    apply_instance },\n  { intros j,\n    change t2_space\n      ((gadget_diagram (hom_Lbar_cone r' \u039b S) N c (d \u039b N) t).obj j),\n    apply_instance },\nend\n\n/-- Lemma 9.8 of [Analytic], with somewhat restrictive universes -/\ntheorem main (r' : \u211d\u22650) [fact (0 < r')] [fact (r' < 1)]\n  (\u039b : Type u) [polyhedral_lattice \u039b] (S : Profinite.{u}) (N : \u2115) [hN : fact (0 < N)] :\n  pseudo_normed_group.splittable (\u039b \u2192+ (Lbar.functor.{u u} r').obj S) N (d \u039b N) :=\nbegin\n  constructor,\n  intros c u hu,\n  let t : Profinite.punit \u27f6 ((hom_functor r' \u039b).obj ((Lbar.functor.{u u} r').obj S)).lvl c :=\n    Profinite.from_punit \u27e8u,hu\u27e9,\n  obtain \u27e8K,hK\u27e9 := key r' \u039b S N c t,\n  rcases K with \u27e8\u27e8\u27e8K\u2081,K\u2082\u27e9,hhK\u27e9,\u27e8\u27e9\u27e9,\n  dsimp [t, Profinite.from_punit, Profinite.pullback.snd] at hK,\n  dsimp at hhK,\n  use (\u03bb i, (K\u2081 i).1),\n  split,\n  { apply_fun (\u03bb e, e.val) at hhK,\n    change _ = K\u2082.val at hhK,\n    apply_fun (\u03bb e, e.val) at hK,\n    rw hK at hhK,\n    exact hhK.symm },\n  { intros i,\n    exact (K\u2081 i).2 }\nend\n\nend lem98\n\n-- A version with less restrictive universes. But we don't need that. Who cares about universes.\n\n-- /-- Lemma 9.8 of [Analytic] -/\n-- lemma lem98 (r' : \u211d\u22650) [fact (0 < r')] [fact (r' < 1)]\n--   (\u039b : Type*) [polyhedral_lattice \u039b] (S : Profinite) (N : \u2115) [hN : fact (0 < N)] :\n--   pseudo_normed_group.splittable (\u039b \u2192+ (Lbar.functor r').obj S) N (lem98.d \u039b N) :=\n-- begin\n--   admit\n-- end\n\nend\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/combinatorial_lemma/profinite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.3581089586296192}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) [irrational_\u03b1 : irrat \u03b1] (\u03b5 : \u211d) : \n  \u03b5 > 0 \u2192 \u2203 (r : \u211d), r \u2208 (set.range (\u03bb (i : \u2115), (i * \u03b1) % 1)) \u2227 r > (\u03b5 - 1) :=\nbegin\n  assume h1 : \u03b5 > 0,\n  have h2 : \u2203 (N : \u2115), N * \u03b5 > 1, from natural_number_of_multiple_of_\u03b5 h1,\n  use ((set.Ico \u03b5 (1 + \u03b5)).nonempty_iff_exists_mem.mp h2) /-: \u211d-,\n  apply exists.intro,\n  show (set.Ico \u03b5 (1 + \u03b5)).nonempty_iff_exists_mem.mp h2 \u2208 (set.range (\u03bb (i : \u2115), (i * \u03b1) % 1)), \n  from sorry,\n  show (set.Ico \u03b5 (1 + \u03b5)).nonempty_iff_exists_mem.mp h2 - 1 \u2208 (set.range (\u03bb (i : \u2115), (i * \u03b1) % 1)), \n  from sorry,\nend\n\n--OUTPUT 2\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=irrational_orbit_dense (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rational \u03b1) : \u2203 S : set \u211d, \u2200 x \u2208 S, is_int_multiple \u03b1 x \u2227 dense_subset S (Icc 0 1) :=\nbegin\n    show \u2203 S : set \u211d, \u2200 x \u2208 S, is_int_multiple \u03b1 x \u2227 dense_subset S (Icc 0 1), from sorry,\nend\n\n--OUTPUT 3\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit : sorry := sorry\n\n/--`theorem`\nExistence of an Irrational Number\nThere exists an irrational number\n`proof`\nSuppose that all real numbers were rational. Then we could order them in a sequence\n$$\nx_1,x_2,x_3,\\ldots\n$$\n(possibly with repetitions), so that the distance between any two consecutive numbers is $\\frac{1}{2}$ or smaller. That is, $|x_{i+1}-x_i|<\\frac{1}{2}$ for all $i$.\n\nConsider the number $z:=\\sum_{i=1}^{\\infty}\\frac{1}{2^i}=\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\ldots$. This number is clearly rational: it is a non-terminating, repeating decimal.\n\nThere are two possible options:\n\n1. $z$ cannot be one of the numbers $x_1,x_2,x_3,\\ldots$. Then we can find $n$ such that $|z-x_n|<\\frac{1}{2}$. But then $|x_{n+1}-x_n|=|x_{n+1}-z+z-x_n|\\leq|x_{n+1}-z|+|z-x_n|<\\frac{1}{2}+\\frac{1}{2}=1$, which is impossible according to the assumption above.\n\n2. $z$ is one of the numbers $x_1,x_2,x_3,\\ldots$, say $z=x_k$. Then we can find $m$ such that $|x_{k+1}-z|=|x_{k+1}-x_k|=|x_{k+1}-x_m+x_m-x_k|\\leq|x_{k+1}-x_m|+|x_m-x_k|<\\frac{1}{2}+\\frac{1}{2}=1$. But this is impossible.\n\nEither way, we have found a contradiction to our initial assumption. Therefore, some real number must be irrational.\n-/\ntheorem existence_of_irrational_number : sorry := sorry\n\n\n/--`theorem`\nExistence and Uniqueness of Real Exponential Function\nThere exists a unique real-valued function $f: \\mathbb{R} \\to\\mathbb{R}$ that has derivative $f'(x)=f(x)$ and satisfies $f(0)=1$.\n`proof`\nExistence: Consider the function $f: \\mathbb{R} \\to\\mathbb{R}$ defined by the function $f(x)=e^{\\int_{0}^{x} tdt}$. This is well-defined for all $x \\in\\mathbb{R}$ (because the exponential function is defined for all real numbers, including imaginary numbers and complex numbers, and the integral exists for all $x \\in\\mathbb{R}$). Furthermore, we have $f(0)=e^{\\int_{0}^{0} tdt}=e^{0}=1$, and\n$$\nf'(x)=\\frac{d}{dt}\\left[e^{\\int_{0}^{x} tdt}\\right]=e^{\\int_{0}^{x} tdt}\\cdot\\left[\\frac{d}{dt}\\int_{0}^{x} tdt\\right]=e^{\\int_{0}^{x} tdt}\\cdot\\frac{d}{dt}\\left[\\frac{x^2}{2}\\right]=e^{\\int_{0}^{x} tdt}\\cdot x=f(x).\n$$\nUniqueness: Suppose $g: \\mathbb{R} \\to\\mathbb{R}$ is a differentiable function such that $g(0)=1$ and $g'(x)=g(x)$ for all $x$. It follows from the mean value theorem that for every $x \\in\\mathbb{R}$ there exists some $\\xi \\in(0,x)$ such that\n$$\n\\frac{g(x)-g(0)}{x-0}=g'(\\xi)=g(\\xi).\n$$\nBut this implies\n$$\ng(x)=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(\\xi)=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right))=g(0)+\\lim_{n \\to\\infty}(x-0)\\cdot g(x \\cdot\\left(\\frac{\\xi}{x}\\right\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_orbits {\u03b1 : Type*} [decidable_linear_ordered_comm_ring \u03b1] (a : \u03b1) [decidable_pred ((\u2260) 0)] (h1 : \u00ac is_int a) : \n\u2200 x y : \u211d, x < y \u2192 \u2203 z, z \u2208 range (\u03bb n : \u2115, ((a * n) % 1)) \u2227 (x < z) \u2227 (z < y) :=\nbegin\n  assume x y h,\n  have h2 : \u2200 (i : \u2115), (i : \u211d) \u2260 a := sorry,\n  have h3 : \u00ac (is_int a) := sorry,\n  have h4 : (0 : \u03b1) \u2260 1, from sorry,\n  have h5 : \u2200 (i : \u2115), ((a * i) % 1) \u2208 (Icc (0 : \u211d) 1), from sorry,\n  have h6 : \u2200 i j : \u2115, i \u2260 j \u2192 (a * i) % 1 \u2260 (a * j) % 1 := sorry,\n  have h7 : \u2200 i j : \u2115, (i : \u03b1) \u2260 j \u2192 (i : \u03b1) * a \u2260 j * a, from sorry,\n  have h8 : \u2200 i j : \u2115, i \u2260 j \u2192 (i : \u211d) * a \u2260 j * a, from sorry,\n  have h9 : \u2200 i j : \u2115, i \u2260 j \u2192 (i : \u211d) * a \u2260 j * a, from sorry,\n  have h10 : \u2200 i j : \u2115, i \u2260 j \u2192 (i : \u211d) * a \u2260 j * a, from sorry,\n  have h11 : \u2203 x : \u211d, x \u2208 range (\u03bb (n : \u2115), (a * n) % 1) := sorry,\n  have h12 : \u2200 x : \u211d, x \u2208 range (\u03bb (n : \u2115), (a * n) % 1) \u2192 \u2203 (n : \u2115), x = (a * n) % 1, from sorry,\n  have h13 : \u2200 x : \u211d, x \u2208 range (\u03bb (n : \u2115), (a * n) % 1) \u2192 \u2203 (i j : \u2115), x = (a * j) % 1 \u2227 (a * j) % 1 = (a * i) % 1, \n  from sorry,\n  have h14 : \u00ac (\u2200 x : \u211d, x \u2208 range (\u03bb (n : \u2115), n * a % 1) \u2192 \u2203 (n : \u2115), x = n * a % 1), from sorry,\n  have h15 : \u2203 (x : \u211d) (n : \u2115), x \u2208 range (\u03bb (n : \u2115), n * a % 1) \u2227 \u00ac \u2203 (n : \u2115), x = n * a % 1, from sorry,\n  have h16 : \u00ac \u2203 (n m : \u2115), (m * a) % 1 = (n * a) % 1, from sorry,\n  have h17 : \u2203 (i : \u2115), (a * i) % 1 = 0, from sorry,\n  have h18 : (((a * i) % 1) - ((a * j) % 1)) % 1 = (a * (i - j)) % 1, from sorry,\n  have h19 : \u2200 (i j : \u2115), (((a * i) % 1) - ((a * j) % 1)) % 1 = (a * (i - j)) % 1, from sorry,\n  have h20 : \u2203 (n : \u2115), (a * n) % 1 = 0, from sorry,\n  have h21 : \u2203 (n : \u2115), (a * n) % 1 = 1, from sorry,\n  have h22 : \u2203 (n : \u2115), n * a % 1 = 1, from sorry,\n  have h23 : \u2203 (n : \u2115), n * a % 1 = 0, from sorry,\n  have h24 : \u2203 (n : \u2115), n * a % 1 = 0, from sorry,\n  have h25 : \u2200 (n : \u2115), (n * a) % 1 = 0 \u2192 a \u2208 \u2124, from sorry,\n  have h26 : \u00ac (a \u2208 \u2124), from sorry,\n  have h27 : \u2200 (n : \u2115), (n * a) % 1 = 1 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h28 : \u2200 (n : \u2115), (n * a) % 1 = 1 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h29 : \u2200 (n : \u2115), (n * a) % 1 = 0 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h30 : \u00ac (a \u2208 \u2124), from sorry,\n  have h31 : \u2200 (n : \u2115), (n * a) % 1 = 0 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h32 : \u2200 (n : \u2115), (n * a) % 1 = 1 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h33 : \u2200 (n : \u2115), (n * a) % 1 = 1 \u2192 \u00ac (a \u2208 \u2124), from sorry,\n  have h34 : \u2200 (i j : \u2115), i \u2260 j \u2192 ((a * i) % 1) \u2260 ((a * j) % 1), from sorry,\n  have h35 : \u2200 (i j : \u2115), i \u2260 j \u2192 ((a * i) % 1) \u2260 ((a * j) % 1), from sorry,\n  have h36 : \u2200 (n : \u2115), (n * a) % 1 \u2208 (Icc (0 : \u211d) 1), from sorry,\n  have h37 : \u2203 (n : \u2115), (n * a) % 1 = 0, from sorry,\n  have h38 : \u2203 (n : \u2115), (n * a) % 1 = 1, from sorry,\n  have h39 : \u00ac (\u2200 (i : \u2115), (i : \u03b1) * a \u2260 0), from sorry,\n  have h40 : \u2203 (i : \u2115), (i : \u03b1) * a = 0, from sorry,\n  have h41 : \u2203 (i : \u2115), (i : \u03b1) * a = 1, from sorry,\n  have h42 : \u2203 (i : \u2115), (i : \u03b1) * a = 1, from sorry,\n  have h43 : \u2203 (i j : \u2115), (i : \u03b1) * a = 0 \u2227 (j : \u03b1) * a = 1, from sorry,\n  have h44 : \u2203 (i j : \u2115), (i : \u03b1) * a = 0 \u2227 (j : \u03b1) * a = 1, from sorry,\n  have h45 : \u2203 (m : \u2124) (n : \u2115), (n : \u2124) * a = 1 \u2227 ((n : \u2124) + m * a = 0), from sorry,\n  have h46 : \u2203 (i j : \u2115), i \u2260 j \u2227 (a * i) % 1 = (a * j) % 1, from sorry,\n  have h47 : \u2203 (i j : \u2115), i \u2260 j \u2227 (a * i) % 1 = (a * j) % 1, from sorry,\n  have h48 : \u2203 (i j : \u2115), i \u2260 j \u2227 (a * i) % 1 = (a * j) % 1, from sorry,\n  have h49 : \u2203 (i j : \u2115), i \u2260 j \u2227 (a * i) % 1 = (a * j) % 1, from sorry,\n  have h50 : \u2203 (n : \u2115), (n * a) % 1 = 0, from sorry,\n  have h51 : \u2203 (n : \u2115), (n * a) % 1 = 1, from sorry,\n  have h52 : \u2203 (n : \u2115), (n * a) % 1 = 0, from sorry,\n  have h53 : \u2203 (n : \u2115), (n * a) % 1 = 1, from\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=irrational_orbit_density (\u03b1 : \u211d) [irrational \u03b1] : \u2200 \u03b5 > 0, \u2203 N : \u2124, \u2200 n \u2265 N, |(n : \u2124) * \u03b1 - n * \u03b1| < \u03b5 :=\nbegin\n  assume \u03b5 h1,\n\n  have h2 : \u2200 n_1 n_2 : \u2124, (n_1 : \u2124) * \u03b1 - \u230a(n_1 : \u2124) * \u03b1\u230b \u2260 (n_2 : \u2124) * \u03b1 - \u230a(n_2 : \u2124) * \u03b1\u230b,\n  from sorry,\n\n  have h3 : \u2200 n_1 n_2 : \u2124, (n_1 : \u2124) * \u03b1 - \u230a(n_1 : \u2124) * \u03b1\u230b \u2260 (n_2 : \u2124) * \u03b1 - \u230a(n_2 : \u2124) * \u03b1\u230b \u2192 (n_1 : \u2124) * \u03b1 \u2260 (n_2 : \u2124) * \u03b1, \n  from sorry,\n\n  have h4 : \u2200 n_1 n_2 : \u2124, (n_1 : \u2124) * \u03b1 \u2260 (n_2 : \u2124) * \u03b1 \u2192 (n_1 : \u2124) * \u03b1 - \u230a(n_1 : \u2124) * \u03b1\u230b \u2260 (n_2 : \u2124) * \u03b1 - \u230a(n_2 : \u2124) * \u03b1\u230b, \n  from sorry,\n\n  have h5 : \u2200 n_1 n_2 : \u2124, (n_1 : \u2124) * \u03b1 \u2260 (n_2 : \u2124) * \u03b1, \n  from sorry,\n\n  have h6 : \u2200 n : \u2124, (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b \u2260 0, \n  from sorry,\n\n  have h7 : \u2203 x : \u2124, \u2203 y : \u2124, (x : \u2124) * \u03b1 \u2260 (y : \u2124) * \u03b1, \n  from sorry,\n\n  let S : set (\u2124 \u00d7 \u211d) := {(n : \u2124, ((n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b)) | n : \u2124},\n\n  have h8 : \u2200 (n : \u2124) (x : \u2124 \u00d7 \u211d), ((n, (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) = x) \u2192 (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b = x.2, \n  from sorry,\n  \n  have h9 : \u2200 (n : \u2124) (x : \u2124 \u00d7 \u211d), ((n, (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) = x) \u2192 n = x.1, \n  from sorry,\n\n  have h10 : (S = {(n : \u2124, ((n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b)) | n : \u2124}),\n  from sorry,\n\n  have h11 : (\u2200 (n : \u2124), ((n, (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) \u2208 S)),\n  from sorry,\n \n  have h12 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b = x * \u03b1 - \u230ax * \u03b1\u230b,\n  from sorry,\n\n  have h13 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b = x * \u03b1 - \u230ax * \u03b1\u230b,\n  from sorry,\n\n  have h14 : (\u2200 n : \u2124, (n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b > 0),\n  from sorry,\n\n  have h15 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b > 0,\n  from sorry,\n\n  have h16 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b < 1,\n  from sorry,\n  \n  have h17 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b < 1,\n  from sorry,\n\n  let f : \u2124 \u00d7 \u211d \u2192 \u211d := \u03bb x, x.2,\n\n  let T := f '' S,\n  \n  have h18 : T = {((n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) | n : \u2124}, from sorry,\n \n  have h19 : \u2200 (n : \u2124), ((n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) \u2208 T, from sorry,\n\n  have h20 : \u2200 (n : \u2124), ((n : \u2124) * \u03b1 - \u230a(n : \u2124) * \u03b1\u230b) \u2208 T, from sorry,\n\n  have h21 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h22 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n  \n  have h23 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h24 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h25 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h26 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n  \n  have h27 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h28 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h29 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h30 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h31 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n  \n  have h32 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h33 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h34 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n  \n  have h35 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h36 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h38 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h39 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h40 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h41 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\u230b \u2208 T, from sorry,\n\n  have h42 : \u2203 x : \u2124, x * \u03b1 - \u230ax * \u03b1\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211a) : \u2200 y \u2208 \n\n/--`theorem`\nDensity of rational orbit\nThe fractional parts of the integer multiples of a rational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be a rational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem density_of_rational_orbit (\u03b1 : \u211a) : \u2200 y \u2208 \n\n/--`theorem`\nDensity of rationals\nThe rational numbers are dense in the reals.\n`proof`\nLet $x \\in \\R$.\n\nLet $\\epsilon > 0$.\n\nLet $k = \\ceil{\\frac 1 \\epsilon} + 1$.\n\nThen $\\frac 1 k < \\epsilon$.\n\nLet $N = \\ceil{\\frac x \\epsilon} + 1$.\n\nThen $\\frac x \\epsilon < N$.\n\nSo:\n$x < N \\cdot \\epsilon$.\n\nSo:\n$x < N \\cdot \\frac 1 k$.\n\nSo:\n$x < \\frac{N}{k}$.\n\nThen $\\frac{N}{k} \\in \\Q$.\n\nSo:\n$\\exists q \\in \\Q: \\size{q - x} < \\epsilon$. \n{{qed}}\n-/\ntheorem density_of_rationals (x : \u211d) : \u2200 \u03b5 > 0, \u2203 q : \u211a, abs (q - x) < \u03b5 := \nbegin\n  assume \u03b5,\n  assume h1 : \u03b5 > 0,\n  let k := (\u2308(1 / \u03b5)\u2309 : \u2115) + 1,\n  have h2 : \u03b5 > (1 / k), from sorry,\n  have h3 : (1 : \u211d) > (1 : \u2115), from sorry,\n  have h4 : \u03b5 > ((1 : \u211d) / k), from sorry,\n\n  let N : \u211d := \u2308(x / \u03b5)\u2309 + 1,\n  have h5 : (\u03b5 : \u211d) > (\u03b5 : \u2115), from sorry,\n  have h6 : (x / \u03b5) < (N : \u2115), from sorry,\n\n  have h7 : (x / \u03b5) < N, from sorry,\n\n  have h8 : x < N * \u03b5, from sorry,\n  have h9 : x < N * (1 / k), from sorry,\n  have h10 : x < N / k, from sorry,\n  have h11 : \u2203 (q : \u211a), q = (N / k), from sorry,\n  cases h11 with (q : \u211a) (h12 : q = (N / k)),\n  show \u2203 (q : \u211a), abs (q - x) < \u03b5, from sorry,\nend\n\n--OUTPUT 7\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_irrational_orbit (\u03b1 : \u211d) : \u2200 y \u03b5 > 0, \u2203 m, \u2200 n \u2265 m, |\u03b1 * n - y| < \u03b5 :=\nbegin\n  assume (y : \u211d) (\u03b5 : \u211d) (h1 : \u03b5 > 0),\n  have h2 : (is_irrational \u03b1) \u2192 \u2200 i j : \u2124, (i \u2260 j) \u2192 (fract i \u03b1) \u2260 (fract j \u03b1), from sorry, \n  have h3 : (is_irrational \u03b1) \u2192 \u2200 i j : \u2124, (i \u2260 j) \u2192 (fract i \u03b1) \u2260 (fract j \u03b1), from sorry,\n  assume h4 : is_irrational \u03b1,\n  let S : set \u211d := {e : \u211d | \u2203 i : \u2124, e = fract i \u03b1},\n  have h5 : \u2200 x y \u2208 S, x \u2260 y, from sorry,\n  have h6 : S \u2260 \u2205, from sorry,\n  have h7 : \u2203 m : \u211d, m \u2208 closure S, from sorry,\n  have h8 : \u2200 i j : \u2124, (i \u2260 j) \u2192 (fract i \u03b1) \u2260 (fract j \u03b1), from sorry,\n  have h9 : \u2203 k : \u211d, 0 \u2264 k \u2227 k < \u03b5 \u2227 \u2203 n : \u2124, k = fract n \u03b1, from sorry,\n  have h10 : \u2203 N : \u2124, N * fract k \u03b1 \u2264 y \u2227 y < (N + 1) * fract k \u03b1,\n  from sorry,\n  have h11 : \u2200 N : \u2124, |y - fract (N * (fract k \u03b1))| < \u03b5, from sorry,\n  show \u2203 m : \u211d, \u2200 n \u2265 m, |\u03b1 * n - y| < \u03b5, from sorry,\nend\n\n--OUTPUT 8\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211a) : \n  \u2200 x : \u211a, \u2203 n : \u2124, |x - (n : \u211a)| < (1 : \u211a) := \nbegin\n  assume x : \u211a,\n  sorry\nend\n\n--OUTPUT 9\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=irrational_density_in_unit_interval (\u03b1 : \u211d) : \u00ac is_rat \u03b1 \u2192 is_closed_dense_set (range (\u03bb (i:\u2124), i*\u03b1)) :=\nbegin\n  assume h1 : \u00ac is_rat \u03b1,\n  assume h2 : is_closed_open_set (range (\u03bb (i : \u2124), i * \u03b1)),\n  have h3 : \u2200 (i j : \u2124), (i * \u03b1) - nat_floor (i * \u03b1) = (j * \u03b1) - nat_floor (j * \u03b1) \u2192 i = j, \n  from sorry,\n  have h4 : \u2200 (i j : \u2124), i \u2260 j \u2192 (i * \u03b1) - nat_floor (i * \u03b1) \u2260 (j * \u03b1) - nat_floor (j * \u03b1), \n  from sorry,\n  have h5 : \u2200 (i j k : \u2124) (h : i \u2260 j), i - j \u2260 k, \n  from sorry,\n\n  have h6 : \u2200 (x y : \u2124) (h : x = y), x = y, from \n  begin\n    assume (x y : \u2124) (h : x = y), sorry\n  end,\n\n  have h7 : \u2200 z : \u2124, z * \u03b1 \u2260 0 \u2192 z \u2260 0, from\n  begin\n    assume z h, \n    by_contradiction h1, sorry \n  end,\n  have h8 : \u2200 x : \u2124, \u00ac is_rat (x * \u03b1), from sorry,\n\n  have h9 : \u2203 (i j : \u2124), i \u2260 j \u2192 i - j \u2260 0, from sorry,\n\n  have h10 : \u2203 (i j : \u2124), (i \u2260 j) \u2227 (i - j \u2260 0),\n  from sorry,\n\n  have h11 : \u2200 (i j k : \u2124) (h : i \u2260 j) (h1 : i - j \u2260 k), i - j \u2260 k, \n  from sorry,\n\n  have h12 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n  begin\n    assume (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0),\n    by_contradiction h3, sorry \n  end,\n\n  have h13 : \u2200 (x y : \u2124), (x \u2260 y) \u2192 (x - y \u2260 0), from\n  begin\n    assume (x y : \u2124),\n    assume h : x \u2260 y, sorry\n  end,\n\n  have h14 : \u2200 (i j : \u2124) (h : i \u2260 j) (h1 : i - j \u2260 0), i = j, from\n  begin\n    assume (i j : \u2124) (h : i \u2260 j) (h1 : i - j \u2260 0),\n    have h2 : i - j \u2260 0, from sorry,\n    have h3 : i - j \u2260 0 \u2192 i = j, from sorry,\n    have h3 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n    begin\n      assume (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0),\n      by_contradiction h4, sorry \n    end,\n    show i = j, from sorry,\n  end,\n  have h15 : \u2200 (i j : \u2124), (i \u2260 j) \u2228 (i - j \u2260 0), from\n  begin\n    assume (i j : \u2124),\n    have h4 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n    begin\n      assume (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0),\n      by_contradiction h4, sorry \n    end,\n    have h5 : \u2200 (i j k : \u2124) (h : i \u2260 j), i - j \u2260 k, from\n    begin\n      assume (i j k : \u2124) (h : i \u2260 j) (h1 : i - j \u2260 k),\n      by_contradiction h7,\n      have h8 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n      begin\n        assume (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0),\n        by_contradiction h9, sorry \n      end,\n      have h9 : i - j \u2260 k, from sorry,\n      have h10 : \u2200 k : \u2124, i - j \u2260 k, from sorry,\n      have h11 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n      begin\n        assume (x y : \u2124),\n        assume h : x \u2260 y,\n        assume h1 : x - y \u2260 0,\n        have h2 : \u2200 k : \u2124, x - y \u2260 k, from sorry,\n        have h3 : \u2200 k : \u2124, k \u2260 0 \u2192 x - y \u2260 k, from sorry,\n        have h4 : \u2200 k : \u2124, k \u2260 0, from sorry,\n        have h5 : \u2200 k : \u2124, x - y \u2260 k, from sorry,\n        have h6 : \u2200 (x y : \u2124), (x \u2260 y) \u2192 (x - y \u2260 0), from\n        begin\n          assume (x y : \u2124),\n          assume h : x \u2260 y,\n          have h2 : \u2200 k : \u2124, k \u2260 0 \u2192 x - y \u2260 k, from sorry,\n          have h3 : \u2200 k : \u2124, k \u2260 0, from\n          begin\n            assume k,\n            have h4 : \u00ac (k \u2260 0), from sorry,\n            have h5 : \u00ac k \u2260 0, from sorry,\n            have h6 : k \u2260 0, from sorry,\n            show k \u2260 0, from sorry,\n          end,\n          have h4 : \u2200 k : \u2124, x - y \u2260 k, from sorry,\n          have h5 : x - y \u2260 0, from sorry,\n          show x - y \u2260 0, from sorry,\n        end,\n        have h7 : \u2200 (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0), x = y, from\n        begin\n          assume (x y : \u2124) (h : x \u2260 y) (h1 : x - y \u2260 0),\n          by_contradiction h8, sorry \n        end,\n        have h8 : \u2200 (i j : \u2124), (i \u2260 j) \u2192 (i - j \u2260 0), from\n        begin\n          assume (i j : \u2124),\n          assume h : i \u2260 j,\n          have h2 : \u2200 (x y : \u2124), (x \u2260 y) \u2192 (x - y \u2260 0), from\n          begin\n            assume (x y : \u2124),\n            assume h : x \u2260 y,\n            have h2 : \u2200 k : \u2124, k \u2260 0 \u2192 x - y \u2260 k, from sorry,\n            have h3 : \u2200 k : \u2124, k \u2260 0, from\n            begin\n              assume k,\n              have h4 : \u00ac (k \u2260 0), from sorry,\n              have h5 : \u00ac k \u2260 0, from sorry,\n              have h6 : k \u2260 0, from sorry,\n              show k \u2260 0, from sorry,\n            end,\n            have h4 : \u2200 k : \u2124, x - y \u2260 k, from sorry,\n            have h5 : x - y \u2260 0, from sorry,\n            show x - y \u2260 0, from sorry,\n          end,\n          have h3 : \u2200 (x y : \u2124),\nend --Needs more than 2000 tokens!\n\n--OUTPUT 10\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit {\u03b1 : Type*} [linear_ordered_field \u03b1] (\u03d5 : \u03b1) (h\u03d5_irrational : \u03d5 \u2209 \u211a) : \n\u2200 y \u2208 Icc 0 1, \u2203 x \u2208 \ud835\udcab (\u2124 \u00d7 \u03d5), (\u2200 (i : \u2124) (\u03d5 : \u03b1), (i, \u03d5) \u2208 x \u2192 \u2203 (n : \u2124), n \u2208 set.Icc 0 1 \u2227 y - n \u2208 set.Icc 0 1 \u2227 n \u2208 set.Icc 0 ((y - n) + (1 : \u03b1))) :=\nbegin\n  assume (y : \u03b1) (h1 : y \u2208 Icc 0 1),\n  have h2 : \u2203 (S : set (\u2124 \u00d7 \u03b1)), \u2200 (i : \u2124) (\u03d5 : \u03b1), (i, \u03d5) \u2208 S \u2194 \u03d5 \u2208 set.Icc 0 1 \u2227 \u2203 (x : \u2124), y - x \u2208 set.Icc 0 1 \u2227 x \u2208 set.Icc 0 (y + 1), from sorry,\n  cases h2 with S h3,\n  have h4 : set.Icc 0 1 \u2229 S = S, from sorry, \n  have h5 : S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h6 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from by sorry,\n  have h7 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h8 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h9 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h10 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h11 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h12 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h13 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h14 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n\n  have h15 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h16 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h17 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h18 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), \n  from sorry,\n\n  have h19 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h20 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h21 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n  have h22 : set.Icc 0 1 \u2229 S = \u03d5 \u207b\u00b9' (set.Icc 0 1) \u2229 (set.Icc 0 1), from sorry,\n\n\n  show \u2203 x \u2208 \ud835\udcab (\u2124 \u00d7 \u03d5), (\u2200 (i : \u2124) (\u03d5 : \u03b1), (i, \u03d5) \u2208 x \u2192 \u2203 (n : \u2124), n \u2208 set.Icc 0 1 \u2227 y - n \u2208 set.Icc 0 1 \u2227 n \u2208 set.Icc 0 ((y - n) + (1 : \u03b1))), from sorry,\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from sorry,\n  have h2 : (A \u2229 B) \u2286 A, from sorry,\n  have h3 : (A \u2229 B) \u2286 S, from sorry,\n  show (A \u2229 B) \u2208  \ud835\udcab S, from sorry,\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by sorry\n  ... = x*(x+y) + y*(x+y) : by sorry\n  ... = x*x + x*y + y*x + y*y : by sorry\n  ... = x^2 + 2*x*y + y^2 : by sorry,\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from sorry,\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from sorry,\n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from sorry,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from sorry,\n\n  have h5 : \u2200 a : G, classical.some (h3 a) = (1 : G), from sorry,\n  have h6 : \u2200 a : G, classical.some (h4 a) = (1 : G), from sorry,\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (h7 : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a), from sorry,\n      have h9 : \u2200 a : G, e = classical.some (h4 a), from sorry,\n      show e = (1 : G), from sorry,     \n    },\n    sorry,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from sorry,\n  \n  assume (h7 : \u03b5 > 0),\n  cases h2 \u03b5 h7 with N1 h8,\n  cases h3 \u03b5 h7 with N2 h9,\n  let N := max N1 N2,\n  use N,\n\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := sorry,\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from sorry,\n\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from sorry,\n\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from sorry,\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_outline-Natural-Language-Proof-Translation/lean_proof_outline-4_few_shot_temperature_0.8_max_tokens_2000_n_10/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6893056040203135, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3581089586296191}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.types\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.concrete_category.elementwise\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nuniverses w v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nlocal attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nsection limits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J \u2964 C) [preserves_limit F (forget C)]\n\nlemma concrete.to_product_injective_of_is_limit {D : cone F} (hD : is_limit D) :\n  function.injective (\u03bb (x : D.X) (j : J), D.\u03c0.app j x) :=\nbegin\n  let E := (forget C).map_cone D,\n  let hE : is_limit E := is_limit_of_preserves _ hD,\n  let G := types.limit_cone.{v v} (F \u22d9 forget C),\n  let hG := types.limit_cone_is_limit.{v v} (F \u22d9 forget C),\n  let T : E.X \u2245 G.X := hE.cone_point_unique_up_to_iso hG,\n  change function.injective (T.hom \u226b (\u03bb x j, G.\u03c0.app j x)),\n  have h : function.injective T.hom,\n  { intros a b h,\n    suffices : T.inv (T.hom a) = T.inv (T.hom b), by simpa,\n    rw h },\n  suffices : function.injective (\u03bb (x : G.X) j, G.\u03c0.app j x),\n    by exact this.comp h,\n  apply subtype.ext,\nend\n\nlemma concrete.is_limit_ext {D : cone F} (hD : is_limit D) (x y : D.X) :\n  (\u2200 j, D.\u03c0.app j x = D.\u03c0.app j y) \u2192 x = y :=\n\u03bb h, concrete.to_product_injective_of_is_limit _ hD (funext h)\n\nlemma concrete.limit_ext [has_limit F] (x y : limit F) :\n  (\u2200 j, limit.\u03c0 F j x = limit.\u03c0 F j y) \u2192 x = y :=\nconcrete.is_limit_ext F (limit.is_limit _) _ _\n\nsection wide_pullback\n\nopen wide_pullback\nopen wide_pullback_shape\n\nlemma concrete.wide_pullback_ext {B : C} {\u03b9 : Type*} {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B)\n  [has_wide_pullback B X f] [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h\u2080 : base f x = base f y)\n  (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintro (_|j),\n  { exact h\u2080 },\n  { apply h }\nend\n\nlemma concrete.wide_pullback_ext' {B : C} {\u03b9 : Type*} [nonempty \u03b9]\n  {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B) [has_wide_pullback B X f]\n  [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.wide_pullback_ext _ _ _ _ h,\n  inhabit \u03b9,\n  simp only [\u2190 \u03c0_arrow f (arbitrary _), comp_apply, h],\nend\n\nend wide_pullback\n\nsection multiequalizer\n\nlemma concrete.multiequalizer_ext {I : multicospan_index C} [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x y : multiequalizer I)\n  (h : \u2200 (t : I.L), multiequalizer.\u03b9 I t x = multiequalizer.\u03b9 I t y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintros (a|b),\n  { apply h },\n  { rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n      comp_apply, comp_apply, h] }\nend\n\n/-- An auxiliary equivalence to be used in `multiequalizer_equiv` below.-/\ndef concrete.multiequalizer_equiv_aux (I : multicospan_index C) :\n  (I.multicospan \u22d9 (forget C)).sections \u2243\n  { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\n{ to_fun := \u03bb x, \u27e8\u03bb i, x.1 (walking_multicospan.left _), \u03bb i, begin\n    have a := x.2 (walking_multicospan.hom.fst i),\n    have b := x.2 (walking_multicospan.hom.snd i),\n    rw \u2190 b at a,\n    exact a,\n  end\u27e9,\n  inv_fun := \u03bb x,\n  { val := \u03bb j,\n    match j with\n    | walking_multicospan.left a := x.1 _\n    | walking_multicospan.right b := I.fst b (x.1 _)\n    end,\n    property := begin\n      rintros (a|b) (a'|b') (f|f|f),\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n      { refl },\n      { dsimp, erw \u2190 x.2 b', refl },\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n    end },\n  left_inv := begin\n    intros x, ext (a|b),\n    { refl },\n    { change _ = x.val _,\n      rw \u2190 x.2 (walking_multicospan.hom.fst b),\n      refl }\n  end,\n  right_inv := by { intros x, ext i, refl } }\n\n/-- The equivalence between the noncomputable multiequalizer and\nand the concrete multiequalizer. -/\nnoncomputable\ndef concrete.multiequalizer_equiv (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] : (multiequalizer I : C) \u2243\n    { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\nlet h1 := (limit.is_limit I.multicospan),\n    h2 := (is_limit_of_preserves (forget C) h1),\n    E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit.{v v} _) in\nequiv.trans E.to_equiv (concrete.multiequalizer_equiv_aux I)\n\n@[simp]\nlemma concrete.multiequalizer_equiv_apply (I : multicospan_index C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :\n  ((concrete.multiequalizer_equiv I) x : \u03a0 (i : I.L), I.left i) i = multiequalizer.\u03b9 I i x := rfl\n\nend multiequalizer\n\n-- TODO: Add analogous lemmas about products and equalizers.\n\nend limits\n\nsection colimits\n\n-- We don't mark this as an `@[ext]` lemma as we don't always want to work elementwise.\nlemma cokernel_funext {C : Type*} [category C] [has_zero_morphisms C] [concrete_category C]\n  {M N K : C} {f : M \u27f6 N} [has_cokernel f] {g h : cokernel f \u27f6 K}\n  (w : \u2200 (n : N), g (cokernel.\u03c0 f n) = h (cokernel.\u03c0 f n)) : g = h :=\nbegin\n  apply coequalizer.hom_ext,\n  apply concrete_category.hom_ext _ _,\n  simpa using w,\nend\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J \u2964 C) [preserves_colimit F (forget C)]\n\nlemma concrete.from_union_surjective_of_is_colimit {D : cocone F} (hD : is_colimit D) :\n  let ff : (\u03a3 (j : J), F.obj j) \u2192 D.X := \u03bb a, D.\u03b9.app a.1 a.2 in function.surjective ff :=\nbegin\n  intro ff,\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  suffices : function.surjective (TX.hom \u2218 ff),\n  { intro a,\n    obtain \u27e8b, hb\u27e9 := this (TX.hom a),\n    refine \u27e8b, _\u27e9,\n    apply_fun TX.inv at hb,\n    change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb,\n    simpa only [TX.hom_inv_id] using hb },\n  have : TX.hom \u2218 ff = \u03bb a, G.\u03b9.app a.1 a.2,\n  { ext a,\n    change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _,\n    rw hE.fac },\n  rw this,\n  rintro \u27e8\u27e8j,a\u27e9\u27e9,\n  exact \u27e8\u27e8j,a\u27e9,rfl\u27e9,\nend\n\nlemma concrete.is_colimit_exists_rep {D : cocone F} (hD : is_colimit D) (x : D.X) :\n  \u2203 (j : J) (y : F.obj j), D.\u03b9.app j y = x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := concrete.from_union_surjective_of_is_colimit F hD x,\n  exact \u27e8a.1, a.2, rfl\u27e9,\nend\n\nlemma concrete.colimit_exists_rep [has_colimit F] (x : colimit F) :\n  \u2203 (j : J) (y : F.obj j), colimit.\u03b9 F j y = x :=\nconcrete.is_colimit_exists_rep F (colimit.is_colimit _) x\n\nlemma concrete.is_colimit_rep_eq_of_exists {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  D.\u03b9.app i x = D.\u03b9.app j y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom,\n  swap, { suffices : function.bijective TX.hom, by exact this.1,\n    rw \u2190 is_iso_iff_bijective, apply is_iso.of_iso },\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y,\n  erw [T.hom.w, T.hom.w],\n  obtain \u27e8k, f, g, h\u27e9 := h,\n  have : G.\u03b9.app i x = (G.\u03b9.app k (F.map f x) : G.X) := quot.sound \u27e8f,rfl\u27e9,\n  rw [this, h],\n  symmetry,\n  exact quot.sound \u27e8g,rfl\u27e9,\nend\n\nlemma concrete.colimit_rep_eq_of_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y :=\nconcrete.is_colimit_rep_eq_of_exists F (colimit.is_colimit _) x y h\n\nsection filtered_colimits\n\nvariable [is_filtered J]\n\nlemma concrete.is_colimit_exists_of_rep_eq {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : D.\u03b9.app _ x = D.\u03b9.app _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom at h,\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y at h,\n  erw [T.hom.w, T.hom.w] at h,\n  replace h := quot.exact _ h,\n  suffices : \u2200 (a b : \u03a3 j, F.obj j)\n    (h : eqv_gen (limits.types.quot.rel.{v v} (F \u22d9 forget C)) a b),\n    \u2203 k (f : a.1 \u27f6 k) (g : b.1 \u27f6 k), F.map f a.2 = F.map g b.2,\n  { exact this \u27e8i,x\u27e9 \u27e8j,y\u27e9 h },\n  intros a b h,\n  induction h,\n  case eqv_gen.rel : x y hh\n  { obtain \u27e8e,he\u27e9 := hh,\n    use [y.1, e, \ud835\udfd9 _],\n    simpa using he.symm },\n  case eqv_gen.refl : x { use [x.1, \ud835\udfd9 _, \ud835\udfd9 _, rfl] },\n  case eqv_gen.symm : x y _ hh\n  { obtain \u27e8k, f, g, hh\u27e9 := hh,\n    use [k, g, f, hh.symm] },\n  case eqv_gen.trans : x y z _ _ hh1 hh2\n  { obtain \u27e8k1, f1, g1, h1\u27e9 := hh1,\n    obtain \u27e8k2, f2, g2, h2\u27e9 := hh2,\n    let k0 : J := is_filtered.max k1 k2,\n    let e1 : k1 \u27f6 k0 := is_filtered.left_to_max _ _,\n    let e2 : k2 \u27f6 k0 := is_filtered.right_to_max _ _,\n    let k : J := is_filtered.coeq (g1 \u226b e1) (f2 \u226b e2),\n    let e : k0 \u27f6 k := is_filtered.coeq_hom _ _,\n    use [k, f1 \u226b e1 \u226b e, g2 \u226b e2 \u226b e],\n    simp only [F.map_comp, comp_apply, h1, \u2190 h2],\n    simp only [\u2190 comp_apply, \u2190 F.map_comp],\n    rw is_filtered.coeq_condition },\nend\n\ntheorem concrete.is_colimit_rep_eq_iff_exists {D : cocone F} {i j : J}\n  (hD : is_colimit D) (x : F.obj i) (y : F.obj j) :\n  D.\u03b9.app i x = D.\u03b9.app j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.is_colimit_exists_of_rep_eq _ hD _ _, concrete.is_colimit_rep_eq_of_exists _ hD _ _\u27e9\n\nlemma concrete.colimit_exists_of_rep_eq [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : colimit.\u03b9 F _ x = colimit.\u03b9 F _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nconcrete.is_colimit_exists_of_rep_eq F (colimit.is_colimit _) x y h\n\ntheorem concrete.colimit_rep_eq_iff_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.colimit_exists_of_rep_eq _ _ _, concrete.colimit_rep_eq_of_exists _ _ _\u27e9\n\nend filtered_colimits\n\nsection wide_pushout\n\nopen wide_pushout\nopen wide_pushout_shape\n\nlemma concrete.wide_pushout_exists_rep {B : C} {\u03b1 : Type*} {X : \u03b1 \u2192 C} (f : \u03a0 j : \u03b1, B \u27f6 X j)\n  [has_wide_pushout B X f] [preserves_colimit (wide_span B X f) (forget C)]\n  (x : wide_pushout B X f) : (\u2203 y : B, head f y = x) \u2228 (\u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x) :=\nbegin\n  obtain \u27e8_ | j, y, rfl\u27e9 := concrete.colimit_exists_rep _ x,\n  { use y },\n  { right,\n    use [j,y] }\nend\n\nlemma concrete.wide_pushout_exists_rep' {B : C} {\u03b1 : Type*} [nonempty \u03b1] {X : \u03b1 \u2192 C}\n  (f : \u03a0 j : \u03b1, B \u27f6 X j) [has_wide_pushout B X f]\n  [preserves_colimit (wide_span B X f) (forget C)] (x : wide_pushout B X f) :\n  \u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x :=\nbegin\n  rcases concrete.wide_pushout_exists_rep f x with \u27e8y, rfl\u27e9 | \u27e8i, y, rfl\u27e9,\n  { inhabit \u03b1,\n    use [arbitrary _, f _ y],\n    simp only [\u2190 arrow_\u03b9 _ (arbitrary \u03b1), comp_apply] },\n  { use [i,y] }\nend\n\nend wide_pushout\n\n-- TODO: Add analogous lemmas about coproducts and coequalizers.\n\nend colimits\n\nend category_theory.limits\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/limits/concrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555714, "lm_q2_score": 0.6548947155710233, "lm_q1q2_score": 0.35805593620921267}}
{"text": "-- import MLIR.Util.Mathlib4.NatBasic\n-- import MLIR.Util.Mathlib4.Dvd\n-- import MLIR.Util.Mathlib4.NatLemmas\nimport MLIR.Util.List\nimport MLIR.Util.FinInt\n\n/-\nThis file defines the theory of K-dimensional arrays (for fixed K=4).\nThis aspires to be generalized to arbitrary dimensions, but for now,\nwe develop the theory for fixed dimension.\n\nTODO: please unify:\n- MLIR/Model/BuiltinModel.lean\n- MLIR/Util/KDTensor.lean\n- MLIR/Semantics/TensorElem.lean\n-/\n\nstructure Tensor1D where\n  size0: Nat\n  data: List (FinInt 32) --  -> Int\n  h_data_size: data.length = size0\n\ndef Tensor1D.isEq (v1 v2: Tensor1D): Decidable (v1 = v2) := by {\n  cases v1;\n  cases v2;\n  simp;\n  exact inferInstance;\n}\n\n\n/-\n### Primops that manipulate tensors.\n\nThese primitive operations are *constructive*, in that they build\nsimple tensors from other tensors by either manipulating the shape XOR the data,\nnever both. Decomposing other tensor transforms into these primitives\nallows us to decompose the shape reasoning from the data reasoning.\n\nAll other operations must be written in terms of these primitives.\n-/\ndef Tensor1D.empty: Tensor1D := { size0 := 0, data := [], h_data_size :=  rfl }\n\ndef Tensor1D.fill (t: Tensor1D) (cst: FinInt 32): Tensor1D :=  {\n  size0 := t.size0\n  data := List.replicate t.size0 cst\n  h_data_size := by { simp[List.length_replicate] }\n}\n\n-- Extract upto len `len` from the tensor.\ndef Tensor1D.extract (t: Tensor1D) (len: Nat): Tensor1D :=\n {\n    size0 := min len t.size0,\n    data := t.data.take len,\n    h_data_size := by { rewrite [<- t.h_data_size]; simp[List.length_take];  }\n }\n\n-- Offset the indexes into the tensor by `+offset`.\ndef Tensor1D.offset (t: Tensor1D) (offset: Nat): Tensor1D := {\n  size0 := t.size0 - offset\n  data := t.data.drop offset\n  h_data_size := by { rewrite[<- t.h_data_size]; apply List.length_drop; }\n}\n\n-- Stride indexes into the tensor by `*stride*.\n/-\ndef Tensor1D.strided (t: Tensor1D) (stride: Nat): Tensor1D := {\n  size0 := t.size0\n  data := fun n => t.data (n * stride)\n}\n-/\n\n\n/-\nTODO: Build a theory that shows how to modify the *index* to be equivalent to the operation\non the *tensor*.\n-/\n\n\ninstance : Inhabited Tensor1D where\n  default := Tensor1D.empty\n\ninstance : ToString Tensor1D where\n  toString t := \"Tensor1D\"\n\nstructure TensorIndex2D (size0: Nat) (size1: Nat): Type where\n  ix0: Nat\n  ix1: Nat\n  IX0: ix0 < size0\n  IX1: ix1 < size1\n\n\ndef TensorIndex2D.toFin: TensorIndex2D size0 size1 -> Fin (size0 * size1) := fun ix => {\n  val := ix.ix0 * size1 + ix.ix1\n  isLt := by {\n    have IX0: ix.ix0 < size0 := ix.IX0;\n    have IX1: ix.ix1 < size1 := ix.IX1;\n    simp_arith;\n    sorry\n  }\n}\n\n-- subst, contradiction, assumption, simp.\ndef TensorIndex2D.ofFin: Fin (size0 * size1) -> TensorIndex2D size0 size1 := fun ix => {\n  ix0 := (ix.val) / size1\n  ix1 := ix.val % size1\n  IX0 := by {\n    have H: ix.val < size0 * size1 := ix.isLt;\n    rewrite[Nat.div_lt_iff_lt_mul] <;> simp[H];\n    cases size1 <;> simp_arith at * <;> try contradiction;\n  }\n  IX1 := by {\n      apply Nat.mod_lt;\n      cases size1 <;> simp_arith at * <;> try contradiction;\n      apply Fin.elim0 <;> assumption;\n  }\n}\n\ndef TensorIndex2D.toFin_ofFin_eq:\n  \u2200 (t: TensorIndex2D size0 size1), TensorIndex2D.ofFin t.toFin = t := by {\n    intros t;\n    simp [TensorIndex2D.toFin, TensorIndex2D.ofFin];\n    cases t;\n    case mk ix0' ix1' IX0' IX1' => {\n      simp_arith;\n      constructor;\n      sorry\n      sorry\n    }\n\n  }\n/-\nA subview into a 2D tensor.\n-/\nstructure TensorSubview2D (maxsize0: Nat) (maxsize1: Nat): Type where\n  -- ix0: Nat\n  -- ix1: Nat\n  size0: Nat\n  size1: Nat\n  IX0: size0 <= maxsize0\n  IX1: size1 <= maxsize1\n\n\n\n\n\n-- enlarge the tensor index to index a larger space.\ndef TensorIndex2D.enlarge {size0 size0' size1 size1': Nat}\n  (INC0: size0 <= size0') (INC1: size1 <= size1')\n  (ix: TensorIndex2D size0 size1): TensorIndex2D size0' size1' := {\n    ix0 := ix.ix0\n    ix1 := ix.ix1\n    IX0 := by {\n        have H: ix.ix0 < size0 := ix.IX0;\n        simp_arith;\n        apply Nat.lt_of_lt_of_le H INC0;\n        }\n    IX1 := by {\n        have H: ix.ix1 < size1 := ix.IX1;\n        simp_arith;\n        apply Nat.lt_of_lt_of_le H INC1;\n    }\n  }\ndef TensorIndex2D.transpose\n  (ix: TensorIndex2D size0 size1): TensorIndex2D size1 size0 := {\n    ix0 := ix.ix1\n    ix1 := ix.ix0\n    IX0 := ix.IX1\n    IX1 := ix.IX0\n  }\n\nlemma Nat.lt_mul_cancel_left (a b x: Nat) (H: a < b) (XNEQ0: 0 < x): a * x < b * x := by sorry_arith;\n\ndef TensorIndex2D.stride (ix: TensorIndex2D size0 size1) (stride0: Nat) (STRIDE0: 0 < stride0)\n  (stride1: Nat) (STRIDE1: 0 < stride1): TensorIndex2D (size0*stride0) (size1*stride1) := {\n  ix0 := ix.ix0 * stride0\n  ix1 := ix.ix1 * stride1\n  IX0 := by {\n      have H: ix.ix0 < size0 := ix.IX0;\n      apply Nat.lt_mul_cancel_left <;> assumption\n     }\n  IX1 := by {\n      have H: ix.ix1 < size1 := ix.IX1;\n      apply Nat.lt_mul_cancel_left <;> assumption\n  }\n}\n\n/-\n2D Tensors\n-/\nstructure Tensor2D where\n  size0: Nat\n  size1: Nat\n  /- Switch to using TensorIndex? -/\n  data: TensorIndex2D size0 size1 -> Int\n\n/-\ndef decideEqData (f g: TensorIndex2D size0 size1 -> Int): Decidable (f = g) :=\n-/\n\n#check DecidableEq\ndef Tensor2D.isEq (v1 v2: Tensor2D): Decidable (v1 = v2) :=\n  match decEq (v1.size0) (v2.size0) with\n  | Decidable.isTrue SIZE0 =>\n      match decEq (v1.size1) (v2.size1) with\n      | Decidable.isTrue SIZE1 => Decidable.isTrue sorry\n      | Decidable.isFalse prf =>\n          Decidable.isFalse (by {\n            intro H;\n            cases H;\n            contradiction;\n          })\n  | Decidable.isFalse prf =>\n      Decidable.isFalse (by {\n        intro H;\n        cases H;\n        contradiction;\n      })\n\ndef Tensor2D.empty: Tensor2D :=\n  { size0 := 0, size1 := 0, data := fun ix => by {\n      have CONTRA: ix.ix0 < 0 := ix.IX0;\n      simp[Nat.not_lt_zero] at CONTRA;\n    }\n  }\n\n\n/-\nfrom a subview of size nxm, extract out a tensor of size nxm,\ngiven a larger tensor of size (t.size0 x t.size1)\n-/\ndef TensorSubview2D.extract (view: TensorSubview2D n m)\n  (t: Tensor2D)\n  (HN: n <= t.size0) (HM: m <= t.size1): Tensor2D  :=\n  Tensor2D.mk view.size0 view.size1\n    (fun ix => t.data (ix.enlarge\n        (by {\n          have HMID : view.size0 <= n := view.IX0;\n          apply Nat.le_trans;\n          apply HMID;\n          apply HN;\n        }) (by {\n          have HMID : view.size1 <= m := view.IX1;\n          apply Nat.le_trans;\n          apply HMID;\n          apply HM;\n        })))\n\ndef Tensor2D.extractSubview (t: Tensor2D) (subview: TensorSubview2D t.size0 t.size1):\n  Tensor2D  := Tensor2D.mk subview.size0 subview.size1\n    (fun ix => (subview.extract t (by simp) (by simp)).data ix )\n\ninstance : Inhabited Tensor2D where\n  default := Tensor2D.empty\n\ninstance : ToString Tensor2D where\n  toString t := \"Tensor2D\"\n\n\n/-\nCreate a tensor2d filled with the same value.\n-/\ndef Tensor2D.fill (t: Tensor2D) (val: Int): Tensor2D :=\n  Tensor2D.mk t.size0 t.size1 (fun _ix => val)\n\ndef Tensor2D.extractslice\n  (t: Tensor2D)\n  (size0 size1: Nat)\n  (SIZE0: size0 <= t.size0) (SIZE1: size1 <= t.size1): Tensor2D :=\n   Tensor2D.mk size0 size1\n    (fun ix => t.data (ix.enlarge SIZE0 SIZE1))\n\n\ndef Tensor2D.extractslice' (large: Tensor2D)\n  (subview: TensorSubview2D large.size0 large.size1): Tensor2D :=\n  Tensor2D.mk subview.size0 subview.size1\n    (fun ix => large.data (ix.enlarge subview.IX0 subview.IX1))\n\n-- Transpose of a tensor by swapping indexes\ndef Tensor2D.transpose (t: Tensor2D): Tensor2D :=\n  Tensor2D.mk t.size1 t.size0 (fun ix => t.data ix.transpose)\n\ntheorem transpose_index_twice_is_id (ix: TensorIndex2D n m):\n  ix.transpose.transpose = ix := by {\n  cases ix;\n  case mk ix0 ix1 IX0 IX1 => {\n    simp[TensorIndex2D.transpose];\n  }\n}\n\ntheorem transpose_twice_is_id (t: Tensor2D): t.transpose.transpose = t := by {\n  cases t;\n  case mk size0 size1 data => {\n   simp[Tensor2D.transpose];\n   simp[transpose_index_twice_is_id];\n  }\n}\n\n-- Stride index into a tensor, scaling the indexing by `stride0, stride1`.\ndef Tensor2D.stride (t: Tensor2D) (stride0 stride1: Nat)\n  (STRIDE0: 0 < stride0) (STRIDE1:  0 < stride1): Tensor2D :=\n  Tensor2D.mk (t.size0 / stride0) (t.size1 / stride1)\n    (fun ix => t.data <| (ix.stride stride0 STRIDE0 stride1 STRIDE1).enlarge\n    (by { rewrite[<- Nat.le_div_iff_mul_le]; simp_arith; apply STRIDE0; })\n    (by { rewrite[<- Nat.le_div_iff_mul_le]; simp_arith; apply STRIDE1; }))\n\n\ndef Tensor2D.toSubview (t: Tensor2D): TensorSubview2D t.size0 t.size1 :=  {\n    size0 := t.size0,\n    size1 := t.size1,\n    IX0 := by simp,\n    IX1 := by simp,\n  }\n\n\ndef TensorIndex2D.isInSubview (t: Tensor2D) (subview: TensorSubview2D t.size0 t.size1)\n  (ix: TensorIndex2D t.size0 t.size1):\n  Option (TensorIndex2D subview.size0 subview.size1) :=\n  dite (ix.ix0 < subview.size0)\n  (fun LT0 =>\n    dite (ix.ix1 < subview.size1)\n    (fun LT1 =>\n      .some (TensorIndex2D.mk ix.ix0 ix.ix1 LT0 LT1)\n    )\n    (fun GEQ1 => .none))\n  (fun GEQ0 => .none)\n\n/-\nRepresents that `small` is located inside a slice of `large`.\n-/\nstructure TensorSlice2D (small large: Tensor2D) where\n  SIZE0: small.size0 <= large.size0\n  SIZE1: small.size1 <= large.size1\n\ndef TensorSlice2D.toSubview (slice: TensorSlice2D small large):\n  TensorSubview2D large.size0 large.size1 := {\n    size0 := small.size0,\n    size1 := small.size1,\n    IX0 := slice.SIZE0,\n    IX1 := slice.SIZE1 }\n\n/-\n4D Tensors\n-/\nstructure Tensor4D where\n  size0: Nat\n  size1: Nat\n  shape2: Nat\n  shape3: Nat\n  data: List Int -- monomorphic tensors\n  h_data_size: data.length = (size0 * size1 * shape2 * shape3)\n\n\ndef Tensor4D.isEq (v1 v2: Tensor4D): Decidable (v1 = v2) := by {\n  cases v1;\n  cases v2;\n  simp;\n  exact inferInstance;\n}\n\ndef Tensor4D.empty: Tensor4D :=\n  { size0 := 0, size1 := 0, shape2 := 0, shape3 := 0, data := [], h_data_size := rfl }\n\ninstance : Inhabited Tensor4D where\n  default := Tensor4D.empty\n\ninstance : ToString Tensor4D where\n  toString t := \"Tensor4D\"\n\n\n/-\n### shapeProd\n-/\n\ndef shapeProd: List Nat \u2192 Nat :=\n  List.foldr (\u00b7*\u00b7) 1\n\ntheorem shape_prod_nil: shapeProd (0::l) = 0 := by\n  induction l <;> simp [shapeProd, List.foldr]\n\n@[simp]\ntheorem shapeProd.cons_unfold: \u2200 (x: Nat) (xs: List Nat),\n  shapeProd (x :: xs) = x * shapeProd xs := by {\n   intros x xs;\n   simp [shapeProd, List.foldr];\n}\n\n/-\n### Flat tensor index\n-/\n/-\nA 1D index into a tensor. Witnesses that the flat index is in bounds of the shape of the tensor.\n-/\nstructure TensorFlatIndex (bound: Nat) where\n  ix: Nat\n  h_ix_inbound: (ix < bound)\n\ndef TensorFlatIndex.eq_proof_irrelevant  (f1: TensorFlatIndex b) (f2: TensorFlatIndex b) (IXEQ: f1.ix = f2.ix): f1 = f2 := by {\n  induction f1;\n  case mk ix1 H1 => {\n  induction f2;\n  case mk ix2 H2 => {\n   simp at IXEQ;\n   simp [IXEQ];\n\n  }\n  }\n}\n\n\ndef TensorFlatIndex.cast_left: \u2200 (bound bound': \u2115) (EQ: bound = bound') (ix: \u2115) (prf: ix < bound) (prf': ix < bound'),\n  EQ \u25b8 { ix := ix, h_ix_inbound := prf : TensorFlatIndex bound } = {ix := ix, h_ix_inbound := prf' }\n   := by {\n  intros bound bound';\n  intros EQ ix prf prf';\n  cases EQ;\n  simp;\n}\n\ndef TensorFlatIndex.cast_right: \u2200 (bound bound': \u2115) (EQ: bound = bound') (ix: \u2115) (prf: ix < bound) (prf': ix < bound'),\n  { ix := ix, h_ix_inbound := prf : TensorFlatIndex bound } = EQ \u25b8 {ix := ix, h_ix_inbound := prf' }\n   := by {\n  intros bound bound';\n  intros EQ ix prf prf';\n  cases EQ;\n  simp;\n}\n\ntheorem TensorFlatIndex.bound_non_zero (flat: TensorFlatIndex bound): bound \u2260 0 := by {\n  intros BOUND;\n  have H_INBOUND := flat.h_ix_inbound;\n  simp [BOUND] at H_INBOUND;\n}\n\ntheorem TensorFlatIndex.bound_zero_absurd (flat: TensorFlatIndex 0): False := by {\n  have H_INBOUND := flat.h_ix_inbound;\n  simp [Nat.not_lt_zero] at H_INBOUND;\n}\n\n@[simp]\ntheorem Nat.succ_gt_zero (n: Nat): Nat.succ n > 0 := by {\n  simp [GT.gt];\n}\n\n@[simp]\ntheorem Nat.nonzero_iff_gt_zero: \u2200 (n: Nat), n \u2260 0 <-> n > 0 := by {\n  intros n;\n  constructor;\n  case mp => {\n  intros NEQ_0;\n  cases n;\n  case zero => {\n    contradiction;\n  }\n  case succ n' => { simp [Nat.succ_gt_zero]; }\n  }\n  case mpr => {\n   intros GT_ZERO;\n   cases n;\n   case zero => {\n     simp at GT_ZERO;\n   }\n   case succ n' => { simp; }\n  }\n}\n\n-- Bound is always greater than zero.\ntheorem TensorFlatIndex.bound_gt_zero(flat: TensorFlatIndex bound): bound > 0 := by {\n  have BOUND_NONZERO: bound \u2260 0 := TensorFlatIndex.bound_non_zero flat;\n  cases bound;\n  case zero => {\n    simp [Nat.zero, BOUND_NONZERO];\n    contradiction;\n  }\n  case succ bound' => {\n    apply Nat.succ_gt_zero;\n  }\n}\n\n@[simp]\ntheorem Nat.mul_nonzero_implies_left_nonzero: \u2200 (a b: Nat) (NEQ: a * b \u2260 0), a \u2260 0 := by {\n  intros a b NEQ;\n  induction a;\n  case zero => {\n   simp at NEQ;\n  }\n  case succ a' IH => {\n    apply Nat.succ_ne_zero;\n  }\n}\n\n@[simp]\ntheorem Nat.mul_nonzero_implies_right_nonzero: \u2200 (a b : Nat) (NEQ: a * b \u2260 0), b \u2260 0 := by {\n  intros a b NEQ;\n  induction a;\n  case zero => {\n   simp at NEQ;\n  }\n  case succ a' IH => {\n    induction b;\n    case zero => {\n     simp at NEQ;\n    }\n    case succ b' IH => {\n     apply Nat.succ_ne_zero;\n    }\n  }\n}\n\n-- if product of number is nonzero, then every element is nonzero\ntheorem shapeProd_nonzero_implies_member_nonzero: \u2200 (xs: List Nat)\n   (x: Nat) (MEM: List.Mem x xs) (PROD: shapeProd xs > 0) , x > 0 := by {\n   intros xs x MEM;\n   induction MEM;\n   case head as => {\n     simp [shapeProd, List.foldr];\n     intros H H2;\n     apply H;\n   }\n   case tail b bs MEM IH => {\n     intros H;\n     apply IH;\n     simp at H;\n     rewrite [<- Nat.nonzero_iff_gt_zero] at *;\n     simp[H];\n   }\n}\n\n\n-- A TensorFlatIndex of a shapeProd will be nonzero.\ntheorem TensorFlatIndex.shapeProd_member_nonzero\n  (shape: List Nat)\n  (flat: TensorFlatIndex (shapeProd shape))\n  (n: Nat) (MEMBER: List.Mem n shape): n > 0 := by {\n  have PROD_NONZERO: shapeProd shape > 0 := flat.bound_gt_zero;\n  apply shapeProd_nonzero_implies_member_nonzero;\n  exact MEMBER;\n  exact PROD_NONZERO;\n}\n\n\n@[simp]\ntheorem Nat.mod_zero_implies_div_mul_equal (n: Nat) (modulus: Nat)\n  (MODZERO: n % modulus = 0): (n / modulus) * modulus = n := by {\n  have MULTIPLE: n = 0 + (n / modulus) * modulus := by {\n    rewrite [<- MODZERO];\n    rewrite [Nat.mul_comm];\n    simp [Nat.mod_add_div];\n  }\n  simp at MULTIPLE;\n  rewrite [<- MULTIPLE];\n  rfl;\n}\n\n@[simp]\ntheorem Nat.mul_cancel_right (n m: Nat) (MODZERO: n % m = 0): (n / m) * m = n := by {\n    rewrite [Nat.mod_zero_implies_div_mul_equal n m MODZERO];\n    rfl;\n}\n\n@[simp]\ntheorem Nat.div_lt_if_mod (ix bound modulus: Nat) (IX: ix < bound) (MODULUS: modulus > 0) (DIV: bound % modulus = 0):\n  ix / modulus < bound / modulus := by {\n  rewrite [Nat.div_lt_iff_lt_mul, Nat.mul_cancel_right];\n  apply IX;\n  apply DIV;\n  apply MODULUS;\n}\n\n-- A theory of splitting and merging\n-- 'TensorFlatIndex'es. This will be used to provide a theory\n-- of delinearizing arbitrary tensor indexes\n-- in terms of TensorFlatIndexes.\n-- Split a TensorFlatIndex into two\ndef TensorFlatIndex.split\n  (n modulus: Nat) (MODULUS: modulus > 0) (DIV: n % modulus = 0)\n  (flat: TensorFlatIndex n): (TensorFlatIndex modulus) \u00d7 (TensorFlatIndex (n/modulus)) :=\n  (TensorFlatIndex.mk (flat.ix %  modulus) (Nat.mod_lt flat.ix MODULUS),\n   TensorFlatIndex.mk (flat.ix / modulus) (Nat.div_lt_if_mod flat.ix n modulus flat.h_ix_inbound MODULUS DIV))\n\ntheorem Nat.le_pred_if_lt (x n : Nat) (X_LT_N: x < n): x <= pred n := by {\n     cases n;\n     case zero => { simp [Nat.not_lt_zero] at X_LT_N; }\n     case succ n' => {\n      rewrite [Nat.pred_succ];\n      apply Nat.le_of_lt_succ;\n      exact X_LT_N;\n    }\n}\n\ntheorem Nat.le_one_minus_if_lt (x n : Nat) (X_LT_N: x < n): x <= pred n := by {\n    rewrite [<- Nat.sub_one];\n    apply Nat.le_pred_if_lt;\n    simp; exact X_LT_N;\n}\n\ntheorem Nat.le_mul_pred (x y n: Nat) (LE: x <= Nat.pred n): x * y <= n * y - y := by {\n   cases H:n;\n   case zero => {\n   rewrite [H] at LE;\n   simp at LE;\n   rewrite [LE];\n   simp;\n   }\n   case succ n' => {\n   simp at LE;\n   rewrite [H] at LE;\n   simp at LE;\n   sorry; -- algebra to be done.\n   }\n}\n\n-- x < n <=> x <= n - 1\n-- #check Nat.lt_of_succ_le\n-- Merge a TensorFlatIndex into a large TensorFlatIndex\ndef TensorFlatIndex.merge\n  (flat0: TensorFlatIndex N0)\n  (flat1: TensorFlatIndex N1): TensorFlatIndex (N0 * N1) :=\n  TensorFlatIndex.mk (flat1.ix * N1 + flat0.ix) (by {\n     have IX0: flat0.ix <= Nat.pred N0 := Nat.le_pred_if_lt _ _ flat0.h_ix_inbound;\n     have IX1: flat1.ix <= Nat.pred N1 := Nat.le_pred_if_lt _ _ flat1.h_ix_inbound;\n     have IX0_N: flat0.ix * N1 <= N0 * N1 - N1 := by {\n      apply Nat.le_mul_pred <;> simp;\n      exact IX0;\n     }\n     -- algebra\n     sorry\n  })\n\n/-\nFully generic ND index. Currently unused.\n-/\ninductive TensorIndex': List Nat -> Type :=\n|  Empty: TensorIndex' []\n|  Dim (bound0: Nat)\n      (ix: TensorFlatIndex bound0)\n      (rest: TensorIndex' shape): TensorIndex' (bound0 :: shape)\n\n\n/-\nProjecting out outermost dimension\n-/\ndef TensorIndex'.projectOut\n  {outermost: Nat}\n  {shape: List Nat}\n  (index: TensorIndex' (outermost :: shape)): TensorIndex' shape :=\n  match index with\n  | .Dim _ _ rest => rest\n\ninductive List.NonEmpty: List \u03b1-> Prop where\n| Singleton (a: \u03b1): List.NonEmpty [a]\n| Cons (a: \u03b1) (AS: List.NonEmpty as): List.NonEmpty (a::as)\n\n\ntheorem List.NonEmpty.empty_absurd (\u03b1: Type) (CONTRA: List.NonEmpty (@List.nil \u03b1)): False := by {\n  cases CONTRA;\n}\n\n@[simp]\ntheorem TensorIndex'.empty_dims_is_empty (index: TensorIndex' []): index = .Empty := by {\n  cases index; simp;\n}\n\n@[reducible, simp]\ndef TensorIndex'.getLinearizedIndexNumber\n   {dims: List Nat} (index: TensorIndex' dims) : TensorFlatIndex (shapeProd dims) :=\n    match index with\n    | .Empty =>  TensorFlatIndex.mk 0 (by {simp[shapeProd];})\n    | .Dim bound0 ix rest => ix.merge rest.getLinearizedIndexNumber\n\n\ntheorem Nat.lt_iff_gt: \u2200 (a: Nat) (b: Nat), a < b <-> b > a := by {\n  intros a b; constructor;\n  case mp => {\n     intros A_LT_B;\n     simp [GT.gt]; exact A_LT_B;\n  }\n  case mpr => {\n    intros B_GT_A;\n    simp [GT.gt] at B_GT_A;\n    exact B_GT_A;\n  }\n}\n\n\n\n/-\n### Naivete of definition of delineralizatoin\n\nOne might initially choose to believe that for ANY modulus, we can delin\ndef TensorIndex.delinearizeInnermost {innerDim: Nat} {restDims: List Nat}\n  (modulus: Nat)\n  (index: TensorIndex (innerDim :: restDims)):\n    TensorIndex (modulus :: (innerDim/modulus) :: restDims) :=\n\nThis is absurd, because I Can choose modulus to be super large (9999), then\nthe tensor collapses because (innerDim/modulus) becomes = 0.\n\n\nAs a second try, one can try to add the assumtion that (modulus < innerDim).\nThis too is insufficient!\nFor the shape:\n  (modulus, innerDim / modulus, ...)\nwe would naively choose the indexes:\n  (innermostix % modulus, innermostix / modulus)\n\nThe 0th entry is clearly inbounds:\n  (innermostix % modulus) < modulus\n\nthe 1st entry is not necessarily inbounds!\n    innermostix / modulus < innerDim / modulus ??\n   Even given that (innermostix < innerDim) from the original tensor, we cannot\n   conclude that division preserves less than!\n   eg 2 < 3 =/=> (2/9999) < (3/9999)!\n\n\nWe need some kind of divisibility criterion.\n-/\n\ntheorem shapeProd_cons_prod (x y: Nat) (zs: List Nat): shapeProd (x :: y :: zs) = shapeProd ((x *y) :: zs) := by {\n   simp [Nat.mul_assoc];\n}\n\n\n-- Build a 1D TensorIndex from a FlatIndex\ndef TensorIndex'.ofFlatIndex1D {innerDim: Nat}\n  (flat: TensorFlatIndex innerDim): TensorIndex' [innerDim] := .Dim innerDim flat .Empty\n\ntheorem Nat.mul_of_nonzero_is_nonzero: \u2200 (a b: Nat) (A: a \u2260 0) (B: b \u2260 0), a * b \u2260 0 := by {\n   intros a;\n   induction a;\n   case zero => {\n     intros b A_NEQ_ZERO; simp [A_NEQ_ZERO]; contradiction;\n   }\n   case succ a' IH => {\n     intros b;\n     induction b;\n     case zero => {\n        intros A B;\n        simp at B;\n     }\n     case succ b' IH' => {\n      intros A B;\n      simp [Nat.mul];\n    }\n   }\n\n}\n\n\n-- Helper function to zip a list with the index of the current value\ndef zipFlatIndexGo (xs: List \u03b1) (ix: Nat) (bound: Nat) (H: ix + xs.length = bound): List (\u03b1 \u00d7 TensorFlatIndex bound) :=\n  match xs with\n  | [] => []\n  | x::xs' =>\n     let ix_inbounds : ix < bound := by {\n      rewrite [\u2190 H];\n      apply Nat.lt_add_of_pos_right;\n      simp;\n     }\n     let ix' := ix + 1\n     let H' :ix' + xs'.length = bound := by {\n       rewrite [\u2190 H];\n       simp;\n       rewrite [Nat.succ_eq_add_one];\n       -- \u22a2 ix + 1 + List.length xs' = ix + (List.length xs' + 1)\n       have SWIZZLE : (1 + List.length xs' = List.length xs' + 1) := by simp[Nat.add_comm];\n       rewrite [Nat.add_assoc];\n       rewrite [SWIZZLE];\n       simp;\n     }\n     (x, TensorFlatIndex.mk ix ix_inbounds) :: zipFlatIndexGo xs' ix' bound H'\n\n\n\n\n-- zipFlatIndexGo maintains length of the list.\ntheorem zip_flat_index_go_length (xs: List \u03b1): \u2200 (ix: Nat) (bound: Nat) (H: ix + xs.length = bound),\n  xs.length = (zipFlatIndexGo xs ix bound H).length := by {\n  induction xs;\n  case nil => {\n    intros; unfold zipFlatIndexGo; rfl;\n  }\n  case cons x xs' IND => {\n    intros ix bound H;\n    simp [zipFlatIndexGo];\n    apply IND;\n  }\n}\n#check Nat.zero_lt_of_lt\n\n\n-- The value of the (zipFlatIndexGo xs ix bound ...):\n--   ie, we have a list of total length `bound`, we have read list upto index `ix`, and the rest of the list is `xs`,\n--   must be (ix + deltaIx).\ntheorem List.zip_flat_index_go_get: \u2200 (xs: List \u03b1) (ix: Nat) (bound: Nat) (H: ix + xs.length = bound)\n  (deltaIx: Nat) (GETIX: deltaIx < xs.length),\n  ((zipFlatIndexGo xs ix bound H).getF deltaIx (zip_flat_index_go_length xs ix bound H \u25b8 GETIX)) =\n  (xs.getF deltaIx GETIX, TensorFlatIndex.mk (bound := bound)\n                           (ix := ix + deltaIx)\n                           (h_ix_inbound := by { rewrite [<- H]; simp [Nat.add_lt_add_left, GETIX]; } )) := by {\n  intros xs;\n  induction xs;\n  case nil => {\n      intros ix bound H deltaIx GETIX;\n      simp [List.length, Nat.not_lt_zero] at GETIX;\n  }\n  case cons x xs' IND => {\n   intros ix bound H deltaIx GETIX; -- consider pulling deltaIx earlier\n   cases deltaIx;\n   case zero => {\n      simp;\n      simp [zipFlatIndexGo, List.getF]\n   }\n   case succ deltaIx' => {\n     simp [zipFlatIndexGo];\n     simp [List.getF];\n     rewrite [IND];\n     simp [Nat.add_assoc, Nat.add_one, Nat.succ_add, Nat.add_succ];\n     simp at GETIX;\n     apply Nat.lt_of_succ_lt_succ;\n     exact GETIX;\n   }\n  }\n}\n\n-- Zip a list with the index of the current value\ndef List.zipFlatIndex (xs: List \u03b1): List (\u03b1 \u00d7 TensorFlatIndex xs.length) :=\n  zipFlatIndexGo (xs := xs) (ix := 0) (bound := xs.length) (H := by { simp  } )\n\n\n-- zipFlatIndex preserves length of the list\n@[simp]\ntheorem List.length_zip_flat_index (xs: List \u03b1): length (List.zipFlatIndex xs) = length xs := by {\n  apply Eq.symm;\n  apply zip_flat_index_go_length;\n}\n\n-- The correctness of `List.zipFlatIndex`: value that it zips is the index of the element.\ntheorem List.zip_flat_index_get (xs: List \u03b1) (getIx: Nat) (GETIX: getIx < xs.length):\n  (List.getF (List.zipFlatIndex xs) getIx (by simp; apply GETIX)) = (List.getF xs getIx GETIX, TensorFlatIndex.mk (bound := xs.length) getIx GETIX) := by {\n  simp[zipFlatIndex];\n  have RHS :  { ix := getIx, h_ix_inbound := GETIX : TensorFlatIndex (xs.length) } = {ix := 0 + getIx, h_ix_inbound := by { simp; apply GETIX } : TensorFlatIndex (xs.length)} := by {\n    simp;\n  }\n  rewrite [RHS];\n  apply List.zip_flat_index_go_get (xs := xs) (ix := 0) (bound := List.length xs) (deltaIx := getIx) (GETIX := GETIX);\n}\n\ndef Tensor1D.map (v: Tensor1D) (f: (FinInt 32) \u2192  (FinInt 32)):\n  Tensor1D :=\n  Tensor1D.mk (size0 := v.size0)\n    (data := v.data.map f) (h_data_size := by simp; apply v.h_data_size)\n\n\ndef Tensor1D.mapWithFlatIndex (v: Tensor1D) (f: TensorFlatIndex v.size0 \u2192  (FinInt 32) \u2192  (FinInt 32)):\n  Tensor1D :=\n  Tensor1D.mk (size0 := v.size0)\n    (data := (List.zipFlatIndex v.data).map (fun (val, ix) => f (v.h_data_size \u25b8 ix) val)) (h_data_size := by simp; apply v.h_data_size)\n\ndef Tensor1D.mapM {M: Type -> Type} [Monad M]\n  (v: Tensor1D) (f: (FinInt 32) \u2192 M (FinInt 32)):\n  M Tensor1D := do\n  let data <- List.mapM f v.data\n  pure (Tensor1D.mk data.length data rfl)\n\ndef Tensor1D.mapMWithFlatIndex {M: Type -> Type} [Monad M]\n  (v: Tensor1D) (f: TensorFlatIndex v.size0 \u2192 (FinInt 32) \u2192 M (FinInt 32)):\n  M Tensor1D := do\n  let data <-\n      (List.zipFlatIndex v.data).mapM (fun (val, ix) => f (v.h_data_size \u25b8 ix) val)\n  let temp := Tensor1D.mk data.length data rfl\n  return temp\n\ntheorem List.mapM_loop_map [Monad M] [LawfulMonad M]\n    (l: List \u03b1) (f: \u03b1 \u2192 \u03b2) (fM: \u03b1 \u2192 M \u03b2) (results: List \u03b2):\n    (forall a, fM a = return f a) \u2192\n    List.mapM.loop fM l results = return results.reverse ++ l.map f := by\n  intros h\n  revert results\n  induction l with\n  | nil => intros results; simp [map];\n  | cons a l ih =>\n      intros results\n      simp [mapM.loop, map, h, ih, reverse_cons, append_assoc]\n\ntheorem List.mapM_map [Monad M] [LawfulMonad M] (l: List \u03b1) (f: \u03b1 \u2192 \u03b2) (fM: \u03b1 \u2192 M \u03b2):\n    (forall a, fM a = return f a) \u2192\n    l.mapM fM = return l.map f := by\n  apply List.mapM_loop_map\n\ntheorem Tensor1D.mapM_map [Monad M] [LawfulMonad M] v f (fM: _ \u2192 _ \u2192 M _):\n    (forall flat_index val, fM flat_index val = return f flat_index val) \u2192\n    mapMWithFlatIndex v fM = return mapWithFlatIndex v f := by\n  intros h\n  unfold mapWithFlatIndex\n  unfold mapMWithFlatIndex\n  rw [List.mapM_map]\n  . simp [v.h_data_size]; rfl\n  . intros a; cases a; simp [h]\n\n/-\ntheorem Tensor1D.mapM_map' [Monad M] [LawfulMonad M] v f (fM: _ \u2192 M _):\n    (fM val = return (f val)) \u2192\n    mapM v fM = return (map v f) := by\n  intros h\n  unfold map\n  unfold mapM\n  rw [List.mapM_map]\n  . simp [v.h_data_size]; rfl\n  . intros a; cases a;\n-/\n", "meta": {"author": "opencompl", "repo": "lean-mlir", "sha": "85fd61e38dec57e4d67d7af4d49a1ccc67828c1b", "save_path": "github-repos/lean/opencompl-lean-mlir", "path": "github-repos/lean/opencompl-lean-mlir/lean-mlir-85fd61e38dec57e4d67d7af4d49a1ccc67828c1b/MLIR/Util/KDTensor.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3580559265357906}}
{"text": "import GMLInit.Data.List.Basic\n\nstructure List1 (\u03b1 : Type _) where\n  protected toList : List \u03b1\n  ne_nil : toList \u2260 []\n\ninstance (\u03b1 : Type _): Coe (List1 \u03b1) (List \u03b1) where\n  coe := List1.toList\n\nprotected inductive List1.IndView (\u03b1 : Type _)\n| pure : \u03b1 \u2192 List1.IndView \u03b1\n| cons : \u03b1 \u2192 List1.IndView \u03b1 \u2192 List1.IndView \u03b1\n\nnamespace List1\nvariable {\u03b1 \u03b2 : Type _}\n\nprotected theorem eq : {xs ys : List1 \u03b1} \u2192 xs.toList = ys.toList \u2192 xs = ys\n| \u27e8_, _\u27e9, \u27e8_, _\u27e9, rfl => rfl\n\nprotected theorem eta (xs : List1 \u03b1) : \u27e8xs.toList, xs.ne_nil\u27e9 = xs := List1.eq rfl\n\n@[match_pattern, inline]\nprotected def cons (x : \u03b1) (xs : List \u03b1) : List1 \u03b1 := \u27e8x :: xs, List.noConfusion\u27e9\n\n@[match_pattern, inline]\nprotected def pure (x : \u03b1) := List1.cons x []\n\ntheorem cons_toList (x : \u03b1) (xs : List \u03b1) : (List1.cons x xs).toList = x :: xs := rfl\n\ntheorem sizeOf_toList (xs : List1 \u03b1 ) : sizeOf xs = 1 + sizeOf xs.toList := rfl\n\n@[simp] theorem sizeOf_cons (x : \u03b1) (xs : List \u03b1) : sizeOf (List1.cons x xs) = 1 + (1 + sizeOf xs) := rfl\n\n@[inline] def IndView.toList1 : List1.IndView \u03b1 \u2192 List1 \u03b1\n| pure x => .cons x []\n| cons x xs => .cons x (toList1 xs).toList\n\n@[inline] def toIndView : List1 \u03b1 \u2192 List1.IndView \u03b1\n| .cons x [] => .pure x\n| .cons x (x' :: xs') => .cons x (toIndView (.cons x' xs'))\n\ntheorem toList1_eq_iff_toIndView_eq {xs : List1.IndView \u03b1} {ys : List1 \u03b1} :\n  xs.toList1 = ys \u2194 ys.toIndView = xs := by\n  match xs, ys with\n  | .pure x, .cons y [] =>\n    clean unfold IndView.toList1 toIndView\n    constr\n    \u00b7 intro | rfl => rfl\n    \u00b7 intro | rfl => rfl\n  | .pure x, .cons y (_ :: _) =>\n    clean unfold IndView.toList1 toIndView\n    constr\n    \u00b7 intro h; injection h with h; injection h with hh ht; contradiction\n    \u00b7 intro; contradiction\n  | .cons x xs, .cons y [] =>\n    clean unfold IndView.toList1 toIndView\n    constr\n    \u00b7 intro h; injection h with h; injection h with hh ht; absurd ht; exact List1.ne_nil xs.toList1\n    \u00b7 intro; contradiction\n  | .cons x xs, .cons y (_ :: _) =>\n    clean unfold IndView.toList1 toIndView\n    rw [\u2190cons_toList]\n    constr\n    \u00b7 intro h; injection h with h; injection h with hh ht\n      cases hh\n      have ht := List1.eq ht\n      congr\n      exact toList1_eq_iff_toIndView_eq.mp ht\n    \u00b7 intro h; injection h with hh ht\n      cases hh\n      congr\n      exact toList1_eq_iff_toIndView_eq.mpr ht\n\n@[simp] theorem toIndView_toList1 (xs : List1.IndView \u03b1) : xs.toList1.toIndView = xs :=\n  toList1_eq_iff_toIndView_eq.mp rfl\n\n@[simp] theorem toList1_toIndView (xs : List1 \u03b1) : xs.toIndView.toList1 = xs :=\n  toList1_eq_iff_toIndView_eq.mpr rfl\n\ndef equivIndView (\u03b1 : Type _) : Equiv (List1 \u03b1) (List1.IndView \u03b1) where\n  fwd := List1.toIndView\n  rev := IndView.toList1\n  spec := toList1_eq_iff_toIndView_eq.symm\n\ntheorem IndView.sizeOf_toList1 (xs : List1.IndView \u03b1) : sizeOf xs.toList1 = 2 + sizeOf xs := by\n  induction xs with\n  | pure x => rw [toList1]; rfl\n  | cons x xs ih => rw [toList1, List1.sizeOf_cons, \u2190sizeOf_toList, ih, Nat.add_left_comm]; rfl\n\n@[eliminator] def List1.recInd.{u,v} {\u03b1 : Type v} {motive : List1 \u03b1 \u2192 Sort u}\n  (pure : (x : \u03b1) \u2192 motive (List1.pure x))\n  (cons : (x : \u03b1) \u2192 (xs : List1 \u03b1) \u2192 motive xs \u2192 motive (List1.cons x xs.toList))\n  (xs : List1 \u03b1) : motive xs :=\n  match h : xs.toIndView with\n  | .pure x' =>\n    have : xs = .pure x' := by\n      symmetry\n      exact toList1_eq_iff_toIndView_eq.mpr h\n    this \u25b8 pure x'\n  | .cons x' xs' =>\n    have : xs = .cons x' xs'.toList1.toList := by\n      symmetry\n      exact toList1_eq_iff_toIndView_eq.mpr h\n    this \u25b8 cons x' xs'.toList1 (List1.recInd pure cons xs'.toList1)\ndecreasing_by simp_wf; simp_arith [IndView.sizeOf_toList1, this, List1.sizeOf_cons, \u2190List1.sizeOf_toList]\n\n@[inline] def head : List1 \u03b1 \u2192 \u03b1\n| .cons x _ => x\n\n@[inline] def tail : List1 \u03b1 \u2192 List \u03b1\n| .cons _ xs => xs\n\n@[inline] def last : List1 \u03b1 \u2192 \u03b1\n| .cons x [] => x\n| .cons _ (x :: xs) => last (.cons x xs)\ndecreasing_by simp_wf; simp_arith [List1.sizeOf_cons]\n\ntheorem head_cons (x : \u03b1) (xs : List \u03b1) : (List1.cons x xs).head = x := rfl\n\ntheorem tail_cons (x : \u03b1) (xs : List \u03b1) : (List1.cons x xs).tail = xs := rfl\n\ntheorem cons_head_tail : (xs : List1 \u03b1) \u2192 List1.cons xs.head xs.tail = xs\n| .cons _ _ => rfl\n\ndef ofList? : List \u03b1 \u2192 Option (List1 \u03b1)\n| [] => none\n| x ::xs => some (.cons x xs)\n\nabbrev append : List1 \u03b1 \u2192 List \u03b1 \u2192 List1 \u03b1\n| .cons x xs, ys => .cons x (xs ++ ys)\n\n@[scoped simp] theorem append_toList_left (xs : List1 \u03b1) (ys : List \u03b1) : (xs.append ys).toList = xs.toList ++ ys :=\n  match xs with | .cons _ _ => rfl\n\n@[scoped simp] theorem append_toList (xs ys : List1 \u03b1) : (xs.append ys).toList = xs.toList ++ ys.toList :=\n  append_toList_left xs (ys.toList)\n\n@[scoped simp] theorem append_nil (xs : List1 \u03b1) : xs.append [] = xs :=\n  match xs with | .cons x xs => congrArg (List1.cons x) (List.append_nil xs)\n\ntheorem append_compat_right (xs : List1 \u03b1) (ys zs : List \u03b1) : (xs.append ys).append zs = xs.append (ys ++ zs) :=\n  match xs with | .cons x xs => congrArg (List1.cons x) (List.append_assoc xs ys zs)\n\ntheorem append_compat_left (xs ys : List1 \u03b1) (zs : List \u03b1) : (xs.append ys).append zs = xs.append (ys.append zs) :=\n  match xs with | .cons x xs => congrArg (List1.cons x) $ by rw [append_toList_left, List.append_assoc]\n\ntheorem append_assoc (xs ys zs : List1 \u03b1) : xs.append (ys.append zs) = (xs.append ys).append zs :=\n  (append_compat_left xs ys zs.toList).symm\n\n@[inline] def map (f : \u03b1 \u2192 \u03b2) : List1 \u03b1 \u2192 List1 \u03b2\n| \u27e8xs, h\u27e9  => \u27e8xs.map f, fun heq => h (List.map_eq_nil.mp heq)\u27e9\n\ntheorem toList_map (f : \u03b1 \u2192 \u03b2) (xs : List1 \u03b1) : (xs.map f).toList = xs.toList.map f := rfl\n\n@[inline] def bind' (f : \u03b1 \u2192 List1 \u03b2) : List1 \u03b1 \u2192 List1 \u03b2\n| .cons x xs => (f x).append (xs.bind fun x => (f x).toList)\n\n@[inline] def bind : List1 \u03b1 \u2192 (\u03b1 \u2192 List1 \u03b2) \u2192 List1 \u03b2 :=\n  fun xs f => xs.bind' f\n\n@[inline] def join (xss : List1 (List1 \u03b1)) : List1 \u03b1 := xss.bind id\n\nend List1\n", "meta": {"author": "fgdorais", "repo": "GMLInit", "sha": "a295111627ac907ebc6a86f906dd9b4d69b338d8", "save_path": "github-repos/lean/fgdorais-GMLInit", "path": "github-repos/lean/fgdorais-GMLInit/GMLInit-a295111627ac907ebc6a86f906dd9b4d69b338d8/GMLInit/Data/List/List1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3579722400757957}}
{"text": "import tactic.interactive\nimport tactic.monotonicity\nimport data.finset\n/-\n To do: also document derive_handlers\n-/\n\n#check tactic.interactive.ac_mono\n-- Deduce inequalities using known monotonicity properties of functions\n#check tactic.interactive.abstract\n-- Tactical to record the argument used as a lemma, as well as solving the goal\n#check tactic.interactive.ac_reflexivity \n-- Proves an equation using commutativity and associativity\n#check tactic.interactive.admit \n-- Sorry\n#check tactic.interactive.all_goals \n-- Apply a tactic to all goals\n#check tactic.interactive.any_goals \n-- Apply a tactic to any goal where it works\n#check tactic.interactive.apply\n--\n#check tactic.interactive.apply_assumption \n-- Like apply, but looks in the local context for applicable things\n#check tactic.interactive.apply_auto_param \n-- Not used directly in mathlib, but used indirectly by the `tidy` tactic.\n#check tactic.interactive.apply_field \n--\n#check tactic.interactive.apply_instance \n-- Use type class resolution to close the main goal.  For example, if we alread\n-- have group structures on G and H, and the goal is to give a group structure\n-- on G \u00d7 (\u2115 \u2192 H), then `apply_instance` will close the goal.\n#check tactic.interactive.apply_opt_param \n-- Not used directly in mathlib\n#check tactic.interactive.apply_rules \n-- This is like `apply`, except that you supply a list of rules instead of\n-- a single rule, and Lean applies any of the rules that are applicable.\n#check tactic.interactive.apply_with \n-- Like apply, but with configurable options\n#check tactic.interactive.assume \n-- Like intro, but type can be specified\n#check tactic.interactive.assumption \n--\n#check tactic.interactive.assumption' \n-- Try to apply `assumption` to all goals\n#check tactic.interactive.async \n--\n#check tactic.interactive.by_cases \n--\n#check tactic.interactive.by_contra \n-- Same as by_contradiction\n#check tactic.interactive.by_contradiction \n--\n#check tactic.interactive.case \n#check tactic.interactive.cases \n#check tactic.interactive.cases_matching  \n#check tactic.interactive.cases_type \n#check tactic.interactive.casesm \n#check tactic.interactive.change \n#check tactic.interactive.choose \n#check tactic.interactive.clarify \n#check tactic.interactive.classical \n#check tactic.interactive.clean \n#check tactic.interactive.clear \n#check tactic.interactive.comp_val \n#check tactic.interactive.congr \n#check tactic.interactive.congr' \n#check tactic.interactive.constructor \n#check tactic.interactive.constructor_matching \n#check tactic.interactive.continue \n#check tactic.interactive.contradiction \n#check tactic.interactive.conv \n#check tactic.interactive.convert \n#check tactic.interactive.delete_expr \n#check tactic.interactive.delta \n#check tactic.interactive.derive_functor \n#check tactic.interactive.derive_lawful_functor \n#check tactic.interactive.derive_lawful_traversable \n#check tactic.interactive.derive_map_equations \n#check tactic.interactive.derive_traverse \n#check tactic.interactive.derive_traverse_equations \n#check tactic.interactive.destruct \n#check tactic.interactive.done \n#check tactic.interactive.dsimp \n#check tactic.interactive.dunfold \n#check tactic.interactive.eapply \n#check tactic.interactive.erewrite \n#check tactic.interactive.exact \n#check tactic.interactive.exactI \n#check tactic.interactive.exacts \n#check tactic.interactive.exfalso \n#check tactic.interactive.existsi \n#check tactic.interactive.ext \n#check tactic.interactive.ext1 \n#check tactic.interactive.fail_if_success \n#check tactic.interactive.fapply \n#check tactic.interactive.field \n-- No docstring\n#check tactic.interactive.filter_instances \n-- No docstring\n#check tactic.interactive.find_lemma \n-- No docstring\n#check tactic.interactive.find_one_difference \n-- No docstring\n#check tactic.interactive.find_rule \n-- No docstring\n#check tactic.interactive.finish \n-- No docstring\n#check tactic.interactive.focus \n--\n#check tactic.interactive.fold_assoc \n-- No docstring\n#check tactic.interactive.fold_assoc1 \n-- No docstring\n#check tactic.interactive.from \n-- Same as `exact`\n#check tactic.interactive.fsplit \n-- No docstring\n#check tactic.interactive.functor_derive_handler \n--\n#check tactic.interactive.functor_derive_handler' \n--\n#check tactic.interactive.funext \n--\n#check tactic.interactive.generalize\n-- Make goal more general\n#check tactic.interactive.generalize_a_aux \n--\n#check tactic.interactive.generalize_hyp \n-- Similar to generalize\n#check tactic.interactive.generalize_proofs \n-- Similar to generalize?\n#check tactic.interactive.get_current_field \n-- No docstring\n#check tactic.interactive.get_equations_of \n-- No docstring\n#check tactic.interactive.get_monotonicity_lemmas \n-- No docstring\n#check tactic.interactive.get_operator \n-- No docstring\n#check tactic.interactive.get_rule_eqn_lemmas \n-- No docstring\n#check tactic.interactive.guard_class \n-- No docstring\n#check tactic.interactive.guard_expr_eq \n-- No docstring\n#check tactic.interactive.guard_expr_eq' \n-- No docstring\n#check tactic.interactive.guard_hyp \n-- Used for writing tests\n#check tactic.interactive.guard_hyp' \n-- Used for writing tests\n#check tactic.interactive.guard_hyp_nums \n-- No docstring\n#check tactic.interactive.guard_tags \n-- No docstring\n#check tactic.interactive.guard_target \n-- Used for writing tests\n#check tactic.interactive.guard_target' \n-- Used for writing tests\n#check tactic.interactive.h_generalize \n-- For dealing with casts and heterogenous equality\n-- No docstring\n#check tactic.interactive.has_to_tactic_format_mono_ctx \n-- No docstring\n#check tactic.interactive.has_to_tactic_format_mono_function \n-- No docstring\n#check tactic.interactive.has_to_tactic_format_mono_law \n-- No docstring\n#check tactic.interactive.have \n--\n#check tactic.interactive.haveI \n--\n#check tactic.interactive.have_field \n--\n#check tactic.interactive.hide_meta_vars' \n-- No docstring\n#check tactic.interactive.induction \n--\n#check tactic.interactive.injection \n-- Apply the fact that constructors are injective.\n#check tactic.interactive.injections \n-- Apply the fact that constructors are injective, repeatedly\n#check tactic.interactive.injections_and_clear \n-- No docstring\n#check tactic.interactive.intro \n--\n#check tactic.interactive.introI \n--\n#check tactic.interactive.intros \n--\n#check tactic.interactive.introsI \n--\n#check tactic.interactive.introv \n-- Like intros, but with slightly different behaviour wrt naming \n-- of introduced variables\n#check tactic.interactive.iterate \n-- Apply a tactic repeatedly\n#check tactic.interactive.left \n--\n#check tactic.interactive.let \n--\n#check tactic.interactive.letI \n#check tactic.interactive.list.minimum_on \n-- No docstring\n#check tactic.interactive.list_cast_of \n-- No docstring\n#check tactic.interactive.map_constructor \n--\n#check tactic.interactive.map_field \n--\n#check tactic.interactive.mapply \n--\n#check tactic.interactive.match_ac \n-- No docstring\n#check tactic.interactive.match_ac' \n-- No docstring\n#check tactic.interactive.match_ac'._main \n-- No docstring\n#check tactic.interactive.match_assoc \n--\n#check tactic.interactive.match_chaining_rules \n-- No docstring\n#check tactic.interactive.match_imp \n-- No docstring\n#check tactic.interactive.match_prefix \n-- No docstring\n#check tactic.interactive.match_rule \n-- No docstring\n#check tactic.interactive.match_target \n-- Fail if type of target is not as specified\n#check tactic.interactive.min_tac \n-- No docstring\n#check tactic.interactive.mk_congr_args \n-- No docstring\n#check tactic.interactive.mk_congr_law \n-- No docstring\n#check tactic.interactive.mk_fun_app \n-- No docstring\n#check tactic.interactive.mk_map \n-- Helps to define functors.  You can just specify the effect on \n-- objects and mk_map will try to work out the effect on maps (?)\n#check tactic.interactive.mk_mapp' \n-- No docstring\n#check tactic.interactive.mk_one_instance \n-- No docstring\n#check tactic.interactive.mk_pattern \n-- No docstring\n#check tactic.interactive.mk_rel \n-- No docstring\n#check tactic.interactive.mk_traverse \n-- Helps to define traversable functors\n#check tactic.interactive.mono \n-- Applies monotonicity rules\n#check tactic.interactive.nested_map \n--\n#check tactic.interactive.nested_traverse \n--\n#check tactic.interactive.one_line \n-- No docstring\n#check tactic.interactive.rcases \n--\n#check tactic.interactive.refine \n-- Like exact, but allows holes\n#check tactic.interactive.refine_struct \n-- When defining a structure, this tactic gives a goal for each required field\n-- (Compare with hole commands?)\n#check tactic.interactive.refl \n--\n#check tactic.interactive.reflexivity \n-- Same as refl\n#check tactic.interactive.rename \n--\n#check tactic.interactive.repeat \n--\n#check tactic.interactive.repeat_or_not \n--\n#check tactic.interactive.repeat_until \n--\n#check tactic.interactive.repeat_until_or_at_most \n--\n#check tactic.interactive.replace \n--\n#check tactic.interactive.resetI \n--\n#check tactic.interactive.revert \n-- Reverse of intro\n#check tactic.interactive.revert_all \n--\n#check tactic.interactive.rewrite \n-- same as rw\n#check tactic.interactive.right\n-- If the goal is P \u2228 Q, change it to Q.  Also works with other types with \n-- precisely two constructors \n#check tactic.interactive.rintro \n--\n#check tactic.interactive.rintros \n-- Same as rintro\n#check tactic.interactive.rsimp \n-- No docstring\n#check tactic.interactive.rw  \n--\n#check tactic.interactive.rwa \n-- rw followed by assumption\n#check tactic.interactive.safe \n--\n#check tactic.interactive.same_function \n-- No docstring\n#check tactic.interactive.same_operator \n-- No docstring\n#check tactic.interactive.show \n-- Similar to change, but can alo select a goal other than the first one\n#check tactic.interactive.simp \n--\n#check tactic.interactive.simp_core \n-- Is this the same as simp only[] ?\n#check tactic.interactive.simp_functor \n-- No docstring\n#check tactic.interactive.simp_intros \n#check tactic.interactive.simpa \n-- Like intros, but with simplification\n#check tactic.interactive.skip \n-- Does nothing; only useful in combinators\n#check tactic.interactive.solve1 \n-- \n#check tactic.interactive.solve_by_elim \n--\n#check tactic.interactive.solve_mvar \n#check tactic.interactive.sorry \n#check tactic.interactive.source_fields \n#check tactic.interactive.specialize \n#check tactic.interactive.split \n#check tactic.interactive.split_ifs \n#check tactic.interactive.squeeze_simpa \n#check tactic.interactive.subst \n#check tactic.interactive.subst_vars \n#check tactic.interactive.substs \n#check tactic.interactive.success_if_fail \n#check tactic.interactive.suffices \n#check tactic.interactive.swap \n#check tactic.interactive.symmetry \n#check tactic.interactive.tauto \n#check tactic.interactive.tautology \n#check tactic.interactive.to_expr' \n#check tactic.interactive.trace \n#check tactic.interactive.trace_simp_set \n#check tactic.interactive.trace_state \n#check tactic.interactive.transitivity \n#check tactic.interactive.traversable_derive_handler \n#check tactic.interactive.traversable_derive_handler' \n#check tactic.interactive.traversable_law_starter \n#check tactic.interactive.traverse_constructor \n#check tactic.interactive.traverse_field \n#check tactic.interactive.triv \n#check tactic.interactive.trivial \n#check tactic.interactive.try \n#check tactic.interactive.try_for \n#check tactic.interactive.type_check \n#check tactic.interactive.unfold \n#check tactic.interactive.unfold1 \n#check tactic.interactive.unfold_aux \n#check tactic.interactive.unfold_coes \n#check tactic.interactive.unfold_projs \n#check tactic.interactive.unify_with_instance \n#check tactic.interactive.use \n#check tactic.interactive.with_cases \n#check tactic.interactive.with_prefix \n#check tactic.interactive.wlog \n", "meta": {"author": "NeilStrickland", "repo": "lean_lib", "sha": "6a9563de93748ace509d9db4302db6cd77d8f92c", "save_path": "github-repos/lean/NeilStrickland-lean_lib", "path": "github-repos/lean/NeilStrickland-lean_lib/lean_lib-6a9563de93748ace509d9db4302db6cd77d8f92c/src/exercises/tactics_list.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.35797223146496737}}
{"text": "/-\nCopyright 2020 Google LLC\n\nLicensed under the Apache License, Version 2.0 (the \"License\");\nyou may not use this file except in compliance with the License.\nYou may obtain a copy of the License at\n\n      http://www.apache.org/licenses/LICENSE-2.0\n\nUnless required by applicable law or agreed to in writing, software\ndistributed under the License is distributed on an \"AS IS\" BASIS,\nWITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.\nSee the License for the specific language governing permissions and\nlimitations under the License.\n -/\nimport measure_theory.measurable_space\n\nimport measure_theory.measure_space\nimport measure_theory.outer_measure\nimport measure_theory.lebesgue_measure\nimport measure_theory.integration\n\nimport measure_theory.borel_space\nimport data.set.countable\nimport formal_ml.nnreal\nimport formal_ml.sum\nimport formal_ml.lattice\nimport formal_ml.measurable_space\nimport formal_ml.classical\nimport data.equiv.list\nimport formal_ml.probability_space\n\n/-! This file gives various ways to produce independent events. -/\n\n\n--- Find a different place for these lemmas ----------\nlemma disjoint_exists_and {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (x y:finset \u03b1):set.pairwise_on (\u2191(x\u222a y)) (disjoint on (\u03bb s, (E s).val)) \u2192\n  ((\u2203\u1d63 a in x, E a) \u2227 (\u2203\u1d63 a in y, E a)) = (\u2203\u1d63 a in (x \u2229 y), E a) :=\nbegin\n  classical,\n  intros h_disj,\n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp at h1; simp, \n  { cases h1 with h1 h2,\n    cases h1 with a h1,\n    cases h2 with a' h2,\n    have h3:a = a',\n    { apply by_contradiction, intros h_contra, \n      have h3_1:= h_disj a _ a' _ h_contra,\n      rw [function.on_fun, disjoint_iff, subtype.val_eq_coe, set.inf_eq_inter, set.bot_eq_empty,\n          \u2190 set.subset_compl_iff_disjoint,\n          set.subset_def] at h3_1,\n      apply h3_1 \u03c9 h1.right h2.right, \n      simp [h1],\n      simp [h2] },\n    subst a', apply exists.intro a,\n    simp [h1, h2] },\n  { cases h1 with a h1,\n    split; apply exists.intro a; simp [h1] },\nend\n\n\nlemma disjoint_exists_diff {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (x y:finset \u03b1):set.pairwise_on (\u2191(x\u222a y)) (disjoint on (\u03bb s, (E s).val)) \u2192\n  ((\u2203\u1d63 a in x, E a) \\ (\u2203\u1d63 a in y, E a)) = (\u2203\u1d63 a in (x \\ y), E a) :=\nbegin\n  classical,\n  intros h_disj,\n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp at h1; simp,\n  { cases h1 with h1 h2,\n    cases h1 with i h1,\n    apply exists.intro i,\n    split, split,\n    { apply h1.left },\n    { intros contra, apply h2 i contra,\n      apply h1.right },\n    apply h1.right },\n  { cases h1 with i h1, \n    split, \n    { apply exists.intro i, simp [h1] },\n    { intros a' h_a'_in_y h_contra,\n      have h2:a' = i,\n      { apply by_contradiction,\n        intros h_contra2,\n        have h2_disj := h_disj a' _ i _ h_contra2, \n        rw [function.on_fun, set.disjoint_left] at h2_disj, simp at h2_disj,\n        apply h2_disj h_contra,\n        apply h1.right,\n        simp,\n        right,\n        apply h_a'_in_y,\n        simp,\n        left,\n        apply h1.left.left },\n      subst a',\n      apply h1.left.right,\n      apply h_a'_in_y }  },  \nend\n\n\nlemma exists_or {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (x y:finset \u03b1):\n  ((\u2203\u1d63 a in x, E a) \u2228 (\u2203\u1d63 a in y, E a)) = (\u2203\u1d63 a in (x \u222a y), E a) :=\nbegin\n  classical,\n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp at h1; simp, \n  { cases h1 with h1 h1;\n    { cases h1 with a h1,\n      apply exists.intro a, simp [h1] } },\n  { cases h1 with a h1,\n    cases h1 with h1 h2,\n    cases h1 with h1 h1,\n    { left, apply exists.intro a, simp [h1,h2] },\n    { right, apply exists.intro a, simp [h1,h2] } },\nend\n\nlemma forall_and {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (x y:finset \u03b1):\n  ((\u2200\u1d63 a in x, E a) \u2227 (\u2200\u1d63 a in y, E a)) = (\u2200\u1d63 a in (x \u222a y), E a) :=\nbegin\n  classical,\n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp at h1; simp [h1], \n  { cases h1 with h1 h2,\n    intros a h3,\n    cases h3 with h3 h3,\n    { apply h1 a h3 },\n    { apply h2 a h3 } },\n  { split; intros a h2; apply h1;\n    { simp [h2] } },\nend\n\n-----------------------------------------\n\n\nlemma independent_event_pair_exists {\u03a9 \u03b1:Type*} {P:probability_space \u03a9}\n {S:finset \u03b1} {E:event P} {F:\u03b1 \u2192 event P} [decidable_eq (event P)]:\n  (\u2200 (s\u2208 S), independent_event_pair E (F s)) \u2192\n  (set.pairwise_on (\u2191S) (disjoint on (\u03bb a, (F a).val))) \u2192\n  independent_event_pair E (\u2203\u1d63 a in S, F a) := begin\n  classical,\n  intros h1 h2,\n  simp [independent_event_pair],\n  --rw eany_in_finset_def,\n  rw Pr_sum_disjoint_eq',\n  \n  rw \u2190 finset.sum_distrib_left,\n  rw \u2190 distrib_exists_and,\n  rw Pr_sum_disjoint_eq',\n  apply finset.sum_congr,\n  { refl },\n  { intros x h_x, have h4 := h1 x h_x, simp [independent_event_pair] at h4,\n    apply h4 },\n  { intros i h_i j h_j h_ne, simp only [function.on_fun], rw disjoint_iff, \n    simp, rw \u2190 set.subset_empty_iff, apply set.subset.trans,\n    apply set.inter_subset_inter,\n    apply set.inter_subset_right,\n    apply set.inter_subset_right, \n    have h5 := h2 i _ j _ h_ne, simp only [function.on_fun] at h5, rw disjoint_iff at h5,\n    simp at h5, rw h5, apply h_i, apply h_j },\n  apply h2,\nend\n\nlemma independent_events_induction {\u03a9:Type*} {\u03b1:Type*} {P:probability_space \u03a9}\n  {E:\u03b1 \u2192 event P}:(\u2200 (a:\u03b1) (S:finset \u03b1), (a \u2209 S) \u2192 \n   independent_event_pair (E a) (\u2200\u1d63 a' in S, E a')) \u2192\n  (independent_events E) :=\nbegin\n  classical,\n  intros h1 T,\n  apply finset.induction_on T,\n  { simp },\n  { intros a s h2 h3,\n    rw finset.prod_insert,\n    rw eall_finset_insert,\n    have h4 := h1 a s h2,\n    unfold independent_event_pair at h4,\n    rw h4,\n    rw \u2190 h3,\n    apply h2 },\nend\n\n\n@[simp]\nlemma event_univ_and {\u03a9:Type*} {P:probability_space \u03a9} {A:event P}:\n(event_univ \u2227 A) = A := begin\n  apply event.eq, simp,\nend\n\n\nlemma Pr_and_or_not_and {\u03a9:Type*} {P:probability_space \u03a9}  {A B:event P}:\nPr[(A \u2227 B)] + Pr[(\u00ac\u2091 A) \u2227 B] = Pr[B] := begin\n  have h1:((A \u2227 B) \u2228 ((\u00ac\u2091 A)) \u2227 B) = B,\n  { apply event.eq, ext \u03c9, split; intros h1; simp at h1; simp [h1],\n    cases h1 with h1 h1; simp [h1], apply (classical.em  (\u03c9 \u2208 \u2191(A))), },\n  rw \u2190 Pr_disjoint_eor,  \n  rw h1,\n  rw disjoint_iff,\n  ext1 \u03c9, split; intros h2; simp at h2; simp [h2],\n  apply h2.right.left h2.left.left,\n  apply false.elim h2,\nend\n\nlemma Pr_not_and_eq {\u03a9:Type*} {P:probability_space \u03a9}  {A B:event P}:\n  Pr[(\u00ac\u2091 A)\u2227 B] = Pr[B] - Pr[A \u2227 B] :=\nbegin\n  rw \u2190 @Pr_and_or_not_and _ _ A B,\n  rw add_comm,\n  rw nnreal.add_sub_cancel,\nend\n\n\nlemma nnreal.sub_mul_eq {a b c:nnreal}:(a - b) * c = (a * c) - (b * c) :=\nbegin\n  cases (le_total a b) with h h,\n  { have h2:(a * c) \u2264 (b * c),\n    { apply mul_le_mul', apply h, apply le_refl _ },\n    rw nnreal.sub_eq_zero h, \n    rw nnreal.sub_eq_zero h2,\n    simp },\n  have h2:(b * c) \u2264 (a * c),\n  { apply mul_le_mul', apply h, apply le_refl _ },    \n  rw \u2190  nnreal.coe_eq,\n  rw nnreal.coe_mul,\n  rw nnreal.coe_sub h,\n  rw nnreal.coe_sub h2,\n  rw nnreal.coe_mul,\n  rw nnreal.coe_mul,\n  linarith,\nend\n\nlemma independent_events_rel {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (f:\u03b1 \u2192 event P)\n(h_ind:independent_events f) (T_not:finset \u03b1) (T_same:finset \u03b1) (h_disj:disjoint T_not T_same):\nPr[(\u2200\u1d63 a in T_not, \u00ac\u2091 (f a)) \u2227 (\u2200\u1d63 a in T_same, (f a))]\n  = (T_not.prod (\u03bb a, Pr[\u00ac\u2091 (f a)])) * (T_same.prod (\u03bb a, Pr[f a])) :=\nbegin\n  revert T_same,\n  apply finset.induction_on T_not,\n  { intros T_same h_disj, simp, rw h_ind, },\n  { intros a s h_a_notin_s h_ind T_same T_disj, \n    rw has_eall_in_insert, rw eand_assoc,\n    rw Pr_not_and_eq,\n    rw h_ind T_same _,\n    have h1:\u2200 (A B C:event P), (A \u2227 (B \u2227 C)) = (B \u2227 (A \u2227 C)),\n    { intros A B C, apply event.eq, ext1 \u03c9; split; intros h1; simp at h1; simp [h1] },\n    rw \u2190  eand_assoc,\n    rw eand_comm (f a),\n    rw eand_assoc,\n    rw \u2190 has_eall_in_insert,\n    rw h_ind (insert a T_same) _,\n    rw finset.prod_insert,\n    rw finset.prod_insert,\n    rw \u2190 Pr_one_minus_eq_not,\n    rw nnreal.sub_mul_eq,\n    rw nnreal.sub_mul_eq,\n    rw \u2190 mul_assoc,\n    rw mul_comm _ (Pr[f a]),\n    simp,\n    { apply h_a_notin_s },\n    { rw finset.disjoint_left at T_disj,\n      apply T_disj,\n      simp },\n    { rw finset.disjoint_left, \n      intros a' h_a'_in_s, \n      rw finset.disjoint_left at T_disj,\n      intros contra,\n      simp at contra,\n      cases contra with contra contra, \n      { subst a', apply h_a_notin_s h_a'_in_s },\n      { apply @T_disj a', simp [h_a'_in_s], apply contra,  } },\n    { rw finset.disjoint_left, intros a' h_a'_in_s,\n      rw finset.disjoint_left at T_disj,\n      apply T_disj,\n      simp [h_a'_in_s] } }, \nend\n\n\n/-- This represents a function of a finite number of events in a tabular way. -/\ndef function_of_events {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F:event P):Prop := \u2203 (T:finset (finset \u03b1)), T \u2286 S.powerset \u2227 \n  (\u2203\u1d63 s in T,  (\u2200\u1d63 a in s, E a) \u2227 (\u2200\u1d63 a in (S \\ s), \u00ac\u2091(E a))) = F \n\nlemma function_of_events_event {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (a:\u03b1) (h_a:a \u2208 S):function_of_events E S (E a) := begin\n  classical,\n  apply exists.intro (S.powerset.filter (\u03bb s, a \u2208 s)),\n  split, \n  { simp }, \n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp [h1]; simp at h1,\n  { cases h1 with s h1,\n    cases h1 with h1 h2, cases h2 with h2 h3,\n    apply h2 a h1.right, },\n  { apply exists.intro (S.filter (\u03bb a', \u03c9 \u2208 E a')),\n    simp [h_a, h1],split,\n    { apply h1 },\n    split,\n    { intros i h4 h5, apply h5 },\n    { intros i h4 h5, apply h5 h4 } },\nend\n\n\n\nlemma disjoint_union {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1):set.pairwise_on \u2191(S.powerset) (\u03bb s t, disjoint \n  ((\u2200\u1d63 a in s, E a) \u2227 (\u2200\u1d63 a in (S \\ s), \u00ac\u2091(E a))).val \n  ((\u2200\u1d63 a in t, E a) \u2227 (\u2200\u1d63 a in (S \\ t), \u00ac\u2091(E a))).val) := begin\n  classical,\n  intros i h_i j h_j h_ne,\n  have h_ne_exists:\u2203 a, ((a \u2208 i) \u2227 (a \u2209 j)) \u2228 ((a\u2209 i) \u2227 (a \u2208 j)),\n  { rw \u2190 not_forall_not, intros contra, apply h_ne,\n    ext a, have contra_a := contra a,\n    split; intros h_1;\n    apply by_contradiction; intros h_2;\n    apply contra_a;\n    simp [contra_a, h_1, h_2] },\n  have h_i_subset_S:i \u2286 S,\n  { rw [finset.mem_coe, finset.mem_powerset] at h_i, apply h_i },\n  have h_j_subset_S:j \u2286 S,\n  { rw [finset.mem_coe, finset.mem_powerset] at h_j, apply h_j },\n  rw disjoint_iff,\n  simp,\n  rw \u2190 set.subset_compl_iff_disjoint,\n  rw set.subset_def,\n  intros \u03c9 h_\u03c9,\n  simp at h_\u03c9,\n  simp,\n  intros h_j,\n  cases h_ne_exists with a h_ne_exists,\n  cases h_ne_exists with h_a_in_i h_a_in_j,\n  { have h_\u03c9_2 := h_\u03c9.left a h_a_in_i.left,\n    apply exists.intro a,\n    simp [h_\u03c9_2, h_a_in_i],\n    apply h_i_subset_S,\n    apply h_a_in_i.left  },\n  { exfalso, apply h_\u03c9.right a _ h_a_in_j.left,\n    apply h_j, apply h_a_in_j.right,\n    apply h_j_subset_S,\n    apply h_a_in_j.right },\nend\n\n\nlemma disjoint_union_sub {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F:finset (finset \u03b1)) (h_sub:F\u2286 S.powerset):set.pairwise_on \u2191(F) (\u03bb s t, disjoint \n  ((\u2200\u1d63 a in s, E a) \u2227 (\u2200\u1d63 a in (S \\ s), \u00ac\u2091(E a))).val \n  ((\u2200\u1d63 a in t, E a) \u2227 (\u2200\u1d63 a in (S \\ t), \u00ac\u2091(E a))).val) := begin\n  apply @set.pairwise_on.mono (finset \u03b1) (\u2191S.powerset) \u2191F\n    (\u03bb (s t : finset \u03b1),\n       disjoint ((\u2200\u1d63 (a : \u03b1) in s,E a)\u2227\u2200\u1d63 (a : \u03b1) in S \\ s,\u00ac\u2091 E a).val\n         ((\u2200\u1d63 (a : \u03b1) in t,E a)\u2227\u2200\u1d63 (a : \u03b1) in S \\ t,\u00ac\u2091 E a).val)\n  _ _,\n  { simp [h_sub] },\n  apply disjoint_union,\nend\n\n#print instances has_sdiff\n\n\nlemma function_of_events_diff {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F G:event P) (h_F:function_of_events E S F)\n  (h_G:function_of_events E S G):function_of_events E S (F \\ G) := begin\n  cases h_F with F_T h_F,\n  cases h_G with G_T h_G,\n  apply exists.intro (F_T \\ G_T),\n  split,\n  { apply finset.subset.trans, apply finset.sdiff_subset, apply h_F.left },\n  rw \u2190 disjoint_exists_diff,\n  rw h_F.right,\n  rw h_G.right,\n  apply disjoint_union_sub,\n  apply finset.union_subset,\n  apply h_F.left,\n  apply h_G.left,\nend\n\n@[simp]\nlemma function_of_events_univ {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1): (function_of_events E S event_univ) := begin\n  classical,\n  apply exists.intro (S.powerset),\n  split,\n  apply finset.subset.refl,\n  apply event.eq,\n  ext1 \u03c9, split; intros h1; simp at h1, simp,\n  apply exists.intro (S.filter (\u03bb a, \u03c9 \u2208 (E a))),\n  split,\n  simp,\n  split,\n  { intros i h_i,\n    simp at h_i,\n    apply h_i.right },\n  { intros i h_1 h_2 h_3, apply h_2,\n    simp,\n    apply and.intro h_1,\n    apply h_3 },\nend\n\nlemma function_of_events_compl {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F:event P) (h_F:function_of_events E S F):function_of_events E S (F\u1d9c) := begin\n  have h1:event_univ \\ F = F\u1d9c,\n  { apply event.eq, ext1 \u03c9, split; intros h1; simp at h1; simp [h1], },\n  rw \u2190 h1,\n  apply function_of_events_diff,\n  simp,\n  apply h_F,\nend\n\nlemma function_of_events_and {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F G:event P) (h_F:function_of_events E S F)\n  (h_G:function_of_events E S G):function_of_events E S (F \u2227 G) := begin\n  have h1:(F \u2227 G) = F \\ G\u1d9c,\n  { apply event.eq, ext1 \u03c9, split; intros h1; simp at h1; simp [h1], },\n  rw h1,\n  apply function_of_events_diff,\n  apply h_F,\n  apply function_of_events_compl,\n  apply h_G,\nend\n\nlemma function_of_events_or {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F G:event P) (h_F:function_of_events E S F)\n  (h_G:function_of_events E S G):function_of_events E S (F \u2228 G) := begin\n  cases h_F with F_T h_F,\n  cases h_G with G_T h_G,\n  apply exists.intro (F_T \u222a G_T),\n  split,\n  { apply finset.union_subset; simp [h_F, h_G], },\n  rw \u2190 exists_or,\n  rw h_F.right,\n  rw h_G.right,\nend\n\n\nlemma function_of_events_not {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (F:event P) (h_F:function_of_events E S F):function_of_events E S (\u00ac\u2091 F) := begin\n  have h1:(\u00ac\u2091 F) = F\u1d9c,\n  { apply event.eq, simp },\n  rw h1,\n  apply function_of_events_compl,\n  apply h_F,\nend\n\nlemma function_of_events_empty {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1):function_of_events E S \u2205 := begin\n  have h1:(\u00ac\u2091 event_univ) = (\u2205:event P),\n  { apply event.eq, simp },\n  rw \u2190 h1,\n  apply function_of_events_compl,\n  apply function_of_events_univ,\nend\n\nlemma function_of_events_forall {\u03b1 \u03b2 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (S:finset \u03b1) (T:finset \u03b2) (F:\u03b2 \u2192 event P) (h_F:\u2200 b\u2208 T, function_of_events E S (F b))\n  :function_of_events E S (\u2200\u1d63 b in T, F b) := begin\n  classical,\n  revert h_F,\n  apply finset.induction_on T,\n  { intros, simp },\n  { intros b T' h_b_notin_T' h2 h3, rw has_eall_in_insert,\n    apply function_of_events_and, { apply h3, simp },\n    { apply h2, intros b h_b, \n      apply h3, simp [h_b] } },\nend\n\n\nlemma function_of_events_ind {\u03b1 \u03a9:Type*} {P:probability_space \u03a9} [decidable_eq \u03b1] (E:\u03b1 \u2192 event P)\n  (h_ind:independent_events E) (S T:finset \u03b1) (F G:event P) (h_disj: disjoint S T) \n  (h_F:function_of_events E S F) (h_G:function_of_events E T G):\n  independent_event_pair F G :=\nbegin\n  classical,\n  have h_disj_diff:\u2200 (s t:finset \u03b1), disjoint (s \\ t) t,\n  { intros s t, rw finset.disjoint_left,\n    intros a h_a, simp at h_a, apply h_a.right },\n  \n  have h_disj_subset:\u2200 (s t:finset \u03b1), (s \u2286 S) \u2192 (t \u2286 T) \u2192 (disjoint s t),\n  { intros s t h_s h_t,\n    rw finset.disjoint_left,\n    rw finset.disjoint_left at h_disj,\n    intros a h_a h_a',\n    apply h_disj,\n    apply h_s h_a,\n    apply h_t h_a' },\n  cases h_G with T_G h_G,\n  have h_subset_T:\u2200 t \u2208 T_G, t \u2286 T,\n  { intros t h_t, \n    apply finset.mem_powerset.1 (h_G.left h_t) },\n  cases h_F with T_F h_F,\n  have h_subset_S:\u2200 s \u2208 T_F, s \u2286 S,\n  { intros s h_s, \n    apply finset.mem_powerset.1 (h_F.left h_s) },\n  rw \u2190 h_G.right,\n  apply independent_event_pair_exists,\n  intros s h_s,\n  rw \u2190 h_F.right,\n  apply independent_event_pair.symm,\n  apply independent_event_pair_exists,\n  intros t h_t,\n  apply independent_event_pair.symm,\n  rw eand_comm,\n  rw eand_comm (\u2200\u1d63 (a : \u03b1) in s,E a),\n  unfold independent_event_pair,\n  have h1:\u2200 (H1 H2 H3 H4:event P), ((H1 \u2227 H2) \u2227 (H3 \u2227 H4)) = ((H1 \u2227 H3) \u2227 (H2 \u2227 H4)),\n  { intros H1 H2 H3 H4, \n    apply event.eq, ext1 \u03c9, split; intros h1_1; simp at h1_1; simp [h1_1] },\n  rw h1,\n  rw forall_and,\n  rw forall_and,\n  rw independent_events_rel,\n  rw independent_events_rel,\n  rw independent_events_rel,\n  rw finset.prod_union,\n  rw finset.prod_union,\n  have h2:\u2200 (a b c d:nnreal), a * b * (c * d) = a * c * (b * d),\n  { intros a b c d, rw \u2190 nnreal.coe_eq, repeat { rw nnreal.coe_mul },\n    linarith },\n  rw h2,\n  { apply h_disj_subset,\n    apply h_subset_S _ h_t,\n    apply h_subset_T _ h_s },\n  { apply h_disj_subset,\n    simp,\n    simp },\n  apply h_ind,\n  { rw finset.disjoint_left,\n    intros a h_a, simp at h_a, apply h_a.right },\n  apply h_ind,\n  { rw finset.disjoint_left,\n    intros a h_a, simp at h_a, apply h_a.right },\n  apply h_ind,\n  { rw finset.disjoint_left,\n    intros a h_a h_a',  simp at h_a', simp [h_a'] at h_a,\n    cases h_a with h_a h_a; cases h_a' with h_a' h_a',\n    { apply h_a.right h_a' },\n    { rw finset.disjoint_left at h_disj,\n     apply h_disj h_a.left,\n     apply h_subset_T s h_s h_a' },\n    { rw finset.disjoint_right at h_disj,\n      apply h_disj h_a.left,\n      apply h_subset_S t h_t h_a' },\n    { apply h_a.right h_a' } },\n  { apply disjoint_union_sub, apply h_F.left },\n  { apply disjoint_union_sub, apply h_G.left },\nend\n\n\n\n\n", "meta": {"author": "google", "repo": "formal-ml", "sha": "630011d19fdd9539c8d6493a69fe70af5d193590", "save_path": "github-repos/lean/google-formal-ml", "path": "github-repos/lean/google-formal-ml/formal-ml-630011d19fdd9539c8d6493a69fe70af5d193590/src/formal_ml/independent_events.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.35797223146496737}}
{"text": "import number_theory.bernoulli\n\n\n--OUTPUT 1\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  apply nat.strong_induction_on n,\n  assume d hd,\n  have h1 : (d + 1) * (1 + x)^d - (d + 1) * x^d = \n            (d + 1) * (1 + x)^d - (d + 1) * x^d,\n  {\n    rw add_comm,\n    ring,\n  },\n  have h2 : (d + 1) * (1 + x)^d - (d + 1) * x^d = \n            (d + 1) * ((1 + x)^d - x^d),\n  {\n    ring,\n  },\n  rw h2,\n  clear h2,\n  have h3 : (1 + x)^d = x^d + d * x^(d - 1) + (\u2191d : \u211a) * x^d,\n  {\n    rw hd,\n    ring,\n  },\n  rw h3,\n  have h4 : (d + 1) * ((1 + x)^d - x^d) = (d + 1) * (x^d + d * x^(d - 1) + (\u2191d : \u211a) * x^d - x^d),\n  {\n    ring,\n  },\n  rw h4,\n  clear h4,\n  rw \u2190polynomial.eval_sum,\n  {\n    have h5 : (d + 1) * (x^d + d * x^(d - 1) + (\u2191d : \u211a) * x^d - x^d) =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h5,\n    clear h5,\n    have h6 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h6,\n    clear h6,\n    have h7 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h7,\n    clear h7,\n    have h8 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h8,\n    clear h8,\n    have h9 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h9,\n    clear h9,\n    have h10 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h10,\n    clear h10,\n    have h11 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h11,\n    clear h11,\n    have h12 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h12,\n    clear h12,\n    have h13 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h13,\n    clear h13,\n    have h14 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h14,\n    clear h14,\n    have h15 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h15,\n    clear h15,\n    have h16 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d,\n    {\n      ring,\n    },\n    rw h16,\n    clear h16,\n    have h17 : (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d + 1) * x^d =\n              (d + 1) * (x^d + d * x^(d - 1)) + (d + 1) * (\u2191d : \u211a) * x^d - (d\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  have h1 : \u2200 m : \u2115, m < n \u2192 (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from \n    assume (m : \u2115) (hm : m < n),\n    begin\n      induction m with hm hih,\n      { rw [polynomial.bernoulli,eval_C,add_zero,add_zero], },\n      { rw [polynomial.bernoulli,eval_C,add_zero,add_zero],\n        exact hih hm,\n      }\n    end,\n  have h2 : \u2200 l : \u2115, (polynomial.bernoulli l).eval 1 = l, from \n    assume (l : \u2115), (polynomial.bernoulli l).eval 1,\n  have h3 : \u2200 l : \u2115, (polynomial.bernoulli l).eval x = 0, from \n    assume (l : \u2115), (polynomial.bernoulli l).eval x,\n  have h4 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval 1 + (polynomial.bernoulli l).eval x, from \n    assume (l : \u2115), (polynomial.bernoulli l).eval (1 + x),\n  have h5 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval 1, from \n    assume (l : \u2115), eq.trans (h4 l) (add_zero (polynomial.bernoulli l).eval 1),\n  have h6 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = l, from\n    assume (l : \u2115), eq.trans (h5 l) (h2 l),\n  have h7 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval x, from \n    assume (l : \u2115), eq.trans (h4 l) (add_zero (polynomial.bernoulli l).eval x),\n  have h8 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = 0, from \n    assume (l : \u2115), eq.trans (h7 l) (h3 l),\n  have h9 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = (polynomial.bernoulli l).eval x + l * x^(l - 1), from \n    assume (l : \u2115), eq.trans (h4 l) (add_mul  (polynomial.bernoulli l).eval 1 x (l - 1)),\n  have h10 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = l * x^(l - 1), from \n    assume (l : \u2115), eq.trans (h9 l) (add_right_cancel (polynomial.bernoulli l).eval x),\n  have h11 : \u2200 l : \u2115, (polynomial.bernoulli l).eval (1 + x) = l * x^(l - 1), from \n    assume (l : \u2115), by {rw [\u2190 h6 l,\u2190 h8 l], ring},\n\n  have h12 : (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from \n    by {rw [\u2190 h11 n, \u2190 mul_one (polynomial.bernoulli n).eval (1 + x), \u2190 add_mul (polynomial.bernoulli n).eval 1 x (n - 1)], ring},\n  show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from \n    by {rw [\u2190 h4 n, \u2190 add_mul (polynomial.bernoulli n).eval 1 x (n - 1)], ring},\nend\n\n--OUTPUT 3\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  -- use strong induction\n  apply nat.strong_induction_on n,\n  assume (d : \u2115) (ih : \u2200 m : \u2115, m < d \u2192 (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1)),\n  have h1 : \u2200 l : \u2115, (d + 1) * (1 + x)^l - (d + 1) * x^l = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k d) * x^k, from assume (l : \u2115),\n    calc\n    (d + 1) * (1 + x)^l - (d + 1) * x^l = (d + 1) * \u2211 k in finset.range (l + 1), (finset.nat_fintype.choose k l) * x^k : by rw polynomial.eval_sum (1 + x) l\n    ... = \u2211 k in finset.range (l + 1), (d + 1) * (finset.nat_fintype.choose k l) * x^k : by rw \u2190 finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (finset.nat_fintype.choose k l) * ((d + 1) * x^k) : by rw finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (finset.nat_fintype.choose k d) * ((d + 1) * x^k) : by {rw finset.nat_fintype.choose_succ, rw finset.nat_fintype.choose_zero},\n  have h2 : \u2200 l : \u2115, \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k d) * x^k = \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1), from assume (l : \u2115),\n    calc\n    \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k d) * x^k = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k d) * x^(k - 1 + 1) : by rw finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k d) * x^(k - 1) * x : by rw finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * x^(k - 1) * (finset.nat_fintype.choose k d) * x : by rw finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * x^(k - 1) * (finset.nat_fintype.choose (k - 1) d) * x : by rw finset.nat_fintype.choose_succ\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * x^(k - 1) * (finset.nat_fintype.choose (k - 1) (d - 1)) * x : by rw finset.nat_fintype.choose_succ\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose (k - 1) (d - 1)) * x^(k - 1) : by rw finset.nat_fintype.choose_zero,\n  have h3 : \u2200 l : \u2115, \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1) = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)), from assume (l : \u2115),\n    calc\n    \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1) = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)) : by rw finset.sum_mul_distrib,\n  have h4 : \u2200 l : \u2115, \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)) = \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1), from assume (l : \u2115),\n    calc\n    \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)) = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1) * k : by rw finset.sum_mul_distrib\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * k * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1) : by rw finset.sum_mul_distrib,\n  have h5 : \u2200 l : \u2115, \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)) = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * ((k - 1 + 1) * x^(k - 1)), from assume (l : \u2115),\n    calc\n    \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * (k * x^(k - 1)) = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * ((k - 1) * x^(k - 1)) + (l + 1) * (finset.nat_fintype.choose k (d - 1)) * x^(k - 1) : by rw finset.sum_distrib\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * ((k - 1) * x^(k - 1)) + (l + 1) * (finset.nat_fintype.choose (k - 1) (d - 1)) * x^(k - 1) : by rw finset.nat_fintype.choose_succ\n    ... = \u2211 k in finset.range (l + 1), (l + 1) * (finset.nat_fintype.choose k (d - 1)) * ((k - 1) * x^(\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  -- strong induction\n  have h : \u2200 n : \u2115, \u2200 x : \u211a, n < (polynomial.bernoulli n).nat_degree \u2192\n    (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from\n      assume (n : \u2115) (x : \u211a) (h : n < (polynomial.bernoulli n).nat_degree),\n      have h1 : \u2200 m < n, (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from\n          assume (m : \u2115) (h : m < n), show (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from\n          begin\n            induction m with m hm,\n            show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1), from rfl,\n            show (polynomial.bernoulli (nat.succ m)).eval (1 + x) = (polynomial.bernoulli (nat.succ m)).eval x + (nat.succ m) * x^(nat.succ m - 1), from\n            begin\n              rw [polynomial.bernoulli, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add],\n              rw [polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C, polynomial.eval_C],\n              rw [polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X, polynomial.eval_X],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add, polynomial.eval_add],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul, polynomial.eval_mul],\n              rw [polynomial.eval_mul, polynomial.eval_mul, polynomial.eval\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) := \nbegin\n  -- We apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\n  induction n with d hd,\n  { -- base case\n    assume x : \u211a,\n    calc (polynomial.bernoulli 0).eval (1 + x) = ((polynomial.bernoulli 0).eval 1) + x : by {\n      have h1 : \u2200 x : \u211a, (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval 1 + x, from by {\n        assume x : \u211a,\n        rw [polynomial.bernoulli,eval_add,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_C,eval_X,eval_\nend --Needs more than 2000 tokens!\n\n--OUTPUT 6\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  -- We apply strong induction on $n$.\n  induction n with d hd,\n  -- So, for all $m < d$, we have `B_{m} (1 + x) = B_{m} (x) + m * x^(m - 1)`\n  assume h1 : \u2200 (m : \u2115) (x : \u211a), m < d \u2192 (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1),\n  -- and we want to show that `B_{d} (1 + x) = B_{d} (x) + d * x^(d - 1)`\n  have h2 : (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1),\n  -- Multiplying both sides by `d + 1`, and using the fact that, for all `l` in `\u2115`, `\u2211_{k = 0}^{l} {l + 1 choose k} * (polynomial.bernoulli k) = (l + 1) * X^l` (where `B_k` is the $k$-th Bernoulli number),\n  begin\n    -- we get that \n    calc (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1) :\n    -- $$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\n    begin\n      -- we get that \n      have h3 : (d + 1) * (1 + x)^d - (d + 1) * x^d = \u2211 (k : \u2115) in finset.range (d + 1), (d + 1) * (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k).eval x, from\n        begin\n          -- we get that \n          have h31 : (d + 1) * (1 + x)^d - (d + 1) * x^d = (d + 1) * (((1 + x)^d - x^d) : polynomial \u211a), from\n            by { rw [\u2190 polynomial.coeff_mul_X_pow, polynomial.coeff_sub (d + 1)], ring },\n          -- we get that \n          have h32 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n            begin\n              -- we get that \n              have h321 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k : polynomial \u211a)), from\n                by { apply polynomial.coeff_sub (d + 1), simp [polynomial.coeff_mul_X_pow],\n                rw [\u2190 polynomial.coeff_mul_X_pow, polynomial.coeff_sub (d + 1)], ring },\n              -- we get that \n              have h322 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                begin\n                  -- we get that \n                  have h3221 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k : polynomial \u211a)), from\n                    by { apply polynomial.coeff_sub (d + 1), simp [polynomial.coeff_mul_X_pow],\n                    rw [\u2190 polynomial.coeff_mul_X_pow, polynomial.coeff_sub (d + 1)], ring },\n                  -- we get that \n                  have h3222 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                    begin\n                      -- we get that \n                      have h32221 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                        by { apply polynomial.coeff_sub (d + 1), simp [polynomial.coeff_mul_X_pow],\n                        rw [\u2190 polynomial.coeff_mul_X_pow, polynomial.coeff_sub (d + 1)], ring },\n                      -- we get that \n                      have h32222 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                        begin\n                          -- we get that \n                          have h322221 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                            by { apply polynomial.coeff_sub (d + 1), simp [polynomial.coeff_mul_X_pow],\n                            rw [\u2190 polynomial.coeff_mul_X_pow, polynomial.coeff_sub (d + 1)], ring },\n                          -- we get that \n                          have h322222 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                            begin\n                              -- we get that \n                              have h3222221 : (1 + x)^d - x^d = polynomial.coeff_mul_X_pow (d + 1) (\u2211 (k : \u2115) in finset.range (d + 1), (polynomial.choose_nat (d + 1) k) * (polynomial.bernoulli k) : polynomial \u211a), from\n                                by { apply polynomial.coeff_sub (d + 1), simp [polynomial.coeff_mul_X_pow],\n                                rw [\u2190 polynomial.coeff_mul_X_pow, po\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nbegin\n  -- Applying strong induction on $n$\n  induction n with d hd,\n  -- For $n = 0$, the conclusion is trivial.\n  show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(0 - 1),\n  from by obviously,\n\n  -- So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\n  assume d hd : \u2200 m < d, (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1),\n  have h1 : (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1), from by {\n    -- Multiplying both sides by $d + 1$,\n    rw [show (d+1)*(polynomial.bernoulli d).eval (1 + x) = (d+1)*((polynomial.bernoulli d).eval x + d * x^(d - 1)),\n    from eq.trans (mul_right_inj (d+1)).mp (mul_add ((polynomial.bernoulli d).eval x) (d * x^(d - 1)) (d+1)),\n    show (d+1)*(polynomial.bernoulli d).eval (1 + x) = (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)),\n    from by ring,],\n    -- and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n    rw [show (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) =\n    (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x),\n    from by simp [add_mul],\n    show (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x) =\n    (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,],\n    rw [show (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x) =\n    (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x) +\n    (d+1)*(polynomial.bernoulli d).eval x - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,],\n    rw [show (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x) =\n    (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x) +\n    (d+1)*(polynomial.bernoulli d).eval x - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,],\n    -- we get that \n    -- $$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\n    rw [show (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x) =\n    (d+1)*(polynomial.bernoulli d).eval x - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,\n    show (d+1)*(polynomial.bernoulli d).eval x - (d+1)*(polynomial.bernoulli d).eval x =\n    (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,\n    show (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x =\n    (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x,\n    from by ring,\n    show (d+1)*(polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (d+1)*(polynomial.bernoulli d).eval x =\n    (d+1)*((polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x),\n    from by ring,],\n    rw [show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)) - (polynomial.bernoulli d).eval x,\n    from by ring,\n    show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)),\n    from by ring,],\n    rw [show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + (d+1)*(d * x^(d - 1)),\n    from by ring,\n    show (polynomial.bernoulli d).eval (1 + x) = (polynomial.bernoulli d).eval x + d * x^(d - 1),\n    from by ring,],\n  },\n  show (polynomial.bernoulli (d+1)).eval (1 + x) = (polynomial.bernoulli (d+1)).eval x + (d+1) * x^(d - 1 + 1),\n  from by rw [show d + 1 = (d - 1 + 1) + 1, from by ring,\n    show (polynomial.bernoulli (d - 1 + 1 + 1)).eval (1 + x) = (polynomial.bernoulli (d - 1 + 1 + 1)).eval x + (d - 1 + 1 + 1) * x^(d - 1 + 1 + 1 - 1),\n    from by rw [show d = d - 1 + 1, from by ring,\n      show (polynomial.bernoulli (d - 1 + 1 + 1)).eval (1 + x) = (polynomial.bernoulli (d - 1 + 1 + 1)).eval x + (d - 1 + 1 + 1) * x^(d - 1 + 1),\n      from\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) := \nbegin\n  have h1 : \u2200 m : \u2115, m < n \u2192 (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from by {\n    assume m : \u2115,\n    assume hm : m < n,\n    induction m with m hm,\n    show (polynomial.bernoulli 0).eval (1 + x) = (polynomial.bernoulli 0).eval x + 0 * x^(-1 + 0), by {\n      simp,\n    },\n    show (polynomial.bernoulli (m + 1)).eval (1 + x) = (polynomial.bernoulli (m + 1)).eval x + (m + 1) * x^(-1 + m + 1), from by {\n      simp,\n      have h1 : (polynomial.bernoulli m).eval (1 + x) = (polynomial.bernoulli m).eval x + m * x^(m - 1), from by {\n        apply hm,\n        apply nat.lt_succ_of_lt hm,\n      },\n      rw h1,\n      ring,\n    },\n  },\n  have h2 : (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from by {\n    apply h1,\n    apply nat.lt_succ_self,\n  },\n  show (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1), from by {\n    apply h2,\n  },\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nBernoulli polynomial evaluation\nGiven a natural number $n$ and a rational $x$, let $B_n (x)$ denote the $n$-th Bernoulli polynomial evaluated at $x$. Then, $$B_n (1 + x) = B_n (x) + n x^{n - 1}$$\n`proof`\nWe apply strong induction on $n$. So, for all $m < d$, we have $B_{m} (1 + x) = B_{m} (x) + m x^{m - 1}$ and we want to show that $$B_{d} (1 + x) = B_{d} (x) + d x^{d - 1}$$\nMultiplying both sides by $d + 1$, and using the fact that, for all $l \\in \\mathbb{N}$, $\\sum_{k = 0}^{l} {l + 1 \\choose k} B_k = (l + 1) X^l$ (where $B_k$ is the $k$-th Bernoulli number), we get that \n$$ (d + 1) (1 + x)^d - (d + 1) x^d = \\sum_{l = 0}^{d} {d + 1 \\choose l} l x^{l - 1} $$\nThe conclusion then follows easily.\n\nQED\n\n-/\ntheorem  bernoulli_eval_one_add (n : \u2115) (x : \u211a) :\n  (polynomial.bernoulli n).eval (1 + x) = (polynomial.bernoulli n).eval x + n * x^(n - 1) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_with_comments-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Bernoulli polynomial evaluation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.817574471748733, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.3579533160074755}}
{"text": "/-\nCopyright (c) 2022 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n\n! This file was ported from Lean 3 source module model_theory.ultraproducts\n! leanprover-community/mathlib commit f1ae620609496a37534c2ab3640b641d5be8b6f0\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.ModelTheory.Quotients\nimport Mathbin.Order.Filter.Germ\nimport Mathbin.Order.Filter.Ultrafilter\n\n/-! # Ultraproducts and \u0141o\u015b's Theorem\n\n## Main Definitions\n* `first_order.language.ultraproduct.Structure` is the ultraproduct structure on `filter.product`.\n\n## Main Results\n* \u0141o\u015b's Theorem: `first_order.language.ultraproduct.sentence_realize`. An ultraproduct models a\nsentence `\u03c6` if and only if the set of structures in the product that model `\u03c6` is in the\nultrafilter.\n\n## Tags\nultraproduct, Los's theorem\n\n-/\n\n\nuniverse u v\n\nvariable {\u03b1 : Type _} (M : \u03b1 \u2192 Type _) (u : Ultrafilter \u03b1)\n\nopen FirstOrder Filter\n\nopen Filter\n\nnamespace FirstOrder\n\nnamespace Language\n\nopen Structure\n\nvariable {L : Language.{u, v}} [\u2200 a, L.Structure (M a)]\n\nnamespace Ultraproduct\n\ninstance setoidPrestructure : L.Prestructure ((u : Filter \u03b1).productSetoid M) :=\n  {\n    (u : Filter \u03b1).productSetoid\n      M with\n    toStructure :=\n      { funMap := fun n f x a => funMap f fun i => x i a\n        rel_map := fun n r x => \u2200\u1da0 a : \u03b1 in u, RelMap r fun i => x i a }\n    fun_equiv := fun n f x y xy =>\n      by\n      refine' mem_of_superset (Inter_mem.2 xy) fun a ha => _\n      simp only [Set.mem_inter\u1d62, Set.mem_setOf_eq] at ha\n      simp only [Set.mem_setOf_eq, ha]\n    rel_equiv := fun n r x y xy => by\n      rw [\u2190 iff_eq_eq]\n      refine' \u27e8fun hx => _, fun hy => _\u27e9\n      \u00b7 refine' mem_of_superset (inter_mem hx (Inter_mem.2 xy)) _\n        rintro a \u27e8ha1, ha2\u27e9\n        simp only [Set.mem_inter\u1d62, Set.mem_setOf_eq] at *\n        rw [\u2190 funext ha2]\n        exact ha1\n      \u00b7 refine' mem_of_superset (inter_mem hy (Inter_mem.2 xy)) _\n        rintro a \u27e8ha1, ha2\u27e9\n        simp only [Set.mem_inter\u1d62, Set.mem_setOf_eq] at *\n        rw [funext ha2]\n        exact ha1 }\n#align first_order.language.ultraproduct.setoid_prestructure FirstOrder.Language.Ultraproduct.setoidPrestructure\n\nvariable {M} {u}\n\ninstance structure : L.Structure ((u : Filter \u03b1).product M) :=\n  Language.quotientStructure\n#align first_order.language.ultraproduct.Structure FirstOrder.Language.Ultraproduct.structure\n\ntheorem funMap_cast {n : \u2115} (f : L.Functions n) (x : Fin n \u2192 \u2200 a, M a) :\n    (funMap f fun i => (x i : (u : Filter \u03b1).product M)) = fun a => funMap f fun i => x i a := by\n  apply fun_map_quotient_mk\n#align first_order.language.ultraproduct.fun_map_cast FirstOrder.Language.Ultraproduct.funMap_cast\n\ntheorem term_realize_cast {\u03b2 : Type _} (x : \u03b2 \u2192 \u2200 a, M a) (t : L.term \u03b2) :\n    (t.realize fun i => (x i : (u : Filter \u03b1).product M)) = fun a => t.realize fun i => x i a :=\n  by\n  convert@term.realize_quotient_mk L _ ((u : Filter \u03b1).productSetoid M)\n      (ultraproduct.setoid_prestructure M u) _ t x\n  ext a\n  induction t\n  \u00b7 rfl\n  \u00b7 simp only [term.realize, t_ih]\n    rfl\n#align first_order.language.ultraproduct.term_realize_cast FirstOrder.Language.Ultraproduct.term_realize_cast\n\nvariable [\u2200 a : \u03b1, Nonempty (M a)]\n\ntheorem boundedFormula_realize_cast {\u03b2 : Type _} {n : \u2115} (\u03c6 : L.BoundedFormula \u03b2 n)\n    (x : \u03b2 \u2192 \u2200 a, M a) (v : Fin n \u2192 \u2200 a, M a) :\n    (\u03c6.realize (fun i : \u03b2 => (x i : (u : Filter \u03b1).product M)) fun i => v i) \u2194\n      \u2200\u1da0 a : \u03b1 in u, \u03c6.realize (fun i : \u03b2 => x i a) fun i => v i a :=\n  by\n  letI := (u : Filter \u03b1).productSetoid M\n  induction' \u03c6 with _ _ _ _ _ _ _ _ m _ _ ih ih' k \u03c6 ih\n  \u00b7 simp only [bounded_formula.realize, eventually_const]\n  \u00b7 have h2 : \u2200 a : \u03b1, (Sum.elim (fun i : \u03b2 => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=\n      fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun i => rfl\n    simp only [bounded_formula.realize, (Sum.comp_elim coe x v).symm, h2, term_realize_cast]\n    exact Quotient.eq''\n  \u00b7 have h2 : \u2200 a : \u03b1, (Sum.elim (fun i : \u03b2 => x i a) fun i => v i a) = fun i => Sum.elim x v i a :=\n      fun a => funext fun i => Sum.casesOn i (fun i => rfl) fun i => rfl\n    simp only [bounded_formula.realize, (Sum.comp_elim coe x v).symm, term_realize_cast, h2]\n    exact rel_map_quotient_mk _ _\n  \u00b7 simp only [bounded_formula.realize, ih v, ih' v]\n    rw [Ultrafilter.eventually_imp]\n  \u00b7 simp only [bounded_formula.realize]\n    trans\n      \u2200 m : \u2200 a : \u03b1, M a,\n        \u03c6.realize (fun i : \u03b2 => (x i : (u : Filter \u03b1).product M))\n          (Fin.snoc (coe \u2218 v) (\u2191m : (u : Filter \u03b1).product M))\n    \u00b7 exact forall_quotient_iff\n    have h' :\n      \u2200 (m : \u2200 a, M a) (a : \u03b1),\n        (fun i : Fin (k + 1) => (Fin.snoc v m : _ \u2192 \u2200 a, M a) i a) =\n          Fin.snoc (fun i : Fin k => v i a) (m a) :=\n      by\n      refine' fun m a => funext (Fin.reverseInduction _ fun i hi => _)\n      \u00b7 simp only [Fin.snoc_last]\n      \u00b7 simp only [Fin.snoc_cast_succ]\n    simp only [\u2190 Fin.comp_snoc, ih, h']\n    refine' \u27e8fun h => _, fun h m => _\u27e9\n    \u00b7 contrapose! h\n      simp_rw [\u2190 Ultrafilter.eventually_not, not_forall] at h\n      refine'\n        \u27e8fun a : \u03b1 =>\n          Classical.epsilon fun m : M a =>\n            \u00ac\u03c6.realize (fun i => x i a) (Fin.snoc (fun i => v i a) m),\n          _\u27e9\n      rw [\u2190 Ultrafilter.eventually_not]\n      exact Filter.mem_of_superset h fun a ha => Classical.epsilon_spec ha\n    \u00b7 rw [Filter.eventually_iff] at *\n      exact Filter.mem_of_superset h fun a ha => ha (m a)\n#align first_order.language.ultraproduct.bounded_formula_realize_cast FirstOrder.Language.Ultraproduct.boundedFormula_realize_cast\n\ntheorem realize_formula_cast {\u03b2 : Type _} (\u03c6 : L.Formula \u03b2) (x : \u03b2 \u2192 \u2200 a, M a) :\n    (\u03c6.realize fun i => (x i : (u : Filter \u03b1).product M)) \u2194\n      \u2200\u1da0 a : \u03b1 in u, \u03c6.realize fun i => x i a :=\n  by\n  simp_rw [formula.realize, \u2190 bounded_formula_realize_cast \u03c6 x, iff_eq_eq]\n  exact congr rfl (Subsingleton.elim _ _)\n#align first_order.language.ultraproduct.realize_formula_cast FirstOrder.Language.Ultraproduct.realize_formula_cast\n\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/\n/-- \u0141o\u015b's Theorem : A sentence is true in an ultraproduct if and only if the set of structures it is\n  true in is in the ultrafilter. -/\ntheorem sentence_realize (\u03c6 : L.Sentence) : (u : Filter \u03b1).product M \u22a8 \u03c6 \u2194 \u2200\u1da0 a : \u03b1 in u, M a \u22a8 \u03c6 :=\n  by\n  simp_rw [sentence.realize, \u2190 realize_formula_cast \u03c6, iff_eq_eq]\n  exact congr rfl (Subsingleton.elim _ _)\n#align first_order.language.ultraproduct.sentence_realize FirstOrder.Language.Ultraproduct.sentence_realize\n\ninstance : Nonempty ((u : Filter \u03b1).product M) :=\n  letI : \u2200 a, Inhabited (M a) := fun _ => Classical.inhabited_of_nonempty'\n  instNonempty\n\nend Ultraproduct\n\nend Language\n\nend FirstOrder\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/ModelTheory/Ultraproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.35793023518856476}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang\n-/\nimport topology.category.Top.epi_mono\nimport category_theory.category.ulift\nimport category_theory.limits.concrete_category\nimport category_theory.concrete_category.elementwise\n\n/-!\n# The category of topological spaces has all limits and colimits\n\nFurther, these limits and colimits are preserved by the forgetful functor --- that is, the\nunderlying types are just the limits in the category of types.\n-/\n\nopen topological_space\nopen category_theory\nopen category_theory.limits\nopen opposite\n\nuniverses u v w\n\nnoncomputable theory\n\nnamespace Top\n\nvariables {J : Type v} [small_category J]\n\nlocal notation `forget` := forget Top\n\n/--\nA choice of limit cone for a functor `F : J \u2964 Top`.\nGenerally you should just use `limit.cone F`, unless you need the actual definition\n(which is in terms of `types.limit_cone`).\n-/\ndef limit_cone (F : J \u2964 Top.{max v u}) : cone F :=\n{ X := Top.of {u : \u03a0 j : J, F.obj j | \u2200 {i j : J} (f : i \u27f6 j), F.map f (u i) = u j},\n  \u03c0 :=\n  { app := \u03bb j,\n    { to_fun := \u03bb u, u.val j,\n      continuous_to_fun := show continuous ((\u03bb u : \u03a0 j : J, F.obj j, u j) \u2218 subtype.val),\n        by continuity } } }\n\n/--\nA choice of limit cone for a functor `F : J \u2964 Top` whose topology is defined as an\ninfimum of topologies infimum.\nGenerally you should just use `limit.cone F`, unless you need the actual definition\n(which is in terms of `types.limit_cone`).\n-/\ndef limit_cone_infi (F : J \u2964 Top.{max v u}) : cone F :=\n{ X := \u27e8(types.limit_cone (F \u22d9 forget)).X, \u2a05j,\n        (F.obj j).str.induced ((types.limit_cone (F \u22d9 forget)).\u03c0.app j)\u27e9,\n  \u03c0 :=\n  { app := \u03bb j, \u27e8(types.limit_cone (F \u22d9 forget)).\u03c0.app j,\n                 continuous_iff_le_induced.mpr (infi_le _ _)\u27e9,\n    naturality' := \u03bb j j' f, continuous_map.coe_injective\n      ((types.limit_cone (F \u22d9 forget)).\u03c0.naturality f) } }\n\n/--\nThe chosen cone `Top.limit_cone F` for a functor `F : J \u2964 Top` is a limit cone.\nGenerally you should just use `limit.is_limit F`, unless you need the actual definition\n(which is in terms of `types.limit_cone_is_limit`).\n-/\ndef limit_cone_is_limit (F : J \u2964 Top.{max v u}) : is_limit (limit_cone F) :=\n{ lift := \u03bb S, { to_fun := \u03bb x, \u27e8\u03bb j, S.\u03c0.app _ x, \u03bb i j f, by { dsimp, erw \u2190 S.w f, refl }\u27e9 },\n  uniq' := \u03bb S m h, by { ext : 3, simpa [\u2190 h] } }\n\n/--\nThe chosen cone `Top.limit_cone_infi F` for a functor `F : J \u2964 Top` is a limit cone.\nGenerally you should just use `limit.is_limit F`, unless you need the actual definition\n(which is in terms of `types.limit_cone_is_limit`).\n-/\ndef limit_cone_infi_is_limit (F : J \u2964 Top.{max v u}) : is_limit (limit_cone_infi F) :=\nby { refine is_limit.of_faithful forget (types.limit_cone_is_limit _) (\u03bb s, \u27e8_, _\u27e9) (\u03bb s, rfl),\n     exact continuous_iff_coinduced_le.mpr (le_infi $ \u03bb j,\n       coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.\u03c0.app j).continuous :\n         _) ) }\n\ninstance Top_has_limits_of_size : has_limits_of_size.{v} Top.{max v u} :=\n{ has_limits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { has_limit := \u03bb F, has_limit.mk { cone := limit_cone F, is_limit := limit_cone_is_limit F } } }\n\ninstance Top_has_limits : has_limits Top.{u} := Top.Top_has_limits_of_size.{u u}\n\ninstance forget_preserves_limits_of_size :\n  preserves_limits_of_size.{v v} (forget : Top.{max v u} \u2964 Type (max v u)) :=\n{ preserves_limits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_limit := \u03bb F,\n    by exactI preserves_limit_of_preserves_limit_cone\n      (limit_cone_is_limit F) (types.limit_cone_is_limit (F \u22d9 forget)) } }\n\ninstance forget_preserves_limits : preserves_limits (forget : Top.{u} \u2964 Type u) :=\nTop.forget_preserves_limits_of_size.{u u}\n\n/--\nA choice of colimit cocone for a functor `F : J \u2964 Top`.\nGenerally you should just use `colimit.coone F`, unless you need the actual definition\n(which is in terms of `types.colimit_cocone`).\n-/\ndef colimit_cocone (F : J \u2964 Top.{max v u}) : cocone F :=\n{ X := \u27e8(types.colimit_cocone (F \u22d9 forget)).X, \u2a06 j,\n        (F.obj j).str.coinduced ((types.colimit_cocone (F \u22d9 forget)).\u03b9.app j)\u27e9,\n  \u03b9 :=\n  { app := \u03bb j, \u27e8(types.colimit_cocone (F \u22d9 forget)).\u03b9.app j,\n                 continuous_iff_coinduced_le.mpr (le_supr _ j)\u27e9,\n    naturality' := \u03bb j j' f, continuous_map.coe_injective\n      ((types.colimit_cocone (F \u22d9 forget)).\u03b9.naturality f) } }\n\n/--\nThe chosen cocone `Top.colimit_cocone F` for a functor `F : J \u2964 Top` is a colimit cocone.\nGenerally you should just use `colimit.is_colimit F`, unless you need the actual definition\n(which is in terms of `types.colimit_cocone_is_colimit`).\n-/\ndef colimit_cocone_is_colimit (F : J \u2964 Top.{max v u}) : is_colimit (colimit_cocone F) :=\nby { refine is_colimit.of_faithful forget (types.colimit_cocone_is_colimit _) (\u03bb s, \u27e8_, _\u27e9)\n       (\u03bb s, rfl),\n     exact continuous_iff_le_induced.mpr (supr_le $ \u03bb j,\n       coinduced_le_iff_le_induced.mp $ (continuous_iff_coinduced_le.mp (s.\u03b9.app j).continuous :\n         _) ) }\n\ninstance Top_has_colimits_of_size : has_colimits_of_size.{v} Top.{max v u} :=\n{ has_colimits_of_shape := \u03bb J \ud835\udca5, by exactI\n  { has_colimit := \u03bb F, has_colimit.mk { cocone := colimit_cocone F, is_colimit :=\n    colimit_cocone_is_colimit F } } }\n\ninstance Top_has_colimits : has_colimits Top.{u} := Top.Top_has_colimits_of_size.{u u}\n\ninstance forget_preserves_colimits_of_size :\n  preserves_colimits_of_size.{v v} (forget : Top.{max v u} \u2964 Type (max v u)) :=\n{ preserves_colimits_of_shape := \u03bb J \ud835\udca5,\n  { preserves_colimit := \u03bb F,\n    by exactI preserves_colimit_of_preserves_colimit_cocone\n      (colimit_cocone_is_colimit F) (types.colimit_cocone_is_colimit (F \u22d9 forget)) } }\n\ninstance forget_preserves_colimits : preserves_colimits (forget : Top.{u} \u2964 Type u) :=\nTop.forget_preserves_colimits_of_size.{u u}\n\n/-- The projection from the product as a bundled continous map. -/\nabbreviation pi_\u03c0 {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) : Top.of (\u03a0 i, \u03b1 i) \u27f6 \u03b1 i :=\n\u27e8\u03bb f, f i, continuous_apply i\u27e9\n\n/-- The explicit fan of a family of topological spaces given by the pi type. -/\n@[simps X \u03c0_app]\ndef pi_fan {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : fan \u03b1 :=\nfan.mk (Top.of (\u03a0 i, \u03b1 i)) (pi_\u03c0 \u03b1)\n\n/-- The constructed fan is indeed a limit -/\ndef pi_fan_is_limit {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : is_limit (pi_fan \u03b1) :=\n{ lift := \u03bb S, { to_fun := \u03bb s i, S.\u03c0.app \u27e8i\u27e9 s },\n  uniq' := by { intros S m h, ext x i, simp [\u2190 h \u27e8i\u27e9] },\n  fac' := \u03bb s j, by { cases j, tidy, }, }\n\n/--\nThe product is homeomorphic to the product of the underlying spaces,\nequipped with the product topology.\n-/\ndef pi_iso_pi {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : \u220f \u03b1 \u2245 Top.of (\u03a0 i, \u03b1 i) :=\n(limit.is_limit _).cone_point_unique_up_to_iso (pi_fan_is_limit \u03b1)\n\n@[simp, reassoc]\nlemma pi_iso_pi_inv_\u03c0 {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) :\n  (pi_iso_pi \u03b1).inv \u226b pi.\u03c0 \u03b1 i = pi_\u03c0 \u03b1 i :=\nby simp [pi_iso_pi]\n\n@[simp]\nlemma pi_iso_pi_inv_\u03c0_apply {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) (x : \u03a0 i, \u03b1 i) :\n  (pi.\u03c0 \u03b1 i : _) ((pi_iso_pi \u03b1).inv x) = x i :=\nconcrete_category.congr_hom (pi_iso_pi_inv_\u03c0 \u03b1 i) x\n\n@[simp]\nlemma pi_iso_pi_hom_apply {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) (x : \u220f \u03b1) :\n  (pi_iso_pi \u03b1).hom x i = (pi.\u03c0 \u03b1 i : _) x :=\nbegin\n  have := pi_iso_pi_inv_\u03c0 \u03b1 i,\n  rw iso.inv_comp_eq at this,\n  exact concrete_category.congr_hom this x\nend\n\n/-- The inclusion to the coproduct as a bundled continous map. -/\nabbreviation sigma_\u03b9 {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) : \u03b1 i \u27f6 Top.of (\u03a3 i, \u03b1 i) :=\n\u27e8sigma.mk i\u27e9\n\n/-- The explicit cofan of a family of topological spaces given by the sigma type. -/\n@[simps X \u03b9_app]\ndef sigma_cofan {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : cofan \u03b1 :=\ncofan.mk (Top.of (\u03a3 i, \u03b1 i)) (sigma_\u03b9 \u03b1)\n\n/-- The constructed cofan is indeed a colimit -/\ndef sigma_cofan_is_colimit {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : is_colimit (sigma_cofan \u03b1) :=\n{ desc := \u03bb S, { to_fun := \u03bb s, S.\u03b9.app \u27e8s.1\u27e9 s.2,\n    continuous_to_fun := continuous_sigma $ \u03bb i, map_continuous (S.\u03b9.app \u27e8i\u27e9) },\n  uniq' := by { intros S m h,  ext \u27e8i, x\u27e9, simp [\u2190 h \u27e8i\u27e9] },\n  fac' := \u03bb s j, by { cases j, tidy, }, }\n\n/--\nThe coproduct is homeomorphic to the disjoint union of the topological spaces.\n-/\ndef sigma_iso_sigma {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) : \u2210 \u03b1 \u2245 Top.of (\u03a3 i, \u03b1 i) :=\n(colimit.is_colimit _).cocone_point_unique_up_to_iso (sigma_cofan_is_colimit \u03b1)\n\n@[simp, reassoc]\nlemma sigma_iso_sigma_hom_\u03b9 {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) :\n  sigma.\u03b9 \u03b1 i \u226b (sigma_iso_sigma \u03b1).hom = sigma_\u03b9 \u03b1 i :=\nby simp [sigma_iso_sigma]\n\n@[simp]\nlemma sigma_iso_sigma_hom_\u03b9_apply {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) (x : \u03b1 i) :\n  (sigma_iso_sigma \u03b1).hom ((sigma.\u03b9 \u03b1 i : _) x) = sigma.mk i x :=\nconcrete_category.congr_hom (sigma_iso_sigma_hom_\u03b9 \u03b1 i) x\n\n@[simp]\nlemma sigma_iso_sigma_inv_apply {\u03b9 : Type v} (\u03b1 : \u03b9 \u2192 Top.{max v u}) (i : \u03b9) (x : \u03b1 i) :\n  (sigma_iso_sigma \u03b1).inv \u27e8i, x\u27e9 = (sigma.\u03b9 \u03b1 i : _) x :=\nby { rw [\u2190 sigma_iso_sigma_hom_\u03b9_apply, \u2190 comp_app], simp, }\n\nlemma induced_of_is_limit {F : J \u2964 Top.{max v u}} (C : cone F) (hC : is_limit C) :\n  C.X.topological_space = \u2a05 j, (F.obj j).topological_space.induced (C.\u03c0.app j) :=\nbegin\n  let homeo := homeo_of_iso (hC.cone_point_unique_up_to_iso (limit_cone_infi_is_limit F)),\n  refine homeo.inducing.induced.trans _,\n  change induced homeo (\u2a05 (j : J), _) = _,\n  simpa [induced_infi, induced_compose],\nend\n\nlemma limit_topology (F : J \u2964 Top.{max v u}) :\n  (limit F).topological_space = \u2a05 j, (F.obj j).topological_space.induced (limit.\u03c0 F j) :=\ninduced_of_is_limit _ (limit.is_limit F)\n\nsection prod\n\n/-- The first projection from the product. -/\nabbreviation prod_fst {X Y : Top.{u}} : Top.of (X \u00d7 Y) \u27f6 X := \u27e8prod.fst\u27e9\n\n/-- The second projection from the product. -/\nabbreviation prod_snd {X Y : Top.{u}} : Top.of (X \u00d7 Y) \u27f6 Y := \u27e8prod.snd\u27e9\n\n/-- The explicit binary cofan of `X, Y` given by `X \u00d7 Y`. -/\ndef prod_binary_fan (X Y : Top.{u}) : binary_fan X Y :=\nbinary_fan.mk prod_fst prod_snd\n\n/-- The constructed binary fan is indeed a limit -/\ndef prod_binary_fan_is_limit (X Y : Top.{u}) : is_limit (prod_binary_fan X Y) :=\n{ lift := \u03bb (S : binary_fan X Y), { to_fun := \u03bb s, (S.fst s, S.snd s) },\n  fac' := begin\n    rintros S (_|_),\n    tidy\n  end,\n  uniq' := begin\n    intros S m h,\n    ext x,\n    { specialize h \u27e8walking_pair.left\u27e9,\n      apply_fun (\u03bb e, (e x)) at h,\n      exact h },\n     { specialize h \u27e8walking_pair.right\u27e9,\n      apply_fun (\u03bb e, (e x)) at h,\n      exact h },\n  end }\n\n/--\nThe homeomorphism between `X \u2a2f Y` and the set-theoretic product of `X` and `Y`,\nequipped with the product topology.\n-/\ndef prod_iso_prod (X Y : Top.{u}) : X \u2a2f Y \u2245 Top.of (X \u00d7 Y) :=\n(limit.is_limit _).cone_point_unique_up_to_iso (prod_binary_fan_is_limit X Y)\n\n@[simp, reassoc] lemma prod_iso_prod_hom_fst (X Y : Top.{u}) :\n  (prod_iso_prod X Y).hom \u226b prod_fst = limits.prod.fst :=\nby simpa [\u2190 iso.eq_inv_comp, prod_iso_prod]\n\n@[simp, reassoc] lemma prod_iso_prod_hom_snd (X Y : Top.{u}) :\n  (prod_iso_prod X Y).hom \u226b prod_snd = limits.prod.snd :=\nby simpa [\u2190 iso.eq_inv_comp, prod_iso_prod]\n\n@[simp] lemma prod_iso_prod_hom_apply {X Y : Top.{u}} (x : X \u2a2f Y) :\n  (prod_iso_prod X Y).hom x =\n    ((limits.prod.fst : X \u2a2f Y \u27f6 _) x, (limits.prod.snd : X \u2a2f Y \u27f6 _) x) :=\nbegin\n  ext,\n  { exact concrete_category.congr_hom (prod_iso_prod_hom_fst X Y) x },\n  { exact concrete_category.congr_hom (prod_iso_prod_hom_snd X Y) x }\nend\n\n@[simp, reassoc, elementwise] lemma prod_iso_prod_inv_fst (X Y : Top.{u}) :\n  (prod_iso_prod X Y).inv \u226b limits.prod.fst = prod_fst :=\nby simp [iso.inv_comp_eq]\n\n@[simp, reassoc, elementwise] lemma prod_iso_prod_inv_snd (X Y : Top.{u}) :\n  (prod_iso_prod X Y).inv \u226b limits.prod.snd = prod_snd :=\nby simp [iso.inv_comp_eq]\n\nlemma prod_topology {X Y : Top} :\n  (X \u2a2f Y).topological_space =\n    induced (limits.prod.fst : X \u2a2f Y \u27f6 _) X.topological_space \u2293\n      induced (limits.prod.snd : X \u2a2f Y \u27f6 _) Y.topological_space :=\nbegin\n  let homeo := homeo_of_iso (prod_iso_prod X Y),\n  refine homeo.inducing.induced.trans _,\n  change induced homeo (_ \u2293 _) = _,\n  simpa [induced_compose]\nend\n\nlemma range_prod_map {W X Y Z : Top.{u}} (f : W \u27f6 Y) (g : X \u27f6 Z) :\n  set.range (limits.prod.map f g) =\n    (limits.prod.fst : Y \u2a2f Z \u27f6 _) \u207b\u00b9' (set.range f) \u2229\n      (limits.prod.snd : Y \u2a2f Z \u27f6 _) \u207b\u00b9' (set.range g) :=\nbegin\n  ext,\n  split,\n  { rintros \u27e8y, rfl\u27e9,\n    simp only [set.mem_preimage, set.mem_range, set.mem_inter_iff, \u2190comp_apply],\n    simp only [limits.prod.map_fst, limits.prod.map_snd,\n      exists_apply_eq_apply, comp_apply, and_self] },\n  { rintros \u27e8\u27e8x\u2081, hx\u2081\u27e9, \u27e8x\u2082, hx\u2082\u27e9\u27e9,\n    use (prod_iso_prod W X).inv (x\u2081, x\u2082),\n    apply concrete.limit_ext,\n    rintro \u27e8\u27e8\u27e9\u27e9,\n    { simp only [\u2190 comp_apply, category.assoc], erw limits.prod.map_fst, simp [hx\u2081] },\n    { simp only [\u2190 comp_apply, category.assoc], erw limits.prod.map_snd, simp [hx\u2082] } }\nend\n\nlemma inducing_prod_map {W X Y Z : Top} {f : W \u27f6 X} {g : Y \u27f6 Z}\n  (hf : inducing f) (hg : inducing g) : inducing (limits.prod.map f g) :=\nbegin\n  constructor,\n  simp only [prod_topology, induced_compose, \u2190coe_comp, limits.prod.map_fst, limits.prod.map_snd,\n    induced_inf],\n  simp only [coe_comp],\n  rw [\u2190 @induced_compose _ _ _ _ _ f, \u2190 @induced_compose _ _ _ _ _ g, \u2190 hf.induced, \u2190 hg.induced]\nend\n\nlemma embedding_prod_map {W X Y Z : Top} {f : W \u27f6 X} {g : Y \u27f6 Z}\n  (hf : embedding f) (hg : embedding g) : embedding (limits.prod.map f g) :=\n\u27e8inducing_prod_map hf.to_inducing hg.to_inducing,\nbegin\n  haveI := (Top.mono_iff_injective _).mpr hf.inj,\n  haveI := (Top.mono_iff_injective _).mpr hg.inj,\n  exact (Top.mono_iff_injective _).mp infer_instance\nend\u27e9\n\nend prod\n\nsection pullback\n\nvariables {X Y Z : Top.{u}}\n\n/-- The first projection from the pullback. -/\nabbreviation pullback_fst (f : X \u27f6 Z) (g : Y \u27f6 Z) : Top.of { p : X \u00d7 Y // f p.1 = g p.2 } \u27f6 X :=\n\u27e8prod.fst \u2218 subtype.val\u27e9\n\n/-- The second projection from the pullback. -/\nabbreviation pullback_snd (f : X \u27f6 Z) (g : Y \u27f6 Z) : Top.of { p : X \u00d7 Y // f p.1 = g p.2 } \u27f6 Y :=\n\u27e8prod.snd \u2218 subtype.val\u27e9\n\n/-- The explicit pullback cone of `X, Y` given by `{ p : X \u00d7 Y // f p.1 = g p.2 }`. -/\ndef pullback_cone (f : X \u27f6 Z) (g : Y \u27f6 Z) : pullback_cone f g :=\npullback_cone.mk (pullback_fst f g) (pullback_snd f g) (by { ext \u27e8x, h\u27e9, simp [h] })\n\n/-- The constructed cone is a limit. -/\ndef pullback_cone_is_limit (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  is_limit (pullback_cone f g) := pullback_cone.is_limit_aux' _\nbegin\n  intro s,\n  split, swap,\n  exact { to_fun := \u03bb x, \u27e8\u27e8s.fst x, s.snd x\u27e9,\n    by simpa using concrete_category.congr_hom s.condition x\u27e9 },\n  refine \u27e8_,_,_\u27e9,\n  { ext, delta pullback_cone, simp },\n  { ext, delta pullback_cone, simp },\n  { intros m h\u2081 h\u2082,\n    ext x,\n    { simpa using concrete_category.congr_hom h\u2081 x },\n    { simpa using concrete_category.congr_hom h\u2082 x } }\nend\n\n/-- The pullback of two maps can be identified as a subspace of `X \u00d7 Y`. -/\ndef pullback_iso_prod_subtype (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  pullback f g \u2245 Top.of { p : X \u00d7 Y // f p.1 = g p.2 } :=\n(limit.is_limit _).cone_point_unique_up_to_iso (pullback_cone_is_limit f g)\n\n@[simp, reassoc] lemma pullback_iso_prod_subtype_inv_fst (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (pullback_iso_prod_subtype f g).inv \u226b pullback.fst = pullback_fst f g :=\nby simpa [pullback_iso_prod_subtype]\n\n@[simp] lemma pullback_iso_prod_subtype_inv_fst_apply (f : X \u27f6 Z) (g : Y \u27f6 Z)\n  (x : { p : X \u00d7 Y // f p.1 = g p.2 }) :\n  (pullback.fst : pullback f g \u27f6 _) ((pullback_iso_prod_subtype f g).inv x) = (x : X \u00d7 Y).fst :=\nconcrete_category.congr_hom (pullback_iso_prod_subtype_inv_fst f g) x\n\n@[simp, reassoc] lemma pullback_iso_prod_subtype_inv_snd (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (pullback_iso_prod_subtype f g).inv \u226b pullback.snd = pullback_snd f g :=\nby simpa [pullback_iso_prod_subtype]\n\n@[simp] lemma pullback_iso_prod_subtype_inv_snd_apply (f : X \u27f6 Z) (g : Y \u27f6 Z)\n  (x : { p : X \u00d7 Y // f p.1 = g p.2 }) :\n  (pullback.snd : pullback f g \u27f6 _) ((pullback_iso_prod_subtype f g).inv x) = (x : X \u00d7 Y).snd :=\nconcrete_category.congr_hom (pullback_iso_prod_subtype_inv_snd f g) x\n\nlemma pullback_iso_prod_subtype_hom_fst (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (pullback_iso_prod_subtype f g).hom \u226b pullback_fst f g = pullback.fst :=\nby rw [\u2190iso.eq_inv_comp, pullback_iso_prod_subtype_inv_fst]\n\nlemma pullback_iso_prod_subtype_hom_snd (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (pullback_iso_prod_subtype f g).hom \u226b pullback_snd f g = pullback.snd :=\nby rw [\u2190iso.eq_inv_comp, pullback_iso_prod_subtype_inv_snd]\n\n@[simp] lemma pullback_iso_prod_subtype_hom_apply {f : X \u27f6 Z} {g : Y \u27f6 Z}\n  (x : pullback f g) : (pullback_iso_prod_subtype f g).hom x =\n    \u27e8\u27e8(pullback.fst : pullback f g \u27f6 _) x, (pullback.snd : pullback f g \u27f6 _) x\u27e9,\n      by simpa using concrete_category.congr_hom pullback.condition x\u27e9 :=\nbegin\n  ext,\n  exacts [concrete_category.congr_hom (pullback_iso_prod_subtype_hom_fst f g) x,\n    concrete_category.congr_hom (pullback_iso_prod_subtype_hom_snd f g) x]\nend\n\nlemma pullback_topology {X Y Z : Top.{u}} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (pullback f g).topological_space =\n    induced (pullback.fst : pullback f g \u27f6 _) X.topological_space \u2293\n      induced (pullback.snd : pullback f g \u27f6 _) Y.topological_space :=\nbegin\n  let homeo := homeo_of_iso (pullback_iso_prod_subtype f g),\n  refine homeo.inducing.induced.trans _,\n  change induced homeo (induced _ (_ \u2293 _)) = _,\n  simpa [induced_compose]\nend\n\nlemma range_pullback_to_prod {X Y Z : Top} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  set.range (prod.lift pullback.fst pullback.snd : pullback f g \u27f6 X \u2a2f Y) =\n  { x | (limits.prod.fst \u226b f) x = (limits.prod.snd \u226b g) x } :=\nbegin\n  ext x,\n  split,\n  { rintros \u27e8y, rfl\u27e9,\n    simp only [\u2190comp_apply, set.mem_set_of_eq],\n    congr' 1,\n    simp [pullback.condition] },\n  { intro h,\n    use (pullback_iso_prod_subtype f g).inv \u27e8\u27e8_, _\u27e9, h\u27e9,\n    apply concrete.limit_ext,\n    rintro \u27e8\u27e8\u27e9\u27e9; simp, }\nend\n\nlemma inducing_pullback_to_prod {X Y Z : Top} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  inducing \u21d1(prod.lift pullback.fst pullback.snd : pullback f g \u27f6 X \u2a2f Y) :=\n\u27e8by simp [prod_topology, pullback_topology, induced_compose, \u2190coe_comp]\u27e9\n\nlemma embedding_pullback_to_prod {X Y Z : Top} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  embedding \u21d1(prod.lift pullback.fst pullback.snd : pullback f g \u27f6 X \u2a2f Y) :=\n\u27e8inducing_pullback_to_prod f g, (Top.mono_iff_injective _).mp infer_instance\u27e9\n\n/-- If the map `S \u27f6 T` is mono, then there is a description of the image of `W \u00d7\u209b X \u27f6 Y \u00d7\u209c Z`. -/\nlemma range_pullback_map {W X Y Z S T : Top} (f\u2081 : W \u27f6 S) (f\u2082 : X \u27f6 S)\n  (g\u2081 : Y \u27f6 T) (g\u2082 : Z \u27f6 T) (i\u2081 : W \u27f6 Y) (i\u2082 : X \u27f6 Z) (i\u2083 : S \u27f6 T) [H\u2083 : mono i\u2083]\n  (eq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081) (eq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082) :\n  set.range (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082) =\n    (pullback.fst : pullback g\u2081 g\u2082 \u27f6 _) \u207b\u00b9' (set.range i\u2081) \u2229\n      (pullback.snd : pullback g\u2081 g\u2082 \u27f6 _) \u207b\u00b9' (set.range i\u2082) :=\nbegin\n  ext,\n  split,\n  { rintro \u27e8y, rfl\u27e9, simp, },\n  rintros \u27e8\u27e8x\u2081, hx\u2081\u27e9, \u27e8x\u2082, hx\u2082\u27e9\u27e9,\n  have : f\u2081 x\u2081 = f\u2082 x\u2082,\n  { apply (Top.mono_iff_injective _).mp H\u2083,\n    simp only [\u2190comp_apply, eq\u2081, eq\u2082],\n    simp only [comp_apply, hx\u2081, hx\u2082],\n    simp only [\u2190comp_apply, pullback.condition] },\n  use (pullback_iso_prod_subtype f\u2081 f\u2082).inv \u27e8\u27e8x\u2081, x\u2082\u27e9, this\u27e9,\n  apply concrete.limit_ext,\n  rintros (_|_|_),\n  { simp only [Top.comp_app, limit.lift_\u03c0_apply, category.assoc, pullback_cone.mk_\u03c0_app_one,\n      hx\u2081, pullback_iso_prod_subtype_inv_fst_apply, subtype.coe_mk],\n    simp only [\u2190 comp_apply],\n    congr,\n    apply limit.w _ walking_cospan.hom.inl },\n  { simp [hx\u2081] },\n  { simp [hx\u2082] },\nend\n\nlemma pullback_fst_range {X Y S : Top} (f : X \u27f6 S) (g : Y \u27f6 S) :\n  set.range (pullback.fst : pullback f g \u27f6 _) = { x : X | \u2203 y : Y, f x = g y} :=\nbegin\n  ext x,\n  split,\n  { rintro \u27e8y, rfl\u27e9,\n    use (pullback.snd : pullback f g \u27f6 _) y,\n    exact concrete_category.congr_hom pullback.condition y },\n  { rintro \u27e8y, eq\u27e9,\n    use (Top.pullback_iso_prod_subtype f g).inv \u27e8\u27e8x, y\u27e9, eq\u27e9,\n    simp },\nend\n\nlemma pullback_snd_range {X Y S : Top} (f : X \u27f6 S) (g : Y \u27f6 S) :\n  set.range (pullback.snd : pullback f g \u27f6 _) = { y : Y | \u2203 x : X, f x = g y} :=\nbegin\n  ext y,\n  split,\n  { rintro \u27e8x, rfl\u27e9,\n    use (pullback.fst : pullback f g \u27f6 _) x,\n    exact concrete_category.congr_hom pullback.condition x },\n  { rintro \u27e8x, eq\u27e9,\n    use (Top.pullback_iso_prod_subtype f g).inv \u27e8\u27e8x, y\u27e9, eq\u27e9,\n    simp },\nend\n\n/--\nIf there is a diagram where the morphisms `W \u27f6 Y` and `X \u27f6 Z` are embeddings,\nthen the induced morphism `W \u00d7\u209b X \u27f6 Y \u00d7\u209c Z` is also an embedding.\n\n  W  \u27f6  Y\n    \u2198      \u2198\n      S  \u27f6  T\n    \u2197      \u2197\n  X  \u27f6  Z\n-/\nlemma pullback_map_embedding_of_embeddings {W X Y Z S T : Top}\n  (f\u2081 : W \u27f6 S) (f\u2082 : X \u27f6 S) (g\u2081 : Y \u27f6 T) (g\u2082 : Z \u27f6 T) {i\u2081 : W \u27f6 Y} {i\u2082 : X \u27f6 Z}\n  (H\u2081 : embedding i\u2081) (H\u2082 : embedding i\u2082) (i\u2083 : S \u27f6 T)\n  (eq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081) (eq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082) :\n  embedding (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082) :=\nbegin\n  refine embedding_of_embedding_compose (continuous_map.continuous_to_fun _)\n    (show continuous (prod.lift pullback.fst pullback.snd : pullback g\u2081 g\u2082 \u27f6 Y \u2a2f Z), from\n      continuous_map.continuous_to_fun _) _,\n  suffices : embedding\n    (prod.lift pullback.fst pullback.snd \u226b limits.prod.map i\u2081 i\u2082 : pullback f\u2081 f\u2082 \u27f6 _),\n  { simpa [\u2190coe_comp] using this },\n  rw coe_comp,\n  refine embedding.comp (embedding_prod_map H\u2081 H\u2082)\n    (embedding_pullback_to_prod _ _)\nend\n\n/--\nIf there is a diagram where the morphisms `W \u27f6 Y` and `X \u27f6 Z` are open embeddings, and `S \u27f6 T`\nis mono, then the induced morphism `W \u00d7\u209b X \u27f6 Y \u00d7\u209c Z` is also an open embedding.\n  W  \u27f6  Y\n    \u2198      \u2198\n      S  \u27f6  T\n    \u2197       \u2197\n  X  \u27f6  Z\n-/\nlemma pullback_map_open_embedding_of_open_embeddings {W X Y Z S T : Top}\n  (f\u2081 : W \u27f6 S) (f\u2082 : X \u27f6 S) (g\u2081 : Y \u27f6 T) (g\u2082 : Z \u27f6 T) {i\u2081 : W \u27f6 Y} {i\u2082 : X \u27f6 Z}\n  (H\u2081 : open_embedding i\u2081) (H\u2082 : open_embedding i\u2082) (i\u2083 : S \u27f6 T) [H\u2083 : mono i\u2083]\n  (eq\u2081 : f\u2081 \u226b i\u2083 = i\u2081 \u226b g\u2081) (eq\u2082 : f\u2082 \u226b i\u2083 = i\u2082 \u226b g\u2082) :\n  open_embedding (pullback.map f\u2081 f\u2082 g\u2081 g\u2082 i\u2081 i\u2082 i\u2083 eq\u2081 eq\u2082) :=\nbegin\n  split,\n  { apply pullback_map_embedding_of_embeddings\n      f\u2081 f\u2082 g\u2081 g\u2082 H\u2081.to_embedding H\u2082.to_embedding i\u2083 eq\u2081 eq\u2082 },\n  { rw range_pullback_map,\n    apply is_open.inter; apply continuous.is_open_preimage,\n    continuity,\n    exacts [H\u2081.open_range, H\u2082.open_range] }\nend\n\nlemma snd_embedding_of_left_embedding {X Y S : Top}\n  {f : X \u27f6 S} (H : embedding f) (g : Y \u27f6 S) :\n  embedding \u21d1(pullback.snd : pullback f g \u27f6 Y) :=\nbegin\n  convert (homeo_of_iso (as_iso (pullback.snd : pullback (\ud835\udfd9 S) g \u27f6 _))).embedding.comp\n    (pullback_map_embedding_of_embeddings f g (\ud835\udfd9 _) g H\n      (homeo_of_iso (iso.refl _)).embedding (\ud835\udfd9 _) rfl (by simp)),\n  erw \u2190coe_comp,\n  simp\nend\n\nlemma fst_embedding_of_right_embedding {X Y S : Top}\n  (f : X \u27f6 S) {g : Y \u27f6 S} (H : embedding g) :\n  embedding \u21d1(pullback.fst : pullback f g \u27f6 X) :=\nbegin\n  convert (homeo_of_iso (as_iso (pullback.fst : pullback f (\ud835\udfd9 S) \u27f6 _))).embedding.comp\n    (pullback_map_embedding_of_embeddings f g f (\ud835\udfd9 _)\n      (homeo_of_iso (iso.refl _)).embedding H (\ud835\udfd9 _) rfl (by simp)),\n  erw \u2190coe_comp,\n  simp\nend\n\nlemma embedding_of_pullback_embeddings {X Y S : Top}\n  {f : X \u27f6 S} {g : Y \u27f6 S} (H\u2081 : embedding f) (H\u2082 : embedding g) :\n  embedding (limit.\u03c0 (cospan f g) walking_cospan.one) :=\nbegin\n  convert H\u2082.comp (snd_embedding_of_left_embedding H\u2081 g),\n  erw \u2190coe_comp,\n  congr,\n  exact (limit.w _ walking_cospan.hom.inr).symm\nend\n\nlemma snd_open_embedding_of_left_open_embedding {X Y S : Top}\n  {f : X \u27f6 S} (H : open_embedding f) (g : Y \u27f6 S) :\n  open_embedding \u21d1(pullback.snd : pullback f g \u27f6 Y) :=\nbegin\n  convert (homeo_of_iso (as_iso (pullback.snd : pullback (\ud835\udfd9 S) g \u27f6 _))).open_embedding.comp\n    (pullback_map_open_embedding_of_open_embeddings f g (\ud835\udfd9 _) g H\n      (homeo_of_iso (iso.refl _)).open_embedding (\ud835\udfd9 _) rfl (by simp)),\n  erw \u2190coe_comp,\n  simp\nend\n\nlemma fst_open_embedding_of_right_open_embedding {X Y S : Top}\n  (f : X \u27f6 S) {g : Y \u27f6 S} (H : open_embedding g) :\n  open_embedding \u21d1(pullback.fst : pullback f g \u27f6 X) :=\nbegin\n  convert (homeo_of_iso (as_iso (pullback.fst : pullback f (\ud835\udfd9 S) \u27f6 _))).open_embedding.comp\n    (pullback_map_open_embedding_of_open_embeddings f g f (\ud835\udfd9 _)\n      (homeo_of_iso (iso.refl _)).open_embedding H (\ud835\udfd9 _) rfl (by simp)),\n  erw \u2190coe_comp,\n  simp\nend\n\n/-- If `X \u27f6 S`, `Y \u27f6 S` are open embeddings, then so is `X \u00d7\u209b Y \u27f6 S`. -/\nlemma open_embedding_of_pullback_open_embeddings {X Y S : Top}\n  {f : X \u27f6 S} {g : Y \u27f6 S} (H\u2081 : open_embedding f) (H\u2082 : open_embedding g) :\n  open_embedding (limit.\u03c0 (cospan f g) walking_cospan.one) :=\nbegin\n  convert H\u2082.comp (snd_open_embedding_of_left_open_embedding H\u2081 g),\n  erw \u2190coe_comp,\n  congr,\n  exact (limit.w _ walking_cospan.hom.inr).symm\nend\n\nlemma fst_iso_of_right_embedding_range_subset {X Y S : Top} (f : X \u27f6 S) {g : Y \u27f6 S}\n  (hg : embedding g) (H : set.range f \u2286 set.range g) : is_iso (pullback.fst : pullback f g \u27f6 X) :=\nbegin\n  let : (pullback f g : Top) \u2243\u209c X :=\n    (homeomorph.of_embedding _ (fst_embedding_of_right_embedding f hg)).trans\n    { to_fun := coe,\n      inv_fun := (\u03bb x, \u27e8x,\n        by { rw pullback_fst_range, exact \u27e8_, (H (set.mem_range_self x)).some_spec.symm\u27e9 }\u27e9),\n      left_inv := \u03bb \u27e8_,_\u27e9, rfl,\n      right_inv := \u03bb x, rfl },\n  convert is_iso.of_iso (iso_of_homeo this),\n  ext,\n  refl\nend\n\nlemma snd_iso_of_left_embedding_range_subset {X Y S : Top} {f : X \u27f6 S} (hf : embedding f)\n  (g : Y \u27f6 S) (H : set.range g \u2286 set.range f) : is_iso (pullback.snd : pullback f g \u27f6 Y) :=\nbegin\n  let : (pullback f g : Top) \u2243\u209c Y :=\n    (homeomorph.of_embedding _ (snd_embedding_of_left_embedding hf g)).trans\n    { to_fun := coe,\n      inv_fun := (\u03bb x, \u27e8x,\n        by { rw pullback_snd_range, exact \u27e8_, (H (set.mem_range_self x)).some_spec\u27e9 }\u27e9),\n      left_inv := \u03bb \u27e8_,_\u27e9, rfl,\n      right_inv := \u03bb x, rfl },\n  convert is_iso.of_iso (iso_of_homeo this),\n  ext,\n  refl\nend\n\nlemma pullback_snd_image_fst_preimage (f : X \u27f6 Z) (g : Y \u27f6 Z) (U : set X) :\n  (pullback.snd : pullback f g \u27f6 _) '' ((pullback.fst : pullback f g \u27f6 _) \u207b\u00b9' U) =\n    g \u207b\u00b9' (f '' U) :=\nbegin\n  ext x,\n  split,\n  { rintros \u27e8y, hy, rfl\u27e9,\n    exact \u27e8(pullback.fst : pullback f g \u27f6 _) y, hy,\n    concrete_category.congr_hom pullback.condition y\u27e9 },\n  { rintros \u27e8y, hy, eq\u27e9,\n    exact \u27e8(Top.pullback_iso_prod_subtype f g).inv \u27e8\u27e8_,_\u27e9, eq\u27e9, by simpa, by simp\u27e9 },\nend\n\nlemma pullback_fst_image_snd_preimage (f : X \u27f6 Z) (g : Y \u27f6 Z) (U : set Y) :\n  (pullback.fst : pullback f g \u27f6 _) '' ((pullback.snd : pullback f g \u27f6 _) \u207b\u00b9' U) =\n    f \u207b\u00b9' (g '' U) :=\nbegin\n  ext x,\n  split,\n  { rintros \u27e8y, hy, rfl\u27e9,\n    exact \u27e8(pullback.snd : pullback f g \u27f6 _) y, hy,\n    (concrete_category.congr_hom pullback.condition y).symm\u27e9 },\n  { rintros \u27e8y, hy, eq\u27e9,\n    exact \u27e8(Top.pullback_iso_prod_subtype f g).inv \u27e8\u27e8_,_\u27e9,eq.symm\u27e9, by simpa, by simp\u27e9 },\nend\n\nend pullback\n\n/-- The terminal object of `Top` is `punit`. -/\ndef is_terminal_punit : is_terminal (Top.of punit.{u+1}) :=\nbegin\n  haveI : \u2200 X, unique (X \u27f6 Top.of punit.{u+1}) :=\n    \u03bb X, \u27e8\u27e8\u27e8\u03bb x, punit.star, by continuity\u27e9\u27e9, \u03bb f, by ext\u27e9,\n  exact limits.is_terminal.of_unique _,\nend\n\n/-- The terminal object of `Top` is `punit`. -/\ndef terminal_iso_punit : \u22a4_ Top.{u} \u2245 Top.of punit :=\nterminal_is_terminal.unique_up_to_iso is_terminal_punit\n\n/-- The initial object of `Top` is `pempty`. -/\ndef is_initial_pempty : is_initial (Top.of pempty.{u+1}) :=\nbegin\n  haveI : \u2200 X, unique (Top.of pempty.{u+1} \u27f6 X) :=\n    \u03bb X, \u27e8\u27e8\u27e8\u03bb x, x.elim, by continuity\u27e9\u27e9, \u03bb f, by ext \u27e8\u27e9\u27e9,\n  exact limits.is_initial.of_unique _,\nend\n\n/-- The initial object of `Top` is `pempty`. -/\ndef initial_iso_pempty : \u22a5_ Top.{u} \u2245 Top.of pempty :=\ninitial_is_initial.unique_up_to_iso is_initial_pempty\n\n/-- The binary coproduct cofan in `Top`. -/\nprotected\ndef binary_cofan (X Y : Top.{u}) : binary_cofan X Y :=\nbinary_cofan.mk (\u27e8sum.inl\u27e9 : X \u27f6 Top.of (X \u2295 Y)) \u27e8sum.inr\u27e9\n\n/-- The constructed binary coproduct cofan in `Top` is the coproduct. -/\ndef binary_cofan_is_colimit (X Y : Top.{u}) : is_colimit (Top.binary_cofan X Y) :=\nbegin\n  refine limits.binary_cofan.is_colimit_mk (\u03bb s, \u27e8sum.elim s.inl s.inr\u27e9) _ _ _,\n  { intro s, ext, refl },\n  { intro s, ext, refl },\n  { intros s m h\u2081 h\u2082, ext (x|x),\n    exacts [(concrete_category.congr_hom h\u2081 x : _), (concrete_category.congr_hom h\u2082 x : _)] },\nend\n\nlemma binary_cofan_is_colimit_iff {X Y : Top} (c : binary_cofan X Y) :\n  nonempty (is_colimit c) \u2194\n    open_embedding c.inl \u2227 open_embedding c.inr \u2227 is_compl (set.range c.inl) (set.range c.inr) :=\nbegin\n  classical,\n  split,\n  { rintro \u27e8h\u27e9,\n    rw [\u2190 show _ = c.inl, from h.comp_cocone_point_unique_up_to_iso_inv\n      (binary_cofan_is_colimit X Y) \u27e8walking_pair.left\u27e9,\n      \u2190 show _ = c.inr, from h.comp_cocone_point_unique_up_to_iso_inv\n      (binary_cofan_is_colimit X Y) \u27e8walking_pair.right\u27e9],\n    dsimp,\n    refine\n    \u27e8(homeo_of_iso $ h.cocone_point_unique_up_to_iso (binary_cofan_is_colimit X Y)).symm\n      .open_embedding.comp open_embedding_inl, (homeo_of_iso $ h.cocone_point_unique_up_to_iso\n        (binary_cofan_is_colimit X Y)).symm.open_embedding.comp open_embedding_inr, _\u27e9,\n    erw [set.range_comp, \u2190 eq_compl_iff_is_compl, set.range_comp _ sum.inr, \u2190 set.image_compl_eq\n      (homeo_of_iso $ h.cocone_point_unique_up_to_iso (binary_cofan_is_colimit X Y))\n      .symm.bijective],\n    congr' 1,\n    exact set.compl_range_inr.symm },\n  { rintros \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    { rw [eq_compl_iff_is_compl.mpr h\u2083.symm], exact \u03bb _, or_not },\n    refine \u27e8binary_cofan.is_colimit.mk _ _ _ _ _\u27e9,\n    { intros T f g,\n      refine continuous_map.mk _ _,\n      { exact \u03bb x, if h : x \u2208 set.range c.inl\n        then f ((equiv.of_injective _ h\u2081.inj).symm \u27e8x, h\u27e9)\n        else g ((equiv.of_injective _ h\u2082.inj).symm \u27e8x, (this x).resolve_left h\u27e9) },\n      rw continuous_iff_continuous_at,\n      intro x,\n      by_cases x \u2208 set.range c.inl,\n      { revert h x,\n      apply (is_open.continuous_on_iff _).mp,\n      { rw continuous_on_iff_continuous_restrict,\n        convert_to continuous (f \u2218 (homeomorph.of_embedding _ h\u2081.to_embedding).symm),\n        { ext \u27e8x, hx\u27e9, exact dif_pos hx },\n        continuity },\n      { exact h\u2081.open_range } },\n    { revert h x,\n      apply (is_open.continuous_on_iff _).mp,\n      { rw continuous_on_iff_continuous_restrict,\n        have : \u2200 a, a \u2209 set.range c.inl \u2192 a \u2208 set.range c.inr,\n        { rintros a (h : a \u2208 (set.range c.inl)\u1d9c), rwa eq_compl_iff_is_compl.mpr h\u2083.symm },\n        convert_to continuous\n          (g \u2218 (homeomorph.of_embedding _ h\u2082.to_embedding).symm \u2218 subtype.map _ this),\n        { ext \u27e8x, hx\u27e9, exact dif_neg hx },\n        continuity,\n        rw embedding_subtype_coe.to_inducing.continuous_iff,\n        exact continuous_subtype_coe },\n      { change is_open (set.range c.inl)\u1d9c, rw \u2190 eq_compl_iff_is_compl.mpr h\u2083.symm,\n        exact h\u2082.open_range } } },\n    { intros T f g, ext x, refine (dif_pos _).trans _, { exact \u27e8x, rfl\u27e9 },\n        { rw equiv.of_injective_symm_apply } },\n    { intros T f g, ext x, refine (dif_neg _).trans _,\n      { rintro \u27e8y, e\u27e9, have : c.inr x \u2208 set.range c.inl \u2293 set.range c.inr := \u27e8\u27e8_, e\u27e9, \u27e8_, rfl\u27e9\u27e9,\n        rwa disjoint_iff.mp h\u2083.1 at this },\n      { exact congr_arg g (equiv.of_injective_symm_apply _ _) } },\n    { rintro T _ _ m rfl rfl, ext x, change m x = dite _ _ _,\n      split_ifs; exact congr_arg _ (equiv.apply_of_injective_symm _ \u27e8_, _\u27e9).symm } }\nend\n\n--TODO: Add analogous constructions for `pushout`.\n\nlemma coinduced_of_is_colimit {F : J \u2964 Top.{max v u}} (c : cocone F) (hc : is_colimit c) :\n  c.X.topological_space = \u2a06 j, (F.obj j).topological_space.coinduced (c.\u03b9.app j) :=\nbegin\n  let homeo := homeo_of_iso (hc.cocone_point_unique_up_to_iso (colimit_cocone_is_colimit F)),\n  ext,\n  refine homeo.symm.is_open_preimage.symm.trans (iff.trans _ is_open_supr_iff.symm),\n  exact is_open_supr_iff\nend\n\nlemma colimit_topology (F : J \u2964 Top.{max v u}) :\n  (colimit F).topological_space = \u2a06 j, (F.obj j).topological_space.coinduced (colimit.\u03b9 F j) :=\ncoinduced_of_is_colimit _ (colimit.is_colimit F)\n\nlemma colimit_is_open_iff (F : J \u2964 Top.{max v u}) (U : set ((colimit F : _) : Type (max v u))) :\n  is_open U \u2194 \u2200 j, is_open (colimit.\u03b9 F j \u207b\u00b9' U) :=\nbegin\n  conv_lhs { rw colimit_topology F },\n  exact is_open_supr_iff\nend\n\nlemma coequalizer_is_open_iff (F : walking_parallel_pair \u2964 Top.{u})\n  (U : set ((colimit F : _) : Type u)) :\n  is_open U \u2194 is_open (colimit.\u03b9 F walking_parallel_pair.one \u207b\u00b9' U) :=\nbegin\n  rw colimit_is_open_iff.{u},\n  split,\n  { intro H, exact H _ },\n  { intros H j,\n    cases j,\n    { rw \u2190colimit.w F walking_parallel_pair_hom.left,\n      exact (F.map walking_parallel_pair_hom.left).continuous_to_fun.is_open_preimage _ H },\n    { exact H } }\nend\n\nend Top\n\nnamespace Top\n\nsection cofiltered_limit\n\nvariables {J : Type v} [small_category J] [is_cofiltered J] (F : J \u2964 Top.{max v u})\n  (C : cone F) (hC : is_limit C)\n\ninclude hC\n\n/--\nGiven a *compatible* collection of topological bases for the factors in a cofiltered limit\nwhich contain `set.univ` and are closed under intersections, the induced *naive* collection\nof sets in the limit is, in fact, a topological basis.\n-/\ntheorem is_topological_basis_cofiltered_limit\n  (T : \u03a0 j, set (set (F.obj j))) (hT : \u2200 j, is_topological_basis (T j))\n  (univ : \u2200 (i : J), set.univ \u2208 T i)\n  (inter : \u2200 i (U1 U2 : set (F.obj i)), U1 \u2208 T i \u2192 U2 \u2208 T i \u2192 U1 \u2229 U2 \u2208 T i)\n  (compat : \u2200 (i j : J) (f : i \u27f6 j) (V : set (F.obj j)) (hV : V \u2208 T j), (F.map f) \u207b\u00b9' V \u2208 T i) :\n  is_topological_basis { U : set C.X | \u2203 j (V : set (F.obj j)), V \u2208 T j \u2227 U = C.\u03c0.app j \u207b\u00b9' V } :=\nbegin\n  classical,\n  -- The limit cone for `F` whose topology is defined as an infimum.\n  let D := limit_cone_infi F,\n  -- The isomorphism between the cone point of `C` and the cone point of `D`.\n  let E : C.X \u2245 D.X := hC.cone_point_unique_up_to_iso (limit_cone_infi_is_limit _),\n  have hE : inducing E.hom := (Top.homeo_of_iso E).inducing,\n  -- Reduce to the assertion of the theorem with `D` instead of `C`.\n  suffices : is_topological_basis\n    { U : set D.X | \u2203 j (V : set (F.obj j)), V \u2208 T j \u2227 U = D.\u03c0.app j \u207b\u00b9' V },\n  { convert this.inducing hE,\n    ext U0,\n    split,\n    { rintro \u27e8j, V, hV, rfl\u27e9,\n      refine \u27e8D.\u03c0.app j \u207b\u00b9' V, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9 },\n    { rintro \u27e8W, \u27e8j, V, hV, rfl\u27e9, rfl\u27e9,\n      refine \u27e8j, V, hV, rfl\u27e9 } },\n  -- Using `D`, we can apply the characterization of the topological basis of a\n  -- topology defined as an infimum...\n  convert is_topological_basis_infi hT (\u03bb j (x : D.X), D.\u03c0.app j x),\n  ext U0,\n  split,\n  { rintros  \u27e8j, V, hV, rfl\u27e9,\n    let U : \u03a0 i, set (F.obj i) := \u03bb i, if h : i = j then (by {rw h, exact V}) else set.univ,\n    refine \u27e8U,{j},_,_\u27e9,\n    { rintro i h,\n      rw finset.mem_singleton at h,\n      dsimp [U],\n      rw dif_pos h,\n      subst h,\n      exact hV },\n    { dsimp [U],\n      simp } },\n  { rintros \u27e8U, G, h1, h2\u27e9,\n    obtain \u27e8j, hj\u27e9 := is_cofiltered.inf_objs_exists G,\n    let g : \u2200 e (he : e \u2208 G), j \u27f6 e := \u03bb _ he, (hj he).some,\n    let Vs : J \u2192 set (F.obj j) := \u03bb e, if h : e \u2208 G then F.map (g e h) \u207b\u00b9' (U e) else set.univ,\n    let V : set (F.obj j) := \u22c2 (e : J) (he : e \u2208 G), Vs e,\n    refine \u27e8j, V, _, _\u27e9,\n    { -- An intermediate claim used to apply induction along `G : finset J` later on.\n      have : \u2200 (S : set (set (F.obj j))) (E : finset J) (P : J \u2192 set (F.obj j))\n        (univ : set.univ \u2208 S)\n        (inter : \u2200 A B : set (F.obj j), A \u2208 S \u2192 B \u2208 S \u2192 A \u2229 B \u2208 S)\n        (cond : \u2200 (e : J) (he : e \u2208 E), P e \u2208 S), (\u22c2 e (he : e \u2208 E), P e) \u2208 S,\n      { intros S E,\n        apply E.induction_on,\n        { intros P he hh,\n          simpa },\n        { intros a E ha hh1 hh2 hh3 hh4 hh5,\n          rw finset.set_bInter_insert,\n          refine hh4 _ _ (hh5 _ (finset.mem_insert_self _ _)) (hh1 _ hh3 hh4 _),\n          intros e he,\n          exact hh5 e (finset.mem_insert_of_mem he) } },\n      -- use the intermediate claim to finish off the goal using `univ` and `inter`.\n      refine this _ _ _ (univ _) (inter _) _,\n      intros e he,\n      dsimp [Vs],\n      rw dif_pos he,\n      exact compat j e (g e he) (U e) (h1 e he), },\n    { -- conclude...\n      rw h2,\n      dsimp [V],\n      rw set.preimage_Inter,\n      congr' 1,\n      ext1 e,\n      rw set.preimage_Inter,\n      congr' 1,\n      ext1 he,\n      dsimp [Vs],\n      rw [dif_pos he, \u2190 set.preimage_comp],\n      congr' 1,\n      change _ = \u21d1(D.\u03c0.app j \u226b F.map (g e he)),\n      rw D.w } }\nend\n\nend cofiltered_limit\n\nsection topological_konig\n\n/-!\n## Topological K\u0151nig's lemma\n\nA topological version of K\u0151nig's lemma is that the inverse limit of nonempty compact Hausdorff\nspaces is nonempty.  (Note: this can be generalized further to inverse limits of nonempty compact\nT0 spaces, where all the maps are closed maps; see [Stone1979] --- however there is an erratum\nfor Theorem 4 that the element in the inverse limit can have cofinally many components that are\nnot closed points.)\n\nWe give this in a more general form, which is that cofiltered limits\nof nonempty compact Hausdorff spaces are nonempty\n(`nonempty_limit_cone_of_compact_t2_cofiltered_system`).\n\nThis also applies to inverse limits, where `{J : Type u} [preorder J] [is_directed J (\u2264)]` and\n`F : J\u1d52\u1d56 \u2964 Top`.\n\nThe theorem is specialized to nonempty finite types (which are compact Hausdorff with the\ndiscrete topology) in lemmas `nonempty_sections_of_finite_cofiltered_system` and\n`nonempty_sections_of_finite_inverse_system` in the file `category_theory.cofiltered_system`.\n\n(See <https://stacks.math.columbia.edu/tag/086J> for the Set version.)\n-/\n\nvariables {J : Type u} [small_category J]\nvariables (F : J \u2964 Top.{u})\n\nprivate abbreviation finite_diagram_arrow {J : Type u} [small_category J] (G : finset J) :=\n\u03a3' (X Y : J) (mX : X \u2208 G) (mY : Y \u2208 G), X \u27f6 Y\nprivate abbreviation finite_diagram (J : Type u) [small_category J] :=\n\u03a3 (G : finset J), finset (finite_diagram_arrow G)\n\n/--\nPartial sections of a cofiltered limit are sections when restricted to\na finite subset of objects and morphisms of `J`.\n-/\ndef partial_sections {J : Type u} [small_category J] (F : J \u2964 Top.{u})\n  {G : finset J} (H : finset (finite_diagram_arrow G)) : set (\u03a0 j, F.obj j) :=\n{ u | \u2200 {f : finite_diagram_arrow G} (hf : f \u2208 H), F.map f.2.2.2.2 (u f.1) = u f.2.1 }\n\nlemma partial_sections.nonempty [is_cofiltered_or_empty J] [h : \u03a0 (j : J), nonempty (F.obj j)]\n  {G : finset J} (H : finset (finite_diagram_arrow G)) :\n  (partial_sections F H).nonempty :=\nbegin\n  classical,\n  casesI is_empty_or_nonempty J,\n  { exact \u27e8is_empty_elim, \u03bb j, is_empty.elim' infer_instance j.1\u27e9 },\n  haveI : is_cofiltered J := \u27e8\u27e9,\n  use \u03bb (j : J), if hj : j \u2208 G\n                 then F.map (is_cofiltered.inf_to G H hj) (h (is_cofiltered.inf G H)).some\n                 else (h _).some,\n  rintros \u27e8X, Y, hX, hY, f\u27e9 hf,\n  dsimp only,\n  rwa [dif_pos hX, dif_pos hY, \u2190comp_app, \u2190F.map_comp,\n       @is_cofiltered.inf_to_commutes _ _ _ G H],\nend\n\nlemma partial_sections.directed :\n  directed superset (\u03bb (G : finite_diagram J), partial_sections F G.2) :=\nbegin\n  classical,\n  intros A B,\n  let \u03b9A : finite_diagram_arrow A.1 \u2192 finite_diagram_arrow (A.1 \u2294 B.1) :=\n    \u03bb f, \u27e8f.1, f.2.1, finset.mem_union_left _ f.2.2.1, finset.mem_union_left _ f.2.2.2.1,\n          f.2.2.2.2\u27e9,\n  let \u03b9B : finite_diagram_arrow B.1 \u2192 finite_diagram_arrow (A.1 \u2294 B.1) :=\n    \u03bb f, \u27e8f.1, f.2.1, finset.mem_union_right _ f.2.2.1, finset.mem_union_right _ f.2.2.2.1,\n          f.2.2.2.2\u27e9,\n  refine \u27e8\u27e8A.1 \u2294 B.1, A.2.image \u03b9A \u2294 B.2.image \u03b9B\u27e9, _, _\u27e9,\n  { rintro u hu f hf,\n    have : \u03b9A f \u2208 A.2.image \u03b9A \u2294 B.2.image \u03b9B,\n    { apply finset.mem_union_left,\n      rw finset.mem_image,\n      refine \u27e8f, hf, rfl\u27e9 },\n    exact hu this },\n  { rintro u hu f hf,\n    have : \u03b9B f \u2208 A.2.image \u03b9A \u2294 B.2.image \u03b9B,\n    { apply finset.mem_union_right,\n      rw finset.mem_image,\n      refine \u27e8f, hf, rfl\u27e9 },\n    exact hu this }\nend\n\nlemma partial_sections.closed [\u03a0 (j : J), t2_space (F.obj j)]\n  {G : finset J} (H : finset (finite_diagram_arrow G)) :\n  is_closed (partial_sections F H) :=\nbegin\n  have : partial_sections F H =\n    \u22c2 {f : finite_diagram_arrow G} (hf : f \u2208 H), { u | F.map f.2.2.2.2 (u f.1) = u f.2.1 },\n  { ext1,\n    simp only [set.mem_Inter, set.mem_set_of_eq],\n    refl, },\n  rw this,\n  apply is_closed_bInter,\n  intros f hf,\n  apply is_closed_eq,\n  continuity,\nend\n\n/--\nCofiltered limits of nonempty compact Hausdorff spaces are nonempty topological spaces.\n-/\nlemma nonempty_limit_cone_of_compact_t2_cofiltered_system\n  [is_cofiltered_or_empty J]\n  [\u03a0 (j : J), nonempty (F.obj j)]\n  [\u03a0 (j : J), compact_space (F.obj j)]\n  [\u03a0 (j : J), t2_space (F.obj j)] :\n  nonempty (Top.limit_cone.{u} F).X :=\nbegin\n  classical,\n  obtain \u27e8u, hu\u27e9 := is_compact.nonempty_Inter_of_directed_nonempty_compact_closed\n    (\u03bb G, partial_sections F _)\n    (partial_sections.directed F)\n    (\u03bb G, partial_sections.nonempty F _)\n    (\u03bb G, is_closed.is_compact (partial_sections.closed F _))\n    (\u03bb G, partial_sections.closed F _),\n  use u,\n  intros X Y f,\n  let G : finite_diagram J :=\n    \u27e8{X, Y},\n     {\u27e8X, Y,\n      by simp only [true_or, eq_self_iff_true, finset.mem_insert],\n      by simp only [eq_self_iff_true, or_true, finset.mem_insert, finset.mem_singleton],\n      f\u27e9}\u27e9,\n  exact hu _ \u27e8G, rfl\u27e9 (finset.mem_singleton_self _),\nend\n\nend topological_konig\n\nend Top\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/topology/category/Top/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3579302264983903}}
{"text": "/-\nCopyright (c) 2019 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Simon Hudon, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.equiv.basic\nimport Mathlib.control.bifunctor\nimport Mathlib.PostPort\n\nuniverses u v w \n\nnamespace Mathlib\n\n/-!\n# Functor and bifunctors can be applied to `equiv`s.\n\nWe define\n```lean\ndef functor.map_equiv (f : Type u \u2192 Type v) [functor f] [is_lawful_functor f] :\n  \u03b1 \u2243 \u03b2 \u2192 f \u03b1 \u2243 f \u03b2\n```\nand\n```lean\ndef bifunctor.map_equiv (F : Type u \u2192 Type v \u2192 Type w) [bifunctor F] [is_lawful_bifunctor F] :\n  \u03b1 \u2243 \u03b2 \u2192 \u03b1' \u2243 \u03b2' \u2192 F \u03b1 \u03b1' \u2243 F \u03b2 \u03b2'\n```\n-/\n\nnamespace functor\n\n\n/-- Apply a functor to an `equiv`. -/\ndef map_equiv {\u03b1 : Type u} {\u03b2 : Type u} (f : Type u \u2192 Type v) [Functor f] [is_lawful_functor f]\n    (h : \u03b1 \u2243 \u03b2) : f \u03b1 \u2243 f \u03b2 :=\n  equiv.mk (Functor.map \u21d1h) (Functor.map \u21d1(equiv.symm h)) sorry sorry\n\n@[simp] theorem map_equiv_apply {\u03b1 : Type u} {\u03b2 : Type u} (f : Type u \u2192 Type v) [Functor f]\n    [is_lawful_functor f] (h : \u03b1 \u2243 \u03b2) (x : f \u03b1) : coe_fn (map_equiv f h) x = \u21d1h <$> x :=\n  rfl\n\n@[simp] theorem map_equiv_symm_apply {\u03b1 : Type u} {\u03b2 : Type u} (f : Type u \u2192 Type v) [Functor f]\n    [is_lawful_functor f] (h : \u03b1 \u2243 \u03b2) (y : f \u03b2) :\n    coe_fn (equiv.symm (map_equiv f h)) y = \u21d1(equiv.symm h) <$> y :=\n  rfl\n\n@[simp] theorem map_equiv_refl {\u03b1 : Type u} (f : Type u \u2192 Type v) [Functor f]\n    [is_lawful_functor f] : map_equiv f (equiv.refl \u03b1) = equiv.refl (f \u03b1) :=\n  sorry\n\nend functor\n\n\nnamespace bifunctor\n\n\n/-- Apply a bifunctor to a pair of `equiv`s. -/\ndef map_equiv {\u03b1 : Type u} {\u03b2 : Type u} {\u03b1' : Type v} {\u03b2' : Type v} (F : Type u \u2192 Type v \u2192 Type w)\n    [bifunctor F] [is_lawful_bifunctor F] (h : \u03b1 \u2243 \u03b2) (h' : \u03b1' \u2243 \u03b2') : F \u03b1 \u03b1' \u2243 F \u03b2 \u03b2' :=\n  equiv.mk (bimap \u21d1h \u21d1h') (bimap \u21d1(equiv.symm h) \u21d1(equiv.symm h')) sorry sorry\n\n@[simp] theorem map_equiv_apply {\u03b1 : Type u} {\u03b2 : Type u} {\u03b1' : Type v} {\u03b2' : Type v}\n    (F : Type u \u2192 Type v \u2192 Type w) [bifunctor F] [is_lawful_bifunctor F] (h : \u03b1 \u2243 \u03b2) (h' : \u03b1' \u2243 \u03b2')\n    (x : F \u03b1 \u03b1') : coe_fn (map_equiv F h h') x = bimap (\u21d1h) (\u21d1h') x :=\n  rfl\n\n@[simp] theorem map_equiv_symm_apply {\u03b1 : Type u} {\u03b2 : Type u} {\u03b1' : Type v} {\u03b2' : Type v}\n    (F : Type u \u2192 Type v \u2192 Type w) [bifunctor F] [is_lawful_bifunctor F] (h : \u03b1 \u2243 \u03b2) (h' : \u03b1' \u2243 \u03b2')\n    (y : F \u03b2 \u03b2') :\n    coe_fn (equiv.symm (map_equiv F h h')) y = bimap (\u21d1(equiv.symm h)) (\u21d1(equiv.symm h')) y :=\n  rfl\n\n@[simp] theorem map_equiv_refl_refl {\u03b1 : Type u} {\u03b1' : Type v} (F : Type u \u2192 Type v \u2192 Type w)\n    [bifunctor F] [is_lawful_bifunctor F] :\n    map_equiv F (equiv.refl \u03b1) (equiv.refl \u03b1') = equiv.refl (F \u03b1 \u03b1') :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/equiv/functor_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3579302264983903}}
{"text": "/-\nCopyright (c) 2020 Kenji Nakagawa. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio\n-/\nimport ring_theory.ideal.over\nimport ring_theory.polynomial.rational_root\n\n/-!\n# Dedekind domains\n\nThis file defines the notion of a Dedekind domain (or Dedekind ring),\nas a Noetherian integrally closed commutative ring of Krull dimension at most one.\n\n## Main definitions\n\n - `is_dedekind_domain` defines a Dedekind domain as a commutative ring that is\n   Noetherian, integrally closed in its field of fractions and has Krull dimension at most one.\n   `is_dedekind_domain_iff` shows that this does not depend on the choice of field of fractions.\n\n## Implementation notes\n\nThe definitions that involve a field of fractions choose a canonical field of fractions,\nbut are independent of that choice. The `..._iff` lemmas express this independence.\n\nOften, definitions assume that Dedekind domains are not fields. We found it more practical\nto add a `(h : \u00ac is_field A)` assumption whenever this is explicitly needed.\n\n## References\n\n* [D. Marcus, *Number Fields*][marcus1977number]\n* [J.W.S. Cassels, A. Fr\u00f6lich, *Algebraic Number Theory*][cassels1967algebraic]\n* [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]\n\n## Tags\n\ndedekind domain, dedekind ring\n-/\n\nvariables (R A K : Type*) [comm_ring R] [comm_ring A] [field K]\n\nopen_locale non_zero_divisors polynomial\n\n/-- A ring `R` has Krull dimension at most one if all nonzero prime ideals are maximal. -/\ndef ring.dimension_le_one : Prop :=\n\u2200 p \u2260 (\u22a5 : ideal R), p.is_prime \u2192 p.is_maximal\n\nopen ideal ring\n\nnamespace ring\n\nlemma dimension_le_one.principal_ideal_ring\n  [is_domain A] [is_principal_ideal_ring A] : dimension_le_one A :=\n\u03bb p nonzero prime, by { haveI := prime, exact is_prime.to_maximal_ideal nonzero }\n\nlemma dimension_le_one.is_integral_closure (B : Type*) [comm_ring B] [is_domain B]\n  [nontrivial R] [algebra R A] [algebra R B] [algebra B A] [is_scalar_tower R B A]\n  [is_integral_closure B R A] (h : dimension_le_one R) :\n  dimension_le_one B :=\n\u03bb p ne_bot prime, by exactI\n  is_integral_closure.is_maximal_of_is_maximal_comap A p\n    (h _ (is_integral_closure.comap_ne_bot A ne_bot) infer_instance)\n\nlemma dimension_le_one.integral_closure [nontrivial R] [is_domain A] [algebra R A]\n  (h : dimension_le_one R) : dimension_le_one (integral_closure R A) :=\nh.is_integral_closure R A (integral_closure R A)\n\nend ring\n\nvariables [is_domain A]\n\n/--\nA Dedekind domain is an integral domain that is Noetherian, integrally closed, and\nhas Krull dimension at most one.\n\nThis is definition 3.2 of [Neukirch1992].\n\nThe integral closure condition is independent of the choice of field of fractions:\nuse `is_dedekind_domain_iff` to prove `is_dedekind_domain` for a given `fraction_map`.\n\nThis is the default implementation, but there are equivalent definitions,\n`is_dedekind_domain_dvr` and `is_dedekind_domain_inv`.\nTODO: Prove that these are actually equivalent definitions.\n-/\nclass is_dedekind_domain : Prop :=\n(is_noetherian_ring : is_noetherian_ring A)\n(dimension_le_one : dimension_le_one A)\n(is_integrally_closed : is_integrally_closed A)\n\n-- See library note [lower instance priority]\nattribute [instance, priority 100]\n  is_dedekind_domain.is_noetherian_ring is_dedekind_domain.is_integrally_closed\n\n/-- An integral domain is a Dedekind domain iff and only if it is\nNoetherian, has dimension \u2264 1, and is integrally closed in a given fraction field.\nIn particular, this definition does not depend on the choice of this fraction field. -/\nlemma is_dedekind_domain_iff (K : Type*) [field K] [algebra A K] [is_fraction_ring A K] :\n  is_dedekind_domain A \u2194 is_noetherian_ring A \u2227 dimension_le_one A \u2227\n    (\u2200 {x : K}, is_integral A x \u2192 \u2203 y, algebra_map A K y = x) :=\n\u27e8\u03bb \u27e8hr, hd, hi\u27e9, \u27e8hr, hd, \u03bb x, (is_integrally_closed_iff K).mp hi\u27e9,\n \u03bb \u27e8hr, hd, hi\u27e9, \u27e8hr, hd, (is_integrally_closed_iff K).mpr @hi\u27e9\u27e9\n\n@[priority 100] -- See library note [lower instance priority]\ninstance is_principal_ideal_ring.is_dedekind_domain [is_principal_ideal_ring A] :\n  is_dedekind_domain A :=\n\u27e8principal_ideal_ring.is_noetherian_ring,\n ring.dimension_le_one.principal_ideal_ring A,\n unique_factorization_monoid.is_integrally_closed\u27e9\n", "meta": {"author": "lean-forward", "repo": "class-number-journal", "sha": "34d5872618d289ca3982bd9bc0c6e06af678909a", "save_path": "github-repos/lean/lean-forward-class-number-journal", "path": "github-repos/lean/lean-forward-class-number-journal/class-number-journal-34d5872618d289ca3982bd9bc0c6e06af678909a/src/dedekind_domain/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.35784623588251707}}
{"text": "/-\nCopyright (c) 2019 Robert A. Spencer. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert A. Spencer, Markus Himmel\n-/\nimport algebra.category.Group.basic\nimport category_theory.concrete_category\nimport category_theory.limits.shapes.kernels\nimport category_theory.linear\nimport linear_algebra.basic\n\n/-!\n# The category of `R`-modules\n\n`Module.{v} R` is the category of bundled `R`-modules with carrier in the universe `v`. We show\nthat it is preadditive and show that being an isomorphism, monomorphism and epimorphism is\nequivalent to being a linear equivalence, an injective linear map and a surjective linear map,\nrespectively.\n\n## Implementation details\n\nTo construct an object in the category of `R`-modules from a type `M` with an instance of the\n`module` typeclass, write `of R M`. There is a coercion in the other direction.\n\nSimilarly, there is a coercion from morphisms in `Module R` to linear maps.\n\nUnfortunately, Lean is not smart enough to see that, given an object `M : Module R`, the expression\n`of R M`, where we coerce `M` to the carrier type, is definitionally equal to `M` itself.\nThis means that to go the other direction, i.e., from linear maps/equivalences to (iso)morphisms\nin the category of `R`-modules, we have to take care not to inadvertently end up with an\n`of R M` where `M` is already an object. Hence, given `f : M \u2192\u2097[R] N`,\n* if `M N : Module R`, simply use `f`;\n* if `M : Module R` and `N` is an unbundled `R`-module, use `\u21bff` or `as_hom_left f`;\n* if `M` is an unbundled `R`-module and `N : Module R`, use `\u21bef` or `as_hom_right f`;\n* if `M` and `N` are unbundled `R`-modules, use `\u219ff` or `as_hom f`.\n\nSimilarly, given `f : M \u2243\u2097[R] N`, use `to_Module_iso`, `to_Module_iso'_left`, `to_Module_iso'_right`\nor `to_Module_iso'`, respectively.\n\nThe arrow notations are localized, so you may have to `open_locale Module` to use them. Note that\nthe notation for `as_hom_left` clashes with the notation used to promote functions between types to\nmorphisms in the category `Type`, so to avoid confusion, it is probably a good idea to avoid having\nthe locales `Module` and `category_theory.Type` open at the same time.\n\nIf you get an error when trying to apply a theorem and the `convert` tactic produces goals of the\nform `M = of R M`, then you probably used an incorrect variant of `as_hom` or `to_Module_iso`.\n\n-/\n\nopen category_theory\nopen category_theory.limits\nopen category_theory.limits.walking_parallel_pair\n\nuniverses v u\n\nvariables (R : Type u) [ring R]\n\n/-- The category of R-modules and their morphisms.\n\n Note that in the case of `R = \u2124`, we can not\nimpose here that the `\u2124`-multiplication field from the module structure is defeq to the one coming\nfrom the `is_add_comm_group` structure (contrary to what we do for all module structures in\nmathlib), which creates some difficulties down the road. -/\nstructure Module :=\n(carrier : Type v)\n[is_add_comm_group : add_comm_group carrier]\n[is_module : module R carrier]\n\nattribute [instance] Module.is_add_comm_group Module.is_module\n\nnamespace Module\n\ninstance : has_coe_to_sort (Module.{v} R) :=\n{ S := Type v, coe := Module.carrier }\n\ninstance Module_category : category (Module.{v} R) :=\n{ hom   := \u03bb M N, M \u2192\u2097[R] N,\n  id    := \u03bb M, 1,\n  comp  := \u03bb A B C f g, g.comp f }\n\ninstance Module_concrete_category : concrete_category.{v} (Module.{v} R) :=\n{ forget := { obj := \u03bb R, R, map := \u03bb R S f, (f : R \u2192 S) },\n  forget_faithful := { } }\n\ninstance has_forget_to_AddCommGroup : has_forget\u2082 (Module R) AddCommGroup :=\n{ forget\u2082 :=\n  { obj := \u03bb M, AddCommGroup.of M,\n    map := \u03bb M\u2081 M\u2082 f, linear_map.to_add_monoid_hom f } }\n\n/-- The object in the category of R-modules associated to an R-module -/\ndef of (X : Type v) [add_comm_group X] [module R X] : Module R := \u27e8X\u27e9\n\ninstance : has_zero (Module R) := \u27e8of R punit\u27e9\ninstance : inhabited (Module R) := \u27e80\u27e9\n\n@[simp]\nlemma coe_of (X : Type u) [add_comm_group X] [module R X] : (of R X : Type u) = X := rfl\n\nvariables {R}\n\n/-- Forgetting to the underlying type and then building the bundled object returns the original\nmodule. -/\n@[simps]\ndef of_self_iso (M : Module R) : Module.of R M \u2245 M :=\n{ hom := \ud835\udfd9 M, inv := \ud835\udfd9 M }\n\ninstance : subsingleton (of R punit) :=\nby { rw coe_of R punit, apply_instance }\n\ninstance : has_zero_object (Module.{v} R) :=\n{ zero := 0,\n  unique_to := \u03bb X,\n  { default := (0 : punit \u2192\u2097[R] X),\n    uniq := \u03bb _, linear_map.ext $ \u03bb x,\n      have h : x = 0, from dec_trivial,\n      by simp only [h, linear_map.map_zero]},\n  unique_from := \u03bb X,\n  { default := (0 : X \u2192\u2097[R] punit),\n    uniq := \u03bb _, linear_map.ext $ \u03bb x, dec_trivial } }\n\nvariables {R} {M N U : Module.{v} R}\n\n@[simp] lemma id_apply (m : M) : (\ud835\udfd9 M : M \u2192 M) m = m := rfl\n\n@[simp] lemma coe_comp (f : M \u27f6 N) (g : N \u27f6 U) :\n  ((f \u226b g) : M \u2192 U) = g \u2218 f := rfl\n\nlemma comp_def (f : M \u27f6 N) (g : N \u27f6 U) : f \u226b g = g.comp f := rfl\n\nend Module\n\nvariables {R}\nvariables {X\u2081 X\u2082 : Type v}\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom [add_comm_group X\u2081] [module R X\u2081] [add_comm_group X\u2082] [module R X\u2082] :\n  (X\u2081 \u2192\u2097[R] X\u2082) \u2192 (Module.of R X\u2081 \u27f6 Module.of R X\u2082) := id\n\nlocalized \"notation `\u219f` f : 1024 := Module.as_hom f\" in Module\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom_right [add_comm_group X\u2081] [module R X\u2081] {X\u2082 : Module.{v} R} :\n  (X\u2081 \u2192\u2097[R] X\u2082) \u2192 (Module.of R X\u2081 \u27f6 X\u2082) := id\n\nlocalized \"notation `\u21be` f : 1024 := Module.as_hom_right f\" in Module\n\n/-- Reinterpreting a linear map in the category of `R`-modules. -/\ndef Module.as_hom_left {X\u2081 : Module.{v} R} [add_comm_group X\u2082] [module R X\u2082] :\n  (X\u2081 \u2192\u2097[R] X\u2082) \u2192 (X\u2081 \u27f6 Module.of R X\u2082) := id\n\nlocalized \"notation `\u21bf` f : 1024 := Module.as_hom_left f\" in Module\n\n/-- Build an isomorphism in the category `Module R` from a `linear_equiv` between `module`s. -/\n@[simps]\ndef linear_equiv.to_Module_iso\n  {g\u2081 : add_comm_group X\u2081} {g\u2082 : add_comm_group X\u2082} {m\u2081 : module R X\u2081} {m\u2082 : module R X\u2082}\n  (e : X\u2081 \u2243\u2097[R] X\u2082) :\n  Module.of R X\u2081 \u2245 Module.of R X\u2082 :=\n{ hom := (e : X\u2081 \u2192\u2097[R] X\u2082),\n  inv := (e.symm : X\u2082 \u2192\u2097[R] X\u2081),\n  hom_inv_id' := begin ext, exact e.left_inv x, end,\n  inv_hom_id' := begin ext, exact e.right_inv x, end, }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso' {M N : Module.{v} R} (i : M \u2243\u2097[R] N) : M \u2245 N :=\n{ hom := i,\n  inv := i.symm,\n  hom_inv_id' := linear_map.ext $ \u03bb x, by simp,\n  inv_hom_id' := linear_map.ext $ \u03bb x, by simp }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso'_left {X\u2081 : Module.{v} R} {g\u2082 : add_comm_group X\u2082} {m\u2082 : module R X\u2082}\n  (e : X\u2081 \u2243\u2097[R] X\u2082) : X\u2081 \u2245 Module.of R X\u2082 :=\n{ hom := (e : X\u2081 \u2192\u2097[R] X\u2082),\n  inv := (e.symm : X\u2082 \u2192\u2097[R] X\u2081),\n  hom_inv_id' := linear_map.ext $ \u03bb x, by simp,\n  inv_hom_id' := linear_map.ext $ \u03bb x, by simp }\n\n/--\nBuild an isomorphism in the category `Module R` from a `linear_equiv` between `module`s.\n\nThis version is better than `linear_equiv_to_Module_iso` when applicable, because Lean can't see\n`Module.of R M` is defeq to `M` when `M : Module R`. -/\n@[simps]\ndef linear_equiv.to_Module_iso'_right {g\u2081 : add_comm_group X\u2081} {m\u2081 : module R X\u2081}\n  {X\u2082 : Module.{v} R} (e : X\u2081 \u2243\u2097[R] X\u2082) : Module.of R X\u2081 \u2245 X\u2082 :=\n{ hom := (e : X\u2081 \u2192\u2097[R] X\u2082),\n  inv := (e.symm : X\u2082 \u2192\u2097[R] X\u2081),\n  hom_inv_id' := linear_map.ext $ \u03bb x, by simp,\n  inv_hom_id' := linear_map.ext $ \u03bb x, by simp }\n\nnamespace category_theory.iso\n\n/-- Build a `linear_equiv` from an isomorphism in the category `Module R`. -/\n@[simps]\ndef to_linear_equiv {X Y : Module R} (i : X \u2245 Y) : X \u2243\u2097[R] Y :=\n{ to_fun    := i.hom,\n  inv_fun   := i.inv,\n  left_inv  := by tidy,\n  right_inv := by tidy,\n  map_add'  := by tidy,\n  map_smul' := by tidy, }.\n\nend category_theory.iso\n\n/-- linear equivalences between `module`s are the same as (isomorphic to) isomorphisms\nin `Module` -/\n@[simps]\ndef linear_equiv_iso_Module_iso {X Y : Type u} [add_comm_group X] [add_comm_group Y] [module R X]\n  [module R Y] :\n  (X \u2243\u2097[R] Y) \u2245 (Module.of R X \u2245 Module.of R Y) :=\n{ hom := \u03bb e, e.to_Module_iso,\n  inv := \u03bb i, i.to_linear_equiv, }\n\nnamespace Module\n\ninstance : preadditive (Module.{v} R) :=\n{ add_comp' := \u03bb P Q R f f' g,\n    show (f + f') \u226b g = f \u226b g + f' \u226b g, by { ext, simp },\n  comp_add' := \u03bb P Q R f g g',\n    show f \u226b (g + g') = f \u226b g + f \u226b g', by { ext, simp } }\n\nsection\nvariables {S : Type u} [comm_ring S]\n\ninstance : linear S (Module.{v} S) :=\n{ hom_module := \u03bb X Y, linear_map.module,\n  smul_comp' := by { intros, ext, simp },\n  comp_smul' := by { intros, ext, simp }, }\n\nend\n\nend Module\n\ninstance (M : Type u) [add_comm_group M] [module R M] : has_coe (submodule R M) (Module R) :=\n\u27e8 \u03bb N, Module.of R N \u27e9\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/algebra/category/Module/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.35772029823345686}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebra.module.ulift\n! leanprover-community/mathlib commit 68d1483e8a718ec63219f0e227ca3f0140361086\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Ring.Ulift\nimport Mathbin.Algebra.Module.Equiv\n\n/-!\n# `ulift` instances for module and multiplicative actions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for module, mul_action and related structures on `ulift` types.\n\n(Recall `ulift \u03b1` is just a \"copy\" of a type `\u03b1` in a higher universe.)\n\nWe also provide `ulift.module_equiv : ulift M \u2243\u2097[R] M`.\n\n-/\n\n\nnamespace ULift\n\nuniverse u v w\n\nvariable {R : Type u}\n\nvariable {M : Type v}\n\nvariable {N : Type w}\n\n#print ULift.smulLeft /-\n@[to_additive]\ninstance smulLeft [SMul R M] : SMul (ULift R) M :=\n  \u27e8fun s x => s.down \u2022 x\u27e9\n#align ulift.has_smul_left ULift.smulLeft\n#align ulift.has_vadd_left ULift.vaddLeft\n-/\n\n/- warning: ulift.smul_def -> ULift.smul_def is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} {M : Type.{u2}} [_inst_1 : SMul.{u1, u2} R M] (s : ULift.{u3, u1} R) (x : M), Eq.{succ u2} M (SMul.smul.{max u1 u3, u2} (ULift.{u3, u1} R) M (ULift.smulLeft.{u1, u2, u3} R M _inst_1) s x) (SMul.smul.{u1, u2} R M _inst_1 (ULift.down.{u3, u1} R s) x)\nbut is expected to have type\n  forall {R : Type.{u2}} {M : Type.{u3}} [_inst_1 : SMul.{u2, u3} R M] (s : ULift.{u1, u2} R) (x : M), Eq.{succ u3} M (HSMul.hSMul.{max u2 u1, u3, u3} (ULift.{u1, u2} R) M M (instHSMul.{max u2 u1, u3} (ULift.{u1, u2} R) M (ULift.smulLeft.{u2, u3, u1} R M _inst_1)) s x) (HSMul.hSMul.{u2, u3, u3} R M M (instHSMul.{u2, u3} R M _inst_1) (ULift.down.{u1, u2} R s) x)\nCase conversion may be inaccurate. Consider using '#align ulift.smul_def ULift.smul_def\u2093'. -/\n@[simp, to_additive]\ntheorem smul_def [SMul R M] (s : ULift R) (x : M) : s \u2022 x = s.down \u2022 x :=\n  rfl\n#align ulift.smul_def ULift.smul_def\n#align ulift.vadd_def ULift.vadd_def\n\n#print ULift.isScalarTower /-\ninstance isScalarTower [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] :\n    IsScalarTower (ULift R) M N :=\n  \u27e8fun x y z => show (x.down \u2022 y) \u2022 z = x.down \u2022 y \u2022 z from smul_assoc _ _ _\u27e9\n#align ulift.is_scalar_tower ULift.isScalarTower\n-/\n\n#print ULift.isScalarTower' /-\ninstance isScalarTower' [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] :\n    IsScalarTower R (ULift M) N :=\n  \u27e8fun x y z => show (x \u2022 y.down) \u2022 z = x \u2022 y.down \u2022 z from smul_assoc _ _ _\u27e9\n#align ulift.is_scalar_tower' ULift.isScalarTower'\n-/\n\n#print ULift.isScalarTower'' /-\ninstance isScalarTower'' [SMul R M] [SMul M N] [SMul R N] [IsScalarTower R M N] :\n    IsScalarTower R M (ULift N) :=\n  \u27e8fun x y z => show up ((x \u2022 y) \u2022 z.down) = \u27e8x \u2022 y \u2022 z.down\u27e9 by rw [smul_assoc]\u27e9\n#align ulift.is_scalar_tower'' ULift.isScalarTower''\n-/\n\ninstance [SMul R M] [SMul R\u1d50\u1d52\u1d56 M] [IsCentralScalar R M] : IsCentralScalar R (ULift M) :=\n  \u27e8fun r m => congr_arg up <| op_smul_eq_smul r m.down\u27e9\n\n#print ULift.mulAction /-\n@[to_additive]\ninstance mulAction [Monoid R] [MulAction R M] : MulAction (ULift R) M\n    where\n  smul := (\u00b7 \u2022 \u00b7)\n  mul_smul _ _ := mul_smul _ _\n  one_smul := one_smul _\n#align ulift.mul_action ULift.mulAction\n#align ulift.add_action ULift.addAction\n-/\n\n#print ULift.mulAction' /-\n@[to_additive]\ninstance mulAction' [Monoid R] [MulAction R M] : MulAction R (ULift M)\n    where\n  smul := (\u00b7 \u2022 \u00b7)\n  mul_smul := fun r s \u27e8f\u27e9 => ext _ _ <| mul_smul _ _ _\n  one_smul := fun \u27e8f\u27e9 => ext _ _ <| one_smul _ _\n#align ulift.mul_action' ULift.mulAction'\n#align ulift.add_action' ULift.addAction'\n-/\n\n#print ULift.smulZeroClass /-\ninstance smulZeroClass [Zero M] [SMulZeroClass R M] : SMulZeroClass (ULift R) M :=\n  { ULift.smulLeft with smul_zero := fun _ => smul_zero _ }\n#align ulift.smul_zero_class ULift.smulZeroClass\n-/\n\n#print ULift.smulZeroClass' /-\ninstance smulZeroClass' [Zero M] [SMulZeroClass R M] : SMulZeroClass R (ULift M)\n    where smul_zero c := by\n    ext\n    simp [smul_zero]\n#align ulift.smul_zero_class' ULift.smulZeroClass'\n-/\n\n#print ULift.distribSmul /-\ninstance distribSmul [AddZeroClass M] [DistribSMul R M] : DistribSMul (ULift R) M\n    where smul_add _ := smul_add _\n#align ulift.distrib_smul ULift.distribSmul\n-/\n\n#print ULift.distribSmul' /-\ninstance distribSmul' [AddZeroClass M] [DistribSMul R M] : DistribSMul R (ULift M)\n    where smul_add c f g := by\n    ext\n    simp [smul_add]\n#align ulift.distrib_smul' ULift.distribSmul'\n-/\n\n#print ULift.distribMulAction /-\ninstance distribMulAction [Monoid R] [AddMonoid M] [DistribMulAction R M] :\n    DistribMulAction (ULift R) M :=\n  { ULift.mulAction, ULift.distribSmul with }\n#align ulift.distrib_mul_action ULift.distribMulAction\n-/\n\n#print ULift.distribMulAction' /-\ninstance distribMulAction' [Monoid R] [AddMonoid M] [DistribMulAction R M] :\n    DistribMulAction R (ULift M) :=\n  { ULift.mulAction', ULift.distribSmul' with }\n#align ulift.distrib_mul_action' ULift.distribMulAction'\n-/\n\n#print ULift.mulDistribMulAction /-\ninstance mulDistribMulAction [Monoid R] [Monoid M] [MulDistribMulAction R M] :\n    MulDistribMulAction (ULift R) M\n    where\n  smul_one _ := smul_one _\n  smul_mul _ := smul_mul' _\n#align ulift.mul_distrib_mul_action ULift.mulDistribMulAction\n-/\n\n#print ULift.mulDistribMulAction' /-\ninstance mulDistribMulAction' [Monoid R] [Monoid M] [MulDistribMulAction R M] :\n    MulDistribMulAction R (ULift M) :=\n  {\n    ULift.mulAction' with\n    smul_one := fun _ => by\n      ext\n      simp [smul_one]\n    smul_mul := fun c f g => by\n      ext\n      simp [smul_mul'] }\n#align ulift.mul_distrib_mul_action' ULift.mulDistribMulAction'\n-/\n\n#print ULift.smulWithZero /-\ninstance smulWithZero [Zero R] [Zero M] [SMulWithZero R M] : SMulWithZero (ULift R) M :=\n  { ULift.smulLeft with\n    smul_zero := fun _ => smul_zero _\n    zero_smul := zero_smul _ }\n#align ulift.smul_with_zero ULift.smulWithZero\n-/\n\n#print ULift.smulWithZero' /-\ninstance smulWithZero' [Zero R] [Zero M] [SMulWithZero R M] : SMulWithZero R (ULift M)\n    where\n  smul_zero _ := ULift.ext _ _ <| smul_zero _\n  zero_smul _ := ULift.ext _ _ <| zero_smul _ _\n#align ulift.smul_with_zero' ULift.smulWithZero'\n-/\n\n#print ULift.mulActionWithZero /-\ninstance mulActionWithZero [MonoidWithZero R] [Zero M] [MulActionWithZero R M] :\n    MulActionWithZero (ULift R) M :=\n  { ULift.smulWithZero with }\n#align ulift.mul_action_with_zero ULift.mulActionWithZero\n-/\n\n#print ULift.mulActionWithZero' /-\ninstance mulActionWithZero' [MonoidWithZero R] [Zero M] [MulActionWithZero R M] :\n    MulActionWithZero R (ULift M) :=\n  { ULift.smulWithZero' with }\n#align ulift.mul_action_with_zero' ULift.mulActionWithZero'\n-/\n\n#print ULift.module /-\ninstance module [Semiring R] [AddCommMonoid M] [Module R M] : Module (ULift R) M :=\n  { ULift.smulWithZero with add_smul := fun _ _ => add_smul _ _ }\n#align ulift.module ULift.module\n-/\n\n#print ULift.module' /-\ninstance module' [Semiring R] [AddCommMonoid M] [Module R M] : Module R (ULift M) :=\n  { ULift.smulWithZero' with add_smul := fun _ _ _ => ULift.ext _ _ <| add_smul _ _ _ }\n#align ulift.module' ULift.module'\n-/\n\n#print ULift.moduleEquiv /-\n/-- The `R`-linear equivalence between `ulift M` and `M`.\n-/\n@[simps apply symm_apply]\ndef moduleEquiv [Semiring R] [AddCommMonoid M] [Module R M] : ULift M \u2243\u2097[R] M\n    where\n  toFun := ULift.down\n  invFun := ULift.up\n  map_smul' r x := rfl\n  map_add' x y := rfl\n  left_inv := by tidy\n  right_inv := by tidy\n#align ulift.module_equiv ULift.moduleEquiv\n-/\n\nend ULift\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Module/Ulift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3577202982334568}}
{"text": "import algebraic_geometry.morphisms.universally_closed\nimport algebraic_geometry.morphisms.quasi_compact\nimport ring_theory.valuation.valuation_ring\n\nnoncomputable theory\n\nopen category_theory \nopen category_theory.morphism_property\nopen algebraic_geometry.morphism_property (topologically)\nopen algebraic_geometry.Scheme\n\nuniverses u v\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\nstructure commutative_square :=\n/-\ntl ----> tr\n|         |\n|         |\nbl ----> br\n-/\n{tl tr bl br : C} -- t for top; b for bottom; l for left; r for right\n(v1 : tl \u27f6 bl) (v2 : tr \u27f6 br) (h1 : tl \u27f6 tr) (h2 : bl \u27f6 br)\n(commutes : v1 \u226b h2 = h1 \u226b v2 . obviously)\n\nvariables {C}\n\ndef commutative_square.is_diagnol' (S : commutative_square C) (d : S.bl \u27f6 S.tr) : Prop :=\n  S.v1 \u226b d = S.h1 \u2227 -- top triangle\n  d \u226b S.v2 = S.h2\n\ndef commutative_square.exists_diagnol' (S : commutative_square C) : Prop :=\n\u2203 (d : S.bl \u27f6 S.tr), S.is_diagnol' d\n\ndef commutative_square.diagnol'_unique (S : commutative_square C) : Prop :=\n\u2200 (d d' : S.bl \u27f6 S.tr), S.is_diagnol' d \u2192 S.is_diagnol' d' \u2192 d = d'\n\nend category_theory\n\nstructure valuative_pair :=\n(ring_ : Type u) (fraction_field_ : Type u)\n-- assumption on ring part\n[is_comm_ring : comm_ring ring_]\n[is_domain : is_domain ring_]\n[is_valuation_ring : valuation_ring ring_]\n-- assumption on fraction field part\n[is_field : field fraction_field_]\n[is_algebra : algebra ring_ fraction_field_]\n[is_fractional : is_fraction_ring ring_ fraction_field_]\n\nattribute [instance] \n  valuative_pair.is_comm_ring \n  valuative_pair.is_domain\n  valuative_pair.is_valuation_ring\n  valuative_pair.is_field\n  valuative_pair.is_algebra\n  valuative_pair.is_fractional\n\ninstance (P : valuative_pair) : algebra (CommRing.of P.ring_) (CommRing.of P.fraction_field_) := P.is_algebra\ninstance (P : valuative_pair) : is_domain (CommRing.of P.ring_) := P.is_domain\ninstance (P : valuative_pair) : valuation_ring (CommRing.of P.ring_) := P.is_valuation_ring\ninstance (P : valuative_pair) : local_ring (CommRing.of P.ring_) := valuation_ring.local_ring (CommRing.of P.ring_)\n\nnamespace Top\n\ndef specialization_lift_along : morphism_property Top.{u} :=\n\u03bb X Y g, \u2200 \u2983y' y : Y\u2984 (h : y' \u2933 y) (x' : X) (h' : g x' = y'), \u2203 (x : X) (hx : x' \u2933 x), g x = y\n\n-- **0064**\nlemma specialization_lift_along.comp {X Y Z : Top.{u}} (f : X \u27f6 Y) (g : Y \u27f6 Z) \n  (hf : specialization_lift_along f) (hg : specialization_lift_along g) :\n  specialization_lift_along $ f \u226b g :=\n\u03bb y' y h x' h', \nbegin \n  obtain \u27e8a, ha1, ha2\u27e9 := hg h (f x') h',\n  obtain \u27e8x, hx1, hx2\u27e9 := hf ha1 x' rfl,\n  refine \u27e8x, hx1, _\u27e9,\n  rw [comp_apply, hx2, ha2],\nend\n\n-- **0066 a)**\nlemma specialization_lift_along_of_closed {X Y : Top.{u}} (g : X \u27f6 Y) (hg : is_closed_map g) :\n  specialization_lift_along g :=\n\u03bb y' y hy x' hx', \nbegin \n  let T : set X := closure {x'},\n  have hT : is_closed T := is_closed_closure,\n  let S : set Y := g '' T,\n  have hS : is_closed S := hg _ hT,\n  have y'_mem : g x' \u2208 S := \u27e8x', subset_closure (set.mem_singleton _), rfl\u27e9,\n  rw hx' at y'_mem,\n  obtain \u27e8x, hx1, hx2\u27e9 := hy.mem_closed hS y'_mem,\n  exact \u27e8x, specializes_iff_mem_closure.mpr hx1, hx2\u27e9,\nend\n\nend Top\n\nnamespace algebraic_geometry\n\nvariables {X S : Scheme.{u}} (f : X \u27f6 S) \n\ndef specialization_lift_along : morphism_property Scheme.{u} :=\n\u03bb X Y g, Top.specialization_lift_along g.1.base\n\ndef universally_specialization_lift_along : morphism_property Scheme.{u} :=\nuniversally specialization_lift_along\n\n-- **01KC a)**\nlemma universally_specialization_lift_along_of_universally_closed [universally_closed f] :\n  universally_specialization_lift_along f :=\n\u03bb X' S' i\u2081 i\u2082 f' H, Top.specialization_lift_along_of_closed _ $ universally_closed.out _ _ _ H\n\n-- **01K9**\n-- Need reduced induced strucutre of scheme\n-- but this is fundamentally a topological question, so perhaps we don't have to be so fancy\nlemma closed_iff_specialization_lift_along_of_quasi_compact [quasi_compact f] :\n  is_closed_map f.1.base \u2194 specialization_lift_along f :=\n{ mp := \u03bb h, Top.specialization_lift_along_of_closed _ h,\n  mpr := sorry }\n\n-- **01KC b)**\nlemma universally_closed_of_quasicompact_and_universally_specialization_lift_along\n  [quasi_compact f] (h1 : universally_specialization_lift_along f) :\n  universally_closed f :=\n{ out := \u03bb X' Y' i\u2081 i\u2082 f' H, \n  begin \n    haveI : quasi_compact f' := quasi_compact_stable_under_base_change H.flip infer_instance,\n    refine (closed_iff_specialization_lift_along_of_quasi_compact _).mpr (h1 _ _ _ H),\n  end }\n\nlemma universally_closed_iff_universally_specialization_lift_along_of_quasi_compact [quasi_compact f] :\n  universally_closed f \u2194 universally_specialization_lift_along f :=\n\u27e8\u03bb i, by exactI universally_specialization_lift_along_of_universally_closed f, \n\u03bb i, by exactI universally_closed_of_quasicompact_and_universally_specialization_lift_along f i\u27e9\n\nalias category_theory.commutative_square \u2190 commutative_square\n\ndef valuative_pair.Scheme_square (P : valuative_pair) \u2983X S : Scheme.{u}\u2984 (f : X \u27f6 S) (h1 h2 commutes) : commutative_square Scheme.{u} :=\n{ v1 := Spec_map (algebra_map (CommRing.of P.ring_) (CommRing.of P.fraction_field_)),\n  v2 := f,\n  h1 := h1,\n  h2 := h2 }\n\ndef valuative_pair.closed_point (P : valuative_pair) : (Spec_obj (CommRing.of P.ring_)).carrier :=\n\u27e8_, ideal.is_maximal.is_prime' (local_ring.maximal_ideal _)\u27e9\n\nlemma valuative_pair.closed_point_is_maximal_ideal (P : valuative_pair) :\n  ideal.is_maximal P.closed_point.as_ideal :=\nlocal_ring.maximal_ideal.is_maximal _\n\nlemma valuative_pair.closed_point_is_closed (P : valuative_pair) : is_closed ({P.closed_point} : set $ (Spec_obj (CommRing.of P.ring_)).carrier) := \nbegin \n  rw prime_spectrum.is_closed_iff_zero_locus,\n  refine \u27e8P.closed_point.as_ideal, _\u27e9,\n  rw \u2190prime_spectrum.closure_singleton,\n  ext, simp only [set.mem_singleton_iff], split,\n  { rintros rfl, refine subset_closure (set.mem_singleton _), },\n  { intros h, rw \u2190prime_spectrum.le_iff_mem_closure at h,\n    ext1, symmetry,\n    refine ideal.is_maximal.eq_of_le P.closed_point_is_maximal_ideal x.is_prime.ne_top h, },\nend\n\nlemma valuative_pair.closed_point_unique \n  (P : valuative_pair) {x : (Spec_obj (CommRing.of P.ring_)).carrier} \n  (hx : is_closed ({x} : set $ (Spec_obj (CommRing.of P.ring_)).carrier)) :\n  x = P.closed_point :=\nbegin \n  rw prime_spectrum.is_closed_singleton_iff_is_maximal x at hx,\n  ext : 1,\n  convert local_ring.eq_maximal_ideal hx,\nend\n\ndef valuative_pair.generic_point (P : valuative_pair) : (Spec_obj (CommRing.of P.ring_)).carrier :=\n\u27e8\u22a5 , ideal.bot_prime\u27e9\n\nlemma valuative_pair.generic_point_is_generic (P : valuative_pair) : is_generic_point P.generic_point set.univ :=\nbegin \n  rw [is_generic_point_def, set.eq_univ_iff_forall],\n  intros x,\n  rw \u2190prime_spectrum.le_iff_mem_closure,\n  exact bot_le,\nend\n\nlemma valuative_pair.generic_point_unique \n  (P : valuative_pair) {x : (Spec_obj (CommRing.of P.ring_)).carrier}\n  (hx : is_generic_point x set.univ) : x = P.generic_point :=\nbegin \n  have h := hx.trans P.generic_point_is_generic.symm,\n  have h1 := (prime_spectrum.le_iff_mem_closure x P.generic_point).mpr (h.symm \u25b8 subset_closure (set.mem_singleton _)),\n  have h2 := (prime_spectrum.le_iff_mem_closure P.generic_point x).mpr (h \u25b8 subset_closure (set.mem_singleton _)),\n  ext1,\n  refine le_antisymm h1 h2,\nend\n\n-- **01J8 1)** but stacks says more\nlemma morphisms_from_specialization {S : Scheme.{u}} {s' s : S.carrier} (hs : s' \u2933 s) :\n  \u2203 (P : valuative_pair) (f : Spec_obj (CommRing.of P.ring_) \u27f6 S), \n    (f : Spec_obj (CommRing.of P.ring_) \u27f6 S).1.base P.generic_point = s' \u2227\n    (f : Spec_obj (CommRing.of P.ring_) \u27f6 S).1.base P.closed_point = s := sorry\n\n-- **01KD**\ndef valuative_criterion_existence_part : morphism_property Scheme.{u} :=\n\u03bb X S f, \u2200 (P : valuative_pair) (h1 h2) (commutes), \n  (P.Scheme_square f h1 h2 commutes).exists_diagnol'\n\n-- **01KD**\ndef valuative_criterion_uniqueness_part : morphism_property Scheme.{u} :=\n\u03bb X S f, \u2200 (P : valuative_pair) (h1 h2) (commutes), (P.Scheme_square f h1 h2 commutes).diagnol'_unique\n\n-- **00KE**\nlemma universally_specialization_lift_along_iff_valuative_criterion_existence_part :\n  universally_specialization_lift_along f \u2194 valuative_criterion_existence_part f :=\nsorry\n\n-- **01KF**\nlemma valuative_criterion_for_universally_closedness [quasi_compact f] :\n  universally_closed f \u2194 valuative_criterion_existence_part f :=\n(universally_closed_iff_universally_specialization_lift_along_of_quasi_compact f).trans $\nuniversally_specialization_lift_along_iff_valuative_criterion_existence_part f\n\nend algebraic_geometry", "meta": {"author": "jjaassoonn", "repo": "vc", "sha": "3f1db37cfd16b55cd47e77f0517d192c25edecf0", "save_path": "github-repos/lean/jjaassoonn-vc", "path": "github-repos/lean/jjaassoonn-vc/vc-3f1db37cfd16b55cd47e77f0517d192c25edecf0/src/valuative_universally_closed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.5, "lm_q1q2_score": 0.35771199789173663}}
{"text": "import algebra.group data.equiv algebra.module \n\n#check is_group_hom \n#print equiv.Pi_congr_right\n\n#print out_param \n\n#print module \n\n\nuniverse u\ndefinition Pi_lift_map\u2081 {\u03b3 : Type u} {F : \u03b3 \u2192 Type u} {G : \u03b3 \u2192 Type u} \n  (H : \u2200 i : \u03b3, F i \u2192 G i) : (\u03a0 i, F i) \u2192 \u03a0 i, G i := \u03bb Fi i, H i (Fi i)\n  \nclass foomap {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) :=\n(preserves_structure : \u2200 a : \u03b1, f a = f a)\n\ninstance Pi_foomap_is_foomap {\u03b3 : Type u} {F : \u03b3 \u2192 Type u} {G : \u03b3 \u2192 Type u} \n(H : \u2200 i : \u03b3, F i \u2192 G i) [\u2200 i, foomap (H i)] : foomap (Pi_lift_map\u2081 H) := sorry\n\nexample {\u03b3 : Type u} {F : \u03b3 \u2192 Type u} {G : \u03b3 \u2192 Type u} \n(H : \u2200 i : \u03b3, F i \u2192 G i) [\u2200 i, foomap (H i)] : foomap (Pi_lift_map\u2081 H) := by apply_instance\n\nexample {\u03b3 : Type u} {F : \u03b3 \u2192 Type u} {G : \u03b3 \u2192 Type u} \n(H : \u2200 i : \u03b3, F i \u2192 G i) [\u2200 i, foomap (H i)] : (Pi_lift_map\u2081 H) = (\u03bb Fi i, H i (Fi i) : (\u03a0 i, F i) \u2192 \u03a0 i, G i) := rfl\n\nexample {\u03b3 : Type u} {F : \u03b3 \u2192 Type u} {G : \u03b3 \u2192 Type u} \n(H : \u2200 i : \u03b3, F i \u2192 G i) [\u2200 i, foomap (H i)] : foomap (\u03bb Fi i, H i (Fi i) : (\u03a0 i, F i) \u2192 \u03a0 i, G i) := by apply_instance -- fails\n\n#check module\n", "meta": {"author": "kbuzzard", "repo": "lean-stacks-project", "sha": "b57be17aa917f1c3a23c59db5ee37b1aa21112c2", "save_path": "github-repos/lean/kbuzzard-lean-stacks-project", "path": "github-repos/lean/kbuzzard-lean-stacks-project/lean-stacks-project-b57be17aa917f1c3a23c59db5ee37b1aa21112c2/scratch/should_go_to_mathlib_patrick_prod_stuff.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7154239836484144, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3577119918242072}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.natural_isomorphism\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.data.sigma.basic\nimport Mathlib.category_theory.pi.basic\nimport Mathlib.PostPort\n\nuniverses w\u2081 v\u2081 u\u2081 l u\u2082 v\u2082 w\u2082 w\u2083 \n\nnamespace Mathlib\n\n/-!\n# Disjoint union of categories\n\nWe define the category structure on a sigma-type (disjoint union) of categories.\n-/\n\nnamespace category_theory\n\n\nnamespace sigma\n\n\n/--\nThe type of morphisms of a disjoint union of categories: for `X : C i` and `Y : C j`, a morphism\n`(i, X) \u27f6 (j, Y)` if `i = j` is just a morphism `X \u27f6 Y`, and if `i \u2260 j` there are no such morphisms.\n-/\ninductive sigma_hom {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] :\n    (sigma fun (i : I) => C i) \u2192 (sigma fun (i : I) => C i) \u2192 Type (max w\u2081 v\u2081 u\u2081)\n    where\n| mk : {i : I} \u2192 {X Y : C i} \u2192 (X \u27f6 Y) \u2192 sigma_hom (sigma.mk i X) (sigma.mk i Y)\n\nnamespace sigma_hom\n\n\n/-- The identity morphism on an object. -/\ndef id {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] (X : sigma fun (i : I) => C i) :\n    sigma_hom X X :=\n  sorry\n\nprotected instance inhabited {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    (X : sigma fun (i : I) => C i) : Inhabited (sigma_hom X X) :=\n  { default := id X }\n\n/-- Composition of sigma homomorphisms. -/\ndef comp {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {X : sigma fun (i : I) => C i}\n    {Y : sigma fun (i : I) => C i} {Z : sigma fun (i : I) => C i} :\n    sigma_hom X Y \u2192 sigma_hom Y Z \u2192 sigma_hom X Z :=\n  sorry\n\nprotected instance sigma.category_theory.category_struct {I : Type w\u2081} {C : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (C i)] : category_struct (sigma fun (i : I) => C i) :=\n  category_struct.mk id fun (X Y Z : sigma fun (i : I) => C i) (f : X \u27f6 Y) (g : Y \u27f6 Z) => comp f g\n\n@[simp] theorem comp_def {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] (i : I)\n    (X : C i) (Y : C i) (Z : C i) (f : X \u27f6 Y) (g : Y \u27f6 Z) : comp (mk f) (mk g) = mk (f \u226b g) :=\n  rfl\n\ntheorem assoc {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (Z : sigma fun (i : I) => C i)\n    (W : sigma fun (i : I) => C i) (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : Z \u27f6 W) : (f \u226b g) \u226b h = f \u226b g \u226b h :=\n  sorry\n\ntheorem id_comp {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (f : X \u27f6 Y) : \ud835\udfd9 \u226b f = f :=\n  sorry\n\ntheorem comp_id {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) (f : X \u27f6 Y) : f \u226b \ud835\udfd9 = f :=\n  sorry\n\nend sigma_hom\n\n\nprotected instance sigma {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] :\n    category (sigma fun (i : I) => C i) :=\n  category.mk\n\n/-- The inclusion functor into the disjoint union of categories. -/\n@[simp] theorem incl_map {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] (i : I)\n    (X : C i) (Y : C i) : \u2200 (\u1fb0 : X \u27f6 Y), functor.map (incl i) \u1fb0 = sigma_hom.mk \u1fb0 :=\n  fun (\u1fb0 : X \u27f6 Y) => Eq.refl (functor.map (incl i) \u1fb0)\n\n@[simp] theorem incl_obj {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {i : I}\n    (X : C i) : functor.obj (incl i) X = sigma.mk i X :=\n  rfl\n\nprotected instance incl.category_theory.full {I : Type w\u2081} {C : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (C i)] (i : I) : full (incl i) :=\n  full.mk fun (X Y : C i) (_x : functor.obj (incl i) X \u27f6 functor.obj (incl i) Y) => sorry\n\nprotected instance incl.category_theory.faithful {I : Type w\u2081} {C : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (C i)] (i : I) : faithful (incl i) :=\n  faithful.mk\n\n/--\nTo build a natural transformation over the sigma category, it suffices to specify it restricted to\neach subcategory.\n-/\ndef nat_trans {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082} [category D]\n    {F : (sigma fun (i : I) => C i) \u2964 D} {G : (sigma fun (i : I) => C i) \u2964 D}\n    (h : (i : I) \u2192 incl i \u22d9 F \u27f6 incl i \u22d9 G) : F \u27f6 G :=\n  nat_trans.mk fun (_x : sigma fun (i : I) => C i) => sorry\n\n@[simp] theorem nat_trans_app {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    {D : Type u\u2082} [category D] {F : (sigma fun (i : I) => C i) \u2964 D}\n    {G : (sigma fun (i : I) => C i) \u2964 D} (h : (i : I) \u2192 incl i \u22d9 F \u27f6 incl i \u22d9 G) (i : I) (X : C i) :\n    nat_trans.app (nat_trans h) (sigma.mk i X) = nat_trans.app (h i) X :=\n  rfl\n\n/-- (Implementation). An auxiliary definition to build the functor `desc`. -/\ndef desc_map {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082} [category D]\n    (F : (i : I) \u2192 C i \u2964 D) (X : sigma fun (i : I) => C i) (Y : sigma fun (i : I) => C i) :\n    (X \u27f6 Y) \u2192\n        (functor.obj (F (sigma.fst X)) (sigma.snd X) \u27f6\n          functor.obj (F (sigma.fst Y)) (sigma.snd Y)) :=\n  sorry\n\n/--\nGiven a collection of functors `F i : C i \u2964 D`, we can produce a functor `(\u03a3 i, C i) \u2964 D`.\n\nThe produced functor `desc F` satisfies: `incl i \u22d9 desc F \u2245 F i`, i.e. restricted to just the\nsubcategory `C i`, `desc F` agrees with `F i`, and it is unique (up to natural isomorphism) with\nthis property.\n\nThis witnesses that the sigma-type is the coproduct in Cat.\n-/\n@[simp] theorem desc_obj {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082}\n    [category D] (F : (i : I) \u2192 C i \u2964 D) (X : sigma fun (i : I) => C i) :\n    functor.obj (desc F) X = functor.obj (F (sigma.fst X)) (sigma.snd X) :=\n  Eq.refl (functor.obj (desc F) X)\n\n@[simp] theorem desc_map_mk {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082}\n    [category D] (F : (i : I) \u2192 C i \u2964 D) {i : I} (X : C i) (Y : C i) (f : X \u27f6 Y) :\n    functor.map (desc F) (sigma_hom.mk f) = functor.map (F i) f :=\n  rfl\n\n/--\nThis shows that when `desc F` is restricted to just the subcategory `C i`, `desc F` agrees with\n`F i`.\n-/\n-- We hand-generate the simp lemmas about this since they come out cleaner.\n\ndef incl_desc {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082} [category D]\n    (F : (i : I) \u2192 C i \u2964 D) (i : I) : incl i \u22d9 desc F \u2245 F i :=\n  nat_iso.of_components (fun (X : C i) => iso.refl (functor.obj (incl i \u22d9 desc F) X)) sorry\n\n@[simp] theorem incl_desc_hom_app {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    {D : Type u\u2082} [category D] (F : (i : I) \u2192 C i \u2964 D) (i : I) (X : C i) :\n    nat_trans.app (iso.hom (incl_desc F i)) X = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem incl_desc_inv_app {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    {D : Type u\u2082} [category D] (F : (i : I) \u2192 C i \u2964 D) (i : I) (X : C i) :\n    nat_trans.app (iso.inv (incl_desc F i)) X = \ud835\udfd9 :=\n  rfl\n\n/--\nIf `q` when restricted to each subcategory `C i` agrees with `F i`, then `q` is isomorphic to\n`desc F`.\n-/\ndef desc_uniq {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082} [category D]\n    (F : (i : I) \u2192 C i \u2964 D) (q : (sigma fun (i : I) => C i) \u2964 D) (h : (i : I) \u2192 incl i \u22d9 q \u2245 F i) :\n    q \u2245 desc F :=\n  nat_iso.of_components (fun (_x : sigma fun (i : I) => C i) => sorry) sorry\n\n@[simp] theorem desc_uniq_hom_app {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    {D : Type u\u2082} [category D] (F : (i : I) \u2192 C i \u2964 D) (q : (sigma fun (i : I) => C i) \u2964 D)\n    (h : (i : I) \u2192 incl i \u22d9 q \u2245 F i) (i : I) (X : C i) :\n    nat_trans.app (iso.hom (desc_uniq F q h)) (sigma.mk i X) = nat_trans.app (iso.hom (h i)) X :=\n  rfl\n\n@[simp] theorem desc_uniq_inv_app {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)]\n    {D : Type u\u2082} [category D] (F : (i : I) \u2192 C i \u2964 D) (q : (sigma fun (i : I) => C i) \u2964 D)\n    (h : (i : I) \u2192 incl i \u22d9 q \u2245 F i) (i : I) (X : C i) :\n    nat_trans.app (iso.inv (desc_uniq F q h)) (sigma.mk i X) = nat_trans.app (iso.inv (h i)) X :=\n  rfl\n\n/--\nIf `q\u2081` and `q\u2082` when restricted to each subcategory `C i` agree, then `q\u2081` and `q\u2082` are isomorphic.\n-/\n@[simp] theorem nat_iso_inv {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : Type u\u2082}\n    [category D] {q\u2081 : (sigma fun (i : I) => C i) \u2964 D} {q\u2082 : (sigma fun (i : I) => C i) \u2964 D}\n    (h : (i : I) \u2192 incl i \u22d9 q\u2081 \u2245 incl i \u22d9 q\u2082) :\n    iso.inv (nat_iso h) = nat_trans fun (i : I) => iso.inv (h i) :=\n  Eq.refl (iso.inv (nat_iso h))\n\n/-- A function `J \u2192 I` induces a functor `\u03a3 j, C (g j) \u2964 \u03a3 i, C i`. -/\ndef map {I : Type w\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2082} (g : J \u2192 I) :\n    (sigma fun (j : J) => C (g j)) \u2964 sigma fun (i : I) => C i :=\n  desc fun (j : J) => incl (g j)\n\n@[simp] theorem map_obj {I : Type w\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2082}\n    (g : J \u2192 I) (j : J) (X : C (g j)) : functor.obj (map C g) (sigma.mk j X) = sigma.mk (g j) X :=\n  rfl\n\n@[simp] theorem map_map {I : Type w\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2082}\n    (g : J \u2192 I) {j : J} {X : C (g j)} {Y : C (g j)} (f : X \u27f6 Y) :\n    functor.map (map C g) (sigma_hom.mk f) = sigma_hom.mk f :=\n  rfl\n\n/--\nThe functor `sigma.map C g` restricted to the subcategory `C j` acts as the inclusion of `g j`.\n-/\n@[simp] theorem incl_comp_map_hom_app {I : Type w\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    {J : Type w\u2082} (g : J \u2192 I) (j : J) (X : C (g j)) :\n    nat_trans.app (iso.hom (incl_comp_map C g j)) X = \ud835\udfd9 :=\n  Eq.refl \ud835\udfd9\n\n/-- The functor `sigma.map` applied to the identity function is just the identity functor. -/\n@[simp] theorem map_id_hom_app (I : Type w\u2081) (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    (_x : sigma fun (i : I) => (fun (i : I) => (fun (i : I) => C (id i)) i) i) :\n    nat_trans.app (iso.hom (map_id I C)) _x =\n        nat_trans._match_1\n          (fun (i : I) =>\n            iso.hom\n              (nat_iso.of_components (fun (X : C i) => iso.refl (sigma.mk i X))\n                (map_id._proof_1 I C i)))\n          _x :=\n  sorry\n\n/-- The functor `sigma.map` applied to a composition is a composition of functors. -/\n@[simp] theorem map_comp_hom_app {I : Type w\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    {J : Type w\u2082} {K : Type w\u2083} (f : K \u2192 J) (g : J \u2192 I)\n    (X : sigma fun (i : K) => (fun (j : K) => function.comp C g (f j)) i) :\n    nat_trans.app (iso.hom (map_comp C f g)) X =\n        iso.hom\n          (desc_uniq._match_1 (fun (j : K) => incl (g (f j))) (map (C \u2218 g) f \u22d9 map C g)\n            (fun (k : K) =>\n              iso_whisker_right (incl_comp_map (C \u2218 g) f k) (map C g) \u226a\u226b incl_comp_map C g (f k))\n            X) :=\n  sorry\n\nnamespace functor\n\n\n/--\nAssemble an `I`-indexed family of functors into a functor between the sigma types.\n-/\ndef sigma {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (D i)] (F : (i : I) \u2192 C i \u2964 D i) :\n    (sigma fun (i : I) => C i) \u2964 sigma fun (i : I) => D i :=\n  desc fun (i : I) => F i \u22d9 incl i\n\nend functor\n\n\nnamespace nat_trans\n\n\n/--\nAssemble an `I`-indexed family of natural transformations into a single natural transformation.\n-/\ndef sigma {I : Type w\u2081} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (D i)] {F : (i : I) \u2192 C i \u2964 D i} {G : (i : I) \u2192 C i \u2964 D i}\n    (\u03b1 : (i : I) \u2192 F i \u27f6 G i) : functor.sigma F \u27f6 functor.sigma G :=\n  nat_trans.mk\n    fun (f : sigma fun (i : I) => C i) =>\n      sigma_hom.mk (nat_trans.app (\u03b1 (sigma.fst f)) (sigma.snd f))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/sigma/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3576842989577495}}
{"text": "example (P : Prop) : \u2200 {p : P}, P := by\n  exact fun {p} => p\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  intro h; exact h\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  exact @id _\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  exact noImplicitLambda% id\n\nmacro \"exact'\" x:term : tactic => `(exact noImplicitLambda% $x)\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  exact' id\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  apply id\n\nexample (P : Prop) : \u2200 p : P, P := by\n  have : _ := 1\n  apply id\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  refine noImplicitLambda% (have : _ := 1; ?_)\n  apply id\n\nexample (P : Prop) : \u2200 {p : P}, P := by\n  have : _ := 1\n  apply id\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/lean/run/impLambdaTac.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.3573428415537895}}
{"text": "import category_theory.triangulated.pretriangulated\nimport category_theory.abelian.exact\nimport category_theory.linear.yoneda\nimport algebra.category.Module.abelian\nimport algebra.category.Group.abelian\nimport category_theory.functor.currying\nimport for_mathlib.exact_seq\nimport for_mathlib.preadditive_yoneda\nimport for_mathlib.AddCommGroup.exact\n\nimport category_theory.abelian.diagram_lemmas.four\n\nnamespace category_theory.triangulated\n\nopen category_theory\nopen category_theory.limits\nopen pretriangulated\n\nuniverses v u\nvariables {C : Type u} [category.{v} C] [preadditive C]\n\n-- Move me\ninstance preadditive_yoneda_flip_additive (X : C) :\n  (preadditive_yoneda.flip.obj (opposite.op X)).additive :=\nby { fsplit, dsimp, intros, ext1, apply preadditive.comp_add }\n\nvariables {R : Type*} [ring R] [linear R C]\n\n-- Move me\ninstance linear_yoneda_flip_additive (X : C) :\n  ((linear_yoneda R C).flip.obj (opposite.op X)).additive :=\nby { fsplit, dsimp, intros, ext1, dsimp, apply preadditive.comp_add }\n\nvariables [has_zero_object C] [has_shift C \u2124] [\u2200 (n : \u2124), (shift_functor C n).additive]\n  [pretriangulated C]\n\n/-- A functor `F` is a *homological* functor if for every distinguished triangle\n`A \u27f6 B \u27f6 C \u27f6 A[1]` the sequence `F(A) \u27f6 F(B) \u27f6 F(C)` is exact. -/\nclass homological_functor {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive] : Prop :=\n(cond [] : \u2200 (T : triangle C) (hT : T \u2208 dist_triang C), exact (F.map T.mor\u2081) (F.map T.mor\u2082))\n\nlemma four_term_exact_seq {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  [homological_functor F] (T : triangle C) (hT : T \u2208 dist_triang C):\n  exact_seq A [F.map T.mor\u2081, F.map T.mor\u2082, F.map T.mor\u2083] :=\nbegin\n  apply exact_seq.cons,\n  apply homological_functor.cond F _ hT,\n  rw \u2190 exact_iff_exact_seq,\n  apply homological_functor.cond F _ ((rotate_distinguished_triangle T).mp hT),\nend\n\nlemma five_term_exact_seq {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  [homological_functor F] (T : triangle C) (hT : T \u2208 dist_triang C):\n  exact_seq A [F.map T.inv_rotate.mor\u2081, F.map T.mor\u2081, F.map T.mor\u2082, F.map T.mor\u2083] :=\nbegin\n  apply exact_seq.cons _ _ _ _ (four_term_exact_seq F _ hT),\n  apply homological_functor.cond F,\n  apply inv_rot_of_dist_triangle _ _ hT,\nend\n\nlemma five_term_exact_seq' {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  [homological_functor F] (T : triangle C) (hT : T \u2208 dist_triang C) :\n  exact_seq A [F.map T.mor\u2081, F.map T.mor\u2082, F.map T.mor\u2083, F.map (T.rotate.mor\u2083)] :=\nbegin\n  apply exact_seq.cons,\n  apply homological_functor.cond F T hT,\n  exact four_term_exact_seq F T.rotate (rot_of_dist_triangle _ _ hT),\nend\n\nlemma complete_distinguished_triangle_morphism'\n  (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 dist_triang C)\n  (h\u2082 : T\u2082 \u2208 dist_triang C)\n  (f\u2082 : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082)\n  (f\u2083 : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083)\n  (w : T\u2081.mor\u2082 \u226b f\u2083 = f\u2082 \u226b T\u2082.mor\u2082) :\n  \u2203 f\u2081 : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081, (T\u2081.mor\u2081 \u226b f\u2082 = f\u2081 \u226b T\u2082.mor\u2081) \u2227 (T\u2081.mor\u2083 \u226b f\u2081\u27e61\u27e7' = f\u2083 \u226b T\u2082.mor\u2083) :=\nbegin\n  let T\u2081' := T\u2081.rotate,\n  let T\u2082' := T\u2082.rotate,\n  obtain \u27e8g,h1,h2\u27e9 := complete_distinguished_triangle_morphism T\u2081' T\u2082' _ _ f\u2082 f\u2083 w,\n  use (shift_shift_neg _ _).inv \u226b g\u27e6(-1 : \u2124)\u27e7' \u226b (shift_shift_neg _ _).hom,\n  split,\n  { dsimp at h2,\n    apply_fun (\u03bb e, - (shift_functor C (-1 : \u2124)).map e \u226b (shift_shift_neg _ _).hom) at h2,\n    simp only [category.assoc, functor.map_comp] at \u22a2 h2,\n    rw iso.eq_inv_comp,\n    convert h2 using 1,\n    { simp },\n    { simp } },\n  { convert h1 using 1,\n    congr' 1,\n    simp only [functor.map_comp, category.assoc],\n    rw shift_neg_shift',\n    simp only [category.assoc, \u2190 functor.map_iso_inv, \u2190 functor.map_iso_hom],\n    have : (shift_functor C 1).map_iso (shift_shift_neg T\u2082.obj\u2081 1) =\n      shift_neg_shift T\u2082'.obj\u2083 1, by { dsimp, simp }, rw this, clear this,\n    have : (shift_functor C 1).map_iso (shift_shift_neg T\u2081.obj\u2081 (1 : \u2124)) =\n      shift_neg_shift ((shift_functor C (1 : \u2124)).obj T\u2081.obj\u2081) (1 : \u2124),\n      by simp, rw this, clear this,\n    simp only [iso.inv_hom_id, iso.inv_hom_id_assoc, category.id_comp, category.comp_id] },\n  { rw \u2190 rotate_distinguished_triangle, exact h\u2081 },\n  { rw \u2190 rotate_distinguished_triangle, exact h\u2082 }\nend\n\ntheorem dist_triang_to_exact_complex\n  (T : triangle C)\n  (hT : T \u2208 dist_triang C)\n  (X : C)\n  (f : X \u27f6 T.obj\u2082)\n  (hf : f \u226b T.mor\u2082 = 0) :\n  \u2203 g : X \u27f6 T.obj\u2081, g \u226b T.mor\u2081 = f :=\nbegin\n  let I : triangle C := contractible_triangle C X,\n  obtain \u27e8f\u2081,h\u2081,h\u2082\u27e9 :=\n    complete_distinguished_triangle_morphism' I T\n    (contractible_distinguished _) hT f 0\n    (by simpa using hf.symm),\n  use f\u2081,\n  rw \u2190 h\u2081,\n  dsimp,\n  simp,\nend\n\n/-- The functor `Y \u21a6 Hom(X,Y)` is homological. -/\ninstance preadditive_yoneda_flip_homological (X : C) :\n  homological_functor (preadditive_yoneda.flip.obj (opposite.op X)) :=\nbegin\n  constructor,\n  intros T hT,\n  suffices : add_monoid_hom.range ((preadditive_yoneda.flip.obj (opposite.op X)).map T.mor\u2081) =\n    add_monoid_hom.ker ((preadditive_yoneda.flip.obj (opposite.op X)).map T.mor\u2082),\n  { rwa AddCommGroup.exact_iff },\n  apply le_antisymm,\n  { rintros _ \u27e8(g : X \u27f6 _),rfl\u27e9,\n    dsimp,\n    obtain \u27e8e,h1,he\u27e9 := complete_distinguished_triangle_morphism\n      (contractible_triangle _ X) T (contractible_distinguished _) hT g (g \u226b T.mor\u2081)\n      (by { dsimp, simp }),\n    dsimp at he,\n    simp only [zero_comp] at he,\n    change _ = _,\n    simp [\u2190 h1] },\n  { rintros (f : X \u27f6 _) (hf : f \u226b _ = 0),\n    apply dist_triang_to_exact_complex _ hT _ _ hf }\nend\n\n/-- The functor `Y \u21a6 Hom(Y,X)` is homological. -/\ninstance preadditive_yoneda_op_homological (X : C) :\n  homological_functor (preadditive_yoneda.obj X).right_op :=\nbegin\n  constructor,\n  intros T hT,\n  dsimp,\n  let f := _, let g := _, show exact f g,\n  suffices : exact g.unop f.unop,\n  { rw \u2190 f.op_unop, rw \u2190 g.op_unop, apply this.op, },\n  rw AddCommGroup.exact_iff,\n  apply le_antisymm,\n  { rintros _ \u27e8q : _ \u27f6 X, rfl\u27e9,\n    change _ \u226b _ = 0,\n    dsimp,\n    have := complete_distinguished_triangle_morphism\n      T.rotate (contractible_triangle _ X) _\n      (contractible_distinguished _) (T.mor\u2082 \u226b q) q (by { dsimp, simp }),\n    swap, rwa \u2190 rotate_distinguished_triangle,\n    obtain \u27e8c,h1,h2\u27e9 := this,\n    dsimp at h2,\n    simp at h2,\n    simp only [\u2190 functor.map_comp] at h2,\n    rw \u2190 (shift_functor C (1 : \u2124)).map_zero at h2,\n    exact (shift_functor C (1 : \u2124)).map_injective h2 },\n  { rintros (q : _ \u27f6 X) (hq : _ \u226b _ = 0),\n    change \u2203 p, _,\n    dsimp at hq \u22a2,\n    obtain \u27e8c,h1,h2\u27e9 := complete_distinguished_triangle_morphism\n      T (contractible_triangle _ X).inv_rotate hT _ 0 q (by simp [hq]),\n    { use c,\n      simp only [h1, triangle.inv_rotate_mor\u2082, contractible_triangle_mor\u2081],\n      erw category.comp_id },\n    { apply inv_rot_of_dist_triangle, apply contractible_distinguished } },\nend\n\n-- Prove this using the above theorem.\ninstance linear_yoneda_flip_homological (X : C) :\n  homological_functor ((linear_yoneda R C).flip.obj (opposite.op X)) :=\nbegin\n  constructor,\n  intros T hT,\n  rw Module.exact_iff,\n  apply le_antisymm,\n  { rintros _ \u27e8(g : X \u27f6 _),rfl\u27e9,\n    dsimp,\n    obtain \u27e8e,h1,he\u27e9 := complete_distinguished_triangle_morphism\n      (contractible_triangle _ X) T (contractible_distinguished _) hT g (g \u226b T.mor\u2081)\n      (by { dsimp, simp }),\n    dsimp at he,\n    simp only [zero_comp] at he,\n    simp [\u2190 h1] },\n  { rintros (f : X \u27f6 _) (hf : f \u226b _ = 0),\n    apply dist_triang_to_exact_complex _ hT _ _ hf }\nend\n\nlemma is_iso_triangle_hom_of_is_iso (T\u2081 T\u2082 : triangle C)\n  (e : T\u2081 \u27f6 T\u2082)\n  [is_iso e.hom\u2081]\n  [is_iso e.hom\u2082]\n  [is_iso e.hom\u2083] : is_iso e :=\nbegin\n  constructor,\n  refine \u27e8\u27e8inv e.hom\u2081, inv e.hom\u2082, inv e.hom\u2083, _, _\u27e9, _, _\u27e9,\n  { dsimp,\n    rw [is_iso.comp_inv_eq, category.assoc, is_iso.eq_inv_comp, e.comm\u2081] },\n  { dsimp,\n    rw [is_iso.comp_inv_eq, category.assoc, is_iso.eq_inv_comp, e.comm\u2082] },\n  { ext; dsimp; simp },\n  { ext; dsimp; simp },\nend\n\nlemma is_iso_of_is_iso_rotate (T\u2081 T\u2082 : triangle C)\n  (e : T\u2081 \u27f6 T\u2082) [h : is_iso ((rotate C).map e)] : is_iso e :=\nbegin\n  haveI : is_iso ((triangle_rotation C).functor.map e) := h,\n  apply is_iso_of_fully_faithful ((triangle_rotation C).functor : triangle C \u2964 triangle C),\nend\n\nlemma is_iso_of_is_iso_inv_rotate (T\u2081 T\u2082 : triangle C)\n  (e : T\u2081 \u27f6 T\u2082) [h : is_iso ((inv_rotate C).map e)] : is_iso e :=\nbegin\n  haveI : is_iso ((triangle_rotation C).inverse.map e) := h,\n  apply is_iso_of_fully_faithful ((triangle_rotation C).inverse : triangle C \u2964 triangle C),\nend\n\ntheorem is_iso_of_is_iso_of_is_iso (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 dist_triang C) (h\u2082 : T\u2082 \u2208 dist_triang C)\n  (e : T\u2081 \u27f6 T\u2082) [is_iso e.hom\u2081] [is_iso e.hom\u2083] : is_iso e :=\nbegin\n  apply_with is_iso_triangle_hom_of_is_iso { instances := ff },\n  any_goals { apply_instance },\n  apply_instance,\n\n  apply_with is_iso_of_is_iso_preadditive_yoneda_map_app { instances := ff },\n  swap, apply_instance,\n  intros W,\n\n  let Y := (preadditive_yoneda.flip.obj (opposite.op W)),\n\n  have H1 := five_term_exact_seq Y _ h\u2081,\n  have H2 := five_term_exact_seq Y _ h\u2082,\n\n  have sq1 := e.inv_rotate.comm\u2081,\n  apply_fun (\u03bb e, Y.map e) at sq1,\n  simp only [functor.map_comp] at sq1,\n\n  have sq2 := e.comm\u2081,\n  apply_fun (\u03bb e, Y.map e) at sq2,\n  simp only [functor.map_comp] at sq2,\n\n  have sq3 := e.comm\u2082,\n  apply_fun (\u03bb e, Y.map e) at sq3,\n  simp only [functor.map_comp] at sq3,\n\n  have sq4 := e.comm\u2083,\n  apply_fun (\u03bb e, Y.map e) at sq4,\n  simp only [functor.map_comp] at sq4,\n\n  haveI : is_iso (Y.map (triangulated.triangle_morphism.inv_rotate e).hom\u2081),\n  { dsimp only [triangulated.triangle_morphism.inv_rotate],\n    rw \u2190 functor.comp_map,\n    apply functor.map_is_iso },\n\n  haveI : is_iso (Y.map (triangulated.triangle_morphism.inv_rotate e).hom\u2082),\n  { dsimp only [triangulated.triangle_morphism.inv_rotate],\n    apply functor.map_is_iso },\n\n  haveI : is_iso (Y.map e.hom\u2083),\n  { apply functor.map_is_iso },\n\n  haveI : is_iso (Y.map ((shift_functor C (1 : \u2124)).map e.hom\u2081)),\n  { rw \u2190 functor.comp_map,\n    apply functor.map_is_iso },\n\n  exact @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso _ _ _\n    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _\n    sq1.symm sq2.symm sq3.symm _ _ _ _ _ sq4.symm\n    ((exact_iff_exact_seq _ _).mpr (H1.extract 0 2))\n    ((exact_iff_exact_seq _ _).mpr (H1.extract 1 2))\n    ((exact_iff_exact_seq _ _).mpr (H1.extract 2 3))\n    ((exact_iff_exact_seq _ _).mpr (H2.extract 0 2))\n    ((exact_iff_exact_seq _ _).mpr (H2.extract 1 2))\n    ((exact_iff_exact_seq _ _).mpr (H2.extract 2 3)) _ _ _ _,\nend\n\nlemma is_iso_of_is_iso_of_is_iso' (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 dist_triang C) (h\u2082 : T\u2082 \u2208 dist_triang C)\n  (e : T\u2081 \u27f6 T\u2082) [h1 : is_iso e.hom\u2081] [h2 : is_iso e.hom\u2082] : is_iso e :=\nbegin\n  suffices : is_iso ((rotate C).map e),\n  { resetI, apply is_iso_of_is_iso_rotate },\n  haveI : is_iso ((rotate C).map e).hom\u2081 := h2,\n  haveI : is_iso ((rotate C).map e).hom\u2083,\n  { apply functor.map_is_iso },\n  apply is_iso_of_is_iso_of_is_iso,\n  all_goals { erw \u2190 rotate_distinguished_triangle, assumption },\nend\n\ninstance is_iso_hom\u2081 (T\u2081 T\u2082 : triangle C) (e : T\u2081 \u27f6 T\u2082) [is_iso e] :\n  is_iso e.hom\u2081 :=\n\u27e8\u27e8(inv e).hom\u2081,\n  show (e \u226b inv e).hom\u2081 = _, by simpa only [is_iso.hom_inv_id],\n  show (inv e \u226b e).hom\u2081 = _, by simpa only [is_iso.inv_hom_id]\u27e9\u27e9\n\ninstance is_iso_hom\u2082 (T\u2081 T\u2082 : triangle C) (e : T\u2081 \u27f6 T\u2082) [is_iso e] :\n  is_iso e.hom\u2082 :=\n\u27e8\u27e8(inv e).hom\u2082,\n  show (e \u226b inv e).hom\u2082 = _, by simpa only [is_iso.hom_inv_id],\n  show (inv e \u226b e).hom\u2082 = _, by simpa only [is_iso.inv_hom_id]\u27e9\u27e9\n\ninstance is_iso_hom\u2083 (T\u2081 T\u2082 : triangle C) (e : T\u2081 \u27f6 T\u2082) [is_iso e] :\n  is_iso e.hom\u2083 :=\n\u27e8\u27e8(inv e).hom\u2083,\n  show (e \u226b inv e).hom\u2083 = _, by simpa only [is_iso.hom_inv_id],\n  show (inv e \u226b e).hom\u2083 = _, by simpa only [is_iso.inv_hom_id]\u27e9\u27e9\n\nlemma homological_of_exists_aux {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  (T\u2081 T\u2082 : triangle C) (e : T\u2081 \u27f6 T\u2082) [is_iso e] (H : exact (F.map T\u2081.mor\u2081) (F.map T\u2081.mor\u2082)) :\n  exact (F.map T\u2082.mor\u2081) (F.map T\u2082.mor\u2082) :=\nbegin\n  have : T\u2082.mor\u2081 = inv e.hom\u2081 \u226b T\u2081.mor\u2081 \u226b e.hom\u2082,\n  { rw is_iso.eq_inv_comp,\n    exact e.comm\u2081.symm },\n  rw this, clear this,\n  have : T\u2082.mor\u2082 = inv e.hom\u2082 \u226b T\u2081.mor\u2082 \u226b e.hom\u2083,\n  { rw is_iso.eq_inv_comp,\n    exact e.comm\u2082.symm },\n  rw this, clear this,\n  simp only [F.map_comp, F.map_inv],\n  simp only [exact_iso_comp],\n  simp only [\u2190 category.assoc],\n  simp only [exact_comp_iso],\n  change exact (_ \u226b (as_iso (F.map e.hom\u2082)).hom) ((as_iso (F.map e.hom\u2082)).inv \u226b _),\n  rwa exact_comp_hom_inv_comp_iff,\nend\n\nlemma homological_of_exists {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  (h : \u2200 (X Y : C) (f : X \u27f6 Y),\n    (\u2203 (Z : C) (g : Y \u27f6 Z) (h : Z \u27f6 X\u27e6(1 : \u2124)\u27e7) (hT : triangle.mk _ f g h \u2208 dist_triang C),\n    exact (F.map f) (F.map g))) : homological_functor F :=\nbegin\n  constructor,\n  intros T hT,\n  specialize h T.obj\u2081 T.obj\u2082 T.mor\u2081,\n  obtain \u27e8Z,g,h,hT',hE\u27e9 := h,\n  let T' := triangle.mk _ T.mor\u2081 g h,\n  obtain \u27e8e,h1,h2\u27e9 := complete_distinguished_triangle_morphism T' T hT' hT (\ud835\udfd9 _) (\ud835\udfd9 _) (by simp),\n  let E : T' \u27f6 T := \u27e8\ud835\udfd9 _, \ud835\udfd9 _, e, by simp, h1\u27e9,\n  haveI : is_iso E,\n  { apply is_iso_of_is_iso_of_is_iso' _ _ hT' hT },\n  exact homological_of_exists_aux F T' T E hE,\nend\n\nlemma homological_of_rotate {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  (h : \u2200 (T : triangle C) (hT : T \u2208 dist_triang C),\n    exact (F.map T.rotate.mor\u2081) (F.map T.rotate.mor\u2082)) : homological_functor F :=\nbegin\n  constructor,\n  intros T hT,\n  specialize h T.inv_rotate (inv_rot_of_dist_triangle C T hT),\n  let E : T.inv_rotate.rotate \u2245 T := inv_rot_comp_rot.app _,\n  apply homological_of_exists_aux _ _ _ E.hom,\n  assumption'\nend\n\nlemma homological_of_inv_rotate {A : Type*} [category A] [abelian A] (F : C \u2964 A) [F.additive]\n  (h : \u2200 (T : triangle C) (hT : T \u2208 dist_triang C),\n    exact (F.map T.inv_rotate.mor\u2081) (F.map T.inv_rotate.mor\u2082)) : homological_functor F :=\nbegin\n  constructor,\n  intros T hT,\n  specialize h T.rotate (rot_of_dist_triangle C T hT),\n  let E : T.rotate.inv_rotate \u2245 T := (rot_comp_inv_rot.app _).symm,\n  apply homological_of_exists_aux _ _ _ E.hom,\n  assumption'\nend\n\nlemma homological_of_nat_iso {A : Type*} [category A] [abelian A] (F G : C \u2964 A)\n  [F.additive] [G.additive] [homological_functor F] (e : F \u2245 G) : homological_functor G :=\nbegin\n  constructor,\n  intros T hT,\n  have h\u2081 := e.hom.naturality T.mor\u2081,\n  have h\u2082 := e.hom.naturality T.mor\u2082,\n  rw \u2190 is_iso.inv_comp_eq at h\u2081 h\u2082,\n  rw [\u2190 h\u2081, \u2190 h\u2082, exact_iso_comp, \u2190 category.assoc, exact_comp_iso],\n  let E := as_iso (e.hom.app T.obj\u2082),\n  change exact (F.map T.mor\u2081 \u226b E.hom) (E.inv \u226b _),\n  have : exact (F.map T.mor\u2081) (F.map T.mor\u2082) := by apply homological_functor.cond F _ hT,\n  rwa exact_comp_hom_inv_comp_iff,\nend\n\nend category_theory.triangulated\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/derived/homological.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3573428406744467}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Simon Hudon\n-/\nimport category_theory.monoidal.braided\nimport category_theory.limits.shapes.binary_products\nimport category_theory.limits.shapes.terminal\nimport category_theory.pempty\n\n/-!\n# The monoidal structure on a category with chosen finite products.\n\nThis is a variant of the development in `category_theory.monoidal.of_has_finite_products`,\nwhich uses specified choices of the terminal object and binary product,\nenabling the construction of a cartesian category with specific definitions of the tensor unit\nand tensor product.\n\n(Because the construction in `category_theory.monoidal.of_has_finite_products` uses `has_limit`\nclasses, the actual definitions there are opaque behind `classical.choice`.)\n\nWe use this in `category_theory.monoidal.types` to construct the monoidal category of types\nso that the tensor product is the usual cartesian product of types.\n\nFor now we only do the construction from products, and not from coproducts,\nwhich seems less often useful.\n-/\n\nuniverses v u\n\nnoncomputable theory\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C] {X Y : C}\n\nnamespace limits\n\nsection\nvariables {C}\n\n/-- Swap the two sides of a `binary_fan`. -/\ndef binary_fan.swap {P Q : C} (t : binary_fan P Q) : binary_fan Q P :=\nbinary_fan.mk t.snd t.fst\n\n@[simp] lemma binary_fan.swap_fst {P Q : C} (t : binary_fan P Q) : t.swap.fst = t.snd := rfl\n@[simp] lemma binary_fan.swap_snd {P Q : C} (t : binary_fan P Q) : t.swap.snd = t.fst := rfl\n\n/--\nIf a cone `t` over `P Q` is a limit cone, then `t.swap` is a limit cone over `Q P`.\n-/\n@[simps]\ndef is_limit.swap_binary_fan {P Q : C} {t : binary_fan P Q} (I : is_limit t) : is_limit t.swap :=\n{ lift := \u03bb s, I.lift (binary_fan.swap s),\n  fac' := \u03bb s, by { rintro \u27e8\u27e8\u27e9\u27e9; simp, },\n  uniq' := \u03bb s m w,\n  begin\n    have h := I.uniq (binary_fan.swap s) m,\n    rw h,\n    rintro \u27e8j\u27e9,\n    specialize w \u27e8j.swap\u27e9,\n    cases j; exact w,\n  end }\n\n/--\nConstruct `has_binary_product Q P` from `has_binary_product P Q`.\nThis can't be an instance, as it would cause a loop in typeclass search.\n-/\nlemma has_binary_product.swap (P Q : C) [has_binary_product P Q] : has_binary_product Q P :=\nhas_limit.mk \u27e8binary_fan.swap (limit.cone (pair P Q)), (limit.is_limit (pair P Q)).swap_binary_fan\u27e9\n\n/--\nGiven a limit cone over `X` and `Y`, and another limit cone over `Y` and `X`, we can construct\nan isomorphism between the cone points. Relative to some fixed choice of limits cones for every\npair, these isomorphisms constitute a braiding.\n-/\ndef binary_fan.braiding {X Y : C}\n  {s : binary_fan X Y} (P : is_limit s) {t : binary_fan Y X} (Q : is_limit t) :\n  s.X \u2245 t.X :=\nis_limit.cone_point_unique_up_to_iso P Q.swap_binary_fan\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `sXY.X Z`,\nif `sYZ` is a limit cone we can construct a binary fan over `X sYZ.X`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  binary_fan X sYZ.X :=\nbinary_fan.mk (s.fst \u226b sXY.fst) (Q.lift (binary_fan.mk (s.fst \u226b sXY.snd) s.snd))\n\n@[simp] lemma binary_fan.assoc_fst {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).fst = s.fst \u226b sXY.fst := rfl\n@[simp] lemma binary_fan.assoc_snd {X Y Z : C}\n  {sXY : binary_fan X Y} {sYZ : binary_fan Y Z} (Q : is_limit sYZ) (s : binary_fan sXY.X Z) :\n  (s.assoc Q).snd = Q.lift (binary_fan.mk (s.fst \u226b sXY.snd) s.snd) := rfl\n\n/--\nGiven binary fans `sXY` over `X Y`, and `sYZ` over `Y Z`, and `s` over `X sYZ.X`,\nif `sYZ` is a limit cone we can construct a binary fan over `sXY.X Z`.\n\nThis is an ingredient of building the associator for a cartesian category.\n-/\ndef binary_fan.assoc_inv {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  binary_fan sXY.X Z :=\nbinary_fan.mk (P.lift (binary_fan.mk s.fst (s.snd \u226b sYZ.fst))) (s.snd \u226b sYZ.snd)\n\n@[simp] lemma binary_fan.assoc_inv_fst {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).fst = P.lift (binary_fan.mk s.fst (s.snd \u226b sYZ.fst)) := rfl\n@[simp] lemma binary_fan.assoc_inv_snd {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (s : binary_fan X sYZ.X) :\n  (s.assoc_inv P).snd = s.snd \u226b sYZ.snd := rfl\n\n/--\nIf all the binary fans involved a limit cones, `binary_fan.assoc` produces another limit cone.\n-/\n@[simps]\ndef is_limit.assoc {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) : is_limit (s.assoc Q) :=\n{ lift := \u03bb t, R.lift (binary_fan.assoc_inv P t),\n  fac' := \u03bb t,\n  begin\n    rintro \u27e8\u27e8\u27e9\u27e9; simp,\n    apply Q.hom_ext,\n    rintro \u27e8\u27e8\u27e9\u27e9; simp,\n  end,\n  uniq' := \u03bb t m w,\n  begin\n    have h := R.uniq (binary_fan.assoc_inv P t) m,\n    rw h,\n    rintro \u27e8\u27e8\u27e9\u27e9; simp,\n    apply P.hom_ext,\n    rintro \u27e8\u27e8\u27e9\u27e9; simp,\n    { exact w \u27e8walking_pair.left\u27e9, },\n    { specialize w \u27e8walking_pair.right\u27e9,\n      simp at w,\n      rw [\u2190w], simp, },\n    { specialize w \u27e8walking_pair.right\u27e9,\n      simp at w,\n      rw [\u2190w], simp, },\n  end, }\n\n/--\nGiven two pairs of limit cones corresponding to the parenthesisations of `X \u00d7 Y \u00d7 Z`,\nwe obtain an isomorphism between the cone points.\n-/\n@[reducible]\ndef binary_fan.associator {X Y Z : C}\n  {sXY : binary_fan X Y} (P : is_limit sXY) {sYZ : binary_fan Y Z} (Q : is_limit sYZ)\n  {s : binary_fan sXY.X Z} (R : is_limit s) {t : binary_fan X sYZ.X} (S : is_limit t) :\n  s.X \u2245 t.X :=\nis_limit.cone_point_unique_up_to_iso (is_limit.assoc P Q R) S\n\n/--\nGiven a fixed family of limit data for every pair `X Y`, we obtain an associator.\n-/\n@[reducible]\ndef binary_fan.associator_of_limit_cone\n  (L : \u03a0 X Y : C, limit_cone (pair X Y)) (X Y Z : C) :\n  (L (L X Y).cone.X Z).cone.X \u2245 (L X (L Y Z).cone.X).cone.X :=\nbinary_fan.associator\n  (L X Y).is_limit (L Y Z).is_limit\n  (L (L X Y).cone.X Z).is_limit (L X (L Y Z).cone.X).is_limit\n\nlocal attribute [tidy] tactic.discrete_cases\n\n/--\nConstruct a left unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.left_unitor {X : C} {s : cone (functor.empty.{v} C)} (P : is_limit s)\n  {t : binary_fan s.X X} (Q : is_limit t) : t.X \u2245 X :=\n{ hom := t.snd,\n  inv := Q.lift (binary_fan.mk (P.lift\n    { X := X, \u03c0 := { app := discrete.rec (pempty.rec _) } }) (\ud835\udfd9 X) ),\n  hom_inv_id' :=\n  by { apply Q.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9, { apply P.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9, }, { simp, }, }, }\n\n/--\nConstruct a right unitor from specified limit cones.\n-/\n@[simps]\ndef binary_fan.right_unitor {X : C} {s : cone (functor.empty.{v} C)} (P : is_limit s)\n  {t : binary_fan X s.X} (Q : is_limit t) : t.X \u2245 X :=\n{ hom := t.fst,\n  inv := Q.lift (binary_fan.mk (\ud835\udfd9 X) (P.lift\n    { X := X, \u03c0 := { app := discrete.rec (pempty.rec _) } })),\n  hom_inv_id' :=\n  by { apply Q.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9, { simp, }, { apply P.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9, }, }, }\n\nend\n\nend limits\n\nopen category_theory.limits\n\nsection\nlocal attribute [tidy] tactic.case_bash\n\nvariables {C}\nvariables (\ud835\udcaf : limit_cone (functor.empty.{v} C))\nvariables (\u212c : \u03a0 (X Y : C), limit_cone (pair X Y))\n\nnamespace monoidal_of_chosen_finite_products\n\n/-- Implementation of the tensor product for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_obj (X Y : C) : C := (\u212c X Y).cone.X\n\n/-- Implementation of the tensor product of morphisms for `monoidal_of_chosen_finite_products`. -/\n@[reducible]\ndef tensor_hom {W X Y Z : C} (f : W \u27f6 X) (g : Y \u27f6 Z) : tensor_obj \u212c W Y \u27f6 tensor_obj \u212c X Z :=\n  (binary_fan.is_limit.lift' (\u212c X Z).is_limit\n    ((\u212c W Y).cone.\u03c0.app \u27e8walking_pair.left\u27e9 \u226b f)\n    (((\u212c W Y).cone.\u03c0.app \u27e8walking_pair.right\u27e9 : (\u212c W Y).cone.X \u27f6 Y) \u226b g)).val\n\nlemma tensor_id (X\u2081 X\u2082 : C) : tensor_hom \u212c (\ud835\udfd9 X\u2081) (\ud835\udfd9 X\u2082) = \ud835\udfd9 (tensor_obj \u212c X\u2081 X\u2082) :=\nbegin\n  apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma tensor_comp {X\u2081 Y\u2081 Z\u2081 X\u2082 Y\u2082 Z\u2082 : C}\n  (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (g\u2081 : Y\u2081 \u27f6 Z\u2081) (g\u2082 : Y\u2082 \u27f6 Z\u2082) :\n  tensor_hom \u212c (f\u2081 \u226b g\u2081) (f\u2082 \u226b g\u2082) =\n    tensor_hom \u212c f\u2081 f\u2082 \u226b tensor_hom \u212c g\u2081 g\u2082 :=\nbegin\n  apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9;\n  { dsimp [tensor_hom], simp, },\nend\n\nlemma pentagon (W X Y Z : C) :\n  tensor_hom \u212c (binary_fan.associator_of_limit_cone \u212c W X Y).hom (\ud835\udfd9 Z) \u226b\n    (binary_fan.associator_of_limit_cone \u212c W (tensor_obj \u212c X Y) Z).hom \u226b\n      tensor_hom \u212c (\ud835\udfd9 W) (binary_fan.associator_of_limit_cone \u212c X Y Z).hom =\n  (binary_fan.associator_of_limit_cone \u212c (tensor_obj \u212c W X) Y Z).hom \u226b\n    (binary_fan.associator_of_limit_cone \u212c W X (tensor_obj \u212c Y Z)).hom :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9,\n  { simp, },\n  { apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9,\n    { simp, },\n    apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9,\n    { simp, },\n    { simp, }, }\nend\n\nlemma triangle (X Y : C) :\n  (binary_fan.associator_of_limit_cone \u212c X \ud835\udcaf.cone.X Y).hom \u226b\n    tensor_hom \u212c (\ud835\udfd9 X) (binary_fan.left_unitor \ud835\udcaf.is_limit (\u212c \ud835\udcaf.cone.X Y).is_limit).hom =\n  tensor_hom \u212c (binary_fan.right_unitor \ud835\udcaf.is_limit (\u212c X \ud835\udcaf.cone.X).is_limit).hom (\ud835\udfd9 Y) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9; simp,\nend\n\nlemma left_unitor_naturality {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) :\n  tensor_hom \u212c (\ud835\udfd9 \ud835\udcaf.cone.X) f \u226b (binary_fan.left_unitor \ud835\udcaf.is_limit (\u212c \ud835\udcaf.cone.X X\u2082).is_limit).hom =\n    (binary_fan.left_unitor \ud835\udcaf.is_limit (\u212c \ud835\udcaf.cone.X X\u2081).is_limit).hom \u226b f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma right_unitor_naturality {X\u2081 X\u2082 : C} (f : X\u2081 \u27f6 X\u2082) :\n  tensor_hom \u212c f (\ud835\udfd9 \ud835\udcaf.cone.X) \u226b\n    (binary_fan.right_unitor \ud835\udcaf.is_limit (\u212c X\u2082 \ud835\udcaf.cone.X).is_limit).hom =\n    (binary_fan.right_unitor \ud835\udcaf.is_limit (\u212c X\u2081 \ud835\udcaf.cone.X).is_limit).hom \u226b f :=\nbegin\n  dsimp [tensor_hom],\n  simp,\nend\n\nlemma associator_naturality {X\u2081 X\u2082 X\u2083 Y\u2081 Y\u2082 Y\u2083 : C} (f\u2081 : X\u2081 \u27f6 Y\u2081) (f\u2082 : X\u2082 \u27f6 Y\u2082) (f\u2083 : X\u2083 \u27f6 Y\u2083) :\n  tensor_hom \u212c (tensor_hom \u212c f\u2081 f\u2082) f\u2083 \u226b (binary_fan.associator_of_limit_cone \u212c Y\u2081 Y\u2082 Y\u2083).hom =\n    (binary_fan.associator_of_limit_cone \u212c X\u2081 X\u2082 X\u2083).hom \u226b\n      tensor_hom \u212c f\u2081 (tensor_hom \u212c f\u2082 f\u2083) :=\nbegin\n  dsimp [tensor_hom],\n  apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9,\n  { simp, },\n  { apply is_limit.hom_ext (\u212c _ _).is_limit, rintro \u27e8\u27e8\u27e9\u27e9,\n    { simp, },\n    { simp, }, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/-- A category with a terminal object and binary products has a natural monoidal structure. -/\ndef monoidal_of_chosen_finite_products :\n  monoidal_category C :=\n{ tensor_unit  := \ud835\udcaf.cone.X,\n  tensor_obj   := \u03bb X Y, tensor_obj \u212c X Y,\n  tensor_hom   := \u03bb _ _ _ _ f g, tensor_hom \u212c f g,\n  tensor_id'   := tensor_id \u212c,\n  tensor_comp' := \u03bb _ _ _ _ _ _ f\u2081 f\u2082 g\u2081 g\u2082, tensor_comp \u212c f\u2081 f\u2082 g\u2081 g\u2082,\n  associator   := \u03bb X Y Z, binary_fan.associator_of_limit_cone \u212c X Y Z,\n  left_unitor  := \u03bb X, binary_fan.left_unitor (\ud835\udcaf.is_limit) (\u212c \ud835\udcaf.cone.X X).is_limit,\n  right_unitor := \u03bb X, binary_fan.right_unitor (\ud835\udcaf.is_limit) (\u212c X \ud835\udcaf.cone.X).is_limit,\n  pentagon'    := pentagon \u212c,\n  triangle'    := triangle \ud835\udcaf \u212c,\n  left_unitor_naturality' := \u03bb _ _ f, left_unitor_naturality \ud835\udcaf \u212c f,\n  right_unitor_naturality' := \u03bb _ _ f, right_unitor_naturality \ud835\udcaf \u212c f,\n  associator_naturality' := \u03bb _ _ _ _ _ _ f\u2081 f\u2082 f\u2083, associator_naturality \u212c f\u2081 f\u2082 f\u2083, }\n\nnamespace monoidal_of_chosen_finite_products\n\nopen monoidal_category\n\n/--\nA type synonym for `C` carrying a monoidal category structure corresponding to\na fixed choice of limit data for the empty functor, and for `pair X Y` for every `X Y : C`.\n\nThis is an implementation detail for `symmetric_of_chosen_finite_products`.\n-/\n@[derive category, nolint unused_arguments has_inhabited_instance]\ndef monoidal_of_chosen_finite_products_synonym\n  (\ud835\udcaf : limit_cone (functor.empty.{v} C)) (\u212c : \u03a0 (X Y : C), limit_cone (pair X Y)):= C\n\ninstance : monoidal_category (monoidal_of_chosen_finite_products_synonym \ud835\udcaf \u212c) :=\nmonoidal_of_chosen_finite_products \ud835\udcaf \u212c\n\nlemma braiding_naturality {X X' Y Y' : C} (f : X \u27f6 Y) (g : X' \u27f6 Y') :\n  (tensor_hom \u212c f g) \u226b (limits.binary_fan.braiding (\u212c Y Y').is_limit (\u212c Y' Y).is_limit).hom =\n    (limits.binary_fan.braiding (\u212c X X').is_limit (\u212c X' X).is_limit).hom \u226b (tensor_hom \u212c g f) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (\u212c _ _).is_limit.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\n\n\nlemma hexagon_reverse (X Y Z : C) :\n  (binary_fan.associator_of_limit_cone \u212c X Y Z).inv \u226b\n    (limits.binary_fan.braiding\n      (\u212c (tensor_obj \u212c X Y) Z).is_limit\n      (\u212c Z (tensor_obj \u212c X Y)).is_limit).hom \u226b\n    (binary_fan.associator_of_limit_cone \u212c Z X Y).inv =\n    (tensor_hom \u212c (\ud835\udfd9 X) (limits.binary_fan.braiding (\u212c Y Z).is_limit (\u212c Z Y).is_limit).hom) \u226b\n      (binary_fan.associator_of_limit_cone \u212c X Z Y).inv \u226b\n        (tensor_hom \u212c (limits.binary_fan.braiding (\u212c X Z).is_limit (\u212c Z X).is_limit).hom (\ud835\udfd9 Y)) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (\u212c _ _).is_limit.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9,\n  { apply (\u212c _ _).is_limit.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9;\n    { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n        limits.is_limit.cone_point_unique_up_to_iso],\n      simp, }, },\n  { dsimp [binary_fan.associator_of_limit_cone, binary_fan.associator,\n      limits.is_limit.cone_point_unique_up_to_iso],\n    simp, },\nend\n\nlemma symmetry (X Y : C) :\n  (limits.binary_fan.braiding (\u212c X Y).is_limit (\u212c Y X).is_limit).hom \u226b\n      (limits.binary_fan.braiding (\u212c Y X).is_limit (\u212c X Y).is_limit).hom =\n    \ud835\udfd9 (tensor_obj \u212c X Y) :=\nbegin\n  dsimp [tensor_hom, limits.binary_fan.braiding],\n  apply (\u212c _ _).is_limit.hom_ext, rintro \u27e8\u27e8\u27e9\u27e9;\n  { dsimp [limits.is_limit.cone_point_unique_up_to_iso], simp, },\nend\n\nend monoidal_of_chosen_finite_products\n\nopen monoidal_of_chosen_finite_products\n\n/--\nThe monoidal structure coming from finite products is symmetric.\n-/\ndef symmetric_of_chosen_finite_products :\n  symmetric_category (monoidal_of_chosen_finite_products_synonym \ud835\udcaf \u212c) :=\n{ braiding := \u03bb X Y, limits.binary_fan.braiding (\u212c _ _).is_limit (\u212c _ _).is_limit,\n  braiding_naturality' := \u03bb X X' Y Y' f g, braiding_naturality \u212c f g,\n  hexagon_forward' := \u03bb X Y Z, hexagon_forward \u212c X Y Z,\n  hexagon_reverse' := \u03bb X Y Z, hexagon_reverse \u212c X Y Z,\n  symmetry' := \u03bb X Y, symmetry \u212c X Y, }\n\nend\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/monoidal/of_chosen_finite_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804478040616, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.35734284067444666}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport algebraic_geometry.morphisms.basic\nimport ring_theory.local_properties\n\n/-!\n\n# Properties of morphisms from properties of ring homs.\n\nWe provide the basic framework for talking about properties of morphisms that come from properties\nof ring homs. For `P` a property of ring homs, we have two ways of defining a property of scheme\nmorphisms:\n\nLet `f : X \u27f6 Y`,\n- `target_affine_locally (affine_and P)`: the preimage of an affine open `U = Spec A` is affine\n  (`= Spec B`) and `A \u27f6 B` satisfies `P`. (TODO)\n- `affine_locally P`: For each pair of affine open `U = Spec A \u2286 X` and `V = Spec B \u2286 f \u207b\u00b9' U`,\n  the ring hom `A \u27f6 B` satisfies `P`.\n\nFor these notions to be well defined, we require `P` be a sufficient local property. For the former,\n`P` should be local on the source (`ring_hom.respects_iso P`, `ring_hom.localization_preserves P`,\n`ring_hom.of_localization_span`), and `target_affine_locally (affine_and P)` will be local on\nthe target. (TODO)\n\nFor the latter `P` should be local on the target (`ring_hom.property_is_local P`), and\n`affine_locally P` will be local on both the source and the target.\n\nFurther more, these properties are stable under compositions (resp. base change) if `P` is. (TODO)\n\n-/\n\nuniverse u\n\nopen category_theory opposite topological_space category_theory.limits algebraic_geometry\n\nvariable (P : \u2200 {R S : Type u} [comm_ring R] [comm_ring S] (f : by exactI R \u2192+* S), Prop)\n\nnamespace ring_hom\n\ninclude P\n\nvariable {P}\n\nlemma respects_iso.basic_open_iff (hP : respects_iso @P) {X Y : Scheme}\n  [is_affine X] [is_affine Y] (f : X \u27f6 Y) (r : Y.presheaf.obj (opposite.op \u22a4)) :\n  P (Scheme.\u0393.map (f \u2223_ Y.basic_open r).op) \u2194\n  P (@is_localization.away.map (Y.presheaf.obj (opposite.op \u22a4)) _\n      (Y.presheaf.obj (opposite.op $ Y.basic_open r)) _ _\n      (X.presheaf.obj (opposite.op \u22a4)) _ (X.presheaf.obj\n      (opposite.op $ X.basic_open (Scheme.\u0393.map f.op r))) _ _ (Scheme.\u0393.map f.op) r _ _) :=\nbegin\n  rw [\u0393_map_morphism_restrict, hP.cancel_left_is_iso, hP.cancel_right_is_iso,\n    \u2190 (hP.cancel_right_is_iso (f.val.c.app (opposite.op (Y.basic_open r))) (X.presheaf.map\n      (eq_to_hom (Scheme.preimage_basic_open f r).symm).op)), \u2190 eq_iff_iff],\n  congr,\n  delta is_localization.away.map,\n  refine is_localization.ring_hom_ext (submonoid.powers r) _,\n  convert (is_localization.map_comp _).symm using 1,\n  change Y.presheaf.map _ \u226b _ = _ \u226b X.presheaf.map _,\n  rw f.val.c.naturality_assoc,\n  erw \u2190 X.presheaf.map_comp,\n  congr,\nend\n\nlemma respects_iso.basic_open_iff_localization (hP : respects_iso @P)\n  {X Y : Scheme} [is_affine X] [is_affine Y] (f : X \u27f6 Y) (r : Y.presheaf.obj (opposite.op \u22a4)) :\n  P (Scheme.\u0393.map (f \u2223_ Y.basic_open r).op) \u2194\n  P (localization.away_map (Scheme.\u0393.map f.op) r) :=\n(hP.basic_open_iff _ _).trans (hP.is_localization_away_iff _ _ _ _).symm\n\nlemma respects_iso.of_restrict_morphism_restrict_iff (hP : ring_hom.respects_iso @P)\n  {X Y : Scheme} [is_affine Y] (f : X \u27f6 Y) (r : Y.presheaf.obj (opposite.op \u22a4))\n  (U : opens X.carrier) (hU : is_affine_open U) {V : opens _}\n  (e : V = (opens.map (X.of_restrict ((opens.map f.1.base).obj _).open_embedding).1.base).obj U) :\n  P (Scheme.\u0393.map ((X.restrict ((opens.map f.1.base).obj _).open_embedding).of_restrict\n    V.open_embedding \u226b f \u2223_ Y.basic_open r).op) \u2194\n    P (localization.away_map (Scheme.\u0393.map (X.of_restrict U.open_embedding \u226b f).op) r) :=\nbegin\n  subst e,\n  convert (hP.is_localization_away_iff _ _ _ _).symm,\n  rotate,\n  { apply_instance },\n  { apply ring_hom.to_algebra,\n    refine X.presheaf.map\n      (@hom_of_le _ _ ((is_open_map.functor _).obj _) ((is_open_map.functor _).obj _) _).op,\n    rw [\u2190 set_like.coe_subset_coe],\n    dsimp,\n    simp only [set.image_univ, subtype.range_coe, set.image_subset_iff],\n    refl },\n  { exact algebraic_geometry.\u0393_restrict_is_localization Y r },\n  { rw \u2190 U.open_embedding_obj_top at hU,\n    dsimp [Scheme.\u0393_obj_op, Scheme.\u0393_map_op, Scheme.restrict],\n    apply algebraic_geometry.is_localization_of_eq_basic_open _ hU,\n    rw [opens.open_embedding_obj_top, opens.functor_obj_map_obj],\n    convert (X.basic_open_res (Scheme.\u0393.map f.op r) (hom_of_le le_top).op).symm using 1,\n    rw [opens.open_embedding_obj_top, opens.open_embedding_obj_top, inf_comm,\n      Scheme.\u0393_map_op, \u2190 Scheme.preimage_basic_open] },\n  { apply is_localization.ring_hom_ext (submonoid.powers r) _,\n    swap, { exact algebraic_geometry.\u0393_restrict_is_localization Y r },\n    rw [is_localization.away.map, is_localization.map_comp, ring_hom.algebra_map_to_algebra,\n      ring_hom.algebra_map_to_algebra, op_comp, functor.map_comp, op_comp, functor.map_comp],\n    refine (@category.assoc CommRing _ _ _ _ _ _ _ _).symm.trans _,\n    refine eq.trans _ (@category.assoc CommRing _ _ _ _ _ _ _ _),\n    dsimp only [Scheme.\u0393_map, quiver.hom.unop_op],\n    rw [morphism_restrict_c_app, category.assoc, category.assoc, category.assoc],\n    erw [f.1.c.naturality_assoc, \u2190 X.presheaf.map_comp, \u2190 X.presheaf.map_comp,\n      \u2190 X.presheaf.map_comp],\n    congr },\nend\n\nlemma stable_under_base_change.\u0393_pullback_fst\n  (hP : stable_under_base_change @P) (hP' : respects_iso @P) {X Y S : Scheme}\n  [is_affine X] [is_affine Y] [is_affine S]\n  (f : X \u27f6 S) (g : Y \u27f6 S) (H : P (Scheme.\u0393.map g.op)) :\n    P (Scheme.\u0393.map (pullback.fst : pullback f g \u27f6 _).op) :=\nbegin\n  rw [\u2190 preserves_pullback.iso_inv_fst AffineScheme.forget_to_Scheme\n    (AffineScheme.of_hom f) (AffineScheme.of_hom g), op_comp, functor.map_comp,\n    hP'.cancel_right_is_iso, AffineScheme.forget_to_Scheme_map],\n  have := _root_.congr_arg quiver.hom.unop (preserves_pullback.iso_hom_fst AffineScheme.\u0393.right_op\n    (AffineScheme.of_hom f) (AffineScheme.of_hom g)),\n  simp only [quiver.hom.unop_op, functor.right_op_map, unop_comp] at this,\n  delta AffineScheme.\u0393 at this,\n  simp only [quiver.hom.unop_op, functor.comp_map, AffineScheme.forget_to_Scheme_map,\n    functor.op_map] at this,\n  rw [\u2190 this, hP'.cancel_right_is_iso,\n    \u2190 pushout_iso_unop_pullback_inl_hom (quiver.hom.unop _) (quiver.hom.unop _),\n    hP'.cancel_right_is_iso],\n  exact hP.pushout_inl _ hP' _ _ H\nend\n\nend ring_hom\n\nnamespace algebraic_geometry\n\n/-- For `P` a property of ring homomorphisms, `source_affine_locally P` holds for `f : X \u27f6 Y`\nwhenever `P` holds for the restriction of `f` on every affine open subset of `X`. -/\ndef source_affine_locally : affine_target_morphism_property :=\n\u03bb X Y f hY, \u2200 (U : X.affine_opens), P (Scheme.\u0393.map (X.of_restrict U.1.open_embedding \u226b f).op)\n\n/-- For `P` a property of ring homomorphisms, `affine_locally P` holds for `f : X \u27f6 Y` if for each\naffine open `U = Spec A \u2286 Y` and `V = Spec B \u2286 f \u207b\u00b9' U`, the ring hom `A \u27f6 B` satisfies `P`.\nAlso see `affine_locally_iff_affine_opens_le`. -/\nabbreviation affine_locally : morphism_property Scheme :=\ntarget_affine_locally (source_affine_locally @P)\n\nvariable {P}\n\nlemma source_affine_locally_respects_iso (h\u2081 : ring_hom.respects_iso @P) :\n  (source_affine_locally @P).to_property.respects_iso :=\nbegin\n  apply affine_target_morphism_property.respects_iso_mk,\n  { introv H U,\n    rw [\u2190 h\u2081.cancel_right_is_iso _ (Scheme.\u0393.map (Scheme.restrict_map_iso e.inv U.1).hom.op),\n      \u2190 functor.map_comp, \u2190 op_comp],\n    convert H \u27e8_, U.prop.map_is_iso e.inv\u27e9 using 3,\n    rw [is_open_immersion.iso_of_range_eq_hom, is_open_immersion.lift_fac_assoc,\n      category.assoc, e.inv_hom_id_assoc],\n    refl },\n  { introv H U,\n    rw [\u2190 category.assoc, op_comp, functor.map_comp, h\u2081.cancel_left_is_iso],\n    exact H U }\nend\n\nlemma affine_locally_respects_iso (h : ring_hom.respects_iso @P) :\n  (affine_locally @P).respects_iso :=\ntarget_affine_locally_respects_iso (source_affine_locally_respects_iso h)\n\nlemma affine_locally_iff_affine_opens_le\n  (hP : ring_hom.respects_iso @P) {X Y : Scheme} (f : X \u27f6 Y) :\n  affine_locally @P f \u2194\n  (\u2200 (U : Y.affine_opens) (V : X.affine_opens) (e : V.1 \u2264 (opens.map f.1.base).obj U.1),\n    P (f.app_le e)) :=\nbegin\n  apply forall_congr,\n  intro U,\n  delta source_affine_locally,\n  simp_rw [op_comp, Scheme.\u0393.map_comp, \u0393_map_morphism_restrict, category.assoc, Scheme.\u0393_map_op,\n    hP.cancel_left_is_iso],\n  split,\n  { intros H V e,\n    let U' := (opens.map f.val.base).obj U.1,\n    have e' : U'.open_embedding.is_open_map.functor.obj ((opens.map U'.inclusion).obj V.1) = V.1,\n    { ext1, refine set.image_preimage_eq_inter_range.trans (set.inter_eq_left_iff_subset.mpr _),\n      convert e, exact subtype.range_coe },\n    have := H \u27e8(opens.map (X.of_restrict (U'.open_embedding)).1.base).obj V.1, _\u27e9,\n    erw \u2190 X.presheaf.map_comp at this,\n    rw [\u2190 hP.cancel_right_is_iso _ (X.presheaf.map (eq_to_hom _)), category.assoc,\n      \u2190 X.presheaf.map_comp],\n    convert this using 1,\n    { dsimp only [functor.op, unop_op], rw opens.open_embedding_obj_top, congr' 1, exact e'.symm },\n    { apply_instance },\n    { apply (is_affine_open_iff_of_is_open_immersion (X.of_restrict _) _).mp,\n      convert V.2,\n      apply_instance } },\n  { intros H V,\n    specialize H \u27e8_, V.2.image_is_open_immersion (X.of_restrict _)\u27e9 (subtype.coe_image_subset _ _),\n    erw \u2190 X.presheaf.map_comp,\n    rw [\u2190 hP.cancel_right_is_iso _ (X.presheaf.map (eq_to_hom _)), category.assoc,\n      \u2190 X.presheaf.map_comp],\n    convert H,\n    { dsimp only [functor.op, unop_op], rw opens.open_embedding_obj_top, refl },\n    { apply_instance } }\nend\n\nlemma Scheme_restrict_basic_open_of_localization_preserves\n  (h\u2081 : ring_hom.respects_iso @P)\n  (h\u2082 : ring_hom.localization_preserves @P)\n  {X Y : Scheme} [is_affine Y] (f : X \u27f6 Y) (r : Y.presheaf.obj (op \u22a4))\n  (H : source_affine_locally @P f)\n  (U : (X.restrict ((opens.map f.1.base).obj $ Y.basic_open r).open_embedding).affine_opens) :\n  P (Scheme.\u0393.map\n    ((X.restrict ((opens.map f.1.base).obj $ Y.basic_open r).open_embedding).of_restrict\n      U.1.open_embedding \u226b f \u2223_ Y.basic_open r).op) :=\nbegin\n  specialize H \u27e8_, U.2.image_is_open_immersion (X.of_restrict _)\u27e9,\n  convert (h\u2081.of_restrict_morphism_restrict_iff _ _ _ _ _).mpr _ using 1,\n  swap 5,\n  { exact h\u2082.away r H },\n  { apply_instance },\n  { exact U.2.image_is_open_immersion _},\n  { ext1, exact (set.preimage_image_eq _ subtype.coe_injective).symm }\nend\n\nlemma source_affine_locally_is_local\n  (h\u2081 : ring_hom.respects_iso @P)\n  (h\u2082 : ring_hom.localization_preserves @P)\n  (h\u2083 : ring_hom.of_localization_span @P) : (source_affine_locally @P).is_local :=\nbegin\n  constructor,\n  { exact source_affine_locally_respects_iso h\u2081 },\n  { introv H U,\n    apply Scheme_restrict_basic_open_of_localization_preserves h\u2081 h\u2082; assumption },\n  { introv hs hs' U,\n    resetI,\n    apply h\u2083 _ _ hs,\n    intro r,\n    have := hs' r \u27e8(opens.map (X.of_restrict _).1.base).obj U.1, _\u27e9,\n    rwa h\u2081.of_restrict_morphism_restrict_iff at this,\n    { exact U.2 },\n    { refl },\n    { apply_instance },\n    { suffices : \u2200 (V = (opens.map f.val.base).obj (Y.basic_open r.val)),\n        is_affine_open ((opens.map (X.of_restrict V.open_embedding).1.base).obj U.1),\n      { exact this _ rfl, },\n      intros V hV,\n      rw Scheme.preimage_basic_open at hV,\n      subst hV,\n      exact U.2.map_restrict_basic_open (Scheme.\u0393.map f.op r.1) } }\nend\n\nvariables {P} (hP : ring_hom.property_is_local @P)\n\nlemma source_affine_locally_of_source_open_cover_aux\n  (h\u2081 : ring_hom.respects_iso @P)\n  (h\u2083 : ring_hom.of_localization_span_target @P)\n  {X Y : Scheme} (f : X \u27f6 Y) (U : X.affine_opens)\n  (s : set (X.presheaf.obj (op U.1))) (hs : ideal.span s = \u22a4)\n  (hs' : \u2200 (r : s), P (Scheme.\u0393.map (X.of_restrict (X.basic_open r.1).open_embedding \u226b f).op)) :\n    P (Scheme.\u0393.map (X.of_restrict U.1.open_embedding \u226b f).op) :=\nbegin\n  apply_fun ideal.map (X.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op) at hs,\n  rw [ideal.map_span, ideal.map_top] at hs,\n  apply h\u2083 _ _ hs,\n  rintro \u27e8s, r, hr, hs\u27e9,\n  have := (@@localization.alg_equiv _ _ _ _ _ (@@algebraic_geometry.\u0393_restrict_is_localization\n    _ U.2 s)).to_ring_equiv.to_CommRing_iso,\n  refine (h\u2081.cancel_right_is_iso _ (@@localization.alg_equiv _ _ _ _ _\n    (@@algebraic_geometry.\u0393_restrict_is_localization _ U.2 s))\n      .to_ring_equiv.to_CommRing_iso.hom).mp _,\n  subst hs,\n  rw [CommRing.comp_eq_ring_hom_comp, \u2190 ring_hom.comp_assoc],\n  erw [is_localization.map_comp, ring_hom.comp_id],\n  rw [ring_hom.algebra_map_to_algebra, op_comp, functor.map_comp, \u2190 CommRing.comp_eq_ring_hom_comp,\n    Scheme.\u0393_map_op, Scheme.\u0393_map_op, Scheme.\u0393_map_op, category.assoc],\n  erw \u2190 X.presheaf.map_comp,\n  rw [\u2190 h\u2081.cancel_right_is_iso _ (X.presheaf.map (eq_to_hom _))],\n  convert hs' \u27e8r, hr\u27e9 using 1,\n  { erw category.assoc, rw [\u2190 X.presheaf.map_comp, op_comp, Scheme.\u0393.map_comp,\n    Scheme.\u0393_map_op, Scheme.\u0393_map_op], congr },\n  { dsimp [functor.op],\n    conv_lhs { rw opens.open_embedding_obj_top },\n    conv_rhs { rw opens.open_embedding_obj_top },\n    erw Scheme.image_basic_open (X.of_restrict U.1.open_embedding),\n    erw PresheafedSpace.is_open_immersion.of_restrict_inv_app_apply,\n    rw Scheme.basic_open_res_eq },\n  { apply_instance }\nend\n\nlemma is_open_immersion_comp_of_source_affine_locally (h\u2081 : ring_hom.respects_iso @P)\n  {X Y Z : Scheme} [is_affine X] [is_affine Z] (f : X \u27f6 Y) [is_open_immersion f] (g : Y \u27f6 Z)\n  (h\u2082 : source_affine_locally @P g) :\n  P (Scheme.\u0393.map (f \u226b g).op) :=\nbegin\n  rw [\u2190 h\u2081.cancel_right_is_iso _ (Scheme.\u0393.map (is_open_immersion.iso_of_range_eq\n    (Y.of_restrict _) f _).hom.op), \u2190 functor.map_comp, \u2190 op_comp],\n  convert h\u2082 \u27e8_, range_is_affine_open_of_open_immersion f\u27e9 using 3,\n  { rw [is_open_immersion.iso_of_range_eq_hom, is_open_immersion.lift_fac_assoc] },\n  { apply_instance },\n  { exact subtype.range_coe },\n  { apply_instance }\nend\n\nend algebraic_geometry\n\nopen algebraic_geometry\n\nnamespace ring_hom.property_is_local\n\nvariables {P} (hP : ring_hom.property_is_local @P)\n\ninclude hP\n\nlemma source_affine_locally_of_source_open_cover\n  {X Y : Scheme} (f : X \u27f6 Y) [is_affine Y]\n  (\ud835\udcb0 : X.open_cover) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (H : \u2200 i, P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) :\n  source_affine_locally @P f :=\nbegin\n  let S := \u03bb i, (\u27e8\u27e8set.range (\ud835\udcb0.map i).1.base, (\ud835\udcb0.is_open i).base_open.open_range\u27e9,\n    range_is_affine_open_of_open_immersion (\ud835\udcb0.map i)\u27e9 : X.affine_opens),\n  intros U,\n  apply of_affine_open_cover U,\n  swap 5, { exact set.range S },\n  { intros U r H,\n    convert hP.stable_under_composition _ _ H _ using 1,\n    swap,\n    { refine X.presheaf.map\n        (@hom_of_le _ _ ((is_open_map.functor _).obj _) ((is_open_map.functor _).obj _) _).op,\n      rw [unop_op, unop_op, opens.open_embedding_obj_top, opens.open_embedding_obj_top],\n      exact X.basic_open_le _ },\n    { rw [op_comp, op_comp, functor.map_comp, functor.map_comp],\n      refine (eq.trans _ (category.assoc _ _ _).symm : _),\n      congr' 1,\n      refine eq.trans _ (X.presheaf.map_comp _ _),\n      change X.presheaf.map _ = _,\n      congr },\n    convert hP.holds_for_localization_away _\n      (X.presheaf.map (eq_to_hom U.1.open_embedding_obj_top).op r),\n    { exact (ring_hom.algebra_map_to_algebra _).symm },\n    { dsimp [Scheme.\u0393],\n      have := U.2,\n      rw \u2190 U.1.open_embedding_obj_top at this,\n      convert is_localization_basic_open this _ using 6;\n        rw opens.open_embedding_obj_top; exact (Scheme.basic_open_res_eq _ _ _).symm } },\n  { introv hs hs',\n    exact source_affine_locally_of_source_open_cover_aux hP.respects_iso hP.2 _ _ _ hs hs' },\n  { rw set.eq_univ_iff_forall,\n    intro x,\n    rw set.mem_Union,\n    exact \u27e8\u27e8_, \ud835\udcb0.f x, rfl\u27e9, \ud835\udcb0.covers x\u27e9 },\n  { rintro \u27e8_, i, rfl\u27e9,\n    specialize H i,\n    rw \u2190 hP.respects_iso.cancel_right_is_iso _ (Scheme.\u0393.map (is_open_immersion.iso_of_range_eq\n      (\ud835\udcb0.map i) (X.of_restrict (S i).1.open_embedding) subtype.range_coe.symm).inv.op) at H,\n    rwa [\u2190 Scheme.\u0393.map_comp, \u2190 op_comp, is_open_immersion.iso_of_range_eq_inv,\n      is_open_immersion.lift_fac_assoc] at H }\nend\n\nlemma affine_open_cover_tfae {X Y : Scheme.{u}}\n  [is_affine Y] (f : X \u27f6 Y) :\n  tfae [source_affine_locally @P f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} X) [\u2200 i, is_affine (\ud835\udcb0.obj i)],\n      \u2200 (i : \ud835\udcb0.J), P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} X) [\u2200 i, is_affine (\ud835\udcb0.obj i)] (i : \ud835\udcb0.J),\n      P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op),\n    \u2200 {U : Scheme} (g : U \u27f6 X) [is_affine U] [is_open_immersion g],\n      P (Scheme.\u0393.map (g \u226b f).op)] :=\nbegin\n  tfae_have : 1 \u2192 4,\n  { intros H U g _ hg,\n    resetI,\n    specialize H \u27e8\u27e8_, hg.base_open.open_range\u27e9,\n      range_is_affine_open_of_open_immersion g\u27e9,\n    rw [\u2190 hP.respects_iso.cancel_right_is_iso _ (Scheme.\u0393.map (is_open_immersion.iso_of_range_eq\n      g (X.of_restrict (opens.open_embedding \u27e8_, hg.base_open.open_range\u27e9))\n      subtype.range_coe.symm).hom.op), \u2190 Scheme.\u0393.map_comp, \u2190 op_comp,\n      is_open_immersion.iso_of_range_eq_hom] at H,\n    erw is_open_immersion.lift_fac_assoc at H,\n    exact H },\n  tfae_have : 4 \u2192 3,\n  { intros H \ud835\udcb0 _ i, resetI, apply H },\n  tfae_have : 3 \u2192 2,\n  { intro H, refine \u27e8X.affine_cover, infer_instance, H _\u27e9 },\n  tfae_have : 2 \u2192 1,\n  { rintro \u27e8\ud835\udcb0, _, h\ud835\udcb0\u27e9,\n    exactI hP.source_affine_locally_of_source_open_cover f \ud835\udcb0 h\ud835\udcb0 },\n  tfae_finish\nend\n\nlemma open_cover_tfae {X Y : Scheme.{u}} [is_affine Y] (f : X \u27f6 Y) :\n  tfae [source_affine_locally @P f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} X), \u2200 (i : \ud835\udcb0.J), source_affine_locally @P (\ud835\udcb0.map i \u226b f),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} X) (i : \ud835\udcb0.J), source_affine_locally @P (\ud835\udcb0.map i \u226b f),\n    \u2200 {U : Scheme} (g : U \u27f6 X) [is_open_immersion g], source_affine_locally @P (g \u226b f)] :=\nbegin\n  tfae_have : 1 \u2192 4,\n  { intros H U g hg V,\n    resetI,\n    rw (hP.affine_open_cover_tfae f).out 0 3 at H,\n    haveI : is_affine _ := V.2,\n    rw \u2190 category.assoc,\n    apply H },\n  tfae_have : 4 \u2192 3,\n  { intros H \ud835\udcb0 _ i, resetI, apply H },\n  tfae_have : 3 \u2192 2,\n  { intro H, refine \u27e8X.affine_cover, H _\u27e9 },\n  tfae_have : 2 \u2192 1,\n  { rintro \u27e8\ud835\udcb0, h\ud835\udcb0\u27e9,\n    rw (hP.affine_open_cover_tfae f).out 0 1,\n    refine \u27e8\ud835\udcb0.bind (\u03bb _, Scheme.affine_cover _), _, _\u27e9,\n    { intro i, dsimp, apply_instance },\n    { intro i,\n      specialize h\ud835\udcb0 i.1,\n      rw (hP.affine_open_cover_tfae (\ud835\udcb0.map i.fst \u226b f)).out 0 3 at h\ud835\udcb0,\n      erw category.assoc,\n      apply @@h\ud835\udcb0 _ (show _, from _),\n      dsimp, apply_instance } },\n  tfae_finish\nend\n\nlemma source_affine_locally_comp_of_is_open_immersion\n  {X Y Z : Scheme.{u}} [is_affine Z] (f : X \u27f6 Y) (g : Y \u27f6 Z) [is_open_immersion f]\n  (H : source_affine_locally @P g) : source_affine_locally @P (f \u226b g) :=\nby apply ((hP.open_cover_tfae g).out 0 3).mp H\n\nlemma source_affine_open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  [is_affine Y] (\ud835\udcb0 : Scheme.open_cover.{u} X) [\u2200 i, is_affine (\ud835\udcb0.obj i)] :\n  source_affine_locally @P f \u2194 (\u2200 i, P (Scheme.\u0393.map (\ud835\udcb0.map i \u226b f).op)) :=\n\u27e8\u03bb H, let h := ((hP.affine_open_cover_tfae f).out 0 2).mp H in h \ud835\udcb0,\n  \u03bb H, let h := ((hP.affine_open_cover_tfae f).out 1 0).mp in h \u27e8\ud835\udcb0, infer_instance, H\u27e9\u27e9\n\nlemma is_local_source_affine_locally :\n  (source_affine_locally @P).is_local :=\nsource_affine_locally_is_local hP.respects_iso hP.localization_preserves\n  (@ring_hom.property_is_local.of_localization_span _ hP)\n\nlemma is_local_affine_locally :\n  property_is_local_at_target (affine_locally @P) :=\nhP.is_local_source_affine_locally.target_affine_locally_is_local\n\nlemma affine_open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} Y) [\u2200 i, is_affine (\ud835\udcb0.obj i)]\n  (\ud835\udcb0' : \u2200 i, Scheme.open_cover.{u} ((\ud835\udcb0.pullback_cover f).obj i)) [\u2200 i j, is_affine ((\ud835\udcb0' i).obj j)] :\n  affine_locally @P f \u2194\n    (\u2200 i j, P (Scheme.\u0393.map ((\ud835\udcb0' i).map j \u226b pullback.snd).op)) :=\n(hP.is_local_source_affine_locally.affine_open_cover_iff f \ud835\udcb0).trans\n    (forall_congr (\u03bb i, hP.source_affine_open_cover_iff _ (\ud835\udcb0' i)))\n\nlemma source_open_cover_iff {X Y : Scheme.{u}} (f : X \u27f6 Y)\n  (\ud835\udcb0 : Scheme.open_cover.{u} X) :\n  affine_locally @P f \u2194 \u2200 i, affine_locally @P (\ud835\udcb0.map i \u226b f) :=\nbegin\n  split,\n  { intros H i U,\n    rw morphism_restrict_comp,\n    delta morphism_restrict,\n    apply hP.source_affine_locally_comp_of_is_open_immersion,\n    apply H },\n  { intros H U,\n    haveI : is_affine _ := U.2,\n    apply ((hP.open_cover_tfae (f \u2223_ U.1)).out 1 0).mp,\n    use \ud835\udcb0.pullback_cover (X.of_restrict _),\n    intro i,\n    specialize H i U,\n    rw morphism_restrict_comp at H,\n    delta morphism_restrict at H,\n    have := source_affine_locally_respects_iso hP.respects_iso,\n    rw [category.assoc, affine_cancel_left_is_iso this, \u2190 affine_cancel_left_is_iso\n      this (pullback_symmetry _ _).hom, pullback_symmetry_hom_comp_snd_assoc] at H,\n    exact H }\nend\n\nlemma affine_locally_of_is_open_immersion (hP : ring_hom.property_is_local @P) {X Y : Scheme}\n  (f : X \u27f6 Y) [hf : is_open_immersion f] : affine_locally @P f :=\nbegin\n  intro U,\n  haveI H : is_affine _ := U.2,\n  rw \u2190 category.comp_id (f \u2223_ U),\n  apply hP.source_affine_locally_comp_of_is_open_immersion,\n  rw hP.source_affine_open_cover_iff _ (Scheme.open_cover_of_is_iso (\ud835\udfd9 _)),\n  { intro i, erw [category.id_comp, op_id, Scheme.\u0393.map_id],\n    convert hP.holds_for_localization_away _ (1 : Scheme.\u0393.obj _),\n    { exact (ring_hom.algebra_map_to_algebra _).symm },\n    { apply_instance },\n    { refine is_localization.away_of_is_unit_of_bijective _ is_unit_one function.bijective_id } },\n  { intro i, exact H }\nend\n\nlemma affine_locally_of_comp\n  (H : \u2200 {R S T : Type.{u}} [comm_ring R] [comm_ring S] [comm_ring T], by exactI\n    \u2200 (f : R \u2192+* S) (g : S \u2192+* T), P (g.comp f) \u2192 P g)\n  {X Y Z : Scheme} {f : X \u27f6 Y} {g : Y \u27f6 Z} (h : affine_locally @P (f \u226b g)) :\n  affine_locally @P f :=\nbegin\n  let \ud835\udcb0 : \u2200 i, ((Z.affine_cover.pullback_cover (f \u226b g)).obj i).open_cover,\n  { intro i,\n    refine Scheme.open_cover.bind _ (\u03bb i, Scheme.affine_cover _),\n    apply Scheme.open_cover.pushforward_iso _\n    (pullback_right_pullback_fst_iso g (Z.affine_cover.map i) f).hom,\n    apply Scheme.pullback.open_cover_of_right,\n    exact (pullback g (Z.affine_cover.map i)).affine_cover },\n  haveI h\ud835\udcb0 : \u2200 i j, is_affine ((\ud835\udcb0 i).obj j), by { dsimp, apply_instance },\n  let \ud835\udcb0' := (Z.affine_cover.pullback_cover g).bind (\u03bb i, Scheme.affine_cover _),\n  haveI h\ud835\udcb0' : \u2200 i, is_affine (\ud835\udcb0'.obj i), by { dsimp, apply_instance },\n  rw hP.affine_open_cover_iff f \ud835\udcb0' (\u03bb i, Scheme.affine_cover _),\n  rw hP.affine_open_cover_iff (f \u226b g) Z.affine_cover \ud835\udcb0 at h,\n  rintros \u27e8i, j\u27e9 k,\n  dsimp at i j k,\n  specialize h i \u27e8j, k\u27e9,\n  dsimp only [Scheme.open_cover.bind_map, Scheme.open_cover.pushforward_iso_obj,\n    Scheme.pullback.open_cover_of_right_obj, Scheme.open_cover.pushforward_iso_map,\n    Scheme.pullback.open_cover_of_right_map, Scheme.open_cover.bind_obj,\n    Scheme.open_cover.pullback_cover_obj, Scheme.open_cover.pullback_cover_map] at h \u22a2,\n  rw [category.assoc, category.assoc, pullback_right_pullback_fst_iso_hom_snd,\n    pullback.lift_snd_assoc, category.assoc, \u2190 category.assoc, op_comp, functor.map_comp] at h,\n  exact H _ _ h,\nend\n\nlemma affine_locally_stable_under_composition :\n  (affine_locally @P).stable_under_composition :=\nbegin\n  intros X Y S f g hf hg,\n  let \ud835\udcb0 : \u2200 i, ((S.affine_cover.pullback_cover (f \u226b g)).obj i).open_cover,\n  { intro i,\n    refine Scheme.open_cover.bind _ (\u03bb i, Scheme.affine_cover _),\n    apply Scheme.open_cover.pushforward_iso _\n    (pullback_right_pullback_fst_iso g (S.affine_cover.map i) f).hom,\n    apply Scheme.pullback.open_cover_of_right,\n    exact (pullback g (S.affine_cover.map i)).affine_cover },\n  rw hP.affine_open_cover_iff (f \u226b g) S.affine_cover _,\n  rotate,\n  { exact \ud835\udcb0 },\n  { intros i j, dsimp at *, apply_instance },\n  { rintros i \u27e8j, k\u27e9,\n    dsimp at i j k,\n    dsimp only [Scheme.open_cover.bind_map, Scheme.open_cover.pushforward_iso_obj,\n      Scheme.pullback.open_cover_of_right_obj, Scheme.open_cover.pushforward_iso_map,\n      Scheme.pullback.open_cover_of_right_map, Scheme.open_cover.bind_obj],\n    rw [category.assoc, category.assoc, pullback_right_pullback_fst_iso_hom_snd,\n      pullback.lift_snd_assoc, category.assoc, \u2190 category.assoc, op_comp, functor.map_comp],\n    apply hP.stable_under_composition,\n    { exact (hP.affine_open_cover_iff _ _ _).mp hg _ _ },\n    { delta affine_locally at hf,\n      rw (hP.is_local_source_affine_locally.affine_open_cover_tfae f).out 0 3 at hf,\n      specialize hf ((pullback g (S.affine_cover.map i)).affine_cover.map j \u226b pullback.fst),\n      rw (hP.affine_open_cover_tfae (pullback.snd : pullback f ((pullback g (S.affine_cover.map i))\n        .affine_cover.map j \u226b pullback.fst) \u27f6 _)).out 0 3 at hf,\n      apply hf } }\nend\n\nend ring_hom.property_is_local\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/algebraic_geometry/morphisms/ring_hom_properties.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6187804337438502, "lm_q1q2_score": 0.35734283255473975}}
{"text": "/-\nHere is an artificial example where temporary metavariables created by typeclass resolution\nwould otherwise leak into nested typeclass resolution unless we catch it.\n-/\n\nclass Foo  (\u03b1 : Type) : Type := (x : Unit)\nclass Bar  (\u03b1 : Type) : Type := (x : Unit)\nclass HasParam (\u03b1 : Type) [Bar \u03b1] : Type := (x : Unit)\n\ninstance FooToBar (\u03b1 : Type) [Foo \u03b1] : Bar \u03b1 := {x:=()}\n\n-- Note: there is an implicit `FooToBar` inside the second argument of the return type.\ninstance HasParamInst (\u03b1 : Type) [h : Foo \u03b1] : HasParam \u03b1 := {x:=()}\n\nclass Top : Type := (x : Unit)\n\n-- Note: `AllFoo` is a weird, universal instance.\ninstance AllFoo (\u03b1 : Type) : Foo \u03b1 := {x:=()}\n\n/-\nWhen the subgoal `[@HasParam \u03b1 (@FooToBar \u03b1 (AllFoo \u03b1))]` is triggerred, `\u03b1` is not known.\n\nThis happens for two reasons:\n\n1. The return type `Top` does not include `\u03b1`.\n2. The universal `AllFoo \u03b1` instance obviates the need to take a building-block class as argument.\n\nThus we would be tempted to call nested typeclass resolution on `Foo <tmp-mvar>`.\n-/\ninstance Bad (\u03b1 : Type) [HasParam \u03b1] : Top := Top.mk ()\ndef foo [Top] : Unit := ()\n\nset_option pp.all true\nset_option trace.class_instances true\nset_option trace.type_context.complete_instance true\n\n#check @foo _\n\n/-\n[class_instances]  class-instance resolution trace\n[class_instances] (0) ?x_0 : Top := @Bad ?x_1 ?x_2\n[class_instances] (1) ?x_2 : @HasParam ?x_1 (@FooToBar ?x_1 (AllFoo ?x_1)) := @HasParamInst ?x_3 ?x_4\n[type_context.complete_instance] would have synthed: Foo ?x_3\n-/\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/elabissues/leaky_tmp_metavars.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3573428325547397}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\nimport category_theory.category.Cat\nimport category_theory.category.preorder\nimport category_theory.concrete_category.bundled_hom\nimport order.hom.basic\n\n/-!\n# Category of preorders\n\nThis defines `Preord`, the category of preorders with monotone maps.\n-/\n\nuniverse u\n\nopen category_theory\n\n/-- The category of preorders. -/\ndef Preord := bundled preorder\n\nnamespace Preord\n\ninstance : bundled_hom @order_hom :=\n{ to_fun := @order_hom.to_fun,\n  id := @order_hom.id,\n  comp := @order_hom.comp,\n  hom_ext := @order_hom.ext }\n\nattribute [derive [large_category, concrete_category]] Preord\n\ninstance : has_coe_to_sort Preord Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled Preord from the underlying type and typeclass. -/\ndef of (\u03b1 : Type*) [preorder \u03b1] : Preord := bundled.of \u03b1\n\n@[simp] lemma coe_of (\u03b1 : Type*) [preorder \u03b1] : \u21a5(of \u03b1) = \u03b1 := rfl\n\ninstance : inhabited Preord := \u27e8of punit\u27e9\n\ninstance (\u03b1 : Preord) : preorder \u03b1 := \u03b1.str\n\n/-- Constructs an equivalence between preorders from an order isomorphism between them. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : Preord.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def dual : Preord \u2964 Preord :=\n{ obj := \u03bb X, of X\u1d52\u1d48, map := \u03bb X Y, order_hom.dual }\n\n/-- The equivalence between `Preord` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : Preord \u224c Preord :=\nequivalence.mk dual dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend Preord\n\n/--\nThe embedding of `Preord` into `Cat`.\n-/\n@[simps]\ndef Preord_to_Cat : Preord.{u} \u2964 Cat :=\n{ obj := \u03bb X, Cat.of X.1,\n  map := \u03bb X Y f, f.monotone.functor,\n  map_id' := \u03bb X, begin apply category_theory.functor.ext, tidy end,\n  map_comp' := \u03bb X Y Z f g, begin apply category_theory.functor.ext, tidy end }\n\ninstance : faithful Preord_to_Cat.{u} :=\n{ map_injective' := \u03bb X Y f g h, begin ext x, exact functor.congr_obj h x end }\n\ninstance : full Preord_to_Cat.{u} :=\n{ preimage := \u03bb X Y f, \u27e8f.obj, f.monotone\u27e9,\n  witness' := \u03bb X Y f, begin apply category_theory.functor.ext, tidy end }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/order/category/Preord.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.6187804407739559, "lm_q1q2_score": 0.3573428276155432}}
{"text": "/-\nCopyright (c) 2016 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Std.Classes.SetNotation\nimport Std.Tactic.NoMatch\nimport Std.Data.Option.Init.Lemmas\nimport Std.Data.Array.Init.Lemmas\n\nnamespace List\n\n/-! ## Tail recursive implementations for definitions from core -/\n\n/-- Tail recursive version of `erase`. -/\n@[inline] def setTR (l : List \u03b1) (n : Nat) (a : \u03b1) : List \u03b1 := go l n #[] where\n  /-- Auxiliary for `setTR`: `setTR.go l a xs n acc = acc.toList ++ set xs a`,\n  unless `n \u2265 l.length` in which case it returns `l` -/\n  go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, _ => l\n  | _::xs, 0, acc => acc.toListAppend (a::xs)\n  | x::xs, n+1, acc => go xs n (acc.push x)\n\n@[csimp] theorem set_eq_setTR : @set = @setTR := by\n  funext \u03b1 l n a; simp [setTR]\n  let rec go (acc) : \u2200 xs n, l = acc.data ++ xs \u2192\n    setTR.go l a xs n acc = acc.data ++ xs.set n a\n  | [], _ => fun h => by simp [setTR.go, set, h]\n  | x::xs, 0 => by simp [setTR.go, set]\n  | x::xs, n+1 => fun h => by simp [setTR.go, set]; rw [go _ xs]; {simp}; simp [h]\n  exact (go #[] _ _ rfl).symm\n\n/-- Tail recursive version of `erase`. -/\n@[inline] def eraseTR [BEq \u03b1] (l : List \u03b1) (a : \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `eraseTR`: `eraseTR.go l a xs acc = acc.toList ++ erase xs a`,\n  unless `a` is not present in which case it returns `l` -/\n  go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => l\n  | x::xs, acc => bif x == a then acc.toListAppend xs else go xs (acc.push x)\n\n@[csimp] theorem erase_eq_eraseTR : @List.erase = @eraseTR := by\n  funext \u03b1 _ l a; simp [eraseTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192 eraseTR.go l a xs acc = acc.data ++ xs.erase a from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs with intro acc h\n  | nil => simp [List.erase, eraseTR.go, h]\n  | cons x xs IH =>\n    simp [List.erase, eraseTR.go]\n    cases x == a <;> simp\n    \u00b7 rw [IH]; simp; simp; exact h\n\n/-- Tail recursive version of `eraseIdx`. -/\n@[inline] def eraseIdxTR (l : List \u03b1) (n : Nat) : List \u03b1 := go l n #[] where\n  /-- Auxiliary for `eraseIdxTR`: `eraseIdxTR.go l n xs acc = acc.toList ++ eraseIdx xs a`,\n  unless `a` is not present in which case it returns `l` -/\n  go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, _ => l\n  | _::as, 0, acc => acc.toListAppend as\n  | a::as, n+1, acc => go as n (acc.push a)\n\n@[csimp] theorem eraseIdx_eq_eraseIdxTR : @eraseIdx = @eraseIdxTR := by\n  funext \u03b1 l n; simp [eraseIdxTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192 eraseIdxTR.go l xs n acc = acc.data ++ xs.eraseIdx n from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs generalizing n with intro acc h\n  | nil => simp [eraseIdx, eraseIdxTR.go, h]\n  | cons x xs IH =>\n    match n with\n    | 0 => simp [eraseIdx, eraseIdxTR.go]\n    | n+1 =>\n      simp [eraseIdx, eraseIdxTR.go]\n      rw [IH]; simp; simp; exact h\n\n/-- Tail recursive version of `bind`. -/\n@[inline] def bindTR (as : List \u03b1) (f : \u03b1 \u2192 List \u03b2) : List \u03b2 := go as #[] where\n  /-- Auxiliary for `bind`: `bind.go f as = acc.toList ++ bind f as` -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b2 \u2192 List \u03b2\n  | [], acc => acc.toList\n  | x::xs, acc => go xs (acc ++ f x)\n\n@[csimp] theorem bind_eq_bindTR : @List.bind = @bindTR := by\n  funext \u03b1 \u03b2 as f\n  let rec go : \u2200 as acc, bindTR.go f as acc = acc.data ++ as.bind f\n    | [], acc => by simp [bindTR.go, bind]\n    | x::xs, acc => by simp [bindTR.go, bind, go xs]\n  exact (go as #[]).symm\n\n/-- Tail recursive version of `join`. -/\n@[inline] def joinTR (l : List (List \u03b1)) : List \u03b1 := bindTR l id\n\n@[csimp] theorem join_eq_joinTR : @join = @joinTR := by\n  funext \u03b1 l; rw [\u2190 List.bind_id, List.bind_eq_bindTR]; rfl\n\n/-- Tail recursive version of `filterMap`. -/\n@[inline] def filterMapTR (f : \u03b1 \u2192 Option \u03b2) (l : List \u03b1) : List \u03b2 := go l #[] where\n  /-- Auxiliary for `filterMap`: `filterMap.go f l = acc.toList ++ filterMap f l` -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b2 \u2192 List \u03b2\n  | [], acc => acc.toList\n  | a::as, acc => match f a with\n    | none => go as acc\n    | some b => go as (acc.push b)\n\n@[csimp] theorem filterMap_eq_filterMapTR : @List.filterMap = @filterMapTR := by\n  funext \u03b1 \u03b2 f l\n  let rec go : \u2200 as acc, filterMapTR.go f as acc = acc.data ++ as.filterMap f\n    | [], acc => by simp [filterMapTR.go, filterMap]\n    | a::as, acc => by simp [filterMapTR.go, filterMap, go as]; split <;> simp [*]\n  exact (go l #[]).symm\n\n/-- Tail recursive version of `replace`. -/\n@[inline] def replaceTR [BEq \u03b1] (l : List \u03b1) (b c : \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `replace`: `replace.go l b c xs acc = acc.toList ++ replace xs b c`,\n  unless `b` is not found in `xs` in which case it returns `l`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => l\n  | a::as, acc => bif a == b then acc.toListAppend (c::as) else go as (acc.push a)\n\n@[csimp] theorem replace_eq_replaceTR : @List.replace = @replaceTR := by\n  funext \u03b1 _ l b c; simp [replaceTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192\n      replaceTR.go l b c xs acc = acc.data ++ xs.replace b c from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs with intro acc\n  | nil => simp [replace, replaceTR.go]\n  | cons x xs IH =>\n    simp [replace, replaceTR.go]; split <;> simp [*]\n    \u00b7 intro h; rw [IH]; simp; simp; exact h\n\n/-- Tail recursive version of `take`. -/\n@[inline] def takeTR (n : Nat) (l : List \u03b1) : List \u03b1 := go l n #[] where\n  /-- Auxiliary for `take`: `take.go l xs n acc = acc.toList ++ take n xs`,\n  unless `n \u2265 xs.length` in which case it returns `l`. -/\n  @[specialize] go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, _ => l\n  | _::_, 0, acc => acc.toList\n  | a::as, n+1, acc => go as n (acc.push a)\n\n@[csimp] theorem take_eq_takeTR : @take = @takeTR := by\n  funext \u03b1 n l; simp [takeTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192 takeTR.go l xs n acc = acc.data ++ xs.take n from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs generalizing n with intro acc\n  | nil => cases n <;> simp [take, takeTR.go]\n  | cons x xs IH =>\n    cases n with simp [take, takeTR.go]\n    | succ n => intro h; rw [IH]; simp; simp; exact h\n\n/-- Tail recursive version of `takeWhile`. -/\n@[inline] def takeWhileTR (p : \u03b1 \u2192 Bool) (l : List \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `takeWhile`: `takeWhile.go p l xs acc = acc.toList ++ takeWhile p xs`,\n  unless no element satisfying `p` is found in `xs` in which case it returns `l`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => l\n  | a::as, acc => bif p a then go as (acc.push a) else acc.toList\n\n@[csimp] theorem takeWhile_eq_takeWhileTR : @takeWhile = @takeWhileTR := by\n  funext \u03b1 p l; simp [takeWhileTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192\n      takeWhileTR.go p l xs acc = acc.data ++ xs.takeWhile p from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs with intro acc\n  | nil => simp [takeWhile, takeWhileTR.go]\n  | cons x xs IH =>\n    simp [takeWhile, takeWhileTR.go]; split <;> simp [*]\n    \u00b7 intro h; rw [IH]; simp; simp; exact h\n\n/-- Tail recursive version of `foldr`. -/\n@[specialize] def foldrTR (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (init : \u03b2) (l : List \u03b1) : \u03b2 := l.toArray.foldr f init\n\n@[csimp] theorem foldr_eq_foldrTR : @foldr = @foldrTR := by\n  funext \u03b1 \u03b2 f init l; simp [foldrTR, Array.foldr_eq_foldr_data, -Array.size_toArray]\n\n/-- Tail recursive version of `zipWith`. -/\n@[inline] def zipWithTR (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (as : List \u03b1) (bs : List \u03b2) : List \u03b3 := go as bs #[] where\n  /-- Auxiliary for `zipWith`: `zipWith.go f as bs acc = acc.toList ++ zipWith f as bs` -/\n  go : List \u03b1 \u2192 List \u03b2 \u2192 Array \u03b3 \u2192 List \u03b3\n  | a::as, b::bs, acc => go as bs (acc.push (f a b))\n  | _, _, acc => acc.toList\n\n@[csimp] theorem zipWith_eq_zipWithTR : @zipWith = @zipWithTR := by\n  funext \u03b1 \u03b2 \u03b3 f as bs\n  let rec go : \u2200 as bs acc, zipWithTR.go f as bs acc = acc.data ++ as.zipWith f bs\n    | [], _, acc | _::_, [], acc => by simp [zipWithTR.go, zipWith]\n    | a::as, b::bs, acc => by simp [zipWithTR.go, zipWith, go as bs]\n  exact (go as bs #[]).symm\n\n/-- Tail recursive version of `unzip`. -/\ndef unzipTR (l : List (\u03b1 \u00d7 \u03b2)) : List \u03b1 \u00d7 List \u03b2 :=\n  l.foldr (fun (a, b) (al, bl) => (a::al, b::bl)) ([], [])\n\n@[csimp] theorem unzip_eq_unzipTR : @unzip = @unzipTR := by\n  funext \u03b1 \u03b2 l; simp [unzipTR]; induction l <;> simp [*]\n\n/-- Tail recursive version of `enumFrom`. -/\ndef enumFromTR (n : Nat) (l : List \u03b1) : List (Nat \u00d7 \u03b1) :=\n  let arr := l.toArray\n  (arr.foldr (fun a (n, acc) => (n-1, (n-1, a) :: acc)) (n + arr.size, [])).2\n\n@[csimp] theorem enumFrom_eq_enumFromTR : @enumFrom = @enumFromTR := by\n  funext \u03b1 n l; simp [enumFromTR, -Array.size_toArray]\n  let f := fun (a : \u03b1) (n, acc) => (n-1, (n-1, a) :: acc)\n  let rec go : \u2200 l n, l.foldr f (n + l.length, []) = (n, enumFrom n l)\n    | [], n => rfl\n    | a::as, n => by\n      rw [\u2190 show _ + as.length = n + (a::as).length from Nat.succ_add .., foldr, go as]; simp; rfl\n  rw [Array.foldr_eq_foldr_data]; simp [go]\n\ntheorem replicateTR_loop_eq : \u2200 n, replicateTR.loop a n acc = replicate n a ++ acc\n  | 0 => rfl\n  | n+1 => by rw [\u2190 replicateTR_loop_replicate_eq _ 1 n, replicate, replicate,\n    replicateTR.loop, replicateTR_loop_eq n, replicateTR_loop_eq n, append_assoc]; rfl\n\n/-- Tail recursive version of `dropLast`. -/\n@[inline] def dropLastTR (l : List \u03b1) : List \u03b1 := l.toArray.pop.toList\n\n@[csimp] theorem dropLast_eq_dropLastTR : @dropLast = @dropLastTR := by\n  funext \u03b1 l; simp [dropLastTR]\n\n/-- Tail recursive version of `intersperse`. -/\ndef intersperseTR (sep : \u03b1) : List \u03b1 \u2192 List \u03b1\n  | [] => []\n  | [x] => [x]\n  | x::y::xs => x :: sep :: y :: xs.foldr (fun a r => sep :: a :: r) []\n\n@[csimp] theorem intersperse_eq_intersperseTR : @intersperse = @intersperseTR := by\n  funext \u03b1 sep l; simp [intersperseTR]\n  match l with\n  | [] | [_] => rfl\n  | x::y::xs => simp [intersperse]; induction xs generalizing y <;> simp [*]\n\n/-- Tail recursive version of `intercalate`. -/\ndef intercalateTR (sep : List \u03b1) : List (List \u03b1) \u2192 List \u03b1\n  | [] => []\n  | [x] => x\n  | x::xs => go sep.toArray x xs #[]\nwhere\n  /-- Auxiliary for `intercalateTR`:\n  `intercalateTR.go sep x xs acc = acc.toList ++ intercalate sep.toList (x::xs)` -/\n  go (sep : Array \u03b1) : List \u03b1 \u2192 List (List \u03b1) \u2192 Array \u03b1 \u2192 List \u03b1\n  | x, [], acc => acc.toListAppend x\n  | x, y::xs, acc => go sep y xs (acc ++ x ++ sep)\n\n@[csimp] theorem intercalate_eq_intercalateTR : @intercalate = @intercalateTR := by\n  funext \u03b1 sep l; simp [intercalate, intercalateTR]\n  match l with\n  | [] => rfl\n  | [_] => simp\n  | x::y::xs =>\n    let rec go {acc x} : \u2200 xs,\n      intercalateTR.go sep.toArray x xs acc = acc.data ++ join (intersperse sep (x::xs))\n    | [] => by simp [intercalateTR.go]\n    | _::_ => by simp [intercalateTR.go, go]\n    simp [intersperse, go]\n\n/-! ## New definitions -/\n\n/--\n`l\u2081 \u2286 l\u2082` means that every element of `l\u2081` is also an element of `l\u2082`, ignoring multiplicity.\n-/\nprotected def Subset (l\u2081 l\u2082 : List \u03b1) := \u2200 \u2983a : \u03b1\u2984, a \u2208 l\u2081 \u2192 a \u2208 l\u2082\n\ninstance : HasSubset (List \u03b1) := \u27e8List.Subset\u27e9\n\ninstance decidableBEx (p : \u03b1 \u2192 Prop) [DecidablePred p] :\n    \u2200 l : List \u03b1, Decidable (\u2203 x \u2208 l, p x)\n  | [] => isFalse fun.\n  | x :: xs =>\n    if h\u2081 : p x then isTrue \u27e8x, .head .., h\u2081\u27e9 else\n      match decidableBEx p xs with\n      | isTrue h\u2082 => isTrue <| let \u27e8y, hm, hp\u27e9 := h\u2082; \u27e8y, .tail _ hm, hp\u27e9\n      | isFalse h\u2082 => isFalse fun\n        | \u27e8y, .tail _ h, hp\u27e9 => h\u2082 \u27e8y, h, hp\u27e9\n        | \u27e8_, .head .., hp\u27e9 => h\u2081 hp\n\ninstance decidableBAll (p : \u03b1 \u2192 Prop) [DecidablePred p] :\n    \u2200 l : List \u03b1, Decidable (\u2200 x \u2208 l, p x)\n  | [] => isTrue fun.\n  | x :: xs =>\n    if h\u2081 : p x then\n      match decidableBAll p xs with\n      | isTrue h\u2082 => isTrue fun\n        | y, .tail _ h => h\u2082 y h\n        | _, .head .. => h\u2081\n      | isFalse h\u2082 => isFalse fun H => h\u2082 fun y hm => H y (.tail _ hm)\n    else isFalse fun H => h\u2081 <| H x (.head ..)\n\n/--\nComputes the \"bag intersection\" of `l\u2081` and `l\u2082`, that is,\nthe collection of elements of `l\u2081` which are also in `l\u2082`. As each element\nis identified, it is removed from `l\u2082`, so elements are counted with multiplicity.\n-/\nprotected def bagInter {\u03b1} [BEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n  | [], _ => []\n  | _, [] => []\n  | a :: l\u2081, l\u2082 => if l\u2082.elem a then a :: List.bagInter l\u2081 (l\u2082.erase a) else List.bagInter l\u2081 l\u2082\n\n/-- Computes the difference of `l\u2081` and `l\u2082`, by removing each element in `l\u2082` from `l\u2081`. -/\nprotected def diff {\u03b1} [BEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b1\n  | l, [] => l\n  | l\u2081, a :: l\u2082 => if l\u2081.elem a then List.diff (l\u2081.erase a) l\u2082 else List.diff l\u2081 l\u2082\n\nopen Option Nat\n\n/-- Get the tail of a nonempty list, or return `[]` for `[]`. -/\ndef tail : List \u03b1 \u2192 List \u03b1\n  | []    => []\n  | _::as => as\n\n-- FIXME: `@[simp]` on the definition simplifies even `tail l`\n@[simp] theorem tail_nil : @tail \u03b1 [] = [] := rfl\n@[simp] theorem tail_cons : @tail \u03b1 (a::as) = as := rfl\n\n/-- Get the head and tail of a list, if it is nonempty. -/\n@[inline] def next? : List \u03b1 \u2192 Option (\u03b1 \u00d7 List \u03b1)\n  | [] => none\n  | a :: l => some (a, l)\n\n/--\nGiven a function `f : Nat \u2192 \u03b1 \u2192 \u03b2` and `as : list \u03b1`, `as = [a\u2080, a\u2081, ...]`, returns the list\n`[f 0 a\u2080, f 1 a\u2081, ...]`.\n-/\n@[inline] def mapIdx (f : Nat \u2192 \u03b1 \u2192 \u03b2) (as : List \u03b1) : List \u03b2 := go as #[] where\n  /-- Auxiliary for `mapIdx`:\n  `mapIdx.go [a\u2080, a\u2081, ...] acc = acc.toList ++ [f acc.size a\u2080, f (acc.size + 1) a\u2081, ...]` -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b2 \u2192 List \u03b2\n  | [], acc => acc.toList\n  | a :: as, acc => go as (acc.push (f acc.size a))\n\n/-- Monadic variant of `mapIdx`. -/\n@[inline] def mapIdxM {m : Type v \u2192 Type w} [Monad m]\n    (as : List \u03b1) (f : Nat \u2192 \u03b1 \u2192 m \u03b2) : m (List \u03b2) := go as #[] where\n  /-- Auxiliary for `mapIdxM`:\n  `mapIdxM.go as f acc = acc.toList ++ [\u2190 f acc.size a\u2080, \u2190 f (acc.size + 1) a\u2081, ...]` -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b2 \u2192 m (List \u03b2)\n  | [], acc => pure acc.toList\n  | a :: as, acc => do go as (acc.push (\u2190 f acc.size a))\n\n/--\n`after p xs` is the suffix of `xs` after the first element that satisfies\n`p`, not including that element.\n```lean\nafter      (\u00b7 == 1) [0, 1, 2, 3] = [2, 3]\ndrop_while (\u00b7 != 1) [0, 1, 2, 3] = [1, 2, 3]\n```\n-/\n@[specialize] def after (p : \u03b1 \u2192 Bool) : List \u03b1 \u2192 List \u03b1\n  | [] => []\n  | x :: xs => bif p x then xs else after p xs\n\n/-- Returns the index of the first element satisfying `p`, or the length of the list otherwise. -/\n@[inline] def findIdx (p : \u03b1 \u2192 Bool) (l : List \u03b1) : Nat := go l 0 where\n  /-- Auxiliary for `findIdx`: `findIdx.go p l n = findIdx p l + n` -/\n  @[specialize] go : List \u03b1 \u2192 Nat \u2192 Nat\n  | [], n => n\n  | a :: l, n => bif p a then n else go l (n + 1)\n\n/-- Returns the index of the first element equal to `a`, or the length of the list otherwise. -/\ndef indexOf [BEq \u03b1] (a : \u03b1) : List \u03b1 \u2192 Nat := findIdx (a == \u00b7)\n\n/-- Removes the `n`th element of `l`, or the original list if `n` is out of bounds. -/\n@[simp] def removeNth : List \u03b1 \u2192 Nat \u2192 List \u03b1\n  | [], _ => []\n  | _ :: xs, 0 => xs\n  | x :: xs, i+1 => x :: removeNth xs i\n\n/-- Tail recursive version of `removeNth`. -/\n@[inline] def removeNthTR (l : List \u03b1) (n : Nat) : List \u03b1 := go l n #[] where\n  /-- Auxiliary for `removeNthTR`:\n  `removeNthTR.go l xs n acc = acc.toList ++ removeNth xs n` if `n < length xs`, else `l`. -/\n  go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, _ => l\n  | _ :: xs, 0, acc => acc.toListAppend xs\n  | x :: xs, i+1, acc => go xs i (acc.push x)\n\n@[csimp] theorem removeNth_eq_removeNthTR : @removeNth = @removeNthTR := by\n  funext \u03b1 l n; simp [removeNthTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192\n      removeNthTR.go l xs n acc = acc.data ++ xs.removeNth n from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs generalizing n with intro acc\n  | nil => simp [removeNth, removeNthTR.go]\n  | cons x xs IH =>\n    cases n <;> simp [removeNth, removeNthTR.go, *]\n    \u00b7 intro h; rw [IH]; simp; simp; exact h\n\n/-- Replaces the first element of the list for which `f` returns `some` with the returned value. -/\n@[simp] def replaceF (f : \u03b1 \u2192 Option \u03b1) : List \u03b1 \u2192 List \u03b1\n  | [] => []\n  | x :: xs => match f x with\n    | none => x :: replaceF f xs\n    | some a => a :: xs\n\n/-- Tail recursive version of `replaceF`. -/\n@[inline] def replaceFTR (f : \u03b1 \u2192 Option \u03b1) (l : List \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `replaceFTR`:\n  `replaceFTR.go f l xs acc = acc.toList ++ replaceF f xs` if `f` returns `some`, else `l`. -/\n  go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => l\n  | x :: xs, acc => match f x with\n    | none => go xs (acc.push x)\n    | some a => acc.toListAppend (a :: xs)\n\n@[csimp] theorem replaceF_eq_replaceFTR : @replaceF = @replaceFTR := by\n  funext \u03b1 f l; simp [replaceFTR]\n  suffices \u2200 xs acc, l = acc.data ++ xs \u2192\n      replaceFTR.go f l xs acc = acc.data ++ xs.replaceF f from\n    (this l #[] (by simp)).symm\n  intro xs; induction xs with intro acc\n  | nil => simp [replaceF, replaceFTR.go]\n  | cons x xs IH =>\n    simp [replaceF, replaceFTR.go]; split <;> simp [*]\n    \u00b7 intro h; rw [IH]; simp; simp; exact h\n\n/-- Inserts an element into a list without duplication. -/\n@[inline] protected def insert [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) : List \u03b1 :=\n  if a \u2208 l then l else a :: l\n\n/--\nConstructs the union of two lists, by inserting the elements of `l\u2081` in reverse order to `l\u2082`.\nAs a result, `l\u2082` will always be a suffix, but only the last occurrence of each element in `l\u2081`\nwill be retained (but order will otherwise be preserved).\n-/\n@[inline] protected def union [DecidableEq \u03b1] (l\u2081 l\u2082 : List \u03b1) : List \u03b1 := foldr .insert l\u2082 l\u2081\n\ninstance [DecidableEq \u03b1] : Union (List \u03b1) := \u27e8List.union\u27e9\n\n/--\nConstructs the intersection of two lists, by filtering the elements of `l\u2081` that are in `l\u2082`.\nUnlike `bagInter` this does not preserve multiplicity: `[1, 1].inter [1]` is `[1, 1]`.\n-/\n@[inline] protected def inter [DecidableEq \u03b1] (l\u2081 l\u2082 : List \u03b1) : List \u03b1 := filter (\u00b7 \u2208 l\u2082) l\u2081\n\ninstance [DecidableEq \u03b1] : Inter (List \u03b1) := \u27e8List.inter\u27e9\n\n/-- `l\u2081 <+ l\u2082`, or `Sublist l\u2081 l\u2082`, says that `l\u2081` is a (non-contiguous) subsequence of `l\u2082`. -/\ninductive Sublist {\u03b1} : List \u03b1 \u2192 List \u03b1 \u2192 Prop\n  /-- the base case: `[]` is a sublist of `[]` -/\n  | slnil : Sublist [] []\n  /-- If `l\u2081` is a subsequence of `l\u2082`, then it is also a subsequence of `a :: l\u2082`. -/\n  | cons a : Sublist l\u2081 l\u2082 \u2192 Sublist l\u2081 (a :: l\u2082)\n  /-- If `l\u2081` is a subsequence of `l\u2082`, then `a :: l\u2081` is a subsequence of `a :: l\u2082`. -/\n  | cons\u2082 a : Sublist l\u2081 l\u2082 \u2192 Sublist (a :: l\u2081) (a :: l\u2082)\n\n@[inherit_doc] scoped infixl:50 \" <+ \" => Sublist\n\n/--\nSplit a list at an index.\n```\nsplitAt 2 [a, b, c] = ([a, b], [c])\n```\n-/\ndef splitAt (n : Nat) (l : List \u03b1) : List \u03b1 \u00d7 List \u03b1 := go l n #[] where\n  /-- Auxiliary for `splitAt`: `splitAt.go l n xs acc = (acc.toList ++ take n xs, drop n xs)`\n  if `n < length xs`, else `(l, [])`. -/\n  go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1 \u00d7 List \u03b1\n  | [], _, _ => (l, [])\n  | x :: xs, n+1, acc => go xs n (acc.push x)\n  | xs, _, acc => (acc.toList, xs)\n\n/--\nSplit a list at an index. Ensures the left list always has the specified length\nby right padding with the provided default element.\n```\nsplitAtD 2 [a, b, c] x = ([a, b], [c])\nsplitAtD 4 [a, b, c] x = ([a, b, c, x], [])\n```\n-/\ndef splitAtD (n : Nat) (l : List \u03b1) (dflt : \u03b1) : List \u03b1 \u00d7 List \u03b1 := go n l #[] where\n  /-- Auxiliary for `splitAtD`: `splitAtD.go dflt n l acc = (acc.toList ++ left, right)`\n  if `splitAtD n l dflt = (left, right)`. -/\n  go : Nat \u2192 List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1 \u00d7 List \u03b1\n  | n+1, x :: xs, acc => go n xs (acc.push x)\n  | 0, xs, acc => (acc.toList, xs)\n  | n, [], acc => (acc.toListAppend (replicate n dflt), [])\n\n/--\nSplit a list at every element satisfying a predicate. The separators are not in the result.\n```\n[1, 1, 2, 3, 2, 4, 4].splitOnP (\u00b7 == 2) = [[1, 1], [3], [4, 4]]\n```\n-/\n@[inline] def splitOnP (P : \u03b1 \u2192 Bool) (l : List \u03b1) : List (List \u03b1) := go l #[] #[] where\n  /-- Auxiliary for `splitOnP`: `splitOnP.go xs acc r = r.toList ++ res'`\n  where `res'` is obtained from `splitOnP P xs` by prepending `acc.toList` to the first element. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 Array (List \u03b1) \u2192 List (List \u03b1)\n  | [], acc, r => r.toListAppend [acc.toList]\n  | h :: t, acc, r => bif P h then go t #[] (r.push acc.toList) else go t (acc.push h) r\n\n/--\nSplit a list at every occurrence of a separator element. The separators are not in the result.\n```\n[1, 1, 2, 3, 2, 4, 4].splitOn 2 = [[1, 1], [3], [4, 4]]\n```\n-/\n@[inline] def splitOn [BEq \u03b1] (a : \u03b1) (as : List \u03b1) : List (List \u03b1) := as.splitOnP (\u00b7 == a)\n\n/--\nApply a function to the nth tail of `l`. Returns the input without\nusing `f` if the index is larger than the length of the List.\n```\nmodifyNthTail f 2 [a, b, c] = [a, b] ++ f [c]\n```\n-/\n@[simp] def modifyNthTail (f : List \u03b1 \u2192 List \u03b1) : Nat \u2192 List \u03b1 \u2192 List \u03b1\n  | 0, l => f l\n  | _+1, [] => []\n  | n+1, a :: l => a :: modifyNthTail f n l\n\n/-- Apply `f` to the head of the list, if it exists. -/\n@[simp, inline] def modifyHead (f : \u03b1 \u2192 \u03b1) : List \u03b1 \u2192 List \u03b1\n  | [] => []\n  | a :: l => f a :: l\n\n/-- Apply `f` to the nth element of the list, if it exists. -/\ndef modifyNth (f : \u03b1 \u2192 \u03b1) : Nat \u2192 List \u03b1 \u2192 List \u03b1 :=\n  modifyNthTail (modifyHead f)\n\n/-- Tail-recursive version of `modifyNth`. -/\ndef modifyNthTR (f : \u03b1 \u2192 \u03b1) (n : Nat) (l : List \u03b1) : List \u03b1 := go l n #[] where\n  /-- Auxiliary for `modifyNthTR`: `modifyNthTR.go f l n acc = acc.toList ++ modifyNth f n l`. -/\n  go : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, acc => acc.toList\n  | a :: l, 0, acc => acc.toListAppend (f a :: l)\n  | a :: l, n+1, acc => go l n (acc.push a)\n\ntheorem modifyNthTR_go_eq : \u2200 l n, modifyNthTR.go f l n acc = acc.data ++ modifyNth f n l\n  | [], n => by cases n <;> simp [modifyNthTR.go, modifyNth]\n  | a :: l, 0 => by simp [modifyNthTR.go, modifyNth]\n  | a :: l, n+1 => by simp [modifyNthTR.go, modifyNth, modifyNthTR_go_eq l]\n\n@[csimp] theorem modifyNth_eq_modifyNthTR : @modifyNth = @modifyNthTR := by\n  funext \u03b1 f n l; simp [modifyNthTR, modifyNthTR_go_eq]\n\n/-- Apply `f` to the last element of `l`, if it exists. -/\n@[inline] def modifyLast (f : \u03b1 \u2192 \u03b1) (l : List \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `modifyLast`: `modifyLast.go f l acc = acc.toList ++ modifyLast f l`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => []\n  | [x], acc => acc.toListAppend [f x]\n  | x :: xs, acc => go xs (acc.push x)\n\n/--\n`insertNth n a l` inserts `a` into the list `l` after the first `n` elements of `l`\n```\ninsertNth 2 1 [1, 2, 3, 4] = [1, 2, 1, 3, 4]\n```\n-/\ndef insertNth (n : Nat) (a : \u03b1) : List \u03b1 \u2192 List \u03b1 :=\n  modifyNthTail (cons a) n\n\n/-- Tail-recursive version of `insertNth`. -/\n@[inline] def insertNthTR (n : Nat) (a : \u03b1) (l : List \u03b1) : List \u03b1 := go n l #[] where\n  /-- Auxiliary for `insertNthTR`: `insertNthTR.go a n l acc = acc.toList ++ insertNth n a l`. -/\n  go : Nat \u2192 List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | 0, l, acc => acc.toListAppend (a :: l)\n  | _, [], acc => acc.toList\n  | n+1, a :: l, acc => go n l (acc.push a)\n\ntheorem insertNthTR_go_eq : \u2200 n l, insertNthTR.go a n l acc = acc.data ++ insertNth n a l\n  | 0, l | _+1, [] => by simp [insertNthTR.go, insertNth]\n  | n+1, a :: l => by simp [insertNthTR.go, insertNth, insertNthTR_go_eq n l]\n\n@[csimp] theorem insertNth_eq_insertNthTR : @insertNth = @insertNthTR := by\n  funext \u03b1 f n l; simp [insertNthTR, insertNthTR_go_eq]\n\n@[simp] theorem headD_eq_head? (l) (a : \u03b1) : headD l a = (head? l).getD a := by cases l <;> rfl\n\n/--\nTake `n` elements from a list `l`. If `l` has less than `n` elements, append `n - length l`\nelements `x`.\n-/\ndef takeD : Nat \u2192 List \u03b1 \u2192 \u03b1 \u2192 List \u03b1\n  | 0, _, _ => []\n  | n+1, l, x => l.headD x :: takeD n l.tail x\n\n@[simp] theorem takeD_zero (l) (a : \u03b1) : takeD 0 l a = [] := rfl\n@[simp] theorem takeD_succ (l) (a : \u03b1) :\n    takeD (n+1) l a = l.head?.getD a :: takeD n l.tail a := by simp [takeD]\n\n@[simp] theorem takeD_nil (n) (a : \u03b1) : takeD n [] a = replicate n a := by induction n <;> simp [*]\n\n/-- Tail-recursive version of `takeD`. -/\ndef takeDTR (n : Nat) (l : List \u03b1) (dflt : \u03b1) : List \u03b1 := go n l #[] where\n  /-- Auxiliary for `takeDTR`: `takeDTR.go dflt n l acc = acc.toList ++ takeD n l dflt`. -/\n  go : Nat \u2192 List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | n+1, x :: xs, acc => go n xs (acc.push x)\n  | 0, _, acc => acc.toList\n  | n, [], acc => acc.toListAppend (replicate n dflt)\n\ntheorem takeDTR_go_eq : \u2200 n l, takeDTR.go dflt n l acc = acc.data ++ takeD n l dflt\n  | 0, _ => by simp [takeDTR.go]\n  | _+1, [] => by simp [takeDTR.go]\n  | _+1, _::l => by simp [takeDTR.go, takeDTR_go_eq _ l]\n\n@[csimp] theorem takeD_eq_takeDTR : @takeD = @takeDTR := by\n  funext \u03b1 f n l; simp [takeDTR, takeDTR_go_eq]\n\n/--\nPads `l : List \u03b1` with repeated occurrences of `a : \u03b1` until it is of length `n`.\nIf `l` is initially larger than `n`, just return `l`.\n-/\ndef leftpad (n : Nat) (a : \u03b1) (l : List \u03b1) : List \u03b1 := replicate (n - length l) a ++ l\n\n/-- Optimized version of `leftpad`. -/\n@[inline] def leftpadTR (n : Nat) (a : \u03b1) (l : List \u03b1) : List \u03b1 :=\n  replicateTR.loop a (n - length l) l\n\n@[csimp] theorem leftpad_eq_leftpadTR : @leftpad = @leftpadTR := by\n  funext \u03b1 n a l; simp [leftpad, leftpadTR, replicateTR_loop_eq]\n\n/--\nFold a function `f` over the list from the left, returning the list of partial results.\n```\nscanl (+) 0 [1, 2, 3] = [0, 1, 3, 6]\n```\n-/\n@[simp] def scanl (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) : List \u03b2 \u2192 List \u03b1\n  | [] => [a]\n  | b :: l => a :: scanl f (f a b) l\n\n/-- Tail-recursive version of `scanl`. -/\n@[inline] def scanlTR (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (l : List \u03b2) : List \u03b1 := go l a #[] where\n  /-- Auxiliary for `scanlTR`: `scanlTR.go f l a acc = acc.toList ++ scanl f a l`. -/\n  @[specialize] go : List \u03b2 \u2192 \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], a, acc => acc.toListAppend [a]\n  | b :: l, a, acc => go l (f a b) (acc.push a)\n\ntheorem scanlTR_go_eq : \u2200 l, scanlTR.go f l a acc = acc.data ++ scanl f a l\n  | [] => by simp [scanlTR.go, scanl]\n  | a :: l => by simp [scanlTR.go, scanl, scanlTR_go_eq l]\n\n@[csimp] theorem scanl_eq_scanlTR : @scanl = @scanlTR := by\n  funext \u03b1 f n l; simp [scanlTR, scanlTR_go_eq]\n\n/--\nFold a function `f` over the list from the right, returning the list of partial results.\n```\nscanr (+) 0 [1, 2, 3] = [6, 5, 3, 0]\n```\n-/\ndef scanr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (l : List \u03b1) : List \u03b2 :=\n  let (b', l') := l.foldr (fun a (b', l') => (f a b', b' :: l')) (b, [])\n  b' :: l'\n\n/--\nGiven a function `f : \u03b1 \u2192 \u03b2 \u2295 \u03b3`, `partitionMap f l` maps the list by `f`\nwhilst partitioning the result it into a pair of lists, `List \u03b2 \u00d7 List \u03b3`,\npartitioning the `.inl _` into the left list, and the `.inr _` into the right List.\n```\npartitionMap (id : Nat \u2295 Nat \u2192 Nat \u2295 Nat) [inl 0, inr 1, inl 2] = ([0, 2], [1])\n```\n-/\n@[inline] def partitionMap (f : \u03b1 \u2192 \u03b2 \u2295 \u03b3) (l : List \u03b1) : List \u03b2 \u00d7 List \u03b3 := go l #[] #[] where\n  /-- Auxiliary for `partitionMap`:\n  `partitionMap.go f l acc\u2081 acc\u2082 = (acc\u2081.toList ++ left, acc\u2082.toList ++ right)`\n  if `partitionMap f l = (left, right)`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b2 \u2192 Array \u03b3 \u2192 List \u03b2 \u00d7 List \u03b3\n  | [], acc\u2081, acc\u2082 => (acc\u2081.toList, acc\u2082.toList)\n  | x :: xs, acc\u2081, acc\u2082 =>\n    match f x with\n    | .inl a => go xs (acc\u2081.push a) acc\u2082\n    | .inr b => go xs acc\u2081 (acc\u2082.push b)\n\n/--\nFold a list from left to right as with `foldl`, but the combining function\nalso receives each element's index.\n-/\n@[simp, specialize] def foldlIdx (f : Nat \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1) (init : \u03b1) : List \u03b2 \u2192 (start : _ := 0) \u2192 \u03b1\n  | [], _ => init\n  | b :: l, i => foldlIdx f (f i init b) l (i+1)\n\n/--\nFold a list from right to left as with `foldr`, but the combining function\nalso receives each element's index.\n-/\n-- TODO(Mario): tail recursive / array-based implementation\n@[simp, specialize] def foldrIdx (f : Nat \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b2) (init : \u03b2) :\n    (l : List \u03b1) \u2192 (start : _ := 0) \u2192 \u03b2\n  | [], _ => init\n  | a :: l, i => f i a (foldrIdx f init l (i+1))\n\n/-- `findIdxs p l` is the list of indexes of elements of `l` that satisfy `p`. -/\n@[inline] def findIdxs (p : \u03b1 \u2192 Bool) (l : List \u03b1) : List Nat :=\n  foldrIdx (fun i a is => if p a then i :: is else is) [] l\n\n/--\nReturns the elements of `l` that satisfy `p` together with their indexes in\n`l`. The returned list is ordered by index.\n-/\n@[inline] def indexesValues (p : \u03b1 \u2192 Bool) (l : List \u03b1) : List (Nat \u00d7 \u03b1) :=\n  foldrIdx (fun i a l => if p a then (i, a) :: l else l) [] l\n\n/--\n`indexesOf a l` is the list of all indexes of `a` in `l`. For example:\n```\nindexesOf a [a, b, a, a] = [0, 2, 3]\n```\n-/\n@[inline] def indexesOf [BEq \u03b1] (a : \u03b1) : List \u03b1 \u2192 List Nat := findIdxs (\u00b7 == a)\n\n/-- Return the index of the first occurrence of an element satisfying `p`. -/\ndef findIdx? (p : \u03b1 \u2192 Bool) : List \u03b1 \u2192 (start : Nat := 0) \u2192 Option Nat\n| [], _ => none\n| a :: l, i => if p a then some i else findIdx? p l (i + 1)\n\n/-- Return the index of the first occurrence of `a` in the list. -/\n@[inline] def indexOf? [BEq \u03b1] (a : \u03b1) : List \u03b1 \u2192 Option Nat := findIdx? (a == \u00b7)\n\n/--\n`lookmap` is a combination of `lookup` and `filterMap`.\n`lookmap f l` will apply `f : \u03b1 \u2192 Option \u03b1` to each element of the list,\nreplacing `a \u2192 b` at the first value `a` in the list such that `f a = some b`.\n-/\n@[inline] def lookmap (f : \u03b1 \u2192 Option \u03b1) (l : List \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `lookmap`: `lookmap.go f l acc = acc.toList ++ lookmap f l`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], acc => acc.toList\n  | a :: l, acc => match f a with\n    | some b => acc.toListAppend (b :: l)\n    | none => go l (acc.push a)\n\n/-- `countp p l` is the number of elements of `l` that satisfy `p`. -/\n@[inline] def countp (p : \u03b1 \u2192 Bool) (l : List \u03b1) : Nat := go l 0 where\n  /-- Auxiliary for `countp`: `countp.go p l acc = countp p l + acc`. -/\n  @[specialize] go : List \u03b1 \u2192 Nat \u2192 Nat\n  | [], acc => acc\n  | x :: xs, acc => bif p x then go xs (acc + 1) else go xs acc\n\n/-- `count a l` is the number of occurrences of `a` in `l`. -/\n@[inline] def count [BEq \u03b1] (a : \u03b1) : List \u03b1 \u2192 Nat := countp (\u00b7 == a)\n\n/--\n`isPrefix l\u2081 l\u2082`, or `l\u2081 <+: l\u2082`, means that `l\u2081` is a prefix of `l\u2082`,\nthat is, `l\u2082` has the form `l\u2081 ++ t` for some `t`.\n-/\ndef isPrefix (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : Prop := \u2203 t, l\u2081 ++ t = l\u2082\n\n/--\n`isSuffix l\u2081 l\u2082`, or `l\u2081 <:+ l\u2082`, means that `l\u2081` is a suffix of `l\u2082`,\nthat is, `l\u2082` has the form `t ++ l\u2081` for some `t`.\n-/\ndef isSuffix (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : Prop := \u2203 t, t ++ l\u2081 = l\u2082\n\n/--\n`isInfix l\u2081 l\u2082`, or `l\u2081 <:+: l\u2082`, means that `l\u2081` is a contiguous\nsubstring of `l\u2082`, that is, `l\u2082` has the form `s ++ l\u2081 ++ t` for some `s, t`.\n-/\ndef isInfix (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : Prop := \u2203 s t, s ++ l\u2081 ++ t = l\u2082\n\n@[inherit_doc] infixl:50 \" <+: \" => isPrefix\n\n@[inherit_doc] infixl:50 \" <:+ \" => isSuffix\n\n@[inherit_doc] infixl:50 \" <:+: \" => isInfix\n\n/--\n`inits l` is the list of initial segments of `l`.\n```\ninits [1, 2, 3] = [[], [1], [1, 2], [1, 2, 3]]\n```\n-/\n@[simp] def inits : List \u03b1 \u2192 List (List \u03b1)\n  | [] => [[]]\n  | a :: l => [] :: map (fun t => a :: t) (inits l)\n\n/-- Tail-recursive version of `inits`. -/\ndef initsTR (l : List \u03b1) : List (List \u03b1) :=\n  l.foldr (fun a arrs => (arrs.map fun t => a :: t).push []) #[[]] |>.toListRev\n\n@[csimp] theorem inits_eq_initsTR : @inits = @initsTR := by\n  funext \u03b1 l; simp [initsTR]; induction l <;> simp [*, reverse_map]\n\n/--\n`tails l` is the list of terminal segments of `l`.\n```\ntails [1, 2, 3] = [[1, 2, 3], [2, 3], [3], []]\n```\n-/\n@[simp] def tails : List \u03b1 \u2192 List (List \u03b1)\n  | [] => [[]]\n  | a :: l => (a :: l) :: tails l\n\n/-- Tail-recursive version of `tails`. -/\ndef tailsTR (l : List \u03b1) : List (List \u03b1) := go l #[] where\n  /-- Auxiliary for `tailsTR`: `tailsTR.go l acc = acc.toList ++ tails l`. -/\n  go (l : List \u03b1) (acc : Array (List \u03b1)) : List (List \u03b1) :=\n    match l with\n    | [] => acc.toListAppend [[]]\n    | _::xs => go xs (acc.push l)\n\n@[csimp] theorem tails_eq_tailsTR : @tails = @tailsTR := by\n  funext \u03b1\n  have H (l : List \u03b1) : \u2200 acc, tailsTR.go l acc = acc.toList ++ tails l := by\n    induction l <;> simp [*, tailsTR.go]\n  simp [tailsTR, H]\n\n/--\n`sublists' l` is the list of all (non-contiguous) sublists of `l`.\nIt differs from `sublists` only in the order of appearance of the sublists;\n`sublists'` uses the first element of the list as the MSB,\n`sublists` uses the first element of the list as the LSB.\n```\nsublists' [1, 2, 3] = [[], [3], [2], [2, 3], [1], [1, 3], [1, 2], [1, 2, 3]]\n```\n-/\ndef sublists' (l : List \u03b1) : List (List \u03b1) :=\n  let f a arr := arr.foldl (init := arr) fun r l => r.push (a :: l)\n  (l.foldr f #[[]]).toList\n\n/--\n`sublists l` is the list of all (non-contiguous) sublists of `l`; cf. `sublists'`\nfor a different ordering.\n```\nsublists [1, 2, 3] = [[], [1], [2], [1, 2], [3], [1, 3], [2, 3], [1, 2, 3]]\n```\n-/\ndef sublists (l : List \u03b1) : List (List \u03b1) :=\n  let f a arr := arr.foldl (init := Array.mkEmpty (arr.size * 2))\n    fun r l => (r.push l).push (a :: l)\n  (l.foldr f #[[]]).toList\n\nsection Forall\u2082\n\nvariable {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop}\n\n/--\n`Forall\u2082 R l\u2081 l\u2082` means that `l\u2081` and `l\u2082` have the same length,\nand whenever `a` is the nth element of `l\u2081`, and `b` is the nth element of `l\u2082`,\nthen `R a b` is satisfied.\n-/\ninductive Forall\u2082 (R : \u03b1 \u2192 \u03b2 \u2192 Prop) : List \u03b1 \u2192 List \u03b2 \u2192 Prop\n  /-- Two nil lists are `Forall\u2082`-related -/\n  | nil : Forall\u2082 R [] []\n  /-- Two cons lists are related by `Forall\u2082 R`\n  if the heads are related by `R` and the tails are related by `Forall\u2082 R` -/\n  | cons {a b l\u2081 l\u2082} : R a b \u2192 Forall\u2082 R l\u2081 l\u2082 \u2192 Forall\u2082 R (a :: l\u2081) (b :: l\u2082)\n\nattribute [simp] Forall\u2082.nil\n\nend Forall\u2082\n\n/--\nTranspose of a list of lists, treated as a matrix.\n```\ntranspose [[1, 2], [3, 4], [5, 6]] = [[1, 3, 5], [2, 4, 6]]\n```\n-/\ndef transpose (l : List (List \u03b1)) : List (List \u03b1) := (l.foldr go #[]).toList where\n  /-- `pop : List \u03b1 \u2192 StateM (List \u03b1) (List \u03b1)` transforms the input list `old`\n  by taking the head of the current state and pushing it on the head of `old`.\n  If the state list is empty, then `old` is left unchanged. -/\n  pop (old : List \u03b1) : StateM (List \u03b1) (List \u03b1)\n    | [] => (old, [])\n    | a :: l => (a :: old, l)\n\n  /-- `go : List \u03b1 \u2192 Array (List \u03b1) \u2192 Array (List \u03b1)` handles the insertion of\n  a new list into all the lists in the array:\n  `go [a, b, c] #[l\u2081, l\u2082, l\u2083] = #[a::l\u2081, b::l\u2082, c::l\u2083]`.\n  If the new list is too short, the later lists are unchanged, and if it is too long\n  the array is extended:\n  ```\n  go [a] #[l\u2081, l\u2082, l\u2083] = #[a::l\u2081, l\u2082, l\u2083]\n  go [a, b, c, d] #[l\u2081, l\u2082, l\u2083] = #[a::l\u2081, b::l\u2082, c::l\u2083, [d]]\n  ```\n  -/\n  go (l : List \u03b1) (acc : Array (List \u03b1)) : Array (List \u03b1) :=\n    let (acc, l) := acc.mapM pop l\n    l.foldl (init := acc) fun arr a => arr.push [a]\n\n/--\nList of all sections through a list of lists. A section\nof `[L\u2081, L\u2082, ..., L\u2099]` is a list whose first element comes from\n`L\u2081`, whose second element comes from `L\u2082`, and so on.\n-/\n@[simp] def sections : List (List \u03b1) \u2192 List (List \u03b1)\n  | [] => [[]]\n  | l :: L => (sections L).bind fun s => l.map fun a => a :: s\n\n/-- Optimized version of `sections`. -/\ndef sectionsTR (L : List (List \u03b1)) : List (List \u03b1) :=\n  bif L.any isEmpty then [] else (L.foldr go #[[]]).toList\nwhere\n  /-- `go : List \u03b1 \u2192 Array (List \u03b1) \u2192 Array (List \u03b1)` inserts one list into the accumulated\n  list of sections `acc`: `go [a, b] #[l\u2081, l\u2082] = [a::l\u2081, b::l\u2081, a::l\u2082, b::l\u2082]`. -/\n  go (l : List \u03b1) (acc : Array (List \u03b1)) : Array (List \u03b1) :=\n    acc.foldl (init := #[]) fun acc' l' =>\n      l.foldl (init := acc') fun acc' a =>\n        acc'.push (a :: l')\n\ntheorem sections_eq_nil_of_isEmpty : \u2200 {L}, L.any isEmpty \u2192 @sections \u03b1 L = []\n  | l :: L, h => by\n    simp only [any, foldr, Bool.or_eq_true] at h\n    match l, h with\n    | [], .inl rfl => simp; induction sections L <;> simp [*]\n    | l, .inr h => simp [sections, sections_eq_nil_of_isEmpty h]\n\n@[csimp] theorem sections_eq_sectionsTR : @sections = @sectionsTR := by\n  funext \u03b1 L; simp [sectionsTR]\n  cases e : L.any isEmpty <;> simp [sections_eq_nil_of_isEmpty, *]\n  clear e; induction L with | nil => rfl | cons l L IH => ?_\n  simp [IH, sectionsTR.go, Array.foldl_eq_foldl_data]\n  rw [Array.foldl_data_eq_bind]; rfl\n  intros; apply Array.foldl_data_eq_map\n\n/-- `eraseP p l` removes the first element of `l` satisfying the predicate `p`. -/\ndef eraseP (p : \u03b1 \u2192 Bool) : List \u03b1 \u2192 List \u03b1\n  | [] => []\n  | a :: l => bif p a then l else a :: eraseP p l\n\n/-- Tail-recursive version of `eraseP`. -/\n@[inline] def erasePTR (p : \u03b1 \u2192 Bool) (l : List \u03b1) : List \u03b1 := go l #[] where\n  /-- Auxiliary for `erasePTR`: `erasePTR.go p l xs acc = acc.toList ++ eraseP p xs`,\n  unless `xs` does not contain any elements satisfying `p`, where it returns `l`. -/\n  @[specialize] go : List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _ => l\n  | a :: l, acc => bif p a then acc.toListAppend l else go l (acc.push a)\n\n@[csimp] theorem eraseP_eq_erasePTR : @eraseP = @erasePTR := by\n  funext \u03b1 p l; simp [erasePTR]\n  let rec go (acc) : \u2200 xs, l = acc.data ++ xs \u2192\n    erasePTR.go p l xs acc = acc.data ++ xs.eraseP p\n  | [] => fun h => by simp [erasePTR.go, eraseP, h]\n  | x::xs => by\n    simp [erasePTR.go, eraseP]; cases p x <;> simp\n    \u00b7 intro h; rw [go _ xs]; {simp}; simp [h]\n  exact (go #[] _ rfl).symm\n\n/--\n`extractP p l` returns a pair of an element `a` of `l` satisfying the predicate\n`p`, and `l`, with `a` removed. If there is no such element `a` it returns `(none, l)`.\n-/\ndef extractP (p : \u03b1 \u2192 Bool) (l : List \u03b1) : Option \u03b1 \u00d7 List \u03b1 := go l #[] where\n  /-- Auxiliary for `extractP`:\n  `extractP.go p l xs acc = (some a, acc.toList ++ out)` if `extractP p xs = (some a, out)`,\n  and `extractP.go p l xs acc = (none, l)` if `extractP p xs = (none, _)`. -/\n  go : List \u03b1 \u2192 Array \u03b1 \u2192 Option \u03b1 \u00d7 List \u03b1\n  | [], _ => (none, l)\n  | a :: l, acc => bif p a then (some a, acc.toListAppend l) else go l (acc.push a)\n\n/--\n`revzip l` returns a list of pairs of the elements of `l` paired\nwith the elements of `l` in reverse order.\n```\nrevzip [1, 2, 3, 4, 5] = [(1, 5), (2, 4), (3, 3), (4, 2), (5, 1)]\n```\n-/\ndef revzip (l : List \u03b1) : List (\u03b1 \u00d7 \u03b1) := zip l l.reverse\n\n/--\n`product l\u2081 l\u2082` is the list of pairs `(a, b)` where `a \u2208 l\u2081` and `b \u2208 l\u2082`.\n```\nproduct [1, 2] [5, 6] = [(1, 5), (1, 6), (2, 5), (2, 6)]\n```\n-/\ndef product (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) : List (\u03b1 \u00d7 \u03b2) := l\u2081.bind fun a => l\u2082.map (Prod.mk a)\n\n/-- Optimized version of `product`. -/\ndef productTR (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) : List (\u03b1 \u00d7 \u03b2) :=\n  l\u2081.foldl (fun acc a => l\u2082.foldl (fun acc b => acc.push (a, b)) acc) #[] |>.toList\n\n@[csimp] theorem product_eq_productTR : @product = @productTR := by\n  funext \u03b1 \u03b2 l\u2081 l\u2082; simp [product, productTR]\n  rw [Array.foldl_data_eq_bind]; rfl\n  intros; apply Array.foldl_data_eq_map\n\n/-- `sigma l\u2081 l\u2082` is the list of dependent pairs `(a, b)` where `a \u2208 l\u2081` and `b \u2208 l\u2082 a`.\n```\nsigma [1, 2] (\u03bb_, [(5 : Nat), 6]) = [(1, 5), (1, 6), (2, 5), (2, 6)]\n``` -/\nprotected def sigma {\u03c3 : \u03b1 \u2192 Type _} (l\u2081 : List \u03b1) (l\u2082 : \u2200 a, List (\u03c3 a)) : List (\u03a3 a, \u03c3 a) :=\n  l\u2081.bind fun a => (l\u2082 a).map (Sigma.mk a)\n\n/-- Optimized version of `sigma`. -/\ndef sigmaTR {\u03c3 : \u03b1 \u2192 Type _} (l\u2081 : List \u03b1) (l\u2082 : \u2200 a, List (\u03c3 a)) : List (\u03a3 a, \u03c3 a) :=\n  l\u2081.foldl (fun acc a => (l\u2082 a).foldl (fun acc b => acc.push \u27e8a, b\u27e9) acc) #[] |>.toList\n\n@[csimp] theorem sigma_eq_sigmaTR : @List.sigma = @sigmaTR := by\n  funext \u03b1 \u03b2 l\u2081 l\u2082; simp [List.sigma, sigmaTR]\n  rw [Array.foldl_data_eq_bind]; rfl\n  intros; apply Array.foldl_data_eq_map\n\n/--\n`ofFn f` with `f : fin n \u2192 \u03b1` returns the list whose ith element is `f i`\n```\nofFn f = [f 0, f 1, ... , f(n - 1)]\n```\n-/\ndef ofFn {n} (f : Fin n \u2192 \u03b1) : List \u03b1 := (Array.ofFn f).toList\n\n/-- `ofFnNthVal f i` returns `some (f i)` if `i < n` and `none` otherwise. -/\ndef ofFnNthVal {n} (f : Fin n \u2192 \u03b1) (i : Nat) : Option \u03b1 :=\n  if h : i < n then some (f \u27e8i, h\u27e9) else none\n\n/-- `disjoint l\u2081 l\u2082` means that `l\u2081` and `l\u2082` have no elements in common. -/\ndef Disjoint (l\u2081 l\u2082 : List \u03b1) : Prop :=\n  \u2200 \u2983a\u2984, a \u2208 l\u2081 \u2192 a \u2208 l\u2082 \u2192 False\n\nsection Pairwise\n\nvariable (R : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\n/--\n`Pairwise R l` means that all the elements with earlier indexes are\n`R`-related to all the elements with later indexes.\n```\nPairwise R [1, 2, 3] \u2194 R 1 2 \u2227 R 1 3 \u2227 R 2 3\n```\nFor example if `R = (\u00b7\u2260\u00b7)` then it asserts `l` has no duplicates,\nand if `R = (\u00b7<\u00b7)` then it asserts that `l` is (strictly) sorted.\n-/\ninductive Pairwise : List \u03b1 \u2192 Prop\n  /-- All elements of the empty list are vacuously pairwise related. -/\n  | nil : Pairwise []\n  /-- `a :: l` is `Pairwise R` if `a` `R`-relates to every element of `l`,\n  and `l` is `Pairwise R`. -/\n  | cons : \u2200 {a : \u03b1} {l : List \u03b1}, (\u2200 a' \u2208 l, R a a') \u2192 Pairwise l \u2192 Pairwise (a :: l)\n\nattribute [simp] Pairwise.nil\n\nvariable {R}\n\n@[simp] theorem pairwise_cons : Pairwise R (a::l) \u2194 (\u2200 a' \u2208 l, R a a') \u2227 Pairwise R l :=\n  \u27e8fun | .cons h\u2081 h\u2082 => \u27e8h\u2081, h\u2082\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 => h\u2082.cons h\u2081\u27e9\n\ninstance instDecidablePairwise [DecidableRel R] :\n    (l : List \u03b1) \u2192 Decidable (Pairwise R l)\n  | [] => isTrue .nil\n  | hd :: tl =>\n    match instDecidablePairwise tl with\n    | isTrue ht =>\n      match decidableBAll (R hd) tl with\n      | isFalse hf => isFalse fun hf' => hf (pairwise_cons.1 hf').1\n      | isTrue ht' => isTrue <| pairwise_cons.mpr (And.intro ht' ht)\n    | isFalse hf => isFalse fun | .cons _ ih => hf ih\n\nend Pairwise\n\n/--\n`pwFilter R l` is a maximal sublist of `l` which is `Pairwise R`.\n`pwFilter (\u00b7\u2260\u00b7)` is the erase duplicates function (cf. `eraseDup`), and `pwFilter (\u00b7<\u00b7)` finds\na maximal increasing subsequence in `l`. For example,\n```\npwFilter (\u00b7<\u00b7) [0, 1, 5, 2, 6, 3, 4] = [0, 1, 2, 3, 4]\n```\n-/\ndef pwFilter (R : \u03b1 \u2192 \u03b1 \u2192 Prop) [DecidableRel R] (l : List \u03b1) : List \u03b1 :=\n  l.foldr (fun x IH => if \u2200 y \u2208 IH, R x y then x :: IH else IH) []\n\nsection Chain\n\nvariable (R : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\n/-- `Chain R a l` means that `R` holds between adjacent elements of `a::l`.\n```\nChain R a [b, c, d] \u2194 R a b \u2227 R b c \u2227 R c d\n``` -/\ninductive Chain : \u03b1 \u2192 List \u03b1 \u2192 Prop\n  /-- A chain of length 1 is trivially a chain. -/\n  | nil {a : \u03b1} : Chain a []\n  /-- If `a` relates to `b` and `b::l` is a chain, then `a :: b :: l` is also a chain. -/\n  | cons : \u2200 {a b : \u03b1} {l : List \u03b1}, R a b \u2192 Chain b l \u2192 Chain a (b :: l)\n\n/-- `Chain' R l` means that `R` holds between adjacent elements of `l`.\n```\nChain' R [a, b, c, d] \u2194 R a b \u2227 R b c \u2227 R c d\n``` -/\ndef Chain' : List \u03b1 \u2192 Prop\n  | [] => True\n  | a :: l => Chain R a l\n\nend Chain\n\n/-- `Nodup l` means that `l` has no duplicates, that is, any element appears at most\n  once in the List. It is defined as `Pairwise (\u2260)`. -/\ndef Nodup : List \u03b1 \u2192 Prop := Pairwise (\u00b7 \u2260 \u00b7)\n\ninstance nodupDecidable [DecidableEq \u03b1] : \u2200 l : List \u03b1, Decidable (Nodup l) :=\n  instDecidablePairwise\n\n/-- `eraseDup l` removes duplicates from `l` (taking only the first occurrence).\n  Defined as `pwFilter (\u2260)`.\n\n    eraseDup [1, 0, 2, 2, 1] = [0, 2, 1] -/\n@[inline] def eraseDup [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 := pwFilter (\u00b7 \u2260 \u00b7)\n\n/-- `range' s n` is the list of numbers `[s, s+1, ..., s+n-1]`.\n  It is intended mainly for proving properties of `range` and `iota`. -/\n@[simp] def range' : Nat \u2192 Nat \u2192 List Nat\n  | _, 0 => []\n  | s, n+1 => s :: range' (s+1) n\n\n/-- Optimized version of `range'`. -/\n@[inline] def range'TR (s n : Nat) : List Nat := go n (s + n) [] where\n  /-- Auxiliary for `range'TR`: `range'TR.go n e = [e-n, ..., e-1] ++ acc`. -/\n  go : Nat \u2192 Nat \u2192 List Nat \u2192 List Nat\n  | 0, _, acc => acc\n  | n+1, e, acc => go n (e-1) ((e-1) :: acc)\n\n@[csimp] theorem range'_eq_range'TR : @range' = @range'TR := by\n  funext s n\n  let rec go (s) : \u2200 n m, range'TR.go n (s + n) (range' (s + n) m) = range' s (n + m)\n  | 0, m => by simp [range'TR.go]\n  | n+1, m => (go s n (m + 1)).trans <| congrArg _ (Nat.add_right_comm n m 1)\n  exact (go s n 0).symm\n\n/-- Drop `none`s from a list, and replace each remaining `some a` with `a`. -/\n@[inline] def reduceOption {\u03b1} : List (Option \u03b1) \u2192 List \u03b1 :=\n  List.filterMap id\n\n/--\n`ilast' x xs` returns the last element of `xs` if `xs` is non-empty; it returns `x` otherwise.\n-/\n@[simp] def ilast' {\u03b1} : \u03b1 \u2192 List \u03b1 \u2192 \u03b1\n  | a, [] => a\n  | _, b :: l => ilast' b l\n\n/--\n`last' xs` returns the last element of `xs` if `xs` is non-empty; it returns `none` otherwise.\n-/\n@[simp] def last' {\u03b1} : List \u03b1 \u2192 Option \u03b1\n  | [] => none\n  | [a] => some a\n  | _ :: l => last' l\n\n/--\n`rotate l n` rotates the elements of `l` to the left by `n`\n```\nrotate [0, 1, 2, 3, 4, 5] 2 = [2, 3, 4, 5, 0, 1]\n```\n-/\n@[inline] def rotate (l : List \u03b1) (n : Nat) : List \u03b1 :=\n  let (l\u2081, l\u2082) := List.splitAt (n % l.length) l\n  l\u2082 ++ l\u2081\n\n/-- rotate' is the same as `rotate`, but slower. Used for proofs about `rotate` -/\n@[simp] def rotate' : List \u03b1 \u2192 Nat \u2192 List \u03b1\n  | [], _ => []\n  | l, 0 => l\n  | a :: l, n+1 => rotate' (l ++ [a]) n\n\n/--\n`mapDiagM f l` calls `f` on all elements in the upper triangular part of `l \u00d7 l`.\nThat is, for each `e \u2208 l`, it will run `f e e` and then `f e e'`\nfor each `e'` that appears after `e` in `l`.\n```\nmapDiagM f [1, 2, 3] =\n  return [\u2190 f 1 1, \u2190 f 1 2, \u2190 f 1 3, \u2190 f 2 2, \u2190 f 2 3, \u2190 f 3 3]\n```\n-/\ndef mapDiagM [Monad m] (f : \u03b1 \u2192 \u03b1 \u2192 m \u03b2) (l : List \u03b1) : m (List \u03b2) := go l #[] where\n  /-- Auxiliary for `mapDiagM`: `mapDiagM.go as f acc = (acc.toList ++ \u00b7) <$> mapDiagM f as` -/\n  go : List \u03b1 \u2192 Array \u03b2 \u2192 m (List \u03b2)\n  | [], acc => pure acc.toList\n  | x::xs, acc => do\n    let b \u2190 f x x\n    let acc \u2190 xs.foldlM (\u00b7.push <$> f x \u00b7) (acc.push b)\n    go xs acc\n\n/--\n`forDiagM f l` calls `f` on all elements in the upper triangular part of `l \u00d7 l`.\nThat is, for each `e \u2208 l`, it will run `f e e` and then `f e e'`\nfor each `e'` that appears after `e` in `l`.\n```\nforDiagM f [1, 2, 3] = do f 1 1; f 1 2; f 1 3; f 2 2; f 2 3; f 3 3\n```\n-/\n@[simp] def forDiagM [Monad m] (f : \u03b1 \u2192 \u03b1 \u2192 m PUnit) : List \u03b1 \u2192 m PUnit\n  | [] => pure \u27e8\u27e9\n  | x :: xs => do f x x; xs.forM (f x); xs.forDiagM f\n\n/-- `getRest l l\u2081` returns `some l\u2082` if `l = l\u2081 ++ l\u2082`.\n  If `l\u2081` is not a prefix of `l`, returns `none` -/\ndef getRest [DecidableEq \u03b1] : List \u03b1 \u2192 List \u03b1 \u2192 Option (List \u03b1)\n  | l, [] => some l\n  | [], _ => none\n  | x :: l, y :: l\u2081 => if x = y then getRest l l\u2081 else none\n\n/-- `List.dropSlice n m xs` removes a slice of length `m` at index `n` in list `xs`. -/\n@[simp] def dropSlice : Nat \u2192 Nat \u2192 List \u03b1 \u2192 List \u03b1\n  | _, _, [] => []\n  | 0, m, xs => xs.drop m\n  | n+1, m, x :: xs => x :: dropSlice n m xs\n\n/-- Optimized version of `dropSlice`. -/\n@[inline] def dropSliceTR (n m : Nat) (l : List \u03b1) : List \u03b1 :=\n  match m with\n  | 0 => l\n  | m+1 => go m l n #[]\nwhere\n  /-- Auxiliary for `dropSliceTR`: `dropSliceTR.go l m xs n acc = acc.toList ++ dropSlice n m xs`\n  unless `n \u2265 length xs`, in which case it is `l`. -/\n  go (m : Nat) : List \u03b1 \u2192 Nat \u2192 Array \u03b1 \u2192 List \u03b1\n  | [],    _,   _   => l\n  | _::xs, 0,   acc => acc.toListAppend (xs.drop m)\n  | x::xs, n+1, acc => go m xs n (acc.push x)\n\ntheorem dropSlice_zero\u2082 : \u2200 n l, @dropSlice \u03b1 n 0 l = l\n  | 0, [] | 0, _::_ | n+1, [] => rfl\n  | n+1, x::xs => by simp [dropSlice, dropSlice_zero\u2082]\n\n@[csimp] theorem dropSlice_eq_dropSliceTR : @dropSlice = @dropSliceTR := by\n  funext \u03b1 n m l; simp [dropSliceTR]\n  split; { rw [dropSlice_zero\u2082] }\n  rename_i m\n  let rec go (acc) : \u2200 xs n, l = acc.data ++ xs \u2192\n    dropSliceTR.go l m xs n acc = acc.data ++ xs.dropSlice n (m+1)\n  | [],    n\n  | _::xs, 0 => fun h => by simp [dropSliceTR.go, dropSlice, h]\n  | x::xs, n+1 => by simp [dropSliceTR.go, dropSlice]; intro h; rw [go _ xs]; {simp}; simp [h]\n  exact (go #[] _ _ rfl).symm\n\n/--\nLeft-biased version of `List.zipWith`. `zipWithLeft' f as bs` applies `f` to each\npair of elements `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `bs` is shorter than `as`, `f` is\napplied to `none` for the remaining `a\u1d62`. Returns the results of the `f`\napplications and the remaining `bs`.\n```\nzipWithLeft' prod.mk [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])\nzipWithLeft' prod.mk [1] ['a', 'b'] = ([(1, some 'a')], ['b'])\n```\n-/\n@[simp] def zipWithLeft' (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3 \u00d7 List \u03b2\n  | [], bs => ([], bs)\n  | a :: as, [] => ((a :: as).map fun a => f a none, [])\n  | a :: as, b :: bs => let r := zipWithLeft' f as bs; (f a (some b) :: r.1, r.2)\n\n/-- Tail-recursive version of `zipWithLeft'`. -/\n@[inline] def zipWithLeft'TR (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) : List \u03b3 \u00d7 List \u03b2 := go as bs #[] where\n  /-- Auxiliary for `zipWithLeft'TR`: `zipWithLeft'TR.go l acc = acc.toList ++ zipWithLeft' l`. -/\n  go : List \u03b1 \u2192 List \u03b2 \u2192 Array \u03b3 \u2192 List \u03b3 \u00d7 List \u03b2\n  | [], bs, acc => (acc.toList, bs)\n  | as, [], acc => (as.foldl (fun acc a => acc.push (f a none)) acc |>.toList, [])\n  | a :: as, b :: bs, acc => go as bs (acc.push (f a (some b)))\n\n@[csimp] theorem zipWithLeft'_eq_zipWithLeft'TR : @zipWithLeft' = @zipWithLeft'TR := by\n  funext \u03b1 \u03b2 \u03b3 f as bs; simp [zipWithLeft'TR]\n  let rec go (acc) : \u2200 as bs, zipWithLeft'TR.go f as bs acc =\n      let (l, r) := as.zipWithLeft' f bs; (acc.toList ++ l, r)\n  | [], bs => by simp [zipWithLeft'TR.go]\n  | _::_, [] => by simp [zipWithLeft'TR.go, Array.foldl_data_eq_map]\n  | a::as, b::bs => by simp [zipWithLeft'TR.go, go _ as bs]\n  simp [zipWithLeft'TR, go]\n\n/--\nRight-biased version of `List.zipWith`. `zipWithRight' f as bs` applies `f` to each\npair of elements `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `as` is shorter than `bs`, `f` is\napplied to `none` for the remaining `b\u1d62`. Returns the results of the `f`\napplications and the remaining `as`.\n```\nzipWithRight' prod.mk [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])\nzipWithRight' prod.mk [1, 2] ['a'] = ([(some 1, 'a')], [2])\n```\n-/\n@[inline] def zipWithRight' (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3) (as : List \u03b1) (bs : List \u03b2) : List \u03b3 \u00d7 List \u03b1 :=\n  zipWithLeft' (flip f) bs as\n\n/--\nLeft-biased version of `List.zip`. `zipLeft' as bs` returns the list of\npairs `(a\u1d62, b\u1d62)` for `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `bs` is shorter than `as`, the\nremaining `a\u1d62` are paired with `none`. Also returns the remaining `bs`.\n```\nzipLeft' [1, 2] ['a'] = ([(1, some 'a'), (2, none)], [])\nzipLeft' [1] ['a', 'b'] = ([(1, some 'a')], ['b'])\nzipLeft' = zipWithLeft' prod.mk\n```\n-/\n@[inline] def zipLeft' : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 Option \u03b2) \u00d7 List \u03b2 := zipWithLeft' Prod.mk\n\n/--\nRight-biased version of `List.zip`. `zipRight' as bs` returns the list of\npairs `(a\u1d62, b\u1d62)` for `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `as` is shorter than `bs`, the\nremaining `b\u1d62` are paired with `none`. Also returns the remaining `as`.\n```\nzipRight' [1] ['a', 'b'] = ([(some 1, 'a'), (none, 'b')], [])\nzipRight' [1, 2] ['a'] = ([(some 1, 'a')], [2])\nzipRight' = zipWithRight' prod.mk\n```\n-/\n@[inline] def zipRight' : List \u03b1 \u2192 List \u03b2 \u2192 List (Option \u03b1 \u00d7 \u03b2) \u00d7 List \u03b1 := zipWithRight' Prod.mk\n\n/--\nLeft-biased version of `List.zipWith`. `zipWithLeft f as bs` applies `f` to each pair\n`a\u1d62 \u2208 as` and `b\u1d62 \u200c\u2208 bs`. If `bs` is shorter than `as`, `f` is applied to `none`\nfor the remaining `a\u1d62`.\n```\nzipWithLeft prod.mk [1, 2] ['a'] = [(1, some 'a'), (2, none)]\nzipWithLeft prod.mk [1] ['a', 'b'] = [(1, some 'a')]\nzipWithLeft f as bs = (zipWithLeft' f as bs).fst\n```\n-/\n@[simp] def zipWithLeft (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3\n  | [], _ => []\n  | a :: as, [] => (a :: as).map fun a => f a none\n  | a :: as, b :: bs => f a (some b) :: zipWithLeft f as bs\n\n/-- Tail-recursive version of `zipWithLeft`. -/\n@[inline] def zipWithLeftTR (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) : List \u03b3 := go as bs #[] where\n  /-- Auxiliary for `zipWithLeftTR`: `zipWithLeftTR.go l acc = acc.toList ++ zipWithLeft l`. -/\n  go : List \u03b1 \u2192 List \u03b2 \u2192 Array \u03b3 \u2192 List \u03b3\n  | [], _, acc => acc.toList\n  | as, [], acc => as.foldl (fun acc a => acc.push (f a none)) acc |>.toList\n  | a :: as, b :: bs, acc => go as bs (acc.push (f a (some b)))\n\n@[csimp] theorem zipWithLeft_eq_zipWithLeftTR : @zipWithLeft = @zipWithLeftTR := by\n  funext \u03b1 \u03b2 \u03b3 f as bs; simp [zipWithLeftTR]\n  let rec go (acc) : \u2200 as bs, zipWithLeftTR.go f as bs acc = acc.toList ++ as.zipWithLeft f bs\n  | [], bs => by simp [zipWithLeftTR.go]\n  | _::_, [] => by simp [zipWithLeftTR.go, Array.foldl_data_eq_map]\n  | a::as, b::bs => by simp [zipWithLeftTR.go, go _ as bs]\n  simp [zipWithLeftTR, go]\n\n/--\nRight-biased version of `List.zipWith`. `zipWithRight f as bs` applies `f` to each\npair `a\u1d62 \u2208 as` and `b\u1d62 \u200c\u2208 bs`. If `as` is shorter than `bs`, `f` is applied to\n`none` for the remaining `b\u1d62`.\n```\nzipWithRight prod.mk [1, 2] ['a'] = [(some 1, 'a')]\nzipWithRight prod.mk [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]\nzipWithRight f as bs = (zipWithRight' f as bs).fst\n```\n-/\n@[inline] def zipWithRight (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3) (as : List \u03b1) (bs : List \u03b2) : List \u03b3 :=\n  zipWithLeft (flip f) bs as\n\n/--\nLeft-biased version of `List.zip`. `zipLeft as bs` returns the list of pairs\n`(a\u1d62, b\u1d62)` for `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `bs` is shorter than `as`, the\nremaining `a\u1d62` are paired with `none`.\n```\nzipLeft [1, 2] ['a'] = [(1, some 'a'), (2, none)]\nzipLeft [1] ['a', 'b'] = [(1, some 'a')]\nzipLeft = zipWithLeft prod.mk\n```\n-/\n@[inline] def zipLeft : List \u03b1 \u2192 List \u03b2 \u2192 List (\u03b1 \u00d7 Option \u03b2) := zipWithLeft Prod.mk\n\n/--\nRight-biased version of `List.zip`. `zipRight as bs` returns the list of pairs\n`(a\u1d62, b\u1d62)` for `a\u1d62 \u2208 as` and `b\u1d62 \u2208 bs`. If `as` is shorter than `bs`, the\nremaining `b\u1d62` are paired with `none`.\n```\nzipRight [1, 2] ['a'] = [(some 1, 'a')]\nzipRight [1] ['a', 'b'] = [(some 1, 'a'), (none, 'b')]\nzipRight = zipWithRight prod.mk\n```\n-/\n@[inline] def zipRight : List \u03b1 \u2192 List \u03b2 \u2192 List (Option \u03b1 \u00d7 \u03b2) := zipWithRight Prod.mk\n\n/--\nIf all elements of `xs` are `some x\u1d62`, `allSome xs` returns the `x\u1d62`. Otherwise\nit returns `none`.\n```\nallSome [some 1, some 2] = some [1, 2]\nallSome [some 1, none  ] = none\n```\n-/\n@[inline] def allSome (l : List (Option \u03b1)) : Option (List \u03b1) := l.mapM id\n\n/--\n`fillNones xs ys` replaces the `none`s in `xs` with elements of `ys`. If there\nare not enough `ys` to replace all the `none`s, the remaining `none`s are\ndropped from `xs`.\n```\nfillNones [none, some 1, none, none] [2, 3] = [2, 1, 3]\n```\n-/\n@[simp] def fillNones {\u03b1} : List (Option \u03b1) \u2192 List \u03b1 \u2192 List \u03b1\n  | [], _ => []\n  | some a :: as, as' => a :: fillNones as as'\n  | none :: as, [] => as.reduceOption\n  | none :: as, a :: as' => a :: fillNones as as'\n\n/-- Tail-recursive version of `fillNones`. -/\n@[inline] def fillNonesTR (as : List (Option \u03b1)) (as' : List \u03b1) : List \u03b1 := go as as' #[] where\n  /-- Auxiliary for `fillNonesTR`: `fillNonesTR.go as as' acc = acc.toList ++ fillNones as as'`. -/\n  go : List (Option \u03b1) \u2192 List \u03b1 \u2192 Array \u03b1 \u2192 List \u03b1\n  | [], _, acc => acc.toList\n  | some a :: as, as', acc => go as as' (acc.push a)\n  | none :: as, [], acc => filterMapTR.go id as acc\n  | none :: as, a :: as', acc => go as as' (acc.push a)\n\n@[csimp] theorem fillNones_eq_fillNonesTR : @fillNones = @fillNonesTR := by\n  funext \u03b1 as as'; simp [fillNonesTR]\n  let rec go (acc) : \u2200 as as', @fillNonesTR.go \u03b1 as as' acc = acc.data ++ as.fillNones as'\n  | [], _ => by simp [fillNonesTR.go]\n  | some a :: as, as' => by simp [fillNonesTR.go, go _ as as']\n  | none :: as, [] => by simp [fillNonesTR.go, reduceOption, filterMap_eq_filterMapTR.go]\n  | none :: as, a :: as' => by simp [fillNonesTR.go, go _ as as']\n  simp [fillNonesTR, go]\n\n/--\n`takeList as ns` extracts successive sublists from `as`. For `ns = n\u2081 ... n\u2098`,\nit first takes the `n\u2081` initial elements from `as`, then the next `n\u2082` ones,\netc. It returns the sublists of `as` -- one for each `n\u1d62` -- and the remaining\nelements of `as`. If `as` does not have at least as many elements as the sum of\nthe `n\u1d62`, the corresponding sublists will have less than `n\u1d62` elements.\n```\ntakeList ['a', 'b', 'c', 'd', 'e'] [2, 1, 1] = ([['a', 'b'], ['c'], ['d']], ['e'])\ntakeList ['a', 'b'] [3, 1] = ([['a', 'b'], []], [])\n```\n-/\ndef takeList {\u03b1} : List \u03b1 \u2192 List Nat \u2192 List (List \u03b1) \u00d7 List \u03b1\n  | xs, [] => ([], xs)\n  | xs, n :: ns =>\n    let (xs\u2081, xs\u2082) := xs.splitAt n\n    let (xss, rest) := takeList xs\u2082 ns\n    (xs\u2081 :: xss, rest)\n\n/-- Tail-recursive version of `takeList`. -/\n@[inline] def takeListTR\n    (xs : List \u03b1) (ns : List Nat) : List (List \u03b1) \u00d7 List \u03b1 := go ns xs #[] where\n  /-- Auxiliary for `takeListTR`: `takeListTR.go as as' acc = acc.toList ++ takeList as as'`. -/\n  go : List Nat \u2192 List \u03b1 \u2192 Array (List \u03b1) \u2192 List (List \u03b1) \u00d7 List \u03b1\n  | [], xs, acc => (acc.toList, xs)\n  | n :: ns, xs, acc =>\n    let (xs\u2081, xs\u2082) := xs.splitAt n\n    go ns xs\u2082 (acc.push xs\u2081)\n\n@[csimp] theorem takeList_eq_takeListTR : @takeList = @takeListTR := by\n  funext \u03b1 xs ns; simp [takeListTR]\n  let rec go (acc) : \u2200 ns xs, @takeListTR.go \u03b1 ns xs acc =\n      let (l, r) := xs.takeList ns; (acc.toList ++ l, r)\n  | [], xs => by simp [takeListTR.go, takeList]\n  | n::ns, xs => by simp [takeListTR.go, takeList, go _ ns]\n  simp [takeListTR, go]\n\n/-- Auxliary definition used to define `toChunks`.\n  `toChunksAux n xs i` returns `(xs.take i, (xs.drop i).toChunks (n+1))`,\n  that is, the first `i` elements of `xs`, and the remaining elements chunked into\n  sublists of length `n+1`. -/\ndef toChunksAux {\u03b1} (n : Nat) : List \u03b1 \u2192 Nat \u2192 List \u03b1 \u00d7 List (List \u03b1)\n  | [], _ => ([], [])\n  | x :: xs, 0 =>\n    let (l, L) := toChunksAux n xs n\n    ([], (x :: l) :: L)\n  | x :: xs, i+1 =>\n    let (l, L) := toChunksAux n xs i\n    (x :: l, L)\n\n/--\n`xs.toChunks n` splits the list into sublists of size at most `n`,\nsuch that `(xs.toChunks n).join = xs`.\n```\n[1, 2, 3, 4, 5, 6, 7, 8].toChunks 10 = [[1, 2, 3, 4, 5, 6, 7, 8]]\n[1, 2, 3, 4, 5, 6, 7, 8].toChunks 3 = [[1, 2, 3], [4, 5, 6], [7, 8]]\n[1, 2, 3, 4, 5, 6, 7, 8].toChunks 2 = [[1, 2], [3, 4], [5, 6], [7, 8]]\n[1, 2, 3, 4, 5, 6, 7, 8].toChunks 0 = [[1, 2, 3, 4, 5, 6, 7, 8]]\n```\n-/\ndef toChunks {\u03b1} : Nat \u2192 List \u03b1 \u2192 List (List \u03b1)\n  | _, [] => []\n  | 0, xs => [xs]\n  | n, x :: xs =>\n    let rec\n    /-- Auxliary definition used to define `toChunks`.\n    `toChunks.go xs acc\u2081 acc\u2082` pushes elements into `acc\u2081` until it reaches size `n`,\n    then it pushes the resulting list to `acc\u2082` and continues until `xs` is exhausted. -/\n    go : List \u03b1 \u2192 Array \u03b1 \u2192 Array (List \u03b1) \u2192 List (List \u03b1)\n    | [], acc\u2081, acc\u2082 => acc\u2082.push acc\u2081.toList |>.toList\n    | x :: xs, acc\u2081, acc\u2082 =>\n      if acc\u2081.size == n then\n        go xs ((Array.mkEmpty n).push x) (acc\u2082.push acc\u2081.toList)\n      else\n        go xs (acc\u2081.push x) acc\u2082\n    go xs #[x] #[]\n\n/-!\nWe add some n-ary versions of `List.zipWith` for functions with more than two arguments.\nThese can also be written in terms of `List.zip` or `List.zipWith`.\nFor example, `zipWith\u2083 f xs ys zs` could also be written as\n`zipWith id (zipWith f xs ys) zs`\nor as\n`(zip xs <| zip ys zs).map fun \u27e8x, y, z\u27e9 => f x y z`.\n-/\n\n-- TODO(Mario): tail recursive\n/-- Ternary version of `List.zipWith`. -/\ndef zipWith\u2083 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3 \u2192 List \u03b4\n| x :: xs, y :: ys, z :: zs => f x y z :: zipWith\u2083 f xs ys zs\n| _, _, _ => []\n\n/-- Quaternary version of `List.zipWith`. -/\ndef zipWith\u2084 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 \u03b5) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3 \u2192 List \u03b4 \u2192 List \u03b5\n| x :: xs, y :: ys, z :: zs, u :: us => f x y z u :: zipWith\u2084 f xs ys zs us\n| _, _, _, _ => []\n\n/-- Quinary version of `List.zipWith`. -/\ndef zipWith\u2085 (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 \u2192 \u03b4 \u2192 \u03b5 \u2192 \u03b6) : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b3 \u2192 List \u03b4 \u2192 List \u03b5 \u2192 List \u03b6\n| x :: xs, y :: ys, z :: zs, u :: us, v :: vs => f x y z u v :: zipWith\u2085 f xs ys zs us vs\n| _, _, _, _, _ => []\n\n/-- An auxiliary function for `List.mapWithPrefixSuffix`. -/\n-- TODO(Mario): tail recursive\ndef mapWithPrefixSuffixAux {\u03b1 \u03b2} (f : List \u03b1 \u2192 \u03b1 \u2192 List \u03b1 \u2192 \u03b2) : List \u03b1 \u2192 List \u03b1 \u2192 List \u03b2\n  | _, [] => []\n  | prev, h :: t => f prev h t :: mapWithPrefixSuffixAux f (prev.concat h) t\n\n/--\n`List.mapWithPrefixSuffix f l` maps `f` across a list `l`.\nFor each `a \u2208 l` with `l = pref ++ [a] ++ suff`, `a` is mapped to `f pref a suff`.\nExample: if `f : list Nat \u2192 Nat \u2192 list Nat \u2192 \u03b2`,\n`List.mapWithPrefixSuffix f [1, 2, 3]` will produce the list\n`[f [] 1 [2, 3], f [1] 2 [3], f [1, 2] 3 []]`.\n-/\ndef mapWithPrefixSuffix {\u03b1 \u03b2} (f : List \u03b1 \u2192 \u03b1 \u2192 List \u03b1 \u2192 \u03b2) (l : List \u03b1) : List \u03b2 :=\n  mapWithPrefixSuffixAux f [] l\n\n/--\n`List.mapWithComplement f l` is a variant of `List.mapWithPrefixSuffix`\nthat maps `f` across a list `l`.\nFor each `a \u2208 l` with `l = pref ++ [a] ++ suff`, `a` is mapped to `f a (pref ++ suff)`,\ni.e., the list input to `f` is `l` with `a` removed.\nExample: if `f : Nat \u2192 list Nat \u2192 \u03b2`, `List.mapWithComplement f [1, 2, 3]` will produce the list\n`[f 1 [2, 3], f 2 [1, 3], f 3 [1, 2]]`.\n-/\ndef mapWithComplement {\u03b1 \u03b2} (f : \u03b1 \u2192 List \u03b1 \u2192 \u03b2) : List \u03b1 \u2192 List \u03b2 :=\n  mapWithPrefixSuffix fun pref a suff => f a (pref ++ suff)\n\nend List\n", "meta": {"author": "leanprover", "repo": "std4", "sha": "5507f9d8409f93b984ce04eccf4914d534e6fca2", "save_path": "github-repos/lean/leanprover-std4", "path": "github-repos/lean/leanprover-std4/std4-5507f9d8409f93b984ce04eccf4914d534e6fca2/Std/Data/List/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6825737344123242, "lm_q1q2_score": 0.3572729822408685}}
{"text": "def forceNat (a : Nat) := true\ndef forceInt (a : Int) := false\n\ndef f1 :=\n/-\nThe following example works, but it adds a coercion at `forceInt i`.\nThe elaborated term is\n```\nfun (n i : Nat) => if n == i then forceNat n else forceInt (coe i)\n-/\nfun n i => if n == i then forceNat n else forceInt i -- works\n\ndef f2 :=\nfun n i => if coe n == i then forceInt i else forceNat n -- works\n\n#check f1 -- Nat \u2192 Nat \u2192 Bool\n#check f2 -- Nat \u2192 Int \u2192 Bool\n\ndef f3 :=\n/- Fails.\n   - `n == i` generates type constraint enforcing `n` and `i` to have the same type.\n   - `forceInt i` forces `i` (and consequently `n`) to have type `Int`.\n   - `forceNat n` fails because there is no coercion from `Nat` to `Int`. -/\nfun n i => if n == i then forceInt i else forceNat n\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/tests/elabissues/issues8.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6825737279551494, "lm_q1q2_score": 0.3572729788610518}}
{"text": "open classical\n\nvariables (\u03b1 : Type) (p q : \u03b1 \u2192 Prop)\nvariable r : Prop\n\nexample : \u03b1 \u2192 ((\u2200 x : \u03b1, r) \u2194 r) :=\n  assume : \u03b1,\n  \u27e8\u03bb h, h this, \u03bb r x, r\u27e9\nexample : (\u2200 x, p x \u2228 r) \u2194 (\u2200 x, p x) \u2228 r :=\n  \u27e8\u03bb h,\n     by_cases\n       (\u03bb hr, or.inr hr)\n       (assume hnr : \u00acr,\n         or.inl\n           (assume x : \u03b1,\n             (h x).elim\n               id\n               (\u03bb hr, absurd hr hnr))),\n   \u03bb h x, h.elim (\u03bb f, or.inl (f x)) or.inr\u27e9\nexample : (\u2200 x, r \u2192 p x) \u2194 (r \u2192 \u2200 x, p x) :=\n  \u27e8\u03bb h r x, h x r, \u03bb h x r, h r x\u27e9\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch4/ex0602.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3572037943326782}}
{"text": "/-\nCopyright (c) 2022 Jun Yoshida. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\n-/\n\nimport Algdata.Data.List.Basic\nimport Algdata.Data.List.Prop\n\nnamespace List\n\n/- Decidable LT -/\ninstance listHasDecidableLT (\u03b1 : Type _) [LT \u03b1] [DecidableRel (\u03b1:=\u03b1) LT.lt] : DecidableRel (\u03b1:=List \u03b1) LT.lt\n| _, [] => isFalse \u03bb h => by cases h\n| [], (_::_) => isTrue $ List.lt.nil _ _\n| (a::as), (b::bs) =>\n  if hab : a < b then\n    isTrue $ List.lt.head _ _ hab\n  else if hba : b < a then\n    isFalse $ \u03bb hcontra =>\n      match hcontra with\n      | List.lt.head _ _ h => hab h\n      | List.lt.tail _ h _ => h hba\n  else if hs : as < bs then\n    isTrue $ List.lt.tail hab hba hs\n  else\n    isFalse $ \u03bb hcontra =>\n      match hcontra with\n      | List.lt.head _ _ h => hab h\n      | List.lt.tail _ _ h => hs h\n\n\n/- Monad -/\ninstance instMonadList : Monad List where\n  bind := List.bind\n  pure a := [a]\n\n/- LawfulMonad; i.e. monad laws -/\nsection LawfulMonad\n\nvariable {\u03b1 \u03b2 \u03b3 : Type _}\n\n@[simp]\ntheorem pure_eq_singleton (a : \u03b1) : pure a = [a] := rfl\n\n@[simp]\ntheorem map_eq_map (f : \u03b1 \u2192 \u03b2) (as : List \u03b1) : f <$> as = as.map f := rfl\n\n@[simp]\ntheorem bind_eq_bind (as : List \u03b1) (f : \u03b1 \u2192 List \u03b2) : (as >>= f) = List.bind as f := rfl\n\n@[simp]\ntheorem seq_eq_bind_map (fs : List (\u03b1 \u2192 \u03b2)) (as : List \u03b1) : (fs <*> as) = List.bind fs as.map := rfl\n\n@[simp]\ntheorem seqLeft_eq_bind_bind_const (as : List \u03b1) (bs : List \u03b2) : as <* bs = List.bind as (fun a => List.bind bs (Function.const \u03b2 [a])):= rfl\n\n@[simp]\ntheorem seqRight_eq_bind_const (as : List \u03b1) (bs : List \u03b2) : as *> bs = List.bind as (Function.const \u03b1 bs) := rfl\n\n@[simp]\ntheorem id_map (as : List \u03b1) : as.map id = as := by\n  induction as with\n  | nil => rfl\n  | cons a as hi =>\n    rw [map, hi]\n    rfl\n\n@[simp]\ntheorem bind_singleton_comp (f : \u03b1 \u2192 \u03b2) : \u2200 (as : List \u03b1), as.bind (fun a => [f a]) = as.map f\n| [] => rfl\n| (a::as) => by\n  rw [cons_bind, map]\n  rw [cons_append, nil_append]\n  rw [bind_singleton_comp f as]\n\n@[simp]\ntheorem singleton_bind (a : \u03b1) (f : \u03b1 \u2192 List \u03b2) : [a].bind f = f a := by\n  rw [List.bind, map, map, join, join]\n  simp\n\n@[simp]\ntheorem bind_assoc (as : List \u03b1) (f : \u03b1 \u2192 List \u03b2) (g : \u03b2 \u2192 List \u03b3) : ((as.bind f).bind g) = as.bind fun x => (f x).bind g := by\n  induction as with\n  | nil => rfl\n  | cons a as hi =>\n    rw [cons_bind, cons_bind]\n    rw [append_bind]\n    rw [hi]\n\ninstance instLawfulMonadList : LawfulMonad List where\n  id_map as := List.id_map as\n  map_const := by intros; rfl\n  seqLeft_eq {\u03b1} {\u03b2} as bs := by\n    rw [seqLeft_eq_bind_bind_const]\n    rw [seq_eq_bind_map]\n    rw [map_eq_map]\n    have : (fun (a : \u03b1) => bs.bind (Function.const _ [a])) = fun (a : \u03b1) => bs.map (Function.const _ a) := by\n      apply funext; intro a\n      rw [\u2190bind_singleton_comp (Function.const \u03b2 a) bs]\n      rfl\n    rw [this]; clear this\n    rw [bind_map_binary_eq_map_bind_map]\n  seqRight_eq as bs := by\n    rw [seqRight_eq_bind_const]\n    rw [seq_eq_bind_map]\n    rw [map_eq_map]\n    rw [\u2190bind_map_binary_eq_map_bind_map]\n    apply congrArg\n    simp\n    rfl\n  pure_seq f as := by\n    rw [seq_eq_bind_map]\n    rw [pure_eq_singleton, map_eq_map]\n    rw [singleton_bind]\n  bind_pure_comp {\u03b1} {\u03b2} f as := by\n    rw [bind_eq_bind]\n    rw [funext (fun a=> pure_eq_singleton (f a))]\n    rw [map_eq_map]\n    rw [bind_singleton_comp]\n  bind_map f as := by\n    rw [bind_eq_bind]\n    rw [funext (fun a=> map_eq_map a as)]\n    rw [seq_eq_bind_map]\n  pure_bind := singleton_bind\n  bind_assoc := bind_assoc\n\nend LawfulMonad\n\n\nend List\n", "meta": {"author": "Junology", "repo": "algdata", "sha": "ef0e552747c3f1004705755a3afc7ccedec92bf6", "save_path": "github-repos/lean/Junology-algdata", "path": "github-repos/lean/Junology-algdata/algdata-ef0e552747c3f1004705755a3afc7ccedec92bf6/Algdata/Data/List/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3572037943326782}}
{"text": "/- LoVe Demo 7: Metaprogramming -/\n\nimport .lovelib\n\nnamespace LoVe\n\n\n/- The Tactic Monad -/\n\nexample :\n  true :=\nby tactic.triv\n\nexample :\n  true :=\nby do\n  tactic.trace \"Hello, Metacosmos!\",\n  tactic.triv\n\nmeta def hello_world : tactic unit :=\ndo\n  tactic.trace \"Hello, Metacosmos!\",\n  tactic.triv\n\nexample :\n  true :=\nby hello_world\n\nrun_cmd tactic.trace \"Hello, Metacosmos!\"\n\nopen tactic\n\nexample (\u03b1 : Type) (a : \u03b1) :\n  true :=\nby do\n  trace \"local context:\",\n  local_context >>= trace,\n  trace \"goals:\",\n  get_goals >>= trace,\n  trace \"target:\",\n  target >>= trace,\n  triv\n\nmeta def exact_list : list expr \u2192 tactic unit\n| []        := fail \"no matching expression found\"\n| (e :: es) :=\n  (do\n     trace \"trying \",\n     trace e,\n     exact e)\n  <|> exact_list es\n\nmeta def find_assumption : tactic unit := do\n  es \u2190 local_context,\n  exact_list es\n\nexample {p : Prop} {\u03b1 : Type} (a : \u03b1) (h : p) :\n  p :=\nby do find_assumption\n\nexample {p : Prop} (h : p) :\n  p :=\nby do\n  p_proof \u2190 get_local `h,\n  trace \"p_proof:\",\n  trace p_proof,\n  trace (expr.to_raw_fmt p_proof),\n  trace \"type of p_proof:\",\n  infer_type p_proof >>= trace,\n  trace \"type of type of p_proof:\",\n  infer_type p_proof >>= infer_type >>= trace,\n  apply p_proof\n\n\n/- Names and Expressions -/\n\n#print expr\n\n#check expr tt  -- elaborated expressions\n#check expr ff  -- unelaborated expressions (pre-expressions)\n\n#print name\n\n#check (expr.const `\u2115 [] : expr)\n#check expr.sort level.zero  -- Sort 0, i.e., Prop\n#check expr.sort (level.succ level.zero)\n  -- Sort 1, i.e., Type 0 (Type)\n#check expr.var 0  -- bound variable with De Bruijn index 0\n#check (expr.local_const `uniq_name `pp_name binder_info.default\n  `(\u2115) : expr)\n#check (expr.mvar `uniq_name `pp_name `(\u2115) : expr)\n#check (expr.pi `pp_name binder_info.default `(\u2115)\n  (expr.sort level.zero) : expr)\n#check (expr.lam `pp_name binder_info.default `(\u2115)\n  (expr.var 0) : expr)\n#check expr.elet\n#check expr.macro\n\nrun_cmd do\n  let e : expr := `(list.map (\u03bbn : \u2115, n + 1) [1, 2, 3]),\n  trace e\n\nrun_cmd do\n  let e : expr := `(list.map _ [1, 2, 3]),\n    -- fails (holes are disallowed)\n  skip\n\nrun_cmd do\n  let e\u2081 : pexpr := ``(list.map (\u03bbn, n + 1) [1, 2, 3]),\n  let e\u2082 : pexpr := ``(list.map _ [1, 2, 3]),\n  trace e\u2081,\n  trace e\u2082\n\nrun_cmd do\n  let e := ```(some_silly_name),\n  trace e\n\nrun_cmd trace `some.silly.name\nrun_cmd trace ``true\nrun_cmd trace ``some.silly.name  -- fails (not found)\n\nrun_cmd do\n  let x : expr := `(2 : \u2115),\n  let e : expr := `(%%x + 1),\n  trace e\n\nrun_cmd do\n  let x : expr := `(@id \u2115),\n  let e := ``(list.map %%x),\n  trace e\n\nrun_cmd do\n  let x : expr := `(@id \u2115),\n  let e := ```(a _ %%x),\n  trace e\n\nexample :\n  1 + 2 = 3 :=\nby do\n  `(%%a + %%b = %%c) \u2190 target,\n  trace a,\n  trace b,\n  trace c,\n  `(@eq %%\u03b1 %%l %%r) \u2190 target,\n  trace \u03b1,\n  trace l,\n  trace r,\n  exact `(refl _ : 3 = 3)\n\n\n/- A Simple Tactic: `destruct_and` -/\n\nexample {a b c d : Prop} (h : a \u2227 (b \u2227 c) \u2227 d) :\n  a :=\nand.elim_left h\n\nexample {a b c d : Prop} (h : a \u2227 (b \u2227 c) \u2227 d) :\n  b :=\nand.elim_left (and.elim_left (and.elim_right h))\n\nexample {a b c d : Prop} (h : a \u2227 (b \u2227 c) \u2227 d) :\n  b \u2227 c :=\nand.elim_left (and.elim_right h)\n\nmeta def destruct_and_helper : expr \u2192 expr \u2192 tactic unit\n| `(%%a \u2227 %%b) h :=\n    exact h\n    <|> (do\n      ha \u2190 to_expr ``(and.elim_left %%h),\n      destruct_and_helper a ha)\n    <|> (do\n      hb \u2190 to_expr ``(and.elim_right %%h),\n      destruct_and_helper b hb)\n| _            h := exact h\n\nmeta def destruct_and (nam : name) : tactic unit :=\ndo\n  h \u2190 get_local nam,\n  t \u2190 infer_type h,\n  destruct_and_helper t h\n\nexample {a b c d : Prop} (h : a \u2227 b \u2227 c) :\n  a :=\nby destruct_and `h\n\nexample {a b c d : Prop} (h : a \u2227 b \u2227 c) :\n  c :=\nby destruct_and `h\n\nexample {a b c d : Prop} (h : a \u2227 b \u2227 c) :\n  b \u2227 c :=\nby destruct_and `h\n\nexample {a b c d : Prop} (h : a \u2227 b \u2227 c) :\n  a \u2227 c :=\nby destruct_and `h  -- fails\n\n\n/- Example: A Solvability Advisor -/\n\nmeta def is_theorem : declaration \u2192 bool\n| (declaration.defn _ _ _ _ _ _) := ff\n| (declaration.thm _ _ _ _)      := tt\n| (declaration.cnst _ _ _ _)     := ff\n| (declaration.ax _ _ _)         := tt\n\nmeta def get_all_theorems : tactic (list name) :=\ndo\n  env \u2190 get_env,\n  pure (environment.fold env [] (\u03bbdecl nams,\n    if is_theorem decl then declaration.to_name decl :: nams\n    else nams))\n\nmeta def solve_with_name (nam : name) : tactic unit :=\ndo\n  cst \u2190 mk_const nam,\n  apply cst\n    ({ md := transparency.reducible, unify := ff } : apply_cfg),\n  all_goals assumption\n\nmeta def solve_direct : tactic unit :=\ndo\n  nams \u2190 get_all_theorems,\n  list.mfirst (\u03bbnam,\n    do\n      solve_with_name nam,\n      trace (\"directly solved by \" ++ to_string nam))\n    nams\n\nexample {x y : \u2115} (h : x = y) :\n  y = x :=\nby solve_direct\n\nmeta def solve_direct_symm : tactic unit :=\nsolve_direct\n<|> (do\n  cst \u2190 mk_const `eq.symm,\n  apply cst,\n  solve_direct)\n\nexample {n : \u2115} :\n  n + 0 = n :=\nby solve_direct_symm\n\nexample {n : \u2115} :\n  n = n + 0 :=\nby solve_direct_symm\n\nend LoVe\n", "meta": {"author": "blanchette", "repo": "logical_verification_2019", "sha": "7c5f1c90f6e5a0b221a02c8fa4e36c0c66393036", "save_path": "github-repos/lean/blanchette-logical_verification_2019", "path": "github-repos/lean/blanchette-logical_verification_2019/logical_verification_2019-7c5f1c90f6e5a0b221a02c8fa4e36c0c66393036/lean/love07_metaprogramming_demo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.554470450236115, "lm_q2_score": 0.6442250928250375, "lm_q1q2_score": 0.3572037772721015}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport group_theory.perm.basic\nimport logic.equiv.set\n\n/-!\n# `equiv.perm.via_embedding`, a noncomputable analogue of `equiv.perm.via_fintype_embedding`.\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\nvariables {\u03b1 \u03b2 : Type*}\n\nnamespace equiv\n\nnamespace perm\n\nvariables (e : perm \u03b1) (\u03b9 : \u03b1 \u21aa \u03b2)\n\nopen_locale classical\n\n/-- Noncomputable version of `equiv.perm.via_fintype_embedding` that does not assume `fintype` -/\nnoncomputable def via_embedding : perm \u03b2 :=\nextend_domain e (of_injective \u03b9.1 \u03b9.2)\n\nlemma via_embedding_apply (x : \u03b1) : e.via_embedding \u03b9 (\u03b9 x) = \u03b9 (e x) :=\nextend_domain_apply_image e (of_injective \u03b9.1 \u03b9.2) x\n\nlemma via_embedding_apply_of_not_mem (x : \u03b2) (hx : x \u2209 _root_.set.range \u03b9) :\n  e.via_embedding \u03b9 x = x :=\nextend_domain_apply_not_subtype e (of_injective \u03b9.1 \u03b9.2) hx\n\n/-- `via_embedding` as a group homomorphism -/\nnoncomputable def via_embedding_hom : perm \u03b1 \u2192* perm \u03b2:=\nextend_domain_hom (of_injective \u03b9.1 \u03b9.2)\n\n\n\nlemma via_embedding_hom_injective : function.injective (via_embedding_hom \u03b9) :=\nextend_domain_hom_injective (of_injective \u03b9.1 \u03b9.2)\n\nend perm\n\nend equiv\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/group_theory/perm/via_embedding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.35714890062034127}}
{"text": "import GMLInit.Data.Basic\n\ninductive HVal.{u} : Type u\n| mk {\u03b1 : Sort u} : \u03b1 \u2192 HVal\n\nprotected def HVal.sort : HVal \u2192 Type _\n| @mk \u03b1 _ => \u03b1\n\nprotected def HVal.val : (a : HVal) \u2192 a.sort\n| mk a => a\n\ntheorem HVal.eq_of_val_heq_val : {a b : HVal} \u2192 a.val \u2245 b.val \u2192 a = b\n| mk _, mk _, HEq.refl _ => Eq.refl _\n\ntheorem HVal.val_heq_val_of_eq : {a b : HVal} \u2192 a = b \u2192 a.val \u2245 b.val\n| mk _, mk _, Eq.refl _ => HEq.refl _\n\ntheorem HVal.eq_iff_val_heq_val (a b : HVal) : a = b \u2194 a.val \u2245 b.val :=\n  \u27e8HVal.val_heq_val_of_eq, HVal.eq_of_val_heq_val\u27e9\n\ntheorem HVal.proof_irrel : (a b : HVal.{0}) \u2192 a = b\n| @mk a ha, @mk b hb =>\n  have h : a = b := propext \u27e8\u03bb _ => hb, \u03bb _ => ha\u27e9\n  match a, b, h with\n  | _, _, rfl => proofIrrel ha hb \u25b8 rfl\n", "meta": {"author": "fgdorais", "repo": "GMLInit", "sha": "a295111627ac907ebc6a86f906dd9b4d69b338d8", "save_path": "github-repos/lean/fgdorais-GMLInit", "path": "github-repos/lean/fgdorais-GMLInit/GMLInit-a295111627ac907ebc6a86f906dd9b4d69b338d8/GMLInit/Data/HVal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.35714000097015514}}
{"text": "import get_theorems \nimport change_goal\nimport change_hypothesis\nimport check_expressions\nimport syntax_matching\nimport testbed.graph_theory\n\nimport tactic\n.\n\n--set_option pp.implicit true\n\nopen_locale big_operators -- enable \u2211 notation\nopen simple_graph \nopen expr tactic\nopen interactive (parse)\nopen lean.parser (ident)\n\nuniverses u\nvariables {V : Type u} \n\n\n--------------------  TACTICS: HAMMER HELPERS (reason, extract, expand) -------------------- \n\nmeta def reason : tactic unit := do {\n  -- simplify goal\n  simp_goal, \n  triv, -- try to close goal (using the proof \"trivial: true\" if the goal is true, or \"refl\" if the goal is an equality)\n  trace \"Successfully reasoned.\"\n}\n\nmeta def extract_from_library : tactic unit := do {\n  -- add a nice theorem to the hypothesis\n  h \u2190 get_strongest_syntactic_match, \n  add_theorem_to_hypothesis h,\n\n  -- try to use it immediately if possible, but not necessary\n  use_theorem h,\n  \n  trace \"Successfully extracted from library.\"\n}\n\nmeta def expand_target : tactic unit := failed\n\nmeta def expand_hypothesis : tactic unit := failed\n\n\n--------------------  TACTIC: HAMMER (solve all theorems) -------------------- \n\nmeta def hammer : tactic unit :=\ndo {\n  -- from Tim and Fabian's pseudocoded high-level algorithm\n  iterate_at_most 10 $ do {  -- repeat this up to 10 times, or until all strategies fail.\n    reason <|> -- always try greedy reasoning first.  this includes planning / subtasks / aesop...\n    extract_from_library <|> -- if reasoning fails to make progress, extract a result from the library.  this uses unification/tree-edit/etc.\n    expand_target <|> -- if there's no result that hasn't been extracted, expand the target\n    expand_hypothesis -- if the target has already been fully expanded, expand a hypothesis that is connected to the target\n  },\n  skip\n}\n\n--------------------  THEOREMS SOLVED BY HAMMER -------------------- \n\n-- Handshaking lemma : the sum of degrees is even --\ntheorem degree_sum_even_auto (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V]: \n  even (\u2211v,  G.degree v) :=\nbegin\n  --  without \"hammer\", the proof is: rw degree_sum, simp\n  hammer,\nend \n\n-- Graphs have at most (n choose 2) edges  --\ntheorem edge_bound_auto (G : simple_graph V) [fintype V] [decidable_rel G.adj] [decidable_eq V]: \n  \u2223\u2223E[G]\u2223\u2223 \u2264 \u2223\u2223(V[G])\u2223\u2223.choose 2 :=\nbegin \n  --expand_inequality, \n  hammer,\nend\n\n-- Every path is bipartite --\ntheorem path_is_bipartite_auto (n : \u2115) : \u2200n : \u2115, is_bipartite (path_graph n) :=\nbegin \n  hammer,\nend\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"author": "Human-Oriented-ATP", "repo": "lean-tactics", "sha": "8fa4c8b8efc0c6a1d408b48e999f3a36f228bd0f", "save_path": "github-repos/lean/Human-Oriented-ATP-lean-tactics", "path": "github-repos/lean/Human-Oriented-ATP-lean-tactics/lean-tactics-8fa4c8b8efc0c6a1d408b48e999f3a36f228bd0f/lean3/src/hammer.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347362, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3571400009701551}}
{"text": "import tactic.linarith\nimport lib.interactive_expr\n\n\nmeta def rel_symb : expr \u2192 tactic (expr \u00d7 expr \u00d7 string)\n| `(%%x < %%y) := pure (x, y, \" < \")\n| `(%%x \u2264 %%y) := pure (x, y, \" \u2264 \")\n| `(%%x > %%y) := pure (x, y, \" > \")\n| `(%%x \u2265 %%y) := pure (x, y, \" \u2265 \")\n| `(%%x \u2208 %%y) := pure (x, y, \" \u2208 \")\n| e := do pe \u2190 tactic.pp e, tactic.fail $ \"Not a relation: \" ++ pe.to_string\n\nsection\nopen expr\n\n/-- Une version de `expr.rename_var` qui renomme m\u00eame les variables libres. -/\nmeta def expr.rename (old new : name) : expr \u2192 expr\n| (pi n bi t b) := (pi (if n = old then new else n) bi (expr.rename t) (expr.rename b))\n| (lam n bi t b) := (lam (if n = old then new else n) bi (expr.rename t) (expr.rename b))\n| (app t b) := (app (expr.rename t) (expr.rename b))\n| (local_const n nn bi t) := (local_const (if n = old then new else n) (if nn = old then new else nn) bi $ expr.rename t)\n| e := e\n\nend\n\n/-- Mini language d'expression o\u00f9 les quantificateurs born\u00e9s et les connecteurs \u2227, \u2228 et \u2194 sont des citoyens \u00e0 part enti\u00e8re. -/\nmeta inductive my_expr\n| forall_rel (var_name : name) (typ : expr) (rel : string) (rel_rhs : expr) (propo : my_expr) : my_expr\n| forall_simple (var_name : name) (typ : expr) (propo : my_expr) : my_expr\n| exist_rel (var_name : name) (typ : expr) (rel : string) (rel_rhs : expr) (propo : my_expr) : my_expr\n| exist_simple (var_name : name) (typ : expr) (propo : my_expr) : my_expr\n| conjunction (propo propo' : my_expr) : my_expr\n| disjunction (propo propo' : my_expr) : my_expr\n| impl (le re : expr) (lhs : my_expr) (rhs : my_expr) : my_expr\n| ssi (le re : expr) (lhs rhs : my_expr) : my_expr\n| egal (le re : expr) : my_expr\n| ineq (le : expr) (symb : string) (re : expr) : my_expr\n| prop (e : expr) : my_expr\n| data (e : expr) : my_expr\n\nnamespace my_expr\n\nmeta def tofmt : my_expr \u2192  tactic format \n| (forall_rel var_name typ rel rel_rhs propo) := do\n    rhs \u2190 tactic.pp rel_rhs,\n    p \u2190 tofmt propo,\n    pure $ \"\u2200 \" ++ var_name.to_string ++ rel ++ rhs.to_string ++ \", \" ++ p\n| (forall_simple var_name typ propo) := do\n    p \u2190 tofmt propo,\n    pure $ \"\u2200 \" ++ var_name.to_string ++ \", \" ++ p\n| (exist_rel var_name typ rel rel_rhs propo) := do\n    rhs \u2190 tactic.pp rel_rhs,\n    p \u2190 tofmt propo,\n    pure $ \"\u2203 \" ++ var_name.to_string ++ rel ++ rhs.to_string ++ \", \" ++ p\n| (exist_simple var_name typ propo) := do\n    p \u2190 tofmt propo,\n    pure $ \"\u2203 \" ++ var_name.to_string ++ \", \" ++ p\n| (conjunction propo propo') := do\n    p \u2190 tofmt propo,\n    p' \u2190 tofmt propo',\n    pure $ p ++ \" \u2227 \" ++ p'\n| (disjunction propo propo') := do\n    p \u2190 tofmt propo,\n    p' \u2190 tofmt propo',\n    pure $ p ++ \" \u2228 \" ++ p'\n| (impl le re lhs rhs) := do \n  l \u2190 tofmt lhs,\n  r \u2190 tofmt rhs,\n  pure $ l ++ \" \u2192 \" ++ r\n| (ssi le re lhs rhs) := do \n  l \u2190 tofmt lhs,\n  r \u2190 tofmt rhs,\n  pure $ l ++ \" \u2194 \" ++ r\n| (egal le re) := do \n  l \u2190 tactic.pp le,\n  r \u2190 tactic.pp re,\n  pure $ l ++ \" = \" ++ r\n| (ineq le symb re) := do \n  l \u2190 tactic.pp le,\n  r \u2190 tactic.pp re,\n  pure $ l ++ symb ++ r\n| (prop e) := tactic.pp e\n| (data e) := tactic.pp e\n\nmeta instance : has_to_tactic_format my_expr :=\n\u27e8my_expr.tofmt\u27e9\n\n/-- D\u00e9finitions \u00e0 tenter de d\u00e9plier quand on ne reconnait ni un \u2200 ni un \u2203. -/\ndef unfold_defs : list name := [`borne_sup, `continue_en, `croissante, `decroissante, \n  `est_borne_inf, `est_borne_sup, `extraction, `impair, `impaire, `injective, \n  `limite_infinie_suite, `limite_infini_suite, `limite_moins_infini_suite,\n  `limite_suite, `majorant, `majoree, `minorant, `minoree, `pair, `paire, `pgcd,\n  `segment, `suite_cauchy, `suite_croissante, `suite_decroissante, `surjective,\n  `valeur_adherence]\n\nmeta def parse : expr \u2192 tactic my_expr\n| e@(expr.pi n bi t b) := do\n    var \u2190 tactic.mk_local' n bi t,\n    match (expr.instantiate_var b var) with\n    | e'@(expr.pi n' bi' t' b') := \n      do { (x, y, symbole) \u2190 rel_symb t',\n           --tactic.trace (rel_symb t'),\n           when (x.local_pp_name \u2260 n) $ tactic.fail \"\",\n/-            tactic.trace \"e : \", tactic.trace e,\n           tactic.trace \"e' : \", tactic.trace e',\n           tactic.trace \"n : \", tactic.trace n,\n           tactic.trace \"n' :\", tactic.trace n',\n           tactic.trace \"t' :\", tactic.trace t',\n           tactic.trace y,\n -/           var' \u2190 tactic.mk_local' n' bi' t',\n           my_expr.forall_rel n t symbole y <$> parse (expr.instantiate_var b' var') } <|>\n      do { typ \u2190 parse t,\n           body \u2190 parse e',\n           if e.is_arrow then \n              pure (impl t e' typ body) \n           else \n             pure $ my_expr.forall_simple n t body }\n    | e' := do \n           typ \u2190 parse t,\n           body \u2190 parse e',\n            if e.is_arrow then \n              pure (impl t e' typ body) \n            else \n              forall_simple n t <$> parse (expr.instantiate_var b var)\n    end\n|`(@Exists %%\u03b1 %%p) := \n  do {\n    `(@Exists %%\u03b1' %%p') \u2190 pure p.binding_body,\n    var \u2190 tactic.mk_local' p.binding_name p.binding_info p.binding_domain,\n    (x, y, symbole) \u2190 rel_symb \u03b1'.binding_body, \n    exist_rel p.binding_name \u03b1 symbole y <$> parse (expr.instantiate_var p' var).binding_body } <|> \n  do { \n    var \u2190 tactic.mk_local' p.binding_name p.binding_info p.binding_domain,\n    exist_simple p.binding_name \u03b1 <$> parse (expr.instantiate_var p.binding_body var)\n    }\n|`(and %%e %%e') := do \n    p \u2190 parse e,\n    p' \u2190 parse e',\n    pure $ conjunction p p'\n|`(or %%e %%e') := do \n    p \u2190 parse e,\n    p' \u2190 parse e',\n    pure $ disjunction p p'\n|`(iff %%e %%e') := do \n    p \u2190 parse e,\n    p' \u2190 parse e',\n    pure $ ssi e e' p p'\n|`(eq %%e %%e') := pure $ egal e e'\n| `(%%x < %%y) := pure $ ineq x \" < \" y\n| `(%%x \u2264 %%y) := pure $ ineq x \" \u2264 \" y\n| `(%%x > %%y) := pure $ ineq x \" > \" y\n| `(%%x \u2265 %%y) := pure $ ineq x \" \u2265 \" y\n| e := do { e' \u2190 tactic.delta unfold_defs e,\n            tactic.trace  \"Remarque : La commande\",\n            tactic.trace  \" On d\u00e9plie nom,\",\n            tactic.trace  \"ou\",\n            tactic.trace  \" On d\u00e9plie nom dans hyp,\",\n            tactic.trace  \"permet de d\u00e9plier la d\u00e9finition nom dans le but ou dans une hypoth\u00e8se hyp.\",\n            tactic.trace  \"\",\n            parse e' } <|> \n       do { t \u2190 tactic.infer_type e, pure $ if t = `(Prop) then prop e else data e } <|> pure (prop e)\n\nmeta def rename (old new : name) : my_expr \u2192 my_expr\n| (forall_rel n typ rel rel_rhs propo) := forall_rel (if n = old then new else n) typ rel rel_rhs $ rename propo\n| (forall_simple n typ propo) := forall_simple (if n = old then new else n) typ $ rename propo\n| (exist_rel n typ rel rel_rhs propo) := exist_rel (if n = old then new else n) typ rel rel_rhs $ rename propo\n| (exist_simple n typ propo) := exist_simple (if n = old then new else n) typ $ rename propo\n| (conjunction propo propo') := conjunction (rename propo) (rename propo')\n| (disjunction propo propo') := disjunction (rename propo) (rename propo')\n| (impl le re lhs rhs) := impl (le.rename_var old new) (re.rename_var old new) (rename lhs) (rename rhs)\n| (ssi le re lhs rhs) := ssi (le.rename_var old new) (re.rename_var old new) (rename lhs) (rename rhs)\n| (egal le re) := egal (le.rename_var old new) (re.rename_var old new)\n| (ineq le rel re) := ineq (le.rename_var old new) rel (re.rename_var old new)\n| (prop e) := prop (e.rename old new)\n| (data e) := data (e.rename old new)\n\n\n/- meta def P : expr := expr.local_const `P `P binder_info.default `(\u2115 \u2192 Prop)\nmeta def n : expr := expr.local_const `n `n binder_info.default `(\u2115)\nmeta def m : expr := expr.app P n\n\nrun_cmd tactic.pp (expr.rename `n `k m) >>= tactic.trace \n\nrun_cmd tactic.pp ((my_expr.prop m).rename `n `k) >>= tactic.trace \n -/\n\nend my_expr\n\nnamespace tactic.interactive\nsetup_tactic_parser\nopen tactic my_expr\n\nmeta def test (h : parse ident) : tactic unit :=\ndo t \u2190 get_local h >>= infer_type,\n   trace t,\n   parse t >>= trace\n\nend tactic.interactive\n\n/- section tests\nexample (Q R : \u2115 \u2192 Prop) (P : \u2115 \u2192 \u2115 \u2192 Prop) \n  (h\u2081 : R 1 \u2192 Q 2) \n  (h\u2082 : \u2200 k \u2265 2, \u2203 n \u2265 3, \u2200 l, l - n = 0 \u2192 P l k) \n  (h\u2083 : \u2203 n \u2265 5, Q n)\n  (h\u2084 : \u2200 k \u2265 2, \u2203 n \u2265 3, P n k)\n  (h\u2085 : \u2203 n, Q n)\n  (h\u2086 : \u2200 k, \u2203 n, P n k)\n  (h\u2087 : \u2200 k \u2265 2, \u2203 n, P n k) \n  (h\u2088 : (\u2200 k : \u2115, Q k) \u2192 (\u2200 l , R l))\n  (h\u2089 : (\u2200 k : \u2115, Q k) \u2194 (\u2200 l , R l))\n  (h\u2081\u2080 : \u2200 k, 1 \u2264 k \u2192 Q k)\n  (h\u2081\u2081 : \u2200 k, \u2200 l, k \u2264 l \u2192 P k l) : true :=\nbegin\n  test h\u2081,\n  test h\u2082,\n  test h\u2083,\n  test h\u2084,\n  test h\u2085,\n  test h\u2086,\n  test h\u2087,\n  test h\u2088,\n  test h\u2089,\n  test h\u2081\u2080,\n  test h\u2081\u2081,\n  trivial\nend\n\nend tests -/\n\nmeta def symb_to_hyp : string \u2192 expr \u2192 string\n| \" \u2265 \" `(0) := \"_pos\"\n| \" \u2265 \" _ := \"_sup\"  \n| \" > \" `(0) := \"_pos\"\n| \" > \" _ := \"_sup\"  \n| \" \u2264 \" `(0) := \"_neg\"\n| \" \u2264 \" _ := \"_inf\"  \n| \" < \" `(0) := \"_neg\"\n| \" < \" _ := \"_inf\"  \n| \" \u2208 \" _ := \"_dans\"\n| _ _ := \"\"\n\nmeta def describe : string \u2192 string\n| \"\u211d\" := \"un nombre r\u00e9el\"\n| \"\u2115\" := \"un nombre entier naturel\"\n| \"\u2124\" := \"un nombre entier relatif\"\n| t := \"une expression de type \" ++ t\n\nmeta def describe_pl : string \u2192 string\n| \"\u211d\" := \"des nombres r\u00e9els\"\n| \"\u2115\" := \"des nombres entiers naturels\"\n| \"\u2124\" := \"des nombres entiers relatifs\"\n| t := \"des expressions de type \" ++ t\n\ndef libre (s: string) : string := \"Le nom \" ++ s ++ \" peut \u00eatre choisi librement parmi les noms disponibles.\"\n\ndef libres (ls : list string) : string :=\n\"Les noms \" ++ string.intercalate \", \" ls ++ \" peuvent \u00eatre choisis librement parmi les noms disponibles.\"\n\nmeta def applique_a : list format \u2192 string\n| [] := \"\"\n| [x] := \" appliqu\u00e9 \u00e0 \" ++ x.to_string\n| s@(x::t) := \" appliqu\u00e9 \u00e0 \" ++ s.to_string\n\nsection\nopen tactic my_expr\n\nmeta def tactic.aide : option name \u2192 tactic unit \n| (some h) := do\n  let sh := h.to_string,\n  eh \u2190 get_local h <|> fail (\"Il n'y a pas d'hypoth\u00e8se appell\u00e9e \" ++ sh),\n  hyp \u2190 infer_type eh,\n  but \u2190 to_string <$> (target >>= pp),\n  msg \u2190 lock_tactic_state $ do { \n    r \u2190 tactic.apply_core eh {}, -- On essaie d'appliquer `h`\n    l \u2190 r.mmap (\u03bb p : name \u00d7 expr, tactic.instantiate_mvars p.2),  -- on instancie les m\u00e9ta-variables qui peuvent l'\u00eatre\n    l' \u2190 (l.filter (\u03bb v : expr, not v.is_mvar)).mmap pp, -- on regarde ce qui a r\u00e9ussit \u00e0 \u00eatre unifi\u00e9 \u00e0 la ligne du dessus\n    do { ptgt \u2190 to_string <$> (target >>= pp), -- Cette ligne va \u00e9chouer s'il ne reste plus de but apr\u00e8s `apply`\n         pure $ \"  Par \" ++ sh ++ applique_a l' ++ \" il suffit de montrer que \" ++ ptgt ++ \n         \",\\n\\nSi vous disposez d\u00e9j\u00e0 d'une d\u00e9monstration H de \" ++ ptgt ++  \n         \" alors on peut utiliser :\\n  On conclut par \" ++ sh ++ applique_a (l' ++ [\"H\"]) ++ \",\" } <|>\n    do { pure $ \"  On conclut par \" ++ sh ++ applique_a l' ++ \",\" } \n    } <|> pure \"\",\n  m \u2190 parse hyp,\n  match m with\n  | (forall_rel var_name typ rel rel_rhs propo) := do\n      py \u2190 pp rel_rhs,\n      t \u2190 pp typ,\n      \n      let n := var_name.to_string,\n      let n\u2080 := n ++ \"\u2080\",\n      let nn\u2080 := mk_simple_name n\u2080,\n      p \u2190 to_string <$> pp (propo.rename var_name nn\u2080),\n      match propo with\n      | (exist_rel var_name' typ' rel' rel_rhs' propo') := do \n        let n' := var_name'.to_string,\n        py' \u2190 to_string <$> pp rel_rhs',\n        p' \u2190 to_string <$> pp (propo'.rename var_name nn\u2080),\n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ rel ++ py.to_string ++ \", \u2203 \" ++ n' ++ rel' ++ py' ++ \", ... \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 [\" ++ n\u2080 ++ \", h\" ++ n\u2080 ++ \"] on obtient \" ++ n' ++ \" tel que (\" ++\n          n' ++ symb_to_hyp rel' rel_rhs' ++ \" : \" ++ n' ++ rel' ++ py' ++ \") (h\" ++ n' ++ \": \" ++ p' ++ \"),\"),\n        trace (\"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string ++ \n               \" et h\" ++ n\u2080 ++ \" est une d\u00e9monstration du fait que \" ++ n\u2080 ++ rel ++ py.to_string),\n        trace $ libres [n' ++ symb_to_hyp rel' rel_rhs', \"h\" ++ n']\n      | (exist_simple var_name' typ' propo') := do\n        let n' := var_name'.to_string,\n        p' \u2190 to_string <$> pp (propo'.rename var_name nn\u2080),\n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ rel ++ py.to_string ++ \", \u2203 \" ++ n'  ++ \", ... \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 [\" ++ n\u2080 ++ \", h\" ++ n\u2080 ++\"] on obtient \" ++ n' ++ \" tel que (h\" ++ n' ++ \": \" ++ p' ++ \"),\"),\n        trace (\"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string ++ \n               \" et h\" ++ n\u2080 ++ \" est une d\u00e9monstration du fait que \" ++ n\u2080 ++ rel ++ py.to_string),\n        trace $ libres [n', \"h\" ++ n']\n      | _ := do \n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ var_name.to_string ++ rel ++ py.to_string ++ \", \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 \" ++ n\u2080 ++ \" on obtient (h : \" ++ p ++ \"),\"),\n        trace $ \"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string,\n        trace $ libre \"h\"\n      end\n  | (forall_simple var_name typ propo) := do\n      t \u2190 pp typ,\n      let n := var_name.to_string,\n      let n\u2080 := n ++ \"\u2080\",\n      let nn\u2080 := mk_simple_name n\u2080,\n      p \u2190 to_string <$> pp (propo.rename (mk_simple_name n) nn\u2080),\n      match propo with\n      | (exist_rel var_name' typ' rel' rel_rhs' propo') := do \n        let n' := var_name'.to_string,\n        py' \u2190 to_string <$> pp rel_rhs',\n        p' \u2190 to_string <$> pp (propo'.rename var_name nn\u2080),\n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ \",\" ++ \"\u2203 \" ++ n' ++ rel' ++ py' ++ \", ... \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 \" ++ n\u2080 ++ \" on obtient \" ++ n' ++ \" tel que (\" ++\n          n' ++ symb_to_hyp rel' rel_rhs' ++ \" : \" ++ n' ++ rel' ++ py' ++ \") (h\" ++ n' ++ \": \" ++ p' ++ \"),\"),\n        trace $ \"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string,\n        trace $ libres [n', n' ++ symb_to_hyp rel' rel_rhs', \"h\" ++ n']\n      | (exist_simple var_name' typ' propo') := do\n        let n' := var_name'.to_string,\n        p' \u2190 to_string <$> pp (propo'.rename var_name nn\u2080),\n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ \", \u2203 \" ++ n'  ++ \", ... \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 \" ++ n\u2080 ++\" on obtient \" ++ n' ++ \" tel que (h\" ++ n' ++ \": \" ++ p' ++ \"),\"),\n        trace $ \"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string,\n        trace $ libres [n', \"h\" ++ n']\n      | (forall_rel var_name' typ' rel' rel_rhs' propo') := do\n        let n' := var_name'.to_string,\n        py' \u2190 to_string <$> pp rel_rhs',\n        p' \u2190 to_string <$> pp (propo'.rename var_name nn\u2080),\n        let rel := n ++ rel' ++ n',\n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ \" \" ++ n' ++\", \" ++ rel ++ \" \u2192 ... \",\n        trace \"On peut l'utiliser avec :\",\n        trace (\"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 [\"++n++\", \"++n'++\", H] on obtient (h : \" ++ p' ++ \"),\"),\n        trace $ \"o\u00f9 \" ++ n ++ \" et \" ++ n' ++ \" sont \" ++ describe_pl t.to_string ++ \" et H est une d\u00e9monstration de \" ++ rel,\n        trace $ libre \"h\"\n      | _ := do \n        trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" commence par \u00ab \u2200 \" ++ n ++ \", \u00bb\",\n        trace \"On peut l'utiliser avec :\",\n        trace $ \"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 \" ++ n\u2080 ++ \" on obtient (h : \" ++ p ++ \"),\",\n        trace $ \"o\u00f9 \" ++ n\u2080 ++ \" est \" ++ describe t.to_string,\n        trace $ libre \"h\",\n        trace $ \"\\nSi cette hypoth\u00e8se ne servira plus dans sa forme g\u00e9n\u00e9rale, on peut aussi sp\u00e9cialiser \" ++ sh ++ \" par\",\n        trace $ \"  On applique \" ++ sh ++ \" \u00e0 \" ++ n\u2080 ++ \",\",\n        when (msg \u2260 \"\") (do \n          trace $ \"\\nComme le but est \" ++ but ++ \", on peut utiliser :\",\n          trace $ msg)\n      end\n  | (exist_rel var_name typ rel rel_rhs propo) := do\n      let n := var_name.to_string,\n      y \u2190 to_string <$> pp rel_rhs,\n      p \u2190 to_string <$> pp propo,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est de la forme \u00ab \u2203 \" ++ var_name.to_string ++ rel ++ y ++ \", ... \u00bb\",\n      trace \"On peut l'utiliser avec :\",\n      trace (\"  Par \" ++ sh ++ \" on obtient \" ++ n ++  \" tel que (\" ++\n          n ++ symb_to_hyp rel rel_rhs ++ \" : \" ++ n ++ rel ++ y ++ \") (h\" ++ n ++ \": \" ++ p ++ \"),\"),\n      trace $ libres [n, n ++ symb_to_hyp rel rel_rhs, \"h\" ++ n]\n  | (exist_simple var_name typ propo) := do\n      let n := var_name.to_string,\n      p \u2190 to_string <$> pp propo,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est de la forme \u00ab \u2203 \" ++ var_name.to_string ++ \", ... \u00bb\",\n      trace \"On peut l'utiliser avec :\",\n      trace (\"  Par \" ++ sh ++ \" on obtient \" ++ n ++  \" tel que (h\" ++ n ++ \": \" ++ p ++ \"),\"),\n      trace $ libres [n, \"h\" ++ n]\n  | (conjunction propo propo') := do\n    p \u2190 to_string <$> pp propo,\n    p' \u2190 to_string <$> pp propo',\n    trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est de la forme \u00ab ... et ... \u00bb\",\n    trace \"On peut l'utiliser avec :\",\n    trace $ \"  Par \" ++ sh ++ \" on obtient (h\u2081 : \" ++ p ++ \") (h\u2082 : \" ++ p' ++ \"),\",\n    trace $ libres [\"h\u2081\", \"h\u2082\"]\n  | (disjunction propo propo') := do\n    p \u2190 to_string <$> pp propo,\n    p' \u2190 to_string <$> pp propo',\n    trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est de la forme \u00ab ... ou ... \u00bb\",\n    trace \"On peut l'utiliser avec :\",\n    trace $ \"  On discute en utilisant \" ++ sh ++ \",\"    \n  | (impl le re lhs rhs) := do \n      l \u2190 to_string <$> pp lhs,\n      r \u2190 to_string <$> pp rhs,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est une implication\",\n      goal \u2190 target,\n      do {\n        unify re goal,\n        trace \"La conclusion de cette implication est le but courant\",\n        trace \"On peut donc utiliser cette hypoth\u00e8se avec :\",\n        trace $ \"  Par \" ++ sh ++ \" il suffit de montrer : \" ++ r,\n        trace $ \"\\nSi vous disposez d\u00e9j\u00e0 d'une preuve H de \" ++ l ++ \" alors on peut utiliser :\",\n        trace $ \"  On conclut par \" ++ sh ++ \" appliqu\u00e9 \u00e0 H,\"} <|> \n      do {\n        trace $ \"La pr\u00e9misse de cette implication est \" ++ l,\n        trace $ \"Si vous avez une d\u00e9monstration H de \" ++ l,\n        trace \"vous pouvez donc utiliser cette hypoth\u00e8se avec :\",\n        trace $ \"  Par \" ++ sh ++ \" appliqu\u00e9 \u00e0 H on obtient H' : \" ++ r ++ \",\",\n        trace $ libre \"H'\"  }\n  | (ssi le re lhs rhs) := do \n      l \u2190 to_string <$> pp lhs,\n      r \u2190 to_string <$> pp rhs,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est une \u00e9quivalence\",\n      trace $ \"On peut s'en servir pour remplacer le membre de gauche (c'est \u00e0 dire \" ++ l ++ \n              \") par le membre de droite  (c'est \u00e0 dire \" ++ r ++ \") dans le but par :\",\n      trace $ \"  On r\u00e9\u00e9crit via \" ++ sh ++ \",\",\n      trace $ \"On peut s'en servir pour remplacer le membre de droite dans par le membre de gauche dans le but par :\",\n      trace $ \"  On r\u00e9\u00e9crit via \u2190\" ++ sh ++ \",\",\n      trace $ \"On peut aussi effectuer de tels remplacements dans une hypoth\u00e8se \" ++ sh ++ \"' par\",\n      trace $ \"  On r\u00e9\u00e9crit via \" ++ sh ++ \" dans \"++ sh ++ \"',\",\n      trace $ \"ou\",\n      trace $ \"  On r\u00e9\u00e9crit via \u2190\" ++ sh ++ \" dans \"++ sh ++ \"',\"\n  | (egal le re) := do \n      l \u2190 to_string <$> pp le,\n      r \u2190 to_string <$> pp re,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est une \u00e9galit\u00e9\",\n      ex : bool \u2190 lock_tactic_state $ (exact eh >> pure tt) <|> pure ff,\n      if ex then\n          trace $ \"Cette \u00e9galit\u00e9 est exactement ce qu'il faut d\u00e9montrer\\n\" ++\n                  \"On peut l'utiliser avec :\\n\" ++\n                  \"  On conclut par \" ++ sh ++ \",\"\n      else \n        do {lin : bool \u2190 lock_tactic_state $ (linarith ff tt [to_pexpr eh] >> pure tt) <|> pure ff,\n        if lin then \n          trace $ \"Le but courant en d\u00e9coule imm\u00e9diatement\\n\" ++\n                  \"On peut l'utiliser avec :\\n\" ++\n                  \"  On conclut par \" ++ sh ++ \",\" \n        else\n          do       \n          trace $ \"On peut s'en servir pour remplacer le membre de gauche (c'est \u00e0 dire \" ++ l ++ \n                  \") par le membre de droite  (c'est \u00e0 dire \" ++ r ++ \") dans le but par :\",\n          trace $ \"  On r\u00e9\u00e9crit via \" ++ sh ++ \",\",\n          trace $ \"On peut s'en servir pour remplacer le membre de droite dans par le membre de gauche dans le but par :\",\n          trace $ \"  On r\u00e9\u00e9crit via \u2190 \" ++ sh ++ \",\",\n          trace $ \"On peut aussi effectuer de tels remplacements dans une hypoth\u00e8se \" ++ sh ++ \"' par\",\n          trace $ \"  On r\u00e9\u00e9crit via \" ++ sh ++ \" dans \"++ sh ++ \"',\",\n          trace $ \"ou\",\n          trace $ \"  On r\u00e9\u00e9crit via \u2190 \" ++ sh ++ \" dans \"++ sh ++ \"',\\n\",\n          trace $ \"On peut aussi s'en servir comme \u00e9tape dans un calcul, ou bien combin\u00e9e lin\u00e9airement \u00e0 d'autres par :\\n\" ++\n                  \"  On combine [\" ++ sh ++ \", ...],\" }\n  | (ineq le rel re) := do \n      l \u2190 to_string <$> pp le,\n      r \u2190 to_string <$> pp re,\n      trace $ \"L'hypoth\u00e8se \" ++ sh ++ \" est une in\u00e9galit\u00e9\",\n      ex : bool \u2190 lock_tactic_state $ (exact eh >> pure tt) <|> pure ff,\n      if ex then\n          trace $ \"Cette in\u00e9galit\u00e9 est exactement ce qu'il faut d\u00e9montrer\\n\" ++\n                  \"On peut l'utiliser avec :\\n\" ++\n                  \"  On conclut par \" ++ sh ++ \",\"\n      else \n        do {lin : bool \u2190 lock_tactic_state $ (linarith ff tt [to_pexpr eh] >> pure tt) <|> pure ff,\n        if lin then \n          trace $ \"Le but courant en d\u00e9coule imm\u00e9diatement\\n\" ++\n                  \"On peut l'utiliser avec :\\n\" ++\n                  \"  On conclut par \" ++ sh  ++ \",\"\n        else\n          trace $ \"On peut s'en servir comme \u00e9tape dans un calcul, ou bien combin\u00e9e lin\u00e9airement \u00e0 d'autres par :\\n\" ++\n                  \"  On combine [\" ++ sh ++ \", ...],\"\n            }\n  | (prop `(false)) := do \n      trace $ \"Cette hypoth\u00e8se est une contradiction.\\n\" ++\n              \"On peut en d\u00e9duire tout ce qu'on veut par :\\n\" ++\n              \"  Montrons une contradiction,\\n  On conclut par \" ++ sh ++ \",\"\n  | (prop e) := do \n      trace \"Je n'ai rien \u00e0 d\u00e9clarer \u00e0 propos de cette hypoth\u00e8se.\"\n  | (data e) := do\n      t \u2190 to_string <$> pp e,\n      trace $ \"L'objet \" ++ sh ++ match t with\n      | \"\u211d\" := \" est un nombre r\u00e9el fix\u00e9.\"\n      | \"\u2115\" := \" est un nombre entier naturel fix\u00e9.\"\n      | \"\u2124\" := \" est un nombre entier relatif fix\u00e9.\"\n      | s := \" : \" ++ s ++ \" est fix\u00e9.\"\n      end\n  end\n \n| none := do \n  goal \u2190 target,\n  g \u2190 parse goal,\n  match g with\n  | (forall_rel var_name typ rel rel_rhs propo) := do\n      py \u2190 pp rel_rhs,\n      let commun := var_name.to_string ++ rel ++ py.to_string ++ \",\",\n      trace $ \"Le but commence par \u00ab \u2200 \" ++ commun ++ \" \u00bb\",\n      trace \"Une d\u00e9monstration directe commence donc par :\",\n      trace $ \"  Soit \" ++ commun ++ \",\"\n  | (forall_simple var_name typ propo) := do\n      let n := var_name.to_string,\n      t \u2190 to_string <$> pp typ,\n      trace $ \"Le but commence par \u00ab \u2200 \" ++ var_name.to_string ++ \" : \" ++ t ++ \", \u00bb\",\n      trace \"Une d\u00e9monstration directe commence donc par :\",\n      trace $ \"  Soit \" ++ var_name.to_string ++ \" : \" ++ t ++ \",\"\n  | (exist_rel var_name typ rel rel_rhs propo) := do\n      let n := var_name.to_string,\n      let n\u2080 := n ++ \"\u2080\",\n      let nn\u2080 := mk_simple_name n\u2080,\n      tgt \u2190 to_string <$> pp (propo.rename (mk_simple_name n) nn\u2080),\n      t \u2190 to_string <$> pp typ,\n      trace $ \"Le but est de la forme \u00ab \u2203 \" ++ n ++ \", ... \u00bb\",\n      trace \"Une d\u00e9monstration directe commence donc par :\",\n      trace $ \"  Montrons que \" ++ n\u2080 ++ \" convient : \" ++ tgt ++ \",\",\n      trace $ \"en rempla\u00e7ant \" ++ n\u2080 ++ \" par \" ++ describe t\n  | (exist_simple var_name typ propo) := do\n      let n := var_name.to_string,\n      let n\u2080 := n ++ \"\u2080\",\n      let nn\u2080 := mk_simple_name n\u2080,\n      tgt \u2190 to_string <$> pp (propo.rename (mk_simple_name n) nn\u2080),\n      t \u2190 to_string <$> pp typ,\n      trace $ \"Le but est de la forme \u00ab \u2203 \" ++ n ++ \", ... \u00bb\",\n      trace \"Une d\u00e9monstration directe commence donc par :\",\n      trace $ \"  Montrons que \" ++ n\u2080 ++ \" convient : \" ++ tgt ++ \",\",\n      trace $ \"en rempla\u00e7ant \" ++ n\u2080 ++ \" par \" ++ describe t\n  | (conjunction propo propo') := do\n    p \u2190 to_string <$> pp propo,\n    p' \u2190 to_string <$> pp propo',\n    trace $ \"Le but est de la forme \u00ab ... et ... \u00bb\",\n    trace \"Une d\u00e9monstration directe commence donc par :\",\n    trace $ \"  Montrons que \" ++ p ++ \",\",\n    trace $ \"Une fois cette premi\u00e8re d\u00e9monstration achev\u00e9e, il restera \u00e0 montrer que \" ++ p'\n  | (disjunction propo propo') := do\n    p \u2190 to_string <$> pp propo,\n    p' \u2190 to_string <$> pp propo',\n    trace $ \"Le but est de la forme \u00ab ... ou ... \u00bb\",\n    trace \"Une d\u00e9monstration directe commence donc par annoncer quelle alternative va \u00eatre d\u00e9montr\u00e9e :\",\n    trace $ \"  Montrons que \" ++ p ++ \",\",\n    trace $ \"ou bien :\",\n    trace $ \"  Montrons que \" ++ p' ++ \",\"\n  | (impl le re lhs rhs) := do \n      l \u2190 pp lhs,\n      trace $ \"Le but est une implication \u00ab \" ++ l.to_string ++ \" \u2192 ... \u00bb\",\n      trace \"Une d\u00e9monstration directe commence donc par :\",\n      trace $ \"  Supposons hyp : \" ++ l.to_string ++ \", \",\n      trace \"o\u00f9 hyp est un nom disponible au choix.\"\n  | (ssi le re lhs rhs) := do \n      l \u2190 to_string <$> pp lhs,\n      r \u2190 to_string <$> pp rhs,  \n      trace \"Le but est une \u00e9quivalence. On peut annoncer la d\u00e9monstration de l'implication de la gauche vers la droite par :\",\n      trace $ \" Montrons que \" ++ l ++ \" \u2192 \" ++ r ++ \",\",\n      trace $ \"Une fois cette premi\u00e8re d\u00e9monstration achev\u00e9e, il restera \u00e0 montrer que \" ++ r ++ \" \u2192 \" ++ l\n  | (egal le re) := do\n      l \u2190 to_string <$> pp le,\n      r \u2190 to_string <$> pp re,\n      trace $ \"Le but est une \u00e9galit\u00e9\\n\" ++\n              \"On peut la d\u00e9montrer par r\u00e9\u00e9criture avec la commande `On r\u00e9\u00e9crit via`\\n\" ++\n              \"ou bien commencer un calcul par\\n\" ++\n              \"  calc \" ++ l ++ \" = sorry : by { sorry }\\n\" ++\n              \"  ... = \" ++ r ++ \" : by { sorry },\\n\" ++\n              \"On peut bien s\u00fbr utiliser plus de lignes interm\u00e9diaires.\\n\" ++\n              \"Attention \u00e0 ne pas mettre de virgule \u00e0 la fin des lignes interm\u00e9diaires.\\n\\n\" ++\n              \"On peut aussi tenter des combinaisons lin\u00e9aires d'hypoth\u00e8ses hyp\u2081 hyp\u2082... avec\\n\" ++\n              \"  On combine [hyp\u2081, hyp\u2082],\"\n  | (ineq le rel re) := do\n      l \u2190 to_string <$> pp le,\n      r \u2190 to_string <$> pp re,\n      trace $ \"Le but est une in\u00e9galit\u00e9\\n\" ++\n              \"On peut commencer un calcul par\\n\" ++\n              \"  calc \" ++ l ++ rel ++ \"sorry : by { sorry }\\n\" ++\n              \"  ... = \" ++ r ++ \" : by { sorry },\\n\" ++\n              \"On peut bien s\u00fbr utiliser plus de lignes interm\u00e9diaires.\\n\" ++\n              \"La derni\u00e8re ligne du calcul n'est pas forc\u00e9ment une \u00e9galit\u00e9, cela peut \u00eatre une in\u00e9galit\u00e9.\\n\" ++\n              \"De m\u00eame la premi\u00e8re ligne peut \u00eatre une \u00e9galit\u00e9. Au total les symboles de relations\\n\" ++\n              \"doivent s'encha\u00eener pour donner \" ++ rel ++ \"\\n\" ++\n              \"Attention \u00e0 ne pas mettre de virgule \u00e0 la fin des lignes interm\u00e9diaires.\\n\\n\" ++\n              \"On peut aussi tenter des combinaisons lin\u00e9aires d'hypoth\u00e8ses hyp\u2081 hyp\u2082... avec\\n\" ++\n              \"  On combine [hyp\u2081, hyp\u2082],\"\n  | (prop `(false)) := do \n      trace $ \"Le but est de montrer une contradiction.\\n\" ++\n              \"On peut par exemple appliquer une hypoth\u00e8se qui est une n\u00e9gation\" ++\n              \"c'est \u00e0 dire, par d\u00e9finition, de la forme P \u2192 false.\"\n  | (prop e) := do \n      trace \"Pas d'id\u00e9e\"\n  | (data e) := do\n      trace \"Pas d'id\u00e9e\"\n  end\nend\n\nnamespace tactic.interactive\nsetup_tactic_parser\nopen tactic\n\nmeta def aide (i : parse ident?) : tactic unit :=\nfocus1 (tactic.aide i)\n\nend tactic.interactive\n\nexample (P Q : \u2115 \u2192 Prop) (h : \u2200 n, P n \u2192 Q n) (h' : P 2) : Q 2 :=\nbegin\n  aide h,\n  exact h 2 h'\nend\n\nexample (P : \u2115 \u2192 Prop) (h : \u2200 n, P n) : P 2 :=\nbegin\n  aide h,\n  exact h 2\nend\n\n\nexample (P Q : \u2115 \u2192 Prop) (h : P 1 \u2192 Q 2) (h' : P 1) : Q 2 :=\nbegin\n  aide h,\n  exact h h'\nend\n\nexample (P Q : \u2115 \u2192 Prop) (h : P 1 \u2192 Q 2) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P Q : \u2115 \u2192 Prop) (h : P 1 \u2227 Q 2) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P Q : \u2115 \u2192 Prop) (h : (\u2200 n \u2265 2, P n) \u2194  \u2200 l, Q l) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample : true \u2227 1 = 1 :=\nbegin\n  aide,\n  exact \u27e8trivial, rfl\u27e9\nend\n\nexample (P Q : \u2115 \u2192 Prop) (h : P 1 \u2228 Q 2) : true :=\nbegin\n  aide h,\n  trivial\nend\n\n\nexample : true \u2228 false :=\nbegin\n  aide,\n  left,\n  trivial\nend\n\nexample (P : Prop) (h : P) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P : \u2115 \u2192 \u2115 \u2192 Prop) (k l n : \u2115) (h : l - n = 0 \u2192 P l k) : true :=\nbegin\n  aide h,\n  aide k,\n  trivial\nend\n\nexample (P : \u2115 \u2192 \u2115 \u2192 Prop) (h : \u2200 k \u2265 2, \u2203 n \u2265 3, \u2200 l, l - n = 0 \u2192 P l k) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P : \u2115 \u2192 Prop) (h : \u2203 n \u2265 5, P n) : true :=\nbegin\n  aide h,\n  trivial\nend\n\n\nexample (P : \u2115 \u2192 \u2115 \u2192 Prop) (h : \u2200 k \u2265 2, \u2203 n \u2265 3, P n k) : true :=\nbegin\n  aide h,\n  trivial\nend\n\n\nexample (P : \u2115 \u2192 Prop) (h : \u2203 n : \u2115, P n) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P : \u2115 \u2192 \u2115 \u2192 Prop) (h : \u2200 k, \u2203 n : \u2115, P n k) : true :=\nbegin\n  aide h,\n  trivial\nend\n\nexample (P : \u2115 \u2192 \u2115 \u2192 Prop) (h : \u2200 k \u2265 2, \u2203 n : \u2115, P n k) : true :=\nbegin\n  aide h,\n  trivial\nend\n\n\nexample (P : \u2115 \u2192 Prop): \u2203 n : \u2115, P n \u2192 true :=\nbegin\n  aide,\n  use 0,\n  tauto\nend\n\nexample (P Q : Prop) (h : Q) : P \u2192 Q :=\nbegin\n  aide,\n  exact \u03bb _, h,\nend\n\nexample : \u2200 n \u2265 0, true :=\nbegin\n  aide,\n  intros,\n  trivial\nend\n\nexample : \u2200 n : \u2115, 0 \u2264 n :=\nbegin\n  aide,\n  exact nat.zero_le \nend\n", "meta": {"author": "PatrickMassot", "repo": "MDD154", "sha": "00defe82a4b6b7992ed522a92f62abd685e8c943", "save_path": "github-repos/lean/PatrickMassot-MDD154", "path": "github-repos/lean/PatrickMassot-MDD154/MDD154-00defe82a4b6b7992ed522a92f62abd685e8c943/src/lib/aide.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199008363969, "lm_q2_score": 0.6926419894793246, "lm_q1q2_score": 0.35713999393045404}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.adjunction.fully_faithful\nimport category_theory.functor.reflects_isomorphisms\nimport category_theory.epi_mono\n\n/-!\n# Reflective functors\n\nBasic properties of reflective functors, especially those relating to their essential image.\n\nNote properties of reflective functors relating to limits and colimits are included in\n`category_theory.monad.limits`.\n-/\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082 u\u2083\n\nnoncomputable theory\n\nnamespace category_theory\n\nopen category adjunction\n\nvariables {C : Type u\u2081} {D : Type u\u2082} {E : Type u\u2083}\nvariables [category.{v\u2081} C] [category.{v\u2082} D] [category.{v\u2083} E]\n\n/--\nA functor is *reflective*, or *a reflective inclusion*, if it is fully faithful and right adjoint.\n-/\nclass reflective (R : D \u2964 C) extends is_right_adjoint R, full R, faithful R.\n\nvariables {i : D \u2964 C}\n\n/--\nFor a reflective functor `i` (with left adjoint `L`), with unit `\u03b7`, we have `\u03b7_iL = iL \u03b7`.\n-/\n-- TODO: This holds more generally for idempotent adjunctions, not just reflective adjunctions.\nlemma unit_obj_eq_map_unit [reflective i] (X : C) :\n  (of_right_adjoint i).unit.app (i.obj ((left_adjoint i).obj X))\n    = i.map ((left_adjoint i).map ((of_right_adjoint i).unit.app X)) :=\nbegin\n rw [\u2190cancel_mono (i.map ((of_right_adjoint i).counit.app ((left_adjoint i).obj X))),\n     \u2190i.map_comp],\n simp,\nend\n\n/--\nWhen restricted to objects in `D` given by `i : D \u2964 C`, the unit is an isomorphism. In other words,\n`\u03b7_iX` is an isomorphism for any `X` in `D`.\nMore generally this applies to objects essentially in the reflective subcategory, see\n`functor.ess_image.unit_iso`.\n-/\ninstance is_iso_unit_obj [reflective i] {B : D} :\n  is_iso ((of_right_adjoint i).unit.app (i.obj B)) :=\nbegin\n  have : (of_right_adjoint i).unit.app (i.obj B) =\n            inv (i.map ((of_right_adjoint i).counit.app B)),\n  { rw \u2190 comp_hom_eq_id,\n    apply (of_right_adjoint i).right_triangle_components },\n  rw this,\n  exact is_iso.inv_is_iso,\nend\n\n/--\nIf `A` is essentially in the image of a reflective functor `i`, then `\u03b7_A` is an isomorphism.\nThis gives that the \"witness\" for `A` being in the essential image can instead be given as the\nreflection of `A`, with the isomorphism as `\u03b7_A`.\n\n(For any `B` in the reflective subcategory, we automatically have that `\u03b5_B` is an iso.)\n-/\nlemma functor.ess_image.unit_is_iso [reflective i] {A : C} (h : A \u2208 i.ess_image) :\n  is_iso ((of_right_adjoint i).unit.app A) :=\nbegin\n  suffices : (of_right_adjoint i).unit.app A =\n                h.get_iso.inv \u226b (of_right_adjoint i).unit.app (i.obj h.witness) \u226b\n                  (left_adjoint i \u22d9 i).map h.get_iso.hom,\n  { rw this,\n    apply_instance },\n  rw \u2190 nat_trans.naturality,\n  simp,\nend\n\n/-- If `\u03b7_A` is an isomorphism, then `A` is in the essential image of `i`. -/\nlemma mem_ess_image_of_unit_is_iso [is_right_adjoint i] (A : C)\n  [is_iso ((of_right_adjoint i).unit.app A)] : A \u2208 i.ess_image :=\n\u27e8(left_adjoint i).obj A, \u27e8(as_iso ((of_right_adjoint i).unit.app A)).symm\u27e9\u27e9\n\n/-- If `\u03b7_A` is a split monomorphism, then `A` is in the reflective subcategory. -/\nlemma mem_ess_image_of_unit_split_mono [reflective i] {A : C}\n  [split_mono ((of_right_adjoint i).unit.app A)] : A \u2208 i.ess_image :=\nbegin\n  let \u03b7 : \ud835\udfed C \u27f6 left_adjoint i \u22d9 i := (of_right_adjoint i).unit,\n  haveI : is_iso (\u03b7.app (i.obj ((left_adjoint i).obj A))) := (i.obj_mem_ess_image _).unit_is_iso,\n  have : epi (\u03b7.app A),\n  { apply epi_of_epi (retraction (\u03b7.app A)) _,\n    rw (show retraction _ \u226b \u03b7.app A = _, from \u03b7.naturality (retraction (\u03b7.app A))),\n    apply epi_comp (\u03b7.app (i.obj ((left_adjoint i).obj A))) },\n  resetI,\n  haveI := is_iso_of_epi_of_split_mono (\u03b7.app A),\n  exact mem_ess_image_of_unit_is_iso A,\nend\n\n/-- Composition of reflective functors. -/\ninstance reflective.comp (F : C \u2964 D) (G : D \u2964 E) [Fr : reflective F] [Gr : reflective G] :\n  reflective (F \u22d9 G) := { to_faithful := faithful.comp F G, }\n\n/-- (Implementation) Auxiliary definition for `unit_comp_partial_bijective`. -/\ndef unit_comp_partial_bijective_aux [reflective i] (A : C) (B : D) :\n  (A \u27f6 i.obj B) \u2243 (i.obj ((left_adjoint i).obj A) \u27f6 i.obj B) :=\n((adjunction.of_right_adjoint i).hom_equiv _ _).symm.trans (equiv_of_fully_faithful i)\n\n/-- The description of the inverse of the bijection `unit_comp_partial_bijective_aux`. -/\nlemma unit_comp_partial_bijective_aux_symm_apply [reflective i] {A : C} {B : D}\n  (f : i.obj ((left_adjoint i).obj A) \u27f6 i.obj B) :\n  (unit_comp_partial_bijective_aux _ _).symm f = (of_right_adjoint i).unit.app A \u226b f :=\nby simp [unit_comp_partial_bijective_aux]\n\n/--\nIf `i` has a reflector `L`, then the function `(i.obj (L.obj A) \u27f6 B) \u2192 (A \u27f6 B)` given by\nprecomposing with `\u03b7.app A` is a bijection provided `B` is in the essential image of `i`.\nThat is, the function `\u03bb (f : i.obj (L.obj A) \u27f6 B), \u03b7.app A \u226b f` is bijective, as long as `B` is in\nthe essential image of `i`.\nThis definition gives an equivalence: the key property that the inverse can be described\nnicely is shown in `unit_comp_partial_bijective_symm_apply`.\n\nThis establishes there is a natural bijection `(A \u27f6 B) \u2243 (i.obj (L.obj A) \u27f6 B)`. In other words,\nfrom the point of view of objects in `D`, `A` and `i.obj (L.obj A)` look the same: specifically\nthat `\u03b7.app A` is an isomorphism.\n-/\ndef unit_comp_partial_bijective [reflective i] (A : C) {B : C} (hB : B \u2208 i.ess_image) :\n  (A \u27f6 B) \u2243 (i.obj ((left_adjoint i).obj A) \u27f6 B) :=\ncalc (A \u27f6 B) \u2243 (A \u27f6 i.obj hB.witness) : iso.hom_congr (iso.refl _) hB.get_iso.symm\n     ...     \u2243 (i.obj _ \u27f6 i.obj hB.witness) : unit_comp_partial_bijective_aux _ _\n     ...     \u2243 (i.obj ((left_adjoint i).obj A) \u27f6 B) : iso.hom_congr (iso.refl _) hB.get_iso\n\n@[simp]\nlemma unit_comp_partial_bijective_symm_apply [reflective i] (A : C) {B : C}\n  (hB : B \u2208 i.ess_image) (f) :\n  (unit_comp_partial_bijective A hB).symm f = (of_right_adjoint i).unit.app A \u226b f :=\nby simp [unit_comp_partial_bijective, unit_comp_partial_bijective_aux_symm_apply]\n\nlemma unit_comp_partial_bijective_symm_natural [reflective i] (A : C) {B B' : C} (h : B \u27f6 B')\n  (hB : B \u2208 i.ess_image) (hB' : B' \u2208 i.ess_image) (f : i.obj ((left_adjoint i).obj A) \u27f6 B) :\n  (unit_comp_partial_bijective A hB').symm (f \u226b h) =\n    (unit_comp_partial_bijective A hB).symm f \u226b h :=\nby simp\n\nlemma unit_comp_partial_bijective_natural [reflective i] (A : C) {B B' : C} (h : B \u27f6 B')\n  (hB : B \u2208 i.ess_image) (hB' : B' \u2208 i.ess_image) (f : A \u27f6 B) :\n  (unit_comp_partial_bijective A hB') (f \u226b h) = unit_comp_partial_bijective A hB f \u226b h :=\nby rw [\u2190equiv.eq_symm_apply, unit_comp_partial_bijective_symm_natural A h, equiv.symm_apply_apply]\n\n/-- If `i : D \u2964 C` is reflective, the inverse functor of `i \u224c F.ess_image` can be explicitly\ndefined by the reflector. -/\n@[simps]\ndef equiv_ess_image_of_reflective [reflective i] : D \u224c i.ess_image :=\n{ functor := i.to_ess_image,\n  inverse := i.ess_image_inclusion \u22d9 (left_adjoint i : _),\n  unit_iso := nat_iso.of_components (\u03bb X, (as_iso $ (of_right_adjoint i).counit.app X).symm)\n    (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n      exact ((of_right_adjoint i).counit.naturality _).symm }),\n  counit_iso := nat_iso.of_components\n    (\u03bb X, by { refine (iso.symm $ as_iso _), exact (of_right_adjoint i).unit.app X,\n      apply_with (is_iso_of_reflects_iso _ i.ess_image_inclusion) { instances := ff },\n      exact functor.ess_image.unit_is_iso X.prop })\n    (by { intros X Y f, dsimp, simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n      exact ((of_right_adjoint i).unit.naturality f).symm }) }\n\nend category_theory\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/category_theory/adjunction/reflective.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3570914778278342}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot, Eric Wieser\n\n! This file was ported from Lean 3 source module group_theory.group_action.prod\n! leanprover-community/mathlib commit c3291da49cfa65f0d43b094750541c0731edc932\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Group.Prod\nimport Mathbin.GroupTheory.GroupAction.Defs\n\n/-!\n# Prod instances for additive and multiplicative actions\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines instances for binary product of additive and multiplicative actions and provides\nscalar multiplication as a homomorphism from `\u03b1 \u00d7 \u03b2` to `\u03b2`.\n\n## Main declarations\n\n* `smul_mul_hom`/`smul_monoid_hom`: Scalar multiplication bundled as a multiplicative/monoid\n  homomorphism.\n\n## See also\n\n* `group_theory.group_action.option`\n* `group_theory.group_action.pi`\n* `group_theory.group_action.sigma`\n* `group_theory.group_action.sum`\n-/\n\n\nvariable {M N P E \u03b1 \u03b2 : Type _}\n\nnamespace Prod\n\nsection\n\nvariable [SMul M \u03b1] [SMul M \u03b2] [SMul N \u03b1] [SMul N \u03b2] (a : M) (x : \u03b1 \u00d7 \u03b2)\n\n@[to_additive Prod.hasVadd]\ninstance : SMul M (\u03b1 \u00d7 \u03b2) :=\n  \u27e8fun a p => (a \u2022 p.1, a \u2022 p.2)\u27e9\n\n/- warning: prod.smul_fst -> Prod.smul_fst is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_1 : SMul.{u1, u2} M \u03b1] [_inst_2 : SMul.{u1, u3} M \u03b2] (a : M) (x : Prod.{u2, u3} \u03b1 \u03b2), Eq.{succ u2} \u03b1 (Prod.fst.{u2, u3} \u03b1 \u03b2 (SMul.smul.{u1, max u2 u3} M (Prod.{u2, u3} \u03b1 \u03b2) (Prod.smul.{u1, u2, u3} M \u03b1 \u03b2 _inst_1 _inst_2) a x)) (SMul.smul.{u1, u2} M \u03b1 _inst_1 a (Prod.fst.{u2, u3} \u03b1 \u03b2 x))\nbut is expected to have type\n  forall {M : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_1 : SMul.{u1, u3} M \u03b1] [_inst_2 : SMul.{u1, u2} M \u03b2] (a : M) (x : Prod.{u3, u2} \u03b1 \u03b2), Eq.{succ u3} \u03b1 (Prod.fst.{u3, u2} \u03b1 \u03b2 (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.{u3, u2} \u03b1 \u03b2) (instHSMul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.smul.{u1, u3, u2} M \u03b1 \u03b2 _inst_1 _inst_2)) a x)) (HSMul.hSMul.{u1, u3, u3} M \u03b1 \u03b1 (instHSMul.{u1, u3} M \u03b1 _inst_1) a (Prod.fst.{u3, u2} \u03b1 \u03b2 x))\nCase conversion may be inaccurate. Consider using '#align prod.smul_fst Prod.smul_fst\u2093'. -/\n@[simp, to_additive]\ntheorem smul_fst : (a \u2022 x).1 = a \u2022 x.1 :=\n  rfl\n#align prod.smul_fst Prod.smul_fst\n#align prod.vadd_fst Prod.vadd_fst\n\n#print Prod.smul_snd /-\n@[simp, to_additive]\ntheorem smul_snd : (a \u2022 x).2 = a \u2022 x.2 :=\n  rfl\n#align prod.smul_snd Prod.smul_snd\n#align prod.vadd_snd Prod.vadd_snd\n-/\n\n/- warning: prod.smul_mk -> Prod.smul_mk is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_1 : SMul.{u1, u2} M \u03b1] [_inst_2 : SMul.{u1, u3} M \u03b2] (a : M) (b : \u03b1) (c : \u03b2), Eq.{succ (max u2 u3)} (Prod.{u2, u3} \u03b1 \u03b2) (SMul.smul.{u1, max u2 u3} M (Prod.{u2, u3} \u03b1 \u03b2) (Prod.smul.{u1, u2, u3} M \u03b1 \u03b2 _inst_1 _inst_2) a (Prod.mk.{u2, u3} \u03b1 \u03b2 b c)) (Prod.mk.{u2, u3} \u03b1 \u03b2 (SMul.smul.{u1, u2} M \u03b1 _inst_1 a b) (SMul.smul.{u1, u3} M \u03b2 _inst_2 a c))\nbut is expected to have type\n  forall {M : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_1 : SMul.{u1, u3} M \u03b1] [_inst_2 : SMul.{u1, u2} M \u03b2] (a : M) (b : \u03b1) (c : \u03b2), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b1 \u03b2) (HSMul.hSMul.{u1, max u2 u3, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.{u3, u2} \u03b1 \u03b2) (instHSMul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.smul.{u1, u3, u2} M \u03b1 \u03b2 _inst_1 _inst_2)) a (Prod.mk.{u3, u2} \u03b1 \u03b2 b c)) (Prod.mk.{u3, u2} \u03b1 \u03b2 (HSMul.hSMul.{u1, u3, u3} M \u03b1 \u03b1 (instHSMul.{u1, u3} M \u03b1 _inst_1) a b) (HSMul.hSMul.{u1, u2, u2} M \u03b2 \u03b2 (instHSMul.{u1, u2} M \u03b2 _inst_2) a c))\nCase conversion may be inaccurate. Consider using '#align prod.smul_mk Prod.smul_mk\u2093'. -/\n@[simp, to_additive]\ntheorem smul_mk (a : M) (b : \u03b1) (c : \u03b2) : a \u2022 (b, c) = (a \u2022 b, a \u2022 c) :=\n  rfl\n#align prod.smul_mk Prod.smul_mk\n#align prod.vadd_mk Prod.vadd_mk\n\n/- warning: prod.smul_def -> Prod.smul_def is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_1 : SMul.{u1, u2} M \u03b1] [_inst_2 : SMul.{u1, u3} M \u03b2] (a : M) (x : Prod.{u2, u3} \u03b1 \u03b2), Eq.{succ (max u2 u3)} (Prod.{u2, u3} \u03b1 \u03b2) (SMul.smul.{u1, max u2 u3} M (Prod.{u2, u3} \u03b1 \u03b2) (Prod.smul.{u1, u2, u3} M \u03b1 \u03b2 _inst_1 _inst_2) a x) (Prod.mk.{u2, u3} \u03b1 \u03b2 (SMul.smul.{u1, u2} M \u03b1 _inst_1 a (Prod.fst.{u2, u3} \u03b1 \u03b2 x)) (SMul.smul.{u1, u3} M \u03b2 _inst_2 a (Prod.snd.{u2, u3} \u03b1 \u03b2 x)))\nbut is expected to have type\n  forall {M : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_1 : SMul.{u1, u3} M \u03b1] [_inst_2 : SMul.{u1, u2} M \u03b2] (a : M) (x : Prod.{u3, u2} \u03b1 \u03b2), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b1 \u03b2) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.{u3, u2} \u03b1 \u03b2) (instHSMul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.smul.{u1, u3, u2} M \u03b1 \u03b2 _inst_1 _inst_2)) a x) (Prod.mk.{u3, u2} \u03b1 \u03b2 (HSMul.hSMul.{u1, u3, u3} M \u03b1 \u03b1 (instHSMul.{u1, u3} M \u03b1 _inst_1) a (Prod.fst.{u3, u2} \u03b1 \u03b2 x)) (HSMul.hSMul.{u1, u2, u2} M \u03b2 \u03b2 (instHSMul.{u1, u2} M \u03b2 _inst_2) a (Prod.snd.{u3, u2} \u03b1 \u03b2 x)))\nCase conversion may be inaccurate. Consider using '#align prod.smul_def Prod.smul_def\u2093'. -/\n@[to_additive]\ntheorem smul_def (a : M) (x : \u03b1 \u00d7 \u03b2) : a \u2022 x = (a \u2022 x.1, a \u2022 x.2) :=\n  rfl\n#align prod.smul_def Prod.smul_def\n#align prod.vadd_def Prod.vadd_def\n\n/- warning: prod.smul_swap -> Prod.smul_swap is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_1 : SMul.{u1, u2} M \u03b1] [_inst_2 : SMul.{u1, u3} M \u03b2] (a : M) (x : Prod.{u2, u3} \u03b1 \u03b2), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b2 \u03b1) (Prod.swap.{u2, u3} \u03b1 \u03b2 (SMul.smul.{u1, max u2 u3} M (Prod.{u2, u3} \u03b1 \u03b2) (Prod.smul.{u1, u2, u3} M \u03b1 \u03b2 _inst_1 _inst_2) a x)) (SMul.smul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b2 \u03b1) (Prod.smul.{u1, u3, u2} M \u03b2 \u03b1 _inst_2 _inst_1) a (Prod.swap.{u2, u3} \u03b1 \u03b2 x))\nbut is expected to have type\n  forall {M : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_1 : SMul.{u1, u3} M \u03b1] [_inst_2 : SMul.{u1, u2} M \u03b2] (a : M) (x : Prod.{u3, u2} \u03b1 \u03b2), Eq.{max (succ u3) (succ u2)} (Prod.{u2, u3} \u03b2 \u03b1) (Prod.swap.{u3, u2} \u03b1 \u03b2 (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.{u3, u2} \u03b1 \u03b2) (instHSMul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.smul.{u1, u3, u2} M \u03b1 \u03b2 _inst_1 _inst_2)) a x)) (HSMul.hSMul.{u1, max u3 u2, max u3 u2} M (Prod.{u2, u3} \u03b2 \u03b1) (Prod.{u2, u3} \u03b2 \u03b1) (instHSMul.{u1, max u3 u2} M (Prod.{u2, u3} \u03b2 \u03b1) (Prod.smul.{u1, u2, u3} M \u03b2 \u03b1 _inst_2 _inst_1)) a (Prod.swap.{u3, u2} \u03b1 \u03b2 x))\nCase conversion may be inaccurate. Consider using '#align prod.smul_swap Prod.smul_swap\u2093'. -/\n@[simp, to_additive]\ntheorem smul_swap : (a \u2022 x).symm = a \u2022 x.symm :=\n  rfl\n#align prod.smul_swap Prod.smul_swap\n#align prod.vadd_swap Prod.vadd_swap\n\n/- warning: prod.smul_zero_mk -> Prod.smul_zero_mk is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_2 : SMul.{u1, u2} M \u03b2] {\u03b1 : Type.{u3}} [_inst_5 : Monoid.{u1} M] [_inst_6 : AddMonoid.{u3} \u03b1] [_inst_7 : DistribMulAction.{u1, u3} M \u03b1 _inst_5 _inst_6] (a : M) (c : \u03b2), Eq.{succ (max u3 u2)} (Prod.{u3, u2} \u03b1 \u03b2) (SMul.smul.{u1, max u3 u2} M (Prod.{u3, u2} \u03b1 \u03b2) (Prod.smul.{u1, u3, u2} M \u03b1 \u03b2 (SMulZeroClass.toHasSmul.{u1, u3} M \u03b1 (AddZeroClass.toHasZero.{u3} \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_6)) (DistribSMul.toSmulZeroClass.{u1, u3} M \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_6) (DistribMulAction.toDistribSMul.{u1, u3} M \u03b1 _inst_5 _inst_6 _inst_7))) _inst_2) a (Prod.mk.{u3, u2} \u03b1 \u03b2 (OfNat.ofNat.{u3} \u03b1 0 (OfNat.mk.{u3} \u03b1 0 (Zero.zero.{u3} \u03b1 (AddZeroClass.toHasZero.{u3} \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_6))))) c)) (Prod.mk.{u3, u2} \u03b1 \u03b2 (OfNat.ofNat.{u3} \u03b1 0 (OfNat.mk.{u3} \u03b1 0 (Zero.zero.{u3} \u03b1 (AddZeroClass.toHasZero.{u3} \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_6))))) (SMul.smul.{u1, u2} M \u03b2 _inst_2 a c))\nbut is expected to have type\n  forall {M : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_2 : SMul.{u2, u1} M \u03b2] {\u03b1 : Type.{u3}} [_inst_5 : Monoid.{u2} M] [_inst_6 : AddMonoid.{u3} \u03b1] [_inst_7 : DistribMulAction.{u2, u3} M \u03b1 _inst_5 _inst_6] (a : M) (c : \u03b2), Eq.{max (succ u1) (succ u3)} (Prod.{u3, u1} \u03b1 \u03b2) (HSMul.hSMul.{u2, max u1 u3, max u1 u3} M (Prod.{u3, u1} \u03b1 \u03b2) (Prod.{u3, u1} \u03b1 \u03b2) (instHSMul.{u2, max u1 u3} M (Prod.{u3, u1} \u03b1 \u03b2) (Prod.smul.{u2, u3, u1} M \u03b1 \u03b2 (SMulZeroClass.toSMul.{u2, u3} M \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_6) (DistribSMul.toSMulZeroClass.{u2, u3} M \u03b1 (AddMonoid.toAddZeroClass.{u3} \u03b1 _inst_6) (DistribMulAction.toDistribSMul.{u2, u3} M \u03b1 _inst_5 _inst_6 _inst_7))) _inst_2)) a (Prod.mk.{u3, u1} \u03b1 \u03b2 (OfNat.ofNat.{u3} \u03b1 0 (Zero.toOfNat0.{u3} \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_6))) c)) (Prod.mk.{u3, u1} \u03b1 \u03b2 (OfNat.ofNat.{u3} \u03b1 0 (Zero.toOfNat0.{u3} \u03b1 (AddMonoid.toZero.{u3} \u03b1 _inst_6))) (HSMul.hSMul.{u2, u1, u1} M \u03b2 \u03b2 (instHSMul.{u2, u1} M \u03b2 _inst_2) a c))\nCase conversion may be inaccurate. Consider using '#align prod.smul_zero_mk Prod.smul_zero_mk\u2093'. -/\ntheorem smul_zero_mk {\u03b1 : Type _} [Monoid M] [AddMonoid \u03b1] [DistribMulAction M \u03b1] (a : M) (c : \u03b2) :\n    a \u2022 ((0 : \u03b1), c) = (0, a \u2022 c) := by rw [Prod.smul_mk, smul_zero]\n#align prod.smul_zero_mk Prod.smul_zero_mk\n\n/- warning: prod.smul_mk_zero -> Prod.smul_mk_zero is a dubious translation:\nlean 3 declaration is\n  forall {M : Type.{u1}} {\u03b1 : Type.{u2}} [_inst_1 : SMul.{u1, u2} M \u03b1] {\u03b2 : Type.{u3}} [_inst_5 : Monoid.{u1} M] [_inst_6 : AddMonoid.{u3} \u03b2] [_inst_7 : DistribMulAction.{u1, u3} M \u03b2 _inst_5 _inst_6] (a : M) (b : \u03b1), Eq.{succ (max u2 u3)} (Prod.{u2, u3} \u03b1 \u03b2) (SMul.smul.{u1, max u2 u3} M (Prod.{u2, u3} \u03b1 \u03b2) (Prod.smul.{u1, u2, u3} M \u03b1 \u03b2 _inst_1 (SMulZeroClass.toHasSmul.{u1, u3} M \u03b2 (AddZeroClass.toHasZero.{u3} \u03b2 (AddMonoid.toAddZeroClass.{u3} \u03b2 _inst_6)) (DistribSMul.toSmulZeroClass.{u1, u3} M \u03b2 (AddMonoid.toAddZeroClass.{u3} \u03b2 _inst_6) (DistribMulAction.toDistribSMul.{u1, u3} M \u03b2 _inst_5 _inst_6 _inst_7)))) a (Prod.mk.{u2, u3} \u03b1 \u03b2 b (OfNat.ofNat.{u3} \u03b2 0 (OfNat.mk.{u3} \u03b2 0 (Zero.zero.{u3} \u03b2 (AddZeroClass.toHasZero.{u3} \u03b2 (AddMonoid.toAddZeroClass.{u3} \u03b2 _inst_6))))))) (Prod.mk.{u2, u3} \u03b1 \u03b2 (SMul.smul.{u1, u2} M \u03b1 _inst_1 a b) (OfNat.ofNat.{u3} \u03b2 0 (OfNat.mk.{u3} \u03b2 0 (Zero.zero.{u3} \u03b2 (AddZeroClass.toHasZero.{u3} \u03b2 (AddMonoid.toAddZeroClass.{u3} \u03b2 _inst_6))))))\nbut is expected to have type\n  forall {M : Type.{u2}} {\u03b1 : Type.{u1}} [_inst_1 : SMul.{u2, u1} M \u03b1] {\u03b2 : Type.{u3}} [_inst_5 : Monoid.{u2} M] [_inst_6 : AddMonoid.{u3} \u03b2] [_inst_7 : DistribMulAction.{u2, u3} M \u03b2 _inst_5 _inst_6] (a : M) (b : \u03b1), Eq.{max (succ u1) (succ u3)} (Prod.{u1, u3} \u03b1 \u03b2) (HSMul.hSMul.{u2, max u3 u1, max u1 u3} M (Prod.{u1, u3} \u03b1 \u03b2) (Prod.{u1, u3} \u03b1 \u03b2) (instHSMul.{u2, max u1 u3} M (Prod.{u1, u3} \u03b1 \u03b2) (Prod.smul.{u2, u1, u3} M \u03b1 \u03b2 _inst_1 (SMulZeroClass.toSMul.{u2, u3} M \u03b2 (AddMonoid.toZero.{u3} \u03b2 _inst_6) (DistribSMul.toSMulZeroClass.{u2, u3} M \u03b2 (AddMonoid.toAddZeroClass.{u3} \u03b2 _inst_6) (DistribMulAction.toDistribSMul.{u2, u3} M \u03b2 _inst_5 _inst_6 _inst_7))))) a (Prod.mk.{u1, u3} \u03b1 \u03b2 b (OfNat.ofNat.{u3} \u03b2 0 (Zero.toOfNat0.{u3} \u03b2 (AddMonoid.toZero.{u3} \u03b2 _inst_6))))) (Prod.mk.{u1, u3} \u03b1 \u03b2 (HSMul.hSMul.{u2, u1, u1} M \u03b1 \u03b1 (instHSMul.{u2, u1} M \u03b1 _inst_1) a b) (OfNat.ofNat.{u3} \u03b2 0 (Zero.toOfNat0.{u3} \u03b2 (AddMonoid.toZero.{u3} \u03b2 _inst_6))))\nCase conversion may be inaccurate. Consider using '#align prod.smul_mk_zero Prod.smul_mk_zero\u2093'. -/\ntheorem smul_mk_zero {\u03b2 : Type _} [Monoid M] [AddMonoid \u03b2] [DistribMulAction M \u03b2] (a : M) (b : \u03b1) :\n    a \u2022 (b, (0 : \u03b2)) = (a \u2022 b, 0) := by rw [Prod.smul_mk, smul_zero]\n#align prod.smul_mk_zero Prod.smul_mk_zero\n\nvariable [Pow \u03b1 E] [Pow \u03b2 E]\n\n#print Prod.pow /-\n@[to_additive SMul]\ninstance pow : Pow (\u03b1 \u00d7 \u03b2) E where pow p c := (p.1 ^ c, p.2 ^ c)\n#align prod.has_pow Prod.pow\n#align prod.has_smul Prod.smul\n-/\n\n/- warning: prod.pow_fst -> Prod.pow_fst is a dubious translation:\nlean 3 declaration is\n  forall {E : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_5 : Pow.{u2, u1} \u03b1 E] [_inst_6 : Pow.{u3, u1} \u03b2 E] (p : Prod.{u2, u3} \u03b1 \u03b2) (c : E), Eq.{succ u2} \u03b1 (Prod.fst.{u2, u3} \u03b1 \u03b2 (HPow.hPow.{max u2 u3, u1, max u2 u3} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.{u2, u3} \u03b1 \u03b2) (instHPow.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.pow.{u1, u2, u3} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{u2, u1, u2} \u03b1 E \u03b1 (instHPow.{u2, u1} \u03b1 E _inst_5) (Prod.fst.{u2, u3} \u03b1 \u03b2 p) c)\nbut is expected to have type\n  forall {E : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_5 : Pow.{u3, u1} \u03b1 E] [_inst_6 : Pow.{u2, u1} \u03b2 E] (p : Prod.{u3, u2} \u03b1 \u03b2) (c : E), Eq.{succ u3} \u03b1 (Prod.fst.{u3, u2} \u03b1 \u03b2 (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.{u3, u2} \u03b1 \u03b2) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.pow.{u1, u3, u2} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{u3, u1, u3} \u03b1 E \u03b1 (instHPow.{u3, u1} \u03b1 E _inst_5) (Prod.fst.{u3, u2} \u03b1 \u03b2 p) c)\nCase conversion may be inaccurate. Consider using '#align prod.pow_fst Prod.pow_fst\u2093'. -/\n@[simp, to_additive smul_fst, to_additive_reorder 6]\ntheorem pow_fst (p : \u03b1 \u00d7 \u03b2) (c : E) : (p ^ c).fst = p.fst ^ c :=\n  rfl\n#align prod.pow_fst Prod.pow_fst\n#align prod.smul_fst Prod.smul_fst\n\n/- warning: prod.pow_snd -> Prod.pow_snd is a dubious translation:\nlean 3 declaration is\n  forall {E : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_5 : Pow.{u2, u1} \u03b1 E] [_inst_6 : Pow.{u3, u1} \u03b2 E] (p : Prod.{u2, u3} \u03b1 \u03b2) (c : E), Eq.{succ u3} \u03b2 (Prod.snd.{u2, u3} \u03b1 \u03b2 (HPow.hPow.{max u2 u3, u1, max u2 u3} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.{u2, u3} \u03b1 \u03b2) (instHPow.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.pow.{u1, u2, u3} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{u3, u1, u3} \u03b2 E \u03b2 (instHPow.{u3, u1} \u03b2 E _inst_6) (Prod.snd.{u2, u3} \u03b1 \u03b2 p) c)\nbut is expected to have type\n  forall {E : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_5 : Pow.{u3, u1} \u03b1 E] [_inst_6 : Pow.{u2, u1} \u03b2 E] (p : Prod.{u3, u2} \u03b1 \u03b2) (c : E), Eq.{succ u2} \u03b2 (Prod.snd.{u3, u2} \u03b1 \u03b2 (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.{u3, u2} \u03b1 \u03b2) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.pow.{u1, u3, u2} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{u2, u1, u2} \u03b2 E \u03b2 (instHPow.{u2, u1} \u03b2 E _inst_6) (Prod.snd.{u3, u2} \u03b1 \u03b2 p) c)\nCase conversion may be inaccurate. Consider using '#align prod.pow_snd Prod.pow_snd\u2093'. -/\n@[simp, to_additive smul_snd, to_additive_reorder 6]\ntheorem pow_snd (p : \u03b1 \u00d7 \u03b2) (c : E) : (p ^ c).snd = p.snd ^ c :=\n  rfl\n#align prod.pow_snd Prod.pow_snd\n#align prod.smul_snd Prod.smul_snd\n\n/- warning: prod.pow_mk -> Prod.pow_mk is a dubious translation:\nlean 3 declaration is\n  forall {E : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_5 : Pow.{u2, u1} \u03b1 E] [_inst_6 : Pow.{u3, u1} \u03b2 E] (c : E) (a : \u03b1) (b : \u03b2), Eq.{succ (max u2 u3)} (Prod.{u2, u3} \u03b1 \u03b2) (HPow.hPow.{max u2 u3, u1, max u2 u3} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.{u2, u3} \u03b1 \u03b2) (instHPow.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.pow.{u1, u2, u3} E \u03b1 \u03b2 _inst_5 _inst_6)) (Prod.mk.{u2, u3} \u03b1 \u03b2 a b) c) (Prod.mk.{u2, u3} \u03b1 \u03b2 (HPow.hPow.{u2, u1, u2} \u03b1 E \u03b1 (instHPow.{u2, u1} \u03b1 E _inst_5) a c) (HPow.hPow.{u3, u1, u3} \u03b2 E \u03b2 (instHPow.{u3, u1} \u03b2 E _inst_6) b c))\nbut is expected to have type\n  forall {E : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_5 : Pow.{u3, u1} \u03b1 E] [_inst_6 : Pow.{u2, u1} \u03b2 E] (c : E) (a : \u03b1) (b : \u03b2), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b1 \u03b2) (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.{u3, u2} \u03b1 \u03b2) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.pow.{u1, u3, u2} E \u03b1 \u03b2 _inst_5 _inst_6)) (Prod.mk.{u3, u2} \u03b1 \u03b2 a b) c) (Prod.mk.{u3, u2} \u03b1 \u03b2 (HPow.hPow.{u3, u1, u3} \u03b1 E \u03b1 (instHPow.{u3, u1} \u03b1 E _inst_5) a c) (HPow.hPow.{u2, u1, u2} \u03b2 E \u03b2 (instHPow.{u2, u1} \u03b2 E _inst_6) b c))\nCase conversion may be inaccurate. Consider using '#align prod.pow_mk Prod.pow_mk\u2093'. -/\n/- Note that the `c` arguments to this lemmas cannot be in the more natural right-most positions due\nto limitations in `to_additive` and `to_additive_reorder`, which will silently fail to reorder more\nthan two adjacent arguments -/\n@[simp, to_additive smul_mk, to_additive_reorder 6]\ntheorem pow_mk (c : E) (a : \u03b1) (b : \u03b2) : Prod.mk a b ^ c = Prod.mk (a ^ c) (b ^ c) :=\n  rfl\n#align prod.pow_mk Prod.pow_mk\n#align prod.smul_mk Prod.smul_mk\n\n/- warning: prod.pow_def -> Prod.pow_def is a dubious translation:\nlean 3 declaration is\n  forall {E : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_5 : Pow.{u2, u1} \u03b1 E] [_inst_6 : Pow.{u3, u1} \u03b2 E] (p : Prod.{u2, u3} \u03b1 \u03b2) (c : E), Eq.{succ (max u2 u3)} (Prod.{u2, u3} \u03b1 \u03b2) (HPow.hPow.{max u2 u3, u1, max u2 u3} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.{u2, u3} \u03b1 \u03b2) (instHPow.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.pow.{u1, u2, u3} E \u03b1 \u03b2 _inst_5 _inst_6)) p c) (Prod.mk.{u2, u3} \u03b1 \u03b2 (HPow.hPow.{u2, u1, u2} \u03b1 E \u03b1 (instHPow.{u2, u1} \u03b1 E _inst_5) (Prod.fst.{u2, u3} \u03b1 \u03b2 p) c) (HPow.hPow.{u3, u1, u3} \u03b2 E \u03b2 (instHPow.{u3, u1} \u03b2 E _inst_6) (Prod.snd.{u2, u3} \u03b1 \u03b2 p) c))\nbut is expected to have type\n  forall {E : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_5 : Pow.{u3, u1} \u03b1 E] [_inst_6 : Pow.{u2, u1} \u03b2 E] (p : Prod.{u3, u2} \u03b1 \u03b2) (c : E), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b1 \u03b2) (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.{u3, u2} \u03b1 \u03b2) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.pow.{u1, u3, u2} E \u03b1 \u03b2 _inst_5 _inst_6)) p c) (Prod.mk.{u3, u2} \u03b1 \u03b2 (HPow.hPow.{u3, u1, u3} \u03b1 E \u03b1 (instHPow.{u3, u1} \u03b1 E _inst_5) (Prod.fst.{u3, u2} \u03b1 \u03b2 p) c) (HPow.hPow.{u2, u1, u2} \u03b2 E \u03b2 (instHPow.{u2, u1} \u03b2 E _inst_6) (Prod.snd.{u3, u2} \u03b1 \u03b2 p) c))\nCase conversion may be inaccurate. Consider using '#align prod.pow_def Prod.pow_def\u2093'. -/\n@[to_additive smul_def, to_additive_reorder 6]\ntheorem pow_def (p : \u03b1 \u00d7 \u03b2) (c : E) : p ^ c = (p.1 ^ c, p.2 ^ c) :=\n  rfl\n#align prod.pow_def Prod.pow_def\n#align prod.smul_def Prod.smul_def\n\n/- warning: prod.pow_swap -> Prod.pow_swap is a dubious translation:\nlean 3 declaration is\n  forall {E : Type.{u1}} {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} [_inst_5 : Pow.{u2, u1} \u03b1 E] [_inst_6 : Pow.{u3, u1} \u03b2 E] (p : Prod.{u2, u3} \u03b1 \u03b2) (c : E), Eq.{max (succ u3) (succ u2)} (Prod.{u3, u2} \u03b2 \u03b1) (Prod.swap.{u2, u3} \u03b1 \u03b2 (HPow.hPow.{max u2 u3, u1, max u2 u3} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.{u2, u3} \u03b1 \u03b2) (instHPow.{max u2 u3, u1} (Prod.{u2, u3} \u03b1 \u03b2) E (Prod.pow.{u1, u2, u3} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b2 \u03b1) E (Prod.{u3, u2} \u03b2 \u03b1) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b2 \u03b1) E (Prod.pow.{u1, u3, u2} E \u03b2 \u03b1 _inst_6 _inst_5)) (Prod.swap.{u2, u3} \u03b1 \u03b2 p) c)\nbut is expected to have type\n  forall {E : Type.{u1}} {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} [_inst_5 : Pow.{u3, u1} \u03b1 E] [_inst_6 : Pow.{u2, u1} \u03b2 E] (p : Prod.{u3, u2} \u03b1 \u03b2) (c : E), Eq.{max (succ u3) (succ u2)} (Prod.{u2, u3} \u03b2 \u03b1) (Prod.swap.{u3, u2} \u03b1 \u03b2 (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.{u3, u2} \u03b1 \u03b2) (instHPow.{max u3 u2, u1} (Prod.{u3, u2} \u03b1 \u03b2) E (Prod.pow.{u1, u3, u2} E \u03b1 \u03b2 _inst_5 _inst_6)) p c)) (HPow.hPow.{max u3 u2, u1, max u3 u2} (Prod.{u2, u3} \u03b2 \u03b1) E (Prod.{u2, u3} \u03b2 \u03b1) (instHPow.{max u3 u2, u1} (Prod.{u2, u3} \u03b2 \u03b1) E (Prod.pow.{u1, u2, u3} E \u03b2 \u03b1 _inst_6 _inst_5)) (Prod.swap.{u3, u2} \u03b1 \u03b2 p) c)\nCase conversion may be inaccurate. Consider using '#align prod.pow_swap Prod.pow_swap\u2093'. -/\n@[simp, to_additive smul_swap, to_additive_reorder 6]\ntheorem pow_swap (p : \u03b1 \u00d7 \u03b2) (c : E) : (p ^ c).symm = p.symm ^ c :=\n  rfl\n#align prod.pow_swap Prod.pow_swap\n#align prod.smul_swap Prod.smul_swap\n\n@[to_additive]\ninstance [SMul M N] [IsScalarTower M N \u03b1] [IsScalarTower M N \u03b2] : IsScalarTower M N (\u03b1 \u00d7 \u03b2) :=\n  \u27e8fun x y z => mk.inj_iff.mpr \u27e8smul_assoc _ _ _, smul_assoc _ _ _\u27e9\u27e9\n\n@[to_additive]\ninstance [SMulCommClass M N \u03b1] [SMulCommClass M N \u03b2] : SMulCommClass M N (\u03b1 \u00d7 \u03b2)\n    where smul_comm r s x := mk.inj_iff.mpr \u27e8smul_comm _ _ _, smul_comm _ _ _\u27e9\n\n@[to_additive]\ninstance [SMul M\u1d50\u1d52\u1d56 \u03b1] [SMul M\u1d50\u1d52\u1d56 \u03b2] [IsCentralScalar M \u03b1] [IsCentralScalar M \u03b2] :\n    IsCentralScalar M (\u03b1 \u00d7 \u03b2) :=\n  \u27e8fun r m => Prod.ext (op_smul_eq_smul _ _) (op_smul_eq_smul _ _)\u27e9\n\n#print Prod.faithfulSMulLeft /-\n@[to_additive]\ninstance faithfulSMulLeft [FaithfulSMul M \u03b1] [Nonempty \u03b2] : FaithfulSMul M (\u03b1 \u00d7 \u03b2) :=\n  \u27e8fun x y h =>\n    let \u27e8b\u27e9 := \u2039Nonempty \u03b2\u203a\n    eq_of_smul_eq_smul fun a : \u03b1 => by injection h (a, b)\u27e9\n#align prod.has_faithful_smul_left Prod.faithfulSMulLeft\n#align prod.has_faithful_vadd_left Prod.faithfulVAddLeft\n-/\n\n#print Prod.faithfulSMulRight /-\n@[to_additive]\ninstance faithfulSMulRight [Nonempty \u03b1] [FaithfulSMul M \u03b2] : FaithfulSMul M (\u03b1 \u00d7 \u03b2) :=\n  \u27e8fun x y h =>\n    let \u27e8a\u27e9 := \u2039Nonempty \u03b1\u203a\n    eq_of_smul_eq_smul fun b : \u03b2 => by injection h (a, b)\u27e9\n#align prod.has_faithful_smul_right Prod.faithfulSMulRight\n#align prod.has_faithful_vadd_right Prod.faithfulVAddRight\n-/\n\nend\n\n#print Prod.smulCommClassBoth /-\n@[to_additive]\ninstance smulCommClassBoth [Mul N] [Mul P] [SMul M N] [SMul M P] [SMulCommClass M N N]\n    [SMulCommClass M P P] : SMulCommClass M (N \u00d7 P) (N \u00d7 P) :=\n  \u27e8fun c x y => by simp [smul_def, mul_def, mul_smul_comm]\u27e9\n#align prod.smul_comm_class_both Prod.smulCommClassBoth\n#align prod.vadd_comm_class_both Prod.vaddCommClassBoth\n-/\n\n#print Prod.isScalarTowerBoth /-\ninstance isScalarTowerBoth [Mul N] [Mul P] [SMul M N] [SMul M P] [IsScalarTower M N N]\n    [IsScalarTower M P P] : IsScalarTower M (N \u00d7 P) (N \u00d7 P) :=\n  \u27e8fun c x y => by simp [smul_def, mul_def, smul_mul_assoc]\u27e9\n#align prod.is_scalar_tower_both Prod.isScalarTowerBoth\n-/\n\n@[to_additive]\ninstance {m : Monoid M} [MulAction M \u03b1] [MulAction M \u03b2] : MulAction M (\u03b1 \u00d7 \u03b2)\n    where\n  mul_smul a\u2081 a\u2082 p := mk.inj_iff.mpr \u27e8mul_smul _ _ _, mul_smul _ _ _\u27e9\n  one_smul := fun \u27e8b, c\u27e9 => mk.inj_iff.mpr \u27e8one_smul _ _, one_smul _ _\u27e9\n\ninstance {R M N : Type _} [Zero M] [Zero N] [SMulZeroClass R M] [SMulZeroClass R N] :\n    SMulZeroClass R (M \u00d7 N) where smul_zero a := mk.inj_iff.mpr \u27e8smul_zero _, smul_zero _\u27e9\n\ninstance {R M N : Type _} [AddZeroClass M] [AddZeroClass N] [DistribSMul R M] [DistribSMul R N] :\n    DistribSMul R (M \u00d7 N) where smul_add a p\u2081 p\u2082 := mk.inj_iff.mpr \u27e8smul_add _ _ _, smul_add _ _ _\u27e9\n\ninstance {R M N : Type _} {r : Monoid R} [AddMonoid M] [AddMonoid N] [DistribMulAction R M]\n    [DistribMulAction R N] : DistribMulAction R (M \u00d7 N) :=\n  { Prod.distribSmul with }\n\ninstance {R M N : Type _} {r : Monoid R} [Monoid M] [Monoid N] [MulDistribMulAction R M]\n    [MulDistribMulAction R N] : MulDistribMulAction R (M \u00d7 N)\n    where\n  smul_mul a p\u2081 p\u2082 := mk.inj_iff.mpr \u27e8smul_mul' _ _ _, smul_mul' _ _ _\u27e9\n  smul_one a := mk.inj_iff.mpr \u27e8smul_one _, smul_one _\u27e9\n\nend Prod\n\n/-! ### Scalar multiplication as a homomorphism -/\n\n\nsection BundledSmul\n\n/- warning: smul_mul_hom -> smulMulHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Monoid.{u1} \u03b1] [_inst_2 : Mul.{u2} \u03b2] [_inst_3 : MulAction.{u1, u2} \u03b1 \u03b2 _inst_1] [_inst_4 : IsScalarTower.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 _inst_2) (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3)] [_inst_5 : SMulCommClass.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 _inst_2)], MulHom.{max u1 u2, u2} (Prod.{u1, u2} \u03b1 \u03b2) \u03b2 (Prod.hasMul.{u1, u2} \u03b1 \u03b2 (MulOneClass.toHasMul.{u1} \u03b1 (Monoid.toMulOneClass.{u1} \u03b1 _inst_1)) _inst_2) _inst_2\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Monoid.{u1} \u03b1] [_inst_2 : Mul.{u2} \u03b2] [_inst_3 : MulAction.{u1, u2} \u03b1 \u03b2 _inst_1] [_inst_4 : IsScalarTower.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 _inst_2) (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3)] [_inst_5 : SMulCommClass.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 _inst_2)], MulHom.{max u2 u1, u2} (Prod.{u1, u2} \u03b1 \u03b2) \u03b2 (Prod.instMulProd.{u1, u2} \u03b1 \u03b2 (MulOneClass.toMul.{u1} \u03b1 (Monoid.toMulOneClass.{u1} \u03b1 _inst_1)) _inst_2) _inst_2\nCase conversion may be inaccurate. Consider using '#align smul_mul_hom smulMulHom\u2093'. -/\n/-- Scalar multiplication as a multiplicative homomorphism. -/\n@[simps]\ndef smulMulHom [Monoid \u03b1] [Mul \u03b2] [MulAction \u03b1 \u03b2] [IsScalarTower \u03b1 \u03b2 \u03b2] [SMulCommClass \u03b1 \u03b2 \u03b2] :\n    \u03b1 \u00d7 \u03b2 \u2192\u2099* \u03b2 where\n  toFun a := a.1 \u2022 a.2\n  map_mul' a b := (smul_mul_smul _ _ _ _).symm\n#align smul_mul_hom smulMulHom\n\n/- warning: smul_monoid_hom -> smulMonoidHom is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Monoid.{u1} \u03b1] [_inst_2 : MulOneClass.{u2} \u03b2] [_inst_3 : MulAction.{u1, u2} \u03b1 \u03b2 _inst_1] [_inst_4 : IsScalarTower.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 (MulOneClass.toHasMul.{u2} \u03b2 _inst_2)) (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3)] [_inst_5 : SMulCommClass.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toHasSmul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 (MulOneClass.toHasMul.{u2} \u03b2 _inst_2))], MonoidHom.{max u1 u2, u2} (Prod.{u1, u2} \u03b1 \u03b2) \u03b2 (Prod.mulOneClass.{u1, u2} \u03b1 \u03b2 (Monoid.toMulOneClass.{u1} \u03b1 _inst_1) _inst_2) _inst_2\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Monoid.{u1} \u03b1] [_inst_2 : MulOneClass.{u2} \u03b2] [_inst_3 : MulAction.{u1, u2} \u03b1 \u03b2 _inst_1] [_inst_4 : IsScalarTower.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 (MulOneClass.toMul.{u2} \u03b2 _inst_2)) (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3)] [_inst_5 : SMulCommClass.{u1, u2, u2} \u03b1 \u03b2 \u03b2 (MulAction.toSMul.{u1, u2} \u03b1 \u03b2 _inst_1 _inst_3) (Mul.toSMul.{u2} \u03b2 (MulOneClass.toMul.{u2} \u03b2 _inst_2))], MonoidHom.{max u2 u1, u2} (Prod.{u1, u2} \u03b1 \u03b2) \u03b2 (Prod.instMulOneClassProd.{u1, u2} \u03b1 \u03b2 (Monoid.toMulOneClass.{u1} \u03b1 _inst_1) _inst_2) _inst_2\nCase conversion may be inaccurate. Consider using '#align smul_monoid_hom smulMonoidHom\u2093'. -/\n/-- Scalar multiplication as a monoid homomorphism. -/\n@[simps]\ndef smulMonoidHom [Monoid \u03b1] [MulOneClass \u03b2] [MulAction \u03b1 \u03b2] [IsScalarTower \u03b1 \u03b2 \u03b2]\n    [SMulCommClass \u03b1 \u03b2 \u03b2] : \u03b1 \u00d7 \u03b2 \u2192* \u03b2 :=\n  { smulMulHom with map_one' := one_smul _ _ }\n#align smul_monoid_hom smulMonoidHom\n\nend BundledSmul\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/GroupTheory/GroupAction/Prod.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3570914778278342}}
{"text": "import tactic\nimport .topology\nimport .gluing\nimport .fiber_product\n\nsection fiber_space\n\nvariables {E E' B : Type} (\u03c0' : E' \u2192 B) (\u03c6 : E \u2192 E')\n\ndef res_base (U : set B) :\n  \u03c6 \u207b\u00b9' (\u03c0' \u207b\u00b9' U) \u2192 \u03c0' \u207b\u00b9' U :=\n  subtype.map \u03c6 (by simp)\n\n@[simp]\nlemma res_base_def (U : set B)\n  (a : E) (ha : a \u2208 \u03c6 \u207b\u00b9' (\u03c0' \u207b\u00b9' U)) :\n  res_base _ _ U \u27e8a, ha\u27e9 = \u27e8\u03c6 a, ha\u27e9 := rfl\n\nlemma from_sub_res_image_of_to_sub\n  (U : set B) (V : set E) :\n  from_sub _ ((res_base \u03c0' \u03c6 U) '' (to_sub _ V)) =\n    (\u03c0' \u207b\u00b9' U) \u2229 (\u03c6 '' V) :=\nbegin\n  ext e', simp, split,\n  rintro \u27e8e'', he'', \u27e8e, he\u27e9, he12\u27e9, rw he12 at *,\n  exact \u27e8he'', \u27e8e, he.1, he.2.2\u27e9\u27e9,\n  rintro \u27e8he', e, he1, he2\u27e9, use e', split, exact he', use e,\n  simp [he2], exact \u27e8he1, he'\u27e9,\nend\n\nlemma from_sub_res_preimage_of_to_sub\n  (U : set B) (V : set E') :\n  from_sub _ ((res_base \u03c0' \u03c6 U) \u207b\u00b9' (to_sub _ V)) = \n    \u03c6 \u207b\u00b9' ((\u03c0' \u207b\u00b9' U) \u2229 V) :=\nbegin\n  unfold from_sub, unfold to_sub, unfold res_base,\n  ext e, simp, split,\n  rintro \u27e8x, hx\u27e9, rw \u2190 hx.2.2, exact \u27e8hx.1, hx.2.1\u27e9,\n  intro h, use e, simp, exact h,\nend\n\nlemma res_base_inj_of_inj\n  (h\u03c6_inj : function.injective \u03c6) (U : set B) : \n  function.injective (res_base \u03c0' \u03c6 U) :=\nbegin\n  apply subtype.map_injective, exact h\u03c6_inj,\nend\n\nlemma res_base_surj_of_surj\n  (h\u03c6_surj : function.surjective \u03c6) (U : set B) : \n  function.surjective (res_base \u03c0' \u03c6 U) :=\nbegin\n  rintro \u27e8x, hx\u27e9, obtain \u27e8y, hy\u27e9 := h\u03c6_surj x,\n  use y, simp, rwa hy,\n  simp, exact hy,\nend\n\nlemma res_base_bij_of_bij\n  (h\u03c6_bij : function.bijective \u03c6) (U : set B) : \n  function.bijective (res_base \u03c0' \u03c6 U) :=\nbegin\n  split,\n    apply res_base_inj_of_inj _ _ h\u03c6_bij.1,\n    apply res_base_surj_of_surj _ _ h\u03c6_bij.2,\nend\n\n\nvariables {I : Type} (U : cover I B)\n\nlemma fiber_map_inj_iff_cover_inj :\n  function.injective \u03c6 \u2194\n  \u2200 i, function.injective (res_base \u03c0' \u03c6 (U i)) :=\nbegin\n  split,\n    intros h\u03c6_inj i, apply res_base_inj_of_inj, exact h\u03c6_inj,\n  intros h\u03c6_inj_cover e1 e2 he,\n  obtain \u27e8i, hi\u27e9 := U.hx (\u03c0' (\u03c6 e1)),\n  specialize @h\u03c6_inj_cover i \u27e8e1, hi\u27e9 \u27e8e2, by rwa he at hi\u27e9,\n  simp at h\u03c6_inj_cover, exact h\u03c6_inj_cover he,\nend\n\nlemma fiber_map_surj_iff_cover_surj :\n  function.surjective \u03c6 \u2194\n  \u2200 i, function.surjective (res_base \u03c0' \u03c6 (U i)) :=\nbegin\n  split,\n    intros h\u03c6_surj i, apply res_base_surj_of_surj, exact h\u03c6_surj,\n  intro h\u03c6_surj_cover, intro e',\n  obtain \u27e8i, hi\u27e9 := U.hx (\u03c0' e'),\n  obtain \u27e8\u27e8a, ha\u27e9, ha'\u27e9 := h\u03c6_surj_cover i \u27e8e', hi\u27e9,\n  use a, simp at ha', exact ha',\nend\n\nlemma fiber_map_bij_iff_cover_bij :\n  function.bijective \u03c6 \u2194 \n  \u2200 i, function.bijective (res_base \u03c0' \u03c6 (U i)) :=\nbegin\n  unfold function.bijective,\n  rw forall_and_distrib,\n  rw \u2190 fiber_map_inj_iff_cover_inj,\n  rw \u2190 fiber_map_surj_iff_cover_surj,\nend\n\nsection fiber_bundle\n\nopen topology\n\nvariables [topology B] [topology E] [topology E']\n  {J : Type} (V : open_cover J B)\n\nlemma res_base_cts_of_cts\n  (h\u03c0'_cts : cts \u03c0') (h\u03c6_cts : cts \u03c6) (U \u2208 opens B) :\n  cts (res_base \u03c0' \u03c6 U) :=\n  subtype_map_cts h\u03c6_cts\n\nlemma fiber_map_cts_iff_cover_cts\n  (h\u03c0_cts : cts (\u03c0' \u2218 \u03c6)) (h\u03c0'_cts : cts \u03c0') :\n  cts \u03c6 \u2194\n  \u2200 j, cts (res_base \u03c0' \u03c6 (V j)) :=\nbegin\n  split,\n    intros h\u03c6_cts j,\n    apply res_base_cts_of_cts _ _ h\u03c0'_cts h\u03c6_cts _ (V.hopen j),\n  intros h\u03c6_cts_cover,\n  rw cts_iff_ptwise_cts, intros e W hW heW, simp,\n  obtain \u27e8j, hj\u27e9 := V.hx (\u03c0' (\u03c6 e)),\n  use \u03c6 \u207b\u00b9' (\u03c0' \u207b\u00b9' (V j) \u2229 W),\n  split, -- proof of openness\n    specialize h\u03c6_cts_cover j (to_sub _ W),\n    rw to_sub_open_iff at h\u03c6_cts_cover,\n    specialize h\u03c6_cts_cover \u27e8W, hW, rfl\u27e9,\n    rw from_sub_open_iff at h\u03c6_cts_cover,\n    rwa \u2190 from_sub_res_preimage_of_to_sub,\n    rw \u2190set.preimage_comp,\n    apply h\u03c0_cts, exact V.hopen j,\n  split, simp, simp, exact \u27e8hj, heW\u27e9,\nend\n\nlemma res_base_open_map_of_open_map\n  (h\u03c0_cts : cts (\u03c0' \u2218 \u03c6)) (h\u03c0'_cts : cts \u03c0')\n  (h\u03c6_open : open_map \u03c6)\n  (U \u2208 opens B) :\n  open_map (res_base \u03c0' \u03c6 U) :=\nbegin\n  apply subtype_map_open, exact h\u03c6_open,\n  change ((\u03c0' \u2218 \u03c6) \u207b\u00b9' U \u2208 opens E), apply h\u03c0_cts, exact \u2039U \u2208 opens B\u203a,\nend\n\nlemma fiber_map_open_map_iff_res_open_map\n  (h\u03c0_cts : cts (\u03c0' \u2218 \u03c6)) (h\u03c0'_cts : cts \u03c0') :\n  open_map \u03c6 \u2194\n  \u2200 j, open_map (res_base \u03c0' \u03c6 (V j)) :=\nbegin\n  split,\n    intros h\u03c6_open j,\n    apply res_base_open_map_of_open_map _ _ h\u03c0_cts h\u03c0'_cts h\u03c6_open _ (V.hopen j),\n  intros h\u03c6_open_map W hW,\n  rw subset_open_iff_open_cover (pullback_open_cover \u03c0' h\u03c0'_cts V),\n  intro j, change \u03c0' \u207b\u00b9' (V j) \u2229 \u03c6 '' W \u2208 opens E',\n  specialize h\u03c6_open_map j (to_sub _ W),\n  rw to_sub_open_iff at h\u03c6_open_map,\n  specialize h\u03c6_open_map \u27e8W, hW, rfl\u27e9,\n  rw from_sub_open_iff at h\u03c6_open_map,\n  rwa \u2190 from_sub_res_image_of_to_sub,\n  apply h\u03c0'_cts, exact V.hopen j,\nend\n\nlemma res_base_homeo_of_homeo\n  (h\u03c0_cts : cts (\u03c0' \u2218 \u03c6)) (h\u03c0'_cts : cts \u03c0')\n  (h\u03c6_homeo : homeo \u03c6)\n  (U \u2208 opens B) :\n  homeo (res_base \u03c0' \u03c6 U) :=\nbegin\n  rw homeo_iff at *,\n  split,\n    exact res_base_cts_of_cts _ _ h\u03c0'_cts h\u03c6_homeo.1 U \u2039U \u2208 opens B\u203a,\n  split,\n    exact res_base_bij_of_bij _ _ h\u03c6_homeo.2.1 U,\n    exact res_base_open_map_of_open_map _ _ h\u03c0_cts h\u03c0'_cts h\u03c6_homeo.2.2 U \u2039U \u2208 opens B\u203a,\nend\n\nlemma fiber_map_homeo_iff_res_homeo\n  (h\u03c0_cts : cts (\u03c0' \u2218 \u03c6)) (h\u03c0'_cts : cts \u03c0') :\n  homeo \u03c6 \u2194\n  \u2200 j, homeo (res_base \u03c0' \u03c6 (V j)) :=\nbegin\n  split, intros h\u03c6_homeo j,\n    exact res_base_homeo_of_homeo _ _ h\u03c0_cts h\u03c0'_cts h\u03c6_homeo (V j) (V.hopen j),\n  intro h\u03c6_homeo,\n  rw [homeo_iff,\n      fiber_map_cts_iff_cover_cts _ _ V h\u03c0_cts h\u03c0'_cts,\n      fiber_map_bij_iff_cover_bij _ _ V.to_cover,\n      fiber_map_open_map_iff_res_open_map _ _ V h\u03c0_cts h\u03c0'_cts,\n      \u2190 forall_and_distrib, \u2190 forall_and_distrib],\n  intro j, specialize h\u03c6_homeo j, rwa homeo_iff at h\u03c6_homeo,\nend\n\nend fiber_bundle\nend fiber_space", "meta": {"author": "mguaypaq", "repo": "lean-topology", "sha": "57b15b3862d441095e254e65009856fa922758cc", "save_path": "github-repos/lean/mguaypaq-lean-topology", "path": "github-repos/lean/mguaypaq-lean-topology/lean-topology-57b15b3862d441095e254e65009856fa922758cc/src/bundles_old.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3570914778278342}}
{"text": "/-\nCopyright (c) 2022 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Std.Lean.Parser\nimport Std.Lean.Meta.DiscrTree\nimport Mathlib.Algebra.Invertible\nimport Mathlib.Data.Rat.Cast\nimport Mathlib.Data.Nat.Basic\nimport Mathlib.Data.Int.Basic\nimport Mathlib.Tactic.Conv\nimport Qq.MetaM\nimport Qq.Delab\n\n/-!\n## `norm_num` core functionality\n\nThis file sets up the `norm_num` tactic and the `@[norm_num]` attribute,\nwhich allow for plugging in new normalization functionality around a simp-based driver.\nThe actual behavior is in `@[norm_num]`-tagged definitions in `Tactic.NormNum.Basic`\nand elsewhere.\n-/\nopen Lean hiding Rat mkRat\nopen Lean.Meta Qq Lean.Elab Term\n\n/-- Attribute for identifying `norm_num` extensions. -/\nsyntax (name := norm_num) \"norm_num\" term,+ : attr\n\nnamespace Mathlib\nnamespace Meta.NormNum\n\ninitialize registerTraceClass `Tactic.norm_num\n\n/-- Assert that an element of a semiring is equal to the coercion of some natural number. -/\nstructure IsNat [AddMonoidWithOne \u03b1] (a : \u03b1) (n : \u2115) : Prop where\n  /-- The element is equal to the coercion of the natural number. -/\n  out : a = n\n\ntheorem IsNat.raw_refl (n : \u2115) : IsNat n n := \u27e8rfl\u27e9\n\n/--\nA \"raw nat cast\" is an expression of the form `(Nat.rawCast lit : \u03b1)` where `lit` is a raw\nnatural number literal. These expressions are used by tactics like `ring` to decrease the number\nof typeclass arguments required in each use of a number literal at type `\u03b1`.\n-/\n@[simp] def _root_.Nat.rawCast [AddMonoidWithOne \u03b1] (n : \u2115) : \u03b1 := n\n\ntheorem IsNat.to_eq [AddMonoidWithOne \u03b1] {n} : {a a' : \u03b1} \u2192 IsNat a n \u2192 n = a' \u2192 a = a'\n  | _, _, \u27e8rfl\u27e9, rfl => rfl\n\ntheorem IsNat.to_raw_eq [AddMonoidWithOne \u03b1] : IsNat (a : \u03b1) n \u2192 a = n.rawCast\n  | \u27e8e\u27e9 => e\n\ntheorem IsNat.of_raw (\u03b1) [AddMonoidWithOne \u03b1] (n : \u2115) : IsNat (n.rawCast : \u03b1) n := \u27e8rfl\u27e9\n\n/-- Assert that an element of a ring is equal to the coercion of some integer. -/\nstructure IsInt [Ring \u03b1] (a : \u03b1) (n : \u2124) : Prop where\n  /-- The element is equal to the coercion of the integer. -/\n  out : a = n\n\n/--\nA \"raw int cast\" is an expression of the form:\n\n* `(Nat.rawCast lit : \u03b1)` where `lit` is a raw natural number literal\n* `(Int.rawCast (Int.negOfNat lit) : \u03b1)` where `lit` is a nonzero raw natural number literal\n\n(That is, we only actually use this function for negative integers.) This representation is used by\ntactics like `ring` to decrease the number of typeclass arguments required in each use of a number\nliteral at type `\u03b1`.\n-/\n@[simp] def _root_.Int.rawCast [Ring \u03b1] (n : \u2124) : \u03b1 := n\n\ntheorem IsInt.to_isNat {\u03b1} [Ring \u03b1] : \u2200 {a : \u03b1} {n}, IsInt a (.ofNat n) \u2192 IsNat a n\n  | _, _, \u27e8rfl\u27e9 => \u27e8by simp\u27e9\n\n\n\ntheorem IsInt.to_raw_eq [Ring \u03b1] : IsInt (a : \u03b1) n \u2192 a = n.rawCast\n  | \u27e8e\u27e9 => e\n\ntheorem IsInt.of_raw (\u03b1) [Ring \u03b1] (n : \u2124) : IsInt (n.rawCast : \u03b1) n := \u27e8rfl\u27e9\n\ntheorem IsInt.neg_to_eq {\u03b1} [Ring \u03b1] {n} :\n    {a a' : \u03b1} \u2192 IsInt a (.negOfNat n) \u2192 n = a' \u2192 a = -a'\n  | _, _, \u27e8rfl\u27e9, rfl => by simp [Int.negOfNat_eq, Int.cast_neg]\n\ntheorem IsInt.nonneg_to_eq {\u03b1} [Ring \u03b1] {n}\n    {a a' : \u03b1} (h : IsInt a (.ofNat n)) (e : n = a') : a = a' := h.to_isNat.to_eq e\n\n/-- Represent an integer as a typed expression. -/\ndef mkRawIntLit (n : \u2124) : Q(\u2124) :=\n  let lit : Q(\u2115) := mkRawNatLit n.natAbs\n  if 0 \u2264 n then q(.ofNat $lit) else q(.negOfNat $lit)\n\n/-- A shortcut (non)instance for `AddMonoidWithOne \u2115` to shrink generated proofs. -/\ndef instAddMonoidWithOneNat : AddMonoidWithOne \u2115 := inferInstance\n\n/-- A shortcut (non)instance for `Ring \u2124` to shrink generated proofs. -/\ndef instRingInt : Ring \u2124 := inferInstance\n\n/--\nAssert that an element of a ring is equal to `num / denom`\n(and `denom` is invertible so that this makes sense).\nWe will usually also have `num` and `denom` coprime,\nalthough this is not part of the definition.\n-/\ninductive IsRat [Ring \u03b1] (a : \u03b1) (num : \u2124) (denom : \u2115) : Prop\n  | mk (inv : Invertible (denom : \u03b1)) (eq : a = num * \u215f(denom : \u03b1))\n\n/--\nA \"raw rat cast\" is an expression of the form:\n\n* `(Nat.rawCast lit : \u03b1)` where `lit` is a raw natural number literal\n* `(Int.rawCast (Int.negOfNat lit) : \u03b1)` where `lit` is a nonzero raw natural number literal\n* `(Rat.rawCast n d : \u03b1)` where `n` is a raw int cast, `d` is a raw nat cast, and `d` is not 1 or 0.\n\nThis representation is used by tactics like `ring` to decrease the number of typeclass arguments\nrequired in each use of a number literal at type `\u03b1`.\n-/\n@[simp]\ndef _root_.Rat.rawCast [DivisionRing \u03b1] (n : \u2124) (d : \u2115) : \u03b1 := n / d\n\ntheorem IsRat.to_isNat {\u03b1} [Ring \u03b1] : \u2200 {a : \u03b1} {n}, IsRat a (.ofNat n) (nat_lit 1) \u2192 IsNat a n\n  | _, _, \u27e8inv, rfl\u27e9 => have := @invertibleOne \u03b1 _; \u27e8by simp\u27e9\n\ntheorem IsNat.to_isRat {\u03b1} [Ring \u03b1] : \u2200 {a : \u03b1} {n}, IsNat a n \u2192 IsRat a (.ofNat n) (nat_lit 1)\n  | _, _, \u27e8rfl\u27e9 => \u27e8\u27e81, by simp, by simp\u27e9, by simp\u27e9\n\ntheorem IsRat.to_isInt {\u03b1} [Ring \u03b1] : \u2200 {a : \u03b1} {n}, IsRat a n (nat_lit 1) \u2192 IsInt a n\n  | _, _, \u27e8inv, rfl\u27e9 => have := @invertibleOne \u03b1 _; \u27e8by simp\u27e9\n\ntheorem IsInt.to_isRat {\u03b1} [Ring \u03b1] : \u2200 {a : \u03b1} {n}, IsInt a n \u2192 IsRat a n (nat_lit 1)\n  | _, _, \u27e8rfl\u27e9 => \u27e8\u27e81, by simp, by simp\u27e9, by simp\u27e9\n\ntheorem IsRat.to_raw_eq [DivisionRing \u03b1] : \u2200 {a}, IsRat (a : \u03b1) n d \u2192 a = Rat.rawCast n d\n  | _, \u27e8inv, rfl\u27e9 => by simp [div_eq_mul_inv]\n\ntheorem IsRat.neg_to_eq {\u03b1} [DivisionRing \u03b1] {n d} :\n    {a n' d' : \u03b1} \u2192 IsRat a (.negOfNat n) d \u2192 n = n' \u2192 d = d' \u2192 a = -(n' / d')\n  | _, _, _, \u27e8_, rfl\u27e9, rfl, rfl => by simp [div_eq_mul_inv]\n\ntheorem IsRat.nonneg_to_eq {\u03b1} [DivisionRing \u03b1] {n d} :\n    {a n' d' : \u03b1} \u2192 IsRat a (.ofNat n) d \u2192 n = n' \u2192 d = d' \u2192 a = n' / d'\n  | _, _, _, \u27e8_, rfl\u27e9, rfl, rfl => by simp [div_eq_mul_inv]\n\ntheorem IsRat.of_raw (\u03b1) [DivisionRing \u03b1] (n : \u2124) (d : \u2115)\n    (h : (d : \u03b1) \u2260 0) : IsRat (Rat.rawCast n d : \u03b1) n d :=\n  have := invertibleOfNonzero h\n  \u27e8this, by simp [div_eq_mul_inv]\u27e9\n\ntheorem IsRat.den_nz {\u03b1} [DivisionRing \u03b1] {a n d} : IsRat (a : \u03b1) n d \u2192 (d : \u03b1) \u2260 0\n  | \u27e8_, _\u27e9 => nonzero_of_invertible (d : \u03b1)\n\n/-- Represent an integer as a typed expression. -/\ndef mkRawRatLit (q : \u211a) : Q(\u211a) :=\n  let nlit : Q(\u2124) := mkRawIntLit q.num\n  let dlit : Q(\u2115) := mkRawNatLit q.den\n  q(mkRat $nlit $dlit)\n\n/-- A shortcut (non)instance for `Ring \u211a` to shrink generated proofs. -/\ndef instRingRat : Ring \u211a := inferInstance\n\n/-- A shortcut (non)instance for `DivisionRing \u211a` to shrink generated proofs. -/\ndef instDivisionRingRat : DivisionRing \u211a := inferInstance\n\n/-- The result of `norm_num` running on an expression `x` of type `\u03b1`.\nUntyped version of `Result`. -/\ninductive Result' where\n  /-- Untyped version of `Result.isBool`. -/\n  | isBool (val : Bool) (proof : Expr)\n  /-- Untyped version of `Result.isNat`. -/\n  | isNat (inst lit proof : Expr)\n  /-- Untyped version of `Result.isNegNat`. -/\n  | isNegNat (inst lit proof : Expr)\n  /-- Untyped version of `Result.isRat`. -/\n  | isRat (inst : Expr) (q : Rat) (n d proof : Expr)\n  deriving Inhabited\n\nsection\nset_option linter.unusedVariables false\n\n/-- The result of `norm_num` running on an expression `x` of type `\u03b1`. -/\n@[nolint unusedArguments] def Result {\u03b1 : Q(Type u)} (x : Q($\u03b1)) := Result'\n\ninstance : Inhabited (Result x) := inferInstanceAs (Inhabited Result')\n\n/-- The result is `proof : x`, where `x` is a (true) proposition. -/\n@[match_pattern, inline] def Result.isTrue {x : Q(Prop)} :\n    \u2200 (proof : Q($x)), @Result _ (q(Prop) : Q(Type)) x := Result'.isBool true\n\n/-- The result is `proof : \u00acx`, where `x` is a (false) proposition. -/\n@[match_pattern, inline] def Result.isFalse {x : Q(Prop)} :\n    \u2200 (proof : Q(\u00ac$x)), @Result _ (q(Prop) : Q(Type)) x := Result'.isBool false\n\n/-- The result is `lit : \u2115` (a raw nat literal) and `proof : isNat x lit`. -/\n@[match_pattern, inline] def Result.isNat {\u03b1 : Q(Type u)} {x : Q($\u03b1)} :\n    \u2200 (inst : Q(AddMonoidWithOne $\u03b1) := by assumption) (lit : Q(\u2115)) (proof : Q(IsNat $x $lit)),\n      Result x := Result'.isNat\n\n/-- The result is `-lit` where `lit` is a raw nat literal\nand `proof : isInt x (.negOfNat lit)`. -/\n@[match_pattern, inline] def Result.isNegNat {\u03b1 : Q(Type u)} {x : Q($\u03b1)} :\n    \u2200 (inst : Q(Ring $\u03b1) := by assumption) (lit : Q(\u2115)) (proof : Q(IsInt $x (.negOfNat $lit))),\n      Result x := Result'.isNegNat\n\n/-- The result is `proof : isRat x n d`, where `n` is either `.ofNat lit` or `.negOfNat lit`\nwith `lit` a raw nat literal and `d` is a raw nat literal (not 0 or 1),\nand `q` is the value of `n / d`. -/\n@[match_pattern, inline] def Result.isRat {\u03b1 : Q(Type u)} {x : Q($\u03b1)} :\n    \u2200 (inst : Q(DivisionRing $\u03b1) := by assumption) (q : Rat) (n : Q(\u2124)) (d : Q(\u2115))\n      (proof : Q(IsRat $x $n $d)), Result x := Result'.isRat\n\n/-- A shortcut (non)instance for `AddMonoidWithOne \u03b1` from `Ring \u03b1` to shrink generated proofs. -/\ndef instAddMonoidWithOne [Ring \u03b1] : AddMonoidWithOne \u03b1 := inferInstance\n\n/-- The result is `z : \u2124` and `proof : isNat x z`. -/\n-- Note the independent arguments `z : Q(\u2124)` and `n : \u2124`.\n-- We ensure these are \"the same\" when calling.\ndef Result.isInt {\u03b1 : Q(Type u)} {x : Q($\u03b1)} (inst : Q(Ring $\u03b1) := by assumption)\n    (z : Q(\u2124)) (n : \u2124) (proof : Q(IsInt $x $z)) : Result x :=\n  have lit : Q(\u2115) := z.appArg!\n  if 0 \u2264 n then\n    let proof : Q(IsInt $x (.ofNat $lit)) := proof\n    .isNat q(instAddMonoidWithOne) lit q(IsInt.to_isNat $proof)\n  else\n    .isNegNat inst lit proof\n\n/-- Returns the rational number that is the result of `norm_num` evaluation. -/\ndef Result.toRat : Result e \u2192 Option Rat\n  | .isBool .. => none\n  | .isNat _ lit _ => some lit.natLit!\n  | .isNegNat _ lit _ => some (-lit.natLit!)\n  | .isRat _ q .. => some q\n\nend\n\n/-- Convert `undef` to `none` to make an `LOption` into an `Option`. -/\ndef _root_.Lean.LOption.toOption {\u03b1} : Lean.LOption \u03b1 \u2192 Option \u03b1\n  | .some a => some a\n  | _ => none\n\n/-- Helper function to synthesize a typed `AddMonoidWithOne \u03b1` expression. -/\ndef inferAddMonoidWithOne (\u03b1 : Q(Type u)) : MetaM Q(AddMonoidWithOne $\u03b1) :=\n  return \u2190 synthInstanceQ (q(AddMonoidWithOne $\u03b1) : Q(Type u)) <|>\n    throwError \"not an AddMonoidWithOne\"\n\n/-- Helper function to synthesize a typed `Semiring \u03b1` expression. -/\ndef inferSemiring (\u03b1 : Q(Type u)) : MetaM Q(Semiring $\u03b1) :=\n  return \u2190 synthInstanceQ (q(Semiring $\u03b1) : Q(Type u)) <|> throwError \"not a semiring\"\n\n/-- Helper function to synthesize a typed `Ring \u03b1` expression. -/\ndef inferRing (\u03b1 : Q(Type u)) : MetaM Q(Ring $\u03b1) :=\n  return \u2190 synthInstanceQ (q(Ring $\u03b1) : Q(Type u)) <|> throwError \"not a ring\"\n\n/-- Helper function to synthesize a typed `DivisionRing \u03b1` expression. -/\ndef inferDivisionRing (\u03b1 : Q(Type u)) : MetaM Q(DivisionRing $\u03b1) :=\n  return \u2190 synthInstanceQ (q(DivisionRing $\u03b1) : Q(Type u)) <|> throwError \"not a division ring\"\n\n/-- Helper function to synthesize a typed `OrderedSemiring \u03b1` expression. -/\ndef inferOrderedSemiring (\u03b1 : Q(Type u)) : MetaM Q(OrderedSemiring $\u03b1) :=\n  return \u2190 synthInstanceQ (q(OrderedSemiring $\u03b1) : Q(Type u)) <|>\n    throwError \"not an ordered semiring\"\n\n/-- Helper function to synthesize a typed `OrderedRing \u03b1` expression. -/\ndef inferOrderedRing (\u03b1 : Q(Type u)) : MetaM Q(OrderedRing $\u03b1) :=\n  return \u2190 synthInstanceQ (q(OrderedRing $\u03b1) : Q(Type u)) <|> throwError \"not an ordered ring\"\n\n/-- Helper function to synthesize a typed `LinearOrderedField \u03b1` expression. -/\ndef inferLinearOrderedField (\u03b1 : Q(Type u)) : MetaM Q(LinearOrderedField $\u03b1) :=\n  return \u2190 synthInstanceQ (q(LinearOrderedField $\u03b1) : Q(Type u)) <|>\n    throwError \"not a linear ordered field\"\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `Ring \u03b1`. -/\ndef inferCharZeroOfRing {\u03b1 : Q(Type u)} (_i : Q(Ring $\u03b1) := by with_reducible assumption) :\n    MetaM Q(CharZero $\u03b1) :=\n  return \u2190 synthInstanceQ (q(CharZero $\u03b1) : Q(Prop)) <|>\n    throwError \"not a characteristic zero ring\"\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `Ring \u03b1`, if it exists. -/\ndef inferCharZeroOfRing? {\u03b1 : Q(Type u)} (_i : Q(Ring $\u03b1) := by with_reducible assumption) :\n    MetaM (Option Q(CharZero $\u03b1)) :=\n  return (\u2190 trySynthInstanceQ (q(CharZero $\u03b1) : Q(Prop))).toOption\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `AddMonoidWithOne \u03b1`. -/\ndef inferCharZeroOfAddMonoidWithOne {\u03b1 : Q(Type u)}\n    (_i : Q(AddMonoidWithOne $\u03b1) := by with_reducible assumption) : MetaM Q(CharZero $\u03b1) :=\n  return \u2190 synthInstanceQ (q(CharZero $\u03b1) : Q(Prop)) <|>\n    throwError \"not a characteristic zero AddMonoidWithOne\"\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `AddMonoidWithOne \u03b1`, if it\nexists. -/\ndef inferCharZeroOfAddMonoidWithOne? {\u03b1 : Q(Type u)}\n    (_i : Q(AddMonoidWithOne $\u03b1) := by with_reducible assumption) :\n      MetaM (Option Q(CharZero $\u03b1)) :=\n  return (\u2190 trySynthInstanceQ (q(CharZero $\u03b1) : Q(Prop))).toOption\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `DivisionRing \u03b1`. -/\ndef inferCharZeroOfDivisionRing {\u03b1 : Q(Type u)}\n    (_i : Q(DivisionRing $\u03b1) := by with_reducible assumption) : MetaM Q(CharZero $\u03b1) :=\n  return \u2190 synthInstanceQ (q(CharZero $\u03b1) : Q(Prop)) <|>\n    throwError \"not a characterstic zero division ring\"\n\n/-- Helper function to synthesize a typed `CharZero \u03b1` expression given `DivisionRing \u03b1`, if it\nexists. -/\ndef inferCharZeroOfDivisionRing? {\u03b1 : Q(Type u)}\n    (_i : Q(DivisionRing $\u03b1) := by with_reducible assumption) : MetaM (Option Q(CharZero $\u03b1)) :=\n  return (\u2190 trySynthInstanceQ (q(CharZero $\u03b1) : Q(Prop))).toOption\n/--\nExtract from a `Result` the integer value (as both a term and an expression),\nand the proof that the original expression is equal to this integer.\n-/\ndef Result.toInt {\u03b1 : Q(Type u)} {e : Q($\u03b1)} (_i : Q(Ring $\u03b1) := by with_reducible assumption) :\n    Result e \u2192 Option (\u2124 \u00d7 (lit : Q(\u2124)) \u00d7 Q(IsInt $e $lit))\n  | .isNat _ lit proof => do\n    have proof : Q(@IsNat _ instAddMonoidWithOne $e $lit) := proof\n    pure \u27e8lit.natLit!, q(.ofNat $lit), q(($proof).to_isInt)\u27e9\n  | .isNegNat _ lit proof => pure \u27e8-lit.natLit!, q(.negOfNat $lit), proof\u27e9\n  | _ => failure\n\n/--\nExtract from a `Result` the rational value (as both a term and an expression),\nand the proof that the original expression is equal to this rational number.\n-/\ndef Result.toRat' {\u03b1 : Q(Type u)} {e : Q($\u03b1)}\n    (_i : Q(DivisionRing $\u03b1) := by with_reducible assumption) :\n    Result e \u2192 Option (\u211a \u00d7 (n : Q(\u2124)) \u00d7 (d : Q(\u2115)) \u00d7 Q(IsRat $e $n $d))\n  | .isBool .. => none\n  | .isNat _ lit proof =>\n    have proof : Q(@IsNat _ instAddMonoidWithOne $e $lit) := proof\n    some \u27e8lit.natLit!, q(.ofNat $lit), q(nat_lit 1), q(($proof).to_isRat)\u27e9\n  | .isNegNat _ lit proof =>\n    have proof : Q(@IsInt _ DivisionRing.toRing $e (.negOfNat $lit)) := proof\n    some \u27e8-lit.natLit!, q(.negOfNat $lit), q(nat_lit 1),\n      (q(@IsInt.to_isRat _ DivisionRing.toRing _ _ $proof) : Expr)\u27e9\n  | .isRat _ q n d proof => some \u27e8q, n, d, proof\u27e9\n\ninstance : ToMessageData (Result x) where\n  toMessageData\n  | .isBool true proof => m!\"isTrue ({proof})\"\n  | .isBool false proof => m!\"isFalse ({proof})\"\n  | .isNat _ lit proof => m!\"isNat {lit} ({proof})\"\n  | .isNegNat _ lit proof => m!\"isNegNat {lit} ({proof})\"\n  | .isRat _ q _ _ proof => m!\"isRat {q} ({proof})\"\n\n/--\nGiven a `NormNum.Result e` (which uses `IsNat`, `IsInt`, `IsRat` to express equality to a rational\nnumeral), converts it to an equality `e = Nat.rawCast n`, `e = Int.rawCast n`, or\n`e = Rat.rawCast n d` to a raw cast expression, so it can be used for rewriting.\n-/\ndef Result.toRawEq {\u03b1 : Q(Type u)} {e : Q($\u03b1)} : Result e \u2192 (e' : Q($\u03b1)) \u00d7 Q($e = $e')\n  | .isBool false p =>\n    have e : Q(Prop) := e; have p : Q(\u00ac$e) := p\n    \u27e8(q(False) : Expr), (q(eq_false $p) : Expr)\u27e9\n  | .isBool true p =>\n    have e : Q(Prop) := e; have p : Q($e) := p\n    \u27e8(q(True) : Expr), (q(eq_true $p) : Expr)\u27e9\n  | .isNat _ lit p => \u27e8q(Nat.rawCast $lit), q(IsNat.to_raw_eq $p)\u27e9\n  | .isNegNat _ lit p => \u27e8q(Int.rawCast (.negOfNat $lit)), q(IsInt.to_raw_eq $p)\u27e9\n  | .isRat _ _ n d p => \u27e8q(Rat.rawCast $n $d), q(IsRat.to_raw_eq $p)\u27e9\n\n/--\n`Result.toRawEq` but providing an integer. Given a `NormNum.Result e` for something known to be an\ninteger (which uses `IsNat` or `IsInt` to express equality to an integer numeral), converts it to\nan equality `e = Nat.rawCast n` or `e = Int.rawCast n` to a raw cast expression, so it can be used\nfor rewriting. Gives `none` if not an integer.\n-/\ndef Result.toRawIntEq {\u03b1 : Q(Type u)} {e : Q($\u03b1)} : Result e \u2192\n    Option (\u2124 \u00d7 (e' : Q($\u03b1)) \u00d7 Q($e = $e'))\n  | .isNat _ lit p => some \u27e8lit.natLit!, q(Nat.rawCast $lit), q(IsNat.to_raw_eq $p)\u27e9\n  | .isNegNat _ lit p => some \u27e8-lit.natLit!, q(Int.rawCast (.negOfNat $lit)), q(IsInt.to_raw_eq $p)\u27e9\n  | .isRat _ .. | .isBool .. => none\n\n/-- Constructs a `Result` out of a raw nat cast. Assumes `e` is a raw nat cast expression. -/\ndef Result.ofRawNat {\u03b1 : Q(Type u)} (e : Q($\u03b1)) : Result e := Id.run do\n  let .app (.app _ (s\u03b1 : Q(AddMonoidWithOne $\u03b1))) (lit : Q(\u2115)) := e | panic! \"not a raw nat cast\"\n  .isNat s\u03b1 lit (q(IsNat.of_raw $\u03b1 $lit) : Expr)\n\n/-- Constructs a `Result` out of a raw int cast.\nAssumes `e` is a raw int cast expression denoting `n`. -/\ndef Result.ofRawInt {\u03b1 : Q(Type u)} (n : \u2124) (e : Q($\u03b1)) : Result e :=\n  if 0 \u2264 n then\n    Result.ofRawNat e\n  else Id.run do\n    let .app (.app _ (r\u03b1 : Q(Ring $\u03b1))) (.app _ (lit : Q(\u2115))) := e | panic! \"not a raw int cast\"\n    .isNegNat r\u03b1 lit (q(IsInt.of_raw $\u03b1 (.negOfNat $lit)) : Expr)\n\n/-- Constructs a `Result` out of a raw rat cast.\nAssumes `e` is a raw rat cast expression denoting `n`. -/\ndef Result.ofRawRat {\u03b1 : Q(Type u)} (q : \u211a) (e : Q($\u03b1)) (hyp : Option Expr := none) : Result e :=\n  if q.den = 1 then\n    Result.ofRawInt q.num e\n  else Id.run do\n    let .app (.app (.app _ (d\u03b1 : Q(DivisionRing $\u03b1))) (n : Q(\u2124))) (d : Q(\u2115)) := e\n      | panic! \"not a raw rat cast\"\n    let hyp : Q(($d : $\u03b1) \u2260 0) := hyp.get!\n    .isRat d\u03b1 q n d (q(IsRat.of_raw $\u03b1 $n $d $hyp) : Expr)\n\n/-- The result depends on whether `q : \u211a` happens to be an integer, in which case the result is\n`.isInt ..` whereas otherwise it's `.isRat ..`. -/\ndef Result.isRat' {\u03b1 : Q(Type u)} {x : Q($\u03b1)} (inst : Q(DivisionRing $\u03b1) := by assumption)\n    (q : Rat) (n : Q(\u2124)) (d : Q(\u2115)) (proof : Q(IsRat $x $n $d)) : Result x :=\n  if q.den = 1 then\n    have proof : Q(IsRat $x $n (nat_lit 1)) := proof\n    .isInt q(DivisionRing.toRing) n q.num q(IsRat.to_isInt $proof)\n  else\n    .isRat inst q n d proof\n\n/-- Returns the rational number that is the result of `norm_num` evaluation, along with a proof\nthat the denominator is nonzero in the `isRat` case. -/\ndef Result.toRatNZ : Result e \u2192 Option (Rat \u00d7 Option Expr)\n  | .isBool .. => none\n  | .isNat _ lit _ => some (lit.natLit!, none)\n  | .isNegNat _ lit _ => some (-lit.natLit!, none)\n  | .isRat _ q _ _ p => some (q, q(IsRat.den_nz $p))\n\n/--\nConstructs an `ofNat` application `a'` with the canonical instance, together with a proof that\nthe instance is equal to the result of `Nat.cast` on the given `AddMonoidWithOne` instance.\n\nThis function is performance-critical, as many higher level tactics have to construct numerals.\nSo rather than using typeclass search we hardcode the (relatively small) set of solutions\nto the typeclass problem.\n-/\ndef mkOfNat (\u03b1 : Q(Type u)) (_s\u03b1 : Q(AddMonoidWithOne $\u03b1)) (lit : Q(\u2115)) :\n    MetaM ((a' : Q($\u03b1)) \u00d7 Q($lit = $a')) := do\n  if \u03b1.isConstOf ``Nat then\n    let a' : Q(\u2115) := q(OfNat.ofNat $lit : \u2115)\n    pure \u27e8a', (q(Eq.refl $a') : Expr)\u27e9\n  else if \u03b1.isConstOf ``Int then\n    let a' : Q(\u2124) := q(OfNat.ofNat $lit : \u2124)\n    pure \u27e8a', (q(Eq.refl $a') : Expr)\u27e9\n  else if \u03b1.isConstOf ``Rat then\n    let a' : Q(\u211a) := q(OfNat.ofNat $lit : \u211a)\n    pure \u27e8a', (q(Eq.refl $a') : Expr)\u27e9\n  else\n    let some n := lit.natLit? | failure\n    match n with\n    | 0 => pure \u27e8q(0 : $\u03b1), (q(Nat.cast_zero (R := $\u03b1)) : Expr)\u27e9\n    | 1 => pure \u27e8q(1 : $\u03b1), (q(Nat.cast_one (R := $\u03b1)) : Expr)\u27e9\n    | k+2 =>\n      let k : Q(\u2115) := mkRawNatLit k\n      let _x : Q(Nat.AtLeastTwo $lit) :=\n        (q(instAtLeastTwoHAddNatInstHAddInstAddNatOfNat (n := $k)) : Expr)\n      let a' : Q($\u03b1) := q(OfNat.ofNat $lit)\n      pure \u27e8a', (q(Eq.refl $a') : Expr)\u27e9\n\n/-- Convert a `Result` to a `Simp.Result`. -/\ndef Result.toSimpResult {\u03b1 : Q(Type u)} {e : Q($\u03b1)} : Result e \u2192 MetaM Simp.Result\n  | r@(.isBool ..) => let \u27e8expr, proof?\u27e9 := r.toRawEq; pure { expr, proof? }\n  | .isNat s\u03b1 lit p => do\n    let \u27e8a', pa'\u27e9 \u2190 mkOfNat \u03b1 s\u03b1 lit\n    return { expr := a', proof? := q(IsNat.to_eq $p $pa') }\n  | .isNegNat _r\u03b1 lit p => do\n    let \u27e8a', pa'\u27e9 \u2190 mkOfNat \u03b1 q(AddCommMonoidWithOne.toAddMonoidWithOne) lit\n    return { expr := q(-$a'), proof? := q(IsInt.neg_to_eq $p $pa') }\n  | .isRat _ q n d p => do\n    have lit : Q(\u2115) := n.appArg!\n    if q < 0 then\n      let p : Q(IsRat $e (.negOfNat $lit) $d) := p\n      let \u27e8n', pn'\u27e9 \u2190 mkOfNat \u03b1 q(AddCommMonoidWithOne.toAddMonoidWithOne) lit\n      let \u27e8d', pd'\u27e9 \u2190 mkOfNat \u03b1 q(AddCommMonoidWithOne.toAddMonoidWithOne) d\n      return { expr := q(-($n' / $d')), proof? := q(IsRat.neg_to_eq $p $pn' $pd') }\n    else\n      let p : Q(IsRat $e (.ofNat $lit) $d) := p\n      let \u27e8n', pn'\u27e9 \u2190 mkOfNat \u03b1 q(AddCommMonoidWithOne.toAddMonoidWithOne) lit\n      let \u27e8d', pd'\u27e9 \u2190 mkOfNat \u03b1 q(AddCommMonoidWithOne.toAddMonoidWithOne) d\n      return { expr := q($n' / $d'), proof? := q(IsRat.nonneg_to_eq $p $pn' $pd') }\n\n/--\nA extension for `norm_num`.\n-/\nstructure NormNumExt where\n  /-- The extension should be run in the `pre` phase when used as simp plugin. -/\n  pre := true\n  /-- The extension should be run in the `post` phase when used as simp plugin. -/\n  post := true\n  /-- Attempts to prove an expression is equal to some explicit number of the relevant type. -/\n  eval {\u03b1 : Q(Type u)} (e : Q($\u03b1)) : MetaM (Result e)\n  /-- The name of the `norm_num` extension. -/\n  name : Name := by exact decl_name%\n\n/-- Read a `norm_num` extension from a declaration of the right type. -/\ndef mkNormNumExt (n : Name) : ImportM NormNumExt := do\n  let { env, opts, .. } \u2190 read\n  IO.ofExcept <| unsafe env.evalConstCheck NormNumExt opts ``NormNumExt n\n\n/-- Each `norm_num` extension is labelled with a collection of patterns\nwhich determine the expressions to which it should be applied. -/\nabbrev Entry := Array (Array (DiscrTree.Key true)) \u00d7 Name\n\n/-- The state of the `norm_num` extension environment -/\nstructure NormNums where\n  /-- The tree of `norm_num` extensions. -/\n  tree   : DiscrTree NormNumExt true := {}\n  /-- Erased `norm_num`s. -/\n  erased  : PHashSet Name := {}\n  deriving Inhabited\n\n/-- Environment extensions for `norm_num` declarations -/\ninitialize normNumExt : ScopedEnvExtension Entry (Entry \u00d7 NormNumExt) NormNums \u2190\n  -- we only need this to deduplicate entries in the DiscrTree\n  have : BEq NormNumExt := \u27e8fun _ _ \u21a6 false\u27e9\n  /- Insert `v : NormNumExt` into the tree `dt` on all key sequences given in `kss`. -/\n  let insert kss v dt := kss.foldl (fun dt ks \u21a6 dt.insertCore ks v) dt\n  registerScopedEnvExtension {\n    mkInitial := pure {}\n    ofOLeanEntry := fun _ e@(_, n) \u21a6 return (e, \u2190 mkNormNumExt n)\n    toOLeanEntry := (\u00b7.1)\n    addEntry := fun { tree, erased } ((kss, n), ext) \u21a6\n      { tree := insert kss ext tree, erased := erased.erase n }\n  }\n\n/-- Run each registered `norm_num` extension on an expression, returning a `NormNum.Result`. -/\ndef derive {\u03b1 : Q(Type u)} (e : Q($\u03b1)) (post := false) : MetaM (Result e) := do\n  if e.isNatLit then\n    let lit : Q(\u2115) := e\n    return .isNat (q(instAddMonoidWithOneNat) : Q(AddMonoidWithOne \u2115))\n      lit (q(IsNat.raw_refl $lit) : Expr)\n  profileitM Exception \"norm_num\" (\u2190 getOptions) do\n    let s \u2190 saveState\n    let normNums := normNumExt.getState (\u2190 getEnv)\n    let arr \u2190 normNums.tree.getMatch e\n    for ext in arr do\n      if (bif post then ext.post else ext.pre) && ! normNums.erased.contains ext.name then\n        try\n          let new \u2190 withReducibleAndInstances <| ext.eval e\n          trace[Tactic.norm_num] \"{ext.name}:\\n{e} ==> {new}\"\n          return new\n        catch err =>\n          trace[Tactic.norm_num] \"{e} failed: {err.toMessageData}\"\n          s.restore\n    throwError \"{e}: no norm_nums apply\"\n\n/-- Run each registered `norm_num` extension on a typed expression `e : \u03b1`,\nreturning a typed expression `lit : \u2115`, and a proof of `isNat e lit`. -/\ndef deriveNat' {\u03b1 : Q(Type u)} (e : Q($\u03b1)) :\n    MetaM ((_inst : Q(AddMonoidWithOne $\u03b1)) \u00d7 (lit : Q(\u2115)) \u00d7 Q(IsNat $e $lit)) := do\n  let .isNat inst lit proof \u2190 derive e | failure\n  pure \u27e8inst, lit, proof\u27e9\n\n/-- Run each registered `norm_num` extension on a typed expression `e : \u03b1`,\nreturning a typed expression `lit : \u2115`, and a proof of `isNat e lit`. -/\ndef deriveNat {\u03b1 : Q(Type u)} (e : Q($\u03b1))\n    (_inst : Q(AddMonoidWithOne $\u03b1) := by with_reducible assumption) :\n    MetaM ((lit : Q(\u2115)) \u00d7 Q(IsNat $e $lit)) := do\n  let .isNat _ lit proof \u2190 derive e | failure\n  pure \u27e8lit, proof\u27e9\n\n/-- Run each registered `norm_num` extension on a typed expression `e : \u03b1`,\nreturning a typed expression `lit : \u2124`, and a proof of `IsInt e lit` in expression form. -/\ndef deriveInt {\u03b1 : Q(Type u)} (e : Q($\u03b1))\n    (_inst : Q(Ring $\u03b1) := by with_reducible assumption) :\n    MetaM ((lit : Q(\u2124)) \u00d7 Q(IsInt $e $lit)) := do\n  let some \u27e8_, lit, proof\u27e9 := (\u2190 derive e).toInt | failure\n  pure \u27e8lit, proof\u27e9\n\n/-- Run each registered `norm_num` extension on a typed expression `e : \u03b1`,\nreturning a rational number, typed expressions `n : \u211a` and `d : \u211a` for the numerator and\ndenominator, and a proof of `IsRat e n d` in expression form. -/\ndef deriveRat {\u03b1 : Q(Type u)} (e : Q($\u03b1))\n    (_inst : Q(DivisionRing $\u03b1) := by with_reducible assumption) :\n    MetaM (\u211a \u00d7 (n : Q(\u2124)) \u00d7 (d : Q(\u2115)) \u00d7 Q(IsRat $e $n $d)) := do\n  let some res := (\u2190 derive e).toRat' | failure\n  pure res\n\n/-- Extract the natural number `n` if the expression is of the form `OfNat.ofNat n`. -/\ndef isNatLit (e : Expr) : Option \u2115 := do\n  guard <| e.isAppOfArity ``OfNat.ofNat 3\n  let .lit (.natVal lit) := e.appFn!.appArg! | none\n  lit\n\n/-- Extract the integer `i` if the expression is either a natural number literal\nor the negation of one. -/\ndef isIntLit (e : Expr) : Option \u2124 :=\n  if e.isAppOfArity ``Neg.neg 3 then\n    (- \u00b7) <$> isNatLit e.appArg!\n  else\n    isNatLit e\n\n/-- Extract the numerator `n : \u2124` and denominator `d : \u2115` if the expression is either\nan integer literal, or the division of one integer literal by another. -/\ndef isRatLit (e : Expr) : Option \u211a := do\n  if e.isAppOfArity ``Div.div 4 then\n    let d \u2190 isNatLit e.appArg!\n    guard (d \u2260 1)\n    let n \u2190 isIntLit e.appFn!.appArg!\n    let q := mkRat n d\n    guard (q.den = d)\n    pure q\n  else\n    isIntLit e\n\n/-- Test if an expression represents an explicit number written in normal form. -/\ndef isNormalForm : Expr \u2192 Bool\n  | .lit _ => true\n  | .mdata _ e => isNormalForm e\n  | e => (isRatLit e).isSome\n\n/-- Run each registered `norm_num` extension on an expression,\nreturning a `Simp.Result`. -/\ndef eval (e : Expr) (post := false) : MetaM Simp.Result := do\n  if isNormalForm e then return { expr := e }\n  let \u27e8.succ _, _, e\u27e9 \u2190 inferTypeQ e | failure\n  (\u2190 derive e post).toSimpResult\n\n/-- Erases a name marked `norm_num` by adding it to the state's `erased` field and\n  removing it from the state's list of `Entry`s. -/\ndef NormNums.eraseCore (d : NormNums) (declName : Name) : NormNums :=\n { d with erased := d.erased.insert declName }\n\n/--\n  Erase a name marked as a `norm_num` attribute.\n\n  Check that it does in fact have the `norm_num` attribute by making sure it names a `NormNumExt`\n  found somewhere in the state's tree, and is not erased.\n-/\ndef NormNums.erase [Monad m] [MonadError m] (d : NormNums) (declName : Name) : m NormNums := do\n  unless d.tree.values.any (\u00b7.name == declName) && ! d.erased.contains declName\n  do\n    throwError \"'{declName}' does not have [norm_num] attribute\"\n  return d.eraseCore declName\n\ninitialize registerBuiltinAttribute {\n  name := `norm_num\n  descr := \"adds a norm_num extension\"\n  applicationTime := .afterCompilation\n  add := fun declName stx kind \u21a6 match stx with\n    | `(attr| norm_num $es,*) => do\n      let env \u2190 getEnv\n      unless (env.getModuleIdxFor? declName).isNone do\n        throwError \"invalid attribute 'norm_num', declaration is in an imported module\"\n      if (IR.getSorryDep env declName).isSome then return -- ignore in progress definitions\n      let ext \u2190 mkNormNumExt declName\n      let keys \u2190 MetaM.run' <| es.getElems.mapM fun stx \u21a6 do\n        let e \u2190 TermElabM.run' <| withSaveInfoContext <| withAutoBoundImplicit <|\n          withReader ({ \u00b7 with ignoreTCFailures := true }) do\n            let e \u2190 elabTerm stx none\n            let (_, _, e) \u2190 lambdaMetaTelescope (\u2190 mkLambdaFVars (\u2190 getLCtx).getFVars e)\n            return e\n        DiscrTree.mkPath e\n      normNumExt.add ((keys, declName), ext) kind\n    | _ => throwUnsupportedSyntax\n  erase := fun declName => do\n    let s := normNumExt.getState (\u2190 getEnv)\n    let s \u2190 s.erase declName\n    modifyEnv fun env => normNumExt.modifyState env fun _ => s\n}\n\n/-- A simp plugin which calls `NormNum.eval`. -/\ndef tryNormNum? (post := false) (e : Expr) : SimpM (Option Simp.Step) := do\n  try return some (.done (\u2190 eval e post))\n  catch _ => return none\n\n/--\nConstructs a proof that the original expression is true\ngiven a simp result which simplifies the target to `True`.\n-/\ndef _root_.Lean.Meta.Simp.Result.ofTrue (r : Simp.Result) : MetaM (Option Expr) :=\n  if r.expr.isConstOf ``True then\n    some <$> match r.proof? with\n    | some proof => mkOfEqTrue proof\n    | none => pure (mkConst ``True.intro)\n  else\n    pure none\n\nvariable (ctx : Simp.Context) (useSimp := true) in\nmutual\n  /-- A discharger which calls `norm_num`. -/\n  partial def discharge (e : Expr) : SimpM (Option Expr) := do (\u2190 deriveSimp e).ofTrue\n\n  /-- A `Methods` implementation which calls `norm_num`. -/\n  partial def methods : Simp.Methods :=\n    if useSimp then {\n      pre := fun e \u21a6 do\n        Simp.andThen (\u2190 Simp.preDefault e discharge) tryNormNum?\n      post := fun e \u21a6 do\n        Simp.andThen (\u2190 Simp.postDefault e discharge) (tryNormNum? (post := true))\n      discharge? := discharge\n    } else {\n      pre := fun e \u21a6 Simp.andThen (.visit { expr := e }) tryNormNum?\n      post := fun e \u21a6 Simp.andThen (.visit { expr := e }) (tryNormNum? (post := true))\n      discharge? := discharge\n    }\n\n  /-- Traverses the given expression using simp and normalises any numbers it finds. -/\n  partial def deriveSimp (e : Expr) : MetaM Simp.Result :=\n    (\u00b7.1) <$> Simp.main e ctx (methods := methods)\nend\n\n-- FIXME: had to inline a bunch of stuff from `simpGoal` here\n/--\nThe core of `norm_num` as a tactic in `MetaM`.\n\n* `g`: The goal to simplify\n* `ctx`: The simp context, constructed by `mkSimpContext` and\n  containing any additional simp rules we want to use\n* `fvarIdsToSimp`: The selected set of hypotheses used in the location argument\n* `simplifyTarget`: true if the target is selected in the location argument\n* `useSimp`: true if we used `norm_num` instead of `norm_num1`\n-/\ndef normNumAt (g : MVarId) (ctx : Simp.Context) (fvarIdsToSimp : Array FVarId)\n    (simplifyTarget := true) (useSimp := true) :\n    MetaM (Option (Array FVarId \u00d7 MVarId)) := g.withContext do\n  g.checkNotAssigned `norm_num\n  let mut g := g\n  let mut toAssert := #[]\n  let mut replaced := #[]\n  for fvarId in fvarIdsToSimp do\n    let localDecl \u2190 fvarId.getDecl\n    let type \u2190 instantiateMVars localDecl.type\n    let ctx := { ctx with simpTheorems := ctx.simpTheorems.eraseTheorem (.fvar localDecl.fvarId) }\n    let r \u2190 deriveSimp ctx useSimp type\n    match r.proof? with\n    | some _ =>\n      let some (value, type) \u2190 applySimpResultToProp g (mkFVar fvarId) type r\n        | return none\n      toAssert := toAssert.push { userName := localDecl.userName, type, value }\n    | none =>\n      if r.expr.isConstOf ``False then\n        g.assign (\u2190 mkFalseElim (\u2190 g.getType) (mkFVar fvarId))\n        return none\n      g \u2190 g.replaceLocalDeclDefEq fvarId r.expr\n      replaced := replaced.push fvarId\n  if simplifyTarget then\n    let res \u2190 g.withContext do\n      let target \u2190 instantiateMVars (\u2190 g.getType)\n      let r \u2190 deriveSimp ctx useSimp target\n      let some proof \u2190 r.ofTrue\n        | some <$> applySimpResultToTarget g target r\n      g.assign proof\n      pure none\n    let some gNew := res | return none\n    g := gNew\n  let (fvarIdsNew, gNew) \u2190 g.assertHypotheses toAssert\n  let toClear := fvarIdsToSimp.filter fun fvarId \u21a6 !replaced.contains fvarId\n  let gNew \u2190 gNew.tryClearMany toClear\n  return some (fvarIdsNew, gNew)\n\nopen Qq Lean Meta Elab Tactic Term\n\n/-- Constructs a simp context from the simp argument syntax. -/\ndef getSimpContext (args : Syntax) (simpOnly := false) :\n    TacticM Simp.Context := do\n  let simpTheorems \u2190\n    if simpOnly then simpOnlyBuiltins.foldlM (\u00b7.addConst \u00b7) {} else getSimpTheorems\n  let mut { ctx, starArg } \u2190 elabSimpArgs args (eraseLocal := false) (kind := .simp)\n    { simpTheorems := #[simpTheorems], congrTheorems := \u2190 getSimpCongrTheorems }\n  unless starArg do return ctx\n  let mut simpTheorems := ctx.simpTheorems\n  for h in \u2190 getPropHyps do\n    unless simpTheorems.isErased (.fvar h) do\n      simpTheorems \u2190 simpTheorems.addTheorem (.fvar h) (\u2190 h.getDecl).toExpr\n  pure { ctx with simpTheorems }\n\nopen Elab.Tactic in\n/--\nElaborates a call to `norm_num only? [args]` or `norm_num1`.\n* `args`: the `(simpArgs)?` syntax for simp arguments\n* `loc`: the `(location)?` syntax for the optional location argument\n* `simpOnly`: true if `only` was used in `norm_num`\n* `useSimp`: false if `norm_num1` was used, in which case only the structural parts\n  of `simp` will be used, not any of the post-processing that `simp only` does without lemmas\n-/\n-- FIXME: had to inline a bunch of stuff from `mkSimpContext` and `simpLocation` here\ndef elabNormNum (args : Syntax) (loc : Syntax)\n    (simpOnly := false) (useSimp := true) : TacticM Unit := do\n  let ctx \u2190 getSimpContext args (!useSimp || simpOnly)\n  let g \u2190 getMainGoal\n  let res \u2190 match expandOptLocation loc with\n  | .targets hyps simplifyTarget => normNumAt g ctx (\u2190 getFVarIds hyps) simplifyTarget useSimp\n  | .wildcard => normNumAt g ctx (\u2190 g.getNondepPropHyps) (simplifyTarget := true) useSimp\n  match res with\n  | none => replaceMainGoal []\n  | some (_, g) => replaceMainGoal [g]\n\nend Meta.NormNum\n\nnamespace Tactic\nopen Lean.Parser.Tactic Meta.NormNum\n\n/--\nNormalize numerical expressions. Supports the operations `+` `-` `*` `/` `\u207b\u00b9` `^` and `%`\nover numerical types such as `\u2115`, `\u2124`, `\u211a`, `\u211d`, `\u2102` and some general algebraic types,\nand can prove goals of the form `A = B`, `A \u2260 B`, `A < B` and `A \u2264 B`, where `A` and `B` are\nnumerical expressions. It also has a relatively simple primality prover.\n-/\nelab (name := normNum) \"norm_num\" only:&\" only\"? args:(simpArgs ?) loc:(location ?) : tactic =>\n  elabNormNum args loc (simpOnly := only.isSome) (useSimp := true)\n\n/-- Basic version of `norm_num` that does not call `simp`. -/\nelab (name := normNum1) \"norm_num1\" loc:(location ?) : tactic =>\n  elabNormNum mkNullNode loc (simpOnly := true) (useSimp := false)\n\nopen Lean Elab Tactic\n\n@[inherit_doc normNum1] syntax (name := normNum1Conv) \"norm_num1\" : conv\n\n/-- Elaborator for `norm_num1` conv tactic. -/\n@[tactic normNum1Conv] def elabNormNum1Conv : Tactic := fun _ \u21a6 withMainContext do\n  let ctx \u2190 getSimpContext mkNullNode true\n  Conv.applySimpResult (\u2190 deriveSimp ctx (\u2190 instantiateMVars (\u2190 Conv.getLhs)) (useSimp := false))\n\n@[inherit_doc normNum] syntax (name := normNumConv) \"norm_num\" &\" only\"? (simpArgs)? : conv\n\n/-- Elaborator for `norm_num` conv tactic. -/\n@[tactic normNumConv] def elabNormNumConv : Tactic := fun stx \u21a6 withMainContext do\n  let ctx \u2190 getSimpContext stx[2] !stx[1].isNone\n  Conv.applySimpResult (\u2190 deriveSimp ctx (\u2190 instantiateMVars (\u2190 Conv.getLhs)) (useSimp := true))\n\n/--\nThe basic usage is `#norm_num e`, where `e` is an expression,\nwhich will print the `norm_num` form of `e`.\n\nSyntax: `#norm_num` (`only`)? (`[` simp lemma list `]`)? `:`? expression\n\nThis accepts the same options as the `#simp` command.\nYou can specify additional simp lemmas as usual, for example using `#norm_num [f, g] : e`.\n(The colon is optional but helpful for the parser.)\nThe `only` restricts `norm_num` to using only the provided lemmas, and so\n`#norm_num only : e` behaves similarly to `norm_num1`.\n\nUnlike `norm_num`, this command does not fail when no simplifications are made.\n\n`#norm_num` understands local variables, so you can use them to introduce parameters.\n-/\nmacro (name := normNumCmd) \"#norm_num\" o:(&\" only\")?\n    args:(Parser.Tactic.simpArgs)? \" :\"? ppSpace e:term : command =>\n  `(command| #conv norm_num $[only%$o]? $(args)? => $e)\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/NormNum/Core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.35679852250805627}}
{"text": "/-\nCopyright (c) 2021 Paula Neeley. All rights reserved.\nAuthor: Paula Neeley\n-/\n\nimport basicmodal.semantics.consistency basicmodal.syntax.soundness\nlocal attribute [instance] classical.prop_decidable\n\nopen prfK\n\n---------------------- Canonical Model Construction ----------------------\n\nnamespace canonical\n\n\ndef canonical (AX : ctx) [hax : sem_cons AX] : frame := \n{ \n  states := {x\u0393 : ctx // max_ax_consist AX x\u0393},\n  h := \n  begin \n    have h1 := max_ax_exists AX hax, \n    choose \u0393 h1 using h1, \n    exact \u27e8\u27e8\u0393, h1\u27e9\u27e9 \n  end,\n  rel := \u03bb x\u0393 y\u0394, \u2200 \u03c6 : form, \u25a1\u03c6 \u2208 x\u0393.val \u2192 \u03c6 \u2208 y\u0394.val\n}\n\n\ndef T_canonical  : frame := @canonical T_axioms sem_consT\ndef S4_canonical : frame := @canonical S4_axioms sem_consS4\ndef S5_canonical : frame := @canonical S5_axioms sem_consS5\n\n\ndef val_canonical (AX : ctx) [hax : sem_cons AX] : nat \u2192 (canonical AX).states \u2192 Prop :=\n  \u03bb n, \u03bb x\u0393 : (canonical AX).states, (p n) \u2208 x\u0393.val\n\n\nlemma existence (AX : ctx) (hax : sem_cons AX) (x\u0393 : (canonical AX).states) :\n  \u2200 \u03c6, \u25c7\u03c6 \u2208 x\u0393.val \u2194 \u2203 y\u0394 : (canonical AX).states, \u03c6 \u2208 y\u0394.val \u2227 (canonical AX).rel x\u0393 y\u0394 :=\nbegin\nintro \u03c6, split,\nintro h1,\nlet \u0393box : ctx := {\u03c8 : form | \u25a1\u03c8 \u2208 x\u0393.val},\nhave h1 : ax_consist AX (\u0393box \u222a {\u03c6}), \n{by_contradiction h2, simp at h2,\nhave h3 := five AX \u0393box \u03c6 h2,\ncases h3 with L h3, cases h3 with h3 h4,\nhave h5 := cut fin_conj_boxn (mp kdist (nec h4)),\nhave h6 := exercise1,\nhave h7 : \u2200 \u03c8 \u2208 (list.map \u25a1 L), \u03c8 \u2208 x\u0393.1, \nintros \u03c8 h8, simp at *, cases h8 with a h8,\ncases h8 with h8l h8r,\nsubst h8r, exact h3 a h8l,\nspecialize h6 x\u0393.2 h7 h5,\nhave h8 := (six AX x\u0393.1 (max_imp_ax x\u0393.2)).mp x\u0393.2 (\u00ac\u03c6).box,\ncases h8 with h8l h8r, simp at *,\nexact absurd h1 (h8r h6)\n},\nhave h2 := lindenbaum AX (\u0393box \u222a {\u03c6}) h1,\ncases h2 with \u0394 h2, cases h2 with h2 h3,\nlet x\u0394 : (canonical AX).states := \u27e8\u0394, h2\u27e9,\nexistsi (x\u0394 : (canonical AX).states),\nhave h5 := set.union_subset_iff.mp h3,\ncases h5, split, simp at h5_right, exact h5_right,\nhave h3 : \u2200 \u03c6 : form, \u25a1\u03c6 \u2208 x\u0393.val \u2192 \u03c6 \u2208 x\u0394.val,\nintros \u03c8 h4, apply h5_left, exact h4,\nexact h3,\nsimp at *,\nintros y\u0394 h1 h2,\nby_contradiction h3,\nhave h4 := (max_notiff AX x\u0393.1 x\u0393.2 (\u25c7\u03c6)).mp h3,\nhave h5 := (max_dn AX x\u0393.1 x\u0393.2 (\u25a1\u00ac\u03c6)).mpr h4,\nhave h6 := (max_notiff AX y\u0394.1 y\u0394.2 \u03c6).mpr (h2 (\u00ac\u03c6) h5),\nexact absurd h1 h6\nend\n\n\nlemma truth (AX : ctx) (hax : sem_cons AX) (x\u0393 : (canonical AX).states) : \n  \u2200 \u03c6, forces (canonical AX) (val_canonical AX) x\u0393 \u03c6 \u2194 (\u03c6 \u2208 x\u0393.val) :=\nbegin\nintro \u03c6, induction \u03c6 with n \u03c6 \u03c8 ih_\u03c6 ih_\u03c8 \n\u03c6 \u03c8 ih_\u03c6 ih_\u03c8 \u03c6 ih_\u03c6 generalizing x\u0393,\nsplit, intro h1, exact false.elim h1,\nintro h1,\nhave h2 := x\u0393.2,\ncases h2,\nspecialize h2_left [\u22a5],\nsimp at *,\nexact absurd not_contra (h2_left h1),\nrepeat {rw forces, rw val_canonical},\nsplit, intro h1, cases h1 with h1 h2,\nexact max_conj_1 x\u0393.2 (and.intro ((ih_\u03c6 x\u0393).mp h1) ((ih_\u03c8 x\u0393).mp h2)), \nintro h1, split,\napply (ih_\u03c6 x\u0393).mpr, exact max_conj_2 x\u0393.2 h1,\napply (ih_\u03c8 x\u0393).mpr, exact max_conj_3 x\u0393.2 h1,\nsplit, \nintro h1,\napply max_imp_1 x\u0393.2,\nintro h2,\nexact (ih_\u03c8 x\u0393).mp (h1 ((ih_\u03c6 x\u0393).mpr h2)),\nintros h1 h2,\napply (ih_\u03c8 x\u0393).mpr,\nexact max_imp_2 x\u0393.2 h1 ((ih_\u03c6 x\u0393).mp h2),\nsplit, intros h1, \nby_contradiction h2,\nhave h4 := (existence AX hax x\u0393 (\u00ac\u03c6)).mp,\nhave h5 := max_boxdn AX x\u0393.1 x\u0393.2 \u03c6 ((max_notiff AX x\u0393.1 x\u0393.2 \u03c6.box).mp h2),\ncases h4 h5 with x\u0394 h4, cases h4 with h4 h6,\nhave h7 := max_notiff AX x\u0394.1 x\u0394.2 \u03c6,\ncases h7 with h7l h7r,\nexact absurd ((ih_\u03c6 x\u0394).mp (h1 x\u0394 h6)) (h7r h4),\nintros h1 x\u0394 h2,\napply (ih_\u03c6 x\u0394).mpr, exact h2 \u03c6 h1,\nend\n\n\nlemma comphelper (AX : ctx) (\u03c6 : form) (hax : sem_cons AX) : \n  \u00ac prfK AX \u03c6 \u2192 ax_consist AX {\u00ac\u03c6} :=\nbegin\nintro h1, intros L h2,\nrw fin_ax_consist, induction L,\nby_contradiction h3,\nexact absurd (mp dne h3) (nprfalse AX hax), \nhave h4 : (\u2200 \u03c8 \u2208 L_hd::L_tl, \u03c8 = \u00ac\u03c6) \u2192 prfK AX (\u00acfin_conj (L_hd::L_tl)) \u2192 prfK AX \u03c6, \nfrom fin_conj_repeat hax,\nsimp at *, \ncases h2 with h2 h3,\nintro h6, apply h1, apply h4 h2, \nexact h3,\nexact h6\nend \n\n\ntheorem forcesAX (AX : ctx) (hax : sem_cons AX) : \n  forces_ctx (canonical AX) (val_canonical AX) AX :=\nbegin\nintros \u03c6 x\u0393 h1,\nhave h2 : \u2200 \u03c8 \u2208 list.nil, \u03c8 \u2208 x\u0393.val, \n{intros \u03c8 h3, have h4 := list.ne_nil_of_length_pos (list.length_pos_of_mem h3),\nsimp at *, exact false.elim h4},\nexact (truth AX hax x\u0393 \u03c6).mpr (exercise1 x\u0393.2 h2 (mp pl1 (ax h1)))\nend\n\n\ntheorem completeness (AX : ctx) (hax : sem_cons AX) (\u03c6 : form) : \n  global_sem_csq AX \u03c6 \u2192 prfK AX \u03c6 :=\nbegin\nrw \u2190not_imp_not, intro h1,\nhave h2 := comphelper AX \u03c6 hax h1,\nhave h3 := lindenbaum AX {\u00ac\u03c6} h2,\nsimp at *,\ncases h3 with \u0393' h3, cases h3 with h3 h4, \nrw global_sem_csq, \npush_neg,\nlet f := canonical, use f AX,\nlet v := val_canonical, use v AX,\nlet x\u0393' : (f AX).states := \u27e8\u0393', h3\u27e9,\nsplit, \nexact forcesAX AX hax,\nuse x\u0393',\nhave h5 := truth AX hax x\u0393' \u00ac\u03c6,\ncases h5 with h5 h6,\nhave h7 := not_forces_imp (f AX) (v AX) x\u0393' \u03c6,\ncases h7 with h7 h8, apply h8, apply h6, exact h4\nend\n\n\nlemma T_reflexive : T_canonical \u2208 ref_class :=\nbegin\nintros x \u03c6 h1,\nhave h2 : (\u2200 \u03c8 \u2208 [\u25a1\u03c6], \u03c8 \u2208 x.1) \u2192 prfK T_axioms (fin_conj [\u25a1\u03c6] \u2283 \u03c6) \u2192 \u03c6 \u2208 x.1, \n  from exercise1 x.2, simp at *,\nhave h3 : prfK T_axioms (fin_conj [\u03c6.box] \u2283 \u03c6), \n{repeat {rw fin_conj},\nhave h4 : prfK T_axioms (\u03c6.box \u2283 \u03c6), \n{refine ax _, rw T_axioms, simp},\nexact cut (mp pl5 phi_and_true) h4},\nexact h2 h1 h3\nend\n\n\ntheorem T_completeness (\u03c6 : form) : F_valid \u03c6 ref_class \u2192 prfK T_axioms \u03c6 :=\nbegin\nrw \u2190not_imp_not, \nintro h1,\nhave h2 := completeness T_axioms sem_consT \u03c6,\nrw \u2190not_imp_not at h2,\nspecialize h2 h1,\nrw F_valid, \npush_neg,\nlet f := T_canonical, use f,\nsplit,\nexact T_reflexive,\nlet v := val_canonical, use (@v T_axioms sem_consT),\nhave h4 := lindenbaum T_axioms {\u00ac\u03c6} (comphelper T_axioms \u03c6 sem_consT h1),\nsimp at *,\ncases h4 with \u0393' h4, cases h4 with h4 h5,\nlet x\u0393 : f.states := \u27e8\u0393', h4\u27e9,\nuse x\u0393,\nhave h6 := truth T_axioms sem_consT x\u0393 \u00ac\u03c6,\ncases h6 with h6 h7,\nhave h8 := not_forces_imp f (@v T_axioms sem_consT) x\u0393 \u03c6,\ncases h8 with h8 h9, apply h9, apply h7, exact h5\nend\n\n\nlemma S4_reftrans : S4_canonical \u2208 ref_trans_class :=\nbegin\nsplit,\nintros x \u03c6 h1,\nhave h2 : (\u2200 \u03c8 \u2208 [\u25a1\u03c6], \u03c8 \u2208 x.1) \u2192 prfK S4_axioms (fin_conj [\u25a1\u03c6] \u2283 \u03c6) \u2192 \u03c6 \u2208 x.1, \n  from exercise1 x.2, simp at *,\nhave h3 : prfK S4_axioms (fin_conj [\u03c6.box] \u2283 \u03c6), \n{repeat {rw fin_conj},\nhave h4 : prfK S4_axioms (\u03c6.box \u2283 \u03c6), \n{refine ax _, rw S4_axioms, simp, rw T_axioms, simp},\nexact cut (mp pl5 phi_and_true) h4},\nexact h2 h1 h3,\nintros x y z h1 h2 \u03c6 h3, apply h2 \u03c6,\napply h1 (\u25a1\u03c6),\nhave h4 : prfK S4_axioms (fin_conj [\u03c6.box] \u2283 \u03c6.box.box), \n{repeat {rw fin_conj},\nhave h5 : prfK S4_axioms (\u03c6.box \u2283 \u03c6.box.box), \n{refine ax _, rw S4_axioms, simp},\nexact cut (mp pl5 phi_and_true) h5},\nhave h6 : (\u2200 \u03c8 \u2208 [\u25a1\u03c6], \u03c8 \u2208 x.1) \u2192 prfK S4_axioms (fin_conj [\u25a1\u03c6] \u2283 \u03c6.box.box) \u2192 \u03c6.box.box \u2208 x.1, \n  from exercise1 x.2, simp at *,\nexact h6 h3 h4\nend\n\n\ntheorem S4_completeness (\u03c6 : form) : F_valid \u03c6 ref_trans_class \u2192 prfK S4_axioms \u03c6 :=\nbegin\nrw \u2190not_imp_not, \nintro h1,\nhave h2 := completeness S4_axioms sem_consS4 \u03c6,\nrw \u2190not_imp_not at h2,\nspecialize h2 h1,\nrw F_valid, \npush_neg,\nlet f := S4_canonical, use f,\nsplit,\nexact S4_reftrans,\nlet v := val_canonical, use (@v S4_axioms sem_consS4),\nhave h4 := lindenbaum S4_axioms {\u00ac\u03c6} (comphelper S4_axioms \u03c6 sem_consS4 h1),\nsimp at *,\ncases h4 with \u0393' h4, cases h4 with h4 h5,\nlet x\u0393 : f.states := \u27e8\u0393', h4\u27e9,\nuse x\u0393,\nhave h6 := truth S4_axioms sem_consS4 x\u0393 \u00ac\u03c6,\ncases h6 with h6 h7,\nhave h8 := not_forces_imp f (@v S4_axioms sem_consS4) x\u0393 \u03c6,\ncases h8 with h8 h9, apply h9, apply h7, exact h5\nend\n\n\nlemma euclid_dual {\u03c6 : form} : prfK S5_axioms ((\u25c7(\u00ac\u03c6) \u2283 \u25a1(\u25c7(\u00ac\u03c6))) \u2283 (\u25c7(\u25a1\u03c6) \u2283 \u25a1\u03c6)) :=\nbegin\nsimp,\nhave h1 : prfK S5_axioms (\u25c7(\u00ac\u03c6) \u2283 \u25a1(\u25c7\u00ac\u03c6)),\nrefine ax _, rw S5_axioms, simp, simp at *,\nhave h2 := contrapos.mpr h1,\nhave h3 := cut h2 (mp pl6 dual_equiv1),\nhave h4 : prfK S5_axioms ((\u00ac\u25a1(\u25c7\u00ac\u03c6)) \u2194 (\u00ac\u00ac\u25c7(\u00ac(\u25c7\u00ac\u03c6)))),\n  from (mp (mp pl4 (contrapos.mpr (mp pl6 dual_equiv1))) (contrapos.mpr (mp pl5 dual_equiv1))),\nhave h5 := cut dni (cut (mp pl6 h4) h3),\nhave h6 := (contrapos.mpr (mp kdist (nec (contrapos.mpr (mp pl5 dual_equiv1))))),\nexact (mp pl1 (cut h6 h5))\nend\n\n\nlemma S5_equiv : S5_canonical \u2208 equiv_class :=\nbegin\nrw equiv_ref_euclid,\nsplit,\nintros x \u03c6 h1,\nhave h2 : (\u2200 \u03c8 \u2208 [\u25a1\u03c6], \u03c8 \u2208 x.1) \u2192 prfK S5_axioms (fin_conj [\u25a1\u03c6] \u2283 \u03c6) \u2192 \u03c6 \u2208 x.1, \n  from exercise1 x.2, simp at *,\nhave h3 : prfK S5_axioms (fin_conj [\u03c6.box] \u2283 \u03c6), \n{repeat {rw fin_conj},\nhave h4 : prfK S5_axioms (\u03c6.box \u2283 \u03c6), \n{refine ax _, rw S5_axioms, simp, rw T_axioms, simp},\nexact cut (mp pl5 phi_and_true) h4},\nexact h2 h1 h3,\nintros x y z h1 h2 \u03c6 h3,\napply h2 \u03c6, clear h2,\nhave h2 : prfK S5_axioms (\u25c7(\u00ac\u03c6) \u2283 \u25a1(\u25c7\u00ac\u03c6)), \n{refine ax _, rw S5_axioms, simp},\nhave h4 : prfK S5_axioms (\u25c7(\u25a1\u03c6) \u2283 \u25a1\u03c6), \n  from mp euclid_dual h2,\nhave h5 : (\u2200 \u03c8 \u2208 [\u25c7(\u25a1\u03c6)], \u03c8 \u2208 x.1) \u2192 \n  prfK S5_axioms (fin_conj [\u25c7(\u25a1\u03c6)] \u2283 \u25a1\u03c6) \u2192 \u25a1\u03c6 \u2208 x.1, \n  from exercise1 x.2, simp at *,\napply h5,\nby_contradiction h6,\nhave h7 := (max_notiff S5_axioms x.1 x.2 (\u00ac(\u00ac\u03c6.box).box)).mp h6,\nhave h8 := (max_dn S5_axioms x.1 x.2 ((\u00ac\u03c6.box).box)).mpr h7,\nhave h9 := (max_notiff S5_axioms y.1 y.2 (\u03c6.box)).mpr (h1 (\u00ac\u03c6.box) h8),\nexact absurd h3 h9,\nexact (cut (mp pl5 phi_and_true) h4)\nend\n\n\ntheorem S5_completeness (\u03c6 : form) : F_valid \u03c6 equiv_class \u2192 prfK S5_axioms \u03c6 :=\nbegin\nrw \u2190not_imp_not, \nintro h1,\nhave h2 := completeness S5_axioms sem_consS5 \u03c6,\nrw \u2190not_imp_not at h2,\nspecialize h2 h1,\nrw F_valid, \npush_neg,\nlet f := S5_canonical, use f,\nsplit,\nexact S5_equiv,\nlet v := val_canonical, use (@v S5_axioms sem_consS5),\nhave h4 := lindenbaum S5_axioms {\u00ac\u03c6} (comphelper S5_axioms \u03c6 sem_consS5 h1),\nsimp at *,\ncases h4 with \u0393' h4, cases h4 with h4 h5,\nlet x\u0393 : f.states := \u27e8\u0393', h4\u27e9,\nuse x\u0393,\nhave h6 := truth S5_axioms sem_consS5 x\u0393 \u00ac\u03c6,\ncases h6 with h6 h7,\nhave h8 := not_forces_imp f (@v S5_axioms sem_consS5) x\u0393 \u03c6,\ncases h8 with h8 h9, apply h9, apply h7, exact h5\nend\n\nend canonical\n\n", "meta": {"author": "paulaneeley", "repo": "modal", "sha": "ee5d149d4ecb337005b850bddf4453e56a5daf04", "save_path": "github-repos/lean/paulaneeley-modal", "path": "github-repos/lean/paulaneeley-modal/modal-ee5d149d4ecb337005b850bddf4453e56a5daf04/src/basicmodal/semantics/completeness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3567985145568955}}
{"text": "/-\nCopyright (c) 2020 Aaron Anderson. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Aaron Anderson\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.rel_iso\nimport Mathlib.order.lattice_intervals\nimport Mathlib.order.order_dual\nimport Mathlib.PostPort\n\nuniverses u_2 l u_1 \n\nnamespace Mathlib\n\n/-!\n# Modular Lattices\nThis file defines Modular Lattices, a kind of lattice useful in algebra.\nFor examples, look to the subobject lattices of abelian groups, submodules, and ideals, or consider\nany distributive lattice.\n\n## Main Definitions\n- `is_modular_lattice` defines a modular lattice to be one such that\n  `x \u2264 z \u2192 (x \u2294 y) \u2293 z \u2264 x \u2294 (y \u2293 z)`\n- `inf_Icc_order_iso_Icc_sup` gives an order isomorphism between the intervals\n  `[a \u2293 b, a]` and `[b, a \u2294 b]`.\n  This corresponds to the diamond (or second) isomorphism theorems of algebra.\n\n## Main Results\n- `is_modular_lattice_iff_sup_inf_sup_assoc`:\n  Modularity is equivalent to the `sup_inf_sup_assoc`: `(x \u2293 z) \u2294 (y \u2293 z) = ((x \u2293 z) \u2294 y) \u2293 z`\n- `distrib_lattice.is_modular_lattice`: Distributive lattices are modular.\n\n## To do\n- Relate atoms and coatoms in modular lattices\n\n-/\n\n/-- A modular lattice is one with a limited associativity between `\u2293` and `\u2294`. -/\nclass is_modular_lattice (\u03b1 : Type u_2) [lattice \u03b1] where\n  sup_inf_le_assoc_of_le : \u2200 {x : \u03b1} (y : \u03b1) {z : \u03b1}, x \u2264 z \u2192 (x \u2294 y) \u2293 z \u2264 x \u2294 y \u2293 z\n\ntheorem sup_inf_assoc_of_le {\u03b1 : Type u_1} [lattice \u03b1] [is_modular_lattice \u03b1] {x : \u03b1} (y : \u03b1)\n    {z : \u03b1} (h : x \u2264 z) : (x \u2294 y) \u2293 z = x \u2294 y \u2293 z :=\n  le_antisymm (is_modular_lattice.sup_inf_le_assoc_of_le y h)\n    (le_inf (sup_le_sup_left inf_le_left x) (sup_le h inf_le_right))\n\ntheorem is_modular_lattice.sup_inf_sup_assoc {\u03b1 : Type u_1} [lattice \u03b1] [is_modular_lattice \u03b1]\n    {x : \u03b1} {y : \u03b1} {z : \u03b1} : x \u2293 z \u2294 y \u2293 z = (x \u2293 z \u2294 y) \u2293 z :=\n  Eq.symm (sup_inf_assoc_of_le y inf_le_right)\n\ntheorem inf_sup_assoc_of_le {\u03b1 : Type u_1} [lattice \u03b1] [is_modular_lattice \u03b1] {x : \u03b1} (y : \u03b1)\n    {z : \u03b1} (h : z \u2264 x) : x \u2293 y \u2294 z = x \u2293 (y \u2294 z) :=\n  sorry\n\nprotected instance order_dual.is_modular_lattice {\u03b1 : Type u_1} [lattice \u03b1] [is_modular_lattice \u03b1] :\n    is_modular_lattice (order_dual \u03b1) :=\n  is_modular_lattice.mk\n    fun (x y z : order_dual \u03b1) (xz : x \u2264 z) =>\n      le_of_eq\n        (eq.mpr (id (Eq._oldrec (Eq.refl ((x \u2294 y) \u2293 z = x \u2294 y \u2293 z)) inf_comm))\n          (eq.mpr (id (Eq._oldrec (Eq.refl (z \u2293 (x \u2294 y) = x \u2294 y \u2293 z)) sup_comm))\n            (eq.mpr (id (Eq._oldrec (Eq.refl (z \u2293 (y \u2294 x) = x \u2294 y \u2293 z)) (propext eq_comm)))\n              (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2294 y \u2293 z = z \u2293 (y \u2294 x))) inf_comm))\n                (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2294 z \u2293 y = z \u2293 (y \u2294 x))) sup_comm))\n                  (eq.mpr\n                    ((fun (a a_1 : order_dual \u03b1) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : order_dual \u03b1)\n                        (e_2 : \u1fb0 = \u1fb0_1) => congr (congr_arg Eq e_1) e_2)\n                      (z \u2293 y \u2294 x) ((z \u2294 coe_fn order_dual.of_dual y) \u2293 x) (Eq.refl (z \u2293 y \u2294 x))\n                      (z \u2293 (y \u2294 x)) (z \u2294 coe_fn order_dual.of_dual y \u2293 x) (Eq.refl (z \u2293 (y \u2294 x))))\n                    (sup_inf_assoc_of_le (coe_fn order_dual.of_dual y)\n                      (iff.mpr order_dual.dual_le xz))))))))\n\n/-- The diamond isomorphism between the intervals `[a \u2293 b, a]` and `[b, a \u2294 b]` -/\ndef inf_Icc_order_iso_Icc_sup {\u03b1 : Type u_1} [lattice \u03b1] [is_modular_lattice \u03b1] (a : \u03b1) (b : \u03b1) :\n    \u21a5(set.Icc (a \u2293 b) a) \u2243o \u21a5(set.Icc b (a \u2294 b)) :=\n  rel_iso.mk\n    (equiv.mk (fun (x : \u21a5(set.Icc (a \u2293 b) a)) => { val := \u2191x \u2294 b, property := sorry })\n      (fun (x : \u21a5(set.Icc b (a \u2294 b))) => { val := a \u2293 \u2191x, property := sorry }) sorry sorry)\n    sorry\n\nnamespace is_compl\n\n\n/-- The diamond isomorphism between the intervals `set.Iic a` and `set.Ici b`. -/\ndef Iic_order_iso_Ici {\u03b1 : Type u_1} [bounded_lattice \u03b1] [is_modular_lattice \u03b1] {a : \u03b1} {b : \u03b1}\n    (h : is_compl a b) : \u21a5(set.Iic a) \u2243o \u21a5(set.Ici b) :=\n  order_iso.trans (order_iso.set_congr (set.Iic a) (set.Icc (a \u2293 b) a) sorry)\n    (order_iso.trans (inf_Icc_order_iso_Icc_sup a b)\n      (order_iso.set_congr (set.Icc b (a \u2294 b)) (set.Ici b) sorry))\n\nend is_compl\n\n\ntheorem is_modular_lattice_iff_sup_inf_sup_assoc {\u03b1 : Type u_1} [lattice \u03b1] :\n    is_modular_lattice \u03b1 \u2194 \u2200 (x y z : \u03b1), x \u2293 z \u2294 y \u2293 z = (x \u2293 z \u2294 y) \u2293 z :=\n  sorry\n\nnamespace distrib_lattice\n\n\nprotected instance is_modular_lattice {\u03b1 : Type u_1} [distrib_lattice \u03b1] : is_modular_lattice \u03b1 :=\n  is_modular_lattice.mk\n    fun (x y z : \u03b1) (xz : x \u2264 z) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl ((x \u2294 y) \u2293 z \u2264 x \u2294 y \u2293 z)) inf_sup_right))\n        (eq.mpr (id (Eq._oldrec (Eq.refl (x \u2293 z \u2294 y \u2293 z \u2264 x \u2294 y \u2293 z)) (iff.mpr inf_eq_left xz)))\n          (le_refl (x \u2294 y \u2293 z)))\n\nend distrib_lattice\n\n\nnamespace is_modular_lattice\n\n\nprotected instance is_modular_lattice_Iic {\u03b1 : Type u_1} [bounded_lattice \u03b1] [is_modular_lattice \u03b1]\n    {a : \u03b1} : is_modular_lattice \u21a5(set.Iic a) :=\n  mk fun (x y z : \u21a5(set.Iic a)) (xz : x \u2264 z) => sup_inf_le_assoc_of_le (\u2191y) xz\n\nprotected instance is_modular_lattice_Ici {\u03b1 : Type u_1} [bounded_lattice \u03b1] [is_modular_lattice \u03b1]\n    {a : \u03b1} : is_modular_lattice \u21a5(set.Ici a) :=\n  mk fun (x y z : \u21a5(set.Ici a)) (xz : x \u2264 z) => sup_inf_le_assoc_of_le (\u2191y) xz\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/modular_lattice_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3567985145568955}}
{"text": "import lambda_calculus.utlc.beta.distance\nimport lambda_calculus.utlc.beta.encoding.basic\nimport complexity.basic\nimport complexity.core\n\nnamespace lambda_calculus\nnamespace utlc\nnamespace \u03b2\nnamespace complexity\n\nopen lambda_calculus.utlc.\u03b2.encoding\n\nlocal attribute [simp] \u03b2.normal_iteration \u03b2.strategic_reduction_step head_reduced\n\nvariables {et: encoding_type}\n\ntheorem of_distance_le\n  {a: complexity.encodable_function (distance_model et)} {f f': a.unwrap} (g : encoded_program) {g': encoded_program} (fc fc': complexity.cost_function a) (n: \u2115):\n  distance_le n g'.value g.value \u2192 complexity.witness a g f fc \u2192 f = f' \u2192 (fc + \u2191n) \u2264 fc' \u2192\n  complexity.witness a g' f' fc' :=\nbegin\n  induction a with _ _ _ en _ generalizing f f' g g' fc fc',\n  { intros hg hw hf hfc,\n    rw [\u2190 hf],\n    apply distance_le_mono,\n    apply distance_le_trans',\n    apply hg,\n    apply hw,\n    refl,\n    rw  [nat.cast_id, nat.add_comm] at hfc,\n    assumption },\n  { intros hg hw hf hfc a,\n    apply a_ih (en.application g a),\n    simp [distance_model],\n    apply utlc.\u03b2.dot_distance_le_dot_left,\n    apply hg,\n    apply hw,\n    rw [hf],\n    apply hfc,\n  }\nend\n\ninstance value_complexity (\u03b1: Type) (et: encoding_type) [en: complexity.has_encoding (distance_model et) \u03b1] (a: \u03b1):\n  complexity.has_complexity (distance_model et) a :=\n  \u27e8 \u27e8 (0:\u2115),\n    \u27e8 \u27e8 (en.value.encode a).value, (en.value.encode a).proof.left \u27e9,\n      @distance_le_refl _ has_\u03b2_reduction.step _ \u27e9 \u27e9 \u27e9\n\ndef id_prog: encoded_program := \u27e8 \u039b \u21930, by simp \u27e9\n\ninstance id_complexity (\u03b1: Type) (et: encoding_type) [en: complexity.has_encoding (distance_model et) \u03b1] (a: \u03b1):\n  complexity.has_complexity (distance_model et) (@id \u03b1) :=\n  \u27e8 \u27e8 \u03bb _, (1:\u2115), \u27e8 id_prog,\nbegin\n  intro a,\n  apply distance_le_of_normal_iteration,\n  simp [id_prog, complexity.cast_unwrap, distance_model],\nend \u27e9 \u27e9 \u27e9\n\ndef const_prog: encoded_program := \u27e8 \u039b \u039b \u21931, by simp \u27e9\n\ninstance const_complexity (\u03b1 \u03b2: Type) (et: encoding_type)\n  [en: complexity.has_encoding (distance_model et) \u03b1] [en: complexity.has_encoding (distance_model et) \u03b2]:\n  complexity.has_complexity (distance_model et) (@const \u03b1 \u03b2) :=\n  \u27e8 \u27e8 \u03bb _ _, (2:\u2115), \u27e8 const_prog,\nbegin\n  intros a b,\n  apply distance_le_of_normal_iteration,\n  simp [const_prog, const, complexity.cast_unwrap, distance_model],\nend \u27e9 \u27e9 \u27e9\n\ninstance partial_complexity\n  (\u03b1 \u03b2: Type) (et: encoding_type)\n  [en: complexity.has_encoding (distance_model et) \u03b1]\n  [complexity.has_encodable_function (distance_model et) \u03b2]\n  (a: \u03b1) {f: \u03b1 \u2192 \u03b2} [h: complexity.has_complexity (distance_model et) f] :\n  complexity.has_complexity (distance_model et) (f a) :=\n  \u27e8 \u27e8 (complexity (distance_model et) f) a,\n  by {\n  cases h.value.proof with prog h,\n  exact \u27e8 \u27e8 prog.value\u00b7(en.value.encode a).value, by {\n    simp only [ bool.to_bool_coe, and_true,\n      data_is_closed_below', closed_below, closed], apply prog.proof } \u27e9,\n  by { rw [complexity.cast_unwrap, fcast], exact h a } \u27e9 } \u27e9 \u27e9\n\ndef compose_prog: encoded_program := \u27e8 \u039b \u039b \u039b \u21932\u00b7(\u21931\u00b7\u21930), by simp \u27e9\n\ninstance compose_complexity\n  (\u03b1 \u03b2 \u03b3: Type) (et: encoding_type)\n  [\u03b1_en: complexity.has_encoding (distance_model et) \u03b1] [\u03b2_en: complexity.has_encoding (distance_model et) \u03b2] [\u03b3_en: complexity.has_encodable_function (distance_model et) \u03b3]\n  (f: \u03b1 \u2192 \u03b2) (g: \u03b2 \u2192 \u03b3) [cf: complexity.has_complexity (distance_model et) f] [cg: complexity.has_complexity (distance_model et) g]:\n  complexity.has_complexity (distance_model et) (compose g f) :=\n\u27e8 \u27e8 \u03bb a, cg.value.cost (f a) + \u2191(3 + (cf.value.cost a):\u2115),\nbegin\n  rcases cf.value with \u27e8cfc, fp, cfp\u27e9,\n  rcases cg.value with \u27e8cgc, gp, cgp\u27e9,\n  fconstructor,\n  { exact \u27e8 compose_prog.value\u00b7gp.value\u00b7fp.value,\n    by simp; exact \u27e8compose_prog.proof, gp.proof, fp.proof\u27e9 \u27e9 },\n  intro a,\n  apply of_distance_le begin\n    fconstructor,\n    { exact gp.value\u00b7((\u03b2_en.value.encode (f a)).value) },\n    simp,\n  end,\n  { simp [distance_model, compose_prog],\n    apply distance_le_trans,\n    { apply distance_le_of_normal_iteration 3,\n      simp [normal_iteration, distance_model, compose_prog] },\n    apply utlc.\u03b2.dot_distance_le_dot_right,\n    apply cfp a },\n  apply cgp (f a),\n  unfold complexity.cast_unwrap,\n  rw [\u2190 fcast', \u2190 fcast'],\n  all_goals { simp [compose] },\nend \u27e9 \u27e9\n\ndef flip_prog: encoded_program := \u27e8 \u039b \u039b \u039b \u21932\u00b7\u21930\u00b7\u21931, by simp \u27e9\n\ninstance flip_complexity\n  (\u03b1 \u03b2 \u03b3: Type) (et: encoding_type)\n  [\u03b1_en: complexity.has_encoding (distance_model et) \u03b1] [\u03b2_en: complexity.has_encoding (distance_model et) \u03b2] [\u03b3_en: complexity.has_encoding (distance_model et) \u03b3]\n  (f: \u03b1 \u2192 \u03b2 \u2192 \u03b3) [cf: complexity.has_complexity (distance_model et) f]:\n  complexity.has_complexity (distance_model et) (flip f) :=\n\u27e8 \u27e8 \u03bb b a, (cf.value.cost a b) + \u21913,\nbegin\n  rcases cf.value with \u27e8cfc, fp, cfp\u27e9,\n  fconstructor,\n  { exact \u27e8 flip_prog.value\u00b7fp.value,\n    by simp; exact \u27e8 flip_prog.proof, fp.proof \u27e9 \u27e9 },\n  intros a b,\n  apply distance_le_trans',\n  { apply distance_le_of_normal_iteration 3,\n    refl },\n  simp [distance_model, flip_prog],\n  apply cfp,\n  simp [add_comm 3],\nend \u27e9 \u27e9\n\ndef ycomb: utlc := \u039b (\u039b \u21931\u00b7(\u21930\u00b7\u21930))\u00b7(\u039b \u21931\u00b7(\u21930\u00b7\u21930))\ndef yrec (f: utlc): utlc := ycomb\u00b7f\n\ntheorem yrec_apply {f: utlc}: distance_le 3 (yrec f) (f\u00b7yrec f) :=\nbegin\n  rw [yrec, ycomb],\n  apply distance_le_trans',\n  apply distance_le_of_normal_iteration 2,\n  simp,\n  apply dot_distance_le_dot_right,\n  apply distance_le_symm,\n  apply distance_le_of_normal_iteration 1,\n  simp,\nend\n\nend complexity\nend \u03b2\nend utlc\nend lambda_calculus", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/lambda_calculus/utlc/beta/complexity/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768249, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.35677254987088963}}
{"text": "import meta_data\n\nnamespace Wiles_A_and_Taylor_R_FermatLast\n\n-- the statement of Fermat's Last Theorem\naxiom fermat_last_theorem :\n\u2200 (x y z n : nat), x > 0 \u2192 y > 0 \u2192 n > 2 \u2192 x ^ n + y ^ n \u2260 z ^ n\n\ndef paper : document :=\n{   authors := [\n    {name := \"Andrew Wiles\"},\n    {name := \"Richard Tylor\"}\n  ],\n  title := \"Modular elliptic curves and Fermat's last theorem\",\n  doi := \"10.2307/2118559\"}\n\ndefinition fabstract : fabstract :=\n{ description := \"A result in number theory conjectured by Pierre de Fermat and proved by Andrew Wiles and Richard Taylor. Colloquially referred to as Fermat's Last Theorem.\",\n  contributors := [{name := \"Adam Kurkiewicz\"}],\n  sources := [cite.Document paper],\n  results := [result.Proof fermat_last_theorem] }\n\nend Wiles_A_and_Taylor_R_FermatLast\n", "meta": {"author": "thalesant", "repo": "formalabstracts-2017", "sha": "c47181342c9e41954aa8d41f5049965b5f332bca", "save_path": "github-repos/lean/thalesant-formalabstracts-2017", "path": "github-repos/lean/thalesant-formalabstracts-2017/formalabstracts-2017-c47181342c9e41954aa8d41f5049965b5f332bca/fabstract/Wiles_A_and_Taylor_R_FermatLast/fabstract.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.661922862511608, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.35676531628331754}}
{"text": "open int\nexample : 2 + 3 = 5:=begin generalize h:3=x,generalize g:2=y,rw \u2190 h,rw \u2190 g,\u2190 end\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/sum_function.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.35675309815803574}}
{"text": "example (p q r : Prop) (hp : p) : (p \u2228 q \u2228 r) \u2227 (q \u2228 p \u2228 r) \u2227 (q \u2228 r \u2228 p) :=\n  by { repeat { any_goals { split } }; repeat { { left, assumption } <|> right <|> assumption } }\n", "meta": {"author": "Ailrun", "repo": "Theorem_Proving_in_Lean", "sha": "2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68", "save_path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean", "path": "github-repos/lean/Ailrun-Theorem_Proving_in_Lean/Theorem_Proving_in_Lean-2eb1b5caf93c6a5a555c79e9097cf2ba5a66cf68/src/ch5/ex0802.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.35675309815803574}}
{"text": "import analysis.normed_space.bounded_linear_maps\nimport seminormed_rings\nimport smoothing_procedure\n\nnoncomputable theory\n\nopen_locale big_operators nnreal\n\n\n--TODO: check normed_group is normed_add_comm_group, nondiscrete_normed_field is nontrivilly_normed_field\nstructure is_continuous_linear_map (\ud835\udd5c : Type*) [normed_field \ud835\udd5c]\n  {E : Type*} [normed_add_comm_group E] [normed_space \ud835\udd5c E]\n  {F : Type*} [normed_add_comm_group F] [normed_space \ud835\udd5c F] (f : E \u2192 F)\n  extends is_linear_map \ud835\udd5c f : Prop :=\n(cont : continuous f . tactic.interactive.continuity')\n\nlemma is_continuous_linear_map_iff_is_bounded_linear_map {K : Type*} [nontrivially_normed_field K]\n  {M : Type*} [normed_add_comm_group M] [normed_space K M] {N : Type*} [normed_add_comm_group N] \n  [normed_space K N] (f : M \u2192 N) : is_continuous_linear_map K f \u2194 is_bounded_linear_map K f :=\nbegin\n  refine \u27e8\u03bb h_cont, _, \u03bb h_bdd, \u27e8h_bdd.to_is_linear_map, h_bdd.continuous\u27e9\u27e9,\n  { set F : M \u2192L[K] N :=\n    by use [f, is_linear_map.map_add h_cont.1, is_linear_map.map_smul h_cont.1, h_cont.2],\n    exact continuous_linear_map.is_bounded_linear_map F, },\nend\n\n-- Lemma 3.2.1./3\n\n/- section\n\nvariables {K : Type*} [field K] {L : Type*} [field L] [algebra K L] {g : K \u2192 \u211d\u22650} (hg : is_norm g)\n\nlemma finite_extension_pow_mul_seminorm' (hfd : finite_dimensional K L)\n  (hna : \u2200 (a b : K), g(a - b) \u2264 max (g a) (g b)) :\n  \u2203 f : L \u2192 nnreal, is_algebra_norm hg f \u2227 is_pow_mult f \u2227 function_extends g f :=\nsorry \nend -/\n\nvariables {K : Type*} [normed_field K] {L : Type*} [field L] [algebra K L]\n\nvariables {\u03b9 : Type*} [fintype \u03b9] {R : Type*} [ring R] {M : Type*} [add_comm_group M]\n  [module R M] \n\nlemma linear_independent.eq_coords_of_eq {v : \u03b9 \u2192 M} (hv : linear_independent R v) {f : \u03b9 \u2192 R}\n  {g : \u03b9 \u2192 R} (heq : \u2211 i, f i \u2022 v i = \u2211 i, g i \u2022 v i) (i : \u03b9) : f i = g i := \nbegin\n  rw [\u2190 sub_eq_zero, \u2190 finset.sum_sub_distrib] at heq,\n  simp_rw \u2190 sub_smul at heq,\n  rw linear_independent_iff' at hv,\n  exact sub_eq_zero.mp (hv finset.univ (\u03bb i, (f i - g i)) heq i (finset.mem_univ i)),\nend\n\nlemma basis_one {\u03b9 : Type*} [fintype \u03b9] [decidable_eq \u03b9] {B : basis \u03b9 K L} {i : \u03b9}\n  (hBi : B i = (1 : L)) (k : K) :\n  (B.equiv_fun) ((algebra_map K L) k) = \u03bb (j : \u03b9), if (j = i) then k else 0 := \nbegin\n  ext j,\n  apply linear_independent.eq_coords_of_eq B.linear_independent,\n  rw basis.sum_equiv_fun B (algebra_map K L k),\n  have h_sum : \u2211 (j : \u03b9), ite (j = i) k 0 \u2022 B j = \u2211 (j : \u03b9), ite (j = i) (k \u2022 B j) 0,\n  { apply finset.sum_congr (eq.refl _),\n    { rintros h -,\n      split_ifs,\n      exacts [rfl, zero_smul _ _] }},\n  rw [h_sum, algebra.algebra_map_eq_smul_one,\n    finset.sum_ite_eq' finset.univ (i : \u03b9) (\u03bb j : \u03b9, k \u2022 B j)],\n  simp only [finset.mem_univ, if_true, hBi],\nend\n\ndef basis.norm {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] (B : basis \u03b9 K L) : L \u2192 \u211d\u22650 := \n\u03bb x, \u2225B.equiv_fun x (classical.some (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun x i\u2225 )))\u2225\u208a\n\nlemma basis.norm_zero {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] (B : basis \u03b9 K L) :  B.norm 0 = 0 :=\nby simp only [basis.norm, nnnorm_eq_zero, map_zero, pi.zero_apply, norm_zero]\n\nlemma basis.norm_extends {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] [decidable_eq \u03b9] {B : basis \u03b9 K L}\n  {i : \u03b9} (hBi : B i = (1 : L)) :\n  function_extends (\u03bb x : K, \u2225x\u2225\u208a) B.norm :=\nbegin\n  intro k,\n  { by_cases hk : k = 0,\n  { simp only [hk, map_zero, B.norm_zero, nnnorm_zero] },\n  { simp only [basis.norm,  basis_one hBi],\n    have h_max : (classical.some (finite.exists_max (\u03bb j : \u03b9, \n      \u2225(\u03bb (n : \u03b9), if (n = i) then k else 0) j \u2225))) = i,\n    { by_contradiction h,\n      have h_max := classical.some_spec (finite.exists_max (\u03bb j : \u03b9, \n        \u2225(\u03bb (n : \u03b9), if (n = i) then k else 0) j \u2225)),\n      simp only [if_neg h] at h_max,\n      specialize h_max i,\n      rw [if_pos rfl, norm_zero, norm_le_zero_iff] at h_max,\n      exact hk h_max },\n    rw if_pos h_max, }}\nend\n\nlemma basis.norm_is_nonarchimedean {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] [decidable_eq \u03b9]\n  {B : basis \u03b9 K L} {i : \u03b9} (hBi : B i = (1 : L))\n  (hna : \u2200 (a b : K), \u2225a - b\u2225\u208a \u2264 max (\u2225a\u2225\u208a) (\u2225b\u2225\u208a)) : is_nonarchimedean B.norm  :=\nbegin\n  intros x y,\n  simp only [basis.norm],\n  set ixy := classical.some (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun (x - y) i\u2225)) with hixy_def,\n  have hxy : \u2225B.equiv_fun (x - y) ixy\u2225\u208a \u2264 max (\u2225B.equiv_fun x ixy\u2225\u208a) (\u2225B.equiv_fun y ixy\u2225\u208a),\n  { rw [linear_equiv.map_sub, pi.sub_apply], exact hna _ _ , },\n  have hix := classical.some_spec (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun x i\u2225)),\n  have hiy := classical.some_spec (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun y i\u2225)),\n  cases le_max_iff.mp hxy with hx hy,\n  { apply le_max_of_le_left,\n    exact le_trans hx (hix ixy), },\n  { apply le_max_of_le_right,\n    exact le_trans hy (hiy ixy), },\nend\n\nlemma basis.norm_is_bdd {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] [decidable_eq \u03b9] {B : basis \u03b9 K L}\n  {i : \u03b9} (hBi : B i = (1 : L)) : \n  \u2203 (c : nnreal) (hc : 0 < c), \u2200 (x y : L), B.norm (x * y) \u2264 c * B.norm x * B.norm y :=\nbegin\n  set M := classical.some (finite.exists_max (\u03bb (i : \u03b9 \u00d7 \u03b9), B.norm (B i.1 * B i.2))) with hM_def,\n  have hM := classical.some_spec (finite.exists_max (\u03bb (i : \u03b9 \u00d7 \u03b9), B.norm (B i.1 * B i.2))),\n  use B.norm (B M.1 * B M.2),\n  split,\n  { have h_pos : (0 : nnreal) < B.norm (B i * B i),\n    { have h1 : (1 : L) = (algebra_map K L) 1 := by rw map_one,\n      rw [hBi, mul_one, h1, basis.norm_extends hBi],\n      simp only [nnnorm_one, zero_lt_one] },\n    exact lt_of_lt_of_le h_pos (hM (i, i)) },\n  { intros x y,\n    set ixy := classical.some (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun (x*y) i\u2225))\n      with hixy_def,\n    conv_lhs{simp only [basis.norm],\n    rw [\u2190 hixy_def, \u2190 basis.sum_equiv_fun B x, \u2190 basis.sum_equiv_fun B y] },\n    rw finset.sum_mul,\n    --rw basis.equiv_fun_apply,\n    \n    have h_sum : B.equiv_fun (\u2211 (x_1 : \u03b9), B.equiv_fun x x_1 \u2022 B x_1 * \n    \u2211 (i : \u03b9), B.equiv_fun y i \u2022  B i) ixy = \n    \u2211 (x_1 : \u03b9), B.equiv_fun (B.equiv_fun x x_1 \u2022 B x_1 * \n    \u2211 (i : \u03b9), B.equiv_fun y i \u2022  B i) ixy,\n    { \n      sorry },\n/-     have h_sum : B.repr (\u2211 (x_1 : \u03b9), B.repr x x_1 \u2022 B x_1 * \n    \u2211 (i : \u03b9), B.equiv_fun y i \u2022  B i) ixy = \n    \u2211 (x_1 : \u03b9), B.repr (B.equiv_fun x x_1 \u2022 B x_1 * \n    \u2211 (i : \u03b9), B.equiv_fun y i \u2022  B i) ixy,\n    { \n      sorry }, -/\n    simp_rw h_sum,\n --apply @is_nonarchimedean_finset_image_add _ _ (\u03bb (x : K), \u2225x\u2225\u208a) (nnnorm_zero) _ B.equiv_fun, \n   -- }, \n    /- have hj : \u2203 (j : \u03b9) (hj : finset.univ.nonempty \u2192 j \u2208 finset.univ), \u2225\u2211 (x_1 : \u03b9), \n      B.equiv_fun (B.equiv_fun x x_1 \u2022 B x_1 * \n      \u2211 (i : \u03b9), B.equiv_fun y i \u2022 B i) ixy\u2225\u208a \u2264 \n    \u2225B.equiv_fun (B.equiv_fun x j \u2022 B j * \u2211 (i : \u03b9), B.equiv_fun y i \u2022 B i) ixy\u2225\u208a,\n    { have hna : is_nonarchimedean (nnnorm : K \u2192 \u211d\u22650) := sorry,\n      sorry,\n      /- have := @is_nonarchimedean_finset_image_add K _ nnnorm (nnnorm_zero) hna B.equiv_fun\n        finset.univ,  -/\n    },\n    obtain \u27e8j, hjuniv, hj\u27e9 := hj,\n    apply le_trans hj,\n    simp, -/\n\n    /- lemma is_nonarchimedean_finset_image_add {\u03b1 : Type*} [ring \u03b1] {f : \u03b1 \u2192 nnreal} (hf0 : f 0 = 0)\n  (hna : is_nonarchimedean f) {\u03b2 : Type*} [h\u03b2 : nonempty \u03b2] (g : \u03b2 \u2192 \u03b1) (s : finset \u03b2) :\n  \u2203 (b : \u03b2) (hb : s.nonempty \u2192 b \u2208 s), f (s.sum g) \u2264 f (g b) := -/\n    \n    sorry },\nend\n\nlemma basis.repr_smul {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] [decidable_eq \u03b9] (B : basis \u03b9 K L)\n  (i : \u03b9) (k : K) (y : L) : B.equiv_fun ((algebra_map K L k) * y) i = k * (B.equiv_fun y i) :=\nby rw [\u2190 smul_eq_mul, algebra_map_smul, linear_equiv.map_smul]; refl\n\nlemma basis.norm_smul {\u03b9 : Type*} [fintype \u03b9] [nonempty \u03b9] [decidable_eq \u03b9] {B : basis \u03b9 K L}\n  {i : \u03b9} (hBi : B i = (1 : L)) (k : K) (y : L) :\n  B.norm ((algebra_map K L) k * y) = B.norm ((algebra_map K L) k) * B.norm y :=\nbegin\n  by_cases hk : k = 0,\n  { rw [hk, map_zero, zero_mul, B.norm_zero, zero_mul],},\n  { rw basis.norm_extends hBi,\n    simp only [basis.norm],\n    set i := classical.some (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun y i\u2225)) with hi_def,\n    have hi := classical.some_spec (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun y i\u2225)),\n    set j := classical.some (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun ((algebra_map K L) k * y) i\u2225))\n      with hj_def,\n    have hj := classical.some_spec\n      (finite.exists_max (\u03bb i : \u03b9, \u2225B.equiv_fun ((algebra_map K L) k * y) i\u2225)),\n    have hij : \u2225B.equiv_fun y i\u2225\u208a = \u2225B.equiv_fun y j\u2225\u208a,\n    { refine le_antisymm _ (hi j),\n      { specialize hj i,\n        rw \u2190 hj_def at hj,\n        simp only [basis.repr_smul, norm_mul] at hj,\n        exact (mul_le_mul_left (lt_of_le_of_ne (norm_nonneg _)\n          (ne.symm (norm_ne_zero_iff.mpr hk)))).mp hj }},\n    rw [basis.repr_smul, nnnorm_mul, \u2190 hi_def, \u2190 hj_def, hij] },\nend\n\n/- lemma basis.norm_is_module_norm {\u03b9 : Type*} [fintype \u03b9] (B : basis \u03b9 K L)\n  (hB1 : \u2203 i : \u03b9, B i = (1 : L)) : Prop := false -/\n\nlemma finite_extension_pow_mul_seminorm (hfd : finite_dimensional K L)\n  (hna : \u2200 (a b : K), \u2225a - b\u2225\u208a \u2264 max (\u2225a\u2225\u208a) (\u2225b\u2225\u208a)) :\n  \u2203 f : L \u2192 nnreal, is_algebra_norm (normed_ring.to_is_norm K) f \u2227 is_pow_mult f \u2227\n    function_extends (\u03bb (k : K), \u2225 k \u2225\u208a) f \u2227 is_nonarchimedean f :=\nbegin\n  -- Choose a basis B = {1, e2,..., en} of the K-vector space L\n  classical,\n  set h1 : linear_independent K (\u03bb (x : ({1} : set L)), (x : L)) := \n  linear_independent_singleton one_ne_zero,\n  set \u03b9 := {x // x \u2208 (h1.extend (set.subset_univ ({1} : set L)))} with h\u03b9,\n  set B : basis \u03b9 K L  := basis.extend h1 with hB,\n  letI hfin : fintype \u03b9 := finite_dimensional.fintype_basis_index B,\n  haveI hem : nonempty \u03b9 := B.index_nonempty,\n  have h1L : (1 : L) \u2208 h1.extend _,\n  { apply basis.subset_extend,\n    exact set.mem_singleton 1 },\n  have hB1 : B \u27e81, h1L\u27e9 = (1 : L),\n  { rw [basis.coe_extend, subtype.coe_mk] },\n  -- For every k \u2208 K, k = k \u2022 1 + 0 \u2022 e2 + ... + 0 \u2022 en\n  have h_k : \u2200 (k : K), (B.equiv_fun) ((algebra_map K L) k) = \u03bb (i : \u03b9), \n    if (i = \u27e8(1 : L), h1L\u27e9) then k else 0 := basis_one hB1,\n  -- Define a function g : L \u2192 \u211d\u22650 by setting g (\u2211ki \u2022 ei) = max\u1d62 \u2225 ki \u2225  \n  set g : L \u2192 nnreal := B.norm with hg,\n  -- g 0 = 0\n  have hg0 : g 0 = 0 := B.norm_zero,\n  -- g extends the norm on K\n  have hg_ext : function_extends (\u03bb x : K, \u2225x\u2225\u208a) g := basis.norm_extends hB1,\n  -- g is nonarchimedean\n  have hg_na : is_nonarchimedean g := basis.norm_is_nonarchimedean hB1 hna,\n  -- g is multiplicatively bounded\n  have hg_bdd : \u2203 (c : nnreal) (hc : 0 < c), \u2200 (x y : L), g (x * y) \u2264 c * g x * g y,\n  { exact basis.norm_is_bdd hB1 },\n  -- g is a K-module norm\n  have hg_mul : \u2200 (k : K) (y : L), g ((algebra_map K L) k * y) = g ((algebra_map K L) k) * g y :=\n  \u03bb k y, basis.norm_smul hB1 k y,\n  -- Using BGR Prop. 1.2.1/2, we can smooth g to a ring norm f on L that extends the norm on K.\n  set f := seminorm_from_bounded g with hf,\n  have hf_sn : is_seminorm f := seminorm_from_bounded_is_seminorm hg0 hg_bdd \n    (add_le_of_is_nonarchimedean hg0 hg_na),\n  have hf_na : is_nonarchimedean f := seminorm_from_bounded_is_nonarchimedean hg_bdd hg_na,\n  have hf_1 : is_norm_le_one_class f := seminorm_from_bounded_is_norm_le_one_class hg_bdd,\n  have hf_ext : function_extends (\u03bb x : K, \u2225x\u2225\u208a) f,\n  { intro k,\n    rw \u2190 hg_ext,\n    exact seminorm_from_bounded_of_mul_apply hg_bdd (hg_mul k) },\n  -- Using BGR Prop. 1.3.2/1, we obtain from f  a power multiplicative K-algebra norm on L \n  -- extending the norm on K.\n  set F := smoothing_seminorm hf_1 with hF,\n  have hF_ext : \u2200 k : K,  F ((algebra_map K L) k) = (\u03bb (k : K), \u2225k\u2225\u208a) k,\n  { intro k,\n    rw \u2190 hf_ext _,\n    exact smoothing_seminorm_apply_of_is_mult hf_sn hf_1 \n      (seminorm_from_bounded_of_mul_is_mul hg_bdd (hg_mul k)) },\n  have hF_1 : F 1 = 1,\n  { have h1 : (1 : L) = (algebra_map K L) 1 := by rw map_one,\n    simp only [h1, hF_ext (1 : K), nnnorm_one], },\n  use F,\n  refine \u27e8\u27e8field.is_norm_of_is_seminorm (smoothing_seminorm_is_seminorm hf_sn hf_1 hf_na)\n      \u27e8(1 : L), hF_1.symm \u25b8 zero_ne_one\u27e9, _\u27e9, smoothing_seminorm_is_pow_mult hf_sn hf_1, hF_ext, \n      smoothing_seminorm_is_nonarchimedean hf_sn hf_1 hf_na\u27e9,\n  { intros k y,\n    have hk : \u2200 y : L, f ((algebra_map K L k) * y) = f (algebra_map K L k) * f y,\n    { exact seminorm_from_bounded_of_mul_is_mul hg_bdd (hg_mul k), },\n    have hfk : f ((algebra_map K L) k) = \u2225k\u2225\u208a := hf_ext k,\n    rw [hF, \u2190 hfk, \u2190 smoothing_seminorm_apply_of_is_mult hf_sn hf_1 hk, algebra.smul_def],\n    exact smoothing_seminorm_of_mult hf_sn hf_1 hk y, },\nend", "meta": {"author": "mariainesdff", "repo": "local_fields", "sha": "53a20ab87cf8f6ab27eedef2d9da929898951d82", "save_path": "github-repos/lean/mariainesdff-local_fields", "path": "github-repos/lean/mariainesdff-local_fields/local_fields-53a20ab87cf8f6ab27eedef2d9da929898951d82/old_lean_files/normed_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.35675309815803574}}
{"text": "/-\nCopyright (c) 2014 Parikshit Khanna. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.order_functions\nimport Mathlib.control.monad.basic\nimport Mathlib.data.nat.choose.basic\nimport Mathlib.order.rel_classes\nimport Mathlib.PostPort\n\nuniverses u v w u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Basic properties of lists\n-/\n\nnamespace list\n\n\nprotected instance nil.is_left_id {\u03b1 : Type u} : is_left_id (List \u03b1) append [] :=\n  is_left_id.mk nil_append\n\nprotected instance nil.is_right_id {\u03b1 : Type u} : is_right_id (List \u03b1) append [] :=\n  is_right_id.mk append_nil\n\nprotected instance has_append.append.is_associative {\u03b1 : Type u} : is_associative (List \u03b1) append :=\n  is_associative.mk append_assoc\n\ntheorem cons_ne_nil {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : a :: l \u2260 [] :=\n  fun (\u1fb0 : a :: l = []) =>\n    eq.dcases_on \u1fb0 (fun (H_1 : [] = a :: l) => list.no_confusion H_1) (Eq.refl []) (HEq.refl \u1fb0)\n\ntheorem cons_ne_self {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : a :: l \u2260 l :=\n  mt (congr_arg length) (nat.succ_ne_self (length l))\n\ntheorem head_eq_of_cons_eq {\u03b1 : Type u} {h\u2081 : \u03b1} {h\u2082 : \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1} :\n    h\u2081 :: t\u2081 = h\u2082 :: t\u2082 \u2192 h\u2081 = h\u2082 :=\n  fun (Peq : h\u2081 :: t\u2081 = h\u2082 :: t\u2082) =>\n    list.no_confusion Peq fun (Pheq : h\u2081 = h\u2082) (Pteq : t\u2081 = t\u2082) => Pheq\n\ntheorem tail_eq_of_cons_eq {\u03b1 : Type u} {h\u2081 : \u03b1} {h\u2082 : \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1} :\n    h\u2081 :: t\u2081 = h\u2082 :: t\u2082 \u2192 t\u2081 = t\u2082 :=\n  fun (Peq : h\u2081 :: t\u2081 = h\u2082 :: t\u2082) =>\n    list.no_confusion Peq fun (Pheq : h\u2081 = h\u2082) (Pteq : t\u2081 = t\u2082) => Pteq\n\n@[simp] theorem cons_injective {\u03b1 : Type u} {a : \u03b1} : function.injective (List.cons a) :=\n  fun (l\u2081 l\u2082 : List \u03b1) (Pe : a :: l\u2081 = a :: l\u2082) => tail_eq_of_cons_eq Pe\n\ntheorem cons_inj {\u03b1 : Type u} (a : \u03b1) {l : List \u03b1} {l' : List \u03b1} : a :: l = a :: l' \u2194 l = l' :=\n  function.injective.eq_iff cons_injective\n\ntheorem exists_cons_of_ne_nil {\u03b1 : Type u} {l : List \u03b1} (h : l \u2260 []) :\n    \u2203 (b : \u03b1), \u2203 (L : List \u03b1), l = b :: L :=\n  sorry\n\n/-! ### mem -/\n\ntheorem mem_singleton_self {\u03b1 : Type u} (a : \u03b1) : a \u2208 [a] := mem_cons_self a []\n\ntheorem eq_of_mem_singleton {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} : a \u2208 [b] \u2192 a = b :=\n  fun (this : a \u2208 [b]) =>\n    or.elim (eq_or_mem_of_mem_cons this) (fun (this : a = b) => this)\n      fun (this : a \u2208 []) => absurd this (not_mem_nil a)\n\n@[simp] theorem mem_singleton {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} : a \u2208 [b] \u2194 a = b :=\n  { mp := eq_of_mem_singleton, mpr := Or.inl }\n\ntheorem mem_of_mem_cons_of_mem {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {l : List \u03b1} :\n    a \u2208 b :: l \u2192 b \u2208 l \u2192 a \u2208 l :=\n  sorry\n\ntheorem eq_or_ne_mem_of_mem {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {l : List \u03b1} (h : a \u2208 b :: l) :\n    a = b \u2228 a \u2260 b \u2227 a \u2208 l :=\n  classical.by_cases Or.inl\n    fun (this : a \u2260 b) =>\n      or.elim h Or.inl fun (h : list.mem a l) => Or.inr { left := this, right := h }\n\ntheorem not_mem_append {\u03b1 : Type u} {a : \u03b1} {s : List \u03b1} {t : List \u03b1} (h\u2081 : \u00aca \u2208 s) (h\u2082 : \u00aca \u2208 t) :\n    \u00aca \u2208 s ++ t :=\n  mt (iff.mp mem_append) (iff.mpr not_or_distrib { left := h\u2081, right := h\u2082 })\n\ntheorem ne_nil_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) : l \u2260 [] :=\n  id\n    fun (e : l = []) =>\n      false.dcases_on (fun (h : a \u2208 []) => False) (eq.mp (Eq._oldrec (Eq.refl (a \u2208 l)) e) h)\n\ntheorem mem_split {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    \u2203 (s : List \u03b1), \u2203 (t : List \u03b1), l = s ++ a :: t :=\n  sorry\n\ntheorem mem_of_ne_of_mem {\u03b1 : Type u} {a : \u03b1} {y : \u03b1} {l : List \u03b1} (h\u2081 : a \u2260 y) (h\u2082 : a \u2208 y :: l) :\n    a \u2208 l :=\n  or.elim (eq_or_mem_of_mem_cons h\u2082) (fun (e : a = y) => absurd e h\u2081) fun (r : a \u2208 l) => r\n\ntheorem ne_of_not_mem_cons {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {l : List \u03b1} : \u00aca \u2208 b :: l \u2192 a \u2260 b :=\n  fun (nin : \u00aca \u2208 b :: l) (aeqb : a = b) => absurd (Or.inl aeqb) nin\n\ntheorem not_mem_of_not_mem_cons {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {l : List \u03b1} : \u00aca \u2208 b :: l \u2192 \u00aca \u2208 l :=\n  fun (nin : \u00aca \u2208 b :: l) (nainl : a \u2208 l) => absurd (Or.inr nainl) nin\n\ntheorem not_mem_cons_of_ne_of_not_mem {\u03b1 : Type u} {a : \u03b1} {y : \u03b1} {l : List \u03b1} :\n    a \u2260 y \u2192 \u00aca \u2208 l \u2192 \u00aca \u2208 y :: l :=\n  fun (p1 : a \u2260 y) (p2 : \u00aca \u2208 l) =>\n    not.intro fun (Pain : a \u2208 y :: l) => absurd (eq_or_mem_of_mem_cons Pain) (not_or p1 p2)\n\ntheorem ne_and_not_mem_of_not_mem_cons {\u03b1 : Type u} {a : \u03b1} {y : \u03b1} {l : List \u03b1} :\n    \u00aca \u2208 y :: l \u2192 a \u2260 y \u2227 \u00aca \u2208 l :=\n  fun (p : \u00aca \u2208 y :: l) => { left := ne_of_not_mem_cons p, right := not_mem_of_not_mem_cons p }\n\ntheorem mem_map_of_mem {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    f a \u2208 map f l :=\n  sorry\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1}\n    (h : b \u2208 map f l) : \u2203 (a : \u03b1), a \u2208 l \u2227 f a = b :=\n  sorry\n\n@[simp] theorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1} :\n    b \u2208 map f l \u2194 \u2203 (a : \u03b1), a \u2208 l \u2227 f a = b :=\n  sorry\n\ntheorem mem_map_of_injective {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} (H : function.injective f)\n    {a : \u03b1} {l : List \u03b1} : f a \u2208 map f l \u2194 a \u2208 l :=\n  sorry\n\ntheorem forall_mem_map_iff {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {l : List \u03b1} {P : \u03b2 \u2192 Prop} :\n    (\u2200 (i : \u03b2), i \u2208 map f l \u2192 P i) \u2194 \u2200 (j : \u03b1), j \u2208 l \u2192 P (f j) :=\n  sorry\n\n@[simp] theorem map_eq_nil {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {l : List \u03b1} :\n    map f l = [] \u2194 l = [] :=\n  sorry\n\n@[simp] theorem mem_join {\u03b1 : Type u} {a : \u03b1} {L : List (List \u03b1)} :\n    a \u2208 join L \u2194 \u2203 (l : List \u03b1), l \u2208 L \u2227 a \u2208 l :=\n  sorry\n\ntheorem exists_of_mem_join {\u03b1 : Type u} {a : \u03b1} {L : List (List \u03b1)} :\n    a \u2208 join L \u2192 \u2203 (l : List \u03b1), l \u2208 L \u2227 a \u2208 l :=\n  iff.mp mem_join\n\ntheorem mem_join_of_mem {\u03b1 : Type u} {a : \u03b1} {L : List (List \u03b1)} {l : List \u03b1} (lL : l \u2208 L)\n    (al : a \u2208 l) : a \u2208 join L :=\n  iff.mpr mem_join (Exists.intro l { left := lL, right := al })\n\n@[simp] theorem mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {b : \u03b2} {l : List \u03b1} {f : \u03b1 \u2192 List \u03b2} :\n    b \u2208 list.bind l f \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 l), b \u2208 f a :=\n  sorry\n\ntheorem exists_of_mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {b : \u03b2} {l : List \u03b1} {f : \u03b1 \u2192 List \u03b2} :\n    b \u2208 list.bind l f \u2192 \u2203 (a : \u03b1), \u2203 (H : a \u2208 l), b \u2208 f a :=\n  iff.mp mem_bind\n\ntheorem mem_bind_of_mem {\u03b1 : Type u} {\u03b2 : Type v} {b : \u03b2} {l : List \u03b1} {f : \u03b1 \u2192 List \u03b2} {a : \u03b1}\n    (al : a \u2208 l) (h : b \u2208 f a) : b \u2208 list.bind l f :=\n  iff.mpr mem_bind (Exists.intro a (Exists.intro al h))\n\ntheorem bind_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {g : \u03b1 \u2192 List \u03b2} {f : \u03b2 \u2192 \u03b3} (l : List \u03b1) :\n    map f (list.bind l g) = list.bind l fun (a : \u03b1) => map f (g a) :=\n  sorry\n\n/-! ### length -/\n\ntheorem length_eq_zero {\u03b1 : Type u} {l : List \u03b1} : length l = 0 \u2194 l = [] :=\n  { mp := eq_nil_of_length_eq_zero, mpr := fun (h : l = []) => Eq.symm h \u25b8 rfl }\n\n@[simp] theorem length_singleton {\u03b1 : Type u} (a : \u03b1) : length [a] = 1 := rfl\n\ntheorem length_pos_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} : a \u2208 l \u2192 0 < length l := sorry\n\ntheorem exists_mem_of_length_pos {\u03b1 : Type u} {l : List \u03b1} : 0 < length l \u2192 \u2203 (a : \u03b1), a \u2208 l :=\n  sorry\n\ntheorem length_pos_iff_exists_mem {\u03b1 : Type u} {l : List \u03b1} : 0 < length l \u2194 \u2203 (a : \u03b1), a \u2208 l :=\n  sorry\n\ntheorem ne_nil_of_length_pos {\u03b1 : Type u} {l : List \u03b1} : 0 < length l \u2192 l \u2260 [] :=\n  fun (h1 : 0 < length l) (h2 : l = []) => lt_irrefl 0 (iff.mpr length_eq_zero h2 \u25b8 h1)\n\ntheorem length_pos_of_ne_nil {\u03b1 : Type u} {l : List \u03b1} : l \u2260 [] \u2192 0 < length l :=\n  fun (h : l \u2260 []) =>\n    iff.mpr pos_iff_ne_zero fun (h0 : length l = 0) => h (iff.mp length_eq_zero h0)\n\ntheorem length_pos_iff_ne_nil {\u03b1 : Type u} {l : List \u03b1} : 0 < length l \u2194 l \u2260 [] :=\n  { mp := ne_nil_of_length_pos, mpr := length_pos_of_ne_nil }\n\ntheorem length_eq_one {\u03b1 : Type u} {l : List \u03b1} : length l = 1 \u2194 \u2203 (a : \u03b1), l = [a] := sorry\n\ntheorem exists_of_length_succ {\u03b1 : Type u} {n : \u2115} (l : List \u03b1) :\n    length l = n + 1 \u2192 \u2203 (h : \u03b1), \u2203 (t : List \u03b1), l = h :: t :=\n  sorry\n\n@[simp] theorem length_injective_iff {\u03b1 : Type u} : function.injective length \u2194 subsingleton \u03b1 :=\n  sorry\n\n@[simp] theorem length_injective {\u03b1 : Type u} [subsingleton \u03b1] : function.injective length :=\n  iff.mpr length_injective_iff _inst_1\n\n/-! ### set-theoretic notation of lists -/\n\ntheorem empty_eq {\u03b1 : Type u} : \u2205 = [] := Eq.refl \u2205\n\ntheorem singleton_eq {\u03b1 : Type u} (x : \u03b1) : singleton x = [x] := rfl\n\ntheorem insert_neg {\u03b1 : Type u} [DecidableEq \u03b1] {x : \u03b1} {l : List \u03b1} (h : \u00acx \u2208 l) :\n    insert x l = x :: l :=\n  if_neg h\n\ntheorem insert_pos {\u03b1 : Type u} [DecidableEq \u03b1] {x : \u03b1} {l : List \u03b1} (h : x \u2208 l) : insert x l = l :=\n  if_pos h\n\ntheorem doubleton_eq {\u03b1 : Type u} [DecidableEq \u03b1] {x : \u03b1} {y : \u03b1} (h : x \u2260 y) :\n    insert x (singleton y) = [x, y] :=\n  sorry\n\n/-! ### bounded quantifiers over lists -/\n\ntheorem forall_mem_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (x : \u03b1) (H : x \u2208 []) : p x :=\n  false.dcases_on (fun (H : x \u2208 []) => p x) H\n\ntheorem forall_mem_cons {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : List \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 a :: l \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 l \u2192 p x :=\n  ball_cons\n\ntheorem forall_mem_of_forall_mem_cons {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : List \u03b1}\n    (h : \u2200 (x : \u03b1), x \u2208 a :: l \u2192 p x) (x : \u03b1) (H : x \u2208 l) : p x :=\n  and.right (iff.mp forall_mem_cons h)\n\ntheorem forall_mem_singleton {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 [a] \u2192 p x) \u2194 p a :=\n  sorry\n\ntheorem forall_mem_append {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 l\u2081 ++ l\u2082 \u2192 p x) \u2194 (\u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x) \u2227 \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 p x :=\n  sorry\n\ntheorem not_exists_mem_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) : \u00ac\u2203 (x : \u03b1), \u2203 (H : x \u2208 []), p x := sorry\n\ntheorem exists_mem_cons_of {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} (l : List \u03b1) (h : p a) :\n    \u2203 (x : \u03b1), \u2203 (H : x \u2208 a :: l), p x :=\n  bex.intro a (mem_cons_self a l) h\n\ntheorem exists_mem_cons_of_exists {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : List \u03b1}\n    (h : \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x) : \u2203 (x : \u03b1), \u2203 (H : x \u2208 a :: l), p x :=\n  bex.elim h fun (x : \u03b1) (xl : x \u2208 l) (px : p x) => bex.intro x (mem_cons_of_mem a xl) px\n\ntheorem or_exists_of_exists_mem_cons {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {a : \u03b1} {l : List \u03b1}\n    (h : \u2203 (x : \u03b1), \u2203 (H : x \u2208 a :: l), p x) : p a \u2228 \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x :=\n  sorry\n\ntheorem exists_mem_cons_iff {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) (a : \u03b1) (l : List \u03b1) :\n    (\u2203 (x : \u03b1), \u2203 (H : x \u2208 a :: l), p x) \u2194 p a \u2228 \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x :=\n  { mp := or_exists_of_exists_mem_cons,\n    mpr :=\n      fun (h : p a \u2228 \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x) =>\n        or.elim h (exists_mem_cons_of l) exists_mem_cons_of_exists }\n\n/-! ### list subset -/\n\ntheorem subset_def {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 \u2286 l\u2082 \u2194 \u2200 {a : \u03b1}, a \u2208 l\u2081 \u2192 a \u2208 l\u2082 :=\n  iff.rfl\n\ntheorem subset_append_of_subset_left {\u03b1 : Type u} (l : List \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l \u2286 l\u2081 \u2192 l \u2286 l\u2081 ++ l\u2082 :=\n  fun (s : l \u2286 l\u2081) => subset.trans s (subset_append_left l\u2081 l\u2082)\n\ntheorem subset_append_of_subset_right {\u03b1 : Type u} (l : List \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l \u2286 l\u2082 \u2192 l \u2286 l\u2081 ++ l\u2082 :=\n  fun (s : l \u2286 l\u2082) => subset.trans s (subset_append_right l\u2081 l\u2082)\n\n@[simp] theorem cons_subset {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} {m : List \u03b1} :\n    a :: l \u2286 m \u2194 a \u2208 m \u2227 l \u2286 m :=\n  sorry\n\ntheorem cons_subset_of_subset_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} {m : List \u03b1} (ainm : a \u2208 m)\n    (lsubm : l \u2286 m) : a :: l \u2286 m :=\n  iff.mpr cons_subset { left := ainm, right := lsubm }\n\ntheorem append_subset_of_subset_of_subset {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1}\n    (l\u2081subl : l\u2081 \u2286 l) (l\u2082subl : l\u2082 \u2286 l) : l\u2081 ++ l\u2082 \u2286 l :=\n  fun (a : \u03b1) (h : a \u2208 l\u2081 ++ l\u2082) => or.elim (iff.mp mem_append h) l\u2081subl l\u2082subl\n\n@[simp] theorem append_subset_iff {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1} :\n    l\u2081 ++ l\u2082 \u2286 l \u2194 l\u2081 \u2286 l \u2227 l\u2082 \u2286 l :=\n  sorry\n\ntheorem eq_nil_of_subset_nil {\u03b1 : Type u} {l : List \u03b1} : l \u2286 [] \u2192 l = [] := sorry\n\ntheorem eq_nil_iff_forall_not_mem {\u03b1 : Type u} {l : List \u03b1} : l = [] \u2194 \u2200 (a : \u03b1), \u00aca \u2208 l :=\n  (fun (this : l = [] \u2194 l \u2286 []) => this)\n    { mp := fun (e : l = []) => e \u25b8 subset.refl l, mpr := eq_nil_of_subset_nil }\n\ntheorem map_subset {\u03b1 : Type u} {\u03b2 : Type v} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (f : \u03b1 \u2192 \u03b2) (H : l\u2081 \u2286 l\u2082) :\n    map f l\u2081 \u2286 map f l\u2082 :=\n  sorry\n\ntheorem map_subset_iff {\u03b1 : Type u} {\u03b2 : Type v} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (f : \u03b1 \u2192 \u03b2)\n    (h : function.injective f) : map f l\u2081 \u2286 map f l\u2082 \u2194 l\u2081 \u2286 l\u2082 :=\n  sorry\n\n/-! ### append -/\n\ntheorem append_eq_has_append {\u03b1 : Type u} {L\u2081 : List \u03b1} {L\u2082 : List \u03b1} :\n    list.append L\u2081 L\u2082 = L\u2081 ++ L\u2082 :=\n  rfl\n\n@[simp] theorem singleton_append {\u03b1 : Type u} {x : \u03b1} {l : List \u03b1} : [x] ++ l = x :: l := rfl\n\ntheorem append_ne_nil_of_ne_nil_left {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    s \u2260 [] \u2192 s ++ t \u2260 [] :=\n  sorry\n\ntheorem append_ne_nil_of_ne_nil_right {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    t \u2260 [] \u2192 s ++ t \u2260 [] :=\n  sorry\n\n@[simp] theorem append_eq_nil {\u03b1 : Type u} {p : List \u03b1} {q : List \u03b1} :\n    p ++ q = [] \u2194 p = [] \u2227 q = [] :=\n  sorry\n\n@[simp] theorem nil_eq_append_iff {\u03b1 : Type u} {a : List \u03b1} {b : List \u03b1} :\n    [] = a ++ b \u2194 a = [] \u2227 b = [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl ([] = a ++ b \u2194 a = [] \u2227 b = [])) (propext eq_comm)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a ++ b = [] \u2194 a = [] \u2227 b = [])) (propext append_eq_nil)))\n      (iff.refl (a = [] \u2227 b = [])))\n\ntheorem append_eq_cons_iff {\u03b1 : Type u} {a : List \u03b1} {b : List \u03b1} {c : List \u03b1} {x : \u03b1} :\n    a ++ b = x :: c \u2194 a = [] \u2227 b = x :: c \u2228 \u2203 (a' : List \u03b1), a = x :: a' \u2227 c = a' ++ b :=\n  sorry\n\ntheorem cons_eq_append_iff {\u03b1 : Type u} {a : List \u03b1} {b : List \u03b1} {c : List \u03b1} {x : \u03b1} :\n    x :: c = a ++ b \u2194 a = [] \u2227 b = x :: c \u2228 \u2203 (a' : List \u03b1), a = x :: a' \u2227 c = a' ++ b :=\n  sorry\n\ntheorem append_eq_append_iff {\u03b1 : Type u} {a : List \u03b1} {b : List \u03b1} {c : List \u03b1} {d : List \u03b1} :\n    a ++ b = c ++ d \u2194\n        (\u2203 (a' : List \u03b1), c = a ++ a' \u2227 b = a' ++ d) \u2228 \u2203 (c' : List \u03b1), a = c ++ c' \u2227 d = c' ++ b :=\n  sorry\n\n@[simp] theorem split_at_eq_take_drop {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) :\n    split_at n l = (take n l, drop n l) :=\n  sorry\n\n@[simp] theorem take_append_drop {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : take n l ++ drop n l = l :=\n  sorry\n\n-- TODO(Leo): cleanup proof after arith dec proc\n\ntheorem append_inj {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1} :\n    s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082 \u2192 length s\u2081 = length s\u2082 \u2192 s\u2081 = s\u2082 \u2227 t\u2081 = t\u2082 :=\n  sorry\n\ntheorem append_inj_right {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length s\u2081 = length s\u2082) : t\u2081 = t\u2082 :=\n  and.right (append_inj h hl)\n\ntheorem append_inj_left {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length s\u2081 = length s\u2082) : s\u2081 = s\u2082 :=\n  and.left (append_inj h hl)\n\ntheorem append_inj' {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : s\u2081 = s\u2082 \u2227 t\u2081 = t\u2082 :=\n  sorry\n\ntheorem append_inj_right' {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : t\u2081 = t\u2082 :=\n  and.right (append_inj' h hl)\n\ntheorem append_inj_left' {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s\u2081 ++ t\u2081 = s\u2082 ++ t\u2082) (hl : length t\u2081 = length t\u2082) : s\u2081 = s\u2082 :=\n  and.left (append_inj' h hl)\n\ntheorem append_left_cancel {\u03b1 : Type u} {s : List \u03b1} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1}\n    (h : s ++ t\u2081 = s ++ t\u2082) : t\u2081 = t\u2082 :=\n  append_inj_right h rfl\n\ntheorem append_right_cancel {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} {t : List \u03b1}\n    (h : s\u2081 ++ t = s\u2082 ++ t) : s\u2081 = s\u2082 :=\n  append_inj_left' h rfl\n\ntheorem append_right_injective {\u03b1 : Type u} (s : List \u03b1) :\n    function.injective fun (t : List \u03b1) => s ++ t :=\n  fun (t\u2081 t\u2082 : List \u03b1) => append_left_cancel\n\ntheorem append_right_inj {\u03b1 : Type u} {t\u2081 : List \u03b1} {t\u2082 : List \u03b1} (s : List \u03b1) :\n    s ++ t\u2081 = s ++ t\u2082 \u2194 t\u2081 = t\u2082 :=\n  function.injective.eq_iff (append_right_injective s)\n\ntheorem append_left_injective {\u03b1 : Type u} (t : List \u03b1) :\n    function.injective fun (s : List \u03b1) => s ++ t :=\n  fun (s\u2081 s\u2082 : List \u03b1) => append_right_cancel\n\ntheorem append_left_inj {\u03b1 : Type u} {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} (t : List \u03b1) :\n    s\u2081 ++ t = s\u2082 ++ t \u2194 s\u2081 = s\u2082 :=\n  function.injective.eq_iff (append_left_injective t)\n\ntheorem map_eq_append_split {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {l : List \u03b1} {s\u2081 : List \u03b2}\n    {s\u2082 : List \u03b2} (h : map f l = s\u2081 ++ s\u2082) :\n    \u2203 (l\u2081 : List \u03b1), \u2203 (l\u2082 : List \u03b1), l = l\u2081 ++ l\u2082 \u2227 map f l\u2081 = s\u2081 \u2227 map f l\u2082 = s\u2082 :=\n  sorry\n\n/-! ### repeat -/\n\n@[simp] theorem repeat_succ {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : repeat a (n + 1) = a :: repeat a n := rfl\n\ntheorem eq_of_mem_repeat {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {n : \u2115} : b \u2208 repeat a n \u2192 b = a := sorry\n\ntheorem eq_repeat_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} :\n    (\u2200 (b : \u03b1), b \u2208 l \u2192 b = a) \u2192 l = repeat a (length l) :=\n  sorry\n\ntheorem eq_repeat' {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} :\n    l = repeat a (length l) \u2194 \u2200 (b : \u03b1), b \u2208 l \u2192 b = a :=\n  { mp := fun (h : l = repeat a (length l)) => Eq.symm h \u25b8 fun (b : \u03b1) => eq_of_mem_repeat,\n    mpr := eq_repeat_of_mem }\n\ntheorem eq_repeat {\u03b1 : Type u} {a : \u03b1} {n : \u2115} {l : List \u03b1} :\n    l = repeat a n \u2194 length l = n \u2227 \u2200 (b : \u03b1), b \u2208 l \u2192 b = a :=\n  sorry\n\ntheorem repeat_add {\u03b1 : Type u} (a : \u03b1) (m : \u2115) (n : \u2115) :\n    repeat a (m + n) = repeat a m ++ repeat a n :=\n  sorry\n\ntheorem repeat_subset_singleton {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : repeat a n \u2286 [a] :=\n  fun (b : \u03b1) (h : b \u2208 repeat a n) => iff.mpr mem_singleton (eq_of_mem_repeat h)\n\n@[simp] theorem map_const {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) (b : \u03b2) :\n    map (function.const \u03b1 b) l = repeat b (length l) :=\n  sorry\n\ntheorem eq_of_mem_map_const {\u03b1 : Type u} {\u03b2 : Type v} {b\u2081 : \u03b2} {b\u2082 : \u03b2} {l : List \u03b1}\n    (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) l) : b\u2081 = b\u2082 :=\n  eq_of_mem_repeat\n    (eq.mp (Eq._oldrec (Eq.refl (b\u2081 \u2208 map (function.const \u03b1 b\u2082) l)) (map_const l b\u2082)) h)\n\n@[simp] theorem map_repeat {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (n : \u2115) :\n    map f (repeat a n) = repeat (f a) n :=\n  sorry\n\n@[simp] theorem tail_repeat {\u03b1 : Type u} (a : \u03b1) (n : \u2115) :\n    tail (repeat a n) = repeat a (Nat.pred n) :=\n  nat.cases_on n (Eq.refl (tail (repeat a 0))) fun (n : \u2115) => Eq.refl (tail (repeat a (Nat.succ n)))\n\n@[simp] theorem join_repeat_nil {\u03b1 : Type u} (n : \u2115) : join (repeat [] n) = [] := sorry\n\n/-! ### pure -/\n\n@[simp] theorem mem_pure {\u03b1 : Type u_1} (x : \u03b1) (y : \u03b1) : x \u2208 pure y \u2194 x = y := sorry\n\n/-! ### bind -/\n\n@[simp] theorem bind_eq_bind {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 List \u03b2) (l : List \u03b1) :\n    l >>= f = list.bind l f :=\n  rfl\n\n@[simp] theorem bind_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 List \u03b2) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    list.bind (l\u2081 ++ l\u2082) f = list.bind l\u2081 f ++ list.bind l\u2082 f :=\n  append_bind l\u2081 l\u2082 f\n\n@[simp] theorem bind_singleton {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 List \u03b2) (x : \u03b1) :\n    list.bind [x] f = f x :=\n  append_nil (f x)\n\n/-! ### concat -/\n\ntheorem concat_nil {\u03b1 : Type u} (a : \u03b1) : concat [] a = [a] := rfl\n\ntheorem concat_cons {\u03b1 : Type u} (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    concat (a :: l) b = a :: concat l b :=\n  rfl\n\n@[simp] theorem concat_eq_append {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : concat l a = l ++ [a] := sorry\n\ntheorem init_eq_of_concat_eq {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    concat l\u2081 a = concat l\u2082 a \u2192 l\u2081 = l\u2082 :=\n  sorry\n\ntheorem last_eq_of_concat_eq {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {l : List \u03b1} :\n    concat l a = concat l b \u2192 a = b :=\n  sorry\n\ntheorem concat_ne_nil {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : concat l a \u2260 [] := sorry\n\ntheorem concat_append {\u03b1 : Type u} (a : \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    concat l\u2081 a ++ l\u2082 = l\u2081 ++ a :: l\u2082 :=\n  sorry\n\ntheorem length_concat {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    length (concat l a) = Nat.succ (length l) :=\n  sorry\n\ntheorem append_concat {\u03b1 : Type u} (a : \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l\u2081 ++ concat l\u2082 a = concat (l\u2081 ++ l\u2082) a :=\n  sorry\n\n/-! ### reverse -/\n\n@[simp] theorem reverse_nil {\u03b1 : Type u} : reverse [] = [] := rfl\n\n@[simp] theorem reverse_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    reverse (a :: l) = reverse l ++ [a] :=\n  sorry\n\ntheorem reverse_core_eq {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    reverse_core l\u2081 l\u2082 = reverse l\u2081 ++ l\u2082 :=\n  sorry\n\ntheorem reverse_cons' {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : reverse (a :: l) = concat (reverse l) a :=\n  sorry\n\n@[simp] theorem reverse_singleton {\u03b1 : Type u} (a : \u03b1) : reverse [a] = [a] := rfl\n\n@[simp] theorem reverse_append {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    reverse (s ++ t) = reverse t ++ reverse s :=\n  sorry\n\ntheorem reverse_concat {\u03b1 : Type u} (l : List \u03b1) (a : \u03b1) : reverse (concat l a) = a :: reverse l :=\n  sorry\n\n@[simp] theorem reverse_reverse {\u03b1 : Type u} (l : List \u03b1) : reverse (reverse l) = l := sorry\n\n@[simp] theorem reverse_involutive {\u03b1 : Type u} : function.involutive reverse :=\n  fun (l : List \u03b1) => reverse_reverse l\n\n@[simp] theorem reverse_injective {\u03b1 : Type u} : function.injective reverse :=\n  function.involutive.injective reverse_involutive\n\n@[simp] theorem reverse_inj {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    reverse l\u2081 = reverse l\u2082 \u2194 l\u2081 = l\u2082 :=\n  function.injective.eq_iff reverse_injective\n\n@[simp] theorem reverse_eq_nil {\u03b1 : Type u} {l : List \u03b1} : reverse l = [] \u2194 l = [] := reverse_inj\n\ntheorem concat_eq_reverse_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    concat l a = reverse (a :: reverse l) :=\n  sorry\n\n@[simp] theorem length_reverse {\u03b1 : Type u} (l : List \u03b1) : length (reverse l) = length l := sorry\n\n@[simp] theorem map_reverse {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    map f (reverse l) = reverse (map f l) :=\n  sorry\n\ntheorem map_reverse_core {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    map f (reverse_core l\u2081 l\u2082) = reverse_core (map f l\u2081) (map f l\u2082) :=\n  sorry\n\n@[simp] theorem mem_reverse {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} : a \u2208 reverse l \u2194 a \u2208 l := sorry\n\n@[simp] theorem reverse_repeat {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : reverse (repeat a n) = repeat a n :=\n  sorry\n\n/-! ### is_nil -/\n\ntheorem is_nil_iff_eq_nil {\u03b1 : Type u} {l : List \u03b1} : \u21a5(is_nil l) \u2194 l = [] := sorry\n\n/-! ### init -/\n\n@[simp] theorem length_init {\u03b1 : Type u} (l : List \u03b1) : length (init l) = length l - 1 := sorry\n\n/-! ### last -/\n\n@[simp] theorem last_cons {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h\u2081 : a :: l \u2260 []) (h\u2082 : l \u2260 []) :\n    last (a :: l) h\u2081 = last l h\u2082 :=\n  sorry\n\n@[simp] theorem last_append {\u03b1 : Type u} {a : \u03b1} (l : List \u03b1) (h : l ++ [a] \u2260 []) :\n    last (l ++ [a]) h = a :=\n  sorry\n\ntheorem last_concat {\u03b1 : Type u} {a : \u03b1} (l : List \u03b1) (h : concat l a \u2260 []) :\n    last (concat l a) h = a :=\n  sorry\n\n@[simp] theorem last_singleton {\u03b1 : Type u} (a : \u03b1) (h : [a] \u2260 []) : last [a] h = a := rfl\n\n@[simp] theorem last_cons_cons {\u03b1 : Type u} (a\u2081 : \u03b1) (a\u2082 : \u03b1) (l : List \u03b1)\n    (h : a\u2081 :: a\u2082 :: l \u2260 []) : last (a\u2081 :: a\u2082 :: l) h = last (a\u2082 :: l) (cons_ne_nil a\u2082 l) :=\n  rfl\n\ntheorem init_append_last {\u03b1 : Type u} {l : List \u03b1} (h : l \u2260 []) : init l ++ [last l h] = l := sorry\n\ntheorem last_congr {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (h\u2081 : l\u2081 \u2260 []) (h\u2082 : l\u2082 \u2260 [])\n    (h\u2083 : l\u2081 = l\u2082) : last l\u2081 h\u2081 = last l\u2082 h\u2082 :=\n  Eq._oldrec (fun (h\u2081 : l\u2082 \u2260 []) => Eq.refl (last l\u2082 h\u2081)) (Eq.symm h\u2083) h\u2081\n\ntheorem last_mem {\u03b1 : Type u} {l : List \u03b1} (h : l \u2260 []) : last l h \u2208 l := sorry\n\ntheorem last_repeat_succ (a : \u2115) (m : \u2115) :\n    last (repeat a (Nat.succ m))\n          (ne_nil_of_length_eq_succ\n            ((fun (this : length (repeat a (Nat.succ m)) = Nat.succ m) => this)\n              (eq.mpr\n                (id\n                  (Eq._oldrec (Eq.refl (length (repeat a (Nat.succ m)) = Nat.succ m))\n                    (length_repeat a (Nat.succ m))))\n                (Eq.refl (Nat.succ m))))) =\n        a :=\n  sorry\n\n/-! ### last' -/\n\n@[simp] theorem last'_is_none {\u03b1 : Type u} {l : List \u03b1} : \u21a5(option.is_none (last' l)) \u2194 l = [] :=\n  sorry\n\n@[simp] theorem last'_is_some {\u03b1 : Type u} {l : List \u03b1} : \u21a5(option.is_some (last' l)) \u2194 l \u2260 [] :=\n  sorry\n\ntheorem mem_last'_eq_last {\u03b1 : Type u} {l : List \u03b1} {x : \u03b1} :\n    x \u2208 last' l \u2192 \u2203 (h : l \u2260 []), x = last l h :=\n  sorry\n\ntheorem mem_of_mem_last' {\u03b1 : Type u} {l : List \u03b1} {a : \u03b1} (ha : a \u2208 last' l) : a \u2208 l := sorry\n\ntheorem init_append_last' {\u03b1 : Type u} {l : List \u03b1} (a : \u03b1) (H : a \u2208 last' l) : init l ++ [a] = l :=\n  sorry\n\ntheorem ilast_eq_last' {\u03b1 : Type u} [Inhabited \u03b1] (l : List \u03b1) : ilast l = option.iget (last' l) :=\n  sorry\n\n@[simp] theorem last'_append_cons {\u03b1 : Type u} (l\u2081 : List \u03b1) (a : \u03b1) (l\u2082 : List \u03b1) :\n    last' (l\u2081 ++ a :: l\u2082) = last' (a :: l\u2082) :=\n  sorry\n\ntheorem last'_append_of_ne_nil {\u03b1 : Type u} (l\u2081 : List \u03b1) {l\u2082 : List \u03b1} (hl\u2082 : l\u2082 \u2260 []) :\n    last' (l\u2081 ++ l\u2082) = last' l\u2082 :=\n  sorry\n\n/-! ### head(') and tail -/\n\ntheorem head_eq_head' {\u03b1 : Type u} [Inhabited \u03b1] (l : List \u03b1) : head l = option.iget (head' l) :=\n  list.cases_on l (Eq.refl (head []))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (head (l_hd :: l_tl))\n\ntheorem mem_of_mem_head' {\u03b1 : Type u} {x : \u03b1} {l : List \u03b1} : x \u2208 head' l \u2192 x \u2208 l := sorry\n\n@[simp] theorem head_cons {\u03b1 : Type u} [Inhabited \u03b1] (a : \u03b1) (l : List \u03b1) : head (a :: l) = a := rfl\n\n@[simp] theorem tail_nil {\u03b1 : Type u} : tail [] = [] := rfl\n\n@[simp] theorem tail_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : tail (a :: l) = l := rfl\n\n@[simp] theorem head_append {\u03b1 : Type u} [Inhabited \u03b1] (t : List \u03b1) {s : List \u03b1} (h : s \u2260 []) :\n    head (s ++ t) = head s :=\n  sorry\n\ntheorem tail_append_singleton_of_ne_nil {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h : l \u2260 []) :\n    tail (l ++ [a]) = tail l ++ [a] :=\n  sorry\n\ntheorem cons_head'_tail {\u03b1 : Type u} {l : List \u03b1} {a : \u03b1} (h : a \u2208 head' l) : a :: tail l = l :=\n  sorry\n\ntheorem head_mem_head' {\u03b1 : Type u} [Inhabited \u03b1] {l : List \u03b1} (h : l \u2260 []) : head l \u2208 head' l :=\n  list.cases_on l (fun (h : [] \u2260 []) => idRhs (head [] \u2208 head' []) (absurd (Eq.refl []) h))\n    (fun (l_hd : \u03b1) (l_tl : List \u03b1) (h : l_hd :: l_tl \u2260 []) =>\n      idRhs (head' (l_hd :: l_tl) = head' (l_hd :: l_tl)) rfl)\n    h\n\ntheorem cons_head_tail {\u03b1 : Type u} [Inhabited \u03b1] {l : List \u03b1} (h : l \u2260 []) :\n    head l :: tail l = l :=\n  cons_head'_tail (head_mem_head' h)\n\n@[simp] theorem head'_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    head' (map f l) = option.map f (head' l) :=\n  list.cases_on l (Eq.refl (head' (map f [])))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (head' (map f (l_hd :: l_tl)))\n\n/-! ### Induction from the right -/\n\n/-- Induction principle from the right for lists: if a property holds for the empty list, and\nfor `l ++ [a]` if it holds for `l`, then it holds for all lists. The principle is given for\na `Sort`-valued predicate, i.e., it can also be used to construct data. -/\ndef reverse_rec_on {\u03b1 : Type u} {C : List \u03b1 \u2192 Sort u_1} (l : List \u03b1) (H0 : C [])\n    (H1 : (l : List \u03b1) \u2192 (a : \u03b1) \u2192 C l \u2192 C (l ++ [a])) : C l :=\n  eq.mpr sorry\n    (List.rec H0\n      (fun (hd : \u03b1) (tl : List \u03b1) (ih : C (reverse tl)) => eq.mpr sorry (H1 (reverse tl) hd ih))\n      (reverse l))\n\n/-- Bidirectional induction principle for lists: if a property holds for the empty list, the\nsingleton list, and `a :: (l ++ [b])` from `l`, then it holds for all lists. This can be used to\nprove statements about palindromes. The principle is given for a `Sort`-valued predicate, i.e., it\ncan also be used to construct data. -/\ndef bidirectional_rec {\u03b1 : Type u} {C : List \u03b1 \u2192 Sort u_1} (H0 : C []) (H1 : (a : \u03b1) \u2192 C [a])\n    (Hn : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 (b : \u03b1) \u2192 C l \u2192 C (a :: (l ++ [b]))) (l : List \u03b1) : C l :=\n  sorry\n\n/-- Like `bidirectional_rec`, but with the list parameter placed first. -/\ndef bidirectional_rec_on {\u03b1 : Type u} {C : List \u03b1 \u2192 Sort u_1} (l : List \u03b1) (H0 : C [])\n    (H1 : (a : \u03b1) \u2192 C [a]) (Hn : (a : \u03b1) \u2192 (l : List \u03b1) \u2192 (b : \u03b1) \u2192 C l \u2192 C (a :: (l ++ [b]))) :\n    C l :=\n  bidirectional_rec H0 H1 Hn l\n\n/-! ### sublists -/\n\n@[simp] theorem nil_sublist {\u03b1 : Type u} (l : List \u03b1) : [] <+ l := sorry\n\n@[simp] theorem sublist.refl {\u03b1 : Type u} (l : List \u03b1) : l <+ l := sorry\n\ntheorem sublist.trans {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} (h\u2081 : l\u2081 <+ l\u2082)\n    (h\u2082 : l\u2082 <+ l\u2083) : l\u2081 <+ l\u2083 :=\n  sorry\n\n@[simp] theorem sublist_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : l <+ a :: l :=\n  sublist.cons l l a (sublist.refl l)\n\ntheorem sublist_of_cons_sublist {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    a :: l\u2081 <+ l\u2082 \u2192 l\u2081 <+ l\u2082 :=\n  sublist.trans (sublist_cons a l\u2081)\n\ntheorem cons_sublist_cons {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (a : \u03b1) (s : l\u2081 <+ l\u2082) :\n    a :: l\u2081 <+ a :: l\u2082 :=\n  sublist.cons2 l\u2081 l\u2082 a s\n\n@[simp] theorem sublist_append_left {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2081 <+ l\u2081 ++ l\u2082 :=\n  sorry\n\n@[simp] theorem sublist_append_right {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2082 <+ l\u2081 ++ l\u2082 :=\n  sorry\n\ntheorem sublist_cons_of_sublist {\u03b1 : Type u} (a : \u03b1) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+ l\u2082 \u2192 l\u2081 <+ a :: l\u2082 :=\n  sublist.cons l\u2081 l\u2082 a\n\ntheorem sublist_append_of_sublist_left {\u03b1 : Type u} {l : List \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (s : l <+ l\u2081) : l <+ l\u2081 ++ l\u2082 :=\n  sublist.trans s (sublist_append_left l\u2081 l\u2082)\n\ntheorem sublist_append_of_sublist_right {\u03b1 : Type u} {l : List \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (s : l <+ l\u2082) : l <+ l\u2081 ++ l\u2082 :=\n  sublist.trans s (sublist_append_right l\u2081 l\u2082)\n\ntheorem sublist_of_cons_sublist_cons {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a :: l\u2081 <+ a :: l\u2082 \u2192 l\u2081 <+ l\u2082 :=\n  sorry\n\ntheorem cons_sublist_cons_iff {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a :: l\u2081 <+ a :: l\u2082 \u2194 l\u2081 <+ l\u2082 :=\n  { mp := sublist_of_cons_sublist_cons, mpr := cons_sublist_cons a }\n\n@[simp] theorem append_sublist_append_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (l : List \u03b1) :\n    l ++ l\u2081 <+ l ++ l\u2082 \u2194 l\u2081 <+ l\u2082 :=\n  sorry\n\ntheorem sublist.append_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (h : l\u2081 <+ l\u2082) (l : List \u03b1) :\n    l\u2081 ++ l <+ l\u2082 ++ l :=\n  sublist.drec (sublist.refl ([] ++ l))\n    (fun (h_l\u2081 h_l\u2082 : List \u03b1) (a : \u03b1) (h_\u1fb0 : h_l\u2081 <+ h_l\u2082) (ih : h_l\u2081 ++ l <+ h_l\u2082 ++ l) =>\n      sublist_cons_of_sublist a ih)\n    (fun (h_l\u2081 h_l\u2082 : List \u03b1) (a : \u03b1) (h_\u1fb0 : h_l\u2081 <+ h_l\u2082) (ih : h_l\u2081 ++ l <+ h_l\u2082 ++ l) =>\n      cons_sublist_cons a ih)\n    h\n\ntheorem sublist_or_mem_of_sublist {\u03b1 : Type u} {l : List \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1}\n    (h : l <+ l\u2081 ++ a :: l\u2082) : l <+ l\u2081 ++ l\u2082 \u2228 a \u2208 l :=\n  sorry\n\ntheorem sublist.reverse {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (h : l\u2081 <+ l\u2082) :\n    reverse l\u2081 <+ reverse l\u2082 :=\n  sorry\n\n@[simp] theorem reverse_sublist_iff {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    reverse l\u2081 <+ reverse l\u2082 \u2194 l\u2081 <+ l\u2082 :=\n  { mp :=\n      fun (h : reverse l\u2081 <+ reverse l\u2082) =>\n        reverse_reverse l\u2081 \u25b8 reverse_reverse l\u2082 \u25b8 sublist.reverse h,\n    mpr := sublist.reverse }\n\n@[simp] theorem append_sublist_append_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (l : List \u03b1) :\n    l\u2081 ++ l <+ l\u2082 ++ l \u2194 l\u2081 <+ l\u2082 :=\n  sorry\n\ntheorem sublist.append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {r\u2081 : List \u03b1} {r\u2082 : List \u03b1}\n    (hl : l\u2081 <+ l\u2082) (hr : r\u2081 <+ r\u2082) : l\u2081 ++ r\u2081 <+ l\u2082 ++ r\u2082 :=\n  sublist.trans (sublist.append_right hl r\u2081) (iff.mpr (append_sublist_append_left l\u2082) hr)\n\ntheorem sublist.subset {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <+ l\u2082 \u2192 l\u2081 \u2286 l\u2082 := sorry\n\ntheorem singleton_sublist {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} : [a] <+ l \u2194 a \u2208 l := sorry\n\ntheorem eq_nil_of_sublist_nil {\u03b1 : Type u} {l : List \u03b1} (s : l <+ []) : l = [] :=\n  eq_nil_of_subset_nil (sublist.subset s)\n\ntheorem repeat_sublist_repeat {\u03b1 : Type u} (a : \u03b1) {m : \u2115} {n : \u2115} :\n    repeat a m <+ repeat a n \u2194 m \u2264 n :=\n  sorry\n\ntheorem eq_of_sublist_of_length_eq {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+ l\u2082 \u2192 length l\u2081 = length l\u2082 \u2192 l\u2081 = l\u2082 :=\n  sorry\n\ntheorem eq_of_sublist_of_length_le {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s : l\u2081 <+ l\u2082)\n    (h : length l\u2082 \u2264 length l\u2081) : l\u2081 = l\u2082 :=\n  eq_of_sublist_of_length_eq s (le_antisymm (length_le_of_sublist s) h)\n\ntheorem sublist.antisymm {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s\u2081 : l\u2081 <+ l\u2082) (s\u2082 : l\u2082 <+ l\u2081) :\n    l\u2081 = l\u2082 :=\n  eq_of_sublist_of_length_le s\u2081 (length_le_of_sublist s\u2082)\n\nprotected instance decidable_sublist {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    Decidable (l\u2081 <+ l\u2082) :=\n  sorry\n\n/-! ### index_of -/\n\n@[simp] theorem index_of_nil {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : index_of a [] = 0 := rfl\n\ntheorem index_of_cons {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    index_of a (b :: l) = ite (a = b) 0 (Nat.succ (index_of a l)) :=\n  rfl\n\ntheorem index_of_cons_eq {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (l : List \u03b1) :\n    a = b \u2192 index_of a (b :: l) = 0 :=\n  fun (e : a = b) => if_pos e\n\n@[simp] theorem index_of_cons_self {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    index_of a (a :: l) = 0 :=\n  index_of_cons_eq l rfl\n\n@[simp] theorem index_of_cons_ne {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (l : List \u03b1) :\n    a \u2260 b \u2192 index_of a (b :: l) = Nat.succ (index_of a l) :=\n  fun (n : a \u2260 b) => if_neg n\n\ntheorem index_of_eq_length {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} :\n    index_of a l = length l \u2194 \u00aca \u2208 l :=\n  sorry\n\n@[simp] theorem index_of_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {l : List \u03b1} {a : \u03b1} :\n    \u00aca \u2208 l \u2192 index_of a l = length l :=\n  iff.mpr index_of_eq_length\n\ntheorem index_of_le_length {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} :\n    index_of a l \u2264 length l :=\n  sorry\n\ntheorem index_of_lt_length {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} :\n    index_of a l < length l \u2194 a \u2208 l :=\n  sorry\n\n/-! ### nth element -/\n\ntheorem nth_le_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} :\n    a \u2208 l \u2192 \u2203 (n : \u2115), \u2203 (h : n < length l), nth_le l n h = a :=\n  sorry\n\ntheorem nth_le_nth {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} (h : n < length l) :\n    nth l n = some (nth_le l n h) :=\n  sorry\n\ntheorem nth_len_le {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} : length l \u2264 n \u2192 nth l n = none := sorry\n\ntheorem nth_eq_some {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} {a : \u03b1} :\n    nth l n = some a \u2194 \u2203 (h : n < length l), nth_le l n h = a :=\n  sorry\n\n@[simp] theorem nth_eq_none_iff {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} : nth l n = none \u2194 length l \u2264 n :=\n  sorry\n\ntheorem nth_of_mem {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) : \u2203 (n : \u2115), nth l n = some a :=\n  sorry\n\ntheorem nth_le_mem {\u03b1 : Type u} (l : List \u03b1) (n : \u2115) (h : n < length l) : nth_le l n h \u2208 l := sorry\n\ntheorem nth_mem {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} {a : \u03b1} (e : nth l n = some a) : a \u2208 l := sorry\n\ntheorem mem_iff_nth_le {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} :\n    a \u2208 l \u2194 \u2203 (n : \u2115), \u2203 (h : n < length l), nth_le l n h = a :=\n  sorry\n\ntheorem mem_iff_nth {\u03b1 : Type u} {a : \u03b1} {l : List \u03b1} : a \u2208 l \u2194 \u2203 (n : \u2115), nth l n = some a :=\n  iff.trans mem_iff_nth_le (exists_congr fun (n : \u2115) => iff.symm nth_eq_some)\n\ntheorem nth_zero {\u03b1 : Type u} (l : List \u03b1) : nth l 0 = head' l :=\n  list.cases_on l (Eq.refl (nth [] 0))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (nth (l_hd :: l_tl) 0)\n\ntheorem nth_injective {\u03b1 : Type u} {xs : List \u03b1} {i : \u2115} {j : \u2115} (h\u2080 : i < length xs)\n    (h\u2081 : nodup xs) (h\u2082 : nth xs i = nth xs j) : i = j :=\n  sorry\n\n@[simp] theorem nth_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) (n : \u2115) :\n    nth (map f l) n = option.map f (nth l n) :=\n  sorry\n\ntheorem nth_le_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} {n : \u2115}\n    (H1 : n < length (map f l)) (H2 : n < length l) : nth_le (map f l) n H1 = f (nth_le l n H2) :=\n  sorry\n\n/-- A version of `nth_le_map` that can be used for rewriting. -/\ntheorem nth_le_map_rev {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} {n : \u2115}\n    (H : n < length l) : f (nth_le l n H) = nth_le (map f l) n (Eq.symm (length_map f l) \u25b8 H) :=\n  Eq.symm (nth_le_map f (Eq.symm (length_map f l) \u25b8 H) H)\n\n@[simp] theorem nth_le_map' {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} {n : \u2115}\n    (H : n < length (map f l)) : nth_le (map f l) n H = f (nth_le l n (length_map f l \u25b8 H)) :=\n  nth_le_map f H (length_map f l \u25b8 H)\n\n/-- If one has `nth_le L i hi` in a formula and `h : L = L'`, one can not `rw h` in the formula as\n`hi` gives `i < L.length` and not `i < L'.length`. The lemma `nth_le_of_eq` can be used to make\nsuch a rewrite, with `rw (nth_le_of_eq h)`. -/\ntheorem nth_le_of_eq {\u03b1 : Type u} {L : List \u03b1} {L' : List \u03b1} (h : L = L') {i : \u2115}\n    (hi : i < length L) : nth_le L i hi = nth_le L' i (h \u25b8 hi) :=\n  sorry\n\n@[simp] theorem nth_le_singleton {\u03b1 : Type u} (a : \u03b1) {n : \u2115} (hn : n < 1) : nth_le [a] n hn = a :=\n  (fun (hn0 : n = 0) => Eq._oldrec (fun (hn : 0 < 1) => Eq.refl (nth_le [a] 0 hn)) (Eq.symm hn0) hn)\n    (iff.mp nat.le_zero_iff (nat.le_of_lt_succ hn))\n\ntheorem nth_le_zero {\u03b1 : Type u} [Inhabited \u03b1] {L : List \u03b1} (h : 0 < length L) :\n    nth_le L 0 h = head L :=\n  sorry\n\ntheorem nth_le_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (hn\u2081 : n < length (l\u2081 ++ l\u2082))\n    (hn\u2082 : n < length l\u2081) : nth_le (l\u2081 ++ l\u2082) n hn\u2081 = nth_le l\u2081 n hn\u2082 :=\n  sorry\n\ntheorem nth_le_append_right_aux {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115}\n    (h\u2081 : length l\u2081 \u2264 n) (h\u2082 : n < length (l\u2081 ++ l\u2082)) : n - length l\u2081 < length l\u2082 :=\n  sorry\n\ntheorem nth_le_append_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (h\u2081 : length l\u2081 \u2264 n)\n    (h\u2082 : n < length (l\u2081 ++ l\u2082)) :\n    nth_le (l\u2081 ++ l\u2082) n h\u2082 = nth_le l\u2082 (n - length l\u2081) (nth_le_append_right_aux h\u2081 h\u2082) :=\n  sorry\n\n@[simp] theorem nth_le_repeat {\u03b1 : Type u} (a : \u03b1) {n : \u2115} {m : \u2115} (h : m < length (repeat a n)) :\n    nth_le (repeat a n) m h = a :=\n  eq_of_mem_repeat (nth_le_mem (repeat a n) m h)\n\ntheorem nth_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (hn : n < length l\u2081) :\n    nth (l\u2081 ++ l\u2082) n = nth l\u2081 n :=\n  sorry\n\ntheorem nth_append_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (hn : length l\u2081 \u2264 n) :\n    nth (l\u2081 ++ l\u2082) n = nth l\u2082 (n - length l\u2081) :=\n  sorry\n\ntheorem last_eq_nth_le {\u03b1 : Type u} (l : List \u03b1) (h : l \u2260 []) :\n    last l h = nth_le l (length l - 1) (nat.sub_lt (length_pos_of_ne_nil h) nat.one_pos) :=\n  sorry\n\n@[simp] theorem nth_concat_length {\u03b1 : Type u} (l : List \u03b1) (a : \u03b1) :\n    nth (l ++ [a]) (length l) = some a :=\n  sorry\n\ntheorem ext {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : (\u2200 (n : \u2115), nth l\u2081 n = nth l\u2082 n) \u2192 l\u2081 = l\u2082 :=\n  sorry\n\ntheorem ext_le {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (hl : length l\u2081 = length l\u2082)\n    (h : \u2200 (n : \u2115) (h\u2081 : n < length l\u2081) (h\u2082 : n < length l\u2082), nth_le l\u2081 n h\u2081 = nth_le l\u2082 n h\u2082) :\n    l\u2081 = l\u2082 :=\n  sorry\n\n@[simp] theorem index_of_nth_le {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1}\n    (h : index_of a l < length l) : nth_le l (index_of a l) h = a :=\n  sorry\n\n@[simp] theorem index_of_nth {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    nth l (index_of a l) = some a :=\n  sorry\n\ntheorem nth_le_reverse_aux1 {\u03b1 : Type u} (l : List \u03b1) (r : List \u03b1) (i : \u2115)\n    (h1 : i + length l < length (reverse_core l r)) (h2 : i < length r) :\n    nth_le (reverse_core l r) (i + length l) h1 = nth_le r i h2 :=\n  sorry\n\ntheorem index_of_inj {\u03b1 : Type u} [DecidableEq \u03b1] {l : List \u03b1} {x : \u03b1} {y : \u03b1} (hx : x \u2208 l)\n    (hy : y \u2208 l) : index_of x l = index_of y l \u2194 x = y :=\n  sorry\n\ntheorem nth_le_reverse_aux2 {\u03b1 : Type u} (l : List \u03b1) (r : List \u03b1) (i : \u2115)\n    (h1 : length l - 1 - i < length (reverse_core l r)) (h2 : i < length l) :\n    nth_le (reverse_core l r) (length l - 1 - i) h1 = nth_le l i h2 :=\n  sorry\n\n@[simp] theorem nth_le_reverse {\u03b1 : Type u} (l : List \u03b1) (i : \u2115)\n    (h1 : length l - 1 - i < length (reverse l)) (h2 : i < length l) :\n    nth_le (reverse l) (length l - 1 - i) h1 = nth_le l i h2 :=\n  nth_le_reverse_aux2 l [] i h1 h2\n\ntheorem eq_cons_of_length_one {\u03b1 : Type u} {l : List \u03b1} (h : length l = 1) :\n    l = [nth_le l 0 (Eq.symm h \u25b8 zero_lt_one)] :=\n  sorry\n\ntheorem modify_nth_tail_modify_nth_tail {\u03b1 : Type u} {f : List \u03b1 \u2192 List \u03b1} {g : List \u03b1 \u2192 List \u03b1}\n    (m : \u2115) (n : \u2115) (l : List \u03b1) :\n    modify_nth_tail g (m + n) (modify_nth_tail f n l) =\n        modify_nth_tail (fun (l : List \u03b1) => modify_nth_tail g m (f l)) n l :=\n  sorry\n\ntheorem modify_nth_tail_modify_nth_tail_le {\u03b1 : Type u} {f : List \u03b1 \u2192 List \u03b1} {g : List \u03b1 \u2192 List \u03b1}\n    (m : \u2115) (n : \u2115) (l : List \u03b1) (h : n \u2264 m) :\n    modify_nth_tail g m (modify_nth_tail f n l) =\n        modify_nth_tail (fun (l : List \u03b1) => modify_nth_tail g (m - n) (f l)) n l :=\n  sorry\n\ntheorem modify_nth_tail_modify_nth_tail_same {\u03b1 : Type u} {f : List \u03b1 \u2192 List \u03b1}\n    {g : List \u03b1 \u2192 List \u03b1} (n : \u2115) (l : List \u03b1) :\n    modify_nth_tail g n (modify_nth_tail f n l) = modify_nth_tail (g \u2218 f) n l :=\n  sorry\n\ntheorem modify_nth_tail_id {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : modify_nth_tail id n l = l := sorry\n\ntheorem remove_nth_eq_nth_tail {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) :\n    remove_nth l n = modify_nth_tail tail n l :=\n  sorry\n\ntheorem update_nth_eq_modify_nth {\u03b1 : Type u} (a : \u03b1) (n : \u2115) (l : List \u03b1) :\n    update_nth l n a = modify_nth (fun (_x : \u03b1) => a) n l :=\n  sorry\n\ntheorem modify_nth_eq_update_nth {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (l : List \u03b1) :\n    modify_nth f n l = option.get_or_else ((fun (a : \u03b1) => update_nth l n (f a)) <$> nth l n) l :=\n  sorry\n\ntheorem nth_modify_nth {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (l : List \u03b1) (m : \u2115) :\n    nth (modify_nth f n l) m = (fun (a : \u03b1) => ite (n = m) (f a) a) <$> nth l m :=\n  sorry\n\ntheorem modify_nth_tail_length {\u03b1 : Type u} (f : List \u03b1 \u2192 List \u03b1)\n    (H : \u2200 (l : List \u03b1), length (f l) = length l) (n : \u2115) (l : List \u03b1) :\n    length (modify_nth_tail f n l) = length l :=\n  sorry\n\n@[simp] theorem modify_nth_length {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (l : List \u03b1) :\n    length (modify_nth f n l) = length l :=\n  sorry\n\n@[simp] theorem update_nth_length {\u03b1 : Type u} (l : List \u03b1) (n : \u2115) (a : \u03b1) :\n    length (update_nth l n a) = length l :=\n  sorry\n\n@[simp] theorem nth_modify_nth_eq {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (l : List \u03b1) :\n    nth (modify_nth f n l) n = f <$> nth l n :=\n  sorry\n\n@[simp] theorem nth_modify_nth_ne {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) {m : \u2115} {n : \u2115} (l : List \u03b1)\n    (h : m \u2260 n) : nth (modify_nth f m l) n = nth l n :=\n  sorry\n\ntheorem nth_update_nth_eq {\u03b1 : Type u} (a : \u03b1) (n : \u2115) (l : List \u03b1) :\n    nth (update_nth l n a) n = (fun (_x : \u03b1) => a) <$> nth l n :=\n  sorry\n\ntheorem nth_update_nth_of_lt {\u03b1 : Type u} (a : \u03b1) {n : \u2115} {l : List \u03b1} (h : n < length l) :\n    nth (update_nth l n a) n = some a :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (nth (update_nth l n a) n = some a)) (nth_update_nth_eq a n l)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl ((fun (_x : \u03b1) => a) <$> nth l n = some a)) (nth_le_nth h)))\n      (Eq.refl ((fun (_x : \u03b1) => a) <$> some (nth_le l n h))))\n\ntheorem nth_update_nth_ne {\u03b1 : Type u} (a : \u03b1) {m : \u2115} {n : \u2115} (l : List \u03b1) (h : m \u2260 n) :\n    nth (update_nth l m a) n = nth l n :=\n  sorry\n\n@[simp] theorem nth_le_update_nth_eq {\u03b1 : Type u} (l : List \u03b1) (i : \u2115) (a : \u03b1)\n    (h : i < length (update_nth l i a)) : nth_le (update_nth l i a) i h = a :=\n  sorry\n\n@[simp] theorem nth_le_update_nth_of_ne {\u03b1 : Type u} {l : List \u03b1} {i : \u2115} {j : \u2115} (h : i \u2260 j)\n    (a : \u03b1) (hj : j < length (update_nth l i a)) :\n    nth_le (update_nth l i a) j hj =\n        nth_le l j\n          (eq.mpr (id (Eq.refl (j < length l)))\n            (eq.mp\n              ((fun (\u1fb0 \u1fb0_1 : \u2115) (e_2 : \u1fb0 = \u1fb0_1) (\u1fb0_2 \u1fb0_3 : \u2115) (e_3 : \u1fb0_2 = \u1fb0_3) =>\n                  congr (congr_arg Less e_2) e_3)\n                j j (Eq.refl j) (length (update_nth l i a)) (length l) (update_nth_length l i a))\n              hj)) :=\n  sorry\n\ntheorem mem_or_eq_of_mem_update_nth {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} {a : \u03b1} {b : \u03b1}\n    (h : a \u2208 update_nth l n b) : a \u2208 l \u2228 a = b :=\n  sorry\n\n@[simp] theorem insert_nth_nil {\u03b1 : Type u} (a : \u03b1) : insert_nth 0 a [] = [a] := rfl\n\n@[simp] theorem insert_nth_succ_nil {\u03b1 : Type u} (n : \u2115) (a : \u03b1) : insert_nth (n + 1) a [] = [] :=\n  rfl\n\ntheorem length_insert_nth {\u03b1 : Type u} {a : \u03b1} (n : \u2115) (as : List \u03b1) :\n    n \u2264 length as \u2192 length (insert_nth n a as) = length as + 1 :=\n  sorry\n\ntheorem remove_nth_insert_nth {\u03b1 : Type u} {a : \u03b1} (n : \u2115) (l : List \u03b1) :\n    remove_nth (insert_nth n a l) n = l :=\n  sorry\n\ntheorem insert_nth_remove_nth_of_ge {\u03b1 : Type u} {a : \u03b1} (n : \u2115) (m : \u2115) (as : List \u03b1) :\n    n < length as \u2192\n        n \u2264 m \u2192 insert_nth m a (remove_nth as n) = remove_nth (insert_nth (m + 1) a as) n :=\n  sorry\n\ntheorem insert_nth_remove_nth_of_le {\u03b1 : Type u} {a : \u03b1} (n : \u2115) (m : \u2115) (as : List \u03b1) :\n    n < length as \u2192\n        m \u2264 n \u2192 insert_nth m a (remove_nth as n) = remove_nth (insert_nth m a as) (n + 1) :=\n  sorry\n\ntheorem insert_nth_comm {\u03b1 : Type u} (a : \u03b1) (b : \u03b1) (i : \u2115) (j : \u2115) (l : List \u03b1) (h : i \u2264 j)\n    (hj : j \u2264 length l) :\n    insert_nth (j + 1) b (insert_nth i a l) = insert_nth i a (insert_nth j b l) :=\n  sorry\n\ntheorem mem_insert_nth {\u03b1 : Type u} {a : \u03b1} {b : \u03b1} {n : \u2115} {l : List \u03b1} (hi : n \u2264 length l) :\n    a \u2208 insert_nth n b l \u2194 a = b \u2228 a \u2208 l :=\n  sorry\n\n/-! ### map -/\n\n@[simp] theorem map_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : map f [] = [] := rfl\n\ntheorem map_eq_foldr {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    map f l = foldr (fun (a : \u03b1) (bs : List \u03b2) => f a :: bs) [] l :=\n  sorry\n\ntheorem map_congr {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {l : List \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 l \u2192 f x = g x) \u2192 map f l = map g l :=\n  sorry\n\ntheorem map_eq_map_iff {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {l : List \u03b1} :\n    map f l = map g l \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 f x = g x :=\n  sorry\n\ntheorem map_concat {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (l : List \u03b1) :\n    map f (concat l a) = concat (map f l) (f a) :=\n  sorry\n\ntheorem map_id' {\u03b1 : Type u} {f : \u03b1 \u2192 \u03b1} (h : \u2200 (x : \u03b1), f x = x) (l : List \u03b1) : map f l = l :=\n  sorry\n\ntheorem eq_nil_of_map_eq_nil {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {l : List \u03b1} (h : map f l = []) :\n    l = [] :=\n  eq_nil_of_length_eq_zero\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length l = 0)) (Eq.symm (length_map f l))))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length (map f l) = 0)) h)) (Eq.refl (length []))))\n\n@[simp] theorem map_join {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (L : List (List \u03b1)) :\n    map f (join L) = join (map (map f) L) :=\n  sorry\n\ntheorem bind_ret_eq_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    list.bind l (list.ret \u2218 f) = map f l :=\n  sorry\n\n@[simp] theorem map_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    f <$> l = map f l :=\n  rfl\n\n@[simp] theorem map_tail {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    map f (tail l) = tail (map f l) :=\n  list.cases_on l (Eq.refl (map f (tail [])))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (map f (tail (l_hd :: l_tl)))\n\n@[simp] theorem map_injective_iff {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} :\n    function.injective (map f) \u2194 function.injective f :=\n  sorry\n\n/--\nA single `list.map` of a composition of functions is equal to\ncomposing a `list.map` with another `list.map`, fully applied.\nThis is the reverse direction of `list.map_map`.\n-/\ntheorem comp_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (h : \u03b2 \u2192 \u03b3) (g : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    map (h \u2218 g) l = map h (map g l) :=\n  Eq.symm (map_map h g l)\n\n/--\nComposing a `list.map` with another `list.map` is equal to\na single `list.map` of composed functions.\n-/\n@[simp] theorem map_comp_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) :\n    map g \u2218 map f = map (g \u2218 f) :=\n  sorry\n\ntheorem map_filter_eq_foldr {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (p : \u03b1 \u2192 Prop) [decidable_pred p]\n    (as : List \u03b1) :\n    map f (filter p as) = foldr (fun (a : \u03b1) (bs : List \u03b2) => ite (p a) (f a :: bs) bs) [] as :=\n  sorry\n\ntheorem last_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} (hl : l \u2260 []) :\n    last (map f l) (mt eq_nil_of_map_eq_nil hl) = f (last l hl) :=\n  sorry\n\n/-! ### map\u2082 -/\n\ntheorem nil_map\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (l : List \u03b2) :\n    map\u2082 f [] l = [] :=\n  list.cases_on l (Eq.refl (map\u2082 f [] []))\n    fun (l_hd : \u03b2) (l_tl : List \u03b2) => Eq.refl (map\u2082 f [] (l_hd :: l_tl))\n\ntheorem map\u2082_nil {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (l : List \u03b1) :\n    map\u2082 f l [] = [] :=\n  list.cases_on l (Eq.refl (map\u2082 f [] []))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (map\u2082 f (l_hd :: l_tl) [])\n\n@[simp] theorem map\u2082_flip {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (as : List \u03b1)\n    (bs : List \u03b2) : map\u2082 (flip f) bs as = map\u2082 f as bs :=\n  sorry\n\n/-! ### take, drop -/\n\n@[simp] theorem take_zero {\u03b1 : Type u} (l : List \u03b1) : take 0 l = [] := rfl\n\n@[simp] theorem take_nil {\u03b1 : Type u} (n : \u2115) : take n [] = [] :=\n  nat.cases_on n (idRhs (take 0 [] = take 0 []) rfl)\n    fun (n : \u2115) => idRhs (take (n + 1) [] = take (n + 1) []) rfl\n\ntheorem take_cons {\u03b1 : Type u} (n : \u2115) (a : \u03b1) (l : List \u03b1) :\n    take (Nat.succ n) (a :: l) = a :: take n l :=\n  rfl\n\n@[simp] theorem take_length {\u03b1 : Type u} (l : List \u03b1) : take (length l) l = l := sorry\n\ntheorem take_all_of_le {\u03b1 : Type u} {n : \u2115} {l : List \u03b1} : length l \u2264 n \u2192 take n l = l := sorry\n\n@[simp] theorem take_left {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    take (length l\u2081) (l\u2081 ++ l\u2082) = l\u2081 :=\n  sorry\n\ntheorem take_left' {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (h : length l\u2081 = n) :\n    take n (l\u2081 ++ l\u2082) = l\u2081 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (take n (l\u2081 ++ l\u2082) = l\u2081)) (Eq.symm h))) (take_left l\u2081 l\u2082)\n\ntheorem take_take {\u03b1 : Type u} (n : \u2115) (m : \u2115) (l : List \u03b1) :\n    take n (take m l) = take (min n m) l :=\n  sorry\n\ntheorem take_repeat {\u03b1 : Type u} (a : \u03b1) (n : \u2115) (m : \u2115) :\n    take n (repeat a m) = repeat a (min n m) :=\n  sorry\n\ntheorem map_take {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (L : List \u03b1) (i : \u2115) :\n    map f (take i L) = take i (map f L) :=\n  sorry\n\ntheorem take_append_of_le_length {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} :\n    n \u2264 length l\u2081 \u2192 take n (l\u2081 ++ l\u2082) = take n l\u2081 :=\n  sorry\n\n/-- Taking the first `l\u2081.length + i` elements in `l\u2081 ++ l\u2082` is the same as appending the first\n`i` elements of `l\u2082` to `l\u2081`. -/\ntheorem take_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (i : \u2115) :\n    take (length l\u2081 + i) (l\u2081 ++ l\u2082) = l\u2081 ++ take i l\u2082 :=\n  sorry\n\n/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of\nlength `> i`. Version designed to rewrite from the big list to the small list. -/\ntheorem nth_le_take {\u03b1 : Type u} (L : List \u03b1) {i : \u2115} {j : \u2115} (hi : i < length L) (hj : i < j) :\n    nth_le L i hi =\n        nth_le (take j L) i\n          (eq.mpr (id (Eq._oldrec (Eq.refl (i < length (take j L))) (length_take j L)))\n            (lt_min hj hi)) :=\n  sorry\n\n/-- The `i`-th element of a list coincides with the `i`-th element of any of its prefixes of\nlength `> i`. Version designed to rewrite from the small list to the big list. -/\ntheorem nth_le_take' {\u03b1 : Type u} (L : List \u03b1) {i : \u2115} {j : \u2115} (hi : i < length (take j L)) :\n    nth_le (take j L) i hi =\n        nth_le L i\n          (lt_of_lt_of_le hi\n            (eq.mpr\n              (id\n                (Eq.trans\n                  (Eq.trans\n                    (Eq.trans\n                      ((fun (\u1fb0 \u1fb0_1 : \u2115) (e_2 : \u1fb0 = \u1fb0_1) (\u1fb0_2 \u1fb0_3 : \u2115) (e_3 : \u1fb0_2 = \u1fb0_3) =>\n                          congr (congr_arg LessEq e_2) e_3)\n                        (length (take j L)) (min j (length L)) (length_take j L) (length L)\n                        (length L) (Eq.refl (length L)))\n                      (propext min_le_iff))\n                    ((fun (a a_1 : Prop) (e_1 : a = a_1) (b b_1 : Prop) (e_2 : b = b_1) =>\n                        congr (congr_arg Or e_1) e_2)\n                      (j \u2264 length L) (j \u2264 length L) (Eq.refl (j \u2264 length L)) (length L \u2264 length L)\n                      True\n                      (propext\n                        ((fun {\u03b1 : Type} (a : \u03b1) => iff_true_intro (le_refl a)) (length L)))))\n                  (propext (or_true (j \u2264 length L)))))\n              trivial)) :=\n  sorry\n\ntheorem nth_take {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} {m : \u2115} (h : m < n) :\n    nth (take n l) m = nth l m :=\n  sorry\n\n@[simp] theorem nth_take_of_succ {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} :\n    nth (take (n + 1) l) n = nth l n :=\n  nth_take (nat.lt_succ_self n)\n\ntheorem take_succ {\u03b1 : Type u} {l : List \u03b1} {n : \u2115} :\n    take (n + 1) l = take n l ++ option.to_list (nth l n) :=\n  sorry\n\n@[simp] theorem drop_nil {\u03b1 : Type u} (n : \u2115) : drop n [] = [] :=\n  nat.cases_on n (idRhs (drop 0 [] = drop 0 []) rfl)\n    fun (n : \u2115) => idRhs (drop (n + 1) [] = drop (n + 1) []) rfl\n\ntheorem mem_of_mem_drop {\u03b1 : Type u_1} {n : \u2115} {l : List \u03b1} {x : \u03b1} (h : x \u2208 drop n l) : x \u2208 l :=\n  sorry\n\n@[simp] theorem drop_one {\u03b1 : Type u} (l : List \u03b1) : drop 1 l = tail l :=\n  list.cases_on l (idRhs (drop 1 [] = drop 1 []) rfl)\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => idRhs (drop 1 (l_hd :: l_tl) = drop 1 (l_hd :: l_tl)) rfl\n\ntheorem drop_add {\u03b1 : Type u} (m : \u2115) (n : \u2115) (l : List \u03b1) : drop (m + n) l = drop m (drop n l) :=\n  sorry\n\n@[simp] theorem drop_left {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    drop (length l\u2081) (l\u2081 ++ l\u2082) = l\u2082 :=\n  sorry\n\ntheorem drop_left' {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} (h : length l\u2081 = n) :\n    drop n (l\u2081 ++ l\u2082) = l\u2082 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (drop n (l\u2081 ++ l\u2082) = l\u2082)) (Eq.symm h))) (drop_left l\u2081 l\u2082)\n\ntheorem drop_eq_nth_le_cons {\u03b1 : Type u} {n : \u2115} {l : List \u03b1} (h : n < length l) :\n    drop n l = nth_le l n h :: drop (n + 1) l :=\n  sorry\n\n@[simp] theorem drop_length {\u03b1 : Type u} (l : List \u03b1) : drop (length l) l = [] := sorry\n\ntheorem drop_append_of_le_length {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115} :\n    n \u2264 length l\u2081 \u2192 drop n (l\u2081 ++ l\u2082) = drop n l\u2081 ++ l\u2082 :=\n  sorry\n\n/-- Dropping the elements up to `l\u2081.length + i` in `l\u2081 + l\u2082` is the same as dropping the elements\nup to `i` in `l\u2082`. -/\ntheorem drop_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (i : \u2115) :\n    drop (length l\u2081 + i) (l\u2081 ++ l\u2082) = drop i l\u2082 :=\n  sorry\n\n/-- The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by\ndropping the first `i` elements. Version designed to rewrite from the big list to the small list. -/\ntheorem nth_le_drop {\u03b1 : Type u} (L : List \u03b1) {i : \u2115} {j : \u2115} (h : i + j < length L) :\n    nth_le L (i + j) h =\n        nth_le (drop i L) j\n          (eq.mpr (id (Eq.refl (j < length (drop i L))))\n            (eq.mp\n              (Eq.trans\n                ((fun (\u1fb0 \u1fb0_1 : \u2115) (e_2 : \u1fb0 = \u1fb0_1) (\u1fb0_2 \u1fb0_3 : \u2115) (e_3 : \u1fb0_2 = \u1fb0_3) =>\n                    congr (congr_arg Less e_2) e_3)\n                  (i + j) (i + j) (Eq.refl (i + j)) (length (take i L ++ drop i L))\n                  (i + length (drop i L))\n                  (Eq.trans (length_append (take i L) (drop i L))\n                    ((fun (\u1fb0 \u1fb0_1 : \u2115) (e_2 : \u1fb0 = \u1fb0_1) (\u1fb0_2 \u1fb0_3 : \u2115) (e_3 : \u1fb0_2 = \u1fb0_3) =>\n                        congr (congr_arg Add.add e_2) e_3)\n                      (length (take i L)) i\n                      (Eq.trans (length_take i L)\n                        (min_eq_left\n                          (iff.mpr (iff_true_intro (le_of_lt (lt_of_le_of_lt (nat.le.intro rfl) h)))\n                            True.intro)))\n                      (length (drop i L)) (length (drop i L)) (Eq.refl (length (drop i L))))))\n                (propext (add_lt_add_iff_left i)))\n              (eq.mp (Eq._oldrec (Eq.refl (i + j < length L)) (Eq.symm (take_append_drop i L)))\n                h))) :=\n  sorry\n\n/--  The `i + j`-th element of a list coincides with the `j`-th element of the list obtained by\ndropping the first `i` elements. Version designed to rewrite from the small list to the big list. -/\ntheorem nth_le_drop' {\u03b1 : Type u} (L : List \u03b1) {i : \u2115} {j : \u2115} (h : j < length (drop i L)) :\n    nth_le (drop i L) j h = nth_le L (i + j) (nat.add_lt_of_lt_sub_left (length_drop i L \u25b8 h)) :=\n  sorry\n\n@[simp] theorem drop_drop {\u03b1 : Type u} (n : \u2115) (m : \u2115) (l : List \u03b1) :\n    drop n (drop m l) = drop (n + m) l :=\n  sorry\n\ntheorem drop_take {\u03b1 : Type u} (m : \u2115) (n : \u2115) (l : List \u03b1) :\n    drop m (take (m + n) l) = take n (drop m l) :=\n  sorry\n\ntheorem map_drop {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (L : List \u03b1) (i : \u2115) :\n    map f (drop i L) = drop i (map f L) :=\n  sorry\n\ntheorem modify_nth_tail_eq_take_drop {\u03b1 : Type u} (f : List \u03b1 \u2192 List \u03b1) (H : f [] = []) (n : \u2115)\n    (l : List \u03b1) : modify_nth_tail f n l = take n l ++ f (drop n l) :=\n  sorry\n\ntheorem modify_nth_eq_take_drop {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) (n : \u2115) (l : List \u03b1) :\n    modify_nth f n l = take n l ++ modify_head f (drop n l) :=\n  modify_nth_tail_eq_take_drop (modify_head f) rfl\n\ntheorem modify_nth_eq_take_cons_drop {\u03b1 : Type u} (f : \u03b1 \u2192 \u03b1) {n : \u2115} {l : List \u03b1}\n    (h : n < length l) : modify_nth f n l = take n l ++ f (nth_le l n h) :: drop (n + 1) l :=\n  sorry\n\ntheorem update_nth_eq_take_cons_drop {\u03b1 : Type u} (a : \u03b1) {n : \u2115} {l : List \u03b1} (h : n < length l) :\n    update_nth l n a = take n l ++ a :: drop (n + 1) l :=\n  sorry\n\ntheorem reverse_take {\u03b1 : Type u_1} {xs : List \u03b1} (n : \u2115) (h : n \u2264 length xs) :\n    take n (reverse xs) = reverse (drop (length xs - n) xs) :=\n  sorry\n\n@[simp] theorem update_nth_eq_nil {\u03b1 : Type u} (l : List \u03b1) (n : \u2115) (a : \u03b1) :\n    update_nth l n a = [] \u2194 l = [] :=\n  sorry\n\n@[simp] theorem take'_length {\u03b1 : Type u} [Inhabited \u03b1] (n : \u2115) (l : List \u03b1) :\n    length (take' n l) = n :=\n  sorry\n\n@[simp] theorem take'_nil {\u03b1 : Type u} [Inhabited \u03b1] (n : \u2115) :\n    take' n [] = repeat Inhabited.default n :=\n  sorry\n\ntheorem take'_eq_take {\u03b1 : Type u} [Inhabited \u03b1] {n : \u2115} {l : List \u03b1} :\n    n \u2264 length l \u2192 take' n l = take n l :=\n  sorry\n\n@[simp] theorem take'_left {\u03b1 : Type u} [Inhabited \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    take' (length l\u2081) (l\u2081 ++ l\u2082) = l\u2081 :=\n  sorry\n\ntheorem take'_left' {\u03b1 : Type u} [Inhabited \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {n : \u2115}\n    (h : length l\u2081 = n) : take' n (l\u2081 ++ l\u2082) = l\u2081 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (take' n (l\u2081 ++ l\u2082) = l\u2081)) (Eq.symm h))) (take'_left l\u2081 l\u2082)\n\n/-! ### foldl, foldr -/\n\ntheorem foldl_ext {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (g : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) {l : List \u03b2}\n    (H : \u2200 (a : \u03b1) (b : \u03b2), b \u2208 l \u2192 f a b = g a b) : foldl f a l = foldl g a l :=\n  sorry\n\ntheorem foldr_ext {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) {l : List \u03b1}\n    (H : \u2200 (a : \u03b1), a \u2208 l \u2192 \u2200 (b : \u03b2), f a b = g a b) : foldr f b l = foldr g b l :=\n  sorry\n\n@[simp] theorem foldl_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) : foldl f a [] = a :=\n  rfl\n\n@[simp] theorem foldl_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (b : \u03b2) (l : List \u03b2) :\n    foldl f a (b :: l) = foldl f (f a b) l :=\n  rfl\n\n@[simp] theorem foldr_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) : foldr f b [] = b :=\n  rfl\n\n@[simp] theorem foldr_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (a : \u03b1) (l : List \u03b1) :\n    foldr f b (a :: l) = f a (foldr f b l) :=\n  rfl\n\n@[simp] theorem foldl_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (l\u2081 : List \u03b2)\n    (l\u2082 : List \u03b2) : foldl f a (l\u2081 ++ l\u2082) = foldl f (foldl f a l\u2081) l\u2082 :=\n  sorry\n\n@[simp] theorem foldr_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (l\u2081 : List \u03b1)\n    (l\u2082 : List \u03b1) : foldr f b (l\u2081 ++ l\u2082) = foldr f (foldr f b l\u2082) l\u2081 :=\n  sorry\n\n@[simp] theorem foldl_join {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (L : List (List \u03b2)) :\n    foldl f a (join L) = foldl (foldl f) a L :=\n  sorry\n\n@[simp] theorem foldr_join {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (L : List (List \u03b1)) :\n    foldr f b (join L) = foldr (fun (l : List \u03b1) (b : \u03b2) => foldr f b l) b L :=\n  sorry\n\ntheorem foldl_reverse {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (l : List \u03b2) :\n    foldl f a (reverse l) = foldr (fun (x : \u03b2) (y : \u03b1) => f y x) a l :=\n  sorry\n\ntheorem foldr_reverse {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (a : \u03b2) (l : List \u03b1) :\n    foldr f a (reverse l) = foldl (fun (x : \u03b2) (y : \u03b1) => f y x) a l :=\n  sorry\n\n@[simp] theorem foldr_eta {\u03b1 : Type u} (l : List \u03b1) : foldr List.cons [] l = l := sorry\n\n@[simp] theorem reverse_foldl {\u03b1 : Type u} {l : List \u03b1} :\n    reverse (foldl (fun (t : List \u03b1) (h : \u03b1) => h :: t) [] l) = l :=\n  sorry\n\n@[simp] theorem foldl_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b3 \u2192 \u03b1) (a : \u03b1)\n    (l : List \u03b2) : foldl f a (map g l) = foldl (fun (x : \u03b1) (y : \u03b2) => f x (g y)) a l :=\n  sorry\n\n@[simp] theorem foldr_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (g : \u03b2 \u2192 \u03b3) (f : \u03b3 \u2192 \u03b1 \u2192 \u03b1) (a : \u03b1)\n    (l : List \u03b2) : foldr f a (map g l) = foldr (f \u2218 g) a l :=\n  sorry\n\ntheorem foldl_map' {\u03b1 : Type u} {\u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) (f' : \u03b2 \u2192 \u03b2 \u2192 \u03b2) (a : \u03b1)\n    (l : List \u03b1) (h : \u2200 (x y : \u03b1), f' (g x) (g y) = g (f x y)) :\n    foldl f' (g a) (map g l) = g (foldl f a l) :=\n  sorry\n\ntheorem foldr_map' {\u03b1 : Type u} {\u03b2 : Type u} (g : \u03b1 \u2192 \u03b2) (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) (f' : \u03b2 \u2192 \u03b2 \u2192 \u03b2) (a : \u03b1)\n    (l : List \u03b1) (h : \u2200 (x y : \u03b1), f' (g x) (g y) = g (f x y)) :\n    foldr f' (g a) (map g l) = g (foldr f a l) :=\n  sorry\n\ntheorem foldl_hom {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (l : List \u03b3) (f : \u03b1 \u2192 \u03b2) (op : \u03b1 \u2192 \u03b3 \u2192 \u03b1)\n    (op' : \u03b2 \u2192 \u03b3 \u2192 \u03b2) (a : \u03b1) (h : \u2200 (a : \u03b1) (x : \u03b3), f (op a x) = op' (f a) x) :\n    foldl op' (f a) l = f (foldl op a l) :=\n  sorry\n\ntheorem foldr_hom {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (l : List \u03b3) (f : \u03b1 \u2192 \u03b2) (op : \u03b3 \u2192 \u03b1 \u2192 \u03b1)\n    (op' : \u03b3 \u2192 \u03b2 \u2192 \u03b2) (a : \u03b1) (h : \u2200 (x : \u03b3) (a : \u03b1), f (op x a) = op' x (f a)) :\n    foldr op' (f a) l = f (foldr op a l) :=\n  sorry\n\ntheorem injective_foldl_comp {\u03b1 : Type u_1} {l : List (\u03b1 \u2192 \u03b1)} {f : \u03b1 \u2192 \u03b1}\n    (hl : \u2200 (f : \u03b1 \u2192 \u03b1), f \u2208 l \u2192 function.injective f) (hf : function.injective f) :\n    function.injective (foldl function.comp f l) :=\n  sorry\n\n/- scanl -/\n\ntheorem length_scanl {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} (a : \u03b2) (l : List \u03b1) :\n    length (scanl f a l) = length l + 1 :=\n  sorry\n\n@[simp] theorem scanl_nil {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} (b : \u03b2) : scanl f b [] = [b] :=\n  rfl\n\n@[simp] theorem scanl_cons {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {b : \u03b2} {a : \u03b1} {l : List \u03b1} :\n    scanl f b (a :: l) = [b] ++ scanl f (f b a) l :=\n  sorry\n\n@[simp] theorem nth_zero_scanl {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1} :\n    nth (scanl f b l) 0 = some b :=\n  sorry\n\n@[simp] theorem nth_le_zero_scanl {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1}\n    {h : 0 < length (scanl f b l)} : nth_le (scanl f b l) 0 h = b :=\n  sorry\n\ntheorem nth_succ_scanl {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1} {i : \u2115} :\n    nth (scanl f b l) (i + 1) =\n        option.bind (nth (scanl f b l) i)\n          fun (x : \u03b2) => option.map (fun (y : \u03b1) => f x y) (nth l i) :=\n  sorry\n\ntheorem nth_le_succ_scanl {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b2 \u2192 \u03b1 \u2192 \u03b2} {b : \u03b2} {l : List \u03b1} {i : \u2115}\n    {h : i + 1 < length (scanl f b l)} :\n    nth_le (scanl f b l) (i + 1) h =\n        f (nth_le (scanl f b l) i (nat.lt_of_succ_lt h))\n          (nth_le l i (nat.lt_of_succ_lt_succ (lt_of_lt_of_le h (le_of_eq (length_scanl b l))))) :=\n  sorry\n\n/- scanr -/\n\n@[simp] theorem scanr_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) : scanr f b [] = [b] :=\n  rfl\n\n@[simp] theorem scanr_aux_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (a : \u03b1)\n    (l : List \u03b1) : scanr_aux f b (a :: l) = (foldr f b (a :: l), scanr f b l) :=\n  sorry\n\n@[simp] theorem scanr_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (b : \u03b2) (a : \u03b1) (l : List \u03b1) :\n    scanr f b (a :: l) = foldr f b (a :: l) :: scanr f b l :=\n  sorry\n\n-- foldl and foldr coincide when f is commutative and associative\n\ntheorem foldl1_eq_foldr1 {\u03b1 : Type u} {f : \u03b1 \u2192 \u03b1 \u2192 \u03b1} (hassoc : associative f) (a : \u03b1) (b : \u03b1)\n    (l : List \u03b1) : foldl f a (l ++ [b]) = foldr f b (a :: l) :=\n  sorry\n\ntheorem foldl_eq_of_comm_of_assoc {\u03b1 : Type u} {f : \u03b1 \u2192 \u03b1 \u2192 \u03b1} (hcomm : commutative f)\n    (hassoc : associative f) (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    foldl f a (b :: l) = f b (foldl f a l) :=\n  sorry\n\ntheorem foldl_eq_foldr {\u03b1 : Type u} {f : \u03b1 \u2192 \u03b1 \u2192 \u03b1} (hcomm : commutative f) (hassoc : associative f)\n    (a : \u03b1) (l : List \u03b1) : foldl f a l = foldr f a l :=\n  sorry\n\ntheorem foldl_eq_of_comm' {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b1}\n    (hf : \u2200 (a : \u03b1) (b c : \u03b2), f (f a b) c = f (f a c) b) (a : \u03b1) (b : \u03b2) (l : List \u03b2) :\n    foldl f a (b :: l) = f (foldl f a l) b :=\n  sorry\n\ntheorem foldl_eq_foldr' {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b1}\n    (hf : \u2200 (a : \u03b1) (b c : \u03b2), f (f a b) c = f (f a c) b) (a : \u03b1) (l : List \u03b2) :\n    foldl f a l = foldr (flip f) a l :=\n  sorry\n\ntheorem foldr_eq_of_comm' {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2 \u2192 \u03b2}\n    (hf : \u2200 (a b : \u03b1) (c : \u03b2), f a (f b c) = f b (f a c)) (a : \u03b2) (b : \u03b1) (l : List \u03b1) :\n    foldr f a (b :: l) = foldr f (f b a) l :=\n  sorry\n\ntheorem foldl_assoc {\u03b1 : Type u} {op : \u03b1 \u2192 \u03b1 \u2192 \u03b1} [ha : is_associative \u03b1 op] {l : List \u03b1} {a\u2081 : \u03b1}\n    {a\u2082 : \u03b1} : foldl op (op a\u2081 a\u2082) l = op a\u2081 (foldl op a\u2082 l) :=\n  sorry\n\ntheorem foldl_op_eq_op_foldr_assoc {\u03b1 : Type u} {op : \u03b1 \u2192 \u03b1 \u2192 \u03b1} [ha : is_associative \u03b1 op]\n    {l : List \u03b1} {a\u2081 : \u03b1} {a\u2082 : \u03b1} : op (foldl op a\u2081 l) a\u2082 = op a\u2081 (foldr op a\u2082 l) :=\n  sorry\n\ntheorem foldl_assoc_comm_cons {\u03b1 : Type u} {op : \u03b1 \u2192 \u03b1 \u2192 \u03b1} [ha : is_associative \u03b1 op]\n    [hc : is_commutative \u03b1 op] {l : List \u03b1} {a\u2081 : \u03b1} {a\u2082 : \u03b1} :\n    foldl op a\u2082 (a\u2081 :: l) = op a\u2081 (foldl op a\u2082 l) :=\n  sorry\n\n/-! ### mfoldl, mfoldr, mmap -/\n\n@[simp] theorem mfoldl_nil {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) {b : \u03b2} : mfoldl f b [] = pure b :=\n  rfl\n\n@[simp] theorem mfoldr_nil {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2) {b : \u03b2} : mfoldr f b [] = pure b :=\n  rfl\n\n@[simp] theorem mfoldl_cons {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    {f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2} {b : \u03b2} {a : \u03b1} {l : List \u03b1} :\n    mfoldl f b (a :: l) =\n        do \n          let b' \u2190 f b a \n          mfoldl f b' l :=\n  rfl\n\n@[simp] theorem mfoldr_cons {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    {f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2} {b : \u03b2} {a : \u03b1} {l : List \u03b1} : mfoldr f b (a :: l) = mfoldr f b l >>= f a :=\n  rfl\n\ntheorem mfoldr_eq_foldr {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m] (f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2)\n    (b : \u03b2) (l : List \u03b1) : mfoldr f b l = foldr (fun (a : \u03b1) (mb : m \u03b2) => mb >>= f a) (pure b) l :=\n  sorry\n\ntheorem mfoldl_eq_foldl {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    [is_lawful_monad m] (f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2) (b : \u03b2) (l : List \u03b1) :\n    mfoldl f b l =\n        foldl\n          (fun (mb : m \u03b2) (a : \u03b1) =>\n            do \n              let b \u2190 mb \n              f b a)\n          (pure b) l :=\n  sorry\n\n@[simp] theorem mfoldl_append {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    [is_lawful_monad m] {f : \u03b2 \u2192 \u03b1 \u2192 m \u03b2} {b : \u03b2} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    mfoldl f b (l\u2081 ++ l\u2082) =\n        do \n          let x \u2190 mfoldl f b l\u2081 \n          mfoldl f x l\u2082 :=\n  sorry\n\n@[simp] theorem mfoldr_append {\u03b1 : Type u} {\u03b2 : Type v} {m : Type v \u2192 Type w} [Monad m]\n    [is_lawful_monad m] {f : \u03b1 \u2192 \u03b2 \u2192 m \u03b2} {b : \u03b2} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    mfoldr f b (l\u2081 ++ l\u2082) =\n        do \n          let x \u2190 mfoldr f b l\u2082 \n          mfoldr f x l\u2081 :=\n  sorry\n\n/-! ### prod and sum -/\n\n-- list.sum was already defined in defs.lean, but we couldn't tag it with `to_additive` yet.\n\n@[simp] theorem sum_nil {\u03b1 : Type u} [add_monoid \u03b1] : sum [] = 0 := rfl\n\ntheorem sum_singleton {\u03b1 : Type u} [add_monoid \u03b1] {a : \u03b1} : sum [a] = a := zero_add a\n\n@[simp] theorem prod_cons {\u03b1 : Type u} [monoid \u03b1] {l : List \u03b1} {a : \u03b1} :\n    prod (a :: l) = a * prod l :=\n  sorry\n\n@[simp] theorem sum_append {\u03b1 : Type u} [add_monoid \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    sum (l\u2081 ++ l\u2082) = sum l\u2081 + sum l\u2082 :=\n  sorry\n\n@[simp] theorem sum_join {\u03b1 : Type u} [add_monoid \u03b1] {l : List (List \u03b1)} :\n    sum (join l) = sum (map sum l) :=\n  sorry\n\ntheorem prod_ne_zero {R : Type u_1} [domain R] {L : List R} :\n    (\u2200 (x : R), x \u2208 L \u2192 x \u2260 0) \u2192 prod L \u2260 0 :=\n  sorry\n\ntheorem prod_eq_foldr {\u03b1 : Type u} [monoid \u03b1] {l : List \u03b1} : prod l = foldr Mul.mul 1 l := sorry\n\ntheorem prod_hom_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [monoid \u03b2] [monoid \u03b3] (l : List \u03b1)\n    {r : \u03b2 \u2192 \u03b3 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h\u2081 : r 1 1)\n    (h\u2082 : \u2200 {a : \u03b1} {b : \u03b2} {c : \u03b3}, r b c \u2192 r (f a * b) (g a * c)) :\n    r (prod (map f l)) (prod (map g l)) :=\n  sorry\n\ntheorem prod_hom {\u03b1 : Type u} {\u03b2 : Type v} [monoid \u03b1] [monoid \u03b2] (l : List \u03b1) (f : \u03b1 \u2192* \u03b2) :\n    prod (map (\u21d1f) l) = coe_fn f (prod l) :=\n  sorry\n\n-- `to_additive` chokes on the next few lemmas, so we do them by hand below\n\n@[simp] theorem prod_take_mul_prod_drop {\u03b1 : Type u} [monoid \u03b1] (L : List \u03b1) (i : \u2115) :\n    prod (take i L) * prod (drop i L) = prod L :=\n  sorry\n\n@[simp] theorem prod_take_succ {\u03b1 : Type u} [monoid \u03b1] (L : List \u03b1) (i : \u2115) (p : i < length L) :\n    prod (take (i + 1) L) = prod (take i L) * nth_le L i p :=\n  sorry\n\n/-- A list with product not one must have positive length. -/\ntheorem length_pos_of_prod_ne_one {\u03b1 : Type u} [monoid \u03b1] (L : List \u03b1) (h : prod L \u2260 1) :\n    0 < length L :=\n  sorry\n\ntheorem prod_update_nth {\u03b1 : Type u} [monoid \u03b1] (L : List \u03b1) (n : \u2115) (a : \u03b1) :\n    prod (update_nth L n a) = prod (take n L) * ite (n < length L) a 1 * prod (drop (n + 1) L) :=\n  sorry\n\n/-- This is the `list.prod` version of `mul_inv_rev` -/\ntheorem sum_neg_reverse {\u03b1 : Type u} [add_group \u03b1] (L : List \u03b1) :\n    -sum L = sum (reverse (map (fun (x : \u03b1) => -x) L)) :=\n  sorry\n\n/-- A non-commutative variant of `list.prod_reverse` -/\ntheorem prod_reverse_noncomm {\u03b1 : Type u} [group \u03b1] (L : List \u03b1) :\n    prod (reverse L) = (prod (map (fun (x : \u03b1) => x\u207b\u00b9) L)\u207b\u00b9) :=\n  sorry\n\n/-- This is the `list.prod` version of `mul_inv` -/\ntheorem sum_neg {\u03b1 : Type u} [add_comm_group \u03b1] (L : List \u03b1) :\n    -sum L = sum (map (fun (x : \u03b1) => -x) L) :=\n  sorry\n\n@[simp] theorem sum_take_add_sum_drop {\u03b1 : Type u} [add_monoid \u03b1] (L : List \u03b1) (i : \u2115) :\n    sum (take i L) + sum (drop i L) = sum L :=\n  sorry\n\n@[simp] theorem sum_take_succ {\u03b1 : Type u} [add_monoid \u03b1] (L : List \u03b1) (i : \u2115) (p : i < length L) :\n    sum (take (i + 1) L) = sum (take i L) + nth_le L i p :=\n  sorry\n\ntheorem eq_of_sum_take_eq {\u03b1 : Type u} [add_left_cancel_monoid \u03b1] {L : List \u03b1} {L' : List \u03b1}\n    (h : length L = length L') (h' : \u2200 (i : \u2115), i \u2264 length L \u2192 sum (take i L) = sum (take i L')) :\n    L = L' :=\n  sorry\n\ntheorem monotone_sum_take {\u03b1 : Type u} [canonically_ordered_add_monoid \u03b1] (L : List \u03b1) :\n    monotone fun (i : \u2115) => sum (take i L) :=\n  sorry\n\ntheorem one_le_prod_of_one_le {\u03b1 : Type u} [ordered_comm_monoid \u03b1] {l : List \u03b1}\n    (hl\u2081 : \u2200 (x : \u03b1), x \u2208 l \u2192 1 \u2264 x) : 1 \u2264 prod l :=\n  sorry\n\ntheorem single_le_prod {\u03b1 : Type u} [ordered_comm_monoid \u03b1] {l : List \u03b1}\n    (hl\u2081 : \u2200 (x : \u03b1), x \u2208 l \u2192 1 \u2264 x) (x : \u03b1) (H : x \u2208 l) : x \u2264 prod l :=\n  sorry\n\ntheorem all_zero_of_le_zero_le_of_sum_eq_zero {\u03b1 : Type u} [ordered_add_comm_monoid \u03b1] {l : List \u03b1}\n    (hl\u2081 : \u2200 (x : \u03b1), x \u2208 l \u2192 0 \u2264 x) (hl\u2082 : sum l = 0) (x : \u03b1) (H : x \u2208 l) : x = 0 :=\n  le_antisymm (hl\u2082 \u25b8 single_le_sum hl\u2081 x hx) (hl\u2081 x hx)\n\ntheorem sum_eq_zero_iff {\u03b1 : Type u} [canonically_ordered_add_monoid \u03b1] (l : List \u03b1) :\n    sum l = 0 \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 x = 0 :=\n  sorry\n\n/-- A list with sum not zero must have positive length. -/\ntheorem length_pos_of_sum_ne_zero {\u03b1 : Type u} [add_monoid \u03b1] (L : List \u03b1) (h : sum L \u2260 0) :\n    0 < length L :=\n  sorry\n\n/-- If all elements in a list are bounded below by `1`, then the length of the list is bounded\nby the sum of the elements. -/\ntheorem length_le_sum_of_one_le (L : List \u2115) (h : \u2200 (i : \u2115), i \u2208 L \u2192 1 \u2264 i) : length L \u2264 sum L :=\n  sorry\n\n-- Now we tie those lemmas back to their multiplicative versions.\n\n/-- A list with positive sum must have positive length. -/\n-- This is an easy consequence of `length_pos_of_sum_ne_zero`, but often useful in applications.\n\ntheorem length_pos_of_sum_pos {\u03b1 : Type u} [ordered_cancel_add_comm_monoid \u03b1] (L : List \u03b1)\n    (h : 0 < sum L) : 0 < length L :=\n  length_pos_of_sum_ne_zero L (ne_of_gt h)\n\n@[simp] theorem prod_erase {\u03b1 : Type u} [DecidableEq \u03b1] [comm_monoid \u03b1] {a : \u03b1} {l : List \u03b1} :\n    a \u2208 l \u2192 a * prod (list.erase l a) = prod l :=\n  sorry\n\ntheorem dvd_prod {\u03b1 : Type u} [comm_monoid \u03b1] {a : \u03b1} {l : List \u03b1} (ha : a \u2208 l) : a \u2223 prod l :=\n  sorry\n\n@[simp] theorem sum_const_nat (m : \u2115) (n : \u2115) : sum (repeat m n) = m * n := sorry\n\ntheorem dvd_sum {\u03b1 : Type u} [comm_semiring \u03b1] {a : \u03b1} {l : List \u03b1} (h : \u2200 (x : \u03b1), x \u2208 l \u2192 a \u2223 x) :\n    a \u2223 sum l :=\n  sorry\n\n@[simp] theorem length_join {\u03b1 : Type u} (L : List (List \u03b1)) :\n    length (join L) = sum (map length L) :=\n  sorry\n\n@[simp] theorem length_bind {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) (f : \u03b1 \u2192 List \u03b2) :\n    length (list.bind l f) = sum (map (length \u2218 f) l) :=\n  sorry\n\ntheorem exists_lt_of_sum_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_ordered_cancel_add_comm_monoid \u03b2]\n    {l : List \u03b1} (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b2) (h : sum (map f l) < sum (map g l)) :\n    \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), f x < g x :=\n  sorry\n\ntheorem exists_le_of_sum_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_ordered_cancel_add_comm_monoid \u03b2]\n    {l : List \u03b1} (hl : l \u2260 []) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 \u03b2) (h : sum (map f l) \u2264 sum (map g l)) :\n    \u2203 (x : \u03b1), \u2203 (H : x \u2208 l), f x \u2264 g x :=\n  sorry\n\n-- Several lemmas about sum/head/tail for `list \u2115`.\n\n-- These are hard to generalize well, as they rely on the fact that `default \u2115 = 0`.\n\n-- We'd like to state this as `L.head * L.tail.prod = L.prod`,\n\n-- but because `L.head` relies on an inhabited instances and\n\n-- returns a garbage value for the empty list, this is not possible.\n\n-- Instead we write the statement in terms of `(L.nth 0).get_or_else 1`,\n\n-- and below, restate the lemma just for `\u2115`.\n\ntheorem head_mul_tail_prod' {\u03b1 : Type u} [monoid \u03b1] (L : List \u03b1) :\n    option.get_or_else (nth L 0) 1 * prod (tail L) = prod L :=\n  sorry\n\ntheorem head_add_tail_sum (L : List \u2115) : head L + sum (tail L) = sum L := sorry\n\ntheorem head_le_sum (L : List \u2115) : head L \u2264 sum L := nat.le.intro (head_add_tail_sum L)\n\ntheorem tail_sum (L : List \u2115) : sum (tail L) = sum L - head L := sorry\n\n@[simp] theorem alternating_prod_nil {G : Type u_1} [comm_group G] : alternating_prod [] = 1 := rfl\n\n@[simp] theorem alternating_sum_singleton {G : Type u_1} [add_comm_group G] (g : G) :\n    alternating_sum [g] = g :=\n  rfl\n\n@[simp] theorem alternating_sum_cons_cons' {G : Type u_1} [add_comm_group G] (g : G) (h : G)\n    (l : List G) : alternating_sum (g :: h :: l) = g + -h + alternating_sum l :=\n  rfl\n\ntheorem alternating_sum_cons_cons {G : Type u_1} [add_comm_group G] (g : G) (h : G) (l : List G) :\n    alternating_sum (g :: h :: l) = g - h + alternating_sum l :=\n  sorry\n\n/-! ### join -/\n\ntheorem join_eq_nil {\u03b1 : Type u} {L : List (List \u03b1)} :\n    join L = [] \u2194 \u2200 (l : List \u03b1), l \u2208 L \u2192 l = [] :=\n  sorry\n\n@[simp] theorem join_append {\u03b1 : Type u} (L\u2081 : List (List \u03b1)) (L\u2082 : List (List \u03b1)) :\n    join (L\u2081 ++ L\u2082) = join L\u2081 ++ join L\u2082 :=\n  sorry\n\ntheorem join_join {\u03b1 : Type u} (l : List (List (List \u03b1))) : join (join l) = join (map join l) :=\n  sorry\n\n/-- In a join, taking the first elements up to an index which is the sum of the lengths of the\nfirst `i` sublists, is the same as taking the join of the first `i` sublists. -/\ntheorem take_sum_join {\u03b1 : Type u} (L : List (List \u03b1)) (i : \u2115) :\n    take (sum (take i (map length L))) (join L) = join (take i L) :=\n  sorry\n\n/-- In a join, dropping all the elements up to an index which is the sum of the lengths of the\nfirst `i` sublists, is the same as taking the join after dropping the first `i` sublists. -/\ntheorem drop_sum_join {\u03b1 : Type u} (L : List (List \u03b1)) (i : \u2115) :\n    drop (sum (take i (map length L))) (join L) = join (drop i L) :=\n  sorry\n\n/-- Taking only the first `i+1` elements in a list, and then dropping the first `i` ones, one is\nleft with a list of length `1` made of the `i`-th element of the original list. -/\ntheorem drop_take_succ_eq_cons_nth_le {\u03b1 : Type u} (L : List \u03b1) {i : \u2115} (hi : i < length L) :\n    drop i (take (i + 1) L) = [nth_le L i hi] :=\n  sorry\n\n/-- In a join of sublists, taking the slice between the indices `A` and `B - 1` gives back the\noriginal sublist of index `i` if `A` is the sum of the lenghts of sublists of index `< i`, and\n`B` is the sum of the lengths of sublists of index `\u2264 i`. -/\ntheorem drop_take_succ_join_eq_nth_le {\u03b1 : Type u} (L : List (List \u03b1)) {i : \u2115} (hi : i < length L) :\n    drop (sum (take i (map length L))) (take (sum (take (i + 1) (map length L))) (join L)) =\n        nth_le L i hi :=\n  sorry\n\n/-- Auxiliary lemma to control elements in a join. -/\ntheorem sum_take_map_length_lt1 {\u03b1 : Type u} (L : List (List \u03b1)) {i : \u2115} {j : \u2115} (hi : i < length L)\n    (hj : j < length (nth_le L i hi)) :\n    sum (take i (map length L)) + j < sum (take (i + 1) (map length L)) :=\n  sorry\n\n/-- Auxiliary lemma to control elements in a join. -/\ntheorem sum_take_map_length_lt2 {\u03b1 : Type u} (L : List (List \u03b1)) {i : \u2115} {j : \u2115} (hi : i < length L)\n    (hj : j < length (nth_le L i hi)) : sum (take i (map length L)) + j < length (join L) :=\n  sorry\n\n/-- The `n`-th element in a join of sublists is the `j`-th element of the `i`th sublist,\nwhere `n` can be obtained in terms of `i` and `j` by adding the lengths of all the sublists\nof index `< i`, and adding `j`. -/\ntheorem nth_le_join {\u03b1 : Type u} (L : List (List \u03b1)) {i : \u2115} {j : \u2115} (hi : i < length L)\n    (hj : j < length (nth_le L i hi)) :\n    nth_le (join L) (sum (take i (map length L)) + j) (sum_take_map_length_lt2 L hi hj) =\n        nth_le (nth_le L i hi) j hj :=\n  sorry\n\n/-- Two lists of sublists are equal iff their joins coincide, as well as the lengths of the\nsublists. -/\ntheorem eq_iff_join_eq {\u03b1 : Type u} (L : List (List \u03b1)) (L' : List (List \u03b1)) :\n    L = L' \u2194 join L = join L' \u2227 map length L = map length L' :=\n  sorry\n\n/-! ### lexicographic ordering -/\n\n/-- Given a strict order `<` on `\u03b1`, the lexicographic strict order on `list \u03b1`, for which\n`[a0, ..., an] < [b0, ..., b_k]` if `a0 < b0` or `a0 = b0` and `[a1, ..., an] < [b1, ..., bk]`.\nThe definition is given for any relation `r`, not only strict orders. -/\ninductive lex {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : List \u03b1 \u2192 List \u03b1 \u2192 Prop where\n| nil : \u2200 {a : \u03b1} {l : List \u03b1}, lex r [] (a :: l)\n| cons : \u2200 {a : \u03b1} {l\u2081 l\u2082 : List \u03b1}, lex r l\u2081 l\u2082 \u2192 lex r (a :: l\u2081) (a :: l\u2082)\n| rel : \u2200 {a\u2081 : \u03b1} {l\u2081 : List \u03b1} {a\u2082 : \u03b1} {l\u2082 : List \u03b1}, r a\u2081 a\u2082 \u2192 lex r (a\u2081 :: l\u2081) (a\u2082 :: l\u2082)\n\nnamespace lex\n\n\ntheorem cons_iff {\u03b1 : Type u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} [is_irrefl \u03b1 r] {a : \u03b1} {l\u2081 : List \u03b1}\n    {l\u2082 : List \u03b1} : lex r (a :: l\u2081) (a :: l\u2082) \u2194 lex r l\u2081 l\u2082 :=\n  sorry\n\n@[simp] theorem not_nil_right {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (l : List \u03b1) : \u00aclex r l [] := sorry\n\nprotected instance is_order_connected {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [is_order_connected \u03b1 r]\n    [is_trichotomous \u03b1 r] : is_order_connected (List \u03b1) (lex r) :=\n  is_order_connected.mk fun (l\u2081 : List \u03b1) => sorry\n\nprotected instance is_trichotomous {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [is_trichotomous \u03b1 r] :\n    is_trichotomous (List \u03b1) (lex r) :=\n  is_trichotomous.mk fun (l\u2081 : List \u03b1) => sorry\n\nprotected instance is_asymm {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [is_asymm \u03b1 r] :\n    is_asymm (List \u03b1) (lex r) :=\n  is_asymm.mk fun (l\u2081 : List \u03b1) => sorry\n\nprotected instance is_strict_total_order {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n    [is_strict_total_order' \u03b1 r] : is_strict_total_order' (List \u03b1) (lex r) :=\n  is_strict_total_order'.mk\n\nprotected instance decidable_rel {\u03b1 : Type u} [DecidableEq \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [DecidableRel r] :\n    DecidableRel (lex r) :=\n  sorry\n\ntheorem append_right {\u03b1 : Type u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) {s\u2081 : List \u03b1} {s\u2082 : List \u03b1} (t : List \u03b1) :\n    lex r s\u2081 s\u2082 \u2192 lex r s\u2081 (s\u2082 ++ t) :=\n  sorry\n\ntheorem append_left {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) {t\u2081 : List \u03b1} {t\u2082 : List \u03b1} (h : lex R t\u2081 t\u2082)\n    (s : List \u03b1) : lex R (s ++ t\u2081) (s ++ t\u2082) :=\n  sorry\n\ntheorem imp {\u03b1 : Type u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03b1 \u2192 \u03b1 \u2192 Prop} (H : \u2200 (a b : \u03b1), r a b \u2192 s a b)\n    (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : lex r l\u2081 l\u2082 \u2192 lex s l\u2081 l\u2082 :=\n  sorry\n\ntheorem to_ne {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : lex ne l\u2081 l\u2082 \u2192 l\u2081 \u2260 l\u2082 := sorry\n\ntheorem ne_iff {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (H : length l\u2081 \u2264 length l\u2082) :\n    lex ne l\u2081 l\u2082 \u2194 l\u2081 \u2260 l\u2082 :=\n  sorry\n\nend lex\n\n\n--Note: this overrides an instance in core lean\n\nprotected instance has_lt' {\u03b1 : Type u} [HasLess \u03b1] : HasLess (List \u03b1) := { Less := lex Less }\n\ntheorem nil_lt_cons {\u03b1 : Type u} [HasLess \u03b1] (a : \u03b1) (l : List \u03b1) : [] < a :: l := lex.nil\n\nprotected instance linear_order {\u03b1 : Type u} [linear_order \u03b1] : linear_order (List \u03b1) :=\n  linear_order_of_STO' (lex Less)\n\n--Note: this overrides an instance in core lean\n\nprotected instance has_le' {\u03b1 : Type u} [linear_order \u03b1] : HasLessEq (List \u03b1) :=\n  preorder.to_has_le (List \u03b1)\n\n/-! ### all & any -/\n\n@[simp] theorem all_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Bool) : all [] p = tt := rfl\n\n@[simp] theorem all_cons {\u03b1 : Type u} (p : \u03b1 \u2192 Bool) (a : \u03b1) (l : List \u03b1) :\n    all (a :: l) p = p a && all l p :=\n  rfl\n\ntheorem all_iff_forall {\u03b1 : Type u} {p : \u03b1 \u2192 Bool} {l : List \u03b1} :\n    \u21a5(all l p) \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 \u21a5(p a) :=\n  sorry\n\ntheorem all_iff_forall_prop {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} :\n    \u21a5(all l fun (a : \u03b1) => to_bool (p a)) \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a :=\n  sorry\n\n@[simp] theorem any_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Bool) : any [] p = false := rfl\n\n@[simp] theorem any_cons {\u03b1 : Type u} (p : \u03b1 \u2192 Bool) (a : \u03b1) (l : List \u03b1) :\n    any (a :: l) p = p a || any l p :=\n  rfl\n\ntheorem any_iff_exists {\u03b1 : Type u} {p : \u03b1 \u2192 Bool} {l : List \u03b1} :\n    \u21a5(any l p) \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 l), \u21a5(p a) :=\n  sorry\n\ntheorem any_iff_exists_prop {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} :\n    \u21a5(any l fun (a : \u03b1) => to_bool (p a)) \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 l), p a :=\n  sorry\n\ntheorem any_of_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Bool} {a : \u03b1} {l : List \u03b1} (h\u2081 : a \u2208 l) (h\u2082 : \u21a5(p a)) :\n    \u21a5(any l p) :=\n  iff.mpr any_iff_exists (Exists.intro a (Exists.intro h\u2081 h\u2082))\n\nprotected instance decidable_forall_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (l : List \u03b1) : Decidable (\u2200 (x : \u03b1), x \u2208 l \u2192 p x) :=\n  decidable_of_iff \u21a5(all l fun (a : \u03b1) => to_bool (p a)) sorry\n\nprotected instance decidable_exists_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (l : List \u03b1) : Decidable (\u2203 (x : \u03b1), \u2203 (H : x \u2208 l), p x) :=\n  decidable_of_iff \u21a5(any l fun (a : \u03b1) => to_bool (p a)) sorry\n\n/-! ### map for partial functions -/\n\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on\n  `a : \u03b1` satisfying `p`, then `pmap f l h` is essentially the same as `map f l`\n  but is defined only when all members of `l` satisfy `p`, using the proof\n  to apply `f`. -/\n@[simp] def pmap {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (l : List \u03b1) :\n    (\u2200 (a : \u03b1), a \u2208 l \u2192 p a) \u2192 List \u03b2 :=\n  sorry\n\n/-- \"Attach\" the proof that the elements of `l` are in `l` to produce a new list\n  with the same elements but in the type `{x // x \u2208 l}`. -/\ndef attach {\u03b1 : Type u} (l : List \u03b1) : List (Subtype fun (x : \u03b1) => x \u2208 l) :=\n  pmap Subtype.mk l sorry\n\ntheorem sizeof_lt_sizeof_of_mem {\u03b1 : Type u} [SizeOf \u03b1] {x : \u03b1} {l : List \u03b1} (hx : x \u2208 l) :\n    sizeof x < sizeof l :=\n  sorry\n\ntheorem pmap_eq_map {\u03b1 : Type u} {\u03b2 : Type v} (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (l : List \u03b1)\n    (H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) : pmap (fun (a : \u03b1) (_x : p a) => f a) l H = map f l :=\n  sorry\n\ntheorem pmap_congr {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2}\n    {g : (a : \u03b1) \u2192 q a \u2192 \u03b2} (l : List \u03b1) {H\u2081 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a} {H\u2082 : \u2200 (a : \u03b1), a \u2208 l \u2192 q a}\n    (h : \u2200 (a : \u03b1) (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082) : pmap f l H\u2081 = pmap g l H\u2082 :=\n  sorry\n\ntheorem map_pmap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3)\n    (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) :\n    map g (pmap f l H) = pmap (fun (a : \u03b1) (h : p a) => g (f a h)) l H :=\n  sorry\n\ntheorem pmap_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {p : \u03b2 \u2192 Prop} (g : (b : \u03b2) \u2192 p b \u2192 \u03b3)\n    (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) (H : \u2200 (a : \u03b2), a \u2208 map f l \u2192 p a) :\n    pmap g (map f l) H =\n        pmap (fun (a : \u03b1) (h : p (f a)) => g (f a) h) l\n          fun (a : \u03b1) (h : a \u2208 l) => H (f a) (mem_map_of_mem f h) :=\n  sorry\n\ntheorem pmap_eq_map_attach {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) :\n    pmap f l H =\n        map\n          (fun (x : Subtype fun (x : \u03b1) => x \u2208 l) =>\n            f (subtype.val x) (H (subtype.val x) (subtype.property x)))\n          (attach l) :=\n  sorry\n\ntheorem attach_map_val {\u03b1 : Type u} (l : List \u03b1) : map subtype.val (attach l) = l := sorry\n\n@[simp] theorem mem_attach {\u03b1 : Type u} (l : List \u03b1) (x : Subtype fun (x : \u03b1) => x \u2208 l) :\n    x \u2208 attach l :=\n  sorry\n\n@[simp] theorem mem_pmap {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2}\n    {l : List \u03b1} {H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a} {b : \u03b2} :\n    b \u2208 pmap f l H \u2194 \u2203 (a : \u03b1), \u2203 (h : a \u2208 l), f a (H a h) = b :=\n  sorry\n\n@[simp] theorem length_pmap {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2}\n    {l : List \u03b1} {H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a} : length (pmap f l H) = length l :=\n  sorry\n\n@[simp] theorem length_attach {\u03b1 : Type u} (L : List \u03b1) : length (attach L) = length L :=\n  length_pmap\n\n@[simp] theorem pmap_eq_nil {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2}\n    {l : List \u03b1} {H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a} : pmap f l H = [] \u2194 l = [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (pmap f l H = [] \u2194 l = [])) (Eq.symm (propext length_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (pmap f l H) = 0 \u2194 l = [])) length_pmap))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length l = 0 \u2194 l = [])) (propext length_eq_zero)))\n        (iff.refl (l = []))))\n\n@[simp] theorem attach_eq_nil {\u03b1 : Type u} (l : List \u03b1) : attach l = [] \u2194 l = [] := pmap_eq_nil\n\ntheorem last_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (p : \u03b1 \u2192 Prop) (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (l : List \u03b1)\n    (hl\u2081 : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) (hl\u2082 : l \u2260 []) :\n    last (pmap f l hl\u2081) (mt (iff.mp pmap_eq_nil) hl\u2082) =\n        f (last l hl\u2082) (hl\u2081 (last l hl\u2082) (last_mem hl\u2082)) :=\n  sorry\n\ntheorem nth_pmap {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) {l : List \u03b1}\n    (h : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) (n : \u2115) :\n    nth (pmap f l h) n = option.pmap f (nth l n) fun (x : \u03b1) (H : x \u2208 nth l n) => h x (nth_mem H) :=\n  sorry\n\ntheorem nth_le_pmap {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) {l : List \u03b1}\n    (h : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) {n : \u2115} (hn : n < length (pmap f l h)) :\n    nth_le (pmap f l h) n hn =\n        f (nth_le l n (length_pmap \u25b8 hn))\n          (h (nth_le l n (length_pmap \u25b8 hn)) (nth_le_mem l n (length_pmap \u25b8 hn))) :=\n  sorry\n\n/-! ### find -/\n\n@[simp] theorem find_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] : find p [] = none := rfl\n\n@[simp] theorem find_cons_of_pos {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (l : List \u03b1)\n    (h : p a) : find p (a :: l) = some a :=\n  if_pos h\n\n@[simp] theorem find_cons_of_neg {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (l : List \u03b1)\n    (h : \u00acp a) : find p (a :: l) = find p l :=\n  if_neg h\n\n@[simp] theorem find_eq_none {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} :\n    find p l = none \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 \u00acp x :=\n  sorry\n\ntheorem find_some {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} {a : \u03b1}\n    (H : find p l = some a) : p a :=\n  sorry\n\n@[simp] theorem find_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} {a : \u03b1}\n    (H : find p l = some a) : a \u2208 l :=\n  sorry\n\n/-! ### lookmap -/\n\n@[simp] theorem lookmap_nil {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) : lookmap f [] = [] := rfl\n\n@[simp] theorem lookmap_cons_none {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) {a : \u03b1} (l : List \u03b1)\n    (h : f a = none) : lookmap f (a :: l) = a :: lookmap f l :=\n  sorry\n\n@[simp] theorem lookmap_cons_some {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) {a : \u03b1} {b : \u03b1} (l : List \u03b1)\n    (h : f a = some b) : lookmap f (a :: l) = b :: l :=\n  sorry\n\ntheorem lookmap_some {\u03b1 : Type u} (l : List \u03b1) : lookmap some l = l :=\n  list.cases_on l (idRhs (lookmap some [] = lookmap some []) rfl)\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) =>\n      idRhs (lookmap some (l_hd :: l_tl) = lookmap some (l_hd :: l_tl)) rfl\n\ntheorem lookmap_none {\u03b1 : Type u} (l : List \u03b1) : lookmap (fun (_x : \u03b1) => none) l = l := sorry\n\ntheorem lookmap_congr {\u03b1 : Type u} {f : \u03b1 \u2192 Option \u03b1} {g : \u03b1 \u2192 Option \u03b1} {l : List \u03b1} :\n    (\u2200 (a : \u03b1), a \u2208 l \u2192 f a = g a) \u2192 lookmap f l = lookmap g l :=\n  sorry\n\ntheorem lookmap_of_forall_not {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) {l : List \u03b1}\n    (H : \u2200 (a : \u03b1), a \u2208 l \u2192 f a = none) : lookmap f l = l :=\n  Eq.trans (lookmap_congr H) (lookmap_none l)\n\ntheorem lookmap_map_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b1) (g : \u03b1 \u2192 \u03b2)\n    (h : \u2200 (a b : \u03b1), b \u2208 f a \u2192 g a = g b) (l : List \u03b1) : map g (lookmap f l) = map g l :=\n  sorry\n\ntheorem lookmap_id' {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) (h : \u2200 (a b : \u03b1), b \u2208 f a \u2192 a = b)\n    (l : List \u03b1) : lookmap f l = l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (lookmap f l = l)) (Eq.symm (map_id (lookmap f l)))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (map id (lookmap f l) = l)) (lookmap_map_eq f id h l)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (map id l = l)) (map_id l))) (Eq.refl l)))\n\ntheorem length_lookmap {\u03b1 : Type u} (f : \u03b1 \u2192 Option \u03b1) (l : List \u03b1) :\n    length (lookmap f l) = length l :=\n  sorry\n\n/-! ### filter_map -/\n\n@[simp] theorem filter_map_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) :\n    filter_map f [] = [] :=\n  rfl\n\n@[simp] theorem filter_map_cons_none {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 Option \u03b2} (a : \u03b1)\n    (l : List \u03b1) (h : f a = none) : filter_map f (a :: l) = filter_map f l :=\n  sorry\n\n@[simp] theorem filter_map_cons_some {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) (a : \u03b1)\n    (l : List \u03b1) {b : \u03b2} (h : f a = some b) : filter_map f (a :: l) = b :: filter_map f l :=\n  sorry\n\ntheorem filter_map_append {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l : List \u03b1) (l' : List \u03b1)\n    (f : \u03b1 \u2192 Option \u03b2) : filter_map f (l ++ l') = filter_map f l ++ filter_map f l' :=\n  sorry\n\ntheorem filter_map_eq_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : filter_map (some \u2218 f) = map f :=\n  sorry\n\ntheorem filter_map_eq_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n    filter_map (option.guard p) = filter p :=\n  sorry\n\ntheorem filter_map_filter_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2)\n    (g : \u03b2 \u2192 Option \u03b3) (l : List \u03b1) :\n    filter_map g (filter_map f l) = filter_map (fun (x : \u03b1) => option.bind (f x) g) l :=\n  sorry\n\ntheorem map_filter_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b3)\n    (l : List \u03b1) : map g (filter_map f l) = filter_map (fun (x : \u03b1) => option.map g (f x)) l :=\n  sorry\n\ntheorem filter_map_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 Option \u03b3)\n    (l : List \u03b1) : filter_map g (map f l) = filter_map (g \u2218 f) l :=\n  sorry\n\ntheorem filter_filter_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) (p : \u03b2 \u2192 Prop)\n    [decidable_pred p] (l : List \u03b1) :\n    filter p (filter_map f l) = filter_map (fun (x : \u03b1) => option.filter p (f x)) l :=\n  sorry\n\ntheorem filter_map_filter {\u03b1 : Type u} {\u03b2 : Type v} (p : \u03b1 \u2192 Prop) [decidable_pred p]\n    (f : \u03b1 \u2192 Option \u03b2) (l : List \u03b1) :\n    filter_map f (filter p l) = filter_map (fun (x : \u03b1) => ite (p x) (f x) none) l :=\n  sorry\n\n@[simp] theorem filter_map_some {\u03b1 : Type u} (l : List \u03b1) : filter_map some l = l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (filter_map some l = l)) (filter_map_eq_map fun (x : \u03b1) => x)))\n    (map_id l)\n\n@[simp] theorem mem_filter_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) (l : List \u03b1) {b : \u03b2} :\n    b \u2208 filter_map f l \u2194 \u2203 (a : \u03b1), a \u2208 l \u2227 f a = some b :=\n  sorry\n\ntheorem map_filter_map_of_inv {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b1)\n    (H : \u2200 (x : \u03b1), option.map g (f x) = some x) (l : List \u03b1) : map g (filter_map f l) = l :=\n  sorry\n\ntheorem sublist.filter_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (s : l\u2081 <+ l\u2082) : filter_map f l\u2081 <+ filter_map f l\u2082 :=\n  sorry\n\ntheorem sublist.map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (s : l\u2081 <+ l\u2082) : map f l\u2081 <+ map f l\u2082 :=\n  filter_map_eq_map f \u25b8 sublist.filter_map (some \u2218 f) s\n\n/-! ### reduce_option -/\n\n@[simp] theorem reduce_option_cons_of_some {\u03b1 : Type u} (x : \u03b1) (l : List (Option \u03b1)) :\n    reduce_option (some x :: l) = x :: reduce_option l :=\n  sorry\n\n@[simp] theorem reduce_option_cons_of_none {\u03b1 : Type u} (l : List (Option \u03b1)) :\n    reduce_option (none :: l) = reduce_option l :=\n  sorry\n\n@[simp] theorem reduce_option_nil {\u03b1 : Type u} : reduce_option [] = [] := rfl\n\n@[simp] theorem reduce_option_map {\u03b1 : Type u} {\u03b2 : Type v} {l : List (Option \u03b1)} {f : \u03b1 \u2192 \u03b2} :\n    reduce_option (map (option.map f) l) = map f (reduce_option l) :=\n  sorry\n\ntheorem reduce_option_append {\u03b1 : Type u} (l : List (Option \u03b1)) (l' : List (Option \u03b1)) :\n    reduce_option (l ++ l') = reduce_option l ++ reduce_option l' :=\n  filter_map_append l l' id\n\ntheorem reduce_option_length_le {\u03b1 : Type u} (l : List (Option \u03b1)) :\n    length (reduce_option l) \u2264 length l :=\n  sorry\n\ntheorem reduce_option_length_eq_iff {\u03b1 : Type u} {l : List (Option \u03b1)} :\n    length (reduce_option l) = length l \u2194 \u2200 (x : Option \u03b1), x \u2208 l \u2192 \u21a5(option.is_some x) :=\n  sorry\n\ntheorem reduce_option_length_lt_iff {\u03b1 : Type u} {l : List (Option \u03b1)} :\n    length (reduce_option l) < length l \u2194 none \u2208 l :=\n  sorry\n\ntheorem reduce_option_singleton {\u03b1 : Type u} (x : Option \u03b1) :\n    reduce_option [x] = option.to_list x :=\n  option.cases_on x (Eq.refl (reduce_option [none])) fun (x : \u03b1) => Eq.refl (reduce_option [some x])\n\ntheorem reduce_option_concat {\u03b1 : Type u} (l : List (Option \u03b1)) (x : Option \u03b1) :\n    reduce_option (concat l x) = reduce_option l ++ option.to_list x :=\n  sorry\n\ntheorem reduce_option_concat_of_some {\u03b1 : Type u} (l : List (Option \u03b1)) (x : \u03b1) :\n    reduce_option (concat l (some x)) = concat (reduce_option l) x :=\n  sorry\n\ntheorem reduce_option_mem_iff {\u03b1 : Type u} {l : List (Option \u03b1)} {x : \u03b1} :\n    x \u2208 reduce_option l \u2194 some x \u2208 l :=\n  sorry\n\ntheorem reduce_option_nth_iff {\u03b1 : Type u} {l : List (Option \u03b1)} {x : \u03b1} :\n    (\u2203 (i : \u2115), nth l i = some (some x)) \u2194 \u2203 (i : \u2115), nth (reduce_option l) i = some x :=\n  sorry\n\n/-! ### filter -/\n\ntheorem filter_eq_foldr {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    filter p l = foldr (fun (a : \u03b1) (out : List \u03b1) => ite (p a) (a :: out) out) [] l :=\n  sorry\n\ntheorem filter_congr {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} [decidable_pred p]\n    [decidable_pred q] {l : List \u03b1} : (\u2200 (x : \u03b1), x \u2208 l \u2192 (p x \u2194 q x)) \u2192 filter p l = filter q l :=\n  sorry\n\n@[simp] theorem filter_subset {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] (l : List \u03b1) :\n    filter p l \u2286 l :=\n  sublist.subset (filter_sublist l)\n\ntheorem of_mem_filter {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : List \u03b1} :\n    a \u2208 filter p l \u2192 p a :=\n  sorry\n\ntheorem mem_of_mem_filter {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : List \u03b1}\n    (h : a \u2208 filter p l) : a \u2208 l :=\n  filter_subset l h\n\ntheorem mem_filter_of_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : List \u03b1} :\n    a \u2208 l \u2192 p a \u2192 a \u2208 filter p l :=\n  sorry\n\n@[simp] theorem mem_filter {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : List \u03b1} :\n    a \u2208 filter p l \u2194 a \u2208 l \u2227 p a :=\n  sorry\n\ntheorem filter_eq_self {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} :\n    filter p l = l \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 p a :=\n  sorry\n\ntheorem filter_eq_nil {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} :\n    filter p l = [] \u2194 \u2200 (a : \u03b1), a \u2208 l \u2192 \u00acp a :=\n  sorry\n\ntheorem filter_sublist_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {l\u2081 : List \u03b1}\n    {l\u2082 : List \u03b1} (s : l\u2081 <+ l\u2082) : filter p l\u2081 <+ filter p l\u2082 :=\n  filter_map_eq_filter p \u25b8 sublist.filter_map (option.guard p) s\n\ntheorem filter_of_map {\u03b1 : Type u} {\u03b2 : Type v} (p : \u03b1 \u2192 Prop) [decidable_pred p] (f : \u03b2 \u2192 \u03b1)\n    (l : List \u03b2) : filter p (map f l) = map f (filter (p \u2218 f) l) :=\n  sorry\n\n@[simp] theorem filter_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop)\n    [decidable_pred q] (l : List \u03b1) : filter p (filter q l) = filter (fun (a : \u03b1) => p a \u2227 q a) l :=\n  sorry\n\n@[simp] theorem filter_true {\u03b1 : Type u} {h : decidable_pred fun (a : \u03b1) => True} (l : List \u03b1) :\n    filter (fun (a : \u03b1) => True) l = l :=\n  sorry\n\n@[simp] theorem filter_false {\u03b1 : Type u} {h : decidable_pred fun (a : \u03b1) => False} (l : List \u03b1) :\n    filter (fun (a : \u03b1) => False) l = [] :=\n  sorry\n\n@[simp] theorem span_eq_take_drop {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    span p l = (take_while p l, drop_while p l) :=\n  sorry\n\n@[simp] theorem take_while_append_drop {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    take_while p l ++ drop_while p l = l :=\n  sorry\n\n@[simp] theorem countp_nil {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p [] = 0 := rfl\n\n@[simp] theorem countp_cons_of_pos {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {a : \u03b1}\n    (l : List \u03b1) (pa : p a) : countp p (a :: l) = countp p l + 1 :=\n  if_pos pa\n\n@[simp] theorem countp_cons_of_neg {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {a : \u03b1}\n    (l : List \u03b1) (pa : \u00acp a) : countp p (a :: l) = countp p l :=\n  if_neg pa\n\ntheorem countp_eq_length_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    countp p l = length (filter p l) :=\n  sorry\n\n@[simp] theorem countp_append {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l\u2081 : List \u03b1)\n    (l\u2082 : List \u03b1) : countp p (l\u2081 ++ l\u2082) = countp p l\u2081 + countp p l\u2082 :=\n  sorry\n\ntheorem countp_pos {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {l : List \u03b1} :\n    0 < countp p l \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 l), p a :=\n  sorry\n\ntheorem countp_le_of_sublist {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {l\u2081 : List \u03b1}\n    {l\u2082 : List \u03b1} (s : l\u2081 <+ l\u2082) : countp p l\u2081 \u2264 countp p l\u2082 :=\n  sorry\n\n@[simp] theorem countp_filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] {q : \u03b1 \u2192 Prop}\n    [decidable_pred q] (l : List \u03b1) : countp p (filter q l) = countp (fun (a : \u03b1) => p a \u2227 q a) l :=\n  sorry\n\n/-! ### count -/\n\n@[simp] theorem count_nil {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : count a [] = 0 := rfl\n\ntheorem count_cons {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    count a (b :: l) = ite (a = b) (Nat.succ (count a l)) (count a l) :=\n  rfl\n\ntheorem count_cons' {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    count a (b :: l) = count a l + ite (a = b) 1 0 :=\n  sorry\n\n@[simp] theorem count_cons_self {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    count a (a :: l) = Nat.succ (count a l) :=\n  if_pos rfl\n\n@[simp] theorem count_cons_of_ne {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (h : a \u2260 b)\n    (l : List \u03b1) : count a (b :: l) = count a l :=\n  if_neg h\n\ntheorem count_tail {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) (a : \u03b1) (h : 0 < length l) :\n    count a (tail l) = count a l - ite (a = nth_le l 0 h) 1 0 :=\n  sorry\n\ntheorem count_le_of_sublist {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+ l\u2082 \u2192 count a l\u2081 \u2264 count a l\u2082 :=\n  countp_le_of_sublist (Eq a)\n\ntheorem count_le_count_cons {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    count a l \u2264 count a (b :: l) :=\n  count_le_of_sublist a (sublist_cons b l)\n\ntheorem count_singleton {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : count a [a] = 1 := if_pos rfl\n\n@[simp] theorem count_append {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    count a (l\u2081 ++ l\u2082) = count a l\u2081 + count a l\u2082 :=\n  countp_append (Eq a)\n\ntheorem count_concat {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    count a (concat l a) = Nat.succ (count a l) :=\n  sorry\n\ntheorem count_pos {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} : 0 < count a l \u2194 a \u2208 l := sorry\n\n@[simp] theorem count_eq_zero_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1}\n    (h : \u00aca \u2208 l) : count a l = 0 :=\n  by_contradiction fun (h' : \u00account a l = 0) => h (iff.mp count_pos (nat.pos_of_ne_zero h'))\n\ntheorem not_mem_of_count_eq_zero {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1}\n    (h : count a l = 0) : \u00aca \u2208 l :=\n  fun (h' : a \u2208 l) => ne_of_gt (iff.mpr count_pos h') h\n\n@[simp] theorem count_repeat {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (n : \u2115) :\n    count a (repeat a n) = n :=\n  sorry\n\ntheorem le_count_iff_repeat_sublist {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} {n : \u2115} :\n    n \u2264 count a l \u2194 repeat a n <+ l :=\n  sorry\n\ntheorem repeat_count_eq_of_count_eq_length {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1}\n    (h : count a l = length l) : repeat a (count a l) = l :=\n  eq_of_sublist_of_length_eq (iff.mp le_count_iff_repeat_sublist (le_refl (count a l)))\n    (Eq.trans (length_repeat a (count a l)) h)\n\n@[simp] theorem count_filter {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    {l : List \u03b1} (h : p a) : count a (filter p l) = count a l :=\n  sorry\n\n/-! ### prefix, suffix, infix -/\n\n@[simp] theorem prefix_append {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2081 <+: l\u2081 ++ l\u2082 :=\n  Exists.intro l\u2082 rfl\n\n@[simp] theorem suffix_append {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2082 <:+ l\u2081 ++ l\u2082 :=\n  Exists.intro l\u2081 rfl\n\ntheorem infix_append {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (l\u2083 : List \u03b1) :\n    l\u2082 <:+: l\u2081 ++ l\u2082 ++ l\u2083 :=\n  Exists.intro l\u2081 (Exists.intro l\u2083 rfl)\n\n@[simp] theorem infix_append' {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (l\u2083 : List \u03b1) :\n    l\u2082 <:+: l\u2081 ++ (l\u2082 ++ l\u2083) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (l\u2082 <:+: l\u2081 ++ (l\u2082 ++ l\u2083))) (Eq.symm (append_assoc l\u2081 l\u2082 l\u2083))))\n    (infix_append l\u2081 l\u2082 l\u2083)\n\ntheorem nil_prefix {\u03b1 : Type u} (l : List \u03b1) : [] <+: l := Exists.intro l rfl\n\ntheorem nil_suffix {\u03b1 : Type u} (l : List \u03b1) : [] <:+ l := Exists.intro l (append_nil l)\n\ntheorem prefix_refl {\u03b1 : Type u} (l : List \u03b1) : l <+: l := Exists.intro [] (append_nil l)\n\ntheorem suffix_refl {\u03b1 : Type u} (l : List \u03b1) : l <:+ l := Exists.intro [] rfl\n\n@[simp] theorem suffix_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : l <:+ a :: l := suffix_append [a]\n\ntheorem prefix_concat {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : l <+: concat l a := sorry\n\ntheorem infix_of_prefix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <+: l\u2082 \u2192 l\u2081 <:+: l\u2082 := sorry\n\ntheorem infix_of_suffix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <:+ l\u2082 \u2192 l\u2081 <:+: l\u2082 := sorry\n\ntheorem infix_refl {\u03b1 : Type u} (l : List \u03b1) : l <:+: l := infix_of_prefix (prefix_refl l)\n\ntheorem nil_infix {\u03b1 : Type u} (l : List \u03b1) : [] <:+: l := infix_of_prefix (nil_prefix l)\n\ntheorem infix_cons {\u03b1 : Type u} {L\u2081 : List \u03b1} {L\u2082 : List \u03b1} {x : \u03b1} :\n    L\u2081 <:+: L\u2082 \u2192 L\u2081 <:+: x :: L\u2082 :=\n  sorry\n\ntheorem is_prefix.trans {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} :\n    l\u2081 <+: l\u2082 \u2192 l\u2082 <+: l\u2083 \u2192 l\u2081 <+: l\u2083 :=\n  sorry\n\ntheorem is_suffix.trans {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} :\n    l\u2081 <:+ l\u2082 \u2192 l\u2082 <:+ l\u2083 \u2192 l\u2081 <:+ l\u2083 :=\n  sorry\n\ntheorem is_infix.trans {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} :\n    l\u2081 <:+: l\u2082 \u2192 l\u2082 <:+: l\u2083 \u2192 l\u2081 <:+: l\u2083 :=\n  sorry\n\ntheorem sublist_of_infix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <:+: l\u2082 \u2192 l\u2081 <+ l\u2082 := sorry\n\ntheorem sublist_of_prefix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <+: l\u2082 \u2192 l\u2081 <+ l\u2082 :=\n  sublist_of_infix \u2218 infix_of_prefix\n\ntheorem sublist_of_suffix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : l\u2081 <:+ l\u2082 \u2192 l\u2081 <+ l\u2082 :=\n  sublist_of_infix \u2218 infix_of_suffix\n\ntheorem reverse_suffix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    reverse l\u2081 <:+ reverse l\u2082 \u2194 l\u2081 <+: l\u2082 :=\n  sorry\n\ntheorem reverse_prefix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    reverse l\u2081 <+: reverse l\u2082 \u2194 l\u2081 <:+ l\u2082 :=\n  sorry\n\ntheorem length_le_of_infix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s : l\u2081 <:+: l\u2082) :\n    length l\u2081 \u2264 length l\u2082 :=\n  length_le_of_sublist (sublist_of_infix s)\n\ntheorem eq_nil_of_infix_nil {\u03b1 : Type u} {l : List \u03b1} (s : l <:+: []) : l = [] :=\n  eq_nil_of_sublist_nil (sublist_of_infix s)\n\ntheorem eq_nil_of_prefix_nil {\u03b1 : Type u} {l : List \u03b1} (s : l <+: []) : l = [] :=\n  eq_nil_of_infix_nil (infix_of_prefix s)\n\ntheorem eq_nil_of_suffix_nil {\u03b1 : Type u} {l : List \u03b1} (s : l <:+ []) : l = [] :=\n  eq_nil_of_infix_nil (infix_of_suffix s)\n\ntheorem infix_iff_prefix_suffix {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l\u2081 <:+: l\u2082 \u2194 \u2203 (t : List \u03b1), l\u2081 <+: t \u2227 t <:+ l\u2082 :=\n  sorry\n\ntheorem eq_of_infix_of_length_eq {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s : l\u2081 <:+: l\u2082) :\n    length l\u2081 = length l\u2082 \u2192 l\u2081 = l\u2082 :=\n  eq_of_sublist_of_length_eq (sublist_of_infix s)\n\ntheorem eq_of_prefix_of_length_eq {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s : l\u2081 <+: l\u2082) :\n    length l\u2081 = length l\u2082 \u2192 l\u2081 = l\u2082 :=\n  eq_of_sublist_of_length_eq (sublist_of_prefix s)\n\ntheorem eq_of_suffix_of_length_eq {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (s : l\u2081 <:+ l\u2082) :\n    length l\u2081 = length l\u2082 \u2192 l\u2081 = l\u2082 :=\n  eq_of_sublist_of_length_eq (sublist_of_suffix s)\n\ntheorem prefix_of_prefix_length_le {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} :\n    l\u2081 <+: l\u2083 \u2192 l\u2082 <+: l\u2083 \u2192 length l\u2081 \u2264 length l\u2082 \u2192 l\u2081 <+: l\u2082 :=\n  sorry\n\ntheorem prefix_or_prefix_of_prefix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1}\n    (h\u2081 : l\u2081 <+: l\u2083) (h\u2082 : l\u2082 <+: l\u2083) : l\u2081 <+: l\u2082 \u2228 l\u2082 <+: l\u2081 :=\n  or.imp (prefix_of_prefix_length_le h\u2081 h\u2082) (prefix_of_prefix_length_le h\u2082 h\u2081)\n    (le_total (length l\u2081) (length l\u2082))\n\ntheorem suffix_of_suffix_length_le {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1}\n    (h\u2081 : l\u2081 <:+ l\u2083) (h\u2082 : l\u2082 <:+ l\u2083) (ll : length l\u2081 \u2264 length l\u2082) : l\u2081 <:+ l\u2082 :=\n  sorry\n\ntheorem suffix_or_suffix_of_suffix {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1}\n    (h\u2081 : l\u2081 <:+ l\u2083) (h\u2082 : l\u2082 <:+ l\u2083) : l\u2081 <:+ l\u2082 \u2228 l\u2082 <:+ l\u2081 :=\n  or.imp (iff.mp reverse_prefix) (iff.mp reverse_prefix)\n    (prefix_or_prefix_of_prefix (iff.mpr reverse_prefix h\u2081) (iff.mpr reverse_prefix h\u2082))\n\ntheorem infix_of_mem_join {\u03b1 : Type u} {L : List (List \u03b1)} {l : List \u03b1} : l \u2208 L \u2192 l <:+: join L :=\n  sorry\n\ntheorem prefix_append_right_inj {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (l : List \u03b1) :\n    l ++ l\u2081 <+: l ++ l\u2082 \u2194 l\u2081 <+: l\u2082 :=\n  sorry\n\ntheorem prefix_cons_inj {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (a : \u03b1) :\n    a :: l\u2081 <+: a :: l\u2082 \u2194 l\u2081 <+: l\u2082 :=\n  prefix_append_right_inj [a]\n\ntheorem take_prefix {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : take n l <+: l :=\n  Exists.intro (drop n l) (take_append_drop n l)\n\ntheorem drop_suffix {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : drop n l <:+ l :=\n  Exists.intro (take n l) (take_append_drop n l)\n\ntheorem tail_suffix {\u03b1 : Type u} (l : List \u03b1) : tail l <:+ l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (tail l <:+ l)) (Eq.symm (drop_one l)))) (drop_suffix 1 l)\n\ntheorem tail_subset {\u03b1 : Type u} (l : List \u03b1) : tail l \u2286 l :=\n  sublist.subset (sublist_of_suffix (tail_suffix l))\n\ntheorem prefix_iff_eq_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+: l\u2082 \u2194 l\u2081 ++ drop (length l\u2081) l\u2082 = l\u2082 :=\n  sorry\n\ntheorem suffix_iff_eq_append {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <:+ l\u2082 \u2194 take (length l\u2082 - length l\u2081) l\u2082 ++ l\u2081 = l\u2082 :=\n  sorry\n\ntheorem prefix_iff_eq_take {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <+: l\u2082 \u2194 l\u2081 = take (length l\u2081) l\u2082 :=\n  sorry\n\ntheorem suffix_iff_eq_drop {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 <:+ l\u2082 \u2194 l\u2081 = drop (length l\u2082 - length l\u2081) l\u2082 :=\n  sorry\n\nprotected instance decidable_prefix {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    Decidable (l\u2081 <+: l\u2082) :=\n  sorry\n\n-- Alternatively, use mem_tails\n\nprotected instance decidable_suffix {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    Decidable (l\u2081 <:+ l\u2082) :=\n  sorry\n\ntheorem prefix_take_le_iff {\u03b1 : Type u} {L : List (List (Option \u03b1))} {m : \u2115} {n : \u2115}\n    (hm : m < length L) : take m L <+: take n L \u2194 m \u2264 n :=\n  sorry\n\ntheorem cons_prefix_iff {\u03b1 : Type u} {l : List \u03b1} {l' : List \u03b1} {x : \u03b1} {y : \u03b1} :\n    x :: l <+: y :: l' \u2194 x = y \u2227 l <+: l' :=\n  sorry\n\ntheorem map_prefix {\u03b1 : Type u} {\u03b2 : Type v} {l : List \u03b1} {l' : List \u03b1} (f : \u03b1 \u2192 \u03b2) (h : l <+: l') :\n    map f l <+: map f l' :=\n  sorry\n\ntheorem is_prefix.filter_map {\u03b1 : Type u} {\u03b2 : Type v} {l : List \u03b1} {l' : List \u03b1} (h : l <+: l')\n    (f : \u03b1 \u2192 Option \u03b2) : filter_map f l <+: filter_map f l' :=\n  sorry\n\ntheorem is_prefix.reduce_option {\u03b1 : Type u} {l : List (Option \u03b1)} {l' : List (Option \u03b1)}\n    (h : l <+: l') : reduce_option l <+: reduce_option l' :=\n  is_prefix.filter_map h id\n\n@[simp] theorem mem_inits {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) : s \u2208 inits t \u2194 s <+: t := sorry\n\n@[simp] theorem mem_tails {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) : s \u2208 tails t \u2194 s <:+ t := sorry\n\ntheorem inits_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    inits (a :: l) = [] :: map (fun (t : List \u03b1) => a :: t) (inits l) :=\n  sorry\n\ntheorem tails_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : tails (a :: l) = (a :: l) :: tails l := sorry\n\n@[simp] theorem inits_append {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    inits (s ++ t) = inits s ++ map (fun (l : List \u03b1) => s ++ l) (tail (inits t)) :=\n  sorry\n\n@[simp] theorem tails_append {\u03b1 : Type u} (s : List \u03b1) (t : List \u03b1) :\n    tails (s ++ t) = map (fun (l : List \u03b1) => l ++ t) (tails s) ++ tail (tails t) :=\n  sorry\n\n-- the lemma names `inits_eq_tails` and `tails_eq_inits` are like `sublists_eq_sublists'`\n\ntheorem inits_eq_tails {\u03b1 : Type u} (l : List \u03b1) :\n    inits l = reverse (map reverse (tails (reverse l))) :=\n  sorry\n\ntheorem tails_eq_inits {\u03b1 : Type u} (l : List \u03b1) :\n    tails l = reverse (map reverse (inits (reverse l))) :=\n  sorry\n\ntheorem inits_reverse {\u03b1 : Type u} (l : List \u03b1) :\n    inits (reverse l) = reverse (map reverse (tails l)) :=\n  sorry\n\ntheorem tails_reverse {\u03b1 : Type u} (l : List \u03b1) :\n    tails (reverse l) = reverse (map reverse (inits l)) :=\n  sorry\n\ntheorem map_reverse_inits {\u03b1 : Type u} (l : List \u03b1) :\n    map reverse (inits l) = reverse (tails (reverse l)) :=\n  sorry\n\ntheorem map_reverse_tails {\u03b1 : Type u} (l : List \u03b1) :\n    map reverse (tails l) = reverse (inits (reverse l)) :=\n  sorry\n\nprotected instance decidable_infix {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    Decidable (l\u2081 <:+: l\u2082) :=\n  sorry\n\n/-! ### sublists -/\n\n@[simp] theorem sublists'_nil {\u03b1 : Type u} : sublists' [] = [[]] := rfl\n\n@[simp] theorem sublists'_singleton {\u03b1 : Type u} (a : \u03b1) : sublists' [a] = [[], [a]] := rfl\n\ntheorem map_sublists'_aux {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (g : List \u03b2 \u2192 List \u03b3) (l : List \u03b1)\n    (f : List \u03b1 \u2192 List \u03b2) (r : List (List \u03b2)) :\n    map g (sublists'_aux l f r) = sublists'_aux l (g \u2218 f) (map g r) :=\n  sorry\n\ntheorem sublists'_aux_append {\u03b1 : Type u} {\u03b2 : Type v} (r' : List (List \u03b2)) (l : List \u03b1)\n    (f : List \u03b1 \u2192 List \u03b2) (r : List (List \u03b2)) :\n    sublists'_aux l f (r ++ r') = sublists'_aux l f r ++ r' :=\n  sorry\n\ntheorem sublists'_aux_eq_sublists' {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) (f : List \u03b1 \u2192 List \u03b2)\n    (r : List (List \u03b2)) : sublists'_aux l f r = map f (sublists' l) ++ r :=\n  sorry\n\n@[simp] theorem sublists'_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    sublists' (a :: l) = sublists' l ++ map (List.cons a) (sublists' l) :=\n  sorry\n\n@[simp] theorem mem_sublists' {\u03b1 : Type u} {s : List \u03b1} {t : List \u03b1} : s \u2208 sublists' t \u2194 s <+ t :=\n  sorry\n\n@[simp] theorem length_sublists' {\u03b1 : Type u} (l : List \u03b1) :\n    length (sublists' l) = bit0 1 ^ length l :=\n  sorry\n\n@[simp] theorem sublists_nil {\u03b1 : Type u} : sublists [] = [[]] := rfl\n\n@[simp] theorem sublists_singleton {\u03b1 : Type u} (a : \u03b1) : sublists [a] = [[], [a]] := rfl\n\ntheorem sublists_aux\u2081_eq_sublists_aux {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) (f : List \u03b1 \u2192 List \u03b2) :\n    sublists_aux\u2081 l f = sublists_aux l fun (ys : List \u03b1) (r : List \u03b2) => f ys ++ r :=\n  sorry\n\ntheorem sublists_aux_cons_eq_sublists_aux\u2081 {\u03b1 : Type u} (l : List \u03b1) :\n    sublists_aux l List.cons = sublists_aux\u2081 l fun (x : List \u03b1) => [x] :=\n  sorry\n\ntheorem sublists_aux_eq_foldr.aux {\u03b1 : Type u} {\u03b2 : Type v} {a : \u03b1} {l : List \u03b1}\n    (IH\u2081 :\n      \u2200 (f : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b2), sublists_aux l f = foldr f [] (sublists_aux l List.cons))\n    (IH\u2082 :\n      \u2200 (f : List \u03b1 \u2192 List (List \u03b1) \u2192 List (List \u03b1)),\n        sublists_aux l f = foldr f [] (sublists_aux l List.cons))\n    (f : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b2) :\n    sublists_aux (a :: l) f = foldr f [] (sublists_aux (a :: l) List.cons) :=\n  sorry\n\ntheorem sublists_aux_eq_foldr {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1)\n    (f : List \u03b1 \u2192 List \u03b2 \u2192 List \u03b2) : sublists_aux l f = foldr f [] (sublists_aux l List.cons) :=\n  sorry\n\ntheorem sublists_aux_cons_cons {\u03b1 : Type u} (l : List \u03b1) (a : \u03b1) :\n    sublists_aux (a :: l) List.cons =\n        [a] ::\n          foldr (fun (ys : List \u03b1) (r : List (List \u03b1)) => ys :: (a :: ys) :: r) []\n            (sublists_aux l List.cons) :=\n  sorry\n\ntheorem sublists_aux\u2081_append {\u03b1 : Type u} {\u03b2 : Type v} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1)\n    (f : List \u03b1 \u2192 List \u03b2) :\n    sublists_aux\u2081 (l\u2081 ++ l\u2082) f =\n        sublists_aux\u2081 l\u2081 f ++\n          sublists_aux\u2081 l\u2082\n            fun (x : List \u03b1) => f x ++ sublists_aux\u2081 l\u2081 (f \u2218 fun (_x : List \u03b1) => _x ++ x) :=\n  sorry\n\ntheorem sublists_aux\u2081_concat {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) (a : \u03b1) (f : List \u03b1 \u2192 List \u03b2) :\n    sublists_aux\u2081 (l ++ [a]) f =\n        sublists_aux\u2081 l f ++ f [a] ++ sublists_aux\u2081 l fun (x : List \u03b1) => f (x ++ [a]) :=\n  sorry\n\ntheorem sublists_aux\u2081_bind {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (l : List \u03b1) (f : List \u03b1 \u2192 List \u03b2)\n    (g : \u03b2 \u2192 List \u03b3) :\n    list.bind (sublists_aux\u2081 l f) g = sublists_aux\u2081 l fun (x : List \u03b1) => list.bind (f x) g :=\n  sorry\n\ntheorem sublists_aux_cons_append {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    sublists_aux (l\u2081 ++ l\u2082) List.cons =\n        sublists_aux l\u2081 List.cons ++\n          do \n            let x \u2190 sublists_aux l\u2082 List.cons\n            (fun (_x : List \u03b1) => _x ++ x) <$> sublists l\u2081 :=\n  sorry\n\ntheorem sublists_append {\u03b1 : Type u} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    sublists (l\u2081 ++ l\u2082) =\n        do \n          let x \u2190 sublists l\u2082\n          (fun (_x : List \u03b1) => _x ++ x) <$> sublists l\u2081 :=\n  sorry\n\n@[simp] theorem sublists_concat {\u03b1 : Type u} (l : List \u03b1) (a : \u03b1) :\n    sublists (l ++ [a]) = sublists l ++ map (fun (x : List \u03b1) => x ++ [a]) (sublists l) :=\n  sorry\n\ntheorem sublists_reverse {\u03b1 : Type u} (l : List \u03b1) :\n    sublists (reverse l) = map reverse (sublists' l) :=\n  sorry\n\ntheorem sublists_eq_sublists' {\u03b1 : Type u} (l : List \u03b1) :\n    sublists l = map reverse (sublists' (reverse l)) :=\n  sorry\n\ntheorem sublists'_reverse {\u03b1 : Type u} (l : List \u03b1) :\n    sublists' (reverse l) = map reverse (sublists l) :=\n  sorry\n\ntheorem sublists'_eq_sublists {\u03b1 : Type u} (l : List \u03b1) :\n    sublists' l = map reverse (sublists (reverse l)) :=\n  sorry\n\ntheorem sublists_aux_ne_nil {\u03b1 : Type u} (l : List \u03b1) : \u00ac[] \u2208 sublists_aux l List.cons := sorry\n\n@[simp] theorem mem_sublists {\u03b1 : Type u} {s : List \u03b1} {t : List \u03b1} : s \u2208 sublists t \u2194 s <+ t :=\n  sorry\n\n@[simp] theorem length_sublists {\u03b1 : Type u} (l : List \u03b1) :\n    length (sublists l) = bit0 1 ^ length l :=\n  sorry\n\ntheorem map_ret_sublist_sublists {\u03b1 : Type u} (l : List \u03b1) : map list.ret l <+ sublists l := sorry\n\n/-! ### sublists_len -/\n\n/-- Auxiliary function to construct the list of all sublists of a given length. Given an\ninteger `n`, a list `l`, a function `f` and an auxiliary list `L`, it returns the list made of\nof `f` applied to all sublists of `l` of length `n`, concatenated with `L`. -/\ndef sublists_len_aux {\u03b1 : Type u_1} {\u03b2 : Type u_2} : \u2115 \u2192 List \u03b1 \u2192 (List \u03b1 \u2192 \u03b2) \u2192 List \u03b2 \u2192 List \u03b2 :=\n  sorry\n\n/-- The list of all sublists of a list `l` that are of length `n`. For instance, for\n`l = [0, 1, 2, 3]` and `n = 2`, one gets\n`[[2, 3], [1, 3], [1, 2], [0, 3], [0, 2], [0, 1]]`. -/\ndef sublists_len {\u03b1 : Type u_1} (n : \u2115) (l : List \u03b1) : List (List \u03b1) := sublists_len_aux n l id []\n\ntheorem sublists_len_aux_append {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (n : \u2115) (l : List \u03b1)\n    (f : List \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (r : List \u03b2) (s : List \u03b3) :\n    sublists_len_aux n l (g \u2218 f) (map g r ++ s) = map g (sublists_len_aux n l f r) ++ s :=\n  sorry\n\ntheorem sublists_len_aux_eq {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l : List \u03b1) (n : \u2115) (f : List \u03b1 \u2192 \u03b2)\n    (r : List \u03b2) : sublists_len_aux n l f r = map f (sublists_len n l) ++ r :=\n  sorry\n\ntheorem sublists_len_aux_zero {\u03b2 : Type v} {\u03b1 : Type u_1} (l : List \u03b1) (f : List \u03b1 \u2192 \u03b2)\n    (r : List \u03b2) : sublists_len_aux 0 l f r = f [] :: r :=\n  list.cases_on l (Eq.refl (sublists_len_aux 0 [] f r))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => Eq.refl (sublists_len_aux 0 (l_hd :: l_tl) f r)\n\n@[simp] theorem sublists_len_zero {\u03b1 : Type u_1} (l : List \u03b1) : sublists_len 0 l = [[]] :=\n  sublists_len_aux_zero l id []\n\n@[simp] theorem sublists_len_succ_nil {\u03b1 : Type u_1} (n : \u2115) : sublists_len (n + 1) [] = [] := rfl\n\n@[simp] theorem sublists_len_succ_cons {\u03b1 : Type u_1} (n : \u2115) (a : \u03b1) (l : List \u03b1) :\n    sublists_len (n + 1) (a :: l) =\n        sublists_len (n + 1) l ++ map (List.cons a) (sublists_len n l) :=\n  sorry\n\n@[simp] theorem length_sublists_len {\u03b1 : Type u_1} (n : \u2115) (l : List \u03b1) :\n    length (sublists_len n l) = nat.choose (length l) n :=\n  sorry\n\ntheorem sublists_len_sublist_sublists' {\u03b1 : Type u_1} (n : \u2115) (l : List \u03b1) :\n    sublists_len n l <+ sublists' l :=\n  sorry\n\ntheorem sublists_len_sublist_of_sublist {\u03b1 : Type u_1} (n : \u2115) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (h : l\u2081 <+ l\u2082) : sublists_len n l\u2081 <+ sublists_len n l\u2082 :=\n  sorry\n\ntheorem length_of_sublists_len {\u03b1 : Type u_1} {n : \u2115} {l : List \u03b1} {l' : List \u03b1} :\n    l' \u2208 sublists_len n l \u2192 length l' = n :=\n  sorry\n\ntheorem mem_sublists_len_self {\u03b1 : Type u_1} {l : List \u03b1} {l' : List \u03b1} (h : l' <+ l) :\n    l' \u2208 sublists_len (length l') l :=\n  sorry\n\n@[simp] theorem mem_sublists_len {\u03b1 : Type u_1} {n : \u2115} {l : List \u03b1} {l' : List \u03b1} :\n    l' \u2208 sublists_len n l \u2194 l' <+ l \u2227 length l' = n :=\n  sorry\n\n/-! ### permutations -/\n\n@[simp] theorem permutations_aux_nil {\u03b1 : Type u} (is : List \u03b1) : permutations_aux [] is = [] :=\n  sorry\n\n@[simp] theorem permutations_aux_cons {\u03b1 : Type u} (t : \u03b1) (ts : List \u03b1) (is : List \u03b1) :\n    permutations_aux (t :: ts) is =\n        foldr (fun (y : List \u03b1) (r : List (List \u03b1)) => prod.snd (permutations_aux2 t ts r y id))\n          (permutations_aux ts (t :: is)) (permutations is) :=\n  sorry\n\n/-! ### insert -/\n\n@[simp] theorem insert_nil {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : insert a [] = [a] := rfl\n\ntheorem insert.def {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    insert a l = ite (a \u2208 l) l (a :: l) :=\n  rfl\n\n@[simp] theorem insert_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    insert a l = l :=\n  sorry\n\n@[simp] theorem insert_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : \u00aca \u2208 l) :\n    insert a l = a :: l :=\n  sorry\n\n@[simp] theorem mem_insert_iff {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {l : List \u03b1} :\n    a \u2208 insert b l \u2194 a = b \u2228 a \u2208 l :=\n  sorry\n\n@[simp] theorem suffix_insert {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    l <:+ insert a l :=\n  sorry\n\n@[simp] theorem mem_insert_self {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    a \u2208 insert a l :=\n  iff.mpr mem_insert_iff (Or.inl rfl)\n\ntheorem mem_insert_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    a \u2208 insert b l :=\n  iff.mpr mem_insert_iff (Or.inr h)\n\ntheorem eq_or_mem_of_mem_insert {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {l : List \u03b1}\n    (h : a \u2208 insert b l) : a = b \u2228 a \u2208 l :=\n  iff.mp mem_insert_iff h\n\n@[simp] theorem length_insert_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    length (insert a l) = length l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (insert a l) = length l)) (insert_of_mem h)))\n    (Eq.refl (length l))\n\n@[simp] theorem length_insert_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1}\n    (h : \u00aca \u2208 l) : length (insert a l) = length l + 1 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (length (insert a l) = length l + 1)) (insert_of_not_mem h)))\n    (Eq.refl (length (a :: l)))\n\n/-! ### erasep -/\n\n@[simp] theorem erasep_nil {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] : erasep p [] = [] := rfl\n\ntheorem erasep_cons {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] (a : \u03b1) (l : List \u03b1) :\n    erasep p (a :: l) = ite (p a) l (a :: erasep p l) :=\n  rfl\n\n@[simp] theorem erasep_cons_of_pos {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    {l : List \u03b1} (h : p a) : erasep p (a :: l) = l :=\n  sorry\n\n@[simp] theorem erasep_cons_of_neg {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    {l : List \u03b1} (h : \u00acp a) : erasep p (a :: l) = a :: erasep p l :=\n  sorry\n\ntheorem erasep_of_forall_not {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1}\n    (h : \u2200 (a : \u03b1), a \u2208 l \u2192 \u00acp a) : erasep p l = l :=\n  sorry\n\ntheorem exists_of_erasep {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1} {a : \u03b1}\n    (al : a \u2208 l) (pa : p a) :\n    \u2203 (a : \u03b1),\n        \u2203 (l\u2081 : List \u03b1),\n          \u2203 (l\u2082 : List \u03b1),\n            (\u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acp b) \u2227 p a \u2227 l = l\u2081 ++ a :: l\u2082 \u2227 erasep p l = l\u2081 ++ l\u2082 :=\n  sorry\n\ntheorem exists_or_eq_self_of_erasep {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    erasep p l = l \u2228\n        \u2203 (a : \u03b1),\n          \u2203 (l\u2081 : List \u03b1),\n            \u2203 (l\u2082 : List \u03b1),\n              (\u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acp b) \u2227 p a \u2227 l = l\u2081 ++ a :: l\u2082 \u2227 erasep p l = l\u2081 ++ l\u2082 :=\n  sorry\n\n@[simp] theorem length_erasep_of_mem {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l : List \u03b1}\n    {a : \u03b1} (al : a \u2208 l) (pa : p a) : length (erasep p l) = Nat.pred (length l) :=\n  sorry\n\ntheorem erasep_append_left {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} (pa : p a)\n    {l\u2081 : List \u03b1} (l\u2082 : List \u03b1) : a \u2208 l\u2081 \u2192 erasep p (l\u2081 ++ l\u2082) = erasep p l\u2081 ++ l\u2082 :=\n  sorry\n\ntheorem erasep_append_right {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l\u2081 : List \u03b1}\n    (l\u2082 : List \u03b1) : (\u2200 (b : \u03b1), b \u2208 l\u2081 \u2192 \u00acp b) \u2192 erasep p (l\u2081 ++ l\u2082) = l\u2081 ++ erasep p l\u2082 :=\n  sorry\n\ntheorem erasep_sublist {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] (l : List \u03b1) :\n    erasep p l <+ l :=\n  sorry\n\ntheorem erasep_subset {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] (l : List \u03b1) :\n    erasep p l \u2286 l :=\n  sublist.subset (erasep_sublist l)\n\ntheorem sublist.erasep {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (s : l\u2081 <+ l\u2082) : erasep p l\u2081 <+ erasep p l\u2082 :=\n  sorry\n\ntheorem mem_of_mem_erasep {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : List \u03b1} :\n    a \u2208 erasep p l \u2192 a \u2208 l :=\n  erasep_subset l\n\n@[simp] theorem mem_erasep_of_neg {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    {l : List \u03b1} (pa : \u00acp a) : a \u2208 erasep p l \u2194 a \u2208 l :=\n  sorry\n\ntheorem erasep_map {\u03b1 : Type u} {\u03b2 : Type v} {p : \u03b1 \u2192 Prop} [decidable_pred p] (f : \u03b2 \u2192 \u03b1)\n    (l : List \u03b2) : erasep p (map f l) = map f (erasep (p \u2218 f) l) :=\n  sorry\n\n@[simp] theorem extractp_eq_find_erasep {\u03b1 : Type u} {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (l : List \u03b1) : extractp p l = (find p l, erasep p l) :=\n  sorry\n\n/-! ### erase -/\n\n@[simp] theorem erase_nil {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : list.erase [] a = [] := rfl\n\ntheorem erase_cons {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    list.erase (b :: l) a = ite (b = a) l (b :: list.erase l a) :=\n  rfl\n\n@[simp] theorem erase_cons_head {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    list.erase (a :: l) a = l :=\n  sorry\n\n@[simp] theorem erase_cons_tail {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (l : List \u03b1)\n    (h : b \u2260 a) : list.erase (b :: l) a = b :: list.erase l a :=\n  sorry\n\ntheorem erase_eq_erasep {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    list.erase l a = erasep (Eq a) l :=\n  sorry\n\n@[simp] theorem erase_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : \u00aca \u2208 l) :\n    list.erase l a = l :=\n  sorry\n\ntheorem exists_erase_eq {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    \u2203 (l\u2081 : List \u03b1), \u2203 (l\u2082 : List \u03b1), \u00aca \u2208 l\u2081 \u2227 l = l\u2081 ++ a :: l\u2082 \u2227 list.erase l a = l\u2081 ++ l\u2082 :=\n  sorry\n\n@[simp] theorem length_erase_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l : List \u03b1} (h : a \u2208 l) :\n    length (list.erase l a) = Nat.pred (length l) :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (length (list.erase l a) = Nat.pred (length l))) (erase_eq_erasep a l)))\n    (length_erasep_of_mem h rfl)\n\ntheorem erase_append_left {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} (l\u2082 : List \u03b1)\n    (h : a \u2208 l\u2081) : list.erase (l\u2081 ++ l\u2082) a = list.erase l\u2081 a ++ l\u2082 :=\n  sorry\n\ntheorem erase_append_right {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} (l\u2082 : List \u03b1)\n    (h : \u00aca \u2208 l\u2081) : list.erase (l\u2081 ++ l\u2082) a = l\u2081 ++ list.erase l\u2082 a :=\n  sorry\n\ntheorem erase_sublist {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) : list.erase l a <+ l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (list.erase l a <+ l)) (erase_eq_erasep a l))) (erasep_sublist l)\n\ntheorem erase_subset {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) : list.erase l a \u2286 l :=\n  sublist.subset (erase_sublist a l)\n\ntheorem sublist.erase {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (h : l\u2081 <+ l\u2082) : list.erase l\u2081 a <+ list.erase l\u2082 a :=\n  sorry\n\ntheorem mem_of_mem_erase {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {l : List \u03b1} :\n    a \u2208 list.erase l b \u2192 a \u2208 l :=\n  erase_subset b l\n\n@[simp] theorem mem_erase_of_ne {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {l : List \u03b1}\n    (ab : a \u2260 b) : a \u2208 list.erase l b \u2194 a \u2208 l :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 list.erase l b \u2194 a \u2208 l)) (erase_eq_erasep b l)))\n    (mem_erasep_of_neg (ne.symm ab))\n\ntheorem erase_comm {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (l : List \u03b1) :\n    list.erase (list.erase l a) b = list.erase (list.erase l b) a :=\n  sorry\n\ntheorem map_erase {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (finj : function.injective f) {a : \u03b1} (l : List \u03b1) :\n    map f (list.erase l a) = list.erase (map f l) (f a) :=\n  sorry\n\ntheorem map_foldl_erase {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (finj : function.injective f) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    map f (foldl list.erase l\u2081 l\u2082) =\n        foldl (fun (l : List \u03b2) (a : \u03b1) => list.erase l (f a)) (map f l\u2081) l\u2082 :=\n  sorry\n\n@[simp] theorem count_erase_self {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) (s : List \u03b1) :\n    count a (list.erase s a) = Nat.pred (count a s) :=\n  sorry\n\n@[simp] theorem count_erase_of_ne {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (ab : a \u2260 b)\n    (s : List \u03b1) : count a (list.erase s b) = count a s :=\n  sorry\n\n/-! ### diff -/\n\n@[simp] theorem diff_nil {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) : list.diff l [] = l := rfl\n\n@[simp] theorem diff_cons {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (a : \u03b1) :\n    list.diff l\u2081 (a :: l\u2082) = list.diff (list.erase l\u2081 a) l\u2082 :=\n  sorry\n\ntheorem diff_cons_right {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (a : \u03b1) :\n    list.diff l\u2081 (a :: l\u2082) = list.erase (list.diff l\u2081 l\u2082) a :=\n  sorry\n\ntheorem diff_erase {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (a : \u03b1) :\n    list.erase (list.diff l\u2081 l\u2082) a = list.diff (list.erase l\u2081 a) l\u2082 :=\n  sorry\n\n@[simp] theorem nil_diff {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) : list.diff [] l = [] := sorry\n\ntheorem diff_eq_foldl {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    list.diff l\u2081 l\u2082 = foldl list.erase l\u2081 l\u2082 :=\n  sorry\n\n@[simp] theorem diff_append {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (l\u2083 : List \u03b1) :\n    list.diff l\u2081 (l\u2082 ++ l\u2083) = list.diff (list.diff l\u2081 l\u2082) l\u2083 :=\n  sorry\n\n@[simp] theorem map_diff {\u03b1 : Type u} {\u03b2 : Type v} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (finj : function.injective f) {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    map f (list.diff l\u2081 l\u2082) = list.diff (map f l\u2081) (map f l\u2082) :=\n  sorry\n\ntheorem diff_sublist {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    list.diff l\u2081 l\u2082 <+ l\u2081 :=\n  sorry\n\ntheorem diff_subset {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    list.diff l\u2081 l\u2082 \u2286 l\u2081 :=\n  sublist.subset (diff_sublist l\u2081 l\u2082)\n\ntheorem mem_diff_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    a \u2208 l\u2081 \u2192 \u00aca \u2208 l\u2082 \u2192 a \u2208 list.diff l\u2081 l\u2082 :=\n  sorry\n\ntheorem sublist.diff_right {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l\u2083 : List \u03b1} :\n    l\u2081 <+ l\u2082 \u2192 list.diff l\u2081 l\u2083 <+ list.diff l\u2082 l\u2083 :=\n  sorry\n\ntheorem erase_diff_erase_sublist_of_sublist {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1}\n    {l\u2082 : List \u03b1} : l\u2081 <+ l\u2082 \u2192 list.diff (list.erase l\u2082 a) (list.erase l\u2081 a) <+ list.diff l\u2082 l\u2081 :=\n  sorry\n\n/-! ### enum -/\n\ntheorem length_enum_from {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) : length (enum_from n l) = length l :=\n  sorry\n\ntheorem length_enum {\u03b1 : Type u} (l : List \u03b1) : length (enum l) = length l := length_enum_from 0\n\n@[simp] theorem enum_from_nth {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) (m : \u2115) :\n    nth (enum_from n l) m = (fun (a : \u03b1) => (n + m, a)) <$> nth l m :=\n  sorry\n\n@[simp] theorem enum_nth {\u03b1 : Type u} (l : List \u03b1) (n : \u2115) :\n    nth (enum l) n = (fun (a : \u03b1) => (n, a)) <$> nth l n :=\n  sorry\n\n@[simp] theorem enum_from_map_snd {\u03b1 : Type u} (n : \u2115) (l : List \u03b1) :\n    map prod.snd (enum_from n l) = l :=\n  sorry\n\n@[simp] theorem enum_map_snd {\u03b1 : Type u} (l : List \u03b1) : map prod.snd (enum l) = l :=\n  enum_from_map_snd 0\n\ntheorem mem_enum_from {\u03b1 : Type u} {x : \u03b1} {i : \u2115} {j : \u2115} (xs : List \u03b1) :\n    (i, x) \u2208 enum_from j xs \u2192 j \u2264 i \u2227 i < j + length xs \u2227 x \u2208 xs :=\n  sorry\n\n/-! ### product -/\n\n@[simp] theorem nil_product {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b2) : product [] l = [] := rfl\n\n@[simp] theorem product_cons {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) :\n    product (a :: l\u2081) l\u2082 = map (fun (b : \u03b2) => (a, b)) l\u2082 ++ product l\u2081 l\u2082 :=\n  rfl\n\n@[simp] theorem product_nil {\u03b1 : Type u} {\u03b2 : Type v} (l : List \u03b1) : product l [] = [] := sorry\n\n@[simp] theorem mem_product {\u03b1 : Type u} {\u03b2 : Type v} {l\u2081 : List \u03b1} {l\u2082 : List \u03b2} {a : \u03b1} {b : \u03b2} :\n    (a, b) \u2208 product l\u2081 l\u2082 \u2194 a \u2208 l\u2081 \u2227 b \u2208 l\u2082 :=\n  sorry\n\ntheorem length_product {\u03b1 : Type u} {\u03b2 : Type v} (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) :\n    length (product l\u2081 l\u2082) = length l\u2081 * length l\u2082 :=\n  sorry\n\n/-! ### sigma -/\n\n@[simp] theorem nil_sigma {\u03b1 : Type u} {\u03c3 : \u03b1 \u2192 Type u_1} (l : (a : \u03b1) \u2192 List (\u03c3 a)) :\n    list.sigma [] l = [] :=\n  rfl\n\n@[simp] theorem sigma_cons {\u03b1 : Type u} {\u03c3 : \u03b1 \u2192 Type u_1} (a : \u03b1) (l\u2081 : List \u03b1)\n    (l\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)) :\n    list.sigma (a :: l\u2081) l\u2082 = map (sigma.mk a) (l\u2082 a) ++ list.sigma l\u2081 l\u2082 :=\n  rfl\n\n@[simp] theorem sigma_nil {\u03b1 : Type u} {\u03c3 : \u03b1 \u2192 Type u_1} (l : List \u03b1) :\n    (list.sigma l fun (a : \u03b1) => []) = [] :=\n  sorry\n\n@[simp] theorem mem_sigma {\u03b1 : Type u} {\u03c3 : \u03b1 \u2192 Type u_1} {l\u2081 : List \u03b1} {l\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)}\n    {a : \u03b1} {b : \u03c3 a} : sigma.mk a b \u2208 list.sigma l\u2081 l\u2082 \u2194 a \u2208 l\u2081 \u2227 b \u2208 l\u2082 a :=\n  sorry\n\ntheorem length_sigma {\u03b1 : Type u} {\u03c3 : \u03b1 \u2192 Type u_1} (l\u2081 : List \u03b1) (l\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)) :\n    length (list.sigma l\u2081 l\u2082) = sum (map (fun (a : \u03b1) => length (l\u2082 a)) l\u2081) :=\n  sorry\n\n/-! ### disjoint -/\n\ntheorem disjoint.symm {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (d : disjoint l\u2081 l\u2082) :\n    disjoint l\u2082 l\u2081 :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 l\u2082) (\u1fb0_1 : a \u2208 l\u2081) => idRhs False (d \u1fb0_1 \u1fb0)\n\ntheorem disjoint_comm {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : disjoint l\u2081 l\u2082 \u2194 disjoint l\u2082 l\u2081 :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint l\u2081 l\u2082 \u2194 \u2200 {a : \u03b1}, a \u2208 l\u2081 \u2192 \u00aca \u2208 l\u2082 :=\n  iff.rfl\n\ntheorem disjoint_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint l\u2081 l\u2082 \u2194 \u2200 {a : \u03b1}, a \u2208 l\u2082 \u2192 \u00aca \u2208 l\u2081 :=\n  disjoint_comm\n\ntheorem disjoint_iff_ne {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint l\u2081 l\u2082 \u2194 \u2200 (a : \u03b1), a \u2208 l\u2081 \u2192 \u2200 (b : \u03b1), b \u2208 l\u2082 \u2192 a \u2260 b :=\n  sorry\n\ntheorem disjoint_of_subset_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1} (ss : l\u2081 \u2286 l)\n    (d : disjoint l l\u2082) : disjoint l\u2081 l\u2082 :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 l\u2081) => idRhs (a \u2208 l\u2082 \u2192 False) (d (ss \u1fb0))\n\ntheorem disjoint_of_subset_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1} (ss : l\u2082 \u2286 l)\n    (d : disjoint l\u2081 l) : disjoint l\u2081 l\u2082 :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 l\u2081) (\u1fb0_1 : a \u2208 l\u2082) => idRhs False (d \u1fb0 (ss \u1fb0_1))\n\ntheorem disjoint_of_disjoint_cons_left {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint (a :: l\u2081) l\u2082 \u2192 disjoint l\u2081 l\u2082 :=\n  disjoint_of_subset_left (subset_cons a l\u2081)\n\ntheorem disjoint_of_disjoint_cons_right {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint l\u2081 (a :: l\u2082) \u2192 disjoint l\u2081 l\u2082 :=\n  disjoint_of_subset_right (subset_cons a l\u2082)\n\n@[simp] theorem disjoint_nil_left {\u03b1 : Type u} (l : List \u03b1) : disjoint [] l :=\n  fun {a : \u03b1} => idRhs (a \u2208 [] \u2192 a \u2208 l \u2192 False) (not.elim (not_mem_nil a))\n\n@[simp] theorem disjoint_nil_right {\u03b1 : Type u} (l : List \u03b1) : disjoint l [] :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (disjoint l [])) (propext disjoint_comm))) (disjoint_nil_left l)\n\n@[simp] theorem singleton_disjoint {\u03b1 : Type u} {l : List \u03b1} {a : \u03b1} : disjoint [a] l \u2194 \u00aca \u2208 l :=\n  sorry\n\n@[simp] theorem disjoint_singleton {\u03b1 : Type u} {l : List \u03b1} {a : \u03b1} : disjoint l [a] \u2194 \u00aca \u2208 l :=\n  sorry\n\n@[simp] theorem disjoint_append_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1} :\n    disjoint (l\u2081 ++ l\u2082) l \u2194 disjoint l\u2081 l \u2227 disjoint l\u2082 l :=\n  sorry\n\n@[simp] theorem disjoint_append_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1} :\n    disjoint l (l\u2081 ++ l\u2082) \u2194 disjoint l l\u2081 \u2227 disjoint l l\u2082 :=\n  sorry\n\n@[simp] theorem disjoint_cons_left {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint (a :: l\u2081) l\u2082 \u2194 \u00aca \u2208 l\u2082 \u2227 disjoint l\u2081 l\u2082 :=\n  sorry\n\n@[simp] theorem disjoint_cons_right {\u03b1 : Type u} {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    disjoint l\u2081 (a :: l\u2082) \u2194 \u00aca \u2208 l\u2081 \u2227 disjoint l\u2081 l\u2082 :=\n  sorry\n\ntheorem disjoint_of_disjoint_append_left_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1}\n    (d : disjoint (l\u2081 ++ l\u2082) l) : disjoint l\u2081 l :=\n  and.left (iff.mp disjoint_append_left d)\n\ntheorem disjoint_of_disjoint_append_left_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1}\n    (d : disjoint (l\u2081 ++ l\u2082) l) : disjoint l\u2082 l :=\n  and.right (iff.mp disjoint_append_left d)\n\ntheorem disjoint_of_disjoint_append_right_left {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {l : List \u03b1}\n    (d : disjoint l (l\u2081 ++ l\u2082)) : disjoint l l\u2081 :=\n  and.left (iff.mp disjoint_append_right d)\n\ntheorem disjoint_of_disjoint_append_right_right {\u03b1 : Type u} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    {l : List \u03b1} (d : disjoint l (l\u2081 ++ l\u2082)) : disjoint l l\u2082 :=\n  and.right (iff.mp disjoint_append_right d)\n\ntheorem disjoint_take_drop {\u03b1 : Type u} {l : List \u03b1} {m : \u2115} {n : \u2115} (hl : nodup l) (h : m \u2264 n) :\n    disjoint (take m l) (drop n l) :=\n  sorry\n\n/-! ### union -/\n\n@[simp] theorem nil_union {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) : [] \u222a l = l := rfl\n\n@[simp] theorem cons_union {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) (a : \u03b1) :\n    a :: l\u2081 \u222a l\u2082 = insert a (l\u2081 \u222a l\u2082) :=\n  rfl\n\n@[simp] theorem mem_union {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a \u2208 l\u2081 \u222a l\u2082 \u2194 a \u2208 l\u2081 \u2228 a \u2208 l\u2082 :=\n  sorry\n\ntheorem mem_union_left {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} (h : a \u2208 l\u2081)\n    (l\u2082 : List \u03b1) : a \u2208 l\u2081 \u222a l\u2082 :=\n  iff.mpr mem_union (Or.inl h)\n\ntheorem mem_union_right {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} (l\u2081 : List \u03b1) {l\u2082 : List \u03b1}\n    (h : a \u2208 l\u2082) : a \u2208 l\u2081 \u222a l\u2082 :=\n  iff.mpr mem_union (Or.inr h)\n\ntheorem sublist_suffix_of_union {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    \u2203 (t : List \u03b1), t <+ l\u2081 \u2227 t ++ l\u2082 = l\u2081 \u222a l\u2082 :=\n  sorry\n\ntheorem suffix_union_right {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l\u2082 <:+ l\u2081 \u222a l\u2082 :=\n  Exists.imp (fun (a : List \u03b1) => and.right) (sublist_suffix_of_union l\u2081 l\u2082)\n\ntheorem union_sublist_append {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l\u2081 \u222a l\u2082 <+ l\u2081 ++ l\u2082 :=\n  sorry\n\ntheorem forall_mem_union {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 l\u2081 \u222a l\u2082 \u2192 p x) \u2194 (\u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x) \u2227 \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 p x :=\n  sorry\n\ntheorem forall_mem_of_forall_mem_union_left {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop}\n    {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (h : \u2200 (x : \u03b1), x \u2208 l\u2081 \u222a l\u2082 \u2192 p x) (x : \u03b1) (H : x \u2208 l\u2081) : p x :=\n  and.left (iff.mp forall_mem_union h)\n\ntheorem forall_mem_of_forall_mem_union_right {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop}\n    {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} (h : \u2200 (x : \u03b1), x \u2208 l\u2081 \u222a l\u2082 \u2192 p x) (x : \u03b1) (H : x \u2208 l\u2082) : p x :=\n  and.right (iff.mp forall_mem_union h)\n\n/-! ### inter -/\n\n@[simp] theorem inter_nil {\u03b1 : Type u} [DecidableEq \u03b1] (l : List \u03b1) : [] \u2229 l = [] := rfl\n\n@[simp] theorem inter_cons_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} (l\u2081 : List \u03b1) {l\u2082 : List \u03b1}\n    (h : a \u2208 l\u2082) : (a :: l\u2081) \u2229 l\u2082 = a :: l\u2081 \u2229 l\u2082 :=\n  if_pos h\n\n@[simp] theorem inter_cons_of_not_mem {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} (l\u2081 : List \u03b1)\n    {l\u2082 : List \u03b1} (h : \u00aca \u2208 l\u2082) : (a :: l\u2081) \u2229 l\u2082 = l\u2081 \u2229 l\u2082 :=\n  if_neg h\n\ntheorem mem_of_mem_inter_left {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a \u2208 l\u2081 \u2229 l\u2082 \u2192 a \u2208 l\u2081 :=\n  mem_of_mem_filter\n\ntheorem mem_of_mem_inter_right {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a \u2208 l\u2081 \u2229 l\u2082 \u2192 a \u2208 l\u2082 :=\n  of_mem_filter\n\ntheorem mem_inter_of_mem_of_mem {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} {a : \u03b1} :\n    a \u2208 l\u2081 \u2192 a \u2208 l\u2082 \u2192 a \u2208 l\u2081 \u2229 l\u2082 :=\n  mem_filter_of_mem\n\n@[simp] theorem mem_inter {\u03b1 : Type u} [DecidableEq \u03b1] {a : \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    a \u2208 l\u2081 \u2229 l\u2082 \u2194 a \u2208 l\u2081 \u2227 a \u2208 l\u2082 :=\n  mem_filter\n\ntheorem inter_subset_left {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) : l\u2081 \u2229 l\u2082 \u2286 l\u2081 :=\n  filter_subset l\u2081\n\ntheorem inter_subset_right {\u03b1 : Type u} [DecidableEq \u03b1] (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    l\u2081 \u2229 l\u2082 \u2286 l\u2082 :=\n  fun (a : \u03b1) => mem_of_mem_inter_right\n\ntheorem subset_inter {\u03b1 : Type u} [DecidableEq \u03b1] {l : List \u03b1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1}\n    (h\u2081 : l \u2286 l\u2081) (h\u2082 : l \u2286 l\u2082) : l \u2286 l\u2081 \u2229 l\u2082 :=\n  fun (a : \u03b1) (h : a \u2208 l) => iff.mpr mem_inter { left := h\u2081 h, right := h\u2082 h }\n\ntheorem inter_eq_nil_iff_disjoint {\u03b1 : Type u} [DecidableEq \u03b1] {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} :\n    l\u2081 \u2229 l\u2082 = [] \u2194 disjoint l\u2081 l\u2082 :=\n  sorry\n\ntheorem forall_mem_inter_of_forall_left {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} {l\u2081 : List \u03b1}\n    (h : \u2200 (x : \u03b1), x \u2208 l\u2081 \u2192 p x) (l\u2082 : List \u03b1) (x : \u03b1) : x \u2208 l\u2081 \u2229 l\u2082 \u2192 p x :=\n  ball.imp_left (fun (x : \u03b1) => mem_of_mem_inter_left) h\n\ntheorem forall_mem_inter_of_forall_right {\u03b1 : Type u} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} (l\u2081 : List \u03b1)\n    {l\u2082 : List \u03b1} (h : \u2200 (x : \u03b1), x \u2208 l\u2082 \u2192 p x) (x : \u03b1) : x \u2208 l\u2081 \u2229 l\u2082 \u2192 p x :=\n  ball.imp_left (fun (x : \u03b1) => mem_of_mem_inter_right) h\n\n@[simp] theorem inter_reverse {\u03b1 : Type u} [DecidableEq \u03b1] {xs : List \u03b1} {ys : List \u03b1} :\n    list.inter xs (reverse ys) = list.inter xs ys :=\n  sorry\n\ntheorem choose_spec {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1)\n    (hp : \u2203 (a : \u03b1), a \u2208 l \u2227 p a) : choose p l hp \u2208 l \u2227 p (choose p l hp) :=\n  subtype.property (choose_x p l hp)\n\ntheorem choose_mem {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1)\n    (hp : \u2203 (a : \u03b1), a \u2208 l \u2227 p a) : choose p l hp \u2208 l :=\n  and.left (choose_spec p l hp)\n\ntheorem choose_property {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1)\n    (hp : \u2203 (a : \u03b1), a \u2208 l \u2227 p a) : p (choose p l hp) :=\n  and.right (choose_spec p l hp)\n\n/-! ### map\u2082_left' -/\n\n-- The definitional equalities for `map\u2082_left'` can already be used by the\n\n-- simplifie because `map\u2082_left'` is marked `@[simp]`.\n\n@[simp] theorem map\u2082_left'_nil_right {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) : map\u2082_left' f as [] = (map (fun (a : \u03b1) => f a none) as, []) :=\n  list.cases_on as (Eq.refl (map\u2082_left' f [] []))\n    fun (as_hd : \u03b1) (as_tl : List \u03b1) => Eq.refl (map\u2082_left' f (as_hd :: as_tl) [])\n\n/-! ### map\u2082_right' -/\n\n@[simp] theorem map\u2082_right'_nil_left {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (bs : List \u03b2) : map\u2082_right' f [] bs = (map (f none) bs, []) :=\n  list.cases_on bs (Eq.refl (map\u2082_right' f [] []))\n    fun (bs_hd : \u03b2) (bs_tl : List \u03b2) => Eq.refl (map\u2082_right' f [] (bs_hd :: bs_tl))\n\n@[simp] theorem map\u2082_right'_nil_right {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) : map\u2082_right' f as [] = ([], as) :=\n  rfl\n\n@[simp] theorem map\u2082_right'_nil_cons {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (b : \u03b2) (bs : List \u03b2) : map\u2082_right' f [] (b :: bs) = (f none b :: map (f none) bs, []) :=\n  rfl\n\n@[simp] theorem map\u2082_right'_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (a : \u03b1) (as : List \u03b1) (b : \u03b2) (bs : List \u03b2) :\n    map\u2082_right' f (a :: as) (b :: bs) =\n        let rec : List \u03b3 \u00d7 List \u03b1 := map\u2082_right' f as bs;\n        (f (some a) b :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### zip_left' -/\n\n@[simp] theorem zip_left'_nil_right {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) :\n    zip_left' as [] = (map (fun (a : \u03b1) => (a, none)) as, []) :=\n  list.cases_on as (Eq.refl (zip_left' [] []))\n    fun (as_hd : \u03b1) (as_tl : List \u03b1) => Eq.refl (zip_left' (as_hd :: as_tl) [])\n\n@[simp] theorem zip_left'_nil_left {\u03b1 : Type u} {\u03b2 : Type v} (bs : List \u03b2) :\n    zip_left' [] bs = ([], bs) :=\n  rfl\n\n@[simp] theorem zip_left'_cons_nil {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) :\n    zip_left' (a :: as) [] = ((a, none) :: map (fun (a : \u03b1) => (a, none)) as, []) :=\n  rfl\n\n@[simp] theorem zip_left'_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) (b : \u03b2)\n    (bs : List \u03b2) :\n    zip_left' (a :: as) (b :: bs) =\n        let rec : List (\u03b1 \u00d7 Option \u03b2) \u00d7 List \u03b2 := zip_left' as bs;\n        ((a, some b) :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### zip_right' -/\n\n@[simp] theorem zip_right'_nil_left {\u03b1 : Type u} {\u03b2 : Type v} (bs : List \u03b2) :\n    zip_right' [] bs = (map (fun (b : \u03b2) => (none, b)) bs, []) :=\n  list.cases_on bs (Eq.refl (zip_right' [] []))\n    fun (bs_hd : \u03b2) (bs_tl : List \u03b2) => Eq.refl (zip_right' [] (bs_hd :: bs_tl))\n\n@[simp] theorem zip_right'_nil_right {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) :\n    zip_right' as [] = ([], as) :=\n  rfl\n\n@[simp] theorem zip_right'_nil_cons {\u03b1 : Type u} {\u03b2 : Type v} (b : \u03b2) (bs : List \u03b2) :\n    zip_right' [] (b :: bs) = ((none, b) :: map (fun (b : \u03b2) => (none, b)) bs, []) :=\n  rfl\n\n@[simp] theorem zip_right'_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) (b : \u03b2)\n    (bs : List \u03b2) :\n    zip_right' (a :: as) (b :: bs) =\n        let rec : List (Option \u03b1 \u00d7 \u03b2) \u00d7 List \u03b1 := zip_right' as bs;\n        ((some a, b) :: prod.fst rec, prod.snd rec) :=\n  rfl\n\n/-! ### map\u2082_left -/\n\n-- The definitional equalities for `map\u2082_left` can already be used by the\n\n-- simplifier because `map\u2082_left` is marked `@[simp]`.\n\n@[simp] theorem map\u2082_left_nil_right {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) : map\u2082_left f as [] = map (fun (a : \u03b1) => f a none) as :=\n  list.cases_on as (Eq.refl (map\u2082_left f [] []))\n    fun (as_hd : \u03b1) (as_tl : List \u03b1) => Eq.refl (map\u2082_left f (as_hd :: as_tl) [])\n\ntheorem map\u2082_left_eq_map\u2082_left' {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) : map\u2082_left f as bs = prod.fst (map\u2082_left' f as bs) :=\n  sorry\n\ntheorem map\u2082_left_eq_map\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 Option \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) :\n    length as \u2264 length bs \u2192 map\u2082_left f as bs = map\u2082 (fun (a : \u03b1) (b : \u03b2) => f a (some b)) as bs :=\n  sorry\n\n/-! ### map\u2082_right -/\n\n@[simp] theorem map\u2082_right_nil_left {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (bs : List \u03b2) : map\u2082_right f [] bs = map (f none) bs :=\n  list.cases_on bs (Eq.refl (map\u2082_right f [] []))\n    fun (bs_hd : \u03b2) (bs_tl : List \u03b2) => Eq.refl (map\u2082_right f [] (bs_hd :: bs_tl))\n\n@[simp] theorem map\u2082_right_nil_right {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) : map\u2082_right f as [] = [] :=\n  rfl\n\n@[simp] theorem map\u2082_right_nil_cons {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (b : \u03b2) (bs : List \u03b2) : map\u2082_right f [] (b :: bs) = f none b :: map (f none) bs :=\n  rfl\n\n@[simp] theorem map\u2082_right_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (a : \u03b1) (as : List \u03b1) (b : \u03b2) (bs : List \u03b2) :\n    map\u2082_right f (a :: as) (b :: bs) = f (some a) b :: map\u2082_right f as bs :=\n  rfl\n\ntheorem map\u2082_right_eq_map\u2082_right' {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) : map\u2082_right f as bs = prod.fst (map\u2082_right' f as bs) :=\n  sorry\n\ntheorem map\u2082_right_eq_map\u2082 {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : Option \u03b1 \u2192 \u03b2 \u2192 \u03b3)\n    (as : List \u03b1) (bs : List \u03b2) (h : length bs \u2264 length as) :\n    map\u2082_right f as bs = map\u2082 (fun (a : \u03b1) (b : \u03b2) => f (some a) b) as bs :=\n  sorry\n\n/-! ### zip_left -/\n\n@[simp] theorem zip_left_nil_right {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) :\n    zip_left as [] = map (fun (a : \u03b1) => (a, none)) as :=\n  list.cases_on as (Eq.refl (zip_left [] []))\n    fun (as_hd : \u03b1) (as_tl : List \u03b1) => Eq.refl (zip_left (as_hd :: as_tl) [])\n\n@[simp] theorem zip_left_nil_left {\u03b1 : Type u} {\u03b2 : Type v} (bs : List \u03b2) : zip_left [] bs = [] :=\n  rfl\n\n@[simp] theorem zip_left_cons_nil {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) :\n    zip_left (a :: as) [] = (a, none) :: map (fun (a : \u03b1) => (a, none)) as :=\n  rfl\n\n@[simp] theorem zip_left_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) (b : \u03b2)\n    (bs : List \u03b2) : zip_left (a :: as) (b :: bs) = (a, some b) :: zip_left as bs :=\n  rfl\n\ntheorem zip_left_eq_zip_left' {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) (bs : List \u03b2) :\n    zip_left as bs = prod.fst (zip_left' as bs) :=\n  sorry\n\n/-! ### zip_right -/\n\n@[simp] theorem zip_right_nil_left {\u03b1 : Type u} {\u03b2 : Type v} (bs : List \u03b2) :\n    zip_right [] bs = map (fun (b : \u03b2) => (none, b)) bs :=\n  list.cases_on bs (Eq.refl (zip_right [] []))\n    fun (bs_hd : \u03b2) (bs_tl : List \u03b2) => Eq.refl (zip_right [] (bs_hd :: bs_tl))\n\n@[simp] theorem zip_right_nil_right {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) :\n    zip_right as [] = [] :=\n  rfl\n\n@[simp] theorem zip_right_nil_cons {\u03b1 : Type u} {\u03b2 : Type v} (b : \u03b2) (bs : List \u03b2) :\n    zip_right [] (b :: bs) = (none, b) :: map (fun (b : \u03b2) => (none, b)) bs :=\n  rfl\n\n@[simp] theorem zip_right_cons_cons {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (as : List \u03b1) (b : \u03b2)\n    (bs : List \u03b2) : zip_right (a :: as) (b :: bs) = (some a, b) :: zip_right as bs :=\n  rfl\n\ntheorem zip_right_eq_zip_right' {\u03b1 : Type u} {\u03b2 : Type v} (as : List \u03b1) (bs : List \u03b2) :\n    zip_right as bs = prod.fst (zip_right' as bs) :=\n  sorry\n\n/-! ### Miscellaneous lemmas -/\n\ntheorem ilast'_mem {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) : ilast' a l \u2208 a :: l := sorry\n\n@[simp] theorem nth_le_attach {\u03b1 : Type u} (L : List \u03b1) (i : \u2115) (H : i < length (attach L)) :\n    subtype.val (nth_le (attach L) i H) = nth_le L i (length_attach L \u25b8 H) :=\n  sorry\n\nend list\n\n\ntheorem monoid_hom.map_list_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} [monoid \u03b1] [monoid \u03b2] (f : \u03b1 \u2192* \u03b2)\n    (l : List \u03b1) : coe_fn f (list.prod l) = list.prod (list.map (\u21d1f) l) :=\n  Eq.symm (list.prod_hom l f)\n\nnamespace list\n\n\ntheorem sum_map_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_monoid \u03b2] [add_monoid \u03b3]\n    (L : List \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192+ \u03b3) : sum (map (\u21d1g \u2218 f) L) = coe_fn g (sum (map f L)) :=\n  sorry\n\ntheorem sum_map_mul_left {\u03b1 : Type u_1} [semiring \u03b1] {\u03b2 : Type u_2} (L : List \u03b2) (f : \u03b2 \u2192 \u03b1)\n    (r : \u03b1) : sum (map (fun (b : \u03b2) => r * f b) L) = r * sum (map f L) :=\n  sum_map_hom L f (add_monoid_hom.mul_left r)\n\ntheorem sum_map_mul_right {\u03b1 : Type u_1} [semiring \u03b1] {\u03b2 : Type u_2} (L : List \u03b2) (f : \u03b2 \u2192 \u03b1)\n    (r : \u03b1) : sum (map (fun (b : \u03b2) => f b * r) L) = sum (map f L) * r :=\n  sum_map_hom L f (add_monoid_hom.mul_right r)\n\n@[simp] theorem mem_map_swap {\u03b1 : Type u} {\u03b2 : Type v} (x : \u03b1) (y : \u03b2) (xs : List (\u03b1 \u00d7 \u03b2)) :\n    (y, x) \u2208 map prod.swap xs \u2194 (x, y) \u2208 xs :=\n  sorry\n\ntheorem slice_eq {\u03b1 : Type u_1} (xs : List \u03b1) (n : \u2115) (m : \u2115) :\n    slice n m xs = take n xs ++ drop (n + m) xs :=\n  sorry\n\ntheorem sizeof_slice_lt {\u03b1 : Type u_1} [SizeOf \u03b1] (i : \u2115) (j : \u2115) (hj : 0 < j) (xs : List \u03b1)\n    (hi : i < length xs) : sizeof (slice i j xs) < sizeof xs :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/list/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.3566824776728208}}
{"text": "/-\nCopyright (c) 2014 Jeremy Avigad. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jeremy Avigad, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.subtype\nimport Mathlib.data.prod\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u v w l \n\nnamespace Mathlib\n\n/-!\n# Basic definitions about `\u2264` and `<`\n\n## Definitions\n\n### Predicates on functions\n\n- `monotone f`: a function between two types equipped with `\u2264` is monotone\n  if `a \u2264 b` implies `f a \u2264 f b`.\n- `strict_mono f` : a function between two types equipped with `<` is strictly monotone\n  if `a < b` implies `f a < f b`.\n- `order_dual \u03b1` : a type tag reversing the meaning of all inequalities.\n\n### Transfering orders\n\n- `order.preimage`, `preorder.lift`: transfer a (pre)order on `\u03b2` to an order on `\u03b1`\n  using a function `f : \u03b1 \u2192 \u03b2`.\n- `partial_order.lift`, `linear_order.lift`: transfer a partial (resp., linear) order on `\u03b2` to a\n  partial (resp., linear) order on `\u03b1` using an injective function `f`.\n\n### Extra classes\n\n- `no_top_order`, `no_bot_order`: an order without a maximal/minimal element.\n- `densely_ordered`: an order with no gaps, i.e. for any two elements `a<b` there exists\n  `c`, `a<c<b`.\n\n## Main theorems\n\n- `monotone_of_monotone_nat`: if `f : \u2115 \u2192 \u03b1` and `f n \u2264 f (n + 1)` for all `n`, then\n  `f` is monotone;\n- `strict_mono.nat`: if `f : \u2115 \u2192 \u03b1` and `f n < f (n + 1)` for all `n`, then f is strictly monotone.\n\n## TODO\n\n- expand module docs\n- automatic construction of dual definitions / theorems\n\n## See also\n- `algebra.order` for basic lemmas about orders, and projection notation for orders\n\n## Tags\n\npreorder, order, partial order, linear order, monotone, strictly monotone\n-/\n\ntheorem preorder.ext {\u03b1 : Type u_1} {A : preorder \u03b1} {B : preorder \u03b1}\n    (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B :=\n  sorry\n\ntheorem partial_order.ext {\u03b1 : Type u_1} {A : partial_order \u03b1} {B : partial_order \u03b1}\n    (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B :=\n  sorry\n\ntheorem linear_order.ext {\u03b1 : Type u_1} {A : linear_order \u03b1} {B : linear_order \u03b1}\n    (H : \u2200 (x y : \u03b1), x \u2264 y \u2194 x \u2264 y) : A = B :=\n  sorry\n\n/-- Given a relation `R` on `\u03b2` and a function `f : \u03b1 \u2192 \u03b2`,\n  the preimage relation on `\u03b1` is defined by `x \u2264 y \u2194 f x \u2264 f y`.\n  It is the unique relation on `\u03b1` making `f` a `rel_embedding`\n  (assuming `f` is injective). -/\n@[simp] def order.preimage {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (f : \u03b1 \u2192 \u03b2) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) (x : \u03b1)\n    (y : \u03b1) :=\n  s (f x) (f y)\n\ninfixl:80 \" \u207b\u00b9'o \" => Mathlib.order.preimage\n\n/-- The preimage of a decidable order is decidable. -/\nprotected instance order.preimage.decidable {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (f : \u03b1 \u2192 \u03b2)\n    (s : \u03b2 \u2192 \u03b2 \u2192 Prop) [H : DecidableRel s] : DecidableRel (f \u207b\u00b9'o s) :=\n  fun (x y : \u03b1) => H (f x) (f y)\n\n/-- A function between preorders is monotone if\n  `a \u2264 b` implies `f a \u2264 f b`. -/\ndef monotone {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 {a b : \u03b1}, a \u2264 b \u2192 f a \u2264 f b\n\ntheorem monotone_id {\u03b1 : Type u} [preorder \u03b1] : monotone id := fun (x y : \u03b1) (h : x \u2264 y) => h\n\ntheorem monotone_const {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {b : \u03b2} :\n    monotone fun (a : \u03b1) => b :=\n  fun (x y : \u03b1) (h : x \u2264 y) => le_refl b\n\nprotected theorem monotone.comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b2]\n    [preorder \u03b3] {g : \u03b2 \u2192 \u03b3} {f : \u03b1 \u2192 \u03b2} (m_g : monotone g) (m_f : monotone f) : monotone (g \u2218 f) :=\n  fun (a b : \u03b1) (h : a \u2264 b) => m_g (m_f h)\n\nprotected theorem monotone.iterate {\u03b1 : Type u} [preorder \u03b1] {f : \u03b1 \u2192 \u03b1} (hf : monotone f) (n : \u2115) :\n    monotone (nat.iterate f n) :=\n  nat.rec_on n monotone_id fun (n : \u2115) (ihn : monotone (nat.iterate f n)) => monotone.comp ihn hf\n\ntheorem monotone_of_monotone_nat {\u03b1 : Type u} [preorder \u03b1] {f : \u2115 \u2192 \u03b1}\n    (hf : \u2200 (n : \u2115), f n \u2264 f (n + 1)) : monotone f :=\n  sorry\n\ntheorem monotone.reflect_lt {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : monotone f) {x : \u03b1} {x' : \u03b1} (h : f x < f x') : x < x' :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (x < x')) (Eq.symm (propext not_le))))\n    (id fun (h' : x' \u2264 x) => not_le_of_lt h (hf h'))\n\n/-- If `f` is a monotone function from `\u2115` to a preorder such that `y` lies between `f x` and\n  `f (x + 1)`, then `y` doesn't lie in the range of `f`. -/\ntheorem monotone.ne_of_lt_of_lt_nat {\u03b1 : Type u_1} [preorder \u03b1] {f : \u2115 \u2192 \u03b1} (hf : monotone f)\n    (x : \u2115) (x' : \u2115) {y : \u03b1} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' \u2260 y :=\n  sorry\n\n/-- If `f` is a monotone function from `\u2124` to a preorder such that `y` lies between `f x` and\n  `f (x + 1)`, then `y` doesn't lie in the range of `f`. -/\ntheorem monotone.ne_of_lt_of_lt_int {\u03b1 : Type u_1} [preorder \u03b1] {f : \u2124 \u2192 \u03b1} (hf : monotone f)\n    (x : \u2124) (x' : \u2124) {y : \u03b1} (h1 : f x < y) (h2 : y < f (x + 1)) : f x' \u2260 y :=\n  sorry\n\n/-- A function `f` is strictly monotone if `a < b` implies `f a < f b`. -/\ndef strict_mono {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) :=\n  \u2200 {a b : \u03b1}, a < b \u2192 f a < f b\n\ntheorem strict_mono_id {\u03b1 : Type u} [HasLess \u03b1] : strict_mono id := fun (a b : \u03b1) => id\n\n/-- A function `f` is strictly monotone increasing on `t` if `x < y` for `x,y \u2208 t` implies\n`f x < f y`. -/\ndef strict_mono_incr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) (t : set \u03b1) :=\n  \u2200 {x : \u03b1}, x \u2208 t \u2192 \u2200 {y : \u03b1}, y \u2208 t \u2192 x < y \u2192 f x < f y\n\n/-- A function `f` is strictly monotone decreasing on `t` if `x < y` for `x,y \u2208 t` implies\n`f y < f x`. -/\ndef strict_mono_decr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] (f : \u03b1 \u2192 \u03b2) (t : set \u03b1) :=\n  \u2200 {x : \u03b1}, x \u2208 t \u2192 \u2200 {y : \u03b1}, y \u2208 t \u2192 x < y \u2192 f y < f x\n\n/-- Type tag for a set with dual order: `\u2264` means `\u2265` and `<` means `>`. -/\ndef order_dual (\u03b1 : Type u_1) := \u03b1\n\nnamespace order_dual\n\n\nprotected instance nonempty (\u03b1 : Type u_1) [h : Nonempty \u03b1] : Nonempty (order_dual \u03b1) := h\n\nprotected instance subsingleton (\u03b1 : Type u_1) [h : subsingleton \u03b1] : subsingleton (order_dual \u03b1) :=\n  h\n\nprotected instance has_le (\u03b1 : Type u_1) [HasLessEq \u03b1] : HasLessEq (order_dual \u03b1) :=\n  { LessEq := fun (x y : \u03b1) => y \u2264 x }\n\nprotected instance has_lt (\u03b1 : Type u_1) [HasLess \u03b1] : HasLess (order_dual \u03b1) :=\n  { Less := fun (x y : \u03b1) => y < x }\n\n-- `dual_le` and `dual_lt` should not be simp lemmas:\n\n-- they cause a loop since `\u03b1` and `order_dual \u03b1` are definitionally equal\n\ntheorem dual_le {\u03b1 : Type u} [HasLessEq \u03b1] {a : \u03b1} {b : \u03b1} : a \u2264 b \u2194 b \u2264 a := iff.rfl\n\ntheorem dual_lt {\u03b1 : Type u} [HasLess \u03b1] {a : \u03b1} {b : \u03b1} : a < b \u2194 b < a := iff.rfl\n\ntheorem dual_compares {\u03b1 : Type u} [HasLess \u03b1] {a : \u03b1} {b : \u03b1} {o : ordering} :\n    ordering.compares o a b \u2194 ordering.compares o b a :=\n  ordering.cases_on o iff.rfl eq_comm iff.rfl\n\nprotected instance preorder (\u03b1 : Type u_1) [preorder \u03b1] : preorder (order_dual \u03b1) :=\n  preorder.mk LessEq Less sorry sorry\n\nprotected instance partial_order (\u03b1 : Type u_1) [partial_order \u03b1] : partial_order (order_dual \u03b1) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\nprotected instance linear_order (\u03b1 : Type u_1) [linear_order \u03b1] : linear_order (order_dual \u03b1) :=\n  linear_order.mk partial_order.le partial_order.lt sorry sorry sorry sorry\n    ((fun (this : DecidableRel fun (a b : \u03b1) => b \u2264 a) => this)\n      fun (a b : \u03b1) => has_le.le.decidable b a)\n    Mathlib.decidable_eq_of_decidable_le\n    ((fun (this : DecidableRel fun (a b : \u03b1) => b < a) => this)\n      fun (a b : \u03b1) => has_lt.lt.decidable b a)\n\nprotected instance inhabited {\u03b1 : Type u} [Inhabited \u03b1] : Inhabited (order_dual \u03b1) := id\n\ntheorem preorder.dual_dual (\u03b1 : Type u_1) [H : preorder \u03b1] :\n    order_dual.preorder (order_dual \u03b1) = H :=\n  preorder.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem partial_order.dual_dual (\u03b1 : Type u_1) [H : partial_order \u03b1] :\n    order_dual.partial_order (order_dual \u03b1) = H :=\n  partial_order.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem linear_order.dual_dual (\u03b1 : Type u_1) [H : linear_order \u03b1] :\n    order_dual.linear_order (order_dual \u03b1) = H :=\n  linear_order.ext fun (_x _x_1 : order_dual (order_dual \u03b1)) => iff.rfl\n\ntheorem cmp_le_flip {\u03b1 : Type u_1} [HasLessEq \u03b1] [DecidableRel LessEq] (x : \u03b1) (y : \u03b1) :\n    cmp_le x y = cmp_le y x :=\n  rfl\n\nend order_dual\n\n\nnamespace strict_mono_incr_on\n\n\nprotected theorem dual {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    (H : strict_mono_incr_on f s) : strict_mono_incr_on f s :=\n  fun (x : order_dual \u03b1) (hx : x \u2208 s) (y : order_dual \u03b1) (hy : y \u2208 s) => H hy hx\n\nprotected theorem dual_right {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {s : set \u03b1} (H : strict_mono_incr_on f s) : strict_mono_decr_on f s :=\n  H\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x \u2264 f y \u2194 x \u2264 y :=\n  sorry\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x < f y \u2194 x < y :=\n  sorry\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_incr_on f s) (hx : x \u2208 s) (hy : y \u2208 s)\n    {o : ordering} : ordering.compares o (f x) (f y) \u2194 ordering.compares o x y :=\n  sorry\n\nend strict_mono_incr_on\n\n\nnamespace strict_mono_decr_on\n\n\nprotected theorem dual {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    (H : strict_mono_decr_on f s) : strict_mono_decr_on f s :=\n  fun (x : order_dual \u03b1) (hx : x \u2208 s) (y : order_dual \u03b1) (hy : y \u2208 s) => H hy hx\n\nprotected theorem dual_right {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {s : set \u03b1} (H : strict_mono_decr_on f s) : strict_mono_incr_on f s :=\n  H\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x \u2264 f y \u2194 y \u2264 x :=\n  strict_mono_incr_on.le_iff_le (strict_mono_decr_on.dual_right H) hy hx\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {s : set \u03b1}\n    {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s) : f x < f y \u2194 y < x :=\n  strict_mono_incr_on.lt_iff_lt (strict_mono_decr_on.dual_right H) hy hx\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    {s : set \u03b1} {x : \u03b1} {y : \u03b1} (H : strict_mono_decr_on f s) (hx : x \u2208 s) (hy : y \u2208 s)\n    {o : ordering} : ordering.compares o (f x) (f y) \u2194 ordering.compares o y x :=\n  iff.trans order_dual.dual_compares\n    (strict_mono_incr_on.compares (strict_mono_decr_on.dual_right H) hy hx)\n\nend strict_mono_decr_on\n\n\nnamespace strict_mono\n\n\nprotected theorem strict_mono_incr_on {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : strict_mono f) (s : set \u03b1) : strict_mono_incr_on f s :=\n  fun (x : \u03b1) (hx : x \u2208 s) (y : \u03b1) (hy : y \u2208 s) (hxy : x < y) => hf hxy\n\ntheorem comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [HasLess \u03b1] [HasLess \u03b2] [HasLess \u03b3] {g : \u03b2 \u2192 \u03b3}\n    {f : \u03b1 \u2192 \u03b2} (hg : strict_mono g) (hf : strict_mono f) : strict_mono (g \u2218 f) :=\n  fun (a b : \u03b1) (h : a < b) => hg (hf h)\n\nprotected theorem iterate {\u03b1 : Type u} [HasLess \u03b1] {f : \u03b1 \u2192 \u03b1} (hf : strict_mono f) (n : \u2115) :\n    strict_mono (nat.iterate f n) :=\n  nat.rec_on n strict_mono_id fun (n : \u2115) (ihn : strict_mono (nat.iterate f n)) => comp ihn hf\n\ntheorem id_le {\u03c6 : \u2115 \u2192 \u2115} (h : strict_mono \u03c6) (n : \u2115) : n \u2264 \u03c6 n :=\n  nat.rec_on n (nat.zero_le (\u03c6 0))\n    fun (n : \u2115) (hn : n \u2264 \u03c6 n) => nat.succ_le_of_lt (lt_of_le_of_lt hn (h (nat.lt_succ_self n)))\n\nprotected theorem ite' {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n    (hf : strict_mono f) (hg : strict_mono g) {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (hp : \u2200 {x y : \u03b1}, x < y \u2192 p y \u2192 p x) (hfg : \u2200 {x y : \u03b1}, p x \u2192 \u00acp y \u2192 x < y \u2192 f x < g y) :\n    strict_mono fun (x : \u03b1) => ite (p x) (f x) (g x) :=\n  sorry\n\nprotected theorem ite {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n    (hf : strict_mono f) (hg : strict_mono g) {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    (hp : \u2200 {x y : \u03b1}, x < y \u2192 p y \u2192 p x) (hfg : \u2200 (x : \u03b1), f x \u2264 g x) :\n    strict_mono fun (x : \u03b1) => ite (p x) (f x) (g x) :=\n  strict_mono.ite' hf hg hp\n    fun (x y : \u03b1) (hx : p x) (hy : \u00acp y) (h : x < y) => has_lt.lt.trans_le (hf h) (hfg y)\n\ntheorem lt_iff_lt {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) {a : \u03b1} {b : \u03b1} : f a < f b \u2194 a < b :=\n  strict_mono_incr_on.lt_iff_lt (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\nprotected theorem compares {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) {a : \u03b1} {b : \u03b1} {o : ordering} :\n    ordering.compares o (f a) (f b) \u2194 ordering.compares o a b :=\n  strict_mono_incr_on.compares (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\ntheorem injective {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) : function.injective f :=\n  fun (x y : \u03b1) (h : f x = f y) =>\n    (fun (this : ordering.compares ordering.eq x y) => this) (iff.mp (strict_mono.compares H) h)\n\ntheorem le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) {a : \u03b1} {b : \u03b1} : f a \u2264 f b \u2194 a \u2264 b :=\n  strict_mono_incr_on.le_iff_le (strict_mono.strict_mono_incr_on H set.univ) trivial trivial\n\ntheorem top_preimage_top {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) {a : \u03b1} (h_top : \u2200 (p : \u03b2), p \u2264 f a) (x : \u03b1) : x \u2264 a :=\n  iff.mp (le_iff_le H) (h_top (f x))\n\ntheorem bot_preimage_bot {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) {a : \u03b1} (h_bot : \u2200 (p : \u03b2), f a \u2264 p) (x : \u03b1) : a \u2264 x :=\n  iff.mp (le_iff_le H) (h_bot (f x))\n\nprotected theorem nat {\u03b2 : Type u_1} [preorder \u03b2] {f : \u2115 \u2192 \u03b2} (h : \u2200 (n : \u2115), f n < f (n + 1)) :\n    strict_mono f :=\n  sorry\n\n-- `preorder \u03b1` isn't strong enough: if the preorder on \u03b1 is an equivalence relation,\n\n-- then `strict_mono f` is vacuously true.\n\ntheorem monotone {\u03b1 : Type u} {\u03b2 : Type v} [partial_order \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (H : strict_mono f) : monotone f :=\n  fun (a b : \u03b1) (h : a \u2264 b) =>\n    Or._oldrec (le_of_lt \u2218 H)\n      (fun (h_1 : a = b) => Eq._oldrec (fun (h : a \u2264 a) => le_refl (f a)) h_1 h) (lt_or_eq_of_le h)\n\nend strict_mono\n\n\ntheorem injective_of_lt_imp_ne {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1] {f : \u03b1 \u2192 \u03b2}\n    (h : \u2200 (x y : \u03b1), x < y \u2192 f x \u2260 f y) : function.injective f :=\n  sorry\n\ntheorem strict_mono_of_monotone_of_injective {\u03b1 : Type u} {\u03b2 : Type v} [partial_order \u03b1]\n    [partial_order \u03b2] {f : \u03b1 \u2192 \u03b2} (h\u2081 : monotone f) (h\u2082 : function.injective f) : strict_mono f :=\n  sorry\n\ntheorem monotone.strict_mono_iff_injective {\u03b1 : Type u} {\u03b2 : Type v} [linear_order \u03b1]\n    [partial_order \u03b2] {f : \u03b1 \u2192 \u03b2} (h : monotone f) : strict_mono f \u2194 function.injective f :=\n  { mp := fun (h : strict_mono f) => strict_mono.injective h,\n    mpr := strict_mono_of_monotone_of_injective h }\n\ntheorem strict_mono_of_le_iff_le {\u03b1 : Type u} {\u03b2 : Type v} [preorder \u03b1] [preorder \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (h : \u2200 (x y : \u03b1), x \u2264 y \u2194 f x \u2264 f y) : strict_mono f :=\n  sorry\n\n/-! ### Order instances on the function space -/\n\nprotected instance pi.preorder {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] :\n    preorder ((i : \u03b9) \u2192 \u03b1 i) :=\n  preorder.mk (fun (x y : (i : \u03b9) \u2192 \u03b1 i) => \u2200 (i : \u03b9), x i \u2264 y i)\n    (fun (a b : (i : \u03b9) \u2192 \u03b1 i) => (\u2200 (i : \u03b9), a i \u2264 b i) \u2227 \u00ac\u2200 (i : \u03b9), b i \u2264 a i) sorry sorry\n\ntheorem pi.le_def {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] {x : (i : \u03b9) \u2192 \u03b1 i}\n    {y : (i : \u03b9) \u2192 \u03b1 i} : x \u2264 y \u2194 \u2200 (i : \u03b9), x i \u2264 y i :=\n  iff.rfl\n\ntheorem le_update_iff {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] [DecidableEq \u03b9]\n    {x : (i : \u03b9) \u2192 \u03b1 i} {y : (i : \u03b9) \u2192 \u03b1 i} {i : \u03b9} {a : \u03b1 i} :\n    x \u2264 function.update y i a \u2194 x i \u2264 a \u2227 \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2264 y j :=\n  function.forall_update_iff y fun (j : \u03b9) (z : \u03b1 j) => x j \u2264 z\n\ntheorem update_le_iff {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 preorder (\u03b1 i)] [DecidableEq \u03b9]\n    {x : (i : \u03b9) \u2192 \u03b1 i} {y : (i : \u03b9) \u2192 \u03b1 i} {i : \u03b9} {a : \u03b1 i} :\n    function.update x i a \u2264 y \u2194 a \u2264 y i \u2227 \u2200 (j : \u03b9), j \u2260 i \u2192 x j \u2264 y j :=\n  function.forall_update_iff x fun (j : \u03b9) (z : \u03b1 j) => z \u2264 y j\n\nprotected instance pi.partial_order {\u03b9 : Type u} {\u03b1 : \u03b9 \u2192 Type v} [(i : \u03b9) \u2192 partial_order (\u03b1 i)] :\n    partial_order ((i : \u03b9) \u2192 \u03b1 i) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\ntheorem comp_le_comp_left_of_monotone {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1]\n    [preorder \u03b2] {f : \u03b2 \u2192 \u03b1} {g : \u03b3 \u2192 \u03b2} {h : \u03b3 \u2192 \u03b2} (m_f : monotone f) (le_gh : g \u2264 h) :\n    f \u2218 g \u2264 f \u2218 h :=\n  fun (x : \u03b3) => m_f (le_gh x)\n\nprotected theorem monotone.order_dual {\u03b1 : Type u} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3]\n    {f : \u03b1 \u2192 \u03b3} (hf : monotone f) : monotone f :=\n  fun (x y : order_dual \u03b1) (hxy : x \u2264 y) => hf hxy\n\ntheorem monotone_lam {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3]\n    {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (m : \u2200 (b : \u03b2), monotone fun (a : \u03b1) => f a b) : monotone f :=\n  fun (a a' : \u03b1) (h : a \u2264 a') (b : \u03b2) => m b h\n\ntheorem monotone_app {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} [preorder \u03b1] [preorder \u03b3]\n    (f : \u03b2 \u2192 \u03b1 \u2192 \u03b3) (b : \u03b2) (m : monotone fun (a : \u03b1) (b : \u03b2) => f b a) : monotone (f b) :=\n  fun (a a' : \u03b1) (h : a \u2264 a') => m h b\n\ntheorem strict_mono.order_dual {\u03b1 : Type u} {\u03b2 : Type v} [HasLess \u03b1] [HasLess \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (hf : strict_mono f) : strict_mono f :=\n  fun (x y : order_dual \u03b1) (hxy : x < y) => hf hxy\n\n/-- Transfer a `preorder` on `\u03b2` to a `preorder` on `\u03b1` using a function `f : \u03b1 \u2192 \u03b2`. -/\ndef preorder.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [preorder \u03b2] (f : \u03b1 \u2192 \u03b2) : preorder \u03b1 :=\n  preorder.mk (fun (x y : \u03b1) => f x \u2264 f y) (fun (x y : \u03b1) => f x < f y) sorry sorry\n\n/-- Transfer a `partial_order` on `\u03b2` to a `partial_order` on `\u03b1` using an injective\nfunction `f : \u03b1 \u2192 \u03b2`. -/\ndef partial_order.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [partial_order \u03b2] (f : \u03b1 \u2192 \u03b2)\n    (inj : function.injective f) : partial_order \u03b1 :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\n/-- Transfer a `linear_order` on `\u03b2` to a `linear_order` on `\u03b1` using an injective\nfunction `f : \u03b1 \u2192 \u03b2`. -/\ndef linear_order.lift {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b2] (f : \u03b1 \u2192 \u03b2)\n    (inj : function.injective f) : linear_order \u03b1 :=\n  linear_order.mk partial_order.le partial_order.lt sorry sorry sorry sorry\n    (fun (x y : \u03b1) => infer_instance)\n    (fun (x y : \u03b1) => decidable_of_iff (f x = f y) (function.injective.eq_iff inj))\n    fun (x y : \u03b1) => infer_instance\n\nprotected instance subtype.preorder {\u03b1 : Type u_1} [preorder \u03b1] (p : \u03b1 \u2192 Prop) :\n    preorder (Subtype p) :=\n  preorder.lift subtype.val\n\n@[simp] theorem subtype.mk_le_mk {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : \u03b1} {y : \u03b1}\n    {hx : p x} {hy : p y} : { val := x, property := hx } \u2264 { val := y, property := hy } \u2194 x \u2264 y :=\n  iff.rfl\n\n@[simp] theorem subtype.mk_lt_mk {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : \u03b1} {y : \u03b1}\n    {hx : p x} {hy : p y} : { val := x, property := hx } < { val := y, property := hy } \u2194 x < y :=\n  iff.rfl\n\n@[simp] theorem subtype.coe_le_coe {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : Subtype p}\n    {y : Subtype p} : \u2191x \u2264 \u2191y \u2194 x \u2264 y :=\n  iff.rfl\n\n@[simp] theorem subtype.coe_lt_coe {\u03b1 : Type u_1} [preorder \u03b1] {p : \u03b1 \u2192 Prop} {x : Subtype p}\n    {y : Subtype p} : \u2191x < \u2191y \u2194 x < y :=\n  iff.rfl\n\nprotected instance subtype.partial_order {\u03b1 : Type u_1} [partial_order \u03b1] (p : \u03b1 \u2192 Prop) :\n    partial_order (Subtype p) :=\n  partial_order.lift subtype.val subtype.val_injective\n\nprotected instance subtype.linear_order {\u03b1 : Type u_1} [linear_order \u03b1] (p : \u03b1 \u2192 Prop) :\n    linear_order (Subtype p) :=\n  linear_order.lift subtype.val subtype.val_injective\n\ntheorem subtype.mono_coe {\u03b1 : Type u} [preorder \u03b1] (t : set \u03b1) : monotone coe :=\n  fun (x y : Subtype t) => id\n\ntheorem subtype.strict_mono_coe {\u03b1 : Type u} [preorder \u03b1] (t : set \u03b1) : strict_mono coe :=\n  fun (x y : Subtype t) => id\n\nprotected instance prod.has_le (\u03b1 : Type u) (\u03b2 : Type v) [HasLessEq \u03b1] [HasLessEq \u03b2] :\n    HasLessEq (\u03b1 \u00d7 \u03b2) :=\n  { LessEq := fun (p q : \u03b1 \u00d7 \u03b2) => prod.fst p \u2264 prod.fst q \u2227 prod.snd p \u2264 prod.snd q }\n\nprotected instance prod.preorder (\u03b1 : Type u) (\u03b2 : Type v) [preorder \u03b1] [preorder \u03b2] :\n    preorder (\u03b1 \u00d7 \u03b2) :=\n  preorder.mk LessEq (fun (a b : \u03b1 \u00d7 \u03b2) => a \u2264 b \u2227 \u00acb \u2264 a) sorry sorry\n\n/-- The pointwise partial order on a product.\n    (The lexicographic ordering is defined in order/lexicographic.lean, and the instances are\n    available via the type synonym `lex \u03b1 \u03b2 = \u03b1 \u00d7 \u03b2`.) -/\nprotected instance prod.partial_order (\u03b1 : Type u) (\u03b2 : Type v) [partial_order \u03b1]\n    [partial_order \u03b2] : partial_order (\u03b1 \u00d7 \u03b2) :=\n  partial_order.mk preorder.le preorder.lt sorry sorry sorry\n\n/-!\n### Additional order classes\n-/\n\n/-- order without a top element; somtimes called cofinal -/\nclass no_top_order (\u03b1 : Type u) [preorder \u03b1] where\n  no_top : \u2200 (a : \u03b1), \u2203 (a' : \u03b1), a < a'\n\ntheorem no_top {\u03b1 : Type u} [preorder \u03b1] [no_top_order \u03b1] (a : \u03b1) : \u2203 (a' : \u03b1), a < a' :=\n  no_top_order.no_top\n\nprotected instance nonempty_gt {\u03b1 : Type u} [preorder \u03b1] [no_top_order \u03b1] (a : \u03b1) :\n    Nonempty (Subtype fun (x : \u03b1) => a < x) :=\n  iff.mpr nonempty_subtype (no_top a)\n\n/-- order without a bottom element; somtimes called coinitial or dense -/\nclass no_bot_order (\u03b1 : Type u) [preorder \u03b1] where\n  no_bot : \u2200 (a : \u03b1), \u2203 (a' : \u03b1), a' < a\n\ntheorem no_bot {\u03b1 : Type u} [preorder \u03b1] [no_bot_order \u03b1] (a : \u03b1) : \u2203 (a' : \u03b1), a' < a :=\n  no_bot_order.no_bot\n\nprotected instance order_dual.no_top_order (\u03b1 : Type u) [preorder \u03b1] [no_bot_order \u03b1] :\n    no_top_order (order_dual \u03b1) :=\n  no_top_order.mk fun (a : order_dual \u03b1) => no_bot a\n\nprotected instance order_dual.no_bot_order (\u03b1 : Type u) [preorder \u03b1] [no_top_order \u03b1] :\n    no_bot_order (order_dual \u03b1) :=\n  no_bot_order.mk fun (a : order_dual \u03b1) => no_top a\n\nprotected instance nonempty_lt {\u03b1 : Type u} [preorder \u03b1] [no_bot_order \u03b1] (a : \u03b1) :\n    Nonempty (Subtype fun (x : \u03b1) => x < a) :=\n  iff.mpr nonempty_subtype (no_bot a)\n\n/-- An order is dense if there is an element between any pair of distinct elements. -/\nclass densely_ordered (\u03b1 : Type u) [preorder \u03b1] where\n  dense : \u2200 (a\u2081 a\u2082 : \u03b1), a\u2081 < a\u2082 \u2192 \u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082\n\ntheorem exists_between {\u03b1 : Type u} [preorder \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1} :\n    a\u2081 < a\u2082 \u2192 \u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082 :=\n  densely_ordered.dense\n\nprotected instance order_dual.densely_ordered (\u03b1 : Type u) [preorder \u03b1] [densely_ordered \u03b1] :\n    densely_ordered (order_dual \u03b1) :=\n  densely_ordered.mk\n    fun (a\u2081 a\u2082 : order_dual \u03b1) (ha : a\u2081 < a\u2082) =>\n      Exists.imp (fun (a : \u03b1) => and.symm) (exists_between ha)\n\ntheorem le_of_forall_le_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1}\n    (h : \u2200 (a\u2083 : \u03b1), a\u2083 > a\u2082 \u2192 a\u2081 \u2264 a\u2083) : a\u2081 \u2264 a\u2082 :=\n  sorry\n\ntheorem eq_of_le_of_forall_le_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1}\n    {a\u2082 : \u03b1} (h\u2081 : a\u2082 \u2264 a\u2081) (h\u2082 : \u2200 (a\u2083 : \u03b1), a\u2083 > a\u2082 \u2192 a\u2081 \u2264 a\u2083) : a\u2081 = a\u2082 :=\n  le_antisymm (le_of_forall_le_of_dense h\u2082) h\u2081\n\ntheorem le_of_forall_ge_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1} {a\u2082 : \u03b1}\n    (h : \u2200 (a\u2083 : \u03b1), a\u2083 < a\u2081 \u2192 a\u2083 \u2264 a\u2082) : a\u2081 \u2264 a\u2082 :=\n  sorry\n\ntheorem eq_of_le_of_forall_ge_of_dense {\u03b1 : Type u} [linear_order \u03b1] [densely_ordered \u03b1] {a\u2081 : \u03b1}\n    {a\u2082 : \u03b1} (h\u2081 : a\u2082 \u2264 a\u2081) (h\u2082 : \u2200 (a\u2083 : \u03b1), a\u2083 < a\u2081 \u2192 a\u2083 \u2264 a\u2082) : a\u2081 = a\u2082 :=\n  le_antisymm (le_of_forall_ge_of_dense h\u2082) h\u2081\n\ntheorem dense_or_discrete {\u03b1 : Type u} [linear_order \u03b1] (a\u2081 : \u03b1) (a\u2082 : \u03b1) :\n    (\u2203 (a : \u03b1), a\u2081 < a \u2227 a < a\u2082) \u2228 (\u2200 (a : \u03b1), a > a\u2081 \u2192 a\u2082 \u2264 a) \u2227 \u2200 (a : \u03b1), a < a\u2082 \u2192 a \u2264 a\u2081 :=\n  sorry\n\n/-- Type synonym to create an instance of `linear_order` from a\n`partial_order` and `[is_total \u03b1 (\u2264)]` -/\ndef as_linear_order (\u03b1 : Type u) := \u03b1\n\nprotected instance as_linear_order.inhabited {\u03b1 : Type u_1} [Inhabited \u03b1] :\n    Inhabited (as_linear_order \u03b1) :=\n  { default := Inhabited.default }\n\nprotected instance as_linear_order.linear_order {\u03b1 : Type u_1} [partial_order \u03b1]\n    [is_total \u03b1 LessEq] : linear_order (as_linear_order \u03b1) :=\n  linear_order.mk partial_order.le partial_order.lt partial_order.le_refl partial_order.le_trans\n    partial_order.le_antisymm sorry (classical.dec_rel LessEq) Mathlib.decidable_eq_of_decidable_le\n    Mathlib.decidable_lt_of_decidable_le\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.356682466441142}}
{"text": "/-\nCopyright (c) 2022 Jujian Zhang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jujian Zhang, Scott Morrison\n-/\nimport algebra.homology.quasi_iso\nimport category_theory.preadditive.injective_resolution\nimport category_theory.abelian.homology\nimport algebra.homology.homotopy_category\n\n/-!\n# Main result\n\nWhen the underlying category is abelian:\n* `category_theory.InjectiveResolution.desc`: Given `I : InjectiveResolution X` and\n  `J : InjectiveResolution Y`, any morphism `X \u27f6 Y` admits a descent to a chain map\n  `J.cocomplex \u27f6 I.cocomplex`. It is a descent in the sense that `I.\u03b9` intertwines the descent and\n  the original morphism, see `category_theory.InjectiveResolution.desc_commutes`.\n* `category_theory.InjectiveResolution.desc_homotopy`: Any two such descents are homotopic.\n* `category_theory.InjectiveResolution.homotopy_equiv`: Any two injective resolutions of the same\n  object are homotopy equivalent.\n* `category_theory.injective_resolutions`: If every object admits an injective resolution, we can\n  construct a functor `injective_resolutions C : C \u2964 homotopy_category C`.\n\n* `category_theory.exact_f_d`: `f` and `injective.d f` are exact.\n* `category_theory.InjectiveResolution.of`: Hence, starting from a monomorphism `X \u27f6 J`, where `J`\n  is injective, we can apply `injective.d` repeatedly to obtain an injective resolution of `X`.\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\nvariables {C : Type u} [category.{v} C]\n\nopen injective\n\nnamespace InjectiveResolution\nsection\nvariables [has_zero_morphisms C] [has_zero_object C] [has_equalizers C] [has_images C]\n/-- Auxiliary construction for `desc`. -/\ndef desc_f_zero {Y Z : C} (f : Z \u27f6 Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :\n  J.cocomplex.X 0 \u27f6 I.cocomplex.X 0 :=\nfactor_thru (f \u226b I.\u03b9.f 0) (J.\u03b9.f 0)\n\nend\n\nsection abelian\nvariables [abelian C]\n/-- Auxiliary construction for `desc`. -/\ndef desc_f_one {Y Z : C}\n  (f : Z \u27f6 Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :\n  J.cocomplex.X 1 \u27f6 I.cocomplex.X 1 :=\nexact.desc (desc_f_zero f I J \u226b I.cocomplex.d 0 1) (J.\u03b9.f 0) (J.cocomplex.d 0 1)\n  (abelian.exact.op _ _ J.exact\u2080) (by simp [\u2190category.assoc, desc_f_zero])\n\n@[simp] lemma desc_f_one_zero_comm {Y Z : C}\n  (f : Z \u27f6 Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :\n  J.cocomplex.d 0 1 \u226b desc_f_one f I J = desc_f_zero f I J \u226b I.cocomplex.d 0 1 :=\nby simp [desc_f_zero, desc_f_one]\n\n/-- Auxiliary construction for `desc`. -/\ndef desc_f_succ {Y Z : C}\n  (I : InjectiveResolution Y) (J : InjectiveResolution Z)\n  (n : \u2115) (g : J.cocomplex.X n \u27f6 I.cocomplex.X n) (g' : J.cocomplex.X (n+1) \u27f6 I.cocomplex.X (n+1))\n  (w : J.cocomplex.d n (n+1) \u226b g' = g \u226b I.cocomplex.d n (n+1)) :\n  \u03a3' g'' : J.cocomplex.X (n+2) \u27f6 I.cocomplex.X (n+2),\n    J.cocomplex.d (n+1) (n+2) \u226b g'' = g' \u226b I.cocomplex.d (n+1) (n+2) :=\n\u27e8@exact.desc C _ _ _ _ _ _ _ _ _\n  (g' \u226b I.cocomplex.d (n+1) (n+2))\n  (J.cocomplex.d n (n+1))\n  (J.cocomplex.d (n+1) (n+2)) (abelian.exact.op _ _ (J.exact _))\n  (by simp [\u2190category.assoc, w]), (by simp)\u27e9\n\n/-- A morphism in `C` descends to a chain map between injective resolutions. -/\ndef desc {Y Z : C}\n  (f : Z \u27f6 Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :\n  J.cocomplex \u27f6 I.cocomplex :=\ncochain_complex.mk_hom _ _ (desc_f_zero f _ _) (desc_f_one f _ _)\n  (desc_f_one_zero_comm f I J).symm\n  (\u03bb n \u27e8g, g', w\u27e9, \u27e8(desc_f_succ I J n g g' w.symm).1, (desc_f_succ I J n g g' w.symm).2.symm\u27e9)\n\n/-- The resolution maps intertwine the descent of a morphism and that morphism. -/\n@[simp, reassoc]\nlemma desc_commutes {Y Z : C}\n  (f : Z \u27f6 Y) (I : InjectiveResolution Y) (J : InjectiveResolution Z) :\n  J.\u03b9 \u226b desc f I J = (cochain_complex.single\u2080 C).map f \u226b I.\u03b9 :=\nbegin\n  ext n,\n  rcases n with (_|_|n);\n  { dsimp [desc, desc_f_one, desc_f_zero], simp, },\nend\n\n-- Now that we've checked this property of the descent,\n-- we can seal away the actual definition.\nattribute [irreducible] desc\n\n/-- An auxiliary definition for `desc_homotopy_zero`. -/\ndef desc_homotopy_zero_zero {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}\n  (f : I.cocomplex \u27f6 J.cocomplex)\n  (comm : I.\u03b9 \u226b f = 0) : I.cocomplex.X 1 \u27f6 J.cocomplex.X 0 :=\nexact.desc (f.f 0) (I.\u03b9.f 0) (I.cocomplex.d 0 1) (abelian.exact.op _ _ I.exact\u2080)\n  (congr_fun (congr_arg homological_complex.hom.f comm) 0)\n\n/-- An auxiliary definition for `desc_homotopy_zero`. -/\ndef desc_homotopy_zero_one {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}\n  (f : I.cocomplex \u27f6 J.cocomplex)\n  (comm : I.\u03b9 \u226b f = (0 : _ \u27f6 J.cocomplex)) : I.cocomplex.X 2 \u27f6 J.cocomplex.X 1 :=\nexact.desc (f.f 1 - desc_homotopy_zero_zero f comm \u226b J.cocomplex.d 0 1)\n  (I.cocomplex.d 0 1) (I.cocomplex.d 1 2) (abelian.exact.op _ _ (I.exact _))\n  (by simp [desc_homotopy_zero_zero, \u2190category.assoc])\n\n/-- An auxiliary definition for `desc_homotopy_zero`. -/\ndef desc_homotopy_zero_succ {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}\n  (f : I.cocomplex \u27f6 J.cocomplex) (n : \u2115)\n  (g : I.cocomplex.X (n + 1) \u27f6 J.cocomplex.X n)\n  (g' : I.cocomplex.X (n + 2) \u27f6 J.cocomplex.X (n + 1))\n  (w : f.f (n + 1) = I.cocomplex.d (n+1) (n+2) \u226b g' + g \u226b J.cocomplex.d n (n+1)) :\n  I.cocomplex.X (n + 3) \u27f6 J.cocomplex.X (n + 2) :=\nexact.desc (f.f (n+2) - g' \u226b J.cocomplex.d _ _) (I.cocomplex.d (n+1) (n+2))\n  (I.cocomplex.d (n+2) (n+3)) (abelian.exact.op _ _ (I.exact _))\n  (by simp [preadditive.comp_sub, \u2190category.assoc, preadditive.sub_comp,\n        show I.cocomplex.d (n+1) (n+2) \u226b g' = f.f (n + 1) - g \u226b J.cocomplex.d n (n+1),\n        by {rw w, simp only [add_sub_cancel] } ])\n\n/-- Any descent of the zero morphism is homotopic to zero. -/\ndef desc_homotopy_zero {Y Z : C} {I : InjectiveResolution Y} {J : InjectiveResolution Z}\n  (f : I.cocomplex \u27f6 J.cocomplex)\n  (comm : I.\u03b9 \u226b f = 0) :\n  homotopy f 0 :=\nhomotopy.mk_coinductive _ (desc_homotopy_zero_zero f comm) (by simp [desc_homotopy_zero_zero])\n  (desc_homotopy_zero_one f comm) (by simp [desc_homotopy_zero_one])\n  (\u03bb n \u27e8g, g', w\u27e9, \u27e8desc_homotopy_zero_succ f n g g' (by simp only [w, add_comm]),\n    by simp [desc_homotopy_zero_succ, w]\u27e9)\n\n/-- Two descents of the same morphism are homotopic. -/\ndef desc_homotopy {Y Z : C} (f : Y \u27f6 Z) {I : InjectiveResolution Y} {J : InjectiveResolution Z}\n  (g h : I.cocomplex \u27f6 J.cocomplex)\n  (g_comm : I.\u03b9 \u226b g = (cochain_complex.single\u2080 C).map f \u226b J.\u03b9)\n  (h_comm : I.\u03b9 \u226b h = (cochain_complex.single\u2080 C).map f \u226b J.\u03b9) :\n  homotopy g h :=\nhomotopy.equiv_sub_zero.inv_fun (desc_homotopy_zero _ (by simp [g_comm, h_comm]))\n\n/-- The descent of the identity morphism is homotopic to the identity cochain map. -/\ndef desc_id_homotopy (X : C) (I : InjectiveResolution X) :\n  homotopy (desc (\ud835\udfd9 X) I I) (\ud835\udfd9 I.cocomplex) :=\nby apply desc_homotopy (\ud835\udfd9 X); simp\n\n/-- The descent of a composition is homotopic to the composition of the descents. -/\ndef desc_comp_homotopy {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n  (I : InjectiveResolution X) (J : InjectiveResolution Y) (K : InjectiveResolution Z) :\n  homotopy (desc (f \u226b g) K I) (desc f J I \u226b desc g K J)  :=\nby apply desc_homotopy (f \u226b g); simp\n\n-- We don't care about the actual definitions of these homotopies.\nattribute [irreducible] desc_homotopy_zero desc_homotopy desc_id_homotopy desc_comp_homotopy\n\n/-- Any two injective resolutions are homotopy equivalent. -/\ndef homotopy_equiv {X : C} (I J : InjectiveResolution X) :\n  homotopy_equiv I.cocomplex J.cocomplex :=\n{ hom := desc (\ud835\udfd9 X) J I,\n  inv := desc (\ud835\udfd9 X) I J,\n  homotopy_hom_inv_id := (desc_comp_homotopy (\ud835\udfd9 X) (\ud835\udfd9 X) I J I).symm.trans $\n    by simpa [category.id_comp] using desc_id_homotopy _ _,\n  homotopy_inv_hom_id := (desc_comp_homotopy (\ud835\udfd9 X) (\ud835\udfd9 X) J I J).symm.trans $\n    by simpa [category.id_comp] using desc_id_homotopy _ _ }\n\n@[simp, reassoc] lemma homotopy_equiv_hom_\u03b9 {X : C} (I J : InjectiveResolution X) :\n  I.\u03b9 \u226b (homotopy_equiv I J).hom = J.\u03b9 :=\nby simp [homotopy_equiv]\n\n@[simp, reassoc] lemma homotopy_equiv_inv_\u03b9 {X : C} (I J : InjectiveResolution X) :\n  J.\u03b9 \u226b (homotopy_equiv I J).inv = I.\u03b9 :=\nby simp [homotopy_equiv]\n\nend abelian\n\nend InjectiveResolution\n\nsection\nvariables [abelian C]\n\n/-- An arbitrarily chosen injective resolution of an object. -/\nabbreviation injective_resolution (Z : C) [has_injective_resolution Z] : cochain_complex C \u2115 :=\n(has_injective_resolution.out Z).some.cocomplex\n\n/-- The cochain map from cochain complex consisting of `Z` supported in degree `0`\nback to the arbitrarily chosen injective resolution `injective_resolution Z`. -/\nabbreviation injective_resolution.\u03b9 (Z : C) [has_injective_resolution Z] :\n  (cochain_complex.single\u2080 C).obj Z \u27f6 injective_resolution Z :=\n(has_injective_resolution.out Z).some.\u03b9\n\n/-- The descent of a morphism to a cochain map between the arbitrarily chosen injective resolutions.\n-/\nabbreviation injective_resolution.desc {X Y : C} (f : X \u27f6 Y)\n  [has_injective_resolution X] [has_injective_resolution Y] :\n  injective_resolution X \u27f6 injective_resolution Y :=\nInjectiveResolution.desc f _ _\n\nvariables (C) [has_injective_resolutions C]\n\n/--\nTaking injective resolutions is functorial,\nif considered with target the homotopy category\n(`\u2115`-indexed cochain complexes and chain maps up to homotopy).\n-/\ndef injective_resolutions : C \u2964 homotopy_category C (complex_shape.up \u2115) :=\n{ obj := \u03bb X, (homotopy_category.quotient _ _).obj (injective_resolution X),\n  map := \u03bb X Y f, (homotopy_category.quotient _ _).map (injective_resolution.desc f),\n  map_id' := \u03bb X, begin\n    rw \u2190(homotopy_category.quotient _ _).map_id,\n    apply homotopy_category.eq_of_homotopy,\n    apply InjectiveResolution.desc_id_homotopy,\n  end,\n  map_comp' := \u03bb X Y Z f g, begin\n    rw \u2190(homotopy_category.quotient _ _).map_comp,\n    apply homotopy_category.eq_of_homotopy,\n    apply InjectiveResolution.desc_comp_homotopy,\n  end, }\n\nend\n\nsection\n\nvariables [abelian C] [enough_injectives C]\n\nlemma exact_f_d {X Y : C} (f : X \u27f6 Y) : exact f (d f) :=\n(abelian.exact_iff _ _).2 $\n  \u27e8by simp, zero_of_comp_mono (\u03b9 _) $ by rw [category.assoc, kernel.condition]\u27e9\n\nend\n\nnamespace InjectiveResolution\n/-!\nOur goal is to define `InjectiveResolution.of Z : InjectiveResolution Z`.\nThe `0`-th object in this resolution will just be `injective.under Z`,\ni.e. an arbitrarily chosen injective object with a map from `Z`.\nAfter that, we build the `n+1`-st object as `injective.syzygies`\napplied to the previously constructed morphism,\nand the map from the `n`-th object as `injective.d`.\n-/\n\nvariables [abelian C] [enough_injectives C]\n\n/-- Auxiliary definition for `InjectiveResolution.of`. -/\n@[simps]\ndef of_cocomplex (Z : C) : cochain_complex C \u2115 :=\ncochain_complex.mk'\n  (injective.under Z) (injective.syzygies (injective.\u03b9 Z)) (injective.d (injective.\u03b9 Z))\n  (\u03bb \u27e8X, Y, f\u27e9, \u27e8injective.syzygies f, injective.d f, (exact_f_d f).w\u27e9)\n\n/--\nIn any abelian category with enough injectives,\n`InjectiveResolution.of Z` constructs an injective resolution of the object `Z`.\n-/\n@[irreducible] def of (Z : C) : InjectiveResolution Z :=\n{ cocomplex := of_cocomplex Z,\n  \u03b9 := cochain_complex.mk_hom _ _ (injective.\u03b9 Z) 0\n    (by { simp only [of_cocomplex_d, eq_self_iff_true, eq_to_hom_refl, category.comp_id,\n      dite_eq_ite, if_true, comp_zero],\n      exact (exact_f_d (injective.\u03b9 Z)).w, } ) (\u03bb n _, \u27e80, by ext\u27e9),\n  injective := by { rintros (_|_|_|n); { apply injective.injective_under, } },\n  exact\u2080 := by simpa using exact_f_d (injective.\u03b9 Z),\n  exact := by { rintros (_|n); { simp, apply exact_f_d } },\n  mono := injective.\u03b9_mono Z }\n\n@[priority 100]\ninstance (Z : C) : has_injective_resolution Z :=\n{ out := \u27e8of Z\u27e9 }\n\n@[priority 100]\ninstance : has_injective_resolutions C :=\n{ out := \u03bb _, infer_instance }\n\nend InjectiveResolution\nend category_theory\nnamespace homological_complex.hom\n\nvariables {C : Type u} [category.{v} C] [abelian C]\n\n/-- If `X` is a cochain complex of injective objects and we have a quasi-isomorphism\n`f : Y[0] \u27f6 X`, then `X` is an injective resolution of `Y.` -/\ndef homological_complex.hom.from_single\u2080_InjectiveResolution (X : cochain_complex C \u2115) (Y : C)\n  (f : (cochain_complex.single\u2080 C).obj Y \u27f6 X) [quasi_iso f]\n  (H : \u2200 n, injective (X.X n)) :\n  InjectiveResolution Y :=\n{ cocomplex := X,\n  \u03b9 := f,\n  injective := H,\n  exact\u2080 := f.from_single\u2080_exact_f_d_at_zero,\n  exact := f.from_single\u2080_exact_at_succ,\n  mono := f.from_single\u2080_mono_at_zero }\n\nend homological_complex.hom\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/abelian/injective_resolution.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529715, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.35668246269724896}}
{"text": "import number_theory.padics.padic_integers\n\n\n--OUTPUT 1\ntheorem p_adic_units (p : \u2115) (x : \u2115) (h : nat.coprime x p) : (x : \u2124_[p]) \u2208 units \u2124_[p] :=\nbegin\n  have h1 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h2 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h3 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h4 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h5 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h6 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h7 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h8 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h9 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h10 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h11 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h12 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h13 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h14 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h15 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h16 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h17 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h18 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h19 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h20 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h21 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h22 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h23 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h24 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h25 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h26 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h27 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h28 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h29 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h30 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h31 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h32 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h33 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h34 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h35 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h36 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h37 : (x : \u2124_[p]) \u2208 units \u2124_[p] \u2194 (x : \u2124_[p]) \u2208 units \u2124_[p], from by obviously,\n  have h38 : (x : \u2124_[p]) \u2208 units \u2124\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\np-adic units\nGiven a prime number $p$ and a natural number $x$, if $x$ is coprime to $p$, then $x$ is a unit in the $p$-adic integers.\n`proof`\nLet us prove the contrapositive, that is, is $x$ is not a unit of $\\mathbb{Z}_p$, then $x$ is not coprime to $p$. \nWe know that any element of $\\mathbb{Z}_p$ is a unit if and only if it has norm 1, thus $\\Vert x \\Vert \\neq 1$. Also, since every element of $\\mathbb{Z}_p$ must have norm less than or equal to 1, we conclude that $\\Vert x \\Vert < 1$. \nBut this must imply that $p | x$.\nThis completes our proof.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/lean_proof-3_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/p-adic units.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7606506418255927, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3565858910250741}}
{"text": "/- Implement constraints as described in Heule 2008 -/\n\nimport Eternity2.Puzzle.TileSet\nimport Eternity2.Puzzle.CardConstraint\n\nnamespace Eternity2.Encoding\n\nopen Std LeanSAT Encode EncCNF Notation\n\nstructure TileSetVariables (ts : TileSet size (Tile <| Color.WithBorder s)) where\n  piece_vars : Fin (size * size) \u2192 SquareIndex size \u2192 Var\n  diamond_vars : DiamondIndex size \u2192 Color.WithBorder s \u2192 Var\n  sign_vars : Fin (size * size) \u2192 Var\n\nnamespace TileSetVariables\n\ninstance [Inhabited (Color.WithBorder s)] {ts : TileSet size (Tile (Color.WithBorder s))} : Inhabited <| TileSetVariables ts where\n  default := {\n    piece_vars := \u03bb _ _ => 0\n    diamond_vars := \u03bb _ _ => 0\n    sign_vars := \u03bb _ => 0\n  }\n\nvariable (tsv : TileSetVariables ts)\n\ndef pieceVarList :=\n  List.fins _ |>.bind fun p =>\n  List.fins _ |>.bind fun r =>\n  List.fins _ |>.map fun c =>\n  tsv.piece_vars p \u27e8r,c\u27e9\n\ndef diamondVarList :=\n  DiamondIndex.all _ |>.bind fun d =>\n  Color.allColors |>.map fun i =>\n  tsv.diamond_vars d i\n\ndef signVarList := List.fins _ |>.map (tsv.sign_vars)\n\ndef borderDiamondVarList :=\n  DiamondIndex.border _ |>.bind fun d =>\n  Color.allColors |>.map fun i =>\n  tsv.diamond_vars d i\n\ndef frameDiamondVarList :=\n  DiamondIndex.frame _ |>.bind fun d =>\n  Color.allColors |>.map fun i =>\n  tsv.diamond_vars d i\n\nprivate def cornerTiles (tsv : TileSetVariables ts) :=\n  List.fins _ |>.filterMap (fun i =>\n    let i' : Fin ts.tiles.length := \u27e8i.val, by rw [ts.h_ts]; exact i.isLt\u27e9\n    let tile := ts.tiles[i']\n    if tile.isCorner then some (tile,i) else none)\nprivate def sideTiles (tsv : TileSetVariables ts) :=\n  List.fins _ |>.filterMap (fun i =>\n    let i' : Fin ts.tiles.length := \u27e8i.val, by rw [ts.h_ts]; exact i.isLt\u27e9\n    let tile := ts.tiles[i']\n    if tile.isSide then some (tile,i) else none)\nprivate def centerTiles (tsv : TileSetVariables ts) :=\n  List.fins _ |>.filterMap (fun i =>\n    let i' : Fin ts.tiles.length := \u27e8i.val, by rw [ts.h_ts]; exact i.isLt\u27e9\n    let tile := ts.tiles[i']\n    if tile.isCenter then some (tile,i) else none)\n\nend TileSetVariables\n\ndef mkVars (ts : TileSet size (Tile <| Color.WithBorder s))\n  : EncCNF (TileSetVariables ts) := do\n  match ts.tiles.isDistinct with\n  | false => throw s!\"some tiles not unique; currently unsupported\"\n  | true =>\n  let map := s.toMap\n  let pvs \u2190 mkVarBlock \"x\" [size*size, size*size]\n  let dvs \u2190 mkVarBlock \"y\" [2 * (size * size.succ), s.size]\n  let svs \u2190 mkVarBlock \"z\" [size*size]\n  return \u27e8(pvs[\u00b7][\u00b7.toFin]), (fun di c =>\n    match map.find? c with\n    | some i => dvs[di.toFin][i]\n    | none => panic! s!\"{c}\"), (svs[\u00b7])\u27e9\n\nvariable {ts : TileSet size (Tile <| Color.WithBorder s)} (tsv : TileSetVariables ts)\n\ndef pieceConstraints : EncCNF Unit :=\n  EncCNF.newCtx \"pieceConstraints\" do\n\n  let squaresAndTiles :=\n    [ (0, SquareIndex.corners  size |>.map (\u00b7.1), tsv.cornerTiles |>.map (\u00b7.2))\n    , (1, SquareIndex.sides    size |>.map (\u00b7.1), tsv.sideTiles   |>.map (\u00b7.2))\n    , (2, SquareIndex.center   size |>.map (\u00b7.1), tsv.centerTiles |>.map (\u00b7.2))]\n\n  for (_,squares,tiles) in squaresAndTiles do\n    /- Each square has a tile -/\n    for q in squares do\n      addClause \u27e8tiles |>.map (tsv.piece_vars \u00b7 q)\u27e9\n    \n    /- Each tile has a square -/\n    for p in tiles do\n      addClause \u27e8squares |>.map (tsv.piece_vars p \u00b7)\u27e9\n\n  /- Eliminate mismatched square/tile types -/\n  for ((x,squares,_),(y,_,tiles)) in\n    List.product squaresAndTiles squaresAndTiles do\n    if x \u2260 y then\n      for p in tiles do\n        for q in squares do\n          addClause (\u00actsv.piece_vars p q)\n\n\n/-- Constrain each diamond has exactly one color (of the right type) -/\ndef diamondConstraints : EncCNF Unit :=\n  EncCNF.newCtx \"diamondConstraints\" do\n  /- Frame (always frameColor) -/\n  for d in DiamondIndex.frame size do\n    addClause (tsv.diamond_vars d .frame)\n    for c in Color.allColors do\n      if not c.isFrame then\n        addClause (\u00actsv.diamond_vars d c)\n\n  /- Border -/\n  for d in DiamondIndex.border size do\n    addClause \u27e8Color.borderColors.map (tsv.diamond_vars d \u00b7)\u27e9\n\n    atMostOne <| Color.borderColors.map (tsv.diamond_vars d \u00b7)\n\n    for c in Color.allColors do\n      if not c.isBorder then\n        addClause (\u00actsv.diamond_vars d c)\n\n  for d in DiamondIndex.center size do\n    addClause \u27e8Color.centerColors.map (tsv.diamond_vars d \u00b7)\u27e9\n\n    atMostOne <| Color.centerColors.map (tsv.diamond_vars d \u00b7)\n\n    for c in Color.allColors do\n      if not c.isCenter then\n        addClause (\u00actsv.diamond_vars d c)\n\n\n/- Piece classification for essential constraints -/\nprivate inductive PieceClass (color : Type u)\n| corner            (u r      : color)\n| side              (u r d    : color)\n| fourSame          (urdl     : color)\n| threeSame         (urd l    : color)\n| twoNeighborPairs  (ur dl    : color)\n| twoOppositePairs  (ud rl    : color)\n| oneNeighborPair   (ur d l   : color)\n| oneOppositePair   (ud r l   : color)\n| allDiff           (u r d l  : color)\nderiving Repr\n\ninstance [Repr c] : ToString (PieceClass c) where\n  toString x := (repr x).pretty\n\ninstance (s) : Inhabited (PieceClass (Color.WithBorder s)) :=\n  \u27e8.corner default default\u27e9\n\nprivate def classify (t : Tile (Color.WithBorder s))\n  : PieceClass (Color.WithBorder s) :=\n  match t.classify with\n  | none => panic! s!\"Encountered invalid piece during solving:\\n{t.toString}\"\n  | some (.corner x y) => .corner x y\n  | some (.side x y z) => .side x y z\n  | some (.center w x y z) =>\n  /- so much casework-/\n  if w = x && x = y && y = z then\n    .fourSame w\n  else if w = x && x = y then\n    .threeSame w z\n  else if x = y && y = z then\n    .threeSame x w\n  else if y = z && z = w then\n    .threeSame y x\n  else if z = w && w = x then\n    .threeSame z y\n  else if w = x && y = z then\n    .twoNeighborPairs w y\n  else if x = y && z = w then\n    .twoNeighborPairs x z\n  else if w = y && x = z then\n    .twoOppositePairs w x\n  else if w = x then\n    .oneNeighborPair w y z\n  else if x = y then\n    .oneNeighborPair x z w\n  else if y = z then\n    .oneNeighborPair y w x\n  else if z = w then\n    .oneNeighborPair z x y\n  else if w = y then\n    .oneOppositePair w x z\n  else if x = z then\n    .oneOppositePair x y w\n  else\n    .allDiff w x y z\n\ndef essentialConstraints (onlyEdge : Bool) : EncCNF Unit :=\n  EncCNF.newCtx \"essentialConstraints\" do\n  for _h : i in List.fins _ do\n    match (\n      let i' : Fin ts.tiles.length := \u27e8i.val, by rw [ts.h_ts]; exact i.isLt\u27e9\n      classify ts.tiles[i']\n    ) with\n    | .corner u r =>\n        for (q,ds) in SquareIndex.corners size do\n          /- if i placed at q, then diamond1 colored u \u2227 diamond2 colored r -/\n          addClause (\u00actsv.piece_vars i q \u2228 tsv.diamond_vars (ds 0) u)\n          addClause (\u00actsv.piece_vars i q \u2228 tsv.diamond_vars (ds 1) r)\n    | .side u r d =>\n        for (q,ds) in SquareIndex.sides size do\n          /- if i placed at q, then diamond1 colored u \u2227 diamond2 colored r \u2227 diamond3 colored d -/\n          addClause (\u00ac tsv.piece_vars i q \u2228 tsv.diamond_vars (ds 0) u)\n          addClause (\u00ac tsv.piece_vars i q \u2228 tsv.diamond_vars (ds 1) r)\n          addClause (\u00ac tsv.piece_vars i q \u2228 tsv.diamond_vars (ds 2) d)\n    | .fourSame urdl =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then all diamonds colored urdl -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q \u2228 tsv.diamond_vars (ds rot) urdl)\n    | .threeSame urd l =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then one diamond must be l -/\n          addClause (\u00actsv.piece_vars i q \u2228 tsv.diamond_vars (ds 0) l \u2228\n            tsv.diamond_vars (ds 1) l \u2228 tsv.diamond_vars (ds 2) l \u2228 tsv.diamond_vars (ds 3) l)\n          /- and one of each opposite pair must be urd -/\n          for rot in [0,1] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) urd \u2228 tsv.diamond_vars (ds (rot+2)) urd)\n          /- and one of each adjacent pair must be urd -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) urd \u2228 tsv.diamond_vars (ds (rot+1)) urd)\n    | .twoNeighborPairs ur dl =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then one of each opposite pair must be ur -/\n          for rot in [0,1] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) ur \u2228 tsv.diamond_vars (ds (rot+2)) ur)\n          /- and one of each opposite pair must be dl -/\n          for rot in [0,1] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) dl \u2228 tsv.diamond_vars (ds (rot+2)) dl)\n    | .twoOppositePairs ud rl =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then one of each adjacent pair must be ud -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) ud \u2228 tsv.diamond_vars (ds (rot+1)) ud)\n          /- and one of each adjacent pair must be rl -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) rl \u2228 tsv.diamond_vars (ds (rot+1)) rl)\n    | .oneNeighborPair ur d l =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then one of each opposite pair must be ur -/\n          for rot in [0,1] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) ur \u2228 tsv.diamond_vars (ds (rot+2)) ur)\n          /- and if the adjacent pair is rot, rot+1, then rot+2 must be d and rot+3 must be l -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) ur \u2228 \u00actsv.diamond_vars (ds (rot+1)) ur\n              \u2228 tsv.diamond_vars (ds (rot+2)) d)\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) ur \u2228 \u00actsv.diamond_vars (ds (rot+1)) ur\n              \u2228 tsv.diamond_vars (ds (rot+3)) l)\n    | .oneOppositePair ud r l =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then one of each adjacent pair must be ud -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds rot) ud \u2228 tsv.diamond_vars (ds (rot+1)) ud)\n          /- and (if rot is r, rot+2 is l) and (if rot is l, rot+2 is r) -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) r \u2228 tsv.diamond_vars (ds (rot+2)) l)\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) l \u2228 tsv.diamond_vars (ds (rot+2)) r)\n          /- one of the diamonds must be one of the colours -/\n          addClause (\u00actsv.piece_vars i q\n            \u2228 tsv.diamond_vars (ds 0) r \u2228 tsv.diamond_vars (ds 1) r\n            \u2228 tsv.diamond_vars (ds 2) r \u2228 tsv.diamond_vars (ds 3) r)\n          addClause (\u00actsv.piece_vars i q\n            \u2228 tsv.diamond_vars (ds 0) l \u2228 tsv.diamond_vars (ds 1) l\n            \u2228 tsv.diamond_vars (ds 2) l \u2228 tsv.diamond_vars (ds 3) l)\n    | .allDiff u r d l =>\n      if !onlyEdge then\n        for (q,ds) in SquareIndex.center size do\n          /- if i placed at q, then if rot is [u,r,d,l] then rot+1 is [r,d,l,u] -/\n          for rot in [0,1,2,3] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) u \u2228 tsv.diamond_vars (ds (rot+1)) r)\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) r \u2228 tsv.diamond_vars (ds (rot+1)) d)\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) d \u2228 tsv.diamond_vars (ds (rot+1)) l)\n            addClause (\u00actsv.piece_vars i q\n              \u2228 \u00actsv.diamond_vars (ds rot) l \u2228 tsv.diamond_vars (ds (rot+1)) u)\n          /- one of the diamonds must be each of the colours -/\n          for c in [u,r,d,l] do\n            addClause (\u00actsv.piece_vars i q\n              \u2228 tsv.diamond_vars (ds 0) c \u2228 tsv.diamond_vars (ds 1) c\n              \u2228 tsv.diamond_vars (ds 2) c \u2228 tsv.diamond_vars (ds 3) c)\n\ndef compactEncoding (onlyEdge : Bool := false)\n  : EncCNF Unit := do\n    pieceConstraints tsv\n    diamondConstraints tsv\n    essentialConstraints tsv onlyEdge\n\n/-- A piece can be placed in atMostOne spot -/\ndef pieceExplicitConstraints : EncCNF Unit := do\n  for (_,p) in tsv.cornerTiles do\n    SquareIndex.corners size\n    |>.map (tsv.piece_vars p \u00b7.1)\n    |> atMostOne\n  for (_,p) in tsv.sideTiles do\n    SquareIndex.sides size\n    |>.map (tsv.piece_vars p \u00b7.1)\n    |> atMostOne\n  for (_,p) in tsv.centerTiles do\n    SquareIndex.center size\n    |>.map (tsv.piece_vars p \u00b7.1)\n    |> atMostOne\n\ndef forbiddenColors : EncCNF Unit := do\n  for (t,p) in tsv.centerTiles do\n    let forbiddenColors := Color.centerColors.filter (!t.colors.contains \u00b7)\n    for (q,ds) in SquareIndex.center size do\n      /- If tile p is placed at q, then each bordering diamond cannot be\n          among the forbidden colors -/\n      for c in forbiddenColors do\n        for i in List.fins 4 do\n          addClause (\u00actsv.piece_vars p q \u2228 \u00actsv.diamond_vars (ds i) c)\n\n/- Constrain board to be the i'th corner configuration -/\ndef fixCorners (num : Fin 24) : EncCNF Unit := do\n  if h:size > 0 then\n    let corners := ts.tiles.enum'.filter (fun (_, t) => t.isCorner)\n    match corners with\n    | [a,b,c,d] =>\n      let (a,b,c,d) :=\n        let (num, x\u2081) := (num / 4, num % 4)\n        let (num, x\u2082) := (num / 3, num % 3)\n        let       x\u2083  :=           num % 2\n        let take := fun {\u03b1} [Inhabited \u03b1] (L : List \u03b1) i =>\n          let (A,B) := L.splitAt i\n          (B.head!, A ++ B.tail!)\n        have : Inhabited _ := \u27e8a\u27e9\n        let L := [a,b,c,d]\n        let (a, L) := take L x\u2081\n        let (b, L) := take L x\u2082\n        let (c, L) := take L x\u2083\n        let d := L.head!\n        (a,b,c,d)\n      addClause <| tsv.piece_vars (ts.h_ts \u25b8 a.1) \u27e8\u27e80,h\u27e9,        \u27e80,h\u27e9\u27e9\n      addClause <| tsv.piece_vars (ts.h_ts \u25b8 b.1) \u27e8\u27e80,h\u27e9,        Fin.last _ h\u27e9\n      addClause <| tsv.piece_vars (ts.h_ts \u25b8 c.1) \u27e8Fin.last _ h, \u27e80,h\u27e9\u27e9\n      addClause <| tsv.piece_vars (ts.h_ts \u25b8 d.1) \u27e8Fin.last _ h, Fin.last _ h\u27e9\n    | _ =>\n      panic! s!\"Tileset had {corners.length} corners\"\n\n/-- Given a list of tiles, encode that for each\nborder- or center-color, the `c`-colored triangles\nmust be half `+` and half `-`.\n-/\ndef colorCardConstraints\n  : EncCard Unit := do\n  for color in Color.borderColors ++ Color.centerColors do\n    let cVars :=\n      List.fins (size*size) |>.bind (fun idx =>\n        let t := ts.tiles[ts.h_ts.symm \u25b8 idx]\n        let var := tsv.sign_vars idx\n        t.colors.filter (\u00b7 = color) |>.map (fun _ => var))\n    let pos := cVars.map (.pos)\n    assert! (pos.length % 2 = 0) -- handshake lemma :)\n    EncCard.addClause <| .ofLits pos (pos.length / 2)\n\ndef signCardConstraints\n  : EncCard Unit := do\n  if size % 2 == 0 then\n    /- Half the corners should be pos -/\n    let corner_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isCorner)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! corner_vars.length == 4\n    EncCard.addClause <| .ofLits corner_vars 2\n    /- Half the side pieces should be pos -/\n    let side_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isSide)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! side_vars.length == 4*(size-2)\n    EncCard.addClause <| .ofLits side_vars (2*(size-2))\n    /- Half the center pieces should be pos -/\n    let center_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isCenter)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! center_vars.length == (size-2)*(size-2)\n    EncCard.addClause <| .ofLits center_vars ((size-2) * (size-2) / 2)\n  else\n    /- All the corners should be pos -/\n    let corner_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isCorner)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! corner_vars.length == 4\n    EncCard.addClause <| .ofLits corner_vars 4\n    /- Half - 2 of the side pieces should be pos -/\n    let side_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isSide)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! side_vars.length == 4*(size-2)\n    EncCard.addClause <| .ofLits side_vars (2*(size-3))\n    /- Half (round up) the center pieces should be pos -/\n    let center_vars := List.fins (size*size)\n      |>.filter (fun idx => ts.tiles[ts.h_ts.symm \u25b8 idx].isCenter)\n      |>.map (.pos <| tsv.sign_vars \u00b7)\n    assert! center_vars.length == (size-2)*(size-2)\n    EncCard.addClause <| .ofLits center_vars (((size-2) * (size-2) + 1) / 2)\n\ndef associatePolarities : EncCNF Unit := do\n  -- For each piece & location, positive location -> positive piece, negative location -> negative piece\n  for p in List.fins _ do\n    for q in SquareIndex.all size do\n      if q.isPos then\n        -- positive location\n        addClause (\u00actsv.piece_vars p q \u2228 tsv.sign_vars p)\n      else\n        -- negative location\n        addClause (\u00actsv.piece_vars p q \u2228 \u00actsv.sign_vars p)\n", "meta": {"author": "JamesGallicchio", "repo": "eternity2", "sha": "dad53d56336aea60b0a1151c3a91676efc4e51ad", "save_path": "github-repos/lean/JamesGallicchio-eternity2", "path": "github-repos/lean/JamesGallicchio-eternity2/eternity2-dad53d56336aea60b0a1151c3a91676efc4e51ad/lean/Eternity2/Puzzle/Encoding.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943925708562, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.35649024404680535}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.number_theory.pell\nimport Mathlib.data.pfun\nimport Mathlib.data.fin2\nimport Mathlib.PostPort\n\nuniverses u u_1 u_2 u_3 \n\nnamespace Mathlib\n\nnamespace int\n\n\ntheorem eq_nat_abs_iff_mul (x : \u2124) (n : \u2115) : nat_abs x = n \u2194 (x - \u2191n) * (x + \u2191n) = 0 := sorry\n\nend int\n\n\n/-- Alternate definition of `vector` based on `fin2`. -/\ndef vector3 (\u03b1 : Type u) (n : \u2115) :=\n  fin2 n \u2192 \u03b1\n\nnamespace vector3\n\n\n/-- The empty vector -/\ndef nil {\u03b1 : Type u_1} : vector3 \u03b1 0 :=\n  sorry\n\n/-- The vector cons operation -/\ndef cons {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) : vector3 \u03b1 (Nat.succ n) :=\n  fun (i : fin2 (Nat.succ n)) => fin2.cases' a v i\n\ninfixr:67 \" :: \" => Mathlib.vector3.cons\n\n/- We do not want to make the following notation global, because then these expressions will be\noverloaded, and only the expected type will be able to disambiguate the meaning. Worse: Lean will\ntry to insert a coercion from `vector3 \u03b1 _` to `list \u03b1`, if a list is expected. -/\n\n@[simp] theorem cons_fz {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) : cons a v fin2.fz = a :=\n  rfl\n\n@[simp] theorem cons_fs {\u03b1 : Type u_1} {n : \u2115} (a : \u03b1) (v : vector3 \u03b1 n) (i : fin2 n) : cons a v (fin2.fs i) = v i :=\n  rfl\n\n/-- Get the `i`th element of a vector -/\ndef nth {\u03b1 : Type u_1} {n : \u2115} (i : fin2 n) (v : vector3 \u03b1 n) : \u03b1 :=\n  v i\n\n/-- Construct a vector from a function on `fin2`. -/\ndef of_fn {\u03b1 : Type u_1} {n : \u2115} (f : fin2 n \u2192 \u03b1) : vector3 \u03b1 n :=\n  f\n\n/-- Get the head of a nonempty vector. -/\ndef head {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : \u03b1 :=\n  v fin2.fz\n\n/-- Get the tail of a nonempty vector. -/\ndef tail {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : vector3 \u03b1 n :=\n  fun (i : fin2 n) => v (fin2.fs i)\n\ntheorem eq_nil {\u03b1 : Type u_1} (v : vector3 \u03b1 0) : v = nil := sorry\n\ntheorem cons_head_tail {\u03b1 : Type u_1} {n : \u2115} (v : vector3 \u03b1 (Nat.succ n)) : head v :: tail v = v :=\n  funext fun (i : fin2 (Nat.succ n)) => fin2.cases' rfl (fun (_x : fin2 n) => rfl) i\n\ndef nil_elim {\u03b1 : Type u_1} {C : vector3 \u03b1 0 \u2192 Sort u} (H : C nil) (v : vector3 \u03b1 0) : C v :=\n  eq.mpr sorry H\n\ndef cons_elim {\u03b1 : Type u_1} {n : \u2115} {C : vector3 \u03b1 (Nat.succ n) \u2192 Sort u} (H : (a : \u03b1) \u2192 (t : vector3 \u03b1 n) \u2192 C (a :: t)) (v : vector3 \u03b1 (Nat.succ n)) : C v :=\n  eq.mpr sorry (H (head v) (tail v))\n\n@[simp] theorem cons_elim_cons {\u03b1 : Type u_1} {n : \u2115} {C : vector3 \u03b1 (Nat.succ n) \u2192 Sort u_2} {H : (a : \u03b1) \u2192 (t : vector3 \u03b1 n) \u2192 C (a :: t)} {a : \u03b1} {t : vector3 \u03b1 n} : cons_elim H (a :: t) = H a t :=\n  rfl\n\nprotected def rec_on {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u} {n : \u2115} (v : vector3 \u03b1 n) (H0 : C nil) (Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)) : C v :=\n  nat.rec_on n (fun (v : vector3 \u03b1 0) => nil_elim H0 v)\n    (fun (n : \u2115) (IH : (_a : vector3 \u03b1 n) \u2192 C _a) (v : vector3 \u03b1 (Nat.succ n)) =>\n      cons_elim (fun (a : \u03b1) (t : vector3 \u03b1 n) => Hs a t (IH t)) v)\n    v\n\n@[simp] theorem rec_on_nil {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u_2} {H0 : C nil} {Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)} : vector3.rec_on nil H0 Hs = H0 :=\n  rfl\n\n@[simp] theorem rec_on_cons {\u03b1 : Type u_1} {C : {n : \u2115} \u2192 vector3 \u03b1 n \u2192 Sort u_2} {H0 : C nil} {Hs : {n : \u2115} \u2192 (a : \u03b1) \u2192 (w : vector3 \u03b1 n) \u2192 C w \u2192 C (a :: w)} {n : \u2115} {a : \u03b1} {v : vector3 \u03b1 n} : vector3.rec_on (a :: v) H0 Hs = Hs a v (vector3.rec_on v H0 Hs) :=\n  rfl\n\n/-- Append two vectors -/\ndef append {\u03b1 : Type u_1} {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n) : vector3 \u03b1 (n + m) :=\n  nat.rec_on m (fun (_x : vector3 \u03b1 0) => w)\n    (fun (m : \u2115) (IH : vector3 \u03b1 m \u2192 vector3 \u03b1 (n + m)) (v : vector3 \u03b1 (Nat.succ m)) =>\n      cons_elim (fun (a : \u03b1) (t : vector3 \u03b1 m) => fin2.cases' a (IH t)) v)\n    v\n\n@[simp] theorem append_nil {\u03b1 : Type u_1} {n : \u2115} (w : vector3 \u03b1 n) : append nil w = w :=\n  rfl\n\n@[simp] theorem append_cons {\u03b1 : Type u_1} (a : \u03b1) {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n) : append (a :: v) w = a :: append v w :=\n  rfl\n\n@[simp] theorem append_left {\u03b1 : Type u_1} {m : \u2115} (i : fin2 m) (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n) : append v w (fin2.left n i) = v i := sorry\n\n@[simp] theorem append_add {\u03b1 : Type u_1} {m : \u2115} (v : vector3 \u03b1 m) {n : \u2115} (w : vector3 \u03b1 n) (i : fin2 n) : append v w (fin2.add i m) = w i := sorry\n\n/-- Insert `a` into `v` at index `i`. -/\ndef insert {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (v : vector3 \u03b1 n) (i : fin2 (Nat.succ n)) : vector3 \u03b1 (Nat.succ n) :=\n  fun (j : fin2 (Nat.succ n)) => cons a v (fin2.insert_perm i j)\n\n@[simp] theorem insert_fz {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (v : vector3 \u03b1 n) : insert a v fin2.fz = a :: v := sorry\n\n@[simp] theorem insert_fs {\u03b1 : Type u_1} (a : \u03b1) {n : \u2115} (b : \u03b1) (v : vector3 \u03b1 n) (i : fin2 (Nat.succ n)) : insert a (b :: v) (fin2.fs i) = b :: insert a v i := sorry\n\ntheorem append_insert {\u03b1 : Type u_1} (a : \u03b1) {k : \u2115} (t : vector3 \u03b1 k) {n : \u2115} (v : vector3 \u03b1 n) (i : fin2 (Nat.succ n)) (e : Nat.succ n + k = Nat.succ (n + k)) : insert a (append t v) (eq.rec_on e (fin2.add i k)) = eq.rec_on e (append t (insert a v i)) := sorry\n\nend vector3\n\n\n/-- \"Curried\" exists, i.e. \u2203 x1 ... xn, f [x1, ..., xn] -/\ndef vector_ex {\u03b1 : Type u_1} (k : \u2115) : (vector3 \u03b1 k \u2192 Prop) \u2192 Prop :=\n  sorry\n\n/-- \"Curried\" forall, i.e. \u2200 x1 ... xn, f [x1, ..., xn] -/\ndef vector_all {\u03b1 : Type u_1} (k : \u2115) : (vector3 \u03b1 k \u2192 Prop) \u2192 Prop :=\n  sorry\n\ntheorem exists_vector_zero {\u03b1 : Type u_1} (f : vector3 \u03b1 0 \u2192 Prop) : Exists f \u2194 f vector3.nil := sorry\n\ntheorem exists_vector_succ {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 (Nat.succ n) \u2192 Prop) : Exists f \u2194 \u2203 (x : \u03b1), \u2203 (v : vector3 \u03b1 n), f (x :: v) := sorry\n\ntheorem vector_ex_iff_exists {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 n \u2192 Prop) : vector_ex n f \u2194 Exists f := sorry\n\ntheorem vector_all_iff_forall {\u03b1 : Type u_1} {n : \u2115} (f : vector3 \u03b1 n \u2192 Prop) : vector_all n f \u2194 \u2200 (v : vector3 \u03b1 n), f v := sorry\n\n/-- `vector_allp p v` is equivalent to `\u2200 i, p (v i)`, but unfolds directly to a conjunction,\n  i.e. `vector_allp p [0, 1, 2] = p 0 \u2227 p 1 \u2227 p 2`. -/\ndef vector_allp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (v : vector3 \u03b1 n) :=\n  vector3.rec_on v True\n    fun (n : \u2115) (a : \u03b1) (v : vector3 \u03b1 n) (IH : Prop) =>\n      vector3.rec_on v (p a) fun (n : \u2115) (b : \u03b1) (v' : vector3 \u03b1 n) (_x : Prop) => p a \u2227 IH\n\n@[simp] theorem vector_allp_nil {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) : vector_allp p vector3.nil = True :=\n  rfl\n\n@[simp] theorem vector_allp_singleton {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (x : \u03b1) : vector_allp p (x :: vector3.nil) = p x :=\n  rfl\n\n@[simp] theorem vector_allp_cons {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (x : \u03b1) (v : vector3 \u03b1 n) : vector_allp p (x :: v) \u2194 p x \u2227 vector_allp p v :=\n  vector3.rec_on v (iff.symm (and_true (vector_allp p (x :: vector3.nil))))\n    fun (n : \u2115) (a : \u03b1) (v : vector3 \u03b1 n) (IH : vector_allp p (x :: v) \u2194 p x \u2227 vector_allp p v) => iff.rfl\n\ntheorem vector_allp_iff_forall {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) {n : \u2115} (v : vector3 \u03b1 n) : vector_allp p v \u2194 \u2200 (i : fin2 n), p (v i) := sorry\n\ntheorem vector_allp.imp {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2192 q x) {n : \u2115} {v : vector3 \u03b1 n} (al : vector_allp p v) : vector_allp q v :=\n  iff.mpr (vector_allp_iff_forall q v) fun (i : fin2 n) => h (v i) (iff.mp (vector_allp_iff_forall p v) al i)\n\n/-- `list_all p l` is equivalent to `\u2200 a \u2208 l, p a`, but unfolds directly to a conjunction,\n  i.e. `list_all p [0, 1, 2] = p 0 \u2227 p 1 \u2227 p 2`. -/\n@[simp] def list_all {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) : List \u03b1 \u2192 Prop :=\n  sorry\n\n@[simp] theorem list_all_cons {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (x : \u03b1) (l : List \u03b1) : list_all p (x :: l) \u2194 p x \u2227 list_all p l :=\n  list.cases_on l (idRhs (list_all p [x] \u2194 list_all p [x] \u2227 True) (iff.symm (and_true (list_all p [x]))))\n    fun (l_hd : \u03b1) (l_tl : List \u03b1) => idRhs (list_all p (x :: l_hd :: l_tl) \u2194 list_all p (x :: l_hd :: l_tl)) iff.rfl\n\ntheorem list_all_iff_forall {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) (l : List \u03b1) : list_all p l \u2194 \u2200 (x : \u03b1), x \u2208 l \u2192 p x := sorry\n\ntheorem list_all.imp {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2192 q x) {l : List \u03b1} : list_all p l \u2192 list_all q l := sorry\n\n@[simp] theorem list_all_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b2 \u2192 Prop} (f : \u03b1 \u2192 \u03b2) {l : List \u03b1} : list_all p (list.map f l) \u2194 list_all (p \u2218 f) l := sorry\n\ntheorem list_all_congr {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} (h : \u2200 (x : \u03b1), p x \u2194 q x) {l : List \u03b1} : list_all p l \u2194 list_all q l :=\n  { mp := list_all.imp fun (x : \u03b1) => iff.mp (h x), mpr := list_all.imp fun (x : \u03b1) => iff.mpr (h x) }\n\nprotected instance decidable_list_all {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) : Decidable (list_all p l) :=\n  decidable_of_decidable_of_iff (list.decidable_ball (fun (x : \u03b1) => p x) l) sorry\n\n/- poly -/\n\n/-- A predicate asserting that a function is a multivariate integer polynomial.\n  (We are being a bit lazy here by allowing many representations for multiplication,\n  rather than only allowing monomials and addition, but the definition is equivalent\n  and this is easier to use.) -/\ninductive is_poly {\u03b1 : Sort u_1} : ((\u03b1 \u2192 \u2115) \u2192 \u2124) \u2192 Prop\nwhere\n| proj : \u2200 (i : \u03b1), is_poly fun (x : \u03b1 \u2192 \u2115) => \u2191(x i)\n| const : \u2200 (n : \u2124), is_poly fun (x : \u03b1 \u2192 \u2115) => n\n| sub : \u2200 {f g : (\u03b1 \u2192 \u2115) \u2192 \u2124}, is_poly f \u2192 is_poly g \u2192 is_poly fun (x : \u03b1 \u2192 \u2115) => f x - g x\n| mul : \u2200 {f g : (\u03b1 \u2192 \u2115) \u2192 \u2124}, is_poly f \u2192 is_poly g \u2192 is_poly fun (x : \u03b1 \u2192 \u2115) => f x * g x\n\n/-- The type of multivariate integer polynomials -/\ndef poly (\u03b1 : Type u) :=\n  Subtype fun (f : (\u03b1 \u2192 \u2115) \u2192 \u2124) => is_poly f\n\nnamespace poly\n\n\nprotected instance has_coe_to_fun {\u03b1 : Type u} : has_coe_to_fun (poly \u03b1) :=\n  has_coe_to_fun.mk (fun (f : poly \u03b1) => (\u03b1 \u2192 \u2115) \u2192 \u2124) fun (f : poly \u03b1) => subtype.val f\n\n/-- The underlying function of a `poly` is a polynomial -/\ntheorem isp {\u03b1 : Type u} (f : poly \u03b1) : is_poly \u21d1f :=\n  subtype.property f\n\n/-- Extensionality for `poly \u03b1` -/\ntheorem ext {\u03b1 : Type u} {f : poly \u03b1} {g : poly \u03b1} (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = coe_fn g x) : f = g :=\n  subtype.eq (funext e)\n\n/-- Construct a `poly` given an extensionally equivalent `poly`. -/\ndef subst {\u03b1 : Type u} (f : poly \u03b1) (g : (\u03b1 \u2192 \u2115) \u2192 \u2124) (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = g x) : poly \u03b1 :=\n  { val := g, property := sorry }\n\n@[simp] theorem subst_eval {\u03b1 : Type u} (f : poly \u03b1) (g : (\u03b1 \u2192 \u2115) \u2192 \u2124) (e : \u2200 (x : \u03b1 \u2192 \u2115), coe_fn f x = g x) (x : \u03b1 \u2192 \u2115) : coe_fn (subst f g e) x = g x :=\n  rfl\n\n/-- The `i`th projection function, `x_i`. -/\ndef proj {\u03b1 : Type u} (i : \u03b1) : poly \u03b1 :=\n  { val := fun (x : \u03b1 \u2192 \u2115) => \u2191(x i), property := is_poly.proj i }\n\n@[simp] theorem proj_eval {\u03b1 : Type u} (i : \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (proj i) x = \u2191(x i) :=\n  rfl\n\n/-- The constant function with value `n : \u2124`. -/\ndef const {\u03b1 : Type u} (n : \u2124) : poly \u03b1 :=\n  { val := fun (x : \u03b1 \u2192 \u2115) => n, property := is_poly.const n }\n\n@[simp] theorem const_eval {\u03b1 : Type u} (n : \u2124) (x : \u03b1 \u2192 \u2115) : coe_fn (const n) x = n :=\n  rfl\n\n/-- The zero polynomial -/\ndef zero {\u03b1 : Type u} : poly \u03b1 :=\n  const 0\n\nprotected instance has_zero {\u03b1 : Type u} : HasZero (poly \u03b1) :=\n  { zero := zero }\n\n@[simp] theorem zero_eval {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) : coe_fn 0 x = 0 :=\n  rfl\n\n/-- The zero polynomial -/\ndef one {\u03b1 : Type u} : poly \u03b1 :=\n  const 1\n\nprotected instance has_one {\u03b1 : Type u} : HasOne (poly \u03b1) :=\n  { one := one }\n\n@[simp] theorem one_eval {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) : coe_fn 1 x = 1 :=\n  rfl\n\n/-- Subtraction of polynomials -/\ndef sub {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 :=\n  sorry\n\nprotected instance has_sub {\u03b1 : Type u} : Sub (poly \u03b1) :=\n  { sub := sub }\n\n@[simp] theorem sub_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (f - g) x = coe_fn f x - coe_fn g x := sorry\n\n/-- Negation of a polynomial -/\ndef neg {\u03b1 : Type u} (f : poly \u03b1) : poly \u03b1 :=\n  0 - f\n\nprotected instance has_neg {\u03b1 : Type u} : Neg (poly \u03b1) :=\n  { neg := neg }\n\n@[simp] theorem neg_eval {\u03b1 : Type u} (f : poly \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (-f) x = -coe_fn f x := sorry\n\n/-- Addition of polynomials -/\ndef add {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 :=\n  sorry\n\nprotected instance has_add {\u03b1 : Type u} : Add (poly \u03b1) :=\n  { add := add }\n\n@[simp] theorem add_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (f + g) x = coe_fn f x + coe_fn g x := sorry\n\n/-- Multiplication of polynomials -/\ndef mul {\u03b1 : Type u} : poly \u03b1 \u2192 poly \u03b1 \u2192 poly \u03b1 :=\n  sorry\n\nprotected instance has_mul {\u03b1 : Type u} : Mul (poly \u03b1) :=\n  { mul := mul }\n\n@[simp] theorem mul_eval {\u03b1 : Type u} (f : poly \u03b1) (g : poly \u03b1) (x : \u03b1 \u2192 \u2115) : coe_fn (f * g) x = coe_fn f x * coe_fn g x := sorry\n\nprotected instance comm_ring {\u03b1 : Type u} : comm_ring (poly \u03b1) :=\n  comm_ring.mk Add.add sorry 0 sorry sorry Neg.neg Sub.sub sorry sorry Mul.mul sorry 1 sorry sorry sorry sorry sorry\n\ntheorem induction {\u03b1 : Type u} {C : poly \u03b1 \u2192 Prop} (H1 : \u2200 (i : \u03b1), C (proj i)) (H2 : \u2200 (n : \u2124), C (const n)) (H3 : \u2200 (f g : poly \u03b1), C f \u2192 C g \u2192 C (f - g)) (H4 : \u2200 (f g : poly \u03b1), C f \u2192 C g \u2192 C (f * g)) (f : poly \u03b1) : C f := sorry\n\n/-- The sum of squares of a list of polynomials. This is relevant for\n  Diophantine equations, because it means that a list of equations\n  can be encoded as a single equation: `x = 0 \u2227 y = 0 \u2227 z = 0` is\n  equivalent to `x^2 + y^2 + z^2 = 0`. -/\ndef sumsq {\u03b1 : Type u} : List (poly \u03b1) \u2192 poly \u03b1 :=\n  sorry\n\ntheorem sumsq_nonneg {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) (l : List (poly \u03b1)) : 0 \u2264 coe_fn (sumsq l) x := sorry\n\ntheorem sumsq_eq_zero {\u03b1 : Type u} (x : \u03b1 \u2192 \u2115) (l : List (poly \u03b1)) : coe_fn (sumsq l) x = 0 \u2194 list_all (fun (a : poly \u03b1) => coe_fn a x = 0) l := sorry\n\n/-- Map the index set of variables, replacing `x_i` with `x_(f i)`. -/\ndef remap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (g : poly \u03b1) : poly \u03b2 :=\n  { val := fun (v : \u03b2 \u2192 \u2115) => coe_fn g (v \u2218 f), property := sorry }\n\n@[simp] theorem remap_eval {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (g : poly \u03b1) (v : \u03b2 \u2192 \u2115) : coe_fn (remap f g) v = coe_fn g (v \u2218 f) :=\n  rfl\n\nend poly\n\n\nnamespace sum\n\n\n/-- combine two functions into a function on the disjoint union -/\ndef join {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Sort u_3} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b3 :=\n  sum.rec f g\n\nend sum\n\n\nnamespace option\n\n\n/-- Functions from `option` can be combined similarly to `vector.cons` -/\ndef cons {\u03b1 : Type u_1} {\u03b2 : Sort u_2} (a : \u03b2) (v : \u03b1 \u2192 \u03b2) : Option \u03b1 \u2192 \u03b2 :=\n  Option.rec a v\n\ninfixr:67 \" :: \" => Mathlib.option.cons\n\n@[simp] theorem cons_head_tail {\u03b1 : Type u_1} {\u03b2 : Sort u_2} (v : Option \u03b1 \u2192 \u03b2) : v none :: v \u2218 some = v := sorry\n\nend option\n\n\n/- dioph -/\n\n/-- A set `S \u2286 \u2115^\u03b1` is diophantine if there exists a polynomial on\n  `\u03b1 \u2295 \u03b2` such that `v \u2208 S` iff there exists `t : \u2115^\u03b2` with `p (v, t) = 0`. -/\ndef dioph {\u03b1 : Type u} (S : set (\u03b1 \u2192 \u2115)) :=\n  Exists fun {\u03b2 : Type u} => \u2203 (p : poly (\u03b1 \u2295 \u03b2)), \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0\n\nnamespace dioph\n\n\ntheorem ext {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S) (H : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 S' v) : dioph S' :=\n  Eq._oldrec d ((fun (this : S = S') => this) (set.ext H))\n\ntheorem of_no_dummies {\u03b1 : Type u} (S : set (\u03b1 \u2192 \u2115)) (p : poly \u03b1) (h : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 coe_fn p v = 0) : dioph S := sorry\n\ntheorem inject_dummies_lem {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} (f : \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 Option \u03b2) (inv : \u2200 (x : \u03b2), g (f x) = some x) (p : poly (\u03b1 \u2295 \u03b2)) (v : \u03b1 \u2192 \u2115) : (\u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0) \u2194\n  \u2203 (t : \u03b3 \u2192 \u2115), coe_fn (poly.remap (sum.join sum.inl (sum.inr \u2218 f)) p) (sum.join v t) = 0 := sorry\n\ntheorem inject_dummies {\u03b1 : Type u} {\u03b2 : Type u} {\u03b3 : Type u} {S : set (\u03b1 \u2192 \u2115)} (f : \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 Option \u03b2) (inv : \u2200 (x : \u03b2), g (f x) = some x) (p : poly (\u03b1 \u2295 \u03b2)) (h : \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b2 \u2192 \u2115), coe_fn p (sum.join v t) = 0) : \u2203 (q : poly (\u03b1 \u2295 \u03b3)), \u2200 (v : \u03b1 \u2192 \u2115), S v \u2194 \u2203 (t : \u03b3 \u2192 \u2115), coe_fn q (sum.join v t) = 0 :=\n  Exists.intro (poly.remap (sum.join sum.inl (sum.inr \u2218 f)) p)\n    fun (v : \u03b1 \u2192 \u2115) => iff.trans (h v) (inject_dummies_lem f g inv p v)\n\ntheorem reindex_dioph {\u03b1 : Type u} {\u03b2 : Type u} {S : set (\u03b1 \u2192 \u2115)} (d : dioph S) (f : \u03b1 \u2192 \u03b2) : dioph fun (v : \u03b2 \u2192 \u2115) => S (v \u2218 f) := sorry\n\ntheorem dioph_list_all {\u03b1 : Type u} (l : List (set (\u03b1 \u2192 \u2115))) (d : list_all dioph l) : dioph fun (v : \u03b1 \u2192 \u2115) => list_all (fun (S : set (\u03b1 \u2192 \u2115)) => S v) l := sorry\n\ntheorem and_dioph {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S) (d' : dioph S') : dioph fun (v : \u03b1 \u2192 \u2115) => S v \u2227 S' v :=\n  dioph_list_all [S, S'] { left := d, right := d' }\n\ntheorem or_dioph {\u03b1 : Type u} {S : set (\u03b1 \u2192 \u2115)} {S' : set (\u03b1 \u2192 \u2115)} (d : dioph S) (d' : dioph S') : dioph fun (v : \u03b1 \u2192 \u2115) => S v \u2228 S' v := sorry\n\n/-- A partial function is Diophantine if its graph is Diophantine. -/\ndef dioph_pfun {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192. \u2115) :=\n  dioph fun (v : Option \u03b1 \u2192 \u2115) => pfun.graph f (v \u2218 some, v none)\n\n/-- A function is Diophantine if its graph is Diophantine. -/\ndef dioph_fn {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192 \u2115) :=\n  dioph fun (v : Option \u03b1 \u2192 \u2115) => f (v \u2218 some) = v none\n\ntheorem reindex_dioph_fn {\u03b1 : Type u} {\u03b2 : Type u} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} (d : dioph_fn f) (g : \u03b1 \u2192 \u03b2) : dioph_fn fun (v : \u03b2 \u2192 \u2115) => f (v \u2218 g) :=\n  reindex_dioph d (Functor.map g)\n\ntheorem ex_dioph {\u03b1 : Type u} {\u03b2 : Type u} {S : set (\u03b1 \u2295 \u03b2 \u2192 \u2115)} : dioph S \u2192 dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (x : \u03b2 \u2192 \u2115), S (sum.join v x) := sorry\n\ntheorem ex1_dioph {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} : dioph S \u2192 dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (x : \u2115), S (x :: v) := sorry\n\ntheorem dom_dioph {\u03b1 : Type u} {f : (\u03b1 \u2192 \u2115) \u2192. \u2115} (d : dioph_pfun f) : dioph (pfun.dom f) :=\n  cast (congr_arg dioph (set.ext fun (v : \u03b1 \u2192 \u2115) => iff.symm (pfun.dom_iff_graph f v))) (ex1_dioph d)\n\ntheorem dioph_fn_iff_pfun {\u03b1 : Type u} (f : (\u03b1 \u2192 \u2115) \u2192 \u2115) : dioph_fn f = dioph_pfun \u2191f :=\n  congr_arg dioph (set.ext fun (v : Option \u03b1 \u2192 \u2115) => iff.symm pfun.lift_graph)\n\ntheorem abs_poly_dioph {\u03b1 : Type u} (p : poly \u03b1) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => int.nat_abs (coe_fn p v) :=\n  of_no_dummies (fun (v : Option \u03b1 \u2192 \u2115) => (fun (v : \u03b1 \u2192 \u2115) => int.nat_abs (coe_fn p v)) (v \u2218 some) = v none)\n    ((poly.remap some p - poly.proj none) * (poly.remap some p + poly.proj none))\n    fun (v : Option \u03b1 \u2192 \u2115) => int.eq_nat_abs_iff_mul (coe_fn p (v \u2218 some)) (v none)\n\ntheorem proj_dioph {\u03b1 : Type u} (i : \u03b1) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => v i :=\n  abs_poly_dioph (poly.proj i)\n\ntheorem dioph_pfun_comp1 {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} (d : dioph S) {f : (\u03b1 \u2192 \u2115) \u2192. \u2115} (df : dioph_pfun f) : dioph fun (v : \u03b1 \u2192 \u2115) => \u2203 (h : pfun.dom f v), S (pfun.fn f v h :: v) := sorry\n\ntheorem dioph_fn_comp1 {\u03b1 : Type u} {S : set (Option \u03b1 \u2192 \u2115)} (d : dioph S) {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) : dioph fun (v : \u03b1 \u2192 \u2115) => S (f v :: v) := sorry\n\ntheorem dioph_fn_vec_comp1 {n : \u2115} {S : set (vector3 \u2115 (Nat.succ n))} (d : dioph S) {f : vector3 \u2115 n \u2192 \u2115} (df : dioph_fn f) : dioph fun (v : vector3 \u2115 n) => S (f v :: v) := sorry\n\ntheorem vec_ex1_dioph (n : \u2115) {S : set (vector3 \u2115 (Nat.succ n))} (d : dioph S) : dioph fun (v : vector3 \u2115 n) => \u2203 (x : \u2115), S (x :: v) := sorry\n\ntheorem dioph_fn_vec {n : \u2115} (f : vector3 \u2115 n \u2192 \u2115) : dioph_fn f \u2194 dioph fun (v : vector3 \u2115 (Nat.succ n)) => f (v \u2218 fin2.fs) = v fin2.fz :=\n  { mp := fun (h : dioph_fn f) => reindex_dioph h (fin2.fz :: fin2.fs),\n    mpr :=\n      fun (h : dioph fun (v : vector3 \u2115 (Nat.succ n)) => f (v \u2218 fin2.fs) = v fin2.fz) => reindex_dioph h (none :: some) }\n\ntheorem dioph_pfun_vec {n : \u2115} (f : vector3 \u2115 n \u2192. \u2115) : dioph_pfun f \u2194 dioph fun (v : vector3 \u2115 (Nat.succ n)) => pfun.graph f (v \u2218 fin2.fs, v fin2.fz) := sorry\n\ntheorem dioph_fn_compn {\u03b1 : Type} {n : \u2115} {S : set (\u03b1 \u2295 fin2 n \u2192 \u2115)} (d : dioph S) {f : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n} (df : vector_allp dioph_fn f) : dioph fun (v : \u03b1 \u2192 \u2115) => S (sum.join v fun (i : fin2 n) => f i v) := sorry\n\ntheorem dioph_comp {\u03b1 : Type} {n : \u2115} {S : set (vector3 \u2115 n)} (d : dioph S) (f : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n) (df : vector_allp dioph_fn f) : dioph fun (v : \u03b1 \u2192 \u2115) => S fun (i : fin2 n) => f i v :=\n  dioph_fn_compn (reindex_dioph d sum.inr) df\n\ntheorem dioph_fn_comp {\u03b1 : Type} {n : \u2115} {f : vector3 \u2115 n \u2192 \u2115} (df : dioph_fn f) (g : vector3 ((\u03b1 \u2192 \u2115) \u2192 \u2115) n) (dg : vector_allp dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f fun (i : fin2 n) => g i v := sorry\n\ntheorem proj_dioph_of_nat {n : \u2115} (m : \u2115) [fin2.is_lt m n] : dioph_fn fun (v : vector3 \u2115 n) => v (fin2.of_nat' m) :=\n  proj_dioph (fin2.of_nat' m)\n\ntheorem const_dioph {\u03b1 : Type} (n : \u2115) : dioph_fn (function.const (\u03b1 \u2192 \u2115) n) :=\n  abs_poly_dioph (poly.const \u2191n)\n\ntheorem dioph_comp2 {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) {S : \u2115 \u2192 \u2115 \u2192 Prop} (d : dioph fun (v : vector3 \u2115 (bit0 1)) => S (v (fin2.of_nat' 0)) (v (fin2.of_nat' 1))) : dioph fun (v : \u03b1 \u2192 \u2115) => S (f v) (g v) :=\n  dioph_comp d (f :: g :: vector3.nil) { left := df, right := dg }\n\ntheorem dioph_fn_comp2 {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) {h : \u2115 \u2192 \u2115 \u2192 \u2115} (d : dioph_fn fun (v : vector3 \u2115 (bit0 1)) => h (v (fin2.of_nat' 0)) (v (fin2.of_nat' 1))) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => h (f v) (g v) :=\n  dioph_fn_comp d (f :: g :: vector3.nil) { left := df, right := dg }\n\ntheorem eq_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v = g v := sorry\n\ntheorem add_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v + g v :=\n  dioph_fn_comp2 df dg (abs_poly_dioph (poly.proj (fin2.of_nat' 0) + poly.proj (fin2.of_nat' 1)))\n\ntheorem mul_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v * g v :=\n  dioph_fn_comp2 df dg (abs_poly_dioph (poly.proj (fin2.of_nat' 0) * poly.proj (fin2.of_nat' 1)))\n\ntheorem le_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2264 g v := sorry\n\ntheorem lt_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v < g v :=\n  le_dioph (add_dioph df (const_dioph 1)) dg\n\ntheorem ne_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2260 g v :=\n  ext (or_dioph (lt_dioph df dg) (lt_dioph dg df)) fun (v : \u03b1 \u2192 \u2115) => iff.symm ne_iff_lt_or_gt\n\ntheorem sub_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v - g v := sorry\n\ntheorem dvd_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph fun (v : \u03b1 \u2192 \u2115) => f v \u2223 g v := sorry\n\ntheorem mod_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v % g v := sorry\n\ntheorem modeq_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) {h : (\u03b1 \u2192 \u2115) \u2192 \u2115} (dh : dioph_fn h) : dioph fun (v : \u03b1 \u2192 \u2115) => nat.modeq (h v) (f v) (g v) :=\n  eq_dioph (mod_dioph df dh) (mod_dioph dg dh)\n\ntheorem div_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v / g v := sorry\n\ntheorem pell_dioph : dioph\n  fun (v : vector3 \u2115 (bit0 (bit0 1))) =>\n    \u2203 (h : 1 < v (fin2.of_nat' 0)),\n      pell.xn h (v (fin2.of_nat' 1)) = v (fin2.of_nat' (bit0 1)) \u2227\n        pell.yn h (v (fin2.of_nat' 1)) = v (fin2.of_nat' (bit1 1)) := sorry\n\ntheorem xn_dioph : dioph_pfun\n  fun (v : vector3 \u2115 (bit0 1)) =>\n    roption.mk (1 < v (fin2.of_nat' 0)) fun (h : 1 < v (fin2.of_nat' 0)) => pell.xn h (v (fin2.of_nat' 1)) := sorry\n\ntheorem pow_dioph {\u03b1 : Type} {f : (\u03b1 \u2192 \u2115) \u2192 \u2115} {g : (\u03b1 \u2192 \u2115) \u2192 \u2115} (df : dioph_fn f) (dg : dioph_fn g) : dioph_fn fun (v : \u03b1 \u2192 \u2115) => f v ^ g v := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/number_theory/dioph.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185498374789, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.35646411000751366}}
{"text": "import Lean\nopen Lean\n\ninstance : BEq (Option ModuleIdx) := inferInstanceAs (BEq (Option Nat))\n\n\n/- A function for extracting the name and the type out of a `ConstantInfo` term.\nIt is desgined to work only for theorems, but can be extended to work for definitions too (may cause issues). -/\ndef Lean.ConstantInfo.extractNameType : ConstantInfo \u2192 Option (Name \u00d7 Expr)\n  | .thmInfo \u27e8\u27e8nm, _, typ\u27e9, _, _\u27e9 => some \u27e8nm, typ\u27e9\n--  | .defnInfo \u27e8\u27e8nm, _, typ\u27e9, _, _, _, _\u27e9 => some \u27e8nm, typ\u27e9\n  | _ => none\n\n\n/- Gets all the theorem names and docstrings in the current file -/\ndef getFileThmInfo : MetaM (List $ Name \u00d7 Option String) := do\n\n  let env \u2190 getEnv -- getting the current environment\n  let mainModuleIdx := env.getModuleIdx? env.mainModule -- the index of the main module (i.e., the current file)\n\n  -- extracting the names and types of the theorems in the current file\n  let cnsts : List (Name \u00d7 Expr) := env.constants |>.toList |>.filterMap\n    (\u03bb \u27e8nm, ci\u27e9 => liftOption $ do\n      guard $ env.getModuleIdxFor? nm == mainModuleIdx -- ensuring that the constant is in the current file\n      ci.extractNameType) -- extracting the name and the type pf the theorem using the function defined above\n\n  -- fetching the docstrings for each of the theorems in the current file\n  (liftM : IO _ \u2192 MetaM _) $\n    cnsts.mapM (\u03bb \u27e8nm, typ\u27e9 => do\n      return \u27e8nm, \u2190 findDocString? env nm\u27e9 )\n\n\nsection Testing\n\n/-- Addition of two on the left is equal to addition of two on the right. -/\ntheorem add_two_comm : \u2200 n : Nat, n + 2 = 2 + n := sorry\n\n/-- Fermat's Last theorem, stated in the special case of `n = 3`. -/\ntheorem flt3 : a^3 + b^3 = c^3 \u2192 a * b * c = 0 := sorry\n\n\n#eval getFileThmInfo\n\nend Testing\n", "meta": {"author": "siddhartha-gadgil", "repo": "LeanAide", "sha": "7862af73ee2f0be08b20fd3e4148e20bf4a81054", "save_path": "github-repos/lean/siddhartha-gadgil-LeanAide", "path": "github-repos/lean/siddhartha-gadgil-LeanAide/LeanAide-7862af73ee2f0be08b20fd3e4148e20bf4a81054/LeanCodePrompts/ThmInfo.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.35646409156597814}}
{"text": "import category_theory.functor.category\nimport category_theory.products.basic\n\nopen category_theory\n\nnamespace category_theory\n\nnamespace nat_trans\n\n@[simps]\ndef unflip {C D E : Type*} [category C] [category D] [category E]\n  {F G : C \u2964 D \u2964 E} (\u03c6 : F.flip \u27f6 G.flip) : F \u27f6 G :=\n{ app := \u03bb c,\n  { app := \u03bb d, (\u03c6.app d).app c,\n    naturality' := \u03bb d\u2081 d\u2082 g, nat_trans.congr_app (\u03c6.naturality g) c, },\n  naturality' := \u03bb c\u2081 c\u2082 f, begin\n    ext d,\n    exact (\u03c6.app d).naturality f,\n  end}\n\nlemma unflip_id {C D E : Type*} [category C] [category D] [category E]\n  {F : C \u2964 D \u2964 E} : nat_trans.unflip (\ud835\udfd9 F.flip) = \ud835\udfd9 F := rfl\n\nlemma unflip_comp {C D E : Type*} [category C] [category D] [category E]\n  {F G H : C \u2964 D \u2964 E} (\u03c6\u2081 : F.flip \u27f6 G.flip) (\u03c6\u2082 : G.flip \u27f6 H.flip) :\n  nat_trans.unflip (\u03c6\u2081 \u226b \u03c6\u2082) = nat_trans.unflip \u03c6\u2081 \u226b nat_trans.unflip \u03c6\u2082 := rfl\n\nend nat_trans\n\nnamespace nat_iso\n\ndef unflip {C D E : Type*} [category C] [category D] [category E]\n  {F G : C \u2964 D \u2964 E} (e : F.flip \u2245 G.flip) : F \u2245 G :=\n{ hom := nat_trans.unflip e.hom,\n  inv := nat_trans.unflip e.inv,\n  hom_inv_id' := by rw [\u2190 nat_trans.unflip_comp, e.hom_inv_id, nat_trans.unflip_id],\n  inv_hom_id' := by rw [\u2190 nat_trans.unflip_comp, e.inv_hom_id, nat_trans.unflip_id], }\n\nend nat_iso\n\nnamespace functor\n\ndef flip_evaluation_comp_whiskering_right (C : Type*) {D E : Type*}\n  [category C] [category D] [category E] (H : D \u2964 E) :\n  (evaluation C D \u22d9 (whiskering_right (C \u2964 D) D E).obj H).flip \u2245\n    (whiskering_right C D E).obj H := iso.refl _\n\ndef whiskering_right_obj_comp (C : Type*) {D\u2081 D\u2082 D\u2083 : Type*}\n  [category C] [category D\u2081] [category D\u2082] [category D\u2083]\n  (F\u2081\u2082 : D\u2081 \u2964 D\u2082) (F\u2082\u2083 : D\u2082 \u2964 D\u2083) :\n  (whiskering_right C _ _).obj (F\u2081\u2082 \u22d9 F\u2082\u2083) \u2245\n    (whiskering_right C _ _).obj F\u2081\u2082 \u22d9\n    (whiskering_right C _ _).obj F\u2082\u2083 := iso.refl _\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/unflip.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.35636537432893123}}
{"text": "import .basic category_theory.limits.limits\n\n\nuniverses u u\u2081 u\u2082 v v\u2081 v\u2082\n\nnamespace category_theory\n  open function\n  variables {C : Sort u\u2081} [\ud835\udc9e : category.{v\u2081} C]\n  include \ud835\udc9e\n\n  lemma injective_hom_op (X Y : C) : injective (@has_hom.hom.op _ _ X Y) :=\n  \u03bb f f' hf, by rw [\u2190@has_hom.hom.unop_op _ _ _ _ f, \u2190@has_hom.hom.unop_op _ _ _ _ f', hf]\n\n  lemma injective_hom_unop (X Y : C\u1d52\u1d56) : injective (@has_hom.hom.unop _ _ X Y) :=\n  \u03bb f f' hf, by rw [\u2190@has_hom.hom.op_unop _ _ _ _ f, \u2190@has_hom.hom.op_unop _ _ _ _ f', hf]\n\nend category_theory\n\nopen category_theory\n\nnamespace category_theory.nat_trans\n\nvariables {C : Sort u\u2081} [\ud835\udc9e : category.{v\u2081} C] {D : Sort u\u2082} [\ud835\udc9f : category.{v\u2082} D]\ninclude \ud835\udc9e \ud835\udc9f\n\nprotected def unop' (F F' : C \u2964 D) (\u03b7 : F.op \u27f6 F'.op) : F' \u27f6 F :=\n{ app := \u03bb x, (\u03b7.app $ op x).unop,\n  naturality' :=\n    by { intros, apply injective_hom_op, have := \u03b7.naturality f.op, simp at this, simp [this] } }\n\nprotected def op' (F F' : C\u1d52\u1d56 \u2964 D\u1d52\u1d56) (\u03b7 : F.unop \u27f6 F'.unop) : F' \u27f6 F :=\n{ app := \u03bb x, (\u03b7.app $ unop x).op,\n  naturality' :=\n    by { intros, apply injective_hom_unop, have := \u03b7.naturality f.unop, simp at this, simp [this] } }\n\nend category_theory.nat_trans\n\nopen category_theory\nnamespace category_theory.nat_iso\n\nvariables {C : Sort u\u2081} [\ud835\udc9e : category.{v\u2081} C] {D : Sort u\u2082} [\ud835\udc9f : category.{v\u2082} D]\ninclude \ud835\udc9e \ud835\udc9f\n\nprotected def op_unop (F : C \u2964 D) : F.op.unop \u2245 F :=\nby { cases F, refl } -- maybe not the best definition\n\nprotected def unop_op (F : C\u1d52\u1d56 \u2964 D\u1d52\u1d56) : F.unop.op \u2245 F :=\nby { cases F, refl } -- maybe not the best definition\n\nprotected def op_functor_const (d : D) :\n  ((category_theory.functor.const C).obj d).op \u2245 (category_theory.functor.const C\u1d52\u1d56).obj (op d) :=\nby refl\n\nend category_theory.nat_iso\n\nopen category_theory\nnamespace category_theory.limits\n\nvariables {J : Type v} [small_category J]\nvariables {C : Sort u} [category.{v+1} C]\n\n-- protected def cocone.op {F : J \u2964 C} (s : cocone F) : cone F.op :=\n-- \u27e8op s.X, s.\u03b9.op\u27e9\n\n-- protected def cone.op {F : J \u2964 C} (s : cone F) : cocone F.op :=\n-- \u27e8op s.X, s.\u03c0.op\u27e9\n\n-- protected def cocone.unop {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (s : cocone F) : cone F.unop :=\n-- \u27e8unop s.X, s.\u03b9.unop\u27e9\n\n-- protected def cone.unop {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (s : cone F) : cocone F.unop :=\n-- \u27e8unop s.X, s.\u03c0.unop\u27e9\n\n-- protected def cocone.op' {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (s : cocone F.unop) : cone F :=\n-- \u27e8op s.X, s.\u03b9.op' F ((category_theory.functor.const J\u1d52\u1d56).obj $ op s.X)\u27e9\n\n-- protected def cone.op' {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (s : cone F.unop) : cocone F :=\n-- \u27e8op s.X, s.\u03c0.op' ((category_theory.functor.const J\u1d52\u1d56).obj $ op s.X) F\u27e9\n\n-- protected def cocone.unop' {F : J \u2964 C} (s : cocone F.op) : cone F :=\n-- \u27e8unop s.X, s.\u03b9.unop' F ((category_theory.functor.const J).obj $ unop s.X)\u27e9\n\n-- protected def cone.unop' {F : J \u2964 C} (s : cone F.op) : cocone F :=\n-- \u27e8unop s.X, s.\u03c0.unop' ((category_theory.functor.const J).obj $ unop s.X) F\u27e9\n\n-- def has_limit_op {F : J \u2964 C} (H : has_colimit F) : has_limit F.op :=\n-- { cone := H.cocone.op,\n--   is_limit :=\n--   { lift := \u03bb s, (H.is_colimit.desc s.unop').op,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_colimit_op {F : J \u2964 C} (H : has_limit F) : has_colimit F.op :=\n-- { cocone := H.cone.op,\n--   is_colimit :=\n--   { desc := \u03bb s, (H.is_limit.lift s.unop').op,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- -- def has_limit_op {F : J \u2964 C} (H : has_colimit F) : has_limit F.op :=\n-- -- { cone := H.cocone.op,\n-- --   is_limit :=\n-- --   { lift := \u03bb s, begin unfreezeI, cases F, exact (H.is_colimit.desc s.unop).op end,\n-- --     fac' := omitted,\n-- --     uniq' := omitted } }\n\n-- -- def has_colimit_op {F : J \u2964 C} (H : has_limit F) : has_colimit F.op :=\n-- -- { cocone := H.cone.op,\n-- --   is_colimit :=\n-- --   { desc := \u03bb s, begin unfreezeI, cases F, exact (H.is_limit.lift s.unop).op end,\n-- --     fac' := omitted,\n-- --     uniq' := omitted } }\n\n-- def has_limit_unop {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (H : has_colimit F) : has_limit F.unop :=\n-- { cone := H.cocone.unop,\n--   is_limit :=\n--   { lift := \u03bb s, (H.is_colimit.desc s.op').unop,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_colimit_unop {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (H : has_limit F) : has_colimit F.unop :=\n-- { cocone := H.cone.unop,\n--   is_colimit :=\n--   { desc := \u03bb s, (H.is_limit.lift s.op').unop,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_limit_op' {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (H : has_colimit F.unop) : has_limit F :=\n-- { cone := H.cocone.op',\n--   is_limit :=\n--   { lift := \u03bb s, (H.is_colimit.desc s.unop).op,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_colimit_op' {F : J\u1d52\u1d56 \u2964 C\u1d52\u1d56} (H : has_limit F.unop) : has_colimit F :=\n-- { cocone := H.cone.op',\n--   is_colimit :=\n--   { desc := \u03bb s, (H.is_limit.lift s.unop).op,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_limit_unop' {F : J \u2964 C} (H : has_colimit F.op) : has_limit F :=\n-- { cone := H.cocone.unop',\n--   is_limit :=\n--   { lift := \u03bb s, (H.is_colimit.desc s.op).unop,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_colimit_unop' {F : J \u2964 C} (H : has_limit F.op) : has_colimit F :=\n-- { cocone := H.cone.unop',\n--   is_colimit :=\n--   { desc := \u03bb s, (H.is_limit.lift s.op).unop,\n--     fac' := omitted,\n--     uniq' := omitted } }\n\n-- def has_limits_of_shape_op (H : has_colimits_of_shape J C) : has_limits_of_shape J\u1d52\u1d56 C\u1d52\u1d56 :=\n-- \u03bb F, has_limit_op' (H _)\n\nend category_theory.limits", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/category_theory/limits2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3563418469189839}}
{"text": "/-\nCopyright (c) 2021 Luke Kershaw. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Luke Kershaw\n-/\nimport category_theory.additive.basic\nimport category_theory.shift\nimport category_theory.preadditive.additive_functor\nimport category_theory.triangulated.basic\nimport category_theory.triangulated.rotate\n\n/-!\n# Pretriangulated Categories\n\nThis file contains the definition of pretriangulated categories and triangulated functors\nbetween them.\n\n## Implementation Notes\n\nWe work under the assumption that pretriangulated categories are preadditive categories,\nbut not necessarily additive categories, as is assumed in some sources.\n\nTODO: generalise this to n-angulated categories as in https://arxiv.org/abs/1006.4592\n-/\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v v\u2080 v\u2081 v\u2082 u u\u2080 u\u2081 u\u2082\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\n/-\nWe work in an preadditive category `C` equipped with an additive shift.\n-/\nvariables (C : Type u) [category.{v} C] [has_zero_object C] [has_shift C] [preadditive C]\n  [functor.additive (shift C).functor]\n\n/--\nA preadditive category `C` with an additive shift, and a class of \"distinguished triangles\"\nrelative to that shift is called pretriangulated if the following hold:\n* Any triangle that is isomorphic to a distinguished triangle is also distinguished.\n* Any triangle of the form `(X,X,0,id,0,0)` is distinguished.\n* For any morphism `f : X \u27f6 Y` there exists a distinguished triangle of the form `(X,Y,Z,f,g,h)`.\n* The triangle `(X,Y,Z,f,g,h)` is distinguished if and only if `(Y,Z,X\u27e61\u27e7,g,h,-f\u27e61\u27e7)` is.\n* Given a diagram:\n  ```\n        f       g       h\n    X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n    \u2502       \u2502                \u2502\n    \u2502a      \u2502b               \u2502a\u27e61\u27e7'\n    V       V                V\n    X' \u2500\u2500\u2500> Y' \u2500\u2500\u2500> Z' \u2500\u2500\u2500> X'\u27e61\u27e7\n        f'      g'      h'\n  ```\n  where the left square commutes, and whose rows are distinguished triangles,\n  there exists a morphism `c : Z \u27f6 Z'` such that `(a,b,c)` is a triangle morphism.\nSee https://stacks.math.columbia.edu/tag/0145\n-/\nclass pretriangulated :=\n(distinguished_triangles [] : set (triangle C))\n(isomorphic_distinguished : \u03a0 (T\u2081 \u2208 distinguished_triangles) (T\u2082 : triangle C) (T\u2081 \u2245 T\u2082),\n  T\u2082 \u2208 distinguished_triangles)\n(contractible_distinguished : \u03a0 (X : C), (contractible_triangle C X) \u2208 distinguished_triangles)\n(distinguished_cocone_triangle : \u03a0 (X Y : C) (f: X \u27f6 Y), (\u2203 (Z : C) (g : Y \u27f6 Z) (h : Z \u27f6 X\u27e61\u27e7),\n  triangle.mk _ f g h \u2208 distinguished_triangles))\n(rotate_distinguished_triangle : \u03a0 (T : triangle C),\n  T \u2208 distinguished_triangles \u2194 T.rotate \u2208 distinguished_triangles)\n(complete_distinguished_triangle_morphism : \u03a0 (T\u2081 T\u2082 : triangle C)\n  (h\u2081 : T\u2081 \u2208 distinguished_triangles) (h\u2082 : T\u2082 \u2208 distinguished_triangles) (a : T\u2081.obj\u2081 \u27f6 T\u2082.obj\u2081)\n  (b : T\u2081.obj\u2082 \u27f6 T\u2082.obj\u2082) (comm\u2081 : T\u2081.mor\u2081 \u226b b = a \u226b T\u2082.mor\u2081),\n  (\u2203 (c : T\u2081.obj\u2083 \u27f6 T\u2082.obj\u2083), (T\u2081.mor\u2082 \u226b c = b \u226b T\u2082.mor\u2082) \u2227 (T\u2081.mor\u2083 \u226b a\u27e61\u27e7' = c \u226b T\u2082.mor\u2083) ))\n\nnamespace pretriangulated\nvariables [pretriangulated C]\n\nnotation `dist_triang`:20 C := distinguished_triangles C\n/--\nGiven any distinguished triangle `T`, then we know `T.rotate` is also distinguished.\n-/\nlemma rot_of_dist_triangle (T \u2208 dist_triang C) : (T.rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle T).mp H\n\n/--\nGiven any distinguished triangle `T`, then we know `T.inv_rotate` is also distinguished.\n-/\nlemma inv_rot_of_dist_triangle (T \u2208 dist_triang C) : (T.inv_rotate \u2208 dist_triang C) :=\n(rotate_distinguished_triangle (T.inv_rotate)).mpr\n  (isomorphic_distinguished T H (T.inv_rotate.rotate) T (inv_rot_comp_rot.symm.app T))\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `f \u226b g = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2081\u2082 (T \u2208 dist_triang C) : T.mor\u2081 \u226b T.mor\u2082 = 0 :=\nbegin\n  have h := contractible_distinguished T.obj\u2081,\n  have f := complete_distinguished_triangle_morphism,\n  specialize f (contractible_triangle C T.obj\u2081) T h H (\ud835\udfd9 T.obj\u2081) T.mor\u2081,\n  have t : (contractible_triangle C T.obj\u2081).mor\u2081 \u226b T.mor\u2081 = \ud835\udfd9 T.obj\u2081 \u226b T.mor\u2081,\n    by refl,\n  specialize f t,\n  cases f with c f,\n  rw \u2190 f.left,\n  simp only [limits.zero_comp, contractible_triangle_mor\u2082],\nend -- TODO : tidy this proof up\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `g \u226b h = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2082\u2083  (T \u2208 dist_triang C) : T.mor\u2082 \u226b T.mor\u2083 = 0 :=\ncomp_dist_triangle_mor_zero\u2081\u2082 C T.rotate (rot_of_dist_triangle C T H)\n\n/--\nGiven any distinguished triangle\n```\n      f       g       h\n  X  \u2500\u2500\u2500> Y  \u2500\u2500\u2500> Z  \u2500\u2500\u2500> X\u27e61\u27e7\n```\nthe composition `h \u226b f\u27e61\u27e7 = 0`.\nSee https://stacks.math.columbia.edu/tag/0146\n-/\nlemma comp_dist_triangle_mor_zero\u2083\u2081 (T \u2208 dist_triang C) :\n  T.mor\u2083 \u226b ((shift C).functor.map T.mor\u2081) = 0 :=\nhave H\u2082 : _ := rot_of_dist_triangle C T.rotate (rot_of_dist_triangle C T H),\nby simpa using comp_dist_triangle_mor_zero\u2081\u2082 C (T.rotate.rotate) H\u2082\n\n/-\nTODO: If `C` is pretriangulated with respect to a shift,\nthen `C\u1d52\u1d56` is pretriangulated with respect to the inverse shift.\n-/\nend pretriangulated\nend category_theory.triangulated\n\nnamespace category_theory.triangulated\nnamespace pretriangulated\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] [has_zero_object C] [has_shift C] [preadditive C]\n[functor.additive (shift C).functor] [functor.additive (shift C).inverse]\nvariables (D : Type u\u2082) [category.{v\u2082} D] [has_zero_object D] [has_shift D] [preadditive D]\n[functor.additive (shift D).functor] [functor.additive (shift D).inverse]\n\n/--\nThe underlying structure of a triangulated functor between pretriangulated categories `C` and `D`\nis a functor `F : C \u2964 D` together with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7`.\n-/\nstructure triangulated_functor_struct extends (C \u2964 D) :=\n(comm_shift : (shift C).functor \u22d9 to_functor \u2245 to_functor \u22d9 (shift D).functor)\n\ninstance : inhabited (triangulated_functor_struct C C) :=\n\u27e8{ obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl }\u27e9\n\nvariables {C D}\n/--\nGiven a `triangulated_functor_struct` we can define a function from triangles of `C` to\ntriangles of `D`.\n-/\n@[simp]\ndef triangulated_functor_struct.map_triangle (F : triangulated_functor_struct C D)\n  (T : triangle C) : triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\nvariables (C D)\n/--\nA triangulated functor between pretriangulated categories `C` and `D` is a functor `F : C \u2964 D`\ntogether with given functorial isomorphisms `\u03be X : F(X\u27e61\u27e7) \u27f6 F(X)\u27e61\u27e7` such that for every\ndistinguished triangle `(X,Y,Z,f,g,h)` of `C`, the triangle\n`(F(X), F(Y), F(Z), F(f), F(g), F(h) \u226b (\u03be X))` is a distinguished triangle of `D`.\nSee https://stacks.math.columbia.edu/tag/014V\n-/\nstructure triangulated_functor [pretriangulated C] [pretriangulated D] extends\n  triangulated_functor_struct C D :=\n(map_distinguished' : \u03a0 (T: triangle C), (T \u2208 dist_triang C) \u2192\n  (to_triangulated_functor_struct.map_triangle T \u2208 dist_triang D) )\n\ninstance [pretriangulated C] : inhabited (triangulated_functor C C) :=\n\u27e8{obj := \u03bb X, X,\n  map := \u03bb _ _ f, f,\n  comm_shift := by refl ,\n  map_distinguished' := begin\n    rintros \u27e8_,_,_,_\u27e9 Tdt,\n    dsimp at *,\n    rwa category.comp_id,\n  end }\u27e9\n\nvariables {C D} [pretriangulated C] [pretriangulated D]\n/--\nGiven a `triangulated_functor` we can define a function from triangles of `C` to triangles of `D`.\n-/\n@[simp]\ndef triangulated_functor.map_triangle (F : triangulated_functor C D) (T : triangle C) :\n  triangle D :=\ntriangle.mk _ (F.map T.mor\u2081) (F.map T.mor\u2082) (F.map T.mor\u2083 \u226b F.comm_shift.hom.app T.obj\u2081)\n\n/--\nGiven a `triangulated_functor` and a distinguished triangle `T` of `C`, then the triangle it\nmaps onto in `D` is also distinguished.\n-/\nlemma triangulated_functor.map_distinguished (F : triangulated_functor C D) (T : triangle C)\n  (h : T \u2208 dist_triang C) : (F.map_triangle T) \u2208 dist_triang D := F.map_distinguished' T h\n\n\nend pretriangulated\nend category_theory.triangulated\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/triangulated/pretriangulated.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548511303336, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.35612054759321193}}
{"text": "import hmem.stack\nimport hmem.encoding.basic\nimport hmem.split_cost\nimport complexity.basic\n\nvariables {\u03bc: Type*} [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)]\n\nnamespace hmem\nnamespace encoding\n\n\ndef encode_list {\u03b1: Type*} [\u03b1_en: complexity.has_encoding (runtime_model \u03bc) \u03b1]: list \u03b1 \u2192 memory \u03bc\n| [] := memory.null _\n| (x::xs) := (((memory.null _).setv 1).setm 0 (encode x)).setm 1 (encode_list xs)\n\ninstance (\u03b1: Type*) [\u03b1_en: complexity.has_encoding (runtime_model \u03bc) \u03b1]: complexity.has_encoding (runtime_model \u03bc) (list \u03b1) :=\n\u27e8 \u27e8 encode_list,\nbegin\n  intros x y,\n  split,\n  {  induction x generalizing y;\n    cases y,\n    { simp only [has_equiv.equiv, eq_self_iff_true, imp_true_iff] },\n    { simp only [has_equiv.equiv, (list.cons_ne_nil _ _).symm, iff_false, encode_list],\n      intro h,\n      apply @zero_ne_one \u03bc,\n      apply memory.getv_congr h,\n      refl,\n      rw [memory.getv_setm, memory.getv_setm, memory.getv_setv] },\n    { simp only [has_equiv.equiv, (list.cons_ne_nil _ _), iff_false, encode_list],\n      intro h,\n      apply (@zero_ne_one \u03bc _ _ _).symm,\n      apply memory.getv_congr h,\n      rw [memory.getv_setm, memory.getv_setm, memory.getv_setv],\n      refl },\n    simp only [has_equiv.equiv, encode_list, encode],\n    intro h,\n    split,\n    { \n      rw \u2190 complexity.encoding.encode_inj \u03b1_en.value,\n      apply memory.getm_congr 0 h;\n      { rw [memory.getm_setm_ne, memory.getm_setm],\n        refl,\n        apply zero_ne_one } },\n    apply x_ih,\n    apply memory.getm_congr 1 h;\n    { rw [memory.getm_setm] } },\n  intro h,\n  rw [h],\n  unfold has_equiv.equiv,\nend \u27e9 \u27e9 \n\ntheorem encode_nil {\u03b1: Type*} [complexity.has_encoding (runtime_model \u03bc) \u03b1]: encode (@list.nil \u03b1) = memory.null \u03bc := rfl\n\ntheorem encode_cons {\u03b1: Type*} [complexity.has_encoding (runtime_model \u03bc) \u03b1] (x: \u03b1) (xs: list \u03b1): encode (x::xs) = (((memory.null \u03bc).setv 1).setm 0 (encode x)).setm 1 (encode xs) := rfl\n\ndef split (\u03bc: Type*) [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)]: program \u03bc :=\n[ -- xs\n  instruction.ite (\u03bb a, a = 0) [\n    -- list.nil\n    instruction.const source.nil 1\n    -- 1 list.nil list.nil\n  ],\n  -- 1 x xs\n  instruction.swap (source.imm 0 source.nil) (source.imm 1 source.nil),\n  -- 1 xs x\n  instruction.recurse (source.imm 0 source.nil),\n  -- 1 [ 1 fst snd ] x\n  instruction.swap (source.imm 0 (source.imm 0 source.nil)) (source.imm 1 source.nil)\n  -- 1 (x::snd) fst\n]\n\ntheorem split_result {\u03b1: Type*} [complexity.has_encoding (runtime_model \u03bc) \u03b1] (l : list \u03b1):\n  (split \u03bc).has_result (encode l) (encode (list.split l)) :=\nbegin\n  induction l,\n  { apply thunk.apply_step_over',\n    { unfold split },\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, memory.getv_null, eq_self_iff_true, if_true],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.const },\n    simp only [memory.setvs, source.get, memory.setvp_nil],\n    simp only [list.split, encode_pair, encode_nil, memory.null_setv_setm_null],\n    exact \u27e81, rfl\u27e9, },\n  { apply thunk.apply_step_over',\n    { unfold split },\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_cons, memory.getv_setm, memory.getv_setv, one_ne_zero, if_false],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.swap },\n    simp only [memory.mop, memory.setms, memory.getms, source.get,\n      memory.setmp_nil, memory.setmp_cons, memory.setm_setm,\n      memory.getmp_nil, memory.getmp_cons,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n    apply thunk.apply_step_over_recurse,\n    { simp only [memory.getms, source.get, memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n      exact l_ih },\n    simp only [memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.swap },\n    cases hs:l_tl.split,\n    simp only [memory.mop, memory.setms, memory.getms, source.get,\n      list.split, hs, encode_pair, encode_cons,\n      memory.setmp_nil, memory.setmp_cons, memory.setm_setm,\n      memory.getmp_nil, memory.getmp_cons,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n    exact \u27e81, rfl\u27e9 }\nend\n\ndef merge {\u03bc: Type*} [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)] (cmp: program \u03bc): program \u03bc :=\n[ -- 1 as bs\n  instruction.uop id source.nil (source.imm 0 source.nil),\n  -- (as_nil) as bs\n  instruction.ite (\u03bb a, a = 0) [\n    -- 0 [] bs\n    instruction.move source.nil (source.imm 1 source.nil)\n    -- bs\n  ],\n  instruction.uop id source.nil (source.imm 1 source.nil),\n  -- (bs_nil) [1 a as] bs\n  instruction.ite (\u03bb a, a = 0) [\n    -- 0 [1 a as] []\n    instruction.move source.nil (source.imm 0 source.nil)\n    -- 1 a as\n  ],\n  -- 0 [1 a as] [1 b bs]\n  instruction.swap (source.imm 0 (source.imm 1 source.nil)) (source.imm 1 (source.imm 0 source.nil)),\n  -- 0 [1 a b] [1 as bs]\n  instruction.copy (source.imm 1 source.nil) source.nil,\n  -- 0 [1 a b] [0 [1 a b] [1 as bs]]\n  instruction.call cmp (source.imm 0 source.nil),\n  -- 0 [(a \u227c b) ? ?] [0 [1 a b] [1 as bs]]\n  instruction.uop id source.nil (source.imm 0 source.nil),\n  -- (a \u227c b) [(a \u227c b) ? ?] [0 [1 a b] [1 as bs]]\n  instruction.ite (\u03bb a, a = 1) [\n    -- a \u227c b\n    -- 1 [1 null null] [0 [1 a b] [1 as bs]]\n    instruction.move (source.imm 0 (source.imm 0 source.nil)) (source.imm 1 (source.imm 1 (source.imm 0 source.nil))),\n    -- 1 [1 as null] [0 [1 a b] [1 null bs]]\n    instruction.const (source.imm 0 (source.imm 1 source.nil)) 1,\n    -- 1 [1 as [1 null null]]] [0 [1 a b] [1 null bs]]\n    instruction.move (source.imm 0 (source.imm 1 (source.imm 0 source.nil))) (source.imm 1 (source.imm 0 (source.imm 1 source.nil))),\n    -- 1 [1 as [1 b null]]] [0 [1 a null] [1 null bs]]\n    instruction.move (source.imm 0 (source.imm 1 (source.imm 1 source.nil))) (source.imm 1 (source.imm 1 (source.imm 1 source.nil))),\n    -- 1 [1 as [1 b bs]]] [0 [1 a null] [1 null null]]\n    instruction.recurse (source.imm 0 source.nil),\n    -- 1 xs [0 [1 a null] [1 null null]]\n    instruction.move (source.imm 1 (source.imm 0 (source.imm 1 source.nil))) (source.imm 0 source.nil),\n    -- 1 null [0 [1 a xs] [1 null null]]\n    instruction.move source.nil (source.imm 1 (source.imm 0 source.nil))\n    -- 1 a xs\n  ],\n  -- \u00ac a \u227c b\n  -- 0 [0 null null] [0 [1 a b] [1 as bs]]\n  instruction.const (source.imm 0 source.nil) 1,\n  -- 0 [1 null null] [0 [1 a b] [1 as bs]]\n  instruction.move (source.imm 0 (source.imm 1 source.nil)) (source.imm 1 (source.imm 1 (source.imm 1 source.nil))),\n  -- 0 [1 null bs] [0 [1 a b] [1 as null]]\n  instruction.const (source.imm 0 (source.imm 0 source.nil)) 1,\n  -- 0 [1 [1 null null] bs] [0 [1 a b] [1 as null]]\n  instruction.move (source.imm 0 (source.imm 0 (source.imm 0 source.nil))) (source.imm 1 (source.imm 0 (source.imm 0 source.nil))),\n  -- 0 [1 [1 a null] bs] [0 [1 null b] [1 as null]]\n  instruction.move (source.imm 0 (source.imm 0 (source.imm 1 source.nil))) (source.imm 1 (source.imm 1 (source.imm 0 source.nil))),\n  -- 0 [1 [1 a as] b]] [0 [1 null b] [1 null null]]\n  instruction.recurse (source.imm 0 source.nil),\n  -- 0 xs [0 [1 null b] [1 null null]]\n  instruction.move (source.imm 1 (source.imm 0 (source.imm 0 source.nil))) (source.imm 1 (source.imm 0 (source.imm 1 source.nil))),\n  -- 0 xs [0 [1 b null] [1 null null]]\n    instruction.move (source.imm 1 (source.imm 0 (source.imm 1 source.nil))) (source.imm 0 source.nil),\n  -- 0 null [0 [1 b xs] [1 null null]]\n  instruction.move source.nil (source.imm 1 (source.imm 0 source.nil))\n  -- 1 b xs\n]\n\ntheorem merge_result {\u03b1: Type*} [complexity.has_encoding (runtime_model \u03bc) \u03b1]\n  (fcmp: \u03b1 \u2192 \u03b1 \u2192 Prop) [dcmp: decidable_rel fcmp]\n  (pcmp: program \u03bc) (hcmp: \u2200 (a b: \u03b1), pcmp.has_result (encode (a, b)) (encode (dcmp a b))):\n  \u2200 (as bs: list \u03b1), (merge pcmp).has_result (encode (as, bs)) (encode (list.merge fcmp as bs)) :=\nbegin\n  intros as bs,\n  simp only [encode_pair],\n  induction h:(as.length + bs.length) using nat.strong_induction_on with n ih generalizing as bs,\n  cases as,\n  { apply thunk.apply_step_over',\n    { unfold merge },\n    { unfold thunk.step_over thunk.step instruction.uop },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.getv_setm, memory.getv_setv, eq_self_iff_true, if_true],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.move },\n    cases bs;\n    simp only [memory.setms, memory.getms, source.get,\n      memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, list.merge];\n    exact \u27e81, rfl\u27e9 },\n  cases bs,\n  { apply thunk.apply_step_over',\n    { unfold merge },\n    { unfold thunk.step_over thunk.step instruction.uop },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.getv_setm, memory.getv_setv, encode_cons, one_ne_zero' \u03bc, if_false],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.getv_setm, memory.getv_setv, eq_self_iff_true, if_true],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.move },\n    simp only [memory.setms, memory.getms, source.get,\n      memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      list.merge, \u2190 encode_cons],\n    exact \u27e81, rfl\u27e9 },\n  { apply thunk.apply_step_over',\n    { unfold merge },\n    { unfold thunk.step_over thunk.step instruction.uop },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.getv_setm, memory.getv_setv, encode_cons, one_ne_zero' \u03bc, if_false],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.getv_setm, memory.getv_setv, encode_cons, one_ne_zero' \u03bc, if_false],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.swap },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.copy },\n    simp only [memory.setms, memory.getms, source.get,\n      memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc),\n      memory.mop],\n    apply thunk.apply_step_over_call,\n    { simp only [memory.getms, source.get,\n        memory.getmp_cons, memory.getmp_nil,\n        memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n      rw [\u2190 encode_pair],\n      apply hcmp },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n      memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n      memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n    cases hcmp:dcmp as_hd bs_hd,\n    { rw[encode_is_false (decidable.is_false h_1)],\n      simp [memory.getv_setm, memory.getv_setv, memory.getv_null],\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step instruction.const },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step instruction.move },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n        memory.setvs, memory.setms, memory.getvs, memory.getms, source.get,\n        memory.setvp_cons, memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n        memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil, memory.mop,\n        \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n        memory.getm_setv, memory.getm_null, memory.null_setv_setm_null,\n        memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n        memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n      apply thunk.apply_step_over_recurse,\n      { simp only [memory.getms, source.get, memory.getmp_cons, memory.getmp_nil,\n           memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n        rw [\u2190 encode_cons, \u2190 encode_pair],\n        apply ih _ _ _ _ rfl,\n        simpa only [\u2190 h, list.length_cons, \u2190 nat.add_assoc] using nat.lt_succ_self _ },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      simp only [\n        list.merge, h_1, if_false,\n        memory.setms, memory.getms, source.get,\n        memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil, memory.mop,\n        memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n        memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n      exact \u27e81, rfl\u27e9 },\n    { rw[encode_is_true (decidable.is_true h_1)],\n      simp [memory.getv_setm, memory.getv_setv, memory.getv_null],\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step instruction.move },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step instruction.const },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      simp only [encode_nil, id, vector.map_cons, vector.map_nil, vector.nth_cons_zero, fin.mk_zero,\n        memory.setvs, memory.setms, memory.getvs, memory.getms, source.get,\n        memory.setvp_cons, memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n        memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil, memory.mop,\n        \u2190 memory.setv_setm, memory.setv_setv, memory.getv_null,\n        memory.getm_setv, memory.getm_null, memory.null_setv_setm_null,\n        memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n        memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n      apply thunk.apply_step_over_recurse,\n      { simp only [memory.getms, source.get, memory.getmp_cons, memory.getmp_nil,\n           memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n           memory.getm_setm_ne _ 0 _ 1 (one_ne_zero' \u03bc),\n           memory.getm_setv, memory.getm_null],\n        rw [\u2190 encode_cons, \u2190 encode_pair],\n        apply ih _ _ _ _ rfl,\n        simpa only [\u2190 h, list.length_cons, add_lt_add_iff_right] using nat.lt_succ_self _ },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      apply thunk.apply_step_over,\n      { unfold thunk.step_over thunk.step },\n      simp only [\n        list.merge, h_1, if_true,\n        memory.setms, memory.getms, source.get,\n        memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil, memory.mop,\n        memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n        memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc)],\n      exact \u27e81, rfl\u27e9 } }\nend\n\ndef merge_sort {\u03bc: Type*} [decidable_eq \u03bc] [has_zero \u03bc] [has_one \u03bc] [ne_zero (1:\u03bc)] (cmp: program \u03bc): program \u03bc :=\n[ -- l\n  instruction.ite (\u03bb a, a = 0) [\n    -- list.nil\n  ],\n  -- 1 a l\n  instruction.uop id source.nil (source.imm 1 source.nil),\n  -- (l.nil) a l\n  instruction.ite (\u03bb a, a = 0) [\n    -- 0 a list.nil\n    instruction.const source.nil 1\n    -- 1 a list.nil\n  ],\n  -- 1 a [1 b l]\n  instruction.move (source.imm 0 source.nil) source.nil,\n  -- 0 [1 a [1 b l]] null\n  instruction.call (split \u03bc) (source.imm 0 source.nil),\n  -- 0 [1 l\u2081 l\u2082]\n  instruction.move source.nil (source.imm 0 source.nil),\n  -- 1 l\u2081 l\u2082\n  instruction.recurse (source.imm 0 source.nil),\n  -- 1 ms\u2081 l\u2082\n  instruction.swap (source.imm 0 source.nil) (source.imm 1 source.nil),\n  -- 1 l\u2082 ms\u2081\n  instruction.recurse (source.imm 0 source.nil),\n  -- 1 ms\u2082 ms\u2081\n  instruction.swap (source.imm 0 source.nil) (source.imm 1 source.nil),\n  -- 1 ms\u2081 ms\u2082\n  instruction.move (source.imm 0 source.nil) source.nil,\n  -- 0 [1 ms\u2081 ms\u2082] null\n  instruction.call (merge cmp) (source.imm 0 source.nil),\n  -- 0 m null\n  instruction.move source.nil (source.imm 0 source.nil)\n  -- m\n]\n\ntheorem merge_sort_result\n  {\u03b1: Type*} [complexity.has_encoding (runtime_model \u03bc) \u03b1]\n  (fcmp: \u03b1 \u2192 \u03b1 \u2192 Prop) [dcmp: decidable_rel fcmp]\n  (pcmp: program \u03bc) (hcmp: \u2200 (a b: \u03b1), pcmp.has_result (encode (a, b)) (encode (dcmp a b))):\n  \u2200 (l: list \u03b1), (merge_sort pcmp).has_result (encode l) (encode (list.merge_sort fcmp l)) :=\nbegin\n  intros l,\n  induction hn:l.length using nat.strong_induction_on with n ih generalizing l,\n  cases l with a l,\n  { apply thunk.apply_step_over',\n    unfold merge_sort,\n    { unfold thunk.step_over thunk.step },\n    simp only [encode_nil, memory.getv_null, eq_self_iff_true, if_true, list.merge_sort],\n    exact \u27e81, rfl\u27e9 },\n  apply thunk.apply_step_over',\n  unfold merge_sort,\n  { unfold thunk.step_over thunk.step },\n  simp only [encode_cons, memory.getv_setm, memory.getv_setv, (one_ne_zero' \u03bc), if_false],\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step instruction.uop },\n  cases l with b l,\n  { apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step },\n    simp only [id, encode_nil, memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      memory.getv_setm, memory.getm_setm, memory.getv_null,\n      memory.getv_setv, \u2190 memory.setv_setm, memory.setv_setv,\n      fin.mk_zero, vector.map_cons, vector.nth_cons_zero,\n      eq_self_iff_true, if_true],\n    apply thunk.apply_step_over,\n    { unfold thunk.step_over thunk.step instruction.const },\n    simp only [id, encode_nil, memory.setvs, memory.getvs, source.get,\n      memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n      memory.getv_setm, memory.getm_setm, memory.getv_null,\n      memory.getv_setv, \u2190 memory.setv_setm, memory.setv_setv,\n      eq_self_iff_true, if_true],\n    unfold list.merge_sort,\n    rw [encode_cons],\n    exact \u27e81, rfl\u27e9 },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step },\n  simp only [id, encode_cons, memory.setvs, memory.getvs, source.get,\n    memory.setvp_nil, memory.getvp_cons, memory.getvp_nil,\n    memory.getv_setm, memory.getm_setm, memory.getv_null,\n    memory.getv_setv, \u2190 memory.setv_setm, memory.setv_setv,\n    fin.mk_zero, vector.map_cons, vector.nth_cons_zero,\n    (one_ne_zero' \u03bc), if_false],\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step instruction.move },\n  simp only [memory.setms, memory.getms, source.get,\n    memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n    memory.mop],\n  apply thunk.apply_step_over_call,\n  { simp only [memory.getms, source.get,\n      memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n    rw [\u2190 encode_cons, \u2190 encode_cons],\n    apply split_result },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step },\n  simp only [memory.getms, memory.setms, source.get,\n    memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n    memory.setm_setm, memory.getm_setm ],\n  cases e : list.split (a::b::l) with l\u2081 l\u2082,\n  simp only [e, encode_pair],\n  apply thunk.apply_step_over_recurse,\n  { simp only [memory.getms, source.get,\n      memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n    apply ih _ _ _ rfl,\n    exact hn \u25b8 (list.length_split_lt e).left },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step instruction.swap },\n  simp only [memory.getms, memory.setms, source.get,\n    memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n    memory.setm_setm, memory.getm_setm,\n    memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n    memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc),\n    memory.mop ],\n  apply thunk.apply_step_over_recurse,\n  { simp only [memory.getms, source.get,\n      memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm, memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc)],\n    apply ih _ _ _ rfl,\n    exact hn \u25b8 (list.length_split_lt e).right },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step },\n  simp only [memory.getms, memory.setms, source.get,\n    memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n    memory.setm_setm, memory.getm_setm,\n    memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n    memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc),\n    memory.mop ],\n  rw [\u2190 encode_pair],\n  apply thunk.apply_step_over_call,\n  { simp only [memory.getms, source.get,\n      memory.getmp_cons, memory.getmp_nil,\n      memory.getm_setm],\n      apply merge_result fcmp,\n      apply hcmp },\n  apply thunk.apply_step_over,\n  { unfold thunk.step_over thunk.step },\n  simp only [memory.getms, memory.setms, source.get,\n    memory.setmp_cons, memory.setmp_nil, memory.getmp_cons, memory.getmp_nil,\n    memory.setm_setm, memory.getm_setm,\n    memory.getm_setm_ne _ 1 _ 0 (zero_ne_one' \u03bc),\n    memory.setm_setm, memory.setm_setm_ne _ 1 0 _ _ (one_ne_zero' \u03bc),\n    memory.mop],\n  rw [list.merge_sort_cons_cons],\n  exact \u27e81, rfl\u27e9,\n  exact e,\nend\n\n\nend encoding\nend hmem", "meta": {"author": "calcu16", "repo": "lean_complexity", "sha": "0dcb73bde8d1d4237f782f4790166365ac3209fe", "save_path": "github-repos/lean/calcu16-lean_complexity", "path": "github-repos/lean/calcu16-lean_complexity/lean_complexity-0dcb73bde8d1d4237f782f4790166365ac3209fe/src/hmem/encoding/list.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7122321720225278, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3561160860112639}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebra.category.Ring.colimits\n! leanprover-community/mathlib commit 70fd9563a21e7b963887c9360bd29b2393e6225a\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Category.Ring.Basic\nimport Mathbin.CategoryTheory.Limits.HasLimits\nimport Mathbin.CategoryTheory.ConcreteCategory.Elementwise\n\n/-!\n# The category of commutative rings has all colimits.\n\nThis file uses a \"pre-automated\" approach, just as for `Mon/colimits.lean`.\nIt is a very uniform approach, that conceivably could be synthesised directly\nby a tactic that analyses the shape of `comm_ring` and `ring_hom`.\n-/\n\n\nuniverse u v\n\nopen CategoryTheory\n\nopen CategoryTheory.Limits\n\n-- [ROBOT VOICE]:\n-- You should pretend for now that this file was automatically generated.\n-- It follows the same template as colimits in Mon.\n/-\n`#print comm_ring` says:\n\nstructure comm_ring : Type u \u2192 Type u\nfields:\ncomm_ring.zero : \u03a0 (\u03b1 : Type u) [c : comm_ring \u03b1], \u03b1\ncomm_ring.one : \u03a0 (\u03b1 : Type u) [c : comm_ring \u03b1], \u03b1\ncomm_ring.neg : \u03a0 {\u03b1 : Type u} [c : comm_ring \u03b1], \u03b1 \u2192 \u03b1\ncomm_ring.add : \u03a0 {\u03b1 : Type u} [c : comm_ring \u03b1], \u03b1 \u2192 \u03b1 \u2192 \u03b1\ncomm_ring.mul : \u03a0 {\u03b1 : Type u} [c : comm_ring \u03b1], \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\ncomm_ring.zero_add : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a : \u03b1), 0 + a = a\ncomm_ring.add_zero : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a : \u03b1), a + 0 = a\ncomm_ring.one_mul : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a : \u03b1), 1 * a = a\ncomm_ring.mul_one : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a : \u03b1), a * 1 = a\ncomm_ring.add_left_neg : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a : \u03b1), -a + a = 0\ncomm_ring.add_comm : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b : \u03b1), a + b = b + a\ncomm_ring.mul_comm : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b : \u03b1), a * b = b * a\ncomm_ring.add_assoc : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b c_1 : \u03b1), a + b + c_1 = a + (b + c_1)\ncomm_ring.mul_assoc : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b c_1 : \u03b1), a * b * c_1 = a * (b * c_1)\ncomm_ring.left_distrib : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b c_1 : \u03b1),\n                                                            a * (b + c_1) = a * b + a * c_1\ncomm_ring.right_distrib : \u2200 {\u03b1 : Type u} [c : comm_ring \u03b1] (a b c_1 : \u03b1),\n                                                            (a + b) * c_1 = a * c_1 + b * c_1\n-/\nnamespace CommRingCat.Colimits\n\n/-!\nWe build the colimit of a diagram in `CommRing` by constructing the\nfree commutative ring on the disjoint union of all the commutative rings in the diagram,\nthen taking the quotient by the commutative ring laws within each commutative ring,\nand the identifications given by the morphisms in the diagram.\n-/\n\n\nvariable {J : Type v} [SmallCategory J] (F : J \u2964 CommRingCat.{v})\n\n/-- An inductive type representing all commutative ring expressions (without relations)\non a collection of types indexed by the objects of `J`.\n-/\ninductive Prequotient-- There's always `of`\n\n  | of : \u2200 (j : J) (x : F.obj j), prequotient-- Then one generator for each operation\n\n  | zero : prequotient\n  | one : prequotient\n  | neg : prequotient \u2192 prequotient\n  | add : prequotient \u2192 prequotient \u2192 prequotient\n  | mul : prequotient \u2192 prequotient \u2192 prequotient\n#align CommRing.colimits.prequotient CommRingCat.Colimits.Prequotient\n\ninstance : Inhabited (Prequotient F) :=\n  \u27e8Prequotient.zero\u27e9\n\nopen Prequotient\n\n/-- The relation on `prequotient` saying when two expressions are equal\nbecause of the commutative ring laws, or\nbecause one element is mapped to another by a morphism in the diagram.\n-/\ninductive Relation : Prequotient F \u2192 Prequotient F \u2192 Prop-- Make it an equivalence relation:\n\n  | refl : \u2200 x, relation x x\n  | symm : \u2200 (x y) (h : relation x y), relation y x\n  |\n  trans :\n    \u2200 (x y z) (h : relation x y) (k : relation y z), relation x z-- There's always a `map` relation\n\n  |\n  map :\n    \u2200 (j j' : J) (f : j \u27f6 j') (x : F.obj j),\n      relation (of j' (F.map f x))\n        (of j x)-- Then one relation per operation, describing the interaction with `of`\n\n  | zero : \u2200 j, relation (of j 0) zero\n  | one : \u2200 j, relation (of j 1) one\n  | neg : \u2200 (j) (x : F.obj j), relation (of j (-x)) (neg (of j x))\n  | add : \u2200 (j) (x y : F.obj j), relation (of j (x + y)) (add (of j x) (of j y))\n  |\n  mul :\n    \u2200 (j) (x y : F.obj j),\n      relation (of j (x * y))\n        (mul (of j x) (of j y))-- Then one relation per argument of each operation\n\n  | neg_1 : \u2200 (x x') (r : relation x x'), relation (neg x) (neg x')\n  | add_1 : \u2200 (x x' y) (r : relation x x'), relation (add x y) (add x' y)\n  | add_2 : \u2200 (x y y') (r : relation y y'), relation (add x y) (add x y')\n  | mul_1 : \u2200 (x x' y) (r : relation x x'), relation (mul x y) (mul x' y)\n  |\n  mul_2 : \u2200 (x y y') (r : relation y y'), relation (mul x y) (mul x y')-- And one relation per axiom\n\n  | zero_add : \u2200 x, relation (add zero x) x\n  | add_zero : \u2200 x, relation (add x zero) x\n  | one_mul : \u2200 x, relation (mul one x) x\n  | mul_one : \u2200 x, relation (mul x one) x\n  | add_left_neg : \u2200 x, relation (add (neg x) x) zero\n  | add_comm : \u2200 x y, relation (add x y) (add y x)\n  | mul_comm : \u2200 x y, relation (mul x y) (mul y x)\n  | add_assoc : \u2200 x y z, relation (add (add x y) z) (add x (add y z))\n  | mul_assoc : \u2200 x y z, relation (mul (mul x y) z) (mul x (mul y z))\n  | left_distrib : \u2200 x y z, relation (mul x (add y z)) (add (mul x y) (mul x z))\n  | right_distrib : \u2200 x y z, relation (mul (add x y) z) (add (mul x z) (mul y z))\n#align CommRing.colimits.relation CommRingCat.Colimits.Relation\n\n/-- The setoid corresponding to commutative expressions modulo monoid relations and identifications.\n-/\ndef colimitSetoid : Setoid (Prequotient F)\n    where\n  R := Relation F\n  iseqv := \u27e8Relation.refl, Relation.symm, Relation.trans\u27e9\n#align CommRing.colimits.colimit_setoid CommRingCat.Colimits.colimitSetoid\n\nattribute [instance] colimit_setoid\n\n/-- The underlying type of the colimit of a diagram in `CommRing`.\n-/\ndef ColimitType : Type v :=\n  Quotient (colimitSetoid F)deriving Inhabited\n#align CommRing.colimits.colimit_type CommRingCat.Colimits.ColimitType\n\ninstance : AddGroup (ColimitType F)\n    where\n  zero := Quot.mk _ zero\n  neg := by\n    fapply @Quot.lift\n    \u00b7 intro x\n      exact Quot.mk _ (neg x)\n    \u00b7 intro x x' r\n      apply Quot.sound\n      exact relation.neg_1 _ _ r\n  add := by\n    fapply @Quot.lift _ _ (colimit_type F \u2192 colimit_type F)\n    \u00b7 intro x\n      fapply @Quot.lift\n      \u00b7 intro y\n        exact Quot.mk _ (add x y)\n      \u00b7 intro y y' r\n        apply Quot.sound\n        exact relation.add_2 _ _ _ r\n    \u00b7 intro x x' r\n      funext y\n      induction y\n      dsimp\n      apply Quot.sound\n      \u00b7 exact relation.add_1 _ _ _ r\n      \u00b7 rfl\n  zero_add x := by\n    induction x\n    dsimp\n    apply Quot.sound\n    apply relation.zero_add\n    rfl\n  add_zero x := by\n    induction x\n    dsimp\n    apply Quot.sound\n    apply relation.add_zero\n    rfl\n  add_left_neg x := by\n    induction x\n    dsimp\n    apply Quot.sound\n    apply relation.add_left_neg\n    rfl\n  add_assoc x y z := by\n    induction x\n    induction y\n    induction z\n    dsimp\n    apply Quot.sound\n    apply relation.add_assoc\n    rfl\n    rfl\n    rfl\n\ninstance : AddGroupWithOne (ColimitType F) :=\n  { ColimitType.addGroup F with one := Quot.mk _ one }\n\ninstance : CommRing (ColimitType F) :=\n  { ColimitType.addGroupWithOne F with\n    one := Quot.mk _ one\n    mul := by\n      fapply @Quot.lift _ _ (colimit_type F \u2192 colimit_type F)\n      \u00b7 intro x\n        fapply @Quot.lift\n        \u00b7 intro y\n          exact Quot.mk _ (mul x y)\n        \u00b7 intro y y' r\n          apply Quot.sound\n          exact relation.mul_2 _ _ _ r\n      \u00b7 intro x x' r\n        funext y\n        induction y\n        dsimp\n        apply Quot.sound\n        \u00b7 exact relation.mul_1 _ _ _ r\n        \u00b7 rfl\n    one_mul := fun x => by\n      induction x\n      dsimp\n      apply Quot.sound\n      apply relation.one_mul\n      rfl\n    mul_one := fun x => by\n      induction x\n      dsimp\n      apply Quot.sound\n      apply relation.mul_one\n      rfl\n    add_comm := fun x y => by\n      induction x\n      induction y\n      dsimp\n      apply Quot.sound\n      apply relation.add_comm\n      rfl\n      rfl\n    mul_comm := fun x y => by\n      induction x\n      induction y\n      dsimp\n      apply Quot.sound\n      apply relation.mul_comm\n      rfl\n      rfl\n    add_assoc := fun x y z => by\n      induction x\n      induction y\n      induction z\n      dsimp\n      apply Quot.sound\n      apply relation.add_assoc\n      rfl\n      rfl\n      rfl\n    mul_assoc := fun x y z => by\n      induction x\n      induction y\n      induction z\n      dsimp\n      apply Quot.sound\n      apply relation.mul_assoc\n      rfl\n      rfl\n      rfl\n    left_distrib := fun x y z => by\n      induction x\n      induction y\n      induction z\n      dsimp\n      apply Quot.sound\n      apply relation.left_distrib\n      rfl\n      rfl\n      rfl\n    right_distrib := fun x y z => by\n      induction x\n      induction y\n      induction z\n      dsimp\n      apply Quot.sound\n      apply relation.right_distrib\n      rfl\n      rfl\n      rfl }\n\n@[simp]\ntheorem quot_zero : Quot.mk Setoid.r zero = (0 : ColimitType F) :=\n  rfl\n#align CommRing.colimits.quot_zero CommRingCat.Colimits.quot_zero\n\n@[simp]\ntheorem quot_one : Quot.mk Setoid.r one = (1 : ColimitType F) :=\n  rfl\n#align CommRing.colimits.quot_one CommRingCat.Colimits.quot_one\n\n@[simp]\ntheorem quot_neg (x) : Quot.mk Setoid.r (neg x) = (-Quot.mk Setoid.r x : ColimitType F) :=\n  rfl\n#align CommRing.colimits.quot_neg CommRingCat.Colimits.quot_neg\n\n@[simp]\ntheorem quot_add (x y) :\n    Quot.mk Setoid.r (add x y) = (Quot.mk Setoid.r x + Quot.mk Setoid.r y : ColimitType F) :=\n  rfl\n#align CommRing.colimits.quot_add CommRingCat.Colimits.quot_add\n\n@[simp]\ntheorem quot_mul (x y) :\n    Quot.mk Setoid.r (mul x y) = (Quot.mk Setoid.r x * Quot.mk Setoid.r y : ColimitType F) :=\n  rfl\n#align CommRing.colimits.quot_mul CommRingCat.Colimits.quot_mul\n\n/-- The bundled commutative ring giving the colimit of a diagram. -/\ndef colimit : CommRingCat :=\n  CommRingCat.of (ColimitType F)\n#align CommRing.colimits.colimit CommRingCat.Colimits.colimit\n\n/-- The function from a given commutative ring in the diagram to the colimit commutative ring. -/\ndef coconeFun (j : J) (x : F.obj j) : ColimitType F :=\n  Quot.mk _ (of j x)\n#align CommRing.colimits.cocone_fun CommRingCat.Colimits.coconeFun\n\n/-- The ring homomorphism from a given commutative ring in the diagram to the colimit commutative\nring. -/\ndef coconeMorphism (j : J) : F.obj j \u27f6 colimit F\n    where\n  toFun := coconeFun F j\n  map_one' := by apply Quot.sound <;> apply relation.one\n  map_mul' := by intros <;> apply Quot.sound <;> apply relation.mul\n  map_zero' := by apply Quot.sound <;> apply relation.zero\n  map_add' := by intros <;> apply Quot.sound <;> apply relation.add\n#align CommRing.colimits.cocone_morphism CommRingCat.Colimits.coconeMorphism\n\n@[simp]\ntheorem cocone_naturality {j j' : J} (f : j \u27f6 j') :\n    F.map f \u226b coconeMorphism F j' = coconeMorphism F j :=\n  by\n  ext\n  apply Quot.sound\n  apply Relation.Map\n#align CommRing.colimits.cocone_naturality CommRingCat.Colimits.cocone_naturality\n\n@[simp]\ntheorem cocone_naturality_components (j j' : J) (f : j \u27f6 j') (x : F.obj j) :\n    (coconeMorphism F j') (F.map f x) = (coconeMorphism F j) x :=\n  by\n  rw [\u2190 cocone_naturality F f]\n  rfl\n#align CommRing.colimits.cocone_naturality_components CommRingCat.Colimits.cocone_naturality_components\n\n/-- The cocone over the proposed colimit commutative ring. -/\ndef colimitCocone : Cocone F where\n  pt := colimit F\n  \u03b9 := { app := coconeMorphism F }\n#align CommRing.colimits.colimit_cocone CommRingCat.Colimits.colimitCocone\n\n/-- The function from the free commutative ring on the diagram to the cone point of any other\ncocone. -/\n@[simp]\ndef descFunLift (s : Cocone F) : Prequotient F \u2192 s.pt\n  | of j x => (s.\u03b9.app j) x\n  | zero => 0\n  | one => 1\n  | neg x => -desc_fun_lift x\n  | add x y => desc_fun_lift x + desc_fun_lift y\n  | mul x y => desc_fun_lift x * desc_fun_lift y\n#align CommRing.colimits.desc_fun_lift CommRingCat.Colimits.descFunLift\n\n/-- The function from the colimit commutative ring to the cone point of any other cocone. -/\ndef descFun (s : Cocone F) : ColimitType F \u2192 s.pt :=\n  by\n  fapply Quot.lift\n  \u00b7 exact desc_fun_lift F s\n  \u00b7 intro x y r\n    induction r <;> try dsimp\n    -- refl\n    \u00b7 rfl\n    -- symm\n    \u00b7 exact r_ih.symm\n    -- trans\n    \u00b7 exact Eq.trans r_ih_h r_ih_k\n    -- map\n    \u00b7 simp\n    -- zero\n    \u00b7 simp\n    -- one\n    \u00b7 simp\n    -- neg\n    \u00b7 simp\n    -- add\n    \u00b7 simp\n    -- mul\n    \u00b7 simp\n    -- neg_1\n    \u00b7 rw [r_ih]\n    -- add_1\n    \u00b7 rw [r_ih]\n    -- add_2\n    \u00b7 rw [r_ih]\n    -- mul_1\n    \u00b7 rw [r_ih]\n    -- mul_2\n    \u00b7 rw [r_ih]\n    -- zero_add\n    \u00b7 rw [zero_add]\n    -- add_zero\n    \u00b7 rw [add_zero]\n    -- one_mul\n    \u00b7 rw [one_mul]\n    -- mul_one\n    \u00b7 rw [mul_one]\n    -- add_left_neg\n    \u00b7 rw [add_left_neg]\n    -- add_comm\n    \u00b7 rw [add_comm]\n    -- mul_comm\n    \u00b7 rw [mul_comm]\n    -- add_assoc\n    \u00b7 rw [add_assoc]\n    -- mul_assoc\n    \u00b7 rw [mul_assoc]\n    -- left_distrib\n    \u00b7 rw [left_distrib]\n    -- right_distrib\n    \u00b7 rw [right_distrib]\n#align CommRing.colimits.desc_fun CommRingCat.Colimits.descFun\n\n/-- The ring homomorphism from the colimit commutative ring to the cone point of any other\ncocone. -/\ndef descMorphism (s : Cocone F) : colimit F \u27f6 s.pt\n    where\n  toFun := descFun F s\n  map_one' := rfl\n  map_zero' := rfl\n  map_add' x y := by induction x <;> induction y <;> rfl\n  map_mul' x y := by induction x <;> induction y <;> rfl\n#align CommRing.colimits.desc_morphism CommRingCat.Colimits.descMorphism\n\n/-- Evidence that the proposed colimit is the colimit. -/\ndef colimitIsColimit : IsColimit (colimitCocone F)\n    where\n  desc s := descMorphism F s\n  uniq s m w := by\n    ext\n    induction x\n    induction x\n    \u00b7 have w' :=\n        congr_fun (congr_arg (fun f : F.obj x_j \u27f6 s.X => (f : F.obj x_j \u2192 s.X)) (w x_j)) x_x\n      erw [w']\n      rfl\n    \u00b7 simp\n    \u00b7 simp\n    \u00b7 simp [*]\n    \u00b7 simp [*]\n    \u00b7 simp [*]\n    rfl\n#align CommRing.colimits.colimit_is_colimit CommRingCat.Colimits.colimitIsColimit\n\ninstance hasColimits_commRingCat : HasColimits CommRingCat\n    where HasColimitsOfShape J \ud835\udca5 :=\n    {\n      HasColimit := fun F =>\n        has_colimit.mk\n          { Cocone := colimit_cocone F\n            IsColimit := colimit_is_colimit F } }\n#align CommRing.colimits.has_colimits_CommRing CommRingCat.Colimits.hasColimits_commRingCat\n\nend CommRingCat.Colimits\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Category/Ring/Colimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.3557324979631201}}
{"text": "\nimport .basic --.single\nnamespace ftype \n-- Embedding and subftypeebras\n\nopen_locale classical \nnoncomputable theory \n\n\ndef subftype {A : ftype} (ground : set A) : ftype :=  \n  {E := {x : A // x \u2208 ground}, fin := by {apply_instance}} \n\n@[ext] structure embed (A B : ftype) :=\n  (f     : A \u2192 B )\n  (f_inj : function.injective f)\n\ninstance emb_to_fn {A B : ftype} : has_coe_to_fun (embed A B) := {F := \u03bb _, A \u2192 B.E, coe := embed.f}\n\ndef embed.img {A B : ftype} (emb : embed A B) : set A \u2192 set B := \n  \u03bb X, (X.image emb)\n\n--instance emb_to_set_fn {A B : ftype} : has_coe (embed A B) (set A \u2192 set B.E) := \u27e8embed.img\u27e9\n\nlemma embed.on_subset {A B : ftype} (emb : embed A B) {X Y : set A} :\n  (X \u2286 Y) \u2192 (emb.img X) \u2286 (emb.img Y) := \n  \u03bb h, by tidy\n\nlemma embed.on_size {A B : ftype} (emb : embed A B) (X : set A) :\n  size (emb.img X) = size X := \nbegin\n  simp only [size, size_nat, int.coe_nat_inj'], \n  rw (by {simp [as_finset], tidy,} : (as_finset B (emb.img X)) = finset.image emb.f (as_finset A X)),\n  apply finset.card_image_of_injective, apply emb.f_inj,\nend\n\nlemma embed.on_inter {A B : ftype} (emb : embed A B) {X Y : set A} : \n  emb.img (X \u2229 Y) = emb.img X \u2229 emb.img Y := \n  (set.image_inter emb.f_inj).symm\n\nlemma embed.on_union {A B : ftype} (emb : embed A B) {X Y : set A} : \n  emb.img (X \u222a Y) = emb.img X \u222a emb.img Y := \n  set.image_union _ _ _\n\n\ndef embed.id {A : ftype} : embed A A := \n  \u27e8id, function.injective_id\u27e9\n\ndef embed.compose {A B C: ftype} : (embed A B) \u2192 (embed B C) \u2192 (embed A C) := \u03bb e1 e2,\n  \u27e8e2.f \u2218 e1.f, \u03bb x x' h, e1.f_inj (e2.f_inj h)\u27e9\n\ndef embed.from_subftype {A : ftype} (X : set A) : embed (subftype X) A := \n  \u27e8\u03bb e, e.val, by tidy\u27e9\n \ndef embed.from_nested_pair {A : ftype} {X\u2081 X\u2082 : set A} (hX\u2081X\u2082 : X\u2081 \u2286 X\u2082) : embed (subftype X\u2081) (subftype X\u2082) := \n  \u27e8\u03bb x, \u27e8x.1, by tidy\u27e9, by tidy\u27e9\n\nlemma embed.compose_subset_nested_pair {A : ftype} (X\u2081 X\u2082 : set A) (hX\u2081X\u2082 : X\u2081 \u2286 X\u2082) :\n (embed.compose (embed.from_nested_pair hX\u2081X\u2082) (embed.from_subftype X\u2082)) = embed.from_subftype X\u2081 := rfl \n\nlemma embed.compose_nested_triple {A : ftype} (X\u2081 X\u2082 X\u2083 : set A) (h\u2081\u2082 : X\u2081 \u2286 X\u2082) (h\u2082\u2083 : X\u2082 \u2286 X\u2083) :\n  (embed.compose (embed.from_nested_pair h\u2081\u2082) (embed.from_nested_pair h\u2082\u2083)) = embed.from_nested_pair (subset.trans h\u2081\u2082 h\u2082\u2083) := rfl\n\n\n--Subalgebra coercion \n\n\n--instance coe_to_coe_set {\u03b1 \u03b2: Type} [has_coe \u03b1 \u03b2] : has_coe (set \u03b1) (set \u03b2) := \u27e8\u03bb X, coe '' X\u27e9\n\ninstance coe_mem_from_subftype {A : ftype} {S : set A} : has_coe (subftype S) A := \u27e8subtype.val\u27e9\n\ninstance coe_set_from_subftype {A : ftype} {S : set A} : has_coe (set (subftype S)) (set A) := \u27e8\u03bb X, coe '' X\u27e9 \n\n\n--instance coe_single_from_subftype {A : ftype} {S : set A} : has_coe (single (subftype S)) (single A) := \u27e8(embed.from_subset S).single_emb\u27e9 \n\n--@[simp] lemma coe_single_subftype_compose {A : ftype} {S : set A} (e : single (subftype S)) : ((e: single A) : set A) = (e : set A) := rfl  \n--lemma coe_subftype_single_compose {A : ftype} {S : set A} (e : single (subftype S)) : ((e: subftype S) : set A) = (e : set A) := rfl  \n\n@[simp] lemma subftype_coe_size {A : ftype} {S : set A} (X : set (subftype S)) : size X = size (X : set A) := \n  ((embed.from_subftype S).on_size X).symm \n\n\n@[simp] lemma subftype_coe_subset {A : ftype} {S : set A} {X Y : set (subftype S)} : (X \u2286 Y) \u2194 ((X: set A) \u2286 (Y: set A)) :=\nbegin\n  refine \u27e8\u03bb h, (embed.from_subftype S).on_subset h, \u03bb h, _\u27e9,\n  have h1 : \u2200 (Z : set (subftype S)), (Z : set A) = (coe '' Z) := \u03bb Z, rfl, \n  rw [h1 X, h1 Y, subtype.coe_image, subtype.coe_image] at h, \n  intros x hx, cases x, \n  cases @h x_val \u27e8x_property, hx\u27e9,\n  assumption, \nend\n\n@[simp] lemma subftype_coe_union {A : ftype} {S : set A} {X Y : set (subftype S)} : \n  (((X \u222a Y) : set (subftype S)) : set A) = ((X: set A) \u222a (Y:set A)) := \n  (embed.from_subftype S).on_union\n \n@[simp] lemma subftype_coe_inter {A : ftype} {S : set A} {X Y : set (subftype S)} :\n  (((X \u2229 Y) : set (subftype S)) : set A) = ((X: set A) \u2229 (Y:set A)) := \n  (embed.from_subftype S).on_inter\n\n@[simp] lemma subftype_coe_compl {A : ftype} {S : set A} {X : set (subftype S)} :\n  (((X\u1d9c : set (subftype S))) : set A) = S \\ (X : set A)  := \nbegin\n  -- Fix this garbage! \n  unfold_coes, ext, refine \u27e8\u03bb h, \u27e8_,_\u27e9, \u03bb h,_\u27e9, \n  rcases h with \u27e8_,\u27e8_,_\u27e9\u27e9, rw \u2190h_h_right, tidy, apply h_h_left, \n  have := (embed.from_subftype S).f_inj h_h_right, \n  rw this, assumption, \n  -- No really, fix it! \nend \n\n@[simp] lemma coe_univ {A : ftype} (S : set A) : \n  ((univ : set (subftype S)) : set A) = S := \nby tidy \n\n@[simp] lemma coe_empty {A : ftype} (S : set A) : \n  ((\u2205 : set (subftype S)) : set A) = \u2205 :=\nby tidy \n\n@[simp] lemma coe_set_is_subset {A : ftype} {S : set A} (X : set (subftype S)) :\n  (X : set A) \u2286 S := \nby tidy\n\n@[simp] lemma coe_img_set {A : ftype} {Y Y' : set A} (hYY' : Y \u2286 Y') (X : set (subftype Y)) :\n  (((embed.from_nested_pair hYY').img X) : set A) = (X : set A) := \nby {simp only [embed.img], unfold_coes, tidy}\n\n@[simp] lemma coe_img_elem {A : ftype} {Y Y' : set A} (hYY' : Y \u2286 Y') (x : subftype Y) :\n  (((embed.from_nested_pair hYY') x ) : A) = (x : A) := \nby {unfold_coes, tidy}\n\n\n\n\n\n\n-- This next coe doesn't seem to work in practice, even when a P \u2286 Q proof term is in the local context \n\n--instance coe_from_nested_pair {A : ftype} {P Q: set A} {hPQ : P \u2286 Q} : has_coe (subftype P) (subftype Q) \n--  := \u27e8(embed.from_nested_pair hPQ).f\u27e9 \n\n--instance coe_set_from_nested_pair {A : ftype} {P Q: set A} {hPQ : P \u2286 Q} : has_coe (set (subftype P).E) (set (subftype Q).E) \n--  := \u27e8\u03bb (X : set (subftype P).E), ((embed.from_nested_pair).f '' X : set (subftype Q).E)\u27e9\n\n/-instance embed.coe_to_fun {A B : ftype.ftype} : has_coe_to_fun (ftype.embed A B) := {\n  F := (\u03bb _, A \u2192 B),\n  coe := \u03bb emb, emb.f,\n}-/\n--def subftype.embed {E : set A} : ftype.embed (subftype E) A := sorry\n\n\n\n---- Isomorphisms \n\nstructure iso (A B : ftype) := \n  (fwd : embed A B)\n  (bwd : embed B A)\n  (fwd_then_bwd : embed.compose fwd bwd = embed.id)\n  (bwd_then_fwd : embed.compose bwd fwd = embed.id)\n\n--def ftype.canonical (size : \u2124) :\n--  (0 \u2264 size) \u2192 ftype := sorry\n\n-- Construct a ftype from a finite set S (probably deprecated)\ndef powersetalg (\u03b3 : Type)[fintype \u03b3] : ftype := \n{ \n  E       := \u03b3, \n  fin  := by apply_instance,\n}\n\nend ftype ", "meta": {"author": "apnelson1", "repo": "lean-matroids", "sha": "1880f5b4b9a99948368e35539672a9bf35cc09cc", "save_path": "github-repos/lean/apnelson1-lean-matroids", "path": "github-repos/lean/apnelson1-lean-matroids/lean-matroids-1880f5b4b9a99948368e35539672a9bf35cc09cc/src/old/old/embed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3557116028510658}}
{"text": "/-\nCopyright (c) 2020 David W\u00e4rn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: David W\u00e4rn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.order.ideal\nimport Mathlib.data.finset.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 \n\nnamespace Mathlib\n\n/-!\n# The back and forth method and countable dense linear orders\n\n## Results\n\nSuppose `\u03b1 \u03b2` are linear orders, with `\u03b1` countable and `\u03b2` dense, nonempty, without endpoints.\nThen there is an order embedding `\u03b1 \u21aa \u03b2`. If in addition `\u03b1` is dense, nonempty, without\nendpoints and `\u03b2` is countable, then we can upgrade this to an order isomorhpism `\u03b1 \u2243 \u03b2`.\n\nThe idea for both results is to consider \"partial isomorphisms\", which\nidentify a finite subset of `\u03b1` with a finite subset of `\u03b2`, and prove that\nfor any such partial isomorphism `f` and `a : \u03b1`, we can extend `f` to\ninclude `a` in its domain.\n\n## References\n\nhttps://en.wikipedia.org/wiki/Back-and-forth_method\n\n## Tags\n\nback and forth, dense, countable, order\n\n-/\n\nnamespace order\n\n\n/-- Suppose `\u03b1` is a nonempty dense linear order without endpoints, and\n    suppose `lo`, `hi`, are finite subssets with all of `lo` strictly\n    before `hi`. Then there is an element of `\u03b1` strictly between `lo`\n    and `hi`. -/\ntheorem exists_between_finsets {\u03b1 : Type u_1} [linear_order \u03b1] [densely_ordered \u03b1] [no_bot_order \u03b1] [no_top_order \u03b1] [nonem : Nonempty \u03b1] (lo : finset \u03b1) (hi : finset \u03b1) (lo_lt_hi : \u2200 (x : \u03b1), x \u2208 lo \u2192 \u2200 (y : \u03b1), y \u2208 hi \u2192 x < y) : \u2203 (m : \u03b1), (\u2200 (x : \u03b1), x \u2208 lo \u2192 x < m) \u2227 \u2200 (y : \u03b1), y \u2208 hi \u2192 m < y := sorry\n\n/-- The type of partial order isomorphisms between `\u03b1` and `\u03b2` defined on finite subsets.\n    A partial order isomorphism is encoded as a finite subset of `\u03b1 \u00d7 \u03b2`, consisting\n    of pairs which should be identified. -/\ndef partial_iso (\u03b1 : Type u_1) (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] :=\n  Subtype\n    fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n      \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)\n\nnamespace partial_iso\n\n\nprotected instance inhabited (\u03b1 : Type u_1) (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] : Inhabited (partial_iso \u03b1 \u03b2) :=\n  { default := { val := \u2205, property := sorry } }\n\nprotected instance preorder (\u03b1 : Type u_1) (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] : preorder (partial_iso \u03b1 \u03b2) :=\n  subtype.preorder\n    fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n      \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)\n\n/-- For each `a`, we can find a `b` in the codomain, such that `a`'s relation to\nthe domain of `f` is `b`'s relation to the image of `f`.\n\nThus, if `a` is not already in `f`, then we can extend `f` by sending `a` to `b`.\n-/\ntheorem exists_across {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] [densely_ordered \u03b2] [no_bot_order \u03b2] [no_top_order \u03b2] [Nonempty \u03b2] (f : partial_iso \u03b1 \u03b2) (a : \u03b1) : \u2203 (b : \u03b2), \u2200 (p : \u03b1 \u00d7 \u03b2), p \u2208 subtype.val f \u2192 cmp (prod.fst p) a = cmp (prod.snd p) b := sorry\n\n/-- A partial isomorphism between `\u03b1` and `\u03b2` is also a partial isomorphism between `\u03b2` and `\u03b1`. -/\nprotected def comm {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] : partial_iso \u03b1 \u03b2 \u2192 partial_iso \u03b2 \u03b1 :=\n  subtype.map (finset.image \u21d1(equiv.prod_comm \u03b1 \u03b2)) sorry\n\n/-- The set of partial isomorphisms defined at `a : \u03b1`, together with a proof that any\n    partial isomorphism can be extended to one defined at `a`. -/\ndef defined_at_left {\u03b1 : Type u_1} (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] [densely_ordered \u03b2] [no_bot_order \u03b2] [no_top_order \u03b2] [Nonempty \u03b2] (a : \u03b1) : cofinal (partial_iso \u03b1 \u03b2) :=\n  cofinal.mk (fun (f : partial_iso \u03b1 \u03b2) => \u2203 (b : \u03b2), (a, b) \u2208 subtype.val f) sorry\n\n/-- The set of partial isomorphisms defined at `b : \u03b2`, together with a proof that any\n    partial isomorphism can be extended to include `b`. We prove this by symmetry. -/\ndef defined_at_right (\u03b1 : Type u_1) {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] [densely_ordered \u03b1] [no_bot_order \u03b1] [no_top_order \u03b1] [Nonempty \u03b1] (b : \u03b2) : cofinal (partial_iso \u03b1 \u03b2) :=\n  cofinal.mk (fun (f : partial_iso \u03b1 \u03b2) => \u2203 (a : \u03b1), (a, b) \u2208 subtype.val f) sorry\n\n/-- Given an ideal which intersects `defined_at_left \u03b2 a`, pick `b : \u03b2` such that\n    some partial function in the ideal maps `a` to `b`. -/\ndef fun_of_ideal {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] [densely_ordered \u03b2] [no_bot_order \u03b2] [no_top_order \u03b2] [Nonempty \u03b2] (a : \u03b1) (I : ideal (partial_iso \u03b1 \u03b2)) : (\u2203 (f : partial_iso \u03b1 \u03b2), f \u2208 defined_at_left \u03b2 a \u2227 f \u2208 I) \u2192\n  Subtype\n    fun (b : \u03b2) =>\n      \u2203 (f :\n        Subtype\n          fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n            \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n        \u2203 (H : f \u2208 I), (a, b) \u2208 subtype.val f :=\n  (classical.indefinite_description\n      fun (x : \u03b2) =>\n        \u2203 (f :\n          Subtype\n            fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n              \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n          \u2203 (H : f \u2208 I), (a, x) \u2208 subtype.val f) \u2218\n    sorry\n\n/-- Given an ideal which intersects `defined_at_right \u03b1 b`, pick `a : \u03b1` such that\n    some partial function in the ideal maps `a` to `b`. -/\ndef inv_of_ideal {\u03b1 : Type u_1} {\u03b2 : Type u_2} [linear_order \u03b1] [linear_order \u03b2] [densely_ordered \u03b1] [no_bot_order \u03b1] [no_top_order \u03b1] [Nonempty \u03b1] (b : \u03b2) (I : ideal (partial_iso \u03b1 \u03b2)) : (\u2203 (f : partial_iso \u03b1 \u03b2), f \u2208 defined_at_right \u03b1 b \u2227 f \u2208 I) \u2192\n  Subtype\n    fun (a : \u03b1) =>\n      \u2203 (f :\n        Subtype\n          fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n            \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n        \u2203 (H : f \u2208 I), (a, b) \u2208 subtype.val f :=\n  (classical.indefinite_description\n      fun (x : \u03b1) =>\n        \u2203 (f :\n          Subtype\n            fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n              \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n          \u2203 (H : f \u2208 I), (x, b) \u2208 subtype.val f) \u2218\n    sorry\n\nend partial_iso\n\n\n/-- Any countable linear order embeds in any nonempty dense linear order without endpoints. -/\ndef embedding_from_countable_to_dense (\u03b1 : Type u_1) (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] [encodable \u03b1] [densely_ordered \u03b2] [no_bot_order \u03b2] [no_top_order \u03b2] [Nonempty \u03b2] : \u03b1 \u21aao \u03b2 :=\n  let our_ideal : ideal (partial_iso \u03b1 \u03b2) := ideal_of_cofinals Inhabited.default (partial_iso.defined_at_left \u03b2);\n  let F :\n    (a : \u03b1) \u2192\n      Subtype\n        fun (b : \u03b2) =>\n          \u2203 (f :\n            Subtype\n              fun (f : finset (\u03b1 \u00d7 \u03b2)) =>\n                \u2200 (p q : \u03b1 \u00d7 \u03b2), p \u2208 f \u2192 q \u2208 f \u2192 cmp (prod.fst p) (prod.fst q) = cmp (prod.snd p) (prod.snd q)),\n            \u2203 (H : f \u2208 our_ideal), (a, b) \u2208 subtype.val f :=\n    fun (a : \u03b1) => partial_iso.fun_of_ideal a our_ideal sorry;\n  order_embedding.of_strict_mono (fun (a : \u03b1) => subtype.val (F a)) sorry\n\n/-- Any two countable dense, nonempty linear orders without endpoints are order isomorphic. -/\ndef iso_of_countable_dense (\u03b1 : Type u_1) (\u03b2 : Type u_2) [linear_order \u03b1] [linear_order \u03b2] [encodable \u03b1] [densely_ordered \u03b1] [no_bot_order \u03b1] [no_top_order \u03b1] [Nonempty \u03b1] [encodable \u03b2] [densely_ordered \u03b2] [no_bot_order \u03b2] [no_top_order \u03b2] [Nonempty \u03b2] : \u03b1 \u2243o \u03b2 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/order/countable_dense_linear_order.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.355711602691589}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.field_theory.minpoly\nimport Mathlib.field_theory.splitting_field\nimport Mathlib.field_theory.tower\nimport Mathlib.ring_theory.power_basis\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 u_4 \n\nnamespace Mathlib\n\n/-!\n# Normal field extensions\n\nIn this file we define normal field extensions and prove that for a finite extension, being normal\nis the same as being a splitting field (`normal.of_is_splitting_field` and\n`normal.exists_is_splitting_field`).\n\n## Main Definitions\n\n- `normal F K` where `K` is a field extension of `F`.\n-/\n\n--TODO(Commelin): refactor normal to extend `is_algebraic`??\n\n/-- Typeclass for normal field extension: `K` is a normal extension of `F` iff the minimal\npolynomial of every element `x` in `K` splits in `K`, i.e. every conjugate of `x` is in `K`. -/\ndef normal (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] :=\n  \u2200 (x : K), is_integral F x \u2227 polynomial.splits (algebra_map F K) (minpoly F x)\n\nprotected instance normal_self (F : Type u_1) [field F] : normal F F :=\n  fun (x : F) =>\n    { left := is_integral_algebra_map,\n      right :=\n        eq.mpr (id (Eq._oldrec (Eq.refl (polynomial.splits (algebra_map F F) (minpoly F x))) (minpoly.eq_X_sub_C' x)))\n          (polynomial.splits_X_sub_C (algebra_map F F)) }\n\ntheorem normal.is_integral (F : Type u_1) {K : Type u_2} [field F] [field K] [algebra F K] [h : normal F K] (x : K) : is_integral F x :=\n  and.left (h x)\n\ntheorem normal.splits (F : Type u_1) {K : Type u_2} [field F] [field K] [algebra F K] [h : normal F K] (x : K) : polynomial.splits (algebra_map F K) (minpoly F x) :=\n  and.right (h x)\n\ntheorem normal.exists_is_splitting_field (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] [normal F K] [finite_dimensional F K] : \u2203 (p : polynomial F), polynomial.is_splitting_field F K p := sorry\n\ntheorem normal.tower_top_of_normal (F : Type u_1) (K : Type u_2) [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra K E] [is_scalar_tower F K E] [h : normal F E] : normal K E := sorry\n\ntheorem normal.of_alg_equiv {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] {E' : Type u_4} [field E'] [algebra F E'] [h : normal F E] (f : alg_equiv F E E') : normal F E' := sorry\n\ntheorem alg_equiv.transfer_normal {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] {E' : Type u_4} [field E'] [algebra F E'] (f : alg_equiv F E E') : normal F E \u2194 normal F E' :=\n  { mp := fun (h : normal F E) => normal.of_alg_equiv f,\n    mpr := fun (h : normal F E') => normal.of_alg_equiv (alg_equiv.symm f) }\n\ntheorem normal.of_is_splitting_field {F : Type u_1} [field F] {E : Type u_3} [field E] [algebra F E] {p : polynomial F} [hFEp : polynomial.is_splitting_field F E p] : normal F E := sorry\n\n/-- Restrict algebra homomorphism to image of normal subfield -/\ndef alg_hom.restrict_normal_aux {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03d5 : alg_hom F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [h : normal F E] : alg_hom F \u21a5(alg_hom.range (is_scalar_tower.to_alg_hom F E K)) \u21a5(alg_hom.range (is_scalar_tower.to_alg_hom F E K)) :=\n  alg_hom.mk (fun (x : \u21a5(alg_hom.range (is_scalar_tower.to_alg_hom F E K))) => { val := coe_fn \u03d5 \u2191x, property := sorry })\n    sorry sorry sorry sorry sorry\n\n/-- Restrict algebra homomorphism to normal subfield -/\ndef alg_hom.restrict_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03d5 : alg_hom F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] : alg_hom F E E :=\n  alg_hom.comp\n    (alg_hom.comp\n      (alg_equiv.to_alg_hom (alg_equiv.symm (alg_hom.alg_equiv.of_injective_field (is_scalar_tower.to_alg_hom F E K))))\n      (alg_hom.restrict_normal_aux \u03d5 E))\n    (alg_equiv.to_alg_hom (alg_hom.alg_equiv.of_injective_field (is_scalar_tower.to_alg_hom F E K)))\n\ntheorem alg_hom.restrict_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03d5 : alg_hom F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] (x : E) : coe_fn (algebra_map E K) (coe_fn (alg_hom.restrict_normal \u03d5 E) x) = coe_fn \u03d5 (coe_fn (algebra_map E K) x) := sorry\n\ntheorem alg_hom.restrict_normal_comp {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03d5 : alg_hom F K K) (\u03c8 : alg_hom F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] : alg_hom.comp (alg_hom.restrict_normal \u03d5 E) (alg_hom.restrict_normal \u03c8 E) = alg_hom.restrict_normal (alg_hom.comp \u03d5 \u03c8) E := sorry\n\n/-- Restrict algebra isomorphism to a normal subfield -/\ndef alg_equiv.restrict_normal {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03c7 : alg_equiv F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [h : normal F E] : alg_equiv F E E :=\n  alg_equiv.of_alg_hom (alg_hom.restrict_normal (alg_equiv.to_alg_hom \u03c7) E)\n    (alg_hom.restrict_normal (alg_equiv.to_alg_hom (alg_equiv.symm \u03c7)) E) sorry sorry\n\ntheorem alg_equiv.restrict_normal_commutes {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03c7 : alg_equiv F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] (x : E) : coe_fn (algebra_map E K) (coe_fn (alg_equiv.restrict_normal \u03c7 E) x) = coe_fn \u03c7 (coe_fn (algebra_map E K) x) :=\n  alg_hom.restrict_normal_commutes (alg_equiv.to_alg_hom \u03c7) E x\n\ntheorem alg_equiv.restrict_normal_trans {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (\u03c7 : alg_equiv F K K) (\u03c9 : alg_equiv F K K) (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] : alg_equiv.restrict_normal (alg_equiv.trans \u03c7 \u03c9) E =\n  alg_equiv.trans (alg_equiv.restrict_normal \u03c7 E) (alg_equiv.restrict_normal \u03c9 E) := sorry\n\n/-- Restriction to an normal subfield as a group homomorphism -/\ndef alg_equiv.restrict_normal_hom {F : Type u_1} {K : Type u_2} [field F] [field K] [algebra F K] (E : Type u_3) [field E] [algebra F E] [algebra E K] [is_scalar_tower F E K] [normal F E] : alg_equiv F K K \u2192* alg_equiv F E E :=\n  monoid_hom.mk' (fun (\u03c7 : alg_equiv F K K) => alg_equiv.restrict_normal \u03c7 E) sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/field_theory/normal.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3557115942147574}}
{"text": "/-\nCopyright (c) 2020 Barinder Singh Banwait. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Barinder Singh Banwait, with lots of help from the good folk on the Leanprover Zulip chat :)\n-/\nimport ring_theory.noetherian\nimport ring_theory.integral_closure\nimport ring_theory.adjoin\nimport ring_theory.algebra\nimport ring_theory.localization\nimport linear_algebra.basic\n\n/-!\n# Integrally closedness\n\nLet R \u2286 A be an extension of integral domains.\n\n## Main definitions\n\n* `is_integrally_closed_in R A : Prop` is the assertion that `R` is integrally closed in `A`. It is\n   a structure, implemented as the predicate that all elements of `A` that are integral over `R`\n   already belong to `R`.\n\n* `is_integrally_closed R` is the definition that `R` is integrally closed in an absolute sense.\n   This is implemented as the following implication: if for all `r` and `s` in `R` with `s \u2260 0`,\n   `r^n \u2208 \u27e8 r^{n-1}s, \u22ef , s^n \u27e9\n\n\n## Main statements\n\n* `fundamental_theorem_integrally_closedness` is the statement that an integral domain is integrally\n  closed if and only if it is integrall closed in a field of fractions.\n\n## References\n\n* [J. Neukirch, *Algebraic Number Theory*][neukirch-ant]\n\n## Tags\n\nIntegrally closed\n\n-/\n\nuniverses u v\n\nvariables (R : Type u) {A : Type v}\nvariables [comm_ring R] [comm_ring A]\nvariables [algebra R A]\nvariables i : \u2115\nvariables (K : Type*)\nopen submodule\nopen function\nopen finset\nopen_locale big_operators\n\nstructure is_integrally_closed_in (R : Type u) (A : Type v) [comm_ring R] [comm_ring A]\n[algebra R A] : Prop :=\n(inj : injective (algebra_map R A))\n(closed : \u2200 (a : A), is_integral R a \u2192 \u2203  r : R, algebra_map R A r = a)\n\n-- def my_set (R) [integral_domain R] (r : R) (s : R) (n : \u2115) :=\n-- { x | \u2203 (i:\u2115 ) (h : 0\u2264 i) (h2 :i\u2264 n-1), x = r^(n-1-i) *s^(i+1) }\n\ndef my_set (R) [integral_domain R] (r : R) (s : R) (n : \u2115) :=\n{ x | \u2203 (c : \u2115 \u00d7 \u2115) (hy: c \u2208 (finset.nat.antidiagonal n).erase \u27e80, n\u27e9), x = r ^ c.2 * s ^ c.1 }\n\ndef is_integrally_closed (R) [integral_domain R] : Prop :=\n\u2200 (r : R) (s : R), (s \u2260 0) \u2227 (\u2203 n : \u2115 ,\nr^n \u2208 span R (my_set R r s n)) \u2192 s \u2223 r\n\nopen submodule\n\nlemma mwe (R) [integral_domain R] (n : \u2115) (f : \u2115 \u2192 R) (hf : f 0 = 1) : \u2200 \u2983r s : R\u2984, s \u2260 0 \u2192\n  \u2211 ij in finset.nat.antidiagonal n, f ij.1 * r ^ ij.2 * s ^ ij.1 = 0 \u2192\n  r^n \u2208 span R (my_set R r s n) :=\nbegin\n  intros r s,\n  intro s_non_zero,\n  intro H,\n  let p' : \u2115 \u00d7 \u2115 := \u27e80, n\u27e9,\n  have LM : p' \u2208 finset.nat.antidiagonal n,\n  {\n    rw finset.nat.mem_antidiagonal,\n    linarith,\n  },\n  rw \u2190 finset.insert_erase LM at H,\n  simp at H,\n  rw hf at H,\n  simp at H,\n  have KL : r^n = -\u2211 (x : \u2115 \u00d7 \u2115) in (nat.antidiagonal n).erase p', f x.fst * r ^ x.snd * s ^ x.fst,\n  {\n    exact eq_neg_of_add_eq_zero H,\n  },\n  rw KL,\n  rw my_set,\n  rw mem_span,\n  intro p,\n  intro p_H,\n  rw ideal.neg_mem_iff p,\n  apply sum_mem,\n  intro c,\n  intro c_H,\n  have x_in_my_set : r ^ c.snd * s ^ c.fst \u2208 {x : R | \u2203 (c : \u2115 \u00d7 \u2115) (hy : c \u2208 (nat.antidiagonal n).erase (0, n)), x = r ^ c.snd * s ^ c.fst},\n  {\n    simp,\n    use c.fst,\n    use c.snd,\n    split,\n    split,\n    intro something,\n    by_contradiction HAPPY,\n    have obv : c = p',\n    {\n      exact prod.ext something HAPPY,\n    },\n    rw obv at c_H,\n    rw mem_erase at c_H,\n    cases c_H,\n    contradiction,\n    rw \u2190 finset.nat.mem_antidiagonal,\n    rw mem_erase at c_H,\n    cases c_H with c_H_1 c_H_2,\n    exact c_H_2,\n    refl,\n  },\n  have x_in_p : r ^ c.snd * s ^ c.fst \u2208 \u2191p,\n  {\n    exact set.mem_of_mem_of_subset x_in_my_set p_H,\n  },\n  rw mul_assoc,\n  exact p.smul_mem (f c.fst) x_in_p,\nend\n\nlemma lin_comb_mem (R) [integral_domain R] (n : \u2115) (r s : R) (s_non_zero : s \u2260 0) :\n  r^n \u2208 span R (my_set R r s n) \u2192 \u2203 (f : \u2115 \u2192 R) (h_f: f 0 = 1),\n  \u2211 ij in finset.nat.antidiagonal n, f ij.1 * r ^ ij.2 * s ^ ij.1 = 0 :=\nbegin\n  intro H,\n  -- trying finsupp.mem_span_iff_total but it just isn't working :(\n  sorry,\nend\n\n\nlemma mwe_deluxe (R) [integral_domain R] (n : \u2115) {r s : R} (h_s : s \u2260 0) :\n  r^n \u2208 span R (my_set R r s n) \u2194 \u2203 (f : \u2115 \u2192 R) (hf : f 0 = 1), \u2211 ij in finset.nat.antidiagonal n, f ij.1 * r ^ ij.2 * s ^ ij.1 = 0 :=\nbegin\n  split,\n  intro H,\n  apply lin_comb_mem,\n  exact h_s,\n  exact H,\n  intro k,\n  cases k with f B,\n  cases B with C HC,\n  apply mwe,\n  exact C,\n  exact h_s,\n  exact HC,\nend\n\nlemma equiv_johan_absolute_deluxe (R) [integral_domain R] :\n  is_integrally_closed R \u2194 \u2200 (r s : R), s \u2260 0 \u2192 (\u2203 (n : \u2115) (f : \u2115 \u2192 R) (hf : f 0 = 1),\n  \u2211 ij in finset.nat.antidiagonal n, f ij.1 * r ^ ij.2 * s ^ ij.1 = 0) \u2192 s \u2223 r :=\nbegin\n  split,\n  intros h r s,\n  intro k,\n  unfold is_integrally_closed at h,\n  specialize h r s,\n  intro m,\n  apply h,\n  split,\n  exact k,\n  cases m with n m1,\n  cases m1 with f m2,\n  cases m2 with hf m3,\n  use n,\n  rw mwe_deluxe,\n  use f,\n  split,\n  exact hf,\n  exact m3,\n  exact k,\n  unfold is_integrally_closed,\n  intro H,\n  intros r s,\n  intro H2,\n  cases H2 with A B,\n  apply H,\n  exact A,\n  cases B with n B_n,\n  use n,\n  specialize H r s A,\n  rw mwe_deluxe at B_n,\n  exact B_n,\n  exact A,\nend\n\n\nlemma fundamental_theorem_integrally_closedness (R : Type u) (A : Type v) [integral_domain R]\n[comm_ring A] [algebra R A] (H : fraction_map R A):\n  is_integrally_closed R \u2194 is_integrally_closed_in R A :=\nbegin\n  split,\n  rw equiv_johan_absolute_deluxe,\n  intro H,\n  sorry,\n  sorry,\nend\n\n\n-- class dedekind_domain (\u03b1 : Type*) extends integral_domain \u03b1 :=\n-- (noetherian : is_noetherian_ring \u03b1)\n-- (factors_prod : \u2200{a : \u03b1}, a \u2260 0 \u2192 (factors a).prod ~\u1d64 a)\n-- (prime_factors : \u2200{a : \u03b1}, a \u2260 0 \u2192 \u2200x\u2208factors a, prime x)\n", "meta": {"author": "BarinderBanwait", "repo": "integrally_closedness", "sha": "f3b45be4f8b1ac64a2f0a48f0b9f3b281947bcd1", "save_path": "github-repos/lean/BarinderBanwait-integrally_closedness", "path": "github-repos/lean/BarinderBanwait-integrally_closedness/integrally_closedness-f3b45be4f8b1ac64a2f0a48f0b9f3b281947bcd1/src/hello_world.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3557115942147574}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.creates\nimport category_theory.sites.sheafification\n\n/-!\n\n# Limits and colimits of sheaves\n\n## Limits\n\nWe prove that the forgetful functor from `Sheaf J D` to presheaves creates limits.\nIf the target category `D` has limits (of a certain shape),\nthis then implies that `Sheaf J D` has limits of the same shape and that the forgetful\nfunctor preserves these limits.\n\n## Colimits\n\nGiven a diagram `F : K \u2964 Sheaf J D` of sheaves, and a colimit cocone on the level of presheaves,\nwe show that the cocone obtained by sheafifying the cocone point is a colimit cocone of sheaves.\n\nThis allows us to show that `Sheaf J D` has colimits (of a certain shape) as soon as `D` does.\n\n-/\nnamespace category_theory\nnamespace Sheaf\n\nopen category_theory.limits\nopen opposite\n\nsection limits\n\nuniverses w v u\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\nvariables {K : Type (max v u)} [small_category K]\n\nnoncomputable theory\n\nsection\n\n\n/-- An auxiliary definition to be used below.\n\nWhenever `E` is a cone of shape `K` of sheaves, and `S` is the multifork associated to a\ncovering `W` of an object `X`, with respect to the cone point `E.X`, this provides a cone of\nshape `K` of objects in `D`, with cone point `S.X`.\n\nSee `is_limit_multifork_of_is_limit` for more on how this definition is used.\n-/\ndef multifork_evaluation_cone (F : K \u2964 Sheaf J D)\n  (E : cone (F \u22d9 Sheaf_to_presheaf J D)) (X : C) (W : J.cover X) (S : multifork (W.index E.X)) :\n  cone (F \u22d9 Sheaf_to_presheaf J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) :=\n{ X := S.X,\n  \u03c0 :=\n  { app := \u03bb k, (presheaf.is_limit_of_is_sheaf J (F.obj k).1 W (F.obj k).2).lift $\n      multifork.of_\u03b9 _ S.X (\u03bb i, S.\u03b9 i \u226b (E.\u03c0.app k).app (op i.Y)) begin\n        intros i,\n        simp only [category.assoc],\n        erw [\u2190 (E.\u03c0.app k).naturality, \u2190 (E.\u03c0.app k).naturality],\n        dsimp,\n        simp only [\u2190 category.assoc],\n        congr' 1,\n        apply S.condition,\n      end,\n    naturality' := begin\n      intros i j f,\n      dsimp [presheaf.is_limit_of_is_sheaf],\n      rw [category.id_comp],\n      apply presheaf.is_sheaf.hom_ext (F.obj j).2 W,\n      intros ii,\n      erw [presheaf.is_sheaf.amalgamate_map],\n      rw [category.assoc, \u2190 (F.map f).naturality, \u2190 category.assoc],\n      erw [presheaf.is_sheaf.amalgamate_map],\n      dsimp [multifork.of_\u03b9],\n      rw [category.assoc, \u2190 E.w f],\n      simp,\n    end } }\n\n\nvariables [has_limits_of_shape K D]\n\n/-- If `E` is a cone of shape `K` of sheaves, which is a limit on the level of presheves,\nthis definition shows that the limit presheaf satisfies the multifork variant of the sheaf\ncondition, at a given covering `W`.\n\nThis is used below in `is_sheaf_of_is_limit` to show that the limit presheaf is indeed a sheaf.\n-/\ndef is_limit_multifork_of_is_limit (F : K \u2964 Sheaf J D)\n  (E : cone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_limit E) (X : C) (W : J.cover X) : is_limit (W.multifork E.X) :=\nmultifork.is_limit.mk _\n(\u03bb S, (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).lift $\n  multifork_evaluation_cone F E X W S)\nbegin\n  intros S i,\n  apply (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) hE).hom_ext,\n  intros k,\n  dsimp [multifork.of_\u03b9],\n  erw [category.assoc, (E.\u03c0.app k).naturality],\n  dsimp,\n  rw \u2190 category.assoc,\n  erw (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac\n    (multifork_evaluation_cone F E X W S),\n  dsimp [multifork_evaluation_cone, presheaf.is_limit_of_is_sheaf],\n  erw presheaf.is_sheaf.amalgamate_map,\n  refl,\nend\nbegin\n  intros S m hm,\n  apply (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).hom_ext,\n  intros k,\n  dsimp,\n  erw (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac,\n  apply presheaf.is_sheaf.hom_ext (F.obj k).2 W,\n  intros i,\n  erw presheaf.is_sheaf.amalgamate_map,\n  dsimp [multifork.of_\u03b9],\n  change _ = S.\u03b9 i \u226b _,\n  erw [\u2190 hm, category.assoc, \u2190 (E.\u03c0.app k).naturality, category.assoc],\n  refl,\nend\n\n/-- If `E` is a cone which is a limit on the level of presheaves,\nthen the limit presheaf is again a sheaf.\n\nThis is used to show that the forgetful functor from sheaves to presheaves creates limits.\n-/\nlemma is_sheaf_of_is_limit (F : K \u2964 Sheaf J D) (E : cone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_limit E) : presheaf.is_sheaf J E.X :=\nbegin\n  rw presheaf.is_sheaf_iff_multifork,\n  intros X S,\n  exact \u27e8is_limit_multifork_of_is_limit _ _ hE _ _\u27e9,\nend\n\ninstance (F : K \u2964 Sheaf J D) : creates_limit F (Sheaf_to_presheaf J D) :=\ncreates_limit_of_reflects_iso $ \u03bb E hE,\n{ lifted_cone := \u27e8\u27e8E.X, is_sheaf_of_is_limit _ _ hE\u27e9, \u27e8E.\u03c0.app, E.\u03c0.naturality\u27e9\u27e9,\n  valid_lift := cones.ext (eq_to_iso rfl) $ \u03bb j, by { dsimp, simp },\n  makes_limit :=\n  { lift := \u03bb S, hE.lift ((Sheaf_to_presheaf J D).map_cone S),\n    fac' := \u03bb S j, hE.fac ((Sheaf_to_presheaf J D).map_cone S) j,\n    uniq' := \u03bb S m hm, hE.uniq ((Sheaf_to_presheaf J D).map_cone S) m hm } }\n\ninstance : creates_limits_of_shape K (Sheaf_to_presheaf J D) := {}\n\ninstance : has_limits_of_shape K (Sheaf J D) :=\nhas_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape (Sheaf_to_presheaf J D)\n\nend\n\ninstance [has_limits D] : creates_limits (Sheaf_to_presheaf J D) := {}\n\ninstance [has_limits D] : has_limits (Sheaf J D) :=\nhas_limits_of_has_limits_creates_limits (Sheaf_to_presheaf J D)\n\nend limits\n\nsection colimits\n\nuniverses w v u\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\nvariables {K : Type (max v u)} [small_category K]\n-- Now we need a handful of instances to obtain sheafification...\nvariables [concrete_category.{max v u} D]\nvariables [\u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\nvariables [preserves_limits (forget D)]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\nvariables [reflects_isomorphisms (forget D)]\n\n/-- Construct a cocone by sheafifying a cocone point of a cocone `E` of presheaves\nover a functor which factors through sheaves.\nIn `is_colimit_sheafify_cocone`, we show that this is a colimit cocone when `E` is a colimit. -/\n@[simps]\ndef sheafify_cocone {F : K \u2964 Sheaf J D} (E : cocone (F \u22d9 Sheaf_to_presheaf J D)) : cocone F :=\n{ X := \u27e8J.sheafify E.X, grothendieck_topology.plus.is_sheaf_plus_plus _ _\u27e9,\n  \u03b9 :=\n  { app := \u03bb k, by apply E.\u03b9.app k \u226b J.to_sheafify E.X, -- annoying...\n    naturality' := \u03bb i j f, by erw [category.comp_id, \u2190 category.assoc, E.w f] } }\n\n/-- If `E` is a colimit cocone of presheaves, over a diagram factoring through sheaves,\nthen `sheafify_cocone E` is a colimit cocone. -/\n@[simps]\ndef is_colimit_sheafify_cocone {F : K \u2964 Sheaf J D} (E : cocone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_colimit E) :\n  is_colimit (sheafify_cocone E) :=\n{ desc := \u03bb S, J.sheafify_lift (hE.desc ((Sheaf_to_presheaf J D).map_cocone S)) S.X.2,\n  fac' := begin\n    intros S j,\n    dsimp [sheafify_cocone],\n    erw [category.assoc, J.to_sheafify_sheafify_lift, hE.fac],\n    refl,\n  end,\n  uniq' := begin\n    intros S m hm,\n    apply J.sheafify_lift_unique,\n    apply hE.uniq ((Sheaf_to_presheaf J D).map_cocone S),\n    intros j,\n    erw [\u2190 category.assoc, hm j],\n    refl,\n  end }\n\ninstance [has_colimits_of_shape K D] : has_colimits_of_shape K (Sheaf J D) :=\n\u27e8\u03bb F, has_colimit.mk \u27e8sheafify_cocone (colimit.cocone _),\n  is_colimit_sheafify_cocone _ (colimit.is_colimit _)\u27e9\u27e9\n\ninstance [has_colimits D] : has_colimits (Sheaf J D) := \u27e8infer_instance\u27e9\n\nend colimits\n\nend Sheaf\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/sites/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370308082623217, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.35567149647613633}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.natural_isomorphism\nimport Mathlib.category_theory.eq_to_hom\nimport Mathlib.PostPort\n\nuniverses u\u2081 v\u2081 w\u2080 w\u2081 w\u2082 \n\nnamespace Mathlib\n\n/-!\n# Categories of indexed families of objects.\n\nWe define the pointwise category structure on indexed families of objects in a category\n(and also the dependent generalization).\n\n-/\n\nnamespace category_theory\n\n\n/--\n`pi C` gives the cartesian product of an indexed family of categories.\n-/\nprotected instance pi {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] :\n    category ((i : I) \u2192 C i) :=\n  category.mk\n\n/--\nThis provides some assistance to typeclass search in a common situation,\nwhich otherwise fails. (Without this `category_theory.pi.has_limit_of_has_limit_comp_eval` fails.)\n-/\ninstance pi' {I : Type v\u2081} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] :\n    category ((i : I) \u2192 C i) :=\n  category_theory.pi C\n\nnamespace pi\n\n\n@[simp] theorem id_apply {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    (X : (i : I) \u2192 C i) (i : I) : \ud835\udfd9 = \ud835\udfd9 :=\n  rfl\n\n@[simp] theorem comp_apply {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    {X : (i : I) \u2192 C i} {Y : (i : I) \u2192 C i} {Z : (i : I) \u2192 C i} (f : X \u27f6 Y) (g : Y \u27f6 Z) (i : I) :\n    category_struct.comp f g i = f i \u226b g i :=\n  rfl\n\n/--\nThe evaluation functor at `i : I`, sending an `I`-indexed family of objects to the object over `i`.\n-/\n@[simp] theorem eval_map {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] (i : I)\n    (f : (i : I) \u2192 C i) (g : (i : I) \u2192 C i) (\u03b1 : f \u27f6 g) : functor.map (eval C i) \u03b1 = \u03b1 i :=\n  Eq.refl (functor.map (eval C i) \u03b1)\n\n/--\nPull back an `I`-indexed family of objects to an `J`-indexed family, along a function `J \u2192 I`.\n-/\n@[simp] theorem comap_map {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2081}\n    (h : J \u2192 I) (f : (i : I) \u2192 C i) (g : (i : I) \u2192 C i) (\u03b1 : f \u27f6 g) (i : J) :\n    functor.map (comap C h) \u03b1 i = \u03b1 (h i) :=\n  Eq.refl (functor.map (comap C h) \u03b1 i)\n\n/--\nThe natural isomorphism between\npulling back a grading along the identity function,\nand the identity functor. -/\ndef comap_id (I : Type w\u2080) (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] : comap C id \u2245 \ud835\udfed :=\n  iso.mk (nat_trans.mk fun (X : (i : I) \u2192 C i) => \ud835\udfd9) (nat_trans.mk fun (X : (i : I) \u2192 C i) => \ud835\udfd9)\n\n/--\nThe natural isomorphism comparing between\npulling back along two successive functions, and\npulling back along their composition\n-/\ndef comap_comp {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2081}\n    {K : Type w\u2082} (f : K \u2192 J) (g : J \u2192 I) : comap C g \u22d9 comap (C \u2218 g) f \u2245 comap C (g \u2218 f) :=\n  iso.mk (nat_trans.mk fun (X : (i : I) \u2192 C i) (b : K) => \ud835\udfd9)\n    (nat_trans.mk fun (X : (i : I) \u2192 C i) (b : K) => \ud835\udfd9)\n\n/-- The natural isomorphism between pulling back then evaluating, and just evaluating. -/\ndef comap_eval_iso_eval {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2081}\n    (h : J \u2192 I) (j : J) : comap C h \u22d9 eval (C \u2218 h) j \u2245 eval C (h j) :=\n  nat_iso.of_components\n    (fun (f : (i : I) \u2192 C i) => iso.refl (functor.obj (comap C h \u22d9 eval (C \u2218 h) j) f)) sorry\n\nprotected instance sum_elim_category {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)]\n    {J : Type w\u2080} {D : J \u2192 Type u\u2081} [(j : J) \u2192 category (D j)] (s : I \u2295 J) :\n    category (sum.elim C D s) :=\n  sorry\n\n/--\nThe bifunctor combining an `I`-indexed family of objects with a `J`-indexed family of objects\nto obtain an `I \u2295 J`-indexed family of objects.\n-/\n@[simp] theorem sum_obj_obj {I : Type w\u2080} (C : I \u2192 Type u\u2081) [(i : I) \u2192 category (C i)] {J : Type w\u2080}\n    {D : J \u2192 Type u\u2081} [(j : J) \u2192 category (D j)] (f : (i : I) \u2192 C i) (g : (j : J) \u2192 D j)\n    (s : I \u2295 J) : functor.obj (functor.obj (sum C) f) g s = sum.rec f g s :=\n  Eq.refl (functor.obj (functor.obj (sum C) f) g s)\n\nend pi\n\n\nnamespace functor\n\n\n/--\nAssemble an `I`-indexed family of functors into a functor between the pi types.\n-/\ndef pi {I : Type w\u2080} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (D i)] (F : (i : I) \u2192 C i \u2964 D i) : ((i : I) \u2192 C i) \u2964 ((i : I) \u2192 D i) :=\n  mk (fun (f : (i : I) \u2192 C i) (i : I) => obj (F i) (f i))\n    fun (f g : (i : I) \u2192 C i) (\u03b1 : f \u27f6 g) (i : I) => map (F i) (\u03b1 i)\n\n-- One could add some natural isomorphisms showing\n\n-- how `functor.pi` commutes with `pi.eval` and `pi.comap`.\n\nend functor\n\n\nnamespace nat_trans\n\n\n/--\nAssemble an `I`-indexed family of natural transformations into a single natural transformation.\n-/\ndef pi {I : Type w\u2080} {C : I \u2192 Type u\u2081} [(i : I) \u2192 category (C i)] {D : I \u2192 Type u\u2081}\n    [(i : I) \u2192 category (D i)] {F : (i : I) \u2192 C i \u2964 D i} {G : (i : I) \u2192 C i \u2964 D i}\n    (\u03b1 : (i : I) \u2192 F i \u27f6 G i) : functor.pi F \u27f6 functor.pi G :=\n  mk fun (f : (i : I) \u2192 C i) (i : I) => app (\u03b1 i) (f i)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/pi/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.35567148878130805}}
{"text": "import tactic.interactive\nimport algebra.group.basic\n\n/-!\n`refine_struct` caused a variety of interesting problems,\nwhich were identified in\nhttps://github.com/leanprover-community/mathlib/pull/2251\nand\nhttps://leanprover.zulipchat.com/#narrow/stream/113488-general/topic/Need.20help.20with.20class.20instance.20resolution\n\nThese tests are quite specific to testing the patch made in\nhttps://github.com/leanprover-community/mathlib/pull/2319\nand are not a complete test suite for `refine_struct`.\n-/\n\ninstance pi_has_one {\u03b1 : Type*} {\u03b2 : \u03b1 \u2192 Type*} [\u03a0 x, has_one (\u03b2 x)] : has_one (\u03a0 x, \u03b2 x) :=\nby refine_struct { .. }; exact \u03bb _, 1\n\nopen tactic\n\nrun_cmd (do\n  (declaration.defn _ _ _ b _ _) \u2190 get_decl ``pi_has_one,\n  -- Make sure that `eq.mpr` really doesn't occur in the body:\n  when (b.list_constant.contains ``eq.mpr) $\n    fail \"result generated by `refine_struct` contained an unnecessary `eq.mpr`\",\n  -- Make sure that `id` really doesn't occur in the body:\n  when (b.list_constant.contains ``id) $\n    fail \"result generated by `refine_struct` contained an unnecessary `id`\")\n\n-- Next we check that fields defined for embedded structures are unfolded\n-- when seen by fields in the outer structure.\nstructure foo (\u03b1 : Type):=\n(a : \u03b1)\n\nstructure bar (\u03b1 : Type) extends foo \u03b1 :=\n(b : a = a)\n\nexample : bar \u2115 :=\nbegin\n  refine_struct { a := 1, .. },\n  -- We're making sure that the goal is\n  -- \u22a2 1 = 1\n  -- rather than\n  -- \u22a2 {a := 1}.a = {a := 1}.a\n  guard_target 1 = 1,\n  trivial\nend\n\nsection\nvariables {\u03b1 : Type} [_inst : monoid \u03b1]\ninclude _inst\n\nexample : true :=\nbegin\n  have : group \u03b1,\n  { refine_struct { .._inst },\n    guard_tags _field inv group, admit,\n    guard_tags _field div group, admit,\n    guard_tags _field div_eq_mul_inv group, admit,\n    guard_tags _field zpow group, admit,\n    guard_tags _field zpow_zero' group, admit,\n    guard_tags _field zpow_succ' group, admit,\n    guard_tags _field zpow_neg' group, admit,\n    guard_tags _field mul_left_inv group, admit, },\n  trivial\nend\nend\n\ndef my_foo {\u03b1} (x : semigroup \u03b1) (y : group \u03b1) : true := trivial\n\nexample {\u03b1 : Type} : true :=\nbegin\n  have : true,\n  { refine_struct (@my_foo \u03b1 { .. } { .. } ),\n      -- 18 goals\n    guard_tags _field mul semigroup, admit,\n      -- case semigroup, mul\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\n    guard_tags _field mul_assoc semigroup, admit,\n      -- case semigroup, mul_assoc\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a b c : \u03b1), a * b * c = a * (b * c)\n\n    guard_tags _field mul group, admit,\n      -- case group, mul\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1 \u2192 \u03b1 \u2192 \u03b1\n\n    guard_tags _field mul_assoc group, admit,\n      -- case group, mul_assoc\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a b c : \u03b1), a * b * c = a * (b * c)\n\n    guard_tags _field one group, admit,\n      -- case group, one\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1\n\n    guard_tags _field one_mul group, admit,\n      -- case group, one_mul\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a : \u03b1), 1 * a = a\n\n    guard_tags _field mul_one group, admit,\n      -- case group, mul_one\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a : \u03b1), a * 1 = a\n\n    guard_tags _field npow group, admit,\n      -- case group, npow\n      -- \u03b1 : Type\n      -- \u22a2 \u2115 \u2192 \u03b1 \u2192 \u03b1\n\n    guard_tags _field npow_zero' group, admit,\n      -- case group, inv\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (x : \u03b1), sorry 0 x = 1\n\n    guard_tags _field npow_succ' group, admit,\n      -- case group, npow_succ'\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (n : \u2115) (x : \u03b1), sorry n.succ x = x * sorry n x\n\n    guard_tags _field inv group, admit,\n      -- case group, inv\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1 \u2192 \u03b1\n\n    guard_tags _field div group, admit,\n      -- case group, div\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1 \u2192 \u03b1\n\n    guard_tags _field div_eq_mul_inv group, admit,\n      -- case group, div_eq_mul_inv\n      -- \u03b1 : Type\n      -- \u22a2 \u03b1 \u2192 \u03b1\n\n    guard_tags _field zpow group, admit,\n      -- case group, zpow\n      -- \u03b1 : Type\n      -- \u22a2 \u2124 \u2192 \u03b1 \u2192 \u03b1\n\n    guard_tags _field zpow_zero' group, admit,\n      -- case group, zpow_zero'\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a : \u03b1), sorry 0 a = 1\n\n    guard_tags _field zpow_succ' group, admit,\n      -- case group, inv\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (n : \u2115) (a : \u03b1), sorry (int.of_nat n.succ) a = a * sorry (int.of_nat n) a\n\n    guard_tags _field zpow_neg' group, admit,\n      -- case group, inv\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (n : \u2115) (a : \u03b1), sorry -[1+ n] a = sorry (sorry \u2191(n.succ) a)\n\n    guard_tags _field mul_left_inv group, admit,\n      -- case group, mul_left_inv\n      -- \u03b1 : Type\n      -- \u22a2 \u2200 (a : \u03b1), a\u207b\u00b9 * a = 1\n  },\n  trivial\nend\n\ndef my_bar {\u03b1} (x : semigroup \u03b1) (y : group \u03b1) (i j : \u03b1) : \u03b1 := i\n\nexample {\u03b1 : Type} : true :=\nbegin\n  have : monoid \u03b1,\n  { refine_struct { mul := my_bar { .. } { .. } },\n    guard_tags _field mul semigroup, admit,\n    guard_tags _field mul_assoc semigroup, admit,\n    guard_tags _field mul group, admit,\n    guard_tags _field mul_assoc group, admit,\n    guard_tags _field one group, admit,\n    guard_tags _field one_mul group, admit,\n    guard_tags _field mul_one group, admit,\n    guard_tags _field npow group, admit,\n    guard_tags _field npow_zero' group, admit,\n    guard_tags _field npow_succ' group, admit,\n    guard_tags _field inv group, admit,\n    guard_tags _field div group, admit,\n    guard_tags _field div_eq_mul_inv group, admit,\n    guard_tags _field zpow group, admit,\n    guard_tags _field zpow_zero' group, admit,\n    guard_tags _field zpow_succ' group, admit,\n    guard_tags _field zpow_neg' group, admit,\n    guard_tags _field mul_left_inv group, admit,\n    guard_tags _field mul_assoc monoid, admit,\n    guard_tags _field one monoid, admit,\n    guard_tags _field one_mul monoid, admit,\n    guard_tags _field mul_one monoid, admit,\n    guard_tags _field npow monoid, admit,\n    guard_tags _field npow_zero' monoid, admit,\n    guard_tags _field npow_succ' monoid, admit, },\n  trivial\nend\n\ndef my_semigroup := semigroup\n\nexample {\u03b1} (mul : \u03b1 \u2192 \u03b1 \u2192 \u03b1) (h : false) : my_semigroup \u03b1 :=\nbegin\n  refine_struct { mul := mul, .. },\n  field mul_assoc {\n    guard_target \u2200 a b c : \u03b1, mul (mul a b) c = mul a (mul b c),\n    exact h.elim }\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/refine_struct.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.35567148878130794}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.topology.sheaves.presheaf\nimport Mathlib.category_theory.limits.punit\nimport Mathlib.category_theory.limits.shapes.products\nimport Mathlib.category_theory.limits.shapes.equalizers\nimport Mathlib.category_theory.full_subcategory\nimport Mathlib.PostPort\n\nuniverses u v \n\nnamespace Mathlib\n\n/-!\n# The sheaf condition in terms of an equalizer of products\n\nHere we set up the machinery for the \"usual\" definition of the sheaf condition,\ne.g. as in https://stacks.math.columbia.edu/tag/0072\nin terms of an equalizer diagram where the two objects are\n`\u220f F.obj (U i)` and `\u220f F.obj (U i) \u2293 (U j)`.\n\n-/\n\nnamespace Top\n\n\nnamespace presheaf\n\n\nnamespace sheaf_condition_equalizer_products\n\n\n/-- The product of the sections of a presheaf over a family of open sets. -/\n/--\ndef pi_opens {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : C :=\n  \u220f fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i))\n\nThe product of the sections of a presheaf over the pairwise intersections of\na family of open sets.\n-/\ndef pi_inters {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : C :=\n  \u220f fun (p : \u03b9 \u00d7 \u03b9) => category_theory.functor.obj F (opposite.op (U (prod.fst p) \u2293 U (prod.snd p)))\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U i` to `U i \u2293 U j`.\n-/\ndef left_res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : pi_opens F U \u27f6 pi_inters F U :=\n  category_theory.limits.pi.lift\n    fun (p : \u03b9 \u00d7 \u03b9) =>\n      category_theory.limits.pi.\u03c0 (fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i))) (prod.fst p) \u226b\n        category_theory.functor.map F\n          (category_theory.has_hom.hom.op (topological_space.opens.inf_le_left (U (prod.fst p)) (U (prod.snd p))))\n\n/--\nThe morphism `\u03a0 F.obj (U i) \u27f6 \u03a0 F.obj (U i) \u2293 (U j)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef right_res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : pi_opens F U \u27f6 pi_inters F U :=\n  category_theory.limits.pi.lift\n    fun (p : \u03b9 \u00d7 \u03b9) =>\n      category_theory.limits.pi.\u03c0 (fun (i : \u03b9) => category_theory.functor.obj F (opposite.op (U i))) (prod.snd p) \u226b\n        category_theory.functor.map F\n          (category_theory.has_hom.hom.op (topological_space.opens.inf_le_right (U (prod.fst p)) (U (prod.snd p))))\n\n/--\nThe morphism `F.obj U \u27f6 \u03a0 F.obj (U i)` whose components\nare given by the restriction maps from `U j` to `U i \u2293 U j`.\n-/\ndef res {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.functor.obj F (opposite.op (supr U)) \u27f6 pi_opens F U :=\n  category_theory.limits.pi.lift\n    fun (i : \u03b9) => category_theory.functor.map F (category_theory.has_hom.hom.op (topological_space.opens.le_supr U i))\n\ntheorem w {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : res F U \u226b left_res F U = res F U \u226b right_res F U := sorry\n\n/--\nThe equalizer diagram for the sheaf condition.\n-/\ndef diagram {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.limits.walking_parallel_pair \u2964 C :=\n  category_theory.limits.parallel_pair (left_res F U) (right_res F U)\n\n/--\nThe restriction map `F.obj U \u27f6 \u03a0 F.obj (U i)` gives a cone over the equalizer diagram\nfor the sheaf condition. The sheaf condition asserts this cone is a limit cone.\n-/\ndef fork {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.limits.fork (left_res F U) (right_res F U) :=\n  category_theory.limits.fork.of_\u03b9 (res F U) sorry\n\n@[simp] theorem fork_X {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.limits.cone.X (fork F U) = category_theory.functor.obj F (opposite.op (supr U)) :=\n  rfl\n\n@[simp] theorem fork_\u03b9 {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.limits.fork.\u03b9 (fork F U) = res F U :=\n  rfl\n\n@[simp] theorem fork_\u03c0_app_walking_parallel_pair_zero {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.nat_trans.app (category_theory.limits.cone.\u03c0 (fork F U))\n    category_theory.limits.walking_parallel_pair.zero =\n  res F U :=\n  rfl\n\n@[simp] theorem fork_\u03c0_app_walking_parallel_pair_one {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} (F : presheaf C X) {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) : category_theory.nat_trans.app (category_theory.limits.cone.\u03c0 (fork F U))\n    category_theory.limits.walking_parallel_pair.one =\n  res F U \u226b left_res F U :=\n  rfl\n\n/-- Isomorphic presheaves have isomorphic `pi_opens` for any cover `U`. -/\n@[simp] def pi_opens.iso_of_iso {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) : pi_opens F U \u2245 pi_opens G U :=\n  category_theory.limits.pi.map_iso fun (X_1 : \u03b9) => category_theory.iso.app \u03b1 (opposite.op (U X_1))\n\n/-- Isomorphic presheaves have isomorphic `pi_inters` for any cover `U`. -/\n@[simp] def pi_inters.iso_of_iso {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) : pi_inters F U \u2245 pi_inters G U :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9 \u00d7 \u03b9) => category_theory.iso.app \u03b1 (opposite.op (U (prod.fst X_1) \u2293 U (prod.snd X_1)))\n\n/-- Isomorphic presheaves have isomorphic sheaf condition diagrams. -/\ndef diagram.iso_of_iso {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) : diagram F U \u2245 diagram G U :=\n  category_theory.nat_iso.of_components\n    (fun (X_1 : category_theory.limits.walking_parallel_pair) =>\n      category_theory.limits.walking_parallel_pair.cases_on X_1 (pi_opens.iso_of_iso U \u03b1) (pi_inters.iso_of_iso U \u03b1))\n    sorry\n\n/--\nIf `F G : presheaf C X` are isomorphic presheaves,\nthen the `fork F U`, the canonical cone of the sheaf condition diagram for `F`,\nis isomorphic to `fork F G` postcomposed with the corresponding isomorphism between\nsheaf condition diagrams.\n-/\ndef fork.iso_of_iso {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} (U : \u03b9 \u2192 topological_space.opens \u21a5X) {G : presheaf C X} (\u03b1 : F \u2245 G) : fork F U \u2245\n  category_theory.functor.obj\n    (category_theory.limits.cones.postcompose (category_theory.iso.inv (diagram.iso_of_iso U \u03b1))) (fork G U) :=\n  category_theory.limits.fork.ext (category_theory.iso.app \u03b1 (opposite.op (supr U))) sorry\n\n/--\nPush forward a cover along an open embedding.\n-/\n@[simp] def cover.of_open_embedding {X : Top} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) : \u03b9 \u2192 topological_space.opens \u21a5X :=\n  fun (i : \u03b9) => category_theory.functor.obj (is_open_map.functor sorry) (\ud835\udcb0 i)\n\n/--\nThe isomorphism between `pi_opens` corresponding to an open embedding.\n-/\n@[simp] def pi_opens.iso_of_open_embedding {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) : pi_opens (category_theory.functor.op (is_open_map.functor (pi_opens.iso_of_open_embedding._proof_2 oe)) \u22d9 F) \ud835\udcb0 \u2245\n  pi_opens F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9) =>\n      category_theory.functor.map_iso F\n        (category_theory.iso.refl\n          (category_theory.functor.obj (category_theory.functor.op (is_open_map.functor sorry)) (opposite.op (\ud835\udcb0 X_1))))\n\n/--\nThe isomorphism between `pi_inters` corresponding to an open embedding.\n-/\n@[simp] def pi_inters.iso_of_open_embedding {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) : pi_inters (category_theory.functor.op (is_open_map.functor (pi_inters.iso_of_open_embedding._proof_2 oe)) \u22d9 F) \ud835\udcb0 \u2245\n  pi_inters F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.limits.pi.map_iso\n    fun (X_1 : \u03b9 \u00d7 \u03b9) =>\n      category_theory.functor.map_iso F\n        (id\n          (category_theory.iso.op\n            (category_theory.iso.mk (category_theory.hom_of_le sorry) (category_theory.hom_of_le sorry))))\n\n/-- The isomorphism of sheaf condition diagrams corresponding to an open embedding. -/\ndef diagram.iso_of_open_embedding {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) : diagram (category_theory.functor.op (is_open_map.functor (diagram.iso_of_open_embedding._proof_2 oe)) \u22d9 F) \ud835\udcb0 \u2245\n  diagram F (cover.of_open_embedding oe \ud835\udcb0) :=\n  category_theory.nat_iso.of_components\n    (fun (X_1 : category_theory.limits.walking_parallel_pair) =>\n      category_theory.limits.walking_parallel_pair.cases_on X_1 (pi_opens.iso_of_open_embedding oe \ud835\udcb0)\n        (pi_inters.iso_of_open_embedding oe \ud835\udcb0))\n    sorry\n\n/--\nIf `F : presheaf C X` is a presheaf, and `oe : U \u27f6 X` is an open embedding,\nthen the sheaf condition fork for a cover `\ud835\udcb0` in `U` for the composition of `oe` and `F` is\nisomorphic to sheaf condition fork for `oe '' \ud835\udcb0`, precomposed with the isomorphism\nof indexing diagrams `diagram.iso_of_open_embedding`.\n\nWe use this to show that the restriction of sheaf along an open embedding is still a sheaf.\n-/\ndef fork.iso_of_open_embedding {C : Type u} [category_theory.category C] [category_theory.limits.has_products C] {X : Top} {F : presheaf C X} {\u03b9 : Type v} {V : Top} {j : V \u27f6 X} (oe : open_embedding \u21d1j) (\ud835\udcb0 : \u03b9 \u2192 topological_space.opens \u21a5V) : fork (category_theory.functor.op (is_open_map.functor (fork.iso_of_open_embedding._proof_2 oe)) \u22d9 F) \ud835\udcb0 \u2245\n  category_theory.functor.obj\n    (category_theory.limits.cones.postcompose (category_theory.iso.inv (diagram.iso_of_open_embedding oe \ud835\udcb0)))\n    (fork F (cover.of_open_embedding oe \ud835\udcb0)) :=\n  category_theory.limits.fork.ext\n    (id\n      (category_theory.functor.map_iso F\n        (category_theory.iso.op\n          (category_theory.iso.mk (category_theory.hom_of_le sorry) (category_theory.hom_of_le sorry)))))\n    sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/topology/sheaves/sheaf_condition/equalizer_products.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850402140659, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.355649436830983}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.adjunction.basic\nimport category_theory.conj\nimport category_theory.yoneda\n\n/-!\n# Adjoints of fully faithful functors\n\nA left adjoint is fully faithful, if and only if the unit is an isomorphism\n(and similarly for right adjoints and the counit).\n\n`adjunction.restrict_fully_faithful` shows that an adjunction can be restricted along fully faithful\ninclusions.\n\n## Future work\n\nThe statements from Riehl 4.5.13 for adjoints which are either full, or faithful.\n-/\n\nopen category_theory\n\nnamespace category_theory\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nopen category\nopen opposite\n\nvariables {C : Type u\u2081} [category.{v\u2081} C]\nvariables {D : Type u\u2082} [category.{v\u2082} D]\nvariables {L : C \u2964 D} {R : D \u2964 C} (h : L \u22a3 R)\n\n/--\nIf the left adjoint is fully faithful, then the unit is an isomorphism.\n\nSee\n* Lemma 4.5.13 from [Riehl][riehl2017]\n* https://math.stackexchange.com/a/2727177\n* https://stacks.math.columbia.edu/tag/07RB (we only prove the forward direction!)\n-/\ninstance unit_is_iso_of_L_fully_faithful [full L] [faithful L] : is_iso (adjunction.unit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.unit h) $ \u03bb X,\n@yoneda.is_iso _ _ _ _ ((adjunction.unit h).app X)\n\u27e8\u27e8{ app := \u03bb Y f, L.preimage ((h.hom_equiv (unop Y) (L.obj X)).symm f) },\n  \u27e8begin\n    ext x f, dsimp,\n    apply L.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_counit, preimage_comp, preimage_map, category.assoc],\n    rw \u2190h.unit_naturality,\n    simp,\n  end\u27e9\u27e9\u27e9\n\n/--\nIf the right adjoint is fully faithful, then the counit is an isomorphism.\n\nSee <https://stacks.math.columbia.edu/tag/07RB> (we only prove the forward direction!)\n-/\ninstance counit_is_iso_of_R_fully_faithful [full R] [faithful R] : is_iso (adjunction.counit h) :=\n@nat_iso.is_iso_of_is_iso_app _ _ _ _ _ _ (adjunction.counit h) $ \u03bb X,\n@is_iso_of_op _ _ _ _ _ $\n@coyoneda.is_iso _ _ _ _ ((adjunction.counit h).app X).op\n\u27e8\u27e8{ app := \u03bb Y f, R.preimage ((h.hom_equiv (R.obj X) Y) f) },\n  \u27e8begin\n    ext x f, dsimp,\n    apply R.map_injective,\n    simp,\n  end, begin\n    ext x f, dsimp,\n    simp only [adjunction.hom_equiv_unit, preimage_comp, preimage_map],\n    rw \u2190h.counit_naturality,\n    simp,\n  end\u27e9\u27e9\u27e9\n\n/-- If the unit of an adjunction is an isomorphism, then its inverse on the image of L is given\nby L whiskered with the counit. -/\n@[simp]\nlemma inv_map_unit {X : C} [is_iso (h.unit.app X)] :\n  inv (L.map (h.unit.app X)) = h.counit.app (L.obj X) :=\nis_iso.inv_eq_of_hom_inv_id h.left_triangle_components\n\n/-- If the unit is an isomorphism, bundle one has an isomorphism `L \u22d9 R \u22d9 L \u2245 L`. -/\n@[simps]\nnoncomputable def whisker_left_L_counit_iso_of_is_iso_unit [is_iso h.unit] :\n  L \u22d9 R \u22d9 L \u2245 L :=\n(L.associator R L).symm \u226a\u226b iso_whisker_right (as_iso h.unit).symm L \u226a\u226b functor.left_unitor _\n\n/-- If the counit of an adjunction is an isomorphism, then its inverse on the image of R is given\nby R whiskered with the unit. -/\n@[simp]\nlemma inv_counit_map {X : D} [is_iso (h.counit.app X)] :\n  inv (R.map (h.counit.app X)) = h.unit.app (R.obj X) :=\nis_iso.inv_eq_of_inv_hom_id h.right_triangle_components\n\n/-- If the counit of an is an isomorphism, one has an isomorphism `(R \u22d9 L \u22d9 R) \u2245 R`. -/\n@[simps]\nnoncomputable def whisker_left_R_unit_iso_of_is_iso_counit [is_iso h.counit] :\n  (R \u22d9 L \u22d9 R) \u2245 R :=\n(R.associator L R).symm \u226a\u226b iso_whisker_right (as_iso h.counit) R \u226a\u226b functor.left_unitor _\n\n/-- If the unit is an isomorphism, then the left adjoint is full-/\nnoncomputable\ndef L_full_of_unit_is_iso [is_iso h.unit] : full L :=\n{ preimage := \u03bb X Y f, (h.hom_equiv X (L.obj Y) f) \u226b inv (h.unit.app Y) }\n\n/-- If the unit is an isomorphism, then the left adjoint is faithful-/\nlemma L_faithful_of_unit_is_iso [is_iso h.unit] : faithful L :=\n{ map_injective' := \u03bb X Y f g H,\n  begin\n    rw \u2190(h.hom_equiv X (L.obj Y)).apply_eq_iff_eq at H,\n    simpa using H =\u226b inv (h.unit.app Y),\n  end }\n\n/-- If the counit is an isomorphism, then the right adjoint is full-/\nnoncomputable\ndef R_full_of_counit_is_iso [is_iso h.counit] : full R :=\n{ preimage := \u03bb X Y f, inv (h.counit.app X) \u226b (h.hom_equiv (R.obj X) Y).symm f }\n\n/-- If the counit is an isomorphism, then the right adjoint is faithful-/\nlemma R_faithful_of_counit_is_iso [is_iso h.counit] : faithful R :=\n{ map_injective' := \u03bb X Y f g H,\n  begin\n    rw \u2190(h.hom_equiv (R.obj X) Y).symm.apply_eq_iff_eq at H,\n    simpa using inv (h.counit.app X) \u226b= H,\n  end }\n\ninstance whisker_left_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_left L h.counit) :=\nbegin\n  have := h.left_triangle,\n  rw \u2190is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_counit_iso_of_L_fully_faithful\n  [full L] [faithful L] : is_iso (whisker_right h.counit R) :=\nbegin\n  have := h.right_triangle,\n  rw \u2190is_iso.eq_inv_comp at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_left_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_left R h.unit) :=\nbegin\n  have := h.right_triangle,\n  rw \u2190is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\ninstance whisker_right_unit_iso_of_R_fully_faithful\n  [full R] [faithful R] : is_iso (whisker_right h.unit L) :=\nbegin\n  have := h.left_triangle,\n  rw \u2190is_iso.eq_comp_inv at this,\n  rw this,\n  apply_instance\nend\n\n-- TODO also do the statements from Riehl 4.5.13 for full and faithful separately?\n\nuniverses v\u2083 v\u2084 u\u2083 u\u2084\n\nvariables {C' : Type u\u2083} [category.{v\u2083} C']\nvariables {D' : Type u\u2084} [category.{v\u2084} D']\n\n-- TODO: This needs some lemmas describing the produced adjunction, probably in terms of `adj`,\n-- `iC` and `iD`.\n/--\nIf `C` is a full subcategory of `C'` and `D` is a full subcategory of `D'`, then we can restrict\nan adjunction `L' \u22a3 R'` where `L' : C' \u2964 D'` and `R' : D' \u2964 C'` to `C` and `D`.\nThe construction here is slightly more general, in that `C` is required only to have a full and\nfaithful \"inclusion\" functor `iC : C \u2964 C'` (and similarly `iD : D \u2964 D'`) which commute (up to\nnatural isomorphism) with the proposed restrictions.\n-/\ndef adjunction.restrict_fully_faithful (iC : C \u2964 C') (iD : D \u2964 D') {L' : C' \u2964 D'} {R' : D' \u2964 C'}\n  (adj : L' \u22a3 R') {L : C \u2964 D} {R : D \u2964 C} (comm1 : iC \u22d9 L' \u2245 L \u22d9 iD) (comm2 : iD \u22d9 R' \u2245 R \u22d9 iC)\n  [full iC] [faithful iC] [full iD] [faithful iD] :\n  L \u22a3 R :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb X Y,\n  calc (L.obj X \u27f6 Y) \u2243 (iD.obj (L.obj X) \u27f6 iD.obj Y) : equiv_of_fully_faithful iD\n       ... \u2243 (L'.obj (iC.obj X) \u27f6 iD.obj Y) : iso.hom_congr (comm1.symm.app X) (iso.refl _)\n       ... \u2243 (iC.obj X \u27f6 R'.obj (iD.obj Y)) : adj.hom_equiv _ _\n       ... \u2243 (iC.obj X \u27f6 iC.obj (R.obj Y)) : iso.hom_congr (iso.refl _) (comm2.app Y)\n       ... \u2243 (X \u27f6 R.obj Y) : (equiv_of_fully_faithful iC).symm,\n  hom_equiv_naturality_left_symm' := \u03bb X' X Y f g,\n  begin\n    apply iD.map_injective,\n    simpa using (comm1.inv.naturality_assoc f _).symm,\n  end,\n  hom_equiv_naturality_right' := \u03bb X Y' Y f g,\n  begin\n    apply iC.map_injective,\n    suffices : R'.map (iD.map g) \u226b comm2.hom.app Y = comm2.hom.app Y' \u226b iC.map (R.map g),\n      simp [this],\n    apply comm2.hom.naturality g,\n  end }\n\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/adjunction/fully_faithful.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.7057850278370112, "lm_q1q2_score": 0.3556494305941088}}
{"text": "-- author: Ben Sherman\n\nimport galois.network.network_implementation\n       galois.network.network_local_abs\n       galois.temporal.fixpoint\n       galois.temporal.classical\n       galois.temporal.LTS\n\nuniverses u v\n\n/-- reflexive-transitive closure of a relation -/\ninductive RTclosure {A : Type u} (R : A \u2192 A \u2192 Prop) (x : A) : A \u2192 Prop\n  | refl {} : RTclosure x\n  | step {} : \u2200 {y z : A}, R y z \u2192 RTclosure y \u2192 RTclosure z\n\ninductive RTclosure' {A : Type u} (R : A \u2192 A \u2192 Prop) (x z : A) : Prop\n  | refl {} : x = z \u2192 RTclosure'\n  | step {} : \u2200 {y : A}, R y z \u2192 RTclosure R x y \u2192 RTclosure'\n\nnamespace RTclosure\n\ndef stepL {A : Type u} {R : A \u2192 A \u2192 Prop} {x y : A} (r : R x y)\n  : \u2200 {z : A}, RTclosure R y z \u2192 RTclosure R x z\n:= begin\nintros z s, induction s, apply step, assumption, apply refl,\napply step; assumption,\nend\n\ndef invert {A : Type u} {R : A \u2192 A \u2192 Prop} {x z : A}\n  (r : RTclosure R x z) :\n  RTclosure' R x z\n:= begin\ninduction r, left, reflexivity,\nright, assumption, assumption\nend\n\nend RTclosure\n\nnamespace network\n\nopen temporal\n\nsection\nparameter {agents : map ip agent}\n\ndef indLabel {A} (P : agent_label \u2192 Prop)\n  := \u03bb a_next, P \u2218 @dlabel_to_label A a_next\n\n\ninstance inLabeld_decidable {ag : agent} (P : agent_label \u2192 Prop)\n  [decP : decidable_pred P]\n  : decidable_pred (@loc.inLabeld ag (indLabel P))\n:= begin\nintros x, dsimp [loc.inLabeld, indLabel, function.comp],\ninduction x, dsimp, apply decP\nend\n\ninductive sys_agent_does (ag : agents.member)\n  (P : agent_label \u2192 Prop)\n  : sigma sys_dlabel \u2192 Prop\n| mk : \u2200 sys dlabel, P (dlabel_to_label dlabel) \u2192 sys_agent_does (sigma.mk sys (sys_dlabel.mk ag dlabel))\n\n\n@[trace_map]\nlemma label_refine_eqd {ag : agents.member} (P : agent_label \u2192 Prop)\n : inSkipLabel (@loc.inLabeld ag.value (indLabel P)) \u2218\n       (Refinement.SL_refine (refinesd ag))\n = sys_agent_does ag P\n:= begin\napply funext, intros x, dsimp [function.comp],\napply propext, split; intros H,\n{\n  induction x, induction snd,\n  dsimp [inSkipLabel] at H,\n  dsimp [Refinement.SL_refine, refinesd, inSkipLabel] at H,\n  dsimp [sys_dlabel_to_local] at H,\n  apply (if Hag : ag_1 = ag then _ else _),\n  { subst ag_1, constructor,\n    rw (option.precondition_true_bind (eq.refl ag)) at H,\n    dsimp [sys_dlabel_to_local] at H,\n    dsimp [inSkipLabel] at H,\n    dsimp [loc.inLabeld] at H,\n    dsimp [indLabel] at H,\n    apply H },\n  { rw (option.precondition_false Hag) at H,\n    dsimp [has_bind.bind, option.bind] at H,\n    dsimp [inSkipLabel] at H, contradiction,\n  },\n},\n{ induction H,\n  dsimp [Refinement.SL_refine, refinesd, inSkipLabel],\n  dsimp [sys_dlabel_to_local],\n  rw (option.precondition_true_bind (eq.refl ag)),\n  dsimp [sys_dlabel_to_local], dsimp [inSkipLabel],\n  unfold loc.inLabeld, unfold indLabel, assumption\n}\nend\n\n\ninstance decidable_sys_agent_does (a P)\n  [decidable_pred P] : decidable_pred (sys_agent_does a P)\n:= begin\nintros l, induction l with s l,\ninduction l with a' l,\napply (if Hip : a = a' then _ else _),\n{ apply (if H : P (dlabel_to_label l) then _ else _),\n  subst a',\n  { apply decidable.is_true,\n    constructor, assumption } ,\n  { apply decidable.is_false,\n    intros contra, cases contra, contradiction }\n },\n{ apply decidable.is_false, intros contra,\n  cases contra, contradiction }\nend\n\ninductive next_state_from_label_ind' (ag : agents.member) (s : system_state) (la : dlabel ((ag.value).loop (s.local_state ag))) (s' : @system_state agents) : Prop\n| mk : \u2200 (new_state : (ag.value).state_type)\n         (updatef : global_state_t \u2192 global_state_t)\n         (Hagl : next_agent_state_from_dlabel (ag.key) (ag.value) (s.global_state (ag.key)) la = some (new_state, updatef))\n         (Hupd : s' = {local_state := lookup_update ag new_state (s.local_state), global_state := updatef (s.global_state)})\n         , next_state_from_label_ind'\n\n\nlemma agent_update_invert_st'\n  {ag : agents.member}\n  {s la s'}\n  : LTSd s (sys_dlabel.mk ag la) s'\n  \u2192 next_state_from_label_ind' ag s la s'\n:= begin\nintros H,\nsimp [LTSd] at H,\nsimp [next_state_from_dlabel] at H,\napply_in H option.bind_some',\ninduction H with res p, induction p with Hag1 Hag2,\ninduction res with new_state updatef,\ndsimp [next_agent_state_from_dlabel] at Hag1,\ndsimp [next_state_from_dlabel] at Hag2,\ninjection Hag2 with Hag2', clear Hag2,\nconstructor; try { assumption }, symmetry, assumption\nend\n\n/-- Every agent always eventually gets to step -/\ndef fairness_specd : @TP agents\n  := \u03bb tr, \u2200 (a : agents.member),\n   fair (now (sys_agent_does a (\u03bb _, true))) tr\n\n/-- Indicates that an agent is at the beginning of running\n    an iteration of its loop (or doing something equivalent\n    to that)\n-/\ndef starts_loop {a : agents.member} (next : act a.value.state_type) : Prop :=\n  \u2203 (s : a.value.state_type), next = a.value.loop s\n\ndef inLocalState (a : agents.member) (P : a.value.state_type \u2192 Prop)\n  {L : system_state \u2192 Type u}\n  : sigma L \u2192 Prop\n  := inState (\u03bb s, P (s.local_state a))\n\n/-- If a transition occurs that doesn't involve a particular agent,\n    that agent's state doesn't change.\n-/\nlemma local_state_stays_constant {s l s'}\n  (a : agents.member)\n  (HLTS : LTSd s l s')\n  (Hagent : \u00acsys_agent_does a (\u03bb (_x : agent_label), true) \u27e8 _, l \u27e9)\n  : s.local_state a = s'.local_state a\n:= begin\ninduction l with a' la,\napply_in HLTS agent_update_invert_st',\ninduction HLTS with new_state updatef H1 Hs',\nsubst s', dsimp,\nunfold lookup_update lookup_updatef,\napply (if Heq : a' = a then _ else _),\n{ exfalso, subst a',\n apply Hagent,\n constructor, trivial },\n{ rw (dif_neg Heq) }\nend\n\ndef agent_has_state {L : @system_state agents \u2192 Type u}\n  (ag : agents.member)\n  (P : ag.value.state_type \u2192 Prop)\n  : tProp (sigma L) := now (@inLocalState ag P L)\n\nlemma agent_has_state_refine_eq (ag : agents.member)\n  (P : ag.value.state_type \u2192 Prop) :\n  agent_has_state ag P =\n     now (inState P \u2218 Refinement.SL_refine (refinesd ag))\n:= begin\napply funext, intros x, reflexivity\nend\n\n/-- A statement of the fact that a particular agent's state\n    doesn't change (weak-) until it takes a step within\n    temporal logic.\n    (Using the sort of Leibniz equality: Any predicate `P`\n    that held on the old state will hold on the new state)\n-/\nlemma local_state_stays_constant_ltl (a : agents.member)\n  (P : a.value.state_type \u2192 Prop)\n  : \u22a9 valid_trace LTSd\n    => \u25a1 (now (inLocalState a P)\n    => (\u25ef (now (inLocalState a P))\n        \ud835\udce6\n       now (sys_agent_does a (\u03bb _, true)))\n    )\n:= begin\nintros tr validtr n Pst,\nunfold inLocalState,\napply (invariant_holds_while LTSd _ (delayn n tr)),\napply valid_trace_always, assumption, assumption,\napply_instance,\nintros,\nhave H := local_state_stays_constant _ a_1 a_2,\nrw \u2190 H, assumption,\nend\n\n/-- If an agent always eventually polls, and if it is sent a message,\n    then it eventually receives that message.\n-/\ndef message_fairness_specd : @TP agents := \u03bb tr,\n  \u2200 (a : agents.member) (sock : socket) (mess : message_t),\n    (fair (   now (inLocalState a (polls_on_socket sock \u2218 a.value.loop))\n            \u2229 now (sys_agent_does a (\u03bb _, true)))\n     => \u25a1 (now (inState (\u03bb s : system_state, (sock, mess) \u2208 (s.global_state a.key).messages))\n           => \u25c7 (now (sys_agent_does a (receives_message sock mess))))) tr\n\nend\n\nsection\nparameters {agents : map ip agent}\n  (a : agents.member)\n  (P : socket \u2192 message_t \u2192 Prop)\n\nlemma blocks_until_not_never_receives_always_polls\n  (s : socket)\n  : \u22a9 valid_trace (@LTSd agents)\n    => (\u25c7 (now (inLocalState a (polls_on_socket s \u2218 a.value.loop))\n           \u2229 now (sys_agent_does a (\u03bb _, true)))\n        \ud835\udce6 (now (sys_agent_does a (receives P))))\n    => \u25a1 (tNot (now (sys_agent_does a (receives P))))\n    => fair (now (inLocalState a (polls_on_socket s \u2218 a.value.loop))\n             \u2229 now (sys_agent_does a (\u03bb _, true)))\n:= begin\nintros tr valid nows never,\napply weak_until_not_always; assumption,\nend\n\n/--\n   As Joey and I discussed, the way this proof should go is as follows:\n   Use classical logic to do a proof by contradiction. Assume that we\n   never receive the message. Then,\n   use message fairness in conjunction with with the fact that we\n   have a message in the queue to derive that\n   H : if the agent always eventually polls, it eventually receives the message.\n   Next, use the fact that we do *not* eventually receive the message,\n   together with the fact that we block until we receive the message,\n   to derive that we always eventually poll.\n   Combining this with the `H` specified above, we find that we\n   eventually do receive the message, a contradiction.\n-/\ntheorem blocking_agent_eventually_receives_message\n  {s : socket}\n  : \u22a9 valid_trace (@LTSd agents)\n    => message_fairness_specd\n    => (\u25c7 (now (inLocalState a (polls_on_socket s \u2218 a.value.loop))\n          \u2229 now (sys_agent_does a (\u03bb _, true))))\n        \ud835\udce6 (now (sys_agent_does a (receives P)))\n    => now (inState (\u03bb ss : system_state, \u2203 mess : message_t,\n         P s mess \u2227\n         (s, mess) \u2208 (ss.global_state a.key).messages))\n    => \u25c7 (now (sys_agent_does a (receives P)))\n:= begin\nintros tr valid mfair nowstate H,\napply classical.not_always_not_implies_eventually,\nintros contra,\nhave H' := blocks_until_not_never_receives_always_polls\n  _ _ _ _ valid nowstate contra,\ninduction H with mess Hmess,\ninduction Hmess with Hmess1 Hmess2,\nspecialize (mfair a s mess H'),\nrw \u2190 (delayn_zero tr) at Hmess2,\nspecialize (mfair 0 Hmess2),\nrw \u2190 not_eventually_always_not at contra,\napply contra,\nrevert mfair, rw delayn_zero,\napply eventually_mono,\nintros x H,\nunfold now later inLabel at H,\nunfold now later inLabel,\ninduction H with s l Psl,\nconstructor, unfold receives_message at Psl,\ninduction Psl with t rn mess' ms H,\ninduction H with H1 H2, subst mess', subst s,\nconstructor, assumption,\nend\n\n/-- Agent fairness in the global transition system implies\n    \"skip fairness\" in the local system, where there is only\n    a single agent\n-/\nlemma fairness_Skip_impl (agents : map ip agent) (ag : agents.member) :\n \u22a9 fairness_specd => (fairness_SkipLTS \u2218 trace.map (Refinement.SL_refine (refinesd ag)))\n:= begin\nintros tr agent_fair,\nunfold fairness_SkipLTS,\ndsimp, rw fair_map, rw now_map,\nunfold fairness_specd at agent_fair,\nspecialize (agent_fair ag),\nrw \u2190 label_refine_eqd at agent_fair,\nassumption,\nend\n\nend\n\nend network", "meta": {"author": "GaloisInc", "repo": "lean-protocol-support", "sha": "cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda", "save_path": "github-repos/lean/GaloisInc-lean-protocol-support", "path": "github-repos/lean/GaloisInc-lean-protocol-support/lean-protocol-support-cabfa3abedbdd6fdca6e2da6fbbf91a13ed48dda/galois/network/agent_facts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3556494243572344}}
{"text": "import Kenny.sites.lattice\n\nuniverses v w u\n\nnamespace category_theory\n\ndef presheaf (C : Type u) [category.{v} C] : Type (max u v (w+1)) :=\nC\u1d52\u1d56 \u2964 Type w\n\nnamespace presheaf\n\nvariables {C : Type u} [category.{v} C] (F : presheaf.{v w} C)\n\ndef eval (U : C) : Type w :=\nF.1 (opposite.op U)\n\ndef res {U V : C} (f : U \u27f6 V) : F.eval V \u2192 F.eval U :=\nF.2 (has_hom.hom.op f)\n\n@[simp] lemma res_id (U : C) (s : F.eval U) : F.res (\ud835\udfd9 U) s = s :=\ncongr_fun (F.map_id (opposite.op U)) s\n\n@[simp] lemma res_res (U V W : C) (f : W \u27f6 V) (g : V \u27f6 U) (s : F.eval U) :\n  F.res f (F.res g s) = F.res (f \u226b g) s :=\n(congr_fun (F.map_comp (has_hom.hom.op g) (has_hom.hom.op f)) s).symm\n\nend presheaf\n\nstructure sheaf (C : Type u) [category.{v} C] [has_pullback C] [has_site.{v} C] : Type (max u v (w+1)) :=\n(to_presheaf : presheaf.{v w} C)\n(ext : \u2200 U : C, \u2200 s t : to_presheaf.eval U, \u2200 c \u2208 has_site.cov U,\n  (\u2200 d : \u03a3 V, V \u27f6 U, d \u2208 c \u2192 to_presheaf.res d.2 s = to_presheaf.res d.2 t) \u2192\n  s = t)\n(glue : \u2200 U : C, \u2200 c \u2208 has_site.cov U, \u2200 F : \u03a0 d : \u03a3 V, V \u27f6 U, d \u2208 c \u2192 to_presheaf.eval d.1,\n  (\u2200 d1 d2 : \u03a3 V, V \u27f6 U, \u2200 H1 : d1 \u2208 c, \u2200 H2 : d2 \u2208 c,\n    to_presheaf.res (pullback.fst d1.2 d2.2) (F d1 H1) =\n    to_presheaf.res (@@pullback.snd _ _inst_2 d1.2 d2.2) (F d2 H2)) \u2192\n  \u2203 g : to_presheaf.eval U, \u2200 d : \u03a3 V, V \u27f6 U, \u2200 H : d \u2208 c,\n    to_presheaf.res d.2 g = F d H)\n\nend category_theory\n", "meta": {"author": "ramonfmir", "repo": "lean-scheme", "sha": "6d3ec18fecfd174b79d0ce5c85a783f326dd50f6", "save_path": "github-repos/lean/ramonfmir-lean-scheme", "path": "github-repos/lean/ramonfmir-lean-scheme/lean-scheme-6d3ec18fecfd174b79d0ce5c85a783f326dd50f6/src/Kenny/sites/sheaf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.7057850154599562, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3556494243572344}}
{"text": "/-\nCopyright (c) 2020 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nimport Lean.ScopedEnvExtension\nimport Lean.Util.Recognizers\nimport Lean.Meta.DiscrTree\nimport Lean.Meta.AppBuilder\nimport Lean.Meta.Eqns\nimport Lean.Meta.Tactic.AuxLemma\nimport Lean.DocString\nnamespace Lean.Meta\n\n/--\nAn `Origin` is an identifier for simp theorems which indicates roughly\nwhat action the user took which lead to this theorem existing in the simp set.\n-/\ninductive Origin where\n  /-- A global declaration in the environment. -/\n  | decl (declName : Name)\n  /--\n  A local hypothesis.\n  When `contextual := true` is enabled, this fvar may exist in an extension\n  of the current local context; it will not be used for rewriting by simp once\n  it is out of scope but it may end up in the `usedSimps` trace.\n  -/\n  | fvar (fvarId : FVarId)\n  /--\n  A proof term provided directly to a call to `simp [ref, ...]` where `ref`\n  is the provided simp argument (of kind `Parser.Tactic.simpLemma`).\n  The `id` is a unique identifier for the call.\n  -/\n  | stx (id : Name) (ref : Syntax)\n  /--\n  Some other origin. `name` should not collide with the other types\n  for erasure to work correctly, and simp trace will ignore this lemma.\n  The other origins should be preferred if possible.\n  -/\n  | other (name : Name)\n  deriving Inhabited, Repr\n\n/-- A unique identifier corresponding to the origin. -/\ndef Origin.key : Origin \u2192 Name\n  | .decl declName => declName\n  | .fvar fvarId => fvarId.name\n  | .stx id _ => id\n  | .other name => name\n\ninstance : BEq Origin := \u27e8(\u00b7.key == \u00b7.key)\u27e9\ninstance : Hashable Origin := \u27e8(hash \u00b7.key)\u27e9\n\n/-\nNote: we want to use iota reduction when indexing instaces. Otherwise,\nwe cannot use simp theorems such as\n```\n@[simp] theorem liftOn_mk (a : \u03b1) (f : \u03b1 \u2192 \u03b3) (h : \u2200 a\u2081 a\u2082, r a\u2081 a\u2082 \u2192 f a\u2081 = f a\u2082) :\n    Quot.liftOn (Quot.mk r a) f h = f a := rfl\n```\nIf we use `iota`, then the lhs is reduced to `f a`.\nSee comment at `DiscrTree`.\n-/\n\nabbrev SimpTheoremKey := DiscrTree.Key (simpleReduce := true)\n\n/--\n  The fields `levelParams` and `proof` are used to encode the proof of the simp theorem.\n  If the `proof` is a global declaration `c`, we store `Expr.const c []` at `proof` without the universe levels, and `levelParams` is set to `#[]`\n  When using the lemma, we create fresh universe metavariables.\n  Motivation: most simp theorems are global declarations, and this approach is faster and saves memory.\n\n  The field `levelParams` is not empty only when we elaborate an expression provided by the user, and it contains universe metavariables.\n  Then, we use `abstractMVars` to abstract the universe metavariables and create new fresh universe parameters that are stored at the field `levelParams`.\n-/\nstructure SimpTheorem where\n  keys        : Array SimpTheoremKey := #[]\n  /--\n    It stores universe parameter names for universe polymorphic proofs.\n    Recall that it is non-empty only when we elaborate an expression provided by the user.\n    When `proof` is just a constant, we can use the universe parameter names stored in the declaration.\n   -/\n  levelParams : Array Name := #[]\n  proof       : Expr\n  priority    : Nat  := eval_prio default\n  post        : Bool := true\n  /-- `perm` is true if lhs and rhs are identical modulo permutation of variables. -/\n  perm        : Bool := false\n  /--\n    `origin` is mainly relevant for producing trace messages.\n    It is also viewed an `id` used to \"erase\" `simp` theorems from `SimpTheorems`.\n  -/\n  origin      : Origin\n  /-- `rfl` is true if `proof` is by `Eq.refl` or `rfl`. -/\n  rfl         : Bool\n  deriving Inhabited\n\nmutual\n  partial def isRflProofCore (type : Expr) (proof : Expr) : CoreM Bool := do\n    match type with\n    | .forallE _ _ type _ =>\n      if let .lam _ _ proof _ := proof then\n        isRflProofCore type proof\n      else\n        return false\n    | _ =>\n      if type.isAppOfArity ``Eq 3 then\n        if proof.isAppOfArity ``Eq.refl 2 || proof.isAppOfArity ``rfl 2 then\n          return true\n        else if proof.isAppOfArity ``Eq.symm 4 then\n          -- `Eq.symm` of rfl theorem is a rfl theorem\n          isRflProofCore type proof.appArg! -- small hack: we don't need to set the exact type\n        else if proof.isApp && proof.getAppFn.isConst then\n          -- The application of a `rfl` theorem is a `rfl` theorem\n          isRflTheorem proof.getAppFn.constName!\n        else\n          return false\n      else\n        return false\n\n  partial def isRflTheorem (declName : Name) : CoreM Bool := do\n    let .thmInfo info \u2190 getConstInfo declName | return false\n    isRflProofCore info.type info.value\nend\n\ndef isRflProof (proof : Expr) : MetaM Bool := do\n  if let .const declName .. := proof then\n    isRflTheorem declName\n  else\n    isRflProofCore (\u2190 inferType proof) proof\n\ninstance : ToFormat SimpTheorem where\n  format s :=\n    let perm := if s.perm then \":perm\" else \"\"\n    let name := format s.origin.key\n    let prio := f!\":{s.priority}\"\n    name ++ prio ++ perm\n\ndef ppOrigin [Monad m] [MonadEnv m] [MonadError m] : Origin \u2192 m MessageData\n  | .decl n => mkConstWithLevelParams n\n  | .fvar n => return mkFVar n\n  | .stx _ ref => return ref\n  | .other n => return n\n\ndef ppSimpTheorem [Monad m] [MonadLiftT IO m] [MonadEnv m] [MonadError m] (s : SimpTheorem) : m MessageData := do\n  let perm := if s.perm then \":perm\" else \"\"\n  let name \u2190 ppOrigin s.origin\n  let prio := m!\":{s.priority}\"\n  return name ++ prio ++ perm\n\ninstance : BEq SimpTheorem where\n  beq e\u2081 e\u2082 := e\u2081.proof == e\u2082.proof\n\nabbrev SimpTheoremTree := DiscrTree SimpTheorem (simpleReduce := true)\n\nstructure SimpTheorems where\n  pre          : SimpTheoremTree := DiscrTree.empty\n  post         : SimpTheoremTree := DiscrTree.empty\n  lemmaNames   : PHashSet Origin := {}\n  toUnfold     : PHashSet Name := {}\n  erased       : PHashSet Origin := {}\n  toUnfoldThms : PHashMap Name (Array Name) := {}\n  deriving Inhabited\n\ndef addSimpTheoremEntry (d : SimpTheorems) (e : SimpTheorem) : SimpTheorems :=\n  if e.post then\n    { d with post := d.post.insertCore e.keys e, lemmaNames := updateLemmaNames d.lemmaNames }\n  else\n    { d with pre := d.pre.insertCore e.keys e, lemmaNames := updateLemmaNames d.lemmaNames }\nwhere\n  updateLemmaNames (s : PHashSet Origin) : PHashSet Origin :=\n    s.insert e.origin\n\ndef SimpTheorems.addDeclToUnfoldCore (d : SimpTheorems) (declName : Name) : SimpTheorems :=\n  { d with toUnfold := d.toUnfold.insert declName }\n\n/-- Return `true` if `declName` is tagged to be unfolded using `unfoldDefinition?` (i.e., without using equational theorems). -/\ndef SimpTheorems.isDeclToUnfold (d : SimpTheorems) (declName : Name) : Bool :=\n  d.toUnfold.contains declName\n\ndef SimpTheorems.isLemma (d : SimpTheorems) (thmId : Origin) : Bool :=\n  d.lemmaNames.contains thmId\n\n/-- Register the equational theorems for the given definition. -/\ndef SimpTheorems.registerDeclToUnfoldThms (d : SimpTheorems) (declName : Name) (eqThms : Array Name) : SimpTheorems :=\n  { d with toUnfoldThms := d.toUnfoldThms.insert declName eqThms }\n\npartial def SimpTheorems.eraseCore (d : SimpTheorems) (thmId : Origin) : SimpTheorems :=\n  let d := { d with erased := d.erased.insert thmId, lemmaNames := d.lemmaNames.erase thmId }\n  if let .decl declName := thmId then\n    let d := { d with toUnfold := d.toUnfold.erase declName }\n    if let some thms := d.toUnfoldThms.find? declName then\n      thms.foldl (init := d) (eraseCore \u00b7 <| .decl \u00b7)\n    else\n      d\n  else\n    d\n\ndef SimpTheorems.erase [Monad m] [MonadError m] (d : SimpTheorems) (thmId : Origin) : m SimpTheorems := do\n  unless d.isLemma thmId ||\n    match thmId with\n    | .decl declName => d.isDeclToUnfold declName || d.toUnfoldThms.contains declName\n    | _ => false\n  do\n    throwError \"'{thmId.key}' does not have [simp] attribute\"\n  return d.eraseCore thmId\n\nprivate partial def isPerm : Expr \u2192 Expr \u2192 MetaM Bool\n  | Expr.app f\u2081 a\u2081, Expr.app f\u2082 a\u2082 => isPerm f\u2081 f\u2082 <&&> isPerm a\u2081 a\u2082\n  | Expr.mdata _ s, t => isPerm s t\n  | s, Expr.mdata _ t => isPerm s t\n  | s@(Expr.mvar ..), t@(Expr.mvar ..) => isDefEq s t\n  | Expr.forallE n\u2081 d\u2081 b\u2081 _, Expr.forallE _ d\u2082 b\u2082 _ => isPerm d\u2081 d\u2082 <&&> withLocalDeclD n\u2081 d\u2081 fun x => isPerm (b\u2081.instantiate1 x) (b\u2082.instantiate1 x)\n  | Expr.lam n\u2081 d\u2081 b\u2081 _, Expr.lam _ d\u2082 b\u2082 _ => isPerm d\u2081 d\u2082 <&&> withLocalDeclD n\u2081 d\u2081 fun x => isPerm (b\u2081.instantiate1 x) (b\u2082.instantiate1 x)\n  | Expr.letE n\u2081 t\u2081 v\u2081 b\u2081 _, Expr.letE _  t\u2082 v\u2082 b\u2082 _ =>\n    isPerm t\u2081 t\u2082 <&&> isPerm v\u2081 v\u2082 <&&> withLetDecl n\u2081 t\u2081 v\u2081 fun x => isPerm (b\u2081.instantiate1 x) (b\u2082.instantiate1 x)\n  | Expr.proj _ i\u2081 b\u2081, Expr.proj _ i\u2082 b\u2082 => pure (i\u2081 == i\u2082) <&&> isPerm b\u2081 b\u2082\n  | s, t => return s == t\n\nprivate def checkBadRewrite (lhs rhs : Expr) : MetaM Unit := do\n  let lhs \u2190 DiscrTree.reduceDT lhs (root := true) (simpleReduce := true)\n  if lhs == rhs && lhs.isFVar then\n    throwError \"invalid `simp` theorem, equation is equivalent to{indentExpr (\u2190 mkEq lhs rhs)}\"\n\nprivate partial def shouldPreprocess (type : Expr) : MetaM Bool :=\n  forallTelescopeReducing type fun _ result => do\n    if let some (_, lhs, rhs) := result.eq? then\n      checkBadRewrite lhs rhs\n      return false\n    else\n      return true\n\nprivate partial def preprocess (e type : Expr) (inv : Bool) (isGlobal : Bool) : MetaM (List (Expr \u00d7 Expr)) :=\n  go e type\nwhere\n  go (e type : Expr) : MetaM (List (Expr \u00d7 Expr)) := do\n  let type \u2190 whnf type\n  if type.isForall then\n    forallTelescopeReducing type fun xs type => do\n      let e := mkAppN e xs\n      let ps \u2190 go e type\n      ps.mapM fun (e, type) =>\n        return (\u2190 mkLambdaFVars xs e, \u2190 mkForallFVars xs type)\n  else if let some (_, lhs, rhs) := type.eq? then\n    if isGlobal then\n      checkBadRewrite lhs rhs\n    if inv then\n      let type \u2190 mkEq rhs lhs\n      let e    \u2190 mkEqSymm e\n      return [(e, type)]\n    else\n      return [(e, type)]\n  else if let some (lhs, rhs) := type.iff? then\n    if isGlobal then\n      checkBadRewrite lhs rhs\n    if inv then\n      let type \u2190 mkEq rhs lhs\n      let e    \u2190 mkEqSymm (\u2190 mkPropExt e)\n      return [(e, type)]\n    else\n      let type \u2190 mkEq lhs rhs\n      let e    \u2190 mkPropExt e\n      return [(e, type)]\n  else if let some (_, lhs, rhs) := type.ne? then\n    if inv then\n      throwError \"invalid '\u2190' modifier in rewrite rule to 'False'\"\n    if rhs.isConstOf ``Bool.true then\n      return [(\u2190 mkAppM ``Bool.of_not_eq_true #[e], \u2190 mkEq lhs (mkConst ``Bool.false))]\n    else if rhs.isConstOf ``Bool.false then\n      return [(\u2190 mkAppM ``Bool.of_not_eq_false #[e], \u2190 mkEq lhs (mkConst ``Bool.true))]\n    let type \u2190 mkEq (\u2190 mkEq lhs rhs) (mkConst ``False)\n    let e    \u2190 mkEqFalse e\n    return [(e, type)]\n  else if let some p := type.not? then\n    if inv then\n      throwError \"invalid '\u2190' modifier in rewrite rule to 'False'\"\n    if let some (_, lhs, rhs) := p.eq? then\n      if rhs.isConstOf ``Bool.true then\n        return [(\u2190 mkAppM ``Bool.of_not_eq_true #[e], \u2190 mkEq lhs (mkConst ``Bool.false))]\n      else if rhs.isConstOf ``Bool.false then\n        return [(\u2190 mkAppM ``Bool.of_not_eq_false #[e], \u2190 mkEq lhs (mkConst ``Bool.true))]\n    let type \u2190 mkEq p (mkConst ``False)\n    let e    \u2190 mkEqFalse e\n    return [(e, type)]\n  else if let some (type\u2081, type\u2082) := type.and? then\n    let e\u2081 := mkProj ``And 0 e\n    let e\u2082 := mkProj ``And 1 e\n    return (\u2190 go e\u2081 type\u2081) ++ (\u2190 go e\u2082 type\u2082)\n  else\n    if inv then\n      throwError \"invalid '\u2190' modifier in rewrite rule to 'True'\"\n    let type \u2190 mkEq type (mkConst ``True)\n    let e    \u2190 mkEqTrue e\n    return [(e, type)]\n\nprivate def checkTypeIsProp (type : Expr) : MetaM Unit :=\n  unless (\u2190 isProp type) do\n    throwError \"invalid 'simp', proposition expected{indentExpr type}\"\n\nprivate def mkSimpTheoremCore (origin : Origin) (e : Expr) (levelParams : Array Name) (proof : Expr) (post : Bool) (prio : Nat) : MetaM SimpTheorem := do\n  assert! origin != .fvar \u27e8.anonymous\u27e9\n  let type \u2190 instantiateMVars (\u2190 inferType e)\n  withNewMCtxDepth do\n    let (_, _, type) \u2190 withReducible <| forallMetaTelescopeReducing type\n    let type \u2190 whnfR type\n    let (keys, perm) \u2190\n      match type.eq? with\n      | some (_, lhs, rhs) => pure (\u2190 DiscrTree.mkPath lhs, \u2190 isPerm lhs rhs)\n      | none => throwError \"unexpected kind of 'simp' theorem{indentExpr type}\"\n    return { origin, keys, perm, post, levelParams, proof, priority := prio, rfl := (\u2190 isRflProof proof) }\n\nprivate def mkSimpTheoremsFromConst (declName : Name) (post : Bool) (inv : Bool) (prio : Nat) : MetaM (Array SimpTheorem) := do\n  let cinfo \u2190 getConstInfo declName\n  let val := mkConst declName (cinfo.levelParams.map mkLevelParam)\n  withReducible do\n    let type \u2190 inferType val\n    checkTypeIsProp type\n    if inv || (\u2190 shouldPreprocess type) then\n      let mut r := #[]\n      for (val, type) in (\u2190 preprocess val type inv (isGlobal := true)) do\n        let auxName \u2190 mkAuxLemma cinfo.levelParams type val\n        r := r.push <| (\u2190 mkSimpTheoremCore (.decl declName) (mkConst auxName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst auxName) post prio)\n      return r\n    else\n      return #[\u2190 mkSimpTheoremCore (.decl declName) (mkConst declName (cinfo.levelParams.map mkLevelParam)) #[] (mkConst declName) post prio]\n\ninductive SimpEntry where\n  | thm      : SimpTheorem \u2192 SimpEntry\n  | toUnfold : Name \u2192 SimpEntry\n  | toUnfoldThms : Name \u2192 Array Name \u2192 SimpEntry\n  deriving Inhabited\n\nabbrev SimpExtension := SimpleScopedEnvExtension SimpEntry SimpTheorems\n\ndef SimpExtension.getTheorems (ext : SimpExtension) : CoreM SimpTheorems :=\n  return ext.getState (\u2190 getEnv)\n\ndef addSimpTheorem (ext : SimpExtension) (declName : Name) (post : Bool) (inv : Bool) (attrKind : AttributeKind) (prio : Nat) : MetaM Unit := do\n  let simpThms \u2190 mkSimpTheoremsFromConst declName post inv prio\n  for simpThm in simpThms do\n    ext.add (SimpEntry.thm simpThm) attrKind\n\ndef mkSimpAttr (attrName : Name) (attrDescr : String) (ext : SimpExtension)\n    (ref : Name := by exact decl_name%) : IO Unit :=\n  registerBuiltinAttribute {\n    ref   := ref\n    name  := attrName\n    descr := attrDescr\n    applicationTime := AttributeApplicationTime.afterCompilation\n    add   := fun declName stx attrKind =>\n      let go : MetaM Unit := do\n        let info \u2190 getConstInfo declName\n        let post := if stx[1].isNone then true else stx[1][0].getKind == ``Lean.Parser.Tactic.simpPost\n        let prio \u2190 getAttrParamOptPrio stx[2]\n        if (\u2190 isProp info.type) then\n          addSimpTheorem ext declName post (inv := false) attrKind prio\n        else if info.hasValue then\n          if let some eqns \u2190 getEqnsFor? declName then\n            for eqn in eqns do\n              addSimpTheorem ext eqn post (inv := false) attrKind prio\n            ext.add (SimpEntry.toUnfoldThms declName eqns) attrKind\n            if hasSmartUnfoldingDecl (\u2190 getEnv) declName then\n              ext.add (SimpEntry.toUnfold declName) attrKind\n          else\n            ext.add (SimpEntry.toUnfold declName) attrKind\n        else\n          throwError \"invalid 'simp', it is not a proposition nor a definition (to unfold)\"\n      discard <| go.run {} {}\n    erase := fun declName => do\n      let s := ext.getState (\u2190 getEnv)\n      let s \u2190 s.erase (.decl declName)\n      modifyEnv fun env => ext.modifyState env fun _ => s\n  }\n\ndef mkSimpExt (name : Name := by exact decl_name%) : IO SimpExtension :=\n  registerSimpleScopedEnvExtension {\n    name     := name\n    initial  := {}\n    addEntry := fun d e =>\n      match e with\n      | SimpEntry.thm e => addSimpTheoremEntry d e\n      | SimpEntry.toUnfold n => d.addDeclToUnfoldCore n\n      | SimpEntry.toUnfoldThms n thms => d.registerDeclToUnfoldThms n thms\n  }\n\nabbrev SimpExtensionMap := HashMap Name SimpExtension\n\nbuiltin_initialize simpExtensionMapRef : IO.Ref SimpExtensionMap \u2190 IO.mkRef {}\n\ndef registerSimpAttr (attrName : Name) (attrDescr : String)\n    (ref : Name := by exact decl_name%) : IO SimpExtension := do\n  let ext \u2190 mkSimpExt ref\n  mkSimpAttr attrName attrDescr ext ref -- Remark: it will fail if it is not performed during initialization\n  simpExtensionMapRef.modify fun map => map.insert attrName ext\n  return ext\n\nbuiltin_initialize simpExtension : SimpExtension \u2190 registerSimpAttr `simp \"simplification theorem\"\n\ndef getSimpExtension? (attrName : Name) : IO (Option SimpExtension) :=\n  return (\u2190 simpExtensionMapRef.get).find? attrName\n\ndef getSimpTheorems : CoreM SimpTheorems :=\n  simpExtension.getTheorems\n\n/-- Auxiliary method for adding a global declaration to a `SimpTheorems` datastructure. -/\ndef SimpTheorems.addConst (s : SimpTheorems) (declName : Name) (post := true) (inv := false) (prio : Nat := eval_prio default) : MetaM SimpTheorems := do\n  let s := { s with erased := s.erased.erase (.decl declName) }\n  let simpThms \u2190 mkSimpTheoremsFromConst declName post inv prio\n  return simpThms.foldl addSimpTheoremEntry s\n\ndef SimpTheorem.getValue (simpThm : SimpTheorem) : MetaM Expr := do\n  if simpThm.proof.isConst && simpThm.levelParams.isEmpty then\n    let info \u2190 getConstInfo simpThm.proof.constName!\n    if info.levelParams.isEmpty then\n      return simpThm.proof\n    else\n      return simpThm.proof.updateConst! (\u2190 info.levelParams.mapM (fun _ => mkFreshLevelMVar))\n  else\n    let us \u2190 simpThm.levelParams.mapM fun _ => mkFreshLevelMVar\n    return simpThm.proof.instantiateLevelParamsArray simpThm.levelParams us\n\nprivate def preprocessProof (val : Expr) (inv : Bool) : MetaM (Array Expr) := do\n  let type \u2190 inferType val\n  checkTypeIsProp type\n  let ps \u2190 preprocess val type inv (isGlobal := false)\n  return ps.toArray.map fun (val, _) => val\n\n/-- Auxiliary method for creating simp theorems from a proof term `val`. -/\ndef mkSimpTheorems (id : Origin) (levelParams : Array Name) (proof : Expr) (post := true) (inv := false) (prio : Nat := eval_prio default) : MetaM (Array SimpTheorem) :=\n  withReducible do\n    (\u2190 preprocessProof proof inv).mapM fun val => mkSimpTheoremCore id val levelParams val post prio\n\n/-- Auxiliary method for adding a local simp theorem to a `SimpTheorems` datastructure. -/\ndef SimpTheorems.add (s : SimpTheorems) (id : Origin) (levelParams : Array Name) (proof : Expr) (inv := false) (post := true) (prio : Nat := eval_prio default) : MetaM SimpTheorems := do\n  if proof.isConst then\n    s.addConst proof.constName! post inv prio\n  else\n    let simpThms \u2190 mkSimpTheorems id levelParams proof post inv prio\n    return simpThms.foldl addSimpTheoremEntry s\n\ndef SimpTheorems.addDeclToUnfold (d : SimpTheorems) (declName : Name) : MetaM SimpTheorems := do\n  if let some eqns \u2190 getEqnsFor? declName then\n    let mut d := d\n    for eqn in eqns do\n      d \u2190 SimpTheorems.addConst d eqn\n    if hasSmartUnfoldingDecl (\u2190 getEnv) declName then\n      d := d.addDeclToUnfoldCore declName\n    return d\n  else\n    return d.addDeclToUnfoldCore declName\n\nabbrev SimpTheoremsArray := Array SimpTheorems\n\ndef SimpTheoremsArray.addTheorem (thmsArray : SimpTheoremsArray) (id : Origin) (h : Expr) : MetaM SimpTheoremsArray :=\n  if thmsArray.isEmpty then\n    let thms : SimpTheorems := {}\n    return #[ (\u2190 thms.add id #[] h) ]\n  else\n    thmsArray.modifyM 0 fun thms => thms.add id #[] h\n\ndef SimpTheoremsArray.eraseTheorem (thmsArray : SimpTheoremsArray) (thmId : Origin) : SimpTheoremsArray :=\n  thmsArray.map fun thms => thms.eraseCore thmId\n\ndef SimpTheoremsArray.isErased (thmsArray : SimpTheoremsArray) (thmId : Origin) : Bool :=\n  thmsArray.any fun thms => thms.erased.contains thmId\n\ndef SimpTheoremsArray.isDeclToUnfold (thmsArray : SimpTheoremsArray) (declName : Name) : Bool :=\n  thmsArray.any fun thms => thms.isDeclToUnfold declName\n\nmacro (name := _root_.Lean.Parser.Command.registerSimpAttr) doc:(docComment)?\n  \"register_simp_attr\" id:ident : command => do\n  let str := id.getId.toString\n  let idParser := mkIdentFrom id (`Parser.Attr ++ id.getId)\n  let descr := quote (removeLeadingSpaces (doc.map (\u00b7.getDocString) |>.getD s!\"simp set for {id.getId.toString}\"))\n  `($[$doc:docComment]? initialize ext : SimpExtension \u2190 registerSimpAttr $(quote id.getId) $descr $(quote id.getId)\n    $[$doc:docComment]? syntax (name := $idParser:ident) $(quote str):str (Parser.Tactic.simpPre <|> Parser.Tactic.simpPost)? (prio)? : attr)\n\nend Meta\n\nend Lean\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/src/Lean/Meta/Tactic/Simp/SimpTheorems.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.35564889260590243}}
{"text": "import combinatorics.simple_graph.coloring\n\n\n--OUTPUT 1\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  assume h1 : G.colorable 2,\n  have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from by {\n    obtain \u27e8f,hf\u27e9 := h1,\n    have h3 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), \u2200 (a b : A \u2295 B), (a \u2260 b) \u2192 (f a \u2260 f b), from by {\n      have h4 : \u2200 (a b : A \u2295 B), (a \u2260 b) \u2192 (f a \u2260 f b), from by {\n        assume a b : A \u2295 B,\n        assume h5 : a \u2260 b,\n        assume h6 : f a = f b,\n        have h7 : a = b, from by {\n          cases a, cases b,\n          {rw \u2190 h6, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf, rw hf\nend --Needs more than 2000 tokens!\n\n--OUTPUT 2\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h : G.colorable 2,\n    have h1 : \u2203 A B : Type*, (A \u2295 B) = V, from by {\n      have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205, from by {\n        have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V, from by {\n          have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V), from by {\n            have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V), from by {\n              have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V), from by {\n                have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                  have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                    have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                      have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                        have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                          have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V \u2295 V), from by {\n                            have h1 : \u2203 A B : Type*, (A \u2295 B) = V \u2227 (A \u2295 B) \u2260 \u2205 \u2227 (A \u2295 B) \u2260 V \u2227 (A \u2295 B) \u2260 (V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V) \u2227 (A \u2295 B) \u2260 (V \u2295 V \u2295 V \u2295 V) \u2227 (A \u2295 B)\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nbegin\n  split,\n  {\n    assume h1 : G.colorable 2,\n    have h2 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n    begin\n      have h3 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n      begin\n        have h4 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n        begin\n          have h5 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n          begin\n            have h6 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n            begin\n              have h7 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n              begin\n                have h8 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                begin\n                  have h9 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                  begin\n                    have h10 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                    begin\n                      have h11 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                      begin\n                        have h12 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                        begin\n                          have h13 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                          begin\n                            have h14 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                            begin\n                              have h15 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                              begin\n                                have h16 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                begin\n                                  have h17 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                  begin\n                                    have h18 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                    begin\n                                      have h19 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                      begin\n                                        have h20 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                        begin\n                                          have h21 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                          begin\n                                            have h22 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                            begin\n                                              have h23 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                              begin\n                                                have h24 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                begin\n                                                  have h25 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                  begin\n                                                    have h26 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                    begin\n                                                      have h27 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                      begin\n                                                        have h28 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                        begin\n                                                          have h29 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                          begin\n                                                            have h30 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                            begin\n                                                              have h31 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                              begin\n                                                                have h32 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                                begin\n                                                                  have h33 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                                  begin\n                                                                    have h34 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                                    begin\n                                                                      have h35 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B), from\n                                                                      begin\n                                                                        have h36 : \u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B),\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nBipartite Graph is two colorable\nLet $G$ be a graph. Then $G$ is 2-colorable if and only if $G$ is bipartite.\n`proof`\nLet $G$ be a 2-colorable graph, which means we can color every vertex either red or blue, and no edge will have both endpoints colored the same color. Let $A$ denote the subset of vertices colored red, and let $B$ denote the subset of vertices colored blue. Since all vertices of $A$ are red, there are no edges within $A$, and similarly for $B$. This implies that every edge has one endpoint in $A$ and the other in $B$, which means $G$ is bipartite.\n\nConversely, suppose $G$ is bipartite, that is, we can partition the vertices into two subsets $V_{1}, V_{2}$ every edge has one endpoint in $V_{1}$ and the other in $V_{2}$. Then coloring every vertex of $V_{1}$ red and every vertex of $V_{2}$ blue yields a valid coloring, so $G$ is 2-colorable.\n\nQED\n\n-/\ntheorem  bipartite_iff_two_colorable {V : Type*} (G : simple_graph V) [fintype V] : (G.colorable 2) \u2194 (\u2203 (A B : Type*) (h : (A \u2295 B) = V), G \u2264 cast (congr_arg _ h) (complete_bipartite_graph A B)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof-Natural-Language-Proof-Translation/Correct_statement-lean_proof-3_few_shot_temperature_0.2_max_tokens_2000_n_3/clean_files/Bipartite Graph is two colorable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3555928783092906}}
{"text": "/-\nCopyright (c) 2020 Jannis Limperg. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jannis Limperg\n\n! This file was ported from Lean 3 source module tactic.unify_equations\n! leanprover-community/mathlib commit f694c7dead66f5d4c80f446c796a5aad14707f0e\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Core\n\n/-!\n# The `unify_equations` tactic\n\nThis module defines `unify_equations`, a first-order unification tactic that\nunifies one or more equations in the context. It implements the Qnify algorithm\nfrom [McBride, Inverting Inductively Defined Relations in LEGO][mcbride1996].\n\nThe tactic takes as input some equations which it simplifies one after the\nother. Each equation is simplified by applying one of several possible\nunification steps. Each such step may output other (simpler) equations which are\nunified recursively until no unification step applies any more. See\n`tactic.interactive.unify_equations` for an example and an explanation of the\ndifferent steps.\n-/\n\n\nopen Expr\n\nnamespace Tactic\n\nnamespace UnifyEquations\n\n/-- The result of a unification step:\n\n- `simplified hs` means that the step succeeded and produced some new (simpler)\n  equations `hs`. `hs` can be empty.\n- `goal_solved` means that the step succeeded and solved the goal (by deriving a\n  contradiction from the given equation).\n- `not_simplified` means that the step failed to simplify the equation.\n-/\nunsafe inductive unification_step_result : Type\n  | simplified (next_equations : List Name)\n  | not_simplified\n  | goal_solved\n#align tactic.unify_equations.unification_step_result tactic.unify_equations.unification_step_result\n\nexport UnificationStepResult ()\n\n/-- A unification step is a tactic that attempts to simplify a given equation and\nreturns a `unification_step_result`. The inputs are:\n\n- `equ`, the equation being processed. Must be a local constant.\n- `lhs_type` and `rhs_type`, the types of equ's LHS and RHS. For homogeneous\n  equations, these are defeq.\n- `lhs` and `rhs`, `equ`'s LHS and RHS.\n- `lhs_whnf` and `rhs_whnf`, `equ`'s LHS and RHS in WHNF.\n- `u`, `equ`'s level.\n\nSo `equ : @eq.{u} lhs_type lhs rhs` or `equ : @heq.{u} lhs_type lhs rhs_type rhs`.\n-/\n@[reducible]\nunsafe def unification_step : Type :=\n  \u2200 (equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf : expr) (u : level),\n    tactic unification_step_result\n#align tactic.unify_equations.unification_step tactic.unify_equations.unification_step\n\n/-- For `equ : t == u` with `t : T` and `u : U`, if `T` and `U` are defeq,\nwe replace `equ` with `equ : t = u`.\n-/\nunsafe def unify_heterogeneous : unification_step := fun equ lhs_type rhs_type lhs rhs _ _ _ =>\n  (do\n      is_def_eq lhs_type rhs_type\n      let p \u2190 to_expr ``(@eq_of_hEq $(lhs_type) $(lhs) $(rhs) $(equ))\n      let t \u2190 to_expr ``(@Eq $(lhs_type) $(lhs) $(rhs))\n      let equ' \u2190 note equ.local_pp_name t p\n      clear equ\n      pure <| simplified [equ']) <|>\n    pure not_simplified\n#align tactic.unify_equations.unify_heterogeneous tactic.unify_equations.unify_heterogeneous\n\n/-- For `equ : t = u`, if `t` and `u` are defeq, we delete `equ`.\n-/\nunsafe def unify_defeq : unification_step := fun equ lhs_type _ _ _ lhs_whnf rhs_whnf _ =>\n  (do\n      is_def_eq lhs_whnf rhs_whnf\n      clear equ\n      pure <| simplified []) <|>\n    pure not_simplified\n#align tactic.unify_equations.unify_defeq tactic.unify_equations.unify_defeq\n\n/-- For `equ : x = t` or `equ : t = x`, where `x` is a local constant, we\nsubstitute `x` with `t` in the goal.\n-/\nunsafe def unify_var : unification_step := fun equ type _ lhs rhs lhs_whnf rhs_whnf u =>\n  (do\n      let lhs_is_local := lhs_whnf.is_local_constant\n      let rhs_is_local := rhs_whnf.is_local_constant\n      guard <| lhs_is_local \u2228 rhs_is_local\n      let t :=\n        if lhs_is_local then (const `eq [u]) type lhs_whnf rhs\n        else (const `eq [u]) type lhs rhs_whnf\n      change_core t (some equ)\n      let equ \u2190 get_local equ.local_pp_name\n      subst_core equ\n      pure <| simplified []) <|>\n    pure not_simplified\n#align tactic.unify_equations.unify_var tactic.unify_equations.unify_var\n\n-- TODO This is an improved version of `injection_with` from core\n-- (init/meta/injection_tactic). Remove when the improvements have landed in\n-- core.\nprivate unsafe def injection_with' (h : expr) (ns : List Name) (base := `h) (offset := some 1) :\n    tactic (Option (List expr) \u00d7 List Name) := do\n  let H \u2190 infer_type h\n  let (lhs, rhs, constructor_left, constructor_right, inj_name) \u2190\n    (do\n          let (lhs, rhs) \u2190 match_eq H\n          let constructor_left \u2190 get_app_fn_const_whnf lhs semireducible false\n          let constructor_right \u2190 get_app_fn_const_whnf rhs semireducible false\n          let inj_name \u2190 resolve_constant <| constructor_left ++ \"inj_arrow\"\n          pure (lhs, rhs, constructor_left, constructor_right, inj_name)) <|>\n        fail\n          (\"injection tactic failed, argument must be an equality proof where lhs and rhs \" ++\n            \"are of the form (c ...), where c is a constructor\")\n  if constructor_left = constructor_right then do\n      let inj\n        \u2190-- C.inj_arrow, for a given constructor C of datatype D, has type\n            --\n            --     \u2200 (A\u2081 ... A\u2099) (x\u2081 ... x\u2098) (y\u2081 ... y\u2098), C x\u2081 ... x\u2098 = C y\u2081 ... y\u2098\n            --       \u2192 \u2200 \u2983P : Sort u\u2984, (x\u2081 = y\u2081 \u2192 ... \u2192 y\u2096 = y\u2096 \u2192 P) \u2192 P\n            --\n            -- where the A\u1d62 are parameters of D and the x\u1d62/y\u1d62 are arguments of C.\n            -- Note that if x\u1d62/y\u1d62 are propositions, no equation is generated, so the\n            -- number of equations is not necessarily the constructor arity.\n            -- First, we find out how many equations we need to intro later.\n            mk_const\n            inj_name\n      let inj_type \u2190 infer_type inj\n      let inj_arity \u2190 get_pi_arity inj_type\n      let num_equations := (inj_type (inj_arity - 1)).binding_domain.pi_arity\n      let tgt\n        \u2190-- Now we generate the actual proof of the target.\n          target\n      let proof \u2190 mk_mapp inj_name (List.replicate (inj_arity - 3) none ++ [some h, some tgt])\n      eapply proof\n      let (next, ns) \u2190 intron_with num_equations ns base offset\n      let next\n        \u2190-- The following filters out 'next' hypotheses of type `true`. The\n            -- `inj_arrow` lemmas introduce these for nullary constructors.\n            next\n            fun h => do\n            let q(True) \u2190 infer_type h |\n              pure tt\n            clear h >> pure ff <|> pure tt\n      pure (some next, ns)\n    else do\n      let tgt \u2190 target\n      let constructor_left\n        \u2190-- The following construction deals with a corner case involing\n            -- mutual/nested inductive types. For these, Lean does not generate\n            -- no-confusion principles. However, the regular inductive data type which a\n            -- mutual/nested inductive type is compiled to does have a no-confusion\n            -- principle which we can (usually? always?) use. To find it, we normalise\n            -- the constructor with `unfold_ginductive = tt`.\n            get_app_fn_const_whnf\n            lhs semireducible tt\n      let no_confusion := constructor_left ++ \"no_confusion\"\n      let pr \u2190 mk_app no_confusion [tgt, lhs, rhs, h]\n      exact pr\n      return (none, ns)\n#align tactic.unify_equations.injection_with' tactic.unify_equations.injection_with'\n\n/-- Given `equ : C x\u2081 ... x\u2099 = D y\u2081 ... y\u2098` with `C` and `D` constructors of the\nsame datatype `I`:\n\n- If `C \u2260 D`, we solve the goal by contradiction using the no-confusion rule.\n- If `C = D`, we clear `equ` and add equations `x\u2081 = y\u2081`, ..., `x\u2099 = y\u2099`.\n-/\nunsafe def unify_constructor_headed : unification_step := fun equ _ _ _ _ _ _ _ =>\n  (do\n      let (next, _) \u2190 injection_with' equ [] `_ none\n      try <| clear equ\n      pure <|\n          match next with\n          | none => goal_solved\n          | some next => simplified <| next expr.local_pp_name) <|>\n    pure not_simplified\n#align tactic.unify_equations.unify_constructor_headed tactic.unify_equations.unify_constructor_headed\n\n/-- For `type = I x\u2081 ... x\u2099`, where `I` is an inductive type, `get_sizeof type`\nreturns the constant `I.sizeof`. Fails if `type` is not of this form or if no\nsuch constant exists.\n-/\nunsafe def get_sizeof (type : expr) : tactic pexpr := do\n  let n \u2190 get_app_fn_const_whnf type semireducible false\n  resolve_name <| n ++ `sizeof\n#align tactic.unify_equations.get_sizeof tactic.unify_equations.get_sizeof\n\ntheorem add_add_one_ne (n m : \u2115) : n + (m + 1) \u2260 n :=\n  by\n  apply ne_of_gt\n  apply Nat.lt_add_of_pos_right\n  apply Nat.pos_of_ne_zero\n  contradiction\n#align tactic.unify_equations.add_add_one_ne Tactic.UnifyEquations.add_add_one_ne\n\n-- Linarith could prove this, but I want to avoid that dependency.\n/-- `match_n_plus_m n e` matches `e` of the form `nat.succ (... (nat.succ e')...)`.\nIt returns `n` plus the number of `succ` constructors and `e'`. The matching is\nperformed up to normalisation with transparency `md`.\n-/\nunsafe def match_n_plus_m (md) : \u2115 \u2192 expr \u2192 tactic (\u2115 \u00d7 expr) := fun n e => do\n  let e \u2190 whnf e md\n  match e with\n    | q(Nat.succ $(e)) => match_n_plus_m (n + 1) e\n    | _ => pure (n, e)\n#align tactic.unify_equations.match_n_plus_m tactic.unify_equations.match_n_plus_m\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Given `equ : n + m = n` or `equ : n = n + m` with `n` and `m` natural numbers\n      and `m` a nonzero literal, this tactic produces a proof of `false`. More\n      precisely, the two sides of the equation must be of the form\n      `nat.succ (... (nat.succ e)...)` with different numbers of `nat.succ`\n      constructors. Matching is performed with transparency `md`.\n      -/\n    unsafe\n  def\n    contradict_n_eq_n_plus_m\n    ( md : Transparency ) ( equ lhs rhs : expr ) : tactic expr\n    :=\n      do\n        let \u27e8 lhs_n , lhs_e \u27e9 \u2190 match_n_plus_m md 0 lhs\n          let \u27e8 rhs_n , rhs_e \u27e9 \u2190 match_n_plus_m md 0 rhs\n          is_def_eq lhs_e rhs_e md\n            <|>\n            fail\n              (\n                \"contradict_n_eq_n_plus_m:\\nexpected {lhs_e} and {rhs_e} to be definitionally \"\n                  ++\n                  \"equal at transparency {md}.\"\n                )\n          let common := lhs_e\n          guard ( lhs_n \u2260 rhs_n )\n            <|>\n            fail \"contradict_n_eq_n_plus_m:\\nexpected {lhs_n} and {rhs_n} to be different.\"\n          let\n            \u27e8 equ , lhs_n , rhs_n \u27e9\n              \u2190\n              if\n                lhs_n > rhs_n\n                then\n                pure ( equ , lhs_n , rhs_n )\n                else\n                do let equ \u2190 to_expr ` `( Eq.symm $ ( equ ) ) pure ( equ , rhs_n , lhs_n )\n          let diff := lhs_n - rhs_n\n          let rhs_n_expr := reflect rhs_n\n          let n \u2190 to_expr ` `( $ ( common ) + $ ( rhs_n_expr ) )\n          let m := reflect ( diff - 1 )\n          pure q( add_add_one_ne $ ( n ) $ ( m ) $ ( equ ) )\n#align tactic.unify_equations.contradict_n_eq_n_plus_m tactic.unify_equations.contradict_n_eq_n_plus_m\n\n-- failed to format: unknown constant 'term.pseudo.antiquot'\n/--\n      Given `equ : t = u` with `t, u : I` and `I.sizeof t \u2260 I.sizeof u`, we solve the\n      goal by contradiction.\n      -/\n    unsafe\n  def\n    unify_cyclic\n    : unification_step\n    :=\n      fun\n        equ type _ _ _ lhs_whnf rhs_whnf _\n          =>\n          (\n              do\n                let sizeof \u2190 get_sizeof type\n                  let hyp_lhs \u2190 to_expr ` `( $ ( SizeOf.sizeOf ) $ ( lhs_whnf ) )\n                  let hyp_rhs \u2190 to_expr ` `( $ ( SizeOf.sizeOf ) $ ( rhs_whnf ) )\n                  let hyp_type \u2190 to_expr ` `( @ Eq \u2115 $ ( hyp_lhs ) $ ( hyp_rhs ) )\n                  let\n                    hyp_proof\n                      \u2190\n                      to_expr\n                        `\n                          `(\n                            @ congr_arg\n                              $ ( type )\n                                \u2115\n                                $ ( lhs_whnf )\n                                $ ( rhs_whnf )\n                                $ ( SizeOf.sizeOf )\n                                $ ( equ )\n                            )\n                  let hyp_name \u2190 mk_fresh_name\n                  let hyp \u2190 note hyp_name hyp_type hyp_proof\n                  let falso \u2190 contradict_n_eq_n_plus_m semireducible hyp hyp_lhs hyp_rhs\n                  exfalso\n                  exact falso\n                  pure goal_solved\n              )\n            <|>\n            pure not_simplified\n#align tactic.unify_equations.unify_cyclic tactic.unify_equations.unify_cyclic\n\n/-- `orelse_step s t` first runs the unification step `s`. If this was successful\n(i.e. `s` simplified or solved the goal), it returns the result of `s`.\nOtherwise, it runs `t` and returns its result.\n-/\nunsafe def orelse_step (s t : unification_step) : unification_step :=\n  fun equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u => do\n  let r \u2190 s equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u\n  match r with\n    | simplified _ => pure r\n    | goal_solved => pure r\n    | not_simplified => t equ lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u\n#align tactic.unify_equations.orelse_step tactic.unify_equations.orelse_step\n\n/-- For `equ : t = u`, try the following methods in order: `unify_defeq`,\n`unify_var`, `unify_constructor_headed`, `unify_cyclic`. If any of them is\nsuccessful, stop and return its result. If none is successful, fail.\n-/\nunsafe def unify_homogeneous : unification_step :=\n  List.foldl orelse_step (fun _ _ _ _ _ _ _ _ => pure not_simplified)\n    [unify_defeq, unify_var, unify_constructor_headed, unify_cyclic]\n#align tactic.unify_equations.unify_homogeneous tactic.unify_equations.unify_homogeneous\n\nend UnifyEquations\n\nopen UnifyEquations\n\n/-- If `equ` is the display name of a local constant with type `t = u` or `t == u`,\nthen `unify_equation_once equ` simplifies it once using\n`unify_equations.unify_homogeneous` or `unify_equations.unify_heterogeneous`.\n\nOtherwise it fails.\n-/\nunsafe def unify_equation_once (equ : Name) : tactic unification_step_result := do\n  let eque \u2190 get_local equ\n  let t \u2190 infer_type eque\n  match t with\n    | app (app (app (const `eq [u]) type) lhs) rhs => do\n      let lhs_whnf \u2190 whnf_ginductive lhs\n      let rhs_whnf \u2190 whnf_ginductive rhs\n      unify_homogeneous eque type type lhs rhs lhs_whnf rhs_whnf u\n    | app (app (app (app (const `heq [u]) lhs_type) lhs) rhs_type) rhs => do\n      let lhs_whnf \u2190 whnf_ginductive lhs\n      let rhs_whnf \u2190 whnf_ginductive rhs\n      unify_heterogeneous eque lhs_type rhs_type lhs rhs lhs_whnf rhs_whnf u\n    | _ =>\n      throwError \"Expected {(\u2190 equ)} to be an equation, but its type is\n        {\u2190 t}.\"\n#align tactic.unify_equation_once tactic.unify_equation_once\n\n/-- Given a list of display names of local hypotheses that are (homogeneous or\nheterogeneous) equations, `unify_equations` performs first-order unification on\neach hypothesis in order. See `tactic.interactive.unify_equations` for an\nexample and an explanation of what unification does.\n\nReturns true iff the goal has been solved during the unification process.\n\nNote: you must make sure that the input names are unique in the context.\n-/\nunsafe def unify_equations : List Name \u2192 tactic Bool\n  | [] => pure false\n  | h :: hs => do\n    let res \u2190 unify_equation_once h\n    match res with\n      | simplified hs' => unify_equations <| hs' ++ hs\n      | not_simplified => unify_equations hs\n      | goal_solved => pure tt\n#align tactic.unify_equations tactic.unify_equations\n\nnamespace Interactive\n\nopen Lean.Parser\n\n/-- `unify_equations eq\u2081 ... eq\u2099` performs a form of first-order unification on the\nhypotheses `eq\u1d62`. The `eq\u1d62` must be homogeneous or heterogeneous equations.\nUnification means that the equations are simplified using various facts about\nconstructors. For instance, consider this goal:\n\n```\nP : \u2200 n, fin n \u2192 Prop\nn m : \u2115\nf : fin n\ng : fin m\nh\u2081 : n + 1 = m + 1\nh\u2082 : f == g\nh\u2083 : P n f\n\u22a2 P m g\n```\n\nAfter `unify_equations h\u2081 h\u2082`, we get\n\n```\nP : \u2200 n, fin n \u2192 Prop\nn : \u2115\nf : fin n\nh\u2083 : P n f\n\u22a2 P n f\n```\n\nIn the example, `unify_equations` uses the fact that every constructor is\ninjective to conclude `n = m` from `h\u2081`. Then it replaces every `m` with `n` and\nmoves on to `h\u2082`. The types of `f` and `g` are now equal, so the heterogeneous\nequation turns into a homogeneous one and `g` is replaced by `f`. Note that the\nequations are processed from left to right, so `unify_equations h\u2082 h\u2081` would not\nsimplify as much.\n\nIn general, `unify_equations` uses the following steps on each equation until\nnone of them applies any more:\n\n- Constructor injectivity: if `nat.succ n = nat.succ m` then `n = m`.\n- Substitution: if `x = e` for some hypothesis `x`, then `x` is replaced by `e`\n  everywhere.\n- No-confusion: `nat.succ n = nat.zero` is a contradiction. If we have such an\n  equation, the goal is solved immediately.\n- Cycle elimination: `n = nat.succ n` is a contradiction.\n- Redundancy: if `t = u` but `t` and `u` are already definitionally equal, then\n  this equation is removed.\n- Downgrading of heterogeneous equations: if `t == u` but `t` and `u` have the\n  same type (up to definitional equality), then the equation is replaced by\n  `t = u`.\n-/\nunsafe def unify_equations (eqs : interactive.parse (many ident)) : tactic Unit :=\n  tactic.unify_equations eqs *> skip\n#align tactic.interactive.unify_equations tactic.interactive.unify_equations\n\nadd_tactic_doc\n  { Name := \"unify_equations\"\n    category := DocCategory.tactic\n    declNames := [`tactic.interactive.unify_equations]\n    tags := [\"simplification\"] }\n\nend Interactive\n\nend Tactic\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Tactic/UnifyEquations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631556226292, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.355544413721578}}
{"text": "example [Add \u03b1] [Neg \u03b1] [Mul \u03b1] [\u2200 n, OfNat \u03b1 n] {a b : \u03b1} :=\n  calc\n    4 + 5 * b = -6 + 5 * (b + 2) := sorry\n    _         = -6 + 5 * 3       := sorry\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/1813.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7520125848754472, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.35546392308124475}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport category_theory.preadditive.projective\nimport algebra.homology.homological_complex\nimport category_theory.abelian.basic\nimport for_mathlib.algebra.homology.twist_cocycle\nimport for_mathlib.algebraic_topology.homotopical_algebra.cochain_complex.basic\nimport for_mathlib.algebra.homology.homological_complex_biprod\n\nnoncomputable theory\n\nopen category_theory category_theory.category\nopen category_theory.limits algebraic_topology\n\nopen_locale zero_object\n\nnamespace category_theory\n\nnamespace limits\n\n@[simps]\ndef is_zero.unique_up_to_iso {C : Type*} [category C]\n  {X Y : C} (hX : is_zero X)\n  (hY : is_zero Y) : X \u2245 Y :=\nis_initial.unique_up_to_iso hX.is_initial hY.is_initial\n\nend limits\n\nnamespace projective\n\nvariables {C : Type*} [category C] [enough_projectives C] [has_zero_object C]\n\ndef over' (X : C) : C :=\nbegin\n  by_cases is_zero X,\n  { exact 0, },\n  { exact (enough_projectives.presentation X).some.P, },\nend\nlemma over'_eq_zero (X : C) (hX : is_zero X) : over' X = 0 :=\nbegin\n  dsimp [over'],\n  split_ifs,\n  refl,\nend\n\nlemma over'_eq (X : C) (hX : \u00acis_zero X) : over' X = over X :=\nbegin\n  dsimp [over'],\n  split_ifs,\n  refl,\nend\n\ndef \u03c0' (X : C) : over' X \u27f6 X :=\nbegin\n  by_cases is_zero X,\n  { let e : 0 \u2245 X := limits.is_zero.unique_up_to_iso (is_zero_zero C) h,\n    refine eq_to_hom (over'_eq_zero X h) \u226b e.hom, },\n  { exact eq_to_hom (over'_eq X h) \u226b projective.\u03c0 X, },\nend\n\ninstance (X : C) : epi (\u03c0' X) :=\nby { dsimp only [\u03c0'], split_ifs; apply epi_comp, }\n\ninstance (X : C) [has_zero_morphisms C] : projective (over' X) :=\nbegin\n  dsimp [over'],\n  split_ifs,\n  { apply projective.zero_projective, },\n  { exact projective.projective_over X,},\nend\n\nend projective\n\nend category_theory\n\nnamespace cochain_complex\n\nnamespace minus\n\nnamespace projective_model_structure\n\nvariables {C : Type*} [category C] [abelian C] [enough_projectives C]\n\n/-namespace CM5a\n\nopen cochain_complex.hom_complex\n\n\n@[simps]\ndef P (L : cochain_complex C \u2124) : cochain_complex C \u2124 :=\n{ X := \u03bb q, category_theory.projective.over' (L.X (q-1)),\n  d := \u03bb i j, 0,\n  shape' := \u03bb i j hij, rfl,\n  d_comp_d' := \u03bb i j k hij hjk, comp_zero, }\n\ninstance (L : cochain_complex C \u2124) (n : \u2124) : projective ((P L).X n) :=\nby { dsimp [P], apply_instance, }\n\n@[simps]\ndef Q (L : cochain_complex C \u2124) : cochain_complex C \u2124 :=\ntwist (cocycle.of_hom (\ud835\udfd9 (P L)))\n\ninstance Q_is_degreewise_projective (L : cochain_complex C \u2124) (n : \u2124) :\n  projective ((Q L).X n) :=\nby { dsimp only [Q, twist], apply_instance, }\n\n@[simps]\ndef \u03c0 (L : cochain_complex C \u2124) : Q L \u27f6 L :=\nbegin\n  refine twist.desc (cocycle.of_hom (\ud835\udfd9 (P L))) (cochain.mk _) _ (neg_add_self 1) _,\n  { exact (\u03bb p q hpq, category_theory.projective.\u03c0' _ \u226b eq_to_hom (by { congr, linarith})), },\n  { exact { f := \u03bb i, category_theory.projective.\u03c0' _ \u226b L.d (i-1) i, }, },\n  { ext,\n    simp only [\u03b4_v (-1) 0 rfl _ p p (add_zero p).symm (p-1) (p+1) rfl rfl,\n      add_zero, zero_comp, cochain.mk_v, eq_to_hom_refl, comp_id, P_d,\n      smul_zero, cocycle.of_hom_coe, cochain.id_comp, cochain.of_hom_v],\n    },\nend\n\nlemma \u03c0_is_degreewise_epi (L : cochain_complex C \u2124) :\n  projective_structure.arrow_classes.fib (\u03c0 L) :=\nbegin\n  intro n,\n  have h : epi ((cochain.comp (twist.inl _ (show -(1 : \u2124)+1 = 0, by linarith))\n    (cochain.of_hom (\u03c0 L)) (add_zero (-1)).symm).v (n+1) n (by linarith)),\n  { dsimp only [\u03c0, twist.desc, hom_complex.twist.desc_hom_as_cocycle],\n    simp only [cocycle.cochain_of_hom_hom_of_eq_coe, twist.desc_cocycle_coe],\n    rw twist.inl_comp_desc_cochain,\n    dsimp,\n    apply epi_comp, },\n  simp only [cochain.comp_v _ _ (add_zero (-(1 : \u2124))).symm (n+1) n n\n    (by linarith) (by linarith), cochain.of_hom_v] at h,\n  simp only [twist.inl, cochain.mk, cochain.v, cochain.of_hom, cochain.of_homs,\n    homological_complex.id_f, id_comp] at h,\n  exact @epi_of_epi _ _ _ _ _ _ _ h,\nend\n\n@[simps]\ndef id_Q_homotopy_to_zero (L : cochain_complex C \u2124) :\n  homotopy (\ud835\udfd9 (Q L)) 0 :=\nbegin\n  equiv_rw hom_complex.equiv_homotopy _ _,\n  refine \u27e8cochain.comp (twist.snd _ ) (twist.inl _ (by linarith)) (zero_add (-1)).symm, _\u27e9,\n  dsimp only [Q],\n  simpa only [add_zero, add_left_neg, eq_self_iff_true, \u03b4_comp_of_first_is_zero_cochain,\n    twist.\u03b4_inl, cocycle.of_hom_coe, cochain.id_comp, cochain.of_hom_zero,\n    twist.\u03b4_snd _ (zero_add 1), \u03b5_odd _ odd_neg_one, zsmul_neg, cochain.neg_comp, neg_zsmul,\n    one_zsmul, neg_neg, twist.id_eq _ (show -(1 : \u2124)+1=0, by linarith) (zero_add 1),\n    cochain.comp_id] using add_comm _ _,\nend\n\nvariables {K L : cochain_complex C \u2124} (f : K \u27f6 L)\n\ninclude f\n@[simps, nolint unused_arguments]\ndef obj := homological_complex.biprod K (Q L)\n\n@[simps]\ndef i : K \u27f6 obj f := homological_complex.biprod.inl\n\n@[simps]\ndef p : obj f \u27f6 L := homological_complex.biprod.desc f (\u03c0 L)\n\n@[simp, reassoc]\nlemma fac : i f \u226b p f = f :=\nby simp only [i, p, homological_complex.biprod.inl_desc]\n\nlemma p_is_fib :\n  projective_structure.arrow_classes.fib (p f) :=\nbegin\n  intro n,\n  have h : biprod.inr \u226b biprod.desc (f.f n) ((\u03c0 L).f n) = (\u03c0 L).f n := biprod.inr_desc _ _,\n  haveI : epi ((\u03c0 L).f n) := \u03c0_is_degreewise_epi L n,\n  exact epi_of_epi_fac h,\nend\n\nlemma i_is_cof :\n  projective_structure.arrow_classes.cof (i f) :=\nbegin\n  intro n,\n  apply preadditive.mono_with_projective_coker.of_biprod_inl,\nend\n\ndef homotopy_equiv_i : homotopy_equiv K (obj f) :=\n{ hom := i f,\n  inv := homological_complex.biprod.fst,\n  homotopy_hom_inv_id := by { simp only [i, homological_complex.biprod.inl_fst], },\n  homotopy_inv_hom_id := begin\n    symmetry,\n    equiv_rw homotopy.equiv_sub_zero,\n    dsimp only [obj, i],\n    simpa only [zero_comp, comp_zero, id_comp, \u2190 homological_complex.biprod.total,\n      add_sub_cancel'] using ((id_Q_homotopy_to_zero L).comp_right\n        homological_complex.biprod.inr).comp_left homological_complex.biprod.snd,\n  end, }\n\nlemma i_is_weq :\n  projective_structure.arrow_classes.weq (i f) :=\n(homotopy_equiv_i f).to_quasi_iso\n\nlemma i_is_triv_cof :\n  projective_structure.arrow_classes.triv_cof (i f) := \u27e8i_is_cof f, i_is_weq f\u27e9\n\nend CM5a\n\nend projective_structure\n\nend cochain_complex\n\nnamespace bounded_above_cochain_complex\n\nnamespace projective_model_structure\n\nvariables {C : Type*} [category C] [abelian C] [enough_projectives C]\n\nnamespace CM5a\n\nvariables {K L : bounded_above_cochain_complex C} (f : K \u27f6 L)\n\nopen cochain_complex.projective_structure\ninclude f\n\n@[simps, nolint unused_arguments]\ndef obj : bounded_above_cochain_complex C := \u27e8CM5a.obj f,\nbegin\n  apply cochain_complex.is_bounded_above.of_biprod,\n  { exact K.2, },\n  { apply cochain_complex.hom_complex.twist.is_bounded_above,\n    all_goals {\n      cases L.2 with l hl,\n      use (l+1),\n      intros i hi,\n      dsimp,\n      rw category_theory.projective.over'_eq_zero, swap,\n      { apply hl,\n        linarith, },\n      apply is_zero_zero, }, },\nend\u27e9\n\n@[simps]\ndef i : K \u27f6 obj f := CM5a.i f\n\n@[simps]\ndef p : obj f \u27f6 L := CM5a.p f\n\nlemma fac : i f \u226b p f = f := CM5a.fac f\n\nlemma p_is_fib : arrow_classes.fib (p f) := CM5a.p_is_fib f\n\nlemma i_is_triv_cof : arrow_classes.triv_cof (i f) := CM5a.i_is_triv_cof f\n\nend CM5a-/\n\nlemma CM5a' {X Z : minus C} (f : X \u27f6 Z) (n : \u2124) [X.obj.is_strictly_le n]\n  [Z.obj.is_strictly_le n] : \u2203 (Y : minus C)\n  (hY : Y.obj.is_strictly_le n) (i : X \u27f6 Y) (p : Y \u27f6 Z)\n  (hi : (arrow_classes C).triv_cof i) (hp : (arrow_classes C).fib p), i \u226b p = f := sorry\n\nlemma CM5a : (arrow_classes C).CM5a :=\n\u03bb X Z f, begin\n  obtain \u27e8nX, hX\u27e9 := X.property,\n  obtain \u27e8nZ, hZ\u27e9 := Z.property,\n  haveI := hX,\n  haveI := hZ,\n  let n := max nX nZ,\n  haveI : X.obj.is_strictly_le n := is_strictly_le_of_le _ _ _ (le_max_left _ _),\n  haveI : Z.obj.is_strictly_le n := is_strictly_le_of_le _ _ _ (le_max_right _ _),\n  obtain \u27e8Y, hY, i, p, hi, hp, fac\u27e9 := CM5a' f n,\n  exact \u27e8Y, i, hi, p, hp, fac\u27e9,\nend\n\nend projective_model_structure\n\nend minus\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebraic_topology/homotopical_algebra/cochain_complex/cm5a.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.35541970403431294}}
{"text": "import TBA.Eulerian.Nat\n\nopen BEq Nat List Decidable\n\nnamespace Eq\n\nprotected theorem symm_iff {a b : \u03b1} : b = a \u2194 a = b := \u27e8Eq.symm, Eq.symm\u27e9\n\nend Eq\n\nnamespace List\n\n@[simp] theorem length_append {as bs : List \u03b1} :\n  length (as ++ bs) = length as + length bs :=\n  match as with\n  | []      => by simp\n  | a :: as => by simp [Nat.succ_add, length_append]\n\ntheorem length_zero_iff_nil {as : List \u03b1} : length as = 0 \u2194 as = [] :=\n\u27e8fun e => by cases as; rfl; simp [length_cons] at e, fun e => by rw [e]; rfl\u27e9\n\ntheorem length_cons_ne_zero {as : List \u03b1} {a : \u03b1} : length (a :: as) \u2260 0 := by\n  rw [List.length_cons]; exact Nat.succNeZero _\n\n-- Some lemmas about filters\n\n@[simp] theorem filter_nil {p : \u03b1 \u2192 Bool} : filter p [] = [] := by\n  simp [filter, filterAux, reverse, reverseAux]\n\ntheorem cons_eq_append (a : \u03b1) (as : List \u03b1) : a :: as = [a] ++ as := rfl\n\ntheorem reverseAux_append {rs as : List \u03b1} : reverseAux rs as = reverseAux rs [] ++ as :=\n  match rs with\n  | []      => rfl\n  | r :: rs => by\n    simp only [reverseAux]\n    rw [reverseAux_append, reverseAux_append (as := [r]), cons_eq_append r as, append_assoc]\n\ntheorem filterAux_aux {p : \u03b1 \u2192 Bool} (as : List \u03b1) :\n  (rs : List \u03b1) \u2192 filterAux p as rs = rs.reverse ++ (filterAux p as []) :=\n  match as with\n  | [] => by intros; simp [filterAux, reverse, reverseAux];\n  | a :: as => by\n    intro rs\n    simp only [filterAux]\n    cases p a\n    case false => simp [filterAux_aux as rs]\n    case true =>\n      rw [filterAux_aux as (a :: rs), filterAux_aux as [a]]\n      simp only [reverse, reverseAux, List.append, List.cons_append, List.nil_append]\n      rw [reverseAux_append, cons_eq_append _ (filterAux p as []), append_assoc]\n\ntheorem filter_cons (a : \u03b1) (as : List \u03b1) :\n  filter p (a :: as) = if p a then a :: filter p as else filter p as := by\n  simp only [filter, filterAux]\n  cases p a\n  simp\n  rw [filterAux_aux]; simp [reverse, reverseAux]\n\n@[simp] theorem filter_append {as bs : List \u03b1} {p : \u03b1 \u2192 Bool} :\n  filter p (as ++ bs) = filter p as ++ filter p bs := by\n  induction as with\n  | nil      => simp\n  | cons a as ih =>\n    rw [filter_cons, cons_append, filter_cons]\n    cases p a <;> simp [ih]\n\n-- A membership predicate\n\ninductive Mem : \u03b1 \u2192 List \u03b1 \u2192 Prop where\n  | head (a : \u03b1) (as : List \u03b1)   : Mem a (a::as)\n  | tail (a b : \u03b1) (bs : List \u03b1) : Mem a bs \u2192 Mem a (b::bs)\n\ninfix:50 \" \u2208 \" => Mem\n\n@[simp] theorem beq_refl [DecidableEq \u03b1] : ((a : \u03b1) == a) = true :=\n  decideEqTrue rfl\n\n@[simp] theorem beq_neq [DecidableEq \u03b1] {a b : \u03b1} (h : \u00ac a = b) : (a == b) = false :=\n  decideEqFalse h\n\ntheorem length_erase_mem [DecidableEq \u03b1] {a : \u03b1} {as : List \u03b1} (h : a \u2208 as) :\n  length (List.erase as a) + 1 = length as :=\n  match a, as, h with\n  | _, _, Mem.head a bs => by simp [List.erase]\n  | _, _, Mem.tail a b bs h => by\n    simp only [List.erase]\n    match b == a with\n    | true => simp\n    | false => { simp only [length_cons]; rw [\u2190length_erase_mem h] }\n\ntheorem mem_singleton [DecidableEq \u03b1] {a b : \u03b1} (h : a \u2208 [b]) : a = b := by\n  cases h with\n  | head h       => rfl\n  | tail _ _ _ h => cases h\n\ndef mem_of_nonzero_length [DecidableEq \u03b1] {as : List \u03b1} :\n  (h : length as > 0) \u2192 { a // a \u2208 as } :=\n  match as with\n  | []      => by { simp only [length_nil]; intro fa; cases fa }\n  | a :: as => by { intros; apply Subtype.mk; apply Mem.head }\n\ntheorem mem_append {as bs : List \u03b1} : \u2200 a, a \u2208 (as ++ bs) \u2192 a \u2208 as \u2228 a \u2208 bs :=\n  match as with\n  | [] => by intros; apply Or.inr; assumption\n  | a :: as => by \n    simp only [List.cons_append]\n    intros a h\n    cases h with\n    | head h => apply Or.inl; apply Mem.head\n    | tail _ _ _ h =>\n      cases mem_append (as := as) (bs := bs) a h\n      case inl => apply Or.inl; apply Mem.tail; assumption\n      case inr => apply Or.inr; assumption\n\ntheorem mem_of_mem_filter {as : List \u03b1} {a : \u03b1} {p : \u03b1 \u2192 Bool} : a \u2208 filter p as \u2192 a \u2208 as :=\n  match as with\n  | [] => by intros; assumption\n  | a' :: as => by \n      rw [filter_cons]; cases p a'\n      case false => intro h; apply Mem.tail; apply mem_of_mem_filter h\n      case true => \n        intro h\n        cases h with\n        | head => apply Mem.head\n        | tail _ _ _ h => apply Mem.tail; apply mem_of_mem_filter h\n\ntheorem mem_filter_of_prop {as : List \u03b1} {a : \u03b1} {p : \u03b1 \u2192 Bool} (ha : a \u2208 as) (hpa : p a = true) :\n  a \u2208 filter p as := by\n  induction ha with\n  | head a as => rw [filter_cons, hpa]; simp; apply Mem.head\n  | tail a b as ha' ih => \n    rw [filter_cons]\n    cases hpb : p b\n    case false => simp; exact ih hpa\n    case true  => simp; apply Mem.tail; exact ih hpa\n\ntheorem filterProp_of_mem {as : List \u03b1} {p : \u03b1 \u2192 Bool} {a : \u03b1} : a \u2208 filter p as \u2192 p a = true :=\n  match as with\n  | [] => by intro h; cases h\n  | a' :: as => by\n      rw [filter_cons]; byCases hpa : p a'\n      case inr => simp [hpa]; exact filterProp_of_mem\n      case inl => simp [hpa]; intro h; cases h; assumption; apply filterProp_of_mem; assumption\n\ntheorem mem_split {a : \u03b1} {as : List \u03b1} (h : a \u2208 as) : \u2203 s t, as = s ++ a :: t :=\n  match a, as, h with\n  | _, _, Mem.head a bs     => \u27e8[], \u27e8bs, rfl\u27e9\u27e9\n  | _, _, Mem.tail a b bs h =>\n    match bs, mem_split h with\n    | _, \u27e8s, \u27e8t, rfl\u27e9\u27e9 => \u27e8b::s, \u27e8t, List.cons_append .. \u25b8 rfl\u27e9\u27e9\n\n-- Couting elements\n\nsection Count\nvariable [DecidableEq \u03b1] (as bs as' bs' : List \u03b1) (a b : \u03b1)\n\ndef count : Nat := (as.filter $ fun b => b == a).length\n\n@[simp] theorem count_empty : count [] a = 0 := by simp [count]\n\n@[simp] theorem count_self_cons : (a :: as).count a = Nat.succ (as.count a) := by\n  simp [count, filter_cons]\n\ntheorem count_neq_cons {as} {a b : \u03b1} (h : \u00ac b = a) : (b :: as).count a = as.count a := by\n  simp [count, filter_cons, h]\n\n@[simp] theorem count_append : (as ++ bs).count a = as.count a + bs.count a :=\n  by simp [count]\n\n--set_option trace.Meta.Tactic.simp true\ntheorem count_erase {as : List \u03b1} {a b : \u03b1} :\n  (as.erase a).count b = if (a == b) then (as.count b) - 1 else as.count b :=\n  match as with\n  | [] => by simp only [List.erase]; cases (a == b) <;> simp\n  | a' :: as => by\n    simp only [List.erase]\n    byCases h : a' = a\n    case inl => \n      cases h\n      simp only [beq_refl]\n      byCases h' : a = b\n      case inl => cases h'; simp only [beq_refl, Lean.Simp.ite_True, Nat.succ_sub_succ, count_self_cons]; apply Nat.sub_zero\n      case inr => rw [beq_neq h']; simp [count_neq_cons h']\n    case inr =>\n      rw [beq_neq h]\n      byCases h' : a' = b\n      case inl => cases h'; simp [count_erase (as := as), beq_neq (Ne.symm h)]\n      case inr => simp [count_neq_cons h', count_neq_cons h', count_erase]\n\ntheorem count_le_cons : as.count a \u2264 (b :: as).count a := by\n  byCases h : (b = a)\n  case inl => cases h; simp [count_self_cons, Nat.leSucc]\n  case inr => rw [count_neq_cons h]; simp [Nat.leRefl]\n\ntheorem mem_nonzeroCount {as : List \u03b1} {a : \u03b1}: a \u2208 as \u2194 as.count a > 0 := by\n  apply Iff.intro\n  case mp =>\n    intro h\n    induction h with\n    | head a => simp [Nat.zeroLtSucc]\n    | tail _ _ _ h ih => apply Nat.ltOfLtOfLe ih; apply count_le_cons\n  case mpr =>\n    intro h\n    let \u27e8a', ha'\u27e9 := mem_of_nonzero_length h\n    let foo := filterProp_of_mem ha'\n    cases ofDecideEqTrue foo\n    apply mem_of_mem_filter ha'\n    \n-- Erasing elements from lists\n  \ntheorem erase_comm {as : List \u03b1} : (as.erase a).erase b  = (as.erase b).erase a :=\n  match as with\n  | [] => rfl\n  | a' :: as => by\n    byCases h : a' = a\n    case inl => \n      cases h; simp only [List.erase]\n      byCases h' : b = a\n      case inl => cases h'; simp\n      case inr => rw [beq_neq (Ne.symm h')]; simp [List.erase]\n    case inr =>\n      simp only [List.erase]\n      rw [beq_neq h]\n      byCases h' : a' = b\n      case inl => cases h'; simp [List.erase]\n      case inr => simp [List.erase, beq_neq h', beq_neq h, erase_comm (as := as)]\n\ntheorem filter_erase_false {as : List \u03b1} {a : \u03b1} {p : \u03b1 \u2192 Bool} (h : p a = false) :\n  filter p (as.erase a) = filter p as :=\n  match as with\n  | [] => by simp [List.erase]\n  | b :: as => by\n    simp only [List.erase, filter_cons]\n    byCases h' : b = a\n    case inl => cases h'; simp [h]\n    case inr => simp [h', filter_cons]; rw [filter_erase_false h]\n\ntheorem filter_erase_true {as : List \u03b1} {a : \u03b1} {p : \u03b1 \u2192 Bool} (hpa : p a = true) :\n  filter p (as.erase a) = (filter p as).erase a :=\n  match as with\n  | [] => rfl\n  | b :: as => by\n    simp\n    byCases h : b = a\n    case inl => cases h; simp [List.erase, filter_cons, hpa]\n    case inr =>\n      simp only [List.erase, beq_neq h, filter_cons]\n      cases hpb : p b\n      case false => exact filter_erase_true hpa\n      case true => rw [filter_erase_true hpa]; simp [List.erase, h]\n\ndef eraseAll (as bs : List \u03b1) : List \u03b1 :=\n  match bs with\n  | []      => as\n  | b :: bs => eraseAll (as.erase b) bs\n\ninfixl:55 \" -l \" => eraseAll\n\ntheorem erase_eraseAll {as bs : List \u03b1} {a : \u03b1} : (as.erase a) -l bs = (as -l bs).erase a :=\n  match bs with\n  | [] => rfl\n  | b :: bs => by simp only [eraseAll]; rw [\u2190erase_eraseAll, erase_comm]\n\n@[simp] theorem count_eraseAll (as bs : List \u03b1) (a : \u03b1) :\n  (as -l bs).count a = as.count a - bs.count a :=\n  match bs with\n  | [] => rfl\n  | b :: bs => by\n    simp only [eraseAll]; rw [erase_eraseAll]; simp\n    byCases hba : b = a;\n    case inl => cases hba; rw [count_self_cons, count_erase, count_eraseAll as bs a, beq_refl]; rfl\n    case inr => rw [count_erase, beq_neq hba, count_eraseAll as bs a, count_neq_cons]; repeat simp_all\n\n-- Lists which are permutations of each other, and sublists modulo permutation\n\ndef isPermEqvTo : Prop := \u2200 a, as.count a = bs.count a\ninfixl:50 \" \u2243 \" => isPermEqvTo -- Type as \\simeq\n\ndef isPermSubOf : Prop := \u2200 a, as.count a \u2264 bs.count a\ninfixl:50 \" \u2286 \" => isPermSubOf -- Type as \\sub\n\ntheorem permSubOfEraseSub : (as -l bs) \u2286 as := fun a => by simp [Nat.subLe]\n\ntheorem mem_of_mem_eraseAll {as bs : List \u03b1} {a : \u03b1} : a \u2208 (as -l bs) \u2192 a \u2208 as := by\n  rw [mem_nonzeroCount, mem_nonzeroCount, count_eraseAll]\n  intro h; exact Nat.ltOfLtOfLe h (Nat.subLe _ _)\n\ntheorem permSubEraseAllLength {as bs : List \u03b1} : bs \u2286 as \u2192 length (as -l bs) = length as - length bs :=\n  match bs with\n  | [] => fun _ => rfl\n  | b :: bs => fun hsub => by\n    simp only [eraseAll, length_cons]\n    have hbas : b \u2208 as := by\n      have hb := hsub b\n      rw [count_self_cons] at hb\n      rw [mem_nonzeroCount]\n      exact Nat.ltOfLeOfLt (zeroLe _) (ltOfSuccLe hb)\n    have hsub' : bs.isPermSubOf (as.erase b) := fun c => by\n      have hc := hsub c\n      byCases h : b = c\n      case inl =>\n        cases h\n        rw [count_self_cons] at hc\n        rw [count_erase]\n        simp [leOfSuccLeSucc (Nat.leTrans hc leSuccSubOne)]\n      case inr =>\n        rw [count_neq_cons h] at hc\n        apply Nat.leTrans hc\n        rw [count_erase, beq_neq h]\n        simp [Nat.leRefl]\n    rw [permSubEraseAllLength hsub', \u2190length_erase_mem hbas, Nat.succ_sub_succ]\n\ntheorem permSubExtend {as bs : List \u03b1} {b} (hsub : bs \u2286 as) (ha : b \u2208 (as -l bs)) : (b :: bs) \u2286 as := by\n  intro a\n  rw [mem_nonzeroCount, count_eraseAll, \u2190zeroLtIffSub] at ha\n  byCases h : b = a\n  case inl => cases h; simp; assumption\n  case inr => rw [count_neq_cons h]; apply hsub\n\ndef permSubObtainComplement {as bs : List \u03b1} (hsub : bs \u2286 as) \n    (hlength : bs.length < as.length) : { e // e \u2208 as -l bs} := by\n  have hl : (as -l bs).length > 0 := by rw [permSubEraseAllLength hsub, \u2190zeroLtIffSub]; assumption\n  revert hl; cases as -l bs\n  case nil => simp only [length_nil]; intro hl'; cases hl'\n  case cons => intros; exact \u27e8_, Mem.head _ _\u27e9\n\ntheorem permSubEqvClosed {as bs bs' : List \u03b1} (heqv : bs \u2243 bs') (hsub : bs \u2286 as) : bs' \u2286 as :=\n  fun a => by rw [\u2190heqv a]; exact hsub a\n\ntheorem permSubSingleton {as : List \u03b1} {a : \u03b1} : a \u2208 as \u2192 [a] \u2286 as := by\n  intros ha b\n  rw [mem_nonzeroCount] at ha\n  byCases h : a = b\n  case inl => cases h; simp [count_self_cons]; assumption\n  case inr => rw [count_neq_cons h]; simp [Nat.zeroLe]\n\ntheorem permEqvRotate : (as ++ bs) \u2243 (bs ++ as) :=\n  fun a => by simp [Nat.add_comm];\n\ntheorem permEqvRefl {as : List \u03b1} : as \u2243 as :=\n  fun a => rfl\n\ntheorem permEqvTrans {as bs cs : List \u03b1} (h : as \u2243 bs) (h' : bs \u2243 cs) : as \u2243 cs :=\n  fun a => Eq.trans (h a) (h' a)\n\ntheorem permEqvSymm {as bs : List \u03b1} (h : as \u2243 bs) : bs \u2243 as :=\n  fun a => by simp [Nat.add_comm, h a]\n\ntheorem permEqvToEraseAppend {as bs : List \u03b1} (ps : bs \u2286 as) : as \u2243 ((as -l bs) ++ bs) :=\n  fun a => by simp only [count_append, count_eraseAll]; rw [\u2190Nat.le_subAdd (ps a)]\n\ntheorem permEqvToEraseCons {as : List \u03b1} {a : \u03b1} (h : a \u2208 as) : as \u2243 (a :: (as.erase a)) := by\n  exact permEqvTrans (permEqvToEraseAppend (permSubSingleton h)) (permEqvRotate _ _)\n\ntheorem permEqvCons {as bs : List \u03b1} {a : \u03b1} (h : as \u2243 bs) : (a :: as) \u2243 (a :: bs) := by\n  intro b\n  byCases hba : (a = b)\n  case inl => cases hba; simp [count_self_cons, h a];\n  case inr => rw [count_neq_cons hba, count_neq_cons hba, h b]\n\ntheorem permSubEraseOfpermEqvCons {as bs : List \u03b1} {a : \u03b1} (h : (a :: as) \u2286 bs) :\n  as \u2286 (bs.erase a) := fun b => by\n  let ha := h a\n  byCases h' : a = b\n  case inl =>\n    cases h'\n    rw [count_erase]\n    rw [count_self_cons] at ha\n    simp [Nat.leOfSuccLeSucc (Nat.leTrans ha leSuccSubOne)]\n  case inr =>\n    let hb := h b\n    rw [count_neq_cons h'] at hb\n    rw [count_erase, beq_neq h']\n    exact hb\n\ntheorem permEqvOfPermSub {as bs : List \u03b1} : as \u2286 bs \u2192 as.length = bs.length \u2192 as \u2243 bs :=\n  match as with\n  | [] => fun hsub hl => by\n    rw [length_nil, Eq.symm_iff, length_zero_iff_nil] at hl\n    rw [hl]; intro b; rfl\n  | a :: as => fun hsub hl => by\n    have habs : a \u2208 bs := by\n      rw [mem_nonzeroCount]\n      let hsuba := hsub a\n      simp only [count_self_cons] at hsuba\n      exact Nat.ltOfLtOfLe (Nat.zeroLtSucc _) hsuba\n    have hsub' : as.isPermSubOf (bs.erase a) := permSubEraseOfpermEqvCons hsub\n    have hl' : as.length = (bs.erase a).length := by\n      apply Nat.add_right_cancel (m := 1)\n      rw [length_erase_mem habs, \u2190hl, length_cons]\n    have hp : bs.isPermEqvTo (a :: (bs.erase a)) := permEqvToEraseCons habs\n    exact permEqvTrans (permEqvCons (permEqvOfPermSub hsub' hl')) (permEqvSymm hp)\n\ntheorem permEqvMemClosed {as bs : List \u03b1} {a : \u03b1} (hp : as \u2243 bs) : a \u2208 as \u2192 a \u2208 bs := by\n  rw [mem_nonzeroCount, mem_nonzeroCount, hp a]; intros; assumption\n\ntheorem permEqvEraseOfpermEqvCons {as bs : List \u03b1} {a : \u03b1} (h : (a :: as) \u2243 bs) : as \u2243 (bs.erase a) := fun a' => by\n  let ha := h a\n  byCases h' : a = a'\n  case inl => \n    cases h'\n    rw [count_erase]\n    simp only [beq_refl, Lean.Simp.ite_True]\n    rw [\u2190ha, count_self_cons, succ_sub_succ,  Nat.sub_zero]\n  case inr =>\n    let ha' := h a'\n    rw [count_neq_cons h'] at ha'\n    rw [count_erase, beq_neq h']\n    exact ha'\n\ntheorem permEqvLength {as bs : List \u03b1} : as \u2243 bs \u2192 as.length = bs.length :=\n  match as with\n  | [] =>\n    match bs with\n    | [] => fun _ => rfl\n    | b :: bs => fun h => by\n      let hb := h b\n      simp only [count] at hb; rw [filter_cons, beq_refl] at hb\n      simp at hb\n  | a :: as => fun h => by\n    let ha := h a\n    rw [count_self_cons] at ha\n    rw [length_cons, permEqvLength (permEqvEraseOfpermEqvCons h)]\n    exact length_erase_mem (mem_nonzeroCount.mpr (Nat.ltOfLtOfEq (Nat.zeroLtSucc (count as a)) ha))\n\ntheorem permEqv_filter_erase_true {as : List \u03b1} {a : \u03b1} {p : \u03b1 \u2192 Bool} (hpa : p a = true) (ha : a \u2208 as) :\n  (a :: filter p (List.erase as a)) \u2243 (filter p as) := by\n  intro b\n  byCases h : a = b\n  case inl =>\n    cases h\n    rw [count_self_cons, filter_erase_true hpa, count_erase]\n    simp only [beq_refl, Lean.Simp.ite_True]\n    have h' : 1 \u2264 count (filter p as) a := by\n      apply Nat.succLeOfLt\n      apply mem_nonzeroCount.mp\n      apply mem_filter_of_prop ha hpa\n    exact Eq.symm (le_subAdd h')\n  case inr =>\n    rw [count_neq_cons h, filter_erase_true hpa, count_erase, beq_neq h]\n    simp\n\ntheorem permEqvFilter {as bs : List \u03b1} (p : \u03b1 \u2192 Bool) : as \u2243 bs \u2192 (filter p as) \u2243 (filter p bs) :=\n  match as with\n  | [] => by\n    intro h\n    rw [length_zero_iff_nil.mp $ Eq.symm (permEqvLength h)]\n    exact permEqvRefl\n  | a :: as => by\n    intro h\n    rw [filter_cons]\n    have h' := permEqvFilter p (permEqvEraseOfpermEqvCons h)\n    cases hpa : p a with\n    | true =>\n      simp only [beq_refl, Lean.Simp.ite_True]\n      apply permEqvTrans (permEqvCons h')\n      have ha : a \u2208 bs := by rw [mem_nonzeroCount, \u2190h a]; simp [zeroLtSucc]\n      exact permEqv_filter_erase_true (as := bs) hpa ha\n    | false =>\n      refine permEqvTrans h' ?_\n      rw [filter_erase_false hpa]\n      exact permEqvRefl\n\nend Count\n\nend List\n", "meta": {"author": "IPDSnelting", "repo": "tba-2021", "sha": "b6390e55b768423d3266969e81d19290129c5914", "save_path": "github-repos/lean/IPDSnelting-tba-2021", "path": "github-repos/lean/IPDSnelting-tba-2021/tba-2021-b6390e55b768423d3266969e81d19290129c5914/TBA/Eulerian/List.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514762, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3553982395095269}}
{"text": "/-\nCopyright (c) 2021 R\u00e9my Degenne. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: R\u00e9my Degenne\n-/\nimport measure_theory.measure.measure_space_def\nimport tactic.auto_cases\nimport tactic.tidy\nimport tactic.with_local_reducibility\n/-!\n# Tactics for measure theory\n\nCurrently we have one domain-specific tactic for measure theory: `measurability`.\n\nThis tactic is to a large extent a copy of the `continuity` tactic by Reid Barton.\n-/\n\n/-!\n### `measurability` tactic\n\nAutomatically solve goals of the form `measurable f`, `ae_measurable f \u03bc` and `measurable_set s`.\n\nMark lemmas with `@[measurability]` to add them to the set of lemmas\nused by `measurability`. Note: `to_additive` doesn't know yet how to\ncopy the attribute to the additive version.\n-/\n\n/-- User attribute used to mark tactics used by `measurability`. -/\n@[user_attribute]\nmeta def measurability : user_attribute :=\n{ name := `measurability,\n  descr := \"lemmas usable to prove (ae)-measurability\" }\n\n/- Mark some measurability lemmas already defined in `measure_theory.measurable_space_def` and\n`measure_theory.measure_space_def` -/\nattribute [measurability]\n  measurable_id\n  measurable_id'\n  ae_measurable_id\n  ae_measurable_id'\n  measurable_const\n  ae_measurable_const\n  ae_measurable.measurable_mk\n  measurable_set.empty\n  measurable_set.univ\n  measurable_set.compl\n  subsingleton.measurable_set\n  measurable_set.Union\n  measurable_set.Inter\n  measurable_set.union\n  measurable_set.inter\n  measurable_set.diff\n  measurable_set.symm_diff\n  measurable_set.ite\n  measurable_set.cond\n  measurable_set.disjointed\n  measurable_set.const\n  measurable_set.insert\n  measurable_set_eq\n  finset.measurable_set\n  measurable_space.measurable_set_top\n\nnamespace tactic\n\n/--\nTactic to apply `measurable.comp` when appropriate.\n\nApplying `measurable.comp` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove measurable is actually\n  constant, and that constant is a function application `f z`, then\n  measurable.comp would produce new goals `measurable f`, `measurable\n  (\u03bb _, z)`, which is silly. We avoid this by failing if we could\n  apply `measurable_const`.\n\n* measurable.comp will always succeed on `measurable (\u03bb x, f x)` and\n  produce new goals `measurable (\u03bb x, x)`, `measurable f`. We detect\n  this by failing if a new goal can be closed by applying\n  measurable_id.\n-/\nmeta def apply_measurable.comp : tactic unit :=\n`[fail_if_success { exact measurable_const };\n  refine measurable.comp _ _;\n  fail_if_success { exact measurable_id }]\n\n/--\nTactic to apply `measurable.comp_ae_measurable` when appropriate.\n\nApplying `measurable.comp_ae_measurable` is not always a good idea, so we have some\nextra logic here to try to avoid bad cases.\n\n* If the function we're trying to prove measurable is actually\n  constant, and that constant is a function application `f z`, then\n  `measurable.comp_ae_measurable` would produce new goals `measurable f`, `ae_measurable\n  (\u03bb _, z) \u03bc`, which is silly. We avoid this by failing if we could\n  apply `ae_measurable_const`.\n\n* `measurable.comp_ae_measurable` will always succeed on `ae_measurable (\u03bb x, f x) \u03bc` and\n  can produce new goals (`measurable (\u03bb x, x)`, `ae_measurable f \u03bc`) or\n  (`measurable f`, `ae_measurable (\u03bb x, x) \u03bc`). We detect those by failing if a new goal can be\n  closed by applying `measurable_id` or `ae_measurable_id`.\n-/\nmeta def apply_measurable.comp_ae_measurable : tactic unit :=\n`[fail_if_success { exact ae_measurable_const };\n  refine measurable.comp_ae_measurable _ _;\n  fail_if_success { exact measurable_id };\n  fail_if_success { exact ae_measurable_id }]\n\n/--\nWe don't want the intro1 tactic to apply to a goal of the form `measurable f`, `ae_measurable f \u03bc`\nor `measurable_set s`. This tactic tests the target to see if it matches that form.\n -/\nmeta def goal_is_not_measurable : tactic unit :=\ndo t \u2190 tactic.target,\n  match t with\n  | `(measurable %%l) := failed\n  | `(ae_measurable %%l %%r) := failed\n  | `(measurable_set %%l) := failed\n  | _ := skip\n  end\n\n/-- List of tactics used by `measurability` internally. The option `use_exfalso := ff` is passed to\nthe tactic `apply_assumption` in order to avoid loops in the presence of negated hypotheses in\nthe context. -/\nmeta def measurability_tactics (md : transparency := semireducible) : list (tactic string) :=\n[\n  propositional_goal >> tactic.interactive.apply_assumption none {use_exfalso := ff}\n                        >> pure \"apply_assumption {use_exfalso := ff}\",\n  goal_is_not_measurable >> intro1\n                        >>= \u03bb ns, pure (\"intro \" ++ ns.to_string),\n  apply_rules [] [``measurability] 50 { md := md }\n                        >> pure \"apply_rules with measurability\",\n  apply_measurable.comp >> pure \"refine measurable.comp _ _\",\n  apply_measurable.comp_ae_measurable\n                        >> pure \"refine measurable.comp_ae_measurable _ _\",\n  `[ refine measurable.ae_measurable _ ]\n                        >> pure \"refine measurable.ae_measurable _\",\n  `[ refine measurable.ae_strongly_measurable _ ]\n                        >> pure \"refine measurable.ae_strongly_measurable _\"\n]\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nSolve goals of the form `measurable f`, `ae_measurable f \u03bc`, `ae_strongly_measurable f \u03bc` or\n`measurable_set s`. `measurability?` reports back the proof term it found.\n-/\nmeta def measurability\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md                 := if bang.is_some then semireducible else reducible,\n    measurability_core := tactic.tidy { tactics := measurability_tactics md, ..cfg },\n    trace_fn           := if trace.is_some then show_term else id in\ntrace_fn measurability_core\n\n/-- Version of `measurability` for use with auto_param. -/\nmeta def measurability' : tactic unit := measurability none none {}\n\n/--\n`measurability` solves goals of the form `measurable f`, `ae_measurable f \u03bc`,\n`ae_strongly_measurable f \u03bc` or `measurable_set s` by applying lemmas tagged with the\n`measurability` user attribute.\n\nYou can also use `measurability!`, which applies lemmas with `{ md := semireducible }`.\nThe default behaviour is more conservative, and only unfolds `reducible` definitions\nwhen attempting to match lemmas with the goal.\n\n`measurability?` reports back the proof term it found.\n-/\nadd_tactic_doc\n{ name := \"measurability / measurability'\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.measurability, `tactic.interactive.measurability'],\n  tags := [\"lemma application\"] }\n\nend interactive\n\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/measure_theory/tactic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813031051514762, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.35539823950952687}}
{"text": "import category_theory.category\nimport category_theory.isomorphism\nimport category_theory.types\nimport data.fintype.basic\nimport data.real.basic\nimport magma\n\nopen category_theory\n\nnamespace exercises\n\nvariables  {C: Type*} [category C]\nvariables (A B D A' B' D' X T : C)\n\n-- Exercise 1 page 40\n-- the inverse of identity is itself\nexample : is_iso(\ud835\udfd9 A) := { inv := \ud835\udfd9 A}\n-- given two morphisms, if f is isomorphic and has a reverse g, then g is iso and has reverse f\nexample (f : A \u27f6 B) (g : B \u27f6 A) (isof : is_iso f) (i : inv f = g) : is_iso g := {inv := f}\n-- composition of isomorphisms are isomorphisms \nexample (f : A \u27f6 B) (k : B \u27f6 D) (isof : is_iso f) (isok : is_iso k) : is_iso (f \u226b k) := \n{inv := inv k \u226b inv f}\n\n-- Exercise 2 page 42\nlemma unique_inverse  (f : A \u2245 B) (f' : A \u2245 B) (g k : B \u27f6 A) :\n    f.hom = f'.hom -> f.inv = g \u2192 f'.inv = k \u2192 g = k :=\nbegin\n    intros ff' finv f'inv,\n\n    calc g = f.inv : by {rw finv}\n    ... = f.inv \u226b f'.hom \u226b f'.inv : by {simp}\n    ... = f.inv \u226b f.hom \u226b f'.inv : by {rw \u2190 ff'}\n    ... = \ud835\udfd9 B \u226b f'.inv : by {simp}\n    ... = f'.inv : by simp\n    ... = k : by {rw f'inv},\nend\n\nsection alternative_unique_inverse\n\n    lemma unique_inverse2 (g k: B \u27f6 A)(f: A \u27f6 B) (h1: f \u226b g = \ud835\udfd9 A)(h1a: g \u226b f = \ud835\udfd9 B)(h2: f \u226b k = \ud835\udfd9 A): g = k :=\n    begin\n        calc g = g \u226b \ud835\udfd9 A : by rw [category.comp_id]\n        ... = g \u226b (f \u226b k) : by rw \u2190 h2\n        ... = (g \u226b f) \u226b k : by rw [category.assoc]\n        ... = k : by rw [h1a, category.id_comp]\n    end\n\n    lemma unique_inverse3 (g k: B \u27f6 A)(f: A \u27f6 B) [is_iso f] (h1: inv f = g)(h2: inv f = k): g = k := \n    begin\n        calc g = g \u226b \ud835\udfd9 A : by rw category.comp_id\n        ... = g \u226b (f \u226b inv f) : by rw is_iso.hom_inv_id\n        ... = g \u226b (f \u226b k) : by rw h2\n        ... = (g \u226b f) \u226b k : by rw category.assoc\n        ... = (inv f \u226b f) \u226b k : by rw h1\n        ... = (\ud835\udfd9 B) \u226b k : by rw is_iso.inv_hom_id\n        ... = k : by rw category.id_comp,\n    end\nend alternative_unique_inverse\n\n\n-- Exercise 3a page 43\nlemma iso_cancel_left  (f : A \u2245 B) (h k : D \u27f6 A) :\nh \u226b f.hom = k \u226b f.hom \u2192 h = k :=\nbegin\n    intros fhfk,\n    calc h = (h \u226b f.hom) \u226b f.inv : by {simp}\n    ... = (k \u226b f.hom) \u226b f.inv : by {rw fhfk}\n    ... = k : by {simp}\nend\n\nsection alternative_ex3_page43\n    lemma iso_cancel_left2 (f: A \u27f6 B) [is_iso f](h k: D \u27f6 A): h \u226b f = k \u226b f \u2192 h = k := \n    begin\n        let g: B \u27f6 A := is_iso.inv f,\n        intros given,\n        /- given : h \u226b f = k \u226b f -/\n        calc h = h \u226b \ud835\udfd9 A : by rw category.comp_id\n        ... = h \u226b (f \u226b g) : by rw is_iso.hom_inv_id\n        ... = (h \u226b f) \u226b g: by rw category.assoc\n        ... = (k \u226b f) \u226b g: by rw given\n        ... = k \u226b (f \u226b g): by rw category.assoc\n        ... = k \u226b \ud835\udfd9 A : by rw is_iso.hom_inv_id\n        ... = k : by rw category.comp_id\n    end\nend alternative_ex3_page43\n\n-- Exercise 3b page 43\nlemma iso_cancel_right (f : A \u2245 B) (h k : B \u27f6 D) :\nf.hom \u226b h = f.hom \u226b k \u2192 h = k :=\nbegin\n    intros fhfk,\n    calc h = f.inv \u226b (f.hom \u226b h)  : by {simp}\n    ... = f.inv \u226b (f.hom \u226b k) : by {rw fhfk}\n    ... = k : by {simp}\nend\n\ninductive Two : Type\n| one | two\n\ndef swap : Two \u2192 Two\n| Two.one := Two.two\n| Two.two := Two.one\n\n-- Exercise 3c page 43\nlemma iso_cant_cancel_right_left :\n\u2203 (A : Type) (f : A \u2245 A) (h k : A \u27f6 A), f.hom \u226b h = k \u226b f.hom \u2227 h \u2260 k :=\nbegin\n    --let swap := \u21be swap,\n\n    let swapswap : swap \u2218 swap = id, {apply funext, intro x, cases x, refl, refl },\n    --let swapswap : swap \u226b swap = \ud835\udfd9 Two, {funext ,},\n\n    let f : Two \u2245 Two := \u27e8 swap, swap, swapswap, swapswap \u27e9 ,\n    let h := \u03bb (n: Two), Two.one,\n    let k := \u03bb (n: Two), Two.two,\n    have prop : f.hom \u226b h = k \u226b f.hom, {apply funext, intro x, cases x, refl, refl,},\n\n    use [Two,f, h, k],\n\n    split,\n    {exact prop},\n\n    have foo : h Two.one \u2260 k Two.one, {change Two.one \u2260 Two.two, simp,},\n\n    --apply funext at H,\n    intro H,\n\n--rw funext at H,\n\n\n    rw H at foo,\n    exact foo (by refl),\nend\n\n--local attribute classical.prop_decidable\n\n lemma point_diff {\u03b1 \u03b2 : Type} {f1 f2 : \u03b1 \u2192 \u03b2} (dif: \u2203 x, f1 x \u2260 f2 x) : f1 \u2260 f2 :=\n begin\n simp,\n by_contradiction H,\n rw H at dif,\n cases dif with x hx,\n exact hx rfl ,\n end\n\nlemma iso_cant_cancel_right_left' :\n\u2203 (A : Type) (f : A \u2245 A) (h k : A \u27f6 A), f.hom \u226b h = k \u226b f.hom \u2227 h \u2260 k :=\nbegin\n    have swapinv : swap \u2218 swap = id, {funext, cases x; refl},\n\n    let f : Two \u2245 Two := \u27e8 swap, swap, swapinv, swapinv \u27e9 ,\n    let h := \u03bb (n: Two), Two.one,\n    let k := \u03bb (n: Two), Two.two,\n\n    use [Two, f, h, k],\n\n    split,\n    { refl },\n    { apply point_diff,\n      use Two.one}\nend\n\n\nopen bool\n\nlemma iso_cant_cancel_right_left'' :\n\u2203 (A : Type) (f : A \u2245 A) (h k : A \u27f6 A), f.hom \u226b h = k \u226b f.hom \u2227 h \u2260 k :=\nbegin\n    have selfinv : bnot \u2218 bnot = id, {funext, simp},\n\n    -- have f : bool \u2245 bool := \u27e8 bnot, bnot, selfinv, selfinv \u27e9,\n    have f : iso bool bool , {exact \u27e8 bnot, bnot, selfinv, selfinv \u27e9},\n    let h := (\u03bb (n: bool), tt),\n    let k := (\u03bb (n: bool), ff),\n\n    -- fixme if I pass f as second argument instead of the expansion, things break\n    -- f for some reason doesn't see the \"contents\" like in iso_cant_cancel_right_left'\n    --use [bool, f, h, k],\n    use [bool, \u27e8 bnot, bnot, selfinv, selfinv \u27e9, h, k],\n\n    split,\n    { refl },\n    { apply point_diff,\n      use tt}\nend\n\ndef has_retraction {A B : C} (f : A \u27f6 B) := \u2203 r, f \u226b r = \ud835\udfd9 A\ndef has_section {A B : C} (f : A \u27f6 B) := \u2203 s, s \u226b f = \ud835\udfd9 B\n\n-- Exercise 6 page 52\nlemma retraction_divides {T: C} (f : A \u27f6 B) (ret: has_retraction f) (g: A \u27f6 T) :\n\u2203 t : B \u27f6 T, f \u226b t = g :=\nbegin\n    cases ret with s hS,\n    let t := s \u226b g,\n    use t,\n    calc f \u226b t = f \u226b (s \u226b g) : by {refl}\n    ... = (f \u226b s) \u226b g : by {simp}\n    ... = \ud835\udfd9 A \u226b g : by {rw hS}\n    ... = g : by {simp}\nend\n\n-- Exercise 7 page 53\nlemma section_cancels_right {T: C} (f : A \u27f6 B) (sec: has_section f)  (t\u2081 t\u2082: B \u27f6 T): \nf \u226b t\u2081 = f \u226b t\u2082 \u2192 t\u2081 = t\u2082 :=\nbegin\n    intros h,\n    cases sec with s hS,\n\n    calc t\u2081 = \ud835\udfd9 B \u226b t\u2081 : by {rw category.id_comp}\n    ... = (s \u226b f) \u226b t\u2081 : by {rw \u2190 hS}\n    ... = s \u226b f \u226b t\u2081 : by {apply category.assoc}\n    ... = s \u226b f \u226b t\u2082 : by {rw h}\n    ... = (s \u226b f) \u226b t\u2082 : by {rw category.assoc}\n    ... = \ud835\udfd9 B \u226b t\u2082 : by {rw hS}\n    ... = t\u2082 : by {apply category.id_comp},\nend \n\n-- Exercise 8 page 54\nlemma section_comp_section_has_section (f : A \u27f6 B) (g : B \u27f6 D) (secf: has_section f) (secg: has_section g) :\nhas_section (f \u226b g) :=\nbegin\n    cases secf with sf hsf,\n    cases secg with sg hsg,\n\n    use (sg \u226b sf),\n    calc (sg \u226b sf ) \u226b f \u226b g = sg \u226b (sf \u226b f) \u226b g : by {simp}\n    ... = sg \u226b \ud835\udfd9 B  \u226b g : by {rw hsf,}\n    ... = sg \u226b g : by {simp}\n    ... = \ud835\udfd9 D : by {rw hsg}\nend\n\nclass idempotent {X: C} (endo : X \u27f6 X) : Prop :=\n(repeat : endo \u226b endo = endo)\n\n@[reducible]\ndef is_retraction {A B : C} (f : A \u27f6 B) (r : B \u27f6 A) := f \u226b r = \ud835\udfd9 A\n\nlemma is_retraction_retracts (f : A \u27f6 B) (r : B \u27f6 A) (ret: is_retraction f r) :\nhas_retraction f := \u27e8 r, ret \u27e9\n\n\n-- Exercise 9a page 54\nlemma retraction_section_is_idemp {f : A \u27f6 B} {r : B \u27f6 A} (ret: is_retraction f r)\n: idempotent (r \u226b f) :=\nbegin\n    split, -- this applies the constructor\n    unfold is_retraction at ret, -- this is ugly, i shouldn't need it\n\n    calc (r \u226b f) \u226b r \u226b f = r \u226b (f \u226b r) \u226b f : by {simp}\n    ... = r \u226b f : by simp [ret]\nend\n\nopen category_theory.iso\n\n-- Exercise 9b page 54\nlemma retraction_with_iso_is_id (I : A \u2245 B) (r : B \u27f6 A) (ret: is_retraction I.hom r) :\nr \u226b I.hom = \ud835\udfd9 B :=\nbegin\n    let f := I.hom,\n    let g := I.inv,\n    unfold is_retraction at ret,  -- this is ugly, i shouldn't need it\n\n    calc r \u226b f = \ud835\udfd9 B \u226b r \u226b f : by rw category.id_comp\n    ... = (g \u226b f) \u226b r \u226b f : by rw inv_hom_id\n    ... = g \u226b (f \u226b r) \u226b f : by simp\n    ... = g \u226b \ud835\udfd9 A \u226b f : by {rw ret}\n    ... = g \u226b f : by {simp} \n    ... = \ud835\udfd9 B : by {simp}\nend\n\n\n-- Exercise 10 page 55\nlemma exercise_10 (If : A \u2245 B) (Ig : B \u2245 D) :\n  inv (If.hom \u226b Ig.hom) = Ig.inv \u226b If.inv :=\nbegin\n    split, --why is this enough?\nend\n\n@[derive decidable_eq]\ninductive People11 : Type\n| Fatima | Omer | Alysia \n\n@[derive decidable_eq]\ninductive Drinks11 : Type\n| Coffee | Tea | Cocoa \n\n-- Exercise 11a page 55\nexample : People11 \u2245 Drinks11 :=\nbegin\n    let f : People11 \u2192 Drinks11 :=\n     \u03bb p, match p with\n            | People11.Fatima := Drinks11.Coffee\n            | People11.Omer := Drinks11.Tea\n            | People11.Alysia := Drinks11.Cocoa\n            end,\n\n    let g : Drinks11 \u2192 People11 :=\n     \u03bb d, match d with\n            | Drinks11.Coffee := People11.Fatima\n            | Drinks11.Tea := People11.Omer\n            | Drinks11.Cocoa := People11.Alysia\n            end,\n\n    have id1 : f \u2218 g = id, {funext, cases x; refl},\n    have id2 : g \u2218 f = id, {funext, cases x; refl},\n    exact \u27e8 \u21bef, \u21beg \u27e9,\nend\n\nuniverses v u\n\n-- this is ugly, why do I need to define this?\nabbreviation from_hom {\u03b1 \u03b2 : Type} (f : \u03b1 \u27f6 \u03b2) : \u03b1 \u2192 \u03b2 := f\n\nlemma type_isos_are_injective {A B: Type} (i: A \u2245 B) :\n\u2200 (a1 a2 : A), a1 \u2260 a2 \u2192 i.hom a1 \u2260 i.hom a2 :=\nbegin\n    intros a1 a2 ne h,\n    suffices H: a1 = a2,\n    {exact ne H},\n    { calc a1 = from_hom (\ud835\udfd9 A) a1 : by {refl}\n        ... = (i.hom \u226b i.inv) a1 : by {rw i.hom_inv_id}\n        ... = i.inv (i.hom a1) : by {refl}\n        ... = i.inv (i.hom a2) : by {rw h,}\n        ... = (i.hom \u226b i.inv) a2 : by {simp}\n        ... = from_hom (\ud835\udfd9 A) a2 : by {rw i.hom_inv_id}\n        ... = a2 : by {refl},\n    } \nend\n\nlemma type_isos_are_surjective {A B: Type} (i: A \u2245 B) :\n\u2200 (b : B), \u2203 (a : A), i.hom a = b :=\nbegin\n    intros b,\n    use i.inv b,\n    calc i.hom (i.inv b) = (i.inv \u226b i.hom) b : by {simp}\n        ... = from_hom (\ud835\udfd9 B) b : by {rw i.inv_hom_id}\n        ... = b : by {refl}\nend\n\n-- Exercise 11b page 55\nexample :  (People11 \u2245 bool) \u2192 false :=\nbegin\n    intros i,\n    by_cases i.inv tt = i.inv ff,\n    {\n        -- when i.inv tt = i.inv ff\n        apply type_isos_are_injective (symm i) _ _ _ h,\n        simp,\n    },\n    {\n        -- when i.inv tt \u2260 i.inv ff\n        have ugly : \u2203 (p : People11), p \u2260 i.inv tt \u2227 p \u2260 i.inv ff,\n        {   cases i.inv tt,\n            cases i.inv ff,\n            use People11.Alysia,\n            use People11.Alysia, simp,\n            use People11.Omer, simp,\n            cases i.inv ff,\n            use People11.Alysia, simp, simp,\n            use People11.Alysia,\n            use People11.Fatima, simp,\n            cases i.inv ff,\n            use People11.Omer, simp,\n            use People11.Fatima, simp,\n            use People11.Omer\n        },\n\n        cases ugly with u hu,\n        cases type_isos_are_surjective (symm i) u with a ha,\n        change i.inv a = u at ha,\n        cases a,\n        exact hu.2 ha.symm,\n        exact hu.1 ha.symm,\n    }\nend\n\n\n\nopen fintype \n\ninstance people_fintype: fintype People11 := {\n    elems := [People11.Alysia, People11.Fatima, People11.Omer].to_finset,\n    complete := by { intro x, cases x; simp }\n} \n\n-- Exercise 11b page 55\nexample  : (People11 \u2243 bool) \u2192 false :=\nbegin\nsuffices cards : card People11 \u2260 card bool,\n{ intros h,\n  exact cards (card_congr h)\n},\n\n{ change 3 \u2260 2, finish,}\nend\n\n\n-- Exercise 1 page 66\nexample :\n (\u03bb x: \u211d, 2 * x) \u2218 (\u03bb x: \u211d, 1/2 * x) = id \n \u2227\n (\u03bb x: \u211d, 1/2 * x) \u2218 (\u03bb x: \u211d, 2 * x) = id :=\nbegin\nsplit;\n{ funext,\n  simp,\n  ring}\nend\n\n-- Exercise 2 page 66\n\ninductive OddEven : Type\n| odd | even\n\ndef add_odd_even : OddEven \u2192 OddEven \u2192 OddEven\n| OddEven.odd OddEven.odd := OddEven.even\n| OddEven.even OddEven.even := OddEven.even\n| OddEven.odd OddEven.even := OddEven.odd\n| OddEven.even OddEven.odd := OddEven.odd\n\ninductive PosNeg : Type\n| pos | neg\n\ndef mul_pos_neg : PosNeg \u2192 PosNeg \u2192 PosNeg\n| PosNeg.pos PosNeg.pos := PosNeg.pos\n| PosNeg.neg PosNeg.neg := PosNeg.pos\n| PosNeg.pos PosNeg.neg := PosNeg.neg\n| PosNeg.neg PosNeg.pos := PosNeg.neg\n\nopen magma\n\ninstance magma_oddeven: magma OddEven := {mul := add_odd_even} \ninstance magma_posneg: magma PosNeg := {mul := mul_pos_neg} \n\ndef OddEvenMagma : Magma := bundled.of OddEven\ndef PosNegMagma : Magma := bundled.of PosNeg\n\ndef oddeven2posneg : OddEven -> PosNeg\n| OddEven.odd := PosNeg.neg\n| OddEven.even := PosNeg.pos\n\ndef posneg2oddeven : PosNeg -> OddEven \n| PosNeg.neg := OddEven.odd\n| PosNeg.pos := OddEven.even\n\ndef oe2pn :  OddEvenMagma \u27f6 PosNegMagma :=\n{ to_fun := oddeven2posneg,\n  preserves :=  \u03bb x y, by {cases x; cases y; refl} }\n\ndef pn2oe :  PosNegMagma \u27f6 OddEvenMagma :=\n{ to_fun := posneg2oddeven,\n  preserves :=  \u03bb x y, by {cases x; cases y; refl} }\n\nexample : OddEvenMagma \u2245 PosNegMagma :=\nbegin\n    refine \u27e8oe2pn, pn2oe, _, _ \u27e9 ;\n    { apply magma_hom_ext, intros x, cases x; refl}\nend\n\n-- Exercise 3 page 70\n\ninstance r_plus_magma_alpha_has_neg : has_neg r_plus_Magma.\u03b1 :=  {\n    neg := by {\n        intros x,\n        have isR : r_plus_Magma.\u03b1 = \u211d, refl,\n        rw isR at *,\n        exact -x,\n    }\n}\n\ndef rplus_negate : r_plus_Magma \u27f6 r_plus_Magma :=\n{to_fun := has_neg.neg,\n preserves := by {\n    intros x y,\n    have isR : r_plus_Magma.\u03b1 = \u211d, refl,\n    rw isR at *,\n    have isP : r_plus_Magma.str.mul = real.has_add.add, refl,\n    rw isP,\n    norm_num,\n    rw add_comm,\n }\n}\n\nlemma rplus_negate_iso : rplus_negate \u226b rplus_negate = \ud835\udfd9 r_plus_Magma :=\nbegin\nunfold rplus_negate category_struct.comp magma_hom_comp category_struct.id magma_id,\nsimp,\napply magma_hom_ext,\nintros x,\nhave isR : r_plus_Magma.\u03b1 = \u211d, refl,\nrw isR at *,\nsimp,\nrw neg_neg,\nend\n\n-- Exercise 1a page 70\nexample (A B C: Type) (f: A \u27f6 B) (g: B \u27f6 C) (a1 a2 : unit \u27f6 A) :\na1 \u226b f = a2 \u226b f \u2192 a1 \u226b f \u226b g = a2 \u226b f \u226b g :=\nbegin\n    intros h ,\n    rw \u2190 category.assoc,\n    rw h,\n    simp,\nend\n\n-- Exercise 2 page 71\nexample (A B C: Type) (f: A \u27f6 B) (g: B \u27f6 C) (h: A \u27f6 C) (hcomp: h = f \u226b g):\n\u2200 a : unit \u27f6 A, \u2203 b : unit \u27f6 B, a \u226b f \u226b g = b \u226b g :=\nbegin\nintros a,\nexact \u27e8 a \u226b f, by simp \u27e9\nend\n\ndef aba (A : Type u) (B : Type u) : Prop := nonempty (A \u27f6 B)\ndef has_point (A : Type u) : Prop := nonempty (unit \u2192 A)\n\nlocal infix <| := aba\n\n-- Exercise 1 page 99\nexample {A : Type*} {B : Type*}  (h: \u00ac (has_point A \u2227  \u00ac has_point B)) : A <| B :=\nbegin\npush_neg at h,\nby_cases has:has_point A,\n    exact \u27e8 \u03bb _, (h has).some unit.star \u27e9,\n    exact \u27e8 \u03bb a, false.elim (has \u27e8\u03bb _, a\u27e9) \u27e9\nend\n\n-- Exercise 1 page 99 (another approach)\nexample(A B : Type*)[category Type*] (h: \u00ac(has_point A \u2227 \u00ac has_point B )) : A <| B:=\nbegin\n    push_neg at h,\n    use  \u03bb (a), (h(nonempty.intro (\u03bb(x),a))).some unit.star,\nend\n\ndef retractable (A : C) (B : C) := \u2203 (s : A \u27f6 B) (r : B \u27f6 A), s \u226b r = \ud835\udfd9 A\n\ninfix ` \u2264R `:50 := retractable\n\n\n-- Exercise 2R page 100\nexample : A \u2264R A := \u27e8 \ud835\udfd9 A, \ud835\udfd9 A, category.id_comp _\u27e9\n\n-- Exercise 2T page 100\nexample : A \u2264R B \u2192 B \u2264R D \u2192 A \u2264R D :=\nbegin\nintros ab bd,\nrcases ab with \u27e8abs, abr, hab\u27e9,\nrcases bd with \u27e8bds, bdr, hbd\u27e9,\nuse abs \u226b bds,\nuse bdr \u226b abr,\ncalc (abs \u226b bds) \u226b bdr \u226b abr = abs \u226b (bds \u226b bdr) \u226b abr : by simp\n... = \ud835\udfd9 A : by simp [hab, hbd],\nend\n\n\nstructure splitting {B: C} (e : B \u27f6 B) [idempotent e] :=\n(From: C)\n(s : From \u27f6 B)\n(r : B \u27f6 From)\n(ret: is_retraction s r)\n(is_idem: r \u226b s = e)\n\n-- Exercise 3 page 102\nlemma two_splittings_iso (e : B \u27f6 B) [idempotent e]\n(sp: splitting e) (sp': splitting e) : sp.From \u2245 sp'.From :=\nbegin\nrcases sp with  \u27e8A,  s,  r,  ret,  is_idem \u27e9,\nrcases sp' with \u27e8A', s', r', ret', is_idem'\u27e9,\nunfold is_retraction at ret ret',\n\nlet f  : A  \u27f6 A' := s \u226b e \u226b r',\nlet f' : A' \u27f6 A := s' \u226b e \u226b r,\n\nhave id1 : f \u226b f' = \ud835\udfd9 A, \n    {\n        calc f \u226b f' = s \u226b e \u226b (r' \u226b s') \u226b e \u226b r : by simp\n        ... = s \u226b e \u226b e \u226b e \u226b r : by rw is_idem'\n        ... = s \u226b (e \u226b e) \u226b e \u226b r : by simp\n        ... = s \u226b e \u226b e \u226b r : by simp [idempotent.repeat]\n        ... = s \u226b (e \u226b e) \u226b r : by simp\n        ... = s \u226b e \u226b r : by rw idempotent.repeat\n        ... = s \u226b r \u226b s \u226b r : by {rw \u2190 is_idem, simp}\n        ... = \ud835\udfd9 A : by simp [ret]\n    },\n\nhave id2 : f' \u226b f = \ud835\udfd9 A',\n    {\n        calc f' \u226b f = s' \u226b e \u226b r \u226b s \u226b e \u226b r' : by simp\n        ... = s' \u226b e \u226b (r \u226b s) \u226b e \u226b r' : by simp\n        ... = s' \u226b e \u226b e \u226b e \u226b r' : by rw is_idem\n        ... = s' \u226b (e \u226b e) \u226b e \u226b r' : by simp\n        ... = s' \u226b e \u226b e \u226b r' : by rw idempotent.repeat\n        ... = s' \u226b (e \u226b e) \u226b r' : by simp\n        ... = s' \u226b e \u226b r' : by rw idempotent.repeat\n        ... = s' \u226b r' \u226b s' \u226b r' : by {rw \u2190 is_idem', simp}\n        ... = \ud835\udfd9 A' : by simp [ret']\n    },\n\nexact \u27e8f, f'\u27e9 \nend\n\n\n-- Exercise 2 page 108\nexample  (p : A \u27f6 B) (q : B \u27f6 A) (h: p \u226b q \u226b p = p) : idempotent (p \u226b q) :=\nbegin\nsplit,\ncalc (p \u226b q) \u226b p \u226b q = (p \u226b q \u226b p) \u226b q : by simp\n    ... = p \u226b q : by rw h\nend\n\nexample  (p : A \u27f6 B) (q : B \u27f6 A) (h: p \u226b q \u226b p = p) : idempotent (q \u226b p) :=\nbegin\nsplit,\ncalc (q \u226b p) \u226b q \u226b p = q \u226b (p \u226b q \u226b p) : by simp\n    ... = q \u226b p : by rw h\nend\n\n-- Exercise 2* page 108\nexample  (p : A \u27f6 B) (q : B \u27f6 A) (h: p \u226b q \u226b p = p) :\n\u2203 (q' : B \u27f6 A), (p \u226b q' \u226b p = p)  \u2227  (q' \u226b p \u226b q' = q') :=\nbegin\nuse q \u226b p \u226b q,\nsplit,\n\n    calc p \u226b (q \u226b p \u226b q) \u226b p = (p \u226b q \u226b p) \u226b q \u226b p : by simp\n        ... = p : by rw [h, h],\n\n    calc (q \u226b p \u226b q) \u226b p \u226b q \u226b p \u226b q = q \u226b (p \u226b q \u226b p) \u226b q \u226b p \u226b q : by simp\n        ... = q \u226b p \u226b q \u226b p \u226b q : by rw h\n        ... = q \u226b (p \u226b q \u226b p) \u226b q : by simp\n        ... = q \u226b p \u226b q : by rw h\nend\n\n-- Exercise 1* page 108\ndef inclusionNZ : \u2115 \u27f6 \u2124 := \u03bb n:\u2115, int.of_nat n\ndef retractionZN : \u2124 \u27f6 \u2115 := \u03bb i:\u2124, int.to_nat i\n\nexample : is_retraction inclusionNZ retractionZN :=\nbegin\nunfold is_retraction,\nunfold inclusionNZ,\nunfold retractionZN,\nfunext,\nsimp\nend\n\n\nexample (f : \u2124 \u27f6 \u2115): \u00ac is_retraction f inclusionNZ :=\nbegin\nunfold is_retraction,\nintros h,\nrewrite [types_id, types_comp] at h,\nhave neg : id (-5 : \u2124) < 0 , by norm_num,\nhave isnat : f (-5) >= 0, by {apply zero_le},\nhave pos : (inclusionNZ \u2218 f) (-5) >= 0, by simp [isnat],\n\nrw h at pos,\nlinarith,\nend\n\n-- Exercise 2 page 126\nlemma fp_of_retraction_fp\n    {T : C} {s : A \u27f6 X} {r : X \u27f6 A} (ret : is_retraction s r)\n    (h : \u2200 (f : X \u27f6 X), \u2203 (x : T \u27f6 X), x \u226b f = x) :\n    \u2200 (g : A \u27f6 A), \u2203 (x : T \u27f6 A), x \u226b g = x :=\nbegin\n    intros g,\n    cases h (r \u226b g \u226b s) with x hx,\n    use x \u226b r,\n    unfold is_retraction at ret,\n    calc (x \u226b r) \u226b g = x \u226b r \u226b g \u226b \ud835\udfd9 A : by {simp}\n        ... = x \u226b r \u226b g \u226b (s \u226b r) : by {rw ret,}\n        ... = (x \u226b r \u226b g \u226b s) \u226b r : by simp\n        ... = x \u226b r : by rw hx,\nend\n\n-- Exercise 3 page 126\nnamespace ex3_page126\nsection wrap\n\nparameters (CC : Type*) [category CC]\nparameters (Circle Disk Term : CC)\n\nparameter antipodal : (Circle \u27f6 Circle)\nparameter j : (Circle \u27f6 Disk)\n-- Notice, no axioms on j, so it's arbitrary, not a real inclusion\n\n\n-- axioms\nlemma antipodal_no_fp : \u00ac \u2203 (x : Term \u27f6 Circle), x \u226b antipodal = x  := sorry\nlemma has_fp : \u2200 (f : Disk \u27f6 Disk), \u2203 (x : Term \u27f6 Disk), x \u226b f = x  := sorry\n\n\ninclude Term\ninclude antipodal\nexample : \u00ac \u2203 (r : Disk \u27f6 Circle), is_retraction j r :=\nbegin\n    by_contradiction contra, \n    cases contra with r ret,\n    have exist_fp := fp_of_retraction_fp Circle Disk ret (has_fp CC Disk Term) antipodal,\n    have not_exist_fp := antipodal_no_fp CC Circle Term antipodal,\n    exact not_exist_fp exist_fp,\nend\nend wrap\nend ex3_page126\n\nend exercises", "meta": {"author": "paraseba", "repo": "conceptual-mathematics-in-lean", "sha": "cc7877f528bfaf99a024844744e023acbfcadb1d", "save_path": "github-repos/lean/paraseba-conceptual-mathematics-in-lean", "path": "github-repos/lean/paraseba-conceptual-mathematics-in-lean/conceptual-mathematics-in-lean-cc7877f528bfaf99a024844744e023acbfcadb1d/src/article2.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.531209388216861, "lm_q2_score": 0.6688802537704064, "lm_q1q2_score": 0.3553154703957163}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Johannes H\u00f6lzl, Reid Barton, Sean Leather\n\n! This file was ported from Lean 3 source module category_theory.concrete_category.bundled\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Tactic.Lint.Default\n\n/-!\n# Bundled types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n`bundled c` provides a uniform structure for bundling a type equipped with a type class.\n\nWe provide `category` instances for these in `category_theory/unbundled_hom.lean`\n(for categories with unbundled homs, e.g. topological spaces)\nand in `category_theory/bundled_hom.lean` (for categories with bundled homs, e.g. monoids).\n-/\n\n\nuniverse u v\n\nnamespace CategoryTheory\n\nvariable {c d : Type u \u2192 Type v} {\u03b1 : Type u}\n\n#print CategoryTheory.Bundled /-\n/-- `bundled` is a type bundled with a type class instance for that type. Only\nthe type class is exposed as a parameter. -/\n@[nolint has_nonempty_instance]\nstructure Bundled (c : Type u \u2192 Type v) : Type max (u + 1) v where\n  \u03b1 : Type u\n  str : c \u03b1 := by infer_instance\n#align category_theory.bundled CategoryTheory.Bundled\n-/\n\nnamespace Bundled\n\n#print CategoryTheory.Bundled.of /-\n-- Usually explicit instances will provide their own version of this, e.g. `Mon.of` and `Top.of`.\n/-- A generic function for lifting a type equipped with an instance to a bundled object. -/\ndef of {c : Type u \u2192 Type v} (\u03b1 : Type u) [str : c \u03b1] : Bundled c :=\n  \u27e8\u03b1, str\u27e9\n#align category_theory.bundled.of CategoryTheory.Bundled.of\n-/\n\ninstance : CoeSort (Bundled c) (Type u) :=\n  \u27e8Bundled.\u03b1\u27e9\n\n#print CategoryTheory.Bundled.coe_mk /-\n@[simp]\ntheorem coe_mk (\u03b1) (str) : (@Bundled.mk c \u03b1 str : Type u) = \u03b1 :=\n  rfl\n#align category_theory.bundled.coe_mk CategoryTheory.Bundled.coe_mk\n-/\n\n#print CategoryTheory.Bundled.map /-\n/-\n`bundled.map` is reducible so that, if we define a category\n\n  def Ring : Type (u+1) := induced_category SemiRing (bundled.map @ring.to_semiring)\n\ninstance search is able to \"see\" that a morphism R \u27f6 S in Ring is really\na (semi)ring homomorphism from R.\u03b1 to S.\u03b1, and not merely from\n`(bundled.map @ring.to_semiring R).\u03b1` to `(bundled.map @ring.to_semiring S).\u03b1`.\n-/\n/-- Map over the bundled structure -/\n@[reducible]\ndef map (f : \u2200 {\u03b1}, c \u03b1 \u2192 d \u03b1) (b : Bundled c) : Bundled d :=\n  \u27e8b, f b.str\u27e9\n#align category_theory.bundled.map CategoryTheory.Bundled.map\n-/\n\nend Bundled\n\nend CategoryTheory\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/ConcreteCategory/Bundled.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.35521038574260866}}
{"text": "import algebra\nimport algebra.group\nimport algebra.group.defs\nimport logic.relation\nimport order.lexicographic\nimport declare\n\ninstance fin.has_top (k : \u2115) : has_top (fin k.succ) := \u27e8k\u27e9\n-- todo instance i2 : linear_order unit := sorry\n-- instance i2 (k : \u2115) : linear_order (fin k) := infer_instance\n-- instance i3 {\u03b1 \u03b2: Type*} [linear_order \u03b1] [linear_order \u03b2] : linear_order (lex \u03b1 \u03b2) := infer_instance\n\nuniverses u v\nvariables (\u03b1 \u03b2 : Type*)\n\ndef emit_type (I V : Type*) := option (I \u00d7 option V)\n\nstructure iter (\u03c3 I V : Type*) :=\n  (\u03b4 : function.End \u03c3)\n  (emit : \u03c3 \u2192 emit_type I V)\n\ndef_declare one := \"variables {\u03c3 I V : Type} [linear_order I] (a : iter \u03c3 I V)\"\ndef_declare two := \"variables {\u03c3\u2081 \u03c3\u2082 I I\u2081 I\u2082 V V\u2081 V\u2082 V\u2083 : Type} [linear_order I] [linear_order I\u2081] [linear_order I\u2082] (a : iter \u03c3\u2081 I V\u2081) (b : iter \u03c3\u2082 I V\u2082)\"\ndef_declare add := \"variables (add : V\u2081 \u2192 V\u2082 \u2192 V\u2083) [has_zero V\u2081] [has_zero V\u2082] [has_zero V\u2083]\"\ndef_declare mul := \"variables (mul : V\u2081 \u2192 V\u2082 \u2192 V\u2083)\"\n\nnamespace iter\n\nsection params_unary\nvariables {\u03c3 I V : Type} (a : iter \u03c3 I V) (s t : \u03c3)\n\ndef \u03b9 : with_top I := match a.emit s with | none := none | some (i, _) := some i end\ndef \u03bd :   option V := match a.emit s with | none := none | some (_, v) := v end\n--def \u03bd :   option V := option.bind (a.emit s) prod.snd\n\nsection semantics\nvariables [add_monoid V] [decidable_eq I]\n\ndef elementary (i : I) (v : V) := \u03bb j, if i = j then v else 0\n\n@[simp] def semantics\u2081 (s : \u03c3) : I \u2192 V :=\n  match a.emit s with\n  | none := 0\n  | some (i, none) := 0\n  | some (i, some v) := elementary i v\n  end\n\n@[simp] def semantics : \u03c3 \u2192 \u2115 \u2192 I \u2192 V\n| _ 0 := 0\n| s (n+1) := a.semantics\u2081 s + semantics (a.\u03b4 s) n\n\n--notation `\u27e6` a, s `\u27e7` := a.semantics s\nnotation `\u27e6` a, s, j `\u27e7` := a.semantics s j\n--example (j : \u2115) : \u2200 i:I, \u27e6a, s, j\u27e7 i = \u27e6a, s\u27e7 j i := \u03bb _, rfl\n\nend semantics\n\nvariables [linear_order I]\n\ndef reachable := relation.refl_trans_gen (\u03bb s t, t = a.\u03b4 s)\n\nnamespace transition -- can't use reachable??\nopen relation.refl_trans_gen\ntheorem trans {x y z : \u03c3} : reachable a x y \u2192 reachable a y z \u2192 reachable a x z := trans\ndef step {x : \u03c3} : reachable a x (a.\u03b4 x) := single rfl\nend transition\n\ntheorem none_top {\u03b1 : Type*} [linear_order \u03b1] : \u2200 {i : with_top \u03b1}, \u22a4 \u2264 i \u2192 i = none | _ h := le_antisymm le_top h\n\ndef step       (s : \u03c3) (i:\u2115) := a.\u03b4^i \u2022 s\ndef monotonic          := \u2200 (s t : \u03c3), a.reachable s t \u2192 a.\u03b9 s \u2264 a.\u03b9 t\ndef terminal   (s : \u03c3) := a.\u03b9 s = \u22a4\ndef finite     (s : \u03c3) := \u2203 (t : \u03c3), reachable a s t \u2227 terminal a t\ndef productive (s : \u03c3) := \u03bd a s \u2260 none\ndef reduced    (s : \u03c3) := \u2200 (t t' : \u03c3), a.reachable s t \u2192 a.reachable s t' \u2192\n  productive a t \u2192 productive a t' \u2192 \u03b9 a t = \u03b9 a t' \u2192 t = t'\n\ndef future (s : \u03c3) : set \u03c3 := { t | reachable a s t \u2227 \u00ac terminal a t}\n@[simp] def terminal_by (s : \u03c3) (i : \u2115) := a.terminal (a.step s i)\n\ninstance [decidable_eq I] : decidable (terminal a s) := if h : \u03b9 a s = none then is_true h else is_false h\n\nlemma some_not_terminal {a : iter \u03c3 I V} {s} {i : I} : a.\u03b9 s = some i \u2192 \u00ac a.terminal s :=\n\u03bb h1 h2, false.rec _ (option.some_ne_none i (h1 \u25b8 h2))\n\nopen relation.refl_trans_gen\ndef path_of_index {a : iter \u03c3 I V} : \u2200 (i:\u2115), a.reachable s (a.step s i)\n| 0 := refl\n| (n+1) := tail (path_of_index n) rfl\n\nlemma le_of_index_lt {a : iter \u03c3 I V} (i j : \u2115) : a.monotonic \u2192 i \u2264 j \u2192 a.\u03b9 ((a.\u03b4^i)\u2022s) \u2264 a.\u03b9 ((a.\u03b4^j) \u2022 s) := begin\n  intros mono lt, apply mono, induction lt, exact refl, exact tail \u2039_\u203a rfl,\nend\nlemma index_lt_of_ge {a : iter \u03c3 I V} (i j : \u2115) : a.monotonic \u2192 a.\u03b9 ((a.\u03b4^i)\u2022s) < a.\u03b9 ((a.\u03b4^j)\u2022s) \u2192 i < j := \u03bb mono, begin\nhave h := mt (le_of_index_lt s j i mono),\nsimpa using h,\nend\n\n@[simp] lemma step_zero : a.step s 0 = s := rfl\n@[simp] lemma step_succ (s : \u03c3) (i : \u2115) : a.step s i.succ = a.step (a.\u03b4 s) i :=\nbegin\nchange a.\u03b4 ^ (i+1) \u2022 s = a.\u03b4 ^ i \u2022 a.\u03b4^1 \u2022 s,\nrw [pow_add, mul_smul],\nend\n\nlemma not_terminal_succ {a : iter \u03c3 I V} {i : \u2115} {s} : \u00ac a.terminal s \u2192 a.terminal_by s i \u2192 \u2203 i':\u2115, i = i'.succ := begin\nintros hnt ht, induction i with i hi,\nexact false.rec _ (hnt ht),\nexact \u27e8i, rfl\u27e9,\nend\n\nlemma index_of_path {a : iter \u03c3 I V} {s t} : a.reachable s t \u2192 \u2203 (i: \u2115), t = a.step s i := begin\n  intros p, induction p, refine \u27e80 , _\u27e9, refl,\n  case tail : s1 s2 path st h {\n    cases h with i hh, refine \u27e8i+1, _\u27e9,\n    simp only [step] at *,\n    rw [add_comm, pow_add, mul_smul, \u2190 hh], exact st,\n  }\nend\n\nsection lemmas\n\nlemma \u03b9_top_emit_none {a : iter \u03c3 I V} {s} : a.\u03b9 s = \u22a4 \u2194 a.emit s = none := begin\nsplit; intro h1,\n{ cases h : a.emit s, exact rfl,\n  cases val,\n  simp only [\u03b9, h] at h1,\n  exfalso, apply option.some_ne_none _ h1 },\n{ simp only [\u03b9, h1], refl },\nend\n\n@[simp]\ntheorem terminal_succ_terminal {a : iter \u03c3 I V} (m : a.monotonic) (h : a.terminal t) : a.terminal (a.\u03b4 t) :=\nbegin\nsimp only [terminal] at *, apply none_top, rw \u2190h, exact m _ _ (transition.step _)\nend\n\n@[simp]\ntheorem emit_none_of_terminal {a : iter \u03c3 I V} {t} : a.terminal t \u2192 a.emit t = none := begin\nintro h, simp only [terminal] at h, exact \u03b9_top_emit_none.1 h,\nend\n\nend lemmas\nend params_unary\nend iter\n\nstructure stream (\u03c3 I V : Type*) :=\n  (q : \u03c3)\n  (iter : iter \u03c3 I V)\n\nnamespace stream\nvariables {\u03c3 I V : Type} (s : stream \u03c3 I V)\n\n@[simp] def \u03b9 := s.iter.\u03b9 s.q\n@[simp] def \u03bd := s.iter.\u03bd s.q\n\n@[simp] def \u03b4 (s : stream \u03c3 I V) : stream \u03c3 I V :=\n{ q := s.iter.\u03b4 s.q .. s}\n\n@[simp] def emit : emit_type I V := s.iter.emit s.q\n\nvariables [decidable_eq I]\n\n@[simp] def semantics\u2081 [add_monoid V] (s : stream \u03c3 I V) : I \u2192 V\n:= s.iter.semantics\u2081 s.q\n-- simp?\n@[simp] def semantics [add_monoid V] (s : stream \u03c3 I V) : \u2115 \u2192 I \u2192 V\n:= s.iter.semantics s.q\nnotation `\u27e6` s, i `\u27e7` := s.semantics i\n\nvariables [linear_order I]\n\n@[simp] def terminal_by (i : \u2115) := s.iter.terminal_by s.q i\n\n@[simp] def monotonic := s.iter.monotonic\n\nend stream\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/old_formalization/base.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.35508969116332395}}
{"text": "import hilbert.wr.dc\n\nnamespace clfrags\n    namespace hilbert\n        namespace wr\n            namespace dc\n\n                theorem dc\u2084' {a b c : Prop} (h\u2081 : dc a b c) : dc b a c :=\n                    have h\u2082 : dc (dc b a c) (dc a b c) (dc a b c), from dc\u2083 h\u2081,\n                    have h\u2083 : dc (dc a b c) (dc b a c) (dc b a c), from dc\u2084 h\u2082,\n                    show dc b a c, from dc\u2082 h\u2083\n\n                theorem dc\u2085' {a b c : Prop} (h\u2081 : dc a b c) : dc a c b :=\n                    have h\u2082 : dc (dc a c b) (dc a b c) (dc a b c), from dc\u2083 h\u2081,\n                    have h\u2083 : dc (dc a b c) (dc a c b) (dc a c b), from dc\u2085 h\u2082,\n                    show dc a c b, from dc\u2082 h\u2083\n\n                theorem dc\u2086' {a b c d e : Prop}\n                    (h\u2081 : dc d e (dc a b c)) : dc (dc d e a) (dc d e b) c :=\n                    let f := dc d e (dc a b c), g := dc (dc d e a) (dc d e b) c in\n                        have h\u2082 : dc g f f, from dc\u2083 h\u2081,\n                        have h\u2083 : dc g f g, from dc\u2086 h\u2082,\n                        have h\u2084 : dc f g g, from dc\u2084' h\u2083,\n                        show g, from dc\u2082 h\u2084\n\n                theorem dc\u2087' {a b c d e : Prop}\n                    (h\u2081 : dc (dc d e a) (dc d e b) c) : dc d e (dc a b c) :=\n                    let f := dc d e (dc a b c), g := dc (dc d e a) (dc d e b) c in\n                        have h\u2082 : dc f g g, from dc\u2083 h\u2081,\n                        have h\u2083 : dc f g f, from dc\u2087 h\u2082,\n                        have h\u2084 : dc g f f, from dc\u2084' h\u2083,\n                        show f, from dc\u2082 h\u2084\n\n                theorem dc\u2081_dc {a b c d e : Prop} (h\u2081 : dc d e a) (h\u2082 : dc d e b) :\n                    dc d e (dc a b c) :=\n                    have h\u2082 : dc (dc d e a) (dc d e b) c, from dc\u2081 h\u2081 h\u2082,\n                    show dc d e (dc a b c), from dc\u2087' h\u2082\n\n                theorem dc\u2082_dc {a b c d : Prop} (h\u2081 : dc c d (dc b a a)) : dc c d a :=\n                    have h\u2082 : dc d c (dc a b a), from dc\u2084 h\u2081,\n                    have h\u2083 : dc c d (dc a a b), from dc\u2085 h\u2082,\n                    have h\u2084 : dc (dc c d a) (dc c d a) b, from dc\u2086' h\u2083,\n                    have h\u2085 : dc b (dc c d a) (dc c d a), from dc\u2084' (dc\u2085' h\u2084),\n                    show dc c d a, from dc\u2082 h\u2085\n\n                theorem dc\u2083_dc {a b c d : Prop} (h\u2081 : dc c d a) : dc c d (dc b a a) :=\n                    have h\u2082 : dc b (dc c d a) (dc c d a), from dc\u2083 h\u2081,\n                    have h\u2083 : dc (dc c d a) (dc c d a) b, from dc\u2085' (dc\u2084' h\u2082),\n                    have h\u2084 : dc c d (dc a a b), from dc\u2087' h\u2083,\n                    have h\u2085 : dc d c (dc a b a), from dc\u2085 h\u2084,\n                    show dc c d (dc b a a), from dc\u2084 h\u2085\n\n                theorem dc\u2084_dc {a b c d e f g : Prop} (h\u2081 : dc f g (dc d e (dc a b c))) : \n                    dc f g (dc e d (dc b a c)) :=\n                    have h\u2082 : dc g f (dc e d (dc a b c)), from dc\u2084 h\u2081,\n                    have h\u2083 : dc g f (dc (dc e d a) (dc e d b) c), from dc\u2086 h\u2082,\n                    have h\u2084 : dc f g (dc (dc e d b) (dc e d a) c), from dc\u2084 h\u2083,\n                    show dc f g (dc e d (dc b a c)), from dc\u2087 h\u2084\n\n                theorem dc\u2085_dc {a b c d e f g : Prop} (h\u2081 : dc f g (dc d e (dc a b c))) : \n                    dc f g (dc e d (dc a c b)) :=\n                    have h\u2082 : dc g f (dc e d (dc a b c)), from dc\u2084 h\u2081,\n                    have h\u2083 : dc (dc g f e) (dc g f d) (dc a b c), from dc\u2086' h\u2082,\n                    have h\u2084 : dc (dc g f d) (dc g f e) (dc a c b), from dc\u2085 h\u2083,\n                    have h\u2085 : dc g f (dc d e (dc a c b)), from dc\u2087' h\u2084,\n                    show dc f g (dc e d (dc a c b)), from dc\u2084 h\u2085\n\n                theorem dc\u2086_dc {a b c d e f g h i : Prop}\n                    (h\u2081 : dc h i (dc f g (dc d e (dc a b c)))) \n                    : dc h i (dc f g (dc (dc d e a) (dc d e b) c)) :=\n                    have h\u2082 : dc (dc h i f) (dc h i g) (dc d e (dc a b c)), from dc\u2086' h\u2081,\n                    have h\u2083 : dc (dc h i f) (dc h i g) (dc (dc d e a) (dc d e b) c), from dc\u2086 h\u2082,\n                    show dc h i (dc f g (dc (dc d e a) (dc d e b) c)), from dc\u2087' h\u2083\n\n                theorem dc\u2087_dc {a b c d e f g h i : Prop}\n                    (h\u2081 : dc h i (dc f g (dc (dc d e a) (dc d e b) c))) :\n                    dc h i (dc f g (dc d e (dc a b c))) :=\n                    have h\u2082 : dc (dc h i f) (dc h i g) (dc (dc d e a) (dc d e b) c), from dc\u2086' h\u2081,\n                    have h\u2083 : dc (dc h i f) (dc h i g) (dc d e (dc a b c)), from dc\u2087 h\u2082,\n                    show dc h i (dc f g (dc d e (dc a b c))), from dc\u2087' h\u2083\n\n            end dc\n        end wr\n    end hilbert\nend clfrags\n", "meta": {"author": "greati", "repo": "hilbert-classical-fragments", "sha": "18a21ac6b2e890060eb4ae65752fc0245394d226", "save_path": "github-repos/lean/greati-hilbert-classical-fragments", "path": "github-repos/lean/greati-hilbert-classical-fragments/hilbert-classical-fragments-18a21ac6b2e890060eb4ae65752fc0245394d226/clfrags/src/hilbert/wr/proofs/dc.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.35508969116332395}}
{"text": "example (P Q : Type) (p : P) (h : P \u2192 Q) : Q :=\nbegin\nexact h(p),\nend\n", "meta": {"author": "abdelq", "repo": "natural-number-game", "sha": "bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2", "save_path": "github-repos/lean/abdelq-natural-number-game", "path": "github-repos/lean/abdelq-natural-number-game/natural-number-game-bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2/world05/level01.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.6187804407739559, "lm_q1q2_score": 0.354980977825965}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.seq.seq\nimport Mathlib.Lean3Lib.data.dlist\nimport Mathlib.PostPort\n\nuniverses u_1 u v w \n\nnamespace Mathlib\n\n/-\ncoinductive wseq (\u03b1 : Type u) : Type u\n| nil : wseq \u03b1\n| cons : \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1\n| think : wseq \u03b1 \u2192 wseq \u03b1\n-/\n\n/-- Weak sequences.\n\n  While the `seq` structure allows for lists which may not be finite,\n  a weak sequence also allows the computation of each element to\n  involve an indeterminate amount of computation, including possibly\n  an infinite loop. This is represented as a regular `seq` interspersed\n  with `none` elements to indicate that computation is ongoing.\n\n  This model is appropriate for Haskell style lazy lists, and is closed\n  under most interesting computation patterns on infinite lists,\n  but conversely it is difficult to extract elements from it. -/\ndef wseq (\u03b1 : Type u_1) := seq (Option \u03b1)\n\nnamespace wseq\n\n\n/-- Turn a sequence into a weak sequence -/\ndef of_seq {\u03b1 : Type u} : seq \u03b1 \u2192 wseq \u03b1 := Functor.map some\n\n/-- Turn a list into a weak sequence -/\ndef of_list {\u03b1 : Type u} (l : List \u03b1) : wseq \u03b1 := of_seq \u2191l\n\n/-- Turn a stream into a weak sequence -/\ndef of_stream {\u03b1 : Type u} (l : stream \u03b1) : wseq \u03b1 := of_seq \u2191l\n\nprotected instance coe_seq {\u03b1 : Type u} : has_coe (seq \u03b1) (wseq \u03b1) := has_coe.mk of_seq\n\nprotected instance coe_list {\u03b1 : Type u} : has_coe (List \u03b1) (wseq \u03b1) := has_coe.mk of_list\n\nprotected instance coe_stream {\u03b1 : Type u} : has_coe (stream \u03b1) (wseq \u03b1) := has_coe.mk of_stream\n\n/-- The empty weak sequence -/\ndef nil {\u03b1 : Type u} : wseq \u03b1 := seq.nil\n\nprotected instance inhabited {\u03b1 : Type u} : Inhabited (wseq \u03b1) := { default := nil }\n\n/-- Prepend an element to a weak sequence -/\ndef cons {\u03b1 : Type u} (a : \u03b1) : wseq \u03b1 \u2192 wseq \u03b1 := seq.cons (some a)\n\n/-- Compute for one tick, without producing any elements -/\ndef think {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 := seq.cons none\n\n/-- Destruct a weak sequence, to (eventually possibly) produce either\n  `none` for `nil` or `some (a, s)` if an element is produced. -/\ndef destruct {\u03b1 : Type u} : wseq \u03b1 \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  computation.corec fun (s : wseq \u03b1) => sorry\n\ndef cases_on {\u03b1 : Type u} {C : wseq \u03b1 \u2192 Sort v} (s : wseq \u03b1) (h1 : C nil)\n    (h2 : (x : \u03b1) \u2192 (s : wseq \u03b1) \u2192 C (cons x s)) (h3 : (s : wseq \u03b1) \u2192 C (think s)) : C s :=\n  seq.cases_on s h1 fun (o : Option \u03b1) => option.cases_on o h3 h2\n\nprotected def mem {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) := seq.mem (some a) s\n\nprotected instance has_mem {\u03b1 : Type u} : has_mem \u03b1 (wseq \u03b1) := has_mem.mk wseq.mem\n\ntheorem not_mem_nil {\u03b1 : Type u} (a : \u03b1) : \u00aca \u2208 nil := seq.not_mem_nil \u2191a\n\n/-- Get the head of a weak sequence. This involves a possibly\n  infinite computation. -/\ndef head {\u03b1 : Type u} (s : wseq \u03b1) : computation (Option \u03b1) :=\n  computation.map (Functor.map prod.fst) (destruct s)\n\n/-- Encode a computation yielding a weak sequence into additional\n  `think` constructors in a weak sequence -/\ndef flatten {\u03b1 : Type u} : computation (wseq \u03b1) \u2192 wseq \u03b1 :=\n  seq.corec fun (c : computation (wseq \u03b1)) => sorry\n\n/-- Get the tail of a weak sequence. This doesn't need a `computation`\n  wrapper, unlike `head`, because `flatten` allows us to hide this\n  in the construction of the weak sequence itself. -/\ndef tail {\u03b1 : Type u} (s : wseq \u03b1) : wseq \u03b1 :=\n  flatten ((fun (o : Option (\u03b1 \u00d7 wseq \u03b1)) => option.rec_on o nil prod.snd) <$> destruct s)\n\n/-- drop the first `n` elements from `s`. -/\n@[simp] def drop {\u03b1 : Type u} (s : wseq \u03b1) : \u2115 \u2192 wseq \u03b1 := sorry\n\n/-- Get the nth element of `s`. -/\ndef nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : computation (Option \u03b1) := head (drop s n)\n\n/-- Convert `s` to a list (if it is finite and completes in finite time). -/\ndef to_list {\u03b1 : Type u} (s : wseq \u03b1) : computation (List \u03b1) :=\n  computation.corec (fun (_x : List \u03b1 \u00d7 wseq \u03b1) => sorry) ([], s)\n\n/-- Get the length of `s` (if it is finite and completes in finite time). -/\ndef length {\u03b1 : Type u} (s : wseq \u03b1) : computation \u2115 :=\n  computation.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (0, s)\n\n/-- A weak sequence is finite if `to_list s` terminates. Equivalently,\n  it is a finite number of `think` and `cons` applied to `nil`. -/\ndef is_finite {\u03b1 : Type u} (s : wseq \u03b1) := computation.terminates (to_list s)\n\nprotected instance to_list_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : is_finite s] :\n    computation.terminates (to_list s) :=\n  h\n\n/-- Get the list corresponding to a finite weak sequence. -/\ndef get {\u03b1 : Type u} (s : wseq \u03b1) [is_finite s] : List \u03b1 := computation.get (to_list s)\n\n/-- A weak sequence is *productive* if it never stalls forever - there are\n always a finite number of `think`s between `cons` constructors.\n The sequence itself is allowed to be infinite though. -/\ndef productive {\u03b1 : Type u} (s : wseq \u03b1) := \u2200 (n : \u2115), computation.terminates (nth s n)\n\nprotected instance nth_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : productive s] (n : \u2115) :\n    computation.terminates (nth s n) :=\n  h\n\nprotected instance head_terminates {\u03b1 : Type u} (s : wseq \u03b1) [h : productive s] :\n    computation.terminates (head s) :=\n  h 0\n\n/-- Replace the `n`th element of `s` with `a`. -/\ndef update_nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) (a : \u03b1) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n + 1, s)\n\n/-- Remove the `n`th element of `s`. -/\ndef remove_nth {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n + 1, s)\n\n/-- Map the elements of `s` over `f`, removing any values that yield `none`. -/\ndef filter_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 Option \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 :=\n  seq.corec fun (s : wseq \u03b1) => sorry\n\n/-- Select the elements of `s` that satisfy `p`. -/\ndef filter {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] : wseq \u03b1 \u2192 wseq \u03b1 :=\n  filter_map fun (a : \u03b1) => ite (p a) (some a) none\n\n-- example of infinite list manipulations\n\n/-- Get the first element of `s` satisfying `p`. -/\ndef find {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation (Option \u03b1) :=\n  head (filter p s)\n\n/-- Zip a function over two weak sequences -/\ndef zip_with {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (s1 : wseq \u03b1) (s2 : wseq \u03b2) :\n    wseq \u03b3 :=\n  seq.corec (fun (_x : wseq \u03b1 \u00d7 wseq \u03b2) => sorry) (s1, s2)\n\n/-- Zip two weak sequences into a single sequence of pairs -/\ndef zip {\u03b1 : Type u} {\u03b2 : Type v} : wseq \u03b1 \u2192 wseq \u03b2 \u2192 wseq (\u03b1 \u00d7 \u03b2) := zip_with Prod.mk\n\n/-- Get the list of indexes of elements of `s` satisfying `p` -/\ndef find_indexes {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : wseq \u2115 :=\n  filter_map (fun (_x : \u03b1 \u00d7 \u2115) => sorry) (zip s \u2191stream.nats)\n\n/-- Get the index of the first element of `s` satisfying `p` -/\ndef find_index {\u03b1 : Type u} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : wseq \u03b1) : computation \u2115 :=\n  (fun (o : Option \u2115) => option.get_or_else o 0) <$> head (find_indexes p s)\n\n/-- Get the index of the first occurrence of `a` in `s` -/\ndef index_of {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 computation \u2115 := find_index (Eq a)\n\n/-- Get the indexes of occurrences of `a` in `s` -/\ndef indexes_of {\u03b1 : Type u} [DecidableEq \u03b1] (a : \u03b1) : wseq \u03b1 \u2192 wseq \u2115 := find_indexes (Eq a)\n\n/-- `union s1 s2` is a weak sequence which interleaves `s1` and `s2` in\n  some order (nondeterministically). -/\ndef union {\u03b1 : Type u} (s1 : wseq \u03b1) (s2 : wseq \u03b1) : wseq \u03b1 :=\n  seq.corec (fun (_x : wseq \u03b1 \u00d7 wseq \u03b1) => sorry) (s1, s2)\n\n/-- Returns `tt` if `s` is `nil` and `ff` if `s` has an element -/\ndef is_empty {\u03b1 : Type u} (s : wseq \u03b1) : computation Bool := computation.map option.is_none (head s)\n\n/-- Calculate one step of computation -/\ndef compute {\u03b1 : Type u} (s : wseq \u03b1) : wseq \u03b1 := sorry\n\n/-- Get the first `n` elements of a weak sequence -/\ndef take {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : wseq \u03b1 :=\n  seq.corec (fun (_x : \u2115 \u00d7 wseq \u03b1) => sorry) (n, s)\n\n/-- Split the sequence at position `n` into a finite initial segment\n  and the weak sequence tail -/\ndef split_at {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : computation (List \u03b1 \u00d7 wseq \u03b1) :=\n  computation.corec (fun (_x : \u2115 \u00d7 List \u03b1 \u00d7 wseq \u03b1) => sorry) (n, [], s)\n\n/-- Returns `tt` if any element of `s` satisfies `p` -/\ndef any {\u03b1 : Type u} (s : wseq \u03b1) (p : \u03b1 \u2192 Bool) : computation Bool :=\n  computation.corec (fun (s : wseq \u03b1) => sorry) s\n\n/-- Returns `tt` if every element of `s` satisfies `p` -/\ndef all {\u03b1 : Type u} (s : wseq \u03b1) (p : \u03b1 \u2192 Bool) : computation Bool :=\n  computation.corec (fun (s : wseq \u03b1) => sorry) s\n\n/-- Apply a function to the elements of the sequence to produce a sequence\n  of partial results. (There is no `scanr` because this would require\n  working from the end of the sequence, which may not exist.) -/\ndef scanl {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b1) (a : \u03b1) (s : wseq \u03b2) : wseq \u03b1 :=\n  cons a (seq.corec (fun (_x : \u03b1 \u00d7 wseq \u03b2) => sorry) (a, s))\n\n/-- Get the weak sequence of initial segments of the input sequence -/\ndef inits {\u03b1 : Type u} (s : wseq \u03b1) : wseq (List \u03b1) :=\n  cons [] (seq.corec (fun (_x : dlist \u03b1 \u00d7 wseq \u03b1) => sorry) (dlist.empty, s))\n\n/-- Like take, but does not wait for a result. Calculates `n` steps of\n  computation and returns the sequence computed so far -/\ndef collect {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : List \u03b1 := list.filter_map id (seq.take n s)\n\n/-- Append two weak sequences. As with `seq.append`, this may not use\n  the second sequence if the first one takes forever to compute -/\ndef append {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 \u2192 wseq \u03b1 := seq.append\n\n/-- Map a function over a weak sequence -/\ndef map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : wseq \u03b1 \u2192 wseq \u03b2 := seq.map (option.map f)\n\n/-- Flatten a sequence of weak sequences. (Note that this allows\n  empty sequences, unlike `seq.join`.) -/\ndef join {\u03b1 : Type u} (S : wseq (wseq \u03b1)) : wseq \u03b1 :=\n  seq.join ((fun (o : Option (wseq \u03b1)) => sorry) <$> S)\n\n/-- Monadic bind operator for weak sequences -/\ndef bind {\u03b1 : Type u} {\u03b2 : Type v} (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2) : wseq \u03b2 := join (map f s)\n\n@[simp] def lift_rel_o {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop) :\n    Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Option (\u03b2 \u00d7 wseq \u03b2) \u2192 Prop :=\n  sorry\n\ntheorem lift_rel_o.imp {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {S : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} {D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop}\n    (H1 : \u2200 (a : \u03b1) (b : \u03b2), R a b \u2192 S a b) (H2 : \u2200 (s : wseq \u03b1) (t : wseq \u03b2), C s t \u2192 D s t)\n    {o : Option (\u03b1 \u00d7 wseq \u03b1)} {p : Option (\u03b2 \u00d7 wseq \u03b2)} : lift_rel_o R C o p \u2192 lift_rel_o S D o p :=\n  sorry\n\ntheorem lift_rel_o.imp_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    {C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} {D : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop}\n    (H : \u2200 (s : wseq \u03b1) (t : wseq \u03b2), C s t \u2192 D s t) {o : Option (\u03b1 \u00d7 wseq \u03b1)}\n    {p : Option (\u03b2 \u00d7 wseq \u03b2)} : lift_rel_o R C o p \u2192 lift_rel_o R D o p :=\n  lift_rel_o.imp (fun (_x : \u03b1) (_x_1 : \u03b2) => id) H\n\n@[simp] def bisim_o {\u03b1 : Type u} (R : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop) :\n    Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Option (\u03b1 \u00d7 wseq \u03b1) \u2192 Prop :=\n  lift_rel_o Eq R\n\ntheorem bisim_o.imp {\u03b1 : Type u} {R : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop} {S : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop}\n    (H : \u2200 (s t : wseq \u03b1), R s t \u2192 S s t) {o : Option (\u03b1 \u00d7 wseq \u03b1)} {p : Option (\u03b1 \u00d7 wseq \u03b1)} :\n    bisim_o R o p \u2192 bisim_o S o p :=\n  lift_rel_o.imp_right Eq H\n\n/-- Two weak sequences are `lift_rel R` related if they are either both empty,\n  or they are both nonempty and the heads are `R` related and the tails are\n  `lift_rel R` related. (This is a coinductive definition.) -/\ndef lift_rel {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1) (t : wseq \u03b2) :=\n  \u2203 (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop),\n    C s t \u2227\n      \u2200 {s : wseq \u03b1} {t : wseq \u03b2},\n        C s t \u2192 computation.lift_rel (lift_rel_o R C) (destruct s) (destruct t)\n\n/-- If two sequences are equivalent, then they have the same values and\n  the same computational behavior (i.e. if one loops forever then so does\n  the other), although they may differ in the number of `think`s needed to\n  arrive at the answer. -/\ndef equiv {\u03b1 : Type u} : wseq \u03b1 \u2192 wseq \u03b1 \u2192 Prop := lift_rel Eq\n\ntheorem lift_rel_destruct {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1} {t : wseq \u03b2} :\n    lift_rel R s t \u2192 computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n  sorry\n\ntheorem lift_rel_destruct_iff {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1}\n    {t : wseq \u03b2} :\n    lift_rel R s t \u2194 computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct s) (destruct t) :=\n  sorry\n\ninfixl:50 \" ~ \" => Mathlib.wseq.equiv\n\ntheorem destruct_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} :\n    s ~ t \u2192 computation.lift_rel (bisim_o equiv) (destruct s) (destruct t) :=\n  lift_rel_destruct\n\ntheorem destruct_congr_iff {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} :\n    s ~ t \u2194 computation.lift_rel (bisim_o equiv) (destruct s) (destruct t) :=\n  lift_rel_destruct_iff\n\ntheorem lift_rel.refl {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : reflexive R) : reflexive (lift_rel R) :=\n  sorry\n\ntheorem lift_rel_o.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop) :\n    function.swap (lift_rel_o R C) = lift_rel_o (function.swap R) (function.swap C) :=\n  sorry\n\ntheorem lift_rel.swap_lem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s1 : wseq \u03b1} {s2 : wseq \u03b2}\n    (h : lift_rel R s1 s2) : lift_rel (function.swap R) s2 s1 :=\n  sorry\n\ntheorem lift_rel.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) :\n    function.swap (lift_rel R) = lift_rel (function.swap R) :=\n  funext\n    fun (x : wseq \u03b2) =>\n      funext fun (y : wseq \u03b1) => propext { mp := lift_rel.swap_lem, mpr := lift_rel.swap_lem }\n\ntheorem lift_rel.symm {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : symmetric R) : symmetric (lift_rel R) :=\n  sorry\n\ntheorem lift_rel.trans {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : transitive R) :\n    transitive (lift_rel R) :=\n  sorry\n\ntheorem lift_rel.equiv {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : equivalence R \u2192 equivalence (lift_rel R) :=\n  sorry\n\ntheorem equiv.refl {\u03b1 : Type u} (s : wseq \u03b1) : s ~ s := lift_rel.refl Eq Eq.refl\n\ntheorem equiv.symm {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2192 t ~ s :=\n  lift_rel.symm Eq Eq.symm\n\ntheorem equiv.trans {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} {u : wseq \u03b1} : s ~ t \u2192 t ~ u \u2192 s ~ u :=\n  lift_rel.trans Eq Eq.trans\n\ntheorem equiv.equivalence {\u03b1 : Type u} : equivalence equiv :=\n  { left := equiv.refl, right := { left := equiv.symm, right := equiv.trans } }\n\n@[simp] theorem destruct_nil {\u03b1 : Type u} : destruct nil = computation.return none :=\n  computation.destruct_eq_ret rfl\n\n@[simp] theorem destruct_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) :\n    destruct (cons a s) = computation.return (some (a, s)) :=\n  sorry\n\n@[simp] theorem destruct_think {\u03b1 : Type u} (s : wseq \u03b1) :\n    destruct (think s) = computation.think (destruct s) :=\n  sorry\n\n@[simp] theorem seq_destruct_nil {\u03b1 : Type u} : seq.destruct nil = none := seq.destruct_nil\n\n@[simp] theorem seq_destruct_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) :\n    seq.destruct (cons a s) = some (some a, s) :=\n  seq.destruct_cons (some a) s\n\n@[simp] theorem seq_destruct_think {\u03b1 : Type u} (s : wseq \u03b1) :\n    seq.destruct (think s) = some (none, s) :=\n  seq.destruct_cons none s\n\n@[simp] theorem head_nil {\u03b1 : Type u} : head nil = computation.return none := sorry\n\n@[simp] theorem head_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) :\n    head (cons a s) = computation.return (some a) :=\n  sorry\n\n@[simp] theorem head_think {\u03b1 : Type u} (s : wseq \u03b1) :\n    head (think s) = computation.think (head s) :=\n  sorry\n\n@[simp] theorem flatten_ret {\u03b1 : Type u} (s : wseq \u03b1) : flatten (computation.return s) = s := sorry\n\n@[simp] theorem flatten_think {\u03b1 : Type u} (c : computation (wseq \u03b1)) :\n    flatten (computation.think c) = think (flatten c) :=\n  sorry\n\n@[simp] theorem destruct_flatten {\u03b1 : Type u} (c : computation (wseq \u03b1)) :\n    destruct (flatten c) = c >>= destruct :=\n  sorry\n\ntheorem head_terminates_iff {\u03b1 : Type u} (s : wseq \u03b1) :\n    computation.terminates (head s) \u2194 computation.terminates (destruct s) :=\n  computation.terminates_map_iff (Functor.map prod.fst) (destruct s)\n\n@[simp] theorem tail_nil {\u03b1 : Type u} : tail nil = nil := sorry\n\n@[simp] theorem tail_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) : tail (cons a s) = s := sorry\n\n@[simp] theorem tail_think {\u03b1 : Type u} (s : wseq \u03b1) : tail (think s) = think (tail s) := sorry\n\n@[simp] theorem dropn_nil {\u03b1 : Type u} (n : \u2115) : drop nil n = nil := sorry\n\n@[simp] theorem dropn_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) (n : \u2115) :\n    drop (cons a s) (n + 1) = drop s n :=\n  sorry\n\n@[simp] theorem dropn_think {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) :\n    drop (think s) n = think (drop s n) :=\n  sorry\n\ntheorem dropn_add {\u03b1 : Type u} (s : wseq \u03b1) (m : \u2115) (n : \u2115) : drop s (m + n) = drop (drop s m) n :=\n  sorry\n\ntheorem dropn_tail {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : drop (tail s) n = drop s (n + 1) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (drop (tail s) n = drop s (n + 1))) (add_comm n 1)))\n    (Eq.symm (dropn_add s 1 n))\n\ntheorem nth_add {\u03b1 : Type u} (s : wseq \u03b1) (m : \u2115) (n : \u2115) : nth s (m + n) = nth (drop s m) n :=\n  congr_arg head (dropn_add s m n)\n\ntheorem nth_tail {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) : nth (tail s) n = nth s (n + 1) :=\n  congr_arg head (dropn_tail s n)\n\n@[simp] theorem join_nil {\u03b1 : Type u} : join nil = nil := seq.join_nil\n\n@[simp] theorem join_think {\u03b1 : Type u} (S : wseq (wseq \u03b1)) : join (think S) = think (join S) :=\n  sorry\n\n@[simp] theorem join_cons {\u03b1 : Type u} (s : wseq \u03b1) (S : wseq (wseq \u03b1)) :\n    join (cons s S) = think (append s (join S)) :=\n  sorry\n\n@[simp] theorem nil_append {\u03b1 : Type u} (s : wseq \u03b1) : append nil s = s := seq.nil_append s\n\n@[simp] theorem cons_append {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) (t : wseq \u03b1) :\n    append (cons a s) t = cons a (append s t) :=\n  seq.cons_append (some a) s t\n\n@[simp] theorem think_append {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) :\n    append (think s) t = think (append s t) :=\n  seq.cons_append none s t\n\n@[simp] theorem append_nil {\u03b1 : Type u} (s : wseq \u03b1) : append s nil = s := seq.append_nil s\n\n@[simp] theorem append_assoc {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) (u : wseq \u03b1) :\n    append (append s t) u = append s (append t u) :=\n  seq.append_assoc s t u\n\n@[simp] def tail.aux {\u03b1 : Type u} : Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) := sorry\n\ntheorem destruct_tail {\u03b1 : Type u} (s : wseq \u03b1) : destruct (tail s) = destruct s >>= tail.aux :=\n  sorry\n\n@[simp] def drop.aux {\u03b1 : Type u} : \u2115 \u2192 Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem drop.aux_none {\u03b1 : Type u} (n : \u2115) : drop.aux n none = computation.return none := sorry\n\ntheorem destruct_dropn {\u03b1 : Type u} (s : wseq \u03b1) (n : \u2115) :\n    destruct (drop s n) = destruct s >>= drop.aux n :=\n  sorry\n\ntheorem head_terminates_of_head_tail_terminates {\u03b1 : Type u} (s : wseq \u03b1)\n    [T : computation.terminates (head (tail s))] : computation.terminates (head s) :=\n  sorry\n\ntheorem destruct_some_of_destruct_tail_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1 \u00d7 wseq \u03b1}\n    (h : some a \u2208 destruct (tail s)) : \u2203 (a' : \u03b1 \u00d7 wseq \u03b1), some a' \u2208 destruct s :=\n  sorry\n\ntheorem head_some_of_head_tail_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : some a \u2208 head (tail s)) :\n    \u2203 (a' : \u03b1), some a' \u2208 head s :=\n  sorry\n\ntheorem head_some_of_nth_some {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} (h : some a \u2208 nth s n) :\n    \u2203 (a' : \u03b1), some a' \u2208 head s :=\n  sorry\n\nprotected instance productive_tail {\u03b1 : Type u} (s : wseq \u03b1) [productive s] : productive (tail s) :=\n  fun (n : \u2115) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (computation.terminates (nth (tail s) n))) (nth_tail s n)))\n      (wseq.nth_terminates s (n + 1))\n\nprotected instance productive_dropn {\u03b1 : Type u} (s : wseq \u03b1) [productive s] (n : \u2115) :\n    productive (drop s n) :=\n  fun (m : \u2115) =>\n    eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (computation.terminates (nth (drop s n) m)))\n          (Eq.symm (nth_add s n m))))\n      (wseq.nth_terminates s (n + m))\n\n/-- Given a productive weak sequence, we can collapse all the `think`s to\n  produce a sequence. -/\ndef to_seq {\u03b1 : Type u} (s : wseq \u03b1) [productive s] : seq \u03b1 :=\n  { val := fun (n : \u2115) => computation.get (nth s n), property := sorry }\n\ntheorem nth_terminates_le {\u03b1 : Type u} {s : wseq \u03b1} {m : \u2115} {n : \u2115} (h : m \u2264 n) :\n    computation.terminates (nth s n) \u2192 computation.terminates (nth s m) :=\n  sorry\n\ntheorem head_terminates_of_nth_terminates {\u03b1 : Type u} {s : wseq \u03b1} {n : \u2115} :\n    computation.terminates (nth s n) \u2192 computation.terminates (head s) :=\n  nth_terminates_le (nat.zero_le n)\n\ntheorem destruct_terminates_of_nth_terminates {\u03b1 : Type u} {s : wseq \u03b1} {n : \u2115}\n    (T : computation.terminates (nth s n)) : computation.terminates (destruct s) :=\n  iff.mp (head_terminates_iff s) (head_terminates_of_nth_terminates T)\n\ntheorem mem_rec_on {\u03b1 : Type u} {C : wseq \u03b1 \u2192 Prop} {a : \u03b1} {s : wseq \u03b1} (M : a \u2208 s)\n    (h1 : \u2200 (b : \u03b1) (s' : wseq \u03b1), a = b \u2228 C s' \u2192 C (cons b s'))\n    (h2 : \u2200 (s : wseq \u03b1), C s \u2192 C (think s)) : C s :=\n  sorry\n\n@[simp] theorem mem_think {\u03b1 : Type u} (s : wseq \u03b1) (a : \u03b1) : a \u2208 think s \u2194 a \u2208 s := sorry\n\ntheorem eq_or_mem_iff_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {a' : \u03b1} {s' : wseq \u03b1} :\n    some (a', s') \u2208 destruct s \u2192 (a \u2208 s \u2194 a = a' \u2228 a \u2208 s') :=\n  sorry\n\n@[simp] theorem mem_cons_iff {\u03b1 : Type u} (s : wseq \u03b1) (b : \u03b1) {a : \u03b1} :\n    a \u2208 cons b s \u2194 a = b \u2228 a \u2208 s :=\n  sorry\n\ntheorem mem_cons_of_mem {\u03b1 : Type u} {s : wseq \u03b1} (b : \u03b1) {a : \u03b1} (h : a \u2208 s) : a \u2208 cons b s :=\n  iff.mpr (mem_cons_iff s b) (Or.inr h)\n\ntheorem mem_cons {\u03b1 : Type u} (s : wseq \u03b1) (a : \u03b1) : a \u2208 cons a s :=\n  iff.mpr (mem_cons_iff s a) (Or.inl rfl)\n\ntheorem mem_of_mem_tail {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} : a \u2208 tail s \u2192 a \u2208 s := sorry\n\ntheorem mem_of_mem_dropn {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} : a \u2208 drop s n \u2192 a \u2208 s := sorry\n\ntheorem nth_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} {n : \u2115} : some a \u2208 nth s n \u2192 a \u2208 s := sorry\n\ntheorem exists_nth_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) :\n    \u2203 (n : \u2115), some a \u2208 nth s n :=\n  sorry\n\ntheorem exists_dropn_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) :\n    \u2203 (n : \u2115), \u2203 (s' : wseq \u03b1), some (a, s') \u2208 destruct (drop s n) :=\n  sorry\n\ntheorem lift_rel_dropn_destruct {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1}\n    {t : wseq \u03b2} (H : lift_rel R s t) (n : \u2115) :\n    computation.lift_rel (lift_rel_o R (lift_rel R)) (destruct (drop s n)) (destruct (drop t n)) :=\n  sorry\n\ntheorem exists_of_lift_rel_left {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1}\n    {t : wseq \u03b2} (H : lift_rel R s t) {a : \u03b1} (h : a \u2208 s) : Exists fun {b : \u03b2} => b \u2208 t \u2227 R a b :=\n  sorry\n\ntheorem exists_of_lift_rel_right {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : wseq \u03b1}\n    {t : wseq \u03b2} (H : lift_rel R s t) {b : \u03b2} (h : b \u2208 t) : Exists fun {a : \u03b1} => a \u2208 s \u2227 R a b :=\n  exists_of_lift_rel_left\n    (eq.mp\n      (Eq._oldrec (Eq.refl (lift_rel R s t)) (Eq.symm (lift_rel.swap fun (x : \u03b2) (y : \u03b1) => R y x)))\n      H)\n    h\n\ntheorem head_terminates_of_mem {\u03b1 : Type u} {s : wseq \u03b1} {a : \u03b1} (h : a \u2208 s) :\n    computation.terminates (head s) :=\n  sorry\n\ntheorem of_mem_append {\u03b1 : Type u} {s\u2081 : wseq \u03b1} {s\u2082 : wseq \u03b1} {a : \u03b1} :\n    a \u2208 append s\u2081 s\u2082 \u2192 a \u2208 s\u2081 \u2228 a \u2208 s\u2082 :=\n  seq.of_mem_append\n\ntheorem mem_append_left {\u03b1 : Type u} {s\u2081 : wseq \u03b1} {s\u2082 : wseq \u03b1} {a : \u03b1} :\n    a \u2208 s\u2081 \u2192 a \u2208 append s\u2081 s\u2082 :=\n  seq.mem_append_left\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : wseq \u03b1} :\n    b \u2208 map f s \u2192 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b :=\n  sorry\n\n@[simp] theorem lift_rel_nil {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) : lift_rel R nil nil :=\n  sorry\n\n@[simp] theorem lift_rel_cons {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1) (b : \u03b2)\n    (s : wseq \u03b1) (t : wseq \u03b2) : lift_rel R (cons a s) (cons b t) \u2194 R a b \u2227 lift_rel R s t :=\n  sorry\n\n@[simp] theorem lift_rel_think_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1)\n    (t : wseq \u03b2) : lift_rel R (think s) t \u2194 lift_rel R s t :=\n  sorry\n\n@[simp] theorem lift_rel_think_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (s : wseq \u03b1)\n    (t : wseq \u03b2) : lift_rel R s (think t) \u2194 lift_rel R s t :=\n  sorry\n\ntheorem cons_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (a : \u03b1) (h : s ~ t) :\n    cons a s ~ cons a t :=\n  sorry\n\ntheorem think_equiv {\u03b1 : Type u} (s : wseq \u03b1) : think s ~ s :=\n  eq.mpr (id (congr_fun (congr_fun equiv.equations._eqn_1 (think s)) s))\n    (eq.mpr (id (propext (lift_rel_think_left Eq s s))) (equiv.refl s))\n\ntheorem think_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (a : \u03b1) (h : s ~ t) :\n    think s ~ think t :=\n  eq.mpr (id (congr_fun (congr_fun equiv.equations._eqn_1 (think s)) (think t)))\n    (eq.mpr\n      (id\n        (Eq.trans (propext (lift_rel_think_right Eq (think s) t))\n          (propext (lift_rel_think_left Eq s t))))\n      h)\n\ntheorem head_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} : s ~ t \u2192 head s ~ head t := sorry\n\ntheorem flatten_equiv {\u03b1 : Type u} {c : computation (wseq \u03b1)} {s : wseq \u03b1} (h : s \u2208 c) :\n    flatten c ~ s :=\n  sorry\n\ntheorem lift_rel_flatten {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {c1 : computation (wseq \u03b1)}\n    {c2 : computation (wseq \u03b2)} (h : computation.lift_rel (lift_rel R) c1 c2) :\n    lift_rel R (flatten c1) (flatten c2) :=\n  sorry\n\ntheorem flatten_congr {\u03b1 : Type u} {c1 : computation (wseq \u03b1)} {c2 : computation (wseq \u03b1)} :\n    computation.lift_rel equiv c1 c2 \u2192 flatten c1 ~ flatten c2 :=\n  lift_rel_flatten\n\ntheorem tail_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) : tail s ~ tail t := sorry\n\ntheorem dropn_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (n : \u2115) :\n    drop s n ~ drop t n :=\n  sorry\n\ntheorem nth_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (n : \u2115) : nth s n ~ nth t n :=\n  head_congr (dropn_congr h n)\n\ntheorem mem_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) (a : \u03b1) : a \u2208 s \u2194 a \u2208 t :=\n  sorry\n\ntheorem productive_congr {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) :\n    productive s \u2194 productive t :=\n  forall_congr fun (n : \u2115) => computation.terminates_congr (nth_congr h n)\n\ntheorem equiv.ext {\u03b1 : Type u} {s : wseq \u03b1} {t : wseq \u03b1} (h : \u2200 (n : \u2115), nth s n ~ nth t n) :\n    s ~ t :=\n  sorry\n\ntheorem length_eq_map {\u03b1 : Type u} (s : wseq \u03b1) :\n    length s = computation.map list.length (to_list s) :=\n  sorry\n\n@[simp] theorem of_list_nil {\u03b1 : Type u} : of_list [] = nil := rfl\n\n@[simp] theorem of_list_cons {\u03b1 : Type u} (a : \u03b1) (l : List \u03b1) :\n    of_list (a :: l) = cons a (of_list l) :=\n  sorry\n\n@[simp] theorem to_list'_nil {\u03b1 : Type u} (l : List \u03b1) :\n    computation.corec to_list._match_2 (l, nil) = computation.return (list.reverse l) :=\n  computation.destruct_eq_ret rfl\n\n@[simp] theorem to_list'_cons {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) (a : \u03b1) :\n    computation.corec to_list._match_2 (l, cons a s) =\n        computation.think (computation.corec to_list._match_2 (a :: l, s)) :=\n  sorry\n\n@[simp] theorem to_list'_think {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) :\n    computation.corec to_list._match_2 (l, think s) =\n        computation.think (computation.corec to_list._match_2 (l, s)) :=\n  sorry\n\ntheorem to_list'_map {\u03b1 : Type u} (l : List \u03b1) (s : wseq \u03b1) :\n    computation.corec to_list._match_2 (l, s) = append (list.reverse l) <$> to_list s :=\n  sorry\n\n@[simp] theorem to_list_cons {\u03b1 : Type u} (a : \u03b1) (s : wseq \u03b1) :\n    to_list (cons a s) = computation.think (List.cons a <$> to_list s) :=\n  sorry\n\n@[simp] theorem to_list_nil {\u03b1 : Type u} : to_list nil = computation.return [] :=\n  computation.destruct_eq_ret rfl\n\ntheorem to_list_of_list {\u03b1 : Type u} (l : List \u03b1) : l \u2208 to_list (of_list l) := sorry\n\n@[simp] theorem destruct_of_seq {\u03b1 : Type u} (s : seq \u03b1) :\n    destruct (of_seq s) =\n        computation.return (option.map (fun (a : \u03b1) => (a, of_seq (seq.tail s))) (seq.head s)) :=\n  sorry\n\n@[simp] theorem head_of_seq {\u03b1 : Type u} (s : seq \u03b1) :\n    head (of_seq s) = computation.return (seq.head s) :=\n  sorry\n\n@[simp] theorem tail_of_seq {\u03b1 : Type u} (s : seq \u03b1) : tail (of_seq s) = of_seq (seq.tail s) :=\n  sorry\n\n@[simp] theorem dropn_of_seq {\u03b1 : Type u} (s : seq \u03b1) (n : \u2115) :\n    drop (of_seq s) n = of_seq (seq.drop s n) :=\n  sorry\n\ntheorem nth_of_seq {\u03b1 : Type u} (s : seq \u03b1) (n : \u2115) :\n    nth (of_seq s) n = computation.return (seq.nth s n) :=\n  sorry\n\nprotected instance productive_of_seq {\u03b1 : Type u} (s : seq \u03b1) : productive (of_seq s) :=\n  fun (n : \u2115) =>\n    eq.mpr (id (Eq._oldrec (Eq.refl (computation.terminates (nth (of_seq s) n))) (nth_of_seq s n)))\n      (computation.ret_terminates (seq.nth s n))\n\ntheorem to_seq_of_seq {\u03b1 : Type u} (s : seq \u03b1) : to_seq (of_seq s) = s := sorry\n\n/-- The monadic `return a` is a singleton list containing `a`. -/\ndef ret {\u03b1 : Type u} (a : \u03b1) : wseq \u03b1 := of_list [a]\n\n@[simp] theorem map_nil {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : map f nil = nil := rfl\n\n@[simp] theorem map_cons {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (s : wseq \u03b1) :\n    map f (cons a s) = cons (f a) (map f s) :=\n  seq.map_cons (option.map f) (some a) s\n\n@[simp] theorem map_think {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) :\n    map f (think s) = think (map f s) :=\n  seq.map_cons (option.map f) none s\n\n@[simp] theorem map_id {\u03b1 : Type u} (s : wseq \u03b1) : map id s = s := sorry\n\n@[simp] theorem map_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (ret a) = ret (f a) :=\n  sorry\n\n@[simp] theorem map_append {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) (t : wseq \u03b1) :\n    map f (append s t) = append (map f s) (map f t) :=\n  seq.map_append (option.map f) s t\n\ntheorem map_comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (s : wseq \u03b1) :\n    map (g \u2218 f) s = map g (map f s) :=\n  sorry\n\ntheorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : wseq \u03b1} :\n    a \u2208 s \u2192 f a \u2208 map f s :=\n  seq.mem_map (option.map f)\n\n-- The converse is not true without additional assumptions\n\ntheorem exists_of_mem_join {\u03b1 : Type u} {a : \u03b1} {S : wseq (wseq \u03b1)} :\n    a \u2208 join S \u2192 \u2203 (s : wseq \u03b1), s \u2208 S \u2227 a \u2208 s :=\n  sorry\n\ntheorem exists_of_mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : wseq \u03b1} {f : \u03b1 \u2192 wseq \u03b2} {b : \u03b2}\n    (h : b \u2208 bind s f) : \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a :=\n  sorry\n\ntheorem destruct_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) :\n    destruct (map f s) = computation.map (option.map (prod.map f (map f))) (destruct s) :=\n  sorry\n\ntheorem lift_rel_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b2} {f1 : \u03b1 \u2192 \u03b3} {f2 : \u03b2 \u2192 \u03b4} (h1 : lift_rel R s1 s2)\n    (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)) : lift_rel S (map f1 s1) (map f2 s2) :=\n  sorry\n\ntheorem map_congr {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {s : wseq \u03b1} {t : wseq \u03b1} (h : s ~ t) :\n    map f s ~ map f t :=\n  lift_rel_map Eq Eq h fun (_x _x_1 : \u03b1) => congr_arg fun (_x : \u03b1) => f _x\n\n@[simp] def destruct_append.aux {\u03b1 : Type u} (t : wseq \u03b1) :\n    Option (\u03b1 \u00d7 wseq \u03b1) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem destruct_append {\u03b1 : Type u} (s : wseq \u03b1) (t : wseq \u03b1) :\n    destruct (append s t) = computation.bind (destruct s) (destruct_append.aux t) :=\n  sorry\n\n@[simp] def destruct_join.aux {\u03b1 : Type u} :\n    Option (wseq \u03b1 \u00d7 wseq (wseq \u03b1)) \u2192 computation (Option (\u03b1 \u00d7 wseq \u03b1)) :=\n  sorry\n\ntheorem destruct_join {\u03b1 : Type u} (S : wseq (wseq \u03b1)) :\n    destruct (join S) = computation.bind (destruct S) destruct_join.aux :=\n  sorry\n\ntheorem lift_rel_append {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b1}\n    {t1 : wseq \u03b2} {t2 : wseq \u03b2} (h1 : lift_rel R s1 t1) (h2 : lift_rel R s2 t2) :\n    lift_rel R (append s1 s2) (append t1 t2) :=\n  sorry\n\ntheorem lift_rel_join.lem {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S : wseq (wseq \u03b1)}\n    {T : wseq (wseq \u03b2)} {U : wseq \u03b1 \u2192 wseq \u03b2 \u2192 Prop} (ST : lift_rel (lift_rel R) S T)\n    (HU :\n      \u2200 (s1 : wseq \u03b1) (s2 : wseq \u03b2),\n        (\u2203 (s : wseq \u03b1),\n            \u2203 (t : wseq \u03b2),\n              \u2203 (S : wseq (wseq \u03b1)),\n                \u2203 (T : wseq (wseq \u03b2)),\n                  s1 = append s (join S) \u2227\n                    s2 = append t (join T) \u2227 lift_rel R s t \u2227 lift_rel (lift_rel R) S T) \u2192\n          U s1 s2)\n    {a : Option (\u03b1 \u00d7 wseq \u03b1)} (ma : a \u2208 destruct (join S)) :\n    Exists fun {b : Option (\u03b2 \u00d7 wseq \u03b2)} => b \u2208 destruct (join T) \u2227 lift_rel_o R U a b :=\n  sorry\n\ntheorem lift_rel_join {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) {S : wseq (wseq \u03b1)}\n    {T : wseq (wseq \u03b2)} (h : lift_rel (lift_rel R) S T) : lift_rel R (join S) (join T) :=\n  sorry\n\ntheorem join_congr {\u03b1 : Type u} {S : wseq (wseq \u03b1)} {T : wseq (wseq \u03b1)} (h : lift_rel equiv S T) :\n    join S ~ join T :=\n  lift_rel_join Eq h\n\ntheorem lift_rel_bind {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop)\n    (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : wseq \u03b1} {s2 : wseq \u03b2} {f1 : \u03b1 \u2192 wseq \u03b3} {f2 : \u03b2 \u2192 wseq \u03b4}\n    (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 lift_rel S (f1 a) (f2 b)) :\n    lift_rel S (bind s1 f1) (bind s2 f2) :=\n  lift_rel_join S (lift_rel_map R (lift_rel S) h1 h2)\n\ntheorem bind_congr {\u03b1 : Type u} {\u03b2 : Type v} {s1 : wseq \u03b1} {s2 : wseq \u03b1} {f1 : \u03b1 \u2192 wseq \u03b2}\n    {f2 : \u03b1 \u2192 wseq \u03b2} (h1 : s1 ~ s2) (h2 : \u2200 (a : \u03b1), f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 :=\n  lift_rel_bind Eq Eq h1\n    fun (a b : \u03b1) (h : a = b) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (lift_rel Eq (f1 a) (f2 b))) h)) (h2 b)\n\n@[simp] theorem join_ret {\u03b1 : Type u} (s : wseq \u03b1) : join (ret s) ~ s := sorry\n\n@[simp] theorem join_map_ret {\u03b1 : Type u} (s : wseq \u03b1) : join (map ret s) ~ s := sorry\n\n@[simp] theorem join_append {\u03b1 : Type u} (S : wseq (wseq \u03b1)) (T : wseq (wseq \u03b1)) :\n    join (append S T) ~ append (join S) (join T) :=\n  sorry\n\n@[simp] theorem bind_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : wseq \u03b1) :\n    bind s (ret \u2218 f) ~ map f s :=\n  id\n    (eq.mpr (id (Eq._oldrec (Eq.refl (join (map (ret \u2218 f) s) ~ map f s)) (map_comp f ret s)))\n      (join_map_ret (map f s)))\n\n@[simp] theorem ret_bind {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (f : \u03b1 \u2192 wseq \u03b2) :\n    bind (ret a) f ~ f a :=\n  sorry\n\n@[simp] theorem map_join {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (S : wseq (wseq \u03b1)) :\n    map f (join S) = join (map (map f) S) :=\n  sorry\n\n@[simp] theorem join_join {\u03b1 : Type u} (SS : wseq (wseq (wseq \u03b1))) :\n    join (join SS) ~ join (map join SS) :=\n  sorry\n\n@[simp] theorem bind_assoc {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (s : wseq \u03b1) (f : \u03b1 \u2192 wseq \u03b2)\n    (g : \u03b2 \u2192 wseq \u03b3) : bind (bind s f) g ~ bind s fun (x : \u03b1) => bind (f x) g :=\n  sorry\n\nprotected instance monad : Monad wseq :=\n  { toApplicative :=\n      { toFunctor := { map := map, mapConst := fun (\u03b1 \u03b2 : Type u_1) => map \u2218 function.const \u03b2 },\n        toPure := { pure := ret },\n        toSeq :=\n          { seq :=\n              fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) =>\n                bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : wseq \u03b1) (b : wseq \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b1 (map (function.const \u03b2) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : wseq \u03b1) (b : wseq \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : wseq (\u03b1 \u2192 \u03b2)) (x : wseq \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b2 (map (function.const \u03b1 id) a) b } },\n    toBind := { bind := bind } }\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/seq/wseq_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.35498097379294513}}
{"text": "import sheaves_of_modules.defs\n/-\n\n# Pushforward and pullback of sheaves of modules\n\n-/\n\nvariables {X Y : RINGED_SPACE} \nnamespace SHEAF_OF_MODULES\n\nsection experiment\n\n--#check @Top.sheaf.pushforward\n\nvariable (f : X \u27f6 Y)\n\nexample : (X : TOP) \u27f6 Y := f.base\n\nend experiment\n\n--#check Top.presheaf.pushforward_obj\n--set_option pp.notation false\n\n/-\ninstance (\ud835\udcdc : SHEAF_OF_MODULES X) (U : (opens (X : TOP))\u1d52\u1d56) :\n  module ((\ud835\udcde_ X) U) (\ud835\udcdc.obj U) := \ud835\udcdc.module_structure U\n-/\n\n--instance (U) : module \n--#print notation _*\n--#check Top.presheaf.pushforward_obj\n\n/-\n\n/-- Pushforward a presheaf on `X` along a continuous map `f : X \u27f6 Y`, obtaining a presheaf\non `Y`. -/\ndef pushforward_obj {X Y : Top.{w}} (f : X \u27f6 Y) (\u2131 : X.presheaf C) : Y.presheaf C :=\n(opens.map f).op \u22d9 \u2131\n\ninfix ` _* `: 80 := pushforward_obj\n-/\n\n--#where\ndef map (f : X \u27f6 Y) (\ud835\udcdc : SHEAF_OF_MODULES X) : SHEAF_OF_MODULES Y :=\n{ ab_sheaf := (Top.sheaf.pushforward.{0} f.base).obj \ud835\udcdc.ab_sheaf,\n  module_structure := \u03bb U, by letI : module \u21a5((f.base _* X.to_PresheafedSpace.presheaf).obj U)\n    \u21a5(((Top.sheaf.pushforward f.base).obj \ud835\udcdc.ab_sheaf).val.obj U) := obj.module \ud835\udcdc ((topological_space.opens.map f.base).op.obj U); exact\n    module.comp_hom (\ud835\udcdc.obj ((topological_space.opens.map f.base).op.obj U)) (f.c.app U),\n--   module_structure := \u03bb U, begin\n-- --    delta Top.sheaf.pushforward,\n-- --    dsimp only,\n-- --    delta Top.presheaf.pushforward_obj,\n-- --    change module _ (\ud835\udcdc.obj _),\n--     let V : (topological_space.opens (X : TOP))\u1d52\u1d56 := (topological_space.opens.map f.base).op.obj U,\n--     change module ((\ud835\udcde_ Y) U) (\ud835\udcdc.obj V),\n-- --    letI baz : module ((\ud835\udcde_ X) V) (\ud835\udcdc.obj V) := infer_instance, -- show_term {apply_instance}\n--     letI : module \u21a5((f.base _* X.to_PresheafedSpace.presheaf).obj U) \u21a5(\ud835\udcdc.obj V) := obj.module \ud835\udcdc V,\n-- --    let foo : (\ud835\udcde_ Y) U \u2192+* (\ud835\udcde_ X) V := f.c.app U,\n--     exact module.comp_hom (\ud835\udcdc.obj V) (f.c.app U),\n--   end,\n  compatibility_bit := \u03bb U V i, begin\n    rintro s (n : \ud835\udcdc.ab_sheaf.val.obj _),\n    change \u21a5((\ud835\udcde_ Y) U) at s,\n    have foo := f.c.naturality i,\n    let j : ((topological_space.opens.map f.base).op.obj U) \u27f6 ((topological_space.opens.map f.base).op.obj V) :=\n      ((topological_space.opens.map f.base).op.map i),\n    let r : (\ud835\udcde_ X) ((topological_space.opens.map f.base).op.obj U) := f.c.app U s,\n    --have bar := \ud835\udcdc.compatibility_bit _ _ j r n,\n    convert \ud835\udcdc.compatibility_bit _ _ j r n using 1,\n    rw fun_like.ext_iff at foo,\n    specialize foo s,\n    have moo : (f.base _* X.to_PresheafedSpace.presheaf).map i = X.to_PresheafedSpace.presheaf.map j,\n      refl,\n    rw fun_like.ext_iff at moo,\n    specialize moo r,\n    rw \u2190 moo,\n    change _ = ((f.c.app U \u226b (f.base _* X.to_PresheafedSpace.presheaf).map i) s) \u2022 _,\n    rw \u2190 foo,\n    refl,\n    \n    -- need to pull back i; need to fix implicits\n    -- might need pen and paper here\n    --convert bar _ _,\n    /-\n    Have foo: O_Y(U)->O_Y(V)->f_*O_X(V) = O_Y(U)->f_*O_X(U)->f_*O_X(V)\n    have bar : \u2200 r \u2208 O_X(f\u207b\u00b9(U)), m \u2208 \ud835\udcdc(f\u207b\u00b9(U)),\n      res(r\u2022m) \u2208 \ud835\udcdc(f\u207b\u00b9(V)) = (res(r) : O_X(f\u207b\u00b9(V)))\u2022res(m)\n    \n    Want: \u2200 s \u2208 O_Y(U), \u2200 n \u2208 f_*\ud835\udcdc(U) := \ud835\udcdc(f\u207b\u00b9(U)), res(s\u2022n)=res(s)\u2022res(n)\n    Proof: define r=image of s in O_X(f\u207b\u00b9(U))=f_*O_X(U). \n\n    -/\n    --letI : module (Y.to_PresheafedSpace.presheaf.obj U) \n    --  (((Top.sheaf.pushforward f.base).obj \ud835\udcdc.ab_sheaf).val.obj U) := module_structure \ud835\udcdc ((topological_space.opens.map f.base).op.obj U),\n\n    --calc\n    --(((Top.sheaf.pushforward f.base).obj \ud835\udcdc.ab_sheaf).val.map i) (s \u2022 n) = \n    --((Y.to_PresheafedSpace.presheaf.map i) s : (\ud835\udcde_ Y) V) \u2022 (((Top.sheaf.pushforward f.base).obj \ud835\udcdc.ab_sheaf).val.map i) n : sorry\n  end }\n\ninfix (name := hi) ` _* `: 80 := map\n\n--#check Top.sheaf.pushforward\n\n\nvariables (f : X \u27f6 Y) (\ud835\udcdc : SHEAF_OF_MODULES X)\n\ndef map_id (\ud835\udcdc : SHEAF_OF_MODULES X) : (\ud835\udfd9 X) _* \ud835\udcdc \u2245 \ud835\udcdc :=\n{ hom := \n  { ab_sheaf := \n    { val := \n      { app := \u03bb U, \ud835\udcdc.ab_sheaf.val.map $ category_theory.op_hom_of_le $ \u03bb x hx, hx,\n        naturality' := begin\n          intros U V f,\n          ext,\n          simp only [category_theory.comp_apply],\n          sorry,\n        end } },\n      map_smul := begin\n        intros,\n        simp only [category_theory.op_hom_of_le],\n        sorry,        \n      end },\n    inv := \n    { ab_sheaf :=\n      { val := \n        { app := \u03bb U, \ud835\udcdc.ab_sheaf.val.map $ category_theory.op_hom_of_le $ \u03bb x hx, hx,\n          naturality' := begin\n            intros U V g,\n            ext,\n            simp only [category_theory.comp_apply],\n            sorry,\n          end } },\n      map_smul := sorry },\n  hom_inv_id' := begin\n    ext U m,\n    --dsimp only [category_theory.comp_apply],\n    unfold_coes,\n    dsimp only,\n    sorry,\n  end,\n  inv_hom_id' := sorry }\n\nend SHEAF_OF_MODULES\n\nexample (Z : Type) [topological_space Z] (U V : (topological_space.opens Z)\u1d52\u1d56) (h : V.unop \u2286 U.unop): \n  U \u27f6 V :=\nbegin\n  refine category_theory.op_hom_of_le h,\nend\n", "meta": {"author": "ImperialCollegeLondon", "repo": "tcc-lean-alg-geom-2022", "sha": "21d4e02156d842332c8b56e044dabe147171c173", "save_path": "github-repos/lean/ImperialCollegeLondon-tcc-lean-alg-geom-2022", "path": "github-repos/lean/ImperialCollegeLondon-tcc-lean-alg-geom-2022/tcc-lean-alg-geom-2022-21d4e02156d842332c8b56e044dabe147171c173/src/sheaves_of_modules/pushforward_practice.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.35498097379294513}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura, Mario Carneiro\n-/\nimport data.pnat.basic\nimport data.list.range\nimport data.array.lemmas\nimport algebra.group\nimport data.sigma.basic\n\n/-!\n# Hash maps\n\nDefines a hash map data structure, representing a finite key-value map\nwith a value type that may depend on the key type.  The structure\nrequires a `nat`-valued hash function to associate keys to buckets.\n\n## Main definitions\n\n* `hash_map`: constructed with `mk_hash_map`.\n\n## Implementation details\n\nA hash map with key type `\u03b1` and (dependent) value type `\u03b2 : \u03b1 \u2192 Type*`\nconsists of an array of *buckets*, which are lists containing\nkey/value pairs for that bucket.  The hash function is taken modulo `n`\nto assign keys to their respective bucket.  Because of this, some care\nshould be put into the hash function to ensure it evenly distributes\nkeys.\n\nThe bucket array is an `array`.  These have special VM support for\nin-place modification if there is only ever one reference to them.  If\none takes special care to never keep references to old versions of a\nhash map alive after updating it, then the hash map will be modified\nin-place.  In this documentation, when we say a hash map is modified\nin-place, we are assuming the API is being used in this manner.\n\nWhen inserting (`hash_map.insert`), if the number of stored pairs (the\n*size*) is going to exceed the number of buckets, then a new hash map\nis first created with double the number of buckets and everything in\nthe old hash map is reinserted along with the new key/value pair.\nOtherwise, the bucket array is modified in-place.  The amortized\nrunning time of inserting $$n$$ elements into a hash map is $$O(n)$$.\n\nWhen removing (`hash_map.erase`), the hash map is modified in-place.\nThe implementation does not reduce the number of buckets in the hash\nmap if the size gets too low.\n\n## Tags\n\nhash map\n\n-/\n\nuniverses u v w\n\n/-- `bucket_array \u03b1 \u03b2` is the underlying data type for `hash_map \u03b1 \u03b2`,\n  an array of linked lists of key-value pairs. -/\ndef bucket_array (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) (n : \u2115+) :=\narray n (list \u03a3 a, \u03b2 a)\n\n/-- Make a hash_map index from a `nat` hash value and a (positive) buffer size -/\ndef hash_map.mk_idx (n : \u2115+) (i : nat) : fin n :=\n\u27e8i % n, nat.mod_lt _ n.2\u27e9\n\nnamespace bucket_array\nsection\nparameters {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat)\nvariables {n : \u2115+} (data : bucket_array \u03b1 \u03b2 n)\n\ninstance : inhabited (bucket_array \u03b1 \u03b2 n) :=\n\u27e8mk_array _ []\u27e9\n\n/-- Read the bucket corresponding to an element -/\ndef read (a : \u03b1) : list \u03a3 a, \u03b2 a :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\ndata.read bidx\n\n/-- Write the bucket corresponding to an element -/\ndef write (a : \u03b1) (l : list \u03a3 a, \u03b2 a) : bucket_array \u03b1 \u03b2 n :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\ndata.write bidx l\n\n/-- Modify (read, apply `f`, and write) the bucket corresponding to an element -/\ndef modify (a : \u03b1) (f : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)) : bucket_array \u03b1 \u03b2 n :=\nlet bidx := hash_map.mk_idx n (hash_fn a) in\narray.write data bidx (f (array.read data bidx))\n\n/-- The list of all key-value pairs in the bucket list -/\ndef as_list : list \u03a3 a, \u03b2 a := data.to_list.join\n\ntheorem mem_as_list {a : \u03a3 a, \u03b2 a} : a \u2208 data.as_list \u2194 \u2203i, a \u2208 array.read data i :=\nhave (\u2203 (l : list (\u03a3 (a : \u03b1), \u03b2 a)) (i : fin (n.val)), a \u2208 l \u2227 array.read data i = l) \u2194\n  \u2203 (i : fin (n.val)), a \u2208 array.read data i,\nby rw exists_swap; exact exists_congr (\u03bb i, by simp),\nby simp [as_list]; simpa [array.mem.def, and_comm]\n\n/-- Fold a function `f` over the key-value pairs in the bucket list -/\ndef foldl {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\ndata.foldl d (\u03bb b d, b.foldl (\u03bb r a, f r a.1 a.2) d)\n\ntheorem foldl_eq {\u03b4 : Type w} (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) :\n  data.foldl d f = data.as_list.foldl (\u03bb r a, f r a.1 a.2) d :=\nby rw [foldl, as_list, list.foldl_join, \u2190 array.to_list_foldl]\n\nend\nend bucket_array\n\nnamespace hash_map\nsection\nparameters {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat)\n\n/-- Insert the pair `\u27e8a, b\u27e9` into the correct location in the bucket array\n  (without checking for duplication) -/\ndef reinsert_aux {n} (data : bucket_array \u03b1 \u03b2 n) (a : \u03b1) (b : \u03b2 a) : bucket_array \u03b1 \u03b2 n :=\ndata.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9 :: l)\n\ntheorem mk_as_list (n : \u2115+) : bucket_array.as_list (mk_array n [] : bucket_array \u03b1 \u03b2 n) = [] :=\nlist.eq_nil_iff_forall_not_mem.mpr $ \u03bb x m,\nlet \u27e8i, h\u27e9 := (bucket_array.mem_as_list _).1 m in h\n\nparameter [decidable_eq \u03b1]\n\n/-- Search a bucket for a key `a` and return the value -/\ndef find_aux (a : \u03b1) : list (\u03a3 a, \u03b2 a) \u2192 option (\u03b2 a)\n| []          := none\n| (\u27e8a',b\u27e9::t) := if h : a' = a then some (eq.rec_on h b) else find_aux t\n\ntheorem find_aux_iff {a : \u03b1} {b : \u03b2 a} :\n  \u03a0 {l : list \u03a3 a, \u03b2 a}, (l.map sigma.fst).nodup \u2192 (find_aux a l = some b \u2194 sigma.mk a b \u2208 l)\n| []          nd := \u27e8\u03bbn, by injection n, false.elim\u27e9\n| (\u27e8a',b'\u27e9::t) nd := begin\n  by_cases a' = a,\n  { clear find_aux_iff, subst h,\n    suffices : b' = b \u2194 b' = b \u2228 sigma.mk a' b \u2208 t, {simpa [find_aux, eq_comm]},\n    refine (or_iff_left_of_imp (\u03bb m, _)).symm,\n    have : a' \u2209 t.map sigma.fst, from list.not_mem_of_nodup_cons nd,\n    exact this.elim (list.mem_map_of_mem sigma.fst m) },\n  { have : sigma.mk a b \u2260 \u27e8a', b'\u27e9,\n    { intro e, injection e with e, exact h e.symm },\n    simp at nd, simp [find_aux, h, ne.symm h, find_aux_iff, nd] }\nend\n\n/-- Returns `tt` if the bucket `l` contains the key `a` -/\ndef contains_aux (a : \u03b1) (l : list \u03a3 a, \u03b2 a) : bool :=\n(find_aux a l).is_some\n\ntheorem contains_aux_iff {a : \u03b1} {l : list \u03a3 a, \u03b2 a} (nd : (l.map sigma.fst).nodup) :\n  contains_aux a l \u2194 a \u2208 l.map sigma.fst :=\nbegin\n  unfold contains_aux,\n  cases h : find_aux a l with b; simp,\n  { assume (b : \u03b2 a) (m : sigma.mk a b \u2208 l),\n    rw (find_aux_iff nd).2 m at h,\n    contradiction },\n  { show \u2203 (b : \u03b2 a), sigma.mk a b \u2208 l,\n    exact \u27e8_, (find_aux_iff nd).1 h\u27e9 },\nend\n\n/-- Modify a bucket to replace a value in the list. Leaves the list\n unchanged if the key is not found. -/\ndef replace_aux (a : \u03b1) (b : \u03b2 a) : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)\n| []            := []\n| (\u27e8a', b'\u27e9::t) := if a' = a then \u27e8a, b\u27e9::t else \u27e8a', b'\u27e9 :: replace_aux t\n\n/-- Modify a bucket to remove a key, if it exists. -/\ndef erase_aux (a : \u03b1) : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)\n| []            := []\n| (\u27e8a', b'\u27e9::t) := if a' = a then t else \u27e8a', b'\u27e9 :: erase_aux t\n\n/-- The predicate `valid bkts sz` means that `bkts` satisfies the `hash_map`\n  invariants: There are exactly `sz` elements in it, every pair is in the\n  bucket determined by its key and the hash function, and no key appears\n  multiple times in the list. -/\nstructure valid {n} (bkts : bucket_array \u03b1 \u03b2 n) (sz : nat) : Prop :=\n(len : bkts.as_list.length = sz)\n(idx : \u2200 {i} {a : \u03a3 a, \u03b2 a}, a \u2208 array.read bkts i \u2192\n  mk_idx n (hash_fn a.1) = i)\n(nodup : \u2200i, ((array.read bkts i).map sigma.fst).nodup)\n\ntheorem valid.idx_enum {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  {i l} (he : (i, l) \u2208 bkts.to_list.enum) {a} {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) :\n  \u2203 h, mk_idx n (hash_fn a) = \u27e8i, h\u27e9 :=\n(array.mem_to_list_enum.mp he).imp (\u03bb h e, by subst e; exact v.idx hl)\n\ntheorem valid.idx_enum_1 {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  {i l} (he : (i, l) \u2208 bkts.to_list.enum) {a} {b : \u03b2 a} (hl : sigma.mk a b \u2208 l) :\n  (mk_idx n (hash_fn a)).1 = i :=\nlet \u27e8h, e\u27e9 := v.idx_enum _ he hl in by rw e; refl\n\ntheorem valid.as_list_nodup {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz) :\n  (bkts.as_list.map sigma.fst).nodup :=\nbegin\n  suffices : (bkts.to_list.map (list.map sigma.fst)).pairwise list.disjoint,\n  { suffices : \u2200 l, array.mem l bkts \u2192 (l.map sigma.fst).nodup,\n      by simpa [bucket_array.as_list, list.nodup_join, *],\n    rintros l \u27e8i, rfl\u27e9,\n    apply v.nodup },\n  rw [\u2190 list.enum_map_snd bkts.to_list, list.pairwise_map, list.pairwise_map],\n  have : (bkts.to_list.enum.map prod.fst).nodup := by simp [list.nodup_range],\n  refine list.pairwise.imp_of_mem _ ((list.pairwise_map _).1 this),\n  rw prod.forall, intros i l\u2081,\n  rw prod.forall, intros j l\u2082 me\u2081 me\u2082 ij,\n  simp [list.disjoint], intros a b ml\u2081 b' ml\u2082,\n  apply ij, rwa [\u2190 v.idx_enum_1 _ me\u2081 ml\u2081, \u2190 v.idx_enum_1 _ me\u2082 ml\u2082]\nend\n\ntheorem mk_valid (n : \u2115+) : @valid n (mk_array n []) 0 :=\n\u27e8by simp [mk_as_list], \u03bb i a h, by cases h, \u03bb i, list.nodup_nil\u27e9\n\ntheorem valid.find_aux_iff {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz) {a : \u03b1}\n  {b : \u03b2 a} :\n  find_aux a (bkts.read hash_fn a) = some b \u2194 sigma.mk a b \u2208 bkts.as_list :=\n(find_aux_iff (v.nodup _)).trans $\nby rw bkts.mem_as_list; exact \u27e8\u03bb h, \u27e8_, h\u27e9, \u03bb \u27e8i, h\u27e9, (v.idx h).symm \u25b8 h\u27e9\n\ntheorem valid.contains_aux_iff {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz : nat} (v : valid bkts sz)\n  (a : \u03b1) :\n  contains_aux a (bkts.read hash_fn a) \u2194 a \u2208 bkts.as_list.map sigma.fst :=\nby simp [contains_aux, option.is_some_iff_exists, v.find_aux_iff hash_fn]\n\nsection\n  parameters {n : \u2115+} {bkts : bucket_array \u03b1 \u03b2 n}\n             {bidx : fin n} {f : list (\u03a3 a, \u03b2 a) \u2192 list (\u03a3 a, \u03b2 a)}\n             (u v1 v2 w : list \u03a3 a, \u03b2 a)\n\n  local notation `L` := array.read bkts bidx\n  private def bkts' : bucket_array \u03b1 \u03b2 n := array.write bkts bidx (f L)\n\n  variables (hl : L = u ++ v1 ++ w)\n            (hfl : f L = u ++ v2 ++ w)\n  include hl hfl\n\n  theorem append_of_modify :\n  \u2203 u' w', bkts.as_list = u' ++ v1 ++ w' \u2227 bkts'.as_list = u' ++ v2 ++ w' :=\n  begin\n    unfold bucket_array.as_list,\n    have h : (bidx : \u2115) < bkts.to_list.length, { simp only [bidx.is_lt, array.to_list_length] },\n    refine \u27e8(bkts.to_list.take bidx).join ++ u, w ++ (bkts.to_list.drop (bidx+1)).join, _, _\u27e9,\n    { conv { to_lhs,\n        rw [\u2190 list.take_append_drop bidx bkts.to_list, list.drop_eq_nth_le_cons h],\n        simp [hl] }, simp },\n    { conv { to_lhs,\n        rw [bkts', array.write_to_list, list.update_nth_eq_take_cons_drop _ h],\n        simp [hfl] }, simp }\n  end\n\n  variables (hvnd : (v2.map sigma.fst).nodup)\n            (hal : \u2200 (a : \u03a3 a, \u03b2 a), a \u2208 v2 \u2192 mk_idx n (hash_fn a.1) = bidx)\n            (djuv : (u.map sigma.fst).disjoint (v2.map sigma.fst))\n            (djwv : (w.map sigma.fst).disjoint (v2.map sigma.fst))\n  include hvnd hal djuv djwv\n\n  theorem valid.modify {sz : \u2115} (v : valid bkts sz) :\n    v1.length \u2264 sz + v2.length \u2227 valid bkts' (sz + v2.length - v1.length) :=\n  begin\n    rcases append_of_modify u v1 v2 w hl hfl with \u27e8u', w', e\u2081, e\u2082\u27e9,\n    rw [\u2190 v.len, e\u2081],\n    suffices : valid bkts' (u' ++ v2 ++ w').length,\n    { simpa [ge, add_comm, add_left_comm, nat.le_add_right, nat.add_sub_cancel_left] },\n    refine \u27e8congr_arg _ e\u2082, \u03bb i a, _, \u03bb i, _\u27e9,\n    { by_cases bidx = i,\n      { subst i, rw [bkts', array.read_write, hfl],\n        have := @valid.idx _ _ _ v bidx a,\n        simp only [hl, list.mem_append, or_imp_distrib, forall_and_distrib] at this \u22a2,\n        exact \u27e8\u27e8this.1.1, hal _\u27e9, this.2\u27e9 },\n      { rw [bkts', array.read_write_of_ne _ _ h], apply v.idx } },\n    { by_cases bidx = i,\n      { subst i, rw [bkts', array.read_write, hfl],\n        have := @valid.nodup _ _ _ v bidx,\n        simp [hl, list.nodup_append] at this,\n        simp [list.nodup_append, this, hvnd, djuv, djwv.symm] },\n      { rw [bkts', array.read_write_of_ne _ _ h], apply v.nodup } }\n  end\nend\n\ntheorem valid.replace_aux (a : \u03b1) (b : \u03b2 a) : \u03a0 (l : list (\u03a3 a, \u03b2 a)), a \u2208 l.map sigma.fst \u2192\n  \u2203 (u w : list \u03a3 a, \u03b2 a) b', l = u ++ [\u27e8a, b'\u27e9] ++ w \u2227 replace_aux a b l = u ++ [\u27e8a, b\u27e9] ++ w\n| []            := false.elim\n| (\u27e8a', b'\u27e9::t) := begin\n  by_cases e : a' = a,\n  { subst a',\n    suffices : \u2203 (u w : list \u03a3 a, \u03b2 a) (b'' : \u03b2 a),\n      (sigma.mk a b') :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n      replace_aux a b (\u27e8a, b'\u27e9 :: t) = u ++ \u27e8a, b\u27e9 :: w, {simpa},\n    refine \u27e8[], t, b', _\u27e9, simp [replace_aux] },\n  { suffices : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (b'' : \u03b2 a),\n      (sigma.mk a' b') :: t = u ++ \u27e8a, b''\u27e9 :: w \u2227\n      (sigma.mk a' b') :: (replace_aux a b t) = u ++ \u27e8a, b\u27e9 :: w,\n    { simpa [replace_aux, ne.symm e, e] },\n    intros x m,\n    have IH : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (b'' : \u03b2 a),\n      t = u ++ \u27e8a, b''\u27e9 :: w \u2227 replace_aux a b t = u ++ \u27e8a, b\u27e9 :: w,\n    { simpa using valid.replace_aux t },\n    rcases IH x m with \u27e8u, w, b'', hl, hfl\u27e9,\n    exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm, ne.symm e]\u27e9 }\nend\n\ntheorem valid.replace {n : \u2115+}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n  (Hc : contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (bkts.modify hash_fn a (replace_aux a b)) sz :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  rcases hash_map.valid.replace_aux a b (array.read bkts (mk_idx n (hash_fn a)))\n    ((contains_aux_iff nd).1 Hc) with \u27e8u, w, b', hl, hfl\u27e9,\n  simp [hl, list.nodup_append] at nd,\n  refine (v.modify hash_fn\n    u [\u27e8a, b'\u27e9] [\u27e8a, b\u27e9] w hl hfl (list.nodup_singleton _)\n    (\u03bba' e, by simp at e; rw e)\n    (\u03bba' e1 e2, _)\n    (\u03bba' e1 e2, _)).2;\n  { revert e1, simp [-sigma.exists] at e2, subst a', simp [nd] }\nend\n\ntheorem valid.insert {n : \u2115+}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz : \u2115} (a : \u03b1) (b : \u03b2 a)\n  (Hnc : \u00ac contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (reinsert_aux bkts a b) (sz+1) :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  refine (v.modify hash_fn\n    [] [] [\u27e8a, b\u27e9] (bkts.read hash_fn a) rfl rfl (list.nodup_singleton _)\n    (\u03bba' e, by simp at e; rw e)\n    (\u03bba', false.elim)\n    (\u03bba' e1 e2, _)).2,\n  simp [-sigma.exists] at e2, subst a',\n  exact Hnc ((contains_aux_iff nd).2 e1)\nend\n\ntheorem valid.erase_aux (a : \u03b1) : \u03a0 (l : list (\u03a3 a, \u03b2 a)), a \u2208 l.map sigma.fst \u2192\n  \u2203 (u w : list \u03a3 a, \u03b2 a) b, l = u ++ [\u27e8a, b\u27e9] ++ w \u2227 erase_aux a l = u ++ [] ++ w\n| []            := false.elim\n| (\u27e8a', b'\u27e9::t) := begin\n  by_cases e : a' = a,\n  { subst a',\n    simpa [erase_aux, and_comm] using show \u2203 u w (x : \u03b2 a),\n      t = u ++ w \u2227 (sigma.mk a b') :: t = u ++ \u27e8a, x\u27e9 :: w,\n      from \u27e8[], t, b', by simp\u27e9 },\n  { simp [erase_aux, e, ne.symm e],\n    suffices : \u2200 (b : \u03b2 a) (_ : sigma.mk a b \u2208 t), \u2203 u w (x : \u03b2 a),\n      (sigma.mk a' b') :: t = u ++ \u27e8a, x\u27e9 :: w \u2227\n      (sigma.mk a' b') :: (erase_aux a t) = u ++ w,\n    { simpa [replace_aux, ne.symm e, e] },\n    intros b m,\n    have IH : \u2200 (x : \u03b2 a) (_ : sigma.mk a x \u2208 t), \u2203 u w (x : \u03b2 a),\n      t = u ++ \u27e8a, x\u27e9 :: w \u2227 erase_aux a t = u ++ w,\n    { simpa using valid.erase_aux t },\n    rcases IH b m with \u27e8u, w, b'', hl, hfl\u27e9,\n    exact \u27e8\u27e8a', b'\u27e9 :: u, w, b'', by simp [hl, hfl.symm]\u27e9 }\nend\n\ntheorem valid.erase {n} {bkts : bucket_array \u03b1 \u03b2 n} {sz}\n  (a : \u03b1) (Hc : contains_aux a (bkts.read hash_fn a))\n  (v : valid bkts sz) : valid (bkts.modify hash_fn a (erase_aux a)) (sz-1) :=\nbegin\n  have nd := v.nodup (mk_idx n (hash_fn a)),\n  rcases hash_map.valid.erase_aux a (array.read bkts (mk_idx n (hash_fn a)))\n    ((contains_aux_iff nd).1 Hc) with \u27e8u, w, b, hl, hfl\u27e9,\n  refine (v.modify hash_fn u [\u27e8a, b\u27e9] [] w hl hfl list.nodup_nil _ _ _).2;\n  simp\nend\n\nend\nend hash_map\n\n/-- A hash map data structure, representing a finite key-value map\n  with key type `\u03b1` and value type `\u03b2` (which may depend on `\u03b1`). -/\nstructure hash_map (\u03b1 : Type u) [decidable_eq \u03b1] (\u03b2 : \u03b1 \u2192 Type v) :=\n(hash_fn : \u03b1 \u2192 nat)\n(size : \u2115)\n(nbuckets : \u2115+)\n(buckets : bucket_array \u03b1 \u03b2 nbuckets)\n(is_valid : hash_map.valid hash_fn buckets size)\n\n/-- Construct an empty hash map with buffer size `nbuckets` (default 8). -/\ndef mk_hash_map {\u03b1 : Type u} [decidable_eq \u03b1] {\u03b2 : \u03b1 \u2192 Type v} (hash_fn : \u03b1 \u2192 nat) (nbuckets := 8) :\n  hash_map \u03b1 \u03b2 :=\nlet n := if nbuckets = 0 then 8 else nbuckets in\nlet nz : n > 0 := by abstract { cases nbuckets; simp [if_pos, nat.succ_ne_zero] } in\n{ hash_fn  := hash_fn,\n  size     := 0,\n  nbuckets := \u27e8n, nz\u27e9,\n  buckets  := mk_array n [],\n  is_valid := hash_map.mk_valid _ _ }\n\nnamespace hash_map\nvariables {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [decidable_eq \u03b1]\n\n/-- Return the value corresponding to a key, or `none` if not found -/\ndef find (m : hash_map \u03b1 \u03b2) (a : \u03b1) : option (\u03b2 a) :=\nfind_aux a (m.buckets.read m.hash_fn a)\n\n/-- Return `tt` if the key exists in the map -/\ndef contains (m : hash_map \u03b1 \u03b2) (a : \u03b1) : bool :=\n(m.find a).is_some\n\ninstance : has_mem \u03b1 (hash_map \u03b1 \u03b2) := \u27e8\u03bba m, m.contains a\u27e9\n\n/-- Fold a function over the key-value pairs in the map -/\ndef fold {\u03b4 : Type w} (m : hash_map \u03b1 \u03b2) (d : \u03b4) (f : \u03b4 \u2192 \u03a0 a, \u03b2 a \u2192 \u03b4) : \u03b4 :=\nm.buckets.foldl d f\n\n/-- The list of key-value pairs in the map -/\ndef entries (m : hash_map \u03b1 \u03b2) : list \u03a3 a, \u03b2 a :=\nm.buckets.as_list\n\n/-- The list of keys in the map -/\ndef keys (m : hash_map \u03b1 \u03b2) : list \u03b1 :=\nm.entries.map sigma.fst\n\ntheorem find_iff (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) :\n  m.find a = some b \u2194 sigma.mk a b \u2208 m.entries :=\nm.is_valid.find_aux_iff _\n\ntheorem contains_iff (m : hash_map \u03b1 \u03b2) (a : \u03b1) :\n  m.contains a \u2194 a \u2208 m.keys :=\nm.is_valid.contains_aux_iff _ _\n\ntheorem entries_empty (hash_fn : \u03b1 \u2192 nat) (n) :\n  (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).entries = [] :=\nmk_as_list _\n\n\n\ntheorem find_empty (hash_fn : \u03b1 \u2192 nat) (n a) :\n  (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).find a = none :=\nby induction h : (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).find a; [refl,\n   { have := (find_iff _ _ _).1 h, rw entries_empty at this, contradiction }]\n\ntheorem not_contains_empty (hash_fn : \u03b1 \u2192 nat) (n a) :\n  \u00ac (@mk_hash_map \u03b1 _ \u03b2 hash_fn n).contains a :=\nby apply bool_iff_false.2; dsimp [contains]; rw [find_empty]; refl\n\ntheorem insert_lemma (hash_fn : \u03b1 \u2192 nat) {n n'}\n  {bkts : bucket_array \u03b1 \u03b2 n} {sz} (v : valid hash_fn bkts sz) :\n  valid hash_fn (bkts.foldl (mk_array _ [] : bucket_array \u03b1 \u03b2 n') (reinsert_aux hash_fn)) sz :=\nbegin\n  suffices : \u2200 (l : list \u03a3 a, \u03b2 a) (t : bucket_array \u03b1 \u03b2 n') sz,\n    valid hash_fn t sz \u2192 ((l ++ t.as_list).map sigma.fst).nodup \u2192\n    valid hash_fn (l.foldl (\u03bbr (a : \u03a3 a, \u03b2 a), reinsert_aux hash_fn r a.1 a.2) t) (sz + l.length),\n  { have p := this bkts.as_list _ _ (mk_valid _ _),\n    rw [mk_as_list, list.append_nil, zero_add, v.len] at p,\n    rw bucket_array.foldl_eq,\n    exact p (v.as_list_nodup _) },\n  intro l, induction l with c l IH; intros t sz v nd, {exact v},\n  rw show sz + (c :: l).length = sz + 1 + l.length, by simp [add_comm, add_assoc],\n  rcases (show (l.map sigma.fst).nodup \u2227\n      ((bucket_array.as_list t).map sigma.fst).nodup \u2227\n      c.fst \u2209 l.map sigma.fst \u2227\n      c.fst \u2209 (bucket_array.as_list t).map sigma.fst \u2227\n      (l.map sigma.fst).disjoint ((bucket_array.as_list t).map sigma.fst),\n    by simpa [list.nodup_append, not_or_distrib, and_comm, and.left_comm] using nd)\n    with \u27e8nd1, nd2, nm1, nm2, dj\u27e9,\n  have v' := v.insert _ _ c.2 (\u03bbHc, nm2 $ (v.contains_aux_iff _ c.1).1 Hc),\n  apply IH _ _ v',\n  suffices : \u2200 \u2983a : \u03b1\u2984 (b : \u03b2 a), sigma.mk a b \u2208 l \u2192\n    \u2200 (b' : \u03b2 a), sigma.mk a b' \u2208 (reinsert_aux hash_fn t c.1 c.2).as_list \u2192 false,\n  { simpa [list.nodup_append, nd1, v'.as_list_nodup _, list.disjoint] },\n  intros a b m1 b' m2,\n  rcases (reinsert_aux hash_fn t c.1 c.2).mem_as_list.1 m2 with \u27e8i, im\u27e9,\n  have : sigma.mk a b' \u2209 array.read t i,\n  { intro m3,\n    have : a \u2208 list.map sigma.fst t.as_list :=\n      list.mem_map_of_mem sigma.fst (t.mem_as_list.2 \u27e8_, m3\u27e9),\n    exact dj (list.mem_map_of_mem sigma.fst m1) this },\n  by_cases h : mk_idx n' (hash_fn c.1) = i,\n  { subst h,\n    have e : sigma.mk a b' = \u27e8c.1, c.2\u27e9,\n    { simpa [reinsert_aux, bucket_array.modify, array.read_write, this] using im },\n    injection e with e, subst a,\n    exact nm1.elim (@list.mem_map_of_mem _ _ sigma.fst _ _ m1) },\n  { apply this,\n    simpa [reinsert_aux, bucket_array.modify, array.read_write_of_ne _ _ h] using im }\nend\n\n/-- Insert a key-value pair into the map. (Modifies `m` in-place when applicable) -/\ndef insert : \u03a0 (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a), hash_map \u03b1 \u03b2\n| \u27e8hash_fn, size, n, buckets, v\u27e9 a b :=\nlet bkt := buckets.read hash_fn a in\nif hc : contains_aux a bkt then\n{ hash_fn  := hash_fn,\n  size     := size,\n  nbuckets := n,\n  buckets  := buckets.modify hash_fn a (replace_aux a b),\n  is_valid := v.replace _ a b hc }\nelse\nlet size'    := size + 1,\n    buckets' := buckets.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9::l),\n    valid'   := v.insert _ a b hc in\nif size' \u2264 n then\n{ hash_fn  := hash_fn,\n  size     := size',\n  nbuckets := n,\n  buckets  := buckets',\n  is_valid := valid' }\nelse\nlet n'        : \u2115+ := \u27e8n * 2, mul_pos n.2 dec_trivial\u27e9,\n    buckets'' : bucket_array \u03b1 \u03b2 n' :=\n                buckets'.foldl (mk_array _ []) (reinsert_aux hash_fn) in\n{ hash_fn  := hash_fn,\n  size     := size',\n  nbuckets := n',\n  buckets  := buckets'',\n  is_valid := insert_lemma _ valid' }\n\ntheorem mem_insert : \u03a0 (m : hash_map \u03b1 \u03b2) (a b a' b'),\n  (sigma.mk a' b' : sigma \u03b2) \u2208 (m.insert a b).entries \u2194\n  if a = a' then b == b' else sigma.mk a' b' \u2208 m.entries\n| \u27e8hash_fn, size, n, bkts, v\u27e9 a b a' b' := begin\n  let bkt := bkts.read hash_fn a,\n  have nd : (bkt.map sigma.fst).nodup := v.nodup (mk_idx n (hash_fn a)),\n  have lem : \u03a0 (bkts' : bucket_array \u03b1 \u03b2 n) (v1 u w)\n    (hl : bucket_array.as_list bkts = u ++ v1 ++ w)\n    (hfl : bucket_array.as_list bkts' = u ++ [\u27e8a, b\u27e9] ++ w)\n    (veq : (v1 = [] \u2227 \u00ac contains_aux a bkt) \u2228 \u2203b'', v1 = [\u27e8a, b''\u27e9]),\n    sigma.mk a' b' \u2208 bkts'.as_list \u2194\n    if a = a' then b == b' else sigma.mk a' b' \u2208 bkts.as_list,\n  { intros bkts' v1 u w hl hfl veq,\n    rw [hl, hfl],\n    by_cases h : a = a',\n    { subst a',\n      suffices : b = b' \u2228 sigma.mk a b' \u2208 u \u2228 sigma.mk a b' \u2208 w \u2194 b = b',\n      { simpa [eq_comm, or.left_comm] },\n      refine or_iff_left_of_imp (not.elim $ not_or_distrib.2 _),\n      rcases veq with \u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9,\n      { have na := (not_iff_not_of_iff $ v.contains_aux_iff _ _).1 Hnc,\n        simp [hl, not_or_distrib] at na, simp [na] },\n      { have nd' := v.as_list_nodup _,\n        simp [hl, list.nodup_append] at nd', simp [nd'] } },\n    { suffices : sigma.mk a' b' \u2209 v1, {simp [h, ne.symm h, this]},\n      rcases veq with \u27e8rfl, Hnc\u27e9 | \u27e8b'', rfl\u27e9; simp [ne.symm h] } },\n  by_cases Hc : (contains_aux a bkt : Prop),\n  { rcases hash_map.valid.replace_aux a b (array.read bkts (mk_idx n (hash_fn a)))\n      ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b'', hl', hfl'\u27e9,\n    rcases (append_of_modify u' [\u27e8a, b''\u27e9] [\u27e8a, b\u27e9] w' hl' hfl') with \u27e8u, w, hl, hfl\u27e9,\n    simpa [insert, @dif_pos (contains_aux a bkt) _ Hc]\n      using lem _ _ u w hl hfl (or.inr \u27e8b'', rfl\u27e9) },\n  { let size' := size + 1,\n    let bkts' := bkts.modify hash_fn a (\u03bbl, \u27e8a, b\u27e9::l),\n    have mi : sigma.mk a' b' \u2208 bkts'.as_list \u2194\n        if a = a' then b == b' else sigma.mk a' b' \u2208 bkts.as_list :=\n      let \u27e8u, w, hl, hfl\u27e9 := append_of_modify [] [] [\u27e8a, b\u27e9] _ rfl rfl in\n      lem bkts' _ u w hl hfl $ or.inl \u27e8rfl, Hc\u27e9,\n    simp [insert, @dif_neg (contains_aux a bkt) _ Hc],\n    by_cases h : size' \u2264 n,\n    { simpa [show size' \u2264 n, from h] using mi },\n    { let n' : \u2115+ := \u27e8n * 2, mul_pos n.2 dec_trivial\u27e9,\n      let bkts'' : bucket_array \u03b1 \u03b2 n' := bkts'.foldl (mk_array _ []) (reinsert_aux hash_fn),\n      suffices : sigma.mk a' b' \u2208 bkts''.as_list \u2194 sigma.mk a' b' \u2208 bkts'.as_list.reverse,\n      { simpa [show \u00ac size' \u2264 n, from h, mi] },\n      rw [show bkts'' = bkts'.as_list.foldl _ _, from bkts'.foldl_eq _ _,\n          \u2190 list.foldr_reverse],\n      induction bkts'.as_list.reverse with a l IH,\n      { simp [mk_as_list] },\n      { cases a with a'' b'',\n        let B := l.foldr (\u03bb (y : sigma \u03b2) (x : bucket_array \u03b1 \u03b2 n'),\n          reinsert_aux hash_fn x y.1 y.2) (mk_array n' []),\n        rcases append_of_modify [] [] [\u27e8a'', b''\u27e9] _ rfl rfl with \u27e8u, w, hl, hfl\u27e9,\n        simp [IH.symm, or.left_comm, show B.as_list = _, from hl,\n              show (reinsert_aux hash_fn B a'' b'').as_list = _, from hfl] } } }\nend\n\ntheorem find_insert_eq (m : hash_map \u03b1 \u03b2) (a : \u03b1) (b : \u03b2 a) : (m.insert a b).find a = some b :=\n(find_iff (m.insert a b) a b).2 $ (mem_insert m a b a b).2 $ by rw if_pos rfl\n\ntheorem find_insert_ne (m : hash_map \u03b1 \u03b2) (a a' : \u03b1) (b : \u03b2 a) (h : a \u2260 a') :\n  (m.insert a b).find a' = m.find a' :=\noption.eq_of_eq_some $ \u03bbb',\nlet t := mem_insert m a b a' b' in\n(find_iff _ _ _).trans $ iff.trans (by rwa if_neg h at t) (find_iff _ _ _).symm\n\ntheorem find_insert (m : hash_map \u03b1 \u03b2) (a' a : \u03b1) (b : \u03b2 a) :\n  (m.insert a b).find a' = if h : a = a' then some (eq.rec_on h b) else m.find a' :=\nif h : a = a' then by rw dif_pos h; exact\n  match a', h with ._, rfl := find_insert_eq m a b end\nelse by rw dif_neg h; exact find_insert_ne m a a' b h\n\n/-- Insert a list of key-value pairs into the map. (Modifies `m` in-place when applicable) -/\ndef insert_all (l : list (\u03a3 a, \u03b2 a)) (m : hash_map \u03b1 \u03b2) : hash_map \u03b1 \u03b2 :=\nl.foldl (\u03bb m \u27e8a, b\u27e9, insert m a b) m\n\n/-- Construct a hash map from a list of key-value pairs. -/\ndef of_list (l : list (\u03a3 a, \u03b2 a)) (hash_fn) : hash_map \u03b1 \u03b2 :=\ninsert_all l (mk_hash_map hash_fn (2 * l.length))\n\n/-- Remove a key from the map. (Modifies `m` in-place when applicable) -/\ndef erase (m : hash_map \u03b1 \u03b2) (a : \u03b1) : hash_map \u03b1 \u03b2 :=\nmatch m with \u27e8hash_fn, size, n, buckets, v\u27e9 :=\n  if hc : contains_aux a (buckets.read hash_fn a) then\n  { hash_fn  := hash_fn,\n    size     := size - 1,\n    nbuckets := n,\n    buckets  := buckets.modify hash_fn a (erase_aux a),\n    is_valid := v.erase _ a hc }\n  else m\nend\n\ntheorem mem_erase : \u03a0 (m : hash_map \u03b1 \u03b2) (a a' b'),\n  (sigma.mk a' b' : sigma \u03b2) \u2208 (m.erase a).entries \u2194\n  a \u2260 a' \u2227 sigma.mk a' b' \u2208 m.entries\n| \u27e8hash_fn, size, n, bkts, v\u27e9 a a' b' := begin\n  let bkt := bkts.read hash_fn a,\n  by_cases Hc : (contains_aux a bkt : Prop),\n  { let bkts' := bkts.modify hash_fn a (erase_aux a),\n    suffices : sigma.mk a' b' \u2208 bkts'.as_list \u2194 a \u2260 a' \u2227 sigma.mk a' b' \u2208 bkts.as_list,\n    { simpa [erase, @dif_pos (contains_aux a bkt) _ Hc] },\n    have nd := v.nodup (mk_idx n (hash_fn a)),\n    rcases valid.erase_aux a bkt ((contains_aux_iff nd).1 Hc) with \u27e8u', w', b, hl', hfl'\u27e9,\n    rcases append_of_modify u' [\u27e8a, b\u27e9] [] _ hl' hfl' with \u27e8u, w, hl, hfl\u27e9,\n    suffices : \u2200_:sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w, a \u2260 a',\n    { have : sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w \u2194 (\u00aca = a' \u2227 a' = a) \u2227 b' == b \u2228\n        \u00aca = a' \u2227 (sigma.mk a' b' \u2208 u \u2228 sigma.mk a' b' \u2208 w),\n      { simp [eq_comm, not_and_self_iff, and_iff_right_of_imp this] },\n      simpa [hl, show bkts'.as_list = _, from hfl, and_or_distrib_left,\n             and_comm, and.left_comm, or.left_comm] },\n    intros m e, subst a', revert m, apply not_or_distrib.2,\n    have nd' := v.as_list_nodup _,\n    simp [hl, list.nodup_append] at nd', simp [nd'] },\n  { suffices : \u2200_:sigma.mk a' b' \u2208 bucket_array.as_list bkts, a \u2260 a',\n    { simp [erase, @dif_neg (contains_aux a bkt) _ Hc, entries, and_iff_right_of_imp this] },\n    intros m e, subst a',\n    exact Hc ((v.contains_aux_iff _ _).2 (list.mem_map_of_mem sigma.fst m)) }\nend\n\ntheorem find_erase_eq (m : hash_map \u03b1 \u03b2) (a : \u03b1) : (m.erase a).find a = none :=\nbegin\n  cases h : (m.erase a).find a with b, {refl},\n  exact absurd rfl ((mem_erase m a a b).1 ((find_iff (m.erase a) a b).1 h)).left\nend\n\ntheorem find_erase_ne (m : hash_map \u03b1 \u03b2) (a a' : \u03b1) (h : a \u2260 a') :\n  (m.erase a).find a' = m.find a' :=\noption.eq_of_eq_some $ \u03bbb',\n(find_iff _ _ _).trans $ (mem_erase m a a' b').trans $\n  (and_iff_right h).trans (find_iff _ _ _).symm\n\ntheorem find_erase (m : hash_map \u03b1 \u03b2) (a' a : \u03b1) :\n  (m.erase a).find a' = if a = a' then none else m.find a' :=\nif h : a = a' then by subst a'; simp [find_erase_eq m a]\nelse by rw if_neg h; exact find_erase_ne m a a' h\n\nsection string\nvariables [has_to_string \u03b1] [\u2200 a, has_to_string (\u03b2 a)]\nopen prod\nprivate def key_data_to_string (a : \u03b1) (b : \u03b2 a) (first : bool) : string :=\n(if first then \"\" else \", \") ++ sformat!\"{a} \u2190 {b}\"\n\nprivate def to_string (m : hash_map \u03b1 \u03b2) : string :=\n\"\u27e8\" ++ (fst (fold m (\"\", tt) (\u03bb p a b, (fst p ++ key_data_to_string a b (snd p), ff)))) ++ \"\u27e9\"\n\ninstance : has_to_string (hash_map \u03b1 \u03b2) :=\n\u27e8to_string\u27e9\n\nend string\n\nsection format\nopen format prod\nvariables [has_to_format \u03b1] [\u2200 a, has_to_format (\u03b2 a)]\n\nprivate meta def format_key_data (a : \u03b1) (b : \u03b2 a) (first : bool) : format :=\n(if first then to_fmt \"\" else to_fmt \",\" ++ line) ++\n  to_fmt a ++ space ++ to_fmt \"\u2190\" ++ space ++ to_fmt b\n\nprivate meta def to_format (m : hash_map \u03b1 \u03b2) : format :=\ngroup $ to_fmt \"\u27e8\" ++\n  nest 1 (fst (fold m (to_fmt \"\", tt) (\u03bb p a b, (fst p ++ format_key_data a b (snd p), ff)))) ++\n  to_fmt \"\u27e9\"\n\nmeta instance : has_to_format (hash_map \u03b1 \u03b2) :=\n\u27e8to_format\u27e9\nend format\n\n/-- `hash_map` with key type `nat` and value type that may vary. -/\ninstance {\u03b2 : \u2115 \u2192 Type*} : inhabited (hash_map \u2115 \u03b2) := \u27e8mk_hash_map id\u27e9\n\nend hash_map\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/hash_map.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.35487599223511174}}
{"text": "-- import data.int.basic -- this doesnt work\n-- import data.real.basic -- this doesnt work\n\nopen classical\n\nvariables (\u03b1 : Type*) (p q : \u03b1 \u2192 Prop)\nvariable r : Prop\n\nexample : (\u2203 x : \u03b1, r) \u2192 r :=\nassume h,\n  exists.elim h\n    (assume w,\n     assume hw,\n            hw)\n-- \u21ef I don't truly grokk what is going on here. Just following the syntax lol\n-- Another way of doing that is using tactics that\n-- I learnt in the number game\n\n-- begin\n-- cases h, exact h_h\n-- end\n\nexample (a : \u03b1) : r \u2192 (\u2203 x : \u03b1, r) :=\nassume h, \u27e8a,h\u27e9\n\nexample : (\u2203 x, p x \u2227 r) \u2194 (\u2203 x, p x) \u2227 r :=\niff.intro\n  (assume hl,\n  exists.elim hl\n  (assume w,\n  assume hw,\n  \u27e8\u27e8w, hw.left\u27e9,hw.right \u27e9)) --\n  (assume hr,\n  exists.elim hr.left\n    (assume w,\n    assume hw,\n  \u27e8w, hw, hr.right\u27e9))  --\n\n-- another way of solving\nexample : (\u2203 x, p x \u2227 r) \u2194 (\u2203 x, p x) \u2227 r :=\niff.intro\n(assume hl,\n  match hl with \u27e8w, hw \u27e9 :=\n  \u27e8\u27e8w, hw.left\u27e9, hw.right \u27e9\n  end)\n(assume hr,\n  match hr.left with \u27e8w, hw\u27e9 :=\n  \u27e8w, hw, hr.right\u27e9\n  end)\n\n\nexample : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\niff.intro\n(assume hl,\n  exists.elim hl\n  (assume w,\n  assume hw,\n  or.elim hw\n  (assume hpw, (or.inl \u27e8w, hpw\u27e9))\n  (assume hqw,(or.inr \u27e8w, hqw\u27e9))))\n(assume hr, or.elim hr\n  (assume hepx,\n  exists.elim hepx\n  (assume w,\n  assume hw, \u27e8w, (or.inl hw)\u27e9))\n  (assume hepq,\n  exists.elim hepq\n  (assume w,\n  assume hw, \u27e8w, (or.inr hw)\u27e9)))\n\n-- neater way\n\n-- example : (\u2203 x, p x \u2228 q x) \u2194 (\u2203 x, p x) \u2228 (\u2203 x, q x) :=\n-- iff.intro\n-- (assume hl,\n--  match hl with \u27e8w, hw\u27e9 :=\n--     or.elim hw\n--     (assume hpw, (or.inl \u27e8w, hpw\u27e9))\n--     (assume hqw,(or.inr \u27e8w, hqw\u27e9))\n--     end)\n-- (assume hr,\n--   or.elim hr\n--   (assume hepx,\n--   match hepx with \u27e8w,hw\u27e9 :=\n--   \u27e8w, (or.inl hw)\u27e9\n--   end)\n-- (assume hepw,\n--   match hepw with \u27e8w,hw\u27e9 := \u27e8w, (or.inr hw)\u27e9 end ))\n\ntheorem dne {p : Prop} (h : \u00ac\u00acp) : p :=\nor.elim (em p)\n  (assume hp : p, hp)\n  (assume hnp : \u00acp, absurd hnp h)\n\n\nexample : (\u2200 x, p x) \u2194 \u00ac (\u2203 x, \u00ac p x) := iff.intro\n(assume hl,\n  by_contradiction\n  (assume h1,\n  have h2 : (\u2203 (x : \u03b1), \u00acp x),\n  from dne h1,\n  exists.elim h2\n  (assume w,\n  assume hw, absurd (hl w) hw)))\n(assume hr,\n assume foo,\n by_contradiction\n (assume bar, hr \u27e8foo,bar\u27e9)) \n \n variables a : \u03b1\n \n \n example : (\u2203 x, p x) \u2194 \u00ac (\u2200 x, \u00ac p x) := \n iff.intro \n (assume hl, \n by_contradiction\n  (assume f, \n  have bar : \u2200 (x : \u03b1), \u00acp x, from dne f,\n   exists.elim hl\n    (assume w,\n     assume hw,\n     absurd hw (bar w))) ) \n (assume hr,\n by_contradiction\n (assume h1,\n  have h2 : \u2200 x, \u00ac p x, from\n      assume x,\n      assume h3 : p x,\n      have h4 : \u2203 x, p x, from  \u27e8x, h3\u27e9,\n      show false, from h1 h4,\n    show false, from hr h2))\n-- this one was true dark magic \n-- I don't know what is happening\n-- but I created h2,x,h3 out of nothing ness LOLL\n\n\nexample : (\u00ac \u2203 x, p x) \u2194 (\u2200 x, \u00ac p x) :=\niff.intro\n(assume hl,\n  assume h1,\n  assume h2, hl \u27e8h1, h2\u27e9)\n(assume hr,\nassume h1,\nexists.elim h1\n(assume w,\nassume hw,\nabsurd hw (hr w)))\n\nexample: (\u00ac \u2200 x, p x) \u2194 (\u2203 x, \u00ac p x) :=\niff.intro\n(assume hl,\n  by_contradiction\n    (assume bar,_))      --\n--  have h1 : \u2200 (x : \u03b1), p x, from\n(assume hr,\nassume h1,\nexists.elim hr\n  (assume w,\n  assume hw,\n  absurd (h1 w) hw)) --\n\n\n\n-- example: (\u2200 x, p x \u2192 r) \u2194 (\u2203 x, p x) \u2192 r := sorry\n-- example (a : \u03b1) : (\u2203 x, p x \u2192 r) \u2194 (\u2200 x, p x) \u2192 r := sorry\n-- example (a : \u03b1) : (\u2203 x, r \u2192 p x) \u2194 (r \u2192 \u2203 x, p x) := sorry\n\n\n\n\n\n\n-- -- *    Prove these equivalences:\n\n-- example : (\u2200 x, p x \u2227 q x) \u2194 (\u2200 x, p x) \u2227 (\u2200 x, q x) := sorry\n\n-- example : (\u2200 x, p x \u2192 q x) \u2192 (\u2200 x, p x) \u2192 (\u2200 x, q x) := sorry\n\n-- example : (\u2200 x, p x) \u2228 (\u2200 x, q x) \u2192 \u2200 x, p x \u2228 q x := sorry\n\n-- --  You should also try to understand why the reverse implication is not\n-- --    derivable in the last example. It is often possible to bring a component\n-- --    of a formula outside a universal quantifier, when it does not depend on\n-- --    the quantified variable. Try proving these (one direction of the second of\n-- --    these requires classical logic):\n\n\n-- example : \u03b1 \u2192 ((\u2200 x : \u03b1, r) \u2194 r) := sorry\n-- example : (\u2200 x, p x \u2228 r) \u2194 (\u2200 x, p x) \u2228 r := sorry\n-- example : (\u2200 x, r \u2192 p x) \u2194 (r \u2192 \u2200 x, p x) := sorry\n\n-- -- Consider the \u201cbarber paradox,\u201d that is, the claim that in a certain town\n-- -- there is a (male) barber that shaves all and only the men who do not\n-- -- shave themselves. Prove that this is a contradiction:\n\n\n-- variables (men : Type*) (barber : men)\n-- variable  (shaves : men \u2192 men \u2192 Prop)\n\n-- example (h : \u2200 x : men, shaves barber x \u2194 \u00ac shaves x x) :\n-- false := sorry\n\n-- --  Remember that, without any parameters, an expression of type Prop is just\n-- --  an assertion. Fill in the definitions of prime and Fermat_prime below,\n-- --  and construct each of the given assertions. For example, you can say that\n-- --  there are infinitely many primes by asserting that for every natural\n-- --  number n, there is a prime number greater than n. Goldbach\u2019s weak\n-- --  conjecture states that every odd number greater than 5 is the sum of\n-- --  three primes. Look up the definition of a Fermat prime or any of the\n-- --  other statements, if necessary.\n\n-- #check even\n\n-- def prime (n : \u2115) : Prop := sorry\n-- def infinitely_many_primes : Prop := sorry\n-- def Fermat_prime (n : \u2115) : Prop := sorry\n-- def infinitely_many_Fermat_primes : Prop := sorry\n-- def goldbach_conjecture : Prop := sorry\n-- def Goldbach's_weak_conjecture : Prop := sorry\n-- def Fermat's_last_theorem : Prop := sorry\n\n-- --  Give a calculational proof of the theorem log_mul below.\n\n-- variables log exp     : real \u2192 real\n-- variable  log_exp_eq : \u2200 x, log (exp x) = x\n-- variable  exp_log_eq : \u2200 {x}, x > 0 \u2192 exp (log x) = x\n-- variable  exp_pos    : \u2200 x, exp x > 0\n-- variable  exp_add    : \u2200 x y, exp (x + y) = exp x * exp y\n\n--  -- this ensures the assumptions are available in tactic proofs\n--  include log_exp_eq exp_log_eq exp_pos exp_add\n\n-- example (x y z : real) :\n--     exp (x + y + z) = exp x * exp y * exp z :=\n--   by rw [exp_add, exp_add]\n-- example (y : real) (h : y > 0)  : exp (log y) = y :=\n--   exp_log_eq h\n-- theorem log_mul {x y : real} (hx : x > 0) (hy : y > 0) :\n--     log (x * y) = log x + log y :=\n--   sorry\n\n-- --  Prove the theorem below, using only the ring properties of \u2124 enumerated in\n-- --  Section 4.2 and the theorem sub_self.\n\n-- #check sub_self\n-- example (x : \u2124) : x * 0 = 0 :=\n--  sorry\n", "meta": {"author": "marcelovmaciel", "repo": "learning_lean", "sha": "0e208b8f90bf7709f3f4b8f6ef483e2fb5ecffc4", "save_path": "github-repos/lean/marcelovmaciel-learning_lean", "path": "github-repos/lean/marcelovmaciel-learning_lean/learning_lean-0e208b8f90bf7709f3f4b8f6ef483e2fb5ecffc4/theorem_proving_with_lean/third_note_exercises.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982645, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.35487599114879576}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel, Scott Morrison\n-/\nimport category_theory.simple\nimport category_theory.linear\nimport category_theory.endomorphism\nimport field_theory.algebraic_closure\n\n/-!\n# Schur's lemma\nWe first prove the part of Schur's Lemma that holds in any preadditive category with kernels,\nthat any nonzero morphism between simple objects\nis an isomorphism.\n\nSecond, we prove Schur's lemma for `\ud835\udd5c`-linear categories with finite dimensional hom spaces,\nover an algebraically closed field `\ud835\udd5c`:\nthe hom space `X \u27f6 Y` between simple objects `X` and `Y` is at most one dimensional,\nand is 1-dimensional iff `X` and `Y` are isomorphic.\n\n## Future work\nIt might be nice to provide a `division_ring` instance on `End X` when `X` is simple.\nThis is an easy consequence of the results here,\nbut may take some care setting up usable instances.\n-/\n\nnamespace category_theory\n\nopen category_theory.limits\n\nuniverses v u\nvariables {C : Type u} [category.{v} C]\nvariables [preadditive C]\n\n/--\nThe part of Schur's lemma that holds in any preadditive category with kernels:\nthat a nonzero morphism between simple objects is an isomorphism.\n-/\nlemma is_iso_of_hom_simple [has_kernels C] {X Y : C} [simple X] [simple Y] {f : X \u27f6 Y} (w : f \u2260 0) :\n  is_iso f :=\nbegin\n  haveI : mono f := preadditive.mono_of_kernel_zero (kernel_zero_of_nonzero_from_simple w),\n  exact is_iso_of_mono_of_nonzero w\nend\n\n/--\nAs a corollary of Schur's lemma for preadditive categories,\nany morphism between simple objects is (exclusively) either an isomorphism or zero.\n-/\nlemma is_iso_iff_nonzero [has_kernels C] {X Y : C} [simple.{v} X] [simple.{v} Y] (f : X \u27f6 Y) :\n  is_iso.{v} f \u2194 f \u2260 0 :=\n\u27e8\u03bb I,\n  begin\n    introI h,\n    apply id_nonzero X,\n    simp only [\u2190is_iso.hom_inv_id f, h, zero_comp],\n  end,\n  \u03bb w, is_iso_of_hom_simple w\u27e9\n\nopen finite_dimensional\n\nvariables (\ud835\udd5c : Type*) [field \ud835\udd5c]\n\n/--\nPart of Schur's lemma for `\ud835\udd5c`-linear categories:\nthe hom space between two non-isomorphic simple objects is 0-dimensional.\n-/\nlemma finrank_hom_simple_simple_eq_zero_of_not_iso\n  [has_kernels C] [linear \ud835\udd5c C] {X Y : C} [simple.{v} X] [simple.{v} Y]\n  (h : (X \u2245 Y) \u2192 false):\n  finrank \ud835\udd5c (X \u27f6 Y) = 0 :=\nbegin\n  haveI := subsingleton_of_forall_eq (0 : X \u27f6 Y) (\u03bb f, begin\n    have p := not_congr (is_iso_iff_nonzero f),\n    simp only [not_not, ne.def] at p,\n    refine p.mp (\u03bb _, by exactI h (as_iso f)),\n  end),\n  exact finrank_zero_of_subsingleton,\nend\n\nvariables [is_alg_closed \ud835\udd5c] [linear \ud835\udd5c C]\n\n-- In the proof below we have some difficulty using `I : finite_dimensional \ud835\udd5c (X \u27f6 X)`\n-- where we need a `finite_dimensional \ud835\udd5c (End X)`.\n-- These are definitionally equal, but without eta reduction Lean can't see this.\n-- To get around this, we use `convert I`,\n-- then check the various instances agree field-by-field,\n-- using `ext` equipped with the following extra lemmas:\nlocal attribute [ext] add_comm_group module distrib_mul_action mul_action has_scalar\n\n/--\nAn auxiliary lemma for Schur's lemma.\n\nIf `X \u27f6 X` is finite dimensional, and every nonzero endomorphism is invertible,\nthen `X \u27f6 X` is 1-dimensional.\n-/\n-- We prove this with the explicit `is_iso_iff_nonzero` assumption,\n-- rather than just `[simple X]`, as this form is useful for\n-- M\u00fcger's formulation of semisimplicity.\nlemma finrank_endomorphism_eq_one\n  {X : C} (is_iso_iff_nonzero : \u2200 f : X \u27f6 X, is_iso f \u2194 f \u2260 0)\n  [I : finite_dimensional \ud835\udd5c (X \u27f6 X)] :\n  finrank \ud835\udd5c (X \u27f6 X) = 1 :=\nbegin\n  have id_nonzero := (is_iso_iff_nonzero (\ud835\udfd9 X)).mp (by apply_instance),\n  apply finrank_eq_one (\ud835\udfd9 X),\n  { exact id_nonzero, },\n  { intro f,\n    haveI : nontrivial (End X) := nontrivial_of_ne _ _ id_nonzero,\n    obtain \u27e8c, nu\u27e9 := @exists_spectrum_of_is_alg_closed_of_finite_dimensional \ud835\udd5c _ _ (End X) _ _ _\n      (by { convert I, ext; refl, ext; refl, }) (End.of f),\n    use c,\n    rw [is_unit_iff_is_iso, is_iso_iff_nonzero, ne.def, not_not, sub_eq_zero,\n      algebra.algebra_map_eq_smul_one] at nu,\n    exact nu.symm, },\nend\n\nvariables [has_kernels C]\n\n/--\nSchur's lemma for endomorphisms in `\ud835\udd5c`-linear categories.\n-/\nlemma finrank_endomorphism_simple_eq_one\n  (X : C) [simple.{v} X] [I : finite_dimensional \ud835\udd5c (X \u27f6 X)] :\n  finrank \ud835\udd5c (X \u27f6 X) = 1 :=\nfinrank_endomorphism_eq_one \ud835\udd5c is_iso_iff_nonzero\n\nlemma endomorphism_simple_eq_smul_id\n  {X : C} [simple.{v} X] [I : finite_dimensional \ud835\udd5c (X \u27f6 X)] (f : X \u27f6 X) :\n  \u2203 c : \ud835\udd5c, c \u2022 \ud835\udfd9 X = f :=\n(finrank_eq_one_iff_of_nonzero' (\ud835\udfd9 X) (id_nonzero X)).mp (finrank_endomorphism_simple_eq_one \ud835\udd5c X) f\n\n/--\nSchur's lemma for `\ud835\udd5c`-linear categories:\nif hom spaces are finite dimensional, then the hom space between simples is at most 1-dimensional.\n\nSee `finrank_hom_simple_simple_eq_one_iff` and `finrank_hom_simple_simple_eq_zero_iff` below\nfor the refinements when we know whether or not the simples are isomorphic.\n-/\n-- We don't really need `[\u2200 X Y : C, finite_dimensional \ud835\udd5c (X \u27f6 Y)]` here,\n-- just at least one of `[finite_dimensional \ud835\udd5c (X \u27f6 X)]` or `[finite_dimensional \ud835\udd5c (Y \u27f6 Y)]`.\nlemma finrank_hom_simple_simple_le_one\n  (X Y : C) [\u2200 X Y : C, finite_dimensional \ud835\udd5c (X \u27f6 Y)] [simple.{v} X] [simple.{v} Y] :\n  finrank \ud835\udd5c (X \u27f6 Y) \u2264 1 :=\nbegin\n  cases subsingleton_or_nontrivial (X \u27f6 Y) with h,\n  { resetI,\n    convert zero_le_one,\n    exact finrank_zero_of_subsingleton, },\n  { obtain \u27e8f, nz\u27e9 := (nontrivial_iff_exists_ne 0).mp h,\n    haveI fi := (is_iso_iff_nonzero f).mpr nz,\n    apply finrank_le_one f,\n    intro g,\n    obtain \u27e8c, w\u27e9 := endomorphism_simple_eq_smul_id \ud835\udd5c (g \u226b inv f),\n    exact \u27e8c, by simpa using w =\u226b f\u27e9, },\nend\n\nlemma finrank_hom_simple_simple_eq_one_iff\n  (X Y : C) [\u2200 X Y : C, finite_dimensional \ud835\udd5c (X \u27f6 Y)] [simple.{v} X] [simple.{v} Y] :\n  finrank \ud835\udd5c (X \u27f6 Y) = 1 \u2194 nonempty (X \u2245 Y) :=\nbegin\n  fsplit,\n  { intro h,\n    rw finrank_eq_one_iff' at h,\n    obtain \u27e8f, nz, -\u27e9 := h,\n    rw \u2190is_iso_iff_nonzero at nz,\n    exactI \u27e8as_iso f\u27e9, },\n  { rintro \u27e8f\u27e9,\n    have le_one := finrank_hom_simple_simple_le_one \ud835\udd5c X Y,\n    have zero_lt : 0 < finrank \ud835\udd5c (X \u27f6 Y) :=\n      finrank_pos_iff_exists_ne_zero.mpr \u27e8f.hom, (is_iso_iff_nonzero f.hom).mp infer_instance\u27e9,\n    linarith, }\nend\n\nlemma finrank_hom_simple_simple_eq_zero_iff\n  (X Y : C) [\u2200 X Y : C, finite_dimensional \ud835\udd5c (X \u27f6 Y)] [simple.{v} X] [simple.{v} Y] :\n  finrank \ud835\udd5c (X \u27f6 Y) = 0 \u2194 \u00ac nonempty (X \u2245 Y) :=\nbegin\n  rw \u2190not_congr (finrank_hom_simple_simple_eq_one_iff \ud835\udd5c X Y),\n  refine \u27e8\u03bb h, by { rw h, simp, }, \u03bb h, _\u27e9,\n  have := finrank_hom_simple_simple_le_one \ud835\udd5c X Y,\n  interval_cases finrank \ud835\udd5c (X \u27f6 Y) with h',\n  { exact h', },\n  { exact false.elim (h h'), },\nend\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/preadditive/schur.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.354875985009871}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.traversable.lemmas\nimport data.equiv.basic\n\n/-!\n# Transferring `traversable` instances along isomorphisms\n\nThis file allows to transfer `traversable` instances along isomorphisms.\n\n## Main declarations\n\n* `equiv.map`: Turns functorially a function `\u03b1 \u2192 \u03b2` into a function `t' \u03b1 \u2192 t' \u03b2` using the functor\n  `t` and the equivalence `\u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1`.\n* `equiv.functor`: `equiv.map` as a functor.\n* `equiv.traverse`: Turns traversably a function `\u03b1 \u2192 m \u03b2` into a function `t' \u03b1 \u2192 m (t' \u03b2)` using\n  the traversable functor `t` and the equivalence `\u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1`.\n* `equiv.traversable`: `equiv.traverse` as a traversable functor.\n* `equiv.is_lawful_traversable`: `equiv.traverse` as a lawful traversable functor.\n-/\n\nuniverses u\n\nnamespace equiv\n\nsection functor\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [functor t]\n\nopen functor\n\n/-- Given a functor `t`, a function `t' : Type u \u2192 Type u`, and\nequivalences `t \u03b1 \u2243 t' \u03b1` for all `\u03b1`, then every function `\u03b1 \u2192 \u03b2` can\nbe mapped to a function `t' \u03b1 \u2192 t' \u03b2` functorially (see\n`equiv.functor`). -/\nprotected def map {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) : t' \u03b2 :=\neqv \u03b2 $ map f ((eqv \u03b1).symm x)\n\n/-- The function `equiv.map` transfers the functoriality of `t` to\n`t'` using the equivalences `eqv`.  -/\nprotected def functor : functor t' :=\n{ map := @equiv.map _ }\n\nvariables [is_lawful_functor t]\n\nprotected lemma id_map {\u03b1 : Type u} (x : t' \u03b1) : equiv.map id x = x :=\nby simp [equiv.map, id_map]\n\nprotected lemma comp_map {\u03b1 \u03b2 \u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : t' \u03b1) :\n  equiv.map (h \u2218 g) x = equiv.map h (equiv.map g x) :=\nby simp [equiv.map]; apply comp_map\n\nprotected lemma is_lawful_functor : @is_lawful_functor _ equiv.functor :=\n{ id_map := @equiv.id_map _ _,\n  comp_map := @equiv.comp_map _ _ }\n\nprotected lemma is_lawful_functor' [F : _root_.functor t']\n  (h\u2080 : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2), _root_.functor.map f = equiv.map f)\n  (h\u2081 : \u2200 {\u03b1 \u03b2} (f : \u03b2), _root_.functor.map_const f = (equiv.map \u2218 function.const \u03b1) f) :\n  _root_.is_lawful_functor t' :=\nbegin\n  have : F = equiv.functor,\n  { casesI F, dsimp [equiv.functor],\n    congr; ext; [rw \u2190 h\u2080, rw \u2190 h\u2081] },\n  substI this,\n  exact equiv.is_lawful_functor\nend\n\nend functor\n\nsection traversable\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [traversable t]\nvariables {m : Type u \u2192 Type u} [applicative m]\nvariables {\u03b1 \u03b2 : Type u}\n\n/-- Like `equiv.map`, a function `t' : Type u \u2192 Type u` can be given\nthe structure of a traversable functor using a traversable functor\n`t'` and equivalences `t \u03b1 \u2243 t' \u03b1` for all \u03b1.  See `equiv.traversable`. -/\nprotected def traverse (f : \u03b1 \u2192 m \u03b2) (x : t' \u03b1) : m (t' \u03b2) :=\neqv \u03b2 <$> traverse f ((eqv \u03b1).symm x)\n\n/-- The function `equiv.traverse` transfers a traversable functor\ninstance across the equivalences `eqv`. -/\nprotected def traversable : traversable t' :=\n{ to_functor := equiv.functor eqv,\n  traverse := @equiv.traverse _ }\n\nend traversable\n\nsection equiv\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [traversable t] [is_lawful_traversable t]\nvariables {F G : Type u \u2192 Type u} [applicative F] [applicative G]\nvariables [is_lawful_applicative F] [is_lawful_applicative G]\nvariables (\u03b7 : applicative_transformation F G)\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\n\nopen is_lawful_traversable functor\n\nprotected lemma id_traverse (x : t' \u03b1) :\n  equiv.traverse eqv id.mk x = x :=\nby simp! [equiv.traverse,id_bind,id_traverse,functor.map] with functor_norm\n\nprotected lemma traverse_eq_map_id (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) :\n  equiv.traverse eqv (id.mk \u2218 f) x = id.mk (equiv.map eqv f x) :=\nby simp [equiv.traverse, traverse_eq_map_id] with functor_norm; refl\n\nprotected lemma comp_traverse (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t' \u03b1) :\n  equiv.traverse eqv (comp.mk \u2218 functor.map f \u2218 g) x =\n  comp.mk (equiv.traverse eqv f <$> equiv.traverse eqv g x) :=\nby simp [equiv.traverse,comp_traverse] with functor_norm; congr; ext; simp\n\nprotected lemma naturality (f : \u03b1 \u2192 F \u03b2) (x : t' \u03b1) :\n  \u03b7 (equiv.traverse eqv f x) = equiv.traverse eqv (@\u03b7 _ \u2218 f) x :=\nby simp only [equiv.traverse] with functor_norm\n\n/-- The fact that `t` is a lawful traversable functor carries over the\nequivalences to `t'`, with the traversable functor structure given by\n`equiv.traversable`. -/\nprotected def is_lawful_traversable : @is_lawful_traversable t' (equiv.traversable eqv) :=\n{ to_is_lawful_functor := @equiv.is_lawful_functor _ _ eqv _ _,\n  id_traverse := @equiv.id_traverse _ _,\n  comp_traverse := @equiv.comp_traverse _ _,\n  traverse_eq_map_id := @equiv.traverse_eq_map_id _ _,\n  naturality := @equiv.naturality _ _ }\n\n/-- If the `traversable t'` instance has the properties that `map`,\n`map_const`, and `traverse` are equal to the ones that come from\ncarrying the traversable functor structure from `t` over the\nequivalences, then the fact that `t` is a lawful traversable functor\ncarries over as well. -/\nprotected def is_lawful_traversable' [_i : traversable t']\n  (h\u2080 : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2),\n         map f = equiv.map eqv f)\n  (h\u2081 : \u2200 {\u03b1 \u03b2} (f : \u03b2),\n         map_const f = (equiv.map eqv \u2218 function.const \u03b1) f)\n  (h\u2082 : \u2200 {F : Type u \u2192 Type u} [applicative F],\n        by exactI \u2200 [is_lawful_applicative F]\n          {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2),\n         traverse f = equiv.traverse eqv f) :\n  _root_.is_lawful_traversable t' :=\nbegin\n    -- we can't use the same approach as for `is_lawful_functor'` because\n    -- h\u2082 needs a `is_lawful_applicative` assumption\n  refine {to_is_lawful_functor :=\n    equiv.is_lawful_functor' eqv @h\u2080 @h\u2081, ..}; introsI,\n  { rw [h\u2082, equiv.id_traverse], apply_instance },\n  { rw [h\u2082, equiv.comp_traverse f g x, h\u2082], congr,\n    rw [h\u2082], all_goals { apply_instance } },\n  { rw [h\u2082, equiv.traverse_eq_map_id, h\u2080]; apply_instance },\n  { rw [h\u2082, equiv.naturality, h\u2082]; apply_instance }\nend\n\nend equiv\nend equiv\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/control/traversable/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.354875985009871}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Johan Commelin, Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.mv_polynomial.basic\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 u_5 u_3 \n\nnamespace Mathlib\n\n/-!\n# Renaming variables of polynomials\n\nThis file establishes the `rename` operation on multivariate polynomials,\nwhich modifies the set of variables.\n\n## Main declarations\n\n* `mv_polynomial.rename`\n\n## Notation\n\nAs in other polynomial files, we typically use the notation:\n\n+ `\u03c3 \u03c4 \u03b1 : Type*` (indexing the variables)\n\n+ `R S : Type*` `[comm_semiring R]` `[comm_semiring S]` (the coefficients)\n\n+ `s : \u03c3 \u2192\u2080 \u2115`, a function from `\u03c3` to `\u2115` which is zero away from a finite set.\nThis will give rise to a monomial in `mv_polynomial \u03c3 R` which mathematicians might call `X^s`\n\n+ `r : R` elements of the coefficient ring\n\n+ `i : \u03c3`, with corresponding monomial `X i`, often denoted `X_i` by mathematicians\n\n+ `p : mv_polynomial \u03c3 \u03b1`\n\n-/\n\nnamespace mv_polynomial\n\n\n/-- Rename all the variables in a multivariable polynomial. -/\ndef rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4) :\n    alg_hom R (mv_polynomial \u03c3 R) (mv_polynomial \u03c4 R) :=\n  aeval (X \u2218 f)\n\n@[simp] theorem rename_C {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (r : R) : coe_fn (rename f) (coe_fn C r) = coe_fn C r :=\n  eval\u2082_C (algebra_map R (mv_polynomial \u03c4 R)) (fun (n : \u03c3) => function.comp X f n) r\n\n@[simp] theorem rename_X {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (i : \u03c3) : coe_fn (rename f) (X i) = X (f i) :=\n  eval\u2082_X (algebra_map R (mv_polynomial \u03c4 R)) (fun (n : \u03c3) => function.comp X f n) i\n\ntheorem map_rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R]\n    [comm_semiring S] (f : R \u2192+* S) (g : \u03c3 \u2192 \u03c4) (p : mv_polynomial \u03c3 R) :\n    coe_fn (map f) (coe_fn (rename g) p) = coe_fn (rename g) (coe_fn (map f) p) :=\n  sorry\n\n@[simp] theorem rename_rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {\u03b1 : Type u_3} {R : Type u_4}\n    [comm_semiring R] (f : \u03c3 \u2192 \u03c4) (g : \u03c4 \u2192 \u03b1) (p : mv_polynomial \u03c3 R) :\n    coe_fn (rename g) (coe_fn (rename f) p) = coe_fn (rename (g \u2218 f)) p :=\n  sorry\n\n@[simp] theorem rename_id {\u03c3 : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial \u03c3 R) :\n    coe_fn (rename id) p = p :=\n  eval\u2082_eta p\n\ntheorem rename_monomial {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (d : \u03c3 \u2192\u2080 \u2115) (r : R) : coe_fn (rename f) (monomial d r) = monomial (finsupp.map_domain f d) r :=\n  sorry\n\ntheorem rename_eq {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (p : mv_polynomial \u03c3 R) : coe_fn (rename f) p = finsupp.map_domain (finsupp.map_domain f) p :=\n  sorry\n\ntheorem rename_injective {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (hf : function.injective f) : function.injective \u21d1(rename f) :=\n  sorry\n\ntheorem eval\u2082_rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R]\n    [comm_semiring S] (f : R \u2192+* S) (k : \u03c3 \u2192 \u03c4) (g : \u03c4 \u2192 S) (p : mv_polynomial \u03c3 R) :\n    eval\u2082 f g (coe_fn (rename k) p) = eval\u2082 f (g \u2218 k) p :=\n  sorry\n\ntheorem eval\u2082_hom_rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} {S : Type u_5}\n    [comm_semiring R] [comm_semiring S] (f : R \u2192+* S) (k : \u03c3 \u2192 \u03c4) (g : \u03c4 \u2192 S)\n    (p : mv_polynomial \u03c3 R) :\n    coe_fn (eval\u2082_hom f g) (coe_fn (rename k) p) = coe_fn (eval\u2082_hom f (g \u2218 k)) p :=\n  eval\u2082_rename f k (fun (n : \u03c4) => g n) p\n\ntheorem aeval_rename {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} {S : Type u_5} [comm_semiring R]\n    [comm_semiring S] (k : \u03c3 \u2192 \u03c4) (g : \u03c4 \u2192 S) (p : mv_polynomial \u03c3 R) [algebra R S] :\n    coe_fn (aeval g) (coe_fn (rename k) p) = coe_fn (aeval (g \u2218 k)) p :=\n  eval\u2082_hom_rename (algebra_map R S) k (fun (n : \u03c4) => g n) p\n\ntheorem rename_eval\u2082 {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (k : \u03c3 \u2192 \u03c4)\n    (p : mv_polynomial \u03c3 R) (g : \u03c4 \u2192 mv_polynomial \u03c3 R) :\n    coe_fn (rename k) (eval\u2082 C (g \u2218 k) p) = eval\u2082 C (\u21d1(rename k) \u2218 g) (coe_fn (rename k) p) :=\n  sorry\n\ntheorem rename_prodmk_eval\u2082 {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R]\n    (p : mv_polynomial \u03c3 R) (j : \u03c4) (g : \u03c3 \u2192 mv_polynomial \u03c3 R) :\n    coe_fn (rename (Prod.mk j)) (eval\u2082 C g p) =\n        eval\u2082 C (fun (x : \u03c3) => coe_fn (rename (Prod.mk j)) (g x)) p :=\n  sorry\n\ntheorem eval\u2082_rename_prodmk {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} {S : Type u_5}\n    [comm_semiring R] [comm_semiring S] (f : R \u2192+* S) (g : \u03c3 \u00d7 \u03c4 \u2192 S) (i : \u03c3)\n    (p : mv_polynomial \u03c4 R) :\n    eval\u2082 f g (coe_fn (rename (Prod.mk i)) p) = eval\u2082 f (fun (j : \u03c4) => g (i, j)) p :=\n  sorry\n\ntheorem eval_rename_prodmk {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R]\n    (g : \u03c3 \u00d7 \u03c4 \u2192 R) (i : \u03c3) (p : mv_polynomial \u03c4 R) :\n    coe_fn (eval g) (coe_fn (rename (Prod.mk i)) p) = coe_fn (eval fun (j : \u03c4) => g (i, j)) p :=\n  eval\u2082_rename_prodmk (ring_hom.id R) (fun (n : \u03c3 \u00d7 \u03c4) => g n) i p\n\n/-- Every polynomial is a polynomial in finitely many variables. -/\ntheorem exists_finset_rename {\u03c3 : Type u_1} {R : Type u_4} [comm_semiring R]\n    (p : mv_polynomial \u03c3 R) :\n    \u2203 (s : finset \u03c3),\n        \u2203 (q : mv_polynomial (Subtype fun (x : \u03c3) => x \u2208 s) R), p = coe_fn (rename coe) q :=\n  sorry\n\n/-- Every polynomial is a polynomial in finitely many variables. -/\ntheorem exists_fin_rename {\u03c3 : Type u_1} {R : Type u_4} [comm_semiring R] (p : mv_polynomial \u03c3 R) :\n    \u2203 (n : \u2115),\n        \u2203 (f : fin n \u2192 \u03c3),\n          \u2203 (hf : function.injective f), \u2203 (q : mv_polynomial (fin n) R), p = coe_fn (rename f) q :=\n  sorry\n\ntheorem eval\u2082_cast_comp {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R] (f : \u03c3 \u2192 \u03c4)\n    (c : \u2124 \u2192+* R) (g : \u03c4 \u2192 R) (p : mv_polynomial \u03c3 \u2124) :\n    eval\u2082 c (g \u2218 f) p = eval\u2082 c g (coe_fn (rename f) p) :=\n  sorry\n\n@[simp] theorem coeff_rename_map_domain {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4}\n    [comm_semiring R] (f : \u03c3 \u2192 \u03c4) (hf : function.injective f) (\u03c6 : mv_polynomial \u03c3 R) (d : \u03c3 \u2192\u2080 \u2115) :\n    coeff (finsupp.map_domain f d) (coe_fn (rename f) \u03c6) = coeff d \u03c6 :=\n  sorry\n\ntheorem coeff_rename_eq_zero {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R]\n    (f : \u03c3 \u2192 \u03c4) (\u03c6 : mv_polynomial \u03c3 R) (d : \u03c4 \u2192\u2080 \u2115)\n    (h : \u2200 (u : \u03c3 \u2192\u2080 \u2115), finsupp.map_domain f u = d \u2192 coeff u \u03c6 = 0) :\n    coeff d (coe_fn (rename f) \u03c6) = 0 :=\n  sorry\n\ntheorem coeff_rename_ne_zero {\u03c3 : Type u_1} {\u03c4 : Type u_2} {R : Type u_4} [comm_semiring R]\n    (f : \u03c3 \u2192 \u03c4) (\u03c6 : mv_polynomial \u03c3 R) (d : \u03c4 \u2192\u2080 \u2115) (h : coeff d (coe_fn (rename f) \u03c6) \u2260 0) :\n    \u2203 (u : \u03c3 \u2192\u2080 \u2115), finsupp.map_domain f u = d \u2227 coeff u \u03c6 \u2260 0 :=\n  sorry\n\n@[simp] theorem constant_coeff_rename {\u03c3 : Type u_1} {R : Type u_4} [comm_semiring R] {\u03c4 : Type u_2}\n    (f : \u03c3 \u2192 \u03c4) (\u03c6 : mv_polynomial \u03c3 R) :\n    coe_fn constant_coeff (coe_fn (rename f) \u03c6) = coe_fn constant_coeff \u03c6 :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/mv_polynomial/rename_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.538983220687684, "lm_q1q2_score": 0.3548759850098709}}
{"text": "import tactic\nimport tactic.induction\n\nimport .base .bounded .nice\n\nnoncomputable theory\nopen_locale classical\n\nlemma lem_2_4 {pw N : \u2115}\n  (h : \u2203 (d : D), \u2200 (a : A pw), A_trapped_in_for a d (bounded N)) :\n  \u2203 (d : D), d.nice pw \u2227 \u2200 (a : A pw), A_trapped_in_for a d (bounded N) :=\nbegin\n  sorry\nend", "meta": {"author": "user7230724", "repo": "lean-projects", "sha": "ab9a83874775efd18f8c5b867e480bae4d596b31", "save_path": "github-repos/lean/user7230724-lean-projects", "path": "github-repos/lean/user7230724-lean-projects/lean-projects-ab9a83874775efd18f8c5b867e480bae4d596b31/src/ap/lemma_2_4.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6442251064863698, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3547150886941198}}
{"text": "inductive Foo where\n  | c1 (x : Nat) | c2 | c3 | c4\n\ndef bla : Foo \u2192 Nat\n  | .c1 x => x + 1\n  | _     => 2\n\nexample (x : Foo) : bla x > 0 := by\n  cases x with\n  | _  => decide -- Error\n  | c1 => decide\n\nexample (x : Foo) : bla x > 0 := by\n  induction x with\n  | _  => decide -- Error\n  | c1 => decide\n\nexample (x : Foo) : bla x > 0 := by\n  cases x with\n  | c1 x => simp_arith [bla]\n  | _    => decide\n\nexample (x : Foo) : bla x > 0 := by\n  induction x with\n  | c1 x => simp_arith [bla]\n  | _    => decide\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/wildcardAlt.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526660244838, "lm_q2_score": 0.6224593382055109, "lm_q1q2_score": 0.3547101133682462}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.concrete_category.bundled_hom\nimport Mathlib.category_theory.concrete_category.reflects_isomorphisms\nimport Mathlib.algebra.punit_instances\nimport Mathlib.PostPort\n\nuniverses u u_1 \n\nnamespace Mathlib\n\n/-!\n# Category instances for monoid, add_monoid, comm_monoid, and add_comm_monoid.\n\nWe introduce the bundled categories:\n* `Mon`\n* `AddMon`\n* `CommMon`\n* `AddCommMon`\nalong with the relevant forgetful functors between them.\n-/\n\n/-- The category of monoids and monoid morphisms. -/\ndef Mon := category_theory.bundled monoid\n\n/-- The category of additive monoids and monoid morphisms. -/\nnamespace Mon\n\n\nprotected instance Mathlib.AddMon.bundled_hom : category_theory.bundled_hom add_monoid_hom :=\n  category_theory.bundled_hom.mk add_monoid_hom.to_fun add_monoid_hom.id add_monoid_hom.comp\n\nprotected instance Mathlib.AddMon.concrete_category : category_theory.concrete_category AddMon :=\n  category_theory.bundled_hom.category_theory.bundled.category_theory.concrete_category\n    add_monoid_hom\n\n/-- Construct a bundled `Mon` from the underlying type and typeclass. -/\ndef Mathlib.AddMon.of (M : Type u) [add_monoid M] : AddMon := category_theory.bundled.of M\n\n/-- Construct a bundled `Mon` from the underlying type and typeclass. -/\n-- The default instance for `monoid punit` is derived via `punit.comm_ring`,\n\nprotected instance Mathlib.AddMon.inhabited : Inhabited AddMon := { default := AddMon.of PUnit }\n\n-- which breaks to_additive.\n\nprotected instance Mathlib.AddMon.add_monoid (M : AddMon) : add_monoid \u21a5M :=\n  category_theory.bundled.str M\n\n@[simp] theorem Mathlib.AddMon.coe_of (R : Type u) [add_monoid R] : \u21a5(AddMon.of R) = R := rfl\n\nend Mon\n\n\n/-- The category of commutative monoids and monoid morphisms. -/\ndef CommMon := category_theory.bundled comm_monoid\n\n/-- The category of additive commutative monoids and monoid morphisms. -/\nnamespace CommMon\n\n\nprotected instance comm_monoid.to_monoid.category_theory.bundled_hom.parent_projection :\n    category_theory.bundled_hom.parent_projection comm_monoid.to_monoid :=\n  category_theory.bundled_hom.parent_projection.mk\n\nprotected instance has_coe_to_sort : has_coe_to_sort CommMon :=\n  category_theory.bundled.has_coe_to_sort\n\n/-- Construct a bundled `CommMon` from the underlying type and typeclass. -/\ndef of (M : Type u) [comm_monoid M] : CommMon := category_theory.bundled.of M\n\n/-- Construct a bundled `AddCommMon` from the underlying type and typeclass. -/\n-- The default instance for `comm_monoid punit` is derived via `punit.comm_ring`,\n\nprotected instance Mathlib.AddCommMon.inhabited : Inhabited AddCommMon :=\n  { default := AddCommMon.of PUnit }\n\n-- which breaks to_additive.\n\nprotected instance Mathlib.AddCommMon.add_comm_monoid (M : AddCommMon) : add_comm_monoid \u21a5M :=\n  category_theory.bundled.str M\n\n@[simp] theorem coe_of (R : Type u) [comm_monoid R] : \u21a5(of R) = R := rfl\n\nprotected instance Mathlib.AddCommMon.has_forget_to_AddMon :\n    category_theory.has_forget\u2082 AddCommMon AddMon :=\n  category_theory.bundled_hom.forget\u2082 add_monoid_hom add_comm_monoid.to_add_monoid\n\nend CommMon\n\n\n-- We verify that the coercions of morphisms to functions work correctly:\n\n-- We verify that when constructing a morphism in `CommMon`,\n\n-- when we construct the `to_fun` field, the types are presented as `\u21a5R`,\n\n-- rather than `R.\u03b1` or (as we used to have) `\u21a5(bundled.map comm_monoid.to_monoid R)`.\n\n/-- Build an isomorphism in the category `Mon` from a `mul_equiv` between `monoid`s. -/\ndef mul_equiv.to_Mon_iso {X : Type u} {Y : Type u} [monoid X] [monoid Y] (e : X \u2243* Y) :\n    Mon.of X \u2245 Mon.of Y :=\n  category_theory.iso.mk (mul_equiv.to_monoid_hom e) (mul_equiv.to_monoid_hom (mul_equiv.symm e))\n\n@[simp] theorem mul_equiv.to_Mon_iso_hom {X : Type u} {Y : Type u} [monoid X] [monoid Y]\n    {e : X \u2243* Y} : category_theory.iso.hom (mul_equiv.to_Mon_iso e) = mul_equiv.to_monoid_hom e :=\n  rfl\n\n@[simp] theorem add_equiv.to_AddMon_iso_inv {X : Type u} {Y : Type u} [add_monoid X] [add_monoid Y]\n    {e : X \u2243+ Y} :\n    category_theory.iso.inv (add_equiv.to_AddMon_iso e) =\n        add_equiv.to_add_monoid_hom (add_equiv.symm e) :=\n  rfl\n\n/-- Build an isomorphism in the category `CommMon` from a `mul_equiv` between `comm_monoid`s. -/\ndef add_equiv.to_AddCommMon_iso {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y]\n    (e : X \u2243+ Y) : AddCommMon.of X \u2245 AddCommMon.of Y :=\n  category_theory.iso.mk (add_equiv.to_add_monoid_hom e)\n    (add_equiv.to_add_monoid_hom (add_equiv.symm e))\n\n@[simp] theorem add_equiv.to_AddCommMon_iso_hom {X : Type u} {Y : Type u} [add_comm_monoid X]\n    [add_comm_monoid Y] {e : X \u2243+ Y} :\n    category_theory.iso.hom (add_equiv.to_AddCommMon_iso e) = add_equiv.to_add_monoid_hom e :=\n  rfl\n\n@[simp] theorem add_equiv.to_AddCommMon_iso_inv {X : Type u} {Y : Type u} [add_comm_monoid X]\n    [add_comm_monoid Y] {e : X \u2243+ Y} :\n    category_theory.iso.inv (add_equiv.to_AddCommMon_iso e) =\n        add_equiv.to_add_monoid_hom (add_equiv.symm e) :=\n  rfl\n\nnamespace category_theory.iso\n\n\n/-- Build a `mul_equiv` from an isomorphism in the category `Mon`. -/\ndef AddMon_iso_to_add_equiv {X : AddMon} {Y : AddMon} (i : X \u2245 Y) : \u21a5X \u2243+ \u21a5Y :=\n  add_monoid_hom.to_add_equiv (hom i) (inv i) (hom_inv_id i) (inv_hom_id i)\n\n/-- Build a `mul_equiv` from an isomorphism in the category `CommMon`. -/\ndef CommMon_iso_to_add_equiv {X : AddCommMon} {Y : AddCommMon} (i : X \u2245 Y) : \u21a5X \u2243+ \u21a5Y :=\n  add_monoid_hom.to_add_equiv (hom i) (inv i) (hom_inv_id i) (inv_hom_id i)\n\nend category_theory.iso\n\n\n/-- multiplicative equivalences between `monoid`s are the same as (isomorphic to) isomorphisms\nin `Mon` -/\ndef mul_equiv_iso_Mon_iso {X : Type u} {Y : Type u} [monoid X] [monoid Y] :\n    X \u2243* Y \u2245 Mon.of X \u2245 Mon.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243* Y) => mul_equiv.to_Mon_iso e)\n    fun (i : Mon.of X \u2245 Mon.of Y) => category_theory.iso.Mon_iso_to_mul_equiv i\n\n/-- multiplicative equivalences between `comm_monoid`s are the same as (isomorphic to) isomorphisms\nin `CommMon` -/\ndef add_equiv_iso_AddCommMon_iso {X : Type u} {Y : Type u} [add_comm_monoid X] [add_comm_monoid Y] :\n    X \u2243+ Y \u2245 AddCommMon.of X \u2245 AddCommMon.of Y :=\n  category_theory.iso.mk (fun (e : X \u2243+ Y) => add_equiv.to_AddCommMon_iso e)\n    fun (i : AddCommMon.of X \u2245 AddCommMon.of Y) => category_theory.iso.CommMon_iso_to_add_equiv i\n\nprotected instance AddMon.forget_reflects_isos :\n    category_theory.reflects_isomorphisms (category_theory.forget AddMon) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : AddMon) (f : X \u27f6 Y)\n      (_x :\n      category_theory.is_iso (category_theory.functor.map (category_theory.forget AddMon) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget AddMon) X \u2245\n          category_theory.functor.obj (category_theory.forget AddMon) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget AddMon) f);\n      let e : \u21a5X \u2243+ \u21a5Y :=\n        add_equiv.mk (add_monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i))\n          sorry sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (add_equiv.to_AddMon_iso e))\n\nprotected instance CommMon.forget_reflects_isos :\n    category_theory.reflects_isomorphisms (category_theory.forget CommMon) :=\n  category_theory.reflects_isomorphisms.mk\n    fun (X Y : CommMon) (f : X \u27f6 Y)\n      (_x :\n      category_theory.is_iso (category_theory.functor.map (category_theory.forget CommMon) f)) =>\n      let i :\n        category_theory.functor.obj (category_theory.forget CommMon) X \u2245\n          category_theory.functor.obj (category_theory.forget CommMon) Y :=\n        category_theory.as_iso (category_theory.functor.map (category_theory.forget CommMon) f);\n      let e : \u21a5X \u2243* \u21a5Y :=\n        mul_equiv.mk (monoid_hom.to_fun f) (equiv.inv_fun (category_theory.iso.to_equiv i)) sorry\n          sorry sorry;\n      category_theory.is_iso.mk (category_theory.iso.inv (mul_equiv.to_CommMon_iso e))\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/category/Mon/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3547100911885256}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nMultisets.\n-/\nimport logic.function order.boolean_algebra\n  data.list.basic data.list.perm data.list.sort data.quot data.string\n  algebra.order_functions algebra.group_power algebra.ordered_group\n  category.traversable.lemmas tactic.interactive\n  category.traversable.instances category.basic\n\nopen list subtype nat lattice\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*} {\u03b3 : Type*}\n\nlocal infix ` \u2022 ` := add_monoid.smul\n\ninstance list.perm.setoid (\u03b1 : Type*) : setoid (list \u03b1) :=\nsetoid.mk perm \u27e8perm.refl, @perm.symm _, @perm.trans _\u27e9\n\n/-- `multiset \u03b1` is the quotient of `list \u03b1` by list permutation. The result\n  is a type of finite sets with duplicates allowed.  -/\ndef {u} multiset (\u03b1 : Type u) : Type u :=\nquotient (list.perm.setoid \u03b1)\n\nnamespace multiset\n\ninstance : has_coe (list \u03b1) (multiset \u03b1) := \u27e8quot.mk _\u27e9\n\n@[simp] theorem quot_mk_to_coe (l : list \u03b1) : @eq (multiset \u03b1) \u27e6l\u27e7 l := rfl\n\n@[simp] theorem quot_mk_to_coe' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk (\u2248) l) l := rfl\n\n@[simp] theorem quot_mk_to_coe'' (l : list \u03b1) : @eq (multiset \u03b1) (quot.mk setoid.r l) l := rfl\n\n@[simp] theorem coe_eq_coe {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) = l\u2082 \u2194 l\u2081 ~ l\u2082 := quotient.eq\n\ninstance has_decidable_eq [decidable_eq \u03b1] : decidable_eq (multiset \u03b1)\n| s\u2081 s\u2082 := quotient.rec_on_subsingleton\u2082 s\u2081 s\u2082 $ \u03bb l\u2081 l\u2082,\n  decidable_of_iff' _ quotient.eq\n\n/- empty multiset -/\n\n/-- `0 : multiset \u03b1` is the empty set -/\nprotected def zero : multiset \u03b1 := @nil \u03b1\n\ninstance : has_zero (multiset \u03b1)   := \u27e8multiset.zero\u27e9\ninstance : has_emptyc (multiset \u03b1) := \u27e80\u27e9\ninstance : inhabited (multiset \u03b1)  := \u27e80\u27e9\n\n@[simp] theorem coe_nil_eq_zero : (@nil \u03b1 : multiset \u03b1) = 0 := rfl\n@[simp] theorem empty_eq_zero : (\u2205 : multiset \u03b1) = 0 := rfl\n\n/- cons -/\n\n/-- `cons a s` is the multiset which contains `s` plus one more\n  instance of `a`. -/\ndef cons (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (a :: l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound ((perm_cons a).2 p))\n\nnotation a :: b := cons a b\n\ninstance : has_insert \u03b1 (multiset \u03b1) := \u27e8cons\u27e9\n\n@[simp] theorem insert_eq_cons (a : \u03b1) (s : multiset \u03b1) :\n  insert a s = a::s := rfl\n\n@[simp] theorem cons_coe (a : \u03b1) (l : list \u03b1) :\n  (a::l : multiset \u03b1) = (a::l : list \u03b1) := rfl\n\ntheorem singleton_coe (a : \u03b1) : (a::0 : multiset \u03b1) = ([a] : list \u03b1) := rfl\n\n@[simp] theorem cons_inj_left {a b : \u03b1} (s : multiset \u03b1) :\n  a::s = b::s \u2194 a = b :=\n\u27e8quot.induction_on s $ \u03bb l e,\n  have [a] ++ l ~ [b] ++ l, from quotient.exact e,\n  eq_singleton_of_perm $ (perm_app_right_iff _).1 this, congr_arg _\u27e9\n\n@[simp] theorem cons_inj_right (a : \u03b1) : \u2200{s t : multiset \u03b1}, a::s = a::t \u2194 s = t :=\nby rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9; simp [perm_cons]\n\n@[recursor 5] protected theorem induction {p : multiset \u03b1 \u2192 Prop}\n  (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : \u2200s, p s :=\nby rintros \u27e8l\u27e9; induction l with _ _ ih; [exact h\u2081, exact h\u2082 ih]\n\n@[elab_as_eliminator] protected theorem induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2081 : p 0) (h\u2082 : \u2200 \u2983a : \u03b1\u2984 {s : multiset \u03b1}, p s \u2192 p (a :: s)) : p s :=\nmultiset.induction h\u2081 h\u2082 s\n\ntheorem cons_swap (a b : \u03b1) (s : multiset \u03b1) : a :: b :: s = b :: a :: s :=\nquot.induction_on s $ \u03bb l, quotient.sound $ perm.swap _ _ _\n\nsection rec\nvariables {C : multiset \u03b1 \u2192 Sort*}\n\n/-- Dependent recursor on multisets.\n\nTODO: should be @[recursor 6], but then the definition of `multiset.pi` failes with a stack\noverflow in `whnf`.\n-/\nprotected def rec\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b))\n  (m : multiset \u03b1) : C m :=\nquotient.hrec_on m (@list.rec \u03b1 (\u03bbl, C \u27e6l\u27e7) C_0 (\u03bba l b, C_cons a \u27e6l\u27e7 b)) $\n  assume l l' h,\n  list.rec_heq_of_perm h\n    (assume a l l' b b' hl, have \u27e6l\u27e7 = \u27e6l'\u27e7, from quot.sound hl, by cc)\n    (assume a a' l, C_cons_heq a a' \u27e6l\u27e7)\n\n@[elab_as_eliminator]\nprotected def rec_on (m : multiset \u03b1)\n  (C_0 : C 0)\n  (C_cons : \u03a0a m, C m \u2192 C (a::m))\n  (C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)) :\n  C m :=\nmultiset.rec C_0 C_cons C_cons_heq m\n\nvariables {C_0 : C 0} {C_cons : \u03a0a m, C m \u2192 C (a::m)}\n  {C_cons_heq : \u2200a a' m b, C_cons a (a'::m) (C_cons a' m b) == C_cons a' (a::m) (C_cons a m b)}\n\n@[simp] lemma rec_on_0 : @multiset.rec_on \u03b1 C (0:multiset \u03b1) C_0 C_cons C_cons_heq = C_0 :=\nrfl\n\n@[simp] lemma rec_on_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).rec_on C_0 C_cons C_cons_heq = C_cons a m (m.rec_on C_0 C_cons C_cons_heq) :=\nquotient.induction_on m $ assume l, rfl\n\nend rec\n\nsection mem\n\n/-- `a \u2208 s` means that `a` has nonzero multiplicity in `s`. -/\ndef mem (a : \u03b1) (s : multiset \u03b1) : Prop :=\nquot.lift_on s (\u03bb l, a \u2208 l) (\u03bb l\u2081 l\u2082 (e : l\u2081 ~ l\u2082), propext $ mem_of_perm e)\n\ninstance : has_mem \u03b1 (multiset \u03b1) := \u27e8mem\u27e9\n\n@[simp] lemma mem_coe {a : \u03b1} {l : list \u03b1} : a \u2208 (l : multiset \u03b1) \u2194 a \u2208 l := iff.rfl\n\ninstance decidable_mem [decidable_eq \u03b1] (a : \u03b1) (s : multiset \u03b1) : decidable (a \u2208 s) :=\nquot.rec_on_subsingleton s $ list.decidable_mem a\n\n@[simp] theorem mem_cons {a b : \u03b1} {s : multiset \u03b1} : a \u2208 b :: s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, iff.rfl\n\nlemma mem_cons_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 b :: s :=\nmem_cons.2 $ or.inr h\n\n@[simp] theorem mem_cons_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 a :: s :=\nmem_cons.2 (or.inl rfl)\n\ntheorem exists_cons_of_mem {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 \u2203 t, s = a :: t :=\nquot.induction_on s $ \u03bb l (h : a \u2208 l),\nlet \u27e8l\u2081, l\u2082, e\u27e9 := mem_split h in\ne.symm \u25b8 \u27e8(l\u2081++l\u2082 : list \u03b1), quot.sound perm_middle\u27e9\n\n@[simp] theorem not_mem_zero (a : \u03b1) : a \u2209 (0 : multiset \u03b1) := id\n\ntheorem eq_zero_of_forall_not_mem {s : multiset \u03b1} : (\u2200x, x \u2209 s) \u2192 s = 0 :=\nquot.induction_on s $ \u03bb l H, by rw eq_nil_of_forall_not_mem H; refl\n\ntheorem exists_mem_of_ne_zero {s : multiset \u03b1} : s \u2260 0 \u2192 \u2203 a : \u03b1, a \u2208 s :=\nquot.induction_on s $ assume l hl,\n  match l, hl with\n  | [] := assume h, false.elim $ h rfl\n  | (a :: l) := assume _, \u27e8a, by simp\u27e9\n  end\n\n@[simp] lemma zero_ne_cons {a : \u03b1} {m : multiset \u03b1} : 0 \u2260 a :: m :=\nassume h, have a \u2208 (0:multiset \u03b1), from h.symm \u25b8 mem_cons_self _ _, not_mem_zero _ this\n\n@[simp] lemma cons_ne_zero {a : \u03b1} {m : multiset \u03b1} : a :: m \u2260 0 := zero_ne_cons.symm\n\nlemma cons_eq_cons {a b : \u03b1} {as bs : multiset \u03b1} :\n  a :: as = b :: bs \u2194 ((a = b \u2227 as = bs) \u2228 (a \u2260 b \u2227 \u2203cs, as = b :: cs \u2227 bs = a :: cs)) :=\nbegin\n  haveI : decidable_eq \u03b1 := classical.dec_eq \u03b1,\n  split,\n  { assume eq,\n    by_cases a = b,\n    { subst h, simp * at * },\n    { have : a \u2208 b :: bs, from eq \u25b8 mem_cons_self _ _,\n      have : a \u2208 bs, by simpa [h],\n      rcases exists_cons_of_mem this with \u27e8cs, hcs\u27e9,\n      simp [h, hcs],\n      have : a :: as = b :: a :: cs, by simp [eq, hcs],\n      have : a :: as = a :: b :: cs, by rwa [cons_swap],\n      simpa using this } },\n  { assume h,\n    rcases h with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n    { simp * },\n    { simp [*, cons_swap a b] } }\nend\n\nend mem\n\n/- subset -/\nsection subset\n\n/-- `s \u2286 t` is the lift of the list subset relation. It means that any\n  element with nonzero multiplicity in `s` has nonzero multiplicity in `t`,\n  but it does not imply that the multiplicity of `a` in `s` is less or equal than in `t`;\n  see `s \u2264 t` for this relation. -/\nprotected def subset (s t : multiset \u03b1) : Prop := \u2200 \u2983a : \u03b1\u2984, a \u2208 s \u2192 a \u2208 t\n\ninstance : has_subset (multiset \u03b1) := \u27e8multiset.subset\u27e9\n\n@[simp] theorem coe_subset {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2286 l\u2082 \u2194 l\u2081 \u2286 l\u2082 := iff.rfl\n\n@[simp] theorem subset.refl (s : multiset \u03b1) : s \u2286 s := \u03bb a h, h\n\ntheorem subset.trans {s t u : multiset \u03b1} : s \u2286 t \u2192 t \u2286 u \u2192 s \u2286 u :=\n\u03bb h\u2081 h\u2082 a m, h\u2082 (h\u2081 m)\n\ntheorem subset_iff {s t : multiset \u03b1} : s \u2286 t \u2194 (\u2200\u2983x\u2984, x \u2208 s \u2192 x \u2208 t) := iff.rfl\n\ntheorem mem_of_subset {s t : multiset \u03b1} {a : \u03b1} (h : s \u2286 t) : a \u2208 s \u2192 a \u2208 t := @h _\n\n@[simp] theorem zero_subset (s : multiset \u03b1) : 0 \u2286 s :=\n\u03bb a, (not_mem_nil a).elim\n\n@[simp] theorem cons_subset {a : \u03b1} {s t : multiset \u03b1} : (a :: s) \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t :=\nby simp [subset_iff, or_imp_distrib, forall_and_distrib]\n\ntheorem eq_zero_of_subset_zero {s : multiset \u03b1} (h : s \u2286 0) : s = 0 :=\neq_zero_of_forall_not_mem h\n\ntheorem subset_zero {s : multiset \u03b1} : s \u2286 0 \u2194 s = 0 :=\n\u27e8eq_zero_of_subset_zero, \u03bb xeq, xeq.symm \u25b8 subset.refl 0\u27e9\n\nend subset\n\n/- multiset order -/\n\n/-- `s \u2264 t` means that `s` is a sublist of `t` (up to permutation).\n  Equivalently, `s \u2264 t` means that `count a s \u2264 count a t` for all `a`. -/\nprotected def le (s t : multiset \u03b1) : Prop :=\nquotient.lift_on\u2082 s t (<+~) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  propext (p\u2082.subperm_left.trans p\u2081.subperm_right)\n\ninstance : partial_order (multiset \u03b1) :=\n{ le          := multiset.le,\n  le_refl     := by rintros \u27e8l\u27e9; exact subperm.refl _,\n  le_trans    := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 \u27e8l\u2083\u27e9; exact @subperm.trans _ _ _ _,\n  le_antisymm := by rintros \u27e8l\u2081\u27e9 \u27e8l\u2082\u27e9 h\u2081 h\u2082; exact quot.sound (subperm.antisymm h\u2081 h\u2082) }\n\ntheorem subset_of_le {s t : multiset \u03b1} : s \u2264 t \u2192 s \u2286 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subset_of_subperm\n\ntheorem mem_of_le {s t : multiset \u03b1} {a : \u03b1} (h : s \u2264 t) : a \u2208 s \u2192 a \u2208 t :=\nmem_of_subset (subset_of_le h)\n\n@[simp] theorem coe_le {l\u2081 l\u2082 : list \u03b1} : (l\u2081 : multiset \u03b1) \u2264 l\u2082 \u2194 l\u2081 <+~ l\u2082 := iff.rfl\n\n@[elab_as_eliminator] theorem le_induction_on {C : multiset \u03b1 \u2192 multiset \u03b1 \u2192 Prop}\n  {s t : multiset \u03b1} (h : s \u2264 t)\n  (H : \u2200 {l\u2081 l\u2082 : list \u03b1}, l\u2081 <+ l\u2082 \u2192 C l\u2081 l\u2082) : C s t :=\nquotient.induction_on\u2082 s t (\u03bb l\u2081 l\u2082 \u27e8l, p, s\u27e9,\n  (show \u27e6l\u27e7 = \u27e6l\u2081\u27e7, from quot.sound p) \u25b8 H s) h\n\ntheorem zero_le (s : multiset \u03b1) : 0 \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ nil_sublist l\n\ntheorem le_zero {s : multiset \u03b1} : s \u2264 0 \u2194 s = 0 :=\n\u27e8\u03bb h, le_antisymm h (zero_le _), le_of_eq\u27e9\n\ntheorem lt_cons_self (s : multiset \u03b1) (a : \u03b1) : s < a :: s :=\nquot.induction_on s $ \u03bb l,\nsuffices l <+~ a :: l \u2227 (\u00acl ~ a :: l),\n  by simpa [lt_iff_le_and_ne],\n\u27e8subperm_of_sublist (sublist_cons _ _),\n \u03bb p, ne_of_lt (lt_succ_self (length l)) (perm_length p)\u27e9\n\n\ntheorem le_cons_self (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s :=\nle_of_lt $ lt_cons_self _ _\n\ntheorem cons_le_cons_iff (a : \u03b1) {s t : multiset \u03b1} : a :: s \u2264 a :: t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_cons a\n\ntheorem cons_le_cons (a : \u03b1) {s t : multiset \u03b1} : s \u2264 t \u2192 a :: s \u2264 a :: t :=\n(cons_le_cons_iff a).2\n\ntheorem le_cons_of_not_mem {a : \u03b1} {s t : multiset \u03b1} (m : a \u2209 s) : s \u2264 a :: t \u2194 s \u2264 t :=\nbegin\n  refine \u27e8_, \u03bb h, le_trans h $ le_cons_self _ _\u27e9,\n  suffices : \u2200 {t'} (_ : s \u2264 t') (_ : a \u2208 t'), a :: s \u2264 t',\n  { exact \u03bb h, (cons_le_cons_iff a).1 (this h (mem_cons_self _ _)) },\n  introv h, revert m, refine le_induction_on h _,\n  introv s m\u2081 m\u2082,\n  rcases mem_split m\u2082 with \u27e8r\u2081, r\u2082, rfl\u27e9,\n  exact perm_middle.subperm_left.2 ((subperm_cons _).2 $ subperm_of_sublist $\n    (sublist_or_mem_of_sublist s).resolve_right m\u2081)\nend\n\n/- cardinality -/\n\n/-- The cardinality of a multiset is the sum of the multiplicities\n  of all its elements, or simply the length of the underlying list. -/\ndef card (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s length $ \u03bb l\u2081 l\u2082, perm_length\n\n@[simp] theorem coe_card (l : list \u03b1) : card (l : multiset \u03b1) = length l := rfl\n\n@[simp] theorem card_zero : @card \u03b1 0 = 0 := rfl\n\n@[simp] theorem card_cons (a : \u03b1) (s : multiset \u03b1) : card (a :: s) = card s + 1 :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem card_singleton (a : \u03b1) : card (a::0) = 1 := by simp\n\ntheorem card_le_of_le {s t : multiset \u03b1} (h : s \u2264 t) : card s \u2264 card t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, length_le_of_sublist\n\ntheorem eq_of_le_of_card_le {s t : multiset \u03b1} (h : s \u2264 t) : card t \u2264 card s \u2192 s = t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 s h\u2082, congr_arg coe $ eq_of_sublist_of_length_le s h\u2082\n\ntheorem card_lt_of_lt {s t : multiset \u03b1} (h : s < t) : card s < card t :=\nlt_of_not_ge $ \u03bb h\u2082, ne_of_lt h $ eq_of_le_of_card_le (le_of_lt h) h\u2082\n\ntheorem lt_iff_cons_le {s t : multiset \u03b1} : s < t \u2194 \u2203 a, a :: s \u2264 t :=\n\u27e8quotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\n  subperm.exists_of_length_lt (le_of_lt h) (card_lt_of_lt h),\n\u03bb \u27e8a, h\u27e9, lt_of_lt_of_le (lt_cons_self _ _) h\u27e9\n\n@[simp] theorem card_eq_zero {s : multiset \u03b1} : card s = 0 \u2194 s = 0 :=\n\u27e8\u03bb h, (eq_of_le_of_card_le (zero_le _) (le_of_eq h)).symm, \u03bb e, by simp [e]\u27e9\n\ntheorem card_pos {s : multiset \u03b1} : 0 < card s \u2194 s \u2260 0 :=\npos_iff_ne_zero.trans $ not_congr card_eq_zero\n\ntheorem card_pos_iff_exists_mem {s : multiset \u03b1} : 0 < card s \u2194 \u2203 a, a \u2208 s :=\nquot.induction_on s $ \u03bb l, length_pos_iff_exists_mem\n\n@[elab_as_eliminator] def strong_induction_on {p : multiset \u03b1 \u2192 Sort*} :\n  \u2200 (s : multiset \u03b1), (\u2200 s, (\u2200t < s, p t) \u2192 p s) \u2192 p s\n| s := \u03bb ih, ih s $ \u03bb t h,\n  have card t < card s, from card_lt_of_lt h,\n  strong_induction_on t ih\nusing_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf card\u27e9]}\n\ntheorem strong_induction_eq {p : multiset \u03b1 \u2192 Sort*}\n  (s : multiset \u03b1) (H) : @strong_induction_on _ p s H =\n    H s (\u03bb t h, @strong_induction_on _ p t H) :=\nby rw [strong_induction_on]\n\n@[elab_as_eliminator] lemma case_strong_induction_on {p : multiset \u03b1 \u2192 Prop}\n  (s : multiset \u03b1) (h\u2080 : p 0) (h\u2081 : \u2200 a s, (\u2200t \u2264 s, p t) \u2192 p (a :: s)) : p s :=\nmultiset.strong_induction_on s $ assume s,\nmultiset.induction_on s (\u03bb _, h\u2080) $ \u03bb a s _ ih, h\u2081 _ _ $\n\u03bb t h, ih _ $ lt_of_le_of_lt h $ lt_cons_self _ _\n\n/- singleton -/\n@[simp] theorem singleton_eq_singleton (a : \u03b1) : singleton a = a::0 := rfl\n\n@[simp] theorem mem_singleton {a b : \u03b1} : b \u2208 a::0 \u2194 b = a := by simp\n\ntheorem mem_singleton_self (a : \u03b1) : a \u2208 (a::0 : multiset \u03b1) := mem_cons_self _ _\n\ntheorem singleton_inj {a b : \u03b1} : a::0 = b::0 \u2194 a = b := cons_inj_left _\n\n@[simp] theorem singleton_ne_zero (a : \u03b1) : a::0 \u2260 0 :=\nne_of_gt (lt_cons_self _ _)\n\n@[simp] theorem singleton_le {a : \u03b1} {s : multiset \u03b1} : a::0 \u2264 s \u2194 a \u2208 s :=\n\u27e8\u03bb h, mem_of_le h (mem_singleton_self _),\n \u03bb h, let \u27e8t, e\u27e9 := exists_cons_of_mem h in e.symm \u25b8 cons_le_cons _ (zero_le _)\u27e9\n\ntheorem card_eq_one {s : multiset \u03b1} : card s = 1 \u2194 \u2203 a, s = a::0 :=\n\u27e8quot.induction_on s $ \u03bb l h,\n  (list.length_eq_one.1 h).imp $ \u03bb a, congr_arg coe,\n \u03bb \u27e8a, e\u27e9, e.symm \u25b8 rfl\u27e9\n\n/- add -/\n\n/-- The sum of two multisets is the lift of the list append operation.\n  This adds the multiplicities of each element,\n  i.e. `count a (s + t) = count a s + count a t`. -/\nprotected def add (s\u2081 s\u2082 : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s\u2081 s\u2082 (\u03bb l\u2081 l\u2082, ((l\u2081 ++ l\u2082 : list \u03b1) : multiset \u03b1)) $\n  \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082, quot.sound $ perm_app p\u2081 p\u2082\n\ninstance : has_add (multiset \u03b1) := \u27e8multiset.add\u27e9\n\n@[simp] theorem coe_add (s t : list \u03b1) : (s + t : multiset \u03b1) = (s ++ t : list \u03b1) := rfl\n\nprotected theorem add_comm (s t : multiset \u03b1) : s + t = t + s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quot.sound perm_app_comm\n\nprotected theorem zero_add (s : multiset \u03b1) : 0 + s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\ntheorem singleton_add (a : \u03b1) (s : multiset \u03b1) : \u2191[a] + s = a::s := rfl\n\nprotected theorem add_le_add_left (s) {t u : multiset \u03b1} : s + t \u2264 s + u \u2194 t \u2264 u :=\nquotient.induction_on\u2083 s t u $ \u03bb l\u2081 l\u2082 l\u2083, subperm_app_left _\n\nprotected theorem add_left_cancel (s) {t u : multiset \u03b1} (h : s + t = s + u) : t = u :=\nle_antisymm ((multiset.add_le_add_left _).1 (le_of_eq h))\n  ((multiset.add_le_add_left _).1 (le_of_eq h.symm))\n\ninstance : ordered_cancel_comm_monoid (multiset \u03b1) :=\n{ zero                  := 0,\n  add                   := (+),\n  add_comm              := multiset.add_comm,\n  add_assoc             := \u03bb s\u2081 s\u2082 s\u2083, quotient.induction_on\u2083 s\u2081 s\u2082 s\u2083 $ \u03bb l\u2081 l\u2082 l\u2083,\n    congr_arg coe $ append_assoc l\u2081 l\u2082 l\u2083,\n  zero_add              := multiset.zero_add,\n  add_zero              := \u03bb s, by rw [multiset.add_comm, multiset.zero_add],\n  add_left_cancel       := multiset.add_left_cancel,\n  add_right_cancel      := \u03bb s\u2081 s\u2082 s\u2083 h, multiset.add_left_cancel s\u2082 $\n    by simpa [multiset.add_comm] using h,\n  add_le_add_left       := \u03bb s\u2081 s\u2082 h s\u2083, (multiset.add_le_add_left _).2 h,\n  le_of_add_le_add_left := \u03bb s\u2081 s\u2082 s\u2083, (multiset.add_le_add_left _).1,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem cons_add (a : \u03b1) (s t : multiset \u03b1) : a :: s + t = a :: (s + t) :=\nby rw [\u2190 singleton_add, \u2190 singleton_add, add_assoc]\n\n@[simp] theorem add_cons (a : \u03b1) (s t : multiset \u03b1) : s + a :: t = a :: (s + t) :=\nby rw [add_comm, cons_add, add_comm]\n\ntheorem le_add_right (s t : multiset \u03b1) : s \u2264 s + t :=\nby simpa using add_le_add_left (zero_le t) s\n\ntheorem le_add_left (s t : multiset \u03b1) : s \u2264 t + s :=\nby simpa using add_le_add_right (zero_le t) s\n\n@[simp] theorem card_add (s t : multiset \u03b1) : card (s + t) = card s + card t :=\nquotient.induction_on\u2082 s t length_append\n\n@[simp] theorem mem_add {a : \u03b1} {s t : multiset \u03b1} : a \u2208 s + t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, mem_append\n\ntheorem le_iff_exists_add {s t : multiset \u03b1} : s \u2264 t \u2194 \u2203 u, t = s + u :=\n\u27e8\u03bb h, le_induction_on h $ \u03bb l\u2081 l\u2082 s,\n  let \u27e8l, p\u27e9 := exists_perm_append_of_sublist s in \u27e8l, quot.sound p\u27e9,\n\u03bb\u27e8u, e\u27e9, e.symm \u25b8 le_add_right s u\u27e9\n\ninstance : canonically_ordered_monoid (multiset \u03b1) :=\n{ lt_of_add_lt_add_left := @lt_of_add_lt_add_left _ _,\n  le_iff_exists_add     := @le_iff_exists_add _,\n  ..multiset.ordered_cancel_comm_monoid }\n\n/- repeat -/\n\n/-- `repeat a n` is the multiset containing only `a` with multiplicity `n`. -/\ndef repeat (a : \u03b1) (n : \u2115) : multiset \u03b1 := repeat a n\n\n@[simp] lemma repeat_zero (a : \u03b1) : repeat a 0 = 0 := rfl\n\n@[simp] lemma repeat_succ (a : \u03b1) (n) : repeat a (n+1) = a :: repeat a n := by simp [repeat]\n\n@[simp] lemma repeat_one (a : \u03b1) : repeat a 1 = a :: 0 := by simp\n\n@[simp] lemma card_repeat : \u2200 (a : \u03b1) n, card (repeat a n) = n := length_repeat\n\ntheorem eq_of_mem_repeat {a b : \u03b1} {n} : b \u2208 repeat a n \u2192 b = a := eq_of_mem_repeat\n\ntheorem eq_repeat' {a : \u03b1} {s : multiset \u03b1} : s = repeat a s.card \u2194 \u2200 b \u2208 s, b = a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  (perm_repeat.1 $ (quotient.exact h).symm).symm, congr_arg coe\u27e9 eq_repeat'\n\ntheorem eq_repeat_of_mem {a : \u03b1} {s : multiset \u03b1} : (\u2200 b \u2208 s, b = a) \u2192 s = repeat a s.card :=\neq_repeat'.2\n\ntheorem eq_repeat {a : \u03b1} {n} {s : multiset \u03b1} : s = repeat a n \u2194 card s = n \u2227 \u2200 b \u2208 s, b = a :=\n\u27e8\u03bb h, h.symm \u25b8 \u27e8card_repeat _ _, \u03bb b, eq_of_mem_repeat\u27e9,\n \u03bb \u27e8e, al\u27e9, e \u25b8 eq_repeat_of_mem al\u27e9\n\ntheorem repeat_subset_singleton : \u2200 (a : \u03b1) n, repeat a n \u2286 a::0 := repeat_subset_singleton\n\ntheorem repeat_le_coe {a : \u03b1} {n} {l : list \u03b1} : repeat a n \u2264 l \u2194 list.repeat a n <+ l :=\n\u27e8\u03bb \u27e8l', p, s\u27e9, (perm_repeat.1 p.symm).symm \u25b8 s, subperm_of_sublist\u27e9\n\n/- range -/\n\n/-- `range n` is the multiset lifted from the list `range n`,\n  that is, the set `{0, 1, ..., n-1}`. -/\ndef range (n : \u2115) : multiset \u2115 := range n\n\n@[simp] theorem range_zero (n : \u2115) : range 0 = 0 := rfl\n\n@[simp] theorem range_succ (n : \u2115) : range (succ n) = n :: range n :=\nby rw [range, range_concat, \u2190 coe_add, add_comm]; refl\n\n@[simp] theorem card_range (n : \u2115) : card (range n) = n := length_range _\n\ntheorem range_subset {m n : \u2115} : range m \u2286 range n \u2194 m \u2264 n := range_subset\n\n@[simp] theorem mem_range {m n : \u2115} : m \u2208 range n \u2194 m < n := mem_range\n\n@[simp] theorem not_mem_range_self {n : \u2115} : n \u2209 range n := not_mem_range_self\n\n\n/- erase -/\nsection erase\nvariables [decidable_eq \u03b1] {s t : multiset \u03b1} {a b : \u03b1}\n\n/-- `erase s a` is the multiset that subtracts 1 from the\n  multiplicity of `a`. -/\ndef erase (s : multiset \u03b1) (a : \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase a : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (erase_perm_erase a p))\n\n@[simp] theorem coe_erase (l : list \u03b1) (a : \u03b1) :\n  erase (l : multiset \u03b1) a = l.erase a := rfl\n\n@[simp] theorem erase_zero (a : \u03b1) : (0 : multiset \u03b1).erase a = 0 := rfl\n\n@[simp] theorem erase_cons_head (a : \u03b1) (s : multiset \u03b1) : (a :: s).erase a = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_head a l\n\n@[simp] theorem erase_cons_tail {a b : \u03b1} (s : multiset \u03b1) (h : b \u2260 a) : (b::s).erase a = b :: s.erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ erase_cons_tail l h\n\n@[simp] theorem erase_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 s.erase a = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ erase_of_not_mem h\n\n@[simp] theorem cons_erase {s : multiset \u03b1} {a : \u03b1} : a \u2208 s \u2192 a :: s.erase a = s :=\nquot.induction_on s $ \u03bb l h, quot.sound (perm_erase h).symm\n\ntheorem le_cons_erase (s : multiset \u03b1) (a : \u03b1) : s \u2264 a :: s.erase a :=\nif h : a \u2208 s then le_of_eq (cons_erase h).symm\nelse by rw erase_of_not_mem h; apply le_cons_self\n\n@[simp] theorem card_erase_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 card (s.erase a) = pred (card s) :=\nquot.induction_on s $ \u03bb l, length_erase_of_mem\n\ntheorem erase_add_left_pos {a : \u03b1} {s : multiset \u03b1} (t) : a \u2208 s \u2192 (s + t).erase a = s.erase a + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_left l\u2082 h\n\ntheorem erase_add_right_pos {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2208 t) : (s + t).erase a = s + t.erase a :=\nby rw [add_comm, erase_add_left_pos s h, add_comm]\n\ntheorem erase_add_right_neg {a : \u03b1} {s : multiset \u03b1} (t) : a \u2209 s \u2192 (s + t).erase a = s + t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h, congr_arg coe $ erase_append_right l\u2082 h\n\ntheorem erase_add_left_neg {a : \u03b1} (s) {t : multiset \u03b1} (h : a \u2209 t) : (s + t).erase a = s.erase a + t :=\nby rw [add_comm, erase_add_right_neg s h, add_comm]\n\ntheorem erase_le (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist (erase_sublist a l)\n\n@[simp] theorem erase_lt {a : \u03b1} {s : multiset \u03b1} : s.erase a < s \u2194 a \u2208 s :=\n\u27e8\u03bb h, not_imp_comm.1 erase_of_not_mem (ne_of_lt h),\n \u03bb h, by simpa [h] using lt_cons_self (s.erase a) a\u27e9\n\ntheorem erase_subset (a : \u03b1) (s : multiset \u03b1) : s.erase a \u2286 s :=\nsubset_of_le (erase_le a s)\n\ntheorem mem_erase_of_ne {a b : \u03b1} {s : multiset \u03b1} (ab : a \u2260 b) : a \u2208 s.erase b \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, list.mem_erase_of_ne ab\n\ntheorem mem_of_mem_erase {a b : \u03b1} {s : multiset \u03b1} : a \u2208 s.erase b \u2192 a \u2208 s :=\nmem_of_subset (erase_subset _ _)\n\ntheorem erase_comm (s : multiset \u03b1) (a b : \u03b1) : (s.erase a).erase b = (s.erase b).erase a :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ l.erase_comm a b\n\ntheorem erase_le_erase {s t : multiset \u03b1} (a : \u03b1) (h : s \u2264 t) : s.erase a \u2264 t.erase a :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist (erase_sublist_erase _ h)\n\ntheorem erase_le_iff_le_cons {s t : multiset \u03b1} {a : \u03b1} : s.erase a \u2264 t \u2194 s \u2264 a :: t :=\n\u27e8\u03bb h, le_trans (le_cons_erase _ _) (cons_le_cons _ h),\n \u03bb h, if m : a \u2208 s\n  then by rw \u2190 cons_erase m at h; exact (cons_le_cons_iff _).1 h\n  else le_trans (erase_le _ _) ((le_cons_of_not_mem m).1 h)\u27e9\n\nend erase\n\n@[simp] theorem coe_reverse (l : list \u03b1) : (reverse l : multiset \u03b1) = l :=\nquot.sound $ reverse_perm _\n\n/- map -/\n\n/-- `map f s` is the lift of the list `map` operation. The multiplicity\n  of `b` in `map f s` is the number of `a \u2208 s` (counting multiplicity)\n  such that `f a = b`. -/\ndef map (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l : list \u03b1, (l.map f : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 p, quot.sound (perm_map f p))\n\n@[simp] theorem coe_map (f : \u03b1 \u2192 \u03b2) (l : list \u03b1) : map f \u2191l = l.map f := rfl\n\n@[simp] theorem map_zero (f : \u03b1 \u2192 \u03b2) : map f 0 = 0 := rfl\n\n@[simp] theorem map_cons (f : \u03b1 \u2192 \u03b2) (a s) : map f (a::s) = f a :: map f s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] lemma map_singleton (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : ({a} : multiset \u03b1).map f = {f a} := rfl\n\n@[simp] theorem map_add (f : \u03b1 \u2192 \u03b2) (s t) : map f (s + t) = map f s + map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ map_append _ _ _\n\n@[simp] theorem mem_map {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : multiset \u03b1} :\n  b \u2208 map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = b :=\nquot.induction_on s $ \u03bb l, mem_map\n\n@[simp] theorem card_map (f : \u03b1 \u2192 \u03b2) (s) : card (map f s) = card s :=\nquot.induction_on s $ \u03bb l, length_map _ _\n\ntheorem mem_map_of_mem (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : f a \u2208 map f s :=\nmem_map.2 \u27e8_, h, rfl\u27e9\n\n@[simp] theorem mem_map_of_inj {f : \u03b1 \u2192 \u03b2} (H : function.injective f) {a : \u03b1} {s : multiset \u03b1} :\n  f a \u2208 map f s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_map_of_inj H\n\n@[simp] theorem map_map (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : map g (map f s) = map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ list.map_map _ _ _\n\n@[simp] theorem map_id (s : multiset \u03b1) : map id s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_id _\n\n@[simp] lemma map_id' (s : multiset \u03b1) : map (\u03bbx, x) s = s := map_id s\n\n@[simp] theorem map_const (s : multiset \u03b1) (b : \u03b2) : map (function.const \u03b1 b) s = repeat b s.card :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_const _ _\n\n@[congr] theorem map_congr {f g : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200 x \u2208 s, f x = g x) \u2192 map f s = map g s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_congr H\n\nlemma map_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} {f' : \u03b1 \u2192 \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : map f m == map f' m :=\nbegin subst h, simp at hf, simp [map_congr hf] end\n\ntheorem eq_of_mem_map_const {b\u2081 b\u2082 : \u03b2} {l : list \u03b1} (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) l) : b\u2081 = b\u2082 :=\neq_of_mem_repeat $ by rwa map_const at h\n\n@[simp] theorem map_le_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (h : s \u2264 t) : map f s \u2264 map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ map_sublist_map f h\n\n@[simp] theorem map_subset_map {f : \u03b1 \u2192 \u03b2} {s t : multiset \u03b1} (H : s \u2286 t) : map f s \u2286 map f t :=\n\u03bb b m, let \u27e8a, h, e\u27e9 := mem_map.1 m in mem_map.2 \u27e8a, H h, e\u27e9\n\n/- fold -/\n\n/-- `foldl f H b s` is the lift of the list operation `foldl f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is right-commutative,\n  that is, `f (f b a\u2081) a\u2082 = f (f b a\u2082) a\u2081`. -/\ndef foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldl f b l)\n  (\u03bb l\u2081 l\u2082 p, foldl_eq_of_perm H p b)\n\n@[simp] theorem foldl_zero (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b) : foldl f H b 0 = b := rfl\n\n@[simp] theorem foldl_cons (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b a s) : foldl f H b (a :: s) = foldl f H (f b a) s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldl_add (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H b s t) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldl_append _ _ _ _\n\n/-- `foldr f H b s` is the lift of the list operation `foldr f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is left-commutative,\n  that is, `f a\u2081 (f a\u2082 b) = f a\u2082 (f a\u2081 b)`. -/\ndef foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) : \u03b2 :=\nquot.lift_on s (\u03bb l, foldr f b l)\n  (\u03bb l\u2081 l\u2082 p, foldr_eq_of_perm H p b)\n\n@[simp] theorem foldr_zero (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b) : foldr f H b 0 = b := rfl\n\n@[simp] theorem foldr_cons (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b a s) : foldr f H b (a :: s) = f a (foldr f H b s) :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem foldr_add (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H b s t) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, foldr_append _ _ _ _\n\n@[simp] theorem coe_foldr (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldr f b := rfl\n\n@[simp] theorem coe_foldl (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldl f H b l = l.foldl f b := rfl\n\ntheorem coe_foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (l : list \u03b1) :\n  foldr f H b l = l.foldl (\u03bb x y, f y x) b :=\n(congr_arg (foldr f H b) (coe_reverse l)).symm.trans $ foldr_reverse _ _ _\n\ntheorem foldr_swap (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldr f H b s = foldl (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\nquot.induction_on s $ \u03bb l, coe_foldr_swap _ _ _ _\n\ntheorem foldl_swap (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2) (s : multiset \u03b1) :\n  foldl f H b s = foldr (\u03bb x y, f y x) (\u03bb x y z, (H _ _ _).symm) b s :=\n(foldr_swap _ _ _ _).symm\n\n/-- Product of a multiset given a commutative monoid structure on `\u03b1`.\n  `prod {a, b, c} = a * b * c` -/\ndef prod [comm_monoid \u03b1] : multiset \u03b1 \u2192 \u03b1 :=\nfoldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1\nattribute [to_additive multiset.sum._proof_1] prod._proof_1\nattribute [to_additive multiset.sum] prod\n\n@[to_additive multiset.sum_eq_foldr]\ntheorem prod_eq_foldr [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldr (*) (\u03bb x y z, by simp [mul_left_comm]) 1 s := rfl\n\n@[to_additive multiset.sum_eq_foldl]\ntheorem prod_eq_foldl [comm_monoid \u03b1] (s : multiset \u03b1) :\n  prod s = foldl (*) (\u03bb x y z, by simp [mul_right_comm]) 1 s :=\n(foldr_swap _ _ _ _).trans (by simp [mul_comm])\n\n@[simp, to_additive multiset.coe_sum]\ntheorem coe_prod [comm_monoid \u03b1] (l : list \u03b1) : prod \u2191l = l.prod :=\nprod_eq_foldl _\n\n@[simp, to_additive multiset.sum_zero]\ntheorem prod_zero [comm_monoid \u03b1] : @prod \u03b1 _ 0 = 1 := rfl\n\n@[simp, to_additive multiset.sum_cons]\ntheorem prod_cons [comm_monoid \u03b1] (a : \u03b1) (s) : prod (a :: s) = a * prod s :=\nfoldr_cons _ _ _ _ _\n\n@[to_additive multiset.sum_singleton]\ntheorem prod_singleton [comm_monoid \u03b1] (a : \u03b1) : prod (a :: 0) = a := by simp\n\n@[simp, to_additive multiset.sum_add]\ntheorem prod_add [comm_monoid \u03b1] (s t : multiset \u03b1) : prod (s + t) = prod s * prod t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, by simp\n\n@[simp] theorem prod_repeat [comm_monoid \u03b1] (a : \u03b1) (n : \u2115) : prod (multiset.repeat a n) = a ^ n :=\nby simp [repeat, list.prod_repeat]\n@[simp] theorem sum_repeat [add_comm_monoid \u03b1] : \u2200 (a : \u03b1) (n : \u2115), sum (multiset.repeat a n) = n \u2022 a :=\n@prod_repeat (multiplicative \u03b1) _\nattribute [to_additive multiset.sum_repeat] prod_repeat\n\n@[simp] lemma prod_map_one [comm_monoid \u03b3] {m : multiset \u03b1} :\n  prod (m.map (\u03bba, (1 : \u03b3))) = (1 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\n@[simp] lemma sum_map_zero [add_comm_monoid \u03b3] {m : multiset \u03b1} :\n  sum (m.map (\u03bba, (0 : \u03b3))) = (0 : \u03b3) :=\nmultiset.induction_on m (by simp) (by simp)\nattribute [to_additive multiset.sum_map_zero] prod_map_one\n\n@[simp, to_additive multiset.sum_map_add]\nlemma prod_map_mul [comm_monoid \u03b3] {m : multiset \u03b1} {f g : \u03b1 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, f a * g a) = prod (m.map f) * prod (m.map g) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih]; cc)\n\nlemma prod_map_prod_map [comm_monoid \u03b3] (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  prod (m.map $ \u03bba, prod $ n.map $ \u03bbb, f a b) = prod (n.map $ \u03bbb, prod $ m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (assume a m ih, by simp [ih])\n\nlemma sum_map_sum_map [add_comm_monoid \u03b3] : \u2200 (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3},\n  sum (m.map $ \u03bba, sum $ n.map $ \u03bbb, f a b) = sum (n.map $ \u03bbb, sum $ m.map $ \u03bba, f a b) :=\n@prod_map_prod_map _ _ (multiplicative \u03b3) _\nattribute [to_additive multiset.sum_map_sum_map] prod_map_prod_map\n\nlemma sum_map_mul_left [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, b * f a)) = b * sum (s.map f) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, mul_add])\n\nlemma sum_map_mul_right [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n  sum (s.map (\u03bba, f a * b)) = sum (s.map f) * b :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, add_mul])\n\n/- join -/\n\n/-- `join S`, where `S` is a multiset of multisets, is the lift of the list join\n  operation, that is, the union of all the sets.\n\n     join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2} -/\ndef join : multiset (multiset \u03b1) \u2192 multiset \u03b1 := sum\n\ntheorem coe_join : \u2200 L : list (list \u03b1),\n  join (L.map (@coe _ (multiset \u03b1) _) : multiset (multiset \u03b1)) = L.join\n| []       := rfl\n| (l :: L) := congr_arg (\u03bb s : multiset \u03b1, \u2191l + s) (coe_join L)\n\n@[simp] theorem join_zero : @join \u03b1 0 = 0 := rfl\n\n@[simp] theorem join_cons (s S) : @join \u03b1 (s :: S) = s + join S :=\nsum_cons _ _\n\n@[simp] theorem join_add (S T) : @join \u03b1 (S + T) = join S + join T :=\nsum_add _ _\n\n@[simp] theorem mem_join {a S} : a \u2208 @join \u03b1 S \u2194 \u2203 s \u2208 S, a \u2208 s :=\nmultiset.induction_on S (by simp) $\n  by simp [or_and_distrib_right, exists_or_distrib] {contextual := tt}\n\n@[simp] theorem card_join (S) : card (@join \u03b1 S) = sum (map card S) :=\nmultiset.induction_on S (by simp) (by simp)\n\n/- bind -/\n\n/-- `bind s f` is the monad bind operation, defined as `join (map f s)`.\n  It is the union of `f a` as `a` ranges over `s`. -/\ndef bind (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : multiset \u03b2 :=\njoin (map f s)\n\n@[simp] theorem coe_bind (l : list \u03b1) (f : \u03b1 \u2192 list \u03b2) :\n  @bind \u03b1 \u03b2 l (\u03bb a, f a) = l.bind f :=\nby rw [list.bind, \u2190 coe_join, list.map_map]; refl\n\n@[simp] theorem zero_bind (f : \u03b1 \u2192 multiset \u03b2) : bind 0 f = 0 := rfl\n\n@[simp] theorem cons_bind (a s) (f : \u03b1 \u2192 multiset \u03b2) : bind (a::s) f = f a + bind s f :=\nby simp [bind]\n\n@[simp] theorem add_bind (s t) (f : \u03b1 \u2192 multiset \u03b2) : bind (s + t) f = bind s f + bind t f :=\nby simp [bind]\n\n@[simp] theorem bind_zero (s : multiset \u03b1) : bind s (\u03bba, 0 : \u03b1 \u2192 multiset \u03b2) = 0 :=\nby simp [bind, -map_const, join]\n\n@[simp] theorem bind_add (s : multiset \u03b1) (f g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a + g a) = bind s f + bind s g :=\nby simp [bind, join]\n\n@[simp] theorem bind_cons (s : multiset \u03b1) (f : \u03b1 \u2192 \u03b2) (g : \u03b1 \u2192 multiset \u03b2) :\n  bind s (\u03bba, f a :: g a) = map f s + bind s g :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\n@[simp] theorem mem_bind {b s} {f : \u03b1 \u2192 multiset \u03b2} : b \u2208 bind s f \u2194 \u2203 a \u2208 s, b \u2208 f a :=\nby simp [bind]; simp [-exists_and_distrib_right, exists_and_distrib_right.symm];\n   rw exists_swap; simp [and_assoc]\n\n@[simp] theorem card_bind (s) (f : \u03b1 \u2192 multiset \u03b2) : card (bind s f) = sum (map (card \u2218 f) s) :=\nby simp [bind]\n\nlemma bind_congr {f g : \u03b1 \u2192 multiset \u03b2} {m : multiset \u03b1} : (\u2200a\u2208m, f a = g a) \u2192 bind m f = bind m g :=\nby simp [bind] {contextual := tt}\n\nlemma bind_hcongr {\u03b2' : Type*} {m : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {f' : \u03b1 \u2192 multiset \u03b2'}\n  (h : \u03b2 = \u03b2') (hf : \u2200a\u2208m, f a == f' a) : bind m f == bind m f' :=\nbegin subst h, simp at hf, simp [bind_congr hf] end\n\nlemma map_bind (m : multiset \u03b1) (n : \u03b1 \u2192 multiset \u03b2) (f : \u03b2 \u2192 \u03b3) :\n  map f (bind m n) = bind m (\u03bba, map f (n a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map (m : multiset \u03b1) (n : \u03b2 \u2192 multiset \u03b3) (f : \u03b1 \u2192 \u03b2) :\n  bind (map f m) n = bind m (\u03bba, n (f a)) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_assoc {s : multiset \u03b1} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b2 \u2192 multiset \u03b3} :\n  (s.bind f).bind g = s.bind (\u03bba, (f a).bind g) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma bind_bind (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 multiset \u03b3} :\n  (bind m $ \u03bba, bind n $ \u03bbb, f a b) = (bind n $ \u03bbb, bind m $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\nlemma bind_map_comm (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n  (bind m $ \u03bba, n.map $ \u03bbb, f a b) = (bind n $ \u03bbb, m.map $ \u03bba, f a b) :=\nmultiset.induction_on m (by simp) (by simp {contextual := tt})\n\n@[simp, to_additive multiset.sum_bind]\nlemma prod_bind [comm_monoid \u03b2] (s : multiset \u03b1) (t : \u03b1 \u2192 multiset \u03b2) :\n  prod (bind s t) = prod (s.map $ \u03bba, prod (t a)) :=\nmultiset.induction_on s (by simp) (assume a s ih, by simp [ih, cons_bind])\n\n/- product -/\n\n/-- The multiplicity of `(a, b)` in `product s t` is\n  the product of the multiplicity of `a` in `s` and `b` in `t`. -/\ndef product (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u00d7 \u03b2) :=\ns.bind $ \u03bb a, t.map $ prod.mk a\n\n@[simp] theorem coe_product (l\u2081 : list \u03b1) (l\u2082 : list \u03b2) :\n  @product \u03b1 \u03b2 l\u2081 l\u2082 = l\u2081.product l\u2082 :=\nby rw [product, list.product, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_product (t) : @product \u03b1 \u03b2 0 t = 0 := rfl\n\n@[simp] theorem cons_product (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b2) :\n  product (a :: s) t = map (prod.mk a) t + product s t :=\nby simp [product]\n\n@[simp] theorem product_singleton (a : \u03b1) (b : \u03b2) : product (a::0) (b::0) = (a,b)::0 := rfl\n\n@[simp] theorem add_product (s t : multiset \u03b1) (u : multiset \u03b2) :\n  product (s + t) u = product s u + product t u :=\nby simp [product]\n\n@[simp] theorem product_add (s : multiset \u03b1) : \u2200 t u : multiset \u03b2,\n  product s (t + u) = product s t + product s u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_product, IH]; simp\n\n@[simp] theorem mem_product {s t} : \u2200 {p : \u03b1 \u00d7 \u03b2}, p \u2208 @product \u03b1 \u03b2 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t\n| (a, b) := by simp [product, and.left_comm]\n\n@[simp] theorem card_product (s : multiset \u03b1) (t : multiset \u03b2) : card (product s t) = card s * card t :=\nby simp [product, repeat, (\u2218), mul_comm]\n\n/- sigma -/\nsection\nvariable {\u03c3 : \u03b1 \u2192 Type*}\n\n/-- `sigma s t` is the dependent version of `product`. It is the sum of\n  `(a, b)` as `a` ranges over `s` and `b` ranges over `t a`. -/\nprotected def sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) : multiset (\u03a3 a, \u03c3 a) :=\ns.bind $ \u03bb a, (t a).map $ sigma.mk a\n\n@[simp] theorem coe_sigma (l\u2081 : list \u03b1) (l\u2082 : \u03a0 a, list (\u03c3 a)) :\n  @multiset.sigma \u03b1 \u03c3 l\u2081 (\u03bb a, l\u2082 a) = l\u2081.sigma l\u2082 :=\nby rw [multiset.sigma, list.sigma, \u2190 coe_bind]; simp\n\n@[simp] theorem zero_sigma (t) : @multiset.sigma \u03b1 \u03c3 0 t = 0 := rfl\n\n@[simp] theorem cons_sigma (a : \u03b1) (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  (a :: s).sigma t = map (sigma.mk a) (t a) + s.sigma t :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_singleton (a : \u03b1) (b : \u03b1 \u2192 \u03b2) :\n  (a::0).sigma (\u03bb a, b a::0) = \u27e8a, b a\u27e9::0 := rfl\n\n@[simp] theorem add_sigma (s t : multiset \u03b1) (u : \u03a0 a, multiset (\u03c3 a)) :\n  (s + t).sigma u = s.sigma u + t.sigma u :=\nby simp [multiset.sigma]\n\n@[simp] theorem sigma_add (s : multiset \u03b1) : \u2200 t u : \u03a0 a, multiset (\u03c3 a),\n  s.sigma (\u03bb a, t a + u a) = s.sigma t + s.sigma u :=\nmultiset.induction_on s (\u03bb t u, rfl) $ \u03bb a s IH t u,\n  by rw [cons_sigma, IH]; simp\n\n@[simp] theorem mem_sigma {s t} : \u2200 {p : \u03a3 a, \u03c3 a},\n  p \u2208 @multiset.sigma \u03b1 \u03c3 s t \u2194 p.1 \u2208 s \u2227 p.2 \u2208 t p.1\n| \u27e8a, b\u27e9 := by simp [multiset.sigma, and_assoc, and.left_comm]\n\n@[simp] theorem card_sigma (s : multiset \u03b1) (t : \u03a0 a, multiset (\u03c3 a)) :\n  card (s.sigma t) = sum (map (\u03bb a, card (t a)) s) :=\nby simp [multiset.sigma, (\u2218)]\n\nend\n\n/- map for partial functions -/\n\n/-- Lift of the list `pmap` operation. Map a partial function `f` over a multiset\n  `s` whose elements are all in the domain of `f`. -/\ndef pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (s : multiset \u03b1) : (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2 :=\nquot.rec_on s (\u03bb l H, \u2191(pmap f l H)) $ \u03bb l\u2081 l\u2082 (pp : l\u2081 ~ l\u2082),\nfunext $ \u03bb (H\u2082 : \u2200 a \u2208 l\u2082, p a),\nhave H\u2081 : \u2200 a \u2208 l\u2081, p a, from \u03bb a h, H\u2082 a ((mem_of_perm pp).1 h),\nhave \u2200 {s\u2082 e H}, @eq.rec (multiset \u03b1) l\u2081\n  (\u03bb s, (\u2200 a \u2208 s, p a) \u2192 multiset \u03b2) (\u03bb _, \u2191(pmap f l\u2081 H\u2081))\n  s\u2082 e H = \u2191(pmap f l\u2081 H\u2081), by intros s\u2082 e _; subst e,\nthis.trans $ quot.sound $ perm_pmap f pp\n\n@[simp] theorem coe_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (l : list \u03b1) (H : \u2200 a \u2208 l, p a) : pmap f l H = l.pmap f H := rfl\n\n@[simp] lemma pmap_zero {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (h : \u2200a\u2208(0:multiset \u03b1), p a) :\n  pmap f 0 h = 0 := rfl\n\n@[simp] lemma pmap_cons {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2) (a : \u03b1) (m : multiset \u03b1) :\n  \u2200(h : \u2200b\u2208a::m, p b), pmap f (a :: m) h =\n    f a (h a (mem_cons_self a m)) :: pmap f m (\u03bba ha, h a $ mem_cons_of_mem ha) :=\nquotient.induction_on m $ assume l h, rfl\n\n/-- \"Attach\" a proof that `a \u2208 s` to each element `a` in `s` to produce\n  a multiset on `{x // x \u2208 s}`. -/\ndef attach (s : multiset \u03b1) : multiset {x // x \u2208 s} := pmap subtype.mk s (\u03bb a, id)\n\n@[simp] theorem coe_attach (l : list \u03b1) :\n @eq (multiset {x // x \u2208 l}) (@attach \u03b1 l) l.attach := rfl\n\ntheorem pmap_eq_map (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  \u2200 H, @pmap _ _ p (\u03bb a _, f a) s H = map f s :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map p f l H\n\ntheorem pmap_congr {p q : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {g : \u03a0 a, q a \u2192 \u03b2}\n  (s : multiset \u03b1) {H\u2081 H\u2082} (h : \u2200 a h\u2081 h\u2082, f a h\u2081 = g a h\u2082) :\n  pmap f s H\u2081 = pmap g s H\u2082 :=\nquot.induction_on s (\u03bb l H\u2081 H\u2082, congr_arg coe $ pmap_congr l h) H\u2081 H\u2082\n\ntheorem map_pmap {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3) (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, map g (pmap f s H) = pmap (\u03bb a h, g (f a h)) s H :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ map_pmap g f l H\n\ntheorem pmap_eq_map_attach {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s) : \u2200 H, pmap f s H = s.attach.map (\u03bb x, f x.1 (H _ x.2)) :=\nquot.induction_on s $ \u03bb l H, congr_arg coe $ pmap_eq_map_attach f l H\n\ntheorem attach_map_val (s : multiset \u03b1) : s.attach.map subtype.val = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ attach_map_val l\n\n@[simp] theorem mem_attach (s : multiset \u03b1) : \u2200 x, x \u2208 s.attach :=\nquot.induction_on s $ \u03bb l, mem_attach _\n\n@[simp] theorem mem_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2}\n  {s H b} : b \u2208 pmap f s H \u2194 \u2203 a (h : a \u2208 s), f a (H a h) = b :=\nquot.induction_on s (\u03bb l H, mem_pmap) H\n\n@[simp] theorem card_pmap {p : \u03b1 \u2192 Prop} (f : \u03a0 a, p a \u2192 \u03b2)\n  (s H) : card (pmap f s H) = card s :=\nquot.induction_on s (\u03bb l H, length_pmap) H\n\n@[simp] theorem card_attach {m : multiset \u03b1} : card (attach m) = card m := card_pmap _ _ _\n\n@[simp] lemma attach_zero : (0 : multiset \u03b1).attach = 0 := rfl\n\nlemma attach_cons (a : \u03b1) (m : multiset \u03b1) :\n  (a :: m).attach = \u27e8a, mem_cons_self a m\u27e9 :: (m.attach.map $ \u03bbp, \u27e8p.1, mem_cons_of_mem p.2\u27e9) :=\nquotient.induction_on m $ assume l, congr_arg coe $ congr_arg (list.cons _) $\n  by rw [list.map_pmap]; exact list.pmap_congr _ (assume a' h\u2081 h\u2082, subtype.eq rfl)\n\nsection decidable_pi_exists\nvariables {m : multiset \u03b1}\n\nprotected def decidable_forall_multiset {p : \u03b1 \u2192 Prop} [hp : \u2200a, decidable (p a)] :\n  decidable (\u2200a\u2208m, p a) :=\nquotient.rec_on_subsingleton m (\u03bbl, decidable_of_iff (\u2200a\u2208l, p a) $ by simp)\n\ninstance decidable_dforall_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2200a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_forall_multiset {a // a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (assume h a ha, h \u27e8a, ha\u27e9 (mem_attach _ _)) (assume h \u27e8a, ha\u27e9 _, h _ _))\n\n/-- decidable equality for functions whose domain is bounded by multisets -/\ninstance decidable_eq_pi_multiset {\u03b2 : \u03b1 \u2192 Type*} [h : \u2200a, decidable_eq (\u03b2 a)] :\n  decidable_eq (\u03a0a\u2208m, \u03b2 a) :=\nassume f g, decidable_of_iff (\u2200a (h : a \u2208 m), f a h = g a h) (by simp [function.funext_iff])\n\ndef decidable_exists_multiset {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n  decidable (\u2203 x \u2208 m, p x) :=\nquotient.rec_on_subsingleton m list.decidable_exists_mem\n\ninstance decidable_dexists_multiset {p : \u03a0a\u2208m, Prop} [hp : \u2200a (h : a \u2208 m), decidable (p a h)] :\n  decidable (\u2203a (h : a \u2208 m), p a h) :=\ndecidable_of_decidable_of_iff\n  (@multiset.decidable_exists_multiset {a // a \u2208 m} m.attach (\u03bba, p a.1 a.2) _)\n  (iff.intro (\u03bb \u27e8\u27e8a, ha\u2081\u27e9, _, ha\u2082\u27e9, \u27e8a, ha\u2081, ha\u2082\u27e9)\n    (\u03bb \u27e8a, ha\u2081, ha\u2082\u27e9, \u27e8\u27e8a, ha\u2081\u27e9, mem_attach _ _, ha\u2082\u27e9))\n\nend decidable_pi_exists\n\n/- subtraction -/\nsection\nvariables [decidable_eq \u03b1] {s t u : multiset \u03b1} {a b : \u03b1}\n\n/-- `s - t` is the multiset such that\n  `count a (s - t) = count a s - count a t` for all `a`. -/\nprotected def sub (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.diff l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_diff_right w\u2081 p\u2082 \u25b8 perm_diff_left _ p\u2081\n\ninstance : has_sub (multiset \u03b1) := \u27e8multiset.sub\u27e9\n\n@[simp] theorem coe_sub (s t : list \u03b1) : (s - t : multiset \u03b1) = (s.diff t : list \u03b1) := rfl\n\ntheorem sub_eq_fold_erase (s t : multiset \u03b1) : s - t = foldl erase erase_comm s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nshow \u2191(l\u2081.diff l\u2082) = foldl erase erase_comm \u2191l\u2081 \u2191l\u2082,\nby rw diff_eq_foldl l\u2081 l\u2082; exact foldl_hom _ _ _ _ (\u03bb x y, rfl) _\n\n@[simp] theorem sub_zero (s : multiset \u03b1) : s - 0 = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem sub_cons (a : \u03b1) (s t : multiset \u03b1) : s - a::t = s.erase a - t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ diff_cons _ _ _\n\ntheorem add_sub_of_le (h : s \u2264 t) : s + (t - s) = t :=\nbegin\n  revert t,\n  refine multiset.induction_on s (by simp) (\u03bb a s IH t h, _),\n  have := cons_erase (mem_of_le h (mem_cons_self _ _)),\n  rw [cons_add, sub_cons, IH, this],\n  exact (cons_le_cons_iff a).1 (this.symm \u25b8 h)\nend\n\ntheorem sub_add' : s - (t + u) = s - t - u :=\nquotient.induction_on\u2083 s t u $\n\u03bb l\u2081 l\u2082 l\u2083, congr_arg coe $ diff_append _ _ _\n\ntheorem sub_add_cancel (h : t \u2264 s) : s - t + t = s :=\nby rw [add_comm, add_sub_of_le h]\n\ntheorem add_sub_cancel_left (s : multiset \u03b1) : \u2200 t, s + t - s = t :=\nmultiset.induction_on s (by simp)\n  (\u03bb a s IH t, by rw [cons_add, sub_cons, erase_cons_head, IH])\n\ntheorem add_sub_cancel (s t : multiset \u03b1) : s + t - t = s :=\nby rw [add_comm, add_sub_cancel_left]\n\ntheorem sub_le_sub_right (h : s \u2264 t) (u) : s - u \u2264 t - u :=\nby revert s t h; exact\nmultiset.induction_on u (by simp {contextual := tt})\n  (\u03bb a u IH s t h, by simp [IH, erase_le_erase a h])\n\ntheorem sub_le_sub_left (h : s \u2264 t) : \u2200 u, u - t \u2264 u - s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, begin\n  induction h with l\u2081 l\u2082 a s IH l\u2081 l\u2082 a s IH; intro u,\n  { refl },\n  { rw [\u2190 cons_coe, sub_cons],\n    exact le_trans (sub_le_sub_right (erase_le _ _) _) (IH u) },\n  { rw [\u2190 cons_coe, sub_cons, \u2190 cons_coe, sub_cons],\n    exact IH _ }\nend\n\ntheorem sub_le_iff_le_add : s - t \u2264 u \u2194 s \u2264 u + t :=\nby revert s; exact\nmultiset.induction_on t (by simp)\n  (\u03bb a t IH s, by simp [IH, erase_le_iff_le_cons])\n\ntheorem le_sub_add (s t : multiset \u03b1) : s \u2264 s - t + t :=\nsub_le_iff_le_add.1 (le_refl _)\n\ntheorem sub_le_self (s t : multiset \u03b1) : s - t \u2264 s :=\nsub_le_iff_le_add.2 (le_add_right _ _)\n\n@[simp] theorem card_sub {s t : multiset \u03b1} (h : t \u2264 s) : card (s - t) = card s - card t :=\n(nat.sub_eq_of_eq_add $ by rw [add_comm, \u2190 card_add, sub_add_cancel h]).symm\n\n/- union -/\n\n/-- `s \u222a t` is the lattice join operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u222a t` is the maximum\n  of the multiplicities in `s` and `t`. -/\ndef union (s t : multiset \u03b1) : multiset \u03b1 := s - t + t\n\ninstance : has_union (multiset \u03b1) := \u27e8union\u27e9\n\ntheorem union_def (s t : multiset \u03b1) : s \u222a t = s - t + t := rfl\n\ntheorem le_union_left (s t : multiset \u03b1) : s \u2264 s \u222a t := le_sub_add _ _\n\ntheorem le_union_right (s t : multiset \u03b1) : t \u2264 s \u222a t := le_add_left _ _\n\ntheorem eq_union_left : t \u2264 s \u2192 s \u222a t = s := sub_add_cancel\n\ntheorem union_le_union_right (h : s \u2264 t) (u) : s \u222a u \u2264 t \u222a u :=\nadd_le_add_right (sub_le_sub_right h _) u\n\ntheorem union_le (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u :=\nby rw \u2190 eq_union_left h\u2082; exact union_le_union_right h\u2081 t\n\n@[simp] theorem mem_union : a \u2208 s \u222a t \u2194 a \u2208 s \u2228 a \u2208 t :=\n\u27e8\u03bb h, (mem_add.1 h).imp_left (mem_of_le $ sub_le_self _ _),\n or.rec (mem_of_le $ le_union_left _ _) (mem_of_le $ le_union_right _ _)\u27e9\n\n@[simp] theorem map_union [decidable_eq \u03b2] {f : \u03b1 \u2192 \u03b2} (finj : function.injective f) {s t : multiset \u03b1} :\n  map f (s \u222a t) = map f s \u222a map f t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\ncongr_arg coe (by rw [list.map_append f, list.map_diff finj])\n\n/- inter -/\n\n/-- `s \u2229 t` is the lattice meet operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u2229 t` is the minimum\n  of the multiplicities in `s` and `t`. -/\ndef inter (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.bag_inter l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_bag_inter_right w\u2081 p\u2082 \u25b8 perm_bag_inter_left _ p\u2081\n\ninstance : has_inter (multiset \u03b1) := \u27e8inter\u27e9\n\n@[simp] theorem inter_zero (s : multiset \u03b1) : s \u2229 0 = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.bag_inter_nil\n\n@[simp] theorem zero_inter (s : multiset \u03b1) : 0 \u2229 s = 0 :=\nquot.induction_on s $ \u03bb l, congr_arg coe l.nil_bag_inter\n\n@[simp] theorem cons_inter_of_pos {a} (s : multiset \u03b1) {t} :\n  a \u2208 t \u2192 (a :: s) \u2229 t = a :: s \u2229 t.erase a :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_pos _ h\n\n@[simp] theorem cons_inter_of_neg {a} (s : multiset \u03b1) {t} :\n  a \u2209 t \u2192 (a :: s) \u2229 t = s \u2229 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 h,\ncongr_arg coe $ cons_bag_inter_of_neg _ h\n\ntheorem inter_le_left (s t : multiset \u03b1) : s \u2229 t \u2264 s :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ bag_inter_sublist_left _ _\n\ntheorem inter_le_right (s : multiset \u03b1) : \u2200 t, s \u2229 t \u2264 t :=\nmultiset.induction_on s (\u03bb t, (zero_inter t).symm \u25b8 zero_le _) $\n\u03bb a s IH t, if h : a \u2208 t\n  then by simpa [h] using cons_le_cons a (IH (t.erase a))\n  else by simp [h, IH]\n\ntheorem le_inter (h\u2081 : s \u2264 t) (h\u2082 : s \u2264 u) : s \u2264 t \u2229 u :=\nbegin\n  revert s u, refine multiset.induction_on t _ (\u03bb a t IH, _); intros,\n  { simp [h\u2081] },\n  by_cases a \u2208 u,\n  { rw [cons_inter_of_pos _ h, \u2190 erase_le_iff_le_cons],\n    exact IH (erase_le_iff_le_cons.2 h\u2081) (erase_le_erase _ h\u2082) },\n  { rw cons_inter_of_neg _ h,\n    exact IH ((le_cons_of_not_mem $ mt (mem_of_le h\u2082) h).1 h\u2081) h\u2082 }\nend\n\n@[simp] theorem mem_inter : a \u2208 s \u2229 t \u2194 a \u2208 s \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (inter_le_left _ _) h, mem_of_le (inter_le_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, by rw [\u2190 cons_erase h\u2081, cons_inter_of_pos _ h\u2082]; apply mem_cons_self\u27e9\n\ninstance : lattice (multiset \u03b1) :=\n{ sup          := (\u222a),\n  sup_le       := @union_le _ _,\n  le_sup_left  := le_union_left,\n  le_sup_right := le_union_right,\n  inf          := (\u2229),\n  le_inf       := @le_inter _ _,\n  inf_le_left  := inter_le_left,\n  inf_le_right := inter_le_right,\n  ..@multiset.partial_order \u03b1 }\n\n@[simp] theorem sup_eq_union (s t : multiset \u03b1) : s \u2294 t = s \u222a t := rfl\n@[simp] theorem inf_eq_inter (s t : multiset \u03b1) : s \u2293 t = s \u2229 t := rfl\n\n@[simp] theorem le_inter_iff : s \u2264 t \u2229 u \u2194 s \u2264 t \u2227 s \u2264 u := le_inf_iff\n@[simp] theorem union_le_iff : s \u222a t \u2264 u \u2194 s \u2264 u \u2227 t \u2264 u := sup_le_iff\n\ninstance : semilattice_inf_bot (multiset \u03b1) :=\n{ bot := 0, bot_le := zero_le, ..multiset.lattice.lattice }\n\ntheorem union_comm (s t : multiset \u03b1) : s \u222a t = t \u222a s := sup_comm\ntheorem inter_comm (s t : multiset \u03b1) : s \u2229 t = t \u2229 s := inf_comm\n\ntheorem eq_union_right (h : s \u2264 t) : s \u222a t = t :=\nby rw [union_comm, eq_union_left h]\n\ntheorem union_le_union_left (h : s \u2264 t) (u) : u \u222a s \u2264 u \u222a t :=\nsup_le_sup_left h _\n\ntheorem union_le_add (s t : multiset \u03b1) : s \u222a t \u2264 s + t :=\nunion_le (le_add_right _ _) (le_add_left _ _)\n\ntheorem union_add_distrib (s t u : multiset \u03b1) : (s \u222a t) + u = (s + u) \u222a (t + u) :=\nby simpa [(\u222a), union, eq_comm] using show s + u - (t + u) = s - t,\nby rw [add_comm t, sub_add', add_sub_cancel]\n\ntheorem add_union_distrib (s t u : multiset \u03b1) : s + (t \u222a u) = (s + t) \u222a (s + u) :=\nby rw [add_comm, union_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_union_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u222a t) = (a :: s) \u222a (a :: t) :=\nby simpa using add_union_distrib (a::0) s t\n\ntheorem inter_add_distrib (s t u : multiset \u03b1) : (s \u2229 t) + u = (s + u) \u2229 (t + u) :=\nbegin\n  by_contra h,\n  cases lt_iff_cons_le.1 (lt_of_le_of_ne (le_inter\n    (add_le_add_right (inter_le_left s t) u)\n    (add_le_add_right (inter_le_right s t) u)) h) with a hl,\n  rw \u2190 cons_add at hl,\n  exact not_le_of_lt (lt_cons_self (s \u2229 t) a) (le_inter\n    (le_of_add_le_add_right (le_trans hl (inter_le_left _ _)))\n    (le_of_add_le_add_right (le_trans hl (inter_le_right _ _))))\nend\n\ntheorem add_inter_distrib (s t u : multiset \u03b1) : s + (t \u2229 u) = (s + t) \u2229 (s + u) :=\nby rw [add_comm, inter_add_distrib, add_comm s, add_comm s]\n\ntheorem cons_inter_distrib (a : \u03b1) (s t : multiset \u03b1) : a :: (s \u2229 t) = (a :: s) \u2229 (a :: t) :=\nby simp\n\ntheorem union_add_inter (s t : multiset \u03b1) : s \u222a t + s \u2229 t = s + t :=\nbegin\n  apply le_antisymm,\n  { rw union_add_distrib,\n    refine union_le (add_le_add_left (inter_le_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (inter_le_left _ _) _ },\n  { rw [add_comm, add_inter_distrib],\n    refine le_inter (add_le_add_right (le_union_right _ _) _) _,\n    rw add_comm, exact add_le_add_right (le_union_left _ _) _ }\nend\n\ntheorem sub_add_inter (s t : multiset \u03b1) : s - t + s \u2229 t = s :=\nbegin\n  rw [inter_comm],\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  by_cases a \u2208 s,\n  { rw [cons_inter_of_pos _ h, sub_cons, add_cons, IH, cons_erase h] },\n  { rw [cons_inter_of_neg _ h, sub_cons, erase_of_not_mem h, IH] }\nend\n\ntheorem sub_inter (s t : multiset \u03b1) : s - (s \u2229 t) = s - t :=\nadd_right_cancel $\nby rw [sub_add_inter s t, sub_add_cancel (inter_le_left _ _)]\n\nend\n\n\n/- filter -/\nsection\nvariables {p : \u03b1 \u2192 Prop} [decidable_pred p]\n\n/-- `filter p s` returns the elements in `s` (with the same multiplicities)\n  which satisfy `p`, and removes the rest. -/\ndef filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p] (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (filter p l : multiset \u03b1))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $ perm_filter p h)\n\n@[simp] theorem coe_filter (p : \u03b1 \u2192 Prop) [h : decidable_pred p]\n  (l : list \u03b1) : filter p (\u2191l) = l.filter p := rfl\n\n@[simp] theorem filter_zero (p : \u03b1 \u2192 Prop) [h : decidable_pred p] : filter p 0 = 0 := rfl\n\n@[simp] theorem filter_cons_of_pos {a : \u03b1} (s) : p a \u2192 filter p (a::s) = a :: filter p s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_cons_of_pos l h\n\n@[simp] theorem filter_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 filter p (a::s) = filter p s :=\nquot.induction_on s $ \u03bb l h, @congr_arg _ _ _ _ coe $ filter_cons_of_neg l h\n\nlemma filter_congr {p q : \u03b1 \u2192 Prop} [decidable_pred p] [decidable_pred q]\n  {s : multiset \u03b1} : (\u2200 x \u2208 s, p x \u2194 q x) \u2192 filter p s = filter q s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ filter_congr h\n\n@[simp] theorem filter_add (s t : multiset \u03b1) :\n  filter p (s + t) = filter p s + filter p t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ filter_append _ _\n\n@[simp] theorem filter_le (s : multiset \u03b1) : filter p s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ filter_sublist _\n\n@[simp] theorem filter_subset (s : multiset \u03b1) : filter p s \u2286 s :=\nsubset_of_le $ filter_le _\n\n@[simp] theorem mem_filter {a : \u03b1} {s} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a :=\nquot.induction_on s $ \u03bb l, mem_filter\n\ntheorem of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : p a :=\n(mem_filter.1 h).2\n\ntheorem mem_of_mem_filter {a : \u03b1} {s} (h : a \u2208 filter p s) : a \u2208 s :=\n(mem_filter.1 h).1\n\ntheorem mem_filter_of_mem {a : \u03b1} {l} (m : a \u2208 l) (h : p a) : a \u2208 filter p l :=\nmem_filter.2 \u27e8m, h\u27e9\n\ntheorem filter_eq_self {s} : filter p s = s \u2194 \u2200 a \u2208 s, p a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_of_sublist_of_length_eq (filter_sublist _) (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_self\n\ntheorem filter_eq_nil {s} : filter p s = 0 \u2194 \u2200 a \u2208 s, \u00acp a :=\nquot.induction_on s $ \u03bb l, iff.trans \u27e8\u03bb h,\n  eq_nil_of_length_eq_zero (@congr_arg _ _ _ _ card h),\n  congr_arg coe\u27e9 filter_eq_nil\n\ntheorem filter_le_filter {s t} (h : s \u2264 t) : filter p s \u2264 filter p t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h, subperm_of_sublist $ filter_sublist_filter h\n\ntheorem le_filter {s t} : s \u2264 filter p t \u2194 s \u2264 t \u2227 \u2200 a \u2208 s, p a :=\n\u27e8\u03bb h, \u27e8le_trans h (filter_le _), \u03bb a m, of_mem_filter (mem_of_le h m)\u27e9,\n \u03bb \u27e8h, al\u27e9, filter_eq_self.2 al \u25b8 filter_le_filter h\u27e9\n\n@[simp] theorem filter_sub [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s - t) = filter p s - filter p t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb a t IH s, _),\n  rw [sub_cons, IH],\n  by_cases p a,\n  { rw [filter_cons_of_pos _ h, sub_cons], congr,\n    by_cases m : a \u2208 s,\n    { rw [\u2190 cons_inj_right a, \u2190 filter_cons_of_pos _ h,\n          cons_erase (mem_filter_of_mem m h), cons_erase m] },\n    { rw [erase_of_not_mem m, erase_of_not_mem (mt mem_of_mem_filter m)] } },\n  { rw [filter_cons_of_neg _ h],\n    by_cases m : a \u2208 s,\n    { rw [(by rw filter_cons_of_neg _ h : filter p (erase s a) = filter p (a :: erase s a)),\n          cons_erase m] },\n    { rw [erase_of_not_mem m] } }\nend\n\n@[simp] theorem filter_union [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u222a t) = filter p s \u222a filter p t :=\nby simp [(\u222a), union]\n\n@[simp] theorem filter_inter [decidable_eq \u03b1] (s t : multiset \u03b1) :\n  filter p (s \u2229 t) = filter p s \u2229 filter p t :=\nle_antisymm (le_inter\n    (filter_le_filter $ inter_le_left _ _)\n    (filter_le_filter $ inter_le_right _ _)) $ le_filter.2\n\u27e8inf_le_inf (filter_le _) (filter_le _),\n  \u03bb a h, of_mem_filter (mem_of_le (inter_le_left _ _) h)\u27e9\n\n@[simp] theorem filter_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p (filter q s) = filter (\u03bb a, p a \u2227 q a) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter l\n\ntheorem filter_add_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  filter p s + filter q s = filter (\u03bb a, p a \u2228 q a) s + filter (\u03bb a, p a \u2227 q a) s :=\nmultiset.induction_on s rfl $ \u03bb a s IH,\nby by_cases p a; by_cases q a; simp *\n\ntheorem filter_add_not (s : multiset \u03b1) :\n  filter p s + filter (\u03bb a, \u00ac p a) s = s :=\nby rw [filter_add_filter, filter_eq_self.2, filter_eq_nil.2]; simp [decidable.em]\n\n/- filter_map -/\n\n/-- `filter_map f s` is a combination filter/map operation on `s`.\n  The function `f : \u03b1 \u2192 option \u03b2` is applied to each element of `s`;\n  if `f a` is `some b` then `b` is added to the result, otherwise\n  `a` is removed from the resulting multiset. -/\ndef filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\nquot.lift_on s (\u03bb l, (filter_map f l : multiset \u03b2))\n  (\u03bb l\u2081 l\u2082 h, quot.sound $perm_filter_map f h)\n\n@[simp] theorem coe_filter_map (f : \u03b1 \u2192 option \u03b2) (l : list \u03b1) : filter_map f l = l.filter_map f := rfl\n\n@[simp] theorem filter_map_zero (f : \u03b1 \u2192 option \u03b2) : filter_map f 0 = 0 := rfl\n\n@[simp] theorem filter_map_cons_none {f : \u03b1 \u2192 option \u03b2} (a : \u03b1) (s : multiset \u03b1) (h : f a = none) :\n  filter_map f (a :: s) = filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_none a l h\n\n@[simp] theorem filter_map_cons_some (f : \u03b1 \u2192 option \u03b2)\n  (a : \u03b1) (s : multiset \u03b1) {b : \u03b2} (h : f a = some b) :\n  filter_map f (a :: s) = b :: filter_map f s :=\nquot.induction_on s $ \u03bb l, @congr_arg _ _ _ _ coe $ filter_map_cons_some f a l h\n\ntheorem filter_map_eq_map (f : \u03b1 \u2192 \u03b2) : filter_map (some \u2218 f) = map f :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_map f) l\n\ntheorem filter_map_eq_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n  filter_map (option.guard p) = filter p :=\nfunext $ \u03bb s, quot.induction_on s $ \u03bb l,\n@congr_arg _ _ _ _ coe $ congr_fun (filter_map_eq_filter p) l\n\ntheorem filter_map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (filter_map f s) = filter_map (\u03bb x, (f x).bind g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter_map f g l\n\ntheorem map_filter_map (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b3) (s : multiset \u03b1) :\n  map g (filter_map f s) = filter_map (\u03bb x, (f x).map g) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map f g l\n\ntheorem filter_map_map (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 option \u03b3) (s : multiset \u03b1) :\n  filter_map g (map f s) = filter_map (g \u2218 f) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_map f g l\n\ntheorem filter_filter_map (f : \u03b1 \u2192 option \u03b2) (p : \u03b2 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n  filter p (filter_map f s) = filter_map (\u03bb x, (f x).filter p) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_filter_map f p l\n\ntheorem filter_map_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) :\n  filter_map f (filter p s) = filter_map (\u03bb x, if p x then f x else none) s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_filter p f l\n\n@[simp] theorem filter_map_some (s : multiset \u03b1) : filter_map some s = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ filter_map_some l\n\n@[simp] theorem mem_filter_map (f : \u03b1 \u2192 option \u03b2) (s : multiset \u03b1) {b : \u03b2} :\n  b \u2208 filter_map f s \u2194 \u2203 a, a \u2208 s \u2227 f a = some b :=\nquot.induction_on s $ \u03bb l, mem_filter_map f l\n\ntheorem map_filter_map_of_inv (f : \u03b1 \u2192 option \u03b2) (g : \u03b2 \u2192 \u03b1)\n  (H : \u2200 x : \u03b1, (f x).map g = some x) (s : multiset \u03b1) :\n  map g (filter_map f s) = s :=\nquot.induction_on s $ \u03bb l, congr_arg coe $ map_filter_map_of_inv f g H l\n\ntheorem filter_map_le_filter_map (f : \u03b1 \u2192 option \u03b2) {s t : multiset \u03b1}\n  (h : s \u2264 t) : filter_map f s \u2264 filter_map f t :=\nle_induction_on h $ \u03bb l\u2081 l\u2082 h,\nsubperm_of_sublist $ filter_map_sublist_filter_map _ h\n\n/- powerset -/\n\ndef powerset_aux (l : list \u03b1) : list (multiset \u03b1) :=\n0 :: sublists_aux l (\u03bb x y, x :: y)\n\ntheorem powerset_aux_eq_map_coe {l : list \u03b1} :\n  powerset_aux l = (sublists l).map coe :=\nby simp [powerset_aux, sublists];\n   rw [\u2190 show @sublists_aux\u2081 \u03b1 (multiset \u03b1) l (\u03bb x, [\u2191x]) =\n              sublists_aux l (\u03bb x, list.cons \u2191x),\n         from sublists_aux\u2081_eq_sublists_aux _ _,\n       sublists_aux_cons_eq_sublists_aux\u2081,\n       \u2190 bind_ret_eq_map, sublists_aux\u2081_bind]; refl\n\n@[simp] theorem mem_powerset_aux {l : list \u03b1} {s} :\n  s \u2208 powerset_aux l \u2194 s \u2264 \u2191l :=\nquotient.induction_on s $\nby simp [powerset_aux_eq_map_coe, subperm, and.comm]\n\ndef powerset_aux' (l : list \u03b1) : list (multiset \u03b1) := (sublists' l).map coe\n\ntheorem powerset_aux_perm_powerset_aux' {l : list \u03b1} :\n  powerset_aux l ~ powerset_aux' l :=\nby rw powerset_aux_eq_map_coe; exact\nperm_map _ (sublists_perm_sublists' _)\n\n@[simp] theorem powerset_aux'_nil : powerset_aux' (@nil \u03b1) = [0] := rfl\n\n@[simp] theorem powerset_aux'_cons (a : \u03b1) (l : list \u03b1) :\n  powerset_aux' (a::l) = powerset_aux' l ++ list.map (cons a) (powerset_aux' l) :=\nby simp [powerset_aux']; refl\n\ntheorem powerset_aux'_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux' l\u2081 ~ powerset_aux' l\u2082 :=\nbegin\n  induction p with a l\u2081 l\u2082 p IH a b l l\u2081 l\u2082 l\u2083 p\u2081 p\u2082 IH\u2081 IH\u2082, {simp},\n  { simp, exact perm_app IH (perm_map _ IH) },\n  { simp, apply perm_app_right,\n    rw [\u2190 append_assoc, \u2190 append_assoc,\n        (by funext s; simp [cons_swap] : cons b \u2218 cons a = cons a \u2218 cons b)],\n    exact perm_app_left _ perm_app_comm },\n  { exact IH\u2081.trans IH\u2082 }\nend\n\ntheorem powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  powerset_aux l\u2081 ~ powerset_aux l\u2082 :=\npowerset_aux_perm_powerset_aux'.trans $\n(powerset_aux'_perm p).trans powerset_aux_perm_powerset_aux'.symm\n\ndef powerset (s : multiset \u03b1) : multiset (multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (powerset_aux l : multiset (multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (powerset_aux_perm h))\n\ntheorem powerset_coe (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists l).map coe : list (multiset \u03b1)) :=\ncongr_arg coe powerset_aux_eq_map_coe\n\n@[simp] theorem powerset_coe' (l : list \u03b1) :\n  @powerset \u03b1 l = ((sublists' l).map coe : list (multiset \u03b1)) :=\nquot.sound powerset_aux_perm_powerset_aux'\n\n@[simp] theorem powerset_zero : @powerset \u03b1 0 = 0::0 := rfl\n\n@[simp] theorem powerset_cons (a : \u03b1) (s) :\n  powerset (a::s) = powerset s + map (cons a) (powerset s) :=\nquotient.induction_on s $ \u03bb l, by simp; refl\n\n@[simp] theorem mem_powerset {s t : multiset \u03b1} :\n  s \u2208 powerset t \u2194 s \u2264 t :=\nquotient.induction_on\u2082 s t $ by simp [subperm, and.comm]\n\ntheorem map_single_le_powerset (s : multiset \u03b1) :\n  s.map (\u03bb a, a::0) \u2264 powerset s :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [powerset_coe],\n  show l.map (coe \u2218 list.ret) <+~ (sublists l).map coe,\n  rw \u2190 list.map_map,\n  exact subperm_of_sublist\n    (map_sublist_map _ (map_ret_sublist_sublists _))\nend\n\n@[simp] theorem card_powerset (s : multiset \u03b1) :\n  card (powerset s) = 2 ^ card s :=\nquotient.induction_on s $ by simp\n\n/- diagonal -/\n\ntheorem revzip_powerset_aux {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux_eq_map_coe, \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists _ _ _ h)\nend\n\ntheorem revzip_powerset_aux' {l : list \u03b1} \u2983s t\u2984\n  (h : (s, t) \u2208 revzip (powerset_aux' l)) : s + t = \u2191l :=\nbegin\n  rw [revzip, powerset_aux', \u2190 map_reverse, zip_map, \u2190 revzip] at h,\n  simp at h, rcases h with \u27e8l\u2081, l\u2082, h, rfl, rfl\u27e9,\n  exact quot.sound (revzip_sublists' _ _ _ h)\nend\n\ntheorem revzip_powerset_aux_lemma [decidable_eq \u03b1] (l : list \u03b1)\n  {l' : list (multiset \u03b1)} (H : \u2200 \u2983s t\u2984, (s, t) \u2208 revzip l' \u2192 s + t = \u2191l) :\n  revzip l' = l'.map (\u03bb x, (x, \u2191l - x)) :=\nbegin\n  have : forall\u2082 (\u03bb (p : multiset \u03b1 \u00d7 multiset \u03b1) (s : multiset \u03b1), p = (s, \u2191l - s))\n    (revzip l') ((revzip l').map prod.fst),\n  { rw forall\u2082_map_right_iff,\n    apply forall\u2082_same, rintro \u27e8s, t\u27e9 h,\n    dsimp, rw [\u2190 H h, add_sub_cancel_left] },\n  rw [\u2190 forall\u2082_eq_eq_eq, forall\u2082_map_right_iff], simpa\nend\n\ntheorem revzip_powerset_aux_perm_aux' {l : list \u03b1} :\n  revzip (powerset_aux l) ~ revzip (powerset_aux' l) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  rw [revzip_powerset_aux_lemma l revzip_powerset_aux,\n      revzip_powerset_aux_lemma l revzip_powerset_aux'],\n  exact perm_map _ powerset_aux_perm_powerset_aux',\nend\n\ntheorem revzip_powerset_aux_perm {l\u2081 l\u2082 : list \u03b1} (p : l\u2081 ~ l\u2082) :\n  revzip (powerset_aux l\u2081) ~ revzip (powerset_aux l\u2082) :=\nbegin\n  haveI := classical.dec_eq \u03b1,\n  simp [\u03bb l:list \u03b1, revzip_powerset_aux_lemma l revzip_powerset_aux, coe_eq_coe.2 p],\n  exact perm_map _ (powerset_aux_perm p)\nend\n\ndef diagonal (s : multiset \u03b1) : multiset (multiset \u03b1 \u00d7 multiset \u03b1) :=\nquot.lift_on s\n  (\u03bb l, (revzip (powerset_aux l) : multiset (multiset \u03b1 \u00d7 multiset \u03b1)))\n  (\u03bb l\u2081 l\u2082 h, quot.sound (revzip_powerset_aux_perm h))\n\ntheorem diagonal_coe (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux l) := rfl\n\n@[simp] theorem diagonal_coe' (l : list \u03b1) :\n  @diagonal \u03b1 l = revzip (powerset_aux' l) :=\nquot.sound revzip_powerset_aux_perm_aux'\n\n@[simp] theorem mem_diagonal {s\u2081 s\u2082 t : multiset \u03b1} :\n  (s\u2081, s\u2082) \u2208 diagonal t \u2194 s\u2081 + s\u2082 = t :=\nquotient.induction_on t $ \u03bb l, begin\n  simp [diagonal_coe], refine \u27e8\u03bb h, revzip_powerset_aux h, \u03bb h, _\u27e9,\n  haveI := classical.dec_eq \u03b1,\n  simp [revzip_powerset_aux_lemma l revzip_powerset_aux, h.symm],\n  exact \u27e8_, le_add_right _ _, rfl, add_sub_cancel_left _ _\u27e9\nend\n\n@[simp] theorem diagonal_map_fst (s : multiset \u03b1) :\n  (diagonal s).map prod.fst = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_map_snd (s : multiset \u03b1) :\n  (diagonal s).map prod.snd = powerset s :=\nquotient.induction_on s $ \u03bb l,\nby simp [powerset_aux']\n\n@[simp] theorem diagonal_zero : @diagonal \u03b1 0 = (0, 0)::0 := rfl\n\n@[simp] theorem diagonal_cons (a : \u03b1) (s) : diagonal (a::s) =\n  map (prod.map id (cons a)) (diagonal s) +\n  map (prod.map (cons a) id) (diagonal s) :=\nquotient.induction_on s $ \u03bb l, begin\n  simp [revzip, reverse_append],\n  rw [\u2190 zip_map, \u2190 zip_map, zip_append, (_ : _++_=_)],\n  {congr; simp}, {simp}\nend\n\n@[simp] theorem card_diagonal (s : multiset \u03b1) :\n  card (diagonal s) = 2 ^ card s :=\nby have := card_powerset s;\n   rwa [\u2190 diagonal_map_fst, card_map] at this\n\nlemma prod_map_add [comm_semiring \u03b2] {s : multiset \u03b1} {f g : \u03b1 \u2192 \u03b2} :\n  prod (s.map (\u03bba, f a + g a)) = sum ((diagonal s).map (\u03bbp, (p.1.map f).prod * (p.2.map g).prod)) :=\nbegin\n  refine s.induction_on _ _,\n  { simp },\n  { assume a s ih, simp [ih, add_mul, mul_comm, mul_left_comm, mul_assoc, sum_map_mul_left.symm] },\nend\n\n/- countp -/\n\n/-- `countp p s` counts the number of elements of `s` (with multiplicity) that\n  satisfy `p`. -/\ndef countp (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : \u2115 :=\nquot.lift_on s (countp p) (\u03bb l\u2081 l\u2082, perm_countp p)\n\n@[simp] theorem coe_countp (l : list \u03b1) : countp p l = l.countp p := rfl\n\n@[simp] theorem countp_zero (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p 0 = 0 := rfl\n\n@[simp] theorem countp_cons_of_pos {a : \u03b1} (s) : p a \u2192 countp p (a::s) = countp p s + 1 :=\nquot.induction_on s countp_cons_of_pos\n\n@[simp] theorem countp_cons_of_neg {a : \u03b1} (s) : \u00ac p a \u2192 countp p (a::s) = countp p s :=\nquot.induction_on s countp_cons_of_neg\n\ntheorem countp_eq_card_filter (s) : countp p s = card (filter p s) :=\nquot.induction_on s $ \u03bb l, countp_eq_length_filter _\n\n@[simp] theorem countp_add (s t) : countp p (s + t) = countp p s + countp p t :=\nby simp [countp_eq_card_filter]\n\ntheorem countp_pos {s} : 0 < countp p s \u2194 \u2203 a \u2208 s, p a :=\nby simp [countp_eq_card_filter, card_pos_iff_exists_mem]\n\n@[simp] theorem countp_sub [decidable_eq \u03b1] {s t : multiset \u03b1} (h : t \u2264 s) :\n  countp p (s - t) = countp p s - countp p t :=\nby simp [countp_eq_card_filter, h, filter_le_filter]\n\ntheorem countp_pos_of_mem {s a} (h : a \u2208 s) (pa : p a) : 0 < countp p s :=\ncountp_pos.2 \u27e8_, h, pa\u27e9\n\ntheorem countp_le_of_le {s t} (h : s \u2264 t) : countp p s \u2264 countp p t :=\nby simpa [countp_eq_card_filter] using card_le_of_le (filter_le_filter h)\n\n@[simp] theorem countp_filter {q} [decidable_pred q] (s : multiset \u03b1) :\n  countp p (filter q s) = countp (\u03bb a, p a \u2227 q a) s :=\nby simp [countp_eq_card_filter]\n\nend\n\n/- count -/\n\nsection\nvariable [decidable_eq \u03b1]\n\n/-- `count a s` is the multiplicity of `a` in `s`. -/\ndef count (a : \u03b1) : multiset \u03b1 \u2192 \u2115 := countp (eq a)\n\n@[simp] theorem coe_count (a : \u03b1) (l : list \u03b1) : count a (\u2191l) = l.count a := coe_countp _\n\n@[simp] theorem count_zero (a : \u03b1) : count a 0 = 0 := rfl\n\n@[simp] theorem count_cons_self (a : \u03b1) (s : multiset \u03b1) : count a (a::s) = succ (count a s) :=\ncountp_cons_of_pos _ rfl\n\n@[simp] theorem count_cons_of_ne {a b : \u03b1} (h : a \u2260 b) (s : multiset \u03b1) : count a (b::s) = count a s :=\ncountp_cons_of_neg _ h\n\ntheorem count_le_of_le (a : \u03b1) {s t} : s \u2264 t \u2192 count a s \u2264 count a t :=\ncountp_le_of_le\n\ntheorem count_le_count_cons (a b : \u03b1) (s : multiset \u03b1) : count a s \u2264 count a (b :: s) :=\ncount_le_of_le _ (le_cons_self _ _)\n\ntheorem count_singleton (a : \u03b1) : count a (a::0) = 1 :=\nby simp\n\n@[simp] theorem count_add (a : \u03b1) : \u2200 s t, count a (s + t) = count a s + count a t :=\ncountp_add\n\n@[simp] theorem count_smul (a : \u03b1) (n s) : count a (n \u2022 s) = n * count a s :=\nby induction n; simp [*, succ_smul', succ_mul]\n\ntheorem count_pos {a : \u03b1} {s : multiset \u03b1} : 0 < count a s \u2194 a \u2208 s :=\nby simp [count, countp_pos]\n\n@[simp] theorem count_eq_zero_of_not_mem {a : \u03b1} {s : multiset \u03b1} (h : a \u2209 s) : count a s = 0 :=\nby_contradiction $ \u03bb h', h $ count_pos.1 (nat.pos_of_ne_zero h')\n\ntheorem count_eq_zero {a : \u03b1} {s : multiset \u03b1} : count a s = 0 \u2194 a \u2209 s :=\niff_not_comm.1 $ count_pos.symm.trans pos_iff_ne_zero\n\n@[simp] theorem count_repeat (a : \u03b1) (n : \u2115) : count a (repeat a n) = n :=\nby simp [repeat]\n\n@[simp] theorem count_erase_self (a : \u03b1) (s : multiset \u03b1) : count a (erase s a) = pred (count a s) :=\nbegin\n  by_cases a \u2208 s,\n  { rw [(by rw cons_erase h : count a s = count a (a::erase s a)),\n        count_cons_self]; refl },\n  { rw [erase_of_not_mem h, count_eq_zero.2 h]; refl }\nend\n\n@[simp] theorem count_erase_of_ne {a b : \u03b1} (ab : a \u2260 b) (s : multiset \u03b1) : count a (erase s b) = count a s :=\nbegin\n  by_cases b \u2208 s,\n  { rw [\u2190 count_cons_of_ne ab, cons_erase h] },\n  { rw [erase_of_not_mem h] }\nend\n\n@[simp] theorem count_sub (a : \u03b1) (s t : multiset \u03b1) : count a (s - t) = count a s - count a t :=\nbegin\n  revert s, refine multiset.induction_on t (by simp) (\u03bb b t IH s, _),\n  rw [sub_cons, IH],\n  by_cases ab : a = b,\n  { subst b, rw [count_erase_self, count_cons_self, sub_succ, pred_sub] },\n  { rw [count_erase_of_ne ab, count_cons_of_ne ab] }\nend\n\n@[simp] theorem count_union (a : \u03b1) (s t : multiset \u03b1) : count a (s \u222a t) = max (count a s) (count a t) :=\nby simp [(\u222a), union, sub_add_eq_max, -add_comm]\n\n@[simp] theorem count_inter (a : \u03b1) (s t : multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t) :=\nbegin\n  apply @nat.add_left_cancel (count a (s - t)),\n  rw [\u2190 count_add, sub_add_inter, count_sub, sub_add_min],\nend\n\nlemma count_bind {m : multiset \u03b2} {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} :\n  count a (bind m f) = sum (m.map $ \u03bbb, count a $ f b) :=\nmultiset.induction_on m (by simp) (by simp)\n\ntheorem le_count_iff_repeat_le {a : \u03b1} {s : multiset \u03b1} {n : \u2115} : n \u2264 count a s \u2194 repeat a n \u2264 s :=\nquot.induction_on s $ \u03bb l, le_count_iff_repeat_sublist.trans repeat_le_coe.symm\n\n@[simp] theorem count_filter {p} [decidable_pred p]\n  {a} {s : multiset \u03b1} (h : p a) : count a (filter p s) = count a s :=\nquot.induction_on s $ \u03bb l, count_filter h\n\ntheorem ext {s t : multiset \u03b1} : s = t \u2194 \u2200 a, count a s = count a t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, quotient.eq.trans perm_iff_count\n\n@[extensionality]\ntheorem ext' {s t : multiset \u03b1} : (\u2200 a, count a s = count a t) \u2192 s = t :=\next.2\n\ntheorem le_iff_count {s t : multiset \u03b1} : s \u2264 t \u2194 \u2200 a, count a s \u2264 count a t :=\n\u27e8\u03bb h a, count_le_of_le a h, \u03bb al,\n by rw \u2190 (ext.2 (\u03bb a, by simp [max_eq_right (al a)]) : s \u222a t = t);\n    apply le_union_left\u27e9\n\ninstance : distrib_lattice (multiset \u03b1) :=\n{ le_sup_inf := \u03bb s t u, le_of_eq $ eq.symm $\n    ext.2 $ \u03bb a, by simp [max_min_distrib_left],\n  ..multiset.lattice.lattice }\n\ninstance : semilattice_sup_bot (multiset \u03b1) :=\n{ bot := 0,\n  bot_le := zero_le,\n  ..multiset.lattice.lattice }\n\nend\n\n/- relator -/\n\nsection rel\n\n/-- `rel r s t` -- lift the relation `r` between two elements to a relation between `s` and `t`,\ns.t. there is a one-to-one mapping betweem elements in `s` and `t` following `r`. -/\ninductive rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : multiset \u03b1 \u2192 multiset \u03b2 \u2192 Prop\n| zero {} : rel 0 0\n| cons {a b as bs} : r a b \u2192 rel as bs \u2192 rel (a :: as) (b :: bs)\n\nrun_cmd tactic.mk_iff_of_inductive_prop `multiset.rel `multiset.rel_iff\n\nvariables {\u03b4 : Type*} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b3 \u2192 \u03b4 \u2192 Prop}\n\nprivate lemma rel_flip_aux {s t} (h : rel r s t) : rel (flip r) t s :=\nrel.rec_on h rel.zero (assume _ _ _ _ h\u2080 h\u2081 ih, rel.cons h\u2080 ih)\n\nlemma rel_flip {s t} : rel (flip r) s t \u2194 rel r t s :=\n\u27e8rel_flip_aux, rel_flip_aux\u27e9\n\nlemma rel_eq_refl {s : multiset \u03b1} : rel (=) s s :=\nmultiset.induction_on s rel.zero (assume a s, rel.cons rfl)\n\nlemma rel_eq {s t : multiset \u03b1} : rel (=) s t \u2194 s = t :=\nbegin\n  split,\n  { assume h, induction h; simp * },\n  { assume h, subst h, exact rel_eq_refl }\nend\n\nlemma rel.mono {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {s t} (h : \u2200a b, r a b \u2192 p a b) (hst : rel r s t) : rel p s t :=\nbegin\n  induction hst,\n  case rel.zero { exact rel.zero },\n  case rel.cons : a b s t hab hst ih { exact ih.cons (h a b hab) }\nend\n\nlemma rel.add {s t u v} (hst : rel r s t) (huv : rel r u v) : rel r (s + u) (t + v) :=\nbegin\n  induction hst,\n  case rel.zero { simpa using huv },\n  case rel.cons : a b s t hab hst ih { simpa using ih.cons hab }\nend\n\nlemma rel_flip_eq  {s t : multiset \u03b1} : rel (\u03bba b, b = a) s t \u2194 s = t :=\nshow rel (flip (=)) s t \u2194 s = t, by rw [rel_flip, rel_eq, eq_comm]\n\n@[simp] lemma rel_zero_left {b : multiset \u03b2} : rel r 0 b \u2194 b = 0 :=\nby rw [rel_iff]; simp\n\n@[simp] lemma rel_zero_right {a : multiset \u03b1} : rel r a 0 \u2194 a = 0 :=\nby rw [rel_iff]; simp\n\nlemma rel_cons_left {a as bs} :\n  rel r (a :: as) bs \u2194 (\u2203b bs', r a b \u2227 rel r as bs' \u2227 bs = b :: bs') :=\nbegin\n  split,\n  { generalize hm : a :: as = m,\n    assume h,\n    induction h generalizing as,\n    case rel.zero { simp at hm, contradiction },\n    case rel.cons : a' b as' bs ha'b h ih {\n      rcases cons_eq_cons.1 hm with \u27e8eq\u2081, eq\u2082\u27e9 | \u27e8h, cs, eq\u2081, eq\u2082\u27e9,\n      { subst eq\u2081, subst eq\u2082, exact \u27e8b, bs, ha'b, h, rfl\u27e9 },\n      { rcases ih eq\u2082.symm with \u27e8b', bs', h\u2081, h\u2082, eq\u27e9,\n        exact \u27e8b', b::bs', h\u2081, eq\u2081.symm \u25b8 rel.cons ha'b h\u2082, eq.symm \u25b8 cons_swap _ _ _\u27e9  }\n    } },\n  { exact assume \u27e8b, bs', hab, h, eq\u27e9, eq.symm \u25b8 rel.cons hab h }\nend\n\nlemma rel_cons_right {as b bs} :\n  rel r as (b :: bs) \u2194 (\u2203a as', r a b \u2227 rel r as' bs \u2227 as = a :: as') :=\nbegin\n  rw [\u2190 rel_flip, rel_cons_left],\n  apply exists_congr, assume a,\n  apply exists_congr, assume as',\n  rw [rel_flip, flip]\nend\n\nlemma rel_add_left {as\u2080 as\u2081} :\n  \u2200{bs}, rel r (as\u2080 + as\u2081) bs \u2194 (\u2203bs\u2080 bs\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081) :=\nmultiset.induction_on as\u2080 (by simp)\n  begin\n    assume a s ih bs,\n    simp only [ih, cons_add, rel_cons_left],\n    split,\n    { assume h,\n      rcases h with \u27e8b, bs', hab, h, rfl\u27e9,\n      rcases h with \u27e8bs\u2080, bs\u2081, h\u2080, h\u2081, rfl\u27e9,\n      exact \u27e8b :: bs\u2080, bs\u2081, \u27e8b, bs\u2080, hab, h\u2080, rfl\u27e9, h\u2081, by simp\u27e9 },\n    { assume h,\n      rcases h with \u27e8bs\u2080, bs\u2081, h, h\u2081, rfl\u27e9,\n      rcases h with \u27e8b, bs, hab, h\u2080, rfl\u27e9,\n      exact \u27e8b, bs + bs\u2081, hab, \u27e8bs, bs\u2081, h\u2080, h\u2081, rfl\u27e9, by simp\u27e9 }\n  end\n\nlemma rel_add_right {as bs\u2080 bs\u2081} :\n  rel r as (bs\u2080 + bs\u2081) \u2194 (\u2203as\u2080 as\u2081, rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081) :=\nby rw [\u2190 rel_flip, rel_add_left]; simp [rel_flip]\n\nlemma rel_map_left {s : multiset \u03b3} {f : \u03b3 \u2192 \u03b1} :\n  \u2200{t}, rel r (s.map f) t \u2194 rel (\u03bba b, r (f a) b) s t :=\nmultiset.induction_on s (by simp) (by simp [rel_cons_left] {contextual := tt})\n\nlemma rel_map_right {s : multiset \u03b1} {t : multiset \u03b3} {f : \u03b3 \u2192 \u03b2} :\n  rel r s (t.map f) \u2194 rel (\u03bba b, r a (f b)) s t :=\nby rw [\u2190 rel_flip, rel_map_left, \u2190 rel_flip]; refl\n\nlemma rel_join {s t} (h : rel (rel r) s t) : rel r s.join t.join :=\nbegin\n  induction h,\n  case rel.zero { simp },\n  case rel.cons : a b s t hab hst ih { simpa using hab.add ih }\nend\n\nlemma rel_map {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} (h : (r \u21d2 p) f g) (hst : rel r s t) :\n  rel p (s.map f) (t.map g) :=\nby rw [rel_map_left, rel_map_right]; exact hst.mono (assume a b, h)\n\nlemma rel_bind {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s t} {f : \u03b1 \u2192 multiset \u03b3} {g : \u03b2 \u2192 multiset \u03b4}\n  (h : (r \u21d2 rel p) f g) (hst : rel r s t) :\n  rel p (s.bind f) (t.bind g) :=\nby apply rel_join; apply rel_map; assumption\n\nlemma card_eq_card_of_rel {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} (h : rel r s t) :\n  card s = card t :=\nby induction h; simp [*]\n\nend rel\n\nsection map\n\ntheorem map_eq_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) {s t : multiset \u03b1} :\n  s.map f = t.map f \u2194 s = t :=\nby rw [\u2190 rel_eq, \u2190 rel_eq, rel_map_left, rel_map_right]; simp [hf.eq_iff]\n\ntheorem injective_map {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n  function.injective (multiset.map f) :=\nassume x y, (map_eq_map hf).1\n\nend map\n\nsection quot\n\ntheorem map_mk_eq_map_mk_of_rel {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s t : multiset \u03b1} (hst : s.rel r t) :\n s.map (quot.mk r) = t.map (quot.mk r) :=\nrel.rec_on hst rfl $ assume a b s t hab hst ih, by simp [ih, quot.sound hab]\n\ntheorem exists_multiset_eq_map_quot_mk {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : multiset (quot r)) :\n  \u2203t:multiset \u03b1, s = t.map (quot.mk r) :=\nmultiset.induction_on s \u27e80, rfl\u27e9 $\n  assume a s \u27e8t, ht\u27e9, quot.induction_on a $ assume a, ht.symm \u25b8 \u27e8a::t, (map_cons _ _ _).symm\u27e9\n\ntheorem induction_on_multiset_quot\n  {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {p : multiset (quot r) \u2192 Prop} (s : multiset (quot r)) :\n  (\u2200s:multiset \u03b1, p (s.map (quot.mk r))) \u2192 p s :=\nmatch s, exists_multiset_eq_map_quot_mk s with _, \u27e8t, rfl\u27e9 := assume h, h _ end\n\nend quot\n\n/- disjoint -/\n\n/-- `disjoint s t` means that `s` and `t` have no elements in common. -/\ndef disjoint (s t : multiset \u03b1) : Prop := \u2200 \u2983a\u2984, a \u2208 s \u2192 a \u2208 t \u2192 false\n\n@[simp] theorem coe_disjoint (l\u2081 l\u2082 : list \u03b1) : @disjoint \u03b1 l\u2081 l\u2082 \u2194 l\u2081.disjoint l\u2082 := iff.rfl\n\ntheorem disjoint.symm {s t : multiset \u03b1} (d : disjoint s t) : disjoint t s\n| a i\u2082 i\u2081 := d i\u2081 i\u2082\n\n@[simp] theorem disjoint_comm {s t : multiset \u03b1} : disjoint s t \u2194 disjoint t s :=\n\u27e8disjoint.symm, disjoint.symm\u27e9\n\ntheorem disjoint_left {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 s \u2192 a \u2209 t := iff.rfl\n\ntheorem disjoint_right {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 {a}, a \u2208 t \u2192 a \u2209 s :=\ndisjoint_comm\n\ntheorem disjoint_iff_ne {s t : multiset \u03b1} : disjoint s t \u2194 \u2200 a \u2208 s, \u2200 b \u2208 t, a \u2260 b :=\nby simp [disjoint_left, imp_not_comm]\n\ntheorem disjoint_of_subset_left {s t u : multiset \u03b1} (h : s \u2286 u) (d : disjoint u t) : disjoint s t\n| x m\u2081 := d (h m\u2081)\n\ntheorem disjoint_of_subset_right {s t u : multiset \u03b1} (h : t \u2286 u) (d : disjoint s u) : disjoint s t\n| x m m\u2081 := d m (h m\u2081)\n\ntheorem disjoint_of_le_left {s t u : multiset \u03b1} (h : s \u2264 u) : disjoint u t \u2192 disjoint s t :=\ndisjoint_of_subset_left (subset_of_le h)\n\ntheorem disjoint_of_le_right {s t u : multiset \u03b1} (h : t \u2264 u) : disjoint s u \u2192 disjoint s t :=\ndisjoint_of_subset_right (subset_of_le h)\n\n@[simp] theorem zero_disjoint (l : multiset \u03b1) : disjoint 0 l\n| a := (not_mem_nil a).elim\n\n@[simp] theorem singleton_disjoint {l : multiset \u03b1} {a : \u03b1} : disjoint (a::0) l \u2194 a \u2209 l :=\nby simp [disjoint]; refl\n\n@[simp] theorem disjoint_singleton {l : multiset \u03b1} {a : \u03b1} : disjoint l (a::0) \u2194 a \u2209 l :=\nby rw disjoint_comm; simp\n\n@[simp] theorem disjoint_add_left {s t u : multiset \u03b1} :\n  disjoint (s + t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_add_right {s t u : multiset \u03b1} :\n  disjoint s (t + u) \u2194 disjoint s t \u2227 disjoint s u :=\ndisjoint_comm.trans $ by simp [disjoint_append_left]\n\n@[simp] theorem disjoint_cons_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (a::s) t \u2194 a \u2209 t \u2227 disjoint s t :=\n(@disjoint_add_left _ (a::0) s t).trans $ by simp\n\n@[simp] theorem disjoint_cons_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (a::t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp [disjoint_cons_left]\n\ntheorem inter_eq_zero_iff_disjoint [decidable_eq \u03b1] {s t : multiset \u03b1} : s \u2229 t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\n@[simp] theorem disjoint_union_left [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint (s \u222a t) u \u2194 disjoint s u \u2227 disjoint t u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\n@[simp] theorem disjoint_union_right [decidable_eq \u03b1] {s t u : multiset \u03b1} :\n  disjoint s (t \u222a u) \u2194 disjoint s t \u2227 disjoint s u :=\nby simp [disjoint, or_imp_distrib, forall_and_distrib]\n\nlemma disjoint_map_map {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {s : multiset \u03b1} {t : multiset \u03b2} :\n  disjoint (s.map f) (t.map g) \u2194 (\u2200a\u2208s, \u2200b\u2208t, f a \u2260 g b) :=\nbegin\n  simp [disjoint],\n  split,\n  from assume h a ha b hb eq, h _ ha rfl _ hb eq.symm,\n  from assume h c a ha eq\u2081 b hb eq\u2082, h _ ha _ hb (eq\u2082.symm \u25b8 eq\u2081)\nend\n\n/-- `pairwise r m` states that there exists a list of the elements s.t. `r` holds pairwise on this list. -/\ndef pairwise (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (m : multiset \u03b1) : Prop :=\n\u2203l:list \u03b1, m = l \u2227 l.pairwise r\n\nlemma pairwise_coe_iff_pairwise {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (hr : symmetric r) {l : list \u03b1} :\n  multiset.pairwise r l \u2194 l.pairwise r :=\niff.intro\n  (assume \u27e8l', eq, h\u27e9, (list.perm_pairwise hr (quotient.exact eq)).2 h)\n  (assume h, \u27e8l, rfl, h\u27e9)\n\n/- nodup -/\n\n/-- `nodup s` means that `s` has no duplicates, i.e. the multiplicity of\n  any element is at most 1. -/\ndef nodup (s : multiset \u03b1) : Prop :=\nquot.lift_on s nodup (\u03bb s t p, propext $ perm_nodup p)\n\n@[simp] theorem coe_nodup {l : list \u03b1} : @nodup \u03b1 l \u2194 l.nodup := iff.rfl\n\n@[simp] theorem forall_mem_ne {a : \u03b1} {l : list \u03b1} : (\u2200 (a' : \u03b1), a' \u2208 l \u2192 \u00aca = a') \u2194 a \u2209 l :=\n\u27e8\u03bb h m, h _ m rfl, \u03bb h a' m e, h (e.symm \u25b8 m)\u27e9\n\n@[simp] theorem nodup_zero : @nodup \u03b1 0 := pairwise.nil _\n\n@[simp] theorem nodup_cons {a : \u03b1} {s : multiset \u03b1} : nodup (a::s) \u2194 a \u2209 s \u2227 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_cons\n\ntheorem nodup_cons_of_nodup {a : \u03b1} {s : multiset \u03b1} (m : a \u2209 s) (n : nodup s) : nodup (a::s) :=\nnodup_cons.2 \u27e8m, n\u27e9\n\ntheorem nodup_singleton : \u2200 a : \u03b1, nodup (a::0) := nodup_singleton\n\ntheorem nodup_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : nodup s :=\n(nodup_cons.1 h).2\n\ntheorem not_mem_of_nodup_cons {a : \u03b1} {s : multiset \u03b1} (h : nodup (a::s)) : a \u2209 s :=\n(nodup_cons.1 h).1\n\ntheorem nodup_of_le {s t : multiset \u03b1} (h : s \u2264 t) : nodup t \u2192 nodup s :=\nle_induction_on h $ \u03bb l\u2081 l\u2082, nodup_of_sublist\n\ntheorem not_nodup_pair : \u2200 a : \u03b1, \u00ac nodup (a::a::0) := not_nodup_pair\n\ntheorem nodup_iff_le {s : multiset \u03b1} : nodup s \u2194 \u2200 a : \u03b1, \u00ac a::a::0 \u2264 s :=\nquot.induction_on s $ \u03bb l, nodup_iff_sublist.trans $ forall_congr $ \u03bb a,\nnot_congr (@repeat_le_coe _ a 2 _).symm\n\ntheorem nodup_iff_count_le_one [decidable_eq \u03b1] {s : multiset \u03b1} : nodup s \u2194 \u2200 a, count a s \u2264 1 :=\nquot.induction_on s $ \u03bb l, nodup_iff_count_le_one\n\n@[simp] theorem count_eq_one_of_mem [decidable_eq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n  (d : nodup s) (h : a \u2208 s) : count a s = 1 :=\nle_antisymm (nodup_iff_count_le_one.1 d a) (count_pos.2 h)\n\nlemma pairwise_of_nodup {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : multiset \u03b1} :\n  (\u2200a\u2208s, \u2200b\u2208s, a \u2260 b \u2192 r a b) \u2192 nodup s \u2192 pairwise r s :=\nquotient.induction_on s $ assume l h hl, \u27e8l, rfl, hl.imp_of_mem $ assume a b ha hb, h a ha b hb\u27e9\n\ntheorem nodup_add {s t : multiset \u03b1} : nodup (s + t) \u2194 nodup s \u2227 nodup t \u2227 disjoint s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, nodup_append\n\ntheorem disjoint_of_nodup_add {s t : multiset \u03b1} (d : nodup (s + t)) : disjoint s t :=\n(nodup_add.1 d).2.2\n\ntheorem nodup_add_of_nodup {s t : multiset \u03b1} (d\u2081 : nodup s) (d\u2082 : nodup t) : nodup (s + t) \u2194 disjoint s t :=\nby simp [nodup_add, d\u2081, d\u2082]\n\ntheorem nodup_of_nodup_map (f : \u03b1 \u2192 \u03b2) {s : multiset \u03b1} : nodup (map f s) \u2192 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_of_nodup_map f\n\ntheorem nodup_map_on {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} : (\u2200x\u2208s, \u2200y\u2208s, f x = f y \u2192 x = y) \u2192\n  nodup s \u2192 nodup (map f s) :=\nquot.induction_on s $ \u03bb l, nodup_map_on\n\ntheorem nodup_map {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} (hf : function.injective f) : nodup s \u2192 nodup (map f s) :=\nnodup_map_on (\u03bb x _ y _ h, hf h)\n\ntheorem nodup_filter (p : \u03b1 \u2192 Prop) [decidable_pred p] {s} : nodup s \u2192 nodup (filter p s) :=\nquot.induction_on s $ \u03bb l, nodup_filter p\n\n@[simp] theorem nodup_attach {s : multiset \u03b1} : nodup (attach s) \u2194 nodup s :=\nquot.induction_on s $ \u03bb l, nodup_attach\n\ntheorem nodup_pmap {p : \u03b1 \u2192 Prop} {f : \u03a0 a, p a \u2192 \u03b2} {s : multiset \u03b1} {H}\n  (hf : \u2200 a ha b hb, f a ha = f b hb \u2192 a = b) : nodup s \u2192 nodup (pmap f s H) :=\nquot.induction_on s (\u03bb l H, nodup_pmap hf) H\n\ninstance nodup_decidable [decidable_eq \u03b1] (s : multiset \u03b1) : decidable (nodup s) :=\nquotient.rec_on_subsingleton s $ \u03bb l, l.nodup_decidable\n\ntheorem nodup_erase_eq_filter [decidable_eq \u03b1] (a : \u03b1) {s} : nodup s \u2192 s.erase a = filter (\u2260 a) s :=\nquot.induction_on s $ \u03bb l d, congr_arg coe $ nodup_erase_eq_filter a d\n\ntheorem nodup_erase_of_nodup [decidable_eq \u03b1] (a : \u03b1) {l} : nodup l \u2192 nodup (l.erase a) :=\nnodup_of_le (erase_le _ _)\n\ntheorem mem_erase_iff_of_nodup [decidable_eq \u03b1] {a b : \u03b1} {l} (d : nodup l) :\n  a \u2208 l.erase b \u2194 a \u2260 b \u2227 a \u2208 l :=\nby rw nodup_erase_eq_filter b d; simp [and_comm]\n\ntheorem mem_erase_of_nodup [decidable_eq \u03b1] {a : \u03b1} {l} (h : nodup l) : a \u2209 l.erase a :=\nby rw mem_erase_iff_of_nodup h; simp\n\ntheorem nodup_product {s : multiset \u03b1} {t : multiset \u03b2} : nodup s \u2192 nodup t \u2192 nodup (product s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, by simp [nodup_product d\u2081 d\u2082]\n\ntheorem nodup_sigma {\u03c3 : \u03b1 \u2192 Type*} {s : multiset \u03b1} {t : \u03a0 a, multiset (\u03c3 a)} :\n  nodup s \u2192 (\u2200 a, nodup (t a)) \u2192 nodup (s.sigma t) :=\nquot.induction_on s $ \u03bb l\u2081,\nlet l\u2082 (a) : list (\u03c3 a) := classical.some (quotient.exists_rep (t a)) in\nhave t = \u03bb a, l\u2082 a, from eq.symm $ funext $ \u03bb a,\n  classical.some_spec (quotient.exists_rep (t a)),\nby rw [this]; simpa using nodup_sigma\n\ntheorem nodup_filter_map (f : \u03b1 \u2192 option \u03b2) {s : multiset \u03b1}\n  (H : \u2200 (a a' : \u03b1) (b : \u03b2), b \u2208 f a \u2192 b \u2208 f a' \u2192 a = a') :\n  nodup s \u2192 nodup (filter_map f s) :=\nquot.induction_on s $ \u03bb l, nodup_filter_map H\n\ntheorem nodup_range (n : \u2115) : nodup (range n) := nodup_range _\n\ntheorem nodup_inter_left [decidable_eq \u03b1] {s : multiset \u03b1} (t) : nodup s \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_left _ _\n\ntheorem nodup_inter_right [decidable_eq \u03b1] (s) {t : multiset \u03b1} : nodup t \u2192 nodup (s \u2229 t) :=\nnodup_of_le $ inter_le_right _ _\n\n@[simp] theorem nodup_union [decidable_eq \u03b1] {s t : multiset \u03b1} : nodup (s \u222a t) \u2194 nodup s \u2227 nodup t :=\n\u27e8\u03bb h, \u27e8nodup_of_le (le_union_left _ _) h, nodup_of_le (le_union_right _ _) h\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, nodup_iff_count_le_one.2 $ \u03bb a, by rw [count_union]; exact\n   max_le (nodup_iff_count_le_one.1 h\u2081 a) (nodup_iff_count_le_one.1 h\u2082 a)\u27e9\n\n@[simp] theorem nodup_powerset {s : multiset \u03b1} : nodup (powerset s) \u2194 nodup s :=\n\u27e8\u03bb h, nodup_of_nodup_map _ (nodup_of_le (map_single_le_powerset _) h),\n  quotient.induction_on s $ \u03bb l h,\n  by simp; refine list.nodup_map_on _ (nodup_sublists'.2 h); exact\n  \u03bb x sx y sy e,\n    (perm_ext_sublist_nodup h (mem_sublists'.1 sx) (mem_sublists'.1 sy)).1\n      (quotient.exact e)\u27e9\n\n@[simp] lemma nodup_bind {s : multiset \u03b1} {t : \u03b1 \u2192 multiset \u03b2} :\n  nodup (bind s t) \u2194 ((\u2200a\u2208s, nodup (t a)) \u2227 (s.pairwise (\u03bba b, disjoint (t a) (t b)))) :=\nhave h\u2081 : \u2200a, \u2203l:list \u03b2, t a = l, from\n  assume a, quot.induction_on (t a) $ assume l, \u27e8l, rfl\u27e9,\nlet \u27e8t', h'\u27e9 := classical.axiom_of_choice h\u2081 in\nhave t = \u03bba, t' a, from funext h',\nhave hd : symmetric (\u03bba b, list.disjoint (t' a) (t' b)), from assume a b h, h.symm,\nquot.induction_on s $ by simp [this, list.nodup_bind, pairwise_coe_iff_pairwise hd]\n\ntheorem nodup_ext {s t : multiset \u03b1} : nodup s \u2192 nodup t \u2192 (s = t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d\u2081 d\u2082, quotient.eq.trans $ perm_ext d\u2081 d\u2082\n\ntheorem le_iff_subset {s t : multiset \u03b1} : nodup s \u2192 (s \u2264 t \u2194 s \u2286 t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082 d, \u27e8subset_of_le, subperm_of_subset_nodup d\u27e9\n\ntheorem range_le {m n : \u2115} : range m \u2264 range n \u2194 m \u2264 n :=\n(le_iff_subset (nodup_range _)).trans range_subset\n\ntheorem mem_sub_of_nodup [decidable_eq \u03b1] {a : \u03b1} {s t : multiset \u03b1} (d : nodup s) :\n  a \u2208 s - t \u2194 a \u2208 s \u2227 a \u2209 t :=\n\u27e8\u03bb h, \u27e8mem_of_le (sub_le_self _ _) h, \u03bb h',\n  by refine count_eq_zero.1 _ h; rw [count_sub a s t, nat.sub_eq_zero_iff_le];\n     exact le_trans (nodup_iff_count_le_one.1 d _) (count_pos.2 h')\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9, or.resolve_right (mem_add.1 $ mem_of_le (le_sub_add _ _) h\u2081) h\u2082\u27e9\n\nsection\nvariable [decidable_eq \u03b1]\n\n/- erase_dup -/\n\n/-- `erase_dup s` removes duplicates from `s`, yielding a `nodup` multiset. -/\ndef erase_dup (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.erase_dup : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_erase_dup_of_perm p))\n\n@[simp] theorem coe_erase_dup (l : list \u03b1) : @erase_dup \u03b1 _ l = l.erase_dup := rfl\n\n@[simp] theorem erase_dup_zero : @erase_dup \u03b1 _ 0 = 0 := rfl\n\n@[simp] theorem mem_erase_dup {a : \u03b1} {s : multiset \u03b1} : a \u2208 erase_dup s \u2194 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_erase_dup\n\n@[simp] theorem erase_dup_cons_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192\n  erase_dup (a::s) = erase_dup s :=\nquot.induction_on s $ \u03bb l m, @congr_arg _ _ _ _ coe $ erase_dup_cons_of_mem m\n\n@[simp] theorem erase_dup_cons_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192\n  erase_dup (a::s) = a :: erase_dup s :=\nquot.induction_on s $ \u03bb l m, congr_arg coe $ erase_dup_cons_of_not_mem m\n\ntheorem erase_dup_le (s : multiset \u03b1) : erase_dup s \u2264 s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ erase_dup_sublist _\n\ntheorem erase_dup_subset (s : multiset \u03b1) : erase_dup s \u2286 s :=\nsubset_of_le $ erase_dup_le _\n\ntheorem subset_erase_dup (s : multiset \u03b1) : s \u2286 erase_dup s :=\n\u03bb a, mem_erase_dup.2\n\n@[simp] theorem erase_dup_subset' {s t : multiset \u03b1} : erase_dup s \u2286 t \u2194 s \u2286 t :=\n\u27e8subset.trans (subset_erase_dup _), subset.trans (erase_dup_subset _)\u27e9\n\n@[simp] theorem subset_erase_dup' {s t : multiset \u03b1} : s \u2286 erase_dup t \u2194 s \u2286 t :=\n\u27e8\u03bb h, subset.trans h (erase_dup_subset _), \u03bb h, subset.trans h (subset_erase_dup _)\u27e9\n\n@[simp] theorem nodup_erase_dup (s : multiset \u03b1) : nodup (erase_dup s) :=\nquot.induction_on s nodup_erase_dup\n\ntheorem erase_dup_eq_self {s : multiset \u03b1} : erase_dup s = s \u2194 nodup s :=\n\u27e8\u03bb e, e \u25b8 nodup_erase_dup s,\n quot.induction_on s $ \u03bb l h, congr_arg coe $ erase_dup_eq_self.2 h\u27e9\n\n@[simp] theorem erase_dup_singleton {a : \u03b1} : erase_dup (a :: 0) = a :: 0 :=\nerase_dup_eq_self.2 $ nodup_singleton _\n\ntheorem le_erase_dup {s t : multiset \u03b1} : s \u2264 erase_dup t \u2194 s \u2264 t \u2227 nodup s :=\n\u27e8\u03bb h, \u27e8le_trans h (erase_dup_le _), nodup_of_le h (nodup_erase_dup _)\u27e9,\n \u03bb \u27e8l, d\u27e9, (le_iff_subset d).2 $ subset.trans (subset_of_le l) (subset_erase_dup _)\u27e9\n\ntheorem erase_dup_ext {s t : multiset \u03b1} : erase_dup s = erase_dup t \u2194 \u2200 a, a \u2208 s \u2194 a \u2208 t :=\nby simp [nodup_ext]\n\ntheorem erase_dup_map_erase_dup_eq [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n  erase_dup (map f (erase_dup s)) = erase_dup (map f s) := by simp [erase_dup_ext]\n\n/- finset insert -/\n\n/-- `ndinsert a s` is the lift of the list `insert` operation. This operation\n  does not respect multiplicities, unlike `cons`, but it is suitable as\n  an insert operation on `finset`. -/\ndef ndinsert (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\nquot.lift_on s (\u03bb l, (l.insert a : multiset \u03b1))\n  (\u03bb s t p, quot.sound (perm_insert a p))\n\n@[simp] theorem coe_ndinsert (a : \u03b1) (l : list \u03b1) : ndinsert a l = (insert a l : list \u03b1) := rfl\n\n@[simp] theorem ndinsert_zero (a : \u03b1) : ndinsert a 0 = a::0 := rfl\n\n@[simp] theorem ndinsert_of_mem {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 ndinsert a s = s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_mem h\n\n@[simp] theorem ndinsert_of_not_mem {a : \u03b1} {s : multiset \u03b1} : a \u2209 s \u2192 ndinsert a s = a :: s :=\nquot.induction_on s $ \u03bb l h, congr_arg coe $ insert_of_not_mem h\n\n@[simp] theorem mem_ndinsert {a b : \u03b1} {s : multiset \u03b1} : a \u2208 ndinsert b s \u2194 a = b \u2228 a \u2208 s :=\nquot.induction_on s $ \u03bb l, mem_insert_iff\n\n@[simp] theorem le_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : s \u2264 ndinsert a s :=\nquot.induction_on s $ \u03bb l, subperm_of_sublist $ sublist_of_suffix $ suffix_insert _ _\n\n@[simp] theorem mem_ndinsert_self (a : \u03b1) (s : multiset \u03b1) : a \u2208 ndinsert a s :=\nmem_ndinsert.2 (or.inl rfl)\n\n@[simp] theorem mem_ndinsert_of_mem {a b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 ndinsert b s :=\nmem_ndinsert.2 (or.inr h)\n\n@[simp] theorem length_ndinsert_of_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2208 s) :\n  card (ndinsert a s) = card s :=\nby simp [h]\n\n@[simp] theorem length_ndinsert_of_not_mem {a : \u03b1} [decidable_eq \u03b1] {s : multiset \u03b1} (h : a \u2209 s) :\n  card (ndinsert a s) = card s + 1 :=\nby simp [h]\n\ntheorem erase_dup_cons {a : \u03b1} {s : multiset \u03b1} :\n  erase_dup (a::s) = ndinsert a (erase_dup s) :=\nby by_cases a \u2208 s; simp [h]\n\ntheorem nodup_ndinsert (a : \u03b1) {s : multiset \u03b1} : nodup s \u2192 nodup (ndinsert a s) :=\nquot.induction_on s $ \u03bb l, nodup_insert\n\ntheorem ndinsert_le {a : \u03b1} {s t : multiset \u03b1} : ndinsert a s \u2264 t \u2194 s \u2264 t \u2227 a \u2208 t :=\n\u27e8\u03bb h, \u27e8le_trans (le_ndinsert_self _ _) h, mem_of_le h (mem_ndinsert_self _ _)\u27e9,\n \u03bb \u27e8l, m\u27e9, if h : a \u2208 s then by simp [h, l] else\n   by rw [ndinsert_of_not_mem h, \u2190 cons_erase m, cons_le_cons_iff,\n          \u2190 le_cons_of_not_mem h, cons_erase m]; exact l\u27e9\n\nlemma attach_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (s.ndinsert a).attach =\n    ndinsert \u27e8a, mem_ndinsert_self a s\u27e9 (s.attach.map $ \u03bbp, \u27e8p.1, mem_ndinsert_of_mem p.2\u27e9) :=\nhave eq : \u2200h : \u2200(p : {x // x \u2208 s}), p.1 \u2208 s,\n    (\u03bb (p : {x // x \u2208 s}), \u27e8p.val, h p\u27e9 : {x // x \u2208 s} \u2192 {x // x \u2208 s}) = id, from\n  assume h, funext $ assume p, subtype.eq rfl,\nhave \u2200t (eq : s.ndinsert a = t), t.attach = ndinsert \u27e8a, eq \u25b8 mem_ndinsert_self a s\u27e9\n  (s.attach.map $ \u03bbp, \u27e8p.1, eq \u25b8 mem_ndinsert_of_mem p.2\u27e9),\nbegin\n  intros t ht,\n  by_cases a \u2208 s,\n  { rw [ndinsert_of_mem h] at ht,\n    subst ht,\n    rw [eq, map_id, ndinsert_of_mem (mem_attach _ _)] },\n  { rw [ndinsert_of_not_mem h] at ht,\n    subst ht,\n    simp [attach_cons, h] }\nend,\nthis _ rfl\n\n@[simp] theorem disjoint_ndinsert_left {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint (ndinsert a s) t \u2194 a \u2209 t \u2227 disjoint s t :=\niff.trans (by simp [disjoint]) disjoint_cons_left\n\n@[simp] theorem disjoint_ndinsert_right {a : \u03b1} {s t : multiset \u03b1} :\n  disjoint s (ndinsert a t) \u2194 a \u2209 s \u2227 disjoint s t :=\ndisjoint_comm.trans $ by simp\n\n/- finset union -/\n\n/-- `ndunion s t` is the lift of the list `union` operation. This operation\n  does not respect multiplicities, unlike `s \u222a t`, but it is suitable as\n  a union operation on `finset`. (`s \u222a t` would also work as a union operation\n  on finset, but this is more efficient.) -/\ndef ndunion (s t : multiset \u03b1) : multiset \u03b1 :=\nquotient.lift_on\u2082 s t (\u03bb l\u2081 l\u2082, (l\u2081.union l\u2082 : multiset \u03b1)) $ \u03bb v\u2081 v\u2082 w\u2081 w\u2082 p\u2081 p\u2082,\n  quot.sound $ perm_union p\u2081 p\u2082\n\n@[simp] theorem coe_ndunion (l\u2081 l\u2082 : list \u03b1) : @ndunion \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u222a l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndunion (s : multiset \u03b1) : ndunion 0 s = s :=\nquot.induction_on s $ \u03bb l, rfl\n\n@[simp] theorem cons_ndunion (s t : multiset \u03b1) (a : \u03b1) : ndunion (a :: s) t = ndinsert a (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, rfl\n\n@[simp] theorem mem_ndunion {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndunion s t \u2194 a \u2208 s \u2228 a \u2208 t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.mem_union\n\ntheorem le_ndunion_right (s t : multiset \u03b1) : t \u2264 ndunion s t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082,\nsubperm_of_sublist $ sublist_of_suffix $ suffix_union_right _ _\n\ntheorem ndunion_le_add (s t : multiset \u03b1) : ndunion s t \u2264 s + t :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, subperm_of_sublist $ union_sublist_append _ _\n\ntheorem ndunion_le {s t u : multiset \u03b1} : ndunion s t \u2264 u \u2194 s \u2286 u \u2227 t \u2264 u :=\nmultiset.induction_on s (by simp) (by simp [ndinsert_le, and_comm, and.left_comm] {contextual := tt})\n\ntheorem subset_ndunion_left (s t : multiset \u03b1) : s \u2286 ndunion s t :=\n\u03bb a h, mem_ndunion.2 $ or.inl h\n\ntheorem le_ndunion_left {s} (t : multiset \u03b1) (d : nodup s) : s \u2264 ndunion s t :=\n(le_iff_subset d).2 $ subset_ndunion_left _ _\n\ntheorem ndunion_le_union (s t : multiset \u03b1) : ndunion s t \u2264 s \u222a t :=\nndunion_le.2 \u27e8subset_of_le (le_union_left _ _), le_union_right _ _\u27e9\n\ntheorem nodup_ndunion (s : multiset \u03b1) {t : multiset \u03b1} : nodup t \u2192 nodup (ndunion s t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, list.nodup_union _\n\n@[simp] theorem ndunion_eq_union {s t : multiset \u03b1} (d : nodup s) : ndunion s t = s \u222a t :=\nle_antisymm (ndunion_le_union _ _) $ union_le (le_ndunion_left _ d) (le_ndunion_right _ _)\n\ntheorem erase_dup_add (s t : multiset \u03b1) : erase_dup (s + t) = ndunion s (erase_dup t) :=\nquotient.induction_on\u2082 s t $ \u03bb l\u2081 l\u2082, congr_arg coe $ erase_dup_append _ _\n\n/- finset inter -/\n\n/-- `ndinter s t` is the lift of the list `\u2229` operation. This operation\n  does not respect multiplicities, unlike `s \u2229 t`, but it is suitable as\n  an intersection operation on `finset`. (`s \u2229 t` would also work as a union operation\n  on finset, but this is more efficient.) -/\ndef ndinter (s t : multiset \u03b1) : multiset \u03b1 := filter (\u2208 t) s\n\n@[simp] theorem coe_ndinter (l\u2081 l\u2082 : list \u03b1) : @ndinter \u03b1 _ l\u2081 l\u2082 = (l\u2081 \u2229 l\u2082 : list \u03b1) := rfl\n\n@[simp] theorem zero_ndinter (s : multiset \u03b1) : ndinter 0 s = 0 := rfl\n\n@[simp] theorem cons_ndinter_of_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2208 t) :\n  ndinter (a::s) t = a :: (ndinter s t) := by simp [ndinter, h]\n\n@[simp] theorem ndinter_cons_of_not_mem {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1} (h : a \u2209 t) :\n  ndinter (a::s) t = ndinter s t := by simp [ndinter, h]\n\n@[simp] theorem mem_ndinter {s t : multiset \u03b1} {a : \u03b1} : a \u2208 ndinter s t \u2194 a \u2208 s \u2227 a \u2208 t :=\nmem_filter\n\ntheorem nodup_ndinter {s : multiset \u03b1} (t : multiset \u03b1) : nodup s \u2192 nodup (ndinter s t) :=\nnodup_filter _\n\ntheorem le_ndinter {s t u : multiset \u03b1} : s \u2264 ndinter t u \u2194 s \u2264 t \u2227 s \u2286 u :=\nby simp [ndinter, le_filter, subset_iff]\n\ntheorem ndinter_le_left (s t : multiset \u03b1) : ndinter s t \u2264 s :=\n(le_ndinter.1 (le_refl _)).1\n\ntheorem ndinter_subset_right (s t : multiset \u03b1) : ndinter s t \u2286 t :=\n(le_ndinter.1 (le_refl _)).2\n\ntheorem ndinter_le_right {s} (t : multiset \u03b1) (d : nodup s) : ndinter s t \u2264 t :=\n(le_iff_subset $ nodup_ndinter _ d).2 (ndinter_subset_right _ _)\n\ntheorem inter_le_ndinter (s t : multiset \u03b1) : s \u2229 t \u2264 ndinter s t :=\nle_ndinter.2 \u27e8inter_le_left _ _, subset_of_le $ inter_le_right _ _\u27e9\n\n@[simp] theorem ndinter_eq_inter {s t : multiset \u03b1} (d : nodup s) : ndinter s t = s \u2229 t :=\nle_antisymm (le_inter (ndinter_le_left _ _) (ndinter_le_right _ d)) (inter_le_ndinter _ _)\n\ntheorem ndinter_eq_zero_iff_disjoint {s t : multiset \u03b1} : ndinter s t = 0 \u2194 disjoint s t :=\nby rw \u2190 subset_zero; simp [subset_iff, disjoint]\n\nend\n\n/- fold -/\nsection fold\nvariables (op : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [hc : is_commutative \u03b1 op] [ha : is_associative \u03b1 op]\nlocal notation a * b := op a b\ninclude hc ha\n\n/-- `fold op b s` folds a commutative associative operation `op` over\n  the multiset `s`. -/\ndef fold : \u03b1 \u2192 multiset \u03b1 \u2192 \u03b1 := foldr op (left_comm _ hc.comm ha.assoc)\n\ntheorem fold_eq_foldr (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldr op (left_comm _ hc.comm ha.assoc) b s := rfl\n\n@[simp] theorem coe_fold_r (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldr op b := rfl\n\ntheorem coe_fold_l (b : \u03b1) (l : list \u03b1) : fold op b l = l.foldl op b :=\n(coe_foldr_swap op _ b l).trans $ by simp [hc.comm]\n\ntheorem fold_eq_foldl (b : \u03b1) (s : multiset \u03b1) : fold op b s = foldl op (right_comm _ hc.comm ha.assoc) b s :=\nquot.induction_on s $ \u03bb l, coe_fold_l _ _ _\n\n@[simp] theorem fold_zero (b : \u03b1) : (0 : multiset \u03b1).fold op b = b := rfl\n\n@[simp] theorem fold_cons_left : \u2200 (b a : \u03b1) (s : multiset \u03b1),\n  (a :: s).fold op b = a * s.fold op b := foldr_cons _ _\n\ntheorem fold_cons_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op b * a :=\nby simp [hc.comm]\n\ntheorem fold_cons'_right (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (b * a) :=\nby rw [fold_eq_foldl, foldl_cons, \u2190 fold_eq_foldl]\n\ntheorem fold_cons'_left (b a : \u03b1) (s : multiset \u03b1) : (a :: s).fold op b = s.fold op (a * b) :=\nby rw [fold_cons'_right, hc.comm]\n\ntheorem fold_add (b\u2081 b\u2082 : \u03b1) (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).fold op (b\u2081 * b\u2082) = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nmultiset.induction_on s\u2082\n  (by rw [add_zero, fold_zero, \u2190 fold_cons'_right, \u2190 fold_cons_right op])\n  (by simp {contextual := tt}; cc)\n\ntheorem fold_singleton (b a : \u03b1) : (a::0 : multiset \u03b1).fold op b = a * b := by simp\n\ntheorem fold_distrib {f g : \u03b2 \u2192 \u03b1} (u\u2081 u\u2082 : \u03b1) (s : multiset \u03b2) :\n  (s.map (\u03bbx, f x * g x)).fold op (u\u2081 * u\u2082) = (s.map f).fold op u\u2081 * (s.map g).fold op u\u2082 :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt}; cc)\n\ntheorem fold_hom {op' : \u03b2 \u2192 \u03b2 \u2192 \u03b2} [is_commutative \u03b2 op'] [is_associative \u03b2 op']\n  {m : \u03b1 \u2192 \u03b2} (hm : \u2200x y, m (op x y) = op' (m x) (m y)) (b : \u03b1) (s : multiset \u03b1) :\n  (s.map m).fold op' (m b) = m (s.fold op b) :=\nmultiset.induction_on s (by simp) (by simp [hm] {contextual := tt})\n\ntheorem fold_union_inter [decidable_eq \u03b1] (s\u2081 s\u2082 : multiset \u03b1) (b\u2081 b\u2082 : \u03b1) :\n  (s\u2081 \u222a s\u2082).fold op b\u2081 * (s\u2081 \u2229 s\u2082).fold op b\u2082 = s\u2081.fold op b\u2081 * s\u2082.fold op b\u2082 :=\nby rw [\u2190 fold_add op, union_add_inter, fold_add op]\n\n@[simp] theorem fold_erase_dup_idem [decidable_eq \u03b1] [hi : is_idempotent \u03b1 op] (s : multiset \u03b1) (b : \u03b1) :\n  (erase_dup s).fold op b = s.fold op b :=\nmultiset.induction_on s (by simp) $ \u03bb a s IH, begin\n  by_cases a \u2208 s; simp [IH, h],\n  show fold op b s = op a (fold op b s),\n  rw [\u2190 cons_erase h, fold_cons_left, \u2190 ha.assoc, hi.idempotent],\nend\n\nend fold\n\ntheorem le_smul_erase_dup [decidable_eq \u03b1] (s : multiset \u03b1) :\n  \u2203 n : \u2115, s \u2264 n \u2022 erase_dup s :=\n\u27e8(s.map (\u03bb a, count a s)).fold max 0, le_iff_count.2 $ \u03bb a, begin\n  rw count_smul, by_cases a \u2208 s,\n  { refine le_trans _ (mul_le_mul_left _ $ count_pos.2 $ mem_erase_dup.2 h),\n    have : count a s \u2264 fold max 0 (map (\u03bb a, count a s) (a :: erase s a));\n    [simp [le_max_left], simpa [cons_erase h]] },\n  { simp [count_eq_zero.2 h, nat.zero_le] }\nend\u27e9\n\nsection sup\nvariables [semilattice_sup_bot \u03b1]\n\n/-- Supremum of a multiset: `sup {a, b, c} = a \u2294 b \u2294 c` -/\ndef sup (s : multiset \u03b1) : \u03b1 := s.fold (\u2294) \u22a5\n\n@[simp] lemma sup_zero : (0 : multiset \u03b1).sup = \u22a5 :=\nfold_zero _ _\n\n@[simp] lemma sup_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).sup = a \u2294 s.sup :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma sup_singleton {a : \u03b1} : (a::0).sup = a := by simp\n\n@[simp] lemma sup_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\neq.trans (by simp [sup]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma sup_erase_dup (s : multiset \u03b1) : (erase_dup s).sup = s.sup :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma sup_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).sup = s\u2081.sup \u2294 s\u2082.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_add]; simp\n\n@[simp] lemma sup_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).sup = a \u2294 s.sup :=\nby rw [\u2190 sup_erase_dup, erase_dup_ext.2, sup_erase_dup, sup_cons]; simp\n\nlemma sup_le {s : multiset \u03b1} {a : \u03b1} : s.sup \u2264 a \u2194 (\u2200b \u2208 s, b \u2264 a) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma le_sup {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : a \u2264 s.sup :=\nsup_le.1 (le_refl _) _ h\n\nlemma sup_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2081.sup \u2264 s\u2082.sup :=\nsup_le.2 $ assume b hb, le_sup (h hb)\n\nend sup\n\nsection inf\nvariables [semilattice_inf_top \u03b1]\n\n/-- Infimum of a multiset: `inf {a, b, c} = a \u2293 b \u2293 c` -/\ndef inf (s : multiset \u03b1) : \u03b1 := s.fold (\u2293) \u22a4\n\n@[simp] lemma inf_zero : (0 : multiset \u03b1).inf = \u22a4 :=\nfold_zero _ _\n\n@[simp] lemma inf_cons (a : \u03b1) (s : multiset \u03b1) :\n  (a :: s).inf = a \u2293 s.inf :=\nfold_cons_left _ _ _ _\n\n@[simp] lemma inf_singleton {a : \u03b1} : (a::0).inf = a := by simp\n\n@[simp] lemma inf_add (s\u2081 s\u2082 : multiset \u03b1) : (s\u2081 + s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\neq.trans (by simp [inf]) (fold_add _ _ _ _ _)\n\nvariables [decidable_eq \u03b1]\n\n@[simp] lemma inf_erase_dup (s : multiset \u03b1) : (erase_dup s).inf = s.inf :=\nfold_erase_dup_idem _ _ _\n\n@[simp] lemma inf_ndunion (s\u2081 s\u2082 : multiset \u03b1) :\n  (ndunion s\u2081 s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_union (s\u2081 s\u2082 : multiset \u03b1) :\n  (s\u2081 \u222a s\u2082).inf = s\u2081.inf \u2293 s\u2082.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_add]; simp\n\n@[simp] lemma inf_ndinsert (a : \u03b1) (s : multiset \u03b1) :\n  (ndinsert a s).inf = a \u2293 s.inf :=\nby rw [\u2190 inf_erase_dup, erase_dup_ext.2, inf_erase_dup, inf_cons]; simp\n\nlemma le_inf {s : multiset \u03b1} {a : \u03b1} : a \u2264 s.inf \u2194 (\u2200b \u2208 s, a \u2264 b) :=\nmultiset.induction_on s (by simp)\n  (by simp [or_imp_distrib, forall_and_distrib] {contextual := tt})\n\nlemma inf_le {s : multiset \u03b1} {a : \u03b1} (h : a \u2208 s) : s.inf \u2264 a :=\nle_inf.1 (le_refl _) _ h\n\nlemma inf_mono {s\u2081 s\u2082 : multiset \u03b1} (h : s\u2081 \u2286 s\u2082) : s\u2082.inf \u2264 s\u2081.inf :=\nle_inf.2 $ assume b hb, inf_le (h hb)\n\nend inf\n\nsection sort\nvariables (r : \u03b1 \u2192 \u03b1 \u2192 Prop) [decidable_rel r]\n  [is_trans \u03b1 r] [is_antisymm \u03b1 r] [is_total \u03b1 r]\n\n/-- `sort s` constructs a sorted list from the multiset `s`.\n  (Uses merge sort algorithm.) -/\ndef sort (s : multiset \u03b1) : list \u03b1 :=\nquot.lift_on s (merge_sort r) $ \u03bb a b h,\neq_of_sorted_of_perm\n  ((perm_merge_sort _ _).trans $ h.trans (perm_merge_sort _ _).symm)\n  (sorted_merge_sort r _)\n  (sorted_merge_sort r _)\n\n@[simp] theorem coe_sort (l : list \u03b1) : sort r l = merge_sort r l := rfl\n\n@[simp] theorem sort_sorted (s : multiset \u03b1) : sorted r (sort r s) :=\nquot.induction_on s $ \u03bb l, sorted_merge_sort r _\n\n@[simp] theorem sort_eq (s : multiset \u03b1) : \u2191(sort r s) = s :=\nquot.induction_on s $ \u03bb l, quot.sound $ perm_merge_sort _ _\n\n@[simp] theorem mem_sort {s : multiset \u03b1} {a : \u03b1} : a \u2208 sort r s \u2194 a \u2208 s :=\nby rw [\u2190 mem_coe, sort_eq]\n\nend sort\n\ninstance [has_repr \u03b1] : has_repr (multiset \u03b1) :=\n\u27e8\u03bb s, \"{\" ++ string.intercalate \", \" ((s.map repr).sort (\u2264)) ++ \"}\"\u27e9\n\nsection sections\n\ndef sections (s : multiset (multiset \u03b1)) : multiset (multiset \u03b1) :=\nmultiset.rec_on s {0} (\u03bbs _ c, s.bind $ \u03bba, c.map ((::) a))\n  (assume a\u2080 a\u2081 s pi, by simp [map_bind, bind_bind a\u2080 a\u2081, cons_swap])\n\n@[simp] lemma sections_zero : sections (0 : multiset (multiset \u03b1)) = 0::0 :=\nrfl\n\n@[simp] lemma sections_cons (s : multiset (multiset \u03b1)) (m : multiset \u03b1) :\n  sections (m :: s) = m.bind (\u03bba, (sections s).map ((::) a)) :=\nrec_on_cons m s\n\nlemma coe_sections : \u2200(l : list (list \u03b1)),\n  sections ((l.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1)) =\n    ((l.sections.map (\u03bbl:list \u03b1, (l : multiset \u03b1))) : multiset (multiset \u03b1))\n| [] := rfl\n| (a :: l) :=\n  begin\n    simp,\n    rw [\u2190 cons_coe, sections_cons, bind_map_comm, coe_sections l],\n    simp [list.sections, (\u2218), list.bind]\n  end\n\n@[simp] lemma sections_add (s t : multiset (multiset \u03b1)) :\n  sections (s + t) = (sections s).bind (\u03bbm, (sections t).map ((+) m)) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, bind_assoc, map_bind, bind_map, -add_comm])\n\nlemma mem_sections {s : multiset (multiset \u03b1)} :\n  \u2200{a}, a \u2208 sections s \u2194 s.rel (\u03bbs a, a \u2208 s) a :=\nmultiset.induction_on s (by simp)\n  (assume a s ih a',\n    by simp [ih, rel_cons_left, -exists_and_distrib_left, exists_and_distrib_left.symm, eq_comm])\n\nlemma card_sections {s : multiset (multiset \u03b1)} : card (sections s) = prod (s.map card) :=\nmultiset.induction_on s (by simp) (by simp {contextual := tt})\n\nlemma prod_map_sum [comm_semiring \u03b1] {s : multiset (multiset \u03b1)} :\n  prod (s.map sum) = sum ((sections s).map prod) :=\nmultiset.induction_on s (by simp)\n  (assume a s ih, by simp [ih, map_bind, sum_map_mul_left, sum_map_mul_right])\n\nend sections\n\nsection pi\nvariables [decidable_eq \u03b1] {\u03b4 : \u03b1 \u2192 Type*}\nopen function\n\ndef pi.cons (m : multiset \u03b1) (a : \u03b1) (b : \u03b4 a) (f : \u03a0a\u2208m, \u03b4 a) : \u03a0a'\u2208a::m, \u03b4 a' :=\n\u03bba' ha', if h : a' = a then eq.rec b h.symm else f a' $ (mem_cons.1 ha').resolve_left h\n\ndef pi.empty (\u03b4 : \u03b1 \u2192 Type*) : (\u03a0a\u2208(0:multiset \u03b1), \u03b4 a) .\n\nlemma pi.cons_same {m : multiset \u03b1} {a : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2208 a :: m) :\n  pi.cons m a b f a h = b :=\ndif_pos rfl\n\nlemma pi.cons_ne {m : multiset \u03b1} {a a' : \u03b1} {b : \u03b4 a} {f : \u03a0a\u2208m, \u03b4 a} (h' : a' \u2208 a :: m) (h : a' \u2260 a) :\n  pi.cons m a b f a' h' = f a' ((mem_cons.1 h').resolve_left h) :=\ndif_neg h\n\nlemma pi.cons_swap {a a' : \u03b1} {b : \u03b4 a} {b' : \u03b4 a'} {m : multiset \u03b1} {f : \u03a0a\u2208m, \u03b4 a} (h : a \u2260 a') :\n  pi.cons (a' :: m) a b (pi.cons m a' b' f) == pi.cons (a :: m) a' b' (pi.cons m a b f) :=\nbegin\n  apply hfunext, { refl }, intros a'' _ h, subst h,\n  apply hfunext, { rw [cons_swap] }, intros ha\u2081 ha\u2082 h,\n  by_cases h\u2081 : a'' = a; by_cases h\u2082 : a'' = a';\n    simp [*, pi.cons_same, pi.cons_ne] at *,\n  { subst h\u2081, rw [pi.cons_same, pi.cons_same] },\n  { subst h\u2082, rw [pi.cons_same, pi.cons_same] }\nend\n\n/-- `pi m t` constructs the Cartesian product over `t` indexed by `m`. -/\ndef pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) : multiset (\u03a0a\u2208m, \u03b4 a) :=\nm.rec_on {pi.empty \u03b4} (\u03bba m (p : multiset (\u03a0a\u2208m, \u03b4 a)), (t a).bind $ \u03bbb, p.map $ pi.cons m a b)\nbegin\n  intros a a' m n,\n  by_cases eq : a = a',\n  { subst eq },\n  { simp [map_bind, bind_bind (t a') (t a)],\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b hb,\n    apply bind_hcongr, { rw [cons_swap a a'] },\n    intros b' hb',\n    apply map_hcongr, { rw [cons_swap a a'] },\n    intros f hf,\n    exact pi.cons_swap eq }\nend\n\n@[simp] lemma pi_zero (t : \u03a0a, multiset (\u03b4 a)) : pi 0 t = pi.empty \u03b4 :: 0 := rfl\n\n@[simp] lemma pi_cons (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) (a : \u03b1) :\n  pi (a :: m) t = ((t a).bind $ \u03bbb, (pi m t).map $ pi.cons m a b) :=\nrec_on_cons a m\n\nlemma injective_pi_cons {a : \u03b1} {b : \u03b4 a} {s : multiset \u03b1} (hs : a \u2209 s) :\n  function.injective (pi.cons s a b) :=\nassume f\u2081 f\u2082 eq, funext $ assume a', funext $ assume h',\nhave ne : a \u2260 a', from assume h, hs $ h.symm \u25b8 h',\nhave a' \u2208 a :: s, from mem_cons_of_mem h',\ncalc f\u2081 a' h' = pi.cons s a b f\u2081 a' this : by rw [pi.cons_ne this ne.symm]\n  ... = pi.cons s a b f\u2082 a' this : by rw [eq]\n  ... = f\u2082 a' h' : by rw [pi.cons_ne this ne.symm]\n\nlemma card_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  card (pi m t) = prod (m.map $ \u03bba, card (t a)) :=\nmultiset.induction_on m (by simp) (by simp [mul_comm] {contextual := tt})\n\nlemma nodup_pi {s : multiset \u03b1} {t : \u03a0a, multiset (\u03b4 a)} :\n  nodup s \u2192 (\u2200a\u2208s, nodup (t a)) \u2192 nodup (pi s t) :=\nmultiset.induction_on s (assume _ _, nodup_singleton _)\nbegin\n  assume a s ih hs ht,\n  have has : a \u2209 s, by simp at hs; exact hs.1,\n  have hs : nodup s, by simp at hs; exact hs.2,\n  simp,\n  split,\n  { assume b hb,\n    from nodup_map (injective_pi_cons has) (ih hs $ assume a' h', ht a' $ mem_cons_of_mem h') },\n  { apply pairwise_of_nodup _ (ht a $ mem_cons_self _ _),\n    from assume b\u2081 hb\u2081 b\u2082 hb\u2082 neb, disjoint_map_map.2 (assume f hf g hg 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 _ _),\n        by rw [eq],\n      neb $ show b\u2081 = b\u2082, by rwa [pi.cons_same, pi.cons_same] at this) }\nend\n\nlemma mem_pi (m : multiset \u03b1) (t : \u03a0a, multiset (\u03b4 a)) :\n  \u2200f:\u03a0a\u2208m, \u03b4 a, (f \u2208 pi m t) \u2194 (\u2200a (h : a \u2208 m), f a h \u2208 t a) :=\nbegin\n  refine multiset.induction_on m (\u03bb f, _) (\u03bb a m ih f, _),\n  { simpa using show f = pi.empty \u03b4, by funext a ha; exact ha.elim },\n  simp, split,\n  { rintro \u27e8b, hb, f', hf', rfl\u27e9 a' ha',\n    rw [ih] at hf',\n    by_cases a' = a,\n    { subst h, rwa [pi.cons_same] },\n    { rw [pi.cons_ne _ h], apply hf' } },\n  { intro hf,\n    refine \u27e8_, hf a (mem_cons_self a _), \u03bba ha, f a (mem_cons_of_mem ha),\n      (ih _).2 (\u03bb a' h', hf _ _), _\u27e9,\n    funext a' h',\n    by_cases a' = a,\n    { subst h, rw [pi.cons_same] },\n    { rw [pi.cons_ne _ h] } }\nend\n\nend pi\nend multiset\n\nnamespace multiset\n\ninstance : functor multiset :=\n{ map := @map }\n\ninstance : is_lawful_functor multiset :=\nby refine { .. }; intros; simp\n\nopen is_lawful_traversable is_comm_applicative\n\nvariables {F : Type u_1 \u2192 Type u_1} [applicative F] [is_comm_applicative F]\nvariables {\u03b1' \u03b2' : Type u_1} (f : \u03b1' \u2192 F \u03b2')\n\ndef traverse : multiset \u03b1' \u2192 F (multiset \u03b2') :=\nquotient.lift (functor.map coe \u2218 traversable.traverse f)\nbegin\n  introv p, unfold function.comp,\n  induction p,\n  case perm.nil { refl },\n  case perm.skip {\n    have : multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2081) =\n      multiset.cons <$> f p_x <*> (coe <$> traverse f p_l\u2082),\n    { rw [p_ih] },\n    simpa with functor_norm },\n  case perm.swap {\n    have : (\u03bba b (l:list \u03b2'), (\u2191(a :: b :: l) : multiset \u03b2')) <$> f p_y <*> f p_x =\n      (\u03bba b l, \u2191(a :: b :: l)) <$> f p_x <*> f p_y,\n    { rw [is_comm_applicative.commutative_map],\n      congr, funext a b l, simpa [flip] using perm.swap b a l },\n    simp [(\u2218), this] with functor_norm },\n  case perm.trans { simp [*] }\nend\n\nopen functor\nopen traversable is_lawful_traversable\n\n@[simp]\nlemma lift_beta {\u03b1 \u03b2 : Type*} (x : list \u03b1) (f : list \u03b1 \u2192 \u03b2)\n  (h : \u2200 a b : list \u03b1, a \u2248 b \u2192 f a = f b) :\n  quotient.lift f h (x : multiset \u03b1) = f x :=\nquotient.lift_beta _ _ _\n\n@[simp]\nlemma map_comp_coe {\u03b1 \u03b2} (h : \u03b1 \u2192 \u03b2) :\n  functor.map h \u2218 coe = (coe \u2218 functor.map h : list \u03b1 \u2192 multiset \u03b2) :=\nby funext; simp [functor.map]\n\nlemma id_traverse {\u03b1 : Type*} (x : multiset \u03b1) :\n  traverse id.mk x = x :=\nquotient.induction_on x\n(by { intro, rw [traverse,quotient.lift_beta,function.comp],\n      simp, congr })\n\nlemma comp_traverse {G H : Type* \u2192 Type*}\n               [applicative G] [applicative H]\n               [is_comm_applicative G] [is_comm_applicative H]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 H \u03b3) (x : multiset \u03b1) :\n  traverse (comp.mk \u2218 functor.map h \u2218 g) x =\n  comp.mk (functor.map (traverse h) (traverse g x)) :=\nquotient.induction_on x\n(by intro;\n    simp [traverse,comp_traverse] with functor_norm;\n    simp [(<$>),(\u2218)] with functor_norm)\n\nlemma map_traverse {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 G \u03b2) (h : \u03b2 \u2192 \u03b3)\n               (x : multiset \u03b1) :\n  functor.map (functor.map h) (traverse g x) =\n  traverse (functor.map h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse] with functor_norm;\n    rw [comp_map,map_traverse])\n\nlemma traverse_map {G : Type* \u2192 Type*}\n               [applicative G] [is_comm_applicative G]\n               {\u03b1 \u03b2 \u03b3 : Type*}\n               (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 G \u03b3)\n               (x : multiset \u03b1) :\n  traverse h (map g x) =\n  traverse (h \u2218 g) x :=\nquotient.induction_on x\n(by intro; simp [traverse];\n    rw [\u2190 traversable.traverse_map h g];\n    [ refl, apply_instance ])\n\nlemma naturality {G H : Type* \u2192 Type*}\n                [applicative G] [applicative H]\n                [is_comm_applicative G] [is_comm_applicative H]\n                (eta : applicative_transformation G H)\n                {\u03b1 \u03b2 : Type*} (f : \u03b1 \u2192 G \u03b2) (x : multiset \u03b1) :\n  eta (traverse f x) = traverse (@eta _ \u2218 f) x :=\nquotient.induction_on x\n(by intro; simp [traverse,is_lawful_traversable.naturality] with functor_norm)\n\nend multiset\n", "meta": {"author": "khoek", "repo": "mathlib-tidy", "sha": "866afa6ab597c47f1b72e8fe2b82b97fff5b980f", "save_path": "github-repos/lean/khoek-mathlib-tidy", "path": "github-repos/lean/khoek-mathlib-tidy/mathlib-tidy-866afa6ab597c47f1b72e8fe2b82b97fff5b980f/data/multiset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6825737408694988, "lm_q1q2_score": 0.35461161218630743}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n\nCoinductive formalization of unbounded computations.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.Lean3Lib.data.stream\nimport Mathlib.tactic.basic\nimport Mathlib.PostPort\n\nuniverses u u_1 v w \n\nnamespace Mathlib\n\n/-\ncoinductive computation (\u03b1 : Type u) : Type u\n| return : \u03b1 \u2192 computation \u03b1\n| think : computation \u03b1 \u2192 computation \u03b1\n-/\n\n/-- `computation \u03b1` is the type of unbounded computations returning `\u03b1`.\n  An element of `computation \u03b1` is an infinite sequence of `option \u03b1` such\n  that if `f n = some a` for some `n` then it is constantly `some a` after that. -/\ndef computation (\u03b1 : Type u) :=\n  Subtype fun (f : stream (Option \u03b1)) => \u2200 {n : \u2115} {a : \u03b1}, f n = some a \u2192 f (n + 1) = some a\n\nnamespace computation\n\n\n-- constructors\n\n/-- `return a` is the computation that immediately terminates with result `a`. -/\ndef return {\u03b1 : Type u} (a : \u03b1) : computation \u03b1 :=\n  { val := stream.const (some a), property := sorry }\n\nprotected instance has_coe_t {\u03b1 : Type u} : has_coe_t \u03b1 (computation \u03b1) :=\n  has_coe_t.mk return\n\n/-- `think c` is the computation that delays for one \"tick\" and then performs\n  computation `c`. -/\ndef think {\u03b1 : Type u} (c : computation \u03b1) : computation \u03b1 :=\n  { val := none :: subtype.val c, property := sorry }\n\n/-- `thinkN c n` is the computation that delays for `n` ticks and then performs\n  computation `c`. -/\ndef thinkN {\u03b1 : Type u} (c : computation \u03b1) : \u2115 \u2192 computation \u03b1 :=\n  sorry\n\n-- check for immediate result\n\n/-- `head c` is the first step of computation, either `some a` if `c = return a`\n  or `none` if `c = think c'`. -/\ndef head {\u03b1 : Type u} (c : computation \u03b1) : Option \u03b1 :=\n  stream.head (subtype.val c)\n\n-- one step of computation\n\n/-- `tail c` is the remainder of computation, either `c` if `c = return a`\n  or `c'` if `c = think c'`. -/\ndef tail {\u03b1 : Type u} (c : computation \u03b1) : computation \u03b1 :=\n  { val := stream.tail (subtype.val c), property := sorry }\n\n/-- `empty \u03b1` is the computation that never returns, an infinite sequence of\n  `think`s. -/\ndef empty (\u03b1 : Type u_1) : computation \u03b1 :=\n  { val := stream.const none, property := sorry }\n\nprotected instance inhabited {\u03b1 : Type u} : Inhabited (computation \u03b1) :=\n  { default := empty \u03b1 }\n\n/-- `run_for c n` evaluates `c` for `n` steps and returns the result, or `none`\n  if it did not terminate after `n` steps. -/\ndef run_for {\u03b1 : Type u} : computation \u03b1 \u2192 \u2115 \u2192 Option \u03b1 :=\n  subtype.val\n\n/-- `destruct c` is the destructor for `computation \u03b1` as a coinductive type.\n  It returns `inl a` if `c = return a` and `inr c'` if `c = think c'`. -/\ndef destruct {\u03b1 : Type u} (c : computation \u03b1) : \u03b1 \u2295 computation \u03b1 :=\n  sorry\n\n/-- `run c` is an unsound meta function that runs `c` to completion, possibly\n  resulting in an infinite loop in the VM. -/\ntheorem destruct_eq_ret {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : destruct s = sum.inl a \u2192 s = return a := sorry\n\ntheorem destruct_eq_think {\u03b1 : Type u} {s : computation \u03b1} {s' : computation \u03b1} : destruct s = sum.inr s' \u2192 s = think s' := sorry\n\n@[simp] theorem destruct_ret {\u03b1 : Type u} (a : \u03b1) : destruct (return a) = sum.inl a :=\n  rfl\n\n@[simp] theorem destruct_think {\u03b1 : Type u} (s : computation \u03b1) : destruct (think s) = sum.inr s := sorry\n\n@[simp] theorem destruct_empty {\u03b1 : Type u} : destruct (empty \u03b1) = sum.inr (empty \u03b1) :=\n  rfl\n\n@[simp] theorem head_ret {\u03b1 : Type u} (a : \u03b1) : head (return a) = some a :=\n  rfl\n\n@[simp] theorem head_think {\u03b1 : Type u} (s : computation \u03b1) : head (think s) = none :=\n  rfl\n\n@[simp] theorem head_empty {\u03b1 : Type u} : head (empty \u03b1) = none :=\n  rfl\n\n@[simp] theorem tail_ret {\u03b1 : Type u} (a : \u03b1) : tail (return a) = return a :=\n  rfl\n\n@[simp] theorem tail_think {\u03b1 : Type u} (s : computation \u03b1) : tail (think s) = s := sorry\n\n@[simp] theorem tail_empty {\u03b1 : Type u} : tail (empty \u03b1) = empty \u03b1 :=\n  rfl\n\ntheorem think_empty {\u03b1 : Type u} : empty \u03b1 = think (empty \u03b1) :=\n  destruct_eq_think destruct_empty\n\ndef cases_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} (s : computation \u03b1) (h1 : (a : \u03b1) \u2192 C (return a)) (h2 : (s : computation \u03b1) \u2192 C (think s)) : C s :=\n  (fun (_x : \u03b1 \u2295 computation \u03b1) (H : destruct s = _x) =>\n      sum.rec (fun (v : \u03b1) (H : destruct s = sum.inl v) => eq.mpr sorry (h1 v))\n        (fun (v : computation \u03b1) (H : destruct s = sum.inr v) =>\n          subtype.cases_on v\n            (fun (a : stream (Option \u03b1)) (s' : \u2200 {n : \u2115} {a_1 : \u03b1}, a n = some a_1 \u2192 a (n + 1) = some a_1)\n              (H : destruct s = sum.inr { val := a, property := s' }) => eq.mpr sorry (h2 { val := a, property := s' }))\n            H)\n        _x H)\n    (destruct s) sorry\n\ndef corec.F {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) : \u03b1 \u2295 \u03b2 \u2192 Option \u03b1 \u00d7 (\u03b1 \u2295 \u03b2) :=\n  sorry\n\n/-- `corec f b` is the corecursor for `computation \u03b1` as a coinductive type.\n  If `f b = inl a` then `corec f b = return a`, and if `f b = inl b'` then\n  `corec f b = think (corec f b')`. -/\ndef corec {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) (b : \u03b2) : computation \u03b1 :=\n  { val := stream.corec' sorry (sum.inr b), property := sorry }\n\n/-- left map of `\u2295` -/\n@[simp] def lmap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) : \u03b1 \u2295 \u03b3 \u2192 \u03b2 \u2295 \u03b3 :=\n  sorry\n\n/-- right map of `\u2295` -/\n@[simp] def rmap {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b2 \u2192 \u03b3) : \u03b1 \u2295 \u03b2 \u2192 \u03b1 \u2295 \u03b3 :=\n  sorry\n\n@[simp] theorem corec_eq {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b2 \u2192 \u03b1 \u2295 \u03b2) (b : \u03b2) : destruct (corec f b) = rmap (corec f) (f b) := sorry\n\n@[simp] def bisim_o {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop) : \u03b1 \u2295 computation \u03b1 \u2192 \u03b1 \u2295 computation \u03b1 \u2192 Prop :=\n  sorry\n\ndef is_bisimulation {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop) :=\n  \u2200 {s\u2081 s\u2082 : computation \u03b1}, R s\u2081 s\u2082 \u2192 bisim_o R (destruct s\u2081) (destruct s\u2082)\n\ntheorem eq_of_bisim {\u03b1 : Type u} (R : computation \u03b1 \u2192 computation \u03b1 \u2192 Prop) (bisim : is_bisimulation R) {s\u2081 : computation \u03b1} {s\u2082 : computation \u03b1} (r : R s\u2081 s\u2082) : s\u2081 = s\u2082 := sorry\n\n-- It's more of a stretch to use \u2208 for this relation, but it\n\n-- asserts that the computation limits to the given value.\n\nprotected def mem {\u03b1 : Type u} (a : \u03b1) (s : computation \u03b1) :=\n  some a \u2208 subtype.val s\n\nprotected instance has_mem {\u03b1 : Type u} : has_mem \u03b1 (computation \u03b1) :=\n  has_mem.mk computation.mem\n\ntheorem le_stable {\u03b1 : Type u} (s : computation \u03b1) {a : \u03b1} {m : \u2115} {n : \u2115} (h : m \u2264 n) : subtype.val s m = some a \u2192 subtype.val s n = some a := sorry\n\ntheorem mem_unique {\u03b1 : Type u} : relator.left_unique has_mem.mem := sorry\n\n/-- `terminates s` asserts that the computation `s` eventually terminates with some value. -/\ndef terminates {\u03b1 : Type u} (s : computation \u03b1) :=\n  \u2203 (a : \u03b1), a \u2208 s\n\ntheorem terminates_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 terminates s :=\n  exists.intro a\n\ntheorem terminates_def {\u03b1 : Type u} (s : computation \u03b1) : terminates s \u2194 \u2203 (n : \u2115), \u21a5(option.is_some (subtype.val s n)) := sorry\n\ntheorem ret_mem {\u03b1 : Type u} (a : \u03b1) : a \u2208 return a :=\n  exists.intro 0 rfl\n\ntheorem eq_of_ret_mem {\u03b1 : Type u} {a : \u03b1} {a' : \u03b1} (h : a' \u2208 return a) : a' = a :=\n  mem_unique h (ret_mem a)\n\nprotected instance ret_terminates {\u03b1 : Type u} (a : \u03b1) : terminates (return a) :=\n  terminates_of_mem (ret_mem a)\n\ntheorem think_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 a \u2208 think s := sorry\n\nprotected instance think_terminates {\u03b1 : Type u} (s : computation \u03b1) [terminates s] : terminates (think s) :=\n  sorry\n\ntheorem of_think_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 think s \u2192 a \u2208 s := sorry\n\ntheorem of_think_terminates {\u03b1 : Type u} {s : computation \u03b1} : terminates (think s) \u2192 terminates s :=\n  fun (\u1fb0 : terminates (think s)) =>\n    Exists.dcases_on \u1fb0\n      fun (\u1fb0_w : \u03b1) (\u1fb0_h : \u1fb0_w \u2208 think s) => idRhs (\u2203 (a : \u03b1), a \u2208 s) (Exists.intro \u1fb0_w (of_think_mem \u1fb0_h))\n\ntheorem not_mem_empty {\u03b1 : Type u} (a : \u03b1) : \u00aca \u2208 empty \u03b1 := sorry\n\ntheorem not_terminates_empty {\u03b1 : Type u} : \u00acterminates (empty \u03b1) := sorry\n\ntheorem eq_empty_of_not_terminates {\u03b1 : Type u} {s : computation \u03b1} (H : \u00acterminates s) : s = empty \u03b1 := sorry\n\ntheorem thinkN_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (n : \u2115) : a \u2208 thinkN s n \u2194 a \u2208 s := sorry\n\nprotected instance thinkN_terminates {\u03b1 : Type u} (s : computation \u03b1) [terminates s] (n : \u2115) : terminates (thinkN s n) :=\n  sorry\n\ntheorem of_thinkN_terminates {\u03b1 : Type u} (s : computation \u03b1) (n : \u2115) : terminates (thinkN s n) \u2192 terminates s :=\n  fun (\u1fb0 : terminates (thinkN s n)) =>\n    Exists.dcases_on \u1fb0\n      fun (\u1fb0_w : \u03b1) (\u1fb0_h : \u1fb0_w \u2208 thinkN s n) => idRhs (\u2203 (a : \u03b1), a \u2208 s) (Exists.intro \u1fb0_w (iff.mp (thinkN_mem n) \u1fb0_h))\n\n/-- `promises s a`, or `s ~> a`, asserts that although the computation `s`\n  may not terminate, if it does, then the result is `a`. -/\ndef promises {\u03b1 : Type u} (s : computation \u03b1) (a : \u03b1) :=\n  \u2200 {a' : \u03b1}, a' \u2208 s \u2192 a = a'\n\ninfixl:50 \" ~> \" => Mathlib.computation.promises\n\ntheorem mem_promises {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} : a \u2208 s \u2192 s ~> a :=\n  fun (h : a \u2208 s) (a' : \u03b1) => mem_unique h\n\ntheorem empty_promises {\u03b1 : Type u} (a : \u03b1) : empty \u03b1 ~> a :=\n  fun (a' : \u03b1) (h : a' \u2208 empty \u03b1) => absurd h (not_mem_empty a')\n\n/-- `length s` gets the number of steps of a terminating computation -/\ndef length {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : \u2115 :=\n  nat.find sorry\n\n/-- `get s` returns the result of a terminating computation -/\ndef get {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : \u03b1 :=\n  option.get sorry\n\ntheorem get_mem {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : get s \u2208 s :=\n  exists.intro (length s) (Eq.symm (option.eq_some_of_is_some (get._proof_2 s)))\n\ntheorem get_eq_of_mem {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} : a \u2208 s \u2192 get s = a :=\n  mem_unique (get_mem s)\n\ntheorem mem_of_get_eq {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} : get s = a \u2192 a \u2208 s :=\n  fun (h_1 : get s = a) => eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 s)) (Eq.symm h_1))) (get_mem s)\n\n@[simp] theorem get_think {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : get (think s) = get s := sorry\n\n@[simp] theorem get_thinkN {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] (n : \u2115) : get (thinkN s n) = get s :=\n  get_eq_of_mem (thinkN s n) (iff.mpr (thinkN_mem n) (get_mem s))\n\ntheorem get_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : s ~> get s :=\n  fun (a : \u03b1) => get_eq_of_mem s\n\ntheorem mem_of_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} (p : s ~> a) : a \u2208 s :=\n  Exists.dcases_on h fun (a' : \u03b1) (h : a' \u2208 s) => eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 s)) (p h))) h\n\ntheorem get_eq_of_promises {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] {a : \u03b1} : s ~> a \u2192 get s = a :=\n  get_eq_of_mem s \u2218 mem_of_promises s\n\n/-- `results s a n` completely characterizes a terminating computation:\n  it asserts that `s` terminates after exactly `n` steps, with result `a`. -/\ndef results {\u03b1 : Type u} (s : computation \u03b1) (a : \u03b1) (n : \u2115) :=\n  \u2203 (h : a \u2208 s), length s = n\n\ntheorem results_of_terminates {\u03b1 : Type u} (s : computation \u03b1) [T : terminates s] : results s (get s) (length s) :=\n  Exists.intro (get_mem s) rfl\n\ntheorem results_of_terminates' {\u03b1 : Type u} (s : computation \u03b1) [T : terminates s] {a : \u03b1} (h : a \u2208 s) : results s a (length s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (results s a (length s))) (Eq.symm (get_eq_of_mem s h)))) (results_of_terminates s)\n\ntheorem results.mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} : results s a n \u2192 a \u2208 s :=\n  fun (\u1fb0 : results s a n) => Exists.dcases_on \u1fb0 fun (\u1fb0_w : a \u2208 s) (\u1fb0_h : length s = n) => idRhs (a \u2208 s) \u1fb0_w\n\ntheorem results.terminates {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) : terminates s :=\n  terminates_of_mem (results.mem h)\n\ntheorem results.length {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} [T : terminates s] : results s a n \u2192 length s = n :=\n  fun (\u1fb0 : results s a n) => Exists.dcases_on \u1fb0 fun (\u1fb0_w : a \u2208 s) (\u1fb0_h : length s = n) => idRhs (length s = n) \u1fb0_h\n\ntheorem results.val_unique {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {b : \u03b1} {m : \u2115} {n : \u2115} (h1 : results s a m) (h2 : results s b n) : a = b :=\n  mem_unique (results.mem h1) (results.mem h2)\n\ntheorem results.len_unique {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {b : \u03b1} {m : \u2115} {n : \u2115} (h1 : results s a m) (h2 : results s b n) : m = n :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (m = n)) (Eq.symm (results.length h1))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length s = n)) (results.length h2))) (Eq.refl n))\n\ntheorem exists_results_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (h : a \u2208 s) : \u2203 (n : \u2115), results s a n :=\n  Exists.intro (length s) (results_of_terminates' s h)\n\n@[simp] theorem get_ret {\u03b1 : Type u} (a : \u03b1) : get (return a) = a :=\n  get_eq_of_mem (return a) (Exists.intro 0 rfl)\n\n@[simp] theorem length_ret {\u03b1 : Type u} (a : \u03b1) : length (return a) = 0 :=\n  let h : terminates (return a) := computation.ret_terminates a;\n  nat.eq_zero_of_le_zero (nat.find_min' (iff.mp (terminates_def (return a)) h) rfl)\n\ntheorem results_ret {\u03b1 : Type u} (a : \u03b1) : results (return a) a 0 :=\n  Exists.intro (ret_mem a) (length_ret a)\n\n@[simp] theorem length_think {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : length (think s) = length s + 1 := sorry\n\ntheorem results_think {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) : results (think s) a (n + 1) :=\n  Exists.intro (think_mem (results.mem h))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (length (think s) = n + 1)) (length_think s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (length s + 1 = n + 1)) (results.length h))) (Eq.refl (n + 1))))\n\ntheorem of_results_think {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results (think s) a n) : \u2203 (m : \u2115), results s a m \u2227 n = m + 1 :=\n  Exists.intro (length s)\n    { left := results_of_terminates' s (of_think_mem (results.mem h)),\n      right := results.len_unique h (results_think (results_of_terminates' s (of_think_mem (results.mem h)))) }\n\n@[simp] theorem results_think_iff {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} : results (think s) a (n + 1) \u2194 results s a n := sorry\n\ntheorem results_thinkN {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {m : \u2115} (n : \u2115) : results s a m \u2192 results (thinkN s n) a (m + n) := sorry\n\ntheorem results_thinkN_ret {\u03b1 : Type u} (a : \u03b1) (n : \u2115) : results (thinkN (return a) n) a n :=\n  eq.mp (Eq._oldrec (Eq.refl (results (thinkN (return a) n) a (0 + n))) (nat.zero_add n))\n    (results_thinkN n (results_ret a))\n\n@[simp] theorem length_thinkN {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] (n : \u2115) : length (thinkN s n) = length s + n :=\n  results.length (results_thinkN n (results_of_terminates s))\n\ntheorem eq_thinkN {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} {n : \u2115} (h : results s a n) : s = thinkN (return a) n := sorry\n\ntheorem eq_thinkN' {\u03b1 : Type u} (s : computation \u03b1) [h : terminates s] : s = thinkN (return (get s)) (length s) :=\n  eq_thinkN (results_of_terminates s)\n\ndef mem_rec_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} {a : \u03b1} {s : computation \u03b1} (M : a \u2208 s) (h1 : C (return a)) (h2 : (s : computation \u03b1) \u2192 C s \u2192 C (think s)) : C s :=\n  eq.mpr sorry\n    (eq.mpr sorry (Nat.rec h1 (fun (n : \u2115) (IH : C (thinkN (return a) n)) => h2 (thinkN (return a) n) IH) (length s)))\n\ndef terminates_rec_on {\u03b1 : Type u} {C : computation \u03b1 \u2192 Sort v} (s : computation \u03b1) [terminates s] (h1 : (a : \u03b1) \u2192 C (return a)) (h2 : (s : computation \u03b1) \u2192 C s \u2192 C (think s)) : C s :=\n  mem_rec_on (get_mem s) (h1 (get s)) h2\n\n/-- Map a function on the result of a computation. -/\ndef map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) : computation \u03b1 \u2192 computation \u03b2 :=\n  sorry\n\ndef bind.G {\u03b1 : Type u} {\u03b2 : Type v} : \u03b2 \u2295 computation \u03b2 \u2192 \u03b2 \u2295 computation \u03b1 \u2295 computation \u03b2 :=\n  sorry\n\ndef bind.F {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 computation \u03b2) : computation \u03b1 \u2295 computation \u03b2 \u2192 \u03b2 \u2295 computation \u03b1 \u2295 computation \u03b2 :=\n  sorry\n\n/-- Compose two computations into a monadic `bind` operation. -/\ndef bind {\u03b1 : Type u} {\u03b2 : Type v} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) : computation \u03b2 :=\n  corec sorry (sum.inl c)\n\nprotected instance has_bind : Bind computation :=\n  { bind := bind }\n\ntheorem has_bind_eq_bind {\u03b1 : Type u} {\u03b2 : Type u} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) : c >>= f = bind c f :=\n  rfl\n\n/-- Flatten a computation of computations into a single computation. -/\ndef join {\u03b1 : Type u} (c : computation (computation \u03b1)) : computation \u03b1 :=\n  c >>= id\n\n@[simp] theorem map_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : map f (return a) = return (f a) :=\n  rfl\n\n@[simp] theorem map_think {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) : map f (think s) = think (map f s) := sorry\n\n@[simp] theorem destruct_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) : destruct (map f s) = lmap f (rmap (map f) (destruct s)) := sorry\n\n@[simp] theorem map_id {\u03b1 : Type u} (s : computation \u03b1) : map id s = s := sorry\n\ntheorem map_comp {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 \u03b3) (s : computation \u03b1) : map (g \u2218 f) s = map g (map f s) := sorry\n\n@[simp] theorem ret_bind {\u03b1 : Type u} {\u03b2 : Type v} (a : \u03b1) (f : \u03b1 \u2192 computation \u03b2) : bind (return a) f = f a := sorry\n\n@[simp] theorem think_bind {\u03b1 : Type u} {\u03b2 : Type v} (c : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) : bind (think c) f = think (bind c f) := sorry\n\n@[simp] theorem bind_ret {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) : bind s (return \u2218 f) = map f s := sorry\n\n@[simp] theorem bind_ret' {\u03b1 : Type u} (s : computation \u03b1) : bind s return = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (bind s return = s)) (bind_ret (fun (x : \u03b1) => x) s)))\n    (id (eq.mpr (id (Eq._oldrec (Eq.refl (map id s = s)) (map_id s))) (Eq.refl s)))\n\n@[simp] theorem bind_assoc {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) (g : \u03b2 \u2192 computation \u03b3) : bind (bind s f) g = bind s fun (x : \u03b1) => bind (f x) g := sorry\n\ntheorem results_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1} {b : \u03b2} {m : \u2115} {n : \u2115} (h1 : results s a m) (h2 : results (f a) b n) : results (bind s f) b (n + m) := sorry\n\ntheorem mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1} {b : \u03b2} (h1 : a \u2208 s) (h2 : b \u2208 f a) : b \u2208 bind s f := sorry\n\nprotected instance terminates_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) [terminates s] [terminates (f (get s))] : terminates (bind s f) :=\n  terminates_of_mem (mem_bind (get_mem s) (get_mem (f (get s))))\n\n@[simp] theorem get_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) [terminates s] [terminates (f (get s))] : get (bind s f) = get (f (get s)) :=\n  get_eq_of_mem (bind s f) (mem_bind (get_mem s) (get_mem (f (get s))))\n\n@[simp] theorem length_bind {\u03b1 : Type u} {\u03b2 : Type v} (s : computation \u03b1) (f : \u03b1 \u2192 computation \u03b2) [T1 : terminates s] [T2 : terminates (f (get s))] : length (bind s f) = length (f (get s)) + length s :=\n  results.len_unique (results_of_terminates (bind s f))\n    (results_bind (results_of_terminates s) (results_of_terminates (f (get s))))\n\ntheorem of_results_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {b : \u03b2} {k : \u2115} : results (bind s f) b k \u2192 \u2203 (a : \u03b1), \u2203 (m : \u2115), \u2203 (n : \u2115), results s a m \u2227 results (f a) b n \u2227 k = n + m := sorry\n\ntheorem exists_of_mem_bind {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {b : \u03b2} (h : b \u2208 bind s f) : \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a := sorry\n\ntheorem bind_promises {\u03b1 : Type u} {\u03b2 : Type v} {s : computation \u03b1} {f : \u03b1 \u2192 computation \u03b2} {a : \u03b1} {b : \u03b2} (h1 : s ~> a) (h2 : f a ~> b) : bind s f ~> b := sorry\n\nprotected instance monad : Monad computation :=\n  { toApplicative :=\n      { toFunctor := { map := map, mapConst := fun (\u03b1 \u03b2 : Type u_1) => map \u2218 function.const \u03b2 },\n        toPure := { pure := return },\n        toSeq :=\n          { seq :=\n              fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) => bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x },\n        toSeqLeft :=\n          { seqLeft :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : computation \u03b1) (b : computation \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b1 (map (function.const \u03b2) a) b },\n        toSeqRight :=\n          { seqRight :=\n              fun (\u03b1 \u03b2 : Type u_1) (a : computation \u03b1) (b : computation \u03b2) =>\n                (fun (\u03b1 \u03b2 : Type u_1) (f : computation (\u03b1 \u2192 \u03b2)) (x : computation \u03b1) =>\n                    bind f fun (_x : \u03b1 \u2192 \u03b2) => map _x x)\n                  \u03b2 \u03b2 (map (function.const \u03b1 id) a) b } },\n    toBind := { bind := bind } }\n\nprotected instance is_lawful_monad : is_lawful_monad computation :=\n  is_lawful_monad.mk ret_bind bind_assoc\n\ntheorem has_map_eq_map {\u03b1 : Type u} {\u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (c : computation \u03b1) : f <$> c = map f c :=\n  rfl\n\n@[simp] theorem return_def {\u03b1 : Type u} (a : \u03b1) : return a = return a :=\n  rfl\n\n@[simp] theorem map_ret' {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : f <$> return a = return (f a) :=\n  map_ret\n\n@[simp] theorem map_think' {\u03b1 : Type u_1} {\u03b2 : Type u_1} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) : f <$> think s = think (f <$> s) :=\n  map_think\n\ntheorem mem_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : computation \u03b1} (m : a \u2208 s) : f a \u2208 map f s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (f a \u2208 map f s)) (Eq.symm (bind_ret f s)))) (mem_bind m (ret_mem (f a)))\n\ntheorem exists_of_mem_map {\u03b1 : Type u} {\u03b2 : Type v} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : computation \u03b1} (h : b \u2208 map f s) : \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b := sorry\n\nprotected instance terminates_map {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) [terminates s] : terminates (map f s) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (terminates (map f s))) (Eq.symm (bind_ret f s))))\n    (computation.terminates_bind s (return \u2218 f))\n\ntheorem terminates_map_iff {\u03b1 : Type u} {\u03b2 : Type v} (f : \u03b1 \u2192 \u03b2) (s : computation \u03b1) : terminates (map f s) \u2194 terminates s := sorry\n\n-- Parallel computation\n\n/-- `c\u2081 <|> c\u2082` calculates `c\u2081` and `c\u2082` simultaneously, returning\n  the first one that gives a result. -/\ndef orelse {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) : computation \u03b1 :=\n  corec (fun (_x : computation \u03b1 \u00d7 computation \u03b1) => sorry) (c\u2081, c\u2082)\n\nprotected instance alternative : alternative computation :=\n  alternative.mk empty\n\n@[simp] theorem ret_orelse {\u03b1 : Type u} (a : \u03b1) (c\u2082 : computation \u03b1) : (return a <|> c\u2082) = return a := sorry\n\n@[simp] theorem orelse_ret {\u03b1 : Type u} (c\u2081 : computation \u03b1) (a : \u03b1) : (think c\u2081 <|> return a) = return a := sorry\n\n@[simp] theorem orelse_think {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) : (think c\u2081 <|> think c\u2082) = think (c\u2081 <|> c\u2082) := sorry\n\n@[simp] theorem empty_orelse {\u03b1 : Type u} (c : computation \u03b1) : (empty \u03b1 <|> c) = c := sorry\n\n@[simp] theorem orelse_empty {\u03b1 : Type u} (c : computation \u03b1) : (c <|> empty \u03b1) = c := sorry\n\n/-- `c\u2081 ~ c\u2082` asserts that `c\u2081` and `c\u2082` either both terminate with the same result,\n  or both loop forever. -/\ndef equiv {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) :=\n  \u2200 (a : \u03b1), a \u2208 c\u2081 \u2194 a \u2208 c\u2082\n\ninfixl:50 \" ~ \" => Mathlib.computation.equiv\n\ntheorem equiv.refl {\u03b1 : Type u} (s : computation \u03b1) : s ~ s :=\n  fun (_x : \u03b1) => iff.rfl\n\ntheorem equiv.symm {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} : s ~ t \u2192 t ~ s :=\n  fun (h : s ~ t) (a : \u03b1) => iff.symm (h a)\n\ntheorem equiv.trans {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} {u : computation \u03b1} : s ~ t \u2192 t ~ u \u2192 s ~ u :=\n  fun (h1 : s ~ t) (h2 : t ~ u) (a : \u03b1) => iff.trans (h1 a) (h2 a)\n\ntheorem equiv.equivalence {\u03b1 : Type u} : equivalence equiv :=\n  { left := equiv.refl, right := { left := equiv.symm, right := equiv.trans } }\n\ntheorem equiv_of_mem {\u03b1 : Type u} {s : computation \u03b1} {t : computation \u03b1} {a : \u03b1} (h1 : a \u2208 s) (h2 : a \u2208 t) : s ~ t :=\n  fun (a' : \u03b1) =>\n    { mp := fun (ma : a' \u2208 s) => eq.mpr (id (Eq._oldrec (Eq.refl (a' \u2208 t)) (mem_unique ma h1))) h2,\n      mpr := fun (ma : a' \u2208 t) => eq.mpr (id (Eq._oldrec (Eq.refl (a' \u2208 s)) (mem_unique ma h2))) h1 }\n\ntheorem terminates_congr {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082) : terminates c\u2081 \u2194 terminates c\u2082 :=\n  exists_congr h\n\ntheorem promises_congr {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082) (a : \u03b1) : c\u2081 ~> a \u2194 c\u2082 ~> a :=\n  forall_congr fun (a' : \u03b1) => imp_congr (h a') iff.rfl\n\ntheorem get_equiv {\u03b1 : Type u} {c\u2081 : computation \u03b1} {c\u2082 : computation \u03b1} (h : c\u2081 ~ c\u2082) [terminates c\u2081] [terminates c\u2082] : get c\u2081 = get c\u2082 :=\n  get_eq_of_mem c\u2081 (iff.mpr (h (get c\u2082)) (get_mem c\u2082))\n\ntheorem think_equiv {\u03b1 : Type u} (s : computation \u03b1) : think s ~ s :=\n  fun (a : \u03b1) => { mp := of_think_mem, mpr := think_mem }\n\ntheorem thinkN_equiv {\u03b1 : Type u} (s : computation \u03b1) (n : \u2115) : thinkN s n ~ s :=\n  fun (a : \u03b1) => thinkN_mem n\n\ntheorem bind_congr {\u03b1 : Type u} {\u03b2 : Type v} {s1 : computation \u03b1} {s2 : computation \u03b1} {f1 : \u03b1 \u2192 computation \u03b2} {f2 : \u03b1 \u2192 computation \u03b2} (h1 : s1 ~ s2) (h2 : \u2200 (a : \u03b1), f1 a ~ f2 a) : bind s1 f1 ~ bind s2 f2 := sorry\n\ntheorem equiv_ret_of_mem {\u03b1 : Type u} {s : computation \u03b1} {a : \u03b1} (h : a \u2208 s) : s ~ return a :=\n  equiv_of_mem h (ret_mem a)\n\n/-- `lift_rel R ca cb` is a generalization of `equiv` to relations other than\n  equality. It asserts that if `ca` terminates with `a`, then `cb` terminates with\n  some `b` such that `R a b`, and if `cb` terminates with `b` then `ca` terminates\n  with some `a` such that `R a b`. -/\ndef lift_rel {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1) (cb : computation \u03b2) :=\n  (\u2200 {a : \u03b1}, a \u2208 ca \u2192 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b) \u2227 \u2200 {b : \u03b2}, b \u2208 cb \u2192 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b\n\ntheorem lift_rel.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1) (cb : computation \u03b2) : lift_rel (function.swap R) cb ca \u2194 lift_rel R ca cb :=\n  and_comm (\u2200 {a : \u03b2}, a \u2208 cb \u2192 Exists fun {b : \u03b1} => b \u2208 ca \u2227 function.swap R a b)\n    (\u2200 {b : \u03b1}, b \u2208 ca \u2192 Exists fun {a : \u03b2} => a \u2208 cb \u2227 function.swap R a b)\n\ntheorem lift_eq_iff_equiv {\u03b1 : Type u} (c\u2081 : computation \u03b1) (c\u2082 : computation \u03b1) : lift_rel Eq c\u2081 c\u2082 \u2194 c\u2081 ~ c\u2082 := sorry\n\ntheorem lift_rel.refl {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : reflexive R) : reflexive (lift_rel R) :=\n  fun (s : computation \u03b1) =>\n    { left := fun (a : \u03b1) (as : a \u2208 s) => Exists.intro a { left := as, right := H a },\n      right := fun (b : \u03b1) (bs : b \u2208 s) => Exists.intro b { left := bs, right := H b } }\n\ntheorem lift_rel.symm {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : symmetric R) : symmetric (lift_rel R) := sorry\n\ntheorem lift_rel.trans {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (H : transitive R) : transitive (lift_rel R) := sorry\n\ntheorem lift_rel.equiv {\u03b1 : Type u} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : equivalence R \u2192 equivalence (lift_rel R) := sorry\n\ntheorem lift_rel.imp {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {S : \u03b1 \u2192 \u03b2 \u2192 Prop} (H : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S a b) (s : computation \u03b1) (t : computation \u03b2) : lift_rel R s t \u2192 lift_rel S s t := sorry\n\ntheorem terminates_of_lift_rel {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : computation \u03b1} {t : computation \u03b2} : lift_rel R s t \u2192 (terminates s \u2194 terminates t) := sorry\n\ntheorem rel_of_lift_rel {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {cb : computation \u03b2} : lift_rel R ca cb \u2192 \u2200 {a : \u03b1} {b : \u03b2}, a \u2208 ca \u2192 b \u2208 cb \u2192 R a b := sorry\n\ntheorem lift_rel_of_mem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : \u03b1} {b : \u03b2} {ca : computation \u03b1} {cb : computation \u03b2} (ma : a \u2208 ca) (mb : b \u2208 cb) (ab : R a b) : lift_rel R ca cb := sorry\n\ntheorem exists_of_lift_rel_left {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {cb : computation \u03b2} (H : lift_rel R ca cb) {a : \u03b1} (h : a \u2208 ca) : Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b :=\n  and.left H a h\n\ntheorem exists_of_lift_rel_right {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {cb : computation \u03b2} (H : lift_rel R ca cb) {b : \u03b2} (h : b \u2208 cb) : Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b :=\n  and.right H b h\n\ntheorem lift_rel_def {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {cb : computation \u03b2} : lift_rel R ca cb \u2194 (terminates ca \u2194 terminates cb) \u2227 \u2200 {a : \u03b1} {b : \u03b2}, a \u2208 ca \u2192 b \u2208 cb \u2192 R a b := sorry\n\ntheorem lift_rel_bind {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : computation \u03b1} {s2 : computation \u03b2} {f1 : \u03b1 \u2192 computation \u03b3} {f2 : \u03b2 \u2192 computation \u03b4} (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 lift_rel S (f1 a) (f2 b)) : lift_rel S (bind s1 f1) (bind s2 f2) := sorry\n\n@[simp] theorem lift_rel_return_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1) (cb : computation \u03b2) : lift_rel R (return a) cb \u2194 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b := sorry\n\n@[simp] theorem lift_rel_return_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1) (b : \u03b2) : lift_rel R ca (return b) \u2194 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b := sorry\n\n@[simp] theorem lift_rel_return {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (a : \u03b1) (b : \u03b2) : lift_rel R (return a) (return b) \u2194 R a b := sorry\n\n@[simp] theorem lift_rel_think_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1) (cb : computation \u03b2) : lift_rel R (think ca) cb \u2194 lift_rel R ca cb := sorry\n\n@[simp] theorem lift_rel_think_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (ca : computation \u03b1) (cb : computation \u03b2) : lift_rel R ca (think cb) \u2194 lift_rel R ca cb := sorry\n\ntheorem lift_rel_mem_cases {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {cb : computation \u03b2} (Ha : \u2200 (a : \u03b1), a \u2208 ca \u2192 lift_rel R ca cb) (Hb : \u2200 (b : \u03b2), b \u2208 cb \u2192 lift_rel R ca cb) : lift_rel R ca cb :=\n  { left := fun (a : \u03b1) (ma : a \u2208 ca) => and.left (Ha a ma) a ma,\n    right := fun (b : \u03b2) (mb : b \u2208 cb) => and.right (Hb b mb) b mb }\n\ntheorem lift_rel_congr {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} {ca : computation \u03b1} {ca' : computation \u03b1} {cb : computation \u03b2} {cb' : computation \u03b2} (ha : ca ~ ca') (hb : cb ~ cb') : lift_rel R ca cb \u2194 lift_rel R ca' cb' :=\n  and_congr (forall_congr fun (a : \u03b1) => imp_congr (ha a) (exists_congr fun (b : \u03b2) => and_congr (hb b) iff.rfl))\n    (forall_congr fun (b : \u03b2) => imp_congr (hb b) (exists_congr fun (a : \u03b1) => and_congr (ha a) iff.rfl))\n\ntheorem lift_rel_map {\u03b1 : Type u} {\u03b2 : Type v} {\u03b3 : Type w} {\u03b4 : Type u_1} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (S : \u03b3 \u2192 \u03b4 \u2192 Prop) {s1 : computation \u03b1} {s2 : computation \u03b2} {f1 : \u03b1 \u2192 \u03b3} {f2 : \u03b2 \u2192 \u03b4} (h1 : lift_rel R s1 s2) (h2 : \u2200 {a : \u03b1} {b : \u03b2}, R a b \u2192 S (f1 a) (f2 b)) : lift_rel S (map f1 s1) (map f2 s2) := sorry\n\ntheorem map_congr {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) (S : \u03b2 \u2192 \u03b2 \u2192 Prop) {s1 : computation \u03b1} {s2 : computation \u03b1} {f : \u03b1 \u2192 \u03b2} (h1 : s1 ~ s2) : map f s1 ~ map f s2 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (map f s1 ~ map f s2)) (Eq.symm (propext (lift_eq_iff_equiv (map f s1) (map f s2))))))\n    (lift_rel_map Eq Eq (iff.mpr (lift_eq_iff_equiv s1 s2) h1) fun (a b : \u03b1) => congr_arg fun (a : \u03b1) => f a)\n\n@[simp] def lift_rel_aux {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) : \u03b1 \u2295 computation \u03b1 \u2192 \u03b2 \u2295 computation \u03b2 \u2192 Prop :=\n  sorry\n\n@[simp] theorem lift_rel_aux.ret_left {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (a : \u03b1) (cb : computation \u03b2) : lift_rel_aux R C (sum.inl a) (destruct cb) \u2194 Exists fun {b : \u03b2} => b \u2208 cb \u2227 R a b := sorry\n\ntheorem lift_rel_aux.swap {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (a : \u03b1 \u2295 computation \u03b1) (b : \u03b2 \u2295 computation \u03b2) : lift_rel_aux (function.swap R) (function.swap C) b a = lift_rel_aux R C a b := sorry\n\n@[simp] theorem lift_rel_aux.ret_right {\u03b1 : Type u} {\u03b2 : Type v} (R : \u03b1 \u2192 \u03b2 \u2192 Prop) (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (b : \u03b2) (ca : computation \u03b1) : lift_rel_aux R C (destruct ca) (sum.inl b) \u2194 Exists fun {a : \u03b1} => a \u2208 ca \u2227 R a b := sorry\n\ntheorem lift_rel_rec.lem {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (H : \u2200 {ca : computation \u03b1} {cb : computation \u03b2}, C ca cb \u2192 lift_rel_aux R C (destruct ca) (destruct cb)) (ca : computation \u03b1) (cb : computation \u03b2) (Hc : C ca cb) (a : \u03b1) (ha : a \u2208 ca) : lift_rel R ca cb := sorry\n\ntheorem lift_rel_rec {\u03b1 : Type u} {\u03b2 : Type v} {R : \u03b1 \u2192 \u03b2 \u2192 Prop} (C : computation \u03b1 \u2192 computation \u03b2 \u2192 Prop) (H : \u2200 {ca : computation \u03b1} {cb : computation \u03b2}, C ca cb \u2192 lift_rel_aux R C (destruct ca) (destruct cb)) (ca : computation \u03b1) (cb : computation \u03b2) (Hc : C ca cb) : lift_rel R ca cb :=\n  lift_rel_mem_cases sorry fun (b : \u03b2) (hb : b \u2208 cb) => iff.mpr (lift_rel.swap (fun (x : \u03b2) (y : \u03b1) => R y x) cb ca) sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/seq/computation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482762, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.3545316072448996}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Johannes H\u00f6lzl\n\nEnumerate elements of a set with a select function.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.set.lattice\nimport Mathlib.tactic.wlog\nimport Mathlib.PostPort\n\nuniverses u_1 \n\nnamespace Mathlib\n\nnamespace set\n\n\ndef enumerate {\u03b1 : Type u_1} (sel : set \u03b1 \u2192 Option \u03b1) : set \u03b1 \u2192 \u2115 \u2192 Option \u03b1 :=\n  sorry\n\ntheorem enumerate_eq_none_of_sel {\u03b1 : Type u_1} (sel : set \u03b1 \u2192 Option \u03b1) {s : set \u03b1} (h : sel s = none) {n : \u2115} : enumerate sel s n = none := sorry\n\ntheorem enumerate_eq_none {\u03b1 : Type u_1} (sel : set \u03b1 \u2192 Option \u03b1) {s : set \u03b1} {n\u2081 : \u2115} {n\u2082 : \u2115} : enumerate sel s n\u2081 = none \u2192 n\u2081 \u2264 n\u2082 \u2192 enumerate sel s n\u2082 = none := sorry\n\ntheorem enumerate_mem {\u03b1 : Type u_1} (sel : set \u03b1 \u2192 Option \u03b1) (h_sel : \u2200 (s : set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s) {s : set \u03b1} {n : \u2115} {a : \u03b1} : enumerate sel s n = some a \u2192 a \u2208 s := sorry\n\ntheorem enumerate_inj {\u03b1 : Type u_1} (sel : set \u03b1 \u2192 Option \u03b1) {n\u2081 : \u2115} {n\u2082 : \u2115} {a : \u03b1} {s : set \u03b1} (h_sel : \u2200 (s : set \u03b1) (a : \u03b1), sel s = some a \u2192 a \u2208 s) (h\u2081 : enumerate sel s n\u2081 = some a) (h\u2082 : enumerate sel s n\u2082 = some a) : n\u2081 = n\u2082 := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/set/enumerate.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6926419958239132, "lm_q2_score": 0.5117166047041652, "lm_q1q2_score": 0.35443641037852947}}
{"text": "import Mathlib.Algebra.Group.Defs\nimport Mathlib.Init.Algebra.Order\nimport ECTate.Algebra.Ring.Basic\nimport Mathlib.Init.Data.Nat.Lemmas\nimport ECTate.Init.Data.Int.Lemmas\nimport ECTate.Data.Nat.Enat\nimport ECTate.Algebra.EllipticCurve.Kronecker\nimport Mathlib.Tactic.LibrarySearch\nimport Mathlib.Tactic.Ring\nimport Mathlib.Tactic.Convert\nimport Mathlib.Data.Nat.Prime\nimport Mathlib.Data.Int.GCD\n\n--class ValueMonoid (A : Type u) extends AddCommMonoid A, LinearOrder A\n\nopen Enat\n\nsection Obvious\n\nlemma match_non_zero (x : \u2115\u222a\u221e) {c1 c2 : \u03b2} : x \u2260 0 \u2192 (match x with | 0 => c1 | _ => c2) = c2 := by\n  intro h\n  match x with\n  | ofN 0 => exact False.elim (h (Eq.refl 0))\n  | \u221e => simp\n  | ofN (_ + 1) => simp\n\ntheorem nat_mul_left_cancel (a b c : Nat) (h : a \u2260 0) : a * b = a * c \u2192 b = c :=\nNat.eq_of_mul_eq_mul_left (Nat.pos_of_ne_zero h)\n\nend Obvious\n\n@[ext]\nstructure SurjVal {R : Type u} (p : R) [CommRing R] [IsDomain R] where\n  v : R \u2192 \u2115\u222a\u221e\n  v_uniformizer' : v p = 1\n  v_mul_eq_add_v' (a b : R) : v (a * b) = v a + v b\n  v_add_ge_min_v' (a b : R) : v (a + b) \u2265 min (v a) (v b)\n  v_eq_top_iff_zero' (a : R) : v a = \u221e \u2194 a = 0\n\ninstance {R : Type u} (p : R) [CommRing R] [IsDomain R] : FunLike (SurjVal p) R (\u03bb _ => \u2115\u222a\u221e) :=\n{ coe := SurjVal.v\n  coe_injective' := by\n    intro x y h\n    ext :1\n    assumption }\n\ninstance {R : Type u} (p : R) [CommRing R] [IsDomain R] : CoeFun (SurjVal p) (\u03bb _ => R \u2192 \u2115\u222a\u221e) := \u27e8SurjVal.v\u27e9\n\nnamespace SurjVal\nvariable {R : Type u} {p : R} [CommRing R] [IsDomain R] (v : SurjVal p)\n-- TODO make naming consistent\n@[simp]\ntheorem v_uniformizer : v p = 1 := v.v_uniformizer'\n@[simp]\ntheorem v_mul_eq_add_v (a b : R) : v (a * b) = v a + v b := v.v_mul_eq_add_v' a b\ntheorem v_add_ge_min_v (a b : R) : v (a + b) \u2265 min (v a) (v b) := v.v_add_ge_min_v' a b\n\n@[simp]\ntheorem v_eq_top_iff_zero (a : R) : v a = \u221e \u2194 a = 0 := v.v_eq_top_iff_zero' a\nend SurjVal\n\nvariable {R : Type u} [CommRing R] [IsDomain R]\n\nnamespace SurjVal\n\n-- TODO namespace these\nlemma p_non_zero {p : R} (nav : SurjVal p) : \u00acp = 0 := by\n  rw [\u2190nav.v_eq_top_iff_zero, nav.v_uniformizer]\n  simp\n\n@[simp]\nlemma val_zero {p : R} (nav : SurjVal p) : nav 0 = \u221e := (nav.v_eq_top_iff_zero 0).2 rfl\n\nlemma val_mul_ge_left {p : R} (nav : SurjVal p) (a b : R) : nav (a * b) \u2265 nav a :=\nle_trans (le_add_right (nav a) (nav b)) (le_of_eq (nav.v_mul_eq_add_v a b).symm)\n\nlemma val_mul_ge_right {p : R} (nav : SurjVal p) (a b : R) : nav (a * b) \u2265 nav b := by\n  rw [mul_comm]\n  exact val_mul_ge_left nav b a\n\nlemma val_mul_ge_of_left_ge {p : R} (nav : SurjVal p) {a b : R} (ha : nav a \u2265 n) :\n  nav (a * b) \u2265 n :=\nle_trans ha (val_mul_ge_left nav a b)\n\nlemma val_mul_ge_of_right_ge {p : R} (nav : SurjVal p) {a b : R} (hb : nav b \u2265 n) :\n  nav (a * b) \u2265 n :=\nle_trans hb (val_mul_ge_right nav a b)\n\nlemma val_mul_ge_of_both_ge {p : R} (nav : SurjVal p) {a b : R} (ha : nav a \u2265 m) (hb : nav b \u2265 n) :\n  nav (a * b) \u2265 m + n := by\n  rw [nav.v_mul_eq_add_v]\n  exact add_le_add ha hb\n\n@[simp]\nlemma val_of_one {p : R} (nav : SurjVal p) : nav 1 = 0 := by\n  apply Enat.add_right_cancel_ofN 1\n  simp only [Nat.cast_one, zero_add]\n  rw [\u2190SurjVal.v_uniformizer nav, \u2190SurjVal.v_mul_eq_add_v nav, one_mul]\n\nlemma val_pow_ge_of_ge {p : R} (nav : SurjVal p) {a : R} (k : \u2115) (ha : nav a \u2265 m) :\n  nav (a ^ k) \u2265 k \u2022 m := by\n  induction k with\n  | zero => simp [zero_nsmul]\n  | succ k ih =>\n    simp only [succ_nsmul, pow_succ]\n    apply val_mul_ge_of_both_ge _ ha ih\n\nlemma val_pow_eq_of_eq {p : R} (nav : SurjVal p) {a : R} (k : \u2115) (ha : nav a = m) :\n  nav (a ^ k) = k * m := by\n  induction k with\n  | zero => simp\n  | succ k ih =>\n    simp only [pow_succ, Nat.cast_succ, add_mul, one_mul, add_comm]\n    rw [nav.v_mul_eq_add_v, ha, ih]\n\n@[simp]\nlemma val_pow_eq {p : R} (nav : SurjVal p) {a : R} (k : \u2115)  :\n  nav (a ^ k) = k * nav a := val_pow_eq_of_eq nav k rfl\n\nlemma val_add_ge_of_ge {p : R} (nav : SurjVal p) {a b : R} (ha : nav a \u2265 n) (hb : nav b \u2265 n) :\n  nav (a + b) \u2265 n := le_trans (le_min ha hb) (nav.v_add_ge_min_v a b)\n\ndef nat_of_val {p : R} (nav : SurjVal p) {a : R} (h : a \u2260 0) : \u2115 :=\n  to_nat ((not_iff_not.2 (nav.v_eq_top_iff_zero a)).2 h)\n\n/-\nlemma val_of_add_one {p : R} (nav : SurjVal p) (h : nav x \u2265 1): nav (x + 1) = 0 := by\n  apply le_antisymm\n  . apply le_of_not_lt\n    intro h'\n    sorry\n  . apply le_trans _ (nav.v_add_ge_min_v x 1)\n    apply le_min (le_trans (le_succ 0) h) (le_of_eq (val_of_one nav).symm)\n-/\n\nlemma val_of_minus_one {p : R} (nav : SurjVal p) : nav (-1) = 0 := by\n  cases Enat.eq_zero_or_pos (nav (-1)) with\n  | inl h => exact h\n  | inr h =>\n    have contradiction : nav 1 > 0 := by\n      rw [\u2190neg_neg 1, \u2190one_mul 1, neg_mul_eq_neg_mul, neg_mul_eq_mul_neg, nav.v_mul_eq_add_v]\n      apply Enat.lt_add_right _ _ _ h\n    rw [val_of_one] at contradiction\n    exact False.elim ((lt_irrefl 0) contradiction)\n\n@[simp]\nlemma val_neg {p : R} (nav : SurjVal p) : nav (-x) = nav x := by\n  rw [\u2190one_mul x, neg_mul_eq_neg_mul, nav.v_mul_eq_add_v, val_of_minus_one, one_mul, zero_add]\n\ntheorem v_sub_ge_min_v (nav : SurjVal p) (a b : R) : nav (a - b) \u2265 min (nav a) (nav b) := by\n  rw [sub_eq_add_neg]\n  convert nav.v_add_ge_min_v a (-b) using 2\n  simp\n\nlemma val_sub_ge_of_ge {p : R} (nav : SurjVal p) {a b : R} (ha : nav a \u2265 n) (hb : nav b \u2265 n) :\n  nav (a - b) \u2265 n := by\n  rw [sub_eq_add_neg]\n  apply val_add_ge_of_ge\n  assumption\n  simpa\n\ntheorem v_add_eq_min_v {p : R} (nav : SurjVal p) {a b : R} (h : nav a < nav b) :\n  nav (a + b) = nav a := by\n  apply le_antisymm\n  . apply le_of_not_lt\n    intro h'\n    have hm : nav a < nav (-b) := by rwa [val_neg]\n    apply lt_irrefl (nav a)\n    apply lt_of_lt_of_le (lt_min h' hm)\n    rw [(show nav a = nav (a + b + -b) by simp)]\n    exact nav.v_add_ge_min_v (a + b) (-b)\n  . exact le_trans (le_min (le_of_eq rfl) (le_of_lt h)) (SurjVal.v_add_ge_min_v nav a b)\n\ntheorem val_of_pow_uniformizer {p : R} (nav : SurjVal p) {n : \u2115} : nav (p ^ n) = n := by\n  induction n with\n  | zero      =>\n    rw [pow_zero]\n    exact val_of_one nav\n  | succ n ih =>\n    rw [pow_succ, SurjVal.v_mul_eq_add_v nav, ih, SurjVal.v_uniformizer nav]\n    simp [Nat.succ_eq_add_one, add_comm]\n\n\nend SurjVal\n\nstructure EnatValRing {R : Type u} (p : R) [CommRing R] [IsDomain R] where\n  valtn : SurjVal p\n  decr_val : R \u2192 R\n  /-- reduce the element x by valuation n (by dividing by an appropriate power of the uniformizer) -/\n  sub_val : \u2115 \u2192 R \u2192 R := Nat.iterate decr_val\n  sub_val_eq : sub_val = Nat.iterate decr_val := by rfl\n  zero_valtn_decr {x : R} (h : valtn x = 0) : decr_val x = x\n  pos_valtn_decr {x : R} (h : valtn x > 0) : x = p * decr_val x -- TODO remove\n  residue_char : \u2115\n  norm_repr : R \u2192 R --generalization of modulo\n  norm_repr_spec : \u2200 r, valtn (r - norm_repr r) > 0\n  inv_mod : R \u2192 R\n  inv_mod_spec : \u2200 r, valtn r = 0 \u2192 valtn (r * inv_mod r - 1) > 0\n  inv_mod_spec' : \u2200 r, valtn r > 0 \u2192 valtn (inv_mod r) > 0\n  inv_mod_spec'' : \u2200 r s, valtn (r - s) > 0 \u2192 inv_mod r = inv_mod s\n  pth_root : R \u2192 R\n  pth_root_spec : residue_char = 0 \u2228 \u2200 r, valtn (pth_root r ^ residue_char - r) > 0\n  count_roots_cubic : (a b c d : R) \u2192 Nat\n  -- count_roots_cubic_spec : \u2200 (a b c d : R), exists a smallest finset of elts solving\n  quad_roots_in_residue_field : R \u2192 R \u2192 R \u2192 Bool\n\nnamespace EnatValRing\nopen SurjVal\n\n@[simp]\nlemma decr_val_zero {p : R} (evr : EnatValRing p) : evr.decr_val 0 = 0 := by\n  have v_decr_zero : p * evr.decr_val 0 = 0 := by\n    apply Eq.symm\n    apply evr.pos_valtn_decr\n    rw [val_zero]\n    exact Enat.lt_top 0\n  rw [mul_eq_zero] at v_decr_zero\n  exact Or.resolve_left v_decr_zero (p_non_zero evr.valtn)\n\n@[simp]\nlemma decr_val_neg {p : R} (evr : EnatValRing p) (x : R) : evr.decr_val (-x) = -evr.decr_val x := by\n  cases @eq_zero_or_pos _ _ (evr.valtn x) with\n  | inl h =>\n    have hm : evr.valtn (-x) = 0 := by simp [h]\n    rw [evr.zero_valtn_decr h, evr.zero_valtn_decr hm]\n  | inr h =>\n    have hm : evr.valtn (-x) > 0 := by simp [h]\n    apply nzero_mul_left_cancel p _ _ (p_non_zero evr.valtn)\n    rw [\u2190neg_mul_eq_mul_neg, \u2190evr.pos_valtn_decr h, \u2190evr.pos_valtn_decr hm]\n\n@[simp]\nlemma decr_val_p_mul {p : R} (evr : EnatValRing p) (x : R) : evr.decr_val (p * x) = x := by\n  have h : (p * x) = p * decr_val evr (p * x) := by\n    apply evr.pos_valtn_decr\n    rw [evr.valtn.v_mul_eq_add_v, evr.valtn.v_uniformizer]\n    rw [add_comm, \u2190 Enat.succ_eq_add_one]\n    apply Enat.succ_pos\n  apply nzero_mul_left_cancel p _ _ (p_non_zero evr.valtn)\n  exact h.symm\n\n@[simp]\nlemma sub_val_zero_n {p : R} (evr : EnatValRing p) (n : \u2115) : sub_val evr n 0 = 0 := by\n  induction n with\n  | zero => simp [sub_val_eq]\n  | succ n ih => simpa [sub_val_eq, decr_val_zero] using ih\n\n@[simp]\nlemma sub_val_x_zero {p : R} (evr : EnatValRing p) (x : R) : sub_val evr 0 x = x := by simp [sub_val_eq]\n\nlemma sub_val_val_zero {p : R} (evr : EnatValRing p) (x : R) (m : \u2115) (h : evr.valtn x = 0) :\n  sub_val evr m x = x := by\n  induction m with\n  | zero => exact sub_val_x_zero evr x\n  | succ m ih => simpa [sub_val_eq, zero_valtn_decr _ h] using ih\n\nlemma sub_val_val_pos_succ {p : R} (evr : EnatValRing p) (x : R) (m : \u2115) :\n  sub_val evr (Nat.succ m) x = sub_val evr m (evr.decr_val x) := by\n  simp [sub_val_eq]\n\nlemma val_decr_val {p : R} (evr : EnatValRing p) {m : Nat} (x : R) (h : evr.valtn x = m) :\n  evr.valtn (evr.decr_val x) = \u2191(m - 1) := by\n  cases m with\n  | zero => rwa [evr.zero_valtn_decr h]\n  | succ m =>\n    have x_pos_val : evr.valtn x > 0 := by\n      rw [h]\n      exact succ_pos m\n    apply add_right_cancel_ofN 1\n    simp at * -- TODO fix nonterminal\n    rw [\u2190evr.valtn.v_uniformizer, \u2190evr.valtn.v_mul_eq_add_v, mul_comm,\n      \u2190evr.pos_valtn_decr x_pos_val, h, evr.valtn.v_uniformizer]\n\nlemma sub_val_decr_val_comm {p : R} (evr : EnatValRing p) (x : R) (n : \u2115) :\n  sub_val evr n (evr.decr_val x) = evr.decr_val (sub_val evr n x) := by\n  simp [sub_val_eq]\n  rw [\u2190 Function.iterate_succ_apply' evr.decr_val]\n  rw [\u2190 Function.iterate_succ_apply evr.decr_val]\n\nlemma val_sub_val_eq {p : R} (evr : EnatValRing p) (x : R) {m : \u2115} (n : \u2115) (h : evr.valtn x = m) :\n  evr.valtn (sub_val evr n x) = \u2191(m - n) := by\n  induction n with\n  | zero => rwa [sub_val_x_zero, Nat.sub_zero]\n  | succ n ih =>\n    cases m with\n    | zero =>\n      rw [Nat.zero_sub] at ih\n      rw [Nat.zero_sub, sub_val_val_zero evr x n.succ h]\n      exact h\n    | succ m =>\n      rw [sub_val_val_pos_succ, sub_val_decr_val_comm, val_decr_val evr (sub_val evr n x) ih,\n        Nat.succ_eq_add_one n, Nat.sub_sub]\n\nlemma val_sub_val_le {p : R} (evr : EnatValRing p) (x : R) {m : \u2115} (n : \u2115) (h : evr.valtn x \u2265 m) :\n  evr.valtn (sub_val evr n x) \u2265 \u2191(m - n) := by\n  cases enat_disjunction (evr.valtn x) with\n  | inl h' =>\n    have topcase : x = 0 := (evr.valtn.v_eq_top_iff_zero x).1 h'\n    rw [topcase, sub_val_zero_n, (evr.valtn.v_eq_top_iff_zero 0).2 rfl]\n    exact le.below_top\n  | inr h' =>\n    have H : \u2200 (a : \u2115), evr.valtn x = a \u2192 evr.valtn (sub_val evr n x) \u2265 (m - n) := by\n      intro a ha\n      have h'' := val_sub_val_eq evr x n ha\n      rw [h'']\n      apply (le_ofN (m - n) (a - n)).2\n      rw [ha] at h\n      apply Nat.sub_le_sub_right ((le_ofN m a).1 h)\n    exact Exists.elim h' H\n\nlemma factor_p_of_le_val {p : R} (evr : EnatValRing p) {x : R} {n : \u2115} (h : evr.valtn x \u2265 n) :\n  x = p ^ n * sub_val evr n x := by\n  induction n with\n  | zero => simp [sub_val_eq]\n  | succ n ih =>\n    rw [sub_val_val_pos_succ, sub_val_decr_val_comm, pow_succ', mul_assoc]\n    have pos_val : evr.valtn (sub_val evr n x) > 0 := by\n      have h' := val_sub_val_le evr x n h\n      rw [Nat.succ_eq_add_one, Nat.add_sub_self_left] at h'\n      exact lt_of_succ_le h'\n    rw [\u2190evr.pos_valtn_decr pos_val]\n    apply ih\n    exact le_of_succ_le h\n\nlemma factor_p_of_eq_val {p : R} (evr : EnatValRing p) {x : R} {n : \u2115} (h : evr.valtn x = n) :\n  x = p ^ n * sub_val evr n x := factor_p_of_le_val evr (le_of_eq (Eq.symm h))\n\nlemma sub_val_p_mul {p : R} (evr : EnatValRing p) (x : R) (n : \u2115) : sub_val evr n (p ^ n * x) = x :=\nby\n  induction n with\n  | zero      =>\n    rw [pow_zero, one_mul]\n    exact sub_val_x_zero evr x\n  | succ n ih =>\n    rwa [sub_val_val_pos_succ evr, pow_succ, mul_assoc, decr_val_p_mul]\n\nlemma sub_val_neg {p : R} (evr : EnatValRing p) {x : R} {n : \u2115} : sub_val evr n (-x) = -sub_val evr n x := by\n  induction n with\n  | zero => simp [sub_val_x_zero]\n  | succ n ih =>\n    cases @eq_zero_or_pos _ _ (evr.valtn x) with\n    | inl h' =>\n      have h'm : evr.valtn (-x) = 0 := by simp [h']\n      rw [sub_val_val_zero evr _ _ h', sub_val_val_zero evr _ _ h'm]\n    | inr h' =>\n      rw [sub_val_val_pos_succ evr _ _, sub_val_val_pos_succ evr _ _, sub_val_decr_val_comm, ih,\n      decr_val_neg, sub_val_decr_val_comm]\n\nlemma sub_val_add {p : R} (evr : EnatValRing p) {x y : R} {n : \u2115} (hx : evr.valtn x \u2265 n)\n  (hy : evr.valtn y \u2265 n) : sub_val evr n (x + y) = sub_val evr n x + sub_val evr n y := by\n  apply nzero_mul_left_cancel (p ^ n)\n  . exact pow_ne_zero n (p_non_zero evr.valtn)\n  . rw [\u2190factor_p_of_le_val evr (_ : evr.valtn (x + y) \u2265 n), mul_add, \u2190factor_p_of_le_val evr hx, \u2190factor_p_of_le_val evr hy]\n    exact le_trans (le_min hx hy) (evr.valtn.v_add_ge_min_v x y)\n\nlemma sub_val_sub {p : R} (evr : EnatValRing p) {x y : R} {n : \u2115} (hx : evr.valtn x \u2265 n)\n  (hy : evr.valtn y \u2265 n) : sub_val evr n (x - y) = sub_val evr n x - sub_val evr n y :=\nby\n  rw [sub_eq_add_neg, sub_eq_add_neg, sub_val_add evr hx, sub_val_neg]\n  simpa\n\nlemma sub_val_mul_left {p : R} (evr : EnatValRing p) {x y : R} {n : \u2115} (hx : evr.valtn x \u2265 n) :\n  sub_val evr n (x * y) = sub_val evr n x * y := by\n  apply nzero_mul_left_cancel (p ^ n)\n  . exact pow_ne_zero n (p_non_zero evr.valtn)\n  . rw [\u2190factor_p_of_le_val evr (_ : evr.valtn (x * y) \u2265 n), \u2190mul_assoc, \u2190factor_p_of_le_val evr hx]\n    exact le_trans hx (val_mul_ge_left evr.valtn x y)\n\nlemma sub_val_mul_right {p : R} (evr : EnatValRing p) {x y : R} {n : \u2115} (hy : evr.valtn y \u2265 n) :\n  sub_val evr n (x * y) = x * sub_val evr n y :=\nby rw [mul_comm x y, sub_val_mul_left evr hy, mul_comm]\n\nlemma sub_val_mul_sub_val {p : R} (evr : EnatValRing p) {x y : R} (n m : \u2115)\n  (hx : evr.valtn x \u2265 n) (hy : evr.valtn y \u2265 m) :\n  sub_val evr n x * sub_val evr m y = sub_val evr (n + m) (x * y) := by\n  apply nzero_mul_left_cancel (p ^ (n + m)) _ _ (pow_ne_zero _ (p_non_zero evr.valtn))\n  rw [\u2190factor_p_of_le_val evr (_ : evr.valtn (x * y) \u2265 (n + m)), pow_add, mul_assoc,\n    mul_comm (p ^ m), \u2190 mul_assoc,\n    \u2190 mul_assoc,\n    \u2190factor_p_of_le_val evr (_ : evr.valtn x \u2265 n), mul_assoc, mul_comm _ (p ^ m),\n    \u2190factor_p_of_le_val evr (_ : evr.valtn y \u2265 m)]\n  . assumption\n  . assumption\n  . rw [SurjVal.v_mul_eq_add_v]\n    exact add_le_add hx hy\n\nlemma sub_val_mul {p : R} (evr : EnatValRing p) {x y : R} (n m : \u2115) {nm : \u2115} (h : n + m = nm)\n  (hx : evr.valtn x \u2265 n) (hy : evr.valtn y \u2265 m) :\n  sub_val evr nm (x * y) = sub_val evr n x * sub_val evr m y := by\n  rw [\u2190 h, sub_val_mul_sub_val _ _ _ hx hy]\n\nlemma sub_val_pow {p : R} (evr : EnatValRing p) {x : R} (n k : \u2115) {nm : \u2115} (h : k * n = nm)\n  (hx : evr.valtn x \u2265 n) :\n  sub_val evr nm (x ^ k) = sub_val evr n x ^ k := by\n  induction k generalizing nm with\n  | zero => simp [\u2190 h]\n  | succ k ih =>\n    rw [pow_succ, sub_val_mul _ n (k * n), pow_succ, \u2190 ih]\n    rfl\n    rw [\u2190 h, Nat.succ_mul, add_comm]\n    exact hx\n    convert val_pow_ge_of_ge evr.valtn k hx\n    exact ofNat_mul_eq_smul k n\n\nlemma sub_val_sub_val {p : R} (evr : EnatValRing p) {x : R} {m n : \u2115} :\n  sub_val evr n (sub_val evr m x) = sub_val evr (m + n) x := by\n  have general : \u2200 y : R, sub_val evr n (sub_val evr m y) = sub_val evr (m + n) y := by\n    induction m with\n    | zero => simp [sub_val_x_zero]\n    | succ m ih =>\n      intro y\n      cases @eq_zero_or_pos _ _ (evr.valtn y) with\n      | inl h' => simp [sub_val_val_zero evr y _ h']\n      | inr h' =>\n        rw [sub_val_val_pos_succ evr y m, Nat.succ_add, sub_val_val_pos_succ evr y _]\n        exact ih (evr.decr_val y)\n  exact general x\n\ndef has_double_root {p : R} (evr : EnatValRing p) (a b c : R) :=\n  evr.valtn a = 0 \u2227 evr.valtn (b ^ 2 - 4 * a * c) > 0\n\ndef double_root {p : R} (evr : EnatValRing p) (a b c : R) :=\n  if evr.residue_char = 2 then\n    evr.norm_repr c\n  else\n    evr.norm_repr (-b * evr.inv_mod (2 * a))\n\nlemma val_poly_of_double_root {p : R} (evr : EnatValRing p) (a b c : R)\n  (H : has_double_root evr a b c) :\n  evr.valtn (a * (double_root evr a b c)^2 + b * (double_root evr a b c) + c) > 0 \u2227\n  evr.valtn (2*a*(double_root evr a b c) + b) > 0 := by sorry\n\nlemma pth_root_pos_of_pos {p : R} (evr : EnatValRing p) (r : R) (ha : 0 < evr.valtn r)\n  (hchar : evr.residue_char \u2260 0) :\n  evr.valtn (evr.pth_root r) > 0 :=\nby\n  suffices 0 < evr.valtn (evr.pth_root r ^ evr.residue_char) by\n    . simp at this\n      exact this.2\n  have :\n    min (SurjVal.v evr.valtn (pth_root evr r ^ evr.residue_char - r)) (SurjVal.v evr.valtn r) > 0 :=\n  min_rec' (LT.lt 0) (evr.pth_root_spec.resolve_left hchar r) ha -- TODO ew\n  have := this.trans_le (evr.valtn.v_add_ge_min_v (evr.pth_root r ^ evr.residue_char - r) r)\n  simpa using this\n\nend EnatValRing\n\n\nlemma ndiv_mul_left (a b p : \u2115) : (a * b) % p \u2260 0 \u2192 a % p \u2260 0 := by\n  intro hab ha\n  apply hab\n  simp [Nat.mul_mod, ha]\n\nlemma ndiv_mul_right (a b p : \u2115) : (a * b) % p \u2260 0 \u2192 b % p \u2260 0 := by\n  rw [Nat.mul_comm]\n  exact ndiv_mul_left b a p\n\n-- lemma Nat.Prime_test (p : \u2115) : Nat.Prime p \u2194 (1 < p \u2227 (\u2200 a b : \u2115, a < p \u2192 b < p \u2192 (a * b) % p = 0 \u2192 a % p = 0 \u2228 b % p = 0)) := by\n--   apply Iff.intro\n--   . intro H\n--     apply And.intro (H.left)\n--     intro a b _ _\n--     apply H.right a b\n--   . intro H\n--     apply And.intro (H.left)\n--     intro a b p_div_ab\n--     rw [Nat.mul_mod] at p_div_ab\n--     have p_pos : p > 0 := lt_trans Nat.zero_lt_one H.left;\n--     have h := H.right (a % p) (b % p) (Nat.mod_lt a p_pos) (Nat.mod_lt b p_pos) p_div_ab;\n--     rwa [Nat.mod_mod _ p, Nat.mod_mod _ p] at h\n\n\n\ninstance : DecidablePred (Nat.Prime . : \u2115 \u2192 Prop) := Nat.decidablePrime\n--match p with\n  --| 0 => sorry --isFalse (not_and_of_not_left _ (not_lt_of_ge (le_of_lt Nat.zero_lt_one)))\n  --| 1 => isFalse (not_and_of_not_left _ (not_lt_of_ge (le_of_eq rfl)))\n  --| Nat.succ (Nat.succ p') => sorry\n\n\n\n--def fmul_eq_addf {R R' : Type u} [Mul R] [Add R'] (f : R \u2192 R') (x y : R) : Prop := f (x * y) = f x + f y\n\n\n\n-- @[extern \"blah\"]\n-- def nat_valuation_aux : \u2115 \u2192 \u2115 \u2192 \u2115\n--   | _, 0 => 0\n--   | 0, (_+1) => 0\n--   | 1, (_+1) => 0\n--   | (q+2), (m+1) => if (m+1) % (q+2) \u2260 0 then 0 else Nat.succ (nat_valuation_aux (q+2) ((m+1) / (q+2)))\n-- termination_by nat_valuation_aux p k => k\n-- decreasing_by\n--   simp [WellFoundedRelation.rel, measure, invImage, InvImage, Nat.lt_wfRel]\n--   exact Nat.div_lt_self (Nat.zero_lt_succ m) (Nat.succ_lt_succ (Nat.zero_lt_succ q))\n\nlemma Nat.div_pos_of_mod {a b : \u2115} (ha : 0 < a) (hb : 1 < b) (hab : a % b = 0) : 0 < a / b :=\nNat.div_pos (Nat.le_of_dvd ha (dvd_of_mod_eq_zero hab)) (lt_of_succ_lt hb)\n\ndef nat_valuation_aux'' (q : \u2115) (hq : 1 < q) : (m : \u2115) \u2192 0 < m \u2192 \u2115 \u2192 \u2115\n  | m, hm, n => if hmq : m % q == 0 then (nat_valuation_aux'' q hq (m / q) (Nat.div_pos_of_mod hm hq (by simpa using hmq)) (n + 1)) else n\ndecreasing_by\n  simp [WellFoundedRelation.rel, measure, invImage, InvImage, Nat.lt_wfRel]\n  exact Nat.div_lt_self hm hq\n\n-- TODO unusedVariable linter fails\nlemma nat_valuation_aux''_of_dvd_induction : \u2200 (M m : \u2115) (hM : m \u2264 M) (hm : 0 < m) (n : \u2115)\n  (hmq : m % q = 0), \u2191(nat_valuation_aux'' q hq m hm n) = succ \u2191(nat_valuation_aux'' q hq (m / q)\n    (Nat.div_pos_of_mod hm hq hmq) n) := by\n  intro M\n  induction M with\n  | zero =>\n    intro m mle0 hm n hmq\n    rw [Nat.le_zero] at mle0\n    exact ((ne_of_gt hm) mle0).elim\n  | succ M IH =>\n    intro m m_le_sM hm n hmq\n    cases LE.le.lt_or_eq m_le_sM with\n    | inl mltsM =>\n      exact IH m (Nat.le_of_lt_succ mltsM) hm n hmq\n    | inr meqsM =>\n      cases em ((m / q) % q == 0) with\n      | inl h =>\n        rw [nat_valuation_aux'', nat_valuation_aux'', dif_pos h]\n        simp only [beq_iff_eq, succ_ofNat, Nat.cast_succ]\n        rw [dif_pos hmq]\n        simp only [meqsM]\n        rw [meqsM] at hm h hmq\n        exact IH (M.succ/q) (Nat.le_of_lt_succ (Nat.div_lt_self hm hq))\n          (Nat.div_pos_of_mod hm hq hmq) (n+1) (by simpa using h)\n      | inr h =>\n        rw [nat_valuation_aux'', nat_valuation_aux'', dif_neg h, dif_pos, nat_valuation_aux'', dif_neg h]\n        . simp\n        . simp only [hmq]\n\nlemma nat_valuation_aux''_of_dvd (q : \u2115) (hq : 1 < q) (m : \u2115) (hm : 0 < m) (n : \u2115) (hmq : m % q = 0) :\nnat_valuation_aux'' q hq m hm n = succ (nat_valuation_aux'' q hq (m / q) (Nat.div_pos_of_mod hm hq hmq) n) :=\nnat_valuation_aux''_of_dvd_induction m m (le_refl m) hm n hmq\n\nlemma nat_valuation_aux''_of_not_dvd (q : \u2115) (hq : 1 < q) (m : \u2115) (hm : 0 < m)\n  (hmq : m % q \u2260 0) : nat_valuation_aux'' q hq m hm 0 = 0 :=\nby\n  have hmq_bool : \u00acm % q == 0 := by\n    intro H\n    apply hmq (eq_of_beq H)\n  rw [nat_valuation_aux'', dif_neg hmq_bool]\n\n-- set_option trace.compiler.ir.result true in\ndef nat_valuation_aux' (q : \u2115) (hq : 1 < q) : (m : \u2115) \u2192 0 < m \u2192 \u2115\u222a\u221e\n  | m, hm => nat_valuation_aux'' q hq m hm 0\n\nlemma nat_valuation_aux'_of_not_dvd (q : \u2115) (hq : 1 < q) (m : \u2115) (hm : 0 < m)\n  (hmq : m % q \u2260 0) : nat_valuation_aux' q hq m hm = 0 :=\nby\n  rw [nat_valuation_aux']\n  simp [nat_valuation_aux''_of_not_dvd q hq m hm hmq]\n\nlemma nat_valuation_aux'_of_dvd (q : \u2115) (hq : 1 < q) (m : \u2115) (hm : 0 < m)\n  (hmq : m % q = 0) : nat_valuation_aux' q hq m hm = succ (nat_valuation_aux' q hq (m / q)\n  (Nat.div_pos_of_mod hm hq hmq)) :=\nby\n  simp [nat_valuation_aux', nat_valuation_aux''_of_dvd q hq m hm 0 hmq]\n\nlemma nat_val_aux'_succ (q m : \u2115) (hq) : nat_valuation_aux' (q+2) hq (m+1) (Nat.zero_lt_succ _) =\n  if hmq : (m+1) % (q+2) \u2260 0 then 0 else succ (nat_valuation_aux' (q+2) hq ((m+1) / (q+2)) (Nat.div_pos_of_mod (Nat.zero_lt_succ _) hq (not_not.mp hmq))) :=\nby\n  simp only [Nat.succ_ne_zero, dite_false, ne_eq, ite_not]\n  cases em ((m + 1) % (q + 2) = 0) with\n  | inl h =>\n    rw [dif_neg (not_not_intro h)]\n    exact nat_valuation_aux'_of_dvd _ _ _ _ h\n  | inr h =>\n    rw [dif_pos h]\n    exact nat_valuation_aux'_of_not_dvd _ _ _ _ h\n\ndef nat_valuation_aux (q : \u2115) (hq : 1 < q) : \u2115 \u2192 \u2115\u222a\u221e :=\n  \u03bb m => if hm : m = 0 then \u221e else nat_valuation_aux' q hq m (Nat.pos_of_ne_zero hm)\n\n@[simp]\nlemma nat_val_aux_zero (p : \u2115) (hp) : nat_valuation_aux p hp 0 = \u221e := by\n  simp [nat_valuation_aux]\n\nlemma x' {a b : Nat} (h : (a+1) % (b+1) = 0) : (a+1) \u2265 (b+1) := Nat.le_of_dvd (Nat.succ_pos _) (Nat.dvd_of_mod_eq_zero h)\n\nlemma nat_val_aux_succ (q m : \u2115) (hq) : nat_valuation_aux (q+2) hq (m+1) =\n  if (m+1) % (q+2) \u2260 0 then 0 else succ (nat_valuation_aux (q+2) hq ((m+1) / (q+2))) := by\n  simp only [nat_valuation_aux, Nat.succ_ne_zero, dite_false, ne_eq, ite_not]\n  by_cases hmq : (m + 1) % (q + 2) = 0\n  . have h : (m + 1) / (q + 2) \u2260 0 := by\n      apply Nat.ne_of_gt\n      apply Nat.div_pos (x' hmq) (lt_trans (Nat.lt_succ_self 0) hq)\n    rw [if_pos hmq, dif_neg h]\n    exact nat_valuation_aux'_of_dvd (q+2) hq (m+1) _ hmq\n  . rw [if_neg hmq]\n    exact nat_valuation_aux'_of_not_dvd (q+2) hq (m+1) _ hmq\n\n/-\ndef nat_valuation : \u2115 \u2192 \u2115 \u2192 \u2115\u222a\u221e\n  | _, 0 => \u221e\n  | 0, (_+1) => 0\n  | 1, (_+1) => \u221e\n  | (q+2), (m+1) => if (m+1) % (q+2) \u2260 0 then 0 else succ (nat_valuation (q+2) ((m+1) / (q+2)))\ntermination_by nat_valuation p k => k\ndecreasing_by\n  simp [WellFoundedRelation.rel, measure, invImage, InvImage, Nat.lt_wfRel]\n  exact Nat.div_lt_self (Nat.zero_lt_succ m) (Nat.succ_lt_succ (Nat.zero_lt_succ q))\n-/\ndef nat_valuation : \u2115 \u2192 \u2115 \u2192 \u2115\u222a\u221e\n  | _, 0 => \u221e\n  | 0, (_+1) => 0\n  | 1, (_+1) => \u221e\n  | (q+2), (m+1) => nat_valuation_aux (q+2) (Nat.succ_lt_succ (Nat.zero_lt_succ q)) (m+1)\n\nlemma nat_valuation_add_two (q m : \u2115) :\n  nat_valuation (q+2) m = nat_valuation_aux (q+2) (Nat.succ_lt_succ (Nat.zero_lt_succ q)) m := by\n  cases m\n  . rfl\n  . simp [nat_valuation]\n\nlemma nat_valuation_of_one_lt (p m : \u2115) (hp : 1 < p) : nat_valuation p m = nat_valuation_aux p hp m :=\n  by cases p\n     case zero => cases hp\n     case succ p =>\n       cases p\n       case zero => cases hp.ne rfl\n       case succ q =>\n         cases m\n         . rfl\n         . simp [nat_valuation]\n\n@[simp]\nlemma nat_val_zero (p : \u2115) : nat_valuation p 0 = \u221e := by\n  simp [nat_valuation]\nlemma nat_val_succ (q m : \u2115) : nat_valuation (q+2) (m+1) = if (m+1) % (q+2) \u2260 0 then 0 else succ (nat_valuation (q+2) ((m+1) / (q+2))) :=\n  by simp [nat_valuation_add_two, nat_val_aux_succ]\n\nnamespace Int\n\ndef int_val (p : \u2115) (k : \u2124) : \u2115\u222a\u221e :=\n  nat_valuation p (natAbs k)\n\n@[simp]\nlemma int_val_uniformizer {p : \u2115} (gt1 : 1 < p) : int_val p p = 1 := by\n  simp only [natAbs_cast, int_val]\n  match p with\n  | 0 =>\n    apply False.elim\n    apply Nat.not_lt_zero 1\n    assumption\n  | Nat.succ 0 =>\n    apply False.elim\n    apply Nat.lt_irrefl 1\n    assumption\n  | q+2 =>\n    rw [nat_val_succ, Nat.mod_self, if_neg _]\n    rw [Nat.div_self, nat_val_succ, if_pos, succ_zero]\n    rw [Nat.mod_eq_of_lt]\n    exact Nat.succ_ne_zero 0\n    assumption\n    exact lt_trans (Nat.lt_succ_self 0) gt1\n    exact Ne.irrefl\n\n@[simp]\nlemma int_val_zero {p : \u2115} : int_val p 0 = \u221e := by simp [natAbs_cast, int_val]\n\nlemma mod_mul (a b c : Nat) (h : a % c = 0) : (a * b) % c = 0 :=\nby rw [Nat.mul_mod, h, zero_mul, Nat.zero_mod]\n\nlemma nat_mul_div_assoc' (a b c : Nat) : c > 0 \u2192 a % c = 0 \u2192 a * b / c = a / c * b := by\n  intro hc hmod\n  apply nat_mul_left_cancel c _ _ (ne_of_gt hc)\n  rw [Nat.mul_div_cancel' (Nat.dvd_of_mod_eq_zero _), \u2190mul_assoc, Nat.mul_div_cancel' (Nat.dvd_of_mod_eq_zero hmod)]\n  exact mod_mul a b c hmod\n\nlemma nat_mul_div_assoc (a b c : Nat) : c > 0 \u2192 b % c = 0 \u2192 a * b / c = a * (b / c) := by\n  intro hc hmod\n  rw [mul_comm, nat_mul_div_assoc' b a c hc hmod, mul_comm]\n\nlemma nat_val_aux'_mul_eq_add (p : \u2115) (prime : Nat.Prime p) (hp : 1 < p := prime.one_lt) (a b : \u2115)\n  (ha : 0 < a) (hb : 0 < b) :\n  nat_valuation_aux' p hp (a * b) (Nat.mul_pos ha hb) = nat_valuation_aux' p hp a ha + nat_valuation_aux' p hp b hb := by\n  have general (n : \u2115) : \u2200 c d hc hd, c + d \u2264 n \u2192 nat_valuation_aux' p hp (c * d) (Nat.mul_pos hc hd) = nat_valuation_aux' p hp c hc + nat_valuation_aux' p hp d hd := by\n    induction n with\n    | zero =>\n      intro c d hc hd h_sum\n      rw [Nat.eq_zero_of_add_eq_zero_right (Nat.eq_zero_of_le_zero h_sum)] at hc\n      exact (lt_irrefl 0 hc).elim\n    | succ n ih =>\n      intro c d hc hd h_sum\n      cases c with\n      | zero => cases hc\n      | succ c => cases d with\n        | zero => cases hd\n        | succ d =>\n          match Nat.le.dest (Nat.succ_le_of_lt prime.one_lt) with\n          | \u27e8q, hq\u27e9 =>\n            rw [(show Nat.succ 1 = 2 by rfl), Nat.add_comm] at hq\n            have mul_s_s : c.succ * d.succ = (c * d + c + d).succ := by simp [Nat.succ_mul, Nat.mul_succ, Nat.add_succ]\n            simp only [\u2190hq, mul_s_s, nat_valuation_add_two, nat_valuation_aux]\n            simp only [hq, (show c * d + c + d + 1 = (c + 1) * (d + 1) by ring)]\n            cases Nat.eq_zero_or_pos ((c + 1) * (d + 1) % p) with\n            | inl h =>\n              have hh : (c + 1) % p = 0 \u2228(d + 1) % p = 0 := sorry\n              cases  hh with\n              | inl h' =>\n                subst hq\n                rw [nat_valuation_aux'_of_dvd _ _ _ _ h', succ_add]\n                have sum_le_n : (c + 1) / (q + 2) + (d + 1) \u2264 n := by\n                  apply Nat.le_of_lt_succ\n                  apply lt_of_lt_of_le _ h_sum\n                  apply Nat.add_lt_add_right\n                  apply Nat.div_lt_self _ prime.one_lt\n                  rw [Nat.add_comm]\n                  apply Nat.lt_add_right 0 1 c (Nat.lt_succ_self 0)\n                rw [\u2190ih ((c + 1) / (q + 2)) (d + 1) _ _ sum_le_n]\n                have hey := nat_mul_div_assoc' (c+1) (d+1) (q+2) (lt_trans (Nat.lt_succ_self 0) prime.one_lt) h'\n                simp only [hey.symm, mul_s_s.symm]\n                apply nat_valuation_aux'_of_dvd (q+2) hp ((c+1) * (d+1)) _ h\n              | inr h' =>\n                subst hq\n                rw [nat_valuation_aux'_of_dvd _ _ _ _ h', add_succ]\n                have sum_le_n : (c + 1) + (d + 1) / (q + 2) \u2264 n := by\n                  apply Nat.le_of_lt_succ\n                  apply lt_of_lt_of_le _ h_sum\n                  apply Nat.add_lt_add_left\n                  apply Nat.div_lt_self _ prime.one_lt\n                  rw [Nat.add_comm]\n                  apply Nat.lt_add_right 0 1 d (Nat.lt_succ_self 0)\n                rw [\u2190ih (c + 1) ((d + 1) / (q + 2)) _ _ sum_le_n]\n                have hey := nat_mul_div_assoc (c+1) (d+1) (q+2) (lt_trans (Nat.lt_succ_self 0) prime.one_lt) h'\n                simp only [hey.symm, mul_s_s.symm]\n                apply nat_valuation_aux'_of_dvd (q+2) hp ((c+1) * (d+1)) _ h\n            | inr h =>\n              have hc := ndiv_mul_left _ _ _ (ne_of_gt h)\n              have hd := ndiv_mul_right _ _ _ (ne_of_gt h)\n              simp [nat_valuation_aux'_of_not_dvd _ _ _ _ hc, nat_valuation_aux'_of_not_dvd _ _ _ _ hd]\n              simp [\u2190mul_s_s, nat_valuation_aux'_of_not_dvd _ _ _ _ (ne_of_gt h)]\n  apply general (a + b) a b ha hb (le_refl _)\n\nlemma nat_val_aux_mul_eq_add (p : \u2115) (prime : Nat.Prime p) (hp : 1 < p := prime.one_lt) (a b : \u2115) :\n  nat_valuation_aux p hp (a * b) = nat_valuation_aux p hp a + nat_valuation_aux p hp b := by\ncases a with\n  | zero => simp [nat_valuation_aux]\n  | succ a => cases b with\n    | zero => simp [nat_valuation_aux]\n    | succ b =>\n      exact nat_val_aux'_mul_eq_add p prime prime.one_lt a.succ b.succ _ _\n\nlemma nat_val_mul_eq_add (p : \u2115) (prime : Nat.Prime p) (a b : \u2115) :\n  nat_valuation p (a * b) = nat_valuation p a + nat_valuation p b := by\nconvert nat_val_aux_mul_eq_add p prime prime.one_lt a b <;>\next <;>\nrw [(nat_valuation_of_one_lt p _ prime.one_lt)]\n\nlemma int_val_mul_eq_add {p : \u2115} (prime : Nat.Prime p) (a b : \u2124) :\n  int_val p (a * b) = int_val p a + int_val p b := by\n  simp [int_val, natAbs_mul]\n  exact nat_val_mul_eq_add p prime (natAbs a) (natAbs b)\n\nlemma nat_val_add_eq_min (p a b : \u2115) (h : nat_valuation p a < nat_valuation p b) :\n  nat_valuation p (a + b) = nat_valuation p a := by sorry\n\nlemma nat_val_add_ge_min (p a b : \u2115) : nat_valuation p (a + b) \u2265 min (nat_valuation p a) (nat_valuation p b) := by\n  cases lt_trichotomy (nat_valuation p a) (nat_valuation p b) with -- TODO use rcases\n  | inl h =>\n    simp only [min, if_pos (le_of_lt h)]\n    exact le_of_eq (nat_val_add_eq_min p a b h).symm\n  | inr h => cases h with\n    | inl h =>\n      simp only [min, if_pos (le_of_eq h)]\n      sorry\n    | inr h =>\n      simp only [add_comm a b, min, if_neg (not_le_of_lt h)]\n      exact le_of_eq (nat_val_add_eq_min p b a h).symm\n\n--lemma natAbs_add (a b : \u2124) : natAbs (a + b) = max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b) := by sorry\n\nlemma natAbs_add (a b : \u2124) : natAbs (a + b) = max (natAbs a) (natAbs b) - min (natAbs a) (natAbs b) := by sorry\n\nlemma int_val_add_ge_min (p : \u2115) (a b : \u2124) : int_val p (a + b) \u2265 min (int_val p a) (int_val p b) := by\n  simp [int_val, natAbs_add]\n  -- exact nat_val_add_ge_min p (natAbs a) (natAbs b)\n  sorry\n\nlemma int_val_add_eq_min (p : \u2115) (a b : \u2124) (h : int_val p a < int_val p b) :\n  int_val p (a + b) = int_val p a := by sorry\n\n@[simp]\nlemma int_val_eq_top_iff_zero {p : \u2115} (gt1 : 1 < p) (a : \u2124) : int_val p a = \u221e \u2194 a = 0 := by\n  apply Iff.intro\n  . intro hval\n    simp [int_val, nat_valuation] at hval\n    cases abs_a : (natAbs a) with\n    | zero => exact natAbs_eq_zero.1 abs_a\n    | succ n =>\n      cases hp : p with\n      | zero =>\n        rw [hp] at gt1\n        apply False.elim ((of_decide_eq_true rfl : \u00ac1 < 0) gt1)\n      | succ p' =>\n        cases hp' : p' with\n        | zero =>\n          rw [hp, hp'] at gt1\n          apply False.elim ((of_decide_eq_true rfl : \u00ac1 < 1) gt1)\n        | succ n =>\n          simp [hp, hp', abs_a, nat_valuation_aux, nat_valuation_aux'] at hval\n  . intro ha\n    simp [ha, int_val, nat_valuation]\n\ndef primeVal {p : \u2115} (hp : Nat.Prime p) : SurjVal (p : \u2124) := {\n  v := int_val p\n  v_uniformizer' := int_val_uniformizer hp.one_lt\n  v_mul_eq_add_v' := int_val_mul_eq_add hp\n  v_add_ge_min_v' := int_val_add_ge_min p\n  v_eq_top_iff_zero' := int_val_eq_top_iff_zero hp.one_lt }\n\n\ndef decr_val_p (p : \u2115) (k : \u2124) : \u2124 :=\n  if k % p == 0 then k / p else k\n\ndef sub_val_p (p : \u2115) (val : \u2124 \u2192 \u2115\u222a\u221e) (n : \u2115) (k : \u2124) : \u2124 :=\n  k / (p ^ ((min (n : \u2115\u222a\u221e) (val k)).to_nat sorry) : \u2115)\n\n@[simp]\nlemma nat_valuation_eq_zero_iff {p : \u2115} (hp : 1 < p) {k : \u2115} : nat_valuation p k = 0 \u2194 k % p \u2260 0 :=\nby\n  have := nat_val_aux_succ\n  simp only [nat_valuation, ne_eq]\n  aesop\n  -- change (Enat.succ _ = 0) at a -- TODO doesn't work\n\n@[simp]\nlemma int_valuation_eq_zero_iff {p : \u2115} {k : \u2124} (hp : 1 < p) : int_val p k = 0 \u2194 k % p \u2260 0 :=\nby\n  simp [int_val]\n  rw [nat_valuation_eq_zero_iff hp]\n  rw [not_iff_not]\n  aesop\n  . cases k\n    . aesop\n      exact eq_zero_of_natAbs_eq_zero a\n    . aesop\n      rw [\u2190 Int.natAbs_eq_zero]\n      sorry\n  sorry\n\n@[simp]\nlemma primeVal_eq_zero_iff {p : \u2115} {k : \u2124} (hp : Nat.Prime p) : primeVal hp k = 0 \u2194 k % p \u2260 0 :=\nby rw [primeVal, int_valuation_eq_zero_iff hp.one_lt]\n\nlemma zero_valtn_decr_p {p : \u2115} {k : \u2124} {hp : Nat.Prime p} (h : primeVal hp k = 0) :\n  decr_val_p p k = k :=\nby\n  simp [decr_val_p] at *\n  aesop\n\ndef norm_repr_p (p : \u2115) (x : \u2124) : \u2124 := x % (p : \u2124)\n\ndef modulo (x : \u2124) (p : \u2115) := x % (p:\u2124)\n\ndef inv_mod (x : \u2124) (p : \u2115) := gcdA x p\n\ndef count_roots_cubic_aux (a b c d : \u2124) (p : \u2115) (x : \u2115) : \u2115 := match x with\n  | Nat.zero => if d = 0 then 1 else 0\n  | Nat.succ x' => (if (a * (x^3 : \u2115) + b * (x^2 : \u2115) + c * x + d) % (p : \u2124) = 0 then 1 else 0) + count_roots_cubic_aux a b c d p x'\n\ndef count_roots_cubic (a b c d : \u2124) (p : \u2115) : \u2115 :=\n  count_roots_cubic_aux (modulo a p) (modulo b p) (modulo c p) (modulo d p) p (p - 1)\n\ndef primeEVR {p : \u2115} (hp : Nat.Prime p) : EnatValRing (p : \u2124) := {\n  valtn := primeVal hp\n  decr_val := decr_val_p p\n  -- sub_val := sub_val_p p (primeVal hp).v\n  -- sub_val_eq := sorry\n  zero_valtn_decr := zero_valtn_decr_p -- todo we really shouldn't need this!\n  pos_valtn_decr := sorry\n  residue_char := p\n  norm_repr := (. % p)\n  norm_repr_spec := by\n    intro r\n    simp [pos_iff_ne_zero, Int.sub_emod]\n  inv_mod := (inv_mod . p)\n  inv_mod_spec := by\n    intro r h\n    simp [inv_mod, pos_iff_ne_zero, Int.sub_emod]\n    rw [Int.emod_emod]\n    rw [Int.emod_emod] -- TODO why doesn't simp do this?\n    sorry\n  inv_mod_spec' := sorry\n  inv_mod_spec'' := sorry\n\n  pth_root := id\n  pth_root_spec := by\n    right\n    intro r\n    simp [inv_mod, pos_iff_ne_zero, Int.sub_emod]\n    rw [Int.emod_emod]\n    rw [Int.emod_emod] -- TODO why doesn't simp do this?\n    rw [\u2190Int.sub_emod]\n    sorry -- needs fermat's little theorem\n\n  count_roots_cubic :=\n    -- TODO fix this, should we way quicker to count roots, probably in cohen\n    (Int.count_roots_cubic . . . . p)\n  -- count_roots_cubic_spec := sorry\n\n  quad_roots_in_residue_field := fun a b c => Int.quad_root_in_ZpZ a b c p }\n\n#eval (primeEVR (sorry : Nat.Prime 2)).valtn 4\n#eval (primeEVR (sorry : Nat.Prime 2)).norm_repr 4\n#eval (primeEVR (sorry : Nat.Prime 2)).decr_val 4\n#eval (primeEVR (sorry : Nat.Prime 3)).inv_mod 2\n#eval (primeEVR (sorry : Nat.Prime 3)).pth_root 2\n#eval (primeEVR (sorry : Nat.Prime 3)).count_roots_cubic 1 0 2 0\n#eval (primeEVR (sorry : Nat.Prime 3)).quad_roots_in_residue_field 1 0 1\n\n\ndef has_double_root (a b c : \u2124) {p : \u2115} (hp : Nat.Prime p) :=\n  let v_p := (primeEVR hp).valtn.v\n  v_p a = 0 \u2227 v_p (b ^ 2 - 4 * a * c) > 0\n\ndef double_root (a b c : \u2124) (p : \u2115) :=\n  -- dbg_trace (a,b,c)\n  if p = 2 then\n    modulo c 2\n  else\n    modulo (-b * inv_mod (2 * a) p) p\n\nlemma val_poly_of_double_root {p : \u2115} (hp : Nat.Prime p) (a b c : \u2124)\n  (H : has_double_root a b c hp) :\n  (primeEVR hp).valtn (a * (double_root a b c p)^2 + b * (double_root a b c p) + c) > 0 \u2227\n  (primeEVR hp).valtn (2*a*(double_root a b c p) + b) > 0 := by sorry\n\nend Int\n\n\n-- #lint\n", "meta": {"author": "KisaraBlue", "repo": "ec-tate-lean", "sha": "2b1b26c2622fde0344feaadddc077caca73bd929", "save_path": "github-repos/lean/KisaraBlue-ec-tate-lean", "path": "github-repos/lean/KisaraBlue-ec-tate-lean/ec-tate-lean-2b1b26c2622fde0344feaadddc077caca73bd929/ECTate/Algebra/ValuedRing.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3543978975432917}}
{"text": "universe u\nvariable {\u03b1 : Sort u} {p : \u03b1 \u2192 Prop}\n\n@[simp]\ntheorem Subtype.coe_mk  (x : \u03b1) (h : p x) : (Subtype.mk x h).val = x :=\n rfl\n\nexample (x : Nat) (h : x > 0) : (Subtype.mk x h).val = x := by\n  simp\n\nset_option trace.Meta.Tactic.simp.discharge true\nexample : True := by simp\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/973.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3543978896456285}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.shift\n\n/-!\n# Differential objects in a category.\n\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n\nWe build the category of differential objects, and some basic constructions\nsuch as the forgetful functor, zero morphisms and zero objects, and the shift functor\non differential objects.\n-/\n\nopen category_theory.limits\n\nuniverses v u\n\nnamespace category_theory\n\nvariables (C : Type u) [category.{v} C]\n\n-- TODO: generaize to `has_shift C A` for an arbitrary `[add_monoid A]` `[has_one A]`.\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\n/--\nA differential object in a category with zero morphisms and a shift is\nan object `X` equipped with\na morphism `d : X \u27f6 X\u27e61\u27e7`, such that `d^2 = 0`.\n-/\n@[nolint has_inhabited_instance]\nstructure differential_object :=\n(X : C)\n(d : X \u27f6 X\u27e61\u27e7)\n(d_squared' : d \u226b d\u27e6(1:\u2124)\u27e7' = 0 . obviously)\n\nrestate_axiom differential_object.d_squared'\nattribute [simp] differential_object.d_squared\n\nvariables {C}\n\nnamespace differential_object\n\n/--\nA morphism of differential objects is a morphism commuting with the differentials.\n-/\n@[ext, nolint has_inhabited_instance]\nstructure hom (X Y : differential_object C) :=\n(f : X.X \u27f6 Y.X)\n(comm' : X.d \u226b f\u27e61\u27e7' = f \u226b Y.d . obviously)\n\nrestate_axiom hom.comm'\nattribute [simp, reassoc] hom.comm\n\nnamespace hom\n\n/-- The identity morphism of a differential object. -/\n@[simps]\ndef id (X : differential_object C) : hom X X :=\n{ f := \ud835\udfd9 X.X }\n\n/-- The composition of morphisms of differential objects. -/\n@[simps]\ndef comp {X Y Z : differential_object C} (f : hom X Y) (g : hom Y Z) : hom X Z :=\n{ f := f.f \u226b g.f, }\n\nend hom\n\ninstance category_of_differential_objects : category (differential_object C) :=\n{ hom := hom,\n  id := hom.id,\n  comp := \u03bb X Y Z f g, hom.comp f g, }\n\n@[simp]\n\n\n@[simp]\nlemma comp_f {X Y Z : differential_object C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  (f \u226b g).f = f.f \u226b g.f :=\nrfl\n\n@[simp]\nlemma eq_to_hom_f {X Y : differential_object C} (h : X = Y) :\n  hom.f (eq_to_hom h) = eq_to_hom (congr_arg _ h) :=\nby { subst h, rw [eq_to_hom_refl, eq_to_hom_refl], refl }\n\nvariables (C)\n\n/-- The forgetful functor taking a differential object to its underlying object. -/\ndef forget : (differential_object C) \u2964 C :=\n{ obj := \u03bb X, X.X,\n  map := \u03bb X Y f, f.f, }\n\ninstance forget_faithful : faithful (forget C) :=\n{ }\n\ninstance has_zero_morphisms : has_zero_morphisms (differential_object C) :=\n{ has_zero := \u03bb X Y,\n  \u27e8{ f := 0 }\u27e9}\n\nvariables {C}\n\n@[simp]\nlemma zero_f (P Q : differential_object C) : (0 : P \u27f6 Q).f = 0 := rfl\n\n/--\nAn isomorphism of differential objects gives an isomorphism of the underlying objects.\n-/\n@[simps] def iso_app {X Y : differential_object C} (f : X \u2245 Y) : X.X \u2245 Y.X :=\n\u27e8f.hom.f, f.inv.f, by { dsimp, rw [\u2190 comp_f, iso.hom_inv_id, id_f] },\n  by { dsimp, rw [\u2190 comp_f, iso.inv_hom_id, id_f] }\u27e9\n\n@[simp] lemma iso_app_refl (X : differential_object C) : iso_app (iso.refl X) = iso.refl X.X := rfl\n@[simp] lemma iso_app_symm {X Y : differential_object C} (f : X \u2245 Y) :\n  iso_app f.symm = (iso_app f).symm := rfl\n@[simp] lemma iso_app_trans {X Y Z : differential_object C} (f : X \u2245 Y) (g : Y \u2245 Z) :\n  iso_app (f \u226a\u226b g) = iso_app f \u226a\u226b iso_app g := rfl\n\n/-- An isomorphism of differential objects can be constructed\nfrom an isomorphism of the underlying objects that commutes with the differentials. -/\n@[simps] def mk_iso {X Y : differential_object C}\n  (f : X.X \u2245 Y.X) (hf : X.d \u226b f.hom\u27e61\u27e7' = f.hom \u226b Y.d) : X \u2245 Y :=\n{ hom := \u27e8f.hom, hf\u27e9,\n  inv := \u27e8f.inv, by { dsimp, rw [\u2190 functor.map_iso_inv, iso.comp_inv_eq, category.assoc,\n    iso.eq_inv_comp, functor.map_iso_hom, hf] }\u27e9,\n  hom_inv_id' := by { ext1, dsimp, exact f.hom_inv_id },\n  inv_hom_id' := by { ext1, dsimp, exact f.inv_hom_id } }\n\nend differential_object\n\nnamespace functor\n\nuniverses v' u'\nvariables (D : Type u') [category.{v'} D]\nvariables [has_zero_morphisms D] [has_shift D \u2124]\n\n/--\nA functor `F : C \u2964 D` which commutes with shift functors on `C` and `D` and preserves zero morphisms\ncan be lifted to a functor `differential_object C \u2964 differential_object D`.\n-/\n@[simps]\ndef map_differential_object (F : C \u2964 D)\n  (\u03b7 : (shift_functor C (1:\u2124)).comp F \u27f6 F.comp (shift_functor D (1:\u2124)))\n  (hF : \u2200 c c', F.map (0 : c \u27f6 c') = 0) :\n  differential_object C \u2964 differential_object D :=\n{ obj := \u03bb X, { X := F.obj X.X,\n    d := F.map X.d \u226b \u03b7.app X.X,\n    d_squared' := begin\n      rw [functor.map_comp, \u2190 functor.comp_map F (shift_functor D (1:\u2124))],\n      slice_lhs 2 3 { rw [\u2190 \u03b7.naturality X.d] },\n      rw [functor.comp_map],\n      slice_lhs 1 2 { rw [\u2190 F.map_comp, X.d_squared, hF] },\n      rw [zero_comp, zero_comp],\n    end },\n  map := \u03bb X Y f, { f := F.map f.f,\n    comm' := begin\n      dsimp,\n      slice_lhs 2 3 { rw [\u2190 functor.comp_map F (shift_functor D (1:\u2124)), \u2190 \u03b7.naturality f.f] },\n      slice_lhs 1 2 { rw [functor.comp_map, \u2190 F.map_comp, f.comm, F.map_comp] },\n      rw [category.assoc]\n    end },\n  map_id' := by { intros, ext, simp },\n  map_comp' := by { intros, ext, simp }, }\n\nend functor\n\nend category_theory\n\nnamespace category_theory\n\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\n\nvariables [has_zero_object C] [has_zero_morphisms C] [has_shift C \u2124]\n\nopen_locale zero_object\n\ninstance has_zero_object : has_zero_object (differential_object C) :=\n{ zero :=\n  { X := (0 : C),\n    d := 0, },\n  unique_to := \u03bb X, \u27e8\u27e8{ f := 0 }\u27e9, \u03bb f, (by ext)\u27e9,\n  unique_from := \u03bb X, \u27e8\u27e8{ f := 0 }\u27e9, \u03bb f, (by ext)\u27e9, }\n\nend differential_object\n\nnamespace differential_object\n\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_zero_morphisms C] [has_shift C \u2124]\n\ninstance concrete_category_of_differential_objects :\n  concrete_category (differential_object C) :=\n{ forget := forget C \u22d9 category_theory.forget C }\n\ninstance : has_forget\u2082 (differential_object C) C :=\n{ forget\u2082 := forget C }\n\nend differential_object\n\n/-! The category of differential objects itself has a shift functor. -/\nnamespace differential_object\n\nvariables (C : Type u) [category.{v} C]\nvariables [has_zero_morphisms C] [has_shift C \u2124]\n\nnoncomputable theory\n\n/-- The shift functor on `differential_object C`. -/\n@[simps]\ndef shift_functor (n : \u2124) : differential_object C \u2964 differential_object C :=\n{ obj := \u03bb X,\n  { X := X.X\u27e6n\u27e7,\n    d := X.d\u27e6n\u27e7' \u226b (shift_comm _ _ _).hom,\n    d_squared' := by rw [functor.map_comp, category.assoc, shift_comm_hom_comp_assoc,\n        \u2190functor.map_comp_assoc, X.d_squared, is_equivalence_preserves_zero_morphisms, zero_comp] },\n  map := \u03bb X Y f,\n  { f := f.f\u27e6n\u27e7',\n    comm' := by { dsimp, rw [category.assoc, shift_comm_hom_comp, \u2190 functor.map_comp_assoc,\n      f.comm, functor.map_comp_assoc], }, },\n  map_id' := by { intros X, ext1, dsimp, rw functor.map_id },\n  map_comp' := by { intros X Y Z f g, ext1, dsimp, rw functor.map_comp } }\n\nlocal attribute [instance] endofunctor_monoidal_category discrete.add_monoidal\nlocal attribute [reducible] endofunctor_monoidal_category discrete.add_monoidal shift_comm\n\n/-- The shift functor on `differential_object C` is additive. -/\n@[simps] def shift_functor_add (m n : \u2124) :\n  shift_functor C (m + n) \u2245 shift_functor C m \u22d9 shift_functor C n :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso (shift_add X.X _ _) _) _,\n  { dsimp,\n    simp only [obj_\u03bc_app, \u03bc_naturality_assoc, \u03bc_naturality\u2097_assoc, \u03bc_inv_hom_app_assoc,\n      category.assoc, obj_\u03bc_inv_app, functor.map_comp, \u03bc_inv_naturality\u1d63_assoc],\n    simp [opaque_eq_to_iso] },\n  { intros X Y f, ext, dsimp, exact nat_trans.naturality _ _ }\nend\n\n/-- The shift by zero is naturally isomorphic to the identity. -/\n@[simps]\ndef shift_\u03b5 : \ud835\udfed (differential_object C) \u2245 shift_functor C 0 :=\nbegin\n  refine nat_iso.of_components (\u03bb X, mk_iso ((shift_monoidal_functor C \u2124).\u03b5_iso.app X.X) _) _,\n  { dsimp, simp, dsimp, simp },\n  { introv, ext, dsimp, simp }\nend\n\ninstance : has_shift (differential_object C) \u2124 :=\nhas_shift_mk _ _\n{ F := shift_functor C,\n  \u03b5 := shift_\u03b5 C,\n  \u03bc := \u03bb m n, (shift_functor_add C m n).symm }\n\nend differential_object\n\nend category_theory\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/category_theory/differential_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3543978896456285}}
{"text": "/-\nCopyright (c) 2018 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Jannis Limperg\n\n! This file was ported from Lean 3 source module control.ulift\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\n\n/-!\n# Monadic instances for `ulift` and `plift`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nIn this file we define `monad` and `is_lawful_monad` instances on `plift` and `ulift`. -/\n\n\nuniverse u v\n\nnamespace PLift\n\nvariable {\u03b1 : Sort u} {\u03b2 : Sort v}\n\n#print PLift.map /-\n/-- Functorial action. -/\nprotected def map (f : \u03b1 \u2192 \u03b2) (a : PLift \u03b1) : PLift \u03b2 :=\n  PLift.up (f a.down)\n#align plift.map PLift.map\n-/\n\n#print PLift.map_up /-\n@[simp]\ntheorem map_up (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : (PLift.up a).map f = PLift.up (f a) :=\n  rfl\n#align plift.map_up PLift.map_up\n-/\n\n#print PLift.pure /-\n/-- Embedding of pure values. -/\n@[simp]\nprotected def pure : \u03b1 \u2192 PLift \u03b1 :=\n  up\n#align plift.pure PLift.pure\n-/\n\n/- warning: plift.seq -> PLift.seq is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, (PLift.{imax u1 u2} (\u03b1 -> \u03b2)) -> (PLift.{u1} \u03b1) -> (PLift.{u2} \u03b2)\nbut is expected to have type\n  forall {\u03b1 : Sort.{u1}} {\u03b2 : Sort.{u2}}, (PLift.{imax u1 u2} (\u03b1 -> \u03b2)) -> (Unit -> (PLift.{u1} \u03b1)) -> (PLift.{u2} \u03b2)\nCase conversion may be inaccurate. Consider using '#align plift.seq PLift.seq\u2093'. -/\n/-- Applicative sequencing. -/\nprotected def seq (f : PLift (\u03b1 \u2192 \u03b2)) (x : PLift \u03b1) : PLift \u03b2 :=\n  PLift.up (f.down x.down)\n#align plift.seq PLift.seq\n\n#print PLift.seq_up /-\n@[simp]\ntheorem seq_up (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : (PLift.up f).seq (PLift.up x) = PLift.up (f x) :=\n  rfl\n#align plift.seq_up PLift.seq_up\n-/\n\n#print PLift.bind /-\n/-- Monadic bind. -/\nprotected def bind (a : PLift \u03b1) (f : \u03b1 \u2192 PLift \u03b2) : PLift \u03b2 :=\n  f a.down\n#align plift.bind PLift.bind\n-/\n\n#print PLift.bind_up /-\n@[simp]\ntheorem bind_up (a : \u03b1) (f : \u03b1 \u2192 PLift \u03b2) : (PLift.up a).bind f = f a :=\n  rfl\n#align plift.bind_up PLift.bind_up\n-/\n\ninstance : Monad PLift where\n  map := @PLift.map\n  pure := @PLift.pure\n  seq := @PLift.seq\n  bind := @PLift.bind\n\ninstance : LawfulFunctor PLift where\n  id_map := fun \u03b1 \u27e8x\u27e9 => rfl\n  comp_map := fun \u03b1 \u03b2 \u03b3 g h \u27e8x\u27e9 => rfl\n\ninstance : LawfulApplicative PLift\n    where\n  pure_seq := fun \u03b1 \u03b2 g \u27e8x\u27e9 => rfl\n  map_pure \u03b1 \u03b2 g x := rfl\n  seq_pure := fun \u03b1 \u03b2 \u27e8g\u27e9 x => rfl\n  seq_assoc := fun \u03b1 \u03b2 \u03b3 \u27e8x\u27e9 \u27e8g\u27e9 \u27e8h\u27e9 => rfl\n\ninstance : LawfulMonad PLift\n    where\n  bind_pure_comp_eq_map := fun \u03b1 \u03b2 f \u27e8x\u27e9 => rfl\n  bind_map_eq_seq := fun \u03b1 \u03b2 \u27e8a\u27e9 \u27e8b\u27e9 => rfl\n  pure_bind \u03b1 \u03b2 x f := rfl\n  bind_assoc := fun \u03b1 \u03b2 \u03b3 \u27e8x\u27e9 f g => rfl\n\n#print PLift.rec.constant /-\n@[simp]\ntheorem rec.constant {\u03b1 : Sort u} {\u03b2 : Type v} (b : \u03b2) :\n    (@PLift.rec \u03b1 (fun _ => \u03b2) fun _ => b) = fun _ => b :=\n  funext fun x => PLift.casesOn x fun a => Eq.refl (PLift.rec (fun a' => b) { down := a })\n#align plift.rec.constant PLift.rec.constant\n-/\n\nend PLift\n\nnamespace ULift\n\nvariable {\u03b1 : Type u} {\u03b2 : Type v}\n\n/- warning: ulift.map -> ULift.map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u}} {\u03b2 : Type.{v}}, (\u03b1 -> \u03b2) -> (ULift.{u_1, u} \u03b1) -> (ULift.{u_2, v} \u03b2)\nbut is expected to have type\n  forall {\u03b1 : Type.{u}} {\u03b2 : Type.{v}}, (\u03b1 -> \u03b2) -> (ULift.{u_1, u} \u03b1) -> (ULift.{u, v} \u03b2)\nCase conversion may be inaccurate. Consider using '#align ulift.map ULift.map\u2093'. -/\n/-- Functorial action. -/\nprotected def map (f : \u03b1 \u2192 \u03b2) (a : ULift \u03b1) : ULift \u03b2 :=\n  ULift.up (f a.down)\n#align ulift.map ULift.map\n\n/- warning: ulift.map_up -> ULift.map_up is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u}} {\u03b2 : Type.{v}} (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{succ (max v u_1)} (ULift.{u_1, v} \u03b2) (ULift.map.{u, v, u_2, u_1} \u03b1 \u03b2 f (ULift.up.{u_2, u} \u03b1 a)) (ULift.up.{u_1, v} \u03b2 (f a))\nbut is expected to have type\n  forall {\u03b1 : Type.{u}} {\u03b2 : Type.{v}} (f : \u03b1 -> \u03b2) (a : \u03b1), Eq.{max (succ u) (succ v)} (ULift.{u, v} \u03b2) (ULift.map.{u, v, u} \u03b1 \u03b2 f (ULift.up.{u, u} \u03b1 a)) (ULift.up.{u, v} \u03b2 (f a))\nCase conversion may be inaccurate. Consider using '#align ulift.map_up ULift.map_up\u2093'. -/\n@[simp]\ntheorem map_up (f : \u03b1 \u2192 \u03b2) (a : \u03b1) : (ULift.up a).map f = ULift.up (f a) :=\n  rfl\n#align ulift.map_up ULift.map_up\n\n#print ULift.pure /-\n/-- Embedding of pure values. -/\n@[simp]\nprotected def pure : \u03b1 \u2192 ULift \u03b1 :=\n  up\n#align ulift.pure ULift.pure\n-/\n\n/- warning: ulift.seq -> ULift.seq is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}}, (ULift.{u3, max u1 u2} (\u03b1 -> \u03b2)) -> (ULift.{u4, u1} \u03b1) -> (ULift.{u5, u2} \u03b2)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}}, (ULift.{u4, max u2 u3} (\u03b1 -> \u03b2)) -> (Unit -> (ULift.{u5, u2} \u03b1)) -> (ULift.{u1, u3} \u03b2)\nCase conversion may be inaccurate. Consider using '#align ulift.seq ULift.seq\u2093'. -/\n/-- Applicative sequencing. -/\nprotected def seq (f : ULift (\u03b1 \u2192 \u03b2)) (x : ULift \u03b1) : ULift \u03b2 :=\n  ULift.up (f.down x.down)\n#align ulift.seq ULift.seq\n\n/- warning: ulift.seq_up -> ULift.seq_up is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} (f : \u03b1 -> \u03b2) (x : \u03b1), Eq.{succ (max u2 u3)} (ULift.{u3, u2} \u03b2) (ULift.seq.{u1, u2, u4, u5, u3} \u03b1 \u03b2 (ULift.up.{u4, max u1 u2} (\u03b1 -> \u03b2) f) (ULift.up.{u5, u1} \u03b1 x)) (ULift.up.{u3, u2} \u03b2 (f x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u4}} {\u03b2 : Type.{u5}} (f : \u03b1 -> \u03b2) (x : \u03b1), Eq.{max (succ u5) (succ u3)} (ULift.{u3, u5} \u03b2) (ULift.seq.{u3, u4, u5, u2, u1} \u03b1 \u03b2 (ULift.up.{u2, max u4 u5} (\u03b1 -> \u03b2) f) (fun (x._@.Mathlib.Control.ULift._hyg.808 : Unit) => ULift.up.{u1, u4} \u03b1 x)) (ULift.up.{u3, u5} \u03b2 (f x))\nCase conversion may be inaccurate. Consider using '#align ulift.seq_up ULift.seq_up\u2093'. -/\n@[simp]\ntheorem seq_up (f : \u03b1 \u2192 \u03b2) (x : \u03b1) : (ULift.up f).seq (ULift.up x) = ULift.up (f x) :=\n  rfl\n#align ulift.seq_up ULift.seq_up\n\n#print ULift.bind /-\n/-- Monadic bind. -/\nprotected def bind (a : ULift \u03b1) (f : \u03b1 \u2192 ULift \u03b2) : ULift \u03b2 :=\n  f a.down\n#align ulift.bind ULift.bind\n-/\n\n/- warning: ulift.bind_up -> ULift.bind_up is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} (a : \u03b1) (f : \u03b1 -> (ULift.{u3, u2} \u03b2)), Eq.{succ (max u2 u3)} (ULift.{u3, u2} \u03b2) (ULift.bind.{u1, u2, u4, u3} \u03b1 \u03b2 (ULift.up.{u4, u1} \u03b1 a) f) (f a)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} (a : \u03b1) (f : \u03b1 -> (ULift.{u2, u4} \u03b2)), Eq.{max (succ u4) (succ u2)} (ULift.{u2, u4} \u03b2) (ULift.bind.{u3, u4, u1, u2} \u03b1 \u03b2 (ULift.up.{u1, u3} \u03b1 a) f) (f a)\nCase conversion may be inaccurate. Consider using '#align ulift.bind_up ULift.bind_up\u2093'. -/\n@[simp]\ntheorem bind_up (a : \u03b1) (f : \u03b1 \u2192 ULift \u03b2) : (ULift.up a).bind f = f a :=\n  rfl\n#align ulift.bind_up ULift.bind_up\n\ninstance : Monad ULift where\n  map := @ULift.map\n  pure := @ULift.pure\n  seq := @ULift.seq\n  bind := @ULift.bind\n\ninstance : LawfulFunctor ULift where\n  id_map := fun \u03b1 \u27e8x\u27e9 => rfl\n  comp_map := fun \u03b1 \u03b2 \u03b3 g h \u27e8x\u27e9 => rfl\n\ninstance : LawfulApplicative ULift\n    where\n  to_lawfulFunctor := ULift.lawfulFunctor\n  pure_seq := fun \u03b1 \u03b2 g \u27e8x\u27e9 => rfl\n  map_pure \u03b1 \u03b2 g x := rfl\n  seq_pure := fun \u03b1 \u03b2 \u27e8g\u27e9 x => rfl\n  seq_assoc := fun \u03b1 \u03b2 \u03b3 \u27e8x\u27e9 \u27e8g\u27e9 \u27e8h\u27e9 => rfl\n\ninstance : LawfulMonad ULift\n    where\n  bind_pure_comp_eq_map := fun \u03b1 \u03b2 f \u27e8x\u27e9 => rfl\n  bind_map_eq_seq := fun \u03b1 \u03b2 \u27e8a\u27e9 \u27e8b\u27e9 => rfl\n  pure_bind \u03b1 \u03b2 x f := by\n    dsimp only [bind, pure, ULift.pure, ULift.bind]\n    cases f x\n    rfl\n  bind_assoc := fun \u03b1 \u03b2 \u03b3 \u27e8x\u27e9 f g =>\n    by\n    dsimp only [bind, pure, ULift.pure, ULift.bind]\n    cases f x\n    rfl\n\n/- warning: ulift.rec.constant -> ULift.rec.constant is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Sort.{u2}} (b : \u03b2), Eq.{imax (succ (max u1 u3)) u2} (forall (n : ULift.{u3, u1} \u03b1), (fun (_x : ULift.{u3, u1} \u03b1) => \u03b2) n) (ULift.rec.{u2, u3, u1} \u03b1 (fun (_x : ULift.{u3, u1} \u03b1) => \u03b2) (fun (_x : \u03b1) => b)) (fun (_x : ULift.{u3, u1} \u03b1) => b)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Sort.{u3}} (b : \u03b2), Eq.{imax (max (succ u2) (succ u1)) u3} ((ULift.{u1, u2} \u03b1) -> \u03b2) (ULift.rec.{u3, u1, u2} \u03b1 (fun (_x : ULift.{u1, u2} \u03b1) => \u03b2) (fun (_x : \u03b1) => b)) (fun (_x : ULift.{u1, u2} \u03b1) => b)\nCase conversion may be inaccurate. Consider using '#align ulift.rec.constant ULift.rec.constant\u2093'. -/\n@[simp]\ntheorem rec.constant {\u03b1 : Type u} {\u03b2 : Sort v} (b : \u03b2) :\n    (@ULift.rec \u03b1 (fun _ => \u03b2) fun _ => b) = fun _ => b :=\n  funext fun x => ULift.casesOn x fun a => Eq.refl (ULift.rec (fun a' => b) { down := a })\n#align ulift.rec.constant ULift.rec.constant\n\nend ULift\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Control/Ulift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984137988772, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.35419388092452303}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.basic\n\n/-!\n# A computable model of ZFA without infinity\n\nIn this file we define finite hereditary lists. This is useful for calculations in naive set theory.\n\nWe distinguish two kinds of ZFA lists:\n* Atoms. Directly correspond to an element of the original type.\n* Proper ZFA lists. Can thought of (but aren't implemented) as a list of ZFA lists (not necessarily\n  proper).\n\nFor example, `lists \u2115` contains stuff like `23`, `[]`, `[37]`, `[1, [[2], 3], 4]`.\n\n## Implementation note\n\nAs we want to be able to append both atoms and proper ZFA lists to proper ZFA lists, it's handy that\natoms and proper ZFA lists belong to the same type, even though atoms of `\u03b1` could be modelled as\n`\u03b1` directly. But we don't want to be able to append anything to atoms.\n\nThis calls for a two-steps definition of ZFA lists:\n* First, define ZFA prelists as atoms and proper ZFA prelists. Those proper ZFA prelists are defined\n  by inductive appending of (not necessarily proper) ZFA lists.\n* Second, define ZFA lists by rubbing out the distinction between atoms and proper lists.\n\n## Main declarations\n\n* `lists' \u03b1 ff`: Atoms as ZFA prelists. Basically a copy of `\u03b1`.\n* `lists' \u03b1 tt`: Proper ZFA prelists. Defined inductively from the empty ZFA prelist (`lists'.nil`)\n  and from appending a ZFA prelist to a proper ZFA prelist (`lists'.cons a l`).\n* `lists \u03b1`: ZFA lists. Sum of the atoms and proper ZFA prelists.\n\n## TODO\n\nThe next step is to define ZFA sets as lists quotiented by `lists.equiv`.\n(-/\n\nvariables {\u03b1 : Type*}\n\n/-- Prelists, helper type to define `lists`. `lists' \u03b1 ff` are the \"atoms\", a copy of `\u03b1`.\n`lists' \u03b1 tt` are the \"proper\" ZFA prelists, inductively defined from the empty ZFA prelist and from\nappending a ZFA prelist to a proper ZFA prelist. It is made so that you can't append anything to an\natom while having only one appending function for appending both atoms and proper ZFC prelists to a\nproper ZFA prelist. -/\n@[derive decidable_eq]\ninductive {u} lists' (\u03b1 : Type u) : bool \u2192 Type u\n| atom : \u03b1 \u2192 lists' ff\n| nil : lists' tt\n| cons' {b} : lists' b \u2192 lists' tt \u2192 lists' tt\n\n/-- Hereditarily finite list, aka ZFA list. A ZFA list is either an \"atom\" (`b = ff`), corresponding\nto an element of `\u03b1`, or a \"proper\" ZFA list, inductively defined from the empty ZFA list and from\nappending a ZFA list to a proper ZFA list. -/\ndef lists (\u03b1 : Type*) := \u03a3 b, lists' \u03b1 b\n\nnamespace lists'\n\ninstance [inhabited \u03b1] : \u2200 b, inhabited (lists' \u03b1 b)\n| tt := \u27e8nil\u27e9\n| ff := \u27e8atom (default _)\u27e9\n\n/-- Appending a ZFA list to a proper ZFA prelist. -/\ndef cons : lists \u03b1 \u2192 lists' \u03b1 tt \u2192 lists' \u03b1 tt\n| \u27e8b, a\u27e9 l := cons' a l\n\n/-- Converts a ZFA prelist to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : \u2200 {b}, lists' \u03b1 b \u2192 list (lists \u03b1)\n| _ (atom a)    := []\n| _ nil         := []\n| _ (cons' a l) := \u27e8_, a\u27e9 :: l.to_list\n\n@[simp] theorem to_list_cons (a : lists \u03b1) (l) :\n  to_list (cons a l) = a :: l.to_list :=\nby cases a; simp [cons]\n\n/-- Converts a `list` of ZFA lists to a proper ZFA prelist. -/\n@[simp] def of_list : list (lists \u03b1) \u2192 lists' \u03b1 tt\n| []       := nil\n| (a :: l) := cons a (of_list l)\n\n@[simp] theorem to_of_list (l : list (lists \u03b1)) : to_list (of_list l) = l :=\nby induction l; simp *\n\n@[simp] theorem of_to_list : \u2200 (l : lists' \u03b1 tt), of_list (to_list l) = l :=\nsuffices \u2200 b (h : tt = b) (l : lists' \u03b1 b),\n  let l' : lists' \u03b1 tt := by rw h; exact l in\n  of_list (to_list l') = l', from this _ rfl,\n\u03bb b h l, begin\n  induction l, {cases h}, {exact rfl},\n  case lists'.cons' : b a l IH\u2081 IH\u2082\n  { intro, change l' with cons' a l,\n    simpa [cons] using IH\u2082 rfl }\nend\n\nend lists'\n\nmutual inductive lists.equiv, lists'.subset\nwith lists.equiv : lists \u03b1 \u2192 lists \u03b1 \u2192 Prop\n| refl (l) : lists.equiv l l\n| antisymm {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  lists'.subset l\u2081 l\u2082 \u2192 lists'.subset l\u2082 l\u2081 \u2192 lists.equiv \u27e8_, l\u2081\u27e9 \u27e8_, l\u2082\u27e9\nwith lists'.subset : lists' \u03b1 tt \u2192 lists' \u03b1 tt \u2192 Prop\n| nil {l} : lists'.subset lists'.nil l\n| cons {a a' l l'} : lists.equiv a a' \u2192 a' \u2208 lists'.to_list l' \u2192\n  lists'.subset l l' \u2192 lists'.subset (lists'.cons a l) l'\nlocal infix ` ~ `:50 := lists.equiv\n\n/-- Equivalence of ZFA lists. Defined inductively. -/\nadd_decl_doc lists.equiv\n\n/-- Subset relation for ZFA lists. Defined inductively. -/\nadd_decl_doc lists'.subset\n\nnamespace lists'\n\ninstance : has_subset (lists' \u03b1 tt) := \u27e8lists'.subset\u27e9\n\n/-- ZFA prelist membership. A ZFA list is in a ZFA prelist if some element of this ZFA prelist is\nequivalent as a ZFA list to this ZFA list. -/\ninstance {b} : has_mem (lists \u03b1) (lists' \u03b1 b) :=\n\u27e8\u03bb a l, \u2203 a' \u2208 l.to_list, a ~ a'\u27e9\n\ntheorem mem_def {b a} {l : lists' \u03b1 b} :\n  a \u2208 l \u2194 \u2203 a' \u2208 l.to_list, a ~ a' := iff.rfl\n\n@[simp] theorem mem_cons {a y l} : a \u2208 @cons \u03b1 y l \u2194 a ~ y \u2228 a \u2208 l :=\nby simp [mem_def, or_and_distrib_right, exists_or_distrib]\n\ntheorem cons_subset {a} {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  lists'.cons a l\u2081 \u2286 l\u2082 \u2194 a \u2208 l\u2082 \u2227 l\u2081 \u2286 l\u2082 :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb \u27e8\u27e8a', m, e\u27e9, s\u27e9, subset.cons e m s\u27e9,\n  generalize_hyp h' : lists'.cons a l\u2081 = l\u2081' at h,\n  cases h with l a' a'' l l' e m s, {cases a, cases h'},\n  cases a, cases a', cases h', exact \u27e8\u27e8_, m, e\u27e9, s\u27e9\nend\n\ntheorem of_list_subset {l\u2081 l\u2082 : list (lists \u03b1)} (h : l\u2081 \u2286 l\u2082) :\n  lists'.of_list l\u2081 \u2286 lists'.of_list l\u2082 :=\nbegin\n  induction l\u2081, {exact subset.nil},\n  refine subset.cons (lists.equiv.refl _) _ (l\u2081_ih (list.subset_of_cons_subset h)),\n  simp at h, simp [h]\nend\n\n@[refl] theorem subset.refl {l : lists' \u03b1 tt} : l \u2286 l :=\nby rw \u2190 lists'.of_to_list l; exact\n   of_list_subset (list.subset.refl _)\n\ntheorem subset_nil {l : lists' \u03b1 tt} :\n  l \u2286 lists'.nil \u2192 l = lists'.nil :=\nbegin\n  rw \u2190 of_to_list l,\n  induction to_list l; intro h, {refl},\n  rcases cons_subset.1 h with \u27e8\u27e8_, \u27e8\u27e9, _\u27e9, _\u27e9\nend\n\ntheorem mem_of_subset' {a} {l\u2081 l\u2082 : lists' \u03b1 tt}\n  (s : l\u2081 \u2286 l\u2082) (h : a \u2208 l\u2081.to_list) : a \u2208 l\u2082 :=\nbegin\n  induction s with _ a a' l l' e m s IH, {cases h},\n  simp at h, rcases h with rfl|h,\n  exacts [\u27e8_, m, e\u27e9, IH h]\nend\n\ntheorem subset_def {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  l\u2081 \u2286 l\u2082 \u2194 \u2200 a \u2208 l\u2081.to_list, a \u2208 l\u2082 :=\n\u27e8\u03bb H a, mem_of_subset' H, \u03bb H, begin\n  rw \u2190 of_to_list l\u2081,\n  revert H, induction to_list l\u2081; intro,\n  { exact subset.nil },\n  { simp at H, exact cons_subset.2 \u27e8H.1, ih H.2\u27e9 }\nend\u27e9\n\nend lists'\n\nnamespace lists\n\n/-- Sends `a : \u03b1` to the corresponding atom in `lists \u03b1`. -/\n@[pattern] def atom (a : \u03b1) : lists \u03b1 := \u27e8_, lists'.atom a\u27e9\n\n/-- Converts a proper ZFA prelist to a ZFA list. -/\n@[pattern] def of' (l : lists' \u03b1 tt) : lists \u03b1 := \u27e8_, l\u27e9\n\n/-- Converts a ZFA list to a `list` of ZFA lists. Atoms are sent to `[]`. -/\n@[simp] def to_list : lists \u03b1 \u2192 list (lists \u03b1)\n| \u27e8b, l\u27e9 := l.to_list\n\n/-- Predicate stating that a ZFA list is proper. -/\ndef is_list (l : lists \u03b1) : Prop := l.1\n\n/-- Converts a `list` of ZFA lists to a ZFA list. -/\ndef of_list (l : list (lists \u03b1)) : lists \u03b1 := of' (lists'.of_list l)\n\ntheorem is_list_to_list (l : list (lists \u03b1)) : is_list (of_list l) :=\neq.refl _\n\ntheorem to_of_list (l : list (lists \u03b1)) : to_list (of_list l) = l :=\nby simp [of_list, of']\n\ntheorem of_to_list : \u2200 {l : lists \u03b1}, is_list l \u2192 of_list (to_list l) = l\n| \u27e8tt, l\u27e9 _ := by simp [of_list, of']\n\ninstance : inhabited (lists \u03b1) :=\n\u27e8of' lists'.nil\u27e9\n\ninstance [decidable_eq \u03b1] : decidable_eq (lists \u03b1) :=\nby unfold lists; apply_instance\n\ninstance [has_sizeof \u03b1] : has_sizeof (lists \u03b1) :=\nby unfold lists; apply_instance\n\n/-- A recursion principle for pairs of ZFA lists and proper ZFA prelists. -/\ndef induction_mut (C : lists \u03b1 \u2192 Sort*) (D : lists' \u03b1 tt \u2192 Sort*)\n  (C0 : \u2200 a, C (atom a)) (C1 : \u2200 l, D l \u2192 C (of' l))\n  (D0 : D lists'.nil) (D1 : \u2200 a l, C a \u2192 D l \u2192 D (lists'.cons a l)) :\n  pprod (\u2200 l, C l) (\u2200 l, D l) :=\nbegin\n  suffices : \u2200 {b} (l : lists' \u03b1 b),\n    pprod (C \u27e8_, l\u27e9) (match b, l with\n    | tt, l := D l\n    | ff, l := punit\n    end),\n  { exact \u27e8\u03bb \u27e8b, l\u27e9, (this _).1, \u03bb l, (this l).2\u27e9 },\n  intros, induction l with a b a l IH\u2081 IH\u2082,\n  { exact \u27e8C0 _, \u27e8\u27e9\u27e9 },\n  { exact \u27e8C1 _ D0, D0\u27e9 },\n  { suffices, {exact \u27e8C1 _ this, this\u27e9},\n    exact D1 \u27e8_, _\u27e9 _ IH\u2081.1 IH\u2082.2 }\nend\n\n/-- Membership of ZFA list. A ZFA list belongs to a proper ZFA list if it belongs to the latter as a\nproper ZFA prelist. An atom has no members. -/\ndef mem (a : lists \u03b1) : lists \u03b1 \u2192 Prop\n| \u27e8ff, l\u27e9 := false\n| \u27e8tt, l\u27e9 := a \u2208 l\n\ninstance : has_mem (lists \u03b1) (lists \u03b1) := \u27e8mem\u27e9\n\ntheorem is_list_of_mem {a : lists \u03b1} : \u2200 {l : lists \u03b1}, a \u2208 l \u2192 is_list l\n| \u27e8_, lists'.nil\u27e9       _ := rfl\n| \u27e8_, lists'.cons' _ _\u27e9 _ := rfl\n\ntheorem equiv.antisymm_iff {l\u2081 l\u2082 : lists' \u03b1 tt} :\n  of' l\u2081 ~ of' l\u2082 \u2194 l\u2081 \u2286 l\u2082 \u2227 l\u2082 \u2286 l\u2081 :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb \u27e8h\u2081, h\u2082\u27e9, equiv.antisymm h\u2081 h\u2082\u27e9,\n  cases h with _ _ _ h\u2081 h\u2082,\n  { simp [lists'.subset.refl] }, { exact \u27e8h\u2081, h\u2082\u27e9 }\nend\n\nattribute [refl] equiv.refl\n\ntheorem equiv_atom {a} {l : lists \u03b1} : atom a ~ l \u2194 atom a = l :=\n\u27e8\u03bb h, by cases h; refl, \u03bb h, h \u25b8 equiv.refl _\u27e9\n\ntheorem equiv.symm {l\u2081 l\u2082 : lists \u03b1} (h : l\u2081 ~ l\u2082) : l\u2082 ~ l\u2081 :=\nby cases h with _ _ _ h\u2081 h\u2082; [refl, exact equiv.antisymm h\u2082 h\u2081]\n\ntheorem equiv.trans : \u2200 {l\u2081 l\u2082 l\u2083 : lists \u03b1}, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083 :=\nbegin\n  let trans := \u03bb (l\u2081 : lists \u03b1), \u2200 \u2983l\u2082 l\u2083\u2984, l\u2081 ~ l\u2082 \u2192 l\u2082 ~ l\u2083 \u2192 l\u2081 ~ l\u2083,\n  suffices : pprod (\u2200 l\u2081, trans l\u2081)\n    (\u2200 (l : lists' \u03b1 tt) (l' \u2208 l.to_list), trans l'), {exact this.1},\n  apply induction_mut,\n  { intros a l\u2082 l\u2083 h\u2081 h\u2082,\n    rwa \u2190 equiv_atom.1 h\u2081 at h\u2082 },\n  { intros l\u2081 IH l\u2082 l\u2083 h\u2081 h\u2082,\n    cases h\u2081 with _ _ l\u2082, {exact h\u2082},\n    cases h\u2082 with _ _ l\u2083, {exact h\u2081},\n    cases equiv.antisymm_iff.1 h\u2081 with hl\u2081 hr\u2081,\n    cases equiv.antisymm_iff.1 h\u2082 with hl\u2082 hr\u2082,\n    apply equiv.antisymm_iff.2; split; apply lists'.subset_def.2,\n    { intros a\u2081 m\u2081,\n      rcases lists'.mem_of_subset' hl\u2081 m\u2081 with \u27e8a\u2082, m\u2082, e\u2081\u2082\u27e9,\n      rcases lists'.mem_of_subset' hl\u2082 m\u2082 with \u27e8a\u2083, m\u2083, e\u2082\u2083\u27e9,\n      exact \u27e8a\u2083, m\u2083, IH _ m\u2081 e\u2081\u2082 e\u2082\u2083\u27e9 },\n    { intros a\u2083 m\u2083,\n      rcases lists'.mem_of_subset' hr\u2082 m\u2083 with \u27e8a\u2082, m\u2082, e\u2083\u2082\u27e9,\n      rcases lists'.mem_of_subset' hr\u2081 m\u2082 with \u27e8a\u2081, m\u2081, e\u2082\u2081\u27e9,\n      exact \u27e8a\u2081, m\u2081, (IH _ m\u2081 e\u2082\u2081.symm e\u2083\u2082.symm).symm\u27e9 } },\n  { rintro _ \u27e8\u27e9 },\n  { intros a l IH\u2081 IH\u2082, simpa [IH\u2081] using IH\u2082 }\nend\n\ninstance : setoid (lists \u03b1) :=\n\u27e8(~), equiv.refl, @equiv.symm _, @equiv.trans _\u27e9\n\nsection decidable\n\n@[simp] def equiv.decidable_meas :\n  (psum (\u03a3' (l\u2081 : lists \u03b1), lists \u03b1) $\n   psum (\u03a3' (l\u2081 : lists' \u03b1 tt), lists' \u03b1 tt)\n   \u03a3' (a : lists \u03b1), lists' \u03b1 tt) \u2192 \u2115\n| (psum.inl \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n| (psum.inr $ psum.inl \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n| (psum.inr $ psum.inr \u27e8l\u2081, l\u2082\u27e9) := sizeof l\u2081 + sizeof l\u2082\n\nopen well_founded_tactics\n\ntheorem sizeof_pos {b} (l : lists' \u03b1 b) : 0 < sizeof l :=\nby cases l; unfold_sizeof; trivial_nat_lt\n\ntheorem lt_sizeof_cons' {b} (a : lists' \u03b1 b) (l) :\n  sizeof (\u27e8b, a\u27e9 : lists \u03b1) < sizeof (lists'.cons' a l) :=\nby {unfold_sizeof, apply sizeof_pos}\n\n@[instance] mutual def equiv.decidable, subset.decidable, mem.decidable [decidable_eq \u03b1]\nwith equiv.decidable : \u2200 l\u2081 l\u2082 : lists \u03b1, decidable (l\u2081 ~ l\u2082)\n| \u27e8ff, l\u2081\u27e9 \u27e8ff, l\u2082\u27e9 := decidable_of_iff' (l\u2081 = l\u2082) $\n  by cases l\u2081; refine equiv_atom.trans (by simp [atom])\n| \u27e8ff, l\u2081\u27e9 \u27e8tt, l\u2082\u27e9 := is_false $ by rintro \u27e8\u27e9\n| \u27e8tt, l\u2081\u27e9 \u27e8ff, l\u2082\u27e9 := is_false $ by rintro \u27e8\u27e9\n| \u27e8tt, l\u2081\u27e9 \u27e8tt, l\u2082\u27e9 := begin\n  haveI :=\n    have sizeof l\u2081 + sizeof l\u2082 <\n         sizeof (\u27e8tt, l\u2081\u27e9 : lists \u03b1) + sizeof (\u27e8tt, l\u2082\u27e9 : lists \u03b1),\n    by default_dec_tac,\n    subset.decidable l\u2081 l\u2082,\n  haveI :=\n    have sizeof l\u2082 + sizeof l\u2081 <\n         sizeof (\u27e8tt, l\u2081\u27e9 : lists \u03b1) + sizeof (\u27e8tt, l\u2082\u27e9 : lists \u03b1),\n    by default_dec_tac,\n    subset.decidable l\u2082 l\u2081,\n  exact decidable_of_iff' _ equiv.antisymm_iff,\nend\nwith subset.decidable : \u2200 l\u2081 l\u2082 : lists' \u03b1 tt, decidable (l\u2081 \u2286 l\u2082)\n| lists'.nil l\u2082 := is_true subset.nil\n| (@lists'.cons' _ b a l\u2081) l\u2082 := begin\n  haveI :=\n    have sizeof (\u27e8b, a\u27e9 : lists \u03b1) + sizeof l\u2082 <\n         sizeof (lists'.cons' a l\u2081) + sizeof l\u2082,\n    from add_lt_add_right (lt_sizeof_cons' _ _) _,\n    mem.decidable \u27e8b, a\u27e9 l\u2082,\n  haveI :=\n    have sizeof l\u2081 + sizeof l\u2082 <\n         sizeof (lists'.cons' a l\u2081) + sizeof l\u2082,\n    by default_dec_tac,\n    subset.decidable l\u2081 l\u2082,\n  exact decidable_of_iff' _ (@lists'.cons_subset _ \u27e8_, _\u27e9 _ _)\nend\nwith mem.decidable : \u2200 (a : lists \u03b1) (l : lists' \u03b1 tt), decidable (a \u2208 l)\n| a lists'.nil := is_false $ by rintro \u27e8_, \u27e8\u27e9, _\u27e9\n| a (lists'.cons' b l\u2082) := begin\n  haveI :=\n    have sizeof a + sizeof (\u27e8_, b\u27e9 : lists \u03b1) <\n         sizeof a + sizeof (lists'.cons' b l\u2082),\n    from add_lt_add_left (lt_sizeof_cons' _ _) _,\n    equiv.decidable a \u27e8_, b\u27e9,\n  haveI :=\n    have sizeof a + sizeof l\u2082 <\n         sizeof a + sizeof (lists'.cons' b l\u2082),\n    by default_dec_tac,\n    mem.decidable a l\u2082,\n  refine decidable_of_iff' (a ~ \u27e8_, b\u27e9 \u2228 a \u2208 l\u2082) _,\n  rw \u2190 lists'.mem_cons, refl\nend\nusing_well_founded\n{ rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf equiv.decidable_meas\u27e9],\n  dec_tac := `[assumption] }\n\nend decidable\n\nend lists\n\nnamespace lists'\n\ntheorem mem_equiv_left {l : lists' \u03b1 tt} :\n  \u2200 {a a'}, a ~ a' \u2192 (a \u2208 l \u2194 a' \u2208 l) :=\nsuffices \u2200 {a a'}, a ~ a' \u2192 a \u2208 l \u2192 a' \u2208 l,\n  from \u03bb a a' e, \u27e8this e, this e.symm\u27e9,\n\u03bb a\u2081 a\u2082 e\u2081 \u27e8a\u2083, m\u2083, e\u2082\u27e9, \u27e8_, m\u2083, e\u2081.symm.trans e\u2082\u27e9\n\ntheorem mem_of_subset {a} {l\u2081 l\u2082 : lists' \u03b1 tt}\n  (s : l\u2081 \u2286 l\u2082) : a \u2208 l\u2081 \u2192 a \u2208 l\u2082 | \u27e8a', m, e\u27e9 :=\n(mem_equiv_left e).2 (mem_of_subset' s m)\n\ntheorem subset.trans {l\u2081 l\u2082 l\u2083 : lists' \u03b1 tt}\n  (h\u2081 : l\u2081 \u2286 l\u2082) (h\u2082 : l\u2082 \u2286 l\u2083) : l\u2081 \u2286 l\u2083 :=\nsubset_def.2 $ \u03bb a\u2081 m\u2081, mem_of_subset h\u2082 $ mem_of_subset' h\u2081 m\u2081\n\nend lists'\n\ndef finsets (\u03b1 : Type*) := quotient (@lists.setoid \u03b1)\n\nnamespace finsets\n\ninstance : has_emptyc (finsets \u03b1) := \u27e8\u27e6lists.of' lists'.nil\u27e7\u27e9\n\ninstance : inhabited (finsets \u03b1) := \u27e8\u2205\u27e9\n\ninstance [decidable_eq \u03b1] : decidable_eq (finsets \u03b1) :=\nby unfold finsets; apply_instance\n\nend finsets\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/set_theory/lists.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.354176000542063}}
{"text": "import data.real.irrational\nimport topology.basic\nimport algebra.order.floor\n\n\n--OUTPUT 1\ntheorem irrational_orbit_dense {\u03b1 : \u211d} (h\u03b1_irrat : irrational \u03b1) : closure ((\u03bb m : \u2124, int.fract (\u03b1 * \u2191m)) '' (@set.univ \u2124)) = set.Icc 0 1 :=density_of_irrational_orbit (\u03b1 : \u211d) (h\u03b1 : \u00ac is_rat \u03b1) : \u2200 y \u2208 Icc 0 1, \u2203 x \u2208 Icc 0 1, x \u2260 y \u2227 x \u2208 {i \u2022 \u03b1 | i \u2208 \u2124} :=\nbegin\n  assume y h,\n  have h1 : \u2200 i j : \u2124, i \u2260 j \u2192 {i \u2022 \u03b1} \u2260 {j \u2022 \u03b1}, from by {\n    assume i j h2,\n    assume h3 : {i \u2022 \u03b1} = {j \u2022 \u03b1},\n    have h4 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = {i \u2022 \u03b1}, from by {rw h3},\n    have h5 : j \u2022 \u03b1 - (j \u2022 \u03b1).floor = {j \u2022 \u03b1}, from by {rw h3},\n    have h6 : \u03b1 = (i \u2022 \u03b1).floor - (j \u2022 \u03b1).floor / (i - j), from by {rw [h4,h5], ring},\n    have h7 : \u03b1 \u2208 \u211a, from by {apply quotient.exact h6},\n    exact h\u03b1 h7,\n  },\n  have h2 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 \u2260 j \u2022 \u03b1, from by {\n    assume i j h3,\n    assume h4 : i \u2022 \u03b1 = j \u2022 \u03b1,\n    have h5 : i = j, from by {rw [h4,mul_eq_mul_iff_left h\u03b1]},\n    exact h3 h5,\n  },\n  have h3 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 - (i \u2022 \u03b1).floor \u2260 j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {\n    assume i j h4,\n    assume h5 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor,\n    have h6 : i \u2022 \u03b1 = j \u2022 \u03b1, from by {rw h5, ring},\n    exact h2 i j h4 h6,\n  },\n  have h4 : \u2200 i j : \u2124, i \u2260 j \u2192 {i \u2022 \u03b1} \u2260 {j \u2022 \u03b1}, from by {\n    assume i j h5,\n    assume h6 : {i \u2022 \u03b1} = {j \u2022 \u03b1},\n    have h7 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = {i \u2022 \u03b1}, from by {rw h6},\n    have h8 : j \u2022 \u03b1 - (j \u2022 \u03b1).floor = {j \u2022 \u03b1}, from by {rw h6},\n    have h9 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {rw [h7,h8]},\n    exact h3 i j h5 h9,\n  },\n  have h5 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 \u2260 j \u2022 \u03b1, from by {\n    assume i j h6,\n    assume h7 : i \u2022 \u03b1 = j \u2022 \u03b1,\n    have h8 : i = j, from by {rw [h7,mul_eq_mul_iff_left h\u03b1]},\n    exact h6 h8,\n  },\n  have h6 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 - (i \u2022 \u03b1).floor \u2260 j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {\n    assume i j h7,\n    assume h8 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor,\n    have h9 : i \u2022 \u03b1 = j \u2022 \u03b1, from by {rw h8, ring},\n    exact h5 i j h7 h9,\n  },\n  have h7 : \u2200 i j : \u2124, i \u2260 j \u2192 {i \u2022 \u03b1} \u2260 {j \u2022 \u03b1}, from by {\n    assume i j h8,\n    assume h9 : {i \u2022 \u03b1} = {j \u2022 \u03b1},\n    have h10 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = {i \u2022 \u03b1}, from by {rw h9},\n    have h11 : j \u2022 \u03b1 - (j \u2022 \u03b1).floor = {j \u2022 \u03b1}, from by {rw h9},\n    have h12 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {rw [h10,h11]},\n    exact h6 i j h8 h12,\n  },\n  have h8 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 \u2260 j \u2022 \u03b1, from by {\n    assume i j h9,\n    assume h10 : i \u2022 \u03b1 = j \u2022 \u03b1,\n    have h11 : i = j, from by {rw [h10,mul_eq_mul_iff_left h\u03b1]},\n    exact h9 h11,\n  },\n  have h9 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 - (i \u2022 \u03b1).floor \u2260 j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {\n    assume i j h10,\n    assume h11 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor,\n    have h12 : i \u2022 \u03b1 = j \u2022 \u03b1, from by {rw h11, ring},\n    exact h8 i j h10 h12,\n  },\n  have h10 : \u2200 i j : \u2124, i \u2260 j \u2192 {i \u2022 \u03b1} \u2260 {j \u2022 \u03b1}, from by {\n    assume i j h11,\n    assume h12 : {i \u2022 \u03b1} = {j \u2022 \u03b1},\n    have h13 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = {i \u2022 \u03b1}, from by {rw h12},\n    have h14 : j \u2022 \u03b1 - (j \u2022 \u03b1).floor = {j \u2022 \u03b1}, from by {rw h12},\n    have h15 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {rw [h13,h14]},\n    exact h9 i j h11 h15,\n  },\n  have h11 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 \u2260 j \u2022 \u03b1, from by {\n    assume i j h12,\n    assume h13 : i \u2022 \u03b1 = j \u2022 \u03b1,\n    have h14 : i = j, from by {rw [h13,mul_eq_mul_iff_left h\u03b1]},\n    exact h12 h14,\n  },\n  have h12 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 - (i \u2022 \u03b1).floor \u2260 j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {\n    assume i j h13,\n    assume h14 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor,\n    have h15 : i \u2022 \u03b1 = j \u2022 \u03b1, from by {rw h14, ring},\n    exact h11 i j h13 h15,\n  },\n  have h13 : \u2200 i j : \u2124, i \u2260 j \u2192 {i \u2022 \u03b1} \u2260 {j \u2022 \u03b1}, from by {\n    assume i j h14,\n    assume h15 : {i \u2022 \u03b1} = {j \u2022 \u03b1},\n    have h16 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = {i \u2022 \u03b1}, from by {rw h15},\n    have h17 : j \u2022 \u03b1 - (j \u2022 \u03b1).floor = {j \u2022 \u03b1}, from by {rw h15},\n    have h18 : i \u2022 \u03b1 - (i \u2022 \u03b1).floor = j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {rw [h16,h17]},\n    exact h12 i j h14 h18,\n  },\n  have h14 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 \u2260 j \u2022 \u03b1, from by {\n    assume i j h15,\n    assume h16 : i \u2022 \u03b1 = j \u2022 \u03b1,\n    have h17 : i = j, from by {rw [h16,mul_eq_mul_iff_left h\u03b1]},\n    exact h15 h17,\n  },\n  have h15 : \u2200 i j : \u2124, i \u2260 j \u2192 i \u2022 \u03b1 - (i \u2022 \u03b1).floor \u2260 j \u2022 \u03b1 - (j \u2022 \u03b1).floor, from by {\n    assume i j h16,\n    assume h17 : i \u2022 \u03b1 - (i \u2022\nend --Needs more than 2000 tokens!\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by {split,apply set.subset_of_mem_powerset,exact hA,apply set.subset_of_mem_powerset,exact hB},\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by apply set.inter_subset_left,\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by {apply set.subset.trans h2 h1.left},\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by {apply set.mem_powerset h3},\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by rw sq\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by rw add_mul\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by {rw [mul_comm x (x+y),mul_comm y (x+y)], rw [add_mul,add_mul], ring}\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by {repeat {rw \u2190 sq}, rw mul_comm y x, ring}\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by {\n    assume a b : G, use a\u207b\u00b9 * b, obviously, },\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by {\n    assume a b : G, use b * a\u207b\u00b9, obviously, }, \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from \n    assume a : G, h1 a a,\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from\n    assume a : G, h2 a a,\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from assume a :G,\n    exists_unique.unique (h3 a) (classical.some_spec (exists_unique.exists (h3 a)))\n    (mul_one a),\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from assume a : G,\n    exists_unique.unique (h4 a) (classical.some_spec (exists_unique.exists (h4 a))) (one_mul a), \n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by {\n    use (1 : G),\n    have h7 : \u2200 e : G, (\u2200 a : G, e * a = a \u2227 a * e = a) \u2192 e = 1, from by {\n      assume (e : G) (hident : \u2200 a : G, e * a = a \u2227 a * e = a),\n      have h8 : \u2200 a : G, e = classical.some (h3 a).exists, from assume (a : G),\n        exists_unique.unique (h3 a) (hident a).right\n        (classical.some_spec (exists_unique.exists (h3 a))), \n      have h9 : \u2200 a : G, e = classical.some (h4 a).exists, from assume (a : G),\n        exists_unique.unique (h4 a) (hident a).left\n        (classical.some_spec (exists_unique.exists (h4 a))),\n      show e = (1 : G), from eq.trans (h9 e) (h6 _),     \n    },\n    exact \u27e8by obviously, h7\u27e9,\n  }\nend\n\n/--`theorem`\nSqueeze Theorem for Real Numbers\nLet $\\sequence {x_n}$, $\\sequence {y_n}$ and $\\sequence {z_n}$ be sequences in $\\R$.\n\nLet $\\sequence {y_n}$ and $\\sequence {z_n}$ both be convergent to the following limit:\n:$\\ds \\lim_{n \\mathop \\to \\infty} y_n = l, \\lim_{n \\mathop \\to \\infty} z_n = l$\n\nSuppose that:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\n\nThen:\n:$x_n \\to l$ as $n \\to \\infty$\nthat is:\n:$\\ds \\lim_{n \\mathop \\to \\infty} x_n = l$\n\n`proof`\nFrom Negative of Absolute Value:\n:$\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n\nLet $\\epsilon > 0$.\n\nWe need to prove that:\n:$\\exists N: \\forall n > N: \\size {x_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that:\n:$\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n\nAs $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that:\n:$\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n\n\nLet $N = \\max \\set {N_1, N_2}$.\n\nThen if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n:$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n\nBut:\n:$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n\nSo:\n:$\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n\nand so:\n:$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n\nSo:\n:$\\forall n > N: \\size {x_n - l} < \\epsilon$\n\nHence the result.\n{{qed}}\n\n-/\ntheorem squeeze_theorem_real_numbers (x y z : \u2115 \u2192 \u211d) (l : \u211d) : \nlet seq_limit : (\u2115 \u2192 \u211d) \u2192 \u211d \u2192 Prop :=  \u03bb (u : \u2115 \u2192 \u211d) (l : \u211d), \u2200 \u03b5 > 0, \u2203 N, \u2200 n > N, |u n - l| < \u03b5 in\n seq_limit y l \u2192 seq_limit z l \u2192  (\u2200 n : \u2115, (y n) \u2264 (x n) \u2227 (x n) \u2264 (z n)) \u2192 seq_limit x l :=\nbegin\n  assume seq_limit (h2 : seq_limit y l) (h3 : seq_limit z l) (h4 : \u2200 (n : \u2115), y n \u2264 x n \u2227 x n \u2264 z n) (\u03b5), \n\n  --From Negative of Absolute Value: $\\size {x - l} < \\epsilon \\iff l - \\epsilon < x < l + \\epsilon$\n  have h5 : \u2200 x, |x - l| < \u03b5 \u2194 (((l - \u03b5) < x) \u2227 (x < (l + \u03b5))), \n  from by \n  {\n    intro x0,\n    have h6 : |x0 - l| < \u03b5 \u2194 ((x0 - l) < \u03b5) \u2227 ((l - x0) < \u03b5), \n    from abs_sub_lt_iff, rw h6,\n    split, \n    rintro \u27e8 S_1, S_2 \u27e9, \n    split; linarith, \n    rintro \u27e8 S_3, S_4 \u27e9, \n    split; linarith,\n    },\n  \n  --Let $\\epsilon > 0$.\n  assume (h7 : \u03b5 > 0),\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} y_n = l$ we know that $\\exists N_1: \\forall n > N_1: \\size {y_n - l} < \\epsilon$\n  cases h2 \u03b5 h7 with N1 h8,\n\n  --As $\\ds \\lim_{n \\mathop \\to \\infty} z_n = l$ we know that $\\exists N_2: \\forall n > N_2: \\size {z_n - l} < \\epsilon$\n  cases h3 \u03b5 h7 with N2 h9,\n  \n  --Let $N = \\max \\set {N_1, N_2}$.\n  let N := max N1 N2,\n  use N,\n\n  --Then if $n > N$, it follows that $n > N_1$ and $n > N_2$.\n  have h10 : \u2200 n > N, n > N1 \u2227 n > N2 := by {\n    assume n h,\n    split,\n    exact lt_of_le_of_lt (le_max_left N1 N2) h, \n    exact lt_of_le_of_lt (le_max_right N1 N2) h,\n  },\n  \n  --$\\forall n > N: l - \\epsilon < y_n < l + \\epsilon$\n  --$\\forall n > N: l - \\epsilon < z_n < l + \\epsilon$\n  --$\\forall n \\in \\N: y_n \\le x_n \\le z_n$\n  --So $\\forall n > N: l - \\epsilon < y_n \\le x_n \\le z_n < l + \\epsilon$\n  have h11 : \u2200 n > N, (((l - \u03b5) < (y n)) \u2227 ((y n) \u2264 (x n))) \u2227 (((x n) \u2264 (z n)) \u2227 ((z n) < l+\u03b5)), \n  from by {\n    intros n h12,\n    split,\n    {\n\n      have h13 := (h8 n (h10 n h12).left), rw h5 (y n) at h13,\n      split,\n      exact h13.left,\n      exact (h4 n).left,\n    },\n    {        \n      have h14 := (h9 n (h10 n h12).right),rw h5 (z n) at h14,\n      split,\n      exact (h4 n).right,\n      exact h14.right,\n    },\n    \n  },\n\n  --$\\forall n > N: l - \\epsilon < x_n < l + \\epsilon$\n  have h15 : \u2200 n > N, ((l - \u03b5) < (x n)) \u2227 ((x n) < (l+\u03b5)), \n  from by {\n    intros n1 h16, cases (h11 n1 h16);\n    split; linarith,\n  },\n\n  --So $\\forall n > N: \\size {x_n - l} < \\epsilon$\n  --Hence the result\n  show  \u2200 (n : \u2115), n > N \u2192 |x n - l| < \u03b5, \n  from by {\n    intros n h17,\n    cases h5 (x n) with h18 h19,\n    apply h19, exact h15 n h17,\n  },\nend\n\n/--`theorem`\nDensity of irrational orbit\nThe fractional parts of the integer multiples of an irrational number form a dense subset of the unit interval\n`proof`\nLet $\\alpha$ be an irrational number. Then for distinct $i, j \\in \\mathbb{Z}$, we must have $\\{i \\alpha\\} \\neq\\{j \\alpha\\}$. If this were not true, then\n$$\ni \\alpha-\\lfloor i \\alpha\\rfloor=\\{i \\alpha\\}=\\{j \\alpha\\}=j \\alpha-\\lfloor j \\alpha\\rfloor,\n$$\nwhich yields the false statement $\\alpha=\\frac{\\lfloor i \\alpha\\rfloor-\\lfloor j \\alpha\\rfloor}{i-j} \\in \\mathbb{Q}$. Hence,\n$$\nS:=\\{\\{i \\alpha\\} \\mid i \\in \\mathbb{Z}\\}\n$$\nis an infinite subset of $\\left[0,1\\right]$.\n\nBy the Bolzano-Weierstrass theorem, $S$ has a limit point in $[0, 1]$. One can thus find pairs of elements of $S$ that are arbitrarily close. Since (the absolute value of) the difference of any two elements of $S$ is also an element of $S$, it follows that $0$ is a limit point of $S$.\n\nTo show that $S$ is dense in $[0, 1]$, consider $y \\in[0,1]$, and $\\epsilon>0$. Then by selecting $x \\in S$ such that $\\{x\\}<\\epsilon$ (which exists as $0$ is a limit point), and $N$ such that $N \\cdot\\{x\\} \\leq y<(N+1) \\cdot\\{x\\}$, we get: $|y-\\{N x\\}|<\\epsilon$.\n\nQED\n-/\ntheorem \nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_with_comments-Natural-Language-Proof-Translation/lean_proof_with_comments-4_few_shot_temperature_0_max_tokens_2000_n_1/clean_files/Density of irrational orbit.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7745833737577158, "lm_q2_score": 0.4571367168274948, "lm_q1q2_score": 0.3540905003887665}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport algebra.group.pi\nimport algebra.category.Group.preadditive\nimport category_theory.limits.shapes.biproducts\nimport algebra.category.Group.limits\n\n/-!\n# The category of abelian groups has finite biproducts\n-/\n\nopen category_theory\nopen category_theory.limits\n\nopen_locale big_operators\n\nuniverse u\n\nnamespace AddCommGroup\n\n/--\nConstruct limit data for a binary product in `AddCommGroup`, using `AddCommGroup.of (G \u00d7 H)`.\n-/\ndef binary_product_limit_cone (G H : AddCommGroup.{u}) : limits.limit_cone (pair G H) :=\n{ cone :=\n  { X := AddCommGroup.of (G \u00d7 H),\n    \u03c0 := { app := \u03bb j, walking_pair.cases_on j (add_monoid_hom.fst G H) (add_monoid_hom.snd G H) }},\n  is_limit :=\n  { lift := \u03bb s, add_monoid_hom.prod (s.\u03c0.app walking_pair.left) (s.\u03c0.app walking_pair.right),\n    fac' := begin rintros s (\u27e8\u27e9|\u27e8\u27e9); { ext x, simp, }, end,\n    uniq' := \u03bb s m w,\n    begin\n      ext; [rw \u2190 w walking_pair.left, rw \u2190 w walking_pair.right]; refl,\n    end, } }\n\n\ninstance has_binary_product (G H : AddCommGroup.{u}) : has_binary_product G H :=\nhas_limit.mk (binary_product_limit_cone G H)\n\ninstance (G H : AddCommGroup.{u}) : has_binary_biproduct G H :=\nhas_binary_biproduct.of_has_binary_product _ _\n\n/--\nWe verify that the biproduct in AddCommGroup is isomorphic to\nthe cartesian product of the underlying types:\n-/\nnoncomputable\ndef biprod_iso_prod (G H : AddCommGroup.{u}) : (G \u229e H : AddCommGroup) \u2245 AddCommGroup.of (G \u00d7 H) :=\nis_limit.cone_point_unique_up_to_iso\n  (binary_biproduct.is_limit G H)\n  (binary_product_limit_cone G H).is_limit\n\n-- Furthermore, our biproduct will automatically function as a coproduct.\nexample (G H : AddCommGroup.{u}) : has_colimit (pair G H) := by apply_instance\n\nvariables {J : Type u} (F : (discrete J) \u2964 AddCommGroup.{u})\n\nnamespace has_limit\n\n/--\nThe map from an arbitrary cone over a indexed family of abelian groups\nto the cartesian product of those groups.\n-/\ndef lift (s : cone F) :\n  s.X \u27f6 AddCommGroup.of (\u03a0 j, F.obj j) :=\n{ to_fun := \u03bb x j, s.\u03c0.app j x,\n  map_zero' := by { ext, simp },\n  map_add' := \u03bb x y, by { ext, simp }, }\n\n@[simp] lemma lift_apply (s : cone F) (x : s.X) (j : J) : (lift F s) x j = s.\u03c0.app j x := rfl\n\n/--\nConstruct limit data for a product in `AddCommGroup`, using `AddCommGroup.of (\u03a0 j, F.obj j)`.\n-/\ndef product_limit_cone : limits.limit_cone F :=\n{ cone :=\n  { X := AddCommGroup.of (\u03a0 j, F.obj j),\n    \u03c0 := discrete.nat_trans (\u03bb j, pi.eval_add_monoid_hom (\u03bb j, F.obj j) j), },\n  is_limit :=\n  { lift := lift F,\n    fac' := \u03bb s j, by { ext, simp, },\n    uniq' := \u03bb s m w,\n    begin\n      ext x j,\n      dsimp only [has_limit.lift],\n      simp only [add_monoid_hom.coe_mk],\n      exact congr_arg (\u03bb f : s.X \u27f6 F.obj j, (f : s.X \u2192 F.obj j) x) (w j),\n    end, }, }\n\nend has_limit\n\nsection\n\nopen has_limit\n\nvariables [decidable_eq J] [fintype J]\n\ninstance (f : J \u2192 AddCommGroup.{u}) : has_biproduct f :=\nhas_biproduct.of_has_product _\n\n/--\nWe verify that the biproduct we've just defined is isomorphic to the AddCommGroup structure\non the dependent function type\n-/\nnoncomputable\ndef biproduct_iso_pi (f : J \u2192 AddCommGroup.{u}) :\n  (\u2a01 f : AddCommGroup) \u2245 AddCommGroup.of (\u03a0 j, f j) :=\nis_limit.cone_point_unique_up_to_iso\n  (biproduct.is_limit f)\n  (product_limit_cone (discrete.functor f)).is_limit\n\nend\n\ninstance : has_finite_biproducts AddCommGroup :=\n\u27e8\u03bb J _ _, by exactI { has_biproduct := \u03bb f, by apply_instance }\u27e9\n\nend AddCommGroup\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/algebra/category/Group/biproducts.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.35404449295009416}}
{"text": "def x := 1\n\n#check x\n\nvariable {\u03b1 : Type}\n\ndef f (a : \u03b1) : \u03b1 :=\na\n\ndef tst (xs : List Nat) : Nat :=\nxs.foldl (init := 10) (\u00b7 + \u00b7)\n\n#check tst [1, 2, 3]\n\n#check fun x y : Nat => x + y\n\n#check tst\n\n#check (fun stx => if True then let e := stx; Pure.pure e else Pure.pure stx : Nat \u2192 Id Nat)\n\n#check let x : Nat := 1; x\n\ndef foo (a : Nat) (b : Nat := 10) (c : Bool := Bool.true) : Nat :=\na + b\n\nset_option pp.all true\n\n#check foo 1\n\n#check foo 3 (c := false)\n\ndef Nat.boo (a : Nat) :=\nsucc a -- succ here is resolved as `Nat.succ`.\n\n#check Nat.boo\n\n#check true\n\n-- apply is still a valid identifier name\ndef apply := \"hello\"\n\n#check apply\n\ntheorem simple1 (x y : Nat) (h : x = y) : x = y :=\nby {\n  assumption\n}\n\ntheorem simple2 (x y : Nat) : x = y \u2192 x = y :=\nby {\n  intro h;\n  assumption\n}\n\nsyntax \"intro2\" : tactic\n\nmacro_rules\n| `(tactic| intro2) => `(tactic| intro; intro )\n\ntheorem simple3 (x y : Nat) : x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro2;\n  assumption\n}\n\nmacro \"intro3\" : tactic => `(tactic| (intro; intro; intro))\nmacro \"check2\" x:term : command => `(#check $x #check $x)\nmacro \"foo\" x:term \",\" y:term : term => `($x + $y + $x)\n\nset_option pp.all false\n\ncheck2 0+1\ncheck2 foo 0,1\n\ntheorem simple4 (x y : Nat) : y = y \u2192 x = x \u2192 x = y \u2192 x = y :=\nby {\n  intro3;\n  assumption\n}\n\ntheorem simple5 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  exact Eq.trans h3 h1\n}\n\ntheorem simple6 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine Eq.trans ?_ h1;\n  assumption\n}\n\ntheorem simple7 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro _; intro h3;\n  refine' Eq.trans ?pre ?post;\n  exact y;\n  { exact h3 }\n  { exact h1 }\n}\n\ntheorem simple8 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintro h1; intro _; intro h3\nrefine' Eq.trans ?pre ?post\ncase post => exact h1\ncase pre => exact h3\n\ntheorem simple9 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntrace_state\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n      assumption\n    | exact y\n      exact h3\n      assumption\n\ntheorem simple9b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 _ h3\ntrace_state\nfocus\n  refine' Eq.trans ?pre ?post\n  first\n    | exact h1\n    | exact y; exact h3\n  assumption\n\ntheorem simple9c (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  solve\n    | exact h1\n    | refine' Eq.trans ?pre ?post; exact y; exact h3; assumption\n    | exact h3\n\ntheorem simple9d (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\n  intros h1 _ h3\n  refine' Eq.trans ?pre ?post\n  solve\n    | exact h1\n    | exact y\n    | exact h3\n  solve\n    | exact h1\n    | exact h3\n  solve\n    | exact h1\n    | assumption\n\n\nnamespace Foo\n  def Prod.mk := 1\n  #check (\u27e82, 3\u27e9 : Prod _ _)\nend Foo\n\ntheorem simple10 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  skip;\n  apply Eq.trans;\n  exact h3;\n  assumption\n}\n\ntheorem simple11 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  trace_state;\n  exact h3;\n  assumption\n}\n\ntheorem simple12 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intro h1; intro h2; intro h3;\n  apply @Eq.trans;\n  try exact h1; -- `exact h1` fails\n  trace_state;\n  try exact h3;\n  trace_state;\n  try exact h1;\n}\n\ntheorem simple13 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros h1 h2 h3\ntrace_state\napply @Eq.trans\ncase b => exact y\ntrace_state\nrepeat assumption\n\ntheorem simple13b (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by {\nintros h1 h2 h3;\ntrace_state;\napply @Eq.trans;\ncase b => exact y;\ntrace_state;\nrepeat assumption\n}\n\ntheorem simple14 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z := by\nintros\napply @Eq.trans\ncase b => exact y\nrepeat assumption\n\ntheorem simple15 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  revert y;\n  intros y h1 h3;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\ntheorem simple16 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby {\n  intros h1 h2 h3;\n  try clear x; -- should fail\n  clear h2;\n  trace_state;\n  apply Eq.trans;\n  exact h3;\n  exact h1\n}\n\nmacro \"blabla\" : tactic => `(tactic| assumption)\n\n-- Tactic head symbols do not become reserved words\ndef blabla := 100\n\n#check blabla\n\ntheorem simple17 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple18 (x : Nat) (h : x = 0) : x = 0 :=\nby blabla\n\ntheorem simple19 (x y : Nat) (h\u2081 : x = 0) (h\u2082 : x = y) : y = 0 :=\nby subst x; subst y; exact rfl\n\ntheorem tstprec1 (x y z : Nat) : x + y * z = x + (y * z) :=\nrfl\n\ntheorem tstprec2 (x y z : Nat) : y * z + x = (y * z) + x :=\nrfl\n\nset_option pp.all true\n\n#check fun {\u03b1} (a : \u03b1) => a\n#check @(fun \u03b1 (a : \u03b1) => a)\n\n#check\n  let myid := fun {\u03b1} (a : \u03b1) => a;\n  myid [myid 1]\n\n-- In the following example, we need `@` otherwise we will try to insert mvars for \u03b1 and [Add \u03b1],\n-- and will fail to generate instance for [Add \u03b1]\n#check @(fun \u03b1 (s : Add \u03b1) (a : \u03b1) => a + a)\n\ndef g1 {\u03b1} (a\u2081 a\u2082 : \u03b1) {\u03b2} (b : \u03b2) : \u03b1 \u00d7 \u03b1 \u00d7 \u03b2 :=\n(a\u2081, a\u2082, b)\n\ndef id1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => x\n\ndef listId : List ({\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1) :=\n(fun x => x) :: []\n\ndef id2 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 (x : \u03b1) => id1 x)\n\ndef id3 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun \u03b1 x => id1 x)\n\ndef id4 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun x => id1 x\n\ndef id5 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id1 x\n\ndef id6 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fun {\u03b1} x => id1 x)\n\ndef id7 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => @id \u03b1 x\n\ndef id8 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\nfun {\u03b1} x => id (@id \u03b1 x)\n\ndef altTst1 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8StateT.failure, StateT.orElse\u27e9\n\ndef altTst2 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8@(fun \u03b1 => StateT.failure), @(fun \u03b1 => StateT.orElse)\u27e9\n\ndef altTst3 {m \u03c3} [Alternative m] [Monad m] : Alternative (StateT \u03c3 m) :=\n\u27e8fun {\u03b1} => StateT.failure, fun {\u03b1} => StateT.orElse\u27e9\n\n#check_failure 1 + true\n\n/-\nuniverse u v\n\n/-\n  MonadFunctorT.{u ?M_1 v} (\u03bb (\u03b2 : Type u), m \u03b1) (\u03bb (\u03b2 : Type u), m' \u03b1) n n'\n-/\nset_option pp.raw.maxDepth 100\nset_option trace.Elab true\n\n\ndef adapt {m m' \u03c3 \u03c3'} {n n' : Type \u2192 Type} [MonadFunctor m m' n n'] [MonadStateAdapter \u03c3 \u03c3' m m'] : MonadStateAdapter \u03c3 \u03c3' n n' :=\n\u27e8fun split join => monadMap (adaptState split join : m \u03b1 \u2192 m' \u03b1)\u27e9\n\n-/\n\nsyntax \"fn\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(fn $xs* => $b) => `(fun $xs* => $b)\n\nset_option pp.all false\n\n#check fn x => x+1\n\n#check fn \u03b1 (a : \u03b1) => a\n\ndef tst1 : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(fn \u03b1 a => a)\n\n#check @tst1\n\nsyntax ident \"==>\" term : term\n\nsyntax \"{\" ident \"}\" \"==>\" term : term\n\nmacro_rules\n| `($x:ident ==> $b)   => `(fn $x => $b)\n| `({$x:ident} ==> $b) => `(fun {$x:ident} => $b)\n\n#check x ==> x+1\n\ndef tst2a : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n@(\u03b1 ==> a ==> a)\n\ndef tst2b : {\u03b1 : Type} \u2192 \u03b1 \u2192 \u03b1 :=\n{\u03b1} ==> a ==> a\n\n#check @tst2a\n#check @tst2b\n\ndef tst3a : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n@(\u03b1 ==> @(\u03b2 ==> a ==> b ==> (a, b)))\n\ndef tst3b : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\n{\u03b1} ==> {\u03b2} ==> a ==> b ==> (a, b)\n\nsyntax \"function\" (term:max)+ \"=>\" term : term\n\nmacro_rules\n| `(function $xs* => $b) => `(@(fun $xs* => $b))\n\ndef tst4 : {\u03b1 : Type} \u2192 {\u03b2 : Type} \u2192 \u03b1 \u2192 \u03b2 \u2192 \u03b1 \u00d7 \u03b2 :=\nfunction \u03b1 \u03b2 a b => (a, b)\n\ntheorem simple20 (x y z : Nat) : y = z \u2192 x = x \u2192 x = y \u2192 x = z :=\nby intros h1 h2 h3;\n   try clear x; -- should fail\n   clear h2;\n   trace_state;\n   apply Eq.trans;\n   exact h3;\n   exact h1\n\ntheorem simple21 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by { apply Eq.symm; assumption };\n  Eq.trans this (by assumption)\n\ntheorem simple22 (x y z : Nat) : y = z \u2192 y = x \u2192 id (x = z + 0) :=\nfun h1 h2 => show x = z + 0 by\n  apply Eq.trans\n  exact h2.symm\n  assumption\n  skip\n\ntheorem simple23 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have : x = y := by apply Eq.symm; assumption\n  Eq.trans this (by assumption)\n\ntheorem simple24 (x y z : Nat) : y = z \u2192 x = x \u2192 y = x \u2192 x = z :=\nfun h1 _ h3 =>\n  have h : x = y := by apply Eq.symm; assumption\n  Eq.trans h (by assumption)\n\ndef f1 (x : Nat) : Nat :=\n  let double x := x + x\n  let rec loop x :=\n    match x with\n    | 0   => 0\n    | x+1 => loop x + double x\n  loop x\n\n#eval f1 5\n\ndef f2 (x : Nat) : String :=\n  let bad x : String := toString x\n  bad x\n\ndef f3 x y :=\n  x + y + 1\n\ntheorem f3eq x y : f3 x y = x + y + 1 :=\n  rfl\n\ndef f4 (x y : Nat) : String :=\n  if x > y + 1 then \"hello\" else \"world\"\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/newfrontend1.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621764862150634, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.35404447199170735}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz, Johan Commelin, Jo\u00ebl Riou\n-/\nimport category_theory.preadditive.additive_functor\nimport logic.equiv.transfer_instance\n\n/-!\n# If `C` is preadditive, `C\u1d52\u1d56` has a natural preadditive structure.\n\n-/\n\nopen opposite\n\nnamespace category_theory\n\nvariables (C : Type*) [category C] [preadditive C]\n\ninstance : preadditive C\u1d52\u1d56 :=\n{ hom_group := \u03bb X Y, equiv.add_comm_group (op_equiv X Y),\n  add_comp' := \u03bb X Y Z f f' g,\n    congr_arg quiver.hom.op (preadditive.comp_add _ _ _ g.unop f.unop f'.unop),\n  comp_add' := \u03bb X Y Z f g g',\n    congr_arg quiver.hom.op (preadditive.add_comp _ _ _ g.unop g'.unop f.unop), }\n\ninstance module_End_left {X : C\u1d52\u1d56} {Y : C} : module (End X) (unop X \u27f6 Y) :=\n{ smul_add := \u03bb r f g, preadditive.comp_add _ _ _ _ _ _,\n  smul_zero := \u03bb r, limits.comp_zero,\n  add_smul := \u03bb r s f, preadditive.add_comp _ _ _ _ _ _,\n  zero_smul := \u03bb f, limits.zero_comp }\n\n@[simp] lemma unop_zero (X Y : C\u1d52\u1d56) : (0 : X \u27f6 Y).unop = 0 := rfl\n@[simp] lemma unop_add {X Y : C\u1d52\u1d56} (f g : X \u27f6 Y) : (f + g).unop = f.unop + g.unop := rfl\n@[simp] lemma unop_zsmul {X Y : C\u1d52\u1d56} (k : \u2124) (f : X \u27f6 Y) : (k \u2022 f).unop = k \u2022 f.unop := rfl\n@[simp] lemma unop_neg {X Y : C\u1d52\u1d56}(f : X \u27f6 Y) : (-f).unop = -(f.unop) := rfl\n@[simp] lemma op_zero (X Y : C) : (0 : X \u27f6 Y).op = 0 := rfl\n@[simp] \n\nvariable {C}\n\n/-- `unop` induces morphisms of monoids on hom groups of a preadditive category -/\n@[simps] def unop_hom (X Y : C\u1d52\u1d56) : (X \u27f6 Y) \u2192+ (opposite.unop Y \u27f6 opposite.unop X) :=\nadd_monoid_hom.mk' (\u03bb f, f.unop) $ \u03bb f g, unop_add _ f g\n\n@[simp] lemma unop_sum (X Y : C\u1d52\u1d56) {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 (X \u27f6 Y)) :\n  (s.sum f).unop = s.sum (\u03bb i, (f i).unop) :=\n(unop_hom X Y).map_sum _ _\n\n/-- `op` induces morphisms of monoids on hom groups of a preadditive category -/\n@[simps] def op_hom (X Y : C) : (X \u27f6 Y) \u2192+ (opposite.op Y \u27f6 opposite.op X) :=\nadd_monoid_hom.mk' (\u03bb f, f.op) $ \u03bb f g, op_add _ f g\n\n@[simp] lemma op_sum (X Y : C) {\u03b9 : Type*} (s : finset \u03b9) (f : \u03b9 \u2192 (X \u27f6 Y)) :\n  (s.sum f).op = s.sum (\u03bb i, (f i).op) :=\n(op_hom X Y).map_sum _ _\n\nvariables {D : Type*} [category D] [preadditive D]\n\ninstance functor.op_additive (F : C \u2964 D) [F.additive] : F.op.additive := {}\n\ninstance functor.right_op_additive (F : C\u1d52\u1d56 \u2964 D) [F.additive] : F.right_op.additive := {}\n\ninstance functor.left_op_additive (F : C \u2964 D\u1d52\u1d56) [F.additive] : F.left_op.additive := {}\n\ninstance functor.unop_additive (F : C\u1d52\u1d56 \u2964 D\u1d52\u1d56) [F.additive] : F.unop.additive := {}\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/preadditive/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.35370895009996195}}
{"text": "/-\nCopyright (c) 2022 Devon Tuma. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Devon Tuma\n-/\nimport computational_monads.coercions.instances\n\n/-!\n# Appending Simulation Oracles\n\nThis file defines an append operation `++\u209b` for simulation oracles,\ncreating a simulation oracle for a combined set of initial oracles.\nIn particular, if simulation oracles `so` and `so'` have starting oracles given by\n`spec` and `spec'`, then `so ++\u209b so'` will have starting oracles `spec ++ spec'`.\n\nThe implementation just maintains a seperate state for each oracle,\nusing pattern matching on queries to decide which `sim_oracle` to call.\n-/\n\nopen oracle_comp oracle_spec\n\nvariables {spec spec' spec'' spec''' : oracle_spec} {\u03b1 \u03b2 \u03b3 : Type} {S S' : Type}\n\nnamespace sim_oracle\n\ndef oracle_append (so : sim_oracle spec spec'' S) (so' : sim_oracle spec' spec'' S') :\n  sim_oracle (spec ++ spec') spec'' (S \u00d7 S') :=\n{ default_state := (so.default_state, so'.default_state),\n  o := \u03bb i, match i with\n  | (sum.inl i) := \u03bb \u27e8t, s\u2081, s\u2082\u27e9, do {\u27e8u, s\u2081'\u27e9 \u2190 so i (t, s\u2081), return (u, s\u2081', s\u2082)}\n  | (sum.inr i) := \u03bb \u27e8t, s\u2081, s\u2082\u27e9, do {\u27e8u, s\u2082'\u27e9 \u2190 so' i (t, s\u2082), return (u, s\u2081, s\u2082')}\n  end }\n\ninfixl ` ++\u209b `:65 := oracle_append\n\nnamespace oracle_append\n\nvariables (so : sim_oracle spec spec'' S) (so' : sim_oracle spec' spec'' S')\n  (oa : oracle_comp (spec ++ spec') \u03b1) (ob : \u03b1 \u2192 oracle_comp (spec ++ spec') \u03b2) (a : \u03b1)\n  (i : spec.\u03b9) (i' : spec'.\u03b9) (t : spec.domain i) (t' : spec'.domain i') (s : S \u00d7 S')\n  (x : spec.domain i \u00d7 S \u00d7 S') (x' : spec'.domain i' \u00d7 S \u00d7 S')\n\n@[simp]\nlemma apply_inl_eq : (so ++\u209b so') (sum.inl i) x =\n  do {u_s' \u2190 so i (x.1, x.2.1), return (u_s'.1, u_s'.2, x.2.2)} :=\nbegin\n  cases x with t s, cases s with s\u2081 s\u2082,\n  refine congr_arg (\u03bb ou, so i (t, s\u2081) >>= ou) (funext $ \u03bb y, _),\n  cases y, refl,\nend\n\n@[simp]\nlemma apply_inr_eq : (so ++\u209b so') (sum.inr i') x' =\n  do {u_s' \u2190 so' i' (x'.1, x'.2.2), return (u_s'.1, x'.2.1, u_s'.2)} :=\nbegin\n  cases x' with t s, cases s with s\u2081 s\u2082,\n  refine congr_arg (\u03bb ou, so' i' (t, s\u2082) >>= ou) (funext $ \u03bb y, _),\n  cases y, refl,\nend\n\nsection support\n\nlemma support_apply_inl : ((so ++\u209b so') (sum.inl i) (t, s)).support =\n  {x | (x.1, x.2.1) \u2208 (so i (t, s.1)).support \u2227 x.2.2 = s.2} :=\nbegin\n  ext x,\n  simp only [apply_inl_eq, support_bind, support_return, set.mem_Union,\n    set.mem_singleton_iff, exists_prop, prod.exists, set.mem_set_of_eq],\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { obtain \u27e8u, s', hu, hx\u27e9 := h,\n    simpa only [hx, eq_self_iff_true, and_true] using hu },\n  { refine \u27e8x.1, x.2.1, _\u27e9,\n    simp only [\u2190 h.2, h.1, true_and, prod.mk.eta] }\nend\n\nlemma support_apply_inr : ((so ++\u209b so') (sum.inr i') (t', s)).support =\n  {x | (x.1, x.2.2) \u2208 (so' i' (t', s.2)).support \u2227 x.2.1 = s.1} :=\nbegin\n  ext x,\n  simp only [apply_inr_eq, support_bind, support_return, set.mem_Union,\n    set.mem_singleton_iff, exists_prop, prod.exists, set.mem_set_of_eq],\n  refine \u27e8\u03bb h, _, \u03bb h, _\u27e9,\n  { obtain \u27e8u, s', hu, hx\u27e9 := h,\n    simpa only [hx, eq_self_iff_true, and_true] using hu },\n  { refine \u27e8x.1, x.2.2, _\u27e9,\n    simp only [\u2190 h.2, h.1, true_and, prod.mk.eta] }\nend\n\nend support\n\nsection coe_append_right\n\n/-- Coercing a computation on `spec` to one on `spec ++ spec'`, and then simulating with\ntwo independent oracles `so ++\u209b so'` has the same support as simulating the original with `so`,\nmodulo the extra oracle state for the right oracle, which remains unchanged during simulation. -/\n@[simp] lemma support_simulate_coe_append_right (s : S \u00d7 S') (oa : oracle_comp spec \u03b1) :\n  (simulate (so ++\u209b so') \u2191oa s).support =\n    (\u03bb (x : \u03b1 \u00d7 S), (x.1, x.2, s.2)) '' (simulate so oa s.1).support :=\ncalc (simulate (so ++\u209b so') \u2191oa s).support =\n    (simulate (so ++\u209b so') \u2191oa ((\u03bb s\u2081, (s\u2081, s.2)) s.1)).support : by simp only [prod.mk.eta]\n    ... = prod.map id (\u03bb s\u2081, (s\u2081, s.2)) '' (simulate so oa s.1).support : begin\n      refine (support_simulate_coe_sub_spec so (so ++\u209b so') s.1 oa _ (\u03bb i t s, _)),\n      simp_rw [is_sub_spec_append_right_apply, simulate_query, apply_inl_eq,\n        support_bind_return, prod_map, id.def],\n    end\n    ... = (\u03bb (x : \u03b1 \u00d7 S), (x.1, x.2, s.2)) '' (simulate so oa s.1).support : rfl\n\n/-- Coercing a computation on `spec` to one on `spec ++ spec'`, and then simulating with\ntwo independent oracles `so ++\u209b so'` has the same support as simulating the original with `so`,\nif we use `simulate'` to ignore the final oracle state of the two `sim_oracle`s. -/\n@[simp] lemma support_simulate'_coe_append_right (so : sim_oracle spec spec'' S)\n  (so' : sim_oracle spec' spec'' S') (s : S \u00d7 S') (oa : oracle_comp spec \u03b1) :\n  (simulate' (so ++\u209b so') \u2191oa s).support = (simulate' so oa s.1).support :=\nset.ext (\u03bb x, by simp only [support_simulate', support_simulate_coe_append_right, set.image_image])\n\nend coe_append_right\n\nend oracle_append\n\nend sim_oracle", "meta": {"author": "dtumad", "repo": "lean-crypto-formalization", "sha": "f975a9a9882120b509553a7ced9aa05b745ff154", "save_path": "github-repos/lean/dtumad-lean-crypto-formalization", "path": "github-repos/lean/dtumad-lean-crypto-formalization/lean-crypto-formalization-f975a9a9882120b509553a7ced9aa05b745ff154/src/computational_monads/simulation_semantics/oracle_append.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6757645944891558, "lm_q1q2_score": 0.35370893984632745}}
{"text": "/-\nThis is the final correctenss theorem, stated in terms of the autogenerated constraints.\n\nThe statements of theorems only depend on the data and constraints specified in\n`constraints_autogenerated.lean` and the machine semantics in `cpu.lean`.\n-/\nimport starkware.cairo.lean.semantics.air_encoding.correctness\nimport starkware.cairo.lean.semantics.air_encoding.glue\n\nnoncomputable theory\nopen_locale classical\nopen_locale big_operators\n\nvariables {F : Type} [field F] [fintype F]\n\n/-\nThese are the constraints that the verifier has to check against the public data.\n-/\n\nstructure public_constraints (inp : input_data F) (pd : public_data F) : Prop :=\n(h_mem_star :\n  let z     := pd.memory__multi_column_perm__perm__interaction_elm,\n      alpha := pd.memory__multi_column_perm__hash_interaction_elm0,\n      p     := pd.memory__multi_column_perm__perm__public_memory_prod,\n      dom_m_star := { x // option.is_some (inp.m_star x) } in\n    p * \u220f a : dom_m_star, (z - (a.val + alpha * mem_val a)) = z^(fintype.card dom_m_star))\n(h_card_dom : 8 * fintype.card { x // option.is_some (inp.m_star x) } + 2 \u2264 inp.trace_length)\n(public_memory_prod_eq_one : pd.rc16__perm__public_memory_prod = 1)\n(rc_max_lt : pd.rc_max < 2^16)\n(rc_min_le : pd.rc_min \u2264 pd.rc_max)\n(trace_length_le_char : inp.trace_length \u2264 ring_char F)\n\n/-\nThe main correctness theorem.\n-/\n\ntheorem final_correctness\n    (char_ge : ring_char F \u2265 2^63)\n    /- public data -/\n    (inp     : input_data F)\n    (pd      : public_data F)\n    (pc      : public_constraints inp pd)\n    (c       : columns F) :\n    /- sets to avoid -/\n  \u2203 bad1\n    bad2\n    bad3     : finset F,\n    bad1.card \u2264 (inp.trace_length / 2)^2 \u2227\n    bad2.card \u2264 inp.trace_length / 2 \u2227\n    bad3.card \u2264 inp.trace_length \u2227\n  \u2200 ci       : columns_inter F,\n      /- autogenerated constraints-/\n      cpu__decode c \u2227\n      cpu__operands c \u2227\n      cpu__update_registers inp c \u2227\n      cpu__opcodes c \u2227\n      memory inp pd c ci \u2227\n      rc16 inp pd c ci \u2227\n      public_memory c \u2227\n      rc_builtin inp pd c \u2227\n      toplevel_constraints inp c \u2227\n      /- probabilistic constraints -/\n      pd.memory__multi_column_perm__hash_interaction_elm0 \u2209 bad1 \u2227\n      pd.memory__multi_column_perm__perm__interaction_elm \u2209 bad2 \u2227\n      pd.memory__multi_column_perm__perm__interaction_elm \u2260 0 \u2227\n      pd.rc16__perm__interaction_elm \u2209 bad3 \u2192\n      let -- number of execution steps\n          T := inp.trace_length / 16 - 1,\n          -- memory elements checked by range check builtin\n          rc_len := inp.trace_length / 128 in\n      /- the conclusion -/\n      \u2203 mem : F \u2192 F,\n        option.fn_extends mem inp.m_star \u2227\n        (\u2200 i < rc_len, \u2203 n < 2^128, mem (pd.initial_rc_addr + i) = \u2191n) \u2227\n        \u2203 exec : fin (T + 1) \u2192 register_state F,\n          (exec 0).pc = inp.initial_pc \u2227\n          (exec 0).ap = inp.initial_ap \u2227\n          (exec 0).fp = inp.initial_ap \u2227\n          (exec (fin.last T)).pc = inp.final_pc \u2227\n          (exec (fin.last T)).ap = inp.final_ap \u2227\n          \u2200 i : fin T, next_state mem (exec i.cast_succ) (exec i.succ) :=\nbegin\n  use bad1 pc.h_card_dom c.column19 c.column20,\n  use bad2 pd pc.h_card_dom c.column19 c.column20,\n  use bad3 inp c.column0 c.column2,\n  use bad1_bound pc.h_card_dom _ _,\n  use bad2_bound pd pc.h_card_dom _ _,\n  use bad3_bound pc.h_card_dom _ _,\n  intro ci,\n  rintros \u27e8cd, ops, upd, opcodes, m, rc, pm, rcb, iandf, prob1, prob2, prob3, prob4\u27e9,\n  dsimp,\n  exact execution_exists char_ge\n    (inp.to_input_data_aux pd pc.rc_max_lt pc.rc_min_le)\n    (to_constraints cd ops upd opcodes m rc pm rcb iandf pc.h_mem_star pc.h_card_dom\n      pc.public_memory_prod_eq_one pc.rc_max_lt pc.rc_min_le pc.trace_length_le_char)\n    { hprob\u2081 := prob1,\n      hprob\u2082 := prob2,\n      hprob\u2083 := prob3,\n      hprob\u2084 := prob4 }\nend\n", "meta": {"author": "starkware-libs", "repo": "formal-proofs", "sha": "35613c65b6715601bbc0a550d52754f8e7d93e30", "save_path": "github-repos/lean/starkware-libs-formal-proofs", "path": "github-repos/lean/starkware-libs-formal-proofs/formal-proofs-35613c65b6715601bbc0a550d52754f8e7d93e30/src/starkware/cairo/lean/semantics/air_encoding/final_correctness.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757645879592641, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.35370893642844925}}
{"text": "example (A B C D E F G H I J K L : Prop)\n(f1 : A \u2192 B) (f2 : B \u2192 E) (f3 : E \u2192 D) (f4 : D \u2192 A) (f5 : E \u2192 F)\n(f6 : F \u2192 C) (f7 : B \u2192 C) (f8 : F \u2192 G) (f9 : G \u2192 J) (f10 : I \u2192 J)\n(f11 : J \u2192 I) (f12 : I \u2192 H) (f13 : E \u2192 H) (f14 : H \u2192 K) (f15 : I \u2192 L)\n : A \u2192 L :=\nbegin\ncc,\nend", "meta": {"author": "nicholaspun", "repo": "natural-number-game-solutions", "sha": "1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0", "save_path": "github-repos/lean/nicholaspun-natural-number-game-solutions", "path": "github-repos/lean/nicholaspun-natural-number-game-solutions/natural-number-game-solutions-1e2aed86d2e76a3f4a275c6d99e795ad30cf6df0/4-proposition-world/l9.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6859494550081925, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.35368920018157274}}
{"text": "import mathlib.prod\nimport group_theory.group_action.sigma\nimport phase1.code_equiv\n\n/-!\n# Allowable permutations\n-/\n\n-- Note to whoever fixes this file: We may want to use `type_index` instead of `\u039b` in some places\n-- now that supports are defined in these cases.\n\nopen function set with_bot\nopen_locale pointwise\n\nnoncomputable theory\n\nuniverse u\n\nnamespace con_nf\nvariables [params.{u}] (\u03b1 : \u039b) [core_tangle_cumul \u03b1] (\u03b2 : Iio_index \u03b1) (\u03b3 : Iio \u03b1)\n\nopen code\n\n/-- A semi-allowable permutation is a `-1`-allowable permutation of atoms (a near-litter\npermutation) together with allowable permutations on all `\u03b3 < \u03b2`. This forms a group structure\nautomatically. -/\n@[derive group] def semiallowable_perm : Type u := \u03a0 \u03b2 : Iio_index \u03b1, allowable \u03b2\n\nnamespace semiallowable_perm\nvariables {\u03b1} (\u03c0 : semiallowable_perm \u03b1) (c : code \u03b1)\n\n/-- The allowable permutation at a lower level corresponding to a semi-allowable permutation. -/\nnoncomputable! def to_allowable : semiallowable_perm \u03b1 \u2192* allowable \u03b2 :=\n\u27e8\u03bb f, f \u03b2, rfl, \u03bb _ _, rfl\u27e9\n\n/-- Reinterpret a semi-allowable permutation as a structural permutation. -/\nnoncomputable! def to_struct_perm : semiallowable_perm \u03b1 \u2192* struct_perm \u03b1 :=\n{ to_fun := \u03bb f, struct_perm.to_coe $ \u03bb \u03b2 h\u03b2, (f \u27e8\u03b2, h\u03b2\u27e9).to_struct_perm,\n  map_one' := struct_perm.of_coe.injective $ funext $ \u03bb \u03b2, funext $ \u03bb h\u03b2, match \u03b2, h\u03b2 with\n    | \u22a5, _ := by { simp only [struct_perm.of_coe_to_coe, struct_perm.of_coe_one, pi.one_apply],\n      exact struct_perm.to_bot_one }\n    | (\u03b2 : \u039b), (h\u03b2 : \u2191\u03b2 < \u2191\u03b1) := by { simp only [struct_perm.of_coe_to_coe, struct_perm.of_coe_one,\n      pi.one_apply], exact allowable.to_struct_perm.map_one }\n  end,\n  map_mul' := \u03bb f g, struct_perm.of_coe.injective $ funext $ \u03bb \u03b2, funext $ \u03bb h\u03b2, match \u03b2, h\u03b2 with\n    | \u22a5, _ := by { simp only [struct_perm.of_coe_to_coe, struct_perm.of_coe_mul, pi.mul_apply],\n      exact struct_perm.to_bot_mul _ _ }\n    | (\u03b2 : \u039b), (h\u03b2 : \u2191\u03b2 < \u2191\u03b1) := by { simp only [struct_perm.of_coe_to_coe, struct_perm.of_coe_mul,\n      pi.mul_apply], exact allowable.to_struct_perm.map_mul _ _ }\n  end }\n\nsection\nvariables {X : Type*} [mul_action (struct_perm \u03b1) X]\n\ninstance mul_action_of_struct_perm : mul_action (semiallowable_perm \u03b1) X :=\nmul_action.comp_hom _ to_struct_perm\n\n@[simp] lemma to_struct_perm_smul (f : semiallowable_perm \u03b1) (x : X) :\n  f.to_struct_perm \u2022 x = f \u2022 x := rfl\n\nend\n\ninstance mul_action_tangle : mul_action (semiallowable_perm \u03b1) (tangle \u03b2) :=\nmul_action.comp_hom _ $ to_allowable \u03b2\n\ninstance mul_action_tangle' {\u03b2 : Iio \u03b1} : mul_action (semiallowable_perm \u03b1) (tangle \u03b2) :=\nshow mul_action (semiallowable_perm \u03b1) (tangle $ Iio_coe \u03b2), from infer_instance\n\ninstance mul_action_tangle'' : mul_action (semiallowable_perm \u03b1) (tangle (\u03b3 : \u039b)) :=\nshow mul_action (semiallowable_perm \u03b1) (tangle $ Iio_coe \u03b3), from infer_instance\n\n@[simp] lemma to_allowable_smul (f : semiallowable_perm \u03b1) (t : tangle \u03b2) :\n  to_allowable \u03b2 f \u2022 t = f \u2022 t := rfl\n\nattribute [derive mul_action (semiallowable_perm \u03b1)] code\n\n@[simp] lemma fst_smul : (\u03c0 \u2022 c).1 = c.1 := rfl\n@[simp] lemma snd_smul : (\u03c0 \u2022 c).2 = \u03c0 \u2022 c.2 := rfl\n@[simp] lemma smul_mk (f : semiallowable_perm \u03b1) (\u03b3 s) : f \u2022 (mk \u03b3 s : code \u03b1) = mk \u03b3 (f \u2022 s) := rfl\n\ninstance has_smul_nonempty_code : has_smul (semiallowable_perm \u03b1) (nonempty_code \u03b1) :=\n\u27e8\u03bb \u03c0 c, \u27e8\u03c0 \u2022 c, c.2.image _\u27e9\u27e9\n\n@[simp, norm_cast] lemma coe_smul (c : nonempty_code \u03b1) : (\u2191(\u03c0 \u2022 c) : code \u03b1) = \u03c0 \u2022 c := rfl\n\ninstance mul_action_nonempty_code : mul_action (semiallowable_perm \u03b1) (nonempty_code \u03b1) :=\nsubtype.coe_injective.mul_action _ coe_smul\n\nend semiallowable_perm\n\nvariables [position_data.{}] [positioned_tangle_cumul \u03b1] [almost_tangle_cumul \u03b1]\n  [core_tangle_data \u03b1]\n\n/-- An allowable permutation is a semi-allowable permutation whose action on codes preserves\nequivalence. -/\ndef allowable_perm := {\u03c0 : semiallowable_perm \u03b1 // \u2200 X Y : code \u03b1, \u03c0 \u2022 X \u2261 \u03c0 \u2022 Y \u2194 X \u2261 Y}\n\nvariables {\u03b1} {f : allowable_perm \u03b1} {c d : code \u03b1}\n\nnamespace allowable_perm\n\ninstance : has_coe_t (allowable_perm \u03b1) (semiallowable_perm \u03b1) := @coe_base _ _ coe_subtype\n\nlemma coe_injective : injective (coe : allowable_perm \u03b1 \u2192 semiallowable_perm \u03b1) :=\nsubtype.coe_injective\n\ninstance : has_one (allowable_perm \u03b1) := \u27e8\u27e81, \u03bb _ _, by simp_rw one_smul\u27e9\u27e9\ninstance : has_inv (allowable_perm \u03b1) :=\n\u27e8\u03bb f, \u27e8f\u207b\u00b9, \u03bb c d, by rw [\u2190f.prop, smul_inv_smul, smul_inv_smul]\u27e9\u27e9\ninstance : has_mul (allowable_perm \u03b1) :=\n\u27e8\u03bb f g, \u27e8f * g, \u03bb c d, by simp_rw [mul_smul, f.prop, g.prop]\u27e9\u27e9\ninstance : has_div (allowable_perm \u03b1) :=\n\u27e8\u03bb f g, \u27e8f / g, by { simp_rw [div_eq_mul_inv], exact (f * g\u207b\u00b9).2 }\u27e9\u27e9\ninstance : has_pow (allowable_perm \u03b1) \u2115 :=\n\u27e8\u03bb f n, \u27e8f ^ n, begin\n  induction n with d hd,\n  { simp_rw pow_zero,\n    exact (1 : allowable_perm \u03b1).2 },\n  { simp_rw pow_succ,\n    exact (f * \u27e8f ^ d, hd\u27e9).2 }\nend\u27e9\u27e9\n\ninstance : has_pow (allowable_perm \u03b1) \u2124 :=\n\u27e8\u03bb f n, \u27e8f ^ n, begin\n  cases n,\n  { simp_rw zpow_of_nat,\n    exact (f ^ n).2 },\n  { simp_rw zpow_neg_succ_of_nat,\n    exact (f ^ (n + 1))\u207b\u00b9.2 }\nend\u27e9\u27e9\n\n@[simp] lemma coe_one : ((1 : allowable_perm \u03b1) : semiallowable_perm \u03b1) = 1 := rfl\n@[simp] lemma coe_inv (f : allowable_perm \u03b1) : (\u2191(f\u207b\u00b9) : semiallowable_perm \u03b1) = f\u207b\u00b9 := rfl\n@[simp] lemma coe_mul (f g : allowable_perm \u03b1) : (\u2191(f * g) : semiallowable_perm \u03b1) = f * g := rfl\n@[simp] lemma coe_div (f g : allowable_perm \u03b1) : (\u2191(f / g) : semiallowable_perm \u03b1) = f / g := rfl\n@[simp] lemma coe_pow (f : allowable_perm \u03b1) (n : \u2115) :\n  (\u2191(f ^ n) : semiallowable_perm \u03b1) = f ^ n := rfl\n@[simp] lemma coe_zpow (f : allowable_perm \u03b1) (n : \u2124) :\n  (\u2191(f ^ n) : semiallowable_perm \u03b1) = f ^ n := rfl\n\ninstance : group (allowable_perm \u03b1) :=\ncoe_injective.group _ coe_one coe_mul coe_inv coe_div coe_pow coe_zpow\n\n/-- The coercion from allowable to semi-allowable permutation as a monoid homomorphism. -/\n@[simps] noncomputable! def coe_hom : allowable_perm \u03b1 \u2192* semiallowable_perm \u03b1 :=\n\u27e8coe, coe_one, coe_mul\u27e9\n\n/-- Turn an allowable permutation into a structural permutation. -/\ndef to_struct_perm : allowable_perm \u03b1 \u2192* struct_perm \u03b1 :=\nsemiallowable_perm.to_struct_perm.comp coe_hom\n\nsection\nvariables {X : Type*} [mul_action (semiallowable_perm \u03b1) X]\n\ninstance mul_action_of_semiallowable_perm : mul_action (allowable_perm \u03b1) X :=\nmul_action.comp_hom _ coe_hom\n\n@[simp] lemma coe_smul (f : allowable_perm \u03b1) (x : X) : (f : semiallowable_perm \u03b1) \u2022 x = f \u2022 x :=\nrfl\n\nend\n\n@[simp] lemma fst_smul_near_litter (f : allowable_perm \u03b1) (N : near_litter) : (f \u2022 N).1 = f \u2022 N.1 :=\nrfl\n@[simp] lemma snd_smul_near_litter (f : allowable_perm \u03b1) (N : near_litter) :\n  ((f \u2022 N).2 : set atom) = f \u2022 \u2191N.2 := rfl\n\n@[simp] lemma smul_typed_near_litter (f : allowable_perm \u03b1) (N : near_litter) :\n  f \u2022 (typed_near_litter N : tangle (\u03b3 : \u039b)) =\n    typed_near_litter ((f : semiallowable_perm \u03b1) \u03b3 \u2022 N) :=\nallowable.smul_typed_near_litter _ _\n\n@[simp] lemma fst_smul (f : allowable_perm \u03b1) (c : code \u03b1) : (f \u2022 c).1 = c.1 := rfl\n@[simp] lemma snd_smul (f : allowable_perm \u03b1) (c : code \u03b1) : (f \u2022 c).2 = f \u2022 c.2 := rfl\n@[simp] lemma smul_mk (f : allowable_perm \u03b1) (\u03b3 s) : f \u2022 (mk \u03b3 s : code \u03b1) = mk \u03b3 (f \u2022 s) := rfl\n\nlemma _root_.con_nf.code.equiv.smul : c \u2261 d \u2192 f \u2022 c \u2261 f \u2022 d := (f.2 _ _).2\n\nend allowable_perm\n\nnamespace allowable_perm\nvariables {\u03b2 \u03b3}\n\nlemma smul_f_map (h\u03b2\u03b3 : \u03b2 \u2260 \u03b3) (\u03c0 : allowable_perm \u03b1) (t : tangle \u03b2) :\n  ((\u03c0 : semiallowable_perm \u03b1) \u03b3) \u2022 f_map (coe_ne h\u03b2\u03b3) t = f_map (coe_ne h\u03b2\u03b3) (\u03c0 \u2022 t) :=\nbegin\n  classical,\n  have equiv := code.equiv.singleton h\u03b2\u03b3 t,\n  rw \u2190 \u03c0.prop at equiv,\n  simp only [subtype.val_eq_coe, rec_bot_coe_coe, image_smul, smul_set_singleton] at equiv,\n  simp only [code.equiv_iff] at equiv,\n  obtain a | \u27e8heven, \u03b5, h\u03b5, hA\u27e9 | \u27e8heven, \u03b5, h\u03b5, hA\u27e9 | \u27e8c, heven, \u03b5, h\u03b5, \u03b6, h\u03b6, h\u2081, h\u2082\u27e9 := equiv,\n  { cases h\u03b2\u03b3.symm (congr_arg sigma.fst a) },\n  { simp_rw [semiallowable_perm.smul_mk, smul_set_singleton] at hA,\n    cases A_map_code_ne_singleton _ hA.symm,\n    exact h\u03b2\u03b3.symm },\n  { have := congr_arg sigma.fst hA,\n    simp only [semiallowable_perm.smul_mk, fst_A_map_code, fst_mk, Iio.coe_inj] at this,\n    subst this,\n    simp only [semiallowable_perm.smul_mk, A_map_code_ne _ (mk \u03b2 _) h\u03b2\u03b3, mk_inj] at hA,\n    simp only [coe_smul, snd_mk, smul_set_singleton, A_map_singleton] at hA,\n    simp only [\u2190 image_smul, image_image, smul_typed_near_litter] at hA,\n    rw \u2190 image_image at hA,\n    rw image_eq_image typed_near_litter.injective at hA,\n    have := litter.to_near_litter_mem_local_cardinal (f_map (coe_ne h\u03b2\u03b3) (\u03c0 \u2022 t)),\n    rw \u2190 hA at this,\n    obtain \u27e8N, hN\u2081, hN\u2082\u27e9 := this,\n    have := congr_arg sigma.fst hN\u2082,\n    simp only [litter.to_near_litter_fst] at this,\n    rw [\u2190 allowable.to_struct_perm_smul, struct_perm.smul_near_litter_fst,\n      allowable.to_struct_perm_smul] at this,\n    rw mem_local_cardinal at hN\u2081,\n    rw hN\u2081 at this,\n    exact this },\n  { have := congr_arg sigma.fst h\u2081,\n    simp only [coe_smul, smul_mk, fst_mk, fst_A_map_code] at this,\n    subst this,\n    simp only [coe_smul, smul_mk, smul_set_singleton] at h\u2081,\n    cases A_map_code_ne_singleton h\u03b5 h\u2081.symm }\nend\n\nlemma smul_A_map (\u03c0 : allowable_perm \u03b1) (s : set (tangle \u03b2)) (h\u03b2\u03b3 : \u03b2 \u2260 \u03b3) :\n  \u03c0 \u2022 A_map h\u03b2\u03b3 s = A_map h\u03b2\u03b3 (\u03c0 \u2022 s) :=\nbegin\n  ext,\n  simp only [A_map, mem_image, mem_Union, mem_local_cardinal, exists_prop, \u2190 image_smul],\n  simp only [exists_exists_and_eq_and, smul_typed_near_litter, \u2190 smul_f_map h\u03b2\u03b3],\n  split,\n  { rintro \u27e8N, \u27e8y, y_mem, y_fmap\u27e9, rfl\u27e9,\n    refine \u27e8(\u03c0 : semiallowable_perm \u03b1) \u03b3 \u2022 N, \u27e8y, y_mem, _\u27e9, rfl\u27e9,\n    rw \u2190 y_fmap,\n    refl },\n  { rintro \u27e8N, \u27e8y, y_mem, y_fmap\u27e9, rfl\u27e9,\n    refine \u27e8((\u03c0 : semiallowable_perm \u03b1) \u03b3)\u207b\u00b9 \u2022 N, \u27e8y, y_mem, _\u27e9, _\u27e9,\n    { change _ \u2022 N.fst = _,\n      simp only [y_fmap, map_inv, inv_smul_eq_iff],\n      refl },\n    { simp only [smul_inv_smul] } },\nend\n\nlemma smul_A_map_code (\u03c0 : allowable_perm \u03b1) (hc : c.1 \u2260 \u03b3) :\n  \u03c0 \u2022 A_map_code \u03b3 c = A_map_code \u03b3 (\u03c0 \u2022 c) :=\nby simp only [A_map_code_ne \u03b3 c hc, A_map_code_ne \u03b3 (\u03c0 \u2022 c) hc, smul_A_map, snd_smul, smul_mk]\n\nend allowable_perm\n\nlemma A_map_rel.smul : c \u219d d \u2192 f \u2022 c \u219d f \u2022 d :=\nby { rintro \u27e8\u03b3, h\u03b3\u27e9, exact (A_map_rel_iff _ _).2 \u27e8_, h\u03b3, f.smul_A_map_code h\u03b3\u27e9 }\n\n@[simp] lemma smul_A_map_rel : f \u2022 c \u219d f \u2022 d \u2194 c \u219d d :=\nby { refine \u27e8\u03bb h, _, A_map_rel.smul\u27e9, rw [\u2190inv_smul_smul f c, \u2190inv_smul_smul f d], exact h.smul }\n\nnamespace code\n\nlemma is_even_smul_nonempty : \u2200 (c : nonempty_code \u03b1), (f \u2022 c.val).is_even \u2194 c.val.is_even\n| \u27e8c, hc\u27e9 := begin\n  simp_rw code.is_even_iff,\n  split; intros h d hd,\n  { have := hd.nonempty_iff.2 hc,\n    let rec : A_map_rel' \u27e8d, this\u27e9 \u27e8c, hc\u27e9 := A_map_rel_coe_coe.1 hd,\n    exact code.not_is_even.1 (\u03bb H, (h _ hd.smul).not_is_even $\n      (is_even_smul_nonempty \u27e8d, this\u27e9).2 H) },\n  { rw \u2190smul_inv_smul f d at hd \u22a2,\n    rw smul_A_map_rel at hd,\n    have := hd.nonempty_iff.2 hc,\n    let rec : A_map_rel' \u27e8_, this\u27e9 \u27e8c, hc\u27e9 := A_map_rel_coe_coe.1 hd,\n    exact code.not_is_even.1 (\u03bb H, (h _ hd).not_is_even $ (is_even_smul_nonempty \u27e8_, this\u27e9).1 H) }\nend\nusing_well_founded { dec_tac := `[assumption] }\n\n@[simp] lemma is_even_smul : (f \u2022 c).is_even \u2194 c.is_even :=\nbegin\n  cases c.2.eq_empty_or_nonempty,\n  { rw [is_empty.is_even_iff h, is_empty.is_even_iff],\n    { refl },\n    simpa [code.is_empty] },\n  { exact is_even_smul_nonempty \u27e8c, h\u27e9 }\nend\n\n@[simp] lemma is_odd_smul : (f \u2022 c).is_odd \u2194 c.is_odd :=\nby simp_rw [\u2190code.not_is_even, is_even_smul]\n\nalias is_even_smul \u2194 _ is_even.smul\nalias is_odd_smul \u2194 _ is_odd.smul\n\nend code\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase1/allowable.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7185944046238982, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.35368364035123157}}
{"text": "import language_extension\n\nnamespace fol\nnamespace Lhom\n\nuniverse u\n\n\nvariables {L L' : Language.{u}} {\u03d5 : L \u2192\u1d38 L'}\n\n/-- restatement of `Lhom.reduct_all_ssatisfied` -/\ndef reduct_Theory_induced {S : Structure L'} {T : Theory L} (h\u03d5 : \u03d5.is_injective)\n  (h : S \u22a8 Theory_induced \u03d5 T) : S[[\u03d5]] \u22a8 T :=\nreduct_all_ssatisfied h\u03d5 h\n\nnamespace sum\n\nlemma is_injective_inl : (@Lhom.sum_inl L L').is_injective :=\n{ on_function := \u03bb n x y hxy, sum.inl.inj hxy,\n  on_relation := \u03bb n x y hxy, sum.inl.inj hxy, }\n\nlemma is_injective_inr : (@Lhom.sum_inr L L').is_injective :=\n{ on_function := \u03bb n x y hxy, sum.inr.inj hxy,\n  on_relation := \u03bb n x y hxy, sum.inr.inj hxy, }\n\nend sum\nend Lhom\nend fol\n", "meta": {"author": "Jlh18", "repo": "ModelTheoryInLean8", "sha": "fbda7d869d4169b6e739bb74165e99ee03ca63d6", "save_path": "github-repos/lean/Jlh18-ModelTheoryInLean8", "path": "github-repos/lean/Jlh18-ModelTheoryInLean8/ModelTheoryInLean8-fbda7d869d4169b6e739bb74165e99ee03ca63d6/src/Rings/ToMathlib/Lhom.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3536836344188672}}
{"text": "/-\nCopyright (c) 2020 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Simon Hudon\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-!\n# `pretty_cases` tactic\n\nWhen using `induction` and `cases`, `pretty_cases` prints a `\"Try\nthis:\"` advice that shows how to structure the proof with\n`case { ... }` commands.  In the following example, we apply induction on a\npermutation assumption about lists. `pretty_cases` gives us a proof\nskeleton that explicit selects the branches and explicit names the\nnew local constants:\n\n```lean\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  induction h,\n  pretty_cases,\n    -- Try this:\n    --   case list.perm.nil :\n    --   { admit },\n    --   case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_a h_ih\n    --   { admit },\n    --   case list.perm.swap : h_x h_y h_l\n    --   { admit },\n    --   case list.perm.trans : h_l\u2081 h_l\u2082 h_l\u2083 h_a h_a_1 h_ih_a h_ih_a_1\n    --   { admit },\nend\n```\n\n## Main definitions\n\n * `pretty_cases_advice` return `pretty_cases` advice without printing it\n * `pretty_cases` main tactic\n-/\n\nnamespace tactic\n\n\n/-- Query the proof goal and print the skeleton of a proof by cases. -/\nnamespace interactive\n\n\n/--\nQuery the proof goal and print the skeleton of a proof by\ncases.\n\nFor example, let us consider the following proof:\n\n```lean\nexample {\u03b1} (xs ys : list \u03b1) (h : xs ~ ys) : true :=\nbegin\n  induction h,\n  pretty_cases,\n    -- Try this:\n    --   case list.perm.nil :\n    --   { admit },\n    --   case list.perm.cons : h_x h_l\u2081 h_l\u2082 h_a h_ih\n    --   { admit },\n    --   case list.perm.swap : h_x h_y h_l\n    --   { admit },\n    --   case list.perm.trans : h_l\u2081 h_l\u2082 h_l\u2083 h_a h_a_1 h_ih_a h_ih_a_1\n    --   { admit },\nend\n```\n\nThe output helps the user layout the cases and rename the\nintroduced variables.\n-/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/pretty_cases.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.6334102705979902, "lm_q1q2_score": 0.35365006187423936}}
{"text": "namespace Ex1\n  variable (a : Nat) (i : Fin a) (h : 1 = a)\n  example : i < a := h \u25b8 i.2 -- `\u25b8` uses `subst` here\nend Ex1\n\nnamespace Ex2\ndef heapifyDown' (a : Array \u03b1) (i : Fin a.size) : Array \u03b1 := sorry\ndef heapifyDown (a : Array \u03b1) (i : Fin a.size) : Array \u03b1 :=\n  heapifyDown' a \u27e8i.1, a.size_swap i i \u25b8 i.2\u27e9 -- Error, failed to compute motive, `subst` is not applicable here\nend Ex2\n\nnamespace Ex3\ndef heapifyDown (a : Array \u03b1) (i : Fin a.size) : Array \u03b1 :=\n  have : i < i := sorry\n  heapifyDown a \u27e8i.1, a.size_swap i i \u25b8 i.2\u27e9 -- Error, failed to compute motive, `subst` is not applicable here\ntermination_by' measure fun \u27e8_, a, i\u27e9 => i.1\ndecreasing_by assumption\nend Ex3\n\nnamespace Ex4\ndef heapifyDown (lt : \u03b1 \u2192 \u03b1 \u2192 Bool) (a : Array \u03b1) (i : Fin a.size) : Array \u03b1 :=\n  let left := 2 * i.1 + 1\n  let right := left + 1\n  have left_le : i \u2264 left := sorry\n  have right_le : i \u2264 right := sorry\n  have i_le : i \u2264 i := Nat.le_refl _\n  have j : {j : Fin a.size // i \u2264 j} := if h : left < a.size then\n    if lt (a.get i) (a.get \u27e8left, h\u27e9) then \u27e8\u27e8left, h\u27e9, left_le\u27e9 else \u27e8i, i_le\u27e9 else \u27e8i, i_le\u27e9\n  have j := if h : right < a.size then\n    if lt (a.get j) (a.get \u27e8right, h\u27e9) then \u27e8\u27e8right, h\u27e9, right_le\u27e9 else j else j\n  if h : i \u2260 j then\n    let a' := a.swap i j\n    have : a'.size - j < a.size - i := sorry\n    heapifyDown lt a' \u27e8j.1.1, a.size_swap i j \u25b8 j.1.2\u27e9 -- Error, failed to compute motive, `subst` is not applicable here\n  else\n    a\ntermination_by' measure fun \u27e8_, _, a, i\u27e9 => a.size - i.1\ndecreasing_by assumption\nend Ex4\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/tests/lean/substBadMotive.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.3536500409730124}}
{"text": "import smt2\nimport .test_tactics\n\nlemma P_and_not_P_false (P : Prop) (p : P) (np : not P) : false :=\nbegin\n    intros,\n    z3\nend\n", "meta": {"author": "leanprover", "repo": "smt2_interface", "sha": "7ff0ce248b68ea4db2a2d4966a97b5786da05ed7", "save_path": "github-repos/lean/leanprover-smt2_interface", "path": "github-repos/lean/leanprover-smt2_interface/smt2_interface-7ff0ce248b68ea4db2a2d4966a97b5786da05ed7/test/inconsistent_context.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.35359664769104254}}
{"text": "import GMLInit.Data.Set.Basic\nimport GMLInit.Data.Set.Insert\n\nset_option checkBinderAnnotations false in\nclass inductive Set.IsSubfinite : Set \u03b1 \u2192 Prop\n| protected empty : IsSubfinite Set.empty\n| protected insertIf (s : Set \u03b1) [inst : IsSubfinite s] (a : \u03b1) (p : Prop := True) : IsSubfinite (s.insertIf a p)\nattribute [instance] Set.IsSubfinite.empty Set.IsSubfinite.insertIf\n\nnamespace Set.IsSubfinite\nopen Set Set.Notation\n\nprotected instance pure (x : \u03b1) : IsSubfinite (Set.pure x) := by\n  have : Set.pure x = Set.empty.insertIf x True := by\n    apply Set.ext\n    intro x\n    constr\n    \u00b7 intro h\n      cases h\n      left\n      constr\n      \u00b7 rfl\n      \u00b7 trivial\n    \u00b7 intro\n      | Or.inl \u27e8h,_\u27e9 => cases h; rfl\n      | Or.inr h => contradiction\n  rw [this]\n  infer_instance\n\nprotected instance union (s t : Set \u03b1) [hs : IsSubfinite s] [ht : IsSubfinite t] : IsSubfinite (Set.union s t) := by\n  induction hs with\n  | empty =>\n    rw [empty_union]\n    exact ht\n  | insertIf s a p H =>\n    rw [insertIf_union_left]\n    exact IsSubfinite.insertIf (inst:=H) (p := p) ..\n\nprotected instance inter_left (s t : Set \u03b1) [hs : IsSubfinite s] : IsSubfinite (Set.inter s t) := by\n  induction hs with\n  | empty =>\n    rw [empty_inter]\n    exact IsSubfinite.empty\n  | insertIf s a p H =>\n    rw [insertIf_inter_left]\n    exact IsSubfinite.insertIf (inst:=H) (p := p \u2227 a \u2208 t) ..\n\nprotected instance inter_right (s t : Set \u03b1) [ht : IsSubfinite t] : IsSubfinite (Set.inter s t) := by\n  induction ht with\n  | empty =>\n    rw [inter_empty]\n    exact IsSubfinite.empty\n  | insertIf t a p H =>\n    rw [insertIf_inter_right]\n    exact IsSubfinite.insertIf (inst:=H) (p := p \u2227 a \u2208 s) ..\n\nprotected instance map (f : \u03b1 \u2192 \u03b2) (s : Set \u03b1) [hs : IsSubfinite s] : IsSubfinite (s.map f) := by\n  induction hs with\n  | empty =>\n    rw [empty_map]\n    exact IsSubfinite.empty\n  | insertIf s a p H =>\n    rw [insertIf_map]\n    exact IsSubfinite.insertIf (inst:=H) (p := p) ..\n\nprotected instance bind (f : \u03b1 \u2192 Set \u03b2) [hf : (x : \u03b1) \u2192 IsSubfinite (f x)] (s : Set \u03b1) [hs : IsSubfinite s] : IsSubfinite (s.bind f) := by\n  induction hs with\n  | empty =>\n    rw [empty_bind]\n    exact IsSubfinite.empty\n  | insertIf s a p H =>\n    rw [insertIf_bind]\n    exact IsSubfinite.union (ht:=H) ..\n\nprotected instance seq (f : Set (\u03b1 \u2192 \u03b2)) [hf : IsSubfinite f] (s : Set \u03b1) [hs : IsSubfinite s] : IsSubfinite (Set.seq f s) := by\n  induction hs with\n  | empty =>\n    rw [empty_seq]\n    exact IsSubfinite.empty\n  | insertIf s a p H =>\n    rw [insertIf_seq]\n    exact IsSubfinite.union (ht:=H) ..\n\nprotected instance seqLeft (s : Set \u03b1) [hs : IsSubfinite s] (t : Set \u03b2) [ht : IsSubfinite t] : IsSubfinite (Set.seqLeft s t) := by\n  clean unfold Set.seqLeft\n  infer_instance\n\nprotected instance seqRight (s : Set \u03b1) [hs : IsSubfinite s] (t : Set \u03b2) [ht : IsSubfinite t] : IsSubfinite (Set.seqRight s t) := by\n  clean unfold Set.seqRight\n  infer_instance\n\nend Set.IsSubfinite\n\nstructure Subfinset (\u03b1) extends Set \u03b1 where\n  isSubfinite : toSet.IsSubfinite\nattribute [instance] Subfinset.isSubfinite\n\ninstance (\u03b1) : CoeSort (Subfinset \u03b1) (Set \u03b1) := \u27e8Subfinset.toSet\u27e9\n\nprotected theorem Subfinset.eq {\u03b1} : {s t : Subfinset \u03b1} \u2192 s.toSet = t.toSet \u2192 s = t\n| \u27e8_,_\u27e9, \u27e8_,_\u27e9, rfl => rfl\n\nprotected theorem Subfinset.ext {\u03b1} {s t : Subfinset \u03b1} : (\u2200 x, s.Mem x \u2194 t.Mem x) \u2192 s = t :=\n  \u03bb h => Subfinset.eq (Set.ext h)\n\ninstance : Monad Subfinset where\n  pure a := \u27e8pure a, Set.IsSubfinite.pure a\u27e9\n  map f s := \u27e8f <$> s.toSet, Set.IsSubfinite.map f s.toSet\u27e9\n  bind s f := \u27e8s.toSet >>= \u03bb x => (f x).toSet, Set.IsSubfinite.bind (\u03bb x => (f x).toSet) s.toSet\u27e9\n  seq f s := \u27e8f.toSet <*> (s ()).toSet, Set.IsSubfinite.seq f.toSet (s ()).toSet\u27e9\n  seqLeft s t := \u27e8s.toSet <* (t ()).toSet, Set.IsSubfinite.seqLeft s.toSet (t ()).toSet\u27e9\n  seqRight s t := \u27e8s.toSet *> (t ()).toSet, Set.IsSubfinite.seqRight s.toSet (t ()).toSet\u27e9\n\ninstance : LawfulMonad Subfinset where\n  id_map := by\n    intro _ \u27e8_,_\u27e9\n    apply Subfinset.eq\n    simp only [Functor.map, Set.id_map]\n  comp_map := by\n    intro _ _ _ f g \u27e8_,_\u27e9\n    apply Subfinset.eq\n    simp only [Functor.map, Set.comp_map]\n  map_const := by intros; rfl\n  map_pure f a := by\n    apply Subfinset.eq\n    simp only [Functor.map, Pure.pure, Set.map_pure]\n  pure_bind := by\n    intros\n    apply Subfinset.eq\n    simp only [Bind.bind, Pure.pure, Set.pure_bind]\n  bind_assoc := by\n    intros\n    apply Subfinset.eq\n    simp only [Bind.bind, Set.bind_assoc]\n  bind_pure_comp := by\n    intros\n    apply Subfinset.eq\n    simp only [Bind.bind, Pure.pure, Functor.map, Set.bind_pure_comp]\n  bind_map := by\n    intros\n    apply Subfinset.eq\n    simp only [Bind.bind, Functor.map, Seq.seq, Set.bind_map]\n  pure_seq f s := by\n    intros\n    apply Subfinset.eq\n    simp only [Seq.seq, Pure.pure, Functor.map, Set.pure_seq]\n  seq_assoc := by\n    intros\n    apply Subfinset.eq\n    simp only [Seq.seq, Functor.map, Set.seq_assoc]\n  seqLeft_eq _ _ := rfl\n  seqRight_eq _ _ := rfl\n", "meta": {"author": "fgdorais", "repo": "GMLInit", "sha": "a295111627ac907ebc6a86f906dd9b4d69b338d8", "save_path": "github-repos/lean/fgdorais-GMLInit", "path": "github-repos/lean/fgdorais-GMLInit/GMLInit-a295111627ac907ebc6a86f906dd9b4d69b338d8/GMLInit/Data/Set/Subfinset.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.35348601347883735}}
{"text": "import analysis.inner_product_space.pi_L2\nimport topology.paracompact\n\n\n--OUTPUT 1\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$.\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  -- First, we define a collection of pen balls.\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  let B0 : set (euclidean_space \u211d (fin n)) := \u2205,\n  have hB0 : is_open B0, from by auto [is_open_empty],\n  let Bm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, ball 0 m,\n  have hBm : \u2200 m : \u2115, is_open (Bm m), from by auto [is_open_ball, zero_lt_one],\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem,\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$,\n  -- and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  let Cm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, cover_ball_of_cover A (Bm m),\n  have hCm : \u2200 m : \u2115, is_open_cover (Cm m), from by auto [cover_ball_of_cover],\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  let C : set (euclidean_space \u211d (fin n)) := \u22c3i, (Cm i),\n  have hC : is_open_cover C, from by auto [is_open_cover_Union],\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$,\n  -- there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$),\n  -- and so $x$ is an element of $\\mathcal{C}_m$.\n  have h1 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, (x \u2208 Bm m) \u2227 (\u2200 m\u2081 : \u2115, (x \u2208 Bm\u2081 m\u2081) \u2192 m \u2264 m\u2081), from by auto [mem_ball],\n  have h2 : \u2200 x : euclidean_space \u211d (fin n), x \u2208 C, from by auto [C, mem_Union, classical.some_spec, h1],\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$.\n  have h3 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, (x \u2208 Bm m) \u2227 (\u2200 m\u2081 : \u2115, (x \u2208 Bm\u2081 m\u2081) \u2192 m \u2264 m\u2081), from by auto [mem_ball],\n  have h4 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 m\u2081 : \u2115, (m\u2081 \u2264 m) \u2192 (card (Bm\u2081 \u2229 C) \u2264 m\u2081 + 1), from by auto [Bm, B0, h3, hCm, is_open_cover_iff', is_open_cover_iff, card_Union_le, hC],\n  have h5 : \u2200 x : euclidean_space \u211d (fin n), locally_finite C, from by auto [locally_finite_iff, h4],\n  -- So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  show paracompact_space (euclidean_space \u211d (fin n)), from by auto [paracompact_iff, hA, hC, h2, h5],\nend\n\n--OUTPUT 2\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  assume (A : set (euclidean_space \u211d (fin n))) (hA : is_open_cover A),\n  let A' := \u22c3\u2080 A,\n  have h1 : A' = univ, from by auto [hA, is_open_cover_iff_univ_subset, univ_subset_iff],\n  let \u27e8r, hr\u27e9 := euclidean_space.exists_euclidean_metric \u211d (fin n),\n  let B := \u03bb (m : \u2115), ball r (0 : euclidean_space \u211d (fin n)) m,\n  let C := \u03bb (m : \u2115), (A' \u2229 (B m)) \\ (B (m - 1)),\n  let C' := \u03bb (m : \u2115), (A' \u2229 (B m)) \\ (B (m - 1)) \u2229 (A \u2229 (B m)),\n  let C'' := \u03bb (m : \u2115), (A \u2229 (B m)) \u2229 (A' \u2229 (B m)) \\ (B (m - 1)),\n  let C''' := \u03bb (m : \u2115), (A \u2229 (B m)) \\ (B (m - 1)),\n  have h2 : \u03bb (m : \u2115), (A \u2229 (B m)) \u2229 (A' \u2229 (B m)) = A \u2229 (B m), from by auto [inter_comm],\n  have h3 : \u2200 (m : \u2115), is_open (C m), from by auto [is_open_inter, is_open_diff, set.diff_eq, set.inter_comm, set.inter_assoc, h1, is_open_inter, is_open_ball],\n  have h4 : \u2200 (m : \u2115), is_open (C' m), from by auto [h3],\n  have h5 : \u2200 (m : \u2115), C' m = C''' m, from by auto [set.diff_eq, set.inter_comm],\n  have h6 : \u2200 (m : \u2115), is_open (C''' m), from by auto [h5, h4, set.inter_comm],\n  have h7 : \u2200 (m : \u2115), C m = C'' m, from by auto [set.diff_eq, h2],\n  have h8 : \u2200 (m : \u2115), is_open (C'' m), from by auto [h3, h7],\n  let C''' := \u22c3\u2080 (\u03bb (m : \u2115), C''' m),\n  have h9 : C''' = A', from by auto [set.union_eq_self_of_subset_left, set.subset_univ],\n  have h10 : C''' = C', from by auto [h5],\n  have h11 : C''' = C, from by auto [h2, h7, set.eq_of_subset_of_subset, set.subset.trans],\n  have h12 : is_open C''', from by auto [h10, h4],\n  have h13 : C''' = univ, from by auto [h9, h1],\n  have h14 : C''' \u2286 A, from by auto [h11, set.subset.trans, set.subset_union_iff],\n  have h15 : is_open_cover C''', from by auto [is_open_cover_iff_univ_subset, h13],\n  have h16 : \u2200 (m : \u2115), B m \u2286 B (m + 1), from by auto [ball_subset_ball],\n  have h17 : \u2200 (m : \u2115), C m \u2286 C (m + 1), from by auto [set.subset.trans, h16],\n  have h18 : \u2200 (m : \u2115), C' m \u2286 C' (m + 1), from by auto [h17],\n  have h19 : \u2200 (m : \u2115), C'' m \u2286 C'' (m + 1), from by auto [h17],\n  have h20 : \u2200 (m : \u2115), C''' m \u2286 C''' (m + 1), from by auto [h17],\n  have h21 : \u2200 (m : \u2115), C m \u2286 B m, from by auto [set.subset.trans, set.subset_diff_right],\n  have h22 : \u2200 (m : \u2115), C' m \u2286 B m, from by auto [h21],\n  have h23 : \u2200 (m : \u2115), C'' m \u2286 B m, from by auto [h21],\n  have h24 : \u2200 (m : \u2115), C''' m \u2286 B m, from by auto [h21],\n  have h25 : \u2200 (m : \u2115), 0 \u2264 m - 1, from by auto [nat.sub_nonneg],\n  have h26 : \u2200 (m : \u2115), B (m - 1) \u2286 B m, from by auto [ball_subset_ball, h25],\n  have h27 : \u2200 (m : \u2115), C m \u2286 B m, from by auto [set.subset.trans, h21],\n  have h28 : \u2200 (m : \u2115), C' m \u2286 B m, from by auto [h27],\n  have h29 : \u2200 (m : \u2115), C'' m \u2286 B m, from by auto [h27],\n  have h30 : \u2200 (m : \u2115), C''' m \u2286 B m, from by auto [h27],\n  have h31 : \u2200 (m : \u2115), C m \u2286 C' m, from by auto [set.subset.trans, h21],\n  have h32 : \u2200 (m : \u2115), C'' m \u2286 C' m, from by auto [h31],\n  have h33 : \u2200 (m : \u2115), C''' m \u2286 C' m, from by auto [h31],\n  have h34 : \u2200 (m : \u2115), C m \u2286 C'' m, from by auto [h31],\n  have h35 : \u2200 (m : \u2115), C''' m \u2286 C'' m, from by auto [h31],\n  have h36 : \u2200 (m : \u2115), C m \u2286 C''' m, from by auto [h31],\n  have h37 : \u2200 (m : \u2115), C' m \u2286 C''' m, from by auto [h31],\n  have h38 : \u2200 (m : \u2115), C'' m \u2286 C''' m, from by auto [h31],\n  have h39 : \u2200 (m : \u2115), C m \u2286 A, from by auto [h14, h36],\n  have h40 : \u2200 (m : \u2115), C' m \u2286 A, from by auto [h14, h37],\n  have h41 : \u2200 (m : \u2115), C'' m \u2286 A, from by auto [h14, h38],\n  have h42 : \u2200 (m : \u2115), C''' m \u2286 A, from by auto [h14, h36],\n  have h43 : \u2200 (m : \u2115), C m \u2286 A', from by auto [h1, h36],\n  have h44 : \u2200 (m : \u2115), C' m \u2286 A', from by auto [h1, h37],\n  have h45 : \u2200 (m : \u2115), C'' m \u2286 A', from by auto [h1, h38],\n  have h46 : \u2200 (m : \u2115), C''' m \u2286 A', from by auto [h1, h36],\n  have h47 : \u2200 (m : \u2115), C m \u2286 B m, from by auto [set.subset.trans, h\nend --Needs more than 2000 tokens!\n\n--OUTPUT 3\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  assume (A : set (euclidean_space \u211d (fin n))) (hA : is_open_cover A),\n  have hB : \u2203 B\u2080 : set (euclidean_space \u211d (fin n)), B\u2080 = \u2205, from exists.intro \u2205 (eq.refl \u2205),\n  have hB\u2081 : \u2200 n : \u2115, \u2203 B\u2099 : set (euclidean_space \u211d (fin n)), B\u2099 = ball 0 n, from\n  begin\n    assume n : \u2115,\n    have hB\u2099 : \u2203 B\u2099 : set (euclidean_space \u211d (fin n)), B\u2099 = ball 0 n, from exists.intro (ball 0 n) (eq.refl (ball 0 n)),\n    show \u2203 B\u2099 : set (euclidean_space \u211d (fin n)), B\u2099 = ball 0 n, from hB\u2099,\n  end,\n  have hB\u2082 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), cl B\u2099 = B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), cl B\u2099 = B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from exists.intro (cl B\u2099) (and.intro (eq.refl (cl B\u2099)) (eq.refl (\u22c3 m \u2264 n, ball 0 m))),\n    show \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), cl B\u2099 = B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from hB\u0305\u2099,\n  end,\n  have hB\u2083 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099, from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099, from exists.intro (cl B\u2099) (is_compact_cl_ball),\n    show \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099, from hB\u0305\u2099,\n  end,\n  have hB\u2084 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from exists.intro (cl B\u2099) (and.intro (is_compact_cl_ball) (eq.refl (\u22c3 m \u2264 n, ball 0 m))),\n    show \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m, from hB\u0305\u2099,\n  end,\n  have hB\u2085 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn, from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn, from exists.intro (cl B\u2099) (and.intro (and.intro (is_compact_cl_ball) (eq.refl (\u22c3 m \u2264 n, ball 0 m))) (subset_union _ _)),\n    show \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn, from hB\u0305\u2099,\n  end,\n  have hB\u2086 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn \u2227 \u2203! A\u2099 : set (euclidean_space \u211d (fin n)), A\u2099 \u2208 A \u2227 A\u2099 \u2229 B\u0305\u2099 \u2260 \u2205, from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn \u2227 \u2203! A\u2099 : set (euclidean_space \u211d (fin n)), A\u2099 \u2208 A \u2227 A\u2099 \u2229 B\u0305\u2099 \u2260 \u2205, from exists.intro (cl B\u2099) (and.intro (and.intro (and.intro (is_compact_cl_ball) (eq.refl (\u22c3 m \u2264 n, ball 0 m))) (subset_union _ _)) (exists_finite_subcover_compact _ _)),\n    show \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn \u2227 \u2203! A\u2099 : set (euclidean_space \u211d (fin n)), A\u2099 \u2208 A \u2227 A\u2099 \u2229 B\u0305\u2099 \u2260 \u2205, from hB\u0305\u2099,\n  end,\n  have hB\u2087 : \u2200 n : \u2115, \u2203 B\u0305\u2099 : set (euclidean_space \u211d (fin n)), is_compact B\u0305\u2099 \u2227 B\u0305\u2099 = \u22c3 m \u2264 n, ball 0 m \u2227 B\u0305\u2099 \u2286 \u211dn \u2227 \u2203! A\u2099 : set (euclidean_space \u211d (fin n)), A\u2099 \u2208 A \u2227 A\u2099 \u2229 B\u0305\u2099 \u2260 \u2205 \u2227 \u2203 C\u2099 : set (euclidean_space \u211d (fin n)), C\u2099 = A\u2099 \u2229 (\u211dn \\ B\u0305\u2099), from\n  begin\n    assume n : \u2115,\n    have hB\u0305\u2099 : \u2203 B\u0305\u2099 : set\nend --Needs more than 2000 tokens!\n\n--OUTPUT 4\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  assume open_covering : \u2200 U \u2208 \ud835\udcaa (euclidean_space \u211d (fin n)), \u2203 V \u2208 open_covering, U \u2286 V,\n  let open_covering := open_covering,\n\n  -- First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  let B0 := {x : euclidean_space \u211d (fin n) | x = 0},\n  let Bm := {x : euclidean_space \u211d (fin n) | \u2203 m : \u2115, \u2200 i : fin n, abs (x i) < m},\n  have Bm_subset_Bmplusone : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone'''''''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [lt_succ_iff],\n  have Bm_subset_Bmplusone''''''''''''''''''''''''''''''''' : \u2200 m : \u2115, Bm m \u2286 Bm (m+1), from by auto using [\nend --Needs more than 2000 tokens!\n\n--OUTPUT 5\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Given an open covering of $\\mathbb{R}^n$, we now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  assume (h1 : \u2200 A : set (euclidean_space \u211d (fin n)), is_open A \u2192 is_open_cover A),\n  have h2 : \u2200 A : set (euclidean_space \u211d (fin n)), is_open A \u2192 is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],\n\n  -- First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\n  -- centered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  have h3 : \u2200 A : set (euclidean_space \u211d (fin n)), is_open A \u2192 is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],\n\n  show \u2200 A : set (euclidean_space \u211d (fin n)), is_open A \u2192 is_locally_finite_open_refinement A, from by auto [paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement, paracompact_space.paracompact_iff_every_open_cover_has_locally_finite_open_refinement_inverse],\nend\n\n--OUTPUT 6\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$\n  assume A : set (set (euclidean_space \u211d (fin n))),\n  assume hA : is_open_cover A,\n  /-\n  We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  -/\n  let B\u2080 : set (euclidean_space \u211d (fin n)) := set.empty,\n  let B := \u03bb m : \u2115, (euclidean_space.ball \u211d (fin n) 0 m),\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem,\n  have h1 : compact_space (closure (B m)), from by auto [compact_closure, compact_univ],\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$\n  have h2 : \u2203 f : set (euclidean_space \u211d (fin n)) \u2192 set (euclidean_space \u211d (fin n)), \n            (\u2200 (x : set (euclidean_space \u211d (fin n))) (hx : x \u2208 \ud835\udcab A), is_open (f x)) \u2227\n            (\u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 \ud835\udcab A \u2192 x \u2286 (f x)) \u2227\n            (\u2200 (x : set (euclidean_space \u211d (fin n))) (hx : x \u2208 \ud835\udcab A), closure (f x) \u2286 closure x) \u2227\n            (\u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 \ud835\udcab A \u2192 f x \u2229 B m \u2286 x) \u2227\n            (\u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 \ud835\udcab A \u2192 closure (f x) \u2286 closure (B m)), from by auto [h1, hA, lebesgue_number_lemma],\n  let f := h2.left,\n  let hf1 := h2.right.left.left.left,\n  let hf2 := h2.right.left.left.right,\n  let hf3 := h2.right.left.right,\n  let hf4 := h2.right.right.left,\n  let hf5 := h2.right.right.right,\n  /-\n  and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, \n  and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$).\n  -/\n  let C := \u03bb m : \u2115, {x : set (euclidean_space \u211d (fin n)) | x \u2208 \ud835\udcab A \u2227 (x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1)))) = f x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1)))},\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  have h3 : \u2203 C : \u2115 \u2192 set (set (euclidean_space \u211d (fin n))), (\u2200 m : \u2115, C m \u2208 \ud835\udcab A) \u2227 (\u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C m \u2192 is_open x), from by auto [C, hf1, hf2],\n  let C' := h3.left,\n  let hC1 := h3.right.left,\n  let hC2 := h3.right.right,\n  have h4 : \u2200 m : \u2115, C' m \u2208 \ud835\udcab A, from by auto [C', hf1, hf2],\n  have h5 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 is_open x, from by auto [C', hf1, hf2],\n  have h6 : \u2200 m : \u2115, C' m \u2208 \ud835\udcab A, from by auto [h4],\n  have h7 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 is_open x, from by auto [h5],\n  have h8 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1))) = f x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1))), from by auto [C', hf1, hf2],\n  have h9 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 f x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1))) \u2286 x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1))), from by auto [C', hf1, hf2],\n  have h10 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 f x \u2229 B m \u2286 x \u2229 B m, from by auto [C', hf1, hf2],\n  have h11 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure (f x) \u2286 closure x, from by auto [C', hf1, hf2],\n  have h12 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure (f x) \u2286 closure (B m), from by auto [C', hf1, hf2],\n  have h13 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure x \u2286 closure (B m), from by auto [h11, h12],\n  have h14 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure (x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1)))) \u2286 closure (B m), from by auto [closure_inter, h13],\n  have h15 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure (f x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1)))) \u2286 closure (B m), from by auto [closure_inter, h12],\n  have h16 : \u2200 m : \u2115, \u2200 x : set (euclidean_space \u211d (fin n)), x \u2208 C' m \u2192 closure (x \u2229 (space (euclidean_space \u211d (fin n)) \u2216 closure (B (m - 1)))) \u2286 closure (f x \u2229 (\nend --Needs more than 2000 tokens!\n\n--OUTPUT 7\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$\n  assume A : set (euclidean_space \u211d (fin n)),\n  assume hA : is_open_cover A,\n\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$\n  have h1 : \u2203 C : set (euclidean_space \u211d (fin n)), is_open_cover C \u2227 is_locally_finite_cover C \u2227 \u2200 a \u2208 A, \u2203 b \u2208 C, a \u2286 b, from by auto using [use A] using [hA],\n\n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0\n  let B0 : set (euclidean_space \u211d (fin n)) := \u2205,\n  let Bm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, ball_of_radius \u211d m 0,\n\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$)\n  let Cm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, {x : euclidean_space \u211d (fin n) | \u2203 a \u2208 A, (x \u2208 a \u2227 x \u2208 (set.univ \\ Bar Bm (m-1)) \u2227 \u2203 f : fin m, x \u2208 ball_of_radius \u211d (m-1) f)},\n\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$\n  let C : set (euclidean_space \u211d (fin n)) := set.union Cm,\n  have h2 : is_open_cover C, from by auto using [is_open_cover_union, Cm],\n\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$\n  have h3 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, x \u2208 Bm m, from by auto [ball_of_radius, set.mem_univ],\n  have h4 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, (\u2200 m' : \u2115, m' < m \u2192 \u00ac x \u2208 Bm m'), from by auto [ball_of_radius, set.mem_univ, nat.find_min],\n  have h5 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, x \u2208 Cm m, from by auto [h3, h4, Cm],\n  have h6 : \u2200 x : euclidean_space \u211d (fin n), \u2203 c : euclidean_space \u211d (fin n), (x \u2208 c \u2227 c \u2208 C), from by auto [C, h5, set.mem_union],\n\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$\n  have h7 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m), from by auto [Cm, h4],\n  have h8 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1)), from by auto [Cm, h4],\n  have h9 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1) \u2227 (\u2200 m' : \u2115, m' < m \u2192 \u00ac c \u2208 Cm m')), from by auto [Cm, h4],\n  have h10 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1) \u2227 (\u2200 m' : \u2115, m' < m \u2192 \u00ac c \u2208 Cm m') \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac x \u2208 Cm m'), from by auto [Cm, h4],\n  have h11 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1) \u2227 (\u2200 m' : \u2115, m' < m \u2192 \u00ac c \u2208 Cm m') \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac x \u2208 Cm m' \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac c \u2208 Cm m'), from by auto [Cm, h4],\n  have h12 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1) \u2227 (\u2200 m' : \u2115, m' < m \u2192 \u00ac c \u2208 Cm m') \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac x \u2208 Cm m' \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac c \u2208 Cm m' \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac c \u2208 Cm m'), from by auto [Cm, h4],\n  have h13 : \u2200 x : euclidean_space \u211d (fin n), \u2203 m : \u2115, \u2200 c : euclidean_space \u211d (fin n), (c \u2208 C \u2192 x \u2208 c \u2192 c \u2208 Cm m \u2227 c \u2209 Cm (m+1) \u2227 (\u2200 m' : \u2115, m' < m \u2192 \u00ac c \u2208 Cm m') \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac x \u2208 Cm m' \u2227 \u2200 m' : \u2115, m' \u2265 m \u2192 \u00ac c \u2208 Cm m' \u2227 \u2200 m\nend --Needs more than 2000 tokens!\n\n--OUTPUT 8\ntheorem \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nbegin\n  -- Let $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. \n  assume (A : set (euclidean_space \u211d (fin n))) (hA : is_open_cover A),\n\n  -- We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$.\n  -- First, we define a collection of pen balls. \n  -- Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$ centered at 0.\n  let B0 : set (euclidean_space \u211d (fin n)) := \u2205,\n  let Bm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, {x : (euclidean_space \u211d (fin n)) | \u2203 (r : \u211d), 0 < r \u2227 r \u2264 m \u2227 x \u2208 (ball r (0 : (euclidean_space \u211d (fin n))))},\n\n  -- Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, \n  let h1 : \u2200 m : \u2115, is_compact (closure (Bm m)), from by auto [is_compact_closure, is_compact_ball],\n  have h2 : \u2200 m : \u2115, \u2203 (C : set (euclidean_space \u211d (fin n))) (hC : finite C) (hC' : C \u2286 A), (closure (Bm m)) \u2286 \u22c3 C, from by auto [hA, h1],\n\n  -- so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). \n  let Cm : \u2115 \u2192 set (euclidean_space \u211d (fin n)) := \u03bb m, (\u03bb (x : (euclidean_space \u211d (fin n))), x \u2229 (set.diff (euclidean_space \u211d (fin n)) (closure (Bm (m-1))))) '' (classical.some (h2 m).exists),\n\n  -- So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$.\n  let C : set (euclidean_space \u211d (fin n)) := \u22c3 m, Cm m,\n  have h3 : \u2200 m : \u2115, Cm m \u2286 A, from by auto [classical.some_spec, exists_unique.exists, h2],\n  have h4 : C \u2286 A, from by auto [h3, set.union_subset],\n\n  -- Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$.\n  have h5 : \u2200 x : (euclidean_space \u211d (fin n)), \u2203 (m : \u2115), x \u2208 Bm m, from by auto [Bm],\n  have h6 : \u2200 x : (euclidean_space \u211d (fin n)), \u2203 (m : \u2115), \u2200 (n : \u2115), n \u2264 m \u2192 x \u2209 Bn n, from by auto [Bm],\n  have h7 : \u2200 x : (euclidean_space \u211d (fin n)), \u2203! (m : \u2115), x \u2208 Bm m, from by auto [h5, h6],\n  have h8 : \u2200 x : (euclidean_space \u211d (fin n)), x \u2208 C, from by auto [Cm, classical.some_spec, exists_unique.exists, h7, classical.some_spec, exists_unique.exists, h2],\n\n  -- Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$.\n  have h9 : \u2200 x : (euclidean_space \u211d (fin n)), \u2200 (m : \u2115), finite {n | n \u2208 C \u2227 n \u2229 Bm m \u2260 \u2205}, from by auto [Cm, classical.some_spec, exists_unique.exists, h2, classical.some_spec, exists_unique.exists, h2, set.inter_diff],\n  have h10 : \u2200 x : (euclidean_space \u211d (fin n)), locally_finite C, from by auto [h9],\n\n  -- So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n  show paracompact_space (euclidean_space \u211d (fin n)), from by auto [C, h4, h10],\nend\n\n/- FEW SHOT PROMPTS TO CODEX(START)\n/--`theorem`\nPower Set is Closed under Intersection\nLet $S$ be a set.\n\nLet $\\powerset S$ be the power set of $S$.\n\n\nThen:\n:$\\forall A, B \\in \\powerset S: A \\cap B \\in \\powerset S$\n`proof`\nLet $A, B \\in \\powerset S$.\n\nThen by the definition of power set, $A \\subseteq S$ and $B \\subseteq S$.\n\nFrom Intersection is Subset we have that $A \\cap B \\subseteq A$.\n\nIt follows from Subset Relation is Transitive that $A \\cap B \\subseteq S$.\n\nThus $A \\cap B \\in \\powerset S$ and closure is proved.\n{{qed}}\n-/\ntheorem power_set_intersection_closed {\u03b1 : Type*} (S : set \u03b1) : \u2200 A B \u2208 \ud835\udcab S, (A \u2229 B) \u2208 \ud835\udcab S :=\nbegin\n  -- $A$ and $B$ are sets. $A$ and $B$ belong to power set of $S$\n  assume (A : set \u03b1) (hA : A \u2208 \ud835\udcab S) (B : set \u03b1) (hB : B \u2208 \ud835\udcab S),\n  -- Then $A \u2286 S$ and $B \u2286 S$, by power set definition\n  have h1 : (A \u2286 S) \u2227 (B \u2286 S), from by auto [set.subset_of_mem_powerset, set.subset_of_mem_powerset],\n  -- Then $(A \u2229 B) \u2286 A$, by intersection of set is a subset\n  have h2 : (A \u2229 B) \u2286 A, from by auto [set.inter_subset_left],\n  -- Then $(A \u2229 B) \u2286 S$, by subset relation is transitive \n  have h3 : (A \u2229 B) \u2286 S, from by auto [set.subset.trans],\n  -- Hence $(A \u2229 B) \u2208  \ud835\udcab S$, by power set definition\n  show (A \u2229 B) \u2208  \ud835\udcab S, from by auto [set.mem_powerset],\nend\n\n/--`theorem`\nSquare of Sum\n :$\\forall x, y \\in \\R: \\paren {x + y}^2 = x^2 + 2 x y + y^2$\n`proof`\nFollows from the distribution of multiplication over addition:\n\n{{begin-eqn}}\n{{eqn | l = \\left({x + y}\\right)^2\n      | r = \\left({x + y}\\right) \\cdot \\left({x + y}\\right)\n}}\n{{eqn | r = x \\cdot \\left({x + y}\\right) + y \\cdot \\left({x + y}\\right)\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x \\cdot x + x \\cdot y + y \\cdot x + y \\cdot y\n      | c = Real Multiplication Distributes over Addition\n}}\n{{eqn | r = x^2 + 2xy + y^2\n      | c = \n}}\n{{end-eqn}}\n{{qed}}\n-/\ntheorem square_of_sum (x y : \u211d) : (x + y)^2 = (x^2 + 2*x*y + y^2) := \nbegin\n  -- expand the power\n  calc (x + y)^2 = (x+y)*(x+y) : by auto [sq]\n  -- distributive property of multiplication over addition gives:\n  ... = x*(x+y) + y*(x+y) : by auto [add_mul]\n  -- applying the above property further gives:\n  ... = x*x + x*y + y*x + y*y : by auto [mul_comm, add_mul] using [ring]\n  -- rearranging the terms using commutativity and adding gives:\n  ... = x^2 + 2*x*y + y^2 : by auto [sq, mul_comm] using [ring]\nend\n\n/--`theorem`\nIdentity of Group is Unique\nLet $\\struct {G, \\circ}$ be a group. Then there is a unique identity element $e \\in G$.\n`proof`\nFrom Group has Latin Square Property, there exists a unique $x \\in G$ such that:\n:$a x = b$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = b$\n\nSetting $b = a$, this becomes:\n\nThere exists a unique $x \\in G$ such that:\n:$a x = a$\n\nand there exists a unique $y \\in G$ such that:\n:$y a = a$\n\nThese $x$ and $y$ are both $e$, by definition of identity element.\n{{qed}}\n-/\ntheorem group_identity_unique {G : Type*} [group G] : \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a :=\nbegin\n  -- Group has Latin Square Property\n  have h1 : \u2200 a b : G, \u2203! x : G, a * x = b, from by auto using [use (a\u207b\u00b9 * b)],\n  have h2 : \u2200 a b : G, \u2203! y : G, y * a = b, from by auto using [use b * a\u207b\u00b9], \n\n  -- Setting $b = a$, this becomes:\n  have h3 : \u2200 a : G, \u2203! x : G, a * x = a, from by auto [h1],\n  have h4 : \u2200 a : G, \u2203! y : G, y * a = a, from by auto [h2],\n\n  -- These $x$ and $y$ are both $(1 : G)$, by definition of identity element\n  have h5 : \u2200 a : G, classical.some (h3 a).exists = (1 : G), from by auto [exists_unique.unique, h3, classical.some_spec, exists_unique.exists, mul_one],\n  have h6 : \u2200 a : G, classical.some (h4 a).exists = (1 : G), from by auto [exists_unique.unique, h4, classical.some_spec, exists_unique.exists, one_mul],\n\n  show \u2203! e : G, \u2200 a : G, e * a = a \u2227 a * e = a, from by auto [h3, h4, exists_unique.unique, classical.some_spec, exists_unique.exists] using [use (1 : G)],\nend\n\n/--`theorem`\n\\mathbb{R}^n is paracompact\n$\\mathbb{R}^n$ is paracompact for all $n$.\n`proof`\nLet $\\mathcal{A}$ be an open covering of $\\mathbb{R}^n$. We now construct a locally finite open refinement $\\mathcal{C}$ of $\\mathcal{A}$ that covers $\\mathbb{R}^n$. First, we define a collection of pen balls. Let $B_0 = \\phi$, and for each $n \\in \\mathbb{N}$, let $B_m$ denote the ball of radius $m$\ncentered at 0. Given $m$, set $\\Bar{B_m}$ is compact in $\\mathbb{R}^n$ by the Heine-Borel theorem, so choose finitely many elements of $\\mathcal{A}$ that cover $\\Bar{B_m}$ and intersect each one with the open set $\\mathbb{R}^n \\setminus \\Bar{B_{m - 1}}$, and let $\\mathcal{C}_{m}$ denote this collection of open sets (each an open subset of an element of $\\mathcal{A}$). So $\\mathcal{C} = \\bigcup_{m = 0}^{\\infty} \\mathcal{C}_m$ is an open refinement of $\\mathcal{A}$. Note that $\\mathcal{C}$ covers $\\mathbb{R}^n$ since for any $x \\in \\mathbb{R}^n$, there is a smallest $m \\in \\mathbb{N}$ such that $x \\in \\Bar{B_{m}}$ (namely, some $m$ where $\\rVert x \\lVert \\leq m \\leq \\rVert x \\lVert + 1$), and so $x$ is an element of $\\mathcal{C}_m$. Now collection $\\mathcal{C}$ is locally finite since for given $x \\in \\mathbb{R}^n$, neighborhood $B_m$ intersects only finitely many elements of $\\mathcal{C}$, namely those elements in collection $\\mathcal{C}_1 \\cup \\mathcal{C}_2 \\cup \\cdots \\mathcal{C}_m$. So $\\mathcal{C}$ is a locally finite open refinement of $\\mathcal{A}$ that covers $\\mathbb{R}^n$, hence $\\mathbb{R}^n$ is paracompact.\n\nQED\n-/\ntheorem  \u211dn_paracompact (n : \u2115) : paracompact_space (euclidean_space \u211d (fin n)) :=\nFEW SHOT PROMPTS TO CODEX(END)-/\n", "meta": {"author": "ayush1801", "repo": "Autoformalisation_benchmarks", "sha": "51e1e942a0314a46684f2521b95b6b091c536051", "save_path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks", "path": "github-repos/lean/ayush1801-Autoformalisation_benchmarks/Autoformalisation_benchmarks-51e1e942a0314a46684f2521b95b6b091c536051/proof/lean_proof_auto_with_comments-Natural-Language-Proof-Translation/Correct_statement-lean_proof_auto_with_comments-3_few_shot_temperature_0.6_max_tokens_2000_n_8/clean_files/Rn is paracompact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6654105587468141, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3534723259481762}}
{"text": "import polycodable_init\n\n@[user_attribute]\nmeta def polyfun : user_attribute :=\n{ name := `polyfun,\n  descr := \"lemmas usable to prove polynomial time\" }\n\nattribute [polyfun]\n  polytime_fun.id\n  polytime_fun.const\n\n@[polyfun]\nlemma polytime_fun.id' {\u03b1} [ptree.pencodable \u03b1] : polytime_fun (\u03bb x : \u03b1, x) := polytime_fun.id\n\n\nnamespace tactic\n\nmeta def polytime_fun_lemmas : list name :=\n[``polytime_fun, ``polytime_fun\u2082, ``polytime_fun\u2083]\n\nmeta def polytime_fun_comp_lemmas : list name :=\n[``polytime_fun.comp, ``polytime_fun.comp\u2082, ``polytime_fun.comp\u2083]\n\nmeta def unfold_polytime (md : transparency) : tactic unit :=\ndo dunfold_target (``function.uncurry :: polytime_fun_lemmas.tail),\n   try dsimp_target\n\n-- In order to help resolve polytime_fun of propositions (which are converted to bool's)\nmeta def simp_to_bool : tactic unit :=\n`[simp only [bool.to_bool_not, bool.to_bool_and, bool.to_bool_or, bool.to_bool_coe]]\n\n-- Please help, idk how to write tactics\nmeta def is_polycodable (e : expr) : tactic bool :=\n(do\n   e' \u2190 infer_type e,\n   cache \u2190 mk_instance_cache e',\n   (cache', s) \u2190 instance_cache.get cache ``ptree.pencodable,\n   return tt) <|> (return ff)\n\nmeta def get_num_params : tactic \u2115 :=\ndo `(polytime_fun %%s) \u2190 target,\n    guard s.is_lambda,\n    mv \u2190 mk_meta_var s.binding_domain,\n    e \u2190  instantiate_mvars (s.instantiate_lambdas [mv]),\n    f \u2190 mfilter is_polycodable e.get_app_args,\n    return f.length\n\nmeta def apply_polyfun.comp (md : transparency) : tactic \u2115 :=\ndo fail_if_success `[exact polytime_fun.const _],\n   fail_if_success (to_expr ``(polytime_fun.pair) >>= \u03bb e, apply e {md := md}),\n   old_goal \u2190 target,\n   n \u2190 get_num_params, guard (0 < n \u2227 n \u2264 polytime_fun_lemmas.length),\n   s \u2190 resolve_name (polytime_fun_comp_lemmas.inth (n-1)),\n   s' \u2190 to_expr s,\n   apply s' {md := md},\n   try `[ any_goals { apply_instance, } ], -- why is this necessary??\n   (fail_if_success (unfold_polytime md >> target >>= \u03bb t, unify t old_goal md)) <|>\n    focus1 (apply_rules [] [``polyfun] 50 { md := md } >> done),\n  return (n-1)\n\nmeta def polyfun_tactics (md : transparency := reducible) : list (tactic string) :=\n[\n  apply_rules [] [``polyfun] 50 { md := md }\n                        >> pure \"apply_rules with polyfun\", \n  unfold_polytime md >> pure \"dunfold_target polytime_fun_lemmas.tail\",\n  simp_to_bool >> pure \"simp only [bool.to_bool_not, bool.to_bool_and, bool.to_bool_or]\",\n  apply_polyfun.comp md >>= \u03bb n, pure (\"apply \" ++ (to_string $ polytime_fun_comp_lemmas.inth (n-1)))\n]\n\nnamespace interactive\nsetup_tactic_parser\n\nmeta def polyfun\n  (bang : parse $ optional (tk \"!\")) (trace : parse $ optional (tk \"?\")) (cfg : tidy.cfg := {}) :\n  tactic unit :=\nlet md              := if bang.is_some then semireducible else reducible,\n    polyfun_core := tactic.tidy { tactics := polyfun_tactics md, ..cfg },\n    trace_fn        := if trace.is_some then show_term else id in\ntrace_fn polyfun_core\n\n\nend interactive\n\nend tactic\n\nsection\n\nattribute [polyfun] \n  polytime_fun.fst\n  polytime_fun.snd\n  polytime_fun.pair\n  polytime_fun.node\n  polytime_fun.polytime_code\n  polytime_fun.ptree_left\n  polytime_fun.ptree_right\n  polytime_fun.encode\n  polytime_fun.decode'\n\n\n-- section\n-- parameters {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} [polycodable \u03b1] [polycodable \u03b2] [polycodable \u03b3] [polycodable \u03b4]\n\n-- example {f : \u03b1 \u2192 \u03b2} (hf : polytime_fun f) : polytime_fun f := by polyfun\n-- example {f : \u03b1 \u2192 \u03b2} : polytime_fun f := by { try { polyfun }, sorry, }\n-- example {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} : polytime_fun\u2082 f := by { polyfun, }\n\n-- @[irreducible]\n-- def f : \u03b1 \u2192 \u03b2 \u2192 \u03b3 := sorry\n-- lemma f_polyfun : polytime_fun\u2082 f := sorry\n-- local attribute [polyfun] f_polyfun\n\n-- example : polytime_fun\u2082 f := by { polyfun, }\n-- example : polytime_fun (\u03bb x : \u03b1 \u00d7 \u03b2, f x.1 x.2) := by { polyfun, }\n\n-- end\nend\n", "meta": {"author": "prakol16", "repo": "lean_complexity_theory_polytime_trees", "sha": "4f478b752a2061cd829bf83a68c77180d1318b62", "save_path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees", "path": "github-repos/lean/prakol16-lean_complexity_theory_polytime_trees/lean_complexity_theory_polytime_trees-4f478b752a2061cd829bf83a68c77180d1318b62/src/polytime_tac.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.665410558746814, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3534723259481761}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport algebra.algebra.operations\nimport algebra.ring.equiv\nimport data.nat.choose.sum\nimport ring_theory.coprime.lemmas\nimport ring_theory.ideal.quotient\nimport ring_theory.non_zero_divisors\n/-!\n# More operations on modules and ideals\n-/\nuniverses u v w x\n\nopen_locale big_operators pointwise\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v} {F : Type*} {G : Type*}\n\nsection comm_semiring\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\nopen_locale pointwise\n\ninstance has_scalar' : has_scalar (ideal R) (submodule R M) :=\n\u27e8submodule.map\u2082 (linear_map.lsmul R M)\u27e9\n\n/-- This duplicates the global `smul_eq_mul`, but doesn't have to unfold anywhere near as much to\napply. -/\nprotected lemma _root_.ideal.smul_eq_mul (I J : ideal R) : I \u2022 J = I * J := rfl\n\n/-- `N.annihilator` is the ideal of all elements `r : R` such that `r \u2022 N = 0`. -/\ndef annihilator (N : submodule R M) : ideal R :=\n(linear_map.lsmul R N).ker\n\nvariables {I J : ideal R} {N P : submodule R M}\n\ntheorem mem_annihilator {r} : r \u2208 N.annihilator \u2194 \u2200 n \u2208 N, r \u2022 n = (0:M) :=\n\u27e8\u03bb hr n hn, congr_arg subtype.val (linear_map.ext_iff.1 (linear_map.mem_ker.1 hr) \u27e8n, hn\u27e9),\n\u03bb h, linear_map.mem_ker.2 $ linear_map.ext $ \u03bb n, subtype.eq $ h n.1 n.2\u27e9\n\ntheorem mem_annihilator' {r} : r \u2208 N.annihilator \u2194 N \u2264 comap (r \u2022 linear_map.id) \u22a5 :=\nmem_annihilator.trans \u27e8\u03bb H n hn, (mem_bot R).2 $ H n hn, \u03bb H n hn, (mem_bot R).1 $ H hn\u27e9\n\nlemma mem_annihilator_span (s : set M) (r : R) :\n  r \u2208 (submodule.span R s).annihilator \u2194 \u2200 n : s, r \u2022 (n : M) = 0 :=\nbegin\n  rw submodule.mem_annihilator,\n  split,\n  { intros h n, exact h _ (submodule.subset_span n.prop) },\n  { intros h n hn,\n    apply submodule.span_induction hn,\n    { intros x hx, exact h \u27e8x, hx\u27e9 },\n    { exact smul_zero _ },\n    { intros x y hx hy, rw [smul_add, hx, hy, zero_add] },\n    { intros a x hx, rw [smul_comm, hx, smul_zero] } }\nend\n\nlemma mem_annihilator_span_singleton (g : M) (r : R) :\n  r \u2208 (submodule.span R ({g} : set M)).annihilator \u2194 r \u2022 g = 0 :=\nby simp [mem_annihilator_span]\n\ntheorem annihilator_bot : (\u22a5 : submodule R M).annihilator = \u22a4 :=\n(ideal.eq_top_iff_one _).2 $ mem_annihilator'.2 bot_le\n\ntheorem annihilator_eq_top_iff : N.annihilator = \u22a4 \u2194 N = \u22a5 :=\n\u27e8\u03bb H, eq_bot_iff.2 $ \u03bb (n:M) hn, (mem_bot R).2 $\n  one_smul R n \u25b8 mem_annihilator.1 ((ideal.eq_top_iff_one _).1 H) n hn,\n  \u03bb H, H.symm \u25b8 annihilator_bot\u27e9\n\ntheorem annihilator_mono (h : N \u2264 P) : P.annihilator \u2264 N.annihilator :=\n\u03bb r hrp, mem_annihilator.2 $ \u03bb n hn, mem_annihilator.1 hrp n $ h hn\n\ntheorem annihilator_supr (\u03b9 : Sort w) (f : \u03b9 \u2192 submodule R M) :\n  (annihilator \u2a06 i, f i) = \u2a05 i, annihilator (f i) :=\nle_antisymm (le_infi $ \u03bb i, annihilator_mono $ le_supr _ _)\n(\u03bb r H, mem_annihilator'.2 $ supr_le $ \u03bb i,\n  have _ := (mem_infi _).1 H i, mem_annihilator'.1 this)\n\ntheorem smul_mem_smul {r} {n} (hr : r \u2208 I) (hn : n \u2208 N) : r \u2022 n \u2208 I \u2022 N := apply_mem_map\u2082 _ hr hn\n\ntheorem smul_le {P : submodule R M} : I \u2022 N \u2264 P \u2194 \u2200 (r \u2208 I) (n \u2208 N), r \u2022 n \u2208 P := map\u2082_le\n\n@[elab_as_eliminator]\ntheorem smul_induction_on {p : M \u2192 Prop} {x} (H : x \u2208 I \u2022 N)\n  (Hb : \u2200 (r \u2208 I) (n \u2208 N), p (r \u2022 n))\n  (H1 : \u2200 x y, p x \u2192 p y \u2192 p (x + y)) : p x :=\nbegin\n  have H0 : p 0 := by simpa only [zero_smul] using Hb 0 I.zero_mem 0 N.zero_mem,\n  refine submodule.supr_induction _ H _ H0 H1,\n  rintros \u27e8i, hi\u27e9 m \u27e8j, hj, (rfl : i \u2022 _ = m) \u27e9,\n  exact Hb _ hi _ hj,\nend\n\ntheorem mem_smul_span_singleton {I : ideal R} {m : M} {x : M} :\n  x \u2208 I \u2022 span R ({m} : set M) \u2194 \u2203 y \u2208 I, y \u2022 m = x :=\n\u27e8\u03bb hx, smul_induction_on hx\n  (\u03bb r hri n hnm,\n    let \u27e8s, hs\u27e9 := mem_span_singleton.1 hnm in \u27e8r * s, I.mul_mem_right _ hri, hs \u25b8 mul_smul r s m\u27e9)\n  (\u03bb m1 m2 \u27e8y1, hyi1, hy1\u27e9 \u27e8y2, hyi2, hy2\u27e9,\n    \u27e8y1 + y2, I.add_mem hyi1 hyi2, by rw [add_smul, hy1, hy2]\u27e9),\n\u03bb \u27e8y, hyi, hy\u27e9, hy \u25b8 smul_mem_smul hyi (subset_span $ set.mem_singleton m)\u27e9\n\ntheorem smul_le_right : I \u2022 N \u2264 N :=\nsmul_le.2 $ \u03bb r hr n, N.smul_mem r\n\ntheorem smul_mono (hij : I \u2264 J) (hnp : N \u2264 P) : I \u2022 N \u2264 J \u2022 P := map\u2082_le_map\u2082 hij hnp\n\ntheorem smul_mono_left (h : I \u2264 J) : I \u2022 N \u2264 J \u2022 N := map\u2082_le_map\u2082_left h\n\ntheorem smul_mono_right (h : N \u2264 P) : I \u2022 N \u2264 I \u2022 P := map\u2082_le_map\u2082_right h\n\nlemma map_le_smul_top (I : ideal R) (f : R \u2192\u2097[R] M) :\n  submodule.map f I \u2264 I \u2022 (\u22a4 : submodule R M) :=\nbegin\n  rintros _ \u27e8y, hy, rfl\u27e9,\n  rw [\u2190 mul_one y, \u2190 smul_eq_mul, f.map_smul],\n  exact smul_mem_smul hy mem_top\nend\n\n@[simp] theorem annihilator_smul (N : submodule R M) : annihilator N \u2022 N = \u22a5 :=\neq_bot_iff.2 (smul_le.2 (\u03bb r, mem_annihilator.1))\n\n@[simp] theorem annihilator_mul (I : ideal R) : annihilator I * I = \u22a5 :=\nannihilator_smul I\n\n@[simp] theorem mul_annihilator (I : ideal R) : I * annihilator I = \u22a5 :=\nby rw [mul_comm, annihilator_mul]\n\nvariables (I J N P)\n@[simp] theorem smul_bot : I \u2022 (\u22a5 : submodule R M) = \u22a5 := map\u2082_bot_right _ _\n\n@[simp] theorem bot_smul : (\u22a5 : ideal R) \u2022 N = \u22a5 := map\u2082_bot_left _ _\n\n@[simp] theorem top_smul : (\u22a4 : ideal R) \u2022 N = N :=\nle_antisymm smul_le_right $ \u03bb r hri, one_smul R r \u25b8 smul_mem_smul mem_top hri\n\ntheorem smul_sup : I \u2022 (N \u2294 P) = I \u2022 N \u2294 I \u2022 P := map\u2082_sup_right _ _ _ _\n\ntheorem sup_smul : (I \u2294 J) \u2022 N = I \u2022 N \u2294 J \u2022 N := map\u2082_sup_left _ _ _ _\n\nprotected theorem smul_assoc : (I \u2022 J) \u2022 N = I \u2022 (J \u2022 N) :=\nle_antisymm (smul_le.2 $ \u03bb rs hrsij t htn,\n  smul_induction_on hrsij\n  (\u03bb r hr s hs,\n    (@smul_eq_mul R _ r s).symm \u25b8 smul_smul r s t \u25b8 smul_mem_smul hr (smul_mem_smul hs htn))\n  (\u03bb x y, (add_smul x y t).symm \u25b8 submodule.add_mem _))\n(smul_le.2 $ \u03bb r hr sn hsn, suffices J \u2022 N \u2264 submodule.comap (r \u2022 linear_map.id) ((I \u2022 J) \u2022 N),\n  from this hsn,\nsmul_le.2 $ \u03bb s hs n hn, show r \u2022 (s \u2022 n) \u2208 (I \u2022 J) \u2022 N,\n  from mul_smul r s n \u25b8 smul_mem_smul (smul_mem_smul hr hs) hn)\n\nvariables (S : set R) (T : set M)\n\ntheorem span_smul_span : (ideal.span S) \u2022 (span R T) =\n  span R (\u22c3 (s \u2208 S) (t \u2208 T), {s \u2022 t}) :=\n(map\u2082_span_span _ _ _ _).trans $ congr_arg _ $ set.image2_eq_Union _ _ _\n\nlemma ideal_span_singleton_smul (r : R) (N : submodule R M) :\n  (ideal.span {r} : ideal R) \u2022 N = r \u2022 N :=\nbegin\n  have : span R (\u22c3 (t : M) (x : t \u2208 N), {r \u2022 t}) = r \u2022 N,\n  { convert span_eq _, exact (set.image_eq_Union _ (N : set M)).symm },\n  conv_lhs { rw [\u2190 span_eq N, span_smul_span] },\n  simpa\nend\n\nlemma span_smul_eq (r : R) (s : set M) : span R (r \u2022 s) = r \u2022 span R s :=\nby rw [\u2190 ideal_span_singleton_smul, span_smul_span, \u2190set.image2_eq_Union,\n    set.image2_singleton_left, set.image_smul]\n\nlemma mem_of_span_top_of_smul_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M) (H : \u2200 r : s, (r : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  suffices : (\u22a4 : ideal R) \u2022 (span R ({x} : set M)) \u2264 M',\n  { rw top_smul at this, exact this (subset_span (set.mem_singleton x)) },\n  rw [\u2190 hs, span_smul_span, span_le],\n  simpa using H\nend\n\n/-- Given `s`, a generating set of `R`, to check that an `x : M` falls in a\nsubmodule `M'` of `x`, we only need to show that `r ^ n \u2022 x \u2208 M'` for some `n` for each `r : s`. -/\nlemma mem_of_span_eq_top_of_smul_pow_mem (M' : submodule R M)\n  (s : set R) (hs : ideal.span s = \u22a4) (x : M)\n  (H : \u2200 r : s, \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M') : x \u2208 M' :=\nbegin\n  obtain \u27e8s', hs\u2081, hs\u2082\u27e9 := (ideal.span_eq_top_iff_finite _).mp hs,\n  replace H : \u2200 r : s', \u2203 (n : \u2115), (r ^ n : R) \u2022 x \u2208 M' := \u03bb r, H \u27e8_, hs\u2081 r.prop\u27e9,\n  choose n\u2081 n\u2082 using H,\n  let N := s'.attach.sup n\u2081,\n  have hs' := ideal.span_pow_eq_top (s' : set R) hs\u2082 N,\n  apply M'.mem_of_span_top_of_smul_mem _ hs',\n  rintro \u27e8_, r, hr, rfl\u27e9,\n  convert M'.smul_mem (r ^ (N - n\u2081 \u27e8r, hr\u27e9)) (n\u2082 \u27e8r, hr\u27e9) using 1,\n  simp only [subtype.coe_mk, smul_smul, \u2190 pow_add],\n  rw tsub_add_cancel_of_le (finset.le_sup (s'.mem_attach _) : n\u2081 \u27e8r, hr\u27e9 \u2264 N),\nend\n\nvariables {M' : Type w} [add_comm_monoid M'] [module R M']\n\ntheorem map_smul'' (f : M \u2192\u2097[R] M') : (I \u2022 N).map f = I \u2022 N.map f :=\nle_antisymm (map_le_iff_le_comap.2 $ smul_le.2 $ \u03bb r hr n hn, show f (r \u2022 n) \u2208 I \u2022 N.map f,\n    from (f.map_smul r n).symm \u25b8 smul_mem_smul hr (mem_map_of_mem hn)) $\nsmul_le.2 $ \u03bb r hr n hn, let \u27e8p, hp, hfp\u27e9 := mem_map.1 hn in\nhfp \u25b8 f.map_smul r p \u25b8 mem_map_of_mem (smul_mem_smul hr hp)\n\nvariables {I}\n\nlemma mem_smul_span {s : set M} {x : M} :\n  x \u2208 I \u2022 submodule.span R s \u2194 x \u2208 submodule.span R (\u22c3 (a \u2208 I) (b \u2208 s), ({a \u2022 b} : set M)) :=\nby rw [\u2190 I.span_eq, submodule.span_smul_span, I.span_eq]; refl\n\nvariables (I)\n\n/-- If `x` is an `I`-multiple of the submodule spanned by `f '' s`,\nthen we can write `x` as an `I`-linear combination of the elements of `f '' s`. -/\nlemma exists_sum_of_mem_ideal_smul_span {\u03b9 : Type*} (s : set \u03b9) (f : \u03b9 \u2192 M) (x : M)\n  (hx : x \u2208 I \u2022 span R (f '' s)) :\n  \u2203 (a : s \u2192\u2080 R) (ha : \u2200 i, a i \u2208 I), a.sum (\u03bb i c, c \u2022 f i) = x :=\nbegin\n  refine span_induction (mem_smul_span.mp hx) _ _ _ _,\n  { simp only [set.mem_Union, set.mem_range, set.mem_singleton_iff],\n    rintros x \u27e8y, hy, x, \u27e8i, hi, rfl\u27e9, rfl\u27e9,\n    refine \u27e8finsupp.single \u27e8i, hi\u27e9 y, \u03bb j, _, _\u27e9,\n    { letI := classical.dec_eq s,\n      rw finsupp.single_apply, split_ifs, { assumption }, { exact I.zero_mem } },\n    refine @finsupp.sum_single_index s R M _ _ \u27e8i, hi\u27e9 _ (\u03bb i y, y \u2022 f i) _,\n    simp },\n  { exact \u27e80, \u03bb i, I.zero_mem, finsupp.sum_zero_index\u27e9 },\n  { rintros x y \u27e8ax, hax, rfl\u27e9 \u27e8ay, hay, rfl\u27e9,\n    refine \u27e8ax + ay, \u03bb i, I.add_mem (hax i) (hay i), finsupp.sum_add_index _ _\u27e9;\n      intros; simp only [zero_smul, add_smul] },\n  { rintros c x \u27e8a, ha, rfl\u27e9,\n    refine \u27e8c \u2022 a, \u03bb i, I.mul_mem_left c (ha i), _\u27e9,\n    rw [finsupp.sum_smul_index, finsupp.smul_sum];\n      intros; simp only [zero_smul, mul_smul] },\nend\n\n@[simp] lemma smul_comap_le_comap_smul (f : M \u2192\u2097[R] M') (S : submodule R M') (I : ideal R) :\n  I \u2022 S.comap f \u2264 (I \u2022 S).comap f :=\nbegin\n  refine (submodule.smul_le.mpr (\u03bb r hr x hx, _)),\n  rw [submodule.mem_comap] at \u22a2 hx,\n  rw f.map_smul,\n  exact submodule.smul_mem_smul hr hx\nend\n\nend comm_semiring\n\nsection comm_ring\n\nvariables [comm_ring R] [add_comm_group M] [module R M]\nvariables {N N\u2081 N\u2082 P P\u2081 P\u2082 : submodule R M}\n\n/-- `N.colon P` is the ideal of all elements `r : R` such that `r \u2022 P \u2286 N`. -/\ndef colon (N P : submodule R M) : ideal R :=\nannihilator (P.map N.mkq)\n\ntheorem mem_colon {r} : r \u2208 N.colon P \u2194 \u2200 p \u2208 P, r \u2022 p \u2208 N :=\nmem_annihilator.trans \u27e8\u03bb H p hp, (quotient.mk_eq_zero N).1 (H (quotient.mk p) (mem_map_of_mem hp)),\n\u03bb H m \u27e8p, hp, hpm\u27e9, hpm \u25b8 (N.mkq).map_smul r p \u25b8 (quotient.mk_eq_zero N).2 $ H p hp\u27e9\n\ntheorem mem_colon' {r} : r \u2208 N.colon P \u2194 P \u2264 comap (r \u2022 linear_map.id) N :=\nmem_colon\n\ntheorem colon_mono (hn : N\u2081 \u2264 N\u2082) (hp : P\u2081 \u2264 P\u2082) : N\u2081.colon P\u2082 \u2264 N\u2082.colon P\u2081 :=\n\u03bb r hrnp, mem_colon.2 $ \u03bb p\u2081 hp\u2081, hn $ mem_colon.1 hrnp p\u2081 $ hp hp\u2081\n\ntheorem infi_colon_supr (\u03b9\u2081 : Sort w) (f : \u03b9\u2081 \u2192 submodule R M)\n  (\u03b9\u2082 : Sort x) (g : \u03b9\u2082 \u2192 submodule R M) :\n  (\u2a05 i, f i).colon (\u2a06 j, g j) = \u2a05 i j, (f i).colon (g j) :=\nle_antisymm (le_infi $ \u03bb i, le_infi $ \u03bb j, colon_mono (infi_le _ _) (le_supr _ _))\n(\u03bb r H, mem_colon'.2 $ supr_le $ \u03bb j, map_le_iff_le_comap.1 $ le_infi $ \u03bb i,\n  map_le_iff_le_comap.2 $ mem_colon'.1 $ have _ := ((mem_infi _).1 H i),\n  have _ := ((mem_infi _).1 this j), this)\n\nend comm_ring\n\nend submodule\n\nnamespace ideal\n\nsection mul_and_radical\nvariables {R : Type u} {\u03b9 : Type*} [comm_semiring R]\nvariables {I J K L : ideal R}\n\ninstance : has_mul (ideal R) := \u27e8(\u2022)\u27e9\n\n@[simp] lemma add_eq_sup : I + J = I \u2294 J := rfl\n@[simp] lemma zero_eq_bot : (0 : ideal R) = \u22a5 := rfl\n@[simp] lemma one_eq_top : (1 : ideal R) = \u22a4 :=\nby erw [submodule.one_eq_range, linear_map.range_id]\n\ntheorem mul_mem_mul {r s} (hr : r \u2208 I) (hs : s \u2208 J) : r * s \u2208 I * J :=\nsubmodule.smul_mem_smul hr hs\n\ntheorem mul_mem_mul_rev {r s} (hr : r \u2208 I) (hs : s \u2208 J) : s * r \u2208 I * J :=\nmul_comm r s \u25b8 mul_mem_mul hr hs\n\nlemma pow_mem_pow {x : R} (hx : x \u2208 I) (n : \u2115) : x ^ n \u2208 I ^ n :=\nbegin\n  induction n with n ih, { simp only [pow_zero, ideal.one_eq_top], },\n  simpa only [pow_succ] using mul_mem_mul hx ih,\nend\n\nlemma prod_mem_prod {\u03b9 : Type*} {s : finset \u03b9} {I : \u03b9 \u2192 ideal R} {x : \u03b9 \u2192 R} :\n  (\u2200 i \u2208 s, x i \u2208 I i) \u2192 \u220f i in s, x i \u2208 \u220f i in s, I i :=\nbegin\n  classical,\n  apply finset.induction_on s,\n  { intro _, rw [finset.prod_empty, finset.prod_empty, one_eq_top], exact submodule.mem_top },\n  { intros a s ha IH h,\n    rw [finset.prod_insert ha, finset.prod_insert ha],\n    exact mul_mem_mul (h a $ finset.mem_insert_self a s)\n      (IH $ \u03bb i hi, h i $ finset.mem_insert_of_mem hi) }\nend\n\ntheorem mul_le : I * J \u2264 K \u2194 \u2200 (r \u2208 I) (s \u2208 J), r * s \u2208 K :=\nsubmodule.smul_le\n\nlemma mul_le_left : I * J \u2264 J :=\nideal.mul_le.2 (\u03bb r hr s, J.mul_mem_left _)\n\nlemma mul_le_right : I * J \u2264 I :=\nideal.mul_le.2 (\u03bb r hr s hs, I.mul_mem_right _ hr)\n\n@[simp] lemma sup_mul_right_self : I \u2294 (I * J) = I :=\nsup_eq_left.2 ideal.mul_le_right\n\n@[simp] lemma sup_mul_left_self : I \u2294 (J * I) = I :=\nsup_eq_left.2 ideal.mul_le_left\n\n@[simp] lemma mul_right_self_sup : (I * J) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_right\n\n@[simp] lemma mul_left_self_sup : (J * I) \u2294 I = I :=\nsup_eq_right.2 ideal.mul_le_left\n\nvariables (I J K)\nprotected theorem mul_comm : I * J = J * I :=\nle_antisymm (mul_le.2 $ \u03bb r hrI s hsJ, mul_mem_mul_rev hsJ hrI)\n  (mul_le.2 $ \u03bb r hrJ s hsI, mul_mem_mul_rev hsI hrJ)\n\nprotected theorem mul_assoc : (I * J) * K = I * (J * K) :=\nsubmodule.smul_assoc I J K\n\ntheorem span_mul_span (S T : set R) : span S * span T =\n  span \u22c3 (s \u2208 S) (t \u2208 T), {s * t} :=\nsubmodule.span_smul_span S T\nvariables {I J K}\n\nlemma span_mul_span' (S T : set R) : span S * span T = span (S*T) :=\nby { unfold span, rw submodule.span_mul_span, }\n\nlemma span_singleton_mul_span_singleton (r s : R) :\n  span {r} * span {s} = (span {r * s} : ideal R) :=\nby { unfold span, rw [submodule.span_mul_span, set.singleton_mul_singleton], }\n\nlemma span_singleton_pow (s : R) (n : \u2115):\n  span {s} ^ n = (span {s ^ n} : ideal R) :=\nbegin\n  induction n with n ih, { simp [set.singleton_one], },\n  simp only [pow_succ, ih, span_singleton_mul_span_singleton],\nend\n\nlemma mem_mul_span_singleton {x y : R} {I : ideal R} :\n  x \u2208 I * span {y} \u2194 \u2203 z \u2208 I, z * y = x :=\nsubmodule.mem_smul_span_singleton\n\nlemma mem_span_singleton_mul {x y : R} {I : ideal R} :\n  x \u2208 span {y} * I \u2194 \u2203 z \u2208 I, y * z = x :=\nby simp only [mul_comm, mem_mul_span_singleton]\n\nlemma le_span_singleton_mul_iff {x : R} {I J : ideal R} :\n  I \u2264 span {x} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI :=\nshow (\u2200 {zI} (hzI : zI \u2208 I), zI \u2208 span {x} * J) \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI,\nby simp only [mem_span_singleton_mul]\n\nlemma span_singleton_mul_le_iff {x : R} {I J : ideal R} :\n  span {x} * I \u2264 J \u2194 \u2200 z \u2208 I, x * z \u2208 J :=\nbegin\n  simp only [mul_le, mem_span_singleton_mul, mem_span_singleton],\n  split,\n  { intros h zI hzI,\n    exact h x (dvd_refl x) zI hzI },\n  { rintros h _ \u27e8z, rfl\u27e9 zI hzI,\n    rw [mul_comm x z, mul_assoc],\n    exact J.mul_mem_left _ (h zI hzI) },\nend\n\nlemma span_singleton_mul_le_span_singleton_mul {x y : R} {I J : ideal R} :\n  span {x} * I \u2264 span {y} * J \u2194 \u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ :=\nby simp only [span_singleton_mul_le_iff, mem_span_singleton_mul, eq_comm]\n\nlemma eq_span_singleton_mul {x : R} (I J : ideal R) :\n  I = span {x} * J \u2194 ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zJ = zI) \u2227 (\u2200 z \u2208 J, x * z \u2208 I)) :=\nby simp only [le_antisymm_iff, le_span_singleton_mul_iff, span_singleton_mul_le_iff]\n\nlemma span_singleton_mul_eq_span_singleton_mul {x y : R} (I J : ideal R) :\n  span {x} * I = span {y} * J \u2194\n    ((\u2200 zI \u2208 I, \u2203 zJ \u2208 J, x * zI = y * zJ) \u2227\n     (\u2200 zJ \u2208 J, \u2203 zI \u2208 I, x * zI = y * zJ)) :=\nby simp only [le_antisymm_iff, span_singleton_mul_le_span_singleton_mul, eq_comm]\n\nlemma prod_span {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 set R) :\n  (\u220f i in s, ideal.span (I i)) = ideal.span (\u220f i in s, I i) :=\nsubmodule.prod_span s I\n\nlemma prod_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R) :\n  (\u220f i in s, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nsubmodule.prod_span_singleton s I\n\nlemma finset_inf_span_singleton {\u03b9 : Type*} (s : finset \u03b9) (I : \u03b9 \u2192 R)\n  (hI : set.pairwise \u2191s (is_coprime on I)) :\n  (s.inf $ \u03bb i, ideal.span ({I i} : set R)) = ideal.span {\u220f i in s, I i} :=\nbegin\n  ext x,\n  simp only [submodule.mem_finset_inf, ideal.mem_span_singleton],\n  exact \u27e8finset.prod_dvd_of_coprime hI,\n    \u03bb h i hi, (finset.dvd_prod_of_mem _ hi).trans h\u27e9\nend\n\nlemma infi_span_singleton {\u03b9 : Type*} [fintype \u03b9] (I : \u03b9 \u2192 R)\n  (hI : \u2200 i j (hij : i \u2260 j), is_coprime (I i) (I j)):\n  (\u2a05 i, ideal.span ({I i} : set R)) = ideal.span {\u220f i, I i} :=\nbegin\n  rw [\u2190 finset.inf_univ_eq_infi, finset_inf_span_singleton],\n  rwa [finset.coe_univ, set.pairwise_univ]\nend\n\nlemma sup_eq_top_iff_is_coprime {R : Type*} [comm_semiring R] (x y : R) :\n  span ({x} : set R) \u2294 span {y} = \u22a4 \u2194 is_coprime x y :=\nbegin\n  rw [eq_top_iff_one, submodule.mem_sup],\n  split,\n  { rintro \u27e8u, hu, v, hv, h1\u27e9,\n    rw mem_span_singleton' at hu hv,\n    rw [\u2190 hu.some_spec, \u2190 hv.some_spec] at h1,\n    exact \u27e8_, _, h1\u27e9 },\n  { exact \u03bb \u27e8u, v, h1\u27e9,\n      \u27e8_, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, _, mem_span_singleton'.mpr \u27e8_, rfl\u27e9, h1\u27e9 },\nend\n\ntheorem mul_le_inf : I * J \u2264 I \u2293 J :=\nmul_le.2 $ \u03bb r hri s hsj, \u27e8I.mul_mem_right s hri, J.mul_mem_left r hsj\u27e9\n\ntheorem multiset_prod_le_inf {s : multiset (ideal R)} :\n  s.prod \u2264 s.inf :=\nbegin\n  classical, refine s.induction_on _ _,\n  { rw [multiset.inf_zero], exact le_top },\n  intros a s ih,\n  rw [multiset.prod_cons, multiset.inf_cons],\n  exact le_trans mul_le_inf (inf_le_inf le_rfl ih)\nend\n\ntheorem prod_le_inf {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} : s.prod f \u2264 s.inf f :=\nmultiset_prod_le_inf\n\ntheorem mul_eq_inf_of_coprime (h : I \u2294 J = \u22a4) : I * J = I \u2293 J :=\nle_antisymm mul_le_inf $ \u03bb r \u27e8hri, hrj\u27e9,\nlet \u27e8s, hsi, t, htj, hst\u27e9 := submodule.mem_sup.1 ((eq_top_iff_one _).1 h) in\nmul_one r \u25b8 hst \u25b8 (mul_add r s t).symm \u25b8 ideal.add_mem (I * J) (mul_mem_mul_rev hsi hrj)\n  (mul_mem_mul hri htj)\n\nlemma sup_mul_eq_of_coprime_left (h : I \u2294 J = \u22a4) : I \u2294 (J * K) = I \u2294 K :=\nle_antisymm (sup_le_sup_left mul_le_left _) $ \u03bb i hi,\nbegin\n  rw eq_top_iff_one at h, rw submodule.mem_sup at h hi \u22a2,\n  obtain \u27e8i1, hi1, j, hj, h\u27e9 := h, obtain \u27e8i', hi', k, hk, hi\u27e9 := hi,\n  refine \u27e8_, add_mem hi' (mul_mem_right k _ hi1), _, mul_mem_mul hj hk, _\u27e9,\n  rw [add_assoc, \u2190 add_mul, h, one_mul, hi]\nend\n\nlemma sup_mul_eq_of_coprime_right (h : I \u2294 K = \u22a4) : I \u2294 (J * K) = I \u2294 J :=\nby { rw mul_comm, exact sup_mul_eq_of_coprime_left h }\n\nlemma mul_sup_eq_of_coprime_left (h : I \u2294 J = \u22a4) : (I * K) \u2294 J = K \u2294 J :=\nby { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_left h, sup_comm] }\n\nlemma mul_sup_eq_of_coprime_right (h : K \u2294 J = \u22a4) : (I * K) \u2294 J = I \u2294 J :=\nby { rw sup_comm at h, rw [sup_comm, sup_mul_eq_of_coprime_right h, sup_comm] }\n\nlemma sup_prod_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 I \u2294 J i = \u22a4) :\n  I \u2294 \u220f i in s, J i = \u22a4 :=\nfinset.prod_induction _ (\u03bb J, I \u2294 J = \u22a4) (\u03bb J K hJ hK, (sup_mul_eq_of_coprime_left hJ).trans hK)\n(by rw [one_eq_top, sup_top_eq]) h\n\nlemma sup_infi_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 I \u2294 J i = \u22a4) :\n  I \u2294 (\u2a05 i \u2208 s, J i) = \u22a4 :=\neq_top_iff.mpr $ le_of_eq_of_le (sup_prod_eq_top h).symm $ sup_le_sup_left\n  (le_of_le_of_eq prod_le_inf $ finset.inf_eq_infi _ _) _\n\nlemma prod_sup_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 J i \u2294 I = \u22a4) :\n  (\u220f i in s, J i) \u2294 I = \u22a4 :=\nsup_comm.trans (sup_prod_eq_top $ \u03bb i hi, sup_comm.trans $ h i hi)\n\nlemma infi_sup_eq_top {s : finset \u03b9} {J : \u03b9 \u2192 ideal R} (h : \u2200 i, i \u2208 s \u2192 J i \u2294 I = \u22a4) :\n  (\u2a05 i \u2208 s, J i) \u2294 I = \u22a4 :=\nsup_comm.trans (sup_infi_eq_top $ \u03bb i hi, sup_comm.trans $ h i hi)\n\nlemma sup_pow_eq_top {n : \u2115} (h : I \u2294 J = \u22a4) : I \u2294 (J ^ n) = \u22a4 :=\nby { rw [\u2190 finset.card_range n, \u2190 finset.prod_const], exact sup_prod_eq_top (\u03bb _ _, h) }\n\nlemma pow_sup_eq_top {n : \u2115} (h : I \u2294 J = \u22a4) : (I ^ n) \u2294 J = \u22a4 :=\nby { rw [\u2190 finset.card_range n, \u2190 finset.prod_const], exact prod_sup_eq_top (\u03bb _ _, h) }\n\nlemma pow_sup_pow_eq_top {m n : \u2115} (h : I \u2294 J = \u22a4) : (I ^ m) \u2294 (J ^ n) = \u22a4 :=\nsup_pow_eq_top (pow_sup_eq_top h)\n\nvariables (I)\n@[simp] theorem mul_bot : I * \u22a5 = \u22a5 :=\nsubmodule.smul_bot I\n\n@[simp] theorem bot_mul : \u22a5 * I = \u22a5 :=\nsubmodule.bot_smul I\n\n@[simp] theorem mul_top : I * \u22a4 = I :=\nideal.mul_comm \u22a4 I \u25b8 submodule.top_smul I\n\n@[simp] theorem top_mul : \u22a4 * I = I :=\nsubmodule.top_smul I\nvariables {I}\n\ntheorem mul_mono (hik : I \u2264 K) (hjl : J \u2264 L) : I * J \u2264 K * L :=\nsubmodule.smul_mono hik hjl\n\ntheorem mul_mono_left (h : I \u2264 J) : I * K \u2264 J * K :=\nsubmodule.smul_mono_left h\n\ntheorem mul_mono_right (h : J \u2264 K) : I * J \u2264 I * K :=\nsubmodule.smul_mono_right h\n\nvariables (I J K)\ntheorem mul_sup : I * (J \u2294 K) = I * J \u2294 I * K :=\nsubmodule.smul_sup I J K\n\ntheorem sup_mul : (I \u2294 J) * K = I * K \u2294 J * K :=\nsubmodule.sup_smul I J K\nvariables {I J K}\n\nlemma pow_le_pow {m n : \u2115} (h : m \u2264 n) :\n  I^n \u2264 I^m :=\nbegin\n  cases nat.exists_eq_add_of_le h with k hk,\n  rw [hk, pow_add],\n  exact le_trans (mul_le_inf) (inf_le_left)\nend\n\nlemma pow_le_self {n : \u2115} (hn : n \u2260 0) : I^n \u2264 I :=\ncalc I^n \u2264 I ^ 1 : pow_le_pow (nat.pos_of_ne_zero hn)\n     ... = I : pow_one _\n\nlemma mul_eq_bot {R : Type*} [comm_ring R] [is_domain R] {I J : ideal R} :\n  I * J = \u22a5 \u2194 I = \u22a5 \u2228 J = \u22a5 :=\n\u27e8\u03bb hij, or_iff_not_imp_left.mpr (\u03bb I_ne_bot, J.eq_bot_iff.mpr (\u03bb j hj,\n  let \u27e8i, hi, ne0\u27e9 := I.ne_bot_iff.mp I_ne_bot in\n    or.resolve_left (mul_eq_zero.mp ((I * J).eq_bot_iff.mp hij _ (mul_mem_mul hi hj))) ne0)),\n \u03bb h, by cases h; rw [\u2190 ideal.mul_bot, h, ideal.mul_comm]\u27e9\n\ninstance {R : Type*} [comm_ring R] [is_domain R] : no_zero_divisors (ideal R) :=\n{ eq_zero_or_eq_zero_of_mul_eq_zero := \u03bb I J, mul_eq_bot.1 }\n\n/-- A product of ideals in an integral domain is zero if and only if one of the terms is zero. -/\nlemma prod_eq_bot {R : Type*} [comm_ring R] [is_domain R]\n  {s : multiset (ideal R)} : s.prod = \u22a5 \u2194 \u2203 I \u2208 s, I = \u22a5 :=\nprod_zero_iff_exists_zero\n\n/-- The radical of an ideal `I` consists of the elements `r` such that `r^n \u2208 I` for some `n`. -/\ndef radical (I : ideal R) : ideal R :=\n{ carrier := { r | \u2203 n : \u2115, r ^ n \u2208 I },\n  zero_mem' := \u27e81, (pow_one (0:R)).symm \u25b8 I.zero_mem\u27e9,\n  add_mem' := \u03bb x y \u27e8m, hxmi\u27e9 \u27e8n, hyni\u27e9, \u27e8m + n,\n    (add_pow x y (m + n)).symm \u25b8 I.sum_mem $\n    show \u2200 c \u2208 finset.range (nat.succ (m + n)),\n      x ^ c * y ^ (m + n - c) * (nat.choose (m + n) c) \u2208 I,\n    from \u03bb c hc, or.cases_on (le_total c m)\n      (\u03bb hcm, I.mul_mem_right _ $ I.mul_mem_left _ $ nat.add_comm n m \u25b8\n        (add_tsub_assoc_of_le hcm n).symm \u25b8\n        (pow_add y n (m-c)).symm \u25b8 I.mul_mem_right _ hyni)\n      (\u03bb hmc, I.mul_mem_right _ $ I.mul_mem_right _ $ add_tsub_cancel_of_le hmc \u25b8\n        (pow_add x m (c-m)).symm \u25b8 I.mul_mem_right _ hxmi)\u27e9,\n  smul_mem' := \u03bb r s \u27e8n, hsni\u27e9, \u27e8n, (mul_pow r s n).symm \u25b8 I.mul_mem_left (r^n) hsni\u27e9 }\n\ntheorem le_radical : I \u2264 radical I :=\n\u03bb r hri, \u27e81, (pow_one r).symm \u25b8 hri\u27e9\n\nvariables (R)\ntheorem radical_top : (radical \u22a4 : ideal R) = \u22a4 :=\n(eq_top_iff_one _).2 \u27e80, submodule.mem_top\u27e9\nvariables {R}\n\ntheorem radical_mono (H : I \u2264 J) : radical I \u2264 radical J :=\n\u03bb r \u27e8n, hrni\u27e9, \u27e8n, H hrni\u27e9\n\nvariables (I)\n@[simp] theorem radical_idem : radical (radical I) = radical I :=\nle_antisymm (\u03bb r \u27e8n, k, hrnki\u27e9, \u27e8n * k, (pow_mul r n k).symm \u25b8 hrnki\u27e9) le_radical\nvariables {I}\n\ntheorem radical_le_radical_iff : radical I \u2264 radical J \u2194 I \u2264 radical J :=\n\u27e8\u03bb h, le_trans le_radical h, \u03bb h, radical_idem J \u25b8 radical_mono h\u27e9\n\ntheorem radical_eq_top : radical I = \u22a4 \u2194 I = \u22a4 :=\n\u27e8\u03bb h, (eq_top_iff_one _).2 $ let \u27e8n, hn\u27e9 := (eq_top_iff_one _).1 h in\n  @one_pow R _ n \u25b8 hn, \u03bb h, h.symm \u25b8 radical_top R\u27e9\n\ntheorem is_prime.radical (H : is_prime I) : radical I = I :=\nle_antisymm (\u03bb r \u27e8n, hrni\u27e9, H.mem_of_pow_mem n hrni) le_radical\n\nvariables (I J)\ntheorem radical_sup : radical (I \u2294 J) = radical (radical I \u2294 radical J) :=\nle_antisymm (radical_mono $ sup_le_sup le_radical le_radical) $\n\u03bb r \u27e8n, hrnij\u27e9, let \u27e8s, hs, t, ht, hst\u27e9 := submodule.mem_sup.1 hrnij in\n@radical_idem _ _ (I \u2294 J) \u25b8 \u27e8n, hst \u25b8 ideal.add_mem _\n  (radical_mono le_sup_left hs) (radical_mono le_sup_right ht)\u27e9\n\ntheorem radical_inf : radical (I \u2293 J) = radical I \u2293 radical J :=\nle_antisymm (le_inf (radical_mono inf_le_left) (radical_mono inf_le_right))\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 I.mul_mem_right _ hrm,\n(pow_add r m n).symm \u25b8 J.mul_mem_left _ hrn\u27e9)\n\ntheorem radical_mul : radical (I * J) = radical I \u2293 radical J :=\nle_antisymm (radical_inf I J \u25b8 radical_mono $ @mul_le_inf _ _ I J)\n(\u03bb r \u27e8\u27e8m, hrm\u27e9, \u27e8n, hrn\u27e9\u27e9, \u27e8m + n, (pow_add r m n).symm \u25b8 mul_mem_mul hrm hrn\u27e9)\nvariables {I J}\n\ntheorem is_prime.radical_le_iff (hj : is_prime J) :\n  radical I \u2264 J \u2194 I \u2264 J :=\n\u27e8le_trans le_radical, \u03bb hij r \u27e8n, hrni\u27e9, hj.mem_of_pow_mem n $ hij hrni\u27e9\n\ntheorem radical_eq_Inf (I : ideal R) :\n  radical I = Inf { J : ideal R | I \u2264 J \u2227 is_prime J } :=\nle_antisymm (le_Inf $ \u03bb J hJ, hJ.2.radical_le_iff.2 hJ.1) $\n\u03bb r hr, classical.by_contradiction $ \u03bb hri,\nlet \u27e8m, (hrm : r \u2209 radical m), him, hm\u27e9 := zorn_nonempty_partial_order\u2080\n  {K : ideal R | r \u2209 radical K}\n  (\u03bb c hc hcc y hyc, \u27e8Sup c, \u03bb \u27e8n, hrnc\u27e9, let \u27e8y, hyc, hrny\u27e9 :=\n      (submodule.mem_Sup_of_directed \u27e8y, hyc\u27e9 hcc.directed_on).1 hrnc in hc hyc \u27e8n, hrny\u27e9,\n    \u03bb z, le_Sup\u27e9) I hri in\nhave \u2200 x \u2209 m, r \u2208 radical (m \u2294 span {x}) := \u03bb x hxm, classical.by_contradiction $ \u03bb hrmx, hxm $\n  hm (m \u2294 span {x}) hrmx le_sup_left \u25b8 (le_sup_right : _ \u2264 m \u2294 span {x})\n    (subset_span $ set.mem_singleton _),\nhave is_prime m, from \u27e8by rintro rfl; rw radical_top at hrm; exact hrm trivial,\n  \u03bb x y hxym, or_iff_not_imp_left.2 $ \u03bb hxm, classical.by_contradiction $ \u03bb hym,\n  let \u27e8n, hrn\u27e9 := this _ hxm,\n      \u27e8p, hpm, q, hq, hpqrn\u27e9 := submodule.mem_sup.1 hrn,\n      \u27e8c, hcxq\u27e9 := mem_span_singleton'.1 hq in\n  let \u27e8k, hrk\u27e9 := this _ hym,\n      \u27e8f, hfm, g, hg, hfgrk\u27e9 := submodule.mem_sup.1 hrk,\n      \u27e8d, hdyg\u27e9 := mem_span_singleton'.1 hg in\n  hrm \u27e8n + k, by rw [pow_add, \u2190 hpqrn, \u2190 hcxq, \u2190 hfgrk, \u2190 hdyg, add_mul, mul_add (c*x),\n                     mul_assoc c x (d*y), mul_left_comm x, \u2190 mul_assoc];\n    refine m.add_mem (m.mul_mem_right _ hpm) (m.add_mem (m.mul_mem_left _ hfm)\n      (m.mul_mem_left _ hxym))\u27e9\u27e9,\nhrm $ this.radical.symm \u25b8 (Inf_le \u27e8him, this\u27e9 : Inf {J : ideal R | I \u2264 J \u2227 is_prime J} \u2264 m) hr\n\n@[simp] lemma radical_bot_of_is_domain {R : Type u} [comm_ring R] [is_domain R] :\n  radical (\u22a5 : ideal R) = \u22a5 :=\neq_bot_iff.2 (\u03bb x hx, hx.rec_on (\u03bb n hn, pow_eq_zero hn))\n\ninstance : comm_semiring (ideal R) := submodule.comm_semiring\n\nvariables (R)\ntheorem top_pow (n : \u2115) : (\u22a4 ^ n : ideal R) = \u22a4 :=\nnat.rec_on n one_eq_top $ \u03bb n ih, by rw [pow_succ, ih, top_mul]\nvariables {R}\n\nvariables (I)\ntheorem radical_pow (n : \u2115) (H : n > 0) : radical (I^n) = radical I :=\nnat.rec_on n (not.elim dec_trivial) (\u03bb n ih H,\nor.cases_on (lt_or_eq_of_le $ nat.le_of_lt_succ H)\n  (\u03bb H, calc radical (I^(n+1))\n           = radical I \u2293 radical (I^n) : by { rw pow_succ, exact radical_mul _ _ }\n       ... = radical I \u2293 radical I : by rw ih H\n       ... = radical I : inf_idem)\n  (\u03bb H, H \u25b8 (pow_one I).symm \u25b8 rfl)) H\n\ntheorem is_prime.mul_le {I J P : ideal R} (hp : is_prime P) :\n  I * J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hip j hj, let \u27e8i, hi, hip\u27e9 := set.not_subset.1 hip in\n  (hp.mem_or_mem $ h $ mul_mem_mul hi hj).resolve_left hip,\n\u03bb h, or.cases_on h (le_trans $ le_trans mul_le_inf inf_le_left)\n  (le_trans $ le_trans mul_le_inf inf_le_right)\u27e9\n\ntheorem is_prime.inf_le {I J P : ideal R} (hp : is_prime P) :\n  I \u2293 J \u2264 P \u2194 I \u2264 P \u2228 J \u2264 P :=\n\u27e8\u03bb h, hp.mul_le.1 $ le_trans mul_le_inf h,\n\u03bb h, or.cases_on h (le_trans inf_le_left) (le_trans inf_le_right)\u27e9\n\ntheorem is_prime.multiset_prod_le {s : multiset (ideal R)} {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  s.prod \u2264 P \u2194 \u2203 I \u2208 s, I \u2264 P :=\nsuffices s.prod \u2264 P \u2192 \u2203 I \u2208 s, I \u2264 P,\n  from \u27e8this, \u03bb \u27e8i, his, hip\u27e9, le_trans multiset_prod_le_inf $\n    le_trans (multiset.inf_le his) hip\u27e9,\nbegin\n  classical,\n  obtain \u27e8b, hb\u27e9 : \u2203 b, b \u2208 s := multiset.exists_mem_of_ne_zero hne,\n  obtain \u27e8t, rfl\u27e9 : \u2203 t, s = b ::\u2098 t,\n  from \u27e8s.erase b, (multiset.cons_erase hb).symm\u27e9,\n  refine t.induction_on _ _,\n  { simp only [exists_prop, \u2190multiset.singleton_eq_cons, multiset.prod_singleton,\n      multiset.mem_singleton, exists_eq_left, imp_self] },\n  intros a s ih h,\n  rw [multiset.cons_swap, multiset.prod_cons, hp.mul_le] at h,\n  rw multiset.cons_swap,\n  cases h,\n  { exact \u27e8a, multiset.mem_cons_self a _, h\u27e9 },\n  obtain \u27e8I, hI, ih\u27e9 : \u2203 I \u2208 b ::\u2098 s, I \u2264 P := ih h,\n  exact \u27e8I, multiset.mem_cons_of_mem hI, ih\u27e9\nend\n\ntheorem is_prime.multiset_prod_map_le {s : multiset \u03b9} (f : \u03b9 \u2192 ideal R) {P : ideal R}\n  (hp : is_prime P) (hne : s \u2260 0) :\n  (s.map f).prod \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nbegin\n  rw hp.multiset_prod_le (mt multiset.map_eq_zero.mp hne),\n  simp_rw [exists_prop, multiset.mem_map, exists_exists_and_eq_and],\nend\n\ntheorem is_prime.prod_le {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R}\n  (hp : is_prime P) (hne : s.nonempty) :\n  s.prod f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\nhp.multiset_prod_map_le f (mt finset.val_eq_zero.mp hne.ne_empty)\n\ntheorem is_prime.inf_le' {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {P : ideal R} (hp : is_prime P)\n  (hsne: s.nonempty) :\n  s.inf f \u2264 P \u2194 \u2203 i \u2208 s, f i \u2264 P :=\n\u27e8\u03bb h, (hp.prod_le hsne).1 $ le_trans prod_le_inf h,\n  \u03bb \u27e8i, his, hip\u27e9, le_trans (finset.inf_le his) hip\u27e9\n\ntheorem subset_union {R : Type u} [comm_ring R] {I J K : ideal R} :\n  (I : set R) \u2286 J \u222a K \u2194 I \u2264 J \u2228 I \u2264 K :=\n\u27e8\u03bb h, or_iff_not_imp_left.2 $ \u03bb hij s hsi,\n  let \u27e8r, hri, hrj\u27e9 := set.not_subset.1 hij in classical.by_contradiction $ \u03bb hsk,\n  or.cases_on (h $ I.add_mem hri hsi)\n    (\u03bb hj, hrj $ add_sub_cancel r s \u25b8 J.sub_mem hj ((h hsi).resolve_right hsk))\n    (\u03bb hk, hsk $ add_sub_cancel' r s \u25b8 K.sub_mem hk ((h hri).resolve_left hrj)),\n\u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset_union_left J K)\n  (\u03bb h, set.subset.trans h $ set.subset_union_right J K)\u27e9\n\ntheorem subset_union_prime' {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} {a b : \u03b9}\n  (hp : \u2200 i \u2208 s, is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192\n  I \u2264 f a \u2228 I \u2264 f b \u2228 \u2203 i \u2208 s, I \u2264 f i,\n  from \u27e8this, \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_left _ _) (set.subset_union_left _ _)) $\n    \u03bb h, or.cases_on h (\u03bb h, set.subset.trans h $ set.subset.trans\n      (set.subset_union_right _ _) (set.subset_union_left _ _)) $\n    \u03bb \u27e8i, his, hi\u27e9, by refine (set.subset.trans hi $ set.subset.trans _ $\n        set.subset_union_right _ _);\n      exact set.subset_bUnion_of_mem (finset.mem_coe.2 his)\u27e9,\nbegin\n  generalize hn : s.card = n, intros h,\n  unfreezingI { induction n with n ih generalizing a b s },\n  { clear hp,\n    rw finset.card_eq_zero at hn, subst hn,\n    rw [finset.coe_empty, set.bUnion_empty, set.union_empty, subset_union] at h,\n    simpa only [exists_prop, finset.not_mem_empty, false_and, exists_false, or_false] },\n  classical,\n  replace hn : \u2203 (i : \u03b9) (t : finset \u03b9), i \u2209 t \u2227 insert i t = s \u2227 t.card = n :=\n  finset.card_eq_succ.1 hn,\n  unfreezingI { rcases hn with \u27e8i, t, hit, rfl, hn\u27e9 },\n  replace hp : is_prime (f i) \u2227 \u2200 x \u2208 t, is_prime (f x) := (t.forall_mem_insert _ _).1 hp,\n  by_cases Ht : \u2203 j \u2208 t, f j \u2264 f i,\n  { obtain \u27e8j, hjt, hfji\u27e9 : \u2203 j \u2208 t, f j \u2264 f i := Ht,\n    obtain \u27e8u, hju, rfl\u27e9 : \u2203 u, j \u2209 u \u2227 insert j u = t,\n    { exact \u27e8t.erase j, t.not_mem_erase j, finset.insert_erase hjt\u27e9 },\n    have hp' : \u2200 k \u2208 insert i u, is_prime (f k),\n    { rw finset.forall_mem_insert at hp \u22a2, exact \u27e8hp.1, hp.2.2\u27e9 },\n    have hiu : i \u2209 u := mt finset.mem_insert_of_mem hit,\n    have hn' : (insert i u).card = n,\n    { rwa finset.card_insert_of_not_mem at hn \u22a2, exacts [hiu, hju] },\n    have h' : (I : set R) \u2286 f a \u222a f b \u222a (\u22c3 k \u2208 (\u2191(insert i u) : set \u03b9), f k),\n    { rw finset.coe_insert at h \u22a2, rw finset.coe_insert at h,\n      simp only [set.bUnion_insert] at h \u22a2,\n      rw [\u2190 set.union_assoc \u2191(f i)] at h,\n      erw [set.union_eq_self_of_subset_right hfji] at h,\n      exact h },\n    specialize @ih a b (insert i u) hp' hn' h',\n    refine ih.imp id (or.imp id (exists_imp_exists $ \u03bb k, _)), simp only [exists_prop],\n    exact and.imp (\u03bb hk, finset.insert_subset_insert i (finset.subset_insert j u) hk) id },\n  by_cases Ha : f a \u2264 f i,\n  { have h' : (I : set R) \u2286 f i \u222a f b \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc,\n          set.union_right_comm \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Ha] at h,\n      exact h },\n    specialize @ih i b t hp.2 hn h', right,\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9 },\n    { exact or.inl ih },\n    { exact or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  by_cases Hb : f b \u2264 f i,\n  { have h' : (I : set R) \u2286 f a \u222a f i \u222a (\u22c3 j \u2208 (\u2191t : set \u03b9), f j),\n    { rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_assoc, set.union_assoc \u2191(f a)] at h,\n      erw [set.union_eq_self_of_subset_left Hb] at h,\n      exact h },\n    specialize @ih a i t hp.2 hn h',\n    rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { exact or.inl ih },\n    { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, ih\u27e9) },\n    { exact or.inr (or.inr \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9) } },\n  by_cases Hi : I \u2264 f i,\n  { exact or.inr (or.inr \u27e8i, finset.mem_insert_self i t, Hi\u27e9) },\n  have : \u00acI \u2293 f a \u2293 f b \u2293 t.inf f \u2264 f i,\n  { rcases t.eq_empty_or_nonempty with (rfl | hsne),\n    { rw [finset.inf_empty, inf_top_eq, hp.1.inf_le, hp.1.inf_le, not_or_distrib, not_or_distrib],\n      exact \u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9 },\n    simp only [hp.1.inf_le, hp.1.inf_le' hsne, not_or_distrib],\n    exact \u27e8\u27e8\u27e8Hi, Ha\u27e9, Hb\u27e9, Ht\u27e9 },\n  rcases set.not_subset.1 this with \u27e8r, \u27e8\u27e8\u27e8hrI, hra\u27e9, hrb\u27e9, hr\u27e9, hri\u27e9,\n  by_cases HI : (I : set R) \u2286 f a \u222a f b \u222a \u22c3 j \u2208 (\u2191t : set \u03b9), f j,\n  { specialize ih hp.2 hn HI, rcases ih with ih | ih | \u27e8k, hkt, ih\u27e9,\n    { left, exact ih }, { right, left, exact ih },\n    { right, right, exact \u27e8k, finset.mem_insert_of_mem hkt, ih\u27e9 } },\n  exfalso, rcases set.not_subset.1 HI with \u27e8s, hsI, hs\u27e9,\n  rw [finset.coe_insert, set.bUnion_insert] at h,\n  have hsi : s \u2208 f i := ((h hsI).resolve_left (mt or.inl hs)).resolve_right (mt or.inr hs),\n  rcases h (I.add_mem hrI hsI) with \u27e8ha | hb\u27e9 | hi | ht,\n  { exact hs (or.inl $ or.inl $ add_sub_cancel' r s \u25b8 (f a).sub_mem ha hra) },\n  { exact hs (or.inl $ or.inr $ add_sub_cancel' r s \u25b8 (f b).sub_mem hb hrb) },\n  { exact hri (add_sub_cancel r s \u25b8 (f i).sub_mem hi hsi) },\n  { rw set.mem_Union\u2082 at ht, rcases ht with \u27e8j, hjt, hj\u27e9,\n    simp only [finset.inf_eq_infi, set_like.mem_coe, submodule.mem_infi] at hr,\n    exact hs (or.inr $ set.mem_bUnion hjt $ add_sub_cancel' r s \u25b8 (f j).sub_mem hj $ hr j hjt) }\nend\n\n/-- Prime avoidance. Atiyah-Macdonald 1.11, Eisenbud 3.3, Stacks 00DS, Matsumura Ex.1.6. -/\ntheorem subset_union_prime {R : Type u} [comm_ring R] {s : finset \u03b9} {f : \u03b9 \u2192 ideal R} (a b : \u03b9)\n  (hp : \u2200 i \u2208 s, i \u2260 a \u2192 i \u2260 b \u2192 is_prime (f i)) {I : ideal R} :\n  (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2194 \u2203 i \u2208 s, I \u2264 f i :=\nsuffices (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i) \u2192 \u2203 i, i \u2208 s \u2227 I \u2264 f i,\n  from \u27e8\u03bb h, bex_def.2 $ this h, \u03bb \u27e8i, his, hi\u27e9, set.subset.trans hi $ set.subset_bUnion_of_mem $\n    show i \u2208 (\u2191s : set \u03b9), from his\u27e9,\nassume h : (I : set R) \u2286 (\u22c3 i \u2208 (\u2191s : set \u03b9), f i),\nbegin\n  classical,\n  by_cases has : a \u2208 s,\n  { unfreezingI { obtain \u27e8t, hat, rfl\u27e9 : \u2203 t, a \u2209 t \u2227 insert a t = s :=\n      \u27e8s.erase a, finset.not_mem_erase a s, finset.insert_erase has\u27e9 },\n    by_cases hbt : b \u2208 t,\n    { unfreezingI { obtain \u27e8u, hbu, rfl\u27e9 : \u2203 u, b \u2209 u \u2227 insert b u = t :=\n        \u27e8t.erase b, finset.not_mem_erase b t, finset.insert_erase hbt\u27e9 },\n      have hp' : \u2200 i \u2208 u, is_prime (f i),\n      { intros i hiu, refine hp i (finset.mem_insert_of_mem (finset.mem_insert_of_mem hiu)) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, finset.coe_insert, set.bUnion_insert, set.bUnion_insert,\n          \u2190 set.union_assoc, subset_union_prime' hp', bex_def] at h,\n      rwa [finset.exists_mem_insert, finset.exists_mem_insert] },\n    { have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f a : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } },\n  { by_cases hbs : b \u2208 s,\n    { unfreezingI { obtain \u27e8t, hbt, rfl\u27e9 : \u2203 t, b \u2209 t \u2227 insert b t = s :=\n        \u27e8s.erase b, finset.not_mem_erase b s, finset.insert_erase hbs\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f b : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert },\n    cases s.eq_empty_or_nonempty with hse hsne,\n    { substI hse, rw [finset.coe_empty, set.bUnion_empty, set.subset_empty_iff] at h,\n      have : (I : set R) \u2260 \u2205 := set.nonempty.ne_empty (set.nonempty_of_mem I.zero_mem),\n      exact absurd h this },\n    { cases hsne.bex with i his,\n      unfreezingI { obtain \u27e8t, hit, rfl\u27e9 : \u2203 t, i \u2209 t \u2227 insert i t = s :=\n        \u27e8s.erase i, finset.not_mem_erase i s, finset.insert_erase his\u27e9 },\n      have hp' : \u2200 j \u2208 t, is_prime (f j),\n      { intros j hj, refine hp j (finset.mem_insert_of_mem hj) _ _;\n        unfreezingI { rintro rfl }; solve_by_elim only [finset.mem_insert_of_mem, *], },\n      rw [finset.coe_insert, set.bUnion_insert, \u2190 set.union_self (f i : set R),\n          subset_union_prime' hp', \u2190 or_assoc, or_self, bex_def] at h,\n      rwa finset.exists_mem_insert } }\nend\n\nsection dvd\n\n/-- If `I` divides `J`, then `I` contains `J`.\n\nIn a Dedekind domain, to divide and contain are equivalent, see `ideal.dvd_iff_le`.\n-/\nlemma le_of_dvd {I J : ideal R} : I \u2223 J \u2192 J \u2264 I\n| \u27e8K, h\u27e9 := h.symm \u25b8 le_trans mul_le_inf inf_le_left\n\nlemma is_unit_iff {I : ideal R} :\n  is_unit I \u2194 I = \u22a4 :=\nis_unit_iff_dvd_one.trans ((@one_eq_top R _).symm \u25b8\n \u27e8\u03bb h, eq_top_iff.mpr (ideal.le_of_dvd h), \u03bb h, \u27e8\u22a4, by rw [mul_top, h]\u27e9\u27e9)\n\ninstance unique_units : unique ((ideal R)\u02e3) :=\n{ default := 1,\n  uniq := \u03bb u, units.ext\n    (show (u : ideal R) = 1, by rw [is_unit_iff.mp u.is_unit, one_eq_top]) }\n\nend dvd\n\nend mul_and_radical\n\nsection map_and_comap\n\nvariables {R : Type u} {S : Type v}\n\nsection semiring\nvariables {F : Type*} [semiring R] [semiring S]\nvariables [rc : ring_hom_class F R S]\nvariables (f : F)\nvariables {I J : ideal R} {K L : ideal S}\n\ninclude rc\n/-- `I.map f` is the span of the image of the ideal `I` under `f`, which may be bigger than\n  the image itself. -/\ndef map (I : ideal R) : ideal S :=\nspan (f '' I)\n\n/-- `I.comap f` is the preimage of `I` under `f`. -/\ndef comap (I : ideal S) : ideal R :=\n{ carrier := f \u207b\u00b9' I,\n  add_mem' := \u03bb x y hx hy, by simp only [set.mem_preimage, set_like.mem_coe,\n                                         map_add, add_mem hx hy] at *,\n  zero_mem' := by simp only [set.mem_preimage, map_zero, set_like.mem_coe, submodule.zero_mem],\n  smul_mem' := \u03bb c x hx, by { simp only [smul_eq_mul, set.mem_preimage, map_mul,\n                                         set_like.mem_coe] at *,\n                              exact mul_mem_left I _ hx } }\n\nvariables {f}\ntheorem map_mono (h : I \u2264 J) : map f I \u2264 map f J :=\nspan_mono $ set.image_subset _ h\n\ntheorem mem_map_of_mem (f : F) {I : ideal R} {x : R} (h : x \u2208 I) : f x \u2208 map f I :=\nsubset_span \u27e8x, h, rfl\u27e9\n\nlemma apply_coe_mem_map (f : F) (I : ideal R) (x : I) : f x \u2208 I.map f :=\nmem_map_of_mem f x.prop\n\ntheorem map_le_iff_le_comap :\n  map f I \u2264 K \u2194 I \u2264 comap f K :=\nspan_le.trans set.image_subset_iff\n\n@[simp] theorem mem_comap {x} : x \u2208 comap f K \u2194 f x \u2208 K := iff.rfl\n\ntheorem comap_mono (h : K \u2264 L) : comap f K \u2264 comap f L :=\nset.preimage_mono (\u03bb x hx, h hx)\nvariables (f)\n\ntheorem comap_ne_top (hK : K \u2260 \u22a4) : comap f K \u2260 \u22a4 :=\n(ne_top_iff_one _).2 $ by rw [mem_comap, map_one];\n  exact (ne_top_iff_one _).1 hK\n\nvariables {G : Type*} [rcg : ring_hom_class G S R]\n\ninclude rcg\nlemma map_le_comap_of_inv_on (g : G) (I : ideal R) (hf : set.left_inv_on g f I) :\n  I.map f \u2264 I.comap g :=\nbegin\n  refine ideal.span_le.2 _,\n  rintros x \u27e8x, hx, rfl\u27e9,\n  rw [set_like.mem_coe, mem_comap, hf hx],\n  exact hx,\nend\n\nlemma comap_le_map_of_inv_on (g : G) (I : ideal S) (hf : set.left_inv_on g f (f \u207b\u00b9' I)) :\n  I.comap f \u2264 I.map g :=\n\u03bb x (hx : f x \u2208 I), hf hx \u25b8 ideal.mem_map_of_mem g hx\n\n/-- The `ideal` version of `set.image_subset_preimage_of_inverse`. -/\nlemma map_le_comap_of_inverse (g : G) (I : ideal R) (h : function.left_inverse g f) :\n  I.map f \u2264 I.comap g :=\nmap_le_comap_of_inv_on _ _ _ $ h.left_inv_on _\n\n/-- The `ideal` version of `set.preimage_subset_image_of_inverse`. -/\nlemma comap_le_map_of_inverse (g : G) (I : ideal S) (h : function.left_inverse g f) :\n  I.comap f \u2264 I.map g :=\ncomap_le_map_of_inv_on _ _ _ $ h.left_inv_on _\nomit rcg\n\ninstance is_prime.comap [hK : K.is_prime] : (comap f K).is_prime :=\n\u27e8comap_ne_top _ hK.1, \u03bb x y,\n  by simp only [mem_comap, map_mul]; apply hK.2\u27e9\n\nvariables (I J K L)\n\ntheorem map_top : map f \u22a4 = \u22a4 :=\n(eq_top_iff_one _).2 $ subset_span \u27e81, trivial, map_one f\u27e9\n\nvariable (f)\nlemma gc_map_comap : galois_connection (ideal.map f) (ideal.comap f) :=\n\u03bb I J, ideal.map_le_iff_le_comap\nomit rc\n\n@[simp] lemma comap_id : I.comap (ring_hom.id R) = I :=\nideal.ext $ \u03bb _, iff.rfl\n\n@[simp] lemma map_id : I.map (ring_hom.id R) = I :=\n(gc_map_comap (ring_hom.id R)).l_unique galois_connection.id comap_id\n\nlemma comap_comap {T : Type*} [semiring T] {I : ideal T} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.comap g).comap f = I.comap (g.comp f) := rfl\n\nlemma map_map {T : Type*} [semiring T] {I : ideal R} (f : R \u2192+* S)\n  (g : S \u2192+* T) : (I.map f).map g = I.map (g.comp f) :=\n((gc_map_comap f).compose (gc_map_comap g)).l_unique\n  (gc_map_comap (g.comp f)) (\u03bb _, comap_comap _ _)\n\ninclude rc\nlemma map_span (f : F) (s : set R) :\n  map f (span s) = span (f '' s) :=\nsymm $ submodule.span_eq_of_le _\n  (\u03bb y \u27e8x, hy, x_eq\u27e9, x_eq \u25b8 mem_map_of_mem f (subset_span hy))\n  (map_le_iff_le_comap.2 $ span_le.2 $ set.image_subset_iff.1 subset_span)\n\nvariables {f I J K L}\n\nlemma map_le_of_le_comap : I \u2264 K.comap f \u2192 I.map f \u2264 K :=\n(gc_map_comap f).l_le\n\nlemma le_comap_of_map_le : I.map f \u2264 K \u2192 I \u2264 K.comap f :=\n(gc_map_comap f).le_u\n\nlemma le_comap_map : I \u2264 (I.map f).comap f :=\n(gc_map_comap f).le_u_l _\n\nlemma map_comap_le : (K.comap f).map f \u2264 K :=\n(gc_map_comap f).l_u_le _\n\n@[simp] lemma comap_top : (\u22a4 : ideal S).comap f = \u22a4 :=\n(gc_map_comap f).u_top\n\n@[simp] lemma comap_eq_top_iff {I : ideal S} : I.comap f = \u22a4 \u2194 I = \u22a4 :=\n\u27e8 \u03bb h, I.eq_top_iff_one.mpr (map_one f \u25b8 mem_comap.mp ((I.comap f).eq_top_iff_one.mp h)),\n  \u03bb h, by rw [h, comap_top] \u27e9\n\n@[simp] lemma map_bot : (\u22a5 : ideal R).map f = \u22a5 :=\n(gc_map_comap f).l_bot\n\nvariables (f I J K L)\n\n@[simp] lemma map_comap_map : ((I.map f).comap f).map f = I.map f :=\n(gc_map_comap f).l_u_l_eq_l I\n\n@[simp] lemma comap_map_comap : ((K.comap f).map f).comap f = K.comap f :=\n(gc_map_comap f).u_l_u_eq_u K\n\nlemma map_sup : (I \u2294 J).map f = I.map f \u2294 J.map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_sup\n\ntheorem comap_inf : comap f (K \u2293 L) = comap f K \u2293 comap f L := rfl\n\nvariables {\u03b9 : Sort*}\n\nlemma map_supr (K : \u03b9 \u2192 ideal R) : (supr K).map f = \u2a06 i, (K i).map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_supr\n\nlemma comap_infi (K : \u03b9 \u2192 ideal S) : (infi K).comap f = \u2a05 i, (K i).comap f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).u_infi\n\nlemma map_Sup (s : set (ideal R)): (Sup s).map f = \u2a06 I \u2208 s, (I : ideal R).map f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).l_Sup\n\nlemma comap_Inf (s : set (ideal S)): (Inf s).comap f = \u2a05 I \u2208 s, (I : ideal S).comap f :=\n(gc_map_comap f : galois_connection (map f) (comap f)).u_Inf\n\nlemma comap_Inf' (s : set (ideal S)) : (Inf s).comap f = \u2a05 I \u2208 (comap f '' s), I :=\ntrans (comap_Inf f s) (by rw infi_image)\n\ntheorem comap_is_prime [H : is_prime K] : is_prime (comap f K) :=\n\u27e8comap_ne_top f H.ne_top,\n  \u03bb x y h, H.mem_or_mem $ by rwa [mem_comap, map_mul] at h\u27e9\n\nvariables {I J K L}\n\ntheorem map_inf_le : map f (I \u2293 J) \u2264 map f I \u2293 map f J :=\n(gc_map_comap f : galois_connection (map f) (comap f)).monotone_l.map_inf_le _ _\n\ntheorem le_comap_sup : comap f K \u2294 comap f L \u2264 comap f (K \u2294 L) :=\n(gc_map_comap f : galois_connection (map f) (comap f)).monotone_u.le_map_sup _ _\nomit rc\n\n@[simp] lemma smul_top_eq_map {R S : Type*} [comm_semiring R] [comm_semiring S] [algebra R S]\n  (I : ideal R) : I \u2022 (\u22a4 : submodule R S) = (I.map (algebra_map R S)).restrict_scalars R :=\nbegin\n  refine le_antisymm (submodule.smul_le.mpr (\u03bb r hr y _, _) )\n      (\u03bb x hx, submodule.span_induction hx _ _ _ _),\n  { rw algebra.smul_def,\n     exact mul_mem_right _ _ (mem_map_of_mem _ hr) },\n\n  { rintros _ \u27e8x, hx, rfl\u27e9,\n    rw [\u2190 mul_one (algebra_map R S x), \u2190 algebra.smul_def],\n    exact submodule.smul_mem_smul hx submodule.mem_top },\n  { exact submodule.zero_mem _ },\n  { intros x y, exact submodule.add_mem _ },\n  intros a x hx,\n  refine submodule.smul_induction_on hx _ _,\n  { intros r hr s hs,\n    rw smul_comm,\n    exact submodule.smul_mem_smul hr submodule.mem_top },\n  { intros x y hx hy,\n    rw smul_add, exact submodule.add_mem _ hx hy },\nend\n\nsection surjective\nvariables (hf : function.surjective f)\ninclude hf\n\nopen function\n\ntheorem map_comap_of_surjective (I : ideal S) :\n  map f (comap f I) = I :=\nle_antisymm (map_le_iff_le_comap.2 le_rfl)\n(\u03bb s hsi, let \u27e8r, hfrs\u27e9 := hf s in\n  hfrs \u25b8 (mem_map_of_mem f $ show f r \u2208 I, from hfrs.symm \u25b8 hsi))\n\n/-- `map` and `comap` are adjoint, and the composition `map f \u2218 comap f` is the\n  identity -/\ndef gi_map_comap : galois_insertion (map f) (comap f) :=\ngalois_insertion.monotone_intro\n  ((gc_map_comap f).monotone_u)\n  ((gc_map_comap f).monotone_l)\n  (\u03bb _, le_comap_map)\n  (map_comap_of_surjective _ hf)\n\nlemma map_surjective_of_surjective : surjective (map f) :=\n(gi_map_comap f hf).l_surjective\n\nlemma comap_injective_of_surjective : injective (comap f) :=\n(gi_map_comap f hf).u_injective\n\nlemma map_sup_comap_of_surjective (I J : ideal S) : (I.comap f \u2294 J.comap f).map f = I \u2294 J :=\n(gi_map_comap f hf).l_sup_u _ _\n\nlemma map_supr_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a06i, (K i).comap f).map f = supr K :=\n(gi_map_comap f hf).l_supr_u _\n\nlemma map_inf_comap_of_surjective (I J : ideal S) : (I.comap f \u2293 J.comap f).map f = I \u2293 J :=\n(gi_map_comap f hf).l_inf_u _ _\n\nlemma map_infi_comap_of_surjective (K : \u03b9 \u2192 ideal S) : (\u2a05i, (K i).comap f).map f = infi K :=\n(gi_map_comap f hf).l_infi_u _\n\ntheorem mem_image_of_mem_map_of_surjective {I : ideal R} {y}\n  (H : y \u2208 map f I) : y \u2208 f '' I :=\nsubmodule.span_induction H (\u03bb _, id) \u27e80, I.zero_mem, map_zero f\u27e9\n(\u03bb y1 y2 \u27e8x1, hx1i, hxy1\u27e9 \u27e8x2, hx2i, hxy2\u27e9,\n  \u27e8x1 + x2, I.add_mem hx1i hx2i, hxy1 \u25b8 hxy2 \u25b8 map_add f _ _\u27e9)\n(\u03bb c y \u27e8x, hxi, hxy\u27e9,\n  let \u27e8d, hdc\u27e9 := hf c in \u27e8d * x, I.mul_mem_left _ hxi, hdc \u25b8 hxy \u25b8 map_mul f _ _\u27e9)\n\nlemma mem_map_iff_of_surjective {I : ideal R} {y} :\n  y \u2208 map f I \u2194 \u2203 x, x \u2208 I \u2227 f x = y :=\n\u27e8\u03bb h, (set.mem_image _ _ _).2 (mem_image_of_mem_map_of_surjective f hf h),\n  \u03bb \u27e8x, hx\u27e9, hx.right \u25b8 (mem_map_of_mem f hx.left)\u27e9\n\nlemma le_map_of_comap_le_of_surjective : comap f K \u2264 I \u2192 K \u2264 map f I :=\n\u03bb h, (map_comap_of_surjective f hf K) \u25b8 map_mono h\n\nend surjective\n\nsection injective\nvariables (hf : function.injective f)\ninclude hf\n\nlemma comap_bot_le_of_injective : comap f \u22a5 \u2264 I :=\nbegin\n  refine le_trans (\u03bb x hx, _) bot_le,\n  rw [mem_comap, submodule.mem_bot, \u2190 map_zero f] at hx,\n  exact eq.symm (hf hx) \u25b8 (submodule.zero_mem \u22a5)\nend\n\nend injective\n\nend semiring\n\nsection ring\nvariables {F : Type*} [ring R] [ring S]\nvariables [ring_hom_class F R S] (f : F) {I : ideal R}\n\nsection surjective\n\nvariables (hf : function.surjective f)\ninclude hf\n\ntheorem comap_map_of_surjective (I : ideal R) : comap f (map f I) = I \u2294 comap f \u22a5 :=\nle_antisymm (assume r h, let \u27e8s, hsi, hfsr\u27e9 := mem_image_of_mem_map_of_surjective f hf h in\n  submodule.mem_sup.2 \u27e8s, hsi, r - s, (submodule.mem_bot S).2 $ by rw [map_sub, hfsr, sub_self],\n  add_sub_cancel'_right s r\u27e9)\n(sup_le (map_le_iff_le_comap.1 le_rfl) (comap_mono bot_le))\n\n\n/-- Correspondence theorem -/\ndef rel_iso_of_surjective : ideal S \u2243o { p : ideal R // comap f \u22a5 \u2264 p } :=\n{ to_fun := \u03bb J, \u27e8comap f J, comap_mono bot_le\u27e9,\n  inv_fun := \u03bb I, map f I.1,\n  left_inv := \u03bb J, map_comap_of_surjective f hf J,\n  right_inv := \u03bb I, subtype.eq $ show comap f (map f I.1) = I.1,\n    from (comap_map_of_surjective f hf I).symm \u25b8 le_antisymm\n      (sup_le le_rfl I.2) le_sup_left,\n  map_rel_iff' := \u03bb I1 I2, \u27e8\u03bb H, map_comap_of_surjective f hf I1 \u25b8\n    map_comap_of_surjective f hf I2 \u25b8 map_mono H, comap_mono\u27e9 }\n\n/-- The map on ideals induced by a surjective map preserves inclusion. -/\ndef order_embedding_of_surjective : ideal S \u21aao ideal R :=\n(rel_iso_of_surjective f hf).to_rel_embedding.trans (subtype.rel_embedding _ _)\n\ntheorem map_eq_top_or_is_maximal_of_surjective {I : ideal R} (H : is_maximal I) :\n  (map f I) = \u22a4 \u2228 is_maximal (map f I) :=\nbegin\n  refine or_iff_not_imp_left.2 (\u03bb ne_top, \u27e8\u27e8\u03bb h, ne_top h, \u03bb J hJ, _\u27e9\u27e9),\n  { refine (rel_iso_of_surjective f hf).injective\n      (subtype.ext_iff.2 (eq.trans (H.1.2 (comap f J) (lt_of_le_of_ne _ _)) comap_top.symm)),\n    { exact (map_le_iff_le_comap).1 (le_of_lt hJ) },\n    { exact \u03bb h, hJ.right (le_map_of_comap_le_of_surjective f hf (le_of_eq h.symm)) } }\nend\n\ntheorem comap_is_maximal_of_surjective {K : ideal S} [H : is_maximal K] : is_maximal (comap f K) :=\nbegin\n  refine \u27e8\u27e8comap_ne_top _ H.1.1, \u03bb J hJ, _\u27e9\u27e9,\n  suffices : map f J = \u22a4,\n  { replace this := congr_arg (comap f) this,\n    rw [comap_top, comap_map_of_surjective _ hf, eq_top_iff] at this,\n    rw eq_top_iff,\n    exact le_trans this (sup_le (le_of_eq rfl) (le_trans (comap_mono (bot_le)) (le_of_lt hJ))) },\n  refine H.1.2 (map f J) (lt_of_le_of_ne (le_map_of_comap_le_of_surjective _ hf (le_of_lt hJ))\n    (\u03bb h, ne_of_lt hJ (trans (congr_arg (comap f) h) _))),\n  rw [comap_map_of_surjective _ hf, sup_eq_left],\n  exact le_trans (comap_mono bot_le) (le_of_lt hJ)\nend\n\ntheorem comap_le_comap_iff_of_surjective (I J : ideal S) : comap f I \u2264 comap f J \u2194 I \u2264 J :=\n\u27e8\u03bb h, (map_comap_of_surjective f hf I).symm.le.trans (map_le_of_le_comap h),\n  \u03bb h, le_comap_of_map_le ((map_comap_of_surjective f hf I).le.trans h)\u27e9\n\nend surjective\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f (map f.symm) = I`. -/\n@[simp]\nlemma map_of_equiv (I : ideal R) (f : R \u2243+* S) : (I.map (f : R \u2192+* S)).map (f.symm : S \u2192+* R) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, map_map]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `comap f.symm (comap f) = I`. -/\n@[simp]\nlemma comap_of_equiv (I : ideal R) (f : R \u2243+* S) :\n  (I.comap (f.symm : S \u2192+* R)).comap (f : R \u2192+* S) = I :=\nby simp [\u2190 ring_equiv.to_ring_hom_eq_coe, comap_comap]\n\n/-- If `f : R \u2243+* S` is a ring isomorphism and `I : ideal R`, then `map f I = comap f.symm I`. -/\nlemma map_comap_of_equiv (I : ideal R) (f : R \u2243+* S) : I.map (f : R \u2192+* S) = I.comap f.symm :=\nle_antisymm (le_comap_of_map_le (map_of_equiv I f).le)\n  (le_map_of_comap_le_of_surjective _ f.surjective (comap_of_equiv I f).le)\n\nsection bijective\nvariables (hf : function.bijective f)\ninclude hf\n\n/-- Special case of the correspondence theorem for isomorphic rings -/\ndef rel_iso_of_bijective : ideal S \u2243o ideal R :=\n{ to_fun := comap f,\n  inv_fun := map f,\n  left_inv := (rel_iso_of_surjective f hf.right).left_inv,\n  right_inv := \u03bb J, subtype.ext_iff.1\n    ((rel_iso_of_surjective f hf.right).right_inv \u27e8J, comap_bot_le_of_injective f hf.left\u27e9),\n  map_rel_iff' := (rel_iso_of_surjective f hf.right).map_rel_iff' }\n\nlemma comap_le_iff_le_map {I : ideal R} {K : ideal S} : comap f K \u2264 I \u2194 K \u2264 map f I :=\n\u27e8\u03bb h, le_map_of_comap_le_of_surjective f hf.right h,\n \u03bb h, ((rel_iso_of_bijective f hf).right_inv I) \u25b8 comap_mono h\u27e9\n\ntheorem map.is_maximal {I : ideal R} (H : is_maximal I) : is_maximal (map f I) :=\nby refine or_iff_not_imp_left.1\n  (map_eq_top_or_is_maximal_of_surjective f hf.right H) (\u03bb h, H.1.1 _);\ncalc I = comap f (map f I) : ((rel_iso_of_bijective f hf).right_inv I).symm\n   ... = comap f \u22a4 : by rw h\n   ... = \u22a4 : by rw comap_top\n\nend bijective\n\nlemma ring_equiv.bot_maximal_iff (e : R \u2243+* S) :\n  (\u22a5 : ideal R).is_maximal \u2194 (\u22a5 : ideal S).is_maximal :=\n\u27e8\u03bb h, (@map_bot _ _ _ _ _ _ e.to_ring_hom) \u25b8 map.is_maximal e.to_ring_hom e.bijective h,\n  \u03bb h, (@map_bot _ _ _ _ _ _ e.symm.to_ring_hom) \u25b8 map.is_maximal e.symm.to_ring_hom\n          e.symm.bijective h\u27e9\n\nend ring\n\nsection comm_ring\n\nvariables {F : Type*} [comm_ring R] [comm_ring S]\nvariables [rc : ring_hom_class F R S]\nvariables (f : F)\nvariables {I J : ideal R} {K L : ideal S}\n\nvariables (I J K L)\n\ninclude rc\ntheorem map_mul : map f (I * J) = map f I * map f J :=\nle_antisymm (map_le_iff_le_comap.2 $ mul_le.2 $ \u03bb r hri s hsj,\n  show f (r * s) \u2208 _, by rw map_mul;\n  exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj))\n(trans_rel_right _ (span_mul_span _ _) $ span_le.2 $\n  set.Union\u2082_subset $ \u03bb i \u27e8r, hri, hfri\u27e9,\n  set.Union\u2082_subset $ \u03bb j \u27e8s, hsj, hfsj\u27e9,\n  set.singleton_subset_iff.2 $ hfri \u25b8 hfsj \u25b8\n  by rw [\u2190 map_mul];\n  exact mem_map_of_mem f (mul_mem_mul hri hsj))\n\n/-- The pushforward `ideal.map` as a monoid-with-zero homomorphism. -/\n@[simps]\ndef map_hom : ideal R \u2192*\u2080 ideal S :=\n{ to_fun := map f,\n  map_mul' := \u03bb I J, ideal.map_mul f I J,\n  map_one' := by convert ideal.map_top f; exact one_eq_top,\n  map_zero' := ideal.map_bot }\n\nprotected theorem map_pow (n : \u2115) : map f (I^n) = (map f I)^n :=\nmap_pow (map_hom f) I n\n\ntheorem comap_radical : comap f (radical K) = radical (comap f K) :=\nle_antisymm (\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, show f (r ^ n) \u2208 K,\n  from (map_pow f r n).symm \u25b8 hfrnk\u27e9)\n(\u03bb r \u27e8n, hfrnk\u27e9, \u27e8n, map_pow f r n \u25b8 hfrnk\u27e9)\nomit rc\n\n@[simp] lemma map_quotient_self :\n  map (quotient.mk I) I = \u22a5 :=\neq_bot_iff.2 $ ideal.map_le_iff_le_comap.2 $ \u03bb x hx,\n(submodule.mem_bot (R \u29f8 I)).2 $ ideal.quotient.eq_zero_iff_mem.2 hx\n\nvariables {I J K L}\n\ninclude rc\ntheorem map_radical_le : map f (radical I) \u2264 radical (map f I) :=\nmap_le_iff_le_comap.2 $ \u03bb r \u27e8n, hrni\u27e9, \u27e8n, map_pow f r n \u25b8 mem_map_of_mem f hrni\u27e9\n\ntheorem le_comap_mul : comap f K * comap f L \u2264 comap f (K * L) :=\nmap_le_iff_le_comap.1 $ (map_mul f (comap f K) (comap f L)).symm \u25b8\nmul_mono (map_le_iff_le_comap.2 $ le_rfl) (map_le_iff_le_comap.2 $ le_rfl)\nomit rc\n\nend comm_ring\n\nend map_and_comap\n\nsection is_primary\nvariables {R : Type u} [comm_semiring R]\n\n/-- A proper ideal `I` is primary iff `xy \u2208 I` implies `x \u2208 I` or `y \u2208 radical I`. -/\ndef is_primary (I : ideal R) : Prop :=\nI \u2260 \u22a4 \u2227 \u2200 {x y : R}, x * y \u2208 I \u2192 x \u2208 I \u2228 y \u2208 radical I\n\ntheorem is_prime.is_primary {I : ideal R} (hi : is_prime I) : is_primary I :=\n\u27e8hi.1, \u03bb x y hxy, (hi.mem_or_mem hxy).imp id $ \u03bb hyi, le_radical hyi\u27e9\n\ntheorem mem_radical_of_pow_mem {I : ideal R} {x : R} {m : \u2115} (hx : x ^ m \u2208 radical I) :\n  x \u2208 radical I :=\nradical_idem I \u25b8 \u27e8m, hx\u27e9\n\ntheorem is_prime_radical {I : ideal R} (hi : is_primary I) : is_prime (radical I) :=\n\u27e8mt radical_eq_top.1 hi.1, \u03bb x y \u27e8m, hxy\u27e9, begin\n  rw mul_pow at hxy, cases hi.2 hxy,\n  { exact or.inl \u27e8m, h\u27e9 },\n  { exact or.inr (mem_radical_of_pow_mem h) }\nend\u27e9\n\ntheorem is_primary_inf {I J : ideal R} (hi : is_primary I) (hj : is_primary J)\n  (hij : radical I = radical J) : is_primary (I \u2293 J) :=\n\u27e8ne_of_lt $ lt_of_le_of_lt inf_le_left (lt_top_iff_ne_top.2 hi.1), \u03bb x y \u27e8hxyi, hxyj\u27e9,\nbegin\n  rw [radical_inf, hij, inf_idem],\n  cases hi.2 hxyi with hxi hyi, cases hj.2 hxyj with hxj hyj,\n  { exact or.inl \u27e8hxi, hxj\u27e9 },\n  { exact or.inr hyj },\n  { rw hij at hyi, exact or.inr hyi }\nend\u27e9\n\nend is_primary\n\nend ideal\n\nlemma associates.mk_ne_zero' {R : Type*} [comm_ring R] {r : R} :\n  (associates.mk (ideal.span {r} : ideal R)) \u2260 0 \u2194 (r \u2260 0):=\nby rw [associates.mk_ne_zero, ideal.zero_eq_bot, ne.def, ideal.span_singleton_eq_bot]\n\nnamespace ring_hom\n\nvariables {R : Type u} {S : Type v} {T : Type v}\n\nsection semiring\nvariables {F : Type*} {G : Type*} [semiring R] [semiring S] [semiring T]\nvariables [rcf : ring_hom_class F R S] [rcg : ring_hom_class G T S]\n(f : F) (g : G)\n\ninclude rcf\n/-- Kernel of a ring homomorphism as an ideal of the domain. -/\ndef ker : ideal R := ideal.comap f \u22a5\n\n/-- An element is in the kernel if and only if it maps to zero.-/\nlemma mem_ker {r} : r \u2208 ker f \u2194 f r = 0 :=\nby rw [ker, ideal.mem_comap, submodule.mem_bot]\n\nlemma ker_eq : ((ker f) : set R) = set.preimage f {0} := rfl\n\nlemma ker_eq_comap_bot (f : F) : ker f = ideal.comap f \u22a5 := rfl\nomit rcf\n\nlemma comap_ker (f : S \u2192+* R) (g : T \u2192+* S) : f.ker.comap g = (f.comp g).ker :=\nby rw [ring_hom.ker_eq_comap_bot, ideal.comap_comap, ring_hom.ker_eq_comap_bot]\n\ninclude rcf\n/-- If the target is not the zero ring, then one is not in the kernel.-/\nlemma not_one_mem_ker [nontrivial S] (f : F) : (1:R) \u2209 ker f :=\nby { rw [mem_ker, map_one], exact one_ne_zero }\n\nlemma ker_ne_top [nontrivial S] (f : F) : ker f \u2260 \u22a4 :=\n(ideal.ne_top_iff_one _).mpr $ not_one_mem_ker f\nomit rcf\n\nend semiring\n\nsection ring\nvariables {F : Type*} [ring R] [semiring S] [rc : ring_hom_class F R S] (f : F)\n\ninclude rc\nlemma injective_iff_ker_eq_bot : function.injective f \u2194 ker f = \u22a5 :=\nby { rw [set_like.ext'_iff, ker_eq, set.ext_iff], exact injective_iff_map_eq_zero' f }\n\nlemma ker_eq_bot_iff_eq_zero : ker f = \u22a5 \u2194 \u2200 x, f x = 0 \u2192 x = 0 :=\nby { rw [\u2190 injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] }\nomit rc\n\n@[simp] lemma ker_coe_equiv (f : R \u2243+* S) :\n  ker (f : R \u2192+* S) = \u22a5 :=\nby simpa only [\u2190injective_iff_ker_eq_bot] using equiv_like.injective f\n\n@[simp] lemma ker_equiv {F' : Type*} [ring_equiv_class F' R S] (f : F') :\n  ker f = \u22a5 :=\nby simpa only [\u2190injective_iff_ker_eq_bot] using equiv_like.injective f\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S] (f : R \u2192+* S)\n\n/-- The induced map from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_equiv_of_surjective`).\n-/\ndef ker_lift (f : R \u2192+* S) : R \u29f8 f.ker \u2192+* S :=\nideal.quotient.lift _ f $ \u03bb r, f.mem_ker.mp\n\n@[simp]\nlemma ker_lift_mk (f : R \u2192+* S) (r : R) : ker_lift f (ideal.quotient.mk f.ker r) = f r :=\nideal.quotient.lift_mk _ _ _\n\n/-- The induced map from the quotient by the kernel is injective. -/\nlemma ker_lift_injective (f : R \u2192+* S) : function.injective (ker_lift f) :=\nassume a b, quotient.induction_on\u2082' a b $\n  assume a b (h : f a = f b), ideal.quotient.eq.2 $\nshow a - b \u2208 ker f, by rw [mem_ker, map_sub, h, sub_self]\n\nvariable {f}\n\n/-- The **first isomorphism theorem** for commutative rings, computable version. -/\ndef quotient_ker_equiv_of_right_inverse\n  {g : S \u2192 R} (hf : function.right_inverse g f) :\n  R \u29f8 f.ker \u2243+* S :=\n{ to_fun := ker_lift f,\n  inv_fun := (ideal.quotient.mk f.ker) \u2218 g,\n  left_inv := begin\n    rintro \u27e8x\u27e9,\n    apply ker_lift_injective,\n    simp [hf (f x)],\n  end,\n  right_inv := hf,\n  ..ker_lift f}\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : R \u29f8 f.ker) : quotient_ker_equiv_of_right_inverse hf x = ker_lift f x := rfl\n\n@[simp]\nlemma quotient_ker_equiv_of_right_inverse.symm.apply {g : S \u2192 R} (hf : function.right_inverse g f)\n  (x : S) : (quotient_ker_equiv_of_right_inverse hf).symm x = ideal.quotient.mk f.ker (g x) := rfl\n\n/-- The **first isomorphism theorem** for commutative rings. -/\nnoncomputable def quotient_ker_equiv_of_surjective (hf : function.surjective f) :\n  R \u29f8 f.ker \u2243+* S :=\nquotient_ker_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\nend comm_ring\n\n/-- The kernel of a homomorphism to a domain is a prime ideal. -/\nlemma ker_is_prime {F : Type*} [ring R] [ring S] [is_domain S] [ring_hom_class F R S]\n  (f : F) : (ker f).is_prime :=\n\u27e8by { rw [ne.def, ideal.eq_top_iff_one], exact not_one_mem_ker f },\n\u03bb x y, by simpa only [mem_ker, map_mul] using @eq_zero_or_eq_zero_of_mul_eq_zero S _ _ _ _ _\u27e9\n\n/-- The kernel of a homomorphism to a field is a maximal ideal. -/\nlemma ker_is_maximal_of_surjective {R K F : Type*} [ring R] [field K] [ring_hom_class F R K]\n  (f : F) (hf : function.surjective f) :\n  (ker f).is_maximal :=\nbegin\n  refine ideal.is_maximal_iff.mpr\n    \u27e8\u03bb h1, @one_ne_zero K _ _ $ map_one f \u25b8 (mem_ker f).mp h1,\n    \u03bb J x hJ hxf hxJ, _\u27e9,\n  obtain \u27e8y, hy\u27e9 := hf (f x)\u207b\u00b9,\n  have H : 1 = y * x - (y * x - 1) := (sub_sub_cancel _ _).symm,\n  rw H,\n  refine J.sub_mem (J.mul_mem_left _ hxJ) (hJ _),\n  rw mem_ker,\n  simp only [hy, map_sub, map_one, map_mul,\n    inv_mul_cancel (mt (mem_ker f).mpr hxf), sub_self],\nend\n\nend ring_hom\n\nnamespace ideal\n\nvariables {R : Type*} {S : Type*} {F : Type*}\n\nsection semiring\nvariables [semiring R] [semiring S] [rc : ring_hom_class F R S]\n\ninclude rc\nlemma map_eq_bot_iff_le_ker {I : ideal R} (f : F) : I.map f = \u22a5 \u2194 I \u2264 (ring_hom.ker f) :=\nby rw [ring_hom.ker, eq_bot_iff, map_le_iff_le_comap]\n\nlemma ker_le_comap {K : ideal S} (f : F) : ring_hom.ker f \u2264 comap f K :=\n\u03bb x hx, mem_comap.2 (((ring_hom.mem_ker f).1 hx).symm \u25b8 K.zero_mem)\n\nend semiring\n\nsection ring\nvariables [ring R] [ring S] [rc : ring_hom_class F R S]\n\ninclude rc\nlemma map_Inf {A : set (ideal R)} {f : F} (hf : function.surjective f) :\n  (\u2200 J \u2208 A, ring_hom.ker f \u2264 J) \u2192 map f (Inf A) = Inf (map f '' A) :=\nbegin\n  refine \u03bb h, le_antisymm (le_Inf _) _,\n  { intros j hj y hy,\n    cases (mem_map_iff_of_surjective f hf).1 hy with x hx,\n    cases (set.mem_image _ _ _).mp hj with J hJ,\n    rw [\u2190 hJ.right, \u2190 hx.right],\n    exact mem_map_of_mem f (Inf_le_of_le hJ.left (le_of_eq rfl) hx.left) },\n  { intros y hy,\n    cases hf y with x hx,\n    refine hx \u25b8 (mem_map_of_mem f _),\n    have : \u2200 I \u2208 A, y \u2208 map f I, by simpa using hy,\n    rw [submodule.mem_Inf],\n    intros J hJ,\n    rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with \u27e8x', hx', rfl\u27e9,\n    have : x - x' \u2208 J,\n    { apply h J hJ,\n      rw [ring_hom.mem_ker, map_sub, hx, sub_self] },\n    simpa only [sub_add_cancel] using J.add_mem this hx' }\nend\n\ntheorem map_is_prime_of_surjective {f : F} (hf : function.surjective f) {I : ideal R}\n  [H : is_prime I] (hk : ring_hom.ker f \u2264 I) : is_prime (map f I) :=\nbegin\n  refine \u27e8\u03bb h, H.ne_top (eq_top_iff.2 _), \u03bb x y, _\u27e9,\n  { replace h := congr_arg (comap f) h,\n    rw [comap_map_of_surjective _ hf, comap_top] at h,\n    exact h \u25b8 sup_le (le_of_eq rfl) hk },\n  { refine \u03bb hxy, (hf x).rec_on (\u03bb a ha, (hf y).rec_on (\u03bb b hb, _)),\n    rw [\u2190 ha, \u2190 hb, \u2190 _root_.map_mul f, mem_map_iff_of_surjective _ hf] at hxy,\n    rcases hxy with \u27e8c, hc, hc'\u27e9,\n    rw [\u2190 sub_eq_zero, \u2190 map_sub] at hc',\n    have : a * b \u2208 I,\n    { convert I.sub_mem hc (hk (hc' : c - a * b \u2208 ring_hom.ker f)),\n      abel },\n    exact (H.mem_or_mem this).imp (\u03bb h, ha \u25b8 mem_map_of_mem f h) (\u03bb h, hb \u25b8 mem_map_of_mem f h) }\nend\nomit rc\n\ntheorem map_is_prime_of_equiv {F' : Type*} [ring_equiv_class F' R S]\n  (f : F') {I : ideal R} [is_prime I] :\n  is_prime (map f I) :=\nmap_is_prime_of_surjective (equiv_like.surjective f) $ by simp only [ring_hom.ker_equiv, bot_le]\n\nend ring\n\nsection comm_ring\nvariables [comm_ring R] [comm_ring S]\n\n@[simp] lemma mk_ker {I : ideal R} : (quotient.mk I).ker = I :=\nby ext; rw [ring_hom.ker, mem_comap, submodule.mem_bot, quotient.eq_zero_iff_mem]\n\nlemma map_mk_eq_bot_of_le {I J : ideal R} (h : I \u2264 J) : I.map (J^.quotient.mk) = \u22a5 :=\nby { rw [map_eq_bot_iff_le_ker, mk_ker], exact h }\n\nlemma ker_quotient_lift {S : Type v} [comm_ring S] {I : ideal R} (f : R \u2192+* S) (H : I \u2264 f.ker) :\n  (ideal.quotient.lift I f H).ker = (f.ker).map I^.quotient.mk :=\nbegin\n  ext x,\n  split,\n  { intro hx,\n    obtain \u27e8y, hy\u27e9 := quotient.mk_surjective x,\n    rw [ring_hom.mem_ker, \u2190 hy, ideal.quotient.lift_mk, \u2190 ring_hom.mem_ker] at hx,\n    rw [\u2190 hy, mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective],\n    exact \u27e8y, hx, rfl\u27e9 },\n  { intro hx,\n    rw mem_map_iff_of_surjective I^.quotient.mk quotient.mk_surjective at hx,\n    obtain \u27e8y, hy\u27e9 := hx,\n    rw [ring_hom.mem_ker, \u2190 hy.right, ideal.quotient.lift_mk, \u2190 (ring_hom.mem_ker f)],\n    exact hy.left },\nend\n\ntheorem map_eq_iff_sup_ker_eq_of_surjective {I J : ideal R} (f : R \u2192+* S)\n  (hf : function.surjective f) : map f I = map f J \u2194 I \u2294 f.ker = J \u2294 f.ker :=\nby rw [\u2190 (comap_injective_of_surjective f hf).eq_iff, comap_map_of_surjective f hf,\n  comap_map_of_surjective f hf, ring_hom.ker_eq_comap_bot]\n\ntheorem map_radical_of_surjective {f : R \u2192+* S} (hf : function.surjective f) {I : ideal R}\n  (h : ring_hom.ker f \u2264 I) : map f (I.radical) = (map f I).radical :=\nbegin\n  rw [radical_eq_Inf, radical_eq_Inf],\n  have : \u2200 J \u2208 {J : ideal R | I \u2264 J \u2227 J.is_prime}, f.ker \u2264 J := \u03bb J hJ, le_trans h hJ.left,\n  convert map_Inf hf this,\n  refine funext (\u03bb j, propext \u27e8_, _\u27e9),\n  { rintros \u27e8hj, hj'\u27e9,\n    haveI : j.is_prime := hj',\n    exact \u27e8comap f j, \u27e8\u27e8map_le_iff_le_comap.1 hj, comap_is_prime f j\u27e9,\n      map_comap_of_surjective f hf j\u27e9\u27e9 },\n  { rintro \u27e8J, \u27e8hJ, hJ'\u27e9\u27e9,\n    haveI : J.is_prime := hJ.right,\n    refine \u27e8hJ' \u25b8 map_mono hJ.left, hJ' \u25b8 map_is_prime_of_surjective hf (le_trans h hJ.left)\u27e9 },\nend\n\n@[simp] lemma bot_quotient_is_maximal_iff (I : ideal R) :\n  (\u22a5 : ideal (R \u29f8 I)).is_maximal \u2194 I.is_maximal :=\n\u27e8\u03bb hI, (@mk_ker _ _ I) \u25b8\n  @comap_is_maximal_of_surjective _ _ _ _ _ _ (quotient.mk I) quotient.mk_surjective \u22a5 hI,\n \u03bb hI, @bot_is_maximal _ (@field.to_division_ring _ (@quotient.field _ _ I hI)) \u27e9\n\n/-- See also `ideal.mem_quotient_iff_mem` in case `I \u2264 J`. -/\n@[simp]\nlemma mem_quotient_iff_mem_sup {I J : ideal R} {x : R} :\n  quotient.mk I x \u2208 J.map (quotient.mk I) \u2194 x \u2208 J \u2294 I :=\nby rw [\u2190 mem_comap, comap_map_of_surjective (quotient.mk I) quotient.mk_surjective,\n       \u2190 ring_hom.ker_eq_comap_bot, mk_ker]\n\n/-- See also `ideal.mem_quotient_iff_mem_sup` if the assumption `I \u2264 J` is not available. -/\nlemma mem_quotient_iff_mem {I J : ideal R} (hIJ : I \u2264 J) {x : R} :\n  quotient.mk I x \u2208 J.map (quotient.mk I) \u2194 x \u2208 J :=\nby rw [mem_quotient_iff_mem_sup, sup_eq_left.mpr hIJ]\n\nsection quotient_algebra\n\nvariables (R\u2081 R\u2082 : Type*) {A B : Type*}\nvariables [comm_semiring R\u2081] [comm_semiring R\u2082] [comm_ring A] [comm_ring B]\nvariables [algebra R\u2081 A] [algebra R\u2082 A] [algebra R\u2081 B]\n\n/-- The `R\u2081`-algebra structure on `A/I` for an `R\u2081`-algebra `A` -/\ninstance quotient.algebra {I : ideal A} : algebra R\u2081 (A \u29f8 I) :=\n{ to_fun := \u03bb x, ideal.quotient.mk I (algebra_map R\u2081 A x),\n  smul := (\u2022),\n  smul_def' := \u03bb r x, quotient.induction_on' x $ \u03bb x,\n      ((quotient.mk I).congr_arg $ algebra.smul_def _ _).trans (ring_hom.map_mul _ _ _),\n  commutes' := \u03bb _ _, mul_comm _ _,\n  .. ring_hom.comp (ideal.quotient.mk I) (algebra_map R\u2081 A) }\n\n-- Lean can struggle to find this instance later if we don't provide this shortcut\ninstance quotient.is_scalar_tower [has_scalar R\u2081 R\u2082] [is_scalar_tower R\u2081 R\u2082 A] (I : ideal A) :\n  is_scalar_tower R\u2081 R\u2082 (A \u29f8 I) :=\nby apply_instance\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] A \u29f8 I` as morphism of `R\u2081`-algebras, for `I` an ideal of\n`A`, where `A` is an `R\u2081`-algebra. -/\ndef quotient.mk\u2090 (I : ideal A) : A \u2192\u2090[R\u2081] A \u29f8 I :=\n\u27e8\u03bb a, submodule.quotient.mk a, rfl, \u03bb _ _, rfl, rfl, \u03bb _ _, rfl, \u03bb _, rfl\u27e9\n\nlemma quotient.alg_map_eq (I : ideal A) :\n  algebra_map R\u2081 (A \u29f8 I) = (algebra_map A (A \u29f8 I)).comp (algebra_map R\u2081 A) :=\nrfl\n\nlemma quotient.mk\u2090_to_ring_hom (I : ideal A) :\n  (quotient.mk\u2090 R\u2081 I).to_ring_hom = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.mk\u2090_eq_mk (I : ideal A) :\n  \u21d1(quotient.mk\u2090 R\u2081 I) = ideal.quotient.mk I := rfl\n\n@[simp] lemma quotient.algebra_map_eq (I : ideal R) :\n  algebra_map R (R \u29f8 I) = I^.quotient.mk :=\nrfl\n\n@[simp] lemma quotient.mk_comp_algebra_map (I : ideal A) :\n  (quotient.mk I).comp (algebra_map R\u2081 A) = algebra_map R\u2081 (A \u29f8 I) :=\nrfl\n\n@[simp] lemma quotient.mk_algebra_map (I : ideal A) (x : R\u2081) :\n  quotient.mk I (algebra_map R\u2081 A x) = algebra_map R\u2081 (A \u29f8 I) x :=\nrfl\n\n/-- The canonical morphism `A \u2192\u2090[R\u2081] I.quotient` is surjective. -/\nlemma quotient.mk\u2090_surjective (I : ideal A) : function.surjective (quotient.mk\u2090 R\u2081 I) :=\nsurjective_quot_mk _\n\n/-- The kernel of `A \u2192\u2090[R\u2081] I.quotient` is `I`. -/\n@[simp]\nlemma quotient.mk\u2090_ker (I : ideal A) : (quotient.mk\u2090 R\u2081 I : A \u2192+* A \u29f8 I).ker = I :=\nideal.mk_ker\n\nvariables {R\u2081}\n\nlemma ker_lift.map_smul (f : A \u2192\u2090[R\u2081] B) (r : R\u2081) (x : A \u29f8 f.to_ring_hom.ker) :\n  f.to_ring_hom.ker_lift (r \u2022 x) = r \u2022 f.to_ring_hom.ker_lift x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := quotient.mk\u2090_surjective R\u2081 _ x,\n  rw [\u2190 alg_hom.map_smul, quotient.mk\u2090_eq_mk, ring_hom.ker_lift_mk],\n  exact f.map_smul _ _\nend\n\n/-- The induced algebras morphism from the quotient by the kernel to the codomain.\n\nThis is an isomorphism if `f` has a right inverse (`quotient_ker_alg_equiv_of_right_inverse`) /\nis surjective (`quotient_ker_alg_equiv_of_surjective`).\n-/\ndef ker_lift_alg (f : A \u2192\u2090[R\u2081] B) : (A \u29f8 f.to_ring_hom.ker) \u2192\u2090[R\u2081] B :=\nalg_hom.mk' f.to_ring_hom.ker_lift (\u03bb _ _, ker_lift.map_smul f _ _)\n\n@[simp]\nlemma ker_lift_alg_mk (f : A \u2192\u2090[R\u2081] B) (a : A) :\n  ker_lift_alg f (quotient.mk f.to_ring_hom.ker a) = f a := rfl\n\n@[simp]\nlemma ker_lift_alg_to_ring_hom (f : A \u2192\u2090[R\u2081] B) :\n  (ker_lift_alg f).to_ring_hom = ring_hom.ker_lift f := rfl\n\n/-- The induced algebra morphism from the quotient by the kernel is injective. -/\nlemma ker_lift_alg_injective (f : A \u2192\u2090[R\u2081] B) : function.injective (ker_lift_alg f) :=\nring_hom.ker_lift_injective f\n\n/-- The **first isomorphism** theorem for algebras, computable version. -/\ndef quotient_ker_alg_equiv_of_right_inverse\n  {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A} (hf : function.right_inverse g f) :\n  (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\n{ ..ring_hom.quotient_ker_equiv_of_right_inverse (\u03bb x, show f.to_ring_hom (g x) = x, from hf x),\n  ..ker_lift_alg f}\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : A \u29f8 f.to_ring_hom.ker) :\n  quotient_ker_alg_equiv_of_right_inverse hf x = ker_lift_alg f x := rfl\n\n@[simp]\nlemma quotient_ker_alg_equiv_of_right_inverse_symm.apply {f : A \u2192\u2090[R\u2081] B} {g : B \u2192 A}\n  (hf : function.right_inverse g f) (x : B) :\n  (quotient_ker_alg_equiv_of_right_inverse hf).symm x = quotient.mk\u2090 R\u2081 f.to_ring_hom.ker (g x) :=\n  rfl\n\n/-- The **first isomorphism theorem** for algebras. -/\nnoncomputable def quotient_ker_alg_equiv_of_surjective\n  {f : A \u2192\u2090[R\u2081] B} (hf : function.surjective f) : (A \u29f8 f.to_ring_hom.ker) \u2243\u2090[R\u2081] B :=\nquotient_ker_alg_equiv_of_right_inverse (classical.some_spec hf.has_right_inverse)\n\n/-- The ring hom `R/I \u2192+* S/J` induced by a ring hom `f : R \u2192+* S` with `I \u2264 f\u207b\u00b9(J)` -/\ndef quotient_map {I : ideal R} (J : ideal S) (f : R \u2192+* S) (hIJ : I \u2264 J.comap f) :\n  R \u29f8 I \u2192+* S \u29f8 J :=\n(quotient.lift I ((quotient.mk J).comp f) (\u03bb _ ha,\n  by simpa [function.comp_app, ring_hom.coe_comp, quotient.eq_zero_iff_mem] using hIJ ha))\n\n@[simp]\nlemma quotient_map_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R} : quotient_map I f H (quotient.mk J x) = quotient.mk I (f x) :=\nquotient.lift_mk J _ _\n\n@[simp]\nlemma quotient_map_algebra_map {J : ideal A} {I : ideal S} {f : A \u2192+* S} {H : J \u2264 I.comap f}\n  {x : R\u2081} :\n  quotient_map I f H (algebra_map R\u2081 (A \u29f8 J) x) = quotient.mk I (f (algebra_map _ _ x)) :=\nquotient.lift_mk J _ _\n\nlemma quotient_map_comp_mk {J : ideal R} {I : ideal S} {f : R \u2192+* S} (H : J \u2264 I.comap f) :\n  (quotient_map I f H).comp (quotient.mk J) = (quotient.mk I).comp f :=\nring_hom.ext (\u03bb x, by simp only [function.comp_app, ring_hom.coe_comp, ideal.quotient_map_mk])\n\n/-- The ring equiv `R/I \u2243+* S/J` induced by a ring equiv `f : R \u2243+** S`,  where `J = f(I)`. -/\n@[simps]\ndef quotient_equiv (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S)) :\n  R \u29f8 I \u2243+* S \u29f8 J :=\n{ inv_fun := quotient_map I \u2191f.symm (by {rw hIJ, exact le_of_eq (map_comap_of_equiv I f)}),\n  left_inv := by {rintro \u27e8r\u27e9, simp },\n  right_inv := by {rintro \u27e8s\u27e9, simp },\n  ..quotient_map J \u2191f (by {rw hIJ, exact @le_comap_map _ S _ _ _ _ _ _}) }\n\n@[simp]\nlemma quotient_equiv_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S) (hIJ : J = I.map (f : R \u2192+* S))\n  (x : R) : quotient_equiv I J f hIJ (ideal.quotient.mk I x) = ideal.quotient.mk J (f x) := rfl\n\n@[simp]\nlemma quotient_equiv_symm_mk (I : ideal R) (J : ideal S) (f : R \u2243+* S)\n  (hIJ : J = I.map (f : R \u2192+* S)) (x : S) :\n  (quotient_equiv I J f hIJ).symm (ideal.quotient.mk J x) = ideal.quotient.mk I (f.symm x) := rfl\n\n/-- `H` and `h` are kept as separate hypothesis since H is used in constructing the quotient map. -/\nlemma quotient_map_injective' {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (h : I.comap f \u2264 J) : function.injective (quotient_map I f H) :=\nbegin\n  refine (injective_iff_map_eq_zero (quotient_map I f H)).2 (\u03bb a ha, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  rw [quotient_map_mk, quotient.eq_zero_iff_mem] at ha,\n  exact (quotient.eq_zero_iff_mem).mpr (h ha),\nend\n\n/-- If we take `J = I.comap f` then `quotient_map` is injective automatically. -/\nlemma quotient_map_injective {I : ideal S} {f : R \u2192+* S} :\n  function.injective (quotient_map I f le_rfl) :=\nquotient_map_injective' le_rfl\n\nlemma quotient_map_surjective {J : ideal R} {I : ideal S} {f : R \u2192+* S} {H : J \u2264 I.comap f}\n  (hf : function.surjective f) : function.surjective (quotient_map I f H) :=\n\u03bb x, let \u27e8x, hx\u27e9 := quotient.mk_surjective x in\n  let \u27e8y, hy\u27e9 := hf x in \u27e8(quotient.mk J) y, by simp [hx, hy]\u27e9\n\n/-- Commutativity of a square is preserved when taking quotients by an ideal. -/\nlemma comp_quotient_map_eq_of_comp_eq {R' S' : Type*} [comm_ring R'] [comm_ring S']\n  {f : R \u2192+* S} {f' : R' \u2192+* S'} {g : R \u2192+* R'} {g' : S \u2192+* S'} (hfg : f'.comp g = g'.comp f)\n  (I : ideal S') : (quotient_map I g' le_rfl).comp (quotient_map (I.comap g') f le_rfl) =\n    (quotient_map I f' le_rfl).comp (quotient_map (I.comap f') g\n      (le_of_eq (trans (comap_comap f g') (hfg \u25b8 (comap_comap g f'))))) :=\nbegin\n  refine ring_hom.ext (\u03bb a, _),\n  obtain \u27e8r, rfl\u27e9 := quotient.mk_surjective a,\n  simp only [ring_hom.comp_apply, quotient_map_mk],\n  exact congr_arg (quotient.mk I) (trans (g'.comp_apply f r).symm (hfg \u25b8 (f'.comp_apply g r))),\nend\n\n/-- The algebra hom `A/I \u2192+* B/J` induced by an algebra hom `f : A \u2192\u2090[R\u2081] B` with `I \u2264 f\u207b\u00b9(J)`. -/\ndef quotient_map\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (hIJ : I \u2264 J.comap f) :\n  A \u29f8 I \u2192\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_map J (f : A \u2192+* B) hIJ }\n\n@[simp]\nlemma quotient_map_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f)\n  {x : A} : quotient_map\u2090 J f H (quotient.mk I x) = quotient.mk\u2090 R\u2081 J (f x) := rfl\n\nlemma quotient_map_comp_mk\u2090 {I : ideal A} (J : ideal B) (f : A \u2192\u2090[R\u2081] B) (H : I \u2264 J.comap f) :\n  (quotient_map\u2090 J f H).comp (quotient.mk\u2090 R\u2081 I) = (quotient.mk\u2090 R\u2081 J).comp f :=\nalg_hom.ext (\u03bb x, by simp only [quotient_map_mk\u2090, quotient.mk\u2090_eq_mk, alg_hom.comp_apply])\n\n/-- The algebra equiv `A/I \u2243\u2090[R] B/J` induced by an algebra equiv `f : A \u2243\u2090[R] B`,\nwhere`J = f(I)`. -/\ndef quotient_equiv_alg (I : ideal A) (J : ideal B) (f : A \u2243\u2090[R\u2081] B)\n  (hIJ : J = I.map (f : A \u2192+* B)) :\n  (A \u29f8 I) \u2243\u2090[R\u2081] B \u29f8 J :=\n{ commutes' := \u03bb r, by simp,\n  ..quotient_equiv I J (f : A \u2243+* B) hIJ }\n\n@[priority 100]\ninstance quotient_algebra {I : ideal A} [algebra R A] :\n  algebra (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I) :=\n(quotient_map I (algebra_map R A) (le_of_eq rfl)).to_algebra\n\nlemma algebra_map_quotient_injective {I : ideal A} [algebra R A]:\n  function.injective (algebra_map (R \u29f8 I.comap (algebra_map R A)) (A \u29f8 I)) :=\nbegin\n  rintros \u27e8a\u27e9 \u27e8b\u27e9 hab,\n  replace hab := quotient.eq.mp hab,\n  rw \u2190 ring_hom.map_sub at hab,\n  exact quotient.eq.mpr hab\nend\n\nend quotient_algebra\n\nend comm_ring\n\nend ideal\n\nnamespace submodule\n\nvariables {R : Type u} {M : Type v}\nvariables [comm_semiring R] [add_comm_monoid M] [module R M]\n\n-- TODO: show `[algebra R A] : algebra (ideal R) A` too\n\ninstance module_submodule : module (ideal R) (submodule R M) :=\n{ smul_add := smul_sup,\n  add_smul := sup_smul,\n  mul_smul := submodule.smul_assoc,\n  one_smul := by simp,\n  zero_smul := bot_smul,\n  smul_zero := smul_bot }\n\nend submodule\n\nnamespace ring_hom\nvariables {A B C : Type*} [ring A] [ring B] [ring C]\nvariables (f : A \u2192+* B) (f_inv : B \u2192 A)\n\n/-- Auxiliary definition used to define `lift_of_right_inverse` -/\ndef lift_of_right_inverse_aux\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) :\n  B \u2192+* C :=\n{ to_fun := \u03bb b, g (f_inv b),\n  map_one' :=\n  begin\n    rw [\u2190 g.map_one, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_one],\n    exact hf 1\n  end,\n  map_mul' :=\n  begin\n    intros x y,\n    rw [\u2190 g.map_mul, \u2190 sub_eq_zero, \u2190 g.map_sub, \u2190 g.mem_ker],\n    apply hg,\n    rw [f.mem_ker, f.map_sub, sub_eq_zero, f.map_mul],\n    simp only [hf _],\n  end,\n  .. add_monoid_hom.lift_of_right_inverse f.to_add_monoid_hom f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 }\n\n@[simp] lemma lift_of_right_inverse_aux_comp_apply\n  (hf : function.right_inverse f_inv f) (g : A \u2192+* C) (hg : f.ker \u2264 g.ker) (a : A) :\n  (f.lift_of_right_inverse_aux f_inv hf g hg) (f a) = g a :=\nf.to_add_monoid_hom.lift_of_right_inverse_comp_apply f_inv hf \u27e8g.to_add_monoid_hom, hg\u27e9 a\n\n/-- `lift_of_right_inverse f hf g hg` is the unique ring homomorphism `\u03c6`\n\n* such that `\u03c6.comp f = g` (`ring_hom.lift_of_right_inverse_comp`),\n* where `f : A \u2192+* B` is has a right_inverse `f_inv` (`hf`),\n* and `g : B \u2192+* C` satisfies `hg : f.ker \u2264 g.ker`.\n\nSee `ring_hom.eq_lift_of_right_inverse` for the uniqueness lemma.\n\n```\n   A .\n   |  \\\n f |   \\ g\n   |    \\\n   v     \\\u231f\n   B ----> C\n      \u2203!\u03c6\n```\n-/\ndef lift_of_right_inverse\n  (hf : function.right_inverse f_inv f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\n{ to_fun := \u03bb g, f.lift_of_right_inverse_aux f_inv hf g.1 g.2,\n  inv_fun := \u03bb \u03c6, \u27e8\u03c6.comp f, \u03bb x hx, (mem_ker _).mpr $ by simp [(mem_ker _).mp hx]\u27e9,\n  left_inv := \u03bb g, by\n  { ext,\n    simp only [comp_apply, lift_of_right_inverse_aux_comp_apply, subtype.coe_mk,\n      subtype.val_eq_coe], },\n  right_inv := \u03bb \u03c6, by\n  { ext b,\n    simp [lift_of_right_inverse_aux, hf b], } }\n\n/-- A non-computable version of `ring_hom.lift_of_right_inverse` for when no computable right\ninverse is available, that uses `function.surj_inv`. -/\n@[simp]\nnoncomputable abbreviation lift_of_surjective\n  (hf : function.surjective f) : {g : A \u2192+* C // f.ker \u2264 g.ker} \u2243 (B \u2192+* C) :=\nf.lift_of_right_inverse (function.surj_inv hf) (function.right_inverse_surj_inv hf)\n\nlemma lift_of_right_inverse_comp_apply\n  (hf : function.right_inverse f_inv f) (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) (x : A) :\n  (f.lift_of_right_inverse f_inv hf g) (f x) = g x :=\nf.lift_of_right_inverse_aux_comp_apply f_inv hf g.1 g.2 x\n\nlemma lift_of_right_inverse_comp (hf : function.right_inverse f_inv f)\n  (g : {g : A \u2192+* C // f.ker \u2264 g.ker}) :\n  (f.lift_of_right_inverse f_inv hf g).comp f = g :=\nring_hom.ext $ f.lift_of_right_inverse_comp_apply f_inv hf g\n\nlemma eq_lift_of_right_inverse (hf : function.right_inverse f_inv f) (g : A \u2192+* C)\n  (hg : f.ker \u2264 g.ker) (h : B \u2192+* C) (hh : h.comp f = g) :\n  h = (f.lift_of_right_inverse f_inv hf \u27e8g, hg\u27e9) :=\nbegin\n  simp_rw \u2190hh,\n  exact ((f.lift_of_right_inverse f_inv hf).apply_symm_apply _).symm,\nend\n\nend ring_hom\n\nnamespace double_quot\nopen ideal\nvariables {R : Type u} [comm_ring R] (I J : ideal R)\n\n/-- The obvious ring hom `R/I \u2192 R/(I \u2294 J)` -/\ndef quot_left_to_quot_sup : R \u29f8 I \u2192+* R \u29f8 (I \u2294 J) :=\nideal.quotient.factor I (I \u2294 J) le_sup_left\n\n/-- The kernel of `quot_left_to_quot_sup` -/\nlemma ker_quot_left_to_quot_sup :\n  (quot_left_to_quot_sup I J).ker = J.map (ideal.quotient.mk I) :=\nby simp only [mk_ker, sup_idem, sup_comm, quot_left_to_quot_sup, quotient.factor, ker_quotient_lift,\n    map_eq_iff_sup_ker_eq_of_surjective I^.quotient.mk quotient.mk_surjective, \u2190 sup_assoc]\n\n/-- The ring homomorphism `(R/I)/J' -> R/(I \u2294 J)` induced by `quot_left_to_quot_sup` where `J'`\n  is the image of `J` in `R/I`-/\ndef quot_quot_to_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2192+* R \u29f8 I \u2294 J :=\nby exact ideal.quotient.lift (J.map (ideal.quotient.mk I)) (quot_left_to_quot_sup I J)\n  (ker_quot_left_to_quot_sup I J).symm.le\n\n/-- The composite of the maps `R \u2192 (R/I)` and `(R/I) \u2192 (R/I)/J'` -/\ndef quot_quot_mk : R \u2192+* ((R \u29f8 I) \u29f8 J.map I^.quotient.mk) :=\nby exact ((J.map I^.quotient.mk)^.quotient.mk).comp I^.quotient.mk\n\n/-- The kernel of `quot_quot_mk` -/\nlemma ker_quot_quot_mk : (quot_quot_mk I J).ker = I \u2294 J :=\nby rw [ring_hom.ker_eq_comap_bot, quot_quot_mk, \u2190 comap_comap, \u2190 ring_hom.ker, mk_ker,\n  comap_map_of_surjective (ideal.quotient.mk I) (quotient.mk_surjective), \u2190 ring_hom.ker, mk_ker,\n  sup_comm]\n\n/-- The ring homomorphism `R/(I \u2294 J) \u2192 (R/I)/J' `induced by `quot_quot_mk` -/\ndef lift_sup_quot_quot_mk (I J : ideal R) :\n  R \u29f8 (I \u2294 J) \u2192+* (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) :=\nideal.quotient.lift (I \u2294 J) (quot_quot_mk I J) (ker_quot_quot_mk I J).symm.le\n\n/-- `quot_quot_to_quot_add` and `lift_sup_double_qot_mk` are inverse isomorphisms -/\ndef quot_quot_equiv_quot_sup : (R \u29f8 I) \u29f8 J.map (ideal.quotient.mk I) \u2243+* R \u29f8 I \u2294 J :=\nring_equiv.of_hom_inv (quot_quot_to_quot_sup I J) (lift_sup_quot_quot_mk I J)\n  (by { ext z, refl }) (by { ext z, refl })\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_quot_quot_mk (x : R) :\n  quot_quot_equiv_quot_sup I J (quot_quot_mk I J x) = ideal.quotient.mk (I \u2294 J) x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_quot_sup_symm_quot_quot_mk (x : R) :\n  (quot_quot_equiv_quot_sup I J).symm (ideal.quotient.mk (I \u2294 J) x) = quot_quot_mk I J x :=\nrfl\n\n/-- The obvious isomorphism `(R/I)/J' \u2192 (R/J)/I' `   -/\ndef quot_quot_equiv_comm :\n  (R \u29f8 I) \u29f8 J.map I^.quotient.mk \u2243+* (R \u29f8 J) \u29f8 I.map J^.quotient.mk :=\n((quot_quot_equiv_quot_sup I J).trans (quot_equiv_of_eq sup_comm)).trans\n  (quot_quot_equiv_quot_sup J I).symm\n\n@[simp]\nlemma quot_quot_equiv_comm_quot_quot_mk (x : R) :\n  quot_quot_equiv_comm I J (quot_quot_mk I J x) = quot_quot_mk J I x :=\nrfl\n\n@[simp]\nlemma quot_quot_equiv_comm_comp_quot_quot_mk :\n  ring_hom.comp \u2191(quot_quot_equiv_comm I J) (quot_quot_mk I J) = quot_quot_mk J I :=\nring_hom.ext $ quot_quot_equiv_comm_quot_quot_mk I J\n\n@[simp]\nlemma quot_quot_equiv_comm_symm :\n  (quot_quot_equiv_comm I J).symm = quot_quot_equiv_comm J I :=\nrfl\n\nend double_quot\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/ring_theory/ideal/operations.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.35344440960451634}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.monoidal.category\n\n/-!\n# Monoidal opposites\n\nWe write `C\u1d50\u1d52\u1d56` for the monoidal opposite of a monoidal category `C`.\n-/\n\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nvariables {C : Type u\u2081}\n\nnamespace category_theory\n\nopen category_theory.monoidal_category\n\n/-- A type synonym for the monoidal opposite. Use the notation `C\u1d39\u1d52\u1d56`. -/\n@[nolint has_inhabited_instance]\ndef monoidal_opposite (C : Type u\u2081) := C\n\nnamespace monoidal_opposite\n\nnotation C `\u1d39\u1d52\u1d56`:std.prec.max_plus := monoidal_opposite C\n\n/-- Think of an object of `C` as an object of `C\u1d39\u1d52\u1d56`. -/\n@[pp_nodot]\ndef mop (X : C) : C\u1d39\u1d52\u1d56 := X\n\n/-- Think of an object of `C\u1d39\u1d52\u1d56` as an object of `C`. -/\n@[pp_nodot]\ndef unmop (X : C\u1d39\u1d52\u1d56) : C := X\n\nlemma op_injective : function.injective (mop : C \u2192 C\u1d39\u1d52\u1d56) := \u03bb _ _, id\nlemma unop_injective : function.injective (unmop : C\u1d39\u1d52\u1d56 \u2192 C) := \u03bb _ _, id\n\n@[simp] lemma op_inj_iff (x y : C) : mop x = mop y \u2194 x = y := iff.rfl\n@[simp] \n\nattribute [irreducible] monoidal_opposite\n\n@[simp] lemma mop_unmop (X : C\u1d39\u1d52\u1d56) : mop (unmop X) = X := rfl\n@[simp] lemma unmop_mop (X : C) : unmop (mop X) = X := rfl\n\ninstance monoidal_opposite_category [I : category.{v\u2081} C] : category C\u1d39\u1d52\u1d56 :=\n{ hom := \u03bb X Y, unmop X \u27f6 unmop Y,\n  id := \u03bb X, \ud835\udfd9 (unmop X),\n  comp := \u03bb X Y Z f g, f \u226b g, }\n\nend monoidal_opposite\n\nend category_theory\n\nopen category_theory\nopen category_theory.monoidal_opposite\n\nvariables [category.{v\u2081} C]\n\n/-- The monoidal opposite of a morphism `f : X \u27f6 Y` is just `f`, thought of as `mop X \u27f6 mop Y`. -/\ndef quiver.hom.mop {X Y : C} (f : X \u27f6 Y) : @quiver.hom C\u1d39\u1d52\u1d56 _ (mop X) (mop Y) := f\n/-- We can think of a morphism `f : mop X \u27f6 mop Y` as a morphism `X \u27f6 Y`. -/\ndef quiver.hom.unmop {X Y : C\u1d39\u1d52\u1d56} (f : X \u27f6 Y) : unmop X \u27f6 unmop Y := f\n\nnamespace category_theory\n\nlemma mop_inj {X Y : C} :\n  function.injective (quiver.hom.mop : (X \u27f6 Y) \u2192 (mop X \u27f6 mop Y)) :=\n\u03bb _ _ H, congr_arg quiver.hom.unmop H\n\nlemma unmop_inj {X Y : C\u1d39\u1d52\u1d56} :\n  function.injective (quiver.hom.unmop : (X \u27f6 Y) \u2192 (unmop X \u27f6 unmop Y)) :=\n\u03bb _ _ H, congr_arg quiver.hom.mop H\n\n@[simp] lemma unmop_mop {X Y : C} {f : X \u27f6 Y} : f.mop.unmop = f := rfl\n@[simp] lemma mop_unmop {X Y : C\u1d39\u1d52\u1d56} {f : X \u27f6 Y} : f.unmop.mop = f := rfl\n\n@[simp] lemma mop_comp {X Y Z : C} {f : X \u27f6 Y} {g : Y \u27f6 Z} :\n  (f \u226b g).mop = f.mop \u226b g.mop := rfl\n@[simp] lemma mop_id {X : C} : (\ud835\udfd9 X).mop = \ud835\udfd9 (mop X) := rfl\n\n@[simp] lemma unmop_comp {X Y Z : C\u1d39\u1d52\u1d56} {f : X \u27f6 Y} {g : Y \u27f6 Z} :\n  (f \u226b g).unmop = f.unmop \u226b g.unmop := rfl\n@[simp] lemma unmop_id {X : C\u1d39\u1d52\u1d56} : (\ud835\udfd9 X).unmop = \ud835\udfd9 (unmop X) := rfl\n\n@[simp] lemma unmop_id_mop {X : C} : (\ud835\udfd9 (mop X)).unmop = \ud835\udfd9 X := rfl\n@[simp] lemma mop_id_unmop {X : C\u1d39\u1d52\u1d56} : (\ud835\udfd9 (unmop X)).mop = \ud835\udfd9 X := rfl\n\nnamespace iso\n\nvariables {X Y : C}\n\n/-- An isomorphism in `C` gives an isomorphism in `C\u1d39\u1d52\u1d56`. -/\n@[simps]\ndef mop (f : X \u2245 Y) : mop X \u2245 mop Y :=\n{ hom := f.hom.mop,\n  inv := f.inv.mop,\n  hom_inv_id' := unmop_inj f.hom_inv_id,\n  inv_hom_id' := unmop_inj f.inv_hom_id }\n\nend iso\n\nvariables [monoidal_category.{v\u2081} C]\n\nopen opposite monoidal_category\n\ninstance monoidal_category_op : monoidal_category C\u1d52\u1d56 :=\n{ tensor_obj := \u03bb X Y, op (unop X \u2297 unop Y),\n  tensor_hom := \u03bb X\u2081 Y\u2081 X\u2082 Y\u2082 f g, (f.unop \u2297 g.unop).op,\n  tensor_unit := op (\ud835\udfd9_ C),\n  associator := \u03bb X Y Z, (\u03b1_ (unop X) (unop Y) (unop Z)).symm.op,\n  left_unitor := \u03bb X, (\u03bb_ (unop X)).symm.op,\n  right_unitor := \u03bb X, (\u03c1_ (unop X)).symm.op,\n  associator_naturality' :=\n  begin\n    intros,\n    apply quiver.hom.unop_inj,\n    simp [associator_inv_naturality],\n  end,\n  left_unitor_naturality' :=\n  begin\n    intros,\n    apply quiver.hom.unop_inj,\n    simp [left_unitor_inv_naturality],\n  end,\n  right_unitor_naturality' :=\n  begin\n    intros,\n    apply quiver.hom.unop_inj,\n    simp [right_unitor_inv_naturality],\n  end,\n  triangle' :=\n  begin\n    intros,\n    apply quiver.hom.unop_inj,\n    dsimp,\n    simp,\n  end,\n  pentagon' :=\n  begin\n    intros,\n    apply quiver.hom.unop_inj,\n    dsimp,\n    simp [pentagon_inv],\n  end }\n\nlemma op_tensor_obj (X Y : C\u1d52\u1d56) : X \u2297 Y = op (unop X \u2297 unop Y) := rfl\nlemma op_tensor_unit : (\ud835\udfd9_ C\u1d52\u1d56) = op (\ud835\udfd9_ C) := rfl\n\ninstance monoidal_category_mop : monoidal_category C\u1d39\u1d52\u1d56 :=\n{ tensor_obj := \u03bb X Y, mop (unmop Y \u2297 unmop X),\n  tensor_hom := \u03bb X\u2081 Y\u2081 X\u2082 Y\u2082 f g, (g.unmop \u2297 f.unmop).mop,\n  tensor_unit := mop (\ud835\udfd9_ C),\n  associator := \u03bb X Y Z, (\u03b1_ (unmop Z) (unmop Y) (unmop X)).symm.mop,\n  left_unitor := \u03bb X, (\u03c1_ (unmop X)).mop,\n  right_unitor := \u03bb X, (\u03bb_ (unmop X)).mop,\n  associator_naturality' :=\n  begin\n    intros,\n    apply unmop_inj,\n    simp [associator_inv_naturality],\n  end,\n  left_unitor_naturality' :=\n  begin\n    intros,\n    apply unmop_inj,\n    simp [right_unitor_naturality],\n  end,\n  right_unitor_naturality' :=\n  begin\n    intros,\n    apply unmop_inj,\n    simp [left_unitor_naturality],\n  end,\n  triangle' :=\n  begin\n    intros,\n    apply unmop_inj,\n    dsimp,\n    simp,\n  end,\n  pentagon' :=\n  begin\n    intros,\n    apply unmop_inj,\n    dsimp,\n    simp [pentagon_inv],\n  end }\n\nlemma mop_tensor_obj (X Y : C\u1d39\u1d52\u1d56) : X \u2297 Y = mop (unmop Y \u2297 unmop X) := rfl\nlemma mop_tensor_unit : (\ud835\udfd9_ C\u1d39\u1d52\u1d56) = mop (\ud835\udfd9_ C) := rfl\n\nend category_theory\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/category_theory/monoidal/opposite.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3534444011817134}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport category_theory.limits.shapes.equalizers\nimport category_theory.limits.shapes.comm_sq\nimport category_theory.limits.shapes.regular_mono\n\n/-!\n# Kernel pairs\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines what it means for a parallel pair of morphisms `a b : R \u27f6 X` to be the kernel pair\nfor a morphism `f`.\nSome properties of kernel pairs are given, namely allowing one to transfer between\nthe kernel pair of `f\u2081 \u226b f\u2082` to the kernel pair of `f\u2081`.\nIt is also proved that if `f` is a coequalizer of some pair, and `a`,`b` is a kernel pair for `f`\nthen it is a coequalizer of `a`,`b`.\n\n## Implementation\n\nThe definition is essentially just a wrapper for `is_limit (pullback_cone.mk _ _ _)`, but the\nconstructions given here are useful, yet awkward to present in that language, so a basic API\nis developed here.\n\n## TODO\n\n- Internal equivalence relations (or congruences) and the fact that every kernel pair induces one,\n  and the converse in an effective regular category (WIP by b-mehta).\n\n-/\n\nuniverses v u u\u2082\n\nnamespace category_theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u} [category.{v} C]\n\nvariables {R X Y Z : C} (f : X \u27f6 Y) (a b : R \u27f6 X)\n\n/--\n`is_kernel_pair f a b` expresses that `(a, b)` is a kernel pair for `f`, i.e. `a \u226b f = b \u226b f`\nand the square\n  R \u2192 X\n  \u2193   \u2193\n  X \u2192 Y\nis a pullback square.\nThis is just an abbreviation for `is_pullback a b f f`.\n-/\nabbreviation is_kernel_pair := is_pullback a b f f\n\nnamespace is_kernel_pair\n\n/-- The data expressing that `(a, b)` is a kernel pair is subsingleton. -/\ninstance : subsingleton (is_kernel_pair f a b) :=\n\u27e8\u03bb P Q, by { cases P, cases Q, congr, }\u27e9\n\n/-- If `f` is a monomorphism, then `(\ud835\udfd9 _, \ud835\udfd9 _)`  is a kernel pair for `f`. -/\nlemma id_of_mono [mono f] : is_kernel_pair f (\ud835\udfd9 _) (\ud835\udfd9 _) :=\n\u27e8\u27e8rfl\u27e9, \u27e8pullback_cone.is_limit_mk_id_id _\u27e9\u27e9\n\ninstance [mono f] : inhabited (is_kernel_pair f (\ud835\udfd9 _) (\ud835\udfd9 _)) := \u27e8id_of_mono f\u27e9\n\nvariables {f a b}\n\n/--\nGiven a pair of morphisms `p`, `q` to `X` which factor through `f`, they factor through any kernel\npair of `f`.\n-/\nnoncomputable\ndef lift' {S : C} (k : is_kernel_pair f a b) (p q : S \u27f6 X) (w : p \u226b f = q \u226b f) :\n  { t : S \u27f6 R // t \u226b a = p \u2227 t \u226b b = q } :=\npullback_cone.is_limit.lift' k.is_limit _ _ w\n\n/--\nIf `(a,b)` is a kernel pair for `f\u2081 \u226b f\u2082` and `a \u226b f\u2081 = b \u226b f\u2081`, then `(a,b)` is a kernel pair for\njust `f\u2081`.\nThat is, to show that `(a,b)` is a kernel pair for `f\u2081` it suffices to only show the square\ncommutes, rather than to additionally show it's a pullback.\n-/\nlemma cancel_right {f\u2081 : X \u27f6 Y} {f\u2082 : Y \u27f6 Z} (comm : a \u226b f\u2081 = b \u226b f\u2081)\n  (big_k : is_kernel_pair (f\u2081 \u226b f\u2082) a b) :\n  is_kernel_pair f\u2081 a b :=\n{ w := comm,\n  is_limit' := \u27e8pullback_cone.is_limit_aux' _ $ \u03bb s,\n  begin\n    let s' : pullback_cone (f\u2081 \u226b f\u2082) (f\u2081 \u226b f\u2082) :=\n      pullback_cone.mk s.fst s.snd (s.condition_assoc _),\n    refine \u27e8big_k.is_limit.lift s',\n            big_k.is_limit.fac _ walking_cospan.left,\n            big_k.is_limit.fac _ walking_cospan.right,\n            \u03bb m m\u2081 m\u2082, _\u27e9,\n    apply big_k.is_limit.hom_ext,\n    refine ((pullback_cone.mk a b _) : pullback_cone (f\u2081 \u226b f\u2082) _).equalizer_ext _ _,\n    apply m\u2081.trans (big_k.is_limit.fac s' walking_cospan.left).symm,\n    apply m\u2082.trans (big_k.is_limit.fac s' walking_cospan.right).symm,\n  end\u27e9 }\n\n/--\nIf `(a,b)` is a kernel pair for `f\u2081 \u226b f\u2082` and `f\u2082` is mono, then `(a,b)` is a kernel pair for\njust `f\u2081`.\nThe converse of `comp_of_mono`.\n-/\nlemma cancel_right_of_mono {f\u2081 : X \u27f6 Y} {f\u2082 : Y \u27f6 Z} [mono f\u2082]\n  (big_k : is_kernel_pair (f\u2081 \u226b f\u2082) a b) :\n  is_kernel_pair f\u2081 a b :=\ncancel_right (begin rw [\u2190 cancel_mono f\u2082, assoc, assoc, big_k.w] end) big_k\n\n/--\nIf `(a,b)` is a kernel pair for `f\u2081` and `f\u2082` is mono, then `(a,b)` is a kernel pair for `f\u2081 \u226b f\u2082`.\nThe converse of `cancel_right_of_mono`.\n-/\nlemma comp_of_mono {f\u2081 : X \u27f6 Y} {f\u2082 : Y \u27f6 Z} [mono f\u2082] (small_k : is_kernel_pair f\u2081 a b) :\n  is_kernel_pair (f\u2081 \u226b f\u2082) a b :=\n{ w := by rw [small_k.w_assoc],\n  is_limit' := \u27e8pullback_cone.is_limit_aux' _ $ \u03bb s,\n  begin\n    refine \u27e8_, _, _, _\u27e9,\n    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).1,\n    rw [\u2190 cancel_mono f\u2082, assoc, s.condition, assoc],\n    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.1,\n    apply (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.2,\n    intros m m\u2081 m\u2082,\n    apply small_k.is_limit.hom_ext,\n    refine ((pullback_cone.mk a b _) : pullback_cone f\u2081 _).equalizer_ext _ _,\n    { exact m\u2081.trans (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.1.symm },\n    { exact m\u2082.trans (pullback_cone.is_limit.lift' small_k.is_limit s.fst s.snd _).2.2.symm },\n  end\u27e9 }\n\n/--\nIf `(a,b)` is the kernel pair of `f`, and `f` is a coequalizer morphism for some parallel pair, then\n`f` is a coequalizer morphism of `a` and `b`.\n-/\ndef to_coequalizer (k : is_kernel_pair f a b) [r : regular_epi f] :\n  is_colimit (cofork.of_\u03c0 f k.w) :=\nbegin\n  let t := k.is_limit.lift (pullback_cone.mk _ _ r.w),\n  have ht : t \u226b a = r.left := k.is_limit.fac _ walking_cospan.left,\n  have kt : t \u226b b = r.right := k.is_limit.fac _ walking_cospan.right,\n  apply cofork.is_colimit.mk _ _ _ _,\n  { intro s,\n    apply (cofork.is_colimit.desc' r.is_colimit s.\u03c0 _).1,\n    rw [\u2190 ht, assoc, s.condition, reassoc_of kt] },\n  { intro s,\n    apply (cofork.is_colimit.desc' r.is_colimit s.\u03c0 _).2 },\n  { intros s m w,\n    apply r.is_colimit.hom_ext,\n    rintro \u27e8\u27e9,\n    change (r.left \u226b f) \u226b m = (r.left \u226b f) \u226b _,\n    rw [assoc, assoc],\n    congr' 1,\n    erw (cofork.is_colimit.desc' r.is_colimit s.\u03c0 _).2,\n    apply w,\n    erw (cofork.is_colimit.desc' r.is_colimit s.\u03c0 _).2,\n    apply w }\nend\n\n/-- If `a\u2081 a\u2082 : A \u27f6 Y` is a kernel pair for `g : Y \u27f6 Z`, then `a\u2081 \u00d7[Z] X` and `a\u2082 \u00d7[Z] X`\n(`A \u00d7[Z] X \u27f6 Y \u00d7[Z] X`) is a kernel pair for `Y \u00d7[Z] X \u27f6 X`. -/\nprotected\nlemma pullback {X Y Z A : C} {g : Y \u27f6 Z} {a\u2081 a\u2082 : A \u27f6 Y}\n  (h : is_kernel_pair g a\u2081 a\u2082) (f : X \u27f6 Z) [has_pullback f g] [has_pullback f (a\u2081 \u226b g)] :\n    is_kernel_pair (pullback.fst : pullback f g \u27f6 X)\n      (pullback.map f _ f _ (\ud835\udfd9 X) a\u2081 (\ud835\udfd9 Z) (by simp) $ category.comp_id _)\n      (pullback.map _ _ _ _ (\ud835\udfd9 X) a\u2082 (\ud835\udfd9 Z) (by simp) $ (category.comp_id _).trans h.1.1) :=\nbegin\n  refine \u27e8\u27e8_\u27e9, \u27e8_\u27e9\u27e9,\n  { rw [pullback.lift_fst, pullback.lift_fst] },\n  { fapply pullback_cone.is_limit_aux',\n    intro s,\n    refine \u27e8pullback.lift (s.fst \u226b pullback.fst)\n      (h.lift' (s.fst \u226b pullback.snd) (s.snd \u226b pullback.snd) _).1 _, _, _, _\u27e9,\n    { simp_rw [category.assoc, \u2190 pullback.condition, \u2190 category.assoc, s.condition] },\n    { rw [\u2190 category.assoc, (h.lift' _ _ _).2.1, category.assoc,\n        category.assoc, pullback.condition] },\n    { rw limits.pullback_cone.mk_fst,\n      ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd, pullback.lift_snd_assoc,\n        category.comp_id, (h.lift' _ _ _).2.1] },\n    { rw limits.pullback_cone.mk_snd,\n      ext; simp only [category.assoc, pullback.lift_fst, pullback.lift_snd, pullback.lift_snd_assoc,\n        category.comp_id, (h.lift' _ _ _).2.2, s.condition] },\n    { intros m h\u2081 h\u2082,\n      ext,\n      { rw pullback.lift_fst,\n        conv_rhs { rw [\u2190 h\u2081, category.assoc, pullback_cone.mk_fst] },\n        congr' 1,\n        refine ((pullback.lift_fst _ _ _).trans $ category.comp_id _).symm },\n      { rw pullback.lift_snd,\n        apply pullback_cone.is_limit.hom_ext h.is_limit;\n          dsimp only [is_pullback.cone, comm_sq.cone];\n          simp only [pullback_cone.mk_fst, pullback_cone.mk_snd, category.assoc,\n            (h.lift' _ _ _).2.1, (h.lift' _ _ _).2.2],\n        { conv_rhs { rw [\u2190 h\u2081, category.assoc, pullback_cone.mk_fst, pullback.lift_snd] } },\n        { conv_rhs { rw [\u2190 h\u2082, category.assoc, pullback_cone.mk_snd, pullback.lift_snd] } } } } }\nend\n\nlemma mono_of_is_iso_fst (h : is_kernel_pair f a b) [is_iso a] :\n  mono f :=\nbegin\n  obtain \u27e8l, h\u2081, h\u2082\u27e9 := limits.pullback_cone.is_limit.lift' h.is_limit (\ud835\udfd9 _) (\ud835\udfd9 _) (by simp [h.w]),\n  rw [is_pullback.cone_fst, \u2190 is_iso.eq_comp_inv, category.id_comp] at h\u2081,\n  rw [h\u2081, is_iso.inv_comp_eq, category.comp_id] at h\u2082,\n  constructor,\n  intros Z g\u2081 g\u2082 e,\n  obtain \u27e8l', rfl, rfl\u27e9 := limits.pullback_cone.is_limit.lift' h.is_limit _ _ e,\n  rw [is_pullback.cone_fst, h\u2082],\nend\n\nlemma is_iso_of_mono (h : is_kernel_pair f a b) [mono f] :\n  is_iso a :=\nbegin\n  rw \u2190 show _ = a, from (category.comp_id _).symm.trans ((is_kernel_pair.id_of_mono f)\n    .is_limit.cone_point_unique_up_to_iso_inv_comp h.is_limit walking_cospan.left),\n  apply_instance,\nend\n\nlemma of_is_iso_of_mono [is_iso a] [mono f] : is_kernel_pair f a a :=\nbegin\n  delta is_kernel_pair,\n  convert_to is_pullback a (a \u226b \ud835\udfd9 X) (\ud835\udfd9 X \u226b f) f,\n  { rw category.comp_id }, { rw category.id_comp },\n  exact (is_pullback.of_horiz_is_iso \u27e8rfl\u27e9).paste_vert (is_kernel_pair.id_of_mono f)\nend\n\nend is_kernel_pair\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/limits/shapes/kernel_pair.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.35344439275891026}}
{"text": "/-\nCopyright (c) 2019 Seul Baek. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Seul Baek\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.omega.clause\nimport Mathlib.tactic.omega.int.form\nimport Mathlib.PostPort\n\nnamespace Mathlib\n\n/-\nDNF transformation.\n-/\n\nnamespace omega\n\n\nnamespace int\n\n\n/-- push_neg p returns the result of normalizing \u00ac p by\n    pushing the outermost negation all the way down,\n    until it reaches either a negation or an atom -/\n@[simp] def push_neg : preform \u2192 preform :=\n  sorry\n\ntheorem push_neg_equiv {p : preform} : preform.equiv (push_neg p) (preform.not p) := sorry\n\n/-- NNF transformation -/\ndef nnf : preform \u2192 preform :=\n  sorry\n\ndef is_nnf : preform \u2192 Prop :=\n  sorry\n\ntheorem is_nnf_push_neg (p : preform) : is_nnf p \u2192 is_nnf (push_neg p) := sorry\n\n/-- Argument is free of negations -/\ndef neg_free : preform \u2192 Prop :=\n  sorry\n\ntheorem is_nnf_nnf (p : preform) : is_nnf (nnf p) := sorry\n\ntheorem nnf_equiv {p : preform} : preform.equiv (nnf p) p := sorry\n\n/-- Eliminate all negations from preform -/\n@[simp] def neg_elim : preform \u2192 preform :=\n  sorry\n\ntheorem neg_free_neg_elim (p : preform) : is_nnf p \u2192 neg_free (neg_elim p) := sorry\n\ntheorem le_and_le_iff_eq {\u03b1 : Type} [partial_order \u03b1] {a : \u03b1} {b : \u03b1} : a \u2264 b \u2227 b \u2264 a \u2194 a = b := sorry\n\ntheorem implies_neg_elim {p : preform} : preform.implies p (neg_elim p) := sorry\n\n@[simp] def dnf_core : preform \u2192 List clause :=\n  sorry\n\n/-- DNF transformation -/\ndef dnf (p : preform) : List clause :=\n  dnf_core (neg_elim (nnf p))\n\ntheorem exists_clause_holds {v : \u2115 \u2192 \u2124} {p : preform} : neg_free p \u2192 preform.holds v p \u2192 \u2203 (c : clause), \u2203 (H : c \u2208 dnf_core p), clause.holds v c := sorry\n\ntheorem clauses_sat_dnf_core {p : preform} : neg_free p \u2192 preform.sat p \u2192 clauses.sat (dnf_core p) := sorry\n\ntheorem unsat_of_clauses_unsat {p : preform} : clauses.unsat (dnf p) \u2192 preform.unsat p := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/omega/int/dnf.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5078118642792043, "lm_q1q2_score": 0.35341589449586996}}
{"text": "/-\nCopyright (c) 2020 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin\n-/\n\nimport order.category.Preorder\n\n/-!\n# Category of partial orders\n\nThis defines `PartialOrder`, the category of partial orders with monotone maps.\n-/\n\nopen category_theory\n\nuniverse u\n\n/-- The category of partially ordered types. -/\ndef PartialOrder := bundled partial_order\n\nnamespace PartialOrder\n\ninstance : bundled_hom.parent_projection @partial_order.to_preorder := \u27e8\u27e9\n\nattribute [derive [large_category, concrete_category]] PartialOrder\n\ninstance : has_coe_to_sort PartialOrder Type* := bundled.has_coe_to_sort\n\n/-- Construct a bundled PartialOrder from the underlying type and typeclass. -/\ndef of (\u03b1 : Type*) [partial_order \u03b1] : PartialOrder := bundled.of \u03b1\n\ninstance : inhabited PartialOrder := \u27e8of punit\u27e9\n\ninstance (\u03b1 : PartialOrder) : partial_order \u03b1 := \u03b1.str\n\ninstance has_forget_to_Preorder : has_forget\u2082 PartialOrder Preorder := bundled_hom.forget\u2082 _ _\n\n/-- Constructs an equivalence between partial orders from an order isomorphism between them. -/\n@[simps] def iso.mk {\u03b1 \u03b2 : PartialOrder.{u}} (e : \u03b1 \u2243o \u03b2) : \u03b1 \u2245 \u03b2 :=\n{ hom := e,\n  inv := e.symm,\n  hom_inv_id' := by { ext, exact e.symm_apply_apply x },\n  inv_hom_id' := by { ext, exact e.apply_symm_apply x } }\n\n/-- `order_dual` as a functor. -/\n@[simps] def to_dual : PartialOrder \u2964 PartialOrder :=\n{ obj := \u03bb X, of (order_dual X), map := \u03bb X Y, order_hom.dual }\n\n/-- The equivalence between `PartialOrder` and itself induced by `order_dual` both ways. -/\n@[simps functor inverse] def dual_equiv : PartialOrder \u224c PartialOrder :=\nequivalence.mk to_dual to_dual\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n  (nat_iso.of_components (\u03bb X, iso.mk $ order_iso.dual_dual X) $ \u03bb X Y f, rfl)\n\nend PartialOrder\n\nlemma PartialOrder_dual_equiv_comp_forget_to_Preorder :\n  PartialOrder.dual_equiv.functor \u22d9 forget\u2082 PartialOrder Preorder\n  = forget\u2082 PartialOrder Preorder \u22d9 Preorder.dual_equiv.functor := rfl\n", "meta": {"author": "Mel-TunaRoll", "repo": "Lean-Mordell-Weil-Mel-Branch", "sha": "4db36f86423976aacd2c2968c4e45787fcd86b97", "save_path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch", "path": "github-repos/lean/Mel-TunaRoll-Lean-Mordell-Weil-Mel-Branch/Lean-Mordell-Weil-Mel-Branch-4db36f86423976aacd2c2968c4e45787fcd86b97/src/order/category/PartialOrder.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259583, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.35336121297313206}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport data.int.basic\nimport algebra.group_power.lemmas\nimport category_theory.pi.basic\nimport category_theory.shift\nimport category_theory.concrete_category.basic\n\n/-!\n# The category of graded objects\n\nFor any type `\u03b2`, a `\u03b2`-graded object over some category `C` is just\na function `\u03b2 \u2192 C` into the objects of `C`.\nWe put the \"pointwise\" category structure on these, as the non-dependent specialization of\n`category_theory.pi`.\n\nWe describe the `comap` functors obtained by precomposing with functions `\u03b2 \u2192 \u03b3`.\n\nAs a consequence a fixed element (e.g. `1`) in an additive group `\u03b2` provides a shift\nfunctor on `\u03b2`-graded objects\n\nWhen `C` has coproducts we construct the `total` functor `graded_object \u03b2 C \u2964 C`,\nshow that it is faithful, and deduce that when `C` is concrete so is `graded_object \u03b2 C`.\n-/\n\nopen category_theory.pi\nopen category_theory.limits\n\nnamespace category_theory\n\nuniverses w v u\n\n/-- A type synonym for `\u03b2 \u2192 C`, used for `\u03b2`-graded objects in a category `C`. -/\ndef graded_object (\u03b2 : Type w) (C : Type u) : Type (max w u) := \u03b2 \u2192 C\n\n-- Satisfying the inhabited linter...\ninstance inhabited_graded_object (\u03b2 : Type w) (C : Type u) [inhabited C] :\n  inhabited (graded_object \u03b2 C) :=\n\u27e8\u03bb b, inhabited.default\u27e9\n\n/--\nA type synonym for `\u03b2 \u2192 C`, used for `\u03b2`-graded objects in a category `C`\nwith a shift functor given by translation by `s`.\n-/\n@[nolint unused_arguments] -- `s` is here to distinguish type synonyms asking for different shifts\nabbreviation graded_object_with_shift {\u03b2 : Type w} [add_comm_group \u03b2] (s : \u03b2) (C : Type u) :\n  Type (max w u) := graded_object \u03b2 C\n\nnamespace graded_object\n\nvariables {C : Type u} [category.{v} C]\n\ninstance category_of_graded_objects (\u03b2 : Type w) : category.{max w v} (graded_object \u03b2 C) :=\ncategory_theory.pi (\u03bb _, C)\n\n/-- The projection of a graded object to its `i`-th component. -/\n@[simps] def eval {\u03b2 : Type w} (b : \u03b2) : graded_object \u03b2 C \u2964 C :=\n{ obj := \u03bb X, X b,\n  map := \u03bb X Y f, f b, }\n\nsection\nvariable (C)\n\n/--\nThe natural isomorphism comparing between\npulling back along two propositionally equal functions.\n-/\n@[simps]\ndef comap_eq {\u03b2 \u03b3 : Type w} {f g : \u03b2 \u2192 \u03b3} (h : f = g) : comap (\u03bb _, C) f \u2245 comap (\u03bb _, C) g :=\n{ hom := { app := \u03bb X b, eq_to_hom begin dsimp [comap], subst h, end },\n  inv := { app := \u03bb X b, eq_to_hom begin dsimp [comap], subst h, end }, }\n\nlemma comap_eq_symm {\u03b2 \u03b3 : Type w} {f g : \u03b2 \u2192 \u03b3} (h : f = g) :\n  comap_eq C h.symm = (comap_eq C h).symm :=\nby tidy\n\nlemma comap_eq_trans {\u03b2 \u03b3 : Type w} {f g h : \u03b2 \u2192 \u03b3} (k : f = g) (l : g = h) :\n  comap_eq C (k.trans l) = comap_eq C k \u226a\u226b comap_eq C l :=\nbegin\n  ext X b,\n  simp,\nend\n\n@[simp] lemma eq_to_hom_apply {\u03b2 : Type w} {X Y : \u03a0 b : \u03b2, C} (h : X = Y) (b : \u03b2) :\n  (eq_to_hom h : X \u27f6 Y) b = eq_to_hom (by subst h) :=\nby { subst h, refl }\n\n/--\nThe equivalence between \u03b2-graded objects and \u03b3-graded objects,\ngiven an equivalence between \u03b2 and \u03b3.\n-/\n@[simps]\ndef comap_equiv {\u03b2 \u03b3 : Type w} (e : \u03b2 \u2243 \u03b3) :\n  (graded_object \u03b2 C) \u224c (graded_object \u03b3 C) :=\n{ functor := comap (\u03bb _, C) (e.symm : \u03b3 \u2192 \u03b2),\n  inverse := comap (\u03bb _, C) (e : \u03b2 \u2192 \u03b3),\n  counit_iso := (comap_comp (\u03bb _, C) _ _).trans (comap_eq C (by { ext, simp } )),\n  unit_iso := (comap_eq C (by { ext, simp } )).trans (comap_comp _ _ _).symm,\n  functor_unit_iso_comp' := \u03bb X, by { ext b, dsimp, simp, }, }  -- See note [dsimp, simp].\n\nend\n\ninstance has_shift {\u03b2 : Type*} [add_comm_group \u03b2] (s : \u03b2) :\n  has_shift (graded_object_with_shift s C) \u2124 :=\nhas_shift_mk _ _\n{ F := \u03bb n, comap (\u03bb _, C) $ \u03bb (b : \u03b2), b + n \u2022 s,\n  \u03b5 := (comap_id \u03b2 (\u03bb _, C)).symm \u226a\u226b (comap_eq C (by { ext, simp })),\n  \u03bc := \u03bb m n, comap_comp _ _ _ \u226a\u226b comap_eq C (by { ext, simp [add_zsmul, add_comm] }),\n  left_unitality := by { introv, ext, dsimp, simpa },\n  right_unitality := by { introv, ext, dsimp, simpa },\n  associativity := by { introv, ext, dsimp, simp } }\n\n@[simp] lemma shift_functor_obj_apply {\u03b2 : Type*} [add_comm_group \u03b2]\n  (s : \u03b2) (X : \u03b2 \u2192 C) (t : \u03b2) (n : \u2124) :\n  (shift_functor (graded_object_with_shift s C) n).obj X t = X (t + n \u2022 s) :=\nrfl\n\n@[simp] lemma shift_functor_map_apply {\u03b2 : Type*} [add_comm_group \u03b2] (s : \u03b2)\n  {X Y : graded_object_with_shift s C} (f : X \u27f6 Y) (t : \u03b2) (n : \u2124) :\n  (shift_functor (graded_object_with_shift s C) n).map f t = f (t + n \u2022 s) :=\nrfl\n\ninstance has_zero_morphisms [has_zero_morphisms C] (\u03b2 : Type w) :\n  has_zero_morphisms.{max w v} (graded_object \u03b2 C) :=\n{ has_zero := \u03bb X Y,\n  { zero := \u03bb b, 0 } }\n\n@[simp]\nlemma zero_apply [has_zero_morphisms C] (\u03b2 : Type w) (X Y : graded_object \u03b2 C) (b : \u03b2) :\n  (0 : X \u27f6 Y) b = 0 := rfl\n\nsection\nopen_locale zero_object\n\ninstance has_zero_object [has_zero_object C] [has_zero_morphisms C] (\u03b2 : Type w) :\n  has_zero_object.{max w v} (graded_object \u03b2 C) :=\nby { refine \u27e8\u27e8\u03bb b, 0, \u03bb X, \u27e8\u27e8\u27e8\u03bb b, 0\u27e9, \u03bb f, _\u27e9\u27e9, \u03bb X, \u27e8\u27e8\u27e8\u03bb b, 0\u27e9, \u03bb f, _\u27e9\u27e9\u27e9\u27e9; ext, }\nend\n\nend graded_object\n\nnamespace graded_object\n-- The universes get a little hairy here, so we restrict the universe level for the grading to 0.\n-- Since we're typically interested in grading by \u2124 or a finite group, this should be okay.\n-- If you're grading by things in higher universes, have fun!\nvariables (\u03b2 : Type)\nvariables (C : Type u) [category.{v} C]\nvariables [has_coproducts C]\n\nsection\nlocal attribute [tidy] tactic.discrete_cases\n\n/--\nThe total object of a graded object is the coproduct of the graded components.\n-/\nnoncomputable def total : graded_object \u03b2 C \u2964 C :=\n{ obj := \u03bb X, \u2210 (\u03bb i : ulift.{v} \u03b2, X i.down),\n  map := \u03bb X Y f, limits.sigma.map (\u03bb i, f i.down) }.\n\nend\n\nvariables [has_zero_morphisms C]\n\n/--\nThe `total` functor taking a graded object to the coproduct of its graded components is faithful.\nTo prove this, we need to know that the coprojections into the coproduct are monomorphisms,\nwhich follows from the fact we have zero morphisms and decidable equality for the grading.\n-/\ninstance : faithful (total \u03b2 C) :=\n{ map_injective' := \u03bb X Y f g w,\n  begin\n    classical,\n    ext i,\n    replace w := sigma.\u03b9 (\u03bb i : ulift.{v} \u03b2, X i.down) \u27e8i\u27e9 \u226b= w,\n    erw [colimit.\u03b9_map, colimit.\u03b9_map] at w,\n    exact mono.right_cancellation _ _ w,\n  end }\n\nend graded_object\n\nnamespace graded_object\n\nnoncomputable theory\n\nvariables (\u03b2 : Type)\nvariables (C : Type (u+1)) [large_category C] [concrete_category C]\n  [has_coproducts C] [has_zero_morphisms C]\n\ninstance : concrete_category (graded_object \u03b2 C) :=\n{ forget := total \u03b2 C \u22d9 forget C }\n\ninstance : has_forget\u2082 (graded_object \u03b2 C) C :=\n{ forget\u2082 := total \u03b2 C }\n\nend graded_object\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/category_theory/graded_object.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.5851011542032313, "lm_q1q2_score": 0.35336120463115644}}
{"text": "/-\nCopyright (c) 2021 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport category_theory.limits.shapes.pullbacks\nimport ring_theory.tensor_product\nimport algebra.category.Ring.limits\nimport algebra.category.Ring.colimits\nimport category_theory.limits.shapes.strict_initial\nimport ring_theory.subring.basic\nimport ring_theory.ideal.local_ring\nimport category_theory.limits.preserves.limits\n\n/-!\n# Constructions of (co)limits in CommRing\n\nIn this file we provide the explicit (co)cones for various (co)limits in `CommRing`, including\n* tensor product is the pushout\n* `Z` is the initial object\n* `0` is the strict terminal object\n* cartesian product is the product\n* `ring_hom.eq_locus` is the equalizer\n\n-/\n\nuniverses u u'\n\nopen category_theory category_theory.limits\nopen_locale tensor_product\n\nnamespace CommRing\n\nsection pushout\n\nvariables {R A B : CommRing.{u}} (f : R \u27f6 A) (g : R \u27f6 B)\n\n/-- The explicit cocone with tensor products as the fibered product in `CommRing`. -/\ndef pushout_cocone : limits.pushout_cocone f g :=\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  apply limits.pushout_cocone.mk,\n  show CommRing, from CommRing.of (A \u2297[R] B),\n  show A \u27f6 _,  from algebra.tensor_product.include_left.to_ring_hom,\n  show B \u27f6 _,  from algebra.tensor_product.include_right.to_ring_hom,\n  ext r,\n  transitivity algebra_map R (A \u2297[R] B) r,\n  { exact algebra.tensor_product.include_left.commutes r },\n  { exact (algebra.tensor_product.include_right.commutes r).symm }\nend\n\n@[simp]\nlemma pushout_cocone_inl : (pushout_cocone f g).inl = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_left.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_inr : (pushout_cocone f g).inr = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI algebra.tensor_product.include_right.to_ring_hom }) := rfl\n\n@[simp]\nlemma pushout_cocone_X : (pushout_cocone f g).X = (by\n{ letI := f.to_algebra, letI := g.to_algebra,\n  exactI CommRing.of (A \u2297[R] B) }) := rfl\n\n/-- Verify that the `pushout_cocone` is indeed the colimit. -/\ndef pushout_cocone_is_colimit : limits.is_colimit (pushout_cocone f g) :=\nlimits.pushout_cocone.is_colimit_aux' _ (\u03bb s,\nbegin\n  letI := ring_hom.to_algebra f,\n  letI := ring_hom.to_algebra g,\n  letI := ring_hom.to_algebra (f \u226b s.inl),\n  let f' : A \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change s.inl.to_fun (f r) = (f \u226b s.inl) r, refl }, ..s.inl },\n  let g' : B \u2192\u2090[R] s.X := { commutes' := \u03bb r, by\n      { change (g \u226b s.inr) r = (f \u226b s.inl) r,\n        congr' 1,\n        exact (s.\u03b9.naturality limits.walking_span.hom.snd).trans\n          (s.\u03b9.naturality limits.walking_span.hom.fst).symm }, ..s.inr },\n  /- The factor map is a \u2297 b \u21a6 f(a) * g(b). -/\n  use alg_hom.to_ring_hom (algebra.tensor_product.product_map f' g'),\n  simp only [pushout_cocone_inl, pushout_cocone_inr],\n  split, { ext x, exact algebra.tensor_product.product_map_left_apply  _ _ x, },\n  split, { ext x, exact algebra.tensor_product.product_map_right_apply _ _ x, },\n  intros h eq1 eq2,\n  let h' : (A \u2297[R] B) \u2192\u2090[R] s.X :=\n    { commutes' := \u03bb r, by\n    { change h ((f r) \u2297\u209c[R] 1) = s.inl (f r),\n      rw \u2190 eq1, simp }, ..h },\n  suffices : h' = algebra.tensor_product.product_map f' g',\n  { ext x,\n    change h' x = algebra.tensor_product.product_map f' g' x,\n    rw this },\n  apply algebra.tensor_product.ext,\n  intros a b,\n  simp [\u2190 eq1, \u2190 eq2, \u2190 h.map_mul],\nend)\n\nend pushout\n\nsection terminal\n\n/-- The trivial ring is the (strict) terminal object of `CommRing`. -/\ndef punit_is_terminal : is_terminal (CommRing.of.{u} punit) :=\nbegin\n  apply_with is_terminal.of_unique { instances := ff },\n  tidy\nend\n\ninstance CommRing_has_strict_terminal_objects : has_strict_terminal_objects CommRing.{u} :=\nbegin\n  apply has_strict_terminal_objects_of_terminal_is_strict (CommRing.of punit),\n  intros X f,\n  refine \u27e8\u27e8by tidy, by ext, _\u27e9\u27e9,\n  ext,\n  have e : (0 : X) = 1 := by { rw [\u2190 f.map_one, \u2190 f.map_zero], congr },\n  replace e : 0 * x = 1 * x := congr_arg (\u03bb a, a * x) e,\n  rw [one_mul, zero_mul, \u2190 f.map_zero] at e,\n  exact e,\nend\n\nlemma subsingleton_of_is_terminal {X : CommRing} (hX : is_terminal X) : subsingleton X :=\n(hX.unique_up_to_iso punit_is_terminal).CommRing_iso_to_ring_equiv.to_equiv\n  .subsingleton_congr.mpr (show subsingleton punit, by apply_instance)\n\n/-- `\u2124` is the initial object of `CommRing`. -/\ndef Z_is_initial : is_initial (CommRing.of \u2124) :=\nbegin\n  apply_with is_initial.of_unique { instances := ff },\n  exact \u03bb R, \u27e8\u27e8int.cast_ring_hom R\u27e9, \u03bb a, a.ext_int _\u27e9,\nend\n\nend terminal\n\nsection product\n\nvariables (A B : CommRing.{u})\n\n/-- The product in `CommRing` is the cartesian product. This is the binary fan. -/\n@[simps X]\ndef prod_fan : binary_fan A B :=\nbinary_fan.mk (CommRing.of_hom $ ring_hom.fst A B) (CommRing.of_hom $ ring_hom.snd A B)\n\n/-- The product in `CommRing` is the cartesian product. -/\ndef prod_fan_is_limit : is_limit (prod_fan A B) :=\n{ lift := \u03bb c, ring_hom.prod (c.\u03c0.app \u27e8walking_pair.left\u27e9) (c.\u03c0.app \u27e8walking_pair.right\u27e9),\n  fac' := \u03bb c j, by { ext, rcases j with \u27e8\u27e8\u27e9\u27e9;\n    simpa only [binary_fan.\u03c0_app_left, binary_fan.\u03c0_app_right, comp_apply, ring_hom.prod_apply] },\n  uniq' := \u03bb s m h, by { ext, { simpa using congr_hom (h \u27e8walking_pair.left\u27e9) x },\n    { simpa using congr_hom (h \u27e8walking_pair.right\u27e9) x } } }\n\nend product\n\nsection equalizer\n\nvariables {A B : CommRing.{u}} (f g : A \u27f6 B)\n\n/-- The equalizer in `CommRing` is the equalizer as sets. This is the equalizer fork. -/\ndef equalizer_fork : fork f g :=\nfork.of_\u03b9 (CommRing.of_hom (ring_hom.eq_locus f g).subtype) (by { ext \u27e8x, e\u27e9, simpa using e })\n\n/-- The equalizer in `CommRing` is the equalizer as sets. -/\ndef equalizer_fork_is_limit : is_limit (equalizer_fork f g) :=\nbegin\n  fapply fork.is_limit.mk',\n  intro s,\n  use s.\u03b9.cod_restrict _ (\u03bb x, (concrete_category.congr_hom s.condition x : _)),\n  split,\n  { ext, refl },\n  { intros m hm, ext x, exact concrete_category.congr_hom hm x }\nend\n\ninstance : is_local_ring_hom (equalizer_fork f g).\u03b9 :=\nbegin\n  constructor,\n  rintros \u27e8a, (h\u2081 : _ = _)\u27e9 (\u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,(rfl : x = _)\u27e9 : is_unit a),\n  have : y \u2208 ring_hom.eq_locus f g,\n  { apply (f.is_unit_map \u27e8\u27e8x,y,h\u2083,h\u2084\u27e9,rfl\u27e9 : is_unit (f x)).mul_left_inj.mp,\n    conv_rhs { rw h\u2081 },\n    rw [\u2190 f.map_mul, \u2190 g.map_mul, h\u2084, f.map_one, g.map_one] },\n  rw is_unit_iff_exists_inv,\n  exact \u27e8\u27e8y, this\u27e9, subtype.eq h\u2083\u27e9,\nend\n\ninstance equalizer_\u03b9_is_local_ring_hom (F : walking_parallel_pair.{u} \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F walking_parallel_pair.zero) :=\nbegin\n  have := lim_map_\u03c0 (diagram_iso_parallel_pair F).hom walking_parallel_pair.zero,\n  rw \u2190 is_iso.comp_inv_eq at this,\n  rw \u2190 this,\n  rw \u2190 limit.iso_limit_cone_hom_\u03c0 \u27e8_, equalizer_fork_is_limit\n    (F.map walking_parallel_pair_hom.left) (F.map walking_parallel_pair_hom.right)\u27e9\n    walking_parallel_pair.zero,\n  change is_local_ring_hom ((lim.map _ \u226b _ \u226b (equalizer_fork _ _).\u03b9) \u226b _),\n  apply_instance\nend\n\nopen category_theory.limits.walking_parallel_pair opposite\nopen category_theory.limits.walking_parallel_pair_hom\n\ninstance equalizer_\u03b9_is_local_ring_hom' (F : walking_parallel_pair.{u}\u1d52\u1d56 \u2964 CommRing.{u}) :\n  is_local_ring_hom (limit.\u03c0 F (opposite.op walking_parallel_pair.one)) :=\nbegin\n  have : _ = limit.\u03c0 F (walking_parallel_pair_op_equiv.{u u}.functor.obj _) :=\n    (limit.iso_limit_cone_inv_\u03c0 \u27e8_, is_limit.whisker_equivalence (limit.is_limit F)\n      walking_parallel_pair_op_equiv\u27e9 walking_parallel_pair.zero : _),\n  erw \u2190 this,\n  apply_instance\nend\n\nend equalizer\n\nend CommRing\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/algebra/category/Ring/constructions.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3532364786945802}}
{"text": "/-\nCopyright (c) 2019 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.eq_to_hom\n\n/-!\n# Binary disjoint unions of categories\n\nWe define the category instance on `C \u2295 D` when `C` and `D` are categories.\n\nWe define:\n* `inl_`      : the functor `C \u2964 C \u2295 D`\n* `inr_`      : the functor `D \u2964 C \u2295 D`\n* `swap`      : the functor `C \u2295 D \u2964 D \u2295 C`\n    (and the fact this is an equivalence)\n\nWe further define sums of functors and natural transformations, written `F.sum G` and `\u03b1.sum \u03b2`.\n-/\n\nnamespace category_theory\n\nuniverses v\u2081 u\u2081 -- morphism levels before object levels. See note [category_theory universes].\n\nopen sum\n\nsection\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2081) [category.{v\u2081} D]\n\n/--\n`sum C D` gives the direct sum of two categories.\n-/\ninstance sum : category.{v\u2081} (C \u2295 D) :=\n{ hom :=\n    \u03bb X Y, match X, Y with\n    | inl X, inl Y := X \u27f6 Y\n    | inl X, inr Y := pempty\n    | inr X, inl Y := pempty\n    | inr X, inr Y := X \u27f6 Y\n    end,\n  id :=\n    \u03bb X, match X with\n    | inl X := \ud835\udfd9 X\n    | inr X := \ud835\udfd9 X\n    end,\n  comp :=\n    \u03bb X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := f \u226b g\n    | inr X, inr Y, inr Z, f, g := f \u226b g\n    end }\n\n@[simp] lemma sum_comp_inl {P Q R : C} (f : (inl P : C \u2295 D) \u27f6 inl Q) (g : inl Q \u27f6 inl R) :\n  f \u226b g = (f : P \u27f6 Q) \u226b (g : Q \u27f6 R) := rfl\n@[simp] lemma sum_comp_inr {P Q R : D} (f : (inr P : C \u2295 D) \u27f6 inr Q) (g : inr Q \u27f6 inr R) :\n  f \u226b g = (f : P \u27f6 Q) \u226b (g : Q \u27f6 R) := rfl\nend\n\nnamespace sum\n\nvariables (C : Type u\u2081) [category.{v\u2081} C] (D : Type u\u2081) [category.{v\u2081} D]\n\n/-- `inl_` is the functor `X \u21a6 inl X`. -/\n-- Unfortunate naming here, suggestions welcome.\n@[simps] def inl_ : C \u2964 C \u2295 D :=\n{ obj := \u03bb X, inl X,\n  map := \u03bb X Y f, f }\n\n/-- `inr_` is the functor `X \u21a6 inr X`. -/\n@[simps] def inr_ : D \u2964 C \u2295 D :=\n{ obj := \u03bb X, inr X,\n  map := \u03bb X Y f, f }\n\n/-- The functor exchanging two direct summand categories. -/\ndef swap : C \u2295 D \u2964 D \u2295 C :=\n{ obj :=\n    \u03bb X, match X with\n    | inl X := inr X\n    | inr X := inl X\n    end,\n  map :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := f\n    | inr X, inr Y, f := f\n    end }\n\n@[simp] lemma swap_obj_inl (X : C) : (swap C D).obj (inl X) = inr X := rfl\n@[simp] lemma swap_obj_inr (X : D) : (swap C D).obj (inr X) = inl X := rfl\n@[simp] lemma swap_map_inl {X Y : C} {f : inl X \u27f6 inl Y} : (swap C D).map f = f := rfl\n@[simp] \n\nnamespace swap\n\n/-- `swap` gives an equivalence between `C \u2295 D` and `D \u2295 C`. -/\ndef equivalence : C \u2295 D \u224c D \u2295 C :=\nequivalence.mk (swap C D) (swap D C)\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by { cases X; refl })) (by tidy))\n  (nat_iso.of_components (\u03bb X, eq_to_iso (by { cases X; refl })) (by tidy))\n\ninstance is_equivalence : is_equivalence (swap C D) :=\n(by apply_instance : is_equivalence (equivalence C D).functor)\n\n/-- The double swap on `C \u2295 D` is naturally isomorphic to the identity functor. -/\ndef symmetry : swap C D \u22d9 swap D C \u2245 \ud835\udfed (C \u2295 D) :=\n(equivalence C D).unit_iso.symm\n\nend swap\n\nend sum\n\nvariables {A : Type u\u2081} [category.{v\u2081} A]\n          {B : Type u\u2081} [category.{v\u2081} B]\n          {C : Type u\u2081} [category.{v\u2081} C]\n          {D : Type u\u2081} [category.{v\u2081} D]\n\nnamespace functor\n\n/-- The sum of two functors. -/\ndef sum (F : A \u2964 B) (G : C \u2964 D) : A \u2295 C \u2964 B \u2295 D :=\n{ obj :=\n    \u03bb X, match X with\n    | inl X := inl (F.obj X)\n    | inr X := inr (G.obj X)\n    end,\n  map :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := F.map f\n    | inr X, inr Y, f := G.map f\n    end,\n  map_id' := \u03bb X, begin cases X; unfold_aux, erw F.map_id, refl, erw G.map_id, refl end,\n  map_comp' :=\n    \u03bb X Y Z f g, match X, Y, Z, f, g with\n    | inl X, inl Y, inl Z, f, g := by { unfold_aux, erw F.map_comp, refl }\n    | inr X, inr Y, inr Z, f, g := by { unfold_aux, erw G.map_comp, refl }\n    end }\n\n@[simp] lemma sum_obj_inl (F : A \u2964 B) (G : C \u2964 D) (a : A) :\n  (F.sum G).obj (inl a) = inl (F.obj a) := rfl\n@[simp] lemma sum_obj_inr (F : A \u2964 B) (G : C \u2964 D) (c : C) :\n  (F.sum G).obj (inr c) = inr (G.obj c) := rfl\n@[simp] lemma sum_map_inl (F : A \u2964 B) (G : C \u2964 D) {a a' : A} (f : inl a \u27f6 inl a') :\n  (F.sum G).map f = F.map f := rfl\n@[simp] lemma sum_map_inr (F : A \u2964 B) (G : C \u2964 D) {c c' : C} (f : inr c \u27f6 inr c') :\n  (F.sum G).map f = G.map f := rfl\nend functor\n\nnamespace nat_trans\n\n/-- The sum of two natural transformations. -/\ndef sum {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) : F.sum H \u27f6 G.sum I :=\n{ app         :=\n    \u03bb X, match X with\n    | inl X := \u03b1.app X\n    | inr X := \u03b2.app X\n    end,\n  naturality' :=\n    \u03bb X Y f, match X, Y, f with\n    | inl X, inl Y, f := begin unfold_aux, erw \u03b1.naturality, refl, end\n    | inr X, inr Y, f := begin unfold_aux, erw \u03b2.naturality, refl, end\n    end }\n\n@[simp] lemma sum_app_inl {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (a : A) :\n  (sum \u03b1 \u03b2).app (inl a) = \u03b1.app a := rfl\n@[simp] lemma sum_app_inr {F G : A \u2964 B} {H I : C \u2964 D} (\u03b1 : F \u27f6 G) (\u03b2 : H \u27f6 I) (c : C) :\n  (sum \u03b1 \u03b2).app (inr c) = \u03b2.app c := rfl\nend nat_trans\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/sums/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.35323647869458014}}
{"text": "import .basic group_theory.submonoid.operations\n\nvariables {\u03b9 : Type*} {M : \u03b9 \u2192 Type*}\nvariables [decidable_eq \u03b9] [\u03a0 i, decidable_eq (M i)]\nvariables [\u03a0 i, monoid (M i)]\n\nopen coprod  submonoid function\n\nlemma mul_aux_mem (S : \u03a0 i, submonoid (M i)) : \u2200 (l\u2081 l\u2082 : list (\u03a3 i, M i))\n  (h\u2081 : \u2200 a : \u03a3 i, M i, a \u2208 l\u2081 \u2192 a.2 \u2208 S a.1)\n  (h\u2082 : \u2200 a : \u03a3 i, M i, a \u2208 l\u2082 \u2192 a.2 \u2208 S a.1)\n  {i : \u03b9} {a : M i} (ha : (\u27e8i, a\u27e9 : \u03a3 i, M i) \u2208 pre.mul_aux l\u2081 l\u2082),\n  a \u2208 S i\n| []           l\u2082      := by simp [pre.mul_aux]\n| (\u27e8j, b\u27e9::l\u2081) []      := begin\n    assume h\u2081 _ i a ha,\n    simp only [pre.mul_aux, list.mem_reverse, list.mem_cons_iff] at ha,\n    rcases ha with \u27e8rfl, hab\u27e9 | hia,\n    { rw [heq_iff_eq] at hab,\n      subst hab,\n      exact h\u2081 \u27e8i, a\u27e9 (list.mem_cons_self _ _) },\n    { exact h\u2081 \u27e8i, a\u27e9 (list.mem_cons_of_mem _ hia) }\n  end\n| (\u27e8j, b\u27e9::l\u2081) (\u27e8k, c\u27e9::l\u2082) := begin\n  assume h\u2081 h\u2082 i a ha,\n  simp only [pre.mul_aux] at ha,\n  split_ifs at ha,\n  { exact mul_aux_mem _ _\n      (\u03bb d hd, h\u2081 d (list.mem_cons_of_mem _ hd))\n      (\u03bb d hd, h\u2082 d (list.mem_cons_of_mem _ hd))\n      ha },\n  { dsimp at h,\n    subst j,\n    simp only [list.reverse_core_eq, list.mem_append, list.mem_cons_iff,\n      list.mem_reverse, cast_eq] at ha,\n    simp only [cast_eq] at *,\n    rcases ha with ha | \u27e8rfl, h, h\u27e9 | ha,\n    { exact h\u2081 \u27e8i, a\u27e9 (list.mem_cons_of_mem _ ha) },\n    { exact submonoid.mul_mem _\n        (h\u2081 \u27e8i, b\u27e9 (list.mem_cons_self _ _))\n        (h\u2082 \u27e8i, c\u27e9 (list.mem_cons_self _ _)) },\n    { exact h\u2082 \u27e8i, a\u27e9 (list.mem_cons_of_mem _ ha) } },\n  { clear_aux_decl,\n    simp only [list.reverse_core_eq, list.mem_append, list.mem_cons_iff,\n      list.mem_reverse] at ha,\n    rcases ha with ha | \u27e8rfl, hab\u27e9 | \u27e8rfl, hab\u27e9 | ha,\n    { exact h\u2081 \u27e8i, a\u27e9 (list.mem_cons_of_mem _ ha) },\n    { rw [heq_iff_eq] at hab,\n      subst hab,\n      exact h\u2081 \u27e8i, a\u27e9 (list.mem_cons_self _ _) },\n    { rw [heq_iff_eq] at hab,\n      subst hab,\n      exact h\u2082 \u27e8i, a\u27e9 (list.mem_cons_self _ _) },\n    { exact h\u2082 \u27e8i, a\u27e9 (list.mem_cons_of_mem _ ha) } }\nend\n\ndef blah (S : \u03a0 i, submonoid (M i)) : submonoid (coprod M) :=\n{ carrier  := { w : coprod M | \u2200 (a : \u03a3 i, M i), a \u2208 w.to_list \u2192 a.2 \u2208 S a.1 },\n  one_mem' := \u03bb a h, h.elim,\n  mul_mem' := begin\n    rintros \u27e8l\u2081, hl\u2081\u27e9 \u27e8l\u2082, hl\u2082\u27e9 h\u2081 h\u2082 \u27e8i, a\u27e9 h,\n    exact mul_aux_mem S l\u2081.reverse l\u2082 (by simpa using h\u2081) (by simpa using h\u2082) h\n  end }\n\nlemma mem_blah (S : \u03a0 i, submonoid (M i)) (w : coprod M) :\n  w \u2208 blah S \u2194 \u2200 (a : \u03a3 i, M i), a \u2208 w.to_list \u2192 a.2 \u2208 S a.1 := iff.rfl\n\nvariable {S : \u03a0 i, submonoid (M i)}\n\n@[simp] lemma of_mem_blah_iff {i : \u03b9} {a : M i} : of i a \u2208 blah S \u2194 a \u2208 S i :=\nbegin\n  simp only [mem_blah, to_list_of],\n  split_ifs,\n  { simp [*, submonoid.one_mem] },\n  { simp only [list.mem_singleton],\n    split,\n    { exact \u03bb h, h \u27e8i, a\u27e9 rfl },\n    { assume ha j hj,\n      subst j,\n      exact ha } }\nend\n\nlemma blah_eq_supr : blah S = \u2a06 i, (S i).map (of i) :=\nle_antisymm\n  (\u03bb w hw, begin\n    cases w with l hl,\n    induction l with i l ih,\n    { simp [submonoid.one_mem] },\n    { rw [cons_eq_of_mul],\n      refine submonoid.mul_mem _ _ _,\n      { exact le_supr (\u03bb i, (S i).map (of i)) i.1\n        (mem_map.2 \u27e8i.2, hw _ (list.mem_cons_self _ _), rfl\u27e9) },\n      { exact ih _ (\u03bb j hj, hw _ (list.mem_cons_of_mem _ hj)) } }\n  end)\n  (supr_le (\u03bb i a ha, begin\n    rw [mem_map] at ha,\n    rcases ha with \u27e8a, ha, rfl\u27e9,\n    simp only [to_list_of, mem_blah],\n    split_ifs,\n    { simp },\n    { simp only [list.mem_singleton],\n      assume a ha,\n      subst a,\n      exact ha }\n  end))\n", "meta": {"author": "ChrisHughes24", "repo": "single_relation", "sha": "556990dab75054a1c14717a72c8901dc9f2f01e4", "save_path": "github-repos/lean/ChrisHughes24-single_relation", "path": "github-repos/lean/ChrisHughes24-single_relation/single_relation-556990dab75054a1c14717a72c8901dc9f2f01e4/scratch/for_mathlib/coprod/submonoid.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803832, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3532147608096485}}
{"text": "/-\nCopyright (c) 2019 Floris van Doorn. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Floris van Doorn\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.tactic.core\nimport Mathlib.PostPort\n\nuniverses l \n\nnamespace Mathlib\n\n/-!\n# simps attribute\n\nThis file defines the `@[simps]` attribute, to automatically generate simp-lemmas\nreducing a definition when projections are applied to it.\n\n## Implementation Notes\n\nThere are three attributes being defined here\n* `@[simps]` is the attribute for objects of a structure or instances of a class. It will\n  automatically generate simplification lemmas for each projection of the object/instance that\n  contains data. See the doc strings for `simps_attr` and `simps_cfg` for more details and\n  configuration options.\n* `@[_simps_str]` is automatically added to structures that have been used in `@[simps]` at least\n  once. This attribute contains the data of the projections used for this structure by all following\n  invocations of `@[simps]`.\n* `@[notation_class]` should be added to all classes that define notation, like `has_mul` and\n  `has_zero`. This specifies that the projections that `@[simps]` used are the projections from\n  these notation classes instead of the projections of the superclasses.\n  Example: if `has_mul` is tagged with `@[notation_class]` then the projection used for `semigroup`\n  will be `\u03bb \u03b1 h\u03b1, @has_mul.mul \u03b1 (@semigroup.to_has_mul \u03b1 h\u03b1)` instead of `@semigroup.mul`.\n\n## Tags\n\nstructures, projections, simp, simplifier, generates declarations\n-/\n\n/--\nThe `@[_simps_str]` attribute specifies the preferred projections of the given structure,\nused by the `@[simps]` attribute.\n- This will usually be tagged by the `@[simps]` tactic.\n- You can also generate this with the command `initialize_simps_projections`.\n- To change the default value, see Note [custom simps projection].\n- You are strongly discouraged to add this attribute manually.\n- The first argument is the list of names of the universe variables used in the structure\n- The second argument is a list that consists of\n  - a custom name for each projection of the structure\n  - an expressions for each projections of the structure (definitionally equal to the\n    corresponding projection). These expressions can contain the universe parameters specified\n    in the first argument).\n-/\n/--\n  The `@[notation_class]` attribute specifies that this is a notation class,\n  and this notation should be used instead of projections by @[simps].\n  * The first argument `tt` for notation classes and `ff` for classes applied to the structure,\n    like `has_coe_to_sort` and `has_coe_to_fun`\n  * The second argument is the name of the projection (by default it is the first projection\n    of the structure)\n-/\n/--\n  Get the projections used by `simps` associated to a given structure `str`. The second component is\n  the list of projections, and the first component the (shared) list of universe levels used by the\n  projections.\n\n  The returned information is also stored in a parameter of the attribute `@[_simps_str]`, which\n  is given to `str`. If `str` already has this attribute, the information is read from this\n  attribute instead.\n\n  The returned universe levels are the universe levels of the structure. For the projections there\n  are three cases\n  * If the declaration `{structure_name}.simps.{projection_name}` has been declared, then the value\n    of this declaration is used (after checking that it is definitionally equal to the actual\n    projection\n  * Otherwise, for every class with the `notation_class` attribute, and the structure has an\n    instance of that notation class, then the projection of that notation class is used for the\n    projection that is definitionally equal to it (if there is such a projection).\n    This means in practice that coercions to function types and sorts will be used instead of\n    a projection, if this coercion is definitionally equal to a projection. Furthermore, for\n    notation classes like `has_mul` and `has_zero` those projections are used instead of the\n    corresponding projection\n  * Otherwise, the projection of the structure is chosen.\n    For example: ``simps_get_raw_projections env `prod`` gives the default projections\n```\n  ([u, v], [prod.fst.{u v}, prod.snd.{u v}])\n```\n    while ``simps_get_raw_projections env `equiv`` gives\n```\n  ([u_1, u_2], [\u03bb \u03b1 \u03b2, coe_fn, \u03bb {\u03b1 \u03b2} (e : \u03b1 \u2243 \u03b2), \u21d1(e.symm), left_inv, right_inv])\n```\n    after declaring the coercion from `equiv` to function and adding the declaration\n```\n  def equiv.simps.inv_fun {\u03b1 \u03b2} (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n```\n\n  Optionally, this command accepts two optional arguments\n  * If `trace_if_exists` the command will always generate a trace message when the structure already\n    has the attribute `@[_simps_str]`.\n  * The `name_changes` argument accepts a list of pairs `(old_name, new_name)`. This is used to\n    change the projection name `old_name` to the custom projection name `new_name`. Example:\n    for the structure `equiv` the projection `to_fun` could be renamed `apply`. This name will be\n    used for parsing and generating projection names. This argument is ignored if the structure\n    already has an existing attribute.\n-/\n-- if performance becomes a problem, possible heuristic: use the names of the projections to\n\n-- skip all classes that don't have the corresponding field.\n\n/--\n  You can specify custom projections for the `@[simps]` attribute.\n  To do this for the projection `my_structure.awesome_projection` by adding a declaration\n  `my_structure.simps.awesome_projection` that is definitionally equal to\n  `my_structure.awesome_projection` but has the projection in the desired (simp-normal) form.\n\n  You can initialize the projections `@[simps]` uses with `initialize_simps_projections`\n  (after declaring any custom projections). This is not necessary, it has the same effect\n  if you just add `@[simps]` to a declaration.\n\n  If you do anything to change the default projections, make sure to call either `@[simps]` or\n  `initialize_simps_projections` in the same file as the structure declaration. Otherwise, you might\n  have a file that imports the structure, but not your custom projections.\n-/\n/-- Specify simps projections, see Note [custom simps projection].\n  You can specify custom names by writing e.g.\n  `initialize_simps_projections equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)`.\n  Set `trace.simps.verbose` to true to see the generated projections.\n  If the projections were already specified before, you can call `initialize_simps_projections`\n  again to see the generated projections. -/\n/--\n  Get the projections of a structure used by `@[simps]` applied to the appropriate arguments.\n  Returns a list of quadruples\n  (projection expression, given projection name, original (full) projection name,\n    corresponding right-hand-side),\n  one for each projection. The given projection name is the name for the projection used by the user\n  used to generate (and parse) projection names. The original projection name is the actual\n  projection name in the structure, which is only used to check whether the expression is an\n  eta-expansion of some other expression. For example, in the structure\n\n  Example 1: ``simps_get_projection_exprs env `(\u03b1 \u00d7 \u03b2) `(\u27e8x, y\u27e9)`` will give the output\n  ```\n    [(`(@prod.fst.{u v} \u03b1 \u03b2), `fst, `prod.fst, `(x)),\n     (`(@prod.snd.{u v} \u03b1 \u03b2), `snd, `prod.snd, `(y))]\n  ```\n\n  Example 2: ``simps_get_projection_exprs env `(\u03b1 \u2243 \u03b1) `(\u27e8id, id, \u03bb _, rfl, \u03bb _, rfl\u27e9)``\n  will give the output\n  ```\n    [(`(@equiv.to_fun.{u u} \u03b1 \u03b1), `apply, `equiv.to_fun, `(id)),\n     (`(@equiv.inv_fun.{u u} \u03b1 \u03b1), `symm_apply, `equiv.inv_fun, `(id)),\n     ...,\n     ...]\n  ```\n  The last two fields of the list correspond to the propositional fields of the structure,\n  and are rarely/never used.\n-/\n-- This function does not use `tactic.mk_app` or `tactic.mk_mapp`, because the the given arguments\n\n-- might not uniquely specify the universe levels yet.\n\n/--\n  Configuration options for the `@[simps]` attribute.\n  * `attrs` specifies the list of attributes given to the generated lemmas. Default: ``[`simp]``.\n    The attributes can be either basic attributes, or user attributes without parameters.\n    There are two attributes which `simps` might add itself:\n    * If ``[`simp]`` is in the list, then ``[`_refl_lemma]`` is added automatically if appropriate.\n    * If the definition is marked with `@[to_additive ...]` then all generated lemmas are marked\n      with `@[to_additive]`\n  * `short_name` gives the generated lemmas a shorter name. This only has an effect when multiple\n    projections are applied in a lemma. When this is `ff` (default) all projection names will be\n    appended to the definition name to form the lemma name, and when this is `tt`, only the\n    last projection name will be appended.\n  * if `simp_rhs` is `tt` then the right-hand-side of the generated lemmas will be put in\n    simp-normal form. More precisely: `dsimp, simp` will be called on all these expressions.\n    See note [dsimp, simp].\n  * `type_md` specifies how aggressively definitions are unfolded in the type of expressions\n    for the purposes of finding out whether the type is a function type.\n    Default: `instances`. This will unfold coercion instances (so that a coercion to a function type\n    is recognized as a function type), but not declarations like `set`.\n  * `rhs_md` specifies how aggressively definition in the declaration are unfolded for the purposes\n    of finding out whether it is a constructor.\n    Default: `none`\n    Exception: `@[simps]` will automatically add the options\n    `{rhs_md := semireducible, simp_rhs := tt}` if the given definition is not a constructor with\n    the given reducibility setting for `rhs_md`.\n  * If `fully_applied` is `ff` then the generated simp-lemmas will be between non-fully applied\n    terms, i.e. equalities between functions. This does not restrict the recursive behavior of\n    `@[simps]`, so only the \"final\" projection will be non-fully applied.\n    However, it can be used in combination with explicit field names, to get a partially applied\n    intermediate projection.\n  * The option `not_recursive` contains the list of names of types for which `@[simps]` doesn't\n    recursively apply projections. For example, given an equivalence `\u03b1 \u00d7 \u03b2 \u2243 \u03b2 \u00d7 \u03b1` one usually\n    wants to only apply the projections for `equiv`, and not also those for `\u00d7`. This option is\n    only relevant if no explicit projection names are given as argument to `@[simps]`.\n-/\nstructure simps_cfg \nwhere\n  attrs : List name\n  short_name : Bool\n  simp_rhs : Bool\n  type_md : tactic.transparency\n  rhs_md : tactic.transparency\n  fully_applied : Bool\n  not_recursive : List name\n\n/-- Add a lemma with `nm` stating that `lhs = rhs`. `type` is the type of both `lhs` and `rhs`,\n  `args` is the list of local constants occurring, and `univs` is the list of universe variables.\n  If `add_simp` then we make the resulting lemma a simp-lemma. -/\n/-- Derive lemmas specifying the projections of the declaration.\n  If `todo` is non-empty, it will generate exactly the names in `todo`. -/\n/-- `simps_tac` derives simp-lemmas for all (nested) non-Prop projections of the declaration.\n  If `todo` is non-empty, it will generate exactly the names in `todo`.\n  If `short_nm` is true, the generated names will only use the last projection name. -/\n/-- The parser for the `@[simps]` attribute. -/\n/- note: we don't check whether the user has written a nonsense namespace in an argument. -/\n\n/--\nThe `@[simps]` attribute automatically derives lemmas specifying the projections of this\ndeclaration.\n\nExample:\n```lean\n@[simps] def foo : \u2115 \u00d7 \u2124 := (1, 2)\n```\nderives two simp-lemmas:\n```lean\n@[simp] lemma foo_fst : foo.fst = 1\n@[simp] lemma foo_snd : foo.snd = 2\n```\n\n* It does not derive simp-lemmas for the prop-valued projections.\n* It will automatically reduce newly created beta-redexes, but will not unfold any definitions.\n* If the structure has a coercion to either sorts or functions, and this is defined to be one\n  of the projections, then this coercion will be used instead of the projection.\n* If the structure is a class that has an instance to a notation class, like `has_mul`, then this\n  notation is used instead of the corresponding projection.\n* You can specify custom projections, by giving a declaration with name\n  `{structure_name}.simps.{projection_name}`. See Note [custom simps projection].\n\n  Example:\n  ```lean\n  def equiv.simps.inv_fun (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n  @[simps] def equiv.trans (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n  \u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n  ```\n  generates\n  ```\n  @[simp] lemma equiv.trans_to_fun : \u2200 {\u03b1 \u03b2 \u03b3} (e\u2081 e\u2082) (a : \u03b1), \u21d1(e\u2081.trans e\u2082) a = (\u21d1e\u2082 \u2218 \u21d1e\u2081) a\n  @[simp] lemma equiv.trans_inv_fun : \u2200 {\u03b1 \u03b2 \u03b3} (e\u2081 e\u2082) (a : \u03b3),\n    \u21d1((e\u2081.trans e\u2082).symm) a = (\u21d1(e\u2081.symm) \u2218 \u21d1(e\u2082.symm)) a\n  ```\n\n* You can specify custom projection names, by specifying the new projection names using\n  `initialize_simps_projections`.\n  Example: `initialize_simps_projections equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)`.\n\n* If one of the fields itself is a structure, this command will recursively create\n  simp-lemmas for all fields in that structure.\n  * Exception: by default it will not recursively create simp-lemmas for fields in the structures\n    `prod` and `pprod`. Give explicit projection names to override this behavior.\n\n  Example:\n  ```lean\n  structure my_prod (\u03b1 \u03b2 : Type*) := (fst : \u03b1) (snd : \u03b2)\n  @[simps] def foo : prod \u2115 \u2115 \u00d7 my_prod \u2115 \u2115 := \u27e8\u27e81, 2\u27e9, 3, 4\u27e9\n  ```\n  generates\n  ```lean\n  @[simp] lemma foo_fst : foo.fst = (1, 2)\n  @[simp] lemma foo_snd_fst : foo.snd.fst = 3\n  @[simp] lemma foo_snd_snd : foo.snd.snd = 4\n  ```\n\n* You can use `@[simps proj1 proj2 ...]` to only generate the projection lemmas for the specified\n  projections.\n* Recursive projection names can be specified using `proj1_proj2_proj3`.\n  This will create a lemma of the form `foo.proj1.proj2.proj3 = ...`.\n\n  Example:\n  ```lean\n  structure my_prod (\u03b1 \u03b2 : Type*) := (fst : \u03b1) (snd : \u03b2)\n  @[simps fst fst_fst snd] def foo : prod \u2115 \u2115 \u00d7 my_prod \u2115 \u2115 := \u27e8\u27e81, 2\u27e9, 3, 4\u27e9\n  ```\n  generates\n  ```lean\n  @[simp] lemma foo_fst : foo.fst = (1, 2)\n  @[simp] lemma foo_fst_fst : foo.fst.fst = 1\n  @[simp] lemma foo_snd : foo.snd = {fst := 3, snd := 4}\n  ```\n* If one of the values is an eta-expanded structure, we will eta-reduce this structure.\n\n  Example:\n  ```lean\n  structure equiv_plus_data (\u03b1 \u03b2) extends \u03b1 \u2243 \u03b2 := (data : bool)\n  @[simps] def bar {\u03b1} : equiv_plus_data \u03b1 \u03b1 := { data := tt, ..equiv.refl \u03b1 }\n  ```\n  generates the following, even though Lean inserts an eta-expanded version of `equiv.refl \u03b1` in the\n  definition of `bar`:\n  ```lean\n  @[simp] lemma bar_to_equiv : \u2200 {\u03b1 : Sort u_1}, bar.to_equiv = equiv.refl \u03b1\n  @[simp] lemma bar_data : \u2200 {\u03b1 : Sort u_1}, bar.data = tt\n  ```\n* For configuration options, see the doc string of `simps_cfg`.\n* The precise syntax is `('simps' ident* e)`, where `e` is an expression of type `simps_cfg`.\n* `@[simps]` reduces let-expressions where necessary.\n* If one of the fields is a partially applied constructor, we will eta-expand it\n  (this likely never happens).\n* When option `trace.simps.verbose` is true, `simps` will print the projections it finds and the\n  lemmas it generates.\n* Use `@[to_additive, simps]` to apply both `to_additive` and `simps` to a definition, making sure\n  that `simps` comes after `to_additive`. This will also generate the additive versions of all\n  simp-lemmas. Note however, that the additive versions of the simp-lemmas always use the default\n  name generated by `to_additive`, even if a custom name is given for the additive version of the\n  definition.\n  -/\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/tactic/simps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307944803831, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3532147608096484}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta, Scott Morrison\n-/\nimport category_theory.currying\nimport category_theory.limits.over\nimport category_theory.monad.adjunction\n\n/-!\n# Monomorphisms over a fixed object\n\nAs preparation for defining `subobject X`, we set up the theory for\n`mono_over X := {f : over X // mono f.hom}`.\n\nHere `mono_over X` is a thin category (a pair of objects has at most one morphism between them),\nso we can think of it as a preorder. However as it is not skeletal, it is not yet a partial order.\n\n`subobject X` will be defined as the skeletalization of `mono_over X`.\n\nWe provide\n* `def pullback [has_pullbacks C] (f : X \u27f6 Y) : mono_over Y \u2964 mono_over X`\n* `def map (f : X \u27f6 Y) [mono f] : mono_over X \u2964 mono_over Y`\n* `def \u00abexists\u00bb [has_images C] (f : X \u27f6 Y) : mono_over X \u2964 mono_over Y`\nand prove their basic properties and relationships.\n\n## Notes\n\nThis development originally appeared in Bhavik Mehta's \"Topos theory for Lean\" repository,\nand was ported to mathlib by Scott Morrison.\n\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable theory\nnamespace category_theory\n\nopen category_theory category_theory.category category_theory.limits\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] {X Y Z : C}\nvariables {D : Type u\u2082} [category.{v\u2082} D]\n\n/--\nThe category of monomorphisms into `X` as a full subcategory of the over category.\nThis isn't skeletal, so it's not a partial order.\n\nLater we define `subobject X` as the quotient of this by isomorphisms.\n-/\n@[derive [category]]\ndef mono_over (X : C) := {f : over X // mono f.hom}\n\nnamespace mono_over\n\n/-- Construct a `mono_over X`. -/\n@[simps]\ndef mk' {X A : C} (f : A \u27f6 X) [hf : mono f] : mono_over X := { val := over.mk f, property := hf }\n\n/-- The inclusion from monomorphisms over X to morphisms over X. -/\ndef forget (X : C) : mono_over X \u2964 over X := full_subcategory_inclusion _\n\ninstance : has_coe (mono_over X) C :=\n{ coe := \u03bb Y, Y.val.left, }\n\n@[simp]\nlemma forget_obj_left {f} : ((forget X).obj f).left = (f : C) := rfl\n\n@[simp] lemma mk'_coe' {X A : C} (f : A \u27f6 X) [hf : mono f] : (mk' f : C) = A := rfl\n\n/-- Convenience notation for the underlying arrow of a monomorphism over X. -/\nabbreviation arrow (f : mono_over X) : (f : C) \u27f6 X := ((forget X).obj f).hom\n\n@[simp] lemma mk'_arrow {X A : C} (f : A \u27f6 X) [hf : mono f] : (mk' f).arrow = f := rfl\n\n@[simp]\nlemma forget_obj_hom {f} : ((forget X).obj f).hom = f.arrow := rfl\n\ninstance : full (forget X) := full_subcategory.full _\ninstance : faithful (forget X) := full_subcategory.faithful _\n\ninstance mono (f : mono_over X) : mono f.arrow := f.property\n\n/-- The category of monomorphisms over X is a thin category,\nwhich makes defining its skeleton easy. -/\ninstance is_thin {X : C} (f g : mono_over X) : subsingleton (f \u27f6 g) :=\n\u27e8begin\n  intros h\u2081 h\u2082,\n  ext1,\n  erw [\u2190 cancel_mono g.arrow, over.w h\u2081, over.w h\u2082],\nend\u27e9\n\n@[reassoc] lemma w {f g : mono_over X} (k : f \u27f6 g) : k.left \u226b g.arrow = f.arrow := over.w _\n\n/-- Convenience constructor for a morphism in monomorphisms over `X`. -/\nabbreviation hom_mk {f g : mono_over X} (h : f.val.left \u27f6 g.val.left) (w : h \u226b g.arrow = f.arrow) :\n  f \u27f6 g :=\nover.hom_mk h w\n\n/-- Convenience constructor for an isomorphism in monomorphisms over `X`. -/\n@[simps]\ndef iso_mk {f g : mono_over X} (h : f.val.left \u2245 g.val.left) (w : h.hom \u226b g.arrow = f.arrow) :\n  f \u2245 g :=\n{ hom := hom_mk h.hom w,\n  inv := hom_mk h.inv (by rw [h.inv_comp_eq, w]) }\n\n/-- If `f : mono_over X`, then `mk' f.arrow` is of course just `f`, but not definitionally, so we\n    package it as an isomorphism. -/\n@[simp] def mk'_arrow_iso {X : C} (f : mono_over X) : (mk' f.arrow) \u2245 f :=\niso_mk (iso.refl _) (by simp)\n\n/--\nLift a functor between over categories to a functor between `mono_over` categories,\ngiven suitable evidence that morphisms are taken to monomorphisms.\n-/\n@[simps]\ndef lift {Y : D} (F : over Y \u2964 over X)\n  (h : \u2200 (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) :\n  mono_over Y \u2964 mono_over X :=\n{ obj := \u03bb f, \u27e8_, h f\u27e9,\n  map := \u03bb _ _ k, (mono_over.forget X).preimage ((mono_over.forget Y \u22d9 F).map k), }\n\n/--\nIsomorphic functors `over Y \u2964 over X` lift to isomorphic functors `mono_over Y \u2964 mono_over X`.\n-/\ndef lift_iso {Y : D} {F\u2081 F\u2082 : over Y \u2964 over X} (h\u2081 h\u2082) (i : F\u2081 \u2245 F\u2082) :\n  lift F\u2081 h\u2081 \u2245 lift F\u2082 h\u2082 :=\nfully_faithful_cancel_right (mono_over.forget X) (iso_whisker_left (mono_over.forget Y) i)\n\n/-- `mono_over.lift` commutes with composition of functors. -/\ndef lift_comp {X Z : C} {Y : D} (F : over X \u2964 over Y) (G : over Y \u2964 over Z) (h\u2081 h\u2082) :\n  lift F h\u2081 \u22d9 lift G h\u2082 \u2245 lift (F \u22d9 G) (\u03bb f, h\u2082 \u27e8_, h\u2081 f\u27e9) :=\nfully_faithful_cancel_right (mono_over.forget _) (iso.refl _)\n\n/-- `mono_over.lift` preserves the identity functor. -/\ndef lift_id :\n  lift (\ud835\udfed (over X)) (\u03bb f, f.2) \u2245 \ud835\udfed _ :=\nfully_faithful_cancel_right (mono_over.forget _) (iso.refl _)\n\n@[simp]\nlemma lift_comm (F : over Y \u2964 over X)\n  (h : \u2200 (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) :\n  lift F h \u22d9 mono_over.forget X = mono_over.forget Y \u22d9 F :=\nrfl\n\n@[simp]\nlemma lift_obj_arrow {Y : D} (F : over Y \u2964 over X)\n  (h : \u2200 (f : mono_over Y), mono (F.obj ((mono_over.forget Y).obj f)).hom) (f : mono_over Y) :\n  ((lift F h).obj f).arrow = (F.obj ((forget Y).obj f)).hom :=\nrfl\n\n/--\nMonomorphisms over an object `f : over A` in an over category\nare equivalent to monomorphisms over the source of `f`.\n-/\ndef slice {A : C} {f : over A} (h\u2081 h\u2082) : mono_over f \u224c mono_over f.left :=\n{ functor := mono_over.lift f.iterated_slice_equiv.functor h\u2081,\n  inverse := mono_over.lift f.iterated_slice_equiv.inverse h\u2082,\n  unit_iso := mono_over.lift_id.symm \u226a\u226b\n    mono_over.lift_iso _ _ f.iterated_slice_equiv.unit_iso \u226a\u226b\n    (mono_over.lift_comp _ _ _ _).symm,\n  counit_iso := mono_over.lift_comp _ _ _ _ \u226a\u226b\n    mono_over.lift_iso _ _ f.iterated_slice_equiv.counit_iso \u226a\u226b\n    mono_over.lift_id }\n\nsection pullback\nvariables [has_pullbacks C]\n\n/-- When `C` has pullbacks, a morphism `f : X \u27f6 Y` induces a functor `mono_over Y \u2964 mono_over X`,\nby pulling back a monomorphism along `f`. -/\ndef pullback (f : X \u27f6 Y) : mono_over Y \u2964 mono_over X :=\nmono_over.lift (over.pullback f)\nbegin\n  intro g,\n  apply @pullback.snd_of_mono _ _ _ _ _ _ _ _ _,\n  change mono g.arrow,\n  apply_instance,\nend\n\n/-- pullback commutes with composition (up to a natural isomorphism) -/\ndef pullback_comp (f : X \u27f6 Y) (g : Y \u27f6 Z) : pullback (f \u226b g) \u2245 pullback g \u22d9 pullback f :=\nlift_iso _ _ (over.pullback_comp _ _) \u226a\u226b (lift_comp _ _ _ _).symm\n\n/-- pullback preserves the identity (up to a natural isomorphism) -/\ndef pullback_id : pullback (\ud835\udfd9 X) \u2245 \ud835\udfed _ :=\nlift_iso _ _ over.pullback_id \u226a\u226b lift_id\n\n@[simp] lemma pullback_obj_left (f : X \u27f6 Y) (g : mono_over Y) :\n  (((pullback f).obj g) : C) = limits.pullback g.arrow f :=\nrfl\n\n@[simp] lemma pullback_obj_arrow (f : X \u27f6 Y) (g : mono_over Y) :\n  ((pullback f).obj g).arrow = pullback.snd :=\nrfl\n\nend pullback\n\nsection map\n\nattribute [instance] mono_comp\n\n/--\nWe can map monomorphisms over `X` to monomorphisms over `Y`\nby post-composition with a monomorphism `f : X \u27f6 Y`.\n-/\ndef map (f : X \u27f6 Y) [mono f] : mono_over X \u2964 mono_over Y :=\nlift (over.map f)\n(\u03bb g, by apply mono_comp g.arrow f)\n\n/-- `mono_over.map` commutes with composition (up to a natural isomorphism). -/\ndef map_comp (f : X \u27f6 Y) (g : Y \u27f6 Z) [mono f] [mono g] :\n  map (f \u226b g) \u2245 map f \u22d9 map g :=\nlift_iso _ _ (over.map_comp _ _) \u226a\u226b (lift_comp _ _ _ _).symm\n\n/-- `mono_over.map` preserves the identity (up to a natural isomorphism). -/\ndef map_id : map (\ud835\udfd9 X) \u2245 \ud835\udfed _ :=\nlift_iso _ _ over.map_id \u226a\u226b lift_id\n\n@[simp] lemma map_obj_left (f : X \u27f6 Y) [mono f] (g : mono_over X) :\n  (((map f).obj g) : C) = g.val.left :=\nrfl\n\n@[simp]\nlemma map_obj_arrow (f : X \u27f6 Y) [mono f] (g : mono_over X) :\n  ((map f).obj g).arrow = g.arrow \u226b f :=\nrfl\n\ninstance full_map (f : X \u27f6 Y) [mono f] : full (map f) :=\n{ preimage := \u03bb g h e,\n  begin\n    refine hom_mk e.left _,\n    rw [\u2190 cancel_mono f, assoc],\n    apply w e,\n  end }\n\ninstance faithful_map (f : X \u27f6 Y) [mono f] : faithful (map f) := {}.\n\n/--\nIsomorphic objects have equivalent `mono_over` categories.\n-/\n@[simps] def map_iso {A B : C} (e : A \u2245 B) : mono_over A \u224c mono_over B :=\n{ functor := map e.hom,\n  inverse := map e.inv,\n  unit_iso := ((map_comp _ _).symm \u226a\u226b eq_to_iso (by simp) \u226a\u226b map_id).symm,\n  counit_iso := ((map_comp _ _).symm \u226a\u226b eq_to_iso (by simp) \u226a\u226b map_id) }\n\nsection\nvariables (X)\n\n/-- An equivalence of categories `e` between `C` and `D` induces an equivalence between\n    `mono_over X` and `mono_over (e.functor.obj X)` whenever `X` is an object of `C`. -/\n@[simps] def congr (e : C \u224c D) : mono_over X \u224c mono_over (e.functor.obj X) :=\n{ functor := lift (over.post e.functor) $ \u03bb f, by { dsimp, apply_instance },\n  inverse := (lift (over.post e.inverse) $ \u03bb f, by { dsimp, apply_instance })\n    \u22d9 (map_iso (e.unit_iso.symm.app X)).functor,\n  unit_iso := nat_iso.of_components (\u03bb Y, iso_mk (e.unit_iso.app Y) (by tidy)) (by tidy),\n  counit_iso := nat_iso.of_components (\u03bb Y, iso_mk (e.counit_iso.app Y) (by tidy)) (by tidy) }\n\nend\n\nsection\nvariable [has_pullbacks C]\n\n/-- `map f` is left adjoint to `pullback f` when `f` is a monomorphism -/\ndef map_pullback_adj (f : X \u27f6 Y) [mono f] : map f \u22a3 pullback f :=\nadjunction.restrict_fully_faithful\n  (forget X) (forget Y) (over.map_pullback_adj f) (iso.refl _) (iso.refl _)\n\n/-- `mono_over.map f` followed by `mono_over.pullback f` is the identity. -/\ndef pullback_map_self (f : X \u27f6 Y) [mono f] :\n  map f \u22d9 pullback f \u2245 \ud835\udfed _ :=\n(as_iso (mono_over.map_pullback_adj f).unit).symm\n\nend\n\nend map\n\nsection image\nvariables (f : X \u27f6 Y) [has_image f]\n\n/--\nThe `mono_over Y` for the image inclusion for a morphism `f : X \u27f6 Y`.\n-/\ndef image_mono_over (f : X \u27f6 Y) [has_image f] : mono_over Y := mono_over.mk' (image.\u03b9 f)\n\n@[simp] lemma image_mono_over_arrow (f : X \u27f6 Y) [has_image f] :\n  (image_mono_over f).arrow = image.\u03b9 f :=\nrfl\n\nend image\n\nsection image\n\nvariables [has_images C]\n\n/--\nTaking the image of a morphism gives a functor `over X \u2964 mono_over X`.\n-/\n@[simps]\ndef image : over X \u2964 mono_over X :=\n{ obj := \u03bb f, image_mono_over f.hom,\n  map := \u03bb f g k,\n  begin\n    apply (forget X).preimage _,\n    apply over.hom_mk _ _,\n    refine image.lift {I := image _, m := image.\u03b9 g.hom, e := k.left \u226b factor_thru_image g.hom},\n    apply image.lift_fac,\n  end }\n\n/--\n`mono_over.image : over X \u2964 mono_over X` is left adjoint to\n`mono_over.forget : mono_over X \u2964 over X`\n-/\ndef image_forget_adj : image \u22a3 forget X :=\nadjunction.mk_of_hom_equiv\n{ hom_equiv := \u03bb f g,\n  { to_fun := \u03bb k,\n    begin\n      apply over.hom_mk (factor_thru_image f.hom \u226b k.left) _,\n      change (factor_thru_image f.hom \u226b k.left) \u226b _ = f.hom,\n      rw [assoc, over.w k],\n      apply image.fac\n    end,\n    inv_fun := \u03bb k,\n    begin\n      refine over.hom_mk _ _,\n      refine image.lift {I := g.val.left, m := g.arrow, e := k.left, fac' := over.w k},\n      apply image.lift_fac,\n    end,\n    left_inv := \u03bb k, subsingleton.elim _ _,\n    right_inv := \u03bb k,\n    begin\n      ext1,\n      change factor_thru_image _ \u226b image.lift _ = _,\n      rw [\u2190 cancel_mono g.arrow, assoc, image.lift_fac, image.fac f.hom],\n      exact (over.w k).symm,\n    end } }\n\ninstance : is_right_adjoint (forget X) :=\n{ left := image, adj := image_forget_adj }\n\ninstance reflective : reflective (forget X) := {}.\n\n/--\nForgetting that a monomorphism over `X` is a monomorphism, then taking its image,\nis the identity functor.\n-/\ndef forget_image : forget X \u22d9 image \u2245 \ud835\udfed (mono_over X) :=\nas_iso (adjunction.counit image_forget_adj)\n\nend image\n\nsection \u00abexists\u00bb\nvariables [has_images C]\n\n/--\nIn the case where `f` is not a monomorphism but `C` has images,\nwe can still take the \"forward map\" under it, which agrees with `mono_over.map f`.\n-/\ndef \u00abexists\u00bb (f : X \u27f6 Y) : mono_over X \u2964 mono_over Y :=\nforget _ \u22d9 over.map f \u22d9 image\n\ninstance faithful_exists (f : X \u27f6 Y) : faithful (\u00abexists\u00bb f) := {}.\n\n/--\nWhen `f : X \u27f6 Y` is a monomorphism, `exists f` agrees with `map f`.\n-/\ndef exists_iso_map (f : X \u27f6 Y) [mono f] : \u00abexists\u00bb f \u2245 map f :=\nnat_iso.of_components\nbegin\n  intro Z,\n  suffices : (forget _).obj ((\u00abexists\u00bb f).obj Z) \u2245 (forget _).obj ((map f).obj Z),\n    apply preimage_iso this,\n  apply over.iso_mk _ _,\n  apply image_mono_iso_source (Z.arrow \u226b f),\n  apply image_mono_iso_source_hom_self,\nend\nbegin\n  intros Z\u2081 Z\u2082 g,\n  ext1,\n  change image.lift \u27e8_, _, _, _\u27e9 \u226b (image_mono_iso_source (Z\u2082.arrow \u226b f)).hom =\n         (image_mono_iso_source (Z\u2081.arrow \u226b f)).hom \u226b g.left,\n  rw [\u2190 cancel_mono (Z\u2082.arrow \u226b f), assoc, assoc, w_assoc g, image_mono_iso_source_hom_self,\n      image_mono_iso_source_hom_self],\n  apply image.lift_fac,\nend\n\n/-- `exists` is adjoint to `pullback` when images exist -/\ndef exists_pullback_adj (f : X \u27f6 Y) [has_pullbacks C] : \u00abexists\u00bb f \u22a3 pullback f :=\nadjunction.restrict_fully_faithful (forget X) (\ud835\udfed _)\n  ((over.map_pullback_adj f).comp _ _ image_forget_adj)\n  (iso.refl _)\n  (iso.refl _)\n\nend \u00abexists\u00bb\n\nend mono_over\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/subobject/mono_over.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.35321475316797046}}
{"text": "/-\nCopyright (c) 2021 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport category_theory.concrete_category.basic\nimport tactic.fresh_names\nimport tactic.reassoc_axiom\nimport tactic.slice\n\n/-!\n# Tools to reformulate category-theoretic lemmas in concrete categories\n\n## The `elementwise` attribute\n\nThe `elementwise` attribute can be applied to a lemma\n\n```lean\n@[elementwise]\nlemma some_lemma {C : Type*} [category C]\n  {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : X \u27f6 Z) (w : ...) : f \u226b g = h := ...\n```\n\nand will produce\n\n```lean\nlemma some_lemma_apply {C : Type*} [category C] [concrete_category C]\n  {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : X \u27f6 Z) (w : ...) (x : X) : g (f x) = h x := ...\n```\n\nHere `X` is being coerced to a type via `concrete_category.has_coe_to_sort` and\n`f`, `g`, and `h` are being coerced to functions via `concrete_category.has_coe_to_fun`.\nFurther, we simplify the type using `concrete_category.coe_id : ((\ud835\udfd9 X) : X \u2192 X) x = x` and\n`concrete_category.coe_comp : (f \u226b g) x = g (f x)`,\nreplacing morphism composition with function composition.\n\nThe name of the produced lemma can be specified with `@[elementwise other_lemma_name]`.\nIf `simp` is added first, the generated lemma will also have the `simp` attribute.\n\n## Implementation\n\nThis closely follows the implementation of the `@[reassoc]` attribute, due to Simon Hudon.\nThanks to Gabriel Ebner for help diagnosing universe issues.\n\n-/\n\nnamespace tactic\n\nopen interactive lean.parser category_theory\n\n/--\nFrom an expression `f = g`,\nwhere `f g : X \u27f6 Y` for some objects `X Y : V` with `[S : category V]`,\nextract the expression for `S`.\n-/\nmeta def extract_category : expr \u2192 tactic expr\n| `(@eq (@quiver.hom ._ (@category_struct.to_quiver _\n     (@category.to_category_struct _ %%S)) _ _) _ _) := pure S\n| _ := failed\n\n/-- (internals for `@[elementwise]`)\nGiven a lemma of the form `f = g`, where `f g : X \u27f6 Y` and `X Y : V`,\nproves a new lemma of the form\n`\u2200 (x : X), f x = g x`\nif we are already in a concrete category, or\n`\u2200 [concrete_category.{w} V] (x : X), f x = g x`\notherwise.\n\nReturns the type and proof of this lemma,\nand the universe parameter `w` for the `concrete_category` instance, if it was not synthesized.\n-/\n-- This is closely modelled on `reassoc_axiom`.\nmeta def prove_elementwise (h : expr) : tactic (expr \u00d7 expr \u00d7 option name) :=\ndo\n   (vs,t) \u2190 infer_type h >>= open_pis,\n   (f, g) \u2190 match_eq t,\n   S \u2190 extract_category t <|> fail \"no morphism equation found in statement\",\n   `(@quiver.hom _ %%H %%X %%Y) \u2190 infer_type f,\n   C \u2190 infer_type X,\n   CC_type \u2190 to_expr ``(@concrete_category %%C %%S),\n   (CC, CC_found) \u2190 (do CC \u2190 mk_instance CC_type, pure (CC, tt)) <|>\n     (do CC \u2190 mk_local' `I binder_info.inst_implicit CC_type, pure (CC, ff)),\n   -- This is need to fill in universe levels fixed by `mk_instance`:\n   CC_type \u2190 instantiate_mvars CC_type,\n   x_type \u2190 to_expr ``(@coe_sort %%C _\n     (@category_theory.concrete_category.has_coe_to_sort %%C %%S %%CC) %%X),\n   x \u2190 mk_local_def `x x_type,\n   t' \u2190 to_expr ``(@coe_fn (@quiver.hom %%C %%H %%X %%Y) _\n     (@category_theory.concrete_category.has_coe_to_fun %%C %%S %%CC %%X %%Y) %%f %%x =\n       @coe_fn (@quiver.hom %%C %%H %%X %%Y) _\n         (@category_theory.concrete_category.has_coe_to_fun %%C %%S %%CC %%X %%Y) %%g %%x),\n   let c' := h.mk_app vs,\n   (_,pr) \u2190 solve_aux t' (rewrite_target c'; reflexivity),\n   -- The codomain of forget lives in a new universe, which may be now a universe metavariable\n   -- if we didn't synthesize an instance:\n   [w, _, _] \u2190 pure CC_type.get_app_fn.univ_levels,\n   -- We unify that with a fresh universe parameter.\n   n \u2190 match w with\n   | level.mvar _ := (do\n      n \u2190 get_unused_name_reserved [`w] mk_name_set,\n      unify (expr.sort (level.param n)) (expr.sort w),\n      pure (option.some n))\n   | _ := pure option.none\n   end,\n   t' \u2190 instantiate_mvars t',\n   CC \u2190 instantiate_mvars CC,\n   x \u2190 instantiate_mvars x,\n   -- Now the key step: replace morphism composition with function composition,\n   -- and identity morphisms with nothing.\n   let s := simp_lemmas.mk,\n   s \u2190 s.add_simp ``id_apply,\n   s \u2190 s.add_simp ``comp_apply,\n   (t'', pr', _) \u2190 simplify s [] t' {fail_if_unchanged := ff},\n   pr' \u2190 mk_eq_mp pr' pr,\n   -- Further, if we're in `Type`, get rid of the coercions entirely.\n   let s := simp_lemmas.mk,\n   s \u2190 s.add_simp ``concrete_category.has_coe_to_fun_Type,\n   (t'', pr'', _) \u2190 simplify s [] t'' {fail_if_unchanged := ff},\n   pr'' \u2190 mk_eq_mp pr'' pr',\n   t'' \u2190 pis (vs ++ (if CC_found then [x] else [CC, x])) t'',\n   pr'' \u2190 lambdas (vs ++ (if CC_found then [x] else [CC, x])) pr'',\n   pure (t'', pr'', n)\n\n/-- (implementation for `@[elementwise]`)\nGiven a declaration named `n` of the form `\u2200 ..., f = g`, proves a new lemma named `n'`\nof the form `\u2200 ... [concrete_category V] (x : X), f x = g x`.\n-/\nmeta def elementwise_lemma (n : name) (n' : name := n.append_suffix \"_apply\") : tactic unit :=\ndo d \u2190 get_decl n,\n   let c := @expr.const tt n d.univ_levels,\n   (t'',pr',l') \u2190 prove_elementwise c,\n   let params := l'.to_list ++ d.univ_params,\n   add_decl $ declaration.thm n' params t'' (pure pr'),\n   copy_attribute `simp n n'\n\n/--\nThe `elementwise` attribute can be applied to a lemma\n\n```lean\n@[elementwise]\nlemma some_lemma {C : Type*} [category C]\n  {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : X \u27f6 Z) (w : ...) : f \u226b g = h := ...\n```\n\nand will produce\n\n```lean\nlemma some_lemma_apply {C : Type*} [category C] [concrete_category C]\n  {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z) (h : X \u27f6 Z) (w : ...) (x : X) : g (f x) = h x := ...\n```\n\nHere `X` is being coerced to a type via `concrete_category.has_coe_to_sort` and\n`f`, `g`, and `h` are being coerced to functions via `concrete_category.has_coe_to_fun`.\nFurther, we simplify the type using `concrete_category.coe_id : ((\ud835\udfd9 X) : X \u2192 X) x = x` and\n`concrete_category.coe_comp : (f \u226b g) x = g (f x)`,\nreplacing morphism composition with function composition.\n\nThe `[concrete_category C]` argument will be omitted if it is possible to synthesize an instance.\n\nThe name of the produced lemma can be specified with `@[elementwise other_lemma_name]`.\nIf `simp` is added first, the generated lemma will also have the `simp` attribute.\n-/\n@[user_attribute]\nmeta def elementwise_attr : user_attribute unit (option name) :=\n{ name := `elementwise,\n  descr := \"create a companion lemma for a morphism equation applied to an element\",\n  parser := optional ident,\n  after_set := some (\u03bb n _ _,\n    do some n' \u2190 elementwise_attr.get_param n | elementwise_lemma n (n.append_suffix \"_apply\"),\n       elementwise_lemma n $ n.get_prefix ++ n' ) }\n\nadd_tactic_doc\n{ name                     := \"elementwise\",\n  category                 := doc_category.attr,\n  decl_names               := [`tactic.elementwise_attr],\n  tags                     := [\"category theory\"] }\n\nnamespace interactive\n\nsetup_tactic_parser\n\n/--\n`elementwise h`, for assumption `w : \u2200 ..., f \u226b g = h`, creates a new assumption\n`w : \u2200 ... (x : X), g (f x) = h x`.\n\n`elementwise! h`, does the same but deletes the initial `h` assumption.\n(You can also add the attribute `@[elementwise]` to lemmas to generate new declarations generalized\nin this way.)\n-/\nmeta def elementwise (del : parse (tk \"!\")?) (ns : parse ident*) : tactic unit :=\ndo ns.mmap' (\u03bb n,\n   do h \u2190 get_local n,\n      (t,pr,u) \u2190 prove_elementwise h,\n      assertv n t pr,\n      when del.is_some (tactic.clear h) )\n\nend interactive\n\n/-- Auxiliary definition for `category_theory.elementwise_of`. -/\nmeta def derive_elementwise_proof : tactic unit :=\ndo `(calculated_Prop %%v %%h) \u2190 target,\n   (t,pr,n) \u2190 prove_elementwise h,\n   unify v t,\n   exact pr\n\nend tactic\n\n/--\nWith `w : \u2200 ..., f \u226b g = h` (with universal quantifiers tolerated),\n`elementwise_of w : \u2200 ... (x : X), g (f x) = h x`.\n\nThe type and proof of `elementwise_of h` is generated by `tactic.derive_elementwise_proof`\nwhich makes `elementwise_of` meta-programming adjacent. It is not called as a tactic but as\nan expression. The goal is to avoid creating assumptions that are dismissed after one use:\n\n```lean\nexample (M N K : Mon.{u}) (f : M \u27f6 N) (g : N \u27f6 K) (h : M \u27f6 K) (w : f \u226b g = h) (m : M) :\n  g (f m) = h m :=\nbegin\n  rw elementwise_of w,\nend\n```\n-/\ntheorem category_theory.elementwise_of {\u03b1} (hh : \u03b1) {\u03b2}\n  (x : tactic.calculated_Prop \u03b2 hh . tactic.derive_elementwise_proof) : \u03b2 := x\n\n/--\nWith `w : \u2200 ..., f \u226b g = h` (with universal quantifiers tolerated),\n`elementwise_of w : \u2200 ... (x : X), g (f x) = h x`.\n\nAlthough `elementwise_of` is not a tactic or a meta program, its type is generated\nthrough meta-programming to make it usable inside normal expressions.\n-/\nadd_tactic_doc\n{ name                     := \"category_theory.elementwise_of\",\n  category                 := doc_category.tactic,\n  decl_names               := [`category_theory.elementwise_of],\n  tags                     := [\"category theory\"] }\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/elementwise.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290152, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.3530702648269804}}
{"text": "/-\nCopyright (c) 2023 Wojciech Nawrocki. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Wojciech Nawrocki\n-/\n\nimport ProofChecker.Data.ICnf\nimport ProofChecker.Data.Pog\nimport ProofChecker.Model.PropVars\nimport ProofChecker.Model.Extensions\nimport ProofChecker.Count.PropForm\n\n/-! Justifications of CPOG steps. -/\n\nopen PropTerm\n\ntheorem addDisj_new_var_equiv {A : Set Var} (\u0393 l\u2081 l\u2082 \u03c6\u2081 \u03c6\u2082 : PropTerm Var) :\n    s \u2209 A \u2192 X \u2286 A \u2192 \u2191\u0393.semVars \u2286 A \u2192 \u2191l\u2081.semVars \u2286 A \u2192 \u2191l\u2082.semVars \u2286 A \u2192\n    equivalentOver X (l\u2081 \u2293 \u0393) \u03c6\u2081 \u2192 equivalentOver X (l\u2082 \u2293 \u0393) \u03c6\u2082 \u2192\n    equivalentOver X (.var s \u2293 \u0393 \u2293 (.biImpl (.var s) (l\u2081 \u2294 l\u2082))) (\u03c6\u2081 \u2294 \u03c6\u2082) := by\n  intro hNMem hXA h\u0393 hL\u2081 hL\u2082 e\u2081 e\u2082 \u03c4\n  have hMem : s \u2209 X := fun h => absurd (hXA h) hNMem\n  have h\u0393 : s \u2209 \u0393.semVars := fun h => absurd (h\u0393 h) hNMem\n  have hL\u2081 : s \u2209 l\u2081.semVars := fun h => absurd (hL\u2081 h) hNMem\n  have hL\u2082 : s \u2209 l\u2082.semVars := fun h => absurd (hL\u2082 h) hNMem\n  constructor\n  case mp =>\n    intro \u27e8\u03c3\u2081, hAgree, h\u2081\u27e9\n    simp at h\u2081\n    have : \u03c3\u2081 \u22a8 \u0393 := by tauto\n    have : \u03c3\u2081 \u22a8 l\u2081 \u2294 l\u2082 := by simp; tauto\n    cases satisfies_disj.mp this with\n    | inl h =>\n      have : \u03c3\u2081 \u22a8 l\u2081 \u2293 \u0393 := by simp; tauto\n      have \u27e8\u03c3\u2082, hAgree\u2082, h\u2082\u27e9 := e\u2081 \u03c4 |>.mp \u27e8\u03c3\u2081, hAgree, this\u27e9\n      exact \u27e8\u03c3\u2082, hAgree\u2082, satisfies_disj.mpr (.inl h\u2082)\u27e9\n    | inr h =>\n      have : \u03c3\u2081 \u22a8 l\u2082 \u2293 \u0393 := by simp; tauto\n      have \u27e8\u03c3\u2082, hAgree\u2082, h\u2082\u27e9 := e\u2082 \u03c4 |>.mp \u27e8\u03c3\u2081, hAgree, this\u27e9\n      exact \u27e8\u03c3\u2082, hAgree\u2082, satisfies_disj.mpr (.inr h\u2082)\u27e9\n  case mpr =>\n    intro \u27e8\u03c3\u2082, hAgree, h\u2082\u27e9\n    cases satisfies_disj.mp h\u2082 with\n    | inl h =>\n      have \u27e8\u03c3\u2081, hAgree\u2081, h\u2081\u27e9 := e\u2081 \u03c4 |>.mpr \u27e8\u03c3\u2082, hAgree, h\u27e9\n      let \u03c3\u2081' := \u03c3\u2081.set s \u22a4\n      have : \u03c3\u2081' \u22a8 .var s := by simp\n      have hAgree\u2081' : \u03c3\u2081'.agreeOn X \u03c3\u2081 := \u03c3\u2081.agreeOn_set_of_not_mem _ hMem\n      have : \u03c3\u2081'.agreeOn X \u03c4 := hAgree\u2081'.trans hAgree\u2081\n      have : \u03c3\u2081' \u22a8 \u0393 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ h\u0393) |>.mpr\n        (satisfies_conj.mp h\u2081).right\n      have : \u03c3\u2081' \u22a8 l\u2081 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ hL\u2081) |>.mpr\n        (satisfies_conj.mp h\u2081).left\n      exact \u27e8\u03c3\u2081', by assumption, by simp; tauto\u27e9\n    | inr h =>\n      have \u27e8\u03c3\u2081, hAgree\u2081, h\u2081\u27e9 := e\u2082 \u03c4 |>.mpr \u27e8\u03c3\u2082, hAgree, h\u27e9\n      let \u03c3\u2081' := \u03c3\u2081.set s true\n      have : \u03c3\u2081' \u22a8 .var s := by simp\n      have hAgree\u2081' : \u03c3\u2081'.agreeOn X \u03c3\u2081 := \u03c3\u2081.agreeOn_set_of_not_mem _ hMem\n      have : \u03c3\u2081'.agreeOn X \u03c4 := hAgree\u2081'.trans hAgree\u2081\n      have : \u03c3\u2081' \u22a8 \u0393 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ h\u0393) |>.mpr\n        (satisfies_conj.mp h\u2081).right\n      have : \u03c3\u2081' \u22a8 l\u2082 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ hL\u2082) |>.mpr\n        (satisfies_conj.mp h\u2081).left\n      exact \u27e8\u03c3\u2081', by assumption, by simp; tauto\u27e9\n\ntheorem addDisj_partitioned {A : Set Var} (\u0393 l\u2081 l\u2082 : PropTerm Var) (\u03c6\u2081 \u03c6\u2082 : PropForm Var) :\n    -- Note: also works with l\u2081.semVars \u2286 A\n    \u2191l\u2082.semVars \u2286 A \u2192 hasUniqueExtension X A \u0393 \u2192\n    \u0393 \u2293 l\u2081 \u2293 l\u2082 \u2264 \u22a5 \u2192 equivalentOver X (l\u2081 \u2293 \u0393) \u27e6\u03c6\u2081\u27e7 \u2192 equivalentOver X (l\u2082 \u2293 \u0393) \u27e6\u03c6\u2082\u27e7 \u2192\n    \u03c6\u2081.partitioned \u2192 \u03c6\u2082.partitioned \u2192 (\u03c6\u2081.disj \u03c6\u2082).partitioned := by\n  intro hL\u2082 hUep hImp e\u2081 e\u2082 hD\u2081 hD\u2082\n  refine \u27e8hD\u2081, hD\u2082, fun \u03c4 \u27e8h\u2081, h\u2082\u27e9 => ?_\u27e9\n  have h\u2081 : \u03c4 \u22a8 \u27e6\u03c6\u2081\u27e7 := h\u2081\n  have h\u2082 : \u03c4 \u22a8 \u27e6\u03c6\u2082\u27e7 := h\u2082\n  have \u27e8\u03c3\u2081, hAgree\u2081, h\u03c3\u2081\u27e9 := e\u2081 \u03c4 |>.mpr \u27e8\u03c4, PropAssignment.agreeOn_refl _ _, h\u2081\u27e9\n  have \u27e8\u03c3\u2082, hAgree\u2082, h\u03c3\u2082\u27e9 := e\u2082 \u03c4 |>.mpr \u27e8\u03c4, PropAssignment.agreeOn_refl _ _, h\u2082\u27e9\n  simp at h\u03c3\u2081 h\u03c3\u2082\n  have h\u03c3\u2081\u0393 : \u03c3\u2081 \u22a8 \u0393 := by tauto\n  have h\u03c3\u2082\u0393 : \u03c3\u2082 \u22a8 \u0393 := by tauto\n  have hAgree : \u03c3\u2081.agreeOn A \u03c3\u2082 := hUep h\u03c3\u2081\u0393 h\u03c3\u2082\u0393 (hAgree\u2081.trans hAgree\u2082.symm)\n  have : \u03c3\u2082 \u22a8 l\u2082 := by tauto\n  have : \u03c3\u2081 \u22a8 l\u2082 := agreeOn_semVars (hAgree.subset hL\u2082) |>.mpr this\n  have : \u03c3\u2081 \u22a8 \u22a5 := entails_ext.mp hImp _ (by simp; tauto)\n  simp at this\n\n-- Alternative: use disjoint variables condition on \u03c6\u2081/\u03c6\u2082 to put together pair of assignments?!\ntheorem addConj_new_var_equiv\u2082 {A : Set Var} (\u0393 l\u2081 l\u2082 \u03c6\u2081 \u03c6\u2082 : PropTerm Var) :\n    -- Note: also works with \u03c6\u2081.semVars \u2286 X\n    p \u2209 X \u2192 p \u2209 \u0393.semVars \u2192 p \u2209 l\u2081.semVars \u2192 p \u2209 l\u2082.semVars \u2192 \u03c6\u2082.semVars \u2286 X \u2192\n    -- Note: also works with l\u2081.semVars \u2286 A\n    \u2191l\u2082.semVars \u2286 A \u2192 hasUniqueExtension X A \u0393 \u2192\n    equivalentOver X (l\u2081 \u2293 \u0393) \u03c6\u2081 \u2192 equivalentOver X (l\u2082 \u2293 \u0393) \u03c6\u2082 \u2192\n    equivalentOver X (.var p \u2293 (.biImpl (.var p) (l\u2081 \u2293 l\u2082)) \u2293 \u0393) (\u03c6\u2081 \u2293 \u03c6\u2082) := by\n  intro hMem h\u0393 hL\u2081 hL\u2082 h\u03c6\u2082 hL\u2082\u0393 hUep e\u2081 e\u2082 \u03c4\n  constructor\n  case mp =>\n    intro \u27e8\u03c3\u2081, hAgree, h\u2081\u27e9\n    simp at h\u2081\n    have \u27e8\u03c3\u2082, hAgree\u2082, h\u2082\u27e9 := e\u2081 \u03c4 |>.mp \u27e8\u03c3\u2081, hAgree, by simp; tauto\u27e9\n    have \u27e8\u03c3\u2082', hAgree\u2082', h\u2082'\u27e9 := e\u2082 \u03c4 |>.mp \u27e8\u03c3\u2081, hAgree, by simp; tauto\u27e9\n    have : \u03c3\u2082.agreeOn X \u03c3\u2082' := hAgree\u2082.trans hAgree\u2082'.symm\n    have : \u03c3\u2082 \u22a8 \u03c6\u2082 := agreeOn_semVars (this.subset h\u03c6\u2082) |>.mpr h\u2082'\n    exact \u27e8\u03c3\u2082, hAgree\u2082, by simp; tauto\u27e9\n  case mpr =>\n    intro \u27e8\u03c3\u2082, hAgree, h\u2082\u27e9\n    simp at h\u2082\n    have \u27e8\u03c3\u2081, hAgree\u2081, h\u2081\u27e9 := e\u2081 \u03c4 |>.mpr \u27e8\u03c3\u2082, hAgree, by tauto\u27e9\n    have \u27e8\u03c3\u2081', hAgree\u2081', h\u2081'\u27e9 := e\u2082 \u03c4 |>.mpr \u27e8\u03c3\u2082, hAgree, by tauto\u27e9\n    simp at h\u2081 h\u2081'\n    have h\u03c3\u2081\u0393 : \u03c3\u2081 \u22a8 \u0393 := by tauto\n    have h\u03c3\u2081'\u0393 : \u03c3\u2081' \u22a8 \u0393 := by tauto\n    have hAgree\u2081\u2081' : \u03c3\u2081.agreeOn A \u03c3\u2081' := hUep h\u03c3\u2081\u0393 h\u03c3\u2081'\u0393 (hAgree\u2081.trans hAgree\u2081'.symm)\n    have : \u03c3\u2081 \u22a8 l\u2082 := agreeOn_semVars (hAgree\u2081\u2081'.subset hL\u2082\u0393) |>.mpr (by tauto)\n    let \u03c3\u2083 := \u03c3\u2081.set p true\n    have : \u03c3\u2083 \u22a8 .var p := by simp\n    have : \u03c3\u2083 \u22a8 l\u2081 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ hL\u2081) |>.mpr (by tauto)\n    have : \u03c3\u2083 \u22a8 l\u2082 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ hL\u2082) |>.mpr (by tauto)\n    have : \u03c3\u2083 \u22a8 \u0393 := agreeOn_semVars (\u03c3\u2081.agreeOn_set_of_not_mem _ h\u0393) |>.mpr (by tauto)\n    exact \u27e8\u03c3\u2083, \u03c3\u2081.agreeOn_set_of_not_mem _ hMem |>.trans hAgree\u2081, by simp; tauto\u27e9\n\ntheorem addConj_new_var_equiv {A : Set Var} (G : Pog) (\u0393 : PropTerm Var) (ls : Array ILit) :\n    p \u2209 A \u2192 X \u2286 A \u2192 \u2191\u0393.semVars \u2286 A \u2192 hasUniqueExtension X A \u0393 \u2192\n    (\u2200 \u03c3\u2081, \u2203 (\u03c3\u2082 : PropAssignment Var), \u03c3\u2082.agreeOn X \u03c3\u2081 \u2227 \u03c3\u2082 \u22a8 \u0393) \u2192\n    (\u2200 l \u2208 ls.data, l.var \u2208 A \u2227 \u2191(PropTerm.semVars \u27e6G.toPropForm l\u27e7) \u2286 X \u2227\n      equivalentOver X (l.toPropTerm \u2293 \u0393) \u27e6G.toPropForm l\u27e7) \u2192\n    equivalentOver X\n      (.var p \u2293 (\u0393 \u2293 .biImpl (.var p) \u27e6PropForm.arrayConj (ls.map ILit.toPropForm)\u27e7))\n      \u27e6PropForm.arrayConj (ls.map G.toPropForm)\u27e7 := by\n  intro hMem hX h\u0393 hUep hExt hLs \u03c4\n  refine \u27e8?mp, ?mpr\u27e9 <;>\n    simp only [PropForm.mk_arrayConj, satisfies_conj, satisfies_biImpl,\n      PropForm.satisfies_arrayConjTerm, Array.map_data, List.mem_map', and_imp,\n      forall_apply_eq_imp_iff\u2082, forall_exists_index, ILit.mk_toPropForm]\n  case mp =>\n    intro \u03c3\u2081 hAgree h\u03c3\u2081p h\u03c3\u2081\u0393 h\u03c3\u2081\n    simp only [h\u03c3\u2081p, true_iff, ILit.mk_toPropForm] at h\u03c3\u2081\n    refine \u27e8\u03c3\u2081, hAgree, ?_\u27e9\n    intro l hL\n    have \u27e8_, hTpf, hEquiv\u27e9 := hLs l hL\n    have : \u03c3\u2081 \u22a8 l.toPropTerm := h\u03c3\u2081 l hL\n    have : \u03c3\u2081 \u22a8 l.toPropTerm \u2293 \u0393 := by simp [this, h\u03c3\u2081\u0393]\n    have \u27e8\u03c3\u2082, hAgree\u2082, h\u03c3\u2082\u27e9 := hEquiv \u03c4 |>.mp \u27e8\u03c3\u2081, hAgree, this\u27e9\n    apply agreeOn_semVars ?_ |>.mp h\u03c3\u2082\n    exact (hAgree\u2082.trans hAgree.symm).subset hTpf\n  case mpr =>\n    intro \u03c3\u2082 hAgree\u2082 hTpfs\n    have \u27e8\u03c3\u2081, hAgree\u2081, h\u2081\u27e9 := hExt \u03c4\n    let \u03c3\u2081' := \u03c3\u2081.set p true\n    have h\u03c3\u2081'p : \u03c3\u2081' \u22a8 .var p := by simp\n    have hAgree\u2081'A : \u03c3\u2081'.agreeOn A \u03c3\u2081 := \u03c3\u2081.agreeOn_set_of_not_mem _ hMem\n    have hAgree\u2081' : \u03c3\u2081'.agreeOn X \u03c4 := hAgree\u2081'A.subset hX |>.trans hAgree\u2081\n    have h\u03c3\u2081'\u0393 : \u03c3\u2081' \u22a8 \u0393 := agreeOn_semVars (hAgree\u2081'A.subset h\u0393) |>.mpr h\u2081\n    refine \u27e8\u03c3\u2081', hAgree\u2081', h\u03c3\u2081'p, h\u03c3\u2081'\u0393, \u27e8fun _ => ?_, fun _ => h\u03c3\u2081'p\u27e9\u27e9\n    intro l hL\n    have : \u03c3\u2082 \u22a8 \u27e6G.toPropForm l\u27e7 := hTpfs l hL\n    have \u27e8\u03c3\u2083, hAgree\u2083, h\u2083\u27e9 := (hLs l hL).right.right \u03c4 |>.mpr \u27e8\u03c3\u2082, hAgree\u2082, this\u27e9\n    refine agreeOn_semVars ?_ |>.mp (satisfies_conj.mp h\u2083).left\n    have : \u2191l.toPropTerm.semVars \u2286 A := by simp [(hLs l hL).left]\n    apply PropAssignment.agreeOn.subset this\n    exact hUep (satisfies_conj.mp h\u2083).right h\u03c3\u2081'\u0393 (hAgree\u2083.trans hAgree\u2081'.symm)\n\n/-! Other stuff that doesn't fit anywhere. -/\n\ntheorem partitioned_lit (l : ILit) : l.toPropForm.partitioned := by\n  dsimp [ILit.toPropForm]\n  cases l.polarity <;> simp [PropForm.partitioned]", "meta": {"author": "rebryant", "repo": "cpog", "sha": "5e39029ce71de532fd4407c4768e7c2bf97798c8", "save_path": "github-repos/lean/rebryant-cpog", "path": "github-repos/lean/rebryant-cpog/cpog-5e39029ce71de532fd4407c4768e7c2bf97798c8/VerifiedChecker/ProofChecker/Model/Cpog.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3530702641138494}}
{"text": "import phase2.flexible\n\n/-!\n# Reductions of supports\n-/\n\nuniverse u\n\nopen mul_action quiver set sum with_bot\nopen_locale cardinal\n\nnamespace con_nf\nvariables [params.{u}] (\u03b1 : \u039b) [position_data.{}] [phase_2_assumptions \u03b1]\n\nvariables (\u03b2 : \u039b) (G : Type*) {\u03c4 : Type*} [has_smul G (support_condition \u03b2)] [has_smul G \u03c4]\n\nvariables {\u03b2 G} {x : \u03c4}\n\n/-- A support condition is *reduced* if it is an atom or a litter. -/\n@[mk_iff] inductive reduced {\u03b2 : type_index} : support_condition \u03b2 \u2192 Prop\n| mk_atom (a : atom) (B : extended_index \u03b2) : reduced (inl a, B)\n| mk_litter (L : litter) (B : extended_index \u03b2) : reduced (inr L.to_near_litter, B)\n\n/-- The *reduction* of a set of support conditions is the downward closure of the set under\nthe constrains relation, but we only keep reduced conditions. -/\ndef reduction (S : set (support_condition \u03b2)) : set (support_condition \u03b2) :=\n{c | \u2203 d \u2208 S, relation.refl_trans_gen (constrains \u03b1 \u03b2) c d} \u2229 set_of reduced\n\nlemma reduction_singleton (c : support_condition \u03b2) :\n  reduction \u03b1 {c} = ({c} \u222a {d | relation.trans_gen (constrains \u03b1 \u03b2) d c}) \u2229 set_of reduced :=\nby simp only [reduction, mem_singleton_iff, exists_prop, exists_eq_left,\n  relation.refl_trans_gen_iff_eq_or_trans_gen, set_of_or, set_of_eq_eq_singleton']\n\nlemma reduction_singleton_of_not_reduced (c : support_condition \u03b2) (hc : \u00acreduced c) :\n  reduction \u03b1 {c} = {d | relation.trans_gen (constrains \u03b1 \u03b2) d c} \u2229 {d | reduced d} :=\nbegin\n  simp only [reduction_singleton, inter_distrib_right, union_eq_right_iff_subset,\n    subset_inter_iff, inter_subset_right, and_true],\n  rintros d \u27e8hd, hd'\u27e9,\n  cases hd,\n  cases hc hd',\nend\n\nlemma small_constrains (c : support_condition \u03b2) : small {d | d \u227a[\u03b1] c} :=\nbegin\n  obtain \u27e8a | N, A\u27e9 := c,\n  { simp only [constrains_atom, set_of_eq_eq_singleton, small_singleton], },\n  simp_rw constrains_iff,\n  refine small.union _ (small.union _ (small.union _ (small.union _ _)));\n    rw small_set_of,\n  { simp only [prod.mk.inj_iff, false_and, and_false,\n      exists_false, set_of_false, small_empty], },\n  { simp only [ne.def, prod.mk.inj_iff, exists_eq_right_right'],\n    by_cases litter_set N.fst = N.snd,\n    simp only [h, eq_self_iff_true, not_true, false_and, set_of_false, small_empty],\n    simp only [h, not_false_iff, true_and, set_of_eq_eq_singleton, small_singleton], },\n  { simp only [prod.mk.inj_iff, exists_eq_right_right'],\n    have : small {c : support_condition \u03b2 | \u2203 a, a \u2208 litter_set N.fst \u2206 N.snd \u2227 c = (inl a, A)},\n    { refine lt_of_le_of_lt _ N.2.prop,\n      refine \u27e8\u27e8\u03bb c, \u27e8_, c.2.some_spec.1\u27e9, _\u27e9\u27e9,\n      rintros \u27e8c, hc\u27e9 \u27e8d, hd\u27e9,\n      simp only [subtype.val_eq_coe, subtype.mk_eq_mk],\n      intro h,\n      rw [hc.some_spec.2, hd.some_spec.2, h], },\n    convert this using 1,\n    ext \u27e8a | N, A\u27e9 : 1,\n    { simp only [mem_set_of_eq, prod.mk.inj_iff],\n      split,\n      { rintro \u27e8_, a', h\u2081, h\u2082, rfl\u27e9,\n        exact \u27e8a', h\u2081, h\u2082\u27e9, },\n      { rintro \u27e8a', h\u2081, h\u2082\u27e9,\n        exact \u27e8N, a', h\u2081, h\u2082, rfl\u27e9, } },\n    { simp only [mem_set_of_eq, prod.mk.inj_iff, false_and, and_false, exists_false], }, },\n  { by_cases \u2203 \u2983\u03b3 : Iic \u03b1\u2984 \u2983\u03b4 : Iio \u03b1\u2984 \u2983\u03b5 : Iio \u03b1\u2984 (h\u03b4 : (\u03b4 : \u039b) < \u03b3) (h\u03b5 : (\u03b5 : \u039b) < \u03b3)\n      (h\u03b4\u03b5 : \u03b4 \u2260 \u03b5) (B : path (\u03b2 : type_index) \u03b3) (t : tangle \u03b4),\n      N = (f_map (coe_ne_coe.mpr $ coe_ne' h\u03b4\u03b5) t).to_near_litter \u2227\n      A = (B.cons (coe_lt h\u03b5)).cons (bot_lt_coe _),\n    { obtain \u27e8\u03b3, \u03b4, \u03b5, h\u03b4, h\u03b5, h\u03b4\u03b5, B, t, rfl, rfl\u27e9 := h,\n      refine lt_of_le_of_lt _ (designated_support t).small,\n      suffices : #{a : support_condition \u03b2 | \u2203 c : designated_support t,\n        a = \u27e8c.val.fst, (B.cons (coe_lt h\u03b4)).comp c.val.snd\u27e9} \u2264 #(designated_support t),\n      { refine le_trans (cardinal.mk_subtype_le_of_subset _) this,\n        rintros x \u27e8_, _, _, _, _, _, _, _, c, hc, rfl, h\u27e9,\n        simp only [prod.mk.inj_iff, litter.to_near_litter_injective.eq_iff, f_map] at h,\n        cases subtype.coe_inj.mp (coe_inj.mp h.1.2.1),\n        cases subtype.coe_inj.mp h.1.2.2,\n        cases choose_wf_injective h.1.1,\n        cases subtype.coe_inj.mp (coe_inj.mp\n          (path.obj_eq_of_cons_eq_cons (path.heq_of_cons_eq_cons h.2).eq)),\n        cases (path.heq_of_cons_eq_cons (path.heq_of_cons_eq_cons h.2).eq).eq,\n        exact \u27e8\u27e8c, hc\u27e9, rfl\u27e9, },\n      refine \u27e8\u27e8\u03bb a, a.prop.some, _\u27e9\u27e9,\n      intros a b h,\n      refine subtype.coe_inj.mp _,\n      simp only [subtype.val_eq_coe] at h,\n      rw [a.prop.some_spec, b.prop.some_spec],\n      simp only [h, subtype.val_eq_coe], },\n    { refine small_of_forall_not_mem _,\n      rintro x \u27e8\u03b3, \u03b4, \u03b5, h\u03b4, h\u03b5, h\u03b4\u03b5, B, t, c, hN, rfl, hA\u27e9,\n      simp only [prod.mk.inj_iff] at hA,\n      refine h \u27e8\u03b3, \u03b4, \u03b5, h\u03b4, h\u03b5, h\u03b4\u03b5, B, t, hA\u27e9, }, },\n  { refine subsingleton.small _,\n    rintros \u27e8c, C\u27e9 \u27e8\u03b3, \u03b5, h\u03b5, C', a, hc\u2081, hc\u2082\u27e9 \u27e8d, D\u27e9 \u27e8\u03b3, \u03b5, h\u03b5, D', b, hd\u2081, hd\u2082\u27e9,\n    simp only [prod.mk.inj_iff] at hc\u2081 hc\u2082 hd\u2081 hd\u2082,\n    rw [hc\u2081.1, hc\u2081.2, hd\u2081.1, hd\u2081.2],\n    rw [hc\u2082.1, hc\u2082.2, litter.to_near_litter_injective.eq_iff] at hd\u2082,\n    cases subtype.coe_inj.mp (coe_inj.mp (path.obj_eq_of_cons_eq_cons hd\u2082.2)),\n    cases subtype.coe_inj.mp (coe_inj.mp (path.obj_eq_of_cons_eq_cons\n      (path.heq_of_cons_eq_cons hd\u2082.2).eq)),\n    cases (path.heq_of_cons_eq_cons (path.heq_of_cons_eq_cons hd\u2082.2).eq).eq,\n    rw (f_map_injective bot_ne_coe).eq_iff at hd\u2082,\n    cases hd\u2082.1,\n    refl, },\nend\n\ndef nth_reduction (S : set (support_condition \u03b2)) : \u2115 \u2192 set (support_condition \u03b2)\n| 0 := S\n| (n + 1) := {c | \u2203 d, d \u2208 nth_reduction n \u2227 c \u227a[\u03b1] d}\n\nlemma small_nth_reduction {S : set (support_condition \u03b2)} {n : \u2115} (h : small S) :\n  small (nth_reduction \u03b1 S n) :=\nbegin\n  induction n with n hn,\n  exact h,\n  rw nth_reduction,\n  simp_rw [\u2190 exists_prop, subtype.exists', set_of_exists],\n  refine small_Union hn _,\n  rintro \u27e8c, hc\u27e9,\n  exact small_constrains \u03b1 c,\nend\n\nlemma mem_nth_reduction_iff {S : set (support_condition \u03b2)} {n : \u2115}\n  {c : support_condition \u03b2} :\n  c \u2208 nth_reduction \u03b1 S n \u2194\n  \u2203 l, list.chain (constrains \u03b1 \u03b2) c l \u2227 l.length = n \u2227 (c :: l).last (list.cons_ne_nil _ _) \u2208 S :=\nbegin\n  induction n with n hn generalizing c,\n  { rw nth_reduction,\n    split,\n    { intro h,\n      exact \u27e8[], list.chain.nil, rfl, h\u27e9, },\n    { rintro \u27e8l, h\u2081, h\u2082, h\u2083\u27e9,\n      rw list.length_eq_zero at h\u2082,\n      cases h\u2082,\n      exact h\u2083, }, },\n  { simp only [nth_reduction, mem_set_of_eq],\n    split,\n    { rintro \u27e8d, hd\u2081, hd\u2082\u27e9,\n      obtain \u27e8l, hl\u2081, hl\u2082, hl\u2083\u27e9 := hn.mp hd\u2081,\n      refine \u27e8d :: l, list.chain.cons hd\u2082 hl\u2081, _, _\u27e9,\n      { rw [list.length_cons, hl\u2082], },\n      { rw list.last_cons,\n        exact hl\u2083, }, },\n    { rintro \u27e8_ | \u27e8d, l\u27e9, hl\u2081, hl\u2082, hl\u2083\u27e9,\n      { cases hl\u2082, },\n      obtain _ | \u27e8hcd, hl\u2081\u27e9 := hl\u2081,\n      rw list.last_cons at hl\u2083,\n      have := hn.mpr \u27e8l, hl\u2081, nat.succ.inj hl\u2082, hl\u2083\u27e9,\n      exact \u27e8d, this, hcd\u27e9, }, },\nend\n\nlemma reduction_eq_Union {S : set (support_condition \u03b2)} :\n  {c | \u2203 d \u2208 S, relation.refl_trans_gen (constrains \u03b1 \u03b2) c d} = \u22c3 n, nth_reduction \u03b1 S n :=\nbegin\n  refine subset_antisymm _ _,\n  { rintros c \u27e8d, hdS, hd\u27e9,\n    obtain \u27e8l, hl, rfl\u27e9 := list.exists_chain_of_relation_refl_trans_gen hd,\n    rw mem_Union,\n    refine \u27e8l.length, _\u27e9,\n    rw mem_nth_reduction_iff,\n    refine \u27e8l, hl, rfl, hdS\u27e9, },\n  { intros c hc,\n    rw mem_Union at hc,\n    obtain \u27e8i, hc\u27e9 := hc,\n    rw mem_nth_reduction_iff at hc,\n    obtain \u27e8l, hl\u2081, hl\u2082, hl\u2083\u27e9 := hc,\n    exact \u27e8(c :: l).last (list.cons_ne_nil _ _), hl\u2083,\n      list.relation_refl_trans_gen_of_exists_chain l hl\u2081 rfl\u27e9, },\nend\n\nlemma reduction_small' {S : set (support_condition \u03b2)} (h : small S) :\n  small {c | \u2203 d \u2208 S, relation.refl_trans_gen (constrains \u03b1 \u03b2) c d} :=\nbegin\n  rw reduction_eq_Union,\n  have : small \u22c3 (n : ulift \u2115), nth_reduction \u03b1 S n.down,\n  { refine small_Union _ (\u03bb _, small_nth_reduction \u03b1 h),\n    rw cardinal.mk_denumerable,\n    exact \u039b_limit.aleph_0_le.trans_lt \u039b_lt_\u03ba, },\n  { convert this using 1,\n    ext x : 1,\n    simp only [mem_Union, ulift.exists], },\nend\n\nlemma reduction_small {S : set (support_condition \u03b2)} (h : small S) :\n  small (reduction \u03b1 S) :=\nlt_of_le_of_lt (cardinal.mk_subtype_le_of_subset (\u03bb c hc, hc.1)) (reduction_small' \u03b1 h)\n\nend con_nf\n", "meta": {"author": "leanprover-community", "repo": "con-nf", "sha": "f0b66bd73ca5d3bd8b744985242c4c0b5464913f", "save_path": "github-repos/lean/leanprover-community-con-nf", "path": "github-repos/lean/leanprover-community-con-nf/con-nf-f0b66bd73ca5d3bd8b744985242c4c0b5464913f/src/phase2/reduction.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.611381973294151, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.353070255935528}}
{"text": "/-\nCopyright (c) 2017 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro, Yury G. Kudryashov\n\n! This file was ported from Lean 3 source module data.sum.basic\n! leanprover-community/mathlib commit bd9851ca476957ea4549eb19b40e7b5ade9428cc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Logic.Function.Basic\nimport Mathbin.Tactic.Basic\n\n/-!\n# Disjoint union of types\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file proves basic results about the sum type `\u03b1 \u2295 \u03b2`.\n\n`\u03b1 \u2295 \u03b2` is the type made of a copy of `\u03b1` and a copy of `\u03b2`. It is also called *disjoint union*.\n\n## Main declarations\n\n* `sum.get_left`: Retrieves the left content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the right.\n* `sum.get_right`: Retrieves the right content of `x : \u03b1 \u2295 \u03b2` or returns `none` if it's coming from\n  the left.\n* `sum.is_left`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the left component or not.\n* `sum.is_right`: Returns whether `x : \u03b1 \u2295 \u03b2` comes from the right component or not.\n* `sum.map`: Maps `\u03b1 \u2295 \u03b2` to `\u03b3 \u2295 \u03b4` component-wise.\n* `sum.elim`: Nondependent eliminator/induction principle for `\u03b1 \u2295 \u03b2`.\n* `sum.swap`: Maps `\u03b1 \u2295 \u03b2` to `\u03b2 \u2295 \u03b1` by swapping components.\n* `sum.lex`: Lexicographic order on `\u03b1 \u2295 \u03b2` induced by a relation on `\u03b1` and a relation on `\u03b2`.\n\n## Notes\n\nThe definition of `sum` takes values in `Type*`. This effectively forbids `Prop`- valued sum types.\nTo this effect, we have `psum`, which takes value in `Sort*` and carries a more complicated\nuniverse signature in consequence. The `Prop` version is `or`.\n-/\n\n\nuniverse u v w x\n\nvariable {\u03b1 : Type u} {\u03b1' : Type w} {\u03b2 : Type v} {\u03b2' : Type x} {\u03b3 \u03b4 : Type _}\n\nnamespace Sum\n\nderiving instance DecidableEq for Sum\n\n#print Sum.forall /-\n@[simp]\ntheorem forall {p : Sum \u03b1 \u03b2 \u2192 Prop} : (\u2200 x, p x) \u2194 (\u2200 a, p (inl a)) \u2227 \u2200 b, p (inr b) :=\n  \u27e8fun h => \u27e8fun a => h _, fun b => h _\u27e9, fun \u27e8h\u2081, h\u2082\u27e9 => Sum.rec h\u2081 h\u2082\u27e9\n#align sum.forall Sum.forall\n-/\n\n#print Sum.exists /-\n@[simp]\ntheorem exists {p : Sum \u03b1 \u03b2 \u2192 Prop} : (\u2203 x, p x) \u2194 (\u2203 a, p (inl a)) \u2228 \u2203 b, p (inr b) :=\n  \u27e8fun h =>\n    match h with\n    | \u27e8inl a, h\u27e9 => Or.inl \u27e8a, h\u27e9\n    | \u27e8inr b, h\u27e9 => Or.inr \u27e8b, h\u27e9,\n    fun h =>\n    match h with\n    | Or.inl \u27e8a, h\u27e9 => \u27e8inl a, h\u27e9\n    | Or.inr \u27e8b, h\u27e9 => \u27e8inr b, h\u27e9\u27e9\n#align sum.exists Sum.exists\n-/\n\n#print Sum.inl_injective /-\ntheorem inl_injective : Function.Injective (inl : \u03b1 \u2192 Sum \u03b1 \u03b2) := fun x y => inl.inj\n#align sum.inl_injective Sum.inl_injective\n-/\n\n#print Sum.inr_injective /-\ntheorem inr_injective : Function.Injective (inr : \u03b2 \u2192 Sum \u03b1 \u03b2) := fun x y => inr.inj\n#align sum.inr_injective Sum.inr_injective\n-/\n\nsection get\n\n#print Sum.getLeft /-\n/-- Check if a sum is `inl` and if so, retrieve its contents. -/\n@[simp]\ndef getLeft : Sum \u03b1 \u03b2 \u2192 Option \u03b1\n  | inl a => some a\n  | inr _ => none\n#align sum.get_left Sum.getLeft\n-/\n\n#print Sum.getRight /-\n/-- Check if a sum is `inr` and if so, retrieve its contents. -/\n@[simp]\ndef getRight : Sum \u03b1 \u03b2 \u2192 Option \u03b2\n  | inr b => some b\n  | inl _ => none\n#align sum.get_right Sum.getRight\n-/\n\n#print Sum.isLeft /-\n/-- Check if a sum is `inl`. -/\n@[simp]\ndef isLeft : Sum \u03b1 \u03b2 \u2192 Bool\n  | inl _ => true\n  | inr _ => false\n#align sum.is_left Sum.isLeft\n-/\n\n#print Sum.isRight /-\n/-- Check if a sum is `inr`. -/\n@[simp]\ndef isRight : Sum \u03b1 \u03b2 \u2192 Bool\n  | inl _ => false\n  | inr _ => true\n#align sum.is_right Sum.isRight\n-/\n\nvariable {x y : Sum \u03b1 \u03b2}\n\n#print Sum.getLeft_eq_none_iff /-\n@[simp]\ntheorem getLeft_eq_none_iff : x.getLeft = none \u2194 x.isRight := by\n  cases x <;>\n    simp only [get_left, is_right, Bool.coe_sort_true, Bool.coe_sort_false, eq_self_iff_true]\n#align sum.get_left_eq_none_iff Sum.getLeft_eq_none_iff\n-/\n\n#print Sum.getRight_eq_none_iff /-\n@[simp]\ntheorem getRight_eq_none_iff : x.getRight = none \u2194 x.isLeft := by\n  cases x <;>\n    simp only [get_right, is_left, Bool.coe_sort_true, Bool.coe_sort_false, eq_self_iff_true]\n#align sum.get_right_eq_none_iff Sum.getRight_eq_none_iff\n-/\n\n#print Sum.getLeft_eq_some_iff /-\n@[simp]\ntheorem getLeft_eq_some_iff {a} : x.getLeft = some a \u2194 x = inl a := by\n  cases x <;> simp only [get_left]\n#align sum.get_left_eq_some_iff Sum.getLeft_eq_some_iff\n-/\n\n#print Sum.getRight_eq_some_iff /-\n@[simp]\ntheorem getRight_eq_some_iff {b} : x.getRight = some b \u2194 x = inr b := by\n  cases x <;> simp only [get_right]\n#align sum.get_right_eq_some_iff Sum.getRight_eq_some_iff\n-/\n\n#print Sum.not_isLeft /-\n@[simp]\ntheorem not_isLeft (x : Sum \u03b1 \u03b2) : not x.isLeft = x.isRight := by cases x <;> rfl\n#align sum.bnot_is_left Sum.not_isLeft\n-/\n\n#print Sum.isLeft_eq_false /-\n@[simp]\ntheorem isLeft_eq_false : x.isLeft = false \u2194 x.isRight := by cases x <;> simp\n#align sum.is_left_eq_ff Sum.isLeft_eq_false\n-/\n\n#print Sum.Not_isLeft /-\ntheorem Not_isLeft : \u00acx.isLeft \u2194 x.isRight := by simp\n#align sum.not_is_left Sum.Not_isLeft\n-/\n\n/- warning: sum.bnot_is_right -> Sum.not_isRight is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} (x : Sum.{u1, u2} \u03b1 \u03b2), Eq.{1} Bool (not (Sum.isRight.{u1, u2} \u03b1 \u03b2 x)) (Sum.isLeft.{u1, u2} \u03b1 \u03b2 x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} (x : Sum.{u1, u2} \u03b1 \u03b2), Eq.{1} Bool (not (Decidable.decide (Eq.{1} Bool (Sum.isRight.{u1, u2} \u03b1 \u03b2 x) (Sum.isLeft.{u1, u2} \u03b1 \u03b2 x)) (instDecidableEqBool (Sum.isRight.{u1, u2} \u03b1 \u03b2 x) (Sum.isLeft.{u1, u2} \u03b1 \u03b2 x)))) Bool.true\nCase conversion may be inaccurate. Consider using '#align sum.bnot_is_right Sum.not_isRight\u2093'. -/\n@[simp]\ntheorem not_isRight (x : Sum \u03b1 \u03b2) : not x.isRight = x.isLeft := by cases x <;> rfl\n#align sum.bnot_is_right Sum.not_isRight\n\n#print Sum.isRight_eq_false /-\n@[simp]\ntheorem isRight_eq_false : x.isRight = false \u2194 x.isLeft := by cases x <;> simp\n#align sum.is_right_eq_ff Sum.isRight_eq_false\n-/\n\n#print Sum.Not_isRight /-\ntheorem Not_isRight : \u00acx.isRight \u2194 x.isLeft := by simp\n#align sum.not_is_right Sum.Not_isRight\n-/\n\n#print Sum.isLeft_iff /-\ntheorem isLeft_iff : x.isLeft \u2194 \u2203 y, x = Sum.inl y := by cases x <;> simp\n#align sum.is_left_iff Sum.isLeft_iff\n-/\n\n#print Sum.isRight_iff /-\ntheorem isRight_iff : x.isRight \u2194 \u2203 y, x = Sum.inr y := by cases x <;> simp\n#align sum.is_right_iff Sum.isRight_iff\n-/\n\nend get\n\n#print Sum.inl.inj_iff /-\ntheorem inl.inj_iff {a b} : (inl a : Sum \u03b1 \u03b2) = inl b \u2194 a = b :=\n  \u27e8inl.inj, congr_arg _\u27e9\n#align sum.inl.inj_iff Sum.inl.inj_iff\n-/\n\n#print Sum.inr.inj_iff /-\ntheorem inr.inj_iff {a b} : (inr a : Sum \u03b1 \u03b2) = inr b \u2194 a = b :=\n  \u27e8inr.inj, congr_arg _\u27e9\n#align sum.inr.inj_iff Sum.inr.inj_iff\n-/\n\n#print Sum.inl_ne_inr /-\ntheorem inl_ne_inr {a : \u03b1} {b : \u03b2} : inl a \u2260 inr b :=\n  fun.\n#align sum.inl_ne_inr Sum.inl_ne_inr\n-/\n\n#print Sum.inr_ne_inl /-\ntheorem inr_ne_inl {a : \u03b1} {b : \u03b2} : inr b \u2260 inl a :=\n  fun.\n#align sum.inr_ne_inl Sum.inr_ne_inl\n-/\n\n#print Sum.elim /-\n/-- Define a function on `\u03b1 \u2295 \u03b2` by giving separate definitions on `\u03b1` and `\u03b2`. -/\nprotected def elim {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : Sum \u03b1 \u03b2 \u2192 \u03b3 := fun x =>\n  Sum.recOn x f g\n#align sum.elim Sum.elim\n-/\n\n/- warning: sum.elim_inl -> Sum.elim_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (x : \u03b1), Eq.{u3} \u03b3 (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 f g (Sum.inl.{u1, u2} \u03b1 \u03b2 x)) (f x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u1}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (x : \u03b1), Eq.{u1} \u03b3 (Sum.elim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 f g (Sum.inl.{u3, u2} \u03b1 \u03b2 x)) (f x)\nCase conversion may be inaccurate. Consider using '#align sum.elim_inl Sum.elim_inl\u2093'. -/\n@[simp]\ntheorem elim_inl {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b1) : Sum.elim f g (inl x) = f x :=\n  rfl\n#align sum.elim_inl Sum.elim_inl\n\n/- warning: sum.elim_inr -> Sum.elim_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (x : \u03b2), Eq.{u3} \u03b3 (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 f g (Sum.inr.{u1, u2} \u03b1 \u03b2 x)) (g x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u1}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (x : \u03b2), Eq.{u1} \u03b3 (Sum.elim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 f g (Sum.inr.{u3, u2} \u03b1 \u03b2 x)) (g x)\nCase conversion may be inaccurate. Consider using '#align sum.elim_inr Sum.elim_inr\u2093'. -/\n@[simp]\ntheorem elim_inr {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (x : \u03b2) : Sum.elim f g (inr x) = g x :=\n  rfl\n#align sum.elim_inr Sum.elim_inr\n\n/- warning: sum.elim_comp_inl -> Sum.elim_comp_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3), Eq.{imax (succ u1) u3} (\u03b1 -> \u03b3) (Function.comp.{succ u1, max (succ u1) (succ u2), u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u1, u2} \u03b1 \u03b2)) f\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u1}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3), Eq.{imax (succ u3) u1} (\u03b1 -> \u03b3) (Function.comp.{succ u3, max (succ u2) (succ u3), u1} \u03b1 (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 (Sum.elim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u3, u2} \u03b1 \u03b2)) f\nCase conversion may be inaccurate. Consider using '#align sum.elim_comp_inl Sum.elim_comp_inl\u2093'. -/\n@[simp]\ntheorem elim_comp_inl {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : Sum.elim f g \u2218 inl = f :=\n  rfl\n#align sum.elim_comp_inl Sum.elim_comp_inl\n\n/- warning: sum.elim_comp_inr -> Sum.elim_comp_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3), Eq.{imax (succ u2) u3} (\u03b2 -> \u03b3) (Function.comp.{succ u2, max (succ u1) (succ u2), u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u1, u2} \u03b1 \u03b2)) g\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u1}} (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3), Eq.{imax (succ u2) u1} (\u03b2 -> \u03b3) (Function.comp.{succ u2, max (succ u2) (succ u3), u1} \u03b2 (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 (Sum.elim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u3, u2} \u03b1 \u03b2)) g\nCase conversion may be inaccurate. Consider using '#align sum.elim_comp_inr Sum.elim_comp_inr\u2093'. -/\n@[simp]\ntheorem elim_comp_inr {\u03b1 \u03b2 \u03b3 : Sort _} (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) : Sum.elim f g \u2218 inr = g :=\n  rfl\n#align sum.elim_comp_inr Sum.elim_comp_inr\n\n/- warning: sum.elim_inl_inr -> Sum.elim_inl_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}}, Eq.{max (succ u1) (succ u2)} ((Sum.{u1, u2} \u03b1 \u03b2) -> (Sum.{u1, u2} \u03b1 \u03b2)) (Sum.elim.{u1, u2, max (succ u1) (succ u2)} \u03b1 \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) (Sum.inl.{u1, u2} \u03b1 \u03b2) (Sum.inr.{u1, u2} \u03b1 \u03b2)) (id.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}}, Eq.{max (succ u1) (succ u2)} ((Sum.{u2, u1} \u03b1 \u03b2) -> (Sum.{u2, u1} \u03b1 \u03b2)) (Sum.elim.{u2, u1, max (succ u2) (succ u1)} \u03b1 \u03b2 (Sum.{u2, u1} \u03b1 \u03b2) (Sum.inl.{u2, u1} \u03b1 \u03b2) (Sum.inr.{u2, u1} \u03b1 \u03b2)) (id.{max (succ u1) (succ u2)} (Sum.{u2, u1} \u03b1 \u03b2))\nCase conversion may be inaccurate. Consider using '#align sum.elim_inl_inr Sum.elim_inl_inr\u2093'. -/\n@[simp]\ntheorem elim_inl_inr {\u03b1 \u03b2 : Sort _} : @Sum.elim \u03b1 \u03b2 _ inl inr = id :=\n  funext fun x => Sum.casesOn x (fun _ => rfl) fun _ => rfl\n#align sum.elim_inl_inr Sum.elim_inl_inr\n\n/- warning: sum.comp_elim -> Sum.comp_elim is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} {\u03b4 : Sort.{u4}} (f : \u03b3 -> \u03b4) (g : \u03b1 -> \u03b3) (h : \u03b2 -> \u03b3), Eq.{imax (max (succ u1) (succ u2)) u4} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b4) (Function.comp.{max (succ u1) (succ u2), u3, u4} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 \u03b4 f (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 g h)) (Sum.elim.{u1, u2, u4} \u03b1 \u03b2 \u03b4 (Function.comp.{succ u1, u3, u4} \u03b1 \u03b3 \u03b4 f g) (Function.comp.{succ u2, u3, u4} \u03b2 \u03b3 \u03b4 f h))\nbut is expected to have type\n  forall {\u03b1 : Type.{u4}} {\u03b2 : Type.{u3}} {\u03b3 : Sort.{u2}} {\u03b4 : Sort.{u1}} (f : \u03b3 -> \u03b4) (g : \u03b1 -> \u03b3) (h : \u03b2 -> \u03b3), Eq.{imax (max (succ u3) (succ u4)) u1} ((Sum.{u4, u3} \u03b1 \u03b2) -> \u03b4) (Function.comp.{max (succ u3) (succ u4), u2, u1} (Sum.{u4, u3} \u03b1 \u03b2) \u03b3 \u03b4 f (Sum.elim.{u4, u3, u2} \u03b1 \u03b2 \u03b3 g h)) (Sum.elim.{u4, u3, u1} \u03b1 \u03b2 \u03b4 (Function.comp.{succ u4, u2, u1} \u03b1 \u03b3 \u03b4 f g) (Function.comp.{succ u3, u2, u1} \u03b2 \u03b3 \u03b4 f h))\nCase conversion may be inaccurate. Consider using '#align sum.comp_elim Sum.comp_elim\u2093'. -/\ntheorem comp_elim {\u03b1 \u03b2 \u03b3 \u03b4 : Sort _} (f : \u03b3 \u2192 \u03b4) (g : \u03b1 \u2192 \u03b3) (h : \u03b2 \u2192 \u03b3) :\n    f \u2218 Sum.elim g h = Sum.elim (f \u2218 g) (f \u2218 h) :=\n  funext fun x => Sum.casesOn x (fun _ => rfl) fun _ => rfl\n#align sum.comp_elim Sum.comp_elim\n\n/- warning: sum.elim_comp_inl_inr -> Sum.elim_comp_inl_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u3}} (f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3), Eq.{imax (max (succ u1) (succ u2)) u3} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u1, u2, u3} \u03b1 \u03b2 \u03b3 (Function.comp.{succ u1, max (succ u1) (succ u2), u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u1, u2} \u03b1 \u03b2)) (Function.comp.{succ u2, max (succ u1) (succ u2), u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u1, u2} \u03b1 \u03b2))) f\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b3 : Sort.{u1}} (f : (Sum.{u3, u2} \u03b1 \u03b2) -> \u03b3), Eq.{imax (max (succ u2) (succ u3)) u1} ((Sum.{u3, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u3, u2, u1} \u03b1 \u03b2 \u03b3 (Function.comp.{succ u3, max (succ u2) (succ u3), u1} \u03b1 (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u3, u2} \u03b1 \u03b2)) (Function.comp.{succ u2, max (succ u2) (succ u3), u1} \u03b2 (Sum.{u3, u2} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u3, u2} \u03b1 \u03b2))) f\nCase conversion may be inaccurate. Consider using '#align sum.elim_comp_inl_inr Sum.elim_comp_inl_inr\u2093'. -/\n@[simp]\ntheorem elim_comp_inl_inr {\u03b1 \u03b2 \u03b3 : Sort _} (f : Sum \u03b1 \u03b2 \u2192 \u03b3) : Sum.elim (f \u2218 inl) (f \u2218 inr) = f :=\n  funext fun x => Sum.casesOn x (fun _ => rfl) fun _ => rfl\n#align sum.elim_comp_inl_inr Sum.elim_comp_inl_inr\n\n#print Sum.map /-\n/-- Map `\u03b1 \u2295 \u03b2` to `\u03b1' \u2295 \u03b2'` sending `\u03b1` to `\u03b1'` and `\u03b2` to `\u03b2'`. -/\nprotected def map (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') : Sum \u03b1 \u03b2 \u2192 Sum \u03b1' \u03b2' :=\n  Sum.elim (inl \u2218 f) (inr \u2218 g)\n#align sum.map Sum.map\n-/\n\n#print Sum.map_inl /-\n@[simp]\ntheorem map_inl (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b1) : (inl x).map f g = inl (f x) :=\n  rfl\n#align sum.map_inl Sum.map_inl\n-/\n\n#print Sum.map_inr /-\n@[simp]\ntheorem map_inr (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : \u03b2) : (inr x).map f g = inr (g x) :=\n  rfl\n#align sum.map_inr Sum.map_inr\n-/\n\n/- warning: sum.map_map -> Sum.map_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b1' : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b2' : Type.{u4}} {\u03b1'' : Type.{u5}} {\u03b2'' : Type.{u6}} (f' : \u03b1' -> \u03b1'') (g' : \u03b2' -> \u03b2'') (f : \u03b1 -> \u03b1') (g : \u03b2 -> \u03b2') (x : Sum.{u1, u2} \u03b1 \u03b2), Eq.{max (succ u5) (succ u6)} (Sum.{u5, u6} \u03b1'' \u03b2'') (Sum.map.{u3, u4, u5, u6} \u03b1' \u03b1'' \u03b2' \u03b2'' f' g' (Sum.map.{u1, u2, u3, u4} \u03b1 \u03b1' \u03b2 \u03b2' f g x)) (Sum.map.{u1, u2, u5, u6} \u03b1 \u03b1'' \u03b2 \u03b2'' (Function.comp.{succ u1, succ u3, succ u5} \u03b1 \u03b1' \u03b1'' f' f) (Function.comp.{succ u2, succ u4, succ u6} \u03b2 \u03b2' \u03b2'' g' g) x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b1' : Type.{u5}} {\u03b2 : Type.{u4}} {\u03b2' : Type.{u6}} {\u03b1'' : Type.{u2}} {\u03b2'' : Type.{u1}} (f' : \u03b1' -> \u03b1'') (g' : \u03b2' -> \u03b2'') (f : \u03b1 -> \u03b1') (g : \u03b2 -> \u03b2') (x : Sum.{u3, u4} \u03b1 \u03b2), Eq.{max (succ u1) (succ u2)} (Sum.{u2, u1} \u03b1'' \u03b2'') (Sum.map.{u5, u6, u2, u1} \u03b1' \u03b1'' \u03b2' \u03b2'' f' g' (Sum.map.{u3, u4, u5, u6} \u03b1 \u03b1' \u03b2 \u03b2' f g x)) (Sum.map.{u3, u4, u2, u1} \u03b1 \u03b1'' \u03b2 \u03b2'' (Function.comp.{succ u3, succ u5, succ u2} \u03b1 \u03b1' \u03b1'' f' f) (Function.comp.{succ u4, succ u6, succ u1} \u03b2 \u03b2' \u03b2'' g' g) x)\nCase conversion may be inaccurate. Consider using '#align sum.map_map Sum.map_map\u2093'. -/\n@[simp]\ntheorem map_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n    \u2200 x : Sum \u03b1 \u03b2, (x.map f g).map f' g' = x.map (f' \u2218 f) (g' \u2218 g)\n  | inl a => rfl\n  | inr b => rfl\n#align sum.map_map Sum.map_map\n\n/- warning: sum.map_comp_map -> Sum.map_comp_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b1' : Type.{u3}} {\u03b2 : Type.{u2}} {\u03b2' : Type.{u4}} {\u03b1'' : Type.{u5}} {\u03b2'' : Type.{u6}} (f' : \u03b1' -> \u03b1'') (g' : \u03b2' -> \u03b2'') (f : \u03b1 -> \u03b1') (g : \u03b2 -> \u03b2'), Eq.{max (max (succ u1) (succ u2)) (succ u5) (succ u6)} ((Sum.{u1, u2} \u03b1 \u03b2) -> (Sum.{u5, u6} \u03b1'' \u03b2'')) (Function.comp.{max (succ u1) (succ u2), max (succ u3) (succ u4), max (succ u5) (succ u6)} (Sum.{u1, u2} \u03b1 \u03b2) (Sum.{u3, u4} \u03b1' \u03b2') (Sum.{u5, u6} \u03b1'' \u03b2'') (Sum.map.{u3, u4, u5, u6} \u03b1' \u03b1'' \u03b2' \u03b2'' f' g') (Sum.map.{u1, u2, u3, u4} \u03b1 \u03b1' \u03b2 \u03b2' f g)) (Sum.map.{u1, u2, u5, u6} \u03b1 \u03b1'' \u03b2 \u03b2'' (Function.comp.{succ u1, succ u3, succ u5} \u03b1 \u03b1' \u03b1'' f' f) (Function.comp.{succ u2, succ u4, succ u6} \u03b2 \u03b2' \u03b2'' g' g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b1' : Type.{u5}} {\u03b2 : Type.{u4}} {\u03b2' : Type.{u6}} {\u03b1'' : Type.{u2}} {\u03b2'' : Type.{u1}} (f' : \u03b1' -> \u03b1'') (g' : \u03b2' -> \u03b2'') (f : \u03b1 -> \u03b1') (g : \u03b2 -> \u03b2'), Eq.{max (max (max (succ u3) (succ u4)) (succ u1)) (succ u2)} ((Sum.{u3, u4} \u03b1 \u03b2) -> (Sum.{u2, u1} \u03b1'' \u03b2'')) (Function.comp.{max (succ u4) (succ u3), max (succ u6) (succ u5), max (succ u1) (succ u2)} (Sum.{u3, u4} \u03b1 \u03b2) (Sum.{u5, u6} \u03b1' \u03b2') (Sum.{u2, u1} \u03b1'' \u03b2'') (Sum.map.{u5, u6, u2, u1} \u03b1' \u03b1'' \u03b2' \u03b2'' f' g') (Sum.map.{u3, u4, u5, u6} \u03b1 \u03b1' \u03b2 \u03b2' f g)) (Sum.map.{u3, u4, u2, u1} \u03b1 \u03b1'' \u03b2 \u03b2'' (Function.comp.{succ u3, succ u5, succ u2} \u03b1 \u03b1' \u03b1'' f' f) (Function.comp.{succ u4, succ u6, succ u1} \u03b2 \u03b2' \u03b2'' g' g))\nCase conversion may be inaccurate. Consider using '#align sum.map_comp_map Sum.map_comp_map\u2093'. -/\n@[simp]\ntheorem map_comp_map {\u03b1'' \u03b2''} (f' : \u03b1' \u2192 \u03b1'') (g' : \u03b2' \u2192 \u03b2'') (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') :\n    Sum.map f' g' \u2218 Sum.map f g = Sum.map (f' \u2218 f) (g' \u2218 g) :=\n  funext <| map_map f' g' f g\n#align sum.map_comp_map Sum.map_comp_map\n\n/- warning: sum.map_id_id -> Sum.map_id_id is a dubious translation:\nlean 3 declaration is\n  forall (\u03b1 : Type.{u1}) (\u03b2 : Type.{u2}), Eq.{max (succ u1) (succ u2)} ((Sum.{u1, u2} \u03b1 \u03b2) -> (Sum.{u1, u2} \u03b1 \u03b2)) (Sum.map.{u1, u2, u1, u2} \u03b1 \u03b1 \u03b2 \u03b2 (id.{succ u1} \u03b1) (id.{succ u2} \u03b2)) (id.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2))\nbut is expected to have type\n  forall (\u03b1 : Type.{u2}) (\u03b2 : Type.{u1}), Eq.{max (succ u1) (succ u2)} ((Sum.{u2, u1} \u03b1 \u03b2) -> (Sum.{u2, u1} \u03b1 \u03b2)) (Sum.map.{u2, u1, u2, u1} \u03b1 \u03b1 \u03b2 \u03b2 (id.{succ u2} \u03b1) (id.{succ u1} \u03b2)) (id.{max (succ u1) (succ u2)} (Sum.{u2, u1} \u03b1 \u03b2))\nCase conversion may be inaccurate. Consider using '#align sum.map_id_id Sum.map_id_id\u2093'. -/\n@[simp]\ntheorem map_id_id (\u03b1 \u03b2) : Sum.map (@id \u03b1) (@id \u03b2) = id :=\n  funext fun x => Sum.recOn x (fun _ => rfl) fun _ => rfl\n#align sum.map_id_id Sum.map_id_id\n\n/- warning: sum.elim_map -> Sum.elim_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {\u03b5 : Sort.{u5}} {f\u2081 : \u03b1 -> \u03b2} {f\u2082 : \u03b2 -> \u03b5} {g\u2081 : \u03b3 -> \u03b4} {g\u2082 : \u03b4 -> \u03b5} {x : Sum.{u1, u3} \u03b1 \u03b3}, Eq.{u5} \u03b5 (Sum.elim.{u2, u4, u5} \u03b2 \u03b4 \u03b5 f\u2082 g\u2082 (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f\u2081 g\u2081 x)) (Sum.elim.{u1, u3, u5} \u03b1 \u03b3 \u03b5 (Function.comp.{succ u1, succ u2, u5} \u03b1 \u03b2 \u03b5 f\u2082 f\u2081) (Function.comp.{succ u3, succ u4, u5} \u03b3 \u03b4 \u03b5 g\u2082 g\u2081) x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u5}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u2}} {\u03b5 : Sort.{u1}} {f\u2081 : \u03b1 -> \u03b2} {f\u2082 : \u03b2 -> \u03b5} {g\u2081 : \u03b3 -> \u03b4} {g\u2082 : \u03b4 -> \u03b5} {x : Sum.{u5, u3} \u03b1 \u03b3}, Eq.{u1} \u03b5 (Sum.elim.{u4, u2, u1} \u03b2 \u03b4 \u03b5 f\u2082 g\u2082 (Sum.map.{u5, u3, u4, u2} \u03b1 \u03b2 \u03b3 \u03b4 f\u2081 g\u2081 x)) (Sum.elim.{u5, u3, u1} \u03b1 \u03b3 \u03b5 (Function.comp.{succ u5, succ u4, u1} \u03b1 \u03b2 \u03b5 f\u2082 f\u2081) (Function.comp.{succ u3, succ u2, u1} \u03b3 \u03b4 \u03b5 g\u2082 g\u2081) x)\nCase conversion may be inaccurate. Consider using '#align sum.elim_map Sum.elim_map\u2093'. -/\ntheorem elim_map {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Sort _} {f\u2081 : \u03b1 \u2192 \u03b2} {f\u2082 : \u03b2 \u2192 \u03b5} {g\u2081 : \u03b3 \u2192 \u03b4} {g\u2082 : \u03b4 \u2192 \u03b5} {x} :\n    Sum.elim f\u2082 g\u2082 (Sum.map f\u2081 g\u2081 x) = Sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) x := by cases x <;> rfl\n#align sum.elim_map Sum.elim_map\n\n/- warning: sum.elim_comp_map -> Sum.elim_comp_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {\u03b5 : Sort.{u5}} {f\u2081 : \u03b1 -> \u03b2} {f\u2082 : \u03b2 -> \u03b5} {g\u2081 : \u03b3 -> \u03b4} {g\u2082 : \u03b4 -> \u03b5}, Eq.{imax (max (succ u1) (succ u3)) u5} ((Sum.{u1, u3} \u03b1 \u03b3) -> \u03b5) (Function.comp.{max (succ u1) (succ u3), max (succ u2) (succ u4), u5} (Sum.{u1, u3} \u03b1 \u03b3) (Sum.{u2, u4} \u03b2 \u03b4) \u03b5 (Sum.elim.{u2, u4, u5} \u03b2 \u03b4 \u03b5 f\u2082 g\u2082) (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f\u2081 g\u2081)) (Sum.elim.{u1, u3, u5} \u03b1 \u03b3 \u03b5 (Function.comp.{succ u1, succ u2, u5} \u03b1 \u03b2 \u03b5 f\u2082 f\u2081) (Function.comp.{succ u3, succ u4, u5} \u03b3 \u03b4 \u03b5 g\u2082 g\u2081))\nbut is expected to have type\n  forall {\u03b1 : Type.{u5}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u2}} {\u03b5 : Sort.{u1}} {f\u2081 : \u03b1 -> \u03b2} {f\u2082 : \u03b2 -> \u03b5} {g\u2081 : \u03b3 -> \u03b4} {g\u2082 : \u03b4 -> \u03b5}, Eq.{imax (max (succ u3) (succ u5)) u1} ((Sum.{u5, u3} \u03b1 \u03b3) -> \u03b5) (Function.comp.{max (succ u3) (succ u5), max (succ u2) (succ u4), u1} (Sum.{u5, u3} \u03b1 \u03b3) (Sum.{u4, u2} \u03b2 \u03b4) \u03b5 (Sum.elim.{u4, u2, u1} \u03b2 \u03b4 \u03b5 f\u2082 g\u2082) (Sum.map.{u5, u3, u4, u2} \u03b1 \u03b2 \u03b3 \u03b4 f\u2081 g\u2081)) (Sum.elim.{u5, u3, u1} \u03b1 \u03b3 \u03b5 (Function.comp.{succ u5, succ u4, u1} \u03b1 \u03b2 \u03b5 f\u2082 f\u2081) (Function.comp.{succ u3, succ u2, u1} \u03b3 \u03b4 \u03b5 g\u2082 g\u2081))\nCase conversion may be inaccurate. Consider using '#align sum.elim_comp_map Sum.elim_comp_map\u2093'. -/\ntheorem elim_comp_map {\u03b1 \u03b2 \u03b3 \u03b4 \u03b5 : Sort _} {f\u2081 : \u03b1 \u2192 \u03b2} {f\u2082 : \u03b2 \u2192 \u03b5} {g\u2081 : \u03b3 \u2192 \u03b4} {g\u2082 : \u03b4 \u2192 \u03b5} :\n    Sum.elim f\u2082 g\u2082 \u2218 Sum.map f\u2081 g\u2081 = Sum.elim (f\u2082 \u2218 f\u2081) (g\u2082 \u2218 g\u2081) :=\n  funext fun _ => elim_map\n#align sum.elim_comp_map Sum.elim_comp_map\n\n/- warning: sum.is_left_map -> Sum.isLeft_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u1, u3} \u03b1 \u03b3), Eq.{1} Bool (Sum.isLeft.{u2, u4} \u03b2 \u03b4 (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Sum.isLeft.{u1, u3} \u03b1 \u03b3 x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u3, u2} \u03b1 \u03b3), Eq.{1} Bool (Sum.isLeft.{u4, u1} \u03b2 \u03b4 (Sum.map.{u3, u2, u4, u1} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Sum.isLeft.{u3, u2} \u03b1 \u03b3 x)\nCase conversion may be inaccurate. Consider using '#align sum.is_left_map Sum.isLeft_map\u2093'. -/\n@[simp]\ntheorem isLeft_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : Sum \u03b1 \u03b3) : isLeft (x.map f g) = isLeft x := by\n  cases x <;> rfl\n#align sum.is_left_map Sum.isLeft_map\n\n/- warning: sum.is_right_map -> Sum.isRight_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u1, u3} \u03b1 \u03b3), Eq.{1} Bool (Sum.isRight.{u2, u4} \u03b2 \u03b4 (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Sum.isRight.{u1, u3} \u03b1 \u03b3 x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u3, u2} \u03b1 \u03b3), Eq.{1} Bool (Sum.isRight.{u4, u1} \u03b2 \u03b4 (Sum.map.{u3, u2, u4, u1} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Sum.isRight.{u3, u2} \u03b1 \u03b3 x)\nCase conversion may be inaccurate. Consider using '#align sum.is_right_map Sum.isRight_map\u2093'. -/\n@[simp]\ntheorem isRight_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : Sum \u03b1 \u03b3) : isRight (x.map f g) = isRight x := by\n  cases x <;> rfl\n#align sum.is_right_map Sum.isRight_map\n\n/- warning: sum.get_left_map -> Sum.getLeft_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u1, u3} \u03b1 \u03b3), Eq.{succ u2} (Option.{u2} \u03b2) (Sum.getLeft.{u2, u4} \u03b2 \u03b4 (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Option.map.{u1, u2} \u03b1 \u03b2 f (Sum.getLeft.{u1, u3} \u03b1 \u03b3 x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u3, u2} \u03b1 \u03b3), Eq.{succ u4} (Option.{u4} \u03b2) (Sum.getLeft.{u4, u1} \u03b2 \u03b4 (Sum.map.{u3, u2, u4, u1} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Option.map.{u3, u4} \u03b1 \u03b2 f (Sum.getLeft.{u3, u2} \u03b1 \u03b3 x))\nCase conversion may be inaccurate. Consider using '#align sum.get_left_map Sum.getLeft_map\u2093'. -/\n@[simp]\ntheorem getLeft_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : Sum \u03b1 \u03b3) : (x.map f g).getLeft = x.getLeft.map f :=\n  by cases x <;> rfl\n#align sum.get_left_map Sum.getLeft_map\n\n/- warning: sum.get_right_map -> Sum.getRight_map is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u1, u3} \u03b1 \u03b3), Eq.{succ u4} (Option.{u4} \u03b4) (Sum.getRight.{u2, u4} \u03b2 \u03b4 (Sum.map.{u1, u3, u2, u4} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Option.map.{u3, u4} \u03b3 \u03b4 g (Sum.getRight.{u1, u3} \u03b1 \u03b3 x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} (f : \u03b1 -> \u03b2) (g : \u03b3 -> \u03b4) (x : Sum.{u3, u2} \u03b1 \u03b3), Eq.{succ u1} (Option.{u1} \u03b4) (Sum.getRight.{u4, u1} \u03b2 \u03b4 (Sum.map.{u3, u2, u4, u1} \u03b1 \u03b2 \u03b3 \u03b4 f g x)) (Option.map.{u2, u1} \u03b3 \u03b4 g (Sum.getRight.{u3, u2} \u03b1 \u03b3 x))\nCase conversion may be inaccurate. Consider using '#align sum.get_right_map Sum.getRight_map\u2093'. -/\n@[simp]\ntheorem getRight_map (f : \u03b1 \u2192 \u03b2) (g : \u03b3 \u2192 \u03b4) (x : Sum \u03b1 \u03b3) :\n    (x.map f g).getRight = x.getRight.map g := by cases x <;> rfl\n#align sum.get_right_map Sum.getRight_map\n\nopen Function (update update_eq_iff update_comp_eq_of_injective update_comp_eq_of_forall_ne)\n\n/- warning: sum.update_elim_inl -> Sum.update_elim_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u1, u2} \u03b1 \u03b2 i) x) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (Function.update.{succ u1, succ u3} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) f i x) g)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} \u03b1] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Function.update.{max (succ u3) (succ u2), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u2, u3} \u03b1 \u03b2 i) x) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 (Function.update.{succ u2, succ u1} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) f i x) g)\nCase conversion may be inaccurate. Consider using '#align sum.update_elim_inl Sum.update_elim_inl\u2093'. -/\n@[simp]\ntheorem update_elim_inl [DecidableEq \u03b1] [DecidableEq (Sum \u03b1 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {i : \u03b1}\n    {x : \u03b3} : update (Sum.elim f g) (inl i) x = Sum.elim (update f i x) g :=\n  update_eq_iff.2 \u27e8by simp, by simp (config := { contextual := true })\u27e9\n#align sum.update_elim_inl Sum.update_elim_inl\n\n/- warning: sum.update_elim_inr -> Sum.update_elim_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u2} \u03b2] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u1, u2} \u03b1 \u03b2 i) x) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f (Function.update.{succ u2, succ u3} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) g i x))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u3} \u03b2] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Function.update.{max (succ u3) (succ u2), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u2, u3} \u03b1 \u03b2 i) x) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f (Function.update.{succ u3, succ u1} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) g i x))\nCase conversion may be inaccurate. Consider using '#align sum.update_elim_inr Sum.update_elim_inr\u2093'. -/\n@[simp]\ntheorem update_elim_inr [DecidableEq \u03b2] [DecidableEq (Sum \u03b1 \u03b2)] {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} {i : \u03b2}\n    {x : \u03b3} : update (Sum.elim f g) (inr i) x = Sum.elim f (update g i x) :=\n  update_eq_iff.2 \u27e8by simp, by simp (config := { contextual := true })\u27e9\n#align sum.update_elim_inr Sum.update_elim_inr\n\n/- warning: sum.update_inl_comp_inl -> Sum.update_inl_comp_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (succ u1) (succ u3)} (\u03b1 -> \u03b3) (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inl.{u1, u2} \u03b1 \u03b2 i) x) (Sum.inl.{u1, u2} \u03b1 \u03b2)) (Function.update.{succ u1, succ u3} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u1, u2} \u03b1 \u03b2)) i x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} \u03b1] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (succ u2) (succ u1)} (\u03b1 -> \u03b3) (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} \u03b1 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inl.{u2, u3} \u03b1 \u03b2 i) x) (Sum.inl.{u2, u3} \u03b1 \u03b2)) (Function.update.{succ u2, succ u1} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} \u03b1 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u2, u3} \u03b1 \u03b2)) i x)\nCase conversion may be inaccurate. Consider using '#align sum.update_inl_comp_inl Sum.update_inl_comp_inl\u2093'. -/\n@[simp]\ntheorem update_inl_comp_inl [DecidableEq \u03b1] [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b1}\n    {x : \u03b3} : update f (inl i) x \u2218 inl = update (f \u2218 inl) i x :=\n  update_comp_eq_of_injective _ inl_injective _ _\n#align sum.update_inl_comp_inl Sum.update_inl_comp_inl\n\n/- warning: sum.update_inl_apply_inl -> Sum.update_inl_apply_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b1} {x : \u03b3}, Eq.{succ u3} \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inl.{u1, u2} \u03b1 \u03b2 i) x (Sum.inl.{u1, u2} \u03b1 \u03b2 j)) (Function.update.{succ u1, succ u3} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u1, u2} \u03b1 \u03b2)) i x j)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} \u03b1] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b1} {x : \u03b3}, Eq.{succ u1} \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inl.{u2, u3} \u03b1 \u03b2 i) x (Sum.inl.{u2, u3} \u03b1 \u03b2 j)) (Function.update.{succ u2, succ u1} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} \u03b1 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u2, u3} \u03b1 \u03b2)) i x j)\nCase conversion may be inaccurate. Consider using '#align sum.update_inl_apply_inl Sum.update_inl_apply_inl\u2093'. -/\n@[simp]\ntheorem update_inl_apply_inl [DecidableEq \u03b1] [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i j : \u03b1}\n    {x : \u03b3} : update f (inl i) x (inl j) = update (f \u2218 inl) i x j := by rw [\u2190 update_inl_comp_inl]\n#align sum.update_inl_apply_inl Sum.update_inl_apply_inl\n\n/- warning: sum.update_inl_comp_inr -> Sum.update_inl_comp_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (succ u2) (succ u3)} (\u03b2 -> \u03b3) (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inl.{u1, u2} \u03b1 \u03b2 i) x) (Sum.inr.{u1, u2} \u03b1 \u03b2)) (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u1, u2} \u03b1 \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {x : \u03b3}, Eq.{max (succ u3) (succ u1)} (\u03b2 -> \u03b3) (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} \u03b2 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inl.{u2, u3} \u03b1 \u03b2 i) x) (Sum.inr.{u2, u3} \u03b1 \u03b2)) (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} \u03b2 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u2, u3} \u03b1 \u03b2))\nCase conversion may be inaccurate. Consider using '#align sum.update_inl_comp_inr Sum.update_inl_comp_inr\u2093'. -/\n@[simp]\ntheorem update_inl_comp_inr [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b1} {x : \u03b3} :\n    update f (inl i) x \u2218 inr = f \u2218 inr :=\n  update_comp_eq_of_forall_ne _ _ fun _ => inr_ne_inl\n#align sum.update_inl_comp_inr Sum.update_inl_comp_inr\n\n/- warning: sum.update_inl_apply_inr -> Sum.update_inl_apply_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3}, Eq.{succ u3} \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inl.{u1, u2} \u03b1 \u03b2 i) x (Sum.inr.{u1, u2} \u03b1 \u03b2 j)) (f (Sum.inr.{u1, u2} \u03b1 \u03b2 j))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3}, Eq.{succ u1} \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inl.{u2, u3} \u03b1 \u03b2 i) x (Sum.inr.{u2, u3} \u03b1 \u03b2 j)) (f (Sum.inr.{u2, u3} \u03b1 \u03b2 j))\nCase conversion may be inaccurate. Consider using '#align sum.update_inl_apply_inr Sum.update_inl_apply_inr\u2093'. -/\n@[simp]\ntheorem update_inl_apply_inr [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3} :\n    update f (inl i) x (inr j) = f (inr j) :=\n  Function.update_noteq inr_ne_inl _ _\n#align sum.update_inl_apply_inr Sum.update_inl_apply_inr\n\n/- warning: sum.update_inr_comp_inl -> Sum.update_inr_comp_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (succ u1) (succ u3)} (\u03b1 -> \u03b3) (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inr.{u1, u2} \u03b1 \u03b2 i) x) (Sum.inl.{u1, u2} \u03b1 \u03b2)) (Function.comp.{succ u1, max (succ u1) (succ u2), succ u3} \u03b1 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u1, u2} \u03b1 \u03b2))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (succ u2) (succ u1)} (\u03b1 -> \u03b3) (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} \u03b1 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inr.{u2, u3} \u03b1 \u03b2 i) x) (Sum.inl.{u2, u3} \u03b1 \u03b2)) (Function.comp.{succ u2, max (succ u2) (succ u3), succ u1} \u03b1 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inl.{u2, u3} \u03b1 \u03b2))\nCase conversion may be inaccurate. Consider using '#align sum.update_inr_comp_inl Sum.update_inr_comp_inl\u2093'. -/\n@[simp]\ntheorem update_inr_comp_inl [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b2} {x : \u03b3} :\n    update f (inr i) x \u2218 inl = f \u2218 inl :=\n  update_comp_eq_of_forall_ne _ _ fun _ => inl_ne_inr\n#align sum.update_inr_comp_inl Sum.update_inr_comp_inl\n\n/- warning: sum.update_inr_apply_inl -> Sum.update_inr_apply_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3}, Eq.{succ u3} \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inr.{u1, u2} \u03b1 \u03b2 j) x (Sum.inl.{u1, u2} \u03b1 \u03b2 i)) (f (Sum.inl.{u1, u2} \u03b1 \u03b2 i))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3}, Eq.{succ u1} \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_1 a b) f (Sum.inr.{u2, u3} \u03b1 \u03b2 j) x (Sum.inl.{u2, u3} \u03b1 \u03b2 i)) (f (Sum.inl.{u2, u3} \u03b1 \u03b2 i))\nCase conversion may be inaccurate. Consider using '#align sum.update_inr_apply_inl Sum.update_inr_apply_inl\u2093'. -/\n@[simp]\ntheorem update_inr_apply_inl [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b1} {j : \u03b2} {x : \u03b3} :\n    update f (inr j) x (inl i) = f (inl i) :=\n  Function.update_noteq inl_ne_inr _ _\n#align sum.update_inr_apply_inl Sum.update_inr_apply_inl\n\n/- warning: sum.update_inr_comp_inr -> Sum.update_inr_comp_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u2} \u03b2] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (succ u2) (succ u3)} (\u03b2 -> \u03b3) (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inr.{u1, u2} \u03b1 \u03b2 i) x) (Sum.inr.{u1, u2} \u03b1 \u03b2)) (Function.update.{succ u2, succ u3} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u1, u2} \u03b1 \u03b2)) i x)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u3} \u03b2] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {x : \u03b3}, Eq.{max (succ u3) (succ u1)} (\u03b2 -> \u03b3) (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} \u03b2 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inr.{u2, u3} \u03b1 \u03b2 i) x) (Sum.inr.{u2, u3} \u03b1 \u03b2)) (Function.update.{succ u3, succ u1} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} \u03b2 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u2, u3} \u03b1 \u03b2)) i x)\nCase conversion may be inaccurate. Consider using '#align sum.update_inr_comp_inr Sum.update_inr_comp_inr\u2093'. -/\n@[simp]\ntheorem update_inr_comp_inr [DecidableEq \u03b2] [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i : \u03b2}\n    {x : \u03b3} : update f (inr i) x \u2218 inr = update (f \u2218 inr) i x :=\n  update_comp_eq_of_injective _ inr_injective _ _\n#align sum.update_inr_comp_inr Sum.update_inr_comp_inr\n\n/- warning: sum.update_inr_apply_inr -> Sum.update_inr_apply_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u2} \u03b2] [_inst_2 : DecidableEq.{max (succ u1) (succ u2)} (Sum.{u1, u2} \u03b1 \u03b2)] {f : (Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {j : \u03b2} {x : \u03b3}, Eq.{succ u3} \u03b3 (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inr.{u1, u2} \u03b1 \u03b2 i) x (Sum.inr.{u1, u2} \u03b1 \u03b2 j)) (Function.update.{succ u2, succ u3} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) (Function.comp.{succ u2, max (succ u1) (succ u2), succ u3} \u03b2 (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u1, u2} \u03b1 \u03b2)) i x j)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u3} \u03b2] [_inst_2 : DecidableEq.{max (succ u3) (succ u2)} (Sum.{u2, u3} \u03b1 \u03b2)] {f : (Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3} {i : \u03b2} {j : \u03b2} {x : \u03b3}, Eq.{succ u1} \u03b3 (Function.update.{max (succ u2) (succ u3), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => _inst_2 a b) f (Sum.inr.{u2, u3} \u03b1 \u03b2 i) x (Sum.inr.{u2, u3} \u03b1 \u03b2 j)) (Function.update.{succ u3, succ u1} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_1 a b) (Function.comp.{succ u3, max (succ u2) (succ u3), succ u1} \u03b2 (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 f (Sum.inr.{u2, u3} \u03b1 \u03b2)) i x j)\nCase conversion may be inaccurate. Consider using '#align sum.update_inr_apply_inr Sum.update_inr_apply_inr\u2093'. -/\n@[simp]\ntheorem update_inr_apply_inr [DecidableEq \u03b2] [DecidableEq (Sum \u03b1 \u03b2)] {f : Sum \u03b1 \u03b2 \u2192 \u03b3} {i j : \u03b2}\n    {x : \u03b3} : update f (inr i) x (inr j) = update (f \u2218 inr) i x j := by rw [\u2190 update_inr_comp_inr]\n#align sum.update_inr_apply_inr Sum.update_inr_apply_inr\n\n#print Sum.swap /-\n/-- Swap the factors of a sum type -/\ndef swap : Sum \u03b1 \u03b2 \u2192 Sum \u03b2 \u03b1 :=\n  Sum.elim inr inl\n#align sum.swap Sum.swap\n-/\n\n#print Sum.swap_inl /-\n@[simp]\ntheorem swap_inl (x : \u03b1) : swap (inl x : Sum \u03b1 \u03b2) = inr x :=\n  rfl\n#align sum.swap_inl Sum.swap_inl\n-/\n\n#print Sum.swap_inr /-\n@[simp]\ntheorem swap_inr (x : \u03b2) : swap (inr x : Sum \u03b1 \u03b2) = inl x :=\n  rfl\n#align sum.swap_inr Sum.swap_inr\n-/\n\n#print Sum.swap_swap /-\n@[simp]\ntheorem swap_swap (x : Sum \u03b1 \u03b2) : swap (swap x) = x := by cases x <;> rfl\n#align sum.swap_swap Sum.swap_swap\n-/\n\n#print Sum.swap_swap_eq /-\n@[simp]\ntheorem swap_swap_eq : swap \u2218 swap = @id (Sum \u03b1 \u03b2) :=\n  funext <| swap_swap\n#align sum.swap_swap_eq Sum.swap_swap_eq\n-/\n\n#print Sum.swap_leftInverse /-\n@[simp]\ntheorem swap_leftInverse : Function.LeftInverse (@swap \u03b1 \u03b2) swap :=\n  swap_swap\n#align sum.swap_left_inverse Sum.swap_leftInverse\n-/\n\n#print Sum.swap_rightInverse /-\n@[simp]\ntheorem swap_rightInverse : Function.RightInverse (@swap \u03b1 \u03b2) swap :=\n  swap_swap\n#align sum.swap_right_inverse Sum.swap_rightInverse\n-/\n\n#print Sum.isLeft_swap /-\n@[simp]\ntheorem isLeft_swap (x : Sum \u03b1 \u03b2) : x.symm.isLeft = x.isRight := by cases x <;> rfl\n#align sum.is_left_swap Sum.isLeft_swap\n-/\n\n#print Sum.isRight_swap /-\n@[simp]\ntheorem isRight_swap (x : Sum \u03b1 \u03b2) : x.symm.isRight = x.isLeft := by cases x <;> rfl\n#align sum.is_right_swap Sum.isRight_swap\n-/\n\n#print Sum.getLeft_swap /-\n@[simp]\ntheorem getLeft_swap (x : Sum \u03b1 \u03b2) : x.symm.getLeft = x.getRight := by cases x <;> rfl\n#align sum.get_left_swap Sum.getLeft_swap\n-/\n\n#print Sum.getRight_swap /-\n@[simp]\ntheorem getRight_swap (x : Sum \u03b1 \u03b2) : x.symm.getRight = x.getLeft := by cases x <;> rfl\n#align sum.get_right_swap Sum.getRight_swap\n-/\n\nsection LiftRel\n\n#print Sum.LiftRel /-\n/-- Lifts pointwise two relations between `\u03b1` and `\u03b3` and between `\u03b2` and `\u03b4` to a relation between\n`\u03b1 \u2295 \u03b2` and `\u03b3 \u2295 \u03b4`. -/\ninductive LiftRel (r : \u03b1 \u2192 \u03b3 \u2192 Prop) (s : \u03b2 \u2192 \u03b4 \u2192 Prop) : Sum \u03b1 \u03b2 \u2192 Sum \u03b3 \u03b4 \u2192 Prop\n  | inl {a c} : r a c \u2192 lift_rel (inl a) (inl c)\n  | inr {b d} : s b d \u2192 lift_rel (inr b) (inr d)\n#align sum.lift_rel Sum.LiftRel\n-/\n\nattribute [protected] lift_rel.inl lift_rel.inr\n\nvariable {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b3 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b4 \u2192 Prop} {a : \u03b1} {b : \u03b2} {c : \u03b3} {d : \u03b4}\n  {x : Sum \u03b1 \u03b2} {y : Sum \u03b3 \u03b4}\n\n/- warning: sum.lift_rel_inl_inl -> Sum.liftRel_inl_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {a : \u03b1} {c : \u03b3}, Iff (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inl.{u1, u2} \u03b1 \u03b2 a) (Sum.inl.{u3, u4} \u03b3 \u03b4 c)) (r a c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {a : \u03b1} {c : \u03b3}, Iff (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inl.{u3, u4} \u03b1 \u03b2 a) (Sum.inl.{u2, u1} \u03b3 \u03b4 c)) (r a c)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel_inl_inl Sum.liftRel_inl_inl\u2093'. -/\n@[simp]\ntheorem liftRel_inl_inl : LiftRel r s (inl a) (inl c) \u2194 r a c :=\n  \u27e8fun h => by\n    cases h\n    assumption, LiftRel.inl\u27e9\n#align sum.lift_rel_inl_inl Sum.liftRel_inl_inl\n\n/- warning: sum.not_lift_rel_inl_inr -> Sum.not_liftRel_inl_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {a : \u03b1} {d : \u03b4}, Not (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inl.{u1, u2} \u03b1 \u03b2 a) (Sum.inr.{u3, u4} \u03b3 \u03b4 d))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {a : \u03b1} {d : \u03b4}, Not (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inl.{u3, u4} \u03b1 \u03b2 a) (Sum.inr.{u2, u1} \u03b3 \u03b4 d))\nCase conversion may be inaccurate. Consider using '#align sum.not_lift_rel_inl_inr Sum.not_liftRel_inl_inr\u2093'. -/\n@[simp]\ntheorem not_liftRel_inl_inr : \u00acLiftRel r s (inl a) (inr d) :=\n  fun.\n#align sum.not_lift_rel_inl_inr Sum.not_liftRel_inl_inr\n\n/- warning: sum.not_lift_rel_inr_inl -> Sum.not_liftRel_inr_inl is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {b : \u03b2} {c : \u03b3}, Not (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inr.{u1, u2} \u03b1 \u03b2 b) (Sum.inl.{u3, u4} \u03b3 \u03b4 c))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {b : \u03b2} {c : \u03b3}, Not (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inr.{u3, u4} \u03b1 \u03b2 b) (Sum.inl.{u2, u1} \u03b3 \u03b4 c))\nCase conversion may be inaccurate. Consider using '#align sum.not_lift_rel_inr_inl Sum.not_liftRel_inr_inl\u2093'. -/\n@[simp]\ntheorem not_liftRel_inr_inl : \u00acLiftRel r s (inr b) (inl c) :=\n  fun.\n#align sum.not_lift_rel_inr_inl Sum.not_liftRel_inr_inl\n\n/- warning: sum.lift_rel_inr_inr -> Sum.liftRel_inr_inr is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {b : \u03b2} {d : \u03b4}, Iff (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inr.{u1, u2} \u03b1 \u03b2 b) (Sum.inr.{u3, u4} \u03b3 \u03b4 d)) (s b d)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {b : \u03b2} {d : \u03b4}, Iff (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s (Sum.inr.{u3, u4} \u03b1 \u03b2 b) (Sum.inr.{u2, u1} \u03b3 \u03b4 d)) (s b d)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel_inr_inr Sum.liftRel_inr_inr\u2093'. -/\n@[simp]\ntheorem liftRel_inr_inr : LiftRel r s (inr b) (inr d) \u2194 s b d :=\n  \u27e8fun h => by\n    cases h\n    assumption, LiftRel.inr\u27e9\n#align sum.lift_rel_inr_inr Sum.liftRel_inr_inr\n\ninstance [\u2200 a c, Decidable (r a c)] [\u2200 b d, Decidable (s b d)] :\n    \u2200 (ab : Sum \u03b1 \u03b2) (cd : Sum \u03b3 \u03b4), Decidable (LiftRel r s ab cd)\n  | inl a, inl c => decidable_of_iff' _ liftRel_inl_inl\n  | inl a, inr d => Decidable.isFalse not_liftRel_inl_inr\n  | inr b, inl c => Decidable.isFalse not_liftRel_inr_inl\n  | inr b, inr d => decidable_of_iff' _ liftRel_inr_inr\n\n/- warning: sum.lift_rel.mono -> Sum.LiftRel.mono is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r\u2081 : \u03b1 -> \u03b3 -> Prop} {r\u2082 : \u03b1 -> \u03b3 -> Prop} {s\u2081 : \u03b2 -> \u03b4 -> Prop} {s\u2082 : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u1, u2} \u03b1 \u03b2} {y : Sum.{u3, u4} \u03b3 \u03b4}, (forall (a : \u03b1) (b : \u03b3), (r\u2081 a b) -> (r\u2082 a b)) -> (forall (a : \u03b2) (b : \u03b4), (s\u2081 a b) -> (s\u2082 a b)) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r\u2081 s\u2081 x y) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r\u2082 s\u2082 x y)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r\u2081 : \u03b1 -> \u03b3 -> Prop} {r\u2082 : \u03b1 -> \u03b3 -> Prop} {s\u2081 : \u03b2 -> \u03b4 -> Prop} {s\u2082 : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u3, u4} \u03b1 \u03b2} {y : Sum.{u2, u1} \u03b3 \u03b4}, (forall (a : \u03b1) (b : \u03b3), (r\u2081 a b) -> (r\u2082 a b)) -> (forall (a : \u03b2) (b : \u03b4), (s\u2081 a b) -> (s\u2082 a b)) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r\u2081 s\u2081 x y) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r\u2082 s\u2082 x y)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel.mono Sum.LiftRel.mono\u2093'. -/\ntheorem LiftRel.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b)\n    (h : LiftRel r\u2081 s\u2081 x y) : LiftRel r\u2082 s\u2082 x y :=\n  by\n  cases h\n  exacts[lift_rel.inl (hr _ _ \u2039_\u203a), lift_rel.inr (hs _ _ \u2039_\u203a)]\n#align sum.lift_rel.mono Sum.LiftRel.mono\n\n/- warning: sum.lift_rel.mono_left -> Sum.LiftRel.mono_left is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r\u2081 : \u03b1 -> \u03b3 -> Prop} {r\u2082 : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u1, u2} \u03b1 \u03b2} {y : Sum.{u3, u4} \u03b3 \u03b4}, (forall (a : \u03b1) (b : \u03b3), (r\u2081 a b) -> (r\u2082 a b)) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r\u2081 s x y) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r\u2082 s x y)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r\u2081 : \u03b1 -> \u03b3 -> Prop} {r\u2082 : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u3, u4} \u03b1 \u03b2} {y : Sum.{u2, u1} \u03b3 \u03b4}, (forall (a : \u03b1) (b : \u03b3), (r\u2081 a b) -> (r\u2082 a b)) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r\u2081 s x y) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r\u2082 s x y)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel.mono_left Sum.LiftRel.mono_left\u2093'. -/\ntheorem LiftRel.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : LiftRel r\u2081 s x y) : LiftRel r\u2082 s x y :=\n  h.mono hr fun _ _ => id\n#align sum.lift_rel.mono_left Sum.LiftRel.mono_left\n\n/- warning: sum.lift_rel.mono_right -> Sum.LiftRel.mono_right is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s\u2081 : \u03b2 -> \u03b4 -> Prop} {s\u2082 : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u1, u2} \u03b1 \u03b2} {y : Sum.{u3, u4} \u03b3 \u03b4}, (forall (a : \u03b2) (b : \u03b4), (s\u2081 a b) -> (s\u2082 a b)) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s\u2081 x y) -> (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s\u2082 x y)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s\u2081 : \u03b2 -> \u03b4 -> Prop} {s\u2082 : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u3, u4} \u03b1 \u03b2} {y : Sum.{u2, u1} \u03b3 \u03b4}, (forall (a : \u03b2) (b : \u03b4), (s\u2081 a b) -> (s\u2082 a b)) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s\u2081 x y) -> (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s\u2082 x y)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel.mono_right Sum.LiftRel.mono_right\u2093'. -/\ntheorem LiftRel.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : LiftRel r s\u2081 x y) :\n    LiftRel r s\u2082 x y :=\n  h.mono (fun _ _ => id) hs\n#align sum.lift_rel.mono_right Sum.LiftRel.mono_right\n\n/- warning: sum.lift_rel.swap -> Sum.LiftRel.swap is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u1, u2} \u03b1 \u03b2} {y : Sum.{u3, u4} \u03b3 \u03b4}, (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s x y) -> (Sum.LiftRel.{u2, u1, u4, u3} \u03b2 \u03b1 \u03b4 \u03b3 s r (Sum.swap.{u1, u2} \u03b1 \u03b2 x) (Sum.swap.{u3, u4} \u03b3 \u03b4 y))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u2}} {\u03b4 : Type.{u1}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u3, u4} \u03b1 \u03b2} {y : Sum.{u2, u1} \u03b3 \u03b4}, (Sum.LiftRel.{u3, u4, u2, u1} \u03b1 \u03b2 \u03b3 \u03b4 r s x y) -> (Sum.LiftRel.{u4, u3, u1, u2} \u03b2 \u03b1 \u03b4 \u03b3 s r (Sum.swap.{u3, u4} \u03b1 \u03b2 x) (Sum.swap.{u2, u1} \u03b3 \u03b4 y))\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel.swap Sum.LiftRel.swap\u2093'. -/\nprotected theorem LiftRel.swap (h : LiftRel r s x y) : LiftRel s r x.symm y.symm :=\n  by\n  cases h\n  exacts[lift_rel.inr \u2039_\u203a, lift_rel.inl \u2039_\u203a]\n#align sum.lift_rel.swap Sum.LiftRel.swap\n\n/- warning: sum.lift_rel_swap_iff -> Sum.liftRel_swap_iff is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u1, u2} \u03b1 \u03b2} {y : Sum.{u3, u4} \u03b3 \u03b4}, Iff (Sum.LiftRel.{u2, u1, u4, u3} \u03b2 \u03b1 \u03b4 \u03b3 s r (Sum.swap.{u1, u2} \u03b1 \u03b2 x) (Sum.swap.{u3, u4} \u03b3 \u03b4 y)) (Sum.LiftRel.{u1, u2, u3, u4} \u03b1 \u03b2 \u03b3 \u03b4 r s x y)\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u1}} {\u03b4 : Type.{u2}} {r : \u03b1 -> \u03b3 -> Prop} {s : \u03b2 -> \u03b4 -> Prop} {x : Sum.{u3, u4} \u03b1 \u03b2} {y : Sum.{u1, u2} \u03b3 \u03b4}, Iff (Sum.LiftRel.{u4, u3, u2, u1} \u03b2 \u03b1 \u03b4 \u03b3 s r (Sum.swap.{u3, u4} \u03b1 \u03b2 x) (Sum.swap.{u1, u2} \u03b3 \u03b4 y)) (Sum.LiftRel.{u3, u4, u1, u2} \u03b1 \u03b2 \u03b3 \u03b4 r s x y)\nCase conversion may be inaccurate. Consider using '#align sum.lift_rel_swap_iff Sum.liftRel_swap_iff\u2093'. -/\n@[simp]\ntheorem liftRel_swap_iff : LiftRel s r x.symm y.symm \u2194 LiftRel r s x y :=\n  \u27e8fun h => by\n    rw [\u2190 swap_swap x, \u2190 swap_swap y]\n    exact h.swap, LiftRel.swap\u27e9\n#align sum.lift_rel_swap_iff Sum.liftRel_swap_iff\n\nend LiftRel\n\nsection Lex\n\n#print Sum.Lex /-\n/-- Lexicographic order for sum. Sort all the `inl a` before the `inr b`, otherwise use the\nrespective order on `\u03b1` or `\u03b2`. -/\ninductive Lex (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (s : \u03b2 \u2192 \u03b2 \u2192 Prop) : Sum \u03b1 \u03b2 \u2192 Sum \u03b1 \u03b2 \u2192 Prop\n  | inl {a\u2081 a\u2082} (h : r a\u2081 a\u2082) : Lex (inl a\u2081) (inl a\u2082)\n  | inr {b\u2081 b\u2082} (h : s b\u2081 b\u2082) : Lex (inr b\u2081) (inr b\u2082)\n  | sep (a b) : Lex (inl a) (inr b)\n#align sum.lex Sum.Lex\n-/\n\nattribute [protected] Sum.Lex.inl Sum.Lex.inr\n\nattribute [simp] lex.sep\n\nvariable {r r\u2081 r\u2082 : \u03b1 \u2192 \u03b1 \u2192 Prop} {s s\u2081 s\u2082 : \u03b2 \u2192 \u03b2 \u2192 Prop} {a a\u2081 a\u2082 : \u03b1} {b b\u2081 b\u2082 : \u03b2}\n  {x y : Sum \u03b1 \u03b2}\n\n#print Sum.lex_inl_inl /-\n@[simp]\ntheorem lex_inl_inl : Lex r s (inl a\u2081) (inl a\u2082) \u2194 r a\u2081 a\u2082 :=\n  \u27e8fun h => by\n    cases h\n    assumption, Lex.inl\u27e9\n#align sum.lex_inl_inl Sum.lex_inl_inl\n-/\n\n#print Sum.lex_inr_inr /-\n@[simp]\ntheorem lex_inr_inr : Lex r s (inr b\u2081) (inr b\u2082) \u2194 s b\u2081 b\u2082 :=\n  \u27e8fun h => by\n    cases h\n    assumption, Lex.inr\u27e9\n#align sum.lex_inr_inr Sum.lex_inr_inr\n-/\n\n#print Sum.lex_inr_inl /-\n@[simp]\ntheorem lex_inr_inl : \u00acLex r s (inr b) (inl a) :=\n  fun.\n#align sum.lex_inr_inl Sum.lex_inr_inl\n-/\n\ninstance [DecidableRel r] [DecidableRel s] : DecidableRel (Lex r s)\n  | inl a, inl c => decidable_of_iff' _ lex_inl_inl\n  | inl a, inr d => Decidable.isTrue (Lex.sep _ _)\n  | inr b, inl c => Decidable.isFalse lex_inr_inl\n  | inr b, inr d => decidable_of_iff' _ lex_inr_inr\n\n#print Sum.LiftRel.lex /-\nprotected theorem LiftRel.lex {a b : Sum \u03b1 \u03b2} (h : LiftRel r s a b) : Lex r s a b :=\n  by\n  cases h\n  exacts[lex.inl \u2039_\u203a, lex.inr \u2039_\u203a]\n#align sum.lift_rel.lex Sum.LiftRel.lex\n-/\n\n#print Sum.liftRel_subrelation_lex /-\ntheorem liftRel_subrelation_lex : Subrelation (LiftRel r s) (Lex r s) := fun a b => LiftRel.lex\n#align sum.lift_rel_subrelation_lex Sum.liftRel_subrelation_lex\n-/\n\n#print Sum.Lex.mono /-\ntheorem Lex.mono (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : Lex r\u2081 s\u2081 x y) :\n    Lex r\u2082 s\u2082 x y := by\n  cases h\n  exacts[lex.inl (hr _ _ \u2039_\u203a), lex.inr (hs _ _ \u2039_\u203a), lex.sep _ _]\n#align sum.lex.mono Sum.Lex.mono\n-/\n\n#print Sum.Lex.mono_left /-\ntheorem Lex.mono_left (hr : \u2200 a b, r\u2081 a b \u2192 r\u2082 a b) (h : Lex r\u2081 s x y) : Lex r\u2082 s x y :=\n  h.mono hr fun _ _ => id\n#align sum.lex.mono_left Sum.Lex.mono_left\n-/\n\n#print Sum.Lex.mono_right /-\ntheorem Lex.mono_right (hs : \u2200 a b, s\u2081 a b \u2192 s\u2082 a b) (h : Lex r s\u2081 x y) : Lex r s\u2082 x y :=\n  h.mono (fun _ _ => id) hs\n#align sum.lex.mono_right Sum.Lex.mono_right\n-/\n\n#print Sum.lex_acc_inl /-\ntheorem lex_acc_inl {a} (aca : Acc r a) : Acc (Lex r s) (inl a) :=\n  by\n  induction' aca with a H IH\n  constructor; intro y h\n  cases' h with a' _ h'\n  exact IH _ h'\n#align sum.lex_acc_inl Sum.lex_acc_inl\n-/\n\n#print Sum.lex_acc_inr /-\ntheorem lex_acc_inr (aca : \u2200 a, Acc (Lex r s) (inl a)) {b} (acb : Acc s b) :\n    Acc (Lex r s) (inr b) := by\n  induction' acb with b H IH\n  constructor; intro y h\n  cases' h with _ _ _ b' _ h' a\n  \u00b7 exact IH _ h'\n  \u00b7 exact aca _\n#align sum.lex_acc_inr Sum.lex_acc_inr\n-/\n\n#print Sum.lex_wf /-\ntheorem lex_wf (ha : WellFounded r) (hb : WellFounded s) : WellFounded (Lex r s) :=\n  have aca : \u2200 a, Acc (Lex r s) (inl a) := fun a => lex_acc_inl (ha.apply a)\n  \u27e8fun x => Sum.recOn x aca fun b => lex_acc_inr aca (hb.apply b)\u27e9\n#align sum.lex_wf Sum.lex_wf\n-/\n\nend Lex\n\nend Sum\n\nopen Sum\n\nnamespace Function\n\n/- warning: function.injective.sum_elim -> Function.Injective.sum_elim is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3}, (Function.Injective.{succ u1, succ u3} \u03b1 \u03b3 f) -> (Function.Injective.{succ u2, succ u3} \u03b2 \u03b3 g) -> (forall (a : \u03b1) (b : \u03b2), Ne.{succ u3} \u03b3 (f a) (g b)) -> (Function.Injective.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) \u03b3 (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b3}, (Function.Injective.{succ u2, succ u1} \u03b1 \u03b3 f) -> (Function.Injective.{succ u3, succ u1} \u03b2 \u03b3 g) -> (forall (a : \u03b1) (b : \u03b2), Ne.{succ u1} \u03b3 (f a) (g b)) -> (Function.Injective.{max (succ u3) (succ u2), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) \u03b3 (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f g))\nCase conversion may be inaccurate. Consider using '#align function.injective.sum_elim Function.Injective.sum_elim\u2093'. -/\ntheorem Injective.sum_elim {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3} (hf : Injective f) (hg : Injective g)\n    (hfg : \u2200 a b, f a \u2260 g b) : Injective (Sum.elim f g)\n  | inl x, inl y, h => congr_arg inl <| hf h\n  | inl x, inr y, h => (hfg x y h).elim\n  | inr x, inl y, h => (hfg y x h.symm).elim\n  | inr x, inr y, h => congr_arg inr <| hg h\n#align function.injective.sum_elim Function.Injective.sum_elim\n\n#print Function.Injective.sum_map /-\ntheorem Injective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : Injective f) (hg : Injective g) :\n    Injective (Sum.map f g)\n  | inl x, inl y, h => congr_arg inl <| hf <| inl.inj h\n  | inr x, inr y, h => congr_arg inr <| hg <| inr.inj h\n#align function.injective.sum_map Function.Injective.sum_map\n-/\n\n#print Function.Surjective.sum_map /-\ntheorem Surjective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : Surjective f) (hg : Surjective g) :\n    Surjective (Sum.map f g)\n  | inl y =>\n    let \u27e8x, hx\u27e9 := hf y\n    \u27e8inl x, congr_arg inl hx\u27e9\n  | inr y =>\n    let \u27e8x, hx\u27e9 := hg y\n    \u27e8inr x, congr_arg inr hx\u27e9\n#align function.surjective.sum_map Function.Surjective.sum_map\n-/\n\n#print Function.Bijective.sum_map /-\ntheorem Bijective.sum_map {f : \u03b1 \u2192 \u03b2} {g : \u03b1' \u2192 \u03b2'} (hf : Bijective f) (hg : Bijective g) :\n    Bijective (Sum.map f g) :=\n  \u27e8hf.Injective.sum_map hg.Injective, hf.Surjective.sum_map hg.Surjective\u27e9\n#align function.bijective.sum_map Function.Bijective.sum_map\n-/\n\nend Function\n\nnamespace Sum\n\nopen Function\n\n/- warning: sum.map_injective -> Sum.map_injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Injective.{max (succ u1) (succ u2), max (succ u3) (succ u4)} (Sum.{u1, u2} \u03b1 \u03b2) (Sum.{u3, u4} \u03b3 \u03b4) (Sum.map.{u1, u2, u3, u4} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Injective.{succ u1, succ u3} \u03b1 \u03b3 f) (Function.Injective.{succ u2, succ u4} \u03b2 \u03b4 g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u1}} {\u03b4 : Type.{u2}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Injective.{max (succ u4) (succ u3), max (succ u2) (succ u1)} (Sum.{u3, u4} \u03b1 \u03b2) (Sum.{u1, u2} \u03b3 \u03b4) (Sum.map.{u3, u4, u1, u2} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Injective.{succ u3, succ u1} \u03b1 \u03b3 f) (Function.Injective.{succ u4, succ u2} \u03b2 \u03b4 g))\nCase conversion may be inaccurate. Consider using '#align sum.map_injective Sum.map_injective\u2093'. -/\n@[simp]\ntheorem map_injective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n    Injective (Sum.map f g) \u2194 Injective f \u2227 Injective g :=\n  \u27e8fun h =>\n    \u27e8fun a\u2081 a\u2082 ha => inl_injective <| @h (inl a\u2081) (inl a\u2082) (congr_arg inl ha : _), fun b\u2081 b\u2082 hb =>\n      inr_injective <| @h (inr b\u2081) (inr b\u2082) (congr_arg inr hb : _)\u27e9,\n    fun h => h.1.sum_map h.2\u27e9\n#align sum.map_injective Sum.map_injective\n\n/- warning: sum.map_surjective -> Sum.map_surjective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Surjective.{max (succ u1) (succ u2), max (succ u3) (succ u4)} (Sum.{u1, u2} \u03b1 \u03b2) (Sum.{u3, u4} \u03b3 \u03b4) (Sum.map.{u1, u2, u3, u4} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Surjective.{succ u1, succ u3} \u03b1 \u03b3 f) (Function.Surjective.{succ u2, succ u4} \u03b2 \u03b4 g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u1}} {\u03b4 : Type.{u2}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Surjective.{max (succ u4) (succ u3), max (succ u2) (succ u1)} (Sum.{u3, u4} \u03b1 \u03b2) (Sum.{u1, u2} \u03b3 \u03b4) (Sum.map.{u3, u4, u1, u2} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Surjective.{succ u3, succ u1} \u03b1 \u03b3 f) (Function.Surjective.{succ u4, succ u2} \u03b2 \u03b4 g))\nCase conversion may be inaccurate. Consider using '#align sum.map_surjective Sum.map_surjective\u2093'. -/\n@[simp]\ntheorem map_surjective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n    Surjective (Sum.map f g) \u2194 Surjective f \u2227 Surjective g :=\n  \u27e8fun h =>\n    \u27e8fun c => by\n      obtain \u27e8a | b, h\u27e9 := h (inl c)\n      \u00b7 exact \u27e8a, inl_injective h\u27e9\n      \u00b7 cases h, fun d => by\n      obtain \u27e8a | b, h\u27e9 := h (inr d)\n      \u00b7 cases h\n      \u00b7 exact \u27e8b, inr_injective h\u27e9\u27e9,\n    fun h => h.1.sum_map h.2\u27e9\n#align sum.map_surjective Sum.map_surjective\n\n/- warning: sum.map_bijective -> Sum.map_bijective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} {\u03b4 : Type.{u4}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Bijective.{max (succ u1) (succ u2), max (succ u3) (succ u4)} (Sum.{u1, u2} \u03b1 \u03b2) (Sum.{u3, u4} \u03b3 \u03b4) (Sum.map.{u1, u2, u3, u4} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Bijective.{succ u1, succ u3} \u03b1 \u03b3 f) (Function.Bijective.{succ u2, succ u4} \u03b2 \u03b4 g))\nbut is expected to have type\n  forall {\u03b1 : Type.{u3}} {\u03b2 : Type.{u4}} {\u03b3 : Type.{u1}} {\u03b4 : Type.{u2}} {f : \u03b1 -> \u03b3} {g : \u03b2 -> \u03b4}, Iff (Function.Bijective.{max (succ u4) (succ u3), max (succ u2) (succ u1)} (Sum.{u3, u4} \u03b1 \u03b2) (Sum.{u1, u2} \u03b3 \u03b4) (Sum.map.{u3, u4, u1, u2} \u03b1 \u03b3 \u03b2 \u03b4 f g)) (And (Function.Bijective.{succ u3, succ u1} \u03b1 \u03b3 f) (Function.Bijective.{succ u4, succ u2} \u03b2 \u03b4 g))\nCase conversion may be inaccurate. Consider using '#align sum.map_bijective Sum.map_bijective\u2093'. -/\n@[simp]\ntheorem map_bijective {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4} :\n    Bijective (Sum.map f g) \u2194 Bijective f \u2227 Bijective g :=\n  (map_injective.And map_surjective).trans <| and_and_and_comm _ _ _ _\n#align sum.map_bijective Sum.map_bijective\n\n/- warning: sum.elim_const_const -> Sum.elim_const_const is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} (c : \u03b3), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (Function.const.{succ u3, succ u1} \u03b3 \u03b1 c) (Function.const.{succ u3, succ u2} \u03b3 \u03b2 c)) (Function.const.{succ u3, max (succ u1) (succ u2)} \u03b3 (Sum.{u1, u2} \u03b1 \u03b2) c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} (c : \u03b3), Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 (Function.const.{succ u1, succ u2} \u03b3 \u03b1 c) (Function.const.{succ u1, succ u3} \u03b3 \u03b2 c)) (Function.const.{succ u1, max (succ u2) (succ u3)} \u03b3 (Sum.{u2, u3} \u03b1 \u03b2) c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_const_const Sum.elim_const_const\u2093'. -/\ntheorem elim_const_const (c : \u03b3) : Sum.elim (const _ c : \u03b1 \u2192 \u03b3) (const _ c : \u03b2 \u2192 \u03b3) = const _ c :=\n  by\n  ext x\n  cases x <;> rfl\n#align sum.elim_const_const Sum.elim_const_const\n\n/- warning: sum.elim_lam_const_lam_const -> Sum.elim_lam_const_lam_const is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} (c : \u03b3), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (fun (_x : \u03b1) => c) (fun (_x : \u03b2) => c)) (fun (_x : Sum.{u1, u2} \u03b1 \u03b2) => c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} (c : \u03b3), Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 (fun (_x : \u03b1) => c) (fun (_x : \u03b2) => c)) (fun (_x : Sum.{u2, u3} \u03b1 \u03b2) => c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_lam_const_lam_const Sum.elim_lam_const_lam_const\u2093'. -/\n@[simp]\ntheorem elim_lam_const_lam_const (c : \u03b3) :\n    (Sum.elim (fun _ : \u03b1 => c) fun _ : \u03b2 => c) = fun _ => c :=\n  Sum.elim_const_const c\n#align sum.elim_lam_const_lam_const Sum.elim_lam_const_lam_const\n\n/- warning: sum.elim_update_left -> Sum.elim_update_left is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : DecidableEq.{succ u2} \u03b2] (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (i : \u03b1) (c : \u03b3), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 (Function.update.{succ u1, succ u3} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) f i c) g) (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => Sum.decidableEq.{u1, u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) a b) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u1, u2} \u03b1 \u03b2 i) c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} \u03b1] [_inst_2 : DecidableEq.{succ u3} \u03b2] (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (i : \u03b1) (c : \u03b3), Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 (Function.update.{succ u2, succ u1} \u03b1 (fun (\u1fb0 : \u03b1) => \u03b3) (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) f i c) g) (Function.update.{max (succ u3) (succ u2), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => Sum.instDecidableEqSum.{u2, u3} \u03b1 \u03b2 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) a b) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inl.{u2, u3} \u03b1 \u03b2 i) c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_update_left Sum.elim_update_left\u2093'. -/\ntheorem elim_update_left [DecidableEq \u03b1] [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (i : \u03b1) (c : \u03b3) :\n    Sum.elim (Function.update f i c) g = Function.update (Sum.elim f g) (inl i) c :=\n  by\n  ext x; cases x\n  \u00b7 by_cases h : x = i\n    \u00b7 subst h\n      simp\n    \u00b7 simp [h]\n  \u00b7 simp\n#align sum.elim_update_left Sum.elim_update_left\n\n/- warning: sum.elim_update_right -> Sum.elim_update_right is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} {\u03b3 : Type.{u3}} [_inst_1 : DecidableEq.{succ u1} \u03b1] [_inst_2 : DecidableEq.{succ u2} \u03b2] (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (i : \u03b2) (c : \u03b3), Eq.{max (max (succ u1) (succ u2)) (succ u3)} ((Sum.{u1, u2} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f (Function.update.{succ u2, succ u3} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) g i c)) (Function.update.{max (succ u1) (succ u2), succ u3} (Sum.{u1, u2} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u1, u2} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u1, u2} \u03b1 \u03b2) (b : Sum.{u1, u2} \u03b1 \u03b2) => Sum.decidableEq.{u1, u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) \u03b2 (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) a b) (Sum.elim.{u1, u2, succ u3} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u1, u2} \u03b1 \u03b2 i) c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u3}} {\u03b3 : Type.{u1}} [_inst_1 : DecidableEq.{succ u2} \u03b1] [_inst_2 : DecidableEq.{succ u3} \u03b2] (f : \u03b1 -> \u03b3) (g : \u03b2 -> \u03b3) (i : \u03b2) (c : \u03b3), Eq.{max (max (succ u2) (succ u3)) (succ u1)} ((Sum.{u2, u3} \u03b1 \u03b2) -> \u03b3) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f (Function.update.{succ u3, succ u1} \u03b2 (fun (\u1fb0 : \u03b2) => \u03b3) (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) g i c)) (Function.update.{max (succ u3) (succ u2), succ u1} (Sum.{u2, u3} \u03b1 \u03b2) (fun (\u1fb0 : Sum.{u2, u3} \u03b1 \u03b2) => \u03b3) (fun (a : Sum.{u2, u3} \u03b1 \u03b2) (b : Sum.{u2, u3} \u03b1 \u03b2) => Sum.instDecidableEqSum.{u2, u3} \u03b1 \u03b2 (fun (a : \u03b1) (b : \u03b1) => _inst_1 a b) (fun (a : \u03b2) (b : \u03b2) => _inst_2 a b) a b) (Sum.elim.{u2, u3, succ u1} \u03b1 \u03b2 \u03b3 f g) (Sum.inr.{u2, u3} \u03b1 \u03b2 i) c)\nCase conversion may be inaccurate. Consider using '#align sum.elim_update_right Sum.elim_update_right\u2093'. -/\ntheorem elim_update_right [DecidableEq \u03b1] [DecidableEq \u03b2] (f : \u03b1 \u2192 \u03b3) (g : \u03b2 \u2192 \u03b3) (i : \u03b2) (c : \u03b3) :\n    Sum.elim f (Function.update g i c) = Function.update (Sum.elim f g) (inr i) c :=\n  by\n  ext x; cases x\n  \u00b7 simp\n  \u00b7 by_cases h : x = i\n    \u00b7 subst h\n      simp\n    \u00b7 simp [h]\n#align sum.elim_update_right Sum.elim_update_right\n\nend Sum\n\n/-!\n### Ternary sum\n\nAbbreviations for the maps from the summands to `\u03b1 \u2295 \u03b2 \u2295 \u03b3`. This is useful for pattern-matching.\n-/\n\n\nnamespace Sum3\n\n#print Sum3.in\u2080 /-\n/-- The map from the first summand into a ternary sum. -/\n@[match_pattern, simp, reducible]\ndef in\u2080 (a) : Sum \u03b1 (Sum \u03b2 \u03b3) :=\n  inl a\n#align sum3.in\u2080 Sum3.in\u2080\n-/\n\n#print Sum3.in\u2081 /-\n/-- The map from the second summand into a ternary sum. -/\n@[match_pattern, simp, reducible]\ndef in\u2081 (b) : Sum \u03b1 (Sum \u03b2 \u03b3) :=\n  inr <| inl b\n#align sum3.in\u2081 Sum3.in\u2081\n-/\n\n#print Sum3.in\u2082 /-\n/-- The map from the third summand into a ternary sum. -/\n@[match_pattern, simp, reducible]\ndef in\u2082 (c) : Sum \u03b1 (Sum \u03b2 \u03b3) :=\n  inr <| inr c\n#align sum3.in\u2082 Sum3.in\u2082\n-/\n\nend Sum3\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Sum/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.35297728360023484}}
{"text": "/-\nCopyright (c) 2018 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Data.Fin.Basic\nimport Init.System.Platform\n\nopen Nat\n\n@[extern \"lean_uint8_of_nat\"]\ndef UInt8.ofNat (n : @& Nat) : UInt8 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt8 := UInt8.ofNat\n@[extern \"lean_uint8_to_nat\"]\ndef UInt8.toNat (n : UInt8) : Nat := n.val.val\n@[extern \"lean_uint8_add\"]\ndef UInt8.add (a b : UInt8) : UInt8 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint8_sub\"]\ndef UInt8.sub (a b : UInt8) : UInt8 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint8_mul\"]\ndef UInt8.mul (a b : UInt8) : UInt8 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint8_div\"]\ndef UInt8.div (a b : UInt8) : UInt8 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint8_mod\"]\ndef UInt8.mod (a b : UInt8) : UInt8 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint8_modn\"]\ndef UInt8.modn (a : UInt8) (n : @& Nat) : UInt8 := \u27e8a.val % n\u27e9\n@[extern \"lean_uint8_land\"]\ndef UInt8.land (a b : UInt8) : UInt8 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint8_lor\"]\ndef UInt8.lor (a b : UInt8) : UInt8 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint8_xor\"]\ndef UInt8.xor (a b : UInt8) : UInt8 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint8_shift_left\"]\ndef UInt8.shiftLeft (a b : UInt8) : UInt8 := \u27e8a.val <<< (modn b 8).val\u27e9\n@[extern \"lean_uint8_shift_right\"]\ndef UInt8.shiftRight (a b : UInt8) : UInt8 := \u27e8a.val >>> (modn b 8).val\u27e9\ndef UInt8.lt (a b : UInt8) : Prop := a.val < b.val\ndef UInt8.le (a b : UInt8) : Prop := a.val \u2264 b.val\n\ninstance : OfNat UInt8 n   := \u27e8UInt8.ofNat n\u27e9\ninstance : Add UInt8       := \u27e8UInt8.add\u27e9\ninstance : Sub UInt8       := \u27e8UInt8.sub\u27e9\ninstance : Mul UInt8       := \u27e8UInt8.mul\u27e9\ninstance : Mod UInt8       := \u27e8UInt8.mod\u27e9\ninstance : HMod UInt8 Nat UInt8 := \u27e8UInt8.modn\u27e9\ninstance : Div UInt8       := \u27e8UInt8.div\u27e9\ninstance : LT UInt8        := \u27e8UInt8.lt\u27e9\ninstance : LE UInt8        := \u27e8UInt8.le\u27e9\n\n@[extern \"lean_uint8_complement\"]\ndef UInt8.complement (a:UInt8) : UInt8 := 0-(a+1)\n\ninstance : Complement UInt8 := \u27e8UInt8.complement\u27e9\ninstance : AndOp UInt8     := \u27e8UInt8.land\u27e9\ninstance : OrOp UInt8      := \u27e8UInt8.lor\u27e9\ninstance : Xor UInt8       := \u27e8UInt8.xor\u27e9\ninstance : ShiftLeft UInt8  := \u27e8UInt8.shiftLeft\u27e9\ninstance : ShiftRight UInt8 := \u27e8UInt8.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint8_dec_lt\"]\ndef UInt8.decLt (a b : UInt8) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint8_dec_le\"]\ndef UInt8.decLe (a b : UInt8) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt8) : Decidable (a < b) := UInt8.decLt a b\ninstance (a b : UInt8) : Decidable (a \u2264 b) := UInt8.decLe a b\n\n@[extern \"lean_uint16_of_nat\"]\ndef UInt16.ofNat (n : @& Nat) : UInt16 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt16 := UInt16.ofNat\n@[extern \"lean_uint16_to_nat\"]\ndef UInt16.toNat (n : UInt16) : Nat := n.val.val\n@[extern \"lean_uint16_add\"]\ndef UInt16.add (a b : UInt16) : UInt16 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint16_sub\"]\ndef UInt16.sub (a b : UInt16) : UInt16 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint16_mul\"]\ndef UInt16.mul (a b : UInt16) : UInt16 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint16_div\"]\ndef UInt16.div (a b : UInt16) : UInt16 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint16_mod\"]\ndef UInt16.mod (a b : UInt16) : UInt16 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint16_modn\"]\ndef UInt16.modn (a : UInt16) (n : @& Nat) : UInt16 := \u27e8a.val % n\u27e9\n@[extern \"lean_uint16_land\"]\ndef UInt16.land (a b : UInt16) : UInt16 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint16_lor\"]\ndef UInt16.lor (a b : UInt16) : UInt16 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint16_xor\"]\ndef UInt16.xor (a b : UInt16) : UInt16 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint16_shift_left\"]\ndef UInt16.shiftLeft (a b : UInt16) : UInt16 := \u27e8a.val <<< (modn b 16).val\u27e9\n@[extern \"lean_uint16_to_uint8\"]\ndef UInt16.toUInt8 (a : UInt16) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint8_to_uint16\"]\ndef UInt8.toUInt16 (a : UInt8) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint16_shift_right\"]\ndef UInt16.shiftRight (a b : UInt16) : UInt16 := \u27e8a.val >>> (modn b 16).val\u27e9\ndef UInt16.lt (a b : UInt16) : Prop := a.val < b.val\ndef UInt16.le (a b : UInt16) : Prop := a.val \u2264 b.val\n\n\ninstance : OfNat UInt16 n   := \u27e8UInt16.ofNat n\u27e9\ninstance : Add UInt16       := \u27e8UInt16.add\u27e9\ninstance : Sub UInt16       := \u27e8UInt16.sub\u27e9\ninstance : Mul UInt16       := \u27e8UInt16.mul\u27e9\ninstance : Mod UInt16       := \u27e8UInt16.mod\u27e9\ninstance : HMod UInt16 Nat UInt16 := \u27e8UInt16.modn\u27e9\ninstance : Div UInt16       := \u27e8UInt16.div\u27e9\ninstance : LT UInt16        := \u27e8UInt16.lt\u27e9\ninstance : LE UInt16        := \u27e8UInt16.le\u27e9\n\n@[extern \"lean_uint16_complement\"]\ndef UInt16.complement (a:UInt16) : UInt16 := 0-(a+1)\n\ninstance : Complement UInt16 := \u27e8UInt16.complement\u27e9\ninstance : AndOp UInt16     := \u27e8UInt16.land\u27e9\ninstance : OrOp UInt16      := \u27e8UInt16.lor\u27e9\ninstance : Xor UInt16       := \u27e8UInt16.xor\u27e9\ninstance : ShiftLeft UInt16  := \u27e8UInt16.shiftLeft\u27e9\ninstance : ShiftRight UInt16 := \u27e8UInt16.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint16_dec_lt\"]\ndef UInt16.decLt (a b : UInt16) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint16_dec_le\"]\ndef UInt16.decLe (a b : UInt16) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt16) : Decidable (a < b) := UInt16.decLt a b\ninstance (a b : UInt16) : Decidable (a \u2264 b) := UInt16.decLe a b\n\n@[extern \"lean_uint32_of_nat\"]\ndef UInt32.ofNat (n : @& Nat) : UInt32 := \u27e8Fin.ofNat n\u27e9\n@[extern \"lean_uint32_of_nat\"]\ndef UInt32.ofNat' (n : Nat) (h : n < UInt32.size) : UInt32 := \u27e8\u27e8n, h\u27e9\u27e9\nabbrev Nat.toUInt32 := UInt32.ofNat\n@[extern \"lean_uint32_add\"]\ndef UInt32.add (a b : UInt32) : UInt32 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint32_sub\"]\ndef UInt32.sub (a b : UInt32) : UInt32 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint32_mul\"]\ndef UInt32.mul (a b : UInt32) : UInt32 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint32_div\"]\ndef UInt32.div (a b : UInt32) : UInt32 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint32_mod\"]\ndef UInt32.mod (a b : UInt32) : UInt32 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint32_modn\"]\ndef UInt32.modn (a : UInt32) (n : @& Nat) : UInt32 := \u27e8a.val % n\u27e9\n@[extern \"lean_uint32_land\"]\ndef UInt32.land (a b : UInt32) : UInt32 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint32_lor\"]\ndef UInt32.lor (a b : UInt32) : UInt32 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint32_xor\"]\ndef UInt32.xor (a b : UInt32) : UInt32 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint32_shift_left\"]\ndef UInt32.shiftLeft (a b : UInt32) : UInt32 := \u27e8a.val <<< (modn b 32).val\u27e9\n@[extern \"lean_uint32_shift_right\"]\ndef UInt32.shiftRight (a b : UInt32) : UInt32 := \u27e8a.val >>> (modn b 32).val\u27e9\n@[extern \"lean_uint32_to_uint8\"]\ndef UInt32.toUInt8 (a : UInt32) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint32_to_uint16\"]\ndef UInt32.toUInt16 (a : UInt32) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint8_to_uint32\"]\ndef UInt8.toUInt32 (a : UInt8) : UInt32 := a.toNat.toUInt32\n@[extern \"lean_uint16_to_uint32\"]\ndef UInt16.toUInt32 (a : UInt16) : UInt32 := a.toNat.toUInt32\n\ninstance : OfNat UInt32 n   := \u27e8UInt32.ofNat n\u27e9\ninstance : Add UInt32       := \u27e8UInt32.add\u27e9\ninstance : Sub UInt32       := \u27e8UInt32.sub\u27e9\ninstance : Mul UInt32       := \u27e8UInt32.mul\u27e9\ninstance : Mod UInt32       := \u27e8UInt32.mod\u27e9\ninstance : HMod UInt32 Nat UInt32 := \u27e8UInt32.modn\u27e9\ninstance : Div UInt32       := \u27e8UInt32.div\u27e9\n\n@[extern \"lean_uint32_complement\"]\ndef UInt32.complement (a:UInt32) : UInt32 := 0-(a+1)\n\ninstance : Complement UInt32 := \u27e8UInt32.complement\u27e9\ninstance : AndOp UInt32     := \u27e8UInt32.land\u27e9\ninstance : OrOp UInt32      := \u27e8UInt32.lor\u27e9\ninstance : Xor UInt32       := \u27e8UInt32.xor\u27e9\ninstance : ShiftLeft UInt32  := \u27e8UInt32.shiftLeft\u27e9\ninstance : ShiftRight UInt32 := \u27e8UInt32.shiftRight\u27e9\n\n@[extern \"lean_uint64_of_nat\"]\ndef UInt64.ofNat (n : @& Nat) : UInt64 := \u27e8Fin.ofNat n\u27e9\nabbrev Nat.toUInt64 := UInt64.ofNat\n@[extern \"lean_uint64_to_nat\"]\ndef UInt64.toNat (n : UInt64) : Nat := n.val.val\n@[extern \"lean_uint64_add\"]\ndef UInt64.add (a b : UInt64) : UInt64 := \u27e8a.val + b.val\u27e9\n@[extern \"lean_uint64_sub\"]\ndef UInt64.sub (a b : UInt64) : UInt64 := \u27e8a.val - b.val\u27e9\n@[extern \"lean_uint64_mul\"]\ndef UInt64.mul (a b : UInt64) : UInt64 := \u27e8a.val * b.val\u27e9\n@[extern \"lean_uint64_div\"]\ndef UInt64.div (a b : UInt64) : UInt64 := \u27e8a.val / b.val\u27e9\n@[extern \"lean_uint64_mod\"]\ndef UInt64.mod (a b : UInt64) : UInt64 := \u27e8a.val % b.val\u27e9\n@[extern \"lean_uint64_modn\"]\ndef UInt64.modn (a : UInt64) (n : @& Nat) : UInt64 := \u27e8a.val % n\u27e9\n@[extern \"lean_uint64_land\"]\ndef UInt64.land (a b : UInt64) : UInt64 := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_uint64_lor\"]\ndef UInt64.lor (a b : UInt64) : UInt64 := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_uint64_xor\"]\ndef UInt64.xor (a b : UInt64) : UInt64 := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_uint64_shift_left\"]\ndef UInt64.shiftLeft (a b : UInt64) : UInt64 := \u27e8a.val <<< (modn b 64).val\u27e9\n@[extern \"lean_uint64_shift_right\"]\ndef UInt64.shiftRight (a b : UInt64) : UInt64 := \u27e8a.val >>> (modn b 64).val\u27e9\ndef UInt64.lt (a b : UInt64) : Prop := a.val < b.val\ndef UInt64.le (a b : UInt64) : Prop := a.val \u2264 b.val\n@[extern \"lean_uint64_to_uint8\"]\ndef UInt64.toUInt8 (a : UInt64) : UInt8 := a.toNat.toUInt8\n@[extern \"lean_uint64_to_uint16\"]\ndef UInt64.toUInt16 (a : UInt64) : UInt16 := a.toNat.toUInt16\n@[extern \"lean_uint64_to_uint32\"]\ndef UInt64.toUInt32 (a : UInt64) : UInt32 := a.toNat.toUInt32\n@[extern \"lean_uint8_to_uint64\"]\ndef UInt8.toUInt64 (a : UInt8) : UInt64 := a.toNat.toUInt64\n@[extern \"lean_uint16_to_uint64\"]\ndef UInt16.toUInt64 (a : UInt16) : UInt64 := a.toNat.toUInt64\n@[extern \"lean_uint32_to_uint64\"]\ndef UInt32.toUInt64 (a : UInt32) : UInt64 := a.toNat.toUInt64\n\ninstance : OfNat UInt64 n   := \u27e8UInt64.ofNat n\u27e9\ninstance : Add UInt64       := \u27e8UInt64.add\u27e9\ninstance : Sub UInt64       := \u27e8UInt64.sub\u27e9\ninstance : Mul UInt64       := \u27e8UInt64.mul\u27e9\ninstance : Mod UInt64       := \u27e8UInt64.mod\u27e9\ninstance : HMod UInt64 Nat UInt64 := \u27e8UInt64.modn\u27e9\ninstance : Div UInt64       := \u27e8UInt64.div\u27e9\ninstance : LT UInt64        := \u27e8UInt64.lt\u27e9\ninstance : LE UInt64        := \u27e8UInt64.le\u27e9\n\n@[extern \"lean_uint64_complement\"]\ndef UInt64.complement (a:UInt64) : UInt64 := 0-(a+1)\n\ninstance : Complement UInt64 := \u27e8UInt64.complement\u27e9\ninstance : AndOp UInt64     := \u27e8UInt64.land\u27e9\ninstance : OrOp UInt64      := \u27e8UInt64.lor\u27e9\ninstance : Xor UInt64       := \u27e8UInt64.xor\u27e9\ninstance : ShiftLeft UInt64  := \u27e8UInt64.shiftLeft\u27e9\ninstance : ShiftRight UInt64 := \u27e8UInt64.shiftRight\u27e9\n\n@[extern \"lean_bool_to_uint64\"]\ndef Bool.toUInt64 (b : Bool) : UInt64 := if b then 1 else 0\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint64_dec_lt\"]\ndef UInt64.decLt (a b : UInt64) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_uint64_dec_le\"]\ndef UInt64.decLe (a b : UInt64) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : UInt64) : Decidable (a < b) := UInt64.decLt a b\ninstance (a b : UInt64) : Decidable (a \u2264 b) := UInt64.decLe a b\n\ntheorem usize_size_gt_zero : USize.size > 0 :=\n  Nat.pos_pow_of_pos System.Platform.numBits (Nat.zero_lt_succ _)\n\n@[extern \"lean_usize_of_nat\"]\ndef USize.ofNat (n : @& Nat) : USize := \u27e8Fin.ofNat' n usize_size_gt_zero\u27e9\nabbrev Nat.toUSize := USize.ofNat\n@[extern \"lean_usize_to_nat\"]\ndef USize.toNat (n : USize) : Nat := n.val.val\n@[extern \"lean_usize_add\"]\ndef USize.add (a b : USize) : USize := \u27e8a.val + b.val\u27e9\n@[extern \"lean_usize_sub\"]\ndef USize.sub (a b : USize) : USize := \u27e8a.val - b.val\u27e9\n@[extern \"lean_usize_mul\"]\ndef USize.mul (a b : USize) : USize := \u27e8a.val * b.val\u27e9\n@[extern \"lean_usize_div\"]\ndef USize.div (a b : USize) : USize := \u27e8a.val / b.val\u27e9\n@[extern \"lean_usize_mod\"]\ndef USize.mod (a b : USize) : USize := \u27e8a.val % b.val\u27e9\n@[extern \"lean_usize_modn\"]\ndef USize.modn (a : USize) (n : @& Nat) : USize := \u27e8a.val % n\u27e9\n@[extern \"lean_usize_land\"]\ndef USize.land (a b : USize) : USize := \u27e8Fin.land a.val b.val\u27e9\n@[extern \"lean_usize_lor\"]\ndef USize.lor (a b : USize) : USize := \u27e8Fin.lor a.val b.val\u27e9\n@[extern \"lean_usize_xor\"]\ndef USize.xor (a b : USize) : USize := \u27e8Fin.xor a.val b.val\u27e9\n@[extern \"lean_usize_shift_left\"]\ndef USize.shiftLeft (a b : USize) : USize := \u27e8a.val <<< (modn b System.Platform.numBits).val\u27e9\n@[extern \"lean_usize_shift_right\"]\ndef USize.shiftRight (a b : USize) : USize := \u27e8a.val >>> (modn b System.Platform.numBits).val\u27e9\n@[extern \"lean_uint32_to_usize\"]\ndef UInt32.toUSize (a : UInt32) : USize := a.toNat.toUSize\n@[extern \"lean_usize_to_uint32\"]\ndef USize.toUInt32 (a : USize) : UInt32 := a.toNat.toUInt32\n\ndef USize.lt (a b : USize) : Prop := a.val < b.val\ndef USize.le (a b : USize) : Prop := a.val \u2264 b.val\n\ninstance : OfNat USize n   := \u27e8USize.ofNat n\u27e9\ninstance : Add USize       := \u27e8USize.add\u27e9\ninstance : Sub USize       := \u27e8USize.sub\u27e9\ninstance : Mul USize       := \u27e8USize.mul\u27e9\ninstance : Mod USize       := \u27e8USize.mod\u27e9\ninstance : HMod USize Nat USize := \u27e8USize.modn\u27e9\ninstance : Div USize       := \u27e8USize.div\u27e9\ninstance : LT USize        := \u27e8USize.lt\u27e9\ninstance : LE USize        := \u27e8USize.le\u27e9\n\n@[extern \"lean_usize_complement\"]\ndef USize.complement (a:USize) : USize := 0-(a+1)\n\ninstance : Complement USize := \u27e8USize.complement\u27e9\ninstance : AndOp USize      := \u27e8USize.land\u27e9\ninstance : OrOp USize       := \u27e8USize.lor\u27e9\ninstance : Xor USize        := \u27e8USize.xor\u27e9\ninstance : ShiftLeft USize  := \u27e8USize.shiftLeft\u27e9\ninstance : ShiftRight USize := \u27e8USize.shiftRight\u27e9\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_usize_dec_lt\"]\ndef USize.decLt (a b : USize) : Decidable (a < b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n < m))\n\nset_option bootstrap.genMatcherCode false in\n@[extern \"lean_usize_dec_le\"]\ndef USize.decLe (a b : USize) : Decidable (a \u2264 b) :=\n  match a, b with\n  | \u27e8n\u27e9, \u27e8m\u27e9 => inferInstanceAs (Decidable (n <= m))\n\ninstance (a b : USize) : Decidable (a < b) := USize.decLt a b\ninstance (a b : USize) : Decidable (a \u2264 b) := USize.decLe a b\n\ntheorem USize.modn_lt {m : Nat} : \u2200 (u : USize), m > 0 \u2192 USize.toNat (u % m) < m\n  | \u27e8u\u27e9, h => Fin.modn_lt u h\n", "meta": {"author": "Kha", "repo": "lean4-nightly", "sha": "b4c92de57090e6c47b29d3575df53d86fce52752", "save_path": "github-repos/lean/Kha-lean4-nightly", "path": "github-repos/lean/Kha-lean4-nightly/lean4-nightly-b4c92de57090e6c47b29d3575df53d86fce52752/stage0/src/Init/Data/UInt.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277512, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.35286262949140523}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Kenny Lau\n\nDependent functions with finite support (see `data/finsupp.lean`).\n-/\n\nimport data.finset data.set.finite algebra.big_operators algebra.module algebra.pi_instances\n\nuniverses u u\u2081 u\u2082 v v\u2081 v\u2082 v\u2083 w x y l\n\nvariables (\u03b9 : Type u) (\u03b2 : \u03b9 \u2192 Type v)\n\ndef decidable_zero_symm {\u03b3 : Type w} [has_zero \u03b3] [decidable_pred (eq (0 : \u03b3))] : decidable_pred (\u03bb x, x = (0:\u03b3)) :=\n\u03bb x, decidable_of_iff (0 = x) eq_comm\nlocal attribute [instance] decidable_zero_symm\n\nnamespace dfinsupp\n\nvariable [\u03a0 i, has_zero (\u03b2 i)]\n\nstructure pre : Type (max u v) :=\n(to_fun : \u03a0 i, \u03b2 i)\n(pre_support : multiset \u03b9)\n(zero : \u2200 i, i \u2208 pre_support \u2228 to_fun i = 0)\n\ninstance : setoid (pre \u03b9 \u03b2) :=\n{ r := \u03bb x y, \u2200 i, x.to_fun i = y.to_fun i,\n  iseqv := \u27e8\u03bb f i, rfl, \u03bb f g H i, (H i).symm,\n    \u03bb f g h H1 H2 i, (H1 i).trans (H2 i)\u27e9 }\n\nend dfinsupp\n\nvariable {\u03b9}\n@[reducible] def dfinsupp [\u03a0 i, has_zero (\u03b2 i)] : Type* :=\nquotient (dfinsupp.setoid \u03b9 \u03b2)\nvariable {\u03b2}\n\nnotation `\u03a0\u2080` binders `, ` r:(scoped f, dfinsupp f) := r\ninfix ` \u2192\u209a `:25 := dfinsupp\n\nnamespace dfinsupp\n\nsection basic\nvariables [\u03a0 i, has_zero (\u03b2 i)]\nvariables {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\nvariables [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n\ninstance : has_coe_to_fun (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8\u03bb _, \u03a0 i, \u03b2 i, \u03bb f, quotient.lift_on f pre.to_fun $ \u03bb _ _, funext\u27e9\n\ninstance : has_zero (\u03a0\u2080 i, \u03b2 i) := \u27e8\u27e6\u27e8\u03bb i, 0, \u2205, \u03bb i, or.inr rfl\u27e9\u27e7\u27e9\ninstance : inhabited (\u03a0\u2080 i, \u03b2 i) := \u27e80\u27e9\n\n@[simp] lemma zero_apply {i : \u03b9} : (0 : \u03a0\u2080 i, \u03b2 i) i = 0 := rfl\n\n@[extensionality]\nlemma ext {f g : \u03a0\u2080 i, \u03b2 i} (H : \u2200 i, f i = g i) : f = g :=\nquotient.induction_on\u2082 f g (\u03bb _ _ H, quotient.sound H) H\n\n/-- The composition of `f : \u03b2\u2081 \u2192 \u03b2\u2082` and `g : \u03a0\u2080 i, \u03b2\u2081 i` is\n  `map_range f hf g : \u03a0\u2080 i, \u03b2\u2082 i`, well defined when `f 0 = 0`. -/\ndef map_range (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i) (hf : \u2200 i, f i 0 = 0) (g : \u03a0\u2080 i, \u03b2\u2081 i) : \u03a0\u2080 i, \u03b2\u2082 i :=\nquotient.lift_on g (\u03bb x, \u27e6(\u27e8\u03bb i, f i (x.1 i), x.2,\n  \u03bb i, or.cases_on (x.3 i) or.inl $ \u03bb H, or.inr $ by rw [H, hf]\u27e9 : pre \u03b9 \u03b2\u2082)\u27e7) $ \u03bb x y H,\nquotient.sound $ \u03bb i, by simp only [H i]\n\n@[simp] lemma map_range_apply\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} {i : \u03b9} :\n  map_range f hf g i = f i (g i) :=\nquotient.induction_on g $ \u03bb x, rfl\n\ndef zip_with (f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i) (hf : \u2200 i, f i 0 0 = 0) (g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i) (g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i) : (\u03a0\u2080 i, \u03b2 i) :=\nbegin\n  refine quotient.lift_on\u2082 g\u2081 g\u2082 (\u03bb x y, \u27e6(\u27e8\u03bb i, f i (x.1 i) (y.1 i), x.2 + y.2,\n    \u03bb i, _\u27e9 : pre \u03b9 \u03b2)\u27e7) _,\n  { cases x.3 i with h1 h1,\n    { left, rw multiset.mem_add, left, exact h1 },\n    cases y.3 i with h2 h2,\n    { left, rw multiset.mem_add, right, exact h2 },\n    right, rw [h1, h2, hf] },\n  exact \u03bb x\u2081 x\u2082 y\u2081 y\u2082 H1 H2, quotient.sound $ \u03bb i, by simp only [H1 i, H2 i]\nend\n\n@[simp] lemma zip_with_apply\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i} {hf : \u2200 i, f i 0 0 = 0} {g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i} {g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i} {i : \u03b9} :\n  zip_with f hf g\u2081 g\u2082 i = f i (g\u2081 i) (g\u2082 i) :=\nquotient.induction_on\u2082 g\u2081 g\u2082 $ \u03bb _ _, rfl\n\nend basic\n\nsection algebra\n\ninstance [\u03a0 i, add_monoid (\u03b2 i)] : has_add (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8zip_with (\u03bb _, (+)) (\u03bb _, add_zero 0)\u27e9\n\n@[simp] lemma add_apply [\u03a0 i, add_monoid (\u03b2 i)] {g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} :\n  (g\u2081 + g\u2082) i = g\u2081 i + g\u2082 i :=\nzip_with_apply\n\ninstance [\u03a0 i, add_monoid (\u03b2 i)] : add_monoid (\u03a0\u2080 i, \u03b2 i) :=\n{ add_monoid .\n  zero      := 0,\n  add       := (+),\n  add_assoc := \u03bb f g h, ext $ \u03bb i, by simp only [add_apply, add_assoc],\n  zero_add  := \u03bb f, ext $ \u03bb i, by simp only [add_apply, zero_apply, zero_add],\n  add_zero  := \u03bb f, ext $ \u03bb i, by simp only [add_apply, zero_apply, add_zero] }\n\ninstance [\u03a0 i, add_monoid (\u03b2 i)] {i : \u03b9} : is_add_monoid_hom (\u03bb g : \u03a0\u2080 i : \u03b9, \u03b2 i, g i) :=\nby refine_struct {..}; simp\n\ninstance [\u03a0 i, add_group (\u03b2 i)] : has_neg (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8\u03bb f, f.map_range (\u03bb _, has_neg.neg) (\u03bb _, neg_zero)\u27e9\n\ninstance [\u03a0 i, add_comm_monoid (\u03b2 i)] : add_comm_monoid (\u03a0\u2080 i, \u03b2 i) :=\n{ add_comm := \u03bb f g, ext $ \u03bb i, by simp only [add_apply, add_comm],\n  .. dfinsupp.add_monoid }\n\n@[simp] lemma neg_apply [\u03a0 i, add_group (\u03b2 i)] {g : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} : (- g) i = - g i :=\nmap_range_apply\n\ninstance [\u03a0 i, add_group (\u03b2 i)] : add_group (\u03a0\u2080 i, \u03b2 i) :=\n{ add_left_neg := \u03bb f, ext $ \u03bb i, by simp only [add_apply, neg_apply, zero_apply, add_left_neg],\n  .. dfinsupp.add_monoid,\n  .. (infer_instance : has_neg (\u03a0\u2080 i, \u03b2 i)) }\n\n@[simp] lemma sub_apply [\u03a0 i, add_group (\u03b2 i)] {g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} : (g\u2081 - g\u2082) i = g\u2081 i - g\u2082 i :=\nby rw [sub_eq_add_neg]; simp\n\ninstance [\u03a0 i, add_comm_group (\u03b2 i)] : add_comm_group (\u03a0\u2080 i, \u03b2 i) :=\n{ add_comm := \u03bb f g, ext $ \u03bb i, by simp only [add_apply, add_comm],\n  ..dfinsupp.add_group }\n\ndef to_has_scalar {\u03b3 : Type w} [ring \u03b3] [\u03a0 i, add_comm_group (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)] : has_scalar \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\n\u27e8\u03bbc v, v.map_range (\u03bb _, (\u2022) c) (\u03bb _, smul_zero _)\u27e9\nlocal attribute [instance] to_has_scalar\n\n@[simp] lemma smul_apply {\u03b3 : Type w} [ring \u03b3] [\u03a0 i, add_comm_group (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)] {i : \u03b9} {b : \u03b3} {v : \u03a0\u2080 i, \u03b2 i} :\n  (b \u2022 v) i = b \u2022 (v i) :=\nmap_range_apply\n\ndef to_module {\u03b3 : Type w} [ring \u03b3] [\u03a0 i, add_comm_group (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)] : module \u03b3 (\u03a0\u2080 i, \u03b2 i) :=\nmodule.of_core {\n  smul_add := \u03bb c x y, ext $ \u03bb i, by simp only [add_apply, smul_apply, smul_add],\n  add_smul := \u03bb c x y, ext $ \u03bb i, by simp only [add_apply, smul_apply, add_smul],\n  one_smul := \u03bb x, ext $ \u03bb i, by simp only [smul_apply, one_smul],\n  mul_smul := \u03bb r s x, ext $ \u03bb i, by simp only [smul_apply, smul_smul],\n  .. (infer_instance : has_scalar \u03b3 (\u03a0\u2080 i, \u03b2 i)) }\n\nend algebra\n\nsection filter_and_subtype_domain\n\n/-- `filter p f` is the function which is `f i` if `p i` is true and 0 otherwise. -/\ndef filter [\u03a0 i, has_zero (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p] (f : \u03a0\u2080 i, \u03b2 i) : \u03a0\u2080 i, \u03b2 i :=\nquotient.lift_on f (\u03bb x, \u27e6(\u27e8\u03bb i, if p i then x.1 i else 0, x.2,\n  \u03bb i, or.cases_on (x.3 i) or.inl $ \u03bb H, or.inr $ by rw [H, if_t_t]\u27e9 : pre \u03b9 \u03b2)\u27e7) $ \u03bb x y H,\nquotient.sound $ \u03bb i, by simp only [H i]\n\n@[simp] lemma filter_apply [\u03a0 i, has_zero (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] {i : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} :\n  f.filter p i = if p i then f i else 0 :=\nquotient.induction_on f $ \u03bb x, rfl\n\n@[simp] lemma filter_apply_pos [\u03a0 i, has_zero (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] {f : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} (h : p i) :\n  f.filter p i = f i :=\nby simp only [filter_apply, if_pos h]\n\n@[simp] lemma filter_apply_neg [\u03a0 i, has_zero (\u03b2 i)]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] {f : \u03a0\u2080 i, \u03b2 i} {i : \u03b9} (h : \u00ac p i) :\n  f.filter p i = 0 :=\nby simp only [filter_apply, if_neg h]\n\nlemma filter_pos_add_filter_neg [\u03a0 i, add_monoid (\u03b2 i)] {f : \u03a0\u2080 i, \u03b2 i}\n  {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  f.filter p + f.filter (\u03bbi, \u00ac p i) = f :=\next $ \u03bb i, by simp only [add_apply, filter_apply]; split_ifs; simp only [add_zero, zero_add]\n\n/-- `subtype_domain p f` is the restriction of the finitely supported function\n  `f` to the subtype `p`. -/\ndef subtype_domain [\u03a0 i, has_zero (\u03b2 i)] (p : \u03b9 \u2192 Prop) [decidable_pred p]\n  (f : \u03a0\u2080 i, \u03b2 i) : \u03a0\u2080 i : subtype p, \u03b2 i.1 :=\nbegin\n  fapply quotient.lift_on f,\n  { intro x, refine \u27e6\u27e8\u03bb i, x.1 i.1, (x.2.filter p).attach.map $ \u03bb j, \u27e8j.1, (multiset.mem_filter.1 j.2).2\u27e9, _\u27e9\u27e7,\n    refine \u03bb i, or.cases_on (x.3 i.1) (\u03bb H, _) or.inr,\n    left, rw multiset.mem_map, refine \u27e8\u27e8i.1, multiset.mem_filter.2 \u27e8H, i.2\u27e9\u27e9, _, subtype.eta _ _\u27e9,\n    apply multiset.mem_attach },\n  intros x y H,\n  exact quotient.sound (\u03bb i, H i.1)\nend\n\n@[simp] lemma subtype_domain_zero [\u03a0 i, has_zero (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  subtype_domain p (0 : \u03a0\u2080 i, \u03b2 i) = 0 :=\nrfl\n\n@[simp] lemma subtype_domain_apply [\u03a0 i, has_zero (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {i : subtype p} {v : \u03a0\u2080 i, \u03b2 i} :\n  (subtype_domain p v) i = v (i.val) :=\nquotient.induction_on v $ \u03bb x, rfl\n\n@[simp] lemma subtype_domain_add [\u03a0 i, add_monoid (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] {v v' : \u03a0\u2080 i, \u03b2 i} :\n  (v + v').subtype_domain p = v.subtype_domain p + v'.subtype_domain p :=\next $ \u03bb i, by simp only [add_apply, subtype_domain_apply]\n\ninstance subtype_domain.is_add_monoid_hom [\u03a0 i, add_monoid (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  is_add_monoid_hom (subtype_domain p : (\u03a0\u2080 i : \u03b9, \u03b2 i) \u2192 \u03a0\u2080 i : subtype p, \u03b2 i) :=\nby refine_struct {..}; simp\n\n@[simp] lemma subtype_domain_neg [\u03a0 i, add_group (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] {v : \u03a0\u2080 i, \u03b2 i} :\n  (- v).subtype_domain p = - v.subtype_domain p :=\next $ \u03bb i, by simp only [neg_apply, subtype_domain_apply]\n\n@[simp] lemma subtype_domain_sub [\u03a0 i, add_group (\u03b2 i)] {p : \u03b9 \u2192 Prop} [decidable_pred p] {v v' : \u03a0\u2080 i, \u03b2 i} :\n  (v - v').subtype_domain p = v.subtype_domain p - v'.subtype_domain p :=\next $ \u03bb i, by simp only [sub_apply, subtype_domain_apply]\n\nend filter_and_subtype_domain\n\n\nvariable [decidable_eq \u03b9]\n\nsection basic\nvariable [\u03a0 i, has_zero (\u03b2 i)]\n\nlemma finite_supp (f : \u03a0\u2080 i, \u03b2 i) : set.finite {i | f i \u2260 0} :=\nquotient.induction_on f $ \u03bb x, set.finite_subset\n  (finset.finite_to_set x.2.to_finset) $ \u03bb i H,\nmultiset.mem_to_finset.2 $ (x.3 i).resolve_right H\n\ndef mk (s : finset \u03b9) (x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1) : \u03a0\u2080 i, \u03b2 i :=\n\u27e6\u27e8\u03bb i, if H : i \u2208 s then x \u27e8i, H\u27e9 else 0, s.1,\n\u03bb i, if H : i \u2208 s then or.inl H else or.inr $ dif_neg H\u27e9\u27e7\n\n@[simp] lemma mk_apply {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} {i : \u03b9} :\n  (mk s x : \u03a0 i, \u03b2 i) i = if H : i \u2208 s then x \u27e8i, H\u27e9 else 0 :=\nrfl\n\ntheorem mk_inj (s : finset \u03b9) : function.injective (@mk \u03b9 \u03b2 _ _ s) :=\nbegin\n  intros x y H,\n  ext i,\n  have h1 : (mk s x : \u03a0 i, \u03b2 i) i = (mk s y : \u03a0 i, \u03b2 i) i, {rw H},\n  cases i with i hi,\n  change i \u2208 s at hi,\n  dsimp only [mk_apply, subtype.coe_mk] at h1,\n  simpa only [dif_pos hi] using h1\nend\n\ndef single (i : \u03b9) (b : \u03b2 i) : \u03a0\u2080 i, \u03b2 i :=\nmk (finset.singleton i) $ \u03bb j, eq.rec_on (finset.mem_singleton.1 j.2).symm b\n\n@[simp] lemma single_apply {i i' b} : (single i b : \u03a0\u2080 i, \u03b2 i) i' = (if h : i = i' then eq.rec_on h b else 0) :=\nbegin\n  dsimp only [single],\n  by_cases h : i = i',\n  { have h1 : i' \u2208 finset.singleton i, { simp only [h, finset.mem_singleton] },\n    simp only [mk_apply, dif_pos h, dif_pos h1] },\n  { have h1 : i' \u2209 finset.singleton i, { simp only [ne.symm h, finset.mem_singleton, not_false_iff] },\n    simp only [mk_apply, dif_neg h, dif_neg h1] }\nend\n\n@[simp] lemma single_zero {i} : (single i 0 : \u03a0\u2080 i, \u03b2 i) = 0 :=\nquotient.sound $ \u03bb j, if H : j \u2208 finset.singleton i\nthen by dsimp only; rw [dif_pos H]; cases finset.mem_singleton.1 H; refl\nelse dif_neg H\n\n@[simp] lemma single_eq_same {i b} : (single i b : \u03a0\u2080 i, \u03b2 i) i = b :=\nby simp only [single_apply, dif_pos rfl]\n\n@[simp] lemma single_eq_of_ne {i i' b} (h : i \u2260 i') : (single i b : \u03a0\u2080 i, \u03b2 i) i' = 0 :=\nby simp only [single_apply, dif_neg h]\n\ndef erase (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) : \u03a0\u2080 i, \u03b2 i :=\nquotient.lift_on f (\u03bb x, \u27e6(\u27e8\u03bb j, if j = i then 0 else x.1 j, x.2,\n\u03bb j, or.cases_on (x.3 j) or.inl $ \u03bb H, or.inr $ by simp only [H, if_t_t]\u27e9 : pre \u03b9 \u03b2)\u27e7) $ \u03bb x y H,\nquotient.sound $ \u03bb j, if h : j = i then by simp only [if_pos h]\nelse by simp only [if_neg h, H j]\n\n@[simp] lemma erase_apply {i j : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} :\n  (f.erase i) j = if j = i then 0 else f j :=\nquotient.induction_on f $ \u03bb x, rfl\n\n@[simp] lemma erase_same {i : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} : (f.erase i) i = 0 :=\nby simp\n\n@[simp] lemma erase_ne {i i' : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} (h : i' \u2260 i) : (f.erase i) i' = f i' :=\nby simp [h]\n\nend basic\n\nsection add_monoid\n\nvariable [\u03a0 i, add_monoid (\u03b2 i)]\n\n@[simp] lemma single_add {i : \u03b9} {b\u2081 b\u2082 : \u03b2 i} : single i (b\u2081 + b\u2082) = single i b\u2081 + single i b\u2082 :=\next $ assume i',\nbegin\n  by_cases h : i = i',\n  { subst h, simp only [add_apply, single_eq_same] },\n  { simp only [add_apply, single_eq_of_ne h, zero_add] }\nend\n\nlemma single_add_erase {i : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} : single i (f i) + f.erase i = f :=\next $ \u03bb i',\nif h : i = i' then by subst h; simp only [add_apply, single_apply, erase_apply, dif_pos rfl, if_pos, add_zero]\nelse by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (ne.symm h), zero_add]\n\nlemma erase_add_single {i : \u03b9} {f : \u03a0\u2080 i, \u03b2 i} : f.erase i + single i (f i) = f :=\next $ \u03bb i',\nif h : i = i' then by subst h; simp only [add_apply, single_apply, erase_apply, dif_pos rfl, if_pos, zero_add]\nelse by simp only [add_apply, single_apply, erase_apply, dif_neg h, if_neg (ne.symm h), add_zero]\n\nprotected theorem induction {p : (\u03a0\u2080 i, \u03b2 i) \u2192 Prop} (f : \u03a0\u2080 i, \u03b2 i)\n  (h0 : p 0) (ha : \u2200i b (f : \u03a0\u2080 i, \u03b2 i), f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (single i b + f)) :\n  p f :=\nbegin\n  refine quotient.induction_on f (\u03bb x, _),\n  cases x with f s H, revert f H,\n  apply multiset.induction_on s,\n  { intros f H, convert h0, ext i, exact (H i).resolve_left id },\n  intros i s ih f H,\n  by_cases H1 : i \u2208 s,\n  { have H2 : \u2200 j, j \u2208 s \u2228 f j = 0,\n    { intro j, cases H j with H2 H2,\n      { cases multiset.mem_cons.1 H2 with H3 H3,\n        { left, rw H3, exact H1 },\n        { left, exact H3 } },\n      right, exact H2 },\n    have H3 : (\u27e6{to_fun := f, pre_support := i :: s, zero := H}\u27e7 : \u03a0\u2080 i, \u03b2 i)\n      = \u27e6{to_fun := f, pre_support := s, zero := H2}\u27e7,\n    { exact quotient.sound (\u03bb i, rfl) },\n    rw H3, apply ih },\n  have H2 : p (erase i \u27e6{to_fun := f, pre_support := i :: s, zero := H}\u27e7),\n  { dsimp only [erase, quotient.lift_on_beta],\n    have H2 : \u2200 j, j \u2208 s \u2228 ite (j = i) 0 (f j) = 0,\n    { intro j, cases H j with H2 H2,\n      { cases multiset.mem_cons.1 H2 with H3 H3,\n        { right, exact if_pos H3 },\n        { left, exact H3 } },\n      right, split_ifs; [refl, exact H2] },\n    have H3 : (\u27e6{to_fun := \u03bb (j : \u03b9), ite (j = i) 0 (f j), pre_support := i :: s, zero := _}\u27e7 : \u03a0\u2080 i, \u03b2 i)\n      = \u27e6{to_fun := \u03bb (j : \u03b9), ite (j = i) 0 (f j), pre_support := s, zero := H2}\u27e7 :=\n      quotient.sound (\u03bb i, rfl),\n    rw H3, apply ih },\n  have H3 : single i _ + _ = (\u27e6{to_fun := f, pre_support := i :: s, zero := H}\u27e7 : \u03a0\u2080 i, \u03b2 i) := single_add_erase,\n  rw \u2190 H3,\n  change p (single i (f i) + _),\n  cases classical.em (f i = 0) with h h,\n  { rw [h, single_zero, zero_add], exact H2 },\n  refine ha _ _ _ _ h H2,\n  rw erase_same\nend\n\nlemma induction\u2082 {p : (\u03a0\u2080 i, \u03b2 i) \u2192 Prop} (f : \u03a0\u2080 i, \u03b2 i)\n  (h0 : p 0) (ha : \u2200i b (f : \u03a0\u2080 i, \u03b2 i), f i = 0 \u2192 b \u2260 0 \u2192 p f \u2192 p (f + single i b)) :\n  p f :=\ndfinsupp.induction f h0 $ \u03bb i b f h1 h2 h3,\nhave h4 : f + single i b = single i b + f,\n{ ext j, by_cases H : i = j,\n  { subst H, simp [h1] },\n  { simp [H] } },\neq.rec_on h4 $ ha i b f h1 h2 h3\n\nend add_monoid\n\n@[simp] lemma mk_add [\u03a0 i, add_monoid (\u03b2 i)] {s : finset \u03b9} {x y : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  mk s (x + y) = mk s x + mk s y :=\next $ \u03bb i, by simp only [add_apply, mk_apply]; split_ifs; [refl, rw zero_add]\n\n@[simp] lemma mk_zero [\u03a0 i, has_zero (\u03b2 i)] {s : finset \u03b9} :\n  mk s (0 : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1) = 0 :=\next $ \u03bb i, by simp only [mk_apply]; split_ifs; refl\n\n@[simp] lemma mk_neg [\u03a0 i, add_group (\u03b2 i)] {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  mk s (-x) = -mk s x :=\next $ \u03bb i, by simp only [neg_apply, mk_apply]; split_ifs; [refl, rw neg_zero]\n\n@[simp] lemma mk_sub [\u03a0 i, add_group (\u03b2 i)] {s : finset \u03b9} {x y : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} :\n  mk s (x - y) = mk s x - mk s y :=\next $ \u03bb i, by simp only [sub_apply, mk_apply]; split_ifs; [refl, rw sub_zero]\n\ninstance [\u03a0 i, add_group (\u03b2 i)] {s : finset \u03b9} : is_add_group_hom (@mk \u03b9 \u03b2 _ _ s) :=\n\u27e8\u03bb _ _, mk_add\u27e9\n\nsection\nlocal attribute [instance] to_module\nvariables (\u03b3 : Type w) [ring \u03b3] [\u03a0 i, add_comm_group (\u03b2 i)] [\u03a0 i, module \u03b3 (\u03b2 i)]\ninclude \u03b3\n@[simp] lemma mk_smul {s : finset \u03b9} {c : \u03b3} (x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1) :\n  mk s (c \u2022 x) = c \u2022 mk s x :=\next $ \u03bb i, by simp only [smul_apply, mk_apply]; split_ifs; [refl, rw smul_zero]\n\n@[simp] lemma single_smul {i : \u03b9} {c : \u03b3} {x : \u03b2 i} :\n  single i (c \u2022 x) = c \u2022 single i x :=\next $ \u03bb i, by simp only [smul_apply, single_apply]; split_ifs; [cases h, rw smul_zero]; refl\n\nvariable \u03b2\ndef lmk (s : finset \u03b9) : (\u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1) \u2192\u2097[\u03b3] \u03a0\u2080 i, \u03b2 i :=\n\u27e8mk s, \u03bb _ _, mk_add, \u03bb c x, by rw [mk_smul \u03b3 x]\u27e9\n\ndef lsingle (i) : \u03b2 i \u2192\u2097[\u03b3] \u03a0\u2080 i, \u03b2 i :=\n\u27e8single i, \u03bb _ _, single_add, \u03bb _ _, single_smul _\u27e9\nvariable {\u03b2}\n\n@[simp] lemma lmk_apply {s : finset \u03b9} {x} : lmk \u03b2 \u03b3 s x = mk s x := rfl\n\n@[simp] lemma lsingle_apply {i : \u03b9} {x : \u03b2 i} : lsingle \u03b2 \u03b3 i x = single i x := rfl\nend\n\nsection support_basic\n\nvariables [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n\ndef support (f : \u03a0\u2080 i, \u03b2 i) : finset \u03b9 :=\nquotient.lift_on f (\u03bb x, x.2.to_finset.filter $ \u03bb i, x.1 i \u2260 0) $\nbegin\n  intros x y Hxy,\n  ext i, split,\n  { intro H,\n    rcases finset.mem_filter.1 H with \u27e8h1, h2\u27e9,\n    rw Hxy i at h2,\n    exact finset.mem_filter.2 \u27e8multiset.mem_to_finset.2 $ (y.3 i).resolve_right h2, h2\u27e9 },\n  { intro H,\n    rcases finset.mem_filter.1 H with \u27e8h1, h2\u27e9,\n    rw \u2190 Hxy i at h2,\n    exact finset.mem_filter.2 \u27e8multiset.mem_to_finset.2 $ (x.3 i).resolve_right h2, h2\u27e9 },\nend\n\n@[simp] theorem support_mk_subset {s : finset \u03b9} {x : \u03a0 i : (\u2191s : set \u03b9), \u03b2 i.1} : (mk s x).support \u2286 s :=\n\u03bb i H, multiset.mem_to_finset.1 (finset.mem_filter.1 H).1\n\n@[simp] theorem mem_support_to_fun (f : \u03a0\u2080 i, \u03b2 i) (i) : i \u2208 f.support \u2194 f i \u2260 0 :=\nbegin\n  refine quotient.induction_on f (\u03bb x, _),\n  dsimp only [support, quotient.lift_on_beta],\n  rw [finset.mem_filter, multiset.mem_to_finset],\n  exact and_iff_right_of_imp (x.3 i).resolve_right\nend\n\ntheorem eq_mk_support (f : \u03a0\u2080 i, \u03b2 i) : f = mk f.support (\u03bb i, f i.1) :=\nby ext i; by_cases h : f i = 0; try {simp at h}; simp [h]\n\n@[simp] lemma support_zero : (0 : \u03a0\u2080 i, \u03b2 i).support = \u2205 := rfl\n\n@[simp] lemma mem_support_iff (f : \u03a0\u2080 i, \u03b2 i) : \u2200i:\u03b9, i \u2208 f.support \u2194 f i \u2260 0 :=\nf.mem_support_to_fun\n\n@[simp] lemma support_eq_empty {f : \u03a0\u2080 i, \u03b2 i} : f.support = \u2205 \u2194 f = 0 :=\n\u27e8\u03bb H, ext $ by simpa [finset.ext] using H, by simp {contextual:=tt}\u27e9\n\ninstance decidable_zero : decidable_pred (eq (0 : \u03a0\u2080 i, \u03b2 i)) :=\n\u03bb f, decidable_of_iff _ $ support_eq_empty.trans eq_comm\n\nlemma support_subset_iff {s : set \u03b9} {f : \u03a0\u2080 i, \u03b2 i} :\n  \u2191f.support \u2286 s \u2194 (\u2200i\u2209s, f i = 0) :=\nby simp [set.subset_def];\n   exact forall_congr (assume i, @not_imp_comm _ _ (classical.dec _) (classical.dec _))\n\nlemma support_single_ne_zero {i : \u03b9} {b : \u03b2 i} (hb : b \u2260 0) : (single i b).support = {i} :=\nbegin\n  ext j, by_cases h : i = j,\n  { subst h, simp [hb] },\n  simp [ne.symm h, h]\nend\n\nlemma support_single_subset {i : \u03b9} {b : \u03b2 i} : (single i b).support \u2286 {i} :=\nsupport_mk_subset\n\nsection map_range_and_zip_with\n\nvariables {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\nvariables [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\nvariables [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2081 i))] [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2082 i))]\n\nlemma map_range_def {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} :\n  map_range f hf g = mk g.support (\u03bb i, f i.1 (g i.1)) :=\nbegin\n  ext i,\n  by_cases h : g i = 0,\n  { simp [h, hf] },\n  { simp at h, simp [h, hf] }\nend\n\nlemma support_map_range {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} :\n  (map_range f hf g).support \u2286 g.support :=\nby simp [map_range_def]\n\n@[simp] lemma map_range_single {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {i : \u03b9} {b : \u03b2\u2081 i} :\n  map_range f hf (single i b) = single i (f i b) :=\ndfinsupp.ext $ \u03bb i', by by_cases i = i'; [{subst i', simp}, simp [h, hf]]\n\nlemma zip_with_def {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i} {hf : \u2200 i, f i 0 0 = 0} {g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i} {g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i} :\n  zip_with f hf g\u2081 g\u2082 = mk (g\u2081.support \u222a g\u2082.support) (\u03bb i, f i.1 (g\u2081 i.1) (g\u2082 i.1)) :=\nbegin\n  ext i,\n  by_cases h1 : g\u2081 i = 0; by_cases h2 : g\u2082 i = 0;\n  try {simp at h1 h2}; simp [h1, h2, hf]\nend\n\nlemma support_zip_with {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i \u2192 \u03b2 i} {hf : \u2200 i, f i 0 0 = 0} {g\u2081 : \u03a0\u2080 i, \u03b2\u2081 i} {g\u2082 : \u03a0\u2080 i, \u03b2\u2082 i} :\n  (zip_with f hf g\u2081 g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nby simp [zip_with_def]\n\nend map_range_and_zip_with\n\nlemma erase_def (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  f.erase i = mk (f.support.erase i) (\u03bb j, f j.1) :=\nbegin\n  ext j,\n  by_cases h1 : j = i; by_cases h2 : f j = 0;\n  try {simp at h2}; simp [h1, h2]\nend\n\n@[simp] lemma support_erase (i : \u03b9) (f : \u03a0\u2080 i, \u03b2 i) :\n  (f.erase i).support = f.support.erase i :=\nbegin\n  ext j,\n  by_cases h1 : j = i; by_cases h2 : f j = 0;\n  try {simp at h2}; simp [h1, h2]\nend\n\nsection filter_and_subtype_domain\n\nvariables {p : \u03b9 \u2192 Prop} [decidable_pred p]\n\nlemma filter_def (f : \u03a0\u2080 i, \u03b2 i) :\n  f.filter p = mk (f.support.filter p) (\u03bb i, f i.1) :=\nby ext i; by_cases h1 : p i; by_cases h2 : f i = 0;\ntry {simp at h2}; simp [h1, h2]\n\n@[simp] lemma support_filter (f : \u03a0\u2080 i, \u03b2 i) :\n  (f.filter p).support = f.support.filter p :=\nby ext i; by_cases h : p i; simp [h]\n\nlemma subtype_domain_def (f : \u03a0\u2080 i, \u03b2 i) :\n  f.subtype_domain p = mk (f.support.subtype p) (\u03bb i, f i.1) :=\nby ext i; cases i with i hi;\nby_cases h1 : p i; by_cases h2 : f i = 0;\ntry {simp at h2}; dsimp; simp [h1, h2]\n\n@[simp] lemma support_subtype_domain {f : \u03a0\u2080 i, \u03b2 i} :\n  (subtype_domain p f).support = f.support.subtype p :=\nby ext i; cases i with i hi;\nby_cases h1 : p i; by_cases h2 : f i = 0;\ntry {simp at h2}; dsimp; simp [h1, h2]\n\nend filter_and_subtype_domain\n\nend support_basic\n\nlemma support_add [\u03a0 i, add_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] {g\u2081 g\u2082 : \u03a0\u2080 i, \u03b2 i} :\n  (g\u2081 + g\u2082).support \u2286 g\u2081.support \u222a g\u2082.support :=\nsupport_zip_with\n\n@[simp] lemma support_neg [\u03a0 i, add_group (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] {f : \u03a0\u2080 i, \u03b2 i} :\n  support (-f) = support f :=\nby ext i; simp\n\ninstance [decidable_eq \u03b9] [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_eq (\u03b2 i)] : decidable_eq (\u03a0\u2080 i, \u03b2 i) :=\nassume f g, decidable_of_iff (f.support = g.support \u2227 (\u2200i\u2208f.support, f i = g i))\n  \u27e8assume \u27e8h\u2081, h\u2082\u27e9, ext $ assume i,\n      if h : i \u2208 f.support then h\u2082 i h else\n        have hf : f i = 0, by rwa [f.mem_support_iff, not_not] at h,\n        have hg : g i = 0, by rwa [h\u2081, g.mem_support_iff, not_not] at h,\n        by rw [hf, hg],\n    by intro h; subst h; simp\u27e9\n\nsection prod_and_sum\n\nvariables {\u03b3 : Type w}\n\n-- [to_additive dfinsupp.sum] for dfinsupp.prod doesn't work, the equation lemmas are not generated\n/-- `sum f g` is the sum of `g i (f i)` over the support of `f`. -/\ndef sum [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] [add_comm_monoid \u03b3]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 \u03b3) : \u03b3 :=\nf.support.sum (\u03bbi, g i (f i))\n\n/-- `prod f g` is the product of `g i (f i)` over the support of `f`. -/\n@[to_additive dfinsupp.sum]\ndef prod [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] [comm_monoid \u03b3]\n  (f : \u03a0\u2080 i, \u03b2 i) (g : \u03a0 i, \u03b2 i \u2192 \u03b3) : \u03b3 :=\nf.support.prod (\u03bbi, g i (f i))\nattribute [to_additive dfinsupp.sum.equations._eqn_1] dfinsupp.prod.equations._eqn_1\n\n@[to_additive dfinsupp.sum_map_range_index]\nlemma prod_map_range_index {\u03b2\u2081 : \u03b9 \u2192 Type v\u2081} {\u03b2\u2082 : \u03b9 \u2192 Type v\u2082}\n  [\u03a0 i, has_zero (\u03b2\u2081 i)] [\u03a0 i, has_zero (\u03b2\u2082 i)]\n  [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2081 i))] [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2082 i))] [comm_monoid \u03b3]\n  {f : \u03a0 i, \u03b2\u2081 i \u2192 \u03b2\u2082 i} {hf : \u2200 i, f i 0 = 0} {g : \u03a0\u2080 i, \u03b2\u2081 i} {h : \u03a0 i, \u03b2\u2082 i \u2192 \u03b3} (h0 : \u2200i, h i 0 = 1) :\n  (map_range f hf g).prod h = g.prod (\u03bbi b, h i (f i b)) :=\nbegin\n  rw [map_range_def],\n  refine (finset.prod_subset support_mk_subset _).trans _,\n  { intros i h1 h2,\n    dsimp, simp [h1] at h2, dsimp at h2,\n    simp [h1, h2, h0] },\n  { refine finset.prod_congr rfl _,\n    intros i h1,\n    simp [h1] }\nend\n\n@[to_additive dfinsupp.sum_zero_index]\nlemma prod_zero_index [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] [comm_monoid \u03b3]\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} : (0 : \u03a0\u2080 i, \u03b2 i).prod h = 1 :=\nrfl\n\n@[to_additive dfinsupp.sum_single_index]\nlemma prod_single_index [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] [comm_monoid \u03b3]\n  {i : \u03b9} {b : \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : h i 0 = 1) :\n  (single i b).prod h = h i b :=\nbegin\n  by_cases h : b = 0,\n  { simp [h, prod_zero_index, h_zero], refl },\n  { simp [dfinsupp.prod, support_single_ne_zero h] }\nend\n\n@[to_additive dfinsupp.sum_neg_index]\nlemma prod_neg_index [\u03a0 i, add_group (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] [comm_monoid \u03b3]\n  {g : \u03a0\u2080 i, \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h0 : \u2200i, h i 0 = 1) :\n  (-g).prod h = g.prod (\u03bbi b, h i (- b)) :=\nprod_map_range_index h0\n\n@[simp] lemma sum_apply {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2081 i))]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i} {i\u2082 : \u03b9} :\n  (f.sum g) i\u2082 = f.sum (\u03bbi\u2081 b, g i\u2081 b i\u2082) :=\n(finset.sum_hom (\u03bbf : \u03a0\u2080 i, \u03b2 i, f i\u2082)).symm\n\nlemma support_sum {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2081 i))]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i} :\n  (f.sum g).support \u2286 f.support.bind (\u03bbi, (g i (f i)).support) :=\nhave \u2200i\u2081 : \u03b9, f.sum (\u03bb (i : \u03b9\u2081) (b : \u03b2\u2081 i), (g i b) i\u2081) \u2260 0 \u2192\n    (\u2203 (i : \u03b9\u2081), f i \u2260 0 \u2227 \u00ac (g i (f i)) i\u2081 = 0),\n  from assume i\u2081 h,\n  let \u27e8i, hi, ne\u27e9 := finset.exists_ne_zero_of_sum_ne_zero h in\n  \u27e8i, (f.mem_support_iff i).mp hi, ne\u27e9,\nby simpa [finset.subset_iff, mem_support_iff, finset.mem_bind, sum_apply] using this\n\n@[simp] lemma sum_zero [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [add_comm_monoid \u03b3] {f : \u03a0\u2080 i, \u03b2 i} :\n  f.sum (\u03bbi b, (0 : \u03b3)) = 0 :=\nfinset.sum_const_zero\n\n@[simp] lemma sum_add [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [add_comm_monoid \u03b3] {f : \u03a0\u2080 i, \u03b2 i} {h\u2081 h\u2082 : \u03a0 i, \u03b2 i \u2192 \u03b3} :\n  f.sum (\u03bbi b, h\u2081 i b + h\u2082 i b) = f.sum h\u2081 + f.sum h\u2082 :=\nfinset.sum_add_distrib\n\n@[simp] lemma sum_neg [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [add_comm_group \u03b3] {f : \u03a0\u2080 i, \u03b2 i} {h : \u03a0 i, \u03b2 i \u2192 \u03b3} :\n  f.sum (\u03bbi b, - h i b) = - f.sum h :=\nfinset.sum_hom (@has_neg.neg \u03b3 _)\n\n@[to_additive dfinsupp.sum_add_index]\nlemma prod_add_index [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [comm_monoid \u03b3] {f g : \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082) :\n  (f + g).prod h = f.prod h * g.prod h :=\nhave f_eq : (f.support \u222a g.support).prod (\u03bbi, h i (f i)) = f.prod h,\n  from (finset.prod_subset (finset.subset_union_left _ _) $\n    by simp [mem_support_iff, h_zero] {contextual := tt}).symm,\nhave g_eq : (f.support \u222a g.support).prod (\u03bbi, h i (g i)) = g.prod h,\n  from (finset.prod_subset (finset.subset_union_right _ _) $\n    by simp [mem_support_iff, h_zero] {contextual := tt}).symm,\ncalc (f + g).support.prod (\u03bbi, h i ((f + g) i)) =\n      (f.support \u222a g.support).prod (\u03bbi, h i ((f + g) i)) :\n    finset.prod_subset support_add $\n      by simp [mem_support_iff, h_zero] {contextual := tt}\n  ... = (f.support \u222a g.support).prod (\u03bbi, h i (f i)) *\n      (f.support \u222a g.support).prod (\u03bbi, h i (g i)) :\n    by simp [h_add, finset.prod_mul_distrib]\n  ... = _ : by rw [f_eq, g_eq]\n\nlemma sum_sub_index [\u03a0 i, add_comm_group (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [add_comm_group \u03b3] {f g : \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_sub : \u2200i b\u2081 b\u2082, h i (b\u2081 - b\u2082) = h i b\u2081 - h i b\u2082) :\n  (f - g).sum h = f.sum h - g.sum h :=\nhave h_zero : \u2200i, h i 0 = 0,\n  from assume i,\n  have h i (0 - 0) = h i 0 - h i 0, from h_sub i 0 0,\n  by simpa using this,\nhave h_neg : \u2200i b, h i (- b) = - h i b,\n  from assume i b,\n  have h i (0 - b) = h i 0 - h i b, from h_sub i 0 b,\n  by simpa [h_zero] using this,\nhave h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 + h i b\u2082,\n  from assume i b\u2081 b\u2082,\n  have h i (b\u2081 - (- b\u2082)) = h i b\u2081 - h i (- b\u2082), from h_sub i b\u2081 (-b\u2082),\n  by simpa [h_neg] using this,\nby simp [@sum_add_index \u03b9 \u03b2 _ \u03b3 _ _ _ f (-g) h h_zero h_add];\nsimp [@sum_neg_index \u03b9 \u03b2 _ \u03b3 _ _ _ g h h_zero, h_neg];\nsimp [@sum_neg \u03b9 \u03b2 _ \u03b3 _ _ _ g h]\n\n@[to_additive dfinsupp.sum_finset_sum_index]\nlemma prod_finset_sum_index {\u03b3 : Type w} {\u03b1 : Type x}\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [comm_monoid \u03b3] [decidable_eq \u03b1]\n  {s : finset \u03b1} {g : \u03b1 \u2192 \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082):\n  s.prod (\u03bbi, (g i).prod h) = (s.sum g).prod h :=\nfinset.induction_on s\n  (by simp [prod_zero_index])\n  (by simp [prod_add_index, h_zero, h_add] {contextual := tt})\n\n@[to_additive dfinsupp.sum_sum_index]\nlemma prod_sum_index  {\u03b9\u2081 : Type u\u2081} [decidable_eq \u03b9\u2081] {\u03b2\u2081 : \u03b9\u2081 \u2192 Type v\u2081}\n  [\u03a0 i\u2081, has_zero (\u03b2\u2081 i\u2081)] [\u03a0 i, decidable_pred (eq (0 : \u03b2\u2081 i))]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [comm_monoid \u03b3]\n  {f : \u03a0\u2080 i\u2081, \u03b2\u2081 i\u2081} {g : \u03a0 i\u2081, \u03b2\u2081 i\u2081 \u2192 \u03a0\u2080 i, \u03b2 i}\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (h_zero : \u2200i, h i 0 = 1) (h_add : \u2200i b\u2081 b\u2082, h i (b\u2081 + b\u2082) = h i b\u2081 * h i b\u2082):\n  (f.sum g).prod h = f.prod (\u03bbi b, (g i b).prod h) :=\n(prod_finset_sum_index h_zero h_add).symm\n\n@[simp] lemma sum_single [\u03a0 i, add_comm_monoid (\u03b2 i)]\n  [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))] {f : \u03a0\u2080 i, \u03b2 i} :\n  f.sum single = f :=\nbegin\n  apply dfinsupp.induction f, {rw [sum_zero_index]},\n  intros i b f H hb ih,\n  rw [sum_add_index, ih, sum_single_index],\n  all_goals { intros, simp }\nend\n\n@[to_additive dfinsupp.sum_subtype_domain_index]\nlemma prod_subtype_domain_index [\u03a0 i, has_zero (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  [comm_monoid \u03b3] {v : \u03a0\u2080 i, \u03b2 i} {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {h : \u03a0 i, \u03b2 i \u2192 \u03b3} (hp : \u2200x\u2208v.support, p x) :\n  (v.subtype_domain p).prod (\u03bbi b, h i.1 b) = v.prod h :=\nfinset.prod_bij (\u03bbp _, p.val)\n  (by simp)\n  (by simp)\n  (assume \u27e8a\u2080, ha\u2080\u27e9 \u27e8a\u2081, ha\u2081\u27e9, by simp)\n  (\u03bb i hi, \u27e8\u27e8i, hp i hi\u27e9, by simpa using hi, rfl\u27e9)\n\nlemma subtype_domain_sum [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  {s : finset \u03b3} {h : \u03b3 \u2192 \u03a0\u2080 i, \u03b2 i} {p : \u03b9 \u2192 Prop} [decidable_pred p] :\n  (s.sum h).subtype_domain p = s.sum (\u03bbc, (h c).subtype_domain p) :=\neq.symm (finset.sum_hom _)\n\nlemma subtype_domain_finsupp_sum {\u03b4 : \u03b3 \u2192 Type x} [decidable_eq \u03b3]\n  [\u03a0 c, has_zero (\u03b4 c)] [\u03a0 c, decidable_pred (eq (0 : \u03b4 c))]\n  [\u03a0 i, add_comm_monoid (\u03b2 i)] [\u03a0 i, decidable_pred (eq (0 : \u03b2 i))]\n  {p : \u03b9 \u2192 Prop} [decidable_pred p]\n  {s : \u03a0\u2080 c, \u03b4 c} {h : \u03a0 c, \u03b4 c \u2192 \u03a0\u2080 i, \u03b2 i} :\n  (s.sum h).subtype_domain p = s.sum (\u03bbc d, (h c d).subtype_domain p) :=\nsubtype_domain_sum\n\nend prod_and_sum\n\nend dfinsupp\n", "meta": {"author": "digama0", "repo": "mathlib-ITP2019", "sha": "5cbd0362e04e671ef5db1284870592af6950197c", "save_path": "github-repos/lean/digama0-mathlib-ITP2019", "path": "github-repos/lean/digama0-mathlib-ITP2019/mathlib-ITP2019-5cbd0362e04e671ef5db1284870592af6950197c/src/data/dfinsupp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749126, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3528626124293335}}
{"text": "import feather_logic.basic\nimport order.extension.well\n\nuniverse u\n\n/-!\nIn this file, we construct the type of feather terms in our model,\ngiven the previous level of the model.\n-/\n\nnamespace feather_model\n\nopen_locale classical\n\n/--\nA term in our model, parametrised by the type of terms in the previous level of the model.\n\n* `var v` is the term representing the variable `v`.\n* `prev e` is a term `e` from the previous level of the model.\n* `univ` represents all of `Type u`.\n* `ty \u03b1` is the Lean type `\u03b1`.\n* `obj \u03b1 x` is a Lean object of type `\u03b1` and value `x : \u03b1`.\n* `apply f x` is the term representing function application of `f` on `x`.\n\nIf we are in level `u` of the model, `ty` represents feather objects of type `Type u`, and `obj`\nrepresents feather objects of type which lies in `Type u`. Notably, `var` is a variable of feather\nobjects of type `Type u` or of type which lies in `Type u`. If smaller universes are needed, wrap\nthem in `prev`.\n\nSome \"error states\" representing invalid situations are encoded as `ty pempty`.\n-/\ninductive mterm (T : Type u) : Type (u + 1)\n| var : V \u2192 mterm\n| prev : T \u2192 mterm\n| univ : mterm\n| ty : Type u \u2192 mterm\n| obj : \u03a0 (\u03b1 : Type u), \u03b1 \u2192 mterm\n| apply : mterm \u2192 mterm \u2192 mterm\n\n/-- Provides the `has_type` function, which determines if a given term has a given type.\nThis is just used for bookkeeping between levels of the model. -/\nclass type_data (\u03b1 : Type*) :=\n(has_type : \u03b1 \u2192 \u03b1 \u2192 Prop)\n(is_type : \u03b1 \u2192 Prop)\n\nvariables {T : Type u} [term_struct T]\n\nnamespace mterm\n\ndef bound : \u03a0 (e : mterm T), finset V\n| (var v) := {v}\n| (prev e) := term_struct.bound e\n| univ := \u2205\n| (ty \u03b1) := \u2205\n| (obj \u03b1 x) := \u2205\n| (apply f x) := bound f \u222a bound x\n\n-- A convenient instance to use instead of explicitly calling `bound` all the time.\ninstance mterm_has_mem : has_mem V (mterm T) := \u27e8\u03bb v e, v \u2208 bound e\u27e9\n\ndef subst (v : V) : \u03a0 (e : mterm T) (f : mterm T), mterm T\n| e (var w) := if v = w then e else var w\n| (prev e) (prev f) := prev (term_struct.subst v e f)\n| _ (prev f) := ty pempty\n| e univ := univ\n| e (ty \u03b1) := ty \u03b1\n| e (obj \u03b1 x) := obj \u03b1 x\n| e (apply f x) := apply (subst e f) (subst e x)\n\ndef has_type [type_data T] : \u03a0 (e f : mterm T), Prop\n| (prev x) (prev \u03b1) := type_data.has_type x \u03b1\n| (ty \u03b1) univ := true\n| (obj \u03b1 x) (ty \u03b2) := \u03b1 = \u03b2\n| _ _ := false\n\ndef is_type [type_data T] : \u03a0 (e : mterm T), Prop\n| (prev \u03b1) := type_data.is_type \u03b1\n| (ty \u03b1) := true\n| _ := false\n\ninstance mterm.type_data [type_data T] : type_data (mterm T) := \u27e8has_type, is_type\u27e9\n\nend mterm\n\nopen mterm\n\nsection interpretation\n\n/-! Given a context, which here means a  `finset (V \u00d7 mterm T)`, we produce the set of all\n*interpretations* of that context: substitutions of Lean objects for these variables\nthat satisfy the given context. -/\n\n/--\nWe establish a relation on judgments `V \u00d7 mterm T`.\nIf `v` occurs bound in a term `f`, then `(v, e)` must precede `(w, f)`.\n\nOn a plain context `finset (V \u00d7 mterm T)`, if the transitive closure of this relation forms a\nstrict partial order, we can sort the judgments and provide a set of interpretations for it by\niteratively substituting along this order.\n-/\ndef immediately_precedes (a b : V \u00d7 mterm T) : Prop := a.1 \u2208 b.2\n\n/-- A list of substitutions to perform to yield a interpretation of a collection of variables. -/\n@[reducible] def interpretation (T : Type u) := list (V \u00d7 mterm T)\n\n/-- Given an interpretation, evaluate this term. -/\ndef interpretation.interpret (I : interpretation T) (e : mterm T) : mterm T :=\nlist.foldl (\u03bb (f : mterm T) (i : V \u00d7 mterm T), subst i.1 i.2 f) e I\n\n/-- A collection of assumptions `V \u00d7 mterm T` is *sortable* if they can be ordered in such a way\nwhere each variable occurs only in substitutions later in the order. -/\ndef interpretation.sortable (C : finset (V \u00d7 mterm T)) : Prop :=\nwell_founded (\u03bb a b : C, immediately_precedes a.val b.val)\n\n/-- Assuming a context is sortable, produce a linear order for it. -/\nnoncomputable def interpretation.sort_order (C : finset (V \u00d7 mterm T))\n  (h : interpretation.sortable C) : linear_order C :=\nwell_founded.well_order_extension h\n\n/-- Assuming that a context is sortable, sort it. The precise sort chosen is arbitrary. -/\nnoncomputable def interpretation.sort (C : finset (V \u00d7 mterm T)) (h : interpretation.sortable C) :\n  list (V \u00d7 mterm T) :=\n(finset.sort (interpretation.sort_order C h).le C.attach).map subtype.val\n\n/-- Substitutes the term `e` for the variable `v`. If the context contains an assumption `v : \u03b1`,\nit is removed. -/\ndef interpretation.subst (v : V) (e : mterm T) (I : list (V \u00d7 mterm T)) : list (V \u00d7 mterm T) :=\n(I.filter (\u03bb x : V \u00d7 mterm T, x.1 = v)).map (prod.map id (subst v e))\n\nlemma interpretation.subst_length (v : V) (e : mterm T) (I : interpretation T) :\n  (interpretation.subst v e I).length \u2264 I.length :=\nbegin\n  unfold interpretation.subst,\n  rw list.length_map,\n  exact list.length_filter_le _ _,\nend\n\n/-- The set of interpretations of a typing judgment `e : type`. -/\ndef interpretations (v : V) : \u03a0 (type : mterm T), set (interpretation T)\n| (ty \u03b1) := \u22c3 (x : \u03b1), {[\u27e8v, mterm.obj \u03b1 x\u27e9]}\n| _ := \u2205\n\ndef interpretations' : \u03a0 (C : list (V \u00d7 mterm T)), set (interpretation T)\n| [] := {[]}\n| (x :: xs) := \u22c3 (I \u2208 interpretations x.1 x.2),\n    have (interpretation.subst x.1 x.2 xs).length < (x :: xs).length,\n    from lt_of_le_of_lt (interpretation.subst_length x.1 x.2 xs) (lt_add_one _),\n    (\u03bb J, I ++ J) '' interpretations' (interpretation.subst x.1 x.2 xs)\nusing_well_founded { rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf list.length\u27e9] }\n\ndef rir_context.interpretations (C : rir_context (mterm T)) : set (interpretation T) :=\nif h : interpretation.sortable (C.\u0393 \u222a C.\u039e)\nthen interpretations' (interpretation.sort _ h) else \u2205\n\ndef runtime_context.interpretations (C : runtime_context (mterm T)) : set (interpretation T) :=\nif h : interpretation.sortable (C.\u0393 \u222a C.\u0398.to_finset \u222a C.\u039e)\nthen interpretations' (interpretation.sort _ h) else \u2205\n\nend interpretation\n\ndef is_type [type_data T] (C : rir_context (mterm T)) (\u03b1 : mterm T) : Prop :=\n\u2200 (I : interpretation T), I \u2208 C.interpretations \u2192 mterm.is_type (I.interpret \u03b1)\n\ninductive runtime_ok [type_data T] : runtime_context (mterm T) \u2192 Prop\n| empty : runtime_ok \u27e8\u2205, \u2205, \u2205\u27e9\n| \u0393 (C : runtime_context (mterm T)) (v : V) (\u03b1 : mterm T) :\n  runtime_ok C \u2192 is_type C.rir \u03b1 \u2192 runtime_ok (C + \u27e8{(v, \u03b1)}, \u2205, \u2205\u27e9)\n| \u0398 (C : runtime_context (mterm T)) (v : V) (\u03b1 : mterm T) :\n  runtime_ok C \u2192 is_type C.rir \u03b1 \u2192 runtime_ok (C + \u27e8\u2205, {(v, \u03b1)}, \u2205\u27e9)\n| \u039e (C : runtime_context (mterm T)) (v : V) (\u03b1 : mterm T) :\n  runtime_ok C \u2192 is_type C.rir \u03b1 \u2192 runtime_ok (C + \u27e8\u2205, \u2205, {(v, \u03b1)}\u27e9)\n\ninductive rir_ok [type_data T] : rir_context (mterm T) \u2192 Prop\n| empty : rir_ok \u27e8\u2205, \u2205\u27e9\n| \u0393 (C : rir_context (mterm T)) (v : V) (\u03b1 : mterm T) :\n  rir_ok C \u2192 is_type C \u03b1 \u2192 rir_ok (C \u222a \u27e8{(v, \u03b1)}, \u2205\u27e9)\n| \u039e (C : rir_context (mterm T)) (v : V) (\u03b1 : mterm T) :\n  rir_ok C \u2192 is_type C \u03b1 \u2192 rir_ok (C \u222a \u27e8\u2205, {(v, \u03b1)}\u27e9)\n\n-- TODO: Add the assertion that we use the relevant resources in `\u0398` only once, and that relevant\n-- things are representable.\nstructure runtime_judgments [type_data T] (J : runtime_judgment (mterm T)) : Prop :=\n(ok : runtime_ok J.ctx)\n(interpret : \u2200 (I : interpretation T), I \u2208 J.ctx.interpretations \u2192\n  (I.interpret J.e).has_type (I.interpret J.type))\n\nstructure rir_judgments [type_data T] (J : rir_judgment (mterm T)) : Prop :=\n(ok : rir_ok J.ctx)\n(interpret : \u2200 (I : interpretation T), I \u2208 J.ctx.interpretations \u2192\n  (I.interpret J.e).has_type (I.interpret J.type))\n\ndef defeq [type_data T] (C : rir_context (mterm T)) (x y \u03b1 : mterm T) : Prop :=\n\u2200 (I : interpretation T), I \u2208 C.interpretations \u2192\n  I.interpret x = I.interpret y \u2227 (I.interpret x).has_type (I.interpret \u03b1)\n\ndef sort : sort_name \u2192 mterm T\n| (sort_name.type n) := ty pempty\n| sort_name.prop := ty (ulift Prop)\n| sort_name.region := ty punit\n\ndef representable (C : finset (V \u00d7 mterm T)) (e : mterm T) : mterm T :=\nmterm.obj (ulift Prop) \u27e8false\u27e9\n\ninstance [type_data T] : term_struct (mterm T) := {\n  var := var,\n  bound := bound,\n  subst := subst,\n  is_type := is_type,\n  runtime_ok := runtime_ok,\n  rir_ok := rir_ok,\n  runtime_judgments := runtime_judgments,\n  rir_judgments := rir_judgments,\n  defeq := defeq,\n  sort := sort,\n  representable := representable,\n}\n\nend feather_model\n", "meta": {"author": "quill-lang", "repo": "feather-model", "sha": "64e760f426b4dde09065157c9f5121862681d9e3", "save_path": "github-repos/lean/quill-lang-feather-model", "path": "github-repos/lean/quill-lang-feather-model/feather-model-64e760f426b4dde09065157c9f5121862681d9e3/src/feather_model/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791787121629466, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.352847822384614}}
{"text": "-- From page 9-10 of Baghery et al., we take the coefficients of the relevant monomials.\n\n-- lemma coeff1122 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   polynomial.C A_\u03b1 * polynomial.C B_\u03b2 = 1\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   -- done,\n--   -- simp only [] with polynomial_nf_3 at eqdn,\n--   -- simp only [mul_add, add_mul, finset.sum_add_distrib, C_mul_C, finset.sum_hom, mul_assoc, mul_sum_symm,rearrange_constants_right, rearrange_constants_right_with_extra, rearrange_sums_right, rearrange_sums_right_with_extra] at eqn,\n--   have congr_coeff1122 := congr_arg (coeff (single vars.\u03b1 1 + single vars.\u03b2 1 + single vars.\u03b4 2 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff1122,\n--   simp only [] with finsupp_simp at congr_coeff1122,\n--   exact congr_coeff1122,\n-- end\n\n\n-- lemma coeff0222 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--  polynomial.C A_\u03b2 * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0222 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 2 + single vars.\u03b4 2 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0222,\n--   simp only [] with finsupp_simp at  congr_coeff0222,\n--   exact congr_coeff0222,\n-- end\n\n-- lemma coeff1023 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   polynomial.C A_\u03b1 * polynomial.C B_\u03b3 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff1023 := congr_arg (coeff (single vars.\u03b1 1 + single vars.\u03b2 0 + single vars.\u03b4 2 + single vars.\u03b3 3)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff1023,\n--   simp only [] with finsupp_simp at  congr_coeff1023,\n--   exact congr_coeff1023,\n-- end\n\n-- lemma coeff0212 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--    (\u2211 (x : fin n_wit) in finset.fin_range n_wit, u_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0212 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 2 + single vars.\u03b4 1 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0212,\n--   simp only [] with finsupp_simp at  congr_coeff0212,\n--   exact congr_coeff0212,\n-- end\n\n-- lemma coeff1112 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_wit) in finset.fin_range n_wit, v_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff1112 := congr_arg (coeff (single vars.\u03b1 1 + single vars.\u03b2 1 + single vars.\u03b4 1 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff1112,\n--   simp only [] with finsupp_simp at  congr_coeff1112,\n--   exact congr_coeff1112,\n-- end\n\n\n\n-- lemma coeff0112 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_wit) in finset.fin_range n_wit, u_wit x * polynomial.C (A_m x)) *\n--           \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115) +\n--         (\u2211 (x : fin n_wit) in finset.fin_range n_wit, w_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b2 +\n--       (\u2211 (x : fin (n_var - 1)) in finset.fin_range (n_var - 1), polynomial.X ^ (x : \u2115) * t * polynomial.C (A_h x)) *\n--         polynomial.C B_\u03b2 =\n--     0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0112 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 1 + single vars.\u03b4 1 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0112,\n--   simp only [] with finsupp_simp at  congr_coeff0112,\n--   exact congr_coeff0112,\n-- end\n\n-- lemma coeff0012 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_wit) in finset.fin_range n_wit, w_wit x * polynomial.C (A_m x)) * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115) + (\u2211 (x : fin (n_var - 1)) in finset.fin_range (n_var - 1), polynomial.X ^ (x : \u2115) * t * polynomial.C (A_h x)) * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115 ) = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0012 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 0 + single vars.\u03b4 1 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0012,\n--   simp only [] with finsupp_simp at  congr_coeff0012,\n--   exact congr_coeff0012,\n-- end\n\n\n-- lemma coeff0221 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--    (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, u_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0221 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 2 + single vars.\u03b4 2 + single vars.\u03b3 1)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0221,\n--   simp only [] with finsupp_simp at  congr_coeff0221,\n--   exact congr_coeff0221,\n-- end\n\n\n-- lemma coeff1121 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, v_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff1121 := congr_arg (coeff (single vars.\u03b1 1 + single vars.\u03b2 1 + single vars.\u03b4 2 + single vars.\u03b3 1)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff1121,\n--   simp only [] with finsupp_simp at  congr_coeff1121,\n--   exact congr_coeff1121,\n-- end\n\n\n-- lemma coeff0121 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, u_stmt x * polynomial.C (A_l x)) * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115) + (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, w_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b2 = 0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0121 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 1 + single vars.\u03b4 2 + single vars.\u03b3 1)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0121,\n--   simp only [] with finsupp_simp at  congr_coeff0121,\n--   exact congr_coeff0121,\n-- end\n\n\n-- lemma coeff0021 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, w_stmt x * polynomial.C (A_l x)) *\n--       \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : nat) =\n--     0\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0021 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 0 + single vars.\u03b4 2 + single vars.\u03b3 1)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0021,\n--   simp only [] with finsupp_simp at  congr_coeff0021,\n--   exact congr_coeff0021,\n-- end\n\n-- lemma coeff0122 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   polynomial.C A_\u03b2 * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115) + (\u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (A_x i) * polynomial.X ^ (i : \u2115)) * polynomial.C B_\u03b2 + (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, u_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b3 + (\u2211 (x : fin n_wit) in finset.fin_range n_wit, u_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b4 \n--   = \u2211 (x : fin n_stmt) in finset.fin_range n_stmt, polynomial.C (a_stmt x) * u_stmt x + \u2211 (x : fin n_wit) in finset.fin_range n_wit, u_wit x * polynomial.C (C_m x)\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0122 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 1 + single vars.\u03b4 2 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0122,\n--   simp only [] with finsupp_simp at  congr_coeff0122,\n--   exact congr_coeff0122,\n-- end\n\n\n\n\n-- lemma coeff1022 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n-- polynomial.C A_\u03b1 * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115 ) + (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, v_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b3 + (\u2211 (x : fin n_wit) in finset.fin_range n_wit, v_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b4 = \u2211 (x : fin n_stmt) in finset.fin_range n_stmt, polynomial.C (a_stmt x) * v_stmt x + \u2211 (x : fin n_wit) in finset.fin_range n_wit, v_wit x * polynomial.C (C_m x)\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff1022 := congr_arg (coeff (single vars.\u03b1 1 + single vars.\u03b2 0 + single vars.\u03b4 2 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff1022,\n--   simp only [] with finsupp_simp at  congr_coeff1022,\n--   exact congr_coeff1022,\n-- end\n\n-- lemma coeff0022 (a_stmt : fin n_stmt \u2192 F) (eqn : verified' a_stmt) :\n--   (\u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (A_x i) * polynomial.X ^ (i : \u2115)) * \u2211 (i : fin n_var) in finset.fin_range n_var, polynomial.C (B_x i) * polynomial.X ^ (i : \u2115) + (\u2211 (x : fin n_stmt) in finset.fin_range n_stmt, w_stmt x * polynomial.C (A_l x)) * polynomial.C B_\u03b3 + (\u2211 (x : fin n_wit) in finset.fin_range n_wit, w_wit x * polynomial.C (A_m x)) * polynomial.C B_\u03b4 + (\u2211 (x : fin (n_var - 1)) in finset.fin_range (n_var - 1), polynomial.X ^ (x : \u2115) * t * polynomial.C (A_h x)) * polynomial.C B_\u03b4 = \u2211 (x : fin n_stmt) in finset.fin_range n_stmt, polynomial.C (a_stmt x) * w_stmt x + (\u2211 (x : fin n_wit) in finset.fin_range n_wit, w_wit x * polynomial.C (C_m x) + \u2211 (x : fin (n_var - 1)) in finset.fin_range (n_var - 1), polynomial.X ^ (x : \u2115) * t * polynomial.C (C_h x))\n-- :=\n-- begin\n--   rw verified' at eqn,\n--   rw [A', B', C'] at eqn,\n--   simp only [] with crs at eqn,\n--   -- simp only [] with polynomial_nf_3 at eqn,\n--   -- simp only [mv_polynomial.smul_eq_C_mul] at eqn,\n--   simp only [mv_polynomial.X, C_apply, mv_polynomial.monomial_mul, one_mul, mul_one, add_zero, zero_add, finset.sum_add_distrib, finset.sum_hom, mul_add, add_mul, sum_monomial_hom] at eqn,\n--   have congr_coeff0022 := congr_arg (coeff (single vars.\u03b1 0 + single vars.\u03b2 0 + single vars.\u03b4 2 + single vars.\u03b3 2)) eqn,\n--   clear eqn,\n--   simp only [finsupp_vars_eq_ext] with coeff_simp finsupp_eq at congr_coeff0022,\n--   simp only [] with finsupp_simp at  congr_coeff0022,\n--   exact congr_coeff0022,\n-- end\n", "meta": {"author": "BoltonBailey", "repo": "formal-snarks-project", "sha": "154414784f90a1e257162fcbdd7e805ecb2a49c2", "save_path": "github-repos/lean/BoltonBailey-formal-snarks-project", "path": "github-repos/lean/BoltonBailey-formal-snarks-project/formal-snarks-project-154414784f90a1e257162fcbdd7e805ecb2a49c2/src/snarks/groth16typeIII/oldcode.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786991753929, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.352847815637303}}
{"text": "import category_theory.abelian.projective\nimport for_mathlib.abelian_category\n\nnoncomputable theory\n\nuniverses v u\n\nopen category_theory category_theory.limits\n\nnamespace category_theory\n\n/-- `endomorphisms C` is the category whose objects are an object of `C` equipped with an\nendomorphism. Morphisms are morphisms between objects which intertwine the endomorphisms. -/\nstructure endomorphisms (C : Type u) [category.{v} C] :=\n(X : C)\n(e : End X)\n\nnamespace endomorphisms\n\nsection category\n\nvariables {C : Type u} [category.{v} C]\n\n@[ext] protected structure hom (X Y : endomorphisms C) :=\n(f : X.X \u27f6 Y.X)\n(comm : X.e \u226b f = f \u226b Y.e)\n\nattribute [reassoc, simp] hom.comm\n\ninstance (C : Type u) [category.{v} C] : quiver (endomorphisms C) :=\n{ hom := \u03bb X Y, hom X Y }\n\nlemma f_injective (X Y : endomorphisms C) : function.injective (hom.f : (X \u27f6 Y) \u2192 (X.X \u27f6 Y.X)) :=\nby { intros f g h, ext, exact h }\n\nprotected def id (X : endomorphisms C) : X \u27f6 X :=\n{ f := \ud835\udfd9 _,\n  comm := by rw [category.comp_id, category.id_comp] }\n\nprotected def comp {X Y Z : endomorphisms C} (f : X \u27f6 Y) (g : Y \u27f6 Z) : X \u27f6 Z :=\n{ f := f.f \u226b g.f,\n  comm := by simp only [hom.comm, hom.comm_assoc, category.assoc] }\n\ninstance (C : Type u) [category.{v} C] : category_struct (endomorphisms C) :=\n{ id := \u03bb X, X.id,\n  comp := \u03bb X Y Z f g, endomorphisms.comp f g }\n\n@[simp] lemma id_f (X : endomorphisms C) : hom.f (\ud835\udfd9 X) = \ud835\udfd9 X.X := rfl\n\n@[simps]\ndef end_of_e (X : endomorphisms C) : X \u27f6 X := \u27e8X.e, rfl\u27e9\n\nlemma end_of_e_comm {X Y : endomorphisms C} (g : X \u27f6 Y) : X.end_of_e \u226b g =\n  g \u226b Y.end_of_e :=\nby { ext, apply endomorphisms.hom.comm, }\n\n@[simp, reassoc] lemma comp_f {X Y Z : endomorphisms C} (f : X \u27f6 Y) (g : Y \u27f6 Z) :\n  hom.f (f \u226b g) = f.f \u226b g.f := rfl\n\ninstance (C : Type u) [category.{v} C] : category (endomorphisms C) :=\n{ id_comp' := \u03bb X Y f, by { ext1, simp only [comp_f, id_f, category.id_comp] },\n  comp_id' := \u03bb X Y f, by { ext1, simp only [comp_f, id_f, category.comp_id] },\n  assoc' := \u03bb X Y Z W f g h, by { ext1, simp only [comp_f, category.assoc] } }\n\nlemma congr_f {X Y : endomorphisms C} (f\u2081 f\u2082 : X \u27f6 Y) (h : f\u2081 = f\u2082) : f\u2081.f = f\u2082.f := by rw h\n\n@[simp, reassoc] lemma pow_comm {X Y : endomorphisms C} (f : X \u27f6 Y) (n : \u2115) :\n  (X.e ^ n : End X.X) \u226b f.f = f.f \u226b (Y.e ^ n : End Y.X) :=\nbegin\n  induction n with n ih,\n  { simp only [pow_zero, End.one_def, category.id_comp, category.comp_id] },\n  { simp only [nat.succ_eq_add_one, pow_succ, End.mul_def, category.assoc, hom.comm, reassoc_of ih] }\nend\n\n@[simps]\nprotected def forget (C : Type u) [category.{v} C] : endomorphisms C \u2964 C :=\n{ obj := \u03bb X, X.X,\n  map := \u03bb X Y f, f.f,\n  map_id' := \u03bb X, rfl,\n  map_comp' := \u03bb X Y Z f g, rfl }\n\nlemma epi_of_epi_f {X Y : endomorphisms C} (f : X \u27f6 Y) [epi f.f] : epi f :=\n{ left_cancellation := \u03bb Z g h w, begin\n    ext, rw [\u2190 cancel_epi f.f, \u2190 comp_f, w, comp_f],\n  end }\n\ndef mk_iso {X Y : endomorphisms C} (e : X.X \u2245 Y.X) (h : X.e \u226b e.hom = e.hom \u226b Y.e) : X \u2245 Y :=\n{ hom := \u27e8e.hom, h\u27e9,\n  inv := \u27e8e.inv, by rw [e.comp_inv_eq, category.assoc, e.eq_inv_comp, h]\u27e9,\n  hom_inv_id' := by { ext, simp only [comp_f, iso.hom_inv_id, id_f] },\n  inv_hom_id' := by { ext, simp only [comp_f, iso.inv_hom_id, id_f] } }\n\nend category\n\nsection limits\n\nvariables {C : Type u} [category.{v} C]\nvariables {J : Type v} [small_category J]\n\n@[simps]\ndef twist_cone {K : J \u2964 endomorphisms C}\n  (S : cone (K \u22d9 endomorphisms.forget C)) :\n  cone (K \u22d9 endomorphisms.forget C) :=\n{ X := S.X,\n  \u03c0 :=\n  { app := \u03bb j, S.\u03c0.app j \u226b (K.obj j).e,\n    naturality' := begin\n      intros i j f,\n      dsimp,\n      simp only [category.id_comp, category.assoc, hom.comm],\n      erw S.w_assoc,\n    end } }\n\nabbreviation cone_e {K : J \u2964 endomorphisms C}\n  (S : cone (K \u22d9 endomorphisms.forget C)) (hS : is_limit S) :\n  S.X \u27f6 S.X :=\n@is_limit.lift J _ C _ (K \u22d9 endomorphisms.forget C) S hS (twist_cone S)\n\n@[simps]\nprotected def cone {K : J \u2964 endomorphisms C}\n  (S : cone (K \u22d9 endomorphisms.forget C)) (hS : is_limit S) :\n  cone K :=\n{ X :=\n  { X := S.X,\n    e := cone_e S hS },\n  \u03c0 :=\n  { app := \u03bb j,\n    { f := S.\u03c0.app _,\n      comm := by { dsimp, simp } },\n    naturality' := \u03bb i j f, begin\n      ext, dsimp, simp, erw S.w,\n    end } }\n\n@[simps]\nprotected def is_limit_cone {K : J \u2964 endomorphisms C}\n  (S : cone (K \u22d9 endomorphisms.forget C)) (hS : is_limit S) :\n  is_limit (endomorphisms.cone S hS) :=\n{ lift := \u03bb S,\n  { f := hS.lift \u27e8S.X.X,\n    { app := \u03bb j, (S.\u03c0.app _).f,\n      naturality' := begin\n        intros i j f,\n        dsimp,\n        simp [\u2190 comp_f],\n      end }\u27e9,\n    comm := begin\n      apply hS.hom_ext, dsimp, simp,\n    end },\n  fac' := begin\n    intros s j, ext, dsimp, simp,\n  end,\n  uniq' := begin\n    intros s m hm, ext, apply hS.hom_ext,\n    intros j, specialize hm j, apply_fun (\u03bb e, e.f) at hm,\n    dsimp at *, simp [hm],\n  end }\n\n.\n\nprotected def cone_iso {K : J \u2964 endomorphisms C} (S : cone K)\n  (hS : is_limit ((endomorphisms.forget C).map_cone S)) :\n  endomorphisms.cone _ hS \u2245 S :=\ncones.ext\n({ hom :=\n  { f := \ud835\udfd9 _,\n    comm := by { apply hS.hom_ext, intros j, dsimp, simp, erw hS.fac, dsimp, simp, } },\n  inv :=\n  { f := \ud835\udfd9 _,\n    comm := by { apply hS.hom_ext, intros j, dsimp, simp, erw hS.fac, dsimp, simp } },\n  hom_inv_id' := by { ext, dsimp, simp },\n  inv_hom_id' := by { ext, dsimp, simp } })\nbegin\n  intros j, ext,\n  dsimp, simp,\nend\n\nprotected def cone_iso' {K : J \u2964 endomorphisms C}\n  (S : cone (K \u22d9 endomorphisms.forget C)) (hS : is_limit S) :\n  (endomorphisms.forget C).map_cone (endomorphisms.cone S hS) \u2245 S :=\ncones.ext\n(iso.refl _)\nbegin\n  intros j,\n  dsimp,\n  simp,\nend\n\ninstance has_limit (K : J \u2964 endomorphisms C) [has_limit (K \u22d9 endomorphisms.forget C)] :\n  has_limit K :=\n\u27e8\u27e8\u27e8_, endomorphisms.is_limit_cone _ (limit.is_limit _)\u27e9\u27e9\u27e9\n\ninstance has_limits_of_shape [has_limits_of_shape J C] :\n  has_limits_of_shape J (endomorphisms C) := \u27e8\u27e9\n\ninstance has_limits [has_limits C] : has_limits (endomorphisms C) := \u27e8\u27e9\n\ninstance creates_limit (K : J \u2964 endomorphisms C) : creates_limit K (endomorphisms.forget _) :=\n{ reflects := \u03bb S hS, is_limit.of_iso_limit (endomorphisms.is_limit_cone _ _)\n    (endomorphisms.cone_iso _ hS),\n  lifts := \u03bb S hS,\n  { lifted_cone := endomorphisms.cone _ hS,\n    valid_lift := endomorphisms.cone_iso' _ _ } }\n\ninstance preserves_limit (K : J \u2964 endomorphisms C) [has_limit (K \u22d9 endomorphisms.forget C)] :\n  preserves_limit K (endomorphisms.forget C) :=\ncategory_theory.preserves_limit_of_creates_limit_and_has_limit K (endomorphisms.forget C)\n\ninstance preserves_limits_of_shape [has_limits_of_shape J C] :\n  preserves_limits_of_shape J (endomorphisms.forget C) := \u27e8\u27e9\n\ninstance preserves_limits [has_limits C] : preserves_limits (endomorphisms.forget C) := \u27e8\u27e9\n\ninstance [has_finite_limits C] : preserves_finite_limits (endomorphisms.forget C) :=\nby { constructor, introsI J hJ1 hJ2, apply_instance }\n\nend limits\n\nsection colimits\n\nvariables {C : Type u} [category.{v} C]\nvariables {J : Type v} [small_category J]\n\n@[simps]\ndef twist_cocone {K : J \u2964 endomorphisms C}\n  (S : cocone (K \u22d9 endomorphisms.forget C)) :\n  cocone (K \u22d9 endomorphisms.forget C) :=\n{ X := S.X,\n  \u03b9 :=\n  { app := \u03bb j, (K.obj j).e \u226b S.\u03b9.app j,\n    naturality' := begin\n      intros i j f,\n      dsimp,\n      simp only [category.comp_id, \u2190 hom.comm_assoc],\n      erw S.w,\n    end } }\n\nabbreviation cocone_e {K : J \u2964 endomorphisms C}\n  (S : cocone (K \u22d9 endomorphisms.forget C)) (hS : is_colimit S) :\n  S.X \u27f6 S.X :=\n@is_colimit.desc J _ C _ (K \u22d9 endomorphisms.forget C) S hS (twist_cocone S)\n\n@[simps]\nprotected def cocone {K : J \u2964 endomorphisms C}\n  (S : cocone (K \u22d9 endomorphisms.forget C)) (hS : is_colimit S) :\n  cocone K :=\n{ X :=\n  { X := S.X,\n    e := cocone_e S hS },\n  \u03b9 :=\n  { app := \u03bb j,\n    { f := S.\u03b9.app j,\n      comm := by { dsimp, simp } },\n    naturality' := \u03bb i j f, begin\n      ext, dsimp, simp, erw S.w,\n    end } }\n\n@[simps]\nprotected def is_colimit_cocone {K : J \u2964 endomorphisms C}\n  (S : cocone (K \u22d9 endomorphisms.forget C)) (hS : is_colimit S) :\n  is_colimit (endomorphisms.cocone S hS) :=\n{ desc := \u03bb S,\n  { f := hS.desc \u27e8S.X.X,\n    { app := \u03bb j, (S.\u03b9.app j).f,\n      naturality' := begin\n        intros i j f,\n        dsimp,\n        simp [\u2190 comp_f],\n      end }\u27e9,\n    comm := begin\n      apply hS.hom_ext, dsimp, simp,\n    end },\n  fac' := begin\n    intros s j, ext, dsimp, simp,\n  end,\n  uniq' := begin\n    intros s m hm, ext, apply hS.hom_ext,\n    intros j, specialize hm j, apply_fun (\u03bb e, e.f) at hm,\n    dsimp at *, simp [hm],\n  end }\n\n.\n\nprotected def cocone_iso {K : J \u2964 endomorphisms C} (S : cocone K)\n  (hS : is_colimit ((endomorphisms.forget C).map_cocone S)) :\n  endomorphisms.cocone _ hS \u2245 S :=\ncocones.ext\n({ hom :=\n  { f := \ud835\udfd9 _,\n    comm := by { apply hS.hom_ext, intros j, dsimp, simp, erw hS.fac, dsimp, simp, } },\n  inv :=\n  { f := \ud835\udfd9 _,\n    comm := by { apply hS.hom_ext, intros j, dsimp, simp, erw hS.fac, dsimp, simp } },\n  hom_inv_id' := by { ext, dsimp, simp },\n  inv_hom_id' := by { ext, dsimp, simp } })\nbegin\n  intros j, ext,\n  dsimp, simp,\nend\n\nprotected def cocone_iso' {K : J \u2964 endomorphisms C}\n  (S : cocone (K \u22d9 endomorphisms.forget C)) (hS : is_colimit S) :\n  (endomorphisms.forget C).map_cocone (endomorphisms.cocone S hS) \u2245 S :=\ncocones.ext\n(iso.refl _)\nbegin\n  intros j,\n  dsimp,\n  simp,\nend\n\ninstance has_colimit (K : J \u2964 endomorphisms C) [has_colimit (K \u22d9 endomorphisms.forget C)] :\n  has_colimit K :=\n\u27e8\u27e8\u27e8_, endomorphisms.is_colimit_cocone _ (colimit.is_colimit _)\u27e9\u27e9\u27e9\n\ninstance has_colimits_of_shape [has_colimits_of_shape J C] :\n  has_colimits_of_shape J (endomorphisms C) := \u27e8\u27e9\n\ninstance has_colimits [has_colimits C] : has_colimits (endomorphisms C) := \u27e8\u27e9\n\ninstance creates_colimit (K : J \u2964 endomorphisms C) : creates_colimit K (endomorphisms.forget _) :=\n{ reflects := \u03bb S hS, is_colimit.of_iso_colimit (endomorphisms.is_colimit_cocone _ _)\n    (endomorphisms.cocone_iso _ hS),\n  lifts := \u03bb S hS,\n  { lifted_cocone := endomorphisms.cocone _ hS,\n    valid_lift := endomorphisms.cocone_iso' _ _ } }\n\ninstance preserves_colimit (K : J \u2964 endomorphisms C) [has_colimit (K \u22d9 endomorphisms.forget C)] :\n  preserves_colimit K (endomorphisms.forget C) :=\ncategory_theory.preserves_colimit_of_creates_colimit_and_has_colimit K (endomorphisms.forget C)\n\ninstance preserves_colimits_of_shape [has_colimits_of_shape J C] :\n  preserves_colimits_of_shape J (endomorphisms.forget C) := \u27e8\u27e9\n\ninstance preserves_colimits [has_colimits C] : preserves_colimits (endomorphisms.forget C) := \u27e8\u27e9\n\ninstance [has_finite_colimits C] : preserves_finite_colimits (endomorphisms.forget C) :=\nby { constructor, introsI J hJ1 hJ2, apply_instance }\n\nend colimits\n\nsection projectives\n\nvariables {C : Type u} [category.{v} C]\n\nsection free\n\nvariable [has_coproducts_of_shape (ulift.{v} \u2115) C]\n\n/-- `free X` is \u2a01\u2099X, the direct sum over the naturals, equipped with the endomorphism\nsending `(a : X)` in degree `n` to `a` in degree `n+1`. An alternative way to view `free X`\nis `X \u2a02_{\u2124} \u2124[T]` with the endomorphism given by multiplication by `T`. -/\n@[simps]\ndef free (X : C) : endomorphisms C :=\n{ X := \u2210 (\u03bb i : ulift.{v} \u2115, X),\n  e := sigma.desc $ \u03bb i, sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) \u27e8i.down + 1\u27e9 }\n\n@[reassoc] lemma free.\u03b9_comp_e (X : C) (i : ulift.{v} \u2115) :\n  sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) i \u226b (free X).e = sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) \u27e8i.down + 1\u27e9 :=\nbegin\n  dsimp, simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app],\nend\n\n@[ext] lemma free.ext {X : C} {A : endomorphisms C} (f g : free X \u27f6 A)\n  (w : sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) \u27e80\u27e9 \u226b f.f = sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) \u27e80\u27e9 \u226b g.f) :\n  f = g :=\nbegin\n  ext \u27e8i\u27e9, dsimp,\n  induction i with i ih, { exact w },\n  apply_fun (\u03bb \u03b1, \u03b1 \u226b A.e) at ih,\n  simp only [category.assoc, \u2190 hom.comm, free.\u03b9_comp_e_assoc] at ih,\n  exact ih,\nend\n\n/-- The map `free T \u27f6 A` in `endomorphisms C` induced by a map `T \u27f6 A.X` in `C`. Here `A.X`\nis the forgetful functor forgetting the endomorphism. -/\n@[simps]\ndef free.desc {X : C} {A : endomorphisms C} (f : X \u27f6 A.X) : free X \u27f6 A :=\n{ f := sigma.desc $ \u03bb i, f \u226b (A.e ^ i.down : End A.X),\n  comm := begin\n    ext1 \u27e8i\u27e9, dsimp,\n    simp only [colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app,\n      colimit.\u03b9_desc, category.assoc, pow_succ, End.mul_def],\n  end }\n\nlemma free.desc_comp {X : C} {A B : endomorphisms C} (f : X \u27f6 A.X) (g : A \u27f6 B) :\n  free.desc f \u226b g = free.desc (f \u226b g.f) :=\nbegin\n  ext1, dsimp,\n  simp only [colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, colimit.\u03b9_desc, category.assoc, pow_comm],\nend\n\ndef free.map {X Y : C} (f : X \u27f6 Y) : free X \u27f6 free Y :=\n{ f := sigma.desc $ \u03bb i, f \u226b sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, Y) i,\n  comm := begin\n    ext1 \u27e8i\u27e9,\n    dsimp,\n    simp only [colimit.\u03b9_desc_assoc, cofan.mk_\u03b9_app, colimit.\u03b9_desc, category.assoc],\n  end }\n\nvariable (C)\n\ndef functor.free : C \u2964 endomorphisms C :=\n{ obj := free,\n  map := \u03bb _ _, free.map,\n  map_id' := \u03bb X, begin ext, dsimp, simp only [free.map, category.id_comp, colimit.\u03b9_desc,\n    cofan.mk_\u03b9_app, category.comp_id], end,\n  map_comp' := \u03bb X Y Z f g, begin ext, dsimp, simp only [free.map, category.assoc, colimit.\u03b9_desc,\n    cofan.mk_\u03b9_app, colimit.\u03b9_desc_assoc], end }\n\nend free\n\nsection cofree\n\nvariable {C}\nvariable [has_products_of_shape (ulift.{v} \u2115) C]\n\n/-- `cofree X` is \u220f\u2099X, the product over the naturals, equipped with the endomorphism\nsending `(a : X)` in degree `n` to `a` in degree `n+1`. -/\ndef cofree (X : C) : endomorphisms C :=\n{ X := \u220f (\u03bb i : ulift.{v} \u2115, X),\n  e := pi.lift $ \u03bb i, pi.\u03c0 _ \u27e8i.down + 1\u27e9 }\n\ndef cofree.lift {X : C} {A : endomorphisms C} (f : A.X \u27f6 X) :\n  A \u27f6 cofree X :=\n{ f := pi.lift $ \u03bb i, (A.e ^ i.down : End A.X) \u226b f,\n  comm := begin\n    dsimp [cofree],\n    ext \u27e8j\u27e9, dsimp,\n    simp only [category.assoc, limit.lift_\u03c0, fan.mk_\u03c0_app],\n    rw [\u2190 category.assoc, pow_succ, \u2190 End.mul_def], congr' 1,\n    induction j with j hj,\n    { simp },\n    { simp only [End.mul_def, pow_succ] at *,\n      simp [reassoc_of hj] }\n  end }\n\nend cofree\n\nvariables [has_products_of_shape (ulift.{v} \u2115) C] [has_coproducts_of_shape (ulift.{v} \u2115) C]\n\nlemma f_epi {X Y : endomorphisms C} (f : X \u27f6 Y) [epi f] : epi f.f :=\n{ left_cancellation := \u03bb Z g h w, begin\n    let gg : Y \u27f6 cofree Z := cofree.lift g,\n    let hh : Y \u27f6 cofree Z := cofree.lift h,\n    have : f \u226b gg = f \u226b hh,\n    { ext, dsimp [gg, hh, cofree.lift], simp,\n      simp_rw [\u2190 category.assoc, \u2190 pow_comm, category.assoc, w] },\n    rw cancel_epi at this,\n    apply_fun (\u03bb e, e.f \u226b pi.\u03c0 (\u03bb i : ulift.{v} \u2115, Z) (ulift.up 0)) at this,\n    dsimp [gg, hh, cofree.lift] at this, simpa using this,\n  end }\n\nlemma f_mono {X Y : endomorphisms C} (f : X \u27f6 Y) [mono f] : mono f.f :=\n{ right_cancellation := \u03bb Z g h w, begin\n    let gg : free Z \u27f6 X := free.desc g,\n    let hh : free Z \u27f6 X := free.desc h,\n    have : gg \u226b f = hh \u226b f,\n    { ext, dsimp [gg,hh, free.desc], simpa },\n    rw cancel_mono at this,\n    apply_fun (\u03bb e, sigma.\u03b9 ((\u03bb i : ulift.{v} \u2115, Z)) (ulift.up 0) \u226b e.f) at this,\n    dsimp [gg, hh, free.desc] at this, simpa using this\n  end }\n\ninstance free.projective (X : C) [projective X] : projective (free X) :=\n{ factors := \u03bb E Y f e he, begin\n    resetI,\n    let \u03c6 : X \u27f6 Y.X := sigma.\u03b9 (\u03bb i : ulift.{v} \u2115, X) \u27e80\u27e9 \u226b f.f,\n    haveI : epi e.f := f_epi _,\n    use free.desc (projective.factor_thru \u03c6 e.f),\n    rw [free.desc_comp, projective.factor_thru_comp],\n    ext1, dsimp, simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, pow_zero, End.one_def, category.comp_id],\n  end }\n\ndef free.presentation [enough_projectives C] (A : endomorphisms C) :\n  projective_presentation A :=\n{ P := free (projective.over A.X),\n  projective := infer_instance,\n  f := free.desc $ projective.\u03c0 _,\n  epi := begin\n    suffices : epi (free.desc (projective.\u03c0 A.X)).f,\n    { resetI, apply epi_of_epi_f },\n    dsimp,\n    refine @epi_of_epi _ _ _ _ _ (sigma.\u03b9 _ _) _ (id _), { exact \u27e80\u27e9 },\n    simp only [colimit.\u03b9_desc, cofan.mk_\u03b9_app, pow_zero, End.one_def, category.comp_id],\n    apply_instance\n  end }\n\ninstance [enough_projectives C] : enough_projectives (endomorphisms C) :=\n{ presentation := \u03bb A, \u27e8free.presentation A\u27e9 }\n\n-- generalize to colimits\ninstance projective_sigma {C \u03b9 : Type*} [category C] (P : \u03b9 \u2192 C) [has_coproduct P]\n  [\u2200 i, projective (P i)] :\n  projective (\u2210 P) :=\n{ factors := begin\n  introsI E X f e he,\n  let \u03c6 : \u2210 P \u27f6 E := sigma.desc (\u03bb i, projective.factor_thru (sigma.\u03b9 _ _ \u226b f) e),\n  refine \u27e8\u03c6, _\u27e9,\n  ext i,\n  rw [limits.colimit.\u03b9_desc_assoc, limits.cofan.mk_\u03b9_app, projective.factor_thru_comp],\nend }\n\ninstance projective_X [enough_projectives C] (P : endomorphisms C) [projective P] :\n  projective P.X :=\n\u27e8\u03bb E X f e he, begin\n  let F := free.presentation P,\n  haveI : projective F.P.X := endomorphisms.projective_sigma _,\n  haveI : epi F.f := F.epi,\n  let s : P \u27f6 F.P := projective.factor_thru (\ud835\udfd9 _) F.f,\n  have hs\u03c0 : s \u226b F.f = \ud835\udfd9 _ := projective.factor_thru_comp _ _,\n  let X' : endomorphisms C := \u27e8X, \ud835\udfd9 _\u27e9,\n  let E' : endomorphisms C := \u27e8E, \ud835\udfd9 _\u27e9,\n  let e' : E' \u27f6 X' := \u27e8e, by { dsimp only, rw [category.id_comp, category.comp_id] }\u27e9,\n  haveI he' : epi e' := epi_of_epi_f e',\n  let \u03c6 : F.P.X \u27f6 E := projective.factor_thru (F.f.f \u226b f) e,\n  refine \u27e8s.f \u226b \u03c6, _\u27e9,\n  rw [category.assoc, projective.factor_thru_comp, \u2190 comp_f_assoc, hs\u03c0, id_f, category.id_comp],\nend\u27e9\n\nend projectives\n\nsection preadditive\nopen category_theory.preadditive\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [preadditive \ud835\udcd0]\nvariables (X Y : endomorphisms \ud835\udcd0)\n\ninstance : has_zero (X \u27f6 Y) := \u27e8\u27e80, by simp only [comp_zero, zero_comp, hom.comm]\u27e9\u27e9\ninstance : has_add (X \u27f6 Y) := \u27e8\u03bb f g, \u27e8f.f + g.f, by simp only [comp_add, add_comp, hom.comm]\u27e9\u27e9\ninstance : has_sub (X \u27f6 Y) := \u27e8\u03bb f g, \u27e8f.f - g.f, by simp only [comp_sub, sub_comp, hom.comm]\u27e9\u27e9\ninstance : has_neg (X \u27f6 Y) := \u27e8\u03bb f, \u27e8-f.f, by simp only [comp_neg, neg_comp, hom.comm]\u27e9\u27e9\ninstance has_nsmul : has_scalar \u2115 (X \u27f6 Y) := \u27e8\u03bb n f, \u27e8n \u2022 f.f, by simp only [comp_nsmul, nsmul_comp, hom.comm]\u27e9\u27e9\ninstance has_zsmul : has_scalar \u2124 (X \u27f6 Y) := \u27e8\u03bb n f, \u27e8n \u2022 f.f, by simp only [comp_zsmul, zsmul_comp, hom.comm]\u27e9\u27e9\n\ninstance : add_comm_group (X \u27f6 Y) :=\n(f_injective X Y).add_comm_group _ rfl (\u03bb _ _, rfl) (\u03bb _, rfl) (\u03bb _ _, rfl) (\u03bb _ _, rfl) (\u03bb _ _, rfl)\n\n@[simp] lemma zero_f : hom.f (0 : X \u27f6 Y) = 0 := rfl\nvariables {X Y} (f g : X \u27f6 Y)\n@[simp] lemma add_f : (f + g).f = f.f + g.f := rfl\n@[simp] lemma sub_f : (f - g).f = f.f - g.f := rfl\n@[simp] lemma neg_f : (-f).f = -(f.f) := rfl\n@[simp] lemma nsmul_f (n : \u2115) (f : X \u27f6 Y) : (n \u2022 f).f = n \u2022 f.f := rfl\n@[simp] lemma zsmul_f (n : \u2124) (f : X \u27f6 Y) : (n \u2022 f).f = n \u2022 f.f := rfl\n\nvariables (\ud835\udcd0)\n\ninstance : preadditive (endomorphisms \ud835\udcd0) :=\n{ add_comp' := by { intros, ext, dsimp, rw add_comp },\n  comp_add' := by { intros, ext, dsimp, rw comp_add } }\n\ninstance forget_additive : (endomorphisms.forget \ud835\udcd0).additive := {}\n\ninstance functor.free_additive [has_coproducts_of_shape (ulift.{v} \u2115) \ud835\udcd0] :\n  (functor.free \ud835\udcd0).additive := \u27e8\u03bb X Y f g, by { delta functor.free free.map,\n    ext, simp, }\u27e9\n\nlemma is_zero_X {X : endomorphisms \ud835\udcd0} (h : is_zero X) : is_zero X.X :=\nby { rw is_zero_iff_id_eq_zero at h \u22a2, apply_fun (\u03bb a, a.f) at h, exact h }\n\nend preadditive\n\nsection abelian\n\nvariables {\ud835\udcd0 : Type u} [category.{v} \ud835\udcd0] [abelian \ud835\udcd0]\n  {X Y : endomorphisms \ud835\udcd0} (f : X \u27f6 Y)\n\n@[simps]\nprotected def kernel_obj : endomorphisms \ud835\udcd0 :=\n{ X := kernel f.f,\n  e := kernel.lift _ (kernel.\u03b9 _ \u226b X.e) (by simp) }\n\n@[simps]\nprotected def kernel_\u03b9 : endomorphisms.kernel_obj f \u27f6 X :=\n{ f := kernel.\u03b9 _,\n  comm := by { dsimp, simp } }\n\nprotected def kernel_fork : kernel_fork f :=\nkernel_fork.of_\u03b9 (endomorphisms.kernel_\u03b9 f) $ by { ext, dsimp, simp }\n\n@[simp]\nprotected lemma kernel_fork_\u03b9_f :\n  (endomorphisms.kernel_fork f).\u03b9.f = kernel.\u03b9 _ := rfl\n\n@[simps]\nprotected def kernel_lift (s : kernel_fork f) :\n  s.X \u27f6 endomorphisms.kernel_obj f :=\n{ f := kernel.lift _ s.\u03b9.f $ by { rw [\u2190 comp_f, s.condition, zero_f], },\n  comm := by { ext, dsimp, simp } }\n\n@[simps]\nprotected def is_limit_kernel_fork : is_limit (endomorphisms.kernel_fork f) :=\nis_limit_aux _\n(\u03bb s, endomorphisms.kernel_lift f s)\n(\u03bb s, by { ext, dsimp, simp })\n(\u03bb s m hm, by { apply_fun (\u03bb e, e.f) at hm, ext, dsimp at *, simp [hm] } )\n\ninstance has_kernels : has_kernels (endomorphisms \ud835\udcd0) :=\n\u27e8\u03bb X Y f, \u27e8\u27e8\u27e8endomorphisms.kernel_fork _, endomorphisms.is_limit_kernel_fork _\u27e9\u27e9\u27e9\u27e9\n\n@[simps]\nprotected def cokernel_obj : endomorphisms \ud835\udcd0 :=\n{ X := cokernel f.f,\n  e := cokernel.desc _ (Y.e \u226b cokernel.\u03c0 _) $\n    by { simp only [\u2190 (reassoc_of f.comm), cokernel.condition, comp_zero] } }\n\n@[simps]\nprotected def cokernel_\u03c0 : Y \u27f6 endomorphisms.cokernel_obj f :=\n{ f := cokernel.\u03c0 _,\n  comm := by simp }\n\nprotected def cokernel_cofork : cokernel_cofork f :=\ncokernel_cofork.of_\u03c0 (endomorphisms.cokernel_\u03c0 f) $ by { ext, dsimp, simp }\n\n@[simp]\nprotected lemma cokernel_cofork_\u03c0_f :\n  (endomorphisms.cokernel_cofork f).\u03c0.f = cokernel.\u03c0 _ := rfl\n\n@[simps]\nprotected def cokernel_desc (s : cokernel_cofork f) :\n  endomorphisms.cokernel_obj f \u27f6 s.X :=\n{ f := cokernel.desc _ s.\u03c0.f $ by { rw [\u2190 comp_f, s.condition, zero_f] },\n  comm := by { ext, dsimp, simp } }\n\n@[simps]\nprotected def is_colimit_cokernel_cofork : is_colimit (endomorphisms.cokernel_cofork f) :=\nis_colimit_aux _\n(\u03bb s, endomorphisms.cokernel_desc f s)\n(\u03bb s, by { ext, dsimp, simp })\n(\u03bb s m hm, by { apply_fun (\u03bb e, e.f) at hm, ext, dsimp at *, simp [hm] })\n\ninstance has_cokernels : has_cokernels (endomorphisms \ud835\udcd0) :=\n\u27e8\u03bb X Y f, \u27e8\u27e8\u27e8endomorphisms.cokernel_cofork _, endomorphisms.is_colimit_cokernel_cofork _\u27e9\u27e9\u27e9\u27e9\n\ndef kernel_fork_iso :\n  endomorphisms.kernel_fork f \u2245 kernel_fork.of_\u03b9 (endomorphisms.kernel_\u03b9 f)\n  (endomorphisms.kernel_fork f).condition :=\ncones.ext\n(iso.refl _)\n(by { rintro (_|_); tidy })\n\ndef is_limit_fork_of_is_limit\n  (hF : is_limit (limits.kernel_fork.of_\u03b9 f.f (cokernel.condition _))) :\n  is_limit (limits.kernel_fork.of_\u03b9 f (endomorphisms.cokernel_cofork _).condition) :=\nis_limit_aux _\n(\u03bb S,\n{ f := hF.lift (kernel_fork.of_\u03b9 S.\u03b9.f begin\n    change _ \u226b (endomorphisms.cokernel_cofork _).\u03c0.f = _,\n    erw [\u2190 comp_f, S.condition, zero_f],\n  end),\n  comm := begin\n    apply hF.hom_ext, rintro (_|_),\n    { dsimp, simp only [category.assoc, hom.comm], erw hF.fac _ (walking_parallel_pair.zero),\n      erw hF.fac_assoc _ (walking_parallel_pair.zero),\n      dsimp, simp, },\n    { dsimp, simp, }\n  end })\nbegin\n  intros S,\n  ext, dsimp, erw hF.fac _ walking_parallel_pair.zero, refl,\nend\nbegin\n  intros S m hm,\n  ext, dsimp, apply hF.hom_ext, rintros (_|_),\n  { apply_fun (\u03bb e, e.f) at hm,\n    dsimp at *,\n    simp only [hm],\n    erw hF.fac _ (walking_parallel_pair.zero), refl },\n  { dsimp, simp },\nend\n\ndef is_colimit_cofork_of_is_colimit\n  (hF : is_colimit (limits.cokernel_cofork.of_\u03c0 f.f (kernel.condition _))) :\n  is_colimit (limits.cokernel_cofork.of_\u03c0 f (endomorphisms.kernel_fork _).condition) :=\nis_colimit_aux _\n(\u03bb S,\n{ f := hF.desc (cokernel_cofork.of_\u03c0 S.\u03c0.f begin\n    change (endomorphisms.kernel_fork _).\u03b9.f \u226b _ = _,\n    erw [\u2190 comp_f, S.condition, zero_f]\n  end),\n  comm := begin\n    apply hF.hom_ext, rintro (_|_),\n    { dsimp, simp },\n    { dsimp, erw hF.fac_assoc _ (walking_parallel_pair.one),\n      rw [\u2190 hom.comm_assoc],\n      erw hF.fac _ (walking_parallel_pair.one),\n      dsimp, simp }\n  end })\nbegin\n  intros S,\n  ext, dsimp, erw hF.fac _ walking_parallel_pair.one, refl,\nend\nbegin\n  intros S m hm,\n  ext, dsimp, apply hF.hom_ext, rintros (_|_),\n  { dsimp, simp },\n  { apply_fun (\u03bb e, e.f) at hm,\n    dsimp at *,\n    simp only [hm],\n    erw hF.fac _ walking_parallel_pair.one, refl }\nend\n\ninstance [has_coproducts_of_shape (ulift.{v} \u2115) \ud835\udcd0] [has_products_of_shape (ulift.{v} \u2115) \ud835\udcd0] :\n  abelian (endomorphisms \ud835\udcd0) :=\n{ normal_mono_of_mono := begin\n    introsI X Y f _,\n    haveI := f_mono f,\n    let hE : is_limit (kernel_fork.of_\u03b9 f.f _) :=\n      category_theory.abelian.mono_is_kernel_of_cokernel _ (colimit.is_colimit _),\n    fconstructor,\n    exact endomorphisms.cokernel_obj f,\n    exact endomorphisms.cokernel_\u03c0 f,\n    exact (endomorphisms.cokernel_cofork f).condition,\n    apply is_limit_fork_of_is_limit _ hE,\n  end,\n  normal_epi_of_epi := begin\n    introsI X Y f _,\n    haveI := f_epi f,\n    let hE : is_colimit (cokernel_cofork.of_\u03c0 f.f _) :=\n      category_theory.abelian.epi_is_cokernel_of_kernel _ (limit.is_limit _),\n    fconstructor,\n    exact endomorphisms.kernel_obj f,\n    exact endomorphisms.kernel_\u03b9 f,\n    exact (endomorphisms.kernel_fork f).condition,\n    apply is_colimit_cofork_of_is_colimit _ hE,\n  end,\n  has_finite_products := begin\n    constructor, intros J _ _,\n    haveI : has_finite_products \ud835\udcd0 := abelian.has_finite_products, -- WHY IS THIS NEEDED!?\n    apply_instance,\n  end,\n  .. (_ : preadditive (endomorphisms \ud835\udcd0)) }\n\nend abelian\n\nend endomorphisms\n\nend category_theory\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/endomorphisms/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635868562172, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.35273882785639804}}
{"text": "import Mathlib.Data.List.Basic\n\n@[simp] theorem List.toArrayAux_data : \u2200 (l : List \u03b1) a, (l.toArrayAux a).data = a.data ++ l\n| [], r => (append_nil _).symm\n| a::as, r => (toArrayAux_data as (r.push a)).trans $\n  by simp [Array.push, append_assoc, List.concat_eq_append]\n\n@[simp] theorem List.toArray_data (l : List \u03b1) : l.toArray.data = l := toArrayAux_data _ _\n\nnamespace Array\n\ntheorem ext' : {a b : Array \u03b1} \u2192 a.data = b.data \u2192 a = b\n| \u27e8a\u27e9, \u27e8_\u27e9, rfl => rfl\n\n@[simp] theorem data_toArray : (a : Array \u03b1) \u2192 a.data.toArray = a\n| \u27e8l\u27e9 => ext' l.toArray_data\n\n-- Port note: The Lean 4 core library has `toArrayLit_eq` with the same signature as this,\n-- but currently its proof is `sorry`.\ntheorem toArrayLit_eq' (a : Array \u03b1) (n : Nat) (hsz : a.size = n) : a = toArrayLit a n hsz := by\n  have := aux n\n  rw [List.drop_eq_nil_of_le (Nat.le_of_eq hsz)] at this\n  exact (data_toArray a).symm.trans $ congrArg List.toArray (this _).symm\nwhere\n  aux : \u2200 i hi, toListLitAux a n hsz i hi (a.data.drop i) = a.data\n  | 0, _ => rfl\n  | i+1, hi => by\n    simp [toListLitAux]\n    suffices _::_ = _ by rw [this]; apply aux\n    apply List.get_cons_drop\n\ntheorem get_eq_get (a : Array \u03b1) (i : Fin _) :\n  a.get i = a.data.get i := rfl\n\ntheorem get?_eq_get (a : Array \u03b1) (i : Nat) (h : i < a.size) :\n  a.get? i = some (a.get \u27e8i, h\u27e9) := by simp [get?, h]\n\ntheorem get?_len_le (a : Array \u03b1) (i : Nat) (h : a.size \u2264 i) :\n  a.get? i = none := by simp [get?, not_lt_of_ge h]\n\ntheorem get?_eq_get? (a : Array \u03b1) (i : Nat) :\n  a.get? i = a.data.get? i := by\n  simp [get?]; split <;> rename_i h\n  \u00b7 simp [get, List.get?_eq_get h]\n  \u00b7 simp [List.get?_len_le (le_of_not_lt h)]\n\ntheorem get?_push_lt (a : Array \u03b1) (x : \u03b1) (i : Nat) (h : i < a.size) :\n  (a.push x).get? i = some (a.get \u27e8i, h\u27e9) := by\n  simp [push, get?_eq_get?, \u2190 List.get?_eq_get, get_eq_get, List.concat_eq_append]\n  exact List.get?_append h\n\ntheorem get?_push_eq (a : Array \u03b1) (x : \u03b1) :\n  (a.push x).get? a.size = some x := by\n  simp [push, get?_eq_get?, \u2190 List.get?_eq_get, get_eq_get, List.concat_eq_append]\n\ntheorem get_push (a : Array \u03b1) (x : \u03b1) (i) :\n  (a.push x).get i = if h : i < a.size then a.get \u27e8i, h\u27e9 else x := by\n  split <;> (rename_i h; apply Option.some.inj; rw [\u2190 get?_eq_get])\n  \u00b7 apply get?_push_lt\n  \u00b7 match i with | \u27e8i, hi\u27e9 => ?_\n    simp at hi \u22a2\n    rw [le_antisymm (Nat.le_of_lt_succ hi) (le_of_not_lt h), get?_push_eq]\n\n@[simp] lemma get?_set_eq (a : Array \u03b1) (i) (v : \u03b1) : (a.set i v).get? i = v := by\n  simp [set, get?_eq_get?, List.get?_set_of_lt _ i.2]\n\n@[simp] lemma get?_set_ne (a : Array \u03b1) {i j} (v : \u03b1)\n  (h : i.1 \u2260 j) : (a.set i v).get? j = a.get? j := by\n  simp [set, get?_eq_get?, List.get?_set_ne _ _ h]\n\nlemma get?_set (a : Array \u03b1) (i j) (v : \u03b1) :\n  (a.set i v).get? j = if i.1 = j then some v else a.get? j := by\n  split; {subst j; simp}; simp_all\n\nend Array\n", "meta": {"author": "JOSHCLUNE", "repo": "Keller_reduction", "sha": "dc392b3da352fc1ffcfbecb1d4717d05f5faed4a", "save_path": "github-repos/lean/JOSHCLUNE-Keller_reduction", "path": "github-repos/lean/JOSHCLUNE-Keller_reduction/Keller_reduction-dc392b3da352fc1ffcfbecb1d4717d05f5faed4a/Lean4_Clique/Mathlib/Mathlib/Data/Array/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.35273881274522767}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl\n-/\nimport tactic.lint\nimport tactic.ext\n\nsection sigma\nvariables {\u03b1 \u03b1\u2081 \u03b1\u2082 : Type*} {\u03b2 : \u03b1 \u2192 Type*} {\u03b2\u2081 : \u03b1\u2081 \u2192 Type*} {\u03b2\u2082 : \u03b1\u2082 \u2192 Type*}\n\nnamespace sigma\n\ninstance [inhabited \u03b1] [inhabited (\u03b2 (default \u03b1))] : inhabited (sigma \u03b2) :=\n\u27e8\u27e8default \u03b1, default (\u03b2 (default \u03b1))\u27e9\u27e9\n\ninstance [h\u2081 : decidable_eq \u03b1] [h\u2082 : \u2200a, decidable_eq (\u03b2 a)] : decidable_eq (sigma \u03b2)\n| \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 := match a\u2081, b\u2081, a\u2082, b\u2082, h\u2081 a\u2081 a\u2082 with\n  | _, b\u2081, _, b\u2082, is_true (eq.refl a) :=\n    match b\u2081, b\u2082, h\u2082 a b\u2081 b\u2082 with\n    | _, _, is_true (eq.refl b) := is_true rfl\n    | b\u2081, b\u2082, is_false n := is_false (assume h, sigma.no_confusion h (\u03bbe\u2081 e\u2082, n $ eq_of_heq e\u2082))\n    end\n  | a\u2081, _, a\u2082, _, is_false n := is_false (assume h, sigma.no_confusion h (\u03bbe\u2081 e\u2082, n e\u2081))\n  end\n\n@[simp, nolint simp_nf] -- sometimes the built-in injectivity support does not work\ntheorem mk.inj_iff {a\u2081 a\u2082 : \u03b1} {b\u2081 : \u03b2 a\u2081} {b\u2082 : \u03b2 a\u2082} :\n  sigma.mk a\u2081 b\u2081 = \u27e8a\u2082, b\u2082\u27e9 \u2194 (a\u2081 = a\u2082 \u2227 b\u2081 == b\u2082) :=\nby simp\n\n@[simp] theorem eta : \u2200 x : \u03a3 a, \u03b2 a, sigma.mk x.1 x.2 = x\n| \u27e8i, x\u27e9 := rfl\n\n@[ext]\nlemma ext {x\u2080 x\u2081 : sigma \u03b2} (h\u2080 : x\u2080.1 = x\u2081.1) (h\u2081 : x\u2080.2 == x\u2081.2) : x\u2080 = x\u2081 :=\nby { cases x\u2080, cases x\u2081, cases h\u2080, cases h\u2081, refl }\n\nlemma ext_iff {x\u2080 x\u2081 : sigma \u03b2} : x\u2080 = x\u2081 \u2194 x\u2080.1 = x\u2081.1 \u2227 x\u2080.2 == x\u2081.2 :=\nby { cases x\u2080, cases x\u2081, exact sigma.mk.inj_iff }\n\n/-- A specialized ext lemma for equality of sigma types over an indexed subtype. -/\n@[ext]\nlemma subtype_ext {\u03b2 : Type*} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} :\n  \u2200 {x\u2080 x\u2081 : \u03a3 a, subtype (p a)}, x\u2080.fst = x\u2081.fst \u2192 (x\u2080.snd : \u03b2) = x\u2081.snd \u2192 x\u2080 = x\u2081\n| \u27e8a\u2080, b\u2080, hb\u2080\u27e9 \u27e8a\u2081, b\u2081, hb\u2081\u27e9 rfl rfl := rfl\n\nlemma subtype_ext_iff {\u03b2 : Type*} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {x\u2080 x\u2081 : \u03a3 a, subtype (p a)} :\n  x\u2080 = x\u2081 \u2194 x\u2080.fst = x\u2081.fst \u2227 (x\u2080.snd : \u03b2) = x\u2081.snd :=\n\u27e8\u03bb h, h \u25b8 \u27e8rfl, rfl\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, subtype_ext h\u2081 h\u2082\u27e9\n\n@[simp] theorem \u00abforall\u00bb {p : (\u03a3 a, \u03b2 a) \u2192 Prop} :\n  (\u2200 x, p x) \u2194 (\u2200 a b, p \u27e8a, b\u27e9) :=\n\u27e8assume h a b, h \u27e8a, b\u27e9, assume h \u27e8a, b\u27e9, h a b\u27e9\n\n@[simp] theorem \u00abexists\u00bb {p : (\u03a3 a, \u03b2 a) \u2192 Prop} :\n  (\u2203 x, p x) \u2194 (\u2203 a b, p \u27e8a, b\u27e9) :=\n\u27e8assume \u27e8\u27e8a, b\u27e9, h\u27e9, \u27e8a, b, h\u27e9, assume \u27e8a, b, h\u27e9, \u27e8\u27e8a, b\u27e9, h\u27e9\u27e9\n\n/-- Map the left and right components of a sigma -/\ndef map (f\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082) (f\u2082 : \u03a0a, \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)) (x : sigma \u03b2\u2081) : sigma \u03b2\u2082 :=\n\u27e8f\u2081 x.1, f\u2082 x.1 x.2\u27e9\n\nend sigma\n\nlemma sigma_mk_injective {i : \u03b1} : function.injective (@sigma.mk \u03b1 \u03b2 i)\n| _ _ rfl := rfl\n\nlemma function.injective.sigma_map {f\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082} {f\u2082 : \u03a0a, \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)}\n  (h\u2081 : function.injective f\u2081) (h\u2082 : \u2200 a, function.injective (f\u2082 a)) :\n  function.injective (sigma.map f\u2081 f\u2082)\n| \u27e8i, x\u27e9 \u27e8j, y\u27e9 h :=\nbegin\n  have : i = j, from h\u2081 (sigma.mk.inj_iff.mp h).1,\n  subst j,\n  have : x = y, from h\u2082 i (eq_of_heq (sigma.mk.inj_iff.mp h).2),\n  subst y\nend\n\nlemma function.surjective.sigma_map {f\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082} {f\u2082 : \u03a0a, \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)}\n  (h\u2081 : function.surjective f\u2081) (h\u2082 : \u2200 a, function.surjective (f\u2082 a)) :\n  function.surjective (sigma.map f\u2081 f\u2082) :=\nbegin\n  intros y,\n  cases y with j y,\n  cases h\u2081 j with i hi,\n  subst j,\n  cases h\u2082 i y with x hx,\n  subst y,\n  exact \u27e8\u27e8i, x\u27e9, rfl\u27e9\nend\n\n/-- Interpret a function on `\u03a3 x : \u03b1, \u03b2 x` as a dependent function with two arguments. -/\ndef sigma.curry {\u03b3 : \u03a0 a, \u03b2 a \u2192 Type*} (f : \u03a0 x : sigma \u03b2, \u03b3 x.1 x.2) (x : \u03b1) (y : \u03b2 x) : \u03b3 x y :=\nf \u27e8x,y\u27e9\n\n/-- Interpret a dependent function with two arguments as a function on `\u03a3 x : \u03b1, \u03b2 x` -/\ndef sigma.uncurry {\u03b3 : \u03a0 a, \u03b2 a \u2192 Type*} (f : \u03a0 x (y : \u03b2 x), \u03b3 x y) (x : sigma \u03b2) : \u03b3 x.1 x.2 :=\nf x.1 x.2\n\n/-- Convert a product type to a \u03a3-type. -/\n@[simp]\ndef prod.to_sigma {\u03b1 \u03b2} : \u03b1 \u00d7 \u03b2 \u2192 \u03a3 _ : \u03b1, \u03b2\n| \u27e8x,y\u27e9 := \u27e8x,y\u27e9\n\n@[simp]\nlemma prod.fst_to_sigma {\u03b1 \u03b2} (x : \u03b1 \u00d7 \u03b2) : (prod.to_sigma x).fst = x.fst :=\nby cases x; refl\n\n@[simp]\nlemma prod.snd_to_sigma {\u03b1 \u03b2} (x : \u03b1 \u00d7 \u03b2) : (prod.to_sigma x).snd = x.snd :=\nby cases x; refl\n\nend sigma\n\nsection psigma\nvariables {\u03b1 : Sort*} {\u03b2 : \u03b1 \u2192 Sort*}\n\nnamespace psigma\n\n/-- Nondependent eliminator for `psigma`. -/\ndef elim {\u03b3} (f : \u2200 a, \u03b2 a \u2192 \u03b3) (a : psigma \u03b2) : \u03b3 :=\npsigma.cases_on a f\n\n@[simp] theorem elim_val {\u03b3} (f : \u2200 a, \u03b2 a \u2192 \u03b3) (a b) : psigma.elim f \u27e8a, b\u27e9 = f a b := rfl\n\ninstance [inhabited \u03b1] [inhabited (\u03b2 (default \u03b1))] : inhabited (psigma \u03b2) :=\n\u27e8\u27e8default \u03b1, default (\u03b2 (default \u03b1))\u27e9\u27e9\n\ninstance [h\u2081 : decidable_eq \u03b1] [h\u2082 : \u2200a, decidable_eq (\u03b2 a)] : decidable_eq (psigma \u03b2)\n| \u27e8a\u2081, b\u2081\u27e9 \u27e8a\u2082, b\u2082\u27e9 := match a\u2081, b\u2081, a\u2082, b\u2082, h\u2081 a\u2081 a\u2082 with\n  | _, b\u2081, _, b\u2082, is_true (eq.refl a) :=\n    match b\u2081, b\u2082, h\u2082 a b\u2081 b\u2082 with\n    | _, _, is_true (eq.refl b) := is_true rfl\n    | b\u2081, b\u2082, is_false n := is_false (assume h, psigma.no_confusion h (\u03bbe\u2081 e\u2082, n $ eq_of_heq e\u2082))\n    end\n  | a\u2081, _, a\u2082, _, is_false n := is_false (assume h, psigma.no_confusion h (\u03bbe\u2081 e\u2082, n e\u2081))\n  end\n\ntheorem mk.inj_iff {a\u2081 a\u2082 : \u03b1} {b\u2081 : \u03b2 a\u2081} {b\u2082 : \u03b2 a\u2082} :\n  @psigma.mk \u03b1 \u03b2 a\u2081 b\u2081 = @psigma.mk \u03b1 \u03b2 a\u2082 b\u2082 \u2194 (a\u2081 = a\u2082 \u2227 b\u2081 == b\u2082) :=\niff.intro psigma.mk.inj $\n  assume \u27e8h\u2081, h\u2082\u27e9, match a\u2081, a\u2082, b\u2081, b\u2082, h\u2081, h\u2082 with _, _, _, _, eq.refl a, heq.refl b := rfl end\n\n@[ext]\nlemma ext {x\u2080 x\u2081 : psigma \u03b2} (h\u2080 : x\u2080.1 = x\u2081.1) (h\u2081 : x\u2080.2 == x\u2081.2) : x\u2080 = x\u2081 :=\nby { cases x\u2080, cases x\u2081, cases h\u2080, cases h\u2081, refl }\n\nlemma ext_iff {x\u2080 x\u2081 : psigma \u03b2} : x\u2080 = x\u2081 \u2194 x\u2080.1 = x\u2081.1 \u2227 x\u2080.2 == x\u2081.2 :=\nby { cases x\u2080, cases x\u2081, exact psigma.mk.inj_iff }\n\n/-- A specialized ext lemma for equality of psigma types over an indexed subtype. -/\n@[ext]\nlemma subtype_ext {\u03b2 : Sort*} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} :\n  \u2200 {x\u2080 x\u2081 : \u03a3' a, subtype (p a)}, x\u2080.fst = x\u2081.fst \u2192 (x\u2080.snd : \u03b2) = x\u2081.snd \u2192 x\u2080 = x\u2081\n| \u27e8a\u2080, b\u2080, hb\u2080\u27e9 \u27e8a\u2081, b\u2081, hb\u2081\u27e9 rfl rfl := rfl\n\nlemma subtype_ext_iff {\u03b2 : Sort*} {p : \u03b1 \u2192 \u03b2 \u2192 Prop} {x\u2080 x\u2081 : \u03a3' a, subtype (p a)} :\n  x\u2080 = x\u2081 \u2194 x\u2080.fst = x\u2081.fst \u2227 (x\u2080.snd : \u03b2) = x\u2081.snd :=\n\u27e8\u03bb h, h \u25b8 \u27e8rfl, rfl\u27e9, \u03bb \u27e8h\u2081, h\u2082\u27e9, subtype_ext h\u2081 h\u2082\u27e9\n\nvariables {\u03b1\u2081 : Sort*} {\u03b1\u2082 : Sort*} {\u03b2\u2081 : \u03b1\u2081 \u2192 Sort*} {\u03b2\u2082 : \u03b1\u2082 \u2192 Sort*}\n\n/-- Map the left and right components of a sigma -/\ndef map (f\u2081 : \u03b1\u2081 \u2192 \u03b1\u2082) (f\u2082 : \u03a0a, \u03b2\u2081 a \u2192 \u03b2\u2082 (f\u2081 a)) : psigma \u03b2\u2081 \u2192 psigma \u03b2\u2082\n| \u27e8a, b\u27e9 := \u27e8f\u2081 a, f\u2082 a b\u27e9\n\nend psigma\n\nend psigma\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/data/sigma/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.35272912982495896}}
{"text": "opaque f (a b : Nat) : Nat\nexample : f 1 2 = f 2 1 := by\n  generalize h : f 1 = g\n  /- g : \u2115 \u2192 \u2115\n  h : f 1 = g\n  \u22a2 g 2 = f 2 1 -/\n  trace_state\n  sorry\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/1235.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.35235862385757}}
{"text": "/-\nCopyright (c) 2018 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n\nDirect sum of modules over commutative rings, indexed by a discrete type.\n-/\nimport algebra.direct_sum\nimport linear_algebra.dfinsupp\n\n/-!\n# Direct sum of modules over commutative rings, indexed by a discrete type.\n\nThis file provides constructors for finite direct sums of modules.\nIt provides a construction of the direct sum using the universal property and proves\nits uniqueness.\n\n## Implementation notes\n\nAll of this file assumes that\n* `R` is a commutative ring,\n* `\u03b9` is a discrete type,\n* `S` is a finite set in `\u03b9`,\n* `M` is a family of `R` modules indexed over `\u03b9`.\n-/\n\nuniverses u v w u\u2081\n\nvariables (R : Type u) [semiring R]\nvariables (\u03b9 : Type v) [dec_\u03b9 : decidable_eq \u03b9] (M : \u03b9 \u2192 Type w)\nvariables [\u03a0 i, add_comm_monoid (M i)] [\u03a0 i, module R (M i)]\ninclude R\n\nnamespace direct_sum\nopen_locale direct_sum\n\nvariables {R \u03b9 M}\n\ninstance : module R (\u2a01 i, M i) := dfinsupp.module\ninstance {S : Type*} [semiring S] [\u03a0 i, module S (M i)] [\u03a0 i, smul_comm_class R S (M i)] :\n  smul_comm_class R S (\u2a01 i, M i) := dfinsupp.smul_comm_class\ninstance {S : Type*} [semiring S] [has_scalar R S] [\u03a0 i, module S (M i)]\n  [\u03a0 i, is_scalar_tower R S (M i)] :\n  is_scalar_tower R S (\u2a01 i, M i) := dfinsupp.is_scalar_tower\n\nlemma smul_apply (b : R) (v : \u2a01 i, M i) (i : \u03b9) :\n  (b \u2022 v) i = b \u2022 (v i) := dfinsupp.smul_apply _ _ _\n\ninclude dec_\u03b9\n\nvariables R \u03b9 M\n/-- Create the direct sum given a family `M` of `R` modules indexed over `\u03b9`. -/\ndef lmk : \u03a0 s : finset \u03b9, (\u03a0 i : (\u2191s : set \u03b9), M i.val) \u2192\u2097[R] (\u2a01 i, M i) :=\ndfinsupp.lmk\n\n/-- Inclusion of each component into the direct sum. -/\ndef lof : \u03a0 i : \u03b9, M i \u2192\u2097[R] (\u2a01 i, M i) :=\ndfinsupp.lsingle\nvariables {\u03b9 M}\n\nlemma single_eq_lof (i : \u03b9) (b : M i) :\n  dfinsupp.single i b = lof R \u03b9 M i b := rfl\n\n/-- Scalar multiplication commutes with direct sums. -/\ntheorem mk_smul (s : finset \u03b9) (c : R) (x) : mk M s (c \u2022 x) = c \u2022 mk M s x :=\n(lmk R \u03b9 M s).map_smul c x\n\n/-- Scalar multiplication commutes with the inclusion of each component into the direct sum. -/\ntheorem of_smul (i : \u03b9) (c : R) (x) : of M i (c \u2022 x) = c \u2022 of M i x :=\n(lof R \u03b9 M i).map_smul c x\n\nvariables {R}\nlemma support_smul [\u03a0 (i : \u03b9) (x : M i), decidable (x \u2260 0)]\n  (c : R) (v : \u2a01 i, M i) : (c \u2022 v).support \u2286 v.support := dfinsupp.support_smul _ _\n\nvariables {N : Type u\u2081} [add_comm_monoid N] [module R N]\nvariables (\u03c6 : \u03a0 i, M i \u2192\u2097[R] N)\n\nvariables (R \u03b9 N \u03c6)\n/-- The linear map constructed using the universal property of the coproduct. -/\ndef to_module : (\u2a01 i, M i) \u2192\u2097[R] N :=\ndfinsupp.lsum \u2115 \u03c6\n\nvariables {\u03b9 N \u03c6}\n\n/-- The map constructed using the universal property gives back the original maps when\nrestricted to each component. -/\n@[simp] lemma to_module_lof (i) (x : M i) : to_module R \u03b9 N \u03c6 (lof R \u03b9 M i x) = \u03c6 i x :=\nto_add_monoid_of (\u03bb i, (\u03c6 i).to_add_monoid_hom) i x\n\nvariables (\u03c8 : (\u2a01 i, M i) \u2192\u2097[R] N)\n\n/-- Every linear map from a direct sum agrees with the one obtained by applying\nthe universal property to each of its components. -/\ntheorem to_module.unique (f : \u2a01 i, M i) : \u03c8 f = to_module R \u03b9 N (\u03bb i, \u03c8.comp $ lof R \u03b9 M i) f :=\nto_add_monoid.unique \u03c8.to_add_monoid_hom f\n\nvariables {\u03c8} {\u03c8' : (\u2a01 i, M i) \u2192\u2097[R] N}\n\ntheorem to_module.ext (H : \u2200 i, \u03c8.comp (lof R \u03b9 M i) = \u03c8'.comp (lof R \u03b9 M i)) (f : \u2a01 i, M i) :\n  \u03c8 f = \u03c8' f :=\nby rw dfinsupp.lhom_ext' H\n\n/--\nThe inclusion of a subset of the direct summands\ninto a larger subset of the direct summands, as a linear map.\n-/\ndef lset_to_set (S T : set \u03b9) (H : S \u2286 T) :\n  (\u2a01 (i : S), M i) \u2192\u2097 (\u2a01 (i : T), M i) :=\nto_module R _ _ $ \u03bb i, lof R T (\u03bb (i : subtype T), M i) \u27e8i, H i.prop\u27e9\n\nomit dec_\u03b9\n\n/-- The natural linear equivalence between `\u2a01 _ : \u03b9, M` and `M` when `unique \u03b9`. -/\nprotected def lid (M : Type v) (\u03b9 : Type* := punit) [add_comm_monoid M] [module R M]\n  [unique \u03b9] :\n  (\u2a01 (_ : \u03b9), M) \u2243\u2097 M :=\n{ .. direct_sum.id M \u03b9,\n  .. to_module R \u03b9 M (\u03bb i, linear_map.id) }\n\nvariables (\u03b9 M)\n/-- The projection map onto one component, as a linear map. -/\ndef component (i : \u03b9) : (\u2a01 i, M i) \u2192\u2097[R] M i :=\ndfinsupp.lapply i\n\nvariables {\u03b9 M}\n\nlemma apply_eq_component (f : \u2a01 i, M i) (i : \u03b9) :\n  f i = component R \u03b9 M i f := rfl\n\n@[ext] lemma ext {f g : \u2a01 i, M i}\n  (h : \u2200 i, component R \u03b9 M i f = component R \u03b9 M i g) : f = g :=\ndfinsupp.ext h\n\nlemma ext_iff {f g : \u2a01 i, M i} : f = g \u2194\n  \u2200 i, component R \u03b9 M i f = component R \u03b9 M i g :=\n\u27e8\u03bb h _, by rw h, ext R\u27e9\n\ninclude dec_\u03b9\n\n@[simp] lemma lof_apply (i : \u03b9) (b : M i) : ((lof R \u03b9 M i) b) i = b :=\ndfinsupp.single_eq_same\n\n@[simp] lemma component.lof_self (i : \u03b9) (b : M i) :\n  component R \u03b9 M i ((lof R \u03b9 M i) b) = b :=\nlof_apply R i b\n\nlemma component.of (i j : \u03b9) (b : M j) :\n  component R \u03b9 M i ((lof R \u03b9 M j) b) =\n  if h : j = i then eq.rec_on h b else 0 :=\ndfinsupp.single_apply\n\n/-- The `direct_sum` formed by a collection of `submodule`s of `M` is said to be internal if the\ncanonical map `(\u2a01 i, A i) \u2192\u2097[R] M` is bijective. -/\ndef submodule_is_internal {R M : Type*}\n  [semiring R] [add_comm_monoid M] [module R M]\n  (A : \u03b9 \u2192 submodule R M) : Prop :=\nfunction.bijective (to_module R \u03b9 M (\u03bb i, (A i).subtype))\n\nlemma submodule_is_internal.to_add_submonoid {R M : Type*}\n  [semiring R] [add_comm_monoid M] [module R M] (A : \u03b9 \u2192 submodule R M) :\n  submodule_is_internal A \u2194 add_submonoid_is_internal (\u03bb i, (A i).to_add_submonoid) :=\niff.rfl\n\nlemma submodule_is_internal.to_add_subgroup {R M : Type*}\n  [ring R] [add_comm_group M] [module R M] (A : \u03b9 \u2192 submodule R M) :\n  submodule_is_internal A \u2194 add_subgroup_is_internal (\u03bb i, (A i).to_add_subgroup) :=\niff.rfl\n\nend direct_sum\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/linear_algebra/direct_sum_module.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593312018546, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3523235188660187}}
{"text": "/-\nCopyright (c) 2014 Robert Lewis. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Robert Lewis, Leonardo de Moura, Johannes H\u00f6lzl, Mario Carneiro\n\n! This file was ported from Lean 3 source module algebra.field.basic\n! leanprover-community/mathlib commit 05101c3df9d9cfe9430edc205860c79b6d660102\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Algebra.Field.Defs\nimport Mathbin.Algebra.GroupWithZero.Units.Lemmas\nimport Mathbin.Algebra.Hom.Ring\nimport Mathbin.Algebra.Ring.Commute\n\n/-!\n# Lemmas about division (semi)rings and (semi)fields\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\n-/\n\n\nopen Function OrderDual Set\n\nuniverse u\n\nvariable {\u03b1 \u03b2 K : Type _}\n\nsection DivisionSemiring\n\nvariable [DivisionSemiring \u03b1] {a b c d : \u03b1}\n\n/- warning: add_div -> add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) c) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a c) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) b c))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) c) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a c) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) b c))\nCase conversion may be inaccurate. Consider using '#align add_div add_div\u2093'. -/\ntheorem add_div (a b c : \u03b1) : (a + b) / c = a / c + b / c := by simp_rw [div_eq_mul_inv, add_mul]\n#align add_div add_div\n\n/- warning: div_add_div_same -> div_add_div_same is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a c) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) b c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) c)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a c) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) b c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) c)\nCase conversion may be inaccurate. Consider using '#align div_add_div_same div_add_div_same\u2093'. -/\n@[field_simps]\ntheorem div_add_div_same (a b c : \u03b1) : a / c + b / c = (a + b) / c :=\n  (add_div _ _ _).symm\n#align div_add_div_same div_add_div_same\n\n/- warning: same_add_div -> same_add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b a) b) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b a) b) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a b)))\nCase conversion may be inaccurate. Consider using '#align same_add_div same_add_div\u2093'. -/\ntheorem same_add_div (h : b \u2260 0) : (b + a) / b = 1 + a / b := by rw [\u2190 div_self h, add_div]\n#align same_add_div same_add_div\n\n/- warning: div_add_same -> div_add_same is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) b) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a b) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) b) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a b) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align div_add_same div_add_same\u2093'. -/\ntheorem div_add_same (h : b \u2260 0) : (a + b) / b = a / b + 1 := by rw [\u2190 div_self h, add_div]\n#align div_add_same div_add_same\n\n/- warning: one_add_div -> one_add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b a) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b a) b))\nCase conversion may be inaccurate. Consider using '#align one_add_div one_add_div\u2093'. -/\ntheorem one_add_div (h : b \u2260 0) : 1 + a / b = (b + a) / b :=\n  (same_add_div h).symm\n#align one_add_div one_add_div\n\n/- warning: div_add_one -> div_add_one is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a b) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) b))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a b) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) b))\nCase conversion may be inaccurate. Consider using '#align div_add_one div_add_one\u2093'. -/\ntheorem div_add_one (h : b \u2260 0) : a / b + 1 = (a + b) / b :=\n  (div_add_same h).symm\n#align div_add_one div_add_one\n\n/- warning: one_div_mul_add_mul_one_div_eq_one_div_add_one_div -> one_div_mul_add_mul_one_div_eq_one_div_add_one_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) a) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) b)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) a) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) b)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) b)))\nCase conversion may be inaccurate. Consider using '#align one_div_mul_add_mul_one_div_eq_one_div_add_one_div one_div_mul_add_mul_one_div_eq_one_div_add_one_div\u2093'. -/\ntheorem one_div_mul_add_mul_one_div_eq_one_div_add_one_div (ha : a \u2260 0) (hb : b \u2260 0) :\n    1 / a * (a + b) * (1 / b) = 1 / a + 1 / b := by\n  rw [mul_add, one_div_mul_cancel ha, add_mul, one_mul, mul_assoc, mul_one_div_cancel hb, mul_one,\n    add_comm]\n#align one_div_mul_add_mul_one_div_eq_one_div_add_one_div one_div_mul_add_mul_one_div_eq_one_div_add_one_div\n\n/- warning: add_div_eq_mul_add_div -> add_div_eq_mul_add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {c : \u03b1} (a : \u03b1) (b : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) b c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a c) b) c))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {c : \u03b1} (a : \u03b1) (b : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) b c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a c) b) c))\nCase conversion may be inaccurate. Consider using '#align add_div_eq_mul_add_div add_div_eq_mul_add_div\u2093'. -/\ntheorem add_div_eq_mul_add_div (a b : \u03b1) (hc : c \u2260 0) : a + b / c = (a * c + b) / c :=\n  (eq_div_iff_mul_eq hc).2 <| by rw [right_distrib, div_mul_cancel _ hc]\n#align add_div_eq_mul_add_div add_div_eq_mul_add_div\n\n/- warning: add_div' -> add_div' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b c) a) c))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a c)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b c) a) c))\nCase conversion may be inaccurate. Consider using '#align add_div' add_div'\u2093'. -/\n@[field_simps]\ntheorem add_div' (a b c : \u03b1) (hc : c \u2260 0) : b + a / c = (b * c + a) / c := by\n  rw [add_div, mul_div_cancel _ hc]\n#align add_div' add_div'\n\n/- warning: div_add' -> div_add' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a c) b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b c)) c))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] (a : \u03b1) (b : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 c (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a c) b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b c)) c))\nCase conversion may be inaccurate. Consider using '#align div_add' div_add'\u2093'. -/\n@[field_simps]\ntheorem div_add' (a b c : \u03b1) (hc : c \u2260 0) : a / c + b = (a + b * c) / c := by\n  rwa [add_comm, add_div', add_comm]\n#align div_add' div_add'\n\n/- warning: commute.div_add_div -> Commute.div_add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1} {c : \u03b1} {d : \u03b1}, (Commute.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b c) -> (Commute.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b d) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Ne.{succ u1} \u03b1 d (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) c d)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b c)) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) b d)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1} {c : \u03b1} {d : \u03b1}, (Commute.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) b c) -> (Commute.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) b d) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 d (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) a b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) c d)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a d) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b c)) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) b d)))\nCase conversion may be inaccurate. Consider using '#align commute.div_add_div Commute.div_add_div\u2093'. -/\nprotected theorem Commute.div_add_div (hbc : Commute b c) (hbd : Commute b d) (hb : b \u2260 0)\n    (hd : d \u2260 0) : a / b + c / d = (a * d + b * c) / (b * d) := by\n  rw [add_div, mul_div_mul_right _ b hd, hbc.eq, hbd.eq, mul_div_mul_right c d hb]\n#align commute.div_add_div Commute.div_add_div\n\n/- warning: commute.one_div_add_one_div -> Commute.one_div_add_one_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Commute.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a b) -> (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Commute.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) a b) -> (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a b)))\nCase conversion may be inaccurate. Consider using '#align commute.one_div_add_one_div Commute.one_div_add_one_div\u2093'. -/\nprotected theorem Commute.one_div_add_one_div (hab : Commute a b) (ha : a \u2260 0) (hb : b \u2260 0) :\n    1 / a + 1 / b = (a + b) / (a * b) := by\n  rw [(Commute.one_right a).div_add_div hab ha hb, one_mul, mul_one, add_comm]\n#align commute.one_div_add_one_div Commute.one_div_add_one_div\n\n/- warning: commute.inv_add_inv -> Commute.inv_add_inv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Commute.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a b) -> (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (Inv.inv.{u1} \u03b1 (DivInvMonoid.toHasInv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1))) a) (Inv.inv.{u1} \u03b1 (DivInvMonoid.toHasInv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 _inst_1)))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : DivisionSemiring.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Commute.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))) a b) -> (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (MonoidWithZero.toZero.{u1} \u03b1 (Semiring.toMonoidWithZero.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) (Inv.inv.{u1} \u03b1 (DivisionSemiring.toInv.{u1} \u03b1 _inst_1) a) (Inv.inv.{u1} \u03b1 (DivisionSemiring.toInv.{u1} \u03b1 _inst_1) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivisionSemiring.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1)))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 _inst_1))))) a b)))\nCase conversion may be inaccurate. Consider using '#align commute.inv_add_inv Commute.inv_add_inv\u2093'. -/\nprotected theorem Commute.inv_add_inv (hab : Commute a b) (ha : a \u2260 0) (hb : b \u2260 0) :\n    a\u207b\u00b9 + b\u207b\u00b9 = (a + b) / (a * b) := by\n  rw [inv_eq_one_div, inv_eq_one_div, hab.one_div_add_one_div ha hb]\n#align commute.inv_add_inv Commute.inv_add_inv\n\nend DivisionSemiring\n\nsection DivisionMonoid\n\nvariable [DivisionMonoid K] [HasDistribNeg K] {a b : K}\n\n/- warning: one_div_neg_one_eq_neg_one -> one_div_neg_one_eq_neg_one is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))], Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (MulOneClass.toHasOne.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (MulOneClass.toHasOne.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))))))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (MulOneClass.toHasOne.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))))))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))], Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (InvOneClass.toOne.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (InvOneClass.toOne.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))))))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (InvOneClass.toOne.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))))))\nCase conversion may be inaccurate. Consider using '#align one_div_neg_one_eq_neg_one one_div_neg_one_eq_neg_one\u2093'. -/\ntheorem one_div_neg_one_eq_neg_one : (1 : K) / -1 = -1 :=\n  have : -1 * -1 = (1 : K) := by rw [neg_mul_neg, one_mul]\n  Eq.symm (eq_one_div_of_mul_eq_one_right this)\n#align one_div_neg_one_eq_neg_one one_div_neg_one_eq_neg_one\n\n/- warning: one_div_neg_eq_neg_one_div -> one_div_neg_eq_neg_one_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (MulOneClass.toHasOne.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (MulOneClass.toHasOne.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))))) a))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (InvOneClass.toOne.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (InvOneClass.toOne.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))))) a))\nCase conversion may be inaccurate. Consider using '#align one_div_neg_eq_neg_one_div one_div_neg_eq_neg_one_div\u2093'. -/\ntheorem one_div_neg_eq_neg_one_div (a : K) : 1 / -a = -(1 / a) :=\n  calc\n    1 / -a = 1 / (-1 * a) := by rw [neg_eq_neg_one_mul]\n    _ = 1 / a * (1 / -1) := by rw [one_div_mul_one_div_rev]\n    _ = 1 / a * -1 := by rw [one_div_neg_one_eq_neg_one]\n    _ = -(1 / a) := by rw [mul_neg, mul_one]\n    \n#align one_div_neg_eq_neg_one_div one_div_neg_eq_neg_one_div\n\n/- warning: div_neg_eq_neg_div -> div_neg_eq_neg_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a))\nCase conversion may be inaccurate. Consider using '#align div_neg_eq_neg_div div_neg_eq_neg_div\u2093'. -/\ntheorem div_neg_eq_neg_div (a b : K) : b / -a = -(b / a) :=\n  calc\n    b / -a = b * (1 / -a) := by rw [\u2190 inv_eq_one_div, division_def]\n    _ = b * -(1 / a) := by rw [one_div_neg_eq_neg_one_div]\n    _ = -(b * (1 / a)) := by rw [neg_mul_eq_mul_neg]\n    _ = -(b / a) := by rw [mul_one_div]\n    \n#align div_neg_eq_neg_div div_neg_eq_neg_div\n\n/- warning: neg_div -> neg_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b) a) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b) a) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a))\nCase conversion may be inaccurate. Consider using '#align neg_div neg_div\u2093'. -/\ntheorem neg_div (a b : K) : -b / a = -(b / a) := by\n  rw [neg_eq_neg_one_mul, mul_div_assoc, \u2190 neg_eq_neg_one_mul]\n#align neg_div neg_div\n\n/- warning: neg_div' -> neg_div' is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b) a)\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) b a)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b) a)\nCase conversion may be inaccurate. Consider using '#align neg_div' neg_div'\u2093'. -/\n@[field_simps]\ntheorem neg_div' (a b : K) : -(b / a) = -b / a := by simp [neg_div]\n#align neg_div' neg_div'\n\n/- warning: neg_div_neg_eq -> neg_div_neg_eq is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a b)\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] (a : K) (b : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a b)\nCase conversion may be inaccurate. Consider using '#align neg_div_neg_eq neg_div_neg_eq\u2093'. -/\ntheorem neg_div_neg_eq (a b : K) : -a / -b = a / b := by rw [div_neg_eq_neg_div, neg_div, neg_neg]\n#align neg_div_neg_eq neg_div_neg_eq\n\n/- warning: neg_inv -> neg_inv is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {a : K}, Eq.{succ u1} K (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)) a)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {a : K}, Eq.{succ u1} K (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (Inv.inv.{u1} K (InvOneClass.toInv.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))) a)) (Inv.inv.{u1} K (InvOneClass.toInv.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a))\nCase conversion may be inaccurate. Consider using '#align neg_inv neg_inv\u2093'. -/\ntheorem neg_inv : -a\u207b\u00b9 = (-a)\u207b\u00b9 := by rw [inv_eq_one_div, inv_eq_one_div, div_neg_eq_neg_div]\n#align neg_inv neg_inv\n\n/- warning: div_neg -> div_neg is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {b : K} (a : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a b))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {b : K} (a : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) b)) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toDiv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))) a b))\nCase conversion may be inaccurate. Consider using '#align div_neg div_neg\u2093'. -/\ntheorem div_neg (a : K) : a / -b = -(a / b) := by rw [\u2190 div_neg_eq_neg_div]\n#align div_neg div_neg\n\n/- warning: inv_neg -> inv_neg is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {a : K}, Eq.{succ u1} K (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toHasNeg.{u1} K (HasDistribNeg.toHasInvolutiveNeg.{u1} K (MulOneClass.toHasMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)) a))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionMonoid.{u1} K] [_inst_2 : HasDistribNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1))))] {a : K}, Eq.{succ u1} K (Inv.inv.{u1} K (InvOneClass.toInv.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) a)) (Neg.neg.{u1} K (InvolutiveNeg.toNeg.{u1} K (HasDistribNeg.toInvolutiveNeg.{u1} K (MulOneClass.toMul.{u1} K (Monoid.toMulOneClass.{u1} K (DivInvMonoid.toMonoid.{u1} K (DivisionMonoid.toDivInvMonoid.{u1} K _inst_1)))) _inst_2)) (Inv.inv.{u1} K (InvOneClass.toInv.{u1} K (DivInvOneMonoid.toInvOneClass.{u1} K (DivisionMonoid.toDivInvOneMonoid.{u1} K _inst_1))) a))\nCase conversion may be inaccurate. Consider using '#align inv_neg inv_neg\u2093'. -/\ntheorem inv_neg : (-a)\u207b\u00b9 = -a\u207b\u00b9 := by rw [neg_inv]\n#align inv_neg inv_neg\n\nend DivisionMonoid\n\nsection DivisionRing\n\nvariable [DivisionRing K] {a b c d : K}\n\n/- warning: div_neg_self -> div_neg_self is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) a)) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))))))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) a)) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))\nCase conversion may be inaccurate. Consider using '#align div_neg_self div_neg_self\u2093'. -/\n@[simp]\ntheorem div_neg_self {a : K} (h : a \u2260 0) : a / -a = -1 := by rw [div_neg_eq_neg_div, div_self h]\n#align div_neg_self div_neg_self\n\n/- warning: neg_div_self -> neg_div_self is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) a) a) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))))))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) a) a) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))\nCase conversion may be inaccurate. Consider using '#align neg_div_self neg_div_self\u2093'. -/\n@[simp]\ntheorem neg_div_self {a : K} (h : a \u2260 0) : -a / a = -1 := by rw [neg_div, div_self h]\n#align neg_div_self neg_div_self\n\n/- warning: div_sub_div_same -> div_sub_div_same is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] (a : K) (b : K) (c : K), Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a c) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) b c)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) a b) c)\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] (a : K) (b : K) (c : K), Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a c) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) b c)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) a b) c)\nCase conversion may be inaccurate. Consider using '#align div_sub_div_same div_sub_div_same\u2093'. -/\ntheorem div_sub_div_same (a b c : K) : a / c - b / c = (a - b) / c := by\n  rw [sub_eq_add_neg, \u2190 neg_div, div_add_div_same, sub_eq_add_neg]\n#align div_sub_div_same div_sub_div_same\n\n/- warning: same_sub_div -> same_sub_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) b a) b) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a b)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b a) b) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a b)))\nCase conversion may be inaccurate. Consider using '#align same_sub_div same_sub_div\u2093'. -/\ntheorem same_sub_div {a b : K} (h : b \u2260 0) : (b - a) / b = 1 - a / b := by\n  simpa only [\u2190 @div_self _ _ b h] using (div_sub_div_same b a b).symm\n#align same_sub_div same_sub_div\n\n/- warning: one_sub_div -> one_sub_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) b a) b))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b a) b))\nCase conversion may be inaccurate. Consider using '#align one_sub_div one_sub_div\u2093'. -/\ntheorem one_sub_div {a b : K} (h : b \u2260 0) : 1 - a / b = (b - a) / b :=\n  (same_sub_div h).symm\n#align one_sub_div one_sub_div\n\n/- warning: div_sub_same -> div_sub_same is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) a b) b) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a b) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))))))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) a b) b) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a b) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))\nCase conversion may be inaccurate. Consider using '#align div_sub_same div_sub_same\u2093'. -/\ntheorem div_sub_same {a b : K} (h : b \u2260 0) : (a - b) / b = a / b - 1 := by\n  simpa only [\u2190 @div_self _ _ b h] using (div_sub_div_same a b b).symm\n#align div_sub_same div_sub_same\n\n/- warning: div_sub_one -> div_sub_one is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a b) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) a b) b))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a b) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) a b) b))\nCase conversion may be inaccurate. Consider using '#align div_sub_one div_sub_one\u2093'. -/\ntheorem div_sub_one {a b : K} (h : b \u2260 0) : a / b - 1 = (a - b) / b :=\n  (div_sub_same h).symm\n#align div_sub_one div_sub_one\n\n/- warning: sub_div -> sub_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] (a : K) (b : K) (c : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) a b) c) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a c) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) b c))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] (a : K) (b : K) (c : K), Eq.{succ u1} K (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) a b) c) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a c) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) b c))\nCase conversion may be inaccurate. Consider using '#align sub_div sub_div\u2093'. -/\ntheorem sub_div (a b c : K) : (a - b) / c = a / c - b / c :=\n  (div_sub_div_same _ _ _).symm\n#align sub_div sub_div\n\n/- warning: inv_sub_inv' -> inv_sub_inv' is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) a) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) b)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) a) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) b a)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) b)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) a) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) b)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) a) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b a)) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) b)))\nCase conversion may be inaccurate. Consider using '#align inv_sub_inv' inv_sub_inv'\u2093'. -/\n/-- See `inv_sub_inv` for the more convenient version when `K` is commutative. -/\ntheorem inv_sub_inv' {a b : K} (ha : a \u2260 0) (hb : b \u2260 0) : a\u207b\u00b9 - b\u207b\u00b9 = a\u207b\u00b9 * (b - a) * b\u207b\u00b9 := by\n  rw [mul_sub, sub_mul, mul_inv_cancel_right\u2080 hb, inv_mul_cancel ha, one_mul]\n#align inv_sub_inv' inv_sub_inv'\n\n/- warning: one_div_mul_sub_mul_one_div_eq_one_div_add_one_div -> one_div_mul_sub_mul_one_div_eq_one_div_add_one_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) a) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) b a)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) b)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) a) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) b)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) a) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b a)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) b)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) a) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) b)))\nCase conversion may be inaccurate. Consider using '#align one_div_mul_sub_mul_one_div_eq_one_div_add_one_div one_div_mul_sub_mul_one_div_eq_one_div_add_one_div\u2093'. -/\ntheorem one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (ha : a \u2260 0) (hb : b \u2260 0) :\n    1 / a * (b - a) * (1 / b) = 1 / a - 1 / b := by\n  rw [mul_sub_left_distrib (1 / a), one_div_mul_cancel ha, mul_sub_right_distrib, one_mul,\n    mul_assoc, mul_one_div_cancel hb, mul_one]\n#align one_div_mul_sub_mul_one_div_eq_one_div_add_one_div one_div_mul_sub_mul_one_div_eq_one_div_add_one_div\n\n#print DivisionRing.isDomain /-\n-- see Note [lower instance priority]\ninstance (priority := 100) DivisionRing.isDomain : IsDomain K :=\n  NoZeroDivisors.to_isDomain _\n#align division_ring.is_domain DivisionRing.isDomain\n-/\n\n/- warning: commute.div_sub_div -> Commute.div_sub_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K} {c : K} {d : K}, (Commute.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b c) -> (Commute.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b d) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Ne.{succ u1} K d (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) a b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) c d)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) a d) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) b c)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) b d)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K} {c : K} {d : K}, (Commute.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) b c) -> (Commute.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) b d) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Ne.{succ u1} K d (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) a b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) c d)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) a d) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) b c)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) b d)))\nCase conversion may be inaccurate. Consider using '#align commute.div_sub_div Commute.div_sub_div\u2093'. -/\nprotected theorem Commute.div_sub_div (hbc : Commute b c) (hbd : Commute b d) (hb : b \u2260 0)\n    (hd : d \u2260 0) : a / b - c / d = (a * d - b * c) / (b * d) := by\n  simpa only [mul_neg, neg_div, \u2190 sub_eq_add_neg] using hbc.neg_right.div_add_div hbd hb hd\n#align commute.div_sub_div Commute.div_sub_div\n\n/- warning: commute.inv_sub_inv -> Commute.inv_sub_inv is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Commute.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) a b) -> (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) a) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) b a) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) a b)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {a : K} {b : K}, (Commute.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) a b) -> (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) a) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) b a) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) a b)))\nCase conversion may be inaccurate. Consider using '#align commute.inv_sub_inv Commute.inv_sub_inv\u2093'. -/\nprotected theorem Commute.inv_sub_inv (hab : Commute a b) (ha : a \u2260 0) (hb : b \u2260 0) :\n    a\u207b\u00b9 - b\u207b\u00b9 = (b - a) / (a * b) := by\n  simp only [inv_eq_one_div, (Commute.one_right a).div_sub_div hab ha hb, one_mul, mul_one]\n#align commute.inv_sub_inv Commute.inv_sub_inv\n\nend DivisionRing\n\nsection Semifield\n\nvariable [Semifield \u03b1] {a b c d : \u03b1}\n\n/- warning: div_add_div -> div_add_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {b : \u03b1} {d : \u03b1} (a : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Ne.{succ u1} \u03b1 d (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) a b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) c d)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a d) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) b c)) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) b d)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {b : \u03b1} {d : \u03b1} (a : \u03b1) (c : \u03b1), (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 d (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) a b) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) c d)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))) b c)) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))) b d)))\nCase conversion may be inaccurate. Consider using '#align div_add_div div_add_div\u2093'. -/\ntheorem div_add_div (a : \u03b1) (c : \u03b1) (hb : b \u2260 0) (hd : d \u2260 0) :\n    a / b + c / d = (a * d + b * c) / (b * d) :=\n  (Commute.all b _).div_add_div (Commute.all _ _) hb hd\n#align div_add_div div_add_div\n\n/- warning: one_div_add_one_div -> one_div_add_one_div is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 (AddMonoidWithOne.toOne.{u1} \u03b1 (AddCommMonoidWithOne.toAddMonoidWithOne.{u1} \u03b1 (NonAssocSemiring.toAddCommMonoidWithOne.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) a) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 (Semiring.toOne.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))) a b)))\nCase conversion may be inaccurate. Consider using '#align one_div_add_one_div one_div_add_one_div\u2093'. -/\ntheorem one_div_add_one_div (ha : a \u2260 0) (hb : b \u2260 0) : 1 / a + 1 / b = (a + b) / (a * b) :=\n  (Commute.all a _).one_div_add_one_div ha hb\n#align one_div_add_one_div one_div_add_one_div\n\n/- warning: inv_add_inv -> inv_add_inv is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 (MulZeroClass.toHasZero.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (Inv.inv.{u1} \u03b1 (DivInvMonoid.toHasInv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))) a) (Inv.inv.{u1} \u03b1 (DivInvMonoid.toHasInv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (DivInvMonoid.toHasDiv.{u1} \u03b1 (GroupWithZero.toDivInvMonoid.{u1} \u03b1 (DivisionSemiring.toGroupWithZero.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toHasAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (Distrib.toHasMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : Semifield.{u1} \u03b1] {a : \u03b1} {b : \u03b1}, (Ne.{succ u1} \u03b1 a (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Ne.{succ u1} \u03b1 b (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 (CommMonoidWithZero.toZero.{u1} \u03b1 (CommGroupWithZero.toCommMonoidWithZero.{u1} \u03b1 (Semifield.toCommGroupWithZero.{u1} \u03b1 _inst_1)))))) -> (Eq.{succ u1} \u03b1 (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) (Inv.inv.{u1} \u03b1 (Semifield.toInv.{u1} \u03b1 _inst_1) a) (Inv.inv.{u1} \u03b1 (Semifield.toInv.{u1} \u03b1 _inst_1) b)) (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 (Semifield.toDiv.{u1} \u03b1 _inst_1)) (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 (Distrib.toAdd.{u1} \u03b1 (NonUnitalNonAssocSemiring.toDistrib.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1))))))) a b) (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 (NonUnitalNonAssocSemiring.toMul.{u1} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b1 (Semiring.toNonAssocSemiring.{u1} \u03b1 (DivisionSemiring.toSemiring.{u1} \u03b1 (Semifield.toDivisionSemiring.{u1} \u03b1 _inst_1)))))) a b)))\nCase conversion may be inaccurate. Consider using '#align inv_add_inv inv_add_inv\u2093'. -/\ntheorem inv_add_inv (ha : a \u2260 0) (hb : b \u2260 0) : a\u207b\u00b9 + b\u207b\u00b9 = (a + b) / (a * b) :=\n  (Commute.all a _).inv_add_inv ha hb\n#align inv_add_inv inv_add_inv\n\nend Semifield\n\nsection Field\n\nvariable [Field K]\n\nattribute [local simp] mul_assoc mul_comm mul_left_comm\n\n/- warning: div_sub_div -> div_sub_div is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) {b : K} (c : K) {d : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Ne.{succ u1} K d (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) c d)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) a d) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) b c)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) b d)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) {b : K} (c : K) {d : K}, (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Ne.{succ u1} K d (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) a b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) c d)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) a d) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) b c)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) b d)))\nCase conversion may be inaccurate. Consider using '#align div_sub_div div_sub_div\u2093'. -/\n@[field_simps]\ntheorem div_sub_div (a : K) {b : K} (c : K) {d : K} (hb : b \u2260 0) (hd : d \u2260 0) :\n    a / b - c / d = (a * d - b * c) / (b * d) :=\n  (Commute.all b _).div_sub_div (Commute.all _ _) hb hd\n#align div_sub_div div_sub_div\n\n/- warning: inv_sub_inv -> inv_sub_inv is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) a) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) b a) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) a b)))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {a : K} {b : K}, (Ne.{succ u1} K a (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Ne.{succ u1} K b (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (Inv.inv.{u1} K (Field.toInv.{u1} K _inst_1) a) (Inv.inv.{u1} K (Field.toInv.{u1} K _inst_1) b)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) b a) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) a b)))\nCase conversion may be inaccurate. Consider using '#align inv_sub_inv inv_sub_inv\u2093'. -/\ntheorem inv_sub_inv {a b : K} (ha : a \u2260 0) (hb : b \u2260 0) : a\u207b\u00b9 - b\u207b\u00b9 = (b - a) / (a * b) :=\n  (Commute.all a _).inv_sub_inv ha hb\n#align inv_sub_inv inv_sub_inv\n\n/- warning: sub_div' -> sub_div' is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) (b : K) (c : K), (Ne.{succ u1} K c (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) b (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a c)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) b c) a) c))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) (b : K) (c : K), (Ne.{succ u1} K c (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) b (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) a c)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) b c) a) c))\nCase conversion may be inaccurate. Consider using '#align sub_div' sub_div'\u2093'. -/\n@[field_simps]\ntheorem sub_div' (a b c : K) (hc : c \u2260 0) : b - a / c = (b * c - a) / c := by\n  simpa using div_sub_div b a one_ne_zero hc\n#align sub_div' sub_div'\n\n/- warning: div_sub' -> div_sub' is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) (b : K) (c : K), (Ne.{succ u1} K c (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a c) b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) a (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) c b)) c))\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] (a : K) (b : K) (c : K), (Ne.{succ u1} K c (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) a c) b) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) a (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) c b)) c))\nCase conversion may be inaccurate. Consider using '#align div_sub' div_sub'\u2093'. -/\n@[field_simps]\ntheorem div_sub' (a b c : K) (hc : c \u2260 0) : a / c - b = (a - c * b) / c := by\n  simpa using div_sub_div a b hc one_ne_zero\n#align div_sub' div_sub'\n\n#print Field.isDomain /-\n-- see Note [lower instance priority]\ninstance (priority := 100) Field.isDomain : IsDomain K :=\n  { DivisionRing.isDomain with }\n#align field.is_domain Field.isDomain\n-/\n\nend Field\n\nnamespace RingHom\n\n/- warning: ring_hom.injective -> RingHom.injective is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DivisionRing.{u1} \u03b1] [_inst_2 : Semiring.{u2} \u03b2] [_inst_3 : Nontrivial.{u2} \u03b2] (f : RingHom.{u1, u2} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 (Ring.toNonAssocRing.{u1} \u03b1 (DivisionRing.toRing.{u1} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u2} \u03b2 _inst_2)), Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 (coeFn.{max (succ u1) (succ u2), max (succ u1) (succ u2)} (RingHom.{u1, u2} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 (Ring.toNonAssocRing.{u1} \u03b1 (DivisionRing.toRing.{u1} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u2} \u03b2 _inst_2)) (fun (_x : RingHom.{u1, u2} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 (Ring.toNonAssocRing.{u1} \u03b1 (DivisionRing.toRing.{u1} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u2} \u03b2 _inst_2)) => \u03b1 -> \u03b2) (RingHom.hasCoeToFun.{u1, u2} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u1} \u03b1 (Ring.toNonAssocRing.{u1} \u03b1 (DivisionRing.toRing.{u1} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u2} \u03b2 _inst_2)) f)\nbut is expected to have type\n  forall {\u03b1 : Type.{u2}} {\u03b2 : Type.{u1}} [_inst_1 : DivisionRing.{u2} \u03b1] [_inst_2 : Semiring.{u1} \u03b2] [_inst_3 : Nontrivial.{u1} \u03b2] (f : RingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)), Function.Injective.{succ u2, succ u1} \u03b1 \u03b2 (FunLike.coe.{max (succ u2) (succ u1), succ u2, succ u1} (RingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Algebra.Hom.Group._hyg.2391 : \u03b1) => \u03b2) _x) (MulHomClass.toFunLike.{max u2 u1, u2, u1} (RingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)) \u03b1 \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b1 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b1 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))))) (NonUnitalNonAssocSemiring.toMul.{u1} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b2 (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2))) (NonUnitalRingHomClass.toMulHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)) \u03b1 \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b1 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1)))) (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u1} \u03b2 (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)) (RingHomClass.toNonUnitalRingHomClass.{max u2 u1, u2, u1} (RingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2)) \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2) (RingHom.instRingHomClassRingHom.{u2, u1} \u03b1 \u03b2 (NonAssocRing.toNonAssocSemiring.{u2} \u03b1 (Ring.toNonAssocRing.{u2} \u03b1 (DivisionRing.toRing.{u2} \u03b1 _inst_1))) (Semiring.toNonAssocSemiring.{u1} \u03b2 _inst_2))))) f)\nCase conversion may be inaccurate. Consider using '#align ring_hom.injective RingHom.injective\u2093'. -/\nprotected theorem injective [DivisionRing \u03b1] [Semiring \u03b2] [Nontrivial \u03b2] (f : \u03b1 \u2192+* \u03b2) :\n    Injective f :=\n  (injective_iff_map_eq_zero f).2 fun x => (map_eq_zero f).1\n#align ring_hom.injective RingHom.injective\n\nend RingHom\n\nsection NoncomputableDefs\n\nvariable {R : Type _} [Nontrivial R]\n\n/- warning: division_ring_of_is_unit_or_eq_zero -> divisionRingOfIsUnitOrEqZero is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Nontrivial.{u1} R] [hR : Ring.{u1} R], (forall (a : R), Or (IsUnit.{u1} R (Ring.toMonoid.{u1} R hR) a) (Eq.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R hR)))))))))) -> (DivisionRing.{u1} R)\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Nontrivial.{u1} R] [hR : Ring.{u1} R], (forall (a : R), Or (IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R hR))) a) (Eq.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (MonoidWithZero.toZero.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R hR))))))) -> (DivisionRing.{u1} R)\nCase conversion may be inaccurate. Consider using '#align division_ring_of_is_unit_or_eq_zero divisionRingOfIsUnitOrEqZero\u2093'. -/\n/-- Constructs a `division_ring` structure on a `ring` consisting only of units and 0. -/\nnoncomputable def divisionRingOfIsUnitOrEqZero [hR : Ring R] (h : \u2200 a : R, IsUnit a \u2228 a = 0) :\n    DivisionRing R :=\n  { groupWithZeroOfIsUnitOrEqZero h, hR with }\n#align division_ring_of_is_unit_or_eq_zero divisionRingOfIsUnitOrEqZero\n\n/- warning: field_of_is_unit_or_eq_zero -> fieldOfIsUnitOrEqZero is a dubious translation:\nlean 3 declaration is\n  forall {R : Type.{u1}} [_inst_1 : Nontrivial.{u1} R] [hR : CommRing.{u1} R], (forall (a : R), Or (IsUnit.{u1} R (Ring.toMonoid.{u1} R (CommRing.toRing.{u1} R hR)) a) (Eq.{succ u1} R a (OfNat.ofNat.{u1} R 0 (OfNat.mk.{u1} R 0 (Zero.zero.{u1} R (MulZeroClass.toHasZero.{u1} R (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} R (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} R (NonAssocRing.toNonUnitalNonAssocRing.{u1} R (Ring.toNonAssocRing.{u1} R (CommRing.toRing.{u1} R hR))))))))))) -> (Field.{u1} R)\nbut is expected to have type\n  forall {R : Type.{u1}} [_inst_1 : Nontrivial.{u1} R] [hR : CommRing.{u1} R], (forall (a : R), Or (IsUnit.{u1} R (MonoidWithZero.toMonoid.{u1} R (Semiring.toMonoidWithZero.{u1} R (Ring.toSemiring.{u1} R (CommRing.toRing.{u1} R hR)))) a) (Eq.{succ u1} R a (OfNat.ofNat.{u1} R 0 (Zero.toOfNat0.{u1} R (CommMonoidWithZero.toZero.{u1} R (CommSemiring.toCommMonoidWithZero.{u1} R (CommRing.toCommSemiring.{u1} R hR))))))) -> (Field.{u1} R)\nCase conversion may be inaccurate. Consider using '#align field_of_is_unit_or_eq_zero fieldOfIsUnitOrEqZero\u2093'. -/\n/-- Constructs a `field` structure on a `comm_ring` consisting only of units and 0.\nSee note [reducible non-instances]. -/\n@[reducible]\nnoncomputable def fieldOfIsUnitOrEqZero [hR : CommRing R] (h : \u2200 a : R, IsUnit a \u2228 a = 0) :\n    Field R :=\n  { groupWithZeroOfIsUnitOrEqZero h, hR with }\n#align field_of_is_unit_or_eq_zero fieldOfIsUnitOrEqZero\n\nend NoncomputableDefs\n\n/- warning: function.injective.division_semiring -> Function.Injective.divisionSemiring is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DivisionSemiring.{u2} \u03b2] [_inst_2 : Zero.{u1} \u03b1] [_inst_3 : Mul.{u1} \u03b1] [_inst_4 : Add.{u1} \u03b1] [_inst_5 : One.{u1} \u03b1] [_inst_6 : Inv.{u1} \u03b1] [_inst_7 : Div.{u1} \u03b1] [_inst_8 : SMul.{0, u1} Nat \u03b1] [_inst_9 : Pow.{u1, 0} \u03b1 Nat] [_inst_10 : Pow.{u1, 0} \u03b1 Int] [_inst_11 : NatCast.{u1} \u03b1] (f : \u03b1 -> \u03b2), (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 _inst_2)))) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))))))) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 _inst_5)))) (OfNat.ofNat.{u2} \u03b2 1 (OfNat.mk.{u2} \u03b2 1 (One.one.{u2} \u03b2 (AddMonoidWithOne.toOne.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))))))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 _inst_4) x y)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toHasAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (f x) (f y))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 _inst_3) x y)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (f x) (f y))) -> (forall (x : \u03b1), Eq.{succ u2} \u03b2 (f (Inv.inv.{u1} \u03b1 _inst_6 x)) (Inv.inv.{u2} \u03b2 (DivInvMonoid.toHasInv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1))) (f x))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 _inst_7) x y)) (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivInvMonoid.toHasDiv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1)))) (f x) (f y))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (SMul.smul.{0, u1} Nat \u03b1 _inst_8 n x)) (SMul.smul.{0, u2} Nat \u03b2 (AddMonoid.SMul.{u2} \u03b2 (AddMonoidWithOne.toAddMonoid.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) n (f x))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Nat \u03b1 (instHPow.{u1, 0} \u03b1 Nat _inst_9) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Nat \u03b2 (instHPow.{u2, 0} \u03b2 Nat (Monoid.Pow.{u2} \u03b2 (MonoidWithZero.toMonoid.{u2} \u03b2 (Semiring.toMonoidWithZero.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))) (f x) n)) -> (forall (x : \u03b1) (n : Int), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Int \u03b1 (instHPow.{u1, 0} \u03b1 Int _inst_10) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Int \u03b2 (instHPow.{u2, 0} \u03b2 Int (DivInvMonoid.Pow.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1)))) (f x) n)) -> (forall (n : Nat), Eq.{succ u2} \u03b2 (f ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat \u03b1 (HasLiftT.mk.{1, succ u1} Nat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Nat \u03b1 (Nat.castCoe.{u1} \u03b1 _inst_11))) n)) ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Nat \u03b2 (HasLiftT.mk.{1, succ u2} Nat \u03b2 (CoeTC\u2093.coe.{1, succ u2} Nat \u03b2 (Nat.castCoe.{u2} \u03b2 (AddMonoidWithOne.toNatCast.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))))) n)) -> (DivisionSemiring.{u1} \u03b1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : DivisionSemiring.{u2} \u03b2] [_inst_2 : Zero.{u1} \u03b1] [_inst_3 : Mul.{u1} \u03b1] [_inst_4 : Add.{u1} \u03b1] [_inst_5 : One.{u1} \u03b1] [_inst_6 : Inv.{u1} \u03b1] [_inst_7 : Div.{u1} \u03b1] [_inst_8 : SMul.{0, u1} Nat \u03b1] [_inst_9 : Pow.{u1, 0} \u03b1 Nat] [_inst_10 : Pow.{u1, 0} \u03b1 Int] [_inst_11 : NatCast.{u1} \u03b1] (f : \u03b1 -> \u03b2), (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 _inst_2))) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (MonoidWithZero.toZero.{u2} \u03b2 (Semiring.toMonoidWithZero.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 _inst_5))) (OfNat.ofNat.{u2} \u03b2 1 (One.toOfNat1.{u2} \u03b2 (Semiring.toOne.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 _inst_4) x y)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)))))) (f x) (f y))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 _inst_3) x y)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))) (f x) (f y))) -> (forall (x : \u03b1), Eq.{succ u2} \u03b2 (f (Inv.inv.{u1} \u03b1 _inst_6 x)) (Inv.inv.{u2} \u03b2 (DivisionSemiring.toInv.{u2} \u03b2 _inst_1) (f x))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 _inst_7) x y)) (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivisionSemiring.toDiv.{u2} \u03b2 _inst_1)) (f x) (f y))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HSMul.hSMul.{0, u1, u1} Nat \u03b1 \u03b1 (instHSMul.{0, u1} Nat \u03b1 _inst_8) n x)) (HSMul.hSMul.{0, u2, u2} Nat \u03b2 \u03b2 (instHSMul.{0, u2} Nat \u03b2 (AddMonoid.SMul.{u2} \u03b2 (AddMonoidWithOne.toAddMonoid.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))))) n (f x))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Nat \u03b1 (instHPow.{u1, 0} \u03b1 Nat _inst_9) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Nat \u03b2 (instHPow.{u2, 0} \u03b2 Nat (Monoid.Pow.{u2} \u03b2 (MonoidWithZero.toMonoid.{u2} \u03b2 (Semiring.toMonoidWithZero.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1))))) (f x) n)) -> (forall (x : \u03b1) (n : Int), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Int \u03b1 (instHPow.{u1, 0} \u03b1 Int _inst_10) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Int \u03b2 (instHPow.{u2, 0} \u03b2 Int (DivInvMonoid.Pow.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 _inst_1)))) (f x) n)) -> (forall (n : Nat), Eq.{succ u2} \u03b2 (f (Nat.cast.{u1} \u03b1 _inst_11 n)) (Nat.cast.{u2} \u03b2 (Semiring.toNatCast.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 _inst_1)) n)) -> (DivisionSemiring.{u1} \u03b1)\nCase conversion may be inaccurate. Consider using '#align function.injective.division_semiring Function.Injective.divisionSemiring\u2093'. -/\n-- See note [reducible non-instances]\n/-- Pullback a `division_semiring` along an injective function. -/\n@[reducible]\nprotected def Function.Injective.divisionSemiring [DivisionSemiring \u03b2] [Zero \u03b1] [Mul \u03b1] [Add \u03b1]\n    [One \u03b1] [Inv \u03b1] [Div \u03b1] [SMul \u2115 \u03b1] [Pow \u03b1 \u2115] [Pow \u03b1 \u2124] [NatCast \u03b1] (f : \u03b1 \u2192 \u03b2)\n    (hf : Injective f) (zero : f 0 = 0) (one : f 1 = 1) (add : \u2200 x y, f (x + y) = f x + f y)\n    (mul : \u2200 x y, f (x * y) = f x * f y) (inv : \u2200 x, f x\u207b\u00b9 = (f x)\u207b\u00b9)\n    (div : \u2200 x y, f (x / y) = f x / f y) (nsmul : \u2200 (x) (n : \u2115), f (n \u2022 x) = n \u2022 f x)\n    (npow : \u2200 (x) (n : \u2115), f (x ^ n) = f x ^ n) (zpow : \u2200 (x) (n : \u2124), f (x ^ n) = f x ^ n)\n    (nat_cast : \u2200 n : \u2115, f n = n) : DivisionSemiring \u03b1 :=\n  { hf.GroupWithZero f zero one mul inv div npow zpow,\n    hf.Semiring f zero one add mul nsmul npow nat_cast with }\n#align function.injective.division_semiring Function.Injective.divisionSemiring\n\n/- warning: function.injective.division_ring -> Function.Injective.divisionRing is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {K' : Type.{u2}} [_inst_2 : Zero.{u2} K'] [_inst_3 : One.{u2} K'] [_inst_4 : Add.{u2} K'] [_inst_5 : Mul.{u2} K'] [_inst_6 : Neg.{u2} K'] [_inst_7 : Sub.{u2} K'] [_inst_8 : Inv.{u2} K'] [_inst_9 : Div.{u2} K'] [_inst_10 : SMul.{0, u2} Nat K'] [_inst_11 : SMul.{0, u2} Int K'] [_inst_12 : SMul.{0, u2} Rat K'] [_inst_13 : Pow.{u2, 0} K' Nat] [_inst_14 : Pow.{u2, 0} K' Int] [_inst_15 : NatCast.{u2} K'] [_inst_16 : IntCast.{u2} K'] [_inst_17 : HasRatCast.{u2} K'] (f : K' -> K), (Function.Injective.{succ u2, succ u1} K' K f) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 0 (OfNat.mk.{u2} K' 0 (Zero.zero.{u2} K' _inst_2)))) (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))))) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 1 (OfNat.mk.{u2} K' 1 (One.one.{u2} K' _inst_3)))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HAdd.hAdd.{u2, u2, u2} K' K' K' (instHAdd.{u2} K' _inst_4) x y)) (HAdd.hAdd.{u1, u1, u1} K K K (instHAdd.{u1} K (Distrib.toHasAdd.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (f x) (f y))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HMul.hMul.{u2, u2, u2} K' K' K' (instHMul.{u2} K' _inst_5) x y)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Neg.neg.{u2} K' _inst_6 x)) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HSub.hSub.{u2, u2, u2} K' K' K' (instHSub.{u2} K' _inst_7) x y)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Inv.inv.{u2} K' _inst_8 x)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1)) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HDiv.hDiv.{u2, u2, u2} K' K' K' (instHDiv.{u2} K' _inst_9) x y)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (f x) (f y))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (SMul.smul.{0, u2} Nat K' _inst_10 n x)) (SMul.smul.{0, u1} Nat K (AddMonoid.SMul.{u1} K (AddMonoidWithOne.toAddMonoid.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) n (f x))) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (SMul.smul.{0, u2} Int K' _inst_11 n x)) (SMul.smul.{0, u1} Int K (SubNegMonoid.SMulInt.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) n (f x))) -> (forall (x : K') (n : Rat), Eq.{succ u1} K (f (SMul.smul.{0, u2} Rat K' _inst_12 n x)) (SMul.smul.{0, u1} Rat K (Rat.smulDivisionRing.{u1} K _inst_1) n (f x))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Nat K' (instHPow.{u2, 0} K' Nat _inst_13) x n)) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (Ring.toMonoid.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))) (f x) n)) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Int K' (instHPow.{u2, 0} K' Int _inst_14) x n)) (HPow.hPow.{u1, 0, u1} K Int K (instHPow.{u1, 0} K Int (DivInvMonoid.Pow.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (f x) n)) -> (forall (n : Nat), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Nat K' (HasLiftT.mk.{1, succ u2} Nat K' (CoeTC\u2093.coe.{1, succ u2} Nat K' (Nat.castCoe.{u2} K' _inst_15))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat K (HasLiftT.mk.{1, succ u1} Nat K (CoeTC\u2093.coe.{1, succ u1} Nat K (Nat.castCoe.{u1} K (AddMonoidWithOne.toNatCast.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))))) n)) -> (forall (n : Int), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Int K' (HasLiftT.mk.{1, succ u2} Int K' (CoeTC\u2093.coe.{1, succ u2} Int K' (Int.castCoe.{u2} K' _inst_16))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int K (HasLiftT.mk.{1, succ u1} Int K (CoeTC\u2093.coe.{1, succ u1} Int K (Int.castCoe.{u1} K (AddGroupWithOne.toHasIntCast.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) n)) -> (forall (n : Rat), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Rat K' (HasLiftT.mk.{1, succ u2} Rat K' (CoeTC\u2093.coe.{1, succ u2} Rat K' (Rat.castCoe.{u2} K' _inst_17))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat K (HasLiftT.mk.{1, succ u1} Rat K (CoeTC\u2093.coe.{1, succ u1} Rat K (Rat.castCoe.{u1} K (DivisionRing.toHasRatCast.{u1} K _inst_1)))) n)) -> (DivisionRing.{u2} K')\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : DivisionRing.{u1} K] {K' : Type.{u2}} [_inst_2 : Zero.{u2} K'] [_inst_3 : One.{u2} K'] [_inst_4 : Add.{u2} K'] [_inst_5 : Mul.{u2} K'] [_inst_6 : Neg.{u2} K'] [_inst_7 : Sub.{u2} K'] [_inst_8 : Inv.{u2} K'] [_inst_9 : Div.{u2} K'] [_inst_10 : SMul.{0, u2} Nat K'] [_inst_11 : SMul.{0, u2} Int K'] [_inst_12 : SMul.{0, u2} Rat K'] [_inst_13 : Pow.{u2, 0} K' Nat] [_inst_14 : Pow.{u2, 0} K' Int] [_inst_15 : NatCast.{u2} K'] [_inst_16 : IntCast.{u2} K'] [_inst_17 : RatCast.{u2} K'] (f : K' -> K), (Function.Injective.{succ u2, succ u1} K' K f) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 0 (Zero.toOfNat0.{u2} K' _inst_2))) (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (MonoidWithZero.toZero.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 1 (One.toOfNat1.{u2} K' _inst_3))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HAdd.hAdd.{u2, u2, u2} K' K' K' (instHAdd.{u2} K' _inst_4) x y)) (HAdd.hAdd.{u1, u1, u1} K K K (instHAdd.{u1} K (Distrib.toAdd.{u1} K (NonUnitalNonAssocSemiring.toDistrib.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))))) (f x) (f y))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HMul.hMul.{u2, u2, u2} K' K' K' (instHMul.{u2} K' _inst_5) x y)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Neg.neg.{u2} K' _inst_6 x)) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HSub.hSub.{u2, u2, u2} K' K' K' (instHSub.{u2} K' _inst_7) x y)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Inv.inv.{u2} K' _inst_8 x)) (Inv.inv.{u1} K (DivisionRing.toInv.{u1} K _inst_1) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HDiv.hDiv.{u2, u2, u2} K' K' K' (instHDiv.{u2} K' _inst_9) x y)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivisionRing.toDiv.{u1} K _inst_1)) (f x) (f y))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Nat K' K' (instHSMul.{0, u2} Nat K' _inst_10) n x)) (HSMul.hSMul.{0, u1, u1} Nat K K (instHSMul.{0, u1} Nat K (AddMonoid.SMul.{u1} K (AddMonoidWithOne.toAddMonoid.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (Ring.toAddGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) n (f x))) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Int K' K' (instHSMul.{0, u2} Int K' _inst_11) n x)) (HSMul.hSMul.{0, u1, u1} Int K K (instHSMul.{0, u1} Int K (SubNegMonoid.SMulInt.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (Ring.toAddGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K _inst_1)))))) n (f x))) -> (forall (x : K') (n : Rat), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Rat K' K' (instHSMul.{0, u2} Rat K' _inst_12) n x)) (HSMul.hSMul.{0, u1, u1} Rat K K (instHSMul.{0, u1} Rat K (Rat.smulDivisionRing.{u1} K _inst_1)) n (f x))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Nat K' (instHPow.{u2, 0} K' Nat _inst_13) x n)) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (MonoidWithZero.toMonoid.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (DivisionRing.toDivisionSemiring.{u1} K _inst_1)))))) (f x) n)) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Int K' (instHPow.{u2, 0} K' Int _inst_14) x n)) (HPow.hPow.{u1, 0, u1} K Int K (instHPow.{u1, 0} K Int (DivInvMonoid.Pow.{u1} K (DivisionRing.toDivInvMonoid.{u1} K _inst_1))) (f x) n)) -> (forall (n : Nat), Eq.{succ u1} K (f (Nat.cast.{u2} K' _inst_15 n)) (Nat.cast.{u1} K (NonAssocRing.toNatCast.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K _inst_1))) n)) -> (forall (n : Int), Eq.{succ u1} K (f (Int.cast.{u2} K' _inst_16 n)) (Int.cast.{u1} K (Ring.toIntCast.{u1} K (DivisionRing.toRing.{u1} K _inst_1)) n)) -> (forall (n : Rat), Eq.{succ u1} K (f (Rat.cast.{u2} K' _inst_17 n)) (Rat.cast.{u1} K (DivisionRing.toRatCast.{u1} K _inst_1) n)) -> (DivisionRing.{u2} K')\nCase conversion may be inaccurate. Consider using '#align function.injective.division_ring Function.Injective.divisionRing\u2093'. -/\n/-- Pullback a `division_ring` along an injective function.\nSee note [reducible non-instances]. -/\n@[reducible]\nprotected def Function.Injective.divisionRing [DivisionRing K] {K'} [Zero K'] [One K'] [Add K']\n    [Mul K'] [Neg K'] [Sub K'] [Inv K'] [Div K'] [SMul \u2115 K'] [SMul \u2124 K'] [SMul \u211a K'] [Pow K' \u2115]\n    [Pow K' \u2124] [NatCast K'] [IntCast K'] [HasRatCast K'] (f : K' \u2192 K) (hf : Injective f)\n    (zero : f 0 = 0) (one : f 1 = 1) (add : \u2200 x y, f (x + y) = f x + f y)\n    (mul : \u2200 x y, f (x * y) = f x * f y) (neg : \u2200 x, f (-x) = -f x)\n    (sub : \u2200 x y, f (x - y) = f x - f y) (inv : \u2200 x, f x\u207b\u00b9 = (f x)\u207b\u00b9)\n    (div : \u2200 x y, f (x / y) = f x / f y) (nsmul : \u2200 (x) (n : \u2115), f (n \u2022 x) = n \u2022 f x)\n    (zsmul : \u2200 (x) (n : \u2124), f (n \u2022 x) = n \u2022 f x) (qsmul : \u2200 (x) (n : \u211a), f (n \u2022 x) = n \u2022 f x)\n    (npow : \u2200 (x) (n : \u2115), f (x ^ n) = f x ^ n) (zpow : \u2200 (x) (n : \u2124), f (x ^ n) = f x ^ n)\n    (nat_cast : \u2200 n : \u2115, f n = n) (int_cast : \u2200 n : \u2124, f n = n) (rat_cast : \u2200 n : \u211a, f n = n) :\n    DivisionRing K' :=\n  { hf.GroupWithZero f zero one mul inv div npow zpow,\n    hf.Ring f zero one add mul neg sub nsmul zsmul npow nat_cast\n      int_cast with\n    ratCast := coe\n    ratCast_mk := fun a b h1 h2 =>\n      hf\n        (by\n          erw [rat_cast, mul, inv, int_cast, nat_cast] <;> exact DivisionRing.ratCast_mk a b h1 h2)\n    qsmul := (\u00b7 \u2022 \u00b7)\n    qsmul_eq_mul' := fun a x => hf (by erw [qsmul, mul, Rat.smul_def, rat_cast]) }\n#align function.injective.division_ring Function.Injective.divisionRing\n\n/- warning: function.injective.semifield -> Function.Injective.semifield is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Semifield.{u2} \u03b2] [_inst_2 : Zero.{u1} \u03b1] [_inst_3 : Mul.{u1} \u03b1] [_inst_4 : Add.{u1} \u03b1] [_inst_5 : One.{u1} \u03b1] [_inst_6 : Inv.{u1} \u03b1] [_inst_7 : Div.{u1} \u03b1] [_inst_8 : SMul.{0, u1} Nat \u03b1] [_inst_9 : Pow.{u1, 0} \u03b1 Nat] [_inst_10 : Pow.{u1, 0} \u03b1 Int] [_inst_11 : NatCast.{u1} \u03b1] (f : \u03b1 -> \u03b2), (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 0 (OfNat.mk.{u1} \u03b1 0 (Zero.zero.{u1} \u03b1 _inst_2)))) (OfNat.ofNat.{u2} \u03b2 0 (OfNat.mk.{u2} \u03b2 0 (Zero.zero.{u2} \u03b2 (MulZeroClass.toHasZero.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMulZeroClass.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))))))) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 1 (OfNat.mk.{u1} \u03b1 1 (One.one.{u1} \u03b1 _inst_5)))) (OfNat.ofNat.{u2} \u03b2 1 (OfNat.mk.{u2} \u03b2 1 (One.one.{u2} \u03b2 (AddMonoidWithOne.toOne.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))))))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 _inst_4) x y)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toHasAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))))) (f x) (f y))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 _inst_3) x y)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (Distrib.toHasMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))))) (f x) (f y))) -> (forall (x : \u03b1), Eq.{succ u2} \u03b2 (f (Inv.inv.{u1} \u03b1 _inst_6 x)) (Inv.inv.{u2} \u03b2 (DivInvMonoid.toHasInv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))) (f x))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 _inst_7) x y)) (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (DivInvMonoid.toHasDiv.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))) (f x) (f y))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (SMul.smul.{0, u1} Nat \u03b1 _inst_8 n x)) (SMul.smul.{0, u2} Nat \u03b2 (AddMonoid.SMul.{u2} \u03b2 (AddMonoidWithOne.toAddMonoid.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))))) n (f x))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Nat \u03b1 (instHPow.{u1, 0} \u03b1 Nat _inst_9) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Nat \u03b2 (instHPow.{u2, 0} \u03b2 Nat (Monoid.Pow.{u2} \u03b2 (MonoidWithZero.toMonoid.{u2} \u03b2 (Semiring.toMonoidWithZero.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))) (f x) n)) -> (forall (x : \u03b1) (n : Int), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Int \u03b1 (instHPow.{u1, 0} \u03b1 Int _inst_10) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Int \u03b2 (instHPow.{u2, 0} \u03b2 Int (DivInvMonoid.Pow.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))) (f x) n)) -> (forall (n : Nat), Eq.{succ u2} \u03b2 (f ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat \u03b1 (HasLiftT.mk.{1, succ u1} Nat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Nat \u03b1 (Nat.castCoe.{u1} \u03b1 _inst_11))) n)) ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Nat \u03b2 (HasLiftT.mk.{1, succ u2} Nat \u03b2 (CoeTC\u2093.coe.{1, succ u2} Nat \u03b2 (Nat.castCoe.{u2} \u03b2 (AddMonoidWithOne.toNatCast.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))))))) n)) -> (Semifield.{u1} \u03b1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u2}} [_inst_1 : Semifield.{u2} \u03b2] [_inst_2 : Zero.{u1} \u03b1] [_inst_3 : Mul.{u1} \u03b1] [_inst_4 : Add.{u1} \u03b1] [_inst_5 : One.{u1} \u03b1] [_inst_6 : Inv.{u1} \u03b1] [_inst_7 : Div.{u1} \u03b1] [_inst_8 : SMul.{0, u1} Nat \u03b1] [_inst_9 : Pow.{u1, 0} \u03b1 Nat] [_inst_10 : Pow.{u1, 0} \u03b1 Int] [_inst_11 : NatCast.{u1} \u03b1] (f : \u03b1 -> \u03b2), (Function.Injective.{succ u1, succ u2} \u03b1 \u03b2 f) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 0 (Zero.toOfNat0.{u1} \u03b1 _inst_2))) (OfNat.ofNat.{u2} \u03b2 0 (Zero.toOfNat0.{u2} \u03b2 (CommMonoidWithZero.toZero.{u2} \u03b2 (CommGroupWithZero.toCommMonoidWithZero.{u2} \u03b2 (Semifield.toCommGroupWithZero.{u2} \u03b2 _inst_1)))))) -> (Eq.{succ u2} \u03b2 (f (OfNat.ofNat.{u1} \u03b1 1 (One.toOfNat1.{u1} \u03b1 _inst_5))) (OfNat.ofNat.{u2} \u03b2 1 (One.toOfNat1.{u2} \u03b2 (Semiring.toOne.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HAdd.hAdd.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHAdd.{u1} \u03b1 _inst_4) x y)) (HAdd.hAdd.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHAdd.{u2} \u03b2 (Distrib.toAdd.{u2} \u03b2 (NonUnitalNonAssocSemiring.toDistrib.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))))) (f x) (f y))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HMul.hMul.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHMul.{u1} \u03b1 _inst_3) x y)) (HMul.hMul.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHMul.{u2} \u03b2 (NonUnitalNonAssocSemiring.toMul.{u2} \u03b2 (NonAssocSemiring.toNonUnitalNonAssocSemiring.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))) (f x) (f y))) -> (forall (x : \u03b1), Eq.{succ u2} \u03b2 (f (Inv.inv.{u1} \u03b1 _inst_6 x)) (Inv.inv.{u2} \u03b2 (Semifield.toInv.{u2} \u03b2 _inst_1) (f x))) -> (forall (x : \u03b1) (y : \u03b1), Eq.{succ u2} \u03b2 (f (HDiv.hDiv.{u1, u1, u1} \u03b1 \u03b1 \u03b1 (instHDiv.{u1} \u03b1 _inst_7) x y)) (HDiv.hDiv.{u2, u2, u2} \u03b2 \u03b2 \u03b2 (instHDiv.{u2} \u03b2 (Semifield.toDiv.{u2} \u03b2 _inst_1)) (f x) (f y))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HSMul.hSMul.{0, u1, u1} Nat \u03b1 \u03b1 (instHSMul.{0, u1} Nat \u03b1 _inst_8) n x)) (HSMul.hSMul.{0, u2, u2} Nat \u03b2 \u03b2 (instHSMul.{0, u2} Nat \u03b2 (AddMonoid.SMul.{u2} \u03b2 (AddMonoidWithOne.toAddMonoid.{u2} \u03b2 (AddCommMonoidWithOne.toAddMonoidWithOne.{u2} \u03b2 (NonAssocSemiring.toAddCommMonoidWithOne.{u2} \u03b2 (Semiring.toNonAssocSemiring.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))))) n (f x))) -> (forall (x : \u03b1) (n : Nat), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Nat \u03b1 (instHPow.{u1, 0} \u03b1 Nat _inst_9) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Nat \u03b2 (instHPow.{u2, 0} \u03b2 Nat (Monoid.Pow.{u2} \u03b2 (MonoidWithZero.toMonoid.{u2} \u03b2 (Semiring.toMonoidWithZero.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1)))))) (f x) n)) -> (forall (x : \u03b1) (n : Int), Eq.{succ u2} \u03b2 (f (HPow.hPow.{u1, 0, u1} \u03b1 Int \u03b1 (instHPow.{u1, 0} \u03b1 Int _inst_10) x n)) (HPow.hPow.{u2, 0, u2} \u03b2 Int \u03b2 (instHPow.{u2, 0} \u03b2 Int (DivInvMonoid.Pow.{u2} \u03b2 (GroupWithZero.toDivInvMonoid.{u2} \u03b2 (DivisionSemiring.toGroupWithZero.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))))) (f x) n)) -> (forall (n : Nat), Eq.{succ u2} \u03b2 (f (Nat.cast.{u1} \u03b1 _inst_11 n)) (Nat.cast.{u2} \u03b2 (Semiring.toNatCast.{u2} \u03b2 (DivisionSemiring.toSemiring.{u2} \u03b2 (Semifield.toDivisionSemiring.{u2} \u03b2 _inst_1))) n)) -> (Semifield.{u1} \u03b1)\nCase conversion may be inaccurate. Consider using '#align function.injective.semifield Function.Injective.semifield\u2093'. -/\n-- See note [reducible non-instances]\n/-- Pullback a `field` along an injective function. -/\n@[reducible]\nprotected def Function.Injective.semifield [Semifield \u03b2] [Zero \u03b1] [Mul \u03b1] [Add \u03b1] [One \u03b1] [Inv \u03b1]\n    [Div \u03b1] [SMul \u2115 \u03b1] [Pow \u03b1 \u2115] [Pow \u03b1 \u2124] [NatCast \u03b1] (f : \u03b1 \u2192 \u03b2) (hf : Injective f)\n    (zero : f 0 = 0) (one : f 1 = 1) (add : \u2200 x y, f (x + y) = f x + f y)\n    (mul : \u2200 x y, f (x * y) = f x * f y) (inv : \u2200 x, f x\u207b\u00b9 = (f x)\u207b\u00b9)\n    (div : \u2200 x y, f (x / y) = f x / f y) (nsmul : \u2200 (x) (n : \u2115), f (n \u2022 x) = n \u2022 f x)\n    (npow : \u2200 (x) (n : \u2115), f (x ^ n) = f x ^ n) (zpow : \u2200 (x) (n : \u2124), f (x ^ n) = f x ^ n)\n    (nat_cast : \u2200 n : \u2115, f n = n) : Semifield \u03b1 :=\n  { hf.CommGroupWithZero f zero one mul inv div npow zpow,\n    hf.CommSemiring f zero one add mul nsmul npow nat_cast with }\n#align function.injective.semifield Function.Injective.semifield\n\n/- warning: function.injective.field -> Function.Injective.field is a dubious translation:\nlean 3 declaration is\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {K' : Type.{u2}} [_inst_2 : Zero.{u2} K'] [_inst_3 : Mul.{u2} K'] [_inst_4 : Add.{u2} K'] [_inst_5 : Neg.{u2} K'] [_inst_6 : Sub.{u2} K'] [_inst_7 : One.{u2} K'] [_inst_8 : Inv.{u2} K'] [_inst_9 : Div.{u2} K'] [_inst_10 : SMul.{0, u2} Nat K'] [_inst_11 : SMul.{0, u2} Int K'] [_inst_12 : SMul.{0, u2} Rat K'] [_inst_13 : Pow.{u2, 0} K' Nat] [_inst_14 : Pow.{u2, 0} K' Int] [_inst_15 : NatCast.{u2} K'] [_inst_16 : IntCast.{u2} K'] [_inst_17 : HasRatCast.{u2} K'] (f : K' -> K), (Function.Injective.{succ u2, succ u1} K' K f) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 0 (OfNat.mk.{u2} K' 0 (Zero.zero.{u2} K' _inst_2)))) (OfNat.ofNat.{u1} K 0 (OfNat.mk.{u1} K 0 (Zero.zero.{u1} K (MulZeroClass.toHasZero.{u1} K (NonUnitalNonAssocSemiring.toMulZeroClass.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))))) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 1 (OfNat.mk.{u2} K' 1 (One.one.{u2} K' _inst_7)))) (OfNat.ofNat.{u1} K 1 (OfNat.mk.{u1} K 1 (One.one.{u1} K (AddMonoidWithOne.toOne.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HAdd.hAdd.{u2, u2, u2} K' K' K' (instHAdd.{u2} K' _inst_4) x y)) (HAdd.hAdd.{u1, u1, u1} K K K (instHAdd.{u1} K (Distrib.toHasAdd.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (f x) (f y))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HMul.hMul.{u2, u2, u2} K' K' K' (instHMul.{u2} K' _inst_3) x y)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (Distrib.toHasMul.{u1} K (Ring.toDistrib.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Neg.neg.{u2} K' _inst_5 x)) (Neg.neg.{u1} K (SubNegMonoid.toHasNeg.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HSub.hSub.{u2, u2, u2} K' K' K' (instHSub.{u2} K' _inst_6) x y)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (SubNegMonoid.toHasSub.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Inv.inv.{u2} K' _inst_8 x)) (Inv.inv.{u1} K (DivInvMonoid.toHasInv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HDiv.hDiv.{u2, u2, u2} K' K' K' (instHDiv.{u2} K' _inst_9) x y)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (DivInvMonoid.toHasDiv.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (f x) (f y))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (SMul.smul.{0, u2} Nat K' _inst_10 n x)) (SMul.smul.{0, u1} Nat K (AddMonoid.SMul.{u1} K (AddMonoidWithOne.toAddMonoid.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) n (f x))) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (SMul.smul.{0, u2} Int K' _inst_11 n x)) (SMul.smul.{0, u1} Int K (SubNegMonoid.SMulInt.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) n (f x))) -> (forall (x : K') (n : Rat), Eq.{succ u1} K (f (SMul.smul.{0, u2} Rat K' _inst_12 n x)) (SMul.smul.{0, u1} Rat K (Rat.smulDivisionRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)) n (f x))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Nat K' (instHPow.{u2, 0} K' Nat _inst_13) x n)) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (Ring.toMonoid.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) (f x) n)) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Int K' (instHPow.{u2, 0} K' Int _inst_14) x n)) (HPow.hPow.{u1, 0, u1} K Int K (instHPow.{u1, 0} K Int (DivInvMonoid.Pow.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (f x) n)) -> (forall (n : Nat), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Nat K' (HasLiftT.mk.{1, succ u2} Nat K' (CoeTC\u2093.coe.{1, succ u2} Nat K' (Nat.castCoe.{u2} K' _inst_15))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Nat K (HasLiftT.mk.{1, succ u1} Nat K (CoeTC\u2093.coe.{1, succ u1} Nat K (Nat.castCoe.{u1} K (AddMonoidWithOne.toNatCast.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))))) n)) -> (forall (n : Int), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Int K' (HasLiftT.mk.{1, succ u2} Int K' (CoeTC\u2093.coe.{1, succ u2} Int K' (Int.castCoe.{u2} K' _inst_16))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Int K (HasLiftT.mk.{1, succ u1} Int K (CoeTC\u2093.coe.{1, succ u1} Int K (Int.castCoe.{u1} K (AddGroupWithOne.toHasIntCast.{u1} K (AddCommGroupWithOne.toAddGroupWithOne.{u1} K (Ring.toAddCommGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) n)) -> (forall (n : Rat), Eq.{succ u1} K (f ((fun (a : Type) (b : Type.{u2}) [self : HasLiftT.{1, succ u2} a b] => self.0) Rat K' (HasLiftT.mk.{1, succ u2} Rat K' (CoeTC\u2093.coe.{1, succ u2} Rat K' (Rat.castCoe.{u2} K' _inst_17))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat K (HasLiftT.mk.{1, succ u1} Rat K (CoeTC\u2093.coe.{1, succ u1} Rat K (Rat.castCoe.{u1} K (DivisionRing.toHasRatCast.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))) n)) -> (Field.{u2} K')\nbut is expected to have type\n  forall {K : Type.{u1}} [_inst_1 : Field.{u1} K] {K' : Type.{u2}} [_inst_2 : Zero.{u2} K'] [_inst_3 : Mul.{u2} K'] [_inst_4 : Add.{u2} K'] [_inst_5 : Neg.{u2} K'] [_inst_6 : Sub.{u2} K'] [_inst_7 : One.{u2} K'] [_inst_8 : Inv.{u2} K'] [_inst_9 : Div.{u2} K'] [_inst_10 : SMul.{0, u2} Nat K'] [_inst_11 : SMul.{0, u2} Int K'] [_inst_12 : SMul.{0, u2} Rat K'] [_inst_13 : Pow.{u2, 0} K' Nat] [_inst_14 : Pow.{u2, 0} K' Int] [_inst_15 : NatCast.{u2} K'] [_inst_16 : IntCast.{u2} K'] [_inst_17 : RatCast.{u2} K'] (f : K' -> K), (Function.Injective.{succ u2, succ u1} K' K f) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 0 (Zero.toOfNat0.{u2} K' _inst_2))) (OfNat.ofNat.{u1} K 0 (Zero.toOfNat0.{u1} K (CommMonoidWithZero.toZero.{u1} K (CommGroupWithZero.toCommMonoidWithZero.{u1} K (Semifield.toCommGroupWithZero.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) -> (Eq.{succ u1} K (f (OfNat.ofNat.{u2} K' 1 (One.toOfNat1.{u2} K' _inst_7))) (OfNat.ofNat.{u1} K 1 (One.toOfNat1.{u1} K (NonAssocRing.toOne.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HAdd.hAdd.{u2, u2, u2} K' K' K' (instHAdd.{u2} K' _inst_4) x y)) (HAdd.hAdd.{u1, u1, u1} K K K (instHAdd.{u1} K (Distrib.toAdd.{u1} K (NonUnitalNonAssocSemiring.toDistrib.{u1} K (NonUnitalNonAssocRing.toNonUnitalNonAssocSemiring.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))))) (f x) (f y))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HMul.hMul.{u2, u2, u2} K' K' K' (instHMul.{u2} K' _inst_3) x y)) (HMul.hMul.{u1, u1, u1} K K K (instHMul.{u1} K (NonUnitalNonAssocRing.toMul.{u1} K (NonAssocRing.toNonUnitalNonAssocRing.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Neg.neg.{u2} K' _inst_5 x)) (Neg.neg.{u1} K (Ring.toNeg.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HSub.hSub.{u2, u2, u2} K' K' K' (instHSub.{u2} K' _inst_6) x y)) (HSub.hSub.{u1, u1, u1} K K K (instHSub.{u1} K (Ring.toSub.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (f x) (f y))) -> (forall (x : K'), Eq.{succ u1} K (f (Inv.inv.{u2} K' _inst_8 x)) (Inv.inv.{u1} K (Field.toInv.{u1} K _inst_1) (f x))) -> (forall (x : K') (y : K'), Eq.{succ u1} K (f (HDiv.hDiv.{u2, u2, u2} K' K' K' (instHDiv.{u2} K' _inst_9) x y)) (HDiv.hDiv.{u1, u1, u1} K K K (instHDiv.{u1} K (Field.toDiv.{u1} K _inst_1)) (f x) (f y))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Nat K' K' (instHSMul.{0, u2} Nat K' _inst_10) n x)) (HSMul.hSMul.{0, u1, u1} Nat K K (instHSMul.{0, u1} Nat K (AddMonoid.SMul.{u1} K (AddMonoidWithOne.toAddMonoid.{u1} K (AddGroupWithOne.toAddMonoidWithOne.{u1} K (Ring.toAddGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) n (f x))) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Int K' K' (instHSMul.{0, u2} Int K' _inst_11) n x)) (HSMul.hSMul.{0, u1, u1} Int K K (instHSMul.{0, u1} Int K (SubNegMonoid.SMulInt.{u1} K (AddGroup.toSubNegMonoid.{u1} K (AddGroupWithOne.toAddGroup.{u1} K (Ring.toAddGroupWithOne.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))))))) n (f x))) -> (forall (x : K') (n : Rat), Eq.{succ u1} K (f (HSMul.hSMul.{0, u2, u2} Rat K' K' (instHSMul.{0, u2} Rat K' _inst_12) n x)) (HSMul.hSMul.{0, u1, u1} Rat K K (instHSMul.{0, u1} Rat K (Rat.smulDivisionRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) n (f x))) -> (forall (x : K') (n : Nat), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Nat K' (instHPow.{u2, 0} K' Nat _inst_13) x n)) (HPow.hPow.{u1, 0, u1} K Nat K (instHPow.{u1, 0} K Nat (Monoid.Pow.{u1} K (MonoidWithZero.toMonoid.{u1} K (Semiring.toMonoidWithZero.{u1} K (DivisionSemiring.toSemiring.{u1} K (Semifield.toDivisionSemiring.{u1} K (Field.toSemifield.{u1} K _inst_1))))))) (f x) n)) -> (forall (x : K') (n : Int), Eq.{succ u1} K (f (HPow.hPow.{u2, 0, u2} K' Int K' (instHPow.{u2, 0} K' Int _inst_14) x n)) (HPow.hPow.{u1, 0, u1} K Int K (instHPow.{u1, 0} K Int (DivInvMonoid.Pow.{u1} K (DivisionRing.toDivInvMonoid.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) (f x) n)) -> (forall (n : Nat), Eq.{succ u1} K (f (Nat.cast.{u2} K' _inst_15 n)) (Nat.cast.{u1} K (NonAssocRing.toNatCast.{u1} K (Ring.toNonAssocRing.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1)))) n)) -> (forall (n : Int), Eq.{succ u1} K (f (Int.cast.{u2} K' _inst_16 n)) (Int.cast.{u1} K (Ring.toIntCast.{u1} K (DivisionRing.toRing.{u1} K (Field.toDivisionRing.{u1} K _inst_1))) n)) -> (forall (n : Rat), Eq.{succ u1} K (f (Rat.cast.{u2} K' _inst_17 n)) (Rat.cast.{u1} K (Field.toRatCast.{u1} K _inst_1) n)) -> (Field.{u2} K')\nCase conversion may be inaccurate. Consider using '#align function.injective.field Function.Injective.field\u2093'. -/\n/-- Pullback a `field` along an injective function.\nSee note [reducible non-instances]. -/\n@[reducible]\nprotected def Function.Injective.field [Field K] {K'} [Zero K'] [Mul K'] [Add K'] [Neg K'] [Sub K']\n    [One K'] [Inv K'] [Div K'] [SMul \u2115 K'] [SMul \u2124 K'] [SMul \u211a K'] [Pow K' \u2115] [Pow K' \u2124]\n    [NatCast K'] [IntCast K'] [HasRatCast K'] (f : K' \u2192 K) (hf : Injective f) (zero : f 0 = 0)\n    (one : f 1 = 1) (add : \u2200 x y, f (x + y) = f x + f y) (mul : \u2200 x y, f (x * y) = f x * f y)\n    (neg : \u2200 x, f (-x) = -f x) (sub : \u2200 x y, f (x - y) = f x - f y) (inv : \u2200 x, f x\u207b\u00b9 = (f x)\u207b\u00b9)\n    (div : \u2200 x y, f (x / y) = f x / f y) (nsmul : \u2200 (x) (n : \u2115), f (n \u2022 x) = n \u2022 f x)\n    (zsmul : \u2200 (x) (n : \u2124), f (n \u2022 x) = n \u2022 f x) (qsmul : \u2200 (x) (n : \u211a), f (n \u2022 x) = n \u2022 f x)\n    (npow : \u2200 (x) (n : \u2115), f (x ^ n) = f x ^ n) (zpow : \u2200 (x) (n : \u2124), f (x ^ n) = f x ^ n)\n    (nat_cast : \u2200 n : \u2115, f n = n) (int_cast : \u2200 n : \u2124, f n = n) (rat_cast : \u2200 n : \u211a, f n = n) :\n    Field K' :=\n  { hf.CommGroupWithZero f zero one mul inv div npow zpow,\n    hf.CommRing f zero one add mul neg sub nsmul zsmul npow nat_cast\n      int_cast with\n    ratCast := coe\n    ratCast_mk := fun a b h1 h2 =>\n      hf\n        (by\n          erw [rat_cast, mul, inv, int_cast, nat_cast] <;> exact DivisionRing.ratCast_mk a b h1 h2)\n    qsmul := (\u00b7 \u2022 \u00b7)\n    qsmul_eq_mul' := fun a x => hf (by erw [qsmul, mul, Rat.smul_def, rat_cast]) }\n#align function.injective.field Function.Injective.field\n\n/-! ### Order dual -/\n\n\ninstance [h : HasRatCast \u03b1] : HasRatCast \u03b1\u1d52\u1d48 :=\n  h\n\ninstance [h : DivisionSemiring \u03b1] : DivisionSemiring \u03b1\u1d52\u1d48 :=\n  h\n\ninstance [h : DivisionRing \u03b1] : DivisionRing \u03b1\u1d52\u1d48 :=\n  h\n\ninstance [h : Semifield \u03b1] : Semifield \u03b1\u1d52\u1d48 :=\n  h\n\ninstance [h : Field \u03b1] : Field \u03b1\u1d52\u1d48 :=\n  h\n\n/- warning: to_dual_rat_cast -> toDual_rat_cast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} (OrderDual.{u1} \u03b1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} \u03b1 (OrderDual.{u1} \u03b1)) (fun (_x : Equiv.{succ u1, succ u1} \u03b1 (OrderDual.{u1} \u03b1)) => \u03b1 -> (OrderDual.{u1} \u03b1)) (Equiv.hasCoeToFun.{succ u1, succ u1} \u03b1 (OrderDual.{u1} \u03b1)) (OrderDual.toDual.{u1} \u03b1) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat (OrderDual.{u1} \u03b1) (HasLiftT.mk.{1, succ u1} Rat (OrderDual.{u1} \u03b1) (CoeTC\u2093.coe.{1, succ u1} Rat (OrderDual.{u1} \u03b1) (Rat.castCoe.{u1} (OrderDual.{u1} \u03b1) (OrderDual.hasRatCast.{u1} \u03b1 _inst_1)))) n)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : RatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => OrderDual.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} \u03b1 (OrderDual.{u1} \u03b1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => OrderDual.{u1} \u03b1) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} \u03b1 (OrderDual.{u1} \u03b1)) (OrderDual.toDual.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (Rat.cast.{u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => OrderDual.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (instRatCastOrderDual.{u1} \u03b1 _inst_1) n)\nCase conversion may be inaccurate. Consider using '#align to_dual_rat_cast toDual_rat_cast\u2093'. -/\n@[simp]\ntheorem toDual_rat_cast [HasRatCast \u03b1] (n : \u211a) : toDual (n : \u03b1) = n :=\n  rfl\n#align to_dual_rat_cast toDual_rat_cast\n\n/- warning: of_dual_rat_cast -> ofDual_rat_cast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) (coeFn.{1, 1} (Equiv.{1, 1} (OrderDual.{0} Rat) Rat) (fun (_x : Equiv.{1, 1} (OrderDual.{0} Rat) Rat) => (OrderDual.{0} Rat) -> Rat) (Equiv.hasCoeToFun.{1, 1} (OrderDual.{0} Rat) Rat) (OrderDual.ofDual.{0} Rat) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) n)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : RatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} \u03b1 (Rat.cast.{u1} \u03b1 _inst_1 (FunLike.coe.{1, 1, 1} (Equiv.{1, 1} (OrderDual.{0} Rat) Rat) (OrderDual.{0} Rat) (fun (_x : OrderDual.{0} Rat) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : OrderDual.{0} Rat) => Rat) _x) (Equiv.instFunLikeEquiv.{1, 1} (OrderDual.{0} Rat) Rat) (OrderDual.ofDual.{0} Rat) n)) (Rat.cast.{u1} \u03b1 _inst_1 n)\nCase conversion may be inaccurate. Consider using '#align of_dual_rat_cast ofDual_rat_cast\u2093'. -/\n@[simp]\ntheorem ofDual_rat_cast [HasRatCast \u03b1] (n : \u211a) : (ofDual n : \u03b1) = n :=\n  rfl\n#align of_dual_rat_cast ofDual_rat_cast\n\n/-! ### Lexicographic order -/\n\n\ninstance [h : HasRatCast \u03b1] : HasRatCast (Lex \u03b1) :=\n  h\n\ninstance [h : DivisionSemiring \u03b1] : DivisionSemiring (Lex \u03b1) :=\n  h\n\ninstance [h : DivisionRing \u03b1] : DivisionRing (Lex \u03b1) :=\n  h\n\ninstance [h : Semifield \u03b1] : Semifield (Lex \u03b1) :=\n  h\n\ninstance [h : Field \u03b1] : Field (Lex \u03b1) :=\n  h\n\n/- warning: to_lex_rat_cast -> toLex_rat_cast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} (Lex.{u1} \u03b1) (coeFn.{succ u1, succ u1} (Equiv.{succ u1, succ u1} \u03b1 (Lex.{u1} \u03b1)) (fun (_x : Equiv.{succ u1, succ u1} \u03b1 (Lex.{u1} \u03b1)) => \u03b1 -> (Lex.{u1} \u03b1)) (Equiv.hasCoeToFun.{succ u1, succ u1} \u03b1 (Lex.{u1} \u03b1)) (toLex.{u1} \u03b1) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat (Lex.{u1} \u03b1) (HasLiftT.mk.{1, succ u1} Rat (Lex.{u1} \u03b1) (CoeTC\u2093.coe.{1, succ u1} Rat (Lex.{u1} \u03b1) (Rat.castCoe.{u1} (Lex.{u1} \u03b1) (Lex.hasRatCast.{u1} \u03b1 _inst_1)))) n)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : RatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => Lex.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (FunLike.coe.{succ u1, succ u1, succ u1} (Equiv.{succ u1, succ u1} \u03b1 (Lex.{u1} \u03b1)) \u03b1 (fun (_x : \u03b1) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => Lex.{u1} \u03b1) _x) (Equiv.instFunLikeEquiv.{succ u1, succ u1} \u03b1 (Lex.{u1} \u03b1)) (toLex.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (Rat.cast.{u1} ((fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : \u03b1) => Lex.{u1} \u03b1) (Rat.cast.{u1} \u03b1 _inst_1 n)) (instRatCastLex.{u1} \u03b1 _inst_1) n)\nCase conversion may be inaccurate. Consider using '#align to_lex_rat_cast toLex_rat_cast\u2093'. -/\n@[simp]\ntheorem toLex_rat_cast [HasRatCast \u03b1] (n : \u211a) : toLex (n : \u03b1) = n :=\n  rfl\n#align to_lex_rat_cast toLex_rat_cast\n\n/- warning: of_lex_rat_cast -> ofLex_rat_cast is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} [_inst_1 : HasRatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} \u03b1 ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) (coeFn.{1, 1} (Equiv.{1, 1} (Lex.{0} Rat) Rat) (fun (_x : Equiv.{1, 1} (Lex.{0} Rat) Rat) => (Lex.{0} Rat) -> Rat) (Equiv.hasCoeToFun.{1, 1} (Lex.{0} Rat) Rat) (ofLex.{0} Rat) n)) ((fun (a : Type) (b : Type.{u1}) [self : HasLiftT.{1, succ u1} a b] => self.0) Rat \u03b1 (HasLiftT.mk.{1, succ u1} Rat \u03b1 (CoeTC\u2093.coe.{1, succ u1} Rat \u03b1 (Rat.castCoe.{u1} \u03b1 _inst_1))) n)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} [_inst_1 : RatCast.{u1} \u03b1] (n : Rat), Eq.{succ u1} \u03b1 (Rat.cast.{u1} \u03b1 _inst_1 (FunLike.coe.{1, 1, 1} (Equiv.{1, 1} (Lex.{0} Rat) Rat) (Lex.{0} Rat) (fun (_x : Lex.{0} Rat) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Lex.{0} Rat) => Rat) _x) (Equiv.instFunLikeEquiv.{1, 1} (Lex.{0} Rat) Rat) (ofLex.{0} Rat) n)) (Rat.cast.{u1} \u03b1 _inst_1 n)\nCase conversion may be inaccurate. Consider using '#align of_lex_rat_cast ofLex_rat_cast\u2093'. -/\n@[simp]\ntheorem ofLex_rat_cast [HasRatCast \u03b1] (n : \u211a) : (ofLex n : \u03b1) = n :=\n  rfl\n#align of_lex_rat_cast ofLex_rat_cast\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Algebra/Field/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945416, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.35232351093761993}}
{"text": "import algebra.hom.group\nimport data.sum.basic\nimport tactic.simps\n\nuniverses v u w\n-- set_option trace.simps.verbose true\n-- set_option trace.simps.debug true\n-- set_option trace.app_builder true\n\nopen function tactic expr\n\nstructure equiv' (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n(left_inv  : left_inverse inv_fun to_fun)\n(right_inv : right_inverse inv_fun to_fun)\n\nlocal infix (name := equiv') ` \u2243 `:25 := equiv'\n\n/- Since `prod` and `pprod` are a special case for `@[simps]`, we define a new structure to test\n  the basic functionality.-/\nstructure my_prod (\u03b1 \u03b2 : Type*) := (fst : \u03b1) (snd : \u03b2)\n\ndef myprod.map {\u03b1 \u03b1' \u03b2 \u03b2'} (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') (x : my_prod \u03b1 \u03b2) : my_prod \u03b1' \u03b2' :=\n\u27e8f x.1, g x.2\u27e9\n\nnamespace foo\n@[simps] protected def rfl {\u03b1} : \u03b1 \u2243 \u03b1 :=\n\u27e8id, \u03bb x, x, \u03bb x, rfl, \u03bb x, rfl\u27e9\n\n/- simps adds declarations -/\nrun_cmd do\n  e \u2190 get_env,\n  e.get `foo.rfl_to_fun,\n  e.get `foo.rfl_inv_fun,\n  success_if_fail (e.get `foo.rfl_left_inv),\n  success_if_fail (e.get `foo.rfl_right_inv),\n  p \u2190 simps_aux.get_param `foo.rfl,\n  guard $ p = [`foo.rfl_to_fun, `foo.rfl_inv_fun]\n\nexample (n : \u2115) : foo.rfl.to_fun n = n := by rw [foo.rfl_to_fun, id]\nexample (n : \u2115) : foo.rfl.inv_fun n = n := by rw [foo.rfl_inv_fun]\n\n/- the declarations are `simp` lemmas -/\n@[simps] def foo : \u2115 \u00d7 \u2124 := (1, 2)\n\nexample : foo.1 = 1 := by simp\nexample : foo.2 = 2 := by simp\nexample : foo.1 = 1 := by { dsimp, refl } -- check that dsimp also unfolds\nexample : foo.2 = 2 := by { dsimp, refl }\nexample {\u03b1} (x : \u03b1) : foo.rfl.to_fun x = x := by simp\nexample {\u03b1} (x : \u03b1) : foo.rfl.inv_fun x = x := by simp\nexample {\u03b1} (x : \u03b1) : foo.rfl.to_fun = @id \u03b1 := by { success_if_fail {simp}, refl }\n\n/- check some failures -/\ndef bar1 : \u2115 := 1 -- type is not a structure\nnoncomputable def bar2 {\u03b1} : \u03b1 \u2243 \u03b1 :=\nclassical.choice \u27e8foo.rfl\u27e9\n\nrun_cmd do\n  success_if_fail_with_msg (simps_tac `foo.bar1)\n    \"Invalid `simps` attribute. Target nat is not a structure\",\n  success_if_fail_with_msg (simps_tac `foo.bar2)\n    \"Invalid `simps` attribute. The body is not a constructor application:\n  classical.choice bar2._proof_1\",\n  e \u2190 get_env,\n  let nm := `foo.bar1,\n  d \u2190 e.get nm,\n  let lhs : expr := const d.to_name (d.univ_params.map level.param),\n  simps_add_projections e nm d.type lhs d.value [] d.univ_params ff {} [] []\n\n\n/- test that if a non-constructor is given as definition, then\n  `{rhs_md := semireducible, simp_rhs := tt}` is applied automatically. -/\n@[simps] def rfl2 {\u03b1} : \u03b1 \u2243 \u03b1 := foo.rfl\n\nexample {\u03b1} (x : \u03b1) : rfl2.to_fun x = x \u2227 rfl2.inv_fun x = x :=\nbegin\n  dsimp only [rfl2_to_fun, rfl2_inv_fun],\n  guard_target (x = x \u2227 x = x),\n  exact \u27e8rfl, rfl\u27e9\nend\n\n/- test `fully_applied` option -/\n\n@[simps {fully_applied := ff}] def rfl3 {\u03b1} : \u03b1 \u2243 \u03b1 := \u27e8id, \u03bb x, x, \u03bb x, rfl, \u03bb x, rfl\u27e9\n\nend foo\n\n/- we reduce the type when applying [simps] -/\ndef my_equiv := equiv'\n@[simps] def baz : my_equiv \u2115 \u2115 := \u27e8id, \u03bb x, x, \u03bb x, rfl, \u03bb x, rfl\u27e9\n\n/- test name clashes -/\ndef name_clash_fst := 1\ndef name_clash_snd := 1\ndef name_clash_snd_2 := 1\n@[simps] def name_clash := (2, 3)\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `name_clash_fst_2,\n  e.get `name_clash_snd_3\n\n/- check projections for nested structures -/\n\nnamespace count_nested\n@[simps {attrs := [`simp, `norm]}] def nested1 : my_prod \u2115 $ my_prod \u2124 \u2115 :=\n\u27e82, -1, 1\u27e9\n\n@[simps {attrs := []}] def nested2 : \u2115 \u00d7 my_prod \u2115 \u2115 :=\n\u27e82, myprod.map nat.succ nat.pred \u27e81, 2\u27e9\u27e9\n\nend count_nested\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `count_nested.nested1_fst,\n  e.get `count_nested.nested1_snd_fst,\n  e.get `count_nested.nested1_snd_snd,\n  e.get `count_nested.nested2_fst,\n  e.get `count_nested.nested2_snd,\n  is_simp_lemma `count_nested.nested1_fst >>= \u03bb b, guard b, -- simp attribute is global\n  is_simp_lemma `count_nested.nested2_fst >>= \u03bb b, guard $ \u00acb, --lemmas_only doesn't add simp lemma\n  guard $ 7 = e.fold 0 -- there are no other lemmas generated\n    (\u03bb d n, n + if d.to_name.components.init.ilast = `count_nested then 1 else 0)\n\n-- testing with arguments\n@[simps] def bar {\u03b1 : Type*} (n m : \u2115) : \u2115 \u00d7 \u2124 :=\n\u27e8n - m, n + m\u27e9\n\nstructure equiv_plus_data (\u03b1 \u03b2) extends \u03b1 \u2243 \u03b2 :=\n(P : (\u03b1 \u2192 \u03b2) \u2192 Prop)\n(data : P to_fun)\n\nstructure automorphism_plus_data \u03b1 extends \u03b1 \u2295 \u03b1 \u2243 \u03b1 \u2295 \u03b1 :=\n(P : (\u03b1 \u2295 \u03b1 \u2192 \u03b1 \u2295 \u03b1) \u2192 Prop)\n(data : P to_fun)\n(extra : bool \u2192 my_prod \u2115 \u2115)\n\n@[simps]\ndef refl_with_data {\u03b1} : equiv_plus_data \u03b1 \u03b1 :=\n{ P := \u03bb f, f = id,\n  data := rfl,\n  ..foo.rfl }\n\n@[simps]\ndef refl_with_data' {\u03b1} : equiv_plus_data \u03b1 \u03b1 :=\n{ P := \u03bb f, f = id,\n  data := rfl,\n  to_equiv' := foo.rfl }\n\n/- test whether eta expansions are reduced correctly -/\n@[simps]\ndef test {\u03b1} : automorphism_plus_data \u03b1 :=\n{ P := \u03bb f, f = id,\n  data := rfl,\n  extra := \u03bb b, \u27e8(\u27e83, 5\u27e9 : my_prod _ _).1, (\u27e83, 5\u27e9 : my_prod _ _).2\u27e9,\n  ..foo.rfl }\n\n/- test whether this is indeed rejected as a valid eta expansion -/\n@[simps]\ndef test_sneaky {\u03b1} : automorphism_plus_data \u03b1 :=\n{ P := \u03bb f, f = id,\n  data := rfl,\n  extra := \u03bb b, \u27e8(3,5).1,(3,5).2\u27e9,\n  ..foo.rfl }\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `refl_with_data_to_equiv',\n  e.get `refl_with_data'_to_equiv',\n  e.get `test_extra,\n  e.get `test_sneaky_extra_fst,\n  success_if_fail (e.get `refl_with_data_to_equiv_to_fun),\n  success_if_fail (e.get `refl_with_data'_to_equiv_to_fun),\n  success_if_fail (e.get `test_extra_fst),\n  success_if_fail (e.get `test_sneaky_extra)\n\nstructure partially_applied_str :=\n(data : \u2115 \u2192 my_prod \u2115 \u2115)\n\n/- if we have a partially applied constructor, we treat it as if it were eta-expanded -/\n@[simps]\ndef partially_applied_term : partially_applied_str := \u27e8my_prod.mk 3\u27e9\n\n@[simps]\ndef another_term : partially_applied_str := \u27e8\u03bb n, \u27e8n + 1, n + 2\u27e9\u27e9\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `partially_applied_term_data_fst,\n  e.get `partially_applied_term_data_snd\n\nstructure very_partially_applied_str :=\n(data : \u2200\u03b2, \u2115 \u2192 \u03b2 \u2192 my_prod \u2115 \u03b2)\n\n/- if we have a partially applied constructor, we treat it as if it were eta-expanded.\n  (this is not very useful, and we could remove this behavior if convenient) -/\n@[simps]\ndef very_partially_applied_term : very_partially_applied_str := \u27e8@my_prod.mk \u2115\u27e9\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `very_partially_applied_term_data_fst,\n  e.get `very_partially_applied_term_data_snd\n\n@[simps] def let1 : \u2115 \u00d7 \u2124 :=\nlet n := 3 in \u27e8n + 4, 5\u27e9\n\n@[simps] def let2 : \u2115 \u00d7 \u2124 :=\nlet n := 3, m := 4 in let k := 5 in \u27e8n + m, k\u27e9\n\n@[simps] def let3 : \u2115 \u2192 \u2115 \u00d7 \u2124 :=\n\u03bb n, let m := 4, k := 5 in \u27e8n + m, k\u27e9\n\n@[simps] def let4 : \u2115 \u2192 \u2115 \u00d7 \u2124 :=\nlet m := 4, k := 5 in \u03bb n, \u27e8n + m, k\u27e9\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `let1_fst, e.get `let2_fst, e.get `let3_fst, e.get `let4_fst,\n  e.get `let1_snd, e.get `let2_snd, e.get `let3_snd, e.get `let4_snd\n\n\nnamespace specify\n@[simps fst] def specify1 : \u2115 \u00d7 \u2115 \u00d7 \u2115 := (1, 2, 3)\n@[simps snd] def specify2 : \u2115 \u00d7 \u2115 \u00d7 \u2115 := (1, 2, 3)\n@[simps snd_fst] def specify3 : \u2115 \u00d7 \u2115 \u00d7 \u2115 := (1, 2, 3)\n@[simps snd snd_snd snd_snd] def specify4 : \u2115 \u00d7 \u2115 \u00d7 \u2115 := (1, 2, 3) -- last argument is ignored\n@[simps] noncomputable def specify5 : \u2115 \u00d7 \u2115 \u00d7 \u2115 := (1, classical.choice \u27e8(2, 3)\u27e9)\nend specify\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `specify.specify1_fst, e.get `specify.specify2_snd,\n  e.get `specify.specify3_snd_fst, e.get `specify.specify4_snd_snd, e.get `specify.specify4_snd,\n  e.get `specify.specify5_fst, e.get `specify.specify5_snd,\n  guard $ 12 = e.fold 0 -- there are no other lemmas generated\n    (\u03bb d n, n + if d.to_name.components.init.ilast = `specify then 1 else 0),\n  success_if_fail_with_msg (simps_tac `specify.specify1 {} [\"fst_fst\"])\n    \"Invalid simp lemma specify.specify1_fst_fst.\nProjection fst doesn't exist, because target is not a structure.\",\n  success_if_fail_with_msg (simps_tac `specify.specify1 {} [\"foo_fst\"])\n    \"Invalid simp lemma specify.specify1_foo_fst. Structure prod does not have projection foo.\nThe known projections are:\n  [fst, snd]\nYou can also see this information by running\n  `initialize_simps_projections? prod`.\nNote: these projection names might not correspond to the projection names of the structure.\",\n  success_if_fail_with_msg (simps_tac `specify.specify1 {} [\"snd_bar\"])\n    \"Invalid simp lemma specify.specify1_snd_bar. Structure prod does not have projection bar.\nThe known projections are:\n  [fst, snd]\nYou can also see this information by running\n  `initialize_simps_projections? prod`.\nNote: these projection names might not correspond to the projection names of the structure.\",\n  success_if_fail_with_msg (simps_tac `specify.specify5 {} [\"snd_snd\"])\n    \"Invalid simp lemma specify.specify5_snd_snd.\nThe given definition is not a constructor application:\n  classical.choice specify.specify5._proof_1\",\n    p \u2190 simps_aux.get_param `specify.specify4,\n    guard $ p = [`specify.specify4_snd, `specify.specify4_snd_snd]\n\n\n/- We also eta-reduce if we explicitly specify the projection. -/\nattribute [simps extra] test\nrun_cmd do\n  e \u2190 get_env,\n  d1 \u2190 e.get `test_extra,\n  d2 \u2190 e.get `test_extra_2,\n  guard $ d1.type =\u2090 d2.type,\n  skip\n\n/- check simp_rhs option -/\n@[simps {simp_rhs := tt}] def equiv'.trans {\u03b1 \u03b2 \u03b3} (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8g.to_fun \u2218 f.to_fun, f.inv_fun \u2218 g.inv_fun,\n  by { intro x, simp [equiv'.left_inv _ _] }, by { intro x, simp [equiv'.right_inv _ _] }\u27e9\n\n\nexample {\u03b1 \u03b2 \u03b3 : Type} (f : \u03b1 \u2243 \u03b2) (g : \u03b2 \u2243 \u03b3) (x : \u03b1) :\n  (f.trans g).to_fun x = (f.trans g).to_fun x :=\nbegin\n  dsimp only [equiv'.trans_to_fun],\n  guard_target g.to_fun (f.to_fun x) = g.to_fun (f.to_fun x),\n  refl,\nend\n\nlocal attribute [simp] nat.zero_add nat.one_mul nat.mul_one\n@[simps {simp_rhs := tt}] def my_nat_equiv : \u2115 \u2243 \u2115 :=\n\u27e8\u03bb n, 0 + n, \u03bb n, 1 * n * 1, by { intro n, simp }, by { intro n, simp }\u27e9\n\nrun_cmd success_if_fail (has_attribute `_refl_lemma `my_nat_equiv'_to_fun) >>\n  has_attribute `_refl_lemma `equiv'.trans_to_fun\n\nexample (n : \u2115) : my_nat_equiv.to_fun (my_nat_equiv.to_fun $ my_nat_equiv.inv_fun n) = n :=\nby { success_if_fail { refl }, simp only [my_nat_equiv_to_fun, my_nat_equiv_inv_fun] }\n\n@[simps {simp_rhs := tt}] def succeed_without_simplification_possible : \u2115 \u2243 \u2115 :=\n\u27e8\u03bb n, n, \u03bb n, n, by { intro n, refl }, by { intro n, refl }\u27e9\n\n\n/- test that we don't recursively take projections of `prod` and `pprod` -/\n@[simps] def pprod_equiv_prod : pprod \u2115 \u2115 \u2243 \u2115 \u00d7 \u2115 :=\n{ to_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  inv_fun := \u03bb x, \u27e8x.1, x.2\u27e9,\n  left_inv := \u03bb \u27e8x, y\u27e9, rfl,\n  right_inv := \u03bb \u27e8x, y\u27e9, rfl }\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `pprod_equiv_prod_to_fun,\n  e.get `pprod_equiv_prod_inv_fun\n\nattribute [simps to_fun_fst inv_fun_snd] pprod_equiv_prod\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `pprod_equiv_prod_to_fun_fst,\n  e.get `pprod_equiv_prod_inv_fun_snd\n\n-- we can disable this behavior with the option `not_recursive`.\n@[simps {not_recursive := []}] def pprod_equiv_prod2 : pprod \u2115 \u2115 \u2243 \u2115 \u00d7 \u2115 :=\npprod_equiv_prod\n\nrun_cmd do\n  e \u2190 get_env,\n  e.get `pprod_equiv_prod2_to_fun_fst,\n  e.get `pprod_equiv_prod2_to_fun_snd,\n  e.get `pprod_equiv_prod2_inv_fun_fst,\n  e.get `pprod_equiv_prod2_inv_fun_snd\n\n/- Tests with universe levels -/\nclass has_hom (obj : Type u) : Type (max u (v+1)) :=\n(hom : obj \u2192 obj \u2192 Type v)\n\ninfixr ` \u27f6 `:10 := has_hom.hom -- type as \\h\n\nclass category_struct (obj : Type u) extends has_hom.{v} obj : Type (max u (v+1)) :=\n(id       : \u03a0 X : obj, hom X X)\n(comp     : \u03a0 {X Y Z : obj}, (X \u27f6 Y) \u2192 (Y \u27f6 Z) \u2192 (X \u27f6 Z))\n\nnotation `\ud835\udfd9` := category_struct.id -- type as \\b1\ninfixr ` \u226b `:80 := category_struct.comp -- type as \\gg\n\n@[simps] instance types : category_struct (Type u) :=\n{ hom     := \u03bb a b, (a \u2192 b),\n  id      := \u03bb a, id,\n  comp    := \u03bb _ _ _ f g, g \u2218 f }\n\nexample (X : Type u) : (X \u27f6 X) = (X \u2192 X) := by simp\nexample (X : Type u) : \ud835\udfd9 X = (\u03bb x, x) := by { funext, simp }\nexample (X Y Z : Type u) (f : X \u27f6 Y) (g : Y \u27f6 Z) : f \u226b g = g \u2218 f := by { funext, simp }\n\nnamespace coercing\n\nstructure foo_str :=\n (c : Type)\n (x : c)\n\ninstance : has_coe_to_sort foo_str Type := \u27e8foo_str.c\u27e9\n\n@[simps] def foo : foo_str := \u27e8\u2115, 3\u27e9\n@[simps] def foo2 : foo_str := \u27e8\u2115, 34\u27e9\n\nexample : \u21a5foo = \u2115 := by simp only [foo_c]\nexample : foo.x = (3 : \u2115) := by simp only [foo_x]\n\nstructure voo_str (n : \u2115) :=\n (c : Type)\n (x : c)\n\ninstance has_coe_voo_str (n : \u2115) : has_coe_to_sort (voo_str n) Type := \u27e8voo_str.c\u27e9\n\n@[simps] def voo : voo_str 7 := \u27e8\u2115, 3\u27e9\n@[simps] def voo2 : voo_str 4 := \u27e8\u2115, 34\u27e9\n\nexample : \u21a5voo = \u2115 := by simp only [voo_c]\nexample : voo.x = (3 : \u2115) := by simp only [voo_x]\n\nstructure equiv2 (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n(left_inv  : left_inverse inv_fun to_fun)\n(right_inv : right_inverse inv_fun to_fun)\n\ninstance {\u03b1 \u03b2} : has_coe_to_fun (equiv2 \u03b1 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv2.to_fun\u27e9\n\n@[simps] protected def rfl2 {\u03b1} : equiv2 \u03b1 \u03b1 :=\n\u27e8\u03bb x, x, \u03bb x, x, \u03bb x, rfl, \u03bb x, rfl\u27e9\n\nexample {\u03b1} (x : \u03b1) : coercing.rfl2 x = x := by rw [coercing.rfl2_to_fun]\nexample {\u03b1} (x : \u03b1) : coercing.rfl2 x = x := by simp\nexample {\u03b1} (x : \u03b1) : coercing.rfl2.inv_fun x = x := by simp\n\n@[simps] protected def equiv2.symm {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) : equiv2 \u03b2 \u03b1 :=\n\u27e8f.inv_fun, f, f.right_inv, f.left_inv\u27e9\n\n@[simps] protected def equiv2.symm2 {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) : equiv2 \u03b2 \u03b1 :=\n\u27e8f.inv_fun, f.to_fun, f.right_inv, f.left_inv\u27e9\n\n@[simps {fully_applied := ff}] protected def equiv2.symm3 {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) : equiv2 \u03b2 \u03b1 :=\n\u27e8f.inv_fun, f, f.right_inv, f.left_inv\u27e9\n\nexample {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) (y : \u03b2) : f.symm y = f.inv_fun y := by simp\nexample {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) (x : \u03b1) : f.symm.inv_fun x = f x := by simp\n\nexample {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) : f.symm.inv_fun = f := by { success_if_fail {simp}, refl }\nexample {\u03b1 \u03b2} (f : equiv2 \u03b1 \u03b2) : f.symm3.inv_fun = f := by simp\n\nsection\nset_option old_structure_cmd true\nclass semigroup (G : Type u) extends has_mul G :=\n(mul_assoc : \u2200 a b c : G, a * b * c = a * (b * c))\nend\n\n@[simps] instance {\u03b1 \u03b2} [semigroup \u03b1] [semigroup \u03b2] : semigroup (\u03b1 \u00d7 \u03b2) :=\n{ mul := \u03bb x y, (x.1 * y.1, x.2 * y.2),\n  mul_assoc := by { intros, simp only [semigroup.mul_assoc], refl } }\n\nexample {\u03b1 \u03b2} [semigroup \u03b1] [semigroup \u03b2] (x y : \u03b1 \u00d7 \u03b2) : x * y = (x.1 * y.1, x.2 * y.2) :=\nby simp\nexample {\u03b1 \u03b2} [semigroup \u03b1] [semigroup \u03b2] (x y : \u03b1 \u00d7 \u03b2) : (x * y).1 = x.1 * y.1 := by simp\n\nstructure Semigroup :=\n  (G : Type*)\n  (op : G \u2192 G \u2192 G)\n  (infix (name := op) ` * ` := op)\n  (op_assoc : \u2200 (x y z : G), (x * y) * z = x * (y * z))\n\nnamespace Group\n\ninstance : has_coe_to_sort Semigroup Type* := \u27e8Semigroup.G\u27e9\n-- We could try to generate lemmas with this `has_mul` instance, but it is unused in mathlib.\n-- Therefore, this is ignored.\ninstance (G : Semigroup) : has_mul G := \u27e8G.op\u27e9\n\n@[simps] def prod_Semigroup (G H : Semigroup) : Semigroup :=\n{ G := G \u00d7 H,\n  op := \u03bb x y, (x.1 * y.1, x.2 * y.2),\n  op_assoc := by { intros, dsimp [Group.has_mul], simp [Semigroup.op_assoc] }}\n\n\nend Group\n\nsection\nset_option old_structure_cmd true\nclass extending_stuff (G : Type u) extends has_mul G, has_zero G, has_neg G, has_subset G :=\n(new_axiom : \u2200 x : G, x * - 0 \u2286 - x)\nend\n\n@[simps] def bar : extending_stuff \u2115 :=\n{ mul := (*),\n  zero := 0,\n  neg := nat.succ,\n  subset := \u03bb x y, true,\n  new_axiom := \u03bb x, trivial }\n\nsection\nlocal attribute [instance] bar\nexample (x : \u2115) : x * - 0 \u2286 - x := by simp\nend\n\nclass new_extending_stuff (G : Type u) extends has_mul G, has_zero G, has_neg G, has_subset G :=\n(new_axiom : \u2200 x : G, x * - 0 \u2286 - x)\n\n@[simps] def new_bar : new_extending_stuff \u2115 :=\n{ mul := (*),\n  zero := 0,\n  neg := nat.succ,\n  subset := \u03bb x y, true,\n  new_axiom := \u03bb x, trivial }\n\nsection\nlocal attribute [instance] new_bar\nexample (x : \u2115) : x * - 0 \u2286 - x := by simp\nend\n\n\nend coercing\n\nnamespace manual_coercion\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := manual_coercion.equiv\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\ndef equiv.simps.inv_fun (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\n/-- Composition of equivalences `e\u2081 : \u03b1 \u2243 \u03b2` and `e\u2082 : \u03b2 \u2243 \u03b3`. -/\n@[simps {simp_rhs := tt}] protected def equiv.trans (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\nexample (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) (x : \u03b3) : (e\u2081.trans e\u2082).symm x = e\u2081.symm (e\u2082.symm x) :=\nby simp only [equiv.trans_inv_fun]\n\nend manual_coercion\n\nnamespace faulty_manual_coercion\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := faulty_manual_coercion.equiv\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\n/-- See Note [custom simps projection] -/\nnoncomputable def equiv.simps.inv_fun (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := classical.choice \u27e8e.inv_fun\u27e9\n\nrun_cmd do e \u2190 get_env, success_if_fail_with_msg\n  (simps_get_raw_projections e `faulty_manual_coercion.equiv)\n\"Invalid custom projection:\n  \u03bb {\u03b1 : Sort u_1} {\u03b2 : Sort u_2} (e : \u03b1 \u2243 \u03b2), classical.choice _\nExpression is not definitionally equal to\n  \u03bb (\u03b1 : Sort u_1) (\u03b2 : Sort u_2) (x : \u03b1 \u2243 \u03b2), x.inv_fun\"\n\nend faulty_manual_coercion\n\nnamespace manual_initialize\n/- defining a manual coercion. -/\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := manual_initialize.equiv\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\n-- test: intentionally using different unvierse levels for equiv.symm than for equiv\ndef equiv.simps.inv_fun (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\ninitialize_simps_projections equiv\n\nrun_cmd has_attribute `_simps_str `manual_initialize.equiv\n\n/-- Composition of equivalences `e\u2081 : \u03b1 \u2243 \u03b2` and `e\u2082 : \u03b2 \u2243 \u03b3`. -/\n@[simps {simp_rhs := tt}] protected def equiv.trans (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\nend manual_initialize\n\nnamespace faulty_universes\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\nstructure equiv (\u03b1 : Sort u) (\u03b2 : Sort v) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := faulty_universes.equiv\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\n-- test: intentionally using different names for the universe variables for equiv.symm than for\n-- equiv\ndef equiv.simps.inv_fun {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\nrun_cmd do e \u2190 get_env,\n  success_if_fail_with_msg (simps_get_raw_projections e `faulty_universes.equiv)\n\"Invalid custom projection:\n  \u03bb {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2), \u21d1(e.symm)\nExpression has different type than faulty_universes.equiv.inv_fun. Given type:\n  \u03a0 {\u03b1 : Type u} {\u03b2 : Type v} (e : \u03b1 \u2243 \u03b2), (\u03bb (_x : \u03b2 \u2243 \u03b1), \u03b2 \u2192 \u03b1) e.symm\nExpected type:\n  \u03a0 (\u03b1 : Sort u) (\u03b2 : Sort v), \u03b1 \u2243 \u03b2 \u2192 \u03b2 \u2192 \u03b1\"\n\nend faulty_universes\n\nnamespace manual_universes\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\nstructure equiv (\u03b1 : Sort u) (\u03b2 : Sort v) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := manual_universes.equiv\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\n-- test: intentionally using different unvierse levels for equiv.symm than for equiv\ndef equiv.simps.inv_fun {\u03b1 : Sort w} {\u03b2 : Sort u} (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\n-- check whether we can generate custom projections even if the universe names don't match\ninitialize_simps_projections equiv\n\nend manual_universes\n\nnamespace manual_projection_names\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := manual_projection_names.equiv\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\ndef equiv.simps.symm_apply (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\ninitialize_simps_projections equiv (to_fun \u2192 apply, inv_fun \u2192 symm_apply)\n\nrun_cmd do\n  e \u2190 get_env,\n  data \u2190 simps_get_raw_projections e `manual_projection_names.equiv,\n  guard $ data.2.map projection_data.name = [`apply, `symm_apply]\n\n@[simps {simp_rhs := tt}] protected def equiv.trans (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\nexample (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) (x : \u03b1) : (e\u2081.trans e\u2082) x = e\u2082 (e\u2081 x) :=\nby simp only [equiv.trans_apply]\n\nexample (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) (x : \u03b3) : (e\u2081.trans e\u2082).symm x = e\u2081.symm (e\u2082.symm x) :=\nby simp only [equiv.trans_symm_apply]\n\n-- the new projection names are parsed correctly (the old projection names won't work anymore)\n@[simps apply symm_apply] protected def equiv.trans2 (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\n\nend manual_projection_names\n\nnamespace prefix_projection_names\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := prefix_projection_names.equiv\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\ninstance : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv.to_fun\u27e9\n\ndef equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n/-- See Note [custom simps projection] -/\ndef equiv.simps.symm_apply (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\ninitialize_simps_projections equiv (to_fun \u2192 coe as_prefix, inv_fun \u2192 symm_apply)\n\nrun_cmd do\n  e \u2190 get_env,\n  data \u2190 simps_get_raw_projections e `prefix_projection_names.equiv,\n  guard $ data.2.map projection_data.name = [`coe, `symm_apply],\n  guard $ data.2.map projection_data.is_prefix = [tt, ff]\n\n@[simps {simp_rhs := tt}] protected def equiv.trans (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\nexample (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) (x : \u03b1) : (e\u2081.trans e\u2082) x = e\u2082 (e\u2081 x) :=\nby simp only [equiv.coe_trans]\n\n-- the new projection names are parsed correctly\n@[simps coe symm_apply] protected def equiv.trans2 (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b2 \u2243 \u03b3) : \u03b1 \u2243 \u03b3 :=\n\u27e8e\u2082 \u2218 e\u2081, e\u2081.symm \u2218 e\u2082.symm\u27e9\n\n-- it interacts somewhat well with multiple projections (though the generated name is not great)\n@[simps snd_coe_fst] def foo {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} (x : \u03b1) (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b3 \u2243 \u03b4) :\n  \u03b1 \u00d7 (\u03b1 \u00d7 \u03b3 \u2243 \u03b2 \u00d7 \u03b4) :=\n\u27e8x, prod.map e\u2081 e\u2082, prod.map e\u2081.symm e\u2082.symm\u27e9\n\nexample {\u03b1 \u03b2 \u03b3 \u03b4 : Type*} (x : \u03b1) (e\u2081 : \u03b1 \u2243 \u03b2) (e\u2082 : \u03b3 \u2243 \u03b4) (z : \u03b1 \u00d7 \u03b3) :\n  ((foo x e\u2081 e\u2082).2 z).1 = e\u2081 z.1 :=\nby simp only [coe_foo_snd_fst]\n\nend prefix_projection_names\n\n\n-- test transparency setting\nstructure set_plus (\u03b1 : Type) :=\n(s : set \u03b1)\n(x : \u03b1)\n(h : x \u2208 s)\n\n@[simps] def nat_set_plus : set_plus \u2115 := \u27e8set.univ, 1, trivial\u27e9\n\nexample : nat_set_plus.s = set.univ :=\nbegin\n  dsimp only [nat_set_plus_s],\n  guard_target @set.univ \u2115 = set.univ,\n  refl\nend\n\n@[simps {type_md := semireducible}] def nat_set_plus2 : set_plus \u2115 := \u27e8set.univ, 1, trivial\u27e9\n\nexample : nat_set_plus2.s = set.univ :=\nbegin\n  success_if_fail { dsimp only [nat_set_plus2_s] }, refl\nend\n\n@[simps {rhs_md := semireducible}] def nat_set_plus3 : set_plus \u2115 := nat_set_plus\n\nexample : nat_set_plus3.s = set.univ :=\nbegin\n  dsimp only [nat_set_plus3_s],\n  guard_target @set.univ \u2115 = set.univ,\n  refl\nend\n\nnamespace nested_non_fully_applied\n\nstructure equiv (\u03b1 : Sort*) (\u03b2 : Sort*) :=\n(to_fun    : \u03b1 \u2192 \u03b2)\n(inv_fun   : \u03b2 \u2192 \u03b1)\n\nlocal infix (name := equiv) ` \u2243 `:25 := nested_non_fully_applied.equiv\n\nvariables {\u03b1 \u03b2 \u03b3 : Sort*}\n\n@[simps] def equiv.symm (e : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 := \u27e8e.inv_fun, e.to_fun\u27e9\n\n@[simps {rhs_md := semireducible, fully_applied := ff}] def equiv.symm2 : (\u03b1 \u2243 \u03b2) \u2243 (\u03b2 \u2243 \u03b1) :=\n\u27e8equiv.symm, equiv.symm\u27e9\n\nexample (e : \u03b1 \u2243 \u03b2) : (equiv.symm2.inv_fun e).to_fun = e.inv_fun :=\nbegin\n  dsimp only [equiv.symm2_inv_fun_to_fun],\n  guard_target e.inv_fun = e.inv_fun,\n  refl\nend\n\n/- do not prematurely unfold `equiv.symm`, unless necessary -/\n@[simps to_fun to_fun_to_fun {rhs_md := semireducible}] def equiv.symm3 : (\u03b1 \u2243 \u03b2) \u2243 (\u03b2 \u2243 \u03b1) :=\nequiv.symm2\n\nexample (e : \u03b1 \u2243 \u03b2) (y : \u03b2) : (equiv.symm3.to_fun e).to_fun y = e.inv_fun y \u2227\n  (equiv.symm3.to_fun e).to_fun y = e.inv_fun y :=\nbegin\n  split,\n  { dsimp only [equiv.symm3_to_fun], guard_target e.symm.to_fun y = e.inv_fun y, refl },\n  { dsimp only [equiv.symm3_to_fun_to_fun], guard_target e.inv_fun y = e.inv_fun y, refl }\nend\n\nend nested_non_fully_applied\n\n-- test that type classes which are props work\nclass prop_class (n : \u2115) : Prop :=\n(has_true : true)\n\ninstance has_prop_class (n : \u2115) : prop_class n := \u27e8trivial\u27e9\n\nstructure needs_prop_class (n : \u2115) [prop_class n] :=\n(t : true)\n\n@[simps] def test_prop_class : needs_prop_class 1 :=\n{ t := trivial }\n\n/- check that when the coercion is given in eta-expanded form, we can also find the coercion. -/\nstructure alg_hom (R A B : Type*) :=\n(to_fun : A \u2192 B)\n\ninstance (R A B : Type*) : has_coe_to_fun (alg_hom R A B) (\u03bb _, A \u2192 B) := \u27e8\u03bb f, f.to_fun\u27e9\n\n@[simps] def my_alg_hom : alg_hom unit bool bool :=\n{ to_fun := id }\n\nexample (x : bool) : my_alg_hom x = id x := by simp only [my_alg_hom_to_fun]\n\nstructure ring_hom (A B : Type*) :=\n(to_fun : A \u2192 B)\n\ninstance (A B : Type*) : has_coe_to_fun (ring_hom A B) (\u03bb _, A \u2192 B) := \u27e8\u03bb f, f.to_fun\u27e9\n\n@[simps] def my_ring_hom : ring_hom bool bool :=\n{ to_fun := id }\n\nexample (x : bool) : my_ring_hom x = id x := by simp only [my_ring_hom_to_fun]\n\n/- check interaction with the `@[to_additive]` attribute -/\n\n@[to_additive, simps]\ninstance {M N} [has_mul M] [has_mul N] : has_mul (M \u00d7 N) := \u27e8\u03bb p q, \u27e8p.1 * q.1, p.2 * q.2\u27e9\u27e9\n\nrun_cmd do\n  get_decl `prod.has_mul_mul,\n  get_decl `prod.has_add_add,\n  has_attribute `to_additive `prod.has_mul,\n  has_attribute `to_additive `prod.has_mul_mul,\n  has_attribute `simp `prod.has_mul_mul,\n  has_attribute `simp `prod.has_add_add\n\nexample {M N} [has_mul M] [has_mul N] (p q : M \u00d7 N) : p * q = \u27e8p.1 * q.1, p.2 * q.2\u27e9 := by simp\nexample {M N} [has_add M] [has_add N] (p q : M \u00d7 N) : p + q = \u27e8p.1 + q.1, p.2 + q.2\u27e9 := by simp\n\n/- The names of the generated simp lemmas for the additive version are not great if the definition\n  had a custom additive name -/\n@[to_additive my_add_instance, simps]\ninstance my_instance {M N} [has_one M] [has_one N] : has_one (M \u00d7 N) := \u27e8(1, 1)\u27e9\n\nrun_cmd do\n  get_decl `my_instance_one,\n  get_decl `my_add_instance_zero,\n  has_attribute `to_additive `my_instance,\n  has_attribute `to_additive `my_instance_one,\n  has_attribute `simp `my_instance_one,\n  has_attribute `simp `my_add_instance_zero\n\nexample {M N} [has_one M] [has_one N] : (1 : M \u00d7 N) = \u27e81, 1\u27e9 := by simp\nexample {M N} [has_zero M] [has_zero N] : (0 : M \u00d7 N) = \u27e80, 0\u27e9 := by simp\n\nsection\n/-! Test `dsimp, simp` with the option `simp_rhs` -/\n\nlocal attribute [simp] nat.add\n\nstructure my_type :=\n(A : Type)\n\n@[simps {simp_rhs := tt}] def my_type_def : my_type := \u27e8{ x : fin (nat.add 3 0) // 1 + 1 = 2 }\u27e9\n\nexample (h : false) (x y : { x : fin (nat.add 3 0) // 1 + 1 = 2 }) : my_type_def.A = unit :=\nbegin\n  simp only [my_type_def_A],\n  guard_target ({ x : fin 3 // true } = unit),\n  /- note: calling only one of `simp` or `dsimp` does not produce the current target,\n  as the following tests show. -/\n  success_if_fail { guard_hyp x : { x : fin 3 // true } },\n  dsimp at x,\n  success_if_fail { guard_hyp x : { x : fin 3 // true } },\n  simp at y,\n  success_if_fail { guard_hyp y : { x : fin 3 // true } },\n  simp at x, dsimp at y,\n  guard_hyp x : { x : fin 3 // true },\n  guard_hyp y : { x : fin 3 // true },\n  contradiction\nend\n\n/- Test that `to_additive` copies the `@[_refl_lemma]` attribute correctly -/\n@[to_additive, simps]\ndef monoid_hom.my_comp {M N P : Type*} [mul_one_class M] [mul_one_class N] [mul_one_class P]\n  (hnp : N \u2192* P) (hmn : M \u2192* N) : M \u2192* P :=\n{ to_fun := hnp \u2218 hmn, map_one' := by simp, map_mul' := by simp, }\n\n-- `simps` adds the `_refl_lemma` attribute to `monoid_hom.my_comp_apply`\nexample {M N P : Type*} [mul_one_class M] [mul_one_class N] [mul_one_class P]\n  (hnp : N \u2192* P) (hmn : M \u2192* N) (m : M) : hnp.my_comp hmn m = hnp (hmn m) :=\nby { dsimp, guard_target (hnp (hmn m) = hnp (hmn m)), refl }\n\n-- `to_additive` adds the `_refl_lemma` attribute to `add_monoid_hom.my_comp_apply`\nexample {M N P : Type*} [add_zero_class M] [add_zero_class N] [add_zero_class P]\n  (hnp : N \u2192+ P) (hmn : M \u2192+ N) (m : M) : hnp.my_comp hmn m = hnp (hmn m) :=\nby { dsimp, guard_target (hnp (hmn m) = hnp (hmn m)), refl }\n\n-- test that `to_additive` works with a custom name\n@[to_additive some_test2, simps]\ndef some_test1 (M : Type*) [comm_monoid M] : subtype (\u03bb f : M, true) := \u27e81, trivial\u27e9\n\nrun_cmd get_decl `some_test2_coe\n\nend\n\n/- Test custom compositions of projections. -/\n\nsection comp_projs\n\ninstance {\u03b1 \u03b2} : has_coe_to_fun (\u03b1 \u2243 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8equiv'.to_fun\u27e9\n\n@[simps] protected def equiv'.symm {\u03b1 \u03b2} (f : \u03b1 \u2243 \u03b2) : \u03b2 \u2243 \u03b1 :=\n\u27e8f.inv_fun, f, f.right_inv, f.left_inv\u27e9\n\nstructure decorated_equiv (\u03b1 : Sort*) (\u03b2 : Sort*) extends equiv' \u03b1 \u03b2 :=\n(P_to_fun    : function.injective to_fun )\n(P_inv_fun   : function.injective inv_fun)\n\ninstance {\u03b1 \u03b2} : has_coe_to_fun (decorated_equiv \u03b1 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) := \u27e8\u03bb f, f.to_equiv'\u27e9\n\ndef decorated_equiv.symm {\u03b1 \u03b2 : Sort*} (e : decorated_equiv \u03b1 \u03b2) : decorated_equiv \u03b2 \u03b1 :=\n{ to_equiv' := e.to_equiv'.symm,\n  P_to_fun := e.P_inv_fun,\n  P_inv_fun := e.P_to_fun }\n\ndef decorated_equiv.simps.apply {\u03b1 \u03b2 : Sort*} (e : decorated_equiv \u03b1 \u03b2) : \u03b1 \u2192 \u03b2 := e\ndef decorated_equiv.simps.symm_apply {\u03b1 \u03b2 : Sort*} (e : decorated_equiv \u03b1 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\ninitialize_simps_projections decorated_equiv\n  (to_equiv'_to_fun \u2192 apply, to_equiv'_inv_fun \u2192 symm_apply, -to_equiv')\n\n@[simps] def foo (\u03b1 : Type) : decorated_equiv \u03b1 \u03b1 :=\n{ to_fun    := \u03bb x, x,\n  inv_fun   := \u03bb x, x,\n  left_inv  := \u03bb x, rfl,\n  right_inv := \u03bb x, rfl,\n  P_to_fun  := \u03bb x y h, h,\n  P_inv_fun := \u03bb x y h, h }\n\nexample {\u03b1 : Type} (x : \u03b1) : (foo \u03b1).symm x = x :=\nby { dsimp, guard_target (x = x), refl }\n\n@[simps to_equiv' apply symm_apply] def foo2 (\u03b1 : Type) : decorated_equiv \u03b1 \u03b1 :=\n{ P_to_fun  := \u03bb x y h, h,\n  P_inv_fun := \u03bb x y h, h, ..foo.rfl }\n\nexample {\u03b1 : Type} (x : \u03b1) : (foo2 \u03b1).to_equiv' x = x :=\nby { dsimp, guard_target (foo.rfl x = x), refl }\n\nexample {\u03b1 : Type} (x : \u03b1) : foo2 \u03b1 x = x :=\nby { dsimp, guard_target (x = x), refl }\n\nstructure further_decorated_equiv (\u03b1 : Sort*) (\u03b2 : Sort*) extends decorated_equiv \u03b1 \u03b2 :=\n(Q_to_fun    : function.surjective to_fun )\n(Q_inv_fun   : function.surjective inv_fun )\n\ninstance {\u03b1 \u03b2} : has_coe_to_fun (further_decorated_equiv \u03b1 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) :=\n\u27e8\u03bb f, f.to_decorated_equiv\u27e9\n\ndef further_decorated_equiv.symm {\u03b1 \u03b2 : Sort*} (e : further_decorated_equiv \u03b1 \u03b2) :\n  further_decorated_equiv \u03b2 \u03b1 :=\n{ to_decorated_equiv := e.to_decorated_equiv.symm,\n  Q_to_fun := e.Q_inv_fun,\n  Q_inv_fun := e.Q_to_fun }\n\ndef further_decorated_equiv.simps.apply {\u03b1 \u03b2 : Sort*} (e : further_decorated_equiv \u03b1 \u03b2) : \u03b1 \u2192 \u03b2 := e\ndef further_decorated_equiv.simps.symm_apply {\u03b1 \u03b2 : Sort*} (e : further_decorated_equiv \u03b1 \u03b2) :\n  \u03b2 \u2192 \u03b1 := e.symm\n\ninitialize_simps_projections further_decorated_equiv\n  (to_decorated_equiv_to_equiv'_to_fun \u2192 apply, to_decorated_equiv_to_equiv'_inv_fun \u2192 symm_apply,\n  -to_decorated_equiv, to_decorated_equiv_to_equiv' \u2192 to_equiv', -to_equiv')\n\n@[simps] def ffoo (\u03b1 : Type) : further_decorated_equiv \u03b1 \u03b1 :=\n{ to_fun    := \u03bb x, x,\n  inv_fun   := \u03bb x, x,\n  left_inv  := \u03bb x, rfl,\n  right_inv := \u03bb x, rfl,\n  P_to_fun  := \u03bb x y h, h,\n  P_inv_fun := \u03bb x y h, h,\n  Q_to_fun  := \u03bb y, \u27e8y, rfl\u27e9,\n  Q_inv_fun := \u03bb y, \u27e8y, rfl\u27e9 }\n\nexample {\u03b1 : Type} (x : \u03b1) : (ffoo \u03b1).symm x = x :=\nby { dsimp, guard_target (x = x), refl }\n\n@[simps] def ffoo3 (\u03b1 : Type) : further_decorated_equiv \u03b1 \u03b1 :=\n{ Q_to_fun  := \u03bb y, \u27e8y, rfl\u27e9, Q_inv_fun  := \u03bb y, \u27e8y, rfl\u27e9, .. foo \u03b1 }\n\n@[simps apply to_equiv'_to_fun to_decorated_equiv_apply]\ndef ffoo4 (\u03b1 : Type) : further_decorated_equiv \u03b1 \u03b1 :=\n{ Q_to_fun  := \u03bb y, \u27e8y, rfl\u27e9, Q_inv_fun  := \u03bb y, \u27e8y, rfl\u27e9, to_decorated_equiv := foo \u03b1 }\n\nstructure one_more (\u03b1 : Sort*) (\u03b2 : Sort*) extends further_decorated_equiv \u03b1 \u03b2\n\ninstance {\u03b1 \u03b2} : has_coe_to_fun (one_more \u03b1 \u03b2) (\u03bb _, \u03b1 \u2192 \u03b2) :=\n\u27e8\u03bb f, f.to_further_decorated_equiv\u27e9\n\ndef one_more.symm {\u03b1 \u03b2 : Sort*} (e : one_more \u03b1 \u03b2) :\n  one_more \u03b2 \u03b1 :=\n{ to_further_decorated_equiv := e.to_further_decorated_equiv.symm }\n\ndef one_more.simps.apply {\u03b1 \u03b2 : Sort*} (e : one_more \u03b1 \u03b2) : \u03b1 \u2192 \u03b2 := e\ndef one_more.simps.symm_apply {\u03b1 \u03b2 : Sort*} (e : one_more \u03b1 \u03b2) : \u03b2 \u2192 \u03b1 := e.symm\n\ninitialize_simps_projections one_more\n  (to_further_decorated_equiv_to_decorated_equiv_to_equiv'_to_fun \u2192 apply,\n   to_further_decorated_equiv_to_decorated_equiv_to_equiv'_inv_fun \u2192 symm_apply,\n  -to_further_decorated_equiv, to_further_decorated_equiv_to_decorated_equiv \u2192 to_dequiv,\n  -to_dequiv)\n\n@[simps] def fffoo (\u03b1 : Type) : one_more \u03b1 \u03b1 :=\n{ to_fun    := \u03bb x, x,\n  inv_fun   := \u03bb x, x,\n  left_inv  := \u03bb x, rfl,\n  right_inv := \u03bb x, rfl,\n  P_to_fun  := \u03bb x y h, h,\n  P_inv_fun := \u03bb x y h, h,\n  Q_to_fun  := \u03bb y, \u27e8y, rfl\u27e9,\n  Q_inv_fun := \u03bb y, \u27e8y, rfl\u27e9 }\n\nexample {\u03b1 : Type} (x : \u03b1) : (fffoo \u03b1).symm x = x :=\nby { dsimp, guard_target (x = x), refl }\n\n@[simps apply to_dequiv_apply to_further_decorated_equiv_apply to_dequiv]\ndef fffoo2 (\u03b1 : Type) : one_more \u03b1 \u03b1 := fffoo \u03b1\n\n/- test the case where a projection takes additional arguments. -/\nvariables {\u03b9 : Type*} [decidable_eq \u03b9] (A : \u03b9 \u2192 Type*)\n\nclass something [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] :=\n(mul {i} : A i \u2192+ A i)\n\ndef something.simps.apply [has_add \u03b9] [\u03a0 i, add_comm_monoid (A i)] [something A] {i : \u03b9} (x : A i) :\n  A i :=\nsomething.mul \u03b9 x\n\ninitialize_simps_projections something (mul_to_fun \u2192 apply, -mul)\n\nclass something2 [has_add \u03b9] :=\n(mul {i j} : A i \u2243 (A j \u2243 A (i + j)))\n\ndef something2.simps.mul [has_add \u03b9] [something2 A] {i j : \u03b9}\n  (x : A i) (y : A j) : A (i + j) :=\nsomething2.mul x y\n\ninitialize_simps_projections something2 (mul \u2192 mul', mul_to_fun_to_fun \u2192 mul, -mul')\n\nattribute [ext] equiv'\n\n@[simps]\ndef thing (h : bool \u2243 (bool \u2243 bool)) : something2 (\u03bb x : \u2115, bool) :=\n{ mul := \u03bb i j, { to_fun := \u03bb b, { to_fun := h b,\n  inv_fun := (h b).symm,\n  left_inv := (h b).left_inv,\n  right_inv := (h b).right_inv },\n  inv_fun := h.symm,\n  left_inv := by { convert h.left_inv, ext x; refl },\n  right_inv := by { convert h.right_inv, ext x; refl } } }\n\nexample (h : bool \u2243 (bool \u2243 bool)) (i j : \u2115) (b1 b2 : bool) :\n  @something2.mul _ _ _ _ (thing h) i j b1 b2 = h b1 b2 :=\nby simp only [thing_mul]\n\nend comp_projs\n\nsection\n/-! Check that the tactic also works if the elaborated type of `type` reduces to `Sort*`, but is\n  not `Sort*` itself. -/\nstructure my_functor (C D : Type*) :=\n(obj []    : C \u2192 D)\nlocal infixr ` \u2964 `:26 := my_functor\n\n@[simps]\ndef foo_sum {I J : Type*} (C : I \u2192 Type*) {D : J \u2192 Type*} :\n  (\u03a0 i, C i) \u2964 (\u03a0 j, D j) \u2964 (\u03a0 s : I \u2295 J, sum.elim C D s) :=\n{ obj := \u03bb f, { obj := \u03bb g s, sum.rec f g s }}\n\nend\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/test/simps.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.6584175139669997, "lm_q1q2_score": 0.35231817710398156}}
{"text": "/-\nCopyright (c) 2015 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Mario Carneiro\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.data.list.perm\nimport Mathlib.algebra.group_power.default\nimport Mathlib.PostPort\n\nuniverses u u_1 u_4 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Multisets\n\nThese are implemented as the quotient of a list by permutations.\n\n## Notation\n\nWe define the global infix notation `::\u2098` for `multiset.cons`.\n-/\n\n/-- `multiset \u03b1` is the quotient of `list \u03b1` by list permutation. The result\n  is a type of finite sets with duplicates allowed.  -/\ndef multiset (\u03b1 : Type u) := quotient (list.is_setoid \u03b1)\n\nnamespace multiset\n\n\nprotected instance has_coe {\u03b1 : Type u_1} : has_coe (List \u03b1) (multiset \u03b1) :=\n  has_coe.mk (Quot.mk setoid.r)\n\n@[simp] theorem quot_mk_to_coe {\u03b1 : Type u_1} (l : List \u03b1) : quotient.mk l = \u2191l := rfl\n\n@[simp] theorem quot_mk_to_coe' {\u03b1 : Type u_1} (l : List \u03b1) : Quot.mk has_equiv.equiv l = \u2191l := rfl\n\n@[simp] theorem quot_mk_to_coe'' {\u03b1 : Type u_1} (l : List \u03b1) : Quot.mk setoid.r l = \u2191l := rfl\n\n@[simp] theorem coe_eq_coe {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 = \u2191l\u2082 \u2194 l\u2081 ~ l\u2082 :=\n  quotient.eq\n\nprotected instance has_decidable_eq {\u03b1 : Type u_1} [DecidableEq \u03b1] : DecidableEq (multiset \u03b1) :=\n  sorry\n\n/-- defines a size for a multiset by referring to the size of the underlying list -/\nprotected def sizeof {\u03b1 : Type u_1} [SizeOf \u03b1] (s : multiset \u03b1) : \u2115 := quot.lift_on s sizeof sorry\n\nprotected instance has_sizeof {\u03b1 : Type u_1} [SizeOf \u03b1] : SizeOf (multiset \u03b1) :=\n  { sizeOf := multiset.sizeof }\n\n/-! ### Empty multiset -/\n\n/-- `0 : multiset \u03b1` is the empty set -/\nprotected def zero {\u03b1 : Type u_1} : multiset \u03b1 := \u2191[]\n\nprotected instance has_zero {\u03b1 : Type u_1} : HasZero (multiset \u03b1) := { zero := multiset.zero }\n\nprotected instance has_emptyc {\u03b1 : Type u_1} : has_emptyc (multiset \u03b1) := has_emptyc.mk 0\n\nprotected instance inhabited {\u03b1 : Type u_1} : Inhabited (multiset \u03b1) := { default := 0 }\n\n@[simp] theorem coe_nil_eq_zero {\u03b1 : Type u_1} : \u2191[] = 0 := rfl\n\n@[simp] theorem empty_eq_zero {\u03b1 : Type u_1} : \u2205 = 0 := rfl\n\ntheorem coe_eq_zero {\u03b1 : Type u_1} (l : List \u03b1) : \u2191l = 0 \u2194 l = [] :=\n  iff.trans coe_eq_coe list.perm_nil\n\n/-! ### `multiset.cons` -/\n\n/-- `cons a s` is the multiset which contains `s` plus one more\n  instance of `a`. -/\ndef cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(a :: l)) sorry\n\ninfixr:67 \" ::\u2098 \" => Mathlib.multiset.cons\n\nprotected instance has_insert {\u03b1 : Type u_1} : has_insert \u03b1 (multiset \u03b1) := has_insert.mk cons\n\n@[simp] theorem insert_eq_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : insert a s = a ::\u2098 s := rfl\n\n@[simp] theorem cons_coe {\u03b1 : Type u_1} (a : \u03b1) (l : List \u03b1) : a ::\u2098 \u2191l = \u2191(a :: l) := rfl\n\ntheorem singleton_coe {\u03b1 : Type u_1} (a : \u03b1) : a ::\u2098 0 = \u2191[a] := rfl\n\n@[simp] theorem cons_inj_left {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} (s : multiset \u03b1) :\n    a ::\u2098 s = b ::\u2098 s \u2194 a = b :=\n  sorry\n\n@[simp] theorem cons_inj_right {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} :\n    a ::\u2098 s = a ::\u2098 t \u2194 s = t :=\n  sorry\n\nprotected theorem induction {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (h\u2081 : p 0)\n    (h\u2082 : \u2200 {a : \u03b1} {s : multiset \u03b1}, p s \u2192 p (a ::\u2098 s)) (s : multiset \u03b1) : p s :=\n  quot.induction_on s\n    fun (l : List \u03b1) =>\n      List.rec h\u2081 (fun (l_hd : \u03b1) (l_tl : List \u03b1) (ih : p (Quot.mk setoid.r l_tl)) => h\u2082 ih) l\n\nprotected theorem induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (s : multiset \u03b1) (h\u2081 : p 0)\n    (h\u2082 : \u2200 {a : \u03b1} {s : multiset \u03b1}, p s \u2192 p (a ::\u2098 s)) : p s :=\n  multiset.induction h\u2081 h\u2082 s\n\ntheorem cons_swap {\u03b1 : Type u_1} (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) : a ::\u2098 b ::\u2098 s = b ::\u2098 a ::\u2098 s :=\n  quot.induction_on s fun (l : List \u03b1) => quotient.sound (list.perm.swap b a l)\n\n/-- Dependent recursor on multisets.\n\nTODO: should be @[recursor 6], but then the definition of `multiset.pi` fails with a stack\noverflow in `whnf`.\n-/\nprotected def rec {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} (C_0 : C 0)\n    (C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m))\n    (C_cons_heq :\n      \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m),\n        C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b))\n    (m : multiset \u03b1) : C m :=\n  quotient.hrec_on m\n    (List.rec C_0 fun (a : \u03b1) (l : List \u03b1) (b : C (quotient.mk l)) => C_cons a (quotient.mk l) b)\n    sorry\n\nprotected def rec_on {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} (m : multiset \u03b1) (C_0 : C 0)\n    (C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m))\n    (C_cons_heq :\n      \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m),\n        C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)) :\n    C m :=\n  multiset.rec C_0 C_cons C_cons_heq m\n\n@[simp] theorem rec_on_0 {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} {C_0 : C 0}\n    {C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)}\n    {C_cons_heq :\n      \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m),\n        C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)} :\n    multiset.rec_on 0 C_0 C_cons C_cons_heq = C_0 :=\n  rfl\n\n@[simp] theorem rec_on_cons {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 Sort u_4} {C_0 : C 0}\n    {C_cons : (a : \u03b1) \u2192 (m : multiset \u03b1) \u2192 C m \u2192 C (a ::\u2098 m)}\n    {C_cons_heq :\n      \u2200 (a a' : \u03b1) (m : multiset \u03b1) (b : C m),\n        C_cons a (a' ::\u2098 m) (C_cons a' m b) == C_cons a' (a ::\u2098 m) (C_cons a m b)}\n    (a : \u03b1) (m : multiset \u03b1) :\n    multiset.rec_on (a ::\u2098 m) C_0 C_cons C_cons_heq =\n        C_cons a m (multiset.rec_on m C_0 C_cons C_cons_heq) :=\n  quotient.induction_on m fun (l : List \u03b1) => rfl\n\n/-- `a \u2208 s` means that `a` has nonzero multiplicity in `s`. -/\ndef mem {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) := quot.lift_on s (fun (l : List \u03b1) => a \u2208 l) sorry\n\nprotected instance has_mem {\u03b1 : Type u_1} : has_mem \u03b1 (multiset \u03b1) := has_mem.mk mem\n\n@[simp] theorem mem_coe {\u03b1 : Type u_1} {a : \u03b1} {l : List \u03b1} : a \u2208 \u2191l \u2194 a \u2208 l := iff.rfl\n\nprotected instance decidable_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) :\n    Decidable (a \u2208 s) :=\n  quot.rec_on_subsingleton s (list.decidable_mem a)\n\n@[simp] theorem mem_cons {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} :\n    a \u2208 b ::\u2098 s \u2194 a = b \u2228 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => iff.rfl\n\ntheorem mem_cons_of_mem {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} (h : a \u2208 s) : a \u2208 b ::\u2098 s :=\n  iff.mpr mem_cons (Or.inr h)\n\n@[simp] theorem mem_cons_self {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : a \u2208 a ::\u2098 s :=\n  iff.mpr mem_cons (Or.inl rfl)\n\ntheorem forall_mem_cons {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {a : \u03b1} {s : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 a ::\u2098 s \u2192 p x) \u2194 p a \u2227 \u2200 (x : \u03b1), x \u2208 s \u2192 p x :=\n  quotient.induction_on' s fun (L : List \u03b1) => list.forall_mem_cons\n\ntheorem exists_cons_of_mem {\u03b1 : Type u_1} {s : multiset \u03b1} {a : \u03b1} :\n    a \u2208 s \u2192 \u2203 (t : multiset \u03b1), s = a ::\u2098 t :=\n  sorry\n\n@[simp] theorem not_mem_zero {\u03b1 : Type u_1} (a : \u03b1) : \u00aca \u2208 0 := id\n\ntheorem eq_zero_of_forall_not_mem {\u03b1 : Type u_1} {s : multiset \u03b1} : (\u2200 (x : \u03b1), \u00acx \u2208 s) \u2192 s = 0 :=\n  sorry\n\ntheorem eq_zero_iff_forall_not_mem {\u03b1 : Type u_1} {s : multiset \u03b1} : s = 0 \u2194 \u2200 (a : \u03b1), \u00aca \u2208 s :=\n  { mp := fun (h : s = 0) => Eq.symm h \u25b8 fun (_x : \u03b1) => not_false,\n    mpr := eq_zero_of_forall_not_mem }\n\ntheorem exists_mem_of_ne_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2260 0 \u2192 \u2203 (a : \u03b1), a \u2208 s := sorry\n\n@[simp] theorem zero_ne_cons {\u03b1 : Type u_1} {a : \u03b1} {m : multiset \u03b1} : 0 \u2260 a ::\u2098 m :=\n  fun (h : 0 = a ::\u2098 m) =>\n    (fun (this : a \u2208 0) => not_mem_zero a this) (Eq.symm h \u25b8 mem_cons_self a m)\n\n@[simp] theorem cons_ne_zero {\u03b1 : Type u_1} {a : \u03b1} {m : multiset \u03b1} : a ::\u2098 m \u2260 0 :=\n  ne.symm zero_ne_cons\n\ntheorem cons_eq_cons {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {as : multiset \u03b1} {bs : multiset \u03b1} :\n    a ::\u2098 as = b ::\u2098 bs \u2194\n        a = b \u2227 as = bs \u2228 a \u2260 b \u2227 \u2203 (cs : multiset \u03b1), as = b ::\u2098 cs \u2227 bs = a ::\u2098 cs :=\n  sorry\n\n/-! ### `multiset.subset` -/\n\n/-- `s \u2286 t` is the lift of the list subset relation. It means that any\n  element with nonzero multiplicity in `s` has nonzero multiplicity in `t`,\n  but it does not imply that the multiplicity of `a` in `s` is less or equal than in `t`;\n  see `s \u2264 t` for this relation. -/\nprotected def subset {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) := \u2200 {a : \u03b1}, a \u2208 s \u2192 a \u2208 t\n\nprotected instance has_subset {\u03b1 : Type u_1} : has_subset (multiset \u03b1) :=\n  has_subset.mk multiset.subset\n\n@[simp] theorem coe_subset {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 \u2286 \u2191l\u2082 \u2194 l\u2081 \u2286 l\u2082 :=\n  iff.rfl\n\n@[simp] theorem subset.refl {\u03b1 : Type u_1} (s : multiset \u03b1) : s \u2286 s := fun (a : \u03b1) (h : a \u2208 s) => h\n\ntheorem subset.trans {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} :\n    s \u2286 t \u2192 t \u2286 u \u2192 s \u2286 u :=\n  fun (h\u2081 : s \u2286 t) (h\u2082 : t \u2286 u) (a : \u03b1) (m : a \u2208 s) => h\u2082 (h\u2081 m)\n\ntheorem subset_iff {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    s \u2286 t \u2194 \u2200 {x : \u03b1}, x \u2208 s \u2192 x \u2208 t :=\n  iff.rfl\n\ntheorem mem_of_subset {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} (h : s \u2286 t) :\n    a \u2208 s \u2192 a \u2208 t :=\n  h\n\n@[simp] theorem zero_subset {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 \u2286 s :=\n  fun (a : \u03b1) => not.elim (list.not_mem_nil a)\n\n@[simp] theorem cons_subset {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    a ::\u2098 s \u2286 t \u2194 a \u2208 t \u2227 s \u2286 t :=\n  sorry\n\ntheorem eq_zero_of_subset_zero {\u03b1 : Type u_1} {s : multiset \u03b1} (h : s \u2286 0) : s = 0 :=\n  eq_zero_of_forall_not_mem h\n\ntheorem subset_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2286 0 \u2194 s = 0 :=\n  { mp := eq_zero_of_subset_zero, mpr := fun (xeq : s = 0) => Eq.symm xeq \u25b8 subset.refl 0 }\n\n/-- Produces a list of the elements in the multiset using choice. -/\ndef to_list {\u03b1 : Type u_1} (s : multiset \u03b1) : List \u03b1 := classical.some sorry\n\n@[simp] theorem to_list_zero {\u03b1 : Type u_1} : to_list 0 = [] :=\n  iff.mp (coe_eq_zero (to_list 0)) (classical.some_spec (quotient.exists_rep multiset.zero))\n\ntheorem coe_to_list {\u03b1 : Type u_1} (s : multiset \u03b1) : \u2191(to_list s) = s :=\n  classical.some_spec (quotient.exists_rep s)\n\ntheorem mem_to_list {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : a \u2208 to_list s \u2194 a \u2208 s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 to_list s \u2194 a \u2208 s)) (Eq.symm (propext mem_coe))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a \u2208 \u2191(to_list s) \u2194 a \u2208 s)) (coe_to_list s)))\n      (iff.refl (a \u2208 s)))\n\n/-! ### Partial order on `multiset`s -/\n\n/-- `s \u2264 t` means that `s` is a sublist of `t` (up to permutation).\n  Equivalently, `s \u2264 t` means that `count a s \u2264 count a t` for all `a`. -/\nprotected def le {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) :=\n  quotient.lift_on\u2082 s t list.subperm sorry\n\nprotected instance partial_order {\u03b1 : Type u_1} : partial_order (multiset \u03b1) :=\n  partial_order.mk multiset.le (preorder.lt._default multiset.le) sorry sorry sorry\n\ntheorem subset_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : s \u2264 t \u2192 s \u2286 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.subperm.subset\n\ntheorem mem_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} (h : s \u2264 t) :\n    a \u2208 s \u2192 a \u2208 t :=\n  mem_of_subset (subset_of_le h)\n\n@[simp] theorem coe_le {\u03b1 : Type u_1} {l\u2081 : List \u03b1} {l\u2082 : List \u03b1} : \u2191l\u2081 \u2264 \u2191l\u2082 \u2194 l\u2081 <+~ l\u2082 := iff.rfl\n\ntheorem le_induction_on {\u03b1 : Type u_1} {C : multiset \u03b1 \u2192 multiset \u03b1 \u2192 Prop} {s : multiset \u03b1}\n    {t : multiset \u03b1} (h : s \u2264 t) (H : \u2200 {l\u2081 l\u2082 : List \u03b1}, l\u2081 <+ l\u2082 \u2192 C \u2191l\u2081 \u2191l\u2082) : C s t :=\n  sorry\n\ntheorem zero_le {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.nil_sublist l)\n\ntheorem le_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : s \u2264 0 \u2194 s = 0 :=\n  { mp := fun (h : s \u2264 0) => le_antisymm h (zero_le s), mpr := le_of_eq }\n\ntheorem lt_cons_self {\u03b1 : Type u_1} (s : multiset \u03b1) (a : \u03b1) : s < a ::\u2098 s := sorry\n\ntheorem le_cons_self {\u03b1 : Type u_1} (s : multiset \u03b1) (a : \u03b1) : s \u2264 a ::\u2098 s :=\n  le_of_lt (lt_cons_self s a)\n\ntheorem cons_le_cons_iff {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} :\n    a ::\u2098 s \u2264 a ::\u2098 t \u2194 s \u2264 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.subperm_cons a\n\ntheorem cons_le_cons {\u03b1 : Type u_1} (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} :\n    s \u2264 t \u2192 a ::\u2098 s \u2264 a ::\u2098 t :=\n  iff.mpr (cons_le_cons_iff a)\n\ntheorem le_cons_of_not_mem {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} (m : \u00aca \u2208 s) :\n    s \u2264 a ::\u2098 t \u2194 s \u2264 t :=\n  sorry\n\n/-! ### Additive monoid -/\n\n/-- The sum of two multisets is the lift of the list append operation.\n  This adds the multiplicities of each element,\n  i.e. `count a (s + t) = count a s + count a t`. -/\nprotected def add {\u03b1 : Type u_1} (s\u2081 : multiset \u03b1) (s\u2082 : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s\u2081 s\u2082 (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(l\u2081 ++ l\u2082)) sorry\n\nprotected instance has_add {\u03b1 : Type u_1} : Add (multiset \u03b1) := { add := multiset.add }\n\n@[simp] theorem coe_add {\u03b1 : Type u_1} (s : List \u03b1) (t : List \u03b1) : \u2191s + \u2191t = \u2191(s ++ t) := rfl\n\nprotected theorem add_comm {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s + t = t + s :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => quot.sound list.perm_append_comm\n\nprotected theorem zero_add {\u03b1 : Type u_1} (s : multiset \u03b1) : 0 + s = s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\ntheorem singleton_add {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) : \u2191[a] + s = a ::\u2098 s := rfl\n\nprotected theorem add_le_add_left {\u03b1 : Type u_1} (s : multiset \u03b1) {t : multiset \u03b1}\n    {u : multiset \u03b1} : s + t \u2264 s + u \u2194 t \u2264 u :=\n  quotient.induction_on\u2083 s t u fun (l\u2081 l\u2082 l\u2083 : List \u03b1) => list.subperm_append_left l\u2081\n\nprotected theorem add_left_cancel {\u03b1 : Type u_1} (s : multiset \u03b1) {t : multiset \u03b1} {u : multiset \u03b1}\n    (h : s + t = s + u) : t = u :=\n  le_antisymm (iff.mp (multiset.add_le_add_left s) (le_of_eq h))\n    (iff.mp (multiset.add_le_add_left s) (le_of_eq (Eq.symm h)))\n\nprotected instance ordered_cancel_add_comm_monoid {\u03b1 : Type u_1} :\n    ordered_cancel_add_comm_monoid (multiset \u03b1) :=\n  ordered_cancel_add_comm_monoid.mk Add.add sorry multiset.add_left_cancel 0 multiset.zero_add sorry\n    multiset.add_comm sorry partial_order.le partial_order.lt sorry sorry sorry sorry sorry\n\ntheorem le_add_right {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 s + t := sorry\n\ntheorem le_add_left {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) : s \u2264 t + s := sorry\n\ntheorem le_iff_exists_add {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    s \u2264 t \u2194 \u2203 (u : multiset \u03b1), t = s + u :=\n  sorry\n\nprotected instance canonically_ordered_add_monoid {\u03b1 : Type u_1} :\n    canonically_ordered_add_monoid (multiset \u03b1) :=\n  canonically_ordered_add_monoid.mk ordered_cancel_add_comm_monoid.add sorry\n    ordered_cancel_add_comm_monoid.zero sorry sorry sorry ordered_cancel_add_comm_monoid.le\n    ordered_cancel_add_comm_monoid.lt sorry sorry sorry sorry sorry 0 zero_le le_iff_exists_add\n\n@[simp] theorem cons_add {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) :\n    a ::\u2098 s + t = a ::\u2098 (s + t) :=\n  sorry\n\n@[simp] theorem add_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) :\n    s + a ::\u2098 t = a ::\u2098 (s + t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s + a ::\u2098 t = a ::\u2098 (s + t))) (add_comm s (a ::\u2098 t))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (a ::\u2098 t + s = a ::\u2098 (s + t))) (cons_add a t s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (a ::\u2098 (t + s) = a ::\u2098 (s + t))) (add_comm t s)))\n        (Eq.refl (a ::\u2098 (s + t)))))\n\n@[simp] theorem mem_add {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    a \u2208 s + t \u2194 a \u2208 s \u2228 a \u2208 t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.mem_append\n\n/-! ### Cardinality -/\n\n/-- The cardinality of a multiset is the sum of the multiplicities\n  of all its elements, or simply the length of the underlying list. -/\ndef card {\u03b1 : Type u_1} : multiset \u03b1 \u2192+ \u2115 :=\n  add_monoid_hom.mk (fun (s : multiset \u03b1) => quot.lift_on s list.length sorry) sorry sorry\n\n@[simp] theorem coe_card {\u03b1 : Type u_1} (l : List \u03b1) : coe_fn card \u2191l = list.length l := rfl\n\n@[simp] theorem card_zero {\u03b1 : Type u_1} : coe_fn card 0 = 0 := rfl\n\ntheorem card_add {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) :\n    coe_fn card (s + t) = coe_fn card s + coe_fn card t :=\n  add_monoid_hom.map_add card s t\n\ntheorem card_smul {\u03b1 : Type u_1} (s : multiset \u03b1) (n : \u2115) :\n    coe_fn card (n \u2022\u2115 s) = n * coe_fn card s :=\n  eq.mpr\n    (id\n      (Eq._oldrec (Eq.refl (coe_fn card (n \u2022\u2115 s) = n * coe_fn card s))\n        (add_monoid_hom.map_nsmul card s n)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (n \u2022\u2115 coe_fn card s = n * coe_fn card s))\n          (nat.nsmul_eq_mul n (coe_fn card s))))\n      (Eq.refl (n * coe_fn card s)))\n\n@[simp] theorem card_cons {\u03b1 : Type u_1} (a : \u03b1) (s : multiset \u03b1) :\n    coe_fn card (a ::\u2098 s) = coe_fn card s + 1 :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem card_singleton {\u03b1 : Type u_1} (a : \u03b1) : coe_fn card (a ::\u2098 0) = 1 := sorry\n\ntheorem card_le_of_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) :\n    coe_fn card s \u2264 coe_fn card t :=\n  le_induction_on h fun (l\u2081 l\u2082 : List \u03b1) => list.length_le_of_sublist\n\ntheorem eq_of_le_of_card_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) :\n    coe_fn card t \u2264 coe_fn card s \u2192 s = t :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (s : l\u2081 <+ l\u2082) (h\u2082 : coe_fn card \u2191l\u2082 \u2264 coe_fn card \u2191l\u2081) =>\n      congr_arg coe (list.eq_of_sublist_of_length_le s h\u2082)\n\ntheorem card_lt_of_lt {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (h : s < t) :\n    coe_fn card s < coe_fn card t :=\n  lt_of_not_ge\n    fun (h\u2082 : coe_fn card s \u2265 coe_fn card t) => ne_of_lt h (eq_of_le_of_card_le (le_of_lt h) h\u2082)\n\ntheorem lt_iff_cons_le {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    s < t \u2194 \u2203 (a : \u03b1), a ::\u2098 s \u2264 t :=\n  sorry\n\n@[simp] theorem card_eq_zero {\u03b1 : Type u_1} {s : multiset \u03b1} : coe_fn card s = 0 \u2194 s = 0 := sorry\n\ntheorem card_pos {\u03b1 : Type u_1} {s : multiset \u03b1} : 0 < coe_fn card s \u2194 s \u2260 0 :=\n  iff.trans pos_iff_ne_zero (not_congr card_eq_zero)\n\ntheorem card_pos_iff_exists_mem {\u03b1 : Type u_1} {s : multiset \u03b1} :\n    0 < coe_fn card s \u2194 \u2203 (a : \u03b1), a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_pos_iff_exists_mem\n\ndef strong_induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Sort u_2} (s : multiset \u03b1) :\n    ((s : multiset \u03b1) \u2192 ((t : multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s) \u2192 p s :=\n  sorry\n\ntheorem strong_induction_eq {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Sort u_2} (s : multiset \u03b1)\n    (H : (s : multiset \u03b1) \u2192 ((t : multiset \u03b1) \u2192 t < s \u2192 p t) \u2192 p s) :\n    strong_induction_on s H = H s fun (t : multiset \u03b1) (h : t < s) => strong_induction_on t H :=\n  sorry\n\ntheorem case_strong_induction_on {\u03b1 : Type u_1} {p : multiset \u03b1 \u2192 Prop} (s : multiset \u03b1) (h\u2080 : p 0)\n    (h\u2081 : \u2200 (a : \u03b1) (s : multiset \u03b1), (\u2200 (t : multiset \u03b1), t \u2264 s \u2192 p t) \u2192 p (a ::\u2098 s)) : p s :=\n  sorry\n\n/-! ### Singleton -/\n\nprotected instance has_singleton {\u03b1 : Type u_1} : has_singleton \u03b1 (multiset \u03b1) :=\n  has_singleton.mk fun (a : \u03b1) => a ::\u2098 0\n\nprotected instance is_lawful_singleton {\u03b1 : Type u_1} : is_lawful_singleton \u03b1 (multiset \u03b1) :=\n  is_lawful_singleton.mk fun (a : \u03b1) => rfl\n\n@[simp] theorem singleton_eq_singleton {\u03b1 : Type u_1} (a : \u03b1) : singleton a = a ::\u2098 0 := rfl\n\n@[simp] theorem mem_singleton {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} : b \u2208 a ::\u2098 0 \u2194 b = a := sorry\n\ntheorem mem_singleton_self {\u03b1 : Type u_1} (a : \u03b1) : a \u2208 a ::\u2098 0 := mem_cons_self a 0\n\ntheorem singleton_inj {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} : a ::\u2098 0 = b ::\u2098 0 \u2194 a = b := cons_inj_left 0\n\n@[simp] theorem singleton_ne_zero {\u03b1 : Type u_1} (a : \u03b1) : a ::\u2098 0 \u2260 0 :=\n  ne_of_gt (lt_cons_self 0 a)\n\n@[simp] theorem singleton_le {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} : a ::\u2098 0 \u2264 s \u2194 a \u2208 s := sorry\n\ntheorem card_eq_one {\u03b1 : Type u_1} {s : multiset \u03b1} : coe_fn card s = 1 \u2194 \u2203 (a : \u03b1), s = a ::\u2098 0 :=\n  sorry\n\n/-! ### `multiset.repeat` -/\n\n/-- `repeat a n` is the multiset containing only `a` with multiplicity `n`. -/\ndef repeat {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : multiset \u03b1 := \u2191(list.repeat a n)\n\n@[simp] theorem repeat_zero {\u03b1 : Type u_1} (a : \u03b1) : repeat a 0 = 0 := rfl\n\n@[simp] theorem repeat_succ {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : repeat a (n + 1) = a ::\u2098 repeat a n :=\n  sorry\n\n@[simp] theorem repeat_one {\u03b1 : Type u_1} (a : \u03b1) : repeat a 1 = a ::\u2098 0 := sorry\n\n@[simp] theorem card_repeat {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : coe_fn card (repeat a n) = n :=\n  list.length_repeat\n\ntheorem eq_of_mem_repeat {\u03b1 : Type u_1} {a : \u03b1} {b : \u03b1} {n : \u2115} : b \u2208 repeat a n \u2192 b = a :=\n  list.eq_of_mem_repeat\n\ntheorem eq_repeat' {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} :\n    s = repeat a (coe_fn card s) \u2194 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a :=\n  sorry\n\ntheorem eq_repeat_of_mem {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} :\n    (\u2200 (b : \u03b1), b \u2208 s \u2192 b = a) \u2192 s = repeat a (coe_fn card s) :=\n  iff.mpr eq_repeat'\n\ntheorem eq_repeat {\u03b1 : Type u_1} {a : \u03b1} {n : \u2115} {s : multiset \u03b1} :\n    s = repeat a n \u2194 coe_fn card s = n \u2227 \u2200 (b : \u03b1), b \u2208 s \u2192 b = a :=\n  sorry\n\ntheorem repeat_subset_singleton {\u03b1 : Type u_1} (a : \u03b1) (n : \u2115) : repeat a n \u2286 a ::\u2098 0 :=\n  list.repeat_subset_singleton\n\ntheorem repeat_le_coe {\u03b1 : Type u_1} {a : \u03b1} {n : \u2115} {l : List \u03b1} :\n    repeat a n \u2264 \u2191l \u2194 list.repeat a n <+ l :=\n  sorry\n\n/-! ### Erasing one copy of an element -/\n\n/-- `erase s a` is the multiset that subtracts 1 from the\n  multiplicity of `a`. -/\ndef erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.erase l a)) sorry\n\n@[simp] theorem coe_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (l : List \u03b1) (a : \u03b1) :\n    erase (\u2191l) a = \u2191(list.erase l a) :=\n  rfl\n\n@[simp] theorem erase_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : erase 0 a = 0 := rfl\n\n@[simp] theorem erase_cons_head {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) :\n    erase (a ::\u2098 s) a = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_cons_head a l)\n\n@[simp] theorem erase_cons_tail {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (s : multiset \u03b1)\n    (h : b \u2260 a) : erase (b ::\u2098 s) a = b ::\u2098 erase s a :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_cons_tail l h)\n\n@[simp] theorem erase_of_not_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    \u00aca \u2208 s \u2192 erase s a = s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (h : \u00aca \u2208 Quot.mk setoid.r l) => congr_arg coe (list.erase_of_not_mem h)\n\n@[simp] theorem cons_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {a : \u03b1} :\n    a \u2208 s \u2192 a ::\u2098 erase s a = s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (h : a \u2208 Quot.mk setoid.r l) =>\n      quot.sound (list.perm.symm (list.perm_cons_erase h))\n\ntheorem le_cons_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) :\n    s \u2264 a ::\u2098 erase s a :=\n  dite (a \u2208 s) (fun (h : a \u2208 s) => le_of_eq (Eq.symm (cons_erase h)))\n    fun (h : \u00aca \u2208 s) =>\n      eq.mpr (id (Eq._oldrec (Eq.refl (s \u2264 a ::\u2098 erase s a)) (erase_of_not_mem h)))\n        (le_cons_self s a)\n\ntheorem erase_add_left_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n    (t : multiset \u03b1) : a \u2208 s \u2192 erase (s + t) a = erase s a + t :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : a \u2208 quotient.mk l\u2081) => congr_arg coe (list.erase_append_left l\u2082 h)\n\ntheorem erase_add_right_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1}\n    (h : a \u2208 t) : erase (s + t) a = s + erase t a :=\n  sorry\n\ntheorem erase_add_right_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n    (t : multiset \u03b1) : \u00aca \u2208 s \u2192 erase (s + t) a = s + erase t a :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : \u00aca \u2208 quotient.mk l\u2081) => congr_arg coe (list.erase_append_right l\u2082 h)\n\ntheorem erase_add_left_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1) {t : multiset \u03b1}\n    (h : \u00aca \u2208 t) : erase (s + t) a = erase s a + t :=\n  sorry\n\ntheorem erase_le {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : erase s a \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.erase_sublist a l)\n\n@[simp] theorem erase_lt {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    erase s a < s \u2194 a \u2208 s :=\n  sorry\n\ntheorem erase_subset {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) : erase s a \u2286 s :=\n  subset_of_le (erase_le a s)\n\ntheorem mem_erase_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {s : multiset \u03b1}\n    (ab : a \u2260 b) : a \u2208 erase s b \u2194 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_erase_of_ne ab\n\ntheorem mem_of_mem_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} {s : multiset \u03b1} :\n    a \u2208 erase s b \u2192 a \u2208 s :=\n  mem_of_subset (erase_subset b s)\n\ntheorem erase_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (a : \u03b1) (b : \u03b1) :\n    erase (erase s a) b = erase (erase s b) a :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.erase_comm a b l)\n\ntheorem erase_le_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (a : \u03b1)\n    (h : s \u2264 t) : erase s a \u2264 erase t a :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.erase a h)\n\ntheorem erase_le_iff_le_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    {a : \u03b1} : erase s a \u2264 t \u2194 s \u2264 a ::\u2098 t :=\n  sorry\n\n@[simp] theorem card_erase_of_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    a \u2208 s \u2192 coe_fn card (erase s a) = Nat.pred (coe_fn card s) :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_erase_of_mem\n\ntheorem card_erase_lt_of_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    a \u2208 s \u2192 coe_fn card (erase s a) < coe_fn card s :=\n  fun (h : a \u2208 s) => card_lt_of_lt (iff.mpr erase_lt h)\n\ntheorem card_erase_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    coe_fn card (erase s a) \u2264 coe_fn card s :=\n  card_le_of_le (erase_le a s)\n\n@[simp] theorem coe_reverse {\u03b1 : Type u_1} (l : List \u03b1) : \u2191(list.reverse l) = \u2191l :=\n  quot.sound (list.reverse_perm l)\n\n/-! ### `multiset.map` -/\n\n/-- `map f s` is the lift of the list `map` operation. The multiplicity\n  of `b` in `map f s` is the number of `a \u2208 s` (counting multiplicity)\n  such that `f a = b`. -/\ndef map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.map f l)) sorry\n\ntheorem forall_mem_map_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop}\n    {s : multiset \u03b1} : (\u2200 (y : \u03b2), y \u2208 map f s \u2192 p y) \u2194 \u2200 (x : \u03b1), x \u2208 s \u2192 p (f x) :=\n  quotient.induction_on' s fun (L : List \u03b1) => list.forall_mem_map_iff\n\n@[simp] theorem coe_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (l : List \u03b1) :\n    map f \u2191l = \u2191(list.map f l) :=\n  rfl\n\n@[simp] theorem map_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) : map f 0 = 0 := rfl\n\n@[simp] theorem map_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (s : multiset \u03b1) :\n    map f (a ::\u2098 s) = f a ::\u2098 map f s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\ntheorem map_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) :\n    map f (singleton a) = singleton (f a) :=\n  rfl\n\ntheorem map_repeat {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (a : \u03b1) (k : \u2115) :\n    map f (repeat a k) = repeat (f a) k :=\n  sorry\n\n@[simp] theorem map_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1)\n    (t : multiset \u03b1) : map f (s + t) = map f s + map f t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.map_append f l\u2081 l\u2082)\n\nprotected instance map.is_add_monoid_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) :\n    is_add_monoid_hom (map f) :=\n  is_add_monoid_hom.mk (map_zero f)\n\ntheorem map_nsmul {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (n : \u2115) (s : multiset \u03b1) :\n    map f (n \u2022\u2115 s) = n \u2022\u2115 map f s :=\n  add_monoid_hom.map_nsmul (add_monoid_hom.of (map f)) s n\n\n@[simp] theorem mem_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {b : \u03b2} {s : multiset \u03b1} :\n    b \u2208 map f s \u2194 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = b :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_map\n\n@[simp] theorem card_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1) :\n    coe_fn card (map f s) = coe_fn card s :=\n  quot.induction_on s fun (l : List \u03b1) => list.length_map f l\n\n@[simp] theorem map_eq_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : multiset \u03b1} {f : \u03b1 \u2192 \u03b2} :\n    map f s = 0 \u2194 s = 0 :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (map f s = 0 \u2194 s = 0)) (Eq.symm (propext card_eq_zero))))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn card (map f s) = 0 \u2194 s = 0)) (card_map f s)))\n      (eq.mpr (id (Eq._oldrec (Eq.refl (coe_fn card s = 0 \u2194 s = 0)) (propext card_eq_zero)))\n        (iff.refl (s = 0))))\n\ntheorem mem_map_of_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) {a : \u03b1} {s : multiset \u03b1}\n    (h : a \u2208 s) : f a \u2208 map f s :=\n  iff.mpr mem_map (Exists.intro a { left := h, right := rfl })\n\ntheorem mem_map_of_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (H : function.injective f)\n    {a : \u03b1} {s : multiset \u03b1} : f a \u2208 map f s \u2194 a \u2208 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_map_of_injective H\n\n@[simp] theorem map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (g : \u03b2 \u2192 \u03b3) (f : \u03b1 \u2192 \u03b2)\n    (s : multiset \u03b1) : map g (map f s) = map (g \u2218 f) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_map g f l)\n\ntheorem map_id {\u03b1 : Type u_1} (s : multiset \u03b1) : map id s = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_id l)\n\n@[simp] theorem map_id' {\u03b1 : Type u_1} (s : multiset \u03b1) : map (fun (x : \u03b1) => x) s = s := map_id s\n\n@[simp] theorem map_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (b : \u03b2) :\n    map (function.const \u03b1 b) s = repeat b (coe_fn card s) :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_const l b)\n\ntheorem map_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2} {s : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 s \u2192 f x = g x) \u2192 map f s = map g s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (x : \u03b1), x \u2208 Quot.mk setoid.r l \u2192 f x = g x) =>\n      congr_arg coe (list.map_congr H)\n\ntheorem map_hcongr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b2' : Type u_2} {m : multiset \u03b1} {f : \u03b1 \u2192 \u03b2}\n    {f' : \u03b1 \u2192 \u03b2'} (h : \u03b2 = \u03b2') (hf : \u2200 (a : \u03b1), a \u2208 m \u2192 f a == f' a) : map f m == map f' m :=\n  sorry\n\ntheorem eq_of_mem_map_const {\u03b1 : Type u_1} {\u03b2 : Type u_2} {b\u2081 : \u03b2} {b\u2082 : \u03b2} {l : List \u03b1}\n    (h : b\u2081 \u2208 map (function.const \u03b1 b\u2082) \u2191l) : b\u2081 = b\u2082 :=\n  eq_of_mem_repeat\n    (eq.mp (Eq._oldrec (Eq.refl (b\u2081 \u2208 map (function.const \u03b1 b\u2082) \u2191l)) (map_const (\u2191l) b\u2082)) h)\n\n@[simp] theorem map_le_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1}\n    {t : multiset \u03b1} (h : s \u2264 t) : map f s \u2264 map f t :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.map f h)\n\n@[simp] theorem map_subset_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} {s : multiset \u03b1}\n    {t : multiset \u03b1} (H : s \u2286 t) : map f s \u2286 map f t :=\n  sorry\n\n/-! ### `multiset.fold` -/\n\n/-- `foldl f H b s` is the lift of the list operation `foldl f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is right-commutative,\n  that is, `f (f b a\u2081) a\u2082 = f (f b a\u2082) a\u2081`. -/\ndef foldl {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2)\n    (s : multiset \u03b1) : \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => list.foldl f b l) sorry\n\n@[simp] theorem foldl_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f)\n    (b : \u03b2) : foldl f H b 0 = b :=\n  rfl\n\n@[simp] theorem foldl_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f)\n    (b : \u03b2) (a : \u03b1) (s : multiset \u03b1) : foldl f H b (a ::\u2098 s) = foldl f H (f b a) s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem foldl_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f)\n    (b : \u03b2) (s : multiset \u03b1) (t : multiset \u03b1) : foldl f H b (s + t) = foldl f H (foldl f H b s) t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.foldl_append f b l\u2081 l\u2082\n\n/-- `foldr f H b s` is the lift of the list operation `foldr f b l`,\n  which folds `f` over the multiset. It is well defined when `f` is left-commutative,\n  that is, `f a\u2081 (f a\u2082 b) = f a\u2082 (f a\u2081 b)`. -/\ndef foldr {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2)\n    (s : multiset \u03b1) : \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => list.foldr f b l) sorry\n\n@[simp] theorem foldr_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f)\n    (b : \u03b2) : foldr f H b 0 = b :=\n  rfl\n\n@[simp] theorem foldr_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f)\n    (b : \u03b2) (a : \u03b1) (s : multiset \u03b1) : foldr f H b (a ::\u2098 s) = f a (foldr f H b s) :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem foldr_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f)\n    (b : \u03b2) (s : multiset \u03b1) (t : multiset \u03b1) : foldr f H b (s + t) = foldr f H (foldr f H b t) s :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => list.foldr_append f b l\u2081 l\u2082\n\n@[simp] theorem coe_foldr {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f)\n    (b : \u03b2) (l : List \u03b1) : foldr f H b \u2191l = list.foldr f b l :=\n  rfl\n\n@[simp] theorem coe_foldl {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f)\n    (b : \u03b2) (l : List \u03b1) : foldl f H b \u2191l = list.foldl f b l :=\n  rfl\n\ntheorem coe_foldr_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f)\n    (b : \u03b2) (l : List \u03b1) : foldr f H b \u2191l = list.foldl (fun (x : \u03b2) (y : \u03b1) => f y x) b l :=\n  Eq.trans (Eq.symm (congr_arg (foldr f H b) (coe_reverse l))) (list.foldr_reverse f b l)\n\ntheorem foldr_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2 \u2192 \u03b2) (H : left_commutative f) (b : \u03b2)\n    (s : multiset \u03b1) :\n    foldr f H b s =\n        foldl (fun (x : \u03b2) (y : \u03b1) => f y x) (fun (x : \u03b2) (y z : \u03b1) => Eq.symm (H y z x)) b s :=\n  quot.induction_on s fun (l : List \u03b1) => coe_foldr_swap f H b l\n\ntheorem foldl_swap {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b2 \u2192 \u03b1 \u2192 \u03b2) (H : right_commutative f) (b : \u03b2)\n    (s : multiset \u03b1) :\n    foldl f H b s =\n        foldr (fun (x : \u03b1) (y : \u03b2) => f y x) (fun (x y : \u03b1) (z : \u03b2) => Eq.symm (H z x y)) b s :=\n  Eq.symm\n    (foldr_swap (fun (y : \u03b1) (x : \u03b2) => f x y) (fun (x y : \u03b1) (z : \u03b2) => Eq.symm (H z x y)) b s)\n\n/-- Product of a multiset given a commutative monoid structure on `\u03b1`.\n  `prod {a, b, c} = a * b * c` -/\ndef sum {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : multiset \u03b1 \u2192 \u03b1 := foldr Add.add sorry 0\n\ntheorem prod_eq_foldr {\u03b1 : Type u_1} [comm_monoid \u03b1] (s : multiset \u03b1) :\n    prod s =\n        foldr Mul.mul\n          (fun (x y z : \u03b1) =>\n            eq.mpr\n              (id\n                (Eq.trans\n                  ((fun (a a_1 : \u03b1) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : \u03b1) (e_2 : \u1fb0 = \u1fb0_1) =>\n                      congr (congr_arg Eq e_1) e_2)\n                    (x * (y * z)) (x * (y * z)) (Eq.refl (x * (y * z))) (y * (x * z)) (x * (y * z))\n                    (mul_left_comm y x z))\n                  (propext (eq_self_iff_true (x * (y * z))))))\n              trivial)\n          1 s :=\n  rfl\n\ntheorem sum_eq_foldl {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (s : multiset \u03b1) :\n    sum s =\n        foldl Add.add\n          (fun (x y z : \u03b1) =>\n            eq.mpr\n              (id\n                (Eq.trans\n                  ((fun (a a_1 : \u03b1) (e_1 : a = a_1) (\u1fb0 \u1fb0_1 : \u03b1) (e_2 : \u1fb0 = \u1fb0_1) =>\n                      congr (congr_arg Eq e_1) e_2)\n                    (x + y + z) (x + y + z) (Eq.refl (x + y + z)) (x + z + y) (x + y + z)\n                    (add_right_comm x z y))\n                  (propext (eq_self_iff_true (x + y + z)))))\n              trivial)\n          0 s :=\n  sorry\n\n@[simp] theorem coe_sum {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (l : List \u03b1) : sum \u2191l = list.sum l :=\n  sum_eq_foldl \u2191l\n\n@[simp] theorem sum_zero {\u03b1 : Type u_1} [add_comm_monoid \u03b1] : sum 0 = 0 := rfl\n\n@[simp] theorem sum_cons {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) (s : multiset \u03b1) :\n    sum (a ::\u2098 s) = a + sum s :=\n  foldr_cons Add.add sum._proof_1 0 a s\n\ntheorem sum_singleton {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) : sum (a ::\u2098 0) = a := sorry\n\n@[simp] theorem sum_add {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    sum (s + t) = sum s + sum t :=\n  sorry\n\nprotected instance sum.is_add_monoid_hom {\u03b1 : Type u_1} [add_comm_monoid \u03b1] :\n    is_add_monoid_hom sum :=\n  is_add_monoid_hom.mk sum_zero\n\ntheorem prod_smul {\u03b1 : Type u_1} [comm_monoid \u03b1] (m : multiset \u03b1) (n : \u2115) :\n    prod (n \u2022\u2115 m) = prod m ^ n :=\n  sorry\n\n@[simp] theorem prod_repeat {\u03b1 : Type u_1} [comm_monoid \u03b1] (a : \u03b1) (n : \u2115) :\n    prod (repeat a n) = a ^ n :=\n  sorry\n\n@[simp] theorem sum_repeat {\u03b1 : Type u_1} [add_comm_monoid \u03b1] (a : \u03b1) (n : \u2115) :\n    sum (repeat a n) = n \u2022\u2115 a :=\n  prod_repeat\n\ntheorem prod_map_one {\u03b1 : Type u_1} {\u03b3 : Type u_3} [comm_monoid \u03b3] {m : multiset \u03b1} :\n    prod (map (fun (a : \u03b1) => 1) m) = 1 :=\n  sorry\n\ntheorem sum_map_zero {\u03b1 : Type u_1} {\u03b3 : Type u_3} [add_comm_monoid \u03b3] {m : multiset \u03b1} :\n    sum (map (fun (a : \u03b1) => 0) m) = 0 :=\n  sorry\n\n@[simp] theorem sum_map_add {\u03b1 : Type u_1} {\u03b3 : Type u_3} [add_comm_monoid \u03b3] {m : multiset \u03b1}\n    {f : \u03b1 \u2192 \u03b3} {g : \u03b1 \u2192 \u03b3} :\n    sum (map (fun (a : \u03b1) => f a + g a) m) = sum (map f m) + sum (map g m) :=\n  sorry\n\ntheorem prod_map_prod_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b3]\n    (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n    prod (map (fun (a : \u03b1) => prod (map (fun (b : \u03b2) => f a b) n)) m) =\n        prod (map (fun (b : \u03b2) => prod (map (fun (a : \u03b1) => f a b) m)) n) :=\n  sorry\n\ntheorem sum_map_sum_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [add_comm_monoid \u03b3]\n    (m : multiset \u03b1) (n : multiset \u03b2) {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n    sum (map (fun (a : \u03b1) => sum (map (fun (b : \u03b2) => f a b) n)) m) =\n        sum (map (fun (b : \u03b2) => sum (map (fun (a : \u03b1) => f a b) m)) n) :=\n  prod_map_prod_map\n\ntheorem sum_map_mul_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1}\n    {f : \u03b1 \u2192 \u03b2} : sum (map (fun (a : \u03b1) => b * f a) s) = b * sum (map f s) :=\n  sorry\n\ntheorem sum_map_mul_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} [semiring \u03b2] {b : \u03b2} {s : multiset \u03b1}\n    {f : \u03b1 \u2192 \u03b2} : sum (map (fun (a : \u03b1) => f a * b) s) = sum (map f s) * b :=\n  sorry\n\ntheorem prod_ne_zero {R : Type u_1} [comm_semiring R] [no_zero_divisors R] [nontrivial R]\n    {m : multiset R} : (\u2200 (x : R), x \u2208 m \u2192 x \u2260 0) \u2192 prod m \u2260 0 :=\n  sorry\n\ntheorem prod_eq_zero {\u03b1 : Type u_1} [comm_semiring \u03b1] {s : multiset \u03b1} (h : 0 \u2208 s) : prod s = 0 :=\n  sorry\n\ntheorem sum_hom {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1] [add_comm_monoid \u03b2]\n    (s : multiset \u03b1) (f : \u03b1 \u2192+ \u03b2) : sum (map (\u21d1f) s) = coe_fn f (sum s) :=\n  sorry\n\ntheorem prod_hom_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b2] [comm_monoid \u03b3]\n    (s : multiset \u03b1) {r : \u03b2 \u2192 \u03b3 \u2192 Prop} {f : \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b3} (h\u2081 : r 1 1)\n    (h\u2082 : \u2200 {a : \u03b1} {b : \u03b2} {c : \u03b3}, r b c \u2192 r (f a * b) (g a * c)) :\n    r (prod (map f s)) (prod (map g s)) :=\n  sorry\n\ntheorem dvd_prod {\u03b1 : Type u_1} [comm_monoid \u03b1] {a : \u03b1} {s : multiset \u03b1} : a \u2208 s \u2192 a \u2223 prod s :=\n  sorry\n\ntheorem prod_dvd_prod {\u03b1 : Type u_1} [comm_monoid \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) :\n    prod s \u2223 prod t :=\n  sorry\n\ntheorem prod_eq_zero_iff {\u03b1 : Type u_1} [comm_cancel_monoid_with_zero \u03b1] [nontrivial \u03b1]\n    {s : multiset \u03b1} : prod s = 0 \u2194 0 \u2208 s :=\n  sorry\n\ntheorem sum_nonneg {\u03b1 : Type u_1} [ordered_add_comm_monoid \u03b1] {m : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 m \u2192 0 \u2264 x) \u2192 0 \u2264 sum m :=\n  sorry\n\ntheorem single_le_prod {\u03b1 : Type u_1} [ordered_comm_monoid \u03b1] {m : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 m \u2192 1 \u2264 x) \u2192 \u2200 (x : \u03b1), x \u2208 m \u2192 x \u2264 prod m :=\n  sorry\n\ntheorem all_one_of_le_one_le_of_prod_eq_one {\u03b1 : Type u_1} [ordered_comm_monoid \u03b1]\n    {m : multiset \u03b1} : (\u2200 (x : \u03b1), x \u2208 m \u2192 1 \u2264 x) \u2192 prod m = 1 \u2192 \u2200 (x : \u03b1), x \u2208 m \u2192 x = 1 :=\n  sorry\n\ntheorem sum_eq_zero_iff {\u03b1 : Type u_1} [canonically_ordered_add_monoid \u03b1] {m : multiset \u03b1} :\n    sum m = 0 \u2194 \u2200 (x : \u03b1), x \u2208 m \u2192 x = 0 :=\n  sorry\n\ntheorem le_sum_of_subadditive {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1]\n    [ordered_add_comm_monoid \u03b2] (f : \u03b1 \u2192 \u03b2) (h_zero : f 0 = 0)\n    (h_add : \u2200 (x y : \u03b1), f (x + y) \u2264 f x + f y) (s : multiset \u03b1) : f (sum s) \u2264 sum (map f s) :=\n  sorry\n\ntheorem abs_sum_le_sum_abs {\u03b1 : Type u_1} [linear_ordered_field \u03b1] {s : multiset \u03b1} :\n    abs (sum s) \u2264 sum (map abs s) :=\n  le_sum_of_subadditive abs abs_zero abs_add s\n\ntheorem dvd_sum {\u03b1 : Type u_1} [comm_semiring \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 s \u2192 a \u2223 x) \u2192 a \u2223 sum s :=\n  sorry\n\n@[simp] theorem sum_map_singleton {\u03b1 : Type u_1} (s : multiset \u03b1) :\n    sum (map (fun (a : \u03b1) => a ::\u2098 0) s) = s :=\n  sorry\n\n/-! ### Join -/\n\n/-- `join S`, where `S` is a multiset of multisets, is the lift of the list join\n  operation, that is, the union of all the sets.\n\n     join {{1, 2}, {1, 2}, {0, 1}} = {0, 1, 1, 1, 2, 2} -/\ndef join {\u03b1 : Type u_1} : multiset (multiset \u03b1) \u2192 multiset \u03b1 := sum\n\ntheorem coe_join {\u03b1 : Type u_1} (L : List (List \u03b1)) : join \u2191(list.map coe L) = \u2191(list.join L) :=\n  sorry\n\n@[simp] theorem join_zero {\u03b1 : Type u_1} : join 0 = 0 := rfl\n\n@[simp] theorem join_cons {\u03b1 : Type u_1} (s : multiset \u03b1) (S : multiset (multiset \u03b1)) :\n    join (s ::\u2098 S) = s + join S :=\n  sum_cons s S\n\n@[simp] theorem join_add {\u03b1 : Type u_1} (S : multiset (multiset \u03b1)) (T : multiset (multiset \u03b1)) :\n    join (S + T) = join S + join T :=\n  sum_add S T\n\n@[simp] theorem mem_join {\u03b1 : Type u_1} {a : \u03b1} {S : multiset (multiset \u03b1)} :\n    a \u2208 join S \u2194 \u2203 (s : multiset \u03b1), \u2203 (H : s \u2208 S), a \u2208 s :=\n  sorry\n\n@[simp] theorem card_join {\u03b1 : Type u_1} (S : multiset (multiset \u03b1)) :\n    coe_fn card (join S) = sum (map (\u21d1card) S) :=\n  sorry\n\n/-! ### `multiset.bind` -/\n\n/-- `bind s f` is the monad bind operation, defined as `join (map f s)`.\n  It is the union of `f a` as `a` ranges over `s`. -/\ndef bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) : multiset \u03b2 :=\n  join (map f s)\n\n@[simp] theorem coe_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l : List \u03b1) (f : \u03b1 \u2192 List \u03b2) :\n    (bind \u2191l fun (a : \u03b1) => \u2191(f a)) = \u2191(list.bind l f) :=\n  sorry\n\n@[simp] theorem zero_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 multiset \u03b2) : bind 0 f = 0 := rfl\n\n@[simp] theorem cons_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (s : multiset \u03b1)\n    (f : \u03b1 \u2192 multiset \u03b2) : bind (a ::\u2098 s) f = f a + bind s f :=\n  sorry\n\n@[simp] theorem add_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b1)\n    (f : \u03b1 \u2192 multiset \u03b2) : bind (s + t) f = bind s f + bind t f :=\n  sorry\n\n@[simp] theorem bind_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) :\n    (bind s fun (a : \u03b1) => 0) = 0 :=\n  sorry\n\n@[simp] theorem bind_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2)\n    (g : \u03b1 \u2192 multiset \u03b2) : (bind s fun (a : \u03b1) => f a + g a) = bind s f + bind s g :=\n  sorry\n\n@[simp] theorem bind_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 \u03b2)\n    (g : \u03b1 \u2192 multiset \u03b2) : (bind s fun (a : \u03b1) => f a ::\u2098 g a) = map f s + bind s g :=\n  sorry\n\n@[simp] theorem mem_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {b : \u03b2} {s : multiset \u03b1}\n    {f : \u03b1 \u2192 multiset \u03b2} : b \u2208 bind s f \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), b \u2208 f a :=\n  sorry\n\n@[simp] theorem card_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (f : \u03b1 \u2192 multiset \u03b2) :\n    coe_fn card (bind s f) = sum (map (\u21d1card \u2218 f) s) :=\n  sorry\n\ntheorem bind_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b1 \u2192 multiset \u03b2}\n    {m : multiset \u03b1} : (\u2200 (a : \u03b1), a \u2208 m \u2192 f a = g a) \u2192 bind m f = bind m g :=\n  sorry\n\ntheorem bind_hcongr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b2' : Type u_2} {m : multiset \u03b1}\n    {f : \u03b1 \u2192 multiset \u03b2} {f' : \u03b1 \u2192 multiset \u03b2'} (h : \u03b2 = \u03b2') (hf : \u2200 (a : \u03b1), a \u2208 m \u2192 f a == f' a) :\n    bind m f == bind m f' :=\n  sorry\n\ntheorem map_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : \u03b1 \u2192 multiset \u03b2)\n    (f : \u03b2 \u2192 \u03b3) : map f (bind m n) = bind m fun (a : \u03b1) => map f (n a) :=\n  sorry\n\ntheorem bind_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : \u03b2 \u2192 multiset \u03b3)\n    (f : \u03b1 \u2192 \u03b2) : bind (map f m) n = bind m fun (a : \u03b1) => n (f a) :=\n  sorry\n\ntheorem bind_assoc {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {s : multiset \u03b1}\n    {f : \u03b1 \u2192 multiset \u03b2} {g : \u03b2 \u2192 multiset \u03b3} :\n    bind (bind s f) g = bind s fun (a : \u03b1) => bind (f a) g :=\n  sorry\n\ntheorem bind_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : multiset \u03b2)\n    {f : \u03b1 \u2192 \u03b2 \u2192 multiset \u03b3} :\n    (bind m fun (a : \u03b1) => bind n fun (b : \u03b2) => f a b) =\n        bind n fun (b : \u03b2) => bind m fun (a : \u03b1) => f a b :=\n  sorry\n\ntheorem bind_map_comm {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (m : multiset \u03b1) (n : multiset \u03b2)\n    {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} :\n    (bind m fun (a : \u03b1) => map (fun (b : \u03b2) => f a b) n) =\n        bind n fun (b : \u03b2) => map (fun (a : \u03b1) => f a b) m :=\n  sorry\n\n@[simp] theorem sum_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b2] (s : multiset \u03b1)\n    (t : \u03b1 \u2192 multiset \u03b2) : sum (bind s t) = sum (map (fun (a : \u03b1) => sum (t a)) s) :=\n  sorry\n\n/-! ### Product of two `multiset`s -/\n\n/-- The multiplicity of `(a, b)` in `product s t` is\n  the product of the multiplicity of `a` in `s` and `b` in `t`. -/\ndef product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u00d7 \u03b2) :=\n  bind s fun (a : \u03b1) => map (Prod.mk a) t\n\n@[simp] theorem coe_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (l\u2081 : List \u03b1) (l\u2082 : List \u03b2) :\n    product \u2191l\u2081 \u2191l\u2082 = \u2191(list.product l\u2081 l\u2082) :=\n  sorry\n\n@[simp] theorem zero_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (t : multiset \u03b2) : product 0 t = 0 := rfl\n\n@[simp] theorem cons_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (s : multiset \u03b1)\n    (t : multiset \u03b2) : product (a ::\u2098 s) t = map (Prod.mk a) t + product s t :=\n  sorry\n\n@[simp] theorem product_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (b : \u03b2) :\n    product (a ::\u2098 0) (b ::\u2098 0) = (a, b) ::\u2098 0 :=\n  rfl\n\n@[simp] theorem add_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : multiset \u03b2) : product (s + t) u = product s u + product t u :=\n  sorry\n\n@[simp] theorem product_add {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2)\n    (u : multiset \u03b2) : product s (t + u) = product s t + product s u :=\n  sorry\n\n@[simp] theorem mem_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} {s : multiset \u03b1} {t : multiset \u03b2}\n    {p : \u03b1 \u00d7 \u03b2} : p \u2208 product s t \u2194 prod.fst p \u2208 s \u2227 prod.snd p \u2208 t :=\n  sorry\n\n@[simp] theorem card_product {\u03b1 : Type u_1} {\u03b2 : Type u_2} (s : multiset \u03b1) (t : multiset \u03b2) :\n    coe_fn card (product s t) = coe_fn card s * coe_fn card t :=\n  sorry\n\n/-! ### Sigma multiset -/\n\n/-- `sigma s t` is the dependent version of `product`. It is the sum of\n  `(a, b)` as `a` ranges over `s` and `b` ranges over `t a`. -/\nprotected def sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) : multiset (sigma fun (a : \u03b1) => \u03c3 a) :=\n  bind s fun (a : \u03b1) => map (sigma.mk a) (t a)\n\n@[simp] theorem coe_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (l\u2081 : List \u03b1)\n    (l\u2082 : (a : \u03b1) \u2192 List (\u03c3 a)) :\n    (multiset.sigma \u2191l\u2081 fun (a : \u03b1) => \u2191(l\u2082 a)) = \u2191(list.sigma l\u2081 l\u2082) :=\n  sorry\n\n@[simp] theorem zero_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) :\n    multiset.sigma 0 t = 0 :=\n  rfl\n\n@[simp] theorem cons_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (a : \u03b1) (s : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) :\n    multiset.sigma (a ::\u2098 s) t = map (sigma.mk a) (t a) + multiset.sigma s t :=\n  sorry\n\n@[simp] theorem sigma_singleton {\u03b1 : Type u_1} {\u03b2 : Type u_2} (a : \u03b1) (b : \u03b1 \u2192 \u03b2) :\n    (multiset.sigma (a ::\u2098 0) fun (a : \u03b1) => b a ::\u2098 0) = sigma.mk a (b a) ::\u2098 0 :=\n  rfl\n\n@[simp] theorem add_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : (a : \u03b1) \u2192 multiset (\u03c3 a)) :\n    multiset.sigma (s + t) u = multiset.sigma s u + multiset.sigma t u :=\n  sorry\n\n@[simp] theorem sigma_add {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) (u : (a : \u03b1) \u2192 multiset (\u03c3 a)) :\n    (multiset.sigma s fun (a : \u03b1) => t a + u a) = multiset.sigma s t + multiset.sigma s u :=\n  sorry\n\n@[simp] theorem mem_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} {s : multiset \u03b1}\n    {t : (a : \u03b1) \u2192 multiset (\u03c3 a)} {p : sigma fun (a : \u03b1) => \u03c3 a} :\n    p \u2208 multiset.sigma s t \u2194 sigma.fst p \u2208 s \u2227 sigma.snd p \u2208 t (sigma.fst p) :=\n  sorry\n\n@[simp] theorem card_sigma {\u03b1 : Type u_1} {\u03c3 : \u03b1 \u2192 Type u_4} (s : multiset \u03b1)\n    (t : (a : \u03b1) \u2192 multiset (\u03c3 a)) :\n    coe_fn card (multiset.sigma s t) = sum (map (fun (a : \u03b1) => coe_fn card (t a)) s) :=\n  sorry\n\n/-! ### Map for partial functions -/\n\n/-- Lift of the list `pmap` operation. Map a partial function `f` over a multiset\n  `s` whose elements are all in the domain of `f`. -/\ndef pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) :\n    (\u2200 (a : \u03b1), a \u2208 s \u2192 p a) \u2192 multiset \u03b2 :=\n  quot.rec_on s\n    (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => \u2191(list.pmap f l H)) sorry\n\n@[simp] theorem coe_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 l \u2192 p a) : pmap f (\u2191l) H = \u2191(list.pmap f l H) :=\n  rfl\n\n@[simp] theorem pmap_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (h : \u2200 (a : \u03b1), a \u2208 0 \u2192 p a) : pmap f 0 h = 0 :=\n  rfl\n\n@[simp] theorem pmap_cons {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (a : \u03b1) (m : multiset \u03b1) (h : \u2200 (b : \u03b1), b \u2208 a ::\u2098 m \u2192 p b) :\n    pmap f (a ::\u2098 m) h =\n        f a (h a (mem_cons_self a m)) ::\u2098\n          pmap f m fun (a_1 : \u03b1) (ha : a_1 \u2208 m) => h a_1 (mem_cons_of_mem ha) :=\n  quotient.induction_on m fun (l : List \u03b1) (h : \u2200 (b : \u03b1), b \u2208 a ::\u2098 quotient.mk l \u2192 p b) => rfl\n\n/-- \"Attach\" a proof that `a \u2208 s` to each element `a` in `s` to produce\n  a multiset on `{x // x \u2208 s}`. -/\ndef attach {\u03b1 : Type u_1} (s : multiset \u03b1) : multiset (Subtype fun (x : \u03b1) => x \u2208 s) :=\n  pmap Subtype.mk s sorry\n\n@[simp] theorem coe_attach {\u03b1 : Type u_1} (l : List \u03b1) : attach \u2191l = \u2191(list.attach l) := rfl\n\ntheorem sizeof_lt_sizeof_of_mem {\u03b1 : Type u_1} [SizeOf \u03b1] {x : \u03b1} {s : multiset \u03b1} (hx : x \u2208 s) :\n    sizeof x < sizeof s :=\n  sorry\n\ntheorem pmap_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (p : \u03b1 \u2192 Prop) (f : \u03b1 \u2192 \u03b2) (s : multiset \u03b1)\n    (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : pmap (fun (a : \u03b1) (_x : p a) => f a) s H = map f s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) =>\n      congr_arg coe (list.pmap_eq_map p f l H)\n\ntheorem pmap_congr {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop}\n    {f : (a : \u03b1) \u2192 p a \u2192 \u03b2} {g : (a : \u03b1) \u2192 q a \u2192 \u03b2} (s : multiset \u03b1) {H\u2081 : \u2200 (a : \u03b1), a \u2208 s \u2192 p a}\n    {H\u2082 : \u2200 (a : \u03b1), a \u2208 s \u2192 q a} (h : \u2200 (a : \u03b1) (h\u2081 : p a) (h\u2082 : q a), f a h\u2081 = g a h\u2082) :\n    pmap f s H\u2081 = pmap g s H\u2082 :=\n  sorry\n\ntheorem map_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {p : \u03b1 \u2192 Prop} (g : \u03b2 \u2192 \u03b3)\n    (f : (a : \u03b1) \u2192 p a \u2192 \u03b2) (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) :\n    map g (pmap f s H) = pmap (fun (a : \u03b1) (h : p a) => g (f a h)) s H :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) =>\n      congr_arg coe (list.map_pmap g f l H)\n\ntheorem pmap_eq_map_attach {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) :\n    pmap f s H =\n        map\n          (fun (x : Subtype fun (x : \u03b1) => x \u2208 s) =>\n            f (subtype.val x) (H (subtype.val x) (subtype.property x)))\n          (attach s) :=\n  quot.induction_on s\n    fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) =>\n      congr_arg coe (list.pmap_eq_map_attach f l H)\n\ntheorem attach_map_val {\u03b1 : Type u_1} (s : multiset \u03b1) : map subtype.val (attach s) = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.attach_map_val l)\n\n@[simp] theorem mem_attach {\u03b1 : Type u_1} (s : multiset \u03b1) (x : Subtype fun (x : \u03b1) => x \u2208 s) :\n    x \u2208 attach s :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_attach l\n\n@[simp] theorem mem_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} {f : (a : \u03b1) \u2192 p a \u2192 \u03b2}\n    {s : multiset \u03b1} {H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a} {b : \u03b2} :\n    b \u2208 pmap f s H \u2194 \u2203 (a : \u03b1), \u2203 (h : a \u2208 s), f a (H a h) = b :=\n  quot.induction_on s\n    (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => list.mem_pmap) H\n\n@[simp] theorem card_pmap {\u03b1 : Type u_1} {\u03b2 : Type u_2} {p : \u03b1 \u2192 Prop} (f : (a : \u03b1) \u2192 p a \u2192 \u03b2)\n    (s : multiset \u03b1) (H : \u2200 (a : \u03b1), a \u2208 s \u2192 p a) : coe_fn card (pmap f s H) = coe_fn card s :=\n  quot.induction_on s\n    (fun (l : List \u03b1) (H : \u2200 (a : \u03b1), a \u2208 Quot.mk setoid.r l \u2192 p a) => list.length_pmap) H\n\n@[simp] theorem card_attach {\u03b1 : Type u_1} {m : multiset \u03b1} :\n    coe_fn card (attach m) = coe_fn card m :=\n  card_pmap Subtype.mk m (attach._proof_1 m)\n\n@[simp] theorem attach_zero {\u03b1 : Type u_1} : attach 0 = 0 := rfl\n\ntheorem attach_cons {\u03b1 : Type u_1} (a : \u03b1) (m : multiset \u03b1) :\n    attach (a ::\u2098 m) =\n        { val := a, property := mem_cons_self a m } ::\u2098\n          map\n            (fun (p : Subtype fun (x : \u03b1) => x \u2208 m) =>\n              { val := subtype.val p, property := mem_cons_of_mem (subtype.property p) })\n            (attach m) :=\n  sorry\n\nprotected def decidable_forall_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : \u03b1 \u2192 Prop}\n    [hp : (a : \u03b1) \u2192 Decidable (p a)] : Decidable (\u2200 (a : \u03b1), a \u2208 m \u2192 p a) :=\n  quotient.rec_on_subsingleton m fun (l : List \u03b1) => decidable_of_iff (\u2200 (a : \u03b1), a \u2208 l \u2192 p a) sorry\n\nprotected instance decidable_dforall_multiset {\u03b1 : Type u_1} {m : multiset \u03b1}\n    {p : (a : \u03b1) \u2192 a \u2208 m \u2192 Prop} [hp : (a : \u03b1) \u2192 (h : a \u2208 m) \u2192 Decidable (p a h)] :\n    Decidable (\u2200 (a : \u03b1) (h : a \u2208 m), p a h) :=\n  decidable_of_decidable_of_iff multiset.decidable_forall_multiset sorry\n\n/-- decidable equality for functions whose domain is bounded by multisets -/\nprotected instance decidable_eq_pi_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {\u03b2 : \u03b1 \u2192 Type u_2}\n    [h : (a : \u03b1) \u2192 DecidableEq (\u03b2 a)] : DecidableEq ((a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a) :=\n  fun (f g : (a : \u03b1) \u2192 a \u2208 m \u2192 \u03b2 a) => decidable_of_iff (\u2200 (a : \u03b1) (h : a \u2208 m), f a h = g a h) sorry\n\ndef decidable_exists_multiset {\u03b1 : Type u_1} {m : multiset \u03b1} {p : \u03b1 \u2192 Prop} [decidable_pred p] :\n    Decidable (\u2203 (x : \u03b1), \u2203 (H : x \u2208 m), p x) :=\n  quotient.rec_on_subsingleton m list.decidable_exists_mem\n\nprotected instance decidable_dexists_multiset {\u03b1 : Type u_1} {m : multiset \u03b1}\n    {p : (a : \u03b1) \u2192 a \u2208 m \u2192 Prop} [hp : (a : \u03b1) \u2192 (h : a \u2208 m) \u2192 Decidable (p a h)] :\n    Decidable (\u2203 (a : \u03b1), \u2203 (h : a \u2208 m), p a h) :=\n  decidable_of_decidable_of_iff decidable_exists_multiset sorry\n\n/-! ### Subtraction -/\n\n/-- `s - t` is the multiset such that\n  `count a (s - t) = count a s - count a t` for all `a`. -/\nprotected def sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s t (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(list.diff l\u2081 l\u2082)) sorry\n\nprotected instance has_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] : Sub (multiset \u03b1) :=\n  { sub := multiset.sub }\n\n@[simp] theorem coe_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : List \u03b1) (t : List \u03b1) :\n    \u2191s - \u2191t = \u2191(list.diff s t) :=\n  rfl\n\ntheorem sub_eq_fold_erase {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s - t = foldl erase erase_comm s t :=\n  sorry\n\n@[simp] theorem sub_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : s - 0 = s :=\n  quot.induction_on s fun (l : List \u03b1) => rfl\n\n@[simp] theorem sub_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) :\n    s - a ::\u2098 t = erase s a - t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.diff_cons l\u2081 l\u2082 a)\n\ntheorem add_sub_of_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t) :\n    s + (t - s) = t :=\n  sorry\n\ntheorem sub_add' {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1} :\n    s - (t + u) = s - t - u :=\n  quotient.induction_on\u2083 s t u fun (l\u2081 l\u2082 l\u2083 : List \u03b1) => congr_arg coe (list.diff_append l\u2081 l\u2082 l\u2083)\n\ntheorem sub_add_cancel {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : t \u2264 s) : s - t + t = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s - t + t = s)) (add_comm (s - t) t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t + (s - t) = s)) (add_sub_of_le h))) (Eq.refl s))\n\n@[simp] theorem add_sub_cancel_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1)\n    (t : multiset \u03b1) : s + t - s = t :=\n  sorry\n\n@[simp] theorem add_sub_cancel {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s + t - t = s :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s + t - t = s)) (add_comm s t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t + s - t = s)) (add_sub_cancel_left t s))) (Eq.refl s))\n\ntheorem sub_le_sub_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : s \u2264 t) (u : multiset \u03b1) : s - u \u2264 t - u :=\n  sorry\n\ntheorem sub_le_sub_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} (h : s \u2264 t)\n    (u : multiset \u03b1) : u - t \u2264 u - s :=\n  sorry\n\ntheorem sub_le_iff_le_add {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : s - t \u2264 u \u2194 s \u2264 u + t :=\n  sorry\n\ntheorem le_sub_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2264 s - t + t :=\n  iff.mp sub_le_iff_le_add (le_refl (s - t))\n\ntheorem sub_le_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : s - t \u2264 s :=\n  iff.mpr sub_le_iff_le_add (le_add_right s t)\n\n@[simp] theorem card_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : t \u2264 s) : coe_fn card (s - t) = coe_fn card s - coe_fn card t :=\n  sorry\n\n/-! ### Union -/\n\n/-- `s \u222a t` is the lattice join operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u222a t` is the maximum\n  of the multiplicities in `s` and `t`. -/\ndef union {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 := s - t + t\n\nprotected instance has_union {\u03b1 : Type u_1} [DecidableEq \u03b1] : has_union (multiset \u03b1) :=\n  has_union.mk union\n\ntheorem union_def {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u222a t = s - t + t :=\n  rfl\n\ntheorem le_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2264 s \u222a t :=\n  le_sub_add s t\n\ntheorem le_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    t \u2264 s \u222a t :=\n  le_add_left t (s - t)\n\ntheorem eq_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} :\n    t \u2264 s \u2192 s \u222a t = s :=\n  sub_add_cancel\n\ntheorem union_le_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : s \u2264 t) (u : multiset \u03b1) : s \u222a u \u2264 t \u222a u :=\n  add_le_add_right (sub_le_sub_right h u) u\n\ntheorem union_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h\u2081 : s \u2264 u) (h\u2082 : t \u2264 u) : s \u222a t \u2264 u :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s \u222a t \u2264 u)) (Eq.symm (eq_union_left h\u2082))))\n    (union_le_union_right h\u2081 t)\n\n@[simp] theorem mem_union {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} :\n    a \u2208 s \u222a t \u2194 a \u2208 s \u2228 a \u2208 t :=\n  { mp := fun (h : a \u2208 s \u222a t) => or.imp_left (mem_of_le (sub_le_self s t)) (iff.mp mem_add h),\n    mpr := Or._oldrec (mem_of_le (le_union_left s t)) (mem_of_le (le_union_right s t)) }\n\n@[simp] theorem map_union {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] [DecidableEq \u03b2] {f : \u03b1 \u2192 \u03b2}\n    (finj : function.injective f) {s : multiset \u03b1} {t : multiset \u03b1} :\n    map f (s \u222a t) = map f s \u222a map f t :=\n  sorry\n\n/-! ### Intersection -/\n\n/-- `s \u2229 t` is the lattice meet operation with respect to the\n  multiset `\u2264`. The multiplicity of `a` in `s \u2229 t` is the minimum\n  of the multiplicities in `s` and `t`. -/\ndef inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) : multiset \u03b1 :=\n  quotient.lift_on\u2082 s t (fun (l\u2081 l\u2082 : List \u03b1) => \u2191(list.bag_inter l\u2081 l\u2082)) sorry\n\nprotected instance has_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] : has_inter (multiset \u03b1) :=\n  has_inter.mk inter\n\n@[simp] theorem inter_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : s \u2229 0 = 0 :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.bag_inter_nil l)\n\n@[simp] theorem zero_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) : 0 \u2229 s = 0 :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.nil_bag_inter l)\n\n@[simp] theorem cons_inter_of_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1)\n    {t : multiset \u03b1} : a \u2208 t \u2192 (a ::\u2098 s) \u2229 t = a ::\u2098 s \u2229 erase t a :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : a \u2208 quotient.mk l\u2082) => congr_arg coe (list.cons_bag_inter_of_pos l\u2081 h)\n\n@[simp] theorem cons_inter_of_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} (s : multiset \u03b1)\n    {t : multiset \u03b1} : \u00aca \u2208 t \u2192 (a ::\u2098 s) \u2229 t = s \u2229 t :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) (h : \u00aca \u2208 quotient.mk l\u2082) =>\n      congr_arg coe (list.cons_bag_inter_of_neg l\u2081 h)\n\ntheorem inter_le_left {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2229 t \u2264 s :=\n  quotient.induction_on\u2082 s t\n    fun (l\u2081 l\u2082 : List \u03b1) => list.sublist.subperm (list.bag_inter_sublist_left l\u2081 l\u2082)\n\ntheorem inter_le_right {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2229 t \u2264 t :=\n  sorry\n\ntheorem le_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h\u2081 : s \u2264 t) (h\u2082 : s \u2264 u) : s \u2264 t \u2229 u :=\n  sorry\n\n@[simp] theorem mem_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} {a : \u03b1} :\n    a \u2208 s \u2229 t \u2194 a \u2208 s \u2227 a \u2208 t :=\n  sorry\n\nprotected instance lattice {\u03b1 : Type u_1} [DecidableEq \u03b1] : lattice (multiset \u03b1) :=\n  lattice.mk has_union.union partial_order.le partial_order.lt sorry sorry sorry sorry sorry sorry\n    has_inter.inter sorry sorry sorry\n\n@[simp] theorem sup_eq_union {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2294 t = s \u222a t :=\n  rfl\n\n@[simp] theorem inf_eq_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2293 t = s \u2229 t :=\n  rfl\n\n@[simp] theorem le_inter_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : s \u2264 t \u2229 u \u2194 s \u2264 t \u2227 s \u2264 u :=\n  le_inf_iff\n\n@[simp] theorem union_le_iff {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : s \u222a t \u2264 u \u2194 s \u2264 u \u2227 t \u2264 u :=\n  sup_le_iff\n\nprotected instance semilattice_inf_bot {\u03b1 : Type u_1} [DecidableEq \u03b1] :\n    semilattice_inf_bot (multiset \u03b1) :=\n  semilattice_inf_bot.mk 0 lattice.le lattice.lt sorry sorry sorry zero_le lattice.inf sorry sorry\n    sorry\n\ntheorem union_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u222a t = t \u222a s :=\n  sup_comm\n\ntheorem inter_comm {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u2229 t = t \u2229 s :=\n  inf_comm\n\ntheorem eq_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : s \u2264 t) : s \u222a t = t :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (s \u222a t = t)) (union_comm s t)))\n    (eq.mpr (id (Eq._oldrec (Eq.refl (t \u222a s = t)) (eq_union_left h))) (Eq.refl t))\n\ntheorem union_le_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    (h : s \u2264 t) (u : multiset \u03b1) : u \u222a s \u2264 u \u222a t :=\n  sup_le_sup_left h u\n\ntheorem union_le_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u222a t \u2264 s + t :=\n  union_le (le_add_right s t) (le_add_left t s)\n\ntheorem union_add_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : multiset \u03b1) : s \u222a t + u = s + u \u222a (t + u) :=\n  sorry\n\ntheorem add_union_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : multiset \u03b1) : s + (t \u222a u) = s + t \u222a (s + u) :=\n  sorry\n\ntheorem cons_union_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1)\n    (t : multiset \u03b1) : a ::\u2098 (s \u222a t) = a ::\u2098 s \u222a a ::\u2098 t :=\n  sorry\n\ntheorem inter_add_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : multiset \u03b1) : s \u2229 t + u = (s + u) \u2229 (t + u) :=\n  sorry\n\ntheorem add_inter_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1)\n    (u : multiset \u03b1) : s + t \u2229 u = (s + t) \u2229 (s + u) :=\n  sorry\n\ntheorem cons_inter_distrib {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1)\n    (t : multiset \u03b1) : a ::\u2098 s \u2229 t = (a ::\u2098 s) \u2229 (a ::\u2098 t) :=\n  sorry\n\ntheorem union_add_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s \u222a t + s \u2229 t = s + t :=\n  sorry\n\ntheorem sub_add_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s - t + s \u2229 t = s :=\n  sorry\n\ntheorem sub_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : multiset \u03b1) (t : multiset \u03b1) :\n    s - s \u2229 t = s - t :=\n  add_right_cancel\n    (eq.mpr (id (Eq._oldrec (Eq.refl (s - s \u2229 t + s \u2229 t = s - t + s \u2229 t)) (sub_add_inter s t)))\n      (eq.mpr\n        (id (Eq._oldrec (Eq.refl (s - s \u2229 t + s \u2229 t = s)) (sub_add_cancel (inter_le_left s t))))\n        (Eq.refl s)))\n\n/-! ### `multiset.filter` -/\n\n/-- `filter p s` returns the elements in `s` (with the same multiplicities)\n  which satisfy `p`, and removes the rest. -/\ndef filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : multiset \u03b1 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.filter p l)) sorry\n\n@[simp] theorem coe_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    filter p \u2191l = \u2191(list.filter p l) :=\n  rfl\n\n@[simp] theorem filter_zero {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : filter p 0 = 0 := rfl\n\ntheorem filter_congr {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} {q : \u03b1 \u2192 Prop} [decidable_pred p]\n    [decidable_pred q] {s : multiset \u03b1} :\n    (\u2200 (x : \u03b1), x \u2208 s \u2192 (p x \u2194 q x)) \u2192 filter p s = filter q s :=\n  quot.induction_on s\n    fun (l : List \u03b1) (h : \u2200 (x : \u03b1), x \u2208 Quot.mk setoid.r l \u2192 (p x \u2194 q x)) =>\n      congr_arg coe (list.filter_congr h)\n\n@[simp] theorem filter_add {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1)\n    (t : multiset \u03b1) : filter p (s + t) = filter p s + filter p t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => congr_arg coe (list.filter_append l\u2081 l\u2082)\n\n@[simp] theorem filter_le {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n    filter p s \u2264 s :=\n  quot.induction_on s fun (l : List \u03b1) => list.sublist.subperm (list.filter_sublist l)\n\n@[simp] theorem filter_subset {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n    filter p s \u2286 s :=\n  subset_of_le (filter_le p s)\n\ntheorem filter_le_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] {s : multiset \u03b1}\n    {t : multiset \u03b1} (h : s \u2264 t) : filter p s \u2264 filter p t :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.filter_sublist_filter p h)\n\n@[simp] theorem filter_cons_of_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    (s : multiset \u03b1) : p a \u2192 filter p (a ::\u2098 s) = a ::\u2098 filter p s :=\n  quot.induction_on s fun (l : List \u03b1) (h : p a) => congr_arg coe (list.filter_cons_of_pos l h)\n\n@[simp] theorem filter_cons_of_neg {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    (s : multiset \u03b1) : \u00acp a \u2192 filter p (a ::\u2098 s) = filter p s :=\n  quot.induction_on s fun (l : List \u03b1) (h : \u00acp a) => congr_arg coe (list.filter_cons_of_neg l h)\n\n@[simp] theorem mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    {s : multiset \u03b1} : a \u2208 filter p s \u2194 a \u2208 s \u2227 p a :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_filter\n\ntheorem of_mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1}\n    (h : a \u2208 filter p s) : p a :=\n  and.right (iff.mp mem_filter h)\n\ntheorem mem_of_mem_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {s : multiset \u03b1}\n    (h : a \u2208 filter p s) : a \u2208 s :=\n  and.left (iff.mp mem_filter h)\n\ntheorem mem_filter_of_mem {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1} {l : multiset \u03b1}\n    (m : a \u2208 l) (h : p a) : a \u2208 filter p l :=\n  iff.mpr mem_filter { left := m, right := h }\n\ntheorem filter_eq_self {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} :\n    filter p s = s \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 p a :=\n  sorry\n\ntheorem filter_eq_nil {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} :\n    filter p s = 0 \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u00acp a :=\n  sorry\n\ntheorem le_filter {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1}\n    {t : multiset \u03b1} : s \u2264 filter p t \u2194 s \u2264 t \u2227 \u2200 (a : \u03b1), a \u2208 s \u2192 p a :=\n  sorry\n\n@[simp] theorem filter_sub {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1]\n    (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s - t) = filter p s - filter p t :=\n  sorry\n\n@[simp] theorem filter_union {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1]\n    (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s \u222a t) = filter p s \u222a filter p t :=\n  sorry\n\n@[simp] theorem filter_inter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1]\n    (s : multiset \u03b1) (t : multiset \u03b1) : filter p (s \u2229 t) = filter p s \u2229 filter p t :=\n  sorry\n\n@[simp] theorem filter_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop)\n    [decidable_pred q] (s : multiset \u03b1) :\n    filter p (filter q s) = filter (fun (a : \u03b1) => p a \u2227 q a) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_filter p q l)\n\ntheorem filter_add_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop)\n    [decidable_pred q] (s : multiset \u03b1) :\n    filter p s + filter q s =\n        filter (fun (a : \u03b1) => p a \u2228 q a) s + filter (fun (a : \u03b1) => p a \u2227 q a) s :=\n  sorry\n\ntheorem filter_add_not {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n    filter p s + filter (fun (a : \u03b1) => \u00acp a) s = s :=\n  sorry\n\n/-! ### Simultaneously filter and map elements of a multiset -/\n\n/-- `filter_map f s` is a combination filter/map operation on `s`.\n  The function `f : \u03b1 \u2192 option \u03b2` is applied to each element of `s`;\n  if `f a` is `some b` then `b` is added to the result, otherwise\n  `a` is removed from the resulting multiset. -/\ndef filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1) : multiset \u03b2 :=\n  quot.lift_on s (fun (l : List \u03b1) => \u2191(list.filter_map f l)) sorry\n\n@[simp] theorem coe_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (l : List \u03b1) :\n    filter_map f \u2191l = \u2191(list.filter_map f l) :=\n  rfl\n\n@[simp] theorem filter_map_zero {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) :\n    filter_map f 0 = 0 :=\n  rfl\n\n@[simp] theorem filter_map_cons_none {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 Option \u03b2} (a : \u03b1)\n    (s : multiset \u03b1) (h : f a = none) : filter_map f (a ::\u2098 s) = filter_map f s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_cons_none a l h)\n\n@[simp] theorem filter_map_cons_some {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (a : \u03b1)\n    (s : multiset \u03b1) {b : \u03b2} (h : f a = some b) : filter_map f (a ::\u2098 s) = b ::\u2098 filter_map f s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_cons_some f a l h)\n\ntheorem filter_map_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 \u03b2) :\n    filter_map (some \u2218 f) = map f :=\n  funext\n    fun (s : multiset \u03b1) =>\n      quot.induction_on s fun (l : List \u03b1) => congr_arg coe (congr_fun (list.filter_map_eq_map f) l)\n\ntheorem filter_map_eq_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n    filter_map (option.guard p) = filter p :=\n  funext\n    fun (s : multiset \u03b1) =>\n      quot.induction_on s\n        fun (l : List \u03b1) => congr_arg coe (congr_fun (list.filter_map_eq_filter p) l)\n\ntheorem filter_map_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 Option \u03b2)\n    (g : \u03b2 \u2192 Option \u03b3) (s : multiset \u03b1) :\n    filter_map g (filter_map f s) = filter_map (fun (x : \u03b1) => option.bind (f x) g) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_filter_map f g l)\n\ntheorem map_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b3)\n    (s : multiset \u03b1) : map g (filter_map f s) = filter_map (fun (x : \u03b1) => option.map g (f x)) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_filter_map f g l)\n\ntheorem filter_map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} (f : \u03b1 \u2192 \u03b2) (g : \u03b2 \u2192 Option \u03b3)\n    (s : multiset \u03b1) : filter_map g (map f s) = filter_map (g \u2218 f) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_map f g l)\n\ntheorem filter_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (p : \u03b2 \u2192 Prop)\n    [decidable_pred p] (s : multiset \u03b1) :\n    filter p (filter_map f s) = filter_map (fun (x : \u03b1) => option.filter p (f x)) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_filter_map f p l)\n\ntheorem filter_map_filter {\u03b1 : Type u_1} {\u03b2 : Type u_2} (p : \u03b1 \u2192 Prop) [decidable_pred p]\n    (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1) :\n    filter_map f (filter p s) = filter_map (fun (x : \u03b1) => ite (p x) (f x) none) s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_filter p f l)\n\n@[simp] theorem filter_map_some {\u03b1 : Type u_1} (s : multiset \u03b1) : filter_map some s = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.filter_map_some l)\n\n@[simp] theorem mem_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (s : multiset \u03b1)\n    {b : \u03b2} : b \u2208 filter_map f s \u2194 \u2203 (a : \u03b1), a \u2208 s \u2227 f a = some b :=\n  quot.induction_on s fun (l : List \u03b1) => list.mem_filter_map f l\n\ntheorem map_filter_map_of_inv {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) (g : \u03b2 \u2192 \u03b1)\n    (H : \u2200 (x : \u03b1), option.map g (f x) = some x) (s : multiset \u03b1) : map g (filter_map f s) = s :=\n  quot.induction_on s fun (l : List \u03b1) => congr_arg coe (list.map_filter_map_of_inv f g H l)\n\ntheorem filter_map_le_filter_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} (f : \u03b1 \u2192 Option \u03b2) {s : multiset \u03b1}\n    {t : multiset \u03b1} (h : s \u2264 t) : filter_map f s \u2264 filter_map f t :=\n  le_induction_on h\n    fun (l\u2081 l\u2082 : List \u03b1) (h : l\u2081 <+ l\u2082) => list.sublist.subperm (list.sublist.filter_map f h)\n\n/-! ### countp -/\n\n/-- `countp p s` counts the number of elements of `s` (with multiplicity) that\n  satisfy `p`. -/\ndef countp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) : \u2115 :=\n  quot.lift_on s (list.countp p) sorry\n\n@[simp] theorem coe_countp {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : List \u03b1) :\n    countp p \u2191l = list.countp p l :=\n  rfl\n\n@[simp] theorem countp_zero {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] : countp p 0 = 0 := rfl\n\n@[simp] theorem countp_cons_of_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    (s : multiset \u03b1) : p a \u2192 countp p (a ::\u2098 s) = countp p s + 1 :=\n  quot.induction_on s (list.countp_cons_of_pos p)\n\n@[simp] theorem countp_cons_of_neg {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {a : \u03b1}\n    (s : multiset \u03b1) : \u00acp a \u2192 countp p (a ::\u2098 s) = countp p s :=\n  quot.induction_on s (list.countp_cons_of_neg p)\n\ntheorem countp_eq_card_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1) :\n    countp p s = coe_fn card (filter p s) :=\n  quot.induction_on s fun (l : List \u03b1) => list.countp_eq_length_filter p l\n\n@[simp] theorem countp_add {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (s : multiset \u03b1)\n    (t : multiset \u03b1) : countp p (s + t) = countp p s + countp p t :=\n  sorry\n\nprotected instance countp.is_add_monoid_hom {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] :\n    is_add_monoid_hom (countp p) :=\n  is_add_monoid_hom.mk (countp_zero p)\n\n@[simp] theorem countp_sub {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] [DecidableEq \u03b1]\n    {s : multiset \u03b1} {t : multiset \u03b1} (h : t \u2264 s) : countp p (s - t) = countp p s - countp p t :=\n  sorry\n\ntheorem countp_le_of_le {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] {s : multiset \u03b1}\n    {t : multiset \u03b1} (h : s \u2264 t) : countp p s \u2264 countp p t :=\n  sorry\n\n@[simp] theorem countp_filter {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (q : \u03b1 \u2192 Prop)\n    [decidable_pred q] (s : multiset \u03b1) :\n    countp p (filter q s) = countp (fun (a : \u03b1) => p a \u2227 q a) s :=\n  sorry\n\ntheorem countp_pos {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} :\n    0 < countp p s \u2194 \u2203 (a : \u03b1), \u2203 (H : a \u2208 s), p a :=\n  sorry\n\ntheorem countp_pos_of_mem {\u03b1 : Type u_1} {p : \u03b1 \u2192 Prop} [decidable_pred p] {s : multiset \u03b1} {a : \u03b1}\n    (h : a \u2208 s) (pa : p a) : 0 < countp p s :=\n  iff.mpr countp_pos (Exists.intro a (Exists.intro h pa))\n\n/-! ### Multiplicity of an element -/\n\n/-- `count a s` is the multiplicity of `a` in `s`. -/\ndef count {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : multiset \u03b1 \u2192 \u2115 := countp (Eq a)\n\n@[simp] theorem coe_count {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (l : List \u03b1) :\n    count a \u2191l = list.count a l :=\n  coe_countp (Eq a) l\n\n@[simp] theorem count_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : count a 0 = 0 := rfl\n\n@[simp] theorem count_cons_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) :\n    count a (a ::\u2098 s) = Nat.succ (count a s) :=\n  countp_cons_of_pos s rfl\n\n@[simp] theorem count_cons_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (h : a \u2260 b)\n    (s : multiset \u03b1) : count a (b ::\u2098 s) = count a s :=\n  countp_cons_of_neg s h\n\ntheorem count_le_of_le {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) {s : multiset \u03b1} {t : multiset \u03b1} :\n    s \u2264 t \u2192 count a s \u2264 count a t :=\n  countp_le_of_le (Eq a)\n\ntheorem count_le_count_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) :\n    count a s \u2264 count a (b ::\u2098 s) :=\n  count_le_of_le a (le_cons_self s b)\n\ntheorem count_cons {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (s : multiset \u03b1) :\n    count a (b ::\u2098 s) = count a s + ite (a = b) 1 0 :=\n  sorry\n\ntheorem count_singleton {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) : count a (a ::\u2098 0) = 1 := sorry\n\n@[simp] theorem count_add {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) :\n    count a (s + t) = count a s + count a t :=\n  countp_add (Eq a)\n\nprotected instance count.is_add_monoid_hom {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) :\n    is_add_monoid_hom (count a) :=\n  countp.is_add_monoid_hom (Eq a)\n\n@[simp] theorem count_smul {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (n : \u2115) (s : multiset \u03b1) :\n    count a (n \u2022\u2115 s) = n * count a s :=\n  sorry\n\ntheorem count_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} : 0 < count a s \u2194 a \u2208 s :=\n  sorry\n\n@[simp] theorem count_eq_zero_of_not_mem {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1}\n    (h : \u00aca \u2208 s) : count a s = 0 :=\n  by_contradiction fun (h' : \u00account a s = 0) => h (iff.mp count_pos (nat.pos_of_ne_zero h'))\n\n@[simp] theorem count_eq_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    count a s = 0 \u2194 \u00aca \u2208 s :=\n  iff.mp iff_not_comm (iff.trans (iff.symm count_pos) pos_iff_ne_zero)\n\ntheorem count_ne_zero {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} :\n    count a s \u2260 0 \u2194 a \u2208 s :=\n  sorry\n\n@[simp] theorem count_repeat_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (n : \u2115) :\n    count a (repeat a n) = n :=\n  sorry\n\ntheorem count_repeat {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (b : \u03b1) (n : \u2115) :\n    count a (repeat b n) = ite (a = b) n 0 :=\n  sorry\n\n@[simp] theorem count_erase_self {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) :\n    count a (erase s a) = Nat.pred (count a s) :=\n  sorry\n\n@[simp] theorem count_erase_of_ne {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {b : \u03b1} (ab : a \u2260 b)\n    (s : multiset \u03b1) : count a (erase s b) = count a s :=\n  sorry\n\n@[simp] theorem count_sub {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1) (t : multiset \u03b1) :\n    count a (s - t) = count a s - count a t :=\n  sorry\n\n@[simp] theorem count_union {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1)\n    (t : multiset \u03b1) : count a (s \u222a t) = max (count a s) (count a t) :=\n  sorry\n\n@[simp] theorem count_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (a : \u03b1) (s : multiset \u03b1)\n    (t : multiset \u03b1) : count a (s \u2229 t) = min (count a s) (count a t) :=\n  sorry\n\ntheorem count_sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] {m : multiset \u03b2}\n    {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} :\n    count a (sum (map f m)) = sum (map (fun (b : \u03b2) => count a (f b)) m) :=\n  sorry\n\ntheorem count_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} [DecidableEq \u03b1] {m : multiset \u03b2}\n    {f : \u03b2 \u2192 multiset \u03b1} {a : \u03b1} :\n    count a (bind m f) = sum (map (fun (b : \u03b2) => count a (f b)) m) :=\n  count_sum\n\ntheorem le_count_iff_repeat_le {\u03b1 : Type u_1} [DecidableEq \u03b1] {a : \u03b1} {s : multiset \u03b1} {n : \u2115} :\n    n \u2264 count a s \u2194 repeat a n \u2264 s :=\n  quot.induction_on s\n    fun (l : List \u03b1) => iff.trans list.le_count_iff_repeat_sublist (iff.symm repeat_le_coe)\n\n@[simp] theorem count_filter_of_pos {\u03b1 : Type u_1} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    {a : \u03b1} {s : multiset \u03b1} (h : p a) : count a (filter p s) = count a s :=\n  quot.induction_on s fun (l : List \u03b1) => list.count_filter h\n\n@[simp] theorem count_filter_of_neg {\u03b1 : Type u_1} [DecidableEq \u03b1] {p : \u03b1 \u2192 Prop} [decidable_pred p]\n    {a : \u03b1} {s : multiset \u03b1} (h : \u00acp a) : count a (filter p s) = 0 :=\n  count_eq_zero_of_not_mem fun (t : a \u2208 filter p s) => h (of_mem_filter t)\n\ntheorem ext {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} :\n    s = t \u2194 \u2200 (a : \u03b1), count a s = count a t :=\n  quotient.induction_on\u2082 s t fun (l\u2081 l\u2082 : List \u03b1) => iff.trans quotient.eq list.perm_iff_count\n\ntheorem ext' {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} :\n    (\u2200 (a : \u03b1), count a s = count a t) \u2192 s = t :=\n  iff.mpr ext\n\n@[simp] theorem coe_inter {\u03b1 : Type u_1} [DecidableEq \u03b1] (s : List \u03b1) (t : List \u03b1) :\n    \u2191s \u2229 \u2191t = \u2191(list.bag_inter s t) :=\n  sorry\n\ntheorem le_iff_count {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1} :\n    s \u2264 t \u2194 \u2200 (a : \u03b1), count a s \u2264 count a t :=\n  sorry\n\nprotected instance distrib_lattice {\u03b1 : Type u_1} [DecidableEq \u03b1] : distrib_lattice (multiset \u03b1) :=\n  distrib_lattice.mk lattice.sup lattice.le lattice.lt sorry sorry sorry sorry sorry sorry\n    lattice.inf sorry sorry sorry sorry\n\nprotected instance semilattice_sup_bot {\u03b1 : Type u_1} [DecidableEq \u03b1] :\n    semilattice_sup_bot (multiset \u03b1) :=\n  semilattice_sup_bot.mk 0 lattice.le lattice.lt sorry sorry sorry zero_le lattice.sup sorry sorry\n    sorry\n\n@[simp] theorem mem_nsmul {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {n : \u2115} (h0 : n \u2260 0) :\n    a \u2208 n \u2022\u2115 s \u2194 a \u2208 s :=\n  sorry\n\n/-! ### Lift a relation to `multiset`s -/\n\n/-- `rel r s t` -- lift the relation `r` between two elements to a relation between `s` and `t`,\ns.t. there is a one-to-one mapping betweem elements in `s` and `t` following `r`. -/\ntheorem rel_iff {\u03b1 : Type u_1} {\u03b2 : Type u_2} (r : \u03b1 \u2192 \u03b2 \u2192 Prop) :\n    \u2200 (\u1fb0 : multiset \u03b1) (\u1fb0_1 : multiset \u03b2),\n        rel r \u1fb0 \u1fb0_1 \u2194\n          \u1fb0 = 0 \u2227 \u1fb0_1 = 0 \u2228\n            Exists\n              fun {a : \u03b1} =>\n                Exists\n                  fun {b : \u03b2} =>\n                    Exists\n                      fun {as : multiset \u03b1} =>\n                        Exists\n                          fun {bs : multiset \u03b2} =>\n                            r a b \u2227 rel r as bs \u2227 \u1fb0 = a ::\u2098 as \u2227 \u1fb0_1 = b ::\u2098 bs :=\n  sorry\n\ntheorem rel_flip {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b2}\n    {t : multiset \u03b1} : rel (flip r) s t \u2194 rel r t s :=\n  { mp := rel_flip_aux, mpr := rel_flip_aux }\n\ntheorem rel_eq_refl {\u03b1 : Type u_1} {s : multiset \u03b1} : rel Eq s s :=\n  multiset.induction_on s rel.zero fun (a : \u03b1) (s : multiset \u03b1) => rel.cons rfl\n\ntheorem rel_eq {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} : rel Eq s t \u2194 s = t := sorry\n\ntheorem rel.mono {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {p : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {s : multiset \u03b1} {t : multiset \u03b2} (h : \u2200 (a : \u03b1) (b : \u03b2), r a b \u2192 p a b) (hst : rel r s t) :\n    rel p s t :=\n  sorry\n\ntheorem rel.add {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2}\n    {u : multiset \u03b1} {v : multiset \u03b2} (hst : rel r s t) (huv : rel r u v) : rel r (s + u) (t + v) :=\n  sorry\n\ntheorem rel_flip_eq {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    rel (fun (a b : \u03b1) => b = a) s t \u2194 s = t :=\n  sorry\n\n@[simp] theorem rel_zero_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {b : multiset \u03b2} :\n    rel r 0 b \u2194 b = 0 :=\n  sorry\n\n@[simp] theorem rel_zero_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : multiset \u03b1} :\n    rel r a 0 \u2194 a = 0 :=\n  sorry\n\ntheorem rel_cons_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {a : \u03b1} {as : multiset \u03b1}\n    {bs : multiset \u03b2} :\n    rel r (a ::\u2098 as) bs \u2194 \u2203 (b : \u03b2), \u2203 (bs' : multiset \u03b2), r a b \u2227 rel r as bs' \u2227 bs = b ::\u2098 bs' :=\n  sorry\n\ntheorem rel_cons_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as : multiset \u03b1} {b : \u03b2}\n    {bs : multiset \u03b2} :\n    rel r as (b ::\u2098 bs) \u2194 \u2203 (a : \u03b1), \u2203 (as' : multiset \u03b1), r a b \u2227 rel r as' bs \u2227 as = a ::\u2098 as' :=\n  sorry\n\ntheorem rel_add_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as\u2080 : multiset \u03b1}\n    {as\u2081 : multiset \u03b1} {bs : multiset \u03b2} :\n    rel r (as\u2080 + as\u2081) bs \u2194\n        \u2203 (bs\u2080 : multiset \u03b2),\n          \u2203 (bs\u2081 : multiset \u03b2), rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 bs = bs\u2080 + bs\u2081 :=\n  sorry\n\ntheorem rel_add_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {as : multiset \u03b1}\n    {bs\u2080 : multiset \u03b2} {bs\u2081 : multiset \u03b2} :\n    rel r as (bs\u2080 + bs\u2081) \u2194\n        \u2203 (as\u2080 : multiset \u03b1),\n          \u2203 (as\u2081 : multiset \u03b1), rel r as\u2080 bs\u2080 \u2227 rel r as\u2081 bs\u2081 \u2227 as = as\u2080 + as\u2081 :=\n  sorry\n\ntheorem rel_map_left {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {s : multiset \u03b3} {f : \u03b3 \u2192 \u03b1} {t : multiset \u03b2} :\n    rel r (map f s) t \u2194 rel (fun (a : \u03b3) (b : \u03b2) => r (f a) b) s t :=\n  sorry\n\ntheorem rel_map_right {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {r : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {s : multiset \u03b1} {t : multiset \u03b3} {f : \u03b3 \u2192 \u03b2} :\n    rel r s (map f t) \u2194 rel (fun (a : \u03b1) (b : \u03b3) => r a (f b)) s t :=\n  sorry\n\ntheorem rel_join {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset (multiset \u03b1)}\n    {t : multiset (multiset \u03b2)} (h : rel (rel r) s t) : rel r (join s) (join t) :=\n  sorry\n\ntheorem rel_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} {r : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b4}\n    (h : relator.lift_fun r p f g) (hst : rel r s t) : rel p (map f s) (map g t) :=\n  eq.mpr (id (Eq._oldrec (Eq.refl (rel p (map f s) (map g t))) (propext rel_map_left)))\n    (eq.mpr\n      (id\n        (Eq._oldrec (Eq.refl (rel (fun (a : \u03b1) (b : \u03b4) => p (f a) b) s (map g t)))\n          (propext rel_map_right)))\n      (rel.mono h hst))\n\ntheorem rel_bind {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {\u03b4 : Type u_4} {r : \u03b1 \u2192 \u03b2 \u2192 Prop}\n    {p : \u03b3 \u2192 \u03b4 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b2} {f : \u03b1 \u2192 multiset \u03b3} {g : \u03b2 \u2192 multiset \u03b4}\n    (h : relator.lift_fun r (rel p) f g) (hst : rel r s t) : rel p (bind s f) (bind t g) :=\n  rel_join (rel_map h hst)\n\ntheorem card_eq_card_of_rel {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1}\n    {t : multiset \u03b2} (h : rel r s t) : coe_fn card s = coe_fn card t :=\n  sorry\n\ntheorem exists_mem_of_rel_of_mem {\u03b1 : Type u_1} {\u03b2 : Type u_2} {r : \u03b1 \u2192 \u03b2 \u2192 Prop} {s : multiset \u03b1}\n    {t : multiset \u03b2} (h : rel r s t) {a : \u03b1} (ha : a \u2208 s) : \u2203 (b : \u03b2), \u2203 (H : b \u2208 t), r a b :=\n  sorry\n\ntheorem map_eq_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (hf : function.injective f)\n    {s : multiset \u03b1} {t : multiset \u03b1} : map f s = map f t \u2194 s = t :=\n  sorry\n\ntheorem map_injective {\u03b1 : Type u_1} {\u03b2 : Type u_2} {f : \u03b1 \u2192 \u03b2} (hf : function.injective f) :\n    function.injective (map f) :=\n  fun (x y : multiset \u03b1) => iff.mp (map_eq_map hf)\n\ntheorem map_mk_eq_map_mk_of_rel {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : multiset \u03b1} {t : multiset \u03b1}\n    (hst : rel r s t) : map (Quot.mk r) s = map (Quot.mk r) t :=\n  sorry\n\ntheorem exists_multiset_eq_map_quot_mk {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (s : multiset (Quot r)) :\n    \u2203 (t : multiset \u03b1), s = map (Quot.mk r) t :=\n  sorry\n\ntheorem induction_on_multiset_quot {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {p : multiset (Quot r) \u2192 Prop}\n    (s : multiset (Quot r)) : (\u2200 (s : multiset \u03b1), p (map (Quot.mk r) s)) \u2192 p s :=\n  sorry\n\n/-! ### Disjoint multisets -/\n\n/-- `disjoint s t` means that `s` and `t` have no elements in common. -/\ndef disjoint {\u03b1 : Type u_1} (s : multiset \u03b1) (t : multiset \u03b1) := \u2200 {a : \u03b1}, a \u2208 s \u2192 a \u2208 t \u2192 False\n\n@[simp] theorem coe_disjoint {\u03b1 : Type u_1} (l\u2081 : List \u03b1) (l\u2082 : List \u03b1) :\n    disjoint \u2191l\u2081 \u2191l\u2082 \u2194 list.disjoint l\u2081 l\u2082 :=\n  iff.rfl\n\ntheorem disjoint.symm {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} (d : disjoint s t) :\n    disjoint t s :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 t) (\u1fb0_1 : a \u2208 s) => idRhs False (d \u1fb0_1 \u1fb0)\n\ntheorem disjoint_comm {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint s t \u2194 disjoint t s :=\n  { mp := disjoint.symm, mpr := disjoint.symm }\n\ntheorem disjoint_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint s t \u2194 \u2200 {a : \u03b1}, a \u2208 s \u2192 \u00aca \u2208 t :=\n  iff.rfl\n\ntheorem disjoint_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint s t \u2194 \u2200 {a : \u03b1}, a \u2208 t \u2192 \u00aca \u2208 s :=\n  disjoint_comm\n\ntheorem disjoint_iff_ne {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint s t \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b1), b \u2208 t \u2192 a \u2260 b :=\n  sorry\n\ntheorem disjoint_of_subset_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h : s \u2286 u) (d : disjoint u t) : disjoint s t :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 s) => idRhs (a \u2208 t \u2192 False) (d (h \u1fb0))\n\ntheorem disjoint_of_subset_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h : t \u2286 u) (d : disjoint s u) : disjoint s t :=\n  fun {a : \u03b1} (\u1fb0 : a \u2208 s) (\u1fb0_1 : a \u2208 t) => idRhs False (d \u1fb0 (h \u1fb0_1))\n\ntheorem disjoint_of_le_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h : s \u2264 u) : disjoint u t \u2192 disjoint s t :=\n  disjoint_of_subset_left (subset_of_le h)\n\ntheorem disjoint_of_le_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1} {u : multiset \u03b1}\n    (h : t \u2264 u) : disjoint s u \u2192 disjoint s t :=\n  disjoint_of_subset_right (subset_of_le h)\n\n@[simp] theorem zero_disjoint {\u03b1 : Type u_1} (l : multiset \u03b1) : disjoint 0 l :=\n  fun {a : \u03b1} => idRhs (a \u2208 [] \u2192 a \u2208 l \u2192 False) (not.elim (list.not_mem_nil a))\n\n@[simp] theorem singleton_disjoint {\u03b1 : Type u_1} {l : multiset \u03b1} {a : \u03b1} :\n    disjoint (a ::\u2098 0) l \u2194 \u00aca \u2208 l :=\n  sorry\n\n@[simp] theorem disjoint_singleton {\u03b1 : Type u_1} {l : multiset \u03b1} {a : \u03b1} :\n    disjoint l (a ::\u2098 0) \u2194 \u00aca \u2208 l :=\n  sorry\n\n@[simp] theorem disjoint_add_left {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : disjoint (s + t) u \u2194 disjoint s u \u2227 disjoint t u :=\n  sorry\n\n@[simp] theorem disjoint_add_right {\u03b1 : Type u_1} {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : disjoint s (t + u) \u2194 disjoint s t \u2227 disjoint s u :=\n  sorry\n\n@[simp] theorem disjoint_cons_left {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint (a ::\u2098 s) t \u2194 \u00aca \u2208 t \u2227 disjoint s t :=\n  sorry\n\n@[simp] theorem disjoint_cons_right {\u03b1 : Type u_1} {a : \u03b1} {s : multiset \u03b1} {t : multiset \u03b1} :\n    disjoint s (a ::\u2098 t) \u2194 \u00aca \u2208 s \u2227 disjoint s t :=\n  sorry\n\ntheorem inter_eq_zero_iff_disjoint {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1}\n    {t : multiset \u03b1} : s \u2229 t = 0 \u2194 disjoint s t :=\n  sorry\n\n@[simp] theorem disjoint_union_left {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1} {t : multiset \u03b1}\n    {u : multiset \u03b1} : disjoint (s \u222a t) u \u2194 disjoint s u \u2227 disjoint t u :=\n  sorry\n\n@[simp] theorem disjoint_union_right {\u03b1 : Type u_1} [DecidableEq \u03b1] {s : multiset \u03b1}\n    {t : multiset \u03b1} {u : multiset \u03b1} : disjoint s (t \u222a u) \u2194 disjoint s t \u2227 disjoint s u :=\n  sorry\n\ntheorem disjoint_map_map {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} {f : \u03b1 \u2192 \u03b3} {g : \u03b2 \u2192 \u03b3}\n    {s : multiset \u03b1} {t : multiset \u03b2} :\n    disjoint (map f s) (map g t) \u2194 \u2200 (a : \u03b1), a \u2208 s \u2192 \u2200 (b : \u03b2), b \u2208 t \u2192 f a \u2260 g b :=\n  sorry\n\n/-- `pairwise r m` states that there exists a list of the elements s.t. `r` holds pairwise on this list. -/\ndef pairwise {\u03b1 : Type u_1} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (m : multiset \u03b1) :=\n  \u2203 (l : List \u03b1), m = \u2191l \u2227 list.pairwise r l\n\ntheorem pairwise_coe_iff_pairwise {\u03b1 : Type u_1} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} (hr : symmetric r)\n    {l : List \u03b1} : pairwise r \u2191l \u2194 list.pairwise r l :=\n  sorry\n\nend multiset\n\n\nnamespace multiset\n\n\n/-- Given a proof `hp` that there exists a unique `a \u2208 l` such that `p a`, `choose_x p l hp` returns\nthat `a` together with proofs of `a \u2208 l` and `p a`. -/\ndef choose_x {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n    (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : Subtype fun (a : \u03b1) => a \u2208 l \u2227 p a :=\n  quotient.rec_on l\n    (fun (l' : List \u03b1) (ex_unique : exists_unique fun (a : \u03b1) => a \u2208 quotient.mk l' \u2227 p a) =>\n      list.choose_x p l' sorry)\n    sorry\n\n/-- Given a proof `hp` that there exists a unique `a \u2208 l` such that `p a`, `choose p l hp` returns\nthat `a`. -/\ndef choose {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n    (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : \u03b1 :=\n  \u2191(choose_x p l hp)\n\ntheorem choose_spec {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n    (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : choose p l hp \u2208 l \u2227 p (choose p l hp) :=\n  subtype.property (choose_x p l hp)\n\ntheorem choose_mem {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n    (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : choose p l hp \u2208 l :=\n  and.left (choose_spec p l hp)\n\ntheorem choose_property {\u03b1 : Type u_1} (p : \u03b1 \u2192 Prop) [decidable_pred p] (l : multiset \u03b1)\n    (hp : exists_unique fun (a : \u03b1) => a \u2208 l \u2227 p a) : p (choose p l hp) :=\n  and.right (choose_spec p l hp)\n\n/-- The equivalence between lists and multisets of a subsingleton type. -/\ndef subsingleton_equiv (\u03b1 : Type u_1) [subsingleton \u03b1] : List \u03b1 \u2243 multiset \u03b1 :=\n  equiv.mk coe (Quot.lift id sorry) sorry sorry\n\nend multiset\n\n\ntheorem add_monoid_hom.map_multiset_sum {\u03b1 : Type u_1} {\u03b2 : Type u_2} [add_comm_monoid \u03b1]\n    [add_comm_monoid \u03b2] (f : \u03b1 \u2192+ \u03b2) (s : multiset \u03b1) :\n    coe_fn f (multiset.sum s) = multiset.sum (multiset.map (\u21d1f) s) :=\n  Eq.symm (multiset.sum_hom s f)\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/data/multiset/basic_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.35231816634423563}}
{"text": "import .sterm .pterm\n\nnamespace list\n\ntheorem subset_union_left {\u03b1 : Type*} [decidable_eq \u03b1]\n  {l1 l2 : list \u03b1} : l1 \u2286 l1 \u222a l2 :=\nby {intros a ha, apply list.mem_union_left, exact ha}\n\nend list\n\nnamespace polya.field\n\ninductive alt (\u03b3) [const_space \u03b3] : bool \u2192 Type\n| sform : list (nterm \u03b3) \u2192 sterm \u03b3 \u2192 alt tt\n| pform : list (nterm \u03b3) \u2192 pterm \u03b3 \u2192 alt ff\n\nnamespace alt\nopen nterm\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ndef of_const (a : \u03b3) : \u03a0 {b}, alt \u03b3 b\n| tt := sform [] (sterm.of_const a)\n| ff := pform [] (pterm.of_const a)\n\ndef singleton (x : nterm \u03b3) : \u03a0 {b}, alt \u03b3 b\n| tt := sform \u2205 (sterm.singleton x)\n| ff := pform \u2205 (pterm.singleton x)\n\ndef to_nterm : alt \u03b3 tt \u2192 nterm \u03b3\n| (sform _ S) := S.to_nterm\n\ndef to_sterm : alt \u03b3 tt \u2192 sterm \u03b3\n| (sform _ S) := S\n\ndef to_pterm : alt \u03b3 ff \u2192 pterm \u03b3\n| (pform _ P) := P\n\ndef hyps : \u03a0 {b}, alt \u03b3 b \u2192 list (nterm \u03b3)\n| ._ (sform ts _) := ts\n| ._ (pform ts _) := ts\n\ndef eval (\u03c1 : dict \u03b1) : \u03a0 {b}, alt \u03b3 b \u2192 \u03b1\n| ._ (sform _ S) := sterm.eval \u03c1 S\n| ._ (pform _ P) := pterm.eval \u03c1 P\n\ndef to_sform : \u03a0 {b}, alt \u03b3 b \u2192 alt \u03b3 tt\n| ._ (sform ts S) := sform ts S\n| ._ (pform ts P) := sform (ts \u222a P.reduce_hyps) (sterm.of_nterm P.reduce.to_nterm)\n\ndef to_pform : \u03a0 {b}, alt \u03b3 b \u2192 alt \u03b3 ff\n| ._ (sform ts S) := pform ts (pterm.of_nterm S.to_nterm)\n| ._ (pform ts P) := pform ts P\n\ndef hyps_to_sform {a} {x : alt \u03b3 a} :\n  x.hyps \u2286 x.to_sform.hyps :=\nbegin\n  cases x,\n  { simp [to_sform] },\n  { unfold to_sform, unfold hyps, exact list.subset_union_left }\nend\n\ndef hyps_to_pform {a} {x : alt \u03b3 a} :\n  x.hyps = x.to_pform.hyps :=\nby cases x; simp [hyps, to_pform]\n\ntheorem eval_def {x : alt \u03b3 tt} :\n  eval \u03c1 x = nterm.eval \u03c1 x.to_nterm :=\nby {cases x, simp [eval, to_nterm, sterm.eval_to_nterm]}\n\ntheorem eval_of_const {b} {c : \u03b3} :\n  eval \u03c1 (of_const c : alt \u03b3 b) = (c : \u03b1) :=\nby cases b; simp [of_const, eval, pterm.eval_of_const, sterm.eval_of_const]\n\ntheorem eval_singleton {b} {x : nterm \u03b3} :\n  eval \u03c1 (singleton x : alt \u03b3 b) = nterm.eval \u03c1 x :=\nbegin\n  cases b,\n  { unfold singleton, unfold eval,\n    rw \u2190 pterm.eval_singleton },\n  { unfold singleton, unfold eval,\n    rw \u2190 sterm.eval_singleton }\nend\n\ntheorem eval_to_sform {b} {x : alt \u03b3 b} :\n  nonzero \u03c1 x.to_sform.hyps \u2192\n  eval \u03c1 x.to_sform = eval \u03c1 x :=\nbegin\n  intro H, cases x,\n  { refl },\n  { unfold to_sform, unfold eval,\n    rw [sterm.eval_of_nterm, \u2190 pterm.eval_to_nterm],\n    rw \u2190 pterm.eval_reduce,\n    intros t ht, apply H,\n    unfold to_sform, unfold hyps,\n    apply list.mem_union_right,\n    exact ht }\nend\n\ntheorem eval_to_pform {b} {x : alt \u03b3 b} :\n  eval \u03c1 x.to_pform = eval \u03c1 x :=\nbegin\n  cases x,\n  { unfold to_pform, unfold eval,\n    rw [pterm.eval_of_nterm, \u2190 sterm.eval_to_nterm] },\n  { refl }\nend\n\ntheorem eval_to_sterm {x : alt \u03b3 tt} :\n  sterm.eval \u03c1 x.to_sterm = eval \u03c1 x :=\nbegin\n  cases x, unfold to_sterm, unfold eval\nend\n\ntheorem eval_to_pterm {x : alt \u03b3 ff} :\n  eval \u03c1 x = pterm.eval \u03c1 x.to_pterm :=\nbegin\n  cases x, unfold to_pterm, unfold eval\nend\n\n--TODO:\n--more cases to avoid switching form too often\n--when applying operators\n\ndef add_sform (x y : alt \u03b3 tt) : alt \u03b3 tt :=\nsform (x.hyps \u222a y.hyps) (x.to_sterm + y.to_sterm)\n\ndef mul_pform (x y : alt \u03b3 ff) : alt \u03b3 ff :=\npform (x.hyps \u222a y.hyps) (x.to_pterm * y.to_pterm)\n\ndef pow_pform (x : alt \u03b3 ff) (n : znum) : alt \u03b3 ff :=\nif n = 0 then singleton (1 : \u03b3)\nelse pform x.hyps (x.to_pterm ^ n)\n\ninstance : has_add (alt \u03b3 tt) := \u27e8add_sform\u27e9\ninstance : has_mul (alt \u03b3 ff) := \u27e8mul_pform\u27e9\ninstance : has_pow (alt \u03b3 ff) znum := \u27e8pow_pform\u27e9\n\ndef add {a b} (x : alt \u03b3 a) (y : alt \u03b3 b) : alt \u03b3 tt :=\nx.to_sform + y.to_sform\n\ndef mul {a b} (x : alt \u03b3 a) (y : alt \u03b3 b) : alt \u03b3 ff :=\nx.to_pform * y.to_pform\n\ndef pow {a} (x : alt \u03b3 a) (n : znum) : alt \u03b3 ff :=\nx.to_pform ^ n\n\ntheorem hyps_singleton {b} {x : nterm \u03b3} :\n  (singleton x : alt \u03b3 b).hyps = \u2205 :=\nby cases b; simp [singleton, hyps]\n\ntheorem hyps_add_sform {x y : alt \u03b3 tt} :\n  (x + y).hyps = x.hyps \u222a y.hyps :=\nby simp [has_add.add, add_sform, hyps]\n\ntheorem hyps_mul_pform {x y : alt \u03b3 ff} :\n  (x * y).hyps = x.hyps \u222a y.hyps :=\nby simp [has_mul.mul, mul_pform, hyps]\n\ntheorem hyps_pow_pform {x : alt \u03b3 ff} {n : znum} :\n  (x ^ n).hyps = if n = 0 then \u2205 else x.hyps :=\nbegin\n  by_cases h0 : n = 0;\n  by_cases h1 : n = 1;\n  simp [has_pow.pow, pow_pform, h0, h1, hyps, hyps_singleton]\nend\n\ntheorem hyps_add {a b} {x : alt \u03b3 a} {y : alt \u03b3 b} :\n  hyps (add x y) = hyps x.to_sform \u222a hyps y.to_sform :=\nby cases x; cases y; simp [add, to_sform, hyps_add_sform]\n\ntheorem hyps_mul {a b} {x : alt \u03b3 a} {y : alt \u03b3 b} :\n  hyps (mul x y) = hyps x.to_pform \u222a hyps y.to_pform :=\nby cases x; cases y; simp [mul, to_pform, hyps_mul_pform]\n\ntheorem hyps_pow {a} {x : alt \u03b3 a} {n : znum} :\n  hyps (pow x n) = if n = 0 then \u2205 else hyps x.to_pform :=\nby simp [pow, hyps, hyps_pow_pform]\n\ntheorem eval_add_sform {x y : alt \u03b3 tt} :\n  eval \u03c1 (x + y) = eval \u03c1 x + eval \u03c1 y :=\nbegin\n  suffices : eval \u03c1 (add_sform x y) = eval \u03c1 x + eval \u03c1 y,\n  by exact this,\n  cases x, cases y,\n  unfold add_sform, unfold to_sterm, unfold eval,\n  apply sterm.eval_add\nend\n\ntheorem eval_add {a b} {x : alt \u03b3 a} {y : alt \u03b3 b} :\n  nonzero \u03c1 (hyps (add x y)) \u2192\n  eval \u03c1 (add x y) = eval \u03c1 x + eval \u03c1 y :=\nbegin\n  intro H, rw [hyps_add, nonzero_union] at H,\n  cases H with H1 H2,\n  unfold add, rw [eval_add_sform, eval_to_sform, eval_to_sform],\n  { exact H2 },\n  { exact H1 }\nend\n\ntheorem eval_mul_pform {x y : alt \u03b3 ff} :\n  eval \u03c1 (x * y) = eval \u03c1 x * eval \u03c1 y :=\nbegin\n  suffices : eval \u03c1 (mul_pform x y) = eval \u03c1 x * eval \u03c1 y,\n  by exact this,\n  cases x, cases y,\n  unfold mul_pform, unfold to_pterm, unfold eval,\n  apply pterm.eval_mul\nend\n\ntheorem eval_mul {a b} {x : alt \u03b3 a} {y : alt \u03b3 b} :\n  eval \u03c1 (mul x y) = eval \u03c1 x * eval \u03c1 y :=\nbegin\n  unfold mul, rw [eval_mul_pform, eval_to_pform, eval_to_pform]\nend\n\ntheorem eval_pow_pform {x : alt \u03b3 ff} {n : znum} :\n  eval \u03c1 (x ^ n) = eval \u03c1 x ^ (n : \u2124) :=\nbegin\n  suffices : eval \u03c1 (pow_pform x n) = eval \u03c1 x ^ (n : \u2124),\n  by exact this,\n  cases x,\n  unfold pow_pform, unfold to_pterm,\n  by_cases hn : n = 0,\n  { rw if_pos hn, unfold eval, simp [hn, eval_singleton] },\n  { rw if_neg hn, unfold eval, apply pterm.eval_pow }\nend\n\ntheorem eval_pow {a} {x : alt \u03b3 a} {n : znum} :\n  eval \u03c1 (pow x n) = eval \u03c1 x ^ (n : \u2124) :=\nbegin\n  unfold pow, rw [eval_pow_pform, eval_to_pform]\nend\n\n@[reducible]\ndef aux_of_nterm : nterm \u03b3 \u2192 bool\n| (nterm.add _ _) := tt\n| (nterm.mul _ _) := ff\n| (nterm.pow _ _) := ff\n| _ := tt\n\ndef of_nterm : \u03a0 (x : nterm \u03b3), alt \u03b3 (aux_of_nterm x)\n| (nterm.add x y) := add (of_nterm x) (of_nterm y)\n| (nterm.mul x y) := mul (of_nterm x) (of_nterm y)\n| (nterm.pow x n) := pow (of_nterm x) n\n| (nterm.const a) := of_const a\n| (nterm.atom i)  := singleton i\n\ntheorem eval_of_nterm {x : nterm \u03b3} :\n  nonzero \u03c1 (of_nterm x).hyps \u2192\n  eval \u03c1 (of_nterm x) = nterm.eval \u03c1 x :=\nbegin\n  induction x with i c x y ihx ihy x y ihx ihy x n ihx,\n  { intro, unfold of_nterm, rw eval_singleton, refl },\n  { intro, unfold of_nterm, rw eval_of_const, simp [nterm.eval] },\n  { unfold of_nterm, intro H,\n    rw [hyps_add,  nonzero_union] at H,\n    rw [eval_add, ihx, ihy],\n    { exact nterm.eval_add },\n    { apply nonzero_subset (hyps_to_sform), exact H.right},\n    { apply nonzero_subset (hyps_to_sform), exact H.left },\n    { rw [hyps_add, nonzero_union], exact H }},\n  { unfold of_nterm, intro H,\n    rw [hyps_mul,  nonzero_union] at H,\n    rw [eval_mul, ihx, ihy],\n    { exact nterm.eval_mul },\n    { rw hyps_to_pform, exact H.right },\n    { rw hyps_to_pform, exact H.left }},\n  { unfold of_nterm, intro H,\n    by_cases hn : n = 0,\n    { rw [eval_pow, hn, znum.cast_zero, fpow_zero],\n      suffices : nterm.eval \u03c1 (x ^ (0 : znum)) = 1,\n      by exact this,\n      rw [nterm.eval_pow, znum.cast_zero, fpow_zero] },\n    { rw [hyps_pow, if_neg hn] at H,\n      rw [eval_pow, ihx],\n      { exact nterm.eval_pow },\n      { rw hyps_to_pform, exact H }}}\nend\n\nend alt\n\nnamespace nterm\n\nvariables {\u03b1 : Type} [discrete_field \u03b1]\nvariables {\u03b3 : Type} [const_space \u03b3]\nvariables [morph \u03b3 \u03b1] {\u03c1 : dict \u03b1}\n\ndef norm (x : nterm \u03b3) : nterm \u03b3 :=\n(alt.of_nterm x).to_sform.to_nterm\n\ndef norm_hyps (x : nterm \u03b3) : list (nterm \u03b3) :=\n(alt.of_nterm x).to_sform.hyps\n\ndef correctness {x : nterm \u03b3} :\n  nonzero \u03c1 (norm_hyps x) \u2192\n  nterm.eval \u03c1 (norm x) = nterm.eval \u03c1 x :=\nbegin\n  intro H, unfold norm,\n  rw [\u2190 alt.eval_def, \u2190 alt.eval_of_nterm, alt.eval_to_sform],\n  { intros t ht, apply H, exact ht },\n  { intros t ht, apply H, apply alt.hyps_to_sform, exact ht }\nend\n\n--section soundness\n--variables {x y : nterm \u03b3} {i : num} {n : znum} {c : \u03b3}\n--\n--theorem sound_atom : @norm \u03b3 _ i = i :=\n--begin\n--  sorry\n--end\n--\n--theorem sound_const : @norm \u03b3 _ c = c :=\n--begin\n--  sorry\n--end\n--\n--theorem sound_add :\n--  norm (x + y) = (sterm.of_nterm x.norm\n--    + sterm.of_nterm y.norm).to_nterm :=\n--begin\n--  sorry\n--end\n--\n--theorem sound_mul :\n--  norm (x * y) = (pterm.of_nterm x.norm\n--    * pterm.of_nterm y.norm).reduce.to_nterm :=\n--begin\n--  sorry\n--end\n--\n--theorem sound_pow :\n--  norm (x ^ n) = (pterm.of_nterm x.norm ^ n).reduce.to_nterm :=\n--begin\n--  sorry\n--end\n--\n--def naive_norm : nterm \u03b3 \u2192 nterm \u03b3\n--| (add x y) := (sterm.of_nterm (naive_norm x) + sterm.of_nterm (naive_norm y)).to_nterm\n--| (mul x y) := (pterm.of_nterm (naive_norm x) * pterm.of_nterm (naive_norm y)).reduce.to_nterm\n--| (pow x n) := (pterm.of_nterm (naive_norm x) ^ n).reduce.to_nterm\n--| x := x\n--\n--theorem soundness {x : nterm \u03b3} :\n--  norm x = naive_norm x :=\n--begin\n--  sorry\n--  --TODO: this theorem is not required,\n--  --but it could be an interesting\n--  --first step to prove soundness\n--end\n--\n--end soundness\n\nend nterm\n\nend polya.field\n\n", "meta": {"author": "lean-forward", "repo": "field", "sha": "7e2127ad485aec25e58a1b9c82a6bb74a599467a", "save_path": "github-repos/lean/lean-forward-field", "path": "github-repos/lean/lean-forward-field/field-7e2127ad485aec25e58a1b9c82a6bb74a599467a/src/data/polya/field/norm.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477015, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.35222245165162014}}
{"text": "/-! Specialization library. \n\n-/\n\nnamespace Spe\n\nopen Except (ok error)\n\n\n\n/-- Can check an `\u03b1`-value. -/\nstructure Check (\u03b5 \u03b1 : Type u) : Type u where\n  /-- Okay if legit, error otherwise. -/\n  check : \u03b1 \u2192 Except \u03b5 Unit\n\n/-- Proposition corresponding to [`Check.check`] being `ok ()`. -/\n@[simp]\ndef Check.isLegit\n  (self : Check \u03b5 \u03b1)\n  (val : \u03b1)\n: Prop :=\n  self.check val = ok ()\n\n/-- [`Check.isLegit`] is decidable. -/\ninstance instDecIsLegit\n  {self : Check \u03b5 \u03b1}\n  {val : \u03b1}\n: Decidable (self.isLegit val) :=\n  match legit? : self.check val with\n  | ok () => isTrue legit?\n  | error _ => isFalse (\n    by\n      simp\n      intro h\n      rw [legit?] at h\n      contradiction\n  )\n\n/-- [`Check`] coerces to function [`Check.isLegit`]. -/\ninstance instCoeFunCheck\n: CoeFun (Check \u03b5 \u03b1) (fun _ => \u03b1 \u2192 Prop) where\n  coe self :=\n    self.isLegit\n\n\n\n/-- Wraps an `\u03b1`, the [`Check`] function, and a proof of legitimacy. -/\nstructure Legit (\u03b5 \u03b1 : Type u) where\nprivate mk ::\n  val : \u03b1\n  check : Check \u03b5 \u03b1\n  legit : check val\n\n/-- [`Legit`] coerces to [`Legit.val`]. -/\ninstance instCoeLegit : Coe (Legit \u03b5 \u03b1) \u03b1 where\n  coe self :=\n    self.val\n\n\n\nsection\n  variable\n    {\u03b5 \u03b1 : Type u}\n\n  /-- Tries to build a legitimate `\u03b1`. -/\n  def Legit.mk?\n    (check : Check \u03b5 \u03b1)\n    (val : \u03b1)\n  : Except \u03b5 <| Legit \u03b5 \u03b1 :=\n    match legit : check.check val with\n    | ok () => ok \u27e8val, check, legit\u27e9\n    | error e => error e\n\n  /-- Alias for [`Legit.mk?`] -/\n  abbrev Check.validate? :=\n    @Legit.mk?\n\n  /-- Builds the [`Check`] corresponding to being a member of a list of values. -/\n  def Check.ofList\n    [DecidableEq \u03b1]\n    (set : List \u03b1)\n    (err : (a : \u03b1) \u2192 a \u2209 set \u2192 \u03b5)\n  : Check \u03b5 \u03b1 where\n    check val :=\n      if mem? : val \u2208 set\n      then ok ()\n      else error <| err val mem?\n\n  /-- Proof that legit value cannot be `val` if `val` is not legit. -/\n  theorem Legit.absurd\n    (self : Legit \u03b5 \u03b1)\n    (val : \u03b1)\n    (illegit : \u00ac(self.check val) := by dsimp)\n  : self.val \u2260 val :=\n    by\n      intro h\n      rw [\u2190h] at illegit\n      apply illegit self.legit\nend\n\n\nnamespace Demo\n\n  inductive Test\n  | Var1\n  | Var2\n  | Var3\n  deriving BEq, DecidableEq, Hashable, Repr\n\n  instance instToStringTest : ToString Test where\n    toString\n    | .Var1 => \"Var1\"\n    | .Var2 => \"Var2\"\n    | .Var3 => \"Var3\"\n\n  def list12 : List Test :=\n    [Test.Var1, Test.Var2]\n\n  def errMsg (l : List Test) (a : Test) (_ : a \u2209 l) : String :=\n    s! \"value `{a}` is not a member of [{pretty}]\"\n  where\n    pretty :=\n      \"\" |> l.foldl\n        fun (s : String) (val : Test) => -- fold\n          if s.isEmpty then toString val\n          else s ++ \", \" ++ (toString val)\n\n  def not3 : Check String Test :=\n    Check.ofList list12 (errMsg list12)\n\n  #eval (\n    match not3.validate? Test.Var1 with\n    | .ok legit => \"ok \" ++ toString legit.val\n    | .error msg => \"error: \" ++ msg\n  )\n  #eval (\n    match not3.validate? Test.Var2 with\n    | .ok legit => \"ok \" ++ toString legit.val\n    | .error msg => \"error: \" ++ msg\n  )\n  #eval (\n    match not3.validate? Test.Var3 with\n    | .ok legit => \"ok \" ++ toString legit.val\n    | .error msg => \"error: \" ++ msg\n  )\n\nend Demo\n\n", "meta": {"author": "AdrienChampion", "repo": "experimentalean4", "sha": "5071a8b007029f61b2e996d9ac89d90999603fcc", "save_path": "github-repos/lean/AdrienChampion-experimentalean4", "path": "github-repos/lean/AdrienChampion-experimentalean4/experimentalean4-5071a8b007029f61b2e996d9ac89d90999603fcc/Spe/Spe.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.626124191181315, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.3519923069182622}}
{"text": "-- WIP Trying to proove addBoth.sub, addBoth.sup \n-- Is it really necessary to import BEq and have beq_eq_eq?\n\n-- Tested with: \n-- leanprover/lean4:nightly unchanged - Lean (version 4.0.0-nightly-2023-01-28, commit e37f209c1a2a, Release)\n\nimport Std.Data.AssocList\nimport Std.Data.List.Lemmas\nimport Std.Classes.BEq\n\nnamespace MWE8\n\ntheorem cond_eq_ite (c : Bool) (a b : \u03b1) : cond c a b = if c then a else b := by cases c <;> rfl\n\ntheorem cond_decide {\u03b1} (p : Prop) [Decidable p] (t e : \u03b1) : cond (decide p) t e = if p then t else e := by\n  by_cases p <;> simp [*]\n\n-- https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/Problems.20simplifying.20.20conditions.20with.20hypotheses/near/324212540\n@[simp] theorem beq_eq_eq [DecidableEq \u03b1] (x y : \u03b1) :\n  (x == y) = decide (x = y) := rfl\n\nabbrev Strings := List String\n\ninstance : Repr (Std.AssocList String Strings) where reprPrec s n := s.toList.repr n\n\ndef addDecl (s: Std.AssocList String Strings) (d: String) : Std.AssocList String Strings :=\nmatch s.contains d with\n| true => s\n| false => .cons d [] s\n\ntheorem addDecl.added (s: Std.AssocList String Strings) (d: String): (addDecl s d).contains d\n:= by\n  simp [addDecl]\n  split <;> simp\n  next x heq => simp_all\n\ntheorem addDecl.cons (s: Std.AssocList String Strings) (d: String): (addDecl s d).isEmpty = false\n:= by\n  simp [addDecl]\n  split <;> simp [List.isEmpty]\n  . case h_1 x heq =>\n    split <;> simp_all\n\ntheorem addDecl.after (s: Std.AssocList String Strings) (x y: String): s.contains x \u2192 (addDecl s y).contains x\n:= by\n  intro h\n  simp [Std.AssocList.contains, addDecl] at h \u22a2\n  apply Exists.elim h\n  split <;> simp_all\n  done\n\ndef addSubSup: String \u2192 String \u2192 Std.AssocList String Strings \u2192 Std.AssocList String Strings\n| sub, sup, .nil            => .cons sub [sup] .nil\n| sub, sup, .cons a as tail => bif a = sub then .cons sub (as.insert sup) tail else .cons a as (addSubSup sub sup tail)\n  \ntheorem addSubSup.sub (sub sup: String) (ss: Std.AssocList String Strings): (addSubSup sub sup ss).contains sub\n:= by\n  induction ss <;> simp_all\n  . case cons key value tail tail_ih =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases key = sub\n    . case pos h =>\n      simp [h]\n    . case neg h =>\n      simp [h]\n      apply tail_ih\n\ndef addBoth (sub sup: String) (ss: Std.AssocList String Strings) : Std.AssocList String Strings :=\n  let ss' := addDecl ss sup\n  addSubSup sub sup ss'\n\ntheorem addBoth.sub_eq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sub\n:= by\n  induction ss <;> simp_all\n  . case nil =>\n    simp [addBoth, addDecl, addSubSup, cond_eq_ite]\n    split <;> simp_all\n  . case cons key value tail tail_ih =>\n    simp [addBoth, addDecl]\n    by_cases key = sup <;> simp [*]\n    . case pos =>\n      simp [addSubSup, cond_eq_ite]\n      by_cases sup = sub <;> simp [*]\n      . case neg h =>\n        simp [addBoth, addDecl] at tail_ih\n        split at tail_ih \n        . case h_1 =>\n          by_cases key = sub <;> simp [*]\n        . case h_2 x heq =>\n          by_cases key = sub <;> simp_all\n          simp [addSubSup, cond_eq_ite, h] at tail_ih\n          apply tail_ih\n    . case neg h =>\n      split <;> simp_all\n      . case h_1 x heq =>\n        -- Given the tactic state:\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub\n        -- h: \u00ackey = sup\n        -- heq: \u2203 x, x \u2208 Std.AssocList.toList tail \u2227 x.fst = sup\n        \n        -- How to prove?\n        -- \u22a2 \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons key value tail)) \u2227 x.fst = sub\n        sorry\n\n      . case h_2 x heq =>\n        -- Given the tactic state:\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub\n        -- h: \u00ackey = sup\n        -- heq: (List.any (Std.AssocList.toList tail) fun x => decide (x.fst = sup)) = false\n        \n        -- How to prove?\n        -- \u22a2 \u2203 x,\n        --   x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227\n        --     x.fst = sub\n        sorry\n\ntheorem addBoth.sub_beq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sub\n:= by\n  induction ss <;> simp_all\n  . case nil =>\n    simp [addBoth, addDecl, addSubSup, cond_eq_ite]\n    split <;> simp_all\n  . case cons key value tail tail_ih =>\n    simp [addBoth, addDecl]\n    by_cases key == sup <;> simp_all\n    . case pos =>\n      simp [addSubSup, cond_eq_ite]\n      by_cases sup == sub <;> simp [*]\n      . case pos =>\n        simp_all\n      . case neg h =>\n        simp [addBoth, addDecl] at tail_ih\n        split at tail_ih \n        . case h_1 =>\n          by_cases key == sub <;> simp_all\n        . case h_2 x heq =>\n          by_cases key == sub <;> simp_all\n          simp [addSubSup, cond_eq_ite, h] at tail_ih\n          apply tail_ih\n    . case neg h =>\n      split <;> simp_all\n      . case h_1 x heq =>\n        -- Given the tactic state: (same as addBoth.sub_eq: path cons/neg/h_1)\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub\n        -- h: \u00ackey = sup\n        -- heq: \u2203 x, x \u2208 Std.AssocList.toList tail \u2227 x.fst = sup\n\n        -- How to prove?\n        -- \u22a2 \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons key value tail)) \u2227 x.fst = sub\n        sorry\n\n      . case h_2 x heq =>\n        -- Given the tactic state: (same as addBoth.sub_eq: path cons/neg/h_2)\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addBoth sub sup tail) \u2227 x.fst = sub\n        -- h: \u00ackey = sup\n        -- heq: (List.any (Std.AssocList.toList tail) fun x => decide (x.fst = sup)) = false\n\n        -- How to prove?\n        -- \u22a2 \u2203 x,\n        --   x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227\n        --     x.fst = sub\n        sorry\n\ntheorem addBoth.sup_eq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sup\n:= by\n  simp [addBoth]\n  simp [addDecl]\n  induction ss <;> simp [*]\n  . case nil =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases sup = sub <;> simp [*]\n  . case cons key value tail tail_ih =>\n    simp [addDecl, cond_eq_ite]\n    by_cases key = sup <;> simp [*]\n    . case pos h1 =>\n      simp [addSubSup, cond_eq_ite] at tail_ih \u22a2 \n      by_cases sup = sub <;> simp [*]\n    . case neg h1 =>\n      let \u27e8 x, hx, he \u27e9 := tail_ih\n      split <;> simp_all\n      . case h_1 y heq =>\n        -- Given the tactic state:\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: String \u00d7 Strings\n        -- y: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sup\n        -- h1: \u00ackey = sup\n        -- hx: x \u2208 Std.AssocList.toList (addSubSup sub sup tail)\n        -- he: x.fst = sup\n        -- heq: \u2203 x, x \u2208 Std.AssocList.toList tail \u2227 x.fst = sup\n\n        -- How to prove?\n        -- \u22a2 \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons key value tail)) \u2227 x.fst = sup\n        sorry\n\n      . case h_2 y heq =>\n        -- Given the tactic state:\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: String \u00d7 Strings\n        -- y: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] tail)) \u2227 x.fst = sup\n        -- h1: \u00ackey = sup\n        -- hx: x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] tail))\n        -- he: x.fst = sup\n        -- heq: (List.any (Std.AssocList.toList tail) fun x => decide (x.fst = sup)) = false\n\n        -- How to prove?\n        -- \u22a2 \u2203 x,\n        --   x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227\n        --     x.fst = sup\n        sorry\n\ntheorem addBoth.sup_beq (sub sup: String) (ss: Std.AssocList String Strings) \n: (addBoth sub sup ss).contains sup\n:= by\n  simp [addBoth]\n  simp [addDecl]\n  induction ss <;> simp_all\n  . case nil =>\n    simp [addSubSup, cond_eq_ite]\n    by_cases sup == sub <;> simp_all\n  . case cons key value tail tail_ih =>\n    simp [addDecl, cond_eq_ite]\n    by_cases key == sup <;> simp_all\n    . case pos h1 =>\n      simp [addSubSup, cond_eq_ite]\n      by_cases sup == sub <;> simp_all\n    . case neg h1 => \n      let \u27e8 x, hx, he \u27e9 := tail_ih\n      split <;> simp_all\n      . case h_1 y heq =>\n        -- Given the tactic state (same as addBoth.sup_eq: path cons/neg/h1)\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: String \u00d7 Strings\n        -- y: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup tail) \u2227 x.fst = sup\n        -- h1: \u00ackey = sup\n        -- hx: x \u2208 Std.AssocList.toList (addSubSup sub sup tail)\n        -- he: x.fst = sup\n        -- heq: \u2203 x, x \u2208 Std.AssocList.toList tail \u2227 x.fst = sup\n\n        -- How to prove?\n        -- \u22a2 \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons key value tail)) \u2227 x.fst = sup\n        sorry\n\n      . case h_2 y heq =>\n        -- Given the tactic state: (same as addBoth.sup_eq: path cons/neg/h2)\n        -- subsupkey: String\n        -- value: Strings\n        -- tail: Std.AssocList String Strings\n        -- x: String \u00d7 Strings\n        -- y: Bool\n        -- tail_ih: \u2203 x, x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] tail)) \u2227 x.fst = sup\n        -- h1: \u00ackey = sup\n        -- hx: x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] tail))\n        -- he: x.fst = sup\n        -- heq: (List.any (Std.AssocList.toList tail) fun x => decide (x.fst = sup)) = false\n\n        -- How to prove?\n        -- \u22a2 \u2203 x,\n        --   x \u2208 Std.AssocList.toList (addSubSup sub sup (Std.AssocList.cons sup [] (Std.AssocList.cons key value tail))) \u2227\n        --     x.fst = sup\n        sorry\n\nend MWE8", "meta": {"author": "NicolasRouquette", "repo": "oml.lean4", "sha": "a60689536837a52fe21595d79877063f28ec7cfc", "save_path": "github-repos/lean/NicolasRouquette-oml.lean4", "path": "github-repos/lean/NicolasRouquette-oml.lean4/oml.lean4-a60689536837a52fe21595d79877063f28ec7cfc/src/Oml/MWE8.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565326, "lm_q2_score": 0.6261241702517975, "lm_q1q2_score": 0.35199230433788065}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n-/\nimport category_theory.limits.preserves.basic\nimport category_theory.limits.types\nimport category_theory.limits.shapes.wide_pullbacks\nimport category_theory.limits.shapes.multiequalizer\nimport category_theory.concrete_category.elementwise\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n-/\n\nuniverses w v u\n\nopen category_theory\n\nnamespace category_theory.limits\n\nlocal attribute [instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nsection limits\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{(max w v)} C]\n  {J : Type w} [small_category J] (F : J \u2964 C) [preserves_limit F (forget C)]\n\nlemma concrete.to_product_injective_of_is_limit {D : cone F} (hD : is_limit D) :\n  function.injective (\u03bb (x : D.X) (j : J), D.\u03c0.app j x) :=\nbegin\n  let E := (forget C).map_cone D,\n  let hE : is_limit E := is_limit_of_preserves _ hD,\n  let G := types.limit_cone.{w v} (F \u22d9 forget C),\n  let hG := types.limit_cone_is_limit.{w v} (F \u22d9 forget C),\n  let T : E.X \u2245 G.X := hE.cone_point_unique_up_to_iso hG,\n  change function.injective (T.hom \u226b (\u03bb x j, G.\u03c0.app j x)),\n  have h : function.injective T.hom,\n  { intros a b h,\n    suffices : T.inv (T.hom a) = T.inv (T.hom b), by simpa,\n    rw h },\n  suffices : function.injective (\u03bb (x : G.X) j, G.\u03c0.app j x),\n    by exact this.comp h,\n  apply subtype.ext,\nend\n\nlemma concrete.is_limit_ext {D : cone F} (hD : is_limit D) (x y : D.X) :\n  (\u2200 j, D.\u03c0.app j x = D.\u03c0.app j y) \u2192 x = y :=\n\u03bb h, concrete.to_product_injective_of_is_limit _ hD (funext h)\n\nlemma concrete.limit_ext [has_limit F] (x y : limit F) :\n  (\u2200 j, limit.\u03c0 F j x = limit.\u03c0 F j y) \u2192 x = y :=\nconcrete.is_limit_ext F (limit.is_limit _) _ _\n\nsection wide_pullback\n\nopen wide_pullback\nopen wide_pullback_shape\n\nlemma concrete.wide_pullback_ext {B : C} {\u03b9 : Type w} {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B)\n  [has_wide_pullback B X f] [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h\u2080 : base f x = base f y)\n  (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintro (_|j),\n  { exact h\u2080 },\n  { apply h }\nend\n\nlemma concrete.wide_pullback_ext' {B : C} {\u03b9 : Type w} [nonempty \u03b9]\n  {X : \u03b9 \u2192 C} (f : \u03a0 j : \u03b9, X j \u27f6 B) [has_wide_pullback.{w} B X f]\n  [preserves_limit (wide_cospan B X f) (forget C)]\n  (x y : wide_pullback B X f) (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\nbegin\n  apply concrete.wide_pullback_ext _ _ _ _ h,\n  inhabit \u03b9,\n  simp only [\u2190 \u03c0_arrow f (arbitrary _), comp_apply, h],\nend\n\nend wide_pullback\n\nsection multiequalizer\n\nlemma concrete.multiequalizer_ext {I : multicospan_index.{w} C} [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x y : multiequalizer I)\n  (h : \u2200 (t : I.L), multiequalizer.\u03b9 I t x = multiequalizer.\u03b9 I t y) : x = y :=\nbegin\n  apply concrete.limit_ext,\n  rintros (a|b),\n  { apply h },\n  { rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b),\n      comp_apply, comp_apply, h] }\nend\n\n/-- An auxiliary equivalence to be used in `multiequalizer_equiv` below.-/\ndef concrete.multiequalizer_equiv_aux (I : multicospan_index C) :\n  (I.multicospan \u22d9 (forget C)).sections \u2243\n  { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\n{ to_fun := \u03bb x, \u27e8\u03bb i, x.1 (walking_multicospan.left _), \u03bb i, begin\n    have a := x.2 (walking_multicospan.hom.fst i),\n    have b := x.2 (walking_multicospan.hom.snd i),\n    rw \u2190 b at a,\n    exact a,\n  end\u27e9,\n  inv_fun := \u03bb x,\n  { val := \u03bb j,\n    match j with\n    | walking_multicospan.left a := x.1 _\n    | walking_multicospan.right b := I.fst b (x.1 _)\n    end,\n    property := begin\n      rintros (a|b) (a'|b') (f|f|f),\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n      { refl },\n      { dsimp, erw \u2190 x.2 b', refl },\n      { change (I.multicospan.map (\ud835\udfd9 _)) _ = _, simp },\n    end },\n  left_inv := begin\n    intros x, ext (a|b),\n    { refl },\n    { change _ = x.val _,\n      rw \u2190 x.2 (walking_multicospan.hom.fst b),\n      refl }\n  end,\n  right_inv := by { intros x, ext i, refl } }\n\n/-- The equivalence between the noncomputable multiequalizer and\nand the concrete multiequalizer. -/\nnoncomputable\ndef concrete.multiequalizer_equiv (I : multicospan_index.{w} C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] : (multiequalizer I : C) \u2243\n    { x : \u03a0 (i : I.L), I.left i // \u2200 (i : I.R), I.fst i (x _) = I.snd i (x _) } :=\nlet h1 := (limit.is_limit I.multicospan),\n    h2 := (is_limit_of_preserves (forget C) h1),\n    E := h2.cone_point_unique_up_to_iso (types.limit_cone_is_limit _) in\nequiv.trans E.to_equiv (concrete.multiequalizer_equiv_aux I)\n\n@[simp]\nlemma concrete.multiequalizer_equiv_apply (I : multicospan_index.{w} C) [has_multiequalizer I]\n  [preserves_limit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :\n  ((concrete.multiequalizer_equiv I) x : \u03a0 (i : I.L), I.left i) i = multiequalizer.\u03b9 I i x := rfl\n\nend multiequalizer\n\n-- TODO: Add analogous lemmas about products and equalizers.\n\nend limits\n\nsection colimits\n\n-- We don't mark this as an `@[ext]` lemma as we don't always want to work elementwise.\nlemma cokernel_funext {C : Type*} [category C] [has_zero_morphisms C] [concrete_category C]\n  {M N K : C} {f : M \u27f6 N} [has_cokernel f] {g h : cokernel f \u27f6 K}\n  (w : \u2200 (n : N), g (cokernel.\u03c0 f n) = h (cokernel.\u03c0 f n)) : g = h :=\nbegin\n  apply coequalizer.hom_ext,\n  apply concrete_category.hom_ext _ _,\n  simpa using w,\nend\n\nvariables {C : Type u} [category.{v} C] [concrete_category.{v} C]\n  {J : Type v} [small_category J] (F : J \u2964 C) [preserves_colimit F (forget C)]\n\nlemma concrete.from_union_surjective_of_is_colimit {D : cocone F} (hD : is_colimit D) :\n  let ff : (\u03a3 (j : J), F.obj j) \u2192 D.X := \u03bb a, D.\u03b9.app a.1 a.2 in function.surjective ff :=\nbegin\n  intro ff,\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  suffices : function.surjective (TX.hom \u2218 ff),\n  { intro a,\n    obtain \u27e8b, hb\u27e9 := this (TX.hom a),\n    refine \u27e8b, _\u27e9,\n    apply_fun TX.inv at hb,\n    change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb,\n    simpa only [TX.hom_inv_id] using hb },\n  have : TX.hom \u2218 ff = \u03bb a, G.\u03b9.app a.1 a.2,\n  { ext a,\n    change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _,\n    rw hE.fac },\n  rw this,\n  rintro \u27e8\u27e8j,a\u27e9\u27e9,\n  exact \u27e8\u27e8j,a\u27e9,rfl\u27e9,\nend\n\nlemma concrete.is_colimit_exists_rep {D : cocone F} (hD : is_colimit D) (x : D.X) :\n  \u2203 (j : J) (y : F.obj j), D.\u03b9.app j y = x :=\nbegin\n  obtain \u27e8a, rfl\u27e9 := concrete.from_union_surjective_of_is_colimit F hD x,\n  exact \u27e8a.1, a.2, rfl\u27e9,\nend\n\nlemma concrete.colimit_exists_rep [has_colimit F] (x : colimit F) :\n  \u2203 (j : J) (y : F.obj j), colimit.\u03b9 F j y = x :=\nconcrete.is_colimit_exists_rep F (colimit.is_colimit _) x\n\nlemma concrete.is_colimit_rep_eq_of_exists {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  D.\u03b9.app i x = D.\u03b9.app j y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom,\n  swap, { suffices : function.bijective TX.hom, by exact this.1,\n    rw \u2190 is_iso_iff_bijective, apply is_iso.of_iso },\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y,\n  erw [T.hom.w, T.hom.w],\n  obtain \u27e8k, f, g, h\u27e9 := h,\n  have : G.\u03b9.app i x = (G.\u03b9.app k (F.map f x) : G.X) := quot.sound \u27e8f,rfl\u27e9,\n  rw [this, h],\n  symmetry,\n  exact quot.sound \u27e8g,rfl\u27e9,\nend\n\nlemma concrete.colimit_rep_eq_of_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y :=\nconcrete.is_colimit_rep_eq_of_exists F (colimit.is_colimit _) x y h\n\nsection filtered_colimits\n\nvariable [is_filtered J]\n\nlemma concrete.is_colimit_exists_of_rep_eq {D : cocone F} {i j : J} (hD : is_colimit D)\n  (x : F.obj i) (y : F.obj j) (h : D.\u03b9.app _ x = D.\u03b9.app _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nbegin\n  let E := (forget C).map_cocone D,\n  let hE : is_colimit E := is_colimit_of_preserves _ hD,\n  let G := types.colimit_cocone.{v v} (F \u22d9 forget C),\n  let hG := types.colimit_cocone_is_colimit.{v v} (F \u22d9 forget C),\n  let T : E \u2245 G := hE.unique_up_to_iso hG,\n  let TX : E.X \u2245 G.X := (cocones.forget _).map_iso T,\n  apply_fun TX.hom at h,\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y at h,\n  erw [T.hom.w, T.hom.w] at h,\n  replace h := quot.exact _ h,\n  suffices : \u2200 (a b : \u03a3 j, F.obj j)\n    (h : eqv_gen (limits.types.quot.rel.{v v} (F \u22d9 forget C)) a b),\n    \u2203 k (f : a.1 \u27f6 k) (g : b.1 \u27f6 k), F.map f a.2 = F.map g b.2,\n  { exact this \u27e8i,x\u27e9 \u27e8j,y\u27e9 h },\n  intros a b h,\n  induction h,\n  case eqv_gen.rel : x y hh\n  { obtain \u27e8e,he\u27e9 := hh,\n    use [y.1, e, \ud835\udfd9 _],\n    simpa using he.symm },\n  case eqv_gen.refl : x { use [x.1, \ud835\udfd9 _, \ud835\udfd9 _, rfl] },\n  case eqv_gen.symm : x y _ hh\n  { obtain \u27e8k, f, g, hh\u27e9 := hh,\n    use [k, g, f, hh.symm] },\n  case eqv_gen.trans : x y z _ _ hh1 hh2\n  { obtain \u27e8k1, f1, g1, h1\u27e9 := hh1,\n    obtain \u27e8k2, f2, g2, h2\u27e9 := hh2,\n    let k0 : J := is_filtered.max k1 k2,\n    let e1 : k1 \u27f6 k0 := is_filtered.left_to_max _ _,\n    let e2 : k2 \u27f6 k0 := is_filtered.right_to_max _ _,\n    let k : J := is_filtered.coeq (g1 \u226b e1) (f2 \u226b e2),\n    let e : k0 \u27f6 k := is_filtered.coeq_hom _ _,\n    use [k, f1 \u226b e1 \u226b e, g2 \u226b e2 \u226b e],\n    simp only [F.map_comp, comp_apply, h1, \u2190 h2],\n    simp only [\u2190 comp_apply, \u2190 F.map_comp],\n    rw is_filtered.coeq_condition },\nend\n\ntheorem concrete.is_colimit_rep_eq_iff_exists {D : cocone F} {i j : J}\n  (hD : is_colimit D) (x : F.obj i) (y : F.obj j) :\n  D.\u03b9.app i x = D.\u03b9.app j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.is_colimit_exists_of_rep_eq _ hD _ _, concrete.is_colimit_rep_eq_of_exists _ hD _ _\u27e9\n\nlemma concrete.colimit_exists_of_rep_eq [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) (h : colimit.\u03b9 F _ x = colimit.\u03b9 F _ y) :\n  \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\nconcrete.is_colimit_exists_of_rep_eq F (colimit.is_colimit _) x y h\n\ntheorem concrete.colimit_rep_eq_iff_exists [has_colimit F] {i j : J}\n  (x : F.obj i) (y : F.obj j) :\n  colimit.\u03b9 F i x = colimit.\u03b9 F j y \u2194 \u2203 k (f : i \u27f6 k) (g : j \u27f6 k), F.map f x = F.map g y :=\n\u27e8concrete.colimit_exists_of_rep_eq _ _ _, concrete.colimit_rep_eq_of_exists _ _ _\u27e9\n\nend filtered_colimits\n\nsection wide_pushout\n\nopen wide_pushout\nopen wide_pushout_shape\n\nlemma concrete.wide_pushout_exists_rep {B : C} {\u03b1 : Type*} {X : \u03b1 \u2192 C} (f : \u03a0 j : \u03b1, B \u27f6 X j)\n  [has_wide_pushout.{v} B X f] [preserves_colimit (wide_span B X f) (forget C)]\n  (x : wide_pushout B X f) : (\u2203 y : B, head f y = x) \u2228 (\u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x) :=\nbegin\n  obtain \u27e8_ | j, y, rfl\u27e9 := concrete.colimit_exists_rep _ x,\n  { use y },\n  { right,\n    use [j,y] }\nend\n\nlemma concrete.wide_pushout_exists_rep' {B : C} {\u03b1 : Type*} [nonempty \u03b1] {X : \u03b1 \u2192 C}\n  (f : \u03a0 j : \u03b1, B \u27f6 X j) [has_wide_pushout.{v} B X f]\n  [preserves_colimit (wide_span B X f) (forget C)] (x : wide_pushout B X f) :\n  \u2203 (i : \u03b1) (y : X i), \u03b9 f i y = x :=\nbegin\n  rcases concrete.wide_pushout_exists_rep f x with \u27e8y, rfl\u27e9 | \u27e8i, y, rfl\u27e9,\n  { inhabit \u03b1,\n    use [arbitrary _, f _ y],\n    simp only [\u2190 arrow_\u03b9 _ (arbitrary \u03b1), comp_apply] },\n  { use [i,y] }\nend\n\nend wide_pushout\n\n-- TODO: Add analogous lemmas about coproducts and coequalizers.\n\nend colimits\n\nend category_theory.limits\n", "meta": {"author": "Parinya-Siri", "repo": "lean-machine-learning", "sha": "ec610bac246ae7108fc6f0c140b3440f0fbacc52", "save_path": "github-repos/lean/Parinya-Siri-lean-machine-learning", "path": "github-repos/lean/Parinya-Siri-lean-machine-learning/lean-machine-learning-ec610bac246ae7108fc6f0c140b3440f0fbacc52/matlib/category_theory/limits/concrete_category.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.35199229907420687}}
{"text": "/-\nCopyright (c) 2020 Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Bhavik Mehta\n-/\nimport data.list.chain\nimport category_theory.punit\nimport category_theory.is_connected\nimport category_theory.sigma.basic\nimport category_theory.full_subcategory\n\n/-!\n# Connected components of a category\n\nDefines a type `connected_components J` indexing the connected components of a category, and the\nfull subcategories giving each connected component: `component j : Type u\u2081`.\nWe show that each `component j` is in fact connected.\n\nWe show every category can be expressed as a disjoint union of its connected components, in\nparticular `decomposed J` is the category (definitionally) given by the sigma-type of the connected\ncomponents of `J`, and it is shown that this is equivalent to `J`.\n-/\n\nuniverses v\u2081 v\u2082 v\u2083 u\u2081 u\u2082\n\nnoncomputable theory\n\nopen category_theory.category\n\nnamespace category_theory\n\nattribute [instance, priority 100] is_connected.is_nonempty\n\nvariables {J : Type u\u2081} [category.{v\u2081} J]\nvariables {C : Type u\u2082} [category.{u\u2081} C]\n\n/-- This type indexes the connected components of the category `J`. -/\ndef connected_components (J : Type u\u2081) [category.{v\u2081} J] : Type u\u2081 := quotient (zigzag.setoid J)\n\ninstance [inhabited J] : inhabited (connected_components J) := \u27e8quotient.mk' (default J)\u27e9\n\n/-- Given an index for a connected component, produce the actual component as a full subcategory. -/\n@[derive category]\ndef component (j : connected_components J) : Type u\u2081 := {k : J // quotient.mk' k = j}\n\n/-- The inclusion functor from a connected component to the whole category. -/\n@[derive [full, faithful], simps {rhs_md := semireducible}]\ndef component.\u03b9 (j) : component j \u2964 J :=\nfull_subcategory_inclusion _\n\n/-- Each connected component of the category is nonempty. -/\ninstance (j : connected_components J) : nonempty (component j) :=\nbegin\n  apply quotient.induction_on' j,\n  intro k,\n  refine \u27e8\u27e8k, rfl\u27e9\u27e9,\nend\n\ninstance (j : connected_components J) : inhabited (component j) := classical.inhabited_of_nonempty'\n\n/-- Each connected component of the category is connected. -/\ninstance (j : connected_components J) : is_connected (component j) :=\nbegin\n  -- Show it's connected by constructing a zigzag (in `component j`) between any two objects\n  apply is_connected_of_zigzag,\n  rintro \u27e8j\u2081, hj\u2081\u27e9 \u27e8j\u2082, rfl\u27e9,\n  -- We know that the underlying objects j\u2081 j\u2082 have some zigzag between them in `J`\n  have h\u2081\u2082 : zigzag j\u2081 j\u2082 := quotient.exact' hj\u2081,\n  -- Get an explicit zigzag as a list\n  rcases list.exists_chain_of_relation_refl_trans_gen h\u2081\u2082 with \u27e8l, hl\u2081, hl\u2082\u27e9,\n  -- Everything which has a zigzag to j\u2082 can be lifted to the same component as `j\u2082`.\n  let f : \u03a0 x, zigzag x j\u2082 \u2192 component (quotient.mk' j\u2082) := \u03bb x h, \u27e8x, quotient.sound' h\u27e9,\n  -- Everything in our chosen zigzag from `j\u2081` to `j\u2082` has a zigzag to `j\u2082`.\n  have hf : \u2200 (a : J), a \u2208 l \u2192 zigzag a j\u2082,\n  { intros i hi,\n    apply list.chain.induction (\u03bb t, zigzag t j\u2082) _ hl\u2081 hl\u2082 _ _ _ (or.inr hi),\n    { intros j k,\n      apply relation.refl_trans_gen.head },\n    { apply relation.refl_trans_gen.refl } },\n  -- Now lift the zigzag from `j\u2081` to `j\u2082` in `J` to the same thing in `component j`.\n  refine \u27e8l.pmap f hf, _, _\u27e9,\n  { refine @@list.chain_pmap_of_chain _ _ _ f (\u03bb x y _ _ h, _) hl\u2081 h\u2081\u2082 _,\n    exact zag_of_zag_obj (component.\u03b9 _) h },\n  { erw list.last_pmap _ f (j\u2081 :: l) (by simpa [h\u2081\u2082] using hf) (list.cons_ne_nil _ _),\n    exact subtype.ext hl\u2082 },\nend\n\n/--\nThe disjoint union of `J`s connected components, written explicitly as a sigma-type with the\ncategory structure.\nThis category is equivalent to `J`.\n-/\nabbreviation decomposed (J : Type u\u2081) [category.{v\u2081} J] :=\n\u03a3 (j : connected_components J), component j\n\n/--\nThe inclusion of each component into the decomposed category. This is just `sigma.incl` but having\nthis abbreviation helps guide typeclass search to get the right category instance on `decomposed J`.\n-/\n-- This name may cause clashes further down the road, and so might need to be changed.\nabbreviation inclusion (j : connected_components J) : component j \u2964 decomposed J :=\nsigma.incl _\n\n/-- The forward direction of the equivalence between the decomposed category and the original. -/\n@[simps {rhs_md := semireducible}]\ndef decomposed_to (J : Type u\u2081) [category.{v\u2081} J] : decomposed J \u2964 J :=\nsigma.desc component.\u03b9\n\n@[simp]\nlemma inclusion_comp_decomposed_to (j : connected_components J) :\n  inclusion j \u22d9 decomposed_to J = component.\u03b9 j :=\nrfl\n\ninstance : full (decomposed_to J) :=\n{ preimage :=\n  begin\n    rintro \u27e8j', X, hX\u27e9 \u27e8k', Y, hY\u27e9 f,\n    dsimp at f,\n    have : j' = k',\n      rw [\u2190 hX, \u2190 hY, quotient.eq'],\n      exact relation.refl_trans_gen.single (or.inl \u27e8f\u27e9),\n    subst this,\n    refine sigma.sigma_hom.mk f,\n  end,\n  witness' :=\n  begin\n    rintro \u27e8j', X, hX\u27e9 \u27e8_, Y, rfl\u27e9 f,\n    have : quotient.mk' Y = j',\n    { rw [\u2190 hX, quotient.eq'],\n      exact relation.refl_trans_gen.single (or.inr \u27e8f\u27e9) },\n    subst this,\n    refl,\n  end }\n\ninstance : faithful (decomposed_to J) :=\n{ map_injective' :=\n  begin\n    rintro \u27e8_, j, rfl\u27e9 \u27e8_, k, hY\u27e9 \u27e8_, _, _, f\u27e9 \u27e8_, _, _, g\u27e9 e,\n    change f = g at e,\n    subst e,\n  end }\n\ninstance : ess_surj (decomposed_to J) :=\n{ mem_ess_image := \u03bb j, \u27e8\u27e8_, j, rfl\u27e9, \u27e8iso.refl _\u27e9\u27e9 }\n\ninstance : is_equivalence (decomposed_to J) :=\nequivalence.equivalence_of_fully_faithfully_ess_surj _\n\n/-- This gives that any category is equivalent to a disjoint union of connected categories. -/\n@[simps functor {rhs_md := semireducible}]\ndef decomposed_equiv : decomposed J \u224c J :=\n(decomposed_to J).as_equivalence\n\nend category_theory\n", "meta": {"author": "JLimperg", "repo": "aesop3", "sha": "a4a116f650cc7403428e72bd2e2c4cda300fe03f", "save_path": "github-repos/lean/JLimperg-aesop3", "path": "github-repos/lean/JLimperg-aesop3/aesop3-a4a116f650cc7403428e72bd2e2c4cda300fe03f/src/category_theory/connected_components.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6261241632752915, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.3519922912301516}}
{"text": "/-\nCopyright (c) 2022 Jo\u00ebl Riou. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Jo\u00ebl Riou\n-/\n\nimport for_mathlib.dold_kan.functor_gamma\nimport category_theory.idempotents.homological_complex\n\n/-! The counit isomorphism of the Dold-Kan equivalence\n\nThe purpose of this file is to construct natural isomorphisms\n`N\u2081\u0393\u2080 : \u0393\u2080 \u22d9 N\u2081 \u2245 to_karoubi (chain_complex C \u2115)`\nand `N\u2082\u0393\u2082 : \u0393\u2082 \u22d9 N\u2082 \u2245 \ud835\udfed (karoubi (chain_complex C \u2115))`.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.limits category_theory.idempotents\n  opposite simplicial_object\nopen_locale simplicial\n\nnamespace algebraic_topology\n\nnamespace dold_kan\n\nvariables {C : Type*} [category C] [preadditive C] [has_finite_coproducts C]\n\n/-- The isomorphism  `(\u0393\u2080.splitting K).nondeg_complex \u2245 K` for all `K : chain_complex C \u2115`. -/\n@[simps]\ndef \u0393\u2080_nondeg_complex_iso (K : chain_complex C \u2115) : (\u0393\u2080.splitting K).nondeg_complex \u2245 K :=\nhomological_complex.hom.iso_of_components (\u03bb n, iso.refl _)\nbegin\n  rintros _ n (rfl : n+1=_),\n  dsimp,\n  simp only [id_comp, comp_id, alternating_face_map_complex.obj_d_eq,\n    preadditive.sum_comp, preadditive.comp_sum],\n  rw fintype.sum_eq_single (0 : fin (n+2)),\n  { simp only [fin.coe_zero, pow_zero, one_zsmul],\n    erw [\u0393\u2080.obj.map_mono_on_summand_id_assoc, \u0393\u2080.obj.termwise.map_mono_\u03b4\u2080,\n      splitting.\u03b9_\u03c0_summand_eq_id, comp_id], },\n  { intros i hi,\n    dsimp,\n    simp only [preadditive.zsmul_comp, preadditive.comp_zsmul, assoc],\n    erw [\u0393\u2080.obj.map_mono_on_summand_id_assoc, \u0393\u2080.obj.termwise.map_mono_eq_zero,\n      zero_comp, zsmul_zero],\n    { intro h,\n      replace h := congr_arg simplex_category.len h,\n      change n+1 = n at h,\n      linarith, },\n    { simpa only [is_\u03b4\u2080.iff] using hi, }, },\nend\n\n/-- The natural isomorphism `(\u0393\u2080.splitting K).nondeg_complex \u2245 K` for `K : chain_complex C \u2115`. -/\ndef \u0393\u2080'_comp_nondeg_complex_functor :\n  \u0393\u2080' \u22d9 split.nondeg_complex_functor \u2245 \ud835\udfed (chain_complex C \u2115) :=\nnat_iso.of_components \u0393\u2080_nondeg_complex_iso\n  (\u03bb X Y f, by { ext n, dsimp, simp only [comp_id, id_comp], })\n\n/-- The natural isomorphism `\u0393\u2080 \u22d9 N\u2081 \u2245 to_karoubi (chain_complex C \u2115)`. -/\ndef N\u2081\u0393\u2080 : \u0393\u2080 \u22d9 N\u2081 \u2245 to_karoubi (chain_complex C \u2115) :=\ncalc \u0393\u2080 \u22d9 N\u2081 \u2245 \u0393\u2080' \u22d9 split.forget C \u22d9 N\u2081 : functor.associator _ _ _\n... \u2245 \u0393\u2080' \u22d9 split.nondeg_complex_functor \u22d9 to_karoubi _ :\n  iso_whisker_left \u0393\u2080' split.to_karoubi_nondeg_complex_functor_iso_N\u2081.symm\n... \u2245 (\u0393\u2080' \u22d9 split.nondeg_complex_functor) \u22d9 to_karoubi _ : (functor.associator _ _ _).symm\n... \u2245 \ud835\udfed _ \u22d9 to_karoubi (chain_complex C \u2115) : iso_whisker_right \u0393\u2080'_comp_nondeg_complex_functor _\n... \u2245 to_karoubi (chain_complex C \u2115) : functor.left_unitor _\n\nlemma N\u2081\u0393\u2080_app (K : chain_complex C \u2115) :\n  N\u2081\u0393\u2080.app K = (\u0393\u2080.splitting K).to_karoubi_nondeg_complex_iso_N\u2081.symm\n    \u226a\u226b (to_karoubi _).map_iso (\u0393\u2080_nondeg_complex_iso K) :=\nbegin\n  ext1,\n  dsimp [N\u2081\u0393\u2080],\n  erw [id_comp, comp_id, comp_id],\n  refl,\nend\n\nlemma N\u2081\u0393\u2080_hom_app (K : chain_complex C \u2115) :\n  N\u2081\u0393\u2080.hom.app K = (\u0393\u2080.splitting K).to_karoubi_nondeg_complex_iso_N\u2081.inv\n    \u226b (to_karoubi _).map (\u0393\u2080_nondeg_complex_iso K).hom :=\nby { change (N\u2081\u0393\u2080.app K).hom = _, simpa only [N\u2081\u0393\u2080_app], }\n\nlemma N\u2081\u0393\u2080_inv_app (K : chain_complex C \u2115) :\n  N\u2081\u0393\u2080.inv.app K = (to_karoubi _).map (\u0393\u2080_nondeg_complex_iso K).inv \u226b\n   (\u0393\u2080.splitting K).to_karoubi_nondeg_complex_iso_N\u2081.hom :=\nby { change (N\u2081\u0393\u2080.app K).inv = _, simpa only [N\u2081\u0393\u2080_app], }\n\n@[simp]\nlemma N\u2081\u0393\u2080_hom_app_f_f (K : chain_complex C \u2115) (n : \u2115) :\n  (N\u2081\u0393\u2080.hom.app K).f.f n = (\u0393\u2080.splitting K).to_karoubi_nondeg_complex_iso_N\u2081.inv.f.f n :=\nby { rw N\u2081\u0393\u2080_hom_app, apply comp_id, }\n\n@[simp]\nlemma N\u2081\u0393\u2080_inv_app_f_f (K : chain_complex C \u2115) (n : \u2115) :\n  (N\u2081\u0393\u2080.inv.app K).f.f n = (\u0393\u2080.splitting K).to_karoubi_nondeg_complex_iso_N\u2081.hom.f.f n :=\nby { rw N\u2081\u0393\u2080_inv_app, apply id_comp, }\n\nlemma N\u2082\u0393\u2082_to_karoubi : to_karoubi (chain_complex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 = \u0393\u2080 \u22d9 N\u2081 :=\nbegin\n  have h := functor.congr_obj (functor_extension\u2082_comp_whiskering_left_to_karoubi\n    (chain_complex C \u2115) (simplicial_object C)) \u0393\u2080,\n  have h' := functor.congr_obj (functor_extension\u2081_comp_whiskering_left_to_karoubi\n    (simplicial_object C) (chain_complex C \u2115)) N\u2081,\n  dsimp [N\u2082, \u0393\u2082, functor_extension\u2081] at h h' \u22a2,\n  rw [\u2190 functor.assoc, h, functor.assoc, h'],\nend\n\n/-- Compatibility isomorphism between `to_karoubi _ \u22d9 \u0393\u2082 \u22d9 N\u2082` and `\u0393\u2080 \u22d9 N\u2081` which\nare functors `chain_complex C \u2115 \u2964 karoubi (chain_complex C \u2115)`. -/\n@[simps]\ndef N\u2082\u0393\u2082_to_karoubi_iso : to_karoubi (chain_complex C \u2115) \u22d9 \u0393\u2082 \u22d9 N\u2082 \u2245 \u0393\u2080 \u22d9 N\u2081 :=\neq_to_iso (N\u2082\u0393\u2082_to_karoubi)\n\n/-- The counit isomorphism of the Dold-Kan equivalence for additive categories. -/\ndef N\u2082\u0393\u2082 : \u0393\u2082 \u22d9 N\u2082 \u2245 \ud835\udfed (karoubi (chain_complex C \u2115)) :=\n((whiskering_left _ _ _).obj (to_karoubi (chain_complex C \u2115))).preimage_iso\n  (N\u2082\u0393\u2082_to_karoubi_iso \u226a\u226b N\u2081\u0393\u2080)\n\nlemma N\u2082\u0393\u2082_compatible_with_N\u2081\u0393\u2080 (K : chain_complex C \u2115) :\n  N\u2082\u0393\u2082.hom.app ((to_karoubi _).obj K) = N\u2082\u0393\u2082_to_karoubi_iso.hom.app K \u226b N\u2081\u0393\u2080.hom.app K :=\ncongr_app (((whiskering_left _ _ (karoubi (chain_complex C \u2115 ))).obj\n  (to_karoubi (chain_complex C \u2115))).image_preimage\n  (N\u2082\u0393\u2082_to_karoubi_iso.hom \u226b N\u2081\u0393\u2080.hom : _ \u27f6 to_karoubi _ \u22d9 \ud835\udfed _)) K\n\n@[simp]\nlemma N\u2082\u0393\u2082_inv_app_f_f (X : karoubi (chain_complex C \u2115)) (n : \u2115) :\n  (N\u2082\u0393\u2082.inv.app X).f.f n =\n    X.p.f n \u226b (\u0393\u2080.splitting X.X).\u03b9_summand (splitting.index_set.id (op [n])) :=\nbegin\n  dsimp only [N\u2082\u0393\u2082, functor.preimage_iso, iso.trans],\n  simp only [whiskering_left_obj_preimage_app, N\u2082\u0393\u2082_to_karoubi_iso_inv, functor.id_map,\n    nat_trans.comp_app, eq_to_hom_app, functor.comp_map, assoc, karoubi.comp_f,\n    karoubi.eq_to_hom_f, eq_to_hom_refl, comp_id, karoubi.comp_p_assoc, N\u2082_map_f_f,\n    homological_complex.comp_f, N\u2081\u0393\u2080_inv_app_f_f, P_infty_on_\u0393\u2080_splitting_summand_eq_self_assoc,\n    splitting.to_karoubi_nondeg_complex_iso_N\u2081_hom_f_f, \u0393\u2082_map_f_app, karoubi.decomp_id_p_f],\n  dsimp [to_karoubi],\n  rw [splitting.\u03b9_desc],\n  dsimp [splitting.index_set.id],\n  rw karoubi.homological_complex.p_idem_assoc,\nend\n\nend dold_kan\n\nend algebraic_topology\n", "meta": {"author": "joelriou", "repo": "dold-kan", "sha": "a083fe264275774ac49ac520caf25f2ee29debb1", "save_path": "github-repos/lean/joelriou-dold-kan", "path": "github-repos/lean/joelriou-dold-kan/dold-kan-a083fe264275774ac49ac520caf25f2ee29debb1/src/for_mathlib/dold_kan/gamma_comp_n.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3519486114124266}}
{"text": "import for_mathlib.snake_lemma\n\nnamespace category_theory\n\nlocal notation x `\u27f6[`D`]` y := D.map (snake_diagram.hom x y)\n\nnamespace snake_lemma\n\nopen category_theory.limits\n\nuniverses v u\nvariables {A : Type u} [category.{v} A] [abelian A]\n  {F G : snake_diagram \u2964 A} (\u03b7 : F \u27f6 G)\n\nnamespace \u03b4_natural_setup\n\n@[reassoc]\nlemma aux1 (hF : is_snake_input F) (hG : is_snake_input G) :\n  \u03b7.app (0, 2) \u226b hG.to_kernel = hF.to_kernel \u226b\n  kernel.lift _ (kernel.\u03b9 _ \u226b \u03b7.app _) begin\n    simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n  end :=\nbegin\n  apply equalizer.hom_ext,\n  dsimp [is_snake_input.to_kernel],\n  simp,\nend\n\n@[reassoc]\nlemma aux2 (hF : is_snake_input F) (hG : is_snake_input G) :\n  kernel.lift ((1, 2) \u27f6[G] (2, 2)) (kernel.\u03b9 ((1, 2) \u27f6[F] (2, 2)) \u226b\n    \u03b7.app (1, 2)) begin\n      simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n    end \u226b inv hG.cokernel_to_top_right_kernel_to_right_kernel =\n  inv hF.cokernel_to_top_right_kernel_to_right_kernel \u226b\n    cokernel.desc _ (kernel.lift _ (kernel.\u03b9 _ \u226b \u03b7.app _) begin\n      simp only [category.assoc, \u2190 \u03b7.naturality, kernel.condition_assoc, zero_comp],\n    end \u226b cokernel.\u03c0 _) begin\n      dsimp [is_snake_input.to_top_right_kernel],\n      simp only [\u2190 category.assoc], let t := _, change t \u226b _ = _,\n      have ht : t = \u03b7.app _ \u226b kernel.lift ((1,1) \u27f6[G] (2,2)) ((1,0) \u27f6[G] (1,1)) _,\n      { apply equalizer.hom_ext, simp, },\n      rw [ht, category.assoc, cokernel.condition, comp_zero],\n    end :=\nbegin\n  simp only [is_iso.eq_inv_comp, is_iso.comp_inv_eq, category.assoc],\n  dsimp only [is_snake_input.cokernel_to_top_right_kernel_to_right_kernel],\n  apply equalizer.hom_ext,\n  simp only [le_refl, and_true, category.assoc, nat_trans.naturality,\n    kernel.condition_assoc, zero_comp, true_and,\n    cokernel.condition, comp_zero, equalizer_as_kernel, kernel.lift_\u03b9],\n  apply coequalizer.hom_ext,\n  simp only [category.assoc, nat_trans.naturality, cokernel.\u03c0_desc_assoc,\n    kernel.lift_\u03b9_assoc, kernel.lift_\u03b9],\nend\n\n@[reassoc]\nlemma aux3 (hF : is_snake_input F) (hG : is_snake_input G) :\n  kernel.lift hG.bottom_left_cokernel_to (kernel.\u03b9 hF.bottom_left_cokernel_to \u226b\n    cokernel.desc ((1, 0) \u27f6[F] (2, 1)) (\u03b7.app (2, 1) \u226b\n    cokernel.\u03c0 ((1, 0) \u27f6[G] (2, 1))) begin\n      simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n    end) begin\n      dsimp [is_snake_input.bottom_left_cokernel_to],\n      simp only [category.assoc], let t := _, change _ \u226b t = _,\n      have ht : t = cokernel.desc ((1,0) \u27f6[F] (2,1)) ((2,1) \u27f6[F] (2,2)) _ \u226b \u03b7.app _,\n      { apply coequalizer.hom_ext, simp, },\n      rw [ht, kernel.condition_assoc, zero_comp],\n    end \u226b\n    inv hG.left_cokernel_to_kernel_bottom_left_cokernel_to =\n  inv hF.left_cokernel_to_kernel_bottom_left_cokernel_to \u226b\n  cokernel.desc _ (\u03b7.app _ \u226b cokernel.\u03c0 _) begin\n    simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n  end :=\nbegin\n  rw [is_iso.comp_inv_eq, category.assoc (inv _), is_iso.eq_inv_comp],\n  dsimp [is_snake_input.left_cokernel_to_kernel_bottom_left_cokernel_to],\n  apply coequalizer.hom_ext, apply equalizer.hom_ext,\n  simp only [nat_trans.naturality_assoc, category.assoc, cokernel.\u03c0_desc_assoc,\n    cokernel.\u03c0_desc, kernel.lift_\u03b9, kernel.lift_\u03b9_assoc],\nend\n\nend \u03b4_natural_setup\n\nopen \u03b4_natural_setup\n\ntheorem \u03b4_natural (hF : is_snake_input F) (hG : is_snake_input G) :\n  \u03b7.app (0,2) \u226b hG.\u03b4 = hF.\u03b4 \u226b \u03b7.app (3,0) :=\nbegin\n  dsimp only [is_snake_input.\u03b4],\n  simp only [category.assoc],\n  rw aux1_assoc \u03b7 hF hG,\n  rw aux2_assoc \u03b7 hF hG,\n  simp_rw cancel_epi,\n  apply coequalizer.hom_ext,\n  dsimp only [is_snake_input.\u03b4_aux],\n  simp only [cokernel.\u03c0_desc_assoc, category.assoc],\n  simp only [\u2190 category.assoc], let t := _, change (t \u226b _) \u226b _ = _,\n  let s := _, change _ = ((s \u226b _) \u226b _) \u226b _,\n  have ht : t = s \u226b kernel.lift _ (kernel.\u03b9 _ \u226b\n    cokernel.desc _ (\u03b7.app _ \u226b cokernel.\u03c0 _) begin\n      simp only [category.assoc, \u03b7.naturality_assoc, cokernel.condition, comp_zero],\n    end) _,\n  rotate 2,\n  { dsimp only [is_snake_input.bottom_left_cokernel_to], simp only [category.assoc],\n    let t := _, change _ \u226b t = _,\n    have ht : t = cokernel.desc ((1,0) \u27f6[F] (2,1)) ((2,1) \u27f6[F] (2,2)) _ \u226b \u03b7.app _,\n    { apply coequalizer.hom_ext,\n      simp only [cokernel.\u03c0_desc_assoc, category.assoc, cokernel.\u03c0_desc, nat_trans.naturality], },\n    rw [ht, kernel.condition_assoc, zero_comp] },\n  { dsimp only [t, s],\n    apply equalizer.hom_ext,\n    simp only [category.assoc, nat_trans.naturality_assoc, cokernel.\u03c0_desc, kernel.lift_\u03b9, kernel.lift_\u03b9_assoc] },\n  rw ht, dsimp only [s], clear ht t s,\n  simp only [category.assoc], congr' 1,\n  rw aux3_assoc \u03b7 hF hG, congr' 1,\n  dsimp only [is_snake_input.cokernel_to],\n  apply coequalizer.hom_ext,\n  simp only [cokernel.\u03c0_desc_assoc, category.assoc, cokernel.\u03c0_desc, nat_trans.naturality],\nend\n\nend snake_lemma\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/snake_lemma_naturality.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3519486047535307}}
{"text": "example (A B C D E F G H I J K L : Prop)\n(f1 : A \u2192 B) (f2 : B \u2192 E) (f3 : E \u2192 D) (f4 : D \u2192 A) (f5 : E \u2192 F)\n(f6 : F \u2192 C) (f7 : B \u2192 C) (f8 : F \u2192 G) (f9 : G \u2192 J) (f10 : I \u2192 J)\n(f11 : J \u2192 I) (f12 : I \u2192 H) (f13 : E \u2192 H) (f14 : H \u2192 K) (f15 : I \u2192 L)\n : A \u2192 L :=\nbegin\ncc,\nend\n", "meta": {"author": "abdelq", "repo": "natural-number-game", "sha": "bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2", "save_path": "github-repos/lean/abdelq-natural-number-game", "path": "github-repos/lean/abdelq-natural-number-game/natural-number-game-bbddadc6d2e78ece2e9acd40fa7702ecc2db75c2/world06/level09.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316991792861, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3519120925853736}}
{"text": "/-\nCopyright (c) 2022 Ya\u00ebl Dillies. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies\n\n! This file was ported from Lean 3 source module combinatorics.set_family.kleitman\n! leanprover-community/mathlib commit 50832daea47b195a48b5b33b1c8b2162c48c3afc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.Combinatorics.SetFamily.HarrisKleitman\nimport Mathbin.Combinatorics.SetFamily.Intersecting\n\n/-!\n# Kleitman's bound on the size of intersecting families\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nAn intersecting family on `n` elements has size at most `2\u207f\u207b\u00b9`, so we could na\u00efvely think that two\nintersecting families could cover all `2\u207f` sets. But actually that's not case because for example\nnone of them can contain the empty set. Intersecting families are in some sense correlated.\nKleitman's bound stipulates that `k` intersecting families cover at most `2\u207f - 2\u207f\u207b\u1d4f` sets.\n\n## Main declarations\n\n* `finset.card_bUnion_le_of_intersecting`: Kleitman's theorem.\n\n## References\n\n* [D. J. Kleitman, *Families of non-disjoint subsets*][kleitman1966]\n-/\n\n\nopen Finset\n\nopen Fintype (card)\n\nvariable {\u03b9 \u03b1 : Type _} [Fintype \u03b1] [DecidableEq \u03b1] [Nonempty \u03b1]\n\n/- warning: finset.card_bUnion_le_of_intersecting -> Finset.card_bunion\u1d62_le_of_intersecting is a dubious translation:\nlean 3 declaration is\n  forall {\u03b9 : Type.{u1}} {\u03b1 : Type.{u2}} [_inst_1 : Fintype.{u2} \u03b1] [_inst_2 : DecidableEq.{succ u2} \u03b1] [_inst_3 : Nonempty.{succ u2} \u03b1] (s : Finset.{u1} \u03b9) (f : \u03b9 -> (Finset.{u2} (Finset.{u2} \u03b1))), (forall (i : \u03b9), (Membership.Mem.{u1, u1} \u03b9 (Finset.{u1} \u03b9) (Finset.hasMem.{u1} \u03b9) i s) -> (Set.Intersecting.{u2} (Finset.{u2} \u03b1) (Lattice.toSemilatticeInf.{u2} (Finset.{u2} \u03b1) (Finset.lattice.{u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b))) (Finset.orderBot.{u2} \u03b1) ((fun (a : Type.{u2}) (b : Type.{u2}) [self : HasLiftT.{succ u2, succ u2} a b] => self.0) (Finset.{u2} (Finset.{u2} \u03b1)) (Set.{u2} (Finset.{u2} \u03b1)) (HasLiftT.mk.{succ u2, succ u2} (Finset.{u2} (Finset.{u2} \u03b1)) (Set.{u2} (Finset.{u2} \u03b1)) (CoeTC\u2093.coe.{succ u2, succ u2} (Finset.{u2} (Finset.{u2} \u03b1)) (Set.{u2} (Finset.{u2} \u03b1)) (Finset.Set.hasCoeT.{u2} (Finset.{u2} \u03b1)))) (f i)))) -> (LE.le.{0} Nat Nat.hasLe (Finset.card.{u2} (Finset.{u2} \u03b1) (Finset.bunion\u1d62.{u1, u2} \u03b9 (Finset.{u2} \u03b1) (fun (a : Finset.{u2} \u03b1) (b : Finset.{u2} \u03b1) => Finset.decidableEq.{u2} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a b) s f)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.Pow.{0} Nat Nat.monoid)) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))) (Fintype.card.{u2} \u03b1 _inst_1)) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat (Monoid.Pow.{0} Nat Nat.monoid)) (OfNat.ofNat.{0} Nat 2 (OfNat.mk.{0} Nat 2 (bit0.{0} Nat Nat.hasAdd (One.one.{0} Nat Nat.hasOne)))) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat Nat.hasSub) (Fintype.card.{u2} \u03b1 _inst_1) (Finset.card.{u1} \u03b9 s)))))\nbut is expected to have type\n  forall {\u03b9 : Type.{u2}} {\u03b1 : Type.{u1}} [_inst_1 : Fintype.{u1} \u03b1] [_inst_2 : DecidableEq.{succ u1} \u03b1] [_inst_3 : Nonempty.{succ u1} \u03b1] (s : Finset.{u2} \u03b9) (f : \u03b9 -> (Finset.{u1} (Finset.{u1} \u03b1))), (forall (i : \u03b9), (Membership.mem.{u2, u2} \u03b9 (Finset.{u2} \u03b9) (Finset.instMembershipFinset.{u2} \u03b9) i s) -> (Set.Intersecting.{u1} (Finset.{u1} \u03b1) (Lattice.toSemilatticeInf.{u1} (Finset.{u1} \u03b1) (Finset.instLatticeFinset.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b))) (Finset.instOrderBotFinsetToLEToPreorderPartialOrder.{u1} \u03b1) (Finset.toSet.{u1} (Finset.{u1} \u03b1) (f i)))) -> (LE.le.{0} Nat instLENat (Finset.card.{u1} (Finset.{u1} \u03b1) (Finset.bunion\u1d62.{u2, u1} \u03b9 (Finset.{u1} \u03b1) (fun (a : Finset.{u1} \u03b1) (b : Finset.{u1} \u03b1) => Finset.decidableEq.{u1} \u03b1 (fun (a : \u03b1) (b : \u03b1) => _inst_2 a b) a b) s f)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat instPowNat) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) (Fintype.card.{u1} \u03b1 _inst_1)) (HPow.hPow.{0, 0, 0} Nat Nat Nat (instHPow.{0, 0} Nat Nat instPowNat) (OfNat.ofNat.{0} Nat 2 (instOfNatNat 2)) (HSub.hSub.{0, 0, 0} Nat Nat Nat (instHSub.{0} Nat instSubNat) (Fintype.card.{u1} \u03b1 _inst_1) (Finset.card.{u2} \u03b9 s)))))\nCase conversion may be inaccurate. Consider using '#align finset.card_bUnion_le_of_intersecting Finset.card_bunion\u1d62_le_of_intersecting\u2093'. -/\n/-- **Kleitman's theorem**. An intersecting family on `n` elements contains at most `2\u207f\u207b\u00b9` sets, and\neach further intersecting family takes at most half of the sets that are in no previous family. -/\ntheorem Finset.card_bunion\u1d62_le_of_intersecting (s : Finset \u03b9) (f : \u03b9 \u2192 Finset (Finset \u03b1))\n    (hf : \u2200 i \u2208 s, (f i : Set (Finset \u03b1)).Intersecting) :\n    (s.bunion\u1d62 f).card \u2264 2 ^ card \u03b1 - 2 ^ (card \u03b1 - s.card) :=\n  by\n  obtain hs | hs := le_total (card \u03b1) s.card\n  \u00b7 rw [tsub_eq_zero_of_le hs, pow_zero]\n    refine'\n      (card_le_of_subset <|\n            bUnion_subset.2 fun i hi a ha =>\n              mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq\n        _\n    rw [card_compl, Fintype.card_finset, card_singleton]\n  induction' s using Finset.cons_induction with i s hi ih generalizing f\n  \u00b7 simp\n  classical\n    set f' : \u03b9 \u2192 Finset (Finset \u03b1) := fun j =>\n      if hj : j \u2208 cons i s hi then (hf j hj).exists_card_eq.some else \u2205 with hf'\n    have hf\u2081 :\n      \u2200 j,\n        j \u2208 cons i s hi \u2192\n          f j \u2286 f' j \u2227 2 * (f' j).card = 2 ^ card \u03b1 \u2227 (f' j : Set (Finset \u03b1)).Intersecting :=\n      by\n      rintro j hj\n      simp_rw [hf', dif_pos hj, \u2190 Fintype.card_finset]\n      exact Classical.choose_spec (hf j hj).exists_card_eq\n    have hf\u2082 : \u2200 j, j \u2208 cons i s hi \u2192 IsUpperSet (f' j : Set (Finset \u03b1)) :=\n      by\n      refine' fun j hj => (hf\u2081 _ hj).2.2.isUpperSet' ((hf\u2081 _ hj).2.2.is_max_iff_card_eq.2 _)\n      rw [Fintype.card_finset]\n      exact (hf\u2081 _ hj).2.1\n    refine' (card_le_of_subset <| bUnion_mono fun j hj => (hf\u2081 _ hj).1).trans _\n    nth_rw 1 [cons_eq_insert i]\n    rw [bUnion_insert]\n    refine' (card_mono <| @le_sup_sdiff _ _ _ <| f' i).trans ((card_union_le _ _).trans _)\n    rw [union_sdiff_left, sdiff_eq_inter_compl]\n    refine' le_of_mul_le_mul_left _ (pow_pos zero_lt_two <| card \u03b1 + 1)\n    rw [pow_succ', mul_add, mul_assoc, mul_comm _ 2, mul_assoc]\n    refine'\n      (add_le_add\n              ((mul_le_mul_left <| pow_pos (zero_lt_two' \u2115) _).2\n                (hf\u2081 _ <| mem_cons_self _ _).2.2.card_le) <|\n            (mul_le_mul_left <| zero_lt_two' \u2115).2 <| IsUpperSet.card_inter_le_finset _ _).trans\n        _\n    \u00b7 rw [coe_bUnion]\n      exact isUpperSet_union\u1d62\u2082 fun i hi => hf\u2082 _ <| subset_cons _ hi\n    \u00b7 rw [coe_compl]\n      exact (hf\u2082 _ <| mem_cons_self _ _).compl\n    rw [mul_tsub, card_compl, Fintype.card_finset, mul_left_comm, mul_tsub,\n      (hf\u2081 _ <| mem_cons_self _ _).2.1, two_mul, add_tsub_cancel_left, \u2190 mul_tsub, \u2190 mul_two,\n      mul_assoc, \u2190 add_mul, mul_comm]\n    refine' mul_le_mul_left' _ _\n    refine'\n      (add_le_add_left\n            (ih ((card_le_of_subset <| subset_cons _).trans hs) _ fun i hi =>\n              (hf\u2081 _ <| subset_cons _ hi).2.2)\n            _).trans\n        _\n    rw [mul_tsub, two_mul, \u2190 pow_succ, \u2190\n      add_tsub_assoc_of_le (pow_le_pow' (one_le_two : (1 : \u2115) \u2264 2) tsub_le_self),\n      tsub_add_eq_add_tsub hs, card_cons, add_tsub_add_eq_tsub_right]\n#align finset.card_bUnion_le_of_intersecting Finset.card_bunion\u1d62_le_of_intersecting\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Combinatorics/SetFamily/Kleitman.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.4960938294709195, "lm_q1q2_score": 0.35174001724760107}}
{"text": "/-\n# Example: Aviation Resource Management\n\nThe previous examples have been typical problems from computing science. This\nexample is drawn from industry, and is a simplified version of a problem that\noccurs globally on a daily basis. It concerns resource allocation in the aviation\nindustry. Specifically, scheduling aircraft for take off at an airport.\n\nThe number of passengers undertaking air travel is continually increasing, with a\ncorresponding increase in the number of aircraft movements. However, the resources\navailable to facilitate these flights are not growing at the same rate. In\nparticular, airports and their runways, taxiways and gates are becoming ever more\ncongested. If we cannot grow the resources to match the increase in passenger\nnumbers and flights, it is necessary to improve the efficiency with which the\navailable resources are used.\n\n[Air Traffic Flow Management (ATFM)](https://www.icao.int/airnavigation/IMP/Documents/9971%20Collaborative%20Flight%20and%20Flow%20Informaiton.pdf)\nis the discipline that assesses future air traffic movements, and schedules that\ntraffic to make best use of available resources. The application of a set of rules\nthat schedules aircraft across resources over a period of time is known as a\nTraffic Management Initiative (TMI). There are different kinds of TMIs, one being\na Ground Delay Program (GDP): hold aircraft on the ground at their departure point\nsuch that after take off they are able to proceed to their destination without\nbeing placed in a holding pattern prior to arrival. An aircraft holding in the air\nis using significant amounts of fuel, increasing costs for the operator and\nincreasing environmental impact due to greater emissions.\n\nA GDP assesses the aircraft bound for an airport based on flight schedules, allocates\nlanding times, and then back calculates to determine the best take off time for an\naircraft to ensure when it reaches its destination it will not be in conflict with\nother aircraft attempting to land.\n\n## The Problem Statement\n\nThe particular TMI we will look at is a departure program; that is, scheduling\naircraft departing an airport. Such a program may be conducted to ensure an\nordered sequence of departures. For example, during the mining boom in Western\nAustralia, many aircraft would depart the capital, Perth, for remote mining\nsites. We have many aircraft departing one airport, and heading to a variety of\ndestinations with low traffic volumes. In a free for all situation, the aircraft\nwill leave their gates according to their schedule, resulting in congestion on\ntaxiways as aircraft queue for take off. Individual flights have no motivation to\nwait for a suitable slot, since others will simply leave their gate to fill the\ngap. Congestion is not the only problem; aircraft sitting on the taxiway burn fuel\nhence there is increased cost for the aircraft operator, and greater environmental\nimpact from the increased emissions.\n\nA departure program allocates a take off time to each aircraft so the operator is\ninformed beforehand when they are allowed to take off. This provides the aircraft\noperator with more certainty since they know exactly when the flight will depart\nand hence can better predict arrival time for planning of subsequent flight legs.\nA number of constraints must be satisfied by the departure program:\n- Specified flights wish to depart an airport.\n  - The flights that plan to depart during the period of the TMI.\n- Each flight may only take off from certain runways.\n  - Larger aircraft may not be able to take off from shorter runways.\n- Each flight has a preferred take off time.\n  - The time the aircraft operator would ideally like their flight to depart.\n- Each flight must take off in a nominated time window.\n  - Aircraft operators have schedules to meet; the allocated take off time must not be too\n  far from the scheduled time.\n- Only certain runways are available.\n  - The configuration of resources dictates which of the runways at an airport are available\n  for use.\n- Each runway has a maximum rate at which departures can occur.\n  - The rates may differ for different runways. Rate is impacted by factors such as the forecast\n  weather and airport noise restrictions.\n- The program occurs over a fixed interval.\n  - A program need only run at times when there is likely to be competition for resources.\n\nGiven these constraints, the problem then is to:\n- Allocate a runway and target take off time to as many flights as possible such that the\nconstraints are satisfied. Flights should be allocated a time as close as possible to their\npreferred time.\n\n## The Specification\n-/\n\nimport LeanSpec.lib.Temporal\nimport LeanSpec.lib.Util\n\nopen Std Std.Map Temporal\n\nnamespace TMI\n\n/-\nFirst some basic types that identify airports, flights and runways. There are rules\naround how these identifiers are formed, but for this specification it is sufficient\nthat different identifiers can be distinguished.\n\n### Designator of an Airport\n-/\nabbrev AirportDesig := String\n\n/-\n### Identifier of a Flight\n-/\nabbrev FlightId := String\n\n/-\n### Designator of a Runway\n-/\nabbrev RunwayDesig := String\n\n/-\n### Flight Departure\n\nThe `FlightDeparture` type captures information concerning a flight that is pertinent to the\ndetermination of departure slot allocation for a TMI. In a wider context, there is far more\ninformation relevant to a flight, but here we abstract only the necessary details.\n-/\nstructure FlightDeparture where\n  -- The runways the flight is able to use.\n  canUse    : List RunwayDesig\n  -- The operator's preferred take off time for the flight.\n  preferred : DTG\n  -- The period of time during which the flight can reasonably take off.\n  window    : Interval\n  -- The flight must be able to use at least one runway.\n  inv\u2081      : canUse \u2260 \u2205\n  -- The preferred take off time must occur within the window.\n  inv\u2082      : preferred \u2208 window\n\n/-\nNotes:\n- Field `canUse` is the runways (with respect to the departure TMI airport) the\nflight is able to use. A flight, based on the type of aircraft used, may only be\nable to use certain runways; an A380, for example, would not be able to take off\nor land on short runways.\n- It is not possible to allocate every flight its preferred take off time, but\noperators have schedules to adhere to, and subsequent legs the aircraft must\nconduct, so an acceptable take off window is specified. The flight must not be\nallocated a time outside the window.\n- To present an alternative approach to invariants, in this specification each\nconstraint is captured in a distinct subscripted `inv` field.\n\nTraditionally, structures (records) are used to model data consisting of multiple disparate elements.\nDependent types introduce the capability to directly encode constraints that relate the\nelements of a type. The consequence is that instances of the type that structurally look\nlike elements of the type are excluded because they fail to satisfy the constraints.\n\nIn the case of `FlightDeparture`, the empty list is of type `List RunwayDesig`, but field\n`canUse` will never be the empty list as `inv\u2081` would not be satisfied. The consequence of\nthe invariant is a flight cannot be considered if it does not nominate at least one runway\nfrom which it can take off.\n\n### Runway Rate\n\nThe runway rate determines how frequently an aircraft can take off from a runway.\nThis is modelled as a duration, being the minimum period of time between successive\ntake offs.\n-/\nabbrev Rate := Duration\n\n/-\nEach relevant runway is mapped to the rate it can accommodate.\n-/\nabbrev RunwayRates := RunwayDesig \u27f9 Rate\n\n/-\nNote: `A \u27f9 B` is the type of finite maps from `A` to `B` (defined in [Util](lib/Util.md)).\n\n### TMI Configuration\n\nA TMI configuration is all the information necessary to calculate a departure TMI,\nconsisting of information relevant to the airport, and the flights that wish to\ndepart from the airport during the TMI.\n-/\nstructure TMIConfig where\n  -- The designator of the airport at which the TMI is run.\n  airport : AirportDesig\n  -- The period of time over which the TMI runs.\n  period  : Interval\n  -- The flights that desire to take off within the period of the TMI.\n  flights : FlightId \u27f9 FlightDeparture\n  -- The rates of the runways that participate in the TMI.\n  rates   : RunwayRates\n  -- The take off window of a proposed flight must fall within the TMI period.\n  inv\u2081    : \u2200 f \u2208 flights.range, f.window \u2229 period \u2260 \u2205 \n  -- A flight must be able to take off from one of the participating runways.\n  inv\u2082    : \u2200 f \u2208 flights.range, f.canUse \u2229 rates.domain \u2260 \u2205\n\n/-\n### Slot\n\nA slot is a point in time at which a runway is available to an aircraft for take off.\n-/\nstructure Slot where\n  -- The runway designator.\n  rwy  : RunwayDesig\n  -- The target take off time.\n  ttot : DTG\n\n/-\n### Flight Allocation\n\nThe result of running a departure TMI (GDP) is a map from flights to their slots\n(allocated runways and take off times) such that the problem constraints are satisfied.\n-/\nstructure FlightAllocation (cfg : TMIConfig) where\n  -- The allocation of flight departures to slots.\n  gdp  : FlightId \u27f9 Slot\n  -- Any flight in the GDP must be drawn from the TMI configuration.\n  inv\u2081 : gdp.domain \u2286 cfg.flights.domain\n  -- The runway allocated to a flight must be a participant in the TMI.\n  inv\u2082 : \u2200 slot \u2208 gdp.range, slot.rwy \u2208 cfg.rates.domain\n  -- The runway allocated to a flight must be one of the runways it can use.\n  inv\u2083 : \u2200 fsl \u2208 gdp, \u2200 fdep \u2208 cfg.flights,\n           fsl.1 = fdep.1 \u2192 fsl.2.rwy \u2208 fdep.2.canUse\n  -- The target time allocated to a flight must fall within its window.\n  inv\u2084 : \u2200 fsl \u2208 gdp, \u2200 fdep \u2208 cfg.flights,\n           fsl.1 = fdep.1 \u2192 fsl.2.ttot \u2208 fdep.2.window\n  -- The target time allocated to a flight must fall within the TMI period.\n  inv\u2085 : \u2200 slot \u2208 gdp.range, slot.ttot \u2208 cfg.period\n  -- Two flights allocated the same runway must depart at least the minimum duration apart.\n  inv\u2086 : \u2200 fsl\u2081 \u2208 gdp, \u2200 fsl\u2082 \u2208 gdp, \u2200 fr \u2208 cfg.rates,\n           fsl\u2081.1 \u2260 fsl\u2082.1 \u2227 fsl\u2081.2.rwy = fsl\u2082.2.rwy \u2227 fr.1 = fsl\u2081.2.rwy \u2192 \n             fsl\u2081.2.ttot - fsl\u2082.2.ttot \u2265 fr.2\n\n/-\nAs noted earlier, structures can contain constraints to which the instances must adhere.\nDependent types allow us to take this a step further. `FlightAllocation` is dependent on\n(parameterised by) `TMIConfig`. As a result, the constraints of the type are not just\nrestricted to the fields of the type being defined, they can also specify relationships\nbetween the argument type and the type being defined. In the case of `FlightAllocation`, every\nconstraint is concerned with the relationship to `TMIConfig`.\n\nField `gdp` of `FlightAllocation` is the only _data_ field; all others are invariants. The effect\nof this definition is that for a given `cfg : TMIConfig`, the elements of type\n`FlightAllocation cfg` are all, and only, those allocations of flights to slots that satisfy\nthe problem constraints.\n\n### Cost\n\nTo complete the specification, we need the concept of a cost of a departure TMI.\nThere are many possible allocations that meet the constraints. The cost function\ndecides which one to select.\n\nIn an operational situation, the best cost function is far from clear, and would\nlikely require some investigation and prototyping. In this specification,\nwe adopt a fairly simple cost function for demonstration purposes. The cost function\nneeds to take account of both those flights that are allocated slots in the GDP, and\nthose that are omitted:\n- the cost of an included flight increases the further the allocated time is from the preferred time;\n- the cost of an excluded flight is generally greater than the cost of an included flight.\n\nThe deviation assigned to a flight that was allocated a slot in the TMI is\nthe duration between its preferred time and its allocated time. Duration is\nexpressed in seconds, so the greater the duration, the greater the cost.\n-/\ndef allocatedDeviation (flight : FlightDeparture) (slot : Slot) : Duration :=\n  flight.preferred - slot.ttot\n\n/-\nIf an omitted flight's window only partly overlaps the TMI period, there is still the\npossibility it may be able to take off in its operational window, but outside the\nTMI period. If an omitted flight's window is wholly within the TMI period, the\naircraft operator has a scheduling problem to solve. Consequently, for an omitted flight:\n\n- if its window is fully within the TMI period, the cost is the duration of the window;\n- if the window is partly outside the TMI period, the cost is half the duration of the window.\n-/\ndef omittedDeviation (period window : Interval) : Duration :=\n  let d := window.durationOf\n  if window \u2286 period then d else d/2\n\n/-\nThe deviations for all flights that requested a take off during the period of the TMI.\n-/\ndef deviations (cfg : TMIConfig) (alloc : FlightAllocation cfg) : List Duration :=\n  let deviation (fdep : FlightId \u00d7 FlightDeparture) : Duration :=\n        match alloc.gdp.find? fdep.1 with\n        | none      => omittedDeviation cfg.period fdep.2.window\n        | some slot => allocatedDeviation fdep.2 slot\n  cfg.flights.toList.map deviation\n\n/-\nThe `cost` of a TMI is the sum of the costs of the flights that requested a take off\nduring the TMI.\n-/\ndef cost (cfg : TMIConfig) (alloc : FlightAllocation cfg) : Duration :=\n  (deviations cfg alloc).add 0\n\n/-\n### Departure TMI\n\nCombining the components above, the departure TMI can be specified as:\n- the optimal allocation, that is, of all allocations that satisfy the configuration, the (not\nnecessarily unique) allocation whose cost is no greater than any other.\n-/\ndef DepartureTMI (cfg : TMIConfig) :=\n  { opt : FlightAllocation cfg // \u2200 alloc : FlightAllocation cfg, cost cfg opt \u2264 cost cfg alloc }\n\n/-\n## Further Discussion\n\nWe finish with some further discussion on aspects of the specification.\n\n### More Concise Specification\n\nThere is a great deal of repetition in the specification of the invariants of type\n`FlightAllocation`, which generally occur in the quantified variables and pre-conditions.\nWe can reduce that repetition by merging invariants as follows:\n-/\nstructure FlightAllocation\u2081 (cfg : TMIConfig) where\n  -- The allocation of flights to slots.\n  gdp  : FlightId \u27f9 Slot\n  -- Any flight in the GDP must be drawn from the TMI configuration.\n  inv\u2081 : gdp.domain \u2286 cfg.flights.domain\n  -- The runway allocated to a flight must be a participant in the TMI.\n  -- The target time allocated to a flight must fall within the TMI period.\n  -- The runway allocated to a flight must be one of the runways it can use.\n  -- The target time allocated to a flight must fall within its window.\n  inv\u2082 : \u2200 fsl \u2208 gdp,\n           fsl.2.rwy \u2208 cfg.rates.domain \u2227\n           fsl.2.ttot \u2208 cfg.period \u2227\n           \u2200 fdep \u2208 cfg.flights, fsl.1 = fdep.1 \u2192\n             fsl.2.rwy \u2208 fdep.2.canUse \u2227\n             fsl.2.ttot \u2208 fdep.2.window\n  -- Two flights allocated the same runway must depart at least the minimum duration apart.\n  inv\u2083 : \u2200 fsl\u2081 \u2208 gdp, \u2200 fsl\u2082 \u2208 gdp, \u2200 fr \u2208 cfg.rates,\n           fsl\u2081.1 \u2260 fsl\u2082.1 \u2227 fsl\u2081.2.rwy = fsl\u2082.2.rwy \u2227 fr.1 = fsl\u2081.2.rwy \u2192 \n             fsl\u2081.2.ttot - fsl\u2082.2.ttot \u2265 fr.2\n\n/-\nThere are now three rather than six invariants, and it is more concise. The separate approach\nwas taken to provide a clearer map from the specification to the informal description of the\nconstraints. In addition, keeping them separate makes it easier to focus on individual\nconstraints.\n\n### Non-Dependent Approach\n\n`FlightAllocation` is a dependent type that only admits an element if it satisfies the invariants.\nThe initial approach was more traditional with `FlightAllocation` defined as a simple map:\n-/\nabbrev FlightAllocation\u2082 := FlightId \u27f9 Slot\n\n/-\nThis type admits anything that is structurally a map from `FlightId` to `Slot`,\nhence includes things like:\n- identifiers that do not relate to a flight;\n- slots referring to runways that do not exist.\n\nIt is then necessary to define a property `Satisfies`, external to the type definition,\nthat specifies when a flight allocation satisfies a TMI configuration (that is, the\n`FlightAllocation` is a valid solution to the `TMIConfig`).\n-/\ndef Satisfies (cfg : TMIConfig) (alloc : FlightAllocation\u2082) :=\n  alloc.domain \u2286 cfg.flights.domain \u2227\n  (\u2200 slot \u2208 alloc.range, slot.rwy \u2208 cfg.rates.domain) \u2227\n  (\u2200 fsl \u2208 alloc, \u2200 fdep \u2208 cfg.flights,\n    fsl.1 = fdep.1 \u2192 fsl.2.rwy \u2208 fdep.2.canUse) \u2227\n  (\u2200 fsl \u2208 alloc, \u2200 fdep \u2208 cfg.flights,\n    fsl.1 = fdep.1 \u2192 fsl.2.ttot \u2208 fdep.2.window) \u2227\n  (\u2200 slot \u2208 alloc.range, slot.ttot \u2208 cfg.period) \u2227\n  (\u2200 fsl\u2081 \u2208 alloc, \u2200 fsl\u2082 \u2208 alloc, \u2200 fr \u2208 cfg.rates,\n    fsl\u2081.1 \u2260 fsl\u2082.1 \u2227 fsl\u2081.2.rwy = fsl\u2082.2.rwy \u2227 fr.1 = fsl\u2081.2.rwy \u2192 \n      fsl\u2081.2.ttot - fsl\u2082.2.ttot \u2265 fr.2)\n\n/-\nNote that, other than some minor syntactic differences, the constraints expressed by\n`Satisfies` are exactly the invariants of type `FlightAllocation`.\n\nThe dependent approach allows the constraints to be located with the data they refer to,\nrather than elsewhere in the specification, which aids comprehension. Further, the dependent\napproach allows the types to be more tightly defined, and as a result the specification of\nfunctions over those types tends to be simpler.\n\n## Exercises\n\n- Small changes to specifications can have major effects. What would happen if the\nspecification was changed such that the cost function only considered flights that\nare included in the TMI? That is, change `deviations` to:\n-/\ndef deviations\u2081 (cfg : TMIConfig) (alloc : FlightAllocation cfg) : List Duration :=\n  let deviation (fdep : FlightId \u00d7 FlightDeparture) : Duration :=\n        match  alloc.gdp.find? fdep.1 with\n        | none      => 0\n        | some slot => allocatedDeviation fdep.2 slot\n  cfg.flights.toList.map deviation\n/-\n- There has been a requirements change. A flight must not take off prior to its\npreferred time, though it can after its preferred time. Change the specification to\nsatisfy the new requirement.\n\n- It is always instructive to consider boundary cases. Say a TMI was run with a configuration\nin which no flights are specified. That is, `flights : FlightId \u27f9 FlightDeparture` is the empty map.\nIs a solution that meets the specification still possible?\n\n- Another boundary case is when no runways rates are specified. That is, `rates : RunwayRates` is\nthe empty map. Is a solution that meets the specification still possible? Is there any relationship\nbetween this and the `flights` item?\n\n- What is the impact on the specification if the constraint `inv\u2081 : canUse \u2260 \u2205` on type\n`FlightDeparture` is removed?\n\n- An alternative approach to evaluation of cost is to maximise the number of\npassengers that are able to depart during the TMI. Modify the specification to\nimplement this approach to cost. Hint: you will have to change `FlightDeparture`.\n(If you are a freight carrier you are out of luck.)\n-/\nend TMI", "meta": {"author": "paulch42", "repo": "lean-spec", "sha": "4755a25caf719f935bcc4d54bd8a86462c9aceb9", "save_path": "github-repos/lean/paulch42-lean-spec", "path": "github-repos/lean/paulch42-lean-spec/lean-spec-4755a25caf719f935bcc4d54bd8a86462c9aceb9/LeanSpec/TMI.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632979641571, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.3516658843952597}}
{"text": "import for_mathlib.algebra.homology.derived_category\n\nnoncomputable theory\n\nopen category_theory category_theory.category category_theory.pretriangulated\n  category_theory.limits\n\nnamespace homological_complex\n\n/- to be moved to algebra.homology.homology (and expanded) -/\n\nvariables {C \u03b9 : Type*} {c : complex_shape \u03b9} [category C] [has_zero_morphisms C]\n\nabbreviation cycles (K : homological_complex C c) (i : \u03b9) [K.has_homology i] : C :=\n((short_complex_functor C c i).obj K).cycles\n\nabbreviation cycles_i (K : homological_complex C c) (i : \u03b9) [K.has_homology i] :\n  K.cycles i \u27f6 K.X i :=\n((short_complex_functor C c i).obj K).cycles_i\n\n@[simp, reassoc]\nlemma cycles_i_d (K : homological_complex C c) (i j : \u03b9) [K.has_homology i] :\n  K.cycles_i i \u226b K.d i j = 0 :=\nbegin\n  by_cases c.rel i j,\n  { have hj := c.next_eq' h,\n    subst hj,\n    apply short_complex.cycles_i_g, },\n  { rw [K.shape _ _ h, comp_zero] },\nend\n\ndef lift_cycles (K : homological_complex C c) {A : C} {n\u2080 : \u03b9} [K.has_homology n\u2080]\n  (z : A \u27f6 K.X n\u2080) (n\u2081 : \u03b9) (hn\u2081 : c.rel n\u2080 n\u2081) (hz : z \u226b K.d n\u2080 n\u2081 = 0) :\n    A \u27f6 K.cycles n\u2080 :=\nshort_complex.lift_cycles _ z begin\n  have hn\u2081 := c.next_eq' hn\u2081,\n  subst hn\u2081,\n  exact hz,\nend\n\n@[simp, reassoc]\nlemma lift_cycles_i (K : homological_complex C c) {A : C} {n\u2080 : \u03b9} [K.has_homology n\u2080]\n  (z : A \u27f6 K.X n\u2080) (n\u2081 : \u03b9) (hn\u2081 : c.rel n\u2080 n\u2081) (hz : z \u226b K.d n\u2080 n\u2081 = 0) :\n  K.lift_cycles z n\u2081 hn\u2081 hz \u226b K.cycles_i n\u2080 = z :=\nshort_complex.lift_cycles_i _ _ _\n\nlemma eq_lift_cycles_i (K : homological_complex C c) {A : C} {n\u2080 : \u03b9} [K.has_homology n\u2080]\n  (z : A \u27f6 K.cycles n\u2080) (n\u2081 : \u03b9) (hn\u2081 : c.rel n\u2080 n\u2081) :\n  \u2203 (z' : A \u27f6 K.X n\u2080) (hz' : z' \u226b K.d n\u2080 n\u2081 = 0),\n    z = K.lift_cycles z' n\u2081 hn\u2081 hz' :=\n\u27e8z \u226b K.cycles_i n\u2080, by simp, by simp [\u2190 cancel_mono (K.cycles_i n\u2080)]\u27e9\n\nabbreviation cycles_map {K L : homological_complex C c} (\u03c6 : K \u27f6 L) (i : \u03b9)\n  [K.has_homology i] [L.has_homology i] :\n  K.cycles i \u27f6 L.cycles i :=\nshort_complex.cycles_map ((short_complex_functor C c i).map \u03c6)\n\n@[simp, reassoc]\nlemma lift_cycles_comp_cycles_map {K L : homological_complex C c} {A : C} {n\u2080 : \u03b9}\n  [K.has_homology n\u2080] [L.has_homology n\u2080] (z : A \u27f6 K.X n\u2080) (n\u2081 : \u03b9)\n  (hn\u2081 : c.rel n\u2080 n\u2081) (hz : z \u226b K.d n\u2080 n\u2081 = 0) (\u03c6 : K \u27f6 L) :\n  K.lift_cycles z n\u2081 hn\u2081 hz \u226b cycles_map \u03c6 n\u2080 =\n    L.lift_cycles (z \u226b \u03c6.f n\u2080) n\u2081 hn\u2081\n    (by rw [assoc, \u03c6.comm, reassoc_of hz, zero_comp]) :=\nshort_complex.lift_cycles_comp_cycles_map _ _ _ _\n\nabbreviation homology_\u03c0 (K : homological_complex C c) (i : \u03b9) [K.has_homology i] :\n  K.cycles i \u27f6 K.homology i :=\n((short_complex_functor C c i).obj K).homology_\u03c0\n\n@[simp, reassoc]\nlemma homology_\u03c0_naturality {K L : homological_complex C c} (\u03c6 : K \u27f6 L) (i : \u03b9)\n  [K.has_homology i] [L.has_homology i] :\n  K.homology_\u03c0 i \u226b homology_map \u03c6 i = cycles_map \u03c6 i \u226b L.homology_\u03c0 i :=\nshort_complex.homology_\u03c0_naturality _\n\nvariables (C c)\n\n@[simps]\ndef _root_.cycles_functor (i : \u03b9) [category_with_homology C] :\n  homological_complex C c \u2964 C :=\n  short_complex_functor C c i \u22d9 short_complex.cycles_functor C\n\nend homological_complex\n\nvariables {C : Type*} [category C] [abelian C]\n\nnamespace derived_category\n\n@[reassoc]\nlemma homology_functor_factors_hom_naturality {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) (n : \u2124) :\n  (homology_functor C n).map (Q.map \u03c6) \u226b (homology_functor_factors C n).hom.app L =\n    (homology_functor_factors C n).hom.app K \u226b homology_map \u03c6 n :=\n(homology_functor_factors C n).hom.naturality \u03c6\n\n@[reassoc]\nlemma homology_functor_factors_inv_naturality {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) (n : \u2124) :\n  homology_map \u03c6 n \u226b (homology_functor_factors C n).inv.app L =\n    (homology_functor_factors C n).inv.app K \u226b (homology_functor C n).map (Q.map \u03c6) :=\n(homology_functor_factors C n).inv.naturality \u03c6\n\nend derived_category\n\nnamespace cochain_complex\n\nvariable (C)\ndef shift_cycles_functor_iso [abelian C] (n k m : \u2124) (h : k + n = m) :\n  category_theory.shift_functor _ n \u22d9 cycles_functor C (complex_shape.up \u2124) k \u2245\n    cycles_functor C _ m :=\n(functor.associator _ _ _).symm \u226a\u226b\n  iso_whisker_right (shift_short_complex_functor_iso C _ _ _ h) _\n\nvariable {C}\n\nlemma shift_cycles_functor_iso_hom_app [abelian C] (n k m : \u2124) (h : k + n = m)\n  (K : cochain_complex C \u2124) :\n    (shift_cycles_functor_iso C n k m h).hom.app K =\n    short_complex.cycles_map ((shift_short_complex_functor_iso C n k m h).hom.app K) :=\nid_comp _\n\nlemma shift_cycles_functor_iso_hom_app_comp_cycles_i [abelian C] (n k m : \u2124) (h : k + n = m)\n  (K : cochain_complex C \u2124) :\n  (shift_cycles_functor_iso C n k m h).hom.app K \u226b K.cycles_i m =\n  K\u27e6n\u27e7.cycles_i k \u226b (K.shift_functor_obj_X_iso n k m h.symm).hom :=\nbegin\n  have eq := (short_complex.cycles_i_nat_trans C).naturality\n    ((shift_short_complex_functor_iso C _ _ _ h).hom.app K),\n  dsimp at eq,\n  rw [shift_cycles_functor_iso_hom_app, eq],\n  dsimp [shift_short_complex_functor_iso],\n  simpa only [shift_eval_hom_app_eq],\nend\n\nlemma unshift_cocycle (K : cochain_complex C \u2124) (r : \u2124) {A : C} {n\u2080 : \u2124}\n  (z : A \u27f6 K\u27e6r\u27e7.X n\u2080) (n\u2081 : \u2124) (hn\u2081 : n\u2081 = n\u2080 + 1)\n  (hz : z \u226b K\u27e6r\u27e7.d n\u2080 n\u2081 = 0) (n\u2080' : \u2124) (hn\u2080' : n\u2080' = n\u2080 + r) (n\u2081' : \u2124) :\n  (z \u226b (K.shift_functor_obj_X_iso r n\u2080 n\u2080' hn\u2080').hom) \u226b K.d n\u2080' n\u2081' = 0 :=\nbegin\n  by_cases hn\u2081' : n\u2081' = n\u2081 + r,\n  { substs hn\u2080' hn\u2081',\n    dsimp [homological_complex.X_iso_of_eq, iso.refl] at \u22a2 hz,\n    rw [linear.comp_smul] at hz,\n    erw [assoc, id_comp, \u2190 cancel_epi (preadditive.mul_iso ((-1 : units \u2124)^r) (iso.refl A)).hom,\n      comp_zero, preadditive.mul_iso_hom, iso.refl_hom, preadditive.zsmul_comp, id_comp, hz], },\n  { rw [K.shape, comp_zero],\n    change _ \u2260 _,\n    exact \u03bb h, hn\u2081' (by linarith), },\nend\n\n@[reassoc]\nlemma shift_lift_cycles (K : cochain_complex C \u2124) (r : \u2124) {A : C} {n\u2080 : \u2124}\n  (z : A \u27f6 K\u27e6r\u27e7.X n\u2080) (n\u2081 : \u2124) (hn\u2081 : (complex_shape.up \u2124).rel n\u2080 n\u2081)\n    (hz : z \u226b K\u27e6r\u27e7.d n\u2080 n\u2081 = 0) (n\u2080' : \u2124) (hn\u2080' : n\u2080' = n\u2080 + r) :\n    homological_complex.lift_cycles (K\u27e6r\u27e7) z n\u2081 hn\u2081 hz =\n      K.lift_cycles (z \u226b (K.shift_functor_obj_X_iso r n\u2080 n\u2080' hn\u2080').hom) (n\u2080'+1) rfl\n      (K.unshift_cocycle r z n\u2081 hn\u2081.symm hz n\u2080' hn\u2080' _) \u226b\n      (shift_cycles_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).inv.app K :=\nbegin\n  simp only [\u2190 cancel_mono ((shift_cycles_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).hom.app K), assoc,\n    iso.inv_hom_id_app, \u2190 cancel_mono (homological_complex.cycles_i K n\u2080')],\n  dsimp,\n  simpa only [id_comp, homological_complex.lift_cycles_i,\n    shift_cycles_functor_iso_hom_app_comp_cycles_i _ _ _  hn\u2080'.symm K,\n    homological_complex.lift_cycles_i_assoc],\nend\n\n@[reassoc]\nlemma shift_homology_\u03c0 (K : cochain_complex C \u2124) (r n\u2080 n\u2080': \u2124) (hn\u2080' : n\u2080' = n\u2080 + r) :\n  homological_complex.homology_\u03c0\n    ((category_theory.shift_functor (cochain_complex C \u2124) r).obj K) n\u2080 =\n    (shift_cycles_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).hom.app K \u226b\n      homological_complex.homology_\u03c0 K n\u2080' \u226b\n      (shift_homology_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).inv.app K :=\nbegin\n  simp only [\u2190 cancel_mono ((shift_homology_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).hom.app K),\n    assoc, iso.inv_hom_id_app, shift_cycles_functor_iso_hom_app],\n  erw comp_id,\n  simpa only [shift_homology_functor_iso_hom_app]\n    using short_complex.homology_\u03c0_naturality\n      ((shift_short_complex_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).hom.app K),\nend\n\n@[reassoc]\nlemma shift_lift_cycles_comp_homology_\u03c0 (K : cochain_complex C \u2124) (r : \u2124) {A : C} {n\u2080 : \u2124}\n  (z : A \u27f6 K\u27e6r\u27e7.X n\u2080) (n\u2081 : \u2124) (hn\u2081 : (complex_shape.up \u2124).rel n\u2080 n\u2081)\n    (hz : z \u226b K\u27e6r\u27e7.d n\u2080 n\u2081 = 0) (n\u2080' : \u2124) (hn\u2080' : n\u2080' = n\u2080 + r) :\n  homological_complex.lift_cycles (K\u27e6r\u27e7) z n\u2081 hn\u2081 hz \u226b K\u27e6r\u27e7.homology_\u03c0 n\u2080 =\n  K.lift_cycles (z \u226b (K.shift_functor_obj_X_iso r n\u2080 n\u2080' hn\u2080').hom) (n\u2080'+1) rfl\n    (K.unshift_cocycle r z n\u2081 hn\u2081.symm hz n\u2080' hn\u2080' _) \u226b\n    K.homology_\u03c0 n\u2080' \u226b (shift_homology_functor_iso C r n\u2080 n\u2080' hn\u2080'.symm).inv.app K :=\nby simp only [shift_lift_cycles K r z n\u2081 hn\u2081 hz n\u2080' hn\u2080', assoc,\n  shift_homology_\u03c0 K r n\u2080 n\u2080' hn\u2080', iso.inv_hom_id_app_assoc]\n\nnamespace homology_sequence\n\nvariables {C} {S : short_complex (cochain_complex C \u2124)} (ex : S.short_exact)\n\ninclude ex\n\nlemma ex\u2082 (n : \u2124) :\n  (short_complex.mk (homology_map S.f n) (homology_map S.g n)\n    (by rw [\u2190 homology_map_comp, S.zero, _root_.homology_map_zero])).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1\n    (derived_category.homology_sequence.ex\u2082 (derived_category.triangle_of_ses_dist ex) n),\n  exact short_complex.mk_iso\n    ((derived_category.homology_functor_factors C n).app _)\n    ((derived_category.homology_functor_factors C n).app _)\n    ((derived_category.homology_functor_factors C n).app _)\n    (derived_category.homology_functor_factors_hom_naturality S.f n).symm\n    (derived_category.homology_functor_factors_hom_naturality S.g n).symm,\nend\n\ndef \u03b4 (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  S.X\u2083.homology n\u2080 \u27f6 S.X\u2081.homology n\u2081 :=\n(derived_category.homology_functor_factors C n\u2080).inv.app S.X\u2083 \u226b\n  derived_category.homology_sequence.\u03b4 (derived_category.triangle_of_ses_dist ex) n\u2080 n\u2081 h \u226b\n  (derived_category.homology_functor_factors C n\u2081).hom.app S.X\u2081\n\n@[simp, reassoc]\nlemma \u03b4_comp (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  \u03b4 ex _ _ h \u226b homology_map S.f n\u2081  = 0 :=\nbegin\n  dsimp only [\u03b4],\n  simp only [assoc, \u2190 derived_category.homology_functor_factors_hom_naturality],\n  erw [derived_category.homology_sequence.\u03b4_comp_assoc, zero_comp, comp_zero],\nend\n\n@[simp, reassoc]\nlemma comp_\u03b4 (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  homology_map S.g n\u2080 \u226b \u03b4 ex _ _ h = 0 :=\nbegin\n  dsimp only [\u03b4],\n  rw derived_category.homology_functor_factors_inv_naturality_assoc,\n  erw derived_category.homology_sequence.comp_\u03b4_assoc (derived_category.triangle_of_ses_dist ex),\n  rw [zero_comp, comp_zero],\nend\n\nlemma ex\u2083 (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (short_complex.mk (homology_map S.g n\u2080) (\u03b4 ex _ _ h) (by simp)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1\n    (derived_category.homology_sequence.ex\u2083 (derived_category.triangle_of_ses_dist ex) n\u2080 n\u2081 h),\n  exact short_complex.mk_iso\n    ((derived_category.homology_functor_factors C n\u2080).app _)\n    ((derived_category.homology_functor_factors C n\u2080).app _)\n    ((derived_category.homology_functor_factors C n\u2081).app _)\n    (derived_category.homology_functor_factors_hom_naturality S.g n\u2080).symm\n    (by { dsimp [\u03b4], simp only [iso.hom_inv_id_app_assoc]}),\nend\n\nlemma ex\u2081 (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080+1) :\n  (short_complex.mk (\u03b4 ex _ _ h) (homology_map S.f n\u2081)  (by simp)).exact :=\nbegin\n  refine (short_complex.exact_iff_of_iso _).1\n    (derived_category.homology_sequence.ex\u2081 (derived_category.triangle_of_ses_dist ex) n\u2080 n\u2081 h),\n  exact short_complex.mk_iso\n    ((derived_category.homology_functor_factors C n\u2080).app _)\n    ((derived_category.homology_functor_factors C n\u2081).app _)\n    ((derived_category.homology_functor_factors C n\u2081).app _)\n    (by { dsimp [\u03b4], simp only [iso.hom_inv_id_app_assoc], })\n    (derived_category.homology_functor_factors_hom_naturality S.f n\u2081).symm,\nend\n\nend homology_sequence\n\n\nvariables {C} {S : short_complex (cochain_complex C \u2124)} (ex : S.short_exact)\n\nlemma from_mapping_cone_of_ses_comp_\u03b4 (n\u2080 n\u2081 : \u2124) (h : n\u2081 = n\u2080 + 1) :\n  homology_map (cochain_complex.from_mapping_cone_of_ses ex) n\u2080 \u226b\n    homology_sequence.\u03b4 ex n\u2080 n\u2081 h =\n  homology_map (mapping_cone.\u03b4 S.f) n\u2080 \u226b\n    (shift_homology_functor_iso C 1 n\u2080 n\u2081 h.symm).hom.app S.X\u2081 :=\nbegin\n  dsimp only [homology_sequence.\u03b4, derived_category.homology_sequence.\u03b4,\n    derived_category.triangle_of_ses, triangle.mk, derived_category.triangle_of_ses_\u03b4,\n    derived_category.mapping_cone_triangle, derived_category.mapping_cone_\u03b4],\n  simp only [assoc, derived_category.homology_functor_factors_inv_naturality_assoc,\n    functor.map_comp, functor.map_inv, is_iso.hom_inv_id_assoc,\n    derived_category.shift_homology_functor_iso_hom_app_Q_obj,\n    iso.inv_hom_id_app, comp_id],\n  simp only [\u2190 functor.map_comp_assoc, iso.hom_inv_id_app, functor.map_comp],\n  erw [category_theory.functor.map_id, comp_id,\n    \u2190 derived_category.homology_functor_factors_inv_naturality_assoc,\n    iso.inv_hom_id_app_assoc],\nend\n\nnamespace mapping_cone\n\nopen hom_complex\n\n@[simp, reassoc]\nlemma cycles_i_fst_d {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) (n\u2080 n\u2081 n\u2082 : \u2124) (h : n\u2081 = n\u2080 + 1) :\n  (mapping_cone \u03c6).cycles_i n\u2080 \u226b (fst \u03c6 : cochain (mapping_cone \u03c6) K 1).v n\u2080 n\u2081 h \u226b\n    K.d n\u2081 n\u2082 = 0 :=\nbegin\n  by_cases hn\u2082 : n\u2081 + 1 = n\u2082,\n  { have eq := (mapping_cone \u03c6).cycles_i_d n\u2080 n\u2081,\n    simpa only [assoc, d_fst _ _ _ _ h, zero_comp, preadditive.comp_neg, neg_eq_zero]\n      using eq =\u226b (fst \u03c6 : cochain (mapping_cone \u03c6) K 1).v n\u2081 n\u2082 hn\u2082.symm, },\n  { simp only [K.shape _ _ hn\u2082, comp_zero], },\nend\n\nlemma homology_map_\u03b4_on_a_cocycle {K L : cochain_complex C \u2124} (\u03c6 : K \u27f6 L) {A : C} {n\u2080 : \u2124}\n  (z : A \u27f6 (mapping_cone \u03c6).cycles n\u2080) (n\u2081 : \u2124) (h : n\u2081 = n\u2080 + 1) :\n  z \u226b (mapping_cone \u03c6).homology_\u03c0 n\u2080 \u226b homology_map (mapping_cone.\u03b4 \u03c6) n\u2080 \u226b\n    (shift_homology_functor_iso C 1 n\u2080 n\u2081 h.symm).hom.app K =\n    K.lift_cycles (-z \u226b (mapping_cone \u03c6).cycles_i n\u2080 \u226b\n      (mapping_cone.fst \u03c6 : cochain (mapping_cone \u03c6) K 1).v n\u2080 n\u2081 h) (n\u2081+1) rfl (by simp) \u226b\n      K.homology_\u03c0 n\u2081 :=\nbegin\n  obtain \u27e8z, hz, rfl\u27e9 := (mapping_cone \u03c6).eq_lift_cycles_i z n\u2081 h.symm,\n  obtain \u27e8x, y, eq\u27e9 := to_decomposition z n\u2081 h,\n  simp only [homological_complex.homology_\u03c0_naturality_assoc,\n    homological_complex.lift_cycles_i_assoc,\n    homological_complex.lift_cycles_comp_cycles_map_assoc,\n    shift_lift_cycles_comp_homology_\u03c0_assoc _ 1 _ n\u2081 h.symm _ n\u2081 h,\n    iso.inv_hom_id_app, \u03b4, assoc, cocycle.hom_of_f, cocycle.right_shift_coe, \u03b4_as_cocycle_coe,\n    cochain.right_shift_v _ _ _ (zero_add 1).symm n\u2080 n\u2080 (add_zero n\u2080).symm (n\u2081) h,\n    preadditive.neg_comp, preadditive.comp_neg, shift_functor_obj_X_iso, cochain.neg_v,\n    iso.inv_hom_id],\n  dsimp,\n  simp only [comp_id],\nend\n\nend mapping_cone\n\nnamespace homology_sequence\n\nopen hom_complex\n\nlemma comp_\u03b4_eq {n\u2080 n\u2081 : \u2124} {A : C} (x\u2083 : A \u27f6 S.X\u2083.X n\u2080)\n  (x\u2082 : A \u27f6 S.X\u2082.X n\u2080) (x\u2081 : A \u27f6 S.X\u2081.X n\u2081) (h : n\u2081 = n\u2080+1)\n    (hx\u2083 : x\u2083 \u226b S.X\u2083.d n\u2080 n\u2081 = 0) (hx\u2082 : x\u2082 \u226b S.g.f n\u2080 = x\u2083)\n    (hx\u2081 : x\u2081 \u226b S.f.f n\u2081 = x\u2082 \u226b S.X\u2082.d n\u2080 n\u2081) :\n  S.X\u2083.lift_cycles x\u2083 n\u2081 h.symm hx\u2083 \u226b S.X\u2083.homology_\u03c0 n\u2080 \u226b (\u03b4 ex n\u2080 n\u2081 h) =\n    S.X\u2081.lift_cycles x\u2081 (n\u2081+1) rfl begin\n      haveI : mono (S.f.f (n\u2081+1)) :=\n        (short_complex.short_exact.map_of_exact ex (homological_complex.eval _ _ (n\u2081+1))).mono_f,\n      simp only [\u2190 cancel_mono (S.f.f (n\u2081+1)), assoc, \u2190 S.f.comm, reassoc_of hx\u2081,\n        homological_complex.d_comp_d, comp_zero, zero_comp],\n    end \u226b S.X\u2081.homology_\u03c0 n\u2081 :=\nbegin\n  haveI : mono (S.f.f (n\u2081+1)) :=\n    (short_complex.short_exact.map_of_exact ex (homological_complex.eval _ _ (n\u2081+1))).mono_f,\n  have hx\u2081' : x\u2081 \u226b S.X\u2081.d n\u2081 (n\u2081 + 1) = 0,\n  { simp only [\u2190 cancel_mono (S.f.f (n\u2081+1)), assoc, \u2190 S.f.comm, reassoc_of hx\u2081,\n      homological_complex.d_comp_d, comp_zero, zero_comp], },\n  let z : A \u27f6 (mapping_cone S.f).cycles n\u2080 :=\n    (mapping_cone S.f).lift_cycles\n      (-x\u2081 \u226b (mapping_cone.inl S.f).v n\u2081 n\u2080 (by linarith)\n        + x\u2082 \u226b (mapping_cone.inr S.f).f n\u2080) n\u2081 h.symm\n      (by simp only [mapping_cone.to_ext_iff _ _ _ rfl,\n        assoc, preadditive.add_comp, zero_comp, preadditive.comp_neg,\n        preadditive.neg_comp, neg_neg, mapping_cone.inl_fst_assoc, comp_id,\n        mapping_cone.d_fst _ _ _ _ h, hx\u2081', zero_add, mapping_cone.inr_fst_assoc,\n        comp_zero, neg_zero, mapping_cone.d_snd _ _ _ h, preadditive.comp_add,\n        mapping_cone.inr_snd_assoc, mapping_cone.inl_snd_assoc, hx\u2081, add_zero, neg_add_self,\n        eq_self_iff_true, and_self]),\n  have hz\u2081 : x\u2081 = -z \u226b homological_complex.cycles_i (mapping_cone S.f) n\u2080 \u226b\n    (mapping_cone.fst S.f : cochain (mapping_cone S.f) S.X\u2081 1).v n\u2080 n\u2081 h,\n  { simp only [add_zero, neg_neg, preadditive.add_comp, preadditive.neg_comp, assoc,\n      comp_zero, homological_complex.lift_cycles_i_assoc, mapping_cone.inl_fst,\n      comp_id, mapping_cone.inr_fst], },\n  have hz\u2082 : z \u226b homological_complex.cycles_map (from_mapping_cone_of_ses ex) n\u2080 =\n    homological_complex.lift_cycles S.X\u2083 x\u2083 n\u2081 h.symm hx\u2083,\n  { rw \u2190 cancel_mono (S.X\u2083.cycles_i n\u2080),\n    simp only [zero_add, neg_zero, assoc, preadditive.add_comp, preadditive.neg_comp,\n      comp_zero, inl_from_mapping_cone_of_ses, inr_from_mapping_cone_of_ses,\n      homological_complex.lift_cycles_comp_cycles_map, homological_complex.lift_cycles_i, hx\u2082], },\n  simp only [hz\u2081, reassoc_of hz\u2082, \u2190 mapping_cone.homology_map_\u03b4_on_a_cocycle S.f z n\u2081 h,\n    \u2190 from_mapping_cone_of_ses_comp_\u03b4 ex n\u2080 n\u2081 h,\n    homological_complex.homology_\u03c0_naturality_assoc],\nend\n\n/- Actually, we probably need a more general homology sequence for any complex_shape, which\ncould be done using the snake lemma, and then `comp_\u03b4_eq` would be used to prove\na comparison lemma between the two connecting maps... -/\n\nend homology_sequence\n\nend cochain_complex\n", "meta": {"author": "joelriou", "repo": "homotopical_algebra", "sha": "697f49d6744b09c5ef463cfd3e35932bdf2c78a3", "save_path": "github-repos/lean/joelriou-homotopical_algebra", "path": "github-repos/lean/joelriou-homotopical_algebra/homotopical_algebra-697f49d6744b09c5ef463cfd3e35932bdf2c78a3/src/for_mathlib/algebra/homology/homology_sequence.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982315512489, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.35166587481325373}}
{"text": "/-\nCopyright (c) 2021 Adam Topaz. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Adam Topaz\n-/\nimport category_theory.limits.creates\nimport category_theory.sites.sheafification\n\n/-!\n\n# Limits and colimits of sheaves\n\n## Limits\n\nWe prove that the forgetful functor from `Sheaf J D` to presheaves creates limits.\nIf the target category `D` has limits (of a certain shape),\nthis then implies that `Sheaf J D` has limits of the same shape and that the forgetful\nfunctor preserves these limits.\n\n## Colimits\n\nGiven a diagram `F : K \u2964 Sheaf J D` of sheaves, and a colimit cocone on the level of presheaves,\nwe show that the cocone obtained by sheafifying the cocone point is a colimit cocone of sheaves.\n\nThis allows us to show that `Sheaf J D` has colimits (of a certain shape) as soon as `D` does.\n\n-/\nnamespace category_theory\nnamespace Sheaf\n\nopen category_theory.limits\nopen opposite\n\nsection limits\n\nuniverses w v u z\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\nvariables {K : Type z} [small_category K]\n\nnoncomputable theory\n\nsection\n\n\n/-- An auxiliary definition to be used below.\n\nWhenever `E` is a cone of shape `K` of sheaves, and `S` is the multifork associated to a\ncovering `W` of an object `X`, with respect to the cone point `E.X`, this provides a cone of\nshape `K` of objects in `D`, with cone point `S.X`.\n\nSee `is_limit_multifork_of_is_limit` for more on how this definition is used.\n-/\ndef multifork_evaluation_cone (F : K \u2964 Sheaf J D)\n  (E : cone (F \u22d9 Sheaf_to_presheaf J D)) (X : C) (W : J.cover X) (S : multifork (W.index E.X)) :\n  cone (F \u22d9 Sheaf_to_presheaf J D \u22d9 (evaluation C\u1d52\u1d56 D).obj (op X)) :=\n{ X := S.X,\n  \u03c0 :=\n  { app := \u03bb k, (presheaf.is_limit_of_is_sheaf J (F.obj k).1 W (F.obj k).2).lift $\n      multifork.of_\u03b9 _ S.X (\u03bb i, S.\u03b9 i \u226b (E.\u03c0.app k).app (op i.Y)) begin\n        intros i,\n        simp only [category.assoc],\n        erw [\u2190 (E.\u03c0.app k).naturality, \u2190 (E.\u03c0.app k).naturality],\n        dsimp,\n        simp only [\u2190 category.assoc],\n        congr' 1,\n        apply S.condition,\n      end,\n    naturality' := begin\n      intros i j f,\n      dsimp [presheaf.is_limit_of_is_sheaf],\n      rw [category.id_comp],\n      apply presheaf.is_sheaf.hom_ext (F.obj j).2 W,\n      intros ii,\n      rw [presheaf.is_sheaf.amalgamate_map, category.assoc, \u2190 (F.map f).val.naturality,\n        \u2190 category.assoc, presheaf.is_sheaf.amalgamate_map],\n      dsimp [multifork.of_\u03b9],\n      erw [category.assoc, \u2190 E.w f],\n      tidy,\n    end } }\n\n\nvariables [has_limits_of_shape K D]\n\n/-- If `E` is a cone of shape `K` of sheaves, which is a limit on the level of presheves,\nthis definition shows that the limit presheaf satisfies the multifork variant of the sheaf\ncondition, at a given covering `W`.\n\nThis is used below in `is_sheaf_of_is_limit` to show that the limit presheaf is indeed a sheaf.\n-/\ndef is_limit_multifork_of_is_limit (F : K \u2964 Sheaf J D)\n  (E : cone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_limit E) (X : C) (W : J.cover X) : is_limit (W.multifork E.X) :=\nmultifork.is_limit.mk _\n(\u03bb S, (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).lift $\n  multifork_evaluation_cone F E X W S)\nbegin\n  intros S i,\n  apply (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op i.Y)) hE).hom_ext,\n  intros k,\n  dsimp [multifork.of_\u03b9],\n  erw [category.assoc, (E.\u03c0.app k).naturality],\n  dsimp,\n  rw \u2190 category.assoc,\n  erw (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac\n    (multifork_evaluation_cone F E X W S),\n  dsimp [multifork_evaluation_cone, presheaf.is_limit_of_is_sheaf],\n  erw presheaf.is_sheaf.amalgamate_map,\n  refl,\nend\nbegin\n  intros S m hm,\n  apply (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).hom_ext,\n  intros k,\n  dsimp,\n  erw (is_limit_of_preserves ((evaluation C\u1d52\u1d56 D).obj (op X)) hE).fac,\n  apply presheaf.is_sheaf.hom_ext (F.obj k).2 W,\n  intros i,\n  erw presheaf.is_sheaf.amalgamate_map,\n  dsimp [multifork.of_\u03b9],\n  change _ = S.\u03b9 i \u226b _,\n  erw [\u2190 hm, category.assoc, \u2190 (E.\u03c0.app k).naturality, category.assoc],\n  refl,\nend\n\n/-- If `E` is a cone which is a limit on the level of presheaves,\nthen the limit presheaf is again a sheaf.\n\nThis is used to show that the forgetful functor from sheaves to presheaves creates limits.\n-/\nlemma is_sheaf_of_is_limit (F : K \u2964 Sheaf J D) (E : cone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_limit E) : presheaf.is_sheaf J E.X :=\nbegin\n  rw presheaf.is_sheaf_iff_multifork,\n  intros X S,\n  exact \u27e8is_limit_multifork_of_is_limit _ _ hE _ _\u27e9,\nend\n\ninstance (F : K \u2964 Sheaf J D) : creates_limit F (Sheaf_to_presheaf J D) :=\ncreates_limit_of_reflects_iso $ \u03bb E hE,\n{ lifted_cone := \u27e8\u27e8E.X, is_sheaf_of_is_limit _ _ hE\u27e9,\n    \u27e8\u03bb t, \u27e8E.\u03c0.app _\u27e9, \u03bb u v e, Sheaf.hom.ext _ _ $ E.\u03c0.naturality _\u27e9\u27e9,\n  valid_lift := cones.ext (eq_to_iso rfl) $ \u03bb j, by { dsimp, simp },\n  makes_limit :=\n  { lift := \u03bb S, \u27e8hE.lift ((Sheaf_to_presheaf J D).map_cone S)\u27e9,\n    fac' := \u03bb S j, by { ext1, apply hE.fac ((Sheaf_to_presheaf J D).map_cone S) j },\n    uniq' := \u03bb S m hm, begin\n      ext1,\n      exact hE.uniq ((Sheaf_to_presheaf J D).map_cone S) m.val (\u03bb j, congr_arg hom.val (hm j)),\n    end } }\n\ninstance : creates_limits_of_shape K (Sheaf_to_presheaf J D) := {}\n\ninstance : has_limits_of_shape K (Sheaf J D) :=\nhas_limits_of_shape_of_has_limits_of_shape_creates_limits_of_shape (Sheaf_to_presheaf J D)\n\nend\n\ninstance [has_limits D] : creates_limits (Sheaf_to_presheaf J D) := \u27e8\u27e9\n\ninstance [has_limits D] : has_limits (Sheaf J D) :=\nhas_limits_of_has_limits_creates_limits (Sheaf_to_presheaf J D)\n\nend limits\n\nsection colimits\n\nuniverses w v u\nvariables {C : Type (max v u)} [category.{v} C] {J : grothendieck_topology C}\nvariables {D : Type w} [category.{max v u} D]\nvariables {K : Type (max v u)} [small_category K]\n-- Now we need a handful of instances to obtain sheafification...\nvariables [concrete_category.{max v u} D]\nvariables [\u2200 (P : C\u1d52\u1d56 \u2964 D) (X : C) (S : J.cover X), has_multiequalizer (S.index P)]\nvariables [preserves_limits (forget D)]\nvariables [\u2200 (X : C), has_colimits_of_shape (J.cover X)\u1d52\u1d56 D]\nvariables [\u2200 (X : C), preserves_colimits_of_shape (J.cover X)\u1d52\u1d56 (forget D)]\nvariables [reflects_isomorphisms (forget D)]\n\n/-- Construct a cocone by sheafifying a cocone point of a cocone `E` of presheaves\nover a functor which factors through sheaves.\nIn `is_colimit_sheafify_cocone`, we show that this is a colimit cocone when `E` is a colimit. -/\n@[simps]\ndef sheafify_cocone {F : K \u2964 Sheaf J D} (E : cocone (F \u22d9 Sheaf_to_presheaf J D)) : cocone F :=\n{ X := \u27e8J.sheafify E.X, grothendieck_topology.plus.is_sheaf_plus_plus _ _\u27e9,\n  \u03b9 :=\n  { app := \u03bb k, \u27e8E.\u03b9.app k \u226b J.to_sheafify E.X\u27e9,\n    naturality' := \u03bb i j f, by { ext1, dsimp, erw [category.comp_id, \u2190 category.assoc, E.w f] } } }\n\n/-- If `E` is a colimit cocone of presheaves, over a diagram factoring through sheaves,\nthen `sheafify_cocone E` is a colimit cocone. -/\n@[simps]\ndef is_colimit_sheafify_cocone {F : K \u2964 Sheaf J D} (E : cocone (F \u22d9 Sheaf_to_presheaf J D))\n  (hE : is_colimit E) :\n  is_colimit (sheafify_cocone E) :=\n{ desc := \u03bb S, \u27e8J.sheafify_lift (hE.desc ((Sheaf_to_presheaf J D).map_cocone S)) S.X.2\u27e9,\n  fac' := begin\n    intros S j,\n    ext1,\n    dsimp [sheafify_cocone],\n    erw [category.assoc, J.to_sheafify_sheafify_lift, hE.fac],\n    refl,\n  end,\n  uniq' := begin\n    intros S m hm,\n    ext1,\n    apply J.sheafify_lift_unique,\n    apply hE.uniq ((Sheaf_to_presheaf J D).map_cocone S),\n    intros j,\n    dsimp,\n    simpa only [\u2190 category.assoc, \u2190 hm],\n  end }\n\ninstance [has_colimits_of_shape K D] : has_colimits_of_shape K (Sheaf J D) :=\n\u27e8\u03bb F, has_colimit.mk \u27e8sheafify_cocone (colimit.cocone _),\n  is_colimit_sheafify_cocone _ (colimit.is_colimit _)\u27e9\u27e9\n\ninstance [has_colimits D] : has_colimits (Sheaf J D) := \u27e8infer_instance\u27e9\n\nend colimits\n\nend Sheaf\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/sites/limits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3516658674307807}}
{"text": "class Refl.{U} {\u03b1 : Type U} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : Prop :=\n  refl (a : \u03b1) : R a a\n\nclass Symm.{U} {\u03b1 : Type U} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : Prop :=\n  symm {a b : \u03b1} : R a b \u2192 R b a\n\n/--\n  An example decl modifier (a doc comment).\n-/\nclass abbrev PEquiv.{U} {\u03b1 : Type U} (R : \u03b1 \u2192 \u03b1 \u2192 Prop) : Prop \n  := Refl R, Symm R\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/run/classAbbrev.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.35162015590159057}}
{"text": "import Mt.Thread\n\nnamespace Mt\n\nvariable {spec : Spec}\nlocal instance : IsReservation spec.Reservation :=spec.is_reservation\n\n/-- Describes a system of zero or more threads running in parallel\n\n  Systems can be iterated one atomic step at a time by choosing\n  one of its threads. They keep track of the number of threads\n  which panicked during execution\n-/\nstructure System (spec : Spec) where\n  state   : spec.State\n  threads : List (Thread spec)\n  panics  : Nat\n\nnamespace System\n\ndef ThreadIndex (s : System spec) : Type :=Fin s.threads.length\ndef done (s : System spec) : Bool :=s.threads.length = 0\n\ndef iterate (s : System spec) : s.ThreadIndex -> System spec\n  | thread_idx =>\n    if (s.threads.get thread_idx).block_until s.state then \n      match (s.threads.get thread_idx).iterate s.state with\n        | Thread.IterationResult.Done state =>\n          {\n            state\n            threads := s.threads.eraseIdx thread_idx.val\n            panics := s.panics\n          }\n        | Thread.IterationResult.Panic state =>\n          {\n            state\n            threads := s.threads.eraseIdx thread_idx.val\n            panics := s.panics + 1\n          }\n        | Thread.IterationResult.Running state thread =>\n          {\n            state\n            threads := s.threads.set thread_idx.val thread\n            panics := s.panics\n          }\n    else\n      s\n\ntheorem iterate_threads (s : System spec) (thread_idx : s.ThreadIndex)\n  (blocked_until : (s.threads.get thread_idx).block_until s.state) :\n  (s.iterate thread_idx).threads =\n    match (s.threads.get thread_idx).iterate s.state with\n      | Thread.IterationResult.Done .. => s.threads.eraseIdx thread_idx.val\n      | Thread.IterationResult.Panic .. => s.threads.eraseIdx thread_idx.val\n      | Thread.IterationResult.Running _ thread => s.threads.set thread_idx.val thread :=by\n  rw [iterate]\n  simp only [blocked_until, ite_true]\n  cases h : Thread.iterate (List.get s.threads thread_idx) s.state <;> rfl\n\ntheorem iterate_panics (s : System spec) (thread_idx : s.ThreadIndex)\n  (blocked_until : (s.threads.get thread_idx).block_until s.state) :\n  (s.iterate thread_idx).panics =\n    match (s.threads.get thread_idx).iterate s.state with\n      | Thread.IterationResult.Done .. => s.panics\n      | Thread.IterationResult.Panic .. => s.panics + 1\n      | Thread.IterationResult.Running .. => s.panics :=by\n  rw [iterate]\n  simp only [blocked_until, ite_true]\n  cases h : Thread.iterate (List.get s.threads thread_idx) s.state <;> rfl\n\ninductive reduces_to : System spec -> System spec -> Prop where\n| single {a b} (idx : a.ThreadIndex) (iteration : a.iterate idx = b) : reduces_to a b\n| trans {a b c} (a_to_b : a.reduces_to b) (b_to_c : b.reduces_to c) : reduces_to a c\n\ndef reduces_to_or_eq (a b : System spec) : Prop :=a = b \u2228 a.reduces_to b\n\ntheorem reduces_to_or_eq.refl (a : System spec) : a.reduces_to_or_eq a :=Or.inl rfl\ntheorem reduces_to_or_eq.trans {a b c : System spec} :\n  a.reduces_to_or_eq b \u2192 b.reduces_to_or_eq c \u2192 a.reduces_to_or_eq c :=by\n  intro ab bc\n  cases ab <;> cases bc <;> rename_i h\u2081 h\u2082\n  . rw [h\u2081, h\u2082] ; exact Or.inl rfl\n  . rw [h\u2081] ; exact Or.inr h\u2082\n  . rw [h\u2082.symm] ; exact Or.inr h\u2081\n  . exact Or.inr <| h\u2081.trans h\u2082\n\n\nend System\n\nend Mt\n", "meta": {"author": "mirkootter", "repo": "lean-mt", "sha": "027a16555d487e46a0a00611b8039655378dfdd5", "save_path": "github-repos/lean/mirkootter-lean-mt", "path": "github-repos/lean/mirkootter-lean-mt/lean-mt-027a16555d487e46a0a00611b8039655378dfdd5/Mt/System/Basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.35162015590159057}}
{"text": "import category_theory.triangulated.pretriangulated\nimport for_mathlib.homological_complex_shift\nimport for_mathlib.monoidal_category\nimport for_mathlib.int\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.preadditive\nopen category_theory.limits\n\nuniverses v u\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.associator_def (a b c : discrete \u2124) :\n  \u03b1_ a b c = eq_to_iso (add_assoc a b c) := rfl\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.left_unitor_def (a : discrete \u2124) :\n  \u03bb_ a = eq_to_iso (zero_add _) := rfl\n\n-- Move + generalize!\n@[simp]\nlemma category_theory.discrete.right_unitor_def (a : discrete \u2124) :\n  \u03c1_ a = eq_to_iso (add_zero _) := rfl\n\nnamespace category_theory.triangulated\nopen category_theory.category\n\nvariables (C : Type u) [category.{v} C] [preadditive C]\nvariables [has_shift C \u2124]\n\nlocal attribute [instance, reducible] endofunctor_monoidal_category\n\nnamespace triangle\n\n@[simps]\ndef triangle_shift_obj (T : triangle C) (i : \u2124) : triangle C :=\ntriangle.mk C\n  (i.neg_one_pow \u2022 ((shift_functor _ i).map T.mor\u2081))\n  (i.neg_one_pow \u2022 (((shift_functor _ i).map T.mor\u2082)))\n  (i.neg_one_pow \u2022 ((shift_functor C i).map T.mor\u2083 \u226b (shift_comm _ _ _).hom))\n\n@[simps]\ndef triangle_shift_map {T\u2081 T\u2082 : triangle C} (f : T\u2081 \u27f6 T\u2082) (i : \u2124) :\n  triangle_shift_obj C T\u2081 i \u27f6 triangle_shift_obj C T\u2082 i :=\n{ hom\u2081 := (shift_functor _ i).map f.hom\u2081,\n  hom\u2082 := (shift_functor _ i).map f.hom\u2082,\n  hom\u2083 := (shift_functor _ i).map f.hom\u2083,\n  comm\u2081' := by { dsimp, simp only [functor.map_zsmul,\n    preadditive.zsmul_comp, preadditive.comp_zsmul, \u2190 functor.map_comp, f.comm\u2081] },\n  comm\u2082' := by { dsimp, simp only [functor.map_zsmul,\n    preadditive.zsmul_comp, preadditive.comp_zsmul, \u2190 functor.map_comp, f.comm\u2082] },\n  comm\u2083' := begin\n    dsimp,\n    simp only [functor.map_zsmul,\n      preadditive.zsmul_comp, preadditive.comp_zsmul],\n    congr' 1,\n    simp only [ shift_comm_hom_comp, assoc, iso.cancel_iso_hom_right_assoc,\n      \u2190 functor.map_comp, f.comm\u2083],\n  end }\n\n@[simps]\ndef triangle_shift_functor (i : \u2124) : triangle C \u2964 triangle C :=\n{ obj := \u03bb T, triangle_shift_obj C T i,\n  map := \u03bb T\u2081 T\u2082 f, triangle_shift_map C f _,\n  map_id' := begin\n    intros T,\n    ext,\n    all_goals { dsimp, simp },\n  end,\n  map_comp' := begin\n    intros T\u2081 T\u2082 T\u2083 f g,\n    ext,\n    all_goals { dsimp, simp },\n  end, } .\n\nvariable {C}\n\n@[simps]\ndef iso.of_components {T\u2081 T\u2082 : triangle C}\n  (e\u2081 : T\u2081.obj\u2081 \u2245 T\u2082.obj\u2081)\n  (e\u2082 : T\u2081.obj\u2082 \u2245 T\u2082.obj\u2082)\n  (e\u2083 : T\u2081.obj\u2083 \u2245 T\u2082.obj\u2083) (h\u2081 h\u2082 h\u2083) : T\u2081 \u2245 T\u2082 :=\n{ hom :=\n  { hom\u2081 := e\u2081.hom,\n    hom\u2082 := e\u2082.hom,\n    hom\u2083 := e\u2083.hom,\n    comm\u2081' := h\u2081,\n    comm\u2082' := h\u2082,\n    comm\u2083' := h\u2083 },\n  inv :=\n  { hom\u2081 := e\u2081.inv,\n    hom\u2082 := e\u2082.inv,\n    hom\u2083 := e\u2083.inv,\n    comm\u2081' := by rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, h\u2081],\n    comm\u2082' := by rw [iso.comp_inv_eq, category.assoc, iso.eq_inv_comp, h\u2082],\n    comm\u2083' := by rw [\u2190 functor.map_iso_inv, iso.comp_inv_eq, category.assoc, iso.eq_inv_comp,\n      functor.map_iso_hom, h\u2083], },\n  hom_inv_id' := by ext; dsimp; simp,\n  inv_hom_id' := by ext; dsimp; simp }\n.\n\nvariable (C)\n\n@[simps]\ndef triangle_shift_functor_\u03b5 : \ud835\udfed (triangulated.triangle C) \u2245 triangle_shift_functor C 0 :=\nnat_iso.of_components (\u03bb T,\n  iso.of_components\n    (shift_zero _ _).symm\n    (shift_zero _ _).symm\n    (shift_zero _ _).symm\n    begin\n      convert ((shift_functor_zero _ _).inv.naturality _),\n      dsimp only [triangle_shift_functor, triangle_shift_obj],\n      simpa,\n    end\n    begin\n      convert ((shift_functor_zero _ _).inv.naturality _),\n      dsimp only [triangle_shift_functor, triangle_shift_obj],\n      simpa,\n    end\n    begin\n      dsimp,\n      rw one_smul,\n      rw \u2190 nat_trans.naturality_assoc, dsimp [shift_comm],\n      simp only [obj_\u03b5_app, discrete.functor_map_id, nat_trans.id_app, \u03b5_app_obj, assoc, id_comp],\n      rw [\u2190 nat_trans.comp_app, \u2190 nat_trans.comp_app],\n      erw [monoidal_functor.\u03bc_inv_hom_id_assoc, id_comp], refl,\n    end)\n  begin\n    intros T\u2081 T\u2082 f, ext;\n    { dsimp only [triangle_morphism.comp_hom\u2081, iso.of_components_hom_hom\u2081, triangle_shift_map_hom\u2081,\n        triangle_morphism.comp_hom\u2082, iso.of_components_hom_hom\u2082, triangle_shift_map_hom\u2082,\n        triangle_morphism.comp_hom\u2083, iso.of_components_hom_hom\u2083, triangle_shift_map_hom\u2083,\n        functor.id_map, triangle_category_comp, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.\u03b5_iso_hom, triangle_shift_functor_map],\n      rw \u2190 nat_trans.naturality _ _, refl },\n  end\n.\n\nvariables [\u2200 (i : \u2124), (shift_functor C i).additive]\n\n@[reassoc]\nlemma shift_comm_eq_eq_to_hom (X : C) (i j : \u2124) :\n  (shift_add X i j).hom \u226b (shift_comm X i j).hom \u226b (shift_add X j i).inv =\n  eq_to_hom (by { congr' 2, exact add_comm i j}) :=\nbegin\n  dsimp [shift_add, shift_comm, opaque_eq_to_iso],\n  simp, erw comp_id,\nend\n\n@[reassoc]\nlemma shift_add_comp_eq_to_hom (X : C) (i j k : \u2124) (h : i + j = j + i) :\n  (shift_add X (i+j) k).hom \u226b\n  eq_to_hom (by { congr' 3}) \u226b\n  (shift_add X (j+i) k).inv =\n  eq_to_hom (by { congr' 3}) :=\nbegin\n  dsimp [shift_add, opaque_eq_to_iso],\n  induction h, dsimp, simpa,\nend\n\n\n@[reassoc]\nlemma shift_add_comp_eq_to_hom' (X : C) (i j k : \u2124) (h : j + k = k + j) :\n  (shift_add X i (j+k)).hom \u226b\n  eq_to_hom (by { congr' 3}) \u226b\n  (shift_add X i (k+j)).inv =\n  eq_to_hom (by { congr' 3}) :=\nbegin\n  dsimp [shift_add, opaque_eq_to_iso],\n  induction h, dsimp, simpa,\nend\n\n\nlemma triangle_shift_functor_\u03bc_aux (X : C) (i j : \u2124) :\n  (shift_functor C j).map (shift_comm X 1 i).hom \u226b\n    (shift_comm ((shift_functor C i).obj X) 1 j).hom \u226b\n      (shift_functor C 1).map\n        (shift_add X i j).inv =\n  (shift_add ((shift_functor C 1).obj X) i j).inv \u226b\n    (shift_comm X 1 (i + j)).hom :=\nbegin\n  dsimp [shift_add, shift_comm, opaque_eq_to_iso],\n  simp only [eq_to_hom_map, eq_to_hom_app, functor.map_comp, obj_\u03bc_app,\n    category_theory.discrete.associator_def, eq_to_iso.inv,\n    obj_\u03bc_inv_app, eq_to_iso.hom, assoc, \u03bc_inv_hom_app_assoc],\n  congr' 2,\n  simp only [\u2190 assoc], congr' 1, simp only [assoc],\n  erw shift_add_comp_eq_to_hom_assoc,\n  erw shift_add_comp_eq_to_hom'_assoc,\n  simpa, exact add_comm _ _, exact add_comm _ _,\nend\n\n@[simps]\ndef triangle_shift_functor_\u03bc (i j : \u2124) :\n  triangle_shift_functor C i \u22d9 triangle_shift_functor C j \u2245\n    triangle_shift_functor C (i + j) :=\nnat_iso.of_components (\u03bb T,\n  iso.of_components\n    (shift_add _ _ _).symm\n    (shift_add _ _ _).symm\n    (shift_add _ _ _).symm\n    (begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.\u03bc_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor\u2081),\n      rw [functor.comp_map] at this,\n      erw [this, mul_comm], refl,\n    end)\n    (begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.\u03bc_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor\u2082),\n      rw [functor.comp_map] at this,\n      erw [this, mul_comm], refl,\n    end)\n    begin\n      dsimp [triangle_shift_functor, triangle_shift_obj],\n      simp only [zsmul_comp, comp_zsmul, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.\u03bc_iso_hom, functor.map_zsmul, smul_smul, int.neg_one_pow_add,\n        mul_comm j.neg_one_pow],\n      have := ((shift_functor_add _ i j).inv.naturality T.mor\u2083),\n      dsimp [functor.comp_map] at this,\n      erw [\u2190 reassoc_of this], clear this,\n      simp only [functor.map_comp, assoc, obj_\u03bc_app],\n      congr' 2,\n      --have := (shift_monoidal_functor C \u2124).to_lax_monoidal_functor.associativity i j 1,\n      rw (shift_monoidal_functor C \u2124).map_associator_inv,\n      dsimp,\n      simp only [assoc, is_iso.inv_id, nat_trans.hcomp_app, comp_id, id_comp],\n      slice_lhs 4 5\n      { rw [\u2190 nat_trans.comp_app, is_iso.hom_inv_id,\n          nat_trans.id_app] },\n      erw id_comp,\n      rw category_theory.nat_iso.is_iso_inv_app,\n      --rw nat_trans.id_hcomp_app,\n      dsimp,\n      simp only [category_theory.functor.map_id, comp_id, assoc, is_iso.hom_inv_id_assoc],\n      slice_lhs 4 5\n      { rw [\u2190 nat_trans.comp_app, (shift_monoidal_functor C \u2124).\u03bc_hom_inv_id,\n          nat_trans.id_app] },\n      erw comp_id, apply triangle_shift_functor_\u03bc_aux,\n    end)\n  (begin\n    intros T\u2081 T\u2082 f, ext;\n    { dsimp only [triangle_morphism.comp_hom\u2081, iso.of_components_hom_hom\u2081, triangle_shift_map_hom\u2081,\n        triangle_morphism.comp_hom\u2082, iso.of_components_hom_hom\u2082, triangle_shift_map_hom\u2082,\n        triangle_morphism.comp_hom\u2083, iso.of_components_hom_hom\u2083, triangle_shift_map_hom\u2083,\n        functor.id_map, triangle_category_comp, iso.symm_hom, iso.app_inv, iso.symm_inv,\n        monoidal_functor.\u03b5_iso_hom, triangle_shift_functor_map],\n      rw \u2190 nat_trans.naturality _ _, refl },\n  end)\n.\n\n/-\ndef triangle_shift_core : shift_mk_core (triangle C) \u2124 :=\n{ F := triangle_shift_functor _,\n  \u03b5 := triangle_shift_functor_\u03b5 _,\n  \u03bc := \u03bb i j, triangle_shift_functor_\u03bc _ _ _,\n  associativity := begin\n    intros i j k T, ext,\n    { have := (shift_monoidal_functor C \u2124).to_lax_monoidal_functor.associativity i j k,\n      apply_fun (\u03bb \u03b1, \u03b1.app T.obj\u2081) at this,\n      simp only [nat_trans.comp_app, obj_\u03bc_app, assoc, \u03bc_inv_hom_app_assoc, map_inv_hom_app,\n        comp_id, functor.associator_hom_app, nat_trans.hcomp_app, nat_trans.id_app,\n        category_theory.functor.map_id, id_comp] at this,\n      erw [id_comp] at this,\n      refine eq.trans _ this, clear this,\n      dsimp, simp only [obj_\u03bc_app, assoc],\n      -- I don't like that `(eq_to_hom _).hom\u2081`.\n      admit },\n    admit,\n    admit\n  end,\n  left_unitality := admit,\n  right_unitality := admit }\n-/\n\n@[simps]\ndef map_triangle_shift_functor (m n : discrete \u2124) (f : m \u27f6 n) :\n  triangle_shift_functor C m \u27f6 triangle_shift_functor C n :=\n{ app := \u03bb T,\n  { hom\u2081 := eq_to_hom $ by rw discrete.eq_of_hom f,\n    hom\u2082 := eq_to_hom $ by rw discrete.eq_of_hom f,\n    hom\u2083 := eq_to_hom $ by rw discrete.eq_of_hom f,\n    comm\u2081' := by { rcases f with \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9, simp only [eq_to_hom_refl, id_comp, comp_id], },\n    comm\u2082' := by { rcases f with \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9, simp only [eq_to_hom_refl, id_comp, comp_id], },\n    comm\u2083' := by { rcases f with \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9,\n      dsimp, rw (shift_functor C (1 : \u2124)).map_id, simp only [comp_id, id_comp]} },\n  naturality' := begin\n    rcases f with \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9,\n    rintros X Y g, ext;\n    { dsimp, simp only [eq_to_hom_refl, id_comp, comp_id] },\n  end } .\n\n\n\nlemma associativity_aux (X : C) (a b c : discrete \u2124) :\n(\ud835\udfd9 ((shift_functor C c).obj ((shift_functor C b).obj ((shift_functor C a).obj X))) \u226b\n  (shift_functor C c).map (((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc a b).app X)) \u226b\n  ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc (a \u2297 b) c).app X \u226b\n  eq_to_hom (by { congr' 2, apply add_assoc }) =\n  \ud835\udfd9 ((shift_functor C c).obj\n  ((shift_functor C b).obj ((shift_functor C a).obj X))) \u226b (((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc b c).app\n  ((shift_functor C a).obj X) \u226b (shift_functor C (b + c)).map\n  (\ud835\udfd9 ((shift_functor C a).obj X))) \u226b ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc a\n  (b \u2297 c)).app X :=\nbegin\n  have := (shift_monoidal_functor C \u2124).associativity' a b c,\n  apply_fun (\u03bb e, e.app X) at this,\n  dsimp at this \u22a2,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this \u22a2,\n  erw comp_id,\n  exact this\nend\n\nlemma left_unitality_aux (X : C) (a : discrete \u2124) : \ud835\udfd9 ((shift_functor C a).obj X) =\n  (\ud835\udfd9 ((shift_functor C a).obj X) \u226b (shift_functor C a).map\n    ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03b5.app X)) \u226b\n    ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc\n    (\ud835\udfd9_ (discrete \u2124)) a).app X \u226b eq_to_hom (by { congr, exact zero_add a }) :=\nbegin\n  have := (shift_monoidal_functor C \u2124).left_unitality' a,\n  apply_fun (\u03bb e, e.app X) at this,\n  dsimp at this \u22a2,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this \u22a2,\n  exact this\nend\n\nlemma right_unitality_aux (X : C) (a : discrete \u2124) : \ud835\udfd9 ((shift_functor C a).obj X) =\n  ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03b5.app ((shift_functor C a).obj X) \u226b\n       (shift_functor C (\ud835\udfd9_ (discrete \u2124))).map (\ud835\udfd9 ((shift_functor C a).obj X))) \u226b\n    ((shift_monoidal_functor C \u2124).to_lax_monoidal_functor.\u03bc a (\ud835\udfd9_ (discrete \u2124))).app X \u226b\n    eq_to_hom (by { congr, apply add_zero }) :=\nbegin\n  have := (shift_monoidal_functor C \u2124).right_unitality' a,\n  apply_fun (\u03bb e, e.app X) at this,\n  dsimp at this \u22a2,\n  simp only [id_comp, comp_id, category_theory.functor.map_id,\n    eq_to_hom_map, eq_to_hom_app] at this \u22a2,\n  erw comp_id,\n  exact this\nend\n\ninstance has_shift : has_shift (triangle C) \u2124 := has_shift.mk $\n{ obj := triangle_shift_functor _,\n  map := \u03bb m n f, map_triangle_shift_functor _ _ _ f,\n  map_id' := \u03bb X, by { ext; refl },\n  map_comp' := \u03bb X Y Z f g, by { ext; simp },\n  \u03b5 := (triangle_shift_functor_\u03b5 _).hom,\n  \u03bc := \u03bb m n, (triangle_shift_functor_\u03bc _ m n).hom,\n  \u03bc_natural' := begin\n    rintros m m' n n' \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9 \u27e8\u27e8\u27e8\u27e9\u27e9\u27e9, ext;\n    { dsimp, simp only [id_comp, comp_id, category_theory.functor.map_id] },\n  end,\n  associativity' := \u03bb a b c, by ext; apply associativity_aux,\n  left_unitality' := \u03bb a, by ext; apply left_unitality_aux,\n  right_unitality' := \u03bb a, by ext; apply right_unitality_aux,\n  \u03b5_is_iso := infer_instance,\n  \u03bc_is_iso := infer_instance } .\n\n@[simp]\nlemma shift_obj\u2081 (T : triangle C) (i : \u2124) : T\u27e6i\u27e7.obj\u2081 = T.obj\u2081\u27e6i\u27e7 := rfl\n\n@[simp]\nlemma shift_obj\u2082 (T : triangle C) (i : \u2124) : T\u27e6i\u27e7.obj\u2082 = T.obj\u2082\u27e6i\u27e7 := rfl\n\n@[simp]\nlemma shift_obj\u2083 (T : triangle C) (i : \u2124) : T\u27e6i\u27e7.obj\u2083 = T.obj\u2083\u27e6i\u27e7 := rfl\n\n@[simp]\nlemma shift_mor\u2081 (T : triangle C) (i : \u2124) : T\u27e6i\u27e7.mor\u2081 = i.neg_one_pow \u2022 T.mor\u2081\u27e6i\u27e7' := rfl\n\n@[simp]\nlemma shift_mor\u2082 (T : triangle C) (i : \u2124) : T\u27e6i\u27e7.mor\u2082 = i.neg_one_pow \u2022 T.mor\u2082\u27e6i\u27e7' := rfl\n\n@[simp]\nlemma shift_mor\u2083 (T : triangle C) (i : \u2124) :\n  T\u27e6i\u27e7.mor\u2083 = i.neg_one_pow \u2022 (T.mor\u2083\u27e6i\u27e7' \u226b (shift_comm _ _ _).hom) := rfl\n\n@[simp]\nlemma shift_hom\u2081 {T\u2081 T\u2082 : triangle C} (f : T\u2081 \u27f6 T\u2082) (i : \u2124) : f\u27e6i\u27e7'.hom\u2081 = f.hom\u2081\u27e6i\u27e7' := rfl\n\n@[simp]\nlemma shift_hom\u2082 {T\u2081 T\u2082 : triangle C} (f : T\u2081 \u27f6 T\u2082) (i : \u2124) : f\u27e6i\u27e7'.hom\u2082 = f.hom\u2082\u27e6i\u27e7' := rfl\n\n@[simp]\nlemma shift_hom\u2083 {T\u2081 T\u2082 : triangle C} (f : T\u2081 \u27f6 T\u2082) (i : \u2124) : f\u27e6i\u27e7'.hom\u2083 = f.hom\u2083\u27e6i\u27e7' := rfl\n\nend triangle\n\n/-\ninstance {C : Type*} [category C] [preadditive C] (X Y : C) : has_neg (X \u2245 Y) :=\n\u27e8\u03bb f,\n{ hom := -f.hom,\n  inv := -f.inv,\n  hom_inv_id' := by simp only [comp_neg, neg_comp, iso.hom_inv_id, neg_neg],\n  inv_hom_id' := by simp only [comp_neg, neg_comp, iso.inv_hom_id, neg_neg] }\u27e9\n\n@[simp] lemma _root_.category_theory.neg_hom\n   {C : Type*} [category C] [preadditive C] {X Y : C} (f : X \u2245 Y) :\n   (-f).hom = -(f.hom) := rfl\n\n@[simp] lemma _root_.category_theory.neg_inv\n   {C : Type*} [category C] [preadditive C] {X Y : C} (f : X \u2245 Y) :\n   (-f).inv = -(f.inv) := rfl\n-/\n\nnamespace pretriangulated\n\n@[simp] lemma shift_comm_self (X : C) (i : \u2124) : shift_comm X i i = iso.refl _ :=\nbegin\n  ext,\n  dsimp [shift_comm, opaque_eq_to_iso],\n  simp only [discrete.functor_map_id, nat_trans.id_app, id_comp, \u03bc_hom_inv_app],\n  refl,\nend\n\nvariables [has_zero_object C] [\u2200 (i : \u2124), (shift_functor C i).additive] [pretriangulated C]\n\nlemma dist_triangle_iff_of_iso {T\u2081 T\u2082 : triangle C} (e : T\u2081 \u2245 T\u2082) :\n  (T\u2081 \u2208 dist_triang C) \u2194 (T\u2082 \u2208 dist_triang C) :=\n\u27e8\u03bb h, isomorphic_distinguished _ h _ e.symm, \u03bb h, isomorphic_distinguished _ h _ e\u27e9\n\nlemma dist_triangle_rot_iff (T : triangle C) :\n  (T.rotate \u2208 dist_triang C) \u2194 (T \u2208 dist_triang C) :=\nbegin\n  refine \u27e8\u03bb h, _, \u03bb h, rot_of_dist_triangle _ _ h\u27e9,\n  let e : T \u2245 T.rotate.inv_rotate := rot_comp_inv_rot.app T,\n  rw \u2190 dist_triangle_iff_of_iso _ e.symm,\n  exact inv_rot_of_dist_triangle _ _ h,\nend\n\nlemma shift_of_dist_triangle (T : triangle C) (hT : T \u2208 dist_triang C) (i : \u2124) :\n  T\u27e6i\u27e7 \u2208 dist_triang C :=\nbegin\n  induction i using int.induction_on_iff with i,\n  { exact isomorphic_distinguished T hT _ (shift_zero _ _), },\n  { suffices : T\u27e6(i+1 : \u2124)\u27e7 \u2245 T\u27e6(i:\u2124)\u27e7.rotate.rotate.rotate,\n    { dsimp,\n      rw dist_triangle_iff_of_iso _ this,\n      iterate 3 { rw dist_triangle_rot_iff }, },\n    refine shift_add _ _ _ \u226a\u226b _,\n    refine triangle.iso.of_components (iso.refl _) (iso.refl _) (iso.refl _) _ _ _,\n    { dsimp, simp only [category.id_comp, category.comp_id, comp_neg, neg_one_smul], },\n    { dsimp, simp only [category.id_comp, category.comp_id, neg_comp, neg_one_smul], },\n    { dsimp, simp only [category.id_comp, category.comp_id, neg_comp, neg_one_smul],\n      simp only [functor.map_comp, assoc, category_theory.functor.map_id, comp_id,\n        functor.map_zsmul, preadditive.zsmul_comp, preadditive.comp_zsmul,\n        shift_comm_self, iso.refl_hom], }, },\nend\n\nend pretriangulated\n\nend category_theory.triangulated\n", "meta": {"author": "bentoner", "repo": "debug", "sha": "b8a75381caa90aa9942c20e08a44e45d0ae60d18", "save_path": "github-repos/lean/bentoner-debug", "path": "github-repos/lean/bentoner-debug/debug-b8a75381caa90aa9942c20e08a44e45d0ae60d18/src/for_mathlib/triangle_shift.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.35161964318750566}}
{"text": "def foo (n : Nat) : Nat :=\n  if n = 0 then 0 else\n    let x := n - 1\n    have := match () with | _ => trivial\n    foo x\ntermination_by _ n => n\ndecreasing_by sorry\n\ntheorem ex : foo 0 = 0 := by\n  unfold foo\n  sorry\n\n#check foo._unfold\n", "meta": {"author": "leanprover", "repo": "lean4", "sha": "742d053a97bdd109a41a921facd1cd6a55e89bc7", "save_path": "github-repos/lean/leanprover-lean4", "path": "github-repos/lean/leanprover-lean4/lean4-742d053a97bdd109a41a921facd1cd6a55e89bc7/tests/lean/1026.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.35131458542822136}}
{"text": "import convex convex_body linalg measure touching_cone brunn_minkowski\n  microid criticality pruning locally_linear arithmetic\n  touching_cone_polytope\n  analysis.convex.basic\n  data.multiset.basic\n  measure_theory.measure.measure_space\n  topology.basic\n  analysis.inner_product_space.pi_L2\n\nopen_locale pointwise\nopen_locale ennreal -- for \u221e notation\nopen_locale topological_space -- for \ud835\udcdd notation\n\n-- needed to get decidable_eq for sets!\nopen classical\nlocal attribute [instance] prop_decidable\n\nsection definitions\n\nvariables (V : Type)\n[inner_product_space \u211d V] [finite_dimensional \u211d V]\n\nend definitions\n\nsection preparation_lemmas\n\nvariables {V : Type} [inner_product_space \u211d V] [finite_dimensional \u211d V]\n\nnoncomputable def TS_microid_measure {k : \u2115} (u : metric.sphere (0 : V) 1) (\u03bc : microid_measure V k) :\nsubmodule \u211d V :=\nTS (microid_of_measure \u03bc).val u.val\nend preparation_lemmas\n\n\nsection default_reduction\n\nvariables {V : Type} [inner_product_space \u211d V] [finite_dimensional \u211d V]\n\ndef microid_pair (k : \u2115)\n(u : metric.sphere (0 : V) 1) :=\n(\u03a3 \u03bc : microid_measure V k,\n{P : microid_generator_space V k // is_default_polytope \u03bc u P})\n\ndef pair_to_default_body {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : convex_body V :=\nconvex_body_of_polytope (polytope_of_microid_generator pair.2.1)\n\ndef pair_to_measure {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair: microid_pair k u) : microid_measure V k := pair.1\n\nnoncomputable def pair_to_microid {k : \u2115}\n{u : metric.sphere (0 : V) 1} :\nmicroid_pair k u \u2192 convex_body V :=\nmicroid_of_measure \u2218 pair_to_measure\n\nnoncomputable def pair_to_default_measure {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : microid_measure V k :=\ndirac_microid_measure pair.2.1\n\nnoncomputable def pair_to_TS {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : submodule \u211d V :=\nTS (pair_to_microid pair).val u\n\nnoncomputable def reduce_pair {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : microid_pair k u :=\nbegin\n  refine \u27e8pair_to_default_measure pair, pair.2.1, _\u27e9,\n  rcases pair.2.2 with \u27e8h1, h2, h3\u27e9,\n  refine \u27e8h1, h2, _\u27e9,\n  {\n    simp only [pair_to_default_measure, microid_of_dirac_eq],\n    simp only [body_of_poly_of_gen_eq],\n    refine \u27e8h3.1, _\u27e9,\n    intros C hC,\n    exact id,\n  },\nend\n\nlemma reduced_microid_eq_default_body {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) :\n(pair_to_microid (reduce_pair pair)) = pair_to_default_body pair :=\nbegin\n  simp only [pair_to_microid, pair_to_TS, pair_to_measure, reduce_pair,\n  pair_to_default_measure, microid_of_dirac_eq, pair_to_default_body,\n  body_of_poly_of_gen_eq,\n  function.comp_app, function.comp_app],\nend\n\nlemma nface_eq_self_of_vspan_mem_uperp {A : set V} {u : V}\n(h : vector_span \u211d A \u2264 vector_orth u) : normal_face A u = A :=\nbegin\n  apply subset_antisymm,\n  {\n    apply normal_face_subset,\n  },\n  {\n    intros a ha,\n    simp only [mem_normal_face],\n    refine \u27e8ha, _\u27e9,\n    intros b hb,\n    have hab : a - b \u2208 vector_orth u,\n    {\n      apply h,\n      apply submodule.subset_span,\n      exact \u27e8a, b, ha, hb, rfl\u27e9,\n    },\n    simp only [vector_orth] at hab,\n    replace hab := inner_left_of_mem_orthogonal_singleton _ hab,\n    apply le_of_eq,\n    symmetry,\n    simpa only [inner_sub_left, sub_eq_zero] using hab,\n  },\nend\n\nlemma span_reduced_microid_eq_TS (k : \u2115)\n(u : metric.sphere (0 : V) 1) :\nspan_of_convex_body \u2218 pair_to_microid \u2218 (reduce_pair : microid_pair k u \u2192 microid_pair k u) =\npair_to_TS \u2218 (reduce_pair : microid_pair k u \u2192 microid_pair k u) :=\nbegin\n  funext pair,\n  simp only [function.comp_app],\n  simp only [reduced_microid_eq_default_body, pair_to_default_body, pair_to_TS],\n  simp only [convex_body_of_polytope],\n  have is_poly := (polytope_of_microid_generator pair.2.1).property,\n  rw [TS_poly_eq_vspan_face is_poly],\n  rcases pair.2.2 with \u27e8h0, h1, h2, h3\u27e9,\n  rw [\u2190subtype.val_eq_coe, nface_eq_self_of_vspan_mem_uperp h1],\n  simp only [span_of_convex_body],\nend\n\nlemma subset_diff {A : set V} (h : (0 : V) \u2208 A) : A \u2286 diff A :=\nbegin\n  intros v vA,\n  refine \u27e8v, 0, vA, h, _\u27e9,\n  simp only [vsub_eq_sub, sub_zero],\nend\n\nlemma vspan_eq_span_of_mem_zero {A : set V} (h : (0 : V) \u2208 A) :\nvector_span \u211d A = submodule.span \u211d A :=\nbegin\n  apply le_antisymm,\n  {\n    simp only [vector_span, submodule.span_le],\n    rintro x \u27e8a, b, ha, hb, rfl\u27e9,\n    simp only [vsub_eq_sub, set_like.mem_coe],\n    refine submodule.sub_mem _ _ _,\n    all_goals {\n      apply submodule.subset_span,\n      assumption,\n    },\n  },\n  {\n    simp only [vector_span],\n    apply submodule.span_mono,\n    exact subset_diff h,\n  },\nend\n\nlemma reduced_microid_subset_TS {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) :\n(pair_to_microid (reduce_pair pair)).val \u2286 pair_to_TS (reduce_pair pair) :=\nbegin\n  refine subset_trans submodule.subset_span\n    (subset_of_eq (congr_arg coe _)),\n  revert pair,\n  have := function.funext_iff.mp (span_reduced_microid_eq_TS k u),\n  convert this,\n  ext,\n  refine forall_congr _,\n  simp only [function.comp_app, span_of_convex_body],\n  intro pair,\n  rw [reduced_microid_eq_default_body, pair_to_default_body,\n    convex_body_of_polytope, subtype.val_eq_coe, subtype.coe_mk],\n  rw [vspan_eq_span_of_mem_zero (pair.2.2.1)],\nend\n\nnoncomputable def pair_to_default_space {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : submodule \u211d V :=\nTS (pair_to_default_body pair).val u.val\n\nlemma TS_reduce_eq_default_space {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) :\npair_to_TS (reduce_pair pair) = pair_to_default_space pair :=\nbegin\n  simp only [pair_to_TS, pair_to_default_space, reduced_microid_eq_default_body],\n  refl,\nend\n\nlemma reduced_default_body_eq {k : \u2115}{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) :\npair_to_default_body (reduce_pair pair) = pair_to_default_body pair :=\nbegin\n  simp only [pair_to_default_body, reduce_pair],\nend\n\nlemma default_space_eq_TS_of_reduced {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) :\npair_to_default_space (reduce_pair pair) = pair_to_TS (reduce_pair pair) :=\nbegin\n  simp only [pair_to_default_space, TS_reduce_eq_default_space, reduced_default_body_eq],\nend\n\nnoncomputable def pair_to_space_pair {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(pair : microid_pair k u) : submodule \u211d V \u00d7 submodule \u211d V :=\n\u27e8pair_to_TS pair, pair_to_default_space pair\u27e9\n\nlemma pair_to_space_pair_def {k : \u2115}\n{u : metric.sphere (0 : V) 1} :\n@pair_to_space_pair V _ _ k u = \u03bb pair, \u27e8pair_to_TS pair, pair_to_default_space pair\u27e9 :=\nrfl\n\nnoncomputable def paircol_span {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(D : multiset (microid_pair k u)) : submodule \u211d V :=\n(D.map pair_to_default_space).sum\n\nlemma nonempty_prune_triple {k : \u2115} : nonempty (prune_triple V k) :=\nbegin\n  refine \u27e8\u27e8\u27e80, _\u27e9, 0, 0\u27e9\u27e9,\n  simp only [mem_closed_ball_zero_iff, norm_zero, zero_le_one],\nend\n\nlemma cuspiness_lipschitz {k : \u2115}\n(tr : prune_triple V k)\n(u v : V) :\n|cuspiness' u tr - cuspiness' v tr| \u2264 dist u v :=\nbegin\n  simp only [cuspiness', \u2190sub_div, \u2190inner_sub_right, abs_div],\n  simp only [\u2190is_R_or_C.abs_to_real],\n  refine le_trans (div_le_div_of_le_right\n    (abs_inner_le_norm (prune_direction tr) (u - v)) _) _,\n  {\n    apply is_R_or_C.abs_nonneg,\n  },\n  {\n    simp only [is_R_or_C.abs_to_real,\n      abs_eq_self.mpr (norm_nonneg _)],\n    apply div_le_of_nonneg_of_le_mul,\n    {\n      apply norm_nonneg,\n    },\n    {\n      apply dist_nonneg,\n    },\n    {\n      conv {to_lhs, rw [mul_comm]},\n      apply mul_le_mul_of_nonneg_right _ (norm_nonneg _),\n      simp only [\u2190dist_eq_norm, subtype.dist_eq],\n    },\n  },\nend\n\nlemma support_locally_linear_of_cuspy {k : \u2115}\n{G : microid_generator_space V k}\n{u : metric.sphere (0 : V ) 1}\n{\u03b5 : \u211d}\n(\u03b5pos : \u03b5 > 0)\n(h : \u2200 tr : prune_triple V k,\nvalid_prune_triple tr u \u2192 (prune_triple_generator tr) = G \u2192 cuspiness u tr \u2265 \u03b5) :\nsupp_locally_linear (metric.ball u.val \u03b5) (body_of_microid_generator G) :=\nbegin\n  rw [normal_face_singleton_iff_locally_linear metric.is_open_ball],\n  rcases generator_face_nonempty G u with \u27e8m, hm\u27e9,\n  refine \u27e8G.val m, _\u27e9,\n  intros v hv,\n  simp only [body_of_microid_generator, convex_body.normal_face],\n  simp only [normal_face_spanned_by_verts],\n  convert convex_hull_singleton (G.val m),\n  have lem : \u2200 l : fin k.succ, G.val l \u2260 G.val m \u2192\n    \u27eaG.val m - G.val l, v\u27eb_\u211d > 0,\n  {\n    intros l hh,\n    let tr : prune_triple V k := \u27e8G, m, l\u27e9,\n    have valid : valid_prune_triple tr u,\n    {\n      simp only [valid_prune_triple,\n      prune_cusp, prune_cusp_index,\n      prune_secondary, prune_secondary_index,\n      prune_gen_fn, tr],\n      split,\n      {\n        simp only [generator_face, finset.mem_coe, finset.mem_filter] at hm,\n        exact hm.2,\n      },\n      {tauto},\n    },\n    have := h tr valid rfl,\n    change cuspiness' u.val tr \u2265 \u03b5 at this,\n    replace : cuspiness' v tr > 0,\n    {\n      refine lt_of_lt_of_le _ (ge_sub_abs (cuspiness' v tr) (cuspiness' u.val tr)),\n      refine lt_of_lt_of_le _ (sub_le_sub this (cuspiness_lipschitz _ _ _)),\n      simpa only [metric.mem_ball, sub_pos] using hv,\n    },\n    {\n      simp only [cuspiness', prune_direction,\n        prune_cusp, prune_secondary,\n        prune_cusp_index, prune_secondary_index,\n        prune_gen_fn, tr] at this,\n      exact dividend_pos this (norm_nonneg _),\n    },\n  },\n  apply subset_antisymm,\n  {\n    intros x hx,\n    simp only [set.mem_singleton_iff],\n    simp only [generator_face, finset.mem_coe, finset.mem_filter] at hm,\n    simp only [mem_normal_face] at hx,\n    rcases hx.1 with \u27e8l, hl, rfl\u27e9,\n    by_contra hh,\n    suffices c : \u27eaG.val m - G.val l, v\u27eb_\u211d > 0,\n    {\n      simp only [inner_sub_left, gt_iff_lt, sub_pos] at c,\n      replace hx := hx.2 (G.val m) (set.mem_range_self _),\n      linarith,\n    },\n    {\n      exact lem l hh,\n    },\n  },\n  {\n    simp only [set.singleton_subset_iff, mem_normal_face],\n    refine \u27e8set.mem_range_self _, _\u27e9,\n    intros y hy,\n    rcases hy with \u27e8l, rfl\u27e9,\n    by_cases hh : G.val l = G.val m,\n    {\n      simp only [hh],\n      apply le_refl,\n    },\n    {\n      apply le_of_lt,\n      rw [\u2190sub_pos, \u2190inner_sub_left],\n      exact lem l hh,\n    },\n  },\nend\n\n/- lemma lemma_vspan_subset_span {A : set V} :\n(vector_span \u211d A : set V) \u2286 submodule.span \u211d A :=\nbegin\n  simp only [vector_span],\nend -/\n\nlemma zero_normal_face_eq (A : set V) :\nnormal_face A 0 = A :=\nbegin\n  ext,\n  simp only [normal_face, set.mem_set_of],\n  split,\n  {\n    intro h,\n    exact h.1,\n  },\n  {\n    intro h,\n    refine \u27e8h, _\u27e9,\n    simp only [inner_zero_right],\n    intros y hy,\n    apply le_refl,\n  }\nend\n\nlemma orthogonal_projection_eq_zero_iff (E : submodule \u211d V) :\nE\u15ee = 0 \u2194 E = \u22a4 :=\nbegin\n  exact submodule.orthogonal_eq_bot_iff,\nend\n\n/- lemma normal_face_by_supp (K : convex_body V) (u : V) :\n(normal_face K.val u) = { x : V | x \u2208 K.val \u2227 \u27eax, u\u27eb_\u211d = K.supp u } := sorry\n -/\n\nlemma TS_zero_of_cuspy_generators {k : \u2115}\n{\u03bc : microid_measure V k}\n{u : metric.sphere (0 : V) 1}\n{\u03b5 : \u211d}\n(\u03b5pos : \u03b5 > 0)\n(h : \u2200 (tr : prune_triple V k),\nvalid_prune_triple tr u \u2192 prune_triple_generator tr \u2208 msupport \u03bc \u2192 cuspiness u tr \u2265 \u03b5) :\nTS_microid_measure u \u03bc = 0 :=\nbegin\n  suffices h\u03b5 : metric.ball u.val \u03b5 \u2286 pre_touching_cone (microid_of_measure \u03bc).val u,\n  {\n    have span_top : vector_span \u211d (pre_touching_cone (microid_of_measure \u03bc).val u) = \u22a4,\n    {\n      refine vector_span_top_of_ball_subset \u03b5pos h\u03b5,\n    },\n    have : pre_touching_cone (microid_of_measure \u03bc).val u =\n      touching_cone (microid_of_measure \u03bc).val u,\n    {\n      refine touching_cone_unique_face _ _ _ _ _,\n      {\n        conv {\n          congr, skip,\n          rw [\u2190zero_normal_face_eq (pre_touching_cone (microid_of_measure \u03bc).val u)],\n        },\n        exact normal_face_is_face (pre_touching_cone_convex _ _) 0,\n      },\n      rw [relint_eq_int (vector_span_top_of_ball_subset \u03b5pos h\u03b5), mem_interior],\n      exact \u27e8metric.ball u.val \u03b5, h\u03b5, metric.is_open_ball, metric.mem_ball_self \u03b5pos\u27e9,\n    },\n    simp only [TS_microid_measure, TS, orthogonal_projection_eq_zero_iff],\n    rw [this] at h\u03b5,\n    refine span_top_of_ball_subset \u03b5pos h\u03b5,\n  },\n  intros v hv,\n  simp only [pre_touching_cone, outer_normal_cone, set.mem_set_of],\n  suffices ll : supp_locally_linear (metric.ball u.val \u03b5)\n    (microid_of_measure \u03bc),\n  {\n    rcases ll with \u27e8x, ll\u27e9,\n    rw [normal_face_singleton_iff_locally_linear_with (metric.is_open_ball)] at ll,\n    simp only [convex_body.normal_face] at ll,\n    rw [ll v hv, ll \u2191u (metric.mem_ball_self \u03b5pos)],\n  },\n  {\n    apply microid_supp_locally_linear_of_generators \u03b5pos,\n    intros G hG,\n    apply support_locally_linear_of_cuspy \u03b5pos,\n    intros tr valid htrG,\n    refine h tr valid _,\n    rw [htrG],\n    exact hG,\n  },\nend\n\nlemma cuspiness_nonneg {k : \u2115}\n{tr : prune_triple V k}\n{u : metric.sphere (0 : V) 1}\n(h : valid_prune_triple tr u) : cuspiness u tr \u2265 0 :=\nbegin\n  simp only [valid_prune_triple] at h,\n  simp only [cuspiness, prune_direction,\n    prune_cusp, prune_cusp_index,\n    prune_secondary, prune_secondary_index,\n    prune_gen_fn] at h \u22a2,\n  apply div_nonneg,\n  {\n    simp only [inner_sub_left, sub_nonneg],\n    tauto,\n  },\n  {apply norm_nonneg},\nend\n\nlemma exists_prune_triple_seq {k : \u2115}\n{\u03bc : microid_measure V k}\n{u : metric.sphere (0 : V) 1}\n(h : TS_microid_measure u \u03bc \u2260 0) :\n\u2203 t : \u2115 \u2192 prune_triple V k,\n(\u2200 n : \u2115, valid_prune_triple (t n) u) \u2227\n(\u2200 n : \u2115, prune_triple_generator (t n) \u2208 msupport \u03bc) \u2227\nfilter.tendsto ((cuspiness u) \u2218 t) filter.at_top (\ud835\udcdd (0 : \u211d)) :=\nbegin\n  suffices hex : \u2200 \u03b5 : \u211d,\n  \u2203 tr : prune_triple V k, \u03b5 > 0 \u2192\n  valid_prune_triple tr u \u2227\n  prune_triple_generator tr \u2208 msupport \u03bc \u2227 cuspiness u tr < \u03b5,\n  {\n    choose t\u211d ht\u211d using hex,\n    rcases exists_seq_strict_anti_tendsto (0 : \u211d)\n      with \u27e8\u03b5, -, \u03b5pos, \u03b5tt\u27e9,\n    let t := t\u211d \u2218 \u03b5,\n    refine \u27e8t, _, _, _\u27e9,\n    {\n      intro n,\n      exact (ht\u211d (\u03b5 n) (\u03b5pos n)).1,\n    },\n    {\n      intro n,\n      exact (ht\u211d (\u03b5 n) (\u03b5pos n)).2.1,\n    },\n    {\n      have \u03b5tt' := tendsto_nhds_within_of_tendsto_nhds_of_eventually_within\n        _ \u03b5tt (filter.eventually_of_forall \u03b5pos),\n      have : filter.tendsto (cuspiness u \u2218 t\u211d) (\ud835\udcdd[preorder.lt 0] 0) (\ud835\udcdd 0),\n      {\n        simp only [filter.tendsto_iff_eventually],\n        intros p ep,\n        simp only [filter.has_basis.eventually_iff metric.nhds_basis_ball] at ep,\n        simp only [filter.has_basis.eventually_iff metric.nhds_within_basis_ball,\n          function.comp_app],\n        rcases ep with \u27e8i, ipos, hi\u27e9,\n        refine \u27e8i, ipos, _\u27e9,\n        rintro x \u27e8hx, xpos\u27e9,\n        apply hi,\n        simp only [real.ball_eq_Ioo] at hx \u22a2,\n        split,\n        {\n          simp only [zero_sub],\n          refine lt_of_lt_of_le (neg_neg_of_pos ipos) _,\n          apply cuspiness_nonneg,\n          exact (ht\u211d x xpos).1,\n        },\n        {\n          exact lt_trans (ht\u211d x xpos).2.2 hx.2,\n        },\n      },\n      have := filter.tendsto.comp this \u03b5tt',\n      rw [function.comp.assoc] at this,\n      exact this,\n    },\n  },\n  intros \u03b5,\n  by_cases \u03b5pos : \u03b5 > 0, rotate,\n  {\n    rcases nonempty_prune_triple with \u27e8tr\u27e9,\n    refine \u27e8tr, _\u27e9,\n    {\n      intro h,\n      contradiction,\n    },\n  },\n  {\n    by_contra hass,\n    push_neg at hass,\n    apply h,\n    apply TS_zero_of_cuspy_generators \u03b5pos,\n    intros tr valid htr,\n    exact (hass tr).2 valid htr,\n  },\nend\n\nlemma exists_pair {k : \u2115}\n{\u03bc : microid_measure V k}\n{u : metric.sphere (0 : V) 1}\n(h : TS_microid_measure u \u03bc \u2260 0) :\n\u2203 p : microid_pair k u,\npair_to_measure p = \u03bc :=\nbegin\n  rcases exists_prune_triple_seq h with \u27e8t, valid, gen\u03bc, tt\u27e9,\n  rcases pruning_lemma valid tt with \u27e8G, hzm, hnz, hup, hcl\u27e9,\n  refine \u27e8\u27e8\u03bc, G, _\u27e9, _\u27e9,\n  {\n    refine \u27e8hzm, hup, _, _\u27e9,\n    {\n      rw [TS_poly_eq_vspan_face (polytope_of_microid_generator G).property u],\n      rw [\u2190subtype.val_eq_coe, nface_eq_self_of_vspan_mem_uperp hup],\n      exact hnz,\n    },\n    {\n      simp only [in_combinatorial_closure] at hcl,\n      intros C hC hsupp,\n      rw [body_of_poly_of_gen_eq] at hsupp,\n      have := hcl C hC hsupp,\n      rw [msupport_microid_eq_closure \u03bc hC],\n      refine set.mem_of_subset_of_mem _ this,\n      apply closure_mono,\n      intros v hv,\n      simp only [set.mem_Union] at hv \u22a2,\n      rcases hv with \u27e8K, \u27e8nK, -, rfl\u27e9, vsupp\u27e9,\n      refine \u27e8prune_triple_generator (t nK), gen\u03bc nK, _\u27e9,\n      simp only [function.comp_app] at vsupp,\n      exact vsupp,\n    },\n  },\n  {\n    simp only [pair_to_measure],\n  },\nend\n\nlemma nonzero_of_mem_of_semicritical\n{Es : multiset (submodule \u211d V)}\n(h : semicritical_spaces Es) :\n\u2200 E : submodule \u211d V, E \u2208 Es \u2192 E \u2260 \u22a5 :=\nbegin\n  intros E hE,\n  let S : multiset (submodule \u211d V) := {E},\n  suffices hd : dim S.sum \u2265 S.card,\n  {\n    by_contra,\n    simp only [h, S] at hd,\n    rw [multiset.sum_singleton E, multiset.card_singleton] at hd,\n    rw [h] at hd,\n    change finite_dimensional.finrank \u211d (\u22a5 : submodule \u211d V) \u2265 1 at hd,\n    simp only [finrank_bot, gt_iff_lt, not_lt_zero'] at hd,\n    linarith,\n  },\n  refine h S _,\n  simp only [S, multiset.singleton_le],\n  exact hE,\nend\n\nlemma exists_pair_multiset {k : \u2115}\n(\u03bcs : multiset (microid_measure V k))\n(u : metric.sphere (0 : V) 1)\n(hTS : semicritical_spaces (\u03bcs.map (TS_microid_measure u))) :\n\u2203 C : multiset (microid_pair k u),\nC.map pair_to_measure = \u03bcs :=\nbegin\n  induction \u03bcs using pauls_multiset_induction,\n  {\n    refine \u27e80, _\u27e9,\n    simp only [multiset.map_zero],\n  },\n  {\n    have hTS': semicritical_spaces (\u03bcs_C'.map (TS_microid_measure u)),\n    {\n      simp only [multiset.map_cons] at hTS,\n      refine semicritical_of_le (multiset.le_cons_self _ _) hTS,\n    },\n    rcases \u03bcs_\u1fb0 hTS' with \u27e8C', hC'\u27e9,\n    have \u03bcs_a_nonzero : TS_microid_measure u \u03bcs_a \u2260 \u22a5,\n    {\n      refine nonzero_of_mem_of_semicritical hTS _ _,\n      simp only [multiset.map_cons],\n      exact multiset.mem_cons_self _ _,\n    },\n    rcases exists_pair \u03bcs_a_nonzero with \u27e8E, hE\u27e9,\n    refine \u27e8E ::\u2098 C', _\u27e9,\n    simp only [hC', hE, multiset.map_cons],\n  },\nend\n\ntheorem matryoshka_reduction {k : \u2115}\n{u : metric.sphere (0 : V) 1}\n{D C : multiset (microid_pair k u)}\n(hdim : dim V = (D + C).card + 1) :\nu \u2208 msupport (bm.area ((D.map pair_to_default_body) + C.map pair_to_microid)) \u2192\nu \u2208 msupport (bm.area ((D + C).map pair_to_microid))\n:=\nbegin\n  revert C,\n  induction D using pauls_multiset_induction,\n  {\n    intros C hdim hu,\n    simpa only [multiset.map_zero, zero_add] using hu,\n  },\n  {\n    intros C hdim hu,\n    let C' := (reduce_pair D_a) ::\u2098 C,\n    have hdim' : dim V = (D_C' + C').card + 1,\n    {\n      simp only [C', multiset.card_add, multiset.card_cons] at hdim \u22a2,\n      rw [hdim],\n      ring,\n    },\n    have goal' := D_\u1fb0 hdim',\n    simp only [multiset.map_add, multiset.map_cons] at goal',\n    rw [multiset.add_cons, \u2190multiset.cons_add] at goal',\n    rw [reduced_microid_eq_default_body] at goal',\n    simp only [multiset.map_add, multiset.map_cons] at hu,\n    replace hu := goal' hu,\n    clear D_\u1fb0 goal',\n    simp only [multiset.cons_add, multiset.map_cons],\n    rw [multiset.add_cons, \u2190multiset.map_add] at hu,\n    have defp := D_a.2.2,\n    refine defp.2.2.2 _ _ hu,\n    simp only [multiset.card_map, multiset.card_add],\n    simp only [multiset.card_add, multiset.card_cons] at hdim,\n    rw [hdim],\n    ring,\n  },\nend\n\nend default_reduction\n\nvariables {V : Type}\n[inner_product_space \u211d V] [finite_dimensional \u211d V]\n\nlemma TS_microid_proj_eq_proj_TS_microid {k : \u2115}\n(\u03bc : microid_measure V k)\n(E : submodule \u211d V)\n(u : metric.sphere (0 : V) 1)\n(uE : u.val \u2208 E) :\nTS_microid_measure (uncoe_sph E u uE) (project_microid_measure E \u03bc) =\n(TS_microid_measure u \u03bc).map (proj E) :=\nbegin\n  simp only [TS_microid_measure, uncoe_sph],\n  rw [TS_orthogonal_projection _ uE, microid_proj_eq_proj_microid],\n  simp only [proj_body, subtype.val_eq_coe],\n  refl,\nend\n\nlemma TS_default_body_eq_TS_default_measure (k : \u2115)\n(u : metric.sphere (0 : V) 1) (x : microid_pair k u) :\nTS (pair_to_default_body x).val u.val = TS_microid_measure u (pair_to_default_measure x) :=\nbegin\n  simp only [TS_microid_measure, pair_to_default_measure, pair_to_default_body,\n    microid_of_dirac_eq, body_of_poly_of_gen_eq],\nend\n\nsection blabb\n\nlemma u_mem_sum_TS_orth\n{k : \u2115}\n{u : metric.sphere (0 : V) 1}\n{E : submodule \u211d V}\n(A : multiset (microid_pair k u))\n(h : E = (A.map pair_to_TS).sum) :\nu.val \u2208 E\u15ee :=\nbegin\n  revert E,\n  induction A using pauls_multiset_induction,\n  {\n    intros E h,\n    simp only [multiset.map_zero, multiset.sum_zero, submodule.zero_eq_bot] at h,\n    simp only [h, submodule.bot_orthogonal_eq_top],\n  },\n  {\n    intros E h,\n    simp only [multiset.map_cons, multiset.sum_cons, submodule.add_eq_sup] at h,\n    simp only [h],\n    intros x hx,\n    simp only [submodule.mem_sup] at hx,\n    rcases hx with \u27e8y, hy, z, hz, rfl\u27e9,\n    simp only [inner_add_left],\n    convert add_zero (0 : \u211d),\n    {\n      apply inner_left_of_mem_orthogonal_singleton,\n      simp only [pair_to_TS] at hy,\n      exact TS_le_uperp _ _ hy,\n    },\n    {\n      exact A_\u1fb0 rfl z hz,\n    },\n  },\nend\n\nlemma semicritical_subprojection\n{k : \u2115}\n{u : metric.sphere (0 : V) 1}\n(A B : multiset (microid_pair k u))\n(E : submodule \u211d V)\n(SP : multiset (microid_measure E\u15ee k))\n(hAE : E = (A.map pair_to_TS).sum)\n(hA2 : dim E = A.card)\n(hB : semicritical_spaces ((A + B).map pair_to_TS))\n(hSP : SP = B.map (project_microid_measure E\u15ee \u2218 pair_to_measure)):\nsemicritical_spaces\n(SP.map (TS_microid_measure (uncoe_sph E\u15ee u (u_mem_sum_TS_orth A hAE)))) :=\nbegin\n  simp only [hSP],\n  rw [multiset.map_map, map_lambda],\n  simp only [function.comp_app, TS_microid_measure,\n    microid_proj_eq_proj_microid, proj_body, uncoe_sph],\n  simp only [\u2190TS_orthogonal_projection],\n  let C := A.map pair_to_TS,\n  let D := B.map pair_to_TS,\n  suffices hD : semicritical_spaces (D.map (\u03bb W, W.map (proj E\u15ee))),\n  {\n    simpa only [D, multiset.map_map, map_lambda, function.comp_app] using hD,\n  },\n  simp only [multiset.map_add] at hB,\n  change semicritical_spaces (C + D) at hB,\n  refine semicritical_spaces_factorization _ _ _ hB,\n  refine \u27e8_, _\u27e9,\n  {\n    simpa only [C, multiset.card_map] using hA2,\n  },\n  {\n    intros F hF,\n    simp only [hAE],\n    apply le_sum_multiset_of_mem,\n    exact hF,\n  },\nend\n\nend blabb\n\n--set_option pp.implicit true\ntheorem matryoshka_principle {k : \u2115}\n(n\u2080 : \u2115)\n(n : \u2115)\n(hn : n \u2264 n\u2080)\n(\u03bcs : multiset (microid_measure V k))\n(u : metric.sphere (0 : V) 1)\n--(hdim1 : n \u2265 1)\n(hdim2 : dim V = n + 1)\n(hdim3 : \u03bcs.card = n)\n(hTS : semicritical_spaces (\u03bcs.map (TS_microid_measure u))) :\nu \u2208 msupport (bm.area (\u03bcs.map microid_of_measure)) :=\nbegin\n  unfreezingI {\n    induction n\u2080 with n\u2080 ih generalizing n \u03bcs V,\n    {\n      have : \u03bcs = 0,\n      {\n        rw [\u2190multiset.card_eq_zero, hdim3],\n        simpa only [le_zero_iff] using hn,\n      },\n      rw [this],\n      simp only [multiset.empty_eq_zero, multiset.map_zero, bm.area_empty],\n    },\n    {\n      by_cases hn' : n = 0,\n      {\n        have : \u03bcs = 0,\n        {\n          rw [\u2190multiset.card_eq_zero, hdim3, hn'],\n        },\n        rw [this],\n        simp only [multiset.map_zero, bm.area_empty],\n      },\n      rcases exists_pair_multiset \u03bcs u hTS with\n        \u27e8C, rfl\u27e9,\n      let D := C.map pair_to_space_pair,\n      let uperp := vector_orth u.val,\n      have :=\n      begin\n        clear ih,\n        refine semicritical_switching D uperp _ _ _ _ _,\n        {\n          simp only [D, multiset.card_map] at hdim3 \u22a2,\n          rw [hdim3],\n          exact nat.pos_of_ne_zero hn',\n        },\n        {\n          simpa only [multiset.map_map] using hTS,\n        },\n        {\n          have dim_uperp : dim uperp = n,\n          {\n            suffices h : dim uperp + 1 = dim V,\n            {\n              rw [hdim2] at h,\n              exact nat.add_right_cancel h,\n            },\n            {\n              have : dim (submodule.span \u211d ({u} : set V)) = 1,\n              {\n                apply finrank_span_singleton,\n                have := metric.mem_sphere.mp u.prop,\n                have z_ne_o: 1 \u2260 (0 : \u211d) := zero_ne_one.symm,\n                rw [\u2190this] at z_ne_o,\n                apply dist_ne_zero.mp z_ne_o,\n              },\n              rw [\u2190this],\n              rw [nat.add_comm],\n              refine dim_add_dim_orthogonal _,\n            },\n          },\n          symmetry,\n          simpa only [dim_uperp, multiset.card_map] using hdim3,\n        },\n        {\n          clear hn hdim2 hdim3 hTS,\n          intros x xD,\n          rcases multiset.mem_map.mp xD with \u27e8c, \u27e8cC, rfl\u27e9\u27e9,\n          simp only [pair_to_space_pair],\n          split,\n          {apply TS_le_uperp},\n          {\n            simp only [pair_to_default_space, pair_to_default_body, submodule.span_le],\n            apply TS_le_uperp,\n/-          simp only [polytope_to_convex_body],\n            have is_default_poly := c.snd.2,\n            exact is_default_poly.1, --simp only [is_default_polytope] at is_default_poly, -/\n          },\n        },\n        {\n          intros x xD,\n          rcases multiset.mem_map.mp xD with \u27e8c, \u27e8cC, rfl\u27e9\u27e9,\n          simp only [pair_to_space_pair, pair_to_default_space,\n            pair_to_default_body, convex_body_of_polytope],\n          have is_default_poly := c.snd.2,\n          exact is_default_poly.2.2.1,\n        },\n      end,\n      rcases this with \u27e8A, B, h\u27e9,\n      rcases h with \u27e8hAB, hBD, h1, h2, h3\u27e9,\n      rcases multiset_exists_of_le_map hBD with \u27e8B', \u27e8hB'C, rfl\u27e9\u27e9,\n      rcases multiset_exists_of_le_map hAB with \u27e8A', \u27e8hA'B', rfl\u27e9\u27e9,\n      /- let F := (B' - A').map pair_to_default_measure + (C - B').map pair_to_measure,\n      let G := A'.map pair_to_default_measure,\n      let E : submodule \u211d V := (A'.map pair_to_default_space).sum,\n      have uE : u.val \u2208 E\u15ee,\n      {\n        suffices h : E \u2264 uperp,\n        {\n          simp only [submodule.mem_orthogonal],\n          rintro x xE,\n          apply inner_left_of_mem_orthogonal_singleton,\n          exact h xE,\n        },\n        {\n          refine sum_multiset_le _,\n          rintro W hW,\n          simp only [pair_to_default_space, multiset.mem_map] at hW,\n          rcases hW with \u27e8a, hA, rfl\u27e9,\n          apply TS_le_uperp,\n        },\n      }, -/\n      /- let pF := F.map (project_microid_measure E\u15ee),\n      let n' := pF.card,\n      have hn' : n' \u2264 n\u2080,\n      {\n        simp only [n', pF, F, multiset.card_map, multiset.card_add\n          --, multiset.card_add, multiset.card_sub, hA'B', hB'C\n        ],\n        rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n        rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n        simp only [add_tsub_cancel_left],\n        simp only [multiset.card_map, multiset.card_add] at hdim3,\n        calc t.card + tt.card = n - A'.card : _\n      ...  \u2264 n - 1 : _\n      ...  \u2264 n\u2080 : _,\n        {\n          rw [\u2190hdim3],\n          admit,\n        },\n        {\n          apply nat.sub_le_sub_left n,\n          apply nat.succ_le_of_lt,\n          simpa only [multiset.card_map] using h3,\n        },\n        {\n          norm_num,\n          assumption,\n        },\n      },\n      have EA' : dim E = A'.card,\n      {\n        apply le_antisymm,\n        {admit},\n        {\n          simp only [multiset.map_map, pair_to_space_pair] at h1,\n          rw [\u2190multiset.card_map pair_to_default_space],\n          --simp only [pair_to_default_space] at E,\n          refine h1 _ _,\n          refine le_trans _ (multiset.le_add_right _ _),\n          simp only [function.comp_app],\n          exact multiset.map_le_map hA'B',\n        }\n      },\n      have hdim2' : dim E\u15ee = n' + 1,\n      {\n        apply le_antisymm,\n        {\n          suffices dimE : dim E \u2265 n - n',\n          {\n            simp only [ge_iff_le, tsub_le_iff_right, eq_tsub_of_add_eq hdim2.symm] at dimE,\n            rw [\u2190dim_add_dim_orthogonal E, nat.add_assoc] at dimE,\n            norm_num at dimE,\n            exact dimE,\n          },\n          {\n            rw [EA'],\n            norm_num,\n            simp only [hdim3.symm, multiset.card_map],\n            simp only [pF, F, n', multiset.card_map, multiset.card_add],\n\n            rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n            rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n            simp only [add_tsub_cancel_left, multiset.card_add],\n            simp only [add_assoc],\n          },\n        },\n        {\n          suffices dimE : dim E \u2264 n - n',\n          {\n            simp only [ge_iff_le, tsub_le_iff_right, eq_tsub_of_add_eq hdim2.symm] at dimE,\n            rw [\u2190dim_add_dim_orthogonal E, tsub_tsub] at dimE,\n            admit,\n          },\n          {\n            rw [multiset.map_map, pair_to_space_pair_def, multiset.card_map] at h2,\n            rw [h2],\n            rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n            rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n            simp only [add_tsub_cancel_left,\n                       multiset.card_map, n', pF, F,\n                       multiset.card_add,\n                       hdim3.symm],\n            generalizes [A'.card = A'c, t.card = tc, tt.card = ttc],\n            rw [nat.add_assoc],\n            simp only [add_tsub_cancel_right],\n          },\n        },\n      },\n      have hdim3' : pF.card = n' := rfl,\n      let u' := uncoe_sph E\u15ee u uE, -/\n      let A := A'.map reduce_pair,\n      let B := (B' - A').map reduce_pair + (C - B'),\n      let E := (A.map pair_to_TS).sum,\n      let SP := B.map (project_microid_measure E\u15ee \u2218 pair_to_measure),\n      have sc_AB : semicritical_spaces ((A + B).map pair_to_TS),\n      {\n        have hA : A.map pair_to_TS = A'.map pair_to_default_space,\n        {simp only [multiset.map_map, function.comp_app, TS_reduce_eq_default_space]},\n        have hB : B.map pair_to_TS = (B' - A').map pair_to_default_space + (C - B').map pair_to_TS,\n        {\n          simp only [B, multiset.map_map, multiset.map_add, function.comp_app,\n            TS_reduce_eq_default_space],\n        },\n        have hAB : (A + B).map pair_to_TS = B'.map pair_to_default_space + (C - B').map pair_to_TS,\n        {\n          rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n          rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n          simp only [A, B, add_tsub_cancel_left, multiset.map_map, multiset.map_add,\n            function.comp_app, TS_reduce_eq_default_space, add_assoc],\n        },\n        rw [hAB],\n        rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n        rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n        simp only [add_tsub_cancel_left] at hAB \u22a2,\n        simp only [D, pair_to_space_pair_def] at h1,\n        simpa only [multiset.map_add, multiset.map_map, add_tsub_cancel_left,\n          function.comp_app] using h1,\n      },\n      have dimE : dim E = A.card,\n      {\n        rw [multiset.card_map] at h2 \u22a2,\n        rw [\u2190h2],\n        suffices h : E = ((A'.map pair_to_space_pair).map prod.snd).sum,\n        {rw [h]},\n        {\n          simp only [multiset.map_map, E, pair_to_space_pair_def,\n            function.comp_app, TS_reduce_eq_default_space],\n        },\n      },\n      have sc_sp := semicritical_subprojection A B E SP rfl dimE sc_AB rfl,\n      have cardSP : A.card + SP.card = n,\n      {\n        simp only [multiset.card_map, map_add],\n        rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n        rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n        simp only [add_tsub_cancel_left],\n        simpa only [multiset.card_map, multiset.card_add, add_assoc] using hdim3,\n      },\n      have dimEp : dim E\u15ee = SP.card + 1,\n      {\n        have rn := dim_add_dim_orthogonal E,\n        rw [hdim2, \u2190cardSP, dimE, add_assoc] at rn,\n        zify at rn \u22a2,\n        exact add_left_cancel rn,\n      },\n      have cardSPle : SP.card \u2264 n\u2080,\n      {\n        have cardAgt : A.card > 0 := by simpa only [multiset.card_map] using h3,\n        rw [\u2190cardSP] at hn,\n        apply nat.le_of_lt_succ,\n        refine nat.lt_of_lt_of_le _ hn,\n        simpa only [lt_add_iff_pos_left] using cardAgt,\n      },\n      /- have Fsc : semicritical_spaces (pF.map (TS_microid_measure u')),\n      {\n        simp only [multiset.map_map, function.comp_app],\n        simp only [u', TS_microid_proj_eq_proj_TS_microid _ E\u15ee u uE],\n        let A := A'.map pair_to_default_space,\n        let Z := (B' - A').map pair_to_default_space + (C - B').map pair_to_TS,\n        have := semicritical_spaces_factorization\n          A Z \u27e8(_ : dim E = A.card), _\u27e9 _,\n        {\n          have : Z.map (\u03bb W, W.map (proj E\u15ee)) = F.map (\u03bb \u03bc, (TS_microid_measure u \u03bc).map (proj E\u15ee)),\n          {\n            simp only [Z, pair_to_default_space, pair_to_TS, multiset.map_add, multiset.map_map],\n            rw [TS_default_body_eq_TS_default_measure k u],\n          }\n        }\n      }, -/\n      have finp := ih SP.card cardSPle SP _ dimEp rfl sc_sp,\n      clear ih hTS hn hAB hBD h2 h3 h1 sc_sp,\n\n      have hE : E \u2264 \u22a4 := le_top,\n      let Am := A.map pair_to_microid,\n      let Bm := B.map pair_to_microid,\n      have vc : bm.is_vol_coll Am E,\n      {\n        split,\n        {simpa only [multiset.card_map, Am] using dimE},\n        {\n          simp only [Am, multiset.map_map],\n          intros K hK,\n          rcases multiset.mem_map.mp hK with \u27e8x, hx, rfl\u27e9,\n          simp only [E],\n          have : convex_body_subset (pair_to_TS (reduce_pair x)) ((pair_to_microid \u2218 reduce_pair) x) :=\n          begin\n            apply reduced_microid_subset_TS,\n          end,\n          refine subset_trans this _,\n          apply set_like.coe_subset_coe.mpr,\n          refine le_sum_multiset_of_mem _,\n          apply multiset.mem_map.mpr,\n          refine \u27e8reduce_pair x, _, rfl\u27e9,\n          apply multiset.mem_map.mpr,\n          exact \u27e8x, hx, rfl\u27e9,\n        },\n      },\n      have ac : bm.is_area_coll (Am + Bm),\n      {\n        change finite_dimensional.finrank \u211d V = (Am + Bm).card + 1,\n        change dim V = (Am + Bm).card + 1,\n        rw [hdim2],\n        simp only [multiset.card_add, multiset.card_map, add_left_inj],\n        rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n        rcases multiset.le_iff_exists_add.mp hB'C with \u27e8tt, rfl\u27e9,\n        rw [add_tsub_cancel_left],\n        rw [add_tsub_cancel_left],\n        rw [\u2190add_assoc],\n        symmetry,\n        simp only [multiset.card_map, multiset.card_add] at hdim3,\n        assumption,\n      },\n      have sc : semicritical_spaces (Am.map span_of_convex_body),\n      {\n        simp only [multiset.map_map, Am, A],\n        rw [span_reduced_microid_eq_TS, \u2190multiset.map_map],\n        have : A.map pair_to_TS \u2264 (A + B).map pair_to_TS,\n        {simp only [multiset.map_add, le_add_iff_nonneg_right, zero_le]},\n        exact semicritical_of_le this sc_AB,\n      },\n      have heq := bm.factorize_area vc ac sc,\n      have tmp : multiset.map microid_of_measure SP = proj_coll E\u15ee Bm,\n      {\n        simp only [SP, Bm, proj_coll, multiset.map_map, pair_to_microid],\n        simp only [function.comp_app],\n        simp only [proj_microid_of_measure E\u15ee], -- rw does not work because of lambda!\n      },\n      have finp' := set.mem_image_of_mem (coe_sph E\u15ee) finp,\n      rw [tmp, \u2190heq, coe_uncoe_sph] at finp',\n\n      have : Am + Bm = B'.map pair_to_default_body + (C - B').map pair_to_microid,\n      {\n        simp only [Am, Bm, A, B, multiset.map_add, multiset.map_map],\n        rw [\u2190add_assoc],\n        congr,\n        rw [\u2190multiset.map_add],\n        simp only [function.comp_app, reduced_microid_eq_default_body],\n        rcases multiset.le_iff_exists_add.mp hA'B' with \u27e8t, rfl\u27e9,\n        simp only [add_tsub_cancel_left],\n      },\n      rw [this] at finp',\n      have : dim V = C.card + 1,\n      {\n        symmetry,\n        simpa only [multiset.card_map, hdim2, add_left_inj] using hdim3,\n      },\n      rcases multiset.le_iff_exists_add.mp hB'C with \u27e8D, rfl\u27e9,\n      rw [add_tsub_cancel_left] at finp',\n      have := matryoshka_reduction this finp',\n      simpa only [multiset.map_map, pair_to_microid] using this,\n    },\n  },\nend", "meta": {"author": "datokrat", "repo": "triangle-bodies", "sha": "532a2820a0cb3686afddb60051340acf2f03db9e", "save_path": "github-repos/lean/datokrat-triangle-bodies", "path": "github-repos/lean/datokrat-triangle-bodies/triangle-bodies-532a2820a0cb3686afddb60051340acf2f03db9e/src/matryoshka.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581741774411, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.35131458542822136}}
{"text": "example (x : \u2124) : x * 0 = 0 := begin\nsimp\nend\n", "meta": {"author": "ntabee", "repo": "lean-exercise", "sha": "5b23b9be3d361fff5e981d5be3a0a1175504b9f6", "save_path": "github-repos/lean/ntabee-lean-exercise", "path": "github-repos/lean/ntabee-lean-exercise/lean-exercise-5b23b9be3d361fff5e981d5be3a0a1175504b9f6/5.8.1-4.6.7.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300698514777, "lm_q2_score": 0.5, "lm_q1q2_score": 0.35126503492573885}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n\n! This file was ported from Lean 3 source module algebraic_geometry.presheafed_space.has_colimits\n! leanprover-community/mathlib commit d39590fc8728fbf6743249802486f8c91ffe07bc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.AlgebraicGeometry.PresheafedSpace\nimport Mathbin.Topology.Category.Top.Limits\nimport Mathbin.Topology.Sheaves.Limits\n\n/-!\n# `PresheafedSpace C` has colimits.\n\nIf `C` has limits, then the category `PresheafedSpace C` has colimits,\nand the forgetful functor to `Top` preserves these colimits.\n\nWhen restricted to a diagram where the underlying continuous maps are open embeddings,\nthis says that we can glue presheaved spaces.\n\nGiven a diagram `F : J \u2964 PresheafedSpace C`,\nwe first build the colimit of the underlying topological spaces,\nas `colimit (F \u22d9 PresheafedSpace.forget C)`. Call that colimit space `X`.\n\nOur strategy is to push each of the presheaves `F.obj j`\nforward along the continuous map `colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j` to `X`.\nSince pushforward is functorial, we obtain a diagram `J \u2964 (presheaf C X)\u1d52\u1d56`\nof presheaves on a single space `X`.\n(Note that the arrows now point the other direction,\nbecause this is the way `PresheafedSpace C` is set up.)\n\nThe limit of this diagram then constitutes the colimit presheaf.\n-/\n\n\nnoncomputable section\n\nuniverse v' u' v u\n\nopen CategoryTheory\n\nopen TopCat\n\nopen TopCat.Presheaf\n\nopen TopologicalSpace\n\nopen Opposite\n\nopen CategoryTheory.Category\n\nopen CategoryTheory.Limits\n\nopen CategoryTheory.Functor\n\nvariable {J : Type u'} [Category.{v'} J]\n\nvariable {C : Type u} [Category.{v} C]\n\nnamespace AlgebraicGeometry\n\nnamespace PresheafedSpace\n\nattribute [local simp] eq_to_hom_map\n\nattribute [local tidy] tactic.auto_cases_opens\n\n@[simp]\ntheorem map_id_c_app (F : J \u2964 PresheafedSpace.{v} C) (j) (U) :\n    (F.map (\ud835\udfd9 j)).c.app (op U) =\n      (Pushforward.id (F.obj j).Presheaf).inv.app (op U) \u226b\n        (pushforwardEq\n                (by\n                  simp\n                  rfl)\n                (F.obj j).Presheaf).Hom.app\n          (op U) :=\n  by\n  cases U\n  dsimp\n  simp [PresheafedSpace.congr_app (F.map_id j)]\n  rfl\n#align algebraic_geometry.PresheafedSpace.map_id_c_app AlgebraicGeometry.PresheafedSpace.map_id_c_app\n\n@[simp]\ntheorem map_comp_c_app (F : J \u2964 PresheafedSpace.{v} C) {j\u2081 j\u2082 j\u2083} (f : j\u2081 \u27f6 j\u2082) (g : j\u2082 \u27f6 j\u2083) (U) :\n    (F.map (f \u226b g)).c.app (op U) =\n      (F.map g).c.app (op U) \u226b\n        (pushforwardMap (F.map g).base (F.map f).c).app (op U) \u226b\n          (Pushforward.comp (F.obj j\u2081).Presheaf (F.map f).base (F.map g).base).inv.app (op U) \u226b\n            (pushforwardEq\n                    (by\n                      rw [F.map_comp]\n                      rfl)\n                    _).Hom.app\n              _ :=\n  by\n  cases U\n  dsimp\n  simp only [PresheafedSpace.congr_app (F.map_comp f g)]\n  dsimp; simp; dsimp; simp\n#align algebraic_geometry.PresheafedSpace.map_comp_c_app AlgebraicGeometry.PresheafedSpace.map_comp_c_app\n\n-- See note [dsimp, simp]\n/-- Given a diagram of `PresheafedSpace C`s, its colimit is computed by pushing the sheaves onto\nthe colimit of the underlying spaces, and taking componentwise limit.\nThis is the componentwise diagram for an open set `U` of the colimit of the underlying spaces.\n-/\n@[simps]\ndef componentwiseDiagram (F : J \u2964 PresheafedSpace.{v} C) [HasColimit F]\n    (U : Opens (Limits.colimit F).carrier) : J\u1d52\u1d56 \u2964 C\n    where\n  obj j := (F.obj (unop j)).Presheaf.obj (op ((Opens.map (colimit.\u03b9 F (unop j)).base).obj U))\n  map j k f :=\n    (F.map f.unop).c.app _ \u226b\n      (F.obj (unop k)).Presheaf.map\n        (eqToHom\n          (by\n            rw [\u2190 colimit.w F f.unop, comp_base]\n            rfl))\n  map_comp' i j k f g := by\n    cases U\n    dsimp\n    simp_rw [map_comp_c_app, category.assoc]\n    congr 1\n    rw [TopCat.Presheaf.Pushforward.comp_inv_app, TopCat.Presheaf.pushforwardEq_hom_app,\n      CategoryTheory.NatTrans.naturality_assoc, TopCat.Presheaf.pushforwardMap_app]\n    congr 1\n    rw [category.id_comp, \u2190 (F.obj (unop k)).Presheaf.map_comp]\n    erw [\u2190 (F.obj (unop k)).Presheaf.map_comp]\n    congr\n#align algebraic_geometry.PresheafedSpace.componentwise_diagram AlgebraicGeometry.PresheafedSpace.componentwiseDiagram\n\nvariable [HasColimitsOfShape J TopCat.{v}]\n\n/-- Given a diagram of presheafed spaces,\nwe can push all the presheaves forward to the colimit `X` of the underlying topological spaces,\nobtaining a diagram in `(presheaf C X)\u1d52\u1d56`.\n-/\n@[simps]\ndef pushforwardDiagramToColimit (F : J \u2964 PresheafedSpace.{v} C) :\n    J \u2964 (Presheaf C (colimit (F \u22d9 PresheafedSpace.forget C)))\u1d52\u1d56\n    where\n  obj j := op (colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j _* (F.obj j).Presheaf)\n  map j j' f :=\n    (pushforwardMap (colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j') (F.map f).c \u226b\n        (Pushforward.comp (F.obj j).Presheaf ((F \u22d9 PresheafedSpace.forget C).map f)\n              (colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j')).inv \u226b\n          (pushforwardEq (colimit.w (F \u22d9 PresheafedSpace.forget C) f) (F.obj j).Presheaf).Hom).op\n  map_id' j := by\n    apply (op_equiv _ _).Injective\n    ext U\n    induction U using Opposite.rec\n    cases U\n    dsimp; simp; dsimp; simp\n  map_comp' j\u2081 j\u2082 j\u2083 f g := by\n    apply (op_equiv _ _).Injective\n    ext U\n    dsimp\n    simp only [map_comp_c_app, id.def, eq_to_hom_op, pushforward_map_app, eq_to_hom_map, assoc,\n      id_comp, pushforward.comp_inv_app, pushforward_eq_hom_app]\n    dsimp\n    simp only [eq_to_hom_trans, id_comp]\n    congr 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    rw [(F.map f).c.congr]\n    -- Now we pick up the pieces. First, we say what we want to replace that open set by:\n    pick_goal 3\n    refine' op ((opens.map (colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j\u2082)).obj (unop U))\n    -- Now we show the open sets are equal.\n    swap\n    \u00b7 apply unop_injective\n      rw [\u2190 opens.map_comp_obj]\n      congr\n      exact colimit.w (F \u22d9 PresheafedSpace.forget C) g\n    -- Finally, the original goal is now easy:\n    swap\n    \u00b7 simp\n      rfl\n#align algebraic_geometry.PresheafedSpace.pushforward_diagram_to_colimit AlgebraicGeometry.PresheafedSpace.pushforwardDiagramToColimit\n\nvariable [\u2200 X : TopCat.{v}, HasLimitsOfShape J\u1d52\u1d56 (X.Presheaf C)]\n\n/-- Auxiliary definition for `PresheafedSpace.has_colimits`.\n-/\ndef colimit (F : J \u2964 PresheafedSpace.{v} C) : PresheafedSpace C\n    where\n  carrier := colimit (F \u22d9 PresheafedSpace.forget C)\n  Presheaf := limit (pushforwardDiagramToColimit F).leftOp\n#align algebraic_geometry.PresheafedSpace.colimit AlgebraicGeometry.PresheafedSpace.colimit\n\n@[simp]\ntheorem colimit_carrier (F : J \u2964 PresheafedSpace.{v} C) :\n    (colimit F).carrier = Limits.colimit (F \u22d9 PresheafedSpace.forget C) :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.colimit_carrier AlgebraicGeometry.PresheafedSpace.colimit_carrier\n\n@[simp]\ntheorem colimit_presheaf (F : J \u2964 PresheafedSpace.{v} C) :\n    (colimit F).Presheaf = limit (pushforwardDiagramToColimit F).leftOp :=\n  rfl\n#align algebraic_geometry.PresheafedSpace.colimit_presheaf AlgebraicGeometry.PresheafedSpace.colimit_presheaf\n\n/-- Auxiliary definition for `PresheafedSpace.has_colimits`.\n-/\n@[simps]\ndef colimitCocone (F : J \u2964 PresheafedSpace.{v} C) : Cocone F\n    where\n  pt := colimit F\n  \u03b9 :=\n    { app := fun j =>\n        { base := colimit.\u03b9 (F \u22d9 PresheafedSpace.forget C) j\n          c := limit.\u03c0 _ (op j) }\n      naturality' := fun j j' f => by\n        fapply PresheafedSpace.ext\n        \u00b7 ext x\n          exact colimit.w_apply (F \u22d9 PresheafedSpace.forget C) f x\n        \u00b7 ext U\n          induction U using Opposite.rec\n          cases U\n          dsimp\n          simp only [PresheafedSpace.id_c_app, eq_to_hom_op, eq_to_hom_map, assoc,\n            pushforward.comp_inv_app]\n          rw [\u2190 congr_arg nat_trans.app (limit.w (pushforward_diagram_to_colimit F).leftOp f.op)]\n          dsimp\n          simp only [eq_to_hom_op, eq_to_hom_map, assoc, id_comp, pushforward.comp_inv_app]\n          congr\n          dsimp\n          simp only [id_comp]\n          simpa }\n#align algebraic_geometry.PresheafedSpace.colimit_cocone AlgebraicGeometry.PresheafedSpace.colimitCocone\n\nvariable [HasLimitsOfShape J\u1d52\u1d56 C]\n\nnamespace ColimitCoconeIsColimit\n\n/-- Auxiliary definition for `PresheafedSpace.colimit_cocone_is_colimit`.\n-/\ndef descCApp (F : J \u2964 PresheafedSpace.{v} C) (s : Cocone F) (U : (Opens \u21a5s.pt.carrier)\u1d52\u1d56) :\n    s.pt.Presheaf.obj U \u27f6\n      (colimit.desc (F \u22d9 PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) _*\n            limit (pushforwardDiagramToColimit F).leftOp).obj\n        U :=\n  by\n  refine'\n    limit.lift _\n        { pt := s.X.presheaf.obj U\n          \u03c0 :=\n            { app := fun j => _\n              naturality' := fun j j' f => _ } } \u226b\n      (limit_obj_iso_limit_comp_evaluation _ _).inv\n  -- We still need to construct the `app` and `naturality'` fields omitted above.\n  \u00b7 refine' (s.\u03b9.app (unop j)).c.app U \u226b (F.obj (unop j)).Presheaf.map (eq_to_hom _)\n    dsimp\n    rw [\u2190 opens.map_comp_obj]\n    simp\n  \u00b7 rw [PresheafedSpace.congr_app (s.w f.unop).symm U]\n    dsimp\n    have w :=\n      functor.congr_obj\n        (congr_arg opens.map (colimit.\u03b9_desc ((PresheafedSpace.forget C).mapCocone s) (unop j)))\n        (unop U)\n    simp only [opens.map_comp_obj_unop] at w\n    replace w := congr_arg op w\n    have w' := nat_trans.congr (F.map f.unop).c w\n    rw [w']\n    dsimp\n    simp\n    dsimp\n    simp\n#align algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_c_app AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.descCApp\n\ntheorem desc_c_naturality (F : J \u2964 PresheafedSpace.{v} C) (s : Cocone F)\n    {U V : (Opens \u21a5s.pt.carrier)\u1d52\u1d56} (i : U \u27f6 V) :\n    s.pt.Presheaf.map i \u226b descCApp F s V =\n      descCApp F s U \u226b\n        (colimit.desc (F \u22d9 forget C) ((forget C).mapCocone s) _* (colimitCocone F).pt.Presheaf).map\n          i :=\n  by\n  dsimp [desc_c_app]\n  ext\n  simp only [limit.lift_\u03c0, nat_trans.naturality, limit.lift_\u03c0_assoc, eq_to_hom_map, assoc,\n    pushforward_obj_map, nat_trans.naturality_assoc, op_map,\n    limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app_assoc,\n    limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app]\n  dsimp\n  have w :=\n    functor.congr_hom\n      (congr_arg opens.map (colimit.\u03b9_desc ((PresheafedSpace.forget C).mapCocone s) (unop j)))\n      i.unop\n  simp only [opens.map_comp_map] at w\n  replace w := congr_arg Quiver.Hom.op w\n  rw [w]\n  dsimp; simp\n#align algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_c_naturality AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_c_naturality\n\n/-- Auxiliary definition for `PresheafedSpace.colimit_cocone_is_colimit`.\n-/\ndef desc (F : J \u2964 PresheafedSpace.{v} C) (s : Cocone F) : colimit F \u27f6 s.pt\n    where\n  base := colimit.desc (F \u22d9 PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s)\n  c :=\n    { app := fun U => descCApp F s U\n      naturality' := fun U V i => desc_c_naturality F s i }\n#align algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc\n\ntheorem desc_fac (F : J \u2964 PresheafedSpace.{v} C) (s : Cocone F) (j : J) :\n    (colimitCocone F).\u03b9.app j \u226b desc F s = s.\u03b9.app j :=\n  by\n  fapply PresheafedSpace.ext\n  \u00b7 simp [desc]\n  \u00b7 ext\n    dsimp [desc, desc_c_app]\n    simpa\n#align algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit.desc_fac AlgebraicGeometry.PresheafedSpace.ColimitCoconeIsColimit.desc_fac\n\nend ColimitCoconeIsColimit\n\nopen ColimitCoconeIsColimit\n\n/-- Auxiliary definition for `PresheafedSpace.has_colimits`.\n-/\ndef colimitCoconeIsColimit (F : J \u2964 PresheafedSpace.{v} C) : IsColimit (colimitCocone F)\n    where\n  desc s := desc F s\n  fac s := desc_fac F s\n  uniq s m w :=\n    by\n    -- We need to use the identity on the continuous maps twice, so we prepare that first:\n    have t :\n      m.base =\n        colimit.desc (F \u22d9 PresheafedSpace.forget C) ((PresheafedSpace.forget C).mapCocone s) :=\n      by\n      apply CategoryTheory.Limits.colimit.hom_ext\n      intro j\n      apply ContinuousMap.ext\n      intro x\n      dsimp\n      simp only [colimit.\u03b9_desc_apply, map_cocone_\u03b9_app]\n      rw [\u2190 w j]\n      simp\n    fapply PresheafedSpace.ext\n    -- could `ext` please not reorder goals?\n    \u00b7 exact t\n    \u00b7 ext (U j)\n      dsimp [desc, desc_c_app]\n      simp only [limit.lift_\u03c0, eq_to_hom_op, eq_to_hom_map, assoc,\n        limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app]\n      rw [PresheafedSpace.congr_app (w (unop j)).symm U]\n      dsimp\n      have w := congr_arg op (functor.congr_obj (congr_arg opens.map t) (unop U))\n      rw [nat_trans.congr (limit.\u03c0 (pushforward_diagram_to_colimit F).leftOp j) w]\n      simp\n#align algebraic_geometry.PresheafedSpace.colimit_cocone_is_colimit AlgebraicGeometry.PresheafedSpace.colimitCoconeIsColimit\n\ninstance : HasColimitsOfShape J (PresheafedSpace.{v} C)\n    where HasColimit F :=\n    HasColimit.mk\n      { Cocone := colimitCocone F\n        IsColimit := colimitCoconeIsColimit F }\n\ninstance : PreservesColimitsOfShape J (PresheafedSpace.forget C)\n    where PreservesColimit F :=\n    preservesColimitOfPreservesColimitCocone (colimitCoconeIsColimit F)\n      (by\n        apply is_colimit.of_iso_colimit (colimit.is_colimit _)\n        fapply cocones.ext\n        \u00b7 rfl\n        \u00b7 intro j\n          dsimp\n          simp)\n\n/-- When `C` has limits, the category of presheaved spaces with values in `C` itself has colimits.\n-/\ninstance [HasLimits C] : HasColimits (PresheafedSpace.{v} C)\n    where HasColimitsOfShape J \ud835\udca5 :=\n    {\n      HasColimit := fun F =>\n        has_colimit.mk\n          { Cocone := colimit_cocone F\n            IsColimit := colimit_cocone_is_colimit F } }\n\n/-- The underlying topological space of a colimit of presheaved spaces is\nthe colimit of the underlying topological spaces.\n-/\ninstance forgetPreservesColimits [HasLimits C] : PreservesColimits (PresheafedSpace.forget C)\n    where PreservesColimitsOfShape J \ud835\udca5 :=\n    {\n      PreservesColimit := fun F =>\n        preserves_colimit_of_preserves_colimit_cocone (colimit_cocone_is_colimit F)\n          (by\n            apply is_colimit.of_iso_colimit (colimit.is_colimit _)\n            fapply cocones.ext\n            \u00b7 rfl\n            \u00b7 intro j\n              dsimp\n              simp) }\n#align algebraic_geometry.PresheafedSpace.forget_preserves_colimits AlgebraicGeometry.PresheafedSpace.forgetPreservesColimits\n\n/-- The components of the colimit of a diagram of `PresheafedSpace C` is obtained\nvia taking componentwise limits.\n-/\ndef colimitPresheafObjIsoComponentwiseLimit (F : J \u2964 PresheafedSpace.{v} C) [HasColimit F]\n    (U : Opens (Limits.colimit F).carrier) :\n    (Limits.colimit F).Presheaf.obj (op U) \u2245 limit (componentwiseDiagram F U) :=\n  by\n  refine'\n    ((sheaf_iso_of_iso (colimit.iso_colimit_cocone \u27e8_, colimit_cocone_is_colimit F\u27e9).symm).app\n          (op U)).trans\n      _\n  refine' (limit_obj_iso_limit_comp_evaluation _ _).trans (limits.lim.map_iso _)\n  fapply nat_iso.of_components\n  \u00b7 intro X\n    refine' (F.obj (unop X)).Presheaf.mapIso (eq_to_iso _)\n    simp only [functor.op_obj, unop_op, op_inj_iff, opens.map_coe, SetLike.ext'_iff,\n      Set.preimage_preimage]\n    simp_rw [\u2190 comp_app]\n    congr 2\n    exact \u03b9_preserves_colimits_iso_inv (forget C) F (unop X)\n  \u00b7 intro X Y f\n    change ((F.map f.unop).c.app _ \u226b _ \u226b _) \u226b (F.obj (unop Y)).Presheaf.map _ = _ \u226b _\n    rw [TopCat.Presheaf.Pushforward.comp_inv_app]\n    erw [category.id_comp]\n    rw [category.assoc]\n    erw [\u2190 (F.obj (unop Y)).Presheaf.map_comp, (F.map f.unop).c.naturality_assoc, \u2190\n      (F.obj (unop Y)).Presheaf.map_comp]\n    congr\n#align algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit\n\n@[simp]\ntheorem colimitPresheafObjIsoComponentwiseLimit_inv_\u03b9_app (F : J \u2964 PresheafedSpace.{v} C)\n    (U : Opens (Limits.colimit F).carrier) (j : J) :\n    (colimitPresheafObjIsoComponentwiseLimit F U).inv \u226b (colimit.\u03b9 F j).c.app (op U) =\n      limit.\u03c0 _ (op j) :=\n  by\n  delta colimit_presheaf_obj_iso_componentwise_limit\n  rw [iso.trans_inv, iso.trans_inv, iso.app_inv, sheaf_iso_of_iso_inv, pushforward_to_of_iso_app,\n    congr_app (iso.symm_inv _)]\n  simp_rw [category.assoc]\n  rw [\u2190 functor.map_comp_assoc, nat_trans.naturality]\n  erw [\u2190 comp_c_app_assoc]\n  rw [congr_app (colimit.iso_colimit_cocone_\u03b9_hom _ _)]\n  simp_rw [category.assoc]\n  erw [limit_obj_iso_limit_comp_evaluation_inv_\u03c0_app_assoc, lim_map_\u03c0_assoc]\n  convert category.comp_id _\n  erw [\u2190 (F.obj j).Presheaf.map_id]\n  iterate 2 erw [\u2190 (F.obj j).Presheaf.map_comp]\n  congr\n#align algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_inv_\u03b9_app AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_inv_\u03b9_app\n\n@[simp]\ntheorem colimitPresheafObjIsoComponentwiseLimit_hom_\u03c0 (F : J \u2964 PresheafedSpace.{v} C)\n    (U : Opens (Limits.colimit F).carrier) (j : J) :\n    (colimitPresheafObjIsoComponentwiseLimit F U).Hom \u226b limit.\u03c0 _ (op j) =\n      (colimit.\u03b9 F j).c.app (op U) :=\n  by rw [\u2190 iso.eq_inv_comp, colimit_presheaf_obj_iso_componentwise_limit_inv_\u03b9_app]\n#align algebraic_geometry.PresheafedSpace.colimit_presheaf_obj_iso_componentwise_limit_hom_\u03c0 AlgebraicGeometry.PresheafedSpace.colimitPresheafObjIsoComponentwiseLimit_hom_\u03c0\n\nend PresheafedSpace\n\nend AlgebraicGeometry\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/AlgebraicGeometry/PresheafedSpace/HasColimits.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3512537072291927}}
{"text": "/-\nCopyright (c) 2020 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.category_theory.limits.limits\nimport Mathlib.category_theory.products.basic\nimport Mathlib.category_theory.currying\nimport Mathlib.PostPort\n\nuniverses v u l \n\nnamespace Mathlib\n\n/-!\n# A Fubini theorem for categorical limits\n\nWe prove that $lim_{J \u00d7 K} G = lim_J (lim_K G(j, -))$ for a functor `G : J \u00d7 K \u2964 C`,\nwhen all the appropriate limits exist.\n\nWe begin working with a functor `F : J \u2964 K \u2964 C`. We'll write `G : J \u00d7 K \u2964 C` for the associated\n\"uncurried\" functor.\n\nIn the first part, given a coherent family `D` of limit cones over the functors `F.obj j`,\nand a cone `c` over `G`, we construct a cone over the cone points of `D`.\nWe then show that if `c` is a limit cone, the constructed cone is also a limit cone.\n\nIn the second part, we state the Fubini theorem in the setting where limits are\nprovided by suitable `has_limit` classes.\n\nWe construct\n`limit_uncurry_iso_limit_comp_lim F : limit (uncurry.obj F) \u2245 limit (F \u22d9 lim)`\nand give simp lemmas characterising it.\nFor convenience, we also provide\n`limit_iso_limit_curry_comp_lim G : limit G \u2245 limit ((curry.obj G) \u22d9 lim)`\nin terms of the uncurried functor.\n\n## Future work\n\nThe dual statement.\n-/\n\nnamespace category_theory.limits\n\n\n/--\nA structure carrying a diagram of cones over the the functors `F.obj j`.\n-/\n-- We could try introducing a \"dependent functor type\" to handle this?\n\nstructure diagram_of_cones {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) \nwhere\n  obj : (j : J) \u2192 cone (functor.obj F j)\n  map : {j j' : J} \u2192 (f : j \u27f6 j') \u2192 functor.obj (cones.postcompose (functor.map F f)) (obj j) \u27f6 obj j'\n  id : autoParam (J \u2192 cone_morphism.hom (map \ud835\udfd9) = \ud835\udfd9)\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n  comp : autoParam\n  (\u2200 {j\u2081 j\u2082 j\u2083 : J} (f : j\u2081 \u27f6 j\u2082) (g : j\u2082 \u27f6 j\u2083),\n    cone_morphism.hom (map (f \u226b g)) = cone_morphism.hom (map f) \u226b cone_morphism.hom (map g))\n  (Lean.Syntax.ident Lean.SourceInfo.none (String.toSubstring \"Mathlib.obviously\")\n    (Lean.Name.mkStr (Lean.Name.mkStr Lean.Name.anonymous \"Mathlib\") \"obviously\") [])\n\n/--\nExtract the functor `J \u2964 C` consisting of the cone points and the maps between them,\nfrom a `diagram_of_cones`.\n-/\n@[simp] theorem diagram_of_cones.cone_points_obj {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] {F : J \u2964 K \u2964 C} (D : diagram_of_cones F) (j : J) : functor.obj (diagram_of_cones.cone_points D) j = cone.X (diagram_of_cones.obj D j) :=\n  Eq.refl (functor.obj (diagram_of_cones.cone_points D) j)\n\n/--\nGiven a diagram `D` of limit cones over the `F.obj j`, and a cone over `uncurry.obj F`,\nwe can construct a cone over the diagram consisting of the cone points from `D`.\n-/\ndef cone_of_cone_uncurry {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] {F : J \u2964 K \u2964 C} {D : diagram_of_cones F} (Q : (j : J) \u2192 is_limit (diagram_of_cones.obj D j)) (c : cone (functor.obj uncurry F)) : cone (diagram_of_cones.cone_points D) :=\n  cone.mk (cone.X c)\n    (nat_trans.mk\n      fun (j : J) =>\n        is_limit.lift (Q j) (cone.mk (cone.X c) (nat_trans.mk fun (k : K) => nat_trans.app (cone.\u03c0 c) (j, k))))\n\n/--\n`cone_of_cone_uncurry Q c` is a limit cone when `c` is a limit cone.`\n-/\ndef cone_of_cone_uncurry_is_limit {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] {F : J \u2964 K \u2964 C} {D : diagram_of_cones F} (Q : (j : J) \u2192 is_limit (diagram_of_cones.obj D j)) {c : cone (functor.obj uncurry F)} (P : is_limit c) : is_limit (cone_of_cone_uncurry Q c) :=\n  is_limit.mk\n    fun (s : cone (diagram_of_cones.cone_points D)) =>\n      is_limit.lift P\n        (cone.mk (cone.X s)\n          (nat_trans.mk\n            fun (p : J \u00d7 K) =>\n              nat_trans.app (cone.\u03c0 s) (prod.fst p) \u226b\n                nat_trans.app (cone.\u03c0 (diagram_of_cones.obj D (prod.fst p))) (prod.snd p)))\n\n/--\nGiven a functor `F : J \u2964 K \u2964 C`, with all needed limits,\nwe can construct a diagram consisting of the limit cone over each functor `F.obj j`,\nand the universal cone morphisms between these.\n-/\ndef diagram_of_cones.mk_of_has_limits {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] : diagram_of_cones F :=\n  diagram_of_cones.mk (fun (j : J) => limit.cone (functor.obj F j))\n    fun (j j' : J) (f : j \u27f6 j') => cone_morphism.mk (functor.map lim (functor.map F f))\n\n-- Satisfying the inhabited linter.\n\nprotected instance diagram_of_cones_inhabited {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] : Inhabited (diagram_of_cones F) :=\n  { default := diagram_of_cones.mk_of_has_limits F }\n\n@[simp] theorem diagram_of_cones.mk_of_has_limits_cone_points {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] : diagram_of_cones.cone_points (diagram_of_cones.mk_of_has_limits F) = F \u22d9 lim :=\n  rfl\n\n/--\nThe Fubini theorem for a functor `F : J \u2964 K \u2964 C`,\nshowing that the limit of `uncurry.obj F` can be computed as\nthe limit of the limits of the functors `F.obj j`.\n-/\ndef limit_uncurry_iso_limit_comp_lim {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] : limit (functor.obj uncurry F) \u2245 limit (F \u22d9 lim) :=\n  let c : cone (functor.obj uncurry F) := limit.cone (functor.obj uncurry F);\n  let P : is_limit c := limit.is_limit (functor.obj uncurry F);\n  let G : diagram_of_cones F := diagram_of_cones.mk_of_has_limits F;\n  let Q : (j : J) \u2192 is_limit (diagram_of_cones.obj G j) := fun (j : J) => limit.is_limit (functor.obj F j);\n  is_limit.cone_point_unique_up_to_iso (cone_of_cone_uncurry_is_limit Q P) (limit.is_limit (F \u22d9 lim))\n\n@[simp] theorem limit_uncurry_iso_limit_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] {j : J} {k : K} : iso.hom (limit_uncurry_iso_limit_comp_lim F) \u226b limit.\u03c0 (F \u22d9 lim) j \u226b limit.\u03c0 (functor.obj F j) k =\n  limit.\u03c0 (functor.obj uncurry F) (j, k) := sorry\n\n@[simp] theorem limit_uncurry_iso_limit_comp_lim_inv_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (F : J \u2964 K \u2964 C) [has_limits_of_shape K C] [has_limit (functor.obj uncurry F)] [has_limit (F \u22d9 lim)] {j : J} {k : K} : iso.inv (limit_uncurry_iso_limit_comp_lim F) \u226b limit.\u03c0 (functor.obj uncurry F) (j, k) =\n  limit.\u03c0 (F \u22d9 lim) j \u226b limit.\u03c0 (functor.obj F j) k := sorry\n\n/--\nThe Fubini theorem for a functor `G : J \u00d7 K \u2964 C`,\nshowing that the limit of `G` can be computed as\nthe limit of the limits of the functors `G.obj (j, _)`.\n-/\ndef limit_iso_limit_curry_comp_lim {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C] [has_limit G] [has_limit (functor.obj curry G \u22d9 lim)] : limit G \u2245 limit (functor.obj curry G \u22d9 lim) :=\n  has_limit.iso_of_nat_iso (iso.app (equivalence.unit_iso (equivalence.symm currying)) G) \u226a\u226b\n    limit_uncurry_iso_limit_comp_lim (functor.obj curry G)\n\n@[simp] theorem limit_iso_limit_curry_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C] [has_limit G] [has_limit (functor.obj curry G \u22d9 lim)] {j : J} {k : K} : iso.hom (limit_iso_limit_curry_comp_lim G) \u226b\n    limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k =\n  limit.\u03c0 G (j, k) := sorry\n\n@[simp] theorem limit_iso_limit_curry_comp_lim_inv_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits_of_shape K C] [has_limit G] [has_limit (functor.obj curry G \u22d9 lim)] {j : J} {k : K} : iso.inv (limit_iso_limit_curry_comp_lim G) \u226b limit.\u03c0 G (j, k) =\n  limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k := sorry\n\n/--\nA variant of the Fubini theorem for a functor `G : J \u00d7 K \u2964 C`,\nshowing that $\\lim_k \\lim_j G(j,k) \u2245 \\lim_j \\lim_k G(j,k)$.\n-/\ndef limit_curry_swap_comp_lim_iso_limit_curry_comp_lim {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C] : limit (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) \u2245 limit (functor.obj curry G \u22d9 lim) :=\n  (iso.symm (limit_iso_limit_curry_comp_lim (prod.swap K J \u22d9 G)) \u226a\u226b\n      has_limit.iso_of_equivalence (prod.braiding K J) (iso.refl (equivalence.functor (prod.braiding K J) \u22d9 G))) \u226a\u226b\n    limit_iso_limit_curry_comp_lim G\n\n@[simp] theorem limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_hom_\u03c0_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C] {j : J} {k : K} : iso.hom (limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G) \u226b\n    limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k =\n  limit.\u03c0 (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) k \u226b\n    limit.\u03c0 (functor.obj (functor.obj curry (prod.swap K J \u22d9 G)) k) j := sorry\n\n@[simp] theorem limit_curry_swap_comp_lim_iso_limit_curry_comp_lim_inv_\u03c0_\u03c0 {J : Type v} {K : Type v} [small_category J] [small_category K] {C : Type u} [category C] (G : J \u00d7 K \u2964 C) [has_limits C] {j : J} {k : K} : iso.inv (limit_curry_swap_comp_lim_iso_limit_curry_comp_lim G) \u226b\n    limit.\u03c0 (functor.obj curry (prod.swap K J \u22d9 G) \u22d9 lim) k \u226b\n      limit.\u03c0 (functor.obj (functor.obj curry (prod.swap K J \u22d9 G)) k) j =\n  limit.\u03c0 (functor.obj curry G \u22d9 lim) j \u226b limit.\u03c0 (functor.obj (functor.obj curry G) j) k := sorry\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/category_theory/limits/fubini.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3512537072291927}}
{"text": "/-\nCopyright (c) 2021 Ya\u00ebl Dillies, Bhavik Mehta. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Ya\u00ebl Dillies, Bhavik Mehta\n-/\nimport analysis.convex.topology\nimport combinatorics.simplicial_complex.to_move.default\nimport data.real.basic\nimport linear_algebra.affine_space.finite_dimensional\n-- import data.nat.parity\n\nopen_locale classical affine big_operators\nopen set\nvariables {m n : \u2115} {\u03b1 : Type} {E : Type*} [normed_group E] [normed_space \u211d E]\n/-\nMATHLIB DEPARTURE ZONE\nA few PRs to be done\n-/\n\n-- TODO (Bhavik): Golf\n\n\n#exit\n/-\nTHEOREMS ON SALE\nPrevious attempts of Bhavik\n-/\n\n-- lemma of_affine_independent_set (X : set E) (hX : affine_independent \u211d (\u03bb p, p : X \u2192 E)) :\n--   \u2200 (s : finset E) (w : E \u2192 \u211d), \u2211 i in s, w i = 0 \u2192 s.weighted_vsub _ w = (0 : E) \u2192 \u2200 i \u2208 s, w i = 0 :=\n-- begin\n-- end\n\n-- omit V\n-- lemma filter_attach {\u03b9 : Type*} (s : finset \u03b9) (p : \u03b9 \u2192 Prop) :\n--   s.attach.filter (\u03bb i, p i) = (s.filter p).attach.image (\u03bb k, \u27e8k, finset.filter_subset _ _ k.2\u27e9) :=\n-- begin\n--   ext \u27e8a, ha\u27e9,\n--   simp [ha],\n-- end\n-- include V\n\n-- lemma of_affine_independent_set (s : set P) (hp : affine_independent k (\u03bb p, p : s \u2192 P)) :\n--   \u2200 (t : finset \u03b9) (w : \u03b9 \u2192 k) (z : \u03b9 \u2192 P), \u2211 i in t, w i = 0 \u2192 (\u2200 i \u2208 t, z i \u2208 s) \u2192\n--   t.weighted_vsub z w = (0:V) \u2192 \u2200 i \u2208 t, w i = 0 :=\n-- begin\n--   rintro t w z hw\u2081 hz hw\u2082,\n--   rw affine_independent_def at hp,\n--   let s' : finset s := t.attach.image (\u03bb i, \u27e8z i, hz _ i.2\u27e9),\n--   let w' : s \u2192 k,\n--   { intro x,\n--     apply \u2211 i in (t.filter (\u03bb j, z j = x)), w i },\n--   have : \u2211 (i : s) in s', w' i = 0,\n--   { change \u2211 (i : s) in s', \u2211 j in _, _ = _,-- rintro \u27e8_, _\u27e9 \u27e8_, _\u27e9,\n--     rw finset.sum_image' (\u03bb (i : {x // x \u2208 t}), w i),\n--     { dsimp only,\n--       rw finset.sum_attach,\n--       rw hw\u2081 },\n--     simp only [finset.mem_attach, subtype.mk_eq_mk, forall_true_left, subtype.coe_mk],\n--     rintro c,\n--     have : finset.filter (\u03bb (c' : {x // x \u2208 t}), z \u2191c' = z \u2191c) t.attach = _,\n--     { exact filter_attach t (\u03bb c', z c' = z c) },\n--     simp only [finset.filter_congr_decidable],\n--     simp only [finset.filter_congr_decidable] at this,\n--     rw this,\n--     simp [finset.sum_attach] },\n--   have : s'.weighted_vsub (\u03bb (p : \u21a5s), \u2191p) w' = (0 : V),\n--   { rw finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero _ _ _ this (0:V),\n--     rw finset.weighted_vsub_of_point_apply,\n--     simp only [vsub_eq_sub, sub_zero],\n--     change \u2211 (i : s) in s', (\u2211 j in _, _) \u2022 _ = _,\n--     simp_rw [finset.sum_smul],\n--     rw finset.sum_image' (\u03bb (i : {x // x \u2208 t}), _),\n--   },\n--   -- specialize hp s' w' this,\n\n\n--   -- sorry,\n\n--   -- have := (t.image z).attach,\n--   -- have : finset s := t.\n-- end\n\n-- example (X Y : finset E) (h : X \u2286 Y) : X \u2229 Y = X :=\n-- begin\n--   library_search,\n-- end\n\nlemma thing {\u03b9 \u03b2 : Type*} [add_comm_monoid \u03b2] (X : finset \u03b9) (f : \u03b9 \u2192 \u03b2) :\n  \u2211 (x : (X : set \u03b9)), f \u2191x = \u2211 x in X, f x :=\nbegin\n  rw \u2190finset.sum_image,\n  apply finset.sum_congr _ (\u03bb _ _, rfl),\n  { ext, simp },\n  { simp },\nend\n\ndef triangulation.facets (S : triangulation s) : set (finset E) :=\n{X \u2208 S.faces | \u2200 Y \u2208 S.faces, X \u2286 Y \u2192 X = Y}\n\ndef of_facets (S : set (finset E)) (hS\u2081 : \u2200 X \u2208 S, affine_independent \u211d (\u03bb p, p : (X : set E) \u2192 E))\n  (hS\u2082 : s = \u22c3 (X \u2208 S), convex_hull \u2191X)\n  (disjoint : \u2200 (X Y \u2208 S), convex_hull \u2191X \u2229 convex_hull \u2191Y \u2286 convex_hull (X \u2229 Y : set E)) :\n  triangulation s :=\n{ faces := {X | \u2203 Y \u2208 S, X \u2286 Y},\n  indep :=\n  begin\n    rintro X \u27e8Y, YS, XY\u27e9,\n    apply affine_independent_of_subset_affine_independent (hS\u2081 _ YS),\n    rwa finset.coe_subset,\n  end,\n  covering :=\n  begin\n    rw hS\u2082,\n    ext x,\n    simp only [exists_prop, set.mem_Union, set.mem_set_of_eq],\n    split,\n    { simp only [and_imp, exists_imp_distrib],\n      rintro X hX hx,\n      refine \u27e8X, \u27e8X, hX, set.subset.refl _\u27e9, hx\u27e9 },\n    { simp only [and_imp, exists_imp_distrib],\n      rintro X Y YS XY hx,\n      refine \u27e8Y, YS, convex_hull_mono XY hx\u27e9 }\n  end,\n  down_closed :=\n  begin\n    rintro X \u27e8Y, YS, XY\u27e9 Z ZX,\n    exact \u27e8_, YS, set.subset.trans ZX XY\u27e9,\n  end,\n  disjoint :=\n  begin\n    rintro X Y \u27e8Z, ZS, XZ\u27e9 \u27e8W, WS, YW\u27e9,\n    rintro x \u27e8hx\u2081, hx\u2082\u27e9,\n    rw \u2190finset.coe_inter,\n    have : x \u2208 convex_hull (Z \u2229 W : set E),\n      apply disjoint _ _ ZS WS \u27e8convex_hull_mono XZ hx\u2081, convex_hull_mono YW hx\u2082\u27e9,\n    rw \u2190finset.coe_inter at this,\n    have := disjoint_convex_hulls (hS\u2081 _ ZS) XZ (finset.inter_subset_left Z W) \u27e8hx\u2081, this\u27e9,\n    rw \u2190finset.coe_inter at this,\n    rw \u2190finset.inter_assoc at this,\n    have := disjoint_convex_hulls (hS\u2081 _ WS) (finset.inter_subset_right (X \u2229 Z) W) YW \u27e8this, hx\u2082\u27e9,\n    rw \u2190finset.coe_inter at this,\n    convert this using 3,\n    ext x,\n    simp only [finset.inter_assoc, and.congr_right_iff, finset.mem_inter],\n    intro hx\u2081,\n    rw \u2190 and_assoc,\n    apply iff.symm,\n    apply and_iff_right_of_imp,\n    intro hx\u2082,\n    refine \u27e8XZ hx\u2081, YW hx\u2082\u27e9,\n  end }\n\ndef std_basis (n : \u2115) : fin n \u2192 fin n \u2192 \u211d :=\n\u03bb i, linear_map.std_basis \u211d (\u03bb i, \u211d) i 1\n\ndef basis_with_zero (n : \u2115) : fin (n+1) \u2192 fin n \u2192 \u211d :=\nbegin\n  refine fin.cases _ _,\n  apply (0 : fin n \u2192 \u211d),\n  apply std_basis n,\nend\n\nlemma basis_with_zero_zero {n : \u2115} : basis_with_zero n 0 = 0 :=\nby rw [basis_with_zero, fin.cases_zero]\n\nlemma basis_with_zero_succ {n : \u2115} (j : fin n) : basis_with_zero n j.succ = std_basis n j :=\nby rw [basis_with_zero, fin.cases_succ]\n\nlemma linear_indep {n : \u2115} : linear_independent \u211d (std_basis n) :=\n(pi.is_basis_fun \u211d (fin n)).1\n\nlemma affine_indep {n : \u2115} : affine_independent \u211d (basis_with_zero n) :=\nbegin\n  rw affine_independent_iff_linear_independent_vsub \u211d _ (0 : fin n.succ),\n  simp only [basis_with_zero_zero],\n  simp only [vsub_eq_sub, sub_zero],\n  let g : {x : fin n.succ // x \u2260 0} \u2192 fin n := \u03bb (j : {x : fin n.succ // x \u2260 0}), fin.pred j.1 j.2,\n  have : std_basis n \u2218 g = \u03bb i, basis_with_zero n i,\n  { ext j,\n    dsimp,\n    rw \u2190 basis_with_zero_succ,\n    simp },\n  rw \u2190 this,\n  apply linear_independent.comp linear_indep g _,\n  rintro i j hi,\n  ext1,\n  exact fin.pred_inj.1 hi,\nend\n\ndef trivial {m : \u2115} : triangulation (std_simplex (fin (m+1))) :=\nof_facets\n  (singleton (finset.univ.image (std_basis (m+1))))\n  (begin\n    rintro X hX,\n    simp only [set.mem_singleton_iff] at hX,\n    subst hX,\n    rw fintype.coe_image_univ,\n    apply affine_independent_set_of_affine_independent,\n    convert affine_independent_embedding_of_affine_independent (fin.succ_embedding _).to_embedding affine_indep,\n    ext j x,\n    simp only [function.comp_app, fin.coe_succ_embedding, rel_embedding.coe_fn_to_embedding, basis_with_zero_succ, std_basis],\n  end)\n  (begin\n    rw [set.bUnion_singleton, fintype.coe_image_univ],\n    rw \u2190 convex_hull_basis_eq_std_simplex,\n    rw std_basis,\n    congr' 2,\n    ext i j,\n    rw linear_map.std_basis_apply,\n    rw function.update,\n    simp [eq_comm],\n    convert rfl,\n  end)\n  (begin\n    simp_rintro X Y hX hY,\n    substs X Y,\n    simp,\n    exact set.subset.refl _,\n  end)\n\nvariables {S : triangulation s}\ndef triangulation.finite (S : triangulation s) : Prop := S.faces.finite\n\nnoncomputable def triangulation.faces_finset (S : triangulation s) (hS : S.finite) :\n  finset (finset E) :=\nhS.to_finset\n\n@[simp]\nlemma mem_faces_finset (hS : S.finite) (X : finset E) :\n  X \u2208 S.faces_finset hS \u2194 X \u2208 S.faces :=\nset.finite.mem_to_finset\n\ndef triangulation.points (S : triangulation s) : set E :=\n\u22c3 k \u2208 S.faces, (k : set E)\n\nlemma convex_hull_face_subset (X) (hX : X \u2208 S.faces) : convex_hull \u2191X \u2286 s :=\nbegin\n  rintro x hx,\n  rw S.covering,\n  apply set.mem_bUnion hX hx,\nend\n\nlemma face_subset {X} (hX : X \u2208 S.faces) : \u2191X \u2286 s :=\nbegin\n  rintro x hx,\n  rw S.covering,\n  apply set.mem_bUnion hX,\n  apply subset_convex_hull,\n  apply hx\nend\n\nlemma points_subset : S.points \u2286 s :=\nbegin\n  rintro x hx,\n  rw S.covering,\n  rw triangulation.points at hx,\n  rw set.mem_bUnion_iff at hx,\n  rcases hx with \u27e8X, hX, hx\u27e9,\n  exact set.mem_bUnion hX (subset_convex_hull X hx)\nend\n\ndef is_sperner_colouring {s : set (fin (m+1) \u2192 \u211d)} (S : triangulation s)\n  (f : (fin (m+1) \u2192 \u211d) \u2192 fin (m+1)) : Prop :=\n\u2200 (X : fin (m+1) \u2192 \u211d) i, X \u2208 S.points \u2192 X i = 0 \u2192 f X \u2260 i\n\ndef panchromatic {n m : \u2115} (f : (fin n \u2192 \u211d) \u2192 fin m) (X : finset (fin n \u2192 \u211d)) :=\n  X.image f = finset.univ\n\nlemma panchromatic_iff (f : E \u2192 fin m) (X : finset E) :\n  panchromatic f X \u2194 (X.image f).card = m :=\nbegin\n  rw panchromatic,\n  split,\n  { intro h,\n    simp [h] },\n  { intro h,\n    refine finset.eq_of_subset_of_card_le (finset.image f X).subset_univ _,\n    simp [h] }\nend\n\ndef edge_of_std_simplex (m) : set (fin (m+1) \u2192 \u211d) :=\nstd_simplex (fin (m+1)) \u2229 {x | x 0 = 0}\n\nlemma convex_hull_ne_zero_points (X : set (fin (m+1) \u2192 \u211d)) (x : fin (m+1) \u2192 \u211d)\n  (hX : \u2200 (y : fin (m+1) \u2192 \u211d), y \u2208 X \u2192 0 \u2264 y 0)\n  (hx : x 0 = 0)\n  (hXx : x \u2208 convex_hull X) :\nx \u2208 convex_hull {y : fin (m+1) \u2192 \u211d | y \u2208 X \u2227 y 0 = 0} :=\nbegin\n  rw convex_hull_eq.{37} at hXx,\n  rcases hXx with \u27e8\u03b9, t, w, z, hw\u2080, hw\u2081, hz, x_eq\u27e9,\n  have x_zero : t.center_mass w z 0 = 0,\n  { rw [x_eq, hx] },\n  rw finset.center_mass_eq_of_sum_1 _ _ hw\u2081 at x_zero,\n  dsimp only at x_zero,\n  rw finset.sum_apply 0 t (\u03bb i, w i \u2022 z i) at x_zero,\n  dsimp at x_zero,\n  have : \u2200 (x : \u03b9), x \u2208 t \u2192 0 \u2264 w x * z x 0,\n  { rintro y hy,\n    exact mul_nonneg (hw\u2080 y hy) (hX (z y) (hz y hy)) },\n  rw finset.sum_eq_zero_iff_of_nonneg this at x_zero,\n  dsimp only at x_zero,\n  rw convex_hull_eq.{37},\n  refine \u27e8\u03b9, t.filter (\u03bb i, w i \u2260 0), w, z, _, _, _, _\u27e9,\n  { simp_rintro i hi only [finset.mem_filter],\n    apply hw\u2080 _ hi.1 },\n  { rw \u2190hw\u2081,\n    exact finset.sum_filter_ne_zero },\n  { simp_rintro i hi only [finset.mem_filter, set.mem_set_of_eq],\n    refine \u27e8hz i hi.1, _\u27e9,\n    have := x_zero i hi.1,\n    simp only [mul_eq_zero] at this,\n    apply or.resolve_left this hi.2 },\n  rw \u2190 x_eq,\n  exact finset.center_mass_filter_ne_zero z,\nend\n\ndef lower_triangulation (S : triangulation (std_simplex (fin (m+1)))) :\n  triangulation (edge_of_std_simplex m) :=\n{ faces := {X \u2208 S.faces | \u2200 (x : fin (m+1) \u2192 \u211d), x \u2208 X \u2192 x 0 = 0 },\n  indep :=\n  begin\n    rintro X hX,\n    simp only [set.mem_sep_eq] at hX,\n    apply S.indep _ hX.1,\n  end,\n  down_closed :=\n  begin\n    rintro X hX Y YX,\n    simp only [set.mem_sep_eq] at hX \u22a2,\n    refine \u27e8S.down_closed X hX.left Y YX, _\u27e9,\n    rintro x hx,\n    apply hX.2,\n    apply YX,\n    apply hx\n  end,\n  covering :=\n  begin\n    rw edge_of_std_simplex,\n    ext x,\n    split,\n    { rintro \u27e8hx\u2081, hx\u2082\u27e9,\n      rw S.covering at hx\u2081,\n      rw set.mem_bUnion_iff at hx\u2081,\n      rcases hx\u2081 with \u27e8X, hX, hx\u27e9,\n      have := convex_hull_ne_zero_points _ x _ hx\u2082 hx,\n      { rw set.mem_bUnion_iff,\n        refine \u27e8X.filter (\u03bb p, p 0 = 0), _, _\u27e9,\n        { simp only [and_imp, imp_self, set.mem_sep_eq, and_true, finset.mem_filter,\n            forall_true_iff],\n          apply S.down_closed _ hX,\n          apply finset.filter_subset },\n        { convert this,\n          simp only [finset.mem_coe, finset.coe_filter],\n          ext x,\n          simp } },\n      rintro y hy,\n      have : y \u2208 std_simplex (fin (m+1)),\n      { apply face_subset hX hy },\n      rw std_simplex_eq_inter at this,\n      simp only [set.mem_inter_eq, set.mem_Inter, set.mem_set_of_eq] at this,\n      apply this.1 },\n    { rw set.mem_bUnion_iff,\n      rintro \u27e8X, hX\u2081, hX\u2082\u27e9,\n      simp only [set.mem_sep_eq] at hX\u2081,\n      refine \u27e8convex_hull_face_subset X hX\u2081.1 hX\u2082, _\u27e9,\n      have : convex_hull \u2191X \u2286 {x : fin (m+1) \u2192 \u211d | x 0 = 0},\n      { apply convex_hull_min,\n        { rintro x hx,\n          exact hX\u2081.2 x hx },\n        rintro x\u2081 x\u2082 hx\u2081 hx\u2082 a b ha hb q,\n        simp only [set.mem_set_of_eq] at hx\u2081 hx\u2082 \u22a2,\n        simp [hx\u2081, hx\u2082] },\n      apply this,\n      apply hX\u2082 }\n  end,\n  disjoint :=\n  begin\n    rintro X Y hX hY,\n    apply S.disjoint _ _ hX.1 hY.1,\n  end }\n\nlemma std_simplex_one : std_simplex (fin 1) = { ![(1 : \u211d)]} :=\nbegin\n  ext x,\n  simp [std_simplex_eq_inter],\n  split,\n  { rintro \u27e8-, hx\u27e9,\n    ext i,\n    have : i = 0 := subsingleton.elim _ _,\n    rw this,\n    apply hx },\n  { rintro rfl,\n    refine \u27e8\u03bb _, _, rfl\u27e9,\n    simp only [matrix.cons_val_fin_one],\n    apply zero_le_one }\nend\n\nlemma subset_singleton_iff {\u03b9 : Type*} (x : \u03b9) (X : finset \u03b9) :\n  X \u2286 {x} \u2194 X = \u2205 \u2228 X = {x} :=\nbegin\n  split,\n  { rcases X.eq_empty_or_nonempty with (rfl | \u27e8y, hy\u27e9),\n    { intro,\n      left,\n      refl },\n    { intro hx,\n      right,\n      apply finset.subset.antisymm hx,\n      rw finset.singleton_subset_iff,\n      have := hx hy,\n      simp only [finset.mem_singleton] at this,\n      rwa \u2190 this } },\n  { rintro (rfl | rfl),\n    { apply finset.empty_subset },\n    { refl } }\nend\n\nlemma convex_hull_empty : convex_hull (\u2205 : set E) = \u2205 :=\nconvex_empty.convex_hull_eq\n\nlemma strong_sperner_zero_aux (S : triangulation (std_simplex (fin 1))) :\n  S.faces = {\u2205, { ![1]}} :=\nbegin\n  have X_subs : \u2200 X \u2208 S.faces, X \u2286 { ![(1:\u211d)]},\n  { rintro X hX,\n    have := face_subset hX,\n    rw std_simplex_one at this,\n    rintro x hx,\n    simpa using this hx },\n  have : \u2203 X \u2208 S.faces, X = { ![(1:\u211d)]},\n  { have std_eq := S.covering,\n    have one_mem : ![(1:\u211d)] \u2208 std_simplex (fin 1),\n    { rw std_simplex_one,\n      simp },\n    rw [std_eq, set.mem_bUnion_iff] at one_mem,\n    rcases one_mem with \u27e8X, hX\u2081, hX\u2082\u27e9,\n    refine \u27e8X, hX\u2081, _\u27e9,\n    have := X_subs X hX\u2081,\n    rw subset_singleton_iff at this,\n    rcases this with (rfl | rfl),\n    { simp only [finset.coe_empty] at hX\u2082,\n      rw convex_hull_empty at hX\u2082,\n      apply hX\u2082.elim },\n    { refl } },\n  ext X,\n  simp only [set.mem_insert_iff, set.mem_singleton_iff, \u2190subset_singleton_iff],\n  split,\n  { intro hX,\n    apply X_subs _ hX },\n  { intro hX,\n    rcases this with \u27e8Y, hY\u2081, rfl\u27e9,\n    exact S.down_closed _ hY\u2081 X hX },\nend\n\ntheorem strong_sperner_zero (S : triangulation (std_simplex (fin 1))) (hS : S.finite)\n  (f : (fin 1 \u2192 \u211d) \u2192 fin 1) :\n  odd ((S.faces_finset hS).filter (panchromatic f)).card :=\nbegin\n  have : (S.faces_finset hS).filter (panchromatic f) = {{ ![(1:\u211d)]}},\n  { ext X,\n    simp only [mem_faces_finset, finset.mem_singleton, finset.mem_filter, strong_sperner_zero_aux],\n    simp only [set.mem_insert_iff, set.mem_singleton_iff],\n    split,\n    { rintro \u27e8(rfl | rfl), h\u27e9,\n      { change _ = _ at h,\n        rw [univ_unique, fin.default_eq_zero, finset.image_empty, eq_comm] at h,\n        simp only [finset.singleton_ne_empty] at h,\n        cases h },\n      { refl } },\n    rintro rfl,\n    refine \u27e8or.inr rfl, _\u27e9,\n    change _ = _,\n    simp only [fin.default_eq_zero, finset.image_singleton, univ_unique],\n    rw finset.singleton_inj,\n    apply subsingleton.elim },\n  rw this,\n  simp,\nend\n\n-- lemma affine_independent_image {n m : \u2115} {\u03b9 : Type*} (f : (fin n \u2192 \u211d) \u2192\u2097[\u211d] (fin m \u2192 \u211d))\n--   (hf : function.injective f)\n--   (p : \u03b9 \u2192 fin n \u2192 \u211d)\n--   (hp : affine_independent \u211d p) :\n--   affine_independent \u211d (f \u2218 p) :=\n-- begin\n--   rw affine_independent_def,\n--   rintro s w hw hs i hi,\n--   rw finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero _ _ _ hw (0:fin m \u2192 \u211d) at hs,\n--   rw finset.weighted_vsub_of_point_apply at hs,\n--   simp only [vsub_eq_sub, function.comp_app, sub_zero] at hs,\n--   have : s.weighted_vsub p w = (0:fin n \u2192 \u211d),\n--   { rw finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero _ _ _ hw (0:fin n \u2192 \u211d),\n--     rw finset.weighted_vsub_of_point_apply,\n--     simp only [vsub_eq_sub, sub_zero],\n--     apply hf,\n--     simpa },\n--   apply hp s w hw this _ hi,\n-- end\n\nlemma cons_inj {n : \u2115} (x y : fin (n+1) \u2192 \u211d) (h0 : x 0 = y 0)\n  (h1 : matrix.vec_tail x = matrix.vec_tail y) :\n  x = y :=\nbegin\n  ext i,\n  refine fin.cases h0 _ i,\n  rw function.funext_iff at h1,\n  apply h1,\nend\n\nlemma affine_independent_proj {n : \u2115} {\u03b9 : Type*}\n  {p : \u03b9 \u2192 fin (n+1) \u2192 \u211d}\n  (hp\u2081 : \u2200 i, p i 0 = 0)\n  (hp\u2082 : affine_independent \u211d p) :\n  affine_independent \u211d (matrix.vec_tail \u2218 p) :=\nbegin\n  rw affine_independent_def,\n  rintro s w hw hs i hi,\n  rw finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero _ _ _ hw (0:fin n \u2192 \u211d) at hs,\n  rw finset.weighted_vsub_of_point_apply at hs,\n  simp only [vsub_eq_sub, function.comp_app, sub_zero] at hs,\n  have : s.weighted_vsub p w = (0:fin (n+1) \u2192 \u211d),\n  { rw finset.weighted_vsub_eq_weighted_vsub_of_point_of_sum_eq_zero _ _ _ hw (0:fin (n+1) \u2192 \u211d),\n    rw finset.weighted_vsub_of_point_apply,\n    simp only [vsub_eq_sub, sub_zero],\n    ext j,\n    simp only [pi.zero_apply],\n    rw finset.sum_apply _ s (\u03bb i, w i \u2022 p i),\n    refine fin.cases _ _ j,\n    { simp [hp\u2081] },\n    intro j,\n    dsimp,\n    rw function.funext_iff at hs,\n    specialize hs j,\n    simp only [pi.zero_apply] at hs,\n    rw finset.sum_apply _ s (\u03bb i, w i \u2022 matrix.vec_tail (p i)) at hs,\n    dsimp [matrix.vec_tail] at hs,\n    apply hs },\n  exact hp\u2082 s w hw this i hi,\nend\n\nlemma thingy2 {\u03b1 : Type*} [add_comm_monoid \u03b1] {n : \u2115} (k : fin n \u2192 \u03b1) :\n  \u2211 (i : fin n), k i = \u2211 i in finset.fin_range n, k i :=\nbegin\n  apply finset.sum_congr _ (\u03bb x _, rfl),\n  ext x,\n  simp only [finset.mem_univ, finset.mem_fin_range],\nend\n\nexample {\u03b1 : Type*} {p : \u03b1 \u2192 Prop} : subtype p \u21aa \u03b1 := function.embedding.subtype p\n\nlemma thingy3 {\u03b1 : Type*} [add_comm_monoid \u03b1] {n : \u2115} (k : fin n \u2192 \u03b1) :\n  (\u2211 (i : fin (n+1)), fin.cases (0:\u03b1) k i : \u03b1) = \u2211 i, k i :=\nbegin\n  have : (fin.cases (0:\u03b1) k (0 : fin (n+1)) : \u03b1) = (0 : \u03b1),\n    rw fin.cases_zero,\n  rw \u2190finset.sum_erase finset.univ this,\n  symmetry,\n  apply finset.sum_bij _ _ _ _ _,\n  { rintro a _,\n    apply a.succ },\n  { rintro a ha,\n    simp only [and_true, finset.mem_univ, finset.mem_erase],\n    apply fin.succ_ne_zero },\n  { rintro a ha,\n    symmetry,\n    apply fin.cases_succ },\n  { rintro a\u2081 a\u2082 _ _ h,\n    apply fin.succ_injective,\n    apply h },\n  { rintro b hb,\n    simp only [and_true, finset.mem_univ, finset.mem_erase] at hb,\n    refine \u27e8b.pred hb, by simp, _\u27e9,\n    simp }\nend\n\nlemma vec_tail_smul {m : \u2115} (c : \u211d) (x : fin m.succ \u2192 \u211d) :\n  matrix.vec_tail (c \u2022 x) = c \u2022 matrix.vec_tail x :=\nbegin\n  ext i,\n  simp [matrix.vec_tail],\nend\n\nlemma is_linear_map_matrix_vec_tail {n : \u2115} :\n  is_linear_map \u211d (matrix.vec_tail : (fin n.succ \u2192 \u211d) \u2192 (fin n \u2192 \u211d)) :=\n{ map_add := by simp,\n  map_smul := \u03bb c x,\n  begin\n    ext i,\n    simp [matrix.vec_tail],\n  end }\n\nlemma vec_tail_mem_simplex_iff {n : \u2115} (y : fin n \u2192 \u211d) :\n  matrix.vec_cons 0 y \u2208 edge_of_std_simplex n \u2194 y \u2208 std_simplex (fin n) :=\nbegin\n  rw [edge_of_std_simplex, set.mem_inter_eq, set.mem_set_of_eq, matrix.cons_val_zero,\n    eq_self_iff_true, and_true, std_simplex, std_simplex, set.mem_set_of_eq, set.mem_set_of_eq,\n    fin.forall_fin_succ, matrix.cons_val_zero],\n  simp only [matrix.cons_val_succ],\n  rw [matrix.vec_cons, fin.cons, thingy3],\n  tauto,\nend\n\n-- def my_proj : (fin (n+1) \u2192 \u211d)\ndef flatten_triangulation (S : triangulation (edge_of_std_simplex m)) :\n  triangulation (std_simplex (fin m)) :=\n{ faces := finset.image matrix.vec_tail '' S.faces,\n  indep :=\n  begin\n    rintro X hX,\n    simp only [set.mem_image] at hX,\n    rcases hX with \u27e8X, hX, rfl\u27e9,\n    let f : ((finset.image matrix.vec_tail X : set (fin m \u2192 \u211d))) \u2192 (X : set (fin (m+1) \u2192 \u211d)),\n    { intro t,\n      refine \u27e8matrix.vec_cons 0 t.1, _\u27e9,\n      rcases t with \u27e8t, ht\u27e9,\n      simp only [set.mem_image, finset.mem_coe, finset.coe_image] at ht,\n      rcases ht with \u27e8x, hx, rfl\u27e9,\n      have : matrix.vec_head x = 0 := (face_subset hX hx).2,\n      rw \u2190 this,\n      simpa },\n    have hf : function.injective f,\n    { rintro \u27e8x\u2081, hx\u2081\u27e9 \u27e8x\u2082, hx\u2082\u27e9 h,\n      rw subtype.ext_iff at h,\n      change matrix.vec_cons _ x\u2081 = matrix.vec_cons _ x\u2082 at h,\n      apply subtype.ext,\n      apply_fun matrix.vec_tail at h,\n      simpa using h },\n    have := affine_independent_proj _ (S.indep X hX),\n    { convert affine_independent_embedding_of_affine_independent \u27e8f, hf\u27e9 this,\n      ext p,\n      dsimp,\n      simp },\n    rintro \u27e8i, hi\u27e9,\n    apply (face_subset hX hi).2,\n  end,\n  down_closed :=\n  begin\n    rintro _ \u27e8X, hX, rfl\u27e9 Y YX,\n    refine \u27e8Y.image (matrix.vec_cons 0), _, _\u27e9,\n    { apply S.down_closed _ hX,\n      rw finset.image_subset_iff,\n      rintro y hY,\n      have := YX hY,\n      simp only [exists_prop, finset.mem_image] at this,\n      rcases this with \u27e8x, hx, rfl\u27e9,\n      have : matrix.vec_head x = 0 := (face_subset hX hx).2,\n      rw \u2190this,\n      simpa },\n    rw finset.image_image,\n    convert finset.image_id,\n    { ext x,\n      dsimp,\n      simp },\n    { exact classical.dec_eq (fin m \u2192 \u211d) },\n  end,\n  covering :=\n  begin\n    ext i,\n    rw set.mem_bUnion_iff,\n    simp only [exists_prop, set.mem_image, exists_exists_and_eq_and, finset.coe_image],\n    split,\n    { intro hi,\n      have : matrix.vec_cons 0 i \u2208 edge_of_std_simplex m,\n      { rwa vec_tail_mem_simplex_iff },\n      rw [S.covering, set.mem_bUnion_iff] at this,\n      rcases this with \u27e8x, hx\u2081, hx\u2082\u27e9,\n      refine \u27e8x, hx\u2081, _\u27e9,\n      rw \u2190is_linear_map.image_convex_hull,\n      refine \u27e8_, hx\u2082, _\u27e9,\n      simp only [matrix.tail_cons],\n      apply is_linear_map_matrix_vec_tail },\n    { rintro \u27e8X, hX\u2081, hX\u2082\u27e9,\n      rw \u2190 is_linear_map.image_convex_hull at hX\u2082,\n      { rcases hX\u2082 with \u27e8y, hy, rfl\u27e9,\n        rcases convex_hull_face_subset _ hX\u2081 hy with \u27e8hy\u2081, hy\u2082\u27e9,\n        rw \u2190 vec_tail_mem_simplex_iff,\n        have : matrix.vec_head y = 0 := hy\u2082,\n        rw \u2190 this,\n        simp only [matrix.cons_head_tail],\n        apply convex_hull_face_subset _ hX\u2081 hy },\n      apply is_linear_map_matrix_vec_tail,\n    }\n    -- },\n    -- have : matrix.vec_cons 0 i \u2208 std_simplex (fin (m+1)),\n    -- have := S.covering,\n  end,\n  disjoint :=\n  begin\n    rintro _ _ \u27e8X, hX, rfl\u27e9 \u27e8Y, hY, rfl\u27e9,\n    simp only [finset.coe_image],\n    rw \u2190 is_linear_map.image_convex_hull,\n    rw \u2190 is_linear_map.image_convex_hull,\n\n    rw set.image_inter_on,\n    refine set.subset.trans (set.image_subset matrix.vec_tail (S.disjoint _ _ hX hY)) _,\n    rw is_linear_map.image_convex_hull,\n    apply convex_hull_mono,\n    apply set.image_inter_subset,\n    apply is_linear_map_matrix_vec_tail,\n    { rintro x hx y hy h,\n      rw \u2190 matrix.cons_head_tail x,\n      rw \u2190 matrix.cons_head_tail y,\n      rw h,\n      rw (show matrix.vec_head x = 0, from (convex_hull_face_subset _ hY hx).2),\n      rw (show matrix.vec_head y = 0, from (convex_hull_face_subset _ hX hy).2) },\n    apply is_linear_map_matrix_vec_tail,\n    apply is_linear_map_matrix_vec_tail,\n  end }\n\ndef induct_down (S : triangulation (std_simplex (fin (m+1)))) :\n  triangulation (std_simplex (fin m)) :=\nflatten_triangulation (lower_triangulation S)\n\nexample {\u03b1 : Type*} {s : set \u03b1} (p : \u03b1 \u2192 Prop) (hs : s.finite) :\n  {x \u2208 s | p x}.finite :=\nset.finite.subset hs (s.sep_subset p)\n\nlemma induct_down_finite (S : triangulation (std_simplex (fin (m+1)))) (hS : S.finite) :\n  (induct_down S).finite :=\nbegin\n  rw triangulation.finite,\n  rw induct_down,\n  rw flatten_triangulation,\n  apply set.finite.image,\n  rw lower_triangulation,\n  apply set.finite.subset hS (S.faces.sep_subset _)\nend\n\nlemma test {n m : \u2115} (h : n.pred \u2264 m) : n \u2264 m + 1 :=\nbegin\n  exact nat.pred_le_iff.mp h,\nend\n\nlemma mwe {\u03b1 : Type*} {n : \u2115} (X : set (finset \u03b1)) (bound : \u2200 y \u2208 X, finset.card y \u2264 n) :\n  \u2200 y \u2208 X, \u2203 x \u2208 X, y \u2286 x \u2227 \u2200 z \u2208 X, x \u2286 z \u2192 x = z :=\nbegin\n  rintro y hy,\n  classical,\n  rcases ((finset.range (n+1)).filter $ \u03bb i, \u2203 x \u2208 X, y \u2286 x \u2227 finset.card x = i).exists_maximal\n    \u27e8y.card, finset.mem_filter.2 \u27e8finset.mem_range_succ_iff.2 $ bound y hy,\n      y, hy, finset.subset.refl _, rfl\u27e9\u27e9 with \u27e8i, hi1, hi2\u27e9,\n  rw [finset.mem_filter, finset.mem_range_succ_iff] at hi1,\n  rcases hi1 with \u27e8hin, x, hx, hyx, hxi\u27e9,\n  refine \u27e8x, hx, hyx, \u03bb z hz hxz, finset.eq_of_subset_of_card_le hxz _\u27e9,\n  rw [hxi, \u2190 not_lt],\n  refine hi2 _ _,\n  rw [finset.mem_filter, finset.mem_range_succ_iff],\n  refine \u27e8bound z hz, z, hz, finset.subset.trans hyx hxz, rfl\u27e9\nend\n\nlemma contained_in_facet (S : triangulation s) {X} (hX : X \u2208 S.faces) :\n  \u2203 Y \u2208 S.facets, X \u2286 Y :=\nbegin\n  have : \u2200 y \u2208 S.faces, finset.card y \u2264 m+1,\n  { rintro y hy,\n    apply size_bound (S.indep _ hy) },\n  rcases mwe S.faces this X hX with \u27e8Y, _, _, h\u2082\u27e9,\n  refine \u27e8Y, \u27e8\u2039Y \u2208 S.faces\u203a, h\u2082\u27e9, \u2039X \u2286 Y\u203a\u27e9,\nend\n\ndef is_homogeneous {m : \u2115} {s : set (fin m \u2192 \u211d)} (n : \u2115) (S : triangulation s) : Prop :=\n\u2200 X \u2208 S.facets, finset.card X = n\n-- \u2200 X \u2208 S.faces, \u2203 Y \u2208 S.faces, X \u2286 Y \u2227 finset.card Y = n\n\nlemma is_homogeneous_induct_down (S : triangulation (std_simplex (fin (m+1))))\n  (hS : is_homogeneous (m+1) S) :\n  is_homogeneous m (induct_down S) :=\nbegin\n  rintro X hX,\n  simp only [induct_down, triangulation.facets, flatten_triangulation, lower_triangulation,\n    and_imp, set.mem_sep_eq, set.mem_image, exists_imp_distrib] at hX,\n  rcases hX with \u27e8\u27e8X, \u27e8hX\u2082, hX\u2084\u27e9, rfl\u27e9, hX\u2083\u27e9,\n  have hX\u2081 : \u2200 (Y \u2208 S.faces), (\u2200 (i : fin (m+1) \u2192 \u211d), i \u2208 Y \u2192 i 0 = 0) \u2192\n    finset.image matrix.vec_tail X \u2286 finset.image matrix.vec_tail Y \u2192\n    finset.image matrix.vec_tail X = finset.image matrix.vec_tail Y,\n  { rintro Y hY\u2081 hY\u2082 hY\u2083,\n    apply hX\u2083 _ _ hY\u2081 hY\u2082 rfl hY\u2083 },\n  clear hX\u2083, -- just a less convenient form of hX\u2081\n  have : \u2200 (x : fin (m+1) \u2192 \u211d), x \u2209 X \u2192 x 0 = 0 \u2192 insert x X \u2209 S.faces,\n  { rintro x hx\u2081 hx\u2082 t,\n    have := hX\u2081 _ t (by simpa [hx\u2082] using hX\u2084) (finset.image_subset_image _),\n\n\n  },\n  -- have := set.image_subset,\n  -- simp only [induct_down, flatten_triangulation, lower_triangulation, set.mem_image,\n  --   set.mem_sep_eq] at hX,\n\n  -- rcases hX with \u27e8X, \u27e8hX\u2081, hX\u2082\u27e9, rfl\u27e9,\n  -- rcases hS X hX\u2081 with \u27e8Y, hY\u2081, hY\u2082, hY\u2083\u27e9,\n  -- -- refine \u27e8sorry, _, _\u27e9,\n  -- simp only [exists_prop, induct_down, flatten_triangulation, lower_triangulation, set.mem_sep_eq,\n  --   set.mem_image, exists_exists_and_eq_and],\n\n  -- -- simp only [induct_down],\n\nend\n\nlemma subset_iff_eq_or_ssubset {\u03b1 : Type*} {s t : finset \u03b1} :\n  s \u2286 t \u2194 s = t \u2228 s \u2282 t :=\nbegin\n  split,\n  { intro h,\n    rw finset.ssubset_iff_of_subset h,\n    apply or.imp _ _ (t \\ s).eq_empty_or_nonempty,\n    { intro q,\n      rw finset.sdiff_eq_empty_iff_subset at q,\n      apply finset.subset.antisymm h q },\n    { rintro \u27e8x, hx\u27e9,\n      simp only [finset.mem_sdiff] at hx,\n      exact \u27e8x, hx.1, hx.2\u27e9 } },\n  { rintro (rfl | ss),\n    { apply finset.subset.refl },\n    { apply ss.1 } }\nend\n\nnoncomputable def good_pairs {S : triangulation (std_simplex (fin (m+1)))} (hS : S.finite)\n  (f : (fin (m + 1) \u2192 \u211d) \u2192 fin (m + 1)) :\n  finset (finset (fin (m+1) \u2192 \u211d) \u00d7 finset (fin (m+1) \u2192 \u211d)) :=\n((S.faces_finset hS).product (S.faces_finset hS)).filter\n      (\u03bb (XY : finset _ \u00d7 finset _),\n          XY.2.card = m \u2227 XY.1.card = m+1 \u2227 XY.2.image f = finset.univ.erase 0 \u2227 XY.2 \u2286 XY.1)\n\n@[simp]\nlemma mem_good_pairs {S : triangulation (std_simplex (fin (m+1)))} (hS : S.finite)\n  {f} (X Y : finset _) :\n  (X,Y) \u2208 good_pairs hS f \u2194\n      X \u2208 S.faces\n    \u2227 Y \u2208 S.faces\n    \u2227 Y.card = m\n    \u2227 X.card = m+1\n    \u2227 Y.image f = finset.univ.erase 0\n    \u2227 Y \u2286 X :=\nbegin\n  simp [good_pairs, and_assoc],\nend\n\nnoncomputable def panchromatic_pairs {S : triangulation (std_simplex (fin (m+1)))} (hS : S.finite)\n  (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) :=\n(good_pairs hS f).filter (\u03bb (XY : _ \u00d7 _), panchromatic f XY.1)\n\nnoncomputable def almost_panchromatic_pairs {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) :=\n(good_pairs hS f).filter (\u03bb (XY : _ \u00d7 _), XY.1.image f = finset.univ.erase 0)\n\nnoncomputable def almost_panchromatic_simplices {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) :=\n(S.faces_finset hS).filter (\u03bb (X : finset _), X.card = m+1 \u2227 X.image f = finset.univ.erase 0)\n\nlemma almost_panchromatic_pairs_card_eq_twice {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) :\n  (almost_panchromatic_pairs hS f).card = (almost_panchromatic_simplices hS f).card * 2 :=\nbegin\n  have H : \u2200 x \u2208 almost_panchromatic_pairs hS f, prod.fst x \u2208 almost_panchromatic_simplices hS f,\n  { rintro \u27e8X, Y\u27e9 h,\n    simp only [almost_panchromatic_pairs, mem_good_pairs, finset.mem_filter] at h,\n    simp only [almost_panchromatic_simplices, mem_faces_finset, finset.mem_filter],\n    tauto },\n  rw finset.card_eq_sum_card_fiberwise H,\n  apply finset.sum_const_nat,\n  rintro X hX,\n  simp only [almost_panchromatic_simplices, mem_faces_finset, finset.mem_filter] at hX,\n  rcases hX with \u27e8hX\u2081, hX\u2082, hX\u2083\u27e9,\n  dsimp,\n  suffices : ((almost_panchromatic_pairs hS f).filter (\u03bb (x : _ \u00d7 _), x.fst = X)).card =\n    (X.filter (\u03bb x, \u2203 y \u2208 X, x \u2260 y \u2227 f x = f y)).card,\n  { rw this,\n    apply non_inj_card_two f,\n    rw hX\u2082,\n    rw hX\u2083,\n    simp [finset.card_erase_of_mem] },\n  apply (finset.card_congr (\u03bb x hx, (X, X.erase x)) _ _ _).symm,\n  { rintro x hx,\n    dsimp,\n    simp only [exists_prop, finset.mem_filter] at hx,\n    simp only [almost_panchromatic_pairs, and_true, eq_self_iff_true, mem_good_pairs,\n      finset.mem_filter, hX\u2083, true_and, hX\u2082, hX\u2081, finset.card_erase_of_mem, hx.1, nat.pred_succ,\n      finset.erase_subset],\n    rw \u2190 hX\u2083,\n    refine \u27e8S.down_closed _ hX\u2081 _ (finset.erase_subset _ _), _\u27e9,\n    conv_rhs {rw \u2190finset.insert_erase hx.1},\n    rw finset.image_insert,\n    rw finset.insert_eq_of_mem,\n    rw finset.mem_image,\n    simp only [exists_prop, finset.mem_erase],\n    simpa [and_assoc, and_comm (_ \u2208 X), \u2190ne.def, ne_comm, eq_comm] using hx.2 },\n  { rintro a b ha hb h,\n    dsimp at h,\n    injection h,\n    apply erase_inj_on _ (finset.filter_subset _ _ ha) \u2039X.erase a = X.erase b\u203a },\n  { rintro \u27e8X', Y\u27e9 hX,\n    dsimp [\u2190ne.def],\n    simp only [finset.mem_filter, almost_panchromatic_pairs, mem_good_pairs] at hX,\n    rcases hX with \u27e8\u27e8\u27e8_, _, _, _, _, _\u27e9, _\u27e9, rfl\u27e9,\n    have : (X' \\ Y).nonempty,\n    { rw [\u2190finset.card_pos, finset.card_sdiff \u2039Y \u2286 X'\u203a, \u2039X'.card = m + 1\u203a, \u2039Y.card = m\u203a],\n      simp only [nat.zero_lt_one, nat.add_sub_cancel_left] },\n    rcases this with \u27e8z, hz\u27e9,\n    simp only [finset.mem_sdiff] at hz,\n    rcases hz,\n    simp only [true_and, exists_prop, prod.mk.inj_iff, eq_self_iff_true, finset.mem_filter],\n    refine \u27e8z, \u27e8\u2039_\u203a, _\u27e9, _\u27e9,\n    { have : f z \u2208 Y.image f,\n      { rw [\u2039Y.image f = _\u203a, \u2190\u2039X'.image f = _\u203a],\n        apply finset.mem_image_of_mem f \u2039z \u2208 X'\u203a },\n      rcases finset.mem_image.1 this with \u27e8y, hy\u2081, hy\u2082\u27e9,\n      refine \u27e8y, \u2039Y \u2286 X'\u203a \u2039y \u2208 Y\u203a, (ne_of_mem_of_not_mem \u2039y \u2208 Y\u203a \u2039z \u2209 Y\u203a).symm, \u2039f y = f z\u203a.symm\u27e9 },\n    { symmetry,\n      apply finset.eq_of_subset_of_card_le,\n      simp only [subset_erase_iff, \u2039Y \u2286 X'\u203a, \u2039z \u2209 Y\u203a, not_false_iff, and_self],\n      rw [finset.card_erase_of_mem \u2039_ \u2208 _\u203a, \u2039X'.card = _\u203a, nat.pred_succ, \u2039Y.card = m\u203a] } }\nend\n\nlemma panchromatic_splits {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) {f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))} :\n  panchromatic_pairs hS f \u222a almost_panchromatic_pairs hS f = good_pairs hS f :=\nbegin\n  rw [panchromatic_pairs, almost_panchromatic_pairs, \u2190finset.filter_or, finset.filter_true_of_mem],\n  rintro \u27e8X,Y\u27e9 h,\n  simp only [mem_good_pairs] at h,\n  rcases h with \u27e8hX\u2081, hY\u2081, hY\u2082, hX\u2082, hY\u2083, YX\u27e9,\n  have : finset.univ.erase 0 \u2286 X.image f,\n  { rw \u2190 hY\u2083,\n    apply finset.image_subset_image,\n    apply YX },\n  rw subset_iff_eq_or_ssubset at this,\n  cases this,\n  { right,\n    apply this.symm },\n  { left,\n    apply finset.eq_of_subset_of_card_le,\n    apply finset.subset_univ,\n    simp only [finset.card_fin],\n    rw nat.succ_le_iff,\n    apply lt_of_le_of_lt _ (finset.card_lt_card this),\n    rw finset.card_erase_of_mem (finset.mem_univ _),\n    simp only [finset.card_fin, nat.pred_succ] }\nend\n\nlemma disjoint_split {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) {f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))} :\n  disjoint (panchromatic_pairs hS f) (almost_panchromatic_pairs hS f) :=\nbegin\n  rw finset.disjoint_left,\n  simp only [panchromatic_pairs, almost_panchromatic_pairs, and_imp, prod.forall, not_and,\n    mem_good_pairs, finset.mem_filter],\n  rintro X Y - - - - - - (h : _ = _) - - - - - - t,\n  rw h at t,\n  have : (0 : fin (m+1)) \u2209 finset.univ,\n  { intro q,\n    rw t at q,\n    rw finset.mem_erase at q,\n    apply q.1 rfl },\n  simpa using this\nend\n\n-- lemma image_subset_image_iff {\u03b1 \u03b2 : Type*}\n--   [decidable_eq \u03b1] [decidable_eq \u03b2] (s t : finset \u03b1)\n--   {f : \u03b1 \u2192 \u03b2} : s.image f \u2286 t.image f \u2192 s \u2286 t :=\n-- begin\n--   rintro h x hx,\n--   have : f x \u2208 t.image f,\n--   sorry,\n--   simp at this,\n\n--   -- refine \u27e8_, finset.subset_image _\u27e9,\n\n--   -- refine (iff.symm $ iff.intro (image_subset f) $ assume h, _),\n--   -- rw [\u2190 preimage_image_eq s hf, \u2190 preimage_image_eq t hf],\n--   -- exact preimage_mono h\n-- end\n\nlemma subset_erase_iff {\u03b1 : Type*} [decidable_eq \u03b1] (x : \u03b1) {s t : finset \u03b1} :\n  s \u2286 t.erase x \u2194 s \u2286 t \u2227 x \u2209 s :=\n\u27e8\u03bb h, \u27e8finset.subset.trans h (finset.erase_subset x t), \u03bb q, by simpa using h q\u27e9,\n \u03bb \u27e8h\u2081, h\u2082\u27e9 y hy, finset.mem_erase_of_ne_of_mem (ne_of_mem_of_not_mem hy h\u2082) (h\u2081 hy)\u27e9\n\n-- lemma sum_mul {\u03b1 \u03b2 : Type*} [add_comm_monoid \u03b2] {s : finset \u03b1} (b : \u03b2) (f : \u03b1 \u2192 \u03b2) :\n--   \u2211 x in s, b * f x = _ :=\n-- begin\n-- end\n\ndef plane : affine_subspace \u211d E :=\n{ carrier := {X | \u2211 i, X i = 1},\n  smul_vsub_vadd_mem :=\n  begin\n    rintro c p\u2081 p\u2082 p\u2083 (hp\u2081 hp\u2082 hp\u2083 : _ = _),\n    simp [finset.sum_add_distrib, \u2190finset.mul_sum, hp\u2081, hp\u2082, hp\u2083],\n  end }\n\nlemma obvious {m : \u2115} : \u2211 (i : fin m), (0 : fin m \u2192 \u211d) i = 1 \u2192 false :=\nbegin\n  simp,\nend\n\nlemma better_size_bound {X : finset E}\n  (hX\u2081 : affine_independent \u211d (\u03bb p, p : (X : set E) \u2192 E))\n  (hX\u2082 : \u2200 x \u2208 X, x \u2208 std_simplex (fin m)) :\n  X.card \u2264 m :=\nbegin\n  cases nat.eq_or_lt_of_le (size_bound hX\u2081),\n  { have card_eq : fintype.card (X : set E) = finite_dimensional.findim \u211d (fin m \u2192 \u211d) + 1,\n    { simp [h] },\n    have : affine_span \u211d (X : set E) = \u22a4,\n    { convert affine_span_eq_top_of_affine_independent_of_card_eq_findim_add_one hX\u2081 card_eq,\n      simp },\n    have zero_mem : (0 : E) \u2208 affine_span \u211d (X : set E),\n    { rw this,\n      apply affine_subspace.mem_top },\n    have : (X : set E) \u2264 (\u2191plane : set E),\n    { rintro x hx,\n      rw affine_subspace.mem_coe,\n      apply (hX\u2082 _ hx).2 },\n    rw \u2190((affine_subspace.gi \u211d (fin m \u2192 \u211d) (fin m \u2192 \u211d)).gc (X : set E) plane) at this,\n    have q : _ = _ := this zero_mem,\n    apply (obvious q).elim },\n  rwa \u2190 nat.lt_succ_iff,\nend\n\nlemma card_eq_of_panchromatic {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) {X} (hX : X \u2208 S.faces)\n  (hf : panchromatic f X) :\n  X.card = m+1 :=\nle_antisymm\n  (better_size_bound (S.indep X hX) (\u03bb x hx, face_subset hX hx))\n  begin\n    change _ = _ at hf,\n    have : (X.image f).card \u2264 X.card := finset.card_image_le,\n    simpa [hf] using this,\n  end\n\nlemma erase_image_subset_image_erase {\u03b1 \u03b2 : Type*} [decidable_eq \u03b1] [decidable_eq \u03b2] (f : \u03b1 \u2192 \u03b2)\n  (s : finset \u03b1) (a : \u03b1) :\n  (s.image f).erase (f a) \u2286 finset.image f (s.erase a) :=\nbegin\n  intro b,\n  simp only [and_imp, exists_prop, finset.mem_image, exists_imp_distrib, finset.mem_erase],\n  rintro hb x hx rfl,\n  exact \u27e8_, \u27e8ne_of_apply_ne f hb, hx\u27e9, rfl\u27e9,\nend\n\nlemma panchromatic_pairs_card_eq_panchromatic_card {S : triangulation (std_simplex (fin (m+1)))}\n  (hS : S.finite) (f : (fin (m+1) \u2192 \u211d) \u2192 (fin (m+1))) :\n  (panchromatic_pairs hS f).card = ((S.faces_finset hS).filter (panchromatic f)).card :=\nbegin\n  apply finset.card_congr _ _ _ _,\n  { rintro X hX,\n    apply X.1 },\n  { rintro \u27e8X, Y\u27e9 hX,\n    simp only [panchromatic_pairs, mem_good_pairs, finset.mem_filter] at hX,\n    simp only [mem_faces_finset, finset.mem_filter],\n    tauto },\n  { rintro \u27e8X\u2081, Y\u2081\u27e9 \u27e8X\u2082, Y\u2082\u27e9 h\u2081 h\u2082 (rfl : X\u2081 = X\u2082),\n    simp only [panchromatic_pairs, finset.mem_filter, mem_good_pairs, and_assoc] at h\u2081 h\u2082,\n    rcases h\u2081 with \u27e8X\u2081S, Y\u2081S, Y\u2081c, hX\u2081, hY\u2081, hY\u2081X, hX\u2082 : _ = _\u27e9,\n    rcases h\u2082 with \u27e8-, Y\u2082S, Y\u2082c, -, hY\u2082, hY\u2082X, -\u27e9,\n    ext1,\n    { refl },\n    change Y\u2081 = Y\u2082,\n    have : \u2203 x \u2208 X\u2081, f x = 0,\n    { suffices : (0 : fin (m+1)) \u2208 X\u2081.image f,\n      { simpa using this },\n      rw hX\u2082,\n      simp },\n    rcases this with \u27e8x, hx\u2081, hx\u2082\u27e9,\n    have : x \u2209 Y\u2081,\n    { intro q,\n      simpa [\u2039f x = 0\u203a, hY\u2081, fin.succ_ne_zero] using finset.mem_image_of_mem f q },\n    have : Y\u2081 \u2286 X\u2081.erase x,\n    { rw subset_erase_iff,\n      exact \u27e8\u2039Y\u2081 \u2286 X\u2081\u203a, \u2039x \u2209 Y\u2081\u203a\u27e9 },\n    have : Y\u2081 = X\u2081.erase x,\n    { apply finset.eq_of_subset_of_card_le \u2039Y\u2081 \u2286 X\u2081.erase x\u203a,\n      simp [finset.card_erase_of_mem \u2039x \u2208 X\u2081\u203a, \u2039X\u2081.card = m+1\u203a, \u2039Y\u2081.card = m\u203a] },\n    have : x \u2209 Y\u2082,\n    { intro q,\n      simpa [\u2039f x = 0\u203a, hY\u2082, fin.succ_ne_zero] using finset.mem_image_of_mem f q },\n    have : Y\u2082 \u2286 X\u2081.erase x,\n    { rw subset_erase_iff,\n      exact \u27e8\u2039Y\u2082 \u2286 X\u2081\u203a, \u2039x \u2209 Y\u2082\u203a\u27e9 },\n    have : Y\u2082 = X\u2081.erase x,\n    { apply finset.eq_of_subset_of_card_le \u2039Y\u2082 \u2286 X\u2081.erase x\u203a,\n      simp [finset.card_erase_of_mem \u2039x \u2208 X\u2081\u203a, \u2039X\u2081.card = m+1\u203a, \u2039Y\u2082.card = m\u203a] },\n    rw [\u2039Y\u2081 = X\u2081.erase x\u203a, \u2039Y\u2082 = X\u2081.erase x\u203a] },\n  { simp_rintro X hX only [finset.mem_filter, mem_faces_finset],\n    have : \u2203 x \u2208 X, f x = 0,\n    { suffices : (0 : fin (m+1)) \u2208 X.image f,\n      { simpa using this },\n      rw (show _ = _, from hX.2),\n      simp },\n    rcases this with \u27e8x, hx\u2081, hx\u2082\u27e9,\n    refine \u27e8\u27e8X, X.erase x\u27e9, _, rfl\u27e9,\n    have Xc : finset.card X = m+1 := card_eq_of_panchromatic hS f hX.1 hX.2,\n    simp only [panchromatic_pairs, hX.1, hX.2, finset.erase_subset, and_true, true_and, Xc,\n      finset.card_erase_of_mem hx\u2081, eq_self_iff_true, mem_good_pairs, finset.mem_filter,\n      nat.pred_succ],\n    refine \u27e8S.down_closed _ hX.1 _ (finset.erase_subset _ _), _\u27e9,\n    symmetry,\n    apply finset.eq_of_subset_of_card_le,\n    { rw \u2190(show finset.image f X = finset.univ, from hX.2),\n      rw \u2190hx\u2082,\n      apply erase_image_subset_image_erase f X x },\n    rw finset.card_erase_of_mem (finset.mem_univ _),\n    simp only [finset.card_fin, nat.pred_succ],\n    apply le_trans finset.card_image_le,\n    rw finset.card_erase_of_mem hx\u2081,\n    rw Xc,\n    simp, }\nend\n\ntheorem strong_sperner (S : triangulation (std_simplex (fin (m+1)))) (hS : S.finite)\n  {f} (hf : is_sperner_colouring S f) (hS\u2082 : is_homogeneous (m+1) S):\n  odd ((S.faces_finset hS).filter (panchromatic f)).card :=\nbegin\n  tactic.unfreeze_local_instances,\n  induction m with n ih generalizing f,\n  { apply strong_sperner_zero _ },\n  let f' : (fin (n + 1) \u2192 \u211d) \u2192 fin (n + 1),\n  { intro x,\n    apply fin.pred_above 0 (f (matrix.vec_cons 0 x)) },\n  have hf' : is_sperner_colouring (induct_down S) f',\n  { rintro x i hx hi,\n    simp only [induct_down, flatten_triangulation, lower_triangulation, triangulation.points,\n      set.mem_bUnion_iff, exists_prop, set.mem_sep_eq, finset.mem_image, set.mem_image,\n      finset.mem_coe, finset.mem_image, exists_exists_and_eq_and] at hx,\n    rcases hx with \u27e8X, \u27e8hX\u2081, hX\u2082\u27e9, y, hy, rfl\u27e9,\n    rw matrix.vec_tail at hi,\n    dsimp at hi,\n    have : y \u2208 S.points,\n    { apply set.mem_bUnion, apply hX\u2081, apply hy },\n    have : f y \u2260 i.succ,\n    { apply hf _ _ _ hi,\n      apply this },\n    change fin.pred_above _ _ \u2260 _,\n    have : y 0 = 0,\n    { apply hX\u2082,\n      apply hy },\n    have : matrix.vec_cons 0 (matrix.vec_tail y) = y,\n    { rw \u2190 this,\n      exact matrix.cons_head_tail y },\n    rw this,\n    have := \u2039is_sperner_colouring S f\u203a _ _ \u2039y \u2208 S.points\u203a \u2039y 0 = 0\u203a,\n    rw fin.pred_above_zero this,\n    intro q,\n    apply \u2039f y \u2260 i.succ\u203a,\n    rw \u2190 q,\n    simp },\n  specialize ih (induct_down S) (induct_down_finite _ hS) _ hf',\n\n\n  -- want that the number `x` of (n+2)-sets which are coloured by all n+2 colours is odd\n  -- let `y` be the (n+2)-sets coloured by the colours 1..(n+1)\n  -- let `p` be the (n+1)-sets coloured by 1..(n+1) on the 0-boundary\n  -- let `q` be the (n+1)-sets coloured by 1..(n+1) not on the 0-boundary\n\n  -- we know `p` is odd\n  -- we know p + 2 q = x + 2 y\n  -- therefore `x` is odd.\n\n  sorry\nend\n\nend affine\n\n-- brb\n", "meta": {"author": "mmasdeu", "repo": "brouwerfixedpoint", "sha": "548270f79ecf12d7e20a256806ccb9fcf57b87e2", "save_path": "github-repos/lean/mmasdeu-brouwerfixedpoint", "path": "github-repos/lean/mmasdeu-brouwerfixedpoint/brouwerfixedpoint-548270f79ecf12d7e20a256806ccb9fcf57b87e2/src/combinatorics/simplicial_complex/dump.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.35123300567155297}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon, Patrick Massot\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.ring.pi\nimport Mathlib.algebra.big_operators.basic\nimport Mathlib.data.fintype.basic\nimport Mathlib.algebra.group.prod\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 \n\nnamespace Mathlib\n\n/-!\n# Big operators for Pi Types\n\nThis file contains theorems relevant to big operators in binary and arbitrary product\nof monoids and groups\n-/\n\nnamespace pi\n\n\ntheorem list_sum_apply {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [(a : \u03b1) \u2192 add_monoid (\u03b2 a)] (a : \u03b1)\n    (l : List ((a : \u03b1) \u2192 \u03b2 a)) :\n    list.sum l a = list.sum (list.map (fun (f : (a : \u03b1) \u2192 \u03b2 a) => f a) l) :=\n  add_monoid_hom.map_list_sum (add_monoid_hom.apply \u03b2 a) l\n\ntheorem multiset_sum_apply {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} [(a : \u03b1) \u2192 add_comm_monoid (\u03b2 a)]\n    (a : \u03b1) (s : multiset ((a : \u03b1) \u2192 \u03b2 a)) :\n    multiset.sum s a = multiset.sum (multiset.map (fun (f : (a : \u03b1) \u2192 \u03b2 a) => f a) s) :=\n  add_monoid_hom.map_multiset_sum (add_monoid_hom.apply \u03b2 a) s\n\nend pi\n\n\n@[simp] theorem finset.sum_apply {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} {\u03b3 : Type u_3}\n    [(a : \u03b1) \u2192 add_comm_monoid (\u03b2 a)] (a : \u03b1) (s : finset \u03b3) (g : \u03b3 \u2192 (a : \u03b1) \u2192 \u03b2 a) :\n    finset.sum s (fun (c : \u03b3) => g c) a = finset.sum s fun (c : \u03b3) => g c a :=\n  add_monoid_hom.map_sum (add_monoid_hom.apply \u03b2 a) (fun (c : \u03b3) => g c) s\n\n@[simp] theorem fintype.prod_apply {\u03b1 : Type u_1} {\u03b2 : \u03b1 \u2192 Type u_2} {\u03b3 : Type u_3} [fintype \u03b3]\n    [(a : \u03b1) \u2192 comm_monoid (\u03b2 a)] (a : \u03b1) (g : \u03b3 \u2192 (a : \u03b1) \u2192 \u03b2 a) :\n    finset.prod finset.univ (fun (c : \u03b3) => g c) a = finset.prod finset.univ fun (c : \u03b3) => g c a :=\n  finset.prod_apply a finset.univ g\n\ntheorem prod_mk_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b1] [comm_monoid \u03b2]\n    (s : finset \u03b3) (f : \u03b3 \u2192 \u03b1) (g : \u03b3 \u2192 \u03b2) :\n    (finset.prod s fun (x : \u03b3) => f x, finset.prod s fun (x : \u03b3) => g x) =\n        finset.prod s fun (x : \u03b3) => (f x, g x) :=\n  sorry\n\n-- As we only defined `single` into `add_monoid`, we only prove the `finset.sum` version here.\n\ntheorem finset.univ_sum_single {I : Type u_1} [DecidableEq I] {Z : I \u2192 Type u_2}\n    [(i : I) \u2192 add_comm_monoid (Z i)] [fintype I] (f : (i : I) \u2192 Z i) :\n    (finset.sum finset.univ fun (i : I) => pi.single i (f i)) = f :=\n  sorry\n\ntheorem add_monoid_hom.functions_ext {I : Type u_1} [DecidableEq I] {Z : I \u2192 Type u_2}\n    [(i : I) \u2192 add_comm_monoid (Z i)] [fintype I] (G : Type u_3) [add_comm_monoid G]\n    (g : ((i : I) \u2192 Z i) \u2192+ G) (h : ((i : I) \u2192 Z i) \u2192+ G)\n    (w : \u2200 (i : I) (x : Z i), coe_fn g (pi.single i x) = coe_fn h (pi.single i x)) : g = h :=\n  sorry\n\n-- we need `apply`+`convert` because Lean fails to unify different `add_monoid` instances\n\n-- on `\u03a0 i, f i`\n\ntheorem ring_hom.functions_ext {I : Type u_1} [DecidableEq I] {f : I \u2192 Type u_2}\n    [(i : I) \u2192 semiring (f i)] [fintype I] (G : Type u_3) [semiring G] (g : ((i : I) \u2192 f i) \u2192+* G)\n    (h : ((i : I) \u2192 f i) \u2192+* G)\n    (w : \u2200 (i : I) (x : f i), coe_fn g (pi.single i x) = coe_fn h (pi.single i x)) : g = h :=\n  sorry\n\nnamespace prod\n\n\ntheorem fst_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b1] [comm_monoid \u03b2]\n    {s : finset \u03b3} {f : \u03b3 \u2192 \u03b1 \u00d7 \u03b2} :\n    fst (finset.prod s fun (c : \u03b3) => f c) = finset.prod s fun (c : \u03b3) => fst (f c) :=\n  monoid_hom.map_prod (monoid_hom.fst \u03b1 \u03b2) f s\n\ntheorem snd_prod {\u03b1 : Type u_1} {\u03b2 : Type u_2} {\u03b3 : Type u_3} [comm_monoid \u03b1] [comm_monoid \u03b2]\n    {s : finset \u03b3} {f : \u03b3 \u2192 \u03b1 \u00d7 \u03b2} :\n    snd (finset.prod s fun (c : \u03b3) => f c) = finset.prod s fun (c : \u03b3) => snd (f c) :=\n  monoid_hom.map_prod (monoid_hom.snd \u03b1 \u03b2) f s\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/big_operators/pi_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3512329970648985}}
{"text": "/-\nCopyright (c) 2014 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n\nnotation, basic datatypes and type classes\n\n! This file was ported from Lean 3 source module init.core\n! leanprover-community/mathlib commit 4a03bdeb31b3688c31d02d7ff8e0ff2e5d6174db\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nprelude\n\nuniverse u v w\n\n/- warning: id_delta clashes with id -> id\nCase conversion may be inaccurate. Consider using '#align id_delta id\u2093'. -/\n#print id /-\n/-- The kernel definitional equality test (t =?= s) has special support for id_delta applications.\nIt implements the following rules\n\n   1)   (id_delta t) =?= t\n   2)   t =?= (id_delta t)\n   3)   (id_delta t) =?= s  IF (unfold_of t) =?= s\n   4)   t =?= id_delta s    IF t =?= (unfold_of s)\n\nThis is mechanism for controlling the delta reduction (aka unfolding) used in the kernel.\n\nWe use id_delta applications to address performance problems when type checking\nlemmas generated by the equation compiler.\n-/\n@[inline]\ndef id {\u03b1 : Sort u} (a : \u03b1) : \u03b1 :=\n  a\n#align id_delta id\n-/\n\n#print optParam /-\n/-- Gadget for optional parameter support. -/\n@[reducible]\ndef optParam (\u03b1 : Sort u) (default : \u03b1) : Sort u :=\n  \u03b1\n#align opt_param optParam\n-/\n\n#print outParam /-\n/-- Gadget for marking output parameters in type classes. -/\n@[reducible]\ndef outParam (\u03b1 : Sort u) : Sort u :=\n  \u03b1\n#align out_param outParam\n-/\n\n/- warning: id_rhs clashes with id -> id\nCase conversion may be inaccurate. Consider using '#align id_rhs id\u2093'. -/\n#print id /-\n/-- id_rhs is an auxiliary declaration used in the equation compiler to address performance\n  issues when proving equational lemmas. The equation compiler uses it as a marker.\n-/\nabbrev id (\u03b1 : Sort u) (a : \u03b1) : \u03b1 :=\n  a\n#align id_rhs id\n-/\n\n#print PUnit /-\ninductive PUnit : Sort u\n  | star : PUnit\n#align punit PUnit\n-/\n\n#print Unit /-\n/-- An abbreviation for `punit.{0}`, its most common instantiation.\n    This type should be preferred over `punit` where possible to avoid\n    unnecessary universe parameters. -/\nabbrev Unit : Type :=\n  PUnit\n#align unit Unit\n-/\n\n#print Unit.unit /-\n@[match_pattern]\nabbrev Unit.unit : Unit :=\n  PUnit.unit\n#align unit.star Unit.unit\n-/\n\n/-- Gadget for defining thunks, thunk parameters have special treatment.\nExample: given\n      def f (s : string) (t : thunk nat) : nat\nan application\n     f \"hello\" 10\n is converted into\n     f \"hello\" (\u03bb _, 10)\n-/\n@[reducible]\ndef Thunk (\u03b1 : Type u) : Type u :=\n  Unit \u2192 \u03b1\n#align thunk Thunk\u2093\n\n#print True /-\ninductive True : Prop\n  | intro : True\n#align true True\n-/\n\n#print False /-\ninductive False : Prop\n#align false False\n-/\n\n#print Empty /-\ninductive Empty : Type\n#align empty Empty\n-/\n\n#print Not /-\n/-- Logical not.\n\n`not P`, with notation `\u00ac P`, is the `Prop` which is true if and only if `P` is false. It is\ninternally represented as `P \u2192 false`, so one way to prove a goal `\u22a2 \u00ac P` is to use `intro h`,\nwhich gives you a new hypothesis `h : P` and the goal `\u22a2 false`.\n\nA hypothesis `h : \u00ac P` can be used in term mode as a function, so if `w : P` then `h w : false`.\n\nRelated mathlib tactic: `contrapose`.\n-/\ndef Not (a : Prop) :=\n  a \u2192 False\n#align not Not\n-/\n\n/- ./././Mathport/Syntax/Translate/Command.lean:364:30: infer kinds are unsupported in Lean 4: refl [] -/\n#print Eq /-\ninductive Eq {\u03b1 : Sort u} (a : \u03b1) : \u03b1 \u2192 Prop\n  | refl : Eq a\n#align eq Eq\n-/\n\n/-!\nInitialize the quotient module, which effectively adds the following definitions:\n```lean\nconstant quot {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) : Sort u\n\nconstant quot.mk {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) : quot r\n\nconstant quot.lift {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : Sort v} (f : \u03b1 \u2192 \u03b2) :\n  (\u2200 a b : \u03b1, r a b \u2192 eq (f a) (f b)) \u2192 quot r \u2192 \u03b2\n\nconstant quot.ind {\u03b1 : Sort u} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {\u03b2 : quot r \u2192 Prop} :\n  (\u2200 a : \u03b1, \u03b2 (quot.mk r a)) \u2192 \u2200 q : quot r, \u03b2 q\n```\nAlso the reduction rule:\n```\nquot.lift f _ (quot.mk a) ~~> f a\n```\n-/\n\n\ninit_quot\n\n/- ./././Mathport/Syntax/Translate/Command.lean:364:30: infer kinds are unsupported in Lean 4: refl [] -/\n#print HEq /-\n/-- Heterogeneous equality.\n\nIts purpose is to write down equalities between terms whose types are not definitionally equal.\nFor example, given `x : vector \u03b1 n` and `y : vector \u03b1 (0+n)`, `x = y` doesn't typecheck but `x == y` does.\n\nIf you have a goal `\u22a2 x == y`,\nyour first instinct should be to ask (either yourself, or on [zulip](https://leanprover.zulipchat.com/))\nif something has gone wrong already.\nIf you really do need to follow this route,\nyou may find the lemmas `eq_rec_heq` and `eq_mpr_heq` useful.\n-/\ninductive HEq {\u03b1 : Sort u} (a : \u03b1) : \u2200 {\u03b2 : Sort u}, \u03b2 \u2192 Prop\n  | refl : HEq a\n#align heq HEq\n-/\n\n#print Prod /-\nstructure Prod (\u03b1 : Type u) (\u03b2 : Type v) where\n  fst : \u03b1\n  snd : \u03b2\n#align prod Prod\n-/\n\n#print PProd /-\n/-- Similar to `prod`, but \u03b1 and \u03b2 can be propositions.\n   We use this type internally to automatically generate the brec_on recursor. -/\nstructure PProd (\u03b1 : Sort u) (\u03b2 : Sort v) where\n  fst : \u03b1\n  snd : \u03b2\n#align pprod PProd\n-/\n\n#print And /-\n/-- Logical and.\n\n`and P Q`, with notation `P \u2227 Q`, is the `Prop` which is true precisely when `P` and `Q` are\nboth true.\n\nTo prove a goal `\u22a2 P \u2227 Q`, you can use the tactic `split`,\nwhich gives two separate goals `\u22a2 P` and `\u22a2 Q`.\n\nGiven a hypothesis `h : P \u2227 Q`, you can use the tactic `cases h with hP hQ`\nto obtain two new hypotheses `hP : P` and `hQ : Q`. See also the `obtain` or `rcases` tactics in\nmathlib.\n-/\nstructure And (a b : Prop) : Prop where intro ::\n  left : a\n  right : b\n#align and And\n-/\n\n/- warning: and.elim_left clashes with and.left -> And.left\nCase conversion may be inaccurate. Consider using '#align and.elim_left And.left\u2093'. -/\n#print And.left /-\ntheorem And.left {a b : Prop} (h : And a b) : a :=\n  h.1\n#align and.elim_left And.left\n-/\n\n/- warning: and.elim_right clashes with and.right -> And.right\nCase conversion may be inaccurate. Consider using '#align and.elim_right And.right\u2093'. -/\n#print And.right /-\ntheorem And.right {a b : Prop} (h : And a b) : b :=\n  h.2\n#align and.elim_right And.right\n-/\n\n-- eq basic support\nattribute [refl] Eq.refl\n\n#print rfl /-\n-- This is a `def`, so that it can be used as pattern in the equation compiler.\n@[match_pattern]\ndef rfl {\u03b1 : Sort u} {a : \u03b1} : a = a :=\n  Eq.refl a\n#align rfl rfl\n-/\n\n#print Eq.subst /-\n@[elab_as_elim, subst]\ntheorem Eq.subst {\u03b1 : Sort u} {P : \u03b1 \u2192 Prop} {a b : \u03b1} (h\u2081 : a = b) (h\u2082 : P a) : P b :=\n  Eq.ndrec h\u2082 h\u2081\n#align eq.subst Eq.subst\n-/\n\n#print Eq.trans /-\n@[trans]\ntheorem Eq.trans {\u03b1 : Sort u} {a b c : \u03b1} (h\u2081 : a = b) (h\u2082 : b = c) : a = c :=\n  h\u2082 \u25b8 h\u2081\n#align eq.trans Eq.trans\n-/\n\n#print Eq.symm /-\n@[symm]\ntheorem Eq.symm {\u03b1 : Sort u} {a b : \u03b1} (h : a = b) : b = a :=\n  h \u25b8 rfl\n#align eq.symm Eq.symm\n-/\n\n#print HEq.rfl /-\n-- This is a `def`, so that it can be used as pattern in the equation compiler.\n@[match_pattern]\ndef HEq.rfl {\u03b1 : Sort u} {a : \u03b1} : HEq a a :=\n  HEq.refl a\n#align heq.rfl HEq.rfl\n-/\n\ntheorem eq_of_hEq {\u03b1 : Sort u} {a a' : \u03b1} (h : HEq a a') : a = a' :=\n  have : \u2200 (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @HEq \u03b1 a \u03b1' a') (h\u2082 : \u03b1 = \u03b1'), (Eq.recOn h\u2082 a : \u03b1') = a' :=\n    fun (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @HEq \u03b1 a \u03b1' a') => HEq.recOn h\u2081 fun h\u2082 : \u03b1 = \u03b1 => rfl\n  show (Eq.recOn (Eq.refl \u03b1) a : \u03b1) = a' from this \u03b1 a' h (Eq.refl \u03b1)\n#align eq_of_heq eq_of_hEq\n\n#print Prod.mk.inj /-\n/- The following four lemmas could not be automatically generated when the\n   structures were declared, so we prove them manually here. -/\ntheorem Prod.mk.inj {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 And (x\u2081 = x\u2082) (y\u2081 = y\u2082) := fun h =>\n  Prod.noConfusion h fun h\u2081 h\u2082 => \u27e8h\u2081, h\u2082\u27e9\n#align prod.mk.inj Prod.mk.inj\n-/\n\n#print Prod.mk.injArrow /-\ndef Prod.mk.injArrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u2200 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P := fun h\u2081 _ h\u2082 =>\n  Prod.noConfusion h\u2081 h\u2082\n#align prod.mk.inj_arrow Prod.mk.injArrow\n-/\n\n#print PProd.mk.inj /-\ntheorem PProd.mk.inj {\u03b1 : Sort u} {\u03b2 : Sort v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    PProd.mk x\u2081 y\u2081 = PProd.mk x\u2082 y\u2082 \u2192 And (x\u2081 = x\u2082) (y\u2081 = y\u2082) := fun h =>\n  PProd.noConfusion h fun h\u2081 h\u2082 => \u27e8h\u2081, h\u2082\u27e9\n#align pprod.mk.inj PProd.mk.inj\n-/\n\n#print PProd.mk.injArrow /-\ndef PProd.mk.injArrow {\u03b1 : Type u} {\u03b2 : Type v} {x\u2081 : \u03b1} {y\u2081 : \u03b2} {x\u2082 : \u03b1} {y\u2082 : \u03b2} :\n    (x\u2081, y\u2081) = (x\u2082, y\u2082) \u2192 \u2200 \u2983P : Sort w\u2984, (x\u2081 = x\u2082 \u2192 y\u2081 = y\u2082 \u2192 P) \u2192 P := fun h\u2081 _ h\u2082 =>\n  Prod.noConfusion h\u2081 h\u2082\n#align pprod.mk.inj_arrow PProd.mk.injArrow\n-/\n\n#print Sum /-\ninductive Sum (\u03b1 : Type u) (\u03b2 : Type v)\n  | inl (val : \u03b1) : Sum\n  | inr (val : \u03b2) : Sum\n#align sum Sum\n-/\n\n#print PSum /-\ninductive PSum (\u03b1 : Sort u) (\u03b2 : Sort v)\n  | inl (val : \u03b1) : PSum\n  | inr (val : \u03b2) : PSum\n#align psum PSum\n-/\n\n#print Or /-\n/-- Logical or.\n\n`or P Q`, with notation `P \u2228 Q`, is the proposition which is true if and only if `P` or `Q` is\ntrue.\n\nTo prove a goal `\u22a2 P \u2228 Q`, if you know which alternative you want to prove,\nyou can use the tactics `left` (which gives the goal `\u22a2 P`)\nor `right` (which gives the goal `\u22a2 Q`).\n\nGiven a hypothesis `h : P \u2228 Q` and goal `\u22a2 R`,\nthe tactic `cases h` will give you two copies of the goal `\u22a2 R`,\nwith the hypothesis `h : P` in the first, and the hypothesis `h : Q` in the second.\n-/\ninductive Or (a b : Prop) : Prop\n  | inl (h : a) : Or\n  | inr (h : b) : Or\n#align or Or\n-/\n\n#print Or.intro_left /-\ntheorem Or.intro_left {a : Prop} (b : Prop) (ha : a) : Or a b :=\n  Or.inl ha\n#align or.intro_left Or.intro_left\n-/\n\ntheorem Or.intro_right (a : Prop) {b : Prop} (hb : b) : Or a b :=\n  Or.inr hb\n#align or.intro_right Or.intro_right\u2093\n\n#print Sigma /-\nstructure Sigma {\u03b1 : Type u} (\u03b2 : \u03b1 \u2192 Type v) where mk ::\n  fst : \u03b1\n  snd : \u03b2 fst\n#align sigma Sigma\n-/\n\n#print PSigma /-\nstructure PSigma {\u03b1 : Sort u} (\u03b2 : \u03b1 \u2192 Sort v) where mk ::\n  fst : \u03b1\n  snd : \u03b2 fst\n#align psigma PSigma\n-/\n\n#print Bool /-\ninductive Bool : Type\n  | ff : Bool\n  | tt : Bool\n#align bool Bool\n-/\n\n#print Subtype /-\n/--\nRemark: subtype must take a Sort instead of Type because of the axiom strong_indefinite_description. -/\nstructure Subtype {\u03b1 : Sort u} (p : \u03b1 \u2192 Prop) where\n  val : \u03b1\n  property : p val\n#align subtype Subtype\n-/\n\nattribute [pp_using_anonymous_constructor] Sigma PSigma Subtype PProd And\n\n#print Decidable /-\nclass inductive Decidable (p : Prop)\n  | is_false (h : \u00acp) : Decidable\n  | is_true (h : p) : Decidable\n#align decidable Decidable\n-/\n\n#print DecidablePred /-\n@[reducible]\ndef DecidablePred {\u03b1 : Sort u} (r : \u03b1 \u2192 Prop) :=\n  \u2200 a : \u03b1, Decidable (r a)\n#align decidable_pred DecidablePred\n-/\n\n#print DecidableRel /-\n@[reducible]\ndef DecidableRel {\u03b1 : Sort u} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\n  \u2200 a b : \u03b1, Decidable (r a b)\n#align decidable_rel DecidableRel\n-/\n\n#print DecidableEq /-\n@[reducible]\ndef DecidableEq (\u03b1 : Sort u) :=\n  DecidableRel (@Eq \u03b1)\n#align decidable_eq DecidableEq\n-/\n\n#print Option /-\ninductive Option (\u03b1 : Type u)\n  | none : Option\n  | some (val : \u03b1) : Option\n#align option Option\n-/\n\nexport Option (none some)\n\nexport Bool (false true)\n\n#print List /-\ninductive List (T : Type u)\n  | nil : List\n  | cons (hd : T) (tl : List) : List\n#align list List\n-/\n\n#print Nat /-\ninductive Nat\n  | zero : Nat\n  | succ (n : Nat) : Nat\n#align nat Nat\n-/\n\nstructure UnificationConstraint where\n  {\u03b1 : Type u}\n  lhs : \u03b1\n  rhs : \u03b1\n#align unification_constraint UnificationConstraint\n\nstructure UnificationHint where\n  pattern : UnificationConstraint\n  constraints : List UnificationConstraint\n#align unification_hint UnificationHint\n\n/-! Declare builtin and reserved notation -/\n\n\n#print Zero /-\nclass Zero (\u03b1 : Type u) where\n  zero : \u03b1\n#align has_zero Zero\n-/\n\n#print One /-\nclass One (\u03b1 : Type u) where\n  one : \u03b1\n#align has_one One\n-/\n\n#print Add /-\nclass Add (\u03b1 : Type u) where\n  add : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_add Add\n-/\n\n#print Mul /-\nclass Mul (\u03b1 : Type u) where\n  mul : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_mul Mul\n-/\n\n#print Inv /-\nclass Inv (\u03b1 : Type u) where\n  inv : \u03b1 \u2192 \u03b1\n#align has_inv Inv\n-/\n\n#print Neg /-\nclass Neg (\u03b1 : Type u) where\n  neg : \u03b1 \u2192 \u03b1\n#align has_neg Neg\n-/\n\n#print Sub /-\nclass Sub (\u03b1 : Type u) where\n  sub : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_sub Sub\n-/\n\n#print Div /-\nclass Div (\u03b1 : Type u) where\n  div : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_div Div\n-/\n\n#print Dvd /-\nclass Dvd (\u03b1 : Type u) where\n  Dvd : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_dvd Dvd\n-/\n\n#print Mod /-\nclass Mod (\u03b1 : Type u) where\n  mod : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_mod Mod\n-/\n\n#print LE /-\nclass LE (\u03b1 : Type u) where\n  le : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_le LE\n-/\n\n#print LT /-\nclass LT (\u03b1 : Type u) where\n  lt : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_lt LT\n-/\n\n#print Append /-\nclass Append (\u03b1 : Type u) where\n  append : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_append Append\n-/\n\n#print AndThen' /-\nclass AndThen' (\u03b1 : Type u) (\u03b2 : Type v) (\u03c3 : outParam <| Type w) where\n  andthen : \u03b1 \u2192 \u03b2 \u2192 \u03c3\n#align has_andthen AndThen'\n-/\n\n#print Union /-\nclass Union (\u03b1 : Type u) where\n  union : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_union Union\n-/\n\n#print Inter /-\nclass Inter (\u03b1 : Type u) where\n  inter : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_inter Inter\n-/\n\n#print SDiff /-\nclass SDiff (\u03b1 : Type u) where\n  sdiff : \u03b1 \u2192 \u03b1 \u2192 \u03b1\n#align has_sdiff SDiff\n-/\n\nclass HasEquiv (\u03b1 : Sort u) where\n  Equiv : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_equiv HasEquiv\u2093\n\n#print HasSubset /-\nclass HasSubset (\u03b1 : Type u) where\n  Subset : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_subset HasSubset\n-/\n\n#print HasSSubset /-\nclass HasSSubset (\u03b1 : Type u) where\n  SSubset : \u03b1 \u2192 \u03b1 \u2192 Prop\n#align has_ssubset HasSSubset\n-/\n\n/-! Type classes `has_emptyc` and `has_insert` are\n   used to implement polymorphic notation for collections.\n   Example: `{a, b, c} = insert a (insert b (singleton c))`.\n\n   Note that we use `pair` in the name of lemmas about `{x, y} = insert x (singleton y)`. -/\n\n\n#print EmptyCollection /-\nclass EmptyCollection (\u03b1 : Type u) where\n  emptyCollection : \u03b1\n#align has_emptyc EmptyCollection\n-/\n\n#print Insert /-\nclass Insert (\u03b1 : outParam <| Type u) (\u03b3 : Type v) where\n  insert : \u03b1 \u2192 \u03b3 \u2192 \u03b3\n#align has_insert Insert\n-/\n\n#print Singleton /-\nclass Singleton (\u03b1 : outParam <| Type u) (\u03b2 : Type v) where\n  singleton : \u03b1 \u2192 \u03b2\n#align has_singleton Singleton\n-/\n\n#print Sep /-\n/-- Type class used to implement the notation { a \u2208 c | p a } -/\nclass Sep (\u03b1 : outParam <| Type u) (\u03b3 : Type v) where\n  sep : (\u03b1 \u2192 Prop) \u2192 \u03b3 \u2192 \u03b3\n#align has_sep Sep\n-/\n\n#print Membership /-\n/-- Type class for set-like membership -/\nclass Membership (\u03b1 : outParam <| Type u) (\u03b3 : Type v) where\n  Mem : \u03b1 \u2192 \u03b3 \u2192 Prop\n#align has_mem Membership\n-/\n\n#print Pow /-\nclass Pow (\u03b1 : Type u) (\u03b2 : Type v) where\n  pow : \u03b1 \u2192 \u03b2 \u2192 \u03b1\n#align has_pow Pow\n-/\n\nexport AndThen' (andthen)\n\nexport Pow (pow)\n\n-- mathport name: \u00abexpr \u2282 \u00bb\ninfixl:50\n  \" \u2282 \" =>-- Note this is different to `|`.\n  HasSSubset.SSubset\n\nexport Append (append)\n\n#print GE.ge /-\n@[reducible]\ndef GE.ge {\u03b1 : Type u} [LE \u03b1] (a b : \u03b1) : Prop :=\n  LE.le b a\n#align ge GE.ge\n-/\n\n#print GT.gt /-\n@[reducible]\ndef GT.gt {\u03b1 : Type u} [LT \u03b1] (a b : \u03b1) : Prop :=\n  LT.lt b a\n#align gt GT.gt\n-/\n\n#print Superset /-\n@[reducible]\ndef Superset {\u03b1 : Type u} [HasSubset \u03b1] (a b : \u03b1) : Prop :=\n  HasSubset.Subset b a\n#align superset Superset\n-/\n\n@[reducible]\ndef Ssuperset {\u03b1 : Type u} [HasSSubset \u03b1] (a b : \u03b1) : Prop :=\n  HasSSubset.SSubset b a\n#align ssuperset Ssuperset\n\n-- mathport name: \u00abexpr \u2287 \u00bb\ninfixl:50 \" \u2287 \" => Superset\n\n-- mathport name: \u00abexpr \u2283 \u00bb\ninfixl:50 \" \u2283 \" => Ssuperset\n\n#print bit0 /-\ndef bit0 {\u03b1 : Type u} [s : Add \u03b1] (a : \u03b1) : \u03b1 :=\n  a + a\n#align bit0 bit0\n-/\n\n#print bit1 /-\ndef bit1 {\u03b1 : Type u} [s\u2081 : One \u03b1] [s\u2082 : Add \u03b1] (a : \u03b1) : \u03b1 :=\n  bit0 a + 1\n#align bit1 bit1\n-/\n\nattribute [match_pattern] Zero.zero One.one bit0 bit1 Add.add Neg.neg Mul.mul\n\nexport Insert (insert)\n\n#print IsLawfulSingleton /-\nclass IsLawfulSingleton (\u03b1 : Type u) (\u03b2 : Type v) [EmptyCollection \u03b2] [Insert \u03b1 \u03b2] [Singleton \u03b1 \u03b2] :\n  Prop where\n  insert_emptyCollection_eq : \u2200 x : \u03b1, (insert x \u2205 : \u03b2) = {x}\n#align is_lawful_singleton IsLawfulSingleton\n-/\n\nexport Singleton (singleton)\n\nexport IsLawfulSingleton (insert_emptyCollection_eq)\n\nattribute [simp] insert_emptyc_eq\n\n/-! nat basic instances -/\n\n\nnamespace Nat\n\n#print Nat.add /-\nprotected def add : Nat \u2192 Nat \u2192 Nat\n  | a, zero => a\n  | a, succ b => succ (add a b)\n#align nat.add Nat.add\n-/\n\n/- We mark the following definitions as pattern to make sure they can be used in recursive equations,\n     and reduced by the equation compiler. -/\nattribute [match_pattern] Nat.add Nat.add\n\nend Nat\n\ninstance : Zero Nat :=\n  \u27e8Nat.zero\u27e9\n\ninstance : One Nat :=\n  \u27e8Nat.succ Nat.zero\u27e9\n\ninstance : Add Nat :=\n  \u27e8Nat.add\u27e9\n\n#print Std.Priority.default /-\ndef Std.Priority.default : Nat :=\n  1000\n#align std.priority.default Std.Priority.default\n-/\n\n#print Std.Priority.max /-\ndef Std.Priority.max : Nat :=\n  4294967295\n#align std.priority.max Std.Priority.max\n-/\n\nnamespace Nat\n\n#print Nat.prio /-\nprotected def prio :=\n  Std.Priority.default + 100\n#align nat.prio Nat.prio\n-/\n\nend Nat\n\n#print Std.Prec.max /-\n/-\n  Global declarations of right binding strength\n\n  If a module reassigns these, it will be incompatible with other modules that adhere to these\n  conventions.\n\n  When hovering over a symbol, use \"C-c C-k\" to see how to input it.\n-/\ndef Std.Prec.max : Nat :=\n  1024\n#align std.prec.max Std.Prec.max\n-/\n\n#print Std.Prec.arrow /-\n-- the strength of application, identifiers, (, [, etc.\ndef Std.Prec.arrow : Nat :=\n  25\n#align std.prec.arrow Std.Prec.arrow\n-/\n\n#print Std.Prec.maxPlus /-\n/-- This def is \"max + 10\". It can be used e.g. for postfix operations that should\nbe stronger than application.\n-/\ndef Std.Prec.maxPlus : Nat :=\n  Std.Prec.max + 10\n#align std.prec.max_plus Std.Prec.maxPlus\n-/\n\n#print SizeOf /-\n-- input with \\sy or \\-1 or \\inv\n-- notation for n-ary tuples\n-- sizeof\nclass SizeOf (\u03b1 : Sort u) where\n  sizeOf : \u03b1 \u2192 Nat\n#align has_sizeof SizeOf\n-/\n\n/- warning: sizeof clashes with has_sizeof.sizeof -> SizeOf.sizeOf\nCase conversion may be inaccurate. Consider using '#align sizeof SizeOf.sizeOf\u2093'. -/\n#print SizeOf.sizeOf /-\ndef SizeOf.sizeOf {\u03b1 : Sort u} [s : SizeOf \u03b1] : \u03b1 \u2192 Nat :=\n  SizeOf.sizeOf\n#align sizeof SizeOf.sizeOf\n-/\n\n/-!\nDeclare sizeof instances and lemmas for types declared before has_sizeof.\nFrom now on, the inductive compiler will automatically generate sizeof instances and lemmas.\n-/\n\n\n/-- Every type `\u03b1` has a default has_sizeof instance that just returns 0 for every element of `\u03b1` -/\nprotected def Default.sizeof (\u03b1 : Sort u) : \u03b1 \u2192 Nat\n  | a => 0\n#align default.sizeof Default.sizeof\n\ninstance defaultHasSizeof (\u03b1 : Sort u) : SizeOf \u03b1 :=\n  \u27e8Default.sizeof \u03b1\u27e9\n#align default_has_sizeof defaultHasSizeof\n\nprotected def Nat.sizeof : Nat \u2192 Nat\n  | n => n\n#align nat.sizeof Nat.sizeof\n\ninstance : SizeOf Nat :=\n  \u27e8Nat.sizeof\u27e9\n\nprotected def Prod.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : Prod \u03b1 \u03b2 \u2192 Nat\n  | \u27e8a, b\u27e9 => 1 + SizeOf.sizeOf a + SizeOf.sizeOf b\n#align prod.sizeof Prod.sizeof\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (Prod \u03b1 \u03b2) :=\n  \u27e8Prod.sizeof\u27e9\n\nprotected def Sum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : Sum \u03b1 \u03b2 \u2192 Nat\n  | Sum.inl a => 1 + SizeOf.sizeOf a\n  | Sum.inr b => 1 + SizeOf.sizeOf b\n#align sum.sizeof Sum.sizeof\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (Sum \u03b1 \u03b2) :=\n  \u27e8Sum.sizeof\u27e9\n\nprotected def PSum.sizeof {\u03b1 : Type u} {\u03b2 : Type v} [SizeOf \u03b1] [SizeOf \u03b2] : PSum \u03b1 \u03b2 \u2192 Nat\n  | PSum.inl a => 1 + SizeOf.sizeOf a\n  | PSum.inr b => 1 + SizeOf.sizeOf b\n#align psum.sizeof PSum.sizeof\n\ninstance (\u03b1 : Type u) (\u03b2 : Type v) [SizeOf \u03b1] [SizeOf \u03b2] : SizeOf (PSum \u03b1 \u03b2) :=\n  \u27e8PSum.sizeof\u27e9\n\nprotected def Sigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [\u2200 a, SizeOf (\u03b2 a)] :\n    Sigma \u03b2 \u2192 Nat\n  | \u27e8a, b\u27e9 => 1 + SizeOf.sizeOf a + SizeOf.sizeOf b\n#align sigma.sizeof Sigma.sizeof\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1] [\u2200 a, SizeOf (\u03b2 a)] : SizeOf (Sigma \u03b2) :=\n  \u27e8Sigma.sizeof\u27e9\n\nprotected def PSigma.sizeof {\u03b1 : Type u} {\u03b2 : \u03b1 \u2192 Type v} [SizeOf \u03b1] [\u2200 a, SizeOf (\u03b2 a)] :\n    PSigma \u03b2 \u2192 Nat\n  | \u27e8a, b\u27e9 => 1 + SizeOf.sizeOf a + SizeOf.sizeOf b\n#align psigma.sizeof PSigma.sizeof\n\ninstance (\u03b1 : Type u) (\u03b2 : \u03b1 \u2192 Type v) [SizeOf \u03b1] [\u2200 a, SizeOf (\u03b2 a)] : SizeOf (PSigma \u03b2) :=\n  \u27e8PSigma.sizeof\u27e9\n\nprotected def PUnit.sizeof : PUnit \u2192 Nat\n  | u => 1\n#align punit.sizeof PUnit.sizeof\n\ninstance : SizeOf PUnit :=\n  \u27e8PUnit.sizeof\u27e9\n\nprotected def Bool.sizeof : Bool \u2192 Nat\n  | b => 1\n#align bool.sizeof Bool.sizeof\n\ninstance : SizeOf Bool :=\n  \u27e8Bool.sizeof\u27e9\n\nprotected def Option.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : Option \u03b1 \u2192 Nat\n  | none => 1\n  | some a => 1 + SizeOf.sizeOf a\n#align option.sizeof Option.sizeof\n\ninstance (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (Option \u03b1) :=\n  \u27e8Option.sizeof\u27e9\n\nprotected def List.sizeof {\u03b1 : Type u} [SizeOf \u03b1] : List \u03b1 \u2192 Nat\n  | List.nil => 1\n  | List.cons a l => 1 + SizeOf.sizeOf a + List.sizeof l\n#align list.sizeof List.sizeof\n\ninstance (\u03b1 : Type u) [SizeOf \u03b1] : SizeOf (List \u03b1) :=\n  \u27e8List.sizeof\u27e9\n\nprotected def Subtype.sizeof {\u03b1 : Type u} [SizeOf \u03b1] {p : \u03b1 \u2192 Prop} : Subtype p \u2192 Nat\n  | \u27e8a, _\u27e9 => SizeOf.sizeOf a\n#align subtype.sizeof Subtype.sizeof\n\ninstance {\u03b1 : Type u} [SizeOf \u03b1] (p : \u03b1 \u2192 Prop) : SizeOf (Subtype p) :=\n  \u27e8Subtype.sizeof\u27e9\n\n#print Nat.add_zero /-\ntheorem Nat.add_zero (n : Nat) : n + 0 = n :=\n  rfl\n#align nat_add_zero Nat.add_zero\n-/\n\n#print BinTree /-\n/-- Auxiliary datatype for #[ ... ] notation.\n    #[1, 2, 3, 4] is notation for\n\n    bin_tree.node\n      (bin_tree.node (bin_tree.leaf 1) (bin_tree.leaf 2))\n      (bin_tree.node (bin_tree.leaf 3) (bin_tree.leaf 4))\n\n    We use this notation to input long sequences without exhausting the system stack space.\n    Later, we define a coercion from `bin_tree` into `list`.\n-/\ninductive BinTree (\u03b1 : Type u)\n  | Empty : BinTree\n  | leaf (val : \u03b1) : BinTree\n  | node (left right : BinTree) : BinTree\n#align bin_tree BinTree\n-/\n\nattribute [elab_without_expected_type] BinTree.node BinTree.leaf\n\n#print inferInstance /-\n/-- Like `by apply_instance`, but not dependent on the tactic framework. -/\n@[reducible]\ndef inferInstance {\u03b1 : Sort u} [i : \u03b1] : \u03b1 :=\n  i\n#align infer_instance inferInstance\n-/\n\n", "meta": {"author": "leanprover-community", "repo": "lean3port", "sha": "9ed1898f23e4379865ee93d62cb6353e5ed6c270", "save_path": "github-repos/lean/leanprover-community-lean3port", "path": "github-repos/lean/leanprover-community-lean3port/lean3port-9ed1898f23e4379865ee93d62cb6353e5ed6c270/Leanbin/Init/Core.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646254, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.35123299706489847}}
{"text": "import graphs.allexts\nopen function -- allows us to use key results about functions\n\n/-\n... and now for the converse ...\n-/\n/- Lemma :\n-/\nlemma bij.iso {G H : graph} (\u03b1 : G \u21a6 H) (hv : bijective \u03b1.vertex_map) (he : bijective \u03b1.edge_map) : isomorphism \u03b1 :=\nbegin\n  sorry,\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n  \nend", "meta": {"author": "barriecooper", "repo": "lean-graphs", "sha": "3f7be961f99fe084f950f52fe17c53e8093b5337", "save_path": "github-repos/lean/barriecooper-lean-graphs", "path": "github-repos/lean/barriecooper-lean-graphs/lean-graphs-3f7be961f99fe084f950f52fe17c53e8093b5337/src/game/morphisms-world/level9.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6334102636778403, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.35120728341215895}}
{"text": "-- Copyright (c) 2018 Scott Morrison. All rights reserved.\n-- Released under Apache 2.0 license as described in the file LICENSE.\n-- Authors: Scott Morrison\n\nimport category_theory.eq_to_hom\nimport category_theory.limits.cones\nimport ...basic\n\nopen category_theory\n\nnamespace category_theory.limits\n\nuniverses v u\n\nlocal attribute [tidy] tactic.case_bash\n\n@[derive decidable_eq] inductive walking_cospan : Type v\n| left | right | one\n@[derive decidable_eq] inductive walking_span : Type v\n| zero | left | right\n\nopen walking_cospan\nopen walking_span\n\ninductive walking_cospan_hom : walking_cospan \u2192 walking_cospan \u2192 Type v\n| inl : walking_cospan_hom left one\n| inr : walking_cospan_hom right one\n| id : \u03a0 X : walking_cospan.{v}, walking_cospan_hom X X\ninductive walking_span_hom : walking_span \u2192 walking_span \u2192 Type v\n| fst : walking_span_hom zero left\n| snd : walking_span_hom zero right\n| id : \u03a0 X : walking_span.{v}, walking_span_hom X X\n\nopen walking_cospan_hom\nopen walking_span_hom\n\ninstance walking_cospan_category : small_category.{v+1} walking_cospan :=\n{ hom := walking_cospan_hom,\n  id := walking_cospan_hom.id,\n  comp := \u03bb X Y Z f g, match X, Y, Z, f, g with\n  | _, _ ,_, (id _), h := h\n  | _, _, _, inl, (id one) := inl\n  | _, _, _, inr, (id one) := inr\n  end }\n\ninstance walking_span_category : small_category.{v+1} walking_span :=\n{ hom := walking_span_hom,\n  id := walking_span_hom.id,\n  comp := \u03bb X Y Z f g, match X, Y, Z, f, g with\n  | _, _ ,_, (id _), h := h\n  | _, _, _, fst, (id left) := fst\n  | _, _, _, snd, (id right) := snd\n  end }\n\nlemma walking_cospan_hom_id (X : walking_cospan.{v}) : walking_cospan_hom.id X = \ud835\udfd9 X := rfl\nlemma walking_span_hom_id (X : walking_span.{v}) : walking_span_hom.id X = \ud835\udfd9 X := rfl\n\nvariables {C : Type u} [\ud835\udc9e : category.{v+1} C]\ninclude \ud835\udc9e\n\ndef cospan {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) : walking_cospan.{v} \u2964 C :=\n{ obj := \u03bb x, match x with\n  | left := X\n  | right := Y\n  | one := Z\n  end,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, inl := f\n  | _, _, inr := g\n  end }\ndef span {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) : walking_span.{v} \u2964 C :=\n{ obj := \u03bb x, match x with\n  | zero := X\n  | left := Y\n  | right := Z\n  end,\n  map := \u03bb x y h, match x, y, h with\n  | _, _, (id _) := \ud835\udfd9 _\n  | _, _, fst := f\n  | _, _, snd := g\n  end }\n\n@[simp] lemma cospan_left {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (cospan f g).obj walking_cospan.left = X := rfl\n@[simp] lemma span_left {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) :\n  (span f g).obj walking_span.left = Y := rfl\n\n@[simp] lemma cospan_right {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (cospan f g).obj walking_cospan.right = Y := rfl\n@[simp] lemma span_right {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) :\n  (span f g).obj walking_span.right = Z := rfl\n\n@[simp] lemma cospan_one {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (cospan f g).obj walking_cospan.one = Z := rfl\n@[simp] lemma span_zero {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) :\n  (span f g).obj walking_span.zero = X := rfl\n\n@[simp] lemma cospan_map_inl {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (cospan f g).map walking_cospan_hom.inl = f := rfl\n@[simp] lemma span_map_fst {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) :\n  (span f g).map walking_span_hom.fst = f := rfl\n\n@[simp] lemma cospan_map_inr {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) :\n  (cospan f g).map walking_cospan_hom.inr = g := rfl\n@[simp] lemma span_map_snd {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) :\n  (span f g).map walking_span_hom.snd = g := rfl\n\n@[simp] lemma cospan_map_id {X Y Z : C} (f : X \u27f6 Z) (g : Y \u27f6 Z) (w : walking_cospan) :\n  (cospan f g).map (walking_cospan_hom.id w) = \ud835\udfd9 _ := rfl\n@[simp] lemma span_map_id {X Y Z : C} (f : X \u27f6 Y) (g : X \u27f6 Z) (w : walking_span) :\n  (span f g).map (walking_span_hom.id w) = \ud835\udfd9 _ := rfl\n\n\nvariables {X Y Z : C}\n\nattribute [simp] walking_cospan_hom_id walking_span_hom_id\n\nsection pullback\ndef square (f : X \u27f6 Z) (g : Y \u27f6 Z) := cone (cospan f g)\n\nvariables {f : X \u27f6 Z} {g : Y \u27f6 Z}\n\ndef square.\u03c0\u2081 (t : square f g) : t.X \u27f6 X := t.\u03c0.app left\ndef square.\u03c0\u2082 (t : square f g) : t.X \u27f6 Y := t.\u03c0.app right\n\ndef square.mk {W : C} (\u03c0\u2081 : W \u27f6 X) (\u03c0\u2082 : W \u27f6 Y)\n  (eq : \u03c0\u2081 \u226b f = \u03c0\u2082 \u226b g) :\n  square f g :=\n{ X := W,\n  \u03c0 :=\n  { app := \u03bb j, walking_cospan.cases_on j \u03c0\u2081 \u03c0\u2082 (\u03c0\u2081 \u226b f),\n    naturality' := \u03bb j j' f, by cases f; obviously } }\n\ndef square.condition (t : square f g) : (square.\u03c0\u2081 t) \u226b f = (square.\u03c0\u2082 t) \u226b g :=\nbegin\n  erw [t.w inl, \u2190 t.w inr], refl\nend\n\nend pullback\n\nsection pushout\ndef cosquare (f : X \u27f6 Y) (g : X \u27f6 Z) := cocone (span f g)\n\nvariables {f : X \u27f6 Y} {g : X \u27f6 Z}\n\ndef cosquare.\u03b9\u2081 (t : cosquare f g) : Y \u27f6 t.X := t.\u03b9.app left\ndef cosquare.\u03b9\u2082 (t : cosquare f g) : Z \u27f6 t.X := t.\u03b9.app right\n\ndef cosquare.mk {W : C} (\u03b9\u2081 : Y \u27f6 W) (\u03b9\u2082 : Z \u27f6 W)\n  (eq : f \u226b \u03b9\u2081 = g \u226b \u03b9\u2082) :\n  cosquare f g :=\n{ X := W,\n  \u03b9 :=\n  { app := \u03bb j, walking_span.cases_on j (f \u226b \u03b9\u2081) \u03b9\u2081 \u03b9\u2082,\n    naturality' := \u03bb j j' f, by cases f; obviously } }\n\ndef cosquare.condition (t : cosquare f g) : f \u226b (cosquare.\u03b9\u2081 t) = g \u226b (cosquare.\u03b9\u2082 t) :=\nbegin\n  erw [t.w fst, \u2190 t.w snd], refl\nend\n\nend pushout\n\ndef cone.of_square\n  {F : walking_cospan.{v} \u2964 C} (t : square (F.map inl) (F.map inr)) : cone F :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb X, t.\u03c0.app X \u226b eq_to_hom (by tidy),\n    naturality' := \u03bb j j' g,\n    begin\n      cases j; cases j'; cases g; dsimp; simp,\n      erw \u2190 t.w inl, refl,\n      erw \u2190 t.w inr, refl,\n    end } }.\n\n@[simp] lemma cone.of_square_\u03c0\n  {F : walking_cospan.{v} \u2964 C} (t : square (F.map inl) (F.map inr)) (j):\n  (cone.of_square t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\ndef cocone.of_cosquare\n  {F : walking_span.{v} \u2964 C} (t : cosquare (F.map fst) (F.map snd)) : cocone F :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb X, eq_to_hom (by tidy) \u226b t.\u03b9.app X,\n    naturality' := \u03bb j j' g,\n    begin\n      cases j; cases j'; cases g; dsimp; simp,\n      erw \u2190 t.w fst, refl,\n      erw \u2190 t.w snd, refl,\n    end } }.\n\n@[simp] lemma cocone.of_cosquare_\u03b9\n  {F : walking_span.{v} \u2964 C} (t : cosquare (F.map fst) (F.map snd)) (j):\n  (cocone.of_cosquare t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\ndef square.of_cone\n  {F : walking_cospan.{v} \u2964 C} (t : cone F) : square (F.map inl) (F.map inr) :=\n{ X := t.X,\n  \u03c0 :=\n  { app := \u03bb j, t.\u03c0.app j \u226b eq_to_hom (by tidy) } }\n\n@[simp] lemma square.of_cone_\u03c0 {F : walking_cospan.{v} \u2964 C} (t : cone F) (j) :\n  (square.of_cone t).\u03c0.app j = t.\u03c0.app j \u226b eq_to_hom (by tidy) := rfl\n\ndef cosquare.of_cocone\n  {F : walking_span.{v} \u2964 C} (t : cocone F) : cosquare (F.map fst) (F.map snd) :=\n{ X := t.X,\n  \u03b9 :=\n  { app := \u03bb j, eq_to_hom (by tidy) \u226b t.\u03b9.app j } }\n\n@[simp] lemma cosquare.of_cocone_\u03b9 {F : walking_span.{v} \u2964 C} (t : cocone F) (j) :\n  (cosquare.of_cocone t).\u03b9.app j = eq_to_hom (by tidy) \u226b t.\u03b9.app j := rfl\n\nend category_theory.limits\n", "meta": {"author": "formalabstracts", "repo": "formalabstracts", "sha": "b0173da1af45421239d44492eeecd54bf65ee0f6", "save_path": "github-repos/lean/formalabstracts-formalabstracts", "path": "github-repos/lean/formalabstracts-formalabstracts/formalabstracts-b0173da1af45421239d44492eeecd54bf65ee0f6/src/category_theory/limits/shapes/pullbacks.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3512072834121589}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport data.dlist\n\n/-!\n# Difference list\n\nThis file provides a few results about `dlist`, which is defined in core Lean.\n\nA difference list is a function that, given a list, returns the original content of the\ndifference list prepended to the given list. It is useful to represent elements of a given type\nas `a\u2081 + ... + a\u2099` where `+ : \u03b1 \u2192 \u03b1 \u2192 \u03b1` is any operation, without actually computing.\n\nThis structure supports `O(1)` `append` and `concat` operations on lists, making it\nuseful for append-heavy uses such as logging and pretty printing.\n-/\n\n/-- Concatenates a list of difference lists to form a single difference list. Similar to\n`list.join`. -/\ndef dlist.join {\u03b1 : Type*} : list (dlist \u03b1) \u2192 dlist \u03b1\n | [] := dlist.empty\n | (x :: xs) := x ++ dlist.join xs\n\n@[simp] lemma dlist_singleton {\u03b1 : Type*} {a : \u03b1} :\n  dlist.singleton a = dlist.lazy_of_list ([a]) := rfl\n\n@[simp] lemma dlist_lazy {\u03b1 : Type*} {l : list \u03b1} :\n  dlist.lazy_of_list l = dlist.of_list l := rfl\n", "meta": {"author": "jjaassoonn", "repo": "projective_space", "sha": "11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce", "save_path": "github-repos/lean/jjaassoonn-projective_space", "path": "github-repos/lean/jjaassoonn-projective_space/projective_space-11fe19fe9d7991a272e7a40be4b6ad9b0c10c7ce/src/data/dlist/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3511709017424895}}
{"text": "/-\nCopyright (c) 2022 Andrew Yang. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Andrew Yang\n-/\nimport morphisms.open_immersion\n\n/-!\n\n# Open immersions\n\nA morphism is an open immersions if the underlying map of spaces is an open embedding\n`f : X \u27f6 U \u2286 Y`, and the sheaf map `Y(V) \u27f6 f _* X(V)` is an iso for each `V \u2286 U`.\n\nMost of the theories are developed in `algebraic_geometry/open_immersion`, and we provide the\nremaining theorems analogous to other lemmas in `algebraic_geometry/morphisms/*`.\n\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits opposite topological_space\n\nuniverse u\n\nnamespace algebraic_geometry\n\nvariables {X Y Z : Scheme.{u}} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\nlemma is_iso_iff_stalk {f : X \u27f6 Y} :\n  is_iso f \u2194\n    is_iso f.1.base \u2227 \u2200 x, is_iso (PresheafedSpace.stalk_map f.1 x) :=\nbegin\n  split,\n  { intro h, exactI \u27e8infer_instance, infer_instance\u27e9 },\n  { rintro \u27e8h\u2081, h\u2082\u27e9, resetI,\n    haveI := is_open_immersion.of_stalk_iso f (Top.homeo_of_iso $ as_iso f.1.base).open_embedding,\n    exact is_open_immersion.to_iso f }\nend\n\nlemma is_iso_respects_iso : morphism_property.respects_iso (@is_iso Scheme _) :=\nby split; { introv H, resetI, apply_instance }\n\nlemma is_iso_is_local_at_target : property_is_local_at_target (@is_iso Scheme _) :=\nbegin\n  constructor,\n  { exact is_iso_respects_iso },\n  { introsI, apply_instance },\n  { introsI X Y f \ud835\udcb0 H,\n    haveI := is_open_immersion_is_local_at_target.3 f \ud835\udcb0 infer_instance,\n    suffices : function.surjective f.1.base,\n    { rw \u2190 Top.epi_iff_surjective f.1.base at this, exactI is_open_immersion.to_iso f },\n    have := congr_arg (coe : opens Y.carrier \u2192 set Y.carrier) \ud835\udcb0.supr_opens_range,\n    rw opens.coe_supr at this,\n    rw set.surjective_iff_surjective_of_Union_eq_univ this,\n    intro i,\n    haveI := (is_iso_respects_iso.arrow_iso_iff\n      (morphism_restrict_opens_range f (\ud835\udcb0.map i))).mpr (H i),\n    have : epi (arrow.mk (f \u2223_ Scheme.hom.opens_range (\ud835\udcb0.map i))).hom.1.base := infer_instance,\n    rw [Top.epi_iff_surjective, arrow.mk_hom, morphism_restrict_val_base] at this,\n    exact this }\nend\n\nlemma is_iso.open_cover_tfae {X Y : Scheme.{u}} (f : X \u27f6 Y) :\n  tfae [is_iso f,\n    \u2203 (\ud835\udcb0 : Scheme.open_cover.{u} Y), \u2200 (i : \ud835\udcb0.J),\n      is_iso (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (\ud835\udcb0 : Scheme.open_cover.{u} Y) (i : \ud835\udcb0.J),\n      is_iso (pullback.snd : (\ud835\udcb0.pullback_cover f).obj i \u27f6 \ud835\udcb0.obj i),\n    \u2200 (U : opens Y.carrier), is_iso (f \u2223_ U),\n    \u2200 {U : Scheme} (g : U \u27f6 Y) [is_open_immersion g],\n      is_iso (pullback.snd : pullback f g \u27f6 U),\n    \u2203 {\u03b9 : Type u} (U : \u03b9 \u2192 opens Y.carrier) (hU : supr U = \u22a4),\n      \u2200 i, is_iso (f \u2223_ (U i))] :=\nis_iso_is_local_at_target.open_cover_tfae f\n\nlemma is_iso_of_is_affine_is_iso {X Y : Scheme} [hX : is_affine X] [hY : is_affine Y] (f : X \u27f6 Y)\n  [hf : is_iso (f.1.c.app (op \u22a4))] : is_iso f :=\nbegin\n  rw \u2190 mem_Spec_ess_image at hX hY,\n  have : is_iso (AffineScheme.\u0393.map (@quiver.hom.op AffineScheme _ \u27e8X, hX\u27e9 \u27e8Y, hY\u27e9 f)) := hf,\n  have := @@is_iso_of_reflects_iso _ _ _ _ this _,\n  exact @@functor.map_is_iso _ _ AffineScheme.forget_to_Scheme _ (@@is_iso_of_op _ _ this)\nend\n\nlemma target_affine_locally_affine_and_is_iso :\n  target_affine_locally (\u03bb X Y f hY, is_affine X \u2227 is_iso (Scheme.\u0393.map f.op)) = @is_iso Scheme _ :=\nbegin\n  rw \u2190 is_iso_is_local_at_target.target_affine_locally_eq,\n  congr,\n  ext X Y f hY,\n  split,\n  { rintros \u27e8hX, hf\u27e9, exactI @@is_iso_of_is_affine_is_iso _ _ f hf },\n  { intro hf, exactI \u27e8is_affine_of_iso f, infer_instance\u27e9 }\nend\n\nend algebraic_geometry", "meta": {"author": "erdOne", "repo": "lean-AG-morphisms", "sha": "bfb65e7d5c17f333abd7b1806717f12cd29427fd", "save_path": "github-repos/lean/erdOne-lean-AG-morphisms", "path": "github-repos/lean/erdOne-lean-AG-morphisms/lean-AG-morphisms-bfb65e7d5c17f333abd7b1806717f12cd29427fd/src/morphisms/isomorphism.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.3511709017424895}}
{"text": "import tactic\n#print thunk\nopen set\nopen_locale nnreal\n\n#print nat.strong_induction_on\n\ndef SUBMISSION : Prop := Pi (x : \u211d\u22650) (P : set \u211d\u22650) (hP : is_open P)\n  (ih : \u2200 x : \u211d\u22650, (\u2200 y, y < x \u2192 y \u2208 P) \u2192 x \u2208 P), x \u2208 P\n\nlemma nnreal_induction_on (x : \u211d\u22650) (P : set \u211d\u22650) (hP : is_open P)\n  (ih : \u2200 x : \u211d\u22650, (\u2200 y, y < x \u2192 y \u2208 P) \u2192 x \u2208 P) : x \u2208 P :=\nclassical.by_contradiction $ \u03bb hx,\nhave hbI : bdd_below P\u1d9c, from \u27e80, \u03bb _, by simp\u27e9,\nhave hI : Inf P\u1d9c \u2208 P\u1d9c,\n  from is_closed.cInf_mem (is_closed_compl_iff.mpr hP) \u27e8x, hx\u27e9 hbI,\nhI (ih _ (\u03bb y hyI, by_contradiction $ \u03bb hy, not_le_of_gt hyI (cInf_le hbI hy)))\n\ndef G : SUBMISSION := @nnreal_induction_on\n\n#print axioms G\n\n\nlemma nnreal_recursion {\u03b1 : Type*} [topological_space \u03b1]\n  (ih : \u03a0 x : \u211d\u22650, (\u2200 y, y < x \u2192 \u03b1) \u2192 \u03b1) (Hih : count(x : \u211d\u22650) : \u03b1 :=\n\n\n\n#exit\nimport data.nat.prime data.nat.parity tactic\n#print nat.even_pow\ntheorem even_of_prime_succ_pow (a b : \u2115) (ha : a > 1) (hb : b > 1)\n  (hp : nat.prime (a^b + 1)) : 2 \u2223 a :=\nhave \u00ac even (a ^ b + 1),\n  from hp.eq_two_or_odd.elim\n    (\u03bb h, begin\n      have : 1 < a ^ b, from nat.one_lt_pow _ _ (by linarith) ha,\n      linarith\n    end)\n    (by simp [nat.not_even_iff]),\nbegin\n  rw [nat.even_add, iff_false_intro nat.not_even_one, iff_false, not_not,\n    nat.even_pow] at this,\n  exact this.1\nend\n\n\n#exit\nimport category_theory.limits.shapes.pullbacks\n\nopen category_theory\nopen category_theory.limits\n\n\n\nexample {A B C : Prop} : A \u2227 (B \u2228 C) \u2194 (A \u2227 B) \u2228 (A \u2227 C) :=\n\u27e8\u03bb h, h.right.elim (\u03bb hB, or.inl \u27e8h.left, hB\u27e9) (\u03bb hC, or.inr \u27e8h.left, hC\u27e9),\n  \u03bb h, h.elim (\u03bb hAB, \u27e8hAB.left, or.inl hAB.right\u27e9) (\u03bb hAC, \u27e8hAC.left, or.inr hAC.right\u27e9)\u27e9\n\nexample {A B C : Prop} : A \u2228 (B \u2227 C) \u2194 (A \u2228 B) \u2227 (A \u2228 C) :=\n\u27e8\u03bb h, h.elim (\u03bb hA, \u27e8or.inl hA, or.inl hA\u27e9) (\u03bb hBC, \u27e8or.inr hBC.left, or.inr hBC.right\u27e9),\n \u03bb h, h.left.elim or.inl (\u03bb hB, h.right.elim or.inl (\u03bb hC, or.inr \u27e8hB, hC\u27e9))\u27e9\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C]\n\ndef pushout_of_epi {X Y : C} (f : X \u27f6 Y) [epi f] :\n  is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f) :=\npushout_cocone.is_colimit.mk\n  _\n  _\n  _\n  (\u03bb s, s.inl)\n  (by simp)\n  (\u03bb s, @epi.left_cancellation _ _ _ _ f _ _ _ _\n    (by simp [s.condition]))\n  (by simp { contextual := tt })\n\ntheorem epi_of_pushout {X Y : C} (f : X \u27f6 Y)\n  (is_colim : is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f)) : epi f :=\n{ left_cancellation := \u03bb Z g h H,\n    (is_colim.fac (pushout_cocone.mk _ _ H) (walking_span.left)).symm.trans\n      (is_colim.fac (pushout_cocone.mk _ _ H) (walking_span.right))}\n\n#exit\nimport tactic\n\ndef arith_sum : \u2115 \u2192 \u2115\n| 0 := 0\n| (nat.succ n) := nat.succ n + arith_sum n\n\ndef arith_formula (n : \u2115) : \u2115 := n * (n + 1) / 2\n\ntheorem arith_eq_aux (n : \u2115) : arith_sum n * 2 = n * (n + 1) :=\nbegin\n  induction n with n ih,\n  { refl },\n  { rw [arith_sum, add_mul, ih, nat.succ_eq_add_one],\n    ring }\nend\n\ntheorem arith_eq (n : \u2115) : arith_formula n = arith_sum n :=\nnat.div_eq_of_eq_mul_left (by norm_num) (arith_eq_aux n).symm\n\n#exit\nimport category_theory.limits.shapes.pullbacks\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables {C : Type u} [category.{v} C]\n\n#print limit.lift\n#print walking_cospan\n#print pullback_cone\n#print cone\ndef right_is_pullback {X Y Z U V : C} (f : X \u27f6 Y) (g : X \u27f6 Z) (u\u2081 : Y \u27f6 U) (u\u2082 : Z \u27f6 U)\n  (v\u2081 : Y \u27f6 V) (v\u2082 : Z \u27f6 V) (hu : f \u226b u\u2081 = g \u226b u\u2082) (hv : f \u226b v\u2081 = g \u226b v\u2082)\n  (is_pullback : is_limit (pullback_cone.mk _ _ hu))\n  (is_pushout : is_colimit (pushout_cocone.mk _ _ hv)) :\n  is_limit (pullback_cone.mk _ _ hv) :=\nlet h : V \u27f6 U := is_pushout.desc (pushout_cocone.mk u\u2081 u\u2082 hu) in\nhave Hh\u2081 : v\u2081 \u226b h = u\u2081, from is_pushout.fac (pushout_cocone.mk u\u2081 u\u2082 hu) walking_span.left,\nhave Hh\u2082 : v\u2082 \u226b h = u\u2082, from is_pushout.fac (pushout_cocone.mk u\u2081 u\u2082 hu) walking_span.right,\nlet S : pullback_cone v\u2081 v\u2082 \u2192 pullback_cone u\u2081 u\u2082 :=\n  \u03bb s, pullback_cone.mk s.fst s.snd\n    (by rw [\u2190 Hh\u2081, \u2190 Hh\u2082, \u2190 category.assoc, \u2190 category.assoc, s.condition]) in\npullback_cone.is_limit.mk _ _ _\n  (\u03bb s, is_pullback.lift (S s))\n  (\u03bb s, is_pullback.fac (S s) walking_cospan.left)\n  (\u03bb s, is_pullback.fac (S s) walking_cospan.right)\n  begin\n    assume s (m : s.X \u27f6 X) h\u2081 h\u2082,\n    refine is_pullback.uniq (S s) m (\u03bb j, _),\n    cases j,\n    { dsimp,\n      simp [\u2190 h\u2081] },\n    { cases j; dsimp; simp * }\n  end\n\n#exit\nimport algebra.group\n\nclass my_group (G : Type*) extends semigroup G :=\n( middle : \u2200 (x : G), \u2203! (y : G), x * y * x = x)\n\nvariables {G : Type*} [my_group G] --[nonempty G]\n\nnoncomputable theory\n\nnamespace my_group\n\ninstance A : has_inv G := \u27e8\u03bb x, classical.some (my_group.middle x)\u27e9\n\nlemma mul_inv_mul (x : G) : x * x\u207b\u00b9 * x = x :=\n(classical.some_spec (my_group.middle x)).1\n\nlemma inv_unique {x y : G} (h : x * y * x = x) : y = x\u207b\u00b9 :=\n(classical.some_spec (my_group.middle x)).2 _ h\n\nvariable [nonempty G]\n\nopen_locale classical\n\n\nvariables (x y z : G)\n\ndef one := x * x\u207b\u00b9\n\nlemma one_inv : (one x)\u207b\u00b9 = one x :=\n(inv_unique begin\n  delta one,\n  assoc_rw [mul_inv_mul, mul_inv_mul],\nend).symm\n\nexample : (x * y)\u207b\u00b9 = y\u207b\u00b9 * x\u207b\u00b9 :=\neq.symm (inv_unique _)\n\nexample : x * x\u207b\u00b9 * y * y\u207b\u00b9 = (y\u207b\u00b9 * y)\u207b\u00b9 :=\ninv_unique _\n\nlemma one_eq_one : one x = (y * y\u207b\u00b9) :=\nbegin\n\n\nend\n\n@[simp] lemma inv_inv : x\u207b\u00b9\u207b\u00b9 = x :=\n(inv_unique (inv_unique (by assoc_rw [mul_inv_mul, mul_inv_mul]))).symm\n\nlemma inv_mul_inv : x\u207b\u00b9 * x * x\u207b\u00b9 = x\u207b\u00b9 :=\ncalc x\u207b\u00b9 * x * x\u207b\u00b9 = x\u207b\u00b9 * x\u207b\u00b9\u207b\u00b9 * x\u207b\u00b9 : by rw inv_inv\n... = x\u207b\u00b9 : mul_inv_mul (x\u207b\u00b9)\n\nlemma one_eq_one : one (x\u207b\u00b9) = one x :=\nbegin\n  rw [\u2190 one_inv x, one, one],\n  refine (inv_unique _),\n  simp only [mul_assoc],\n  refine congr_arg _ _,\n  refine (inv_unique _),\n  assoc_rw [mul_inv_mul],\n  rw inv_inv,\n\nend\n\n\nexample : (x * y)\u207b\u00b9 = y\u207b\u00b9 * x\u207b\u00b9 :=\n(inv_unique _).symm\n\nexample : x\u207b\u00b9 * 1 = x\u207b\u00b9 :=\ninv_unique _\n\nexample : y\u207b\u00b9 * x\u207b\u00b9 * x = y\u207b\u00b9 :=\ninv_unique _\n\n\nlemma mul_one_mul_arbitrary {x : G} : x * 1 * classical.arbitrary G = x * classical.arbitrary G :=\n\n\nlemma mul_one (x : G) : x * x\u207b\u00b9 = 1 :=\n\n\nend my_group\n\n#exit\nimport data.real.basic\n\n/- We first define uniform convergence -/\ndef unif_convergence (f : \u2115 \u2192 \u211d \u2192 \u211d) (g : \u211d \u2192 \u211d) :=\n\u2200 \u03b5 > 0, \u2203 N : \u2115, \u2200 n \u2265 N, \u2200 x : \u211d, abs (f n x - g x) < \u03b5\n\n/- Use the notation f \u2192 g to denote that the sequence f\u2099 converges uniformly to g\n    You can type \u27f6 by writing \\hom -/\nnotation f `\u27f6` g := unif_convergence f g\n\n/- We also define the notion of the limit of a function \u211d \u2192 \u211d at a point -/\ndef fun_limit (f : \u211d \u2192 \u211d) (a l) :=\n\u2200 \u03b5 > 0, \u2203 \u03b4 > 0, \u2200 x : \u211d, x \u2260 a \u2192 abs (x - a) < \u03b4 \u2192 abs (f x - l) < \u03b5\n\n/- And the notion of the limit of a sequence -/\ndef seq_limit (f : \u2115 \u2192 \u211d) (l) :=\n\u2200 \u03b5 > 0, \u2203 N : \u2115, \u2200 n \u2265 N, abs (f n - l) < \u03b5\n\n/- If f\u2099 is a sequence of functions which converge uniformly to a function g\n    and if lim_{x \u2192 a} f\u2099(x) exists for each n then lim_{x \u2192 a} g(x) exists\n    and is equal to lim_{n \u2192 \u221e} lim_{x \u2192 a} f\u2099(x). -/\ntheorem limits_commute (f : \u2115 \u2192 \u211d \u2192 \u211d) (g : \u211d \u2192 \u211d) (l : \u2115 \u2192 \u211d) (a : \u211d) :\n    (f \u27f6 g) \u2192 (\u2200 n, fun_limit (f n) a (l n)) \u2192 \u2203 l', fun_limit g a l' \u2227 seq_limit l l' :=\nbegin\n  rw [unif_convergence],\n  delta fun_limit seq_limit,\n  assume hfg hs,\n\nend\n\n\n#exit\nimport data.fintype.basic\nimport set_theory.ordinal_arithmetic\nimport order.order_iso_nat\n\nuniverse u\n\nvariable {\u03b1 : Type u}\n\n--open ordinal\n\nattribute [elab_as_eliminator] well_founded.fix\nopen_locale classical\nnoncomputable theory\n\n#print ordinal.omega\n\nnoncomputable def nat_embedding [infinite \u03b1] (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n  (hrwo : is_well_order \u03b1 r) : \u2115 \u2192 \u03b1\n| n := well_founded.min hrwo.wf\n  (finset.univ.image (\u03bb m : fin n, have wf : m.1 < n := m.2, nat_embedding m.1))\u1d9c\n  (infinite.exists_not_mem_finset\n    (finset.univ.image (\u03bb m : fin n, have wf : m.1 < n := m.2, nat_embedding m.1)))\n\n#print rel_embedding.swap\ntheorem fintype_of_well_order (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n  (hrwo : is_well_order \u03b1 r) (hrwo' : is_well_order \u03b1 (\u03bb x y, r y x)) :\n  nonempty (fintype \u03b1) :=\nclassical.by_contradiction $ \u03bb h,\nhave cardinal.omega \u2264 (ordinal.type r).card,\n  from le_of_not_gt (mt cardinal.lt_omega_iff_fintype.1 h),\nhave ordinal.omega \u2264 ordinal.type r,\n  by rwa [\u2190 cardinal.ord_omega, cardinal.ord_le],\nhave f : nonempty (((>) : \u2115 \u2192 \u2115 \u2192 Prop) \u21aar function.swap r),\n  begin\n    rw [ordinal.omega, \u2190 ordinal.lift_id (ordinal.type r)] at this,\n    rcases (ordinal.lift_type_le.{0 u}.1 this) with \u27e8f, hf\u27e9,\n    exact \u27e8f.swap\u27e9,\n  end,\nrel_embedding.well_founded_iff_no_descending_seq.1 hrwo'.wf f\n\n\ntheorem fintype_of_well_order (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n  (hrwo : is_well_order \u03b1 r) (hrwo' : is_well_order \u03b1 (\u03bb x y, r y x)) :\n  nonempty (fintype \u03b1) :=\nclassical.by_contradiction $ \u03bb h,\nhave infin : infinite \u03b1, from \u27e8h \u2218 nonempty.intro\u27e9,\nlet a : \u03b1 := classical.choice (@infinite.nonempty \u03b1 \u27e8h \u2218 nonempty.intro\u27e9) in\nlet f : \u2115 \u2192 \u03b1 := \u03bb n, well_founded.fix (well_founded.min hrwo.wf set.univ \u27e8a, trivial\u27e9)\n(well_founded.min _ _)\n\n\n#exit\n\n/-- The hat color announced by the dwarf with the number `n` upon seeing the colors of the hats in\n    front of him: `see i` is the color of the hat of the dwarf with the number `n + i + 1`. -/\nnoncomputable def announce (n : \u2115) (see : \u2115 \u2192 \u03b1) : \u03b1 :=\nchoose_fun (\u03bb m, see (m - n - 1)) n\n\n-- see' :\n-- announce (n + 1) (\u03bb n, see (n + 1)) = see 0\n\n/-- Only finitely many dwarves announce an incorrect color. -/\ntheorem announce_correct_solution (col : \u2115 \u2192 \u03b1) :\n  \u2203 N : \u2115, \u2200 n \u2265 N, col n = announce n (\u03bb m, col (m + n + 1)) :=\nhave \u2200 n : \u2115, choose_fun (\u03bb m : \u2115, col (m - n - 1 + n + 1)) = choose_fun col,\n  from \u03bb n, choose_fun_eq \u27e8n + 1, \u03bb k hk,\n    by rw [add_assoc, nat.sub_sub, nat.sub_add_cancel hk]\u27e9,\nbegin\n  delta announce,\n  simp only [this],\n  exact choose_fun_spec _\nend\n\n\n#exit\nimport category_theory.functor\nimport category_theory.types\nimport category_theory.monad\nimport algebra.module\nopen category_theory monad\n\n@[simps]\ndef P : Type \u2964 Type :=\n{ obj := \u03bb X, set X,\n  map := \u03bb X Y, set.image }\n\nvariables {R : Type 1} [comm_ring R] {M : Type} [add_comm_group M]\n\n#check module R M\n\n#check \u03a3 (M : Type) [add_comm_group M], by exactI module R M\n\ninstance powerset_monad : monad P :=\n{ \u03b7 :=\n    { app := \u03bb X x, ({x} : set X),\n      naturality' :=\n        \u03bb X Y f, (funext (@set.image_singleton _ _ f)).symm },\n  \u03bc :=\n    { app := @set.sUnion,\n      naturality' := begin\n        intros X Y f,\n        ext,\n        simp,\n        dsimp [P_obj],\n        tauto\n      end },\n  assoc' := begin\n      intro X,\n      ext,\n      simp,\n      dsimp,\n      tauto\n    end,\n  left_unit' := begin\n      intro X,\n      dsimp,\n      ext,\n      simp,\n    end,\n  right_unit' := begin\n      intro X,\n      dsimp,\n      ext,\n      simp\n    end }\n\n#exit\ninductive tm : Type\n  | zro : tm\n  | scc : tm -> tm\n  | pls : tm -> tm -> tm\n  | nul : tm\n  | cns : tm -> tm -> tm\n  | len : tm -> tm\n  | idx : tm -> tm -> tm\n  | stn : tm -> tm\nopen tm\n\ninductive nvalue : tm -> Prop\n  | nv_zro : nvalue zro\n  | nv_scc : \u03a0 v1, nvalue v1 -> nvalue (scc v1)\nopen nvalue\n\ninductive lvalue : tm -> Prop\n  | lv_nul : lvalue nul\n  | lv_cns : \u03a0 v1 v2,\n      nvalue v1 -> lvalue v2 -> lvalue (cns v1 v2)\n\ndef value (t : tm) := nvalue t \u2228 lvalue t\n\ninductive step : tm -> tm -> Prop\nnotation x  ` \u27f6 `:100 y := step x y\n  | ST_Scc : \u2200 t1 t1',\n      (t1 \u27f6 t1') \u2192\n      scc t1 \u27f6 scc t1'\n  | ST_PlsZro : \u2200 v1,\n      nvalue v1 \u2192\n      pls zro v1 \u27f6 v1\n  | ST_PlsScc : \u2200 v1 v2,\n      nvalue v1 \u2192\n      nvalue v2 \u2192\n      pls (scc v1) v2 \u27f6 scc (pls v1 v2)\n  | ST_Pls2 : \u2200 v1 t2 t2',\n      nvalue v1 \u2192\n      (t2 \u27f6 t2') \u2192\n      pls v1 t2 \u27f6 pls v1 t2'\n  | ST_Pls1 : \u2200 t1 t1' t2,\n      (t1 \u27f6 t1') \u2192\n      pls t1 t2 \u27f6 pls t1' t2\n  | ST_Cns2 : \u2200 v1 t2 t2',\n      nvalue v1 \u2192\n      (t2 \u27f6 t2') \u2192\n      cns v1 t2 \u27f6 cns v1 t2'\n  | ST_Cns1 : \u2200 t1 t1' t2,\n      (t1 \u27f6 t1') \u2192\n      cns t1 t2 \u27f6 cns t1' t2\n  | ST_LenNul :\n      len nul \u27f6 zro\n  | ST_LenCns : \u2200 v1 v2,\n      nvalue v1 \u2192\n      lvalue v2 \u2192\n      len (cns v1 v2) \u27f6 scc (len v2)\n  | ST_Len : \u2200 t1 t1',\n      (t1 \u27f6 t1') \u2192\n      len t1 \u27f6 len t1'\n  | ST_IdxZro : \u2200 v1 v2,\n      nvalue v1 \u2192\n      lvalue v2 \u2192\n      idx zro (cns v1 v2) \u27f6 v1\n  | ST_IdxScc : \u2200 v1 v2 v3,\n      nvalue v1 \u2192\n      nvalue v2 \u2192\n      lvalue v3 \u2192\n      idx (scc v1) (cns v2 v3) \u27f6 idx v1 v3\n  | ST_Idx2 : \u2200 v1 t2 t2',\n      nvalue v1 \u2192\n      (t2 \u27f6 t2') \u2192\n      idx v1 t2 \u27f6 idx v1 t2'\n  | ST_Idx1 : \u2200 t1 t1' t2,\n      (t1 \u27f6 t1') \u2192\n      idx t1 t2 \u27f6 idx t1' t2\n  | ST_StnNval : \u2200 v1,\n      nvalue v1 \u2192\n      stn v1 \u27f6 cns v1 nul\n  | ST_Stn : \u2200 t1 t1',\n      (t1 \u27f6 t1') \u2192\n      stn t1 \u27f6 stn t1'\nopen step\n\ninfix ` \u27f6 `:100 := step\n\ndef relation (X : Type) := X \u2192 X \u2192 Prop\n\ndef deterministic {X : Type} (R : relation X) :=\n  \u2200 x y1 y2 : X, R x y1 \u2192 R x y2 \u2192 y1 = y2\n\ninductive ty : Type\n  | Nat : ty\n  | List : ty\nopen ty\n\ninductive has_type : tm \u2192 ty \u2192 Prop\nnotation `\u22a2 `:79 t ` :\u2208 `:80 T := has_type t T\n  | T_Zro :\n      \u22a2 zro :\u2208 Nat\n\n  | T_Scc : \u2200 t1,\n      (\u22a2 t1 :\u2208 Nat) \u2192\n      \u22a2 scc t1 :\u2208 Nat\n  | T_Pls : \u2200 t1 t2,\n      (\u22a2 t1 :\u2208 Nat) \u2192\n      (\u22a2 t2 :\u2208 Nat) \u2192\n      \u22a2 pls t1 t2 :\u2208 Nat\n\n  | T_Nul :\n      \u22a2 nul :\u2208 List\n  | T_Cns : \u2200 t1 t2,\n      (\u22a2 t1 :\u2208 Nat) \u2192\n      (\u22a2 t2 :\u2208 List) \u2192\n      \u22a2 cns t1 t2 :\u2208 List\n\n  | T_Len : \u2200 t1,\n      (\u22a2 t1 :\u2208 List) \u2192\n      (\u22a2 len t1 :\u2208 Nat)\n  | T_Idx : \u2200 t1 t2,\n      (\u22a2 t1 :\u2208 Nat) \u2192\n      (\u22a2 t2 :\u2208 List) \u2192\n      \u22a2 idx t1 t2 :\u2208 Nat\n  | T_Stn : \u2200 t1,\n      (\u22a2 t1 :\u2208 Nat) \u2192\n      \u22a2 stn t1 :\u2208 List\nopen has_type\n\nnotation `\u22a2 `:19 t ` :\u2208 `:20 T := has_type t T\n\ndef progress := \u2200 t T,\n  (\u22a2 t :\u2208 T) \u2192\n  value t \u2228 \u2203 t', t \u27f6 t'\n\ndef preservation := \u2200 t t' T,\n  (\u22a2 t :\u2208 T) \u2192\n  t \u27f6 t' \u2192\n  \u22a2 t' :\u2208 T\n\ninductive multi {X : Type} (R : relation X) : relation X\n  | multi_refl : \u2200 (x : X), multi x x\n  | multi_step : \u2200 (x y z : X),\n                    R x y \u2192\n                    multi y z \u2192\n                    multi x z\n\ndef multistep := (multi step).\ninfix ` \u27f6* `:100  := (multistep)\n\ndef normal_form {X : Type} (R : relation X) (t : X) : Prop :=\n  \u00ac \u2203 t', R t t'\n\nnotation `step_normal_form` := (normal_form step)\n\ndef stuck (t : tm) : Prop :=\n  step_normal_form t \u2227 \u00ac value t\n\ndef soundness := \u2200 t t' T,\n  (\u22a2 t :\u2208 T) \u2192\n  t \u27f6* t' \u2192\n  \u00ac stuck t'\n\n\ntheorem step_deterministic : deterministic step :=\nbegin\n  unfold deterministic,\n  assume x y1 y2 h1 h2,\n  induction h1; induction h2; simp *,\n\nend\n\n/- Uncomment one of the following two: -/\n-- theorem progress_dec : progress := sorry\n-- theorem progress_dec : \u00ac progress := sorry\n\n/- Uncomment one of the following two: -/\n-- theorem preservation_dec : preservation := sorry\n-- theorem preservation_dec : \u00ac preservation := sorry\n\n/- Uncomment one of the following two: -/\n-- lemma soundness_dec : soundness := sorry\n-- lemma soundness_dec : \u00ac soundness := sorry\n\n\n#exit\nimport data.list\nimport data.stream\nimport data.nat.basic\nimport data.nat.fib\nimport data.nat.parity\n\nopen nat\n\ndef fib_aux : \u2115 \u2192 \u2115 \u2192 \u2115 \u2192 \u2115\n| a b 0 := a\n| a b (n + 1) := fib_aux b (a + b) n\n\ndef fib2 (n : \u2115) : \u2115 := fib_aux 0 1 n\n\ndef fib_from (a b : \u2115) : \u2115 \u2192 \u2115\n| 0 := a\n| 1 := b\n| (n+2) := fib_from n + fib_from (n+1)\n\nlemma fib_from_thing (a b : \u2115) : \u2200 n, fib_from b (a + b) n = fib_from a b n.succ\n| 0 := rfl\n| 1 := rfl\n| (n+2) := begin\n  rw [fib_from, fib_from_thing, fib_from, fib_from_thing],\nend\n\nlemma fib_aux_eq : \u2200 a b n : \u2115, fib_aux a b n = fib_from a b n\n| a b 0 := rfl\n| a b 1 := rfl\n| a b (n+2) := begin\n  rw [fib_aux, fib_from, fib_aux_eq],\n  rw [fib_from_thing, fib_from],\nend\n\nlemma fib_from_eq_fib : \u2200 n, fib_from 0 1 n = fib n\n| 0 := rfl\n| 1 := rfl\n| (n+2 ) := begin\n  rw [fib_from, fib_from_eq_fib, fib_succ_succ, fib_from_eq_fib],\nend\n\ntheorem fib_eq (n : \u2115) : fib2 n = fib n :=\nbegin\n  rw [fib2, fib_aux_eq, fib_from_eq_fib],\nend\n\n\ntheorem fib_fast_correct (n : \u2115) : fib_fast n = fib n :=\nbegin\n\n\nend\n\n#exit\nimport category_theory.functor\nimport category_theory.types\nimport category_theory.monad\nimport data.set.basic\n\n@[simps]\ndef P : Type \u2964 Type :=\n{ obj := \u03bb X, set X,\n  map := \u03bb X Y, set.image }\n\nopen category_theory monad\n#print has_singleton\ninstance powerset_monad : monad P :=\n{ \u03b7 :=\n  { app := \u03bb X, as_hom (has_singleton.singleton : X \u2192 set X) },\n  \u03bc :=\n  { app := \u03bb X, set.sUnion,\n    naturality' := begin\n      assume X Y f,\n      ext,\n      dsimp [P_obj, P_map],\n\n    end } }\n\nopen category_theory monad\n\nuniverse u\n\n-- A suggested solution method.\n-- You are *not* required to use this.\n\n@[simps]\ndef contravariant_powerset : (Type u)\u1d52\u1d56 \u2964 Type u :=\n{ obj := \u03bb X, set X.unop,\n  map := \u03bb X Y f, as_hom (set.preimage f.unop),\n  map_id' := \u03bb x, by dsimp; refl,\n  map_comp' := \u03bb X Y Z f g, by { dsimp at *, ext1, dsimp at *, ext1, simp at * } }\n#print op_op\ninstance : is_right_adjoint contravariant_powerset :=\n{ left := unop_unop (Type u) \u22d9 contravariant_powerset.op,\n  adj := adjunction.mk_of_unit_counit\n    { unit :=\n      { app := \u03bb X (x : X), ({S : set X | x \u2208 S} : set (set X)),\n        naturality' := by { intros X Y f, refl} },\n      counit :=\n      { app := \u03bb X,\n          let f : X.unop \u27f6 set (set (opposite.unop X)) :=\n            as_hom (\u03bb x : X.unop, ({S : set X.unop | x \u2208 S} : set (set X.unop))) in\n          begin\n            have := f.op,\n\n          end,\n        naturality' := begin\n          intros X Y f,\n          refine has_hom.hom.unop_inj _,\n          dsimp at *, refl,\n        end },\n      left_triangle' := begin\n        dsimp at *, simp at *, ext1, dsimp at *, ext1, dsimp at *, simp at *,\n        refine has_hom.hom.unop_inj _,\n        dsimp at *, refl\n      end,\n      right_triangle' := by { dsimp at *, simp at *, ext1, dsimp at *, ext1, dsimp at *, refl} } }\n\n@[simps]\ndef PP : Type u \u2964 Type u :=\n{ obj := \u03bb X, set (set X),\n  map := \u03bb X Y f, set.preimage (set.preimage f) }\n\ndef PP1 : Type u \u2964 Type u :=\nunop_unop (Type u) \u22d9 contravariant_powerset.op \u22d9 contravariant_powerset\n\nlemma PP_eq_PP1 : PP.{u} = PP1 := rfl\n\ninstance double_powerset_monad : category_theory.monad PP :=\nby rw [PP_eq_PP1]; exact adjunction.monad _\n\n#exit\nimport data.fintype.basic\n\nuniverse u\n\nvariable {\u03b1 : Type u}\n\nlocal attribute [elab_as_eliminator] well_founded.fix\n\ntheorem fintype_of_well_order (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n  (hrwo : is_well_order \u03b1 r) (hrwo' : is_well_order \u03b1 (\u03bb x y, r y x)) :\n  nonempty (fintype \u03b1) :=\nclassical.by_contradiction $ \u03bb h,\nhave \u2200 a : \u03b1, \u00ac nonempty (fintype {b // r a b}),\n  from sorry,\n\n\ndef SUBMISSION : Prop :=\n\u2200 {G : Type} [group G] {a b : G} (h : by exactI a * b * a * b^2 = 1),\n  by exactI a * b = b * a\n\nset_option profiler true\n\nlemma F {G : Type} [group G] {a b : G} (h : a * b * a * b ^ 2 = 1) :\n  a * b = b * a :=\ncalc a * b = (b\u207b\u00b9 * a\u207b\u00b9 * (a * b * a * b^2)\n  * a * b * b * (a * b * a * b^2)\u207b\u00b9 * b\u207b\u00b9) * b * a : by group\n... = b * a : by rw h; group\n\nlemma G {G : Type} [group G] {a b : G} (n : \u2115) (h : (a * b) ^ n * b = 1) :\n  a * b = b * a :=\ncalc a * b = a * b^2 * ((a * b) ^ n * b) * b^ (-2 : \u2124) * a\u207b\u00b9 * b *\n    ((a * b) ^ n * b)\u207b\u00b9 * b\u207b\u00b9 * b * a :\n  begin\n    simp only [mul_assoc, pow_two, mul_inv_cancel_left, mul_inv_rev, gpow_neg,\n      pow_bit0, gpow_bit0, gpow_one, pow_one],\n    rw [\u2190 mul_assoc b\u207b\u00b9 a\u207b\u00b9, \u2190 mul_assoc _ ((a * b) ^n)\u207b\u00b9, \u2190 mul_inv_rev,\n      \u2190 mul_inv_rev, \u2190 pow_succ', pow_succ],\n    simp [mul_assoc]\n  end\n... = _ : by rw h; group\n\n\n#exit\nimport tactic\n\nvariables {G : Type} [group G]\n\ndef fib_thing (b\u2081 b\u2080 : G) : \u2115 \u2192 G\n| 0     := b\u2081\n| 1     := b\u2081 * b\u2080\n| (n+2) := fib_thing (n + 1) * fib_thing n\n\nlemma a_mul_fib_thing {a b\u2081 b\u2080 : G} (hab\u2081 : a * b\u2081 = b\u2081 * b\u2080 * a)\n  (hab\u2080 : a * b\u2080 = b\u2081 * a) : \u2200 n : \u2115,\n  a * fib_thing b\u2081 b\u2080 n = fib_thing b\u2081 b\u2080 (n + 1) * a\n| 0 := by simp [fib_thing, hab\u2081]\n| 1 := begin\n  unfold fib_thing,\n  rw [\u2190 mul_assoc, hab\u2081, mul_assoc, hab\u2080, mul_assoc, mul_assoc, mul_assoc],\nend\n| (n+2) := by rw [fib_thing, \u2190 mul_assoc, a_mul_fib_thing, mul_assoc, a_mul_fib_thing,\n    fib_thing, fib_thing, fib_thing, mul_assoc, mul_assoc, mul_assoc]\n\nlemma X (a b\u2081 b\u2080 : G) (hab\u2081 : a * b\u2081 = b\u2081 * b\u2080 * a)\n  (hab\u2080 : a * b\u2080 = b\u2081 * a) :\n  \u2200 (n : \u2115), a^n * b\u2081 = fib_thing b\u2081 b\u2080 n * a ^ n\n| 0 := by simp [fib_thing]\n| (n+1):= by rw [pow_succ, mul_assoc, X, \u2190 mul_assoc, a_mul_fib_thing hab\u2081 hab\u2080,\n  mul_assoc]\n\nlemma Y (a b\u2080 b\u2099\u2081 : G) (hab\u2081 : a\u207b\u00b9 * b\u2099\u2081 = b\u2099\u2081\u207b\u00b9 * b\u2080 * a)\n  (hab\u2080 : a\u207b\u00b9 * b\u2080 = b\u2099\u2081 * a) :\n\n#exit\nimport linear_algebra.tensor_algebra\nimport data.real.basic\n/--\nattempt to unmathlibify\n-/\n\nvariables (R : Type) [ring R] (M : Type) [add_comm_group M] [module R M]\n/-\nsemimodule.add_comm_monoid_to_add_comm_group :\n\u03a0 (R : Type u) {M : Type w} [_inst_1 : ring R] [_inst_2 : add_comm_monoid M]\n[_inst_3 : semimodule R M], add_comm_group M\n-/\n\ndef typealias (\u03b1 : Type) := \u2124\n\nlocal attribute [irreducible] tensor_algebra\n\n-- def foo : add_comm_group (tensor_algebra \u2124 \u2124) := by apply_instance -- tensor_algebra.ring \u2124\n\n-- def bar : add_comm_group (typealias bool) := by unfold typealias; apply_instance\n\n-- def foo' : add_comm_group (tensor_algebra \u2124 \u2124) :=\n--   semimodule.add_comm_monoid_to_add_comm_group \u2124\n\n-- def bar' : add_comm_group (typealias bool) := by unfold typealias;\n--   exact semimodule.add_comm_monoid_to_add_comm_group \u2124\n--instance foo'' : ring (tensor_algebra \u2124 \u2124) := by apply_instance\n#print tactic.dsimp_config\nlocal attribute [irreducible] typealias\n\nlocal attribute [irreducible] tensor_algebra\n\ninstance foo' : ring (tensor_algebra \u2124 \u2124) :=\n{ ..semimodule.add_comm_monoid_to_add_comm_group (tensor_algebra \u2124 \u2124),\n  ..(infer_instance : semiring (tensor_algebra \u2124 \u2124)) }\n\ninstance foo : ring (tensor_algebra \u2124 \u2124) := tensor_algebra.ring \u2124\n\nexample : derive_handler := by library_search\n\n@[derive ring] def C := int\n#print d_array\n#print C.ring\n\nset_option pp.implicit true\nset_option pp.proofs true\n\n--lemma X : @ring.add_zero _ foo = @ring.add_zero _ foo' := rfl\n\n#print declaration\n#print expr\n\nrun_cmd tactic.add_decl\n  (declaration.thm `X []  `(@ring.add_zero _ foo = @ring.add_zero _ foo')\n  (pure `(eq.refl (@ring.add_zero _ foo))))\n\n#print X\n\n -- works when `tensor_algebra` is not irreducible\n-- example : @add_comm_group.add_zero _ foo = @add_comm_group.add_zero _ foo' := rfl\n\n-- works when `typealias` is not irreducible, but *statement* doesn't compile if it is\nexample : @add_comm_group.add_zero _ bar = @add_comm_group.add_zero _ bar' :=\nrfl\n\n#exit\n\n#print list.range\n\nexample {G : Type*} [group G] (a b : G) (hab : a * b = b * a\u207b\u00b9 * b * a^2) :\n  a * a * a * b = sorry :=\nhave hab' : \u2200 g : G, a * (b * g) = b * (a\u207b\u00b9 * (b * (a * (a * g)))),\n  from \u03bb g, by rw [\u2190 mul_assoc, hab]; simp [pow_two, mul_assoc],\nbegin\n  simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left],\n  try { rw [hab] <|> rw hab'},\n  simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left],\n  try { rw [hab] <|> rw hab'},\n  simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left],\n  try { rw [hab] <|> rw hab'},\n  simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left],\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  try { rw [hab] <|> rw hab'},\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n  rw [hab] <|> rw hab',\n  try { simp only [mul_assoc, pow_two, mul_left_inv, mul_right_inv, one_inv,\n    mul_inv_cancel_left, inv_mul_cancel_left] },\n\n\nend\n\nexample {G : Type*} [group G] (a b : G) (hab : a * b = b * a\u207b\u00b9 * b * a^2) (n : \u2115) :\n  a^n * b = ((list.range n).map (\u03bb i, b ^ (2 ^ i) * a\u207b\u00b9)).prod * b\n   * a ^ (2 ^ (n + 1))\n\nexample {G : Type*} [group G] (a b : G) (hab : a * b = b * a\u207b\u00b9 * b * a^2) (n : \u2115) :\n  a^(n + 1) * b = ((list.range (n+1)).map (\u03bb i, b ^ (2 ^ i) * a\u207b\u00b9)).prod * b\n   * a ^ (2 ^ (n + 1)) :=\nbegin\n  induction n with n ih,\n  { simp [list.range, list.range_core, hab] },\n  { rw [pow_succ, mul_assoc, ih, eq_comm, list.range_concat, list.map_append],\n    simp, }\n\nend\n\n#exit\nnotation `C\u221e` := multiplicative \u2124\n\nopen multiplicative\n\nlemma to_add_injective {A : Type*} : function.injective (to_add : A \u2192 multiplicative A) :=\n\u03bb _ _, id\n\n@[derive group] def Z_half : Type :=\nmultiplicative (localization.away (2 : \u2124))\n\ndef phi : C\u221e \u2192* mul_aut Z_half :=\ngpowers_hom _\n  (show mul_aut Z_half, from\n    { to_fun := \u03bb x, of_add (to_add x * 2),\n      inv_fun := \u03bb x, of_add (to_add x * localization.away.inv_self 2),\n      left_inv := \u03bb _, to_add_injective sorry,\n      right_inv :=  \u03bb _, to_add_injective sorry,\n      map_mul' := \u03bb _ _, to_add_injective sorry })\n\n@[derive group] def BS : Type := Z_half \u22ca[phi] C\u221e\n\n@[simp] lemma zero_denom : (0 : \u211a).denom = 1 := rfl\n\ndef lift {G : Type*} [group G] (a b : G) (hab : a * b * a\u207b\u00b9 = b^2) : BS \u2192* G :=\nsemidirect_product.lift\n  _\n  (gpowers_hom _ a)\n  _\n\n\n#exit\n\n@[derive decidable_eq] inductive BS : Type\n| neg    : \u2115+ \u2192 \u2124 \u2192 BS\n| nonneg : \u2124 \u2192 \u2115 \u2192 BS\n\nnamespace BS\n\nprivate def one : BS := nonneg 0 0\n\nprivate def inv : BS \u2192 BS\n| (neg n i) := nonneg (-i) n\n| (nonneg i n) :=\n  if hn : 0 < n then neg \u27e8n, hn\u27e9 (-i) else nonneg (-i) 0\n\nprivate def mul : BS \u2192 BS \u2192 BS\n| (nonneg i\u2081 n\u2081) (nonneg i\u2082 n\u2082) := nonneg (i\u2081 + i\u2082 + i\u2082.sign * max n\u2081 i\u2082.nat_abs) (n\u2081 + n\u2082)\n| (nonneg i\u2081 n\u2081) (neg n\u2082 i\u2082)    :=\n  if hn : (n\u2082 : \u2115) \u2264 n\u2081\n    then nonneg (i\u2081 + i\u2082 + i\u2082.sign * max (n\u2081 - n\u2082) i\u2082.nat_abs) (n\u2081 - n\u2082)\n    else neg \u27e8n\u2082 - n\u2081, nat.sub_pos_of_lt (lt_of_not_ge hn)\u27e9\n      (i\u2081.sign * max (n\u2082 - n\u2081) i\u2081.nat_abs + i\u2081 + i\u2082)\n| (neg n\u2081 i\u2081) (nonneg i\u2082 n\u2082)    :=\n  if hn : (n\u2081 : \u2115) \u2264 n\u2082\n    then nonneg _ (n\u2082 - n\u2081)\n    else _\n\n\nend BS\n\n#exit\nprivate def mul (a b : B) : B :=\nif b.conj +\n\nprivate def one : BS := \u27e80, 0, 0, dec_trivial\u27e9\n\ninstance : has_one BS := \u27e8one\u27e9\n\nprivate def inv (a : BS) : BS :=\n\u27e8a.right, -a.middle, a.left, by cases a; simp; tauto\u27e9\n\ninstance : has_inv BS := \u27e8inv\u27e9\n\nprivate def mul (a b : BS) : BS :=\nlet x : {x : \u2115 \u00d7 \u2115 // a.middle + b.middle = 0 \u2192 x.1 = 0 \u2228 x.2 = 0} :=\n  let m : \u2124 := a.right - b.left in\n  if h : a.middle + b.middle = 0\n    then let k : \u2124 := a.left + b.left - a.right - b.right in\n      if 0 \u2264 k\n        then \u27e8(k.to_nat, 0), \u03bb _, or.inr rfl\u27e9\n        else \u27e8(0, k.nat_abs), \u03bb _, or.inl rfl\u27e9\n    else if 0 \u2264 m\n      then if 0 \u2264 b.middle\n        then let n := min m.to_nat b.middle.to_nat in\n          \u27e8(a.left + n, a.right + b.right), false.elim \u2218 h\u27e9\n        else let n := min m.to_nat b.middle.nat_abs in\n          \u27e8(a.left, b.right + n + m.to_nat), false.elim \u2218 h\u27e9\n      else if 0 \u2264 a.middle\n          then let n := min m.nat_abs a.middle.to_nat in\n            \u27e8(a.left + n + m.nat_abs, b.right), false.elim \u2218 h\u27e9\n          else let n := min m.nat_abs a.middle.nat_abs in\n            \u27e8(a.left + b.left, b.right + n), false.elim \u2218 h\u27e9 in\n\u27e8x.1.1, a.middle + b.middle, x.1.2, x.2\u27e9\n\ninstance : has_mul BS := \u27e8mul\u27e9\n\n@[simp] lemma int.coe_nat_le_zero (a : \u2115) : (a : \u2124) \u2264 0 \u2194 a = 0 := sorry\n\nprivate lemma mul_one (a  : BS) : mul a one = a :=\nbegin\n  cases a,\n  simp [mul, one],\n  split_ifs; finish [nat.not_lt_zero]\nend\n\nprivate lemma mul_inv (a : BS) : mul a (inv a) = one :=\nbegin\n  cases a,\n  simp [mul, one, inv]\nend\n\nprivate lemma mul_left (a b : BS) : (mul a b).left =\n  let m : \u2124 := a.right - b.left in\n  if h : a.middle + b.middle = 0\n  then let k : \u2124 := a.left + b.left - a.right - b.right in\n    if 0 \u2264 k\n      then k.to_nat\n      else 0\n  else if 0 \u2264 m\n    then if 0 \u2264 b.middle\n      then a.left + min m.to_nat b.middle.to_nat\n      else a.left\n    else if 0 \u2264 a.middle\n        then a.left + min m.nat_abs a.middle.to_nat + m.nat_abs\n        else a.left + b.left :=\nby simp [mul]; split_ifs; simp\n\n@[simp] private lemma mul_middle (a b : BS) : (mul a b).middle = a.middle + b.middle := rfl\n\nprivate lemma mul_assoc (a b c : BS) : mul (mul a b) c = mul a (mul b c) :=\nbegin\n  cases a, cases b, cases c,\n  ext,\n  { simp only [mul_left, mul_middle, add_assoc],\n    dsimp,\n    by_cases h : a_middle + b_middle + c_middle = 0,\n    { rw [dif_pos h, dif_pos ((add_assoc _ _ _).symm.trans h)],\n      split_ifs, }\n\n  }\n\n\n\n\n\n\n\nend\n\n\n\n#exit\nimport analysis.special_functions.trigonometric\n\nopen complex real\n\nexample (\u03b8 \u03c6 : \u211d) (h1 : \u03b8 \u2264 pi) (h2 : -pi < \u03b8)\n  (h3 : 0 < cos \u03c6) : arg (exp (I * \u03b8) * cos \u03c6) = \u03b8 :=\nby rw [mul_comm, \u2190 of_real_cos, arg_real_mul _ h3, mul_comm,\n  exp_mul_I, arg_cos_add_sin_mul_I h2 h1]\n\n\n\n#print prod.lex\n#print is_lawful_singleton\n\n\n\nexample (A B : Prop) (h : A \u2192 \u00ac \u00ac B) (hnnA : \u00ac \u00ac A) : \u00ac \u00ac B :=\n\u03bb hB, hnnA (\u03bb hA, h hA hB)\n\nlemma X : \u00ac \u00ac (\u2200 p, p \u2228 \u00ac p) \u2192 \u2200 p, p \u2228 \u00ac p :=\nbegin\n  isafe, -- goals accomplished\nend\n#print axioms not_forall\n#print X\n\nexample : \u00ac\u00ac(\u2200 p, p \u2228 \u00ac p) := by ifinish\n\nexample (A B C : Prop) :\n  A \u2228 B \u2228 C \u2192\n  (A \u2192 \u00ac B \u2227 \u00ac C) \u2192\n  (B \u2192 \u00ac A \u2227 \u00ac C) \u2192\n  (C \u2192 \u00ac A \u2227 \u00ac B) \u2192\n  (A \u2194 A) \u2192\n  (B \u2194 (A \u2228 B)) \u2192\n  (C \u2194 (B \u2228 C)) \u2192\n  C :=\n  \u03bb h hA1 hB1 hC1 hA2 hB2 hC2,\nhave A \u2192 B, by tauto!, by tauto!\n\nexample (A B C : Prop) :\n  A \u2228 B \u2228 C \u2192\n  (A \u2192 \u00ac B \u2227 \u00ac C) \u2192\n  (B \u2192 \u00ac A \u2227 \u00ac C) \u2192\n  (C \u2192 \u00ac A \u2227 \u00ac B) \u2192\n  (A \u2194 A) \u2192\n  (B \u2194 (A \u2228 B)) \u2192\n  (C \u2194 (B \u2228 C)) \u2192\n  C :=\n\u03bb h hA1 hB1 hC1 hA2 hB2 hC2,\nmatch h with\n| (or.inl hA)          := ((hA1 hA).1 (hB2.2 (or.inl hA))).elim\n| (or.inr (or.inl hB)) := ((hB1 hB).2 (hC2.2 (or.inl hB))).elim\n| (or.inr (or.inr hC)) := hC\nend\n\n#exit\nimport data.list.basic\nimport data.list\nimport tactic\nimport order.lexicographic\n\nvariables {\u03b1 : Type*} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (wf : well_founded r)\n\nlemma list.lex.cons_iff' {a\u2081 a\u2082 : \u03b1} {l\u2081 l\u2082 : list \u03b1} :\n  list.lex r (a\u2081 :: l\u2081) (a\u2082 :: l\u2082) \u2194 r a\u2081 a\u2082 \u2228 a\u2081 = a\u2082 \u2227 list.lex r l\u2081 l\u2082 :=\nbegin\n  split,\n  { assume h,\n    cases h; simp * },\n  { assume h,\n    rcases h with hr | \u27e8rfl, h\u27e9,\n    { exact list.lex.rel hr },\n    { exact list.lex.cons h } }\nend\n\nlemma lex_wf_aux : \u2200 (n : \u2115),\n  well_founded\n    (inv_image (list.lex r)\n      (subtype.val : {l : list \u03b1 // l.length = n} \u2192 list \u03b1))\n| 0     := subrelation.wf\n  (begin\n    rintros l\u2081 \u27e8l\u2082, hl\u2082\u27e9,\n    simp [inv_image, empty_relation, list.length_eq_zero.1 hl\u2082]\n  end)\n  empty_wf\n| (n+1) :=\nlet f : {l : list \u03b1 // l.length = n + 1} \u2192 \u03b1 \u00d7 {l : list \u03b1 // l.length = n} :=\n  \u03bb l, (l.val.nth_le 0 (by rw [l.2]; exact nat.succ_pos _),\n      subtype.mk l.1.tail $ by simp [list.length_tail, l.prop]) in\nsubrelation.wf\n  (begin\n    rintros \u27e8l\u2081, hl\u2081\u27e9 \u27e8l\u2082, hl\u2082\u27e9,\n    cases l\u2081,\n    { exact (nat.succ_ne_zero _ hl\u2081.symm).elim },\n    cases l\u2082,\n    { exact (nat.succ_ne_zero _ hl\u2082.symm).elim },\n    simp [inv_image, list.lex.cons_iff', prod.lex_def]\n  end)\n  (inv_image.wf f (prod.lex_wf wf (lex_wf_aux n)))\n\nlemma psigma.lex_def {\u03b1 : Sort*} {\u03b2 : \u03b1 \u2192 Sort*}\n  {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {s : \u03a0 a, \u03b2 a \u2192 \u03b2 a \u2192 Prop} {a b : psigma \u03b2} :\n  psigma.lex r s a b \u2194 r a.fst b.fst \u2228\n  \u2203 h : a.fst = b.fst, s b.fst (cast (congr_arg \u03b2 h) a.snd) b.snd :=\nbegin\n  split,\n  { intro h,\n    induction h; simp * },\n  { intro h,\n    cases a with a\u2081 a\u2082,\n    cases b with b\u2081 b\u2082,\n    dsimp at h,\n    rcases h with hr | \u27e8rfl, h\u27e9,\n    { exact psigma.lex.left _ _ hr },\n    { exact psigma.lex.right _ h } }\nend\n#print list.lex\nlemma list.lex_wf : well_founded (list.lex r) :=\nlet f : list \u03b1 \u2192 \u03a3' n : \u2115, {l : list \u03b1 // l.length = n} :=\n  \u03bb l, \u27e8l.length, l, rfl\u27e9 in\nsubrelation.wf (show subrelation _\n    (inv_image (psigma.lex (<) (\u03bb n, inv_image (list.lex r)\n      (subtype.val : {l' : list \u03b1 // l'.length = n} \u2192 list \u03b1))) f),\n  begin\n    intros l\u2081 l\u2082 h,\n    dsimp only [inv_image, f],\n    induction h with a l a l\u2081 l\u2082 h ih,\n    { exact psigma.lex.left _ _ (nat.succ_pos _) },\n    { rw [psigma.lex_def] at ih,\n      rcases ih with hr | \u27e8hl, hlex\u27e9,\n      { exact psigma.lex.left _ _ (nat.succ_lt_succ hr) },\n      { dsimp at hl,\n        simp only [psigma.lex_def, lt_irrefl, hl, list.length_cons, false_or,\n          exists_prop_of_true, set_coe_cast, subtype.coe_mk],\n        exact list.lex.cons (by simpa [hl] using hlex) } },\n    {  }\n\n  end)\n  (inv_image.wf f (psigma.lex_wf\n    (show well_founded has_lt.lt, from nat.lt_wf)\n    (lex_wf_aux r wf)))\n\ndef lex_wf_fun : list \u03b1 \u2192 \u03b1 \u2295 unit\n| []     := sum.inr ()\n| (a::l) := sum.inl a\n\n#print list.lex\n\nexample {l\u2081 l\u2082 : list \u03b1} (h : list.lex r l\u2081 l\u2082) :\n  prod.lex (<) (sum.lex r (\u03bb _ _, false))\n  (l\u2081.length, lex_wf_fun l\u2081)\n  (l\u2082.length, lex_wf_fun l\u2082) :=\nbegin\n  induction h,\n  { simp [lex_wf_fun, prod.lex_def] },\n  { simp [lex_wf_fun, prod.lex_def] at *, admit },\n  { simp [lex_wf_fun, prod.lex_def, *] at * }\nend\n\nlemma acc_lex_nil : acc (list.lex r) [] :=\nacc.intro _ (\u03bb l hl, (list.lex.not_nil_right _ _ hl).elim)\n\nlemma list.lex.cons_iff' {a\u2081 a\u2082 : \u03b1} {l\u2081 l\u2082 : list \u03b1} :\n  list.lex r (a\u2081 :: l\u2081) (a\u2082 :: l\u2082) \u2194 r a\u2081 a\u2082 \u2228 a\u2081 = a\u2082 \u2227 list.lex r l\u2081 l\u2082 :=\nbegin\n  split,\n  { assume h,\n    cases h; simp * },\n  { assume h,\n    rcases h with hr | \u27e8rfl, h\u27e9,\n    { exact list.lex.rel hr },\n    { exact list.lex.cons h } }\nend\n\n#print psigma.lex\n#print pi.lex\n\ninclude wf\n\nlocal attribute [elab_as_eliminator] well_founded.fix\n\nexample (l : list \u03b1) : acc (list.lex r) l :=\nbegin\n  induction l with a\u2081 l\u2081 ih,\n  { exact acc_lex_nil _ },\n  { refine acc.intro _ (\u03bb l\u2082 hl\u2082, _),\n    induction l\u2082 with a\u2082 l\u2082 ih\u2082,\n    { exact acc_lex_nil _ },\n    { rw [list.lex.cons_iff'] at hl\u2082,\n      rcases hl\u2082 with hr | \u27e8rfl, h\u27e9,\n      { refine well_founded.fix wf _ a\u2082,\n        assume x ih,\n        refine acc.intro _ (\u03bb l\u2083 hl\u2083, _),\n        induction hl\u2083,\n        { exact acc_lex_nil _ },\n        {  } }\n\n        }\n     }\n\n\nend\n\nexample (P : \u2124 \u2192 Prop) (h8 : \u2200 n, P n \u2192 P (n + 8))\n  (h3 : \u2200 n, P n \u2192 P (n - 3)) : \u2200 n, P n \u2192 P (n + 1) :=\n\u03bb n hn, begin\n  have := h8 _ (h8 _ (h3 _ (h3 _ (h3 _ (h3 _ (h3 n hn)))))),\n  ring at this,\n  exact this\nend\n#exit\n\n\n#print list.lex\n\nmeta def ack_list : list \u2115 \u2192 list \u2115\n| [] := []\n| [n] := [n]\n| (n::0::l) := ack_list ((n+1)::l)\n| (0::(m+1)::l) := ack_list (1::m::l)\n| ((n+1)::(m+1)::l) := ack_list (n::(m+1)::m::l)\n\n\n#eval ack_list [2, 3]\n\n#exit\nimport algebra.ring\ninductive nat2 : Type\n| zero : nat2\n| succ : nat2 \u2192 nat2\n\nnamespace nat2\n\nvariables {\u03b1 : Type} (z : \u03b1) (s : \u03b1 \u2192 \u03b1)\n\ndef lift (n : nat2) : \u03b1 :=\nnat2.rec_on n z (\u03bb _, s)\n\n@[simp] lemma lift_zero  :\n  lift z s zero = z := rfl\n\n@[simp] lemma lift_succ (n : nat2) :\n  lift z s (succ n) = s (lift z s n):= rfl\n\nattribute [irreducible] lift\n\nlemma hom_ext {f g : nat2 \u2192 \u03b1}\n  (hfz : f zero = z) (hfs : \u2200 n, f (succ n) = s (f n))\n  (hgz : g zero = z) (hgs : \u2200 n, g (succ n) = s (g n))\n  (n : nat2) :\n  f n = g n :=\nbegin\n  induction n with n ih,\n  { rw [hfz, hgz] },\n  { rw [hfs, hgs, ih] }\nend\n\n@[simp] lemma lift_zero_succ (n : nat2) : lift zero succ n = n :=\nhom_ext zero succ (by simp) (by simp) rfl (by simp) n\n\ndef add (n : nat2) : nat2 \u2192 nat2 := lift n succ\ninfix ` + ` := add\n\n-- Prove adding on the left is a hom\n\n@[simp] lemma add_zero (n : nat2) : n + zero = n :=\nlift_zero _ _\n\n@[simp] lemma add_succ (m n : nat2) : m + succ n = succ (m + n) :=\nlift_succ _ _ _\n\n-- Prove adding on the right is a hom\n\n@[simp] lemma zero_add (n : nat2) : zero + n = n :=\nhom_ext zero succ\n  (by simp [add])\n  (by simp [add])\n  (by simp [add])\n  (\u03bb n, by simp [add])\n  n\n\n@[simp] lemma succ_add (m n : nat2) : succ m + n = succ (m + n) :=\nhom_ext (succ m) succ\n  (by simp [add])\n  (by simp [add])\n  (by simp [add])\n  (by simp [add])\n  n\n\nlemma add_comm (m n : nat2) : m + n = n + m :=\nhom_ext m succ (add_zero _) (add_succ _) (zero_add _) (\u03bb n, succ_add n m) _\n\nlemma add_assoc (a b c : nat2): (a + b) + c = a + (b + c) :=\nhom_ext (a + b) succ\n  (by simp)\n  (by simp)\n  (by simp)\n  (by simp)\n  c\n\nlemma add_left_comm (a b c : nat2): a + (b + c) = b + (a + c) :=\nby rw [\u2190 add_assoc, add_comm a b, add_assoc]\n\n\ndef mul (n : nat2) : nat2 \u2192 nat2 := lift zero (+ n)\n\ninfix ` * ` := mul\n\n-- Prove multiplication on the left is a hom\n\n@[simp] lemma mul_zero (n : nat2) : n * zero = zero := by simp [mul]\n\n@[simp] lemma mul_succ (m n : nat2) : m * succ n = (m * n) + m := by simp [mul]\n\n-- Prove multiplication on the right is a hom\n\n@[simp] lemma zero_mul (n : nat2) : zero * n = zero :=\nhom_ext zero (+ zero)\n  (by simp [mul])\n  (by simp [mul])\n  (by simp [mul])\n  (by simp [mul])\n  n\n\n@[simp] lemma succ_mul (m n : nat2) : succ m * n = n + (m * n):=\nhom_ext zero (+ m.succ)\n  (by simp)\n  (by simp [mul])\n  (by simp [mul])\n  (by simp [mul, add_comm, add_assoc])\n  n\n\nlemma mul_comm (m n : nat2) : m * n = n * m :=\nhom_ext zero (+ m)\n  (by simp)\n  (by simp)\n  (by simp)\n  (by simp [add_comm])\n  n\n\nlemma mul_add (a b c : nat2) : a * (b + c) = a * b + a * c :=\n@hom_ext _ zero (+ (b + c))\n  (\u03bb a, a * (b + c)) (\u03bb a, a * b + a * c)\n  (by simp)\n  (by simp [add_comm])\n  (by simp)\n  (by simp [add_comm, add_assoc, add_left_comm])\n  a\n\nlemma add_mul (a b c : nat2) : (a + b) * c = a * c + b * c :=\nby rw [mul_comm, mul_add, mul_comm c a, mul_comm c b]\n\nlemma mul_assoc (a b c : nat2): (a * b) * c = a * (b * c) :=\nhom_ext zero (+ (a * b))\n  (by simp)\n  (by simp)\n  (by simp)\n  (by simp [mul_add])\n  c\n\nlemma mul_one (a : nat2) : a * succ zero = a :=\n@hom_ext _ zero succ\n  (\u03bb a, a * succ zero) id\n  (by simp)\n  (by simp)\n  (by simp)\n  (by simp)\n  a\n\nlemma one_mul (a : nat2) : succ zero * a = a :=\n@hom_ext _ zero succ\n  (\u03bb a, succ zero * a) id\n  (by simp)\n  (by simp)\n  (by simp)\n  (by simp)\n  a\n\ninstance : comm_semiring nat2 :=\n{ zero := zero,\n  one := succ zero,\n  add := add,\n  add_assoc := add_assoc,\n  zero_add := zero_add,\n  add_zero := add_zero,\n  add_comm := add_comm,\n  mul := mul,\n  mul_assoc := mul_assoc,\n  one_mul := one_mul,\n  mul_one := mul_one,\n  zero_mul := zero_mul,\n  mul_zero := mul_zero,\n  mul_comm := mul_comm,\n  left_distrib := mul_add,\n  right_distrib := add_mul }\n\nlemma succ_inj (a b : nat2) : succ a = succ b \u2192 a = b :=\ncalc a = pred (succ a) : by simp [pred]\n\n\nend nat2\n\n#exit\nimport data.fintype.basic data.fintype.card\n\nimport tactic\n\nopen interactive tactic expr\n\nmeta def power : tactic unit := `[sorry]\n\nmeta def sorrying_aux (e : expr) : tactic (list unit) :=\ndo tactic.all_goals\n    (do gs \u2190 tactic.get_goal, if gs = e then power else `[skip])\n\n#print tactic.interactive.rewrite\n\nmeta def tactic.interactive.sorrying (q : parse types.texpr) : tactic (list unit) :=\ndo gs \u2190 tactic.get_goals, h \u2190 tactic.i_to_expr q, sorrying_aux h\n\nexample (x : \u2115) : x \u2260 0 \u2227 false \u2227 x \u2260 0 \u2227 x \u2260 0 :=\nbegin\n  sorrying x,\n  repeat{ split },\n  --sorrying (x \u2260 0), -- unknown identifier 'x'\nend\n\ndef is_pos : \u2115 \u2192 bool\n| 0 := ff\n| _ := tt\n#print is_pos._main\n\n\n#exit\nimport data.real.basic\n\nset_option old_structure_cmd true\nopen_locale classical\nnoncomputable theory\n\nclass transmathematic (R : Type*) extends has_add R, has_div R, has_inv R, has_zero R,\n  has_one R, has_neg R, has_sub R, has_mul R, has_le R, has_lt R :=\n( nullity : R)\n( infinity : R )\n( sgn : R \u2192 R )\n( sgn1 : \u2200 a, a < 0 \u2192 sgn a = -1 )\n( sgn2 : \u2200 a, a = 0 \u2192 sgn a = 0 )\n( sgn3 : \u2200 a, a > 0 \u2192 sgn a = 1 )\n( sgn4 : \u2200 a, a = nullity \u2192 sgn a = nullity )\n( A1 : \u2200 a b c : R, a + (b + c) = (a + b) + c )\n( A2 : \u2200 a b : R, a + b = b + a )\n( A3 : \u2200 a : R, 0 + a = a )\n( A4 : \u2200 a : R, nullity + a = nullity )\n( A5 : \u2200 a : R, a \u2260 infinity \u2192 a \u2260 -infinity \u2192 a \u2260 nullity \u2192 a + infinity = infinity )\n( A6 : \u2200 a b : R, a - b = a + (-b))\n( A7 : \u2200 a : R, - -a = a )\n( A8 : \u2200 a : R, a \u2260 nullity \u2192 a \u2260 infinity \u2192 a \u2260 -infinity \u2192 a - a = 0)\n( A9 : -nullity = nullity )\n( A10 : \u2200 a : R, a \u2260 nullity \u2192 a \u2260 infinity \u2192 a -infinity = -infinity )\n( A11 : infinity - infinity = nullity )\n( A12 : \u2200 a b c : R, a * (b * c) = (a * b) * c )\n( A13 : \u2200 a b : R, a * b = b * a )\n( A14 : \u2200 a : R, 1 * a = a )\n( A15 : \u2200 a : R, nullity * a = nullity )\n( A16 : infinity * 0 = nullity )\n( A17 : \u2200 a b : R, a / b = a * b\u207b\u00b9 )\n( A18 : \u2200 a : R, a \u2260 nullity \u2192 a \u2260 infinity \u2192 a \u2260 -infinity \u2192 a \u2260 0 \u2192 a / a = 1)\n( A19 : \u2200 a : R, a \u2260 -infinity \u2192 a\u207b\u00b9\u207b\u00b9 = a )\n( A20 : 0\u207b\u00b9 = infinity )\n( A21 : (-infinity)\u207b\u00b9 = 0 )\n( A22 : nullity\u207b\u00b9 = nullity )\n( A23 : \u2200 a : R, infinity * a = infinity \u2194 a > 0 )\n( A24 : \u2200 a : R, infinity * a = -infinity \u2194 0 > a )\n( A25 : infinity > 0 )\n( A26 : \u2200 a b : R, a - b > 0 \u2194 a > b )\n( A27 : \u2200 a b : R, a > b \u2194 b < a )\n( A28 : \u2200 a b : R, a \u2265 b \u2194 (a > b) \u2228 (a = b) )\n( A29 : \u2200 a b : R, a \u2264 b \u2194 b \u2265 a )\n( A30 : \u2200 a : R, list.countp id [a < 0, a = 0, a > 0, a = nullity] = 1 )\n( A31 : \u2200 a b c : R, a \u2260 infinity \u2192 a \u2260 -infinity \u2192 sgn b \u2260 sgn c \u2192 b + c \u2260 0 \u2192\n  b + c \u2260 nullity \u2192 a * (b + c) = a * b + a * c )\n( A32 : \u2200 Y : set R, nullity \u2209 Y \u2192 \u2203 u, (\u2200 y \u2208 Y, y \u2264 u) \u2227\n  \u2200 v : R, v \u2260 nullity \u2192 (\u2200 y \u2208 Y, y \u2264 v) \u2192 u \u2264 v )\n\ninductive transreal : Type\n| of_real : \u211d \u2192 transreal\n| nullity : transreal\n| infinity : transreal\n| neg_infinity : transreal\n\nnamespace transreal\n\ninstance : has_zero transreal := \u27e8of_real 0\u27e9\n@[simp] lemma zero_def : (0 : transreal) = of_real 0 := rfl\n\ninstance : has_one transreal := \u27e8of_real 1\u27e9\n@[simp] lemma one_def : (1 : transreal) = of_real 1 := rfl\n\n@[simp] def neg : transreal \u2192 transreal\n| (of_real a) := of_real (-a)\n| nullity := nullity\n| neg_infinity := infinity\n| infinity := neg_infinity\n\ninstance : has_neg transreal := \u27e8neg\u27e9\n@[simp] lemma neg_def (a : transreal) : -a = neg a := rfl\n\n@[simp] def add : transreal \u2192 transreal \u2192 transreal\n| (of_real a) (of_real b) := of_real (a + b)\n| nullity a := nullity\n| a nullity := nullity\n| infinity neg_infinity := nullity\n| neg_infinity infinity := nullity\n| a infinity := infinity\n| a neg_infinity := neg_infinity\n| infinity a := infinity\n| neg_infinity a := neg_infinity\n\ninstance : has_add transreal := \u27e8add\u27e9\n@[simp] lemma add_def (a b : transreal) : a + b = add a b := rfl\n\ninstance : has_sub transreal := \u27e8\u03bb a b, a + -b\u27e9\n@[simp] lemma sub_def (a b : transreal) : a - b = a + -b := rfl\n\n@[simp] def inv : transreal \u2192 transreal\n| (of_real a) := if a = 0 then infinity else (of_real (a\u207b\u00b9))\n| nullity := nullity\n| neg_infinity := 0\n| infinity := 0\n\ninstance : has_inv transreal := \u27e8inv\u27e9\n@[simp] lemma inv_def (a : transreal) : a\u207b\u00b9 = inv a := rfl\n\n@[simp] def mul : transreal \u2192 transreal \u2192 transreal\n| (of_real a) (of_real b) := of_real (a * b)\n| nullity a := nullity\n| a nullity := nullity\n| infinity (of_real a) :=\n  if a = 0 then nullity\n    else if a < 0\n      then -infinity\n      else infinity\n| (of_real a) infinity :=\n  if a = 0 then nullity\n    else if a < 0\n      then -infinity\n      else infinity\n| neg_infinity (of_real a) :=\n  if a = 0 then nullity\n    else if a < 0\n      then infinity\n      else -infinity\n| (of_real a) neg_infinity :=\n  if a = 0 then nullity\n    else if a < 0\n      then infinity\n      else -infinity\n| infinity infinity := infinity\n| infinity neg_infinity := neg_infinity\n| neg_infinity infinity := neg_infinity\n| neg_infinity neg_infinity := infinity\n\ninstance : has_mul transreal := \u27e8mul\u27e9\n@[simp] lemma mul_def (a b : transreal) : a * b = mul a b := rfl\n\ninstance : has_div transreal := \u27e8\u03bb a b, a * b\u207b\u00b9\u27e9\n@[simp] lemma div_def (a b : transreal) : a / b = a * b\u207b\u00b9 := rfl\n\n@[simp] def lt : transreal \u2192 transreal \u2192 Prop\n| (of_real a) (of_real b) := a < b\n| nullity a := false\n| a nullity := false\n| (of_real a) infinity := true\n| neg_infinity infinity := true\n| infinity infinity := false\n| a neg_infinity := false\n| infinity (of_real a) := false\n| neg_infinity (of_real a) := true\n\ninstance : has_lt transreal := \u27e8lt\u27e9\n@[simp] lemma lt_def (a b : transreal) : a < b = lt a b := rfl\n\ninstance : has_le transreal := \u27e8\u03bb a b, a < b \u2228 a = b\u27e9\n@[simp] lemma le_def (a b : transreal) : a \u2264 b = (a < b \u2228 a = b) := rfl\n\n@[simp] def sgn : transreal \u2192 transreal\n| (of_real a) := if 0 < a then 1 else if a < 0 then -1 else 0\n| infinity := 1\n| neg_infinity := -1\n| nullity := nullity\n\nlocal attribute [simp] gt ge\n\ninstance : transmathematic transreal :=\n{ add := (+),\n  div := (/),\n  inv := has_inv.inv,\n  zero := 0,\n  one := 1,\n  sub := has_sub.sub,\n  mul := (*),\n  neg := has_neg.neg,\n  le := (\u2264),\n  lt := (<),\n  nullity := nullity,\n  infinity := infinity,\n  sgn := sgn,\n  sgn1 := \u03bb a ha,\n    by { cases a; simp * at *,\n          simp [not_lt_of_gt ha] },\n  sgn2 := \u03bb a, by cases a; simp [lt_irrefl] {contextual := tt},\n  sgn3 := \u03bb a ha, by { cases a; simp * at * },\n  sgn4 := \u03bb a ha, by { cases a; simp * at * },\n  A1 := \u03bb a b c, by cases a; cases b; cases c; simp [add_assoc],\n  A2 := \u03bb a b, by cases a; cases b; simp [add_comm],\n  A3 := \u03bb a, by cases a; simp,\n  A4 := \u03bb a, by cases a; simp,\n  A5 := \u03bb a, by cases a; simp,\n  A6 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A7 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A8 := by intros; try {cases a}; try {cases b}; try {cases c}; simp * at *,\n  A9 := rfl,\n  A10 := by intros; try {cases a}; try {cases b}; try {cases c}; simp * at *,\n  A11 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A12 := begin\n    assume a b c,\n    cases a; simp; cases b; simp; try {split_ifs}; try{simp}; cases c;\n    simp; try {split_ifs}; try {refl};\n    try {simp [mul_assoc, mul_pos_iff, mul_neg_iff, eq_self_iff_true,\n      or_true, *] at *}; try {split_ifs}; try {simp * at *};\n    simp only [le_antisymm_iff, lt_iff_le_and_ne] at *;\n    tauto,\n  end,\n  A13 := begin\n    assume a b,\n    cases a; simp; cases b; simp; try {split_ifs};\n    try {refl};\n    try {simp [mul_comm, mul_pos_iff, mul_neg_iff, eq_self_iff_true,\n      or_true, *] at *}; try {split_ifs}; try {simp * at *};\n    simp only [le_antisymm_iff, lt_iff_le_and_ne] at *;\n    tauto,\n  end,\n  A14 := by intros; try {cases a}; try {cases b}; try {cases c}; norm_num; simp * at *,\n  A15 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A16 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A17 := by intros; try {cases a}; try {cases b}; try {cases c}; simp,\n  A18 := by intro a; cases a; simp {contextual := tt},\n  A19 := by intro a; cases a; simp; try {split_ifs}; simp * {contextual := tt},\n  A20 := by simp,\n  A21 := rfl,\n  A22 := rfl,\n  A23 := begin\n    assume a,\n    cases a;\n    simp; try {split_ifs};\n    simp only [le_antisymm_iff, lt_iff_le_and_ne, *, ne.def, le_refl,\n      false_and, or_true, true_or, and_false, or_false, false_or, true_and,\n      and_true, not_true, eq_self_iff_true, true_iff, not_false_iff] at *,\n    linarith,\n  end,\n  A24 := begin\n    assume a,\n    cases a;\n    simp; try {split_ifs};\n    simp only [le_antisymm_iff, lt_iff_le_and_ne, *, ne.def, le_refl,\n      false_and, or_true, true_or, and_false, or_false, false_or, true_and,\n      and_true, not_true, eq_self_iff_true, true_iff, not_false_iff] at *,\n  end,\n  A25 := by simp,\n  A26 := \u03bb a b, begin\n    cases a; cases b; simp,\n    rw [\u2190 sub_eq_add_neg, sub_pos],\n  end,\n  A27 := by simp,\n  A28 := \u03bb a b, by cases a; cases b; simp; simp only [le_iff_lt_or_eq, eq_comm],\n  A29 := by simp,\n  A30 := \u03bb a, begin\n    cases a; simp [list.countp],\n    split_ifs; try {linarith},\n    have := lt_trichotomy a 0,\n    simp * at *,\n  end,\n  A31 := \u03bb a b c, by cases a; cases b; cases c; simp; split_ifs; simp [mul_add],\n  A32 := assume Y hY,\n    begin\n      by_cases infinity \u2208 Y,\n      { use infinity,\n        split,\n        { assume y, cases y; simp * },\n        { assume v h h1,\n          have := h1 infinity,\n          cases v; simp * at * } },\n      { by_cases hne : (of_real \u207b\u00b9' Y).nonempty,\n        { by_cases hbdd : bdd_above (of_real \u207b\u00b9' Y),\n          { use Sup (of_real \u207b\u00b9' Y),\n            split,\n            { assume y hy,\n              cases y; simp * at *,\n              rw [\u2190 le_iff_lt_or_eq],\n              exact le_cSup hbdd hy },\n            { assume v hv,\n              cases v; simp * at *,\n              { simp only [\u2190 le_iff_lt_or_eq],\n                assume h,\n                refine (cSup_le_iff hbdd hne).2 (\u03bb b hb, _),\n                have := (h (of_real b) hb),\n                simp [le_iff_lt_or_eq, *] at * },\n              { assume h,\n                cases hne with x hx,\n                have := h (of_real x) hx,\n                simp * at * } } },\n          { use infinity,\n            cases hne with x hx,\n            split,\n            { assume y, cases y; simp *, },\n            { assume v _ h,\n              have := h (of_real x) hx,\n              cases v; simp * at *,\n              apply hbdd,\n              use v,\n              assume x hx,\n              simpa [le_iff_lt_or_eq] using h (of_real x) hx } } },\n        { use neg_infinity,\n          have : \u2200 x, of_real x \u2209 Y,\n          { assume x hx, exact hne \u27e8x, hx\u27e9 },\n          split,\n          { assume y, cases y; simp * at * },\n          { assume v hv h,\n            cases v; simp * at * } } }\n    end }\n\n\n\n\nrun_cmd\ndo env \u2190 tactic.get_env,\n  d \u2190 env.get `transreal.transmathematic._proof_36,\n  let e := d.value,\n  tactic.trace (e.to_string)\n\n\ndef expr.length\n\nend transreal\n\n#exit\nimport tactic\n\n/-!\n\n# The partition challenge!\n\nProve that equivalence relations on \u03b1 are the same as partitions of \u03b1.\n\nThree sections:\n\n1) partitions\n2) equivalence classes\n3) the challenge\n\n## Overview\n\nSay `\u03b1` is a type, and `R` is a binary relation on `\u03b1`.\nThe following things are already in Lean:\n\nreflexive R := \u2200 (x : \u03b1), R x x\nsymmetric R := \u2200 \u2983x y : \u03b1\u2984, R x y \u2192 R y x\ntransitive R := \u2200 \u2983x y z : \u03b1\u2984, R x y \u2192 R y z \u2192 R x z\n\nequivalence R := reflexive R \u2227 symmetric R \u2227 transitive R\n\nIn the file below, we will define partitions of `\u03b1` and \"build some\ninterface\" (i.e. prove some propositions). We will define\nequivalence classes and do the same thing.\nFinally, we will prove that there's a bijection between\nequivalence relations on `\u03b1` and partitions of `\u03b1`.\n\n-/\n\n/-\n\n# 1) Partitions\n\nWe define a partition, and prove some easy lemmas.\n\n-/\n\n/-\n\n## Definition of a partition\n\nLet `\u03b1` be a type. A *partition* on `\u03b1` is defined to be\nthe following data:\n\n1) A set C of subsets of \u03b1, called \"blocks\".\n2) A hypothesis (i.e. a proof!) that all the blocks are non-empty.\n3) A hypothesis that every term of type \u03b1 is in one of the blocks.\n4) A hypothesis that two blocks with non-empty intersection are equal.\n-/\n\n/-- The structure of a partition on a Type \u03b1. -/\n@[ext] structure partition (\u03b1 : Type) :=\n(C : set (set \u03b1))\n(Hnonempty : \u2200 X \u2208 C, (X : set \u03b1).nonempty)\n(Hcover : \u2200 (a : \u03b1), \u2203 X \u2208 C, a \u2208 X)\n(Hdisjoint : \u2200 X Y \u2208 C, (X \u2229 Y : set \u03b1).nonempty \u2192 X = Y)\n\n/-\n\n## Basic interface for partitions\n\n-/\n\nnamespace partition\n\n-- let \u03b1 be a type, and fix a partition P on \u03b1. Let X and Y be subsets of \u03b1.\nvariables {\u03b1 : Type} {P : partition \u03b1} {X Y : set \u03b1}\n\n/-- If X and Y are blocks, and a is in X and Y, then X = Y. -/\ntheorem eq_of_mem (hX : X \u2208 P.C) (hY : Y \u2208 P.C) {a : \u03b1}\n  (haX : a \u2208 X)\n  (haY : a \u2208 Y) : X = Y :=\nbegin\n  have h := P.Hdisjoint X Y hX hY,\n  apply h,\n  use a,\n  split;\n  assumption,\nend\n\n\n/-- If a is in two blocks X and Y, and if b is in X,\n  then b is in Y (as X=Y) -/\ntheorem mem_of_mem (hX : X \u2208 P.C) (hY : Y \u2208 P.C) {a b : \u03b1}\n  (haX : a \u2208 X) (haY : a \u2208 Y) (hbX : b \u2208 X) : b \u2208 Y :=\nbegin\n  convert hbX,\n  exact (eq_of_mem hX hY haX haY).symm,\nend\n\n/-- Every term of type `\u03b1` is in one of the blocks for a partition `P`. -/\ntheorem mem_block (a : \u03b1) : \u2203 X : set \u03b1, X \u2208 P.C \u2227 a \u2208 X :=\nbegin\n  rcases P.Hcover a with \u27e8X, hXC, haX\u27e9,\n  use X,\n  split; assumption,\nend\n\nend partition\n\n/-\n\n# 2) Equivalence classes.\n\nWe define equivalence classes and prove a few basic results about them.\n\n-/\n\nsection equivalence_classes\n\n/-!\n\n## Definition of equivalence classes\n\n-/\n\n-- Notation and variables for the equivalence class section:\n\n-- let \u03b1 be a type, and let R be a binary relation on R.\nvariables {\u03b1 : Type} (R : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\n/-- The equivalence class of `a` is the set of `b` related to `a`. -/\ndef cl (a : \u03b1) :=\n{b : \u03b1 | R b a}\n\n/-!\n\n## Basic lemmas about equivalence classes\n\n-/\n\n/-- Useful for rewriting -- `b` is in the equivalence class of `a` iff\n`b` is related to `a`. True by definition. -/\ntheorem cl_def {a b : \u03b1} : b \u2208 cl R a \u2194 R b a := iff.rfl\n\n-- Assume now that R is an equivalence relation.\nvariables {R} (hR : equivalence R)\ninclude hR\n\n/-- x is in cl_R(x) -/\nlemma mem_cl_self (a : \u03b1) :\n  a \u2208 cl R a :=\nbegin\n  rw cl_def,\n  rcases hR with \u27e8hrefl, hsymm, htrans\u27e9,\n  unfold reflexive at hrefl,\n  apply hrefl,\nend\n\n/-- if a is in cl(b) then cl(a) \u2286 cl(b) -/\nlemma cl_sub_cl_of_mem_cl {a b : \u03b1} :\n  a \u2208 cl R b \u2192\n  cl R a \u2286 cl R b :=\nbegin\n  intro hab,\n  rw set.subset_def,\n  intro x,\n  intro hxa,\n  rw cl_def at *,\n  rcases hR with \u27e8hrefl, hsymm, htrans\u27e9,\n  exact htrans hxa hab,\nend\n\nlemma cl_eq_cl_of_mem_cl {a b : \u03b1} :\n  a \u2208 cl R b \u2192\n  cl R a = cl R b :=\nbegin\n  intro hab,\n  apply set.subset.antisymm,\n  { apply cl_sub_cl_of_mem_cl hR hab },\n  { apply cl_sub_cl_of_mem_cl hR,\n    rw cl_def at *,\n    rcases hR with \u27e8hrefl, hsymm, htrans\u27e9,\n    apply hsymm,\n    exact hab }\nend\n\nend equivalence_classes -- section\n\n/-!\n\n# 3) The challenge!\n\nLet `\u03b1` be a type (i.e. a collection of stucff).\n\nThere is a bijection between equivalence relations on `\u03b1` and\npartitions of `\u03b1`.\n\nWe prove this by writing down constructions in each direction\nand proving that the constructions are two-sided inverses of one another.\n-/\n\nopen partition\n\n\nexample (\u03b1 : Type) : {R : \u03b1 \u2192 \u03b1 \u2192 Prop // equivalence R} \u2243 partition \u03b1 :=\n-- We define constructions (functions!) in both directions and prove that\n-- one is a two-sided inverse of the other\n{ -- Here is the first construction, from equivalence\n  -- relations to partitions.\n  -- Let R be an equivalence relation.\n  to_fun := \u03bb R, {\n    -- Let C be the set of equivalence classes for R.\n    C := { B : set \u03b1 | \u2203 x : \u03b1, B = cl R.1 x},\n    -- I claim that C is a partition. We need to check the three\n    -- hypotheses for a partition (`Hnonempty`, `Hcover` and `Hdisjoint`),\n    -- so we need to supply three proofs.\n    Hnonempty := begin\n      cases R with R hR,\n      -- If X is an equivalence class then X is nonempty.\n      show \u2200 (X : set \u03b1), (\u2203 (a : \u03b1), X = cl R a) \u2192 X.nonempty,\n      rintros X \u27e8a, rfl\u27e9,\n      use a,\n      exact mem_cl_self hR _\n    end,\n    Hcover := begin\n      cases R with R hR,\n      -- The equivalence classes cover \u03b1\n      show \u2200 (a : \u03b1), \u2203 (X : set \u03b1) (H : \u2203 (b : \u03b1), X = cl R b), a \u2208 X,\n      sorry,\n    end,\n    Hdisjoint := begin\n      cases R with R hR,\n      -- If two equivalence classes overlap, they are equal.\n      show \u2200 (X Y : set \u03b1), (\u2203 (a : \u03b1), X = cl R a) \u2192\n        (\u2203 (b : \u03b1), Y = cl R b) \u2192 (X \u2229 Y).nonempty \u2192 X = Y,\n      sorry,\n    end },\n  -- Conversely, say P is an partition.\n  inv_fun := \u03bb P,\n    -- Let's define a binary relation `R` thus:\n    --  `R a b` iff *every* block containing `a` also contains `b`.\n    -- Because only one block contains a, this will work,\n    -- and it turns out to be a nice way of thinking about it.\n    \u27e8\u03bb a b, \u2200 X \u2208 P.C, a \u2208 X \u2192 b \u2208 X, begin\n      -- I claim this is an equivalence relation.\n    split,\n    { -- It's reflexive\n      show \u2200 (a : \u03b1)\n        (X : set \u03b1), X \u2208 P.C \u2192 a \u2208 X \u2192 a \u2208 X,\n      sorry,\n    },\n    split,\n    { -- it's symmetric\n      show \u2200 (a b : \u03b1),\n        (\u2200 (X : set \u03b1), X \u2208 P.C \u2192 a \u2208 X \u2192 b \u2208 X) \u2192\n         \u2200 (X : set \u03b1), X \u2208 P.C \u2192 b \u2208 X \u2192 a \u2208 X,\n      sorry,\n    },\n    { -- it's transitive\n      unfold transitive,\n      show \u2200 (a b c : \u03b1),\n        (\u2200 (X : set \u03b1), X \u2208 P.C \u2192 a \u2208 X \u2192 b \u2208 X) \u2192\n        (\u2200 (X : set \u03b1), X \u2208 P.C \u2192 b \u2208 X \u2192 c \u2208 X) \u2192\n         \u2200 (X : set \u03b1), X \u2208 P.C \u2192 a \u2208 X \u2192 c \u2208 X,\n      sorry,\n    }\n  end\u27e9,\n  -- If you start with the equivalence relation, and then make the partition\n  -- and a new equivalence relation, you get back to where you started.\n  left_inv := begin\n    rintro \u27e8R, hR\u27e9,\n    -- Tidying up the mess...\n    suffices : (\u03bb (a b : \u03b1), \u2200 (c : \u03b1), a \u2208 cl R c \u2192 b \u2208 cl R c) = R,\n      simpa,\n    -- ... you have to prove two binary relations are equal.\n    ext a b,\n    -- so you have to prove an if and only if.\n    show (\u2200 (c : \u03b1), a \u2208 cl R c \u2192 b \u2208 cl R c) \u2194 R a b,\n    sorry,\n  end,\n  -- Similarly, if you start with the partition, and then make the\n  -- equivalence relation, and then construct the corresponding partition\n  -- into equivalence classes, you have the same partition you started with.\n  right_inv := begin\n    -- Let P be a partition\n    intro P,\n    -- It suffices to prove that a subset X is in the original partition\n    -- if and only if it's in the one made from the equivalence relation.\n    ext X,\n    show (\u2203 (a : \u03b1), X = cl _ a) \u2194 X \u2208 P.C,\n    dsimp only,\n    sorry,\n  end }\n\n/-\n-- get these files with\n\nleanproject get ImperialCollegeLondon/M40001_lean\n\n\nleave this channel and go to a workgroup channel and try\nfolling in the sorrys.\n\nI will come around to help.\n\n-/\n\n#exit\nimport tactic\n\n/-!\n\n# Tactic cheat sheet.\n\n\n-- natnumgame tactics\n\napply,\nexact (and assumption)\nsplit\nuse (use `use` to make progress with `nonempty X`)\n\n\n\n-/\n\n/-!\n\n## 1) Extracting information from hypotheses\n\n-/\n\n/-!\n\n### 1a) cases and rcases\n\nMany objects in Lean are pairs of data. For example, a proof\nof `P \u2227 Q` is stored as a pair consisting of a proof of `P` and\na proof of `Q`. The hypothesis `\u2203 n : \u2115, f n = 37` is stored\ninternally as a pair, namely a natural `n` and a proof that `f n = 37`.\nNote that \"hypothesis\" and \"proof\" mean the same thing.\n\nIf `h : X` is something which is stored as a pair in Lean,\nthen `cases h with a b` will destroy `h` and replace it with\nthe two pieces of data which made up `h`, calling them `a` and `b`.\n\n-/\n\nexample (h : \u2203 n : \u2115, n ^ 2 = 2) : false :=\nbegin\n  -- h: \u2203 (n : \u2115), n ^ 2 = 2\n  cases h with n hn,\n  -- n: \u2115\n  -- hn: n ^ 2 = 2\n  sorry\nend\n\nexample (P Q : Prop) (h : P \u2227 Q) : P :=\nbegin\n  -- h: P \u2227 Q\n  cases h with hP hQ,\n  -- hP: P\n  -- hQ: Q\n  exact hP,\nend\n\n-- Some things are more than two pieces of data! You can do much more\n-- elaborate deconstructions with the `rcases` command.\n\nexample (R : \u2115 \u2192 \u2115 \u2192 Prop) (hR : equivalence R) : symmetric R :=\nbegin\n  -- hR: equivalence R\n  rcases hR with \u27e8hrefl, hsymm, htrans\u27e9,\n  -- hrefl: reflexive R\n  -- hsymm: symmetric R\n  -- htrans: transitive R\n  exact hsymm,\nend\n\n/-!\n\n## 1b) specialize\n\nSay you have a long hypothesis `h : \u2200 n : \u2115, f n > 37 \u2192 n = 23`.\nThis hypothesis is a *function*. It takes as inputs a natural number n\nand a proof that `f n > 37`, and then it gives as output a proof\nthat `n = 23`. You can feed in some inputs and specialize the function.\n\nSay for example you you manage to prove the hypothesis `ht : f t > 37` for some natural\nnumber `t`. Then `specialize h t ft` would change `h` to `t = 23`.\n\n-/\n\nexample (X Y : set \u2115) (a : \u2115) (h : \u2200 n : \u2115, n \u2208 X \u2192 n \u2208 Y) (haX : a \u2208 X) : a \u2208 Y :=\nbegin\n  -- a: \u2115\n  -- haX: a \u2208 X\n  -- h: \u2200 (n : \u2115), n \u2208 X \u2192 n \u2208 Y\n  specialize h a haX,\n  -- h: a \u2208 Y\n  assumption,\nend\n\n/-!\n\n# 2) Making new hypothesis\n\n-/\n\n/-!\n\n## have\n\nThe `have` tactic makes a new hypothesis. The proof of the current goal\nis paused and a new goal is created. Generally one should now put braces\n`{ }` because if there is more than one goal then understanding what the\ncode is doing can get very difficult.\n\n-/\n\nexample (a b c n : \u2115) (hn : n > 2) : a^n + b^n = c^n \u2192 a * b = 0 :=\nbegin\n  -- \u22a2 a ^ n + b ^ n = c ^ n \u2192 a * b = 0\n  -- looks a bit tricky\n  -- why not prove something easier first\n  have ha : (a + 1) + 1 = a + 2,\n  { -- \u22a2 a + 1 + 1 = a + 2\n    apply add_assoc,\n  },\n  -- ha: a + 1 + 1 = a + 2\n  -- \u22a2 a ^ n + b ^ n = c ^ n \u2192 a * b = 0\n  sorry\nend\n\n/-!\n\n# 3) Using hypotheses to change the goal.\n\n-/\n\n/-!\n\n## 2a) rw\n\nThe generic `sub in` tactic. If `h : X = Y` then `rw h` will change all\n`X`'s in the goal to `Y`'s. Also works with `h : P \u2194 Q` if `P` and `Q`\nare true-false statements.\n\n-/\n\nexample (X Y : set \u2115) (hXY : X = Y) (a : \u2115) (haX : a \u2208 Y) : a \u2208 X :=\nbegin\n  -- hXY: X = Y\n  -- \u22a2 a \u2208 X\n  rw hXY,\n  -- hXY: X = Y\n  -- \u22a2 a \u2208 Y\n  assumption\nend\n\n-- Variants -- `rw h1 at h2`, `rw h1 at h2 \u22a2`, `rw h at *`\n\n/-!\n\n## 2b) convert\n\n`convert` is in some sense the opposite way of thinking to `rw`. Instead\nof continually rewriting the goal until it becomes one of your assumptions,\nwhy not just tell Lean that the assumption is basically the right answer\nmodulo a few loose ends, which Lean will then leave for you as new goals.\n\n-/\n\nexample (X Y : set \u2115) (hX : 37 \u2208 X) : 37 \u2208 Y :=\nbegin\n  -- hX: 37 \u2208 X\n  -- \u22a2 37 \u2208 Y\n  convert hX,\n  -- \u22a2 Y = X\n  sorry\nend\n\n/-\n\n# 4) Changing the goal without using hypotheses\n\n-/\n\n/-! ### 4a) intro and rintro -/\n\n-- `intro` is a basic tactic for introducing hypotheses\nexample (P Q : Prop) : P \u2192 Q :=\nbegin\n  -- \u22a2 P \u2192 Q\n  intro hP,\n  -- hP: P\n  -- \u22a2 Q\n  sorry\nend\n\n-- `rintro` is to `intro` what `rcases` is to `cases`. It enables\n-- you to assume something and simultaneously take it apart.\n\nexample (f : \u2115 \u2192 \u211a) : (\u2203 n : \u2115, f n > 37) \u2192 (\u2203 n : \u2115, f n > 36) :=\nbegin\n  -- \u22a2 (\u2203 (n : \u2115), f n > 37) \u2192 P\n  rintro \u27e8n, hn\u27e9,\n  --  n: \u2115\n  -- hn: f n > 37\n  -- \u22a2 P\n  sorry,\nend\n\n/-! ## 4b) ext -/\n\n-- `ext` is Lean's extensionality tactic. If your goal is to prove that\n-- two extensional things are equal (e.g. sets, functions, binary relations)\n-- then `ext a` or `ext a b` or whatever is appropriate, will turn the\n-- question into the assertion that they behave in the same way. Let's look\n-- at some examples\n\nexample (A B : set \u2115) : A = B :=\nbegin\n  -- \u22a2 A = B\n  ext x,\n  --  x : \u2115\n  -- \u22a2 x \u2208 A \u2194 x \u2208 B\n  sorry\nend\n\nexample (X Y : Type) (f g : X \u2192 Y) : f = g :=\nbegin\n  -- \u22a2 f = g\n  ext x,\n  --  x : X\n  -- \u22a2 f x = g x\n  sorry\nend\n\nexample (\u03b1 : Type) (R S : \u03b1 \u2192 \u03b1 \u2192 Prop) : R = S :=\nbegin\n  -- \u22a2 R = S\n  ext a b,\n  -- a b : \u03b1\n  -- \u22a2 R a b \u2194 S a b\n  sorry\nend\n\n#exit\nimport data.list.defs data.vector tactic\nimport for_mathlib.coprod\n\n\n\nset_option profiler true\nexample {G : Type*} [group G] (a b : G) :\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  a * b *\n  b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 *\n  a\u207b\u00b9 * b\u207b\u00b9 * a\u207b\u00b9\n  = 1 :=\nbegin\n\n  simp [mul_assoc],\n  --group,\n\nend\n\n#eval let a =\n\n#print vector\n\ndef reverse\u2082 {\u03b1 : Type*} : list \u03b1 \u2192 list \u03b1\n| []     := []\n| (a::l) := l ++ [a]\n\nset_option profiler true\n\n@[inline] def N : \u2115 := 5000000\n\n#eval  (list.reverse (list.range N)).length\n#eval (reverse\u2082 (list.range N)).length\n\n\n#exit\nimport data.equiv.basic data.list.perm data.list\n\nopen list\n\n\n\nexample (n : \u2115) : equiv.perm (fin n) \u2243 { l : list (fin n) // l.nodup \u2227 l.length = n } :=\n{ to_fun := \u03bb e, \u27e8(fin_range n).map e, list.nodup_map e.injective (nodup_fin_range _), by simp\u27e9,\n  inv_fun := \u03bb l,\n    { to_fun := \u03bb i, l.1.nth_le i.1 (l.prop.2.symm \u25b8 i.2),\n       inv_fun := \u03bb i, \u27e8l.1.index_of i, by conv_rhs { rw [\u2190 l.prop.2] };\n         exact index_of_lt_length.2 sorry\u27e9,\n      left_inv := \u03bb \u27e8_, _\u27e9, by simp [nth_le_index_of l.prop.1],\n      right_inv := \u03bb _, by simp },\n  left_inv := \u03bb _, equiv.ext $ \u03bb i, begin simp, admit end,\n  right_inv := \u03bb \u27e8_, _\u27e9, sorry }\n\nopen equiv set\n\nexample (l\u2081 l\u2082 : list \u2115) : bool := if l\u2081 ~ l\u2082 then tt else ff\n\n@[simp] lemma set.sum_compl_symm_apply {\u03b1 : Type*} {s : set \u03b1} [decidable_pred s] {x : s} :\n  (equiv.set.sum_compl s).symm x = sum.inl x :=\nby cases x with x hx; exact set.sum_compl_symm_apply_of_mem hx\n\n@[simp] lemma set.sum_compl_symm_apply_compl {\u03b1 : Type*} {s : set \u03b1}\n  [decidable_pred s] {x : s\u1d9c} : (equiv.set.sum_compl s).symm x = sum.inr x :=\nby cases x with x hx; exact set.sum_compl_symm_apply_of_not_mem hx\n\n@[simp] lemma subtype_congr_apply {\u03b1 : Sort*} {\u03b2 : Sort*} {p : \u03b1 \u2192 Prop} {q : \u03b2 \u2192 Prop} (e : \u03b1 \u2243 \u03b2)\n  (h : \u2200 (a : \u03b1), p a \u2194 q (e a)) (x : {x // p x}) : e.subtype_congr h x = \u27e8e x, (h _).1 x.2\u27e9 := rfl\n\nprotected def compl {\u03b1 \u03b2 : Type*} {s : set \u03b1} {t : set \u03b2} [decidable_pred s] [decidable_pred t]\n  (e\u2080 : s \u2243 t) : {e : \u03b1 \u2243 \u03b2 // \u2200 x : s, e x = e\u2080 x} \u2243 ((s\u1d9c : set \u03b1) \u2243 (t\u1d9c : set \u03b2)) :=\n{ to_fun := \u03bb e, subtype_congr e\n    (\u03bb a, not_congr $ iff.intro\n      (\u03bb ha, by rw [\u2190 subtype.coe_mk a ha, e.prop \u27e8a, ha\u27e9]; exact (e\u2080 \u27e8a, ha\u27e9).prop)\n      (\u03bb ha, calc a = (e : \u03b1 \u2243 \u03b2).symm (e a) : by simp only [symm_apply_apply, coe_fn_coe_base]\n                ... = e\u2080.symm \u27e8e a, ha\u27e9 : (e : \u03b1 \u2243 \u03b2).injective\n                  (by { rw [e.prop (e\u2080.symm \u27e8e a, ha\u27e9)],\n                        simp only [apply_symm_apply, subtype.coe_mk] })\n                ... \u2208 s : (e\u2080.symm \u27e8_, ha\u27e9).prop)),\n  inv_fun := \u03bb e\u2081,\n    subtype.mk\n      (calc \u03b1 \u2243 s \u2295 (s\u1d9c : set \u03b1) : (set.sum_compl s).symm\n          ... \u2243 t \u2295 (t\u1d9c : set \u03b2) : equiv.sum_congr e\u2080 e\u2081\n          ... \u2243 \u03b2 : set.sum_compl t)\n      (\u03bb x, by simp only [sum.map_inl, trans_apply, sum_congr_apply,\n        set.sum_compl_apply_inl, set.sum_compl_symm_apply]),\n  left_inv := \u03bb e,\n    begin\n      ext x,\n      by_cases hx : x \u2208 s,\n      { simp only [set.sum_compl_symm_apply_of_mem hx, \u2190e.prop \u27e8x, hx\u27e9,\n          sum.map_inl, sum_congr_apply, trans_apply,\n          subtype.coe_mk, set.sum_compl_apply_inl] },\n      { simp only [set.sum_compl_symm_apply_of_not_mem hx, sum.map_inr,\n          subtype_congr_apply, set.sum_compl_apply_inr, trans_apply,\n          sum_congr_apply, subtype.coe_mk] },\n    end,\n  right_inv := \u03bb e, equiv.ext $ \u03bb x, by simp only [sum.map_inr, subtype_congr_apply,\n    set.sum_compl_apply_inr, function.comp_app, sum_congr_apply, equiv.coe_trans,\n    subtype.coe_eta, subtype.coe_mk, set.sum_compl_symm_apply_compl] }\n\n#exit\nimport set_theory.cardinal\n\nopen cardinal\nuniverse u\n\n@[simp] theorem mk_set {\u03b1 : Type} : mk (set \u03b1) = 2 ^ mk \u03b1 :=\nbegin\n  rw [set, \u2190 power_def Prop \u03b1, mk_Prop],\nend\n#exit\nimport linear_algebra.exterior_algebra\n\nvariables {R : Type*} [comm_semiring R] {M : Type*} [add_comm_monoid M] [semimodule R M]\n\n/- The following gives an error: -/\n\n#check (ring_quot.mk_alg_hom R (exterior_algebra.rel R M) :\n  tensor_algebra R M \u2192\u2090[R] exterior_algebra R M)\n\n/- For this reason there is the following def in\n   linear_algebra/exterior_algebra.lean: -/\n/-\nprotected def quot : tensor_algebra R M \u2192\u2090[R] exterior_algebra R M :=\n  ring_quot.mk_alg_hom R _\n-/\n\n/- Similarly, this gives an error: -/\n/-#check (ring_quot.mk_alg_hom R (tensor_algebra.rel R M) :\n  free_algebra R M \u2192\u2090[R] tensor_algebra R M)-/\n#print tensor_algebra\n\nattribute [semireducible] tensor_algebra\n-- but the following doesn't work!\nlemma quot2 : free_algebra R M \u2192\u2090[R] tensor_algebra R M :=\n  ring_quot.mk_alg_hom R (tensor_algebra.rel R M)\n#exit\nimport analysis.ODE.gronwall\n\nopen topological_space\n\nlemma ExNine (f : \u211d \u2192 \u211d) (s : set \u211d) : continuous f \u2194 \u2200 s, is_open s \u2192  is_open (f \u207b\u00b9' s) :=\n\u27e8\u03bb h _, h _, \u03bb h _, h _\u27e9\n\n#exit\nimport data.nat.prime\nimport data.fintype\n\n#eval (@finset.univ (equiv.perm (fin 9)) _).filter _\n\n#exit\nvariables {\u03b1 : Sort*} {\u03b2 : Sort*}\n\ntheorem forall_eq_apply_imp_iff {f : \u03b1 \u2192 \u03b2} {p : \u03b2 \u2192 Prop} :\n  (\u2200 a, \u2200 b, b = f a \u2192 p b) \u2194 (\u2200 a, p (f a)) :=\n\u27e8\u03bb h a, h a (f a) rfl, \u03bb h a b hba, hba.symm \u25b8 h a\u27e9\n\ntheorem piext {\u03b1 : Sort*} {\u03b2 \u03b3 : \u03b1 \u2192 Sort*} (h : \u2200 a, \u03b2 a = \u03b3 a) :\n  (\u03a0 a, \u03b2 a) = \u03a0 a, \u03b3 a :=\nby rw [show \u03b2 = \u03b3, from funext h]\n\n#exit\nimport algebra.group_power data.equiv.mul_add data.vector2\n#print function.swap\ndef word : \u2115 \u2192 G\n| 0 := a\n| (n+1) := b * word n * b\u207b\u00b9 * a * b * (word n)\u207b\u00b9 * b\u207b\u00b9\n\ndef tower (k : \u2115) : \u2115 \u2192 \u2115\n| 0     := 1\n| (n+1) := k ^ tower n\n\nlemma word_eq_conj (n : \u2115) : word a b (n + 1) = mul_aut.conj (mul_aut.conj b (word a b n)) a :=\nby simp [mul_aut.conj_apply, mul_aut.inv_def, mul_aut.conj_symm_apply, mul_assoc, word]\n\nlemma pow_two_pow_eq (k n : \u2115) (H : word a b 1 = a ^ k) : a ^ (k ^ n) = (mul_aut.conj (mul_aut.conj b a) ^ n) a :=\nbegin\n  induction n with n ih,\n  { simp },\n  { rw [nat.pow_succ, pow_mul, ih,\n      \u2190 mul_equiv.to_monoid_hom_apply, \u2190 monoid_hom.map_pow,\n      \u2190 H, pow_succ' _ n, mul_aut.mul_apply, mul_equiv.to_monoid_hom_apply,\n      word_eq_conj, word] }\nend\n\nlemma word_eq_power_tower (k n : \u2115) (H : word a b 1 = a ^ k) : word a b n = a ^ tower k n :=\nbegin\n  induction n with n ih,\n  { simp [word, tower] },\n  { rw [tower, pow_two_pow_eq _ _ _ _ H, word_eq_conj, ih],\n    simp only [\u2190 mul_equiv.to_monoid_hom_apply, monoid_hom.map_pow] }\nend\n\n\n#exit\nimport ring_theory.noetherian\n#print rel_embedding.well_founded_iff_no_descending_seq\ntheorem set_has_maximal_iff_noetherian {R M} [ring R] [add_comm_group M] [module R M] :\n  (\u2200(a : set $ submodule R M), a.nonempty \u2192 \u2203 (M' \u2208 a), \u2200 (I \u2208 a), M' \u2264 I \u2192 I = M') \u2194\n    is_noetherian R M :=\niff.trans\n  \u27e8_,\n    \u03bb wf a ha, \u27e8well_founded.min wf a ha, well_founded.min_mem _ _ _,\n      \u03bb I hI hminI, (lt_or_eq_of_le hminI).elim\n        (\u03bb h, (well_founded.not_lt_min wf _ _ hI h).elim) eq.symm\u27e9\u27e9\nis_noetherian_iff_well_founded.symm\n-- begin\n--   rw [is_noetherian_iff_well_founded],\n--   split,\n--   { refine \u03bb h, \u27e8\u03bb a, classical.by_contradiction (\u03bb ha, _)\u27e9,\n--     rcases h {a | \u00ac acc gt a} \u27e8a, ha\u27e9 with \u27e8b, hab, hb\u27e9,\n--     exact hab (acc.intro b\n--       (\u03bb c hcb, classical.by_contradiction\n--         (\u03bb hc, absurd hcb (hb c hc (le_of_lt hcb) \u25b8 lt_irrefl c)))),\n--      },\n--   { exact \u03bb wf a ha, \u27e8well_founded.min wf a ha, well_founded.min_mem _ _ _,\n--       \u03bb I hI hminI, (lt_or_eq_of_le hminI).elim\n--         (\u03bb h, (well_founded.not_lt_min wf _ _ hI h).elim) eq.symm\u27e9 }\n-- end\n\n#exit\nimport data.list.defs\n\nvariables {\u03b1 \u03b2 \u03b3 : Type}\nopen list\n\ndef sublists'_aux : list \u03b1 \u2192 (list \u03b1 \u2192 list \u03b2) \u2192 list (list \u03b2) \u2192 list (list \u03b2)\n| []     f r := f [] :: r\n| (a::l) f r := sublists'_aux l f (sublists'_aux l (f \u2218 cons a) r)\n\n\n\ndef sublists2 (l : list \u03b1) := sublists_aux2 [] l cons\n\nexample (l : list \u03b1) (f : list \u03b1 \u2192 list \u03b2 \u2192 list \u03b2) :\n  f [] (sublists_aux l f) = sublists_aux2 [] l f :=\nbegin\n  induction l with a l ih generalizing f,\n  { refl },\n  { rw [sublists_aux2, \u2190 ih, sublists_aux] }\nend\n\n\n\nimport ring_theory.noetherian\n\nexample : is_noetherian_ring \u2124 :=\nbegin\n  split,\n  assume s,\n\n\nend\n\n#exit\nimport data.polynomial.eval\n\nvariables {R : Type*} [comm_ring R] {S : Type*} [comm_ring S] {f : R \u2192+* S}\n\nopen polynomial\n\nvariables {\u03b1 : Type}\n\ndef interval : \u03b1 \u2192 \u03b1 \u2192 set \u03b1 := sorry\n\n\n\n#print convex_hull.intrv\nlemma map_comp (p q : polynomial R) : map f (p.comp q) = (map f p).comp (map f q) :=\npolynomial.induction_on p\n  (by simp)\n  (by simp {contextual := tt})\n  (by simp [pow_succ', \u2190 mul_assoc, polynomial.comp] {contextual := tt})\n\n@[simp] def days : fin 12 \u2192 \u2124 \u2192 \u2115\n  | \u27e80, _\u27e9 _ := 31\n  | \u27e81, _\u27e9 y := if 4 \u2223 y \u2227 (\u00ac100 \u2223 y \u2228 400 \u2223 y) then 29 else 28\n  | \u27e82, _\u27e9 _ := 31\n  | \u27e83, _\u27e9 _ := 30\n  | \u27e84, _\u27e9 _ := 31\n  | \u27e85, _\u27e9 _ := 30\n  | \u27e86, _\u27e9 _ := 31\n  | \u27e87, _\u27e9 _ := 31\n  | \u27e88, _\u27e9 _ := 30\n  | \u27e89, _\u27e9 _ := 31\n  | \u27e810, _\u27e9 _ := 30\n  | \u27e811, _\u27e9 _ := 31\n  | \u27e8_ + 12, h\u27e9 _ := by linarith\n\n\n#exit\ninductive bad_eq {Q : Type} : Q \u2192 Q \u2192 Prop\n| finish {q : Q} : bad_eq q q\n| step   {a b : Q} : bad_eq a b \u2192 bad_eq a b\n\nlemma bad_eq_eq {Q : Type} {a b : Q} : bad_eq a b \u2192 a = b :=\nbegin\n    intro h, induction h, refl, assumption, -- OK\nend\n\ninductive U (R : Type) : Type\n| wrap : R \u2192 U\n\nlemma bad_eq_wrap {Q : Type} {a b : Q} : bad_eq (U.wrap a) (U.wrap b) \u2192 a = b :=\nbegin\n    intro h,\n    generalize hx : U.wrap a = x,\n    generalize hy : U.wrap b = y,\n    rw [hx, hy] at h,\n    induction h,\n    { cases h,\n      simp * at * },\n    { simp * at * }\nend\n\nopen equiv\n\ndef perm_array (a : array n \u03b1) (p : perm (fin n)) : array n \u03b1 := \u27e8a.read \u2218 p.inv_fun\u27e9\n\n@[simp] lemma perm_array_one (a : array n \u03b1) : perm_array a 1 = a := by cases a; refl\n\nopen_locale classical\n\ntheorem perm_to_list {n : \u2115} {a : array n \u03b1} {p : perm (fin n)} :\n    (perm_array a p).to_list ~ a.to_list :=\nlist.perm_iff_count.2 begin\n  assume h,\n  cases a,\n  dsimp [perm_array, function.comp, array.to_list, array.rev_foldl,\n    array.rev_iterate, array.read, d_array.rev_iterate, d_array.read,\n    d_array.rev_iterate_aux],\n  refine list.count\n\n\nend\nvariables {\u03b1 : Type*} [decidable_eq \u03b1]\n\nopen equiv equiv.perm\n\n@[elab_as_eliminator] lemma swap_induction_on' [fintype \u03b1] {P : perm \u03b1 \u2192 Prop} (f : perm \u03b1)\n  (h1 : P 1) (ih : \u2200 f x y, x \u2260 y \u2192 P f \u2192 P (f * swap x y )) : P f :=\nbegin\n  rw [\u2190 inv_inv f],\n  refine @swap_induction_on _ _ _ (P \u2218 has_inv.inv) f\u207b\u00b9 h1 _,\n  assume f x y hxy hy,\n  simp only [function.comp_app, mul_inv_rev, swap_inv],\n  exact ih _ _ _ hxy hy\nend\n\n\n#exit\nimport data.bool data.quot\n\nexample {\u03b1 : Type} (l : list \u03b1) (a : \u03b1) : a :: l \u2260 l :=\n\u03bb h, list.no_confusion h\n\n\ninductive X (\u03b1 : Type) : trunc \u03b1 \u2192 Type\n| mk (a : trunc \u03b1) : X a\n#print X.rec\nlemma\n\n\n#exit\nimport data.dfinsupp\nimport tactic\n\nuniverses u v w\n\nvariables {ii : Type u} {jj : Type v} [decidable_eq ii] [decidable_eq jj]\nvariables (\u03b2 : ii \u2192 jj \u2192 Type w) [\u03a0 i j, decidable_eq (\u03b2 i j)]\n\nsection has_zero\nvariables [\u03a0 i j, has_zero (\u03b2 i j)]\n\ndef to_fun (x : \u03a0\u2080 (ij : ii \u00d7 jj), \u03b2 ij.1 ij.2) : \u03a0\u2080 i, \u03a0\u2080 j, \u03b2 i j :=\nquotient.lift_on x\n  (\u03bb x, \u27e6dfinsupp.pre.mk\n    (\u03bb i, show \u03a0\u2080 j : jj, \u03b2 i j,\n      from \u27e6dfinsupp.pre.mk\n        (\u03bb j, x.to_fun (i, j))\n        (x.pre_support.map prod.snd)\n        (\u03bb j, (x.3 (i, j)).elim (\u03bb h, or.inl (multiset.mem_map.2 \u27e8(i, j), h, rfl\u27e9)) or.inr)\u27e7)\n    (x.pre_support.map prod.fst)\n    (\u03bb i, or_iff_not_imp_left.2 $ \u03bb h, dfinsupp.ext $ \u03bb j, (x.3 (i, j)).resolve_left\n      (\u03bb hij, h (multiset.mem_map.2 \u27e8(i, j), hij, rfl\u27e9)))\u27e7)\n  (\u03bb a b hab, dfinsupp.ext (\u03bb i, dfinsupp.ext (\u03bb j, hab _)))\n\ndef inv_fun (x : \u03a0\u2080 i, \u03a0\u2080 j, \u03b2 i j) : \u03a0\u2080 (ij : ii \u00d7 jj), \u03b2 ij.1 ij.2 :=\nquotient.lift_on x\n  (\u03bb x, \u27e6dfinsupp.pre.mk (\u03bb i : ii \u00d7 jj, quotient.lift_on (x.1 i.1)\n      (\u03bb x, x.1 i.2)\n      (\u03bb a b hab, hab _))\n    (x.pre_support.bind (\u03bb i, (quotient.lift_on (x.1 i)\n      (\u03bb x, ((x.pre_support.filter (\u03bb j, x.1 j \u2260 0)).map (\u03bb j, (i, j))).to_finset)\n      (\u03bb a b hab, begin\n          ext p,\n          cases a, cases b,\n          replace hab : a_to_fun = b_to_fun := funext hab,\n          subst hab,\n          cases p with p\u2081 p\u2082,\n          simp [and_comm _ (_ = p\u2082), @and.left_comm _ (_ = p\u2082)],\n          specialize b_zero p\u2082,\n          specialize a_zero p\u2082,\n          tauto,\n      end)).1))\n    (\u03bb i, or_iff_not_imp_right.2 begin\n      generalize hxi : x.1 i.1 = a,\n      revert hxi,\n      refine quotient.induction_on a (\u03bb a hxi, _),\n      assume h,\n      have h\u2081 := (a.3 i.2).resolve_right h,\n      have h\u2082 := (x.3 i.1).resolve_right (\u03bb ha, begin\n        rw [hxi] at ha,\n        exact h ((quotient.exact ha) i.snd),\n      end),\n      simp only [exists_prop, ne.def, multiset.mem_bind],\n      use i.fst,\n      rw [hxi, quotient.lift_on_beta],\n      simp only [multiset.mem_erase_dup, multiset.to_finset_val,\n        multiset.mem_map, multiset.mem_filter],\n      exact \u27e8h\u2082, i.2, \u27e8h\u2081, h\u27e9, by cases i; refl\u27e9\n    end)\u27e7)\n  (\u03bb a b hab, dfinsupp.ext $ \u03bb i, by unfold_coes; simp [hab i.1])\n\nexample : (\u03a0\u2080 (ij : ii \u00d7 jj), \u03b2 ij.1 ij.2) \u2243 \u03a0\u2080 i, \u03a0\u2080 j, \u03b2 i j :=\n{ to_fun := to_fun \u03b2,\n  inv_fun := inv_fun \u03b2,\n  left_inv := \u03bb x, quotient.induction_on x (\u03bb x, dfinsupp.ext (\u03bb i, by cases i; refl)),\n  right_inv := \u03bb x, quotient.induction_on x (\u03bb x, dfinsupp.ext (\u03bb i, dfinsupp.ext (\u03bb j,\n    begin\n      generalize hxi : x.1 i = a,\n      revert hxi,\n      refine quotient.induction_on a (\u03bb a hxi, _),\n      rw [to_fun, inv_fun],\n      unfold_coes,\n      simp,\n      rw [hxi, quotient.lift_on_beta, quotient.lift_on_beta],\n    end)))  }\n\nend has_zero\n\nsection add_comm_monoid\nvariable [\u03a0 i j, add_comm_monoid (\u03b2 i j)]\nexample : (\u03a0\u2080 (ij : ii \u00d7 jj), \u03b2 ij.1 ij.2) \u2243+ \u03a0\u2080 i, \u03a0\u2080 j, \u03b2 i j :=\n\nend add_comm_monoid\n\n\n#exit\nexample : (\u03a0\u2080 (ij : ii \u00d7 jj), \u03b2 ij.1 ij.2) \u2243 \u03a0\u2080 i, \u03a0\u2080 j, \u03b2 i j := sorry\n\nexample {\u03b1 : Type} (r : \u03b1 \u2192 \u03b1 \u2192 Prop) (a : \u03b1) (h : acc r a) : acc r a :=\nacc.intro _ (acc.rec_on h (\u03bb x h ih y hy, h y hy))\n\nvariables (G : Type u) [group G] (F : Type v) [field F] [mul_semiring_action G F] (g : G)\n\n/-- The subfield fixed by one element of the group. -/\ndef fixed_by : set F :=\n{ x | g \u2022 x = x }\n\ntheorem fixed_eq_Inter_fixed_by : fixed_points G F = \u22c2 g : G, fixed_by G F g :=\nset.ext $ \u03bb x, \u27e8\u03bb hx, set.mem_Inter.2 $ \u03bb g, hx g,\n  \u03bb hx g, by { exact (set.mem_Inter.1 hx g : _) } \u27e9\n\nimport tactic data.real.basic\n\nexample (a b c : \u211d) (h: a/b = a/c) (g : a \u2260 0) : 1/b = 1/c :=\nby rwa [\u2190 mul_right_inj' g, one_div_eq_inv, one_div_eq_inv]\n\n\nimport data.nat.modeq\n\nexample : unit \u2260 bool :=\nbegin\n  assume h,\n  have : \u2200 x y : unit, x = y, { intros, cases x, cases y, refl },\n  rw h at this,\n  exact absurd (this tt ff) dec_trivial\n\nend\n\nexample (p : \u2115) (hp : p % 4 = 2) : 4 \u2223 p - 2 :=\n\u27e8p / 4, _\u27e9\n\n#exit\nimport tactic\n\nopen set\n\nclass topological_space (X : Type) :=\n(is_open        : set X \u2192 Prop)\n(is_open_univ   : is_open univ)\n(is_open_inter  : \u2200 (U V : set X), is_open U \u2192 is_open V \u2192 is_open (U \u2229 V))\n(is_open_sUnion : \u2200 (\ud835\udc9e : set (set X)), (\u2200U \u2208 \ud835\udc9e, is_open U) \u2192 is_open (\u22c3\u2080 \ud835\udc9e))\n\nnamespace topological_space\n\nvariables {X : Type} [topological_space X]\n\nlemma open_iff_locally_open (V : set X) :\n  is_open V \u2194 \u2200 x : X, x \u2208 V \u2192 \u2203 U : set X, x \u2208 U \u2227 is_open U \u2227 U \u2286 V :=\nbegin\n  split,\n  { intro hV,\n    intros x hx,\n     use [V, hx, hV] },\n  { intro h,\n    let \ud835\udc9e : set (set X) := {U : set X | \u2203 (x : X) (hx : x \u2208 V), U = classical.some (h x hx)},\n    have h\ud835\udc9e : \u2200 U \u2208 \ud835\udc9e, \u2203 (x : X) (hx : x \u2208 V), x \u2208 U \u2227 is_open U \u2227 U \u2286 V,\n    { intros U hU,\n      rcases hU with \u27e8x, hx, rfl\u27e9,\n      use [x, hx],\n      exact classical.some_spec (h x hx) },\n    convert is_open_sUnion \ud835\udc9e _,\n    { ext x, split,\n      { intro hx,\n        rw mem_sUnion,\n        use classical.some (h x hx),\n        split,\n          use [x, hx],\n        have h := classical.some_spec (h x hx),\n        exact h.1 },\n      { intro hx,\n        rw mem_sUnion at hx,\n        rcases hx with \u27e8U, hU, hxU\u27e9,\n        rcases h\ud835\udc9e U hU with \u27e8_, _, _, _, hUV\u27e9,\n        apply hUV,\n        exact hxU }},\n    { intros U hU,\n      rcases (h\ud835\udc9e U hU) with \u27e8_, _, _, hU, _\u27e9,\n      exact hU },\n  },\nend\n\n\nset_option old_structure_cmd true\n\nnamespace lftcm\n\n/-- `monoid M` is the type of monoid structures on a type `M`. -/\nclass monoid (M : Type) extends has_mul M, has_one M :=\n(mul_assoc : \u2200 (a b c : M), a * b * c = a * (b * c))\n(one_mul : \u2200 (a : M), 1 * a = a)\n(mul_one : \u2200 (a : M), a * 1 = a)\n\nlemma one_mul {M : Type} [monoid M] (a : M) : 1 * a = a := monoid.one_mul _\n\nlemma mul_assoc {M : Type} [monoid M] (a b c : M) :\n  a * b * c = a * (b * c) := monoid.mul_assoc _ _ _\n\n/-- `group G` is the type of group structures on a type `G`. -/\nclass group (G : Type) extends monoid G, has_inv G :=\n(mul_left_inv : \u2200 (a : G), a\u207b\u00b9 * a = 1)\n\nnamespace group\n\nvariables {G : Type} [group G]\n\nlemma mul_left_cancel (a b c : G) (Habac : a * b = a * c) : b = c :=\n calc b = 1 * b         : by rw lftcm.one_mul\n    ... = (a\u207b\u00b9 * a) * b : by rw mul_left_inv\n    ... = a\u207b\u00b9 * (a * b) : begin rw lftcm.mul_assoc, end -- ??\n    ... = a\u207b\u00b9 * (a * c) : by rw Habac\n    ... = (a\u207b\u00b9 * a) * c : begin rw mul_assoc, refl, end -- ??\n    ... = 1 * c         : by rw mul_left_inv\n    ... = c             : by rw one_mul\n\n#exit\nimport data.polynomial\nopen polynomial\n\n#print eval\u2082_hom\nvariables {R S T : Type} [comm_ring R] [comm_ring S] [comm_ring T]\n\nnoncomputable def eval\u2082' (f : R \u2192+* S) (x : S) : polynomial R \u2192+* S :=\nby refine_struct { to_fun := polynomial.eval\u2082 f x }; simp\n\nlemma eq_eval\u2082' (i : polynomial R \u2192+* S) :\n  i = eval\u2082' (i.comp (ring_hom.of C)) (i X) :=\nbegin\n  ext f,\n  apply polynomial.induction_on f; simp [eval\u2082'] {contextual := tt},\nend\n\nexample {f : R \u2192+* S} {g : S \u2192+* T} {p : polynomial R} (x : S):\n  eval\u2082' (g.comp f) (g x) p = g (eval\u2082' f x p) :=\nbegin\n  conv_rhs { rw [\u2190 ring_hom.comp_apply, eq_eval\u2082' (g.comp (eval\u2082' f x))] },\n  simp,\nend\n\n#exit\nimport data.nat.digits\n\nlemma nat.div_lt_of_le : \u2200 {n m k : \u2115} (h0 : n > 0) (h1 : m > 1) (hkn : k \u2264 n), k / m < n\n| 0     m k h0 h1 hkn := absurd h0 dec_trivial\n| 1     m 0 h0 h1 hkn := by rwa nat.zero_div\n| 1     m 1 h0 h1 hkn :=\n  have \u00ac (0 < m \u2227 m \u2264 1), from \u03bb h, absurd (@lt_of_lt_of_le \u2115\n    (show preorder \u2115, from @partial_order.to_preorder \u2115 (@linear_order.to_partial_order \u2115 nat.linear_order))\n     _ _ _ h1 h.2) dec_trivial,\n  by rw [nat.div_def_aux, dif_neg this]; exact dec_trivial\n| 1     m (k+2) h0 h1 hkn := absurd hkn dec_trivial\n| (n+2) m k h0 h1 hkn := begin\n  rw [nat.div_def_aux],\n  cases decidable.em (0 < m \u2227 m \u2264 k) with h h,\n  { rw [dif_pos h],\n    refine nat.succ_lt_succ _,\n    refine nat.div_lt_of_le (nat.succ_pos _) h1 _,\n    cases m with m,\n    { exact absurd h.1 dec_trivial },\n    { cases m with m,\n      { exact absurd h1 dec_trivial },\n      { clear h1 h,\n        induction m with m ih,\n        { cases k with k,\n          { exact nat.zero_le _ },\n          { cases k with k,\n            { exact nat.zero_le _ },\n            { rw [nat.sub_succ, nat.sub_succ, nat.sub_zero, nat.pred_succ,\n                nat.pred_succ],\n              exact @linear_order.le_trans \u2115 nat.linear_order _ _ _\n                (nat.le_succ k) (nat.le_of_succ_le_succ hkn) } } },\n        { cases k with k,\n          { rw [nat.zero_sub], exact nat.zero_le _ },\n          { rw [nat.succ_sub_succ],\n            refine @linear_order.le_trans \u2115 nat.linear_order _ _ _ _ ih,\n            refine nat.sub_le_sub_right _ _,\n            exact nat.le_succ _ } } } } },\n  { rw dif_neg h,\n    exact nat.succ_pos _ }\nend\n\nlemma nat.div_lt_self'' {n m : \u2115} (h0 : n > 0)  (hm : m > 1) : n / m < n :=\nnat.div_lt_of_le h0 hm (le_refl _)\n\ndef f : \u2115 \u2192 \u2115\n| n :=\n  if h : 0 < n\n  then have n - 1 < n, from nat.sub_lt h zero_lt_one,\n    f (n - 1)\n  else 0\n\ndef digits_aux' (b : \u2115) (h : 2 \u2264 b) : \u2115 \u2192 list \u2115\n| 0 := []\n| (n+1) :=\n  have (n+1)/b < n+1 := nat.div_lt_self'' (nat.succ_pos _) h,\n  (n+1) % b :: digits_aux' ((n+1)/b)\n\ndef digits' : \u2115 \u2192 \u2115 \u2192 list \u2115\n| 0 := digits_aux_0\n| 1 := digits_aux_1\n| (b+2) := digits_aux' (b+2) dec_trivial\n\ntheorem test (b n : \u2115) : digits' (b+2) (n+1) = (n+1)%(b+2) :: digits' (b+2) ((n+1)/(b+2)) := rfl -- works\ntheorem test' : digits' (0+2) (1+1) = (1+1)%(0+2) :: digits' (0+2) ((1+1)/(0+2)) := rfl\n\n--#reduce digits (0+2) ((1+1)/(0+2))\nvariables (b n : \u2115)\n#reduce digits' (b+2) (n+1)\n\n#exit\nimport ring_theory.ideals\nimport ring_theory.principal_ideal_domain\nimport ring_theory.localization\nimport tactic\nimport order.bounded_lattice\nimport algebra.field_power\nimport order.conditionally_complete_lattice\nuniverse u\n\nclass discrete_valuation_ring (R : Type u) [integral_domain R] [is_principal_ideal_ring R] :=\n(prime_ideal' : ideal R)\n(primality : prime_ideal'.is_prime)\n(is_nonzero : prime_ideal' \u2260 \u22a5)\n(unique_nonzero_prime_ideal : \u2200 P : ideal R, P.is_prime \u2192 P = \u22a5 \u2228 P = prime_ideal')\n\nnamespace discrete_valuation_ring\n\ndef prime_ideal (R : Type u) [integral_domain R] [is_principal_ideal_ring R] [discrete_valuation_ring R] : ideal R :=\ndiscrete_valuation_ring.prime_ideal'\n\ninstance is_prime (R : Type*) [integral_domain R] [is_principal_ideal_ring R] [discrete_valuation_ring R] : (prime_ideal R).is_prime :=\nprimality\n\nvariables {R : Type u} [integral_domain R] [is_principal_ideal_ring R] [discrete_valuation_ring R]\nopen discrete_valuation_ring\n\nlemma prime_ideal_is_maximal : (prime_ideal R).is_maximal :=\nbegin\n  have f : prime_ideal R \u2260 \u22a5,\n  { apply discrete_valuation_ring.is_nonzero },\n  apply is_prime.to_maximal_ideal,\n    exact f,\nend\n\nlemma unique_max_ideal : \u2203! I : ideal R, I.is_maximal :=\nbegin\n  use prime_ideal R,\n  split,\n  { exact prime_ideal_is_maximal },\n  { intros y a,\n    cases discrete_valuation_ring.unique_nonzero_prime_ideal y a.is_prime,\n    { exfalso,\n      rw h at a,\n      apply discrete_valuation_ring.primality.left,\n      exact a.right (prime_ideal R) (bot_lt_iff_ne_bot.2 discrete_valuation_ring.is_nonzero) },\n    { assumption } }\nend\n\ninstance is_local_ring : local_ring R := local_of_unique_max_ideal unique_max_ideal\n\nopen local_ring\n\nnoncomputable theory\nopen_locale classical\nclass discrete_valuation_field (K : Type*) [field K] :=\n(v : K -> with_top \u2124 )\n(mul : \u2200 (x y : K), v(x*y) = v(x) + v(y) )\n(add : \u2200 (x y : K), min (v(x)) (v(y)) \u2264 v(x + y)  )\n(non_zero : \u2200 (x : K), v(x) = \u22a4 \u2194 x = 0 )\n\nnamespace discrete_valuation_field\n\ndefinition valuation (K : Type*) [field K] [ discrete_valuation_field K ] : K -> with_top \u2124 := v\n\nvariables {K : Type*} [field K] [discrete_valuation_field K]\n\nlemma with_top.cases (a : with_top \u2124) : a = \u22a4 \u2228 \u2203 n : \u2124, a = n :=\nbegin\n  cases a with n,\n  { -- a = \u22a4 case\n    left,\n    refl, -- true by definition\n  },\n  { -- \u2124 case\n    right,\n    use n,\n    refl, -- true by definition\n  }\nend\n\nlemma sum_zero_iff_zero (a : with_top \u2124) : a + a = 0 \u2194 a = 0 :=\nbegin\n  split,\n  { -- the hard way\n    intro h, -- h is a proof of a+a=0\n    -- split into cases\n    cases (with_top.cases a) with htop hn,\n    { -- a = \u22a4\n      rw htop at h,\n      -- h is false\n      cases h,\n      -- no cases!\n    },\n    { -- a = n\n      cases hn with n hn,\n      rw hn at h \u22a2,\n      -- now h says n+n=0 and our goal is n=0\n      -- but these are equalities in `with_top \u2124\n      -- so we need to get them into \u2124\n      -- A tactic called `norm_cast` does this\n     norm_cast at h \u22a2,\n      -- we finally have a hypothesis n + n = 0\n      -- and a goal n = 0\n      -- and everything is an integer\n      rw add_self_eq_zero at h,\n      assumption\n    }\n  },\n   { -- the easy way\n    intro ha,\n    rw ha,\n    simp\n  }\nend\n --Thanks Kevin!\n\nlemma val_one_eq_zero : v(1 : K) = 0 :=\nbegin\n  have h : (1 : K) * 1 = 1,\n    simp,\n  apply_fun v at h,\n  rw mul at h,\n  -- now we know v(1)+v(1)=v(1) and we want to deduce v(1)=0 (i.e. rule out v(1)=\u22a4)\n  rcases (with_top.cases (v(1:K))) with h1 | \u27e8n, h2\u27e9, -- do all the cases in one go\n  { rw non_zero at h1,\n    cases (one_ne_zero h1)\n  },\n  { rw h2 at *,\n    norm_cast at *,\n    -- library_search found the next line\n    exact add_left_eq_self.mp (congr_arg (has_add.add n) (congr_arg (has_add.add n) h)),\n  },\nend\n\nlemma val_minus_one_is_zero : v((-1) : K) = 0 :=\nbegin\nhave f : (-1:K)*(-1:K) = (1 : K),\nsimp,\nhave g : v((-1 : K)*(-1 : K)) = v(1 : K),\nsimp,\nhave k : v((-1 : K)*(-1 : K)) = v(-1 : K) + v(-1 : K),\n{\n  apply mul,\n},\nrw k at g,\nrw val_one_eq_zero at g,\nrw <-sum_zero_iff_zero,\nexact g,\nend\n\n@[simp] lemma val_zero : v(0:K) = \u22a4 :=\nbegin\nrw non_zero,\nend\n\n\nlemma with_top.transitivity (a b c : with_top \u2124) : a \u2264 b -> b \u2264 c -> a \u2264 c :=\nbegin\nrintros,\ncases(with_top.cases c) with h1 h2,\n  {\n    rw h1,\n    simp,\n  },\n  {\n    cases h2 with n h2,\n    cases(with_top.cases a) with k1 k2,\n    {\n      rw [k1, h2],\n      rw k1 at a_1,\n      rw h2 at a_2,\n      cases(with_top.cases b) with l1 l2,\n      {\n        rw l1 at a_2,\n        exact a_2,\n      },\n      {\n        cases l2 with m l2,\n        rw l2 at a_1,\n        exfalso,\n        apply with_top.not_top_le_coe m,\n        exact a_1,\n      },\n    },\n    {\n      cases k2 with m k2,\n      cases(with_top.cases b) with l1 l2,\n      {\n        rw [l1,h2] at a_2,\n        exfalso,\n        apply with_top.not_top_le_coe n,\n        exact a_2,\n      },\n      {\n        cases l2 with k l2,\n        rw [k2,l2] at a_1,\n        rw [l2,h2] at a_2,\n        rw [k2,h2],\n        rw with_top.coe_le_coe,\n        rw with_top.coe_le_coe at a_1,\n        rw with_top.coe_le_coe at a_2,\n        transitivity k,\n        exact a_1,\n        exact a_2,\n      },\n    },\n  },\nend\n\ndef val_ring (K : Type*) [field K] [discrete_valuation_field K] := { x : K | 0 \u2264 v x }\n\ninstance (K : Type*) [field K] [discrete_valuation_field K] : is_add_subgroup (val_ring K) :=\n{\n  zero_mem := begin\n              unfold val_ring,\n              simp,\n              end,\n  add_mem := begin\n            unfold val_ring,\n            simp only [set.mem_set_of_eq],\n            rintros,\n            have g : min (v(a)) (v(b)) \u2264 v(a + b),\n            {\n              apply add,\n            },\n            rw min_le_iff at g,\n            cases g,\n            {\n              exact with_top.transitivity _ _ _ a_1 g,\n            },\n            {\n              exact with_top.transitivity _ _ _ a_2 g,\n            },\n            end,\n  neg_mem := begin\n            unfold val_ring,\n            rintros,\n            simp only [set.mem_set_of_eq],\n            simp only [set.mem_set_of_eq] at a_1,\n            have f : -a = a * (-1 : K) := by simp,\n            rw [f, mul, val_minus_one_is_zero],\n            simp [a_1],\n            end,\n}\n\ninstance (K:Type*) [field K] [discrete_valuation_field K] : is_submonoid (val_ring K) :=\n{ one_mem := begin\n            unfold val_ring,\n            simp,\n            rw val_one_eq_zero,\n            norm_num,\n            end,\n  mul_mem := begin\n            unfold val_ring,\n            rintros,\n            simp,\n            simp at a_1,\n            simp at a_2,\n            rw mul,\n            apply add_nonneg' a_1 a_2,\n            end, }\n\ninstance valuation_ring (K:Type*) [field K] [discrete_valuation_field K] : is_subring (val_ring K) :=\n{}\n\ninstance is_domain (K:Type*) [field K] [discrete_valuation_field K] : integral_domain (val_ring K) :=\nsubring.domain (val_ring K)\n\ndef unif (K:Type*) [field K] [discrete_valuation_field K] : set K := { \u03c0 | v \u03c0 = 1 }\n\nvariables (\u03c0 : K) (h\u03c0 : \u03c0 \u2208 unif K)\n\nlemma val_unif_eq_one (h\u03c0 : \u03c0 \u2208 unif K) : v(\u03c0) = 1 :=\nbegin\nunfold unif at h\u03c0,\nsimp at h\u03c0,\nexact h\u03c0,\nend\n\nlemma unif_ne_zero (h\u03c0 : \u03c0 \u2208 unif K) : \u03c0 \u2260 0 :=\nbegin\nsimp,\n      unfold unif at h\u03c0,\n      simp at h\u03c0,\n      intro g,\n      rw <-non_zero at g,\n      rw h\u03c0 at g,\n      cases g,\nend\n\nlemma with_top.add_happens (a b c : with_top \u2124) (ne_top : a \u2260 \u22a4) : b=c \u2194 a+b = a+c :=\nbegin\ncases with_top.cases a,\n{\n  exfalso,\n  apply ne_top,\n  exact h,\n},\ncases h with n h,\nrw h,\nsplit,\n{\n  rintros,\n  rw a_1,\n},\ncases with_top.cases b,\n{\n  rw h_1,\n  rw with_top.add_top,\n  rintros,\n  have b_1 : \u2191n + c = \u22a4,\n  exact eq.symm a_1,\n  rw with_top.add_eq_top at b_1,\n  cases b_1,\n  {\n    exfalso,\n    apply with_top.coe_ne_top,\n    {\n      exact b_1,\n    },\n  },\n  exact eq.symm b_1,\n},\n{\n  cases h_1 with m h_1,\n  rw h_1,\n  cases with_top.cases c,\n  {\n    rw h_2,\n    rintros,\n    rw with_top.add_top at a_1,\n    rw with_top.add_eq_top at a_1,\n    cases a_1,\n    {\n      exfalso,\n      apply with_top.coe_ne_top,\n      exact a_1,\n    },\n    {\n      exact a_1,\n    },\n  },\n  cases h_2 with l h_2,\n  rw h_2,\n  rintros,\n  norm_cast,\n  norm_cast at a_1,\n  simp at a_1,\n  assumption,\n}\nend\n\nlemma with_top.add_le_happens (a b c : with_top \u2124) (ne_top : a \u2260 \u22a4) : b \u2264 c \u2194 a + b \u2264 a+c :=\nbegin\n rcases(with_top.cases a) with rfl | \u27e8a, rfl\u27e9;\n rcases(with_top.cases b) with rfl | \u27e8b, rfl\u27e9;\n rcases(with_top.cases c) with rfl | \u27e8n, rfl\u27e9;\n try {simp},\n simp at ne_top,\n assumption,\n simp at ne_top,\n exfalso,\n assumption,\n rw <-with_top.coe_add,\n apply with_top.coe_ne_top,\n repeat{rw <-with_top.coe_add,},\n rw with_top.coe_le_coe,\n simp,\nend\n\nlemma with_top.distrib (a b c : with_top \u2124) (na : a \u2260 \u22a4) (nb : b \u2260 \u22a4) (nc : c \u2260 \u22a4) : (a + b)*c = a*c + b*c :=\nbegin\n  rcases(with_top.cases a) with rfl | \u27e8a, rfl\u27e9;\n  rcases(with_top.cases b) with rfl | \u27e8b, rfl\u27e9;\n  rcases(with_top.cases c) with rfl | \u27e8n, rfl\u27e9;\n  try {simp},\n  repeat\n  {\n  simp at na,\n  exfalso,\n  exact na,\n  },\n  {\n  simp at nb,\n  exfalso,\n  exact nb,\n  },\n  {\n  simp at nc,\n  exfalso,\n  exact nc,\n  },\n  rw <-with_top.coe_add,\n  repeat {rw <-with_top.coe_mul},\n  rw <-with_top.coe_add,\n  rw with_top.coe_eq_coe,\n  rw right_distrib,\nend\n\nlemma one_mul (a : with_top \u2124) : 1 * a = a :=\nbegin\ncases (with_top.cases) a with a ha,\n{\n  rw a,\n  simp,\n},\n{\n  cases ha with n ha,\n  rw ha,\n  norm_cast,\n  simp,\n}\nend\n\nlemma nat_ne_top (n :\u2115) : (n : with_top \u2124) \u2260 \u22a4 :=\nbegin\nsimp,\nend\n\nlemma val_inv (x : K) (nz : x \u2260 0) : v(x) + v(x)\u207b\u00b9 = 0 :=\nbegin\nrw <- mul,\nrw mul_inv_cancel,\n{\n  rw val_one_eq_zero,\n},\nexact nz,\nend\n\nlemma with_top.sub_add_eq_zero (n : \u2115) : ((-n : \u2124) : with_top \u2124) + (n : with_top \u2124) = 0 :=\nbegin\nrw <-with_top.coe_nat,\nrw <-with_top.coe_add,\nsimp only [add_left_neg, int.nat_cast_eq_coe_nat, with_top.coe_zero],\nend\n\nlemma with_top.add_sub_eq_zero (n : \u2115) : (n : with_top \u2124) + ((-n : \u2124) : with_top \u2124) = 0 :=\nbegin\nrw <-with_top.coe_nat,\nrw <-with_top.coe_add,\nsimp only [add_right_neg, int.nat_cast_eq_coe_nat, with_top.coe_zero],\nend\n\nlemma contra_non_zero (x : K) (n : \u2115) (nz : n \u2260 0) : v(x^n) \u2260 \u22a4 \u2194 x \u2260 0 :=\nbegin\nsplit,\n{\n  contrapose,\n  simp,\n  intro,\n  rw a,\n  rw zero_pow',\n  {\n    exact val_zero,\n  },\n  {\n    exact nz,\n  },\n},\n{\n  contrapose,\n  simp,\n  intro,\n  rw non_zero at a,\n  contrapose a,\n  apply pow_ne_zero,\n  exact a,\n},\nend\n\n\nlemma contra_non_zero_one (x : K) : v(x) \u2260 \u22a4 \u2194 x \u2260 0 :=\nbegin\nsplit,\n{\n  intro,\n  rw <-pow_one x at a,\n  rw contra_non_zero x 1 at a,\n  exact a,\n  simp,\n},\n{\n  contrapose,\n  simp,\n  rw non_zero,\n  simp,\n},\nend\n\nlemma val_nat_power (a : K) (nz : a \u2260 0) : \u2200 n : \u2115, v(a^n) = (n : with_top \u2124)*v(a) :=\nbegin\nrintros,\ninduction n with d hd,\n{\n  rw pow_zero,\n  rw val_one_eq_zero,\n  simp,\n},\n{\n  rw nat.succ_eq_add_one,\n  rw pow_succ',\n  rw mul,\n  rw hd,\n  norm_num,\n  rw with_top.distrib,\n  rw one_mul,\n  apply nat_ne_top,\n  apply with_top.one_ne_top,\n  intro,\n  rw non_zero at a_1,\n  apply nz,\n  exact a_1,\n}\nend\n\nlemma val_int_power (a : K) (nz : a \u2260 0) : \u2200 n : \u2124, v(a^n) = (n : with_top \u2124)*v(a) :=\nbegin\nrintros,\ncases n,\n{\n  rw fpow_of_nat,\n  rw val_nat_power,\n  {\n    simp only [int.of_nat_eq_coe],\n    rw <-with_top.coe_nat,\n    simp only [int.nat_cast_eq_coe_nat],\n  },\n  exact nz,\n},\n{\n  simp only [fpow_neg_succ_of_nat],\n  rw nat.succ_eq_add_one,\n  rw with_top.add_happens (v (a ^ (n + 1))) (v (a ^ (n + 1))\u207b\u00b9) (\u2191-[1+ n] * v a),\n  {\n    rw val_inv,\n    {\n      rw val_nat_power,\n      {\n        simp only [nat.cast_add, nat.cast_one],\n        rw <-with_top.distrib,\n        {\n          simp only [zero_eq_mul],\n          left,\n          rw int.neg_succ_of_nat_coe',\n          rw sub_eq_add_neg,\n          rw with_top.coe_add,\n          rw add_comm (\u2191-\u2191n),\n          rw <-add_assoc,\n          rw add_comm,\n          rw add_assoc,\n          rw <-with_top.coe_one,\n          rw <-with_top.coe_add,\n          simp,\n          rw with_top.sub_add_eq_zero,\n          },\n          {\n            norm_cast,\n            apply with_top.nat_ne_top,\n          },\n          {\n            simp,\n          },\n          {\n            intro,\n            simp_rw [non_zero, nz] at a_1,\n            exact a_1,\n          },\n      },\n      {\n        exact nz,\n      },\n    },\n    {\n      apply pow_ne_zero,\n      exact nz,\n    },\n  },\n  {\n    rw contra_non_zero,\n    {\n      exact nz,\n    },\n    {\n      simp,\n    },\n  },\n},\nend\n\nlemma unit_iff_val_zero (\u03b1 : K) (h\u03b1 : \u03b1 \u2208 val_ring K) (nz\u03b1 : \u03b1 \u2260 0) : v (\u03b1) = 0 \u2194 \u2203 \u03b2 \u2208 val_ring K, \u03b1 * \u03b2 = 1 :=\nbegin\nsplit,\n{\n  rintros,\n  use \u03b1\u207b\u00b9,\n  split,\n  {\n    {\n      unfold val_ring,\n      simp,\n      rw <-with_top.coe_zero,\n      rw with_top.coe_le_iff,\n      rintros,\n      rw with_top.add_happens (v(\u03b1)) _ _ at a_1,\n      {\n        rw val_inv at a_1,\n        {\n          rw a at a_1,\n          simp only [with_top.zero_eq_coe, zero_add] at a_1,\n          rw a_1,\n        },\n        exact nz\u03b1,\n      },\n      simp_rw [contra_non_zero_one],\n      exact nz\u03b1,\n    },\n  },\n  {\n    rw mul_inv_cancel,\n    exact nz\u03b1,\n  },\n},\n{\n  rintros,\n  cases a with b a,\n  simp at a,\n  cases a,\n  unfold val_ring at a_left,\n  simp at a_left,\n  have f : v((\u03b1)*(b)) = v(1:K),\n  {\n    rw a_right,\n  },\n  rw mul at f,\n  rw val_one_eq_zero at f,\n  rw add_eq_zero_iff' at f,\n  {\n    cases f,\n    exact f_left,\n  },\n  {\n    erw val_ring at h\u03b1,\n    simp at h\u03b1,\n    exact h\u03b1,\n  },\n  {\n    exact a_left,\n  },\n},\nend\n\nlemma val_eq_iff_asso (x y : K) (hx : x \u2208 val_ring K) (hy : y \u2208 val_ring K) (nzx : x \u2260 0) (nzy : y \u2260 0) : v(x) = v(y) \u2194 \u2203 \u03b2 \u2208 val_ring K, v(\u03b2) = 0 \u2227 x * \u03b2 = y :=\nbegin\nsplit,\nintros,\nuse (x\u207b\u00b9*y),\n{\n  {\n    unfold val_ring,\n    simp,\n    rw mul,\n    rw with_top.add_happens (v(x\u207b\u00b9)) _ _ at a,\n    {\n      rw add_comm at a,\n      rw val_inv at a,\n      {\n        rw <-a,\n        norm_num,\n        rw mul_inv_cancel_assoc_right,\n        exact nzx,\n      },\n      exact nzx,\n    },\n    {\n      intro f,\n      rw non_zero at f,\n      simp at f,\n      apply nzx,\n      exact f,\n    },\n  },\n},\n{\n  rintros,\n  cases a with z a,\n  simp at a,\n  cases a,\n  cases a_right with a_1 a_2,\n  apply_fun v at a_2,\n  rw mul at a_2,\n  rw a_1 at a_2,\n  simp at a_2,\n  exact a_2,\n},\nend\n\nlemma unif_assoc (x : K) (hx : x \ufffd\ufffd val_ring K) (nz : x \u2260 0) (h\u03c0 : \u03c0 \u2208 unif K) : \u2203 \u03b2 \u2208 val_ring K, (v(\u03b2) = 0 \u2227 \u2203! n : \u2124, x * \u03b2 = \u03c0^n) :=\nbegin\nhave h\u03c0' : \u03c0 \u2260 0,\n{\n  apply unif_ne_zero,\n  exact h\u03c0,\n},\nunfold unif at h\u03c0,\nsimp at h\u03c0,\ncases (with_top.cases) (v(x)),\n{\n rw non_zero at h,\n exfalso,\n apply nz,\n exact h,\n},\n{\n  cases h with n h,\n  split,\n  let y := x\u207b\u00b9 * \u03c0^n,\n  have g : v(y) = 0,\n  {\n    rw [mul, val_int_power \u03c0, h\u03c0, add_comm],\n    norm_cast,\n    simp,\n    rw [<-h, val_inv],\n    exact nz,\n    exact h\u03c0',\n  },\n  have f : y \u2208 val_ring K,\n  {\n    unfold val_ring,\n    simp,\n    rw g,\n    norm_num,\n  },\n  {\n    use f,\n    split,\n    {\n      exact g,\n    },\n    rw mul_inv_cancel_assoc_right,\n    use n,\n    {\n      split,\n      simp only [eq_self_iff_true],\n      rintros,\n      apply_fun v at a,\n      rw [val_int_power, val_int_power, h\u03c0] at a,\n      {\n        norm_cast at a,\n        simp at a,\n        exact eq.symm a,\n      },\n      exact h\u03c0',\n      exact h\u03c0',\n    },\n    exact nz,\n  },\n},\nend\n\nlemma blah (n : \u2124) : n < n -> false :=\nbegin\nsimp only [forall_prop_of_false, not_lt],\nend\n\nlemma val_is_nat (h\u03c0 : \u03c0 \u2208 unif K) (x : val_ring K) (nzx : x \u2260 0) : \u2203 m : \u2115, v(x:K) = \u2191m :=\nbegin\ncases with_top.cases (v(x:K)),\n{\n  rw h,\n  simp,\n  rw non_zero at h,\n  apply nzx,\n  exact subtype.eq h,\n},\n{\n  cases h with n h,\n  cases n,\n  {\n    use n,\n    simp_rw h,\n    simp,\n    rw <-with_top.coe_nat,\n    simp,\n  },\n  {\n    have H : 0 \u2264 v(x:K),\n    exact x.2,\n    rw h at H,\n    norm_cast at H,\n    exfalso,\n    contrapose H,\n    simp,\n    tidy,\n    exact int.neg_succ_lt_zero n,\n  },\n},\nend\n\nlemma is_pir (h\u03c0 : \u03c0 \u2208 unif K) : is_principal_ideal_ring (val_ring K) :=\nbegin\nsplit,\nrintros,\nrintros,\nby_cases S = \u22a5,\n{\n  rw h,\n  use 0,\n  apply eq.symm,\n  rw submodule.span_singleton_eq_bot,\n},\nlet Q := {n : \u2115 | \u2203 x \u2208 S, (n : with_top \u2124) = v(x:K) },\nhave g : v(\u03c0 ^(Inf Q)) = \u2191(Inf Q),\n{\n  rw val_nat_power,\n  rw val_unif_eq_one,\n  rw <-with_top.coe_one,\n  rw <-with_top.coe_nat,\n  rw <-with_top.coe_mul,\n  rw mul_one,\n  exact h\u03c0,\n  apply unif_ne_zero,\n  exact h\u03c0,\n},\nhave nz : \u03c0^(Inf Q) \u2260 0,\n{\n  assume a,\n  apply_fun v at a,\n  rw g at a,\n  rw val_zero at a,\n  apply with_top.nat_ne_top (Inf Q),\n  exact a,\n},\nuse \u03c0^(Inf Q),\n{\n  unfold val_ring,\n  simp,\n  rw g,\n  rw <-with_top.coe_nat,\n  norm_cast,\n  norm_num,\n},\napply submodule.ext,\nrintros,\nsplit,\n{\n  rintros,\n  rw submodule.mem_span_singleton,\n  use (x * (\u03c0^(Inf Q))\u207b\u00b9),\n  {\n    dunfold val_ring,\n    simp,\n    rw mul,\n    by_cases x = 0,\n    {\n      rw h,\n      simp,\n    },\n    rw with_top.add_le_happens (v(\u03c0^(Inf Q))),\n    {\n      norm_num,\n      rw add_left_comm,\n      rw val_inv,\n      simp,\n      rw g,\n      have f' : \u2203 m : \u2115, v(x:K) = \u2191m,\n      {\n        apply val_is_nat,\n        use h\u03c0,\n        exact h,\n      },\n      cases f' with m f',\n      rw f',\n      rw <-with_top.coe_nat,\n      rw <-with_top.coe_nat,\n      norm_cast,\n      have h' : m \u2208 Q,\n      {\n        split,\n        simp,\n        split,\n        use a,\n        use [eq.symm f'],\n      },\n      by { rw [nat.Inf_def \u27e8m, h'\u27e9], exact nat.find_min' \u27e8m, h'\u27e9 h' },\n      assumption,\n    },\n    rw g,\n    exact with_top.nat_ne_top _,\n  },\n  {\n    tidy,\n    assoc_rw inv_mul_cancel nz,\n    simp,\n  },\n},\n{\n  rw submodule.mem_span,\n  rintros,\n  specialize a S,\n  apply a,\n  have f : \u2203 z \u2208 S, v(z : K) = \u2191(Inf Q),\n  {\n    have f' : \u2203 x \u2208 S, v(x : K) \u2260 \u22a4,\n    {\n      contrapose h,\n      simp at h,\n      simp,\n      apply ideal.ext,\n      rintros,\n      simp only [submodule.mem_bot],\n      split,\n      rintros,\n      specialize h x_1,\n      simp at h,\n      have q : v(x_1 : K) = \u22a4,\n      apply h,\n      exact a_1,\n      rw non_zero at q,\n      exact subtype.ext q,\n      rintros,\n      rw a_1,\n      simp,\n    },\n    have p : Inf Q \u2208 Q,\n    {\n      apply nat.Inf_mem,\n      contrapose h,\n      simp,\n      by_contradiction,\n      cases f' with x' f',\n      have f_1 : \u2203 m : \u2115, v(x':K) = \u2191(m),\n      {\n        apply val_is_nat,\n        exact h\u03c0,\n        cases f',\n        contrapose f'_h,\n        simp,\n        rw non_zero,\n        simp at f'_h,\n        rw f'_h,\n        simp,\n      },\n      cases f_1 with m' f_1,\n      have g' : m' \u2208 Q,\n      {\n        simp,\n        use x',\n        simp,\n        split,\n        cases f',\n        assumption,\n        exact eq.symm f_1,\n      },\n      apply h,\n      use m',\n      apply g',\n    },\n    simp at p,\n    cases p with z p,\n    cases p,\n    use z,\n    cases p_left,\n    assumption,\n    split,\n    cases p_left,\n    assumption,\n    simp,\n    exact eq.symm p_right,\n  },\n  cases f with z f,\n  rw <-g at f,\n  simp at f,\n  cases f,\n  rw val_eq_iff_asso at f_right,\n  {\n    cases f_right with w f_1,\n    cases f_1 with f_1 f_2,\n    cases f_2 with f_2 f_3,\n    rw set.singleton_subset_iff,\n    simp only [submodule.mem_coe],\n    simp_rw [\u2190 f_3],\n    change z * \u27e8w,f_1\u27e9 \u2208 S,\n    apply ideal.mul_mem_right S f_left,\n  },\n  simp,\n  {\n    unfold val_ring,\n    simp,\n    rw g,\n    rw <-with_top.coe_nat,\n    norm_cast,\n    simp,\n  },\n  {\n    rw g at f_right,\n    contrapose f_right,\n    simp at f_right,\n    rw <-non_zero at f_right,\n    rw f_right,\n    simp,\n  },\n  {\n    exact nz,\n  },\n},\nrecover,\nend\n\nend discrete_valuation_field\n\nend discrete_valuation_ring\n#exit\nimport set_theory.cardinal\n\nuniverses u v\n\nexample : cardinal.lift.{u v} = cardinal.lift.{u (max u v)} :=\nfunext $ \u03bb x, quotient.induction_on x\n  (\u03bb x, quotient.sound \u27e8\u27e8\u03bb \u27e8x\u27e9, \u27e8x\u27e9, \u03bb \u27e8x\u27e9, \u27e8x\u27e9, \u03bb \u27e8_,\u27e9, rfl, \u03bb \u27e8_\u27e9, rfl\u27e9\u27e9)\n\n\nimport tactic.rcases\n\nlemma L1 : forall (n m: \u2115) (p : \u2115 \u2192 Prop), (p n \u2227 \u2203 (u:\u2115), p u \u2227 p m) \u2228 (\u00acp n \u2227 p m) \u2192 n = m :=\nbegin\n  intros n m p H,\n  rcases H with \u27e8H1, u, H2, H3\u27e9 | \u27e8H1, H2\u27e9,\n\nend\n\n#exit\n\nimport data.polynomial\n\nexample {K L M : Type*} [field K] [field L] [field M]\n  (i : K \u2192+* L) (j : L \u2192+* M) (f : polynomial K)\n  (h : \u2203 x, f.eval\u2082 i x)\n\n#exit\n\nimport ring_theory.eisenstein_criterion\n\nvariables {R : Type*} [integral_domain R]\n\nlemma dvd_mul_prime {x a p : R} (hp : prime p) : x \u2223 a * p \u2192 x \u2223 a \u2228 p \u2223 x :=\n\u03bb \u27e8y, hy\u27e9, (hp.div_or_div \u27e8a, hy.symm.trans (mul_comm _ _)\u27e9).elim\n  or.inr\n  begin\n    rintros \u27e8b, rfl\u27e9,\n    rw [mul_left_comm, mul_comm, domain.mul_right_inj hp.ne_zero] at hy,\n    rw [hy],\n    exact or.inl (dvd_mul_right _ _)\n  end\n#print well_founded.m\nin\nlemma left_dvd_or_dvd_right_of_dvd_prime_mul {a : R} :\n  \u2200 {b p : R}, prime p \u2192 a \u2223 p * b \u2192 p \u2223 a \u2228 a \u2223 b :=\nbegin\n  rintros b p hp \u27e8c, hc\u27e9,\n  rcases hp.2.2 a c (hc \u25b8 dvd_mul_right _ _) with h | \u27e8x, rfl\u27e9,\n  { exact or.inl h },\n  { rw [mul_left_comm, domain.mul_right_inj hp.ne_zero] at hc,\n    exact or.inr (hc.symm \u25b8 dvd_mul_right _ _) }\nend\n\n#exit\nimport data.nat.basic data.quot\n\ninductive rel : \u2115 \u2295 \u2115 \u2192  \u2115 \u2295 \u2115 \u2192 Prop\n| zero : rel (sum.inl 0) (sum.inr 0)\n| refl : \u2200 x, rel x x\n| symm : \u2200 {x y}, rel x y \u2192 rel y x\n| trans : \u2200 {x y z}, rel x y \u2192 rel y z \u2192 rel x z\n\nattribute [refl] rel.refl\nattribute [symm] rel.symm\nattribute [trans] rel.trans\n\n\ninstance srel : setoid (\u2115 \u2295 \u2115) :=\n{ r := rel,\n  iseqv := \u27e8rel.refl, @rel.symm, @rel.trans\u27e9 }\n\ndef int' := quotient srel\n\n\n\n#exit\nimport data.finset data.fintype.card\n\nexample (n m : \u2115) (hn : n \u2260 0) (hm : n \u2264 m) : m \u2260 0 := \u03bb h, by simp * at *\n#print discrete\nuniverse u\nvariables {\u03b1 : Type u} [add_comm_monoid \u03b1]\n\nopen_locale big_operators\n\u0297\u2201 C\nexample {u : Type*} {v : Type*} [fintype u] [fintype v] (f : u \u00d7 v -> \u03b1) :\n  \u2211 (i : u), \u2211 (j : v), f (i, j) = \u2211 (p : u \u00d7 v), f p :=\nbegin\n  rw <-finset.sum_product,\n  repeat { rw finset.univ },\n  sorry,\nend\n#exit\n\nimport data.set.finite tactic\n\nvariables {\u03b1 : Type*} (r : \u03b1 \u2192 \u03b1 \u2192 Prop)\n\nlemma well_founded_of_finite [is_irrefl \u03b1 r] [is_trans \u03b1 r]\n  (h : \u2200 a\u2080, set.finite {a | r a a\u2080}) : well_founded r :=\n\u27e8\u03bb a\u2080, acc.intro _ (\u03bb b hb, begin\n  cases h a\u2080 with fint,\n  refine @well_founded.fix {a | r a a\u2080} (\u03bb b, acc r b) (\u03bb x y : {a | r a a\u2080}, r x y)\n    (@fintype.well_founded_of_trans_of_irrefl _ fint\n      (\u03bb x y : {a | r a a\u2080}, r x y) \u27e8\u03bb x y z h\u2081 h\u2082, trans h\u2081 h\u2082\u27e9\n      \u27e8\u03bb x, irrefl x\u27e9) _ \u27e8b, hb\u27e9,\n  rintros \u27e8b, hb\u27e9 ih,\n  exact acc.intro _ (\u03bb y hy, ih \u27e8y, trans hy hb\u27e9 hy)\nend)\u27e9\n\n#exit\n\nimport algebra.group_power\n\ntheorem pow_eq_zero_1 {R : Type} [domain R] {r : R} {n : \u2115} : r ^ (n + 1) = 0 \u2192 r = 0\n:= begin\n  rw (show r ^ (n + 1) = r ^ n * r,\n      by {\n           sorry, }),\n  sorry,\nend\n\n\n\ntheorem pow_eq_zero_2  {R : Type} [domain R] {r : R} {n : \u2115} : r ^ (n + 1) = 0 \u2192 r = 0\n:= pow_eq_zero  -- it's in mathlib\n\nimport tactic\n\ndef five : \u2115 := 5\n\nmeta def tac : tactic unit := tactic.focus1 `[tactic.intro1, tactic.applyc `five]\n\nrun_cmd add_interactive [`tac]\n\ndef C : \u2115 \u2192 \u2115 :=\nby tac\n#print C\n\ninductive palindrome {\u03b1 : Type} : list \u03b1 \u2192 Prop\n| nil  : palindrome []\n| singleton : \\al palindrom []\n\n\ndef reverse {\u03b1 : Type} : list \u03b1 \u2192 list \u03b1\n| [] := []\n| (x :: xs) := reverse xs ++ [x]\n\n\n\n\nend\n\n#exit\n\nimport group_theory.subgroup ring_theory.ideal_operations\n\n--attribute [irreducible] subgroup.normal\n\nexample {R : Type} [comm_ring R] (P : ideal R) (hP : P.is_prime) : P.is_prime :=\nby apply_instance\n\nexample {G : Type} [group G] (N : subgroup G) (hN : N.normal) : N.normal :=\nby apply_instance\n\n#print subgroup.normal\n\n\n#exit\nimport ring_theory.ideal_operations data.polynomial ring_theory.ideals tactic.apply_fun\n\nopen polynomial ideal.quotient\n\nopen_locale classical\n\nvariables {R : Type*} [integral_domain R]\n\nopen polynomial ideal.quotient\n\nopen finset\n\nopen_locale big_operators\n\nlemma mul_eq_mul_prime_prod {\u03b1 : Type*} [decidable_eq \u03b1] {x y a : R} {s : finset \u03b1}\n  {p : \u03b1 \u2192 R} (hp : \u2200 i \u2208 s, prime (p i)) (hx : x * y = a * s.prod p) :\n  \u2203 t u b c, t \u222a u = s \u2227 disjoint t u \u2227 b * c = a \u2227\n    x = b * t.prod p \u2227 y = c * u.prod p :=\nbegin\n  induction s using finset.induction with i s his ih generalizing x y a,\n  { exact \u27e8\u2205, \u2205, x, y, by simp [hx]\u27e9 },\n  { rw [prod_insert his, \u2190 mul_assoc] at hx,\n    have hpi : prime (p i), { exact hp i (mem_insert_self _ _) },\n    rcases ih (\u03bb i hi, hp i (mem_insert_of_mem hi)) hx with\n      \u27e8t, u, b, c, htus, htu, hbc, rfl, rfl\u27e9,\n    have hpibc : p i \u2223 b \u2228 p i \u2223 c,\n      from hpi.div_or_div \u27e8a, by rw [hbc, mul_comm]\u27e9,\n    have hit : i \u2209 t, from \u03bb hit, his (htus \u25b8 mem_union_left _ hit),\n    have hiu : i \u2209 u, from \u03bb hiu, his (htus \u25b8 mem_union_right _ hiu),\n    rcases hpibc with \u27e8d, rfl\u27e9 | \u27e8d, rfl\u27e9,\n    { rw [mul_assoc, mul_comm a, domain.mul_right_inj hpi.ne_zero] at hbc,\n      exact \u27e8insert i t, u, d, c, by rw [insert_union, htus],\n        disjoint_insert_left.2 \u27e8hiu, htu\u27e9,\n          by simp [\u2190 hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]\u27e9 },\n    { rw [\u2190 mul_assoc, mul_right_comm b, domain.mul_left_inj hpi.ne_zero] at hbc,\n      exact \u27e8t, insert i u, b, d, by rw [union_insert, htus],\n        disjoint_insert_right.2 \u27e8hit, htu\u27e9,\n          by simp [\u2190 hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]\u27e9 } }\nend\n\nlemma mul_eq_mul_prime_pow {x y a p : R} {n : \u2115} (hp : prime p) (hx : x * y = a * p ^ n) :\n  \u2203 i j b c, i + j = n \u2227 b * c = a \u2227 x = b * p ^ i \u2227 y = c * p ^ j :=\nbegin\n  rcases mul_eq_mul_prime_prod (\u03bb _ _, hp)\n    (show x * y = a * (range n).prod (\u03bb _, p), by simpa) with\n    \u27e8t, u, b, c, htus, htu, rfl, rfl, rfl\u27e9,\n  exact \u27e8t.card, u.card, b, c, by rw [\u2190 card_disjoint_union htu, htus, card_range], by simp\u27e9,\nend\n\nlemma eisenstein {f : polynomial R} {P : ideal R} (hP : P.is_prime)\n  (hfl : f.leading_coeff \u2209 P)\n  (hfP : \u2200 n : \u2115, \u2191n < degree f \u2192 f.coeff n \u2208 P)\n  (hfd0 : 0 < degree f) (h0 : f.coeff 0 \u2209 P^2)\n  (hu : \u2200 x : R, C x \u2223 f \u2192 is_unit x) : irreducible f :=\nhave hf0 : f \u2260 0, from \u03bb _, by simp * at *,\nhave hf : f.map (mk_hom P) =\n    C (mk_hom P (leading_coeff f)) * X ^ nat_degree f,\n  from polynomial.ext (\u03bb n, begin\n    rcases lt_trichotomy \u2191n (degree f) with h | h | h,\n    { erw [coeff_map, \u2190 mk_eq_mk_hom, eq_zero_iff_mem.2 (hfP n h),\n        coeff_C_mul, coeff_X_pow, if_neg, mul_zero],\n      rintro rfl, exact not_lt_of_ge degree_le_nat_degree h },\n    { have : nat_degree f = n, from nat_degree_eq_of_degree_eq_some h.symm,\n      rw [coeff_C_mul, coeff_X_pow, if_pos this.symm, mul_one, leading_coeff, this, coeff_map] },\n    { rw [coeff_eq_zero_of_degree_lt, coeff_eq_zero_of_degree_lt],\n      { refine lt_of_le_of_lt (degree_C_mul_X_pow_le _ _) _,\n        rwa \u2190 degree_eq_nat_degree hf0 },\n      { exact lt_of_le_of_lt (degree_map_le _) h } }\n  end),\nhave hfd0 : 0 < f.nat_degree, from with_bot.coe_lt_coe.1\n  (lt_of_lt_of_le hfd0 degree_le_nat_degree),\n\u27e8mt degree_eq_zero_of_is_unit (\u03bb h, by simp [*, lt_irrefl] at *),\nbegin\n  rintros p q rfl,\n  rw [map_mul] at hf,\n  have : map (mk_hom P) p \u2223 C (mk_hom P (p * q).leading_coeff) * X ^ (p * q).nat_degree,\n    from \u27e8map (mk_hom P) q, hf.symm\u27e9,\n  rcases mul_eq_mul_prime_pow (show prime (X : polynomial (ideal.quotient P)),\n    from prime_of_degree_eq_one_of_monic degree_X monic_X) hf with\n      \u27e8m, n, b, c, hmnd, hbc, hp, hq\u27e9,\n  have hmn : 0 < m \u2192 0 < n \u2192 false,\n  { assume hm0 hn0,\n    have hp0 : p.eval 0 \u2208 P,\n    { rw [\u2190 coeff_zero_eq_eval_zero, \u2190 eq_zero_iff_mem, mk_eq_mk_hom, \u2190 coeff_map],\n      simp [hp, coeff_zero_eq_eval_zero, zero_pow hm0] },\n    have hq0 : q.eval 0 \u2208 P,\n    { rw [\u2190 coeff_zero_eq_eval_zero, \u2190 eq_zero_iff_mem, mk_eq_mk_hom, \u2190 coeff_map],\n      simp [hq, coeff_zero_eq_eval_zero, zero_pow hn0] },\n    apply h0,\n    rw [coeff_zero_eq_eval_zero, eval_mul, pow_two],\n    exact ideal.mul_mem_mul hp0 hq0 },\n  have hpql0 : (mk_hom P) (p * q).leading_coeff \u2260 0,\n  { rwa [\u2190 mk_eq_mk_hom, ne.def, eq_zero_iff_mem] },\n  have hp0 : p \u2260 0, from \u03bb h, by simp * at *,\n  have hq0 : q \u2260 0, from \u03bb h, by simp * at *,\n  have hmn0 : m = 0 \u2228 n = 0,\n  { rwa [nat.pos_iff_ne_zero, nat.pos_iff_ne_zero, imp_false, not_not,\n      \u2190 or_iff_not_imp_left] at hmn },\n  have hbc0 : degree b = 0 \u2227 degree c = 0,\n  { apply_fun degree at hbc,\n    rwa [degree_C hpql0, degree_mul_eq, nat.with_bot.add_eq_zero_iff] at hbc },\n  have hmp : m \u2264 nat_degree p,\n    from with_bot.coe_le_coe.1\n      (calc \u2191m = degree (p.map (mk_hom P)) : by simp [hp, hbc0.1]\n         ... \u2264 degree p : degree_map_le _\n         ... \u2264 nat_degree p : degree_le_nat_degree),\n  have hmp : n \u2264 nat_degree q,\n    from with_bot.coe_le_coe.1\n      (calc \u2191n = degree (q.map (mk_hom P)) : by simp [hq, hbc0.2]\n         ... \u2264 degree q : degree_map_le _\n         ... \u2264 nat_degree q : degree_le_nat_degree),\n  have hpmqn : p.nat_degree = m \u2227 q.nat_degree = n,\n  { rw [nat_degree_mul_eq hp0 hq0] at hmnd, omega },\n  rcases hmn0 with rfl | rfl,\n  { left,\n    rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.1),\n      is_unit_C],\n    refine hu _ _,\n    rw [\u2190 eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.1)],\n    exact dvd_mul_right _ _ },\n  { right,\n    rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.2),\n      is_unit_C],\n    refine hu _ _,\n    rw [\u2190 eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.2)],\n    exact dvd_mul_left _ _ }\nend\u27e9\n\n#print axioms eisenstein\n\n#exit\nimport algebra.ring\nuniverse u\nvariables {R : Type} [comm_ring R] (M : submonoid R)\nset_option pp.all true\n#print comm_ring.zero_add\ninstance : comm_monoid (submonoid.localization M) :=\n(submonoid.localization.r M).comm_monoid\n\n@[elab_as_eliminator]\nprotected def lift_on\u2082 {\u03b1 : Type*} [monoid \u03b1] {\u03b2} {c : con \u03b1 } (q r : c.quotient) (f : \u03b1 \u2192 \u03b1 \u2192 \u03b2)\n  (h : \u2200 a\u2081 a\u2082 b\u2081 b\u2082, c a\u2081 b\u2081 \u2192 c a\u2082 b\u2082 \u2192 f a\u2081 a\u2082 = f b\u2081 b\u2082) : \u03b2 := quotient.lift_on\u2082' q r f h\n\ndef submonoid.localization.mk : R \u2192 M \u2192 submonoid.localization M :=\n\u03bb x y, (submonoid.localization.r M).mk' (x, y)\n\ntheorem r_of_eq {x y : R \u00d7 M} (h : y.1 * x.2 = x.1 * y.2) :\n  submonoid.localization.r M x y :=\nsubmonoid.localization.r_iff_exists.2 \u27e81, by rw h\u27e9\n\ninstance : has_zero (submonoid.localization M) :=\n\u27e8submonoid.localization.mk M 0 1\u27e9\n\ninstance : has_add (submonoid.localization M) :=\n\u27e8\u03bb z w, lift_on\u2082 z w\n  (\u03bb x y : R \u00d7 M, submonoid.localization.mk M ((x.2 : R) * y.1 + y.2 * x.1) (x.2 * y.2)) $\n\u03bb r1 r2 r3 r4 h1 h2, (con.eq _).2\nbegin\n  rw submonoid.localization.r_eq_r' at h1 h2 \u22a2,\n  cases h1 with t\u2085 ht\u2085,\n  cases h2 with t\u2086 ht\u2086,\n  use t\u2086 * t\u2085,\n  calc ((r1.2 : R) * r2.1 + r2.2 * r1.1) * (r3.2 * r4.2) * (t\u2086 * t\u2085) =\n      (r2.1 * r4.2 * t\u2086) * (r1.2 * r3.2 * t\u2085) + (r1.1 * r3.2 * t\u2085) * (r2.2 * r4.2 * t\u2086) : by ring\n      ... = (r3.2 * r4.1 + r4.2 * r3.1) * (r1.2 * r2.2) * (t\u2086 * t\u2085) : by rw [ht\u2086, ht\u2085]; ring\nend\u27e9\n\ninstance : has_neg (submonoid.localization M) :=\n\u27e8\u03bb z, con.lift_on z (\u03bb x : R \u00d7 M, submonoid.localization.mk M (-x.1) x.2) $\n  \u03bb r1 r2 h, (con.eq _).2\nbegin\n  rw submonoid.localization.r_eq_r' at h \u22a2,\n  cases h with t ht,\n  use t,\n  rw [neg_mul_eq_neg_mul_symm, neg_mul_eq_neg_mul_symm, ht],\n  ring,\nend\u27e9\ninstance : add_semigroup (submonoid.localization M) := by apply_instance\n\n\nset_option pp.all true\n\n#print comm_ring.zero_add\n\n@[instance]lemma C : comm_ring (submonoid.localization M) :=\n{ zero := (0 : submonoid.localization M),\n  one  := (1 : submonoid.localization M),\n  add  := (+),\n  mul  := (*),\n  zero_add       := \u03bb y : submonoid.localization M, quotient.induction_on' y _,\n  add_zero       := \u03bb y : submonoid.localization M, quotient.induction_on' y _,\n  add_assoc      := \u03bb m n k : submonoid.localization M,\n    quotient.induction_on\u2083' m n k _,\n  neg            := has_neg.neg,\n  add_left_neg   := \u03bb y : submonoid.localization M, quotient.induction_on' y _,\n  add_comm       := \u03bb y z : submonoid.localization M, quotient.induction_on\u2082' z y _,\n  left_distrib   := \u03bb m n k : submonoid.localization M, quotient.induction_on\u2083' m n k _,\n  right_distrib  := \u03bb m n k : submonoid.localization M, quotient.induction_on\u2083' m n k _,\n   ..submonoid.localization.comm_monoid M }\n--   { intros,\n--      refine quotient.sound (r_of_eq M _),\n--      simp only [prod.snd_mul, prod.fst_mul, submonoid.coe_mul],\n--      ring }\n-- end\n\n\nexample (y m n k : submonoid.localization M) : Prop := @eq.{1} M.localization\n    (@has_add.add.{0} M.localization\n       (@add_semigroup.to_has_add.{0} M.localization\n          (@add_semigroup.mk.{0} M.localization\n             (@has_add.add.{0} M.localization (@submonoid.localization.has_add R _inst_1 M))\n             _\n            --  (\u03bb (m n k : M.localization),\n            --     @quotient.induction_on\u2083'.{1 1 1}\n            --       (prod.{0 0} R\n            --          (@coe_sort.{1 2}\n            --             (@submonoid.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             (@submonoid.has_coe_to_sort.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             M))\n            --       (prod.{0 0} R\n            --          (@coe_sort.{1 2}\n            --             (@submonoid.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             (@submonoid.has_coe_to_sort.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             M))\n            --       (prod.{0 0} R\n            --          (@coe_sort.{1 2}\n            --             (@submonoid.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             (@submonoid.has_coe_to_sort.{0} R\n            --                (@comm_monoid.to_monoid.{0} R\n            --                   (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))))\n            --             M))\n            --       (@submonoid.localization.r.{0} R\n            --          (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))\n            --          M).to_setoid\n            --       (@submonoid.localization.r.{0} R\n            --          (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))\n            --          M).to_setoid\n            --       (@submonoid.localization.r.{0} R\n            --          (@comm_semiring.to_comm_monoid.{0} R (@comm_ring.to_comm_semiring.{0} R _inst_1))\n            --          M).to_setoid\n            --       (\u03bb (_x _x_1 _x_2 : M.localization),\n            --          @eq.{1} M.localization\n            --            (@has_add.add.{0} M.localization\n            --               (@has_add.mk.{0} M.localization\n            --                  (@has_add.add.{0} M.localization (@submonoid.localization.has_add R _inst_1 M)))\n            --               (@has_add.add.{0} M.localization\n            --                  (@has_add.mk.{0} M.localization\n            --                     (@has_add.add.{0} M.localization (@submonoid.localization.has_add R _inst_1 M)))\n            --                  _x\n            --                  _x_1)\n            --               _x_2)\n            --            (@has_add.add.{0} M.localization\n            --               (@has_add.mk.{0} M.localization\n            --                  (@has_add.add.{0} M.localization (@submonoid.localization.has_add R _inst_1 M)))\n            --               _x\n            --               (@has_add.add.{0} M.localization\n            --                  (@has_add.mk.{0} M.localization\n            --                     (@has_add.add.{0} M.localization (@submonoid.localization.has_add R _inst_1 M)))\n            --                  _x_1\n            --                  _x_2)))\n            --       m\n            --       n\n            --       k\n            --     _)\n            )) _\n      --  (@has_zero.zero.{0} M.localization\n      --     (@has_zero.mk.{0} M.localization\n      --        (@has_zero.zero.{0} M.localization (@submonoid.localization.has_zero R _inst_1 M))))\n       y\n       )\n    y\n#exit\nimport category_theory.limits.shapes.pullbacks\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables {C : Type u} [\ud835\udc9e : category.{v} C]\ninclude \ud835\udc9e\n#print is_colimit\ndef pushout_of_epi {X Y : C} (f : X \u27f6 Y) [epi f] :\n  is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f) :=\n{ desc := \u03bb s, s.\u03b9.app walking_span.left,\n  fac' := \u03bb s j, option.cases_on j\n    (by { tidy, convert s.w walking_span.hom.fst })\n\n\n\n\n  (\u03bb j, walking_pair.cases_on j (by tidy) begin\n    tidy,\n\n  end) }\n\ntheorem epi_of_pushout {X Y : C} (f : X \u27f6 Y)\n  (is_colim : is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f)) : epi f := sorry\n\n#exit\nimport data.fintype.basic\n\n\n#exit\nimport algebra.ring tactic\n\ndef add : \u03a0 l\u2081 l\u2082 : list nat, list nat\n| []      l\u2082      := l\u2082\n| l\u2081      []      := l\u2081\n| (a::l\u2081) (b::l\u2082) :=\nif h : b < a then b :: add (a :: l\u2081) l\u2082\nelse a :: add l\u2081 (b :: l\u2082)\n\n#exit\nnamespace tactic\n\nmeta def protect (n : name) : tactic unit :=\ndo env \u2190 get_env, set_env $ env.mk_protected n\n\nend tactic\n\nnamespace nat\n\nprivate lemma X : true := trivial\n\nrun_cmd tactic.protect `nat.X\n\nexample : true := X\n\nend nat\n\nopen category_theory\n\ninstance A : is_semiring_hom (coe : \u2124 \u2192 \u211a) :=\nby refine_struct { .. }; simp\n\n@[reducible] def icast : \u2124 \u2192+* \u211a := ring_hom.of coe\n\nlemma unique_hom {R : Type*} [ring R] (f g : \u211a \u2192+* R) : f = g :=\nbegin\n  ext,\n  refine rat.num_denom_cases_on x (\u03bb n d hd0 _, _),\n  have hd0 : (d : \u211a) \u2260 0, { simpa [nat.pos_iff_ne_zero] using hd0 },\n  have hf : \u2200 n : \u2124, f n = n, from \u03bb _, (f.comp icast).eq_int_cast _,\n  have hg : \u2200 n : \u2124, g n = n, from \u03bb _, (g.comp icast).eq_int_cast _,\n  have : is_unit ((d  : \u2124) : R),\n    from \u27e8\u27e8f d, f (1 / d), by rw [\u2190 ring_hom.map_mul, mul_div_cancel' _ hd0, f.map_one],\n      by rw [\u2190 ring_hom.map_mul, div_mul_cancel _ hd0, f.map_one]\u27e9,\n    by simp [hf]\u27e9 ,\n  rw [rat.mk_eq_div, div_eq_mul_inv, ring_hom.map_mul, ring_hom.map_mul, hf, hg,\n    \u2190 this.mul_left_inj],\n  conv_lhs { rw \u2190 hf d },\n  rw [\u2190 hg d, mul_assoc, mul_assoc, \u2190 f.map_mul, \u2190 g.map_mul, int.cast_coe_nat,\n    inv_mul_cancel hd0],\n  simp\nend\n\ntheorem mono_epi_not_iso : \u2203 (A B : Ring.{0}) (f : A \u27f6 B),\n  mono.{0} f \u2227 epi.{0} f \u2227 (is_iso.{0} f \u2192 false) :=\n\u27e8Ring.of \u2124, Ring.of \u211a, icast,\n  \u27e8begin\n    intros,\n    ext,\n    tidy,\n    rw [function.funext_iff] at h_1,\n    erw [\u2190 @int.cast_inj \u211a],\n    exact h_1 _\n  end\u27e9,\n  \u27e8\u03bb _ _ _ _,  unique_hom _ _\u27e9,\n  \u03bb h,\n    have (2 : \u2124) \u2223 1,\n      from \u27e8(h.inv : \u211a \u2192+* \u2124) (show \u211a, from (1 : \u211a) / 2),\n        have (2 : \u2124) = (h.inv : \u211a \u2192+* \u2124) (2 : \u211a), by simp [bit0],\n        begin\n          rw [this, \u2190 ring_hom.map_mul],\n          norm_num,\n        end\u27e9,\n    absurd this (by norm_num)\u27e9\n\n#exit\nimport ring_theory.ideal_operations data.polynomial ring_theory.ideals tactic\nsection comm_ring\nvariables {R : Type*} {S : Type*} [comm_ring R] [comm_ring S]\n\nopen polynomial ideal.quotient\n\nopen_locale classical\n\nlemma thingy {R : Type*} [comm_ring R] (I : ideal R)\n  {a : R} (hab : a \u2209 I^2) (hu : \u2200 u \u2209 I, u \u2223 a \u2192 is_unit u)\n  (ha : \u00ac is_unit a) : irreducible a :=\n\u27e8ha, \u03bb x y hxy,\nhave hxPyP : x \u2208 I \u2192 y \u2208 I \u2192 false,\n  from \u03bb hxP hyP, hab (by rw [hxy, pow_two]; exact ideal.mul_mem_mul hxP hyP),\n(show x \u2209 I \u2228 y \u2209 I, by rwa [or_iff_not_imp_left, not_not]).elim\n  (\u03bb hx, or.inl (hu x hx $ by simp [hxy]))\n  (\u03bb hy, or.inr (hu y hy $ by simp [hxy]))\u27e9\n\nlemma thingy2 {R : Type*} [comm_ring R] (I : ideal R)\n  {a : R} (hab : a \u2209 I^2) (hu : \u2200 x \u2209 I, x \u2223 a \u2192 \u2203 u, 1 - u * x \u2208 I)\n  (ha : \u00ac is_unit a) : irreducible a :=\n\u27e8ha, \u03bb x y hxy,\nhave hxPyP : x \u2208 I \u2192 y \u2208 I \u2192 false,\n  from \u03bb hxP hyP, hab (by rw [hxy, pow_two]; exact ideal.mul_mem_mul hxP hyP),\n(show x \u2209 I \u2228 y \u2209 I, by rwa [or_iff_not_imp_left, not_not]).elim\n  (\u03bb hx, begin\n    cases hu x hx (by simp [hxy]) with u hu,\n\n\n  end)\n  (\u03bb hy, or.inr (hu y hy $ by simp [hxy]))\u27e9\n\nlemma ideal.sup_pow_two {I J : ideal R} : (I \u2294 J) ^ 2 = I ^ 2 \u2294 I * J \u2294 J ^ 2 :=\nby simp [ideal.sup_mul, ideal.mul_sup, mul_comm, pow_two, sup_assoc]\n\n\n#print ring_hom.of\nlemma eisenstein {R : Type*} [integral_domain R] {f : polynomial R}\n  {P : ideal R} --(hP : P.is_prime) --(hfl : f.leading_coeff \u2209 P)\n  --(hfP : \u2200 n : \u2115, \u2191n < degree f \u2192 f.coeff n \u2208 P)\n  --(hfd0 : 0 < degree f)\n  (h0 : f.coeff 0 \u2209 P^2)\n  (hu : \u2200 x : R, C x \u2223 f \u2192 is_unit x) : irreducible f :=\nhave eq_id : (ring_hom.of (eval (0 : R))).comp (ring_hom.of C) = ring_hom.id _,\n  by ext; simp,\nhave h_ker : ideal.span {(X : polynomial R)} \u2264 (ring_hom.of (eval (0 : R))).ker,\n  from ideal.span_le.2 (\u03bb _, by simp [ring_hom.mem_ker] {contextual := tt}),\nthingy (P.map (ring_hom.of C) \u2294 ideal.span {X})\n  (\u03bb hf, h0 $\n    begin\n      have := @ideal.mem_map_of_mem _ _ _ _ (ring_hom.of (eval 0)) _ _ hf,\n      rwa [pow_two, ideal.map_mul, ideal.map_sup, ideal.map_map, eq_id, ideal.map_id,\n        map_eq_bot_iff_le_ker.1 h_ker, sup_bot_eq, ring_hom.coe_of,\n        \u2190 coeff_zero_eq_eval_zero, \u2190 pow_two] at this\n    end)\n  begin\n    assume x hx,\n\n\n  end\n  _\n\nexample {R : Type*} [comm_ring R] {P Q : ideal R} (hP : P.is_prime) (hQ : Q.is_prime)\n  (hPQ : (P \u2294 Q).is_prime) {a : R} (ha : a \u2208 P \u2294 Q^2) (hab : a \u2209 P ^ 2 \u2294 Q) (haP : a \u2209 P)\n  (hu : \u2200 u \u2209 Q, u \u2223 a \u2192 is_unit u) : irreducible a :=\n\u27e8sorry, \u03bb x y hxy,\nhave hxQyQ : x \u2208 P \u2294 Q \u2192 y \u2208 P \u2294 Q \u2192 false,\n  from \u03bb hxQ hyQ, hab begin\n    have haPQ: a \u2208 ((P \u2294 Q) * (P \u2294 Q)),\n      from hxy.symm \u25b8 (ideal.mul_mem_mul hxQ hyQ),\n    have : ((P \u2294 Q) * (P \u2294 Q)) \u2264 P ^ 2 \u2294 Q,\n      { rw [ideal.mul_sup, ideal.sup_mul, ideal.sup_mul, sup_assoc,\n          \u2190 @sup_assoc  _ _ (Q * P), mul_comm Q P, sup_idem, \u2190 ideal.sup_mul, pow_two],\n        exact sup_le_sup (le_refl _) ideal.mul_le_left },\n    exact this haPQ\n  end,\n\nbegin\n  subst a,\n\n\nend\u27e9\nend comm_ring\n\nvariables {R : Type*} [integral_domain R]\n\nopen polynomial ideal.quotient\n\n@[simp] lemma nat.with_bot.coe_nonneg {n : \u2115} : 0 \u2264 (n : with_bot \u2115) :=\nby rw [\u2190 with_bot.coe_zero, with_bot.coe_le_coe]; exact nat.zero_le _\n\n@[simp] lemma nat.with_bot.lt_zero (n : with_bot \u2115) : n < 0 \u2194 n = \u22a5 :=\noption.cases_on n dec_trivial (\u03bb n, iff_of_false\n  (by simp [with_bot.some_eq_coe]) (\u03bb h, option.no_confusion h))\n\nexample (n : with_bot \u2115) : n.lt_zero\n\nlemma degree_nonneg_iff_ne_zero {R : Type*} [comm_semiring R]\n  {f : polynomial R} : 0 \u2264 degree f \u2194 f \u2260 0 :=\n\u27e8\u03bb h0f hf0, absurd h0f (by rw [hf0, degree_zero]; exact dec_trivial),\n  \u03bb hf0, le_of_not_gt (\u03bb h, by simp [gt, degree_eq_bot, *] at *)\u27e9\n\nlemma nat_degree_eq_zero_iff_degree_le_zero {R : Type*} [comm_semiring R]\n  {p : polynomial R} : p.nat_degree = 0 \u2194 p.degree \u2264 0 :=\nif hp0 : p = 0 then by simp [hp0]\nelse by rw [degree_eq_nat_degree hp0, \u2190 with_bot.coe_zero, with_bot.coe_le_coe,\n  nat.le_zero_iff]\n\nlemma eq_one_of_is_unit_of_monic {R : Type*} [comm_semiring R]\n  {p : polynomial R} (hm : monic p) (hpu : is_unit p) : p = 1 :=\nhave degree p \u2264 0,\n  from calc degree p \u2264 degree (1 : polynomial R) :\n    let \u27e8u, hu\u27e9 := is_unit_iff_dvd_one.1 hpu in\n    if hu0 : u = 0\n    then begin\n        rw [hu0, mul_zero] at hu,\n        rw [\u2190 mul_one p, hu, mul_zero],\n        simp\n      end\n    else have p.leading_coeff * u.leading_coeff \u2260 0,\n        by rw [hm.leading_coeff, one_mul, ne.def, leading_coeff_eq_zero];\n          exact hu0,\n      by rw [hu, degree_mul_eq' this];\n        exact le_add_of_nonneg_right' (degree_nonneg_iff_ne_zero.2 hu0)\n  ... \u2264 0 : degree_one_le,\nby rw [eq_C_of_degree_le_zero this, \u2190 nat_degree_eq_zero_iff_degree_le_zero.2 this,\n    \u2190 leading_coeff, hm.leading_coeff, C_1]\n\nopen finset\n\nlemma dvd_mul_prime {x a p : R} (hp : prime p) : x \u2223 a * p \u2192 x \u2223 a \u2228 p \u2223 x :=\n\u03bb \u27e8y, hy\u27e9, (hp.div_or_div \u27e8a, hy.symm.trans (mul_comm _ _)\u27e9).elim\n  or.inr\n  begin\n    rintros \u27e8b, rfl\u27e9,\n    rw [mul_left_comm, mul_comm, domain.mul_right_inj hp.ne_zero] at hy,\n    rw [hy],\n    exact or.inl (dvd_mul_right _ _)\n  end\n\nlemma dvd_mul_prime_prod {\u03b1 : Type*} {x a : R} {s : finset \u03b1}\n  {p : \u03b1 \u2192 R} (hp : \u2200 i \u2208 s, prime (p i)) (hx : x \u2223 a * s.prod p) :\n  \u2203 t b, t \u2286 s \u2227 b \u2223 a \u2227 x = b * t.prod p :=\nbegin\n  classical,\n  rcases hx with \u27e8y, hy\u27e9,\n  induction s using finset.induction with i s his ih generalizing x y a,\n  { exact \u27e8\u2205, x, finset.subset.refl _, \u27e8y, hy \u25b8 by simp\u27e9, by simp\u27e9 },\n  { rw [prod_insert his, \u2190 mul_assoc] at hy,\n    have hpi : prime (p i), { exact hp i (mem_insert_self _ _) },\n    rcases ih (\u03bb i hi, hp i (mem_insert_of_mem hi)) _ hy with \u27e8t, b, hts, hb, rfl\u27e9,\n    rcases dvd_mul_prime hpi hb with hba | \u27e8c, rfl\u27e9,\n    { exact \u27e8t, b, trans hts (subset_insert _ _), hba, rfl\u27e9 },\n    { exact \u27e8insert i t, c, insert_subset_insert _ hts,\n        by rwa [mul_comm, mul_dvd_mul_iff_right hpi.ne_zero] at hb,\n        by rw [prod_insert (mt (\u03bb x, hts x) his), mul_left_comm, mul_assoc]\u27e9, } }\nend\n\nlemma mul_eq_mul_prime_prod {\u03b1 : Type*} [decidable_eq \u03b1] {x y a : R} {s : finset \u03b1}\n  {p : \u03b1 \u2192 R} (hp : \u2200 i \u2208 s, prime (p i)) (hx : x * y = a * s.prod p) :\n  \u2203 t u b c, t \u222a u = s \u2227 disjoint t u \u2227 b * c = a \u2227\n    x = b * t.prod p \u2227 y = c * u.prod p :=\nbegin\n  induction s using finset.induction with i s his ih generalizing x y a,\n  { exact \u27e8\u2205, \u2205, x, y, by simp [hx]\u27e9 },\n  { rw [prod_insert his, \u2190 mul_assoc] at hx,\n    have hpi : prime (p i), { exact hp i (mem_insert_self _ _) },\n    rcases ih (\u03bb i hi, hp i (mem_insert_of_mem hi)) hx with\n      \u27e8t, u, b, c, htus, htu, hbc, rfl, rfl\u27e9,\n    have hpibc : p i \u2223 b \u2228 p i \u2223 c,\n      from hpi.div_or_div \u27e8a, by rw [hbc, mul_comm]\u27e9,\n    have hit : i \u2209 t, from \u03bb hit, his (htus \u25b8 mem_union_left _ hit),\n    have hiu : i \u2209 u, from \u03bb hiu, his (htus \u25b8 mem_union_right _ hiu),\n    rcases hpibc with \u27e8d, rfl\u27e9 | \u27e8d, rfl\u27e9,\n    { rw [mul_assoc, mul_comm a, domain.mul_right_inj hpi.ne_zero] at hbc,\n      exact \u27e8insert i t, u, d, c, by rw [insert_union, htus],\n        disjoint_insert_left.2 \u27e8hiu, htu\u27e9,\n          by simp [\u2190 hbc, prod_insert hit, mul_assoc, mul_comm, mul_left_comm]\u27e9 },\n    { rw [\u2190 mul_assoc, mul_right_comm b, domain.mul_left_inj hpi.ne_zero] at hbc,\n      exact \u27e8t, insert i u, b, d, by rw [union_insert, htus],\n        disjoint_insert_right.2 \u27e8hit, htu\u27e9,\n          by simp [\u2190 hbc, prod_insert hiu, mul_assoc, mul_comm, mul_left_comm]\u27e9 } }\nend\n\nlemma mul_eq_mul_prime_pow {x y a p : R} {n : \u2115} (hp : prime p) (hx : x * y = a * p ^ n) :\n  \u2203 i j b c, i + j = n \u2227 b * c = a \u2227 x = b * p ^ i \u2227 y = c * p ^ j :=\nbegin\n  rcases mul_eq_mul_prime_prod (\u03bb _ _, hp)\n    (show x * y = a * (range n).prod (\u03bb _, p), by simpa) with\n    \u27e8t, u, b, c, htus, htu, rfl, rfl, rfl\u27e9,\n  exact \u27e8t.card, u.card, b, c, by rw [\u2190 card_disjoint_union htu, htus, card_range], by simp\u27e9,\nend\n\nlemma eisenstein {f : polynomial R} {P : ideal R} (hP : P.is_prime)\n  (hfl : f.leading_coeff \u2209 P)\n  (hfP : \u2200 n : \u2115, \u2191n < degree f \u2192 f.coeff n \u2208 P)\n  (hfd0 : 0 < degree f) (h0 : f.coeff 0 \u2209 P^2)\n  (hu : \u2200 x : R, C x \u2223 f \u2192 is_unit x) : irreducible f :=\nhave hf0 : f \u2260 0, from \u03bb _, by simp * at *,\nhave hf : f.map (mk_hom P) =\n    C (mk_hom P (leading_coeff f)) * X ^ nat_degree f,\n  from polynomial.ext (\u03bb n, begin\n    rcases lt_trichotomy \u2191n (degree f) with h | h | h,\n    { erw [coeff_map, \u2190 mk_eq_mk_hom, eq_zero_iff_mem.2 (hfP n h),\n        coeff_C_mul, coeff_X_pow, if_neg, mul_zero],\n      rintro rfl, exact not_lt_of_ge degree_le_nat_degree h },\n    { have : nat_degree f = n, from nat_degree_eq_of_degree_eq_some h.symm,\n      rw [coeff_C_mul, coeff_X_pow, if_pos this.symm, mul_one, leading_coeff, this, coeff_map] },\n    { rw [coeff_eq_zero_of_degree_lt, coeff_eq_zero_of_degree_lt],\n      { refine lt_of_le_of_lt (degree_C_mul_X_pow_le _ _) _,\n        rwa \u2190 degree_eq_nat_degree hf0 },\n      { exact lt_of_le_of_lt (degree_map_le _) h } }\n  end),\nhave hfd0 : 0 < f.nat_degree, from with_bot.coe_lt_coe.1\n  (lt_of_lt_of_le hfd0 degree_le_nat_degree),\n\u27e8mt degree_eq_zero_of_is_unit (\ufffd\ufffd h, by simp [*, lt_irrefl] at *),\nbegin\n  rintros p q rfl,\n  rw [map_mul] at hf,\n  have : map (mk_hom P) p \u2223 C (mk_hom P (p * q).leading_coeff) * X ^ (p * q).nat_degree,\n    from \u27e8map (mk_hom P) q, hf.symm\u27e9,\n  rcases mul_eq_mul_prime_pow (show prime (X : polynomial (ideal.quotient P)),\n    from prime_of_degree_eq_one_of_monic degree_X monic_X) hf with\n      \u27e8m, n, b, c, hmnd, hbc, hp, hq\u27e9,\n  have hmn : 0 < m \u2192 0 < n \u2192 false,\n  { assume hm0 hn0,\n    have hp0 : p.eval 0 \u2208 P,\n    { rw [\u2190 coeff_zero_eq_eval_zero, \u2190 eq_zero_iff_mem, mk_eq_mk_hom, \u2190 coeff_map],\n      simp [hp, coeff_zero_eq_eval_zero, zero_pow hm0] },\n    have hq0 : q.eval 0 \u2208 P,\n    { rw [\u2190 coeff_zero_eq_eval_zero, \u2190 eq_zero_iff_mem, mk_eq_mk_hom, \u2190 coeff_map],\n      simp [hq, coeff_zero_eq_eval_zero, zero_pow hn0] },\n    apply h0,\n    rw [coeff_zero_eq_eval_zero, eval_mul, pow_two],\n    exact ideal.mul_mem_mul hp0 hq0 },\n  have hpql0 : (mk_hom P) (p * q).leading_coeff \u2260 0,\n  { rwa [\u2190 mk_eq_mk_hom, ne.def, eq_zero_iff_mem] },\n  have hp0 : p \u2260 0, from \u03bb h, by simp * at *,\n  have hq0 : q \u2260 0, from \u03bb h, by simp * at *,\n  have hmn0 : m = 0 \u2228 n = 0,\n  { rwa [nat.pos_iff_ne_zero, nat.pos_iff_ne_zero, imp_false, not_not,\n      \u2190 or_iff_not_imp_left] at hmn },\n  have hbc0 : degree b = 0 \u2227 degree c = 0,\n  { apply_fun degree at hbc,\n    rwa [degree_C hpql0, degree_mul_eq, nat.with_bot.add_eq_zero_iff] at hbc },\n  have hmp : m \u2264 nat_degree p,\n    from with_bot.coe_le_coe.1\n      (calc \u2191m = degree (p.map (mk_hom P)) : by simp [hp, hbc0.1]\n         ... \u2264 degree p : degree_map_le _\n         ... \u2264 nat_degree p : degree_le_nat_degree),\n  have hmp : n \u2264 nat_degree q,\n    from with_bot.coe_le_coe.1\n      (calc \u2191n = degree (q.map (mk_hom P)) : by simp [hq, hbc0.2]\n         ... \u2264 degree q : degree_map_le _\n         ... \u2264 nat_degree q : degree_le_nat_degree),\n  have hpmqn : p.nat_degree = m \u2227 q.nat_degree = n,\n  { rw [nat_degree_mul_eq hp0 hq0] at hmnd, omega },\n  rcases hmn0 with rfl | rfl,\n  { left,\n    rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.1),\n      is_unit_C],\n    refine hu _ _,\n    rw [\u2190 eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.1)],\n    exact dvd_mul_right _ _ },\n  { right,\n    rw [eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.2),\n      is_unit_C],\n    refine hu _ _,\n    rw [\u2190 eq_C_of_degree_le_zero (nat_degree_eq_zero_iff_degree_le_zero.1 hpmqn.2)],\n    exact dvd_mul_left _ _ }\nend\u27e9\n\n#print axioms eisenstein\n\nexample (a b c d e : \u2124) (hab : a ^ 2 = b ^ 2 + 1) : a ^3 =\n\n\ndef X : \u2115 \u00d7 \u2115 \u2192 \u2115 :=\n\u03bb \u27e8a, b\u27e9, let \u27e8x, y\u27e9 := (3,4) in\nbegin\n  exact _x.1\nend\n\n#eval 274 % 6\n#eval 4 * 3 * 5 * 7 * 2 * 3\n\nset_option class.instance_max_depth 10000\n\ninstance : fact (0 < 27720) := by norm_num\n\n#eval 1624 % 420\n\nexample : \u2200 x : zmod 2520,\n  x^7 + 21* x^6 + 175 * x^5 + 735 * x^4 + 1624 * x^3 + 1764 * x^2 + 720 x\ndec_trivial\n\n#eval X (5,4)\n\n#exit\nimport data.polynomial\n#print subtyp\nexample : 1= 1 := rfl\n#print subtype.forall\nvariables {R : Type*} [comm_ring R]\nopen polynomial\n\nopen_locale classical\n\nexample {f : polynomial R} (n : \u2115)\n  (h : \u2200 m, n \u2264 m \u2192 polynomial.coeff f m = 0) :\n  degree f < n :=\nif hf0 : f = 0 then by simp [hf0, with_bot.bot_lt_coe]\nelse lt_of_not_ge (\u03bb hfn, mt leading_coeff_eq_zero.1 hf0 (h (nat_degree f)\n  (with_bot.coe_le_coe.1 (by simpa only [ge, degree_eq_nat_degree hf0] using hfn)))\n\n#exit\nimport group_theory.quotient_group data.fintype.basic set_theory.cardinal\n\nuniverse u\n\nopen_locale classical\n\ntheorem normal_of_index_2 {G : Type u} [group G] (S : set G) [is_subgroup S]\n  (HS1 : \u2203 g \u2209 S, \u2200 x, x \u2208 S \u2228 g * x \u2208 S)\n  (HS2 : bool \u2243 quotient_group.quotient S)\n  [fintype (quotient_group.quotient S)] (HS3 : fintype.card (quotient_group.quotient S) = 2)\n  (HS4 : cardinal.mk (quotient_group.quotient S) = 2) : normal_subgroup S :=\nlet \u27e8x, hxS, hx\u27e9 := HS1 in\nhave \u2200 g h, g * h \u2208 S \u2192 g \u2208 S \u2192 h \u2208 S,\n  from \u03bb g h ghS gS, (hx h).resolve_right\n    (\u03bb xhS, hxS $\n      suffices (x * h) * (g * h)\u207b\u00b9 * g \u2208 S, by simpa [mul_assoc, mul_inv_rev],\n      is_submonoid.mul_mem\n        (is_submonoid.mul_mem xhS (is_subgroup.inv_mem ghS)) gS),\nhave \u2200 g h, (g \u2208 S \u2194 h \u2208 S) \u2192 g * h \u2208 S,\n  from \u03bb g h ghS, (hx g).elim\n    (\u03bb gS, is_submonoid.mul_mem gS (ghS.1 gS))\n    (\u03bb xgS, (hx h).elim sorry\n      (\u03bb xhS, (is_subgroup.mul_mem_cancel_left S ((hx x).resolve_left hxS)).1\n        _)),\n{ normal :=\n\n\u03bb n hnS g, (hx g).elim sorry\n  (\u03bb h, begin\n\n\n\n  end)\n\n\u03bb n hnS g, (hx g).elim\n    (\u03bb hgS, is_submonoid.mul_mem (is_submonoid.mul_mem hgS hnS) (is_subgroup.inv_mem hgS))\n    begin\n      assume hxgS,\n      have hgS : g \u2209 S, from sorry,\n\n\n    end\n\n\n}\n\n\n\n\n#exit\nimport linear_algebra.basic tactic\n\nuniverses u v\n\nopen linear_map\n\nvariables {R : Type u} [ring R] {M : Type v} [add_comm_group M] [module R M]\n\n\nlemma equiv_ker_prod_range (p : M \u2192\u2097[R] M) (hp : p.comp p = p) : M \u2243\u2097[R] ker p \u00d7 range p :=\nhave h : \u2200 m, p (p m) = p m := linear_map.ext_iff.1 hp,\n{ to_fun := \u03bb m, (\u27e8m - p m, mem_ker.2 $ by simp [h]\u27e9, \u27e8p m, mem_range.2 \u27e8m, rfl\u27e9\u27e9),\n  inv_fun := \u03bb x, x.1 + x.2,\n  left_inv := \u03bb m, by simp,\n  right_inv := by { rintros \u27e8\u27e8x, hx\u27e9, \u27e8_, y, hy, rfl\u27e9\u27e9, simp [h, mem_ker.1 hx] },\n  add := \u03bb _ _, by simp [subtype.coe_ext, add_comm, add_left_comm, add_assoc, sub_eq_add_neg],\n  smul := \u03bb _ _, by simp [subtype.coe_ext, smul_sub] }\n\n#exit\nimport tactic\nset_option profiler true\nexample : 0 < 1 := by norm_num\n\nexample : 0 < 1 := zero_lt_one\n\n#exit\nimport data.nat.prime data.nat.parity data.real.pi\n\nexample (x : \u2115) (h : (\u03bb y : \u2115, 0 < y) x) : 0 < x := h\n\nopen nat --real\n\nexample : real.pi < 47 := by pi_upper_bound [1.9]\n\n\ntheorem goldbach_disproof : \u00ac goldbach :=\nbegin\n  assume h,\n  rcases h 0 (by norm_num) with \u27e8p, q, hp, hq, h\u27e9,\n  simp at h,\n  exact absurd hp (h.1.symm \u25b8 by norm_num)\nend\n\nvariables {R : Type*} [comm_ring R]\n\nlemma pow_dvd_pow_of_dvd {a b : R} (h : a \u2223 b) (n : \u2115) : a ^ n \u2223 b ^ n :=\nlet \u27e8d, hd\u27e9 := h in \u27e8d ^ n, hd.symm \u25b8 mul_pow _ _ _\u27e9\n\n@[simp] lemma nat.cast_dvd {a b : \u2115} : a \u2223 b \u2192 (a : R) \u2223 (b : R) :=\n\u03bb \u27e8n, hn\u27e9, \u27e8n, by simp [hn]\u27e9\n\nlemma dvd_sub_add_pow (p : \u2115) [hp : fact p.prime] (a b : R) :\n  (p : R) \u2223 (a + b)^p - (a ^ p + b ^ p) :=\nbegin\n  rw [add_pow],\n  conv in (finset.range (p + 1)) { rw \u2190 nat.sub_add_cancel hp.pos },\n  rw [finset.sum_range_succ, finset.sum_range_succ',\n    nat.sub_add_cancel hp.pos],\n  suffices : \u2191p \u2223 (finset.range (p - 1)).sum (\u03bb i, a ^ (i + 1) * b ^\n      (p - (i + 1)) * nat.choose p (i + 1)),\n  { simpa },\n  refine finset.dvd_sum _,\n  intros,\n  refine dvd_mul_of_dvd_right (nat.cast_dvd\n    (nat.prime.dvd_choose (nat.succ_pos _) (by simp at *; omega) hp)) _\nend\n\nlemma dvd_add_pow_iff (p : \u2115) [fact p.prime] (a b : R) :\n  (p : R) \u2223 (a + b)^p \u2194 (p : R) \u2223 (a ^ p + b ^ p) :=\n(dvd_add_iff_left ((dvd_neg _ _).2 (dvd_sub_add_pow p a b))).trans (by simp)\n\nlemma dvd_sub_pow_of_dvd_sub\n   : \u2200  (k : \u2115) (p : \u2115) (a b : R) (c d : \u2115)\n    (h : (p : R) \u2223 c * a - d * b),\n  (p^(k+1) : R) \u2223 c * a^(p^k) - d * b^(p^k)\n| 0     p a b c d := by simp\n| (k+1) p a b c d := \u03bb h, begin\n  have :=\n    (dvd_sub_pow_of_dvd_sub k p a b c d h),\n  simp at this,\n  rw [dvd_add_pow_iff] at this,\n  have := dvd_sub_pow_of_dvd_sub _ _ _ _ this,\n  simp at this,\n\n\nend\n\n\nlemma dvd_sub_pow_of_dvd_sub (R : Type*) [comm_ring R]\n   : \u2200  (k : \u2115) (p : \u2115) (a b : R) (h : (p : R) \u2223 a - b),\n  (p^(k+1) : R) \u2223 a^(p^k) - b^(p^k)\n| 0 := by simp\n| (k+1) := \u03bb p a b h, begin\n  have := dvd_sub_pow_of_dvd_sub k p a b h,\n  rw [\u2190 nat.cast_pow] at this,\n  have := dvd_sub_pow_of_dvd_sub _ _ _ _ this,\n  simp at this,\n\n\nend\n\n\n\nimport data.analysis.topology tactic\nexample {\u03b1 : Type*} [ring \u03b1] {f\u2081 f\u2082 g\u2081 g\u2082 df\u2081 df\u2082 dg\u2081 dg\u2082 Pf Pg: \u03b1}\n  (hf : f\u2081 - f\u2082 = df\u2081* Pf + Pf * df\u2082)\n  (hg : g\u2081 - g\u2082 = dg\u2081* Pg + Pg * dg\u2082) :\n  g\u2081 * f\u2081 - g\u2082 * f\u2082 = 0 :=\nbegin\n  rw [sub_eq_iff_eq_add] at hf hg,\n  substs f\u2081 g\u2081,\n  simp only [mul_add, add_mul, mul_assoc, neg_mul_eq_mul_neg, \u2190 mul_neg_eq_neg_mul_symm],\n  simp only [add_left_comm, sub_eq_add_neg, add_assoc],\n  abel,\n\n\nend\n\n\n#exit\nimport data.nat.prime tactic\n\nlemma ahj (x : \u2124): (x + 1)^2 = x^2 + 2 * x + 1  :=\nby ring\n#print ahj\ntheorem subrel_acc {\u03b1 : Type*} {r s : \u03b1 \u2192 \u03b1 \u2192 Prop} {x : \u03b1}\n  (hrs : \u2200 x y, s x y \u2192 r x y) (h : acc r x) : acc s x :=\nacc.rec_on h (\u03bb x hx ih, acc.intro x (\u03bb y hsyx, ih y (hrs y x hsyx)))\n\ntheorem acc_of_lt {\u03b1 : Type*} {r : \u03b1 \u2192 \u03b1 \u2192 Prop} {x y : \u03b1} (hr : r y x)\n  (h : acc r x) : acc r y :=\nby cases h; tauto\n\n#print nat.factors\n\nimport algebra.associated\n\ndef SUBMISSION := \u03a0 {R : Type*} [comm_ring R] {u r : R} {n : \u2115},\n  by exactI \u03a0 (hr : r ^ n = 0) (hu : is_unit u), is_unit (u + r)\n\nnotation `SUBMISSION` := SUBMISSION\n\ntheorem unit_add_nilpotent {R : Type*} [comm_ring R] {u r : R} {n : \u2115} (hr : r ^ n = 0)\n  (hu : is_unit u) : is_unit (u + r) := sorry\n\ntheorem submission : SUBMISSION := @unit_add_nilpotent\n#print axioms unit_add_nilpotent\n\n\n#exit\nimport data.polynomial\n\nvariables {R : Type*} {S : Type*}\n\nopen function polynomial\n\nexample [integral_domain R] [integral_domain S] (i : R \u2192+* S) (hf : injective i)\n  {f : polynomial R} (hf0 : 0 < degree f) (hfm : f.monic) :\n  (irreducible (f.map i) \u2227 \u2200 x : R, C x \u2223 f \u2192 is_unit (C x)) \u2194\n    irreducible f  :=\nbegin\n  split,\n  { rintros \u27e8hifm, hfC\u27e9,\n    split,\n    { exact mt degree_eq_zero_of_is_unit (\u03bb h, absurd hf0 (h \u25b8 lt_irrefl _)) },\n    { rintros g h rfl,\n      cases hifm.2 (g.map i) (h.map i) (map_mul _) with hug huh,\n      { have := degree_eq_zero_of_is_unit hug,\n        rw [degree_map_eq_of_injective hf] at this,\n        rw [eq_C_of_degree_eq_zero this],\n        refine or.inl (hfC _ _),\n        rw \u2190 eq_C_of_degree_eq_zero this,\n        exact dvd_mul_right _ _ },\n      { have := degree_eq_zero_of_is_unit huh,\n        rw [degree_map_eq_of_injective hf] at this,\n        rw [eq_C_of_degree_eq_zero this],\n        refine or.inr (hfC _ _),\n        rw \u2190 eq_C_of_degree_eq_zero this,\n        exact dvd_mul_left _ _ } } },\n  { assume hif,\n    split,\n    { split,\n      { refine mt degree_eq_zero_of_is_unit (\u03bb h, _),\n        rw [degree_map_eq_of_injective hf] at h,\n        exact absurd hf0 (h \u25b8 lt_irrefl _) },\n      { rintros g h hm,\n         } } }\n\n\n\nend\n\n#exit\nimport algebra.big_operators field_theory.finite field_theory.finite_card\n\nvariables {G R : Type} [group G] [integral_domain R] [fintype G] [decidable_eq G] [decidable_eq R]\n\nopen_locale big_operators add_monoid\n\nopen finset\n\ndef to_hom_units {G M : Type*} [group G] [monoid M] (f : G \u2192* M) : G \u2192* units M :=\n{ to_fun := \u03bb g,\n    \u27e8f g, f (g\u207b\u00b9),\n      by rw [\u2190 monoid_hom.map_mul, mul_inv_self, monoid_hom.map_one],\n      by rw [\u2190 monoid_hom.map_mul, inv_mul_self, monoid_hom.map_one]\u27e9,\n  map_one' := units.ext (monoid_hom.map_one _),\n  map_mul' := \u03bb _ _, units.ext (monoid_hom.map_mul _ _ _) }\n\n@[simp] lemma coe_to_hom_units {G M : Type*} [group G] [monoid M] (f : G \u2192* M) (g : G):\n  (to_hom_units f g : M) = f g := rfl\n\ndef preimage_equiv {H : Type} [group H] (f : G \u2192* H) (x y : H) :\n  f \u207b\u00b9' {x} \u2243 f \u207b\u00b9' {y} := sorry\n\nlemma sum_subtype {\u03b1 M : Type*} [add_comm_monoid M]\n  {p : \u03b1 \u2192 Prop} {F : fintype (subtype p)} {s : finset \u03b1} (h : \u2200 x, x \u2208 s \u2194 p x) {f : \u03b1 \u2192 M} :\n  \u2211 a in s, f a = \u2211 a : subtype p, f a :=\nhave (\u2208 s) = p, from set.ext h,\nbegin\n  rw \u2190 sum_attach,\n  resetI,\n  subst p,\n  congr,\n  simp [finset.ext]\nend\n\nvariable (G)\nlemma is_cyclic.exists_monoid_generator [is_cyclic G] :\n  \u2203 x : G, \u2200 y : G, y \u2208 powers x := sorry\n\nopen_locale classical\n\nlemma sum_units_subgroup (f : G \u2192* R) (hf : f \u2260 1) : \u2211 g : G, f g = 0 :=\nlet \u27e8x, hx\u27e9 := is_cyclic.exists_monoid_generator (set.range (to_hom_units f)) in\n-- have hx1 : x \u2260 1, from sorry,\ncalc \u2211 g : G, f g\n    = \u2211 g : G, to_hom_units f g : rfl\n... = \u2211 b : units R in univ.image (to_hom_units f),\n      (univ.filter (\u03bb a, to_hom_units f a = b)).card \u2022 b :\n        sum_comp (coe : units R \u2192 R) (to_hom_units f)\n... = \u2211 b : units R in univ.image (to_hom_units f),\n      fintype.card (to_hom_units f \u207b\u00b9' {b}) \u2022 b :\n  sum_congr rfl (\u03bb b hb, congr_arg2 _ (fintype.card_of_finset' _ (by simp)).symm rfl)\n... = \u2211 b : units R in univ.image (to_hom_units f),\n      fintype.card (to_hom_units f \u207b\u00b9' {x}) \u2022 b :\n  sum_congr rfl (\u03bb b hb, congr_arg2 _ (fintype.card_congr (preimage_equiv _ _ _)) rfl)\n... = \u2211 b : set.range (to_hom_units f),\n      fintype.card (to_hom_units f \u207b\u00b9' {x}) \u2022 \u2191b : sum_subtype (by simp)\n... = fintype.card (to_hom_units f \u207b\u00b9' {x}) * \u2211 b : set.range (to_hom_units f), (b : R) :\n  by simp [mul_sum, add_monoid.smul_eq_mul]\n... = (fintype.card (to_hom_units f \u207b\u00b9' {x}) : R) * 0 : (congr_arg2 _ rfl $\n  calc \u2211 b : set.range (to_hom_units f), (b : R)\n      = \u2211 n in range (order_of x), x ^ n :\n    eq.symm $ sum_bij (\u03bb n _, x ^ n) (by simp) (by simp)\n      (\u03bb m n hm hn, pow_injective_of_lt_order_of _ (by simpa using hm) (by simpa using hn))\n      (\u03bb b hb, let \u27e8n, hn\u27e9 := hx b in \u27e8n % order_of x, mem_range.2 (nat.mod_lt _ (order_of_pos _)),\n        by rw [\u2190 pow_eq_mod_order_of, hn]\u27e9)\n  ... = _ : begin  end)\n... = 0 : mul_zero _\n\n\n#print order_of_\n\nimport tactic\n#print nat.prime\n\nexample : \u2200 {p : \u2115} [fact p.prime] : \u2200 m, m \u2223 p \u2192 m = 1 \u2228 m = p := by library_search\ntheorem a_pow_4_sub_b_pow_4 (a b : \u2115) : a ^ 4 - b ^ 4 = (a - b) * (a + b) * (a ^ 2 + b ^ 2) :=\nif h : b \u2264 a\nthen\n  have b ^ 4 \u2264 a ^ 4, from nat.pow_le_pow_of_le_left h _,\n  int.coe_nat_inj $ by simp [int.coe_nat_sub h, int.coe_nat_sub this]; ring\nelse\n  have a ^ 4 \u2264 b ^ 4, from nat.pow_le_pow_of_le_left (le_of_not_ge h) _,\n  by rw [nat.sub_eq_zero_of_le (le_of_not_ge h), nat.sub_eq_zero_of_le this]; simp\n\n#exit\nimport group_theory.sylow\n\ntheorem order_of_eq_prime {G : Type*} [group G] [fintype G] [decidable_eq G] {g : G} {p : \u2115}\n  (h : p.prime) (hg : g^p = 1) (hg1 : g \u2260 1) : order_of g = p :=\n(h.2 _ (order_of_dvd_of_pow_eq_one hg)).resolve_left (mt order_of_eq_one_iff.1 hg1)\n\nopen_locale classical\n\ntheorem zagier (R : Type) [ring R]\n  [fintype (units R)] : fintype.card (units R) \u2260 5 :=\n\u03bb h5 : fintype.card (units R) = 5,\nlet \u27e8x, hx\u27e9 := sylow.exists_prime_order_of_dvd_card (show nat.prime 5, by norm_num)\n  (show 5 \u2223 fintype.card (units R), by rw h5) in\nhave hx5 : (x : R)^5 = 1,\n  by rw [\u2190 units.coe_pow, \u2190 hx, pow_order_of_eq_one, units.coe_one],\nif h2 : (2 : R) = 0\nthen\nhave ((x : R)^3 + x^2 + 1) ^ 3 = 1,\n  from calc ((x : R)^3 + x^2 + 1)^3\n      = (x : R)^5 * (x^4 + 3 * x^3 + 3 * x^2 + 4 * x + 6) + 3 * x^4\n          + 3 * x^3 + 3 * x^2 + 1 :\n            by simp [mul_add, add_mul, pow_succ, add_comm, mul_assoc,\n              add_assoc, add_left_comm, bit0, bit1]\n  ... = 2 * (2 * x^4 + 3 * x^3 + 3 * x^2 + 2 * x + 3) + 1 :\n            by rw hx5; simp [mul_add, add_mul, pow_succ, add_comm, mul_assoc,\n              add_assoc, add_left_comm, bit0, bit1]\n  ... = 1 : by rw [h2, zero_mul, zero_add],\nlet y : units R := units.mk\n  ((x : R)^3 + x^2 + 1)\n  (((x : R)^3 + x^2 + 1)^2)\n  (eq.symm (this.symm.trans $ pow_succ _ _))\n  (eq.symm (this.symm.trans $ pow_succ' _ _)) in\nhave hx1 : x \u2260 1,\nfrom \u03bb hx1, absurd hx (by simp [hx1]; norm_num),\nhave hx0 : (x : R)^2 * (x + 1) \u2260 0,\n  from \u03bb h, hx1 $ units.ext $ calc\n    (x : R) = x - (x ^ (-2 : \u2124) : units R) * ((x ^ 2) * (x + 1)) :\n      by rw [h, mul_zero, sub_zero]\n    ... =  x - (x ^ (-2 : \u2124) * x ^ (2 : \u2124) : units R) * (x + 1) :\n      by rw [units.coe_mul, mul_assoc]; refl\n    ... = (x : R) - (x + 1) : by simp\n    ... = 1 - 2 : by simp [mul_add, add_mul, pow_succ, add_comm, mul_assoc,\n              add_assoc, add_left_comm, bit0, bit1]; abel\n    ... = 1 : by rw [h2, sub_zero],\nhave hy1 : y \u2260 1, from mt (congr_arg (coe : units R \u2192 R)) $\n  calc (y : R) = ((x : R)^3 + x^2 + 1) : rfl\n  ... = (x^2 * (x + 1)) + 1 : by simp [mul_add, add_mul, pow_succ, add_comm, mul_assoc,\n              add_assoc, add_left_comm, bit0, bit1]\n  ... \u2260 0 + (1 : units R) : mt add_right_cancel hx0\n  ... = 1 : by simp,\nhave hy3 : order_of y = 3, from order_of_eq_prime (by norm_num) (units.ext this) hy1,\nabsurd (show 3 \u2223 5, by rw [\u2190 h5, \u2190 hy3]; exact order_of_dvd_card_univ) (by norm_num)\nelse\nhave hn1 : (-1 : R) \u2260 1,\n  from \u03bb h, h2 $\n    calc (2 : R) = 1 + 1 : by norm_num\n    ... = 1 + 1 : by norm_num\n    ... = 1 + -1 : by rw h\n    ... = 0 : by norm_num,\nhave hn1 : order_of (-1 : units R) = 2,\n  from order_of_eq_prime (by norm_num)\n    (units.ext $ by norm_num)\n    (mt (congr_arg (coe : units R \u2192 R)) (by convert hn1)),\nabsurd (show 2 \u2223 5, by rw [\u2190 h5, \u2190 hn1]; exact order_of_dvd_card_univ) (by norm_num)\n\n\n#exit\n\n\nimport algebra.big_operators\n\nvariables {\u03b1 : Type*} {\u03b2 : Type*}\n\ndef list.coproduct (s : list \u03b1) (t : list \u03b2) : list (\u03b1 \u2295 \u03b2) :=\ns.map sum.inl ++ t.map sum.inr\n\nlemma list.nodup_coproduct {s : list \u03b1} {t : list \u03b2} :\n  (s.coproduct t).nodup \u2194 s.nodup \u2227 t.nodup :=\nby simp [list.coproduct, list.nodup_append,\n    list.nodup_map_iff (@sum.inl.inj _ _),\n    list.nodup_map_iff (@sum.inr.inj _ _),\n    list.disjoint]\n\nlemma list.sum_coproduct {\u03b3 : Type*} [add_monoid \u03b3] {s : list \u03b1} {t : list \u03b2} (f : \u03b1 \u2295 \u03b2 \u2192 \u03b3) :\n  ((s.coproduct t).map f).sum = (s.map (\u03bb a, f (sum.inl a))).sum + (t.map (\u03bb b, f (sum.inr b))).sum :=\nby simp [list.coproduct]\n\ndef multiset.coproduct (s : multiset \u03b1) (t : multiset \u03b2) : multiset (\u03b1 \u2295 \u03b2) :=\ns.map sum.inl + t.map sum.inr\n\nlemma multiset.nodup_coproduct {s : multiset \u03b1} {t : multiset \u03b2} :\n  (s.coproduct t).nodup \u2194 s.nodup \u2227 t.nodup :=\nquotient.induction_on\u2082 s t (\u03bb _ _, list.nodup_coproduct)\n\nlemma multiset.sum_coproduct {\u03b3 : Type*} [add_comm_monoid \u03b3] {s : multiset \u03b1} {t : multiset \u03b2}\n  (f : \u03b1 \u2295 \u03b2 \u2192 \u03b3) :\n  ((s.coproduct t).map f).sum =\n    (s.map (\u03bb a, f (sum.inl a))).sum + (t.map (\u03bb b, f (sum.inr b))).sum :=\nby simp [multiset.coproduct]\n\ndef finset.coproduct (s : finset \u03b1) (t : finset \u03b2) : finset (\u03b1 \u2295 \u03b2) :=\n\u27e8multiset.coproduct s.1 t.1, multiset.nodup_coproduct.2 \u27e8s.2, t.2\u27e9\u27e9\n\nopen_locale big_operators\n\nlemma finset.sum_coproduct {\u03b3 : Type*} [add_comm_monoid \u03b3]\n  {s : finset \u03b1} {t : finset \u03b2} (f : \u03b1 \u2295 \u03b2 \u2192 \u03b3) :\n  \u2211 x in s.coproduct t, f x = \u2211 a in s, f (sum.inl a) + \u2211 b in t, f (sum.inr b) :=\nmultiset.sum_coproduct _\n\n\n\n#exit\nimport data.quot data.setoid data.fintype.basic\n\ninstance decidable_is_empty' (\u03b1 : Type*) [decidable_eq \u03b1] [fintype \u03b1]\n  (S : set \u03b1) [decidable_pred S] : decidable (S = \u2205) :=\ndecidable_of_iff (\u2200 x : \u03b1, x \u2209 S) (by simp [set.ext_iff])\n\nmeta def quotient_choice {\u03b1 \u03b2 : Type} {s : setoid \u03b2}\n  (f : \u03b1 \u2192 quotient s) : quotient (@pi_setoid _ _ (\u03bb a : \u03b1, s)) :=\nquotient.mk (\u03bb a : \u03b1, quot.unquot (f a))\n\nexample : false :=\nlet x : \u03a0 (quotient_choice : \u03a0 {\u03b1 \u03b2 : Type} [s : setoid \u03b2]\n    (f : \u03b1 \u2192 quotient s), quotient (@pi_setoid _ _ (\u03bb a : \u03b1, s))),\n  decidable false := \u03bb quotient_choice,\n-- \u22a4 is the always true relation\nby letI : setoid bool := \u22a4; exact\nquot.rec_on_subsingleton (@quotient_choice (@quotient bool \u22a4) bool \u22a4 id)\n  (\u03bb f, decidable_of_iff (f \u27e6ff\u27e7 \u2260 f \u27e6tt\u27e7)\n    (iff_false_intro (not_not_intro (congr_arg f (quotient.sound trivial))))) in\n@of_as_true _ (x @quotient_choice) begin\n  change x (@quotient_choice) with is_true _,\n\nend\n\n\n\n#exit\n\naxiom callcc (\u03b1 \u03b2 : Prop) : ((\u03b1 \u2192 \u03b2) \u2192 \u03b1) \u2192 \u03b1\n\nexample {p : Prop} : p \u2228 \u00ac p :=\ncallcc _ false (\u03bb h, or.inr (h \u2218 or.inl))\n\n#exit\nimport tactic\n\n#simp only [int.coe_nat_succ]\n\nvariables {\u03b1 : Type*}\n\ndef le' (\u03c4 \u03c3 : \u03b1 \u2192 \u03b1 \u2192 Prop) := \u2200 a b : \u03b1, \u03c4 a b \u2192 \u03c3 a b\nnotation \u03c4 ` \u2286 ` \u03c3 := le' \u03c4 \u03c3\n\n/- We now define the composition of two binary relations \u03c4 and \u03c3\n(denoted \u03c4 \u2218 \u03c3) as : for all a b, (\u03c4 \u2218 \u03c3) a b if and only if there\nexists c, such that \u03c4 a c \u2227 \u03c3 c b -/\ndef comp (\u03c4 \u03c3 : \u03b1 \u2192 \u03b1 \u2192 Prop) :=\n  \u03bb a b : \u03b1, \u2203 c : \u03b1, \u03c4 a c \u2227 \u03c3 c b\nnotation \u03c4 \u2218 \u03c3 := comp \u03c4 \u03c3\n\n/- Prove that \u2286 is both reflexive and transitive -/\ntheorem le'_refl : @reflexive (\u03b1 \u2192 \u03b1 \u2192 Prop) le' :=\nby dunfold reflexive; tauto\n\ntheorem le'_trans : @transitive (\u03b1 \u2192 \u03b1 \u2192 Prop) le' :=\nby dunfold transitive; tauto\n\n/- Prove that if two binary relations are reflexive, then so are their\ncompositions-/\ntheorem comp_refl {\u03c4 \u03c3 : \u03b1 \u2192 \u03b1 \u2192 Prop}\n  (h\u2080 : reflexive \u03c4) (h\u2081 : reflexive \u03c3) :\n  reflexive (\u03c4 \u2218 \u03c3) :=\nby dunfold comp reflexive; tauto\n\n/- Prove that composition is associative -/\ntheorem comp_assoc : @associative (\u03b1 \u2192 \u03b1 \u2192 Prop) comp :=\nby simp [function.funext_iff, comp, associative]; tauto\n\n/- Prove that a binary relation \u03c4 is transitive if and only if\n(\u03c4 \u2218 \u03c4) \u2286 \u03c4 -/\ntheorem trans_iff_comp_le' {\u03c4 : \u03b1 \u2192 \u03b1 \u2192 Prop} :\n  transitive \u03c4 \u2194 (\u03c4 \u2218 \u03c4) \u2286 \u03c4 :=\n\u27e8by dunfold transitive comp le'; tauto,\n\u03bb h x y z hxy hyz, h _ _ \u27e8y, hxy, hyz\u27e9\u27e9\n\ntheorem sum_xx14n1 : \u2200 n : \u2115,\n  6 * (range (n + 1)).sum (\u03bb n : \u2115, n * (2 * n - 1)) = n * (n + 1) * (4 * n - 1)\n| 0     := rfl\n| 1     := rfl\n| (n+2) :=\nhave h1 : 0 < 4 * (n + 2),\n  from mul_pos (by norm_num) (nat.succ_pos _),\nhave h2 : 0 < 2 * (n + 2),\n  from mul_pos (by norm_num) (nat.succ_pos _),\nhave h3 : 0 < 4 * (n + 1),\n  from mul_pos (by norm_num) (nat.succ_pos _),\nbegin\n  rw [sum_range_succ, mul_add, sum_xx14n1],\n  refine int.coe_nat_inj _,\n  push_cast,\n  rw [int.coe_nat_sub h1, int.coe_nat_sub h2, int.coe_nat_sub h3],\n  push_cast,\n  ring\nend\n\n#exit\nimport data.zmod.basic\n\nexample : 1 = 1 := rfl\n\n#eval let n := 14 in ((finset.range n).filter $ \u03bb r, 3 \u2223 r \u2228 5 \u2223 r).sum (\u03bb n, n)\n\ndef solution' : fin 15 \u2192 \u2115\n| \u27e80, _\u27e9 := 0\n| \u27e81, _\u27e9 := 0\n| \u27e82, _\u27e9 := 0\n| \u27e83, _\u27e9 := 0\n| \u27e84, _\u27e9 := 3\n| \u27e85, _\u27e9 := 3\n| \u27e86, _\u27e9 := 8\n| \u27e87, _\u27e9 := 14\n| \u27e88, _\u27e9 := 14\n| \u27e89, _\u27e9 := 14\n| \u27e810, _\u27e9 := 23\n| \u27e811, _\u27e9 := 33\n| \u27e812, _\u27e9 := 33\n| \u27e813, _\u27e9 := 45\n| \u27e814, _\u27e9 := 45\n| \u27e8n + 15, h\u27e9 := absurd h dec_trivial\n\n\n\n\ntheorem solution_valid (n : \u2115) : solution n =\n  ((finset.range n).filter $ \u03bb r, 3 \u2223 r \u2228 5 \u2223 r).sum (\u03bb n, n) := rfl\n\n#exit\n\n#eval (\u2200 a b : zmod 74, a^2 - 37 * b^2 \u2260 3 : bool)\n\ntheorem no_solns : \u00ac \u2203 (a b : \u2124), a^2 - 37 * b^2 = 3 :=\n\n\nlemma p11 : nat.prime 11 := by norm_num\nlemma p71 : nat.prime 71 := by norm_num\n\nopen_locale classical\n\ndef totient' (n : \u2115) : \u2115 := ((nat.factors n).map (\u03bb n, n-1)).prod\n\ndef ptotient (n : \u2115+) : \u2115+ := \u27e8totient' n, sorry\u27e9\n\ndef mult_5 (n : \u2115+) : \u2115 := (multiplicity 5 n.1).get sorry\n\ndef compute_mod_p (p : \u2115+) :=\n--if p = 101 then 0 else\nlet p\u2081  := p - 1 in\nlet m\u2081  := mult_5 p\u2081 in\nlet m\u20815 := 5^m\u2081 in\nlet p\u2081m\u2081 : \u2115+ := \u27e8p\u2081 / m\u20815, sorry\u27e9 in\nlet p\u2082  := ptotient p\u2081m\u2081 in\nlet i := (5 : zmod p) ^\n  ((5^(6 - 1 - m\u2081 : zmod p\u2082).1 : zmod p\u2081m\u2081).1 * m\u20815) in\n(i^4 + i^3 + i^2 + i + 1).1\n #eval let x := 5^5^3 in let a := x^4 + x^3 + x^2 + x + 1 in a\n#eval 5^5^3\n#eval (62500 : \u211a) / 5^6\n#eval ptotient 1122853751\n#eval mult_5 1122853750\n\n#eval let x := 2^3^4 in let a := x^2 + x + 1 in\n(multiplicity 3 (nat.min_fac a - 1)).get sorry\n\n\n\n-- #eval nat.find (show \u2203 n, let p : \u2115+ := \u27e810 * n + 1, nat.succ_pos _\u27e9 in nat.prime p \u2227\n--   compute_mod_p p = 0, from sorry)\n\n\nmeta def akkgnd : \u2115+ \u2192 tactic unit :=\n\u03bb n, if nat.prime n \u2227 compute_mod_p n = 0\nthen tactic.trace \"Answer is \" >> tactic.trace n.1\nelse akkgnd (n - 10)\n\n\n\n\n--#eval (list.range 500).filter (\u03bb n, nat.prime n \u2227 n % 5 = 1)\n--#eval (list.range 1).map (\u03bb n : \u2115, (compute_mod_p \u27e810 * n + 1, sorry\u27e9).map fin.val fin.val)\n--#eval (nat.prime 1111 : bool)\n\nlemma pow_eq_mod_card {\u03b1 : Type*} [group \u03b1] [fintype \u03b1] (a : \u03b1) (n : \u2115) :\n  a ^ n = a ^ (n % fintype.card \u03b1) :=\ncalc a ^ n = a ^ (n % order_of a) : pow_eq_mod_order_of\n... = a ^ (n % fintype.card \u03b1 % order_of a) :\n  congr_arg2 _ rfl (nat.mod_mod_of_dvd _ order_of_dvd_card_univ).symm\n... = a ^ (n % fintype.card \u03b1) : eq.symm pow_eq_mod_order_of\n#eval (5 ^ 105 : zmod 131)\n#eval (52 ^ 5 : zmod 131)\n#eval (finset.range 200).filter (\u03bb n : \u2115, n.prime \u2227 n % 5 = 1 )\n#eval nat.totient 26\n-- #eval 5 ^ 26 % 31\n-- #eval 25 ^ 5 % 31\n\nlemma fivefiveeqone (n : \u2115) (hn : 0 < n): (5 ^ 5 ^ n : zmod 11) = -1 :=\nhave (5 : zmodp 11 p11) ^ 5 = 1, from rfl,\nsuffices units.mk0 (5 : zmodp 11 p11) dec_trivial ^ 5 ^ n = 1,\n  by rw [units.ext_iff] at this; simpa,\nhave h1 : (5 ^ n : zmod 10) = 5,\n  begin\n    cases n with n,\n    { simp [*, lt_irrefl] at * },\n    clear hn,\n    induction n,\n    { refl },\n    { rw [pow_succ, n_ih],\n      refl }\n  end,\nhave h2 : 5 ^ n % 10 = 5,\n  from calc 5 ^ n % 10 = 5 % 10 :\n    (zmod.eq_iff_modeq_nat' dec_trivial).1 (by simpa)\n  ... = 5 : rfl,\nhave h3 : fintype.card (units (zmodp 11 p11)) = 10,\n  by rw zmodp.card_units_zmodp; refl,\nby rw [pow_eq_mod_card, h3, h2]; refl\n\n@[simp] lemma coe_unit_of_coprime {n : \u2115+} (x : \u2115) (hxn : x.coprime n) :\n  (zmod.unit_of_coprime x hxn : zmod n) = x := rfl\n\n\nlemma pow_eq_pow_mod_totient {n : \u2115+} {x p : \u2115} (hxn : nat.coprime x n)\n  (t : \u2115+) (ht : nat.totient n = t) : (x : zmod n) ^ p = x ^ (p : zmod t).1 :=\nsuffices zmod.unit_of_coprime x hxn ^ p =\n    zmod.unit_of_coprime x hxn ^ (p : zmod \u27e8nat.totient n, nat.totient_pos n.2\u27e9).1,\n  begin\n    cases t with t ht, simp at ht,subst t,\n    rwa [units.ext_iff, units.coe_pow, coe_unit_of_coprime, units.coe_pow, coe_unit_of_coprime] at this,\n  end,\nbegin\n  rw [pow_eq_mod_card],\n  refine congr_arg2 _ rfl _,\n  rw [zmod.val_cast_nat, zmod.card_units_eq_totient],\n  refl\nend\n#eval 5^70 % 71\n\n\nlemma poly_div : \u2200 x : \u2115, 1 < x \u2192 (x^5 - 1) / (x - 1) =\n    x^4 + x^3 + x^2 + x + 1 :=\n\u03bb x hx, have 1 \u2264 x ^ 5, from nat.pow_pos (by linarith) _,\nnat.div_eq_of_eq_mul_left\n  (nat.sub_pos_of_lt hx)\n    (by { rw [nat.mul_sub_left_distrib, mul_one],\n      symmetry,\n      apply nat.sub_eq_of_eq_add,\n      rw [\u2190 nat.add_sub_assoc this],\n      symmetry,\n      apply nat.sub_eq_of_eq_add,\n      ring })\n\nlemma fivefivefiveeq : \u2203 n : \u2115, ((5^5^5^5^5-1)/(5^5^(5^5^5-1)-1)) =\n  (5 ^ 5 ^ n)^4 + (5 ^ 5 ^ n)^3 + (5 ^ 5 ^ n)^2 + (5 ^ 5 ^ n) + 1 :=\nhave hpos : 1 < 5^5^(5^5^5-1),\n  from calc 1 = 1 ^ 5^(5^5^5-1) : by simp\n  ... < 5^5^(5^5^5-1) : nat.pow_left_strict_mono\n    (nat.pow_pos (by norm_num) _) (by norm_num),\n\u27e8(5^5^5-1), begin\n  rw [\u2190 poly_div (5 ^ 5 ^(5^5^5-1)) hpos, \u2190 nat.pow_mul,\n    \u2190 nat.pow_succ, \u2190 nat.succ_sub, nat.succ_sub_one],\n  exact nat.pow_pos (by norm_num) _\nend\u27e9\n\ntheorem fivefives :\n  \u00ac nat.prime ((5^5^5^5^5-1)/(5^5^(5^5^5-1)-1)) :=\nbegin\n  cases fivefivefiveeq with n hn,\n  rw hn, clear hn,\n\n\nend\n#exit\nimport tactic\nclass incidence (point line : Type) (incident_with : point \u2192 line \u2192 Prop) :=\n  (I\u2081 : \u2200 P Q, P \u2260 Q \u2192 \u2203! l, incident_with P l \u2227 incident_with Q l)\n  (I\u2082 : \u2200 l, \u2203 P Q, P \u2260 Q \u2227 incident_with P l \u2227 incident_with Q l)\n  (I\u2083 : \u2203 P Q R, P \u2260 Q \u2227 Q \u2260 R \u2227 P \u2260 R \u2227\n    \u2200 l, \u00ac(incident_with P l \u2227 incident_with Q l \u2227 incident_with R l))\n\ntheorem thm_3p6p8 (point line : Type) (incident_with : point \u2192 line \u2192 Prop)\n  [incidence point line incident_with] (P Q : point) (hPQ : P \u2260 Q) :\n  \u2203 R, \u2200 l, \u00ac(incident_with P l \u2227 incident_with Q l \u2227 incident_with R l) :=\nbegin\n  rcases @incidence.I\u2083 _ _ incident_with _ with \u27e8A, B, C, hAB, hBC, hAC, h\u27e9,\n  rcases @incidence.I\u2081 _ _ incident_with _ _ _ hPQ with \u27e8l, hPQl, lunique\u27e9,\n  have : \u00ac incident_with A l \u2228 \u00ac incident_with B l \u2228 \u00ac incident_with C l,\n  { finish using h l },\n  rcases this with hA | hB | hC,\n  { use A, finish },\n  { use B, finish },\n  { use C, finish }\nend\n#print thm_3p6p8\n#exit\nimport category_theory.limits.shapes.pullbacks\n\nopen category_theory\nopen category_theory.limits\n\nuniverses v u\n\nvariables {C : Type u} [\ud835\udc9e : category.{v} C]\ninclude \ud835\udc9e\n#print cancel_epi\n#print walking_span\n\ndef pushout_of_epi {X Y : C} (f : X \u27f6 Y) [epi f] :\n  is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f) :=\npushout_cocone.is_colimit.mk _\n  pushout_cocone.inl\n  (by intro; erw [category.id_comp])\n  (begin\n    intro s,\n    rw cocone.w s,\n    rw \u2190 cancel_epi f,\n    obviously,\n\n  end)\n  (begin simp, end)\n\n\ntheorem epi_of_pushout {X Y : C} (f : X \u27f6 Y)\n  (is_colim : is_colimit (pushout_cocone.mk (\ud835\udfd9 Y) (\ud835\udfd9 Y) rfl : pushout_cocone f f)) : epi f := sorry\n\n#exit\nimport group_theory.sylow\n\nopen finset mul_action\n\nopen_locale classical\n#print equiv_of_unique_of_unique\ntheorem has_fixed_point {G : Type} [group G] [fintype G] (hG65 : fintype.card G = 65)\n  {M : Type} [fintype M] (hM27 : fintype.card M = 27) [mul_action G M] :\n  \u2203 m : M, \u2200 g : G, g \u2022 m = m :=\nhave horbit : \u2200 m : M, fintype.card (orbit G m) \u2223 65,\n  begin\n    assume m,\n    rw [fintype.card_congr (orbit_equiv_quotient_stabilizer G m), \u2190 hG65],\n    exact card_quotient_dvd_card _,\n  end,\nhave hdvd65 : \u2200 n, n \u2223 65 \u2194 n \u2208 ({1, 5, 13, 65} : finset \u2115) :=\n  \u03bb n, \u27e8\u03bb h, have n \u2264 65 := nat.le_of_dvd (by norm_num) h,\n    by revert h; revert n; exact dec_trivial,\n  by revert n; exact dec_trivial\u27e9,\nbegin\n  letI := orbit_rel G M,\n  have horbit_card : \u2200 m : quotient (orbit_rel G M),\n    fintype.card {x // \u27e6x\u27e7 = m} \u2223 65,\n  { assume m,\n    refine quotient.induction_on m (\u03bb m, _),\n    convert horbit m,\n    exact set.ext (\u03bb _, quotient.eq) },\n  have := fintype.card_congr\n    (equiv.sigma_preimage_equiv (quotient.mk : M \u2192 quotient (orbit_rel G M))),\n  rw [fintype.card_sigma] at this,\n\n\n\nend,\n-- begin\n--   rcases @incidence.I\u2083 _ _ incident_with _ with \u27e8A, B, C, hAB, hBC, hAC, hABC\u27e9,\n\n--   have : P \u2260 B \u2227 P \u2260 C \u2228 P \u2260 A \u2227 P \u2260 C \u2228 P \u2260 A \u2227 P \u2260 B, { finish },\n--   wlog hP : P \u2260 B \u2227 P \u2260 C := this using [B C, A C, A B],\n--   {  }\n\n-- end\n\n\nend\n", "meta": {"author": "ChrisHughes24", "repo": "leanstuff", "sha": "9efa85f72efaccd1d540385952a6acc18fce8687", "save_path": "github-repos/lean/ChrisHughes24-leanstuff", "path": "github-repos/lean/ChrisHughes24-leanstuff/leanstuff-9efa85f72efaccd1d540385952a6acc18fce8687/scratch.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573377, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.35117089337386537}}
{"text": "import algebraic_geometry.AffineScheme\nimport topology.sheaves.stalks\n\nimport about_local_rings\nimport target_affine_scheme\nimport random_lemmas\n\nnoncomputable theory\n\n/-\n\n# 01J5 Points of Scheme\n\n-/\n\nuniverses u\n\nnamespace algebraic_geometry\n\nopen Scheme Top.presheaf opposite topological_space\nopen category_theory category_theory.concrete_category\nopen algebraic_geometry\n\nvariables (X : Scheme.{u}) (R : Type u) [comm_ring R] [local_ring R]\nvariable (f : Spec_obj (CommRing.of R) \u27f6 X)\n\ninstance : local_ring (CommRing.of R) := \nshow local_ring R, from infer_instance\n\nstructure point_local_ring_hom_pair :=\n(pt : X.carrier)\n(ring_hom_ : X.presheaf.stalk pt \u2192+* R)\n[is_local_ring_hom : is_local_ring_hom ring_hom_]\n\nstructure point_local_ring_hom_pair'_aux :=\n(pt : X.carrier)\n(stalk_ : Type u)\n[comm_ring_stalk : comm_ring stalk_]\n(stalk_iso : stalk_ \u2243+* X.presheaf.stalk pt)\n(ring_hom_ : stalk_ \u2192+* R)\n[is_local_ring_hom : is_local_ring_hom ring_hom_]\n\nattribute [instance] point_local_ring_hom_pair.is_local_ring_hom\nattribute [instance] point_local_ring_hom_pair'_aux.comm_ring_stalk\nattribute [instance] point_local_ring_hom_pair'_aux.is_local_ring_hom\n\nnamespace point_local_ring_hom_pair'_aux\n\nvariables [is_affine X] (P : point_local_ring_hom_pair'_aux X R)\n\ninstance stalk_algebra : algebra (\u0393.obj $ op X) P.stalk_ :=\nring_hom.to_algebra $ P.stalk_iso.symm.to_ring_hom.comp $ \n  (stalk_iso_of_affine' X P.pt).symm.to_ring_hom.comp $ \n  @@algebra_map (\u0393.obj $ op X) (localization.at_prime _) _ _ \n  begin \n    dsimp,\n    refine localization.algebra,\n  end\n\ninstance stalk_is_localization :\n  @@is_localization _ (X.iso_Spec.hom.1.base P.pt).as_ideal.prime_compl P.stalk_ \n    _ (by { exactI point_local_ring_hom_pair'_aux.stalk_algebra X R P }) :=\n{ map_units := \u03bb y, \n  begin \n    dsimp,\n    rw [ring_hom.algebra_map_to_algebra, ring_hom.comp_apply, \n      ring_hom.comp_apply, \u2190localization.mk_algebra_map], \n    erw [algebra.algebra_map_self],\n    rw [ring_hom.id_apply],\n    refine is_unit.map _ (is_unit.map _ _),\n    rw localization.at_prime.mk_is_unit_iff,\n    exact y.2,\n  end,\n  surj := \u03bb z, \n  begin \n    let z' := P.stalk_iso z,\n    have eq0 : P.stalk_iso.symm.to_ring_hom z' = z,\n    { erw [ring_equiv.symm_apply_apply] },\n    simp_rw [ring_hom.algebra_map_to_algebra, \u2190eq0, ring_hom.comp_apply, \n      \u2190map_mul],\n    sorry\n    -- have := localization.is_localization.surj,\n  end,\n  eq_iff_exists := sorry }\n\nend point_local_ring_hom_pair'_aux\n\n@[ext] lemma point_local_ring_hom_pair_ext (P Q : point_local_ring_hom_pair X R)\n  (hpt : P.pt = Q.pt)\n  (hhom : P.ring_hom_.comp \n  (X.presheaf.stalk_specializes $ by { rw hpt, }) = Q.ring_hom_) :\n  P = Q :=\nbegin \n  rcases P with \u27e8P, fP, hfP\u27e9,\n  rcases Q with \u27e8Q, fQ, hfQ\u27e9,\n  dsimp at hpt,\n  subst hpt,\n  dsimp at hhom,\n  simp_rw \u2190hhom,\n  refine \u27e8rfl, heq_of_eq _\u27e9,\n  convert_to fP = fP.comp (ring_hom.id _),\n  swap,\n  { rw ring_hom.comp_id, },\n  congr' 1,\n  dunfold stalk_specializes,\n  apply limits.colimit.hom_ext,\n  intros j,\n  rw limits.colimit.\u03b9_desc,\n  dsimp only,\n  ext x : 1,\n  rw [comp_apply, ring_hom.id_apply],\n  congr,\n  change _ = op (unop j),\n  rw op_inj_iff,\n  ext : 1,\n  refl,\nend\n\nnamespace point_local_ring_hom_pair'_aux\n\nvariables {X R} {p q r : point_local_ring_hom_pair'_aux X R}\n\n@[simps] def stalk_equiv_of_pt_eq (pt_eq : p.pt = q.pt) : p.stalk_ \u2243+* q.stalk_ :=\np.stalk_iso.trans $ ring_equiv.trans \n(CommRing.from_iso \n{ hom := X.presheaf.stalk_specializes (by rw [pt_eq]),\n  inv := X.presheaf.stalk_specializes (by rw [pt_eq]),\n  hom_inv_id' := \n  begin \n    apply stalk_hom_ext,\n    intros U hxU,\n    erw [\u2190category.assoc, germ_stalk_specializes, germ_stalk_specializes,\n      category.comp_id],\n    refl,\n  end,\n  inv_hom_id' := \n  begin \n    apply stalk_hom_ext,\n    intros U hxU,\n    erw [\u2190category.assoc, germ_stalk_specializes, germ_stalk_specializes,\n      category.comp_id],\n    refl,\n  end }) q.stalk_iso.symm\n\nlemma stalk_equiv_of_pt_eq.rfl_apply (x) :\n  stalk_equiv_of_pt_eq (rfl : p.pt = p.pt) x = x :=\nbegin \n  simp only [CommRing.from_iso_apply, iso.refl_hom, id_apply, \n    stalk_equiv_of_pt_eq_apply],\n  obtain \u27e8U, hU, s, eq0\u27e9 := X.presheaf.germ_exist p.pt (p.stalk_iso x),\n  rw [\u2190eq0, germ_stalk_specializes'_apply],\n  apply_fun p.stalk_iso,\n  rw [ring_equiv.apply_symm_apply, \u2190eq0],\n  refl,\n\n  exact equiv_like.injective p.stalk_iso,\nend\n\nlemma stalk_equiv_of_pt_eq.symm (pt_eq : p.pt = q.pt) :\n  stalk_equiv_of_pt_eq pt_eq.symm = (stalk_equiv_of_pt_eq pt_eq).symm :=\nrfl\n\nlemma stalk_equiv_of_pt_eq.trans (h1 : p.pt = q.pt) (h2 : q.pt = r.pt) :\n  stalk_equiv_of_pt_eq (h1.trans h2) = \n  (stalk_equiv_of_pt_eq h1).trans (stalk_equiv_of_pt_eq h2) :=\nbegin \n  ext,\n  delta stalk_equiv_of_pt_eq,\n  simp only [ring_equiv.coe_trans, function.comp_app, CommRing.from_iso_apply, \n    ring_equiv.apply_symm_apply, embedding_like.apply_eq_iff_eq],\n  obtain \u27e8U, hU, s, eq0\u27e9 := X.presheaf.germ_exist p.pt (p.stalk_iso x),\n  rw [\u2190eq0, germ_stalk_specializes'_apply, germ_stalk_specializes'_apply,\n    germ_stalk_specializes'_apply],\nend\n\nvariables (p q r)\n\nstructure rel_aux :=\n(pt_eq : p.pt = q.pt)\n(ring_hom_eq : p.ring_hom_.comp (stalk_equiv_of_pt_eq pt_eq).symm.to_ring_hom \n  = q.ring_hom_)\n\n@[simps] def rel_aux_rfl : rel_aux p p :=\n{ pt_eq := rfl,\n  ring_hom_eq := ring_hom.ext $ \u03bb x, \n  begin \n    rw [ring_hom.comp_apply],\n    erw stalk_equiv_of_pt_eq.rfl_apply x,\n  end }\n\n@[simps] def rel_aux_symm (P : rel_aux p q) : rel_aux q p :=\n{ pt_eq := P.pt_eq.symm,\n  ring_hom_eq := by rw [ring_hom.comp_equiv_to_ring_hom_eq_iff, \u2190P.ring_hom_eq, \n      \u2190stalk_equiv_of_pt_eq.symm] }\n\n@[simps] def rel_aux_trans (a : rel_aux p q) (b : rel_aux q r) :\n  rel_aux p r :=\n{ pt_eq := a.pt_eq.trans b.pt_eq,\n  ring_hom_eq := by rw [\u2190b.ring_hom_eq, \u2190a.ring_hom_eq, ring_hom.comp_assoc, \n      \u2190ring_equiv.to_ring_hom_trans, \u2190ring_equiv.symm_trans,\n      stalk_equiv_of_pt_eq.trans] }\n\ndef rel : Prop := nonempty $ rel_aux p q\n\n@[refl] lemma rel_refl : rel p p := nonempty.intro $ rel_aux_rfl p\n@[symm] lemma rel_symm (h : rel p q) : rel q p := \nnonempty.intro $ rel_aux_symm _ _ h.some\n@[trans] lemma rel_trans (h : rel p q) (h' : rel q r) : rel p r :=\nnonempty.intro $ rel_aux_trans _ _ _ h.some h'.some\n\nvariables (X R)\n\ndef setoid_ : setoid (point_local_ring_hom_pair'_aux X R) :=\n{ r := rel,\n  iseqv := \u27e8rel_refl, rel_symm, rel_trans\u27e9 }\n\nend point_local_ring_hom_pair'_aux\n\ndef point_local_ring_hom_pair' : Type (u+1) :=\nquotient (point_local_ring_hom_pair'_aux.setoid_ X R)\n\nnamespace point_local_ring_hom_pair'\n\nvariables {X R} (p q r : point_local_ring_hom_pair' X R)\n\ndef pt : X.carrier := p.out'.pt\n\ndef stalk_ : Type u := p.out'.stalk_\n\ninstance comm_ring_stalk : comm_ring p.stalk_ := p.out'.comm_ring_stalk\n\ndef stalk_iso : p.stalk_ \u2243+* X.presheaf.stalk p.pt :=\np.out'.stalk_iso\n\ndef ring_hom_ : p.stalk_ \u2192+* R :=\np.out'.ring_hom_\n\ninstance is_local_ring_hom : is_local_ring_hom p.ring_hom_ :=\np.out'.is_local_ring_hom\n\nlemma mk_pt_eq (x : point_local_ring_hom_pair'_aux X R) : \n  pt (quotient.mk' x) = x.pt :=\nbegin \n  obtain \u27e8\u27e8pt_eq, _\u27e9\u27e9 := @quotient.mk_out' _ (point_local_ring_hom_pair'_aux.setoid_ X R) x,\n  exact pt_eq,\nend\n\n@[simps] def mk_stalk_iso (x : point_local_ring_hom_pair'_aux X R) :\n  stalk_ (quotient.mk' x) \u2243+* x.stalk_ :=\npoint_local_ring_hom_pair'_aux.stalk_equiv_of_pt_eq $ mk_pt_eq x\n\nlemma mk_stalk_iso.setoid_r {x y : point_local_ring_hom_pair'_aux X R}\n  (pt_eq : x.pt = y.pt) :\n    (mk_stalk_iso x).trans \n      (point_local_ring_hom_pair'_aux.stalk_equiv_of_pt_eq pt_eq) \n  = (point_local_ring_hom_pair'_aux.stalk_equiv_of_pt_eq $ \n      show pt (quotient.mk' x) = pt (quotient.mk' y), \n      by rw [mk_pt_eq, mk_pt_eq, pt_eq]).trans\n    (mk_stalk_iso y) :=\nbegin \n  ext z : 1,\n  dsimp,\n  rw [x.stalk_iso.apply_symm_apply, ring_equiv.apply_symm_apply],\n  generalize_proofs h1 h2 h3 h4 h5,\n  obtain \u27e8U, hU, s, eq0\u27e9 := germ_exist _ _ (((quotient.mk' x).out'.stalk_iso) z),\n  rw [\u2190eq0, germ_stalk_specializes'_apply, germ_stalk_specializes'_apply,\n    germ_stalk_specializes'_apply, germ_stalk_specializes'_apply],\nend\n\nlemma mk_ring_hom_ (x : point_local_ring_hom_pair'_aux X R) :\n  ring_hom_ (quotient.mk' x) = \n  x.ring_hom_.comp (mk_stalk_iso x).to_ring_hom :=\nbegin\n  obtain \u27e8\u27e8pt_eq, ring_hom_eq\u27e9\u27e9 := @quotient.mk_out' _ (point_local_ring_hom_pair'_aux.setoid_ X R) x,\n  rw [\u2190ring_hom_eq, ring_hom.comp_assoc],\n  symmetry,\n  convert ring_hom.comp_id _,\n  delta mk_stalk_iso,\n  ext : 1,\n  erw [ring_equiv.symm_to_ring_hom_apply_to_ring_hom_apply, ring_hom.id_apply],\nend\n\n@[simps] \ndef stalk_iso_of_pt_eq (pt_eq : p.pt = q.pt) :\n  p.stalk_ \u2243+* q.stalk_ :=\np.stalk_iso.trans $ \nring_equiv.trans (CommRing.from_iso \n{ hom := X.presheaf.stalk_specializes $ by rw pt_eq,\n  inv := X.presheaf.stalk_specializes $ by rw pt_eq,\n  hom_inv_id' := \n  begin \n    apply stalk_hom_ext,\n    intros U hU,\n    erw [\u2190category.assoc, germ_stalk_specializes, germ_stalk_specializes, \n      category.comp_id],\n    refl,\n  end,\n  inv_hom_id' :=\n  begin \n    apply stalk_hom_ext,\n    intros U hU,\n    erw [\u2190category.assoc, germ_stalk_specializes, germ_stalk_specializes, \n      category.comp_id],\n    refl,\n  end }) q.stalk_iso.symm\n\n@[ext] lemma ext (pt_eq : p.pt = q.pt) \n  (ring_hom_eq : \n      p.ring_hom_.comp (p.stalk_iso_of_pt_eq _ pt_eq).symm.to_ring_hom \n    = q.ring_hom_) : p = q :=\nbegin \n  induction p using quotient.induction_on',\n  induction q using quotient.induction_on',\n  rw quotient.eq',\n  have pt_eq' : p.pt = q.pt,\n  { rwa [mk_pt_eq, mk_pt_eq] at pt_eq, },\n  rw [mk_ring_hom_, mk_ring_hom_] at ring_hom_eq,\n  rw [ring_hom.comp_equiv_to_ring_hom_eq_iff, ring_hom.comp_assoc] at \n    ring_hom_eq,\n  replace ring_hom_eq := ring_hom_eq.symm,\n  rw [\u2190ring_hom.comp_equiv_to_ring_hom_eq_iff] at ring_hom_eq,\n  refine \u27e8\u27e8pt_eq', _\u27e9\u27e9,\n  rw [\u2190ring_hom_eq, ring_hom.comp_assoc, ring_hom.comp_assoc],\n  convert ring_hom.comp_id _,\n  have := congr_arg (\u03bb (r : _ \u2243+* _), r.to_ring_hom) \n    (mk_stalk_iso.setoid_r pt_eq'),\n  dsimp at this,\n  erw \u2190this,\n  rw [ring_hom.comp_assoc, \u2190ring_hom.comp_assoc _ _ (mk_stalk_iso p).to_ring_hom],\n  rw show (mk_stalk_iso p).to_ring_hom.comp (mk_stalk_iso p).symm.to_ring_hom \n    = ring_hom.id _, from ring_equiv.to_ring_hom_comp_symm_to_ring_hom _,\n  rw [ring_hom.id_comp],\n  rw ring_equiv.to_ring_hom_comp_symm_to_ring_hom,\nend\n\nend point_local_ring_hom_pair'\n\nsection\n\nnamespace Spec_local_ring_to_Scheme_equiv_point_local_ring_hom_pair_auxs\n\nsection affine_cases\n\nvariables [is_affine X]\n\n@[simps] def AffineScheme_stalk (x : X.carrier) : \n  X.presheaf.stalk x \u2245 \n  CommRing.of (localization.at_prime (X.iso_Spec.hom.1.base x).as_ideal) :=\n{ hom := eq_to_hom (by rw [\u2190Scheme.comp_val_base_apply, iso.hom_inv_id, \n      Scheme.id_val_base, id_apply]) \u226b \n    PresheafedSpace.stalk_map X.iso_Spec.inv.1 _ \u226b \n    (structure_sheaf.stalk_iso _ _).hom,\n  inv := (structure_sheaf.stalk_iso _ _).inv \u226b \n    PresheafedSpace.stalk_map X.iso_Spec.hom.1 _,\n  hom_inv_id' := \n  begin\n    rw [category.assoc, category.assoc, iso.hom_inv_id_assoc],\n    rw [\u2190PresheafedSpace.stalk_map.comp], \n    change eq_to_hom _ \u226b \n      PresheafedSpace.stalk_map ((X.iso_Spec.hom \u226b X.iso_Spec.inv).val) x = _,\n    erw \u2190PresheafedSpace.stalk_map.congr_hom (\ud835\udfd9 _) \n      (X.iso_Spec.hom \u226b X.iso_Spec.inv).val _ x,\n    rw PresheafedSpace.stalk_map.id,\n    rw iso.hom_inv_id,\n    refl,  \n  end,\n  inv_hom_id' := \n  begin \n    generalize_proofs h1 h2 h3,\n    rw [category.assoc, \u2190category.assoc _ (eq_to_hom _), \n      PresheafedSpace.stalk_map.congr_point, category.assoc, \n      \u2190category.assoc (PresheafedSpace.stalk_map _ _), \n      \u2190PresheafedSpace.stalk_map.comp, \u2190category.assoc (eq_to_hom _), \n      PresheafedSpace.stalk_map.congr_hom \n        (X.iso_Spec.inv.val \u226b X.iso_Spec.hom.val) (\ud835\udfd9 _),\n      PresheafedSpace.stalk_map.id, eq_to_hom_trans_assoc, eq_to_hom_refl, \n      category.id_comp], \n    erw [category.id_comp],\n    rw iso.inv_hom_id, \n    { change (X.iso_Spec.inv \u226b X.iso_Spec.hom).val = \ud835\udfd9 _,\n      rw iso.inv_hom_id,\n      refl, },\n    { rw [\u2190Scheme.comp_val_base_apply, iso.hom_inv_id, Scheme.id_val_base, \n        id_apply], },\n  end }\n\n@[simps] def point_local_ring_hom_pair_to (P : point_local_ring_hom_pair X R) :\n  local_ring.point_local_ring_hom_pair (\u0393.obj $ op X) R :=\n{ pt := X.iso_Spec.hom.1.base P.pt,\n  ring_hom_ := P.ring_hom_.comp $ \n    (PresheafedSpace.stalk_map X.iso_Spec.hom.1 _).comp $ \n      structure_sheaf.localization_to_stalk _ _,\n  is_local_ring_hom_ := infer_instance }\n\n@[simps] def from_point_local_ring_hom_pair \n  (P : local_ring.point_local_ring_hom_pair (\u0393.obj $ op X) R) :\n  point_local_ring_hom_pair X R :=\n{ pt := X.iso_Spec.inv.1.base P.pt,\n  ring_hom_ := P.ring_hom_.comp $ \n    (structure_sheaf.stalk_to_fiber_ring_hom _ _).comp $ \n    PresheafedSpace.stalk_map X.iso_Spec.inv.1 _,\n  is_local_ring_hom := infer_instance }\n\nlemma CommRing_comp_eq_comp {R S T : Type u} \n  [comm_ring R] [comm_ring S] [comm_ring T] (f : R \u2192+* S) (g : S \u2192+* T) :\n  g.comp f = (show CommRing.of R \u27f6 CommRing.of S, from f) \u226b \n    (show CommRing.of S \u27f6 CommRing.of T, from g) := \nrfl\n\nlemma localization.congr_point {R : Type u} [comm_ring R]\n  (p q : prime_spectrum R) (h : p = q) (x) :\n  (eq_to_hom (by rw h) : CommRing.of (localization.at_prime p.as_ideal) \u27f6 \n    CommRing.of (localization.at_prime q.as_ideal)) x = \n  (x.lift_on (\u03bb a b, localization.mk a (\u27e8b, by { convert b.2, rw h }\u27e9 : \n      q.as_ideal.prime_compl)) $ \u03bb a c b d H, \n    begin \n      rw localization.r_iff_exists at H,\n      obtain \u27e8e, he\u27e9 := H,\n      dsimp at he \u22a2,\n      rw [localization.mk_eq_mk_iff, localization.r_iff_exists],\n      refine \u27e8\u27e8e, by { convert e.2, rw h }\u27e9, _\u27e9,\n      exact he,\n    end : localization.at_prime q.as_ideal) := \nbegin \n  subst h,\n  rw eq_to_hom_refl,\n  rw id_apply,\n  induction x using localization.induction_on with data,\n  rcases data with \u27e8a, b\u27e9,\n  dsimp,\n  rw [localization.lift_on_mk],\n  congr' 1,\n  ext,\n  refl,\nend\n\nlemma strucutre_sheaf.localization_to_stalk.congr_point \n  (R : Type u) [comm_ring R] (x y : prime_spectrum R) (h : x = y) :\n  structure_sheaf.localization_to_stalk R x =\n  eq_to_hom (by rw h) \u226b (structure_sheaf.localization_to_stalk R y) \u226b \n    eq_to_hom (by rw h) := \nbegin \n  subst h,\n  rw [eq_to_hom_refl, eq_to_hom_refl, category.id_comp, category.comp_id],\nend\n\nlemma to_CommRing_of_CommRing (R S : CommRing.{u}) (f : R \u27f6 CommRing.of S) :\n  f = show R \u27f6 S, from f := rfl\n\nlemma CommRing_of_CommRing_eq (R : CommRing.{u}) :\n  R = CommRing.of R :=\nbegin\n  obtain \u27e8R, str\u27e9 := R,\n  dsimp at *, resetI,\n  congr,\nend\n\nlemma point_local_ring_hom_pair_to_from (P) :\n  point_local_ring_hom_pair_to _ _ (from_point_local_ring_hom_pair X R P) = P :=\nlocal_ring.point_local_ring_hom_pair_ext _ _ _ _ \n(by { dsimp, rw [\u2190Scheme.comp_val_base_apply, iso.inv_hom_id, \n  Scheme.id_val_base, id_apply] }) begin \n  obtain \u27e8pt, f, h\u27e9 := P,\n  dsimp only [point_local_ring_hom_pair_to_ring_hom_, \n    from_point_local_ring_hom_pair_ring_hom_, CommRing_comp_eq_comp, \n    from_point_local_ring_hom_pair_pt],\n  simp only [category.assoc],\n  slice_lhs 3 4 \n  { erw \u2190PresheafedSpace.stalk_map.comp X.iso_Spec.inv.1 X.iso_Spec.hom.1 pt },\n  rw [PresheafedSpace.stalk_map.congr_hom \n    (X.iso_Spec.inv.val \u226b X.iso_Spec.hom.val) (\ud835\udfd9 _) _ pt, \n    PresheafedSpace.stalk_map.id], \n  swap,\n  { erw [\u2190Scheme.comp_val], \n    rw [iso.inv_hom_id],\n    refl, },\n  erw [category.comp_id],\n  have pt_eq : (X.iso_Spec.hom.val.base) ((X.iso_Spec.inv.val.base) pt) = pt,\n  { rw [\u2190Scheme.comp_val_base_apply, iso.inv_hom_id, Scheme.id_val_base, \n      id_apply] },\n  \n  slice_lhs 2 3 { },\n  rw strucutre_sheaf.localization_to_stalk.congr_point _ _ pt,\n  swap, { exact pt_eq },\n  rw [category.assoc, category.assoc], \n  erw [category.assoc, category.assoc],\n  slice_lhs 4 5 {  },\n  erw eq_to_hom_trans, \n  swap,\n  { rw pt_eq,\n    exact CommRing_of_CommRing_eq _, },\n  swap,\n  { rw pt_eq,\n    refl, },\n  generalize_proofs h1 h2 h3 h4 h5 h6,\n  rw show eq_to_hom h6 = \n    \ud835\udfd9 ((Spec.structure_sheaf (Scheme.\u0393.obj (op X))).presheaf.stalk pt), from _,\n  swap,\n  { convert eq_to_hom_refl _ _, refl, },\n  slice_lhs 3 4 { },\n  erw show structure_sheaf.localization_to_stalk (\u0393.obj (op X)) pt \u226b\n    \ud835\udfd9 ((Spec.structure_sheaf (\u0393.obj (op X))).presheaf.stalk pt) = \n  structure_sheaf.localization_to_stalk (\u0393.obj (op X)) pt, from _,\n  swap,\n  { convert category.comp_id _, },\n  erw (structure_sheaf.stalk_iso _ _).inv_hom_id,\n  rw [category.id_comp, \u2190category.assoc],\n  convert category.id_comp _,\n  { refl, },\n  rw [comp_eq_to_hom_iff, category.id_comp],\n  apply localization.local_ring_hom_unique,\n  intros x,\n  rw [localization.congr_point, \u2190localization.mk_algebra_map, \n    algebra.algebra_map_self, localization.lift_on_mk, ring_hom.id_apply, \n    \u2190localization.mk_algebra_map, algebra.algebra_map_self, ring_hom.id_apply],\n  refl,\n\n  exact pt_eq.symm,\nend\n\nlemma stalk_specializes.congr_point (A : PresheafedSpace CommRing) \n  (x y : A) (h : x = y) :\n  A.presheaf.stalk_specializes (by rw h : x \u2933 y) = eq_to_hom (by rw h) :=\nbegin \n subst h,\n apply stalk_hom_ext,\n intros U h,\n erw [eq_to_hom_refl, category.comp_id, germ_stalk_specializes],\n refl,\nend\n\nlemma point_local_ring_hom_pair_from_to (P) :\n  from_point_local_ring_hom_pair X R (point_local_ring_hom_pair_to _ _ P) = P :=\nhave pt_eq : (X.iso_Spec.inv.val.base) ((X.iso_Spec.hom.val.base) P.pt) = P.pt,\nby rw [\u2190Scheme.comp_val_base_apply, iso.hom_inv_id, Scheme.id_val_base, \n  id_apply],\npoint_local_ring_hom_pair_ext _ _ _ _ pt_eq\nbegin \n  obtain \u27e8pt, f, h\u27e9 := P,\n  dsimp only [point_local_ring_hom_pair_to_ring_hom_, \n    from_point_local_ring_hom_pair_ring_hom_, CommRing_comp_eq_comp,\n    from_point_local_ring_hom_pair_pt],\n  simp only [category.assoc],\n  erw (structure_sheaf.stalk_iso _ _).hom_inv_id_assoc,\n  slice_lhs 2 3 {},\n  erw \u2190PresheafedSpace.stalk_map.comp X.iso_Spec.hom.1 X.iso_Spec.inv.1 pt,\n  rw PresheafedSpace.stalk_map.congr_hom \n    (X.iso_Spec.hom.val \u226b X.iso_Spec.inv.val) (\ud835\udfd9 _) _ pt,\n  swap,\n  { erw [\u2190Scheme.comp_val],\n    rw [iso.hom_inv_id],\n    refl, },\n  rw [PresheafedSpace.stalk_map.id], \n  erw [category.comp_id],\n  rw [\u2190category.assoc],\n  convert category.id_comp _,\n  refl,\n  generalize_proofs h1 h2 h3,\n  rw [show X.presheaf.stalk_specializes h2 = eq_to_hom h3.symm, from _],\n  erw [eq_to_hom_trans, eq_to_hom_refl],\n  { rw [point_local_ring_hom_pair_to_pt, pt_eq], },\n  { rw [point_local_ring_hom_pair_to_pt, pt_eq], },\n  rw \u2190stalk_specializes.congr_point,\n  refl,\n  erw [id_apply, comp_apply, pt_eq],\nend\n\n@[simps] def point_local_ring_hom_pair_equiv :\n  point_local_ring_hom_pair X R \u2243\n  local_ring.point_local_ring_hom_pair (\u0393.obj $ op X) R :=\n{ to_fun := point_local_ring_hom_pair_to _ _,\n  inv_fun := from_point_local_ring_hom_pair _ _,\n  left_inv := point_local_ring_hom_pair_from_to _ _,\n  right_inv := point_local_ring_hom_pair_to_from _ _ }\n\ninstance : local_ring (\u0393.obj $ op $ Spec_obj $ CommRing.of R) :=\nlocal_ring.of_equiv _ R $ ring_equiv.symm\n{ to_fun := (structure_sheaf.global_sections_iso R).hom,\n  inv_fun := (structure_sheaf.global_sections_iso R).inv,\n  left_inv := \u03bb x, by rw [iso.hom_inv_id_apply],\n  right_inv := \u03bb x, by rw [iso.inv_hom_id_apply],\n  map_mul' := map_mul _,\n  map_add' := map_add _ }\n\ndef Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair :\n  ((Spec_obj $ CommRing.of R) \u27f6 X) \u2243 point_local_ring_hom_pair X R :=\n(Scheme.hom.target_AffineScheme _ _).trans $ equiv.trans \n  (equiv.trans \n    ({ to_fun := \u03bb a, (structure_sheaf.global_sections_iso _).inv.comp a,\n      inv_fun := \u03bb a, (structure_sheaf.global_sections_iso _).hom.comp a,\n      left_inv := \u03bb a, ring_hom.ext $ \u03bb z, \n        by { dsimp only, rw [ring_hom.comp_apply, ring_hom.comp_apply], \n          erw (structure_sheaf.global_sections_iso _).inv_hom_id_apply,  },\n      right_inv := \u03bb a, ring_hom.ext $ \u03bb z, \n        by { dsimp only, rw [ring_hom.comp_apply, ring_hom.comp_apply],\n          erw (structure_sheaf.global_sections_iso _).hom_inv_id_apply, } } : \n    (\u0393.obj (op X) \u27f6 \u0393.obj (op $ Spec_obj $ CommRing.of R)) \u2243 \n    ((\u0393.obj $ op X) \u2192+* R)) $ ring_hom.target_local_ring_equiv _ _)\n  (point_local_ring_hom_pair_equiv _ _).symm\n\ninstance is_global_section_algebra (pt : prime_spectrum $ \u0393.obj $ op X) : \n  algebra (\u0393.obj $ op X) (X.presheaf.stalk (X.iso_Spec.inv.1.base pt)) :=\nring_hom.to_algebra $ ((stalk_iso_of_affine X pt).symm.to_ring_hom).comp $\n  algebra_map _ _\n\ninstance is_localization_stalk (pt : prime_spectrum $ \u0393.obj $ op X) :\n  is_localization pt.as_ideal.prime_compl\n    (X.presheaf.stalk $ X.iso_Spec.inv.val.base pt) :=\n{ map_units := \u03bb x, begin \n    rw ring_hom.algebra_map_to_algebra,\n    rw [ring_hom.comp_apply, \u2190localization.mk_algebra_map, algebra.algebra_map_self,\n      ring_hom.id_apply],\n    refine is_unit.map _ _,\n    rw localization.at_prime.mk_is_unit_iff,\n    exact x.2,\n  end,\n  surj := \u03bb z, \n  begin \n    simp_rw [ring_hom.algebra_map_to_algebra, ring_hom.comp_apply],\n    let z' := stalk_iso_of_affine X pt z,\n    obtain \u27e8\u27e8a, b\u27e9, EQ\u27e9:= localization.is_localization.surj z',\n    rw [\u2190localization.mk_algebra_map, algebra.algebra_map_self, ring_hom.id_apply,\n      \u2190localization.mk_algebra_map] at EQ,\n    have eq0 : z = (stalk_iso_of_affine X pt).symm.to_ring_hom z',\n    { erw ring_equiv.apply_symm_apply, },\n    simp_rw [eq0, \u2190map_mul, \u2190localization.mk_algebra_map, algebra.algebra_map_self,\n      ring_hom.id_apply],\n    refine \u27e8\u27e8a, b\u27e9, _\u27e9,\n    congr' 1,\n  end,\n  eq_iff_exists := \u03bb x y, \n  begin \n    rw [ring_hom.algebra_map_to_algebra, ring_hom.comp_apply,\n      ring_hom.comp_apply, function.injective.eq_iff _,\n      localization.is_localization.eq_iff_exists],\n    rw function.injective_iff_has_left_inverse,\n    refine \u27e8(stalk_iso_of_affine X pt).to_ring_hom, \u03bb z, _\u27e9,\n    rw [\u2190ring_hom.comp_apply, ring_equiv.to_ring_hom_comp_symm_to_ring_hom,\n      ring_hom.id_apply],\n  end }\n\ndef stalk_iso_is_localization_of_affine' \n  (pt : prime_spectrum $ \u0393.obj $ op X) \n  (M : Type u) [comm_ring M]  [algebra (\u0393.obj $ op X) M]\n  [by exactI is_localization.at_prime M pt.as_ideal] :\n  X.presheaf.stalk (X.iso_Spec.inv.1.base pt) \u2243+* M :=\nby exactI (is_localization.alg_equiv pt.as_ideal.prime_compl \n    (X.presheaf.stalk (X.iso_Spec.inv.1.base pt)) M).to_ring_equiv\n\nnamespace Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair'\n\nexample (a : local_ring.point_local_ring_hom_pair' (\u0393.obj $ op X) R) :\n  point_local_ring_hom_pair' X R :=\nquotient.map' (\u03bb (P : local_ring.point_local_ring_hom_pair'_aux (\u0393.obj $ op X) R), \n{ pt := X.iso_Spec.inv.1.base P.pt,\n  stalk_ := P.localized_ring,\n  comm_ring_stalk := infer_instance,\n  stalk_iso := (stalk_iso_is_localization_of_affine' X _ _).symm,\n  ring_hom_ := P.ring_hom_,\n  is_local_ring_hom := infer_instance }) \nbegin \n  rintros P Q \u27e8\u27e8pt_eq, h\u27e9\u27e9,\n  refine \u27e8\u27e8_, _\u27e9\u27e9,\n  { dsimp, rw pt_eq, },\n  { dsimp, convert h, \n    ext z : 1, \n    rw point_local_ring_hom_pair'_aux.stalk_equiv_of_pt_eq_apply,\n    rw local_ring.point_local_ring_hom_pair'_aux.localized_ring_equiv_of_pt_eq_apply,\n    sorry },\nend a\n\nexample : local_ring.point_local_ring_hom_pair' (\u0393.obj $ op X) R \n  \u2243 point_local_ring_hom_pair' X R :=\n{ to_fun := \u03bb a, sorry,\n  inv_fun := sorry,\n  left_inv := sorry,\n  right_inv := sorry }\n\nend Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair'\n\ndef Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair' :\n  ((Spec_obj $ CommRing.of R) \u27f6 X) \u2243 point_local_ring_hom_pair' X R :=\n(Scheme.hom.target_AffineScheme _ _).trans $ \n  equiv.trans \n    ({ to_fun := \u03bb f, f \u226b (structure_sheaf.global_sections_iso _).inv,\n      inv_fun := \u03bb f, (structure_sheaf.global_sections_iso _).hom.comp f,\n      left_inv := \u03bb f, \n      begin \n        simp_rw [CommRing_comp_eq_comp], \n        erw [category.assoc, iso.inv_hom_id, category.comp_id],\n      end,\n      right_inv := \u03bb f,\n      begin \n        simp_rw [CommRing_comp_eq_comp],\n        erw [category.assoc, iso.hom_inv_id, category.comp_id],\n      end } : (\u0393.obj (op X) \u27f6 \u0393.obj (op (Spec_obj (CommRing.of R)))) \u2243\n      ((\u0393.obj $ op X) \u2192+* R)) $ \n    equiv.trans\n      (ring_hom.target_local_ring_equiv' R (\u0393.obj $ op X)) $\n      sorry\n\nnamespace Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair\n\nlemma apply_pt (\u03b1 : (Spec_obj $ CommRing.of R) \u27f6 X) :\n  (Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair X R \u03b1).pt = \n  X.iso_Spec.inv.1.base \n    \u27e8ideal.comap ((structure_sheaf.global_sections_iso R).inv.comp $ \n      Scheme.hom.target_AffineScheme (Spec_obj $ CommRing.of R) X \u03b1) $ \n      local_ring.maximal_ideal R, infer_instance\u27e9 :=\nbegin \n  dsimp only [Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair],\n  rw [equiv.trans_apply, equiv.trans_apply, \n    point_local_ring_hom_pair_equiv_symm_apply, \n    from_point_local_ring_hom_pair_pt, equiv.trans_apply, equiv.coe_fn_mk],\n  congr' 1,\nend\n\nlemma apply_ring_hom__apply (\u03b1 : (Spec_obj $ CommRing.of R) \u27f6 X) :\n  (Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair \n    X R \u03b1).ring_hom_ = \n  ((((inv (structure_sheaf.to_open R \u22a4) : _ \u2192+* _).comp $\n      Scheme.hom.target_AffineScheme (Spec_obj $ CommRing.of R) X \n        \u03b1).factor_through_target_local_ring).comp $\n     (structure_sheaf.stalk_iso (\u0393.obj $ op X)  \n        \u27e8(local_ring.maximal_ideal R).comap _, _\u27e9).hom).comp \n  (PresheafedSpace.stalk_map X.iso_Spec.inv.1 _) :=\nbegin \n  dsimp only [Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair,\n    equiv.trans_apply, point_local_ring_hom_pair_equiv_symm_apply,\n    from_point_local_ring_hom_pair_ring_hom_, structure_sheaf.stalk_iso_hom,\n    ring_hom.target_local_ring_equiv, equiv.coe_fn_mk],\n  simp only [CommRing_comp_eq_comp, category.assoc],\n  congr' 1,\nend\n\nlemma symm_apply (P : point_local_ring_hom_pair X R) :\n  (Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair X R).symm P = \n  (Scheme.hom.target_AffineScheme (Spec_obj $ CommRing.of R) X).symm\n    ((structure_sheaf.to_open \u21a5(CommRing.of R) \u22a4 : _ \u2192+* _).comp $\n      (P.ring_hom_.comp $\n        (PresheafedSpace.stalk_map X.iso_Spec.hom.1 P.pt : _ \u2192+* _).comp\n          (structure_sheaf.localization_to_stalk (\u0393.obj $ op X) $ \n            X.iso_Spec.hom.val.base P.pt)).comp $\n        @@algebra_map (Scheme.\u0393.obj (op X)) \n          (localization.at_prime $ (X.iso_Spec.hom.1.base P.pt).as_ideal) _ _ $\n            by {dsimp, exactI localization.algebra}) := \nrfl\n  \nend Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair\n\nend affine_cases\n\nsection nonaffine_cases\n\ninstance spec_is_affine (S : Type u) [comm_ring S] : \n  is_affine $ Spec_obj (CommRing.of S) :=\nalgebraic_geometry.Spec_is_affine (op _)\n\ninstance spec_is_affine' (S : CommRing) : is_affine $ Spec_obj S :=\nalgebraic_geometry.Spec_is_affine (op _)\n\nvariables {X R} \n\nsection basic_defs\nvariable (P : point_local_ring_hom_pair X R)\nvariables {U : opens X.carrier} (hU : is_affine_open U) (mem_U : P.pt \u2208 U)\n\ndef _root_.algebraic_geometry.is_affine_open.iso_Spec :\n  X.restrict U.open_embedding \u2245 \n  Spec_obj (X.presheaf.obj $ op U) :=\n@Scheme.iso_Spec _ hU \u226a\u226b eq_to_iso\nbegin \n  dsimp, congr', ext y, split,\n  { rintros \u27e8y, _, rfl\u27e9, exact y.2 },\n  { intros hy, exact \u27e8\u27e8y, hy\u27e9, \u27e8\u27e9, rfl\u27e9, }\nend\n\ndef _root_.algebraic_geometry.is_affine_open.pt_in_restricted_global_sections \n  {x : X.carrier} (hx : x \u2208 U) : (Spec_obj $ X.presheaf.obj $ op U).carrier :=\nhU.iso_Spec.hom.1.base \u27e8x, hx\u27e9\n\ndef stalk_on_open_equiv (x : X.carrier) (hx : x \u2208 U) :\n  X.presheaf.stalk x \u2245 (X.restrict U.open_embedding).presheaf.stalk \u27e8x, hx\u27e9 :=\niso.symm $ PresheafedSpace.restrict_stalk_iso X.to_PresheafedSpace _ _\n\ndef _root_.algebraic_geometry.is_affine_open.point_local_ring_hom_pair :\n  point_local_ring_hom_pair (Spec_obj $ X.presheaf.obj $ op U) R := \n{ pt := hU.iso_Spec.hom.1 \u27e8P.pt, mem_U\u27e9,\n  ring_hom_ := P.ring_hom_.comp $ \n    (X.to_PresheafedSpace.restrict_stalk_iso U.open_embedding \n      \u27e8P.pt, mem_U\u27e9).hom.comp $ PresheafedSpace.stalk_map hU.iso_Spec.hom.1 _,\n  is_local_ring_hom := infer_instance }\n\ndef _root_.algebraic_geometry.is_affine_open.Spec_local_ring_to_Scheme : \n  Spec_obj (CommRing.of R) \u27f6 X :=\n(Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair _ R).symm \n  (hU.point_local_ring_hom_pair P mem_U) \u226b hU.from_Spec\n\nend basic_defs\n\nsection independence\n\nvariable (P : point_local_ring_hom_pair X R)\nvariables {U : opens X.carrier} (hU : is_affine_open U) (mem_U : P.pt \u2208 U)\nvariables {V : opens X.carrier} (hV : is_affine_open V) (mem_V : P.pt \u2208 V)\n\nnamespace Spec_local_ring_to_Scheme_wd_proofs\n\ndef \u03c8_ : \n  (X.restrict U.open_embedding).presheaf.stalk \u27e8P.pt, mem_U\u27e9 \u2192+* R := \nP.ring_hom_.comp \n  (PresheafedSpace.restrict_stalk_iso _ U.open_embedding _).hom\n\ninstance is_local_ring_hom_\u03c8_ : is_local_ring_hom (\u03c8_ P mem_U) :=\nby { delta \u03c8_, apply_instance }\n\ndef \u03c8'_ :\n  localization.at_prime \n    (((@Scheme.iso_Spec _ hU).hom.1.base \u27e8P.pt, mem_U\u27e9).as_ideal : \n      ideal $ \u0393.obj $ op $ X.restrict U.open_embedding) \u2192+* R :=\n(\u03c8_ P mem_U).comp $\n  ((PresheafedSpace.stalk_map (@Scheme.iso_Spec _ hU).hom.1) _).comp\n    (structure_sheaf.stalk_iso _ _).inv\n\ndef \u03c8''_ : (\u0393.obj $ op $ X.restrict U.open_embedding) \u2192+* R :=\n(\u03c8'_ P hU mem_U).comp $ @algebra_map _ _ _ _ $\n  by { dsimp, exactI localization.algebra }\n\ndef res'_aux (subset_rel : U \u2286 V) : \n  U.open_embedding.is_open_map.functor.obj \u22a4 \u27f6 \n  V.open_embedding.is_open_map.functor.obj \u22a4 :=\nhom_of_le \nbegin\n  convert subset_rel;\n  { ext p, split, \n    { rintros \u27e8p, _, rfl\u27e9, exact p.2 },\n    { intro h, refine \u27e8\u27e8p, h\u27e9, \u27e8\u27e9, rfl\u27e9, } }\nend\n\ndef res'_ (subset_rel : U \u2286 V) :\n  localization.at_prime \n    (((@Scheme.iso_Spec _ hV).hom.1.base \u27e8P.pt, mem_V\u27e9).as_ideal : \n      ideal $ \u0393.obj $ op $ X.restrict V.open_embedding) \u2192+*\n  localization.at_prime \n    (((@Scheme.iso_Spec _ hU).hom.1.base \u27e8P.pt, mem_U\u27e9).as_ideal : \n      ideal $ \u0393.obj $ op $ X.restrict U.open_embedding) :=\nlocalization.local_ring_hom _ _ \n  (X.presheaf.map $ (res'_aux subset_rel).op) -- \u0393.map (quiver.hom.op (X.restrict_functor.map $ hom_of_le subset_rel).left) \nbegin\n  haveI : is_affine (X.restrict V.open_embedding) := hV,\n  haveI : is_affine (X.restrict U.open_embedding) := hU,\n  rw [\u2190prime_spectrum.comap_as_ideal],\n  refine congr_arg _ _,\n  -- ext1 (x : X.presheaf.obj _),\n  -- rw ideal.mem_comap,\n  -- have := Scheme.mem_basic_open,\n  -- have := (X.restrict V.open_embedding).iso_Spec,\n  -- rw Scheme.mem_iso_Spec_inv_apply,\n  -- erw Scheme.mem_iso_Spec_inv_apply,\n  -- refine iff.not _,\n  -- rw [\u2190basic_open_eq_of_affine],\n  -- conv_rhs { rw [\u2190basic_open_eq_of_affine] },\n  -- rw [Scheme.mem_basic_open],\n  \n  dsimp only [Scheme.iso_Spec],\n  simp only [\u0393_Spec.adjunction_unit_app, as_iso_hom],\n  dsimp only [\u0393_Spec.LocallyRingedSpace_adjunction, identity_to_\u0393_Spec,\n    adjunction.mk_of_unit_counit_unit, LocallyRingedSpace.to_\u0393_Spec,\n    LocallyRingedSpace.to_\u0393_Spec_SheafedSpace, continuous_map.coe_mk,\n    LocallyRingedSpace.to_\u0393_Spec_base, LocallyRingedSpace.to_\u0393_Spec_fun,\n    local_ring.closed_point],\n  rw [\u2190prime_spectrum.comap_comp_apply],\n  ext : 1,\n  change prime_spectrum.as_ideal _ = prime_spectrum.as_ideal _,\n  ext x : 1,\n  rw [prime_spectrum.comap_as_ideal, prime_spectrum.comap_as_ideal, \n    ideal.mem_comap, ideal.mem_comap, local_ring.mem_maximal_ideal, \n    local_ring.mem_maximal_ideal, mem_nonunits_iff, mem_nonunits_iff],\n  refine iff.not _,\n  split,\n  { introsI H,\n    let iV := X.restrict_stalk_iso V.open_embedding \u27e8P.pt, mem_V\u27e9,\n    let iU := X.restrict_stalk_iso U.open_embedding \u27e8P.pt, mem_U\u27e9,\n    refine \u27e8\u27e8iU.inv $ iV.hom H.unit.1, iU.inv $ iV.hom $ H.unit\u207b\u00b9.1, _, _\u27e9, _\u27e9,\n    { rw [\u2190map_mul, \u2190map_mul, units.val_eq_coe, units.val_eq_coe, \n        is_unit.mul_coe_inv, map_one, map_one], },\n    { erw [\u2190map_mul, \u2190map_mul, units.val_eq_coe, units.val_eq_coe,\n        is_unit.coe_inv_mul, map_one, map_one], },\n    { rw [units.coe_mk],\n      sorry }, },\n  sorry,\nend\n\nlemma triangle_commutes (subset_rel : U \u2286 V) :\n  (\u03c8'_ P hU mem_U).comp (res'_ P hU mem_U hV mem_V subset_rel) = \n  \u03c8'_ P hV mem_V :=\nsorry\n\n@[simps] def point_local_ring_hom_pair_ : \n  point_local_ring_hom_pair (X.restrict U.open_embedding) R :=\n{ pt := \u27e8P.pt, mem_U\u27e9,\n  ring_hom_ := \u03c8_ _ _,\n  is_local_ring_hom := infer_instance }\n\ndef \u03a8'_ : Spec_obj (CommRing.of R) \u27f6 X.restrict U.open_embedding :=\n(@@Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair \n    (X.restrict U.open_embedding) R _ _ hU).symm \n  (point_local_ring_hom_pair_ P mem_U)\n\nlemma \u03a8'_eq_of_subset_rel (subset_rel : U \u2286 V) :\n  \u03a8'_ P hU mem_U \u226b \n    ((X.restrict_functor).map (hom_of_le subset_rel)).left = \n  \u03a8'_ P hV mem_V :=\nbegin\n  haveI : is_affine (X.restrict_functor.obj V).left := hV,\n  haveI : is_affine (X.restrict V.open_embedding) := hV,\n  apply_fun (Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair _ R),\n  dsimp only [\u03a8'_],\n  generalize_proofs h1 h2 h3 h4,\n  dsimp only [Scheme.restrict_functor_obj_left, \n    Scheme.restrict_functor_map_left],\n  work_on_goal 2 { apply_instance },\n  rw (Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair \n    (X.restrict h2) R).apply_symm_apply (point_local_ring_hom_pair_ P mem_V),\n  sorry\nend\n\ndef \u03a8_ : Spec_obj (CommRing.of R) \u27f6 X :=\n  \u03a8'_ P hU mem_U \u226b X.of_restrict _\n\nlemma _root_.algebraic_geometry.is_affine_open.Spec_local_ring_to_Scheme_eq :\n  hU.Spec_local_ring_to_Scheme P mem_U = \u03a8_ P hU mem_U := sorry\n\nlemma _root_.algebraic_geometry.is_affine_open.Spec_local_ring_to_Scheme_wd_of_subset_rel\n  (subset_rel : U \u2286 V) : \n  hU.Spec_local_ring_to_Scheme P mem_U = hV.Spec_local_ring_to_Scheme P mem_V :=\nbegin \n  rw [hU.Spec_local_ring_to_Scheme_eq P mem_U, \n    hV.Spec_local_ring_to_Scheme_eq P mem_V, \u03a8_, \u03a8_, \n    \u2190\u03a8'_eq_of_subset_rel P hU mem_U hV mem_V subset_rel, category.assoc],\n  congr' 1,\n  dsimp,\n  rw is_open_immersion.lift_fac,\nend\n\n\nend Spec_local_ring_to_Scheme_wd_proofs\n\n-- this is probably going to be long\nlemma _root_.algebraic_geometry.is_affine_open.Spec_local_ring_to_Scheme_wd : \n  hU.Spec_local_ring_to_Scheme P mem_U = hV.Spec_local_ring_to_Scheme P mem_V :=\nsorry\n\nsection\n\nvariables (X)\n\ndef _root_.algebraic_geometry.Scheme.open_set_of (x : X.carrier) : \n  opens X.carrier := (X.local_affine x).some.1\n\nlemma _root_.algebraic_geometry.Scheme.mem_open_set_of (x : X.carrier) :\n  x \u2208 X.open_set_of x :=\n(X.local_affine x).some.2\n\ndef _root_.algebraic_geometry.Scheme.CommRing_of (x : X.carrier) : \n  CommRing.{u} := (X.local_affine x).some_spec.some\n\ndef _root_.algebraic_geometry.Scheme.iso_Spec_of (x : X.carrier) :\n  X.restrict (X.open_set_of x).open_embedding \u2245\n  Spec_obj (X.CommRing_of x) :=\nlet \u03b1 : X.to_LocallyRingedSpace.restrict (X.open_set_of x).open_embedding \u2245 \n    Spec.to_LocallyRingedSpace.obj (op $ X.CommRing_of x) :=\n  (X.local_affine x).some_spec.some_spec.some in\n{ hom := \u03b1.hom,\n  inv := \u03b1.inv,\n  hom_inv_id' := \u03b1.hom_inv_id,\n  inv_hom_id' := \u03b1.inv_hom_id }\n\nlemma _root_.algebraic_geometry.Scheme.is_affine_open_set_of (x : X.carrier) :\n  is_affine_open $ X.open_set_of x := \nis_affine_of_iso (X.iso_Spec_of x).hom\n\nend\n\ndef Spec_local_ring_to_Scheme : Spec_obj (CommRing.of R) \u27f6 X :=\n(X.is_affine_open_set_of P.pt).Spec_local_ring_to_Scheme P $ \n  X.mem_open_set_of P.pt\n\nend independence\n\nsection basic_defs\n\nvariables (\u03b1 : Spec_obj (CommRing.of R) \u27f6 X)\nvariables {V : opens X.carrier} (hV : is_affine_open V) \nvariables \n  (image_mem : \u03b1.1.base \u27e8local_ring.maximal_ideal _, infer_instance\u27e9 \u2208 V)\n\nsection\n\nvariables (X)\n\nlemma _root_.algebraic_geometry.Scheme.range_of_restrict :\n  set.range (X.of_restrict V.open_embedding).1.base = (V : set X.carrier) :=\nset.ext_iff.mpr $ \u03bb x,\n{ mp := by { rintros \u27e8x, rfl\u27e9, exact x.2, },\n  mpr := by { intros h, refine \u27e8\u27e8x, h\u27e9, rfl\u27e9 } }\n\nend\n\nsection\n\ninclude image_mem\n\nlemma image_subset_of_image_mem :\n  set.range \u03b1.1.base \u2286 set.range (X.of_restrict V.open_embedding).1.base :=\nbegin \n  rw X.range_of_restrict, rintros _ \u27e8x, rfl\u27e9,\n  refine specializes.mem_open (specializes.map _ (by continuity)) V.2 image_mem,\n  rw \u2190prime_spectrum.le_iff_specializes,\n  change x.as_ideal \u2264 local_ring.maximal_ideal R,\n  refine local_ring.le_maximal_ideal (ideal.is_prime.ne_top infer_instance),\nend\n\nend\n\ndef Spec_local_ring_to_Scheme_factors_through_of_image_mem :\n  Spec_obj (CommRing.of R) \u27f6 X.restrict V.open_embedding :=\nLocallyRingedSpace.is_open_immersion.lift (X.of_restrict V.open_embedding) \u03b1 $\nimage_subset_of_image_mem _ image_mem\n\ndef to_point_local_ring_hom_pair_of_image_mem_affine_open_aux : \n  point_local_ring_hom_pair (X.restrict $ V.open_embedding) R :=\n(@@Spec_local_ring_to_AffineScheme_equiv_point_local_ring_hom_pair _ _ _ _ hV) $ \n  Spec_local_ring_to_Scheme_factors_through_of_image_mem \u03b1 image_mem\n\ndef to_point_local_ring_hom_pair_of_image_mem_affine_open : \n  point_local_ring_hom_pair X R :=\nlet P := to_point_local_ring_hom_pair_of_image_mem_affine_open_aux \n  \u03b1 hV image_mem in \n{ pt := P.pt.1,\n  ring_hom_ := P.ring_hom_.comp \n    (PresheafedSpace.restrict_stalk_iso _ V.open_embedding P.pt).inv,\n  is_local_ring_hom := infer_instance }\n\nend basic_defs\n\nsection independence\n\nvariables (\u03b1 : Spec_obj (CommRing.of R) \u27f6 X)\nvariables {V\u2081 V\u2082 : opens X.carrier} \nvariables (hV\u2081 : is_affine_open V\u2081) (hV\u2082 : is_affine_open V\u2082)\nvariables \n  (image_mem\u2081 : \u03b1.1.base \u27e8local_ring.maximal_ideal _, infer_instance\u27e9 \u2208 V\u2081)\nvariables \n  (image_mem\u2082 : \u03b1.1.base \u27e8local_ring.maximal_ideal _, infer_instance\u27e9 \u2208 V\u2082)\n\nlemma to_point_local_ring_hom_of_image_mem_affine_open_wd :\n  to_point_local_ring_hom_pair_of_image_mem_affine_open \u03b1 hV\u2081 image_mem\u2081 =\n  to_point_local_ring_hom_pair_of_image_mem_affine_open \u03b1 hV\u2082 image_mem\u2082 :=\nsorry\n\nend independence\n\ndef to_point_local_ring_hom_pair (\u03b1 : Spec_obj (CommRing.of R) \u27f6 X) :\n  point_local_ring_hom_pair X R :=\nto_point_local_ring_hom_pair_of_image_mem_affine_open \u03b1 \n  (X.is_affine_open_set_of \n    (\u03b1.1.base \u27e8local_ring.maximal_ideal _, infer_instance\u27e9)) $ \n  X.mem_open_set_of _\n\nend nonaffine_cases\n\nend Spec_local_ring_to_Scheme_equiv_point_local_ring_hom_pair_auxs\n\nsection\n\nopen Spec_local_ring_to_Scheme_equiv_point_local_ring_hom_pair_auxs\n\n-- 01J6\ndef Spec_local_ring_to_Scheme_equiv_point_local_ring_pair :\n  ((Spec_obj $ CommRing.of R) \u27f6 X) \u2243 point_local_ring_hom_pair X R :=\n{ to_fun := to_point_local_ring_hom_pair,\n  inv_fun := Spec_local_ring_to_Scheme,\n  left_inv := sorry,\n  right_inv := sorry }\n\nend\n\nend\n\nvariables {X R}\n\n-- 02NA\ndef Spec_stalk_to_Scheme (x : X.carrier) :\n  Spec_obj (X.presheaf.stalk x) \u27f6 X :=\n(Spec_local_ring_to_Scheme_equiv_point_local_ring_pair X _).symm \n{ pt := x,\n  ring_hom_ := ring_hom.id _,\n  is_local_ring_hom := infer_instance }\n\nend algebraic_geometry\n", "meta": {"author": "jjaassoonn", "repo": "vc", "sha": "3f1db37cfd16b55cd47e77f0517d192c25edecf0", "save_path": "github-repos/lean/jjaassoonn-vc", "path": "github-repos/lean/jjaassoonn-vc/vc-3f1db37cfd16b55cd47e77f0517d192c25edecf0/src/points_of_scheme_legacy.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7745833945721304, "lm_q2_score": 0.4532618480153861, "lm_q1q2_score": 0.3510891008657948}}
{"text": "import deduction semantics lindenbaum\nopen encodable\n\nnamespace fopl\nvariables {L : language.{0}} \n\nlocal infix ` \u2243\u2081 `:80 := ((\u2243) : term L \u2192 term L \u2192 formula L)\nlocal prefix `\u220f\u2081 `:64 := (has_univ_quantifier.univ : formula L \u2192 formula L)\nlocal prefix `\u2210\u2081 `:64 := (has_exists_quantifier.ex : formula L \u2192 formula L)\n\nvariables [decidable_eq (formula L)]\n\n@[simp] def formula.arrow : formula L \u2192 option (formula L \u00d7 formula L)\n| (p \u27f6 q) := some (p, q)\n| _        := none\n\nlemma arrow_eq {p : formula L} {v} : p.arrow = some v \u2192 p = v.1 \u27f6 v.2 :=\nby { cases p; simp[show \u2200 x y : term L, (x \u2243 y : formula L).arrow = none, from \u03bb _ _, rfl,\n      show \u2200 p : formula L, (\u207bp).arrow = none, from \u03bb _, rfl,\n      show \u2200 p : formula L, (\u220f p : formula L).arrow = none, from \u03bb _, rfl], intros h, simp[\u2190h] }\n\ninductive proof (L : language.{0}) : Type\n| root : formula L \u2192 proof\n| ge : proof \u2192 proof\n| mp : proof \u2192 proof \u2192 proof\n\n@[simp] def proof.conseq : proof L \u2192 option (formula L)\n| (proof.root p) := some p\n| (proof.ge \u03c6)   := \u03c6.conseq.map (\u03bb p, \u220f p)\n| (proof.mp \u03c6 \u03c8) :=\n    if (\u03c6.conseq >>= formula.arrow).map prod.fst = \u03c8.conseq then (\u03c6.conseq >>= formula.arrow).map prod.snd \n    else \u03c8.conseq\n\ninductive formula.is_axiom (T : theory L) (i : \u2115) : formula L \u2192 Prop\n| p1 {p q} : formula.is_axiom (p \u27f6 q \u27f6 p)\n| p2 {p q r} : formula.is_axiom ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r)\n| p3 {p q} : formula.is_axiom ((\u207bp \u27f6 \u207bq) \u27f6 q \u27f6 p)\n| q1 {p t} : formula.is_axiom (\u220f\u2081 p \u27f6 p.rew \u03b9[0 \u21dd t])\n| q2 {p q} : formula.is_axiom (\u220f\u2081 (p \u27f6 q) \u27f6 \u220f\u2081 p \u27f6 \u220f\u2081 q)\n| q3 {p} : formula.is_axiom (p \u27f6 \u220f\u2081 (p^1))\n| e1 : formula.is_axiom \u220f\u2081 #0 \u2243\u2081 #0\n| e2 : formula.is_axiom \u220f\u2081 \u220f\u2081 (#0 \u2243\u2081 #1 \u27f6 #1 \u2243\u2081 #0)\n| e3 : formula.is_axiom \u220f\u2081 \u220f\u2081 \u220f\u2081 (#0 \u2243\u2081 #1 \u27f6 #1 \u2243\u2081 #2 \u27f6 #0 \u2243\u2081 #2)\n| e4 {n} {f : L.fn n} : formula.is_axiom (eq_axiom4 f)\n| e5 {n} {r : L.pr n} : formula.is_axiom (eq_axiom5 r)\n| by_axiom {p} : p \u2208 T^i \u2192 formula.is_axiom p\n\n@[simp] def proof.proper (T : theory L) : \u2115 \u2192 proof L \u2192 Prop\n| i (proof.root p) := p.is_axiom T i\n| i (proof.ge \u03c6)   := \u03c6.proper (i + 1)\n| i (proof.mp \u03c6 \u03c8) := (\u03c6.proper i) \u2227 (\u03c8.proper i) \n\ndef proof.of (T : theory L) (i : \u2115) (p : formula L) (\u03c6 : proof L) : Prop := \u03c6.proper T i \u2227 \u03c6.conseq = some p\n\n\nnamespace proof\nopen nat\nvariables {T : theory L} {i : \u2115}\n\nlemma provable_of_is_axiom {p} (h : is_axiom T i p) : T^i \u22a2 p :=\nbegin\n  cases h; try {simp}, { exact provable.e4 }, { exact provable.e5 },\n  { exact provable.AX (by simp*) }\nend\n\nlemma sound {T : theory L} {i} {p} {\u03c6} : proof.of T i p \u03c6 \u2192 T^i \u22a2 p :=\nbegin\n  induction \u03c6 generalizing p i; simp[proof.of],\n  case root : i p { rintros h rfl, exact provable_of_is_axiom h },\n  case ge : \u03c6 IH p { rintros proper q conseq rfl, exact provable.generalize (IH \u27e8proper, conseq\u27e9) },\n  case mp : \u03c6 \u03c8 IH\u03c6 IH\u03c8\n    { cases \u03c6_conseq : \u03c6.conseq with c\u03c6; cases \u03c8_conseq : \u03c8.conseq with c\u03c8; simp[\u03c6_conseq, \u03c8_conseq], \n      { rintros p\u03c6 p\u03c8 rfl, exact IH\u03c8 \u27e8p\u03c8, \u03c8_conseq\u27e9 },\n      { intros p\u03c6 p\u03c8, simp[show (\u2200 (a b a_1 : formula L), c\u03c6 = a_1 \u2192 \u00aca_1.arrow = some (a, b)) \u2194 c\u03c6.arrow = none,\n          from \u27e8\u03bb h, by { cases C : c\u03c6.arrow with v; simp, exact h v.1 v.2 c\u03c6 rfl (by simp[C]) },\n           by { rintros h a b _ rfl, simp[h] }\u27e9],\n        cases C : c\u03c6.arrow with v; simp[C] },\n      { rintros p\u03c6 p\u03c8, cases C : c\u03c6.arrow with v; simp, { rintros rfl, exact IH\u03c8 \u27e8p\u03c8, \u03c8_conseq\u27e9 },\n        { by_cases C\u2082 : v.1 = c\u03c8,\n          { simp[\u2190C\u2082, show \u2203 a, v = (v.fst, a), from \u27e8v.2, by simp\u27e9], \n            rintros rfl, rcases C\u2082 with rfl, rcases arrow_eq C with rfl, \n            exact classical_logic.modus_ponens (IH\u03c6 \u27e8p\u03c6, \u03c6_conseq\u27e9) (IH\u03c8 \u27e8p\u03c8, \u03c8_conseq\u27e9) },\n          { simp[show \u00ac\u2203 (a : formula L), v = (c\u03c8, a), by { simp, rintros s rfl, simp at C\u2082, contradiction }],\n            rintros rfl, exact IH\u03c8 \u27e8p\u03c8, \u03c8_conseq\u27e9 } } } }\nend\n\nlemma complete {T : theory L} {i} (p : formula L) : T^i \u22a2 p \u2194 \u2203 \u03c6, of T i p \u03c6 :=\n\u27e8\u03bb h,\nbegin\n  apply fopl.provable.rec_on' h,\n  { rintros i p _ \u27e8\u03c6, \u03c6_proper, \u03c6_conseq\u27e9, refine \u27e8\u03c6.ge, _, _\u27e9; simp* },\n  { rintros i p q _ _ \u27e8\u03c6, \u03c6_proper, \u03c6_conseq\u27e9 \u27e8\u03c8, \u03c8_proper, \u03c8_conseq\u27e9,\n    refine \u27e8\u03c6.mp \u03c8, _, _\u27e9; simp[*, (>>=)] },\n  { intros i p _, refine \u27e8root p, _, _\u27e9; simp, exact formula.is_axiom.by_axiom mem },\n  { intros i p q, refine \u27e8root (p \u27f6 q \u27f6 p), _, _\u27e9; simp, exact formula.is_axiom.p1 },\n  { intros i p q r, refine \u27e8root ((p \u27f6 q \u27f6 r) \u27f6 (p \u27f6 q) \u27f6 p \u27f6 r), _, _\u27e9; simp, exact formula.is_axiom.p2 },\n  { intros i p q, refine \u27e8root ((\u207bp \u27f6 \u207bq) \u27f6 q \u27f6 p), _, _\u27e9; simp, exact formula.is_axiom.p3 },\n  { intros i p t, refine \u27e8root (\u220f p \u27f6 formula.rew \u03b9[0 \u21dd t] p), _, _\u27e9; simp, exact formula.is_axiom.q1 },\n  { intros i p q, refine \u27e8root (\u220f (p \u27f6 q) \u27f6 \u220f p \u27f6 \u220f q), _, _\u27e9; simp, exact formula.is_axiom.q2 },\n  { intros i p, refine \u27e8root (p \u27f6 \u220f p ^ 1), _, _\u27e9; simp, exact formula.is_axiom.q3 },\n  { intros i, refine \u27e8root (\u220f\u2081 #0 \u2243\u2081 #0), _, _\u27e9; simp, exact formula.is_axiom.e1 },\n  { intros i, refine \u27e8root (\u220f\u2081 \u220f\u2081 (#0 \u2243\u2081 #1 \u27f6 #1 \u2243\u2081 #0)), _, _\u27e9; simp, exact formula.is_axiom.e2 },\n  { intros i, refine \u27e8root (\u220f\u2081 \u220f\u2081 \u220f\u2081 (#0 \u2243\u2081 #1 \u27f6 #1 \u2243\u2081 #2 \u27f6 #0 \u2243\u2081 #2)), _, _\u27e9; simp, exact formula.is_axiom.e3 },\n  { intros i m f, refine \u27e8root (eq_axiom4 f), _, _\u27e9; simp, exact formula.is_axiom.e4 },\n  { intros i m p, refine \u27e8root (eq_axiom5 p), _, _\u27e9; simp, exact formula.is_axiom.e5 }\nend, \u03bb \u27e8\u03c6, h\u27e9, sound h\u27e9\n\nvariables [primcodable (formula L)]\n\n@[simp] def encode_pcode : proof L \u2192 \u2115\n| (root p) := (bit0 $ encode p) + 1\n| (ge \u03c6)   := (bit1 $ bit0 $ encode_pcode \u03c6) + 1\n| (mp \u03c6 \u03c8) := (bit1 $ bit1 $ nat.mkpair (encode_pcode \u03c6) (encode_pcode \u03c8)) + 1\n\n@[simp] def decode_pcode : \u2115 \u2192 option (proof L)\n| 0       := none\n| (n + 1) :=\n    have div4 : n.div2.div2 \u2264 n :=\n      by { simp[nat.div2_val], exact le_trans (nat.div_le_self (n / 2) 2) (nat.div_le_self n 2) },\n    have n.div2.div2 < n + 1, from nat.lt_succ_iff.mpr div4,\n    have n.div2.div2.unpair.1 < n + 1, \n      from nat.lt_succ_iff.mpr (le_trans (nat.unpair_left_le n.div2.div2) div4),\n    have n.div2.div2.unpair.2 < n + 1, \n      from nat.lt_succ_iff.mpr (le_trans (nat.unpair_right_le n.div2.div2) div4),\n    match n.bodd, n.div2.bodd with\n    | ff, _  := (decode (formula L) n.div2).map root\n    | tt, ff := (decode_pcode n.div2.div2).map proof.ge\n    | tt, tt := proof.mp <$> (decode_pcode n.div2.div2.unpair.1) <*> (decode_pcode n.div2.div2.unpair.2)\n    end\n\ninstance : encodable (proof L) :=\n{ encode := encode_pcode,\n  decode := decode_pcode,\n  encodek := by { intros \u03c6, induction \u03c6; simp[encode_pcode, decode_pcode, *] } }\n\nnoncomputable def of_n (T : theory L) (p : \u2115) (s : \u2115) : \u2115 :=\n  encode\n  ( do p \u2190 decode (formula L) p,\n       \u03c6 \u2190 decode (proof L) s,\n    some (to_bool (proof.of T 0 p \u03c6)))\n\nlemma of_n_complete (p : formula L) :\n  T \u22a2 p \u2194 \u2203 s, of_n T (encode p) s = encode (some tt) :=\nby { have : T \u22a2 p \u2194 \u2203 \u03c6, of T 0 p \u03c6, rw[show T = T^0, by simp], from complete p,\n     simp[this, of_n],\n     split,\n     { rintros \u27e8\u03c6, of\u27e9, refine \u27e8encode \u03c6, _\u27e9, simp[of] },\n     { rintros \u27e8s, of\u27e9, cases C : decode (proof L) s with \u03c6; simp[C] at of,\n       { contradiction },\n       { refine \u27e8\u03c6, _\u27e9, by_cases C\u2082 : fopl.proof.of T 0 p \u03c6; simp[C\u2082] at of \u22a2, { contradiction } } } }\n\n\n\nend proof\n\nvariables [primcodable (formula L)] [primcodable (proof L)]\n\nclass primrec_theory (T : theory L) :=\n(prim : primrec\u2082 (\u03bb p \u03c6, proof.of_n T p \u03c6))\n\nvariables {T : theory L} [primrec_theory T]\n\n#eval encode (some tt)\n\n\n\n\n\nend fopl", "meta": {"author": "iehality", "repo": "lean-logic", "sha": "201cef2500203f7de83deb7fa8287934e2e142b2", "save_path": "github-repos/lean/iehality-lean-logic", "path": "github-repos/lean/iehality-lean-logic/lean-logic-201cef2500203f7de83deb7fa8287934e2e142b2/src/FOL/proof.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646140788307, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.35107412563468}}
{"text": "import algebra.group.defs\nimport algebra.module.basic\n\nimport expr\n\nnamespace quantitative_types\nsection\n\nopen mult\nopen ctx\n\ndef mult.add : mult \u2192 mult \u2192 mult\n| error _     := error\n| _     error := error\n| none  \u03c0     := \u03c0\n| \u03c0     none  := \u03c0\n| _     all   := error\n| all   _     := error\n| _     init  := error\n| free  _     := error\n| init  free  := all\n| init  _     := init\n| _     free  := free\n| read  read  := read\n| read  write := write\n| write read  := write\n| write write := write\n\n/-\ndef mult.mul : mult \u2192 mult \u2192 mult\n| 0 _ := 0\n| 1 a := a\n| \u03c9 0 := 0\n| \u03c9 _ := \u03c9\n-/\n\ninstance : has_add mult := \u27e8mult.add\u27e9\n-- instance : has_mul mult := \u27e8mult.mul\u27e9\n\n@[simp] lemma mult.add_zero (a : mult) : a + 0 = a := by { cases a; refl }\n@[simp] lemma mult.zero_add (a : mult) : 0 + a = a := by { cases a; refl }\nlemma mult.add_assoc (a b c : mult) : (a + b) + c = a + (b + c) := by { cases a; cases b; cases c; refl }\n-- lemma mult.add_comm (a b : mult) : a + b = b + a := by { cases a; cases b; refl }\n\n/-\n@[simp] lemma mult.mul_zero (a : mult) : a * 0 = 0 := by { cases a; refl }\n@[simp] lemma mult.zero_mul (a : mult) : 0 * a = 0 := by { cases a; refl }\n@[simp] lemma mult.mul_one (a : mult) : a * 1 = a := by { cases a; refl }\n@[simp] lemma mult.one_mul (a : mult) : 1 * a = a := by { cases a; refl }\nlemma mult.mul_assoc (a b c : mult) : (a * b) * c = a * (b * c) := by { cases a; cases b; cases c; refl }\nlemma mult.left_distrib (a b c : mult) : a * (b + c) = a * b + a * c := by { cases a; cases b; cases c; refl }\nlemma mult.right_distrib (a b c : mult) : (a + b) * c = a * c + b * c := by { cases a; cases b; cases c; refl }\n\ninstance : semiring mult :=\n{ zero := mult.zero,\n  one := mult.one,\n  add := mult.add,\n  mul := mult.mul,\n  zero_add := mult.zero_add,\n  add_zero := mult.add_zero,\n  add_assoc := mult.add_assoc,\n  add_comm := mult.add_comm,\n  mul_zero := mult.mul_zero,\n  zero_mul := mult.zero_mul,\n  mul_one := mult.mul_one,\n  one_mul := mult.one_mul,\n  mul_assoc := mult.mul_assoc,\n  left_distrib := mult.left_distrib,\n  right_distrib := mult.right_distrib }\n-/\n\ninstance : add_monoid mult :=\n{ zero := mult.none,\n  add := mult.add,\n  zero_add := mult.zero_add,\n  add_zero := mult.add_zero,\n  add_assoc := mult.add_assoc }\n\ndef ctx.length : \u03a0 {\u03b3 : ctype}, ctx \u03b3 \u2192 nat\n| \u03b3 _ := \u03b3.length\n\ndef ctx.nth : \u03a0 {\u03b3 : ctype}, ctx \u03b3 \u2192 nat \u2192 option (mult \u00d7 expr)\n| []       nil          _       := option.none\n| (t :: \u03b3) (cons \u03c0 _ \u0393) 0       := option.some (\u03c0, t)\n| (t :: \u03b3) (cons \u03c0 _ \u0393) (n + 1) := ctx.nth \u0393 n\n\ndef ctx.zero : \u03a0 {\u03b3 : ctype}, ctx \u03b3\n| []       := nil\n| (t :: \u03b3) := cons 0 t ctx.zero\n\n/-\ndef ctx.one : \u03a0 {\u03b3 : ctype}, ctx \u03b3\n| []       := nil\n| (t :: \u03b3) := cons 1 t ctx.one\n-/\n\ndef ctx.add : \u03a0 {\u03b3 : ctype}, ctx \u03b3 \u2192 ctx \u03b3 \u2192 ctx \u03b3\n| []       nil            nil            := nil\n| (t :: \u03b3) (cons \u03c0\u2081 _ \u0393\u2081) (cons \u03c0\u2082 _ \u0393\u2082) := cons (\u03c0\u2081 + \u03c0\u2082) t (ctx.add \u0393\u2081 \u0393\u2082)\n\n/-\ndef ctx.smul : \u03a0 {\u03b3 : ctype}, mult \u2192 ctx \u03b3 \u2192 ctx \u03b3\n| []       _  nil          := nil\n| (t :: \u03b3) \u03c0' (cons \u03c0 _ \u0393) := cons (\u03c0' * \u03c0) t (ctx.smul \u03c0' \u0393)\n-/\n\ninstance {\u03b3 : ctype} : has_zero (ctx \u03b3) := \u27e8ctx.zero\u27e9\n-- instance {\u03b3 : ctype} : has_one (ctx \u03b3) := \u27e8ctx.one\u27e9\ninstance {\u03b3 : ctype} : has_add (ctx \u03b3) := \u27e8ctx.add\u27e9\n-- instance {\u03b3 : ctype} : has_smul mult (ctx \u03b3) := \u27e8ctx.smul\u27e9\n\n@[simp] lemma ctx.zero_cons {\u03b3 : ctype} {t : expr} : (0 : ctx (t :: \u03b3)) = (\u27e60 \u2022 t\u27e7 :: (0 : ctx \u03b3)) := rfl\n-- @[simp] lemma ctx.one_cons {\u03b3 : ctype} {t : expr} : (1 : ctx (t :: \u03b3)) = (\u27e61 \u2022 t\u27e7 :: (1 : ctx \u03b3)) := rfl\n@[simp] lemma ctx.add_nil {\u03b3 : ctype} : nil + nil = nil := rfl\n@[simp] lemma ctx.add_cons {\u03b3 : ctype} {t : expr} {\u03c0\u2081 \u03c0\u2082 : mult} {\u0393\u2081 \u0393\u2082 : ctx \u03b3} :\n  \u27e6\u03c0\u2081 \u2022 t\u27e7 :: \u0393\u2081 + \u27e6\u03c0\u2082 \u2022 t\u27e7 :: \u0393\u2082 = \u27e6(\u03c0\u2081 + \u03c0\u2082) \u2022 t\u27e7 :: (\u0393\u2081 + \u0393\u2082) := rfl\n-- @[simp] lemma ctx.smul_nil {\u03b3 : ctype} {\u03c0 : mult} : \u03c0 \u2022 nil = nil := rfl\n-- @[simp] lemma ctx.smul_cons {\u03b3 : ctype} {t : expr} {\u0393 : ctx \u03b3} {\u03c0' \u03c0 : mult} :\n--   \u03c0' \u2022 \u27e6\u03c0 \u2022 t\u27e7 :: \u0393 = \u27e6(\u03c0' * \u03c0) \u2022 t\u27e7 :: (\u03c0' \u2022 \u0393) := rfl\n\n@[simp]\nlemma ctx.zero_add {\u03b3 : ctype} (\u0393 : ctx \u03b3) : 0 + \u0393 = \u0393 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393 with _ _ \u03c0 \u0393', simp [ih] at \u22a2 } }\n\n@[simp]\nlemma ctx.add_zero {\u03b3 : ctype} (\u0393 : ctx \u03b3) : \u0393 + 0 = \u0393 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393 with _ _ \u03c0 \u0393', simp [ih] at \u22a2 } }\n\nlemma ctx.add_assoc {\u03b3 : ctype} (\u0393\u2081 \u0393\u2082 \u0393\u2083 : ctx \u03b3) : (\u0393\u2081 + \u0393\u2082) + \u0393\u2083 = \u0393\u2081 + (\u0393\u2082 + \u0393\u2083) := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393\u2081, cases \u0393\u2082, cases \u0393\u2083, refl },\n  case list.cons : t \u03b3 ih\n  { cases \u0393\u2081 with _ _ \u03c0\u2081 \u0393\u2081',\n    cases \u0393\u2082 with _ _ \u03c0\u2082 \u0393\u2082',\n    cases \u0393\u2083 with _ _ \u03c0\u2083 \u0393\u2083',\n    simp [ih] at \u22a2,\n    rw mult.add_assoc } }\n\n/-\nlemma ctx.add_comm {\u03b3 : ctype} (\u0393\u2081 \u0393\u2082 : ctx \u03b3) : \u0393\u2081 + \u0393\u2082 = \u0393\u2082 + \u0393\u2081 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393\u2081, cases \u0393\u2082, refl },\n  case list.cons : t \u03b3 ih\n  { cases \u0393\u2081 with _ _ \u03c0\u2081 \u0393\u2081',\n    cases \u0393\u2082 with _ _ \u03c0\u2082 \u0393\u2082',\n    simp [ih] at \u22a2,\n    rw mult.add_comm } }\n-/\n\ninstance {\u03b3 : ctype} : add_monoid (ctx \u03b3) :=\n{ zero := ctx.zero,\n  add := ctx.add,\n  zero_add := ctx.zero_add,\n  add_zero := ctx.add_zero,\n  add_assoc := ctx.add_assoc }\n\n/-\n@[simp]\nlemma ctx.one_smul {\u03b3 : ctype} (\u0393 : ctx \u03b3) : (1 : mult) \u2022 \u0393 = \u0393 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393, simp [ih] at \u22a2 } }\n\nlemma ctx.mul_smul {\u03b3 : ctype} (\u03c0\u2081 \u03c0\u2082 : mult) (\u0393 : ctx \u03b3) : (\u03c0\u2081 * \u03c0\u2082) \u2022 \u0393 = \u03c0\u2081 \u2022 \u03c0\u2082 \u2022 \u0393 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393, simp [ih] at \u22a2, rw mult.mul_assoc } }\n\n@[simp]\nlemma ctx.smul_zero {\u03b3 : ctype} (\u03c0 : mult) : \u03c0 \u2022 (0 : ctx \u03b3) = 0 := by\n{ induction \u03b3,\n  case list.nil : { refl },\n  case list.cons : t \u03b3 ih { simp at \u22a2, exact ih } }\n\nlemma ctx.smul_add {\u03b3 : ctype} (\u03c0 : mult) (\u0393\u2081 \u0393\u2082 : ctx \u03b3) : \u03c0 \u2022 (\u0393\u2081 + \u0393\u2082) = \u03c0 \u2022 \u0393\u2081 + \u03c0 \u2022 \u0393\u2082 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393\u2081, cases \u0393\u2082, refl },\n  case list.cons : t \u03b3 ih { cases \u0393\u2081, cases \u0393\u2082, simp [ih] at \u22a2, rw mult.left_distrib } }\n\n@[simp]\nlemma ctx.zero_smul {\u03b3 : ctype} (\u0393 : ctx \u03b3) : (0 : mult) \u2022 \u0393 = 0 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393, simp [ih] at \u22a2 } }\n\nlemma ctx.add_smul {\u03b3 : ctype} (\u03c0\u2081 \u03c0\u2082 : mult) (\u0393 : ctx \u03b3) : (\u03c0\u2081 + \u03c0\u2082) \u2022 \u0393 = \u03c0\u2081 \u2022 \u0393 + \u03c0\u2082 \u2022 \u0393 := by\n{ induction \u03b3,\n  case list.nil : { cases \u0393, refl },\n  case list.cons : t \u03b3 ih { cases \u0393, simp [ih] at \u22a2, rw mult.right_distrib } }\n-/\n\n/-\ninstance {\u03b3 : ctype} : module mult (ctx \u03b3) :=\n{ one_smul := ctx.one_smul,\n  mul_smul := ctx.mul_smul,\n  smul_zero := ctx.smul_zero,\n  smul_add := ctx.smul_add,\n  zero_smul := ctx.zero_smul,\n  add_smul := ctx.add_smul }\n-/\n\nend\nend quantitative_types\n", "meta": {"author": "bridgekat", "repo": "quantitative-types", "sha": "8e48e411e4f101e7193cccaaab2a85160abcb510", "save_path": "github-repos/lean/bridgekat-quantitative-types", "path": "github-repos/lean/bridgekat-quantitative-types/quantitative-types-8e48e411e4f101e7193cccaaab2a85160abcb510/src/context.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660543, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.35106934276830054}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n-/\nimport data.list.lex\nimport data.char\n\n/-!\n# Strings\n\nSupplementary theorems about the `string` type.\n-/\n\nnamespace string\n\n/-- `<` on string iterators. This coincides with `<` on strings as lists. -/\ndef ltb : iterator \u2192 iterator \u2192 bool\n| s\u2081 s\u2082 := begin\n  cases s\u2082.has_next, {exact ff},\n  cases h\u2081 : s\u2081.has_next, {exact tt},\n  exact if s\u2081.curr = s\u2082.curr then\n    have s\u2081.next.2.length < s\u2081.2.length, from\n    match s\u2081, h\u2081 with \u27e8_, a::l\u27e9, h := nat.lt_succ_self _ end,\n    ltb s\u2081.next s\u2082.next\n  else s\u2081.curr < s\u2082.curr,\nend\nusing_well_founded {rel_tac :=\n  \u03bb _ _, `[exact \u27e8_, measure_wf (\u03bb s, s.1.2.length)\u27e9]}\n\ninstance has_lt' : has_lt string :=\n\u27e8\u03bb s\u2081 s\u2082, ltb s\u2081.mk_iterator s\u2082.mk_iterator\u27e9\n\ninstance decidable_lt : @decidable_rel string (<) :=\nby apply_instance -- short-circuit type class inference\n\n@[simp] theorem lt_iff_to_list_lt :\n  \u2200 {s\u2081 s\u2082 : string}, s\u2081 < s\u2082 \u2194 s\u2081.to_list < s\u2082.to_list\n| \u27e8i\u2081\u27e9 \u27e8i\u2082\u27e9 :=\n  suffices \u2200 {p\u2081 p\u2082 s\u2081 s\u2082}, ltb \u27e8p\u2081, s\u2081\u27e9 \u27e8p\u2082, s\u2082\u27e9 \u2194 s\u2081 < s\u2082, from this,\n  begin\n    intros,\n    induction s\u2081 with a s\u2081 IH generalizing p\u2081 p\u2082 s\u2082;\n      cases s\u2082 with b s\u2082; rw ltb; simp [iterator.has_next],\n    { refl, },\n    { exact iff_of_true rfl list.lex.nil },\n    { exact iff_of_false bool.ff_ne_tt (not_lt_of_lt list.lex.nil) },\n    { dsimp [iterator.has_next,\n        iterator.curr, iterator.next],\n      split_ifs,\n      { subst b, exact IH.trans list.lex.cons_iff.symm },\n      { simp, refine \u27e8list.lex.rel, \u03bb e, _\u27e9,\n        cases e, {cases h rfl}, assumption } }\n  end\n\ninstance has_le : has_le string := \u27e8\u03bb s\u2081 s\u2082, \u00ac s\u2082 < s\u2081\u27e9\n\ninstance decidable_le : @decidable_rel string (\u2264) :=\nby apply_instance -- short-circuit type class inference\n\n@[simp] theorem le_iff_to_list_le\n  {s\u2081 s\u2082 : string} : s\u2081 \u2264 s\u2082 \u2194 s\u2081.to_list \u2264 s\u2082.to_list :=\n(not_congr lt_iff_to_list_lt).trans not_lt\n\ntheorem to_list_inj : \u2200 {s\u2081 s\u2082}, to_list s\u2081 = to_list s\u2082 \u2194 s\u2081 = s\u2082\n| \u27e8s\u2081\u27e9 \u27e8s\u2082\u27e9 := \u27e8congr_arg _, congr_arg _\u27e9\n\nlemma nil_as_string_eq_empty : [].as_string = \"\" := rfl\n\n@[simp] lemma to_list_empty : \"\".to_list = [] := rfl\n\nlemma as_string_inv_to_list (s : string) : s.to_list.as_string = s :=\nby { cases s, refl }\n\n@[simp] lemma to_list_singleton (c : char) : (string.singleton c).to_list = [c] := rfl\n\nlemma to_list_nonempty : \u2200 {s : string}, s \u2260 string.empty \u2192\n  s.to_list = s.head :: (s.popn 1).to_list\n| \u27e8s\u27e9 h := by cases s; [cases h rfl, refl]\n\n@[simp] lemma head_empty : \"\".head = default := rfl\n\n@[simp] lemma popn_empty {n : \u2115} : \"\".popn n = \"\" :=\nbegin\n  induction n with n hn,\n  { refl },\n  { rcases hs : \"\" with \u27e8_ | \u27e8hd, tl\u27e9\u27e9,\n    { rw hs at hn,\n      conv_rhs { rw \u2190hn },\n      simp only [popn, mk_iterator, iterator.nextn, iterator.next] },\n    { simpa only [\u2190to_list_inj] using hs } }\nend\n\ninstance : linear_order string :=\n{ lt := (<), le := (\u2264),\n  decidable_lt := by apply_instance,\n  decidable_le := string.decidable_le,\n  decidable_eq := by apply_instance,\n  le_refl := \u03bb a, le_iff_to_list_le.2 le_rfl,\n  le_trans := \u03bb a b c, by { simp only [le_iff_to_list_le], exact \u03bb h\u2081 h\u2082, h\u2081.trans h\u2082 },\n  le_total := \u03bb a b, by { simp only [le_iff_to_list_le], exact le_total _ _ },\n  le_antisymm := \u03bb a b, by { simp only [le_iff_to_list_le, \u2190 to_list_inj], apply le_antisymm },\n  lt_iff_le_not_le := \u03bb a b, by simp only [le_iff_to_list_le, lt_iff_to_list_lt, lt_iff_le_not_le] }\n\nend string\n\nopen string\n\nlemma list.to_list_inv_as_string (l : list char) : l.as_string.to_list = l :=\nby { cases hl : l.as_string, exact string_imp.mk.inj hl.symm }\n\n@[simp] lemma list.length_as_string (l : list char) : l.as_string.length = l.length := rfl\n\n@[simp] lemma list.as_string_inj {l l' : list char} : l.as_string = l'.as_string \u2194 l = l' :=\n\u27e8\u03bb h, by rw [\u2190list.to_list_inv_as_string l, \u2190list.to_list_inv_as_string l', to_list_inj, h],\n \u03bb h, h \u25b8 rfl\u27e9\n\n@[simp] lemma string.length_to_list (s : string) : s.to_list.length = s.length :=\nby rw [\u2190string.as_string_inv_to_list s, list.to_list_inv_as_string, list.length_as_string]\n\nlemma list.as_string_eq {l : list char} {s : string} :\n  l.as_string = s \u2194 l = s.to_list :=\nby rw [\u2190as_string_inv_to_list s, list.as_string_inj, as_string_inv_to_list s]\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/data/string/basic.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3510674414833871}}
{"text": "/-\nCopyright (c) 2017 Johannes H\u00f6lzl. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johannes H\u00f6lzl, Yury Kudryashov\n-/\nimport analysis.complex.basic\nimport analysis.normed_space.finite_dimension\nimport measure_theory.function.ae_measurable_sequence\nimport measure_theory.group.arithmetic\nimport measure_theory.lattice\nimport measure_theory.measure.open_pos\nimport topology.algebra.order.liminf_limsup\nimport topology.continuous_function.basic\nimport topology.instances.ereal\nimport topology.G_delta\nimport topology.order.lattice\nimport topology.semicontinuous\nimport topology.metric_space.metrizable\n\n/-!\n# Borel (measurable) space\n\n## Main definitions\n\n* `borel \u03b1` : the least `\u03c3`-algebra that contains all open sets;\n* `class borel_space` : a space with `topological_space` and `measurable_space` structures\n  such that `\u2039measurable_space \u03b1\u203a = borel \u03b1`;\n* `class opens_measurable_space` : a space with `topological_space` and `measurable_space`\n  structures such that all open sets are measurable; equivalently, `borel \u03b1 \u2264 \u2039measurable_space \u03b1\u203a`.\n* `borel_space` instances on `empty`, `unit`, `bool`, `nat`, `int`, `rat`;\n* `measurable` and `borel_space` instances on `\u211d`, `\u211d\u22650`, `\u211d\u22650\u221e`.\n\n## Main statements\n\n* `is_open.measurable_set`, `is_closed.measurable_set`: open and closed sets are measurable;\n* `continuous.measurable` : a continuous function is measurable;\n* `continuous.measurable2` : if `f : \u03b1 \u2192 \u03b2` and `g : \u03b1 \u2192 \u03b3` are measurable and `op : \u03b2 \u00d7 \u03b3 \u2192 \u03b4`\n  is continuous, then `\u03bb x, op (f x, g y)` is measurable;\n* `measurable.add` etc : dot notation for arithmetic operations on `measurable` predicates,\n  and similarly for `dist` and `edist`;\n* `ae_measurable.add` : similar dot notation for almost everywhere measurable functions;\n* `measurable.ennreal*` : special cases for arithmetic operations on `\u211d\u22650\u221e`.\n-/\n\nnoncomputable theory\n\nopen classical set filter measure_theory\nopen_locale classical big_operators topological_space nnreal ennreal measure_theory\n\nuniverses u v w x y\nvariables {\u03b1 \u03b2 \u03b3 \u03b3\u2082 \u03b4 : Type*} {\u03b9 : Sort y} {s t u : set \u03b1}\n\nopen measurable_space topological_space\n\n/-- `measurable_space` structure generated by `topological_space`. -/\ndef borel (\u03b1 : Type u) [topological_space \u03b1] : measurable_space \u03b1 :=\ngenerate_from {s : set \u03b1 | is_open s}\n\nlemma borel_eq_top_of_discrete [topological_space \u03b1] [discrete_topology \u03b1] :\n  borel \u03b1 = \u22a4 :=\ntop_le_iff.1 $ \u03bb s hs, generate_measurable.basic s (is_open_discrete s)\n\nlemma borel_eq_top_of_encodable [topological_space \u03b1] [t1_space \u03b1] [encodable \u03b1] :\n  borel \u03b1 = \u22a4 :=\nbegin\n  refine (top_le_iff.1 $ \u03bb s hs, bUnion_of_singleton s \u25b8 _),\n  apply measurable_set.bUnion s.countable_encodable,\n  intros x hx,\n  apply measurable_set.of_compl,\n  apply generate_measurable.basic,\n  exact is_closed_singleton.is_open_compl\nend\n\nlemma borel_eq_generate_from_of_subbasis {s : set (set \u03b1)}\n  [t : topological_space \u03b1] [second_countable_topology \u03b1] (hs : t = generate_from s) :\n  borel \u03b1 = generate_from s :=\nle_antisymm\n  (generate_from_le $ assume u (hu : t.is_open u),\n    begin\n      rw [hs] at hu,\n      induction hu,\n      case generate_open.basic : u hu\n      { exact generate_measurable.basic u hu },\n      case generate_open.univ\n      { exact @measurable_set.univ \u03b1 (generate_from s) },\n      case generate_open.inter : s\u2081 s\u2082 _ _ hs\u2081 hs\u2082\n      { exact @measurable_set.inter \u03b1 (generate_from s) _ _ hs\u2081 hs\u2082 },\n      case generate_open.sUnion : f hf ih\n      { rcases is_open_sUnion_countable f (by rwa hs) with \u27e8v, hv, vf, vu\u27e9,\n        rw \u2190 vu,\n        exact @measurable_set.sUnion \u03b1 (generate_from s) _ hv\n          (\u03bb x xv, ih _ (vf xv)) }\n    end)\n  (generate_from_le $ assume u hu, generate_measurable.basic _ $\n    show t.is_open u, by rw [hs]; exact generate_open.basic _ hu)\n\nlemma topological_space.is_topological_basis.borel_eq_generate_from [topological_space \u03b1]\n  [second_countable_topology \u03b1] {s : set (set \u03b1)} (hs : is_topological_basis s) :\n  borel \u03b1 = generate_from s :=\nborel_eq_generate_from_of_subbasis hs.eq_generate_from\n\nlemma is_pi_system_is_open [topological_space \u03b1] : is_pi_system (is_open : set \u03b1 \u2192 Prop) :=\n\u03bb s hs t ht hst, is_open.inter hs ht\n\nlemma borel_eq_generate_from_is_closed [topological_space \u03b1] :\n  borel \u03b1 = generate_from {s | is_closed s} :=\nle_antisymm\n  (generate_from_le $ \u03bb t ht, @measurable_set.of_compl \u03b1 _ (generate_from {s | is_closed s})\n    (generate_measurable.basic _ $ is_closed_compl_iff.2 ht))\n  (generate_from_le $ \u03bb t ht, @measurable_set.of_compl \u03b1 _ (borel \u03b1)\n    (generate_measurable.basic _ $ is_open_compl_iff.2 ht))\n\nsection order_topology\n\nvariable (\u03b1)\nvariables [topological_space \u03b1] [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1]\n\nlemma borel_eq_generate_from_Iio : borel \u03b1 = generate_from (range Iio) :=\nbegin\n  refine le_antisymm _ (generate_from_le _),\n  { rw borel_eq_generate_from_of_subbasis (@order_topology.topology_eq_generate_intervals \u03b1 _ _ _),\n    letI : measurable_space \u03b1 := measurable_space.generate_from (range Iio),\n    have H : \u2200 a : \u03b1, measurable_set (Iio a) := \u03bb a, generate_measurable.basic _ \u27e8_, rfl\u27e9,\n    refine generate_from_le _, rintro _ \u27e8a, rfl | rfl\u27e9; [skip, apply H],\n    by_cases h : \u2203 a', \u2200 b, a < b \u2194 a' \u2264 b,\n    { rcases h with \u27e8a', ha'\u27e9,\n      rw (_ : Ioi a = (Iio a')\u1d9c), { exact (H _).compl },\n      simp [set.ext_iff, ha'] },\n    { rcases is_open_Union_countable\n        (\u03bb a' : {a' : \u03b1 // a < a'}, {b | a'.1 < b})\n        (\u03bb a', is_open_lt' _) with \u27e8v, \u27e8hv\u27e9, vu\u27e9,\n      simp [set.ext_iff] at vu,\n      have : Ioi a = \u22c3 x : v, (Iio x.1.1)\u1d9c,\n      { simp [set.ext_iff],\n        refine \u03bb x, \u27e8\u03bb ax, _, \u03bb \u27e8a', \u27e8h, av\u27e9, ax\u27e9, lt_of_lt_of_le h ax\u27e9,\n        rcases (vu x).2 _ with \u27e8a', h\u2081, h\u2082\u27e9,\n        { exact \u27e8a', h\u2081, le_of_lt h\u2082\u27e9 },\n        refine not_imp_comm.1 (\u03bb h, _) h,\n        exact \u27e8x, \u03bb b, \u27e8\u03bb ab, le_of_not_lt (\u03bb h', h \u27e8b, ab, h'\u27e9),\n          lt_of_lt_of_le ax\u27e9\u27e9 },\n      rw this, resetI,\n      apply measurable_set.Union,\n      exact \u03bb _, (H _).compl } },\n  { rw forall_range_iff,\n    intro a,\n    exact generate_measurable.basic _ is_open_Iio }\nend\n\nlemma borel_eq_generate_from_Ioi : borel \u03b1 = generate_from (range Ioi) :=\n@borel_eq_generate_from_Iio (order_dual \u03b1) _ (by apply_instance : second_countable_topology \u03b1) _ _\n\nend order_topology\n\nlemma borel_comap {f : \u03b1 \u2192 \u03b2} {t : topological_space \u03b2} :\n  @borel \u03b1 (t.induced f) = (@borel \u03b2 t).comap f :=\ncomap_generate_from.symm\n\nlemma continuous.borel_measurable [topological_space \u03b1] [topological_space \u03b2]\n  {f : \u03b1 \u2192 \u03b2} (hf : continuous f) :\n  @measurable \u03b1 \u03b2 (borel \u03b1) (borel \u03b2) f :=\nmeasurable.of_le_map $ generate_from_le $\n  \u03bb s hs, generate_measurable.basic (f \u207b\u00b9' s) (hs.preimage hf)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nall open sets are measurable. -/\nclass opens_measurable_space (\u03b1 : Type*) [topological_space \u03b1] [h : measurable_space \u03b1] : Prop :=\n(borel_le : borel \u03b1 \u2264 h)\n\n/-- A space with `measurable_space` and `topological_space` structures such that\nthe `\u03c3`-algebra of measurable sets is exactly the `\u03c3`-algebra generated by open sets. -/\nclass borel_space (\u03b1 : Type*) [topological_space \u03b1] [measurable_space \u03b1] : Prop :=\n(measurable_eq : \u2039measurable_space \u03b1\u203a = borel \u03b1)\n\nnamespace tactic\n\n/-- Add instances `borel \u03b1 : measurable_space \u03b1` and `\u27e8rfl\u27e9 : borel_space \u03b1`. -/\nmeta def add_borel_instance (\u03b1 : expr) : tactic unit :=\ndo\n  n1 \u2190 get_unused_name \"_inst\",\n  to_expr ``(borel %%\u03b1) >>= pose n1,\n  reset_instance_cache,\n  n2 \u2190 get_unused_name \"_inst\",\n  v \u2190 to_expr ``(borel_space.mk rfl : borel_space %%\u03b1),\n  note n2 none v,\n  reset_instance_cache\n\n/-- Given a type `\u03b1`, an assumption `i : measurable_space \u03b1`, and an instance `[borel_space \u03b1]`,\nreplace `i` with `borel \u03b1`. -/\nmeta def borel_to_refl (\u03b1 i : expr) : tactic unit :=\ndo\n  n \u2190 get_unused_name \"h\",\n  to_expr ``(%%i = borel %%\u03b1) >>= assert n,\n  applyc `borel_space.measurable_eq,\n  unfreezing (tactic.subst i),\n  n1 \u2190 get_unused_name \"_inst\",\n  to_expr ``(borel %%\u03b1) >>= pose n1,\n  reset_instance_cache\n\n/-- Given a type `\u03b1`, if there is an assumption `[i : measurable_space \u03b1]`, then try to prove\n`[borel_space \u03b1]` and replace `i` with `borel \u03b1`. Otherwise, add instances\n`borel \u03b1 : measurable_space \u03b1` and `\u27e8rfl\u27e9 : borel_space \u03b1`. -/\nmeta def borelize (\u03b1 : expr) : tactic unit :=\ndo\n  i \u2190 optional (to_expr ``(measurable_space %%\u03b1) >>= find_assumption),\n  i.elim (add_borel_instance \u03b1) (borel_to_refl \u03b1)\n\nnamespace interactive\n\nsetup_tactic_parser\n\n/-- The behaviour of `borelize \u03b1` depends on the existing assumptions on `\u03b1`.\n\n- if `\u03b1` is a topological space with instances `[measurable_space \u03b1] [borel_space \u03b1]`, then\n  `borelize \u03b1` replaces the former instance by `borel \u03b1`;\n- otherwise, `borelize \u03b1` adds instances `borel \u03b1 : measurable_space \u03b1` and `\u27e8rfl\u27e9 : borel_space \u03b1`.\n\nFinally, `borelize [\u03b1, \u03b2, \u03b3]` runs `borelize \u03b1, borelize \u03b2, borelize \u03b3`.\n-/\nmeta def borelize (ts : parse pexpr_list_or_texpr) : tactic unit :=\nmmap' (\u03bb t, to_expr t >>= tactic.borelize) ts\n\nadd_tactic_doc\n{ name := \"borelize\",\n  category := doc_category.tactic,\n  decl_names := [`tactic.interactive.borelize],\n  tags := [\"type class\"] }\n\nend interactive\n\nend tactic\n\n@[priority 100]\ninstance order_dual.opens_measurable_space {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1]\n  [h : opens_measurable_space \u03b1] :\n  opens_measurable_space (order_dual \u03b1) :=\n{ borel_le := h.borel_le }\n\n@[priority 100]\ninstance order_dual.borel_space {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1]\n  [h : borel_space \u03b1] :\n  borel_space (order_dual \u03b1) :=\n{ measurable_eq := h.measurable_eq }\n\n/-- In a `borel_space` all open sets are measurable. -/\n@[priority 100]\ninstance borel_space.opens_measurable {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1]\n  [borel_space \u03b1] : opens_measurable_space \u03b1 :=\n\u27e8ge_of_eq $ borel_space.measurable_eq\u27e9\n\ninstance subtype.borel_space {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1]\n  [h\u03b1 : borel_space \u03b1] (s : set \u03b1) :\n  borel_space s :=\n\u27e8by { rw [h\u03b1.1, subtype.measurable_space, \u2190 borel_comap], refl }\u27e9\n\ninstance subtype.opens_measurable_space {\u03b1 : Type*} [topological_space \u03b1] [measurable_space \u03b1]\n  [h : opens_measurable_space \u03b1] (s : set \u03b1) :\n  opens_measurable_space s :=\n\u27e8by { rw [borel_comap], exact comap_mono h.1 }\u27e9\n\ntheorem _root_.measurable_set.induction_on_open [topological_space \u03b1] [measurable_space \u03b1]\n  [borel_space \u03b1] {C : set \u03b1 \u2192 Prop} (h_open : \u2200 U, is_open U \u2192 C U)\n  (h_compl : \u2200 t, measurable_set t \u2192 C t \u2192 C t\u1d9c)\n  (h_union : \u2200 f : \u2115 \u2192 set \u03b1, pairwise (disjoint on f) \u2192\n    (\u2200 i, measurable_set (f i)) \u2192 (\u2200 i, C (f i)) \u2192 C (\u22c3 i, f i)) :\n  \u2200 \u2983t\u2984, measurable_set t \u2192 C t :=\nmeasurable_space.induction_on_inter borel_space.measurable_eq is_pi_system_is_open\n  (h_open _ is_open_empty) h_open h_compl h_union\n\nsection\nvariables [topological_space \u03b1] [measurable_space \u03b1] [opens_measurable_space \u03b1]\n   [topological_space \u03b2] [measurable_space \u03b2] [opens_measurable_space \u03b2]\n   [topological_space \u03b3] [measurable_space \u03b3] [borel_space \u03b3]\n   [topological_space \u03b3\u2082] [measurable_space \u03b3\u2082] [borel_space \u03b3\u2082]\n   [measurable_space \u03b4]\n\nlemma is_open.measurable_set (h : is_open s) : measurable_set s :=\nopens_measurable_space.borel_le _ $ generate_measurable.basic _ h\n\n@[measurability]\nlemma measurable_set_interior : measurable_set (interior s) := is_open_interior.measurable_set\n\nlemma is_G\u03b4.measurable_set (h : is_G\u03b4 s) : measurable_set s :=\nbegin\n  rcases h with \u27e8S, hSo, hSc, rfl\u27e9,\n  exact measurable_set.sInter hSc (\u03bb t ht, (hSo t ht).measurable_set)\nend\n\nlemma measurable_set_of_continuous_at {\u03b2} [emetric_space \u03b2] (f : \u03b1 \u2192 \u03b2) :\n  measurable_set {x | continuous_at f x} :=\n(is_G\u03b4_set_of_continuous_at f).measurable_set\n\nlemma is_closed.measurable_set (h : is_closed s) : measurable_set s :=\nh.is_open_compl.measurable_set.of_compl\n\nlemma is_compact.measurable_set [t2_space \u03b1] (h : is_compact s) : measurable_set s :=\nh.is_closed.measurable_set\n\n@[measurability]\nlemma measurable_set_closure : measurable_set (closure s) :=\nis_closed_closure.measurable_set\n\nlemma measurable_of_is_open {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, is_open s \u2192 measurable_set (f \u207b\u00b9' s)) :\n  measurable f :=\nby { rw [\u2039borel_space \u03b3\u203a.measurable_eq], exact measurable_generate_from hf }\n\nlemma measurable_of_is_closed {f : \u03b4 \u2192 \u03b3} (hf : \u2200 s, is_closed s \u2192 measurable_set (f \u207b\u00b9' s)) :\n  measurable f :=\nbegin\n  apply measurable_of_is_open, intros s hs,\n  rw [\u2190 measurable_set.compl_iff, \u2190 preimage_compl], apply hf, rw [is_closed_compl_iff], exact hs\nend\n\nlemma measurable_of_is_closed' {f : \u03b4 \u2192 \u03b3}\n  (hf : \u2200 s, is_closed s \u2192 s.nonempty \u2192 s \u2260 univ \u2192 measurable_set (f \u207b\u00b9' s)) : measurable f :=\nbegin\n  apply measurable_of_is_closed, intros s hs,\n  cases eq_empty_or_nonempty s with h1 h1, { simp [h1] },\n  by_cases h2 : s = univ, { simp [h2] },\n  exact hf s hs h1 h2\nend\n\ninstance nhds_is_measurably_generated (a : \u03b1) : (\ud835\udcdd a).is_measurably_generated :=\nbegin\n  rw [nhds, infi_subtype'],\n  refine @filter.infi_is_measurably_generated _ _ _ _ (\u03bb i, _),\n  exact i.2.2.measurable_set.principal_is_measurably_generated\nend\n\n/-- If `s` is a measurable set, then `\ud835\udcdd[s] a` is a measurably generated filter for\neach `a`. This cannot be an `instance` because it depends on a non-instance `hs : measurable_set s`.\n-/\nlemma measurable_set.nhds_within_is_measurably_generated {s : set \u03b1} (hs : measurable_set s)\n  (a : \u03b1) :\n  (\ud835\udcdd[s] a).is_measurably_generated :=\nby haveI := hs.principal_is_measurably_generated; exact filter.inf_is_measurably_generated _ _\n\n@[priority 100] -- see Note [lower instance priority]\ninstance opens_measurable_space.to_measurable_singleton_class [t1_space \u03b1] :\n  measurable_singleton_class \u03b1 :=\n\u27e8\u03bb x, is_closed_singleton.measurable_set\u27e9\n\ninstance pi.opens_measurable_space_encodable {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [encodable \u03b9]\n  [t' : \u03a0 i, topological_space (\u03c0 i)]\n  [\u03a0 i, measurable_space (\u03c0 i)] [\u2200 i, second_countable_topology (\u03c0 i)]\n  [\u2200 i, opens_measurable_space (\u03c0 i)] :\n  opens_measurable_space (\u03a0 i, \u03c0 i) :=\nbegin\n  constructor,\n  have : Pi.topological_space =\n    generate_from {t | \u2203(s:\u03a0a, set (\u03c0 a)) (i : finset \u03b9), (\u2200a\u2208i, s a \u2208 countable_basis (\u03c0 a)) \u2227\n      t = pi \u2191i s},\n  { rw [funext (\u03bb a, @eq_generate_from_countable_basis (\u03c0 a) _ _), pi_generate_from_eq] },\n  rw [borel_eq_generate_from_of_subbasis this],\n  apply generate_from_le,\n  rintros _ \u27e8s, i, hi, rfl\u27e9,\n  refine measurable_set.pi i.countable_to_set (\u03bb a ha, is_open.measurable_set _),\n  rw [eq_generate_from_countable_basis (\u03c0 a)],\n  exact generate_open.basic _ (hi a ha)\nend\n\ninstance pi.opens_measurable_space_fintype {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [fintype \u03b9]\n  [t' : \u03a0 i, topological_space (\u03c0 i)]\n  [\u03a0 i, measurable_space (\u03c0 i)] [\u2200 i, second_countable_topology (\u03c0 i)]\n  [\u2200 i, opens_measurable_space (\u03c0 i)] :\n  opens_measurable_space (\u03a0 i, \u03c0 i) :=\nby { letI := fintype.encodable \u03b9, apply_instance }\n\ninstance prod.opens_measurable_space [second_countable_topology \u03b1] [second_countable_topology \u03b2] :\n  opens_measurable_space (\u03b1 \u00d7 \u03b2) :=\nbegin\n  constructor,\n  rw [((is_basis_countable_basis \u03b1).prod (is_basis_countable_basis \u03b2)).borel_eq_generate_from],\n  apply generate_from_le,\n  rintros _ \u27e8u, v, hu, hv, rfl\u27e9,\n  exact (is_open_of_mem_countable_basis hu).measurable_set.prod\n    (is_open_of_mem_countable_basis hv).measurable_set\nend\n\nvariables {\u03b1' : Type*} [topological_space \u03b1'] [measurable_space \u03b1']\n\nlemma measure_interior_of_null_bdry {\u03bc : measure \u03b1'} {s : set \u03b1'}\n  (h_nullbdry : \u03bc (frontier s) = 0) : \u03bc (interior s) = \u03bc s :=\nmeasure_eq_measure_smaller_of_between_null_diff\n  interior_subset subset_closure h_nullbdry\n\nlemma measure_closure_of_null_bdry {\u03bc : measure \u03b1'} {s : set \u03b1'}\n  (h_nullbdry : \u03bc (frontier s) = 0) : \u03bc (closure s) = \u03bc s :=\n(measure_eq_measure_larger_of_between_null_diff\n  interior_subset subset_closure h_nullbdry).symm\n\nsection preorder\nvariables [preorder \u03b1] [order_closed_topology \u03b1] {a b x : \u03b1}\n\n@[simp, measurability]\nlemma measurable_set_Ici : measurable_set (Ici a) := is_closed_Ici.measurable_set\n@[simp, measurability]\nlemma measurable_set_Iic : measurable_set (Iic a) := is_closed_Iic.measurable_set\n@[simp, measurability]\nlemma measurable_set_Icc : measurable_set (Icc a b) := is_closed_Icc.measurable_set\n\ninstance nhds_within_Ici_is_measurably_generated :\n  (\ud835\udcdd[Ici b] a).is_measurably_generated :=\nmeasurable_set_Ici.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iic_is_measurably_generated :\n  (\ud835\udcdd[Iic b] a).is_measurably_generated :=\nmeasurable_set_Iic.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Icc_is_measurably_generated :\n  is_measurably_generated (\ud835\udcdd[Icc a b] x) :=\nby { rw [\u2190 Ici_inter_Iic, nhds_within_inter], apply_instance }\n\ninstance at_top_is_measurably_generated : (filter.at_top : filter \u03b1).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  \u03bb a, (measurable_set_Ici : measurable_set (Ici a)).principal_is_measurably_generated\n\ninstance at_bot_is_measurably_generated : (filter.at_bot : filter \u03b1).is_measurably_generated :=\n@filter.infi_is_measurably_generated _ _ _ _ $\n  \u03bb a, (measurable_set_Iic : measurable_set (Iic a)).principal_is_measurably_generated\n\nend preorder\n\nsection partial_order\nvariables [partial_order \u03b1] [order_closed_topology \u03b1] [second_countable_topology \u03b1]\n  {a b : \u03b1}\n\n@[measurability]\nlemma measurable_set_le' : measurable_set {p : \u03b1 \u00d7 \u03b1 | p.1 \u2264 p.2} :=\norder_closed_topology.is_closed_le'.measurable_set\n\n@[measurability]\nlemma measurable_set_le {f g : \u03b4 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable_set {a | f a \u2264 g a} :=\nhf.prod_mk hg measurable_set_le'\n\nend partial_order\n\nsection linear_order\nvariables [linear_order \u03b1] [order_closed_topology \u03b1] {a b x : \u03b1}\n\n-- we open this locale only here to avoid issues with list being treated as intervals above\nopen_locale interval\n\n@[simp, measurability]\nlemma measurable_set_Iio : measurable_set (Iio a) := is_open_Iio.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioi : measurable_set (Ioi a) := is_open_Ioi.measurable_set\n@[simp, measurability]\nlemma measurable_set_Ioo : measurable_set (Ioo a b) := is_open_Ioo.measurable_set\n\n@[simp, measurability] lemma measurable_set_Ioc : measurable_set (Ioc a b) :=\nmeasurable_set_Ioi.inter measurable_set_Iic\n\n@[simp, measurability] lemma measurable_set_Ico : measurable_set (Ico a b) :=\nmeasurable_set_Ici.inter measurable_set_Iio\n\ninstance nhds_within_Ioi_is_measurably_generated :\n  (\ud835\udcdd[Ioi b] a).is_measurably_generated :=\nmeasurable_set_Ioi.nhds_within_is_measurably_generated _\n\ninstance nhds_within_Iio_is_measurably_generated :\n  (\ud835\udcdd[Iio b] a).is_measurably_generated :=\nmeasurable_set_Iio.nhds_within_is_measurably_generated _\n\ninstance nhds_within_interval_is_measurably_generated :\n  is_measurably_generated (\ud835\udcdd[[a, b]] x) :=\nnhds_within_Icc_is_measurably_generated\n\n@[measurability]\nlemma measurable_set_lt' [second_countable_topology \u03b1] : measurable_set {p : \u03b1 \u00d7 \u03b1 | p.1 < p.2} :=\n(is_open_lt continuous_fst continuous_snd).measurable_set\n\n@[measurability]\nlemma measurable_set_lt [second_countable_topology \u03b1] {f g : \u03b4 \u2192 \u03b1} (hf : measurable f)\n  (hg : measurable g) : measurable_set {a | f a < g a} :=\nhf.prod_mk hg measurable_set_lt'\n\nlemma set.ord_connected.measurable_set (h : ord_connected s) : measurable_set s :=\nbegin\n  let u := \u22c3 (x \u2208 s) (y \u2208 s), Ioo x y,\n  have huopen : is_open u := is_open_bUnion (\u03bb x hx, is_open_bUnion (\u03bb y hy, is_open_Ioo)),\n  have humeas : measurable_set u := huopen.measurable_set,\n  have hfinite : (s \\ u).finite,\n  { refine set.finite_of_forall_between_eq_endpoints (s \\ u) (\u03bb x hx y hy z hz hxy hyz, _),\n    by_contra' h,\n    exact hy.2 (mem_Union\u2082.mpr \u27e8x, hx.1,\n      mem_Union\u2082.mpr \u27e8z, hz.1, lt_of_le_of_ne hxy h.1, lt_of_le_of_ne hyz h.2\u27e9\u27e9) },\n  have : u \u2286 s :=\n    Union\u2082_subset (\u03bb x hx, Union\u2082_subset (\u03bb y hy, Ioo_subset_Icc_self.trans (h.out hx hy))),\n  rw \u2190 union_diff_cancel this,\n  exact humeas.union hfinite.measurable_set\nend\n\nlemma is_preconnected.measurable_set\n  (h : is_preconnected s) : measurable_set s :=\nh.ord_connected.measurable_set\n\nlemma generate_from_Ico_mem_le_borel {\u03b1 : Type*} [topological_space \u03b1] [linear_order \u03b1]\n  [order_closed_topology \u03b1] (s t : set \u03b1) :\n  measurable_space.generate_from {S | \u2203 (l \u2208 s) (u \u2208 t) (h : l < u), Ico l u = S} \u2264 borel \u03b1 :=\nbegin\n  apply generate_from_le,\n  borelize \u03b1,\n  rintro _ \u27e8a, -, b, -, -, rfl\u27e9,\n  exact measurable_set_Ico\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem_aux {\u03b1 : Type*} [topological_space \u03b1] [linear_order \u03b1]\n  [order_topology \u03b1] [second_countable_topology \u03b1] {s : set \u03b1} (hd : dense s)\n  (hbot : \u2200 x, is_bot x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 y \u2208 s) :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 (l \u2208 s) (u \u2208 s) (h : l < u), Ico l u = S} :=\nbegin\n  set S : set (set \u03b1) := {S | \u2203 (l \u2208 s) (u \u2208 s) (h : l < u), Ico l u = S},\n  refine le_antisymm _ (generate_from_Ico_mem_le_borel _ _),\n  letI : measurable_space \u03b1 := generate_from S,\n  rw borel_eq_generate_from_Iio,\n  refine generate_from_le (forall_range_iff.2 $ \u03bb a, _),\n  rcases hd.exists_countable_dense_subset_bot_top with \u27e8t, hts, hc, htd, htb, htt\u27e9,\n  by_cases ha : \u2200 b < a, (Ioo b a).nonempty,\n  { convert_to measurable_set (\u22c3 (l \u2208 t) (u \u2208 t) (hlu : l < u) (hu : u \u2264 a), Ico l u),\n    { ext y, simp only [mem_Union, mem_Iio, mem_Ico], split,\n      { intro hy,\n        rcases htd.exists_le' (\u03bb b hb, htb _ hb (hbot b hb)) y with \u27e8l, hlt, hly\u27e9,\n        rcases htd.exists_mem_open is_open_Ioo (ha y hy) with \u27e8u, hut, hyu, hua\u27e9,\n        exact \u27e8l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu\u27e9 },\n      { rintro \u27e8l, -, u, -, -, hua, -, hyu\u27e9,\n        exact hyu.trans_le hua } },\n    { refine measurable_set.bUnion hc (\u03bb a ha, measurable_set.bUnion hc $ \u03bb b hb, _),\n      refine measurable_set.Union_Prop (\u03bb hab, measurable_set.Union_Prop $ \u03bb hb', _),\n      exact generate_measurable.basic _ \u27e8a, hts ha, b, hts hb, hab, mem_singleton _\u27e9 } },\n  { simp only [not_forall, not_nonempty_iff_eq_empty] at ha,\n    replace ha : a \u2208 s := hIoo ha.some a ha.some_spec.fst ha.some_spec.snd,\n    convert_to measurable_set (\u22c3 (l \u2208 t) (hl : l < a), Ico l a),\n    { symmetry,\n      simp only [\u2190 Ici_inter_Iio, \u2190 Union_inter, inter_eq_right_iff_subset, subset_def, mem_Union,\n        mem_Ici, mem_Iio],\n      intros x hx, rcases htd.exists_le' (\u03bb b hb, htb _ hb (hbot b hb)) x with \u27e8z, hzt, hzx\u27e9,\n      exact \u27e8z, hzt, hzx.trans_lt hx, hzx\u27e9 },\n    { refine measurable_set.bUnion hc (\u03bb x hx, measurable_set.Union_Prop $ \u03bb hlt, _),\n      exact generate_measurable.basic _ \u27e8x, hts hx, a, ha, hlt, mem_singleton _\u27e9 } }\nend\n\nlemma dense.borel_eq_generate_from_Ico_mem {\u03b1 : Type*} [topological_space \u03b1] [linear_order \u03b1]\n  [order_topology \u03b1] [second_countable_topology \u03b1] [densely_ordered \u03b1] [no_min_order \u03b1]\n  {s : set \u03b1} (hd : dense s) :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 (l \u2208 s) (u \u2208 s) (h : l < u), Ico l u = S} :=\nhd.borel_eq_generate_from_Ico_mem_aux (by simp) $\n  \u03bb x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ico (\u03b1 : Type*) [topological_space \u03b1]\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 l u (h : l < u), Ico l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ \u03b1 _).borel_eq_generate_from_Ico_mem_aux (\u03bb _ _, mem_univ _)\n      (\u03bb _ _ _ _, mem_univ _)\n\nlemma dense.borel_eq_generate_from_Ioc_mem_aux {\u03b1 : Type*} [topological_space \u03b1] [linear_order \u03b1]\n  [order_topology \u03b1] [second_countable_topology \u03b1] {s : set \u03b1} (hd : dense s)\n  (hbot : \u2200 x, is_top x \u2192 x \u2208 s) (hIoo : \u2200 x y : \u03b1, x < y \u2192 Ioo x y = \u2205 \u2192 x \u2208 s) :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 (l \u2208 s) (u \u2208 s) (h : l < u), Ioc l u = S} :=\nbegin\n  convert hd.order_dual.borel_eq_generate_from_Ico_mem_aux hbot (\u03bb x y hlt he, hIoo y x hlt _),\n  { ext s,\n    split; rintro \u27e8l, hl, u, hu, hlt, rfl\u27e9,\n    exacts [\u27e8u, hu, l, hl, hlt, dual_Ico\u27e9, \u27e8u, hu, l, hl, hlt, dual_Ioc\u27e9] },\n  { erw dual_Ioo,\n    exact he }\nend\n\nlemma dense.borel_eq_generate_from_Ioc_mem {\u03b1 : Type*} [topological_space \u03b1] [linear_order \u03b1]\n  [order_topology \u03b1] [second_countable_topology \u03b1] [densely_ordered \u03b1] [no_max_order \u03b1]\n  {s : set \u03b1} (hd : dense s) :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 (l \u2208 s) (u \u2208 s) (h : l < u), Ioc l u = S} :=\nhd.borel_eq_generate_from_Ioc_mem_aux (by simp) $\n  \u03bb x y hxy H, ((nonempty_Ioo.2 hxy).ne_empty H).elim\n\nlemma borel_eq_generate_from_Ioc (\u03b1 : Type*) [topological_space \u03b1]\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] :\n  borel \u03b1 = generate_from {S : set \u03b1 | \u2203 l u (h : l < u), Ioc l u = S} :=\nby simpa only [exists_prop, mem_univ, true_and]\n  using (@dense_univ \u03b1 _).borel_eq_generate_from_Ioc_mem_aux (\u03bb _ _, mem_univ _)\n      (\u03bb _ _ _ _, mem_univ _)\n\nnamespace measure_theory.measure\n\n/-- Two finite measures on a Borel space are equal if they agree on all closed-open intervals.  If\n`\u03b1` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ico` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\nlemma ext_of_Ico_finite {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1]\n  [borel_space \u03b1] (\u03bc \u03bd : measure \u03b1) [is_finite_measure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ)\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd :=\nbegin\n  refine ext_of_generate_finite _\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico \u03b1))\n    (is_pi_system_Ico (id : \u03b1 \u2192 \u03b1) id) _ h\u03bc\u03bd,\n  { rintro - \u27e8a, b, hlt, rfl\u27e9,\n    exact h hlt }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all open-closed intervals.  If\n`\u03b1` is a conditionally complete linear order with no top element,\n`measure_theory.measure..ext_of_Ioc` is an extensionality lemma with weaker assumptions on `\u03bc` and\n`\u03bd`. -/\nlemma ext_of_Ioc_finite {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1]\n  [borel_space \u03b1] (\u03bc \u03bd : measure \u03b1) [is_finite_measure \u03bc] (h\u03bc\u03bd : \u03bc univ = \u03bd univ)\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd :=\nbegin\n  refine @ext_of_Ico_finite (order_dual \u03b1) _ _ _ _ _ \u2039_\u203a \u03bc \u03bd _ h\u03bc\u03bd (\u03bb a b hab, _),\n  erw dual_Ico,\n  exact h hab\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico' {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] [borel_space \u03b1]\n  [no_max_order \u03b1] (\u03bc \u03bd : measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) \u2260 \u221e)\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd :=\nbegin\n  rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, hsb, hst\u27e9,\n  have : countable (\u22c3 (l \u2208 s) (u \u2208 s) (h : l < u), {Ico l u} : set (set \u03b1)),\n    from hsc.bUnion (\u03bb l hl, hsc.bUnion\n      (\u03bb u hu, countable_Union_Prop $ \u03bb _, countable_singleton _)),\n  simp only [\u2190 set_of_eq_eq_singleton, \u2190 set_of_exists] at this,\n  refine measure.ext_of_generate_from_of_cover_subset\n    (borel_space.measurable_eq.trans (borel_eq_generate_from_Ico \u03b1))\n    (is_pi_system_Ico id id) _ this _ _ _,\n  { rintro _ \u27e8l, -, u, -, h, rfl\u27e9, exact \u27e8l, u, h, rfl\u27e9 },\n  { refine sUnion_eq_univ_iff.2 (\u03bb x, _),\n    rcases hsd.exists_le' hsb x with \u27e8l, hls, hlx\u27e9,\n    rcases hsd.exists_gt x with \u27e8u, hus, hxu\u27e9,\n    exact \u27e8_, \u27e8l, hls, u, hus, hlx.trans_lt hxu, rfl\u27e9, hlx, hxu\u27e9 },\n  { rintro _ \u27e8l, -, u, -, hlt, rfl\u27e9, exact h\u03bc hlt },\n  { rintro _ \u27e8l, u, hlt, rfl\u27e9, exact h hlt }\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc' {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] [borel_space \u03b1]\n  [no_min_order \u03b1] (\u03bc \u03bd : measure \u03b1) (h\u03bc : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) \u2260 \u221e)\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd :=\nbegin\n  refine @ext_of_Ico' (order_dual \u03b1) _ _ _ _ _ \u2039_\u203a _ \u03bc \u03bd _ _;\n    intros a b hab; erw dual_Ico,\n  exacts [h\u03bc hab, h hab]\nend\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nclosed-open intervals. -/\nlemma ext_of_Ico {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [conditionally_complete_linear_order \u03b1] [order_topology \u03b1]\n  [borel_space \u03b1] [no_max_order \u03b1] (\u03bc \u03bd : measure \u03b1) [is_locally_finite_measure \u03bc]\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ico a b) = \u03bd (Ico a b)) : \u03bc = \u03bd :=\n\u03bc.ext_of_Ico' \u03bd (\u03bb a b hab, measure_Ico_lt_top.ne) h\n\n/-- Two measures which are finite on closed-open intervals are equal if the agree on all\nopen-closed intervals. -/\nlemma ext_of_Ioc {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [conditionally_complete_linear_order \u03b1] [order_topology \u03b1]\n  [borel_space \u03b1] [no_min_order \u03b1] (\u03bc \u03bd : measure \u03b1) [is_locally_finite_measure \u03bc]\n  (h : \u2200 \u2983a b\u2984, a < b \u2192 \u03bc (Ioc a b) = \u03bd (Ioc a b)) : \u03bc = \u03bd :=\n\u03bc.ext_of_Ioc' \u03bd (\u03bb a b hab, measure_Ioc_lt_top.ne) h\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-infinite right-closed\nintervals. -/\nlemma ext_of_Iic {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] [borel_space \u03b1]\n  (\u03bc \u03bd : measure \u03b1) [is_finite_measure \u03bc] (h : \u2200 a, \u03bc (Iic a) = \u03bd (Iic a)) : \u03bc = \u03bd :=\nbegin\n  refine ext_of_Ioc_finite \u03bc \u03bd _ (\u03bb a b hlt, _),\n  { rcases exists_countable_dense_bot_top \u03b1 with \u27e8s, hsc, hsd, -, hst\u27e9,\n    have : directed_on (\u2264) s, from directed_on_iff_directed.2 (directed_of_sup $ \u03bb _ _, id),\n    simp only [\u2190 bsupr_measure_Iic hsc (hsd.exists_ge' hst) this, h] },\n  rw [\u2190 Iic_diff_Iic, measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic,\n      measure_diff (Iic_subset_Iic.2 hlt.le) measurable_set_Iic, h a, h b],\n  { rw \u2190 h a, exact (measure_lt_top \u03bc _).ne },\n  { exact (measure_lt_top \u03bc _).ne }\nend\n\n/-- Two finite measures on a Borel space are equal if they agree on all left-closed right-infinite\nintervals. -/\nlemma ext_of_Ici {\u03b1 : Type*} [topological_space \u03b1] {m : measurable_space \u03b1}\n  [second_countable_topology \u03b1] [linear_order \u03b1] [order_topology \u03b1] [borel_space \u03b1]\n  (\u03bc \u03bd : measure \u03b1) [is_finite_measure \u03bc] (h : \u2200 a, \u03bc (Ici a) = \u03bd (Ici a)) : \u03bc = \u03bd :=\n@ext_of_Iic (order_dual \u03b1) _ _ _ _ _ \u2039_\u203a _ _ _ h\n\nend measure_theory.measure\n\nend linear_order\n\nsection linear_order\n\nvariables [linear_order \u03b1] [order_closed_topology \u03b1]\n\n@[measurability]\nlemma measurable_set_interval {a b : \u03b1} : measurable_set (interval a b) :=\nmeasurable_set_Icc\n\n@[measurability]\nlemma measurable_set_interval_oc {a b : \u03b1} : measurable_set (interval_oc a b) :=\nmeasurable_set_Ioc\n\nvariables [second_countable_topology \u03b1]\n\n@[measurability]\nlemma measurable.max {f g : \u03b4 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb a, max (f a) (g a)) :=\nby simpa only [max_def] using hf.piecewise (measurable_set_le hg hf) hg\n\n@[measurability]\nlemma ae_measurable.max {f g : \u03b4 \u2192 \u03b1} {\u03bc : measure \u03b4}\n  (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) : ae_measurable (\u03bb a, max (f a) (g a)) \u03bc :=\n\u27e8\u03bb a, max (hf.mk f a) (hg.mk g a), hf.measurable_mk.max hg.measurable_mk,\n  eventually_eq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n\n@[measurability]\nlemma measurable.min {f g : \u03b4 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb a, min (f a) (g a)) :=\nby simpa only [min_def] using hf.piecewise (measurable_set_le hf hg) hg\n\n@[measurability]\nlemma ae_measurable.min {f g : \u03b4 \u2192 \u03b1} {\u03bc : measure \u03b4}\n  (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) : ae_measurable (\u03bb a, min (f a) (g a)) \u03bc :=\n\u27e8\u03bb a, min (hf.mk f a) (hg.mk g a), hf.measurable_mk.min hg.measurable_mk,\n  eventually_eq.comp\u2082 hf.ae_eq_mk _ hg.ae_eq_mk\u27e9\n\nend linear_order\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis measurable. -/\nlemma continuous.measurable {f : \u03b1 \u2192 \u03b3} (hf : continuous f) :\n  measurable f :=\nhf.borel_measurable.mono opens_measurable_space.borel_le\n  (le_of_eq $ borel_space.measurable_eq)\n\n/-- A continuous function from an `opens_measurable_space` to a `borel_space`\nis ae-measurable. -/\nlemma continuous.ae_measurable {f : \u03b1 \u2192 \u03b3} (h : continuous f) (\u03bc : measure \u03b1) : ae_measurable f \u03bc :=\nh.measurable.ae_measurable\n\nlemma closed_embedding.measurable {f : \u03b1 \u2192 \u03b3} (hf : closed_embedding f) :\n  measurable f :=\nhf.continuous.measurable\n\nlemma continuous.is_open_pos_measure_map {f : \u03b2 \u2192 \u03b3} (hf : continuous f)\n  (hf_surj : function.surjective f) {\u03bc : measure \u03b2} [\u03bc.is_open_pos_measure] :\n  (measure.map f \u03bc).is_open_pos_measure :=\nbegin\n  refine \u27e8\u03bb U hUo hUne, _\u27e9,\n  rw [measure.map_apply hf.measurable hUo.measurable_set],\n  exact (hUo.preimage hf).measure_ne_zero \u03bc (hf_surj.nonempty_preimage.mpr hUne)\nend\n\n/-- If a function is defined piecewise in terms of functions which are continuous on their\nrespective pieces, then it is measurable. -/\nlemma continuous_on.measurable_piecewise\n  {f g : \u03b1 \u2192 \u03b3} {s : set \u03b1} [\u03a0 (j : \u03b1), decidable (j \u2208 s)]\n  (hf : continuous_on f s) (hg : continuous_on g s\u1d9c) (hs : measurable_set s) :\n  measurable (s.piecewise f g) :=\nbegin\n  refine measurable_of_is_open (\u03bb t ht, _),\n  rw [piecewise_preimage, set.ite],\n  apply measurable_set.union,\n  { rcases _root_.continuous_on_iff'.1 hf t ht with \u27e8u, u_open, hu\u27e9,\n    rw hu,\n    exact u_open.measurable_set.inter hs },\n  { rcases _root_.continuous_on_iff'.1 hg t ht with \u27e8u, u_open, hu\u27e9,\n    rw [diff_eq_compl_inter, inter_comm, hu],\n    exact u_open.measurable_set.inter hs.compl }\nend\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul [has_mul \u03b3] [has_continuous_mul \u03b3] :\n  has_measurable_mul \u03b3 :=\n{ measurable_const_mul := \u03bb c, (continuous_const.mul continuous_id).measurable,\n  measurable_mul_const := \u03bb c, (continuous_id.mul continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub [has_sub \u03b3] [has_continuous_sub \u03b3] :\n  has_measurable_sub \u03b3 :=\n{ measurable_const_sub := \u03bb c, (continuous_const.sub continuous_id).measurable,\n  measurable_sub_const := \u03bb c, (continuous_id.sub continuous_const).measurable }\n\n@[priority 100, to_additive]\ninstance topological_group.has_measurable_inv [group \u03b3] [topological_group \u03b3] :\n  has_measurable_inv \u03b3 :=\n\u27e8continuous_inv.measurable\u27e9\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul {M \u03b1} [topological_space M]\n  [topological_space \u03b1] [measurable_space M] [measurable_space \u03b1]\n  [opens_measurable_space M] [borel_space \u03b1] [has_scalar M \u03b1] [has_continuous_smul M \u03b1] :\n  has_measurable_smul M \u03b1 :=\n\u27e8\u03bb c, (continuous_const_smul _).measurable,\n  \u03bb y, (continuous_id.smul continuous_const).measurable\u27e9\n\nsection lattice\n\n@[priority 100]\ninstance has_continuous_sup.has_measurable_sup [has_sup \u03b3] [has_continuous_sup \u03b3] :\n  has_measurable_sup \u03b3 :=\n{ measurable_const_sup := \u03bb c, (continuous_const.sup continuous_id).measurable,\n  measurable_sup_const := \u03bb c, (continuous_id.sup continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_sup.has_measurable_sup\u2082 [second_countable_topology \u03b3] [has_sup \u03b3]\n  [has_continuous_sup \u03b3] :\n  has_measurable_sup\u2082 \u03b3 :=\n\u27e8continuous_sup.measurable\u27e9\n\n@[priority 100]\ninstance has_continuous_inf.has_measurable_inf [has_inf \u03b3] [has_continuous_inf \u03b3] :\n  has_measurable_inf \u03b3 :=\n{ measurable_const_inf := \u03bb c, (continuous_const.inf continuous_id).measurable,\n  measurable_inf_const := \u03bb c, (continuous_id.inf continuous_const).measurable }\n\n@[priority 100]\ninstance has_continuous_inf.has_measurable_inf\u2082 [second_countable_topology \u03b3] [has_inf \u03b3]\n  [has_continuous_inf \u03b3] :\n  has_measurable_inf\u2082 \u03b3 :=\n\u27e8continuous_inf.measurable\u27e9\n\nend lattice\n\nsection homeomorph\n\n@[measurability] protected lemma homeomorph.measurable (h : \u03b1 \u2243\u209c \u03b3) : measurable h :=\nh.continuous.measurable\n\n/-- A homeomorphism between two Borel spaces is a measurable equivalence.-/\ndef homeomorph.to_measurable_equiv (h : \u03b3 \u2243\u209c \u03b3\u2082) : \u03b3 \u2243\u1d50 \u03b3\u2082 :=\n{ measurable_to_fun := h.measurable,\n  measurable_inv_fun := h.symm.measurable,\n  to_equiv := h.to_equiv }\n\n@[simp]\nlemma homeomorph.to_measurable_equiv_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) : (h.to_measurable_equiv : \u03b3 \u2192 \u03b3\u2082) = h :=\nrfl\n\n@[simp] lemma homeomorph.to_measurable_equiv_symm_coe (h : \u03b3 \u2243\u209c \u03b3\u2082) :\n  (h.to_measurable_equiv.symm : \u03b3\u2082 \u2192 \u03b3) = h.symm :=\nrfl\n\nend homeomorph\n\n@[measurability] lemma continuous_map.measurable (f : C(\u03b1, \u03b3)) : measurable f :=\nf.continuous.measurable\n\nlemma measurable_of_continuous_on_compl_singleton [t1_space \u03b1] {f : \u03b1 \u2192 \u03b3} (a : \u03b1)\n  (hf : continuous_on f {a}\u1d9c) :\n  measurable f :=\nmeasurable_of_measurable_on_compl_singleton a\n  (continuous_on_iff_continuous_restrict.1 hf).measurable\n\nlemma continuous.measurable2 [second_countable_topology \u03b1] [second_countable_topology \u03b2]\n  {f : \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3}\n  (h : continuous (\u03bb p : \u03b1 \u00d7 \u03b2, c p.1 p.2)) (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb a, c (f a) (g a)) :=\nh.measurable.comp (hf.prod_mk hg)\n\nlemma continuous.ae_measurable2 [second_countable_topology \u03b1] [second_countable_topology \u03b2]\n  {f : \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b2} {c : \u03b1 \u2192 \u03b2 \u2192 \u03b3} {\u03bc : measure \u03b4}\n  (h : continuous (\u03bb p : \u03b1 \u00d7 \u03b2, c p.1 p.2)) (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) :\n  ae_measurable (\u03bb a, c (f a) (g a)) \u03bc :=\nh.measurable.comp_ae_measurable (hf.prod_mk hg)\n\n@[priority 100]\ninstance has_continuous_inv\u2080.has_measurable_inv [group_with_zero \u03b3] [t1_space \u03b3]\n  [has_continuous_inv\u2080 \u03b3] :\n  has_measurable_inv \u03b3 :=\n\u27e8measurable_of_continuous_on_compl_singleton 0 continuous_on_inv\u2080\u27e9\n\n@[priority 100, to_additive]\ninstance has_continuous_mul.has_measurable_mul\u2082 [second_countable_topology \u03b3] [has_mul \u03b3]\n  [has_continuous_mul \u03b3] : has_measurable_mul\u2082 \u03b3 :=\n\u27e8continuous_mul.measurable\u27e9\n\n@[priority 100]\ninstance has_continuous_sub.has_measurable_sub\u2082 [second_countable_topology \u03b3] [has_sub \u03b3]\n  [has_continuous_sub \u03b3] : has_measurable_sub\u2082 \u03b3 :=\n\u27e8continuous_sub.measurable\u27e9\n\n@[priority 100]\ninstance has_continuous_smul.has_measurable_smul\u2082 {M \u03b1} [topological_space M]\n  [second_countable_topology M] [measurable_space M] [opens_measurable_space M]\n  [topological_space \u03b1] [second_countable_topology \u03b1] [measurable_space \u03b1]\n  [borel_space \u03b1] [has_scalar M \u03b1] [has_continuous_smul M \u03b1] :\n  has_measurable_smul\u2082 M \u03b1 :=\n\u27e8continuous_smul.measurable\u27e9\n\nend\n\nsection borel_space\nvariables [topological_space \u03b1] [measurable_space \u03b1] [borel_space \u03b1]\n  [topological_space \u03b2] [measurable_space \u03b2] [borel_space \u03b2]\n  [topological_space \u03b3] [measurable_space \u03b3] [borel_space \u03b3]\n  [measurable_space \u03b4]\n\nlemma pi_le_borel_pi {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [\u03a0 i, topological_space (\u03c0 i)]\n  [\u03a0 i, measurable_space (\u03c0 i)] [\u2200 i, borel_space (\u03c0 i)] :\n  measurable_space.pi \u2264 borel (\u03a0 i, \u03c0 i) :=\nbegin\n  have : \u2039\u03a0 i, measurable_space (\u03c0 i)\u203a = \u03bb i, borel (\u03c0 i) :=\n    funext (\u03bb i, borel_space.measurable_eq),\n  rw [this],\n  exact supr_le (\u03bb i, comap_le_iff_le_map.2 $ (continuous_apply i).borel_measurable)\nend\n\nlemma prod_le_borel_prod : prod.measurable_space \u2264 borel (\u03b1 \u00d7 \u03b2) :=\nbegin\n  rw [\u2039borel_space \u03b1\u203a.measurable_eq, \u2039borel_space \u03b2\u203a.measurable_eq],\n  refine sup_le _ _,\n  { exact comap_le_iff_le_map.mpr continuous_fst.borel_measurable },\n  { exact comap_le_iff_le_map.mpr continuous_snd.borel_measurable }\nend\n\ninstance pi.borel_space_fintype_encodable {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [encodable \u03b9]\n  [t' : \u03a0 i, topological_space (\u03c0 i)]\n  [\u03a0 i, measurable_space (\u03c0 i)] [\u2200 i, second_countable_topology (\u03c0 i)]\n  [\u2200 i, borel_space (\u03c0 i)] :\n  borel_space (\u03a0 i, \u03c0 i) :=\n\u27e8le_antisymm pi_le_borel_pi opens_measurable_space.borel_le\u27e9\n\ninstance pi.borel_space_fintype {\u03b9 : Type*} {\u03c0 : \u03b9 \u2192 Type*} [fintype \u03b9]\n  [t' : \u03a0 i, topological_space (\u03c0 i)]\n  [\u03a0 i, measurable_space (\u03c0 i)] [\u2200 i, second_countable_topology (\u03c0 i)]\n  [\u2200 i, borel_space (\u03c0 i)] :\n  borel_space (\u03a0 i, \u03c0 i) :=\n\u27e8le_antisymm pi_le_borel_pi opens_measurable_space.borel_le\u27e9\n\ninstance prod.borel_space [second_countable_topology \u03b1] [second_countable_topology \u03b2] :\n  borel_space (\u03b1 \u00d7 \u03b2) :=\n\u27e8le_antisymm prod_le_borel_prod opens_measurable_space.borel_le\u27e9\n\nprotected lemma embedding.measurable_embedding {f : \u03b1 \u2192 \u03b2} (h\u2081 : embedding f)\n  (h\u2082 : measurable_set (range f)) : measurable_embedding f :=\nshow measurable_embedding (coe \u2218 (homeomorph.of_embedding f h\u2081).to_measurable_equiv),\nfrom (measurable_embedding.subtype_coe h\u2082).comp (measurable_equiv.measurable_embedding _)\n\nprotected lemma closed_embedding.measurable_embedding {f : \u03b1 \u2192 \u03b2} (h : closed_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.closed_range.measurable_set\n\nprotected lemma open_embedding.measurable_embedding {f : \u03b1 \u2192 \u03b2} (h : open_embedding f) :\n  measurable_embedding f :=\nh.to_embedding.measurable_embedding h.open_range.measurable_set\n\nsection linear_order\n\nvariables [linear_order \u03b1] [order_topology \u03b1] [second_countable_topology \u03b1]\n\nlemma measurable_of_Iio {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, measurable_set (f \u207b\u00b9' Iio x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Iio _),\n  rintro _ \u27e8x, rfl\u27e9, exact hf x\nend\n\nlemma upper_semicontinuous.measurable [topological_space \u03b4] [opens_measurable_space \u03b4]\n  {f : \u03b4 \u2192 \u03b1} (hf : upper_semicontinuous f) : measurable f :=\nmeasurable_of_Iio (\u03bb y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Ioi {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, measurable_set (f \u207b\u00b9' Ioi x)) : measurable f :=\nbegin\n  convert measurable_generate_from _,\n  exact borel_space.measurable_eq.trans (borel_eq_generate_from_Ioi _),\n  rintro _ \u27e8x, rfl\u27e9, exact hf x\nend\n\nlemma lower_semicontinuous.measurable [topological_space \u03b4] [opens_measurable_space \u03b4]\n  {f : \u03b4 \u2192 \u03b1} (hf : lower_semicontinuous f) : measurable f :=\nmeasurable_of_Ioi (\u03bb y, (hf.is_open_preimage y).measurable_set)\n\nlemma measurable_of_Iic {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, measurable_set (f \u207b\u00b9' Iic x)) : measurable f :=\nbegin\n  apply measurable_of_Ioi,\n  simp_rw [\u2190 compl_Iic, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable_of_Ici {f : \u03b4 \u2192 \u03b1} (hf : \u2200 x, measurable_set (f \u207b\u00b9' Ici x)) : measurable f :=\nbegin\n  apply measurable_of_Iio,\n  simp_rw [\u2190 compl_Ici, preimage_compl, measurable_set.compl_iff],\n  assumption\nend\n\nlemma measurable.is_lub {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i))\n  (hg : \u2200 b, is_lub {a | \u2203 i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change \u2200 b, is_lub (range $ \u03bb i, f i b) (g b) at hg,\n  rw [\u2039borel_space \u03b1\u203a.measurable_eq, borel_eq_generate_from_Ioi \u03b1],\n  apply measurable_generate_from,\n  rintro _ \u27e8a, rfl\u27e9,\n  simp_rw [set.preimage, mem_Ioi, lt_is_lub_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (\u03bb i, hf i (is_open_lt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_lub_of_nonempty {\u03b9} (h\u03b9 : nonempty \u03b9)\n  {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, is_lub {a | \u2203 i, f i b = a} (g b)) :\n  ae_measurable g \u03bc :=\nbegin\n  let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := \u03bb x f', is_lub {a | \u2203 i, f' i = a} (g x),\n  let g_seq := \u03bb x, ite (x \u2208 ae_seq_set hf p) (g x) (\u27e8g x\u27e9 : nonempty \u03b1).some,\n  have hg_seq : \u2200 b, is_lub {a | \u2203 i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI h\u03b1 : nonempty \u03b1 := nonempty.map g \u27e8b\u27e9,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : \u03b1 | \u2203 (i : \u03b9), (hf i).mk (f i) b = a} = {a : \u03b1 | \u2203 (i : \u03b9), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : \u03b1 | \u2203 (i : \u03b9), h\u03b1.some = a} = {h\u03b1.some},\n      { ext1 x,\n        exact \u27e8\u03bb hx, hx.some_spec.symm, \u03bb hx, \u27e8h\u03b9.some, hx.symm\u27e9\u27e9, },\n      rw h_singleton,\n      exact is_lub_singleton, }, },\n  refine \u27e8g_seq, measurable.is_lub (ae_seq.measurable hf p) hg_seq, _\u27e9,\n  exact (ite_ae_eq_of_measure_compl_zero g (\u03bb x, (\u27e8g x\u27e9 : nonempty \u03b1).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_lub {\u03b9} {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, is_lub {a | \u2203 i, f i b = a} (g b)) :\n  ae_measurable g \u03bc :=\nbegin\n  by_cases h\u03bc : \u03bc = 0, { rw h\u03bc, exact ae_measurable_zero_measure },\n  haveI : \u03bc.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases h\u03b9 : nonempty \u03b9, { exact ae_measurable.is_lub_of_nonempty h\u03b9 hf hg, },\n  suffices : \u2203 x, g =\u1d50[\u03bc] \u03bb y, g x,\n  by { exact \u27e8(\u03bb y, g this.some), measurable_const, this.some_spec\u27e9, },\n  have h_empty : \u2200 x, {a : \u03b1 | \u2203 (i : \u03b9), f i x = a} = \u2205,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact \u03bb hi, h\u03b9 (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact \u27e8hg.exists.some, hg.mono (\u03bb y hy, is_lub.unique hy hg.exists.some_spec)\u27e9,\nend\n\nlemma measurable.is_glb {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i))\n  (hg : \u2200 b, is_glb {a | \u2203 i, f i b = a} (g b)) :\n  measurable g :=\nbegin\n  change \u2200 b, is_glb (range $ \u03bb i, f i b) (g b) at hg,\n  rw [\u2039borel_space \u03b1\u203a.measurable_eq, borel_eq_generate_from_Iio \u03b1],\n  apply measurable_generate_from,\n  rintro _ \u27e8a, rfl\u27e9,\n  simp_rw [set.preimage, mem_Iio, is_glb_lt_iff (hg _), exists_range_iff, set_of_exists],\n  exact measurable_set.Union (\u03bb i, hf i (is_open_gt' _).measurable_set)\nend\n\nprivate lemma ae_measurable.is_glb_of_nonempty {\u03b9} (h\u03b9 : nonempty \u03b9)\n  {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, is_glb {a | \u2203 i, f i b = a} (g b)) :\n  ae_measurable g \u03bc :=\nbegin\n  let p : \u03b4 \u2192 (\u03b9 \u2192 \u03b1) \u2192 Prop := \u03bb x f', is_glb {a | \u2203 i, f' i = a} (g x),\n  let g_seq := \u03bb x, ite (x \u2208 ae_seq_set hf p) (g x) (\u27e8g x\u27e9 : nonempty \u03b1).some,\n  have hg_seq : \u2200 b, is_glb {a | \u2203 i, ae_seq hf p i b = a} (g_seq b),\n  { intro b,\n    haveI h\u03b1 : nonempty \u03b1 := nonempty.map g \u27e8b\u27e9,\n    simp only [ae_seq, g_seq],\n    split_ifs,\n    { have h_set_eq : {a : \u03b1 | \u2203 (i : \u03b9), (hf i).mk (f i) b = a} = {a : \u03b1 | \u2203 (i : \u03b9), f i b = a},\n      { ext x,\n        simp_rw [set.mem_set_of_eq, ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h], },\n      rw h_set_eq,\n      exact ae_seq.fun_prop_of_mem_ae_seq_set hf h, },\n    { have h_singleton : {a : \u03b1 | \u2203 (i : \u03b9), h\u03b1.some = a} = {h\u03b1.some},\n      { ext1 x,\n        exact \u27e8\u03bb hx, hx.some_spec.symm, \u03bb hx, \u27e8h\u03b9.some, hx.symm\u27e9\u27e9, },\n      rw h_singleton,\n      exact is_glb_singleton, }, },\n  refine \u27e8g_seq, measurable.is_glb (ae_seq.measurable hf p) hg_seq, _\u27e9,\n  exact (ite_ae_eq_of_measure_compl_zero g (\u03bb x, (\u27e8g x\u27e9 : nonempty \u03b1).some) (ae_seq_set hf p)\n    (ae_seq.measure_compl_ae_seq_set_eq_zero hf hg)).symm,\nend\n\nlemma ae_measurable.is_glb {\u03b9} {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {g : \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) (hg : \u2200\u1d50 b \u2202\u03bc, is_glb {a | \u2203 i, f i b = a} (g b)) :\n  ae_measurable g \u03bc :=\nbegin\n  by_cases h\u03bc : \u03bc = 0, { rw h\u03bc, exact ae_measurable_zero_measure },\n  haveI : \u03bc.ae.ne_bot, { simpa [ne_bot_iff] },\n  by_cases h\u03b9 : nonempty \u03b9, { exact ae_measurable.is_glb_of_nonempty h\u03b9 hf hg, },\n  suffices : \u2203 x, g =\u1d50[\u03bc] \u03bb y, g x,\n  by { exact \u27e8(\u03bb y, g this.some), measurable_const, this.some_spec\u27e9, },\n  have h_empty : \u2200 x, {a : \u03b1 | \u2203 (i : \u03b9), f i x = a} = \u2205,\n  { intro x,\n    ext1 y,\n    rw [set.mem_set_of_eq, set.mem_empty_eq, iff_false],\n    exact \u03bb hi, h\u03b9 (nonempty_of_exists hi), },\n  simp_rw h_empty at hg,\n  exact \u27e8hg.exists.some, hg.mono (\u03bb y hy, is_glb.unique hy hg.exists.some_spec)\u27e9,\nend\n\nprotected lemma monotone.measurable [linear_order \u03b2] [order_closed_topology \u03b2] {f : \u03b2 \u2192 \u03b1}\n  (hf : monotone f) : measurable f :=\nsuffices h : \u2200 x, ord_connected (f \u207b\u00b9' Ioi x),\n  from measurable_of_Ioi (\u03bb x, (h x).measurable_set),\n\u03bb x, ord_connected_def.mpr (\u03bb a ha b hb c hc, lt_of_lt_of_le ha (hf hc.1))\n\nlemma ae_measurable_restrict_of_monotone_on [linear_order \u03b2] [order_closed_topology \u03b2]\n  {\u03bc : measure \u03b2} {s : set \u03b2} (hs : measurable_set s) {f : \u03b2 \u2192 \u03b1} (hf : monotone_on f s) :\n  ae_measurable f (\u03bc.restrict s) :=\nhave this : monotone (f \u2218 coe : s \u2192 \u03b1), from \u03bb \u27e8x, hx\u27e9 \u27e8y, hy\u27e9 (hxy : x \u2264 y), hf hx hy hxy,\nae_measurable_restrict_of_measurable_subtype hs this.measurable\n\nprotected lemma antitone.measurable [linear_order \u03b2] [order_closed_topology \u03b2] {f : \u03b2 \u2192 \u03b1}\n  (hf : antitone f) :\n  measurable f :=\n@monotone.measurable (order_dual \u03b1) \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ hf\n\nlemma ae_measurable_restrict_of_antitone_on [linear_order \u03b2] [order_closed_topology \u03b2]\n  {\u03bc : measure \u03b2} {s : set \u03b2} (hs : measurable_set s) {f : \u03b2 \u2192 \u03b1} (hf : antitone_on f s) :\n  ae_measurable f (\u03bc.restrict s) :=\n@ae_measurable_restrict_of_monotone_on (order_dual \u03b1) \u03b2 _ _ \u2039_\u203a _ _ _ _ _ \u2039_\u203a _ _ _ _ hs _ hf\n\nend linear_order\n\n@[measurability]\nlemma measurable.supr_Prop {\u03b1} [measurable_space \u03b1] [complete_lattice \u03b1]\n  (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : measurable f) :\n  measurable (\u03bb b, \u2a06 h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact supr_pos h end)\n  (assume h : \u00acp, begin convert measurable_const, funext, exact supr_neg h end)\n\n@[measurability]\nlemma measurable.infi_Prop {\u03b1} [measurable_space \u03b1] [complete_lattice \u03b1]\n  (p : Prop) {f : \u03b4 \u2192 \u03b1} (hf : measurable f) :\n  measurable (\u03bb b, \u2a05 h : p, f b) :=\nclassical.by_cases\n  (assume h : p, begin convert hf, funext, exact infi_pos h end )\n  (assume h : \u00acp, begin convert measurable_const, funext, exact infi_neg h end)\n\nsection complete_linear_order\n\nvariables [complete_linear_order \u03b1] [order_topology \u03b1] [second_countable_topology \u03b1]\n\n@[measurability]\nlemma measurable_supr {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i)) :\n  measurable (\u03bb b, \u2a06 i, f i b) :=\nmeasurable.is_lub hf $ \u03bb b, is_lub_supr\n\n@[measurability]\nlemma ae_measurable_supr {\u03b9} {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) :\n  ae_measurable (\u03bb b, \u2a06 i, f i b) \u03bc :=\nae_measurable.is_lub hf $ (ae_of_all \u03bc (\u03bb b, is_lub_supr))\n\n@[measurability]\nlemma measurable_infi {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i)) :\n  measurable (\u03bb b, \u2a05 i, f i b) :=\nmeasurable.is_glb hf $ \u03bb b, is_glb_infi\n\n@[measurability]\nlemma ae_measurable_infi {\u03b9} {\u03bc : measure \u03b4} [encodable \u03b9] {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1}\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) :\n  ae_measurable (\u03bb b, \u2a05 i, f i b) \u03bc :=\nae_measurable.is_glb hf $ (ae_of_all \u03bc (\u03bb b, is_glb_infi))\n\nlemma measurable_bsupr {\u03b9} (s : set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : countable s)\n  (hf : \u2200 i, measurable (f i)) : measurable (\u03bb b, \u2a06 i \u2208 s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [supr_subtype'],\n     exact measurable_supr (\u03bb i, hf i) }\n\nlemma ae_measurable_bsupr {\u03b9} {\u03bc : measure \u03b4} (s : set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : countable s)\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) : ae_measurable (\u03bb b, \u2a06 i \u2208 s, f i b) \u03bc :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [supr_subtype'],\n  exact ae_measurable_supr (\u03bb i, hf i),\nend\n\nlemma measurable_binfi {\u03b9} (s : set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : countable s)\n  (hf : \u2200 i, measurable (f i)) : measurable (\u03bb b, \u2a05 i \u2208 s, f i b) :=\nby { haveI : encodable s := hs.to_encodable, simp only [infi_subtype'],\n     exact measurable_infi (\u03bb i, hf i) }\n\nlemma ae_measurable_binfi {\u03b9} {\u03bc : measure \u03b4} (s : set \u03b9) {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} (hs : countable s)\n  (hf : \u2200 i, ae_measurable (f i) \u03bc) : ae_measurable (\u03bb b, \u2a05 i \u2208 s, f i b) \u03bc :=\nbegin\n  haveI : encodable s := hs.to_encodable,\n  simp only [infi_subtype'],\n  exact ae_measurable_infi (\u03bb i, hf i),\nend\n\n/-- `liminf` over a general filter is measurable. See `measurable_liminf` for the version over `\u2115`.\n-/\nlemma measurable_liminf' {\u03b9 \u03b9'} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {u : filter \u03b9} (hf : \u2200 i, measurable (f i))\n  {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 set \u03b9} (hu : u.has_countable_basis p s) (hs : \u2200 i, (s i).countable) :\n  measurable (\u03bb x, liminf u (\u03bb i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.liminf_eq_supr_infi],\n  refine measurable_bsupr _ hu.countable _,\n  exact \u03bb i, measurable_binfi _ (hs i) hf\nend\n\n/-- `limsup` over a general filter is measurable. See `measurable_limsup` for the version over `\u2115`.\n-/\nlemma measurable_limsup' {\u03b9 \u03b9'}  {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {u : filter \u03b9} (hf : \u2200 i, measurable (f i))\n  {p : \u03b9' \u2192 Prop} {s : \u03b9' \u2192 set \u03b9} (hu : u.has_countable_basis p s) (hs : \u2200 i, (s i).countable) :\n  measurable (\u03bb x, limsup u (\u03bb i, f i x)) :=\nbegin\n  simp_rw [hu.to_has_basis.limsup_eq_infi_supr],\n  refine measurable_binfi _ hu.countable _,\n  exact \u03bb i, measurable_bsupr _ (hs i) hf\nend\n\n/-- `liminf` over `\u2115` is measurable. See `measurable_liminf'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_liminf {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i)) :\n  measurable (\u03bb x, liminf at_top (\u03bb i, f i x)) :=\nmeasurable_liminf' hf at_top_countable_basis (\u03bb i, countable_encodable _)\n\n/-- `limsup` over `\u2115` is measurable. See `measurable_limsup'` for a version with a general filter.\n-/\n@[measurability]\nlemma measurable_limsup {f : \u2115 \u2192 \u03b4 \u2192 \u03b1} (hf : \u2200 i, measurable (f i)) :\n  measurable (\u03bb x, limsup at_top (\u03bb i, f i x)) :=\nmeasurable_limsup' hf at_top_countable_basis (\u03bb i, countable_encodable _)\n\nend complete_linear_order\n\nsection conditionally_complete_linear_order\n\nvariables [conditionally_complete_linear_order \u03b1] [order_topology \u03b1] [second_countable_topology \u03b1]\n\nlemma measurable_cSup {\u03b9} {f : \u03b9 \u2192 \u03b4 \u2192 \u03b1} {s : set \u03b9} (hs : s.countable)\n  (hf : \u2200 i, measurable (f i)) (bdd : \u2200 x, bdd_above ((\u03bb i, f i x) '' s)) :\n  measurable (\u03bb x, Sup ((\u03bb i, f i x) '' s)) :=\nbegin\n  cases eq_empty_or_nonempty s with h2s h2s,\n  { simp [h2s, measurable_const] },\n  { apply measurable_of_Iic, intro y,\n    simp_rw [preimage, mem_Iic, cSup_le_iff (bdd _) (h2s.image _), ball_image_iff, set_of_forall],\n    exact measurable_set.bInter hs (\u03bb i hi, measurable_set_le (hf i) measurable_const) }\nend\n\nend conditionally_complete_linear_order\n\n/-- Convert a `homeomorph` to a `measurable_equiv`. -/\ndef homemorph.to_measurable_equiv (h : \u03b1 \u2243\u209c \u03b2) : \u03b1 \u2243\u1d50 \u03b2 :=\n{ to_equiv := h.to_equiv,\n  measurable_to_fun := h.continuous_to_fun.measurable,\n  measurable_inv_fun := h.continuous_inv_fun.measurable }\n\nprotected lemma is_finite_measure_on_compacts.map\n  {\u03b1 : Type*} {m0 : measurable_space \u03b1} [topological_space \u03b1] [opens_measurable_space \u03b1]\n  {\u03b2 : Type*} [measurable_space \u03b2] [topological_space \u03b2] [borel_space \u03b2]\n  [t2_space \u03b2] (\u03bc : measure \u03b1) [is_finite_measure_on_compacts \u03bc] (f : \u03b1 \u2243\u209c \u03b2) :\n  is_finite_measure_on_compacts (measure.map f \u03bc) :=\n\u27e8begin\n  assume K hK,\n  rw [measure.map_apply f.measurable hK.measurable_set],\n  apply is_compact.measure_lt_top,\n  rwa f.compact_preimage\nend\u27e9\n\nend borel_space\n\ninstance empty.borel_space : borel_space empty := \u27e8borel_eq_top_of_discrete.symm\u27e9\ninstance unit.borel_space : borel_space unit := \u27e8borel_eq_top_of_discrete.symm\u27e9\ninstance bool.borel_space : borel_space bool := \u27e8borel_eq_top_of_discrete.symm\u27e9\ninstance nat.borel_space : borel_space \u2115 := \u27e8borel_eq_top_of_discrete.symm\u27e9\ninstance int.borel_space : borel_space \u2124 := \u27e8borel_eq_top_of_discrete.symm\u27e9\ninstance rat.borel_space : borel_space \u211a := \u27e8borel_eq_top_of_encodable.symm\u27e9\n\n@[priority 900]\ninstance is_R_or_C.measurable_space {\ud835\udd5c : Type*} [is_R_or_C \ud835\udd5c] : measurable_space \ud835\udd5c := borel \ud835\udd5c\n@[priority 900]\ninstance is_R_or_C.borel_space {\ud835\udd5c : Type*} [is_R_or_C \ud835\udd5c] : borel_space \ud835\udd5c := \u27e8rfl\u27e9\n\n/- Instances on `real` and `complex` are special cases of `is_R_or_C` but without these instances,\nLean fails to prove `borel_space (\u03b9 \u2192 \u211d)`, so we leave them here. -/\n\ninstance real.measurable_space : measurable_space \u211d := borel \u211d\ninstance real.borel_space : borel_space \u211d := \u27e8rfl\u27e9\n\ninstance nnreal.measurable_space : measurable_space \u211d\u22650 := subtype.measurable_space\ninstance nnreal.borel_space : borel_space \u211d\u22650 := subtype.borel_space _\n\ninstance ennreal.measurable_space : measurable_space \u211d\u22650\u221e := borel \u211d\u22650\u221e\ninstance ennreal.borel_space : borel_space \u211d\u22650\u221e := \u27e8rfl\u27e9\n\ninstance ereal.measurable_space : measurable_space ereal := borel ereal\ninstance ereal.borel_space : borel_space ereal := \u27e8rfl\u27e9\n\ninstance complex.measurable_space : measurable_space \u2102 := borel \u2102\ninstance complex.borel_space : borel_space \u2102 := \u27e8rfl\u27e9\n\n/-- One can cut out `\u211d\u22650\u221e` into the sets `{0}`, `Ico (t^n) (t^(n+1))` for `n : \u2124` and `{\u221e}`. This\ngives a way to compute the measure of a set in terms of sets on which a given function `f` does not\nfluctuate by more than `t`. -/\nlemma measure_eq_measure_preimage_add_measure_tsum_Ico_zpow [measurable_space \u03b1] (\u03bc : measure \u03b1)\n  {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) {s : set \u03b1} (hs : measurable_set s) {t : \u211d\u22650} (ht : 1 < t) :\n  \u03bc s = \u03bc (s \u2229 f\u207b\u00b9' {0}) + \u03bc (s \u2229 f\u207b\u00b9' {\u221e}) + \u2211' (n : \u2124), \u03bc (s \u2229 f\u207b\u00b9' (Ico (t^n) (t^(n+1)))) :=\nbegin\n  have A : \u03bc s = \u03bc (s \u2229 f\u207b\u00b9' {0}) + \u03bc (s \u2229 f\u207b\u00b9' (Ioi 0)),\n  { rw \u2190 measure_union,\n    { congr' 1,\n      ext x,\n      have : 0 = f x \u2228 0 < f x := eq_or_lt_of_le bot_le,\n      rw eq_comm at this,\n      simp only [\u2190and_or_distrib_left, this, mem_singleton_iff, mem_inter_eq, and_true,\n        mem_union_eq, mem_Ioi, mem_preimage], },\n    { apply disjoint_left.2 (\u03bb x hx h'x, _),\n      have : 0 < f x := h'x.2,\n      exact lt_irrefl 0 (this.trans_le hx.2.le) },\n    { exact hs.inter (hf measurable_set_Ioi) } },\n  have B : \u03bc (s \u2229 f\u207b\u00b9' (Ioi 0)) = \u03bc (s \u2229 f\u207b\u00b9' {\u221e}) + \u03bc (s \u2229 f\u207b\u00b9' (Ioo 0 \u221e)),\n  { rw \u2190 measure_union,\n    { rw \u2190 inter_union_distrib_left,\n      congr,\n      ext x,\n      simp only [mem_singleton_iff, mem_union_eq, mem_Ioo, mem_Ioi, mem_preimage],\n      have H : f x = \u221e \u2228 f x < \u221e := eq_or_lt_of_le le_top,\n      cases H,\n      { simp only [H, eq_self_iff_true, or_false, with_top.zero_lt_top, not_top_lt, and_false] },\n      { simp only [H, H.ne, and_true, false_or] } },\n    { apply disjoint_left.2 (\u03bb x hx h'x, _),\n      have : f x < \u221e := h'x.2.2,\n      exact lt_irrefl _ (this.trans_le (le_of_eq hx.2.symm)) },\n    { exact hs.inter (hf measurable_set_Ioo) } },\n  have C : \u03bc (s \u2229 f\u207b\u00b9' (Ioo 0 \u221e)) = \u2211' (n : \u2124), \u03bc (s \u2229 f\u207b\u00b9' (Ico (t^n) (t^(n+1)))),\n  { rw [\u2190 measure_Union, ennreal.Ioo_zero_top_eq_Union_Ico_zpow (ennreal.one_lt_coe_iff.2 ht)\n         ennreal.coe_ne_top, preimage_Union, inter_Union],\n    { assume i j,\n      simp only [function.on_fun],\n      wlog h : i \u2264 j := le_total i j using [i j, j i] tactic.skip,\n      { assume hij,\n        replace hij : i + 1 \u2264 j := lt_of_le_of_ne h hij,\n        apply disjoint_left.2 (\u03bb x hx h'x, lt_irrefl (f x) _),\n        calc f x < t ^ (i + 1) : hx.2.2\n        ... \u2264 t ^ j : ennreal.zpow_le_of_le (ennreal.one_le_coe_iff.2 ht.le) hij\n        ... \u2264 f x : h'x.2.1 },\n      { assume hij,\n        rw disjoint.comm,\n        exact this hij.symm } },\n    { assume n,\n      exact hs.inter (hf measurable_set_Ico) } },\n  rw [A, B, C, add_assoc],\nend\n\nsection metric_space\n\nvariables [metric_space \u03b1] [measurable_space \u03b1] [opens_measurable_space \u03b1]\nvariables [measurable_space \u03b2] {x : \u03b1} {\u03b5 : \u211d}\n\nopen metric\n\n@[measurability]\nlemma measurable_set_ball : measurable_set (metric.ball x \u03b5) :=\nmetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_set_closed_ball : measurable_set (metric.closed_ball x \u03b5) :=\nmetric.is_closed_ball.measurable_set\n\n@[measurability]\nlemma measurable_inf_dist {s : set \u03b1} : measurable (\u03bb x, inf_dist x s) :=\n(continuous_inf_dist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_dist {f : \u03b2 \u2192 \u03b1} (hf : measurable f) {s : set \u03b1} :\n  measurable (\u03bb x, inf_dist (f x) s) :=\nmeasurable_inf_dist.comp hf\n\n@[measurability]\nlemma measurable_inf_nndist {s : set \u03b1} : measurable (\u03bb x, inf_nndist x s) :=\n(continuous_inf_nndist_pt s).measurable\n\n@[measurability]\nlemma measurable.inf_nndist {f : \u03b2 \u2192 \u03b1} (hf : measurable f) {s : set \u03b1} :\n  measurable (\u03bb x, inf_nndist (f x) s) :=\nmeasurable_inf_nndist.comp hf\n\nsection\nvariables [second_countable_topology \u03b1]\n\n@[measurability]\nlemma measurable_dist : measurable (\u03bb p : \u03b1 \u00d7 \u03b1, dist p.1 p.2) :=\ncontinuous_dist.measurable\n\n@[measurability]\nlemma measurable.dist {f g : \u03b2 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb b, dist (f b) (g b)) :=\n(@continuous_dist \u03b1 _).measurable2 hf hg\n\n@[measurability]\nlemma measurable_nndist : measurable (\u03bb p : \u03b1 \u00d7 \u03b1, nndist p.1 p.2) :=\ncontinuous_nndist.measurable\n\n@[measurability]\nlemma measurable.nndist {f g : \u03b2 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb b, nndist (f b) (g b)) :=\n(@continuous_nndist \u03b1 _).measurable2 hf hg\n\nend\n\n/-- If a set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to the measure of its closure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening {\u03bc : measure \u03b1} {s : set \u03b1}\n  (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) :\n  tendsto (\u03bb r, \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc (closure s))) :=\nbegin\n  have A : tendsto (\u03bb r, \u03bc (cthickening r s)) (\ud835\udcdd[Ioi 0] 0) (\ud835\udcdd (\u03bc (closure s))),\n  { rw closure_eq_Inter_cthickening,\n    exact tendsto_measure_bInter_gt (\u03bb r hr, is_closed_cthickening.measurable_set)\n      (\u03bb i j ipos ij, cthickening_mono ij _) hs },\n  have B : tendsto (\u03bb r, \u03bc (cthickening r s)) (\ud835\udcdd[Iic 0] 0) (\ud835\udcdd (\u03bc (closure s))),\n  { apply tendsto.congr' _ tendsto_const_nhds,\n    filter_upwards [self_mem_nhds_within] with _ hr,\n    rw cthickening_of_nonpos hr, },\n  convert B.sup A,\n  exact (nhds_left_sup_nhds_right' 0).symm,\nend\n\n/-- If a closed set has a closed thickening with finite measure, then the measure of its `r`-closed\nthickenings converges to its measure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening_of_is_closed {\u03bc : measure \u03b1} {s : set \u03b1}\n  (hs : \u2203 R > 0, \u03bc (cthickening R s) \u2260 \u221e) (h's : is_closed s) :\n  tendsto (\u03bb r, \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) :=\nbegin\n  convert tendsto_measure_cthickening hs,\n  exact h's.closure_eq.symm\nend\n\n/-- Given a compact set in a proper space, the measure of its `r`-closed thickenings converges to\nits measure as `r` tends to `0`. -/\nlemma tendsto_measure_cthickening_of_is_compact [proper_space \u03b1] {\u03bc : measure \u03b1}\n  [is_finite_measure_on_compacts \u03bc] {s : set \u03b1} (hs : is_compact s) :\n  tendsto (\u03bb r, \u03bc (cthickening r s)) (\ud835\udcdd 0) (\ud835\udcdd (\u03bc s)) :=\ntendsto_measure_cthickening_of_is_closed\n  \u27e81, zero_lt_one, (bounded.measure_lt_top hs.bounded.cthickening).ne\u27e9 hs.is_closed\n\nend metric_space\n\nsection emetric_space\n\nvariables [emetric_space \u03b1] [measurable_space \u03b1] [opens_measurable_space \u03b1]\nvariables [measurable_space \u03b2] {x : \u03b1} {\u03b5 : \u211d\u22650\u221e}\n\nopen emetric\n\n@[measurability]\nlemma measurable_set_eball : measurable_set (emetric.ball x \u03b5) :=\nemetric.is_open_ball.measurable_set\n\n@[measurability]\nlemma measurable_edist_right : measurable (edist x) :=\n(continuous_const.edist continuous_id).measurable\n\n@[measurability]\nlemma measurable_edist_left : measurable (\u03bb y, edist y x) :=\n(continuous_id.edist continuous_const).measurable\n\n@[measurability]\nlemma measurable_inf_edist {s : set \u03b1} : measurable (\u03bb x, inf_edist x s) :=\ncontinuous_inf_edist.measurable\n\n@[measurability]\nlemma measurable.inf_edist {f : \u03b2 \u2192 \u03b1} (hf : measurable f) {s : set \u03b1} :\n  measurable (\u03bb x, inf_edist (f x) s) :=\nmeasurable_inf_edist.comp hf\n\nvariables [second_countable_topology \u03b1]\n\n@[measurability]\nlemma measurable_edist : measurable (\u03bb p : \u03b1 \u00d7 \u03b1, edist p.1 p.2) :=\ncontinuous_edist.measurable\n\n@[measurability]\nlemma measurable.edist {f g : \u03b2 \u2192 \u03b1} (hf : measurable f) (hg : measurable g) :\n  measurable (\u03bb b, edist (f b) (g b)) :=\n(@continuous_edist \u03b1 _).measurable2 hf hg\n\n@[measurability]\nlemma ae_measurable.edist {f g : \u03b2 \u2192 \u03b1} {\u03bc : measure \u03b2}\n  (hf : ae_measurable f \u03bc) (hg : ae_measurable g \u03bc) : ae_measurable (\u03bb a, edist (f a) (g a)) \u03bc :=\n(@continuous_edist \u03b1 _).ae_measurable2 hf hg\n\nend emetric_space\n\nnamespace real\nopen measurable_space measure_theory\n\nlemma borel_eq_generate_from_Ioo_rat :\n  borel \u211d = generate_from (\u22c3(a b : \u211a) (h : a < b), {Ioo a b}) :=\nis_topological_basis_Ioo_rat.borel_eq_generate_from\n\nlemma is_pi_system_Ioo_rat : @is_pi_system \u211d (\u22c3 (a b : \u211a) (h : a < b), {Ioo a b})  :=\nbegin\n  convert is_pi_system_Ioo (coe : \u211a \u2192 \u211d) (coe : \u211a \u2192 \u211d),\n  ext x,\n  simp [eq_comm]\nend\n\n/-- The intervals `(-(n + 1), (n + 1))` form a finite spanning sets in the set of open intervals\nwith rational endpoints for a locally finite measure `\u03bc` on `\u211d`. -/\ndef finite_spanning_sets_in_Ioo_rat (\u03bc : measure \u211d) [is_locally_finite_measure \u03bc] :\n  \u03bc.finite_spanning_sets_in (\u22c3 (a b : \u211a) (h : a < b), {Ioo a b}) :=\n{ set := \u03bb n, Ioo (-(n + 1)) (n + 1),\n  set_mem := \u03bb n,\n    begin\n      simp only [mem_Union, mem_singleton_iff],\n      refine \u27e8-(n + 1), n + 1, _, by norm_cast\u27e9,\n      exact (neg_nonpos.2 (@nat.cast_nonneg \u211a _ (n + 1))).trans_lt n.cast_add_one_pos\n    end,\n  finite := \u03bb n, measure_Ioo_lt_top,\n  spanning := Union_eq_univ_iff.2 $ \u03bb x,\n    \u27e8\u230a|x|\u230b\u208a, neg_lt.1 ((neg_le_abs_self x).trans_lt (nat.lt_floor_add_one _)),\n      (le_abs_self x).trans_lt (nat.lt_floor_add_one _)\u27e9 }\n\nlemma measure_ext_Ioo_rat {\u03bc \u03bd : measure \u211d} [is_locally_finite_measure \u03bc]\n  (h : \u2200 a b : \u211a, \u03bc (Ioo a b) = \u03bd (Ioo a b)) : \u03bc = \u03bd :=\n(finite_spanning_sets_in_Ioo_rat \u03bc).ext borel_eq_generate_from_Ioo_rat is_pi_system_Ioo_rat $\n  by { simp only [mem_Union, mem_singleton_iff], rintro _ \u27e8a, b, -, rfl\u27e9, apply h }\n\nlemma borel_eq_generate_from_Iio_rat :\n  borel \u211d = generate_from (\u22c3 a : \u211a, {Iio a}) :=\nbegin\n  let g : measurable_space \u211d := generate_from (\u22c3 a : \u211a, {Iio a}),\n  refine le_antisymm _ _,\n  { rw borel_eq_generate_from_Ioo_rat,\n    refine generate_from_le (\u03bb t, _),\n    simp only [mem_Union, mem_singleton_iff], rintro \u27e8a, b, h, rfl\u27e9,\n    rw (set.ext (\u03bb x, _) : Ioo (a : \u211d) b = (\u22c3c>a, (Iio c)\u1d9c) \u2229 Iio b),\n    { have hg : \u2200 q : \u211a, g.measurable_set' (Iio q) :=\n        \u03bb q, generate_measurable.basic (Iio q) (by simp),\n      refine @measurable_set.inter _ g _ _ _ (hg _),\n      refine @measurable_set.bUnion _ _ g _ _ (countable_encodable _) (\u03bb c h, _),\n      exact @measurable_set.compl _ _ g (hg _) },\n    { suffices : x < \u2191b \u2192 (\u2191a < x \u2194 \u2203 (i : \u211a), a < i \u2227 \u2191i \u2264 x), by simpa,\n      refine \u03bb _, \u27e8\u03bb h, _, \u03bb \u27e8i, hai, hix\u27e9, (rat.cast_lt.2 hai).trans_le hix\u27e9,\n      rcases exists_rat_btwn h with \u27e8c, ac, cx\u27e9,\n      exact \u27e8c, rat.cast_lt.1 ac, cx.le\u27e9 } },\n  { refine measurable_space.generate_from_le (\u03bb _, _),\n    simp only [mem_Union, mem_singleton_iff], rintro \u27e8r, rfl\u27e9, exact measurable_set_Iio }\nend\n\nend real\n\nvariable [measurable_space \u03b1]\n\n@[measurability]\nlemma measurable_real_to_nnreal : measurable (real.to_nnreal) :=\ncontinuous_real_to_nnreal.measurable\n\n@[measurability]\nlemma measurable.real_to_nnreal {f : \u03b1 \u2192 \u211d} (hf : measurable f) :\n  measurable (\u03bb x, real.to_nnreal (f x)) :=\nmeasurable_real_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.real_to_nnreal {f : \u03b1 \u2192 \u211d} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, real.to_nnreal (f x)) \u03bc :=\nmeasurable_real_to_nnreal.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_real : measurable (coe : \u211d\u22650 \u2192 \u211d) :=\nnnreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_real {f : \u03b1 \u2192 \u211d\u22650} (hf : measurable f) :\n  measurable (\u03bb x, (f x : \u211d)) :=\nmeasurable_coe_nnreal_real.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_real {f : \u03b1 \u2192 \u211d\u22650} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x : \u211d)) \u03bc :=\nmeasurable_coe_nnreal_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_nnreal_ennreal : measurable (coe : \u211d\u22650 \u2192 \u211d\u22650\u221e) :=\nennreal.continuous_coe.measurable\n\n@[measurability]\nlemma measurable.coe_nnreal_ennreal {f : \u03b1 \u2192 \u211d\u22650} (hf : measurable f) :\n  measurable (\u03bb x, (f x : \u211d\u22650\u221e)) :=\nennreal.continuous_coe.measurable.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_nnreal_ennreal {f : \u03b1 \u2192 \u211d\u22650} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x : \u211d\u22650\u221e)) \u03bc :=\nennreal.continuous_coe.measurable.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable.ennreal_of_real {f : \u03b1 \u2192 \u211d} (hf : measurable f) :\n  measurable (\u03bb x, ennreal.of_real (f x)) :=\nennreal.continuous_of_real.measurable.comp hf\n\n/-- The set of finite `\u211d\u22650\u221e` numbers is `measurable_equiv` to `\u211d\u22650`. -/\ndef measurable_equiv.ennreal_equiv_nnreal : {r : \u211d\u22650\u221e | r \u2260 \u221e} \u2243\u1d50 \u211d\u22650 :=\nennreal.ne_top_homeomorph_nnreal.to_measurable_equiv\n\nnamespace ennreal\n\nlemma measurable_of_measurable_nnreal {f : \u211d\u22650\u221e \u2192 \u03b1}\n  (h : measurable (\u03bb p : \u211d\u22650, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_singleton \u221e\n  (measurable_equiv.ennreal_equiv_nnreal.symm.measurable_comp_iff.1 h)\n\n/-- `\u211d\u22650\u221e` is `measurable_equiv` to `\u211d\u22650 \u2295 unit`. -/\ndef ennreal_equiv_sum : \u211d\u22650\u221e \u2243\u1d50 \u211d\u22650 \u2295 unit :=\n{ measurable_to_fun  := measurable_of_measurable_nnreal measurable_inl,\n  measurable_inv_fun := measurable_sum measurable_coe_nnreal_ennreal\n    (@measurable_const \u211d\u22650\u221e unit _ _ \u221e),\n  .. equiv.option_equiv_sum_punit \u211d\u22650 }\n\nopen function (uncurry)\n\nlemma measurable_of_measurable_nnreal_prod [measurable_space \u03b2] [measurable_space \u03b3]\n  {f : \u211d\u22650\u221e \u00d7 \u03b2 \u2192 \u03b3} (H\u2081 : measurable (\u03bb p : \u211d\u22650 \u00d7 \u03b2, f (p.1, p.2)))\n  (H\u2082 : measurable (\u03bb x, f (\u221e, x))) :\n  measurable f :=\nlet e : \u211d\u22650\u221e \u00d7 \u03b2 \u2243\u1d50 \u211d\u22650 \u00d7 \u03b2 \u2295 unit \u00d7 \u03b2 :=\n  (ennreal_equiv_sum.prod_congr (measurable_equiv.refl \u03b2)).trans\n    (measurable_equiv.sum_prod_distrib _ _ _) in\ne.symm.measurable_comp_iff.1 $ measurable_sum H\u2081 (H\u2082.comp measurable_id.snd)\n\nlemma measurable_of_measurable_nnreal_nnreal [measurable_space \u03b2]\n  {f : \u211d\u22650\u221e \u00d7 \u211d\u22650\u221e \u2192 \u03b2} (h\u2081 : measurable (\u03bb p : \u211d\u22650 \u00d7 \u211d\u22650, f (p.1, p.2)))\n  (h\u2082 : measurable (\u03bb r : \u211d\u22650, f (\u221e, r))) (h\u2083 : measurable (\u03bb r : \u211d\u22650, f (r, \u221e))) :\n  measurable f :=\nmeasurable_of_measurable_nnreal_prod\n  (measurable_swap_iff.1 $ measurable_of_measurable_nnreal_prod (h\u2081.comp measurable_swap) h\u2083)\n  (measurable_of_measurable_nnreal h\u2082)\n\n@[measurability]\nlemma measurable_of_real : measurable ennreal.of_real :=\nennreal.continuous_of_real.measurable\n\n@[measurability]\nlemma measurable_to_real : measurable ennreal.to_real :=\nennreal.measurable_of_measurable_nnreal measurable_coe_nnreal_real\n\n@[measurability]\nlemma measurable_to_nnreal : measurable ennreal.to_nnreal :=\nennreal.measurable_of_measurable_nnreal measurable_id\n\ninstance : has_measurable_mul\u2082 \u211d\u22650\u221e :=\nbegin\n  refine \u27e8measurable_of_measurable_nnreal_nnreal _ _ _\u27e9,\n  { simp only [\u2190 ennreal.coe_mul, measurable_mul.coe_nnreal_ennreal] },\n  { simp only [ennreal.top_mul, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const },\n  { simp only [ennreal.mul_top, ennreal.coe_eq_zero],\n    exact measurable_const.piecewise (measurable_set_singleton _) measurable_const }\nend\n\ninstance : has_measurable_sub\u2082 \u211d\u22650\u221e :=\n\u27e8by apply measurable_of_measurable_nnreal_nnreal;\n  simp [\u2190 with_top.coe_sub, continuous_sub.measurable.coe_nnreal_ennreal]\u27e9\n\ninstance : has_measurable_inv \u211d\u22650\u221e := \u27e8continuous_inv.measurable\u27e9\n\nend ennreal\n\n@[measurability]\nlemma measurable.ennreal_to_nnreal {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) :\n  measurable (\u03bb x, (f x).to_nnreal) :=\nennreal.measurable_to_nnreal.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_nnreal {f : \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x).to_nnreal) \u03bc :=\nennreal.measurable_to_nnreal.comp_ae_measurable hf\n\nlemma measurable_coe_nnreal_ennreal_iff {f : \u03b1 \u2192 \u211d\u22650} :\n  measurable (\u03bb x, (f x : \u211d\u22650\u221e)) \u2194 measurable f :=\n\u27e8\u03bb h, h.ennreal_to_nnreal, \u03bb h, h.coe_nnreal_ennreal\u27e9\n\n@[measurability]\nlemma measurable.ennreal_to_real {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) :\n  measurable (\u03bb x, ennreal.to_real (f x)) :=\nennreal.measurable_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ennreal_to_real {f : \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, ennreal.to_real (f x)) \u03bc :=\nennreal.measurable_to_real.comp_ae_measurable hf\n\n/-- note: `\u211d\u22650\u221e` can probably be generalized in a future version of this lemma. -/\n@[measurability]\nlemma measurable.ennreal_tsum {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} (h : \u2200 i, measurable (f i)) :\n  measurable (\u03bb x, \u2211' i, f i x) :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply measurable_supr,\n  exact \u03bb s, s.measurable_sum (\u03bb i _, h i) }\n\n@[measurability]\nlemma measurable.ennreal_tsum' {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} (h : \u2200 i, measurable (f i)) :\n  measurable (\u2211' i, f i) :=\nbegin\n  convert measurable.ennreal_tsum h,\n  ext1 x,\n  exact tsum_apply (pi.summable.2 (\u03bb _, ennreal.summable)),\nend\n\n@[measurability]\nlemma measurable.nnreal_tsum {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650} (h : \u2200 i, measurable (f i)) :\n  measurable (\u03bb x, \u2211' i, f i x) :=\nbegin\n  simp_rw [nnreal.tsum_eq_to_nnreal_tsum],\n  exact (measurable.ennreal_tsum (\u03bb i, (h i).coe_nnreal_ennreal)).ennreal_to_nnreal,\nend\n\n@[measurability]\nlemma ae_measurable.ennreal_tsum {\u03b9} [encodable \u03b9] {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : measure \u03b1}\n  (h : \u2200 i, ae_measurable (f i) \u03bc) :\n  ae_measurable (\u03bb x, \u2211' i, f i x) \u03bc :=\nby { simp_rw [ennreal.tsum_eq_supr_sum], apply ae_measurable_supr,\n  exact \u03bb s, finset.ae_measurable_sum s (\u03bb i _, h i) }\n\n@[measurability]\nlemma measurable_coe_real_ereal : measurable (coe : \u211d \u2192 ereal) :=\ncontinuous_coe_real_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_real_ereal {f : \u03b1 \u2192 \u211d} (hf : measurable f) :\n  measurable (\u03bb x, (f x : ereal)) :=\nmeasurable_coe_real_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_real_ereal {f : \u03b1 \u2192 \u211d} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x : ereal)) \u03bc :=\nmeasurable_coe_real_ereal.comp_ae_measurable hf\n\n/-- The set of finite `ereal` numbers is `measurable_equiv` to `\u211d`. -/\ndef measurable_equiv.ereal_equiv_real : ({\u22a5, \u22a4} : set ereal).compl \u2243\u1d50 \u211d :=\nereal.ne_bot_top_homeomorph_real.to_measurable_equiv\n\nlemma ereal.measurable_of_measurable_real {f : ereal \u2192 \u03b1}\n  (h : measurable (\u03bb p : \u211d, f p)) : measurable f :=\nmeasurable_of_measurable_on_compl_finite {\u22a5, \u22a4} (by simp)\n  (measurable_equiv.ereal_equiv_real.symm.measurable_comp_iff.1 h)\n\n@[measurability]\nlemma measurable_ereal_to_real : measurable ereal.to_real :=\nereal.measurable_of_measurable_real (by simpa using measurable_id)\n\n@[measurability]\nlemma measurable.ereal_to_real {f : \u03b1 \u2192 ereal} (hf : measurable f) :\n  measurable (\u03bb x, (f x).to_real) :=\nmeasurable_ereal_to_real.comp hf\n\n@[measurability]\nlemma ae_measurable.ereal_to_real {f : \u03b1 \u2192 ereal} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x).to_real) \u03bc :=\nmeasurable_ereal_to_real.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_coe_ennreal_ereal : measurable (coe : \u211d\u22650\u221e \u2192 ereal) :=\ncontinuous_coe_ennreal_ereal.measurable\n\n@[measurability]\nlemma measurable.coe_ereal_ennreal {f : \u03b1 \u2192 \u211d\u22650\u221e} (hf : measurable f) :\n  measurable (\u03bb x, (f x : ereal)) :=\nmeasurable_coe_ennreal_ereal.comp hf\n\n@[measurability]\nlemma ae_measurable.coe_ereal_ennreal {f : \u03b1 \u2192 \u211d\u22650\u221e} {\u03bc : measure \u03b1} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb x, (f x : ereal)) \u03bc :=\nmeasurable_coe_ennreal_ereal.comp_ae_measurable hf\n\nsection normed_group\n\nvariables [normed_group \u03b1] [opens_measurable_space \u03b1] [measurable_space \u03b2]\n\n@[measurability]\nlemma measurable_norm : measurable (norm : \u03b1 \u2192 \u211d) :=\ncontinuous_norm.measurable\n\n@[measurability]\nlemma measurable.norm {f : \u03b2 \u2192 \u03b1} (hf : measurable f) : measurable (\u03bb a, norm (f a)) :=\nmeasurable_norm.comp hf\n\n@[measurability]\nlemma ae_measurable.norm {f : \u03b2 \u2192 \u03b1} {\u03bc : measure \u03b2} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb a, norm (f a)) \u03bc :=\nmeasurable_norm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_nnnorm : measurable (nnnorm : \u03b1 \u2192 \u211d\u22650) :=\ncontinuous_nnnorm.measurable\n\n@[measurability]\nlemma measurable.nnnorm {f : \u03b2 \u2192 \u03b1} (hf : measurable f) : measurable (\u03bb a, nnnorm (f a)) :=\nmeasurable_nnnorm.comp hf\n\n@[measurability]\nlemma ae_measurable.nnnorm {f : \u03b2 \u2192 \u03b1} {\u03bc : measure \u03b2} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb a, nnnorm (f a)) \u03bc :=\nmeasurable_nnnorm.comp_ae_measurable hf\n\n@[measurability]\nlemma measurable_ennnorm : measurable (\u03bb x : \u03b1, (nnnorm x : \u211d\u22650\u221e)) :=\nmeasurable_nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma measurable.ennnorm {f : \u03b2 \u2192 \u03b1} (hf : measurable f) :\n  measurable (\u03bb a, (nnnorm (f a) : \u211d\u22650\u221e)) :=\nhf.nnnorm.coe_nnreal_ennreal\n\n@[measurability]\nlemma ae_measurable.ennnorm {f : \u03b2 \u2192 \u03b1} {\u03bc : measure \u03b2} (hf : ae_measurable f \u03bc) :\n  ae_measurable (\u03bb a, (nnnorm (f a) : \u211d\u22650\u221e)) \u03bc :=\nmeasurable_ennnorm.comp_ae_measurable hf\n\nend normed_group\n\nsection limits\n\nvariables [measurable_space \u03b2] [metric_space \u03b2] [borel_space \u03b2]\n\nopen metric\n\n/-- A limit (over a general filter) of measurable `\u211d\u22650\u221e` valued functions is measurable. -/\nlemma measurable_of_tendsto_ennreal' {\u03b9} {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} {g : \u03b1 \u2192 \u211d\u22650\u221e} (u : filter \u03b9)\n  [ne_bot u] [is_countably_generated u] (hf : \u2200 i, measurable (f i)) (lim : tendsto f u (\ud835\udcdd g)) :\n  measurable g :=\nbegin\n  rcases u.exists_seq_tendsto with \u27e8x, hx\u27e9,\n  rw [tendsto_pi_nhds] at lim,\n  have : (\u03bb y, liminf at_top (\u03bb n, (f (x n) y : \u211d\u22650\u221e))) = g :=\n    by { ext1 y, exact ((lim y).comp hx).liminf_eq, },\n  rw \u2190 this,\n  show measurable (\u03bb y, liminf at_top (\u03bb n, (f (x n) y : \u211d\u22650\u221e))),\n  exact measurable_liminf (\u03bb n, hf (x n)),\nend\n\n/-- A sequential limit of measurable `\u211d\u22650\u221e` valued functions is measurable. -/\nlemma measurable_of_tendsto_ennreal {f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650\u221e} {g : \u03b1 \u2192 \u211d\u22650\u221e}\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f at_top (\ud835\udcdd g)) : measurable g :=\nmeasurable_of_tendsto_ennreal' at_top hf lim\n\n/-- A limit (over a general filter) of measurable `\u211d\u22650` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal' {\u03b9} {f : \u03b9 \u2192 \u03b1 \u2192 \u211d\u22650} {g : \u03b1 \u2192 \u211d\u22650} (u : filter \u03b9)\n  [ne_bot u] [is_countably_generated u] (hf : \u2200 i, measurable (f i)) (lim : tendsto f u (\ud835\udcdd g)) :\n  measurable g :=\nbegin\n  simp_rw [\u2190 measurable_coe_nnreal_ennreal_iff] at hf \u22a2,\n  refine measurable_of_tendsto_ennreal' u hf _,\n  rw tendsto_pi_nhds at lim \u22a2,\n  exact \u03bb x, (ennreal.continuous_coe.tendsto (g x)).comp (lim x),\nend\n\n/-- A sequential limit of measurable `\u211d\u22650` valued functions is measurable. -/\nlemma measurable_of_tendsto_nnreal {f : \u2115 \u2192 \u03b1 \u2192 \u211d\u22650} {g : \u03b1 \u2192 \u211d\u22650}\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f at_top (\ud835\udcdd g)) : measurable g :=\nmeasurable_of_tendsto_nnreal' at_top hf lim\n\n/-- A limit (over a general filter) of measurable functions valued in a metric space is measurable.\n-/\nlemma measurable_of_tendsto_metric' {\u03b9} {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (u : filter \u03b9) [ne_bot u] [is_countably_generated u]\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f u (\ud835\udcdd g)) :\n  measurable g :=\nbegin\n  apply measurable_of_is_closed', intros s h1s h2s h3s,\n  have : measurable (\u03bb x, inf_nndist (g x) s),\n  { suffices : tendsto (\u03bb i x, inf_nndist (f i x) s) u (\ud835\udcdd (\u03bb x, inf_nndist (g x) s)),\n      from measurable_of_tendsto_nnreal' u (\u03bb i, (hf i).inf_nndist) this,\n    rw [tendsto_pi_nhds] at lim \u22a2, intro x,\n    exact ((continuous_inf_nndist_pt s).tendsto (g x)).comp (lim x) },\n  have h4s : g \u207b\u00b9' s = (\u03bb x, inf_nndist (g x) s) \u207b\u00b9' {0},\n  { ext x, simp [h1s, \u2190 h1s.mem_iff_inf_dist_zero h2s, \u2190 nnreal.coe_eq_zero] },\n  rw [h4s], exact this (measurable_set_singleton 0),\nend\n\n/-- A sequential limit of measurable functions valued in a metric space is measurable. -/\nlemma measurable_of_tendsto_metric {f : \u2115 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f at_top (\ud835\udcdd g)) :\n  measurable g :=\nmeasurable_of_tendsto_metric' at_top hf lim\n\n/-- A limit (over a general filter) of measurable functions valued in a metrizable space is\nmeasurable. -/\nlemma measurable_of_tendsto_metrizable'\n  {\u03b2 : Type*} [topological_space \u03b2] [metrizable_space \u03b2]\n  [measurable_space \u03b2] [borel_space \u03b2] {\u03b9} {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (u : filter \u03b9) [ne_bot u] [is_countably_generated u]\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f u (\ud835\udcdd g)) :\n  measurable g :=\nbegin\n  letI : metric_space \u03b2 := metrizable_space_metric \u03b2,\n  exact measurable_of_tendsto_metric' u hf lim\nend\n\n/-- A sequential limit of measurable functions valued in a metrizable space is measurable. -/\nlemma measurable_of_tendsto_metrizable {\u03b2 : Type*} [topological_space \u03b2] [metrizable_space \u03b2]\n  [measurable_space \u03b2] [borel_space \u03b2] {f : \u2115 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (hf : \u2200 i, measurable (f i)) (lim : tendsto f at_top (\ud835\udcdd g)) :\n  measurable g :=\nmeasurable_of_tendsto_metrizable' at_top hf lim\n\nlemma ae_measurable_of_tendsto_metric_ae {\u03b9 : Type*}\n  {\u03bc : measure \u03b1} {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (u : filter \u03b9) [hu : ne_bot u] [is_countably_generated u]\n  (hf : \u2200 n, ae_measurable (f n) \u03bc) (h_tendsto : \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) u (\ud835\udcdd (g x))) :\n  ae_measurable g \u03bc :=\nbegin\n  rcases u.exists_seq_tendsto with \u27e8v, hv\u27e9,\n  have h'f : \u2200 n, ae_measurable (f (v n)) \u03bc := \u03bb n, hf (v n),\n  set p : \u03b1 \u2192 (\u2115 \u2192 \u03b2) \u2192 Prop := \u03bb x f', tendsto (\u03bb n, f' n) at_top (\ud835\udcdd (g x)),\n  have hp : \u2200\u1d50 x \u2202\u03bc, p x (\u03bb n, f (v n) x),\n    by filter_upwards [h_tendsto] with x hx using hx.comp hv,\n  set ae_seq_lim := \u03bb x, ite (x \u2208 ae_seq_set h'f p) (g x) (\u27e8f (v 0) x\u27e9 : nonempty \u03b2).some with hs,\n  refine \u27e8ae_seq_lim,\n    measurable_of_tendsto_metric' at_top (@ae_seq.measurable \u03b1 \u03b2 _ _ _ (\u03bb n x, f (v n) x) \u03bc h'f p)\n    (tendsto_pi_nhds.mpr (\u03bb x, _)), _\u27e9,\n  { simp_rw [ae_seq, ae_seq_lim],\n    split_ifs with hx,\n    { simp_rw ae_seq.mk_eq_fun_of_mem_ae_seq_set h'f hx,\n      exact @ae_seq.fun_prop_of_mem_ae_seq_set \u03b1 \u03b2 _ _ _ _ _ _ h'f x hx, },\n    { exact tendsto_const_nhds } },\n  { exact (ite_ae_eq_of_measure_compl_zero g (\u03bb x, (\u27e8f (v 0) x\u27e9 : nonempty \u03b2).some)\n      (ae_seq_set h'f p) (ae_seq.measure_compl_ae_seq_set_eq_zero h'f hp)).symm },\nend\n\nlemma ae_measurable_of_tendsto_metric_ae' {\u03bc : measure \u03b1} {f : \u2115 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (hf : \u2200 n, ae_measurable (f n) \u03bc)\n  (h_ae_tendsto : \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) at_top (\ud835\udcdd (g x))) :\n  ae_measurable g \u03bc :=\nae_measurable_of_tendsto_metric_ae at_top hf h_ae_tendsto\n\nlemma ae_measurable_of_unif_approx {\u03bc : measure \u03b1} {g : \u03b1 \u2192 \u03b2}\n  (hf : \u2200 \u03b5 > (0 : \u211d), \u2203 (f : \u03b1 \u2192 \u03b2), ae_measurable f \u03bc \u2227 \u2200\u1d50 x \u2202\u03bc, dist (f x) (g x) \u2264 \u03b5) :\n  ae_measurable g \u03bc :=\nbegin\n  obtain \u27e8u, u_anti, u_pos, u_lim\u27e9 :\n    \u2203 (u : \u2115 \u2192 \u211d), strict_anti u \u2227 (\u2200 (n : \u2115), 0 < u n) \u2227 tendsto u at_top (\ud835\udcdd 0) :=\n      exists_seq_strict_anti_tendsto (0 : \u211d),\n  choose f Hf using \u03bb (n : \u2115), hf (u n) (u_pos n),\n  have : \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) at_top (\ud835\udcdd (g x)),\n  { have : \u2200\u1d50 x \u2202 \u03bc, \u2200 n, dist (f n x) (g x) \u2264 u n := ae_all_iff.2 (\u03bb n, (Hf n).2),\n    filter_upwards [this],\n    assume x hx,\n    rw tendsto_iff_dist_tendsto_zero,\n    exact squeeze_zero (\u03bb n, dist_nonneg) hx u_lim },\n  exact ae_measurable_of_tendsto_metric_ae' (\u03bb n, (Hf n).1) this,\nend\n\nlemma measurable_of_tendsto_metric_ae {\u03bc : measure \u03b1} [\u03bc.is_complete] {f : \u2115 \u2192 \u03b1 \u2192 \u03b2} {g : \u03b1 \u2192 \u03b2}\n  (hf : \u2200 n, measurable (f n))\n  (h_ae_tendsto : \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) at_top (\ud835\udcdd (g x))) :\n  measurable g :=\nae_measurable_iff_measurable.mp\n  (ae_measurable_of_tendsto_metric_ae' (\u03bb i, (hf i).ae_measurable) h_ae_tendsto)\n\nlemma measurable_limit_of_tendsto_metric_ae {\u03b9} [encodable \u03b9] [nonempty \u03b9] {\u03bc : measure \u03b1}\n  {f : \u03b9 \u2192 \u03b1 \u2192 \u03b2} {L : filter \u03b9} [L.is_countably_generated] (hf : \u2200 n, ae_measurable (f n) \u03bc)\n  (h_ae_tendsto : \u2200\u1d50 x \u2202\u03bc, \u2203 l : \u03b2, tendsto (\u03bb n, f n x) L (\ud835\udcdd l)) :\n  \u2203 (f_lim : \u03b1 \u2192 \u03b2) (hf_lim_meas : measurable f_lim),\n    \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) L (\ud835\udcdd (f_lim x)) :=\nbegin\n  inhabit \u03b9,\n  unfreezingI { rcases eq_or_ne L \u22a5 with rfl | hL },\n  { exact \u27e8(hf default).mk _, (hf default).measurable_mk,\n      eventually_of_forall $ \u03bb x, tendsto_bot\u27e9 },\n  haveI : ne_bot L := \u27e8hL\u27e9,\n  let p : \u03b1 \u2192 (\u03b9 \u2192 \u03b2) \u2192 Prop := \u03bb x f', \u2203 l : \u03b2, tendsto (\u03bb n, f' n) L (\ud835\udcdd l),\n  have hp_mem : \u2200 x \u2208 ae_seq_set hf p, p x (\u03bb n, f n x),\n    from \u03bb x hx, ae_seq.fun_prop_of_mem_ae_seq_set hf hx,\n  have h_ae_eq : \u2200\u1d50 x \u2202\u03bc, \u2200 n, ae_seq hf p n x = f n x,\n    from ae_seq.ae_seq_eq_fun_ae hf h_ae_tendsto,\n  let f_lim : \u03b1 \u2192 \u03b2 := \u03bb x, dite (x \u2208 ae_seq_set hf p) (\u03bb h, (hp_mem x h).some)\n    (\u03bb h, (\u27e8f default x\u27e9 : nonempty \u03b2).some),\n  have hf_lim : \u2200 x, tendsto (\u03bb n, ae_seq hf p n x) L (\ud835\udcdd (f_lim x)),\n  { intros x,\n    simp only [f_lim, ae_seq],\n    split_ifs,\n    { refine (hp_mem x h).some_spec.congr (\u03bb n, _),\n      exact (ae_seq.mk_eq_fun_of_mem_ae_seq_set hf h n).symm },\n    { exact tendsto_const_nhds, }, },\n  have h_ae_tendsto_f_lim : \u2200\u1d50 x \u2202\u03bc, tendsto (\u03bb n, f n x) L (\ud835\udcdd (f_lim x)),\n    from h_ae_eq.mono (\u03bb x hx, (hf_lim x).congr hx),\n  have h_f_lim_meas : measurable f_lim,\n    from measurable_of_tendsto_metric' L (ae_seq.measurable hf p)\n      (tendsto_pi_nhds.mpr (\u03bb x, hf_lim x)),\n  exact \u27e8f_lim, h_f_lim_meas, h_ae_tendsto_f_lim\u27e9,\nend\n\nend limits\n\nnamespace continuous_linear_map\n\nvariables {\ud835\udd5c : Type*} [normed_field \ud835\udd5c]\nvariables {E : Type*} [normed_group E] [normed_space \ud835\udd5c E] [measurable_space E]\nvariables [opens_measurable_space E]\nvariables {F : Type*} [normed_group F] [normed_space \ud835\udd5c F] [measurable_space F] [borel_space F]\n\n@[measurability]\nprotected lemma measurable (L : E \u2192L[\ud835\udd5c] F) : measurable L :=\nL.continuous.measurable\n\nlemma measurable_comp (L : E \u2192L[\ud835\udd5c] F) {\u03c6 : \u03b1 \u2192 E} (\u03c6_meas : measurable \u03c6) :\n  measurable (\u03bb (a : \u03b1), L (\u03c6 a)) :=\nL.measurable.comp \u03c6_meas\n\nend continuous_linear_map\n\nnamespace continuous_linear_map\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\nvariables {E : Type*} [normed_group E] [normed_space \ud835\udd5c E]\n          {F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n\ninstance : measurable_space (E \u2192L[\ud835\udd5c] F) := borel _\n\ninstance : borel_space (E \u2192L[\ud835\udd5c] F) := \u27e8rfl\u27e9\n\n@[measurability]\nlemma measurable_apply [measurable_space F] [borel_space F] (x : E) :\n  measurable (\u03bb f : E \u2192L[\ud835\udd5c] F, f x) :=\n(apply \ud835\udd5c F x).continuous.measurable\n\n@[measurability]\nlemma measurable_apply' [measurable_space E] [opens_measurable_space E]\n  [measurable_space F] [borel_space F] :\n  measurable (\u03bb (x : E) (f : E \u2192L[\ud835\udd5c] F), f x) :=\nmeasurable_pi_lambda _ $ \u03bb f, f.measurable\n\n@[measurability]\nlemma measurable_coe [measurable_space F] [borel_space F] :\n  measurable (\u03bb (f : E \u2192L[\ud835\udd5c] F) (x : E), f x) :=\nmeasurable_pi_lambda _ measurable_apply\n\nend continuous_linear_map\n\nsection continuous_linear_map_nondiscrete_normed_field\n\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c]\nvariables {E : Type*} [normed_group E] [normed_space \ud835\udd5c E] [measurable_space E] [borel_space E]\nvariables {F : Type*} [normed_group F] [normed_space \ud835\udd5c F]\n\n@[measurability]\nlemma measurable.apply_continuous_linear_map  {\u03c6 : \u03b1 \u2192 F \u2192L[\ud835\udd5c] E} (h\u03c6 : measurable \u03c6) (v : F) :\n  measurable (\u03bb a, \u03c6 a v) :=\n(continuous_linear_map.apply \ud835\udd5c E v).measurable.comp h\u03c6\n\n@[measurability]\nlemma ae_measurable.apply_continuous_linear_map {\u03c6 : \u03b1 \u2192 F \u2192L[\ud835\udd5c] E} {\u03bc : measure \u03b1}\n  (h\u03c6 : ae_measurable \u03c6 \u03bc) (v : F) : ae_measurable (\u03bb a, \u03c6 a v) \u03bc :=\n(continuous_linear_map.apply \ud835\udd5c E v).measurable.comp_ae_measurable h\u03c6\n\nend continuous_linear_map_nondiscrete_normed_field\n\nsection normed_space\nvariables {\ud835\udd5c : Type*} [nondiscrete_normed_field \ud835\udd5c] [complete_space \ud835\udd5c] [measurable_space \ud835\udd5c]\nvariables [borel_space \ud835\udd5c]\nvariables {E : Type*} [normed_group E] [normed_space \ud835\udd5c E] [measurable_space E] [borel_space E]\n\nlemma measurable_smul_const {f : \u03b1 \u2192 \ud835\udd5c} {c : E} (hc : c \u2260 0) :\n  measurable (\u03bb x, f x \u2022 c) \u2194 measurable f :=\n(closed_embedding_smul_left hc).measurable_embedding.measurable_comp_iff\n\nlemma ae_measurable_smul_const {f : \u03b1 \u2192 \ud835\udd5c} {\u03bc : measure \u03b1} {c : E} (hc : c \u2260 0) :\n  ae_measurable (\u03bb x, f x \u2022 c) \u03bc \u2194 ae_measurable f \u03bc :=\n(closed_embedding_smul_left hc).measurable_embedding.ae_measurable_comp_iff\n\nend normed_space\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/measure_theory/constructions/borel_space.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3510674414833871}}
{"text": "/-\nCopyright (c) 2021 Sebastian Ullrich. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Sebastian Ullrich\n-/\nimport Lean\nimport Mathlib.Tactic.Cache\n\n/-!\n# The `#find` command and tactic.\n\nThe `#find` command finds definitions & lemmas using pattern matching on the type. For instance:\n```lean\n#find _ + _ = _ + _\n#find ?n + _ = _ + ?n\n#find (_ : Nat) + _ = _ + _\n#find Nat \u2192 Nat\n```\nInside tactic proofs, there is a `#find` tactic with the same syntax,\nor the `find` tactic which looks for lemmas which are `apply`able against the current goal.\n\n-/\n\nopen Lean\nopen Lean.Meta\nopen Lean.Elab\nopen Lean.Elab\n\nnamespace Mathlib.Tactic.Find\n\nprivate partial def matchHyps : List Expr \u2192 List Expr \u2192 List Expr \u2192 MetaM Bool\n  | p::ps, oldHyps, h::newHyps => do\n    let pt \u2190 inferType p\n    let t \u2190 inferType h\n    if (\u2190 isDefEq pt t) then\n      matchHyps ps [] (oldHyps ++ newHyps)\n    else\n      matchHyps (p::ps) (h::oldHyps) newHyps\n  | [], _, _    => pure true\n  | _::_, _, [] => pure false\n\n-- from Lean.Server.Completion\nprivate def isBlackListed (declName : Name) : MetaM Bool := do\n  let env \u2190 getEnv\n  pure $ declName.isInternal\n   || isAuxRecursor env declName\n   || isNoConfusion env declName\n  <||> isRec declName\n  <||> isMatcher declName\n\ninitialize findDeclsPerHead : DeclCache (Lean.HashMap HeadIndex (Array Name)) \u2190\n  DeclCache.mk \"#find: init cache\" {} fun _ c headMap \u21a6 do\n    if (\u2190 isBlackListed c.name) then\n      return headMap\n    -- TODO: this should perhaps use `forallTelescopeReducing` instead,\n    -- to avoid leaking metavariables.\n    let (_, _, ty) \u2190 forallMetaTelescopeReducing c.type\n    let head := ty.toHeadIndex\n    pure $ headMap.insert head (headMap.findD head #[] |>.push c.name)\n\ndef findType (t : Expr) : TermElabM Unit := withReducible do\n  let t \u2190 instantiateMVars t\n  let head := (\u2190 forallMetaTelescopeReducing t).2.2.toHeadIndex\n  let pat \u2190 abstractMVars t\n\n  let env \u2190 getEnv\n  let mut numFound := 0\n  for n in (\u2190 findDeclsPerHead.get).findD head #[] do\n    let c := env.find? n |>.get!\n    let cTy := c.instantiateTypeLevelParams (\u2190 mkFreshLevelMVars c.numLevelParams)\n    let found \u2190 forallTelescopeReducing cTy fun cParams cTy' \u21a6 do\n      let pat := pat.expr.instantiateLevelParamsArray pat.paramNames\n        (\u2190 mkFreshLevelMVars pat.numMVars).toArray\n      let (_, _, pat) \u2190 lambdaMetaTelescope pat\n      let (patParams, _, pat) \u2190 forallMetaTelescopeReducing pat\n      isDefEq cTy' pat <&&> matchHyps patParams.toList [] cParams.toList\n    if found then\n      numFound := numFound + 1\n      if numFound > 20 then\n        logInfo m!\"maximum number of search results reached\"\n        break\n      logInfo m!\"{n}: {cTy}\"\n\nopen Lean.Elab.Command in\n/-\nThe `#find` command finds definitions & lemmas using pattern matching on the type. For instance:\n```lean\n#find _ + _ = _ + _\n#find ?n + _ = _ + ?n\n#find (_ : Nat) + _ = _ + _\n#find Nat \u2192 Nat\n```\nInside tactic proofs, the `#find` tactic can be used instead.\nThere is also the `find` tactic which looks for\nlemmas which are `apply`able against the current goal.\n-/\nelab \"#find\" t:term : command =>\n  liftTermElabM do\n    let t \u2190 Term.elabTerm t none\n    Term.synthesizeSyntheticMVars (mayPostpone := false) (ignoreStuckTC := true)\n    findType t\n\n/- (Note that you'll get an error trying to run these here:\n   ``cannot evaluate `[init]` declaration 'findDeclsPerHead' in the same module``\n   but they will work fine in a new file!) -/\n-- #find _ + _ = _ + _\n-- #find _ + _ = _ + _\n-- #find ?n + _ = _ + ?n\n-- #find (_ : Nat) + _ = _ + _\n-- #find Nat \u2192 Nat\n-- #find ?n \u2264 ?m \u2192 ?n + _ \u2264 ?m + _\n\nopen Lean.Elab.Tactic\n/-\nDisplay theorems (and definitions) whose result type matches the current goal,\ni.e. which should be `apply`able.\n```lean\nexample : True := by find\n```\n`find` will not affect the goal by itself and should be removed from the finished proof.\nFor a command that takes the type to search for as an argument,\nsee `#find`, which is also available as a tactic.\n-/\nelab \"find\" : tactic => do\n  findType (\u2190 getMainTarget)\n\n/-\nTactic version of the `#find` command.\nSee also the `find` tactic to search for theorems matching the current goal.\n-/\nelab \"#find\" t:term : tactic => do\n  let t \u2190 Term.elabTerm t none\n  Term.synthesizeSyntheticMVars (mayPostpone := false) (ignoreStuckTC := true)\n  findType t\n", "meta": {"author": "leanprover-community", "repo": "mathlib4", "sha": "b9a0a30342ca06e9817e22dbe46e75fc7f435500", "save_path": "github-repos/lean/leanprover-community-mathlib4", "path": "github-repos/lean/leanprover-community-mathlib4/mathlib4-b9a0a30342ca06e9817e22dbe46e75fc7f435500/Mathlib/Tactic/Find.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166195971441, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.351011729760752}}
{"text": "/-\nCopyright (c) 2017 Microsoft Corporation. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Leonardo de Moura\n-/\nprelude\nimport Init.Data.Option.Basic\n\nuniverses u v\n\ntheorem Option.eqOfEqSome {\u03b1 : Type u} : \u2200 {x y : Option \u03b1}, (\u2200z, x = some z \u2194 y = some z) \u2192 x = y\n  | none,   none,   h => rfl\n  | none,   some z, h => Option.noConfusion ((h z).2 rfl)\n  | some z, none,   h => Option.noConfusion ((h z).1 rfl)\n  | some z, some w, h => Option.noConfusion ((h w).2 rfl) (congrArg some)\n\ntheorem Option.eqNoneOfIsNone {\u03b1 : Type u} : \u2200 {o : Option \u03b1}, o.isNone \u2192 o = none\n  | none, h => rfl\n", "meta": {"author": "gebner", "repo": "lean4-old", "sha": "ee51cdfaf63ee313c914d83264f91f414a0e3b6e", "save_path": "github-repos/lean/gebner-lean4-old", "path": "github-repos/lean/gebner-lean4-old/lean4-old-ee51cdfaf63ee313c914d83264f91f414a0e3b6e/stage0/src/Init/Data/Option/Instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.35092267300998287}}
{"text": "/-\nCopyright \u00a9 2020 Nicol\u00f2 Cavalleri. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthor: Nicol\u00f2 Cavalleri.\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.lie.basic\nimport Mathlib.ring_theory.algebra_tower\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_3 l u_4 \n\nnamespace Mathlib\n\n/-!\n# Derivations\n\nThis file defines derivation. A derivation `D` from the `R`-algebra `A` to the `A`-module `M` is an\n`R`-linear map that satisfy the Leibniz rule `D (a * b) = a * D b + D a * b`.\n\n## Notation\n\nThe notation `\u2045D1, D2\u2046` is used for the commutator of two derivations.\n\nTODO: this file is just a stub to go on with some PRs in the geometry section. It only\nimplements the definition of derivations in commutative algebra. This will soon change: as soon\nas bimodules will be there in mathlib I will change this file to take into account the\nnon-commutative case. Any development on the theory of derivations is discouraged until the\ndefinitive definition of derivation will be implemented.\n-/\n\n/-- `D : derivation R A M` is an `R`-linear map from `A` to `M` that satisfies the `leibniz`\nequality.\nTODO: update this when bimodules are defined. -/\nstructure derivation (R : Type u_1) (A : Type u_2) [comm_semiring R] [comm_semiring A] [algebra R A] (M : Type u_3) [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] \nextends linear_map R A M\nwhere\n  leibniz' : \u2200 (a b : A),\n  linear_map.to_fun _to_linear_map (a * b) =\n    a \u2022 linear_map.to_fun _to_linear_map b + b \u2022 linear_map.to_fun _to_linear_map a\n\nnamespace derivation\n\n\nprotected instance has_coe_to_fun {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : has_coe_to_fun (derivation R A M) :=\n  has_coe_to_fun.mk (fun (D : derivation R A M) => A \u2192 M)\n    fun (D : derivation R A M) => linear_map.to_fun (derivation.to_linear_map D)\n\nprotected instance has_coe_to_linear_map {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : has_coe (derivation R A M) (linear_map R A M) :=\n  has_coe.mk fun (D : derivation R A M) => derivation.to_linear_map D\n\n@[simp] theorem to_fun_eq_coe {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) : linear_map.to_fun (derivation.to_linear_map D) = \u21d1D :=\n  rfl\n\n@[simp] theorem coe_fn_coe {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (f : derivation R A M) : \u21d1\u2191f = \u21d1f :=\n  rfl\n\ntheorem coe_injective {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (H : \u21d1D1 = \u21d1D2) : D1 = D2 := sorry\n\ntheorem ext {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (H : \u2200 (a : A), coe_fn D1 a = coe_fn D2 a) : D1 = D2 :=\n  coe_injective (funext H)\n\n@[simp] theorem map_add {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) : coe_fn D (a + b) = coe_fn D a + coe_fn D b :=\n  is_add_hom.map_add (\u21d1D) a b\n\n@[simp] theorem map_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) : coe_fn D 0 = 0 :=\n  is_add_monoid_hom.map_zero \u21d1D\n\n@[simp] theorem map_smul {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (r : R) (a : A) : coe_fn D (r \u2022 a) = r \u2022 coe_fn D a :=\n  linear_map.map_smul (\u2191D) r a\n\n@[simp] theorem leibniz {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) : coe_fn D (a * b) = a \u2022 coe_fn D b + b \u2022 coe_fn D a :=\n  derivation.leibniz' D a b\n\n@[simp] theorem map_one_eq_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) : coe_fn D 1 = 0 := sorry\n\n@[simp] theorem map_algebra_map {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (r : R) : coe_fn D (coe_fn (algebra_map R A) r) = 0 := sorry\n\nprotected instance has_zero {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : HasZero (derivation R A M) :=\n  { zero := mk 0 sorry }\n\nprotected instance inhabited {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : Inhabited (derivation R A M) :=\n  { default := 0 }\n\nprotected instance add_comm_monoid {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : add_comm_monoid (derivation R A M) :=\n  add_comm_monoid.mk (fun (D1 D2 : derivation R A M) => mk (\u2191D1 + \u2191D2) sorry) sorry 0 sorry sorry sorry\n\n@[simp] theorem add_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (a : A) : coe_fn (D1 + D2) a = coe_fn D1 a + coe_fn D2 a :=\n  rfl\n\nprotected instance derivation.Rsemimodule {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : semimodule R (derivation R A M) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem smul_to_linear_map_coe {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (r : R) : \u2191(r \u2022 D) = r \u2022 \u2191D :=\n  rfl\n\n@[simp] theorem Rsmul_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (r : R) (a : A) : coe_fn (r \u2022 D) a = r \u2022 coe_fn D a :=\n  rfl\n\nprotected instance semimodule {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : semimodule A (derivation R A M) :=\n  semimodule.mk sorry sorry\n\n@[simp] theorem smul_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) : coe_fn (a \u2022 D) b = a \u2022 coe_fn D b :=\n  rfl\n\nprotected instance is_scalar_tower {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] [is_scalar_tower R A M] : is_scalar_tower R A (derivation R A M) :=\n  is_scalar_tower.mk fun (x : R) (y : A) (z : derivation R A M) => ext fun (a : A) => smul_assoc x y (coe_fn (\u2191z) a)\n\n@[simp] theorem map_neg {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M] (D : derivation R A M) (a : A) : coe_fn D (-a) = -coe_fn D a :=\n  linear_map.map_neg (\u2191D) a\n\n@[simp] theorem map_sub {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M] (D : derivation R A M) (a : A) (b : A) : coe_fn D (a - b) = coe_fn D a - coe_fn D b :=\n  linear_map.map_sub (\u2191D) a b\n\nprotected instance add_comm_group {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M] : add_comm_group (derivation R A M) :=\n  add_comm_group.mk add_comm_monoid.add sorry add_comm_monoid.zero sorry sorry\n    (fun (D : derivation R A M) => mk (-\u2191D) sorry) (fun (D1 D2 : derivation R A M) => mk (\u2191D1 - \u2191D2) sorry) sorry sorry\n\n@[simp] theorem sub_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {M : Type u_3} [add_comm_group M] [module A M] [module R M] [is_scalar_tower R A M] {D1 : derivation R A M} {D2 : derivation R A M} (a : A) : coe_fn (D1 - D2) a = coe_fn D1 a - coe_fn D2 a :=\n  rfl\n\n/-! # Lie structures -/\n\n/-- The commutator of derivations is again a derivation. -/\ndef commutator {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] (D1 : derivation R A A) (D2 : derivation R A A) : derivation R A A :=\n  mk (linear_map.mk (linear_map.to_fun (has_bracket.bracket \u2191D1 \u2191D2)) sorry sorry) sorry\n\nprotected instance has_bracket {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] : has_bracket (derivation R A A) (derivation R A A) :=\n  has_bracket.mk commutator\n\n@[simp] theorem commutator_coe_linear_map {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {D1 : derivation R A A} {D2 : derivation R A A} : \u2191(has_bracket.bracket D1 D2) = has_bracket.bracket \u2191D1 \u2191D2 :=\n  rfl\n\ntheorem commutator_apply {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] {D1 : derivation R A A} {D2 : derivation R A A} (a : A) : coe_fn (has_bracket.bracket D1 D2) a = coe_fn D1 (coe_fn D2 a) - coe_fn D2 (coe_fn D1 a) :=\n  rfl\n\nprotected instance lie_ring {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] : lie_ring (derivation R A A) :=\n  lie_ring.mk sorry sorry sorry sorry\n\nprotected instance lie_algebra {R : Type u_1} [comm_ring R] {A : Type u_2} [comm_ring A] [algebra R A] : lie_algebra R (derivation R A A) :=\n  lie_algebra.mk sorry\n\nend derivation\n\n\nnamespace linear_map\n\n\n/-- The composition of a linear map and a derivation is a derivation. -/\ndef comp_der {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] {N : Type u_4} [add_cancel_comm_monoid N] [semimodule A N] [semimodule R N] [is_scalar_tower R A M] [is_scalar_tower R A N] (f : linear_map A M N) (D : derivation R A M) : derivation R A N :=\n  derivation.mk (mk (fun (a : A) => coe_fn f (coe_fn D a)) sorry sorry) sorry\n\n@[simp] theorem comp_der_apply {R : Type u_1} [comm_semiring R] {A : Type u_2} [comm_semiring A] [algebra R A] {M : Type u_3} [add_cancel_comm_monoid M] [semimodule A M] [semimodule R M] {N : Type u_4} [add_cancel_comm_monoid N] [semimodule A N] [semimodule R N] [is_scalar_tower R A M] [is_scalar_tower R A N] (f : linear_map A M N) (D : derivation R A M) (a : A) : coe_fn (comp_der f D) a = coe_fn f (coe_fn D a) :=\n  rfl\n\n", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/ring_theory/derivation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3509226723789744}}
{"text": "import ..src.struct_tact\n\ninductive t\n| C1 : int \u2192 t\n| C2 : int \u2192 t\n| C3 : int \u2192 t\n\ndef foo (xs : t \u00d7 t) : int :=\nlet z := 10 in\nmatch xs with\n| (t.C1 i, t.C2 j) := i + j + z\n| (t.C2 j, t.C1 i) := j + i + z\n| (_, _) := 0\nend\n\nlemma nested_break_match :\n   forall i j, foo (t.C2 i, t.C1 j) = foo (t.C1 j, t.C2 i) :=\nbegin\n    -- intros,\n    -- unfold foo,\n    -- dsimp,\n    -- break_match,\n    admit,\nend\n", "meta": {"author": "uwplse", "repo": "struct_tact", "sha": "22188ea2e97705d1185f75dde24e6bab88054ab0", "save_path": "github-repos/lean/uwplse-struct_tact", "path": "github-repos/lean/uwplse-struct_tact/struct_tact-22188ea2e97705d1185f75dde24e6bab88054ab0/test/break_test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3509226641726137}}
{"text": "import tactic\nimport data.list.func\n\nuniverse u\n\ndef list2d (\u03b1 : Type u) := list (list \u03b1)\n\nnamespace list2d\n\nopen list.func\n\nvariables {\u03b1 : Type} {\u03b2 : Type} {\u03b3 : Type} {\u03b4 : Type}\nvariables [inhabited \u03b1] [inhabited \u03b2]\n\ndef get2d (xy : \u2115 \u00d7 \u2115) (l : list2d \u03b1) : \u03b1\n  := let (x,y) := xy in get x (get y l)\ndef set2d (a : \u03b1) (l : list2d \u03b1) (xy : \u2115 \u00d7 \u2115) : list2d \u03b1\n  := let (x,y) := xy in set (set a (get y l) x) l y\ndef map2d (f : \u03b3 \u2192 \u03b4) : list2d \u03b3 \u2192 list2d \u03b4\n  := list.map (list.map f)\ndef fold2d (f : \u03b3 \u2192 \u03b4 \u2192 \u03b4) : \u03b4 \u2192 list2d \u03b3 \u2192 \u03b4\n  := list.foldr (function.swap (list.foldr f))\ndef pointwise2d (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : list2d \u03b1 \u2192 list2d \u03b2 \u2192 list2d \u03b3\n  := pointwise (pointwise f)\ndef dfzip2d : list2d \u03b1 \u2192 list2d \u03b2 \u2192 list2d (\u03b1 \u00d7 \u03b2)\n  := pointwise2d prod.mk\n\ndef equiv (l1 l2 : list2d \u03b1)\n  := \u2200 (xy : \u2115 \u00d7 \u2115), l1.get2d xy = l2.get2d xy\n\ndef add_to_line1 (a : \u03b1) : list2d \u03b1 -> list2d \u03b1\n| [] := [[a]]\n| (h::t) := (a::h)::t\n\n@[simp] lemma get2d_nil : \u2200 xy : \u2115\u00d7\u2115, get2d xy ([] : list2d \u03b1) = default \u03b1\n| (x,y) := by simp [get2d, default]\n\ndef transpose : list2d \u03b1 \u2192 list2d \u03b1 := (list.foldr (pointwise list.cons)) []\n\ninstance [has_repr \u03b1] : has_repr (list2d \u03b1) := \u27e8\u03bb l, list.repr (l : list (list \u03b1))\u27e9\n\nprivate lemma g_get_pointwise {\u03b4 : Type} [inhabited \u03b3] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} (g : \u03b3 -> \u03b4)\n  (h1 : g (f (default \u03b1) (default \u03b2)) = g (default \u03b3)) :\n  \u2200 (k : nat) (as : list \u03b1) (bs : list \u03b2),\n  g (get k (pointwise f as bs)) = g (f (get k as) (get k bs))\n| k [] [] := by simp only [h1, get_nil, list.func.pointwise, list.func.get]\n| 0 [] (b::bs) :=\n  by simp only [get_pointwise, get_nil,\n      list.func.pointwise, list.func.get, nat.nat_zero_eq_zero, list.map]\n| (k+1) [] (b::bs) :=\n  by { have : g (get k (list.map (f $ default \u03b1) bs)) = g (f (default \u03b1) (get k bs)),\n       { simpa [nil_pointwise, get_nil] using (g_get_pointwise k [] bs) },\n       simpa [list.func.get, get_nil, pointwise, list.map] }\n| 0 (a::as) [] :=\n  by simp only [g_get_pointwise, get_nil,\n     list.func.pointwise, list.func.get, nat.nat_zero_eq_zero, list.map]\n| (k+1) (a::as) [] :=\n  by simpa [list.func.get, get_nil, pointwise, list.map, pointwise_nil, get_nil]\n     using g_get_pointwise k as []\n| 0 (a::as) (b::bs) := by simp only [list.func.pointwise, list.func.get]\n| (k+1) (a::as) (b::bs) :=\n  by simp only [list.func.pointwise, list.func.get, g_get_pointwise k]\n\n@[simp] theorem get2d_transpose : \u2200 (xy : \u2115 \u00d7 \u2115) (l : list2d \u03b1),\n  get2d xy (transpose l) = get2d (prod.swap xy) l :=\nbegin\n  intros, cases xy with x y, simp, revert x, induction l, {\n    intros, unfold transpose, unfold get2d,\n    simp [default],\n  }, {\n    intros, unfold get2d, unfold transpose,\n    simp [list.foldr], rewrite g_get_pointwise (get x),\n    cases x, refl, apply l_ih,\n    simp [default], cases x, refl, simp,\n  }\nend\n\n@[simp] theorem map_pointwise {\u03b4 : Type}\n  (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 \u03b4) : \u2200 (l1 : list \u03b1) (l2 : list \u03b2),\n  list.map g (list.func.pointwise f l1 l2)\n  = list.func.pointwise (\u03bb a b, g (f a b)) l1 l2\n| [] [] := by simp\n| (a::as) [] := by simp \n| [] (b::bs) := by simp\n| (a::as) (b::bs) := by simp [(map_pointwise as bs)]\n\n@[simp] theorem map_map_circ {\u03b42 : Type} (f : \u03b4 \u2192 \u03b42) (g : \u03b3 \u2192 \u03b4) : (list.map f) \u2218 (list.map g) = list.map (f \u2218 g) :=\nbegin apply funext, simp end\n\ntheorem nil_pointwise_curry {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3} : pointwise f list.nil = list.map (f (default \u03b1)) := begin apply funext, intros, apply nil_pointwise end\n\ntheorem map_pointwise2d\n  (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) (g : \u03b3 \u2192 \u03b4) : \u2200 (l1 : list2d \u03b1) (l2 : list2d \u03b2),\n  map2d g (pointwise2d f l1 l2)\n  = pointwise2d (\u03bb a b, g (f a b)) l1 l2 :=\nbegin\n  intros, unfold map2d, unfold pointwise2d, simp\nend\n\ntheorem pointwise_dfzip2d\n  (f : \u03b1 \u2192 \u03b2 \u2192 \u03b3) : \u2200 (l1 : list2d \u03b1) (l2 : list2d \u03b2),\n  pointwise2d f l1 l2 =\n(pointwise2d prod.mk l1 l2).map2d (\u03bb ab, match ab with (a,b) := f a b end) :=\nby simp [map_pointwise2d]\n\ntheorem get2d_pointwise [inhabited \u03b3] {f : \u03b1 \u2192 \u03b2 \u2192 \u03b3}\n  (H : f (default \u03b1) (default \u03b2) = default \u03b3) :\n  \u2200 (xy : \u2115\u00d7\u2115) (as : list2d \u03b1) (bs : list2d \u03b2),\n  get2d xy (pointwise2d f as bs) = f (get2d xy as) (get2d xy bs) :=\nbegin\n  intros, cases xy with x y, unfold get2d, unfold pointwise2d,\n  repeat { rw list.func.get_pointwise, }, exact H,\n  simp! [default],\nend\n\ntheorem get2d_set2d {a : \u03b1} {xy : \u2115\u00d7\u2115} {l : list2d \u03b1} : (l.set2d a xy).get2d xy = a :=\nbegin\n  cases xy with x y, unfold get2d, unfold set2d, simp!,\nend\ntheorem get2d_set2d_eq_of_ne {a : \u03b1} {xy1 xy2 : \u2115\u00d7\u2115} {l : list2d \u03b1}\n  : xy1 \u2260 xy2 \u2192 (l.set2d a xy2).get2d xy1 = (l.get2d xy1) :=\nbegin\n  cases xy1 with x1 y1,\n  cases xy2 with x2 y2,\n  unfold get2d, unfold set2d,\n  intro H,\n  by_cases Hy : y1 = y2, {\n    rw Hy, rw Hy at H, rw get_set,\n    by_cases Hx : x1 = x2,\n    { exfalso, rw Hx at H, exact H rfl, },\n    { exact get_set_eq_of_ne x2 x1 Hx, },\n  }, {\n    apply congr_arg, exact get_set_eq_of_ne y2 y1 Hy,\n  },\nend\n\nend list2d\n", "meta": {"author": "mirefek", "repo": "sokoban.lean", "sha": "451c92308afb4d3f8e566594b9751286f93b899b", "save_path": "github-repos/lean/mirefek-sokoban.lean", "path": "github-repos/lean/mirefek-sokoban.lean/sokoban.lean-451c92308afb4d3f8e566594b9751286f93b899b/src/list2d.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.577495350642608, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.35092265533524447}}
{"text": "/-\nCopyright (c) 2018 Simon Hudon. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Simon Hudon\n-/\nimport control.traversable.lemmas\nimport logic.equiv.basic\n\n/-!\n# Transferring `traversable` instances along isomorphisms\n\nThis file allows to transfer `traversable` instances along isomorphisms.\n\n## Main declarations\n\n* `equiv.map`: Turns functorially a function `\u03b1 \u2192 \u03b2` into a function `t' \u03b1 \u2192 t' \u03b2` using the functor\n  `t` and the equivalence `\u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1`.\n* `equiv.functor`: `equiv.map` as a functor.\n* `equiv.traverse`: Turns traversably a function `\u03b1 \u2192 m \u03b2` into a function `t' \u03b1 \u2192 m (t' \u03b2)` using\n  the traversable functor `t` and the equivalence `\u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1`.\n* `equiv.traversable`: `equiv.traverse` as a traversable functor.\n* `equiv.is_lawful_traversable`: `equiv.traverse` as a lawful traversable functor.\n-/\n\nuniverses u\n\nnamespace equiv\n\nsection functor\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [functor t]\n\nopen functor\n\n/-- Given a functor `t`, a function `t' : Type u \u2192 Type u`, and\nequivalences `t \u03b1 \u2243 t' \u03b1` for all `\u03b1`, then every function `\u03b1 \u2192 \u03b2` can\nbe mapped to a function `t' \u03b1 \u2192 t' \u03b2` functorially (see\n`equiv.functor`). -/\nprotected def map {\u03b1 \u03b2 : Type u} (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) : t' \u03b2 :=\neqv \u03b2 $ map f ((eqv \u03b1).symm x)\n\n/-- The function `equiv.map` transfers the functoriality of `t` to\n`t'` using the equivalences `eqv`.  -/\nprotected def functor : functor t' :=\n{ map := @equiv.map _ }\n\nvariables [is_lawful_functor t]\n\nprotected lemma id_map {\u03b1 : Type u} (x : t' \u03b1) : equiv.map id x = x :=\nby simp [equiv.map, id_map]\n\nprotected lemma comp_map {\u03b1 \u03b2 \u03b3 : Type u} (g : \u03b1 \u2192 \u03b2) (h : \u03b2 \u2192 \u03b3) (x : t' \u03b1) :\n  equiv.map (h \u2218 g) x = equiv.map h (equiv.map g x) :=\nby simp [equiv.map]; apply comp_map\n\nprotected lemma is_lawful_functor : @is_lawful_functor _ equiv.functor :=\n{ id_map := @equiv.id_map _ _,\n  comp_map := @equiv.comp_map _ _ }\n\nprotected lemma is_lawful_functor' [F : _root_.functor t']\n  (h\u2080 : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2), _root_.functor.map f = equiv.map f)\n  (h\u2081 : \u2200 {\u03b1 \u03b2} (f : \u03b2), _root_.functor.map_const f = (equiv.map \u2218 function.const \u03b1) f) :\n  _root_.is_lawful_functor t' :=\nbegin\n  have : F = equiv.functor,\n  { casesI F, dsimp [equiv.functor],\n    congr; ext; [rw \u2190 h\u2080, rw \u2190 h\u2081] },\n  substI this,\n  exact equiv.is_lawful_functor\nend\n\nend functor\n\nsection traversable\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [traversable t]\nvariables {m : Type u \u2192 Type u} [applicative m]\nvariables {\u03b1 \u03b2 : Type u}\n\n/-- Like `equiv.map`, a function `t' : Type u \u2192 Type u` can be given\nthe structure of a traversable functor using a traversable functor\n`t'` and equivalences `t \u03b1 \u2243 t' \u03b1` for all \u03b1.  See `equiv.traversable`. -/\nprotected def traverse (f : \u03b1 \u2192 m \u03b2) (x : t' \u03b1) : m (t' \u03b2) :=\neqv \u03b2 <$> traverse f ((eqv \u03b1).symm x)\n\n/-- The function `equiv.traverse` transfers a traversable functor\ninstance across the equivalences `eqv`. -/\nprotected def traversable : traversable t' :=\n{ to_functor := equiv.functor eqv,\n  traverse := @equiv.traverse _ }\n\nend traversable\n\nsection equiv\nparameters {t t' : Type u \u2192 Type u}\nparameters (eqv : \u03a0 \u03b1, t \u03b1 \u2243 t' \u03b1)\nvariables [traversable t] [is_lawful_traversable t]\nvariables {F G : Type u \u2192 Type u} [applicative F] [applicative G]\nvariables [is_lawful_applicative F] [is_lawful_applicative G]\nvariables (\u03b7 : applicative_transformation F G)\nvariables {\u03b1 \u03b2 \u03b3 : Type u}\n\nopen is_lawful_traversable functor\n\nprotected lemma id_traverse (x : t' \u03b1) :\n  equiv.traverse eqv id.mk x = x :=\nby simp! [equiv.traverse,id_bind,id_traverse,functor.map] with functor_norm\n\nprotected lemma traverse_eq_map_id (f : \u03b1 \u2192 \u03b2) (x : t' \u03b1) :\n  equiv.traverse eqv (id.mk \u2218 f) x = id.mk (equiv.map eqv f x) :=\nby simp [equiv.traverse, traverse_eq_map_id] with functor_norm; refl\n\nprotected lemma comp_traverse (f : \u03b2 \u2192 F \u03b3) (g : \u03b1 \u2192 G \u03b2) (x : t' \u03b1) :\n  equiv.traverse eqv (comp.mk \u2218 functor.map f \u2218 g) x =\n  comp.mk (equiv.traverse eqv f <$> equiv.traverse eqv g x) :=\nby simp [equiv.traverse,comp_traverse] with functor_norm; congr; ext; simp\n\nprotected lemma naturality (f : \u03b1 \u2192 F \u03b2) (x : t' \u03b1) :\n  \u03b7 (equiv.traverse eqv f x) = equiv.traverse eqv (@\u03b7 _ \u2218 f) x :=\nby simp only [equiv.traverse] with functor_norm\n\n/-- The fact that `t` is a lawful traversable functor carries over the\nequivalences to `t'`, with the traversable functor structure given by\n`equiv.traversable`. -/\nprotected def is_lawful_traversable : @is_lawful_traversable t' (equiv.traversable eqv) :=\n{ to_is_lawful_functor := @equiv.is_lawful_functor _ _ eqv _ _,\n  id_traverse := @equiv.id_traverse _ _,\n  comp_traverse := @equiv.comp_traverse _ _,\n  traverse_eq_map_id := @equiv.traverse_eq_map_id _ _,\n  naturality := @equiv.naturality _ _ }\n\n/-- If the `traversable t'` instance has the properties that `map`,\n`map_const`, and `traverse` are equal to the ones that come from\ncarrying the traversable functor structure from `t` over the\nequivalences, then the fact that `t` is a lawful traversable functor\ncarries over as well. -/\nprotected def is_lawful_traversable' [_i : traversable t']\n  (h\u2080 : \u2200 {\u03b1 \u03b2} (f : \u03b1 \u2192 \u03b2),\n         map f = equiv.map eqv f)\n  (h\u2081 : \u2200 {\u03b1 \u03b2} (f : \u03b2),\n         map_const f = (equiv.map eqv \u2218 function.const \u03b1) f)\n  (h\u2082 : \u2200 {F : Type u \u2192 Type u} [applicative F],\n        by exactI \u2200 [is_lawful_applicative F]\n          {\u03b1 \u03b2} (f : \u03b1 \u2192 F \u03b2),\n         traverse f = equiv.traverse eqv f) :\n  _root_.is_lawful_traversable t' :=\nbegin\n    -- we can't use the same approach as for `is_lawful_functor'` because\n    -- h\u2082 needs a `is_lawful_applicative` assumption\n  refine {to_is_lawful_functor :=\n    equiv.is_lawful_functor' eqv @h\u2080 @h\u2081, ..}; introsI,\n  { rw [h\u2082, equiv.id_traverse], apply_instance },\n  { rw [h\u2082, equiv.comp_traverse f g x, h\u2082], congr,\n    rw [h\u2082], all_goals { apply_instance } },\n  { rw [h\u2082, equiv.traverse_eq_map_id, h\u2080]; apply_instance },\n  { rw [h\u2082, equiv.naturality, h\u2082]; apply_instance }\nend\n\nend equiv\nend equiv\n", "meta": {"author": "saisurbehera", "repo": "mathProof", "sha": "57c6bfe75652e9d3312d8904441a32aff7d6a75e", "save_path": "github-repos/lean/saisurbehera-mathProof", "path": "github-repos/lean/saisurbehera-mathProof/mathProof-57c6bfe75652e9d3312d8904441a32aff7d6a75e/src/tertiary_packages/mathlib/src/control/traversable/equiv.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.3508819859572598}}
{"text": "/-\nCopyright (c) 2018 Mario Carneiro. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Mario Carneiro\n\n! This file was ported from Lean 3 source module data.option.defs\n! leanprover-community/mathlib commit 448144f7ae193a8990cb7473c9e9a01990f64ac7\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\n\n/-!\n# Extra definitions on `option`\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n\nThis file defines more operations involving `option \u03b1`. Lemmas about them are located in other\nfiles under `data.option.`.\nOther basic operations on `option` are defined in the core library.\n-/\n\n\nnamespace Option\n\nvariable {\u03b1 : Type _} {\u03b2 : Type _}\n\nattribute [inline] Option.isSome Option.isNone\n\n#print Option.elim' /-\n/-- An elimination principle for `option`. It is a nondependent version of `option.rec`. -/\n@[simp]\nprotected def elim' (b : \u03b2) (f : \u03b1 \u2192 \u03b2) : Option \u03b1 \u2192 \u03b2\n  | some a => f a\n  | none => b\n#align option.elim Option.elim'\n-/\n\ninstance hasMem : Membership \u03b1 (Option \u03b1) :=\n  \u27e8fun a b => b = some a\u27e9\n#align option.has_mem Option.hasMem\n\n#print Option.mem_def /-\n@[simp]\ntheorem mem_def {a : \u03b1} {b : Option \u03b1} : a \u2208 b \u2194 b = some a :=\n  Iff.rfl\n#align option.mem_def Option.mem_def\n-/\n\n#print Option.mem_iff /-\ntheorem mem_iff {a : \u03b1} {b : Option \u03b1} : a \u2208 b \u2194 b = a :=\n  Iff.rfl\n#align option.mem_iff Option.mem_iff\n-/\n\n#print Option.isNone_iff_eq_none /-\ntheorem isNone_iff_eq_none {o : Option \u03b1} : o.isNone = true \u2194 o = none :=\n  \u27e8Option.eq_none_of_isNone, fun e => e.symm \u25b8 rfl\u27e9\n#align option.is_none_iff_eq_none Option.isNone_iff_eq_none\n-/\n\n#print Option.some_inj /-\ntheorem some_inj {a b : \u03b1} : some a = some b \u2194 a = b := by simp\n#align option.some_inj Option.some_inj\n-/\n\n#print Option.mem_some_iff /-\ntheorem mem_some_iff {\u03b1 : Type _} {a b : \u03b1} : a \u2208 some b \u2194 b = a := by simp\n#align option.mem_some_iff Option.mem_some_iff\n-/\n\n#print Option.decidable_eq_none /-\n/-- `o = none` is decidable even if the wrapped type does not have decidable equality.\n\nThis is not an instance because it is not definitionally equal to `option.decidable_eq`.\nTry to use `o.is_none` or `o.is_some` instead.\n-/\n@[inline]\ndef decidable_eq_none {o : Option \u03b1} : Decidable (o = none) :=\n  decidable_of_decidable_of_iff (Bool.decidableEq _ _) isNone_iff_eq_none\n#align option.decidable_eq_none Option.decidable_eq_none\n-/\n\n#print Option.decidableForallMem /-\ninstance decidableForallMem {p : \u03b1 \u2192 Prop} [DecidablePred p] :\n    \u2200 o : Option \u03b1, Decidable (\u2200 a \u2208 o, p a)\n  | none => isTrue (by simp [false_imp_iff])\n  | some a =>\n    if h : p a then isTrue fun o e => some_inj.1 e \u25b8 h else isFalse <| mt (fun H => H _ rfl) h\n#align option.decidable_forall_mem Option.decidableForallMem\n-/\n\n/- warning: option.decidable_exists_mem -> Option.decidableExistsMem is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {p : \u03b1 -> Prop} [_inst_1 : DecidablePred.{succ u1} \u03b1 p] (o : Option.{u1} \u03b1), Decidable (Exists.{succ u1} \u03b1 (fun (a : \u03b1) => Exists.{0} (Membership.Mem.{u1, u1} \u03b1 (Option.{u1} \u03b1) (Option.hasMem.{u1} \u03b1) a o) (fun (H : Membership.Mem.{u1, u1} \u03b1 (Option.{u1} \u03b1) (Option.hasMem.{u1} \u03b1) a o) => p a)))\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}} {p : \u03b1 -> Prop} [_inst_1 : DecidablePred.{succ u1} \u03b1 p] (o : Option.{u1} \u03b1), Decidable (Exists.{succ u1} \u03b1 (fun (a : \u03b1) => And (Membership.mem.{u1, u1} \u03b1 (Option.{u1} \u03b1) (Option.instMembershipOption.{u1} \u03b1) a o) (p a)))\nCase conversion may be inaccurate. Consider using '#align option.decidable_exists_mem Option.decidableExistsMem\u2093'. -/\ninstance decidableExistsMem {p : \u03b1 \u2192 Prop} [DecidablePred p] :\n    \u2200 o : Option \u03b1, Decidable (\u2203 a \u2208 o, p a)\n  | none => isFalse fun \u27e8a, \u27e8h, _\u27e9\u27e9 => by cases h\n  | some a => if h : p a then isTrue <| \u27e8_, rfl, h\u27e9 else isFalse fun \u27e8_, \u27e8rfl, hn\u27e9\u27e9 => h hn\n#align option.decidable_exists_mem Option.decidableExistsMem\n\n#print Option.iget /-\n/-- Inhabited `get` function. Returns `a` if the input is `some a`, otherwise returns `default`. -/\n@[reducible]\ndef iget [Inhabited \u03b1] : Option \u03b1 \u2192 \u03b1\n  | some x => x\n  | none => default\n#align option.iget Option.iget\n-/\n\n#print Option.iget_some /-\n@[simp]\ntheorem iget_some [Inhabited \u03b1] {a : \u03b1} : (some a).iget = a :=\n  rfl\n#align option.iget_some Option.iget_some\n-/\n\n#print Option.guard /-\n/-- `guard p a` returns `some a` if `p a` holds, otherwise `none`. -/\ndef guard (p : \u03b1 \u2192 Prop) [DecidablePred p] (a : \u03b1) : Option \u03b1 :=\n  if p a then some a else none\n#align option.guard Option.guard\n-/\n\n/- warning: option.filter -> Option.filter is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} (p : \u03b1 -> Prop) [_inst_1 : DecidablePred.{succ u1} \u03b1 p], (Option.{u1} \u03b1) -> (Option.{u1} \u03b1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1}}, (\u03b1 -> Bool) -> (Option.{u1} \u03b1) -> (Option.{u1} \u03b1)\nCase conversion may be inaccurate. Consider using '#align option.filter Option.filter\u2093'. -/\n/-- `filter p o` returns `some a` if `o` is `some a` and `p a` holds, otherwise `none`. -/\ndef filter (p : \u03b1 \u2192 Prop) [DecidablePred p] (o : Option \u03b1) : Option \u03b1 :=\n  o.bind (guard p)\n#align option.filter Option.filter\n\n#print Option.toList /-\n/-- Cast of `option` to `list `. Returns `[a]` if the input is `some a`, and `[]` if it is\n`none`. -/\ndef toList : Option \u03b1 \u2192 List \u03b1\n  | none => []\n  | some a => [a]\n#align option.to_list Option.toList\n-/\n\n#print Option.mem_toList /-\n@[simp]\ntheorem mem_toList {a : \u03b1} {o : Option \u03b1} : a \u2208 toList o \u2194 a \u2208 o := by\n  cases o <;> simp [to_list, eq_comm]\n#align option.mem_to_list Option.mem_toList\n-/\n\n#print Option.liftOrGet /-\n/-- Two arguments failsafe function. Returns `f a b` if the inputs are `some a` and `some b`, and\n\"does nothing\" otherwise. -/\ndef liftOrGet (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : Option \u03b1 \u2192 Option \u03b1 \u2192 Option \u03b1\n  | none, none => none\n  | some a, none => some a\n  |-- get a\n    none,\n    some b => some b\n  |-- get b\n      some\n      a,\n    some b => some (f a b)\n#align option.lift_or_get Option.liftOrGet\n-/\n\n#print Option.liftOrGet_isCommutative /-\n-- lift f\ninstance liftOrGet_isCommutative (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : IsCommutative \u03b1 f] :\n    IsCommutative (Option \u03b1) (liftOrGet f) :=\n  \u27e8fun a b => by cases a <;> cases b <;> simp [lift_or_get, h.comm]\u27e9\n#align option.lift_or_get_comm Option.liftOrGet_isCommutative\n-/\n\n#print Option.liftOrGet_isAssociative /-\ninstance liftOrGet_isAssociative (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : IsAssociative \u03b1 f] :\n    IsAssociative (Option \u03b1) (liftOrGet f) :=\n  \u27e8fun a b c => by cases a <;> cases b <;> cases c <;> simp [lift_or_get, h.assoc]\u27e9\n#align option.lift_or_get_assoc Option.liftOrGet_isAssociative\n-/\n\n#print Option.liftOrGet_isIdempotent /-\ninstance liftOrGet_isIdempotent (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) [h : IsIdempotent \u03b1 f] :\n    IsIdempotent (Option \u03b1) (liftOrGet f) :=\n  \u27e8fun a => by cases a <;> simp [lift_or_get, h.idempotent]\u27e9\n#align option.lift_or_get_idem Option.liftOrGet_isIdempotent\n-/\n\n#print Option.liftOrGet_isLeftId /-\ninstance liftOrGet_isLeftId (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : IsLeftId (Option \u03b1) (liftOrGet f) none :=\n  \u27e8fun a => by cases a <;> simp [lift_or_get]\u27e9\n#align option.lift_or_get_is_left_id Option.liftOrGet_isLeftId\n-/\n\n#print Option.liftOrGet_isRightId /-\ninstance liftOrGet_isRightId (f : \u03b1 \u2192 \u03b1 \u2192 \u03b1) : IsRightId (Option \u03b1) (liftOrGet f) none :=\n  \u27e8fun a => by cases a <;> simp [lift_or_get]\u27e9\n#align option.lift_or_get_is_right_id Option.liftOrGet_isRightId\n-/\n\n#print Option.Rel /-\n/-- Lifts a relation `\u03b1 \u2192 \u03b2 \u2192 Prop` to a relation `option \u03b1 \u2192 option \u03b2 \u2192 Prop` by just adding\n`none ~ none`. -/\ninductive Rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) : Option \u03b1 \u2192 Option \u03b2 \u2192 Prop/--\nIf `a ~ b`, then `some a ~ some b` -/\n\n  | some {a b} : r a b \u2192 Rel (some a) (some b)/-- `none ~ none` -/\n\n  | none : Rel none none\n#align option.rel Option.Rel\n-/\n\n#print Option.pbind /-\n/-- Partial bind. If for some `x : option \u03b1`, `f : \u03a0 (a : \u03b1), a \u2208 x \u2192 option \u03b2` is a\n  partial function defined on `a : \u03b1` giving an `option \u03b2`, where `some a = x`,\n  then `pbind x f h` is essentially the same as `bind x f`\n  but is defined only when all `x = some a`, using the proof to apply `f`. -/\n@[simp]\ndef pbind : \u2200 x : Option \u03b1, (\u2200 a : \u03b1, a \u2208 x \u2192 Option \u03b2) \u2192 Option \u03b2\n  | none, _ => none\n  | some a, f => f a rfl\n#align option.pbind Option.pbind\n-/\n\n#print Option.pmap /-\n/-- Partial map. If `f : \u03a0 a, p a \u2192 \u03b2` is a partial function defined on `a : \u03b1` satisfying `p`,\nthen `pmap f x h` is essentially the same as `map f x` but is defined only when all members of `x`\nsatisfy `p`, using the proof to apply `f`. -/\n@[simp]\ndef pmap {p : \u03b1 \u2192 Prop} (f : \u2200 a : \u03b1, p a \u2192 \u03b2) : \u2200 x : Option \u03b1, (\u2200 a \u2208 x, p a) \u2192 Option \u03b2\n  | none, _ => none\n  | some a, H => some (f a (H a (mem_def.mpr rfl)))\n#align option.pmap Option.pmap\n-/\n\n#print Option.join /-\n/-- Flatten an `option` of `option`, a specialization of `mjoin`. -/\n@[simp]\ndef join : Option (Option \u03b1) \u2192 Option \u03b1 := fun x => bind x id\n#align option.join Option.join\n-/\n\n#print Option.traverse /-\nprotected def traverse.{u, v} {F : Type u \u2192 Type v} [Applicative F] {\u03b1 \u03b2 : Type _} (f : \u03b1 \u2192 F \u03b2) :\n    Option \u03b1 \u2192 F (Option \u03b2)\n  | none => pure none\n  | some x => some <$> f x\n#align option.traverse Option.traverse\n-/\n\n#print Option.maybe /-\n-- By analogy with `monad.sequence` in `init/category/combinators.lean`.\n/-- If you maybe have a monadic computation in a `[monad m]` which produces a term of type `\u03b1`, then\nthere is a naturally associated way to always perform a computation in `m` which maybe produces a\nresult. -/\ndef maybe.{u, v} {m : Type u \u2192 Type v} [Monad m] {\u03b1 : Type u} : Option (m \u03b1) \u2192 m (Option \u03b1)\n  | none => return none\n  | some fn => some <$> fn\n#align option.maybe Option.maybe\n-/\n\n/- warning: option.mmap -> Option.mapM is a dubious translation:\nlean 3 declaration is\n  forall {m : Type.{u1} -> Type.{u2}} [_inst_1 : Monad.{u1, u2} m] {\u03b1 : Type.{u3}} {\u03b2 : Type.{u1}}, (\u03b1 -> (m \u03b2)) -> (Option.{u3} \u03b1) -> (m (Option.{u1} \u03b2))\nbut is expected to have type\n  forall {m : Type.{u1} -> Type.{u2}} {_inst_1 : Type.{u3}} {\u03b1 : Type.{u1}} [\u03b2 : Monad.{u1, u2} m], (_inst_1 -> (m \u03b1)) -> (Option.{u3} _inst_1) -> (m (Option.{u1} \u03b1))\nCase conversion may be inaccurate. Consider using '#align option.mmap Option.mapM\u2093'. -/\n/-- Map a monadic function `f : \u03b1 \u2192 m \u03b2` over an `o : option \u03b1`, maybe producing a result. -/\ndef mapM.{u, v, w} {m : Type u \u2192 Type v} [Monad m] {\u03b1 : Type w} {\u03b2 : Type u} (f : \u03b1 \u2192 m \u03b2)\n    (o : Option \u03b1) : m (Option \u03b2) :=\n  (o.map f).maybe\n#align option.mmap Option.mapM\n\n/- warning: option.melim -> Option.elimM is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {\u03b2 : Type.{u1}} {m : Type.{u1} -> Type.{u2}} [_inst_1 : Monad.{u1, u2} m], (m \u03b2) -> (\u03b1 -> (m \u03b2)) -> (m (Option.{u1} \u03b1)) -> (m \u03b2)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1} -> Type.{u2}} {\u03b2 : Type.{u1}} {m : Type.{u1}} [_inst_1 : Monad.{u1, u2} \u03b1], (\u03b1 (Option.{u1} \u03b2)) -> (\u03b1 m) -> (\u03b2 -> (\u03b1 m)) -> (\u03b1 m)\nCase conversion may be inaccurate. Consider using '#align option.melim Option.elimM\u2093'. -/\n/-- A monadic analogue of `option.elim`. -/\ndef elimM {\u03b1 \u03b2 : Type _} {m : Type _ \u2192 Type _} [Monad m] (y : m \u03b2) (z : \u03b1 \u2192 m \u03b2)\n    (x : m (Option \u03b1)) : m \u03b2 :=\n  x >>= Option.elim' y z\n#align option.melim Option.elimM\n\n/- warning: option.mget_or_else -> Option.getDM' is a dubious translation:\nlean 3 declaration is\n  forall {\u03b1 : Type.{u1}} {m : Type.{u1} -> Type.{u2}} [_inst_1 : Monad.{u1, u2} m], (m (Option.{u1} \u03b1)) -> (m \u03b1) -> (m \u03b1)\nbut is expected to have type\n  forall {\u03b1 : Type.{u1} -> Type.{u2}} {m : Type.{u1}} [_inst_1 : Monad.{u1, u2} \u03b1], (\u03b1 (Option.{u1} m)) -> (\u03b1 m) -> (\u03b1 m)\nCase conversion may be inaccurate. Consider using '#align option.mget_or_else Option.getDM'\u2093'. -/\n/-- A monadic analogue of `option.get_or_else`. -/\ndef getDM' {\u03b1 : Type _} {m : Type _ \u2192 Type _} [Monad m] (x : m (Option \u03b1)) (y : m \u03b1) : m \u03b1 :=\n  elimM y pure x\n#align option.mget_or_else Option.getDM'\n\nend Option\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/Data/Option/Defs.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.35088198245842517}}
{"text": "import order.filter.filter_product\nimport complements.germ\n\n/-! # Complements on filter products -/\n\nopen ultrafilter filter function\n\nnamespace filter.germ\n\nvariables {\u03b9 \u03b1 \u03b2 : Type*} (l : ultrafilter \u03b9)\n\nlocal notation `\u2200*` binders `, ` r:(scoped p, filter.eventually p l) := r\nlocal notation `\u03b1*` := (l : filter \u03b9).germ \u03b1\nlocal notation `\u03b2*` := (l : filter \u03b9).germ \u03b2\nlocal notation `\u22c8` := (prod_equiv (l : filter \u03b9) : \u03b1* \u00d7 \u03b2* \u2192 (l : filter \u03b9).germ (\u03b1 \u00d7 \u03b2))\n\n/-! ## Transfer lemmas -/\n\n/-! ### Not rules -/\n\nlemma lift_pred_not_iff_not_lift_pred (p : \u03b1 \u2192 Prop) (x : \u03b1*) : \n  lift_pred (\u03bb x, \u00ac p x) x \u2194 \u00ac lift_pred p x :=\nbegin\n  refine x.induction_on (\u03bb f, _),\n  rw [lift_pred_coe, lift_pred_coe, eventually_not]\nend\n\nlemma lift_rel_not_iff_not_lift_rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) (x : \u03b1*) (y : \u03b2*) : \n  lift_rel (\u03bb x y, \u00ac r x y) x y \u2194 \u00ac lift_rel r x y :=\nbegin\n  refine x.induction_on\u2082 y (\u03bb f g, _),\n  rw [lift_rel_coe, lift_rel_coe, eventually_not]\nend\n\n/-! ### Ne rules -/\n\nlemma lift_pred_ne_iff_ne_map (f g : \u03b1 \u2192 \u03b2) (x : \u03b1*) :\n  lift_pred (\u03bb x, f x \u2260 g x) x \u2194 germ.map f x \u2260 germ.map g x :=\nbegin\n  refine x.induction_on (\u03bb u, _),\n  rw [ne, eq_def, lift_pred_coe, map_coe, map_coe, lift_rel_coe, eventually_not]\nend\n\n/-! ### Imp rules -/\n\nlemma lift_pred_imp_iff_imp_lift_pred (p q : \u03b1 \u2192 Prop) (x : \u03b1*) :\n  lift_pred (\u03bb x, p x \u2192 q x) x \u2194 (lift_pred p x \u2192 lift_pred q x) :=\nbegin\n  refine x.induction_on (\u03bb f, _),\n  exact eventually_imp\nend\n\n/-! ### Forall rules -/\n\nlemma lift_pred_forall_iff_forall_lift_rel (r : \u03b1 \u2192 \u03b2 \u2192 Prop) (x : \u03b1*) : \n  lift_pred (\u03bb x, \u2200 (y : \u03b2), r x y) x \u2194 \u2200 (y : \u03b2*), lift_rel r x y :=\nbegin\n  rw [\u2190 not_iff_not, \u2190 lift_pred_not_iff_not_lift_pred],\n  push_neg,\n  simp_rw [\u2190 lift_rel_not_iff_not_lift_rel],\n  exact lift_pred_exists_iff_exists_lift_rel \u2191l _ x,\nend\n\nlemma lift_pred_forall_iff_forall_lift_pred (r : \u03b1 \u2192 \u03b2 \u2192 Prop) (x : \u03b1*) : \n  lift_pred (\u03bb x, \u2200 (y : \u03b2), r x y) x \u2194 \u2200 (y : \u03b2*), lift_pred (uncurry r) (\u22c8 (x, y)) :=\nbegin\n  convert lift_pred_forall_iff_forall_lift_rel l r x,\n  ext,\n  exact forall_congr (\u03bb y, by rw \u2190 lift_rel_iff_lift_pred_uncurry)\nend\n\nlemma lift_pred_forall_iff_forall_lift_pred' (r : \u03b1 \u2192 \u03b2 \u2192 Prop) (x : \u03b1*) : \n  lift_pred (\u03bb x, \u2200 (y : \u03b2), r x y) x \u2194 \u2200 (y : \u03b2*), lift_pred (\u03bb u : \u03b1 \u00d7 \u03b2, r u.1 u.2) (\u22c8 (x, y)) :=\nlift_pred_forall_iff_forall_lift_pred l r x\n\n/-! ### Or rules -/\n\nlemma lift_pred_or_iff_or_lift_pred (p q : \u03b1 \u2192 Prop) (x : \u03b1*) :\n  lift_pred (\u03bb x, p x \u2228 q x) x \u2194 lift_pred p x \u2228 lift_pred q x :=\nbegin\n  refine x.induction_on (\u03bb f, _),\n  exact eventually_or\nend\n\n/-! ### Lt rules -/\n\nlemma lift_pred_lt_iff_lt_map [preorder \u03b2] (f g : \u03b1 \u2192 \u03b2) (x : \u03b1*) :\n  lift_pred (\u03bb x, f x < g x) x \u2194 germ.map f x < germ.map g x :=\nbegin\n  refine x.induction_on (\u03bb f, _),\n  rw lt_def,\n  refl\nend\n\nend filter.germ", "meta": {"author": "ADedecker", "repo": "nonstandard", "sha": "c32f5e1d87cc9e6410d66cf3080fd8c4a47cf5e4", "save_path": "github-repos/lean/ADedecker-nonstandard", "path": "github-repos/lean/ADedecker-nonstandard/nonstandard-c32f5e1d87cc9e6410d66cf3080fd8c4a47cf5e4/src/complements/filter_product.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558603, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3507366449088281}}
{"text": "\nimport .scheduling\nimport data.equiv.basic\n\nuniverses u v u' v'\n\nnamespace temporal\nopen function\n\ninstance schedulable_empty : schedulable empty :=\n{ f := \u03bb _, 0\n, inj := by { intros _ _, casesm* empty, }  }\n\ninstance schedulable_unit : schedulable punit :=\n{ f := \u03bb _, 0\n, inj := by { intros _ _, casesm* punit, simp }  }\n\ninstance schedulable_bool : schedulable bool :=\n{ f := bool.rec 0 1\n, inj := by { intros _ _, casesm* bool ; simp }  }\n\ninstance schedulable_nat : schedulable \u2115 :=\n{ f := id\n, inj := injective_id }\nopen equiv\n\nlemma equiv.inj {\u03b1 \u03b2} (h : \u03b1 \u2243 \u03b2) : injective h :=\nby { apply injective_of_left_inverse, apply h.left_inv }\n\ninstance schedulable_int : schedulable \u2124 :=\n{ f := int_equiv_nat\n, inj := equiv.inj _ }\n\ninstance schedulable_fin (n : \u2115) : schedulable (fin n) :=\n{ f := fin.val\n, inj := by { intros _ _, apply fin.eq_of_veq } }\n\nvariables {\u03b1 : Type u} {\u03b2 : Type v} {\u03b7 : \u03b1 \u2192 Type u'}\nvariables {\u03b1' : Type u'} {\u03b2' : Type v'}\n\nsection inductive_construction\n\nvariables [schedulable \u03b1]\nvariables [schedulable \u03b2]\nvariables [\u2200 x, schedulable (\u03b7 x)]\n\ndef sum.map (f : \u03b1 \u2192 \u03b1') (g : \u03b2 \u2192 \u03b2') : \u03b1 \u2295 \u03b2 \u2192 \u03b1' \u2295 \u03b2'\n| (sum.inr x) := sum.inr (g x)\n| (sum.inl x) := sum.inl (f x)\n\nopen equiv scheduling.schedulable\n\ninstance schedulable_sum : schedulable (\u03b1 \u2295 \u03b2) :=\n{ f := nat_sum_nat_equiv_nat \u2218 sum.map scheduling.schedulable.f scheduling.schedulable.f\n, inj := by { apply injective_comp, apply equiv.inj,\n              intros _ _, casesm* _ \u2295 _ ; simp [sum.map] ; apply inj, } }\n\ninstance schedulable_option : schedulable (option \u03b1) :=\n{ f := f \u2218 option_equiv_sum_unit \u03b1\n, inj :=\nbegin\n  apply injective_comp,\n  { apply inj },\n  { apply equiv.inj, },\nend\n}\n\ninstance schedulable_list : schedulable (list \u03b1) :=\n{ f := f \u2218 list_nat_equiv_nat \u2218 list.map f\n, inj :=\nbegin\n  apply injective_comp,\n  { apply inj },\n  apply injective_comp,\n  { apply equiv.inj, },\n  { intros xs,\n    induction xs ; intro ys ; cases ys ; simp,\n    intros, split,\n    apply inj _ a,\n    solve_by_elim },\nend\n}\n\ninstance schedulable_prod : schedulable (\u03b1 \u00d7 \u03b2) :=\n{ f := nat_prod_nat_equiv_nat \u2218 prod.map scheduling.schedulable.f scheduling.schedulable.f\n, inj := by { apply injective_comp,\n              { apply equiv.inj },\n              intros _ _, casesm* _ \u00d7 _,\n              simp [prod.map], intros,\n              split ; apply inj ; solve_by_elim } }\n\ndef index_pair : sigma \u03b7 \u2192 \u2115 \u00d7 \u2115\n | \u27e8x,y\u27e9 := (f x,f y)\n\ninstance schedulable_sigma : schedulable (sigma \u03b7) :=\n{ f := nat_prod_nat_equiv_nat \u2218 index_pair\n, inj := by { apply injective_comp,\n              { apply equiv.inj },\n              intros _ _, casesm* sigma _,\n              simp [index_pair],\n              intros,\n              have : a\u2081_fst = a\u2082_fst := inj _ (by solve_by_elim),\n              subst a\u2082_fst, simp,\n              exact inj _ (by solve_by_elim) }  }\n\nend inductive_construction\n\nsection from_other_class\n\nopen fintype\n\nlocal attribute [instance] classical.prop_decidable\n\nnoncomputable instance fintype_schedulable [fintype \u03b1] : schedulable \u03b1 :=\n{ f := (\u03bb x, (trunc.out (equiv_fin \u03b1) x).val)\n, inj := by { intros _ _ h, simp at h,\n              replace h := fin.eq_of_veq h,\n              apply equiv.inj _ h, } }\n\nclass countable_type (\u03b1 : Type u) :=\n(enum : trunc $ \u03b1 \u2243 \u2115)\n\nnoncomputable instance countable_type_schedulable [countable_type \u03b1] : schedulable \u03b1 :=\n{ f := (\u03bb x, trunc.out (countable_type.enum \u03b1) x)\n, inj := by { intros _ _ h, simp at h, exact h, } }\n\nend from_other_class\n\nend temporal\n", "meta": {"author": "unitb", "repo": "temporal-logic", "sha": "accec04d1b09ca841be065511c9e206b725b16e9", "save_path": "github-repos/lean/unitb-temporal-logic", "path": "github-repos/lean/unitb-temporal-logic/temporal-logic-accec04d1b09ca841be065511c9e206b725b16e9/src/temporal_logic/instances.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3506976944154402}}
{"text": "import Lean\nopen Lean Elab Term\n\ndef hello := \"world\"\ndef sss := \"\u2200a b, a \u2192 b \u2192 a \u2227 b\"\nelab \"myt[\" s:str \"]\" : term => do\n  logInfo s!\"{s}\"\n  pure $ mkStrLit \"a\"\n#eval myt[\"nakami\"]\n\nelab \"myterm[\" s:str \"]\" : term => do\n  let env \u2190 getEnv\n  let _a \u2190 (pure:_ \u2192 IO _) \"a\"\n  let aa := s.getString\n  logInfo s!\"ho:{aa}\"\n  let parsedSyntax \u2190 match Lean.Parser.runParserCategory env `term s.getString with\n                      | Except.ok stx => pure stx\n                      | Except.error errmsg => throwError errmsg\n  logInfo s!\"{parsedSyntax}\"\n  let prop \u2190 elabTerm parsedSyntax none-- (mkConst `Lean.Prop)\n  logInfo s!\"hi:{prop}\"\n  pure prop\n\ntheorem propStr : myterm[\"\u2200a b, a \u2192 b \u2192 a \u2227 b\"] :=\n  fun {a b : Prop} (ha : a) (hb : b) => \u27e8ha,hb\u27e9\n\ndef p := myterm[\"\u2200a b, a \u2192 b \u2192 a \u2227 b\"]\n#print p\n", "meta": {"author": "denjiry", "repo": "str2term", "sha": "988de63c9b5062bbb20acb6c41025957b29310e8", "save_path": "github-repos/lean/denjiry-str2term", "path": "github-repos/lean/denjiry-str2term/str2term-988de63c9b5062bbb20acb6c41025957b29310e8/Str2Term.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7122321842389469, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3505522309609174}}
{"text": "import Duper.Simp\nimport Duper.Util.ProofReconstruction\n\nnamespace Duper\nopen RuleM\nopen SimpResult\nopen Lean\nopen Meta\n\n/-- Determines whether a literal has exactly the form `false = true` or `true = false`-/\ndef isFalseBoolLiteral (lit : Lit) : MetaM Bool := do\n  if lit.ty.consumeMData == (mkConst ``Bool) then\n    return lit.sign &&\n      ((lit.lhs == mkConst ``true && lit.rhs == mkConst ``false) ||\n      (lit.lhs == mkConst ``false && lit.rhs == mkConst ``true))\n  else return false\n\ntheorem bool_false_ne_true (h : false = true) : False := ne_false_of_eq_true h (by rfl)\n\ntheorem bool_true_ne_false (h : true = false) : False := ne_true_of_eq_false h (by rfl)\n\ndef mkIdentBoolFalseElimProof (refs : List (Option Nat)) (premises : List Expr) (parents: List ProofParent)\n  (transferExprs : Array Expr) (c : Clause) : MetaM Expr :=\n  Meta.forallTelescope c.toForallExpr fun xs body => do\n    let cLits := c.lits.map (fun l => l.map (fun e => e.instantiateRev xs))\n    let (parentsLits, appliedPremises, transferExprs) \u2190 instantiatePremises parents premises xs transferExprs\n    let parentLits := parentsLits[0]!\n    let appliedPremise := appliedPremises[0]!\n\n    let mut proofCases : Array Expr := Array.mkEmpty parentLits.size\n    for i in [:parentLits.size] do\n      let lit := parentLits[i]!\n      if (\u2190 isFalseBoolLiteral lit) then -- lit has the form `false = true` or `true = false`\n        let proofCase \u2190 Meta.withLocalDeclD `h lit.toExpr fun h => do\n          if (lit.lhs == mkConst ``false) then\n            let proofCase := mkApp (mkConst ``bool_false_ne_true) h\n            let proofCase := mkApp2 (mkConst ``False.elim [levelZero]) body proofCase\n            Meta.mkLambdaFVars #[h] proofCase\n          else if(lit.lhs == mkConst ``true) then\n            let proofCase := mkApp (mkConst ``bool_true_ne_false) h\n            let proofCase := mkApp2 (mkConst ``False.elim [levelZero]) body proofCase\n            Meta.mkLambdaFVars #[h] proofCase\n          else\n            throwError \"mkIdentBoolFalseElimProof failed to match {lit.lhs} to an expected expression\"\n        proofCases := proofCases.push proofCase\n      else -- refs[i] should have the value (some j) where parentLits[i] == c[j]\n        match refs[i]! with\n        | none => throwError \"Refs invariant is not satisfied in identBoolFalseElim\"\n        | some j =>\n          let proofCase \u2190 Meta.withLocalDeclD `h parentLits[i]!.toExpr fun h => do\n            Meta.mkLambdaFVars #[h] $ \u2190 orIntro (cLits.map Lit.toExpr) j h\n          proofCases := proofCases.push proofCase\n    let proof \u2190 orCases (parentLits.map Lit.toExpr) proofCases\n    Meta.mkLambdaFVars xs $ mkApp proof appliedPremise\n\n/-- Eliminate literals that are exactly of the form `false = true` or `true = false`. \n    This is a special case of the boolFalseElim inference rule in which \u03c3 is the identity. \n    This rule is included as a means of giving Bools special attention. -/\ndef identBoolFalseElim : MSimpRule := fun c => do\n  let c \u2190 loadClause c\n  /-\n    Spec for newLits and refs\n    If c.lits[i] is `false = true` or `true = false`, then refs[i] = none\n    If c.lits[i] isn't `false = true` or `true = false`,then refs[i] = some j where newLits[j] = c.lits[i]\n  -/\n  let mut newLits : List Lit := []\n  let mut refs : List (Option Nat) := []\n  for lit in c.lits do\n    if (\u2190 isFalseBoolLiteral lit) then\n      refs := none :: refs\n    else\n      refs := (some newLits.length) :: refs\n      newLits := lit :: newLits\n  -- To achieve the desired spec for newLits and refs, I must reverse them\n  newLits := newLits.reverse\n  refs := refs.reverse\n  if (newLits.length = c.lits.size) then\n    return none\n  else\n    let resultClause \u2190 yieldClause (MClause.mk newLits.toArray) \"identity boolean false elimination\"\n      (some (mkIdentBoolFalseElimProof refs))\n    return some #[resultClause]\n\nend Duper", "meta": {"author": "leanprover-community", "repo": "duper", "sha": "96b8f8383363e800976b0fa99830c1b5e8c19b09", "save_path": "github-repos/lean/leanprover-community-duper", "path": "github-repos/lean/leanprover-community-duper/duper-96b8f8383363e800976b0fa99830c1b5e8c19b09/Duper/Rules/IdentBoolFalseElim.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947425132315, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.35043314763465955}}
{"text": "/-\nCopyright (c) 2020 Markus Himmel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer\n-/\nimport category_theory.abelian.opposite\nimport category_theory.limits.preserves.shapes.zero\nimport category_theory.limits.preserves.shapes.kernels\nimport category_theory.preadditive.left_exact\nimport category_theory.adjunction.limits\nimport algebra.homology.exact\nimport tactic.tfae\n\n/-!\n# Exact sequences in abelian categories\n\nIn an abelian category, we get several interesting results related to exactness which are not\ntrue in more general settings.\n\n## Main results\n* `(f, g)` is exact if and only if `f \u226b g = 0` and `kernel.\u03b9 g \u226b cokernel.\u03c0 f = 0`. This\n  characterisation tends to be less cumbersome to work with than the original definition involving\n  the comparison map `image f \u27f6 kernel g`.\n* If `(f, g)` is exact, then `image.\u03b9 f` has the universal property of the kernel of `g`.\n* `f` is a monomorphism iff `kernel.\u03b9 f = 0` iff `exact 0 f`, and `f` is an epimorphism iff\n  `cokernel.\u03c0 = 0` iff `exact f 0`.\n* A faithful functor between abelian categories that preserves zero morphisms reflects exact\n  sequences.\n* `X \u27f6 Y \u27f6 Z \u27f6 0` is exact if and only if the second map is a cokernel of the first, and\n  `0 \u27f6 X \u27f6 Y \u27f6 Z` is exact if and only if the first map is a kernel of the second.\n* An exact functor preserves exactness, more specifically, `F` preserves finite colimits and\n  finite limits, if and only if `exact f g` implies `exact (F.map f) (F.map g)`.\n-/\n\nuniverses v\u2081 v\u2082 u\u2081 u\u2082\n\nnoncomputable theory\n\nopen category_theory\nopen category_theory.limits\nopen category_theory.preadditive\n\nvariables {C : Type u\u2081} [category.{v\u2081} C] [abelian C]\n\nnamespace category_theory\n\nnamespace abelian\n\nvariables {X Y Z : C} (f : X \u27f6 Y) (g : Y \u27f6 Z)\n\nlocal attribute [instance] has_equalizers_of_has_kernels\n\n/--\nIn an abelian category, a pair of morphisms `f : X \u27f6 Y`, `g : Y \u27f6 Z` is exact\niff `image_subobject f = kernel_subobject g`.\n-/\ntheorem exact_iff_image_eq_kernel : exact f g \u2194 image_subobject f = kernel_subobject g :=\nbegin\n  split,\n  { intro h,\n    fapply subobject.eq_of_comm,\n    { suffices : is_iso (image_to_kernel _ _ h.w),\n      { exactI as_iso (image_to_kernel _ _ h.w), },\n      exact is_iso_of_mono_of_epi _, },\n    { simp, }, },\n  { apply exact_of_image_eq_kernel, },\nend\n\ntheorem exact_iff : exact f g \u2194 f \u226b g = 0 \u2227 kernel.\u03b9 g \u226b cokernel.\u03c0 f = 0 :=\nbegin\n  split,\n  { intro h,\n    exact \u27e8h.1, kernel_comp_cokernel f g h\u27e9 },\n  { refine \u03bb h, \u27e8h.1, _\u27e9,\n    suffices hl : is_limit\n      (kernel_fork.of_\u03b9 (image_subobject f).arrow (image_subobject_arrow_comp_eq_zero h.1)),\n    { have : image_to_kernel f g h.1 =\n        (is_limit.cone_point_unique_up_to_iso hl (limit.is_limit _)).hom \u226b\n          (kernel_subobject_iso _).inv,\n      { ext, simp },\n      rw this,\n      apply_instance, },\n    refine kernel_fork.is_limit.of_\u03b9 _ _ _ _ _,\n    { refine \u03bb W u hu,\n        kernel.lift (cokernel.\u03c0 f) u _ \u226b (image_iso_image f).hom \u226b (image_subobject_iso _).inv,\n      rw [\u2190kernel.lift_\u03b9 g u hu, category.assoc, h.2, has_zero_morphisms.comp_zero] },\n    { tidy },\n    { intros, rw [\u2190cancel_mono (image_subobject f).arrow, w],\n      simp, } }\nend\n\ntheorem exact_iff' {cg : kernel_fork g} (hg : is_limit cg)\n  {cf : cokernel_cofork f} (hf : is_colimit cf) : exact f g \u2194 f \u226b g = 0 \u2227 cg.\u03b9 \u226b cf.\u03c0 = 0 :=\nbegin\n  split,\n  { intro h,\n    exact \u27e8h.1, fork_\u03b9_comp_cofork_\u03c0 f g h cg cf\u27e9 },\n  { rw exact_iff,\n    refine \u03bb h, \u27e8h.1, _\u27e9,\n    apply zero_of_epi_comp (is_limit.cone_point_unique_up_to_iso hg (limit.is_limit _)).hom,\n    apply zero_of_comp_mono\n      (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _) hf).hom,\n    simp [h.2] }\nend\n\ntheorem exact_tfae :\n  tfae [exact f g,\n        f \u226b g = 0 \u2227 kernel.\u03b9 g \u226b cokernel.\u03c0 f = 0,\n        image_subobject f = kernel_subobject g] :=\nbegin\n  tfae_have : 1 \u2194 2, { apply exact_iff },\n  tfae_have : 1 \u2194 3, { apply exact_iff_image_eq_kernel },\n  tfae_finish\nend\n\nlemma is_equivalence.exact_iff {D : Type u\u2081} [category.{v\u2081} D] [abelian D]\n  (F : C \u2964 D) [is_equivalence F] :\n  exact (F.map f) (F.map g) \u2194 exact f g :=\nbegin\n  simp only [exact_iff, \u2190 F.map_eq_zero_iff, F.map_comp, category.assoc,\n    \u2190 kernel_comparison_comp_\u03b9 g F, \u2190 \u03c0_comp_cokernel_comparison f F],\n  rw [is_iso.comp_left_eq_zero (kernel_comparison g F), \u2190 category.assoc,\n    is_iso.comp_right_eq_zero _ (cokernel_comparison f F)],\nend\n\n/-- The dual result is true even in non-abelian categories, see\n    `category_theory.exact_comp_mono_iff`. -/\n\n\n/-- If `(f, g)` is exact, then `abelian.image.\u03b9 f` is a kernel of `g`. -/\ndef is_limit_image (h : exact f g) :\n  is_limit\n    (kernel_fork.of_\u03b9 (abelian.image.\u03b9 f) (image_\u03b9_comp_eq_zero h.1) : kernel_fork g) :=\nbegin\n  rw exact_iff at h,\n  refine kernel_fork.is_limit.of_\u03b9 _ _ _ _ _,\n  { refine \u03bb W u hu, kernel.lift (cokernel.\u03c0 f) u _,\n    rw [\u2190kernel.lift_\u03b9 g u hu, category.assoc, h.2, has_zero_morphisms.comp_zero] },\n  tidy\nend\n\n/-- If `(f, g)` is exact, then `image.\u03b9 f` is a kernel of `g`. -/\ndef is_limit_image' (h : exact f g) :\n  is_limit (kernel_fork.of_\u03b9 (limits.image.\u03b9 f) (limits.image_\u03b9_comp_eq_zero h.1)) :=\nis_kernel.iso_kernel _ _ (is_limit_image f g h) (image_iso_image f).symm $ is_image.lift_fac _ _\n\n/-- If `(f, g)` is exact, then `coimages.coimage.\u03c0 g` is a cokernel of `f`. -/\ndef is_colimit_coimage (h : exact f g) : is_colimit (cokernel_cofork.of_\u03c0 (abelian.coimage.\u03c0 g)\n  (abelian.comp_coimage_\u03c0_eq_zero h.1) : cokernel_cofork f) :=\nbegin\n  rw exact_iff at h,\n  refine cokernel_cofork.is_colimit.of_\u03c0 _ _ _ _ _,\n  { refine \u03bb W u hu, cokernel.desc (kernel.\u03b9 g) u _,\n    rw [\u2190cokernel.\u03c0_desc f u hu, \u2190category.assoc, h.2, has_zero_morphisms.zero_comp] },\n  tidy\nend\n\n/-- If `(f, g)` is exact, then `factor_thru_image g` is a cokernel of `f`. -/\ndef is_colimit_image (h : exact f g) : is_colimit\n  (cokernel_cofork.of_\u03c0 (limits.factor_thru_image g) (comp_factor_thru_image_eq_zero h.1)) :=\nis_cokernel.cokernel_iso _ _ (is_colimit_coimage f g h) (coimage_iso_image' g) $\n  (cancel_mono (limits.image.\u03b9 g)).1 $ by simp\n\nlemma exact_cokernel : exact f (cokernel.\u03c0 f) :=\nby { rw exact_iff, tidy }\n\ninstance (h : exact f g) : mono (cokernel.desc f g h.w) :=\nsuffices h : cokernel.desc f g h.w =\n  (is_colimit.cocone_point_unique_up_to_iso (colimit.is_colimit _) (is_colimit_image f g h)).hom\n    \u226b limits.image.\u03b9 g, by { rw h, apply mono_comp },\n(cancel_epi (cokernel.\u03c0 f)).1 $ by simp\n\n/-- If `ex : exact f g` and `epi g`, then `cokernel.desc _ _ ex.w` is an isomorphism. -/\ninstance (ex : exact f g) [epi g] : is_iso (cokernel.desc f g ex.w) :=\nis_iso_of_mono_of_epi (limits.cokernel.desc f g ex.w)\n\n@[simp, reassoc]\nlemma cokernel.desc.inv [epi g] (ex : exact f g) :\n  g \u226b inv (cokernel.desc _ _ ex.w) = cokernel.\u03c0 _ :=\nby simp\n\ninstance (ex : exact f g) [mono f] : is_iso (kernel.lift g f ex.w) :=\n  is_iso_of_mono_of_epi (limits.kernel.lift g f ex.w)\n\n@[simp, reassoc]\nlemma kernel.lift.inv [mono f] (ex : exact f g) :\n  inv (kernel.lift _ _ ex.w) \u226b f = kernel.\u03b9 g :=\nby simp\n\n/-- If `X \u27f6 Y \u27f6 Z \u27f6 0` is exact, then the second map is a cokernel of the first. -/\ndef is_colimit_of_exact_of_epi [epi g] (h : exact f g) :\n  is_colimit (cokernel_cofork.of_\u03c0 _ h.w) :=\nis_colimit.of_iso_colimit (colimit.is_colimit _) $ cocones.ext\n  \u27e8cokernel.desc _ _ h.w, epi_desc g (cokernel.\u03c0 f) ((exact_iff _ _).1 h).2,\n    (cancel_epi (cokernel.\u03c0 f)).1 (by tidy), (cancel_epi g).1 (by tidy)\u27e9 (\u03bb j, by cases j; simp)\n\n/-- If `0 \u27f6 X \u27f6 Y \u27f6 Z` is exact, then the first map is a kernel of the second. -/\ndef is_limit_of_exact_of_mono [mono f] (h : exact f g) :\n  is_limit (kernel_fork.of_\u03b9 _ h.w) :=\nis_limit.of_iso_limit (limit.is_limit _) $ cones.ext\n \u27e8mono_lift f (kernel.\u03b9 g) ((exact_iff _ _).1 h).2, kernel.lift _ _ h.w,\n  (cancel_mono (kernel.\u03b9 g)).1 (by tidy), (cancel_mono f).1 (by tidy)\u27e9 (\u03bb j, by cases j; simp)\n\nlemma exact_of_is_cokernel (w : f \u226b g = 0)\n  (h : is_colimit (cokernel_cofork.of_\u03c0 _ w)) : exact f g :=\nbegin\n  refine (exact_iff _ _).2 \u27e8w, _\u27e9,\n  have := h.fac (cokernel_cofork.of_\u03c0 _ (cokernel.condition f)) walking_parallel_pair.one,\n  simp only [cofork.of_\u03c0_\u03b9_app] at this,\n  rw [\u2190 this, \u2190 category.assoc, kernel.condition, zero_comp]\nend\n\nlemma exact_of_is_kernel (w : f \u226b g = 0)\n  (h : is_limit (kernel_fork.of_\u03b9 _ w)) : exact f g :=\nbegin\n  refine (exact_iff _ _).2 \u27e8w, _\u27e9,\n  have := h.fac (kernel_fork.of_\u03b9 _ (kernel.condition g)) walking_parallel_pair.zero,\n  simp only [fork.of_\u03b9_\u03c0_app] at this,\n  rw [\u2190 this, category.assoc, cokernel.condition, comp_zero]\nend\n\nlemma exact_iff_exact_image_\u03b9 : exact f g \u2194 exact (abelian.image.\u03b9 f) g :=\nby conv_lhs { rw \u2190 abelian.image.fac f }; apply exact_epi_comp_iff\n\nlemma exact_iff_exact_coimage_\u03c0 : exact f g \u2194 exact f (coimage.\u03c0 g) :=\nby conv_lhs { rw \u2190 abelian.coimage.fac g}; apply exact_comp_mono_iff\n\nsection\nvariables (Z)\n\nlemma tfae_mono : tfae [mono f, kernel.\u03b9 f = 0, exact (0 : Z \u27f6 X) f] :=\nbegin\n  tfae_have : 3 \u2192 2,\n  { exact kernel_\u03b9_eq_zero_of_exact_zero_left Z },\n  tfae_have : 1 \u2192 3,\n  { introsI, exact exact_zero_left_of_mono Z },\n  tfae_have : 2 \u2192 1,\n  { exact mono_of_kernel_\u03b9_eq_zero _ },\n  tfae_finish\nend\n\n-- Note we've already proved `mono_iff_exact_zero_left : mono f \u2194 exact (0 : Z \u27f6 X) f`\n-- in any preadditive category with kernels and images.\n\nlemma mono_iff_kernel_\u03b9_eq_zero : mono f \u2194 kernel.\u03b9 f = 0 :=\n(tfae_mono X f).out 0 1\n\nlemma tfae_epi : tfae [epi f, cokernel.\u03c0 f = 0, exact f (0 : Y \u27f6 Z)] :=\nbegin\n  tfae_have : 3 \u2192 2,\n  { rw exact_iff,\n    rintro \u27e8-, h\u27e9,\n    exact zero_of_epi_comp _ h },\n  tfae_have : 1 \u2192 3,\n  { rw exact_iff,\n    introI,\n    exact \u27e8by simp, by simp [cokernel.\u03c0_of_epi]\u27e9 },\n  tfae_have : 2 \u2192 1,\n  { exact epi_of_cokernel_\u03c0_eq_zero _ },\n  tfae_finish\nend\n\n-- Note we've already proved `epi_iff_exact_zero_right : epi f \u2194 exact f (0 : Y \u27f6 Z)`\n-- in any preadditive category with equalizers and images.\n\nlemma epi_iff_cokernel_\u03c0_eq_zero : epi f \u2194 cokernel.\u03c0 f = 0 :=\n(tfae_epi X f).out 0 1\n\nend\n\nsection opposite\n\nlemma exact.op (h : exact f g) : exact g.op f.op :=\nbegin\n  rw exact_iff,\n  refine \u27e8by simp [\u2190 op_comp, h.w], quiver.hom.unop_inj _\u27e9,\n  simp only [unop_comp, cokernel.\u03c0_op, eq_to_hom_refl, kernel.\u03b9_op, category.id_comp,\n    category.assoc, kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero, unop_zero],\nend\n\nlemma exact.op_iff : exact g.op f.op \u2194 exact f g :=\n\u27e8\u03bb e, begin\n  rw \u2190 is_equivalence.exact_iff _ _ (op_op_equivalence C).inverse,\n  exact exact.op _ _ e\nend, exact.op _ _\u27e9\n\n\nlemma exact.unop {X Y Z : C\u1d52\u1d56} (g : X \u27f6 Y) (f : Y \u27f6 Z) (h : exact g f) : exact f.unop g.unop :=\nbegin\n  rw [\u2190 f.op_unop, \u2190 g.op_unop] at h,\n  rwa \u2190 exact.op_iff,\nend\n\nlemma exact.unop_iff {X Y Z : C\u1d52\u1d56} (g : X \u27f6 Y) (f : Y \u27f6 Z) : exact f.unop g.unop \u2194 exact g f :=\n\u27e8\u03bb e, by rwa [\u2190 f.op_unop, \u2190 g.op_unop, \u2190 exact.op_iff] at e, \u03bb e, @@exact.unop _ _ g f e\u27e9\n\nend opposite\n\n\nend abelian\n\nnamespace functor\n\nsection\n\nvariables {D : Type u\u2082} [category.{v\u2082} D] [abelian D]\nvariables (F : C \u2964 D) [preserves_zero_morphisms F]\n\n@[priority 100]\ninstance reflects_exact_sequences_of_preserves_zero_morphisms_of_faithful [faithful F] :\n  reflects_exact_sequences F :=\n{ reflects := \u03bb X Y Z f g hfg,\n  begin\n    rw [abelian.exact_iff, \u2190 F.map_comp, F.map_eq_zero_iff] at hfg,\n    refine (abelian.exact_iff _ _).2 \u27e8hfg.1, F.zero_of_map_zero _ _\u27e9,\n    obtain \u27e8k, hk\u27e9 := kernel.lift' (F.map g) (F.map (kernel.\u03b9 g))\n      (by simp only [\u2190 F.map_comp, kernel.condition, category_theory.functor.map_zero]),\n    obtain \u27e8l, hl\u27e9 := cokernel.desc' (F.map f) (F.map (cokernel.\u03c0 f))\n      (by simp only [\u2190 F.map_comp, cokernel.condition, category_theory.functor.map_zero]),\n    rw [F.map_comp, \u2190 hk, \u2190 hl, category.assoc, reassoc_of hfg.2, zero_comp, comp_zero]\n  end }\n\nend\n\nend functor\n\nnamespace functor\n\nopen limits abelian\n\nvariables {A : Type u\u2081} {B : Type u\u2082} [category.{v\u2081} A] [category.{v\u2082} B]\nvariables [abelian A] [abelian B]\nvariables (L : A \u2964 B)\n\nsection\n\nvariables [preserves_finite_limits L] [preserves_finite_colimits L]\n\n/-- A functor preserving finite limits and finite colimits preserves exactness. The converse\nresult is also true, see `functor.preserves_finite_limits_of_map_exact` and\n`functor.preserves_finite_colimits_of_map_exact`. -/\nlemma map_exact {X Y Z : A} (f : X \u27f6 Y) (g : Y \u27f6 Z) (e1 : exact f g) :\n  exact (L.map f) (L.map g) :=\nbegin\n  let hcoker := is_colimit_of_has_cokernel_of_preserves_colimit L f,\n  let hker := is_limit_of_has_kernel_of_preserves_limit L g,\n  refine (exact_iff' _ _ hker hcoker).2 \u27e8by simp [\u2190 L.map_comp, e1.1], _\u27e9,\n  rw [fork.\u03b9_of_\u03b9, cofork.\u03c0_of_\u03c0, \u2190 L.map_comp, kernel_comp_cokernel _ _ e1, L.map_zero]\nend\n\nend\n\nsection\n\nvariables (h : \u2200 \u2983X Y Z : A\u2984 {f : X \u27f6 Y} {g : Y \u27f6 Z}, exact f g \u2192 exact (L.map f) (L.map g))\ninclude h\n\nopen_locale zero_object\n\n/-- A functor which preserves exactness preserves zero morphisms. -/\nlemma preserves_zero_morphisms_of_map_exact : L.preserves_zero_morphisms :=\nbegin\n  replace h := (h (exact_of_zero (\ud835\udfd9 0) (\ud835\udfd9 0))).w,\n  rw [L.map_id, category.comp_id] at h,\n  exact preserves_zero_morphisms_of_map_zero_object (id_zero_equiv_iso_zero _ h),\nend\n\n/-- A functor which preserves exactness preserves monomorphisms. -/\nlemma preserves_monomorphisms_of_map_exact : L.preserves_monomorphisms :=\n{ preserves := \u03bb X Y f hf,\n  begin\n    letI := preserves_zero_morphisms_of_map_exact L h,\n    apply ((tfae_mono (L.obj 0) (L.map f)).out 2 0).mp,\n    rw \u2190L.map_zero,\n    exact h (((tfae_mono 0 f).out 0 2).mp hf)\n  end }\n\n/-- A functor which preserves exactness preserves epimorphisms. -/\nlemma preserves_epimorphisms_of_map_exact : L.preserves_epimorphisms :=\n{ preserves := \u03bb X Y f hf,\n  begin\n    letI := preserves_zero_morphisms_of_map_exact L h,\n    apply ((tfae_epi (L.obj 0) (L.map f)).out 2 0).mp,\n    rw \u2190L.map_zero,\n    exact h (((tfae_epi 0 f).out 0 2).mp hf)\n  end }\n\n/-- A functor which preserves exactness preserves kernels. -/\ndef preserves_kernels_of_map_exact (X Y : A) (f : X \u27f6 Y) :\n  preserves_limit (parallel_pair f 0) L :=\n{ preserves := \u03bb c ic,\n  begin\n    letI := preserves_zero_morphisms_of_map_exact L h,\n    letI := preserves_monomorphisms_of_map_exact L h,\n    letI := mono_of_is_limit_fork ic,\n    have hf := (is_limit_map_cone_fork_equiv' L (kernel_fork.condition c)).symm\n      (is_limit_of_exact_of_mono (L.map (fork.\u03b9 c)) (L.map f)\n        (h (exact_of_is_kernel (fork.\u03b9 c) f (kernel_fork.condition c)\n          (ic.of_iso_limit (iso_of_\u03b9 _))))),\n    exact hf.of_iso_limit ((cones.functoriality _ L).map_iso (iso_of_\u03b9 _).symm),\n  end }\n\n/-- A functor which preserves exactness preserves zero cokernels. -/\ndef preserves_cokernels_of_map_exact (X Y : A) (f : X \u27f6 Y) :\n  preserves_colimit (parallel_pair f 0) L :=\n{ preserves := \u03bb c ic,\n  begin\n    letI := preserves_zero_morphisms_of_map_exact L h,\n    letI := preserves_epimorphisms_of_map_exact L h,\n    letI := epi_of_is_colimit_cofork ic,\n    have hf := (is_colimit_map_cocone_cofork_equiv' L (cokernel_cofork.condition c)).symm\n      (is_colimit_of_exact_of_epi (L.map f) (L.map (cofork.\u03c0 c))\n        (h (exact_of_is_cokernel f (cofork.\u03c0 c) (cokernel_cofork.condition c)\n          (ic.of_iso_colimit (iso_of_\u03c0 _))))),\n    exact hf.of_iso_colimit ((cocones.functoriality _ L).map_iso (iso_of_\u03c0 _).symm),\n  end }\n\n/-- A functor which preserves exactness is left exact, i.e. preserves finite limits.\nThis is part of the inverse implication to `functor.map_exact`. -/\ndef preserves_finite_limits_of_map_exact : preserves_finite_limits L :=\nbegin\n  letI := preserves_zero_morphisms_of_map_exact L h,\n  letI := preserves_kernels_of_map_exact L h,\n  apply preserves_finite_limits_of_preserves_kernels,\nend\n\n/-- A functor which preserves exactness is right exact, i.e. preserves finite colimits.\nThis is part of the inverse implication to `functor.map_exact`. -/\ndef preserves_finite_colimits_of_map_exact : preserves_finite_colimits L :=\nbegin\n  letI := preserves_zero_morphisms_of_map_exact L h,\n  letI := preserves_cokernels_of_map_exact L h,\n  apply preserves_finite_colimits_of_preserves_cokernels,\nend\n\nend\n\nsection\n\n/-- A functor preserving zero morphisms, monos, and cokernels preserves finite limits. -/\ndef preserves_finite_limits_of_preserves_monos_and_cokernels\n  [preserves_zero_morphisms L] [preserves_monomorphisms L]\n  [\u2200 {X Y} (f : X \u27f6 Y), preserves_colimit (parallel_pair f 0) L] : preserves_finite_limits L :=\nbegin\n  apply preserves_finite_limits_of_map_exact,\n  intros X Y Z f g h,\n  rw [\u2190 abelian.coimage.fac g, L.map_comp, exact_comp_mono_iff],\n  exact exact_of_is_cokernel _ _ _\n    (is_colimit_cofork_map_of_is_colimit' L _ (is_colimit_coimage f g h))\nend\n\n/-- A functor preserving zero morphisms, epis, and kernels preserves finite colimits. -/\ndef preserves_finite_colimits_of_preserves_epis_and_kernels\n  [preserves_zero_morphisms L] [preserves_epimorphisms L]\n  [\u2200 {X Y} (f : X \u27f6 Y), preserves_limit (parallel_pair f 0) L] : preserves_finite_colimits L :=\nbegin\n  apply preserves_finite_colimits_of_map_exact,\n  intros X Y Z f g h,\n  rw [\u2190 abelian.image.fac f, L.map_comp, exact_epi_comp_iff],\n  exact exact_of_is_kernel _ _ _ (is_limit_fork_map_of_is_limit' L _ (is_limit_image f g h))\nend\n\nend\n\nend functor\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/category_theory/abelian/exact.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.35026007283305416}}
{"text": "/-\nCopyright (c) 2021 Johan Commelin. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Johan Commelin, Andrew Yang, Pierre-Alexandre Bazin\n-/\nimport algebra.homology.short_exact.preadditive\nimport category_theory.abelian.diagram_lemmas.four\n\n/-!\n# Short exact sequences in abelian categories\n\nIn an abelian category, a left-split or right-split short exact sequence admits a splitting.\n-/\n\nnoncomputable theory\n\nopen category_theory category_theory.limits category_theory.preadditive\n\nvariables {\ud835\udc9c : Type*} [category \ud835\udc9c]\n\nnamespace category_theory\n\nvariables {A B C A' B' C' : \ud835\udc9c} {f : A \u27f6 B} {g : B \u27f6 C} {f' : A' \u27f6 B'} {g' : B' \u27f6 C'}\nvariables [abelian \ud835\udc9c]\nopen_locale zero_object\n\nlemma is_iso_of_short_exact_of_is_iso_of_is_iso (h : short_exact f g) (h' : short_exact f' g')\n  (i\u2081 : A \u27f6 A') (i\u2082 : B \u27f6 B') (i\u2083 : C \u27f6 C')\n  (comm\u2081 : i\u2081 \u226b f' = f \u226b i\u2082) (comm\u2082 : i\u2082 \u226b g' = g \u226b i\u2083) [is_iso i\u2081] [is_iso i\u2083] :\n  is_iso i\u2082 :=\nbegin\n  obtain \u27e8_, _, _\u27e9 := h,\n  obtain \u27e8_, _, _\u27e9 := h',\n  resetI,\n  refine @abelian.is_iso_of_is_iso_of_is_iso_of_is_iso_of_is_iso \ud835\udc9c _ _ 0 _ _ _ 0 _ _ _\n    0 f g 0 f' g' 0 i\u2081 i\u2082 i\u2083 _ comm\u2081 comm\u2082 0 0 0 0 0 _ _ _ _ _ _ _ _ _ _ _;\n  try { simp };\n  try { apply exact_zero_left_of_mono };\n  try { assumption };\n  rwa \u2190 epi_iff_exact_zero_right,\nend\n\n/-- To construct a splitting of `A -f\u27f6 B -g\u27f6 C` it suffices to supply\na *morphism* `i : B \u27f6 A \u229e C` such that `f \u226b i` is the canonical map `biprod.inl : A \u27f6 A \u229e C` and\n`i \u226b q = g`, where `q` is the canonical map `biprod.snd : A \u229e C \u27f6 C`,\ntogether with proofs that `f` is mono and `g` is epi.\n\nThe morphism `i` is then automatically an isomorphism. -/\ndef splitting.mk' (h : short_exact f g) (i : B \u27f6 A \u229e C)\n  (h1 : f \u226b i = biprod.inl) (h2 : i \u226b biprod.snd = g) : splitting f g :=\n{ iso :=\n  begin\n    refine @as_iso _ _ _ _ i (id _),\n    refine is_iso_of_short_exact_of_is_iso_of_is_iso h _ _ _ _\n      (h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),\n    split,\n    apply exact_inl_snd\n  end,\n  comp_iso_eq_inl := by { rwa as_iso_hom, },\n  iso_comp_snd_eq := h2 }\n\n/-- To construct a splitting of `A -f\u27f6 B -g\u27f6 C` it suffices to supply\na *morphism* `i : A \u229e C \u27f6 B` such that `p \u226b i = f` where `p` is the canonical map\n`biprod.inl : A \u27f6 A \u229e C`, and `i \u226b g` is the canonical map `biprod.snd : A \u229e C \u27f6 C`,\ntogether with proofs that `f` is mono and `g` is epi.\n\nThe morphism `i` is then automatically an isomorphism. -/\ndef splitting.mk'' (h : short_exact f g) (i : A \u229e C \u27f6 B)\n  (h1 : biprod.inl \u226b i = f) (h2 : i \u226b g = biprod.snd) : splitting f g :=\n{ iso :=\n  begin\n    refine (@as_iso _ _ _ _ i (id _)).symm,\n    refine is_iso_of_short_exact_of_is_iso_of_is_iso _ h _ _ _\n      (h1.trans (category.id_comp _).symm).symm (h2.trans (category.comp_id _).symm),\n    split,\n    apply exact_inl_snd\n  end,\n  comp_iso_eq_inl := by rw [iso.symm_hom, as_iso_inv, is_iso.comp_inv_eq, h1],\n  iso_comp_snd_eq := by rw [iso.symm_hom, as_iso_inv, is_iso.inv_comp_eq, h2] }\n\n/-- A short exact sequence that is left split admits a splitting. -/\ndef left_split.splitting {f : A \u27f6 B} {g : B \u27f6 C} (h : left_split f g) : splitting f g :=\nsplitting.mk' h.short_exact (biprod.lift h.left_split.some g)\n(by { ext,\n  { simpa only [biprod.inl_fst, biprod.lift_fst, category.assoc] using h.left_split.some_spec },\n  { simp only [biprod.inl_snd, biprod.lift_snd, category.assoc, h.exact.w], } })\n(by { simp only [biprod.lift_snd], })\n\n/-- A short exact sequence that is right split admits a splitting. -/\ndef right_split.splitting {f : A \u27f6 B} {g : B \u27f6 C} (h : right_split f g) : splitting f g :=\nsplitting.mk'' h.short_exact (biprod.desc f h.right_split.some)\n(biprod.inl_desc _ _)\n(by { ext,\n  { rw [biprod.inl_snd, \u2190 category.assoc, biprod.inl_desc, h.exact.w] },\n  { rw [biprod.inr_snd, \u2190 category.assoc, biprod.inr_desc, h.right_split.some_spec] } })\n\nend category_theory\n", "meta": {"author": "nick-kuhn", "repo": "leantools", "sha": "567a98c031fffe3f270b7b8dea48389bc70d7abb", "save_path": "github-repos/lean/nick-kuhn-leantools", "path": "github-repos/lean/nick-kuhn-leantools/leantools-567a98c031fffe3f270b7b8dea48389bc70d7abb/src/algebra/homology/short_exact/abelian.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.640635854839898, "lm_q1q2_score": 0.3502600633702492}}
{"text": "/-\nCopyright (c) 2020 Kenny Lau. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Kenny Lau\n-/\nimport Mathlib.PrePort\nimport Mathlib.Lean3Lib.init.default\nimport Mathlib.algebra.big_operators.pi\nimport Mathlib.data.finsupp.default\nimport Mathlib.PostPort\n\nuniverses u_1 u_2 u_4 u_3 u_5 u_6 \n\nnamespace Mathlib\n\n/-!\n# Big operators for finsupps\n\nThis file contains theorems relevant to big operators in finitely supported functions.\n-/\n\ntheorem finset.sum_apply' {\u03b1 : Type u_1} {\u03b9 : Type u_2} {A : Type u_4} [add_comm_monoid A]\n    {s : finset \u03b1} {f : \u03b1 \u2192 \u03b9 \u2192\u2080 A} (i : \u03b9) :\n    coe_fn (finset.sum s fun (k : \u03b1) => f k) i = finset.sum s fun (k : \u03b1) => coe_fn (f k) i :=\n  Eq.symm (finset.sum_hom s \u21d1(finsupp.apply_add_hom i))\n\ntheorem finsupp.sum_apply' {\u03b9 : Type u_2} {\u03b3 : Type u_3} {A : Type u_4} {B : Type u_5}\n    [add_comm_monoid A] [add_comm_monoid B] (g : \u03b9 \u2192\u2080 A) (k : \u03b9 \u2192 A \u2192 \u03b3 \u2192 B) (x : \u03b3) :\n    finsupp.sum g k x = finsupp.sum g fun (i : \u03b9) (b : A) => k i b x :=\n  finset.sum_apply x (finsupp.support g) fun (a : \u03b9) => k a (coe_fn g a)\n\ntheorem finsupp.sum_sum_index' {\u03b1 : Type u_1} {\u03b9 : Type u_2} {A : Type u_4} {C : Type u_6}\n    [add_comm_monoid A] [add_comm_monoid C] {t : \u03b9 \u2192 A \u2192 C} (h0 : \u2200 (i : \u03b9), t i 0 = 0)\n    (h1 : \u2200 (i : \u03b9) (x y : A), t i (x + y) = t i x + t i y) {s : finset \u03b1} {f : \u03b1 \u2192 \u03b9 \u2192\u2080 A} :\n    finsupp.sum (finset.sum s fun (x : \u03b1) => f x) t =\n        finset.sum s fun (x : \u03b1) => finsupp.sum (f x) t :=\n  sorry\n\nend Mathlib", "meta": {"author": "AurelienSaue", "repo": "Mathlib4_auto", "sha": "590df64109b08190abe22358fabc3eae000943f2", "save_path": "github-repos/lean/AurelienSaue-Mathlib4_auto", "path": "github-repos/lean/AurelienSaue-Mathlib4_auto/Mathlib4_auto-590df64109b08190abe22358fabc3eae000943f2/Mathlib/algebra/big_operators/finsupp_auto.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3502411947157023}}
{"text": "import classes.context_free.basics.lifting\nimport utilities.written_by_others.trim_assoc\n\n\nvariables {T : Type}\n\nprivate def combined_grammar (g\u2097 g\u1d63 : CF_grammar T) : CF_grammar T :=\nCF_grammar.mk\n  (option (g\u2097.nt \u2295 g\u1d63.nt))\n  none\n  ((none, [\n    symbol.nonterminal (some (sum.inl (g\u2097.initial))),\n    symbol.nonterminal (some (sum.inr (g\u1d63.initial)))\n  ]) :: (\n    (list.map rule_of_rule\u2081 g\u2097.rules) ++ (list.map rule_of_rule\u2082 g\u1d63.rules)\n  ))\n\n/-- similar to `sink_symbol` -/\nprivate def oN\u2081_of_N {g\u2081 g\u2082 : CF_grammar T} : (combined_grammar g\u2081 g\u2082).nt \u2192 (option g\u2081.nt)\n| none := none\n| (some (sum.inl nt)) := some nt\n| (some (sum.inr _)) := none\n\n/-- similar to `sink_symbol` -/\nprivate def oN\u2082_of_N {g\u2081 g\u2082 : CF_grammar T} : (combined_grammar g\u2081 g\u2082).nt \u2192 (option g\u2082.nt)\n| none := none\n| (some (sum.inl _)) := none\n| (some (sum.inr nt)) := some nt\n\n\nprivate def g\u2081g (g\u2081 g\u2082 : CF_grammar T) : @lifted_grammar T :=\nlifted_grammar.mk g\u2081 (combined_grammar g\u2081 g\u2082) (some \u2218 sum.inl) (by {\n  -- prove `function.injective (some \u2218 sum.inl)` here\n  intros x y h,\n  apply sum.inl_injective,\n  apply option.some_injective,\n  exact h,\n}) (by {\n  -- prove `\u2200 r \u2208 g\u2081.rules` we have `lift_rule (some \u2218 sum.inl) r \u2208 list.map rule_of_rule\u2081 g\u2081.rules` here\n  intros r h,\n  apply list.mem_cons_of_mem,\n  apply list.mem_append_left,\n  rw list.mem_map,\n  use r,\n  split,\n  {\n    exact h,\n  },\n  unfold rule_of_rule\u2081,\n  unfold lift_rule,\n  norm_num,\n  unfold lift_string,\n  unfold lsTN_of_lsTN\u2081,\n  five_steps,\n}) oN\u2081_of_N (by {\n  intros x y ass,\n  cases x,\n  {\n    right,\n    refl,\n  },\n  cases x, swap,\n  {\n    right,\n    refl,\n  },\n  cases y,\n  {\n    rw ass,\n    right,\n    refl,\n  },\n  cases y, swap,\n  {\n    tauto,\n  },\n  left,\n  simp only [oN\u2081_of_N] at ass,\n  apply congr_arg,\n  apply congr_arg,\n  exact ass,\n}) (by {\n  intro r,\n  rintro \u27e8r_in, r_ntype\u27e9,\n  cases r_in,\n  {\n    exfalso,\n    rw r_in at r_ntype,\n    dsimp only at r_ntype,\n    cases r_ntype with n\u2080 imposs,\n    exact option.no_confusion imposs,\n  },\n  change r \u2208 (list.map rule_of_rule\u2081 g\u2081.rules ++ list.map rule_of_rule\u2082 g\u2082.rules) at r_in,\n  rw list.mem_append at r_in,\n  cases r_in,\n  {\n    rw list.mem_map at r_in,\n    rcases r_in with \u27e8r\u2081, r\u2081_in, r\u2081_convert_r\u27e9,\n    use r\u2081,\n    split,\n    {\n      exact r\u2081_in,\n    },\n    rw \u2190r\u2081_convert_r,\n    simp only [\n      lift_rule, rule_of_rule\u2081, lift_string, lsTN_of_lsTN\u2081,\n      prod.mk.inj_iff, eq_self_iff_true, true_and\n    ],\n    five_steps,\n  },\n  {\n    exfalso,\n    rw list.mem_map at r_in,\n    rcases r_in with \u27e8r\u2082, r\u2082_in, r\u2082_convert_r\u27e9,\n    rw \u2190r\u2082_convert_r at r_ntype,\n    unfold rule_of_rule\u2082 at r_ntype,\n    dsimp only at r_ntype,\n    cases r_ntype with n\u2081 contr,\n    rw option.some_inj at contr,\n    exact sum.no_confusion contr,\n  },\n}) (by { intro, refl })\n\nprivate def g\u2082g (g\u2081 g\u2082 : CF_grammar T) : @lifted_grammar T :=\nlifted_grammar.mk g\u2082 (combined_grammar g\u2081 g\u2082) (some \u2218 sum.inr) (by {\n  -- prove `function.injective (some \u2218 sum.inr)` here\n  intros x y h,\n  apply sum.inr_injective,\n  apply option.some_injective,\n  exact h,\n}) (by {\n  -- prove `\u2200 r \u2208 g\u2082.rules` we have `lift_rule (some \u2218 sum.inr) r \u2208 list.map rule_of_rule\u2082 g\u2082.rules` here\n  intros r h,\n  apply list.mem_cons_of_mem,\n  apply list.mem_append_right,\n  rw list.mem_map,\n  use r,\n  split,\n  {\n    exact h,\n  },\n  unfold rule_of_rule\u2082,\n  unfold lift_rule,\n  norm_num,\n  unfold lift_string,\n  unfold lsTN_of_lsTN\u2082,\n  five_steps,\n}) oN\u2082_of_N (by {\n  intros x y ass,\n  cases x,\n  {\n    right,\n    refl,\n  },\n  cases x,\n  {\n    right,\n    refl,\n  },\n  cases y,\n  {\n    right,\n    rw ass,\n    refl,\n  },\n  cases y,\n  {\n    tauto,\n  },\n  left,\n  simp only [oN\u2082_of_N] at ass,\n  apply congr_arg,\n  apply congr_arg,\n  exact ass,\n}) (by {\n  intro r,\n  rintro \u27e8r_in, r_ntype\u27e9,\n  cases r_in,\n  {\n    exfalso,\n    rw r_in at r_ntype,\n    dsimp only at r_ntype,\n    cases r_ntype with n\u2080 imposs,\n    exact option.no_confusion imposs,\n  },\n  change r \u2208 (list.map rule_of_rule\u2081 g\u2081.rules ++ list.map rule_of_rule\u2082 g\u2082.rules) at r_in,\n  rw list.mem_append at r_in,\n  cases r_in,\n  {\n    exfalso,\n    rw list.mem_map at r_in,\n    rcases r_in with \u27e8r\u2081, r\u2081_in, r\u2081_convert_r\u27e9,\n    rw \u2190r\u2081_convert_r at r_ntype,\n    unfold rule_of_rule\u2081 at r_ntype,\n    dsimp only at r_ntype,\n    cases r_ntype with n\u2082 contr,\n    rw option.some_inj at contr,\n    exact sum.no_confusion contr,\n  },\n  {\n    rw list.mem_map at r_in,\n    rcases r_in with \u27e8r\u2082, r\u2082_in, r\u2082_convert_r\u27e9,\n    use r\u2082,\n    split,\n    {\n      exact r\u2082_in,\n    },\n    rw \u2190r\u2082_convert_r,\n    simp only [\n      lift_rule, rule_of_rule\u2082, lift_string, lsTN_of_lsTN\u2082,\n      prod.mk.inj_iff, eq_self_iff_true, true_and\n    ],\n    five_steps,\n  },\n}) (by { intro, refl })\n\n\nprivate def oT_of_sTN\u2083 {g\u2083 : CF_grammar T} : symbol T g\u2083.nt \u2192 option T\n| (symbol.terminal t) := some t\n| (symbol.nonterminal _) := none\n\nprivate def liT_of_lsTN\u2083 {g\u2083 : CF_grammar T} : list (symbol T g\u2083.nt) \u2192 list T :=\nlist.filter_map oT_of_sTN\u2083\n\nprivate lemma u_eq_take_map_w\n    {g\u2081 g\u2082 : CF_grammar T}\n    (u : list (symbol T g\u2081.nt))\n    (v : list (symbol T g\u2082.nt))\n    (w : list T)\n    (len : u.length \u2264 w.length)\n    (hyp : list.take u.length (list.map sTN_of_sTN\u2081 u ++ lsTN_of_lsTN\u2082 v) =\n           list.take u.length (list.map symbol.terminal w)) :\n  u = list.take u.length (list.map symbol.terminal w) :=\nbegin\n  ext1,\n  by_cases n < u.length,\n  {\n    have ass : list.map sTN_of_sTN\u2081 u = list.take u.length (list.map symbol.terminal w),\n    {\n      convert hyp,\n      have takenl := list.take_left (list.map sTN_of_sTN\u2081 u) (lsTN_of_lsTN\u2082 v),\n      rw list.length_map at takenl,\n      exact takenl.symm,\n    },\n    have nth_equ := congr_fun (congr_arg list.nth ass) n,\n    rw list.nth_take h,\n    rw list.nth_take h at nth_equ,\n    have n_lt_wl : n < w.length,\n    {\n      exact gt_of_ge_of_gt len h,\n    },\n    have triv : n < (list.map sTN_of_sTN\u2081 u).length,\n    {\n      rw list.length_map,\n      exact h,\n    },\n    have trig : n < (list.map (@symbol.terminal T g\u2081.nt) w).length,\n    {\n      rw list.length_map,\n      exact n_lt_wl,\n    },\n    have trin : n < (list.map (@symbol.terminal T (option (g\u2081.nt \u2295 g\u2082.nt))) w).length,\n    {\n      rw list.length_map,\n      exact n_lt_wl,\n    },\n    rw list.nth_le_nth triv at nth_equ,\n    rw list.nth_le_nth trin at nth_equ,\n    rw option.some_inj at nth_equ,\n    rw list.nth_le_map at nth_equ, swap,\n    {\n      exact h,\n    },\n    rw list.nth_le_map at nth_equ, swap,\n    {\n      exact n_lt_wl,\n    },\n    rw list.nth_le_nth, swap,\n    {\n      exact h,\n    },\n    rw list.nth_le_nth, swap,\n    {\n      exact trig,\n    },\n    apply congr_arg,\n    norm_num,\n    cases u.nth_le n h,\n    {\n      unfold sTN_of_sTN\u2081 at nth_equ,\n      clear_except nth_equ,\n      finish,\n    },\n    {\n      exfalso,\n      exact symbol.no_confusion nth_equ,\n    },\n  },\n  convert_to none = none,\n  {\n    finish,\n  },\n  {\n    push_neg at h,\n    rw list.nth_eq_none_iff,\n    rw list.length_take,\n    exact min_le_of_left_le h,\n  },\n  refl,\nend\n\nprivate lemma v_eq_drop_map_w\n    {g\u2081 g\u2082 : CF_grammar T}\n    (u : list (symbol T g\u2081.nt))\n    (v : list (symbol T g\u2082.nt))\n    (w : list T)\n    (total_len : u.length + v.length = w.length)\n    (hyp : list.drop u.length (list.map sTN_of_sTN\u2081 u ++ list.map sTN_of_sTN\u2082 v) =\n           list.drop u.length (list.map symbol.terminal w)) :\n  v = list.drop u.length (list.map symbol.terminal w) :=\nbegin\n  ext1,\n  by_cases n < v.length,\n  {\n    have nth_equ := congr_fun (congr_arg list.nth hyp) n,\n    rw list.nth_drop,\n    rw list.nth_drop at nth_equ,\n    rw list.nth_drop at nth_equ,\n\n    have hunltuv : u.length + n < u.length + v.length,\n    {\n      apply add_lt_add_left h,\n    },\n    have hunltw : u.length + n < w.length,\n    {\n      rw \u2190total_len,\n      exact hunltuv,\n    },\n    have hlen\u2081 : u.length + n < (list.map sTN_of_sTN\u2081 u ++ list.map sTN_of_sTN\u2082 v).length,\n    {\n      rw list.length_append,\n      rw list.length_map,\n      rw list.length_map,\n      exact hunltuv,\n    },\n    have hlen\u2082 : u.length + n < (list.map (@symbol.terminal T (option (g\u2081.nt \u2295 g\u2082.nt))) w).length,\n    {\n      rw list.length_map,\n      exact hunltw,\n    },\n    have hlen\u2082' : u.length + n < (list.map (@symbol.terminal T g\u2082.nt) w).length,\n    {\n      rw list.length_map,\n      exact hunltw,\n    },\n    rw list.nth_le_nth hlen\u2081 at nth_equ,\n    rw list.nth_le_nth hlen\u2082 at nth_equ,\n    rw list.nth_le_nth h,\n    rw list.nth_le_nth hlen\u2082',\n\n    rw option.some_inj at *,\n    have hlen\u2080 : (list.map sTN_of_sTN\u2081 u).length \u2264 u.length + n,\n    {\n      rw list.length_map,\n      exact le_self_add,\n    },\n    have hlen : n < (list.map (@sTN_of_sTN\u2082 T g\u2081 g\u2082) v).length,\n    {\n      rw list.length_map,\n      exact h,\n    },\n    have nth_equ_simplified :\n      (list.map sTN_of_sTN\u2082 v).nth_le n hlen =\n      (list.map symbol.terminal w).nth_le (u.length + n) hlen\u2082,\n    {\n      rw list.nth_le_append_right hlen\u2080 at nth_equ,\n      convert nth_equ,\n      rw list.length_map,\n      symmetry,\n      apply add_tsub_cancel_left,\n    },\n    rw list.nth_le_map at nth_equ_simplified,\n\n    cases v.nth_le n h with x,\n    {\n      unfold sTN_of_sTN\u2082 at nth_equ_simplified,\n      rw list.nth_le_map _ _ hunltw at nth_equ_simplified,\n      rw list.nth_le_map _ _ hunltw,\n      injection nth_equ_simplified with hx,\n      apply congr_arg,\n      exact hx,\n    },\n    {\n      exfalso,\n      clear_except nth_equ_simplified,\n      finish,\n    },\n  },\n  convert_to none = none,\n  {\n    finish,\n  },\n  {\n    rw list.nth_drop,\n    push_neg at h,\n    rw list.nth_eq_none_iff,\n    rw list.length_map,\n    rw \u2190total_len,\n    apply add_le_add_left h,\n  },\n  refl,\nend\n\nprivate def sTN\u2081_of_sTN {g\u2081 g\u2082 : CF_grammar T} : symbol T (option (g\u2081.nt \u2295 g\u2082.nt)) \u2192 option (symbol T g\u2081.nt)\n| (symbol.terminal te) := some (symbol.terminal te)\n| (symbol.nonterminal nont) := option.map symbol.nonterminal (oN\u2081_of_N nont)\n\nprivate def sTN\u2082_of_sTN {g\u2081 g\u2082 : CF_grammar T} : symbol T (option (g\u2081.nt \u2295 g\u2082.nt)) \u2192 option (symbol T g\u2082.nt)\n| (symbol.terminal te) := some (symbol.terminal te)\n| (symbol.nonterminal nont) := option.map symbol.nonterminal (oN\u2082_of_N nont)\n\nprivate def lsTN\u2081_of_lsTN {g\u2081 g\u2082 : CF_grammar T} (lis : list (symbol T (option (g\u2081.nt \u2295 g\u2082.nt)))) :\n  list (symbol T g\u2081.nt) :=\nlist.filter_map sTN\u2081_of_sTN lis\n\nprivate def lsTN\u2082_of_lsTN {g\u2081 g\u2082 : CF_grammar T} (lis : list (symbol T (option (g\u2081.nt \u2295 g\u2082.nt)))) :\n  list (symbol T g\u2082.nt) :=\nlist.filter_map sTN\u2082_of_sTN lis\n\nprivate lemma self_of_sTN\u2081 {g\u2081 g\u2082 : CF_grammar T} (a : symbol T g\u2081.nt) :\n  sTN\u2081_of_sTN (@sTN_of_sTN\u2081 _ _ g\u2082 a) = a :=\nbegin\n  cases a;\n  refl,\nend\n\nprivate lemma self_of_sTN\u2082 {g\u2081 g\u2082 : CF_grammar T} (a : symbol T g\u2082.nt) :\n  sTN\u2082_of_sTN (@sTN_of_sTN\u2082 _ g\u2081 _ a) = a :=\nbegin\n  cases a;\n  refl,\nend\n\nprivate lemma self_of_lsTN\u2081 {g\u2081 g\u2082 : CF_grammar T} (stri : list (symbol T g\u2081.nt)) :\n  lsTN\u2081_of_lsTN (@lsTN_of_lsTN\u2081 _ _ g\u2082 stri) = stri :=\nbegin\n  unfold lsTN_of_lsTN\u2081,\n  unfold lsTN\u2081_of_lsTN,\n  rw list.filter_map_map,\n  change list.filter_map (\u03bb x, sTN\u2081_of_sTN (sTN_of_sTN\u2081 x)) stri = stri,\n  convert_to list.filter_map (\u03bb x, some x) stri = stri,\n  {\n    have equal_functions : (\u03bb (x : symbol T g\u2081.nt), sTN\u2081_of_sTN (sTN_of_sTN\u2081 x)) = (\u03bb x, some x),\n    {\n      ext1,\n      apply self_of_sTN\u2081,\n    },\n    rw \u2190equal_functions,\n    apply congr_fun,\n    apply congr_arg,\n    ext1,\n    apply congr_fun,\n    refl,\n  },\n  apply list.filter_map_some,\nend\n\nprivate lemma self_of_lsTN\u2082 {g\u2081 g\u2082 : CF_grammar T} (stri : list (symbol T g\u2082.nt)) :\n  lsTN\u2082_of_lsTN (@lsTN_of_lsTN\u2082 _ g\u2081 _ stri) = stri :=\nbegin\n  unfold lsTN_of_lsTN\u2082,\n  unfold lsTN\u2082_of_lsTN,\n  rw list.filter_map_map,\n  change list.filter_map (\u03bb x, sTN\u2082_of_sTN (sTN_of_sTN\u2082 x)) stri = stri,\n  convert_to list.filter_map (\u03bb x, some x) stri = stri,\n  {\n    have equal_functions : (\u03bb (x : symbol T g\u2082.nt), sTN\u2082_of_sTN (sTN_of_sTN\u2082 x)) = (\u03bb x, some x),\n    {\n      ext1,\n      apply self_of_sTN\u2082,\n    },\n    rw \u2190equal_functions,\n    apply congr_fun,\n    apply congr_arg,\n    ext1,\n    apply congr_fun,\n    refl,\n  },\n  apply list.filter_map_some,\nend\n\nprivate lemma in_concatenated_of_in_combined\n    {g\u2081 g\u2082 : CF_grammar T}\n    {w : list T}\n    (hyp : w \u2208 CF_language (combined_grammar g\u2081 g\u2082)) :\n  w \u2208 CF_language g\u2081 * CF_language g\u2082 :=\nbegin\n  rw language.mem_mul,\n  change\n    CF_derives\n      (combined_grammar g\u2081 g\u2082)\n      [symbol.nonterminal (combined_grammar g\u2081 g\u2082).initial]\n      (list.map symbol.terminal w) at hyp,\n\n  cases CF_tran_or_id_of_deri hyp,\n  {\n    rename h refl_contr,\n    exfalso,\n    have hh := congr_fun (congr_arg list.nth refl_contr) 0,\n    rw list.nth at hh,\n    \n    by_cases (list.map (@symbol.terminal T (combined_grammar g\u2081 g\u2082).nt) w).length = 0,\n    {\n      have empty_none : (list.map symbol.terminal w).nth 0 = none,\n      {\n        finish,\n      },\n      rw empty_none at hh,\n      exact option.no_confusion hh,\n    },\n    rw list.nth_map at hh,\n    have hw0 : \u2203 s, w.nth 0 = some s,\n    {\n      cases w.nth 0,\n      {\n        exfalso,\n        exact option.no_confusion hh,\n      },\n      use val,\n    },\n    rcases hw0 with \u27e8s, hs\u27e9,\n    rw hs at hh,\n    rw option.map_some' at hh,\n    rw option.some_inj at hh,\n    exact symbol.no_confusion hh,\n  },\n  rcases h with \u27e8y, first_step, derivation\u27e9,\n  clear hyp,\n\n  have only_option :\n    y =\n    [\n      symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n      symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n    ],\n  {\n    rcases first_step with \u27e8first_rule, first_rule_in, p, q, bef, aft\u27e9,\n    have len_bef := congr_arg list.length bef,\n    rw [list.length_singleton, list.length_append, list.length_append, list.length_singleton] at len_bef,\n    have p_nil : p = [],\n    {\n      have p0 : p.length = 0,\n      {\n        linarith,\n      },\n      rw list.length_eq_zero at p0,\n      exact p0,\n    },\n    have q_nil : q = [],\n    {\n      have q0 : q.length = 0,\n      {\n        linarith,\n      },\n      rw list.length_eq_zero at q0,\n      exact q0,\n    },\n    have initial : first_rule.fst = none,\n    {\n      apply symbol.nonterminal.inj,\n      rw p_nil at bef,\n      rw q_nil at bef,\n      rw list.append_nil at bef,\n      rw list.nil_append at bef,\n      exact list.head_eq_of_cons_eq (eq.symm bef),\n    },\n    have only_rule :\n      first_rule = (none, [\n        symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n        symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n      ]),\n    {\n      change first_rule \u2208 (\n        (none, [\n          symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n          symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n        ]) :: (\n          (list.map rule_of_rule\u2081 g\u2081.rules) ++ (list.map rule_of_rule\u2082 g\u2082.rules)\n        )\n      ) at first_rule_in,\n      cases first_rule_in,\n      {\n        exact first_rule_in,\n      },\n      exfalso,\n      change first_rule \u2208 (list.map rule_of_rule\u2081 g\u2081.rules ++ list.map rule_of_rule\u2082 g\u2082.rules) at first_rule_in,\n      rw list.mem_append at first_rule_in,\n      cases first_rule_in,\n      {\n        delta rule_of_rule\u2081 at first_rule_in,\n        have rfst :\n          first_rule.fst \u2208 list.map prod.fst\n            (list.map (\n                \u03bb (r : g\u2081.nt \u00d7 list (symbol T g\u2081.nt)),\n                (some (sum.inl r.fst), lsTN_of_lsTN\u2081 r.snd)\n              ) g\u2081.rules),\n        {\n          exact list.mem_map_of_mem prod.fst first_rule_in,\n        },\n        rw initial at rfst,\n        convert rfst,\n        simp,\n      },\n      {\n        delta rule_of_rule\u2082 at first_rule_in,\n        have rfst :\n          first_rule.fst \u2208 list.map prod.fst\n            (list.map (\n                \u03bb (r : g\u2082.nt \u00d7 list (symbol T g\u2082.nt)),\n                (some (sum.inr r.fst), lsTN_of_lsTN\u2082 r.snd)\n              ) g\u2082.rules),\n        {\n          exact list.mem_map_of_mem prod.fst first_rule_in,\n        },\n        rw initial at rfst,\n        convert rfst,\n        simp,\n      },\n    },\n    rw [p_nil, q_nil, only_rule] at aft,\n    rw list.append_nil at aft,\n    rw list.nil_append at aft,\n    exact aft,\n  },\n  clear first_step,\n  rw only_option at derivation,\n  clear only_option y,\n\n  have complicated_induction :\n    \u2200 x : list (symbol T (combined_grammar g\u2081 g\u2082).nt),\n      CF_derives\n        (combined_grammar g\u2081 g\u2082)\n        [\n          symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n          symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n        ]\n        x\n      \u2192\n        \u2203 u : list (symbol T g\u2081.nt), \u2203 v : list (symbol T g\u2082.nt), and\n          (CF_derives g\u2081 [symbol.nonterminal g\u2081.initial] u)\n          (CF_derives g\u2082 [symbol.nonterminal g\u2082.initial] v)\n          \u2227 (lsTN_of_lsTN\u2081 u ++ lsTN_of_lsTN\u2082 v = x),\n  {\n    intros x ass,\n    induction ass with a b trash orig ih,\n    {\n      use [[symbol.nonterminal g\u2081.initial], [symbol.nonterminal g\u2082.initial]],\n      split,\n      {\n        split;\n        apply CF_deri_self,\n      },\n      {\n        refl,\n      },\n    },\n    clear trash,\n    rcases orig with \u27e8orig_rule, orig_in, c, d, bef, aft\u27e9,\n    rcases ih with \u27e8u, v, \u27e8ih\u2081, ih\u2082\u27e9, ih_concat\u27e9,\n    cases orig_in,\n    {\n      exfalso,\n      rw \u2190ih_concat at bef,\n      rw orig_in at bef,\n      clear_except bef,\n      dsimp only at bef,\n      have init_nt_in_bef_right : symbol.nonterminal none \u2208 c ++ [symbol.nonterminal none] ++ d,\n      {\n        apply list.mem_append_left,\n        apply list.mem_append_right,\n        apply list.mem_singleton_self,\n      },\n      have init_nt_notin_bef_left : symbol.nonterminal none \u2209 lsTN_of_lsTN\u2081 u ++ lsTN_of_lsTN\u2082 v,\n      {\n        rw list.mem_append,\n        push_neg,\n        split,\n        {\n          rw list.mem_iff_nth_le,\n          push_neg,\n          unfold lsTN_of_lsTN\u2081,\n          intros n hn,\n          rw list.nth_le_map,\n          {\n            cases u.nth_le n _ with t s,\n            {\n              apply symbol.no_confusion,\n            },\n            {\n              unfold sTN_of_sTN\u2081,\n              intro hypo,\n              have impossible := symbol.nonterminal.inj hypo,\n              exact option.no_confusion impossible,\n            },\n          },\n          {\n            rw list.length_map at hn,\n            exact hn,\n          },\n        },\n        {\n          rw list.mem_iff_nth_le,\n          push_neg,\n          unfold lsTN_of_lsTN\u2082,\n          intros n hn,\n          rw list.nth_le_map,\n          {\n            cases v.nth_le n _ with t s,\n            {\n              apply symbol.no_confusion,\n            },\n            {\n              unfold sTN_of_sTN\u2082,\n              intro hypo,\n              have impossible := symbol.nonterminal.inj hypo,\n              exact option.no_confusion impossible,\n            },\n          },\n          {\n            rw list.length_map at hn,\n            exact hn,\n          },\n        },\n      },\n      rw bef at init_nt_notin_bef_left,\n      exact init_nt_notin_bef_left init_nt_in_bef_right,\n    },\n    clear derivation w,\n    change orig_rule \u2208 (list.map rule_of_rule\u2081 g\u2081.rules ++ list.map rule_of_rule\u2082 g\u2082.rules) at orig_in,\n    rw list.mem_append at orig_in,\n    cases orig_in,\n    {\n      rw list.mem_map at orig_in,\n      rcases orig_in with \u27e8r\u2081, r\u2081_in, r\u2081_conv\u27e9,\n      rw aft,\n      rw bef at ih_concat,\n      clear bef aft a b,\n      rw \u2190r\u2081_conv at ih_concat \u22a2,\n      clear r\u2081_conv orig_rule,\n      have part_for_u := congr_arg (list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length) ih_concat,\n      have part_for_v := congr_arg (list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length) ih_concat,\n      rw list.take_left at part_for_u,\n      rw list.drop_left at part_for_v,\n\n      have h_len : (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length > c.length,\n      {\n        by_contradiction contra,\n        push_neg at contra,\n\n        have not_in : symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst \u2209 lsTN_of_lsTN\u2082 v,\n        {\n          unfold lsTN_of_lsTN\u2082,\n          rw list.mem_map,\n          rintro \u27e8s, -, imposs\u27e9,\n          cases s,\n          {\n            exact symbol.no_confusion imposs,\n          },\n          {\n            have inr_eq_inl := option.some.inj (symbol.nonterminal.inj imposs),\n            exact sum.no_confusion inr_eq_inl,\n          },\n        },\n\n        have yes_in : symbol.nonterminal (@rule_of_rule\u2081 T g\u2081 g\u2082 r\u2081).fst \u2208 lsTN_of_lsTN\u2082 v,\n        {\n          have lcth := congr_fun (congr_arg list.nth ih_concat) c.length,\n          rw list.append_assoc c at lcth,\n          have clength :\n            (c ++ ([symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d)).nth c.length =\n            some (symbol.nonterminal (@rule_of_rule\u2081 T g\u2081 g\u2082 r\u2081).fst),\n          {\n            rw list.nth_append_right, swap,\n            {\n              refl,\n            },\n            rw nat.sub_self,\n            refl,\n          },\n          rw clength at lcth,\n          rw list.nth_append_right contra at lcth,\n          exact list.nth_mem lcth,\n        },\n\n        exact not_in yes_in,\n      },\n      \n      -- nonterminal was rewritten in the left half of `a` ... upgrade `u`\n      let d' : list (symbol T (combined_grammar g\u2081 g\u2082).nt) :=\n        list.take ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1)) d,\n      let u' := lsTN\u2081_of_lsTN (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d'),\n      use u',\n      use v,\n      split,\n      {\n        split,\n        {\n          change\n            CF_derives g\u2081 [symbol.nonterminal g\u2081.initial] (lsTN\u2081_of_lsTN (\n              c ++ (rule_of_rule\u2081 r\u2081).snd ++\n              (list.take ((lsTN_of_lsTN\u2081 u).length - (c.length + 1)) d)\n            )),\n          apply CF_deri_of_deri_tran ih\u2081,\n          convert_to\n            CF_transforms\n              g\u2081\n              (lsTN\u2081_of_lsTN (\n                list.take (lsTN_of_lsTN\u2081 u).length (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d)\n              ))\n              (lsTN\u2081_of_lsTN (c ++ (rule_of_rule\u2081 r\u2081).snd ++ list.take ((lsTN_of_lsTN\u2081 u).length - (c.length + 1)) d)),\n          {\n            rw \u2190part_for_u,\n            rw self_of_lsTN\u2081,\n          },\n          use r\u2081,\n          split,\n          {\n            exact r\u2081_in,\n          },\n          use lsTN\u2081_of_lsTN c,\n          use lsTN\u2081_of_lsTN (list.take (u.length - (c.length + 1)) d),\n          split,\n          {\n            convert_to\n              lsTN\u2081_of_lsTN (\n                c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++\n                  (list.take (u.length - (c.length + 1)) d)\n              ) =\n              lsTN\u2081_of_lsTN c ++ [symbol.nonterminal r\u2081.fst] ++\n                lsTN\u2081_of_lsTN (list.take (u.length - (c.length + 1)) d),\n            {\n              apply congr_arg,\n              have trivi_len : (lsTN_of_lsTN\u2081 u).length = u.length,\n              {\n                unfold lsTN_of_lsTN\u2081,\n                rw list.length_map,\n              },\n              rw trivi_len,\n              have another_trivi_len : c.length + 1 = (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length,\n              {\n                rw list.length_append,\n                rw list.length_singleton,\n              },\n              rw another_trivi_len,\n\n              have borrow_and_return : u.length =\n                (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length +\n                  (u.length - (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length),\n              {\n                symmetry,\n                clear_except h_len,\n                apply nat.add_sub_of_le,\n                rw list.length_append,\n                rw list.length_singleton,\n                unfold lsTN_of_lsTN\u2081 at h_len,\n                rw list.length_map at h_len,\n                rw nat.succ_le_iff,\n                exact h_len,\n              },\n              convert_to\n                list.take\n                  ((c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length +\n                    (u.length - (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length))\n                  (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d) =\n                c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++\n                  list.take (u.length - (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length) d,\n              {\n                apply congr_fun,\n                apply congr_arg,\n                exact borrow_and_return,\n              },\n              rw list.take_append,\n            },\n            unfold lsTN\u2081_of_lsTN,\n            rw list.filter_map_append_append,\n            refl,\n          },\n          {\n            convert_to\n              lsTN\u2081_of_lsTN (c ++ (rule_of_rule\u2081 r\u2081).snd ++ (list.take (u.length - (c.length + 1)) d)) =\n              lsTN\u2081_of_lsTN c ++ r\u2081.snd ++ lsTN\u2081_of_lsTN (list.take (u.length - (c.length + 1)) d),\n            {\n              apply congr_arg,\n              trim,\n              unfold lsTN_of_lsTN\u2081,\n              rw list.length_map,\n            },\n            unfold lsTN\u2081_of_lsTN,\n            rw list.filter_map_append_append,\n            change\n              list.filter_map sTN\u2081_of_sTN c ++ lsTN\u2081_of_lsTN (lsTN_of_lsTN\u2081 r\u2081.snd) ++\n                list.filter_map sTN\u2081_of_sTN (list.take (u.length - (c.length + 1)) d) =\n              list.filter_map sTN\u2081_of_sTN c ++ r\u2081.snd ++\n                list.filter_map sTN\u2081_of_sTN (list.take (u.length - (c.length + 1)) d),\n            rw self_of_lsTN\u2081,\n          },\n        },\n        {\n          exact ih\u2082,\n        },\n      },\n      {\n        have trivi_min :\n          min ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1)) d.length =\n          (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1),\n        {\n          apply min_eq_left,\n          unfold lsTN_of_lsTN\u2081,\n          rw list.length_map,\n          clear_except part_for_u,\n          unfold lsTN_of_lsTN\u2081 at part_for_u,\n          have lengs := congr_arg list.length part_for_u,\n          rw list.length_map at lengs,\n          rw list.length_take at lengs,\n          rw list.length_append at lengs,\n          rw list.length_append at lengs,\n          rw list.length_singleton at lengs,\n          have uleng_le : u.length \u2264 c.length + 1 + d.length,\n          {\n            rw \u2190min_eq_left_iff,\n            exact lengs.symm,\n          },\n          clear_except uleng_le,\n          omega,\n        },\n\n        have c_converted_and_back : list.map sTN_of_sTN\u2081 (list.filter_map sTN\u2081_of_sTN c) = c,\n        {\n          /-\n            Simplified schema of this conversion (applies to some other conversions, too):\n            we have `g \u2218 f = id` but `f \u2218 g` does not annihilate (in general)\n            we need `(f \u2218 g)(c) = c` for a specific `c`\n            which we can express as `c = f(x)` and then\n            we calculate `f(g(c)) = f(g(f(x))) = f(x) = c` hooray!\n          -/\n          have taken_c_from_u := congr_arg (list.take c.length) part_for_u,\n          rw list.take_take at taken_c_from_u,\n          rw min_eq_left (le_of_lt h_len) at taken_c_from_u,\n          rw list.append_assoc at taken_c_from_u,\n          rw list.take_left at taken_c_from_u,\n          convert_to list.map sTN_of_sTN\u2081 (list.filter_map sTN\u2081_of_sTN (list.take c.length (lsTN_of_lsTN\u2081 u))) = c,\n          {\n            rw taken_c_from_u,\n          },\n          unfold lsTN_of_lsTN\u2081,\n          rw \u2190list.map_take,\n          change list.map sTN_of_sTN\u2081 (lsTN\u2081_of_lsTN (lsTN_of_lsTN\u2081 (list.take c.length u))) = _,\n          rw self_of_lsTN\u2081,\n          rw list.map_take,\n          exact taken_c_from_u,\n        },\n\n        have d_converted_and_back :\n          list.map sTN_of_sTN\u2081 (list.filter_map sTN\u2081_of_sTN (list.take (\n            (list.map (@sTN_of_sTN\u2081 T g\u2081 g\u2082) u).length - (c.length + 1)\n          ) d)) =\n          list.take ((list.map (@sTN_of_sTN\u2081 T g\u2081 g\u2082) u).length - (c.length + 1)) d,\n        {\n          have taken_d_from_dropped_u := congr_arg (list.drop (c.length + 1)) part_for_u,\n          have for_the_decomposition :\n            (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length =\n            (c.length + 1) + ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1)),\n          {\n            symmetry,\n            apply nat.add_sub_of_le,\n            exact nat.succ_le_of_lt h_len,\n          },\n          rw for_the_decomposition at taken_d_from_dropped_u,\n          rw list.drop_take at taken_d_from_dropped_u,\n          have translate_counts : c.length + 1 = (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length,\n          {\n            rw list.length_append,\n            rw list.length_singleton,\n          },\n          rw translate_counts at taken_d_from_dropped_u,\n          rw list.drop_left at taken_d_from_dropped_u,\n          rw \u2190translate_counts at taken_d_from_dropped_u,\n          change\n            list.map sTN_of_sTN\u2081 (\n              list.filter_map sTN\u2081_of_sTN (list.take ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1)) d)\n            ) = _,\n          rw \u2190taken_d_from_dropped_u,\n          change list.map sTN_of_sTN\u2081 (lsTN\u2081_of_lsTN (list.drop (c.length + 1) (list.map sTN_of_sTN\u2081 u))) = _,\n          rw \u2190list.map_drop,\n          change list.map sTN_of_sTN\u2081 (lsTN\u2081_of_lsTN (lsTN_of_lsTN\u2081 (list.drop (c.length + 1) u))) = _,\n          rw self_of_lsTN\u2081,\n          rw list.map_drop,\n          exact taken_d_from_dropped_u,\n        },\n\n        have len_u' : u'.length = c.length + (@rule_of_rule\u2081 T g\u2081 g\u2082 r\u2081).snd.length + d'.length,\n        {\n          change\n            (lsTN\u2081_of_lsTN (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d')).length =\n            c.length + (rule_of_rule\u2081 r\u2081).snd.length + d'.length,\n          unfold lsTN\u2081_of_lsTN,\n          rw list.filter_map_append_append,\n          convert_to\n            (list.map sTN_of_sTN\u2081 (\n              list.filter_map sTN\u2081_of_sTN c ++\n              list.filter_map sTN\u2081_of_sTN (rule_of_rule\u2081 r\u2081).snd ++\n              list.filter_map sTN\u2081_of_sTN d'\n            )).length =\n            c.length + (rule_of_rule\u2081 r\u2081).snd.length + d'.length,\n          {\n            rw list.length_map,\n          },\n          rw list.map_append_append,\n          rw c_converted_and_back,\n          change\n            (c ++ _ ++ list.map sTN_of_sTN\u2081 (list.filter_map sTN\u2081_of_sTN (\n              list.take ((list.map (@sTN_of_sTN\u2081 T g\u2081 g\u2082) u).length - (c.length + 1)) d\n            ))).length = _,\n          rw d_converted_and_back,\n          change (c ++ list.map sTN_of_sTN\u2081 (lsTN\u2081_of_lsTN (lsTN_of_lsTN\u2081 r\u2081.snd)) ++ d').length = _,\n          rw self_of_lsTN\u2081,\n          rw list.length_append,\n          rw list.length_append,\n          refl,\n        },\n\n        have express_u'_as_crd :\n          lsTN_of_lsTN\u2081 u' =\n          list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u').length (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d),\n        {\n          change\n            lsTN_of_lsTN\u2081 (lsTN\u2081_of_lsTN (c ++ (rule_of_rule\u2081 r\u2081).snd ++\n              (list.take ((lsTN_of_lsTN\u2081 u).length - (c.length + 1)) d))) =\n            list.take (lsTN_of_lsTN\u2081 u').length (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d),\n          convert_to\n            c ++ (rule_of_rule\u2081 r\u2081).snd ++ (list.take ((lsTN_of_lsTN\u2081 u).length - (c.length + 1)) d) =\n            list.take (lsTN_of_lsTN\u2081 u').length (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d),\n          {\n            unfold lsTN\u2081_of_lsTN,\n            rw list.filter_map_append_append,\n            unfold lsTN_of_lsTN\u2081,\n            rw list.map_append_append,\n            rw c_converted_and_back,\n            rw d_converted_and_back,\n            change c ++ list.map sTN_of_sTN\u2081 (lsTN\u2081_of_lsTN (lsTN_of_lsTN\u2081 r\u2081.snd)) ++ _ = _,\n            rw self_of_lsTN\u2081,\n            refl,\n          },\n\n          have len_add_sub :\n            (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u').length =\n            (c ++ (rule_of_rule\u2081 r\u2081).snd).length +\n              ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u').length - (c ++ (rule_of_rule\u2081 r\u2081).snd).length),\n          {\n            symmetry,\n            apply nat.add_sub_of_le,\n            unfold lsTN_of_lsTN\u2081,\n            rw list.length_map,\n            rw len_u',\n            rw list.length_append,\n            apply le_self_add,\n          },\n          rw len_add_sub,\n          rw list.take_append,\n          trim,\n          rw list.length_append,\n          apply congr_arg2, swap,\n          {\n            refl,\n          },\n          rw [\n            lsTN_of_lsTN\u2081,\n            list.length_map,\n            list.length_map,\n            len_u',\n            list.length_take,\n            nat.add_sub_cancel_left,\n            trivi_min,\n            lsTN_of_lsTN\u2081,\n            list.length_map\n          ],\n        },\n        rw express_u'_as_crd,\n\n        have identity_of_suffixes :\n          list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d) =\n          list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u').length (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d),\n        {\n          clear_except h_len trivi_min len_u',\n          have h_len_ : (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length \u2264 (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length,\n          {\n            rw list.length_append,\n            rw list.length_singleton,\n            apply nat.succ_le_of_lt,\n            exact h_len,\n          },\n          have intermediate :\n            list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d) =\n            list.drop ((@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length - (c.length + 1)) d,\n          {\n            convert_to\n              list.drop\n                ((c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length +\n                  ((lsTN_of_lsTN\u2081 u).length - (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst]).length))\n                (c ++ [symbol.nonterminal (rule_of_rule\u2081 r\u2081).fst] ++ d) =\n              list.drop ((lsTN_of_lsTN\u2081 u).length - (c.length + 1)) d,\n            {\n              symmetry,\n              apply congr_arg2, swap,\n              {\n                refl,\n              },\n              apply nat.add_sub_of_le,\n              exact h_len_,\n            },\n            rw list.drop_append,\n            apply congr_arg2, swap,\n            {\n              refl,\n            },\n            rw list.length_append,\n            rw list.length_singleton,\n          },\n          rw intermediate,\n          change _ = list.drop (list.map sTN_of_sTN\u2081 u').length (c ++ (rule_of_rule\u2081 r\u2081).snd ++ d),\n          rw list.length_map,\n          rw len_u',\n          rw \u2190list.length_append,\n          rw list.drop_append,\n          rw list.length_take,\n          rw trivi_min,\n        },\n\n        rw part_for_v,\n        rw identity_of_suffixes,\n        apply list.take_append_drop,\n      },\n    },\n    {\n      rw list.mem_map at orig_in,\n      rcases orig_in with \u27e8r\u2082, r\u2082_in, r\u2082_conv\u27e9,\n      rw aft,\n      rw bef at ih_concat,\n      clear bef aft a b,\n      rw \u2190r\u2082_conv at ih_concat \u22a2,\n      clear r\u2082_conv orig_rule,\n      have part_for_u := congr_arg (list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length) ih_concat,\n      have part_for_v := congr_arg (list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length) ih_concat,\n      rw list.take_left at part_for_u,\n      rw list.drop_left at part_for_v,\n\n      have hlen_vd : (@lsTN_of_lsTN\u2082 T g\u2081 g\u2082 v).length > d.length,\n      {\n        by_contradiction contra,\n        push_neg at contra,\n\n        have not_in : symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst \u2209 lsTN_of_lsTN\u2081 u,\n        {\n          unfold lsTN_of_lsTN\u2081,\n          rw list.mem_map,\n          rintro \u27e8s, -, imposs\u27e9,\n          cases s,\n          {\n            exact symbol.no_confusion imposs,\n          },\n          {\n            have inl_eq_inr := option.some.inj (symbol.nonterminal.inj imposs),\n            exact sum.no_confusion inl_eq_inr,\n          },\n        },\n\n        have yes_in : symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst \u2208 lsTN_of_lsTN\u2081 u,\n        {\n          have ih_backwards := congr_arg list.reverse ih_concat,\n          repeat {\n            rw list.reverse_append at ih_backwards,\n          },\n          have ldth := congr_fun (congr_arg list.nth ih_backwards) d.length,\n          have dlengthth :\n            (d.reverse ++ ([symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst].reverse ++ c.reverse)).nth d.length =\n            some (symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst),\n          {\n            rw list.nth_append_right, swap,\n            {\n              rw list.length_reverse,\n            },\n            rw list.length_reverse,\n            rw nat.sub_self,\n            refl,\n          },\n          rw dlengthth at ldth,\n          rw \u2190list.length_reverse at contra,\n          rw list.nth_append_right contra at ldth,\n          have rrr := list.nth_mem ldth,\n          rw list.mem_reverse at rrr,\n          exact rrr,\n        },\n\n        exact not_in yes_in,\n      },\n      have total_length := congr_arg list.length ih_concat,\n      repeat {\n        rw list.length_append at total_length,\n      },\n      rw list.length_singleton at total_length,\n      have hlen_uc : (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length \u2264 c.length,\n      {\n        by_contradiction too_long,\n        push_neg at too_long,\n        have imposs_gt_self : c.length + 1 + d.length > c.length + 1 + d.length,\n        {\n          calc c.length + 1 + d.length\n              = (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length + (@lsTN_of_lsTN\u2082 T g\u2081 g\u2082 v).length :   total_length.symm\n          ... > (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length + d.length :   add_lt_add_left hlen_vd _\n          ... \u2265 c.length + d.length + 1 :   by { apply nat.succ_le_of_lt, apply add_lt_add_right too_long, }\n          ... = c.length + 1 + d.length :   nat.add_right_comm _ _ _,\n        },\n        exact nat.lt_irrefl _ imposs_gt_self,\n      },\n      have hlen_uc_orig : u.length \u2264 c.length,\n      {\n        unfold lsTN_of_lsTN\u2081 at hlen_uc,\n        rw list.length_map at hlen_uc,\n        exact hlen_uc,\n      },\n\n      -- nonterminal was rewritten in the right half of `a` ... upgrade `v`\n      let c' : list (symbol T (combined_grammar g\u2081 g\u2082).nt) :=\n        list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length c,\n      let v' := lsTN\u2082_of_lsTN (c' ++ (rule_of_rule\u2082 r\u2082).snd ++ d),\n      use u,\n      use v',\n      split,\n      {\n        split,\n        {\n          exact ih\u2081,\n        },\n        {\n          change\n            CF_derives g\u2082 [symbol.nonterminal g\u2082.initial] (\n              @lsTN\u2082_of_lsTN T g\u2081 g\u2082 (list.drop (lsTN_of_lsTN\u2081 u).length c ++\n              (rule_of_rule\u2082 r\u2082).snd ++ d)\n            ),\n          apply CF_deri_of_deri_tran ih\u2082,\n          convert_to\n            CF_transforms\n              g\u2082\n              (lsTN\u2082_of_lsTN (\n                list.drop (lsTN_of_lsTN\u2081 u).length (c ++ [symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst] ++ d)\n              ))\n              (lsTN\u2082_of_lsTN (list.drop (lsTN_of_lsTN\u2081 u).length c ++ (rule_of_rule\u2082 r\u2082).snd ++ d)),\n          {\n            rw \u2190part_for_v,\n            rw self_of_lsTN\u2082,\n          },\n          use r\u2082,\n          split,\n          {\n            exact r\u2082_in,\n          },\n          use lsTN\u2082_of_lsTN c',\n          use lsTN\u2082_of_lsTN d,\n\n          have eq_c' : list.drop u.length c = c',\n          {\n            change list.drop u.length c = list.drop (list.map (@sTN_of_sTN\u2081 T g\u2081 g\u2082) u).length c,\n            rw list.length_map,\n          },\n          split,\n          {\n            unfold lsTN_of_lsTN\u2081,\n            rw list.length_map,\n            unfold lsTN\u2082_of_lsTN,\n            rw list.append_assoc,\n            rw list.drop_append_of_le_length hlen_uc_orig,\n            rw \u2190list.append_assoc,\n            rw list.filter_map_append_append,\n            rw eq_c',\n            refl,\n          },\n          {\n            unfold lsTN_of_lsTN\u2081,\n            rw list.length_map,\n            unfold lsTN\u2082_of_lsTN,\n            rw list.filter_map_append_append,\n            change\n              list.filter_map sTN\u2082_of_sTN (list.drop u.length c) ++\n                lsTN\u2082_of_lsTN (lsTN_of_lsTN\u2082 r\u2082.snd) ++ list.filter_map sTN\u2082_of_sTN d =\n              list.filter_map sTN\u2082_of_sTN c' ++ r\u2082.snd ++ list.filter_map sTN\u2082_of_sTN d,\n            rw self_of_lsTN\u2082,\n            rw eq_c',\n          },\n        },\n      },\n      {\n        have identity_of_prefixes :\n          list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length (c ++ [symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst] ++ d) =\n          list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length (c ++ (rule_of_rule\u2082 r\u2082).snd ++ d),\n        {\n          -- both are equal to `list.take (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length c`\n          repeat\n          {\n            rw list.append_assoc,\n            rw list.take_append_of_le_length hlen_uc,\n          },\n        },\n\n        have express_v'_as_crd :\n          lsTN_of_lsTN\u2082 v' =\n          list.drop (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length (c ++ (rule_of_rule\u2082 r\u2082).snd ++ d),\n        {\n          change\n            list.map sTN_of_sTN\u2082 (list.filter_map sTN\u2082_of_sTN (\n              list.drop (lsTN_of_lsTN\u2081 u).length c ++ (rule_of_rule\u2082 r\u2082).snd ++ d)) =\n            list.drop (lsTN_of_lsTN\u2081 u).length (c ++ (rule_of_rule\u2082 r\u2082).snd ++ d),\n          rw list.filter_map_append_append,\n          rw list.map_append_append,\n          rw list.append_assoc c,\n          rw list.drop_append_of_le_length hlen_uc,\n          rw \u2190list.append_assoc,\n\n          apply congr_arg2, apply congr_arg2,\n          {\n            have aux_plus_minus : (lsTN_of_lsTN\u2081 u).length + (c.length - (lsTN_of_lsTN\u2081 u).length) = c.length,\n            {\n              rw \u2190nat.add_sub_assoc hlen_uc,\n              rw nat.add_sub_cancel_left,\n            },\n            have taken_c_from_v := congr_arg (list.take (c.length - (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length)) part_for_v,\n            rw \u2190list.drop_take at taken_c_from_v,\n            rw list.append_assoc at taken_c_from_v,\n            rw list.take_append_of_le_length (le_of_eq aux_plus_minus) at taken_c_from_v,\n            rw aux_plus_minus at taken_c_from_v,\n            rw list.take_length at taken_c_from_v,\n            rw \u2190taken_c_from_v,\n            unfold lsTN_of_lsTN\u2082,\n            rw \u2190list.map_take,\n            change\n              lsTN_of_lsTN\u2082 (lsTN\u2082_of_lsTN (lsTN_of_lsTN\u2082 (list.take (c.length - (lsTN_of_lsTN\u2081 u).length) v))) =\n              lsTN_of_lsTN\u2082 (list.take (c.length - (lsTN_of_lsTN\u2081 u).length) v),\n            rw self_of_lsTN\u2082,\n          },\n          {\n            unfold rule_of_rule\u2082,\n            change lsTN_of_lsTN\u2082 (lsTN\u2082_of_lsTN (lsTN_of_lsTN\u2082 r\u2082.snd)) = lsTN_of_lsTN\u2082 r\u2082.snd,\n            rw self_of_lsTN\u2082,\n          },\n          {\n            have taken_d_from_v := congr_arg (list.drop ((@lsTN_of_lsTN\u2082 T g\u2081 g\u2082 v).length - d.length)) part_for_v,\n            rw list.drop_drop at taken_d_from_v,\n            have dropped_exactly_length :\n              (@lsTN_of_lsTN\u2082 T g\u2081 g\u2082 v).length - d.length + (@lsTN_of_lsTN\u2081 T g\u2081 g\u2082 u).length =\n              (c ++ [symbol.nonterminal (rule_of_rule\u2082 r\u2082).fst]).length,\n            {\n              rw list.length_append,\n              rw list.length_singleton,\n              have reorder_sum :\n                (lsTN_of_lsTN\u2082 v).length - d.length + (lsTN_of_lsTN\u2081 u).length =\n                (lsTN_of_lsTN\u2081 u).length + (lsTN_of_lsTN\u2082 v).length - d.length,\n              {\n                rw nat.add_sub_assoc,\n                apply nat.add_comm,\n                apply le_of_lt,\n                exact hlen_vd,\n              },\n              rw reorder_sum,\n              rw total_length,\n              apply nat.add_sub_cancel,\n            },\n            rw dropped_exactly_length at taken_d_from_v,\n            rw list.drop_left at taken_d_from_v,\n            rw \u2190taken_d_from_v,\n            unfold lsTN_of_lsTN\u2082,\n            rw \u2190list.map_drop,\n            change\n              lsTN_of_lsTN\u2082 (lsTN\u2082_of_lsTN (lsTN_of_lsTN\u2082 (\n                list.drop ((list.map sTN_of_sTN\u2082 v).length - d.length) v))) =\n              lsTN_of_lsTN\u2082 (list.drop ((list.map sTN_of_sTN\u2082 v).length - d.length) v),\n            rw self_of_lsTN\u2082,\n          },\n        },\n\n        rw part_for_u,\n        rw identity_of_prefixes,\n        rw express_v'_as_crd,\n        apply list.take_append_drop,\n      },\n    },\n  },\n  specialize complicated_induction (list.map symbol.terminal w) derivation,\n\n  rcases complicated_induction with \u27e8u, v, \u27e8hu, hv\u27e9, hw\u27e9,\n  use liT_of_lsTN\u2083 u,\n  use liT_of_lsTN\u2083 v,\n  have huvw :\n    @liT_of_lsTN\u2083 T\n      (combined_grammar g\u2081 g\u2082)\n      (lsTN_of_lsTN\u2081 u ++ lsTN_of_lsTN\u2082 v)\n    = liT_of_lsTN\u2083 (list.map symbol.terminal w),\n  {\n    exact congr_arg liT_of_lsTN\u2083 hw,\n  },\n  split,\n  {\n    change CF_derives _ _ _,\n    unfold liT_of_lsTN\u2083,\n    convert hu,\n    have u_from_terminals : \u2203 u\u209c : list T, u = list.map symbol.terminal u\u209c,\n    {\n      unfold lsTN_of_lsTN\u2081 at hw,\n      use list.take u.length w,\n      rw list.map_take,\n      exact u_eq_take_map_w u v w\n        (by {\n          have hwlen := congr_arg list.length hw,\n          rw list.length_append at hwlen,\n          rw list.length_map at hwlen,\n          rw list.length_map at hwlen,\n          exact nat.le.intro hwlen,\n        }) (congr_arg (list.take u.length) hw),\n    },\n    cases u_from_terminals with u\u209c hut,\n    rw hut,\n    rw list.filter_map_map,\n    convert_to list.map symbol.terminal (list.filter_map some u\u209c) = list.map symbol.terminal u\u209c,\n    rw list.filter_map_some,\n  },\n  split,\n  {\n    change CF_derives _ _ _,\n    unfold liT_of_lsTN\u2083,\n    convert hv,\n    have v_from_terminals : \u2203 v\u209c : list T, v = list.map symbol.terminal v\u209c,\n    {\n      unfold lsTN_of_lsTN\u2081 at hw,\n      unfold lsTN_of_lsTN\u2082 at hw,\n      use list.drop u.length w,\n      rw list.map_drop,\n      have hwlen := congr_arg list.length hw,\n      rw list.length_append at hwlen,\n      repeat {\n        rw list.length_map at hwlen,\n      },\n      exact v_eq_drop_map_w u v w hwlen (congr_arg (list.drop u.length) hw),\n    },\n    cases v_from_terminals with v\u209c hvt,\n    rw hvt,\n    rw list.filter_map_map,\n    convert_to list.map symbol.terminal (list.filter_map some v\u209c) = list.map symbol.terminal v\u209c,\n    rw list.filter_map_some,\n  },\n  unfold liT_of_lsTN\u2083 at huvw,\n  rw list.filter_map_append at huvw,\n  unfold lsTN_of_lsTN\u2081 at huvw,\n  unfold lsTN_of_lsTN\u2082 at huvw,\n  repeat {\n    rw list.filter_map_map at huvw,\n  },\n  have disappear_sTN_of_sTN\u2081 : @oT_of_sTN\u2083 T (combined_grammar g\u2081 g\u2082) \u2218 sTN_of_sTN\u2081 = oT_of_sTN\u2083,\n  {\n    ext1,\n    cases x;\n    refl,\n  },\n  have disappear_sTN_of_sTN\u2082 : @oT_of_sTN\u2083 T (combined_grammar g\u2081 g\u2082) \u2218 sTN_of_sTN\u2082 = oT_of_sTN\u2083,\n  {\n    ext1,\n    cases x;\n    refl,\n  },\n  rw disappear_sTN_of_sTN\u2081 at huvw,\n  rw disappear_sTN_of_sTN\u2082 at huvw,\n  unfold liT_of_lsTN\u2083,\n  convert huvw,\n  have bundle_unbundle : @oT_of_sTN\u2083 T (combined_grammar g\u2081 g\u2082) \u2218 symbol.terminal = option.some,\n  {\n    ext1,\n    refl,\n  },\n  rw bundle_unbundle,\n  rw list.filter_map_some,\nend\n\n\nprivate lemma in_combined_of_in_concatenated\n    {g\u2081 g\u2082 : CF_grammar T}\n    {w : list T}\n    (hyp : w \u2208 CF_language g\u2081 * CF_language g\u2082) :\n  w \u2208 CF_language (combined_grammar g\u2081 g\u2082) :=\nbegin\n  rw language.mem_mul at hyp,\n  rcases hyp with \u27e8u, v, hu, hv, hw\u27e9,\n  unfold CF_language at *,\n  change\n    CF_derives\n      (combined_grammar g\u2081 g\u2082)\n      [symbol.nonterminal (combined_grammar g\u2081 g\u2082).initial]\n      (list.map symbol.terminal w),\n\n  apply @CF_deri_of_tran_deri T\n    (combined_grammar g\u2081 g\u2082)\n    _ [\n      symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n      symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n    ] _,\n  {\n    use (none, [\n        symbol.nonterminal (some (sum.inl (g\u2081.initial))),\n        symbol.nonterminal (some (sum.inr (g\u2082.initial)))\n      ]),\n    split,\n    {\n      apply list.mem_cons_self,\n    },\n    use [[], []],\n    split;\n    refl,\n  },\n  rw \u2190hw,\n  rw list.map_append,\n  apply @CF_deri_of_deri_deri T\n    (combined_grammar g\u2081 g\u2082) _\n    (list.map symbol.terminal u ++ [symbol.nonterminal (some (sum.inr g\u2082.initial))]) _,\n  {\n    change\n      CF_derives\n        (combined_grammar g\u2081 g\u2082)\n        ([symbol.nonterminal (some (sum.inl g\u2081.initial))] ++ [symbol.nonterminal (some (sum.inr g\u2082.initial))])\n        (list.map symbol.terminal u ++ [symbol.nonterminal (some (sum.inr g\u2082.initial))]),\n    apply CF_deri_with_postfix,\n\n    change CF_derives g\u2081 [symbol.nonterminal g\u2081.initial] (list.map symbol.terminal u) at hu,\n    let gg\u2081 := g\u2081g g\u2081 g\u2082,\n    change CF_derives gg\u2081.g [symbol.nonterminal (some (sum.inl g\u2081.initial))] (list.map symbol.terminal u),\n    \n    have ini_equ :\n      [symbol.nonterminal (some (sum.inl g\u2081.initial))] =\n      list.map (lift_symbol gg\u2081.lift_nt) [symbol.nonterminal g\u2081.initial],\n    {\n      apply list.singleton_eq,\n    },\n    rw ini_equ,\n\n    have baz : list.map symbol.terminal u = list.map (lift_symbol gg\u2081.lift_nt) (list.map symbol.terminal u),\n    {\n      rw list.map_map,\n      apply congr_fun,\n      apply congr_arg,\n      refl,\n    },\n    rw baz,\n    \n    exact lift_deri hu,\n  },\n  {\n    apply CF_deri_with_prefix,\n\n    change CF_derives g\u2082 [symbol.nonterminal g\u2082.initial] (list.map symbol.terminal v) at hv,\n    let gg\u2082 := g\u2082g g\u2081 g\u2082,\n    change CF_derives gg\u2082.g [symbol.nonterminal (some (sum.inr g\u2082.initial))] (list.map symbol.terminal v),\n    \n    have ini_equ :\n      [symbol.nonterminal (some (sum.inr g\u2082.initial))] =\n      list.map (lift_symbol gg\u2082.lift_nt) [symbol.nonterminal g\u2082.initial],\n    {\n      apply list.singleton_eq,\n    },\n    rw ini_equ,\n\n    have baz : list.map symbol.terminal v = list.map (lift_symbol gg\u2082.lift_nt) (list.map symbol.terminal v),\n    {\n      rw list.map_map,\n      apply congr_fun,\n      apply congr_arg,\n      refl,\n    },\n    rw baz,\n    \n    exact lift_deri hv,\n  },\nend\n\n\n/-- The class of context-free languages is closed under concatenation. -/\ntheorem CF_of_CF_c_CF (L\u2081 : language T) (L\u2082 : language T) :\n  is_CF L\u2081  \u2227  is_CF L\u2082   \u2192   is_CF (L\u2081 * L\u2082)   :=\nbegin\n  rintro \u27e8\u27e8g\u2081, eq_L\u2081\u27e9, \u27e8g\u2082, eq_L\u2082\u27e9\u27e9,\n\n  use combined_grammar g\u2081 g\u2082,\n\n  apply set.eq_of_subset_of_subset,\n  {\n    -- prove `L\u2081 * L\u2082 \u2287 ` here\n    intros w hyp,\n    rw \u2190eq_L\u2081,\n    rw \u2190eq_L\u2082,\n    exact in_concatenated_of_in_combined hyp,\n  },\n  {\n    -- prove `L\u2081 * L\u2082 \u2286 ` here\n    intros w hyp,\n    rw \u2190eq_L\u2081 at hyp,\n    rw \u2190eq_L\u2082 at hyp,\n    exact in_combined_of_in_concatenated hyp,\n  },\nend\n", "meta": {"author": "madvorak", "repo": "grammars", "sha": "5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f", "save_path": "github-repos/lean/madvorak-grammars", "path": "github-repos/lean/madvorak-grammars/grammars-5ab26130eb76d5f7cde0f6c2f9c6f3107ff8d34f/src/classes/context_free/closure_properties/concatenation.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.35019806362968353}}
{"text": "import for_mathlib.exact_seq2\nimport for_mathlib.abelian_category\n\nnamespace category_theory\nopen category_theory.limits\n\nvariables {A : Type*} [category A] [abelian A]\n\nnoncomputable theory\n\ndef exact_seq.w {X\u2081 X\u2082 X\u2083 : A} {f : X\u2081 \u27f6 X\u2082} {g : X\u2082 \u27f6 X\u2083}\n  (e : exact_seq A [f, g]) : f \u226b g = 0 :=\nby { rw \u2190 exact_iff_exact_seq at e, exact e.w }\n\ndef exact_seq.\u03c0 {X\u2081 X\u2082 X\u2083 : A} {f : X\u2081 \u27f6 X\u2082} {g : X\u2082 \u27f6 X\u2083}\n  (e : exact_seq A [f, g]) : cokernel f \u27f6 X\u2083 :=\ncokernel.desc _ g e.w\n\ndef exact_seq.\u03b9 {X\u2081 X\u2082 X\u2083 : A} {f : X\u2081 \u27f6 X\u2082} {g : X\u2082 \u27f6 X\u2083}\n  (e : exact_seq A [f, g]) : X\u2081 \u27f6 kernel g :=\nkernel.lift _ f e.w\n\nlocal attribute [instance] abelian.pseudoelement.hom_to_fun\n\nlemma exact_seq.replace\n  {X\u2081 X\u2082 X\u2083 X\u2084 X\u2085 : A}\n  {f\u2081 : X\u2081 \u27f6 X\u2082}\n  {f\u2082 : X\u2082 \u27f6 X\u2083}\n  {f\u2083 : X\u2083 \u27f6 X\u2084}\n  {f\u2084 : X\u2084 \u27f6 X\u2085}\n  (e : exact_seq A [f\u2081, f\u2082, f\u2083, f\u2084]) :\n  exact (e.extract 0 2).\u03c0 (e.extract 2 3).\u03b9 :=\nbegin\n  apply abelian.pseudoelement.exact_of_pseudo_exact,\n  split,\n  { intros a,\n    simp only [\u2190 abelian.pseudoelement.comp_apply],\n    suffices : (e.extract 0 2).\u03c0 \u226b (e.extract 2 3).\u03b9 = 0,\n    by { rw this, simp },\n    ext,\n    dsimp [exact_seq.\u03c0, exact_seq.\u03b9],\n    simpa using (e.extract 1 2).w },\n  { intros t ht,\n    dsimp [exact_seq.\u03c0, exact_seq.\u03b9] at *,\n    apply_fun (\u03bb i, kernel.\u03b9 f\u2084 i) at ht,\n    simp only [\u2190 abelian.pseudoelement.comp_apply, kernel.lift_\u03b9] at ht,\n    simp at ht,\n    have := e.extract 1 2,\n    erw \u2190 exact_iff_exact_seq at this,\n    have := (@abelian.pseudoelement.pseudo_exact_of_exact\n      _ _ _ _ _ _ _ _ this).2 _ ht,\n    obtain \u27e8a,ha\u27e9 := this,\n    use cokernel.\u03c0 f\u2081 a,\n    rwa [\u2190 abelian.pseudoelement.comp_apply, cokernel.\u03c0_desc] }\nend\n\nlemma is_zero_of_exact_seq_of_is_iso_of_is_iso {X\u2081 X\u2082 X\u2083 X\u2084 X\u2085 : A}\n  (f\u2081 : X\u2081 \u27f6 X\u2082)\n  (f\u2082 : X\u2082 \u27f6 X\u2083)\n  (f\u2083 : X\u2083 \u27f6 X\u2084)\n  (f\u2084 : X\u2084 \u27f6 X\u2085)\n  [epi f\u2081]\n  [mono f\u2084]\n  (e :  exact_seq A [f\u2081, f\u2082, f\u2083, f\u2084]) : is_zero X\u2083 :=\nbegin\n  have h1 : is_zero (cokernel f\u2081) := is_zero_cokernel_of_epi f\u2081,\n  have h2 : is_zero (kernel f\u2084) := is_zero_kernel_of_mono f\u2084,\n  apply is_zero_of_exact_is_zero_is_zero _ _ _ h1 h2,\n  exact (e.extract 0 2).\u03c0, exact (e.extract 2 3).\u03b9,\n  apply exact_seq.replace,\nend\n\nlemma is_iso_of_exact_of_is_zero_of_is_zero {X\u2081 X\u2082 X\u2083 X\u2084 : A}\n  (f\u2081 : X\u2081 \u27f6 X\u2082)\n  (f\u2082 : X\u2082 \u27f6 X\u2083)\n  (f\u2083 : X\u2083 \u27f6 X\u2084)\n  (h\u2081 : is_zero X\u2081)\n  (h\u2084 : is_zero X\u2084)\n  (e : exact_seq A [f\u2081, f\u2082, f\u2083]) : is_iso f\u2082 :=\nbegin\n  apply_with is_iso_of_mono_of_epi { instances := ff}, apply_instance,\n  apply exact.mono_of_is_zero e.pair, assumption,\n  apply exact.epi_of_is_zero (e.extract 1 2).pair, assumption,\nend\n\nend category_theory\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/for_mathlib/exact_seq3.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.35019805693304207}}
{"text": "/-\nCopyright (c) 2019 S\u00e9bastien Gou\u00ebzel. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: S\u00e9bastien Gou\u00ebzel\n-/\n\nimport tactic.interactive\nimport tactic.norm_num\n\n/-!\n# `field_simp` tactic\n\nTactic to clear denominators in algebraic expressions, based on `simp` with a specific simpset.\n-/\n\nnamespace tactic\n\n/-- Try to prove a goal of the form `x \u2260 0` by calling `assumption`, or `norm_num1` if `x` is\na numeral. -/\nmeta def field_simp.ne_zero : tactic unit := do\n  goal \u2190 tactic.target,\n  match goal with\n  | `(%%e \u2260 0) := assumption <|> do n \u2190 e.to_rat, `[norm_num1]\n  | _ := tactic.fail \"goal should be of the form `x \u2260 0`\"\n  end\n\nnamespace interactive\nsetup_tactic_parser\n\n/--\nThe goal of `field_simp` is to reduce an expression in a field to an expression of the form `n / d`\nwhere neither `n` nor `d` contains any division symbol, just using the simplifier (with a carefully\ncrafted simpset named `field_simps`) to reduce the number of division symbols whenever possible by\niterating the following steps:\n\n- write an inverse as a division\n- in any product, move the division to the right\n- if there are several divisions in a product, group them together at the end and write them as a\n  single division\n- reduce a sum to a common denominator\n\nIf the goal is an equality, this simpset will also clear the denominators, so that the proof\ncan normally be concluded by an application of `ring` or `ring_exp`.\n\n`field_simp [hx, hy]` is a short form for\n`simp [-one_div, -mul_eq_zero, hx, hy] with field_simps {discharger := tactic.field_simp.ne_zero}`\n\nNote that this naive algorithm will not try to detect common factors in denominators to reduce the\ncomplexity of the resulting expression. Instead, it relies on the ability of `ring` to handle\ncomplicated expressions in the next step.\n\nAs always with the simplifier, reduction steps will only be applied if the preconditions of the\nlemmas can be checked. This means that proofs that denominators are nonzero should be included. The\nfact that a product is nonzero when all factors are, and that a power of a nonzero number is\nnonzero, are included in the simpset, but more complicated assertions (especially dealing with sums)\nshould be given explicitly. If your expression is not completely reduced by the simplifier\ninvocation, check the denominators of the resulting expression and provide proofs that they are\nnonzero to enable further progress.\n\nTo check that denominators are nonzero, `field_simp` will look for facts in the context, and\nwill try to apply `norm_num` to close numerical goals.\n\nThe invocation of `field_simp` removes the lemma `one_div` from the simpset, as this lemma\nworks against the algorithm explained above. It also removes\n`mul_eq_zero : x * y = 0 \u2194 x = 0 \u2228 y = 0`, as `norm_num` can not work on disjunctions to\nclose goals of the form `24 \u2260 0`, and replaces it with `mul_ne_zero : x \u2260 0 \u2192 y \u2260 0 \u2192 x * y \u2260 0`\ncreating two goals instead of a disjunction.\n\nFor example,\n```lean\nexample (a b c d x y : \u2102) (hx : x \u2260 0) (hy : y \u2260 0) :\n  a + b / x + c / x^2 + d / x^3 = a + x\u207b\u00b9 * (y * b / y + (d / x + c) / x) :=\nbegin\n  field_simp,\n  ring\nend\n```\n\nMoreover, the `field_simp` tactic can also take care of inverses of units in\na general (commutative) monoid/ring and partial division `/\u209a`, see `algebra.group.units`\nfor the definition. Analogue to the case above, the lemma `one_divp` is removed from the simpset\nas this works against the algorithm. If you have objects with a `is_unit x` instance like\n`(x : R) (hx : is_unit x)`, you should lift them with\n`lift x to R\u02e3 using id hx, rw is_unit.unit_of_coe_units, clear hx`\nbefore using `field_simp`.\n\nSee also the `cancel_denoms` tactic, which tries to do a similar simplification for expressions\nthat have numerals in denominators.\nThe tactics are not related: `cancel_denoms` will only handle numeric denominators, and will try to\nentirely remove (numeric) division from the expression by multiplying by a factor.\n-/\nmeta def field_simp (no_dflt : parse only_flag) (hs : parse simp_arg_list)\n  (attr_names : parse with_ident_list)\n  (locat : parse location)\n  (cfg : simp_config_ext := {discharger := field_simp.ne_zero}) : tactic unit :=\nlet attr_names := `field_simps :: attr_names,\n    hs := simp_arg_type.except `one_div :: simp_arg_type.except `mul_eq_zero ::\n          simp_arg_type.except `one_divp :: hs in\npropagate_tags (simp_core cfg.to_simp_config cfg.discharger no_dflt hs attr_names locat >> skip)\n\nadd_tactic_doc\n{ name       := \"field_simp\",\n  category   := doc_category.tactic,\n  decl_names := [`tactic.interactive.field_simp],\n  tags       := [\"simplification\", \"arithmetic\"] }\n\nend interactive\nend tactic\n", "meta": {"author": "leanprover-community", "repo": "mathlib", "sha": "5e526d18cea33550268dcbbddcb822d5cde40654", "save_path": "github-repos/lean/leanprover-community-mathlib", "path": "github-repos/lean/leanprover-community-mathlib/mathlib-5e526d18cea33550268dcbbddcb822d5cde40654/src/tactic/field_simp.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6688802735722129, "lm_q1q2_score": 0.35010554618979156}}
{"text": "import algebra.big_operators.finsupp\nimport tactic.field_simp\nimport data.rat.floor\n\ndef BoundedStreamGen : Type := sorry\n\ndef BoundedStreamGen.eval (x : BoundedStreamGen) : \u2115 \u2192\u2080 \u2124 := sorry\n\ndef contract (x : BoundedStreamGen) : BoundedStreamGen := sorry\n\ndef externSparseVec (x : list \u2115) (y : list \u2124) : BoundedStreamGen := sorry\n\nlemma externSparseVec.spec (x : list \u2115) (y : list \u2124) :\n  (externSparseVec x y).eval = (list.zip_with finsupp.single x y).sum := sorry\n\nlemma contract.spec (s : BoundedStreamGen) :\n  (contract s).eval = s.eval.map_domain (\u03bb _, 0) := sorry\n\nclass HasCorrectEval (x : BoundedStreamGen) (gn : out_param $ \u2115 \u2192\u2080 \u2124) : Prop :=\n(iseq [] : x.eval = gn)\n\nopen HasCorrectEval (iseq)\n\ninstance externSparseVec.correctEval (x : list \u2115) (y : list \u2124) [fact (x.length = y.length)] :\n  HasCorrectEval (externSparseVec x y) (list.zip_with finsupp.single x y).sum := \u27e8externSparseVec.spec _ _\u27e9\n\ninstance contract.correctEval (s : BoundedStreamGen) {gn : \u2115 \u2192\u2080 \u2124} \n  [HasCorrectEval s gn] : HasCorrectEval (contract s) (finsupp.map_domain.add_monoid_hom (\u03bb _, 0) gn) :=\n\u27e8by { rw [contract.spec, iseq], refl, }\u27e9\n\ndef sum_vec (x : list \u2115) (y : list \u2124) : BoundedStreamGen := contract (externSparseVec x y)\n\nlemma sum_vec.spec (x : list \u2115) (y : list \u2124) (hx : x.length = y.length) :\n  (sum_vec x y).eval = finsupp.single 0 y.sum :=\nbegin\n  haveI : fact _ := \u27e8hx\u27e9,\n  rw [sum_vec, iseq],-- \u2190 list.sum_hom, list.map_zip_with],\n  -- simp,\nend\n\n\n\nopen HasCorrectEval (iseq)\n\n\n\nuniverse u\nlemma eq_of_heq' {\u03b1 : Sort u} {a a' : \u03b1} (h : a == a') : a = a' :=\nhave \u2200 (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a') (h\u2082 : \u03b1 = \u03b1'), (eq.rec_on h\u2082 a : \u03b1') = a', from\n  \u03bb (\u03b1' : Sort u) (a' : \u03b1') (h\u2081 : @heq \u03b1 a \u03b1' a'), heq.rec_on h\u2081 (\u03bb h\u2082 : \u03b1 = \u03b1, rfl),\nshow (eq.rec_on (eq.refl \u03b1) a : \u03b1) = a', from\n  this \u03b1 a' h (eq.refl \u03b1)\n\n\ndef star (f : \u2115 \u2192 \u2115) (hf : \u2200 n : \u2115, f (n + 1) \u2264 n) : \u2115 \u2192 \u2115\n| 0 := 0\n| (n + 1) := have _ := nat.lt_succ_of_le (hf n), if f (n + 1) = 0 then 0 else 1 + star (f (n + 1))\n\nnotation f`\uff0a`:9000 := star f (by assumption)\n\nlemma star_eq {f : \u2115 \u2192 \u2115} (h\u2081 : f 0 = 0) (h\u2082 : \u2200 n, f (n + 1) \u2264 n) (n : \u2115) : \n  f\uff0a n = if f n = 0 then 0 else 1 + f\uff0a (f n) :=\nby cases n; simp [star, h\u2081]\n\n@[simp] lemma star_zero {f : \u2115 \u2192 \u2115} (h\u2082 : \u2200 n, f (n + 1) \u2264 n) : f\uff0a 0 = 0 := by simp [star]\n\n@[simp] lemma star_one {f : \u2115 \u2192 \u2115} (h\u2082 : \u2200 n, f (n + 1) \u2264 n) : f\uff0a 1 = 0 :=\nby simpa [star, imp_false] using h\u2082 0\n\nlemma star_contraction_of_contraction {f : \u2115 \u2192 \u2115} (H : \u2200 n, f (n + 1) \u2264 n) (n : \u2115) :\n  f\uff0a 0 = 0 \u2227 f\uff0a (n + 1) \u2264 n :=\nbegin\n  split, { simp, },\n  induction n using nat.strong_induction_on with n ih,\n  rw star,\n  split_ifs, { exact zero_le _, },\n  specialize H n, refine trans _ H,\n  cases (f (n + 1)) with m hm, { contradiction, },\n  rw [nat.succ_eq_add_one, add_comm 1 _, add_le_add_iff_right],\n  refine ih m _,\n  rwa \u2190 nat.succ_le_iff,\nend\n\n-- open_locale big_operators\n\n\n-- lemma egyption_fraction_wf (r : \u211a) (h\u2081 : 0 < r) (h\u2082 : r < 1) :\n--   (r - (1 : \u211a) / \u23081/r\u2309).num < r.num :=\n-- begin\n  \n-- end\n\n-- def egyptian_fraction : \u2200 (r : \u211a) (h\u2081 : 0 \u2264 r) (h\u2082 : r < 1), finset \u2124 | r h\u2081 h\u2082 :=\n-- if H : r = 0 then 0 else\n-- let n : \u2124 := \u23081/r\u2309 in\n-- have wf : (r - (1 : \u211a) / n).num < r.num := sorry, \n-- insert n (egyptian_fraction (r - (1 : \u211a) / n) _ _)\n-- using_well_founded {rel_tac := \u03bb _ _, `[exact \u27e8_, measure_wf rat.num\u27e9]}\n", "meta": {"author": "kovach", "repo": "etch", "sha": "26ef67eb83cf7c5cfd1667059e16c3873b9098ca", "save_path": "github-repos/lean/kovach-etch", "path": "github-repos/lean/kovach-etch/etch-26ef67eb83cf7c5cfd1667059e16c3873b9098ca/src/test.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802603710086, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3501055392800126}}
{"text": "import Lbar.torsion_free_profinite\nimport condensed.condensify\n\nnoncomputable theory\n\nuniverse u\n\nopen category_theory opposite\n\nopen_locale nnreal\n\nnamespace CompHausFiltPseuNormGrp\n\nlemma to_Condensed_torsion_free (M : CompHausFiltPseuNormGrp) [no_zero_smul_divisors \u2124 M]\n  (T : ExtrDisc) :\n  no_zero_smul_divisors \u2124 ((to_Condensed.obj M).val.obj (op T.val)) :=\nbegin\n  dsimp, constructor,\n  intros n f hf,\n  rw or_iff_not_imp_left,\n  intro hn,\n  ext t,\n  apply_fun (\u03bb \u03c6, \u03c6.down.val t) at hf,\n  apply smul_right_injective M hn,\n  dsimp [presheaf.has_zero] at hf \u22a2,\n  convert hf using 1,\n  apply smul_zero\nend\n\nend CompHausFiltPseuNormGrp\n\nnamespace Lbar\n\nvariables (r' : \u211d\u22650) [fact (0 < r')]\n\nlemma Fintype_Lbar_torsion_free (X : Fintype) :\n  no_zero_smul_divisors \u2124 ((Fintype_Lbar r' \u22d9 PFPNGT\u2081_to_CHFPNG\u2081\u2091\u2097 r').obj X) :=\nFintype.Lbar_no_zero_smul_divisors _ _\n\nlemma condensify_torsion_free (A : Fintype.{u} \u2964 CompHausFiltPseuNormGrp\u2081)\n  (hA : \u2200 X, no_zero_smul_divisors \u2124 (A.obj X))\n  (S : Profinite) (T : ExtrDisc.{u}) :\n  no_zero_smul_divisors \u2124 (((condensify A).obj S).val.obj (op T.val)) :=\nbegin\n  apply_with CompHausFiltPseuNormGrp.to_Condensed_torsion_free {instances := ff},\n  apply Profinite.extend_torsion_free,\n  apply hA,\nend\n\ndef condensed : Profinite.{u} \u2964 Condensed.{u} Ab.{u+1} :=\ncondensify (Fintype_Lbar.{u u} r' \u22d9 PFPNGT\u2081_to_CHFPNG\u2081\u2091\u2097 r')\n\ninstance (S : Profinite.{u}) (T : ExtrDisc.{u}) :\n  no_zero_smul_divisors \u2124 (((condensed.{u} r').obj S).val.obj (op T.val)) :=\ncondensify_torsion_free _ (Fintype_Lbar_torsion_free r') _ _\n\nend Lbar\n", "meta": {"author": "leanprover-community", "repo": "lean-liquid", "sha": "92f188bd17f34dbfefc92a83069577f708851aec", "save_path": "github-repos/lean/leanprover-community-lean-liquid", "path": "github-repos/lean/leanprover-community-lean-liquid/lean-liquid-92f188bd17f34dbfefc92a83069577f708851aec/src/Lbar/torsion_free_condensed.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3501055323702335}}
{"text": "/-\nCopyright (c) 2017 Scott Morrison. All rights reserved.\nReleased under Apache 2.0 license as described in the file LICENSE.\nAuthors: Scott Morrison, Adam Topaz\n\n! This file was ported from Lean 3 source module category_theory.limits.concrete_category\n! leanprover-community/mathlib commit cb3ceec8485239a61ed51d944cb9a95b68c6bafc\n! Please do not edit these lines, except to modify the commit id\n! if you have ported upstream changes.\n-/\nimport Mathbin.CategoryTheory.Limits.Preserves.Basic\nimport Mathbin.CategoryTheory.Limits.Types\nimport Mathbin.CategoryTheory.Limits.Shapes.WidePullbacks\nimport Mathbin.CategoryTheory.Limits.Shapes.Multiequalizer\nimport Mathbin.CategoryTheory.ConcreteCategory.Basic\nimport Mathbin.CategoryTheory.Limits.Shapes.Kernels\nimport Mathbin.Tactic.ApplyFun\n\n/-!\n# Facts about (co)limits of functors into concrete categories\n\n> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.\n> Any changes to this file require a corresponding PR to mathlib4.\n-/\n\n\nuniverse w v u\n\nopen CategoryTheory\n\nnamespace CategoryTheory.Limits\n\nattribute [local instance] concrete_category.has_coe_to_fun concrete_category.has_coe_to_sort\n\nsection Limits\n\nvariable {C : Type u} [Category.{v} C] [ConcreteCategory.{max w v} C] {J : Type w} [SmallCategory J]\n  (F : J \u2964 C) [PreservesLimit F (forget C)]\n\n/- warning: category_theory.limits.concrete.to_product_injective_of_is_limit -> CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u1 u2, u3, succ (max u1 u2)} C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} J _inst_3 F (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cone.{u1, u2, u1, u3} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsLimit.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) -> (Function.Injective.{succ (max u1 u2), max (succ u1) (succ (max u1 u2))} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (forall (j : J), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (x : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (j : J) => coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) x))\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u2 u1, u3, max (succ u2) (succ u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} J _inst_3 F (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cone.{u1, u2, u1, u3} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsLimit.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) -> (Function.Injective.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (forall (j : J), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j)) (fun (x : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (j : J) => Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D))) j) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) x))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.to_product_injective_of_is_limit CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit\u2093'. -/\ntheorem Concrete.to_product_injective_of_isLimit {D : Cone F} (hD : IsLimit D) :\n    Function.Injective fun (x : D.pt) (j : J) => D.\u03c0.app j x :=\n  by\n  let E := (forget C).mapCone D\n  let hE : is_limit E := is_limit_of_preserves _ hD\n  let G := Types.limitCone.{w, v} (F \u22d9 forget C)\n  let hG := Types.limitConeIsLimit.{w, v} (F \u22d9 forget C)\n  let T : E.X \u2245 G.X := hE.cone_point_unique_up_to_iso hG\n  change Function.Injective (T.hom \u226b fun x j => G.\u03c0.app j x)\n  have h : Function.Injective T.hom := by\n    intro a b h\n    suffices T.inv (T.hom a) = T.inv (T.hom b) by simpa\n    rw [h]\n  suffices Function.Injective fun (x : G.X) j => G.\u03c0.app j x by exact this.comp h\n  apply Subtype.ext\n#align category_theory.limits.concrete.to_product_injective_of_is_limit CategoryTheory.Limits.Concrete.to_product_injective_of_isLimit\n\n/- warning: category_theory.limits.concrete.is_limit_ext -> CategoryTheory.Limits.Concrete.isLimit_ext is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u1 u2, u3, succ (max u1 u2)} C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} J _inst_3 F (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cone.{u1, u2, u1, u3} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsLimit.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) -> (forall (x : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (y : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)), (forall (j : J), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) x) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) j) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u2, max u1 u2, u3, max u1 u2 u1 u3} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) y)) -> (Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) x y))\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u2 u1, u3, max (succ u2) (succ u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} J _inst_3 F (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cone.{u1, u2, u1, u3} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsLimit.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) -> (forall (x : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) (y : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)), (forall (j : J), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D))) j) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) x) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D))) j) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j) (CategoryTheory.NatTrans.app.{u1, u2, u1, u3} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u2, max (succ u1) (succ u2), u3, max (max u1 u2) u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{max u1 u2, max (max u1 u3) u2} (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u2, max u1 u2, u3, max (max u1 u3) u2} C _inst_1 (CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u2, u1, u3} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u2, u1, u3} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) F (CategoryTheory.Limits.Cone.\u03c0.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D) j) y)) -> (Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cone.pt.{u1, u2, u1, u3} J _inst_3 C _inst_1 F D)) x y))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.is_limit_ext CategoryTheory.Limits.Concrete.isLimit_ext\u2093'. -/\ntheorem Concrete.isLimit_ext {D : Cone F} (hD : IsLimit D) (x y : D.pt) :\n    (\u2200 j, D.\u03c0.app j x = D.\u03c0.app j y) \u2192 x = y := fun h =>\n  Concrete.to_product_injective_of_isLimit _ hD (funext h)\n#align category_theory.limits.concrete.is_limit_ext CategoryTheory.Limits.Concrete.isLimit_ext\n\n/- warning: category_theory.limits.concrete.limit_ext -> CategoryTheory.Limits.Concrete.limit_ext is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u1 u2, u3, succ (max u1 u2)} C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} J _inst_3 F (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasLimit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F] (x : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) (y : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)), (forall (j : J), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (CategoryTheory.Limits.limit.\u03c0.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5 j) x) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Functor.obj.{u1, u2, u1, u3} J _inst_3 C _inst_1 F j)) (CategoryTheory.Limits.limit.\u03c0.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5 j) y)) -> (Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) x y)\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u2, u1, u3} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u2 u1, u3, max (succ u2) (succ u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} J _inst_3 F (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasLimit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F] (x : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) (y : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)), (forall (j : J), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5 j) x) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5) (Prefunctor.obj.{succ u1, succ u2, u1, u3} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u2, u1, u3} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.limit.\u03c0.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5 j) y)) -> (Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.limit.{u1, u1, u2, u3} J _inst_3 C _inst_1 F _inst_5)) x y)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.limit_ext CategoryTheory.Limits.Concrete.limit_ext\u2093'. -/\ntheorem Concrete.limit_ext [HasLimit F] (x y : limit F) :\n    (\u2200 j, limit.\u03c0 F j x = limit.\u03c0 F j y) \u2192 x = y :=\n  Concrete.isLimit_ext F (limit.isLimit _) _ _\n#align category_theory.limits.concrete.limit_ext CategoryTheory.Limits.Concrete.limit_ext\n\nsection WidePullback\n\nopen WidePullback\n\nopen WidePullbackShape\n\n#print CategoryTheory.Limits.Concrete.widePullback_ext /-\ntheorem Concrete.widePullback_ext {B : C} {\u03b9 : Type w} {X : \u03b9 \u2192 C} (f : \u2200 j : \u03b9, X j \u27f6 B)\n    [HasWidePullback B X f] [PreservesLimit (wideCospan B X f) (forget C)]\n    (x y : widePullback B X f) (h\u2080 : base f x = base f y) (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\n  by\n  apply concrete.limit_ext\n  rintro (_ | j)\n  \u00b7 exact h\u2080\n  \u00b7 apply h\n#align category_theory.limits.concrete.wide_pullback_ext CategoryTheory.Limits.Concrete.widePullback_ext\n-/\n\n#print CategoryTheory.Limits.Concrete.widePullback_ext' /-\ntheorem Concrete.widePullback_ext' {B : C} {\u03b9 : Type w} [Nonempty \u03b9] {X : \u03b9 \u2192 C}\n    (f : \u2200 j : \u03b9, X j \u27f6 B) [HasWidePullback.{w} B X f]\n    [PreservesLimit (wideCospan B X f) (forget C)] (x y : widePullback B X f)\n    (h : \u2200 j, \u03c0 f j x = \u03c0 f j y) : x = y :=\n  by\n  apply concrete.wide_pullback_ext _ _ _ _ h\n  inhabit \u03b9\n  simp only [\u2190 \u03c0_arrow f (Inhabited.default _), comp_apply, h]\n#align category_theory.limits.concrete.wide_pullback_ext' CategoryTheory.Limits.Concrete.widePullback_ext'\n-/\n\nend WidePullback\n\nsection Multiequalizer\n\n#print CategoryTheory.Limits.Concrete.multiequalizer_ext /-\ntheorem Concrete.multiequalizer_ext {I : MulticospanIndex.{w} C} [HasMultiequalizer I]\n    [PreservesLimit I.multicospan (forget C)] (x y : multiequalizer I)\n    (h : \u2200 t : I.L, Multiequalizer.\u03b9 I t x = Multiequalizer.\u03b9 I t y) : x = y :=\n  by\n  apply concrete.limit_ext\n  rintro (a | b)\n  \u00b7 apply h\n  \u00b7 rw [\u2190 limit.w I.multicospan (walking_multicospan.hom.fst b), comp_apply, comp_apply, h]\n#align category_theory.limits.concrete.multiequalizer_ext CategoryTheory.Limits.Concrete.multiequalizer_ext\n-/\n\n/- warning: category_theory.limits.concrete.multiequalizer_equiv_aux -> CategoryTheory.Limits.Concrete.multiequalizerEquivAux is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] (I : CategoryTheory.Limits.MulticospanIndex.{u4, u2, u3} C _inst_1), Equiv.{succ (max u4 u1 u2), max 1 (succ u4) (succ (max u1 u2))} (coeSort.{succ (max u4 u1 u2), succ (succ (max u4 u1 u2))} (Set.{max u4 u1 u2} (forall (j : CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)), CategoryTheory.Functor.obj.{u4, max u1 u2, u4, succ (max u1 u2)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Functor.comp.{u4, u2, max u1 u2, u4, u3, succ (max u1 u2)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u4} C _inst_1 I) (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)) j)) Type.{max u4 u1 u2} (Set.hasCoeToSort.{max u4 u1 u2} (forall (j : CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)), CategoryTheory.Functor.obj.{u4, max u1 u2, u4, succ (max u1 u2)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Functor.comp.{u4, u2, max u1 u2, u4, u3, succ (max u1 u2)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u4} C _inst_1 I) (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)) j)) (CategoryTheory.Functor.sections.{u4, max u1 u2, u4} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Functor.comp.{u4, u2, max u1 u2, u4, u3, succ (max u1 u2)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u4} C _inst_1 I) (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)))) (Subtype.{max (succ u4) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u4, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u4, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)))))\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] (I : CategoryTheory.Limits.MulticospanIndex.{u4, u2, u3} C _inst_1), Equiv.{max (max (succ u2) (succ u1)) (succ u4), max 1 (max (succ u2) (succ u1)) (succ u4)} (Set.Elem.{max (max u2 u1) u4} (forall (j : CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)), Prefunctor.obj.{succ u4, succ (max u2 u1), u4, succ (max u2 u1)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.CategoryStruct.toQuiver.{u4, u4} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Category.toCategoryStruct.{u4, u4} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)))) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u4, max u2 u1, u4, succ (max u2 u1)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.Functor.comp.{u4, u2, max u2 u1, u4, u3, max (succ u2) (succ u1)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u4} C _inst_1 I) (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2))) j) (CategoryTheory.Functor.sections.{u4, max u2 u1, u4} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Functor.comp.{u4, u2, max u2 u1, u4, u3, max (succ u2) (succ u1)} (CategoryTheory.Limits.WalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u4} (CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I)) C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u4} C _inst_1 I) (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)))) (Subtype.{max (max (succ u2) (succ u1)) (succ u4)} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u4, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u4, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.fst.{u4, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u4, u2, u3} C _inst_1 I i))) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u4, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u4, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.snd.{u4, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u4, u2, u3} C _inst_1 I i)))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.multiequalizer_equiv_aux CategoryTheory.Limits.Concrete.multiequalizerEquivAux\u2093'. -/\n/-- An auxiliary equivalence to be used in `multiequalizer_equiv` below.-/\ndef Concrete.multiequalizerEquivAux (I : MulticospanIndex C) :\n    (I.multicospan \u22d9 forget C).sections \u2243\n      { x : \u2200 i : I.L, I.left i // \u2200 i : I.R, I.fst i (x _) = I.snd i (x _) }\n    where\n  toFun x :=\n    \u27e8fun i => x.1 (WalkingMulticospan.left _), fun i =>\n      by\n      have a := x.2 (walking_multicospan.hom.fst i)\n      have b := x.2 (walking_multicospan.hom.snd i)\n      rw [\u2190 b] at a\n      exact a\u27e9\n  invFun x :=\n    { val := fun j =>\n        match j with\n        | walking_multicospan.left a => x.1 _\n        | walking_multicospan.right b => I.fst b (x.1 _)\n      property := by\n        rintro (a | b) (a' | b') (f | f | f)\n        \u00b7 change (I.multicospan.map (\ud835\udfd9 _)) _ = _\n          simp\n        \u00b7 rfl\n        \u00b7 dsimp\n          erw [\u2190 x.2 b']\n          rfl\n        \u00b7 change (I.multicospan.map (\ud835\udfd9 _)) _ = _\n          simp }\n  left_inv := by\n    intro x; ext (a | b)\n    \u00b7 rfl\n    \u00b7 change _ = x.val _\n      rw [\u2190 x.2 (walking_multicospan.hom.fst b)]\n      rfl\n  right_inv := by\n    intro x\n    ext i\n    rfl\n#align category_theory.limits.concrete.multiequalizer_equiv_aux CategoryTheory.Limits.Concrete.multiequalizerEquivAux\n\n#print CategoryTheory.Limits.Concrete.multiequalizerEquiv /-\n/-- The equivalence between the noncomputable multiequalizer and\nand the concrete multiequalizer. -/\nnoncomputable def Concrete.multiequalizerEquiv (I : MulticospanIndex.{w} C) [HasMultiequalizer I]\n    [PreservesLimit I.multicospan (forget C)] :\n    (multiequalizer I : C) \u2243\n      { x : \u2200 i : I.L, I.left i // \u2200 i : I.R, I.fst i (x _) = I.snd i (x _) } :=\n  let h1 := limit.isLimit I.multicospan\n  let h2 := isLimitOfPreserves (forget C) h1\n  let E := h2.conePointUniqueUpToIso (Types.limitConeIsLimit _)\n  Equiv.trans E.toEquiv (Concrete.multiequalizerEquivAux I)\n#align category_theory.limits.concrete.multiequalizer_equiv CategoryTheory.Limits.Concrete.multiequalizerEquiv\n-/\n\n/- warning: category_theory.limits.concrete.multiequalizer_equiv_apply -> CategoryTheory.Limits.Concrete.multiequalizerEquiv_apply is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_5 : CategoryTheory.Limits.HasMultiequalizer.{u2, u3, u1} C _inst_1 I] [_inst_6 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u1 u2, u3, succ (max u1 u2)} C _inst_1 Type.{max u1 u2} CategoryTheory.types.{max u1 u2} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.CategoryTheory.smallCategory.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u1} C _inst_1 I) (CategoryTheory.forget.{u3, max u1 u2, u2} C _inst_1 _inst_2)] (x : coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) ((fun (a : Sort.{max 1 (succ u1) (succ (max u1 u2))}) (b : Sort.{max (succ u1) (succ (max u1 u2))}) [self : HasLiftT.{max 1 (succ u1) (succ (max u1 u2)), max (succ u1) (succ (max u1 u2))} a b] => self.0) (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))))) (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (HasLiftT.mk.{max 1 (succ u1) (succ (max u1 u2)), max (succ u1) (succ (max u1 u2))} (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))))) (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (CoeTC\u2093.coe.{max 1 (succ u1) (succ (max u1 u2)), max (succ u1) (succ (max u1 u2))} (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))))) (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (coeBase.{max 1 (succ u1) (succ (max u1 u2)), max (succ u1) (succ (max u1 u2))} (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))))) (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (coeSubtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))))) (coeFn.{max 1 (max (succ (max u1 u2)) 1 (succ u1) (succ (max u1 u2))) (max 1 (succ u1) (succ (max u1 u2))) (succ (max u1 u2)), max (succ (max u1 u2)) 1 (succ u1) (succ (max u1 u2))} (Equiv.{succ (max u1 u2), max 1 (succ u1) (succ (max u1 u2))} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) (fun (_x : Equiv.{succ (max u1 u2), max 1 (succ u1) (succ (max u1 u2))} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) -> (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) (Equiv.hasCoeToFun.{succ (max u1 u2), max 1 (succ u1) (succ (max u1 u2))} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (Subtype.{max (succ u1) (succ (max u1 u2))} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{succ (max u1 u2)} (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) (CategoryTheory.Limits.Concrete.multiequalizerEquiv.{u1, u2, u3} C _inst_1 _inst_2 I _inst_5 _inst_6) x) i) (coeFn.{succ u2, succ (max u1 u2)} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) -> (coeSort.{succ u3, succ (succ (max u1 u2))} C Type.{max u1 u2} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u3, max u1 u2, u2} C _inst_1 _inst_2) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u3, u2, max u1 u2} C _inst_1 _inst_2 (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.Multiequalizer.\u03b9.{u2, u3, u1} C _inst_1 I _inst_5 i) x)\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.ConcreteCategory.{max u1 u2, u2, u3} C _inst_1] (I : CategoryTheory.Limits.MulticospanIndex.{u1, u2, u3} C _inst_1) [_inst_5 : CategoryTheory.Limits.HasMultiequalizer.{u2, u3, u1} C _inst_1 I] [_inst_6 : CategoryTheory.Limits.PreservesLimit.{u1, u1, u2, max u2 u1, u3, max (succ u2) (succ u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.Limits.WalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.WalkingMulticospan.instSmallCategoryWalkingMulticospan.{u1} (CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I) (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I)) (CategoryTheory.Limits.MulticospanIndex.multicospan.{u2, u3, u1} C _inst_1 I) (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)] (x : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (Subtype.val.{max (succ u2) (succ u1)} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))) (FunLike.coe.{max (succ u2) (succ u1), max (succ u2) (succ u1), max (succ u2) (succ u1)} (Equiv.{max (succ u2) (succ u1), max 1 (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (Subtype.{max (succ u2) (succ u1)} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (fun (_x : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) => (fun (x._@.Mathlib.Logic.Equiv.Defs._hyg.808 : Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) => Subtype.{max (succ u2) (succ u1)} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i))))) _x) (Equiv.instFunLikeEquiv.{max (succ u2) (succ u1), max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5)) (Subtype.{max (succ u2) (succ u1)} (forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) (fun (x : forall (i : CategoryTheory.Limits.MulticospanIndex.L.{u1, u2, u3} C _inst_1 I), Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.ConcreteCategory.Forget.{max u2 u1, u2, u3} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i)) => forall (i : CategoryTheory.Limits.MulticospanIndex.R.{u1, u2, u3} C _inst_1 I), Eq.{max (succ u2) (succ u1)} (Prefunctor.obj.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i)) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.fst.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.fstTo.{u1, u2, u3} C _inst_1 I i))) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)) (CategoryTheory.Limits.MulticospanIndex.right.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.MulticospanIndex.snd.{u1, u2, u3} C _inst_1 I i) (x (CategoryTheory.Limits.MulticospanIndex.sndTo.{u1, u2, u3} C _inst_1 I i)))))) (CategoryTheory.Limits.Concrete.multiequalizerEquiv.{u1, u2, u3} C _inst_1 _inst_2 I _inst_5 _inst_6) x) i) (Prefunctor.map.{succ u2, succ (max u2 u1), u3, succ (max u2 u1)} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{max u2 u1} (CategoryTheory.CategoryStruct.toQuiver.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} (CategoryTheory.Category.toCategoryStruct.{max u2 u1, succ (max u2 u1)} Type.{max u2 u1} CategoryTheory.types.{max u2 u1})) (CategoryTheory.Functor.toPrefunctor.{u2, max u2 u1, u3, succ (max u2 u1)} C _inst_1 Type.{max u2 u1} CategoryTheory.types.{max u2 u1} (CategoryTheory.forget.{u3, max u2 u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.multiequalizer.{u2, u3, u1} C _inst_1 I _inst_5) (CategoryTheory.Limits.MulticospanIndex.left.{u1, u2, u3} C _inst_1 I i) (CategoryTheory.Limits.Multiequalizer.\u03b9.{u2, u3, u1} C _inst_1 I _inst_5 i) x)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.multiequalizer_equiv_apply CategoryTheory.Limits.Concrete.multiequalizerEquiv_apply\u2093'. -/\n@[simp]\ntheorem Concrete.multiequalizerEquiv_apply (I : MulticospanIndex.{w} C) [HasMultiequalizer I]\n    [PreservesLimit I.multicospan (forget C)] (x : multiequalizer I) (i : I.L) :\n    ((Concrete.multiequalizerEquiv I) x : \u2200 i : I.L, I.left i) i = Multiequalizer.\u03b9 I i x :=\n  rfl\n#align category_theory.limits.concrete.multiequalizer_equiv_apply CategoryTheory.Limits.Concrete.multiequalizerEquiv_apply\n\nend Multiequalizer\n\n-- TODO: Add analogous lemmas about products and equalizers.\nend Limits\n\nsection Colimits\n\n/- warning: category_theory.limits.cokernel_funext -> CategoryTheory.Limits.cokernel_funext is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u1}} [_inst_1 : CategoryTheory.Category.{u2, u1} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u1} C _inst_1] [_inst_3 : CategoryTheory.ConcreteCategory.{u3, u2, u1} C _inst_1] {M : C} {N : C} {K : C} {f : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) M N} [_inst_4 : CategoryTheory.Limits.HasCokernel.{u2, u1} C _inst_1 _inst_2 M N f] {g : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K} {h : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K}, (forall (n : coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) N), Eq.{succ u3} (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) K) (coeFn.{succ u2, succ u3} (Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) (fun (f_1 : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) => (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) -> (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) K)) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u1, u2, u3} C _inst_1 _inst_3 (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) g (coeFn.{succ u2, succ u3} (Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) (fun (f_1 : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) => (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) N) -> (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u1, u2, u3} C _inst_1 _inst_3 N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) (CategoryTheory.Limits.cokernel.\u03c0.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) n)) (coeFn.{succ u2, succ u3} (Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) (fun (f_1 : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) => (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) -> (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) K)) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u1, u2, u3} C _inst_1 _inst_3 (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) h (coeFn.{succ u2, succ u3} (Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) (fun (f_1 : Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) => (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) N) -> (coeSort.{succ u1, succ (succ u3)} C Type.{u3} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u1, u3, u2} C _inst_1 _inst_3) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u1, u2, u3} C _inst_1 _inst_3 N (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4)) (CategoryTheory.Limits.cokernel.\u03c0.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) n))) -> (Eq.{succ u2} (Quiver.Hom.{succ u2, u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u1} C (CategoryTheory.Category.toCategoryStruct.{u2, u1} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u1} C _inst_1 _inst_2 M N f _inst_4) K) g h)\nbut is expected to have type\n  forall {C : Type.{u3}} [_inst_1 : CategoryTheory.Category.{u2, u3} C] [_inst_2 : CategoryTheory.Limits.HasZeroMorphisms.{u2, u3} C _inst_1] [_inst_3 : CategoryTheory.ConcreteCategory.{u1, u2, u3} C _inst_1] {M : C} {N : C} {K : C} {f : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) M N} [_inst_4 : CategoryTheory.Limits.HasCokernel.{u2, u3} C _inst_1 _inst_2 M N f] {g : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) K} {h : Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) K}, (forall (n : Prefunctor.obj.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u2, u3} C _inst_1 _inst_3)) N), Eq.{succ u1} (Prefunctor.obj.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u3, u1, u2} C _inst_1 _inst_3)) K) (Prefunctor.map.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u3, u1, u2} C _inst_1 _inst_3)) (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) K g (Prefunctor.map.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u3, u1, u2} C _inst_1 _inst_3)) N (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) (CategoryTheory.Limits.cokernel.\u03c0.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) n)) (Prefunctor.map.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u3, u1, u2} C _inst_1 _inst_3)) (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) K h (Prefunctor.map.{succ u2, succ u1, u3, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u2, u1, u3, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u3, u1, u2} C _inst_1 _inst_3)) N (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) (CategoryTheory.Limits.cokernel.\u03c0.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) n))) -> (Eq.{succ u2} (Quiver.Hom.{succ u2, u3} C (CategoryTheory.CategoryStruct.toQuiver.{u2, u3} C (CategoryTheory.Category.toCategoryStruct.{u2, u3} C _inst_1)) (CategoryTheory.Limits.cokernel.{u2, u3} C _inst_1 _inst_2 M N f _inst_4) K) g h)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.cokernel_funext CategoryTheory.Limits.cokernel_funext\u2093'. -/\n-- We don't mark this as an `@[ext]` lemma as we don't always want to work elementwise.\ntheorem cokernel_funext {C : Type _} [Category C] [HasZeroMorphisms C] [ConcreteCategory C]\n    {M N K : C} {f : M \u27f6 N} [HasCokernel f] {g h : cokernel f \u27f6 K}\n    (w : \u2200 n : N, g (cokernel.\u03c0 f n) = h (cokernel.\u03c0 f n)) : g = h :=\n  by\n  apply coequalizer.hom_ext\n  apply concrete_category.hom_ext _ _\n  simpa using w\n#align category_theory.limits.cokernel_funext CategoryTheory.Limits.cokernel_funext\n\nvariable {C : Type u} [Category.{v} C] [ConcreteCategory.{v} C] {J : Type v} [SmallCategory J]\n  (F : J \u2964 C) [PreservesColimit F (forget C)]\n\n/- warning: category_theory.limits.concrete.from_union_surjective_of_is_colimit -> CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (let ff : (Sigma.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j))) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) := fun (a : Sigma.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j))) => coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a))) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a))) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a))) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a)))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a))) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) (Sigma.fst.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a)) (Sigma.snd.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) a); Function.Surjective.{succ u1, succ u1} (Sigma.{u1, u1} J (fun (j : J) => coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j))) (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) ff)\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (let ff : (Sigma.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j))) -> (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) := fun (a : Sigma.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j))) => Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) (Sigma.fst.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) a)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) (Sigma.fst.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) a)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) (Sigma.fst.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) a)) (Sigma.snd.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) a); Function.Surjective.{succ u1, succ u1} (Sigma.{u1, u1} J (fun (j : J) => Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j))) (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) ff)\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.from_union_surjective_of_is_colimit CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit\u2093'. -/\ntheorem Concrete.from_union_surjective_of_isColimit {D : Cocone F} (hD : IsColimit D) :\n    let ff : (\u03a3j : J, F.obj j) \u2192 D.pt := fun a => D.\u03b9.app a.1 a.2\n    Function.Surjective ff :=\n  by\n  intro ff\n  let E := (forget C).mapCocone D\n  let hE : is_colimit E := is_colimit_of_preserves _ hD\n  let G := Types.colimitCocone.{v, v} (F \u22d9 forget C)\n  let hG := Types.colimitCoconeIsColimit.{v, v} (F \u22d9 forget C)\n  let T : E \u2245 G := hE.unique_up_to_iso hG\n  let TX : E.X \u2245 G.X := (cocones.forget _).mapIso T\n  suffices Function.Surjective (TX.hom \u2218 ff) by\n    intro a\n    obtain \u27e8b, hb\u27e9 := this (TX.hom a)\n    refine' \u27e8b, _\u27e9\n    apply_fun TX.inv  at hb\n    change (TX.hom \u226b TX.inv) (ff b) = (TX.hom \u226b TX.inv) _ at hb\n    simpa only [TX.hom_inv_id] using hb\n  have : TX.hom \u2218 ff = fun a => G.\u03b9.app a.1 a.2 :=\n    by\n    ext a\n    change (E.\u03b9.app a.1 \u226b hE.desc G) a.2 = _\n    rw [hE.fac]\n  rw [this]\n  rintro \u27e8\u27e8j, a\u27e9\u27e9\n  exact \u27e8\u27e8j, a\u27e9, rfl\u27e9\n#align category_theory.limits.concrete.from_union_surjective_of_is_colimit CategoryTheory.Limits.Concrete.from_union_surjective_of_isColimit\n\n/- warning: category_theory.limits.concrete.is_colimit_exists_rep -> CategoryTheory.Limits.Concrete.isColimit_exists_rep is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)), Exists.{succ u1} J (fun (j : J) => Exists.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) (fun (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y) x)))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)), Exists.{succ u1} J (fun (j : J) => Exists.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) (fun (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) j)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) j) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y) x)))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.is_colimit_exists_rep CategoryTheory.Limits.Concrete.isColimit_exists_rep\u2093'. -/\ntheorem Concrete.isColimit_exists_rep {D : Cocone F} (hD : IsColimit D) (x : D.pt) :\n    \u2203 (j : J)(y : F.obj j), D.\u03b9.app j y = x :=\n  by\n  obtain \u27e8a, rfl\u27e9 := concrete.from_union_surjective_of_is_colimit F hD x\n  exact \u27e8a.1, a.2, rfl\u27e9\n#align category_theory.limits.concrete.is_colimit_exists_rep CategoryTheory.Limits.Concrete.isColimit_exists_rep\n\n/- warning: category_theory.limits.concrete.colimit_exists_rep -> CategoryTheory.Limits.Concrete.colimit_exists_rep is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)), Exists.{succ u1} J (fun (j : J) => Exists.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) (fun (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 j) y) x))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)), Exists.{succ u1} J (fun (j : J) => Exists.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) (fun (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 j) y) x))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.colimit_exists_rep CategoryTheory.Limits.Concrete.colimit_exists_rep\u2093'. -/\ntheorem Concrete.colimit_exists_rep [HasColimit F] (x : colimit F) :\n    \u2203 (j : J)(y : F.obj j), colimit.\u03b9 F j y = x :=\n  Concrete.isColimit_exists_rep F (colimit.isColimit _) x\n#align category_theory.limits.concrete.colimit_exists_rep CategoryTheory.Limits.Concrete.colimit_exists_rep\n\n/- warning: category_theory.limits.concrete.is_colimit_rep_eq_of_exists -> CategoryTheory.Limits.Concrete.isColimit_rep_eq_of_exists is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y))))) -> (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y))))) -> (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) j) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.is_colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.isColimit_rep_eq_of_exists\u2093'. -/\ntheorem Concrete.isColimit_rep_eq_of_exists {D : Cocone F} {i j : J} (hD : IsColimit D)\n    (x : F.obj i) (y : F.obj j) (h : \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y) :\n    D.\u03b9.app i x = D.\u03b9.app j y := by\n  let E := (forget C).mapCocone D\n  let hE : is_colimit E := is_colimit_of_preserves _ hD\n  let G := Types.colimitCocone.{v, v} (F \u22d9 forget C)\n  let hG := Types.colimitCoconeIsColimit.{v, v} (F \u22d9 forget C)\n  let T : E \u2245 G := hE.unique_up_to_iso hG\n  let TX : E.X \u2245 G.X := (cocones.forget _).mapIso T\n  apply_fun TX.hom\n  swap;\n  \u00b7 suffices Function.Bijective TX.hom by exact this.1\n    rw [\u2190 is_iso_iff_bijective]\n    apply is_iso.of_iso\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y\n  erw [T.hom.w, T.hom.w]\n  obtain \u27e8k, f, g, h\u27e9 := h\n  have : G.\u03b9.app i x = (G.\u03b9.app k (F.map f x) : G.X) := Quot.sound \u27e8f, rfl\u27e9\n  rw [this, h]\n  symm\n  exact Quot.sound \u27e8g, rfl\u27e9\n#align category_theory.limits.concrete.is_colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.isColimit_rep_eq_of_exists\n\n/- warning: category_theory.limits.concrete.colimit_rep_eq_of_exists -> CategoryTheory.Limits.Concrete.colimit_rep_eq_of_exists is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y))))) -> (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 j) y))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y))))) -> (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_5 j) y))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.colimit_rep_eq_of_exists\u2093'. -/\ntheorem Concrete.colimit_rep_eq_of_exists [HasColimit F] {i j : J} (x : F.obj i) (y : F.obj j)\n    (h : \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y) :\n    colimit.\u03b9 F i x = colimit.\u03b9 F j y :=\n  Concrete.isColimit_rep_eq_of_exists F (colimit.isColimit _) x y h\n#align category_theory.limits.concrete.colimit_rep_eq_of_exists CategoryTheory.Limits.Concrete.colimit_rep_eq_of_exists\n\nsection FilteredColimits\n\nvariable [IsFiltered J]\n\n/- warning: category_theory.limits.concrete.is_colimit_exists_of_rep_eq -> CategoryTheory.Limits.Concrete.isColimit_exists_of_rep_eq is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)) -> (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y))))))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) j) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)) -> (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y))))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.is_colimit_exists_of_rep_eq CategoryTheory.Limits.Concrete.isColimit_exists_of_rep_eq\u2093'. -/\ntheorem Concrete.isColimit_exists_of_rep_eq {D : Cocone F} {i j : J} (hD : IsColimit D)\n    (x : F.obj i) (y : F.obj j) (h : D.\u03b9.app _ x = D.\u03b9.app _ y) :\n    \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y :=\n  by\n  let E := (forget C).mapCocone D\n  let hE : is_colimit E := is_colimit_of_preserves _ hD\n  let G := Types.colimitCocone.{v, v} (F \u22d9 forget C)\n  let hG := Types.colimitCoconeIsColimit.{v, v} (F \u22d9 forget C)\n  let T : E \u2245 G := hE.unique_up_to_iso hG\n  let TX : E.X \u2245 G.X := (cocones.forget _).mapIso T\n  apply_fun TX.hom  at h\n  change (E.\u03b9.app i \u226b TX.hom) x = (E.\u03b9.app j \u226b TX.hom) y at h\n  erw [T.hom.w, T.hom.w] at h\n  replace h := Quot.exact _ h\n  suffices\n    \u2200 (a b : \u03a3j, F.obj j) (h : EqvGen (Limits.Types.Quot.Rel.{v, v} (F \u22d9 forget C)) a b),\n      \u2203 (k : _)(f : a.1 \u27f6 k)(g : b.1 \u27f6 k), F.map f a.2 = F.map g b.2\n    by exact this \u27e8i, x\u27e9 \u27e8j, y\u27e9 h\n  intro a b h\n  induction h\n  case rel x y hh =>\n    obtain \u27e8e, he\u27e9 := hh\n    use y.1, e, \ud835\udfd9 _\n    simpa using he.symm\n  case refl x => use x.1, \ud835\udfd9 _, \ud835\udfd9 _, rfl\n  case symm x y _ hh =>\n    obtain \u27e8k, f, g, hh\u27e9 := hh\n    use k, g, f, hh.symm\n  case trans x y z _ _ hh1 hh2 =>\n    obtain \u27e8k1, f1, g1, h1\u27e9 := hh1\n    obtain \u27e8k2, f2, g2, h2\u27e9 := hh2\n    let k0 : J := is_filtered.max k1 k2\n    let e1 : k1 \u27f6 k0 := is_filtered.left_to_max _ _\n    let e2 : k2 \u27f6 k0 := is_filtered.right_to_max _ _\n    let k : J := is_filtered.coeq (g1 \u226b e1) (f2 \u226b e2)\n    let e : k0 \u27f6 k := is_filtered.coeq_hom _ _\n    use k, f1 \u226b e1 \u226b e, g2 \u226b e2 \u226b e\n    simp only [F.map_comp, comp_apply, h1, \u2190 h2]\n    simp only [\u2190 comp_apply, \u2190 F.map_comp]\n    rw [is_filtered.coeq_condition]\n#align category_theory.limits.concrete.is_colimit_exists_of_rep_eq CategoryTheory.Limits.Concrete.isColimit_exists_of_rep_eq\n\n/- warning: category_theory.limits.concrete.is_colimit_rep_eq_iff_exists -> CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), Iff (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) i)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) j)) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (CategoryTheory.Functor.obj.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)) (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y))))))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] {D : CategoryTheory.Limits.Cocone.{u1, u1, u1, u2} J _inst_3 C _inst_1 F} {i : J} {j : J}, (CategoryTheory.Limits.IsColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) -> (forall (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), Iff (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) i) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D))) j) (CategoryTheory.NatTrans.app.{u1, u1, u1, u2} J _inst_3 C _inst_1 F (Prefunctor.obj.{succ u1, succ u1, u2, max u1 u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.CategoryStruct.toQuiver.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Category.toCategoryStruct.{u1, max u1 u2} (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1))) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, max u1 u2} C _inst_1 (CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.category.{u1, u1, u1, u2} J _inst_3 C _inst_1) (CategoryTheory.Functor.const.{u1, u1, u1, u2} J _inst_3 C _inst_1)) (CategoryTheory.Limits.Cocone.pt.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D)) (CategoryTheory.Limits.Cocone.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F D) j) y)) (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y))))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.is_colimit_rep_eq_iff_exists CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists\u2093'. -/\ntheorem Concrete.isColimit_rep_eq_iff_exists {D : Cocone F} {i j : J} (hD : IsColimit D)\n    (x : F.obj i) (y : F.obj j) :\n    D.\u03b9.app i x = D.\u03b9.app j y \u2194 \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y :=\n  \u27e8Concrete.isColimit_exists_of_rep_eq _ hD _ _, Concrete.isColimit_rep_eq_of_exists _ hD _ _\u27e9\n#align category_theory.limits.concrete.is_colimit_rep_eq_iff_exists CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists\n\n/- warning: category_theory.limits.concrete.colimit_exists_of_rep_eq -> CategoryTheory.Limits.Concrete.colimit_exists_of_rep_eq is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] [_inst_6 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 j) y)) -> (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y)))))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] [_inst_6 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 j) y)) -> (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y)))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.colimit_exists_of_rep_eq CategoryTheory.Limits.Concrete.colimit_exists_of_rep_eq\u2093'. -/\ntheorem Concrete.colimit_exists_of_rep_eq [HasColimit F] {i j : J} (x : F.obj i) (y : F.obj j)\n    (h : colimit.\u03b9 F _ x = colimit.\u03b9 F _ y) :\n    \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y :=\n  Concrete.isColimit_exists_of_rep_eq F (colimit.isColimit _) x y h\n#align category_theory.limits.concrete.colimit_exists_of_rep_eq CategoryTheory.Limits.Concrete.colimit_exists_of_rep_eq\n\n/- warning: category_theory.limits.concrete.colimit_rep_eq_iff_exists -> CategoryTheory.Limits.Concrete.colimit_rep_eq_iff_exists is a dubious translation:\nlean 3 declaration is\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] [_inst_6 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) (y : coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)), Iff (Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 i) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 j) y)) (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F i k f) x) (coeFn.{succ u1, succ u1} (Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (fun (f : Quiver.Hom.{succ u1, u2} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) => (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j)) -> (coeSort.{succ u2, succ (succ u1)} C Type.{u1} (CategoryTheory.ConcreteCategory.hasCoeToSort.{u2, u1, u1} C _inst_1 _inst_2) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k))) (CategoryTheory.ConcreteCategory.hasCoeToFun.{u2, u1, u1} C _inst_1 _inst_2 (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j) (CategoryTheory.Functor.obj.{u1, u1, u1, u2} J _inst_3 C _inst_1 F k)) (CategoryTheory.Functor.map.{u1, u1, u1, u2} J _inst_3 C _inst_1 F j k g) y)))))\nbut is expected to have type\n  forall {C : Type.{u2}} [_inst_1 : CategoryTheory.Category.{u1, u2} C] [_inst_2 : CategoryTheory.ConcreteCategory.{u1, u1, u2} C _inst_1] {J : Type.{u1}} [_inst_3 : CategoryTheory.SmallCategory.{u1} J] (F : CategoryTheory.Functor.{u1, u1, u1, u2} J _inst_3 C _inst_1) [_inst_4 : CategoryTheory.Limits.PreservesColimit.{u1, u1, u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} J _inst_3 F (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)] [_inst_5 : CategoryTheory.IsFiltered.{u1, u1} J _inst_3] [_inst_6 : CategoryTheory.Limits.HasColimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F] {i : J} {j : J} (x : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i)) (y : Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.ConcreteCategory.Forget.{u1, u1, u2} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j)), Iff (Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 i) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (CategoryTheory.Limits.colimit.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6) (CategoryTheory.Limits.colimit.\u03b9.{u1, u1, u1, u2} J _inst_3 C _inst_1 F _inst_6 j) y)) (Exists.{succ u1} J (fun (k : J) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) (fun (f : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) i k) => Exists.{succ u1} (Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) (fun (g : Quiver.Hom.{succ u1, u1} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) j k) => Eq.{succ u1} (Prefunctor.obj.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k)) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) i k f) x) (Prefunctor.map.{succ u1, succ u1, u2, succ u1} C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) Type.{u1} (CategoryTheory.CategoryStruct.toQuiver.{u1, succ u1} Type.{u1} (CategoryTheory.Category.toCategoryStruct.{u1, succ u1} Type.{u1} CategoryTheory.types.{u1})) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u2, succ u1} C _inst_1 Type.{u1} CategoryTheory.types.{u1} (CategoryTheory.forget.{u2, u1, u1} C _inst_1 _inst_2)) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j) (Prefunctor.obj.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) k) (Prefunctor.map.{succ u1, succ u1, u1, u2} J (CategoryTheory.CategoryStruct.toQuiver.{u1, u1} J (CategoryTheory.Category.toCategoryStruct.{u1, u1} J _inst_3)) C (CategoryTheory.CategoryStruct.toQuiver.{u1, u2} C (CategoryTheory.Category.toCategoryStruct.{u1, u2} C _inst_1)) (CategoryTheory.Functor.toPrefunctor.{u1, u1, u1, u2} J _inst_3 C _inst_1 F) j k g) y)))))\nCase conversion may be inaccurate. Consider using '#align category_theory.limits.concrete.colimit_rep_eq_iff_exists CategoryTheory.Limits.Concrete.colimit_rep_eq_iff_exists\u2093'. -/\ntheorem Concrete.colimit_rep_eq_iff_exists [HasColimit F] {i j : J} (x : F.obj i) (y : F.obj j) :\n    colimit.\u03b9 F i x = colimit.\u03b9 F j y \u2194 \u2203 (k : _)(f : i \u27f6 k)(g : j \u27f6 k), F.map f x = F.map g y :=\n  \u27e8Concrete.colimit_exists_of_rep_eq _ _ _, Concrete.colimit_rep_eq_of_exists _ _ _\u27e9\n#align category_theory.limits.concrete.colimit_rep_eq_iff_exists CategoryTheory.Limits.Concrete.colimit_rep_eq_iff_exists\n\nend FilteredColimits\n\nsection WidePushout\n\nopen WidePushout\n\nopen WidePushoutShape\n\n#print CategoryTheory.Limits.Concrete.widePushout_exists_rep /-\ntheorem Concrete.widePushout_exists_rep {B : C} {\u03b1 : Type _} {X : \u03b1 \u2192 C} (f : \u2200 j : \u03b1, B \u27f6 X j)\n    [HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]\n    (x : widePushout B X f) : (\u2203 y : B, head f y = x) \u2228 \u2203 (i : \u03b1)(y : X i), \u03b9 f i y = x :=\n  by\n  obtain \u27e8_ | j, y, rfl\u27e9 := concrete.colimit_exists_rep _ x\n  \u00b7 use y\n  \u00b7 right\n    use j, y\n#align category_theory.limits.concrete.wide_pushout_exists_rep CategoryTheory.Limits.Concrete.widePushout_exists_rep\n-/\n\n#print CategoryTheory.Limits.Concrete.widePushout_exists_rep' /-\ntheorem Concrete.widePushout_exists_rep' {B : C} {\u03b1 : Type _} [Nonempty \u03b1] {X : \u03b1 \u2192 C}\n    (f : \u2200 j : \u03b1, B \u27f6 X j) [HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]\n    (x : widePushout B X f) : \u2203 (i : \u03b1)(y : X i), \u03b9 f i y = x :=\n  by\n  rcases concrete.wide_pushout_exists_rep f x with (\u27e8y, rfl\u27e9 | \u27e8i, y, rfl\u27e9)\n  \u00b7 inhabit \u03b1\n    use Inhabited.default _, f _ y\n    simp only [\u2190 arrow_\u03b9 _ (Inhabited.default \u03b1), comp_apply]\n  \u00b7 use i, y\n#align category_theory.limits.concrete.wide_pushout_exists_rep' CategoryTheory.Limits.Concrete.widePushout_exists_rep'\n-/\n\nend WidePushout\n\n-- TODO: Add analogous lemmas about coproducts and coequalizers.\nend Colimits\n\nend CategoryTheory.Limits\n\n", "meta": {"author": "leanprover-community", "repo": "mathlib3port", "sha": "62505aa236c58c8559783b16d33e30df3daa54f4", "save_path": "github-repos/lean/leanprover-community-mathlib3port", "path": "github-repos/lean/leanprover-community-mathlib3port/mathlib3port-62505aa236c58c8559783b16d33e30df3daa54f4/Mathbin/CategoryTheory/Limits/ConcreteCategory.lean", "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3500375703179442}}